[{"data":1,"prerenderedAt":782685},["ShallowReactive",2],{"library-notes":3},[4,91,128,5990,8342,11705,14184,17327,20016,21653,23688,25641,27295,28881,29907,30734,31417,31524,31554,45636,51541,58145,90810,91074,91356,91642,91900,92183,92436,93927,94383,95041,97232,98729,99544,100414,100440,101063,102413,103734,104276,104545,114746,118538,119852,120068,120496,121725,123283,123862,124281,124840,125409,126357,126372,161999,162028,175221,193842,194804,195525,195725,199905,200418,208638,214273,233077,233107,237709,240791,241328,241804,242347,242827,243383,243703,244070,245992,246491,248249,257997,262978,264272,283803,285727,286642,287556,290004,291929,293357,294856,296320,297760,299058,300486,301785,303061,304092,305356,306664,308469,308702,308951,309239,309529,309818,309830,309842,310362,310911,311455,311917,312380,312666,313069,313122,313427,313644,313893,314393,314499,314514,314525,314548,314578,314606,330679,339106,365355,371602,402624,417970,438830,463163,495547,512611,546231,546251,587743,658086,693475,699538,742675,768262,775848],{"id":5,"title":6,"body":7,"description":17,"extension":85,"meta":86,"navigation":52,"path":87,"seo":88,"stem":89,"__hash__":90},"notes\u002Fsource\u002Fcode\u002Fcpp-start.md","C++ 学习笔记",{"type":8,"value":9,"toc":82},"minimark",[10,14,18,23,78],[11,12,6],"h1",{"id":13},"c-学习笔记",[15,16,17],"p",{},"这是我的第一篇 C++ 笔记。",[19,20,22],"h2",{"id":21},"hello-world","Hello World",[24,25,30],"pre",{"className":26,"code":27,"language":28,"meta":29,"style":29},"language-cpp shiki shiki-themes github-light github-dark","#include \u003Ciostream>\nusing namespace std;\n\nint main() {\n cout \u003C\u003C \"Hello VitePress!\" \u003C\u003C endl;\n return 0;\n}\n","cpp","",[31,32,33,41,47,54,60,66,72],"code",{"__ignoreMap":29},[34,35,38],"span",{"class":36,"line":37},"line",1,[34,39,40],{},"#include \u003Ciostream>\n",[34,42,44],{"class":36,"line":43},2,[34,45,46],{},"using namespace std;\n",[34,48,50],{"class":36,"line":49},3,[34,51,53],{"emptyLinePlaceholder":52},true,"\n",[34,55,57],{"class":36,"line":56},4,[34,58,59],{},"int main() {\n",[34,61,63],{"class":36,"line":62},5,[34,64,65],{}," cout \u003C\u003C \"Hello VitePress!\" \u003C\u003C endl;\n",[34,67,69],{"class":36,"line":68},6,[34,70,71],{}," return 0;\n",[34,73,75],{"class":36,"line":74},7,[34,76,77],{},"}\n",[79,80,81],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":29,"searchDepth":43,"depth":43,"links":83},[84],{"id":21,"depth":43,"text":22},"md",{},"\u002Fsource\u002Fcode\u002Fcpp-start",{"title":6,"description":17},"source\u002Fcode\u002Fcpp-start","AOo3R2lm79o2_4IMNflwAFwxl882LtIy4CgCaeBs9bk",{"id":92,"title":93,"body":94,"description":100,"extension":85,"meta":123,"navigation":52,"path":124,"seo":125,"stem":126,"__hash__":127},"notes\u002Fsource\u002Fcode\u002Findex.md","開発",{"type":8,"value":95,"toc":120},[96,98,101,104],[11,97,93],{"id":93},[15,99,100],{},"编程与前端开发相关笔记。",[19,102,103],{"id":103},"目录",[105,106,107,114],"ul",{},[108,109,110],"li",{},[111,112,6],"a",{"href":113},".\u002Fcpp-start",[108,115,116],{},[111,117,119],{"href":118},".\u002Fvue\u002F","Vue.js 完整课程",{"title":29,"searchDepth":43,"depth":43,"links":121},[122],{"id":103,"depth":43,"text":103},{"order":37},"\u002Fsource\u002Fcode",{"title":93,"description":100},"source\u002Fcode\u002Findex","xQk-c00Vej6Zv5BXtMbstgg_EhfF2ZGYZx_8xxaEim4",{"id":129,"title":130,"body":131,"description":5984,"extension":85,"meta":5985,"navigation":52,"path":5986,"seo":5987,"stem":5988,"__hash__":5989},"notes\u002Fsource\u002Fcode\u002Fvue\u002F01-js-from-zero.md","01 JavaScript 必备基础",{"type":8,"value":132,"toc":5929},[133,136,154,157,161,164,229,232,252,262,267,303,308,337,340,360,364,367,414,417,488,494,520,524,527,653,656,659,720,723,803,806,810,813,870,873,898,901,918,921,1008,1011,1014,1051,1054,1118,1121,1124,1175,1185,1189,1192,1222,1225,1247,1250,1270,1273,1292,1296,1302,1340,1351,1357,1386,1392,1429,1435,1465,1471,1528,1535,1538,1541,1587,1590,1649,1652,1670,1674,1677,1754,1756,1783,1786,1789,1822,1825,1861,1864,1893,1896,1900,2043,2056,2060,2152,2155,2226,2229,2247,2251,2254,2319,2322,2351,2354,2402,2405,2423,2433,2437,2440,2444,2447,2606,2609,2623,2630,2633,2641,2655,2659,2665,2705,2710,2741,2756,2762,2765,2771,2793,2816,2819,2827,2835,2877,2880,2896,2902,2953,2962,2965,3003,3024,3033,3036,3039,3050,3059,3062,3099,3108,3182,3190,3193,3199,3206,3209,3261,3264,3291,3327,3330,3372,3408,3411,3478,3483,3487,3584,3587,3663,3666,3683,3690,3694,3700,3748,3759,3763,3766,3846,3849,3922,3925,3933,3940,3943,3987,3990,4032,4035,4043,4045,4088,4091,4095,4098,4138,4158,4161,4174,4177,4204,4222,4224,4235,4238,4269,4276,4292,4300,4303,4369,4379,4386,4403,4413,4417,4420,4463,4499,4509,4515,4518,4521,4525,4528,4530,4581,4638,4641,4675,4678,4682,4685,4688,4704,4707,4726,4729,4748,4751,4770,4773,4792,4799,4803,4806,4844,4864,4867,4903,4906,4946,4966,4968,4984,4987,5029,5049,5051,5068,5071,5092,5094,5110,5113,5131,5138,5142,5149,5152,5269,5275,5278,5345,5348,5637,5640,5658,5662,5665,5671,5674,5688,5691,5694,5698,5701,5785,5788,5814,5818,5821,5827,5832,5846,5851,5854,5882,5885,5889,5892,5926],[11,134,130],{"id":135},"_01-javascript-必备基础",[15,137,138,139,142,143,146,147,146,150,153],{},"Vue 不是一门脱离 JavaScript 的新语言。Vue 组件里的 ",[31,140,141],{},"\u003Cscript setup>"," 本质上就是 JavaScript，只是 Vue 提供了一些特殊函数，例如 ",[31,144,145],{},"ref","、",[31,148,149],{},"computed",[31,151,152],{},"watch","。所以学 Vue 前，你至少要能看懂变量、数组、对象、函数、模块和异步。",[15,155,156],{},"本章不追求讲完 JavaScript 的全部细节，而是讲 Vue 开发每天都会用到的部分，并且尽量回答“为什么要这么写”。",[19,158,160],{"id":159},"_1-变量给数据起名字","1. 变量：给数据起名字",[15,162,163],{},"变量就是给一个值起名字，后面可以通过这个名字使用它。",[24,165,169],{"className":166,"code":167,"language":168,"meta":29,"style":29},"language-js shiki shiki-themes github-light github-dark","const courseName = \"Vue 完整课程\";\nlet progress = 0;\n\nprogress = progress + 1;\n","js",[31,170,171,192,208,212],{"__ignoreMap":29},[34,172,173,177,181,184,188],{"class":36,"line":37},[34,174,176],{"class":175},"szBVR","const",[34,178,180],{"class":179},"sj4cs"," courseName",[34,182,183],{"class":175}," =",[34,185,187],{"class":186},"sZZnC"," \"Vue 完整课程\"",[34,189,191],{"class":190},"sVt8B",";\n",[34,193,194,197,200,203,206],{"class":36,"line":43},[34,195,196],{"class":175},"let",[34,198,199],{"class":190}," progress ",[34,201,202],{"class":175},"=",[34,204,205],{"class":179}," 0",[34,207,191],{"class":190},[34,209,210],{"class":36,"line":49},[34,211,53],{"emptyLinePlaceholder":52},[34,213,214,217,219,221,224,227],{"class":36,"line":56},[34,215,216],{"class":190},"progress ",[34,218,202],{"class":175},[34,220,199],{"class":190},[34,222,223],{"class":175},"+",[34,225,226],{"class":179}," 1",[34,228,191],{"class":190},[15,230,231],{},"这段代码里：",[105,233,234,240,246],{},[108,235,236,239],{},[31,237,238],{},"courseName"," 保存字符串。",[108,241,242,245],{},[31,243,244],{},"progress"," 保存数字。",[108,247,248,251],{},[31,249,250],{},"progress = progress + 1"," 表示取出原来的进度，加 1，再放回去。",[253,254,256,258,259,261],"h3",{"id":255},"const-和-let-的区别",[31,257,176],{}," 和 ",[31,260,196],{}," 的区别",[15,263,264,266],{},[31,265,176],{}," 表示这个变量名不能重新指向另一个值：",[24,268,270],{"className":166,"code":269,"language":168,"meta":29,"style":29},"const title = \"Vue\";\ntitle = \"React\"; \u002F\u002F 报错\n",[31,271,272,286],{"__ignoreMap":29},[34,273,274,276,279,281,284],{"class":36,"line":37},[34,275,176],{"class":175},[34,277,278],{"class":179}," title",[34,280,183],{"class":175},[34,282,283],{"class":186}," \"Vue\"",[34,285,191],{"class":190},[34,287,288,291,293,296,299],{"class":36,"line":43},[34,289,290],{"class":190},"title ",[34,292,202],{"class":175},[34,294,295],{"class":186}," \"React\"",[34,297,298],{"class":190},"; ",[34,300,302],{"class":301},"sJ8bj","\u002F\u002F 报错\n",[15,304,305,307],{},[31,306,196],{}," 表示这个变量名以后可以重新赋值：",[24,309,311],{"className":166,"code":310,"language":168,"meta":29,"style":29},"let count = 0;\ncount = 1;\n",[31,312,313,326],{"__ignoreMap":29},[34,314,315,317,320,322,324],{"class":36,"line":37},[34,316,196],{"class":175},[34,318,319],{"class":190}," count ",[34,321,202],{"class":175},[34,323,205],{"class":179},[34,325,191],{"class":190},[34,327,328,331,333,335],{"class":36,"line":43},[34,329,330],{"class":190},"count ",[34,332,202],{"class":175},[34,334,226],{"class":179},[34,336,191],{"class":190},[15,338,339],{},"初学时可以记住：",[105,341,342,348,353],{},[108,343,344,345,347],{},"不需要重新赋值：用 ",[31,346,176],{},"。",[108,349,350,351,347],{},"需要重新赋值：用 ",[31,352,196],{},[108,354,355,356,359],{},"不用 ",[31,357,358],{},"var","，因为它的作用域规则更容易制造困惑。",[253,361,363],{"id":362},"vue-里为什么普通变量不够用","Vue 里为什么普通变量不够用",[15,365,366],{},"普通变量变了，Vue 不知道界面需要更新：",[24,368,370],{"className":166,"code":369,"language":168,"meta":29,"style":29},"let count = 0;\n\nfunction increase() {\n count++;\n}\n",[31,371,372,384,388,400,410],{"__ignoreMap":29},[34,373,374,376,378,380,382],{"class":36,"line":37},[34,375,196],{"class":175},[34,377,319],{"class":190},[34,379,202],{"class":175},[34,381,205],{"class":179},[34,383,191],{"class":190},[34,385,386],{"class":36,"line":43},[34,387,53],{"emptyLinePlaceholder":52},[34,389,390,393,397],{"class":36,"line":49},[34,391,392],{"class":175},"function",[34,394,396],{"class":395},"sScJk"," increase",[34,398,399],{"class":190},"() {\n",[34,401,402,405,408],{"class":36,"line":56},[34,403,404],{"class":190}," count",[34,406,407],{"class":175},"++",[34,409,191],{"class":190},[34,411,412],{"class":36,"line":62},[34,413,77],{"class":190},[15,415,416],{},"这只是 JavaScript 变量变化。Vue 要自动更新界面，需要响应式状态：",[24,418,420],{"className":166,"code":419,"language":168,"meta":29,"style":29},"import { ref } from \"vue\";\n\nconst count = ref(0);\n\nfunction increase() {\n count.value++;\n}\n",[31,421,422,438,442,463,467,475,484],{"__ignoreMap":29},[34,423,424,427,430,433,436],{"class":36,"line":37},[34,425,426],{"class":175},"import",[34,428,429],{"class":190}," { ref } ",[34,431,432],{"class":175},"from",[34,434,435],{"class":186}," \"vue\"",[34,437,191],{"class":190},[34,439,440],{"class":36,"line":43},[34,441,53],{"emptyLinePlaceholder":52},[34,443,444,446,449,451,454,457,460],{"class":36,"line":49},[34,445,176],{"class":175},[34,447,448],{"class":179}," count",[34,450,183],{"class":175},[34,452,453],{"class":395}," ref",[34,455,456],{"class":190},"(",[34,458,459],{"class":179},"0",[34,461,462],{"class":190},");\n",[34,464,465],{"class":36,"line":56},[34,466,53],{"emptyLinePlaceholder":52},[34,468,469,471,473],{"class":36,"line":62},[34,470,392],{"class":175},[34,472,396],{"class":395},[34,474,399],{"class":190},[34,476,477,480,482],{"class":36,"line":68},[34,478,479],{"class":190}," count.value",[34,481,407],{"class":175},[34,483,191],{"class":190},[34,485,486],{"class":36,"line":74},[34,487,77],{"class":190},[15,489,490,491,493],{},"先不用急着理解 ",[31,492,145],{}," 的全部原理，只要记住：",[105,495,496,502,508,511],{},[108,497,498,501],{},[31,499,500],{},"ref(0)"," 创建了一个响应式盒子。",[108,503,504,505,347],{},"盒子里的真实值在 ",[31,506,507],{},".value",[108,509,510],{},"Vue 能追踪这个盒子的变化，所以界面会更新。",[108,512,513,514,517,518,347],{},"模板中写 ",[31,515,516],{},"{{ count }}"," 时，Vue 会自动帮你取 ",[31,519,507],{},[19,521,523],{"id":522},"_2-数据类型页面数据长什么样","2. 数据类型：页面数据长什么样",[15,525,526],{},"常见数据类型：",[24,528,530],{"className":166,"code":529,"language":168,"meta":29,"style":29},"const title = \"任务列表\"; \u002F\u002F string 字符串\nconst total = 10; \u002F\u002F number 数字\nconst done = false; \u002F\u002F boolean 布尔值\nconst user = null; \u002F\u002F 暂时没有值\nconst tags = [\"Vue\", \"JavaScript\"]; \u002F\u002F array 数组\nconst task = { id: 1, text: \"学 ref\" }; \u002F\u002F object 对象\n",[31,531,532,548,565,582,599,626],{"__ignoreMap":29},[34,533,534,536,538,540,543,545],{"class":36,"line":37},[34,535,176],{"class":175},[34,537,278],{"class":179},[34,539,183],{"class":175},[34,541,542],{"class":186}," \"任务列表\"",[34,544,298],{"class":190},[34,546,547],{"class":301},"\u002F\u002F string 字符串\n",[34,549,550,552,555,557,560,562],{"class":36,"line":43},[34,551,176],{"class":175},[34,553,554],{"class":179}," total",[34,556,183],{"class":175},[34,558,559],{"class":179}," 10",[34,561,298],{"class":190},[34,563,564],{"class":301},"\u002F\u002F number 数字\n",[34,566,567,569,572,574,577,579],{"class":36,"line":49},[34,568,176],{"class":175},[34,570,571],{"class":179}," done",[34,573,183],{"class":175},[34,575,576],{"class":179}," false",[34,578,298],{"class":190},[34,580,581],{"class":301},"\u002F\u002F boolean 布尔值\n",[34,583,584,586,589,591,594,596],{"class":36,"line":56},[34,585,176],{"class":175},[34,587,588],{"class":179}," user",[34,590,183],{"class":175},[34,592,593],{"class":179}," null",[34,595,298],{"class":190},[34,597,598],{"class":301},"\u002F\u002F 暂时没有值\n",[34,600,601,603,606,608,611,614,617,620,623],{"class":36,"line":62},[34,602,176],{"class":175},[34,604,605],{"class":179}," tags",[34,607,183],{"class":175},[34,609,610],{"class":190}," [",[34,612,613],{"class":186},"\"Vue\"",[34,615,616],{"class":190},", ",[34,618,619],{"class":186},"\"JavaScript\"",[34,621,622],{"class":190},"]; ",[34,624,625],{"class":301},"\u002F\u002F array 数组\n",[34,627,628,630,633,635,638,641,644,647,650],{"class":36,"line":68},[34,629,176],{"class":175},[34,631,632],{"class":179}," task",[34,634,183],{"class":175},[34,636,637],{"class":190}," { id: ",[34,639,640],{"class":179},"1",[34,642,643],{"class":190},", text: ",[34,645,646],{"class":186},"\"学 ref\"",[34,648,649],{"class":190}," }; ",[34,651,652],{"class":301},"\u002F\u002F object 对象\n",[15,654,655],{},"Vue 页面里的绝大多数状态都可以用这些类型表示。",[15,657,658],{},"例如一个任务：",[24,660,662],{"className":166,"code":661,"language":168,"meta":29,"style":29},"const task = {\n id: 1,\n title: \"学习 v-for\",\n done: false,\n priority: \"normal\",\n};\n",[31,663,664,675,685,695,705,715],{"__ignoreMap":29},[34,665,666,668,670,672],{"class":36,"line":37},[34,667,176],{"class":175},[34,669,632],{"class":179},[34,671,183],{"class":175},[34,673,674],{"class":190}," {\n",[34,676,677,680,682],{"class":36,"line":43},[34,678,679],{"class":190}," id: ",[34,681,640],{"class":179},[34,683,684],{"class":190},",\n",[34,686,687,690,693],{"class":36,"line":49},[34,688,689],{"class":190}," title: ",[34,691,692],{"class":186},"\"学习 v-for\"",[34,694,684],{"class":190},[34,696,697,700,703],{"class":36,"line":56},[34,698,699],{"class":190}," done: ",[34,701,702],{"class":179},"false",[34,704,684],{"class":190},[34,706,707,710,713],{"class":36,"line":62},[34,708,709],{"class":190}," priority: ",[34,711,712],{"class":186},"\"normal\"",[34,714,684],{"class":190},[34,716,717],{"class":36,"line":68},[34,718,719],{"class":190},"};\n",[15,721,722],{},"例如一组任务：",[24,724,726],{"className":166,"code":725,"language":168,"meta":29,"style":29},"const tasks = [\n { id: 1, title: \"学习 ref\", done: true },\n { id: 2, title: \"学习 computed\", done: false },\n { id: 3, title: \"学习组件通信\", done: false },\n];\n",[31,727,728,740,762,780,798],{"__ignoreMap":29},[34,729,730,732,735,737],{"class":36,"line":37},[34,731,176],{"class":175},[34,733,734],{"class":179}," tasks",[34,736,183],{"class":175},[34,738,739],{"class":190}," [\n",[34,741,742,745,747,750,753,756,759],{"class":36,"line":43},[34,743,744],{"class":190}," { id: ",[34,746,640],{"class":179},[34,748,749],{"class":190},", title: ",[34,751,752],{"class":186},"\"学习 ref\"",[34,754,755],{"class":190},", done: ",[34,757,758],{"class":179},"true",[34,760,761],{"class":190}," },\n",[34,763,764,766,769,771,774,776,778],{"class":36,"line":49},[34,765,744],{"class":190},[34,767,768],{"class":179},"2",[34,770,749],{"class":190},[34,772,773],{"class":186},"\"学习 computed\"",[34,775,755],{"class":190},[34,777,702],{"class":179},[34,779,761],{"class":190},[34,781,782,784,787,789,792,794,796],{"class":36,"line":56},[34,783,744],{"class":190},[34,785,786],{"class":179},"3",[34,788,749],{"class":190},[34,790,791],{"class":186},"\"学习组件通信\"",[34,793,755],{"class":190},[34,795,702],{"class":179},[34,797,761],{"class":190},[34,799,800],{"class":36,"line":62},[34,801,802],{"class":190},"];\n",[15,804,805],{},"做前端时，你要经常问自己：这个界面背后的数据结构是什么？数据结构清晰，界面代码才会清晰。",[19,807,809],{"id":808},"_3-对象描述一个东西","3. 对象：描述一个东西",[15,811,812],{},"对象适合描述“一个有多个属性的东西”。",[24,814,816],{"className":166,"code":815,"language":168,"meta":29,"style":29},"const lesson = {\n id: \"core\",\n title: \"Vue 核心语法\",\n minutes: 90,\n completed: false,\n};\n",[31,817,818,829,838,847,857,866],{"__ignoreMap":29},[34,819,820,822,825,827],{"class":36,"line":37},[34,821,176],{"class":175},[34,823,824],{"class":179}," lesson",[34,826,183],{"class":175},[34,828,674],{"class":190},[34,830,831,833,836],{"class":36,"line":43},[34,832,679],{"class":190},[34,834,835],{"class":186},"\"core\"",[34,837,684],{"class":190},[34,839,840,842,845],{"class":36,"line":49},[34,841,689],{"class":190},[34,843,844],{"class":186},"\"Vue 核心语法\"",[34,846,684],{"class":190},[34,848,849,852,855],{"class":36,"line":56},[34,850,851],{"class":190}," minutes: ",[34,853,854],{"class":179},"90",[34,856,684],{"class":190},[34,858,859,862,864],{"class":36,"line":62},[34,860,861],{"class":190}," completed: ",[34,863,702],{"class":179},[34,865,684],{"class":190},[34,867,868],{"class":36,"line":68},[34,869,719],{"class":190},[15,871,872],{},"读取属性：",[24,874,876],{"className":166,"code":875,"language":168,"meta":29,"style":29},"console.log(lesson.title);\nconsole.log(lesson.minutes);\n",[31,877,878,889],{"__ignoreMap":29},[34,879,880,883,886],{"class":36,"line":37},[34,881,882],{"class":190},"console.",[34,884,885],{"class":395},"log",[34,887,888],{"class":190},"(lesson.title);\n",[34,890,891,893,895],{"class":36,"line":43},[34,892,882],{"class":190},[34,894,885],{"class":395},[34,896,897],{"class":190},"(lesson.minutes);\n",[15,899,900],{},"修改属性：",[24,902,904],{"className":166,"code":903,"language":168,"meta":29,"style":29},"lesson.completed = true;\n",[31,905,906],{"__ignoreMap":29},[34,907,908,911,913,916],{"class":36,"line":37},[34,909,910],{"class":190},"lesson.completed ",[34,912,202],{"class":175},[34,914,915],{"class":179}," true",[34,917,191],{"class":190},[15,919,920],{},"在 Vue 模板中显示：",[24,922,926],{"className":923,"code":924,"language":925,"meta":29,"style":29},"language-vue shiki shiki-themes github-light github-dark","\u003Ctemplate>\n \u003Carticle>\n \u003Ch2>{{ lesson.title }}\u003C\u002Fh2>\n \u003Cp>{{ lesson.minutes }} 分钟\u003C\u002Fp>\n \u003Cp>{{ lesson.completed ? \"已完成\" : \"未完成\" }}\u003C\u002Fp>\n \u003C\u002Farticle>\n\u003C\u002Ftemplate>\n","vue",[31,927,928,940,950,964,977,990,999],{"__ignoreMap":29},[34,929,930,933,937],{"class":36,"line":37},[34,931,932],{"class":190},"\u003C",[34,934,936],{"class":935},"s9eBZ","template",[34,938,939],{"class":190},">\n",[34,941,942,945,948],{"class":36,"line":43},[34,943,944],{"class":190}," \u003C",[34,946,947],{"class":935},"article",[34,949,939],{"class":190},[34,951,952,955,957,960,962],{"class":36,"line":49},[34,953,954],{"class":190}," \u003C",[34,956,19],{"class":935},[34,958,959],{"class":190},">{{ lesson.title }}\u003C\u002F",[34,961,19],{"class":935},[34,963,939],{"class":190},[34,965,966,968,970,973,975],{"class":36,"line":56},[34,967,954],{"class":190},[34,969,15],{"class":935},[34,971,972],{"class":190},">{{ lesson.minutes }} 分钟\u003C\u002F",[34,974,15],{"class":935},[34,976,939],{"class":190},[34,978,979,981,983,986,988],{"class":36,"line":62},[34,980,954],{"class":190},[34,982,15],{"class":935},[34,984,985],{"class":190},">{{ lesson.completed ? \"已完成\" : \"未完成\" }}\u003C\u002F",[34,987,15],{"class":935},[34,989,939],{"class":190},[34,991,992,995,997],{"class":36,"line":68},[34,993,994],{"class":190}," \u003C\u002F",[34,996,947],{"class":935},[34,998,939],{"class":190},[34,1000,1001,1004,1006],{"class":36,"line":74},[34,1002,1003],{"class":190},"\u003C\u002F",[34,1005,936],{"class":935},[34,1007,939],{"class":190},[253,1009,1010],{"id":1010},"对象的常见误区",[15,1012,1013],{},"误区一：属性名写错。",[24,1015,1017],{"className":166,"code":1016,"language":168,"meta":29,"style":29},"const lesson = { title: \"Vue\" };\n\nconsole.log(lesson.titel); \u002F\u002F undefined，因为 title 拼错了\n",[31,1018,1019,1035,1039],{"__ignoreMap":29},[34,1020,1021,1023,1025,1027,1030,1032],{"class":36,"line":37},[34,1022,176],{"class":175},[34,1024,824],{"class":179},[34,1026,183],{"class":175},[34,1028,1029],{"class":190}," { title: ",[34,1031,613],{"class":186},[34,1033,1034],{"class":190}," };\n",[34,1036,1037],{"class":36,"line":43},[34,1038,53],{"emptyLinePlaceholder":52},[34,1040,1041,1043,1045,1048],{"class":36,"line":49},[34,1042,882],{"class":190},[34,1044,885],{"class":395},[34,1046,1047],{"class":190},"(lesson.titel); ",[34,1049,1050],{"class":301},"\u002F\u002F undefined，因为 title 拼错了\n",[15,1052,1053],{},"误区二：以为对象会自动复制。",[24,1055,1057],{"className":166,"code":1056,"language":168,"meta":29,"style":29},"const a = { title: \"Vue\" };\nconst b = a;\n\nb.title = \"JavaScript\";\n\nconsole.log(a.title); \u002F\u002F JavaScript\n",[31,1058,1059,1074,1086,1090,1102,1106],{"__ignoreMap":29},[34,1060,1061,1063,1066,1068,1070,1072],{"class":36,"line":37},[34,1062,176],{"class":175},[34,1064,1065],{"class":179}," a",[34,1067,183],{"class":175},[34,1069,1029],{"class":190},[34,1071,613],{"class":186},[34,1073,1034],{"class":190},[34,1075,1076,1078,1081,1083],{"class":36,"line":43},[34,1077,176],{"class":175},[34,1079,1080],{"class":179}," b",[34,1082,183],{"class":175},[34,1084,1085],{"class":190}," a;\n",[34,1087,1088],{"class":36,"line":49},[34,1089,53],{"emptyLinePlaceholder":52},[34,1091,1092,1095,1097,1100],{"class":36,"line":56},[34,1093,1094],{"class":190},"b.title ",[34,1096,202],{"class":175},[34,1098,1099],{"class":186}," \"JavaScript\"",[34,1101,191],{"class":190},[34,1103,1104],{"class":36,"line":62},[34,1105,53],{"emptyLinePlaceholder":52},[34,1107,1108,1110,1112,1115],{"class":36,"line":68},[34,1109,882],{"class":190},[34,1111,885],{"class":395},[34,1113,1114],{"class":190},"(a.title); ",[34,1116,1117],{"class":301},"\u002F\u002F JavaScript\n",[15,1119,1120],{},"对象赋值给另一个变量时，复制的是引用，不是完整复制一个新对象。这在组件 props 和状态修改中很重要。",[15,1122,1123],{},"如果你想创建一个新对象：",[24,1125,1127],{"className":166,"code":1126,"language":168,"meta":29,"style":29},"const oldTask = { id: 1, title: \"学习 Vue\", done: false };\nconst newTask = { ...oldTask, done: true };\n",[31,1128,1129,1153],{"__ignoreMap":29},[34,1130,1131,1133,1136,1138,1140,1142,1144,1147,1149,1151],{"class":36,"line":37},[34,1132,176],{"class":175},[34,1134,1135],{"class":179}," oldTask",[34,1137,183],{"class":175},[34,1139,637],{"class":190},[34,1141,640],{"class":179},[34,1143,749],{"class":190},[34,1145,1146],{"class":186},"\"学习 Vue\"",[34,1148,755],{"class":190},[34,1150,702],{"class":179},[34,1152,1034],{"class":190},[34,1154,1155,1157,1160,1162,1165,1168,1171,1173],{"class":36,"line":43},[34,1156,176],{"class":175},[34,1158,1159],{"class":179}," newTask",[34,1161,183],{"class":175},[34,1163,1164],{"class":190}," { ",[34,1166,1167],{"class":175},"...",[34,1169,1170],{"class":190},"oldTask, done: ",[34,1172,758],{"class":179},[34,1174,1034],{"class":190},[15,1176,1177,1180,1181,1184],{},[31,1178,1179],{},"...oldTask"," 表示把旧对象的属性展开，再用 ",[31,1182,1183],{},"done: true"," 覆盖旧的 done。",[19,1186,1188],{"id":1187},"_4-数组描述一组东西","4. 数组：描述一组东西",[15,1190,1191],{},"数组适合描述“多个同类东西”。",[24,1193,1195],{"className":166,"code":1194,"language":168,"meta":29,"style":29},"const lessons = [\"JS 基础\", \"DOM\", \"Vue\"];\n",[31,1196,1197],{"__ignoreMap":29},[34,1198,1199,1201,1204,1206,1208,1211,1213,1216,1218,1220],{"class":36,"line":37},[34,1200,176],{"class":175},[34,1202,1203],{"class":179}," lessons",[34,1205,183],{"class":175},[34,1207,610],{"class":190},[34,1209,1210],{"class":186},"\"JS 基础\"",[34,1212,616],{"class":190},[34,1214,1215],{"class":186},"\"DOM\"",[34,1217,616],{"class":190},[34,1219,613],{"class":186},[34,1221,802],{"class":190},[15,1223,1224],{},"读取第一项：",[24,1226,1228],{"className":166,"code":1227,"language":168,"meta":29,"style":29},"console.log(lessons[0]); \u002F\u002F JS 基础\n",[31,1229,1230],{"__ignoreMap":29},[34,1231,1232,1234,1236,1239,1241,1244],{"class":36,"line":37},[34,1233,882],{"class":190},[34,1235,885],{"class":395},[34,1237,1238],{"class":190},"(lessons[",[34,1240,459],{"class":179},[34,1242,1243],{"class":190},"]); ",[34,1245,1246],{"class":301},"\u002F\u002F JS 基础\n",[15,1248,1249],{},"追加一项：",[24,1251,1253],{"className":166,"code":1252,"language":168,"meta":29,"style":29},"lessons.push(\"Router\");\n",[31,1254,1255],{"__ignoreMap":29},[34,1256,1257,1260,1263,1265,1268],{"class":36,"line":37},[34,1258,1259],{"class":190},"lessons.",[34,1261,1262],{"class":395},"push",[34,1264,456],{"class":190},[34,1266,1267],{"class":186},"\"Router\"",[34,1269,462],{"class":190},[15,1271,1272],{},"数组长度：",[24,1274,1276],{"className":166,"code":1275,"language":168,"meta":29,"style":29},"console.log(lessons.length);\n",[31,1277,1278],{"__ignoreMap":29},[34,1279,1280,1282,1284,1287,1290],{"class":36,"line":37},[34,1281,882],{"class":190},[34,1283,885],{"class":395},[34,1285,1286],{"class":190},"(lessons.",[34,1288,1289],{"class":179},"length",[34,1291,462],{"class":190},[253,1293,1295],{"id":1294},"vue-中最常见的数组方法","Vue 中最常见的数组方法",[15,1297,1298,1301],{},[31,1299,1300],{},"filter","：筛选。",[24,1303,1305],{"className":166,"code":1304,"language":168,"meta":29,"style":29},"const unfinishedTasks = tasks.filter((task) => !task.done);\n",[31,1306,1307],{"__ignoreMap":29},[34,1308,1309,1311,1314,1316,1319,1321,1324,1328,1331,1334,1337],{"class":36,"line":37},[34,1310,176],{"class":175},[34,1312,1313],{"class":179}," unfinishedTasks",[34,1315,183],{"class":175},[34,1317,1318],{"class":190}," tasks.",[34,1320,1300],{"class":395},[34,1322,1323],{"class":190},"((",[34,1325,1327],{"class":1326},"s4XuR","task",[34,1329,1330],{"class":190},") ",[34,1332,1333],{"class":175},"=>",[34,1335,1336],{"class":175}," !",[34,1338,1339],{"class":190},"task.done);\n",[15,1341,1342,1343,1346,1347,1350],{},"意思是：遍历 ",[31,1344,1345],{},"tasks","，只留下 ",[31,1348,1349],{},"done"," 为 false 的任务。",[15,1352,1353,1356],{},[31,1354,1355],{},"map","：把每一项变成另一种样子。",[24,1358,1360],{"className":166,"code":1359,"language":168,"meta":29,"style":29},"const titles = tasks.map((task) => task.title);\n",[31,1361,1362],{"__ignoreMap":29},[34,1363,1364,1366,1369,1371,1373,1375,1377,1379,1381,1383],{"class":36,"line":37},[34,1365,176],{"class":175},[34,1367,1368],{"class":179}," titles",[34,1370,183],{"class":175},[34,1372,1318],{"class":190},[34,1374,1355],{"class":395},[34,1376,1323],{"class":190},[34,1378,1327],{"class":1326},[34,1380,1330],{"class":190},[34,1382,1333],{"class":175},[34,1384,1385],{"class":190}," task.title);\n",[15,1387,1388,1391],{},[31,1389,1390],{},"find","：找到第一个符合条件的项。",[24,1393,1395],{"className":166,"code":1394,"language":168,"meta":29,"style":29},"const target = tasks.find((task) => task.id === 2);\n",[31,1396,1397],{"__ignoreMap":29},[34,1398,1399,1401,1404,1406,1408,1410,1412,1414,1416,1418,1421,1424,1427],{"class":36,"line":37},[34,1400,176],{"class":175},[34,1402,1403],{"class":179}," target",[34,1405,183],{"class":175},[34,1407,1318],{"class":190},[34,1409,1390],{"class":395},[34,1411,1323],{"class":190},[34,1413,1327],{"class":1326},[34,1415,1330],{"class":190},[34,1417,1333],{"class":175},[34,1419,1420],{"class":190}," task.id ",[34,1422,1423],{"class":175},"===",[34,1425,1426],{"class":179}," 2",[34,1428,462],{"class":190},[15,1430,1431,1434],{},[31,1432,1433],{},"some","：判断有没有至少一项符合条件。",[24,1436,1438],{"className":166,"code":1437,"language":168,"meta":29,"style":29},"const hasUnfinished = tasks.some((task) => !task.done);\n",[31,1439,1440],{"__ignoreMap":29},[34,1441,1442,1444,1447,1449,1451,1453,1455,1457,1459,1461,1463],{"class":36,"line":37},[34,1443,176],{"class":175},[34,1445,1446],{"class":179}," hasUnfinished",[34,1448,183],{"class":175},[34,1450,1318],{"class":190},[34,1452,1433],{"class":395},[34,1454,1323],{"class":190},[34,1456,1327],{"class":1326},[34,1458,1330],{"class":190},[34,1460,1333],{"class":175},[34,1462,1336],{"class":175},[34,1464,1339],{"class":190},[15,1466,1467,1470],{},[31,1468,1469],{},"reduce","：累积统计。",[24,1472,1474],{"className":166,"code":1473,"language":168,"meta":29,"style":29},"const totalMinutes = lessons.reduce((sum, lesson) => {\n return sum + lesson.minutes;\n}, 0);\n",[31,1475,1476,1506,1519],{"__ignoreMap":29},[34,1477,1478,1480,1483,1485,1488,1490,1492,1495,1497,1500,1502,1504],{"class":36,"line":37},[34,1479,176],{"class":175},[34,1481,1482],{"class":179}," totalMinutes",[34,1484,183],{"class":175},[34,1486,1487],{"class":190}," lessons.",[34,1489,1469],{"class":395},[34,1491,1323],{"class":190},[34,1493,1494],{"class":1326},"sum",[34,1496,616],{"class":190},[34,1498,1499],{"class":1326},"lesson",[34,1501,1330],{"class":190},[34,1503,1333],{"class":175},[34,1505,674],{"class":190},[34,1507,1508,1511,1514,1516],{"class":36,"line":43},[34,1509,1510],{"class":175}," return",[34,1512,1513],{"class":190}," sum ",[34,1515,223],{"class":175},[34,1517,1518],{"class":190}," lesson.minutes;\n",[34,1520,1521,1524,1526],{"class":36,"line":49},[34,1522,1523],{"class":190},"}, ",[34,1525,459],{"class":179},[34,1527,462],{"class":190},[253,1529,1531,1532],{"id":1530},"为什么数组适合配合-v-for","为什么数组适合配合 ",[31,1533,1534],{},"v-for",[15,1536,1537],{},"页面上的列表，本质上经常就是数组的可视化。",[15,1539,1540],{},"数据：",[24,1542,1544],{"className":166,"code":1543,"language":168,"meta":29,"style":29},"const lessons = [\n { id: \"js\", title: \"JS 基础\" },\n { id: \"dom\", title: \"DOM 到 Vue\" },\n];\n",[31,1545,1546,1556,1569,1583],{"__ignoreMap":29},[34,1547,1548,1550,1552,1554],{"class":36,"line":37},[34,1549,176],{"class":175},[34,1551,1203],{"class":179},[34,1553,183],{"class":175},[34,1555,739],{"class":190},[34,1557,1558,1560,1563,1565,1567],{"class":36,"line":43},[34,1559,744],{"class":190},[34,1561,1562],{"class":186},"\"js\"",[34,1564,749],{"class":190},[34,1566,1210],{"class":186},[34,1568,761],{"class":190},[34,1570,1571,1573,1576,1578,1581],{"class":36,"line":49},[34,1572,744],{"class":190},[34,1574,1575],{"class":186},"\"dom\"",[34,1577,749],{"class":190},[34,1579,1580],{"class":186},"\"DOM 到 Vue\"",[34,1582,761],{"class":190},[34,1584,1585],{"class":36,"line":56},[34,1586,802],{"class":190},[15,1588,1589],{},"模板：",[24,1591,1593],{"className":923,"code":1592,"language":925,"meta":29,"style":29},"\u003Cli v-for=\"lesson in lessons\" :key=\"lesson.id\">\n {{ lesson.title }}\n\u003C\u002Fli>\n",[31,1594,1595,1636,1641],{"__ignoreMap":29},[34,1596,1597,1599,1601,1604,1606,1609,1612,1615,1617,1619,1622,1625,1627,1629,1632,1634],{"class":36,"line":37},[34,1598,932],{"class":190},[34,1600,108],{"class":935},[34,1602,1603],{"class":175}," v-for",[34,1605,202],{"class":190},[34,1607,1608],{"class":186},"\"",[34,1610,1611],{"class":190},"lesson ",[34,1613,1614],{"class":175},"in",[34,1616,1203],{"class":190},[34,1618,1608],{"class":186},[34,1620,1621],{"class":190}," :",[34,1623,1624],{"class":395},"key",[34,1626,202],{"class":190},[34,1628,1608],{"class":186},[34,1630,1631],{"class":190},"lesson.id",[34,1633,1608],{"class":186},[34,1635,939],{"class":190},[34,1637,1638],{"class":36,"line":43},[34,1639,1640],{"class":190}," {{ lesson.title 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add",[34,1690,456],{"class":190},[34,1692,111],{"class":1326},[34,1694,616],{"class":190},[34,1696,1697],{"class":1326},"b",[34,1699,1700],{"class":190},") {\n",[34,1702,1703,1705,1708,1710],{"class":36,"line":43},[34,1704,1510],{"class":175},[34,1706,1707],{"class":190}," a ",[34,1709,223],{"class":175},[34,1711,1712],{"class":190}," b;\n",[34,1714,1715],{"class":36,"line":49},[34,1716,77],{"class":190},[34,1718,1719],{"class":36,"line":56},[34,1720,53],{"emptyLinePlaceholder":52},[34,1722,1723,1725,1728,1730,1732,1734,1736,1738,1740],{"class":36,"line":62},[34,1724,176],{"class":175},[34,1726,1727],{"class":179}," result",[34,1729,183],{"class":175},[34,1731,1688],{"class":395},[34,1733,456],{"class":190},[34,1735,768],{"class":179},[34,1737,616],{"class":190},[34,1739,786],{"class":179},[34,1741,462],{"class":190},[34,1743,1744,1746,1748,1751],{"class":36,"line":68},[34,1745,882],{"class":190},[34,1747,885],{"class":395},[34,1749,1750],{"class":190},"(result); ",[34,1752,1753],{"class":301},"\u002F\u002F 5\n",[15,1755,231],{},[105,1757,1758,1764,1771,1777],{},[108,1759,1760,1763],{},[31,1761,1762],{},"add"," 是函数名。",[108,1765,1766,258,1768,1770],{},[31,1767,111],{},[31,1769,1697],{}," 是参数。",[108,1772,1773,1776],{},[31,1774,1775],{},"return"," 是返回结果。",[108,1778,1779,1782],{},[31,1780,1781],{},"add(2, 3)"," 是调用函数。",[253,1784,1785],{"id":1785},"普通函数和箭头函数",[15,1787,1788],{},"普通函数：",[24,1790,1792],{"className":166,"code":1791,"language":168,"meta":29,"style":29},"function toggleTask(id) {\n console.log(id);\n}\n",[31,1793,1794,1808,1818],{"__ignoreMap":29},[34,1795,1796,1798,1801,1803,1806],{"class":36,"line":37},[34,1797,392],{"class":175},[34,1799,1800],{"class":395}," toggleTask",[34,1802,456],{"class":190},[34,1804,1805],{"class":1326},"id",[34,1807,1700],{"class":190},[34,1809,1810,1813,1815],{"class":36,"line":43},[34,1811,1812],{"class":190}," console.",[34,1814,885],{"class":395},[34,1816,1817],{"class":190},"(id);\n",[34,1819,1820],{"class":36,"line":49},[34,1821,77],{"class":190},[15,1823,1824],{},"箭头函数：",[24,1826,1828],{"className":166,"code":1827,"language":168,"meta":29,"style":29},"const toggleTask = (id) => {\n console.log(id);\n};\n",[31,1829,1830,1849,1857],{"__ignoreMap":29},[34,1831,1832,1834,1836,1838,1841,1843,1845,1847],{"class":36,"line":37},[34,1833,176],{"class":175},[34,1835,1800],{"class":395},[34,1837,183],{"class":175},[34,1839,1840],{"class":190}," (",[34,1842,1805],{"class":1326},[34,1844,1330],{"class":190},[34,1846,1333],{"class":175},[34,1848,674],{"class":190},[34,1850,1851,1853,1855],{"class":36,"line":43},[34,1852,1812],{"class":190},[34,1854,885],{"class":395},[34,1856,1817],{"class":190},[34,1858,1859],{"class":36,"line":49},[34,1860,719],{"class":190},[15,1862,1863],{},"数组方法里经常用箭头函数：",[24,1865,1867],{"className":166,"code":1866,"language":168,"meta":29,"style":29},"const completedTasks = tasks.filter((task) => task.done);\n",[31,1868,1869],{"__ignoreMap":29},[34,1870,1871,1873,1876,1878,1880,1882,1884,1886,1888,1890],{"class":36,"line":37},[34,1872,176],{"class":175},[34,1874,1875],{"class":179}," completedTasks",[34,1877,183],{"class":175},[34,1879,1318],{"class":190},[34,1881,1300],{"class":395},[34,1883,1323],{"class":190},[34,1885,1327],{"class":1326},[34,1887,1330],{"class":190},[34,1889,1333],{"class":175},[34,1891,1892],{"class":190}," task.done);\n",[15,1894,1895],{},"你可以读成：“对每个 task，返回它的 done 是否为真。”",[253,1897,1899],{"id":1898},"vue-中函数通常用来处理事件","Vue 中函数通常用来处理事件",[24,1901,1903],{"className":923,"code":1902,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { ref } from \"vue\";\n\nconst count = ref(0);\n\nfunction increase() {\n count.value++;\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cbutton @click=\"increase\">增加\u003C\u002Fbutton>\n \u003Cp>{{ count }}\u003C\u002Fp>\n\u003C\u002Ftemplate>\n",[31,1904,1905,1917,1929,1933,1949,1953,1961,1969,1974,1983,1988,1997,2020,2034],{"__ignoreMap":29},[34,1906,1907,1909,1912,1915],{"class":36,"line":37},[34,1908,932],{"class":190},[34,1910,1911],{"class":935},"script",[34,1913,1914],{"class":395}," setup",[34,1916,939],{"class":190},[34,1918,1919,1921,1923,1925,1927],{"class":36,"line":43},[34,1920,426],{"class":175},[34,1922,429],{"class":190},[34,1924,432],{"class":175},[34,1926,435],{"class":186},[34,1928,191],{"class":190},[34,1930,1931],{"class":36,"line":49},[34,1932,53],{"emptyLinePlaceholder":52},[34,1934,1935,1937,1939,1941,1943,1945,1947],{"class":36,"line":56},[34,1936,176],{"class":175},[34,1938,448],{"class":179},[34,1940,183],{"class":175},[34,1942,453],{"class":395},[34,1944,456],{"class":190},[34,1946,459],{"class":179},[34,1948,462],{"class":190},[34,1950,1951],{"class":36,"line":62},[34,1952,53],{"emptyLinePlaceholder":52},[34,1954,1955,1957,1959],{"class":36,"line":68},[34,1956,392],{"class":175},[34,1958,396],{"class":395},[34,1960,399],{"class":190},[34,1962,1963,1965,1967],{"class":36,"line":74},[34,1964,479],{"class":190},[34,1966,407],{"class":175},[34,1968,191],{"class":190},[34,1970,1972],{"class":36,"line":1971},8,[34,1973,77],{"class":190},[34,1975,1977,1979,1981],{"class":36,"line":1976},9,[34,1978,1003],{"class":190},[34,1980,1911],{"class":935},[34,1982,939],{"class":190},[34,1984,1986],{"class":36,"line":1985},10,[34,1987,53],{"emptyLinePlaceholder":52},[34,1989,1991,1993,1995],{"class":36,"line":1990},11,[34,1992,932],{"class":190},[34,1994,936],{"class":935},[34,1996,939],{"class":190},[34,1998,2000,2002,2005,2008,2010,2013,2016,2018],{"class":36,"line":1999},12,[34,2001,944],{"class":190},[34,2003,2004],{"class":935},"button",[34,2006,2007],{"class":395}," @click",[34,2009,202],{"class":190},[34,2011,2012],{"class":186},"\"increase\"",[34,2014,2015],{"class":190},">增加\u003C\u002F",[34,2017,2004],{"class":935},[34,2019,939],{"class":190},[34,2021,2023,2025,2027,2030,2032],{"class":36,"line":2022},13,[34,2024,944],{"class":190},[34,2026,15],{"class":935},[34,2028,2029],{"class":190},">{{ count }}\u003C\u002F",[34,2031,15],{"class":935},[34,2033,939],{"class":190},[34,2035,2037,2039,2041],{"class":36,"line":2036},14,[34,2038,1003],{"class":190},[34,2040,936],{"class":935},[34,2042,939],{"class":190},[15,2044,2045,2046,347,2049,2051,2052,2055],{},"点击按钮时，Vue 会调用 ",[31,2047,2048],{},"increase",[31,2050,2048],{}," 改变 ",[31,2053,2054],{},"count.value","，Vue 发现响应式状态变了，于是更新页面。",[19,2057,2059],{"id":2058},"_6-条件判断根据情况走不同逻辑","6. 条件判断：根据情况走不同逻辑",[24,2061,2063],{"className":166,"code":2062,"language":168,"meta":29,"style":29},"if (score >= 90) {\n console.log(\"优秀\");\n} else if (score >= 60) {\n console.log(\"及格\");\n} else {\n console.log(\"继续练习\");\n}\n",[31,2064,2065,2081,2094,2114,2127,2135,2148],{"__ignoreMap":29},[34,2066,2067,2070,2073,2076,2079],{"class":36,"line":37},[34,2068,2069],{"class":175},"if",[34,2071,2072],{"class":190}," (score ",[34,2074,2075],{"class":175},">=",[34,2077,2078],{"class":179}," 90",[34,2080,1700],{"class":190},[34,2082,2083,2085,2087,2089,2092],{"class":36,"line":43},[34,2084,1812],{"class":190},[34,2086,885],{"class":395},[34,2088,456],{"class":190},[34,2090,2091],{"class":186},"\"优秀\"",[34,2093,462],{"class":190},[34,2095,2096,2099,2102,2105,2107,2109,2112],{"class":36,"line":49},[34,2097,2098],{"class":190},"} ",[34,2100,2101],{"class":175},"else",[34,2103,2104],{"class":175}," if",[34,2106,2072],{"class":190},[34,2108,2075],{"class":175},[34,2110,2111],{"class":179}," 60",[34,2113,1700],{"class":190},[34,2115,2116,2118,2120,2122,2125],{"class":36,"line":56},[34,2117,1812],{"class":190},[34,2119,885],{"class":395},[34,2121,456],{"class":190},[34,2123,2124],{"class":186},"\"及格\"",[34,2126,462],{"class":190},[34,2128,2129,2131,2133],{"class":36,"line":62},[34,2130,2098],{"class":190},[34,2132,2101],{"class":175},[34,2134,674],{"class":190},[34,2136,2137,2139,2141,2143,2146],{"class":36,"line":68},[34,2138,1812],{"class":190},[34,2140,885],{"class":395},[34,2142,456],{"class":190},[34,2144,2145],{"class":186},"\"继续练习\"",[34,2147,462],{"class":190},[34,2149,2150],{"class":36,"line":74},[34,2151,77],{"class":190},[15,2153,2154],{},"Vue 模板中的条件渲染：",[24,2156,2158],{"className":923,"code":2157,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"loading\">加载中...\u003C\u002Fp>\n\u003Cp v-else-if=\"error\">{{ error }}\u003C\u002Fp>\n\u003Cp v-else>加载完成\u003C\u002Fp>\n",[31,2159,2160,2185,2210],{"__ignoreMap":29},[34,2161,2162,2164,2166,2169,2171,2173,2176,2178,2181,2183],{"class":36,"line":37},[34,2163,932],{"class":190},[34,2165,15],{"class":935},[34,2167,2168],{"class":175}," v-if",[34,2170,202],{"class":190},[34,2172,1608],{"class":186},[34,2174,2175],{"class":190},"loading",[34,2177,1608],{"class":186},[34,2179,2180],{"class":190},">加载中...\u003C\u002F",[34,2182,15],{"class":935},[34,2184,939],{"class":190},[34,2186,2187,2189,2191,2194,2196,2198,2201,2203,2206,2208],{"class":36,"line":43},[34,2188,932],{"class":190},[34,2190,15],{"class":935},[34,2192,2193],{"class":175}," v-else-if",[34,2195,202],{"class":190},[34,2197,1608],{"class":186},[34,2199,2200],{"class":190},"error",[34,2202,1608],{"class":186},[34,2204,2205],{"class":190},">{{ error }}\u003C\u002F",[34,2207,15],{"class":935},[34,2209,939],{"class":190},[34,2211,2212,2214,2216,2219,2222,2224],{"class":36,"line":49},[34,2213,932],{"class":190},[34,2215,15],{"class":935},[34,2217,2218],{"class":175}," v-else",[34,2220,2221],{"class":190},">加载完成\u003C\u002F",[34,2223,15],{"class":935},[34,2225,939],{"class":190},[15,2227,2228],{},"这不是三段都显示，而是按顺序判断：",[2230,2231,2232,2238,2244],"ol",{},[108,2233,2234,2235,2237],{},"如果 ",[31,2236,2175],{}," 为 true，显示加载中。",[108,2239,2240,2241,2243],{},"否则如果 ",[31,2242,2200],{}," 有值，显示错误。",[108,2245,2246],{},"否则显示加载完成。",[19,2248,2250],{"id":2249},"_7-解构从对象或模块里取东西","7. 解构：从对象或模块里取东西",[15,2252,2253],{},"对象解构：",[24,2255,2257],{"className":166,"code":2256,"language":168,"meta":29,"style":29},"const user = {\n name: \"Ada\",\n role: \"student\",\n};\n\nconst { name, role } = user;\n",[31,2258,2259,2269,2279,2289,2293,2297],{"__ignoreMap":29},[34,2260,2261,2263,2265,2267],{"class":36,"line":37},[34,2262,176],{"class":175},[34,2264,588],{"class":179},[34,2266,183],{"class":175},[34,2268,674],{"class":190},[34,2270,2271,2274,2277],{"class":36,"line":43},[34,2272,2273],{"class":190}," name: ",[34,2275,2276],{"class":186},"\"Ada\"",[34,2278,684],{"class":190},[34,2280,2281,2284,2287],{"class":36,"line":49},[34,2282,2283],{"class":190}," role: ",[34,2285,2286],{"class":186},"\"student\"",[34,2288,684],{"class":190},[34,2290,2291],{"class":36,"line":56},[34,2292,719],{"class":190},[34,2294,2295],{"class":36,"line":62},[34,2296,53],{"emptyLinePlaceholder":52},[34,2298,2299,2301,2303,2306,2308,2311,2314,2316],{"class":36,"line":68},[34,2300,176],{"class":175},[34,2302,1164],{"class":190},[34,2304,2305],{"class":179},"name",[34,2307,616],{"class":190},[34,2309,2310],{"class":179},"role",[34,2312,2313],{"class":190}," } ",[34,2315,202],{"class":175},[34,2317,2318],{"class":190}," user;\n",[15,2320,2321],{},"等价于：",[24,2323,2325],{"className":166,"code":2324,"language":168,"meta":29,"style":29},"const name = user.name;\nconst role = user.role;\n",[31,2326,2327,2339],{"__ignoreMap":29},[34,2328,2329,2331,2334,2336],{"class":36,"line":37},[34,2330,176],{"class":175},[34,2332,2333],{"class":179}," name",[34,2335,183],{"class":175},[34,2337,2338],{"class":190}," user.name;\n",[34,2340,2341,2343,2346,2348],{"class":36,"line":43},[34,2342,176],{"class":175},[34,2344,2345],{"class":179}," role",[34,2347,183],{"class":175},[34,2349,2350],{"class":190}," user.role;\n",[15,2352,2353],{},"数组解构：",[24,2355,2357],{"className":166,"code":2356,"language":168,"meta":29,"style":29},"const colors = [\"red\", \"green\"];\nconst [firstColor, secondColor] = colors;\n",[31,2358,2359,2380],{"__ignoreMap":29},[34,2360,2361,2363,2366,2368,2370,2373,2375,2378],{"class":36,"line":37},[34,2362,176],{"class":175},[34,2364,2365],{"class":179}," colors",[34,2367,183],{"class":175},[34,2369,610],{"class":190},[34,2371,2372],{"class":186},"\"red\"",[34,2374,616],{"class":190},[34,2376,2377],{"class":186},"\"green\"",[34,2379,802],{"class":190},[34,2381,2382,2384,2386,2389,2391,2394,2397,2399],{"class":36,"line":43},[34,2383,176],{"class":175},[34,2385,610],{"class":190},[34,2387,2388],{"class":179},"firstColor",[34,2390,616],{"class":190},[34,2392,2393],{"class":179},"secondColor",[34,2395,2396],{"class":190},"] ",[34,2398,202],{"class":175},[34,2400,2401],{"class":190}," colors;\n",[15,2403,2404],{},"Vue 中经常看到：",[24,2406,2408],{"className":166,"code":2407,"language":168,"meta":29,"style":29},"import { ref, computed } from \"vue\";\n",[31,2409,2410],{"__ignoreMap":29},[34,2411,2412,2414,2417,2419,2421],{"class":36,"line":37},[34,2413,426],{"class":175},[34,2415,2416],{"class":190}," { ref, computed } ",[34,2418,432],{"class":175},[34,2420,435],{"class":186},[34,2422,191],{"class":190},[15,2424,2425,2426,2428,2429,258,2431,347],{},"这也是一种“按名字取出”的感觉：从 ",[31,2427,925],{}," 这个包导出的内容里，取出 ",[31,2430,145],{},[31,2432,149],{},[19,2434,2436],{"id":2435},"_8-模块为什么要导出和导入","8. 模块：为什么要导出和导入",[15,2438,2439],{},"这是很多初学者会卡住的地方。我们慢慢讲。",[253,2441,2443],{"id":2442},"_81-为什么需要模块","8.1 为什么需要模块",[15,2445,2446],{},"如果所有代码都写在一个文件里，文件会越来越长：",[24,2448,2450],{"className":166,"code":2449,"language":168,"meta":29,"style":29},"function add(a, b) {\n return a + b;\n}\n\nfunction subtract(a, b) {\n return a - b;\n}\n\nfunction formatMoney(value) {\n return `¥${value.toFixed(2)}`;\n}\n\nfunction createTask(title) {\n return { id: Date.now(), title, done: false };\n}\n",[31,2451,2452,2468,2478,2482,2486,2503,2514,2518,2522,2536,2563,2567,2571,2584,2601],{"__ignoreMap":29},[34,2453,2454,2456,2458,2460,2462,2464,2466],{"class":36,"line":37},[34,2455,392],{"class":175},[34,2457,1688],{"class":395},[34,2459,456],{"class":190},[34,2461,111],{"class":1326},[34,2463,616],{"class":190},[34,2465,1697],{"class":1326},[34,2467,1700],{"class":190},[34,2469,2470,2472,2474,2476],{"class":36,"line":43},[34,2471,1510],{"class":175},[34,2473,1707],{"class":190},[34,2475,223],{"class":175},[34,2477,1712],{"class":190},[34,2479,2480],{"class":36,"line":49},[34,2481,77],{"class":190},[34,2483,2484],{"class":36,"line":56},[34,2485,53],{"emptyLinePlaceholder":52},[34,2487,2488,2490,2493,2495,2497,2499,2501],{"class":36,"line":62},[34,2489,392],{"class":175},[34,2491,2492],{"class":395}," subtract",[34,2494,456],{"class":190},[34,2496,111],{"class":1326},[34,2498,616],{"class":190},[34,2500,1697],{"class":1326},[34,2502,1700],{"class":190},[34,2504,2505,2507,2509,2512],{"class":36,"line":68},[34,2506,1510],{"class":175},[34,2508,1707],{"class":190},[34,2510,2511],{"class":175},"-",[34,2513,1712],{"class":190},[34,2515,2516],{"class":36,"line":74},[34,2517,77],{"class":190},[34,2519,2520],{"class":36,"line":1971},[34,2521,53],{"emptyLinePlaceholder":52},[34,2523,2524,2526,2529,2531,2534],{"class":36,"line":1976},[34,2525,392],{"class":175},[34,2527,2528],{"class":395}," formatMoney",[34,2530,456],{"class":190},[34,2532,2533],{"class":1326},"value",[34,2535,1700],{"class":190},[34,2537,2538,2540,2543,2545,2548,2551,2553,2555,2558,2561],{"class":36,"line":1985},[34,2539,1510],{"class":175},[34,2541,2542],{"class":186}," `¥${",[34,2544,2533],{"class":190},[34,2546,2547],{"class":186},".",[34,2549,2550],{"class":395},"toFixed",[34,2552,456],{"class":186},[34,2554,768],{"class":179},[34,2556,2557],{"class":186},")",[34,2559,2560],{"class":186},"}`",[34,2562,191],{"class":190},[34,2564,2565],{"class":36,"line":1990},[34,2566,77],{"class":190},[34,2568,2569],{"class":36,"line":1999},[34,2570,53],{"emptyLinePlaceholder":52},[34,2572,2573,2575,2578,2580,2582],{"class":36,"line":2022},[34,2574,392],{"class":175},[34,2576,2577],{"class":395}," createTask",[34,2579,456],{"class":190},[34,2581,1665],{"class":1326},[34,2583,1700],{"class":190},[34,2585,2586,2588,2591,2594,2597,2599],{"class":36,"line":2036},[34,2587,1510],{"class":175},[34,2589,2590],{"class":190}," { id: Date.",[34,2592,2593],{"class":395},"now",[34,2595,2596],{"class":190},"(), title, done: ",[34,2598,702],{"class":179},[34,2600,1034],{"class":190},[34,2602,2604],{"class":36,"line":2603},15,[34,2605,77],{"class":190},[15,2607,2608],{},"一开始没问题，但项目变大后会出现几个问题：",[105,2610,2611,2614,2617,2620],{},[108,2612,2613],{},"文件太长，找代码困难。",[108,2615,2616],{},"工具函数、组件、请求逻辑混在一起。",[108,2618,2619],{},"不知道哪些函数是给外部用的，哪些只是当前文件内部用的。",[108,2621,2622],{},"多个页面都想用同一个函数，只能复制粘贴。",[15,2624,2625,2626],{},"模块的作用就是：",[2627,2628,2629],"strong",{},"把代码拆成多个文件，并明确哪些内容可以给别的文件使用。",[15,2631,2632],{},"例如：",[24,2634,2639],{"className":2635,"code":2637,"language":2638,"meta":29},[2636],"language-text","src\u002F\n math.js\n format.js\n main.js\n","text",[31,2640,2637],{"__ignoreMap":29},[15,2642,2643,2646,2647,2650,2651,2654],{},[31,2644,2645],{},"math.js"," 放数学函数，",[31,2648,2649],{},"format.js"," 放格式化函数，",[31,2652,2653],{},"main.js"," 使用它们。",[253,2656,2658],{"id":2657},"_82-每个模块都有自己的作用域","8.2 每个模块都有自己的作用域",[15,2660,2661,2662,2664],{},"假设有一个文件 ",[31,2663,2645],{},"：",[24,2666,2668],{"className":166,"code":2667,"language":168,"meta":29,"style":29},"\u002F\u002F math.js\nfunction add(a, b) {\n return a + b;\n}\n",[31,2669,2670,2675,2691,2701],{"__ignoreMap":29},[34,2671,2672],{"class":36,"line":37},[34,2673,2674],{"class":301},"\u002F\u002F math.js\n",[34,2676,2677,2679,2681,2683,2685,2687,2689],{"class":36,"line":43},[34,2678,392],{"class":175},[34,2680,1688],{"class":395},[34,2682,456],{"class":190},[34,2684,111],{"class":1326},[34,2686,616],{"class":190},[34,2688,1697],{"class":1326},[34,2690,1700],{"class":190},[34,2692,2693,2695,2697,2699],{"class":36,"line":49},[34,2694,1510],{"class":175},[34,2696,1707],{"class":190},[34,2698,223],{"class":175},[34,2700,1712],{"class":190},[34,2702,2703],{"class":36,"line":56},[34,2704,77],{"class":190},[15,2706,2707,2708,2664],{},"再有一个文件 ",[31,2709,2653],{},[24,2711,2713],{"className":166,"code":2712,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nconsole.log(add(1, 2));\n",[31,2714,2715,2720],{"__ignoreMap":29},[34,2716,2717],{"class":36,"line":37},[34,2718,2719],{"class":301},"\u002F\u002F main.js\n",[34,2721,2722,2724,2726,2728,2730,2732,2734,2736,2738],{"class":36,"line":43},[34,2723,882],{"class":190},[34,2725,885],{"class":395},[34,2727,456],{"class":190},[34,2729,1762],{"class":395},[34,2731,456],{"class":190},[34,2733,640],{"class":179},[34,2735,616],{"class":190},[34,2737,768],{"class":179},[34,2739,2740],{"class":190},"));\n",[15,2742,2743,2744,2746,2747,2749,2750,2752,2753,2755],{},"这通常不能工作，因为 ",[31,2745,1762],{}," 只存在于 ",[31,2748,2645],{}," 这个文件内部。",[31,2751,2653],{}," 不会自动知道 ",[31,2754,2645],{}," 里有什么。",[15,2757,2758,2759],{},"模块的默认规则是：",[2627,2760,2761],{},"一个文件里的变量和函数，不会自动暴露给另一个文件。",[15,2763,2764],{},"这很好，因为它避免了名字互相污染。",[15,2766,2767,2768,2664],{},"例如两个文件里都可以有自己的 ",[31,2769,2770],{},"count",[24,2772,2774],{"className":166,"code":2773,"language":168,"meta":29,"style":29},"\u002F\u002F a.js\nconst count = 1;\n",[31,2775,2776,2781],{"__ignoreMap":29},[34,2777,2778],{"class":36,"line":37},[34,2779,2780],{"class":301},"\u002F\u002F a.js\n",[34,2782,2783,2785,2787,2789,2791],{"class":36,"line":43},[34,2784,176],{"class":175},[34,2786,448],{"class":179},[34,2788,183],{"class":175},[34,2790,226],{"class":179},[34,2792,191],{"class":190},[24,2794,2796],{"className":166,"code":2795,"language":168,"meta":29,"style":29},"\u002F\u002F b.js\nconst count = 100;\n",[31,2797,2798,2803],{"__ignoreMap":29},[34,2799,2800],{"class":36,"line":37},[34,2801,2802],{"class":301},"\u002F\u002F b.js\n",[34,2804,2805,2807,2809,2811,2814],{"class":36,"line":43},[34,2806,176],{"class":175},[34,2808,448],{"class":179},[34,2810,183],{"class":175},[34,2812,2813],{"class":179}," 100",[34,2815,191],{"class":190},[15,2817,2818],{},"它们互不影响。",[253,2820,2822,2823,2826],{"id":2821},"_83-export-是什么意思","8.3 ",[31,2824,2825],{},"export"," 是什么意思",[15,2828,2829,2831,2832],{},[31,2830,2825],{}," 的意思是：",[2627,2833,2834],{},"我允许别的模块导入这个东西。",[24,2836,2838],{"className":166,"code":2837,"language":168,"meta":29,"style":29},"\u002F\u002F math.js\nexport function add(a, b) {\n return a + b;\n}\n",[31,2839,2840,2844,2863,2873],{"__ignoreMap":29},[34,2841,2842],{"class":36,"line":37},[34,2843,2674],{"class":301},[34,2845,2846,2848,2851,2853,2855,2857,2859,2861],{"class":36,"line":43},[34,2847,2825],{"class":175},[34,2849,2850],{"class":175}," function",[34,2852,1688],{"class":395},[34,2854,456],{"class":190},[34,2856,111],{"class":1326},[34,2858,616],{"class":190},[34,2860,1697],{"class":1326},[34,2862,1700],{"class":190},[34,2864,2865,2867,2869,2871],{"class":36,"line":49},[34,2866,1510],{"class":175},[34,2868,1707],{"class":190},[34,2870,223],{"class":175},[34,2872,1712],{"class":190},[34,2874,2875],{"class":36,"line":56},[34,2876,77],{"class":190},[15,2878,2879],{},"这句话做了两件事：",[2230,2881,2882,2890],{},[108,2883,2884,2885,2887,2888,347],{},"在 ",[31,2886,2645],{}," 里声明一个函数 ",[31,2889,1762],{},[108,2891,2892,2893,2895],{},"把 ",[31,2894,1762],{}," 标记为“可以被别的文件导入”。",[15,2897,2898,2899,2901],{},"然后 ",[31,2900,2653],{}," 才能写：",[24,2903,2905],{"className":166,"code":2904,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nimport { add } from \".\u002Fmath.js\";\n\nconsole.log(add(1, 2)); \u002F\u002F 3\n",[31,2906,2907,2911,2925,2929],{"__ignoreMap":29},[34,2908,2909],{"class":36,"line":37},[34,2910,2719],{"class":301},[34,2912,2913,2915,2918,2920,2923],{"class":36,"line":43},[34,2914,426],{"class":175},[34,2916,2917],{"class":190}," { add } ",[34,2919,432],{"class":175},[34,2921,2922],{"class":186}," \".\u002Fmath.js\"",[34,2924,191],{"class":190},[34,2926,2927],{"class":36,"line":49},[34,2928,53],{"emptyLinePlaceholder":52},[34,2930,2931,2933,2935,2937,2939,2941,2943,2945,2947,2950],{"class":36,"line":56},[34,2932,882],{"class":190},[34,2934,885],{"class":395},[34,2936,456],{"class":190},[34,2938,1762],{"class":395},[34,2940,456],{"class":190},[34,2942,640],{"class":179},[34,2944,616],{"class":190},[34,2946,768],{"class":179},[34,2948,2949],{"class":190},")); ",[34,2951,2952],{"class":301},"\u002F\u002F 3\n",[253,2954,2956,2957,2959,2960],{"id":2955},"_84-不写-export-能不能-import","8.4 不写 ",[31,2958,2825],{}," 能不能 ",[31,2961,426],{},[15,2963,2964],{},"不能按这种方式导入。",[24,2966,2967],{"className":166,"code":2667,"language":168,"meta":29,"style":29},[31,2968,2969,2973,2989,2999],{"__ignoreMap":29},[34,2970,2971],{"class":36,"line":37},[34,2972,2674],{"class":301},[34,2974,2975,2977,2979,2981,2983,2985,2987],{"class":36,"line":43},[34,2976,392],{"class":175},[34,2978,1688],{"class":395},[34,2980,456],{"class":190},[34,2982,111],{"class":1326},[34,2984,616],{"class":190},[34,2986,1697],{"class":1326},[34,2988,1700],{"class":190},[34,2990,2991,2993,2995,2997],{"class":36,"line":49},[34,2992,1510],{"class":175},[34,2994,1707],{"class":190},[34,2996,223],{"class":175},[34,2998,1712],{"class":190},[34,3000,3001],{"class":36,"line":56},[34,3002,77],{"class":190},[24,3004,3006],{"className":166,"code":3005,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nimport { add } from \".\u002Fmath.js\";\n",[31,3007,3008,3012],{"__ignoreMap":29},[34,3009,3010],{"class":36,"line":37},[34,3011,2719],{"class":301},[34,3013,3014,3016,3018,3020,3022],{"class":36,"line":43},[34,3015,426],{"class":175},[34,3017,2917],{"class":190},[34,3019,432],{"class":175},[34,3021,2922],{"class":186},[34,3023,191],{"class":190},[15,3025,3026,3027,3029,3030,3032],{},"这会报错，意思大概是：",[31,3028,2645],{}," 没有导出名为 ",[31,3031,1762],{}," 的内容。",[15,3034,3035],{},"为什么？因为模块系统只允许你导入对方明确导出的内容。它不会偷偷把对方文件里的所有变量都拿出来。",[15,3037,3038],{},"这是一种保护：",[105,3040,3041,3044,3047],{},[108,3042,3043],{},"文件内部的临时变量不会暴露。",[108,3045,3046],{},"外部文件只能使用公开 API。",[108,3048,3049],{},"改内部实现时，不容易影响外部。",[253,3051,3053,3054,3056,3057],{"id":3052},"_85-export-会不会影响原来的-mathjs","8.5 ",[31,3055,2825],{}," 会不会影响原来的 ",[31,3058,2645],{},[15,3060,3061],{},"会有影响，但不是“改变函数功能”的影响。",[24,3063,3065],{"className":166,"code":3064,"language":168,"meta":29,"style":29},"export function add(a, b) {\n return a + b;\n}\n",[31,3066,3067,3085,3095],{"__ignoreMap":29},[34,3068,3069,3071,3073,3075,3077,3079,3081,3083],{"class":36,"line":37},[34,3070,2825],{"class":175},[34,3072,2850],{"class":175},[34,3074,1688],{"class":395},[34,3076,456],{"class":190},[34,3078,111],{"class":1326},[34,3080,616],{"class":190},[34,3082,1697],{"class":1326},[34,3084,1700],{"class":190},[34,3086,3087,3089,3091,3093],{"class":36,"line":43},[34,3088,1510],{"class":175},[34,3090,1707],{"class":190},[34,3092,223],{"class":175},[34,3094,1712],{"class":190},[34,3096,3097],{"class":36,"line":49},[34,3098,77],{"class":190},[15,3100,3101,3102,3104,3105,3107],{},"对 ",[31,3103,2645],{}," 自己来说，",[31,3106,1762],{}," 仍然可以正常使用：",[24,3109,3111],{"className":166,"code":3110,"language":168,"meta":29,"style":29},"\u002F\u002F math.js\nexport function add(a, b) {\n return a + b;\n}\n\nconst result = add(1, 2);\nconsole.log(result);\n",[31,3112,3113,3117,3135,3145,3149,3153,3173],{"__ignoreMap":29},[34,3114,3115],{"class":36,"line":37},[34,3116,2674],{"class":301},[34,3118,3119,3121,3123,3125,3127,3129,3131,3133],{"class":36,"line":43},[34,3120,2825],{"class":175},[34,3122,2850],{"class":175},[34,3124,1688],{"class":395},[34,3126,456],{"class":190},[34,3128,111],{"class":1326},[34,3130,616],{"class":190},[34,3132,1697],{"class":1326},[34,3134,1700],{"class":190},[34,3136,3137,3139,3141,3143],{"class":36,"line":49},[34,3138,1510],{"class":175},[34,3140,1707],{"class":190},[34,3142,223],{"class":175},[34,3144,1712],{"class":190},[34,3146,3147],{"class":36,"line":56},[34,3148,77],{"class":190},[34,3150,3151],{"class":36,"line":62},[34,3152,53],{"emptyLinePlaceholder":52},[34,3154,3155,3157,3159,3161,3163,3165,3167,3169,3171],{"class":36,"line":68},[34,3156,176],{"class":175},[34,3158,1727],{"class":179},[34,3160,183],{"class":175},[34,3162,1688],{"class":395},[34,3164,456],{"class":190},[34,3166,640],{"class":179},[34,3168,616],{"class":190},[34,3170,768],{"class":179},[34,3172,462],{"class":190},[34,3174,3175,3177,3179],{"class":36,"line":74},[34,3176,882],{"class":190},[34,3178,885],{"class":395},[34,3180,3181],{"class":190},"(result);\n",[15,3183,3184,3186,3187,3189],{},[31,3185,2825],{}," 不会改变 ",[31,3188,1762],{}," 的计算逻辑。它只是多加了一个身份：这个函数可以被别的模块导入。",[15,3191,3192],{},"你可以把它想成：",[24,3194,3197],{"className":3195,"code":3196,"language":2638,"meta":29},[2636],"没有 export：这是当前文件自己的函数。\n有 export：这是当前文件自己的函数，同时也公开给外部使用。\n",[31,3198,3196],{"__ignoreMap":29},[253,3200,3202,3203,3205],{"id":3201},"_86-import-之后怎么调用","8.6 ",[31,3204,426],{}," 之后怎么调用",[15,3207,3208],{},"导入后，就像使用当前文件里的普通函数一样调用。",[24,3210,3212],{"className":166,"code":3211,"language":168,"meta":29,"style":29},"import { add } from \".\u002Fmath.js\";\n\nconst total = add(10, 20);\nconsole.log(total);\n",[31,3213,3214,3226,3230,3252],{"__ignoreMap":29},[34,3215,3216,3218,3220,3222,3224],{"class":36,"line":37},[34,3217,426],{"class":175},[34,3219,2917],{"class":190},[34,3221,432],{"class":175},[34,3223,2922],{"class":186},[34,3225,191],{"class":190},[34,3227,3228],{"class":36,"line":43},[34,3229,53],{"emptyLinePlaceholder":52},[34,3231,3232,3234,3236,3238,3240,3242,3245,3247,3250],{"class":36,"line":49},[34,3233,176],{"class":175},[34,3235,554],{"class":179},[34,3237,183],{"class":175},[34,3239,1688],{"class":395},[34,3241,456],{"class":190},[34,3243,3244],{"class":179},"10",[34,3246,616],{"class":190},[34,3248,3249],{"class":179},"20",[34,3251,462],{"class":190},[34,3253,3254,3256,3258],{"class":36,"line":56},[34,3255,882],{"class":190},[34,3257,885],{"class":395},[34,3259,3260],{"class":190},"(total);\n",[15,3262,3263],{},"如果导入的是变量：",[24,3265,3267],{"className":166,"code":3266,"language":168,"meta":29,"style":29},"\u002F\u002F config.js\nexport const appName = \"Vue Tutorial\";\n",[31,3268,3269,3274],{"__ignoreMap":29},[34,3270,3271],{"class":36,"line":37},[34,3272,3273],{"class":301},"\u002F\u002F config.js\n",[34,3275,3276,3278,3281,3284,3286,3289],{"class":36,"line":43},[34,3277,2825],{"class":175},[34,3279,3280],{"class":175}," const",[34,3282,3283],{"class":179}," appName",[34,3285,183],{"class":175},[34,3287,3288],{"class":186}," \"Vue Tutorial\"",[34,3290,191],{"class":190},[24,3292,3294],{"className":166,"code":3293,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nimport { appName } from \".\u002Fconfig.js\";\n\nconsole.log(appName);\n",[31,3295,3296,3300,3314,3318],{"__ignoreMap":29},[34,3297,3298],{"class":36,"line":37},[34,3299,2719],{"class":301},[34,3301,3302,3304,3307,3309,3312],{"class":36,"line":43},[34,3303,426],{"class":175},[34,3305,3306],{"class":190}," { appName } ",[34,3308,432],{"class":175},[34,3310,3311],{"class":186}," \".\u002Fconfig.js\"",[34,3313,191],{"class":190},[34,3315,3316],{"class":36,"line":49},[34,3317,53],{"emptyLinePlaceholder":52},[34,3319,3320,3322,3324],{"class":36,"line":56},[34,3321,882],{"class":190},[34,3323,885],{"class":395},[34,3325,3326],{"class":190},"(appName);\n",[15,3328,3329],{},"如果导入的是对象：",[24,3331,3333],{"className":166,"code":3332,"language":168,"meta":29,"style":29},"\u002F\u002F user.js\nexport const user = {\n name: \"Ada\",\n role: \"student\",\n};\n",[31,3334,3335,3340,3352,3360,3368],{"__ignoreMap":29},[34,3336,3337],{"class":36,"line":37},[34,3338,3339],{"class":301},"\u002F\u002F user.js\n",[34,3341,3342,3344,3346,3348,3350],{"class":36,"line":43},[34,3343,2825],{"class":175},[34,3345,3280],{"class":175},[34,3347,588],{"class":179},[34,3349,183],{"class":175},[34,3351,674],{"class":190},[34,3353,3354,3356,3358],{"class":36,"line":49},[34,3355,2273],{"class":190},[34,3357,2276],{"class":186},[34,3359,684],{"class":190},[34,3361,3362,3364,3366],{"class":36,"line":56},[34,3363,2283],{"class":190},[34,3365,2286],{"class":186},[34,3367,684],{"class":190},[34,3369,3370],{"class":36,"line":62},[34,3371,719],{"class":190},[24,3373,3375],{"className":166,"code":3374,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nimport { user } from \".\u002Fuser.js\";\n\nconsole.log(user.name);\n",[31,3376,3377,3381,3395,3399],{"__ignoreMap":29},[34,3378,3379],{"class":36,"line":37},[34,3380,2719],{"class":301},[34,3382,3383,3385,3388,3390,3393],{"class":36,"line":43},[34,3384,426],{"class":175},[34,3386,3387],{"class":190}," { user } ",[34,3389,432],{"class":175},[34,3391,3392],{"class":186}," \".\u002Fuser.js\"",[34,3394,191],{"class":190},[34,3396,3397],{"class":36,"line":49},[34,3398,53],{"emptyLinePlaceholder":52},[34,3400,3401,3403,3405],{"class":36,"line":56},[34,3402,882],{"class":190},[34,3404,885],{"class":395},[34,3406,3407],{"class":190},"(user.name);\n",[15,3409,3410],{},"如果导入的是 Vue 组件：",[24,3412,3414],{"className":923,"code":3413,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport LessonCard from \".\u002Fcomponents\u002FLessonCard.vue\";\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003CLessonCard \u002F>\n\u003C\u002Ftemplate>\n",[31,3415,3416,3426,3440,3448,3452,3460,3470],{"__ignoreMap":29},[34,3417,3418,3420,3422,3424],{"class":36,"line":37},[34,3419,932],{"class":190},[34,3421,1911],{"class":935},[34,3423,1914],{"class":395},[34,3425,939],{"class":190},[34,3427,3428,3430,3433,3435,3438],{"class":36,"line":43},[34,3429,426],{"class":175},[34,3431,3432],{"class":190}," LessonCard ",[34,3434,432],{"class":175},[34,3436,3437],{"class":186}," \".\u002Fcomponents\u002FLessonCard.vue\"",[34,3439,191],{"class":190},[34,3441,3442,3444,3446],{"class":36,"line":49},[34,3443,1003],{"class":190},[34,3445,1911],{"class":935},[34,3447,939],{"class":190},[34,3449,3450],{"class":36,"line":56},[34,3451,53],{"emptyLinePlaceholder":52},[34,3453,3454,3456,3458],{"class":36,"line":62},[34,3455,932],{"class":190},[34,3457,936],{"class":935},[34,3459,939],{"class":190},[34,3461,3462,3464,3467],{"class":36,"line":68},[34,3463,944],{"class":190},[34,3465,3466],{"class":935},"LessonCard",[34,3468,3469],{"class":190}," \u002F>\n",[34,3471,3472,3474,3476],{"class":36,"line":74},[34,3473,1003],{"class":190},[34,3475,936],{"class":935},[34,3477,939],{"class":190},[15,3479,2884,3480,3482],{},[31,3481,141],{}," 中，导入的组件可以直接在模板中使用。",[253,3484,3486],{"id":3485},"_87-命名导出一个文件导出多个东西","8.7 命名导出：一个文件导出多个东西",[24,3488,3490],{"className":166,"code":3489,"language":168,"meta":29,"style":29},"\u002F\u002F math.js\nexport function add(a, b) {\n return a + b;\n}\n\nexport function subtract(a, b) {\n return a - b;\n}\n\nexport const PI = 3.14159;\n",[31,3491,3492,3496,3514,3524,3528,3532,3550,3560,3564,3568],{"__ignoreMap":29},[34,3493,3494],{"class":36,"line":37},[34,3495,2674],{"class":301},[34,3497,3498,3500,3502,3504,3506,3508,3510,3512],{"class":36,"line":43},[34,3499,2825],{"class":175},[34,3501,2850],{"class":175},[34,3503,1688],{"class":395},[34,3505,456],{"class":190},[34,3507,111],{"class":1326},[34,3509,616],{"class":190},[34,3511,1697],{"class":1326},[34,3513,1700],{"class":190},[34,3515,3516,3518,3520,3522],{"class":36,"line":49},[34,3517,1510],{"class":175},[34,3519,1707],{"class":190},[34,3521,223],{"class":175},[34,3523,1712],{"class":190},[34,3525,3526],{"class":36,"line":56},[34,3527,77],{"class":190},[34,3529,3530],{"class":36,"line":62},[34,3531,53],{"emptyLinePlaceholder":52},[34,3533,3534,3536,3538,3540,3542,3544,3546,3548],{"class":36,"line":68},[34,3535,2825],{"class":175},[34,3537,2850],{"class":175},[34,3539,2492],{"class":395},[34,3541,456],{"class":190},[34,3543,111],{"class":1326},[34,3545,616],{"class":190},[34,3547,1697],{"class":1326},[34,3549,1700],{"class":190},[34,3551,3552,3554,3556,3558],{"class":36,"line":74},[34,3553,1510],{"class":175},[34,3555,1707],{"class":190},[34,3557,2511],{"class":175},[34,3559,1712],{"class":190},[34,3561,3562],{"class":36,"line":1971},[34,3563,77],{"class":190},[34,3565,3566],{"class":36,"line":1976},[34,3567,53],{"emptyLinePlaceholder":52},[34,3569,3570,3572,3574,3577,3579,3582],{"class":36,"line":1985},[34,3571,2825],{"class":175},[34,3573,3280],{"class":175},[34,3575,3576],{"class":179}," PI",[34,3578,183],{"class":175},[34,3580,3581],{"class":179}," 3.14159",[34,3583,191],{"class":190},[15,3585,3586],{},"导入：",[24,3588,3590],{"className":166,"code":3589,"language":168,"meta":29,"style":29},"import { add, subtract, PI } from \".\u002Fmath.js\";\n\nconsole.log(add(3, 2));\nconsole.log(subtract(3, 2));\nconsole.log(PI);\n",[31,3591,3592,3605,3609,3629,3650],{"__ignoreMap":29},[34,3593,3594,3596,3599,3601,3603],{"class":36,"line":37},[34,3595,426],{"class":175},[34,3597,3598],{"class":190}," { add, subtract, PI } ",[34,3600,432],{"class":175},[34,3602,2922],{"class":186},[34,3604,191],{"class":190},[34,3606,3607],{"class":36,"line":43},[34,3608,53],{"emptyLinePlaceholder":52},[34,3610,3611,3613,3615,3617,3619,3621,3623,3625,3627],{"class":36,"line":49},[34,3612,882],{"class":190},[34,3614,885],{"class":395},[34,3616,456],{"class":190},[34,3618,1762],{"class":395},[34,3620,456],{"class":190},[34,3622,786],{"class":179},[34,3624,616],{"class":190},[34,3626,768],{"class":179},[34,3628,2740],{"class":190},[34,3630,3631,3633,3635,3637,3640,3642,3644,3646,3648],{"class":36,"line":56},[34,3632,882],{"class":190},[34,3634,885],{"class":395},[34,3636,456],{"class":190},[34,3638,3639],{"class":395},"subtract",[34,3641,456],{"class":190},[34,3643,786],{"class":179},[34,3645,616],{"class":190},[34,3647,768],{"class":179},[34,3649,2740],{"class":190},[34,3651,3652,3654,3656,3658,3661],{"class":36,"line":62},[34,3653,882],{"class":190},[34,3655,885],{"class":395},[34,3657,456],{"class":190},[34,3659,3660],{"class":179},"PI",[34,3662,462],{"class":190},[15,3664,3665],{},"命名导出必须使用花括号，并且名字要对应。",[24,3667,3669],{"className":166,"code":3668,"language":168,"meta":29,"style":29},"import { add } from \".\u002Fmath.js\";\n",[31,3670,3671],{"__ignoreMap":29},[34,3672,3673,3675,3677,3679,3681],{"class":36,"line":37},[34,3674,426],{"class":175},[34,3676,2917],{"class":190},[34,3678,432],{"class":175},[34,3680,2922],{"class":186},[34,3682,191],{"class":190},[15,3684,3685,3686,3689],{},"这里的 ",[31,3687,3688],{},"{ add }"," 不是对象字面量，而是“我要导入名为 add 的导出”。",[253,3691,3693],{"id":3692},"_88-导入时可以改名","8.8 导入时可以改名",[15,3695,3696,3697,347],{},"如果名字冲突，或者你想让名字更清楚，可以用 ",[31,3698,3699],{},"as",[24,3701,3703],{"className":166,"code":3702,"language":168,"meta":29,"style":29},"import { add as addNumbers } from \".\u002Fmath.js\";\n\nconsole.log(addNumbers(1, 2));\n",[31,3704,3705,3723,3727],{"__ignoreMap":29},[34,3706,3707,3709,3712,3714,3717,3719,3721],{"class":36,"line":37},[34,3708,426],{"class":175},[34,3710,3711],{"class":190}," { add ",[34,3713,3699],{"class":175},[34,3715,3716],{"class":190}," addNumbers } ",[34,3718,432],{"class":175},[34,3720,2922],{"class":186},[34,3722,191],{"class":190},[34,3724,3725],{"class":36,"line":43},[34,3726,53],{"emptyLinePlaceholder":52},[34,3728,3729,3731,3733,3735,3738,3740,3742,3744,3746],{"class":36,"line":49},[34,3730,882],{"class":190},[34,3732,885],{"class":395},[34,3734,456],{"class":190},[34,3736,3737],{"class":395},"addNumbers",[34,3739,456],{"class":190},[34,3741,640],{"class":179},[34,3743,616],{"class":190},[34,3745,768],{"class":179},[34,3747,2740],{"class":190},[15,3749,3750,3751,3753,3754,3756,3757,347],{},"这表示：从 ",[31,3752,2645],{}," 导入 ",[31,3755,1762],{},"，但在当前文件叫 ",[31,3758,3737],{},[253,3760,3762],{"id":3761},"_89-统一导出写法","8.9 统一导出写法",[15,3764,3765],{},"有时你会看到这种写法：",[24,3767,3769],{"className":166,"code":3768,"language":168,"meta":29,"style":29},"function add(a, b) {\n return a + b;\n}\n\nfunction subtract(a, b) {\n return a - b;\n}\n\nexport { add, subtract };\n",[31,3770,3771,3787,3797,3801,3805,3821,3831,3835,3839],{"__ignoreMap":29},[34,3772,3773,3775,3777,3779,3781,3783,3785],{"class":36,"line":37},[34,3774,392],{"class":175},[34,3776,1688],{"class":395},[34,3778,456],{"class":190},[34,3780,111],{"class":1326},[34,3782,616],{"class":190},[34,3784,1697],{"class":1326},[34,3786,1700],{"class":190},[34,3788,3789,3791,3793,3795],{"class":36,"line":43},[34,3790,1510],{"class":175},[34,3792,1707],{"class":190},[34,3794,223],{"class":175},[34,3796,1712],{"class":190},[34,3798,3799],{"class":36,"line":49},[34,3800,77],{"class":190},[34,3802,3803],{"class":36,"line":56},[34,3804,53],{"emptyLinePlaceholder":52},[34,3806,3807,3809,3811,3813,3815,3817,3819],{"class":36,"line":62},[34,3808,392],{"class":175},[34,3810,2492],{"class":395},[34,3812,456],{"class":190},[34,3814,111],{"class":1326},[34,3816,616],{"class":190},[34,3818,1697],{"class":1326},[34,3820,1700],{"class":190},[34,3822,3823,3825,3827,3829],{"class":36,"line":68},[34,3824,1510],{"class":175},[34,3826,1707],{"class":190},[34,3828,2511],{"class":175},[34,3830,1712],{"class":190},[34,3832,3833],{"class":36,"line":74},[34,3834,77],{"class":190},[34,3836,3837],{"class":36,"line":1971},[34,3838,53],{"emptyLinePlaceholder":52},[34,3840,3841,3843],{"class":36,"line":1976},[34,3842,2825],{"class":175},[34,3844,3845],{"class":190}," { add, subtract };\n",[15,3847,3848],{},"它和下面这种效果一样：",[24,3850,3852],{"className":166,"code":3851,"language":168,"meta":29,"style":29},"export function add(a, b) {\n return a + b;\n}\n\nexport function subtract(a, b) {\n return a - b;\n}\n",[31,3853,3854,3872,3882,3886,3890,3908,3918],{"__ignoreMap":29},[34,3855,3856,3858,3860,3862,3864,3866,3868,3870],{"class":36,"line":37},[34,3857,2825],{"class":175},[34,3859,2850],{"class":175},[34,3861,1688],{"class":395},[34,3863,456],{"class":190},[34,3865,111],{"class":1326},[34,3867,616],{"class":190},[34,3869,1697],{"class":1326},[34,3871,1700],{"class":190},[34,3873,3874,3876,3878,3880],{"class":36,"line":43},[34,3875,1510],{"class":175},[34,3877,1707],{"class":190},[34,3879,223],{"class":175},[34,3881,1712],{"class":190},[34,3883,3884],{"class":36,"line":49},[34,3885,77],{"class":190},[34,3887,3888],{"class":36,"line":56},[34,3889,53],{"emptyLinePlaceholder":52},[34,3891,3892,3894,3896,3898,3900,3902,3904,3906],{"class":36,"line":62},[34,3893,2825],{"class":175},[34,3895,2850],{"class":175},[34,3897,2492],{"class":395},[34,3899,456],{"class":190},[34,3901,111],{"class":1326},[34,3903,616],{"class":190},[34,3905,1697],{"class":1326},[34,3907,1700],{"class":190},[34,3909,3910,3912,3914,3916],{"class":36,"line":68},[34,3911,1510],{"class":175},[34,3913,1707],{"class":190},[34,3915,2511],{"class":175},[34,3917,1712],{"class":190},[34,3919,3920],{"class":36,"line":74},[34,3921,77],{"class":190},[15,3923,3924],{},"区别只是风格：",[105,3926,3927,3930],{},[108,3928,3929],{},"写在声明前：看到函数时就知道它被导出。",[108,3931,3932],{},"统一写在末尾：文件底部集中列出公开内容。",[253,3934,3936,3937],{"id":3935},"_810-默认导出-export-default","8.10 默认导出 ",[31,3938,3939],{},"export default",[15,3941,3942],{},"一个模块可以有一个默认导出。",[24,3944,3946],{"className":166,"code":3945,"language":168,"meta":29,"style":29},"\u002F\u002F math.js\nexport default function add(a, b) {\n return a + b;\n}\n",[31,3947,3948,3952,3973,3983],{"__ignoreMap":29},[34,3949,3950],{"class":36,"line":37},[34,3951,2674],{"class":301},[34,3953,3954,3956,3959,3961,3963,3965,3967,3969,3971],{"class":36,"line":43},[34,3955,2825],{"class":175},[34,3957,3958],{"class":175}," default",[34,3960,2850],{"class":175},[34,3962,1688],{"class":395},[34,3964,456],{"class":190},[34,3966,111],{"class":1326},[34,3968,616],{"class":190},[34,3970,1697],{"class":1326},[34,3972,1700],{"class":190},[34,3974,3975,3977,3979,3981],{"class":36,"line":49},[34,3976,1510],{"class":175},[34,3978,1707],{"class":190},[34,3980,223],{"class":175},[34,3982,1712],{"class":190},[34,3984,3985],{"class":36,"line":56},[34,3986,77],{"class":190},[15,3988,3989],{},"导入时不需要花括号：",[24,3991,3993],{"className":166,"code":3992,"language":168,"meta":29,"style":29},"import add from \".\u002Fmath.js\";\n\nconsole.log(add(1, 2));\n",[31,3994,3995,4008,4012],{"__ignoreMap":29},[34,3996,3997,3999,4002,4004,4006],{"class":36,"line":37},[34,3998,426],{"class":175},[34,4000,4001],{"class":190}," add ",[34,4003,432],{"class":175},[34,4005,2922],{"class":186},[34,4007,191],{"class":190},[34,4009,4010],{"class":36,"line":43},[34,4011,53],{"emptyLinePlaceholder":52},[34,4013,4014,4016,4018,4020,4022,4024,4026,4028,4030],{"class":36,"line":49},[34,4015,882],{"class":190},[34,4017,885],{"class":395},[34,4019,456],{"class":190},[34,4021,1762],{"class":395},[34,4023,456],{"class":190},[34,4025,640],{"class":179},[34,4027,616],{"class":190},[34,4029,768],{"class":179},[34,4031,2740],{"class":190},[15,4033,4034],{},"默认导出的特点：",[105,4036,4037,4040],{},[108,4038,4039],{},"一个文件最多只能有一个默认导出。",[108,4041,4042],{},"导入时名字可以自己取。",[15,4044,2632],{},[24,4046,4048],{"className":166,"code":4047,"language":168,"meta":29,"style":29},"import whateverName from \".\u002Fmath.js\";\n\nconsole.log(whateverName(1, 2));\n",[31,4049,4050,4063,4067],{"__ignoreMap":29},[34,4051,4052,4054,4057,4059,4061],{"class":36,"line":37},[34,4053,426],{"class":175},[34,4055,4056],{"class":190}," whateverName ",[34,4058,432],{"class":175},[34,4060,2922],{"class":186},[34,4062,191],{"class":190},[34,4064,4065],{"class":36,"line":43},[34,4066,53],{"emptyLinePlaceholder":52},[34,4068,4069,4071,4073,4075,4078,4080,4082,4084,4086],{"class":36,"line":49},[34,4070,882],{"class":190},[34,4072,885],{"class":395},[34,4074,456],{"class":190},[34,4076,4077],{"class":395},"whateverName",[34,4079,456],{"class":190},[34,4081,640],{"class":179},[34,4083,616],{"class":190},[34,4085,768],{"class":179},[34,4087,2740],{"class":190},[15,4089,4090],{},"因为默认导出没有强制使用原名。",[253,4092,4094],{"id":4093},"_811-命名导出和默认导出的区别","8.11 命名导出和默认导出的区别",[15,4096,4097],{},"命名导出：",[24,4099,4100],{"className":166,"code":2837,"language":168,"meta":29,"style":29},[31,4101,4102,4106,4124,4134],{"__ignoreMap":29},[34,4103,4104],{"class":36,"line":37},[34,4105,2674],{"class":301},[34,4107,4108,4110,4112,4114,4116,4118,4120,4122],{"class":36,"line":43},[34,4109,2825],{"class":175},[34,4111,2850],{"class":175},[34,4113,1688],{"class":395},[34,4115,456],{"class":190},[34,4117,111],{"class":1326},[34,4119,616],{"class":190},[34,4121,1697],{"class":1326},[34,4123,1700],{"class":190},[34,4125,4126,4128,4130,4132],{"class":36,"line":49},[34,4127,1510],{"class":175},[34,4129,1707],{"class":190},[34,4131,223],{"class":175},[34,4133,1712],{"class":190},[34,4135,4136],{"class":36,"line":56},[34,4137,77],{"class":190},[24,4139,4140],{"className":166,"code":3005,"language":168,"meta":29,"style":29},[31,4141,4142,4146],{"__ignoreMap":29},[34,4143,4144],{"class":36,"line":37},[34,4145,2719],{"class":301},[34,4147,4148,4150,4152,4154,4156],{"class":36,"line":43},[34,4149,426],{"class":175},[34,4151,2917],{"class":190},[34,4153,432],{"class":175},[34,4155,2922],{"class":186},[34,4157,191],{"class":190},[15,4159,4160],{},"特点：",[105,4162,4163,4166,4171],{},[108,4164,4165],{},"导入时需要花括号。",[108,4167,4168,4169,347],{},"名字必须和导出名一致，除非用 ",[31,4170,3699],{},[108,4172,4173],{},"一个文件可以有多个命名导出。",[15,4175,4176],{},"默认导出：",[24,4178,4180],{"className":166,"code":4179,"language":168,"meta":29,"style":29},"\u002F\u002F LessonCard.vue\nexport default {\n \u002F\u002F 组件配置\n};\n",[31,4181,4182,4187,4195,4200],{"__ignoreMap":29},[34,4183,4184],{"class":36,"line":37},[34,4185,4186],{"class":301},"\u002F\u002F LessonCard.vue\n",[34,4188,4189,4191,4193],{"class":36,"line":43},[34,4190,2825],{"class":175},[34,4192,3958],{"class":175},[34,4194,674],{"class":190},[34,4196,4197],{"class":36,"line":49},[34,4198,4199],{"class":301}," \u002F\u002F 组件配置\n",[34,4201,4202],{"class":36,"line":56},[34,4203,719],{"class":190},[24,4205,4207],{"className":166,"code":4206,"language":168,"meta":29,"style":29},"import LessonCard from \".\u002FLessonCard.vue\";\n",[31,4208,4209],{"__ignoreMap":29},[34,4210,4211,4213,4215,4217,4220],{"class":36,"line":37},[34,4212,426],{"class":175},[34,4214,3432],{"class":190},[34,4216,432],{"class":175},[34,4218,4219],{"class":186}," \".\u002FLessonCard.vue\"",[34,4221,191],{"class":190},[15,4223,4160],{},[105,4225,4226,4229,4232],{},[108,4227,4228],{},"导入时不用花括号。",[108,4230,4231],{},"名字可以自己取。",[108,4233,4234],{},"一个文件只能有一个默认导出。",[15,4236,4237],{},"Vue 单文件组件在构建后本质上会提供默认导出，所以你常看到：",[24,4239,4241],{"className":166,"code":4240,"language":168,"meta":29,"style":29},"import App from \".\u002FApp.vue\";\nimport LessonCard from \".\u002Fcomponents\u002FLessonCard.vue\";\n",[31,4242,4243,4257],{"__ignoreMap":29},[34,4244,4245,4247,4250,4252,4255],{"class":36,"line":37},[34,4246,426],{"class":175},[34,4248,4249],{"class":190}," App ",[34,4251,432],{"class":175},[34,4253,4254],{"class":186}," \".\u002FApp.vue\"",[34,4256,191],{"class":190},[34,4258,4259,4261,4263,4265,4267],{"class":36,"line":43},[34,4260,426],{"class":175},[34,4262,3432],{"class":190},[34,4264,432],{"class":175},[34,4266,3437],{"class":186},[34,4268,191],{"class":190},[253,4270,4272,4273],{"id":4271},"_812-导入路径为什么要写","8.12 导入路径为什么要写 ",[31,4274,4275],{},".\u002F",[24,4277,4278],{"className":166,"code":3668,"language":168,"meta":29,"style":29},[31,4279,4280],{"__ignoreMap":29},[34,4281,4282,4284,4286,4288,4290],{"class":36,"line":37},[34,4283,426],{"class":175},[34,4285,2917],{"class":190},[34,4287,432],{"class":175},[34,4289,2922],{"class":186},[34,4291,191],{"class":190},[15,4293,4294,4297,4298,347],{},[31,4295,4296],{},".\u002Fmath.js"," 的意思是：从当前文件所在目录找 ",[31,4299,2645],{},[15,4301,4302],{},"常见路径：",[24,4304,4306],{"className":166,"code":4305,"language":168,"meta":29,"style":29},"import { add } from \".\u002Fmath.js\"; \u002F\u002F 当前目录\nimport { add } from \"..\u002Fmath.js\"; \u002F\u002F 上一级目录\nimport App from \".\u002FApp.vue\"; \u002F\u002F 当前目录里的 App.vue\nimport { ref } from \"vue\"; \u002F\u002F 从 node_modules 包导入\n",[31,4307,4308,4323,4339,4354],{"__ignoreMap":29},[34,4309,4310,4312,4314,4316,4318,4320],{"class":36,"line":37},[34,4311,426],{"class":175},[34,4313,2917],{"class":190},[34,4315,432],{"class":175},[34,4317,2922],{"class":186},[34,4319,298],{"class":190},[34,4321,4322],{"class":301},"\u002F\u002F 当前目录\n",[34,4324,4325,4327,4329,4331,4334,4336],{"class":36,"line":43},[34,4326,426],{"class":175},[34,4328,2917],{"class":190},[34,4330,432],{"class":175},[34,4332,4333],{"class":186}," \"..\u002Fmath.js\"",[34,4335,298],{"class":190},[34,4337,4338],{"class":301},"\u002F\u002F 上一级目录\n",[34,4340,4341,4343,4345,4347,4349,4351],{"class":36,"line":49},[34,4342,426],{"class":175},[34,4344,4249],{"class":190},[34,4346,432],{"class":175},[34,4348,4254],{"class":186},[34,4350,298],{"class":190},[34,4352,4353],{"class":301},"\u002F\u002F 当前目录里的 App.vue\n",[34,4355,4356,4358,4360,4362,4364,4366],{"class":36,"line":56},[34,4357,426],{"class":175},[34,4359,429],{"class":190},[34,4361,432],{"class":175},[34,4363,435],{"class":186},[34,4365,298],{"class":190},[34,4367,4368],{"class":301},"\u002F\u002F 从 node_modules 包导入\n",[15,4370,4371,4372,4374,4375,4378],{},"有 ",[31,4373,4275],{}," 或 ",[31,4376,4377],{},"..\u002F","：相对路径，找你项目里的文件。",[15,4380,4381,4382,4374,4384,2664],{},"没有 ",[31,4383,4275],{},[31,4385,4377],{},[24,4387,4389],{"className":166,"code":4388,"language":168,"meta":29,"style":29},"import { ref } from \"vue\";\n",[31,4390,4391],{"__ignoreMap":29},[34,4392,4393,4395,4397,4399,4401],{"class":36,"line":37},[34,4394,426],{"class":175},[34,4396,429],{"class":190},[34,4398,432],{"class":175},[34,4400,435],{"class":186},[34,4402,191],{"class":190},[15,4404,4405,4406,4408,4409,4412],{},"这通常表示从安装的包里导入。",[31,4407,925],{}," 会在 ",[31,4410,4411],{},"node_modules"," 中找到。",[253,4414,4416],{"id":4415},"_813-模块会在什么时候执行","8.13 模块会在什么时候执行",[15,4418,4419],{},"当一个模块第一次被导入时，它会执行一次。",[24,4421,4423],{"className":166,"code":4422,"language":168,"meta":29,"style":29},"\u002F\u002F log.js\nconsole.log(\"log.js 执行了\");\n\nexport const message = \"hello\";\n",[31,4424,4425,4430,4443,4447],{"__ignoreMap":29},[34,4426,4427],{"class":36,"line":37},[34,4428,4429],{"class":301},"\u002F\u002F log.js\n",[34,4431,4432,4434,4436,4438,4441],{"class":36,"line":43},[34,4433,882],{"class":190},[34,4435,885],{"class":395},[34,4437,456],{"class":190},[34,4439,4440],{"class":186},"\"log.js 执行了\"",[34,4442,462],{"class":190},[34,4444,4445],{"class":36,"line":49},[34,4446,53],{"emptyLinePlaceholder":52},[34,4448,4449,4451,4453,4456,4458,4461],{"class":36,"line":56},[34,4450,2825],{"class":175},[34,4452,3280],{"class":175},[34,4454,4455],{"class":179}," message",[34,4457,183],{"class":175},[34,4459,4460],{"class":186}," \"hello\"",[34,4462,191],{"class":190},[24,4464,4466],{"className":166,"code":4465,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nimport { message } from \".\u002Flog.js\";\n\nconsole.log(message);\n",[31,4467,4468,4472,4486,4490],{"__ignoreMap":29},[34,4469,4470],{"class":36,"line":37},[34,4471,2719],{"class":301},[34,4473,4474,4476,4479,4481,4484],{"class":36,"line":43},[34,4475,426],{"class":175},[34,4477,4478],{"class":190}," { message } ",[34,4480,432],{"class":175},[34,4482,4483],{"class":186}," \".\u002Flog.js\"",[34,4485,191],{"class":190},[34,4487,4488],{"class":36,"line":49},[34,4489,53],{"emptyLinePlaceholder":52},[34,4491,4492,4494,4496],{"class":36,"line":56},[34,4493,882],{"class":190},[34,4495,885],{"class":395},[34,4497,4498],{"class":190},"(message);\n",[15,4500,4501,4502,4504,4505,4508],{},"运行 ",[31,4503,2653],{}," 时，",[31,4506,4507],{},"log.js"," 会先执行，所以会打印：",[24,4510,4513],{"className":4511,"code":4512,"language":2638,"meta":29},[2636],"log.js 执行了\nhello\n",[31,4514,4512],{"__ignoreMap":29},[15,4516,4517],{},"这说明模块不只是“拿函数”，导入时也会执行模块顶层代码。",[15,4519,4520],{},"所以不要在模块顶层写太多副作用代码，例如随便发请求、改 DOM。更好的方式是导出函数，由需要的地方调用。",[253,4522,4524],{"id":4523},"_814-导入的是活的绑定","8.14 导入的是“活的绑定”",[15,4526,4527],{},"命名导入不是简单复制一份值，它和原模块的导出绑定有关。初学阶段不必深挖规范，但要知道：模块系统比普通对象拷贝更严格。",[15,4529,2632],{},[24,4531,4533],{"className":166,"code":4532,"language":168,"meta":29,"style":29},"\u002F\u002F counter.js\nexport let count = 0;\n\nexport function increase() {\n count++;\n}\n",[31,4534,4535,4540,4555,4559,4569,4577],{"__ignoreMap":29},[34,4536,4537],{"class":36,"line":37},[34,4538,4539],{"class":301},"\u002F\u002F counter.js\n",[34,4541,4542,4544,4547,4549,4551,4553],{"class":36,"line":43},[34,4543,2825],{"class":175},[34,4545,4546],{"class":175}," let",[34,4548,319],{"class":190},[34,4550,202],{"class":175},[34,4552,205],{"class":179},[34,4554,191],{"class":190},[34,4556,4557],{"class":36,"line":49},[34,4558,53],{"emptyLinePlaceholder":52},[34,4560,4561,4563,4565,4567],{"class":36,"line":56},[34,4562,2825],{"class":175},[34,4564,2850],{"class":175},[34,4566,396],{"class":395},[34,4568,399],{"class":190},[34,4570,4571,4573,4575],{"class":36,"line":62},[34,4572,404],{"class":190},[34,4574,407],{"class":175},[34,4576,191],{"class":190},[34,4578,4579],{"class":36,"line":68},[34,4580,77],{"class":190},[24,4582,4584],{"className":166,"code":4583,"language":168,"meta":29,"style":29},"\u002F\u002F main.js\nimport { count, increase } from \".\u002Fcounter.js\";\n\nconsole.log(count); \u002F\u002F 0\nincrease();\nconsole.log(count); \u002F\u002F 1\n",[31,4585,4586,4590,4604,4608,4620,4627],{"__ignoreMap":29},[34,4587,4588],{"class":36,"line":37},[34,4589,2719],{"class":301},[34,4591,4592,4594,4597,4599,4602],{"class":36,"line":43},[34,4593,426],{"class":175},[34,4595,4596],{"class":190}," { count, increase } ",[34,4598,432],{"class":175},[34,4600,4601],{"class":186}," \".\u002Fcounter.js\"",[34,4603,191],{"class":190},[34,4605,4606],{"class":36,"line":49},[34,4607,53],{"emptyLinePlaceholder":52},[34,4609,4610,4612,4614,4617],{"class":36,"line":56},[34,4611,882],{"class":190},[34,4613,885],{"class":395},[34,4615,4616],{"class":190},"(count); ",[34,4618,4619],{"class":301},"\u002F\u002F 0\n",[34,4621,4622,4624],{"class":36,"line":62},[34,4623,2048],{"class":395},[34,4625,4626],{"class":190},"();\n",[34,4628,4629,4631,4633,4635],{"class":36,"line":68},[34,4630,882],{"class":190},[34,4632,885],{"class":395},[34,4634,4616],{"class":190},[34,4636,4637],{"class":301},"\u002F\u002F 1\n",[15,4639,4640],{},"但导入的变量不能在导入方直接重新赋值：",[24,4642,4644],{"className":166,"code":4643,"language":168,"meta":29,"style":29},"import { count } from \".\u002Fcounter.js\";\n\ncount = 10; \u002F\u002F 报错\n",[31,4645,4646,4659,4663],{"__ignoreMap":29},[34,4647,4648,4650,4653,4655,4657],{"class":36,"line":37},[34,4649,426],{"class":175},[34,4651,4652],{"class":190}," { count } ",[34,4654,432],{"class":175},[34,4656,4601],{"class":186},[34,4658,191],{"class":190},[34,4660,4661],{"class":36,"line":43},[34,4662,53],{"emptyLinePlaceholder":52},[34,4664,4665,4667,4669,4671,4673],{"class":36,"line":49},[34,4666,330],{"class":190},[34,4668,202],{"class":175},[34,4670,559],{"class":179},[34,4672,298],{"class":190},[34,4674,302],{"class":301},[15,4676,4677],{},"如果要修改，应该通过导出方提供的函数修改。",[253,4679,4681],{"id":4680},"_815-vue-项目中模块怎么用","8.15 Vue 项目中模块怎么用",[15,4683,4684],{},"Vue 项目里到处都是模块：",[15,4686,4687],{},"导入 Vue API：",[24,4689,4690],{"className":166,"code":2407,"language":168,"meta":29,"style":29},[31,4691,4692],{"__ignoreMap":29},[34,4693,4694,4696,4698,4700,4702],{"class":36,"line":37},[34,4695,426],{"class":175},[34,4697,2416],{"class":190},[34,4699,432],{"class":175},[34,4701,435],{"class":186},[34,4703,191],{"class":190},[15,4705,4706],{},"导入组件：",[24,4708,4710],{"className":166,"code":4709,"language":168,"meta":29,"style":29},"import TodoItem from \"..\u002Fcomponents\u002FTodoItem.vue\";\n",[31,4711,4712],{"__ignoreMap":29},[34,4713,4714,4716,4719,4721,4724],{"class":36,"line":37},[34,4715,426],{"class":175},[34,4717,4718],{"class":190}," TodoItem ",[34,4720,432],{"class":175},[34,4722,4723],{"class":186}," \"..\u002Fcomponents\u002FTodoItem.vue\"",[34,4725,191],{"class":190},[15,4727,4728],{},"导入组合式函数：",[24,4730,4732],{"className":166,"code":4731,"language":168,"meta":29,"style":29},"import { useCounter } from \"..\u002Fcomposables\u002FuseCounter\";\n",[31,4733,4734],{"__ignoreMap":29},[34,4735,4736,4738,4741,4743,4746],{"class":36,"line":37},[34,4737,426],{"class":175},[34,4739,4740],{"class":190}," { useCounter } ",[34,4742,432],{"class":175},[34,4744,4745],{"class":186}," \"..\u002Fcomposables\u002FuseCounter\"",[34,4747,191],{"class":190},[15,4749,4750],{},"导入 Pinia store：",[24,4752,4754],{"className":166,"code":4753,"language":168,"meta":29,"style":29},"import { useTodoStore } from \"..\u002Fstores\u002FuseTodoStore\";\n",[31,4755,4756],{"__ignoreMap":29},[34,4757,4758,4760,4763,4765,4768],{"class":36,"line":37},[34,4759,426],{"class":175},[34,4761,4762],{"class":190}," { useTodoStore } ",[34,4764,432],{"class":175},[34,4766,4767],{"class":186}," \"..\u002Fstores\u002FuseTodoStore\"",[34,4769,191],{"class":190},[15,4771,4772],{},"导入本地数据：",[24,4774,4776],{"className":166,"code":4775,"language":168,"meta":29,"style":29},"import { lessons } from \"..\u002Fdata\u002Flessons\";\n",[31,4777,4778],{"__ignoreMap":29},[34,4779,4780,4782,4785,4787,4790],{"class":36,"line":37},[34,4781,426],{"class":175},[34,4783,4784],{"class":190}," { lessons } ",[34,4786,432],{"class":175},[34,4788,4789],{"class":186}," \"..\u002Fdata\u002Flessons\"",[34,4791,191],{"class":190},[15,4793,4794,4795,4798],{},"你可以把 ",[31,4796,4797],{},"src\u002F"," 看成很多模块拼起来的应用。",[253,4800,4802],{"id":4801},"_816-常见模块错误","8.16 常见模块错误",[15,4804,4805],{},"错误一：忘记导出。",[24,4807,4808],{"className":166,"code":2667,"language":168,"meta":29,"style":29},[31,4809,4810,4814,4830,4840],{"__ignoreMap":29},[34,4811,4812],{"class":36,"line":37},[34,4813,2674],{"class":301},[34,4815,4816,4818,4820,4822,4824,4826,4828],{"class":36,"line":43},[34,4817,392],{"class":175},[34,4819,1688],{"class":395},[34,4821,456],{"class":190},[34,4823,111],{"class":1326},[34,4825,616],{"class":190},[34,4827,1697],{"class":1326},[34,4829,1700],{"class":190},[34,4831,4832,4834,4836,4838],{"class":36,"line":49},[34,4833,1510],{"class":175},[34,4835,1707],{"class":190},[34,4837,223],{"class":175},[34,4839,1712],{"class":190},[34,4841,4842],{"class":36,"line":56},[34,4843,77],{"class":190},[24,4845,4847],{"className":166,"code":4846,"language":168,"meta":29,"style":29},"import { add } from \".\u002Fmath.js\"; \u002F\u002F 报错：没有导出 add\n",[31,4848,4849],{"__ignoreMap":29},[34,4850,4851,4853,4855,4857,4859,4861],{"class":36,"line":37},[34,4852,426],{"class":175},[34,4854,2917],{"class":190},[34,4856,432],{"class":175},[34,4858,2922],{"class":186},[34,4860,298],{"class":190},[34,4862,4863],{"class":301},"\u002F\u002F 报错：没有导出 add\n",[15,4865,4866],{},"修复：",[24,4868,4869],{"className":166,"code":3064,"language":168,"meta":29,"style":29},[31,4870,4871,4889,4899],{"__ignoreMap":29},[34,4872,4873,4875,4877,4879,4881,4883,4885,4887],{"class":36,"line":37},[34,4874,2825],{"class":175},[34,4876,2850],{"class":175},[34,4878,1688],{"class":395},[34,4880,456],{"class":190},[34,4882,111],{"class":1326},[34,4884,616],{"class":190},[34,4886,1697],{"class":1326},[34,4888,1700],{"class":190},[34,4890,4891,4893,4895,4897],{"class":36,"line":43},[34,4892,1510],{"class":175},[34,4894,1707],{"class":190},[34,4896,223],{"class":175},[34,4898,1712],{"class":190},[34,4900,4901],{"class":36,"line":49},[34,4902,77],{"class":190},[15,4904,4905],{},"错误二：命名导出却不用花括号。",[24,4907,4908],{"className":166,"code":2837,"language":168,"meta":29,"style":29},[31,4909,4910,4914,4932,4942],{"__ignoreMap":29},[34,4911,4912],{"class":36,"line":37},[34,4913,2674],{"class":301},[34,4915,4916,4918,4920,4922,4924,4926,4928,4930],{"class":36,"line":43},[34,4917,2825],{"class":175},[34,4919,2850],{"class":175},[34,4921,1688],{"class":395},[34,4923,456],{"class":190},[34,4925,111],{"class":1326},[34,4927,616],{"class":190},[34,4929,1697],{"class":1326},[34,4931,1700],{"class":190},[34,4933,4934,4936,4938,4940],{"class":36,"line":49},[34,4935,1510],{"class":175},[34,4937,1707],{"class":190},[34,4939,223],{"class":175},[34,4941,1712],{"class":190},[34,4943,4944],{"class":36,"line":56},[34,4945,77],{"class":190},[24,4947,4949],{"className":166,"code":4948,"language":168,"meta":29,"style":29},"import add from \".\u002Fmath.js\"; \u002F\u002F 错误，add 不是默认导出\n",[31,4950,4951],{"__ignoreMap":29},[34,4952,4953,4955,4957,4959,4961,4963],{"class":36,"line":37},[34,4954,426],{"class":175},[34,4956,4001],{"class":190},[34,4958,432],{"class":175},[34,4960,2922],{"class":186},[34,4962,298],{"class":190},[34,4964,4965],{"class":301},"\u002F\u002F 错误，add 不是默认导出\n",[15,4967,4866],{},[24,4969,4970],{"className":166,"code":3668,"language":168,"meta":29,"style":29},[31,4971,4972],{"__ignoreMap":29},[34,4973,4974,4976,4978,4980,4982],{"class":36,"line":37},[34,4975,426],{"class":175},[34,4977,2917],{"class":190},[34,4979,432],{"class":175},[34,4981,2922],{"class":186},[34,4983,191],{"class":190},[15,4985,4986],{},"错误三：默认导出却用了花括号。",[24,4988,4989],{"className":166,"code":3945,"language":168,"meta":29,"style":29},[31,4990,4991,4995,5015,5025],{"__ignoreMap":29},[34,4992,4993],{"class":36,"line":37},[34,4994,2674],{"class":301},[34,4996,4997,4999,5001,5003,5005,5007,5009,5011,5013],{"class":36,"line":43},[34,4998,2825],{"class":175},[34,5000,3958],{"class":175},[34,5002,2850],{"class":175},[34,5004,1688],{"class":395},[34,5006,456],{"class":190},[34,5008,111],{"class":1326},[34,5010,616],{"class":190},[34,5012,1697],{"class":1326},[34,5014,1700],{"class":190},[34,5016,5017,5019,5021,5023],{"class":36,"line":49},[34,5018,1510],{"class":175},[34,5020,1707],{"class":190},[34,5022,223],{"class":175},[34,5024,1712],{"class":190},[34,5026,5027],{"class":36,"line":56},[34,5028,77],{"class":190},[24,5030,5032],{"className":166,"code":5031,"language":168,"meta":29,"style":29},"import { add } from \".\u002Fmath.js\"; \u002F\u002F 错误\n",[31,5033,5034],{"__ignoreMap":29},[34,5035,5036,5038,5040,5042,5044,5046],{"class":36,"line":37},[34,5037,426],{"class":175},[34,5039,2917],{"class":190},[34,5041,432],{"class":175},[34,5043,2922],{"class":186},[34,5045,298],{"class":190},[34,5047,5048],{"class":301},"\u002F\u002F 错误\n",[15,5050,4866],{},[24,5052,5054],{"className":166,"code":5053,"language":168,"meta":29,"style":29},"import add from \".\u002Fmath.js\";\n",[31,5055,5056],{"__ignoreMap":29},[34,5057,5058,5060,5062,5064,5066],{"class":36,"line":37},[34,5059,426],{"class":175},[34,5061,4001],{"class":190},[34,5063,432],{"class":175},[34,5065,2922],{"class":186},[34,5067,191],{"class":190},[15,5069,5070],{},"错误四：路径写错。",[24,5072,5074],{"className":166,"code":5073,"language":168,"meta":29,"style":29},"import { add } from \"math.js\"; \u002F\u002F 这会被当成包名\n",[31,5075,5076],{"__ignoreMap":29},[34,5077,5078,5080,5082,5084,5087,5089],{"class":36,"line":37},[34,5079,426],{"class":175},[34,5081,2917],{"class":190},[34,5083,432],{"class":175},[34,5085,5086],{"class":186}," \"math.js\"",[34,5088,298],{"class":190},[34,5090,5091],{"class":301},"\u002F\u002F 这会被当成包名\n",[15,5093,4866],{},[24,5095,5096],{"className":166,"code":3668,"language":168,"meta":29,"style":29},[31,5097,5098],{"__ignoreMap":29},[34,5099,5100,5102,5104,5106,5108],{"class":36,"line":37},[34,5101,426],{"class":175},[34,5103,2917],{"class":190},[34,5105,432],{"class":175},[34,5107,2922],{"class":186},[34,5109,191],{"class":190},[15,5111,5112],{},"错误五：大小写不一致。",[24,5114,5116],{"className":166,"code":5115,"language":168,"meta":29,"style":29},"import LessonCard from \".\u002Fcomponents\u002FlessonCard.vue\";\n",[31,5117,5118],{"__ignoreMap":29},[34,5119,5120,5122,5124,5126,5129],{"class":36,"line":37},[34,5121,426],{"class":175},[34,5123,3432],{"class":190},[34,5125,432],{"class":175},[34,5127,5128],{"class":186}," \".\u002Fcomponents\u002FlessonCard.vue\"",[34,5130,191],{"class":190},[15,5132,5133,5134,5137],{},"如果文件实际叫 ",[31,5135,5136],{},"LessonCard.vue","，在某些系统或部署环境会失败。建议文件名和导入路径大小写完全一致。",[19,5139,5141],{"id":5140},"_9-异步等待未来的结果","9. 异步：等待未来的结果",[15,5143,5144,5145,5148],{},"浏览器请求接口、读取文件、等待定时器，都不是立刻得到结果。JavaScript 用 Promise 和 ",[31,5146,5147],{},"async\u002Fawait"," 处理异步。",[15,5150,5151],{},"最小例子：",[24,5153,5155],{"className":166,"code":5154,"language":168,"meta":29,"style":29},"function wait(ms) {\n return new Promise((resolve) => {\n setTimeout(resolve, ms);\n });\n}\n\nasync function run() {\n console.log(\"开始\");\n await wait(1000);\n console.log(\"1 秒后\");\n}\n",[31,5156,5157,5171,5192,5200,5205,5209,5213,5225,5238,5252,5265],{"__ignoreMap":29},[34,5158,5159,5161,5164,5166,5169],{"class":36,"line":37},[34,5160,392],{"class":175},[34,5162,5163],{"class":395}," wait",[34,5165,456],{"class":190},[34,5167,5168],{"class":1326},"ms",[34,5170,1700],{"class":190},[34,5172,5173,5175,5178,5181,5183,5186,5188,5190],{"class":36,"line":43},[34,5174,1510],{"class":175},[34,5176,5177],{"class":175}," new",[34,5179,5180],{"class":179}," Promise",[34,5182,1323],{"class":190},[34,5184,5185],{"class":1326},"resolve",[34,5187,1330],{"class":190},[34,5189,1333],{"class":175},[34,5191,674],{"class":190},[34,5193,5194,5197],{"class":36,"line":49},[34,5195,5196],{"class":395}," setTimeout",[34,5198,5199],{"class":190},"(resolve, ms);\n",[34,5201,5202],{"class":36,"line":56},[34,5203,5204],{"class":190}," });\n",[34,5206,5207],{"class":36,"line":62},[34,5208,77],{"class":190},[34,5210,5211],{"class":36,"line":68},[34,5212,53],{"emptyLinePlaceholder":52},[34,5214,5215,5218,5220,5223],{"class":36,"line":74},[34,5216,5217],{"class":175},"async",[34,5219,2850],{"class":175},[34,5221,5222],{"class":395}," run",[34,5224,399],{"class":190},[34,5226,5227,5229,5231,5233,5236],{"class":36,"line":1971},[34,5228,1812],{"class":190},[34,5230,885],{"class":395},[34,5232,456],{"class":190},[34,5234,5235],{"class":186},"\"开始\"",[34,5237,462],{"class":190},[34,5239,5240,5243,5245,5247,5250],{"class":36,"line":1976},[34,5241,5242],{"class":175}," await",[34,5244,5163],{"class":395},[34,5246,456],{"class":190},[34,5248,5249],{"class":179},"1000",[34,5251,462],{"class":190},[34,5253,5254,5256,5258,5260,5263],{"class":36,"line":1985},[34,5255,1812],{"class":190},[34,5257,885],{"class":395},[34,5259,456],{"class":190},[34,5261,5262],{"class":186},"\"1 秒后\"",[34,5264,462],{"class":190},[34,5266,5267],{"class":36,"line":1990},[34,5268,77],{"class":190},[15,5270,5271,5274],{},[31,5272,5273],{},"await"," 的意思是：等 Promise 完成后，再继续往下执行。",[253,5276,5277],{"id":5277},"请求数据",[24,5279,5281],{"className":166,"code":5280,"language":168,"meta":29,"style":29},"async function loadUser() {\n const response = await fetch(\"\u002Fapi\u002Fuser\");\n const user = await response.json();\n return user;\n}\n",[31,5282,5283,5294,5317,5335,5341],{"__ignoreMap":29},[34,5284,5285,5287,5289,5292],{"class":36,"line":37},[34,5286,5217],{"class":175},[34,5288,2850],{"class":175},[34,5290,5291],{"class":395}," loadUser",[34,5293,399],{"class":190},[34,5295,5296,5299,5302,5304,5307,5310,5312,5315],{"class":36,"line":43},[34,5297,5298],{"class":175}," const",[34,5300,5301],{"class":179}," response",[34,5303,183],{"class":175},[34,5305,5306],{"class":175}," await",[34,5308,5309],{"class":395}," fetch",[34,5311,456],{"class":190},[34,5313,5314],{"class":186},"\"\u002Fapi\u002Fuser\"",[34,5316,462],{"class":190},[34,5318,5319,5321,5323,5325,5327,5330,5333],{"class":36,"line":49},[34,5320,5298],{"class":175},[34,5322,588],{"class":179},[34,5324,183],{"class":175},[34,5326,5306],{"class":175},[34,5328,5329],{"class":190}," response.",[34,5331,5332],{"class":395},"json",[34,5334,4626],{"class":190},[34,5336,5337,5339],{"class":36,"line":56},[34,5338,1510],{"class":175},[34,5340,2318],{"class":190},[34,5342,5343],{"class":36,"line":62},[34,5344,77],{"class":190},[15,5346,5347],{},"真实项目不能只写成功路径，还要写失败和加载状态：",[24,5349,5351],{"className":166,"code":5350,"language":168,"meta":29,"style":29},"import { ref } from \"vue\";\n\nconst loading = ref(false);\nconst error = ref(\"\");\nconst data = ref(null);\n\nasync function loadData() {\n loading.value = true;\n error.value = \"\";\n\n try {\n const response = await fetch(\"\u002Fapi\u002Fitems\");\n\n if (!response.ok) {\n throw new Error(`请求失败：${response.status}`);\n }\n\n data.value = await response.json();\n } catch (err) {\n error.value = err instanceof Error ? err.message : \"未知错误\";\n } finally {\n loading.value = false;\n }\n}\n",[31,5352,5353,5365,5369,5386,5404,5422,5426,5437,5448,5460,5464,5471,5491,5495,5508,5535,5541,5546,5562,5574,5604,5614,5626,5632],{"__ignoreMap":29},[34,5354,5355,5357,5359,5361,5363],{"class":36,"line":37},[34,5356,426],{"class":175},[34,5358,429],{"class":190},[34,5360,432],{"class":175},[34,5362,435],{"class":186},[34,5364,191],{"class":190},[34,5366,5367],{"class":36,"line":43},[34,5368,53],{"emptyLinePlaceholder":52},[34,5370,5371,5373,5376,5378,5380,5382,5384],{"class":36,"line":49},[34,5372,176],{"class":175},[34,5374,5375],{"class":179}," loading",[34,5377,183],{"class":175},[34,5379,453],{"class":395},[34,5381,456],{"class":190},[34,5383,702],{"class":179},[34,5385,462],{"class":190},[34,5387,5388,5390,5393,5395,5397,5399,5402],{"class":36,"line":56},[34,5389,176],{"class":175},[34,5391,5392],{"class":179}," error",[34,5394,183],{"class":175},[34,5396,453],{"class":395},[34,5398,456],{"class":190},[34,5400,5401],{"class":186},"\"\"",[34,5403,462],{"class":190},[34,5405,5406,5408,5411,5413,5415,5417,5420],{"class":36,"line":62},[34,5407,176],{"class":175},[34,5409,5410],{"class":179}," data",[34,5412,183],{"class":175},[34,5414,453],{"class":395},[34,5416,456],{"class":190},[34,5418,5419],{"class":179},"null",[34,5421,462],{"class":190},[34,5423,5424],{"class":36,"line":68},[34,5425,53],{"emptyLinePlaceholder":52},[34,5427,5428,5430,5432,5435],{"class":36,"line":74},[34,5429,5217],{"class":175},[34,5431,2850],{"class":175},[34,5433,5434],{"class":395}," loadData",[34,5436,399],{"class":190},[34,5438,5439,5442,5444,5446],{"class":36,"line":1971},[34,5440,5441],{"class":190}," loading.value ",[34,5443,202],{"class":175},[34,5445,915],{"class":179},[34,5447,191],{"class":190},[34,5449,5450,5453,5455,5458],{"class":36,"line":1976},[34,5451,5452],{"class":190}," error.value ",[34,5454,202],{"class":175},[34,5456,5457],{"class":186}," \"\"",[34,5459,191],{"class":190},[34,5461,5462],{"class":36,"line":1985},[34,5463,53],{"emptyLinePlaceholder":52},[34,5465,5466,5469],{"class":36,"line":1990},[34,5467,5468],{"class":175}," try",[34,5470,674],{"class":190},[34,5472,5473,5476,5478,5480,5482,5484,5486,5489],{"class":36,"line":1999},[34,5474,5475],{"class":175}," const",[34,5477,5301],{"class":179},[34,5479,183],{"class":175},[34,5481,5306],{"class":175},[34,5483,5309],{"class":395},[34,5485,456],{"class":190},[34,5487,5488],{"class":186},"\"\u002Fapi\u002Fitems\"",[34,5490,462],{"class":190},[34,5492,5493],{"class":36,"line":2022},[34,5494,53],{"emptyLinePlaceholder":52},[34,5496,5497,5500,5502,5505],{"class":36,"line":2036},[34,5498,5499],{"class":175}," if",[34,5501,1840],{"class":190},[34,5503,5504],{"class":175},"!",[34,5506,5507],{"class":190},"response.ok) {\n",[34,5509,5510,5513,5515,5518,5520,5523,5526,5528,5531,5533],{"class":36,"line":2603},[34,5511,5512],{"class":175}," throw",[34,5514,5177],{"class":175},[34,5516,5517],{"class":395}," Error",[34,5519,456],{"class":190},[34,5521,5522],{"class":186},"`请求失败：${",[34,5524,5525],{"class":190},"response",[34,5527,2547],{"class":186},[34,5529,5530],{"class":190},"status",[34,5532,2560],{"class":186},[34,5534,462],{"class":190},[34,5536,5538],{"class":36,"line":5537},16,[34,5539,5540],{"class":190}," }\n",[34,5542,5544],{"class":36,"line":5543},17,[34,5545,53],{"emptyLinePlaceholder":52},[34,5547,5549,5552,5554,5556,5558,5560],{"class":36,"line":5548},18,[34,5550,5551],{"class":190}," data.value ",[34,5553,202],{"class":175},[34,5555,5306],{"class":175},[34,5557,5329],{"class":190},[34,5559,5332],{"class":395},[34,5561,4626],{"class":190},[34,5563,5565,5568,5571],{"class":36,"line":5564},19,[34,5566,5567],{"class":190}," } ",[34,5569,5570],{"class":175},"catch",[34,5572,5573],{"class":190}," (err) {\n",[34,5575,5577,5580,5582,5585,5588,5590,5593,5596,5599,5602],{"class":36,"line":5576},20,[34,5578,5579],{"class":190}," error.value ",[34,5581,202],{"class":175},[34,5583,5584],{"class":190}," err ",[34,5586,5587],{"class":175},"instanceof",[34,5589,5517],{"class":395},[34,5591,5592],{"class":175}," ?",[34,5594,5595],{"class":190}," err.message ",[34,5597,5598],{"class":175},":",[34,5600,5601],{"class":186}," \"未知错误\"",[34,5603,191],{"class":190},[34,5605,5607,5609,5612],{"class":36,"line":5606},21,[34,5608,5567],{"class":190},[34,5610,5611],{"class":175},"finally",[34,5613,674],{"class":190},[34,5615,5617,5620,5622,5624],{"class":36,"line":5616},22,[34,5618,5619],{"class":190}," loading.value ",[34,5621,202],{"class":175},[34,5623,576],{"class":179},[34,5625,191],{"class":190},[34,5627,5629],{"class":36,"line":5628},23,[34,5630,5631],{"class":190}," }\n",[34,5633,5635],{"class":36,"line":5634},24,[34,5636,77],{"class":190},[15,5638,5639],{},"你要记住三件事：",[105,5641,5642,5648,5653],{},[108,5643,5644,5647],{},[31,5645,5646],{},"try"," 里放可能失败的代码。",[108,5649,5650,5652],{},[31,5651,5570],{}," 里处理失败。",[108,5654,5655,5657],{},[31,5656,5611],{}," 里放无论成功失败都要执行的代码，比如关闭 loading。",[19,5659,5661],{"id":5660},"_10-学-vue-前必须形成的观念","10. 学 Vue 前必须形成的观念",[15,5663,5664],{},"页面开发的核心不是“把 HTML 写出来”，而是这条链路：",[24,5666,5669],{"className":5667,"code":5668,"language":2638,"meta":29},[2636],"数据 -> 根据数据渲染界面 -> 用户操作 -> 修改数据 -> 界面更新\n",[31,5670,5668],{"__ignoreMap":29},[15,5672,5673],{},"Vue 做的事情就是让这条链路更稳定：",[105,5675,5676,5679,5682,5685],{},[108,5677,5678],{},"你声明数据。",[108,5680,5681],{},"你写模板描述界面。",[108,5683,5684],{},"你写函数响应用户操作。",[108,5686,5687],{},"Vue 负责把状态变化同步到 DOM。",[15,5689,5690],{},"如果你能一直围绕“数据是什么、数据在哪里变、界面如何从数据得到”思考，Vue 会越来越清晰。",[19,5692,5693],{"id":5693},"本章练习",[253,5695,5697],{"id":5696},"练习-1课程数组","练习 1：课程数组",[15,5699,5700],{},"创建下面数组：",[24,5702,5704],{"className":166,"code":5703,"language":168,"meta":29,"style":29},"const lessons = [\n { id: 1, title: \"JS 基础\", minutes: 40, done: false },\n { id: 2, title: \"DOM\", minutes: 35, done: true },\n { id: 3, title: \"Vue 响应式\", minutes: 50, done: false },\n];\n",[31,5705,5706,5716,5738,5759,5781],{"__ignoreMap":29},[34,5707,5708,5710,5712,5714],{"class":36,"line":37},[34,5709,176],{"class":175},[34,5711,1203],{"class":179},[34,5713,183],{"class":175},[34,5715,739],{"class":190},[34,5717,5718,5720,5722,5724,5726,5729,5732,5734,5736],{"class":36,"line":43},[34,5719,744],{"class":190},[34,5721,640],{"class":179},[34,5723,749],{"class":190},[34,5725,1210],{"class":186},[34,5727,5728],{"class":190},", minutes: ",[34,5730,5731],{"class":179},"40",[34,5733,755],{"class":190},[34,5735,702],{"class":179},[34,5737,761],{"class":190},[34,5739,5740,5742,5744,5746,5748,5750,5753,5755,5757],{"class":36,"line":49},[34,5741,744],{"class":190},[34,5743,768],{"class":179},[34,5745,749],{"class":190},[34,5747,1215],{"class":186},[34,5749,5728],{"class":190},[34,5751,5752],{"class":179},"35",[34,5754,755],{"class":190},[34,5756,758],{"class":179},[34,5758,761],{"class":190},[34,5760,5761,5763,5765,5767,5770,5772,5775,5777,5779],{"class":36,"line":56},[34,5762,744],{"class":190},[34,5764,786],{"class":179},[34,5766,749],{"class":190},[34,5768,5769],{"class":186},"\"Vue 响应式\"",[34,5771,5728],{"class":190},[34,5773,5774],{"class":179},"50",[34,5776,755],{"class":190},[34,5778,702],{"class":179},[34,5780,761],{"class":190},[34,5782,5783],{"class":36,"line":62},[34,5784,802],{"class":190},[15,5786,5787],{},"要求：",[2230,5789,5790,5796,5801,5808],{},[108,5791,5792,5793,5795],{},"用 ",[31,5794,1300],{}," 找出未完成课程。",[108,5797,5792,5798,5800],{},[31,5799,1469],{}," 统计总时长。",[108,5802,5803,5804,5807],{},"写 ",[31,5805,5806],{},"toggleDone(id)"," 切换课程完成状态。",[108,5809,5803,5810,5813],{},[31,5811,5812],{},"addLesson(title, minutes)"," 新增课程。",[253,5815,5817],{"id":5816},"练习-2模块导入导出","练习 2：模块导入导出",[15,5819,5820],{},"创建两个文件：",[24,5822,5825],{"className":5823,"code":5824,"language":2638,"meta":29},[2636],"math.js\nmain.js\n",[31,5826,5824],{"__ignoreMap":29},[15,5828,2884,5829,5831],{},[31,5830,2645],{}," 中导出：",[105,5833,5834,5838,5842],{},[108,5835,5836],{},[31,5837,1762],{},[108,5839,5840],{},[31,5841,3639],{},[108,5843,5844],{},[31,5845,3660],{},[15,5847,2884,5848,5850],{},[31,5849,2653],{}," 中导入并调用它们。",[15,5852,5853],{},"然后故意做下面实验：",[2230,5855,5856,5865,5875],{},[108,5857,5858,5859,5861,5862,5864],{},"去掉 ",[31,5860,2825],{},"，看看 ",[31,5863,426],{}," 报什么错。",[108,5866,2892,5867,5870,5871,5874],{},[31,5868,5869],{},"import { add }"," 改成 ",[31,5872,5873],{},"import add","，看看报什么错。",[108,5876,5877,5878,5870,5880,5874],{},"把路径 ",[31,5879,4296],{},[31,5881,2645],{},[15,5883,5884],{},"这些错误都很有价值，因为它们能帮你理解模块系统。",[253,5886,5888],{"id":5887},"练习-3用自己的话回答","练习 3：用自己的话回答",[15,5890,5891],{},"请写出下面问题的答案：",[2230,5893,5894,5899,5905,5910,5917],{},[108,5895,5896,5898],{},[31,5897,2825],{}," 是干嘛的？",[108,5900,5901,5902,5904],{},"不加 ",[31,5903,2825],{}," 能不能被别的文件按名字导入？",[108,5906,5907,5909],{},[31,5908,2825],{}," 会不会改变原函数自己的功能？",[108,5911,5912,5913,5916],{},"命名导出为什么导入时要加 ",[31,5914,5915],{},"{}","？",[108,5918,5919,258,5922,5925],{},[31,5920,5921],{},"import { ref } from 'vue'",[31,5923,5924],{},"import App from '.\u002FApp.vue'"," 的区别是什么？",[79,5927,5928],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":5930},[5931,5936,5937,5940,5945,5949,5950,5951,5975,5978,5979],{"id":159,"depth":43,"text":160,"children":5932},[5933,5935],{"id":255,"depth":49,"text":5934},"const 和 let 的区别",{"id":362,"depth":49,"text":363},{"id":522,"depth":43,"text":523},{"id":808,"depth":43,"text":809,"children":5938},[5939],{"id":1010,"depth":49,"text":1010},{"id":1187,"depth":43,"text":1188,"children":5941},[5942,5943],{"id":1294,"depth":49,"text":1295},{"id":1530,"depth":49,"text":5944},"为什么数组适合配合 v-for",{"id":1672,"depth":43,"text":1673,"children":5946},[5947,5948],{"id":1785,"depth":49,"text":1785},{"id":1898,"depth":49,"text":1899},{"id":2058,"depth":43,"text":2059},{"id":2249,"depth":43,"text":2250},{"id":2435,"depth":43,"text":2436,"children":5952},[5953,5954,5955,5957,5959,5961,5963,5964,5965,5966,5968,5969,5971,5972,5973,5974],{"id":2442,"depth":49,"text":2443},{"id":2657,"depth":49,"text":2658},{"id":2821,"depth":49,"text":5956},"8.3 export 是什么意思",{"id":2955,"depth":49,"text":5958},"8.4 不写 export 能不能 import",{"id":3052,"depth":49,"text":5960},"8.5 export 会不会影响原来的 math.js",{"id":3201,"depth":49,"text":5962},"8.6 import 之后怎么调用",{"id":3485,"depth":49,"text":3486},{"id":3692,"depth":49,"text":3693},{"id":3761,"depth":49,"text":3762},{"id":3935,"depth":49,"text":5967},"8.10 默认导出 export default",{"id":4093,"depth":49,"text":4094},{"id":4271,"depth":49,"text":5970},"8.12 导入路径为什么要写 .\u002F",{"id":4415,"depth":49,"text":4416},{"id":4523,"depth":49,"text":4524},{"id":4680,"depth":49,"text":4681},{"id":4801,"depth":49,"text":4802},{"id":5140,"depth":43,"text":5141,"children":5976},[5977],{"id":5277,"depth":49,"text":5277},{"id":5660,"depth":43,"text":5661},{"id":5693,"depth":43,"text":5693,"children":5980},[5981,5982,5983],{"id":5696,"depth":49,"text":5697},{"id":5816,"depth":49,"text":5817},{"id":5887,"depth":49,"text":5888},"Vue 不是一门脱离 JavaScript 的新语言。Vue 组件里的 \u003Cscript setup> 本质上就是 JavaScript，只是 Vue 提供了一些特殊函数，例如 ref、computed、watch。所以学 Vue 前，你至少要能看懂变量、数组、对象、函数、模块和异步。",{"order":37},"\u002Fsource\u002Fcode\u002Fvue\u002F01-js-from-zero",{"title":130,"description":5984},"source\u002Fcode\u002Fvue\u002F01-js-from-zero","tZtUg_yX3zClDfoiUzYDKiS_tHFJhohENckcpDAcr6g",{"id":5991,"title":5992,"body":5993,"description":6000,"extension":85,"meta":8337,"navigation":52,"path":8338,"seo":8339,"stem":8340,"__hash__":8341},"notes\u002Fsource\u002Fcode\u002Fvue\u002F02-dom-to-vue.md","02 DOM 到 Vue 的心智模型",{"type":8,"value":5994,"toc":8315},[5995,5998,6001,6004,6008,6011,6014,6072,6075,6081,6084,6154,6158,6427,6430,6458,6464,6467,6504,6507,6511,6514,6570,6573,6598,6836,6839,6859,6862,6866,7133,7136,7163,7166,7213,7216,7220,7223,7264,7267,7321,7324,7327,7338,7342,7345,7351,7354,7357,7461,7464,7491,7495,7498,7501,7525,7528,7575,7601,7604,7621,7624,7744,7757,7761,7764,7789,7792,7803,7806,7817,7820,7845,7856,7860,7863,7902,7905,7921,7924,7927,7949,7952,7990,7993,7995,8046,8048,8075,8079,8083,8098,8101,8122,8126,8129,8146,8149,8153,8156,8180,8183,8187,8190,8226,8229,8231,8235,8238,8252,8258,8262,8265,8288,8292,8295,8312],[11,5996,5992],{"id":5997},"_02-dom-到-vue-的心智模型",[15,5999,6000],{},"你略懂 DOM，这是学习 Vue 的好起点。Vue 并不是让你忘掉 HTML、CSS、JavaScript，而是帮你少写那些容易出错的“手动同步 DOM”的代码。",[15,6002,6003],{},"本章核心目标：理解为什么原生 DOM 在复杂交互里会越来越乱，以及 Vue 如何用“状态驱动界面”的方式解决这个问题。",[19,6005,6007],{"id":6006},"_1-dom-是什么","1. DOM 是什么",[15,6009,6010],{},"浏览器把 HTML 解析成一棵对象树，这棵树叫 DOM。",[15,6012,6013],{},"HTML：",[24,6015,6019],{"className":6016,"code":6017,"language":6018,"meta":29,"style":29},"language-html shiki shiki-themes github-light github-dark","\u003Cmain>\n \u003Ch1>任务列表\u003C\u002Fh1>\n \u003Cbutton id=\"add\">新增\u003C\u002Fbutton>\n\u003C\u002Fmain>\n","html",[31,6020,6021,6030,6043,6064],{"__ignoreMap":29},[34,6022,6023,6025,6028],{"class":36,"line":37},[34,6024,932],{"class":190},[34,6026,6027],{"class":935},"main",[34,6029,939],{"class":190},[34,6031,6032,6034,6036,6039,6041],{"class":36,"line":43},[34,6033,944],{"class":190},[34,6035,11],{"class":935},[34,6037,6038],{"class":190},">任务列表\u003C\u002F",[34,6040,11],{"class":935},[34,6042,939],{"class":190},[34,6044,6045,6047,6049,6052,6054,6057,6060,6062],{"class":36,"line":49},[34,6046,944],{"class":190},[34,6048,2004],{"class":935},[34,6050,6051],{"class":395}," id",[34,6053,202],{"class":190},[34,6055,6056],{"class":186},"\"add\"",[34,6058,6059],{"class":190},">新增\u003C\u002F",[34,6061,2004],{"class":935},[34,6063,939],{"class":190},[34,6065,6066,6068,6070],{"class":36,"line":56},[34,6067,1003],{"class":190},[34,6069,6027],{"class":935},[34,6071,939],{"class":190},[15,6073,6074],{},"浏览器内部会把它变成类似这样的结构：",[24,6076,6079],{"className":6077,"code":6078,"language":2638,"meta":29},[2636],"document\n main\n h1\n button#add\n",[31,6080,6078],{"__ignoreMap":29},[15,6082,6083],{},"JavaScript 可以通过 DOM API 找到元素、读取内容、修改内容、监听事件。",[24,6085,6087],{"className":166,"code":6086,"language":168,"meta":29,"style":29},"const button = document.querySelector('#add')\n\nbutton.addEventListener('click', () => {\n console.log('用户点击了新增')\n})\n",[31,6088,6089,6112,6116,6136,6149],{"__ignoreMap":29},[34,6090,6091,6093,6096,6098,6101,6104,6106,6109],{"class":36,"line":37},[34,6092,176],{"class":175},[34,6094,6095],{"class":179}," button",[34,6097,183],{"class":175},[34,6099,6100],{"class":190}," document.",[34,6102,6103],{"class":395},"querySelector",[34,6105,456],{"class":190},[34,6107,6108],{"class":186},"'#add'",[34,6110,6111],{"class":190},")\n",[34,6113,6114],{"class":36,"line":43},[34,6115,53],{"emptyLinePlaceholder":52},[34,6117,6118,6121,6124,6126,6129,6132,6134],{"class":36,"line":49},[34,6119,6120],{"class":190},"button.",[34,6122,6123],{"class":395},"addEventListener",[34,6125,456],{"class":190},[34,6127,6128],{"class":186},"'click'",[34,6130,6131],{"class":190},", () ",[34,6133,1333],{"class":175},[34,6135,674],{"class":190},[34,6137,6138,6140,6142,6144,6147],{"class":36,"line":56},[34,6139,1812],{"class":190},[34,6141,885],{"class":395},[34,6143,456],{"class":190},[34,6145,6146],{"class":186},"'用户点击了新增'",[34,6148,6111],{"class":190},[34,6150,6151],{"class":36,"line":62},[34,6152,6153],{"class":190},"})\n",[19,6155,6157],{"id":6156},"_2-原生-dom-写计数器","2. 原生 DOM 写计数器",[24,6159,6161],{"className":6016,"code":6160,"language":6018,"meta":29,"style":29},"\u003Cbutton id=\"decrease\">-\u003C\u002Fbutton>\n\u003Cspan id=\"count\">0\u003C\u002Fspan>\n\u003Cbutton id=\"increase\">+\u003C\u002Fbutton>\n\n\u003Cscript>\n let count = 0\n\n const countEl = document.querySelector('#count')\n const increaseBtn = document.querySelector('#increase')\n const decreaseBtn = document.querySelector('#decrease')\n\n function render() {\n countEl.textContent = count\n }\n\n increaseBtn.addEventListener('click', () => {\n count++\n render()\n })\n\n decreaseBtn.addEventListener('click', () => {\n count--\n render()\n })\n\u003C\u002Fscript>\n",[31,6162,6163,6183,6203,6222,6226,6234,6246,6250,6270,6290,6310,6314,6324,6334,6338,6342,6359,6367,6375,6380,6384,6401,6408,6414,6418],{"__ignoreMap":29},[34,6164,6165,6167,6169,6171,6173,6176,6179,6181],{"class":36,"line":37},[34,6166,932],{"class":190},[34,6168,2004],{"class":935},[34,6170,6051],{"class":395},[34,6172,202],{"class":190},[34,6174,6175],{"class":186},"\"decrease\"",[34,6177,6178],{"class":190},">-\u003C\u002F",[34,6180,2004],{"class":935},[34,6182,939],{"class":190},[34,6184,6185,6187,6189,6191,6193,6196,6199,6201],{"class":36,"line":43},[34,6186,932],{"class":190},[34,6188,34],{"class":935},[34,6190,6051],{"class":395},[34,6192,202],{"class":190},[34,6194,6195],{"class":186},"\"count\"",[34,6197,6198],{"class":190},">0\u003C\u002F",[34,6200,34],{"class":935},[34,6202,939],{"class":190},[34,6204,6205,6207,6209,6211,6213,6215,6218,6220],{"class":36,"line":49},[34,6206,932],{"class":190},[34,6208,2004],{"class":935},[34,6210,6051],{"class":395},[34,6212,202],{"class":190},[34,6214,2012],{"class":186},[34,6216,6217],{"class":190},">+\u003C\u002F",[34,6219,2004],{"class":935},[34,6221,939],{"class":190},[34,6223,6224],{"class":36,"line":56},[34,6225,53],{"emptyLinePlaceholder":52},[34,6227,6228,6230,6232],{"class":36,"line":62},[34,6229,932],{"class":190},[34,6231,1911],{"class":935},[34,6233,939],{"class":190},[34,6235,6236,6239,6241,6243],{"class":36,"line":68},[34,6237,6238],{"class":175}," let",[34,6240,319],{"class":190},[34,6242,202],{"class":175},[34,6244,6245],{"class":179}," 0\n",[34,6247,6248],{"class":36,"line":74},[34,6249,53],{"emptyLinePlaceholder":52},[34,6251,6252,6254,6257,6259,6261,6263,6265,6268],{"class":36,"line":1971},[34,6253,5298],{"class":175},[34,6255,6256],{"class":179}," countEl",[34,6258,183],{"class":175},[34,6260,6100],{"class":190},[34,6262,6103],{"class":395},[34,6264,456],{"class":190},[34,6266,6267],{"class":186},"'#count'",[34,6269,6111],{"class":190},[34,6271,6272,6274,6277,6279,6281,6283,6285,6288],{"class":36,"line":1976},[34,6273,5298],{"class":175},[34,6275,6276],{"class":179}," increaseBtn",[34,6278,183],{"class":175},[34,6280,6100],{"class":190},[34,6282,6103],{"class":395},[34,6284,456],{"class":190},[34,6286,6287],{"class":186},"'#increase'",[34,6289,6111],{"class":190},[34,6291,6292,6294,6297,6299,6301,6303,6305,6308],{"class":36,"line":1985},[34,6293,5298],{"class":175},[34,6295,6296],{"class":179}," decreaseBtn",[34,6298,183],{"class":175},[34,6300,6100],{"class":190},[34,6302,6103],{"class":395},[34,6304,456],{"class":190},[34,6306,6307],{"class":186},"'#decrease'",[34,6309,6111],{"class":190},[34,6311,6312],{"class":36,"line":1990},[34,6313,53],{"emptyLinePlaceholder":52},[34,6315,6316,6319,6322],{"class":36,"line":1999},[34,6317,6318],{"class":175}," function",[34,6320,6321],{"class":395}," render",[34,6323,399],{"class":190},[34,6325,6326,6329,6331],{"class":36,"line":2022},[34,6327,6328],{"class":190}," countEl.textContent ",[34,6330,202],{"class":175},[34,6332,6333],{"class":190}," count\n",[34,6335,6336],{"class":36,"line":2036},[34,6337,5631],{"class":190},[34,6339,6340],{"class":36,"line":2603},[34,6341,53],{"emptyLinePlaceholder":52},[34,6343,6344,6347,6349,6351,6353,6355,6357],{"class":36,"line":5537},[34,6345,6346],{"class":190}," increaseBtn.",[34,6348,6123],{"class":395},[34,6350,456],{"class":190},[34,6352,6128],{"class":186},[34,6354,6131],{"class":190},[34,6356,1333],{"class":175},[34,6358,674],{"class":190},[34,6360,6361,6364],{"class":36,"line":5543},[34,6362,6363],{"class":190}," count",[34,6365,6366],{"class":175},"++\n",[34,6368,6369,6372],{"class":36,"line":5548},[34,6370,6371],{"class":395}," render",[34,6373,6374],{"class":190},"()\n",[34,6376,6377],{"class":36,"line":5564},[34,6378,6379],{"class":190}," })\n",[34,6381,6382],{"class":36,"line":5576},[34,6383,53],{"emptyLinePlaceholder":52},[34,6385,6386,6389,6391,6393,6395,6397,6399],{"class":36,"line":5606},[34,6387,6388],{"class":190}," decreaseBtn.",[34,6390,6123],{"class":395},[34,6392,456],{"class":190},[34,6394,6128],{"class":186},[34,6396,6131],{"class":190},[34,6398,1333],{"class":175},[34,6400,674],{"class":190},[34,6402,6403,6405],{"class":36,"line":5616},[34,6404,6363],{"class":190},[34,6406,6407],{"class":175},"--\n",[34,6409,6410,6412],{"class":36,"line":5628},[34,6411,6371],{"class":395},[34,6413,6374],{"class":190},[34,6415,6416],{"class":36,"line":5634},[34,6417,6379],{"class":190},[34,6419,6421,6423,6425],{"class":36,"line":6420},25,[34,6422,1003],{"class":190},[34,6424,1911],{"class":935},[34,6426,939],{"class":190},[15,6428,6429],{},"逐行理解：",[105,6431,6432,6438,6444,6450],{},[108,6433,6434,6437],{},[31,6435,6436],{},"let count = 0","：真正的数据。",[108,6439,6440,6443],{},[31,6441,6442],{},"countEl","：页面上显示数字的 DOM 元素。",[108,6445,6446,6449],{},[31,6447,6448],{},"render()","：把数据同步到 DOM。",[108,6451,6452,6453,6455,6456,347],{},"点击按钮后先修改 ",[31,6454,2770],{},"，再调用 ",[31,6457,6448],{},[15,6459,6460,6461],{},"这里有一个关键问题：",[2627,6462,6463],{},"你必须记得每次数据变化后手动更新 DOM。",[15,6465,6466],{},"如果你忘了：",[24,6468,6470],{"className":166,"code":6469,"language":168,"meta":29,"style":29},"increaseBtn.addEventListener('click', () => {\n count++\n \u002F\u002F 忘记 render()\n})\n",[31,6471,6472,6489,6495,6500],{"__ignoreMap":29},[34,6473,6474,6477,6479,6481,6483,6485,6487],{"class":36,"line":37},[34,6475,6476],{"class":190},"increaseBtn.",[34,6478,6123],{"class":395},[34,6480,456],{"class":190},[34,6482,6128],{"class":186},[34,6484,6131],{"class":190},[34,6486,1333],{"class":175},[34,6488,674],{"class":190},[34,6490,6491,6493],{"class":36,"line":43},[34,6492,404],{"class":190},[34,6494,6366],{"class":175},[34,6496,6497],{"class":36,"line":49},[34,6498,6499],{"class":301}," \u002F\u002F 忘记 render()\n",[34,6501,6502],{"class":36,"line":56},[34,6503,6153],{"class":190},[15,6505,6506],{},"数据已经变了，但页面没变。用户看到的界面和真实数据不一致。",[19,6508,6510],{"id":6509},"_3-原生-dom-写列表为什么容易变复杂","3. 原生 DOM 写列表为什么容易变复杂",[15,6512,6513],{},"假设你有任务数组：",[24,6515,6517],{"className":166,"code":6516,"language":168,"meta":29,"style":29},"let tasks = [\n { id: 1, title: '学习 JS', done: false },\n { id: 2, title: '学习 DOM', done: true }\n]\n",[31,6518,6519,6530,6547,6565],{"__ignoreMap":29},[34,6520,6521,6523,6526,6528],{"class":36,"line":37},[34,6522,196],{"class":175},[34,6524,6525],{"class":190}," tasks ",[34,6527,202],{"class":175},[34,6529,739],{"class":190},[34,6531,6532,6534,6536,6538,6541,6543,6545],{"class":36,"line":43},[34,6533,744],{"class":190},[34,6535,640],{"class":179},[34,6537,749],{"class":190},[34,6539,6540],{"class":186},"'学习 JS'",[34,6542,755],{"class":190},[34,6544,702],{"class":179},[34,6546,761],{"class":190},[34,6548,6549,6551,6553,6555,6558,6560,6562],{"class":36,"line":49},[34,6550,744],{"class":190},[34,6552,768],{"class":179},[34,6554,749],{"class":190},[34,6556,6557],{"class":186},"'学习 DOM'",[34,6559,755],{"class":190},[34,6561,758],{"class":179},[34,6563,6564],{"class":190}," }\n",[34,6566,6567],{"class":36,"line":56},[34,6568,6569],{"class":190},"]\n",[15,6571,6572],{},"你要渲染列表：",[24,6574,6576],{"className":6016,"code":6575,"language":6018,"meta":29,"style":29},"\u003Cul id=\"task-list\">\u003C\u002Ful>\n",[31,6577,6578],{"__ignoreMap":29},[34,6579,6580,6582,6584,6586,6588,6591,6594,6596],{"class":36,"line":37},[34,6581,932],{"class":190},[34,6583,105],{"class":935},[34,6585,6051],{"class":395},[34,6587,202],{"class":190},[34,6589,6590],{"class":186},"\"task-list\"",[34,6592,6593],{"class":190},">\u003C\u002F",[34,6595,105],{"class":935},[34,6597,939],{"class":190},[24,6599,6601],{"className":166,"code":6600,"language":168,"meta":29,"style":29},"const listEl = document.querySelector('#task-list')\n\nfunction renderTasks() {\n listEl.innerHTML = ''\n\n tasks.forEach((task) => {\n const item = document.createElement('li')\n item.textContent = task.done ? `✅ ${task.title}` : task.title\n\n const button = document.createElement('button')\n button.textContent = '切换'\n button.addEventListener('click', () => {\n task.done = !task.done\n renderTasks()\n })\n\n item.appendChild(button)\n listEl.appendChild(item)\n })\n}\n\nrenderTasks()\n",[31,6602,6603,6623,6627,6636,6646,6650,6668,6689,6718,6722,6741,6751,6768,6780,6787,6792,6796,6807,6817,6821,6825,6829],{"__ignoreMap":29},[34,6604,6605,6607,6610,6612,6614,6616,6618,6621],{"class":36,"line":37},[34,6606,176],{"class":175},[34,6608,6609],{"class":179}," listEl",[34,6611,183],{"class":175},[34,6613,6100],{"class":190},[34,6615,6103],{"class":395},[34,6617,456],{"class":190},[34,6619,6620],{"class":186},"'#task-list'",[34,6622,6111],{"class":190},[34,6624,6625],{"class":36,"line":43},[34,6626,53],{"emptyLinePlaceholder":52},[34,6628,6629,6631,6634],{"class":36,"line":49},[34,6630,392],{"class":175},[34,6632,6633],{"class":395}," renderTasks",[34,6635,399],{"class":190},[34,6637,6638,6641,6643],{"class":36,"line":56},[34,6639,6640],{"class":190}," listEl.innerHTML ",[34,6642,202],{"class":175},[34,6644,6645],{"class":186}," ''\n",[34,6647,6648],{"class":36,"line":62},[34,6649,53],{"emptyLinePlaceholder":52},[34,6651,6652,6655,6658,6660,6662,6664,6666],{"class":36,"line":68},[34,6653,6654],{"class":190}," tasks.",[34,6656,6657],{"class":395},"forEach",[34,6659,1323],{"class":190},[34,6661,1327],{"class":1326},[34,6663,1330],{"class":190},[34,6665,1333],{"class":175},[34,6667,674],{"class":190},[34,6669,6670,6672,6675,6677,6679,6682,6684,6687],{"class":36,"line":74},[34,6671,5475],{"class":175},[34,6673,6674],{"class":179}," item",[34,6676,183],{"class":175},[34,6678,6100],{"class":190},[34,6680,6681],{"class":395},"createElement",[34,6683,456],{"class":190},[34,6685,6686],{"class":186},"'li'",[34,6688,6111],{"class":190},[34,6690,6691,6694,6696,6699,6702,6705,6707,6709,6711,6713,6715],{"class":36,"line":1971},[34,6692,6693],{"class":190}," item.textContent ",[34,6695,202],{"class":175},[34,6697,6698],{"class":190}," task.done ",[34,6700,6701],{"class":175},"?",[34,6703,6704],{"class":186}," `✅ ${",[34,6706,1327],{"class":190},[34,6708,2547],{"class":186},[34,6710,1665],{"class":190},[34,6712,2560],{"class":186},[34,6714,1621],{"class":175},[34,6716,6717],{"class":190}," task.title\n",[34,6719,6720],{"class":36,"line":1976},[34,6721,53],{"emptyLinePlaceholder":52},[34,6723,6724,6726,6728,6730,6732,6734,6736,6739],{"class":36,"line":1985},[34,6725,5475],{"class":175},[34,6727,6095],{"class":179},[34,6729,183],{"class":175},[34,6731,6100],{"class":190},[34,6733,6681],{"class":395},[34,6735,456],{"class":190},[34,6737,6738],{"class":186},"'button'",[34,6740,6111],{"class":190},[34,6742,6743,6746,6748],{"class":36,"line":1990},[34,6744,6745],{"class":190}," button.textContent ",[34,6747,202],{"class":175},[34,6749,6750],{"class":186}," '切换'\n",[34,6752,6753,6756,6758,6760,6762,6764,6766],{"class":36,"line":1999},[34,6754,6755],{"class":190}," button.",[34,6757,6123],{"class":395},[34,6759,456],{"class":190},[34,6761,6128],{"class":186},[34,6763,6131],{"class":190},[34,6765,1333],{"class":175},[34,6767,674],{"class":190},[34,6769,6770,6773,6775,6777],{"class":36,"line":2022},[34,6771,6772],{"class":190}," task.done ",[34,6774,202],{"class":175},[34,6776,1336],{"class":175},[34,6778,6779],{"class":190},"task.done\n",[34,6781,6782,6785],{"class":36,"line":2036},[34,6783,6784],{"class":395}," renderTasks",[34,6786,6374],{"class":190},[34,6788,6789],{"class":36,"line":2603},[34,6790,6791],{"class":190}," })\n",[34,6793,6794],{"class":36,"line":5537},[34,6795,53],{"emptyLinePlaceholder":52},[34,6797,6798,6801,6804],{"class":36,"line":5543},[34,6799,6800],{"class":190}," item.",[34,6802,6803],{"class":395},"appendChild",[34,6805,6806],{"class":190},"(button)\n",[34,6808,6809,6812,6814],{"class":36,"line":5548},[34,6810,6811],{"class":190}," listEl.",[34,6813,6803],{"class":395},[34,6815,6816],{"class":190},"(item)\n",[34,6818,6819],{"class":36,"line":5564},[34,6820,6379],{"class":190},[34,6822,6823],{"class":36,"line":5576},[34,6824,77],{"class":190},[34,6826,6827],{"class":36,"line":5606},[34,6828,53],{"emptyLinePlaceholder":52},[34,6830,6831,6834],{"class":36,"line":5616},[34,6832,6833],{"class":395},"renderTasks",[34,6835,6374],{"class":190},[15,6837,6838],{},"这段代码能工作，但你要自己处理很多细节：",[105,6840,6841,6844,6847,6850,6853,6856],{},[108,6842,6843],{},"清空旧列表。",[108,6845,6846],{},"创建每个 DOM 元素。",[108,6848,6849],{},"填入文字。",[108,6851,6852],{},"给每个按钮绑定事件。",[108,6854,6855],{},"数据变化后重新渲染。",[108,6857,6858],{},"删除任务、筛选任务、编辑任务时还要继续扩展。",[15,6860,6861],{},"项目越大，手动 DOM 同步越容易出现问题。",[19,6863,6865],{"id":6864},"_4-vue-怎么写同一个列表","4. Vue 怎么写同一个列表",[24,6867,6869],{"className":923,"code":6868,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { ref } from 'vue'\n\nconst tasks = ref([\n { id: 1, title: '学习 JS', done: false },\n { id: 2, title: '学习 DOM', done: true }\n])\n\nfunction toggleTask(id) {\n const task = tasks.value.find((item) => item.id === id)\n\n if (task) {\n task.done = !task.done\n }\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cul>\n \u003Cli v-for=\"task in tasks\" :key=\"task.id\">\n \u003Cspan>{{ task.done ? '✅' : '⬜' }} {{ task.title }}\u003C\u002Fspan>\n \u003Cbutton @click=\"toggleTask(task.id)\">切换\u003C\u002Fbutton>\n \u003C\u002Fli>\n \u003C\u002Ful>\n\u003C\u002Ftemplate>\n",[31,6870,6871,6881,6892,6896,6909,6925,6941,6946,6950,6962,6992,6996,7004,7015,7019,7023,7031,7035,7043,7051,7074,7088,7108,7117,7125],{"__ignoreMap":29},[34,6872,6873,6875,6877,6879],{"class":36,"line":37},[34,6874,932],{"class":190},[34,6876,1911],{"class":935},[34,6878,1914],{"class":395},[34,6880,939],{"class":190},[34,6882,6883,6885,6887,6889],{"class":36,"line":43},[34,6884,426],{"class":175},[34,6886,429],{"class":190},[34,6888,432],{"class":175},[34,6890,6891],{"class":186}," 'vue'\n",[34,6893,6894],{"class":36,"line":49},[34,6895,53],{"emptyLinePlaceholder":52},[34,6897,6898,6900,6902,6904,6906],{"class":36,"line":56},[34,6899,176],{"class":175},[34,6901,734],{"class":179},[34,6903,183],{"class":175},[34,6905,453],{"class":395},[34,6907,6908],{"class":190},"([\n",[34,6910,6911,6913,6915,6917,6919,6921,6923],{"class":36,"line":62},[34,6912,744],{"class":190},[34,6914,640],{"class":179},[34,6916,749],{"class":190},[34,6918,6540],{"class":186},[34,6920,755],{"class":190},[34,6922,702],{"class":179},[34,6924,761],{"class":190},[34,6926,6927,6929,6931,6933,6935,6937,6939],{"class":36,"line":68},[34,6928,744],{"class":190},[34,6930,768],{"class":179},[34,6932,749],{"class":190},[34,6934,6557],{"class":186},[34,6936,755],{"class":190},[34,6938,758],{"class":179},[34,6940,6564],{"class":190},[34,6942,6943],{"class":36,"line":74},[34,6944,6945],{"class":190},"])\n",[34,6947,6948],{"class":36,"line":1971},[34,6949,53],{"emptyLinePlaceholder":52},[34,6951,6952,6954,6956,6958,6960],{"class":36,"line":1976},[34,6953,392],{"class":175},[34,6955,1800],{"class":395},[34,6957,456],{"class":190},[34,6959,1805],{"class":1326},[34,6961,1700],{"class":190},[34,6963,6964,6966,6968,6970,6973,6975,6977,6980,6982,6984,6987,6989],{"class":36,"line":1985},[34,6965,5298],{"class":175},[34,6967,632],{"class":179},[34,6969,183],{"class":175},[34,6971,6972],{"class":190}," tasks.value.",[34,6974,1390],{"class":395},[34,6976,1323],{"class":190},[34,6978,6979],{"class":1326},"item",[34,6981,1330],{"class":190},[34,6983,1333],{"class":175},[34,6985,6986],{"class":190}," item.id ",[34,6988,1423],{"class":175},[34,6990,6991],{"class":190}," id)\n",[34,6993,6994],{"class":36,"line":1990},[34,6995,53],{"emptyLinePlaceholder":52},[34,6997,6998,7001],{"class":36,"line":1999},[34,6999,7000],{"class":175}," if",[34,7002,7003],{"class":190}," (task) {\n",[34,7005,7006,7009,7011,7013],{"class":36,"line":2022},[34,7007,7008],{"class":190}," task.done ",[34,7010,202],{"class":175},[34,7012,1336],{"class":175},[34,7014,6779],{"class":190},[34,7016,7017],{"class":36,"line":2036},[34,7018,5631],{"class":190},[34,7020,7021],{"class":36,"line":2603},[34,7022,77],{"class":190},[34,7024,7025,7027,7029],{"class":36,"line":5537},[34,7026,1003],{"class":190},[34,7028,1911],{"class":935},[34,7030,939],{"class":190},[34,7032,7033],{"class":36,"line":5543},[34,7034,53],{"emptyLinePlaceholder":52},[34,7036,7037,7039,7041],{"class":36,"line":5548},[34,7038,932],{"class":190},[34,7040,936],{"class":935},[34,7042,939],{"class":190},[34,7044,7045,7047,7049],{"class":36,"line":5564},[34,7046,944],{"class":190},[34,7048,105],{"class":935},[34,7050,939],{"class":190},[34,7052,7053,7055,7057,7059,7061,7064,7067,7069,7072],{"class":36,"line":5576},[34,7054,954],{"class":190},[34,7056,108],{"class":935},[34,7058,1603],{"class":395},[34,7060,202],{"class":190},[34,7062,7063],{"class":186},"\"task in tasks\"",[34,7065,7066],{"class":395}," :key",[34,7068,202],{"class":190},[34,7070,7071],{"class":186},"\"task.id\"",[34,7073,939],{"class":190},[34,7075,7076,7079,7081,7084,7086],{"class":36,"line":5606},[34,7077,7078],{"class":190}," \u003C",[34,7080,34],{"class":935},[34,7082,7083],{"class":190},">{{ task.done ? '✅' : '⬜' }} {{ task.title }}\u003C\u002F",[34,7085,34],{"class":935},[34,7087,939],{"class":190},[34,7089,7090,7092,7094,7096,7098,7101,7104,7106],{"class":36,"line":5616},[34,7091,7078],{"class":190},[34,7093,2004],{"class":935},[34,7095,2007],{"class":395},[34,7097,202],{"class":190},[34,7099,7100],{"class":186},"\"toggleTask(task.id)\"",[34,7102,7103],{"class":190},">切换\u003C\u002F",[34,7105,2004],{"class":935},[34,7107,939],{"class":190},[34,7109,7110,7113,7115],{"class":36,"line":5628},[34,7111,7112],{"class":190}," \u003C\u002F",[34,7114,108],{"class":935},[34,7116,939],{"class":190},[34,7118,7119,7121,7123],{"class":36,"line":5634},[34,7120,994],{"class":190},[34,7122,105],{"class":935},[34,7124,939],{"class":190},[34,7126,7127,7129,7131],{"class":36,"line":6420},[34,7128,1003],{"class":190},[34,7130,936],{"class":935},[34,7132,939],{"class":190},[15,7134,7135],{},"这里发生了什么：",[105,7137,7138,7143,7148,7154,7160],{},[108,7139,7140,7142],{},[31,7141,1345],{}," 是响应式数据。",[108,7144,7145,7147],{},[31,7146,1534],{}," 根据数组渲染列表。",[108,7149,7150,7153],{},[31,7151,7152],{},"@click"," 绑定点击事件。",[108,7155,7156,7159],{},[31,7157,7158],{},"toggleTask"," 只负责改数据。",[108,7161,7162],{},"数据变化后，Vue 自动更新 DOM。",[15,7164,7165],{},"你不再手动写：",[24,7167,7169],{"className":166,"code":7168,"language":168,"meta":29,"style":29},"document.createElement(...)\nappendChild(...)\ntextContent = ...\ninnerHTML = ...\n",[31,7170,7171,7184,7194,7204],{"__ignoreMap":29},[34,7172,7173,7176,7178,7180,7182],{"class":36,"line":37},[34,7174,7175],{"class":190},"document.",[34,7177,6681],{"class":395},[34,7179,456],{"class":190},[34,7181,1167],{"class":175},[34,7183,6111],{"class":190},[34,7185,7186,7188,7190,7192],{"class":36,"line":43},[34,7187,6803],{"class":395},[34,7189,456],{"class":190},[34,7191,1167],{"class":175},[34,7193,6111],{"class":190},[34,7195,7196,7199,7201],{"class":36,"line":49},[34,7197,7198],{"class":190},"textContent ",[34,7200,202],{"class":175},[34,7202,7203],{"class":175}," ...\n",[34,7205,7206,7209,7211],{"class":36,"line":56},[34,7207,7208],{"class":190},"innerHTML ",[34,7210,202],{"class":175},[34,7212,7203],{"class":175},[15,7214,7215],{},"Vue 的价值不是“少写几个字符”，而是让你把注意力从 DOM 操作转回数据和业务逻辑。",[19,7217,7219],{"id":7218},"_5-命令式和声明式","5. 命令式和声明式",[15,7221,7222],{},"原生 DOM 常见写法是命令式：你一步步命令浏览器怎么做。",[24,7224,7226],{"className":166,"code":7225,"language":168,"meta":29,"style":29},"const item = document.createElement('li')\nitem.textContent = task.title\nlistEl.appendChild(item)\n",[31,7227,7228,7246,7255],{"__ignoreMap":29},[34,7229,7230,7232,7234,7236,7238,7240,7242,7244],{"class":36,"line":37},[34,7231,176],{"class":175},[34,7233,6674],{"class":179},[34,7235,183],{"class":175},[34,7237,6100],{"class":190},[34,7239,6681],{"class":395},[34,7241,456],{"class":190},[34,7243,6686],{"class":186},[34,7245,6111],{"class":190},[34,7247,7248,7251,7253],{"class":36,"line":43},[34,7249,7250],{"class":190},"item.textContent ",[34,7252,202],{"class":175},[34,7254,6717],{"class":190},[34,7256,7257,7260,7262],{"class":36,"line":49},[34,7258,7259],{"class":190},"listEl.",[34,7261,6803],{"class":395},[34,7263,6816],{"class":190},[15,7265,7266],{},"Vue 模板是声明式：你描述“数据处于某种状态时，页面应该是什么样”。",[24,7268,7270],{"className":923,"code":7269,"language":925,"meta":29,"style":29},"\u003Cli v-for=\"task in tasks\" :key=\"task.id\">\n {{ task.title }}\n\u003C\u002Fli>\n",[31,7271,7272,7308,7313],{"__ignoreMap":29},[34,7273,7274,7276,7278,7280,7282,7284,7287,7289,7291,7293,7295,7297,7299,7301,7304,7306],{"class":36,"line":37},[34,7275,932],{"class":190},[34,7277,108],{"class":935},[34,7279,1603],{"class":175},[34,7281,202],{"class":190},[34,7283,1608],{"class":186},[34,7285,7286],{"class":190},"task ",[34,7288,1614],{"class":175},[34,7290,734],{"class":190},[34,7292,1608],{"class":186},[34,7294,1621],{"class":190},[34,7296,1624],{"class":395},[34,7298,202],{"class":190},[34,7300,1608],{"class":186},[34,7302,7303],{"class":190},"task.id",[34,7305,1608],{"class":186},[34,7307,939],{"class":190},[34,7309,7310],{"class":36,"line":43},[34,7311,7312],{"class":190}," {{ task.title }}\n",[34,7314,7315,7317,7319],{"class":36,"line":49},[34,7316,1003],{"class":190},[34,7318,108],{"class":935},[34,7320,939],{"class":190},[15,7322,7323],{},"命令式关注“怎么做”，声明式关注“是什么”。",[15,7325,7326],{},"在复杂页面里，声明式更容易维护，因为你能直接看到：",[105,7328,7329,7332,7335],{},[108,7330,7331],{},"数据在哪里。",[108,7333,7334],{},"页面如何由数据得到。",[108,7336,7337],{},"用户操作如何修改数据。",[19,7339,7341],{"id":7340},"_6-vue-的核心循环","6. Vue 的核心循环",[15,7343,7344],{},"Vue 最重要的心智模型：",[24,7346,7349],{"className":7347,"code":7348,"language":2638,"meta":29},[2636],"状态 state\n ↓\n模板 template 描述界面\n ↓\n页面 UI\n ↓\n用户点击、输入、请求返回\n ↓\n修改状态 state\n ↓\nVue 自动更新页面\n",[31,7350,7348],{"__ignoreMap":29},[15,7352,7353],{},"这条链路如果想清楚，很多 Vue 问题都会变简单。",[15,7355,7356],{},"例如输入框：",[24,7358,7360],{"className":923,"code":7359,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { ref } from 'vue'\n\nconst name = ref('')\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cinput v-model=\"name\">\n \u003Cp>你好，{{ name }}\u003C\u002Fp>\n\u003C\u002Ftemplate>\n",[31,7361,7362,7372,7382,7386,7403,7411,7415,7423,7440,7453],{"__ignoreMap":29},[34,7363,7364,7366,7368,7370],{"class":36,"line":37},[34,7365,932],{"class":190},[34,7367,1911],{"class":935},[34,7369,1914],{"class":395},[34,7371,939],{"class":190},[34,7373,7374,7376,7378,7380],{"class":36,"line":43},[34,7375,426],{"class":175},[34,7377,429],{"class":190},[34,7379,432],{"class":175},[34,7381,6891],{"class":186},[34,7383,7384],{"class":36,"line":49},[34,7385,53],{"emptyLinePlaceholder":52},[34,7387,7388,7390,7392,7394,7396,7398,7401],{"class":36,"line":56},[34,7389,176],{"class":175},[34,7391,2333],{"class":179},[34,7393,183],{"class":175},[34,7395,453],{"class":395},[34,7397,456],{"class":190},[34,7399,7400],{"class":186},"''",[34,7402,6111],{"class":190},[34,7404,7405,7407,7409],{"class":36,"line":62},[34,7406,1003],{"class":190},[34,7408,1911],{"class":935},[34,7410,939],{"class":190},[34,7412,7413],{"class":36,"line":68},[34,7414,53],{"emptyLinePlaceholder":52},[34,7416,7417,7419,7421],{"class":36,"line":74},[34,7418,932],{"class":190},[34,7420,936],{"class":935},[34,7422,939],{"class":190},[34,7424,7425,7427,7430,7433,7435,7438],{"class":36,"line":1971},[34,7426,944],{"class":190},[34,7428,7429],{"class":935},"input",[34,7431,7432],{"class":395}," v-model",[34,7434,202],{"class":190},[34,7436,7437],{"class":186},"\"name\"",[34,7439,939],{"class":190},[34,7441,7442,7444,7446,7449,7451],{"class":36,"line":1976},[34,7443,944],{"class":190},[34,7445,15],{"class":935},[34,7447,7448],{"class":190},">你好，{{ name }}\u003C\u002F",[34,7450,15],{"class":935},[34,7452,939],{"class":190},[34,7454,7455,7457,7459],{"class":36,"line":1985},[34,7456,1003],{"class":190},[34,7458,936],{"class":935},[34,7460,939],{"class":190},[15,7462,7463],{},"用户输入后：",[2230,7465,7466,7469,7477,7482,7488],{},[108,7467,7468],{},"输入框内容变化。",[108,7470,7471,7474,7475,347],{},[31,7472,7473],{},"v-model"," 更新 ",[31,7476,2305],{},[108,7478,7479,7481],{},[31,7480,2305],{}," 是响应式状态。",[108,7483,7484,7485,347],{},"Vue 重新渲染 ",[31,7486,7487],{},"{{ name }}",[108,7489,7490],{},"页面显示新内容。",[19,7492,7494],{"id":7493},"_7-vue-不是完全不能操作-dom","7. Vue 不是完全不能操作 DOM",[15,7496,7497],{},"初学阶段建议：能用状态表达，就不要直接操作 DOM。",[15,7499,7500],{},"不推荐：",[24,7502,7504],{"className":166,"code":7503,"language":168,"meta":29,"style":29},"document.querySelector('.message').textContent = '保存成功'\n",[31,7505,7506],{"__ignoreMap":29},[34,7507,7508,7510,7512,7514,7517,7520,7522],{"class":36,"line":37},[34,7509,7175],{"class":190},[34,7511,6103],{"class":395},[34,7513,456],{"class":190},[34,7515,7516],{"class":186},"'.message'",[34,7518,7519],{"class":190},").textContent ",[34,7521,202],{"class":175},[34,7523,7524],{"class":186}," '保存成功'\n",[15,7526,7527],{},"推荐：",[24,7529,7531],{"className":166,"code":7530,"language":168,"meta":29,"style":29},"const message = ref('')\n\nfunction save() {\n message.value = '保存成功'\n}\n",[31,7532,7533,7549,7553,7562,7571],{"__ignoreMap":29},[34,7534,7535,7537,7539,7541,7543,7545,7547],{"class":36,"line":37},[34,7536,176],{"class":175},[34,7538,4455],{"class":179},[34,7540,183],{"class":175},[34,7542,453],{"class":395},[34,7544,456],{"class":190},[34,7546,7400],{"class":186},[34,7548,6111],{"class":190},[34,7550,7551],{"class":36,"line":43},[34,7552,53],{"emptyLinePlaceholder":52},[34,7554,7555,7557,7560],{"class":36,"line":49},[34,7556,392],{"class":175},[34,7558,7559],{"class":395}," save",[34,7561,399],{"class":190},[34,7563,7564,7567,7569],{"class":36,"line":56},[34,7565,7566],{"class":190}," message.value ",[34,7568,202],{"class":175},[34,7570,7524],{"class":186},[34,7572,7573],{"class":36,"line":62},[34,7574,77],{"class":190},[24,7576,7578],{"className":923,"code":7577,"language":925,"meta":29,"style":29},"\u003Cp class=\"message\">{{ message }}\u003C\u002Fp>\n",[31,7579,7580],{"__ignoreMap":29},[34,7581,7582,7584,7586,7589,7591,7594,7597,7599],{"class":36,"line":37},[34,7583,932],{"class":190},[34,7585,15],{"class":935},[34,7587,7588],{"class":395}," class",[34,7590,202],{"class":190},[34,7592,7593],{"class":186},"\"message\"",[34,7595,7596],{"class":190},">{{ message }}\u003C\u002F",[34,7598,15],{"class":935},[34,7600,939],{"class":190},[15,7602,7603],{},"但确实有些场景需要 DOM：",[105,7605,7606,7609,7612,7615,7618],{},[108,7607,7608],{},"页面加载后让输入框自动聚焦。",[108,7610,7611],{},"测量元素宽高。",[108,7613,7614],{},"集成第三方图表库。",[108,7616,7617],{},"操作 canvas。",[108,7619,7620],{},"调用浏览器原生 API。",[15,7622,7623],{},"Vue 提供模板引用：",[24,7625,7627],{"className":923,"code":7626,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { onMounted, ref } from 'vue'\n\nconst inputEl = ref(null)\n\nonMounted(() => {\n inputEl.value.focus()\n})\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cinput ref=\"inputEl\">\n\u003C\u002Ftemplate>\n",[31,7628,7629,7639,7650,7654,7671,7675,7687,7697,7701,7709,7713,7721,7736],{"__ignoreMap":29},[34,7630,7631,7633,7635,7637],{"class":36,"line":37},[34,7632,932],{"class":190},[34,7634,1911],{"class":935},[34,7636,1914],{"class":395},[34,7638,939],{"class":190},[34,7640,7641,7643,7646,7648],{"class":36,"line":43},[34,7642,426],{"class":175},[34,7644,7645],{"class":190}," { onMounted, ref } ",[34,7647,432],{"class":175},[34,7649,6891],{"class":186},[34,7651,7652],{"class":36,"line":49},[34,7653,53],{"emptyLinePlaceholder":52},[34,7655,7656,7658,7661,7663,7665,7667,7669],{"class":36,"line":56},[34,7657,176],{"class":175},[34,7659,7660],{"class":179}," inputEl",[34,7662,183],{"class":175},[34,7664,453],{"class":395},[34,7666,456],{"class":190},[34,7668,5419],{"class":179},[34,7670,6111],{"class":190},[34,7672,7673],{"class":36,"line":62},[34,7674,53],{"emptyLinePlaceholder":52},[34,7676,7677,7680,7683,7685],{"class":36,"line":68},[34,7678,7679],{"class":395},"onMounted",[34,7681,7682],{"class":190},"(() ",[34,7684,1333],{"class":175},[34,7686,674],{"class":190},[34,7688,7689,7692,7695],{"class":36,"line":74},[34,7690,7691],{"class":190}," inputEl.value.",[34,7693,7694],{"class":395},"focus",[34,7696,6374],{"class":190},[34,7698,7699],{"class":36,"line":1971},[34,7700,6153],{"class":190},[34,7702,7703,7705,7707],{"class":36,"line":1976},[34,7704,1003],{"class":190},[34,7706,1911],{"class":935},[34,7708,939],{"class":190},[34,7710,7711],{"class":36,"line":1985},[34,7712,53],{"emptyLinePlaceholder":52},[34,7714,7715,7717,7719],{"class":36,"line":1990},[34,7716,932],{"class":190},[34,7718,936],{"class":935},[34,7720,939],{"class":190},[34,7722,7723,7725,7727,7729,7731,7734],{"class":36,"line":1999},[34,7724,944],{"class":190},[34,7726,7429],{"class":935},[34,7728,453],{"class":395},[34,7730,202],{"class":190},[34,7732,7733],{"class":186},"\"inputEl\"",[34,7735,939],{"class":190},[34,7737,7738,7740,7742],{"class":36,"line":2022},[34,7739,1003],{"class":190},[34,7741,936],{"class":935},[34,7743,939],{"class":190},[15,7745,3685,7746,7749,7750,7753,7754,347],{},[31,7747,7748],{},"ref=\"inputEl\""," 是模板引用，不是响应式 ",[31,7751,7752],{},"ref()"," 本身，但它们配合使用：Vue 会在组件挂载后把真实 DOM 元素放进 ",[31,7755,7756],{},"inputEl.value",[19,7758,7760],{"id":7759},"_8-cdn-示例和工程化项目的区别","8. CDN 示例和工程化项目的区别",[15,7762,7763],{},"你可以用 CDN 直接在 HTML 中使用 Vue：",[24,7765,7767],{"className":6016,"code":7766,"language":6018,"meta":29,"style":29},"\u003Cscript src=\"https:\u002F\u002Funpkg.com\u002Fvue@3\u002Fdist\u002Fvue.global.js\">\u003C\u002Fscript>\n",[31,7768,7769],{"__ignoreMap":29},[34,7770,7771,7773,7775,7778,7780,7783,7785,7787],{"class":36,"line":37},[34,7772,932],{"class":190},[34,7774,1911],{"class":935},[34,7776,7777],{"class":395}," src",[34,7779,202],{"class":190},[34,7781,7782],{"class":186},"\"https:\u002F\u002Funpkg.com\u002Fvue@3\u002Fdist\u002Fvue.global.js\"",[34,7784,6593],{"class":190},[34,7786,1911],{"class":935},[34,7788,939],{"class":190},[15,7790,7791],{},"这适合入门理解：",[105,7793,7794,7797,7800],{},[108,7795,7796],{},"不需要安装 Node。",[108,7798,7799],{},"不需要 Vite。",[108,7801,7802],{},"一个 HTML 文件就能运行。",[15,7804,7805],{},"但正式项目通常使用 Vite：",[24,7807,7811],{"className":7808,"code":7809,"language":7810,"meta":29,"style":29},"language-powershell shiki shiki-themes github-light github-dark","npm create vue@latest\n","powershell",[31,7812,7813],{"__ignoreMap":29},[34,7814,7815],{"class":36,"line":37},[34,7816,7809],{},[15,7818,7819],{},"原因：",[105,7821,7822,7829,7836,7839,7842],{},[108,7823,7824,7825,7828],{},"可以写 ",[31,7826,7827],{},".vue"," 单文件组件。",[108,7830,7831,7832,7835],{},"可以用 ",[31,7833,7834],{},"import\u002Fexport"," 拆文件。",[108,7837,7838],{},"可以安装 Router、Pinia、测试工具。",[108,7840,7841],{},"可以构建生产产物。",[108,7843,7844],{},"开发时有热更新和错误提示。",[15,7846,7847,7848,7851,7852,7855],{},"本课程的 ",[31,7849,7850],{},"examples\u002Fcdn-basics\u002Findex.html"," 用来理解概念，",[31,7853,7854],{},"course-app"," 用来学习真实项目结构。",[19,7857,7859],{"id":7858},"_9-从-dom-思维迁移到-vue-思维","9. 从 DOM 思维迁移到 Vue 思维",[15,7861,7862],{},"当你想写：",[24,7864,7866],{"className":166,"code":7865,"language":168,"meta":29,"style":29},"document.querySelector(...)\nelement.textContent = ...\nelement.classList.add(...)\n",[31,7867,7868,7880,7889],{"__ignoreMap":29},[34,7869,7870,7872,7874,7876,7878],{"class":36,"line":37},[34,7871,7175],{"class":190},[34,7873,6103],{"class":395},[34,7875,456],{"class":190},[34,7877,1167],{"class":175},[34,7879,6111],{"class":190},[34,7881,7882,7885,7887],{"class":36,"line":43},[34,7883,7884],{"class":190},"element.textContent ",[34,7886,202],{"class":175},[34,7888,7203],{"class":175},[34,7890,7891,7894,7896,7898,7900],{"class":36,"line":49},[34,7892,7893],{"class":190},"element.classList.",[34,7895,1762],{"class":395},[34,7897,456],{"class":190},[34,7899,1167],{"class":175},[34,7901,6111],{"class":190},[15,7903,7904],{},"先停一下，问：",[2230,7906,7907,7910,7918],{},[108,7908,7909],{},"我真正想表达的状态是什么？",[108,7911,7912,7913,4374,7915,7917],{},"这个状态能不能用 ",[31,7914,145],{},[31,7916,149],{}," 表示？",[108,7919,7920],{},"模板能不能根据这个状态显示不同内容或 class？",[15,7922,7923],{},"例如按钮禁用：",[15,7925,7926],{},"原生 DOM 思维：",[24,7928,7930],{"className":166,"code":7929,"language":168,"meta":29,"style":29},"button.disabled = title.length === 0\n",[31,7931,7932],{"__ignoreMap":29},[34,7933,7934,7937,7939,7942,7944,7947],{"class":36,"line":37},[34,7935,7936],{"class":190},"button.disabled ",[34,7938,202],{"class":175},[34,7940,7941],{"class":190}," title.",[34,7943,1289],{"class":179},[34,7945,7946],{"class":175}," ===",[34,7948,6245],{"class":179},[15,7950,7951],{},"Vue 思维：",[24,7953,7955],{"className":923,"code":7954,"language":925,"meta":29,"style":29},"\u003Cbutton :disabled=\"title.length === 0\">提交\u003C\u002Fbutton>\n",[31,7956,7957],{"__ignoreMap":29},[34,7958,7959,7961,7963,7965,7968,7970,7972,7975,7977,7979,7981,7983,7986,7988],{"class":36,"line":37},[34,7960,932],{"class":190},[34,7962,2004],{"class":935},[34,7964,1621],{"class":190},[34,7966,7967],{"class":395},"disabled",[34,7969,202],{"class":190},[34,7971,1608],{"class":186},[34,7973,7974],{"class":190},"title.",[34,7976,1289],{"class":179},[34,7978,7946],{"class":175},[34,7980,205],{"class":179},[34,7982,1608],{"class":186},[34,7984,7985],{"class":190},">提交\u003C\u002F",[34,7987,2004],{"class":935},[34,7989,939],{"class":190},[15,7991,7992],{},"例如错误样式：",[15,7994,7926],{},[24,7996,7998],{"className":166,"code":7997,"language":168,"meta":29,"style":29},"if (hasError) {\n input.classList.add('error')\n} else {\n input.classList.remove('error')\n}\n",[31,7999,8000,8007,8021,8029,8042],{"__ignoreMap":29},[34,8001,8002,8004],{"class":36,"line":37},[34,8003,2069],{"class":175},[34,8005,8006],{"class":190}," (hasError) {\n",[34,8008,8009,8012,8014,8016,8019],{"class":36,"line":43},[34,8010,8011],{"class":190}," input.classList.",[34,8013,1762],{"class":395},[34,8015,456],{"class":190},[34,8017,8018],{"class":186},"'error'",[34,8020,6111],{"class":190},[34,8022,8023,8025,8027],{"class":36,"line":49},[34,8024,2098],{"class":190},[34,8026,2101],{"class":175},[34,8028,674],{"class":190},[34,8030,8031,8033,8036,8038,8040],{"class":36,"line":56},[34,8032,8011],{"class":190},[34,8034,8035],{"class":395},"remove",[34,8037,456],{"class":190},[34,8039,8018],{"class":186},[34,8041,6111],{"class":190},[34,8043,8044],{"class":36,"line":62},[34,8045,77],{"class":190},[15,8047,7951],{},[24,8049,8051],{"className":923,"code":8050,"language":925,"meta":29,"style":29},"\u003Cinput :class=\"{ error: hasError }\">\n",[31,8052,8053],{"__ignoreMap":29},[34,8054,8055,8057,8059,8061,8064,8066,8068,8071,8073],{"class":36,"line":37},[34,8056,932],{"class":190},[34,8058,7429],{"class":935},[34,8060,1621],{"class":190},[34,8062,8063],{"class":395},"class",[34,8065,202],{"class":190},[34,8067,1608],{"class":186},[34,8069,8070],{"class":190},"{ error: hasError }",[34,8072,1608],{"class":186},[34,8074,939],{"class":190},[19,8076,8078],{"id":8077},"_10-常见误区","10. 常见误区",[253,8080,8082],{"id":8081},"误区一vue-会自动理解普通变量","误区一：Vue 会自动理解普通变量",[24,8084,8086],{"className":166,"code":8085,"language":168,"meta":29,"style":29},"let count = 0\n",[31,8087,8088],{"__ignoreMap":29},[34,8089,8090,8092,8094,8096],{"class":36,"line":37},[34,8091,196],{"class":175},[34,8093,319],{"class":190},[34,8095,202],{"class":175},[34,8097,6245],{"class":179},[15,8099,8100],{},"普通变量变化，Vue 不会自动更新模板。需要：",[24,8102,8104],{"className":166,"code":8103,"language":168,"meta":29,"style":29},"const count = ref(0)\n",[31,8105,8106],{"__ignoreMap":29},[34,8107,8108,8110,8112,8114,8116,8118,8120],{"class":36,"line":37},[34,8109,176],{"class":175},[34,8111,448],{"class":179},[34,8113,183],{"class":175},[34,8115,453],{"class":395},[34,8117,456],{"class":190},[34,8119,459],{"class":179},[34,8121,6111],{"class":190},[253,8123,8125],{"id":8124},"误区二一边用-vue一边大量-queryselector","误区二：一边用 Vue，一边大量 querySelector",[15,8127,8128],{},"如果你在 Vue 组件里到处写：",[24,8130,8132],{"className":166,"code":8131,"language":168,"meta":29,"style":29},"document.querySelector(...)\n",[31,8133,8134],{"__ignoreMap":29},[34,8135,8136,8138,8140,8142,8144],{"class":36,"line":37},[34,8137,7175],{"class":190},[34,8139,6103],{"class":395},[34,8141,456],{"class":190},[34,8143,1167],{"class":175},[34,8145,6111],{"class":190},[15,8147,8148],{},"往往说明你还没有用状态表达界面。只有真实需要 DOM 本身时才这样做。",[253,8150,8152],{"id":8151},"误区三以为模板只是-html","误区三：以为模板只是 HTML",[15,8154,8155],{},"Vue 模板看起来像 HTML，但它多了指令：",[105,8157,8158,8163,8167,8171,8176],{},[108,8159,8160],{},[31,8161,8162],{},"v-if",[108,8164,8165],{},[31,8166,1534],{},[108,8168,8169],{},[31,8170,7473],{},[108,8172,8173],{},[31,8174,8175],{},":class",[108,8177,8178],{},[31,8179,7152],{},[15,8181,8182],{},"这些指令让模板和响应式状态连接起来。",[253,8184,8186],{"id":8185},"误区四数据结构不清晰就开始写界面","误区四：数据结构不清晰就开始写界面",[15,8188,8189],{},"先想数据：",[24,8191,8193],{"className":166,"code":8192,"language":168,"meta":29,"style":29},"const tasks = [\n { id: 1, title: '学习 Vue', done: false }\n]\n",[31,8194,8195,8205,8222],{"__ignoreMap":29},[34,8196,8197,8199,8201,8203],{"class":36,"line":37},[34,8198,176],{"class":175},[34,8200,734],{"class":179},[34,8202,183],{"class":175},[34,8204,739],{"class":190},[34,8206,8207,8209,8211,8213,8216,8218,8220],{"class":36,"line":43},[34,8208,744],{"class":190},[34,8210,640],{"class":179},[34,8212,749],{"class":190},[34,8214,8215],{"class":186},"'学习 Vue'",[34,8217,755],{"class":190},[34,8219,702],{"class":179},[34,8221,6564],{"class":190},[34,8223,8224],{"class":36,"line":49},[34,8225,6569],{"class":190},[15,8227,8228],{},"再想界面如何显示这些数据。不要反过来先堆 HTML，再到处找地方塞逻辑。",[19,8230,5693],{"id":5693},[253,8232,8234],{"id":8233},"练习-1原生-dom-任务列表","练习 1：原生 DOM 任务列表",[15,8236,8237],{},"用原生 DOM 写一个页面：",[105,8239,8240,8243,8246,8249],{},[108,8241,8242],{},"输入框输入任务标题。",[108,8244,8245],{},"点击按钮新增任务。",[108,8247,8248],{},"列表显示所有任务。",[108,8250,8251],{},"点击任务按钮切换完成状态。",[15,8253,8254,8255,8257],{},"要求你必须写一个 ",[31,8256,6448],{}," 函数，每次数据变化后调用。",[253,8259,8261],{"id":8260},"练习-2vue-任务列表","练习 2：Vue 任务列表",[15,8263,8264],{},"用 Vue 写同样功能：",[105,8266,8267,8275,8280,8285],{},[108,8268,8269,8271,8272,347],{},[31,8270,1345],{}," 使用 ",[31,8273,8274],{},"ref([])",[108,8276,8277,8278,347],{},"表单输入使用 ",[31,8279,7473],{},[108,8281,8282,8283,347],{},"列表使用 ",[31,8284,1534],{},[108,8286,8287],{},"点击按钮只修改数据，不手动创建 DOM。",[253,8289,8291],{"id":8290},"练习-3对比总结","练习 3：对比总结",[15,8293,8294],{},"回答：",[2230,8296,8297,8300,8303,8309],{},[108,8298,8299],{},"原生 DOM 版本中，数据在哪里？DOM 更新在哪里？",[108,8301,8302],{},"Vue 版本中，数据在哪里？DOM 更新是谁做的？",[108,8304,8305,8306,8308],{},"为什么忘记调用 ",[31,8307,6448],{}," 会导致界面和数据不同步？",[108,8310,8311],{},"Vue 中什么时候仍然需要模板引用拿 DOM？",[79,8313,8314],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":8316},[8317,8318,8319,8320,8321,8322,8323,8324,8325,8326,8332],{"id":6006,"depth":43,"text":6007},{"id":6156,"depth":43,"text":6157},{"id":6509,"depth":43,"text":6510},{"id":6864,"depth":43,"text":6865},{"id":7218,"depth":43,"text":7219},{"id":7340,"depth":43,"text":7341},{"id":7493,"depth":43,"text":7494},{"id":7759,"depth":43,"text":7760},{"id":7858,"depth":43,"text":7859},{"id":8077,"depth":43,"text":8078,"children":8327},[8328,8329,8330,8331],{"id":8081,"depth":49,"text":8082},{"id":8124,"depth":49,"text":8125},{"id":8151,"depth":49,"text":8152},{"id":8185,"depth":49,"text":8186},{"id":5693,"depth":43,"text":5693,"children":8333},[8334,8335,8336],{"id":8233,"depth":49,"text":8234},{"id":8260,"depth":49,"text":8261},{"id":8290,"depth":49,"text":8291},{"order":43},"\u002Fsource\u002Fcode\u002Fvue\u002F02-dom-to-vue",{"title":5992,"description":6000},"source\u002Fcode\u002Fvue\u002F02-dom-to-vue","llNR4TmRVo9u4Gcp0XmoIoqlnl1gdOCrG5Sy82vCeJc",{"id":8343,"title":8344,"body":8345,"description":8352,"extension":85,"meta":11700,"navigation":52,"path":11701,"seo":11702,"stem":11703,"__hash__":11704},"notes\u002Fsource\u002Fcode\u002Fvue\u002F03-vue-core.md","03 Vue 核心语法和响应式",{"type":8,"value":8346,"toc":11655},[8347,8350,8353,8366,8370,8376,8489,8492,8515,8518,8521,8532,8538,8552,8559,8562,8621,8626,8691,8696,8698,8827,8834,8837,8880,8883,8894,8896,8914,8919,9004,9022,9029,9154,9159,9164,9191,9194,9205,9210,9219,9222,9226,9229,9243,9252,9260,9265,9328,9333,9358,9360,9389,9397,9400,9427,9429,9477,9484,9492,9544,9550,9576,9585,9588,9600,9607,9615,9672,9681,9684,9716,9719,9744,9776,9779,9811,9817,9853,9859,9865,9927,9932,9938,9969,9972,9984,9992,9998,10055,10058,10078,10084,10089,10092,10094,10144,10146,10199,10202,10209,10214,10316,10318,10336,10339,10342,10379,10391,10394,10396,10429,10436,10438,10483,10490,10495,10553,10556,10576,10583,10588,10639,10644,10683,10691,10695,10700,10747,10750,10761,10767,10877,10880,10884,10887,10999,11001,11020,11024,11569,11572,11610,11612,11652],[11,8348,8344],{"id":8349},"_03-vue-核心语法和响应式",[15,8351,8352],{},"本章开始正式写 Vue。你需要掌握 Vue 单文件组件、响应式状态、模板语法、计算属性、监听器和生命周期。",[15,8354,8355,8356,8358,8359,146,8361,146,8363,8365],{},"核心目标：你看到一个 ",[31,8357,7827],{}," 文件时，能知道每个区域负责什么；看到 ",[31,8360,145],{},[31,8362,149],{},[31,8364,152],{}," 时，知道为什么用它们。",[19,8367,8369],{"id":8368},"_1-单文件组件是什么","1. 单文件组件是什么",[15,8371,8372,8373,8375],{},"Vue 项目里最常见的文件是 ",[31,8374,7827],{}," 文件，也叫单文件组件，英文是 Single File Component，简称 SFC。",[24,8377,8379],{"className":923,"code":8378,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nconst title = 'Vue 核心语法'\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Ch1>{{ title }}\u003C\u002Fh1>\n\u003C\u002Ftemplate>\n\n\u003Cstyle scoped>\nh1 {\n color: #2563eb;\n}\n\u003C\u002Fstyle>\n",[31,8380,8381,8391,8402,8410,8414,8422,8435,8443,8447,8458,8464,8477,8481],{"__ignoreMap":29},[34,8382,8383,8385,8387,8389],{"class":36,"line":37},[34,8384,932],{"class":190},[34,8386,1911],{"class":935},[34,8388,1914],{"class":395},[34,8390,939],{"class":190},[34,8392,8393,8395,8397,8399],{"class":36,"line":43},[34,8394,176],{"class":175},[34,8396,278],{"class":179},[34,8398,183],{"class":175},[34,8400,8401],{"class":186}," 'Vue 核心语法'\n",[34,8403,8404,8406,8408],{"class":36,"line":49},[34,8405,1003],{"class":190},[34,8407,1911],{"class":935},[34,8409,939],{"class":190},[34,8411,8412],{"class":36,"line":56},[34,8413,53],{"emptyLinePlaceholder":52},[34,8415,8416,8418,8420],{"class":36,"line":62},[34,8417,932],{"class":190},[34,8419,936],{"class":935},[34,8421,939],{"class":190},[34,8423,8424,8426,8428,8431,8433],{"class":36,"line":68},[34,8425,944],{"class":190},[34,8427,11],{"class":935},[34,8429,8430],{"class":190},">{{ title }}\u003C\u002F",[34,8432,11],{"class":935},[34,8434,939],{"class":190},[34,8436,8437,8439,8441],{"class":36,"line":74},[34,8438,1003],{"class":190},[34,8440,936],{"class":935},[34,8442,939],{"class":190},[34,8444,8445],{"class":36,"line":1971},[34,8446,53],{"emptyLinePlaceholder":52},[34,8448,8449,8451,8453,8456],{"class":36,"line":1976},[34,8450,932],{"class":190},[34,8452,79],{"class":935},[34,8454,8455],{"class":395}," scoped",[34,8457,939],{"class":190},[34,8459,8460,8462],{"class":36,"line":1985},[34,8461,11],{"class":935},[34,8463,674],{"class":190},[34,8465,8466,8469,8472,8475],{"class":36,"line":1990},[34,8467,8468],{"class":179}," color",[34,8470,8471],{"class":190},": ",[34,8473,8474],{"class":179},"#2563eb",[34,8476,191],{"class":190},[34,8478,8479],{"class":36,"line":1999},[34,8480,77],{"class":190},[34,8482,8483,8485,8487],{"class":36,"line":2022},[34,8484,1003],{"class":190},[34,8486,79],{"class":935},[34,8488,939],{"class":190},[15,8490,8491],{},"三个区域：",[105,8493,8494,8499,8505],{},[108,8495,8496,8498],{},[31,8497,141],{},"：写 JavaScript 逻辑，例如状态、函数、导入组件。",[108,8500,8501,8504],{},[31,8502,8503],{},"\u003Ctemplate>","：写界面结构，并通过 Vue 指令连接状态。",[108,8506,8507,8510,8511,8514],{},[31,8508,8509],{},"\u003Cstyle scoped>","：写当前组件样式，",[31,8512,8513],{},"scoped"," 表示尽量只影响当前组件。",[253,8516,8517],{"id":8517},"为什么要单文件组件",[15,8519,8520],{},"一个组件通常包含三类内容：",[105,8522,8523,8526,8529],{},[108,8524,8525],{},"数据和行为。",[108,8527,8528],{},"HTML 结构。",[108,8530,8531],{},"样式。",[15,8533,8534,8535,8537],{},"把它们放在同一个 ",[31,8536,7827],{}," 文件里，有几个好处：",[105,8539,8540,8543,8546,8549],{},[108,8541,8542],{},"组件的所有内容集中，容易理解。",[108,8544,8545],{},"可以直接导入另一个组件。",[108,8547,8548],{},"样式可以限制在组件内部。",[108,8550,8551],{},"构建工具能检查语法并优化代码。",[19,8553,8555,8556,8558],{"id":8554},"_2-script-setup-为什么不用-return","2. ",[31,8557,141],{}," 为什么不用 return",[15,8560,8561],{},"普通 Vue 组件可能这样写：",[24,8563,8565],{"className":166,"code":8564,"language":168,"meta":29,"style":29},"export default {\n setup() {\n const title = 'Vue'\n\n return {\n title\n }\n }\n}\n",[31,8566,8567,8575,8582,8593,8597,8604,8609,8613,8617],{"__ignoreMap":29},[34,8568,8569,8571,8573],{"class":36,"line":37},[34,8570,2825],{"class":175},[34,8572,3958],{"class":175},[34,8574,674],{"class":190},[34,8576,8577,8580],{"class":36,"line":43},[34,8578,8579],{"class":395}," setup",[34,8581,399],{"class":190},[34,8583,8584,8586,8588,8590],{"class":36,"line":49},[34,8585,5475],{"class":175},[34,8587,278],{"class":179},[34,8589,183],{"class":175},[34,8591,8592],{"class":186}," 'Vue'\n",[34,8594,8595],{"class":36,"line":56},[34,8596,53],{"emptyLinePlaceholder":52},[34,8598,8599,8602],{"class":36,"line":62},[34,8600,8601],{"class":175}," return",[34,8603,674],{"class":190},[34,8605,8606],{"class":36,"line":68},[34,8607,8608],{"class":190}," title\n",[34,8610,8611],{"class":36,"line":74},[34,8612,5540],{"class":190},[34,8614,8615],{"class":36,"line":1971},[34,8616,5631],{"class":190},[34,8618,8619],{"class":36,"line":1976},[34,8620,77],{"class":190},[15,8622,8623,8625],{},[31,8624,141],{}," 是更简洁的写法：",[24,8627,8629],{"className":923,"code":8628,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nconst title = 'Vue'\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Ch1>{{ title }}\u003C\u002Fh1>\n\u003C\u002Ftemplate>\n",[31,8630,8631,8641,8651,8659,8663,8671,8683],{"__ignoreMap":29},[34,8632,8633,8635,8637,8639],{"class":36,"line":37},[34,8634,932],{"class":190},[34,8636,1911],{"class":935},[34,8638,1914],{"class":395},[34,8640,939],{"class":190},[34,8642,8643,8645,8647,8649],{"class":36,"line":43},[34,8644,176],{"class":175},[34,8646,278],{"class":179},[34,8648,183],{"class":175},[34,8650,8592],{"class":186},[34,8652,8653,8655,8657],{"class":36,"line":49},[34,8654,1003],{"class":190},[34,8656,1911],{"class":935},[34,8658,939],{"class":190},[34,8660,8661],{"class":36,"line":56},[34,8662,53],{"emptyLinePlaceholder":52},[34,8664,8665,8667,8669],{"class":36,"line":62},[34,8666,932],{"class":190},[34,8668,936],{"class":935},[34,8670,939],{"class":190},[34,8672,8673,8675,8677,8679,8681],{"class":36,"line":68},[34,8674,944],{"class":190},[34,8676,11],{"class":935},[34,8678,8430],{"class":190},[34,8680,11],{"class":935},[34,8682,939],{"class":190},[34,8684,8685,8687,8689],{"class":36,"line":74},[34,8686,1003],{"class":190},[34,8688,936],{"class":935},[34,8690,939],{"class":190},[15,8692,2884,8693,8695],{},[31,8694,141],{}," 里，顶层声明的变量、函数、导入的组件，都可以直接在模板里使用。",[15,8697,2632],{},[24,8699,8701],{"className":923,"code":8700,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport LessonCard from '.\u002FLessonCard.vue'\n\nconst lesson = { title: '组件基础' }\n\nfunction save() {\n console.log('保存')\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003CLessonCard \u002F>\n \u003Cbutton @click=\"save\">{{ lesson.title }}\u003C\u002Fbutton>\n\u003C\u002Ftemplate>\n",[31,8702,8703,8713,8724,8728,8743,8747,8755,8768,8772,8780,8784,8792,8800,8819],{"__ignoreMap":29},[34,8704,8705,8707,8709,8711],{"class":36,"line":37},[34,8706,932],{"class":190},[34,8708,1911],{"class":935},[34,8710,1914],{"class":395},[34,8712,939],{"class":190},[34,8714,8715,8717,8719,8721],{"class":36,"line":43},[34,8716,426],{"class":175},[34,8718,3432],{"class":190},[34,8720,432],{"class":175},[34,8722,8723],{"class":186}," '.\u002FLessonCard.vue'\n",[34,8725,8726],{"class":36,"line":49},[34,8727,53],{"emptyLinePlaceholder":52},[34,8729,8730,8732,8734,8736,8738,8741],{"class":36,"line":56},[34,8731,176],{"class":175},[34,8733,824],{"class":179},[34,8735,183],{"class":175},[34,8737,1029],{"class":190},[34,8739,8740],{"class":186},"'组件基础'",[34,8742,6564],{"class":190},[34,8744,8745],{"class":36,"line":62},[34,8746,53],{"emptyLinePlaceholder":52},[34,8748,8749,8751,8753],{"class":36,"line":68},[34,8750,392],{"class":175},[34,8752,7559],{"class":395},[34,8754,399],{"class":190},[34,8756,8757,8759,8761,8763,8766],{"class":36,"line":74},[34,8758,1812],{"class":190},[34,8760,885],{"class":395},[34,8762,456],{"class":190},[34,8764,8765],{"class":186},"'保存'",[34,8767,6111],{"class":190},[34,8769,8770],{"class":36,"line":1971},[34,8771,77],{"class":190},[34,8773,8774,8776,8778],{"class":36,"line":1976},[34,8775,1003],{"class":190},[34,8777,1911],{"class":935},[34,8779,939],{"class":190},[34,8781,8782],{"class":36,"line":1985},[34,8783,53],{"emptyLinePlaceholder":52},[34,8785,8786,8788,8790],{"class":36,"line":1990},[34,8787,932],{"class":190},[34,8789,936],{"class":935},[34,8791,939],{"class":190},[34,8793,8794,8796,8798],{"class":36,"line":1999},[34,8795,944],{"class":190},[34,8797,3466],{"class":935},[34,8799,3469],{"class":190},[34,8801,8802,8804,8806,8808,8810,8813,8815,8817],{"class":36,"line":2022},[34,8803,944],{"class":190},[34,8805,2004],{"class":935},[34,8807,2007],{"class":395},[34,8809,202],{"class":190},[34,8811,8812],{"class":186},"\"save\"",[34,8814,959],{"class":190},[34,8816,2004],{"class":935},[34,8818,939],{"class":190},[34,8820,8821,8823,8825],{"class":36,"line":2036},[34,8822,1003],{"class":190},[34,8824,936],{"class":935},[34,8826,939],{"class":190},[19,8828,8830,8831],{"id":8829},"_3-插值表达式","3. 插值表达式 ",[31,8832,8833],{},"{{ }}",[15,8835,8836],{},"插值表达式用于显示文本。",[24,8838,8840],{"className":923,"code":8839,"language":925,"meta":29,"style":29},"\u003Cp>{{ message }}\u003C\u002Fp>\n\u003Cp>{{ count + 1 }}\u003C\u002Fp>\n\u003Cp>{{ user.name }}\u003C\u002Fp>\n",[31,8841,8842,8854,8867],{"__ignoreMap":29},[34,8843,8844,8846,8848,8850,8852],{"class":36,"line":37},[34,8845,932],{"class":190},[34,8847,15],{"class":935},[34,8849,7596],{"class":190},[34,8851,15],{"class":935},[34,8853,939],{"class":190},[34,8855,8856,8858,8860,8863,8865],{"class":36,"line":43},[34,8857,932],{"class":190},[34,8859,15],{"class":935},[34,8861,8862],{"class":190},">{{ count + 1 }}\u003C\u002F",[34,8864,15],{"class":935},[34,8866,939],{"class":190},[34,8868,8869,8871,8873,8876,8878],{"class":36,"line":49},[34,8870,932],{"class":190},[34,8872,15],{"class":935},[34,8874,8875],{"class":190},">{{ user.name }}\u003C\u002F",[34,8877,15],{"class":935},[34,8879,939],{"class":190},[15,8881,8882],{},"注意：",[105,8884,8885,8888,8891],{},[108,8886,8887],{},"插值会被当成文本显示。",[108,8889,8890],{},"Vue 会默认转义内容，减少 XSS 风险。",[108,8892,8893],{},"不要在插值里写很复杂的逻辑。",[15,8895,7500],{},[24,8897,8899],{"className":923,"code":8898,"language":925,"meta":29,"style":29},"\u003Cp>{{ tasks.filter(task => !task.done).map(task => task.title).join(', ') }}\u003C\u002Fp>\n",[31,8900,8901],{"__ignoreMap":29},[34,8902,8903,8905,8907,8910,8912],{"class":36,"line":37},[34,8904,932],{"class":190},[34,8906,15],{"class":935},[34,8908,8909],{"class":190},">{{ tasks.filter(task => !task.done).map(task => task.title).join(', ') }}\u003C\u002F",[34,8911,15],{"class":935},[34,8913,939],{"class":190},[15,8915,8916,8917,2664],{},"推荐用 ",[31,8918,149],{},[24,8920,8922],{"className":166,"code":8921,"language":168,"meta":29,"style":29},"const activeTaskTitles = computed(() => {\n return tasks.value\n .filter((task) => !task.done)\n .map((task) => task.title)\n .join(', ')\n})\n",[31,8923,8924,8942,8949,8969,8986,9000],{"__ignoreMap":29},[34,8925,8926,8928,8931,8933,8936,8938,8940],{"class":36,"line":37},[34,8927,176],{"class":175},[34,8929,8930],{"class":179}," activeTaskTitles",[34,8932,183],{"class":175},[34,8934,8935],{"class":395}," computed",[34,8937,7682],{"class":190},[34,8939,1333],{"class":175},[34,8941,674],{"class":190},[34,8943,8944,8946],{"class":36,"line":43},[34,8945,1510],{"class":175},[34,8947,8948],{"class":190}," tasks.value\n",[34,8950,8951,8954,8956,8958,8960,8962,8964,8966],{"class":36,"line":49},[34,8952,8953],{"class":190}," .",[34,8955,1300],{"class":395},[34,8957,1323],{"class":190},[34,8959,1327],{"class":1326},[34,8961,1330],{"class":190},[34,8963,1333],{"class":175},[34,8965,1336],{"class":175},[34,8967,8968],{"class":190},"task.done)\n",[34,8970,8971,8973,8975,8977,8979,8981,8983],{"class":36,"line":56},[34,8972,8953],{"class":190},[34,8974,1355],{"class":395},[34,8976,1323],{"class":190},[34,8978,1327],{"class":1326},[34,8980,1330],{"class":190},[34,8982,1333],{"class":175},[34,8984,8985],{"class":190}," task.title)\n",[34,8987,8988,8990,8993,8995,8998],{"class":36,"line":62},[34,8989,8953],{"class":190},[34,8991,8992],{"class":395},"join",[34,8994,456],{"class":190},[34,8996,8997],{"class":186},"', '",[34,8999,6111],{"class":190},[34,9001,9002],{"class":36,"line":68},[34,9003,6153],{"class":190},[24,9005,9007],{"className":923,"code":9006,"language":925,"meta":29,"style":29},"\u003Cp>{{ activeTaskTitles }}\u003C\u002Fp>\n",[31,9008,9009],{"__ignoreMap":29},[34,9010,9011,9013,9015,9018,9020],{"class":36,"line":37},[34,9012,932],{"class":190},[34,9014,15],{"class":935},[34,9016,9017],{"class":190},">{{ activeTaskTitles }}\u003C\u002F",[34,9019,15],{"class":935},[34,9021,939],{"class":190},[19,9023,9025,9026,9028],{"id":9024},"_4-ref最常用的响应式状态","4. ",[31,9027,145],{},"：最常用的响应式状态",[24,9030,9032],{"className":923,"code":9031,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { ref } from 'vue'\n\nconst count = ref(0)\n\nfunction increase() {\n count.value++\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cbutton @click=\"increase\">增加\u003C\u002Fbutton>\n \u003Cp>{{ count }}\u003C\u002Fp>\n\u003C\u002Ftemplate>\n",[31,9033,9034,9044,9054,9058,9074,9078,9086,9092,9096,9104,9108,9116,9134,9146],{"__ignoreMap":29},[34,9035,9036,9038,9040,9042],{"class":36,"line":37},[34,9037,932],{"class":190},[34,9039,1911],{"class":935},[34,9041,1914],{"class":395},[34,9043,939],{"class":190},[34,9045,9046,9048,9050,9052],{"class":36,"line":43},[34,9047,426],{"class":175},[34,9049,429],{"class":190},[34,9051,432],{"class":175},[34,9053,6891],{"class":186},[34,9055,9056],{"class":36,"line":49},[34,9057,53],{"emptyLinePlaceholder":52},[34,9059,9060,9062,9064,9066,9068,9070,9072],{"class":36,"line":56},[34,9061,176],{"class":175},[34,9063,448],{"class":179},[34,9065,183],{"class":175},[34,9067,453],{"class":395},[34,9069,456],{"class":190},[34,9071,459],{"class":179},[34,9073,6111],{"class":190},[34,9075,9076],{"class":36,"line":62},[34,9077,53],{"emptyLinePlaceholder":52},[34,9079,9080,9082,9084],{"class":36,"line":68},[34,9081,392],{"class":175},[34,9083,396],{"class":395},[34,9085,399],{"class":190},[34,9087,9088,9090],{"class":36,"line":74},[34,9089,479],{"class":190},[34,9091,6366],{"class":175},[34,9093,9094],{"class":36,"line":1971},[34,9095,77],{"class":190},[34,9097,9098,9100,9102],{"class":36,"line":1976},[34,9099,1003],{"class":190},[34,9101,1911],{"class":935},[34,9103,939],{"class":190},[34,9105,9106],{"class":36,"line":1985},[34,9107,53],{"emptyLinePlaceholder":52},[34,9109,9110,9112,9114],{"class":36,"line":1990},[34,9111,932],{"class":190},[34,9113,936],{"class":935},[34,9115,939],{"class":190},[34,9117,9118,9120,9122,9124,9126,9128,9130,9132],{"class":36,"line":1999},[34,9119,944],{"class":190},[34,9121,2004],{"class":935},[34,9123,2007],{"class":395},[34,9125,202],{"class":190},[34,9127,2012],{"class":186},[34,9129,2015],{"class":190},[34,9131,2004],{"class":935},[34,9133,939],{"class":190},[34,9135,9136,9138,9140,9142,9144],{"class":36,"line":2022},[34,9137,944],{"class":190},[34,9139,15],{"class":935},[34,9141,2029],{"class":190},[34,9143,15],{"class":935},[34,9145,939],{"class":190},[34,9147,9148,9150,9152],{"class":36,"line":2036},[34,9149,1003],{"class":190},[34,9151,936],{"class":935},[34,9153,939],{"class":190},[15,9155,9156,9157,347],{},"这里最容易疑惑的是 ",[31,9158,507],{},[15,9160,9161,9163],{},[31,9162,500],{}," 返回的不是数字 0，而是一个响应式对象，大概可以想成：",[24,9165,9167],{"className":166,"code":9166,"language":168,"meta":29,"style":29},"const count = {\n value: 0\n}\n",[31,9168,9169,9179,9187],{"__ignoreMap":29},[34,9170,9171,9173,9175,9177],{"class":36,"line":37},[34,9172,176],{"class":175},[34,9174,448],{"class":179},[34,9176,183],{"class":175},[34,9178,674],{"class":190},[34,9180,9181,9184],{"class":36,"line":43},[34,9182,9183],{"class":190}," value: ",[34,9185,9186],{"class":179},"0\n",[34,9188,9189],{"class":36,"line":49},[34,9190,77],{"class":190},[15,9192,9193],{},"所以在脚本里要写：",[24,9195,9197],{"className":166,"code":9196,"language":168,"meta":29,"style":29},"count.value++\n",[31,9198,9199],{"__ignoreMap":29},[34,9200,9201,9203],{"class":36,"line":37},[34,9202,2054],{"class":190},[34,9204,6366],{"class":175},[15,9206,9207,9208,2664],{},"模板里不用写 ",[31,9209,507],{},[24,9211,9213],{"className":923,"code":9212,"language":925,"meta":29,"style":29},"{{ count }}\n",[31,9214,9215],{"__ignoreMap":29},[34,9216,9217],{"class":36,"line":37},[34,9218,9212],{"class":190},[15,9220,9221],{},"因为 Vue 模板会自动解包 ref。",[253,9223,9225],{"id":9224},"为什么-vue-不直接让普通变量响应式","为什么 Vue 不直接让普通变量响应式",[15,9227,9228],{},"普通变量：",[24,9230,9231],{"className":166,"code":8085,"language":168,"meta":29,"style":29},[31,9232,9233],{"__ignoreMap":29},[34,9234,9235,9237,9239,9241],{"class":36,"line":37},[34,9236,196],{"class":175},[34,9238,319],{"class":190},[34,9240,202],{"class":175},[34,9242,6245],{"class":179},[15,9244,9245,9246,347,9249,9251],{},"JavaScript 没有办法让 Vue 自动知道你什么时候执行了 ",[31,9247,9248],{},"count++",[31,9250,145],{}," 提供了一个可追踪的容器，Vue 可以知道容器里的值变了。",[19,9253,9255,9256,9259],{"id":9254},"_5-reactive响应式对象","5. ",[31,9257,9258],{},"reactive","：响应式对象",[15,9261,9262,9264],{},[31,9263,9258],{}," 适合对象：",[24,9266,9268],{"className":166,"code":9267,"language":168,"meta":29,"style":29},"import { reactive } from 'vue'\n\nconst form = reactive({\n title: '',\n minutes: 30,\n priority: 'normal'\n})\n",[31,9269,9270,9281,9285,9300,9308,9317,9324],{"__ignoreMap":29},[34,9271,9272,9274,9277,9279],{"class":36,"line":37},[34,9273,426],{"class":175},[34,9275,9276],{"class":190}," { reactive } ",[34,9278,432],{"class":175},[34,9280,6891],{"class":186},[34,9282,9283],{"class":36,"line":43},[34,9284,53],{"emptyLinePlaceholder":52},[34,9286,9287,9289,9292,9294,9297],{"class":36,"line":49},[34,9288,176],{"class":175},[34,9290,9291],{"class":179}," form",[34,9293,183],{"class":175},[34,9295,9296],{"class":395}," reactive",[34,9298,9299],{"class":190},"({\n",[34,9301,9302,9304,9306],{"class":36,"line":56},[34,9303,689],{"class":190},[34,9305,7400],{"class":186},[34,9307,684],{"class":190},[34,9309,9310,9312,9315],{"class":36,"line":62},[34,9311,851],{"class":190},[34,9313,9314],{"class":179},"30",[34,9316,684],{"class":190},[34,9318,9319,9321],{"class":36,"line":68},[34,9320,709],{"class":190},[34,9322,9323],{"class":186},"'normal'\n",[34,9325,9326],{"class":36,"line":74},[34,9327,6153],{"class":190},[15,9329,9330,9331,2664],{},"使用时不需要 ",[31,9332,507],{},[24,9334,9336],{"className":166,"code":9335,"language":168,"meta":29,"style":29},"form.title = '学习 ref'\nform.minutes = 45\n",[31,9337,9338,9348],{"__ignoreMap":29},[34,9339,9340,9343,9345],{"class":36,"line":37},[34,9341,9342],{"class":190},"form.title ",[34,9344,202],{"class":175},[34,9346,9347],{"class":186}," '学习 ref'\n",[34,9349,9350,9353,9355],{"class":36,"line":43},[34,9351,9352],{"class":190},"form.minutes ",[34,9354,202],{"class":175},[34,9356,9357],{"class":179}," 45\n",[15,9359,1589],{},[24,9361,9363],{"className":923,"code":9362,"language":925,"meta":29,"style":29},"\u003Cinput v-model=\"form.title\">\n\u003Cinput v-model.number=\"form.minutes\" type=\"number\">\n",[31,9364,9365,9384],{"__ignoreMap":29},[34,9366,9367,9369,9371,9373,9375,9377,9380,9382],{"class":36,"line":37},[34,9368,932],{"class":190},[34,9370,7429],{"class":935},[34,9372,7432],{"class":395},[34,9374,202],{"class":190},[34,9376,1608],{"class":186},[34,9378,9379],{"class":190},"form.title",[34,9381,1608],{"class":186},[34,9383,939],{"class":190},[34,9385,9386],{"class":36,"line":43},[34,9387,9388],{"class":190},"\u003Cinput v-model.number=\"form.minutes\" type=\"number\">\n",[253,9390,9392,258,9394,9396],{"id":9391},"ref-和-reactive-怎么选",[31,9393,145],{},[31,9395,9258],{}," 怎么选",[15,9398,9399],{},"初学建议：",[105,9401,9402,9415,9421],{},[108,9403,9404,9405,9407,9408,146,9410,146,9412,347],{},"单个值：用 ",[31,9406,145],{},"，例如 ",[31,9409,2770],{},[31,9411,2175],{},[31,9413,9414],{},"keyword",[108,9416,9417,9418,9420],{},"数组：也可以用 ",[31,9419,8274],{},"，因为替换数组很方便。",[108,9422,9423,9424,347],{},"表单对象：可以用 ",[31,9425,9426],{},"reactive({ ... })",[15,9428,2632],{},[24,9430,9432],{"className":166,"code":9431,"language":168,"meta":29,"style":29},"const tasks = ref([])\nconst form = reactive({\n title: '',\n priority: 'normal'\n})\n",[31,9433,9434,9447,9459,9467,9473],{"__ignoreMap":29},[34,9435,9436,9438,9440,9442,9444],{"class":36,"line":37},[34,9437,176],{"class":175},[34,9439,734],{"class":179},[34,9441,183],{"class":175},[34,9443,453],{"class":395},[34,9445,9446],{"class":190},"([])\n",[34,9448,9449,9451,9453,9455,9457],{"class":36,"line":43},[34,9450,176],{"class":175},[34,9452,9291],{"class":179},[34,9454,183],{"class":175},[34,9456,9296],{"class":395},[34,9458,9299],{"class":190},[34,9460,9461,9463,9465],{"class":36,"line":49},[34,9462,689],{"class":190},[34,9464,7400],{"class":186},[34,9466,684],{"class":190},[34,9468,9469,9471],{"class":36,"line":56},[34,9470,709],{"class":190},[34,9472,9323],{"class":186},[34,9474,9475],{"class":36,"line":62},[34,9476,6153],{"class":190},[19,9478,9480,9481],{"id":9479},"_6-属性绑定-v-bind","6. 属性绑定 ",[31,9482,9483],{},"v-bind",[15,9485,9486,9487,9489,9490,347],{},"HTML 属性如果需要使用变量，就用 ",[31,9488,9483],{},"，简写是 ",[31,9491,5598],{},[24,9493,9495],{"className":923,"code":9494,"language":925,"meta":29,"style":29},"\u003Cimg :src=\"avatarUrl\" :alt=\"user.name\">\n\u003Cbutton :disabled=\"loading\">保存\u003C\u002Fbutton>\n\u003Ca :href=\"link\">查看详情\u003C\u002Fa>\n",[31,9496,9497,9534,9539],{"__ignoreMap":29},[34,9498,9499,9501,9504,9506,9509,9511,9513,9516,9518,9520,9523,9525,9527,9530,9532],{"class":36,"line":37},[34,9500,932],{"class":190},[34,9502,9503],{"class":935},"img",[34,9505,1621],{"class":190},[34,9507,9508],{"class":395},"src",[34,9510,202],{"class":190},[34,9512,1608],{"class":186},[34,9514,9515],{"class":190},"avatarUrl",[34,9517,1608],{"class":186},[34,9519,1621],{"class":190},[34,9521,9522],{"class":395},"alt",[34,9524,202],{"class":190},[34,9526,1608],{"class":186},[34,9528,9529],{"class":190},"user.name",[34,9531,1608],{"class":186},[34,9533,939],{"class":190},[34,9535,9536],{"class":36,"line":43},[34,9537,9538],{"class":190},"\u003Cbutton :disabled=\"loading\">保存\u003C\u002Fbutton>\n",[34,9540,9541],{"class":36,"line":49},[34,9542,9543],{"class":190},"\u003Ca :href=\"link\">查看详情\u003C\u002Fa>\n",[15,9545,9546,9547,9549],{},"如果不加 ",[31,9548,5598],{},"，就是普通字符串。",[24,9551,9553],{"className":923,"code":9552,"language":925,"meta":29,"style":29},"\u003Cbutton disabled=\"loading\">保存\u003C\u002Fbutton>\n",[31,9554,9555],{"__ignoreMap":29},[34,9556,9557,9559,9561,9564,9566,9569,9572,9574],{"class":36,"line":37},[34,9558,932],{"class":190},[34,9560,2004],{"class":935},[34,9562,9563],{"class":395}," disabled",[34,9565,202],{"class":190},[34,9567,9568],{"class":186},"\"loading\"",[34,9570,9571],{"class":190},">保存\u003C\u002F",[34,9573,2004],{"class":935},[34,9575,939],{"class":190},[15,9577,3685,9578,9581,9582,9584],{},[31,9579,9580],{},"disabled=\"loading\""," 不是读取变量 ",[31,9583,2175],{},"，而是一个普通 HTML 属性，结果按钮可能一直禁用。",[15,9586,9587],{},"所以你要问：这个属性值是固定文字，还是来自变量？",[105,9589,9590,9595],{},[108,9591,9592,9593],{},"固定文字：不用 ",[31,9594,5598],{},[108,9596,9597,9598],{},"来自变量或表达式：用 ",[31,9599,5598],{},[19,9601,9603,9604],{"id":9602},"_7-事件绑定-v-on","7. 事件绑定 ",[31,9605,9606],{},"v-on",[15,9608,9609,9610,9489,9612,347],{},"事件绑定用 ",[31,9611,9606],{},[31,9613,9614],{},"@",[24,9616,9618],{"className":923,"code":9617,"language":925,"meta":29,"style":29},"\u003Cbutton @click=\"increase\">增加\u003C\u002Fbutton>\n\u003Cinput @input=\"handleInput\">\n\u003Cform @submit.prevent=\"submit\">...\u003C\u002Fform>\n",[31,9619,9620,9646,9667],{"__ignoreMap":29},[34,9621,9622,9624,9626,9629,9632,9634,9636,9638,9640,9642,9644],{"class":36,"line":37},[34,9623,932],{"class":190},[34,9625,2004],{"class":935},[34,9627,9628],{"class":190}," @",[34,9630,9631],{"class":395},"click",[34,9633,202],{"class":190},[34,9635,1608],{"class":186},[34,9637,2048],{"class":190},[34,9639,1608],{"class":186},[34,9641,2015],{"class":190},[34,9643,2004],{"class":935},[34,9645,939],{"class":190},[34,9647,9648,9650,9652,9654,9656,9658,9660,9663,9665],{"class":36,"line":43},[34,9649,932],{"class":190},[34,9651,7429],{"class":935},[34,9653,9628],{"class":190},[34,9655,7429],{"class":395},[34,9657,202],{"class":190},[34,9659,1608],{"class":186},[34,9661,9662],{"class":190},"handleInput",[34,9664,1608],{"class":186},[34,9666,939],{"class":190},[34,9668,9669],{"class":36,"line":49},[34,9670,9671],{"class":190},"\u003Cform @submit.prevent=\"submit\">...\u003C\u002Fform>\n",[15,9673,9674,9677,9678,9680],{},[31,9675,9676],{},"@click=\"increase\""," 表示点击时调用 ",[31,9679,2048],{}," 函数。",[15,9682,9683],{},"如果需要传参数：",[24,9685,9687],{"className":923,"code":9686,"language":925,"meta":29,"style":29},"\u003Cbutton @click=\"toggleTask(task.id)\">切换\u003C\u002Fbutton>\n",[31,9688,9689],{"__ignoreMap":29},[34,9690,9691,9693,9695,9697,9699,9701,9703,9705,9708,9710,9712,9714],{"class":36,"line":37},[34,9692,932],{"class":190},[34,9694,2004],{"class":935},[34,9696,9628],{"class":190},[34,9698,9631],{"class":395},[34,9700,202],{"class":190},[34,9702,1608],{"class":186},[34,9704,7158],{"class":395},[34,9706,9707],{"class":190},"(task.id)",[34,9709,1608],{"class":186},[34,9711,7103],{"class":190},[34,9713,2004],{"class":935},[34,9715,939],{"class":190},[15,9717,9718],{},"如果需要事件对象：",[24,9720,9722],{"className":923,"code":9721,"language":925,"meta":29,"style":29},"\u003Cinput @input=\"handleInput\">\n",[31,9723,9724],{"__ignoreMap":29},[34,9725,9726,9728,9730,9732,9734,9736,9738,9740,9742],{"class":36,"line":37},[34,9727,932],{"class":190},[34,9729,7429],{"class":935},[34,9731,9628],{"class":190},[34,9733,7429],{"class":395},[34,9735,202],{"class":190},[34,9737,1608],{"class":186},[34,9739,9662],{"class":190},[34,9741,1608],{"class":186},[34,9743,939],{"class":190},[24,9745,9747],{"className":166,"code":9746,"language":168,"meta":29,"style":29},"function handleInput(event) {\n console.log(event.target.value)\n}\n",[31,9748,9749,9763,9772],{"__ignoreMap":29},[34,9750,9751,9753,9756,9758,9761],{"class":36,"line":37},[34,9752,392],{"class":175},[34,9754,9755],{"class":395}," handleInput",[34,9757,456],{"class":190},[34,9759,9760],{"class":1326},"event",[34,9762,1700],{"class":190},[34,9764,9765,9767,9769],{"class":36,"line":43},[34,9766,1812],{"class":190},[34,9768,885],{"class":395},[34,9770,9771],{"class":190},"(event.target.value)\n",[34,9773,9774],{"class":36,"line":49},[34,9775,77],{"class":190},[253,9777,9778],{"id":9778},"事件修饰符",[24,9780,9782],{"className":923,"code":9781,"language":925,"meta":29,"style":29},"\u003Cform @submit.prevent=\"submit\">\n",[31,9783,9784],{"__ignoreMap":29},[34,9785,9786,9788,9791,9793,9796,9798,9801,9803,9805,9807,9809],{"class":36,"line":37},[34,9787,932],{"class":190},[34,9789,9790],{"class":935},"form",[34,9792,9628],{"class":190},[34,9794,9795],{"class":395},"submit",[34,9797,2547],{"class":190},[34,9799,9800],{"class":395},"prevent",[34,9802,202],{"class":190},[34,9804,1608],{"class":186},[34,9806,9795],{"class":190},[34,9808,1608],{"class":186},[34,9810,939],{"class":190},[15,9812,9813,9816],{},[31,9814,9815],{},".prevent"," 表示阻止浏览器默认行为。表单默认提交会刷新页面，Vue 单页应用里通常不希望这样。",[24,9818,9820],{"className":923,"code":9819,"language":925,"meta":29,"style":29},"\u003Cbutton @click.stop=\"openMenu\">打开\u003C\u002Fbutton>\n",[31,9821,9822],{"__ignoreMap":29},[34,9823,9824,9826,9828,9830,9832,9834,9837,9839,9841,9844,9846,9849,9851],{"class":36,"line":37},[34,9825,932],{"class":190},[34,9827,2004],{"class":935},[34,9829,9628],{"class":190},[34,9831,9631],{"class":395},[34,9833,2547],{"class":190},[34,9835,9836],{"class":395},"stop",[34,9838,202],{"class":190},[34,9840,1608],{"class":186},[34,9842,9843],{"class":190},"openMenu",[34,9845,1608],{"class":186},[34,9847,9848],{"class":190},">打开\u003C\u002F",[34,9850,2004],{"class":935},[34,9852,939],{"class":190},[15,9854,9855,9858],{},[31,9856,9857],{},".stop"," 表示阻止事件继续冒泡。",[19,9860,9862,9863],{"id":9861},"_8-条件渲染-v-if","8. 条件渲染 ",[31,9864,8162],{},[24,9866,9867],{"className":923,"code":2157,"language":925,"meta":29,"style":29},[31,9868,9869,9891,9913],{"__ignoreMap":29},[34,9870,9871,9873,9875,9877,9879,9881,9883,9885,9887,9889],{"class":36,"line":37},[34,9872,932],{"class":190},[34,9874,15],{"class":935},[34,9876,2168],{"class":175},[34,9878,202],{"class":190},[34,9880,1608],{"class":186},[34,9882,2175],{"class":190},[34,9884,1608],{"class":186},[34,9886,2180],{"class":190},[34,9888,15],{"class":935},[34,9890,939],{"class":190},[34,9892,9893,9895,9897,9899,9901,9903,9905,9907,9909,9911],{"class":36,"line":43},[34,9894,932],{"class":190},[34,9896,15],{"class":935},[34,9898,2193],{"class":175},[34,9900,202],{"class":190},[34,9902,1608],{"class":186},[34,9904,2200],{"class":190},[34,9906,1608],{"class":186},[34,9908,2205],{"class":190},[34,9910,15],{"class":935},[34,9912,939],{"class":190},[34,9914,9915,9917,9919,9921,9923,9925],{"class":36,"line":49},[34,9916,932],{"class":190},[34,9918,15],{"class":935},[34,9920,2218],{"class":175},[34,9922,2221],{"class":190},[34,9924,15],{"class":935},[34,9926,939],{"class":190},[15,9928,9929,9931],{},[31,9930,8162],{}," 会真正创建或销毁元素。",[15,9933,9934,9935,2664],{},"如果只是频繁显示隐藏，可以用 ",[31,9936,9937],{},"v-show",[24,9939,9941],{"className":923,"code":9940,"language":925,"meta":29,"style":29},"\u003Caside v-show=\"sidebarOpen\">侧边栏\u003C\u002Faside>\n",[31,9942,9943],{"__ignoreMap":29},[34,9944,9945,9947,9950,9953,9955,9957,9960,9962,9965,9967],{"class":36,"line":37},[34,9946,932],{"class":190},[34,9948,9949],{"class":935},"aside",[34,9951,9952],{"class":395}," v-show",[34,9954,202],{"class":190},[34,9956,1608],{"class":186},[34,9958,9959],{"class":190},"sidebarOpen",[34,9961,1608],{"class":186},[34,9963,9964],{"class":190},">侧边栏\u003C\u002F",[34,9966,9949],{"class":935},[34,9968,939],{"class":190},[15,9970,9971],{},"区别：",[105,9973,9974,9979],{},[108,9975,9976,9978],{},[31,9977,8162],{},"：条件不满足时，元素不在 DOM 中。",[108,9980,9981,9983],{},[31,9982,9937],{},"：元素还在 DOM 中，只是用 CSS 隐藏。",[15,9985,9986,9987,9989,9990,347],{},"初学时默认用 ",[31,9988,8162],{},"，遇到频繁切换且不希望销毁状态时再考虑 ",[31,9991,9937],{},[19,9993,9995,9996],{"id":9994},"_9-列表渲染-v-for","9. 列表渲染 ",[31,9997,1534],{},[24,9999,10001],{"className":923,"code":10000,"language":925,"meta":29,"style":29},"\u003Carticle v-for=\"lesson in lessons\" :key=\"lesson.id\">\n \u003Ch2>{{ lesson.title }}\u003C\u002Fh2>\n \u003Cp>{{ lesson.minutes }} 分钟\u003C\u002Fp>\n\u003C\u002Farticle>\n",[31,10002,10003,10037,10042,10047],{"__ignoreMap":29},[34,10004,10005,10007,10009,10011,10013,10015,10017,10019,10021,10023,10025,10027,10029,10031,10033,10035],{"class":36,"line":37},[34,10006,932],{"class":190},[34,10008,947],{"class":935},[34,10010,1603],{"class":175},[34,10012,202],{"class":190},[34,10014,1608],{"class":186},[34,10016,1611],{"class":190},[34,10018,1614],{"class":175},[34,10020,1203],{"class":190},[34,10022,1608],{"class":186},[34,10024,1621],{"class":190},[34,10026,1624],{"class":395},[34,10028,202],{"class":190},[34,10030,1608],{"class":186},[34,10032,1631],{"class":190},[34,10034,1608],{"class":186},[34,10036,939],{"class":190},[34,10038,10039],{"class":36,"line":43},[34,10040,10041],{"class":190}," \u003Ch2>{{ lesson.title }}\u003C\u002Fh2>\n",[34,10043,10044],{"class":36,"line":49},[34,10045,10046],{"class":190}," \u003Cp>{{ lesson.minutes }} 分钟\u003C\u002Fp>\n",[34,10048,10049,10051,10053],{"class":36,"line":56},[34,10050,1003],{"class":190},[34,10052,947],{"class":935},[34,10054,939],{"class":190},[15,10056,10057],{},"读法：",[105,10059,10060,10068,10073],{},[108,10061,3101,10062,10065,10066,347],{},[31,10063,10064],{},"lessons"," 数组里的每个 ",[31,10067,1499],{},[108,10069,10070,10071,347],{},"渲染一个 ",[31,10072,947],{},[108,10074,5792,10075,10077],{},[31,10076,1631],{}," 作为身份标识。",[253,10079,10081,10082],{"id":10080},"为什么必须写-key","为什么必须写 ",[31,10083,1624],{},[15,10085,10086,10088],{},[31,10087,1624],{}," 告诉 Vue：列表里的每一项是谁。",[15,10090,10091],{},"如果没有稳定 key，或者用数组下标当 key，当列表增删排序时，Vue 可能复用错 DOM，导致输入框内容、勾选状态等错位。",[15,10093,7527],{},[24,10095,10096],{"className":923,"code":7269,"language":925,"meta":29,"style":29},[31,10097,10098,10132,10136],{"__ignoreMap":29},[34,10099,10100,10102,10104,10106,10108,10110,10112,10114,10116,10118,10120,10122,10124,10126,10128,10130],{"class":36,"line":37},[34,10101,932],{"class":190},[34,10103,108],{"class":935},[34,10105,1603],{"class":175},[34,10107,202],{"class":190},[34,10109,1608],{"class":186},[34,10111,7286],{"class":190},[34,10113,1614],{"class":175},[34,10115,734],{"class":190},[34,10117,1608],{"class":186},[34,10119,1621],{"class":190},[34,10121,1624],{"class":395},[34,10123,202],{"class":190},[34,10125,1608],{"class":186},[34,10127,7303],{"class":190},[34,10129,1608],{"class":186},[34,10131,939],{"class":190},[34,10133,10134],{"class":36,"line":43},[34,10135,7312],{"class":190},[34,10137,10138,10140,10142],{"class":36,"line":49},[34,10139,1003],{"class":190},[34,10141,108],{"class":935},[34,10143,939],{"class":190},[15,10145,7500],{},[24,10147,10149],{"className":923,"code":10148,"language":925,"meta":29,"style":29},"\u003Cli v-for=\"(task, index) in tasks\" :key=\"index\">\n {{ task.title }}\n\u003C\u002Fli>\n",[31,10150,10151,10187,10191],{"__ignoreMap":29},[34,10152,10153,10155,10157,10159,10161,10163,10166,10168,10170,10172,10174,10176,10178,10180,10183,10185],{"class":36,"line":37},[34,10154,932],{"class":190},[34,10156,108],{"class":935},[34,10158,1603],{"class":175},[34,10160,202],{"class":190},[34,10162,1608],{"class":186},[34,10164,10165],{"class":190},"(task, index) ",[34,10167,1614],{"class":175},[34,10169,734],{"class":190},[34,10171,1608],{"class":186},[34,10173,1621],{"class":190},[34,10175,1624],{"class":395},[34,10177,202],{"class":190},[34,10179,1608],{"class":186},[34,10181,10182],{"class":190},"index",[34,10184,1608],{"class":186},[34,10186,939],{"class":190},[34,10188,10189],{"class":36,"line":43},[34,10190,7312],{"class":190},[34,10192,10193,10195,10197],{"class":36,"line":49},[34,10194,1003],{"class":190},[34,10196,108],{"class":935},[34,10198,939],{"class":190},[15,10200,10201],{},"只有当列表永远不会增删、排序、过滤时，index 才勉强可用。",[19,10203,10205,10206,10208],{"id":10204},"_10-computed从已有状态推导新值","10. ",[31,10207,149],{},"：从已有状态推导新值",[15,10210,10211,10212,347],{},"当一个值可以由其他状态计算出来，不要手动存第二份，用 ",[31,10213,149],{},[24,10215,10217],{"className":166,"code":10216,"language":168,"meta":29,"style":29},"const tasks = ref([\n { id: 1, title: '学习 ref', done: true },\n { id: 2, title: '学习 computed', done: false }\n])\n\nconst completedCount = computed(() => {\n return tasks.value.filter((task) => task.done).length\n})\n",[31,10218,10219,10231,10248,10265,10269,10273,10290,10312],{"__ignoreMap":29},[34,10220,10221,10223,10225,10227,10229],{"class":36,"line":37},[34,10222,176],{"class":175},[34,10224,734],{"class":179},[34,10226,183],{"class":175},[34,10228,453],{"class":395},[34,10230,6908],{"class":190},[34,10232,10233,10235,10237,10239,10242,10244,10246],{"class":36,"line":43},[34,10234,744],{"class":190},[34,10236,640],{"class":179},[34,10238,749],{"class":190},[34,10240,10241],{"class":186},"'学习 ref'",[34,10243,755],{"class":190},[34,10245,758],{"class":179},[34,10247,761],{"class":190},[34,10249,10250,10252,10254,10256,10259,10261,10263],{"class":36,"line":49},[34,10251,744],{"class":190},[34,10253,768],{"class":179},[34,10255,749],{"class":190},[34,10257,10258],{"class":186},"'学习 computed'",[34,10260,755],{"class":190},[34,10262,702],{"class":179},[34,10264,6564],{"class":190},[34,10266,10267],{"class":36,"line":56},[34,10268,6945],{"class":190},[34,10270,10271],{"class":36,"line":62},[34,10272,53],{"emptyLinePlaceholder":52},[34,10274,10275,10277,10280,10282,10284,10286,10288],{"class":36,"line":68},[34,10276,176],{"class":175},[34,10278,10279],{"class":179}," completedCount",[34,10281,183],{"class":175},[34,10283,8935],{"class":395},[34,10285,7682],{"class":190},[34,10287,1333],{"class":175},[34,10289,674],{"class":190},[34,10291,10292,10294,10296,10298,10300,10302,10304,10306,10309],{"class":36,"line":74},[34,10293,1510],{"class":175},[34,10295,6972],{"class":190},[34,10297,1300],{"class":395},[34,10299,1323],{"class":190},[34,10301,1327],{"class":1326},[34,10303,1330],{"class":190},[34,10305,1333],{"class":175},[34,10307,10308],{"class":190}," task.done).",[34,10310,10311],{"class":179},"length\n",[34,10313,10314],{"class":36,"line":1971},[34,10315,6153],{"class":190},[15,10317,1589],{},[24,10319,10321],{"className":923,"code":10320,"language":925,"meta":29,"style":29},"\u003Cp>已完成 {{ completedCount }} 个\u003C\u002Fp>\n",[31,10322,10323],{"__ignoreMap":29},[34,10324,10325,10327,10329,10332,10334],{"class":36,"line":37},[34,10326,932],{"class":190},[34,10328,15],{"class":935},[34,10330,10331],{"class":190},">已完成 {{ completedCount }} 个\u003C\u002F",[34,10333,15],{"class":935},[34,10335,939],{"class":190},[253,10337,10338],{"id":10338},"为什么不用普通函数",[15,10340,10341],{},"普通函数也能算：",[24,10343,10345],{"className":166,"code":10344,"language":168,"meta":29,"style":29},"function completedCount() {\n return tasks.value.filter((task) => task.done).length\n}\n",[31,10346,10347,10355,10375],{"__ignoreMap":29},[34,10348,10349,10351,10353],{"class":36,"line":37},[34,10350,392],{"class":175},[34,10352,10279],{"class":395},[34,10354,399],{"class":190},[34,10356,10357,10359,10361,10363,10365,10367,10369,10371,10373],{"class":36,"line":43},[34,10358,1510],{"class":175},[34,10360,6972],{"class":190},[34,10362,1300],{"class":395},[34,10364,1323],{"class":190},[34,10366,1327],{"class":1326},[34,10368,1330],{"class":190},[34,10370,1333],{"class":175},[34,10372,10308],{"class":190},[34,10374,10311],{"class":179},[34,10376,10377],{"class":36,"line":49},[34,10378,77],{"class":190},[15,10380,10381,10382,10384,10385,10387,10388,10390],{},"但 ",[31,10383,149],{}," 有缓存。只要依赖的 ",[31,10386,1345],{}," 没变，Vue 可以复用上次结果。更重要的是，",[31,10389,149],{}," 表达了语义：这是一个“由状态推导出的值”。",[253,10392,10393],{"id":10393},"不要重复存可推导状态",[15,10395,7500],{},[24,10397,10399],{"className":166,"code":10398,"language":168,"meta":29,"style":29},"const tasks = ref([])\nconst completedCount = ref(0)\n",[31,10400,10401,10413],{"__ignoreMap":29},[34,10402,10403,10405,10407,10409,10411],{"class":36,"line":37},[34,10404,176],{"class":175},[34,10406,734],{"class":179},[34,10408,183],{"class":175},[34,10410,453],{"class":395},[34,10412,9446],{"class":190},[34,10414,10415,10417,10419,10421,10423,10425,10427],{"class":36,"line":43},[34,10416,176],{"class":175},[34,10418,10279],{"class":179},[34,10420,183],{"class":175},[34,10422,453],{"class":395},[34,10424,456],{"class":190},[34,10426,459],{"class":179},[34,10428,6111],{"class":190},[15,10430,10431,10432,10435],{},"每次任务变化你都要记得更新 ",[31,10433,10434],{},"completedCount","，容易不同步。",[15,10437,7527],{},[24,10439,10441],{"className":166,"code":10440,"language":168,"meta":29,"style":29},"const completedCount = computed(() => {\n return tasks.value.filter((task) => task.done).length\n})\n",[31,10442,10443,10459,10479],{"__ignoreMap":29},[34,10444,10445,10447,10449,10451,10453,10455,10457],{"class":36,"line":37},[34,10446,176],{"class":175},[34,10448,10279],{"class":179},[34,10450,183],{"class":175},[34,10452,8935],{"class":395},[34,10454,7682],{"class":190},[34,10456,1333],{"class":175},[34,10458,674],{"class":190},[34,10460,10461,10463,10465,10467,10469,10471,10473,10475,10477],{"class":36,"line":43},[34,10462,1510],{"class":175},[34,10464,6972],{"class":190},[34,10466,1300],{"class":395},[34,10468,1323],{"class":190},[34,10470,1327],{"class":1326},[34,10472,1330],{"class":190},[34,10474,1333],{"class":175},[34,10476,10308],{"class":190},[34,10478,10311],{"class":179},[34,10480,10481],{"class":36,"line":49},[34,10482,6153],{"class":190},[19,10484,10486,10487,10489],{"id":10485},"_11-watch状态变化后做一件事","11. ",[31,10488,152],{},"：状态变化后做一件事",[15,10491,10492,10494],{},[31,10493,152],{}," 用来观察一个响应式状态，当它变化后执行副作用。",[24,10496,10498],{"className":166,"code":10497,"language":168,"meta":29,"style":29},"watch(keyword, (newValue, oldValue) => {\n console.log('旧值：', oldValue)\n console.log('新值：', newValue)\n})\n",[31,10499,10500,10521,10535,10549],{"__ignoreMap":29},[34,10501,10502,10504,10507,10510,10512,10515,10517,10519],{"class":36,"line":37},[34,10503,152],{"class":395},[34,10505,10506],{"class":190},"(keyword, (",[34,10508,10509],{"class":1326},"newValue",[34,10511,616],{"class":190},[34,10513,10514],{"class":1326},"oldValue",[34,10516,1330],{"class":190},[34,10518,1333],{"class":175},[34,10520,674],{"class":190},[34,10522,10523,10525,10527,10529,10532],{"class":36,"line":43},[34,10524,1812],{"class":190},[34,10526,885],{"class":395},[34,10528,456],{"class":190},[34,10530,10531],{"class":186},"'旧值：'",[34,10533,10534],{"class":190},", oldValue)\n",[34,10536,10537,10539,10541,10543,10546],{"class":36,"line":49},[34,10538,1812],{"class":190},[34,10540,885],{"class":395},[34,10542,456],{"class":190},[34,10544,10545],{"class":186},"'新值：'",[34,10547,10548],{"class":190},", newValue)\n",[34,10550,10551],{"class":36,"line":56},[34,10552,6153],{"class":190},[15,10554,10555],{},"常见副作用：",[105,10557,10558,10561,10567,10570,10573],{},[108,10559,10560],{},"发送请求。",[108,10562,10563,10564,347],{},"写入 ",[31,10565,10566],{},"localStorage",[108,10568,10569],{},"修改网页标题。",[108,10571,10572],{},"调用第三方库。",[108,10574,10575],{},"打日志。",[253,10577,10579,258,10581,261],{"id":10578},"computed-和-watch-的区别",[31,10580,149],{},[31,10582,152],{},[15,10584,10585,10587],{},[31,10586,149],{},"：我要得到一个新值。",[24,10589,10591],{"className":166,"code":10590,"language":168,"meta":29,"style":29},"const filteredTasks = computed(() => {\n return tasks.value.filter((task) => task.title.includes(keyword.value))\n})\n",[31,10592,10593,10610,10635],{"__ignoreMap":29},[34,10594,10595,10597,10600,10602,10604,10606,10608],{"class":36,"line":37},[34,10596,176],{"class":175},[34,10598,10599],{"class":179}," filteredTasks",[34,10601,183],{"class":175},[34,10603,8935],{"class":395},[34,10605,7682],{"class":190},[34,10607,1333],{"class":175},[34,10609,674],{"class":190},[34,10611,10612,10614,10616,10618,10620,10622,10624,10626,10629,10632],{"class":36,"line":43},[34,10613,1510],{"class":175},[34,10615,6972],{"class":190},[34,10617,1300],{"class":395},[34,10619,1323],{"class":190},[34,10621,1327],{"class":1326},[34,10623,1330],{"class":190},[34,10625,1333],{"class":175},[34,10627,10628],{"class":190}," task.title.",[34,10630,10631],{"class":395},"includes",[34,10633,10634],{"class":190},"(keyword.value))\n",[34,10636,10637],{"class":36,"line":49},[34,10638,6153],{"class":190},[15,10640,10641,10643],{},[31,10642,152],{},"：某个值变了，我要执行动作。",[24,10645,10647],{"className":166,"code":10646,"language":168,"meta":29,"style":29},"watch(keyword, (value) => {\n localStorage.setItem('keyword', value)\n})\n",[31,10648,10649,10663,10679],{"__ignoreMap":29},[34,10650,10651,10653,10655,10657,10659,10661],{"class":36,"line":37},[34,10652,152],{"class":395},[34,10654,10506],{"class":190},[34,10656,2533],{"class":1326},[34,10658,1330],{"class":190},[34,10660,1333],{"class":175},[34,10662,674],{"class":190},[34,10664,10665,10668,10671,10673,10676],{"class":36,"line":43},[34,10666,10667],{"class":190}," localStorage.",[34,10669,10670],{"class":395},"setItem",[34,10672,456],{"class":190},[34,10674,10675],{"class":186},"'keyword'",[34,10677,10678],{"class":190},", value)\n",[34,10680,10681],{"class":36,"line":49},[34,10682,6153],{"class":190},[15,10684,10685,10686,10688,10689,347],{},"如果你只是为了显示一个结果，优先用 ",[31,10687,149],{},"。如果你要“做一件事”，才用 ",[31,10690,152],{},[19,10692,10694],{"id":10693},"_12-生命周期","12. 生命周期",[15,10696,10697,10698,347],{},"组件从创建到销毁，有不同阶段。最常用的是 ",[31,10699,7679],{},[24,10701,10703],{"className":166,"code":10702,"language":168,"meta":29,"style":29},"import { onMounted } from 'vue'\n\nonMounted(() => {\n console.log('组件已经出现在页面上')\n})\n",[31,10704,10705,10716,10720,10730,10743],{"__ignoreMap":29},[34,10706,10707,10709,10712,10714],{"class":36,"line":37},[34,10708,426],{"class":175},[34,10710,10711],{"class":190}," { onMounted } ",[34,10713,432],{"class":175},[34,10715,6891],{"class":186},[34,10717,10718],{"class":36,"line":43},[34,10719,53],{"emptyLinePlaceholder":52},[34,10721,10722,10724,10726,10728],{"class":36,"line":49},[34,10723,7679],{"class":395},[34,10725,7682],{"class":190},[34,10727,1333],{"class":175},[34,10729,674],{"class":190},[34,10731,10732,10734,10736,10738,10741],{"class":36,"line":56},[34,10733,1812],{"class":190},[34,10735,885],{"class":395},[34,10737,456],{"class":190},[34,10739,10740],{"class":186},"'组件已经出现在页面上'",[34,10742,6111],{"class":190},[34,10744,10745],{"class":36,"line":62},[34,10746,6153],{"class":190},[15,10748,10749],{},"常见用途：",[105,10751,10752,10755,10758],{},[108,10753,10754],{},"页面进入后请求数据。",[108,10756,10757],{},"DOM 出现后聚焦输入框。",[108,10759,10760],{},"初始化第三方库。",[15,10762,10763,10764,2664],{},"清理副作用用 ",[31,10765,10766],{},"onUnmounted",[24,10768,10770],{"className":166,"code":10769,"language":168,"meta":29,"style":29},"import { onMounted, onUnmounted } from 'vue'\n\nlet timerId\n\nonMounted(() => {\n timerId = setInterval(() => {\n console.log('tick')\n }, 1000)\n})\n\nonUnmounted(() => {\n clearInterval(timerId)\n})\n",[31,10771,10772,10783,10787,10794,10798,10808,10824,10838,10847,10851,10855,10865,10873],{"__ignoreMap":29},[34,10773,10774,10776,10779,10781],{"class":36,"line":37},[34,10775,426],{"class":175},[34,10777,10778],{"class":190}," { onMounted, onUnmounted } ",[34,10780,432],{"class":175},[34,10782,6891],{"class":186},[34,10784,10785],{"class":36,"line":43},[34,10786,53],{"emptyLinePlaceholder":52},[34,10788,10789,10791],{"class":36,"line":49},[34,10790,196],{"class":175},[34,10792,10793],{"class":190}," timerId\n",[34,10795,10796],{"class":36,"line":56},[34,10797,53],{"emptyLinePlaceholder":52},[34,10799,10800,10802,10804,10806],{"class":36,"line":62},[34,10801,7679],{"class":395},[34,10803,7682],{"class":190},[34,10805,1333],{"class":175},[34,10807,674],{"class":190},[34,10809,10810,10813,10815,10818,10820,10822],{"class":36,"line":68},[34,10811,10812],{"class":190}," timerId ",[34,10814,202],{"class":175},[34,10816,10817],{"class":395}," setInterval",[34,10819,7682],{"class":190},[34,10821,1333],{"class":175},[34,10823,674],{"class":190},[34,10825,10826,10829,10831,10833,10836],{"class":36,"line":74},[34,10827,10828],{"class":190}," console.",[34,10830,885],{"class":395},[34,10832,456],{"class":190},[34,10834,10835],{"class":186},"'tick'",[34,10837,6111],{"class":190},[34,10839,10840,10843,10845],{"class":36,"line":1971},[34,10841,10842],{"class":190}," }, ",[34,10844,5249],{"class":179},[34,10846,6111],{"class":190},[34,10848,10849],{"class":36,"line":1976},[34,10850,6153],{"class":190},[34,10852,10853],{"class":36,"line":1985},[34,10854,53],{"emptyLinePlaceholder":52},[34,10856,10857,10859,10861,10863],{"class":36,"line":1990},[34,10858,10766],{"class":395},[34,10860,7682],{"class":190},[34,10862,1333],{"class":175},[34,10864,674],{"class":190},[34,10866,10867,10870],{"class":36,"line":1999},[34,10868,10869],{"class":395}," clearInterval",[34,10871,10872],{"class":190},"(timerId)\n",[34,10874,10875],{"class":36,"line":2022},[34,10876,6153],{"class":190},[15,10878,10879],{},"如果不清理定时器，组件离开后定时器还会继续跑。",[19,10881,10883],{"id":10882},"_13-模板引用","13. 模板引用",[15,10885,10886],{},"当你真的需要 DOM 元素：",[24,10888,10889],{"className":923,"code":7626,"language":925,"meta":29,"style":29},[31,10890,10891,10901,10911,10915,10931,10935,10945,10953,10957,10965,10969,10977,10991],{"__ignoreMap":29},[34,10892,10893,10895,10897,10899],{"class":36,"line":37},[34,10894,932],{"class":190},[34,10896,1911],{"class":935},[34,10898,1914],{"class":395},[34,10900,939],{"class":190},[34,10902,10903,10905,10907,10909],{"class":36,"line":43},[34,10904,426],{"class":175},[34,10906,7645],{"class":190},[34,10908,432],{"class":175},[34,10910,6891],{"class":186},[34,10912,10913],{"class":36,"line":49},[34,10914,53],{"emptyLinePlaceholder":52},[34,10916,10917,10919,10921,10923,10925,10927,10929],{"class":36,"line":56},[34,10918,176],{"class":175},[34,10920,7660],{"class":179},[34,10922,183],{"class":175},[34,10924,453],{"class":395},[34,10926,456],{"class":190},[34,10928,5419],{"class":179},[34,10930,6111],{"class":190},[34,10932,10933],{"class":36,"line":62},[34,10934,53],{"emptyLinePlaceholder":52},[34,10936,10937,10939,10941,10943],{"class":36,"line":68},[34,10938,7679],{"class":395},[34,10940,7682],{"class":190},[34,10942,1333],{"class":175},[34,10944,674],{"class":190},[34,10946,10947,10949,10951],{"class":36,"line":74},[34,10948,7691],{"class":190},[34,10950,7694],{"class":395},[34,10952,6374],{"class":190},[34,10954,10955],{"class":36,"line":1971},[34,10956,6153],{"class":190},[34,10958,10959,10961,10963],{"class":36,"line":1976},[34,10960,1003],{"class":190},[34,10962,1911],{"class":935},[34,10964,939],{"class":190},[34,10966,10967],{"class":36,"line":1985},[34,10968,53],{"emptyLinePlaceholder":52},[34,10970,10971,10973,10975],{"class":36,"line":1990},[34,10972,932],{"class":190},[34,10974,936],{"class":935},[34,10976,939],{"class":190},[34,10978,10979,10981,10983,10985,10987,10989],{"class":36,"line":1999},[34,10980,944],{"class":190},[34,10982,7429],{"class":935},[34,10984,453],{"class":395},[34,10986,202],{"class":190},[34,10988,7733],{"class":186},[34,10990,939],{"class":190},[34,10992,10993,10995,10997],{"class":36,"line":2022},[34,10994,1003],{"class":190},[34,10996,936],{"class":935},[34,10998,939],{"class":190},[15,11000,8882],{},[105,11002,11003,11011,11014],{},[108,11004,11005,11006,11008,11009,347],{},"初始时 ",[31,11007,7756],{}," 是 ",[31,11010,5419],{},[108,11012,11013],{},"组件挂载后才会变成真实 DOM。",[108,11015,11016,11017,11019],{},"所以通常在 ",[31,11018,7679],{}," 里使用。",[19,11021,11023],{"id":11022},"_14-本章完整小例子","14. 本章完整小例子",[24,11025,11027],{"className":923,"code":11026,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { computed, ref, watch } from 'vue'\n\nconst keyword = ref('')\nconst tasks = ref([\n { id: 1, title: '学习 ref', done: true },\n { id: 2, title: '学习 computed', done: false },\n { id: 3, title: '学习 watch', done: false }\n])\n\nconst filteredTasks = computed(() => {\n return tasks.value.filter((task) => {\n return task.title.includes(keyword.value)\n })\n})\n\nconst remainingCount = computed(() => {\n return tasks.value.filter((task) => !task.done).length\n})\n\nwatch(keyword, (value) => {\n console.log('搜索词变化：', value)\n})\n\nfunction toggleTask(id) {\n const task = tasks.value.find((item) => item.id === id)\n\n if (task) {\n task.done = !task.done\n }\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cinput v-model.trim=\"keyword\" placeholder=\"搜索任务\">\n\n \u003Cp>剩余 {{ remainingCount }} 个任务\u003C\u002Fp>\n\n \u003Cp v-if=\"filteredTasks.length === 0\">没有匹配任务\u003C\u002Fp>\n\n \u003Cul v-else>\n \u003Cli v-for=\"task in filteredTasks\" :key=\"task.id\">\n \u003Clabel>\n \u003Cinput type=\"checkbox\" :checked=\"task.done\" @change=\"toggleTask(task.id)\">\n {{ task.title }}\n \u003C\u002Flabel>\n \u003C\u002Fli>\n \u003C\u002Ful>\n\u003C\u002Ftemplate>\n",[31,11028,11029,11039,11050,11054,11071,11083,11099,11115,11132,11136,11140,11156,11174,11185,11189,11193,11197,11214,11237,11241,11245,11259,11272,11276,11280,11292,11319,11324,11331,11342,11347,11352,11361,11366,11375,11400,11405,11419,11424,11445,11450,11461,11483,11493,11526,11532,11542,11551,11560],{"__ignoreMap":29},[34,11030,11031,11033,11035,11037],{"class":36,"line":37},[34,11032,932],{"class":190},[34,11034,1911],{"class":935},[34,11036,1914],{"class":395},[34,11038,939],{"class":190},[34,11040,11041,11043,11046,11048],{"class":36,"line":43},[34,11042,426],{"class":175},[34,11044,11045],{"class":190}," { computed, ref, watch } ",[34,11047,432],{"class":175},[34,11049,6891],{"class":186},[34,11051,11052],{"class":36,"line":49},[34,11053,53],{"emptyLinePlaceholder":52},[34,11055,11056,11058,11061,11063,11065,11067,11069],{"class":36,"line":56},[34,11057,176],{"class":175},[34,11059,11060],{"class":179}," keyword",[34,11062,183],{"class":175},[34,11064,453],{"class":395},[34,11066,456],{"class":190},[34,11068,7400],{"class":186},[34,11070,6111],{"class":190},[34,11072,11073,11075,11077,11079,11081],{"class":36,"line":62},[34,11074,176],{"class":175},[34,11076,734],{"class":179},[34,11078,183],{"class":175},[34,11080,453],{"class":395},[34,11082,6908],{"class":190},[34,11084,11085,11087,11089,11091,11093,11095,11097],{"class":36,"line":68},[34,11086,744],{"class":190},[34,11088,640],{"class":179},[34,11090,749],{"class":190},[34,11092,10241],{"class":186},[34,11094,755],{"class":190},[34,11096,758],{"class":179},[34,11098,761],{"class":190},[34,11100,11101,11103,11105,11107,11109,11111,11113],{"class":36,"line":74},[34,11102,744],{"class":190},[34,11104,768],{"class":179},[34,11106,749],{"class":190},[34,11108,10258],{"class":186},[34,11110,755],{"class":190},[34,11112,702],{"class":179},[34,11114,761],{"class":190},[34,11116,11117,11119,11121,11123,11126,11128,11130],{"class":36,"line":1971},[34,11118,744],{"class":190},[34,11120,786],{"class":179},[34,11122,749],{"class":190},[34,11124,11125],{"class":186},"'学习 watch'",[34,11127,755],{"class":190},[34,11129,702],{"class":179},[34,11131,6564],{"class":190},[34,11133,11134],{"class":36,"line":1976},[34,11135,6945],{"class":190},[34,11137,11138],{"class":36,"line":1985},[34,11139,53],{"emptyLinePlaceholder":52},[34,11141,11142,11144,11146,11148,11150,11152,11154],{"class":36,"line":1990},[34,11143,176],{"class":175},[34,11145,10599],{"class":179},[34,11147,183],{"class":175},[34,11149,8935],{"class":395},[34,11151,7682],{"class":190},[34,11153,1333],{"class":175},[34,11155,674],{"class":190},[34,11157,11158,11160,11162,11164,11166,11168,11170,11172],{"class":36,"line":1999},[34,11159,1510],{"class":175},[34,11161,6972],{"class":190},[34,11163,1300],{"class":395},[34,11165,1323],{"class":190},[34,11167,1327],{"class":1326},[34,11169,1330],{"class":190},[34,11171,1333],{"class":175},[34,11173,674],{"class":190},[34,11175,11176,11178,11180,11182],{"class":36,"line":2022},[34,11177,8601],{"class":175},[34,11179,10628],{"class":190},[34,11181,10631],{"class":395},[34,11183,11184],{"class":190},"(keyword.value)\n",[34,11186,11187],{"class":36,"line":2036},[34,11188,6379],{"class":190},[34,11190,11191],{"class":36,"line":2603},[34,11192,6153],{"class":190},[34,11194,11195],{"class":36,"line":5537},[34,11196,53],{"emptyLinePlaceholder":52},[34,11198,11199,11201,11204,11206,11208,11210,11212],{"class":36,"line":5543},[34,11200,176],{"class":175},[34,11202,11203],{"class":179}," remainingCount",[34,11205,183],{"class":175},[34,11207,8935],{"class":395},[34,11209,7682],{"class":190},[34,11211,1333],{"class":175},[34,11213,674],{"class":190},[34,11215,11216,11218,11220,11222,11224,11226,11228,11230,11232,11235],{"class":36,"line":5548},[34,11217,1510],{"class":175},[34,11219,6972],{"class":190},[34,11221,1300],{"class":395},[34,11223,1323],{"class":190},[34,11225,1327],{"class":1326},[34,11227,1330],{"class":190},[34,11229,1333],{"class":175},[34,11231,1336],{"class":175},[34,11233,11234],{"class":190},"task.done).",[34,11236,10311],{"class":179},[34,11238,11239],{"class":36,"line":5564},[34,11240,6153],{"class":190},[34,11242,11243],{"class":36,"line":5576},[34,11244,53],{"emptyLinePlaceholder":52},[34,11246,11247,11249,11251,11253,11255,11257],{"class":36,"line":5606},[34,11248,152],{"class":395},[34,11250,10506],{"class":190},[34,11252,2533],{"class":1326},[34,11254,1330],{"class":190},[34,11256,1333],{"class":175},[34,11258,674],{"class":190},[34,11260,11261,11263,11265,11267,11270],{"class":36,"line":5616},[34,11262,1812],{"class":190},[34,11264,885],{"class":395},[34,11266,456],{"class":190},[34,11268,11269],{"class":186},"'搜索词变化：'",[34,11271,10678],{"class":190},[34,11273,11274],{"class":36,"line":5628},[34,11275,6153],{"class":190},[34,11277,11278],{"class":36,"line":5634},[34,11279,53],{"emptyLinePlaceholder":52},[34,11281,11282,11284,11286,11288,11290],{"class":36,"line":6420},[34,11283,392],{"class":175},[34,11285,1800],{"class":395},[34,11287,456],{"class":190},[34,11289,1805],{"class":1326},[34,11291,1700],{"class":190},[34,11293,11295,11297,11299,11301,11303,11305,11307,11309,11311,11313,11315,11317],{"class":36,"line":11294},26,[34,11296,5298],{"class":175},[34,11298,632],{"class":179},[34,11300,183],{"class":175},[34,11302,6972],{"class":190},[34,11304,1390],{"class":395},[34,11306,1323],{"class":190},[34,11308,6979],{"class":1326},[34,11310,1330],{"class":190},[34,11312,1333],{"class":175},[34,11314,6986],{"class":190},[34,11316,1423],{"class":175},[34,11318,6991],{"class":190},[34,11320,11322],{"class":36,"line":11321},27,[34,11323,53],{"emptyLinePlaceholder":52},[34,11325,11327,11329],{"class":36,"line":11326},28,[34,11328,7000],{"class":175},[34,11330,7003],{"class":190},[34,11332,11334,11336,11338,11340],{"class":36,"line":11333},29,[34,11335,7008],{"class":190},[34,11337,202],{"class":175},[34,11339,1336],{"class":175},[34,11341,6779],{"class":190},[34,11343,11345],{"class":36,"line":11344},30,[34,11346,5631],{"class":190},[34,11348,11350],{"class":36,"line":11349},31,[34,11351,77],{"class":190},[34,11353,11355,11357,11359],{"class":36,"line":11354},32,[34,11356,1003],{"class":190},[34,11358,1911],{"class":935},[34,11360,939],{"class":190},[34,11362,11364],{"class":36,"line":11363},33,[34,11365,53],{"emptyLinePlaceholder":52},[34,11367,11369,11371,11373],{"class":36,"line":11368},34,[34,11370,932],{"class":190},[34,11372,936],{"class":935},[34,11374,939],{"class":190},[34,11376,11378,11380,11382,11385,11387,11390,11393,11395,11398],{"class":36,"line":11377},35,[34,11379,944],{"class":190},[34,11381,7429],{"class":935},[34,11383,11384],{"class":395}," v-model.trim",[34,11386,202],{"class":190},[34,11388,11389],{"class":186},"\"keyword\"",[34,11391,11392],{"class":395}," placeholder",[34,11394,202],{"class":190},[34,11396,11397],{"class":186},"\"搜索任务\"",[34,11399,939],{"class":190},[34,11401,11403],{"class":36,"line":11402},36,[34,11404,53],{"emptyLinePlaceholder":52},[34,11406,11408,11410,11412,11415,11417],{"class":36,"line":11407},37,[34,11409,944],{"class":190},[34,11411,15],{"class":935},[34,11413,11414],{"class":190},">剩余 {{ remainingCount }} 个任务\u003C\u002F",[34,11416,15],{"class":935},[34,11418,939],{"class":190},[34,11420,11422],{"class":36,"line":11421},38,[34,11423,53],{"emptyLinePlaceholder":52},[34,11425,11427,11429,11431,11433,11435,11438,11441,11443],{"class":36,"line":11426},39,[34,11428,944],{"class":190},[34,11430,15],{"class":935},[34,11432,2168],{"class":395},[34,11434,202],{"class":190},[34,11436,11437],{"class":186},"\"filteredTasks.length === 0\"",[34,11439,11440],{"class":190},">没有匹配任务\u003C\u002F",[34,11442,15],{"class":935},[34,11444,939],{"class":190},[34,11446,11448],{"class":36,"line":11447},40,[34,11449,53],{"emptyLinePlaceholder":52},[34,11451,11453,11455,11457,11459],{"class":36,"line":11452},41,[34,11454,944],{"class":190},[34,11456,105],{"class":935},[34,11458,2218],{"class":395},[34,11460,939],{"class":190},[34,11462,11464,11466,11468,11470,11472,11475,11477,11479,11481],{"class":36,"line":11463},42,[34,11465,954],{"class":190},[34,11467,108],{"class":935},[34,11469,1603],{"class":395},[34,11471,202],{"class":190},[34,11473,11474],{"class":186},"\"task in filteredTasks\"",[34,11476,7066],{"class":395},[34,11478,202],{"class":190},[34,11480,7071],{"class":186},[34,11482,939],{"class":190},[34,11484,11486,11488,11491],{"class":36,"line":11485},43,[34,11487,7078],{"class":190},[34,11489,11490],{"class":935},"label",[34,11492,939],{"class":190},[34,11494,11496,11499,11501,11504,11506,11509,11512,11514,11517,11520,11522,11524],{"class":36,"line":11495},44,[34,11497,11498],{"class":190}," \u003C",[34,11500,7429],{"class":935},[34,11502,11503],{"class":395}," type",[34,11505,202],{"class":190},[34,11507,11508],{"class":186},"\"checkbox\"",[34,11510,11511],{"class":395}," :checked",[34,11513,202],{"class":190},[34,11515,11516],{"class":186},"\"task.done\"",[34,11518,11519],{"class":395}," @change",[34,11521,202],{"class":190},[34,11523,7100],{"class":186},[34,11525,939],{"class":190},[34,11527,11529],{"class":36,"line":11528},45,[34,11530,11531],{"class":190}," {{ task.title }}\n",[34,11533,11535,11538,11540],{"class":36,"line":11534},46,[34,11536,11537],{"class":190}," \u003C\u002F",[34,11539,11490],{"class":935},[34,11541,939],{"class":190},[34,11543,11545,11547,11549],{"class":36,"line":11544},47,[34,11546,7112],{"class":190},[34,11548,108],{"class":935},[34,11550,939],{"class":190},[34,11552,11554,11556,11558],{"class":36,"line":11553},48,[34,11555,994],{"class":190},[34,11557,105],{"class":935},[34,11559,939],{"class":190},[34,11561,11563,11565,11567],{"class":36,"line":11562},49,[34,11564,1003],{"class":190},[34,11566,936],{"class":935},[34,11568,939],{"class":190},[15,11570,11571],{},"这个例子包含：",[105,11573,11574,11579,11584,11589,11594,11599,11604],{},[108,11575,11576,11578],{},[31,11577,145],{}," 保存状态。",[108,11580,11581,11583],{},[31,11582,149],{}," 推导筛选列表和剩余数量。",[108,11585,11586,11588],{},[31,11587,152],{}," 观察搜索词变化。",[108,11590,11591,11593],{},[31,11592,7473],{}," 处理输入。",[108,11595,11596,11598],{},[31,11597,8162],{}," 处理空状态。",[108,11600,11601,11603],{},[31,11602,1534],{}," 渲染列表。",[108,11605,11606,11609],{},[31,11607,11608],{},"@change"," 处理事件。",[19,11611,5693],{"id":5693},[2230,11613,11614,11620,11628,11631,11636,11642,11649],{},[108,11615,11616,11617,11619],{},"创建一个 ",[31,11618,10064],{}," 数组，包含课程标题、时长、完成状态。",[108,11621,5792,11622,11624,11625,11627],{},[31,11623,1534],{}," 渲染课程列表，",[31,11626,1624],{}," 使用课程 id。",[108,11629,11630],{},"点击按钮切换完成状态。",[108,11632,5792,11633,11635],{},[31,11634,149],{}," 统计完成数量和总时长。",[108,11637,11638,11639,11641],{},"加一个搜索框，用 ",[31,11640,149],{}," 筛选课程。",[108,11643,5792,11644,11646,11647,347],{},[31,11645,152],{}," 把搜索词保存到 ",[31,11648,10566],{},[108,11650,11651],{},"用模板引用让页面加载后自动聚焦搜索框。",[79,11653,11654],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":11656},[11657,11660,11662,11664,11668,11673,11675,11679,11681,11686,11691,11696,11697,11698,11699],{"id":8368,"depth":43,"text":8369,"children":11658},[11659],{"id":8517,"depth":49,"text":8517},{"id":8554,"depth":43,"text":11661},"2. \u003Cscript setup> 为什么不用 return",{"id":8829,"depth":43,"text":11663},"3. 插值表达式 {{ }}",{"id":9024,"depth":43,"text":11665,"children":11666},"4. ref：最常用的响应式状态",[11667],{"id":9224,"depth":49,"text":9225},{"id":9254,"depth":43,"text":11669,"children":11670},"5. reactive：响应式对象",[11671],{"id":9391,"depth":49,"text":11672},"ref 和 reactive 怎么选",{"id":9479,"depth":43,"text":11674},"6. 属性绑定 v-bind",{"id":9602,"depth":43,"text":11676,"children":11677},"7. 事件绑定 v-on",[11678],{"id":9778,"depth":49,"text":9778},{"id":9861,"depth":43,"text":11680},"8. 条件渲染 v-if",{"id":9994,"depth":43,"text":11682,"children":11683},"9. 列表渲染 v-for",[11684],{"id":10080,"depth":49,"text":11685},"为什么必须写 key",{"id":10204,"depth":43,"text":11687,"children":11688},"10. computed：从已有状态推导新值",[11689,11690],{"id":10338,"depth":49,"text":10338},{"id":10393,"depth":49,"text":10393},{"id":10485,"depth":43,"text":11692,"children":11693},"11. watch：状态变化后做一件事",[11694],{"id":10578,"depth":49,"text":11695},"computed 和 watch 的区别",{"id":10693,"depth":43,"text":10694},{"id":10882,"depth":43,"text":10883},{"id":11022,"depth":43,"text":11023},{"id":5693,"depth":43,"text":5693},{"order":49},"\u002Fsource\u002Fcode\u002Fvue\u002F03-vue-core",{"title":8344,"description":8352},"source\u002Fcode\u002Fvue\u002F03-vue-core","S96myrdolc7aMSJM7WerYC83HKuZ6AH9YiqI9w63g-g",{"id":11706,"title":11707,"body":11708,"description":11715,"extension":85,"meta":14179,"navigation":52,"path":14180,"seo":14181,"stem":14182,"__hash__":14183},"notes\u002Fsource\u002Fcode\u002Fvue\u002F04-components.md","04 组件、Props、事件和插槽",{"type":8,"value":11709,"toc":14157},[11710,11713,11716,11719,11723,11726,11830,11833,11847,11850,11950,11955,11959,11962,11985,11988,11999,12002,12006,12009,12034,12037,12153,12158,12161,12188,12191,12195,12198,12200,12273,12276,12279,12282,12288,12291,12295,12298,12300,12450,12452,12495,12498,12574,12577,12612,12616,12619,12622,12660,12663,12714,12717,12721,12726,12933,12935,13029,13032,13157,13160,13164,13167,13173,13231,13234,13266,13273,13276,13282,13286,13289,13440,13442,13506,13515,13521,13524,13547,13550,13573,13579,13663,13665,13687,13690,13702,13705,13709,13712,13718,13727,13769,13812,13815,13818,13832,13836,13902,13906,13910,13913,13916,13920,13922,13948,13950,13969,13981,13983,14007,14011,14013,14028,14030,14056,14065,14067,14092,14096,14099,14110,14112,14154],[11,11711,11707],{"id":11712},"_04-组件props事件和插槽",[15,11714,11715],{},"组件是 Vue 项目最重要的组织方式。你可以把组件理解成“一个有名字、有输入、有输出的界面单元”。",[15,11717,11718],{},"本章目标：你能判断什么时候该拆组件，能正确使用 props 和 emit，能理解插槽为什么存在，能避免子组件乱改父组件数据。",[19,11720,11722],{"id":11721},"_1-为什么需要组件","1. 为什么需要组件",[15,11724,11725],{},"假设一个页面里有三张课程卡片：",[24,11727,11729],{"className":923,"code":11728,"language":925,"meta":29,"style":29},"\u003Carticle>\n \u003Ch2>JavaScript 基础\u003C\u002Fh2>\n \u003Cp>80 分钟\u003C\u002Fp>\n \u003Cbutton>标记完成\u003C\u002Fbutton>\n\u003C\u002Farticle>\n\n\u003Carticle>\n \u003Ch2>Vue 核心语法\u003C\u002Fh2>\n \u003Cp>90 分钟\u003C\u002Fp>\n \u003Cbutton>标记完成\u003C\u002Fbutton>\n\u003C\u002Farticle>\n\n\u003Carticle>\n \u003Ch2>Pinia 状态管理\u003C\u002Fh2>\n \u003Cp>70 分钟\u003C\u002Fp>\n \u003Cbutton>标记完成\u003C\u002Fbutton>\n\u003C\u002Farticle>\n",[31,11730,11731,11739,11744,11749,11754,11762,11766,11774,11779,11784,11788,11796,11800,11808,11813,11818,11822],{"__ignoreMap":29},[34,11732,11733,11735,11737],{"class":36,"line":37},[34,11734,932],{"class":190},[34,11736,947],{"class":935},[34,11738,939],{"class":190},[34,11740,11741],{"class":36,"line":43},[34,11742,11743],{"class":190}," \u003Ch2>JavaScript 基础\u003C\u002Fh2>\n",[34,11745,11746],{"class":36,"line":49},[34,11747,11748],{"class":190}," \u003Cp>80 分钟\u003C\u002Fp>\n",[34,11750,11751],{"class":36,"line":56},[34,11752,11753],{"class":190}," \u003Cbutton>标记完成\u003C\u002Fbutton>\n",[34,11755,11756,11758,11760],{"class":36,"line":62},[34,11757,1003],{"class":190},[34,11759,947],{"class":935},[34,11761,939],{"class":190},[34,11763,11764],{"class":36,"line":68},[34,11765,53],{"emptyLinePlaceholder":52},[34,11767,11768,11770,11772],{"class":36,"line":74},[34,11769,932],{"class":190},[34,11771,947],{"class":935},[34,11773,939],{"class":190},[34,11775,11776],{"class":36,"line":1971},[34,11777,11778],{"class":190}," \u003Ch2>Vue 核心语法\u003C\u002Fh2>\n",[34,11780,11781],{"class":36,"line":1976},[34,11782,11783],{"class":190}," \u003Cp>90 分钟\u003C\u002Fp>\n",[34,11785,11786],{"class":36,"line":1985},[34,11787,11753],{"class":190},[34,11789,11790,11792,11794],{"class":36,"line":1990},[34,11791,1003],{"class":190},[34,11793,947],{"class":935},[34,11795,939],{"class":190},[34,11797,11798],{"class":36,"line":1999},[34,11799,53],{"emptyLinePlaceholder":52},[34,11801,11802,11804,11806],{"class":36,"line":2022},[34,11803,932],{"class":190},[34,11805,947],{"class":935},[34,11807,939],{"class":190},[34,11809,11810],{"class":36,"line":2036},[34,11811,11812],{"class":190}," \u003Ch2>Pinia 状态管理\u003C\u002Fh2>\n",[34,11814,11815],{"class":36,"line":2603},[34,11816,11817],{"class":190}," \u003Cp>70 分钟\u003C\u002Fp>\n",[34,11819,11820],{"class":36,"line":5537},[34,11821,11753],{"class":190},[34,11823,11824,11826,11828],{"class":36,"line":5543},[34,11825,1003],{"class":190},[34,11827,947],{"class":935},[34,11829,939],{"class":190},[15,11831,11832],{},"问题：",[105,11834,11835,11838,11841,11844],{},[108,11836,11837],{},"结构重复。",[108,11839,11840],{},"样式重复。",[108,11842,11843],{},"按钮逻辑重复。",[108,11845,11846],{},"以后改卡片结构要改很多处。",[15,11848,11849],{},"组件化后：",[24,11851,11853],{"className":923,"code":11852,"language":925,"meta":29,"style":29},"\u003CLessonCard\n v-for=\"lesson in lessons\"\n :key=\"lesson.id\"\n :lesson=\"lesson\"\n :completed=\"isCompleted(lesson.id)\"\n @toggle=\"toggleLesson\"\n\u002F>\n",[31,11854,11855,11862,11880,11895,11909,11928,11945],{"__ignoreMap":29},[34,11856,11857,11859],{"class":36,"line":37},[34,11858,932],{"class":190},[34,11860,11861],{"class":935},"LessonCard\n",[34,11863,11864,11867,11869,11871,11873,11875,11877],{"class":36,"line":43},[34,11865,11866],{"class":175}," v-for",[34,11868,202],{"class":190},[34,11870,1608],{"class":186},[34,11872,1611],{"class":190},[34,11874,1614],{"class":175},[34,11876,1203],{"class":190},[34,11878,11879],{"class":186},"\"\n",[34,11881,11882,11885,11887,11889,11891,11893],{"class":36,"line":49},[34,11883,11884],{"class":190}," :",[34,11886,1624],{"class":395},[34,11888,202],{"class":190},[34,11890,1608],{"class":186},[34,11892,1631],{"class":190},[34,11894,11879],{"class":186},[34,11896,11897,11899,11901,11903,11905,11907],{"class":36,"line":56},[34,11898,11884],{"class":190},[34,11900,1499],{"class":395},[34,11902,202],{"class":190},[34,11904,1608],{"class":186},[34,11906,1499],{"class":190},[34,11908,11879],{"class":186},[34,11910,11911,11913,11916,11918,11920,11923,11926],{"class":36,"line":62},[34,11912,11884],{"class":190},[34,11914,11915],{"class":395},"completed",[34,11917,202],{"class":190},[34,11919,1608],{"class":186},[34,11921,11922],{"class":395},"isCompleted",[34,11924,11925],{"class":190},"(lesson.id)",[34,11927,11879],{"class":186},[34,11929,11930,11933,11936,11938,11940,11943],{"class":36,"line":68},[34,11931,11932],{"class":190}," @",[34,11934,11935],{"class":395},"toggle",[34,11937,202],{"class":190},[34,11939,1608],{"class":186},[34,11941,11942],{"class":190},"toggleLesson",[34,11944,11879],{"class":186},[34,11946,11947],{"class":36,"line":74},[34,11948,11949],{"class":190},"\u002F>\n",[15,11951,11952,11954],{},[31,11953,3466],{}," 负责显示一张课程卡片。父组件只负责准备数据和处理事件。",[19,11956,11958],{"id":11957},"_2-什么时候应该拆组件","2. 什么时候应该拆组件",[15,11960,11961],{},"适合拆：",[105,11963,11964,11967,11979,11982],{},[108,11965,11966],{},"一块 UI 重复出现。",[108,11968,11969,11970,146,11973,146,11976,347],{},"一块 UI 有清晰含义，例如 ",[31,11971,11972],{},"TodoItem",[31,11974,11975],{},"SearchBox",[31,11977,11978],{},"UserMenu",[108,11980,11981],{},"一个文件太长，逻辑混在一起。",[108,11983,11984],{},"你能清楚说出组件的输入和输出。",[15,11986,11987],{},"不急着拆：",[105,11989,11990,11993,11996],{},[108,11991,11992],{},"只出现一次且很短。",[108,11994,11995],{},"拆出去后要传很多零碎 props。",[108,11997,11998],{},"组件职责说不清。",[15,12000,12001],{},"判断标准不是“越拆越高级”，而是“拆完后边界更清楚”。",[19,12003,12005],{"id":12004},"_3-父组件向子组件传数据props","3. 父组件向子组件传数据：Props",[15,12007,12008],{},"父组件：",[24,12010,12012],{"className":923,"code":12011,"language":925,"meta":29,"style":29},"\u003CLessonCard :lesson=\"lesson\" \u002F>\n",[31,12013,12014],{"__ignoreMap":29},[34,12015,12016,12018,12020,12022,12024,12026,12028,12030,12032],{"class":36,"line":37},[34,12017,932],{"class":190},[34,12019,3466],{"class":935},[34,12021,1621],{"class":190},[34,12023,1499],{"class":395},[34,12025,202],{"class":190},[34,12027,1608],{"class":186},[34,12029,1499],{"class":190},[34,12031,1608],{"class":186},[34,12033,3469],{"class":190},[15,12035,12036],{},"子组件：",[24,12038,12040],{"className":923,"code":12039,"language":925,"meta":29,"style":29},"\u003Cscript setup>\ndefineProps({\n lesson: {\n type: Object,\n required: true\n }\n})\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Carticle>\n \u003Ch2>{{ lesson.title }}\u003C\u002Fh2>\n \u003Cp>{{ lesson.minutes }} 分钟\u003C\u002Fp>\n \u003C\u002Farticle>\n\u003C\u002Ftemplate>\n",[31,12041,12042,12052,12059,12064,12069,12077,12081,12085,12093,12097,12105,12113,12125,12137,12145],{"__ignoreMap":29},[34,12043,12044,12046,12048,12050],{"class":36,"line":37},[34,12045,932],{"class":190},[34,12047,1911],{"class":935},[34,12049,1914],{"class":395},[34,12051,939],{"class":190},[34,12053,12054,12057],{"class":36,"line":43},[34,12055,12056],{"class":395},"defineProps",[34,12058,9299],{"class":190},[34,12060,12061],{"class":36,"line":49},[34,12062,12063],{"class":190}," lesson: {\n",[34,12065,12066],{"class":36,"line":56},[34,12067,12068],{"class":190}," type: Object,\n",[34,12070,12071,12074],{"class":36,"line":62},[34,12072,12073],{"class":190}," required: ",[34,12075,12076],{"class":179},"true\n",[34,12078,12079],{"class":36,"line":68},[34,12080,5631],{"class":190},[34,12082,12083],{"class":36,"line":74},[34,12084,6153],{"class":190},[34,12086,12087,12089,12091],{"class":36,"line":1971},[34,12088,1003],{"class":190},[34,12090,1911],{"class":935},[34,12092,939],{"class":190},[34,12094,12095],{"class":36,"line":1976},[34,12096,53],{"emptyLinePlaceholder":52},[34,12098,12099,12101,12103],{"class":36,"line":1985},[34,12100,932],{"class":190},[34,12102,936],{"class":935},[34,12104,939],{"class":190},[34,12106,12107,12109,12111],{"class":36,"line":1990},[34,12108,944],{"class":190},[34,12110,947],{"class":935},[34,12112,939],{"class":190},[34,12114,12115,12117,12119,12121,12123],{"class":36,"line":1999},[34,12116,954],{"class":190},[34,12118,19],{"class":935},[34,12120,959],{"class":190},[34,12122,19],{"class":935},[34,12124,939],{"class":190},[34,12126,12127,12129,12131,12133,12135],{"class":36,"line":2022},[34,12128,954],{"class":190},[34,12130,15],{"class":935},[34,12132,972],{"class":190},[34,12134,15],{"class":935},[34,12136,939],{"class":190},[34,12138,12139,12141,12143],{"class":36,"line":2036},[34,12140,994],{"class":190},[34,12142,947],{"class":935},[34,12144,939],{"class":190},[34,12146,12147,12149,12151],{"class":36,"line":2603},[34,12148,1003],{"class":190},[34,12150,936],{"class":935},[34,12152,939],{"class":190},[15,12154,12155,12157],{},[31,12156,12056],{}," 声明这个组件需要外部传什么数据。",[15,12159,12160],{},"这就像函数参数：",[24,12162,12164],{"className":166,"code":12163,"language":168,"meta":29,"style":29},"function LessonCard(lesson) {\n \u002F\u002F 使用 lesson\n}\n",[31,12165,12166,12179,12184],{"__ignoreMap":29},[34,12167,12168,12170,12173,12175,12177],{"class":36,"line":37},[34,12169,392],{"class":175},[34,12171,12172],{"class":395}," LessonCard",[34,12174,456],{"class":190},[34,12176,1499],{"class":1326},[34,12178,1700],{"class":190},[34,12180,12181],{"class":36,"line":43},[34,12182,12183],{"class":301}," \u002F\u002F 使用 lesson\n",[34,12185,12186],{"class":36,"line":49},[34,12187,77],{"class":190},[15,12189,12190],{},"组件的 props 是组件的输入。",[19,12192,12194],{"id":12193},"_4-为什么-props-应该只读","4. 为什么 props 应该只读",[15,12196,12197],{},"子组件不应该直接修改 props。",[15,12199,7500],{},[24,12201,12203],{"className":923,"code":12202,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nconst props = defineProps({\n lesson: Object\n})\n\nfunction complete() {\n props.lesson.completed = true\n}\n\u003C\u002Fscript>\n",[31,12204,12205,12215,12229,12234,12238,12242,12251,12261,12265],{"__ignoreMap":29},[34,12206,12207,12209,12211,12213],{"class":36,"line":37},[34,12208,932],{"class":190},[34,12210,1911],{"class":935},[34,12212,1914],{"class":395},[34,12214,939],{"class":190},[34,12216,12217,12219,12222,12224,12227],{"class":36,"line":43},[34,12218,176],{"class":175},[34,12220,12221],{"class":179}," props",[34,12223,183],{"class":175},[34,12225,12226],{"class":395}," defineProps",[34,12228,9299],{"class":190},[34,12230,12231],{"class":36,"line":49},[34,12232,12233],{"class":190}," lesson: Object\n",[34,12235,12236],{"class":36,"line":56},[34,12237,6153],{"class":190},[34,12239,12240],{"class":36,"line":62},[34,12241,53],{"emptyLinePlaceholder":52},[34,12243,12244,12246,12249],{"class":36,"line":68},[34,12245,392],{"class":175},[34,12247,12248],{"class":395}," complete",[34,12250,399],{"class":190},[34,12252,12253,12256,12258],{"class":36,"line":74},[34,12254,12255],{"class":190}," props.lesson.completed ",[34,12257,202],{"class":175},[34,12259,12260],{"class":179}," true\n",[34,12262,12263],{"class":36,"line":1971},[34,12264,77],{"class":190},[34,12266,12267,12269,12271],{"class":36,"line":1976},[34,12268,1003],{"class":190},[34,12270,1911],{"class":935},[34,12272,939],{"class":190},[15,12274,12275],{},"为什么？",[15,12277,12278],{},"因为数据是父组件传下来的。如果子组件直接改，父组件会很难追踪数据到底是谁改的。",[15,12280,12281],{},"更推荐的模式：",[24,12283,12286],{"className":12284,"code":12285,"language":2638,"meta":29},[2636],"父组件拥有数据\n ↓ props\n子组件显示数据\n ↓ emit\n父组件收到事件后修改数据\n",[31,12287,12285],{"__ignoreMap":29},[15,12289,12290],{},"这叫单向数据流。",[19,12292,12294],{"id":12293},"_5-子组件通知父组件emit","5. 子组件通知父组件：emit",[15,12296,12297],{},"子组件不能直接决定父组件的数据怎么改，但可以通知父组件“用户做了什么”。",[15,12299,12036],{},[24,12301,12303],{"className":923,"code":12302,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nconst props = defineProps({\n lesson: {\n type: Object,\n required: true\n }\n})\n\nconst emit = defineEmits(['toggle'])\n\nfunction handleClick() {\n emit('toggle', props.lesson.id)\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cbutton @click=\"handleClick\">切换完成\u003C\u002Fbutton>\n\u003C\u002Ftemplate>\n",[31,12304,12305,12315,12327,12331,12335,12341,12345,12349,12353,12373,12377,12386,12398,12402,12410,12414,12422,12442],{"__ignoreMap":29},[34,12306,12307,12309,12311,12313],{"class":36,"line":37},[34,12308,932],{"class":190},[34,12310,1911],{"class":935},[34,12312,1914],{"class":395},[34,12314,939],{"class":190},[34,12316,12317,12319,12321,12323,12325],{"class":36,"line":43},[34,12318,176],{"class":175},[34,12320,12221],{"class":179},[34,12322,183],{"class":175},[34,12324,12226],{"class":395},[34,12326,9299],{"class":190},[34,12328,12329],{"class":36,"line":49},[34,12330,12063],{"class":190},[34,12332,12333],{"class":36,"line":56},[34,12334,12068],{"class":190},[34,12336,12337,12339],{"class":36,"line":62},[34,12338,12073],{"class":190},[34,12340,12076],{"class":179},[34,12342,12343],{"class":36,"line":68},[34,12344,5631],{"class":190},[34,12346,12347],{"class":36,"line":74},[34,12348,6153],{"class":190},[34,12350,12351],{"class":36,"line":1971},[34,12352,53],{"emptyLinePlaceholder":52},[34,12354,12355,12357,12360,12362,12365,12368,12371],{"class":36,"line":1976},[34,12356,176],{"class":175},[34,12358,12359],{"class":179}," emit",[34,12361,183],{"class":175},[34,12363,12364],{"class":395}," defineEmits",[34,12366,12367],{"class":190},"([",[34,12369,12370],{"class":186},"'toggle'",[34,12372,6945],{"class":190},[34,12374,12375],{"class":36,"line":1985},[34,12376,53],{"emptyLinePlaceholder":52},[34,12378,12379,12381,12384],{"class":36,"line":1990},[34,12380,392],{"class":175},[34,12382,12383],{"class":395}," handleClick",[34,12385,399],{"class":190},[34,12387,12388,12391,12393,12395],{"class":36,"line":1999},[34,12389,12390],{"class":395}," emit",[34,12392,456],{"class":190},[34,12394,12370],{"class":186},[34,12396,12397],{"class":190},", props.lesson.id)\n",[34,12399,12400],{"class":36,"line":2022},[34,12401,77],{"class":190},[34,12403,12404,12406,12408],{"class":36,"line":2036},[34,12405,1003],{"class":190},[34,12407,1911],{"class":935},[34,12409,939],{"class":190},[34,12411,12412],{"class":36,"line":2603},[34,12413,53],{"emptyLinePlaceholder":52},[34,12415,12416,12418,12420],{"class":36,"line":5537},[34,12417,932],{"class":190},[34,12419,936],{"class":935},[34,12421,939],{"class":190},[34,12423,12424,12426,12428,12430,12432,12435,12438,12440],{"class":36,"line":5543},[34,12425,944],{"class":190},[34,12427,2004],{"class":935},[34,12429,2007],{"class":395},[34,12431,202],{"class":190},[34,12433,12434],{"class":186},"\"handleClick\"",[34,12436,12437],{"class":190},">切换完成\u003C\u002F",[34,12439,2004],{"class":935},[34,12441,939],{"class":190},[34,12443,12444,12446,12448],{"class":36,"line":5548},[34,12445,1003],{"class":190},[34,12447,936],{"class":935},[34,12449,939],{"class":190},[15,12451,12008],{},[24,12453,12455],{"className":923,"code":12454,"language":925,"meta":29,"style":29},"\u003CLessonCard\n :lesson=\"lesson\"\n @toggle=\"toggleLesson\"\n\u002F>\n",[31,12456,12457,12463,12477,12491],{"__ignoreMap":29},[34,12458,12459,12461],{"class":36,"line":37},[34,12460,932],{"class":190},[34,12462,11861],{"class":935},[34,12464,12465,12467,12469,12471,12473,12475],{"class":36,"line":43},[34,12466,11884],{"class":190},[34,12468,1499],{"class":395},[34,12470,202],{"class":190},[34,12472,1608],{"class":186},[34,12474,1499],{"class":190},[34,12476,11879],{"class":186},[34,12478,12479,12481,12483,12485,12487,12489],{"class":36,"line":49},[34,12480,11932],{"class":190},[34,12482,11935],{"class":395},[34,12484,202],{"class":190},[34,12486,1608],{"class":186},[34,12488,11942],{"class":190},[34,12490,11879],{"class":186},[34,12492,12493],{"class":36,"line":56},[34,12494,11949],{"class":190},[15,12496,12497],{},"父组件函数：",[24,12499,12501],{"className":166,"code":12500,"language":168,"meta":29,"style":29},"function toggleLesson(id) {\n const lesson = lessons.value.find((item) => item.id === id)\n\n if (lesson) {\n lesson.completed = !lesson.completed\n }\n}\n",[31,12502,12503,12516,12543,12547,12554,12566,12570],{"__ignoreMap":29},[34,12504,12505,12507,12510,12512,12514],{"class":36,"line":37},[34,12506,392],{"class":175},[34,12508,12509],{"class":395}," toggleLesson",[34,12511,456],{"class":190},[34,12513,1805],{"class":1326},[34,12515,1700],{"class":190},[34,12517,12518,12520,12522,12524,12527,12529,12531,12533,12535,12537,12539,12541],{"class":36,"line":43},[34,12519,5298],{"class":175},[34,12521,824],{"class":179},[34,12523,183],{"class":175},[34,12525,12526],{"class":190}," lessons.value.",[34,12528,1390],{"class":395},[34,12530,1323],{"class":190},[34,12532,6979],{"class":1326},[34,12534,1330],{"class":190},[34,12536,1333],{"class":175},[34,12538,6986],{"class":190},[34,12540,1423],{"class":175},[34,12542,6991],{"class":190},[34,12544,12545],{"class":36,"line":49},[34,12546,53],{"emptyLinePlaceholder":52},[34,12548,12549,12551],{"class":36,"line":56},[34,12550,7000],{"class":175},[34,12552,12553],{"class":190}," (lesson) {\n",[34,12555,12556,12559,12561,12563],{"class":36,"line":62},[34,12557,12558],{"class":190}," lesson.completed ",[34,12560,202],{"class":175},[34,12562,1336],{"class":175},[34,12564,12565],{"class":190},"lesson.completed\n",[34,12567,12568],{"class":36,"line":68},[34,12569,5631],{"class":190},[34,12571,12572],{"class":36,"line":74},[34,12573,77],{"class":190},[15,12575,12576],{},"流程：",[2230,12578,12579,12585,12588,12591,12597,12606,12609],{},[108,12580,12581,12582,12584],{},"父组件把 ",[31,12583,1499],{}," 传给子组件。",[108,12586,12587],{},"子组件显示课程。",[108,12589,12590],{},"用户点击子组件按钮。",[108,12592,12593,12594,347],{},"子组件 ",[31,12595,12596],{},"emit('toggle', id)",[108,12598,12599,12600,12603,12604,347],{},"父组件监听 ",[31,12601,12602],{},"@toggle","，调用 ",[31,12605,11942],{},[108,12607,12608],{},"父组件修改自己的数据。",[108,12610,12611],{},"数据变化后，新的 props 再传给子组件。",[19,12613,12615],{"id":12614},"_6-事件名怎么设计","6. 事件名怎么设计",[15,12617,12618],{},"事件名应该表达“发生了什么”，不要太模糊。",[15,12620,12621],{},"不清晰：",[24,12623,12625],{"className":166,"code":12624,"language":168,"meta":29,"style":29},"emit('change')\nemit('click')\nemit('update')\n",[31,12626,12627,12639,12649],{"__ignoreMap":29},[34,12628,12629,12632,12634,12637],{"class":36,"line":37},[34,12630,12631],{"class":395},"emit",[34,12633,456],{"class":190},[34,12635,12636],{"class":186},"'change'",[34,12638,6111],{"class":190},[34,12640,12641,12643,12645,12647],{"class":36,"line":43},[34,12642,12631],{"class":395},[34,12644,456],{"class":190},[34,12646,6128],{"class":186},[34,12648,6111],{"class":190},[34,12650,12651,12653,12655,12658],{"class":36,"line":49},[34,12652,12631],{"class":395},[34,12654,456],{"class":190},[34,12656,12657],{"class":186},"'update'",[34,12659,6111],{"class":190},[15,12661,12662],{},"更清晰：",[24,12664,12666],{"className":166,"code":12665,"language":168,"meta":29,"style":29},"emit('toggle', id)\nemit('remove', id)\nemit('submit', form)\nemit('select', option)\n",[31,12667,12668,12679,12690,12702],{"__ignoreMap":29},[34,12669,12670,12672,12674,12676],{"class":36,"line":37},[34,12671,12631],{"class":395},[34,12673,456],{"class":190},[34,12675,12370],{"class":186},[34,12677,12678],{"class":190},", id)\n",[34,12680,12681,12683,12685,12688],{"class":36,"line":43},[34,12682,12631],{"class":395},[34,12684,456],{"class":190},[34,12686,12687],{"class":186},"'remove'",[34,12689,12678],{"class":190},[34,12691,12692,12694,12696,12699],{"class":36,"line":49},[34,12693,12631],{"class":395},[34,12695,456],{"class":190},[34,12697,12698],{"class":186},"'submit'",[34,12700,12701],{"class":190},", form)\n",[34,12703,12704,12706,12708,12711],{"class":36,"line":56},[34,12705,12631],{"class":395},[34,12707,456],{"class":190},[34,12709,12710],{"class":186},"'select'",[34,12712,12713],{"class":190},", option)\n",[15,12715,12716],{},"组件事件不是 DOM 事件，它是你定义的业务事件。",[19,12718,12720],{"id":12719},"_7-完整例子todoitem","7. 完整例子：TodoItem",[15,12722,12593,12723,2664],{},[31,12724,12725],{},"TodoItem.vue",[24,12727,12729],{"className":923,"code":12728,"language":925,"meta":29,"style":29},"\u003Cscript setup>\ndefineProps({\n todo: {\n type: Object,\n required: true\n }\n})\n\nconst emit = defineEmits(['toggle', 'remove'])\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cli>\n \u003Clabel>\n \u003Cinput\n type=\"checkbox\"\n :checked=\"todo.done\"\n @change=\"emit('toggle', todo.id)\"\n >\n \u003Cspan>{{ todo.title }}\u003C\u002Fspan>\n \u003C\u002Flabel>\n\n \u003Cbutton @click=\"emit('remove', todo.id)\">删除\u003C\u002Fbutton>\n \u003C\u002Fli>\n\u003C\u002Ftemplate>\n",[31,12730,12731,12741,12747,12752,12756,12762,12766,12770,12774,12794,12802,12806,12814,12822,12830,12837,12847,12857,12867,12872,12885,12893,12897,12917,12925],{"__ignoreMap":29},[34,12732,12733,12735,12737,12739],{"class":36,"line":37},[34,12734,932],{"class":190},[34,12736,1911],{"class":935},[34,12738,1914],{"class":395},[34,12740,939],{"class":190},[34,12742,12743,12745],{"class":36,"line":43},[34,12744,12056],{"class":395},[34,12746,9299],{"class":190},[34,12748,12749],{"class":36,"line":49},[34,12750,12751],{"class":190}," todo: {\n",[34,12753,12754],{"class":36,"line":56},[34,12755,12068],{"class":190},[34,12757,12758,12760],{"class":36,"line":62},[34,12759,12073],{"class":190},[34,12761,12076],{"class":179},[34,12763,12764],{"class":36,"line":68},[34,12765,5631],{"class":190},[34,12767,12768],{"class":36,"line":74},[34,12769,6153],{"class":190},[34,12771,12772],{"class":36,"line":1971},[34,12773,53],{"emptyLinePlaceholder":52},[34,12775,12776,12778,12780,12782,12784,12786,12788,12790,12792],{"class":36,"line":1976},[34,12777,176],{"class":175},[34,12779,12359],{"class":179},[34,12781,183],{"class":175},[34,12783,12364],{"class":395},[34,12785,12367],{"class":190},[34,12787,12370],{"class":186},[34,12789,616],{"class":190},[34,12791,12687],{"class":186},[34,12793,6945],{"class":190},[34,12795,12796,12798,12800],{"class":36,"line":1985},[34,12797,1003],{"class":190},[34,12799,1911],{"class":935},[34,12801,939],{"class":190},[34,12803,12804],{"class":36,"line":1990},[34,12805,53],{"emptyLinePlaceholder":52},[34,12807,12808,12810,12812],{"class":36,"line":1999},[34,12809,932],{"class":190},[34,12811,936],{"class":935},[34,12813,939],{"class":190},[34,12815,12816,12818,12820],{"class":36,"line":2022},[34,12817,944],{"class":190},[34,12819,108],{"class":935},[34,12821,939],{"class":190},[34,12823,12824,12826,12828],{"class":36,"line":2036},[34,12825,954],{"class":190},[34,12827,11490],{"class":935},[34,12829,939],{"class":190},[34,12831,12832,12834],{"class":36,"line":2603},[34,12833,7078],{"class":190},[34,12835,12836],{"class":935},"input\n",[34,12838,12839,12842,12844],{"class":36,"line":5537},[34,12840,12841],{"class":395}," type",[34,12843,202],{"class":190},[34,12845,12846],{"class":186},"\"checkbox\"\n",[34,12848,12849,12852,12854],{"class":36,"line":5543},[34,12850,12851],{"class":395}," :checked",[34,12853,202],{"class":190},[34,12855,12856],{"class":186},"\"todo.done\"\n",[34,12858,12859,12862,12864],{"class":36,"line":5548},[34,12860,12861],{"class":395}," @change",[34,12863,202],{"class":190},[34,12865,12866],{"class":186},"\"emit('toggle', todo.id)\"\n",[34,12868,12869],{"class":36,"line":5564},[34,12870,12871],{"class":190}," >\n",[34,12873,12874,12876,12878,12881,12883],{"class":36,"line":5576},[34,12875,7078],{"class":190},[34,12877,34],{"class":935},[34,12879,12880],{"class":190},">{{ todo.title }}\u003C\u002F",[34,12882,34],{"class":935},[34,12884,939],{"class":190},[34,12886,12887,12889,12891],{"class":36,"line":5606},[34,12888,7112],{"class":190},[34,12890,11490],{"class":935},[34,12892,939],{"class":190},[34,12894,12895],{"class":36,"line":5616},[34,12896,53],{"emptyLinePlaceholder":52},[34,12898,12899,12901,12903,12905,12907,12910,12913,12915],{"class":36,"line":5628},[34,12900,954],{"class":190},[34,12902,2004],{"class":935},[34,12904,2007],{"class":395},[34,12906,202],{"class":190},[34,12908,12909],{"class":186},"\"emit('remove', todo.id)\"",[34,12911,12912],{"class":190},">删除\u003C\u002F",[34,12914,2004],{"class":935},[34,12916,939],{"class":190},[34,12918,12919,12921,12923],{"class":36,"line":5634},[34,12920,994],{"class":190},[34,12922,108],{"class":935},[34,12924,939],{"class":190},[34,12926,12927,12929,12931],{"class":36,"line":6420},[34,12928,1003],{"class":190},[34,12930,936],{"class":935},[34,12932,939],{"class":190},[15,12934,12008],{},[24,12936,12938],{"className":923,"code":12937,"language":925,"meta":29,"style":29},"\u003CTodoItem\n v-for=\"todo in todos\"\n :key=\"todo.id\"\n :todo=\"todo\"\n @toggle=\"toggleTodo\"\n @remove=\"removeTodo\"\n\u002F>\n",[31,12939,12940,12947,12965,12980,12995,13010,13025],{"__ignoreMap":29},[34,12941,12942,12944],{"class":36,"line":37},[34,12943,932],{"class":190},[34,12945,12946],{"class":935},"TodoItem\n",[34,12948,12949,12951,12953,12955,12958,12960,12963],{"class":36,"line":43},[34,12950,11866],{"class":175},[34,12952,202],{"class":190},[34,12954,1608],{"class":186},[34,12956,12957],{"class":190},"todo ",[34,12959,1614],{"class":175},[34,12961,12962],{"class":190}," todos",[34,12964,11879],{"class":186},[34,12966,12967,12969,12971,12973,12975,12978],{"class":36,"line":49},[34,12968,11884],{"class":190},[34,12970,1624],{"class":395},[34,12972,202],{"class":190},[34,12974,1608],{"class":186},[34,12976,12977],{"class":190},"todo.id",[34,12979,11879],{"class":186},[34,12981,12982,12984,12987,12989,12991,12993],{"class":36,"line":56},[34,12983,11884],{"class":190},[34,12985,12986],{"class":395},"todo",[34,12988,202],{"class":190},[34,12990,1608],{"class":186},[34,12992,12986],{"class":190},[34,12994,11879],{"class":186},[34,12996,12997,12999,13001,13003,13005,13008],{"class":36,"line":62},[34,12998,11932],{"class":190},[34,13000,11935],{"class":395},[34,13002,202],{"class":190},[34,13004,1608],{"class":186},[34,13006,13007],{"class":190},"toggleTodo",[34,13009,11879],{"class":186},[34,13011,13012,13014,13016,13018,13020,13023],{"class":36,"line":68},[34,13013,11932],{"class":190},[34,13015,8035],{"class":395},[34,13017,202],{"class":190},[34,13019,1608],{"class":186},[34,13021,13022],{"class":190},"removeTodo",[34,13024,11879],{"class":186},[34,13026,13027],{"class":36,"line":74},[34,13028,11949],{"class":190},[15,13030,13031],{},"父组件逻辑：",[24,13033,13035],{"className":166,"code":13034,"language":168,"meta":29,"style":29},"function toggleTodo(id) {\n todos.value = todos.value.map((todo) => {\n return todo.id === id ? { ...todo, done: !todo.done } : todo\n })\n}\n\nfunction removeTodo(id) {\n todos.value = todos.value.filter((todo) => todo.id !== id)\n}\n",[31,13036,13037,13050,13072,13103,13107,13111,13115,13128,13153],{"__ignoreMap":29},[34,13038,13039,13041,13044,13046,13048],{"class":36,"line":37},[34,13040,392],{"class":175},[34,13042,13043],{"class":395}," toggleTodo",[34,13045,456],{"class":190},[34,13047,1805],{"class":1326},[34,13049,1700],{"class":190},[34,13051,13052,13055,13057,13060,13062,13064,13066,13068,13070],{"class":36,"line":43},[34,13053,13054],{"class":190}," todos.value ",[34,13056,202],{"class":175},[34,13058,13059],{"class":190}," todos.value.",[34,13061,1355],{"class":395},[34,13063,1323],{"class":190},[34,13065,12986],{"class":1326},[34,13067,1330],{"class":190},[34,13069,1333],{"class":175},[34,13071,674],{"class":190},[34,13073,13074,13076,13079,13081,13084,13086,13088,13090,13093,13095,13098,13100],{"class":36,"line":49},[34,13075,8601],{"class":175},[34,13077,13078],{"class":190}," todo.id ",[34,13080,1423],{"class":175},[34,13082,13083],{"class":190}," id ",[34,13085,6701],{"class":175},[34,13087,1164],{"class":190},[34,13089,1167],{"class":175},[34,13091,13092],{"class":190},"todo, done: ",[34,13094,5504],{"class":175},[34,13096,13097],{"class":190},"todo.done } ",[34,13099,5598],{"class":175},[34,13101,13102],{"class":190}," todo\n",[34,13104,13105],{"class":36,"line":56},[34,13106,6379],{"class":190},[34,13108,13109],{"class":36,"line":62},[34,13110,77],{"class":190},[34,13112,13113],{"class":36,"line":68},[34,13114,53],{"emptyLinePlaceholder":52},[34,13116,13117,13119,13122,13124,13126],{"class":36,"line":74},[34,13118,392],{"class":175},[34,13120,13121],{"class":395}," removeTodo",[34,13123,456],{"class":190},[34,13125,1805],{"class":1326},[34,13127,1700],{"class":190},[34,13129,13130,13132,13134,13136,13138,13140,13142,13144,13146,13148,13151],{"class":36,"line":1971},[34,13131,13054],{"class":190},[34,13133,202],{"class":175},[34,13135,13059],{"class":190},[34,13137,1300],{"class":395},[34,13139,1323],{"class":190},[34,13141,12986],{"class":1326},[34,13143,1330],{"class":190},[34,13145,1333],{"class":175},[34,13147,13078],{"class":190},[34,13149,13150],{"class":175},"!==",[34,13152,6991],{"class":190},[34,13154,13155],{"class":36,"line":1976},[34,13156,77],{"class":190},[15,13158,13159],{},"注意：子组件只负责“告诉父组件用户点击了哪个任务”，真正修改数组的是父组件。",[19,13161,13163],{"id":13162},"_8-插槽让父组件决定子组件内部内容","8. 插槽：让父组件决定子组件内部内容",[15,13165,13166],{},"有些组件不应该关心具体内容，只负责外壳。",[15,13168,13169,13170,2664],{},"例如 ",[31,13171,13172],{},"BasePanel.vue",[24,13174,13176],{"className":923,"code":13175,"language":925,"meta":29,"style":29},"\u003Ctemplate>\n \u003Csection class=\"panel\">\n \u003Cslot \u002F>\n \u003C\u002Fsection>\n\u003C\u002Ftemplate>\n",[31,13177,13178,13186,13202,13215,13223],{"__ignoreMap":29},[34,13179,13180,13182,13184],{"class":36,"line":37},[34,13181,932],{"class":190},[34,13183,936],{"class":935},[34,13185,939],{"class":190},[34,13187,13188,13190,13193,13195,13197,13200],{"class":36,"line":43},[34,13189,944],{"class":190},[34,13191,13192],{"class":935},"section",[34,13194,7588],{"class":395},[34,13196,202],{"class":190},[34,13198,13199],{"class":186},"\"panel\"",[34,13201,939],{"class":190},[34,13203,13204,13206,13209,13213],{"class":36,"line":49},[34,13205,954],{"class":190},[34,13207,13208],{"class":935},"slot",[34,13210,13212],{"class":13211},"s7hpK"," \u002F",[34,13214,939],{"class":190},[34,13216,13217,13219,13221],{"class":36,"line":56},[34,13218,994],{"class":190},[34,13220,13192],{"class":935},[34,13222,939],{"class":190},[34,13224,13225,13227,13229],{"class":36,"line":62},[34,13226,1003],{"class":190},[34,13228,936],{"class":935},[34,13230,939],{"class":190},[15,13232,13233],{},"使用：",[24,13235,13237],{"className":923,"code":13236,"language":925,"meta":29,"style":29},"\u003CBasePanel>\n \u003Ch2>学习进度\u003C\u002Fh2>\n \u003Cp>你已经完成 3 章。\u003C\u002Fp>\n\u003C\u002FBasePanel>\n",[31,13238,13239,13248,13253,13258],{"__ignoreMap":29},[34,13240,13241,13243,13246],{"class":36,"line":37},[34,13242,932],{"class":190},[34,13244,13245],{"class":935},"BasePanel",[34,13247,939],{"class":190},[34,13249,13250],{"class":36,"line":43},[34,13251,13252],{"class":190}," \u003Ch2>学习进度\u003C\u002Fh2>\n",[34,13254,13255],{"class":36,"line":49},[34,13256,13257],{"class":190}," \u003Cp>你已经完成 3 章。\u003C\u002Fp>\n",[34,13259,13260,13262,13264],{"class":36,"line":56},[34,13261,1003],{"class":190},[34,13263,13245],{"class":935},[34,13265,939],{"class":190},[15,13267,13268,13269,13272],{},"渲染时，",[31,13270,13271],{},"\u003Cslot \u002F>"," 会被父组件传进来的内容替换。",[15,13274,13275],{},"这就像：",[24,13277,13280],{"className":13278,"code":13279,"language":2638,"meta":29},[2636],"BasePanel 提供盒子\n父组件提供盒子里面放什么\n",[31,13281,13279],{"__ignoreMap":29},[19,13283,13285],{"id":13284},"_9-具名插槽","9. 具名插槽",[15,13287,13288],{},"如果一个组件有多个可填区域，用具名插槽。",[24,13290,13292],{"className":923,"code":13291,"language":925,"meta":29,"style":29},"\u003C!-- CourseCard.vue -->\n\u003Ctemplate>\n \u003Carticle class=\"card\">\n \u003Cheader>\n \u003Cslot name=\"title\" \u002F>\n \u003C\u002Fheader>\n\n \u003Cmain>\n \u003Cslot \u002F>\n \u003C\u002Fmain>\n\n \u003Cfooter>\n \u003Cslot name=\"actions\" \u002F>\n \u003C\u002Ffooter>\n \u003C\u002Farticle>\n\u003C\u002Ftemplate>\n",[31,13293,13294,13299,13307,13322,13331,13348,13356,13360,13368,13378,13386,13390,13399,13416,13424,13432],{"__ignoreMap":29},[34,13295,13296],{"class":36,"line":37},[34,13297,13298],{"class":301},"\u003C!-- CourseCard.vue -->\n",[34,13300,13301,13303,13305],{"class":36,"line":43},[34,13302,932],{"class":190},[34,13304,936],{"class":935},[34,13306,939],{"class":190},[34,13308,13309,13311,13313,13315,13317,13320],{"class":36,"line":49},[34,13310,944],{"class":190},[34,13312,947],{"class":935},[34,13314,7588],{"class":395},[34,13316,202],{"class":190},[34,13318,13319],{"class":186},"\"card\"",[34,13321,939],{"class":190},[34,13323,13324,13326,13329],{"class":36,"line":56},[34,13325,954],{"class":190},[34,13327,13328],{"class":935},"header",[34,13330,939],{"class":190},[34,13332,13333,13335,13337,13339,13341,13344,13346],{"class":36,"line":62},[34,13334,7078],{"class":190},[34,13336,13208],{"class":935},[34,13338,2333],{"class":395},[34,13340,202],{"class":190},[34,13342,13343],{"class":186},"\"title\"",[34,13345,13212],{"class":13211},[34,13347,939],{"class":190},[34,13349,13350,13352,13354],{"class":36,"line":68},[34,13351,7112],{"class":190},[34,13353,13328],{"class":935},[34,13355,939],{"class":190},[34,13357,13358],{"class":36,"line":74},[34,13359,53],{"emptyLinePlaceholder":52},[34,13361,13362,13364,13366],{"class":36,"line":1971},[34,13363,954],{"class":190},[34,13365,6027],{"class":935},[34,13367,939],{"class":190},[34,13369,13370,13372,13374,13376],{"class":36,"line":1976},[34,13371,7078],{"class":190},[34,13373,13208],{"class":935},[34,13375,13212],{"class":13211},[34,13377,939],{"class":190},[34,13379,13380,13382,13384],{"class":36,"line":1985},[34,13381,7112],{"class":190},[34,13383,6027],{"class":935},[34,13385,939],{"class":190},[34,13387,13388],{"class":36,"line":1990},[34,13389,53],{"emptyLinePlaceholder":52},[34,13391,13392,13394,13397],{"class":36,"line":1999},[34,13393,954],{"class":190},[34,13395,13396],{"class":935},"footer",[34,13398,939],{"class":190},[34,13400,13401,13403,13405,13407,13409,13412,13414],{"class":36,"line":2022},[34,13402,7078],{"class":190},[34,13404,13208],{"class":935},[34,13406,2333],{"class":395},[34,13408,202],{"class":190},[34,13410,13411],{"class":186},"\"actions\"",[34,13413,13212],{"class":13211},[34,13415,939],{"class":190},[34,13417,13418,13420,13422],{"class":36,"line":2036},[34,13419,7112],{"class":190},[34,13421,13396],{"class":935},[34,13423,939],{"class":190},[34,13425,13426,13428,13430],{"class":36,"line":2603},[34,13427,994],{"class":190},[34,13429,947],{"class":935},[34,13431,939],{"class":190},[34,13433,13434,13436,13438],{"class":36,"line":5537},[34,13435,1003],{"class":190},[34,13437,936],{"class":935},[34,13439,939],{"class":190},[15,13441,13233],{},[24,13443,13445],{"className":923,"code":13444,"language":925,"meta":29,"style":29},"\u003CCourseCard>\n \u003Ctemplate #title>\n \u003Ch2>Vue Router\u003C\u002Fh2>\n \u003C\u002Ftemplate>\n\n \u003Cp>学习页面跳转和动态路由。\u003C\u002Fp>\n\n \u003Ctemplate #actions>\n \u003Cbutton>开始学习\u003C\u002Fbutton>\n \u003C\u002Ftemplate>\n\u003C\u002FCourseCard>\n",[31,13446,13447,13456,13461,13466,13471,13475,13480,13484,13489,13494,13498],{"__ignoreMap":29},[34,13448,13449,13451,13454],{"class":36,"line":37},[34,13450,932],{"class":190},[34,13452,13453],{"class":935},"CourseCard",[34,13455,939],{"class":190},[34,13457,13458],{"class":36,"line":43},[34,13459,13460],{"class":190}," \u003Ctemplate #title>\n",[34,13462,13463],{"class":36,"line":49},[34,13464,13465],{"class":190}," \u003Ch2>Vue Router\u003C\u002Fh2>\n",[34,13467,13468],{"class":36,"line":56},[34,13469,13470],{"class":190}," \u003C\u002Ftemplate>\n",[34,13472,13473],{"class":36,"line":62},[34,13474,53],{"emptyLinePlaceholder":52},[34,13476,13477],{"class":36,"line":68},[34,13478,13479],{"class":190}," \u003Cp>学习页面跳转和动态路由。\u003C\u002Fp>\n",[34,13481,13482],{"class":36,"line":74},[34,13483,53],{"emptyLinePlaceholder":52},[34,13485,13486],{"class":36,"line":1971},[34,13487,13488],{"class":190}," \u003Ctemplate #actions>\n",[34,13490,13491],{"class":36,"line":1976},[34,13492,13493],{"class":190}," \u003Cbutton>开始学习\u003C\u002Fbutton>\n",[34,13495,13496],{"class":36,"line":1985},[34,13497,13470],{"class":190},[34,13499,13500,13502,13504],{"class":36,"line":1990},[34,13501,1003],{"class":190},[34,13503,13453],{"class":935},[34,13505,939],{"class":190},[15,13507,13508,11008,13511,13514],{},[31,13509,13510],{},"#title",[31,13512,13513],{},"v-slot:title"," 的简写。",[19,13516,13518,13519],{"id":13517},"_10-组件-v-model","10. 组件 ",[31,13520,7473],{},[15,13522,13523],{},"普通输入框：",[24,13525,13527],{"className":923,"code":13526,"language":925,"meta":29,"style":29},"\u003Cinput v-model=\"keyword\">\n",[31,13528,13529],{"__ignoreMap":29},[34,13530,13531,13533,13535,13537,13539,13541,13543,13545],{"class":36,"line":37},[34,13532,932],{"class":190},[34,13534,7429],{"class":935},[34,13536,7432],{"class":395},[34,13538,202],{"class":190},[34,13540,1608],{"class":186},[34,13542,9414],{"class":190},[34,13544,1608],{"class":186},[34,13546,939],{"class":190},[15,13548,13549],{},"如果你自己写一个搜索框组件，也希望这样用：",[24,13551,13553],{"className":923,"code":13552,"language":925,"meta":29,"style":29},"\u003CSearchBox v-model=\"keyword\" \u002F>\n",[31,13554,13555],{"__ignoreMap":29},[34,13556,13557,13559,13561,13563,13565,13567,13569,13571],{"class":36,"line":37},[34,13558,932],{"class":190},[34,13560,11975],{"class":935},[34,13562,7432],{"class":395},[34,13564,202],{"class":190},[34,13566,1608],{"class":186},[34,13568,9414],{"class":190},[34,13570,1608],{"class":186},[34,13572,3469],{"class":190},[15,13574,13575,13576,2664],{},"可以在子组件中使用 ",[31,13577,13578],{},"defineModel",[24,13580,13582],{"className":923,"code":13581,"language":925,"meta":29,"style":29},"\u003C!-- SearchBox.vue -->\n\u003Cscript setup>\nconst model = defineModel()\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cinput v-model=\"model\" placeholder=\"搜索课程\">\n\u003C\u002Ftemplate>\n",[31,13583,13584,13589,13599,13613,13621,13625,13633,13655],{"__ignoreMap":29},[34,13585,13586],{"class":36,"line":37},[34,13587,13588],{"class":301},"\u003C!-- SearchBox.vue -->\n",[34,13590,13591,13593,13595,13597],{"class":36,"line":43},[34,13592,932],{"class":190},[34,13594,1911],{"class":935},[34,13596,1914],{"class":395},[34,13598,939],{"class":190},[34,13600,13601,13603,13606,13608,13611],{"class":36,"line":49},[34,13602,176],{"class":175},[34,13604,13605],{"class":179}," model",[34,13607,183],{"class":175},[34,13609,13610],{"class":395}," defineModel",[34,13612,6374],{"class":190},[34,13614,13615,13617,13619],{"class":36,"line":56},[34,13616,1003],{"class":190},[34,13618,1911],{"class":935},[34,13620,939],{"class":190},[34,13622,13623],{"class":36,"line":62},[34,13624,53],{"emptyLinePlaceholder":52},[34,13626,13627,13629,13631],{"class":36,"line":68},[34,13628,932],{"class":190},[34,13630,936],{"class":935},[34,13632,939],{"class":190},[34,13634,13635,13637,13639,13641,13643,13646,13648,13650,13653],{"class":36,"line":74},[34,13636,944],{"class":190},[34,13638,7429],{"class":935},[34,13640,7432],{"class":395},[34,13642,202],{"class":190},[34,13644,13645],{"class":186},"\"model\"",[34,13647,11392],{"class":395},[34,13649,202],{"class":190},[34,13651,13652],{"class":186},"\"搜索课程\"",[34,13654,939],{"class":190},[34,13656,13657,13659,13661],{"class":36,"line":1971},[34,13658,1003],{"class":190},[34,13660,936],{"class":935},[34,13662,939],{"class":190},[15,13664,12008],{},[24,13666,13667],{"className":923,"code":13552,"language":925,"meta":29,"style":29},[31,13668,13669],{"__ignoreMap":29},[34,13670,13671,13673,13675,13677,13679,13681,13683,13685],{"class":36,"line":37},[34,13672,932],{"class":190},[34,13674,11975],{"class":935},[34,13676,7432],{"class":395},[34,13678,202],{"class":190},[34,13680,1608],{"class":186},[34,13682,9414],{"class":190},[34,13684,1608],{"class":186},[34,13686,3469],{"class":190},[15,13688,13689],{},"含义：",[105,13691,13692,13697],{},[108,13693,13694,13695,12584],{},"父组件的 ",[31,13696,9414],{},[108,13698,13699,13700,347],{},"子组件输入变化时，通知父组件更新 ",[31,13701,9414],{},[15,13703,13704],{},"本质上仍然是 props + emit，只是 Vue 提供了简洁语法。",[19,13706,13708],{"id":13707},"_11-provide-inject","11. Provide \u002F Inject",[15,13710,13711],{},"如果数据要跨很多层组件传递，层层 props 会很麻烦。",[24,13713,13716],{"className":13714,"code":13715,"language":2638,"meta":29},[2636],"App\n Layout\n Sidebar\n UserMenu\n",[31,13717,13715],{"__ignoreMap":29},[15,13719,2234,13720,13723,13724,13726],{},[31,13721,13722],{},"App"," 里的主题要给 ",[31,13725,11978],{}," 用，可以：",[24,13728,13730],{"className":166,"code":13729,"language":168,"meta":29,"style":29},"\u002F\u002F 上层组件\nimport { provide } from 'vue'\n\nprovide('theme', 'dark')\n",[31,13731,13732,13737,13748,13752],{"__ignoreMap":29},[34,13733,13734],{"class":36,"line":37},[34,13735,13736],{"class":301},"\u002F\u002F 上层组件\n",[34,13738,13739,13741,13744,13746],{"class":36,"line":43},[34,13740,426],{"class":175},[34,13742,13743],{"class":190}," { provide } ",[34,13745,432],{"class":175},[34,13747,6891],{"class":186},[34,13749,13750],{"class":36,"line":49},[34,13751,53],{"emptyLinePlaceholder":52},[34,13753,13754,13757,13759,13762,13764,13767],{"class":36,"line":56},[34,13755,13756],{"class":395},"provide",[34,13758,456],{"class":190},[34,13760,13761],{"class":186},"'theme'",[34,13763,616],{"class":190},[34,13765,13766],{"class":186},"'dark'",[34,13768,6111],{"class":190},[24,13770,13772],{"className":166,"code":13771,"language":168,"meta":29,"style":29},"\u002F\u002F 下层组件\nimport { inject } from 'vue'\n\nconst theme = inject('theme')\n",[31,13773,13774,13779,13790,13794],{"__ignoreMap":29},[34,13775,13776],{"class":36,"line":37},[34,13777,13778],{"class":301},"\u002F\u002F 下层组件\n",[34,13780,13781,13783,13786,13788],{"class":36,"line":43},[34,13782,426],{"class":175},[34,13784,13785],{"class":190}," { inject } ",[34,13787,432],{"class":175},[34,13789,6891],{"class":186},[34,13791,13792],{"class":36,"line":49},[34,13793,53],{"emptyLinePlaceholder":52},[34,13795,13796,13798,13801,13803,13806,13808,13810],{"class":36,"line":56},[34,13797,176],{"class":175},[34,13799,13800],{"class":179}," theme",[34,13802,183],{"class":175},[34,13804,13805],{"class":395}," inject",[34,13807,456],{"class":190},[34,13809,13761],{"class":186},[34,13811,6111],{"class":190},[15,13813,13814],{},"但不要滥用。普通业务共享状态通常更适合 Pinia。",[15,13816,13817],{},"Provide\u002FInject 更常用于：",[105,13819,13820,13823,13826,13829],{},[108,13821,13822],{},"主题。",[108,13824,13825],{},"表单上下文。",[108,13827,13828],{},"UI 组件库内部。",[108,13830,13831],{},"深层组件共享少量上下文。",[19,13833,13835],{"id":13834},"_12-组件通信选择表","12. 组件通信选择表",[13837,13838,13839,13852],"table",{},[13840,13841,13842],"thead",{},[13843,13844,13845,13849],"tr",{},[13846,13847,13848],"th",{},"场景",[13846,13850,13851],{},"推荐方式",[13853,13854,13855,13864,13871,13878,13886,13894],"tbody",{},[13843,13856,13857,13861],{},[13858,13859,13860],"td",{},"父组件给子组件展示数据",[13858,13862,13863],{},"props",[13843,13865,13866,13869],{},[13858,13867,13868],{},"子组件告诉父组件用户操作",[13858,13870,12631],{},[13843,13872,13873,13876],{},[13858,13874,13875],{},"父组件决定子组件内部某块内容",[13858,13877,13208],{},[13843,13879,13880,13883],{},[13858,13881,13882],{},"表单组件双向绑定",[13858,13884,13885],{},"组件 v-model",[13843,13887,13888,13891],{},[13858,13889,13890],{},"跨很多层传少量上下文",[13858,13892,13893],{},"provide\u002Finject",[13843,13895,13896,13899],{},[13858,13897,13898],{},"多页面共享业务状态",[13858,13900,13901],{},"Pinia",[19,13903,13905],{"id":13904},"_13-常见错误","13. 常见错误",[253,13907,13909],{"id":13908},"错误一子组件直接改-props","错误一：子组件直接改 props",[15,13911,13912],{},"现象：数据流混乱，父组件难以追踪。",[15,13914,13915],{},"修复：子组件 emit，父组件修改。",[253,13917,13919],{"id":13918},"错误二父组件传错-prop-名","错误二：父组件传错 prop 名",[15,13921,12008],{},[24,13923,13925],{"className":923,"code":13924,"language":925,"meta":29,"style":29},"\u003CLessonCard :course=\"lesson\" \u002F>\n",[31,13926,13927],{"__ignoreMap":29},[34,13928,13929,13931,13933,13935,13938,13940,13942,13944,13946],{"class":36,"line":37},[34,13930,932],{"class":190},[34,13932,3466],{"class":935},[34,13934,1621],{"class":190},[34,13936,13937],{"class":395},"course",[34,13939,202],{"class":190},[34,13941,1608],{"class":186},[34,13943,1499],{"class":190},[34,13945,1608],{"class":186},[34,13947,3469],{"class":190},[15,13949,12036],{},[24,13951,13953],{"className":166,"code":13952,"language":168,"meta":29,"style":29},"defineProps({\n lesson: Object\n})\n",[31,13954,13955,13961,13965],{"__ignoreMap":29},[34,13956,13957,13959],{"class":36,"line":37},[34,13958,12056],{"class":395},[34,13960,9299],{"class":190},[34,13962,13963],{"class":36,"line":43},[34,13964,12233],{"class":190},[34,13966,13967],{"class":36,"line":49},[34,13968,6153],{"class":190},[15,13970,13971,13972,13974,13975,13977,13978,13980],{},"子组件要的是 ",[31,13973,1499],{},"，父组件传的是 ",[31,13976,13937],{},"，所以 ",[31,13979,1499],{}," 为空。",[15,13982,4866],{},[24,13984,13985],{"className":923,"code":12011,"language":925,"meta":29,"style":29},[31,13986,13987],{"__ignoreMap":29},[34,13988,13989,13991,13993,13995,13997,13999,14001,14003,14005],{"class":36,"line":37},[34,13990,932],{"class":190},[34,13992,3466],{"class":935},[34,13994,1621],{"class":190},[34,13996,1499],{"class":395},[34,13998,202],{"class":190},[34,14000,1608],{"class":186},[34,14002,1499],{"class":190},[34,14004,1608],{"class":186},[34,14006,3469],{"class":190},[253,14008,14010],{"id":14009},"错误三emit-名和监听名不一致","错误三：emit 名和监听名不一致",[15,14012,12036],{},[24,14014,14016],{"className":166,"code":14015,"language":168,"meta":29,"style":29},"emit('remove')\n",[31,14017,14018],{"__ignoreMap":29},[34,14019,14020,14022,14024,14026],{"class":36,"line":37},[34,14021,12631],{"class":395},[34,14023,456],{"class":190},[34,14025,12687],{"class":186},[34,14027,6111],{"class":190},[15,14029,12008],{},[24,14031,14033],{"className":923,"code":14032,"language":925,"meta":29,"style":29},"\u003CTodoItem @delete=\"removeTodo\" \u002F>\n",[31,14034,14035],{"__ignoreMap":29},[34,14036,14037,14039,14041,14043,14046,14048,14050,14052,14054],{"class":36,"line":37},[34,14038,932],{"class":190},[34,14040,11972],{"class":935},[34,14042,9628],{"class":190},[34,14044,14045],{"class":395},"delete",[34,14047,202],{"class":190},[34,14049,1608],{"class":186},[34,14051,13022],{"class":190},[34,14053,1608],{"class":186},[34,14055,3469],{"class":190},[15,14057,14058,14059,14061,14062,14064],{},"父组件监听的是 ",[31,14060,14045],{},"，子组件发的是 ",[31,14063,8035],{},"，不会触发。",[15,14066,4866],{},[24,14068,14070],{"className":923,"code":14069,"language":925,"meta":29,"style":29},"\u003CTodoItem @remove=\"removeTodo\" \u002F>\n",[31,14071,14072],{"__ignoreMap":29},[34,14073,14074,14076,14078,14080,14082,14084,14086,14088,14090],{"class":36,"line":37},[34,14075,932],{"class":190},[34,14077,11972],{"class":935},[34,14079,9628],{"class":190},[34,14081,8035],{"class":395},[34,14083,202],{"class":190},[34,14085,1608],{"class":186},[34,14087,13022],{"class":190},[34,14089,1608],{"class":186},[34,14091,3469],{"class":190},[253,14093,14095],{"id":14094},"错误四为了拆而拆","错误四：为了拆而拆",[15,14097,14098],{},"如果一个组件需要 15 个 props 才能工作，可能说明它边界不清。可以考虑：",[105,14100,14101,14104,14107],{},[108,14102,14103],{},"是否拆得太细。",[108,14105,14106],{},"是否应该把相关数据组合成对象。",[108,14108,14109],{},"是否应该让某些逻辑留在父组件。",[19,14111,5693],{"id":5693},[2230,14113,14114,14124,14132,14139,14144,14151],{},[108,14115,14116,14117,14119,14120,258,14122,347],{},"写一个 ",[31,14118,3466],{}," 组件，props 接收 ",[31,14121,1499],{},[31,14123,11915],{},[108,14125,14126,14128,14129,14131],{},[31,14127,3466],{}," 点击按钮后 emit ",[31,14130,11935],{},"，参数是课程 id。",[108,14133,14134,14135,14138],{},"父组件维护 ",[31,14136,14137],{},"completedLessonIds","，收到事件后修改数组。",[108,14140,14116,14141,14143],{},[31,14142,13245],{},"，通过默认插槽显示内容。",[108,14145,14116,14146,14148,14149,347],{},[31,14147,11975],{},"，支持 ",[31,14150,7473],{},[108,14152,14153],{},"回答：为什么子组件不应该直接改 props？",[79,14155,14156],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s7hpK, html code.shiki .s7hpK{--shiki-default:#B31D28;--shiki-default-font-style:italic;--shiki-dark:#FDAEB7;--shiki-dark-font-style:italic}",{"title":29,"searchDepth":43,"depth":43,"links":14158},[14159,14160,14161,14162,14163,14164,14165,14166,14167,14168,14170,14171,14172,14178],{"id":11721,"depth":43,"text":11722},{"id":11957,"depth":43,"text":11958},{"id":12004,"depth":43,"text":12005},{"id":12193,"depth":43,"text":12194},{"id":12293,"depth":43,"text":12294},{"id":12614,"depth":43,"text":12615},{"id":12719,"depth":43,"text":12720},{"id":13162,"depth":43,"text":13163},{"id":13284,"depth":43,"text":13285},{"id":13517,"depth":43,"text":14169},"10. 组件 v-model",{"id":13707,"depth":43,"text":13708},{"id":13834,"depth":43,"text":13835},{"id":13904,"depth":43,"text":13905,"children":14173},[14174,14175,14176,14177],{"id":13908,"depth":49,"text":13909},{"id":13918,"depth":49,"text":13919},{"id":14009,"depth":49,"text":14010},{"id":14094,"depth":49,"text":14095},{"id":5693,"depth":43,"text":5693},{"order":56},"\u002Fsource\u002Fcode\u002Fvue\u002F04-components",{"title":11707,"description":11715},"source\u002Fcode\u002Fvue\u002F04-components","7P7ESU--_Jm1AoGRs3XmVG8MdZfk5Vz3h6y-3fklpOM",{"id":14185,"title":14186,"body":14187,"description":14194,"extension":85,"meta":17322,"navigation":52,"path":17323,"seo":17324,"stem":17325,"__hash__":17326},"notes\u002Fsource\u002Fcode\u002Fvue\u002F05-forms-lists-styles.md","05 列表、表单、样式和用户输入",{"type":8,"value":14188,"toc":17304},[14189,14192,14195,14198,14202,14205,14262,14265,14315,14320,14323,14365,14368,14372,14374,14439,14442,14444,14490,14541,14544,14558,14565,14570,14670,14673,14687,14693,14735,14740,14746,14749,14772,14775,14804,14807,14855,14858,14889,14892,14953,14971,14974,14978,14981,15009,15012,15048,15051,15079,15082,15110,15113,15117,15120,15141,15146,15241,15244,15269,15272,15366,15368,15425,15429,15432,15461,15464,15467,15535,15538,15542,15544,15567,15570,15683,15685,15810,15815,15819,15822,15868,15871,15948,15951,15995,15998,16002,16005,16050,16057,16061,16064,16180,16183,16197,16200,16204,16207,16209,16229,16231,16288,16290,16301,16304,16339,16345,16349,17268,17270,17301],[11,14190,14186],{"id":14191},"_05-列表表单样式和用户输入",[15,14193,14194],{},"真实应用不是只展示数据，还要让用户输入、修改、筛选、提交。表单和列表是 Vue 项目最常见的业务界面。",[15,14196,14197],{},"本章目标：你能处理用户输入，能写基本校验，能稳定渲染列表，能根据状态切换样式，并能考虑空状态、错误状态和可访问性。",[19,14199,14201],{"id":14200},"_1-列表数据应该有稳定-id","1. 列表数据应该有稳定 id",[15,14203,14204],{},"假设任务列表：",[24,14206,14208],{"className":166,"code":14207,"language":168,"meta":29,"style":29},"const tasks = ref([\n { id: 'task-1', title: '学习 v-for', done: false },\n { id: 'task-2', title: '学习表单', done: true }\n])\n",[31,14209,14210,14222,14240,14258],{"__ignoreMap":29},[34,14211,14212,14214,14216,14218,14220],{"class":36,"line":37},[34,14213,176],{"class":175},[34,14215,734],{"class":179},[34,14217,183],{"class":175},[34,14219,453],{"class":395},[34,14221,6908],{"class":190},[34,14223,14224,14226,14229,14231,14234,14236,14238],{"class":36,"line":43},[34,14225,744],{"class":190},[34,14227,14228],{"class":186},"'task-1'",[34,14230,749],{"class":190},[34,14232,14233],{"class":186},"'学习 v-for'",[34,14235,755],{"class":190},[34,14237,702],{"class":179},[34,14239,761],{"class":190},[34,14241,14242,14244,14247,14249,14252,14254,14256],{"class":36,"line":49},[34,14243,744],{"class":190},[34,14245,14246],{"class":186},"'task-2'",[34,14248,749],{"class":190},[34,14250,14251],{"class":186},"'学习表单'",[34,14253,755],{"class":190},[34,14255,758],{"class":179},[34,14257,6564],{"class":190},[34,14259,14260],{"class":36,"line":56},[34,14261,6945],{"class":190},[15,14263,14264],{},"渲染：",[24,14266,14267],{"className":923,"code":7269,"language":925,"meta":29,"style":29},[31,14268,14269,14303,14307],{"__ignoreMap":29},[34,14270,14271,14273,14275,14277,14279,14281,14283,14285,14287,14289,14291,14293,14295,14297,14299,14301],{"class":36,"line":37},[34,14272,932],{"class":190},[34,14274,108],{"class":935},[34,14276,1603],{"class":175},[34,14278,202],{"class":190},[34,14280,1608],{"class":186},[34,14282,7286],{"class":190},[34,14284,1614],{"class":175},[34,14286,734],{"class":190},[34,14288,1608],{"class":186},[34,14290,1621],{"class":190},[34,14292,1624],{"class":395},[34,14294,202],{"class":190},[34,14296,1608],{"class":186},[34,14298,7303],{"class":190},[34,14300,1608],{"class":186},[34,14302,939],{"class":190},[34,14304,14305],{"class":36,"line":43},[34,14306,7312],{"class":190},[34,14308,14309,14311,14313],{"class":36,"line":49},[34,14310,1003],{"class":190},[34,14312,108],{"class":935},[34,14314,939],{"class":190},[15,14316,14317,14319],{},[31,14318,1805],{}," 是每一项的身份。不要把标题当 id，因为标题可能重复或修改。",[15,14321,14322],{},"前端新增时可以生成 id：",[24,14324,14326],{"className":166,"code":14325,"language":168,"meta":29,"style":29},"const task = {\n id: crypto.randomUUID(),\n title: newTitle,\n done: false\n}\n",[31,14327,14328,14338,14349,14354,14361],{"__ignoreMap":29},[34,14329,14330,14332,14334,14336],{"class":36,"line":37},[34,14331,176],{"class":175},[34,14333,632],{"class":179},[34,14335,183],{"class":175},[34,14337,674],{"class":190},[34,14339,14340,14343,14346],{"class":36,"line":43},[34,14341,14342],{"class":190}," id: crypto.",[34,14344,14345],{"class":395},"randomUUID",[34,14347,14348],{"class":190},"(),\n",[34,14350,14351],{"class":36,"line":49},[34,14352,14353],{"class":190}," title: newTitle,\n",[34,14355,14356,14358],{"class":36,"line":56},[34,14357,699],{"class":190},[34,14359,14360],{"class":179},"false\n",[34,14362,14363],{"class":36,"line":62},[34,14364,77],{"class":190},[15,14366,14367],{},"如果后端返回数据，通常使用后端的 id。",[19,14369,14371],{"id":14370},"_2-不要在模板里写复杂过滤","2. 不要在模板里写复杂过滤",[15,14373,7500],{},[24,14375,14377],{"className":923,"code":14376,"language":925,"meta":29,"style":29},"\u003Cli v-for=\"task in tasks.filter(task => !task.done)\" :key=\"task.id\">\n {{ task.title }}\n\u003C\u002Fli>\n",[31,14378,14379,14427,14431],{"__ignoreMap":29},[34,14380,14381,14383,14385,14387,14389,14391,14393,14395,14397,14399,14401,14403,14406,14408,14411,14413,14415,14417,14419,14421,14423,14425],{"class":36,"line":37},[34,14382,932],{"class":190},[34,14384,108],{"class":935},[34,14386,1603],{"class":175},[34,14388,202],{"class":190},[34,14390,1608],{"class":186},[34,14392,7286],{"class":190},[34,14394,1614],{"class":175},[34,14396,1318],{"class":190},[34,14398,1300],{"class":395},[34,14400,456],{"class":190},[34,14402,1327],{"class":1326},[34,14404,14405],{"class":175}," =>",[34,14407,1336],{"class":175},[34,14409,14410],{"class":190},"task.done)",[34,14412,1608],{"class":186},[34,14414,1621],{"class":190},[34,14416,1624],{"class":395},[34,14418,202],{"class":190},[34,14420,1608],{"class":186},[34,14422,7303],{"class":190},[34,14424,1608],{"class":186},[34,14426,939],{"class":190},[34,14428,14429],{"class":36,"line":43},[34,14430,7312],{"class":190},[34,14432,14433,14435,14437],{"class":36,"line":49},[34,14434,1003],{"class":190},[34,14436,108],{"class":935},[34,14438,939],{"class":190},[15,14440,14441],{},"这能运行，但不好读。模板应该尽量描述结构，不要塞复杂逻辑。",[15,14443,7527],{},[24,14445,14447],{"className":166,"code":14446,"language":168,"meta":29,"style":29},"const activeTasks = computed(() => {\n return tasks.value.filter((task) => !task.done)\n})\n",[31,14448,14449,14466,14486],{"__ignoreMap":29},[34,14450,14451,14453,14456,14458,14460,14462,14464],{"class":36,"line":37},[34,14452,176],{"class":175},[34,14454,14455],{"class":179}," activeTasks",[34,14457,183],{"class":175},[34,14459,8935],{"class":395},[34,14461,7682],{"class":190},[34,14463,1333],{"class":175},[34,14465,674],{"class":190},[34,14467,14468,14470,14472,14474,14476,14478,14480,14482,14484],{"class":36,"line":43},[34,14469,1510],{"class":175},[34,14471,6972],{"class":190},[34,14473,1300],{"class":395},[34,14475,1323],{"class":190},[34,14477,1327],{"class":1326},[34,14479,1330],{"class":190},[34,14481,1333],{"class":175},[34,14483,1336],{"class":175},[34,14485,8968],{"class":190},[34,14487,14488],{"class":36,"line":49},[34,14489,6153],{"class":190},[24,14491,14493],{"className":923,"code":14492,"language":925,"meta":29,"style":29},"\u003Cli v-for=\"task in activeTasks\" :key=\"task.id\">\n {{ task.title }}\n\u003C\u002Fli>\n",[31,14494,14495,14529,14533],{"__ignoreMap":29},[34,14496,14497,14499,14501,14503,14505,14507,14509,14511,14513,14515,14517,14519,14521,14523,14525,14527],{"class":36,"line":37},[34,14498,932],{"class":190},[34,14500,108],{"class":935},[34,14502,1603],{"class":175},[34,14504,202],{"class":190},[34,14506,1608],{"class":186},[34,14508,7286],{"class":190},[34,14510,1614],{"class":175},[34,14512,14455],{"class":190},[34,14514,1608],{"class":186},[34,14516,1621],{"class":190},[34,14518,1624],{"class":395},[34,14520,202],{"class":190},[34,14522,1608],{"class":186},[34,14524,7303],{"class":190},[34,14526,1608],{"class":186},[34,14528,939],{"class":190},[34,14530,14531],{"class":36,"line":43},[34,14532,7312],{"class":190},[34,14534,14535,14537,14539],{"class":36,"line":49},[34,14536,1003],{"class":190},[34,14538,108],{"class":935},[34,14540,939],{"class":190},[15,14542,14543],{},"优点：",[105,14545,14546,14549,14552],{},[108,14547,14548],{},"模板更干净。",[108,14550,14551],{},"过滤逻辑有名字。",[108,14553,14554,14555,347],{},"调试时可以单独看 ",[31,14556,14557],{},"activeTasks",[19,14559,14561,14562,14564],{"id":14560},"_3-v-model-是什么","3. ",[31,14563,7473],{}," 是什么",[15,14566,14567,14568,347],{},"输入框的值和响应式状态同步，最常用 ",[31,14569,7473],{},[24,14571,14573],{"className":923,"code":14572,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { ref } from 'vue'\n\nconst title = ref('')\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cinput v-model=\"title\">\n \u003Cp>你输入的是：{{ title }}\u003C\u002Fp>\n\u003C\u002Ftemplate>\n",[31,14574,14575,14585,14595,14599,14615,14623,14627,14635,14649,14662],{"__ignoreMap":29},[34,14576,14577,14579,14581,14583],{"class":36,"line":37},[34,14578,932],{"class":190},[34,14580,1911],{"class":935},[34,14582,1914],{"class":395},[34,14584,939],{"class":190},[34,14586,14587,14589,14591,14593],{"class":36,"line":43},[34,14588,426],{"class":175},[34,14590,429],{"class":190},[34,14592,432],{"class":175},[34,14594,6891],{"class":186},[34,14596,14597],{"class":36,"line":49},[34,14598,53],{"emptyLinePlaceholder":52},[34,14600,14601,14603,14605,14607,14609,14611,14613],{"class":36,"line":56},[34,14602,176],{"class":175},[34,14604,278],{"class":179},[34,14606,183],{"class":175},[34,14608,453],{"class":395},[34,14610,456],{"class":190},[34,14612,7400],{"class":186},[34,14614,6111],{"class":190},[34,14616,14617,14619,14621],{"class":36,"line":62},[34,14618,1003],{"class":190},[34,14620,1911],{"class":935},[34,14622,939],{"class":190},[34,14624,14625],{"class":36,"line":68},[34,14626,53],{"emptyLinePlaceholder":52},[34,14628,14629,14631,14633],{"class":36,"line":74},[34,14630,932],{"class":190},[34,14632,936],{"class":935},[34,14634,939],{"class":190},[34,14636,14637,14639,14641,14643,14645,14647],{"class":36,"line":1971},[34,14638,944],{"class":190},[34,14640,7429],{"class":935},[34,14642,7432],{"class":395},[34,14644,202],{"class":190},[34,14646,13343],{"class":186},[34,14648,939],{"class":190},[34,14650,14651,14653,14655,14658,14660],{"class":36,"line":1976},[34,14652,944],{"class":190},[34,14654,15],{"class":935},[34,14656,14657],{"class":190},">你输入的是：{{ title }}\u003C\u002F",[34,14659,15],{"class":935},[34,14661,939],{"class":190},[34,14663,14664,14666,14668],{"class":36,"line":1985},[34,14665,1003],{"class":190},[34,14667,936],{"class":935},[34,14669,939],{"class":190},[15,14671,14672],{},"用户输入时：",[105,14674,14675,14677,14682],{},[108,14676,7468],{},[108,14678,14679,14681],{},[31,14680,1665],{}," 自动更新。",[108,14683,14684,14681],{},[31,14685,14686],{},"{{ title }}",[15,14688,14689,14690,14692],{},"如果不用 ",[31,14691,7473],{},"，你要手动写：",[24,14694,14696],{"className":923,"code":14695,"language":925,"meta":29,"style":29},"\u003Cinput :value=\"title\" @input=\"title = $event.target.value\">\n",[31,14697,14698],{"__ignoreMap":29},[34,14699,14700,14702,14704,14706,14708,14710,14712,14714,14716,14718,14720,14722,14724,14726,14728,14731,14733],{"class":36,"line":37},[34,14701,932],{"class":190},[34,14703,7429],{"class":935},[34,14705,1621],{"class":190},[34,14707,2533],{"class":395},[34,14709,202],{"class":190},[34,14711,1608],{"class":186},[34,14713,1665],{"class":190},[34,14715,1608],{"class":186},[34,14717,9628],{"class":190},[34,14719,7429],{"class":395},[34,14721,202],{"class":190},[34,14723,1608],{"class":186},[34,14725,290],{"class":190},[34,14727,202],{"class":175},[34,14729,14730],{"class":190}," $event.target.value",[34,14732,1608],{"class":186},[34,14734,939],{"class":190},[15,14736,14737,14739],{},[31,14738,7473],{}," 就是把这种常见模式封装起来。",[19,14741,14743,14744],{"id":14742},"_4-不同表单控件的-v-model","4. 不同表单控件的 ",[31,14745,7473],{},[15,14747,14748],{},"文本输入：",[24,14750,14752],{"className":923,"code":14751,"language":925,"meta":29,"style":29},"\u003Cinput v-model=\"title\">\n",[31,14753,14754],{"__ignoreMap":29},[34,14755,14756,14758,14760,14762,14764,14766,14768,14770],{"class":36,"line":37},[34,14757,932],{"class":190},[34,14759,7429],{"class":935},[34,14761,7432],{"class":395},[34,14763,202],{"class":190},[34,14765,1608],{"class":186},[34,14767,1665],{"class":190},[34,14769,1608],{"class":186},[34,14771,939],{"class":190},[15,14773,14774],{},"多行文本：",[24,14776,14778],{"className":923,"code":14777,"language":925,"meta":29,"style":29},"\u003Ctextarea v-model=\"description\">\u003C\u002Ftextarea>\n",[31,14779,14780],{"__ignoreMap":29},[34,14781,14782,14784,14787,14789,14791,14793,14796,14798,14800,14802],{"class":36,"line":37},[34,14783,932],{"class":190},[34,14785,14786],{"class":935},"textarea",[34,14788,7432],{"class":395},[34,14790,202],{"class":190},[34,14792,1608],{"class":186},[34,14794,14795],{"class":190},"description",[34,14797,1608],{"class":186},[34,14799,6593],{"class":190},[34,14801,14786],{"class":935},[34,14803,939],{"class":190},[15,14805,14806],{},"选择框：",[24,14808,14810],{"className":923,"code":14809,"language":925,"meta":29,"style":29},"\u003Cselect v-model=\"priority\">\n \u003Coption value=\"low\">低\u003C\u002Foption>\n \u003Coption value=\"normal\">普通\u003C\u002Foption>\n \u003Coption value=\"high\">高\u003C\u002Foption>\n\u003C\u002Fselect>\n",[31,14811,14812,14832,14837,14842,14847],{"__ignoreMap":29},[34,14813,14814,14816,14819,14821,14823,14825,14828,14830],{"class":36,"line":37},[34,14815,932],{"class":190},[34,14817,14818],{"class":935},"select",[34,14820,7432],{"class":395},[34,14822,202],{"class":190},[34,14824,1608],{"class":186},[34,14826,14827],{"class":190},"priority",[34,14829,1608],{"class":186},[34,14831,939],{"class":190},[34,14833,14834],{"class":36,"line":43},[34,14835,14836],{"class":190}," \u003Coption value=\"low\">低\u003C\u002Foption>\n",[34,14838,14839],{"class":36,"line":49},[34,14840,14841],{"class":190}," \u003Coption value=\"normal\">普通\u003C\u002Foption>\n",[34,14843,14844],{"class":36,"line":56},[34,14845,14846],{"class":190}," \u003Coption value=\"high\">高\u003C\u002Foption>\n",[34,14848,14849,14851,14853],{"class":36,"line":62},[34,14850,1003],{"class":190},[34,14852,14818],{"class":935},[34,14854,939],{"class":190},[15,14856,14857],{},"单个复选框：",[24,14859,14861],{"className":923,"code":14860,"language":925,"meta":29,"style":29},"\u003Clabel>\n \u003Cinput v-model=\"accepted\" type=\"checkbox\">\n 我已阅读规则\n\u003C\u002Flabel>\n",[31,14862,14863,14871,14876,14881],{"__ignoreMap":29},[34,14864,14865,14867,14869],{"class":36,"line":37},[34,14866,932],{"class":190},[34,14868,11490],{"class":935},[34,14870,939],{"class":190},[34,14872,14873],{"class":36,"line":43},[34,14874,14875],{"class":190}," \u003Cinput v-model=\"accepted\" type=\"checkbox\">\n",[34,14877,14878],{"class":36,"line":49},[34,14879,14880],{"class":190}," 我已阅读规则\n",[34,14882,14883,14885,14887],{"class":36,"line":56},[34,14884,1003],{"class":190},[34,14886,11490],{"class":935},[34,14888,939],{"class":190},[15,14890,14891],{},"多个复选框绑定数组：",[24,14893,14895],{"className":923,"code":14894,"language":925,"meta":29,"style":29},"\u003Clabel>\n \u003Cinput v-model=\"selectedTags\" type=\"checkbox\" value=\"vue\">\n Vue\n\u003C\u002Flabel>\n\n\u003Clabel>\n \u003Cinput v-model=\"selectedTags\" type=\"checkbox\" value=\"router\">\n Router\n\u003C\u002Flabel>\n",[31,14896,14897,14905,14910,14915,14923,14927,14935,14940,14945],{"__ignoreMap":29},[34,14898,14899,14901,14903],{"class":36,"line":37},[34,14900,932],{"class":190},[34,14902,11490],{"class":935},[34,14904,939],{"class":190},[34,14906,14907],{"class":36,"line":43},[34,14908,14909],{"class":190}," \u003Cinput v-model=\"selectedTags\" type=\"checkbox\" value=\"vue\">\n",[34,14911,14912],{"class":36,"line":49},[34,14913,14914],{"class":190}," Vue\n",[34,14916,14917,14919,14921],{"class":36,"line":56},[34,14918,1003],{"class":190},[34,14920,11490],{"class":935},[34,14922,939],{"class":190},[34,14924,14925],{"class":36,"line":62},[34,14926,53],{"emptyLinePlaceholder":52},[34,14928,14929,14931,14933],{"class":36,"line":68},[34,14930,932],{"class":190},[34,14932,11490],{"class":935},[34,14934,939],{"class":190},[34,14936,14937],{"class":36,"line":74},[34,14938,14939],{"class":190}," \u003Cinput v-model=\"selectedTags\" type=\"checkbox\" value=\"router\">\n",[34,14941,14942],{"class":36,"line":1971},[34,14943,14944],{"class":190}," Router\n",[34,14946,14947,14949,14951],{"class":36,"line":1976},[34,14948,1003],{"class":190},[34,14950,11490],{"class":935},[34,14952,939],{"class":190},[24,14954,14956],{"className":166,"code":14955,"language":168,"meta":29,"style":29},"const selectedTags = ref([])\n",[31,14957,14958],{"__ignoreMap":29},[34,14959,14960,14962,14965,14967,14969],{"class":36,"line":37},[34,14961,176],{"class":175},[34,14963,14964],{"class":179}," selectedTags",[34,14966,183],{"class":175},[34,14968,453],{"class":395},[34,14970,9446],{"class":190},[15,14972,14973],{},"选中后，数组里会有对应 value。",[19,14975,14977],{"id":14976},"_5-表单修饰符","5. 表单修饰符",[15,14979,14980],{},"去掉前后空格：",[24,14982,14984],{"className":923,"code":14983,"language":925,"meta":29,"style":29},"\u003Cinput v-model.trim=\"title\">\n",[31,14985,14986],{"__ignoreMap":29},[34,14987,14988,14990,14992,14994,14996,14999,15001,15003,15005,15007],{"class":36,"line":37},[34,14989,932],{"class":190},[34,14991,7429],{"class":935},[34,14993,7432],{"class":395},[34,14995,2547],{"class":190},[34,14997,14998],{"class":395},"trim",[34,15000,202],{"class":190},[34,15002,1608],{"class":186},[34,15004,1665],{"class":190},[34,15006,1608],{"class":186},[34,15008,939],{"class":190},[15,15010,15011],{},"转换数字：",[24,15013,15015],{"className":923,"code":15014,"language":925,"meta":29,"style":29},"\u003Cinput v-model.number=\"minutes\" type=\"number\">\n",[31,15016,15017],{"__ignoreMap":29},[34,15018,15019,15021,15023,15025,15027,15030,15032,15034,15037,15039,15041,15043,15046],{"class":36,"line":37},[34,15020,932],{"class":190},[34,15022,7429],{"class":935},[34,15024,7432],{"class":395},[34,15026,2547],{"class":190},[34,15028,15029],{"class":395},"number",[34,15031,202],{"class":190},[34,15033,1608],{"class":186},[34,15035,15036],{"class":190},"minutes",[34,15038,1608],{"class":186},[34,15040,11503],{"class":395},[34,15042,202],{"class":190},[34,15044,15045],{"class":186},"\"number\"",[34,15047,939],{"class":190},[15,15049,15050],{},"输入框失焦后才同步：",[24,15052,15054],{"className":923,"code":15053,"language":925,"meta":29,"style":29},"\u003Cinput v-model.lazy=\"title\">\n",[31,15055,15056],{"__ignoreMap":29},[34,15057,15058,15060,15062,15064,15066,15069,15071,15073,15075,15077],{"class":36,"line":37},[34,15059,932],{"class":190},[34,15061,7429],{"class":935},[34,15063,7432],{"class":395},[34,15065,2547],{"class":190},[34,15067,15068],{"class":395},"lazy",[34,15070,202],{"class":190},[34,15072,1608],{"class":186},[34,15074,1665],{"class":190},[34,15076,1608],{"class":186},[34,15078,939],{"class":190},[15,15080,15081],{},"表单提交阻止刷新：",[24,15083,15084],{"className":923,"code":9781,"language":925,"meta":29,"style":29},[31,15085,15086],{"__ignoreMap":29},[34,15087,15088,15090,15092,15094,15096,15098,15100,15102,15104,15106,15108],{"class":36,"line":37},[34,15089,932],{"class":190},[34,15091,9790],{"class":935},[34,15093,9628],{"class":190},[34,15095,9795],{"class":395},[34,15097,2547],{"class":190},[34,15099,9800],{"class":395},[34,15101,202],{"class":190},[34,15103,1608],{"class":186},[34,15105,9795],{"class":190},[34,15107,1608],{"class":186},[34,15109,939],{"class":190},[15,15111,15112],{},"这些修饰符不是必须用，但它们能减少重复代码。",[19,15114,15116],{"id":15115},"_6-基础表单校验","6. 基础表单校验",[15,15118,15119],{},"状态：",[24,15121,15123],{"className":166,"code":15122,"language":168,"meta":29,"style":29},"const title = ref('')\n",[31,15124,15125],{"__ignoreMap":29},[34,15126,15127,15129,15131,15133,15135,15137,15139],{"class":36,"line":37},[34,15128,176],{"class":175},[34,15130,278],{"class":179},[34,15132,183],{"class":175},[34,15134,453],{"class":395},[34,15136,456],{"class":190},[34,15138,7400],{"class":186},[34,15140,6111],{"class":190},[15,15142,15143,15144,2664],{},"错误信息用 ",[31,15145,149],{},[24,15147,15149],{"className":166,"code":15148,"language":168,"meta":29,"style":29},"const titleError = computed(() => {\n if (!title.value.trim()) {\n return '标题不能为空'\n }\n\n if (title.value.length > 30) {\n return '标题不能超过 30 个字符'\n }\n\n return ''\n})\n",[31,15150,15151,15168,15184,15191,15195,15199,15216,15223,15227,15231,15237],{"__ignoreMap":29},[34,15152,15153,15155,15158,15160,15162,15164,15166],{"class":36,"line":37},[34,15154,176],{"class":175},[34,15156,15157],{"class":179}," titleError",[34,15159,183],{"class":175},[34,15161,8935],{"class":395},[34,15163,7682],{"class":190},[34,15165,1333],{"class":175},[34,15167,674],{"class":190},[34,15169,15170,15172,15174,15176,15179,15181],{"class":36,"line":43},[34,15171,7000],{"class":175},[34,15173,1840],{"class":190},[34,15175,5504],{"class":175},[34,15177,15178],{"class":190},"title.value.",[34,15180,14998],{"class":395},[34,15182,15183],{"class":190},"()) {\n",[34,15185,15186,15188],{"class":36,"line":49},[34,15187,8601],{"class":175},[34,15189,15190],{"class":186}," '标题不能为空'\n",[34,15192,15193],{"class":36,"line":56},[34,15194,5631],{"class":190},[34,15196,15197],{"class":36,"line":62},[34,15198,53],{"emptyLinePlaceholder":52},[34,15200,15201,15203,15206,15208,15211,15214],{"class":36,"line":68},[34,15202,7000],{"class":175},[34,15204,15205],{"class":190}," (title.value.",[34,15207,1289],{"class":179},[34,15209,15210],{"class":175}," >",[34,15212,15213],{"class":179}," 30",[34,15215,1700],{"class":190},[34,15217,15218,15220],{"class":36,"line":74},[34,15219,8601],{"class":175},[34,15221,15222],{"class":186}," '标题不能超过 30 个字符'\n",[34,15224,15225],{"class":36,"line":1971},[34,15226,5631],{"class":190},[34,15228,15229],{"class":36,"line":1976},[34,15230,53],{"emptyLinePlaceholder":52},[34,15232,15233,15235],{"class":36,"line":1985},[34,15234,1510],{"class":175},[34,15236,6645],{"class":186},[34,15238,15239],{"class":36,"line":1990},[34,15240,6153],{"class":190},[15,15242,15243],{},"提交按钮是否可用：",[24,15245,15247],{"className":166,"code":15246,"language":168,"meta":29,"style":29},"const canSubmit = computed(() => !titleError.value)\n",[31,15248,15249],{"__ignoreMap":29},[34,15250,15251,15253,15256,15258,15260,15262,15264,15266],{"class":36,"line":37},[34,15252,176],{"class":175},[34,15254,15255],{"class":179}," canSubmit",[34,15257,183],{"class":175},[34,15259,8935],{"class":395},[34,15261,7682],{"class":190},[34,15263,1333],{"class":175},[34,15265,1336],{"class":175},[34,15267,15268],{"class":190},"titleError.value)\n",[15,15270,15271],{},"提交：",[24,15273,15275],{"className":166,"code":15274,"language":168,"meta":29,"style":29},"function submit() {\n if (!canSubmit.value) {\n return\n }\n\n tasks.value.unshift({\n id: crypto.randomUUID(),\n title: title.value.trim(),\n done: false\n })\n\n title.value = ''\n}\n",[31,15276,15277,15286,15297,15302,15306,15310,15320,15329,15338,15345,15349,15353,15362],{"__ignoreMap":29},[34,15278,15279,15281,15284],{"class":36,"line":37},[34,15280,392],{"class":175},[34,15282,15283],{"class":395}," submit",[34,15285,399],{"class":190},[34,15287,15288,15290,15292,15294],{"class":36,"line":43},[34,15289,7000],{"class":175},[34,15291,1840],{"class":190},[34,15293,5504],{"class":175},[34,15295,15296],{"class":190},"canSubmit.value) {\n",[34,15298,15299],{"class":36,"line":49},[34,15300,15301],{"class":175}," return\n",[34,15303,15304],{"class":36,"line":56},[34,15305,5631],{"class":190},[34,15307,15308],{"class":36,"line":62},[34,15309,53],{"emptyLinePlaceholder":52},[34,15311,15312,15315,15318],{"class":36,"line":68},[34,15313,15314],{"class":190}," tasks.value.",[34,15316,15317],{"class":395},"unshift",[34,15319,9299],{"class":190},[34,15321,15322,15325,15327],{"class":36,"line":74},[34,15323,15324],{"class":190}," id: crypto.",[34,15326,14345],{"class":395},[34,15328,14348],{"class":190},[34,15330,15331,15334,15336],{"class":36,"line":1971},[34,15332,15333],{"class":190}," title: title.value.",[34,15335,14998],{"class":395},[34,15337,14348],{"class":190},[34,15339,15340,15343],{"class":36,"line":1976},[34,15341,15342],{"class":190}," done: ",[34,15344,14360],{"class":179},[34,15346,15347],{"class":36,"line":1985},[34,15348,6379],{"class":190},[34,15350,15351],{"class":36,"line":1990},[34,15352,53],{"emptyLinePlaceholder":52},[34,15354,15355,15358,15360],{"class":36,"line":1999},[34,15356,15357],{"class":190}," title.value ",[34,15359,202],{"class":175},[34,15361,6645],{"class":186},[34,15363,15364],{"class":36,"line":2022},[34,15365,77],{"class":190},[15,15367,1589],{},[24,15369,15371],{"className":923,"code":15370,"language":925,"meta":29,"style":29},"\u003Cform @submit.prevent=\"submit\">\n \u003Clabel for=\"task-title\">任务标题\u003C\u002Flabel>\n \u003Cinput id=\"task-title\" v-model.trim=\"title\">\n \u003Cp v-if=\"titleError\" role=\"alert\">{{ titleError }}\u003C\u002Fp>\n \u003Cbutton :disabled=\"!canSubmit\">新增\u003C\u002Fbutton>\n\u003C\u002Fform>\n",[31,15372,15373,15397,15402,15407,15412,15417],{"__ignoreMap":29},[34,15374,15375,15377,15379,15381,15383,15385,15387,15389,15391,15393,15395],{"class":36,"line":37},[34,15376,932],{"class":190},[34,15378,9790],{"class":935},[34,15380,9628],{"class":190},[34,15382,9795],{"class":395},[34,15384,2547],{"class":190},[34,15386,9800],{"class":395},[34,15388,202],{"class":190},[34,15390,1608],{"class":186},[34,15392,9795],{"class":190},[34,15394,1608],{"class":186},[34,15396,939],{"class":190},[34,15398,15399],{"class":36,"line":43},[34,15400,15401],{"class":190}," \u003Clabel for=\"task-title\">任务标题\u003C\u002Flabel>\n",[34,15403,15404],{"class":36,"line":49},[34,15405,15406],{"class":190}," \u003Cinput id=\"task-title\" v-model.trim=\"title\">\n",[34,15408,15409],{"class":36,"line":56},[34,15410,15411],{"class":190}," \u003Cp v-if=\"titleError\" role=\"alert\">{{ titleError }}\u003C\u002Fp>\n",[34,15413,15414],{"class":36,"line":62},[34,15415,15416],{"class":190}," \u003Cbutton :disabled=\"!canSubmit\">新增\u003C\u002Fbutton>\n",[34,15418,15419,15421,15423],{"class":36,"line":68},[34,15420,1003],{"class":190},[34,15422,9790],{"class":935},[34,15424,939],{"class":190},[19,15426,15428],{"id":15427},"_7-为什么错误信息用字符串而不是布尔值","7. 为什么错误信息用字符串而不是布尔值",[15,15430,15431],{},"你可能想写：",[24,15433,15435],{"className":166,"code":15434,"language":168,"meta":29,"style":29},"const hasError = computed(() => !title.value.trim())\n",[31,15436,15437],{"__ignoreMap":29},[34,15438,15439,15441,15444,15446,15448,15450,15452,15454,15456,15458],{"class":36,"line":37},[34,15440,176],{"class":175},[34,15442,15443],{"class":179}," hasError",[34,15445,183],{"class":175},[34,15447,8935],{"class":395},[34,15449,7682],{"class":190},[34,15451,1333],{"class":175},[34,15453,1336],{"class":175},[34,15455,15178],{"class":190},[34,15457,14998],{"class":395},[34,15459,15460],{"class":190},"())\n",[15,15462,15463],{},"这只能告诉你“有没有错”，不能告诉用户“错在哪里”。",[15,15465,15466],{},"更好的方式是返回错误文本：",[24,15468,15470],{"className":166,"code":15469,"language":168,"meta":29,"style":29},"const titleError = computed(() => {\n if (!title.value.trim()) return '标题不能为空'\n if (title.value.length > 30) return '标题不能超过 30 个字符'\n return ''\n})\n",[31,15471,15472,15488,15507,15525,15531],{"__ignoreMap":29},[34,15473,15474,15476,15478,15480,15482,15484,15486],{"class":36,"line":37},[34,15475,176],{"class":175},[34,15477,15157],{"class":179},[34,15479,183],{"class":175},[34,15481,8935],{"class":395},[34,15483,7682],{"class":190},[34,15485,1333],{"class":175},[34,15487,674],{"class":190},[34,15489,15490,15492,15494,15496,15498,15500,15503,15505],{"class":36,"line":43},[34,15491,7000],{"class":175},[34,15493,1840],{"class":190},[34,15495,5504],{"class":175},[34,15497,15178],{"class":190},[34,15499,14998],{"class":395},[34,15501,15502],{"class":190},"()) ",[34,15504,1775],{"class":175},[34,15506,15190],{"class":186},[34,15508,15509,15511,15513,15515,15517,15519,15521,15523],{"class":36,"line":49},[34,15510,7000],{"class":175},[34,15512,15205],{"class":190},[34,15514,1289],{"class":179},[34,15516,15210],{"class":175},[34,15518,15213],{"class":179},[34,15520,1330],{"class":190},[34,15522,1775],{"class":175},[34,15524,15222],{"class":186},[34,15526,15527,15529],{"class":36,"line":56},[34,15528,1510],{"class":175},[34,15530,6645],{"class":186},[34,15532,15533],{"class":36,"line":62},[34,15534,6153],{"class":190},[15,15536,15537],{},"这样同一个值既能判断是否有错，也能显示错误。",[19,15539,15541],{"id":15540},"_8-列表筛选","8. 列表筛选",[15,15543,15119],{},[24,15545,15547],{"className":166,"code":15546,"language":168,"meta":29,"style":29},"const filter = ref('all')\n",[31,15548,15549],{"__ignoreMap":29},[34,15550,15551,15553,15556,15558,15560,15562,15565],{"class":36,"line":37},[34,15552,176],{"class":175},[34,15554,15555],{"class":179}," filter",[34,15557,183],{"class":175},[34,15559,453],{"class":395},[34,15561,456],{"class":190},[34,15563,15564],{"class":186},"'all'",[34,15566,6111],{"class":190},[15,15568,15569],{},"筛选结果：",[24,15571,15573],{"className":166,"code":15572,"language":168,"meta":29,"style":29},"const filteredTasks = computed(() => {\n if (filter.value === 'active') {\n return tasks.value.filter((task) => !task.done)\n }\n\n if (filter.value === 'completed') {\n return tasks.value.filter((task) => task.done)\n }\n\n return tasks.value\n})\n",[31,15574,15575,15591,15605,15625,15629,15633,15646,15665,15669,15673,15679],{"__ignoreMap":29},[34,15576,15577,15579,15581,15583,15585,15587,15589],{"class":36,"line":37},[34,15578,176],{"class":175},[34,15580,10599],{"class":179},[34,15582,183],{"class":175},[34,15584,8935],{"class":395},[34,15586,7682],{"class":190},[34,15588,1333],{"class":175},[34,15590,674],{"class":190},[34,15592,15593,15595,15598,15600,15603],{"class":36,"line":43},[34,15594,7000],{"class":175},[34,15596,15597],{"class":190}," (filter.value ",[34,15599,1423],{"class":175},[34,15601,15602],{"class":186}," 'active'",[34,15604,1700],{"class":190},[34,15606,15607,15609,15611,15613,15615,15617,15619,15621,15623],{"class":36,"line":49},[34,15608,8601],{"class":175},[34,15610,6972],{"class":190},[34,15612,1300],{"class":395},[34,15614,1323],{"class":190},[34,15616,1327],{"class":1326},[34,15618,1330],{"class":190},[34,15620,1333],{"class":175},[34,15622,1336],{"class":175},[34,15624,8968],{"class":190},[34,15626,15627],{"class":36,"line":56},[34,15628,5631],{"class":190},[34,15630,15631],{"class":36,"line":62},[34,15632,53],{"emptyLinePlaceholder":52},[34,15634,15635,15637,15639,15641,15644],{"class":36,"line":68},[34,15636,7000],{"class":175},[34,15638,15597],{"class":190},[34,15640,1423],{"class":175},[34,15642,15643],{"class":186}," 'completed'",[34,15645,1700],{"class":190},[34,15647,15648,15650,15652,15654,15656,15658,15660,15662],{"class":36,"line":74},[34,15649,8601],{"class":175},[34,15651,6972],{"class":190},[34,15653,1300],{"class":395},[34,15655,1323],{"class":190},[34,15657,1327],{"class":1326},[34,15659,1330],{"class":190},[34,15661,1333],{"class":175},[34,15663,15664],{"class":190}," task.done)\n",[34,15666,15667],{"class":36,"line":1971},[34,15668,5631],{"class":190},[34,15670,15671],{"class":36,"line":1976},[34,15672,53],{"emptyLinePlaceholder":52},[34,15674,15675,15677],{"class":36,"line":1985},[34,15676,1510],{"class":175},[34,15678,8948],{"class":190},[34,15680,15681],{"class":36,"line":1990},[34,15682,6153],{"class":190},[15,15684,1589],{},[24,15686,15688],{"className":923,"code":15687,"language":925,"meta":29,"style":29},"\u003Cbutton @click=\"filter = 'all'\">全部\u003C\u002Fbutton>\n\u003Cbutton @click=\"filter = 'active'\">未完成\u003C\u002Fbutton>\n\u003Cbutton @click=\"filter = 'completed'\">已完成\u003C\u002Fbutton>\n\n\u003Cul>\n \u003Cli v-for=\"task in filteredTasks\" :key=\"task.id\">\n {{ task.title }}\n \u003C\u002Fli>\n\u003C\u002Ful>\n",[31,15689,15690,15719,15747,15775,15779,15787,15792,15797,15802],{"__ignoreMap":29},[34,15691,15692,15694,15696,15698,15700,15702,15704,15707,15709,15712,15715,15717],{"class":36,"line":37},[34,15693,932],{"class":190},[34,15695,2004],{"class":935},[34,15697,9628],{"class":190},[34,15699,9631],{"class":395},[34,15701,202],{"class":190},[34,15703,1608],{"class":186},[34,15705,15706],{"class":190},"filter ",[34,15708,202],{"class":175},[34,15710,15711],{"class":186}," 'all'\"",[34,15713,15714],{"class":190},">全部\u003C\u002F",[34,15716,2004],{"class":935},[34,15718,939],{"class":190},[34,15720,15721,15723,15725,15727,15729,15731,15733,15735,15737,15740,15743,15745],{"class":36,"line":43},[34,15722,932],{"class":190},[34,15724,2004],{"class":935},[34,15726,9628],{"class":190},[34,15728,9631],{"class":395},[34,15730,202],{"class":190},[34,15732,1608],{"class":186},[34,15734,15706],{"class":190},[34,15736,202],{"class":175},[34,15738,15739],{"class":186}," 'active'\"",[34,15741,15742],{"class":190},">未完成\u003C\u002F",[34,15744,2004],{"class":935},[34,15746,939],{"class":190},[34,15748,15749,15751,15753,15755,15757,15759,15761,15763,15765,15768,15771,15773],{"class":36,"line":49},[34,15750,932],{"class":190},[34,15752,2004],{"class":935},[34,15754,9628],{"class":190},[34,15756,9631],{"class":395},[34,15758,202],{"class":190},[34,15760,1608],{"class":186},[34,15762,15706],{"class":190},[34,15764,202],{"class":175},[34,15766,15767],{"class":186}," 'completed'\"",[34,15769,15770],{"class":190},">已完成\u003C\u002F",[34,15772,2004],{"class":935},[34,15774,939],{"class":190},[34,15776,15777],{"class":36,"line":56},[34,15778,53],{"emptyLinePlaceholder":52},[34,15780,15781,15783,15785],{"class":36,"line":62},[34,15782,932],{"class":190},[34,15784,105],{"class":935},[34,15786,939],{"class":190},[34,15788,15789],{"class":36,"line":68},[34,15790,15791],{"class":190}," \u003Cli v-for=\"task in filteredTasks\" :key=\"task.id\">\n",[34,15793,15794],{"class":36,"line":74},[34,15795,15796],{"class":190}," {{ task.title }}\n",[34,15798,15799],{"class":36,"line":1971},[34,15800,15801],{"class":190}," \u003C\u002Fli>\n",[34,15803,15804,15806,15808],{"class":36,"line":1976},[34,15805,1003],{"class":190},[34,15807,105],{"class":935},[34,15809,939],{"class":190},[15,15811,3685,15812,15814],{},[31,15813,1300],{}," 是页面局部状态。如果只是当前组件使用，不需要放 Pinia。",[19,15816,15818],{"id":15817},"_9-动态-class","9. 动态 class",[15,15820,15821],{},"根据状态切换样式：",[24,15823,15825],{"className":923,"code":15824,"language":925,"meta":29,"style":29},"\u003Cli :class=\"{ done: task.done, urgent: task.priority === 'high' }\">\n {{ task.title }}\n\u003C\u002Fli>\n",[31,15826,15827,15856,15860],{"__ignoreMap":29},[34,15828,15829,15831,15833,15835,15837,15839,15841,15844,15846,15849,15852,15854],{"class":36,"line":37},[34,15830,932],{"class":190},[34,15832,108],{"class":935},[34,15834,1621],{"class":190},[34,15836,8063],{"class":395},[34,15838,202],{"class":190},[34,15840,1608],{"class":186},[34,15842,15843],{"class":190},"{ done: task.done, urgent: task.priority ",[34,15845,1423],{"class":175},[34,15847,15848],{"class":186}," 'high'",[34,15850,15851],{"class":190}," }",[34,15853,1608],{"class":186},[34,15855,939],{"class":190},[34,15857,15858],{"class":36,"line":43},[34,15859,7312],{"class":190},[34,15861,15862,15864,15866],{"class":36,"line":49},[34,15863,1003],{"class":190},[34,15865,108],{"class":935},[34,15867,939],{"class":190},[15,15869,15870],{},"CSS：",[24,15872,15876],{"className":15873,"code":15874,"language":15875,"meta":29,"style":29},"language-css shiki shiki-themes github-light github-dark",".done {\n color: #64748b;\n text-decoration: line-through;\n}\n\n.urgent {\n border-left: 4px solid #dc2626;\n}\n","css",[31,15877,15878,15885,15896,15908,15912,15916,15923,15944],{"__ignoreMap":29},[34,15879,15880,15883],{"class":36,"line":37},[34,15881,15882],{"class":395},".done",[34,15884,674],{"class":190},[34,15886,15887,15889,15891,15894],{"class":36,"line":43},[34,15888,8468],{"class":179},[34,15890,8471],{"class":190},[34,15892,15893],{"class":179},"#64748b",[34,15895,191],{"class":190},[34,15897,15898,15901,15903,15906],{"class":36,"line":49},[34,15899,15900],{"class":179}," text-decoration",[34,15902,8471],{"class":190},[34,15904,15905],{"class":179},"line-through",[34,15907,191],{"class":190},[34,15909,15910],{"class":36,"line":56},[34,15911,77],{"class":190},[34,15913,15914],{"class":36,"line":62},[34,15915,53],{"emptyLinePlaceholder":52},[34,15917,15918,15921],{"class":36,"line":68},[34,15919,15920],{"class":395},".urgent",[34,15922,674],{"class":190},[34,15924,15925,15928,15930,15933,15936,15939,15942],{"class":36,"line":74},[34,15926,15927],{"class":179}," border-left",[34,15929,8471],{"class":190},[34,15931,15932],{"class":179},"4",[34,15934,15935],{"class":175},"px",[34,15937,15938],{"class":179}," solid",[34,15940,15941],{"class":179}," #dc2626",[34,15943,191],{"class":190},[34,15945,15946],{"class":36,"line":1971},[34,15947,77],{"class":190},[15,15949,15950],{},"数组写法：",[24,15952,15954],{"className":923,"code":15953,"language":925,"meta":29,"style":29},"\u003Cbutton :class=\"['button', variant, { loading }]\">\n 保存\n\u003C\u002Fbutton>\n",[31,15955,15956,15982,15987],{"__ignoreMap":29},[34,15957,15958,15960,15962,15964,15966,15968,15970,15973,15975,15978,15980],{"class":36,"line":37},[34,15959,932],{"class":190},[34,15961,2004],{"class":935},[34,15963,1621],{"class":190},[34,15965,8063],{"class":395},[34,15967,202],{"class":190},[34,15969,1608],{"class":186},[34,15971,15972],{"class":190},"[",[34,15974,6738],{"class":186},[34,15976,15977],{"class":190},", variant, { loading }]",[34,15979,1608],{"class":186},[34,15981,939],{"class":190},[34,15983,15984],{"class":36,"line":43},[34,15985,15986],{"class":190}," 保存\n",[34,15988,15989,15991,15993],{"class":36,"line":49},[34,15990,1003],{"class":190},[34,15992,2004],{"class":935},[34,15994,939],{"class":190},[15,15996,15997],{},"对象适合开关类名，数组适合组合多个类名。",[19,15999,16001],{"id":16000},"_10-动态-style","10. 动态 style",[15,16003,16004],{},"动态 style 适合少量数值样式：",[24,16006,16008],{"className":923,"code":16007,"language":925,"meta":29,"style":29},"\u003Cdiv class=\"progress-fill\" :style=\"{ width: progress + '%' }\">\u003C\u002Fdiv>\n",[31,16009,16010],{"__ignoreMap":29},[34,16011,16012,16014,16017,16019,16021,16024,16026,16028,16030,16032,16035,16037,16040,16042,16044,16046,16048],{"class":36,"line":37},[34,16013,932],{"class":190},[34,16015,16016],{"class":935},"div",[34,16018,7588],{"class":395},[34,16020,202],{"class":190},[34,16022,16023],{"class":186},"\"progress-fill\"",[34,16025,1621],{"class":190},[34,16027,79],{"class":395},[34,16029,202],{"class":190},[34,16031,1608],{"class":186},[34,16033,16034],{"class":190},"{ width: progress ",[34,16036,223],{"class":175},[34,16038,16039],{"class":186}," '%'",[34,16041,15851],{"class":190},[34,16043,1608],{"class":186},[34,16045,6593],{"class":190},[34,16047,16016],{"class":935},[34,16049,939],{"class":190},[15,16051,16052,16053,16056],{},"不建议把大量样式都写到 ",[31,16054,16055],{},":style"," 里。复杂样式优先用 class。",[19,16058,16060],{"id":16059},"_11-空状态加载和错误","11. 空状态、加载和错误",[15,16062,16063],{},"任何真实列表都不应该只写成功状态。",[24,16065,16067],{"className":923,"code":16066,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"loading\">加载中...\u003C\u002Fp>\n\u003Cp v-else-if=\"error\" role=\"alert\">{{ error }}\u003C\u002Fp>\n\u003Cp v-else-if=\"filteredTasks.length === 0\">暂无任务\u003C\u002Fp>\n\u003Cul v-else>\n \u003Cli v-for=\"task in filteredTasks\" :key=\"task.id\">\n {{ task.title }}\n \u003C\u002Fli>\n\u003C\u002Ful>\n",[31,16068,16069,16091,16120,16150,16160,16164,16168,16172],{"__ignoreMap":29},[34,16070,16071,16073,16075,16077,16079,16081,16083,16085,16087,16089],{"class":36,"line":37},[34,16072,932],{"class":190},[34,16074,15],{"class":935},[34,16076,2168],{"class":175},[34,16078,202],{"class":190},[34,16080,1608],{"class":186},[34,16082,2175],{"class":190},[34,16084,1608],{"class":186},[34,16086,2180],{"class":190},[34,16088,15],{"class":935},[34,16090,939],{"class":190},[34,16092,16093,16095,16097,16099,16101,16103,16105,16107,16109,16111,16114,16116,16118],{"class":36,"line":43},[34,16094,932],{"class":190},[34,16096,15],{"class":935},[34,16098,2193],{"class":175},[34,16100,202],{"class":190},[34,16102,1608],{"class":186},[34,16104,2200],{"class":190},[34,16106,1608],{"class":186},[34,16108,2345],{"class":395},[34,16110,202],{"class":190},[34,16112,16113],{"class":186},"\"alert\"",[34,16115,2205],{"class":190},[34,16117,15],{"class":935},[34,16119,939],{"class":190},[34,16121,16122,16124,16126,16128,16130,16132,16135,16137,16139,16141,16143,16146,16148],{"class":36,"line":49},[34,16123,932],{"class":190},[34,16125,15],{"class":935},[34,16127,2193],{"class":175},[34,16129,202],{"class":190},[34,16131,1608],{"class":186},[34,16133,16134],{"class":190},"filteredTasks.",[34,16136,1289],{"class":179},[34,16138,7946],{"class":175},[34,16140,205],{"class":179},[34,16142,1608],{"class":186},[34,16144,16145],{"class":190},">暂无任务\u003C\u002F",[34,16147,15],{"class":935},[34,16149,939],{"class":190},[34,16151,16152,16154,16156,16158],{"class":36,"line":56},[34,16153,932],{"class":190},[34,16155,105],{"class":935},[34,16157,2218],{"class":175},[34,16159,939],{"class":190},[34,16161,16162],{"class":36,"line":62},[34,16163,15791],{"class":190},[34,16165,16166],{"class":36,"line":68},[34,16167,15796],{"class":190},[34,16169,16170],{"class":36,"line":74},[34,16171,15801],{"class":190},[34,16173,16174,16176,16178],{"class":36,"line":1971},[34,16175,1003],{"class":190},[34,16177,105],{"class":935},[34,16179,939],{"class":190},[15,16181,16182],{},"四种状态：",[105,16184,16185,16188,16191,16194],{},[108,16186,16187],{},"loading：正在加载。",[108,16189,16190],{},"error：加载失败。",[108,16192,16193],{},"empty：加载成功但没有数据。",[108,16195,16196],{},"success：有数据。",[15,16198,16199],{},"初学者经常只写 success，真实项目必须补全其他状态。",[19,16201,16203],{"id":16202},"_12-表单可访问性","12. 表单可访问性",[15,16205,16206],{},"不要只用 placeholder 当标签。",[15,16208,7500],{},[24,16210,16212],{"className":923,"code":16211,"language":925,"meta":29,"style":29},"\u003Cinput placeholder=\"任务标题\">\n",[31,16213,16214],{"__ignoreMap":29},[34,16215,16216,16218,16220,16222,16224,16227],{"class":36,"line":37},[34,16217,932],{"class":190},[34,16219,7429],{"class":935},[34,16221,11392],{"class":395},[34,16223,202],{"class":190},[34,16225,16226],{"class":186},"\"任务标题\"",[34,16228,939],{"class":190},[15,16230,7527],{},[24,16232,16234],{"className":923,"code":16233,"language":925,"meta":29,"style":29},"\u003Clabel for=\"task-title\">任务标题\u003C\u002Flabel>\n\u003Cinput id=\"task-title\" v-model=\"title\" placeholder=\"例如：学习组件通信\">\n",[31,16235,16236,16257],{"__ignoreMap":29},[34,16237,16238,16240,16242,16245,16247,16250,16253,16255],{"class":36,"line":37},[34,16239,932],{"class":190},[34,16241,11490],{"class":935},[34,16243,16244],{"class":395}," for",[34,16246,202],{"class":190},[34,16248,16249],{"class":186},"\"task-title\"",[34,16251,16252],{"class":190},">任务标题\u003C\u002F",[34,16254,11490],{"class":935},[34,16256,939],{"class":190},[34,16258,16259,16261,16263,16265,16267,16269,16271,16273,16275,16277,16279,16281,16283,16286],{"class":36,"line":43},[34,16260,932],{"class":190},[34,16262,7429],{"class":935},[34,16264,6051],{"class":395},[34,16266,202],{"class":190},[34,16268,16249],{"class":186},[34,16270,7432],{"class":395},[34,16272,202],{"class":190},[34,16274,1608],{"class":186},[34,16276,1665],{"class":190},[34,16278,1608],{"class":186},[34,16280,11392],{"class":395},[34,16282,202],{"class":190},[34,16284,16285],{"class":186},"\"例如：学习组件通信\"",[34,16287,939],{"class":190},[15,16289,7819],{},[105,16291,16292,16295,16298],{},[108,16293,16294],{},"placeholder 输入后会消失。",[108,16296,16297],{},"屏幕阅读器更需要 label。",[108,16299,16300],{},"点击 label 可以聚焦对应输入框。",[15,16302,16303],{},"错误提示：",[24,16305,16307],{"className":923,"code":16306,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"titleError\" role=\"alert\">{{ titleError }}\u003C\u002Fp>\n",[31,16308,16309],{"__ignoreMap":29},[34,16310,16311,16313,16315,16317,16319,16321,16324,16326,16328,16330,16332,16335,16337],{"class":36,"line":37},[34,16312,932],{"class":190},[34,16314,15],{"class":935},[34,16316,2168],{"class":175},[34,16318,202],{"class":190},[34,16320,1608],{"class":186},[34,16322,16323],{"class":190},"titleError",[34,16325,1608],{"class":186},[34,16327,2345],{"class":395},[34,16329,202],{"class":190},[34,16331,16113],{"class":186},[34,16333,16334],{"class":190},">{{ titleError }}\u003C\u002F",[34,16336,15],{"class":935},[34,16338,939],{"class":190},[15,16340,16341,16344],{},[31,16342,16343],{},"role=\"alert\""," 能帮助辅助技术知道这里出现了重要反馈。",[19,16346,16348],{"id":16347},"_13-完整任务表单例子","13. 完整任务表单例子",[24,16350,16352],{"className":923,"code":16351,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { computed, ref } from 'vue'\n\nconst title = ref('')\nconst priority = ref('normal')\nconst filter = ref('all')\nconst tasks = ref([])\n\nconst titleError = computed(() => {\n if (!title.value.trim()) return '标题不能为空'\n if (title.value.length > 30) return '标题不能超过 30 个字符'\n return ''\n})\n\nconst canSubmit = computed(() => !titleError.value)\n\nconst filteredTasks = computed(() => {\n if (filter.value === 'active') return tasks.value.filter((task) => !task.done)\n if (filter.value === 'completed') return tasks.value.filter((task) => task.done)\n return tasks.value\n})\n\nfunction addTask() {\n if (!canSubmit.value) return\n\n tasks.value.unshift({\n id: crypto.randomUUID(),\n title: title.value.trim(),\n priority: priority.value,\n done: false\n })\n\n title.value = ''\n priority.value = 'normal'\n}\n\nfunction toggleTask(id) {\n const task = tasks.value.find((item) => item.id === id)\n\n if (task) {\n task.done = !task.done\n }\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cform @submit.prevent=\"addTask\">\n \u003Clabel for=\"title\">任务标题\u003C\u002Flabel>\n \u003Cinput id=\"title\" v-model.trim=\"title\">\n \u003Cp v-if=\"titleError\" role=\"alert\">{{ titleError }}\u003C\u002Fp>\n\n \u003Clabel for=\"priority\">优先级\u003C\u002Flabel>\n \u003Cselect id=\"priority\" v-model=\"priority\">\n \u003Coption value=\"low\">低\u003C\u002Foption>\n \u003Coption value=\"normal\">普通\u003C\u002Foption>\n \u003Coption value=\"high\">高\u003C\u002Foption>\n \u003C\u002Fselect>\n\n \u003Cbutton :disabled=\"!canSubmit\">新增任务\u003C\u002Fbutton>\n \u003C\u002Fform>\n\n \u003Cbutton @click=\"filter = 'all'\">全部\u003C\u002Fbutton>\n \u003Cbutton @click=\"filter = 'active'\">未完成\u003C\u002Fbutton>\n \u003Cbutton @click=\"filter = 'completed'\">已完成\u003C\u002Fbutton>\n\n \u003Cp v-if=\"filteredTasks.length === 0\">暂无任务\u003C\u002Fp>\n\n \u003Cul v-else>\n \u003Cli\n v-for=\"task in filteredTasks\"\n :key=\"task.id\"\n :class=\"{ done: task.done, urgent: task.priority === 'high' }\"\n >\n \u003Clabel>\n \u003Cinput type=\"checkbox\" :checked=\"task.done\" @change=\"toggleTask(task.id)\">\n {{ task.title }}\n \u003C\u002Flabel>\n \u003C\u002Fli>\n 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>\n",[34,17192,17194,17196,17198],{"class":36,"line":17193},74,[34,17195,7078],{"class":190},[34,17197,11490],{"class":935},[34,17199,939],{"class":190},[34,17201,17203,17205,17207,17209,17211,17213,17215,17217,17219,17221,17223,17225],{"class":36,"line":17202},75,[34,17204,11498],{"class":190},[34,17206,7429],{"class":935},[34,17208,11503],{"class":395},[34,17210,202],{"class":190},[34,17212,11508],{"class":186},[34,17214,11511],{"class":395},[34,17216,202],{"class":190},[34,17218,11516],{"class":186},[34,17220,11519],{"class":395},[34,17222,202],{"class":190},[34,17224,7100],{"class":186},[34,17226,939],{"class":190},[34,17228,17230],{"class":36,"line":17229},76,[34,17231,11531],{"class":190},[34,17233,17235,17237,17239],{"class":36,"line":17234},77,[34,17236,11537],{"class":190},[34,17238,11490],{"class":935},[34,17240,939],{"class":190},[34,17242,17244,17246,17248],{"class":36,"line":17243},78,[34,17245,7112],{"class":190},[34,17247,108],{"class":935},[34,17249,939],{"class":190},[34,17251,17253,17255,17257],{"class":36,"line":17252},79,[34,17254,994],{"class":190},[34,17256,105],{"class":935},[34,17258,939],{"class":190},[34,17260,17262,17264,17266],{"class":36,"line":17261},80,[34,17263,1003],{"class":190},[34,17265,936],{"class":935},[34,17267,939],{"class":190},[19,17269,5693],{"id":5693},[2230,17271,17272,17275,17278,17281,17286,17289,17292,17298],{},[108,17273,17274],{},"写一个任务创建表单。",[108,17276,17277],{},"标题不能为空，不能超过 30 字。",[108,17279,17280],{},"增加优先级选择。",[108,17282,5792,17283,17285],{},[31,17284,149],{}," 渲染“全部 \u002F 未完成 \u002F 已完成”三种列表。",[108,17287,17288],{},"给已完成任务加删除线样式。",[108,17290,17291],{},"给高优先级任务加醒目边框。",[108,17293,17294,17295,17297],{},"给输入框补全 ",[31,17296,11490],{}," 和错误提示。",[108,17299,17300],{},"回答：哪些状态应该留在当前组件内？为什么？",[79,17302,17303],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":17305},[17306,17307,17308,17310,17312,17313,17314,17315,17316,17317,17318,17319,17320,17321],{"id":14200,"depth":43,"text":14201},{"id":14370,"depth":43,"text":14371},{"id":14560,"depth":43,"text":17309},"3. v-model 是什么",{"id":14742,"depth":43,"text":17311},"4. 不同表单控件的 v-model",{"id":14976,"depth":43,"text":14977},{"id":15115,"depth":43,"text":15116},{"id":15427,"depth":43,"text":15428},{"id":15540,"depth":43,"text":15541},{"id":15817,"depth":43,"text":15818},{"id":16000,"depth":43,"text":16001},{"id":16059,"depth":43,"text":16060},{"id":16202,"depth":43,"text":16203},{"id":16347,"depth":43,"text":16348},{"id":5693,"depth":43,"text":5693},{"order":62},"\u002Fsource\u002Fcode\u002Fvue\u002F05-forms-lists-styles",{"title":14186,"description":14194},"source\u002Fcode\u002Fvue\u002F05-forms-lists-styles","yPwjzmLkorpp8I12OtFtCz7t7uHCERHTPDWnrApPk58",{"id":17328,"title":17329,"body":17330,"description":17337,"extension":85,"meta":20011,"navigation":52,"path":20012,"seo":20013,"stem":20014,"__hash__":20015},"notes\u002Fsource\u002Fcode\u002Fvue\u002F06-composition-composables.md","06 Composition API 和组合式函数",{"type":8,"value":17331,"toc":19986},[17332,17335,17338,17348,17352,17355,17484,17487,17506,17509,17608,17611,17615,17618,17635,17650,17709,17712,17718,17866,17869,17893,17897,17904,18071,18074,18230,18236,18242,18274,18277,18292,18301,18304,18308,18311,18314,18382,18391,18422,18431,18434,18502,18508,18515,18683,18685,18707,18709,18764,18771,18776,19012,19015,19050,19052,19075,19078,19107,19110,19117,19120,19366,19368,19456,19458,19530,19534,19537,19551,19554,19583,19586,19597,19601,19604,19640,19643,19646,19685,19688,19691,19707,19711,19713,19724,19727,19738,19740,19791,19793,19797,19834,19840,19849,19853,19888,19894,19906,19910,19916,19932,19935,19937,19983],[11,17333,17329],{"id":17334},"_06-composition-api-和组合式函数",[15,17336,17337],{},"Composition API 是 Vue 3 的核心写法之一。它不是为了让代码看起来高级，而是为了把同一件事相关的状态和逻辑放在一起，并让逻辑可以复用。",[15,17339,17340,17341,17343,17344,17347],{},"本章目标：你能看懂 ",[31,17342,141],{}," 中的 Composition API，能写自己的组合式函数，能判断什么时候该抽 ",[31,17345,17346],{},"useXxx","，什么时候不该抽。",[19,17349,17351],{"id":17350},"_1-options-api-和-composition-api-的区别","1. Options API 和 Composition API 的区别",[15,17353,17354],{},"早期 Vue 常见 Options API：",[24,17356,17358],{"className":923,"code":17357,"language":925,"meta":29,"style":29},"\u003Cscript>\nexport default {\n data() {\n return {\n count: 0\n }\n },\n computed: {\n doubleCount() {\n return this.count * 2\n }\n },\n methods: {\n increase() {\n this.count++\n }\n }\n}\n\u003C\u002Fscript>\n",[31,17359,17360,17368,17376,17383,17389,17396,17400,17405,17410,17417,17434,17438,17442,17447,17454,17464,17468,17472,17476],{"__ignoreMap":29},[34,17361,17362,17364,17366],{"class":36,"line":37},[34,17363,932],{"class":190},[34,17365,1911],{"class":935},[34,17367,939],{"class":190},[34,17369,17370,17372,17374],{"class":36,"line":43},[34,17371,2825],{"class":175},[34,17373,3958],{"class":175},[34,17375,674],{"class":190},[34,17377,17378,17381],{"class":36,"line":49},[34,17379,17380],{"class":395}," data",[34,17382,399],{"class":190},[34,17384,17385,17387],{"class":36,"line":56},[34,17386,8601],{"class":175},[34,17388,674],{"class":190},[34,17390,17391,17394],{"class":36,"line":62},[34,17392,17393],{"class":190}," count: ",[34,17395,9186],{"class":179},[34,17397,17398],{"class":36,"line":68},[34,17399,5540],{"class":190},[34,17401,17402],{"class":36,"line":74},[34,17403,17404],{"class":190}," },\n",[34,17406,17407],{"class":36,"line":1971},[34,17408,17409],{"class":190}," computed: {\n",[34,17411,17412,17415],{"class":36,"line":1976},[34,17413,17414],{"class":395}," doubleCount",[34,17416,399],{"class":190},[34,17418,17419,17422,17425,17428,17431],{"class":36,"line":1985},[34,17420,17421],{"class":175}," return",[34,17423,17424],{"class":179}," this",[34,17426,17427],{"class":190},".count ",[34,17429,17430],{"class":175},"*",[34,17432,17433],{"class":179}," 2\n",[34,17435,17436],{"class":36,"line":1990},[34,17437,5540],{"class":190},[34,17439,17440],{"class":36,"line":1999},[34,17441,17404],{"class":190},[34,17443,17444],{"class":36,"line":2022},[34,17445,17446],{"class":190}," methods: {\n",[34,17448,17449,17452],{"class":36,"line":2036},[34,17450,17451],{"class":395}," increase",[34,17453,399],{"class":190},[34,17455,17456,17459,17462],{"class":36,"line":2603},[34,17457,17458],{"class":179}," this",[34,17460,17461],{"class":190},".count",[34,17463,6366],{"class":175},[34,17465,17466],{"class":36,"line":5537},[34,17467,5540],{"class":190},[34,17469,17470],{"class":36,"line":5543},[34,17471,5631],{"class":190},[34,17473,17474],{"class":36,"line":5548},[34,17475,77],{"class":190},[34,17477,17478,17480,17482],{"class":36,"line":5564},[34,17479,1003],{"class":190},[34,17481,1911],{"class":935},[34,17483,939],{"class":190},[15,17485,17486],{},"逻辑按选项分类：",[105,17488,17489,17495,17500],{},[108,17490,17491,17494],{},[31,17492,17493],{},"data"," 放状态。",[108,17496,17497,17499],{},[31,17498,149],{}," 放计算属性。",[108,17501,17502,17505],{},[31,17503,17504],{},"methods"," 放方法。",[15,17507,17508],{},"Composition API：",[24,17510,17512],{"className":923,"code":17511,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { computed, ref } from 'vue'\n\nconst count = ref(0)\nconst doubleCount = computed(() => count.value * 2)\n\nfunction increase() {\n count.value++\n}\n\u003C\u002Fscript>\n",[31,17513,17514,17524,17534,17538,17554,17578,17582,17590,17596,17600],{"__ignoreMap":29},[34,17515,17516,17518,17520,17522],{"class":36,"line":37},[34,17517,932],{"class":190},[34,17519,1911],{"class":935},[34,17521,1914],{"class":395},[34,17523,939],{"class":190},[34,17525,17526,17528,17530,17532],{"class":36,"line":43},[34,17527,426],{"class":175},[34,17529,16370],{"class":190},[34,17531,432],{"class":175},[34,17533,6891],{"class":186},[34,17535,17536],{"class":36,"line":49},[34,17537,53],{"emptyLinePlaceholder":52},[34,17539,17540,17542,17544,17546,17548,17550,17552],{"class":36,"line":56},[34,17541,176],{"class":175},[34,17543,448],{"class":179},[34,17545,183],{"class":175},[34,17547,453],{"class":395},[34,17549,456],{"class":190},[34,17551,459],{"class":179},[34,17553,6111],{"class":190},[34,17555,17556,17558,17561,17563,17565,17567,17569,17572,17574,17576],{"class":36,"line":62},[34,17557,176],{"class":175},[34,17559,17560],{"class":179}," doubleCount",[34,17562,183],{"class":175},[34,17564,8935],{"class":395},[34,17566,7682],{"class":190},[34,17568,1333],{"class":175},[34,17570,17571],{"class":190}," count.value ",[34,17573,17430],{"class":175},[34,17575,1426],{"class":179},[34,17577,6111],{"class":190},[34,17579,17580],{"class":36,"line":68},[34,17581,53],{"emptyLinePlaceholder":52},[34,17583,17584,17586,17588],{"class":36,"line":74},[34,17585,392],{"class":175},[34,17587,396],{"class":395},[34,17589,399],{"class":190},[34,17591,17592,17594],{"class":36,"line":1971},[34,17593,479],{"class":190},[34,17595,6366],{"class":175},[34,17597,17598],{"class":36,"line":1976},[34,17599,77],{"class":190},[34,17601,17602,17604,17606],{"class":36,"line":1985},[34,17603,1003],{"class":190},[34,17605,1911],{"class":935},[34,17607,939],{"class":190},[15,17609,17610],{},"逻辑按功能组织。和计数器有关的状态、计算属性、函数放在一起。",[19,17612,17614],{"id":17613},"_2-为什么-composition-api-适合复杂组件","2. 为什么 Composition API 适合复杂组件",[15,17616,17617],{},"假设一个组件同时有：",[105,17619,17620,17623,17626,17629,17632],{},[108,17621,17622],{},"搜索逻辑。",[108,17624,17625],{},"分页逻辑。",[108,17627,17628],{},"表单校验。",[108,17630,17631],{},"弹窗状态。",[108,17633,17634],{},"请求状态。",[15,17636,17637,17638,146,17640,146,17642,146,17644,17646,17647,347],{},"Options API 里，搜索相关代码可能分散在 ",[31,17639,17493],{},[31,17641,149],{},[31,17643,17504],{},[31,17645,152],{},"。Composition API 可以把搜索相关内容写在一起，甚至抽成 ",[31,17648,17649],{},"useSearch",[24,17651,17653],{"className":166,"code":17652,"language":168,"meta":29,"style":29},"const keyword = ref('')\nconst filteredItems = computed(...)\nwatch(keyword, ...)\nfunction clearKeyword() {}\n",[31,17654,17655,17671,17688,17699],{"__ignoreMap":29},[34,17656,17657,17659,17661,17663,17665,17667,17669],{"class":36,"line":37},[34,17658,176],{"class":175},[34,17660,11060],{"class":179},[34,17662,183],{"class":175},[34,17664,453],{"class":395},[34,17666,456],{"class":190},[34,17668,7400],{"class":186},[34,17670,6111],{"class":190},[34,17672,17673,17675,17678,17680,17682,17684,17686],{"class":36,"line":43},[34,17674,176],{"class":175},[34,17676,17677],{"class":179}," filteredItems",[34,17679,183],{"class":175},[34,17681,8935],{"class":395},[34,17683,456],{"class":190},[34,17685,1167],{"class":175},[34,17687,6111],{"class":190},[34,17689,17690,17692,17695,17697],{"class":36,"line":49},[34,17691,152],{"class":395},[34,17693,17694],{"class":190},"(keyword, ",[34,17696,1167],{"class":175},[34,17698,6111],{"class":190},[34,17700,17701,17703,17706],{"class":36,"line":56},[34,17702,392],{"class":175},[34,17704,17705],{"class":395}," clearKeyword",[34,17707,17708],{"class":190},"() {}\n",[15,17710,17711],{},"这些代码挨在一起，读起来更像“一个功能块”。",[19,17713,14561,17715,17717],{"id":17714},"_3-script-setup-中的基本写法",[31,17716,141],{}," 中的基本写法",[24,17719,17721],{"className":923,"code":17720,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { computed, ref } from 'vue'\n\nconst count = ref(0)\nconst doubleCount = computed(() => count.value * 2)\n\nfunction increase() {\n count.value++\n}\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cbutton @click=\"increase\">{{ count }}\u003C\u002Fbutton>\n \u003Cp>两倍：{{ doubleCount }}\u003C\u002Fp>\n\u003C\u002Ftemplate>\n",[31,17722,17723,17733,17743,17747,17763,17785,17789,17797,17803,17807,17815,17819,17827,17845,17858],{"__ignoreMap":29},[34,17724,17725,17727,17729,17731],{"class":36,"line":37},[34,17726,932],{"class":190},[34,17728,1911],{"class":935},[34,17730,1914],{"class":395},[34,17732,939],{"class":190},[34,17734,17735,17737,17739,17741],{"class":36,"line":43},[34,17736,426],{"class":175},[34,17738,16370],{"class":190},[34,17740,432],{"class":175},[34,17742,6891],{"class":186},[34,17744,17745],{"class":36,"line":49},[34,17746,53],{"emptyLinePlaceholder":52},[34,17748,17749,17751,17753,17755,17757,17759,17761],{"class":36,"line":56},[34,17750,176],{"class":175},[34,17752,448],{"class":179},[34,17754,183],{"class":175},[34,17756,453],{"class":395},[34,17758,456],{"class":190},[34,17760,459],{"class":179},[34,17762,6111],{"class":190},[34,17764,17765,17767,17769,17771,17773,17775,17777,17779,17781,17783],{"class":36,"line":62},[34,17766,176],{"class":175},[34,17768,17560],{"class":179},[34,17770,183],{"class":175},[34,17772,8935],{"class":395},[34,17774,7682],{"class":190},[34,17776,1333],{"class":175},[34,17778,17571],{"class":190},[34,17780,17430],{"class":175},[34,17782,1426],{"class":179},[34,17784,6111],{"class":190},[34,17786,17787],{"class":36,"line":68},[34,17788,53],{"emptyLinePlaceholder":52},[34,17790,17791,17793,17795],{"class":36,"line":74},[34,17792,392],{"class":175},[34,17794,396],{"class":395},[34,17796,399],{"class":190},[34,17798,17799,17801],{"class":36,"line":1971},[34,17800,479],{"class":190},[34,17802,6366],{"class":175},[34,17804,17805],{"class":36,"line":1976},[34,17806,77],{"class":190},[34,17808,17809,17811,17813],{"class":36,"line":1985},[34,17810,1003],{"class":190},[34,17812,1911],{"class":935},[34,17814,939],{"class":190},[34,17816,17817],{"class":36,"line":1990},[34,17818,53],{"emptyLinePlaceholder":52},[34,17820,17821,17823,17825],{"class":36,"line":1999},[34,17822,932],{"class":190},[34,17824,936],{"class":935},[34,17826,939],{"class":190},[34,17828,17829,17831,17833,17835,17837,17839,17841,17843],{"class":36,"line":2022},[34,17830,944],{"class":190},[34,17832,2004],{"class":935},[34,17834,2007],{"class":395},[34,17836,202],{"class":190},[34,17838,2012],{"class":186},[34,17840,2029],{"class":190},[34,17842,2004],{"class":935},[34,17844,939],{"class":190},[34,17846,17847,17849,17851,17854,17856],{"class":36,"line":2036},[34,17848,944],{"class":190},[34,17850,15],{"class":935},[34,17852,17853],{"class":190},">两倍：{{ doubleCount }}\u003C\u002F",[34,17855,15],{"class":935},[34,17857,939],{"class":190},[34,17859,17860,17862,17864],{"class":36,"line":2603},[34,17861,1003],{"class":190},[34,17863,936],{"class":935},[34,17865,939],{"class":190},[15,17867,17868],{},"重要规则：",[105,17870,17871,17879,17887,17890],{},[108,17872,17873,17874,17876,17877,347],{},"脚本中访问 ",[31,17875,145],{}," 要 ",[31,17878,507],{},[108,17880,17881,17882,17884,17885,347],{},"模板中访问 ",[31,17883,145],{}," 不用 ",[31,17886,507],{},[108,17888,17889],{},"顶层声明的变量和函数可以直接在模板使用。",[108,17891,17892],{},"导入的组件也可以直接在模板使用。",[19,17894,17896],{"id":17895},"_4-组合式函数是什么","4. 组合式函数是什么",[15,17898,17899,17900,17903],{},"组合式函数通常是一个以 ",[31,17901,17902],{},"use"," 开头的函数，它使用 Vue 的响应式 API，并返回状态和方法。",[24,17905,17907],{"className":166,"code":17906,"language":168,"meta":29,"style":29},"\u002F\u002F useCounter.js\nimport { computed, ref } from 'vue'\n\nexport function useCounter(initialValue = 0) {\n const count = ref(initialValue)\n const doubleCount = computed(() => count.value * 2)\n\n function increase() {\n count.value++\n }\n\n function reset() {\n count.value = initialValue\n }\n\n return {\n count,\n doubleCount,\n increase,\n reset\n }\n}\n",[31,17908,17909,17914,17924,17928,17948,17961,17983,17987,17995,18002,18006,18010,18019,18029,18033,18037,18043,18048,18053,18058,18063,18067],{"__ignoreMap":29},[34,17910,17911],{"class":36,"line":37},[34,17912,17913],{"class":301},"\u002F\u002F useCounter.js\n",[34,17915,17916,17918,17920,17922],{"class":36,"line":43},[34,17917,426],{"class":175},[34,17919,16370],{"class":190},[34,17921,432],{"class":175},[34,17923,6891],{"class":186},[34,17925,17926],{"class":36,"line":49},[34,17927,53],{"emptyLinePlaceholder":52},[34,17929,17930,17932,17934,17937,17939,17942,17944,17946],{"class":36,"line":56},[34,17931,2825],{"class":175},[34,17933,2850],{"class":175},[34,17935,17936],{"class":395}," useCounter",[34,17938,456],{"class":190},[34,17940,17941],{"class":1326},"initialValue",[34,17943,183],{"class":175},[34,17945,205],{"class":179},[34,17947,1700],{"class":190},[34,17949,17950,17952,17954,17956,17958],{"class":36,"line":62},[34,17951,5298],{"class":175},[34,17953,448],{"class":179},[34,17955,183],{"class":175},[34,17957,453],{"class":395},[34,17959,17960],{"class":190},"(initialValue)\n",[34,17962,17963,17965,17967,17969,17971,17973,17975,17977,17979,17981],{"class":36,"line":68},[34,17964,5298],{"class":175},[34,17966,17560],{"class":179},[34,17968,183],{"class":175},[34,17970,8935],{"class":395},[34,17972,7682],{"class":190},[34,17974,1333],{"class":175},[34,17976,17571],{"class":190},[34,17978,17430],{"class":175},[34,17980,1426],{"class":179},[34,17982,6111],{"class":190},[34,17984,17985],{"class":36,"line":74},[34,17986,53],{"emptyLinePlaceholder":52},[34,17988,17989,17991,17993],{"class":36,"line":1971},[34,17990,6318],{"class":175},[34,17992,396],{"class":395},[34,17994,399],{"class":190},[34,17996,17997,18000],{"class":36,"line":1976},[34,17998,17999],{"class":190}," count.value",[34,18001,6366],{"class":175},[34,18003,18004],{"class":36,"line":1985},[34,18005,5631],{"class":190},[34,18007,18008],{"class":36,"line":1990},[34,18009,53],{"emptyLinePlaceholder":52},[34,18011,18012,18014,18017],{"class":36,"line":1999},[34,18013,6318],{"class":175},[34,18015,18016],{"class":395}," reset",[34,18018,399],{"class":190},[34,18020,18021,18024,18026],{"class":36,"line":2022},[34,18022,18023],{"class":190}," count.value ",[34,18025,202],{"class":175},[34,18027,18028],{"class":190}," initialValue\n",[34,18030,18031],{"class":36,"line":2036},[34,18032,5631],{"class":190},[34,18034,18035],{"class":36,"line":2603},[34,18036,53],{"emptyLinePlaceholder":52},[34,18038,18039,18041],{"class":36,"line":5537},[34,18040,1510],{"class":175},[34,18042,674],{"class":190},[34,18044,18045],{"class":36,"line":5543},[34,18046,18047],{"class":190}," count,\n",[34,18049,18050],{"class":36,"line":5548},[34,18051,18052],{"class":190}," doubleCount,\n",[34,18054,18055],{"class":36,"line":5564},[34,18056,18057],{"class":190}," increase,\n",[34,18059,18060],{"class":36,"line":5576},[34,18061,18062],{"class":190}," reset\n",[34,18064,18065],{"class":36,"line":5606},[34,18066,5631],{"class":190},[34,18068,18069],{"class":36,"line":5616},[34,18070,77],{"class":190},[15,18072,18073],{},"组件使用：",[24,18075,18077],{"className":923,"code":18076,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { useCounter } from '..\u002Fcomposables\u002FuseCounter'\n\nconst { count, doubleCount, increase, reset } = useCounter(5)\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cp>{{ count }}\u003C\u002Fp>\n \u003Cp>{{ doubleCount }}\u003C\u002Fp>\n \u003Cbutton @click=\"increase\">增加\u003C\u002Fbutton>\n \u003Cbutton @click=\"reset\">重置\u003C\u002Fbutton>\n\u003C\u002Ftemplate>\n",[31,18078,18079,18089,18100,18104,18139,18147,18151,18159,18171,18184,18202,18222],{"__ignoreMap":29},[34,18080,18081,18083,18085,18087],{"class":36,"line":37},[34,18082,932],{"class":190},[34,18084,1911],{"class":935},[34,18086,1914],{"class":395},[34,18088,939],{"class":190},[34,18090,18091,18093,18095,18097],{"class":36,"line":43},[34,18092,426],{"class":175},[34,18094,4740],{"class":190},[34,18096,432],{"class":175},[34,18098,18099],{"class":186}," '..\u002Fcomposables\u002FuseCounter'\n",[34,18101,18102],{"class":36,"line":49},[34,18103,53],{"emptyLinePlaceholder":52},[34,18105,18106,18108,18110,18112,18114,18117,18119,18121,18123,18126,18128,18130,18132,18134,18137],{"class":36,"line":56},[34,18107,176],{"class":175},[34,18109,1164],{"class":190},[34,18111,2770],{"class":179},[34,18113,616],{"class":190},[34,18115,18116],{"class":179},"doubleCount",[34,18118,616],{"class":190},[34,18120,2048],{"class":179},[34,18122,616],{"class":190},[34,18124,18125],{"class":179},"reset",[34,18127,2313],{"class":190},[34,18129,202],{"class":175},[34,18131,17936],{"class":395},[34,18133,456],{"class":190},[34,18135,18136],{"class":179},"5",[34,18138,6111],{"class":190},[34,18140,18141,18143,18145],{"class":36,"line":62},[34,18142,1003],{"class":190},[34,18144,1911],{"class":935},[34,18146,939],{"class":190},[34,18148,18149],{"class":36,"line":68},[34,18150,53],{"emptyLinePlaceholder":52},[34,18152,18153,18155,18157],{"class":36,"line":74},[34,18154,932],{"class":190},[34,18156,936],{"class":935},[34,18158,939],{"class":190},[34,18160,18161,18163,18165,18167,18169],{"class":36,"line":1971},[34,18162,944],{"class":190},[34,18164,15],{"class":935},[34,18166,2029],{"class":190},[34,18168,15],{"class":935},[34,18170,939],{"class":190},[34,18172,18173,18175,18177,18180,18182],{"class":36,"line":1976},[34,18174,944],{"class":190},[34,18176,15],{"class":935},[34,18178,18179],{"class":190},">{{ doubleCount }}\u003C\u002F",[34,18181,15],{"class":935},[34,18183,939],{"class":190},[34,18185,18186,18188,18190,18192,18194,18196,18198,18200],{"class":36,"line":1985},[34,18187,944],{"class":190},[34,18189,2004],{"class":935},[34,18191,2007],{"class":395},[34,18193,202],{"class":190},[34,18195,2012],{"class":186},[34,18197,2015],{"class":190},[34,18199,2004],{"class":935},[34,18201,939],{"class":190},[34,18203,18204,18206,18208,18210,18212,18215,18218,18220],{"class":36,"line":1990},[34,18205,944],{"class":190},[34,18207,2004],{"class":935},[34,18209,2007],{"class":395},[34,18211,202],{"class":190},[34,18213,18214],{"class":186},"\"reset\"",[34,18216,18217],{"class":190},">重置\u003C\u002F",[34,18219,2004],{"class":935},[34,18221,939],{"class":190},[34,18223,18224,18226,18228],{"class":36,"line":1999},[34,18225,1003],{"class":190},[34,18227,936],{"class":935},[34,18229,939],{"class":190},[19,18231,18233,18234],{"id":18232},"_5-为什么组合式函数需要-export","5. 为什么组合式函数需要 ",[31,18235,2825],{},[15,18237,18238,18241],{},[31,18239,18240],{},"useCounter.js"," 是一个单独模块。组件要导入它，就必须从模块中导出。",[24,18243,18245],{"className":166,"code":18244,"language":168,"meta":29,"style":29},"export function useCounter(initialValue = 0) {\n \u002F\u002F ...\n}\n",[31,18246,18247,18265,18270],{"__ignoreMap":29},[34,18248,18249,18251,18253,18255,18257,18259,18261,18263],{"class":36,"line":37},[34,18250,2825],{"class":175},[34,18252,2850],{"class":175},[34,18254,17936],{"class":395},[34,18256,456],{"class":190},[34,18258,17941],{"class":1326},[34,18260,183],{"class":175},[34,18262,205],{"class":179},[34,18264,1700],{"class":190},[34,18266,18267],{"class":36,"line":43},[34,18268,18269],{"class":301}," \u002F\u002F ...\n",[34,18271,18272],{"class":36,"line":49},[34,18273,77],{"class":190},[15,18275,18276],{},"然后组件才能：",[24,18278,18280],{"className":166,"code":18279,"language":168,"meta":29,"style":29},"import { useCounter } from '..\u002Fcomposables\u002FuseCounter'\n",[31,18281,18282],{"__ignoreMap":29},[34,18283,18284,18286,18288,18290],{"class":36,"line":37},[34,18285,426],{"class":175},[34,18287,4740],{"class":190},[34,18289,432],{"class":175},[34,18291,18099],{"class":186},[15,18293,18294,18295,18297,18298,18300],{},"如果不写 ",[31,18296,2825],{},"，这个函数只属于 ",[31,18299,18240],{}," 文件内部，其他文件不能按名字导入。",[15,18302,18303],{},"这和第 1 章讲的模块规则完全一致。",[19,18305,18307],{"id":18306},"_6-每次调用组合式函数都会创建新状态吗","6. 每次调用组合式函数都会创建新状态吗",[15,18309,18310],{},"看函数内部状态写在哪里。",[15,18312,18313],{},"如果状态写在函数内部：",[24,18315,18317],{"className":166,"code":18316,"language":168,"meta":29,"style":29},"export function useCounter() {\n const count = ref(0)\n\n function increase() {\n count.value++\n }\n\n return { count, increase }\n}\n",[31,18318,18319,18329,18345,18349,18357,18363,18367,18371,18378],{"__ignoreMap":29},[34,18320,18321,18323,18325,18327],{"class":36,"line":37},[34,18322,2825],{"class":175},[34,18324,2850],{"class":175},[34,18326,17936],{"class":395},[34,18328,399],{"class":190},[34,18330,18331,18333,18335,18337,18339,18341,18343],{"class":36,"line":43},[34,18332,5298],{"class":175},[34,18334,448],{"class":179},[34,18336,183],{"class":175},[34,18338,453],{"class":395},[34,18340,456],{"class":190},[34,18342,459],{"class":179},[34,18344,6111],{"class":190},[34,18346,18347],{"class":36,"line":49},[34,18348,53],{"emptyLinePlaceholder":52},[34,18350,18351,18353,18355],{"class":36,"line":56},[34,18352,6318],{"class":175},[34,18354,396],{"class":395},[34,18356,399],{"class":190},[34,18358,18359,18361],{"class":36,"line":62},[34,18360,17999],{"class":190},[34,18362,6366],{"class":175},[34,18364,18365],{"class":36,"line":68},[34,18366,5631],{"class":190},[34,18368,18369],{"class":36,"line":74},[34,18370,53],{"emptyLinePlaceholder":52},[34,18372,18373,18375],{"class":36,"line":1971},[34,18374,1510],{"class":175},[34,18376,18377],{"class":190}," { count, increase }\n",[34,18379,18380],{"class":36,"line":1976},[34,18381,77],{"class":190},[15,18383,18384,18385,18388,18389,347],{},"每个组件调用 ",[31,18386,18387],{},"useCounter()"," 都会得到自己的 ",[31,18390,2770],{},[24,18392,18394],{"className":166,"code":18393,"language":168,"meta":29,"style":29},"const counterA = useCounter()\nconst counterB = useCounter()\n",[31,18395,18396,18409],{"__ignoreMap":29},[34,18397,18398,18400,18403,18405,18407],{"class":36,"line":37},[34,18399,176],{"class":175},[34,18401,18402],{"class":179}," counterA",[34,18404,183],{"class":175},[34,18406,17936],{"class":395},[34,18408,6374],{"class":190},[34,18410,18411,18413,18416,18418,18420],{"class":36,"line":43},[34,18412,176],{"class":175},[34,18414,18415],{"class":179}," counterB",[34,18417,183],{"class":175},[34,18419,17936],{"class":395},[34,18421,6374],{"class":190},[15,18423,18424,258,18427,18430],{},[31,18425,18426],{},"counterA.count",[31,18428,18429],{},"counterB.count"," 是两份状态。",[15,18432,18433],{},"如果状态写在函数外部：",[24,18435,18437],{"className":166,"code":18436,"language":168,"meta":29,"style":29},"const count = ref(0)\n\nexport function useSharedCounter() {\n function increase() {\n count.value++\n }\n\n return { count, increase }\n}\n",[31,18438,18439,18455,18459,18470,18478,18484,18488,18492,18498],{"__ignoreMap":29},[34,18440,18441,18443,18445,18447,18449,18451,18453],{"class":36,"line":37},[34,18442,176],{"class":175},[34,18444,448],{"class":179},[34,18446,183],{"class":175},[34,18448,453],{"class":395},[34,18450,456],{"class":190},[34,18452,459],{"class":179},[34,18454,6111],{"class":190},[34,18456,18457],{"class":36,"line":43},[34,18458,53],{"emptyLinePlaceholder":52},[34,18460,18461,18463,18465,18468],{"class":36,"line":49},[34,18462,2825],{"class":175},[34,18464,2850],{"class":175},[34,18466,18467],{"class":395}," useSharedCounter",[34,18469,399],{"class":190},[34,18471,18472,18474,18476],{"class":36,"line":56},[34,18473,6318],{"class":175},[34,18475,396],{"class":395},[34,18477,399],{"class":190},[34,18479,18480,18482],{"class":36,"line":62},[34,18481,17999],{"class":190},[34,18483,6366],{"class":175},[34,18485,18486],{"class":36,"line":68},[34,18487,5631],{"class":190},[34,18489,18490],{"class":36,"line":74},[34,18491,53],{"emptyLinePlaceholder":52},[34,18493,18494,18496],{"class":36,"line":1971},[34,18495,1510],{"class":175},[34,18497,18377],{"class":190},[34,18499,18500],{"class":36,"line":1976},[34,18501,77],{"class":190},[15,18503,18504,18505,18507],{},"所有调用都会共享同一个 ",[31,18506,2770],{},"。这有时有用，但要小心，因为它更像全局状态。",[19,18509,18511,18512],{"id":18510},"_7-示例usetoggle","7. 示例：",[31,18513,18514],{},"useToggle",[24,18516,18518],{"className":166,"code":18517,"language":168,"meta":29,"style":29},"import { ref } from 'vue'\n\nexport function useToggle(initialValue = false) {\n const value = ref(initialValue)\n\n function toggle() {\n value.value = !value.value\n }\n\n function setTrue() {\n value.value = true\n }\n\n function setFalse() {\n value.value = false\n }\n\n return {\n value,\n toggle,\n setTrue,\n setFalse\n }\n}\n",[31,18519,18520,18530,18534,18553,18565,18569,18578,18590,18594,18598,18607,18615,18619,18623,18632,18641,18645,18649,18655,18660,18665,18670,18675,18679],{"__ignoreMap":29},[34,18521,18522,18524,18526,18528],{"class":36,"line":37},[34,18523,426],{"class":175},[34,18525,429],{"class":190},[34,18527,432],{"class":175},[34,18529,6891],{"class":186},[34,18531,18532],{"class":36,"line":43},[34,18533,53],{"emptyLinePlaceholder":52},[34,18535,18536,18538,18540,18543,18545,18547,18549,18551],{"class":36,"line":49},[34,18537,2825],{"class":175},[34,18539,2850],{"class":175},[34,18541,18542],{"class":395}," useToggle",[34,18544,456],{"class":190},[34,18546,17941],{"class":1326},[34,18548,183],{"class":175},[34,18550,576],{"class":179},[34,18552,1700],{"class":190},[34,18554,18555,18557,18559,18561,18563],{"class":36,"line":56},[34,18556,5298],{"class":175},[34,18558,16940],{"class":179},[34,18560,183],{"class":175},[34,18562,453],{"class":395},[34,18564,17960],{"class":190},[34,18566,18567],{"class":36,"line":62},[34,18568,53],{"emptyLinePlaceholder":52},[34,18570,18571,18573,18576],{"class":36,"line":68},[34,18572,6318],{"class":175},[34,18574,18575],{"class":395}," toggle",[34,18577,399],{"class":190},[34,18579,18580,18583,18585,18587],{"class":36,"line":74},[34,18581,18582],{"class":190}," value.value ",[34,18584,202],{"class":175},[34,18586,1336],{"class":175},[34,18588,18589],{"class":190},"value.value\n",[34,18591,18592],{"class":36,"line":1971},[34,18593,5631],{"class":190},[34,18595,18596],{"class":36,"line":1976},[34,18597,53],{"emptyLinePlaceholder":52},[34,18599,18600,18602,18605],{"class":36,"line":1985},[34,18601,6318],{"class":175},[34,18603,18604],{"class":395}," setTrue",[34,18606,399],{"class":190},[34,18608,18609,18611,18613],{"class":36,"line":1990},[34,18610,18582],{"class":190},[34,18612,202],{"class":175},[34,18614,12260],{"class":179},[34,18616,18617],{"class":36,"line":1999},[34,18618,5631],{"class":190},[34,18620,18621],{"class":36,"line":2022},[34,18622,53],{"emptyLinePlaceholder":52},[34,18624,18625,18627,18630],{"class":36,"line":2036},[34,18626,6318],{"class":175},[34,18628,18629],{"class":395}," setFalse",[34,18631,399],{"class":190},[34,18633,18634,18636,18638],{"class":36,"line":2603},[34,18635,18582],{"class":190},[34,18637,202],{"class":175},[34,18639,18640],{"class":179}," false\n",[34,18642,18643],{"class":36,"line":5537},[34,18644,5631],{"class":190},[34,18646,18647],{"class":36,"line":5543},[34,18648,53],{"emptyLinePlaceholder":52},[34,18650,18651,18653],{"class":36,"line":5548},[34,18652,1510],{"class":175},[34,18654,674],{"class":190},[34,18656,18657],{"class":36,"line":5564},[34,18658,18659],{"class":190}," value,\n",[34,18661,18662],{"class":36,"line":5576},[34,18663,18664],{"class":190}," toggle,\n",[34,18666,18667],{"class":36,"line":5606},[34,18668,18669],{"class":190}," setTrue,\n",[34,18671,18672],{"class":36,"line":5616},[34,18673,18674],{"class":190}," setFalse\n",[34,18676,18677],{"class":36,"line":5628},[34,18678,5631],{"class":190},[34,18680,18681],{"class":36,"line":5634},[34,18682,77],{"class":190},[15,18684,18073],{},[24,18686,18688],{"className":166,"code":18687,"language":168,"meta":29,"style":29},"const modal = useToggle(false)\n",[31,18689,18690],{"__ignoreMap":29},[34,18691,18692,18694,18697,18699,18701,18703,18705],{"class":36,"line":37},[34,18693,176],{"class":175},[34,18695,18696],{"class":179}," modal",[34,18698,183],{"class":175},[34,18700,18542],{"class":395},[34,18702,456],{"class":190},[34,18704,702],{"class":179},[34,18706,6111],{"class":190},[15,18708,1589],{},[24,18710,18712],{"className":923,"code":18711,"language":925,"meta":29,"style":29},"\u003Cbutton @click=\"modal.setTrue\">打开\u003C\u002Fbutton>\n\u003Cdialog v-if=\"modal.value\">...\u003C\u002Fdialog>\n",[31,18713,18714,18739],{"__ignoreMap":29},[34,18715,18716,18718,18720,18722,18724,18726,18728,18731,18733,18735,18737],{"class":36,"line":37},[34,18717,932],{"class":190},[34,18719,2004],{"class":935},[34,18721,9628],{"class":190},[34,18723,9631],{"class":395},[34,18725,202],{"class":190},[34,18727,1608],{"class":186},[34,18729,18730],{"class":190},"modal.setTrue",[34,18732,1608],{"class":186},[34,18734,9848],{"class":190},[34,18736,2004],{"class":935},[34,18738,939],{"class":190},[34,18740,18741,18743,18746,18748,18750,18752,18755,18757,18760,18762],{"class":36,"line":43},[34,18742,932],{"class":190},[34,18744,18745],{"class":935},"dialog",[34,18747,2168],{"class":175},[34,18749,202],{"class":190},[34,18751,1608],{"class":186},[34,18753,18754],{"class":190},"modal.value",[34,18756,1608],{"class":186},[34,18758,18759],{"class":190},">...\u003C\u002F",[34,18761,18745],{"class":935},[34,18763,939],{"class":190},[19,18765,18767,18768],{"id":18766},"_8-示例uselocalstorage","8. 示例：",[31,18769,18770],{},"useLocalStorage",[15,18772,18773,18774,347],{},"目标：让一个响应式状态自动保存到浏览器 ",[31,18775,10566],{},[24,18777,18779],{"className":166,"code":18778,"language":168,"meta":29,"style":29},"import { ref, watch } from 'vue'\n\nfunction readStorage(key, defaultValue) {\n try {\n const rawValue = localStorage.getItem(key)\n return rawValue ? JSON.parse(rawValue) : defaultValue\n } catch {\n return defaultValue\n }\n}\n\nexport function useLocalStorage(key, defaultValue) {\n const state = ref(readStorage(key, defaultValue))\n\n watch(\n state,\n (value) => {\n localStorage.setItem(key, JSON.stringify(value))\n },\n { deep: true }\n )\n\n return state\n}\n",[31,18780,18781,18792,18796,18814,18820,18838,18863,18871,18877,18881,18885,18889,18908,18927,18931,18939,18944,18957,18978,18983,18992,18997,19001,19008],{"__ignoreMap":29},[34,18782,18783,18785,18788,18790],{"class":36,"line":37},[34,18784,426],{"class":175},[34,18786,18787],{"class":190}," { ref, watch } ",[34,18789,432],{"class":175},[34,18791,6891],{"class":186},[34,18793,18794],{"class":36,"line":43},[34,18795,53],{"emptyLinePlaceholder":52},[34,18797,18798,18800,18803,18805,18807,18809,18812],{"class":36,"line":49},[34,18799,392],{"class":175},[34,18801,18802],{"class":395}," readStorage",[34,18804,456],{"class":190},[34,18806,1624],{"class":1326},[34,18808,616],{"class":190},[34,18810,18811],{"class":1326},"defaultValue",[34,18813,1700],{"class":190},[34,18815,18816,18818],{"class":36,"line":56},[34,18817,5468],{"class":175},[34,18819,674],{"class":190},[34,18821,18822,18824,18827,18829,18832,18835],{"class":36,"line":62},[34,18823,5475],{"class":175},[34,18825,18826],{"class":179}," rawValue",[34,18828,183],{"class":175},[34,18830,18831],{"class":190}," localStorage.",[34,18833,18834],{"class":395},"getItem",[34,18836,18837],{"class":190},"(key)\n",[34,18839,18840,18842,18845,18847,18850,18852,18855,18858,18860],{"class":36,"line":68},[34,18841,8601],{"class":175},[34,18843,18844],{"class":190}," rawValue ",[34,18846,6701],{"class":175},[34,18848,18849],{"class":179}," JSON",[34,18851,2547],{"class":190},[34,18853,18854],{"class":395},"parse",[34,18856,18857],{"class":190},"(rawValue) ",[34,18859,5598],{"class":175},[34,18861,18862],{"class":190}," defaultValue\n",[34,18864,18865,18867,18869],{"class":36,"line":74},[34,18866,5567],{"class":190},[34,18868,5570],{"class":175},[34,18870,674],{"class":190},[34,18872,18873,18875],{"class":36,"line":1971},[34,18874,8601],{"class":175},[34,18876,18862],{"class":190},[34,18878,18879],{"class":36,"line":1976},[34,18880,5631],{"class":190},[34,18882,18883],{"class":36,"line":1985},[34,18884,77],{"class":190},[34,18886,18887],{"class":36,"line":1990},[34,18888,53],{"emptyLinePlaceholder":52},[34,18890,18891,18893,18895,18898,18900,18902,18904,18906],{"class":36,"line":1999},[34,18892,2825],{"class":175},[34,18894,2850],{"class":175},[34,18896,18897],{"class":395}," useLocalStorage",[34,18899,456],{"class":190},[34,18901,1624],{"class":1326},[34,18903,616],{"class":190},[34,18905,18811],{"class":1326},[34,18907,1700],{"class":190},[34,18909,18910,18912,18915,18917,18919,18921,18924],{"class":36,"line":2022},[34,18911,5298],{"class":175},[34,18913,18914],{"class":179}," state",[34,18916,183],{"class":175},[34,18918,453],{"class":395},[34,18920,456],{"class":190},[34,18922,18923],{"class":395},"readStorage",[34,18925,18926],{"class":190},"(key, defaultValue))\n",[34,18928,18929],{"class":36,"line":2036},[34,18930,53],{"emptyLinePlaceholder":52},[34,18932,18933,18936],{"class":36,"line":2603},[34,18934,18935],{"class":395}," watch",[34,18937,18938],{"class":190},"(\n",[34,18940,18941],{"class":36,"line":5537},[34,18942,18943],{"class":190}," state,\n",[34,18945,18946,18949,18951,18953,18955],{"class":36,"line":5543},[34,18947,18948],{"class":190}," (",[34,18950,2533],{"class":1326},[34,18952,1330],{"class":190},[34,18954,1333],{"class":175},[34,18956,674],{"class":190},[34,18958,18959,18962,18964,18967,18970,18972,18975],{"class":36,"line":5548},[34,18960,18961],{"class":190}," localStorage.",[34,18963,10670],{"class":395},[34,18965,18966],{"class":190},"(key, ",[34,18968,18969],{"class":179},"JSON",[34,18971,2547],{"class":190},[34,18973,18974],{"class":395},"stringify",[34,18976,18977],{"class":190},"(value))\n",[34,18979,18980],{"class":36,"line":5564},[34,18981,18982],{"class":190}," },\n",[34,18984,18985,18988,18990],{"class":36,"line":5576},[34,18986,18987],{"class":190}," { deep: ",[34,18989,758],{"class":179},[34,18991,6564],{"class":190},[34,18993,18994],{"class":36,"line":5606},[34,18995,18996],{"class":190}," )\n",[34,18998,18999],{"class":36,"line":5616},[34,19000,53],{"emptyLinePlaceholder":52},[34,19002,19003,19005],{"class":36,"line":5628},[34,19004,1510],{"class":175},[34,19006,19007],{"class":190}," state\n",[34,19009,19010],{"class":36,"line":5634},[34,19011,77],{"class":190},[15,19013,19014],{},"逐步理解：",[105,19016,19017,19022,19025,19030,19037,19044],{},[108,19018,19019,19021],{},[31,19020,18923],{}," 先尝试从浏览器读取已有值。",[108,19023,19024],{},"读不到就使用默认值。",[108,19026,19027,7481],{},[31,19028,19029],{},"state",[108,19031,19032,19034,19035,347],{},[31,19033,152],{}," 观察 ",[31,19036,19029],{},[108,19038,19039,19041,19042,347],{},[31,19040,19029],{}," 一变化，就写回 ",[31,19043,10566],{},[108,19045,19046,19049],{},[31,19047,19048],{},"{ deep: true }"," 用来监听数组或对象内部变化。",[15,19051,18073],{},[24,19053,19055],{"className":166,"code":19054,"language":168,"meta":29,"style":29},"const todos = useLocalStorage('todos', [])\n",[31,19056,19057],{"__ignoreMap":29},[34,19058,19059,19061,19063,19065,19067,19069,19072],{"class":36,"line":37},[34,19060,176],{"class":175},[34,19062,12962],{"class":179},[34,19064,183],{"class":175},[34,19066,18897],{"class":395},[34,19068,456],{"class":190},[34,19070,19071],{"class":186},"'todos'",[34,19073,19074],{"class":190},", [])\n",[15,19076,19077],{},"之后你像普通 ref 一样使用：",[24,19079,19081],{"className":166,"code":19080,"language":168,"meta":29,"style":29},"todos.value.push({ id: 1, title: '学习持久化', done: false })\n",[31,19082,19083],{"__ignoreMap":29},[34,19084,19085,19088,19090,19093,19095,19097,19100,19102,19104],{"class":36,"line":37},[34,19086,19087],{"class":190},"todos.value.",[34,19089,1262],{"class":395},[34,19091,19092],{"class":190},"({ id: ",[34,19094,640],{"class":179},[34,19096,749],{"class":190},[34,19098,19099],{"class":186},"'学习持久化'",[34,19101,755],{"class":190},[34,19103,702],{"class":179},[34,19105,19106],{"class":190}," })\n",[15,19108,19109],{},"刷新后数据还在。",[19,19111,19113,19114],{"id":19112},"_9-示例usefetch","9. 示例：",[31,19115,19116],{},"useFetch",[15,19118,19119],{},"目标：封装请求的 loading、error、data。",[24,19121,19123],{"className":166,"code":19122,"language":168,"meta":29,"style":29},"import { ref } from 'vue'\n\nexport function useFetch(requester) {\n const data = ref(null)\n const loading = ref(false)\n const error = ref('')\n\n async function execute(...args) {\n loading.value = true\n error.value = ''\n\n try {\n data.value = await requester(...args)\n } catch (err) {\n error.value = err instanceof Error ? err.message : '请求失败'\n } finally {\n loading.value = false\n }\n }\n\n return {\n data,\n loading,\n error,\n execute\n }\n}\n",[31,19124,19125,19135,19139,19155,19171,19187,19203,19207,19226,19234,19242,19246,19253,19272,19281,19303,19311,19320,19324,19328,19332,19338,19343,19348,19353,19358,19362],{"__ignoreMap":29},[34,19126,19127,19129,19131,19133],{"class":36,"line":37},[34,19128,426],{"class":175},[34,19130,429],{"class":190},[34,19132,432],{"class":175},[34,19134,6891],{"class":186},[34,19136,19137],{"class":36,"line":43},[34,19138,53],{"emptyLinePlaceholder":52},[34,19140,19141,19143,19145,19148,19150,19153],{"class":36,"line":49},[34,19142,2825],{"class":175},[34,19144,2850],{"class":175},[34,19146,19147],{"class":395}," useFetch",[34,19149,456],{"class":190},[34,19151,19152],{"class":1326},"requester",[34,19154,1700],{"class":190},[34,19156,19157,19159,19161,19163,19165,19167,19169],{"class":36,"line":56},[34,19158,5298],{"class":175},[34,19160,5410],{"class":179},[34,19162,183],{"class":175},[34,19164,453],{"class":395},[34,19166,456],{"class":190},[34,19168,5419],{"class":179},[34,19170,6111],{"class":190},[34,19172,19173,19175,19177,19179,19181,19183,19185],{"class":36,"line":62},[34,19174,5298],{"class":175},[34,19176,5375],{"class":179},[34,19178,183],{"class":175},[34,19180,453],{"class":395},[34,19182,456],{"class":190},[34,19184,702],{"class":179},[34,19186,6111],{"class":190},[34,19188,19189,19191,19193,19195,19197,19199,19201],{"class":36,"line":68},[34,19190,5298],{"class":175},[34,19192,5392],{"class":179},[34,19194,183],{"class":175},[34,19196,453],{"class":395},[34,19198,456],{"class":190},[34,19200,7400],{"class":186},[34,19202,6111],{"class":190},[34,19204,19205],{"class":36,"line":74},[34,19206,53],{"emptyLinePlaceholder":52},[34,19208,19209,19212,19214,19217,19219,19221,19224],{"class":36,"line":1971},[34,19210,19211],{"class":175}," async",[34,19213,2850],{"class":175},[34,19215,19216],{"class":395}," execute",[34,19218,456],{"class":190},[34,19220,1167],{"class":175},[34,19222,19223],{"class":1326},"args",[34,19225,1700],{"class":190},[34,19227,19228,19230,19232],{"class":36,"line":1976},[34,19229,5619],{"class":190},[34,19231,202],{"class":175},[34,19233,12260],{"class":179},[34,19235,19236,19238,19240],{"class":36,"line":1985},[34,19237,5579],{"class":190},[34,19239,202],{"class":175},[34,19241,6645],{"class":186},[34,19243,19244],{"class":36,"line":1990},[34,19245,53],{"emptyLinePlaceholder":52},[34,19247,19248,19251],{"class":36,"line":1999},[34,19249,19250],{"class":175}," try",[34,19252,674],{"class":190},[34,19254,19255,19258,19260,19262,19265,19267,19269],{"class":36,"line":2022},[34,19256,19257],{"class":190}," data.value ",[34,19259,202],{"class":175},[34,19261,5306],{"class":175},[34,19263,19264],{"class":395}," requester",[34,19266,456],{"class":190},[34,19268,1167],{"class":175},[34,19270,19271],{"class":190},"args)\n",[34,19273,19274,19277,19279],{"class":36,"line":2036},[34,19275,19276],{"class":190}," } ",[34,19278,5570],{"class":175},[34,19280,5573],{"class":190},[34,19282,19283,19286,19288,19290,19292,19294,19296,19298,19300],{"class":36,"line":2603},[34,19284,19285],{"class":190}," error.value ",[34,19287,202],{"class":175},[34,19289,5584],{"class":190},[34,19291,5587],{"class":175},[34,19293,5517],{"class":395},[34,19295,5592],{"class":175},[34,19297,5595],{"class":190},[34,19299,5598],{"class":175},[34,19301,19302],{"class":186}," '请求失败'\n",[34,19304,19305,19307,19309],{"class":36,"line":5537},[34,19306,19276],{"class":190},[34,19308,5611],{"class":175},[34,19310,674],{"class":190},[34,19312,19313,19316,19318],{"class":36,"line":5543},[34,19314,19315],{"class":190}," loading.value ",[34,19317,202],{"class":175},[34,19319,18640],{"class":179},[34,19321,19322],{"class":36,"line":5548},[34,19323,5540],{"class":190},[34,19325,19326],{"class":36,"line":5564},[34,19327,5631],{"class":190},[34,19329,19330],{"class":36,"line":5576},[34,19331,53],{"emptyLinePlaceholder":52},[34,19333,19334,19336],{"class":36,"line":5606},[34,19335,1510],{"class":175},[34,19337,674],{"class":190},[34,19339,19340],{"class":36,"line":5616},[34,19341,19342],{"class":190}," data,\n",[34,19344,19345],{"class":36,"line":5628},[34,19346,19347],{"class":190}," loading,\n",[34,19349,19350],{"class":36,"line":5634},[34,19351,19352],{"class":190}," error,\n",[34,19354,19355],{"class":36,"line":6420},[34,19356,19357],{"class":190}," execute\n",[34,19359,19360],{"class":36,"line":11294},[34,19361,5631],{"class":190},[34,19363,19364],{"class":36,"line":11321},[34,19365,77],{"class":190},[15,19367,13233],{},[24,19369,19371],{"className":166,"code":19370,"language":168,"meta":29,"style":29},"import { useFetch } from '..\u002Fcomposables\u002FuseFetch'\nimport { getMockLessons } from '..\u002Fdata\u002FmockApi'\n\nconst { data, loading, error, execute } = useFetch(getMockLessons)\n\nonMounted(() => {\n execute()\n})\n",[31,19372,19373,19385,19397,19401,19431,19435,19445,19452],{"__ignoreMap":29},[34,19374,19375,19377,19380,19382],{"class":36,"line":37},[34,19376,426],{"class":175},[34,19378,19379],{"class":190}," { useFetch } ",[34,19381,432],{"class":175},[34,19383,19384],{"class":186}," '..\u002Fcomposables\u002FuseFetch'\n",[34,19386,19387,19389,19392,19394],{"class":36,"line":43},[34,19388,426],{"class":175},[34,19390,19391],{"class":190}," { getMockLessons } ",[34,19393,432],{"class":175},[34,19395,19396],{"class":186}," '..\u002Fdata\u002FmockApi'\n",[34,19398,19399],{"class":36,"line":49},[34,19400,53],{"emptyLinePlaceholder":52},[34,19402,19403,19405,19407,19409,19411,19413,19415,19417,19419,19422,19424,19426,19428],{"class":36,"line":56},[34,19404,176],{"class":175},[34,19406,1164],{"class":190},[34,19408,17493],{"class":179},[34,19410,616],{"class":190},[34,19412,2175],{"class":179},[34,19414,616],{"class":190},[34,19416,2200],{"class":179},[34,19418,616],{"class":190},[34,19420,19421],{"class":179},"execute",[34,19423,2313],{"class":190},[34,19425,202],{"class":175},[34,19427,19147],{"class":395},[34,19429,19430],{"class":190},"(getMockLessons)\n",[34,19432,19433],{"class":36,"line":62},[34,19434,53],{"emptyLinePlaceholder":52},[34,19436,19437,19439,19441,19443],{"class":36,"line":68},[34,19438,7679],{"class":395},[34,19440,7682],{"class":190},[34,19442,1333],{"class":175},[34,19444,674],{"class":190},[34,19446,19447,19450],{"class":36,"line":74},[34,19448,19449],{"class":395}," execute",[34,19451,6374],{"class":190},[34,19453,19454],{"class":36,"line":1971},[34,19455,6153],{"class":190},[15,19457,1589],{},[24,19459,19461],{"className":923,"code":19460,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"loading\">加载中...\u003C\u002Fp>\n\u003Cp v-else-if=\"error\">{{ error }}\u003C\u002Fp>\n\u003CLessonList v-else :lessons=\"data\" \u002F>\n",[31,19462,19463,19485,19507],{"__ignoreMap":29},[34,19464,19465,19467,19469,19471,19473,19475,19477,19479,19481,19483],{"class":36,"line":37},[34,19466,932],{"class":190},[34,19468,15],{"class":935},[34,19470,2168],{"class":175},[34,19472,202],{"class":190},[34,19474,1608],{"class":186},[34,19476,2175],{"class":190},[34,19478,1608],{"class":186},[34,19480,2180],{"class":190},[34,19482,15],{"class":935},[34,19484,939],{"class":190},[34,19486,19487,19489,19491,19493,19495,19497,19499,19501,19503,19505],{"class":36,"line":43},[34,19488,932],{"class":190},[34,19490,15],{"class":935},[34,19492,2193],{"class":175},[34,19494,202],{"class":190},[34,19496,1608],{"class":186},[34,19498,2200],{"class":190},[34,19500,1608],{"class":186},[34,19502,2205],{"class":190},[34,19504,15],{"class":935},[34,19506,939],{"class":190},[34,19508,19509,19511,19514,19516,19518,19520,19522,19524,19526,19528],{"class":36,"line":49},[34,19510,932],{"class":190},[34,19512,19513],{"class":935},"LessonList",[34,19515,2218],{"class":175},[34,19517,1621],{"class":190},[34,19519,10064],{"class":395},[34,19521,202],{"class":190},[34,19523,1608],{"class":186},[34,19525,17493],{"class":190},[34,19527,1608],{"class":186},[34,19529,3469],{"class":190},[19,19531,19533],{"id":19532},"_10-什么时候应该抽组合式函数","10. 什么时候应该抽组合式函数",[15,19535,19536],{},"适合抽：",[105,19538,19539,19542,19545,19548],{},[108,19540,19541],{},"多个组件使用同样逻辑。",[108,19543,19544],{},"逻辑和 UI 没有强绑定。",[108,19546,19547],{},"组件太长，其中一组逻辑可以独立命名。",[108,19549,19550],{},"你能清楚描述这个函数负责什么。",[15,19552,19553],{},"例子：",[105,19555,19556,19561,19565,19569,19573,19578],{},[108,19557,19558],{},[31,19559,19560],{},"useCounter",[108,19562,19563],{},[31,19564,18514],{},[108,19566,19567],{},[31,19568,18770],{},[108,19570,19571],{},[31,19572,19116],{},[108,19574,19575],{},[31,19576,19577],{},"useDebounce",[108,19579,19580],{},[31,19581,19582],{},"useMousePosition",[15,19584,19585],{},"不适合抽：",[105,19587,19588,19591,19594],{},[108,19589,19590],{},"只用一次，而且抽出来更难读。",[108,19592,19593],{},"需要依赖很多组件内部变量。",[108,19595,19596],{},"抽出来只是为了“看起来像高级代码”。",[19,19598,19600],{"id":19599},"_11-组合式函数和普通工具函数区别","11. 组合式函数和普通工具函数区别",[15,19602,19603],{},"普通工具函数：",[24,19605,19607],{"className":166,"code":19606,"language":168,"meta":29,"style":29},"export function formatMinutes(minutes) {\n return `${minutes} 分钟`\n}\n",[31,19608,19609,19624,19636],{"__ignoreMap":29},[34,19610,19611,19613,19615,19618,19620,19622],{"class":36,"line":37},[34,19612,2825],{"class":175},[34,19614,2850],{"class":175},[34,19616,19617],{"class":395}," formatMinutes",[34,19619,456],{"class":190},[34,19621,15036],{"class":1326},[34,19623,1700],{"class":190},[34,19625,19626,19628,19631,19633],{"class":36,"line":43},[34,19627,1510],{"class":175},[34,19629,19630],{"class":186}," `${",[34,19632,15036],{"class":190},[34,19634,19635],{"class":186},"} 分钟`\n",[34,19637,19638],{"class":36,"line":49},[34,19639,77],{"class":190},[15,19641,19642],{},"它不使用 Vue 响应式 API，只是输入一个值，返回一个值。",[15,19644,19645],{},"组合式函数：",[24,19647,19649],{"className":166,"code":19648,"language":168,"meta":29,"style":29},"export function useCounter() {\n const count = ref(0)\n \u002F\u002F ...\n}\n",[31,19650,19651,19661,19677,19681],{"__ignoreMap":29},[34,19652,19653,19655,19657,19659],{"class":36,"line":37},[34,19654,2825],{"class":175},[34,19656,2850],{"class":175},[34,19658,17936],{"class":395},[34,19660,399],{"class":190},[34,19662,19663,19665,19667,19669,19671,19673,19675],{"class":36,"line":43},[34,19664,5298],{"class":175},[34,19666,448],{"class":179},[34,19668,183],{"class":175},[34,19670,453],{"class":395},[34,19672,456],{"class":190},[34,19674,459],{"class":179},[34,19676,6111],{"class":190},[34,19678,19679],{"class":36,"line":49},[34,19680,18269],{"class":301},[34,19682,19683],{"class":36,"line":56},[34,19684,77],{"class":190},[15,19686,19687],{},"它内部有响应式状态或生命周期，返回给组件使用。",[15,19689,19690],{},"通常：",[105,19692,19693,19702],{},[108,19694,19695,146,19698,19701],{},[31,19696,19697],{},"formatXxx",[31,19699,19700],{},"calculateXxx"," 是普通工具函数。",[108,19703,19704,19706],{},[31,19705,17346],{}," 是组合式函数。",[19,19708,19710],{"id":19709},"_12-组合式函数和-pinia-的区别","12. 组合式函数和 Pinia 的区别",[15,19712,19645],{},[105,19714,19715,19718,19721],{},[108,19716,19717],{},"复用逻辑。",[108,19719,19720],{},"可以每次调用创建独立状态。",[108,19722,19723],{},"适合组件级或功能级逻辑。",[15,19725,19726],{},"Pinia：",[105,19728,19729,19732,19735],{},[108,19730,19731],{},"管理全局或跨页面共享状态。",[108,19733,19734],{},"多个组件看到同一份 store。",[108,19736,19737],{},"适合业务状态，例如登录用户、购物车、任务列表。",[15,19739,19553],{},[13837,19741,19742,19752],{},[13840,19743,19744],{},[13843,19745,19746,19749],{},[13846,19747,19748],{},"状态\u002F逻辑",[13846,19750,19751],{},"推荐",[13853,19753,19754,19762,19770,19777,19784],{},[13843,19755,19756,19759],{},[13858,19757,19758],{},"鼠标位置",[13858,19760,19761],{},"组合式函数",[13843,19763,19764,19767],{},[13858,19765,19766],{},"弹窗开关",[13858,19768,19769],{},"组件内或组合式函数",[13843,19771,19772,19775],{},[13858,19773,19774],{},"任务列表跨页面共享",[13858,19776,13901],{},[13843,19778,19779,19782],{},[13858,19780,19781],{},"本地存储同步逻辑",[13858,19783,19761],{},[13843,19785,19786,19789],{},[13858,19787,19788],{},"当前登录用户",[13858,19790,13901],{},[19,19792,13905],{"id":13904},[253,19794,19796],{"id":19795},"错误一解构-reactive-导致响应丢失","错误一：解构 reactive 导致响应丢失",[24,19798,19800],{"className":166,"code":19799,"language":168,"meta":29,"style":29},"const form = reactive({ title: '' })\nconst { title } = form\n",[31,19801,19802,19819],{"__ignoreMap":29},[34,19803,19804,19806,19808,19810,19812,19815,19817],{"class":36,"line":37},[34,19805,176],{"class":175},[34,19807,9291],{"class":179},[34,19809,183],{"class":175},[34,19811,9296],{"class":395},[34,19813,19814],{"class":190},"({ title: ",[34,19816,7400],{"class":186},[34,19818,19106],{"class":190},[34,19820,19821,19823,19825,19827,19829,19831],{"class":36,"line":43},[34,19822,176],{"class":175},[34,19824,1164],{"class":190},[34,19826,1665],{"class":179},[34,19828,2313],{"class":190},[34,19830,202],{"class":175},[34,19832,19833],{"class":190}," form\n",[15,19835,19836,19837,19839],{},"这样解构出来的 ",[31,19838,1665],{}," 不是响应式 ref。初学阶段避免这样写。",[15,19841,19842,19843,19846,19847,347],{},"如果需要解构，使用 ",[31,19844,19845],{},"toRefs","，或者直接使用 ",[31,19848,9379],{},[253,19850,19852],{"id":19851},"错误二忘记返回状态或方法","错误二：忘记返回状态或方法",[24,19854,19856],{"className":166,"code":19855,"language":168,"meta":29,"style":29},"export function useCounter() {\n const count = ref(0)\n}\n",[31,19857,19858,19868,19884],{"__ignoreMap":29},[34,19859,19860,19862,19864,19866],{"class":36,"line":37},[34,19861,2825],{"class":175},[34,19863,2850],{"class":175},[34,19865,17936],{"class":395},[34,19867,399],{"class":190},[34,19869,19870,19872,19874,19876,19878,19880,19882],{"class":36,"line":43},[34,19871,5298],{"class":175},[34,19873,448],{"class":179},[34,19875,183],{"class":175},[34,19877,453],{"class":395},[34,19879,456],{"class":190},[34,19881,459],{"class":179},[34,19883,6111],{"class":190},[34,19885,19886],{"class":36,"line":49},[34,19887,77],{"class":190},[15,19889,19890,19891,19893],{},"组件拿不到 ",[31,19892,2770],{},"。需要：",[24,19895,19897],{"className":166,"code":19896,"language":168,"meta":29,"style":29},"return { count }\n",[31,19898,19899],{"__ignoreMap":29},[34,19900,19901,19903],{"class":36,"line":37},[34,19902,1775],{"class":175},[34,19904,19905],{"class":190}," { count }\n",[253,19907,19909],{"id":19908},"错误三组合式函数命名不清","错误三：组合式函数命名不清",[15,19911,19912,19915],{},[31,19913,19914],{},"useSomething"," 太模糊。更好的命名：",[105,19917,19918,19923,19927],{},[108,19919,19920],{},[31,19921,19922],{},"useTodoFilter",[108,19924,19925],{},[31,19926,18770],{},[108,19928,19929],{},[31,19930,19931],{},"useCourseProgress",[15,19933,19934],{},"名字应该告诉别人这个函数负责什么。",[19,19936,5693],{"id":5693},[2230,19938,19939,19953,19968,19974,19980],{},[108,19940,5803,19941,19944,19945,146,19947,146,19949,146,19951,347],{},[31,19942,19943],{},"useCounter(initialValue)","，返回 ",[31,19946,2770],{},[31,19948,18116],{},[31,19950,2048],{},[31,19952,18125],{},[108,19954,5803,19955,19944,19958,146,19960,146,19962,146,19965,347],{},[31,19956,19957],{},"useToggle(initialValue)",[31,19959,2533],{},[31,19961,11935],{},[31,19963,19964],{},"setTrue",[31,19966,19967],{},"setFalse",[108,19969,5803,19970,19973],{},[31,19971,19972],{},"useLocalStorage(key, defaultValue)","，保存数组状态。",[108,19975,19976,19977,19979],{},"写一个页面调用两个 ",[31,19978,18387],{},"，观察它们是否互相影响。",[108,19981,19982],{},"回答：组合式函数和 Pinia 最大区别是什么？",[79,19984,19985],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":19987},[19988,19989,19990,19992,19993,19995,19996,19998,20000,20002,20003,20004,20005,20010],{"id":17350,"depth":43,"text":17351},{"id":17613,"depth":43,"text":17614},{"id":17714,"depth":43,"text":19991},"3. \u003Cscript setup> 中的基本写法",{"id":17895,"depth":43,"text":17896},{"id":18232,"depth":43,"text":19994},"5. 为什么组合式函数需要 export",{"id":18306,"depth":43,"text":18307},{"id":18510,"depth":43,"text":19997},"7. 示例：useToggle",{"id":18766,"depth":43,"text":19999},"8. 示例：useLocalStorage",{"id":19112,"depth":43,"text":20001},"9. 示例：useFetch",{"id":19532,"depth":43,"text":19533},{"id":19599,"depth":43,"text":19600},{"id":19709,"depth":43,"text":19710},{"id":13904,"depth":43,"text":13905,"children":20006},[20007,20008,20009],{"id":19795,"depth":49,"text":19796},{"id":19851,"depth":49,"text":19852},{"id":19908,"depth":49,"text":19909},{"id":5693,"depth":43,"text":5693},{"order":68},"\u002Fsource\u002Fcode\u002Fvue\u002F06-composition-composables",{"title":17329,"description":17337},"source\u002Fcode\u002Fvue\u002F06-composition-composables","7DjhOqqgZV1VWd-kl7xrmlW8k4KhiBaBt2UCTmblQe8",{"id":20017,"title":20018,"body":20019,"description":20026,"extension":85,"meta":21648,"navigation":52,"path":21649,"seo":21650,"stem":21651,"__hash__":21652},"notes\u002Fsource\u002Fcode\u002Fvue\u002F07-router.md","07 Vue Router 路由",{"type":8,"value":20020,"toc":21623},[20021,20024,20027,20030,20034,20037,20043,20046,20049,20055,20062,20068,20072,20075,20224,20227,20298,20301,20336,20341,20348,20392,20411,20414,20420,20427,20473,20476,20500,20510,20513,20519,20545,20548,20552,20555,20561,20564,20606,20612,20615,20718,20720,20773,20777,20780,20786,20789,20803,20806,20859,20862,20911,20914,20918,20925,20988,20991,21030,21036,21040,21043,21085,21088,21092,21095,21149,21159,21162,21170,21174,21177,21287,21290,21351,21354,21358,21361,21364,21380,21383,21386,21400,21403,21414,21418,21424,21431,21444,21448,21451,21475,21478,21503,21509,21513,21515,21529,21531,21544,21547,21556,21560,21569,21572,21578,21581,21583,21620],[11,20022,20018],{"id":20023},"_07-vue-router-路由",[15,20025,20026],{},"Vue Router 负责“当前 URL 对应哪个页面组件”。没有 Router，Vue 也能写组件；有了 Router，才方便做首页、列表页、详情页、设置页等多页面结构的单页应用。",[15,20028,20029],{},"本章目标：你能创建路由、使用链接跳转、读取动态参数、处理查询参数、写 404 页面，并理解 Router 和普通组件状态的边界。",[19,20031,20033],{"id":20032},"_1-什么是单页应用里的路由","1. 什么是单页应用里的路由",[15,20035,20036],{},"传统多页网站：",[24,20038,20041],{"className":20039,"code":20040,"language":2638,"meta":29},[2636],"\u002Findex.html\n\u002Fabout.html\n\u002Fproducts.html\n",[31,20042,20040],{"__ignoreMap":29},[15,20044,20045],{},"点击链接时，浏览器向服务器请求一个新的 HTML 页面。",[15,20047,20048],{},"Vue 单页应用：",[24,20050,20053],{"className":20051,"code":20052,"language":2638,"meta":29},[2636],"\u002F\n\u002Flessons\n\u002Flessons\u002Fcore\n\u002Fsettings\n",[31,20054,20052],{"__ignoreMap":29},[15,20056,20057,20058,20061],{},"浏览器通常只加载一个 ",[31,20059,20060],{},"index.html","。URL 变化后，Vue Router 根据路径决定显示哪个组件。",[24,20063,20066],{"className":20064,"code":20065,"language":2638,"meta":29},[2636],"URL 变化 -> Router 匹配 routes -> 渲染对应 view 组件\n",[31,20067,20065],{"__ignoreMap":29},[19,20069,20071],{"id":20070},"_2-安装和注册-router","2. 安装和注册 Router",[15,20073,20074],{},"创建路由：",[24,20076,20078],{"className":166,"code":20077,"language":168,"meta":29,"style":29},"\u002F\u002F src\u002Frouter\u002Findex.js\nimport { createRouter, createWebHistory } from 'vue-router'\nimport HomeView from '..\u002Fviews\u002FHomeView.vue'\nimport AboutView from '..\u002Fviews\u002FAboutView.vue'\n\nconst router = createRouter({\n history: createWebHistory(import.meta.env.BASE_URL),\n routes: [\n { path: '\u002F', name: 'home', component: HomeView },\n { path: '\u002Fabout', name: 'about', component: AboutView }\n ]\n})\n\nexport default router\n",[31,20079,20080,20085,20097,20109,20121,20125,20139,20165,20170,20187,20202,20207,20211,20215],{"__ignoreMap":29},[34,20081,20082],{"class":36,"line":37},[34,20083,20084],{"class":301},"\u002F\u002F src\u002Frouter\u002Findex.js\n",[34,20086,20087,20089,20092,20094],{"class":36,"line":43},[34,20088,426],{"class":175},[34,20090,20091],{"class":190}," { createRouter, createWebHistory } ",[34,20093,432],{"class":175},[34,20095,20096],{"class":186}," 'vue-router'\n",[34,20098,20099,20101,20104,20106],{"class":36,"line":49},[34,20100,426],{"class":175},[34,20102,20103],{"class":190}," HomeView ",[34,20105,432],{"class":175},[34,20107,20108],{"class":186}," '..\u002Fviews\u002FHomeView.vue'\n",[34,20110,20111,20113,20116,20118],{"class":36,"line":56},[34,20112,426],{"class":175},[34,20114,20115],{"class":190}," AboutView ",[34,20117,432],{"class":175},[34,20119,20120],{"class":186}," '..\u002Fviews\u002FAboutView.vue'\n",[34,20122,20123],{"class":36,"line":62},[34,20124,53],{"emptyLinePlaceholder":52},[34,20126,20127,20129,20132,20134,20137],{"class":36,"line":68},[34,20128,176],{"class":175},[34,20130,20131],{"class":179}," router",[34,20133,183],{"class":175},[34,20135,20136],{"class":395}," createRouter",[34,20138,9299],{"class":190},[34,20140,20141,20144,20147,20149,20151,20153,20156,20159,20162],{"class":36,"line":74},[34,20142,20143],{"class":190}," history: ",[34,20145,20146],{"class":395},"createWebHistory",[34,20148,456],{"class":190},[34,20150,426],{"class":175},[34,20152,2547],{"class":190},[34,20154,20155],{"class":179},"meta",[34,20157,20158],{"class":190},".env.",[34,20160,20161],{"class":179},"BASE_URL",[34,20163,20164],{"class":190},"),\n",[34,20166,20167],{"class":36,"line":1971},[34,20168,20169],{"class":190}," routes: [\n",[34,20171,20172,20175,20178,20181,20184],{"class":36,"line":1976},[34,20173,20174],{"class":190}," { path: ",[34,20176,20177],{"class":186},"'\u002F'",[34,20179,20180],{"class":190},", name: ",[34,20182,20183],{"class":186},"'home'",[34,20185,20186],{"class":190},", component: HomeView },\n",[34,20188,20189,20191,20194,20196,20199],{"class":36,"line":1985},[34,20190,20174],{"class":190},[34,20192,20193],{"class":186},"'\u002Fabout'",[34,20195,20180],{"class":190},[34,20197,20198],{"class":186},"'about'",[34,20200,20201],{"class":190},", component: AboutView }\n",[34,20203,20204],{"class":36,"line":1990},[34,20205,20206],{"class":190}," ]\n",[34,20208,20209],{"class":36,"line":1999},[34,20210,6153],{"class":190},[34,20212,20213],{"class":36,"line":2022},[34,20214,53],{"emptyLinePlaceholder":52},[34,20216,20217,20219,20221],{"class":36,"line":2036},[34,20218,2825],{"class":175},[34,20220,3958],{"class":175},[34,20222,20223],{"class":190}," router\n",[15,20225,20226],{},"注册到应用：",[24,20228,20230],{"className":166,"code":20229,"language":168,"meta":29,"style":29},"\u002F\u002F src\u002Fmain.js\nimport { createApp } from 'vue'\nimport App from '.\u002FApp.vue'\nimport router from '.\u002Frouter'\n\ncreateApp(App).use(router).mount('#app')\n",[31,20231,20232,20237,20248,20259,20271,20275],{"__ignoreMap":29},[34,20233,20234],{"class":36,"line":37},[34,20235,20236],{"class":301},"\u002F\u002F src\u002Fmain.js\n",[34,20238,20239,20241,20244,20246],{"class":36,"line":43},[34,20240,426],{"class":175},[34,20242,20243],{"class":190}," { createApp } ",[34,20245,432],{"class":175},[34,20247,6891],{"class":186},[34,20249,20250,20252,20254,20256],{"class":36,"line":49},[34,20251,426],{"class":175},[34,20253,4249],{"class":190},[34,20255,432],{"class":175},[34,20257,20258],{"class":186}," '.\u002FApp.vue'\n",[34,20260,20261,20263,20266,20268],{"class":36,"line":56},[34,20262,426],{"class":175},[34,20264,20265],{"class":190}," router ",[34,20267,432],{"class":175},[34,20269,20270],{"class":186}," '.\u002Frouter'\n",[34,20272,20273],{"class":36,"line":62},[34,20274,53],{"emptyLinePlaceholder":52},[34,20276,20277,20280,20283,20285,20288,20291,20293,20296],{"class":36,"line":68},[34,20278,20279],{"class":395},"createApp",[34,20281,20282],{"class":190},"(App).",[34,20284,17902],{"class":395},[34,20286,20287],{"class":190},"(router).",[34,20289,20290],{"class":395},"mount",[34,20292,456],{"class":190},[34,20294,20295],{"class":186},"'#app'",[34,20297,6111],{"class":190},[15,20299,20300],{},"根组件放出口：",[24,20302,20304],{"className":923,"code":20303,"language":925,"meta":29,"style":29},"\u003C!-- App.vue -->\n\u003Ctemplate>\n \u003CRouterView \u002F>\n\u003C\u002Ftemplate>\n",[31,20305,20306,20311,20319,20328],{"__ignoreMap":29},[34,20307,20308],{"class":36,"line":37},[34,20309,20310],{"class":301},"\u003C!-- App.vue -->\n",[34,20312,20313,20315,20317],{"class":36,"line":43},[34,20314,932],{"class":190},[34,20316,936],{"class":935},[34,20318,939],{"class":190},[34,20320,20321,20323,20326],{"class":36,"line":49},[34,20322,944],{"class":190},[34,20324,20325],{"class":935},"RouterView",[34,20327,3469],{"class":190},[34,20329,20330,20332,20334],{"class":36,"line":56},[34,20331,1003],{"class":190},[34,20333,936],{"class":935},[34,20335,939],{"class":190},[15,20337,20338,20340],{},[31,20339,20325],{}," 是一个占位区域。当前路径匹配到哪个组件，哪个组件就显示在这里。",[19,20342,14561,20344,20347],{"id":20343},"_3-routes-每个字段什么意思",[31,20345,20346],{},"routes"," 每个字段什么意思",[24,20349,20351],{"className":166,"code":20350,"language":168,"meta":29,"style":29},"{\n path: '\u002Fabout',\n name: 'about',\n component: AboutView\n}\n",[31,20352,20353,20358,20369,20380,20388],{"__ignoreMap":29},[34,20354,20355],{"class":36,"line":37},[34,20356,20357],{"class":190},"{\n",[34,20359,20360,20363,20365,20367],{"class":36,"line":43},[34,20361,20362],{"class":395}," path",[34,20364,8471],{"class":190},[34,20366,20193],{"class":186},[34,20368,684],{"class":190},[34,20370,20371,20374,20376,20378],{"class":36,"line":49},[34,20372,20373],{"class":395}," name",[34,20375,8471],{"class":190},[34,20377,20198],{"class":186},[34,20379,684],{"class":190},[34,20381,20382,20385],{"class":36,"line":56},[34,20383,20384],{"class":395}," component",[34,20386,20387],{"class":190},": AboutView\n",[34,20389,20390],{"class":36,"line":62},[34,20391,77],{"class":190},[105,20393,20394,20400,20405],{},[108,20395,20396,20399],{},[31,20397,20398],{},"path","：URL 路径。",[108,20401,20402,20404],{},[31,20403,2305],{},"：路由名字，方便编程式跳转。",[108,20406,20407,20410],{},[31,20408,20409],{},"component","：这个路径要渲染的组件。",[15,20412,20413],{},"你可以理解为：",[24,20415,20418],{"className":20416,"code":20417,"language":2638,"meta":29},[2636],"当 URL 是 \u002Fabout 时，把 AboutView 放进 RouterView。\n",[31,20419,20417],{"__ignoreMap":29},[19,20421,9025,20423,20426],{"id":20422},"_4-routerlink-跳转",[31,20424,20425],{},"RouterLink"," 跳转",[24,20428,20430],{"className":923,"code":20429,"language":925,"meta":29,"style":29},"\u003CRouterLink to=\"\u002F\">首页\u003C\u002FRouterLink>\n\u003CRouterLink to=\"\u002Fabout\">关于\u003C\u002FRouterLink>\n",[31,20431,20432,20453],{"__ignoreMap":29},[34,20433,20434,20436,20438,20441,20443,20446,20449,20451],{"class":36,"line":37},[34,20435,932],{"class":190},[34,20437,20425],{"class":935},[34,20439,20440],{"class":395}," to",[34,20442,202],{"class":190},[34,20444,20445],{"class":186},"\"\u002F\"",[34,20447,20448],{"class":190},">首页\u003C\u002F",[34,20450,20425],{"class":935},[34,20452,939],{"class":190},[34,20454,20455,20457,20459,20461,20463,20466,20469,20471],{"class":36,"line":43},[34,20456,932],{"class":190},[34,20458,20425],{"class":935},[34,20460,20440],{"class":395},[34,20462,202],{"class":190},[34,20464,20465],{"class":186},"\"\u002Fabout\"",[34,20467,20468],{"class":190},">关于\u003C\u002F",[34,20470,20425],{"class":935},[34,20472,939],{"class":190},[15,20474,20475],{},"不要在应用内部优先用：",[24,20477,20479],{"className":923,"code":20478,"language":925,"meta":29,"style":29},"\u003Ca href=\"\u002Fabout\">关于\u003C\u002Fa>\n",[31,20480,20481],{"__ignoreMap":29},[34,20482,20483,20485,20487,20490,20492,20494,20496,20498],{"class":36,"line":37},[34,20484,932],{"class":190},[34,20486,111],{"class":935},[34,20488,20489],{"class":395}," href",[34,20491,202],{"class":190},[34,20493,20465],{"class":186},[34,20495,20468],{"class":190},[34,20497,111],{"class":935},[34,20499,939],{"class":190},[15,20501,20502,20503,20506,20507,20509],{},"普通 ",[31,20504,20505],{},"\u003Ca>"," 可能触发整页刷新。",[31,20508,20425],{}," 会交给 Vue Router 处理，保持单页应用体验。",[253,20511,20512],{"id":20512},"激活样式",[15,20514,20515,20516,20518],{},"当前链接匹配时，",[31,20517,20425],{}," 会自动添加类名：",[24,20520,20522],{"className":15873,"code":20521,"language":15875,"meta":29,"style":29},".router-link-active {\n color: #2563eb;\n}\n",[31,20523,20524,20531,20541],{"__ignoreMap":29},[34,20525,20526,20529],{"class":36,"line":37},[34,20527,20528],{"class":395},".router-link-active",[34,20530,674],{"class":190},[34,20532,20533,20535,20537,20539],{"class":36,"line":43},[34,20534,8468],{"class":179},[34,20536,8471],{"class":190},[34,20538,8474],{"class":179},[34,20540,191],{"class":190},[34,20542,20543],{"class":36,"line":49},[34,20544,77],{"class":190},[15,20546,20547],{},"也可以配置自定义类名。",[19,20549,20551],{"id":20550},"_5-动态路由","5. 动态路由",[15,20553,20554],{},"课程详情页通常有多个：",[24,20556,20559],{"className":20557,"code":20558,"language":2638,"meta":29},[2636],"\u002Flessons\u002Fjs\n\u002Flessons\u002Fcore\n\u002Flessons\u002Frouter\n",[31,20560,20558],{"__ignoreMap":29},[15,20562,20563],{},"不可能给每个课程都写一条路由。用动态参数：",[24,20565,20567],{"className":166,"code":20566,"language":168,"meta":29,"style":29},"{\n path: '\u002Flessons\u002F:id',\n name: 'lesson-detail',\n component: LessonDetailView\n}\n",[31,20568,20569,20573,20584,20595,20602],{"__ignoreMap":29},[34,20570,20571],{"class":36,"line":37},[34,20572,20357],{"class":190},[34,20574,20575,20577,20579,20582],{"class":36,"line":43},[34,20576,20362],{"class":395},[34,20578,8471],{"class":190},[34,20580,20581],{"class":186},"'\u002Flessons\u002F:id'",[34,20583,684],{"class":190},[34,20585,20586,20588,20590,20593],{"class":36,"line":49},[34,20587,20373],{"class":395},[34,20589,8471],{"class":190},[34,20591,20592],{"class":186},"'lesson-detail'",[34,20594,684],{"class":190},[34,20596,20597,20599],{"class":36,"line":56},[34,20598,20384],{"class":395},[34,20600,20601],{"class":190},": LessonDetailView\n",[34,20603,20604],{"class":36,"line":62},[34,20605,77],{"class":190},[15,20607,20608,20611],{},[31,20609,20610],{},":id"," 表示这一段是变量。",[15,20613,20614],{},"读取参数：",[24,20616,20618],{"className":166,"code":20617,"language":168,"meta":29,"style":29},"import { computed } from 'vue'\nimport { useRoute } from 'vue-router'\nimport { lessons } from '..\u002Fdata\u002Flessons'\n\nconst route = useRoute()\n\nconst lesson = computed(() => {\n return lessons.find((item) => item.id === route.params.id)\n})\n",[31,20619,20620,20631,20642,20653,20657,20671,20675,20691,20714],{"__ignoreMap":29},[34,20621,20622,20624,20627,20629],{"class":36,"line":37},[34,20623,426],{"class":175},[34,20625,20626],{"class":190}," { computed } ",[34,20628,432],{"class":175},[34,20630,6891],{"class":186},[34,20632,20633,20635,20638,20640],{"class":36,"line":43},[34,20634,426],{"class":175},[34,20636,20637],{"class":190}," { useRoute } ",[34,20639,432],{"class":175},[34,20641,20096],{"class":186},[34,20643,20644,20646,20648,20650],{"class":36,"line":49},[34,20645,426],{"class":175},[34,20647,4784],{"class":190},[34,20649,432],{"class":175},[34,20651,20652],{"class":186}," '..\u002Fdata\u002Flessons'\n",[34,20654,20655],{"class":36,"line":56},[34,20656,53],{"emptyLinePlaceholder":52},[34,20658,20659,20661,20664,20666,20669],{"class":36,"line":62},[34,20660,176],{"class":175},[34,20662,20663],{"class":179}," route",[34,20665,183],{"class":175},[34,20667,20668],{"class":395}," useRoute",[34,20670,6374],{"class":190},[34,20672,20673],{"class":36,"line":68},[34,20674,53],{"emptyLinePlaceholder":52},[34,20676,20677,20679,20681,20683,20685,20687,20689],{"class":36,"line":74},[34,20678,176],{"class":175},[34,20680,824],{"class":179},[34,20682,183],{"class":175},[34,20684,8935],{"class":395},[34,20686,7682],{"class":190},[34,20688,1333],{"class":175},[34,20690,674],{"class":190},[34,20692,20693,20695,20697,20699,20701,20703,20705,20707,20709,20711],{"class":36,"line":1971},[34,20694,1510],{"class":175},[34,20696,1487],{"class":190},[34,20698,1390],{"class":395},[34,20700,1323],{"class":190},[34,20702,6979],{"class":1326},[34,20704,1330],{"class":190},[34,20706,1333],{"class":175},[34,20708,6986],{"class":190},[34,20710,1423],{"class":175},[34,20712,20713],{"class":190}," route.params.id)\n",[34,20715,20716],{"class":36,"line":1976},[34,20717,6153],{"class":190},[15,20719,1589],{},[24,20721,20723],{"className":923,"code":20722,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"!lesson\">课程不存在\u003C\u002Fp>\n\u003Carticle v-else>\n \u003Ch1>{{ lesson.title }}\u003C\u002Fh1>\n\u003C\u002Farticle>\n",[31,20724,20725,20750,20760,20765],{"__ignoreMap":29},[34,20726,20727,20729,20731,20733,20735,20737,20739,20741,20743,20746,20748],{"class":36,"line":37},[34,20728,932],{"class":190},[34,20730,15],{"class":935},[34,20732,2168],{"class":175},[34,20734,202],{"class":190},[34,20736,1608],{"class":186},[34,20738,5504],{"class":175},[34,20740,1499],{"class":190},[34,20742,1608],{"class":186},[34,20744,20745],{"class":190},">课程不存在\u003C\u002F",[34,20747,15],{"class":935},[34,20749,939],{"class":190},[34,20751,20752,20754,20756,20758],{"class":36,"line":43},[34,20753,932],{"class":190},[34,20755,947],{"class":935},[34,20757,2218],{"class":175},[34,20759,939],{"class":190},[34,20761,20762],{"class":36,"line":49},[34,20763,20764],{"class":190}," \u003Ch1>{{ lesson.title }}\u003C\u002Fh1>\n",[34,20766,20767,20769,20771],{"class":36,"line":56},[34,20768,1003],{"class":190},[34,20770,947],{"class":935},[34,20772,939],{"class":190},[19,20774,20776],{"id":20775},"_6-查询参数","6. 查询参数",[15,20778,20779],{},"查询参数长这样：",[24,20781,20784],{"className":20782,"code":20783,"language":2638,"meta":29},[2636],"\u002Flessons?keyword=vue&page=2\n",[31,20785,20783],{"__ignoreMap":29},[15,20787,20788],{},"它适合表达：",[105,20790,20791,20794,20797,20800],{},[108,20792,20793],{},"搜索词。",[108,20795,20796],{},"筛选条件。",[108,20798,20799],{},"分页页码。",[108,20801,20802],{},"排序方式。",[15,20804,20805],{},"读取：",[24,20807,20809],{"className":166,"code":20808,"language":168,"meta":29,"style":29},"const route = useRoute()\n\nconst keyword = computed(() => {\n return route.query.keyword ?? ''\n})\n",[31,20810,20811,20823,20827,20843,20855],{"__ignoreMap":29},[34,20812,20813,20815,20817,20819,20821],{"class":36,"line":37},[34,20814,176],{"class":175},[34,20816,20663],{"class":179},[34,20818,183],{"class":175},[34,20820,20668],{"class":395},[34,20822,6374],{"class":190},[34,20824,20825],{"class":36,"line":43},[34,20826,53],{"emptyLinePlaceholder":52},[34,20828,20829,20831,20833,20835,20837,20839,20841],{"class":36,"line":49},[34,20830,176],{"class":175},[34,20832,11060],{"class":179},[34,20834,183],{"class":175},[34,20836,8935],{"class":395},[34,20838,7682],{"class":190},[34,20840,1333],{"class":175},[34,20842,674],{"class":190},[34,20844,20845,20847,20850,20853],{"class":36,"line":56},[34,20846,1510],{"class":175},[34,20848,20849],{"class":190}," route.query.keyword ",[34,20851,20852],{"class":175},"??",[34,20854,6645],{"class":186},[34,20856,20857],{"class":36,"line":62},[34,20858,6153],{"class":190},[15,20860,20861],{},"跳转时带查询参数：",[24,20863,20865],{"className":166,"code":20864,"language":168,"meta":29,"style":29},"router.push({\n name: 'lessons',\n query: {\n keyword: keyword.value,\n page: 1\n }\n})\n",[31,20866,20867,20876,20885,20890,20895,20903,20907],{"__ignoreMap":29},[34,20868,20869,20872,20874],{"class":36,"line":37},[34,20870,20871],{"class":190},"router.",[34,20873,1262],{"class":395},[34,20875,9299],{"class":190},[34,20877,20878,20880,20883],{"class":36,"line":43},[34,20879,2273],{"class":190},[34,20881,20882],{"class":186},"'lessons'",[34,20884,684],{"class":190},[34,20886,20887],{"class":36,"line":49},[34,20888,20889],{"class":190}," query: {\n",[34,20891,20892],{"class":36,"line":56},[34,20893,20894],{"class":190}," keyword: keyword.value,\n",[34,20896,20897,20900],{"class":36,"line":62},[34,20898,20899],{"class":190}," page: ",[34,20901,20902],{"class":179},"1\n",[34,20904,20905],{"class":36,"line":68},[34,20906,5631],{"class":190},[34,20908,20909],{"class":36,"line":74},[34,20910,6153],{"class":190},[15,20912,20913],{},"查询参数的好处：用户可以复制 URL，把当前筛选状态分享给别人。",[19,20915,20917],{"id":20916},"_7-编程式导航","7. 编程式导航",[15,20919,20920,20921,20924],{},"除了 ",[31,20922,20923],{},"\u003CRouterLink>","，你也可以在 JS 中跳转。",[24,20926,20928],{"className":166,"code":20927,"language":168,"meta":29,"style":29},"import { useRouter } from 'vue-router'\n\nconst router = useRouter()\n\nfunction goHome() {\n router.push('\u002F')\n}\n",[31,20929,20930,20941,20945,20958,20962,20971,20984],{"__ignoreMap":29},[34,20931,20932,20934,20937,20939],{"class":36,"line":37},[34,20933,426],{"class":175},[34,20935,20936],{"class":190}," { useRouter } ",[34,20938,432],{"class":175},[34,20940,20096],{"class":186},[34,20942,20943],{"class":36,"line":43},[34,20944,53],{"emptyLinePlaceholder":52},[34,20946,20947,20949,20951,20953,20956],{"class":36,"line":49},[34,20948,176],{"class":175},[34,20950,20131],{"class":179},[34,20952,183],{"class":175},[34,20954,20955],{"class":395}," useRouter",[34,20957,6374],{"class":190},[34,20959,20960],{"class":36,"line":56},[34,20961,53],{"emptyLinePlaceholder":52},[34,20963,20964,20966,20969],{"class":36,"line":62},[34,20965,392],{"class":175},[34,20967,20968],{"class":395}," goHome",[34,20970,399],{"class":190},[34,20972,20973,20976,20978,20980,20982],{"class":36,"line":68},[34,20974,20975],{"class":190}," router.",[34,20977,1262],{"class":395},[34,20979,456],{"class":190},[34,20981,20177],{"class":186},[34,20983,6111],{"class":190},[34,20985,20986],{"class":36,"line":74},[34,20987,77],{"class":190},[15,20989,20990],{},"按路由名字跳转：",[24,20992,20994],{"className":166,"code":20993,"language":168,"meta":29,"style":29},"router.push({\n name: 'lesson-detail',\n params: {\n id: lesson.id\n }\n})\n",[31,20995,20996,21004,21012,21017,21022,21026],{"__ignoreMap":29},[34,20997,20998,21000,21002],{"class":36,"line":37},[34,20999,20871],{"class":190},[34,21001,1262],{"class":395},[34,21003,9299],{"class":190},[34,21005,21006,21008,21010],{"class":36,"line":43},[34,21007,2273],{"class":190},[34,21009,20592],{"class":186},[34,21011,684],{"class":190},[34,21013,21014],{"class":36,"line":49},[34,21015,21016],{"class":190}," params: {\n",[34,21018,21019],{"class":36,"line":56},[34,21020,21021],{"class":190}," id: lesson.id\n",[34,21023,21024],{"class":36,"line":62},[34,21025,5631],{"class":190},[34,21027,21028],{"class":36,"line":68},[34,21029,6153],{"class":190},[15,21031,21032,21033,21035],{},"推荐在复杂跳转中使用 ",[31,21034,2305],{},"，因为路径以后改了，名字可以不变。",[19,21037,21039],{"id":21038},"_8-404-页面","8. 404 页面",[15,21041,21042],{},"用户访问不存在的路径时，应该显示 404 页面。",[24,21044,21046],{"className":166,"code":21045,"language":168,"meta":29,"style":29},"{\n path: '\u002F:pathMatch(.*)*',\n name: 'not-found',\n component: NotFoundView\n}\n",[31,21047,21048,21052,21063,21074,21081],{"__ignoreMap":29},[34,21049,21050],{"class":36,"line":37},[34,21051,20357],{"class":190},[34,21053,21054,21056,21058,21061],{"class":36,"line":43},[34,21055,20362],{"class":395},[34,21057,8471],{"class":190},[34,21059,21060],{"class":186},"'\u002F:pathMatch(.*)*'",[34,21062,684],{"class":190},[34,21064,21065,21067,21069,21072],{"class":36,"line":49},[34,21066,20373],{"class":395},[34,21068,8471],{"class":190},[34,21070,21071],{"class":186},"'not-found'",[34,21073,684],{"class":190},[34,21075,21076,21078],{"class":36,"line":56},[34,21077,20384],{"class":395},[34,21079,21080],{"class":190},": NotFoundView\n",[34,21082,21083],{"class":36,"line":62},[34,21084,77],{"class":190},[15,21086,21087],{},"这条路由应该放在最后。因为它会匹配所有前面没匹配到的路径。",[19,21089,21091],{"id":21090},"_9-路由懒加载","9. 路由懒加载",[15,21093,21094],{},"页面很多时，不必一开始加载所有页面组件。",[24,21096,21098],{"className":166,"code":21097,"language":168,"meta":29,"style":29},"{\n path: '\u002Fsettings',\n name: 'settings',\n component: () => import('..\u002Fviews\u002FSettingsView.vue')\n}\n",[31,21099,21100,21104,21115,21126,21145],{"__ignoreMap":29},[34,21101,21102],{"class":36,"line":37},[34,21103,20357],{"class":190},[34,21105,21106,21108,21110,21113],{"class":36,"line":43},[34,21107,20362],{"class":395},[34,21109,8471],{"class":190},[34,21111,21112],{"class":186},"'\u002Fsettings'",[34,21114,684],{"class":190},[34,21116,21117,21119,21121,21124],{"class":36,"line":49},[34,21118,20373],{"class":395},[34,21120,8471],{"class":190},[34,21122,21123],{"class":186},"'settings'",[34,21125,684],{"class":190},[34,21127,21128,21130,21133,21135,21138,21140,21143],{"class":36,"line":56},[34,21129,20384],{"class":395},[34,21131,21132],{"class":190},": () ",[34,21134,1333],{"class":175},[34,21136,21137],{"class":175}," import",[34,21139,456],{"class":190},[34,21141,21142],{"class":186},"'..\u002Fviews\u002FSettingsView.vue'",[34,21144,6111],{"class":190},[34,21146,21147],{"class":36,"line":62},[34,21148,77],{"class":190},[15,21150,21151,21152,21155,21156,347],{},"这表示访问 ",[31,21153,21154],{},"\u002Fsettings"," 时才加载 ",[31,21157,21158],{},"SettingsView.vue",[15,21160,21161],{},"好处：",[105,21163,21164,21167],{},[108,21165,21166],{},"减少首屏加载体积。",[108,21168,21169],{},"用户不访问的页面不急着下载。",[19,21171,21173],{"id":21172},"_10-路由守卫","10. 路由守卫",[15,21175,21176],{},"如果某些页面需要登录后才能访问：",[24,21178,21180],{"className":166,"code":21179,"language":168,"meta":29,"style":29},"router.beforeEach((to) => {\n const isLoggedIn = Boolean(localStorage.getItem('token'))\n\n if (to.meta.requiresAuth && !isLoggedIn) {\n return {\n name: 'login',\n query: {\n redirect: to.fullPath\n }\n }\n }\n})\n",[31,21181,21182,21200,21225,21229,21244,21250,21260,21265,21270,21275,21279,21283],{"__ignoreMap":29},[34,21183,21184,21186,21189,21191,21194,21196,21198],{"class":36,"line":37},[34,21185,20871],{"class":190},[34,21187,21188],{"class":395},"beforeEach",[34,21190,1323],{"class":190},[34,21192,21193],{"class":1326},"to",[34,21195,1330],{"class":190},[34,21197,1333],{"class":175},[34,21199,674],{"class":190},[34,21201,21202,21204,21207,21209,21212,21215,21217,21219,21222],{"class":36,"line":43},[34,21203,5298],{"class":175},[34,21205,21206],{"class":179}," isLoggedIn",[34,21208,183],{"class":175},[34,21210,21211],{"class":395}," Boolean",[34,21213,21214],{"class":190},"(localStorage.",[34,21216,18834],{"class":395},[34,21218,456],{"class":190},[34,21220,21221],{"class":186},"'token'",[34,21223,21224],{"class":190},"))\n",[34,21226,21227],{"class":36,"line":49},[34,21228,53],{"emptyLinePlaceholder":52},[34,21230,21231,21233,21236,21239,21241],{"class":36,"line":56},[34,21232,7000],{"class":175},[34,21234,21235],{"class":190}," (to.meta.requiresAuth ",[34,21237,21238],{"class":175},"&&",[34,21240,1336],{"class":175},[34,21242,21243],{"class":190},"isLoggedIn) {\n",[34,21245,21246,21248],{"class":36,"line":62},[34,21247,8601],{"class":175},[34,21249,674],{"class":190},[34,21251,21252,21255,21258],{"class":36,"line":68},[34,21253,21254],{"class":190}," name: ",[34,21256,21257],{"class":186},"'login'",[34,21259,684],{"class":190},[34,21261,21262],{"class":36,"line":74},[34,21263,21264],{"class":190}," query: {\n",[34,21266,21267],{"class":36,"line":1971},[34,21268,21269],{"class":190}," redirect: to.fullPath\n",[34,21271,21272],{"class":36,"line":1976},[34,21273,21274],{"class":190}," }\n",[34,21276,21277],{"class":36,"line":1985},[34,21278,5540],{"class":190},[34,21280,21281],{"class":36,"line":1990},[34,21282,5631],{"class":190},[34,21284,21285],{"class":36,"line":1999},[34,21286,6153],{"class":190},[15,21288,21289],{},"路由配置：",[24,21291,21293],{"className":166,"code":21292,"language":168,"meta":29,"style":29},"{\n path: '\u002Fsettings',\n name: 'settings',\n component: SettingsView,\n meta: {\n requiresAuth: true\n }\n}\n",[31,21294,21295,21299,21309,21319,21326,21334,21343,21347],{"__ignoreMap":29},[34,21296,21297],{"class":36,"line":37},[34,21298,20357],{"class":190},[34,21300,21301,21303,21305,21307],{"class":36,"line":43},[34,21302,20362],{"class":395},[34,21304,8471],{"class":190},[34,21306,21112],{"class":186},[34,21308,684],{"class":190},[34,21310,21311,21313,21315,21317],{"class":36,"line":49},[34,21312,20373],{"class":395},[34,21314,8471],{"class":190},[34,21316,21123],{"class":186},[34,21318,684],{"class":190},[34,21320,21321,21323],{"class":36,"line":56},[34,21322,20384],{"class":395},[34,21324,21325],{"class":190},": SettingsView,\n",[34,21327,21328,21331],{"class":36,"line":62},[34,21329,21330],{"class":395}," meta",[34,21332,21333],{"class":190},": {\n",[34,21335,21336,21339,21341],{"class":36,"line":68},[34,21337,21338],{"class":395}," requiresAuth",[34,21340,8471],{"class":190},[34,21342,12076],{"class":179},[34,21344,21345],{"class":36,"line":74},[34,21346,5631],{"class":190},[34,21348,21349],{"class":36,"line":1971},[34,21350,77],{"class":190},[15,21352,21353],{},"守卫适合判断“能不能进入页面”，不适合塞所有业务逻辑。",[19,21355,21357],{"id":21356},"_11-router-和-pinia-的边界","11. Router 和 Pinia 的边界",[15,21359,21360],{},"Router 管页面位置和 URL 状态。",[15,21362,21363],{},"适合放在 URL：",[105,21365,21366,21369,21372,21374,21377],{},[108,21367,21368],{},"当前页面。",[108,21370,21371],{},"详情 id。",[108,21373,20793],{},[108,21375,21376],{},"分页。",[108,21378,21379],{},"排序。",[15,21381,21382],{},"Pinia 管业务状态。",[15,21384,21385],{},"适合放 Pinia：",[105,21387,21388,21391,21394,21397],{},[108,21389,21390],{},"当前登录用户。",[108,21392,21393],{},"购物车。",[108,21395,21396],{},"跨页面任务列表。",[108,21398,21399],{},"课程完成进度。",[15,21401,21402],{},"组件内状态：",[105,21404,21405,21408,21411],{},[108,21406,21407],{},"弹窗是否打开。",[108,21409,21410],{},"输入框正在输入但不需要分享的文字。",[108,21412,21413],{},"当前组件内部 tab。",[19,21415,21417],{"id":21416},"_12-常见错误","12. 常见错误",[253,21419,21421,21422],{"id":21420},"错误一忘记放-routerview","错误一：忘记放 ",[31,21423,20325],{},[15,21425,21426,21427,21430],{},"配置了路由，但页面不显示，检查 ",[31,21428,21429],{},"App.vue"," 是否有：",[24,21432,21434],{"className":923,"code":21433,"language":925,"meta":29,"style":29},"\u003CRouterView \u002F>\n",[31,21435,21436],{"__ignoreMap":29},[34,21437,21438,21440,21442],{"class":36,"line":37},[34,21439,932],{"class":190},[34,21441,20325],{"class":935},[34,21443,3469],{"class":190},[253,21445,21447],{"id":21446},"错误二路由路径和链接不一致","错误二：路由路径和链接不一致",[15,21449,21450],{},"配置：",[24,21452,21454],{"className":166,"code":21453,"language":168,"meta":29,"style":29},"{ path: '\u002Flessons', component: LessonsView }\n",[31,21455,21456],{"__ignoreMap":29},[34,21457,21458,21461,21463,21465,21468,21470,21472],{"class":36,"line":37},[34,21459,21460],{"class":190},"{ ",[34,21462,20398],{"class":395},[34,21464,8471],{"class":190},[34,21466,21467],{"class":186},"'\u002Flessons'",[34,21469,616],{"class":190},[34,21471,20409],{"class":395},[34,21473,21474],{"class":190},": LessonsView }\n",[15,21476,21477],{},"链接：",[24,21479,21481],{"className":923,"code":21480,"language":925,"meta":29,"style":29},"\u003CRouterLink to=\"\u002Flesson\">课程\u003C\u002FRouterLink>\n",[31,21482,21483],{"__ignoreMap":29},[34,21484,21485,21487,21489,21491,21493,21496,21499,21501],{"class":36,"line":37},[34,21486,932],{"class":190},[34,21488,20425],{"class":935},[34,21490,20440],{"class":395},[34,21492,202],{"class":190},[34,21494,21495],{"class":186},"\"\u002Flesson\"",[34,21497,21498],{"class":190},">课程\u003C\u002F",[34,21500,20425],{"class":935},[34,21502,939],{"class":190},[15,21504,21505,21508],{},[31,21506,21507],{},"\u002Flesson"," 少了 s，会进入 404。",[253,21510,21512],{"id":21511},"错误三动态参数名字写错","错误三：动态参数名字写错",[15,21514,21450],{},[24,21516,21518],{"className":166,"code":21517,"language":168,"meta":29,"style":29},"path: '\u002Flessons\u002F:id'\n",[31,21519,21520],{"__ignoreMap":29},[34,21521,21522,21524,21526],{"class":36,"line":37},[34,21523,20398],{"class":395},[34,21525,8471],{"class":190},[34,21527,21528],{"class":186},"'\u002Flessons\u002F:id'\n",[15,21530,20805],{},[24,21532,21534],{"className":166,"code":21533,"language":168,"meta":29,"style":29},"route.params.lessonId \u002F\u002F undefined\n",[31,21535,21536],{"__ignoreMap":29},[34,21537,21538,21541],{"class":36,"line":37},[34,21539,21540],{"class":190},"route.params.lessonId ",[34,21542,21543],{"class":301},"\u002F\u002F undefined\n",[15,21545,21546],{},"应该：",[24,21548,21550],{"className":166,"code":21549,"language":168,"meta":29,"style":29},"route.params.id\n",[31,21551,21552],{"__ignoreMap":29},[34,21553,21554],{"class":36,"line":37},[34,21555,21549],{"class":190},[253,21557,21559],{"id":21558},"错误四部署后刷新详情页-404","错误四：部署后刷新详情页 404",[15,21561,21562,21563,21566,21567,347],{},"本地没问题，部署后访问 ",[31,21564,21565],{},"\u002Flessons\u002Fjs"," 刷新 404。原因是服务器不知道这个路径应该返回 ",[31,21568,20060],{},[15,21570,21571],{},"SPA 部署需要配置：",[24,21573,21576],{"className":21574,"code":21575,"language":2638,"meta":29},[2636],"所有未知路径 -> \u002Findex.html\n",[31,21577,21575],{"__ignoreMap":29},[15,21579,21580],{},"这样刷新后先回到 Vue 应用，再由 Router 匹配页面。",[19,21582,5693],{"id":5693},[2230,21584,21585,21588,21595,21598,21604,21610,21617],{},[108,21586,21587],{},"创建首页、课程列表页、课程详情页、404 页。",[108,21589,21590,21591,21594],{},"配置 ",[31,21592,21593],{},"\u002Flessons\u002F:id"," 动态路由。",[108,21596,21597],{},"从课程列表点击进入详情页。",[108,21599,21600,21601,347],{},"在详情页读取 ",[31,21602,21603],{},"route.params.id",[108,21605,21606,21607,21609],{},"给课程列表加 ",[31,21608,9414],{}," 查询参数。",[108,21611,21612,21613,21616],{},"写一个按钮，用 ",[31,21614,21615],{},"router.push"," 跳转首页。",[108,21618,21619],{},"解释：为什么刷新动态路由可能在部署后 404？",[79,21621,21622],{},"html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":21624},[21625,21626,21627,21629,21633,21634,21635,21636,21637,21638,21639,21640,21647],{"id":20032,"depth":43,"text":20033},{"id":20070,"depth":43,"text":20071},{"id":20343,"depth":43,"text":21628},"3. routes 每个字段什么意思",{"id":20422,"depth":43,"text":21630,"children":21631},"4. RouterLink 跳转",[21632],{"id":20512,"depth":49,"text":20512},{"id":20550,"depth":43,"text":20551},{"id":20775,"depth":43,"text":20776},{"id":20916,"depth":43,"text":20917},{"id":21038,"depth":43,"text":21039},{"id":21090,"depth":43,"text":21091},{"id":21172,"depth":43,"text":21173},{"id":21356,"depth":43,"text":21357},{"id":21416,"depth":43,"text":21417,"children":21641},[21642,21644,21645,21646],{"id":21420,"depth":49,"text":21643},"错误一：忘记放 RouterView",{"id":21446,"depth":49,"text":21447},{"id":21511,"depth":49,"text":21512},{"id":21558,"depth":49,"text":21559},{"id":5693,"depth":43,"text":5693},{"order":74},"\u002Fsource\u002Fcode\u002Fvue\u002F07-router",{"title":20018,"description":20026},"source\u002Fcode\u002Fvue\u002F07-router","p1KiBx1agC3GzZQAYW7ShxdF1Weax7ZxBb394xOs8Ls",{"id":21654,"title":21655,"body":21656,"description":21663,"extension":85,"meta":23683,"navigation":52,"path":23684,"seo":23685,"stem":23686,"__hash__":23687},"notes\u002Fsource\u002Fcode\u002Fvue\u002F08-pinia.md","08 Pinia 状态管理",{"type":8,"value":21657,"toc":23663},[21658,21661,21664,21667,21671,21674,21691,21694,21708,21711,21717,21720,21726,21730,21733,21736,21750,21753,21764,21768,21771,21840,21846,21850,22056,22059,22095,22099,22202,22208,22236,22240,22243,22339,22342,22353,22356,22360,22936,22940,22943,22946,22963,22966,22980,22983,22989,22992,22996,22999,23002,23008,23010,23016,23019,23023,23028,23147,23150,23158,23166,23170,23173,23386,23392,23448,23451,23469,23472,23476,23480,23483,23501,23504,23508,23511,23531,23541,23545,23548,23566,23569,23586,23589,23593,23596,23602,23605,23607,23660],[11,21659,21655],{"id":21660},"_08-pinia-状态管理",[15,21662,21663],{},"Pinia 是 Vue 官方推荐的状态管理库。它解决的问题是：多个组件或多个页面需要共享同一份业务状态时，应该把状态放在哪里、怎么修改、怎么读取。",[15,21665,21666],{},"本章目标：你能创建 store，理解 state\u002Fgetter\u002Faction，知道哪些状态应该放 Pinia，哪些不应该放。",[19,21668,21670],{"id":21669},"_1-为什么需要状态管理","1. 为什么需要状态管理",[15,21672,21673],{},"如果一个任务列表只在一个组件里使用，可以放组件内：",[24,21675,21677],{"className":166,"code":21676,"language":168,"meta":29,"style":29},"const todos = ref([])\n",[31,21678,21679],{"__ignoreMap":29},[34,21680,21681,21683,21685,21687,21689],{"class":36,"line":37},[34,21682,176],{"class":175},[34,21684,12962],{"class":179},[34,21686,183],{"class":175},[34,21688,453],{"class":395},[34,21690,9446],{"class":190},[15,21692,21693],{},"但如果多个地方都需要同一份数据：",[105,21695,21696,21699,21702,21705],{},[108,21697,21698],{},"首页显示未完成任务数。",[108,21700,21701],{},"任务页新增、删除任务。",[108,21703,21704],{},"导航栏显示提醒数量。",[108,21706,21707],{},"详情页修改任务状态。",[15,21709,21710],{},"如果只靠 props 一层层传，会变得很麻烦：",[24,21712,21715],{"className":21713,"code":21714,"language":2638,"meta":29},[2636],"App -> Layout -> Sidebar -> TodoBadge\nApp -> TodoPage -> TodoList -> TodoItem\n",[31,21716,21714],{"__ignoreMap":29},[15,21718,21719],{},"Pinia 提供一个共享 store：",[24,21721,21724],{"className":21722,"code":21723,"language":2638,"meta":29},[2636],"多个组件 -> 使用同一个 todoStore\n",[31,21725,21723],{"__ignoreMap":29},[19,21727,21729],{"id":21728},"_2-store-是什么","2. Store 是什么",[15,21731,21732],{},"store 可以理解成某个业务领域的状态中心。",[15,21734,21735],{},"例如任务 store：",[105,21737,21738,21741,21744,21747],{},[108,21739,21740],{},"保存任务数组。",[108,21742,21743],{},"提供新增任务方法。",[108,21745,21746],{},"提供删除任务方法。",[108,21748,21749],{},"提供未完成数量。",[15,21751,21752],{},"课程 store：",[105,21754,21755,21758,21761],{},[108,21756,21757],{},"保存课程完成状态。",[108,21759,21760],{},"提供切换完成方法。",[108,21762,21763],{},"提供整体学习进度。",[19,21765,21767],{"id":21766},"_3-创建-pinia","3. 创建 Pinia",[15,21769,21770],{},"在入口文件安装：",[24,21772,21774],{"className":166,"code":21773,"language":168,"meta":29,"style":29},"\u002F\u002F src\u002Fmain.js\nimport { createApp } from 'vue'\nimport { createPinia } from 'pinia'\nimport App from '.\u002FApp.vue'\n\ncreateApp(App).use(createPinia()).mount('#app')\n",[31,21775,21776,21780,21790,21802,21812,21816],{"__ignoreMap":29},[34,21777,21778],{"class":36,"line":37},[34,21779,20236],{"class":301},[34,21781,21782,21784,21786,21788],{"class":36,"line":43},[34,21783,426],{"class":175},[34,21785,20243],{"class":190},[34,21787,432],{"class":175},[34,21789,6891],{"class":186},[34,21791,21792,21794,21797,21799],{"class":36,"line":49},[34,21793,426],{"class":175},[34,21795,21796],{"class":190}," { createPinia } ",[34,21798,432],{"class":175},[34,21800,21801],{"class":186}," 'pinia'\n",[34,21803,21804,21806,21808,21810],{"class":36,"line":56},[34,21805,426],{"class":175},[34,21807,4249],{"class":190},[34,21809,432],{"class":175},[34,21811,20258],{"class":186},[34,21813,21814],{"class":36,"line":62},[34,21815,53],{"emptyLinePlaceholder":52},[34,21817,21818,21820,21822,21824,21826,21829,21832,21834,21836,21838],{"class":36,"line":68},[34,21819,20279],{"class":395},[34,21821,20282],{"class":190},[34,21823,17902],{"class":395},[34,21825,456],{"class":190},[34,21827,21828],{"class":395},"createPinia",[34,21830,21831],{"class":190},"()).",[34,21833,20290],{"class":395},[34,21835,456],{"class":190},[34,21837,20295],{"class":186},[34,21839,6111],{"class":190},[15,21841,4381,21842,21845],{},[31,21843,21844],{},".use(createPinia())","，组件里使用 store 会出问题。",[19,21847,21849],{"id":21848},"_4-创建一个-store","4. 创建一个 store",[24,21851,21853],{"className":166,"code":21852,"language":168,"meta":29,"style":29},"\u002F\u002F src\u002Fstores\u002FuseTodoStore.js\nimport { computed, ref } from 'vue'\nimport { defineStore } from 'pinia'\n\nexport const useTodoStore = defineStore('todos', () => {\n const todos = ref([])\n\n const remainingCount = computed(() => {\n return todos.value.filter((todo) => !todo.done).length\n })\n\n function addTodo(title) {\n todos.value.push({\n id: crypto.randomUUID(),\n title,\n done: false\n })\n }\n\n return {\n todos,\n remainingCount,\n addTodo\n }\n})\n",[31,21854,21855,21860,21870,21881,21885,21909,21921,21925,21941,21964,21968,21972,21985,21994,22003,22008,22015,22019,22023,22027,22033,22038,22043,22048,22052],{"__ignoreMap":29},[34,21856,21857],{"class":36,"line":37},[34,21858,21859],{"class":301},"\u002F\u002F src\u002Fstores\u002FuseTodoStore.js\n",[34,21861,21862,21864,21866,21868],{"class":36,"line":43},[34,21863,426],{"class":175},[34,21865,16370],{"class":190},[34,21867,432],{"class":175},[34,21869,6891],{"class":186},[34,21871,21872,21874,21877,21879],{"class":36,"line":49},[34,21873,426],{"class":175},[34,21875,21876],{"class":190}," { defineStore } ",[34,21878,432],{"class":175},[34,21880,21801],{"class":186},[34,21882,21883],{"class":36,"line":56},[34,21884,53],{"emptyLinePlaceholder":52},[34,21886,21887,21889,21891,21894,21896,21899,21901,21903,21905,21907],{"class":36,"line":62},[34,21888,2825],{"class":175},[34,21890,3280],{"class":175},[34,21892,21893],{"class":179}," useTodoStore",[34,21895,183],{"class":175},[34,21897,21898],{"class":395}," defineStore",[34,21900,456],{"class":190},[34,21902,19071],{"class":186},[34,21904,6131],{"class":190},[34,21906,1333],{"class":175},[34,21908,674],{"class":190},[34,21910,21911,21913,21915,21917,21919],{"class":36,"line":68},[34,21912,5298],{"class":175},[34,21914,12962],{"class":179},[34,21916,183],{"class":175},[34,21918,453],{"class":395},[34,21920,9446],{"class":190},[34,21922,21923],{"class":36,"line":74},[34,21924,53],{"emptyLinePlaceholder":52},[34,21926,21927,21929,21931,21933,21935,21937,21939],{"class":36,"line":1971},[34,21928,5298],{"class":175},[34,21930,11203],{"class":179},[34,21932,183],{"class":175},[34,21934,8935],{"class":395},[34,21936,7682],{"class":190},[34,21938,1333],{"class":175},[34,21940,674],{"class":190},[34,21942,21943,21945,21947,21949,21951,21953,21955,21957,21959,21962],{"class":36,"line":1976},[34,21944,8601],{"class":175},[34,21946,13059],{"class":190},[34,21948,1300],{"class":395},[34,21950,1323],{"class":190},[34,21952,12986],{"class":1326},[34,21954,1330],{"class":190},[34,21956,1333],{"class":175},[34,21958,1336],{"class":175},[34,21960,21961],{"class":190},"todo.done).",[34,21963,10311],{"class":179},[34,21965,21966],{"class":36,"line":1985},[34,21967,6379],{"class":190},[34,21969,21970],{"class":36,"line":1990},[34,21971,53],{"emptyLinePlaceholder":52},[34,21973,21974,21976,21979,21981,21983],{"class":36,"line":1999},[34,21975,6318],{"class":175},[34,21977,21978],{"class":395}," addTodo",[34,21980,456],{"class":190},[34,21982,1665],{"class":1326},[34,21984,1700],{"class":190},[34,21986,21987,21990,21992],{"class":36,"line":2022},[34,21988,21989],{"class":190}," todos.value.",[34,21991,1262],{"class":395},[34,21993,9299],{"class":190},[34,21995,21996,21999,22001],{"class":36,"line":2036},[34,21997,21998],{"class":190}," id: crypto.",[34,22000,14345],{"class":395},[34,22002,14348],{"class":190},[34,22004,22005],{"class":36,"line":2603},[34,22006,22007],{"class":190}," title,\n",[34,22009,22010,22013],{"class":36,"line":5537},[34,22011,22012],{"class":190}," done: ",[34,22014,14360],{"class":179},[34,22016,22017],{"class":36,"line":5543},[34,22018,6791],{"class":190},[34,22020,22021],{"class":36,"line":5548},[34,22022,5631],{"class":190},[34,22024,22025],{"class":36,"line":5564},[34,22026,53],{"emptyLinePlaceholder":52},[34,22028,22029,22031],{"class":36,"line":5576},[34,22030,1510],{"class":175},[34,22032,674],{"class":190},[34,22034,22035],{"class":36,"line":5606},[34,22036,22037],{"class":190}," todos,\n",[34,22039,22040],{"class":36,"line":5616},[34,22041,22042],{"class":190}," remainingCount,\n",[34,22044,22045],{"class":36,"line":5628},[34,22046,22047],{"class":190}," addTodo\n",[34,22049,22050],{"class":36,"line":5634},[34,22051,5631],{"class":190},[34,22053,22054],{"class":36,"line":6420},[34,22055,6153],{"class":190},[15,22057,22058],{},"解释：",[105,22060,22061,22067,22072,22078,22084,22090],{},[108,22062,22063,22066],{},[31,22064,22065],{},"defineStore"," 创建 store。",[108,22068,22069,22071],{},[31,22070,19071],{}," 是 store id，要保持唯一。",[108,22073,22074,22077],{},[31,22075,22076],{},"todos"," 是 state。",[108,22079,22080,22083],{},[31,22081,22082],{},"remainingCount"," 是 getter。",[108,22085,22086,22089],{},[31,22087,22088],{},"addTodo"," 是 action。",[108,22091,22092,22094],{},[31,22093,1775],{}," 里写哪些内容，组件才能用哪些内容。",[19,22096,22098],{"id":22097},"_5-在组件中使用-store","5. 在组件中使用 store",[24,22100,22102],{"className":923,"code":22101,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { useTodoStore } from '..\u002Fstores\u002FuseTodoStore'\n\nconst todoStore = useTodoStore()\n\u003C\u002Fscript>\n\n\u003Ctemplate>\n \u003Cp>未完成：{{ todoStore.remainingCount }}\u003C\u002Fp>\n \u003Cbutton @click=\"todoStore.addTodo('学习 Pinia')\">新增\u003C\u002Fbutton>\n\u003C\u002Ftemplate>\n",[31,22103,22104,22114,22125,22129,22142,22150,22154,22162,22175,22194],{"__ignoreMap":29},[34,22105,22106,22108,22110,22112],{"class":36,"line":37},[34,22107,932],{"class":190},[34,22109,1911],{"class":935},[34,22111,1914],{"class":395},[34,22113,939],{"class":190},[34,22115,22116,22118,22120,22122],{"class":36,"line":43},[34,22117,426],{"class":175},[34,22119,4762],{"class":190},[34,22121,432],{"class":175},[34,22123,22124],{"class":186}," '..\u002Fstores\u002FuseTodoStore'\n",[34,22126,22127],{"class":36,"line":49},[34,22128,53],{"emptyLinePlaceholder":52},[34,22130,22131,22133,22136,22138,22140],{"class":36,"line":56},[34,22132,176],{"class":175},[34,22134,22135],{"class":179}," todoStore",[34,22137,183],{"class":175},[34,22139,21893],{"class":395},[34,22141,6374],{"class":190},[34,22143,22144,22146,22148],{"class":36,"line":62},[34,22145,1003],{"class":190},[34,22147,1911],{"class":935},[34,22149,939],{"class":190},[34,22151,22152],{"class":36,"line":68},[34,22153,53],{"emptyLinePlaceholder":52},[34,22155,22156,22158,22160],{"class":36,"line":74},[34,22157,932],{"class":190},[34,22159,936],{"class":935},[34,22161,939],{"class":190},[34,22163,22164,22166,22168,22171,22173],{"class":36,"line":1971},[34,22165,944],{"class":190},[34,22167,15],{"class":935},[34,22169,22170],{"class":190},">未完成：{{ todoStore.remainingCount }}\u003C\u002F",[34,22172,15],{"class":935},[34,22174,939],{"class":190},[34,22176,22177,22179,22181,22183,22185,22188,22190,22192],{"class":36,"line":1976},[34,22178,944],{"class":190},[34,22180,2004],{"class":935},[34,22182,2007],{"class":395},[34,22184,202],{"class":190},[34,22186,22187],{"class":186},"\"todoStore.addTodo('学习 Pinia')\"",[34,22189,6059],{"class":190},[34,22191,2004],{"class":935},[34,22193,939],{"class":190},[34,22195,22196,22198,22200],{"class":36,"line":1985},[34,22197,1003],{"class":190},[34,22199,936],{"class":935},[34,22201,939],{"class":190},[15,22203,22204,22207],{},[31,22205,22206],{},"useTodoStore()"," 返回 store 实例。你可以像普通对象一样访问：",[24,22209,22211],{"className":166,"code":22210,"language":168,"meta":29,"style":29},"todoStore.todos\ntodoStore.remainingCount\ntodoStore.addTodo(...)\n",[31,22212,22213,22218,22223],{"__ignoreMap":29},[34,22214,22215],{"class":36,"line":37},[34,22216,22217],{"class":190},"todoStore.todos\n",[34,22219,22220],{"class":36,"line":43},[34,22221,22222],{"class":190},"todoStore.remainingCount\n",[34,22224,22225,22228,22230,22232,22234],{"class":36,"line":49},[34,22226,22227],{"class":190},"todoStore.",[34,22229,22088],{"class":395},[34,22231,456],{"class":190},[34,22233,1167],{"class":175},[34,22235,6111],{"class":190},[19,22237,22239],{"id":22238},"_6-state-getter-action","6. State \u002F Getter \u002F Action",[15,22241,22242],{},"在组合式 store 写法中：",[24,22244,22246],{"className":166,"code":22245,"language":168,"meta":29,"style":29},"const todos = ref([]) \u002F\u002F state\n\nconst remainingCount = computed(() => {\n return todos.value.filter((todo) => !todo.done).length\n}) \u002F\u002F getter\n\nfunction addTodo(title) {\n \u002F\u002F action\n}\n",[31,22247,22248,22264,22268,22284,22306,22314,22318,22330,22335],{"__ignoreMap":29},[34,22249,22250,22252,22254,22256,22258,22261],{"class":36,"line":37},[34,22251,176],{"class":175},[34,22253,12962],{"class":179},[34,22255,183],{"class":175},[34,22257,453],{"class":395},[34,22259,22260],{"class":190},"([]) ",[34,22262,22263],{"class":301},"\u002F\u002F state\n",[34,22265,22266],{"class":36,"line":43},[34,22267,53],{"emptyLinePlaceholder":52},[34,22269,22270,22272,22274,22276,22278,22280,22282],{"class":36,"line":49},[34,22271,176],{"class":175},[34,22273,11203],{"class":179},[34,22275,183],{"class":175},[34,22277,8935],{"class":395},[34,22279,7682],{"class":190},[34,22281,1333],{"class":175},[34,22283,674],{"class":190},[34,22285,22286,22288,22290,22292,22294,22296,22298,22300,22302,22304],{"class":36,"line":56},[34,22287,1510],{"class":175},[34,22289,13059],{"class":190},[34,22291,1300],{"class":395},[34,22293,1323],{"class":190},[34,22295,12986],{"class":1326},[34,22297,1330],{"class":190},[34,22299,1333],{"class":175},[34,22301,1336],{"class":175},[34,22303,21961],{"class":190},[34,22305,10311],{"class":179},[34,22307,22308,22311],{"class":36,"line":62},[34,22309,22310],{"class":190},"}) ",[34,22312,22313],{"class":301},"\u002F\u002F getter\n",[34,22315,22316],{"class":36,"line":68},[34,22317,53],{"emptyLinePlaceholder":52},[34,22319,22320,22322,22324,22326,22328],{"class":36,"line":74},[34,22321,392],{"class":175},[34,22323,21978],{"class":395},[34,22325,456],{"class":190},[34,22327,1665],{"class":1326},[34,22329,1700],{"class":190},[34,22331,22332],{"class":36,"line":1971},[34,22333,22334],{"class":301}," \u002F\u002F action\n",[34,22336,22337],{"class":36,"line":1976},[34,22338,77],{"class":190},[15,22340,22341],{},"三者职责：",[105,22343,22344,22347,22350],{},[108,22345,22346],{},"state：保存数据。",[108,22348,22349],{},"getter：从 state 推导数据。",[108,22351,22352],{},"action：修改 state 或执行业务动作。",[15,22354,22355],{},"不要在组件里到处手写复杂修改逻辑。把跨组件共享的业务修改放到 action 中，组件只负责调用。",[19,22357,22359],{"id":22358},"_7-完整任务-store","7. 完整任务 store",[24,22361,22363],{"className":166,"code":22362,"language":168,"meta":29,"style":29},"import { computed, ref } from 'vue'\nimport { defineStore } from 'pinia'\n\nexport const useTodoStore = defineStore('todos', () => {\n const todos = ref([])\n const filter = ref('all')\n\n const completedCount = computed(() => {\n return todos.value.filter((todo) => todo.done).length\n })\n\n const remainingCount = computed(() => {\n return todos.value.length - completedCount.value\n })\n\n const filteredTodos = computed(() => {\n if (filter.value === 'active') {\n return todos.value.filter((todo) => !todo.done)\n }\n\n if (filter.value === 'completed') {\n return todos.value.filter((todo) => todo.done)\n }\n\n return todos.value\n })\n\n function addTodo(title) {\n const trimmedTitle = title.trim()\n\n if (!trimmedTitle) {\n return\n }\n\n todos.value.unshift({\n id: crypto.randomUUID(),\n title: trimmedTitle,\n done: false\n })\n }\n\n function toggleTodo(id) {\n todos.value = todos.value.map((todo) => {\n return todo.id === id ? { ...todo, done: !todo.done } : todo\n })\n }\n\n function removeTodo(id) {\n todos.value = todos.value.filter((todo) => todo.id !== id)\n }\n\n function setFilter(nextFilter) {\n filter.value = nextFilter\n }\n\n return {\n todos,\n filter,\n completedCount,\n remainingCount,\n filteredTodos,\n addTodo,\n toggleTodo,\n removeTodo,\n setFilter\n }\n})\n",[31,22364,22365,22375,22385,22389,22411,22423,22439,22443,22459,22480,22484,22488,22504,22518,22522,22526,22543,22555,22576,22580,22584,22596,22615,22619,22623,22630,22634,22638,22650,22665,22669,22680,22685,22689,22693,22701,22709,22714,22720,22724,22728,22732,22744,22765,22791,22795,22799,22803,22815,22839,22843,22847,22861,22871,22875,22879,22885,22889,22894,22899,22903,22908,22913,22918,22923,22928,22932],{"__ignoreMap":29},[34,22366,22367,22369,22371,22373],{"class":36,"line":37},[34,22368,426],{"class":175},[34,22370,16370],{"class":190},[34,22372,432],{"class":175},[34,22374,6891],{"class":186},[34,22376,22377,22379,22381,22383],{"class":36,"line":43},[34,22378,426],{"class":175},[34,22380,21876],{"class":190},[34,22382,432],{"class":175},[34,22384,21801],{"class":186},[34,22386,22387],{"class":36,"line":49},[34,22388,53],{"emptyLinePlaceholder":52},[34,22390,22391,22393,22395,22397,22399,22401,22403,22405,22407,22409],{"class":36,"line":56},[34,22392,2825],{"class":175},[34,22394,3280],{"class":175},[34,22396,21893],{"class":179},[34,22398,183],{"class":175},[34,22400,21898],{"class":395},[34,22402,456],{"class":190},[34,22404,19071],{"class":186},[34,22406,6131],{"class":190},[34,22408,1333],{"class":175},[34,22410,674],{"class":190},[34,22412,22413,22415,22417,22419,22421],{"class":36,"line":62},[34,22414,5298],{"class":175},[34,22416,12962],{"class":179},[34,22418,183],{"class":175},[34,22420,453],{"class":395},[34,22422,9446],{"class":190},[34,22424,22425,22427,22429,22431,22433,22435,22437],{"class":36,"line":68},[34,22426,5298],{"class":175},[34,22428,15555],{"class":179},[34,22430,183],{"class":175},[34,22432,453],{"class":395},[34,22434,456],{"class":190},[34,22436,15564],{"class":186},[34,22438,6111],{"class":190},[34,22440,22441],{"class":36,"line":74},[34,22442,53],{"emptyLinePlaceholder":52},[34,22444,22445,22447,22449,22451,22453,22455,22457],{"class":36,"line":1971},[34,22446,5298],{"class":175},[34,22448,10279],{"class":179},[34,22450,183],{"class":175},[34,22452,8935],{"class":395},[34,22454,7682],{"class":190},[34,22456,1333],{"class":175},[34,22458,674],{"class":190},[34,22460,22461,22463,22465,22467,22469,22471,22473,22475,22478],{"class":36,"line":1976},[34,22462,8601],{"class":175},[34,22464,13059],{"class":190},[34,22466,1300],{"class":395},[34,22468,1323],{"class":190},[34,22470,12986],{"class":1326},[34,22472,1330],{"class":190},[34,22474,1333],{"class":175},[34,22476,22477],{"class":190}," todo.done).",[34,22479,10311],{"class":179},[34,22481,22482],{"class":36,"line":1985},[34,22483,6379],{"class":190},[34,22485,22486],{"class":36,"line":1990},[34,22487,53],{"emptyLinePlaceholder":52},[34,22489,22490,22492,22494,22496,22498,22500,22502],{"class":36,"line":1999},[34,22491,5298],{"class":175},[34,22493,11203],{"class":179},[34,22495,183],{"class":175},[34,22497,8935],{"class":395},[34,22499,7682],{"class":190},[34,22501,1333],{"class":175},[34,22503,674],{"class":190},[34,22505,22506,22508,22510,22512,22515],{"class":36,"line":2022},[34,22507,8601],{"class":175},[34,22509,13059],{"class":190},[34,22511,1289],{"class":179},[34,22513,22514],{"class":175}," -",[34,22516,22517],{"class":190}," completedCount.value\n",[34,22519,22520],{"class":36,"line":2036},[34,22521,6379],{"class":190},[34,22523,22524],{"class":36,"line":2603},[34,22525,53],{"emptyLinePlaceholder":52},[34,22527,22528,22530,22533,22535,22537,22539,22541],{"class":36,"line":5537},[34,22529,5298],{"class":175},[34,22531,22532],{"class":179}," filteredTodos",[34,22534,183],{"class":175},[34,22536,8935],{"class":395},[34,22538,7682],{"class":190},[34,22540,1333],{"class":175},[34,22542,674],{"class":190},[34,22544,22545,22547,22549,22551,22553],{"class":36,"line":5543},[34,22546,5499],{"class":175},[34,22548,15597],{"class":190},[34,22550,1423],{"class":175},[34,22552,15602],{"class":186},[34,22554,1700],{"class":190},[34,22556,22557,22559,22561,22563,22565,22567,22569,22571,22573],{"class":36,"line":5548},[34,22558,17421],{"class":175},[34,22560,13059],{"class":190},[34,22562,1300],{"class":395},[34,22564,1323],{"class":190},[34,22566,12986],{"class":1326},[34,22568,1330],{"class":190},[34,22570,1333],{"class":175},[34,22572,1336],{"class":175},[34,22574,22575],{"class":190},"todo.done)\n",[34,22577,22578],{"class":36,"line":5564},[34,22579,5540],{"class":190},[34,22581,22582],{"class":36,"line":5576},[34,22583,53],{"emptyLinePlaceholder":52},[34,22585,22586,22588,22590,22592,22594],{"class":36,"line":5606},[34,22587,5499],{"class":175},[34,22589,15597],{"class":190},[34,22591,1423],{"class":175},[34,22593,15643],{"class":186},[34,22595,1700],{"class":190},[34,22597,22598,22600,22602,22604,22606,22608,22610,22612],{"class":36,"line":5616},[34,22599,17421],{"class":175},[34,22601,13059],{"class":190},[34,22603,1300],{"class":395},[34,22605,1323],{"class":190},[34,22607,12986],{"class":1326},[34,22609,1330],{"class":190},[34,22611,1333],{"class":175},[34,22613,22614],{"class":190}," todo.done)\n",[34,22616,22617],{"class":36,"line":5628},[34,22618,5540],{"class":190},[34,22620,22621],{"class":36,"line":5634},[34,22622,53],{"emptyLinePlaceholder":52},[34,22624,22625,22627],{"class":36,"line":6420},[34,22626,8601],{"class":175},[34,22628,22629],{"class":190}," todos.value\n",[34,22631,22632],{"class":36,"line":11294},[34,22633,6379],{"class":190},[34,22635,22636],{"class":36,"line":11321},[34,22637,53],{"emptyLinePlaceholder":52},[34,22639,22640,22642,22644,22646,22648],{"class":36,"line":11326},[34,22641,6318],{"class":175},[34,22643,21978],{"class":395},[34,22645,456],{"class":190},[34,22647,1665],{"class":1326},[34,22649,1700],{"class":190},[34,22651,22652,22654,22657,22659,22661,22663],{"class":36,"line":11333},[34,22653,5475],{"class":175},[34,22655,22656],{"class":179}," trimmedTitle",[34,22658,183],{"class":175},[34,22660,7941],{"class":190},[34,22662,14998],{"class":395},[34,22664,6374],{"class":190},[34,22666,22667],{"class":36,"line":11344},[34,22668,53],{"emptyLinePlaceholder":52},[34,22670,22671,22673,22675,22677],{"class":36,"line":11349},[34,22672,5499],{"class":175},[34,22674,1840],{"class":190},[34,22676,5504],{"class":175},[34,22678,22679],{"class":190},"trimmedTitle) {\n",[34,22681,22682],{"class":36,"line":11354},[34,22683,22684],{"class":175}," return\n",[34,22686,22687],{"class":36,"line":11363},[34,22688,5540],{"class":190},[34,22690,22691],{"class":36,"line":11368},[34,22692,53],{"emptyLinePlaceholder":52},[34,22694,22695,22697,22699],{"class":36,"line":11377},[34,22696,21989],{"class":190},[34,22698,15317],{"class":395},[34,22700,9299],{"class":190},[34,22702,22703,22705,22707],{"class":36,"line":11402},[34,22704,21998],{"class":190},[34,22706,14345],{"class":395},[34,22708,14348],{"class":190},[34,22710,22711],{"class":36,"line":11407},[34,22712,22713],{"class":190}," title: trimmedTitle,\n",[34,22715,22716,22718],{"class":36,"line":11421},[34,22717,22012],{"class":190},[34,22719,14360],{"class":179},[34,22721,22722],{"class":36,"line":11426},[34,22723,6791],{"class":190},[34,22725,22726],{"class":36,"line":11447},[34,22727,5631],{"class":190},[34,22729,22730],{"class":36,"line":11452},[34,22731,53],{"emptyLinePlaceholder":52},[34,22733,22734,22736,22738,22740,22742],{"class":36,"line":11463},[34,22735,6318],{"class":175},[34,22737,13043],{"class":395},[34,22739,456],{"class":190},[34,22741,1805],{"class":1326},[34,22743,1700],{"class":190},[34,22745,22746,22749,22751,22753,22755,22757,22759,22761,22763],{"class":36,"line":11485},[34,22747,22748],{"class":190}," todos.value ",[34,22750,202],{"class":175},[34,22752,13059],{"class":190},[34,22754,1355],{"class":395},[34,22756,1323],{"class":190},[34,22758,12986],{"class":1326},[34,22760,1330],{"class":190},[34,22762,1333],{"class":175},[34,22764,674],{"class":190},[34,22766,22767,22769,22771,22773,22775,22777,22779,22781,22783,22785,22787,22789],{"class":36,"line":11495},[34,22768,17421],{"class":175},[34,22770,13078],{"class":190},[34,22772,1423],{"class":175},[34,22774,13083],{"class":190},[34,22776,6701],{"class":175},[34,22778,1164],{"class":190},[34,22780,1167],{"class":175},[34,22782,13092],{"class":190},[34,22784,5504],{"class":175},[34,22786,13097],{"class":190},[34,22788,5598],{"class":175},[34,22790,13102],{"class":190},[34,22792,22793],{"class":36,"line":11528},[34,22794,6791],{"class":190},[34,22796,22797],{"class":36,"line":11534},[34,22798,5631],{"class":190},[34,22800,22801],{"class":36,"line":11544},[34,22802,53],{"emptyLinePlaceholder":52},[34,22804,22805,22807,22809,22811,22813],{"class":36,"line":11553},[34,22806,6318],{"class":175},[34,22808,13121],{"class":395},[34,22810,456],{"class":190},[34,22812,1805],{"class":1326},[34,22814,1700],{"class":190},[34,22816,22817,22819,22821,22823,22825,22827,22829,22831,22833,22835,22837],{"class":36,"line":11562},[34,22818,22748],{"class":190},[34,22820,202],{"class":175},[34,22822,13059],{"class":190},[34,22824,1300],{"class":395},[34,22826,1323],{"class":190},[34,22828,12986],{"class":1326},[34,22830,1330],{"class":190},[34,22832,1333],{"class":175},[34,22834,13078],{"class":190},[34,22836,13150],{"class":175},[34,22838,6991],{"class":190},[34,22840,22841],{"class":36,"line":16859},[34,22842,5631],{"class":190},[34,22844,22845],{"class":36,"line":16885},[34,22846,53],{"emptyLinePlaceholder":52},[34,22848,22849,22851,22854,22856,22859],{"class":36,"line":16890},[34,22850,6318],{"class":175},[34,22852,22853],{"class":395}," setFilter",[34,22855,456],{"class":190},[34,22857,22858],{"class":1326},"nextFilter",[34,22860,1700],{"class":190},[34,22862,22863,22866,22868],{"class":36,"line":16911},[34,22864,22865],{"class":190}," filter.value ",[34,22867,202],{"class":175},[34,22869,22870],{"class":190}," nextFilter\n",[34,22872,22873],{"class":36,"line":16932},[34,22874,5631],{"class":190},[34,22876,22877],{"class":36,"line":16955},[34,22878,53],{"emptyLinePlaceholder":52},[34,22880,22881,22883],{"class":36,"line":16975},[34,22882,1510],{"class":175},[34,22884,674],{"class":190},[34,22886,22887],{"class":36,"line":16996},[34,22888,22037],{"class":190},[34,22890,22891],{"class":36,"line":17005},[34,22892,22893],{"class":190}," filter,\n",[34,22895,22896],{"class":36,"line":17010},[34,22897,22898],{"class":190}," completedCount,\n",[34,22900,22901],{"class":36,"line":17032},[34,22902,22042],{"class":190},[34,22904,22905],{"class":36,"line":17041},[34,22906,22907],{"class":190}," filteredTodos,\n",[34,22909,22910],{"class":36,"line":17046},[34,22911,22912],{"class":190}," addTodo,\n",[34,22914,22915],{"class":36,"line":17066},[34,22916,22917],{"class":190}," toggleTodo,\n",[34,22919,22920],{"class":36,"line":17086},[34,22921,22922],{"class":190}," removeTodo,\n",[34,22924,22925],{"class":36,"line":17106},[34,22926,22927],{"class":190}," setFilter\n",[34,22929,22930],{"class":36,"line":17111},[34,22931,5631],{"class":190},[34,22933,22934],{"class":36,"line":17130},[34,22935,6153],{"class":190},[19,22937,22939],{"id":22938},"_8-store-不应该放什么","8. Store 不应该放什么",[15,22941,22942],{},"不要把所有状态都放到 Pinia。",[15,22944,22945],{},"不适合放：",[105,22947,22948,22951,22954,22957,22960],{},[108,22949,22950],{},"输入框里正在输入、但只在当前组件用的文字。",[108,22952,22953],{},"某个弹窗是否打开。",[108,22955,22956],{},"某个按钮是否 hover。",[108,22958,22959],{},"单个组件内部的临时 tab。",[108,22961,22962],{},"只用于局部 UI 的 loading。",[15,22964,22965],{},"适合放：",[105,22967,22968,22970,22973,22975,22977],{},[108,22969,21390],{},[108,22971,22972],{},"跨页面共享的任务列表。",[108,22974,21393],{},[108,22976,21399],{},[108,22978,22979],{},"用户设置。",[15,22981,22982],{},"判断问题：",[24,22984,22987],{"className":22985,"code":22986,"language":2638,"meta":29},[2636],"这个状态是否被多个远离的组件使用？\n刷新或切换页面后是否还需要？\n它是否代表业务数据，而不是单个 UI 的临时状态？\n",[31,22988,22986],{"__ignoreMap":29},[15,22990,22991],{},"如果答案多为“是”，可以考虑 Pinia。",[19,22993,22995],{"id":22994},"_9-store-拆分","9. Store 拆分",[15,22997,22998],{},"不要把整个应用塞进一个 store。",[15,23000,23001],{},"推荐按业务领域拆：",[24,23003,23006],{"className":23004,"code":23005,"language":2638,"meta":29},[2636],"useUserStore 登录用户、权限\nuseCourseStore 课程、进度\nuseTodoStore 任务\nuseSettingsStore 用户设置\n",[31,23007,23005],{"__ignoreMap":29},[15,23009,7500],{},[24,23011,23014],{"className":23012,"code":23013,"language":2638,"meta":29},[2636],"useAppStore 什么都放\n",[31,23015,23013],{"__ignoreMap":29},[15,23017,23018],{},"一个 store 太大后，会变得和一个巨大组件一样难维护。",[19,23020,23022],{"id":23021},"_10-pinia-和-localstorage","10. Pinia 和 localStorage",[15,23024,23025,23026,347],{},"Pinia 默认是内存状态，刷新页面会丢失。如果要保留，可以配合 ",[31,23027,10566],{},[24,23029,23031],{"className":166,"code":23030,"language":168,"meta":29,"style":29},"const saved = localStorage.getItem('todos')\nconst todos = ref(saved ? JSON.parse(saved) : [])\n\nwatch(\n todos,\n (value) => {\n localStorage.setItem('todos', JSON.stringify(value))\n },\n { deep: true }\n)\n",[31,23032,23033,23052,23081,23085,23091,23096,23109,23130,23134,23143],{"__ignoreMap":29},[34,23034,23035,23037,23040,23042,23044,23046,23048,23050],{"class":36,"line":37},[34,23036,176],{"class":175},[34,23038,23039],{"class":179}," saved",[34,23041,183],{"class":175},[34,23043,18831],{"class":190},[34,23045,18834],{"class":395},[34,23047,456],{"class":190},[34,23049,19071],{"class":186},[34,23051,6111],{"class":190},[34,23053,23054,23056,23058,23060,23062,23065,23067,23069,23071,23073,23076,23078],{"class":36,"line":43},[34,23055,176],{"class":175},[34,23057,12962],{"class":179},[34,23059,183],{"class":175},[34,23061,453],{"class":395},[34,23063,23064],{"class":190},"(saved ",[34,23066,6701],{"class":175},[34,23068,18849],{"class":179},[34,23070,2547],{"class":190},[34,23072,18854],{"class":395},[34,23074,23075],{"class":190},"(saved) ",[34,23077,5598],{"class":175},[34,23079,23080],{"class":190}," [])\n",[34,23082,23083],{"class":36,"line":49},[34,23084,53],{"emptyLinePlaceholder":52},[34,23086,23087,23089],{"class":36,"line":56},[34,23088,152],{"class":395},[34,23090,18938],{"class":190},[34,23092,23093],{"class":36,"line":62},[34,23094,23095],{"class":190}," todos,\n",[34,23097,23098,23101,23103,23105,23107],{"class":36,"line":68},[34,23099,23100],{"class":190}," (",[34,23102,2533],{"class":1326},[34,23104,1330],{"class":190},[34,23106,1333],{"class":175},[34,23108,674],{"class":190},[34,23110,23111,23114,23116,23118,23120,23122,23124,23126,23128],{"class":36,"line":74},[34,23112,23113],{"class":190}," localStorage.",[34,23115,10670],{"class":395},[34,23117,456],{"class":190},[34,23119,19071],{"class":186},[34,23121,616],{"class":190},[34,23123,18969],{"class":179},[34,23125,2547],{"class":190},[34,23127,18974],{"class":395},[34,23129,18977],{"class":190},[34,23131,23132],{"class":36,"line":1971},[34,23133,17404],{"class":190},[34,23135,23136,23139,23141],{"class":36,"line":1976},[34,23137,23138],{"class":190}," { deep: ",[34,23140,758],{"class":179},[34,23142,6564],{"class":190},[34,23144,23145],{"class":36,"line":1985},[34,23146,6111],{"class":190},[15,23148,23149],{},"这段代码做了两件事：",[2230,23151,23152,23155],{},[108,23153,23154],{},"初始化时从 localStorage 读取。",[108,23156,23157],{},"todos 变化时写回 localStorage。",[15,23159,7847,23160,23162,23163,23165],{},[31,23161,7854],{}," 把这段逻辑抽成了 ",[31,23164,18770],{}," 组合式函数。",[19,23167,23169],{"id":23168},"_11-store-中可以请求数据吗","11. Store 中可以请求数据吗",[15,23171,23172],{},"可以。",[24,23174,23176],{"className":166,"code":23175,"language":168,"meta":29,"style":29},"const todos = ref([])\nconst loading = ref(false)\nconst error = ref('')\n\nasync function loadTodos() {\n loading.value = true\n error.value = ''\n\n try {\n const response = await fetch('\u002Fapi\u002Ftodos')\n\n if (!response.ok) {\n throw new Error('任务加载失败')\n }\n\n todos.value = await response.json()\n } catch (err) {\n error.value = err instanceof Error ? err.message : '加载失败'\n } finally {\n loading.value = false\n }\n}\n",[31,23177,23178,23190,23206,23222,23226,23237,23245,23253,23257,23263,23282,23286,23296,23311,23315,23319,23333,23341,23362,23370,23378,23382],{"__ignoreMap":29},[34,23179,23180,23182,23184,23186,23188],{"class":36,"line":37},[34,23181,176],{"class":175},[34,23183,12962],{"class":179},[34,23185,183],{"class":175},[34,23187,453],{"class":395},[34,23189,9446],{"class":190},[34,23191,23192,23194,23196,23198,23200,23202,23204],{"class":36,"line":43},[34,23193,176],{"class":175},[34,23195,5375],{"class":179},[34,23197,183],{"class":175},[34,23199,453],{"class":395},[34,23201,456],{"class":190},[34,23203,702],{"class":179},[34,23205,6111],{"class":190},[34,23207,23208,23210,23212,23214,23216,23218,23220],{"class":36,"line":49},[34,23209,176],{"class":175},[34,23211,5392],{"class":179},[34,23213,183],{"class":175},[34,23215,453],{"class":395},[34,23217,456],{"class":190},[34,23219,7400],{"class":186},[34,23221,6111],{"class":190},[34,23223,23224],{"class":36,"line":56},[34,23225,53],{"emptyLinePlaceholder":52},[34,23227,23228,23230,23232,23235],{"class":36,"line":62},[34,23229,5217],{"class":175},[34,23231,2850],{"class":175},[34,23233,23234],{"class":395}," loadTodos",[34,23236,399],{"class":190},[34,23238,23239,23241,23243],{"class":36,"line":68},[34,23240,5441],{"class":190},[34,23242,202],{"class":175},[34,23244,12260],{"class":179},[34,23246,23247,23249,23251],{"class":36,"line":74},[34,23248,5452],{"class":190},[34,23250,202],{"class":175},[34,23252,6645],{"class":186},[34,23254,23255],{"class":36,"line":1971},[34,23256,53],{"emptyLinePlaceholder":52},[34,23258,23259,23261],{"class":36,"line":1976},[34,23260,5468],{"class":175},[34,23262,674],{"class":190},[34,23264,23265,23267,23269,23271,23273,23275,23277,23280],{"class":36,"line":1985},[34,23266,5475],{"class":175},[34,23268,5301],{"class":179},[34,23270,183],{"class":175},[34,23272,5306],{"class":175},[34,23274,5309],{"class":395},[34,23276,456],{"class":190},[34,23278,23279],{"class":186},"'\u002Fapi\u002Ftodos'",[34,23281,6111],{"class":190},[34,23283,23284],{"class":36,"line":1990},[34,23285,53],{"emptyLinePlaceholder":52},[34,23287,23288,23290,23292,23294],{"class":36,"line":1999},[34,23289,5499],{"class":175},[34,23291,1840],{"class":190},[34,23293,5504],{"class":175},[34,23295,5507],{"class":190},[34,23297,23298,23300,23302,23304,23306,23309],{"class":36,"line":2022},[34,23299,5512],{"class":175},[34,23301,5177],{"class":175},[34,23303,5517],{"class":395},[34,23305,456],{"class":190},[34,23307,23308],{"class":186},"'任务加载失败'",[34,23310,6111],{"class":190},[34,23312,23313],{"class":36,"line":2036},[34,23314,5540],{"class":190},[34,23316,23317],{"class":36,"line":2603},[34,23318,53],{"emptyLinePlaceholder":52},[34,23320,23321,23323,23325,23327,23329,23331],{"class":36,"line":5537},[34,23322,22748],{"class":190},[34,23324,202],{"class":175},[34,23326,5306],{"class":175},[34,23328,5329],{"class":190},[34,23330,5332],{"class":395},[34,23332,6374],{"class":190},[34,23334,23335,23337,23339],{"class":36,"line":5543},[34,23336,5567],{"class":190},[34,23338,5570],{"class":175},[34,23340,5573],{"class":190},[34,23342,23343,23345,23347,23349,23351,23353,23355,23357,23359],{"class":36,"line":5548},[34,23344,5579],{"class":190},[34,23346,202],{"class":175},[34,23348,5584],{"class":190},[34,23350,5587],{"class":175},[34,23352,5517],{"class":395},[34,23354,5592],{"class":175},[34,23356,5595],{"class":190},[34,23358,5598],{"class":175},[34,23360,23361],{"class":186}," '加载失败'\n",[34,23363,23364,23366,23368],{"class":36,"line":5564},[34,23365,5567],{"class":190},[34,23367,5611],{"class":175},[34,23369,674],{"class":190},[34,23371,23372,23374,23376],{"class":36,"line":5576},[34,23373,5619],{"class":190},[34,23375,202],{"class":175},[34,23377,18640],{"class":179},[34,23379,23380],{"class":36,"line":5606},[34,23381,5631],{"class":190},[34,23383,23384],{"class":36,"line":5616},[34,23385,77],{"class":190},[15,23387,23388,23389,2664],{},"但复杂项目通常会把请求函数拆到 ",[31,23390,23391],{},"services\u002F",[24,23393,23395],{"className":166,"code":23394,"language":168,"meta":29,"style":29},"\u002F\u002F services\u002FtodoService.js\nexport async function getTodos() {\n const response = await fetch('\u002Fapi\u002Ftodos')\n return response.json()\n}\n",[31,23396,23397,23402,23416,23434,23444],{"__ignoreMap":29},[34,23398,23399],{"class":36,"line":37},[34,23400,23401],{"class":301},"\u002F\u002F services\u002FtodoService.js\n",[34,23403,23404,23406,23409,23411,23414],{"class":36,"line":43},[34,23405,2825],{"class":175},[34,23407,23408],{"class":175}," async",[34,23410,2850],{"class":175},[34,23412,23413],{"class":395}," getTodos",[34,23415,399],{"class":190},[34,23417,23418,23420,23422,23424,23426,23428,23430,23432],{"class":36,"line":49},[34,23419,5298],{"class":175},[34,23421,5301],{"class":179},[34,23423,183],{"class":175},[34,23425,5306],{"class":175},[34,23427,5309],{"class":395},[34,23429,456],{"class":190},[34,23431,23279],{"class":186},[34,23433,6111],{"class":190},[34,23435,23436,23438,23440,23442],{"class":36,"line":56},[34,23437,1510],{"class":175},[34,23439,5329],{"class":190},[34,23441,5332],{"class":395},[34,23443,6374],{"class":190},[34,23445,23446],{"class":36,"line":62},[34,23447,77],{"class":190},[15,23449,23450],{},"store 调用 service：",[24,23452,23454],{"className":166,"code":23453,"language":168,"meta":29,"style":29},"todos.value = await getTodos()\n",[31,23455,23456],{"__ignoreMap":29},[34,23457,23458,23461,23463,23465,23467],{"class":36,"line":37},[34,23459,23460],{"class":190},"todos.value ",[34,23462,202],{"class":175},[34,23464,5306],{"class":175},[34,23466,23413],{"class":395},[34,23468,6374],{"class":190},[15,23470,23471],{},"这样 store 专注业务状态，service 专注请求细节。",[19,23473,23475],{"id":23474},"_12-pinia-常见错误","12. Pinia 常见错误",[253,23477,23479],{"id":23478},"错误一忘记注册-pinia","错误一：忘记注册 Pinia",[15,23481,23482],{},"如果没有：",[24,23484,23486],{"className":166,"code":23485,"language":168,"meta":29,"style":29},"app.use(createPinia())\n",[31,23487,23488],{"__ignoreMap":29},[34,23489,23490,23493,23495,23497,23499],{"class":36,"line":37},[34,23491,23492],{"class":190},"app.",[34,23494,17902],{"class":395},[34,23496,456],{"class":190},[34,23498,21828],{"class":395},[34,23500,15460],{"class":190},[15,23502,23503],{},"store 无法正常工作。",[253,23505,23507],{"id":23506},"错误二把局部表单状态放进-store","错误二：把局部表单状态放进 store",[15,23509,23510],{},"例如新增任务输入框：",[24,23512,23513],{"className":166,"code":15122,"language":168,"meta":29,"style":29},[31,23514,23515],{"__ignoreMap":29},[34,23516,23517,23519,23521,23523,23525,23527,23529],{"class":36,"line":37},[34,23518,176],{"class":175},[34,23520,278],{"class":179},[34,23522,183],{"class":175},[34,23524,453],{"class":395},[34,23526,456],{"class":190},[34,23528,7400],{"class":186},[34,23530,6111],{"class":190},[15,23532,23533,23534,23537,23538,347],{},"如果只在 ",[31,23535,23536],{},"TodoForm"," 使用，放组件内即可。提交后调用 store 的 ",[31,23539,23540],{},"addTodo(title)",[253,23542,23544],{"id":23543},"错误三直接在很多组件里修改同一份数组","错误三：直接在很多组件里修改同一份数组",[15,23546,23547],{},"如果多个组件都写：",[24,23549,23551],{"className":166,"code":23550,"language":168,"meta":29,"style":29},"todoStore.todos.push(...)\n",[31,23552,23553],{"__ignoreMap":29},[34,23554,23555,23558,23560,23562,23564],{"class":36,"line":37},[34,23556,23557],{"class":190},"todoStore.todos.",[34,23559,1262],{"class":395},[34,23561,456],{"class":190},[34,23563,1167],{"class":175},[34,23565,6111],{"class":190},[15,23567,23568],{},"逻辑会散落。更好的方式是：",[24,23570,23572],{"className":166,"code":23571,"language":168,"meta":29,"style":29},"todoStore.addTodo(...)\n",[31,23573,23574],{"__ignoreMap":29},[34,23575,23576,23578,23580,23582,23584],{"class":36,"line":37},[34,23577,22227],{"class":190},[34,23579,22088],{"class":395},[34,23581,456],{"class":190},[34,23583,1167],{"class":175},[34,23585,6111],{"class":190},[15,23587,23588],{},"这样修改规则集中在 action。",[253,23590,23592],{"id":23591},"错误四store-名和文件名混乱","错误四：store 名和文件名混乱",[15,23594,23595],{},"建议：",[24,23597,23600],{"className":23598,"code":23599,"language":2638,"meta":29},[2636],"stores\u002FuseTodoStore.js\nexport const useTodoStore = defineStore('todos', ...)\n",[31,23601,23599],{"__ignoreMap":29},[15,23603,23604],{},"保持命名一致，方便查找。",[19,23606,5693],{"id":5693},[2230,23608,23609,23615,23622,23630,23639,23642],{},[108,23610,23611,23612,347],{},"创建 ",[31,23613,23614],{},"useTodoStore",[108,23616,23617,23618,146,23620,347],{},"实现 ",[31,23619,22076],{},[31,23621,1300],{},[108,23623,23617,23624,146,23626,146,23628,347],{},[31,23625,22088],{},[31,23627,13007],{},[31,23629,13022],{},[108,23631,23617,23632,146,23634,146,23636,347],{},[31,23633,22082],{},[31,23635,10434],{},[31,23637,23638],{},"filteredTodos",[108,23640,23641],{},"在组件中使用 store 渲染任务列表。",[108,23643,23644,23645],{},"判断下面状态是否应该放 Pinia，并说明理由：\n",[105,23646,23647,23650,23653,23656,23658],{},[108,23648,23649],{},"登录用户。",[108,23651,23652],{},"新增任务输入框文字。",[108,23654,23655],{},"购物车商品。",[108,23657,21407],{},[108,23659,21399],{},[79,23661,23662],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}",{"title":29,"searchDepth":43,"depth":43,"links":23664},[23665,23666,23667,23668,23669,23670,23671,23672,23673,23674,23675,23676,23682],{"id":21669,"depth":43,"text":21670},{"id":21728,"depth":43,"text":21729},{"id":21766,"depth":43,"text":21767},{"id":21848,"depth":43,"text":21849},{"id":22097,"depth":43,"text":22098},{"id":22238,"depth":43,"text":22239},{"id":22358,"depth":43,"text":22359},{"id":22938,"depth":43,"text":22939},{"id":22994,"depth":43,"text":22995},{"id":23021,"depth":43,"text":23022},{"id":23168,"depth":43,"text":23169},{"id":23474,"depth":43,"text":23475,"children":23677},[23678,23679,23680,23681],{"id":23478,"depth":49,"text":23479},{"id":23506,"depth":49,"text":23507},{"id":23543,"depth":49,"text":23544},{"id":23591,"depth":49,"text":23592},{"id":5693,"depth":43,"text":5693},{"order":1971},"\u002Fsource\u002Fcode\u002Fvue\u002F08-pinia",{"title":21655,"description":21663},"source\u002Fcode\u002Fvue\u002F08-pinia","YZUjQNc0txBGAENQGHrvhMz3A_OyZyoLxzvgm6dqiDQ",{"id":23689,"title":23690,"body":23691,"description":23698,"extension":85,"meta":25636,"navigation":52,"path":25637,"seo":25638,"stem":25639,"__hash__":25640},"notes\u002Fsource\u002Fcode\u002Fvue\u002F09-http-async.md","09 异步请求和错误处理",{"type":8,"value":23692,"toc":25609},[23693,23696,23699,23709,23713,23716,23732,23735,23738,23760,23765,23788,23796,23834,23838,23898,23901,23926,23933,23936,23975,23981,24073,24078,24082,24352,24354,24527,24530,24544,24550,24596,24599,24616,24621,24657,24666,24670,24695,24700,24703,24756,24762,24766,24769,24772,24775,24786,24790,24881,24884,24902,24904,24912,24916,24918,24926,24929,24933,24936,25102,25105,25109,25112,25118,25121,25279,25282,25286,25289,25364,25367,25378,25381,25385,25388,25402,25405,25452,25455,25466,25468,25475,25494,25500,25521,25525,25528,25532,25535,25570,25574,25577,25579,25606],[11,23694,23690],{"id":23695},"_09-异步请求和错误处理",[15,23697,23698],{},"真实 Vue 项目几乎都会请求后端数据。请求不是立刻完成的，所以你必须处理异步、加载中、失败、空数据、重试等状态。",[15,23700,23701,23702,23705,23706,23708],{},"本章目标：你能用 ",[31,23703,23704],{},"fetch"," 请求数据，能写 ",[31,23707,5147],{},"，能处理错误，能把请求逻辑拆到合适位置。",[19,23710,23712],{"id":23711},"_1-为什么请求是异步的","1. 为什么请求是异步的",[15,23714,23715],{},"当你请求接口：",[24,23717,23719],{"className":166,"code":23718,"language":168,"meta":29,"style":29},"fetch('\u002Fapi\u002Flessons')\n",[31,23720,23721],{"__ignoreMap":29},[34,23722,23723,23725,23727,23730],{"class":36,"line":37},[34,23724,23704],{"class":395},[34,23726,456],{"class":190},[34,23728,23729],{"class":186},"'\u002Fapi\u002Flessons'",[34,23731,6111],{"class":190},[15,23733,23734],{},"浏览器要通过网络和服务器通信。这个过程可能需要几十毫秒，也可能几秒，还可能失败。",[15,23736,23737],{},"JavaScript 不能停在那里什么都不做，所以请求会返回 Promise：",[24,23739,23741],{"className":166,"code":23740,"language":168,"meta":29,"style":29},"const promise = fetch('\u002Fapi\u002Flessons')\n",[31,23742,23743],{"__ignoreMap":29},[34,23744,23745,23747,23750,23752,23754,23756,23758],{"class":36,"line":37},[34,23746,176],{"class":175},[34,23748,23749],{"class":179}," promise",[34,23751,183],{"class":175},[34,23753,5309],{"class":395},[34,23755,456],{"class":190},[34,23757,23729],{"class":186},[34,23759,6111],{"class":190},[15,23761,5792,23762,23764],{},[31,23763,5273],{}," 等结果：",[24,23766,23768],{"className":166,"code":23767,"language":168,"meta":29,"style":29},"const response = await fetch('\u002Fapi\u002Flessons')\n",[31,23769,23770],{"__ignoreMap":29},[34,23771,23772,23774,23776,23778,23780,23782,23784,23786],{"class":36,"line":37},[34,23773,176],{"class":175},[34,23775,5301],{"class":179},[34,23777,183],{"class":175},[34,23779,5306],{"class":175},[34,23781,5309],{"class":395},[34,23783,456],{"class":190},[34,23785,23729],{"class":186},[34,23787,6111],{"class":190},[15,23789,23790,23792,23793,23795],{},[31,23791,5273],{}," 只能写在 ",[31,23794,5217],{}," 函数里：",[24,23797,23799],{"className":166,"code":23798,"language":168,"meta":29,"style":29},"async function loadLessons() {\n const response = await fetch('\u002Fapi\u002Flessons')\n}\n",[31,23800,23801,23812,23830],{"__ignoreMap":29},[34,23802,23803,23805,23807,23810],{"class":36,"line":37},[34,23804,5217],{"class":175},[34,23806,2850],{"class":175},[34,23808,23809],{"class":395}," loadLessons",[34,23811,399],{"class":190},[34,23813,23814,23816,23818,23820,23822,23824,23826,23828],{"class":36,"line":43},[34,23815,5298],{"class":175},[34,23817,5301],{"class":179},[34,23819,183],{"class":175},[34,23821,5306],{"class":175},[34,23823,5309],{"class":395},[34,23825,456],{"class":190},[34,23827,23729],{"class":186},[34,23829,6111],{"class":190},[34,23831,23832],{"class":36,"line":49},[34,23833,77],{"class":190},[19,23835,23837],{"id":23836},"_2-最基础的-fetch","2. 最基础的 fetch",[24,23839,23841],{"className":166,"code":23840,"language":168,"meta":29,"style":29},"async function loadLessons() {\n const response = await fetch('\u002Fapi\u002Flessons')\n const lessons = await response.json()\n return lessons\n}\n",[31,23842,23843,23853,23871,23887,23894],{"__ignoreMap":29},[34,23844,23845,23847,23849,23851],{"class":36,"line":37},[34,23846,5217],{"class":175},[34,23848,2850],{"class":175},[34,23850,23809],{"class":395},[34,23852,399],{"class":190},[34,23854,23855,23857,23859,23861,23863,23865,23867,23869],{"class":36,"line":43},[34,23856,5298],{"class":175},[34,23858,5301],{"class":179},[34,23860,183],{"class":175},[34,23862,5306],{"class":175},[34,23864,5309],{"class":395},[34,23866,456],{"class":190},[34,23868,23729],{"class":186},[34,23870,6111],{"class":190},[34,23872,23873,23875,23877,23879,23881,23883,23885],{"class":36,"line":49},[34,23874,5298],{"class":175},[34,23876,1203],{"class":179},[34,23878,183],{"class":175},[34,23880,5306],{"class":175},[34,23882,5329],{"class":190},[34,23884,5332],{"class":395},[34,23886,6374],{"class":190},[34,23888,23889,23891],{"class":36,"line":56},[34,23890,1510],{"class":175},[34,23892,23893],{"class":190}," lessons\n",[34,23895,23896],{"class":36,"line":62},[34,23897,77],{"class":190},[15,23899,23900],{},"逐行解释：",[105,23902,23903,23909,23914,23920],{},[108,23904,23905,23908],{},[31,23906,23907],{},"fetch('\u002Fapi\u002Flessons')"," 发起请求。",[108,23910,23911,23913],{},[31,23912,5525],{}," 是响应对象，不是最终数据。",[108,23915,23916,23919],{},[31,23917,23918],{},"response.json()"," 把响应体解析成 JavaScript 数据。",[108,23921,23922,23925],{},[31,23923,23924],{},"return lessons"," 把数据返回给调用者。",[19,23927,23929,23930],{"id":23928},"_3-为什么要检查-responseok","3. 为什么要检查 ",[31,23931,23932],{},"response.ok",[15,23934,23935],{},"初学者常写：",[24,23937,23939],{"className":166,"code":23938,"language":168,"meta":29,"style":29},"const response = await fetch('\u002Fapi\u002Flessons')\nconst lessons = await response.json()\n",[31,23940,23941,23959],{"__ignoreMap":29},[34,23942,23943,23945,23947,23949,23951,23953,23955,23957],{"class":36,"line":37},[34,23944,176],{"class":175},[34,23946,5301],{"class":179},[34,23948,183],{"class":175},[34,23950,5306],{"class":175},[34,23952,5309],{"class":395},[34,23954,456],{"class":190},[34,23956,23729],{"class":186},[34,23958,6111],{"class":190},[34,23960,23961,23963,23965,23967,23969,23971,23973],{"class":36,"line":43},[34,23962,176],{"class":175},[34,23964,1203],{"class":179},[34,23966,183],{"class":175},[34,23968,5306],{"class":175},[34,23970,5329],{"class":190},[34,23972,5332],{"class":395},[34,23974,6374],{"class":190},[15,23976,23977,23978,23980],{},"但如果服务器返回 404 或 500，",[31,23979,23704],{}," 不一定自动抛错。你应该检查：",[24,23982,23984],{"className":166,"code":23983,"language":168,"meta":29,"style":29},"async function loadLessons() {\n const response = await fetch('\u002Fapi\u002Flessons')\n\n if (!response.ok) {\n throw new Error(`请求失败：${response.status}`)\n }\n\n return response.json()\n}\n",[31,23985,23986,23996,24014,24018,24028,24051,24055,24059,24069],{"__ignoreMap":29},[34,23987,23988,23990,23992,23994],{"class":36,"line":37},[34,23989,5217],{"class":175},[34,23991,2850],{"class":175},[34,23993,23809],{"class":395},[34,23995,399],{"class":190},[34,23997,23998,24000,24002,24004,24006,24008,24010,24012],{"class":36,"line":43},[34,23999,5298],{"class":175},[34,24001,5301],{"class":179},[34,24003,183],{"class":175},[34,24005,5306],{"class":175},[34,24007,5309],{"class":395},[34,24009,456],{"class":190},[34,24011,23729],{"class":186},[34,24013,6111],{"class":190},[34,24015,24016],{"class":36,"line":49},[34,24017,53],{"emptyLinePlaceholder":52},[34,24019,24020,24022,24024,24026],{"class":36,"line":56},[34,24021,7000],{"class":175},[34,24023,1840],{"class":190},[34,24025,5504],{"class":175},[34,24027,5507],{"class":190},[34,24029,24030,24033,24035,24037,24039,24041,24043,24045,24047,24049],{"class":36,"line":62},[34,24031,24032],{"class":175}," throw",[34,24034,5177],{"class":175},[34,24036,5517],{"class":395},[34,24038,456],{"class":190},[34,24040,5522],{"class":186},[34,24042,5525],{"class":190},[34,24044,2547],{"class":186},[34,24046,5530],{"class":190},[34,24048,2560],{"class":186},[34,24050,6111],{"class":190},[34,24052,24053],{"class":36,"line":68},[34,24054,5631],{"class":190},[34,24056,24057],{"class":36,"line":74},[34,24058,53],{"emptyLinePlaceholder":52},[34,24060,24061,24063,24065,24067],{"class":36,"line":1971},[34,24062,1510],{"class":175},[34,24064,5329],{"class":190},[34,24066,5332],{"class":395},[34,24068,6374],{"class":190},[34,24070,24071],{"class":36,"line":1976},[34,24072,77],{"class":190},[15,24074,24075,24077],{},[31,24076,23932],{}," 表示 HTTP 状态码是否在 200 到 299。",[19,24079,24081],{"id":24080},"_4-组件中的请求状态","4. 组件中的请求状态",[24,24083,24085],{"className":923,"code":24084,"language":925,"meta":29,"style":29},"\u003Cscript setup>\nimport { onMounted, ref } from 'vue'\n\nconst lessons = ref([])\nconst loading = ref(false)\nconst error = ref('')\n\nasync function fetchLessons() {\n loading.value = true\n error.value = ''\n\n try {\n const response = await fetch('\u002Fapi\u002Flessons')\n\n if (!response.ok) {\n throw new Error('课程加载失败')\n }\n\n lessons.value = await response.json()\n } catch (err) {\n error.value = err instanceof Error ? err.message : '未知错误'\n } finally {\n loading.value = false\n }\n}\n\nonMounted(() => {\n fetchLessons()\n})\n\u003C\u002Fscript>\n",[31,24086,24087,24097,24107,24111,24123,24139,24155,24159,24170,24178,24186,24190,24196,24214,24218,24228,24243,24247,24251,24266,24274,24295,24303,24311,24315,24319,24323,24333,24340,24344],{"__ignoreMap":29},[34,24088,24089,24091,24093,24095],{"class":36,"line":37},[34,24090,932],{"class":190},[34,24092,1911],{"class":935},[34,24094,1914],{"class":395},[34,24096,939],{"class":190},[34,24098,24099,24101,24103,24105],{"class":36,"line":43},[34,24100,426],{"class":175},[34,24102,7645],{"class":190},[34,24104,432],{"class":175},[34,24106,6891],{"class":186},[34,24108,24109],{"class":36,"line":49},[34,24110,53],{"emptyLinePlaceholder":52},[34,24112,24113,24115,24117,24119,24121],{"class":36,"line":56},[34,24114,176],{"class":175},[34,24116,1203],{"class":179},[34,24118,183],{"class":175},[34,24120,453],{"class":395},[34,24122,9446],{"class":190},[34,24124,24125,24127,24129,24131,24133,24135,24137],{"class":36,"line":62},[34,24126,176],{"class":175},[34,24128,5375],{"class":179},[34,24130,183],{"class":175},[34,24132,453],{"class":395},[34,24134,456],{"class":190},[34,24136,702],{"class":179},[34,24138,6111],{"class":190},[34,24140,24141,24143,24145,24147,24149,24151,24153],{"class":36,"line":68},[34,24142,176],{"class":175},[34,24144,5392],{"class":179},[34,24146,183],{"class":175},[34,24148,453],{"class":395},[34,24150,456],{"class":190},[34,24152,7400],{"class":186},[34,24154,6111],{"class":190},[34,24156,24157],{"class":36,"line":74},[34,24158,53],{"emptyLinePlaceholder":52},[34,24160,24161,24163,24165,24168],{"class":36,"line":1971},[34,24162,5217],{"class":175},[34,24164,2850],{"class":175},[34,24166,24167],{"class":395}," fetchLessons",[34,24169,399],{"class":190},[34,24171,24172,24174,24176],{"class":36,"line":1976},[34,24173,5441],{"class":190},[34,24175,202],{"class":175},[34,24177,12260],{"class":179},[34,24179,24180,24182,24184],{"class":36,"line":1985},[34,24181,5452],{"class":190},[34,24183,202],{"class":175},[34,24185,6645],{"class":186},[34,24187,24188],{"class":36,"line":1990},[34,24189,53],{"emptyLinePlaceholder":52},[34,24191,24192,24194],{"class":36,"line":1999},[34,24193,5468],{"class":175},[34,24195,674],{"class":190},[34,24197,24198,24200,24202,24204,24206,24208,24210,24212],{"class":36,"line":2022},[34,24199,5475],{"class":175},[34,24201,5301],{"class":179},[34,24203,183],{"class":175},[34,24205,5306],{"class":175},[34,24207,5309],{"class":395},[34,24209,456],{"class":190},[34,24211,23729],{"class":186},[34,24213,6111],{"class":190},[34,24215,24216],{"class":36,"line":2036},[34,24217,53],{"emptyLinePlaceholder":52},[34,24219,24220,24222,24224,24226],{"class":36,"line":2603},[34,24221,5499],{"class":175},[34,24223,1840],{"class":190},[34,24225,5504],{"class":175},[34,24227,5507],{"class":190},[34,24229,24230,24232,24234,24236,24238,24241],{"class":36,"line":5537},[34,24231,5512],{"class":175},[34,24233,5177],{"class":175},[34,24235,5517],{"class":395},[34,24237,456],{"class":190},[34,24239,24240],{"class":186},"'课程加载失败'",[34,24242,6111],{"class":190},[34,24244,24245],{"class":36,"line":5543},[34,24246,5540],{"class":190},[34,24248,24249],{"class":36,"line":5548},[34,24250,53],{"emptyLinePlaceholder":52},[34,24252,24253,24256,24258,24260,24262,24264],{"class":36,"line":5564},[34,24254,24255],{"class":190}," lessons.value ",[34,24257,202],{"class":175},[34,24259,5306],{"class":175},[34,24261,5329],{"class":190},[34,24263,5332],{"class":395},[34,24265,6374],{"class":190},[34,24267,24268,24270,24272],{"class":36,"line":5576},[34,24269,5567],{"class":190},[34,24271,5570],{"class":175},[34,24273,5573],{"class":190},[34,24275,24276,24278,24280,24282,24284,24286,24288,24290,24292],{"class":36,"line":5606},[34,24277,5579],{"class":190},[34,24279,202],{"class":175},[34,24281,5584],{"class":190},[34,24283,5587],{"class":175},[34,24285,5517],{"class":395},[34,24287,5592],{"class":175},[34,24289,5595],{"class":190},[34,24291,5598],{"class":175},[34,24293,24294],{"class":186}," '未知错误'\n",[34,24296,24297,24299,24301],{"class":36,"line":5616},[34,24298,5567],{"class":190},[34,24300,5611],{"class":175},[34,24302,674],{"class":190},[34,24304,24305,24307,24309],{"class":36,"line":5628},[34,24306,5619],{"class":190},[34,24308,202],{"class":175},[34,24310,18640],{"class":179},[34,24312,24313],{"class":36,"line":5634},[34,24314,5631],{"class":190},[34,24316,24317],{"class":36,"line":6420},[34,24318,77],{"class":190},[34,24320,24321],{"class":36,"line":11294},[34,24322,53],{"emptyLinePlaceholder":52},[34,24324,24325,24327,24329,24331],{"class":36,"line":11321},[34,24326,7679],{"class":395},[34,24328,7682],{"class":190},[34,24330,1333],{"class":175},[34,24332,674],{"class":190},[34,24334,24335,24338],{"class":36,"line":11326},[34,24336,24337],{"class":395}," fetchLessons",[34,24339,6374],{"class":190},[34,24341,24342],{"class":36,"line":11333},[34,24343,6153],{"class":190},[34,24345,24346,24348,24350],{"class":36,"line":11344},[34,24347,1003],{"class":190},[34,24349,1911],{"class":935},[34,24351,939],{"class":190},[15,24353,1589],{},[24,24355,24357],{"className":923,"code":24356,"language":925,"meta":29,"style":29},"\u003Ctemplate>\n \u003Cp v-if=\"loading\">加载中...\u003C\u002Fp>\n \u003Cdiv v-else-if=\"error\" role=\"alert\">\n \u003Cp>{{ error }}\u003C\u002Fp>\n \u003Cbutton @click=\"fetchLessons\">重试\u003C\u002Fbutton>\n \u003C\u002Fdiv>\n \u003Cp v-else-if=\"lessons.length === 0\">暂无课程\u003C\u002Fp>\n \u003Cul v-else>\n \u003Cli v-for=\"lesson in lessons\" :key=\"lesson.id\">\n {{ lesson.title }}\n \u003C\u002Fli>\n \u003C\u002Ful>\n\u003C\u002Ftemplate>\n",[31,24358,24359,24367,24385,24406,24418,24438,24446,24466,24476,24498,24503,24511,24519],{"__ignoreMap":29},[34,24360,24361,24363,24365],{"class":36,"line":37},[34,24362,932],{"class":190},[34,24364,936],{"class":935},[34,24366,939],{"class":190},[34,24368,24369,24371,24373,24375,24377,24379,24381,24383],{"class":36,"line":43},[34,24370,944],{"class":190},[34,24372,15],{"class":935},[34,24374,2168],{"class":395},[34,24376,202],{"class":190},[34,24378,9568],{"class":186},[34,24380,2180],{"class":190},[34,24382,15],{"class":935},[34,24384,939],{"class":190},[34,24386,24387,24389,24391,24393,24395,24398,24400,24402,24404],{"class":36,"line":49},[34,24388,944],{"class":190},[34,24390,16016],{"class":935},[34,24392,2193],{"class":395},[34,24394,202],{"class":190},[34,24396,24397],{"class":186},"\"error\"",[34,24399,2345],{"class":395},[34,24401,202],{"class":190},[34,24403,16113],{"class":186},[34,24405,939],{"class":190},[34,24407,24408,24410,24412,24414,24416],{"class":36,"line":56},[34,24409,954],{"class":190},[34,24411,15],{"class":935},[34,24413,2205],{"class":190},[34,24415,15],{"class":935},[34,24417,939],{"class":190},[34,24419,24420,24422,24424,24426,24428,24431,24434,24436],{"class":36,"line":62},[34,24421,954],{"class":190},[34,24423,2004],{"class":935},[34,24425,2007],{"class":395},[34,24427,202],{"class":190},[34,24429,24430],{"class":186},"\"fetchLessons\"",[34,24432,24433],{"class":190},">重试\u003C\u002F",[34,24435,2004],{"class":935},[34,24437,939],{"class":190},[34,24439,24440,24442,24444],{"class":36,"line":68},[34,24441,994],{"class":190},[34,24443,16016],{"class":935},[34,24445,939],{"class":190},[34,24447,24448,24450,24452,24454,24456,24459,24462,24464],{"class":36,"line":74},[34,24449,944],{"class":190},[34,24451,15],{"class":935},[34,24453,2193],{"class":395},[34,24455,202],{"class":190},[34,24457,24458],{"class":186},"\"lessons.length === 0\"",[34,24460,24461],{"class":190},">暂无课程\u003C\u002F",[34,24463,15],{"class":935},[34,24465,939],{"class":190},[34,24467,24468,24470,24472,24474],{"class":36,"line":1971},[34,24469,944],{"class":190},[34,24471,105],{"class":935},[34,24473,2218],{"class":395},[34,24475,939],{"class":190},[34,24477,24478,24480,24482,24484,24486,24489,24491,24493,24496],{"class":36,"line":1976},[34,24479,954],{"class":190},[34,24481,108],{"class":935},[34,24483,1603],{"class":395},[34,24485,202],{"class":190},[34,24487,24488],{"class":186},"\"lesson in lessons\"",[34,24490,7066],{"class":395},[34,24492,202],{"class":190},[34,24494,24495],{"class":186},"\"lesson.id\"",[34,24497,939],{"class":190},[34,24499,24500],{"class":36,"line":1985},[34,24501,24502],{"class":190}," {{ lesson.title }}\n",[34,24504,24505,24507,24509],{"class":36,"line":1990},[34,24506,7112],{"class":190},[34,24508,108],{"class":935},[34,24510,939],{"class":190},[34,24512,24513,24515,24517],{"class":36,"line":1999},[34,24514,994],{"class":190},[34,24516,105],{"class":935},[34,24518,939],{"class":190},[34,24520,24521,24523,24525],{"class":36,"line":2022},[34,24522,1003],{"class":190},[34,24524,936],{"class":935},[34,24526,939],{"class":190},[15,24528,24529],{},"这才是完整请求界面：",[105,24531,24532,24535,24538,24541],{},[108,24533,24534],{},"加载中。",[108,24536,24537],{},"失败。",[108,24539,24540],{},"空数据。",[108,24542,24543],{},"成功。",[19,24545,9255,24547],{"id":24546},"_5-trycatchfinally",[31,24548,24549],{},"try\u002Fcatch\u002Ffinally",[24,24551,24553],{"className":166,"code":24552,"language":168,"meta":29,"style":29},"try {\n \u002F\u002F 可能失败的代码\n} catch (err) {\n \u002F\u002F 失败后执行\n} finally {\n \u002F\u002F 无论成功失败都执行\n}\n",[31,24554,24555,24561,24566,24574,24579,24587,24592],{"__ignoreMap":29},[34,24556,24557,24559],{"class":36,"line":37},[34,24558,5646],{"class":175},[34,24560,674],{"class":190},[34,24562,24563],{"class":36,"line":43},[34,24564,24565],{"class":301}," \u002F\u002F 可能失败的代码\n",[34,24567,24568,24570,24572],{"class":36,"line":49},[34,24569,2098],{"class":190},[34,24571,5570],{"class":175},[34,24573,5573],{"class":190},[34,24575,24576],{"class":36,"line":56},[34,24577,24578],{"class":301}," \u002F\u002F 失败后执行\n",[34,24580,24581,24583,24585],{"class":36,"line":62},[34,24582,2098],{"class":190},[34,24584,5611],{"class":175},[34,24586,674],{"class":190},[34,24588,24589],{"class":36,"line":68},[34,24590,24591],{"class":301}," \u002F\u002F 无论成功失败都执行\n",[34,24593,24594],{"class":36,"line":74},[34,24595,77],{"class":190},[15,24597,24598],{},"在请求里：",[105,24600,24601,24606,24611],{},[108,24602,24603,24605],{},[31,24604,5646],{},"：发请求、解析数据。",[108,24607,24608,24610],{},[31,24609,5570],{},"：保存错误信息。",[108,24612,24613,24615],{},[31,24614,5611],{},"：关闭 loading。",[15,24617,23533,24618,24620],{},[31,24619,5646],{}," 最后关闭 loading：",[24,24622,24624],{"className":166,"code":24623,"language":168,"meta":29,"style":29},"try {\n loading.value = true\n \u002F\u002F 如果这里失败\n loading.value = false\n}\n",[31,24625,24626,24632,24640,24645,24653],{"__ignoreMap":29},[34,24627,24628,24630],{"class":36,"line":37},[34,24629,5646],{"class":175},[34,24631,674],{"class":190},[34,24633,24634,24636,24638],{"class":36,"line":43},[34,24635,5441],{"class":190},[34,24637,202],{"class":175},[34,24639,12260],{"class":179},[34,24641,24642],{"class":36,"line":49},[34,24643,24644],{"class":301}," \u002F\u002F 如果这里失败\n",[34,24646,24647,24649,24651],{"class":36,"line":56},[34,24648,5441],{"class":190},[34,24650,202],{"class":175},[34,24652,18640],{"class":179},[34,24654,24655],{"class":36,"line":62},[34,24656,77],{"class":190},[15,24658,24659,24660,24663,24664,347],{},"一旦中途抛错，",[31,24661,24662],{},"loading.value = false"," 不会执行，页面可能一直显示加载中。所以用 ",[31,24665,5611],{},[19,24667,24669],{"id":24668},"_6-onmounted-请求","6. onMounted 请求",[24,24671,24673],{"className":166,"code":24672,"language":168,"meta":29,"style":29},"onMounted(() => {\n fetchLessons()\n})\n",[31,24674,24675,24685,24691],{"__ignoreMap":29},[34,24676,24677,24679,24681,24683],{"class":36,"line":37},[34,24678,7679],{"class":395},[34,24680,7682],{"class":190},[34,24682,1333],{"class":175},[34,24684,674],{"class":190},[34,24686,24687,24689],{"class":36,"line":43},[34,24688,24337],{"class":395},[34,24690,6374],{"class":190},[34,24692,24693],{"class":36,"line":49},[34,24694,6153],{"class":190},[15,24696,24697,24699],{},[31,24698,7679],{}," 表示组件已经挂载到页面后执行。进入页面加载数据时常用。",[15,24701,24702],{},"但如果请求依赖路由参数，参数变化时也要重新请求：",[24,24704,24706],{"className":166,"code":24705,"language":168,"meta":29,"style":29},"watch(\n () => route.params.id,\n () => {\n fetchLesson()\n },\n { immediate: true }\n)\n",[31,24707,24708,24714,24724,24732,24739,24743,24752],{"__ignoreMap":29},[34,24709,24710,24712],{"class":36,"line":37},[34,24711,152],{"class":395},[34,24713,18938],{"class":190},[34,24715,24716,24719,24721],{"class":36,"line":43},[34,24717,24718],{"class":190}," () ",[34,24720,1333],{"class":175},[34,24722,24723],{"class":190}," route.params.id,\n",[34,24725,24726,24728,24730],{"class":36,"line":49},[34,24727,24718],{"class":190},[34,24729,1333],{"class":175},[34,24731,674],{"class":190},[34,24733,24734,24737],{"class":36,"line":56},[34,24735,24736],{"class":395}," fetchLesson",[34,24738,6374],{"class":190},[34,24740,24741],{"class":36,"line":62},[34,24742,17404],{"class":190},[34,24744,24745,24748,24750],{"class":36,"line":68},[34,24746,24747],{"class":190}," { immediate: ",[34,24749,758],{"class":179},[34,24751,6564],{"class":190},[34,24753,24754],{"class":36,"line":74},[34,24755,6111],{"class":190},[15,24757,24758,24761],{},[31,24759,24760],{},"immediate: true"," 表示一开始也执行一次。",[19,24763,24765],{"id":24764},"_7-请求逻辑放在哪里","7. 请求逻辑放在哪里",[15,24767,24768],{},"三种常见位置：",[253,24770,24771],{"id":24771},"放组件里",[15,24773,24774],{},"适合：",[105,24776,24777,24780,24783],{},[108,24778,24779],{},"只有当前页面用。",[108,24781,24782],{},"逻辑简单。",[108,24784,24785],{},"不需要多个组件共享。",[253,24787,24789],{"id":24788},"放-service-里","放 service 里",[24,24791,24793],{"className":166,"code":24792,"language":168,"meta":29,"style":29},"\u002F\u002F services\u002FlessonService.js\nexport async function getLessons() {\n const response = await fetch('\u002Fapi\u002Flessons')\n\n if (!response.ok) {\n throw new Error('课程加载失败')\n }\n\n return response.json()\n}\n",[31,24794,24795,24800,24813,24831,24835,24845,24859,24863,24867,24877],{"__ignoreMap":29},[34,24796,24797],{"class":36,"line":37},[34,24798,24799],{"class":301},"\u002F\u002F services\u002FlessonService.js\n",[34,24801,24802,24804,24806,24808,24811],{"class":36,"line":43},[34,24803,2825],{"class":175},[34,24805,23408],{"class":175},[34,24807,2850],{"class":175},[34,24809,24810],{"class":395}," getLessons",[34,24812,399],{"class":190},[34,24814,24815,24817,24819,24821,24823,24825,24827,24829],{"class":36,"line":49},[34,24816,5298],{"class":175},[34,24818,5301],{"class":179},[34,24820,183],{"class":175},[34,24822,5306],{"class":175},[34,24824,5309],{"class":395},[34,24826,456],{"class":190},[34,24828,23729],{"class":186},[34,24830,6111],{"class":190},[34,24832,24833],{"class":36,"line":56},[34,24834,53],{"emptyLinePlaceholder":52},[34,24836,24837,24839,24841,24843],{"class":36,"line":62},[34,24838,7000],{"class":175},[34,24840,1840],{"class":190},[34,24842,5504],{"class":175},[34,24844,5507],{"class":190},[34,24846,24847,24849,24851,24853,24855,24857],{"class":36,"line":68},[34,24848,24032],{"class":175},[34,24850,5177],{"class":175},[34,24852,5517],{"class":395},[34,24854,456],{"class":190},[34,24856,24240],{"class":186},[34,24858,6111],{"class":190},[34,24860,24861],{"class":36,"line":74},[34,24862,5631],{"class":190},[34,24864,24865],{"class":36,"line":1971},[34,24866,53],{"emptyLinePlaceholder":52},[34,24868,24869,24871,24873,24875],{"class":36,"line":1976},[34,24870,1510],{"class":175},[34,24872,5329],{"class":190},[34,24874,5332],{"class":395},[34,24876,6374],{"class":190},[34,24878,24879],{"class":36,"line":1985},[34,24880,77],{"class":190},[15,24882,24883],{},"组件：",[24,24885,24887],{"className":166,"code":24886,"language":168,"meta":29,"style":29},"lessons.value = await getLessons()\n",[31,24888,24889],{"__ignoreMap":29},[34,24890,24891,24894,24896,24898,24900],{"class":36,"line":37},[34,24892,24893],{"class":190},"lessons.value ",[34,24895,202],{"class":175},[34,24897,5306],{"class":175},[34,24899,24810],{"class":395},[34,24901,6374],{"class":190},[15,24903,24774],{},[105,24905,24906,24909],{},[108,24907,24908],{},"多处复用同一个接口。",[108,24910,24911],{},"想把 URL、请求头、错误处理集中。",[253,24913,24915],{"id":24914},"放-pinia-action-里","放 Pinia action 里",[15,24917,24774],{},[105,24919,24920,24923],{},[108,24921,24922],{},"请求结果是全局共享业务状态。",[108,24924,24925],{},"多个组件都要使用同一份数据。",[15,24927,24928],{},"例如登录用户、任务列表。",[19,24930,24932],{"id":24931},"_8-模拟-api","8. 模拟 API",[15,24934,24935],{},"学习阶段可以不接真实后端，用 mock API：",[24,24937,24939],{"className":166,"code":24938,"language":168,"meta":29,"style":29},"export function wait(ms) {\n return new Promise((resolve) => {\n setTimeout(resolve, ms)\n })\n}\n\nexport async function getMockLessons(shouldFail = false) {\n await wait(500)\n\n if (shouldFail) {\n throw new Error('模拟请求失败')\n }\n\n return [\n { id: 1, title: '响应式基础' },\n { id: 2, title: '组件通信' }\n ]\n}\n",[31,24940,24941,24955,24973,24980,24984,24988,24992,25014,25027,25031,25038,25053,25057,25061,25067,25081,25094,25098],{"__ignoreMap":29},[34,24942,24943,24945,24947,24949,24951,24953],{"class":36,"line":37},[34,24944,2825],{"class":175},[34,24946,2850],{"class":175},[34,24948,5163],{"class":395},[34,24950,456],{"class":190},[34,24952,5168],{"class":1326},[34,24954,1700],{"class":190},[34,24956,24957,24959,24961,24963,24965,24967,24969,24971],{"class":36,"line":43},[34,24958,1510],{"class":175},[34,24960,5177],{"class":175},[34,24962,5180],{"class":179},[34,24964,1323],{"class":190},[34,24966,5185],{"class":1326},[34,24968,1330],{"class":190},[34,24970,1333],{"class":175},[34,24972,674],{"class":190},[34,24974,24975,24977],{"class":36,"line":49},[34,24976,5196],{"class":395},[34,24978,24979],{"class":190},"(resolve, ms)\n",[34,24981,24982],{"class":36,"line":56},[34,24983,6379],{"class":190},[34,24985,24986],{"class":36,"line":62},[34,24987,77],{"class":190},[34,24989,24990],{"class":36,"line":68},[34,24991,53],{"emptyLinePlaceholder":52},[34,24993,24994,24996,24998,25000,25003,25005,25008,25010,25012],{"class":36,"line":74},[34,24995,2825],{"class":175},[34,24997,23408],{"class":175},[34,24999,2850],{"class":175},[34,25001,25002],{"class":395}," getMockLessons",[34,25004,456],{"class":190},[34,25006,25007],{"class":1326},"shouldFail",[34,25009,183],{"class":175},[34,25011,576],{"class":179},[34,25013,1700],{"class":190},[34,25015,25016,25018,25020,25022,25025],{"class":36,"line":1971},[34,25017,5242],{"class":175},[34,25019,5163],{"class":395},[34,25021,456],{"class":190},[34,25023,25024],{"class":179},"500",[34,25026,6111],{"class":190},[34,25028,25029],{"class":36,"line":1976},[34,25030,53],{"emptyLinePlaceholder":52},[34,25032,25033,25035],{"class":36,"line":1985},[34,25034,7000],{"class":175},[34,25036,25037],{"class":190}," (shouldFail) {\n",[34,25039,25040,25042,25044,25046,25048,25051],{"class":36,"line":1990},[34,25041,24032],{"class":175},[34,25043,5177],{"class":175},[34,25045,5517],{"class":395},[34,25047,456],{"class":190},[34,25049,25050],{"class":186},"'模拟请求失败'",[34,25052,6111],{"class":190},[34,25054,25055],{"class":36,"line":1999},[34,25056,5631],{"class":190},[34,25058,25059],{"class":36,"line":2022},[34,25060,53],{"emptyLinePlaceholder":52},[34,25062,25063,25065],{"class":36,"line":2036},[34,25064,1510],{"class":175},[34,25066,739],{"class":190},[34,25068,25069,25072,25074,25076,25079],{"class":36,"line":2603},[34,25070,25071],{"class":190}," { id: ",[34,25073,640],{"class":179},[34,25075,749],{"class":190},[34,25077,25078],{"class":186},"'响应式基础'",[34,25080,761],{"class":190},[34,25082,25083,25085,25087,25089,25092],{"class":36,"line":5537},[34,25084,25071],{"class":190},[34,25086,768],{"class":179},[34,25088,749],{"class":190},[34,25090,25091],{"class":186},"'组件通信'",[34,25093,6564],{"class":190},[34,25095,25096],{"class":36,"line":5543},[34,25097,20206],{"class":190},[34,25099,25100],{"class":36,"line":5548},[34,25101,77],{"class":190},[15,25103,25104],{},"这样你可以练习 loading 和 error，而不依赖后端。",[19,25106,25108],{"id":25107},"_9-取消请求-abortcontroller","9. 取消请求 AbortController",[15,25110,25111],{},"搜索框场景：",[24,25113,25116],{"className":25114,"code":25115,"language":2638,"meta":29},[2636],"用户输入 v -> 发请求 1\n用户输入 vu -> 发请求 2\n用户输入 vue -> 发请求 3\n",[31,25117,25115],{"__ignoreMap":29},[15,25119,25120],{},"如果请求 1 最后才返回，可能覆盖请求 3 的结果。可以取消旧请求：",[24,25122,25124],{"className":166,"code":25123,"language":168,"meta":29,"style":29},"let controller\n\nasync function search(keyword) {\n if (controller) {\n controller.abort()\n }\n\n controller = new AbortController()\n\n const response = await fetch(`\u002Fapi\u002Fsearch?q=${keyword}`, {\n signal: controller.signal\n })\n\n if (!response.ok) {\n throw new Error('搜索失败')\n }\n\n return response.json()\n}\n",[31,25125,25126,25133,25137,25152,25159,25169,25173,25177,25191,25195,25219,25224,25228,25232,25242,25257,25261,25265,25275],{"__ignoreMap":29},[34,25127,25128,25130],{"class":36,"line":37},[34,25129,196],{"class":175},[34,25131,25132],{"class":190}," controller\n",[34,25134,25135],{"class":36,"line":43},[34,25136,53],{"emptyLinePlaceholder":52},[34,25138,25139,25141,25143,25146,25148,25150],{"class":36,"line":49},[34,25140,5217],{"class":175},[34,25142,2850],{"class":175},[34,25144,25145],{"class":395}," search",[34,25147,456],{"class":190},[34,25149,9414],{"class":1326},[34,25151,1700],{"class":190},[34,25153,25154,25156],{"class":36,"line":56},[34,25155,7000],{"class":175},[34,25157,25158],{"class":190}," (controller) {\n",[34,25160,25161,25164,25167],{"class":36,"line":62},[34,25162,25163],{"class":190}," controller.",[34,25165,25166],{"class":395},"abort",[34,25168,6374],{"class":190},[34,25170,25171],{"class":36,"line":68},[34,25172,5631],{"class":190},[34,25174,25175],{"class":36,"line":74},[34,25176,53],{"emptyLinePlaceholder":52},[34,25178,25179,25182,25184,25186,25189],{"class":36,"line":1971},[34,25180,25181],{"class":190}," controller ",[34,25183,202],{"class":175},[34,25185,5177],{"class":175},[34,25187,25188],{"class":395}," AbortController",[34,25190,6374],{"class":190},[34,25192,25193],{"class":36,"line":1976},[34,25194,53],{"emptyLinePlaceholder":52},[34,25196,25197,25199,25201,25203,25205,25207,25209,25212,25214,25216],{"class":36,"line":1985},[34,25198,5298],{"class":175},[34,25200,5301],{"class":179},[34,25202,183],{"class":175},[34,25204,5306],{"class":175},[34,25206,5309],{"class":395},[34,25208,456],{"class":190},[34,25210,25211],{"class":186},"`\u002Fapi\u002Fsearch?q=${",[34,25213,9414],{"class":190},[34,25215,2560],{"class":186},[34,25217,25218],{"class":190},", {\n",[34,25220,25221],{"class":36,"line":1990},[34,25222,25223],{"class":190}," signal: controller.signal\n",[34,25225,25226],{"class":36,"line":1999},[34,25227,6379],{"class":190},[34,25229,25230],{"class":36,"line":2022},[34,25231,53],{"emptyLinePlaceholder":52},[34,25233,25234,25236,25238,25240],{"class":36,"line":2036},[34,25235,7000],{"class":175},[34,25237,1840],{"class":190},[34,25239,5504],{"class":175},[34,25241,5507],{"class":190},[34,25243,25244,25246,25248,25250,25252,25255],{"class":36,"line":2603},[34,25245,24032],{"class":175},[34,25247,5177],{"class":175},[34,25249,5517],{"class":395},[34,25251,456],{"class":190},[34,25253,25254],{"class":186},"'搜索失败'",[34,25256,6111],{"class":190},[34,25258,25259],{"class":36,"line":5537},[34,25260,5631],{"class":190},[34,25262,25263],{"class":36,"line":5543},[34,25264,53],{"emptyLinePlaceholder":52},[34,25266,25267,25269,25271,25273],{"class":36,"line":5548},[34,25268,1510],{"class":175},[34,25270,5329],{"class":190},[34,25272,5332],{"class":395},[34,25274,6374],{"class":190},[34,25276,25277],{"class":36,"line":5564},[34,25278,77],{"class":190},[15,25280,25281],{},"如果组件卸载，也可以取消正在进行的请求，避免无意义更新。",[19,25283,25285],{"id":25284},"_10-防抖-debounce","10. 防抖 debounce",[15,25287,25288],{},"搜索时不应该每输入一个字都请求，可以等待用户停顿：",[24,25290,25292],{"className":166,"code":25291,"language":168,"meta":29,"style":29},"let timerId\n\nfunction debounceSearch(keyword) {\n clearTimeout(timerId)\n\n timerId = setTimeout(() => {\n search(keyword)\n }, 300)\n}\n",[31,25293,25294,25300,25304,25317,25324,25328,25343,25351,25360],{"__ignoreMap":29},[34,25295,25296,25298],{"class":36,"line":37},[34,25297,196],{"class":175},[34,25299,10793],{"class":190},[34,25301,25302],{"class":36,"line":43},[34,25303,53],{"emptyLinePlaceholder":52},[34,25305,25306,25308,25311,25313,25315],{"class":36,"line":49},[34,25307,392],{"class":175},[34,25309,25310],{"class":395}," debounceSearch",[34,25312,456],{"class":190},[34,25314,9414],{"class":1326},[34,25316,1700],{"class":190},[34,25318,25319,25322],{"class":36,"line":56},[34,25320,25321],{"class":395}," clearTimeout",[34,25323,10872],{"class":190},[34,25325,25326],{"class":36,"line":62},[34,25327,53],{"emptyLinePlaceholder":52},[34,25329,25330,25332,25334,25337,25339,25341],{"class":36,"line":68},[34,25331,10812],{"class":190},[34,25333,202],{"class":175},[34,25335,25336],{"class":395}," setTimeout",[34,25338,7682],{"class":190},[34,25340,1333],{"class":175},[34,25342,674],{"class":190},[34,25344,25345,25348],{"class":36,"line":74},[34,25346,25347],{"class":395}," search",[34,25349,25350],{"class":190},"(keyword)\n",[34,25352,25353,25355,25358],{"class":36,"line":1971},[34,25354,10842],{"class":190},[34,25356,25357],{"class":179},"300",[34,25359,6111],{"class":190},[34,25361,25362],{"class":36,"line":1976},[34,25363,77],{"class":190},[15,25365,25366],{},"意思：",[105,25368,25369,25372,25375],{},[108,25370,25371],{},"每次输入先清除上一个定时器。",[108,25373,25374],{},"重新等待 300ms。",[108,25376,25377],{},"300ms 内没有新输入，才执行搜索。",[15,25379,25380],{},"真实项目可以使用成熟工具库，但理解原理很重要。",[19,25382,25384],{"id":25383},"_11-错误信息应该给用户看","11. 错误信息应该给用户看",[15,25386,25387],{},"不要只写：",[24,25389,25391],{"className":166,"code":25390,"language":168,"meta":29,"style":29},"console.error(err)\n",[31,25392,25393],{"__ignoreMap":29},[34,25394,25395,25397,25399],{"class":36,"line":37},[34,25396,882],{"class":190},[34,25398,2200],{"class":395},[34,25400,25401],{"class":190},"(err)\n",[15,25403,25404],{},"用户看不到控制台。页面应该显示：",[24,25406,25408],{"className":923,"code":25407,"language":925,"meta":29,"style":29},"\u003Cdiv v-if=\"error\" role=\"alert\">\n \u003Cp>{{ error }}\u003C\u002Fp>\n \u003Cbutton @click=\"fetchLessons\">重试\u003C\u002Fbutton>\n\u003C\u002Fdiv>\n",[31,25409,25410,25434,25439,25444],{"__ignoreMap":29},[34,25411,25412,25414,25416,25418,25420,25422,25424,25426,25428,25430,25432],{"class":36,"line":37},[34,25413,932],{"class":190},[34,25415,16016],{"class":935},[34,25417,2168],{"class":175},[34,25419,202],{"class":190},[34,25421,1608],{"class":186},[34,25423,2200],{"class":190},[34,25425,1608],{"class":186},[34,25427,2345],{"class":395},[34,25429,202],{"class":190},[34,25431,16113],{"class":186},[34,25433,939],{"class":190},[34,25435,25436],{"class":36,"line":43},[34,25437,25438],{"class":190}," \u003Cp>{{ error }}\u003C\u002Fp>\n",[34,25440,25441],{"class":36,"line":49},[34,25442,25443],{"class":190}," \u003Cbutton @click=\"fetchLessons\">重试\u003C\u002Fbutton>\n",[34,25445,25446,25448,25450],{"class":36,"line":56},[34,25447,1003],{"class":190},[34,25449,16016],{"class":935},[34,25451,939],{"class":190},[15,25453,25454],{},"错误信息应该：",[105,25456,25457,25460,25463],{},[108,25458,25459],{},"简短。",[108,25461,25462],{},"告诉用户发生了什么。",[108,25464,25465],{},"尽量提供下一步，例如重试。",[19,25467,21417],{"id":21416},[253,25469,25471,25472],{"id":25470},"错误一忘记-await-responsejson","错误一：忘记 ",[31,25473,25474],{},"await response.json()",[24,25476,25478],{"className":166,"code":25477,"language":168,"meta":29,"style":29},"const data = response.json()\n",[31,25479,25480],{"__ignoreMap":29},[34,25481,25482,25484,25486,25488,25490,25492],{"class":36,"line":37},[34,25483,176],{"class":175},[34,25485,5410],{"class":179},[34,25487,183],{"class":175},[34,25489,5329],{"class":190},[34,25491,5332],{"class":395},[34,25493,6374],{"class":190},[15,25495,25496,25497,25499],{},"这里 ",[31,25498,17493],{}," 是 Promise，不是数据。应该：",[24,25501,25503],{"className":166,"code":25502,"language":168,"meta":29,"style":29},"const data = await response.json()\n",[31,25504,25505],{"__ignoreMap":29},[34,25506,25507,25509,25511,25513,25515,25517,25519],{"class":36,"line":37},[34,25508,176],{"class":175},[34,25510,5410],{"class":179},[34,25512,183],{"class":175},[34,25514,5306],{"class":175},[34,25516,5329],{"class":190},[34,25518,5332],{"class":395},[34,25520,6374],{"class":190},[253,25522,25524],{"id":25523},"错误二没有处理失败状态","错误二：没有处理失败状态",[15,25526,25527],{},"只写成功时，一旦接口失败，页面空白或一直 loading。",[253,25529,25531],{"id":25530},"错误三请求返回空数组时显示错误","错误三：请求返回空数组时显示错误",[15,25533,25534],{},"空数据不是错误。应该用 empty 状态：",[24,25536,25538],{"className":923,"code":25537,"language":925,"meta":29,"style":29},"\u003Cp v-else-if=\"items.length === 0\">暂无数据\u003C\u002Fp>\n",[31,25539,25540],{"__ignoreMap":29},[34,25541,25542,25544,25546,25548,25550,25552,25555,25557,25559,25561,25563,25566,25568],{"class":36,"line":37},[34,25543,932],{"class":190},[34,25545,15],{"class":935},[34,25547,2193],{"class":175},[34,25549,202],{"class":190},[34,25551,1608],{"class":186},[34,25553,25554],{"class":190},"items.",[34,25556,1289],{"class":179},[34,25558,7946],{"class":175},[34,25560,205],{"class":179},[34,25562,1608],{"class":186},[34,25564,25565],{"class":190},">暂无数据\u003C\u002F",[34,25567,15],{"class":935},[34,25569,939],{"class":190},[253,25571,25573],{"id":25572},"错误四组件卸载后还更新状态","错误四：组件卸载后还更新状态",[15,25575,25576],{},"复杂请求场景可能需要取消请求或检查组件是否仍然有效。初学阶段先知道有这个风险即可。",[19,25578,5693],{"id":5693},[2230,25580,25581,25586,25591,25594,25597,25600,25603],{},[108,25582,5803,25583,347],{},[31,25584,25585],{},"wait(ms)",[108,25587,5803,25588,347],{},[31,25589,25590],{},"getMockLessons({ shouldFail, empty })",[108,25592,25593],{},"页面进入时请求课程。",[108,25595,25596],{},"显示 loading、error、empty、success。",[108,25598,25599],{},"错误状态提供重试按钮。",[108,25601,25602],{},"搜索框加 debounce。",[108,25604,25605],{},"回答：请求逻辑什么时候放组件，什么时候放 service，什么时候放 Pinia？",[79,25607,25608],{},"html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":25610},[25611,25612,25613,25615,25616,25618,25619,25624,25625,25626,25627,25628,25635],{"id":23711,"depth":43,"text":23712},{"id":23836,"depth":43,"text":23837},{"id":23928,"depth":43,"text":25614},"3. 为什么要检查 response.ok",{"id":24080,"depth":43,"text":24081},{"id":24546,"depth":43,"text":25617},"5. try\u002Fcatch\u002Ffinally",{"id":24668,"depth":43,"text":24669},{"id":24764,"depth":43,"text":24765,"children":25620},[25621,25622,25623],{"id":24771,"depth":49,"text":24771},{"id":24788,"depth":49,"text":24789},{"id":24914,"depth":49,"text":24915},{"id":24931,"depth":43,"text":24932},{"id":25107,"depth":43,"text":25108},{"id":25284,"depth":43,"text":25285},{"id":25383,"depth":43,"text":25384},{"id":21416,"depth":43,"text":21417,"children":25629},[25630,25632,25633,25634],{"id":25470,"depth":49,"text":25631},"错误一：忘记 await response.json()",{"id":25523,"depth":49,"text":25524},{"id":25530,"depth":49,"text":25531},{"id":25572,"depth":49,"text":25573},{"id":5693,"depth":43,"text":5693},{"order":1976},"\u002Fsource\u002Fcode\u002Fvue\u002F09-http-async",{"title":23690,"description":23698},"source\u002Fcode\u002Fvue\u002F09-http-async","a2iTRYc-YxUPiT8GEDSWor4Zm1AoTelxDHRxKgXjgQg",{"id":25642,"title":25643,"body":25644,"description":25651,"extension":85,"meta":27290,"navigation":52,"path":27291,"seo":27292,"stem":27293,"__hash__":27294},"notes\u002Fsource\u002Fcode\u002Fvue\u002F10-testing-debugging.md","10 测试、调试和质量保障",{"type":8,"value":25645,"toc":27270},[25646,25649,25652,25655,25659,25662,25668,25671,25691,25696,25699,25705,25712,25721,25728,25731,25742,25769,25773,25776,25826,25829,25833,25836,25839,25859,25862,25873,25880,25884,25887,25892,25963,25966,26088,26091,26095,26098,26153,26155,26205,26208,26291,26295,26298,26544,26549,26553,26556,26724,26733,26737,26740,26912,26915,26968,26971,26982,26986,26989,26992,27006,27009,27020,27024,27027,27048,27051,27071,27074,27078,27081,27084,27103,27106,27108,27127,27130,27132,27147,27150,27152,27180,27185,27189,27192,27207,27210,27219,27222,27236,27238,27267],[11,25647,25643],{"id":25648},"_10-测试调试和质量保障",[15,25650,25651],{},"测试不是为了追求形式，而是为了让你修改代码时知道有没有把旧功能弄坏。调试也不是靠猜，而是有步骤地定位问题。",[15,25653,25654],{},"本章目标：你能读懂常见报错，能用 Vue DevTools 和控制台观察状态，能用 Vitest 测试组合式函数和组件事件。",[19,25656,25658],{"id":25657},"_1-先学会读报错","1. 先学会读报错",[15,25660,25661],{},"报错一般包含几类信息：",[24,25663,25666],{"className":25664,"code":25665,"language":2638,"meta":29},[2636],"ReferenceError: count is not defined\n at Counter.vue:12:5\n",[31,25667,25665],{"__ignoreMap":29},[15,25669,25670],{},"你要看：",[105,25672,25673,25679,25685],{},[108,25674,25675,25676,347],{},"错误类型：",[31,25677,25678],{},"ReferenceError",[108,25680,25681,25682,347],{},"错误内容：",[31,25683,25684],{},"count is not defined",[108,25686,25687,25688,347],{},"文件和行号：",[31,25689,25690],{},"Counter.vue:12:5",[15,25692,25693,25694,347],{},"这句话意思是：第 12 行用了一个不存在的变量 ",[31,25695,2770],{},[15,25697,25698],{},"常见错误：",[24,25700,25703],{"className":25701,"code":25702,"language":2638,"meta":29},[2636],"Cannot read properties of undefined\n",[31,25704,25702],{"__ignoreMap":29},[15,25706,25707,25708,25711],{},"意思是你在 ",[31,25709,25710],{},"undefined"," 上读属性，例如：",[24,25713,25715],{"className":166,"code":25714,"language":168,"meta":29,"style":29},"user.name\n",[31,25716,25717],{"__ignoreMap":29},[34,25718,25719],{"class":36,"line":37},[34,25720,25714],{"class":190},[15,25722,10381,25723,11008,25726,347],{},[31,25724,25725],{},"user",[31,25727,25710],{},[15,25729,25730],{},"修复思路：",[105,25732,25733,25736,25739],{},[108,25734,25735],{},"检查数据是否还没加载。",[108,25737,25738],{},"给初始值。",[108,25740,25741],{},"用条件渲染。",[24,25743,25745],{"className":923,"code":25744,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"user\">{{ user.name }}\u003C\u002Fp>\n",[31,25746,25747],{"__ignoreMap":29},[34,25748,25749,25751,25753,25755,25757,25759,25761,25763,25765,25767],{"class":36,"line":37},[34,25750,932],{"class":190},[34,25752,15],{"class":935},[34,25754,2168],{"class":175},[34,25756,202],{"class":190},[34,25758,1608],{"class":186},[34,25760,25725],{"class":190},[34,25762,1608],{"class":186},[34,25764,8875],{"class":190},[34,25766,15],{"class":935},[34,25768,939],{"class":190},[19,25770,25772],{"id":25771},"_2-调试的基本流程","2. 调试的基本流程",[15,25774,25775],{},"遇到问题不要马上乱改。按步骤：",[2230,25777,25778,25784,25790,25796,25802,25808,25820],{},[108,25779,25780,25783],{},[2627,25781,25782],{},"复现问题","：能稳定触发吗？",[108,25785,25786,25789],{},[2627,25787,25788],{},"缩小范围","：是页面没进入、数据没来、事件没触发，还是渲染条件错？",[108,25791,25792,25795],{},[2627,25793,25794],{},"观察状态","：用 Vue DevTools 看组件 state、props、Pinia。",[108,25797,25798,25801],{},[2627,25799,25800],{},"检查事件","：按钮点击函数有没有执行？",[108,25803,25804,25807],{},[2627,25805,25806],{},"检查数据变化","：状态有没有真的改变？",[108,25809,25810,2664,25813,146,25815,146,25817,25819],{},[2627,25811,25812],{},"检查模板条件",[31,25814,8162],{},[31,25816,1534],{},[31,25818,1624],{}," 有没有写错？",[108,25821,25822,25825],{},[2627,25823,25824],{},"修复后验证","：重新跑同样操作。",[15,25827,25828],{},"调试不是“试试这个、试试那个”，而是不断缩小问题范围。",[19,25830,25832],{"id":25831},"_3-vue-devtools-能看什么","3. Vue DevTools 能看什么",[15,25834,25835],{},"推荐安装 Vue 官方 DevTools 浏览器扩展。",[15,25837,25838],{},"它能看：",[105,25840,25841,25844,25847,25850,25853,25856],{},[108,25842,25843],{},"组件树。",[108,25845,25846],{},"当前组件 props。",[108,25848,25849],{},"当前组件 state。",[108,25851,25852],{},"computed 值。",[108,25854,25855],{},"Pinia store。",[108,25857,25858],{},"事件。",[15,25860,25861],{},"例如子组件没有显示数据，先看：",[105,25863,25864,25867,25870],{},[108,25865,25866],{},"父组件是否传了 prop？",[108,25868,25869],{},"子组件 props 面板里有没有收到？",[108,25871,25872],{},"prop 名是否拼错？",[15,25874,25875,25876,25879],{},"这比到处 ",[31,25877,25878],{},"console.log"," 更清楚。",[19,25881,25883],{"id":25882},"_4-什么是测试","4. 什么是测试",[15,25885,25886],{},"测试就是用代码验证另一个代码的行为。",[15,25888,25889,25890,2664],{},"例如你写了 ",[31,25891,19560],{},[24,25893,25895],{"className":166,"code":25894,"language":168,"meta":29,"style":29},"export function useCounter(initialValue = 0) {\n const count = ref(initialValue)\n\n function increase() {\n count.value++\n }\n\n return { count, increase }\n}\n",[31,25896,25897,25915,25927,25931,25939,25945,25949,25953,25959],{"__ignoreMap":29},[34,25898,25899,25901,25903,25905,25907,25909,25911,25913],{"class":36,"line":37},[34,25900,2825],{"class":175},[34,25902,2850],{"class":175},[34,25904,17936],{"class":395},[34,25906,456],{"class":190},[34,25908,17941],{"class":1326},[34,25910,183],{"class":175},[34,25912,205],{"class":179},[34,25914,1700],{"class":190},[34,25916,25917,25919,25921,25923,25925],{"class":36,"line":43},[34,25918,5298],{"class":175},[34,25920,448],{"class":179},[34,25922,183],{"class":175},[34,25924,453],{"class":395},[34,25926,17960],{"class":190},[34,25928,25929],{"class":36,"line":49},[34,25930,53],{"emptyLinePlaceholder":52},[34,25932,25933,25935,25937],{"class":36,"line":56},[34,25934,6318],{"class":175},[34,25936,396],{"class":395},[34,25938,399],{"class":190},[34,25940,25941,25943],{"class":36,"line":62},[34,25942,17999],{"class":190},[34,25944,6366],{"class":175},[34,25946,25947],{"class":36,"line":68},[34,25948,5631],{"class":190},[34,25950,25951],{"class":36,"line":74},[34,25952,53],{"emptyLinePlaceholder":52},[34,25954,25955,25957],{"class":36,"line":1971},[34,25956,1510],{"class":175},[34,25958,18377],{"class":190},[34,25960,25961],{"class":36,"line":1976},[34,25962,77],{"class":190},[15,25964,25965],{},"测试：",[24,25967,25969],{"className":166,"code":25968,"language":168,"meta":29,"style":29},"import { describe, expect, it } from 'vitest'\nimport { useCounter } from '..\u002Fcomposables\u002FuseCounter'\n\ndescribe('useCounter', () => {\n it('increases count', () => {\n const counter = useCounter(1)\n\n counter.increase()\n\n expect(counter.count.value).toBe(2)\n })\n})\n",[31,25970,25971,25983,25993,25997,26013,26029,26046,26050,26059,26063,26080,26084],{"__ignoreMap":29},[34,25972,25973,25975,25978,25980],{"class":36,"line":37},[34,25974,426],{"class":175},[34,25976,25977],{"class":190}," { describe, expect, it } ",[34,25979,432],{"class":175},[34,25981,25982],{"class":186}," 'vitest'\n",[34,25984,25985,25987,25989,25991],{"class":36,"line":43},[34,25986,426],{"class":175},[34,25988,4740],{"class":190},[34,25990,432],{"class":175},[34,25992,18099],{"class":186},[34,25994,25995],{"class":36,"line":49},[34,25996,53],{"emptyLinePlaceholder":52},[34,25998,25999,26002,26004,26007,26009,26011],{"class":36,"line":56},[34,26000,26001],{"class":395},"describe",[34,26003,456],{"class":190},[34,26005,26006],{"class":186},"'useCounter'",[34,26008,6131],{"class":190},[34,26010,1333],{"class":175},[34,26012,674],{"class":190},[34,26014,26015,26018,26020,26023,26025,26027],{"class":36,"line":62},[34,26016,26017],{"class":395}," it",[34,26019,456],{"class":190},[34,26021,26022],{"class":186},"'increases count'",[34,26024,6131],{"class":190},[34,26026,1333],{"class":175},[34,26028,674],{"class":190},[34,26030,26031,26033,26036,26038,26040,26042,26044],{"class":36,"line":68},[34,26032,5475],{"class":175},[34,26034,26035],{"class":179}," counter",[34,26037,183],{"class":175},[34,26039,17936],{"class":395},[34,26041,456],{"class":190},[34,26043,640],{"class":179},[34,26045,6111],{"class":190},[34,26047,26048],{"class":36,"line":74},[34,26049,53],{"emptyLinePlaceholder":52},[34,26051,26052,26055,26057],{"class":36,"line":1971},[34,26053,26054],{"class":190}," counter.",[34,26056,2048],{"class":395},[34,26058,6374],{"class":190},[34,26060,26061],{"class":36,"line":1976},[34,26062,53],{"emptyLinePlaceholder":52},[34,26064,26065,26068,26071,26074,26076,26078],{"class":36,"line":1985},[34,26066,26067],{"class":395}," expect",[34,26069,26070],{"class":190},"(counter.count.value).",[34,26072,26073],{"class":395},"toBe",[34,26075,456],{"class":190},[34,26077,768],{"class":179},[34,26079,6111],{"class":190},[34,26081,26082],{"class":36,"line":1990},[34,26083,6379],{"class":190},[34,26085,26086],{"class":36,"line":1999},[34,26087,6153],{"class":190},[15,26089,26090],{},"这段测试表达的需求是：从 1 开始，调用 increase 后应该变成 2。",[19,26092,26094],{"id":26093},"_5-vitest-基础","5. Vitest 基础",[15,26096,26097],{},"常见结构：",[24,26099,26101],{"className":166,"code":26100,"language":168,"meta":29,"style":29},"describe('一组测试', () => {\n it('描述一个行为', () => {\n expect(实际结果).toBe(期望结果)\n })\n})\n",[31,26102,26103,26118,26133,26145,26149],{"__ignoreMap":29},[34,26104,26105,26107,26109,26112,26114,26116],{"class":36,"line":37},[34,26106,26001],{"class":395},[34,26108,456],{"class":190},[34,26110,26111],{"class":186},"'一组测试'",[34,26113,6131],{"class":190},[34,26115,1333],{"class":175},[34,26117,674],{"class":190},[34,26119,26120,26122,26124,26127,26129,26131],{"class":36,"line":43},[34,26121,26017],{"class":395},[34,26123,456],{"class":190},[34,26125,26126],{"class":186},"'描述一个行为'",[34,26128,6131],{"class":190},[34,26130,1333],{"class":175},[34,26132,674],{"class":190},[34,26134,26135,26137,26140,26142],{"class":36,"line":49},[34,26136,26067],{"class":395},[34,26138,26139],{"class":190},"(实际结果).",[34,26141,26073],{"class":395},[34,26143,26144],{"class":190},"(期望结果)\n",[34,26146,26147],{"class":36,"line":56},[34,26148,6379],{"class":190},[34,26150,26151],{"class":36,"line":62},[34,26152,6153],{"class":190},[15,26154,2632],{},[24,26156,26158],{"className":166,"code":26157,"language":168,"meta":29,"style":29},"it('adds two numbers', () => {\n expect(1 + 2).toBe(3)\n})\n",[31,26159,26160,26176,26201],{"__ignoreMap":29},[34,26161,26162,26165,26167,26170,26172,26174],{"class":36,"line":37},[34,26163,26164],{"class":395},"it",[34,26166,456],{"class":190},[34,26168,26169],{"class":186},"'adds two numbers'",[34,26171,6131],{"class":190},[34,26173,1333],{"class":175},[34,26175,674],{"class":190},[34,26177,26178,26181,26183,26185,26188,26190,26193,26195,26197,26199],{"class":36,"line":43},[34,26179,26180],{"class":395}," expect",[34,26182,456],{"class":190},[34,26184,640],{"class":179},[34,26186,26187],{"class":175}," +",[34,26189,1426],{"class":179},[34,26191,26192],{"class":190},").",[34,26194,26073],{"class":395},[34,26196,456],{"class":190},[34,26198,786],{"class":179},[34,26200,6111],{"class":190},[34,26202,26203],{"class":36,"line":49},[34,26204,6153],{"class":190},[15,26206,26207],{},"常见断言：",[24,26209,26211],{"className":166,"code":26210,"language":168,"meta":29,"style":29},"expect(value).toBe(1) \u002F\u002F 基本值相等\nexpect(object).toEqual({ a: 1 }) \u002F\u002F 对象或数组内容相等\nexpect(text).toContain('Vue') \u002F\u002F 包含文本\nexpect(fn).toThrow() \u002F\u002F 函数会抛错\n",[31,26212,26213,26233,26254,26275],{"__ignoreMap":29},[34,26214,26215,26218,26221,26223,26225,26227,26230],{"class":36,"line":37},[34,26216,26217],{"class":395},"expect",[34,26219,26220],{"class":190},"(value).",[34,26222,26073],{"class":395},[34,26224,456],{"class":190},[34,26226,640],{"class":179},[34,26228,26229],{"class":190},") ",[34,26231,26232],{"class":301},"\u002F\u002F 基本值相等\n",[34,26234,26235,26237,26240,26243,26246,26248,26251],{"class":36,"line":43},[34,26236,26217],{"class":395},[34,26238,26239],{"class":190},"(object).",[34,26241,26242],{"class":395},"toEqual",[34,26244,26245],{"class":190},"({ a: ",[34,26247,640],{"class":179},[34,26249,26250],{"class":190}," }) ",[34,26252,26253],{"class":301},"\u002F\u002F 对象或数组内容相等\n",[34,26255,26256,26258,26261,26264,26266,26269,26272],{"class":36,"line":49},[34,26257,26217],{"class":395},[34,26259,26260],{"class":190},"(text).",[34,26262,26263],{"class":395},"toContain",[34,26265,456],{"class":190},[34,26267,26268],{"class":186},"'Vue'",[34,26270,26271],{"class":190},") ",[34,26273,26274],{"class":301},"\u002F\u002F 包含文本\n",[34,26276,26277,26279,26282,26285,26288],{"class":36,"line":56},[34,26278,26217],{"class":395},[34,26280,26281],{"class":190},"(fn).",[34,26283,26284],{"class":395},"toThrow",[34,26286,26287],{"class":190},"() ",[34,26289,26290],{"class":301},"\u002F\u002F 函数会抛错\n",[19,26292,26294],{"id":26293},"_6-测试组合式函数","6. 测试组合式函数",[15,26296,26297],{},"组合式函数通常没有 DOM，最适合初学者练测试。",[24,26299,26301],{"className":166,"code":26300,"language":168,"meta":29,"style":29},"import { describe, expect, it } from 'vitest'\nimport { useCounter } from '..\u002Fcomposables\u002FuseCounter'\n\ndescribe('useCounter', () => {\n it('starts from initial value', () => {\n const counter = useCounter(5)\n\n expect(counter.count.value).toBe(5)\n })\n\n it('increases by one', () => {\n const counter = useCounter(0)\n\n counter.increase()\n\n expect(counter.count.value).toBe(1)\n })\n\n it('resets to initial value', () => {\n const counter = useCounter(3)\n\n counter.increase()\n counter.reset()\n\n expect(counter.count.value).toBe(3)\n })\n})\n",[31,26302,26303,26313,26323,26327,26341,26356,26372,26376,26390,26394,26398,26413,26429,26433,26441,26445,26459,26463,26467,26482,26498,26502,26510,26518,26522,26536,26540],{"__ignoreMap":29},[34,26304,26305,26307,26309,26311],{"class":36,"line":37},[34,26306,426],{"class":175},[34,26308,25977],{"class":190},[34,26310,432],{"class":175},[34,26312,25982],{"class":186},[34,26314,26315,26317,26319,26321],{"class":36,"line":43},[34,26316,426],{"class":175},[34,26318,4740],{"class":190},[34,26320,432],{"class":175},[34,26322,18099],{"class":186},[34,26324,26325],{"class":36,"line":49},[34,26326,53],{"emptyLinePlaceholder":52},[34,26328,26329,26331,26333,26335,26337,26339],{"class":36,"line":56},[34,26330,26001],{"class":395},[34,26332,456],{"class":190},[34,26334,26006],{"class":186},[34,26336,6131],{"class":190},[34,26338,1333],{"class":175},[34,26340,674],{"class":190},[34,26342,26343,26345,26347,26350,26352,26354],{"class":36,"line":62},[34,26344,26017],{"class":395},[34,26346,456],{"class":190},[34,26348,26349],{"class":186},"'starts from initial value'",[34,26351,6131],{"class":190},[34,26353,1333],{"class":175},[34,26355,674],{"class":190},[34,26357,26358,26360,26362,26364,26366,26368,26370],{"class":36,"line":68},[34,26359,5475],{"class":175},[34,26361,26035],{"class":179},[34,26363,183],{"class":175},[34,26365,17936],{"class":395},[34,26367,456],{"class":190},[34,26369,18136],{"class":179},[34,26371,6111],{"class":190},[34,26373,26374],{"class":36,"line":74},[34,26375,53],{"emptyLinePlaceholder":52},[34,26377,26378,26380,26382,26384,26386,26388],{"class":36,"line":1971},[34,26379,26067],{"class":395},[34,26381,26070],{"class":190},[34,26383,26073],{"class":395},[34,26385,456],{"class":190},[34,26387,18136],{"class":179},[34,26389,6111],{"class":190},[34,26391,26392],{"class":36,"line":1976},[34,26393,6379],{"class":190},[34,26395,26396],{"class":36,"line":1985},[34,26397,53],{"emptyLinePlaceholder":52},[34,26399,26400,26402,26404,26407,26409,26411],{"class":36,"line":1990},[34,26401,26017],{"class":395},[34,26403,456],{"class":190},[34,26405,26406],{"class":186},"'increases by one'",[34,26408,6131],{"class":190},[34,26410,1333],{"class":175},[34,26412,674],{"class":190},[34,26414,26415,26417,26419,26421,26423,26425,26427],{"class":36,"line":1999},[34,26416,5475],{"class":175},[34,26418,26035],{"class":179},[34,26420,183],{"class":175},[34,26422,17936],{"class":395},[34,26424,456],{"class":190},[34,26426,459],{"class":179},[34,26428,6111],{"class":190},[34,26430,26431],{"class":36,"line":2022},[34,26432,53],{"emptyLinePlaceholder":52},[34,26434,26435,26437,26439],{"class":36,"line":2036},[34,26436,26054],{"class":190},[34,26438,2048],{"class":395},[34,26440,6374],{"class":190},[34,26442,26443],{"class":36,"line":2603},[34,26444,53],{"emptyLinePlaceholder":52},[34,26446,26447,26449,26451,26453,26455,26457],{"class":36,"line":5537},[34,26448,26067],{"class":395},[34,26450,26070],{"class":190},[34,26452,26073],{"class":395},[34,26454,456],{"class":190},[34,26456,640],{"class":179},[34,26458,6111],{"class":190},[34,26460,26461],{"class":36,"line":5543},[34,26462,6379],{"class":190},[34,26464,26465],{"class":36,"line":5548},[34,26466,53],{"emptyLinePlaceholder":52},[34,26468,26469,26471,26473,26476,26478,26480],{"class":36,"line":5564},[34,26470,26017],{"class":395},[34,26472,456],{"class":190},[34,26474,26475],{"class":186},"'resets to initial value'",[34,26477,6131],{"class":190},[34,26479,1333],{"class":175},[34,26481,674],{"class":190},[34,26483,26484,26486,26488,26490,26492,26494,26496],{"class":36,"line":5576},[34,26485,5475],{"class":175},[34,26487,26035],{"class":179},[34,26489,183],{"class":175},[34,26491,17936],{"class":395},[34,26493,456],{"class":190},[34,26495,786],{"class":179},[34,26497,6111],{"class":190},[34,26499,26500],{"class":36,"line":5606},[34,26501,53],{"emptyLinePlaceholder":52},[34,26503,26504,26506,26508],{"class":36,"line":5616},[34,26505,26054],{"class":190},[34,26507,2048],{"class":395},[34,26509,6374],{"class":190},[34,26511,26512,26514,26516],{"class":36,"line":5628},[34,26513,26054],{"class":190},[34,26515,18125],{"class":395},[34,26517,6374],{"class":190},[34,26519,26520],{"class":36,"line":5634},[34,26521,53],{"emptyLinePlaceholder":52},[34,26523,26524,26526,26528,26530,26532,26534],{"class":36,"line":6420},[34,26525,26067],{"class":395},[34,26527,26070],{"class":190},[34,26529,26073],{"class":395},[34,26531,456],{"class":190},[34,26533,786],{"class":179},[34,26535,6111],{"class":190},[34,26537,26538],{"class":36,"line":11294},[34,26539,6379],{"class":190},[34,26541,26542],{"class":36,"line":11321},[34,26543,6153],{"class":190},[15,26545,26546,26547,347],{},"注意：测试里访问 ref 仍然要 ",[31,26548,507],{},[19,26550,26552],{"id":26551},"_7-组件测试","7. 组件测试",[15,26554,26555],{},"组件测试需要把组件挂载起来。",[24,26557,26559],{"className":166,"code":26558,"language":168,"meta":29,"style":29},"import { mount } from '@vue\u002Ftest-utils'\nimport { describe, expect, it } from 'vitest'\nimport TodoItem from '..\u002Fcomponents\u002FTodoItem.vue'\n\ndescribe('TodoItem', () => {\n it('renders title', () => {\n const wrapper = mount(TodoItem, {\n props: {\n todo: {\n id: '1',\n title: '学习组件测试',\n done: false\n }\n }\n })\n\n expect(wrapper.text()).toContain('学习组件测试')\n })\n})\n",[31,26560,26561,26573,26583,26594,26598,26613,26628,26643,26648,26653,26663,26673,26680,26685,26689,26693,26697,26716,26720],{"__ignoreMap":29},[34,26562,26563,26565,26568,26570],{"class":36,"line":37},[34,26564,426],{"class":175},[34,26566,26567],{"class":190}," { mount } ",[34,26569,432],{"class":175},[34,26571,26572],{"class":186}," '@vue\u002Ftest-utils'\n",[34,26574,26575,26577,26579,26581],{"class":36,"line":43},[34,26576,426],{"class":175},[34,26578,25977],{"class":190},[34,26580,432],{"class":175},[34,26582,25982],{"class":186},[34,26584,26585,26587,26589,26591],{"class":36,"line":49},[34,26586,426],{"class":175},[34,26588,4718],{"class":190},[34,26590,432],{"class":175},[34,26592,26593],{"class":186}," '..\u002Fcomponents\u002FTodoItem.vue'\n",[34,26595,26596],{"class":36,"line":56},[34,26597,53],{"emptyLinePlaceholder":52},[34,26599,26600,26602,26604,26607,26609,26611],{"class":36,"line":62},[34,26601,26001],{"class":395},[34,26603,456],{"class":190},[34,26605,26606],{"class":186},"'TodoItem'",[34,26608,6131],{"class":190},[34,26610,1333],{"class":175},[34,26612,674],{"class":190},[34,26614,26615,26617,26619,26622,26624,26626],{"class":36,"line":68},[34,26616,26017],{"class":395},[34,26618,456],{"class":190},[34,26620,26621],{"class":186},"'renders title'",[34,26623,6131],{"class":190},[34,26625,1333],{"class":175},[34,26627,674],{"class":190},[34,26629,26630,26632,26635,26637,26640],{"class":36,"line":74},[34,26631,5475],{"class":175},[34,26633,26634],{"class":179}," wrapper",[34,26636,183],{"class":175},[34,26638,26639],{"class":395}," mount",[34,26641,26642],{"class":190},"(TodoItem, {\n",[34,26644,26645],{"class":36,"line":1971},[34,26646,26647],{"class":190}," props: {\n",[34,26649,26650],{"class":36,"line":1976},[34,26651,26652],{"class":190}," todo: {\n",[34,26654,26655,26658,26661],{"class":36,"line":1985},[34,26656,26657],{"class":190}," id: ",[34,26659,26660],{"class":186},"'1'",[34,26662,684],{"class":190},[34,26664,26665,26668,26671],{"class":36,"line":1990},[34,26666,26667],{"class":190}," title: ",[34,26669,26670],{"class":186},"'学习组件测试'",[34,26672,684],{"class":190},[34,26674,26675,26678],{"class":36,"line":1999},[34,26676,26677],{"class":190}," done: ",[34,26679,14360],{"class":179},[34,26681,26682],{"class":36,"line":2022},[34,26683,26684],{"class":190}," }\n",[34,26686,26687],{"class":36,"line":2036},[34,26688,21274],{"class":190},[34,26690,26691],{"class":36,"line":2603},[34,26692,6791],{"class":190},[34,26694,26695],{"class":36,"line":5537},[34,26696,53],{"emptyLinePlaceholder":52},[34,26698,26699,26701,26704,26706,26708,26710,26712,26714],{"class":36,"line":5543},[34,26700,26067],{"class":395},[34,26702,26703],{"class":190},"(wrapper.",[34,26705,2638],{"class":395},[34,26707,21831],{"class":190},[34,26709,26263],{"class":395},[34,26711,456],{"class":190},[34,26713,26670],{"class":186},[34,26715,6111],{"class":190},[34,26717,26718],{"class":36,"line":5548},[34,26719,6379],{"class":190},[34,26721,26722],{"class":36,"line":5564},[34,26723,6153],{"class":190},[15,26725,26726,26728,26729,26732],{},[31,26727,20290],{}," 会创建组件实例。",[31,26730,26731],{},"wrapper"," 是测试工具提供的包装对象，可以读取文本、查找按钮、触发事件。",[19,26734,26736],{"id":26735},"_8-测试-emit","8. 测试 emit",[15,26738,26739],{},"子组件点击后应该发事件：",[24,26741,26743],{"className":166,"code":26742,"language":168,"meta":29,"style":29},"it('emits toggle event', async () => {\n const wrapper = mount(TodoItem, {\n props: {\n todo: {\n id: '1',\n title: '学习测试',\n done: false\n }\n }\n })\n\n await wrapper.get('[data-test=\"toggle\"]').trigger('click')\n\n expect(wrapper.emitted('toggle')).toBeTruthy()\n expect(wrapper.emitted('toggle')[0]).toEqual(['1'])\n})\n",[31,26744,26745,26765,26777,26782,26787,26796,26806,26813,26817,26821,26825,26829,26855,26859,26880,26908],{"__ignoreMap":29},[34,26746,26747,26749,26751,26754,26756,26758,26761,26763],{"class":36,"line":37},[34,26748,26164],{"class":395},[34,26750,456],{"class":190},[34,26752,26753],{"class":186},"'emits toggle event'",[34,26755,616],{"class":190},[34,26757,5217],{"class":175},[34,26759,26760],{"class":190}," () ",[34,26762,1333],{"class":175},[34,26764,674],{"class":190},[34,26766,26767,26769,26771,26773,26775],{"class":36,"line":43},[34,26768,5298],{"class":175},[34,26770,26634],{"class":179},[34,26772,183],{"class":175},[34,26774,26639],{"class":395},[34,26776,26642],{"class":190},[34,26778,26779],{"class":36,"line":49},[34,26780,26781],{"class":190}," props: {\n",[34,26783,26784],{"class":36,"line":56},[34,26785,26786],{"class":190}," todo: {\n",[34,26788,26789,26792,26794],{"class":36,"line":62},[34,26790,26791],{"class":190}," id: ",[34,26793,26660],{"class":186},[34,26795,684],{"class":190},[34,26797,26798,26801,26804],{"class":36,"line":68},[34,26799,26800],{"class":190}," title: ",[34,26802,26803],{"class":186},"'学习测试'",[34,26805,684],{"class":190},[34,26807,26808,26811],{"class":36,"line":74},[34,26809,26810],{"class":190}," done: ",[34,26812,14360],{"class":179},[34,26814,26815],{"class":36,"line":1971},[34,26816,21274],{"class":190},[34,26818,26819],{"class":36,"line":1976},[34,26820,5540],{"class":190},[34,26822,26823],{"class":36,"line":1985},[34,26824,6379],{"class":190},[34,26826,26827],{"class":36,"line":1990},[34,26828,53],{"emptyLinePlaceholder":52},[34,26830,26831,26833,26836,26839,26841,26844,26846,26849,26851,26853],{"class":36,"line":1999},[34,26832,5242],{"class":175},[34,26834,26835],{"class":190}," wrapper.",[34,26837,26838],{"class":395},"get",[34,26840,456],{"class":190},[34,26842,26843],{"class":186},"'[data-test=\"toggle\"]'",[34,26845,26192],{"class":190},[34,26847,26848],{"class":395},"trigger",[34,26850,456],{"class":190},[34,26852,6128],{"class":186},[34,26854,6111],{"class":190},[34,26856,26857],{"class":36,"line":2022},[34,26858,53],{"emptyLinePlaceholder":52},[34,26860,26861,26863,26865,26868,26870,26872,26875,26878],{"class":36,"line":2036},[34,26862,26180],{"class":395},[34,26864,26703],{"class":190},[34,26866,26867],{"class":395},"emitted",[34,26869,456],{"class":190},[34,26871,12370],{"class":186},[34,26873,26874],{"class":190},")).",[34,26876,26877],{"class":395},"toBeTruthy",[34,26879,6374],{"class":190},[34,26881,26882,26884,26886,26888,26890,26892,26895,26897,26900,26902,26904,26906],{"class":36,"line":2603},[34,26883,26180],{"class":395},[34,26885,26703],{"class":190},[34,26887,26867],{"class":395},[34,26889,456],{"class":190},[34,26891,12370],{"class":186},[34,26893,26894],{"class":190},")[",[34,26896,459],{"class":179},[34,26898,26899],{"class":190},"]).",[34,26901,26242],{"class":395},[34,26903,12367],{"class":190},[34,26905,26660],{"class":186},[34,26907,6945],{"class":190},[34,26909,26910],{"class":36,"line":5537},[34,26911,6153],{"class":190},[15,26913,26914],{},"模板中建议给测试钩子：",[24,26916,26918],{"className":923,"code":26917,"language":925,"meta":29,"style":29},"\u003Cbutton data-test=\"toggle\" @click=\"emit('toggle', todo.id)\">\n 切换\n\u003C\u002Fbutton>\n",[31,26919,26920,26955,26960],{"__ignoreMap":29},[34,26921,26922,26924,26926,26929,26931,26934,26936,26938,26940,26942,26944,26946,26948,26951,26953],{"class":36,"line":37},[34,26923,932],{"class":190},[34,26925,2004],{"class":935},[34,26927,26928],{"class":395}," data-test",[34,26930,202],{"class":190},[34,26932,26933],{"class":186},"\"toggle\"",[34,26935,9628],{"class":190},[34,26937,9631],{"class":395},[34,26939,202],{"class":190},[34,26941,1608],{"class":186},[34,26943,12631],{"class":395},[34,26945,456],{"class":190},[34,26947,12370],{"class":186},[34,26949,26950],{"class":190},", todo.id)",[34,26952,1608],{"class":186},[34,26954,939],{"class":190},[34,26956,26957],{"class":36,"line":43},[34,26958,26959],{"class":190}," 切换\n",[34,26961,26962,26964,26966],{"class":36,"line":49},[34,26963,1003],{"class":190},[34,26965,2004],{"class":935},[34,26967,939],{"class":190},[15,26969,26970],{},"为什么不用 class？",[105,26972,26973,26976],{},[108,26974,26975],{},"class 主要服务样式，可能会改。",[108,26977,26978,26981],{},[31,26979,26980],{},"data-test"," 明确表示给测试使用。",[19,26983,26985],{"id":26984},"_9-测试应该关注什么","9. 测试应该关注什么",[15,26987,26988],{},"好测试关注行为，不关注实现细节。",[15,26990,26991],{},"推荐测试：",[105,26993,26994,26997,27000,27003],{},[108,26995,26996],{},"给定 props，组件显示正确文本。",[108,26998,26999],{},"用户点击按钮，组件 emit 正确事件。",[108,27001,27002],{},"组合式函数调用方法后状态改变。",[108,27004,27005],{},"store action 修改状态。",[15,27007,27008],{},"不推荐过度测试：",[105,27010,27011,27014,27017],{},[108,27012,27013],{},"某个内部变量名字。",[108,27015,27016],{},"某个 class 的存在，除非它代表重要状态。",[108,27018,27019],{},"完全重复 Vue 自己已经保证的行为。",[19,27021,27023],{"id":27022},"_10-测试命名","10. 测试命名",[15,27025,27026],{},"好的测试名像需求：",[24,27028,27030],{"className":166,"code":27029,"language":168,"meta":29,"style":29},"it('emits remove event with todo id when remove button is clicked', () => {})\n",[31,27031,27032],{"__ignoreMap":29},[34,27033,27034,27036,27038,27041,27043,27045],{"class":36,"line":37},[34,27035,26164],{"class":395},[34,27037,456],{"class":190},[34,27039,27040],{"class":186},"'emits remove event with todo id when remove button is clicked'",[34,27042,6131],{"class":190},[34,27044,1333],{"class":175},[34,27046,27047],{"class":190}," {})\n",[15,27049,27050],{},"不好的测试名：",[24,27052,27054],{"className":166,"code":27053,"language":168,"meta":29,"style":29},"it('test 1', () => {})\n",[31,27055,27056],{"__ignoreMap":29},[34,27057,27058,27060,27062,27065,27067,27069],{"class":36,"line":37},[34,27059,26164],{"class":395},[34,27061,456],{"class":190},[34,27063,27064],{"class":186},"'test 1'",[34,27066,6131],{"class":190},[34,27068,1333],{"class":175},[34,27070,27047],{"class":190},[15,27072,27073],{},"读测试的人应该不看实现也知道这条测试验证什么。",[19,27075,27077],{"id":27076},"_11-常见-vue-调试场景","11. 常见 Vue 调试场景",[253,27079,27080],{"id":27080},"页面空白",[15,27082,27083],{},"检查：",[105,27085,27086,27089,27092,27095,27100],{},[108,27087,27088],{},"控制台是否有报错。",[108,27090,27091],{},"根组件是否挂载。",[108,27093,27094],{},"Router 是否有匹配页面。",[108,27096,27097,27099],{},[31,27098,8162],{}," 条件是否一直为 false。",[108,27101,27102],{},"请求是否一直 loading。",[253,27104,27105],{"id":27105},"点击按钮没反应",[15,27107,27083],{},[105,27109,27110,27113,27118,27121,27124],{},[108,27111,27112],{},"按钮是否真的可点击，是否 disabled。",[108,27114,27115,27117],{},[31,27116,7152],{}," 函数名是否拼错。",[108,27119,27120],{},"函数里是否改了响应式状态。",[108,27122,27123],{},"子组件是否 emit。",[108,27125,27126],{},"父组件监听事件名是否一致。",[253,27128,27129],{"id":27129},"列表状态错位",[15,27131,27083],{},[105,27133,27134,27141,27144],{},[108,27135,27136,27138,27139,347],{},[31,27137,1534],{}," 是否有稳定 ",[31,27140,1624],{},[108,27142,27143],{},"是否用 index 当 key。",[108,27145,27146],{},"是否对数组做了增删排序。",[253,27148,27149],{"id":27149},"表单提交刷新页面",[15,27151,27083],{},[24,27153,27154],{"className":923,"code":9781,"language":925,"meta":29,"style":29},[31,27155,27156],{"__ignoreMap":29},[34,27157,27158,27160,27162,27164,27166,27168,27170,27172,27174,27176,27178],{"class":36,"line":37},[34,27159,932],{"class":190},[34,27161,9790],{"class":935},[34,27163,9628],{"class":190},[34,27165,9795],{"class":395},[34,27167,2547],{"class":190},[34,27169,9800],{"class":395},[34,27171,202],{"class":190},[34,27173,1608],{"class":186},[34,27175,9795],{"class":190},[34,27177,1608],{"class":186},[34,27179,939],{"class":190},[15,27181,27182,27183,347],{},"是否忘记 ",[31,27184,9815],{},[19,27186,27188],{"id":27187},"_12-质量保障清单","12. 质量保障清单",[15,27190,27191],{},"提交前至少运行：",[24,27193,27195],{"className":7808,"code":27194,"language":7810,"meta":29,"style":29},"npm run test\nnpm run build\n",[31,27196,27197,27202],{"__ignoreMap":29},[34,27198,27199],{"class":36,"line":37},[34,27200,27201],{},"npm run test\n",[34,27203,27204],{"class":36,"line":43},[34,27205,27206],{},"npm run build\n",[15,27208,27209],{},"如果有 lint：",[24,27211,27213],{"className":7808,"code":27212,"language":7810,"meta":29,"style":29},"npm run lint\n",[31,27214,27215],{"__ignoreMap":29},[34,27216,27217],{"class":36,"line":37},[34,27218,27212],{},[15,27220,27221],{},"如果改了 UI，最好浏览器实际点一遍：",[105,27223,27224,27227,27230,27233],{},[108,27225,27226],{},"首页是否能打开。",[108,27228,27229],{},"关键按钮是否有反应。",[108,27231,27232],{},"表单错误是否显示。",[108,27234,27235],{},"移动端是否明显错位。",[19,27237,5693],{"id":5693},[2230,27239,27240,27246,27251,27258,27261,27264],{},[108,27241,27242,27243,27245],{},"为 ",[31,27244,19560],{}," 写 3 个测试：初始值、增加、重置。",[108,27247,27242,27248,27250],{},[31,27249,11972],{}," 写测试：能显示标题。",[108,27252,27242,27253,27255,27256,347],{},[31,27254,11972],{}," 写测试：点击切换按钮 emit ",[31,27257,11935],{},[108,27259,27260],{},"故意把事件名改错，确认测试失败。",[108,27262,27263],{},"修复事件名，确认测试通过。",[108,27265,27266],{},"写一份你自己的调试清单。",[79,27268,27269],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}",{"title":29,"searchDepth":43,"depth":43,"links":27271},[27272,27273,27274,27275,27276,27277,27278,27279,27280,27281,27282,27288,27289],{"id":25657,"depth":43,"text":25658},{"id":25771,"depth":43,"text":25772},{"id":25831,"depth":43,"text":25832},{"id":25882,"depth":43,"text":25883},{"id":26093,"depth":43,"text":26094},{"id":26293,"depth":43,"text":26294},{"id":26551,"depth":43,"text":26552},{"id":26735,"depth":43,"text":26736},{"id":26984,"depth":43,"text":26985},{"id":27022,"depth":43,"text":27023},{"id":27076,"depth":43,"text":27077,"children":27283},[27284,27285,27286,27287],{"id":27080,"depth":49,"text":27080},{"id":27105,"depth":49,"text":27105},{"id":27129,"depth":49,"text":27129},{"id":27149,"depth":49,"text":27149},{"id":27187,"depth":43,"text":27188},{"id":5693,"depth":43,"text":5693},{"order":1985},"\u002Fsource\u002Fcode\u002Fvue\u002F10-testing-debugging",{"title":25643,"description":25651},"source\u002Fcode\u002Fvue\u002F10-testing-debugging","Bf9uJ-uJRAOO7iH0RGCOfQ2D7UrudiEhxBY2t9l-qkk",{"id":27296,"title":27297,"body":27298,"description":27305,"extension":85,"meta":28876,"navigation":52,"path":28877,"seo":28878,"stem":28879,"__hash__":28880},"notes\u002Fsource\u002Fcode\u002Fvue\u002F11-typescript-production.md","11 TypeScript 和生产级代码",{"type":8,"value":27299,"toc":28856},[27300,27303,27306,27309,27313,27316,27373,27376,27431,27435,27485,27488,27546,27549,27609,27611,27666,27670,27673,27719,27725,27728,27756,27759,27782,27786,27898,27901,27906,27931,27934,27955,27961,27965,28021,28024,28039,28042,28058,28061,28064,28084,28087,28091,28094,28137,28140,28205,28211,28219,28222,28273,28277,28416,28422,28425,28429,28432,28451,28454,28460,28510,28512,28531,28537,28541,28544,28547,28576,28580,28586,28589,28627,28631,28634,28656,28659,28661,28665,28710,28713,28720,28741,28746,28750,28795,28797,28822,28824,28853],[11,27301,27297],{"id":27302},"_11-typescript-和生产级代码",[15,27304,27305],{},"你可以先用 JavaScript 学 Vue，再逐步引入 TypeScript。TypeScript 的价值不是“让代码更复杂”，而是让错误更早暴露，尤其适合多人协作和长期维护的项目。",[15,27307,27308],{},"本章目标：你理解 TypeScript 能解决什么问题，能给 Vue props、emits、ref、API 数据加基础类型，并知道生产级代码应该具备哪些习惯。",[19,27310,27312],{"id":27311},"_1-javascript-中的问题","1. JavaScript 中的问题",[15,27314,27315],{},"JavaScript 很灵活：",[24,27317,27319],{"className":166,"code":27318,"language":168,"meta":29,"style":29},"function formatMinutes(minutes) {\n return `${minutes} 分钟`\n}\n\nformatMinutes(30)\nformatMinutes('abc')\n",[31,27320,27321,27333,27343,27347,27351,27362],{"__ignoreMap":29},[34,27322,27323,27325,27327,27329,27331],{"class":36,"line":37},[34,27324,392],{"class":175},[34,27326,19617],{"class":395},[34,27328,456],{"class":190},[34,27330,15036],{"class":1326},[34,27332,1700],{"class":190},[34,27334,27335,27337,27339,27341],{"class":36,"line":43},[34,27336,1510],{"class":175},[34,27338,19630],{"class":186},[34,27340,15036],{"class":190},[34,27342,19635],{"class":186},[34,27344,27345],{"class":36,"line":49},[34,27346,77],{"class":190},[34,27348,27349],{"class":36,"line":56},[34,27350,53],{"emptyLinePlaceholder":52},[34,27352,27353,27356,27358,27360],{"class":36,"line":62},[34,27354,27355],{"class":395},"formatMinutes",[34,27357,456],{"class":190},[34,27359,9314],{"class":179},[34,27361,6111],{"class":190},[34,27363,27364,27366,27368,27371],{"class":36,"line":68},[34,27365,27355],{"class":395},[34,27367,456],{"class":190},[34,27369,27370],{"class":186},"'abc'",[34,27372,6111],{"class":190},[15,27374,27375],{},"第二次调用语法上没问题，但业务上可能错了。TypeScript 可以提前提醒：这里应该传数字。",[24,27377,27381],{"className":27378,"code":27379,"language":27380,"meta":29,"style":29},"language-ts shiki shiki-themes github-light github-dark","function formatMinutes(minutes: number) {\n return `${minutes} 分钟`\n}\n\nformatMinutes('abc') \u002F\u002F 类型错误\n","ts",[31,27382,27383,27400,27410,27414,27418],{"__ignoreMap":29},[34,27384,27385,27387,27389,27391,27393,27395,27398],{"class":36,"line":37},[34,27386,392],{"class":175},[34,27388,19617],{"class":395},[34,27390,456],{"class":190},[34,27392,15036],{"class":1326},[34,27394,5598],{"class":175},[34,27396,27397],{"class":179}," number",[34,27399,1700],{"class":190},[34,27401,27402,27404,27406,27408],{"class":36,"line":43},[34,27403,1510],{"class":175},[34,27405,19630],{"class":186},[34,27407,15036],{"class":190},[34,27409,19635],{"class":186},[34,27411,27412],{"class":36,"line":49},[34,27413,77],{"class":190},[34,27415,27416],{"class":36,"line":56},[34,27417,53],{"emptyLinePlaceholder":52},[34,27419,27420,27422,27424,27426,27428],{"class":36,"line":62},[34,27421,27355],{"class":395},[34,27423,456],{"class":190},[34,27425,27370],{"class":186},[34,27427,1330],{"class":190},[34,27429,27430],{"class":301},"\u002F\u002F 类型错误\n",[19,27432,27434],{"id":27433},"_2-基础类型","2. 基础类型",[24,27436,27438],{"className":27378,"code":27437,"language":27380,"meta":29,"style":29},"const title: string = 'Vue'\nconst minutes: number = 45\nconst done: boolean = false\n",[31,27439,27440,27455,27470],{"__ignoreMap":29},[34,27441,27442,27444,27446,27448,27451,27453],{"class":36,"line":37},[34,27443,176],{"class":175},[34,27445,278],{"class":179},[34,27447,5598],{"class":175},[34,27449,27450],{"class":179}," string",[34,27452,183],{"class":175},[34,27454,8592],{"class":186},[34,27456,27457,27459,27462,27464,27466,27468],{"class":36,"line":43},[34,27458,176],{"class":175},[34,27460,27461],{"class":179}," minutes",[34,27463,5598],{"class":175},[34,27465,27397],{"class":179},[34,27467,183],{"class":175},[34,27469,9357],{"class":179},[34,27471,27472,27474,27476,27478,27481,27483],{"class":36,"line":49},[34,27473,176],{"class":175},[34,27475,571],{"class":179},[34,27477,5598],{"class":175},[34,27479,27480],{"class":179}," boolean",[34,27482,183],{"class":175},[34,27484,18640],{"class":179},[15,27486,27487],{},"数组：",[24,27489,27491],{"className":27378,"code":27490,"language":27380,"meta":29,"style":29},"const tags: string[] = ['vue', 'router']\nconst scores: number[] = [80, 90]\n",[31,27492,27493,27520],{"__ignoreMap":29},[34,27494,27495,27497,27499,27501,27503,27506,27508,27510,27513,27515,27518],{"class":36,"line":37},[34,27496,176],{"class":175},[34,27498,605],{"class":179},[34,27500,5598],{"class":175},[34,27502,27450],{"class":179},[34,27504,27505],{"class":190},"[] ",[34,27507,202],{"class":175},[34,27509,610],{"class":190},[34,27511,27512],{"class":186},"'vue'",[34,27514,616],{"class":190},[34,27516,27517],{"class":186},"'router'",[34,27519,6569],{"class":190},[34,27521,27522,27524,27527,27529,27531,27533,27535,27537,27540,27542,27544],{"class":36,"line":43},[34,27523,176],{"class":175},[34,27525,27526],{"class":179}," scores",[34,27528,5598],{"class":175},[34,27530,27397],{"class":179},[34,27532,27505],{"class":190},[34,27534,202],{"class":175},[34,27536,610],{"class":190},[34,27538,27539],{"class":179},"80",[34,27541,616],{"class":190},[34,27543,854],{"class":179},[34,27545,6569],{"class":190},[15,27547,27548],{},"对象类型：",[24,27550,27552],{"className":27378,"code":27551,"language":27380,"meta":29,"style":29},"type Lesson = {\n id: string\n title: string\n minutes: number\n done: boolean\n}\n",[31,27553,27554,27566,27576,27585,27595,27605],{"__ignoreMap":29},[34,27555,27556,27559,27562,27564],{"class":36,"line":37},[34,27557,27558],{"class":175},"type",[34,27560,27561],{"class":395}," Lesson",[34,27563,183],{"class":175},[34,27565,674],{"class":190},[34,27567,27568,27571,27573],{"class":36,"line":43},[34,27569,27570],{"class":1326}," id",[34,27572,5598],{"class":175},[34,27574,27575],{"class":179}," string\n",[34,27577,27578,27581,27583],{"class":36,"line":49},[34,27579,27580],{"class":1326}," title",[34,27582,5598],{"class":175},[34,27584,27575],{"class":179},[34,27586,27587,27590,27592],{"class":36,"line":56},[34,27588,27589],{"class":1326}," minutes",[34,27591,5598],{"class":175},[34,27593,27594],{"class":179}," number\n",[34,27596,27597,27600,27602],{"class":36,"line":62},[34,27598,27599],{"class":1326}," done",[34,27601,5598],{"class":175},[34,27603,27604],{"class":179}," boolean\n",[34,27606,27607],{"class":36,"line":68},[34,27608,77],{"class":190},[15,27610,13233],{},[24,27612,27614],{"className":27378,"code":27613,"language":27380,"meta":29,"style":29},"const lesson: Lesson = {\n id: 'core',\n title: 'Vue 核心语法',\n minutes: 90,\n done: false\n}\n",[31,27615,27616,27630,27639,27648,27656,27662],{"__ignoreMap":29},[34,27617,27618,27620,27622,27624,27626,27628],{"class":36,"line":37},[34,27619,176],{"class":175},[34,27621,824],{"class":179},[34,27623,5598],{"class":175},[34,27625,27561],{"class":395},[34,27627,183],{"class":175},[34,27629,674],{"class":190},[34,27631,27632,27634,27637],{"class":36,"line":43},[34,27633,679],{"class":190},[34,27635,27636],{"class":186},"'core'",[34,27638,684],{"class":190},[34,27640,27641,27643,27646],{"class":36,"line":49},[34,27642,689],{"class":190},[34,27644,27645],{"class":186},"'Vue 核心语法'",[34,27647,684],{"class":190},[34,27649,27650,27652,27654],{"class":36,"line":56},[34,27651,851],{"class":190},[34,27653,854],{"class":179},[34,27655,684],{"class":190},[34,27657,27658,27660],{"class":36,"line":62},[34,27659,699],{"class":190},[34,27661,14360],{"class":179},[34,27663,27664],{"class":36,"line":68},[34,27665,77],{"class":190},[19,27667,27669],{"id":27668},"_3-可选属性和联合类型","3. 可选属性和联合类型",[15,27671,27672],{},"可选属性：",[24,27674,27676],{"className":27378,"code":27675,"language":27380,"meta":29,"style":29},"type User = {\n id: string\n name: string\n avatarUrl?: string\n}\n",[31,27677,27678,27689,27697,27705,27715],{"__ignoreMap":29},[34,27679,27680,27682,27685,27687],{"class":36,"line":37},[34,27681,27558],{"class":175},[34,27683,27684],{"class":395}," User",[34,27686,183],{"class":175},[34,27688,674],{"class":190},[34,27690,27691,27693,27695],{"class":36,"line":43},[34,27692,27570],{"class":1326},[34,27694,5598],{"class":175},[34,27696,27575],{"class":179},[34,27698,27699,27701,27703],{"class":36,"line":49},[34,27700,20373],{"class":1326},[34,27702,5598],{"class":175},[34,27704,27575],{"class":179},[34,27706,27707,27710,27713],{"class":36,"line":56},[34,27708,27709],{"class":1326}," avatarUrl",[34,27711,27712],{"class":175},"?:",[34,27714,27575],{"class":179},[34,27716,27717],{"class":36,"line":62},[34,27718,77],{"class":190},[15,27720,27721,27724],{},[31,27722,27723],{},"avatarUrl?"," 表示可能存在，也可能没有。",[15,27726,27727],{},"联合类型：",[24,27729,27731],{"className":27378,"code":27730,"language":27380,"meta":29,"style":29},"type Priority = 'low' | 'normal' | 'high'\n",[31,27732,27733],{"__ignoreMap":29},[34,27734,27735,27737,27740,27742,27745,27748,27751,27753],{"class":36,"line":37},[34,27736,27558],{"class":175},[34,27738,27739],{"class":395}," Priority",[34,27741,183],{"class":175},[34,27743,27744],{"class":186}," 'low'",[34,27746,27747],{"class":175}," |",[34,27749,27750],{"class":186}," 'normal'",[34,27752,27747],{"class":175},[34,27754,27755],{"class":186}," 'high'\n",[15,27757,27758],{},"这样可以限制优先级只能是这三个值。",[24,27760,27762],{"className":27378,"code":27761,"language":27380,"meta":29,"style":29},"const priority: Priority = 'urgent' \u002F\u002F 报错\n",[31,27763,27764],{"__ignoreMap":29},[34,27765,27766,27768,27770,27772,27774,27776,27779],{"class":36,"line":37},[34,27767,176],{"class":175},[34,27769,16401],{"class":179},[34,27771,5598],{"class":175},[34,27773,27739],{"class":395},[34,27775,183],{"class":175},[34,27777,27778],{"class":186}," 'urgent'",[34,27780,27781],{"class":301}," \u002F\u002F 报错\n",[19,27783,27785],{"id":27784},"_4-vue-中给-props-加类型","4. Vue 中给 props 加类型",[24,27787,27789],{"className":923,"code":27788,"language":925,"meta":29,"style":29},"\u003Cscript setup lang=\"ts\">\ntype Lesson = {\n id: string\n title: string\n minutes: number\n done: boolean\n}\n\ndefineProps\u003C{\n lesson: Lesson\n completed: boolean\n}>()\n\u003C\u002Fscript>\n",[31,27790,27791,27809,27819,27827,27835,27843,27851,27855,27859,27866,27876,27885,27890],{"__ignoreMap":29},[34,27792,27793,27795,27797,27799,27802,27804,27807],{"class":36,"line":37},[34,27794,932],{"class":190},[34,27796,1911],{"class":935},[34,27798,1914],{"class":395},[34,27800,27801],{"class":395}," lang",[34,27803,202],{"class":190},[34,27805,27806],{"class":186},"\"ts\"",[34,27808,939],{"class":190},[34,27810,27811,27813,27815,27817],{"class":36,"line":43},[34,27812,27558],{"class":175},[34,27814,27561],{"class":395},[34,27816,183],{"class":175},[34,27818,674],{"class":190},[34,27820,27821,27823,27825],{"class":36,"line":49},[34,27822,27570],{"class":1326},[34,27824,5598],{"class":175},[34,27826,27575],{"class":179},[34,27828,27829,27831,27833],{"class":36,"line":56},[34,27830,27580],{"class":1326},[34,27832,5598],{"class":175},[34,27834,27575],{"class":179},[34,27836,27837,27839,27841],{"class":36,"line":62},[34,27838,27589],{"class":1326},[34,27840,5598],{"class":175},[34,27842,27594],{"class":179},[34,27844,27845,27847,27849],{"class":36,"line":68},[34,27846,27599],{"class":1326},[34,27848,5598],{"class":175},[34,27850,27604],{"class":179},[34,27852,27853],{"class":36,"line":74},[34,27854,77],{"class":190},[34,27856,27857],{"class":36,"line":1971},[34,27858,53],{"emptyLinePlaceholder":52},[34,27860,27861,27863],{"class":36,"line":1976},[34,27862,12056],{"class":395},[34,27864,27865],{"class":190},"\u003C{\n",[34,27867,27868,27871,27873],{"class":36,"line":1985},[34,27869,27870],{"class":1326}," lesson",[34,27872,5598],{"class":175},[34,27874,27875],{"class":395}," Lesson\n",[34,27877,27878,27881,27883],{"class":36,"line":1990},[34,27879,27880],{"class":1326}," completed",[34,27882,5598],{"class":175},[34,27884,27604],{"class":179},[34,27886,27887],{"class":36,"line":1999},[34,27888,27889],{"class":190},"}>()\n",[34,27891,27892,27894,27896],{"class":36,"line":2022},[34,27893,1003],{"class":190},[34,27895,1911],{"class":935},[34,27897,939],{"class":190},[15,27899,27900],{},"这样父组件传错类型时，编辑器会提示。",[15,27902,13169,27903,27905],{},[31,27904,11915],{}," 应该是布尔值：",[24,27907,27909],{"className":923,"code":27908,"language":925,"meta":29,"style":29},"\u003CLessonCard :completed=\"true\" \u002F>\n",[31,27910,27911],{"__ignoreMap":29},[34,27912,27913,27915,27917,27919,27921,27923,27925,27927,27929],{"class":36,"line":37},[34,27914,932],{"class":190},[34,27916,3466],{"class":935},[34,27918,1621],{"class":190},[34,27920,11915],{"class":395},[34,27922,202],{"class":190},[34,27924,1608],{"class":186},[34,27926,758],{"class":179},[34,27928,1608],{"class":186},[34,27930,3469],{"class":190},[15,27932,27933],{},"如果写成：",[24,27935,27937],{"className":923,"code":27936,"language":925,"meta":29,"style":29},"\u003CLessonCard completed=\"true\" \u002F>\n",[31,27938,27939],{"__ignoreMap":29},[34,27940,27941,27943,27945,27948,27950,27953],{"class":36,"line":37},[34,27942,932],{"class":190},[34,27944,3466],{"class":935},[34,27946,27947],{"class":395}," completed",[34,27949,202],{"class":190},[34,27951,27952],{"class":186},"\"true\"",[34,27954,3469],{"class":190},[15,27956,27957,27958,27960],{},"这里传的是字符串 ",[31,27959,27952],{},"，TypeScript 和 Vue 工具会更容易发现问题。",[19,27962,27964],{"id":27963},"_5-给-emits-加类型","5. 给 emits 加类型",[24,27966,27968],{"className":27378,"code":27967,"language":27380,"meta":29,"style":29},"const emit = defineEmits\u003C{\n toggle: [id: string]\n remove: [id: string]\n}>()\n",[31,27969,27970,27982,28000,28017],{"__ignoreMap":29},[34,27971,27972,27974,27976,27978,27980],{"class":36,"line":37},[34,27973,176],{"class":175},[34,27975,12359],{"class":179},[34,27977,183],{"class":175},[34,27979,12364],{"class":395},[34,27981,27865],{"class":190},[34,27983,27984,27987,27989,27991,27993,27995,27998],{"class":36,"line":43},[34,27985,27986],{"class":1326}," toggle",[34,27988,5598],{"class":175},[34,27990,610],{"class":190},[34,27992,1805],{"class":395},[34,27994,8471],{"class":190},[34,27996,27997],{"class":179},"string",[34,27999,6569],{"class":190},[34,28001,28002,28005,28007,28009,28011,28013,28015],{"class":36,"line":49},[34,28003,28004],{"class":1326}," remove",[34,28006,5598],{"class":175},[34,28008,610],{"class":190},[34,28010,1805],{"class":395},[34,28012,8471],{"class":190},[34,28014,27997],{"class":179},[34,28016,6569],{"class":190},[34,28018,28019],{"class":36,"line":56},[34,28020,27889],{"class":190},[15,28022,28023],{},"调用：",[24,28025,28027],{"className":27378,"code":28026,"language":27380,"meta":29,"style":29},"emit('toggle', props.lesson.id)\n",[31,28028,28029],{"__ignoreMap":29},[34,28030,28031,28033,28035,28037],{"class":36,"line":37},[34,28032,12631],{"class":395},[34,28034,456],{"class":190},[34,28036,12370],{"class":186},[34,28038,12397],{"class":190},[15,28040,28041],{},"如果写错事件名：",[24,28043,28045],{"className":27378,"code":28044,"language":27380,"meta":29,"style":29},"emit('toggel', props.lesson.id)\n",[31,28046,28047],{"__ignoreMap":29},[34,28048,28049,28051,28053,28056],{"class":36,"line":37},[34,28050,12631],{"class":395},[34,28052,456],{"class":190},[34,28054,28055],{"class":186},"'toggel'",[34,28057,12397],{"class":190},[15,28059,28060],{},"会提示错误。",[15,28062,28063],{},"如果参数类型错：",[24,28065,28067],{"className":27378,"code":28066,"language":27380,"meta":29,"style":29},"emit('toggle', 123)\n",[31,28068,28069],{"__ignoreMap":29},[34,28070,28071,28073,28075,28077,28079,28082],{"class":36,"line":37},[34,28072,12631],{"class":395},[34,28074,456],{"class":190},[34,28076,12370],{"class":186},[34,28078,616],{"class":190},[34,28080,28081],{"class":179},"123",[34,28083,6111],{"class":190},[15,28085,28086],{},"也会提示错误。",[19,28088,28090],{"id":28089},"_6-给-ref-加类型","6. 给 ref 加类型",[15,28092,28093],{},"多数时候 TypeScript 能推断：",[24,28095,28097],{"className":27378,"code":28096,"language":27380,"meta":29,"style":29},"const count = ref(0) \u002F\u002F Ref\u003Cnumber>\nconst title = ref('') \u002F\u002F Ref\u003Cstring>\n",[31,28098,28099,28118],{"__ignoreMap":29},[34,28100,28101,28103,28105,28107,28109,28111,28113,28115],{"class":36,"line":37},[34,28102,176],{"class":175},[34,28104,448],{"class":179},[34,28106,183],{"class":175},[34,28108,453],{"class":395},[34,28110,456],{"class":190},[34,28112,459],{"class":179},[34,28114,1330],{"class":190},[34,28116,28117],{"class":301},"\u002F\u002F Ref\u003Cnumber>\n",[34,28119,28120,28122,28124,28126,28128,28130,28132,28134],{"class":36,"line":43},[34,28121,176],{"class":175},[34,28123,278],{"class":179},[34,28125,183],{"class":175},[34,28127,453],{"class":395},[34,28129,456],{"class":190},[34,28131,7400],{"class":186},[34,28133,1330],{"class":190},[34,28135,28136],{"class":301},"\u002F\u002F Ref\u003Cstring>\n",[15,28138,28139],{},"需要手动写类型的情况：",[24,28141,28143],{"className":27378,"code":28142,"language":27380,"meta":29,"style":29},"type Lesson = {\n id: string\n title: string\n}\n\nconst lesson = ref\u003CLesson | null>(null)\n",[31,28144,28145,28155,28163,28171,28175,28179],{"__ignoreMap":29},[34,28146,28147,28149,28151,28153],{"class":36,"line":37},[34,28148,27558],{"class":175},[34,28150,27561],{"class":395},[34,28152,183],{"class":175},[34,28154,674],{"class":190},[34,28156,28157,28159,28161],{"class":36,"line":43},[34,28158,27570],{"class":1326},[34,28160,5598],{"class":175},[34,28162,27575],{"class":179},[34,28164,28165,28167,28169],{"class":36,"line":49},[34,28166,27580],{"class":1326},[34,28168,5598],{"class":175},[34,28170,27575],{"class":179},[34,28172,28173],{"class":36,"line":56},[34,28174,77],{"class":190},[34,28176,28177],{"class":36,"line":62},[34,28178,53],{"emptyLinePlaceholder":52},[34,28180,28181,28183,28185,28187,28189,28191,28194,28196,28198,28201,28203],{"class":36,"line":68},[34,28182,176],{"class":175},[34,28184,824],{"class":179},[34,28186,183],{"class":175},[34,28188,453],{"class":395},[34,28190,932],{"class":190},[34,28192,28193],{"class":395},"Lesson",[34,28195,27747],{"class":175},[34,28197,593],{"class":179},[34,28199,28200],{"class":190},">(",[34,28202,5419],{"class":179},[34,28204,6111],{"class":190},[15,28206,28207,28208,5916],{},"为什么要 ",[31,28209,28210],{},"Lesson | null",[15,28212,28213,28214,28216,28217,347],{},"因为一开始可能还没请求到课程详情，所以是 ",[31,28215,5419],{},"。请求成功后才是 ",[31,28218,28193],{},[15,28220,28221],{},"模板中要处理：",[24,28223,28225],{"className":923,"code":28224,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"!lesson\">加载中...\u003C\u002Fp>\n\u003Carticle v-else>\n \u003Ch1>{{ lesson.title }}\u003C\u002Fh1>\n\u003C\u002Farticle>\n",[31,28226,28227,28251,28261,28265],{"__ignoreMap":29},[34,28228,28229,28231,28233,28235,28237,28239,28241,28243,28245,28247,28249],{"class":36,"line":37},[34,28230,932],{"class":190},[34,28232,15],{"class":935},[34,28234,2168],{"class":175},[34,28236,202],{"class":190},[34,28238,1608],{"class":186},[34,28240,5504],{"class":175},[34,28242,1499],{"class":190},[34,28244,1608],{"class":186},[34,28246,2180],{"class":190},[34,28248,15],{"class":935},[34,28250,939],{"class":190},[34,28252,28253,28255,28257,28259],{"class":36,"line":43},[34,28254,932],{"class":190},[34,28256,947],{"class":935},[34,28258,2218],{"class":175},[34,28260,939],{"class":190},[34,28262,28263],{"class":36,"line":49},[34,28264,20764],{"class":190},[34,28266,28267,28269,28271],{"class":36,"line":56},[34,28268,1003],{"class":190},[34,28270,947],{"class":935},[34,28272,939],{"class":190},[19,28274,28276],{"id":28275},"_7-api-数据类型","7. API 数据类型",[24,28278,28280],{"className":27378,"code":28279,"language":27380,"meta":29,"style":29},"type ApiLesson = {\n id: string\n title: string\n completed: boolean\n}\n\nasync function getLessons(): Promise\u003CApiLesson[]> {\n const response = await fetch('\u002Fapi\u002Flessons')\n\n if (!response.ok) {\n throw new Error('课程加载失败')\n }\n\n return response.json()\n}\n",[31,28281,28282,28293,28301,28309,28317,28321,28325,28348,28366,28370,28380,28394,28398,28402,28412],{"__ignoreMap":29},[34,28283,28284,28286,28289,28291],{"class":36,"line":37},[34,28285,27558],{"class":175},[34,28287,28288],{"class":395}," ApiLesson",[34,28290,183],{"class":175},[34,28292,674],{"class":190},[34,28294,28295,28297,28299],{"class":36,"line":43},[34,28296,27570],{"class":1326},[34,28298,5598],{"class":175},[34,28300,27575],{"class":179},[34,28302,28303,28305,28307],{"class":36,"line":49},[34,28304,27580],{"class":1326},[34,28306,5598],{"class":175},[34,28308,27575],{"class":179},[34,28310,28311,28313,28315],{"class":36,"line":56},[34,28312,27880],{"class":1326},[34,28314,5598],{"class":175},[34,28316,27604],{"class":179},[34,28318,28319],{"class":36,"line":62},[34,28320,77],{"class":190},[34,28322,28323],{"class":36,"line":68},[34,28324,53],{"emptyLinePlaceholder":52},[34,28326,28327,28329,28331,28333,28336,28338,28340,28342,28345],{"class":36,"line":74},[34,28328,5217],{"class":175},[34,28330,2850],{"class":175},[34,28332,24810],{"class":395},[34,28334,28335],{"class":190},"()",[34,28337,5598],{"class":175},[34,28339,5180],{"class":395},[34,28341,932],{"class":190},[34,28343,28344],{"class":395},"ApiLesson",[34,28346,28347],{"class":190},"[]> {\n",[34,28349,28350,28352,28354,28356,28358,28360,28362,28364],{"class":36,"line":1971},[34,28351,5298],{"class":175},[34,28353,5301],{"class":179},[34,28355,183],{"class":175},[34,28357,5306],{"class":175},[34,28359,5309],{"class":395},[34,28361,456],{"class":190},[34,28363,23729],{"class":186},[34,28365,6111],{"class":190},[34,28367,28368],{"class":36,"line":1976},[34,28369,53],{"emptyLinePlaceholder":52},[34,28371,28372,28374,28376,28378],{"class":36,"line":1985},[34,28373,7000],{"class":175},[34,28375,1840],{"class":190},[34,28377,5504],{"class":175},[34,28379,5507],{"class":190},[34,28381,28382,28384,28386,28388,28390,28392],{"class":36,"line":1990},[34,28383,24032],{"class":175},[34,28385,5177],{"class":175},[34,28387,5517],{"class":395},[34,28389,456],{"class":190},[34,28391,24240],{"class":186},[34,28393,6111],{"class":190},[34,28395,28396],{"class":36,"line":1999},[34,28397,5631],{"class":190},[34,28399,28400],{"class":36,"line":2022},[34,28401,53],{"emptyLinePlaceholder":52},[34,28403,28404,28406,28408,28410],{"class":36,"line":2036},[34,28405,1510],{"class":175},[34,28407,5329],{"class":190},[34,28409,5332],{"class":395},[34,28411,6374],{"class":190},[34,28413,28414],{"class":36,"line":2603},[34,28415,77],{"class":190},[15,28417,28418,28421],{},[31,28419,28420],{},"Promise\u003CApiLesson[]>"," 表示这个异步函数最终返回课程数组。",[15,28423,28424],{},"注意：TypeScript 只能检查你写的代码，不能保证后端真实返回一定符合类型。后端返回脏数据时，TypeScript 不会自动拦截。真实项目如果需要运行时校验，可以使用 zod 等库。",[19,28426,28428],{"id":28427},"_8-类型放在哪里","8. 类型放在哪里",[15,28430,28431],{},"小项目可以写在组件里：",[24,28433,28435],{"className":27378,"code":28434,"language":27380,"meta":29,"style":29},"type Lesson = { ... }\n",[31,28436,28437],{"__ignoreMap":29},[34,28438,28439,28441,28443,28445,28447,28449],{"class":36,"line":37},[34,28440,27558],{"class":175},[34,28442,27561],{"class":395},[34,28444,183],{"class":175},[34,28446,1164],{"class":190},[34,28448,1167],{"class":175},[34,28450,6564],{"class":190},[15,28452,28453],{},"多个文件共用时，可以放：",[24,28455,28458],{"className":28456,"code":28457,"language":2638,"meta":29},[2636],"src\u002Ftypes\u002Flesson.ts\n",[31,28459,28457],{"__ignoreMap":29},[24,28461,28463],{"className":27378,"code":28462,"language":27380,"meta":29,"style":29},"\u002F\u002F src\u002Ftypes\u002Flesson.ts\nexport type Lesson = {\n id: string\n title: string\n minutes: number\n}\n",[31,28464,28465,28470,28482,28490,28498,28506],{"__ignoreMap":29},[34,28466,28467],{"class":36,"line":37},[34,28468,28469],{"class":301},"\u002F\u002F src\u002Ftypes\u002Flesson.ts\n",[34,28471,28472,28474,28476,28478,28480],{"class":36,"line":43},[34,28473,2825],{"class":175},[34,28475,11503],{"class":175},[34,28477,27561],{"class":395},[34,28479,183],{"class":175},[34,28481,674],{"class":190},[34,28483,28484,28486,28488],{"class":36,"line":49},[34,28485,27570],{"class":1326},[34,28487,5598],{"class":175},[34,28489,27575],{"class":179},[34,28491,28492,28494,28496],{"class":36,"line":56},[34,28493,27580],{"class":1326},[34,28495,5598],{"class":175},[34,28497,27575],{"class":179},[34,28499,28500,28502,28504],{"class":36,"line":62},[34,28501,27589],{"class":1326},[34,28503,5598],{"class":175},[34,28505,27594],{"class":179},[34,28507,28508],{"class":36,"line":68},[34,28509,77],{"class":190},[15,28511,3586],{},[24,28513,28515],{"className":27378,"code":28514,"language":27380,"meta":29,"style":29},"import type { Lesson } from '..\u002Ftypes\u002Flesson'\n",[31,28516,28517],{"__ignoreMap":29},[34,28518,28519,28521,28523,28526,28528],{"class":36,"line":37},[34,28520,426],{"class":175},[34,28522,11503],{"class":175},[34,28524,28525],{"class":190}," { Lesson } ",[34,28527,432],{"class":175},[34,28529,28530],{"class":186}," '..\u002Ftypes\u002Flesson'\n",[15,28532,28533,28536],{},[31,28534,28535],{},"import type"," 表示只导入类型，不导入运行时代码。",[19,28538,28540],{"id":28539},"_9-生产级代码习惯","9. 生产级代码习惯",[15,28542,28543],{},"生产级不是指代码很复杂，而是指它能被维护。",[15,28545,28546],{},"好习惯：",[105,28548,28549,28552,28555,28558,28561,28564,28567,28570,28573],{},[108,28550,28551],{},"文件名和组件名清晰。",[108,28553,28554],{},"组件职责单一。",[108,28556,28557],{},"状态放在合适位置。",[108,28559,28560],{},"API 请求有错误处理。",[108,28562,28563],{},"表单有校验。",[108,28565,28566],{},"列表有空状态。",[108,28568,28569],{},"关键逻辑有测试。",[108,28571,28572],{},"不把所有逻辑塞进一个巨大组件。",[108,28574,28575],{},"不依赖“我记得这里不会出错”。",[19,28577,28579],{"id":28578},"_10-目录建议","10. 目录建议",[24,28581,28584],{"className":28582,"code":28583,"language":2638,"meta":29},[2636],"src\u002F\n assets\u002F\n components\u002F\n composables\u002F\n data\u002F\n router\u002F\n services\u002F\n stores\u002F\n types\u002F\n views\u002F\n",[31,28585,28583],{"__ignoreMap":29},[15,28587,28588],{},"职责：",[105,28590,28591,28597,28603,28609,28615,28621],{},[108,28592,28593,28596],{},[31,28594,28595],{},"components","：可复用组件。",[108,28598,28599,28602],{},[31,28600,28601],{},"views","：页面组件。",[108,28604,28605,28608],{},[31,28606,28607],{},"composables","：组合式函数。",[108,28610,28611,28614],{},[31,28612,28613],{},"stores","：Pinia。",[108,28616,28617,28620],{},[31,28618,28619],{},"services","：请求封装。",[108,28622,28623,28626],{},[31,28624,28625],{},"types","：共享类型。",[19,28628,28630],{"id":28629},"_11-初学者引入-typescript-的节奏","11. 初学者引入 TypeScript 的节奏",[15,28632,28633],{},"不要一开始就被类型吓住。推荐顺序：",[2230,28635,28636,28639,28647,28650,28653],{},[108,28637,28638],{},"先用 JavaScript 学会 Vue 核心。",[108,28640,28641,28642,146,28644,347],{},"给数据模型写类型，例如 ",[31,28643,28193],{},[31,28645,28646],{},"Todo",[108,28648,28649],{},"给 props 和 emits 加类型。",[108,28651,28652],{},"给 API 函数返回值加类型。",[108,28654,28655],{},"给复杂组合式函数加类型。",[15,28657,28658],{},"TypeScript 是工具，不是学习 Vue 的前置障碍。",[19,28660,21417],{"id":21416},[253,28662,28664],{"id":28663},"错误一把类型当运行时变量用","错误一：把类型当运行时变量用",[24,28666,28668],{"className":27378,"code":28667,"language":27380,"meta":29,"style":29},"type Lesson = { title: string }\n\nconsole.log(Lesson) \u002F\u002F 错误，Lesson 是类型，不是值\n",[31,28669,28670,28688,28692],{"__ignoreMap":29},[34,28671,28672,28674,28676,28678,28680,28682,28684,28686],{"class":36,"line":37},[34,28673,27558],{"class":175},[34,28675,27561],{"class":395},[34,28677,183],{"class":175},[34,28679,1164],{"class":190},[34,28681,1665],{"class":1326},[34,28683,5598],{"class":175},[34,28685,27450],{"class":179},[34,28687,6564],{"class":190},[34,28689,28690],{"class":36,"line":43},[34,28691,53],{"emptyLinePlaceholder":52},[34,28693,28694,28697,28699,28701,28703,28705,28707],{"class":36,"line":49},[34,28695,28696],{"class":395},"console",[34,28698,2547],{"class":190},[34,28700,885],{"class":395},[34,28702,456],{"class":190},[34,28704,28193],{"class":395},[34,28706,1330],{"class":190},[34,28708,28709],{"class":301},"\u002F\u002F 错误，Lesson 是类型，不是值\n",[15,28711,28712],{},"类型只在开发和编译时存在，运行时没有。",[253,28714,28716,28717],{"id":28715},"错误二过度使用-any","错误二：过度使用 ",[31,28718,28719],{},"any",[24,28721,28723],{"className":27378,"code":28722,"language":27380,"meta":29,"style":29},"const lesson: any = {}\n",[31,28724,28725],{"__ignoreMap":29},[34,28726,28727,28729,28731,28733,28736,28738],{"class":36,"line":37},[34,28728,176],{"class":175},[34,28730,824],{"class":179},[34,28732,5598],{"class":175},[34,28734,28735],{"class":179}," any",[34,28737,183],{"class":175},[34,28739,28740],{"class":190}," {}\n",[15,28742,28743,28745],{},[31,28744,28719],{}," 会关闭类型检查。初学时可以偶尔救急，但不要大量使用。",[253,28747,28749],{"id":28748},"错误三忽略-null","错误三：忽略 null",[24,28751,28753],{"className":27378,"code":28752,"language":27380,"meta":29,"style":29},"const lesson = ref\u003CLesson | null>(null)\n\nconsole.log(lesson.value.title) \u002F\u002F 可能报错，因为 lesson.value 可能是 null\n",[31,28754,28755,28779,28783],{"__ignoreMap":29},[34,28756,28757,28759,28761,28763,28765,28767,28769,28771,28773,28775,28777],{"class":36,"line":37},[34,28758,176],{"class":175},[34,28760,824],{"class":179},[34,28762,183],{"class":175},[34,28764,453],{"class":395},[34,28766,932],{"class":190},[34,28768,28193],{"class":395},[34,28770,27747],{"class":175},[34,28772,593],{"class":179},[34,28774,28200],{"class":190},[34,28776,5419],{"class":179},[34,28778,6111],{"class":190},[34,28780,28781],{"class":36,"line":43},[34,28782,53],{"emptyLinePlaceholder":52},[34,28784,28785,28787,28789,28792],{"class":36,"line":49},[34,28786,882],{"class":190},[34,28788,885],{"class":395},[34,28790,28791],{"class":190},"(lesson.value.title) ",[34,28793,28794],{"class":301},"\u002F\u002F 可能报错，因为 lesson.value 可能是 null\n",[15,28796,4866],{},[24,28798,28800],{"className":27378,"code":28799,"language":27380,"meta":29,"style":29},"if (lesson.value) {\n console.log(lesson.value.title)\n}\n",[31,28801,28802,28809,28818],{"__ignoreMap":29},[34,28803,28804,28806],{"class":36,"line":37},[34,28805,2069],{"class":175},[34,28807,28808],{"class":190}," (lesson.value) {\n",[34,28810,28811,28813,28815],{"class":36,"line":43},[34,28812,1812],{"class":190},[34,28814,885],{"class":395},[34,28816,28817],{"class":190},"(lesson.value.title)\n",[34,28819,28820],{"class":36,"line":49},[34,28821,77],{"class":190},[19,28823,5693],{"id":5693},[2230,28825,28826,28833,28838,28845,28850],{},[108,28827,27242,28828,258,28830,28832],{},[31,28829,28193],{},[31,28831,28646],{}," 写类型。",[108,28834,2892,28835,28837],{},[31,28836,3466],{}," 的 props 改成 TypeScript 写法。",[108,28839,27242,28840,258,28842,28844],{},[31,28841,11935],{},[31,28843,8035],{}," emits 加类型。",[108,28846,5803,28847,347],{},[31,28848,28849],{},"getLessons(): Promise\u003CLesson[]>",[108,28851,28852],{},"回答：TypeScript 能防止哪些错误？不能防止哪些错误？",[79,28854,28855],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}",{"title":29,"searchDepth":43,"depth":43,"links":28857},[28858,28859,28860,28861,28862,28863,28864,28865,28866,28867,28868,28869,28875],{"id":27311,"depth":43,"text":27312},{"id":27433,"depth":43,"text":27434},{"id":27668,"depth":43,"text":27669},{"id":27784,"depth":43,"text":27785},{"id":27963,"depth":43,"text":27964},{"id":28089,"depth":43,"text":28090},{"id":28275,"depth":43,"text":28276},{"id":28427,"depth":43,"text":28428},{"id":28539,"depth":43,"text":28540},{"id":28578,"depth":43,"text":28579},{"id":28629,"depth":43,"text":28630},{"id":21416,"depth":43,"text":21417,"children":28870},[28871,28872,28874],{"id":28663,"depth":49,"text":28664},{"id":28715,"depth":49,"text":28873},"错误二：过度使用 any",{"id":28748,"depth":49,"text":28749},{"id":5693,"depth":43,"text":5693},{"order":1990},"\u002Fsource\u002Fcode\u002Fvue\u002F11-typescript-production",{"title":27297,"description":27305},"source\u002Fcode\u002Fvue\u002F11-typescript-production","85woldebFyvlgo5LVjJm4ySPNvUt5y-AnLMdgWsqBOI",{"id":28882,"title":28883,"body":28884,"description":28891,"extension":85,"meta":29902,"navigation":52,"path":29903,"seo":29904,"stem":29905,"__hash__":29906},"notes\u002Fsource\u002Fcode\u002Fvue\u002F12-performance-a11y-security.md","12 性能、可访问性和安全",{"type":8,"value":28885,"toc":29883},[28886,28889,28892,28899,28903,28906,28923,28926,28930,28932,28963,28965,29056,29058,29089,29094,29098,29104,29107,29118,29121,29125,29128,29167,29170,29174,29177,29191,29194,29198,29201,29215,29218,29222,29224,29244,29246,29302,29304,29315,29319,29321,29350,29352,29388,29393,29407,29412,29416,29449,29454,29457,29459,29465,29467,29473,29477,29480,29510,29513,29581,29584,29588,29606,29609,29638,29641,29647,29650,29679,29684,29687,29698,29703,29707,29710,29724,29727,29734,29745,29747,29758,29761,29765,29767,29781,29790,29793,29797,29800,29814,29817,29834,29837,29853,29855,29880],[11,28887,28883],{"id":28888},"_12-性能可访问性和安全",[15,28890,28891],{},"页面能跑只是第一步。真实项目还要考虑速度、可访问性和安全。初学阶段不必追求极限优化，但必须避免明显问题。",[15,28893,28894,28895,28898],{},"本章目标：你能识别常见性能浪费，能写基础可访问表单，能知道 ",[31,28896,28897],{},"v-html"," 和前端密钥为什么危险。",[19,28900,28902],{"id":28901},"_1-性能优化先从不浪费开始","1. 性能优化先从不浪费开始",[15,28904,28905],{},"常见浪费：",[105,28907,28908,28911,28914,28917,28920],{},[108,28909,28910],{},"模板中写复杂计算。",[108,28912,28913],{},"列表没有稳定 key。",[108,28915,28916],{},"一次性渲染超大列表。",[108,28918,28919],{},"首页加载所有页面代码。",[108,28921,28922],{},"状态放得太高，导致大范围组件更新。",[15,28924,28925],{},"性能不是一开始就上复杂工具，而是先写清楚、写稳定。",[19,28927,28929],{"id":28928},"_2-用-computed-避免重复计算","2. 用 computed 避免重复计算",[15,28931,7500],{},[24,28933,28935],{"className":923,"code":28934,"language":925,"meta":29,"style":29},"\u003Cp>{{ tasks.filter(task => !task.done).length }}\u003C\u002Fp>\n\u003Cp>{{ tasks.filter(task => task.done).length }}\u003C\u002Fp>\n",[31,28936,28937,28950],{"__ignoreMap":29},[34,28938,28939,28941,28943,28946,28948],{"class":36,"line":37},[34,28940,932],{"class":190},[34,28942,15],{"class":935},[34,28944,28945],{"class":190},">{{ tasks.filter(task => !task.done).length }}\u003C\u002F",[34,28947,15],{"class":935},[34,28949,939],{"class":190},[34,28951,28952,28954,28956,28959,28961],{"class":36,"line":43},[34,28953,932],{"class":190},[34,28955,15],{"class":935},[34,28957,28958],{"class":190},">{{ tasks.filter(task => task.done).length }}\u003C\u002F",[34,28960,15],{"class":935},[34,28962,939],{"class":190},[15,28964,7527],{},[24,28966,28968],{"className":166,"code":28967,"language":168,"meta":29,"style":29},"const remainingCount = computed(() => {\n return tasks.value.filter((task) => !task.done).length\n})\n\nconst completedCount = computed(() => {\n return tasks.value.filter((task) => task.done).length\n})\n",[31,28969,28970,28986,29008,29012,29016,29032,29052],{"__ignoreMap":29},[34,28971,28972,28974,28976,28978,28980,28982,28984],{"class":36,"line":37},[34,28973,176],{"class":175},[34,28975,11203],{"class":179},[34,28977,183],{"class":175},[34,28979,8935],{"class":395},[34,28981,7682],{"class":190},[34,28983,1333],{"class":175},[34,28985,674],{"class":190},[34,28987,28988,28990,28992,28994,28996,28998,29000,29002,29004,29006],{"class":36,"line":43},[34,28989,1510],{"class":175},[34,28991,6972],{"class":190},[34,28993,1300],{"class":395},[34,28995,1323],{"class":190},[34,28997,1327],{"class":1326},[34,28999,1330],{"class":190},[34,29001,1333],{"class":175},[34,29003,1336],{"class":175},[34,29005,11234],{"class":190},[34,29007,10311],{"class":179},[34,29009,29010],{"class":36,"line":49},[34,29011,6153],{"class":190},[34,29013,29014],{"class":36,"line":56},[34,29015,53],{"emptyLinePlaceholder":52},[34,29017,29018,29020,29022,29024,29026,29028,29030],{"class":36,"line":62},[34,29019,176],{"class":175},[34,29021,10279],{"class":179},[34,29023,183],{"class":175},[34,29025,8935],{"class":395},[34,29027,7682],{"class":190},[34,29029,1333],{"class":175},[34,29031,674],{"class":190},[34,29033,29034,29036,29038,29040,29042,29044,29046,29048,29050],{"class":36,"line":68},[34,29035,1510],{"class":175},[34,29037,6972],{"class":190},[34,29039,1300],{"class":395},[34,29041,1323],{"class":190},[34,29043,1327],{"class":1326},[34,29045,1330],{"class":190},[34,29047,1333],{"class":175},[34,29049,10308],{"class":190},[34,29051,10311],{"class":179},[34,29053,29054],{"class":36,"line":74},[34,29055,6153],{"class":190},[15,29057,1589],{},[24,29059,29061],{"className":923,"code":29060,"language":925,"meta":29,"style":29},"\u003Cp>未完成 {{ remainingCount }}\u003C\u002Fp>\n\u003Cp>已完成 {{ completedCount }}\u003C\u002Fp>\n",[31,29062,29063,29076],{"__ignoreMap":29},[34,29064,29065,29067,29069,29072,29074],{"class":36,"line":37},[34,29066,932],{"class":190},[34,29068,15],{"class":935},[34,29070,29071],{"class":190},">未完成 {{ remainingCount }}\u003C\u002F",[34,29073,15],{"class":935},[34,29075,939],{"class":190},[34,29077,29078,29080,29082,29085,29087],{"class":36,"line":43},[34,29079,932],{"class":190},[34,29081,15],{"class":935},[34,29083,29084],{"class":190},">已完成 {{ completedCount }}\u003C\u002F",[34,29086,15],{"class":935},[34,29088,939],{"class":190},[15,29090,29091,29093],{},[31,29092,149],{}," 让逻辑有名字，并且根据依赖缓存结果。",[19,29095,29097],{"id":29096},"_3-大列表不要一次性硬渲染","3. 大列表不要一次性硬渲染",[15,29099,29100,29101,29103],{},"如果列表只有几十项，普通 ",[31,29102,1534],{}," 没问题。",[15,29105,29106],{},"如果有几千上万项：",[105,29108,29109,29112,29115],{},[108,29110,29111],{},"考虑分页。",[108,29113,29114],{},"考虑搜索筛选。",[108,29116,29117],{},"考虑虚拟列表库。",[15,29119,29120],{},"不要一开始就把所有数据都塞到页面上。",[19,29122,29124],{"id":29123},"_4-路由懒加载","4. 路由懒加载",[15,29126,29127],{},"页面多时：",[24,29129,29131],{"className":166,"code":29130,"language":168,"meta":29,"style":29},"{\n path: '\u002Fsettings',\n component: () => import('..\u002Fviews\u002FSettingsView.vue')\n}\n",[31,29132,29133,29137,29147,29163],{"__ignoreMap":29},[34,29134,29135],{"class":36,"line":37},[34,29136,20357],{"class":190},[34,29138,29139,29141,29143,29145],{"class":36,"line":43},[34,29140,20362],{"class":395},[34,29142,8471],{"class":190},[34,29144,21112],{"class":186},[34,29146,684],{"class":190},[34,29148,29149,29151,29153,29155,29157,29159,29161],{"class":36,"line":49},[34,29150,20384],{"class":395},[34,29152,21132],{"class":190},[34,29154,1333],{"class":175},[34,29156,21137],{"class":175},[34,29158,456],{"class":190},[34,29160,21142],{"class":186},[34,29162,6111],{"class":190},[34,29164,29165],{"class":36,"line":56},[34,29166,77],{"class":190},[15,29168,29169],{},"访问时才加载。这样首页不必下载用户暂时不用的页面。",[19,29171,29173],{"id":29172},"_5-图片和资源","5. 图片和资源",[15,29175,29176],{},"常见建议：",[105,29178,29179,29182,29185,29188],{},[108,29180,29181],{},"图片尺寸不要远大于显示尺寸。",[108,29183,29184],{},"大图使用压缩格式。",[108,29186,29187],{},"列表图片懒加载。",[108,29189,29190],{},"不要把无关大资源放进首屏。",[15,29192,29193],{},"Vue 本身不能替你解决所有资源问题，工程上也要注意。",[19,29195,29197],{"id":29196},"_6-可访问性是什么","6. 可访问性是什么",[15,29199,29200],{},"可访问性是让更多用户能使用你的应用，包括：",[105,29202,29203,29206,29209,29212],{},[108,29204,29205],{},"使用键盘的人。",[108,29207,29208],{},"使用屏幕阅读器的人。",[108,29210,29211],{},"视力较弱的人。",[108,29213,29214],{},"临时受限的人，例如鼠标坏了。",[15,29216,29217],{},"可访问性不是额外功能，而是基础质量。",[19,29219,29221],{"id":29220},"_7-表单必须有-label","7. 表单必须有 label",[15,29223,7500],{},[24,29225,29227],{"className":923,"code":29226,"language":925,"meta":29,"style":29},"\u003Cinput placeholder=\"邮箱\">\n",[31,29228,29229],{"__ignoreMap":29},[34,29230,29231,29233,29235,29237,29239,29242],{"class":36,"line":37},[34,29232,932],{"class":190},[34,29234,7429],{"class":935},[34,29236,11392],{"class":395},[34,29238,202],{"class":190},[34,29240,29241],{"class":186},"\"邮箱\"",[34,29243,939],{"class":190},[15,29245,7527],{},[24,29247,29249],{"className":923,"code":29248,"language":925,"meta":29,"style":29},"\u003Clabel for=\"email\">邮箱\u003C\u002Flabel>\n\u003Cinput id=\"email\" v-model=\"email\" type=\"email\">\n",[31,29250,29251,29271],{"__ignoreMap":29},[34,29252,29253,29255,29257,29259,29261,29264,29267,29269],{"class":36,"line":37},[34,29254,932],{"class":190},[34,29256,11490],{"class":935},[34,29258,16244],{"class":395},[34,29260,202],{"class":190},[34,29262,29263],{"class":186},"\"email\"",[34,29265,29266],{"class":190},">邮箱\u003C\u002F",[34,29268,11490],{"class":935},[34,29270,939],{"class":190},[34,29272,29273,29275,29277,29279,29281,29283,29285,29287,29289,29292,29294,29296,29298,29300],{"class":36,"line":43},[34,29274,932],{"class":190},[34,29276,7429],{"class":935},[34,29278,6051],{"class":395},[34,29280,202],{"class":190},[34,29282,29263],{"class":186},[34,29284,7432],{"class":395},[34,29286,202],{"class":190},[34,29288,1608],{"class":186},[34,29290,29291],{"class":190},"email",[34,29293,1608],{"class":186},[34,29295,11503],{"class":395},[34,29297,202],{"class":190},[34,29299,29263],{"class":186},[34,29301,939],{"class":190},[15,29303,7819],{},[105,29305,29306,29309,29312],{},[108,29307,29308],{},"屏幕阅读器能读出输入框含义。",[108,29310,29311],{},"点击 label 可以聚焦输入框。",[108,29313,29314],{},"placeholder 不是稳定标签。",[19,29316,29318],{"id":29317},"_8-按钮就用-button","8. 按钮就用 button",[15,29320,7500],{},[24,29322,29324],{"className":923,"code":29323,"language":925,"meta":29,"style":29},"\u003Cdiv @click=\"submit\">提交\u003C\u002Fdiv>\n",[31,29325,29326],{"__ignoreMap":29},[34,29327,29328,29330,29332,29334,29336,29338,29340,29342,29344,29346,29348],{"class":36,"line":37},[34,29329,932],{"class":190},[34,29331,16016],{"class":935},[34,29333,9628],{"class":190},[34,29335,9631],{"class":395},[34,29337,202],{"class":190},[34,29339,1608],{"class":186},[34,29341,9795],{"class":190},[34,29343,1608],{"class":186},[34,29345,7985],{"class":190},[34,29347,16016],{"class":935},[34,29349,939],{"class":190},[15,29351,7527],{},[24,29353,29355],{"className":923,"code":29354,"language":925,"meta":29,"style":29},"\u003Cbutton type=\"button\" @click=\"submit\">提交\u003C\u002Fbutton>\n",[31,29356,29357],{"__ignoreMap":29},[34,29358,29359,29361,29363,29365,29367,29370,29372,29374,29376,29378,29380,29382,29384,29386],{"class":36,"line":37},[34,29360,932],{"class":190},[34,29362,2004],{"class":935},[34,29364,11503],{"class":395},[34,29366,202],{"class":190},[34,29368,29369],{"class":186},"\"button\"",[34,29371,9628],{"class":190},[34,29373,9631],{"class":395},[34,29375,202],{"class":190},[34,29377,1608],{"class":186},[34,29379,9795],{"class":190},[34,29381,1608],{"class":186},[34,29383,7985],{"class":190},[34,29385,2004],{"class":935},[34,29387,939],{"class":190},[15,29389,29390,29392],{},[31,29391,2004],{}," 天然支持：",[105,29394,29395,29398,29401,29404],{},[108,29396,29397],{},"键盘操作。",[108,29399,29400],{},"焦点。",[108,29402,29403],{},"语义。",[108,29405,29406],{},"disabled 状态。",[15,29408,5792,29409,29411],{},[31,29410,16016],{}," 冒充按钮，需要自己补很多行为，容易漏。",[19,29413,29415],{"id":29414},"_9-错误提示","9. 错误提示",[24,29417,29419],{"className":923,"code":29418,"language":925,"meta":29,"style":29},"\u003Cp v-if=\"error\" role=\"alert\">{{ error }}\u003C\u002Fp>\n",[31,29420,29421],{"__ignoreMap":29},[34,29422,29423,29425,29427,29429,29431,29433,29435,29437,29439,29441,29443,29445,29447],{"class":36,"line":37},[34,29424,932],{"class":190},[34,29426,15],{"class":935},[34,29428,2168],{"class":175},[34,29430,202],{"class":190},[34,29432,1608],{"class":186},[34,29434,2200],{"class":190},[34,29436,1608],{"class":186},[34,29438,2345],{"class":395},[34,29440,202],{"class":190},[34,29442,16113],{"class":186},[34,29444,2205],{"class":190},[34,29446,15],{"class":935},[34,29448,939],{"class":190},[15,29450,29451,29453],{},[31,29452,16343],{}," 告诉辅助技术这里是重要变化。",[15,29455,29456],{},"表单错误应该具体：",[15,29458,7500],{},[24,29460,29463],{"className":29461,"code":29462,"language":2638,"meta":29},[2636],"输入错误\n",[31,29464,29462],{"__ignoreMap":29},[15,29466,7527],{},[24,29468,29471],{"className":29469,"code":29470,"language":2638,"meta":29},[2636],"标题不能为空\n标题不能超过 30 个字符\n",[31,29472,29470],{"__ignoreMap":29},[19,29474,29476],{"id":29475},"_10-焦点样式","10. 焦点样式",[15,29478,29479],{},"不要随便去掉：",[24,29481,29483],{"className":15873,"code":29482,"language":15875,"meta":29,"style":29},"button:focus {\n outline: none;\n}\n",[31,29484,29485,29494,29506],{"__ignoreMap":29},[34,29486,29487,29489,29492],{"class":36,"line":37},[34,29488,2004],{"class":935},[34,29490,29491],{"class":395},":focus",[34,29493,674],{"class":190},[34,29495,29496,29499,29501,29504],{"class":36,"line":43},[34,29497,29498],{"class":179}," outline",[34,29500,8471],{"class":190},[34,29502,29503],{"class":179},"none",[34,29505,191],{"class":190},[34,29507,29508],{"class":36,"line":49},[34,29509,77],{"class":190},[15,29511,29512],{},"如果你移除默认焦点样式，必须提供新的明显焦点样式：",[24,29514,29516],{"className":15873,"code":29515,"language":15875,"meta":29,"style":29},"button:focus-visible {\n outline: 3px solid rgba(37, 99, 235, 0.35);\n outline-offset: 2px;\n}\n",[31,29517,29518,29527,29564,29577],{"__ignoreMap":29},[34,29519,29520,29522,29525],{"class":36,"line":37},[34,29521,2004],{"class":935},[34,29523,29524],{"class":395},":focus-visible",[34,29526,674],{"class":190},[34,29528,29529,29531,29533,29535,29537,29539,29542,29544,29547,29549,29552,29554,29557,29559,29562],{"class":36,"line":43},[34,29530,29498],{"class":179},[34,29532,8471],{"class":190},[34,29534,786],{"class":179},[34,29536,15935],{"class":175},[34,29538,15938],{"class":179},[34,29540,29541],{"class":179}," rgba",[34,29543,456],{"class":190},[34,29545,29546],{"class":179},"37",[34,29548,616],{"class":190},[34,29550,29551],{"class":179},"99",[34,29553,616],{"class":190},[34,29555,29556],{"class":179},"235",[34,29558,616],{"class":190},[34,29560,29561],{"class":179},"0.35",[34,29563,462],{"class":190},[34,29565,29566,29569,29571,29573,29575],{"class":36,"line":49},[34,29567,29568],{"class":179}," outline-offset",[34,29570,8471],{"class":190},[34,29572,768],{"class":179},[34,29574,15935],{"class":175},[34,29576,191],{"class":190},[34,29578,29579],{"class":36,"line":56},[34,29580,77],{"class":190},[15,29582,29583],{},"键盘用户需要知道当前焦点在哪里。",[19,29585,29587],{"id":29586},"_11-安全vue-默认会转义文本","11. 安全：Vue 默认会转义文本",[24,29589,29591],{"className":923,"code":29590,"language":925,"meta":29,"style":29},"\u003Cp>{{ userInput }}\u003C\u002Fp>\n",[31,29592,29593],{"__ignoreMap":29},[34,29594,29595,29597,29599,29602,29604],{"class":36,"line":37},[34,29596,932],{"class":190},[34,29598,15],{"class":935},[34,29600,29601],{"class":190},">{{ userInput }}\u003C\u002F",[34,29603,15],{"class":935},[34,29605,939],{"class":190},[15,29607,29608],{},"如果用户输入：",[24,29610,29612],{"className":6016,"code":29611,"language":6018,"meta":29,"style":29},"\u003Cscript>alert('hack')\u003C\u002Fscript>\n",[31,29613,29614],{"__ignoreMap":29},[34,29615,29616,29618,29620,29623,29626,29628,29631,29634,29636],{"class":36,"line":37},[34,29617,932],{"class":190},[34,29619,1911],{"class":935},[34,29621,29622],{"class":190},">",[34,29624,29625],{"class":395},"alert",[34,29627,456],{"class":190},[34,29629,29630],{"class":186},"'hack'",[34,29632,29633],{"class":190},")\u003C\u002F",[34,29635,1911],{"class":935},[34,29637,939],{"class":190},[15,29639,29640],{},"Vue 会把它当文本显示，而不是执行脚本。这是安全的。",[19,29642,29644,29645],{"id":29643},"_12-谨慎使用-v-html","12. 谨慎使用 ",[31,29646,28897],{},[15,29648,29649],{},"危险写法：",[24,29651,29653],{"className":923,"code":29652,"language":925,"meta":29,"style":29},"\u003Cdiv v-html=\"userInput\">\u003C\u002Fdiv>\n",[31,29654,29655],{"__ignoreMap":29},[34,29656,29657,29659,29661,29664,29666,29668,29671,29673,29675,29677],{"class":36,"line":37},[34,29658,932],{"class":190},[34,29660,16016],{"class":935},[34,29662,29663],{"class":395}," v-html",[34,29665,202],{"class":190},[34,29667,1608],{"class":186},[34,29669,29670],{"class":190},"userInput",[34,29672,1608],{"class":186},[34,29674,6593],{"class":190},[34,29676,16016],{"class":935},[34,29678,939],{"class":190},[15,29680,29681,29683],{},[31,29682,28897],{}," 会把字符串当 HTML 插入页面。如果内容来自用户，可能导致 XSS。",[15,29685,29686],{},"除非你非常确定：",[105,29688,29689,29692,29695],{},[108,29690,29691],{},"HTML 来源可信。",[108,29693,29694],{},"已经过安全清洗。",[108,29696,29697],{},"业务确实需要渲染 HTML。",[15,29699,29700,29701,347],{},"否则不要用 ",[31,29702,28897],{},[19,29704,29706],{"id":29705},"_13-前端不能保存真正秘密","13. 前端不能保存真正秘密",[15,29708,29709],{},"不要把这些放前端：",[105,29711,29712,29715,29718,29721],{},[108,29713,29714],{},"数据库密码。",[108,29716,29717],{},"后端私密 API key。",[108,29719,29720],{},"管理员 token。",[108,29722,29723],{},"支付平台密钥。",[15,29725,29726],{},"原因：前端代码会被下载到用户浏览器。即使你写在环境变量里，只要打包进前端，用户就可能看到。",[15,29728,29729,29730,29733],{},"Vite 中以 ",[31,29731,29732],{},"VITE_"," 开头的环境变量会暴露给前端：",[24,29735,29739],{"className":29736,"code":29737,"language":29738,"meta":29,"style":29},"language-env shiki shiki-themes github-light github-dark","VITE_API_BASE_URL=https:\u002F\u002Fapi.example.com\n","env",[31,29740,29741],{"__ignoreMap":29},[34,29742,29743],{"class":36,"line":37},[34,29744,29737],{},[15,29746,22965],{},[105,29748,29749,29752,29755],{},[108,29750,29751],{},"公开 API 地址。",[108,29753,29754],{},"公开功能开关。",[108,29756,29757],{},"公开项目配置。",[15,29759,29760],{},"不适合放秘密。",[19,29762,29764],{"id":29763},"_14-依赖安全","14. 依赖安全",[15,29766,23595],{},[105,29768,29769,29772,29775,29778],{},[108,29770,29771],{},"使用维护活跃的库。",[108,29773,29774],{},"不随便复制陌生脚本。",[108,29776,29777],{},"定期运行依赖审计。",[108,29779,29780],{},"遇到安全公告及时升级。",[24,29782,29784],{"className":7808,"code":29783,"language":7810,"meta":29,"style":29},"npm audit\n",[31,29785,29786],{"__ignoreMap":29},[34,29787,29788],{"class":36,"line":37},[34,29789,29783],{},[15,29791,29792],{},"但也不要看到任何 warning 就恐慌。要看是否影响生产、是否可利用、是否有修复版本。",[19,29794,29796],{"id":29795},"_15-常见检查清单","15. 常见检查清单",[15,29798,29799],{},"性能：",[105,29801,29802,29805,29808,29811],{},[108,29803,29804],{},"模板里有没有复杂计算？",[108,29806,29807],{},"列表有没有稳定 key？",[108,29809,29810],{},"是否一次渲染太多数据？",[108,29812,29813],{},"页面组件是否可以懒加载？",[15,29815,29816],{},"可访问性：",[105,29818,29819,29822,29825,29828,29831],{},[108,29820,29821],{},"输入框有没有 label？",[108,29823,29824],{},"按钮是否用 button？",[108,29826,29827],{},"错误是否有明确文本？",[108,29829,29830],{},"键盘能否操作主要流程？",[108,29832,29833],{},"焦点是否可见？",[15,29835,29836],{},"安全：",[105,29838,29839,29844,29847,29850],{},[108,29840,29841,29842,5916],{},"是否使用了 ",[31,29843,28897],{},[108,29845,29846],{},"前端是否暴露了秘密？",[108,29848,29849],{},"用户输入是否被直接拼进 HTML？",[108,29851,29852],{},"依赖是否来自可信来源？",[19,29854,5693],{"id":5693},[2230,29856,29857,29860,29863,29868,29871,29877],{},[108,29858,29859],{},"把一个模板里的复杂表达式改成 computed。",[108,29861,29862],{},"给任务表单补全 label。",[108,29864,29865,29866,347],{},"给错误提示加 ",[31,29867,16343],{},[108,29869,29870],{},"用键盘 Tab 操作页面，检查焦点是否可见。",[108,29872,29873,29874,29876],{},"解释为什么 ",[31,29875,28897],{}," 危险。",[108,29878,29879],{},"判断哪些环境变量能放前端，哪些不能。",[79,29881,29882],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s9eBZ, html code.shiki .s9eBZ{--shiki-default:#22863A;--shiki-dark:#85E89D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}",{"title":29,"searchDepth":43,"depth":43,"links":29884},[29885,29886,29887,29888,29889,29890,29891,29892,29893,29894,29895,29896,29898,29899,29900,29901],{"id":28901,"depth":43,"text":28902},{"id":28928,"depth":43,"text":28929},{"id":29096,"depth":43,"text":29097},{"id":29123,"depth":43,"text":29124},{"id":29172,"depth":43,"text":29173},{"id":29196,"depth":43,"text":29197},{"id":29220,"depth":43,"text":29221},{"id":29317,"depth":43,"text":29318},{"id":29414,"depth":43,"text":29415},{"id":29475,"depth":43,"text":29476},{"id":29586,"depth":43,"text":29587},{"id":29643,"depth":43,"text":29897},"12. 谨慎使用 v-html",{"id":29705,"depth":43,"text":29706},{"id":29763,"depth":43,"text":29764},{"id":29795,"depth":43,"text":29796},{"id":5693,"depth":43,"text":5693},{"order":1999},"\u002Fsource\u002Fcode\u002Fvue\u002F12-performance-a11y-security",{"title":28883,"description":28891},"source\u002Fcode\u002Fvue\u002F12-performance-a11y-security","NO2axRe45mmMTiRkl0oMXrGatzpNywjmaCcEN66JqC0",{"id":29908,"title":29909,"body":29910,"description":29917,"extension":85,"meta":30729,"navigation":52,"path":30730,"seo":30731,"stem":30732,"__hash__":30733},"notes\u002Fsource\u002Fcode\u002Fvue\u002F13-build-deploy-engineering.md","13 构建、部署和工程化",{"type":8,"value":29911,"toc":30709},[29912,29915,29918,29921,29925,29928,29937,29940,29957,29960,29968,29971,29991,29995,30001,30114,30117,30137,30140,30148,30151,30160,30164,30173,30176,30194,30199,30204,30217,30221,30229,30232,30236,30244,30249,30253,30262,30268,30275,30279,30282,30288,30291,30297,30301,30307,30315,30317,30342,30345,30348,30354,30360,30364,30367,30373,30379,30382,30387,30393,30396,30407,30411,30413,30418,30420,30466,30470,30473,30499,30502,30579,30582,30586,30589,30601,30604,30612,30615,30629,30633,30637,30640,30644,30653,30657,30660,30664,30670,30672,30706],[11,29913,29909],{"id":29914},"_13-构建部署和工程化",[15,29916,29917],{},"工程化不是高级项目才需要。只要你希望项目能稳定运行、多人协作、可以上线，就需要理解基本工具链。",[15,29919,29920],{},"本章目标：你能理解 Vite、package.json、开发服务器、生产构建、环境变量、部署刷新 404 和项目目录组织。",[19,29922,29924],{"id":29923},"_1-vite-做了什么","1. Vite 做了什么",[15,29926,29927],{},"开发时：",[24,29929,29931],{"className":7808,"code":29930,"language":7810,"meta":29,"style":29},"npm run dev\n",[31,29932,29933],{"__ignoreMap":29},[34,29934,29935],{"class":36,"line":37},[34,29936,29930],{},[15,29938,29939],{},"Vite 提供：",[105,29941,29942,29945,29948,29951,29954],{},[108,29943,29944],{},"本地开发服务器。",[108,29946,29947],{},"热更新。",[108,29949,29950],{},"Vue 单文件组件编译。",[108,29952,29953],{},"模块导入处理。",[108,29955,29956],{},"错误提示。",[15,29958,29959],{},"生产构建时：",[24,29961,29962],{"className":7808,"code":27206,"language":7810,"meta":29,"style":29},[31,29963,29964],{"__ignoreMap":29},[34,29965,29966],{"class":36,"line":37},[34,29967,27206],{},[15,29969,29970],{},"Vite 会：",[105,29972,29973,29979,29982,29985],{},[108,29974,29975,29976,29978],{},"编译 ",[31,29977,7827],{}," 文件。",[108,29980,29981],{},"压缩 JS 和 CSS。",[108,29983,29984],{},"拆分资源。",[108,29986,29987,29988,347],{},"生成 ",[31,29989,29990],{},"dist\u002F",[19,29992,29994],{"id":29993},"_2-packagejson-是什么","2. package.json 是什么",[15,29996,29997,30000],{},[31,29998,29999],{},"package.json"," 是项目说明书和命令表。",[24,30002,30005],{"className":30003,"code":30004,"language":5332,"meta":29,"style":29},"language-json shiki shiki-themes github-light github-dark","{\n \"scripts\": {\n \"dev\": \"vite\",\n \"build\": \"vite build\",\n \"preview\": \"vite preview\",\n \"test\": \"vitest run\"\n },\n \"dependencies\": {\n \"vue\": \"^3.5.35\"\n },\n \"devDependencies\": {\n \"vite\": \"^8.0.16\"\n }\n}\n",[31,30006,30007,30011,30018,30030,30042,30054,30064,30068,30075,30085,30089,30096,30106,30110],{"__ignoreMap":29},[34,30008,30009],{"class":36,"line":37},[34,30010,20357],{"class":190},[34,30012,30013,30016],{"class":36,"line":43},[34,30014,30015],{"class":179}," \"scripts\"",[34,30017,21333],{"class":190},[34,30019,30020,30023,30025,30028],{"class":36,"line":49},[34,30021,30022],{"class":179}," \"dev\"",[34,30024,8471],{"class":190},[34,30026,30027],{"class":186},"\"vite\"",[34,30029,684],{"class":190},[34,30031,30032,30035,30037,30040],{"class":36,"line":56},[34,30033,30034],{"class":179}," \"build\"",[34,30036,8471],{"class":190},[34,30038,30039],{"class":186},"\"vite build\"",[34,30041,684],{"class":190},[34,30043,30044,30047,30049,30052],{"class":36,"line":62},[34,30045,30046],{"class":179}," \"preview\"",[34,30048,8471],{"class":190},[34,30050,30051],{"class":186},"\"vite preview\"",[34,30053,684],{"class":190},[34,30055,30056,30059,30061],{"class":36,"line":68},[34,30057,30058],{"class":179}," \"test\"",[34,30060,8471],{"class":190},[34,30062,30063],{"class":186},"\"vitest run\"\n",[34,30065,30066],{"class":36,"line":74},[34,30067,17404],{"class":190},[34,30069,30070,30073],{"class":36,"line":1971},[34,30071,30072],{"class":179}," \"dependencies\"",[34,30074,21333],{"class":190},[34,30076,30077,30080,30082],{"class":36,"line":1976},[34,30078,30079],{"class":179}," \"vue\"",[34,30081,8471],{"class":190},[34,30083,30084],{"class":186},"\"^3.5.35\"\n",[34,30086,30087],{"class":36,"line":1985},[34,30088,17404],{"class":190},[34,30090,30091,30094],{"class":36,"line":1990},[34,30092,30093],{"class":179}," \"devDependencies\"",[34,30095,21333],{"class":190},[34,30097,30098,30101,30103],{"class":36,"line":1999},[34,30099,30100],{"class":179}," \"vite\"",[34,30102,8471],{"class":190},[34,30104,30105],{"class":186},"\"^8.0.16\"\n",[34,30107,30108],{"class":36,"line":2022},[34,30109,5631],{"class":190},[34,30111,30112],{"class":36,"line":2036},[34,30113,77],{"class":190},[15,30115,30116],{},"字段解释：",[105,30118,30119,30125,30131],{},[108,30120,30121,30124],{},[31,30122,30123],{},"scripts","：可运行命令。",[108,30126,30127,30130],{},[31,30128,30129],{},"dependencies","：生产运行需要的依赖。",[108,30132,30133,30136],{},[31,30134,30135],{},"devDependencies","：开发和构建需要的依赖。",[15,30138,30139],{},"运行：",[24,30141,30142],{"className":7808,"code":29930,"language":7810,"meta":29,"style":29},[31,30143,30144],{"__ignoreMap":29},[34,30145,30146],{"class":36,"line":37},[34,30147,29930],{},[15,30149,30150],{},"本质上会执行：",[24,30152,30154],{"className":7808,"code":30153,"language":7810,"meta":29,"style":29},"vite\n",[31,30155,30156],{"__ignoreMap":29},[34,30157,30158],{"class":36,"line":37},[34,30159,30153],{},[19,30161,30163],{"id":30162},"_3-npm-install-做了什么","3. npm install 做了什么",[24,30165,30167],{"className":7808,"code":30166,"language":7810,"meta":29,"style":29},"npm install\n",[31,30168,30169],{"__ignoreMap":29},[34,30170,30171],{"class":36,"line":37},[34,30172,30166],{},[15,30174,30175],{},"它会：",[105,30177,30178,30187,30191],{},[108,30179,30180,30181,258,30183,30186],{},"根据 ",[31,30182,29999],{},[31,30184,30185],{},"package-lock.json"," 安装依赖。",[108,30188,29987,30189,347],{},[31,30190,4411],{},[108,30192,30193],{},"确保项目能运行。",[15,30195,30196,30198],{},[31,30197,4411],{}," 很大，不应该手动编辑，也不应该提交到 Git。",[15,30200,30201,30203],{},[31,30202,30185],{}," 记录具体安装版本，团队项目应该提交它，保证大家依赖一致。",[19,30205,9025,30207,146,30210,146,30213,30216],{"id":30206},"_4-devbuildpreview-区别",[31,30208,30209],{},"dev",[31,30211,30212],{},"build",[31,30214,30215],{},"preview"," 区别",[15,30218,30219,2664],{},[31,30220,30209],{},[24,30222,30223],{"className":7808,"code":29930,"language":7810,"meta":29,"style":29},[31,30224,30225],{"__ignoreMap":29},[34,30226,30227],{"class":36,"line":37},[34,30228,29930],{},[15,30230,30231],{},"用于开发。速度快，有热更新，不是生产产物。",[15,30233,30234,2664],{},[31,30235,30212],{},[24,30237,30238],{"className":7808,"code":27206,"language":7810,"meta":29,"style":29},[31,30239,30240],{"__ignoreMap":29},[34,30241,30242],{"class":36,"line":37},[34,30243,27206],{},[15,30245,30246,30247,347],{},"生成生产文件到 ",[31,30248,29990],{},[15,30250,30251,2664],{},[31,30252,30215],{},[24,30254,30256],{"className":7808,"code":30255,"language":7810,"meta":29,"style":29},"npm run preview\n",[31,30257,30258],{"__ignoreMap":29},[34,30259,30260],{"class":36,"line":37},[34,30261,30255],{},[15,30263,30264,30265,30267],{},"在本地预览 ",[31,30266,29990],{},"，检查构建产物能否正常运行。",[15,30269,30270,30271,30274],{},"不要把 ",[31,30272,30273],{},"npm run dev"," 当部署方式。上线应该部署构建产物。",[19,30276,30278],{"id":30277},"_5-dist-是什么","5. dist 是什么",[15,30280,30281],{},"构建后：",[24,30283,30286],{"className":30284,"code":30285,"language":2638,"meta":29},[2636],"dist\u002F\n index.html\n assets\u002F\n index-xxxx.js\n index-xxxx.css\n",[31,30287,30285],{"__ignoreMap":29},[15,30289,30290],{},"这些文件就是浏览器最终需要的静态资源。",[15,30292,30293,30294,30296],{},"部署静态站点时，通常把 ",[31,30295,29990],{}," 交给平台。",[19,30298,30300],{"id":30299},"_6-环境变量","6. 环境变量",[15,30302,30303,30304,30306],{},"Vite 暴露给前端的变量通常以 ",[31,30305,29732],{}," 开头：",[24,30308,30309],{"className":29736,"code":29737,"language":29738,"meta":29,"style":29},[31,30310,30311],{"__ignoreMap":29},[34,30312,30313],{"class":36,"line":37},[34,30314,29737],{},[15,30316,13233],{},[24,30318,30320],{"className":166,"code":30319,"language":168,"meta":29,"style":29},"const baseUrl = import.meta.env.VITE_API_BASE_URL\n",[31,30321,30322],{"__ignoreMap":29},[34,30323,30324,30326,30329,30331,30333,30335,30337,30339],{"class":36,"line":37},[34,30325,176],{"class":175},[34,30327,30328],{"class":179}," baseUrl",[34,30330,183],{"class":175},[34,30332,21137],{"class":175},[34,30334,2547],{"class":190},[34,30336,20155],{"class":179},[34,30338,20158],{"class":190},[34,30340,30341],{"class":179},"VITE_API_BASE_URL\n",[15,30343,30344],{},"注意：前端环境变量不是秘密。只要被打包进前端，用户就可能看到。",[15,30346,30347],{},"常见文件：",[24,30349,30352],{"className":30350,"code":30351,"language":2638,"meta":29},[2636],".env\n.env.development\n.env.production\n",[31,30353,30351],{"__ignoreMap":29},[15,30355,30356,30357,347],{},"不要提交包含秘密的 ",[31,30358,30359],{},".env.local",[19,30361,30363],{"id":30362},"_7-单页应用刷新-404","7. 单页应用刷新 404",[15,30365,30366],{},"Vue Router 使用 history 模式时，URL 可能是：",[24,30368,30371],{"className":30369,"code":30370,"language":2638,"meta":29},[2636],"\u002Flessons\u002Fcore\n",[31,30372,30370],{"__ignoreMap":29},[15,30374,30375,30376,347],{},"如果用户直接刷新，浏览器会向服务器请求 ",[31,30377,30378],{},"\u002Flessons\u002Fcore",[15,30380,30381],{},"服务器如果不知道这个路径，就返回 404。",[15,30383,30384,30385,347],{},"解决：服务器把未知路径都返回 ",[31,30386,20060],{},[24,30388,30391],{"className":30389,"code":30390,"language":2638,"meta":29},[2636],"未知路径 -> index.html -> Vue Router 接管\n",[31,30392,30390],{"__ignoreMap":29},[15,30394,30395],{},"很多部署平台叫：",[105,30397,30398,30401,30404],{},[108,30399,30400],{},"fallback。",[108,30402,30403],{},"rewrite。",[108,30405,30406],{},"SPA fallback。",[19,30408,30410],{"id":30409},"_8-项目目录组织","8. 项目目录组织",[15,30412,7527],{},[24,30414,30416],{"className":30415,"code":28583,"language":2638,"meta":29},[2636],[31,30417,28583],{"__ignoreMap":29},[15,30419,28588],{},[105,30421,30422,30428,30432,30437,30441,30446,30452,30457,30461],{},[108,30423,30424,30427],{},[31,30425,30426],{},"assets","：图片、全局 CSS 等静态资源。",[108,30429,30430,28596],{},[31,30431,28595],{},[108,30433,30434,30436],{},[31,30435,28601],{},"：路由页面组件。",[108,30438,30439,28608],{},[31,30440,28607],{},[108,30442,30443,30445],{},[31,30444,17493],{},"：本地静态数据。",[108,30447,30448,30451],{},[31,30449,30450],{},"router","：路由配置。",[108,30453,30454,30456],{},[31,30455,28619],{},"：请求接口封装。",[108,30458,30459,28614],{},[31,30460,28613],{},[108,30462,30463,30465],{},[31,30464,28625],{},"：共享 TypeScript 类型。",[19,30467,30469],{"id":30468},"_9-readme-应该写什么","9. README 应该写什么",[15,30471,30472],{},"一个项目 README 至少写：",[105,30474,30475,30478,30481,30484,30487,30490,30493,30496],{},[108,30476,30477],{},"项目做什么。",[108,30479,30480],{},"技术栈。",[108,30482,30483],{},"如何安装。",[108,30485,30486],{},"如何运行。",[108,30488,30489],{},"如何测试。",[108,30491,30492],{},"如何构建。",[108,30494,30495],{},"目录结构。",[108,30497,30498],{},"环境变量说明。",[15,30500,30501],{},"示例：",[24,30503,30506],{"className":30504,"code":30505,"language":85,"meta":29,"style":29},"language-md shiki shiki-themes github-light github-dark","# Project Name\n\n## Setup\n\n```powershell\nnpm install\nnpm run dev\n```\n\n## Scripts\n\n- `npm run dev`\n- `npm run test`\n- `npm run build`\n",[31,30507,30508,30514,30518,30523,30527,30532,30536,30540,30545,30549,30554,30558,30565,30572],{"__ignoreMap":29},[34,30509,30510],{"class":36,"line":37},[34,30511,30513],{"class":30512},"sq-ep","# Project Name\n",[34,30515,30516],{"class":36,"line":43},[34,30517,53],{"emptyLinePlaceholder":52},[34,30519,30520],{"class":36,"line":49},[34,30521,30522],{"class":30512},"## Setup\n",[34,30524,30525],{"class":36,"line":56},[34,30526,53],{"emptyLinePlaceholder":52},[34,30528,30529],{"class":36,"line":62},[34,30530,30531],{"class":190},"```powershell\n",[34,30533,30534],{"class":36,"line":68},[34,30535,30166],{"class":190},[34,30537,30538],{"class":36,"line":74},[34,30539,29930],{"class":190},[34,30541,30542],{"class":36,"line":1971},[34,30543,30544],{"class":190},"```\n",[34,30546,30547],{"class":36,"line":1976},[34,30548,53],{"emptyLinePlaceholder":52},[34,30550,30551],{"class":36,"line":1985},[34,30552,30553],{"class":30512},"## Scripts\n",[34,30555,30556],{"class":36,"line":1990},[34,30557,53],{"emptyLinePlaceholder":52},[34,30559,30560,30562],{"class":36,"line":1999},[34,30561,2511],{"class":1326},[34,30563,30564],{"class":179}," `npm run dev`\n",[34,30566,30567,30569],{"class":36,"line":2022},[34,30568,2511],{"class":1326},[34,30570,30571],{"class":179}," `npm run test`\n",[34,30573,30574,30576],{"class":36,"line":2036},[34,30575,2511],{"class":1326},[34,30577,30578],{"class":179}," `npm run build`\n",[15,30580,30581],{},"好的 README 能让新同事少问很多重复问题。",[19,30583,30585],{"id":30584},"_10-提交前检查","10. 提交前检查",[15,30587,30588],{},"最小检查：",[24,30590,30591],{"className":7808,"code":27194,"language":7810,"meta":29,"style":29},[31,30592,30593,30597],{"__ignoreMap":29},[34,30594,30595],{"class":36,"line":37},[34,30596,27201],{},[34,30598,30599],{"class":36,"line":43},[34,30600,27206],{},[15,30602,30603],{},"如果项目有 lint：",[24,30605,30606],{"className":7808,"code":27212,"language":7810,"meta":29,"style":29},[31,30607,30608],{"__ignoreMap":29},[34,30609,30610],{"class":36,"line":37},[34,30611,27212],{},[15,30613,30614],{},"如果改了页面：",[105,30616,30617,30620,30623,30626],{},[108,30618,30619],{},"打开本地页面。",[108,30621,30622],{},"点一遍关键流程。",[108,30624,30625],{},"检查控制台错误。",[108,30627,30628],{},"检查移动端布局。",[19,30630,30632],{"id":30631},"_11-常见工程化错误","11. 常见工程化错误",[253,30634,30636],{"id":30635},"错误一删除-package-lock","错误一：删除 package-lock",[15,30638,30639],{},"这会让团队依赖版本更不稳定。一般不要随便删。",[253,30641,30643],{"id":30642},"错误二提交-node_modules","错误二：提交 node_modules",[15,30645,30646,30648,30649,30652],{},[31,30647,4411],{}," 应该由 ",[31,30650,30651],{},"npm install"," 生成，不应该提交。",[253,30654,30656],{"id":30655},"错误三构建前没跑测试","错误三：构建前没跑测试",[15,30658,30659],{},"测试能提前发现逻辑问题，构建能发现语法和打包问题，两者都重要。",[253,30661,30663],{"id":30662},"错误四把开发服务器地址当生产地址","错误四：把开发服务器地址当生产地址",[15,30665,30666,30669],{},[31,30667,30668],{},"http:\u002F\u002Flocalhost:5173"," 只适合本地开发。线上应该使用部署平台提供的域名。",[19,30671,5693],{"id":5693},[2230,30673,30674,30683,30691,30697,30700,30703],{},[108,30675,30676,30677,258,30679,30682],{},"解释 ",[31,30678,30273],{},[31,30680,30681],{},"npm run build"," 区别。",[108,30684,4501,30685,30687,30688,347],{},[31,30686,30681],{},"，观察 ",[31,30689,30690],{},"dist",[108,30692,4501,30693,30696],{},[31,30694,30695],{},"npm run preview","，预览构建产物。",[108,30698,30699],{},"写一份项目 README 大纲。",[108,30701,30702],{},"解释 SPA 刷新 404 的原因和解决思路。",[108,30704,30705],{},"判断哪些文件应该提交，哪些不应该提交。",[79,30707,30708],{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sq-ep, html code.shiki .sq-ep{--shiki-default:#005CC5;--shiki-default-font-weight:bold;--shiki-dark:#79B8FF;--shiki-dark-font-weight:bold}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":30710},[30711,30712,30713,30714,30716,30717,30718,30719,30720,30721,30722,30728],{"id":29923,"depth":43,"text":29924},{"id":29993,"depth":43,"text":29994},{"id":30162,"depth":43,"text":30163},{"id":30206,"depth":43,"text":30715},"4. dev、build、preview 区别",{"id":30277,"depth":43,"text":30278},{"id":30299,"depth":43,"text":30300},{"id":30362,"depth":43,"text":30363},{"id":30409,"depth":43,"text":30410},{"id":30468,"depth":43,"text":30469},{"id":30584,"depth":43,"text":30585},{"id":30631,"depth":43,"text":30632,"children":30723},[30724,30725,30726,30727],{"id":30635,"depth":49,"text":30636},{"id":30642,"depth":49,"text":30643},{"id":30655,"depth":49,"text":30656},{"id":30662,"depth":49,"text":30663},{"id":5693,"depth":43,"text":5693},{"order":2022},"\u002Fsource\u002Fcode\u002Fvue\u002F13-build-deploy-engineering",{"title":29909,"description":29917},"source\u002Fcode\u002Fvue\u002F13-build-deploy-engineering","ZS17nJNFdfVW2ovxUw4Wy4geAmOmbvyyTXQAt2rcqIk",{"id":30735,"title":30736,"body":30737,"description":30744,"extension":85,"meta":31412,"navigation":52,"path":31413,"seo":31414,"stem":31415,"__hash__":31416},"notes\u002Fsource\u002Fcode\u002Fvue\u002F14-capstone.md","14 毕业项目：课程任务管理器",{"type":8,"value":30738,"toc":31381},[30739,30742,30745,30748,30752,30755,30775,30782,30786,30789,30791,30866,30869,30886,30889,30891,30952,30954,30971,30975,30977,30983,30986,30999,31003,31006,31012,31015,31026,31030,31036,31040,31046,31050,31052,31063,31066,31072,31074,31080,31083,31087,31091,31094,31097,31108,31112,31118,31129,31132,31136,31139,31153,31156,31160,31163,31167,31170,31174,31179,31187,31191,31194,31211,31215,31218,31238,31242,31244,31256,31259,31263,31266,31289,31293,31297,31300,31304,31307,31311,31314,31318,31321,31325,31328,31332,31335,31369,31371,31378],[11,30740,30736],{"id":30741},"_14-毕业项目课程任务管理器",[15,30743,30744],{},"毕业项目的目标是把前面所有知识串起来。你要做一个“课程任务管理器”：能展示课程、记录完成进度、创建学习任务、筛选任务、保存数据、模拟请求、写测试并构建。",[15,30746,30747],{},"本章不是只列功能，而是教你怎么从需求拆到组件、状态、路由和测试。",[19,30749,30751],{"id":30750},"_1-项目目标","1. 项目目标",[15,30753,30754],{},"最终应用应该包含：",[105,30756,30757,30760,30763,30766,30769,30772],{},[108,30758,30759],{},"首页：学习统计和下一步建议。",[108,30761,30762],{},"课程页：课程列表、搜索、阶段筛选。",[108,30764,30765],{},"课程详情页：课程说明、完成状态。",[108,30767,30768],{},"任务页：新增任务、完成任务、删除任务、筛选任务。",[108,30770,30771],{},"异步页：模拟加载课程，显示 loading\u002Ferror\u002Fempty\u002Fsuccess。",[108,30773,30774],{},"设置或说明页：展示项目说明。",[15,30776,30777,30778,30781],{},"你可以参考 ",[31,30779,30780],{},"D:\\vue-tutorial\\course-app","，但建议自己从空白项目重新做一次。",[19,30783,30785],{"id":30784},"_2-需求拆解","2. 需求拆解",[253,30787,30788],{"id":30788},"课程功能",[15,30790,1540],{},[24,30792,30794],{"className":166,"code":30793,"language":168,"meta":29,"style":29},"const lessons = [\n {\n id: 'js',\n phase: '基础',\n title: 'JavaScript 必备基础',\n minutes: 80,\n summary: '变量、类型、函数、模块和异步。'\n }\n]\n",[31,30795,30796,30806,30811,30821,30831,30841,30850,30858,30862],{"__ignoreMap":29},[34,30797,30798,30800,30802,30804],{"class":36,"line":37},[34,30799,176],{"class":175},[34,30801,1203],{"class":179},[34,30803,183],{"class":175},[34,30805,739],{"class":190},[34,30807,30808],{"class":36,"line":43},[34,30809,30810],{"class":190}," {\n",[34,30812,30813,30816,30819],{"class":36,"line":49},[34,30814,30815],{"class":190}," id: ",[34,30817,30818],{"class":186},"'js'",[34,30820,684],{"class":190},[34,30822,30823,30826,30829],{"class":36,"line":56},[34,30824,30825],{"class":190}," phase: ",[34,30827,30828],{"class":186},"'基础'",[34,30830,684],{"class":190},[34,30832,30833,30836,30839],{"class":36,"line":62},[34,30834,30835],{"class":190}," title: ",[34,30837,30838],{"class":186},"'JavaScript 必备基础'",[34,30840,684],{"class":190},[34,30842,30843,30846,30848],{"class":36,"line":68},[34,30844,30845],{"class":190}," minutes: ",[34,30847,27539],{"class":179},[34,30849,684],{"class":190},[34,30851,30852,30855],{"class":36,"line":74},[34,30853,30854],{"class":190}," summary: ",[34,30856,30857],{"class":186},"'变量、类型、函数、模块和异步。'\n",[34,30859,30860],{"class":36,"line":1971},[34,30861,5631],{"class":190},[34,30863,30864],{"class":36,"line":1976},[34,30865,6569],{"class":190},[15,30867,30868],{},"功能：",[105,30870,30871,30874,30877,30880,30883],{},[108,30872,30873],{},"显示课程列表。",[108,30875,30876],{},"搜索标题。",[108,30878,30879],{},"按阶段筛选。",[108,30881,30882],{},"标记完成。",[108,30884,30885],{},"查看详情。",[253,30887,30888],{"id":30888},"任务功能",[15,30890,1540],{},[24,30892,30894],{"className":166,"code":30893,"language":168,"meta":29,"style":29},"const todos = [\n {\n id: 'todo-1',\n title: '完成组件通信练习',\n course: '组件',\n done: false\n }\n]\n",[31,30895,30896,30906,30910,30919,30928,30938,30944,30948],{"__ignoreMap":29},[34,30897,30898,30900,30902,30904],{"class":36,"line":37},[34,30899,176],{"class":175},[34,30901,12962],{"class":179},[34,30903,183],{"class":175},[34,30905,739],{"class":190},[34,30907,30908],{"class":36,"line":43},[34,30909,30810],{"class":190},[34,30911,30912,30914,30917],{"class":36,"line":49},[34,30913,30815],{"class":190},[34,30915,30916],{"class":186},"'todo-1'",[34,30918,684],{"class":190},[34,30920,30921,30923,30926],{"class":36,"line":56},[34,30922,30835],{"class":190},[34,30924,30925],{"class":186},"'完成组件通信练习'",[34,30927,684],{"class":190},[34,30929,30930,30933,30936],{"class":36,"line":62},[34,30931,30932],{"class":190}," course: ",[34,30934,30935],{"class":186},"'组件'",[34,30937,684],{"class":190},[34,30939,30940,30942],{"class":36,"line":68},[34,30941,15342],{"class":190},[34,30943,14360],{"class":179},[34,30945,30946],{"class":36,"line":74},[34,30947,5631],{"class":190},[34,30949,30950],{"class":36,"line":1971},[34,30951,6569],{"class":190},[15,30953,30868],{},[105,30955,30956,30959,30962,30965,30968],{},[108,30957,30958],{},"新增任务。",[108,30960,30961],{},"切换完成。",[108,30963,30964],{},"删除任务。",[108,30966,30967],{},"全部\u002F未完成\u002F已完成筛选。",[108,30969,30970],{},"统计未完成数量。",[19,30972,30974],{"id":30973},"_3-路由设计","3. 路由设计",[15,30976,7527],{},[24,30978,30981],{"className":30979,"code":30980,"language":2638,"meta":29},[2636],"\u002F 首页\n\u002Flessons 课程列表\n\u002Flessons\u002F:id 课程详情\n\u002Ftodos 任务管理\n\u002Fasync 异步请求演示\n\u002Fabout 项目说明\n",[31,30982,30980],{"__ignoreMap":29},[15,30984,30985],{},"路由职责：",[105,30987,30988,30991,30996],{},[108,30989,30990],{},"URL 表示当前页面。",[108,30992,30993,30995],{},[31,30994,20610],{}," 表示详情页参数。",[108,30997,30998],{},"搜索和筛选可以用查询参数，也可以先用组件状态，进阶时再同步到 URL。",[19,31000,31002],{"id":31001},"_4-组件拆分","4. 组件拆分",[15,31004,31005],{},"建议组件：",[24,31007,31010],{"className":31008,"code":31009,"language":2638,"meta":29},[2636],"AppHeader\nLessonCard\nLessonList\nLessonFilter\nTodoForm\nTodoItem\nTodoStats\nEmptyState\nBaseButton\nBasePanel\n",[31,31011,31009],{"__ignoreMap":29},[15,31013,31014],{},"每个组件要回答：",[105,31016,31017,31020,31023],{},[108,31018,31019],{},"它负责显示什么？",[108,31021,31022],{},"它需要哪些 props？",[108,31024,31025],{},"它会 emit 哪些事件？",[15,31027,13169,31028,2664],{},[31,31029,11972],{},[24,31031,31034],{"className":31032,"code":31033,"language":2638,"meta":29},[2636],"职责：显示单条任务，提供完成和删除操作。\nprops：todo。\nemits：toggle(id)、remove(id)。\n",[31,31035,31033],{"__ignoreMap":29},[15,31037,13169,31038,2664],{},[31,31039,23536],{},[24,31041,31044],{"className":31042,"code":31043,"language":2638,"meta":29},[2636],"职责：收集任务标题和所属阶段。\nprops：可选的阶段列表。\nemits：submit(payload)。\n内部状态：输入框文字、错误信息。\n",[31,31045,31043],{"__ignoreMap":29},[19,31047,31049],{"id":31048},"_5-状态设计","5. 状态设计",[15,31051,21402],{},[105,31053,31054,31057,31060],{},[108,31055,31056],{},"任务表单输入框文字。",[108,31058,31059],{},"当前弹窗开关。",[108,31061,31062],{},"当前页面局部筛选，如果不需要共享。",[15,31064,31065],{},"Pinia 状态：",[24,31067,31070],{"className":31068,"code":31069,"language":2638,"meta":29},[2636],"useCourseStore\n completedLessonIds\n progressPercent\n toggleLesson(id)\n isCompleted(id)\n\nuseTodoStore\n todos\n filter\n filteredTodos\n remainingCount\n addTodo(payload)\n toggleTodo(id)\n removeTodo(id)\n",[31,31071,31069],{"__ignoreMap":29},[15,31073,19645],{},[24,31075,31078],{"className":31076,"code":31077,"language":2638,"meta":29},[2636],"useLocalStorage\nuseFetch\nuseCounter\n",[31,31079,31077],{"__ignoreMap":29},[15,31081,31082],{},"不要把所有状态都塞进一个 store。按业务领域拆。",[19,31084,31086],{"id":31085},"_6-开发顺序","6. 开发顺序",[253,31088,31090],{"id":31089},"第一步静态页面","第一步：静态页面",[15,31092,31093],{},"先写 HTML 结构和样式，不要急着做交互。",[15,31095,31096],{},"目标：",[105,31098,31099,31102,31105],{},[108,31100,31101],{},"能看到首页布局。",[108,31103,31104],{},"能看到课程卡片。",[108,31106,31107],{},"能看到任务表单和任务列表。",[253,31109,31111],{"id":31110},"第二步局部交互","第二步：局部交互",[15,31113,31114,31115,31117],{},"先在页面组件里用 ",[31,31116,145],{}," 实现：",[105,31119,31120,31122,31124,31126],{},[108,31121,30958],{},[108,31123,30964],{},[108,31125,30961],{},[108,31127,31128],{},"筛选任务。",[15,31130,31131],{},"这一步不要急着 Pinia。先确认数据和交互正确。",[253,31133,31135],{"id":31134},"第三步拆组件","第三步：拆组件",[15,31137,31138],{},"把重复 UI 拆出去：",[105,31140,31141,31145,31149],{},[108,31142,31143,347],{},[31,31144,11972],{},[108,31146,31147,347],{},[31,31148,23536],{},[108,31150,31151,347],{},[31,31152,3466],{},[15,31154,31155],{},"拆完后检查 props 和 emit 是否清晰。",[253,31157,31159],{"id":31158},"第四步迁移到-pinia","第四步：迁移到 Pinia",[15,31161,31162],{},"当多个页面都需要任务或课程进度时，再迁移到 store。",[253,31164,31166],{"id":31165},"第五步加-router","第五步：加 Router",[15,31168,31169],{},"配置页面，处理详情参数和 404。",[253,31171,31173],{"id":31172},"第六步加持久化","第六步：加持久化",[15,31175,5792,31176,31178],{},[31,31177,18770],{}," 保存：",[105,31180,31181,31184],{},[108,31182,31183],{},"任务列表。",[108,31185,31186],{},"课程完成 id。",[253,31188,31190],{"id":31189},"第七步加异步请求","第七步：加异步请求",[15,31192,31193],{},"用 mock API 模拟课程加载：",[105,31195,31196,31199,31202,31205,31208],{},[108,31197,31198],{},"loading。",[108,31200,31201],{},"error。",[108,31203,31204],{},"empty。",[108,31206,31207],{},"success。",[108,31209,31210],{},"retry。",[253,31212,31214],{"id":31213},"第八步写测试","第八步：写测试",[15,31216,31217],{},"至少测试：",[105,31219,31220,31225,31231],{},[108,31221,31222,31223,347],{},"一个组合式函数，例如 ",[31,31224,19560],{},[108,31226,31227,31228,31230],{},"一个组件事件，例如 ",[31,31229,11972],{}," emit。",[108,31232,31233,31234,4374,31236,347],{},"一个 store action，例如 ",[31,31235,22088],{},[31,31237,13007],{},[253,31239,31241],{"id":31240},"第九步构建验证","第九步：构建验证",[15,31243,30139],{},[24,31245,31246],{"className":7808,"code":27194,"language":7810,"meta":29,"style":29},[31,31247,31248,31252],{"__ignoreMap":29},[34,31249,31250],{"class":36,"line":37},[34,31251,27201],{},[34,31253,31254],{"class":36,"line":43},[34,31255,27206],{},[15,31257,31258],{},"再在浏览器里手动验证关键流程。",[19,31260,31262],{"id":31261},"_7-验收标准","7. 验收标准",[15,31264,31265],{},"你完成后应该能做到：",[105,31267,31268,31271,31274,31277,31280,31283,31286],{},[108,31269,31270],{},"不看答案解释 props 和 emit 的数据流。",[108,31272,31273],{},"说清楚哪些状态放组件内，哪些放 Pinia。",[108,31275,31276],{},"解释 Router 和 Pinia 的边界。",[108,31278,31279],{},"请求失败时页面有错误和重试。",[108,31281,31282],{},"刷新页面后任务和进度保留。",[108,31284,31285],{},"测试能证明核心逻辑。",[108,31287,31288],{},"构建成功。",[19,31290,31292],{"id":31291},"_8-常见项目陷阱","8. 常见项目陷阱",[253,31294,31296],{"id":31295},"陷阱一一开始就写-store","陷阱一：一开始就写 store",[15,31298,31299],{},"如果你还没想清楚功能，过早写 store 会把简单问题复杂化。先用局部状态跑通。",[253,31301,31303],{"id":31302},"陷阱二组件拆得太碎","陷阱二：组件拆得太碎",[15,31305,31306],{},"拆组件不是目标。清晰边界才是目标。",[253,31308,31310],{"id":31309},"陷阱三只写成功状态","陷阱三：只写成功状态",[15,31312,31313],{},"请求页必须有 loading、error、empty、success。",[253,31315,31317],{"id":31316},"陷阱四没有测试就大改","陷阱四：没有测试就大改",[15,31319,31320],{},"没有测试时，改一个地方很容易弄坏另一个地方。至少给核心逻辑补测试。",[253,31322,31324],{"id":31323},"陷阱五不会复述","陷阱五：不会复述",[15,31326,31327],{},"如果你只能复制代码，却说不清为什么这样写，说明还没掌握。完成项目后必须能复述设计。",[19,31329,31331],{"id":31330},"_9-最终复盘问题","9. 最终复盘问题",[15,31333,31334],{},"写完项目后回答：",[2230,31336,31337,31340,31343,31346,31349,31352,31355,31358,31363,31366],{},[108,31338,31339],{},"你的路由有哪些？每个页面负责什么？",[108,31341,31342],{},"你的组件有哪些？每个组件的 props 和 emits 是什么？",[108,31344,31345],{},"哪些状态在组件内？为什么？",[108,31347,31348],{},"哪些状态在 Pinia？为什么？",[108,31350,31351],{},"哪些逻辑抽成了组合式函数？",[108,31353,31354],{},"请求失败时用户看到什么？",[108,31356,31357],{},"你的测试验证了哪些行为？",[108,31359,31360,31362],{},[31,31361,30681],{}," 生成了什么？",[108,31364,31365],{},"如果要加登录功能，你会新增哪个 store？",[108,31367,31368],{},"如果任务来自真实后端，你会把 API 逻辑放哪里？",[19,31370,5693],{"id":5693},[15,31372,31373,31374,31377],{},"按 ",[31,31375,31376],{},"D:\\vue-tutorial\\projects\\todo-workshop\\README.md"," 的阶段完成项目。完成后，把上面的复盘问题写进你自己的项目 README。",[79,31379,31380],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: 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JavaScript 基础、DOM 心智模型到 Vue 核心语法、工程化、测试和毕业项目的完整学习路径。",[19,31428,31429],{"id":31429},"章节",[2230,31431,31432,31438,31444,31450,31456,31462,31468,31474,31480,31486,31492,31498,31504,31510],{},[108,31433,31434],{},[111,31435,31437],{"href":31436},".\u002F01-js-from-zero","JavaScript 必备基础",[108,31439,31440],{},[111,31441,31443],{"href":31442},".\u002F02-dom-to-vue","DOM 到 Vue 的心智模型",[108,31445,31446],{},[111,31447,31449],{"href":31448},".\u002F03-vue-core","Vue 核心语法和响应式",[108,31451,31452],{},[111,31453,31455],{"href":31454},".\u002F04-components","组件、Props、事件和插槽",[108,31457,31458],{},[111,31459,31461],{"href":31460},".\u002F05-forms-lists-styles","列表、表单、样式和用户输入",[108,31463,31464],{},[111,31465,31467],{"href":31466},".\u002F06-composition-composables","Composition API 和组合式函数",[108,31469,31470],{},[111,31471,31473],{"href":31472},".\u002F07-router","Vue Router 路由",[108,31475,31476],{},[111,31477,31479],{"href":31478},".\u002F08-pinia","Pinia 状态管理",[108,31481,31482],{},[111,31483,31485],{"href":31484},".\u002F09-http-async","异步请求和错误处理",[108,31487,31488],{},[111,31489,31491],{"href":31490},".\u002F10-testing-debugging","测试、调试和质量保障",[108,31493,31494],{},[111,31495,31497],{"href":31496},".\u002F11-typescript-production","TypeScript 和生产级代码",[108,31499,31500],{},[111,31501,31503],{"href":31502},".\u002F12-performance-a11y-security","性能、可访问性和安全",[108,31505,31506],{},[111,31507,31509],{"href":31508},".\u002F13-build-deploy-engineering","构建、部署和工程化",[108,31511,31512],{},[111,31513,31515],{"href":31514},".\u002F14-capstone","毕业项目：课程任务管理器",{"title":29,"searchDepth":43,"depth":43,"links":31517},[31518],{"id":31429,"depth":43,"text":31429},{"order":37},"\u002Fsource\u002Fcode\u002Fvue",{"title":119,"description":31426},"source\u002Fcode\u002Fvue\u002Findex","KcMxylblM9g8ymvngOHLDEw8lc1CrYAm9zk-cgLC6hc",{"id":31525,"title":31526,"body":31527,"description":31548,"extension":85,"meta":31549,"navigation":52,"path":31550,"seo":31551,"stem":31552,"__hash__":31553},"notes\u002Fsource\u002Fcs\u002Falgo\u002Findex.md","Algorithm Design & Analysis",{"type":8,"value":31528,"toc":31546},[31529,31542],[16016,31530,31533,31534,31533,31537],{"className":31531},[31532],"page-header","\n ",[11,31535,31526],{"id":31536},"algorithm-design-analysis",[15,31538,31541],{"className":31539},[31540],"page-subtitle","算法设计与分析",[31543,31544],"auto-index",{"path":31545,"unwrap-single-group":29},"\u002Fcs\u002Falgo\u002F",{"title":29,"searchDepth":43,"depth":43,"links":31547},[],"算法设计与分析学习笔记，涵盖递归、分治、动态规划等核心算法策略，附历年期末考试真题。",{},"\u002Fsource\u002Fcs\u002Falgo",{"title":31526,"description":31548},"source\u002Fcs\u002Falgo\u002Findex","TLVEE4YoslAcT3jedLv0b3QseDvz7sRc4UkebDOGpHM",{"id":31555,"title":31556,"body":31557,"description":29,"extension":85,"meta":45631,"navigation":52,"path":45632,"seo":45633,"stem":45634,"__hash__":45635},"notes\u002Fsource\u002Fcs\u002Falgo\u002Fnotes\u002Fchap2.md","第二章：递归（Recursion）",{"type":8,"value":31558,"toc":45625},[31559,31562,31566,31571,31599,31604,31626,31631,31704,31709,32187,32190,32194,32489,32494,33172,33176,33491,33495,34333,34337,35193,35197,36964,36968,37349,37351,37355,37377,37379,37383,37433,37437,38666,38670,41834,41838,43014,43018],[11,31560,31556],{"id":31561},"第二章递归recursion",[253,31563,31565],{"id":31564},"一-什么是递归-知识点","一、 什么是递归 (知识点)",[15,31567,31568],{},[2627,31569,31570],{},"1. 递归的定义",[105,31572,31573,31579,31593],{},[108,31574,31575,31578],{},[2627,31576,31577],{},"概念","：在定义一个过程或函数时出现调用本过程或本函数的成分，称为递归。",[108,31580,31581,31584,31585,31588,31589,31592],{},[2627,31582,31583],{},"分类","：调用自身称为",[2627,31586,31587],{},"直接递归","；过程p调用q，q又调用p称为",[2627,31590,31591],{},"间接递归","（任何间接递归都可等价转换为直接递归）。",[108,31594,31595,31598],{},[2627,31596,31597],{},"尾递归","：如果递归调用语句是函数中的最后一条执行语句，则称为尾递归。",[15,31600,31601],{},[2627,31602,31603],{},"2. 使用递归解决问题的三个条件",[105,31605,31606,31612,31619],{},[108,31607,31608,31609,347],{},"问题可以转化为一个或多个子问题求解，且子问题的求解方法与原问题完全相同，只是在",[2627,31610,31611],{},"数量规模上不同",[108,31613,31614,31615,31618],{},"递归调用的次数必须是",[2627,31616,31617],{},"有限","的。",[108,31620,31621,31622,31625],{},"必须有",[2627,31623,31624],{},"结束递归的条件","（递归出口）来终止递归。",[15,31627,31628],{},[2627,31629,31630],{},"3. 何时使用递归（三大场景）",[105,31632,31633,31692,31698],{},[108,31634,31635,31638,31639,31691],{},[2627,31636,31637],{},"定义的数学公式是递归的","：如求阶乘 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递归模型与执行过程",[105,31710,31711,32169],{},[108,31712,31713,31716,31717],{},[2627,31714,31715],{},"递归模型","由两部分组成：\n",[105,31718,31719,31902],{},[108,31720,31721,31724,31725,31901],{},[2627,31722,31723],{},"递归出口","：确定递归到何时结束（如 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c_j)",[34,31975,31977,32052],{"className":31976,"ariaHidden":758},[31672],[34,31978,31980,31983,31986,31989,32040,32043,32046,32049],{"className":31979},[31676],[34,31981],{"className":31982,"style":31774},[31680],[34,31984,31739],{"className":31985,"style":31778},[31685,31686],[34,31987,456],{"className":31988},[31782],[34,31990,31992,31995],{"className":31991},[31685],[34,31993,31747],{"className":31994},[31685,31686],[34,31996,31998],{"className":31997},[31792],[34,31999,32001,32031],{"className":32000},[31796,31797],[34,32002,32004,32028],{"className":32003},[31801],[34,32005,32007],{"className":32006,"style":31806},[31805],[34,32008,32009,32012],{"style":31809},[34,32010],{"className":32011,"style":31814},[31813],[34,32013,32015],{"className":32014},[31818,31819,31820,31821],[34,32016,32018,32021,32025],{"className":32017},[31685,31821],[34,32019,31660],{"className":32020},[31685,31686,31821],[34,32022,223],{"className":32023},[32024,31821],"mbin",[34,32026,640],{"className":32027},[31685,31821],[34,32029,31829],{"className":32030},[31828],[34,32032,32034],{"className":32033},[31801],[34,32035,32038],{"className":32036,"style":32037},[31805],"height:0.2083em;",[34,32039],{},[34,32041,2557],{"className":32042},[31690],[34,32044],{"className":32045,"style":31846},[31845],[34,32047,202],{"className":32048},[31850],[34,32050],{"className":32051,"style":31846},[31845],[34,32053,32055,32059,32063,32066,32069,32072,32113,32116,32120,32124,32166],{"className":32054},[31676],[34,32056],{"className":32057,"style":32058},[31680],"height:1.0361em;vertical-align:-0.2861em;",[34,32060,31942],{"className":32061,"style":32062},[31685,31686],"margin-right:0.0359em;",[34,32064,456],{"className":32065},[31782],[34,32067,31739],{"className":32068,"style":31778},[31685,31686],[34,32070,456],{"className":32071},[31782],[34,32073,32075,32078],{"className":32074},[31685],[34,32076,31747],{"className":32077},[31685,31686],[34,32079,32081],{"className":32080},[31792],[34,32082,32084,32105],{"className":32083},[31796,31797],[34,32085,32087,32102],{"className":32086},[31801],[34,32088,32091],{"className":32089,"style":32090},[31805],"height:0.3117em;",[34,32092,32093,32096],{"style":31809},[34,32094],{"className":32095,"style":31814},[31813],[34,32097,32099],{"className":32098},[31818,31819,31820,31821],[34,32100,31955],{"className":32101},[31685,31686,31821],[34,32103,31829],{"className":32104},[31828],[34,32106,32108],{"className":32107},[31801],[34,32109,32111],{"className":32110,"style":31836},[31805],[34,32112],{},[34,32114,2557],{"className":32115},[31690],[34,32117,31960],{"className":32118},[32119],"mpunct",[34,32121],{"className":32122,"style":32123},[31845],"margin-right:0.1667em;",[34,32125,32127,32130],{"className":32126},[31685],[34,32128,31965],{"className":32129},[31685,31686],[34,32131,32133],{"className":32132},[31792],[34,32134,32136,32157],{"className":32135},[31796,31797],[34,32137,32139,32154],{"className":32138},[31801],[34,32140,32142],{"className":32141,"style":32090},[31805],[34,32143,32144,32147],{"style":31809},[34,32145],{"className":32146,"style":31814},[31813],[34,32148,32150],{"className":32149},[31818,31819,31820,31821],[34,32151,31968],{"className":32152,"style":32153},[31685,31686,31821],"margin-right:0.0572em;",[34,32155,31829],{"className":32156},[31828],[34,32158,32160],{"className":32159},[31801],[34,32161,32164],{"className":32162,"style":32163},[31805],"height:0.2861em;",[34,32165],{},[34,32167,2557],{"className":32168},[31690],[108,32170,32171,32174,32175,32186],{},[2627,32172,32173],{},"底层执行过程","：系统利用",[2627,32176,32177,32178,32185],{},"栈（系统栈）",[2627,32179,32180,32181,32184],{},"来为每一次调用开辟存储单元，保存返回地址和被中断函数的参量值。执行分为两步：",[2627,32182,32183],{},"分解过程","（大问题化小问题直到出口）和","求值过程","（退栈计算并返回结果）。递归深度越深，开辟的栈空间越大。",[32188,32189],"hr",{},[253,32191,32193],{"id":32192},"二-递归算法设计方法与经典例题","二、 递归算法设计方法与经典例题",[15,32195,32196,32199,32200,32301,32302,32388,32389,32488],{},[2627,32197,32198],{},"设计步骤","：①抽象出合理的小问题 ",[34,32201,32203,32234],{"className":32202},[31642],[34,32204,32206],{"className":32205},[31646],[31648,32207,32208],{"xmlns":31650},[31652,32209,32210,32231],{},[31655,32211,32212,32214,32216,32229],{},[31658,32213,31739],{},[31662,32215,456],{"stretchy":702},[31743,32217,32218,32220],{},[31658,32219,31747],{},[31655,32221,32222,32224,32227],{},[31658,32223,31660],{},[31662,32225,32226],{},"−",[31749,32228,640],{},[31662,32230,2557],{"stretchy":702},[31665,32232,32233],{"encoding":31667},"f(s_{n-1})",[34,32235,32237],{"className":32236,"ariaHidden":758},[31672],[34,32238,32240,32243,32246,32249,32298],{"className":32239},[31676],[34,32241],{"className":32242,"style":31774},[31680],[34,32244,31739],{"className":32245,"style":31778},[31685,31686],[34,32247,456],{"className":32248},[31782],[34,32250,32252,32255],{"className":32251},[31685],[34,32253,31747],{"className":32254},[31685,31686],[34,32256,32258],{"className":32257},[31792],[34,32259,32261,32290],{"className":32260},[31796,31797],[34,32262,32264,32287],{"className":32263},[31801],[34,32265,32267],{"className":32266,"style":31806},[31805],[34,32268,32269,32272],{"style":31809},[34,32270],{"className":32271,"style":31814},[31813],[34,32273,32275],{"className":32274},[31818,31819,31820,31821],[34,32276,32278,32281,32284],{"className":32277},[31685,31821],[34,32279,31660],{"className":32280},[31685,31686,31821],[34,32282,32226],{"className":32283},[32024,31821],[34,32285,640],{"className":32286},[31685,31821],[34,32288,31829],{"className":32289},[31828],[34,32291,32293],{"className":32292},[31801],[34,32294,32296],{"className":32295,"style":32037},[31805],[34,32297],{},[34,32299,2557],{"className":32300},[31690],"；②在此基础上确定 ",[34,32303,32305,32329],{"className":32304},[31642],[34,32306,32308],{"className":32307},[31646],[31648,32309,32310],{"xmlns":31650},[31652,32311,32312,32326],{},[31655,32313,32314,32316,32318,32324],{},[31658,32315,31739],{},[31662,32317,456],{"stretchy":702},[31743,32319,32320,32322],{},[31658,32321,31747],{},[31658,32323,31660],{},[31662,32325,2557],{"stretchy":702},[31665,32327,32328],{"encoding":31667},"f(s_n)",[34,32330,32332],{"className":32331,"ariaHidden":758},[31672],[34,32333,32335,32338,32341,32344,32385],{"className":32334},[31676],[34,32336],{"className":32337,"style":31774},[31680],[34,32339,31739],{"className":32340,"style":31778},[31685,31686],[34,32342,456],{"className":32343},[31782],[34,32345,32347,32350],{"className":32346},[31685],[34,32348,31747],{"className":32349},[31685,31686],[34,32351,32353],{"className":32352},[31792],[34,32354,32356,32377],{"className":32355},[31796,31797],[34,32357,32359,32374],{"className":32358},[31801],[34,32360,32363],{"className":32361,"style":32362},[31805],"height:0.1514em;",[34,32364,32365,32368],{"style":31809},[34,32366],{"className":32367,"style":31814},[31813],[34,32369,32371],{"className":32370},[31818,31819,31820,31821],[34,32372,31660],{"className":32373},[31685,31686,31821],[34,32375,31829],{"className":32376},[31828],[34,32378,32380],{"className":32379},[31801],[34,32381,32383],{"className":32382,"style":31836},[31805],[34,32384],{},[34,32386,2557],{"className":32387},[31690]," 的解与 ",[34,32390,32392,32421],{"className":32391},[31642],[34,32393,32395],{"className":32394},[31646],[31648,32396,32397],{"xmlns":31650},[31652,32398,32399,32419],{},[31655,32400,32401,32403,32405,32417],{},[31658,32402,31739],{},[31662,32404,456],{"stretchy":702},[31743,32406,32407,32409],{},[31658,32408,31747],{},[31655,32410,32411,32413,32415],{},[31658,32412,31660],{},[31662,32414,32226],{},[31749,32416,640],{},[31662,32418,2557],{"stretchy":702},[31665,32420,32233],{"encoding":31667},[34,32422,32424],{"className":32423,"ariaHidden":758},[31672],[34,32425,32427,32430,32433,32436,32485],{"className":32426},[31676],[34,32428],{"className":32429,"style":31774},[31680],[34,32431,31739],{"className":32432,"style":31778},[31685,31686],[34,32434,456],{"className":32435},[31782],[34,32437,32439,32442],{"className":32438},[31685],[34,32440,31747],{"className":32441},[31685,31686],[34,32443,32445],{"className":32444},[31792],[34,32446,32448,32477],{"className":32447},[31796,31797],[34,32449,32451,32474],{"className":32450},[31801],[34,32452,32454],{"className":32453,"style":31806},[31805],[34,32455,32456,32459],{"style":31809},[34,32457],{"className":32458,"style":31814},[31813],[34,32460,32462],{"className":32461},[31818,31819,31820,31821],[34,32463,32465,32468,32471],{"className":32464},[31685,31821],[34,32466,31660],{"className":32467},[31685,31686,31821],[34,32469,32226],{"className":32470},[32024,31821],[34,32472,640],{"className":32473},[31685,31821],[34,32475,31829],{"className":32476},[31828],[34,32478,32480],{"className":32479},[31801],[34,32481,32483],{"className":32482,"style":32037},[31805],[34,32484],{},[34,32486,2557],{"className":32487},[31690]," 的关系；③确定特定情况下的解作为递归出口。",[32490,32491,32493],"h4",{"id":32492},"例题-1求数组的最大元素","例题 1：求数组的最大元素",[105,32495,32496,32561,32910],{},[108,32497,32498,32501,32502,32531,32532,32560],{},[2627,32499,32500],{},"题意","：求包含 ",[34,32503,32505,32518],{"className":32504},[31642],[34,32506,32508],{"className":32507},[31646],[31648,32509,32510],{"xmlns":31650},[31652,32511,32512,32516],{},[31655,32513,32514],{},[31658,32515,31660],{},[31665,32517,31660],{"encoding":31667},[34,32519,32521],{"className":32520,"ariaHidden":758},[31672],[34,32522,32524,32528],{"className":32523},[31676],[34,32525],{"className":32526,"style":32527},[31680],"height:0.4306em;",[34,32529,31660],{"className":32530},[31685,31686]," 个元素的整数数组 ",[34,32533,32535,32548],{"className":32534},[31642],[34,32536,32538],{"className":32537},[31646],[31648,32539,32540],{"xmlns":31650},[31652,32541,32542,32546],{},[31655,32543,32544],{},[31658,32545,111],{},[31665,32547,111],{"encoding":31667},[34,32549,32551],{"className":32550,"ariaHidden":758},[31672],[34,32552,32554,32557],{"className":32553},[31676],[34,32555],{"className":32556,"style":32527},[31680],[34,32558,111],{"className":32559},[31685,31686]," 的最大元素。",[108,32562,32563,32566,32567,32621,32622,32650,32651,32829,32830,32881,32882,347],{},[2627,32564,32565],{},"推导思路 (方法一)","：将问题分为前 ",[34,32568,32570,32588],{"className":32569},[31642],[34,32571,32573],{"className":32572},[31646],[31648,32574,32575],{"xmlns":31650},[31652,32576,32577,32585],{},[31655,32578,32579,32581,32583],{},[31658,32580,31660],{},[31662,32582,32226],{},[31749,32584,640],{},[31665,32586,32587],{"encoding":31667},"n-1",[34,32589,32591,32611],{"className":32590,"ariaHidden":758},[31672],[34,32592,32594,32598,32601,32605,32608],{"className":32593},[31676],[34,32595],{"className":32596,"style":32597},[31680],"height:0.6667em;vertical-align:-0.0833em;",[34,32599,31660],{"className":32600},[31685,31686],[34,32602],{"className":32603,"style":32604},[31845],"margin-right:0.2222em;",[34,32606,32226],{"className":32607},[32024],[34,32609],{"className":32610,"style":32604},[31845],[34,32612,32614,32618],{"className":32613},[31676],[34,32615],{"className":32616,"style":32617},[31680],"height:0.6444em;",[34,32619,640],{"className":32620},[31685]," 个元素的最大值（小问题）和第 ",[34,32623,32625,32638],{"className":32624},[31642],[34,32626,32628],{"className":32627},[31646],[31648,32629,32630],{"xmlns":31650},[31652,32631,32632,32636],{},[31655,32633,32634],{},[31658,32635,31660],{},[31665,32637,31660],{"encoding":31667},[34,32639,32641],{"className":32640,"ariaHidden":758},[31672],[34,32642,32644,32647],{"className":32643},[31676],[34,32645],{"className":32646,"style":32527},[31680],[34,32648,31660],{"className":32649},[31685,31686]," 个元素。递推关系为 ",[34,32652,32654,32719],{"className":32653},[31642],[34,32655,32657],{"className":32656},[31646],[31648,32658,32659],{"xmlns":31650},[31652,32660,32661,32716],{},[31655,32662,32663,32665,32667,32669,32671,32673,32675,32677,32680,32683,32686,32688,32690,32692,32694,32696,32698,32700,32702,32704,32706,32708,32710,32713],{},[31658,32664,31739],{},[31662,32666,456],{"stretchy":702},[31658,32668,111],{},[31662,32670,31960],{"separator":758},[31658,32672,31660],{},[31662,32674,2557],{"stretchy":702},[31662,32676,202],{},[31658,32678,32679],{},"max",[31662,32681,32682],{},"⁡",[31662,32684,32685],{"stretchy":702},"{",[31658,32687,31739],{},[31662,32689,456],{"stretchy":702},[31658,32691,111],{},[31662,32693,31960],{"separator":758},[31658,32695,31660],{},[31662,32697,32226],{},[31749,32699,640],{},[31662,32701,2557],{"stretchy":702},[31662,32703,31960],{"separator":758},[31658,32705,111],{},[31662,32707,15972],{"stretchy":702},[31658,32709,31660],{},[31662,32711,32712],{"stretchy":702},"]",[31662,32714,32715],{"stretchy":702},"}",[31665,32717,32718],{"encoding":31667},"f(a, n) = \\max\\{f(a, n-1), a[n]\\}",[34,32720,32722,32758,32798],{"className":32721,"ariaHidden":758},[31672],[34,32723,32725,32728,32731,32734,32737,32740,32743,32746,32749,32752,32755],{"className":32724},[31676],[34,32726],{"className":32727,"style":31774},[31680],[34,32729,31739],{"className":32730,"style":31778},[31685,31686],[34,32732,456],{"className":32733},[31782],[34,32735,111],{"className":32736},[31685,31686],[34,32738,31960],{"className":32739},[32119],[34,32741],{"className":32742,"style":32123},[31845],[34,32744,31660],{"className":32745},[31685,31686],[34,32747,2557],{"className":32748},[31690],[34,32750],{"className":32751,"style":31846},[31845],[34,32753,202],{"className":32754},[31850],[34,32756],{"className":32757,"style":31846},[31845],[34,32759,32761,32764,32768,32771,32774,32777,32780,32783,32786,32789,32792,32795],{"className":32760},[31676],[34,32762],{"className":32763,"style":31774},[31680],[34,32765,32679],{"className":32766},[32767],"mop",[34,32769,32685],{"className":32770},[31782],[34,32772,31739],{"className":32773,"style":31778},[31685,31686],[34,32775,456],{"className":32776},[31782],[34,32778,111],{"className":32779},[31685,31686],[34,32781,31960],{"className":32782},[32119],[34,32784],{"className":32785,"style":32123},[31845],[34,32787,31660],{"className":32788},[31685,31686],[34,32790],{"className":32791,"style":32604},[31845],[34,32793,32226],{"className":32794},[32024],[34,32796],{"className":32797,"style":32604},[31845],[34,32799,32801,32804,32807,32810,32813,32816,32819,32822,32825],{"className":32800},[31676],[34,32802],{"className":32803,"style":31774},[31680],[34,32805,640],{"className":32806},[31685],[34,32808,2557],{"className":32809},[31690],[34,32811,31960],{"className":32812},[32119],[34,32814],{"className":32815,"style":32123},[31845],[34,32817,111],{"className":32818},[31685,31686],[34,32820,15972],{"className":32821},[31782],[34,32823,31660],{"className":32824},[31685,31686],[34,32826,32828],{"className":32827},[31690],"]}","，出口为 ",[34,32831,32833,32851],{"className":32832},[31642],[34,32834,32836],{"className":32835},[31646],[31648,32837,32838],{"xmlns":31650},[31652,32839,32840,32848],{},[31655,32841,32842,32844,32846],{},[31658,32843,31660],{},[31662,32845,202],{},[31749,32847,640],{},[31665,32849,32850],{"encoding":31667},"n=1",[34,32852,32854,32872],{"className":32853,"ariaHidden":758},[31672],[34,32855,32857,32860,32863,32866,32869],{"className":32856},[31676],[34,32858],{"className":32859,"style":32527},[31680],[34,32861,31660],{"className":32862},[31685,31686],[34,32864],{"className":32865,"style":31846},[31845],[34,32867,202],{"className":32868},[31850],[34,32870],{"className":32871,"style":31846},[31845],[34,32873,32875,32878],{"className":32874},[31676],[34,32876],{"className":32877,"style":32617},[31680],[34,32879,640],{"className":32880},[31685]," 时返回 ",[34,32883,32885,32898],{"className":32884},[31642],[34,32886,32888],{"className":32887},[31646],[31648,32889,32890],{"xmlns":31650},[31652,32891,32892,32896],{},[31655,32893,32894],{},[31658,32895,111],{},[31665,32897,111],{"encoding":31667},[34,32899,32901],{"className":32900,"ariaHidden":758},[31672],[34,32902,32904,32907],{"className":32903},[31676],[34,32905],{"className":32906,"style":32527},[31680],[34,32908,111],{"className":32909},[31685,31686],[108,32911,32912,32915,32916,32995,32996,33073,33074,33127,33128,347],{},[2627,32913,32914],{},"推导思路 (方法二：分治思想)","：采用二分法，求出左半段最大值 ",[34,32917,32919,32942],{"className":32918},[31642],[34,32920,32922],{"className":32921},[31646],[31648,32923,32924],{"xmlns":31650},[31652,32925,32926,32939],{},[31655,32927,32928,32930,32932],{},[31658,32929,31759],{},[31658,32931,111],{},[31743,32933,32934,32937],{},[31658,32935,32936],{},"x",[31749,32938,640],{},[31665,32940,32941],{"encoding":31667},"max_1",[34,32943,32945],{"className":32944,"ariaHidden":758},[31672],[34,32946,32948,32951,32955],{"className":32947},[31676],[34,32949],{"className":32950,"style":31860},[31680],[34,32952,32954],{"className":32953},[31685,31686],"ma",[34,32956,32958,32961],{"className":32957},[31685],[34,32959,32936],{"className":32960},[31685,31686],[34,32962,32964],{"className":32963},[31792],[34,32965,32967,32987],{"className":32966},[31796,31797],[34,32968,32970,32984],{"className":32969},[31801],[34,32971,32973],{"className":32972,"style":31806},[31805],[34,32974,32975,32978],{"style":31809},[34,32976],{"className":32977,"style":31814},[31813],[34,32979,32981],{"className":32980},[31818,31819,31820,31821],[34,32982,640],{"className":32983},[31685,31821],[34,32985,31829],{"className":32986},[31828],[34,32988,32990],{"className":32989},[31801],[34,32991,32993],{"className":32992,"style":31836},[31805],[34,32994],{}," 和右半段最大值 ",[34,32997,32999,33021],{"className":32998},[31642],[34,33000,33002],{"className":33001},[31646],[31648,33003,33004],{"xmlns":31650},[31652,33005,33006,33018],{},[31655,33007,33008,33010,33012],{},[31658,33009,31759],{},[31658,33011,111],{},[31743,33013,33014,33016],{},[31658,33015,32936],{},[31749,33017,768],{},[31665,33019,33020],{"encoding":31667},"max_2",[34,33022,33024],{"className":33023,"ariaHidden":758},[31672],[34,33025,33027,33030,33033],{"className":33026},[31676],[34,33028],{"className":33029,"style":31860},[31680],[34,33031,32954],{"className":33032},[31685,31686],[34,33034,33036,33039],{"className":33035},[31685],[34,33037,32936],{"className":33038},[31685,31686],[34,33040,33042],{"className":33041},[31792],[34,33043,33045,33065],{"className":33044},[31796,31797],[34,33046,33048,33062],{"className":33047},[31801],[34,33049,33051],{"className":33050,"style":31806},[31805],[34,33052,33053,33056],{"style":31809},[34,33054],{"className":33055,"style":31814},[31813],[34,33057,33059],{"className":33058},[31818,31819,31820,31821],[34,33060,768],{"className":33061},[31685,31821],[34,33063,31829],{"className":33064},[31828],[34,33066,33068],{"className":33067},[31801],[34,33069,33071],{"className":33070,"style":31836},[31805],[34,33072],{},"，取两者的较大者。出口为 ",[34,33075,33077,33095],{"className":33076},[31642],[34,33078,33080],{"className":33079},[31646],[31648,33081,33082],{"xmlns":31650},[31652,33083,33084,33092],{},[31655,33085,33086,33088,33090],{},[31658,33087,31955],{},[31662,33089,202],{},[31658,33091,31968],{},[31665,33093,33094],{"encoding":31667},"i=j",[34,33096,33098,33117],{"className":33097,"ariaHidden":758},[31672],[34,33099,33101,33105,33108,33111,33114],{"className":33100},[31676],[34,33102],{"className":33103,"style":33104},[31680],"height:0.6595em;",[34,33106,31955],{"className":33107},[31685,31686],[34,33109],{"className":33110,"style":31846},[31845],[34,33112,202],{"className":33113},[31850],[34,33115],{"className":33116,"style":31846},[31845],[34,33118,33120,33124],{"className":33119},[31676],[34,33121],{"className":33122,"style":33123},[31680],"height:0.854em;vertical-align:-0.1944em;",[34,33125,31968],{"className":33126,"style":32153},[31685,31686],"（只有一个元素）时返回 ",[34,33129,33131,33151],{"className":33130},[31642],[34,33132,33134],{"className":33133},[31646],[31648,33135,33136],{"xmlns":31650},[31652,33137,33138,33148],{},[31655,33139,33140,33142,33144,33146],{},[31658,33141,111],{},[31662,33143,15972],{"stretchy":702},[31658,33145,31955],{},[31662,33147,32712],{"stretchy":702},[31665,33149,33150],{"encoding":31667},"a[i]",[34,33152,33154],{"className":33153,"ariaHidden":758},[31672],[34,33155,33157,33160,33163,33166,33169],{"className":33156},[31676],[34,33158],{"className":33159,"style":31774},[31680],[34,33161,111],{"className":33162},[31685,31686],[34,33164,15972],{"className":33165},[31782],[34,33167,31955],{"className":33168},[31685,31686],[34,33170,32712],{"className":33171},[31690],[32490,33173,33175],{"id":33174},"例题-2递归数据结构的算法设计单链表","例题 2：递归数据结构的算法设计（单链表）",[105,33177,33178,33398],{},[108,33179,33180,33183,33184],{},[2627,33181,33182],{},"题意1（释放单链表所有结点）","：\n",[105,33185,33186],{},[108,33187,33188,33191,33192,33222,33223,33293,33294,33322,33323,33326,33327,33397],{},[2627,33189,33190],{},"推导","：要释放首结点为 ",[34,33193,33195,33209],{"className":33194},[31642],[34,33196,33198],{"className":33197},[31646],[31648,33199,33200],{"xmlns":31650},[31652,33201,33202,33207],{},[31655,33203,33204],{},[31658,33205,33206],{},"L",[31665,33208,33206],{"encoding":31667},[34,33210,33212],{"className":33211,"ariaHidden":758},[31672],[34,33213,33215,33219],{"className":33214},[31676],[34,33216],{"className":33217,"style":33218},[31680],"height:0.6833em;",[34,33220,33206],{"className":33221},[31685,31686]," 的单链表，先递归释放以 ",[34,33224,33226,33253],{"className":33225},[31642],[34,33227,33229],{"className":33228},[31646],[31648,33230,33231],{"xmlns":31650},[31652,33232,33233,33250],{},[31655,33234,33235,33237,33240,33242,33245,33247],{},[31658,33236,33206],{},[31662,33238,33239],{},"→",[31658,33241,31660],{},[31658,33243,33244],{},"e",[31658,33246,32936],{},[31658,33248,33249],{},"t",[31665,33251,33252],{"encoding":31667},"L\\rightarrow next",[34,33254,33256,33274],{"className":33255,"ariaHidden":758},[31672],[34,33257,33259,33262,33265,33268,33271],{"className":33258},[31676],[34,33260],{"className":33261,"style":33218},[31680],[34,33263,33206],{"className":33264},[31685,31686],[34,33266],{"className":33267,"style":31846},[31845],[34,33269,33239],{"className":33270},[31850],[34,33272],{"className":33273,"style":31846},[31845],[34,33275,33277,33281,33284,33287,33290],{"className":33276},[31676],[34,33278],{"className":33279,"style":33280},[31680],"height:0.6151em;",[34,33282,31660],{"className":33283},[31685,31686],[34,33285,33244],{"className":33286},[31685,31686],[34,33288,32936],{"className":33289},[31685,31686],[34,33291,33249],{"className":33292},[31685,31686]," 为首的剩余链表，然后再释放 ",[34,33295,33297,33310],{"className":33296},[31642],[34,33298,33300],{"className":33299},[31646],[31648,33301,33302],{"xmlns":31650},[31652,33303,33304,33308],{},[31655,33305,33306],{},[31658,33307,33206],{},[31665,33309,33206],{"encoding":31667},[34,33311,33313],{"className":33312,"ariaHidden":758},[31672],[34,33314,33316,33319],{"className":33315},[31676],[34,33317],{"className":33318,"style":33218},[31680],[34,33320,33206],{"className":33321},[31685,31686]," 结点本身（",[31,33324,33325],{},"free(L)","）。出口是 ",[34,33328,33330,33358],{"className":33329},[31642],[34,33331,33333],{"className":33332},[31646],[31648,33334,33335],{"xmlns":31650},[31652,33336,33337,33355],{},[31655,33338,33339,33341,33343,33345,33348,33351,33353],{},[31658,33340,33206],{},[31662,33342,202],{},[31662,33344,202],{},[31658,33346,33347],{},"N",[31658,33349,33350],{},"U",[31658,33352,33206],{},[31658,33354,33206],{},[31665,33356,33357],{"encoding":31667},"L == NULL",[34,33359,33361,33380],{"className":33360,"ariaHidden":758},[31672],[34,33362,33364,33367,33370,33373,33377],{"className":33363},[31676],[34,33365],{"className":33366,"style":33218},[31680],[34,33368,33206],{"className":33369},[31685,31686],[34,33371],{"className":33372,"style":31846},[31845],[34,33374,33376],{"className":33375},[31850],"==",[34,33378],{"className":33379,"style":31846},[31845],[34,33381,33383,33386,33390,33393],{"className":33382},[31676],[34,33384],{"className":33385,"style":33218},[31680],[34,33387,33347],{"className":33388,"style":33389},[31685,31686],"margin-right:0.109em;",[34,33391,33350],{"className":33392,"style":33389},[31685,31686],[34,33394,33396],{"className":33395},[31685,31686],"LL"," 时不做任何事。",[108,33399,33400,33183,33403],{},[2627,33401,33402],{},"题意2（逆序输出单链表的值）",[105,33404,33405],{},[108,33406,33407,33409,33410,33413,33414,33417,33418,33421,33422,33490],{},[2627,33408,33190],{},"：要逆序输出，必须先通过递归调用 ",[31,33411,33412],{},"OutputFromTail(L->next)"," 输出后续结点，",[2627,33415,33416],{},"递归调用返回后","，再执行 ",[31,33419,33420],{},"printf(L->data)"," 输出当前结点的值。出口为 ",[34,33423,33425,33451],{"className":33424},[31642],[34,33426,33428],{"className":33427},[31646],[31648,33429,33430],{"xmlns":31650},[31652,33431,33432,33448],{},[31655,33433,33434,33436,33438,33440,33442,33444,33446],{},[31658,33435,33206],{},[31662,33437,5504],{"stretchy":702},[31662,33439,202],{},[31658,33441,33347],{},[31658,33443,33350],{},[31658,33445,33206],{},[31658,33447,33206],{},[31665,33449,33450],{"encoding":31667},"L!=NULL",[34,33452,33454,33475],{"className":33453,"ariaHidden":758},[31672],[34,33455,33457,33460,33463,33466,33469,33472],{"className":33456},[31676],[34,33458],{"className":33459,"style":31681},[31680],[34,33461,33206],{"className":33462},[31685,31686],[34,33464,5504],{"className":33465},[31690],[34,33467],{"className":33468,"style":31846},[31845],[34,33470,202],{"className":33471},[31850],[34,33473],{"className":33474,"style":31846},[31845],[34,33476,33478,33481,33484,33487],{"className":33477},[31676],[34,33479],{"className":33480,"style":33218},[31680],[34,33482,33347],{"className":33483,"style":33389},[31685,31686],[34,33485,33350],{"className":33486,"style":33389},[31685,31686],[34,33488,33396],{"className":33489},[31685,31686]," 的取反。",[32490,33492,33494],{"id":33493},"例题-3由先序和中序序列创建二叉树","例题 3：由先序和中序序列创建二叉树",[105,33496,33497,33560],{},[108,33498,33499,33501,33502,33530,33531,33559],{},[2627,33500,32500],{},"：给定二叉树的先序序列 ",[34,33503,33505,33518],{"className":33504},[31642],[34,33506,33508],{"className":33507},[31646],[31648,33509,33510],{"xmlns":31650},[31652,33511,33512,33516],{},[31655,33513,33514],{},[31658,33515,111],{},[31665,33517,111],{"encoding":31667},[34,33519,33521],{"className":33520,"ariaHidden":758},[31672],[34,33522,33524,33527],{"className":33523},[31676],[34,33525],{"className":33526,"style":32527},[31680],[34,33528,111],{"className":33529},[31685,31686]," 和中序序列 ",[34,33532,33534,33547],{"className":33533},[31642],[34,33535,33537],{"className":33536},[31646],[31648,33538,33539],{"xmlns":31650},[31652,33540,33541,33545],{},[31655,33542,33543],{},[31658,33544,1697],{},[31665,33546,1697],{"encoding":31667},[34,33548,33550],{"className":33549,"ariaHidden":758},[31672],[34,33551,33553,33556],{"className":33552},[31676],[34,33554],{"className":33555,"style":31681},[31680],[34,33557,1697],{"className":33558},[31685,31686],"，构造二叉链表。",[108,33561,33562,33183,33564],{},[2627,33563,33190],{},[105,33565,33566,33643,33911,34322],{},[108,33567,33568,33569,33639,33640,347],{},"先序序列的第一个元素 ",[34,33570,33572,33590],{"className":33571},[31642],[34,33573,33575],{"className":33574},[31646],[31648,33576,33577],{"xmlns":31650},[31652,33578,33579,33587],{},[31655,33580,33581],{},[31743,33582,33583,33585],{},[31658,33584,111],{},[31749,33586,459],{},[31665,33588,33589],{"encoding":31667},"a_0",[34,33591,33593],{"className":33592,"ariaHidden":758},[31672],[34,33594,33596,33599],{"className":33595},[31676],[34,33597],{"className":33598,"style":31860},[31680],[34,33600,33602,33605],{"className":33601},[31685],[34,33603,111],{"className":33604},[31685,31686],[34,33606,33608],{"className":33607},[31792],[34,33609,33611,33631],{"className":33610},[31796,31797],[34,33612,33614,33628],{"className":33613},[31801],[34,33615,33617],{"className":33616,"style":31806},[31805],[34,33618,33619,33622],{"style":31809},[34,33620],{"className":33621,"style":31814},[31813],[34,33623,33625],{"className":33624},[31818,31819,31820,31821],[34,33626,459],{"className":33627},[31685,31821],[34,33629,31829],{"className":33630},[31828],[34,33632,33634],{"className":33633},[31801],[34,33635,33637],{"className":33636,"style":31836},[31805],[34,33638],{}," 必定是",[2627,33641,33642],{},"根结点",[108,33644,33645,33646,33674,33675,33744,33745,33775,33776,31901],{},"在中序序列 ",[34,33647,33649,33662],{"className":33648},[31642],[34,33650,33652],{"className":33651},[31646],[31648,33653,33654],{"xmlns":31650},[31652,33655,33656,33660],{},[31655,33657,33658],{},[31658,33659,1697],{},[31665,33661,1697],{"encoding":31667},[34,33663,33665],{"className":33664,"ariaHidden":758},[31672],[34,33666,33668,33671],{"className":33667},[31676],[34,33669],{"className":33670,"style":31681},[31680],[34,33672,1697],{"className":33673},[31685,31686]," 中找到 ",[34,33676,33678,33695],{"className":33677},[31642],[34,33679,33681],{"className":33680},[31646],[31648,33682,33683],{"xmlns":31650},[31652,33684,33685,33693],{},[31655,33686,33687],{},[31743,33688,33689,33691],{},[31658,33690,111],{},[31749,33692,459],{},[31665,33694,33589],{"encoding":31667},[34,33696,33698],{"className":33697,"ariaHidden":758},[31672],[34,33699,33701,33704],{"className":33700},[31676],[34,33702],{"className":33703,"style":31860},[31680],[34,33705,33707,33710],{"className":33706},[31685],[34,33708,111],{"className":33709},[31685,31686],[34,33711,33713],{"className":33712},[31792],[34,33714,33716,33736],{"className":33715},[31796,31797],[34,33717,33719,33733],{"className":33718},[31801],[34,33720,33722],{"className":33721,"style":31806},[31805],[34,33723,33724,33727],{"style":31809},[34,33725],{"className":33726,"style":31814},[31813],[34,33728,33730],{"className":33729},[31818,31819,31820,31821],[34,33731,459],{"className":33732},[31685,31821],[34,33734,31829],{"className":33735},[31828],[34,33737,33739],{"className":33738},[31801],[34,33740,33742],{"className":33741,"style":31836},[31805],[34,33743],{}," 的位置 ",[34,33746,33748,33762],{"className":33747},[31642],[34,33749,33751],{"className":33750},[31646],[31648,33752,33753],{"xmlns":31650},[31652,33754,33755,33760],{},[31655,33756,33757],{},[31658,33758,33759],{},"k",[31665,33761,33759],{"encoding":31667},[34,33763,33765],{"className":33764,"ariaHidden":758},[31672],[34,33766,33768,33771],{"className":33767},[31676],[34,33769],{"className":33770,"style":31681},[31680],[34,33772,33759],{"className":33773,"style":33774},[31685,31686],"margin-right:0.0315em;","（即 ",[34,33777,33779,33805],{"className":33778},[31642],[34,33780,33782],{"className":33781},[31646],[31648,33783,33784],{"xmlns":31650},[31652,33785,33786,33802],{},[31655,33787,33788,33794,33796],{},[31743,33789,33790,33792],{},[31658,33791,1697],{},[31658,33793,33759],{},[31662,33795,202],{},[31743,33797,33798,33800],{},[31658,33799,111],{},[31749,33801,459],{},[31665,33803,33804],{"encoding":31667},"b_k = a_0",[34,33806,33808,33865],{"className":33807,"ariaHidden":758},[31672],[34,33809,33811,33815,33856,33859,33862],{"className":33810},[31676],[34,33812],{"className":33813,"style":33814},[31680],"height:0.8444em;vertical-align:-0.15em;",[34,33816,33818,33821],{"className":33817},[31685],[34,33819,1697],{"className":33820},[31685,31686],[34,33822,33824],{"className":33823},[31792],[34,33825,33827,33848],{"className":33826},[31796,31797],[34,33828,33830,33845],{"className":33829},[31801],[34,33831,33834],{"className":33832,"style":33833},[31805],"height:0.3361em;",[34,33835,33836,33839],{"style":31809},[34,33837],{"className":33838,"style":31814},[31813],[34,33840,33842],{"className":33841},[31818,31819,31820,31821],[34,33843,33759],{"className":33844,"style":33774},[31685,31686,31821],[34,33846,31829],{"className":33847},[31828],[34,33849,33851],{"className":33850},[31801],[34,33852,33854],{"className":33853,"style":31836},[31805],[34,33855],{},[34,33857],{"className":33858,"style":31846},[31845],[34,33860,202],{"className":33861},[31850],[34,33863],{"className":33864,"style":31846},[31845],[34,33866,33868,33871],{"className":33867},[31676],[34,33869],{"className":33870,"style":31860},[31680],[34,33872,33874,33877],{"className":33873},[31685],[34,33875,111],{"className":33876},[31685,31686],[34,33878,33880],{"className":33879},[31792],[34,33881,33883,33903],{"className":33882},[31796,31797],[34,33884,33886,33900],{"className":33885},[31801],[34,33887,33889],{"className":33888,"style":31806},[31805],[34,33890,33891,33894],{"style":31809},[34,33892],{"className":33893,"style":31814},[31813],[34,33895,33897],{"className":33896},[31818,31819,31820,31821],[34,33898,459],{"className":33899},[31685,31821],[34,33901,31829],{"className":33902},[31828],[34,33904,33906],{"className":33905},[31801],[34,33907,33909],{"className":33908,"style":31836},[31805],[34,33910],{},[108,33912,33913,33914,34059,34060,34088,34089,34246,34247,34321],{},"由此可知，中序序列中 ",[34,33915,33917,33950],{"className":33916},[31642],[34,33918,33920],{"className":33919},[31646],[31648,33921,33922],{"xmlns":31650},[31652,33923,33924,33947],{},[31655,33925,33926,33932,33935],{},[31743,33927,33928,33930],{},[31658,33929,1697],{},[31749,33931,459],{},[31662,33933,33934],{},"…",[31743,33936,33937,33939],{},[31658,33938,1697],{},[31655,33940,33941,33943,33945],{},[31658,33942,33759],{},[31662,33944,32226],{},[31749,33946,640],{},[31665,33948,33949],{"encoding":31667},"b_0 \\dots b_{k-1}",[34,33951,33953],{"className":33952,"ariaHidden":758},[31672],[34,33954,33956,33960,34000,34003,34007,34010],{"className":33955},[31676],[34,33957],{"className":33958,"style":33959},[31680],"height:0.9028em;vertical-align:-0.2083em;",[34,33961,33963,33966],{"className":33962},[31685],[34,33964,1697],{"className":33965},[31685,31686],[34,33967,33969],{"className":33968},[31792],[34,33970,33972,33992],{"className":33971},[31796,31797],[34,33973,33975,33989],{"className":33974},[31801],[34,33976,33978],{"className":33977,"style":31806},[31805],[34,33979,33980,33983],{"style":31809},[34,33981],{"className":33982,"style":31814},[31813],[34,33984,33986],{"className":33985},[31818,31819,31820,31821],[34,33987,459],{"className":33988},[31685,31821],[34,33990,31829],{"className":33991},[31828],[34,33993,33995],{"className":33994},[31801],[34,33996,33998],{"className":33997,"style":31836},[31805],[34,33999],{},[34,34001],{"className":34002,"style":32123},[31845],[34,34004,33934],{"className":34005},[34006],"minner",[34,34008],{"className":34009,"style":32123},[31845],[34,34011,34013,34016],{"className":34012},[31685],[34,34014,1697],{"className":34015},[31685,31686],[34,34017,34019],{"className":34018},[31792],[34,34020,34022,34051],{"className":34021},[31796,31797],[34,34023,34025,34048],{"className":34024},[31801],[34,34026,34028],{"className":34027,"style":33833},[31805],[34,34029,34030,34033],{"style":31809},[34,34031],{"className":34032,"style":31814},[31813],[34,34034,34036],{"className":34035},[31818,31819,31820,31821],[34,34037,34039,34042,34045],{"className":34038},[31685,31821],[34,34040,33759],{"className":34041,"style":33774},[31685,31686,31821],[34,34043,32226],{"className":34044},[32024,31821],[34,34046,640],{"className":34047},[31685,31821],[34,34049,31829],{"className":34050},[31828],[34,34052,34054],{"className":34053},[31801],[34,34055,34057],{"className":34056,"style":32037},[31805],[34,34058],{}," 为左子树（共 ",[34,34061,34063,34076],{"className":34062},[31642],[34,34064,34066],{"className":34065},[31646],[31648,34067,34068],{"xmlns":31650},[31652,34069,34070,34074],{},[31655,34071,34072],{},[31658,34073,33759],{},[31665,34075,33759],{"encoding":31667},[34,34077,34079],{"className":34078,"ariaHidden":758},[31672],[34,34080,34082,34085],{"className":34081},[31676],[34,34083],{"className":34084,"style":31681},[31680],[34,34086,33759],{"className":34087,"style":33774},[31685,31686]," 个结点），",[34,34090,34092,34130],{"className":34091},[31642],[34,34093,34095],{"className":34094},[31646],[31648,34096,34097],{"xmlns":31650},[31652,34098,34099,34127],{},[31655,34100,34101,34113,34115],{},[31743,34102,34103,34105],{},[31658,34104,1697],{},[31655,34106,34107,34109,34111],{},[31658,34108,33759],{},[31662,34110,223],{},[31749,34112,640],{},[31662,34114,33934],{},[31743,34116,34117,34119],{},[31658,34118,1697],{},[31655,34120,34121,34123,34125],{},[31658,34122,31660],{},[31662,34124,32226],{},[31749,34126,640],{},[31665,34128,34129],{"encoding":31667},"b_{k+1} \\dots b_{n-1}",[34,34131,34133],{"className":34132,"ariaHidden":758},[31672],[34,34134,34136,34139,34188,34191,34194,34197],{"className":34135},[31676],[34,34137],{"className":34138,"style":33959},[31680],[34,34140,34142,34145],{"className":34141},[31685],[34,34143,1697],{"className":34144},[31685,31686],[34,34146,34148],{"className":34147},[31792],[34,34149,34151,34180],{"className":34150},[31796,31797],[34,34152,34154,34177],{"className":34153},[31801],[34,34155,34157],{"className":34156,"style":33833},[31805],[34,34158,34159,34162],{"style":31809},[34,34160],{"className":34161,"style":31814},[31813],[34,34163,34165],{"className":34164},[31818,31819,31820,31821],[34,34166,34168,34171,34174],{"className":34167},[31685,31821],[34,34169,33759],{"className":34170,"style":33774},[31685,31686,31821],[34,34172,223],{"className":34173},[32024,31821],[34,34175,640],{"className":34176},[31685,31821],[34,34178,31829],{"className":34179},[31828],[34,34181,34183],{"className":34182},[31801],[34,34184,34186],{"className":34185,"style":32037},[31805],[34,34187],{},[34,34189],{"className":34190,"style":32123},[31845],[34,34192,33934],{"className":34193},[34006],[34,34195],{"className":34196,"style":32123},[31845],[34,34198,34200,34203],{"className":34199},[31685],[34,34201,1697],{"className":34202},[31685,31686],[34,34204,34206],{"className":34205},[31792],[34,34207,34209,34238],{"className":34208},[31796,31797],[34,34210,34212,34235],{"className":34211},[31801],[34,34213,34215],{"className":34214,"style":31806},[31805],[34,34216,34217,34220],{"style":31809},[34,34218],{"className":34219,"style":31814},[31813],[34,34221,34223],{"className":34222},[31818,31819,31820,31821],[34,34224,34226,34229,34232],{"className":34225},[31685,31821],[34,34227,31660],{"className":34228},[31685,31686,31821],[34,34230,32226],{"className":34231},[32024,31821],[34,34233,640],{"className":34234},[31685,31821],[34,34236,31829],{"className":34237},[31828],[34,34239,34241],{"className":34240},[31801],[34,34242,34244],{"className":34243,"style":32037},[31805],[34,34245],{}," 为右子树（共 ",[34,34248,34250,34272],{"className":34249},[31642],[34,34251,34253],{"className":34252},[31646],[31648,34254,34255],{"xmlns":31650},[31652,34256,34257,34269],{},[31655,34258,34259,34261,34263,34265,34267],{},[31658,34260,31660],{},[31662,34262,32226],{},[31658,34264,33759],{},[31662,34266,32226],{},[31749,34268,640],{},[31665,34270,34271],{"encoding":31667},"n-k-1",[34,34273,34275,34293,34312],{"className":34274,"ariaHidden":758},[31672],[34,34276,34278,34281,34284,34287,34290],{"className":34277},[31676],[34,34279],{"className":34280,"style":32597},[31680],[34,34282,31660],{"className":34283},[31685,31686],[34,34285],{"className":34286,"style":32604},[31845],[34,34288,32226],{"className":34289},[32024],[34,34291],{"className":34292,"style":32604},[31845],[34,34294,34296,34300,34303,34306,34309],{"className":34295},[31676],[34,34297],{"className":34298,"style":34299},[31680],"height:0.7778em;vertical-align:-0.0833em;",[34,34301,33759],{"className":34302,"style":33774},[31685,31686],[34,34304],{"className":34305,"style":32604},[31845],[34,34307,32226],{"className":34308},[32024],[34,34310],{"className":34311,"style":32604},[31845],[34,34313,34315,34318],{"className":34314},[31676],[34,34316],{"className":34317,"style":32617},[31680],[34,34319,640],{"className":34320},[31685]," 个结点）。",[108,34323,34324,34325,34328,34329,34332],{},"递归创建左子树 ",[31,34326,34327],{},"CreateBTree(pre+1, in, k)"," 和右子树 ",[31,34330,34331],{},"CreateBTree(pre+k+1, in+k+1, n-k-1)","，分别挂在根结点的左右指针上。",[32490,34334,34336],{"id":34335},"例题-4集合的全排列问题","例题 4：集合的全排列问题",[105,34338,34339,34550],{},[108,34340,34341,32501,34343,34371,34372,34549],{},[2627,34342,32500],{},[34,34344,34346,34359],{"className":34345},[31642],[34,34347,34349],{"className":34348},[31646],[31648,34350,34351],{"xmlns":31650},[31652,34352,34353,34357],{},[31655,34354,34355],{},[31658,34356,31660],{},[31665,34358,31660],{"encoding":31667},[34,34360,34362],{"className":34361,"ariaHidden":758},[31672],[34,34363,34365,34368],{"className":34364},[31676],[34,34366],{"className":34367,"style":32527},[31680],[34,34369,31660],{"className":34370},[31685,31686]," 个元素的集合 ",[34,34373,34375,34415],{"className":34374},[31642],[34,34376,34378],{"className":34377},[31646],[31648,34379,34380],{"xmlns":31650},[31652,34381,34382,34412],{},[31655,34383,34384,34387,34389,34391,34398,34400,34402,34404,34410],{},[31658,34385,34386],{},"R",[31662,34388,202],{},[31662,34390,32685],{"stretchy":702},[31743,34392,34393,34396],{},[31658,34394,34395],{},"r",[31749,34397,640],{},[31662,34399,31960],{"separator":758},[31662,34401,33934],{},[31662,34403,31960],{"separator":758},[31743,34405,34406,34408],{},[31658,34407,34395],{},[31658,34409,31660],{},[31662,34411,32715],{"stretchy":702},[31665,34413,34414],{"encoding":31667},"R=\\{r_1, \\dots, r_n\\}",[34,34416,34418,34437],{"className":34417,"ariaHidden":758},[31672],[34,34419,34421,34424,34428,34431,34434],{"className":34420},[31676],[34,34422],{"className":34423,"style":33218},[31680],[34,34425,34386],{"className":34426,"style":34427},[31685,31686],"margin-right:0.0077em;",[34,34429],{"className":34430,"style":31846},[31845],[34,34432,202],{"className":34433},[31850],[34,34435],{"className":34436,"style":31846},[31845],[34,34438,34440,34443,34446,34488,34491,34494,34497,34500,34503,34506,34546],{"className":34439},[31676],[34,34441],{"className":34442,"style":31774},[31680],[34,34444,32685],{"className":34445},[31782],[34,34447,34449,34453],{"className":34448},[31685],[34,34450,34395],{"className":34451,"style":34452},[31685,31686],"margin-right:0.0278em;",[34,34454,34456],{"className":34455},[31792],[34,34457,34459,34480],{"className":34458},[31796,31797],[34,34460,34462,34477],{"className":34461},[31801],[34,34463,34465],{"className":34464,"style":31806},[31805],[34,34466,34468,34471],{"style":34467},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[34,34469],{"className":34470,"style":31814},[31813],[34,34472,34474],{"className":34473},[31818,31819,31820,31821],[34,34475,640],{"className":34476},[31685,31821],[34,34478,31829],{"className":34479},[31828],[34,34481,34483],{"className":34482},[31801],[34,34484,34486],{"className":34485,"style":31836},[31805],[34,34487],{},[34,34489,31960],{"className":34490},[32119],[34,34492],{"className":34493,"style":32123},[31845],[34,34495,33934],{"className":34496},[34006],[34,34498],{"className":34499,"style":32123},[31845],[34,34501,31960],{"className":34502},[32119],[34,34504],{"className":34505,"style":32123},[31845],[34,34507,34509,34512],{"className":34508},[31685],[34,34510,34395],{"className":34511,"style":34452},[31685,31686],[34,34513,34515],{"className":34514},[31792],[34,34516,34518,34538],{"className":34517},[31796,31797],[34,34519,34521,34535],{"className":34520},[31801],[34,34522,34524],{"className":34523,"style":32362},[31805],[34,34525,34526,34529],{"style":34467},[34,34527],{"className":34528,"style":31814},[31813],[34,34530,34532],{"className":34531},[31818,31819,31820,31821],[34,34533,31660],{"className":34534},[31685,31686,31821],[34,34536,31829],{"className":34537},[31828],[34,34539,34541],{"className":34540},[31801],[34,34542,34544],{"className":34543,"style":31836},[31805],[34,34545],{},[34,34547,32715],{"className":34548},[31690]," 的所有排列。",[108,34551,34552,34554,34555,34612,34613,34641,34642,34692,34693,34781,34782,34834,34835,34905,34906,35009,35010,347],{},[2627,34553,33190],{},"：设 ",[34,34556,34558,34584],{"className":34557},[31642],[34,34559,34561],{"className":34560},[31646],[31648,34562,34563],{"xmlns":31650},[31652,34564,34565,34581],{},[31655,34566,34567,34569,34571,34573,34575,34577,34579],{},[31658,34568,15],{},[31658,34570,33244],{},[31658,34572,34395],{},[31658,34574,31759],{},[31662,34576,456],{"stretchy":702},[31658,34578,34386],{},[31662,34580,2557],{"stretchy":702},[31665,34582,34583],{"encoding":31667},"perm(R)",[34,34585,34587],{"className":34586,"ariaHidden":758},[31672],[34,34588,34590,34593,34596,34600,34603,34606,34609],{"className":34589},[31676],[34,34591],{"className":34592,"style":31774},[31680],[34,34594,15],{"className":34595},[31685,31686],[34,34597,34599],{"className":34598,"style":34452},[31685,31686],"er",[34,34601,31759],{"className":34602},[31685,31686],[34,34604,456],{"className":34605},[31782],[34,34607,34386],{"className":34608,"style":34427},[31685,31686],[34,34610,2557],{"className":34611},[31690]," 为集合 ",[34,34614,34616,34629],{"className":34615},[31642],[34,34617,34619],{"className":34618},[31646],[31648,34620,34621],{"xmlns":31650},[31652,34622,34623,34627],{},[31655,34624,34625],{},[31658,34626,34386],{},[31665,34628,34386],{"encoding":31667},[34,34630,34632],{"className":34631,"ariaHidden":758},[31672],[34,34633,34635,34638],{"className":34634},[31676],[34,34636],{"className":34637,"style":33218},[31680],[34,34639,34386],{"className":34640,"style":34427},[31685,31686]," 的全排列。当 ",[34,34643,34645,34662],{"className":34644},[31642],[34,34646,34648],{"className":34647},[31646],[31648,34649,34650],{"xmlns":31650},[31652,34651,34652,34660],{},[31655,34653,34654,34656,34658],{},[31658,34655,31660],{},[31662,34657,202],{},[31749,34659,640],{},[31665,34661,32850],{"encoding":31667},[34,34663,34665,34683],{"className":34664,"ariaHidden":758},[31672],[34,34666,34668,34671,34674,34677,34680],{"className":34667},[31676],[34,34669],{"className":34670,"style":32527},[31680],[34,34672,31660],{"className":34673},[31685,31686],[34,34675],{"className":34676,"style":31846},[31845],[34,34678,202],{"className":34679},[31850],[34,34681],{"className":34682,"style":31846},[31845],[34,34684,34686,34689],{"className":34685},[31676],[34,34687],{"className":34688,"style":32617},[31680],[34,34690,640],{"className":34691},[31685]," 时 ",[34,34694,34696,34730],{"className":34695},[31642],[34,34697,34699],{"className":34698},[31646],[31648,34700,34701],{"xmlns":31650},[31652,34702,34703,34727],{},[31655,34704,34705,34707,34709,34711,34713,34715,34717,34719,34721,34723,34725],{},[31658,34706,15],{},[31658,34708,33244],{},[31658,34710,34395],{},[31658,34712,31759],{},[31662,34714,456],{"stretchy":702},[31658,34716,34386],{},[31662,34718,2557],{"stretchy":702},[31662,34720,202],{},[31662,34722,456],{"stretchy":702},[31658,34724,34395],{},[31662,34726,2557],{"stretchy":702},[31665,34728,34729],{"encoding":31667},"perm(R)=(r)",[34,34731,34733,34766],{"className":34732,"ariaHidden":758},[31672],[34,34734,34736,34739,34742,34745,34748,34751,34754,34757,34760,34763],{"className":34735},[31676],[34,34737],{"className":34738,"style":31774},[31680],[34,34740,15],{"className":34741},[31685,31686],[34,34743,34599],{"className":34744,"style":34452},[31685,31686],[34,34746,31759],{"className":34747},[31685,31686],[34,34749,456],{"className":34750},[31782],[34,34752,34386],{"className":34753,"style":34427},[31685,31686],[34,34755,2557],{"className":34756},[31690],[34,34758],{"className":34759,"style":31846},[31845],[34,34761,202],{"className":34762},[31850],[34,34764],{"className":34765,"style":31846},[31845],[34,34767,34769,34772,34775,34778],{"className":34768},[31676],[34,34770],{"className":34771,"style":31774},[31680],[34,34773,456],{"className":34774},[31782],[34,34776,34395],{"className":34777,"style":34452},[31685,31686],[34,34779,2557],{"className":34780},[31690],"。当 ",[34,34783,34785,34803],{"className":34784},[31642],[34,34786,34788],{"className":34787},[31646],[31648,34789,34790],{"xmlns":31650},[31652,34791,34792,34800],{},[31655,34793,34794,34796,34798],{},[31658,34795,31660],{},[31662,34797,29622],{},[31749,34799,640],{},[31665,34801,34802],{"encoding":31667},"n>1",[34,34804,34806,34825],{"className":34805,"ariaHidden":758},[31672],[34,34807,34809,34813,34816,34819,34822],{"className":34808},[31676],[34,34810],{"className":34811,"style":34812},[31680],"height:0.5782em;vertical-align:-0.0391em;",[34,34814,31660],{"className":34815},[31685,31686],[34,34817],{"className":34818,"style":31846},[31845],[34,34820,29622],{"className":34821},[31850],[34,34823],{"className":34824,"style":31846},[31845],[34,34826,34828,34831],{"className":34827},[31676],[34,34829],{"className":34830,"style":32617},[31680],[34,34832,640],{"className":34833},[31685]," 时，全排列由分别以每个元素 ",[34,34836,34838,34856],{"className":34837},[31642],[34,34839,34841],{"className":34840},[31646],[31648,34842,34843],{"xmlns":31650},[31652,34844,34845,34853],{},[31655,34846,34847],{},[31743,34848,34849,34851],{},[31658,34850,34395],{},[31658,34852,31955],{},[31665,34854,34855],{"encoding":31667},"r_i",[34,34857,34859],{"className":34858,"ariaHidden":758},[31672],[34,34860,34862,34865],{"className":34861},[31676],[34,34863],{"className":34864,"style":31860},[31680],[34,34866,34868,34871],{"className":34867},[31685],[34,34869,34395],{"className":34870,"style":34452},[31685,31686],[34,34872,34874],{"className":34873},[31792],[34,34875,34877,34897],{"className":34876},[31796,31797],[34,34878,34880,34894],{"className":34879},[31801],[34,34881,34883],{"className":34882,"style":32090},[31805],[34,34884,34885,34888],{"style":34467},[34,34886],{"className":34887,"style":31814},[31813],[34,34889,34891],{"className":34890},[31818,31819,31820,31821],[34,34892,31955],{"className":34893},[31685,31686,31821],[34,34895,31829],{"className":34896},[31828],[34,34898,34900],{"className":34899},[31801],[34,34901,34903],{"className":34902,"style":31836},[31805],[34,34904],{}," 作为前缀，加上剩余元素集合 ",[34,34907,34909,34935],{"className":34908},[31642],[34,34910,34912],{"className":34911},[31646],[31648,34913,34914],{"xmlns":31650},[31652,34915,34916,34932],{},[31655,34917,34918,34920,34922,34924,34930],{},[31658,34919,34386],{},[31662,34921,32226],{},[31662,34923,32685],{"stretchy":702},[31743,34925,34926,34928],{},[31658,34927,34395],{},[31658,34929,31955],{},[31662,34931,32715],{"stretchy":702},[31665,34933,34934],{"encoding":31667},"R-\\{r_i\\}",[34,34936,34938,34957],{"className":34937,"ariaHidden":758},[31672],[34,34939,34941,34945,34948,34951,34954],{"className":34940},[31676],[34,34942],{"className":34943,"style":34944},[31680],"height:0.7667em;vertical-align:-0.0833em;",[34,34946,34386],{"className":34947,"style":34427},[31685,31686],[34,34949],{"className":34950,"style":32604},[31845],[34,34952,32226],{"className":34953},[32024],[34,34955],{"className":34956,"style":32604},[31845],[34,34958,34960,34963,34966,35006],{"className":34959},[31676],[34,34961],{"className":34962,"style":31774},[31680],[34,34964,32685],{"className":34965},[31782],[34,34967,34969,34972],{"className":34968},[31685],[34,34970,34395],{"className":34971,"style":34452},[31685,31686],[34,34973,34975],{"className":34974},[31792],[34,34976,34978,34998],{"className":34977},[31796,31797],[34,34979,34981,34995],{"className":34980},[31801],[34,34982,34984],{"className":34983,"style":32090},[31805],[34,34985,34986,34989],{"style":34467},[34,34987],{"className":34988,"style":31814},[31813],[34,34990,34992],{"className":34991},[31818,31819,31820,31821],[34,34993,31955],{"className":34994},[31685,31686,31821],[34,34996,31829],{"className":34997},[31828],[34,34999,35001],{"className":35000},[31801],[34,35002,35004],{"className":35003,"style":31836},[31805],[34,35005],{},[34,35007,32715],{"className":35008},[31690]," 的全排列构成，即组合 ",[34,35011,35013,35061],{"className":35012},[31642],[34,35014,35016],{"className":35015},[31646],[31648,35017,35018],{"xmlns":31650},[31652,35019,35020,35058],{},[31655,35021,35022,35024,35030,35032,35034,35036,35038,35040,35042,35044,35046,35048,35054,35056],{},[31662,35023,456],{"stretchy":702},[31743,35025,35026,35028],{},[31658,35027,34395],{},[31658,35029,31955],{},[31662,35031,2557],{"stretchy":702},[31658,35033,15],{},[31658,35035,33244],{},[31658,35037,34395],{},[31658,35039,31759],{},[31662,35041,456],{"stretchy":702},[31658,35043,34386],{},[31662,35045,32226],{},[31662,35047,32685],{"stretchy":702},[31743,35049,35050,35052],{},[31658,35051,34395],{},[31658,35053,31955],{},[31662,35055,32715],{"stretchy":702},[31662,35057,2557],{"stretchy":702},[31665,35059,35060],{"encoding":31667},"(r_i)perm(R-\\{r_i\\})",[34,35062,35064,35140],{"className":35063,"ariaHidden":758},[31672],[34,35065,35067,35070,35073,35113,35116,35119,35122,35125,35128,35131,35134,35137],{"className":35066},[31676],[34,35068],{"className":35069,"style":31774},[31680],[34,35071,456],{"className":35072},[31782],[34,35074,35076,35079],{"className":35075},[31685],[34,35077,34395],{"className":35078,"style":34452},[31685,31686],[34,35080,35082],{"className":35081},[31792],[34,35083,35085,35105],{"className":35084},[31796,31797],[34,35086,35088,35102],{"className":35087},[31801],[34,35089,35091],{"className":35090,"style":32090},[31805],[34,35092,35093,35096],{"style":34467},[34,35094],{"className":35095,"style":31814},[31813],[34,35097,35099],{"className":35098},[31818,31819,31820,31821],[34,35100,31955],{"className":35101},[31685,31686,31821],[34,35103,31829],{"className":35104},[31828],[34,35106,35108],{"className":35107},[31801],[34,35109,35111],{"className":35110,"style":31836},[31805],[34,35112],{},[34,35114,2557],{"className":35115},[31690],[34,35117,15],{"className":35118},[31685,31686],[34,35120,34599],{"className":35121,"style":34452},[31685,31686],[34,35123,31759],{"className":35124},[31685,31686],[34,35126,456],{"className":35127},[31782],[34,35129,34386],{"className":35130,"style":34427},[31685,31686],[34,35132],{"className":35133,"style":32604},[31845],[34,35135,32226],{"className":35136},[32024],[34,35138],{"className":35139,"style":32604},[31845],[34,35141,35143,35146,35149,35189],{"className":35142},[31676],[34,35144],{"className":35145,"style":31774},[31680],[34,35147,32685],{"className":35148},[31782],[34,35150,35152,35155],{"className":35151},[31685],[34,35153,34395],{"className":35154,"style":34452},[31685,31686],[34,35156,35158],{"className":35157},[31792],[34,35159,35161,35181],{"className":35160},[31796,31797],[34,35162,35164,35178],{"className":35163},[31801],[34,35165,35167],{"className":35166,"style":32090},[31805],[34,35168,35169,35172],{"style":34467},[34,35170],{"className":35171,"style":31814},[31813],[34,35173,35175],{"className":35174},[31818,31819,31820,31821],[34,35176,31955],{"className":35177},[31685,31686,31821],[34,35179,31829],{"className":35180},[31828],[34,35182,35184],{"className":35183},[31801],[34,35185,35187],{"className":35186,"style":31836},[31805],[34,35188],{},[34,35190,35192],{"className":35191},[31690],"})",[32490,35194,35196],{"id":35195},"例题-5整数划分问题️极高频考点","例题 5：整数划分问题（⭐️极高频考点）",[105,35198,35199,35319,36633],{},[108,35200,35201,35203,35204,35232,35233,35261,35262,347],{},[2627,35202,32500],{},"：将正整数 ",[34,35205,35207,35220],{"className":35206},[31642],[34,35208,35210],{"className":35209},[31646],[31648,35211,35212],{"xmlns":31650},[31652,35213,35214,35218],{},[31655,35215,35216],{},[31658,35217,31660],{},[31665,35219,31660],{"encoding":31667},[34,35221,35223],{"className":35222,"ariaHidden":758},[31672],[34,35224,35226,35229],{"className":35225},[31676],[34,35227],{"className":35228,"style":32527},[31680],[34,35230,31660],{"className":35231},[31685,31686]," 无序拆分成最大加数不超过 ",[34,35234,35236,35249],{"className":35235},[31642],[34,35237,35239],{"className":35238},[31646],[31648,35240,35241],{"xmlns":31650},[31652,35242,35243,35247],{},[31655,35244,35245],{},[31658,35246,31759],{},[31665,35248,31759],{"encoding":31667},[34,35250,35252],{"className":35251,"ariaHidden":758},[31672],[34,35253,35255,35258],{"className":35254},[31676],[34,35256],{"className":35257,"style":32527},[31680],[34,35259,31759],{"className":35260},[31685,31686]," 的几种方案，记为 ",[34,35263,35265,35289],{"className":35264},[31642],[34,35266,35268],{"className":35267},[31646],[31648,35269,35270],{"xmlns":31650},[31652,35271,35272,35286],{},[31655,35273,35274,35276,35278,35280,35282,35284],{},[31658,35275,31739],{},[31662,35277,456],{"stretchy":702},[31658,35279,31660],{},[31662,35281,31960],{"separator":758},[31658,35283,31759],{},[31662,35285,2557],{"stretchy":702},[31665,35287,35288],{"encoding":31667},"f(n, m)",[34,35290,35292],{"className":35291,"ariaHidden":758},[31672],[34,35293,35295,35298,35301,35304,35307,35310,35313,35316],{"className":35294},[31676],[34,35296],{"className":35297,"style":31774},[31680],[34,35299,31739],{"className":35300,"style":31778},[31685,31686],[34,35302,456],{"className":35303},[31782],[34,35305,31660],{"className":35306},[31685,31686],[34,35308,31960],{"className":35309},[32119],[34,35311],{"className":35312,"style":32123},[31845],[34,35314,31759],{"className":35315},[31685,31686],[34,35317,2557],{"className":35318},[31690],[108,35320,35321,33183,35324],{},[2627,35322,35323],{},"推导（分五种情况讨论）",[2230,35325,35326,35510,35650,36000],{},[108,35327,35328,35329,4374,35379,35430,35431,347],{},"当 ",[34,35330,35332,35349],{"className":35331},[31642],[34,35333,35335],{"className":35334},[31646],[31648,35336,35337],{"xmlns":31650},[31652,35338,35339,35347],{},[31655,35340,35341,35343,35345],{},[31658,35342,31660],{},[31662,35344,202],{},[31749,35346,640],{},[31665,35348,32850],{"encoding":31667},[34,35350,35352,35370],{"className":35351,"ariaHidden":758},[31672],[34,35353,35355,35358,35361,35364,35367],{"className":35354},[31676],[34,35356],{"className":35357,"style":32527},[31680],[34,35359,31660],{"className":35360},[31685,31686],[34,35362],{"className":35363,"style":31846},[31845],[34,35365,202],{"className":35366},[31850],[34,35368],{"className":35369,"style":31846},[31845],[34,35371,35373,35376],{"className":35372},[31676],[34,35374],{"className":35375,"style":32617},[31680],[34,35377,640],{"className":35378},[31685],[34,35380,35382,35400],{"className":35381},[31642],[34,35383,35385],{"className":35384},[31646],[31648,35386,35387],{"xmlns":31650},[31652,35388,35389,35397],{},[31655,35390,35391,35393,35395],{},[31658,35392,31759],{},[31662,35394,202],{},[31749,35396,640],{},[31665,35398,35399],{"encoding":31667},"m=1",[34,35401,35403,35421],{"className":35402,"ariaHidden":758},[31672],[34,35404,35406,35409,35412,35415,35418],{"className":35405},[31676],[34,35407],{"className":35408,"style":32527},[31680],[34,35410,31759],{"className":35411},[31685,31686],[34,35413],{"className":35414,"style":31846},[31845],[34,35416,202],{"className":35417},[31850],[34,35419],{"className":35420,"style":31846},[31845],[34,35422,35424,35427],{"className":35423},[31676],[34,35425],{"className":35426,"style":32617},[31680],[34,35428,640],{"className":35429},[31685]," 时：无论对方是多少，都只有 1 种划分（全由1组成或本身为1），",[34,35432,35434,35462],{"className":35433},[31642],[34,35435,35437],{"className":35436},[31646],[31648,35438,35439],{"xmlns":31650},[31652,35440,35441,35459],{},[31655,35442,35443,35445,35447,35449,35451,35453,35455,35457],{},[31658,35444,31739],{},[31662,35446,456],{"stretchy":702},[31658,35448,31660],{},[31662,35450,31960],{"separator":758},[31658,35452,31759],{},[31662,35454,2557],{"stretchy":702},[31662,35456,202],{},[31749,35458,640],{},[31665,35460,35461],{"encoding":31667},"f(n,m)=1",[34,35463,35465,35501],{"className":35464,"ariaHidden":758},[31672],[34,35466,35468,35471,35474,35477,35480,35483,35486,35489,35492,35495,35498],{"className":35467},[31676],[34,35469],{"className":35470,"style":31774},[31680],[34,35472,31739],{"className":35473,"style":31778},[31685,31686],[34,35475,456],{"className":35476},[31782],[34,35478,31660],{"className":35479},[31685,31686],[34,35481,31960],{"className":35482},[32119],[34,35484],{"className":35485,"style":32123},[31845],[34,35487,31759],{"className":35488},[31685,31686],[34,35490,2557],{"className":35491},[31690],[34,35493],{"className":35494,"style":31846},[31845],[34,35496,202],{"className":35497},[31850],[34,35499],{"className":35500,"style":31846},[31845],[34,35502,35504,35507],{"className":35503},[31676],[34,35505],{"className":35506,"style":32617},[31680],[34,35508,640],{"className":35509},[31685],[108,35511,35328,35512,35563,35564,35592,35593,347],{},[34,35513,35515,35533],{"className":35514},[31642],[34,35516,35518],{"className":35517},[31646],[31648,35519,35520],{"xmlns":31650},[31652,35521,35522,35530],{},[31655,35523,35524,35526,35528],{},[31658,35525,31660],{},[31662,35527,932],{},[31658,35529,31759],{},[31665,35531,35532],{"encoding":31667},"n\u003Cm",[34,35534,35536,35554],{"className":35535,"ariaHidden":758},[31672],[34,35537,35539,35542,35545,35548,35551],{"className":35538},[31676],[34,35540],{"className":35541,"style":34812},[31680],[34,35543,31660],{"className":35544},[31685,31686],[34,35546],{"className":35547,"style":31846},[31845],[34,35549,932],{"className":35550},[31850],[34,35552],{"className":35553,"style":31846},[31845],[34,35555,35557,35560],{"className":35556},[31676],[34,35558],{"className":35559,"style":32527},[31680],[34,35561,31759],{"className":35562},[31685,31686]," 时：最大加数不可能超过 ",[34,35565,35567,35580],{"className":35566},[31642],[34,35568,35570],{"className":35569},[31646],[31648,35571,35572],{"xmlns":31650},[31652,35573,35574,35578],{},[31655,35575,35576],{},[31658,35577,31660],{},[31665,35579,31660],{"encoding":31667},[34,35581,35583],{"className":35582,"ariaHidden":758},[31672],[34,35584,35586,35589],{"className":35585},[31676],[34,35587],{"className":35588,"style":32527},[31680],[34,35590,31660],{"className":35591},[31685,31686],"，所以等价于 ",[34,35594,35596,35620],{"className":35595},[31642],[34,35597,35599],{"className":35598},[31646],[31648,35600,35601],{"xmlns":31650},[31652,35602,35603,35617],{},[31655,35604,35605,35607,35609,35611,35613,35615],{},[31658,35606,31739],{},[31662,35608,456],{"stretchy":702},[31658,35610,31660],{},[31662,35612,31960],{"separator":758},[31658,35614,31660],{},[31662,35616,2557],{"stretchy":702},[31665,35618,35619],{"encoding":31667},"f(n, n)",[34,35621,35623],{"className":35622,"ariaHidden":758},[31672],[34,35624,35626,35629,35632,35635,35638,35641,35644,35647],{"className":35625},[31676],[34,35627],{"className":35628,"style":31774},[31680],[34,35630,31739],{"className":35631,"style":31778},[31685,31686],[34,35633,456],{"className":35634},[31782],[34,35636,31660],{"className":35637},[31685,31686],[34,35639,31960],{"className":35640},[32119],[34,35642],{"className":35643,"style":32123},[31845],[34,35645,31660],{"className":35646},[31685,31686],[34,35648,2557],{"className":35649},[31690],[108,35651,35328,35652,35703,35704,35735,35736,35764,35765,35796,35797,35847,35848,347],{},[34,35653,35655,35673],{"className":35654},[31642],[34,35656,35658],{"className":35657},[31646],[31648,35659,35660],{"xmlns":31650},[31652,35661,35662,35670],{},[31655,35663,35664,35666,35668],{},[31658,35665,31660],{},[31662,35667,202],{},[31658,35669,31759],{},[31665,35671,35672],{"encoding":31667},"n=m",[34,35674,35676,35694],{"className":35675,"ariaHidden":758},[31672],[34,35677,35679,35682,35685,35688,35691],{"className":35678},[31676],[34,35680],{"className":35681,"style":32527},[31680],[34,35683,31660],{"className":35684},[31685,31686],[34,35686],{"className":35687,"style":31846},[31845],[34,35689,202],{"className":35690},[31850],[34,35692],{"className":35693,"style":31846},[31845],[34,35695,35697,35700],{"className":35696},[31676],[34,35698],{"className":35699,"style":32527},[31680],[34,35701,31759],{"className":35702},[31685,31686]," 时：划分方案中",[2627,35705,35706,35707],{},"包含 ",[34,35708,35710,35723],{"className":35709},[31642],[34,35711,35713],{"className":35712},[31646],[31648,35714,35715],{"xmlns":31650},[31652,35716,35717,35721],{},[31655,35718,35719],{},[31658,35720,31759],{},[31665,35722,31759],{"encoding":31667},[34,35724,35726],{"className":35725,"ariaHidden":758},[31672],[34,35727,35729,35732],{"className":35728},[31676],[34,35730],{"className":35731,"style":32527},[31680],[34,35733,31759],{"className":35734},[31685,31686]," 的只有1种（即 ",[34,35737,35739,35752],{"className":35738},[31642],[34,35740,35742],{"className":35741},[31646],[31648,35743,35744],{"xmlns":31650},[31652,35745,35746,35750],{},[31655,35747,35748],{},[31658,35749,31660],{},[31665,35751,31660],{"encoding":31667},[34,35753,35755],{"className":35754,"ariaHidden":758},[31672],[34,35756,35758,35761],{"className":35757},[31676],[34,35759],{"className":35760,"style":32527},[31680],[34,35762,31660],{"className":35763},[31685,31686]," 本身）；",[2627,35766,35767,35768],{},"不包含 ",[34,35769,35771,35784],{"className":35770},[31642],[34,35772,35774],{"className":35773},[31646],[31648,35775,35776],{"xmlns":31650},[31652,35777,35778,35782],{},[31655,35779,35780],{},[31658,35781,31759],{},[31665,35783,31759],{"encoding":31667},[34,35785,35787],{"className":35786,"ariaHidden":758},[31672],[34,35788,35790,35793],{"className":35789},[31676],[34,35791],{"className":35792,"style":32527},[31680],[34,35794,31759],{"className":35795},[31685,31686]," 的方案最大加数不超过 ",[34,35798,35800,35817],{"className":35799},[31642],[34,35801,35803],{"className":35802},[31646],[31648,35804,35805],{"xmlns":31650},[31652,35806,35807,35815],{},[31655,35808,35809,35811,35813],{},[31658,35810,31660],{},[31662,35812,32226],{},[31749,35814,640],{},[31665,35816,32587],{"encoding":31667},[34,35818,35820,35838],{"className":35819,"ariaHidden":758},[31672],[34,35821,35823,35826,35829,35832,35835],{"className":35822},[31676],[34,35824],{"className":35825,"style":32597},[31680],[34,35827,31660],{"className":35828},[31685,31686],[34,35830],{"className":35831,"style":32604},[31845],[34,35833,32226],{"className":35834},[32024],[34,35836],{"className":35837,"style":32604},[31845],[34,35839,35841,35844],{"className":35840},[31676],[34,35842],{"className":35843,"style":32617},[31680],[34,35845,640],{"className":35846},[31685],"。故 ",[34,35849,35851,35897],{"className":35850},[31642],[34,35852,35854],{"className":35853},[31646],[31648,35855,35856],{"xmlns":31650},[31652,35857,35858,35894],{},[31655,35859,35860,35862,35864,35866,35868,35870,35872,35874,35876,35878,35880,35882,35884,35886,35888,35890,35892],{},[31658,35861,31739],{},[31662,35863,456],{"stretchy":702},[31658,35865,31660],{},[31662,35867,31960],{"separator":758},[31658,35869,31660],{},[31662,35871,2557],{"stretchy":702},[31662,35873,202],{},[31749,35875,640],{},[31662,35877,223],{},[31658,35879,31739],{},[31662,35881,456],{"stretchy":702},[31658,35883,31660],{},[31662,35885,31960],{"separator":758},[31658,35887,31660],{},[31662,35889,32226],{},[31749,35891,640],{},[31662,35893,2557],{"stretchy":702},[31665,35895,35896],{"encoding":31667},"f(n, n) = 1 + f(n, n-1)",[34,35898,35900,35936,35955,35988],{"className":35899,"ariaHidden":758},[31672],[34,35901,35903,35906,35909,35912,35915,35918,35921,35924,35927,35930,35933],{"className":35902},[31676],[34,35904],{"className":35905,"style":31774},[31680],[34,35907,31739],{"className":35908,"style":31778},[31685,31686],[34,35910,456],{"className":35911},[31782],[34,35913,31660],{"className":35914},[31685,31686],[34,35916,31960],{"className":35917},[32119],[34,35919],{"className":35920,"style":32123},[31845],[34,35922,31660],{"className":35923},[31685,31686],[34,35925,2557],{"className":35926},[31690],[34,35928],{"className":35929,"style":31846},[31845],[34,35931,202],{"className":35932},[31850],[34,35934],{"className":35935,"style":31846},[31845],[34,35937,35939,35943,35946,35949,35952],{"className":35938},[31676],[34,35940],{"className":35941,"style":35942},[31680],"height:0.7278em;vertical-align:-0.0833em;",[34,35944,640],{"className":35945},[31685],[34,35947],{"className":35948,"style":32604},[31845],[34,35950,223],{"className":35951},[32024],[34,35953],{"className":35954,"style":32604},[31845],[34,35956,35958,35961,35964,35967,35970,35973,35976,35979,35982,35985],{"className":35957},[31676],[34,35959],{"className":35960,"style":31774},[31680],[34,35962,31739],{"className":35963,"style":31778},[31685,31686],[34,35965,456],{"className":35966},[31782],[34,35968,31660],{"className":35969},[31685,31686],[34,35971,31960],{"className":35972},[32119],[34,35974],{"className":35975,"style":32123},[31845],[34,35977,31660],{"className":35978},[31685,31686],[34,35980],{"className":35981,"style":32604},[31845],[34,35983,32226],{"className":35984},[32024],[34,35986],{"className":35987,"style":32604},[31845],[34,35989,35991,35994,35997],{"className":35990},[31676],[34,35992],{"className":35993,"style":31774},[31680],[34,35995,640],{"className":35996},[31685],[34,35998,2557],{"className":35999},[31690],[108,36001,35328,36002,36075,36076,36107,36108,36159,36160,36188,36189,36268,36269,36300,36301,36188,36352,36431,36432,347],{},[34,36003,36005,36027],{"className":36004},[31642],[34,36006,36008],{"className":36007},[31646],[31648,36009,36010],{"xmlns":31650},[31652,36011,36012,36024],{},[31655,36013,36014,36016,36018,36020,36022],{},[31658,36015,31660],{},[31662,36017,29622],{},[31658,36019,31759],{},[31662,36021,29622],{},[31749,36023,640],{},[31665,36025,36026],{"encoding":31667},"n>m>1",[34,36028,36030,36048,36066],{"className":36029,"ariaHidden":758},[31672],[34,36031,36033,36036,36039,36042,36045],{"className":36032},[31676],[34,36034],{"className":36035,"style":34812},[31680],[34,36037,31660],{"className":36038},[31685,31686],[34,36040],{"className":36041,"style":31846},[31845],[34,36043,29622],{"className":36044},[31850],[34,36046],{"className":36047,"style":31846},[31845],[34,36049,36051,36054,36057,36060,36063],{"className":36050},[31676],[34,36052],{"className":36053,"style":34812},[31680],[34,36055,31759],{"className":36056},[31685,31686],[34,36058],{"className":36059,"style":31846},[31845],[34,36061,29622],{"className":36062},[31850],[34,36064],{"className":36065,"style":31846},[31845],[34,36067,36069,36072],{"className":36068},[31676],[34,36070],{"className":36071,"style":32617},[31680],[34,36073,640],{"className":36074},[31685]," 时：划分为两部分。",[2627,36077,36078,36079],{},"包含至少一个 ",[34,36080,36082,36095],{"className":36081},[31642],[34,36083,36085],{"className":36084},[31646],[31648,36086,36087],{"xmlns":31650},[31652,36088,36089,36093],{},[31655,36090,36091],{},[31658,36092,31759],{},[31665,36094,31759],{"encoding":31667},[34,36096,36098],{"className":36097,"ariaHidden":758},[31672],[34,36099,36101,36104],{"className":36100},[31676],[34,36102],{"className":36103,"style":32527},[31680],[34,36105,31759],{"className":36106},[31685,31686]," 的情况，相当于把剩下的 ",[34,36109,36111,36129],{"className":36110},[31642],[34,36112,36114],{"className":36113},[31646],[31648,36115,36116],{"xmlns":31650},[31652,36117,36118,36126],{},[31655,36119,36120,36122,36124],{},[31658,36121,31660],{},[31662,36123,32226],{},[31658,36125,31759],{},[31665,36127,36128],{"encoding":31667},"n-m",[34,36130,36132,36150],{"className":36131,"ariaHidden":758},[31672],[34,36133,36135,36138,36141,36144,36147],{"className":36134},[31676],[34,36136],{"className":36137,"style":32597},[31680],[34,36139,31660],{"className":36140},[31685,31686],[34,36142],{"className":36143,"style":32604},[31845],[34,36145,32226],{"className":36146},[32024],[34,36148],{"className":36149,"style":32604},[31845],[34,36151,36153,36156],{"className":36152},[31676],[34,36154],{"className":36155,"style":32527},[31680],[34,36157,31759],{"className":36158},[31685,31686]," 继续用不超过 ",[34,36161,36163,36176],{"className":36162},[31642],[34,36164,36166],{"className":36165},[31646],[31648,36167,36168],{"xmlns":31650},[31652,36169,36170,36174],{},[31655,36171,36172],{},[31658,36173,31759],{},[31665,36175,31759],{"encoding":31667},[34,36177,36179],{"className":36178,"ariaHidden":758},[31672],[34,36180,36182,36185],{"className":36181},[31676],[34,36183],{"className":36184,"style":32527},[31680],[34,36186,31759],{"className":36187},[31685,31686]," 的数划分，即 ",[34,36190,36192,36220],{"className":36191},[31642],[34,36193,36195],{"className":36194},[31646],[31648,36196,36197],{"xmlns":31650},[31652,36198,36199,36217],{},[31655,36200,36201,36203,36205,36207,36209,36211,36213,36215],{},[31658,36202,31739],{},[31662,36204,456],{"stretchy":702},[31658,36206,31660],{},[31662,36208,32226],{},[31658,36210,31759],{},[31662,36212,31960],{"separator":758},[31658,36214,31759],{},[31662,36216,2557],{"stretchy":702},[31665,36218,36219],{"encoding":31667},"f(n-m, m)",[34,36221,36223,36247],{"className":36222,"ariaHidden":758},[31672],[34,36224,36226,36229,36232,36235,36238,36241,36244],{"className":36225},[31676],[34,36227],{"className":36228,"style":31774},[31680],[34,36230,31739],{"className":36231,"style":31778},[31685,31686],[34,36233,456],{"className":36234},[31782],[34,36236,31660],{"className":36237},[31685,31686],[34,36239],{"className":36240,"style":32604},[31845],[34,36242,32226],{"className":36243},[32024],[34,36245],{"className":36246,"style":32604},[31845],[34,36248,36250,36253,36256,36259,36262,36265],{"className":36249},[31676],[34,36251],{"className":36252,"style":31774},[31680],[34,36254,31759],{"className":36255},[31685,31686],[34,36257,31960],{"className":36258},[32119],[34,36260],{"className":36261,"style":32123},[31845],[34,36263,31759],{"className":36264},[31685,31686],[34,36266,2557],{"className":36267},[31690],"；",[2627,36270,36271,36272],{},"绝对不包含 ",[34,36273,36275,36288],{"className":36274},[31642],[34,36276,36278],{"className":36277},[31646],[31648,36279,36280],{"xmlns":31650},[31652,36281,36282,36286],{},[31655,36283,36284],{},[31658,36285,31759],{},[31665,36287,31759],{"encoding":31667},[34,36289,36291],{"className":36290,"ariaHidden":758},[31672],[34,36292,36294,36297],{"className":36293},[31676],[34,36295],{"className":36296,"style":32527},[31680],[34,36298,31759],{"className":36299},[31685,31686]," 的情况，相当于用不超过 ",[34,36302,36304,36322],{"className":36303},[31642],[34,36305,36307],{"className":36306},[31646],[31648,36308,36309],{"xmlns":31650},[31652,36310,36311,36319],{},[31655,36312,36313,36315,36317],{},[31658,36314,31759],{},[31662,36316,32226],{},[31749,36318,640],{},[31665,36320,36321],{"encoding":31667},"m-1",[34,36323,36325,36343],{"className":36324,"ariaHidden":758},[31672],[34,36326,36328,36331,36334,36337,36340],{"className":36327},[31676],[34,36329],{"className":36330,"style":32597},[31680],[34,36332,31759],{"className":36333},[31685,31686],[34,36335],{"className":36336,"style":32604},[31845],[34,36338,32226],{"className":36339},[32024],[34,36341],{"className":36342,"style":32604},[31845],[34,36344,36346,36349],{"className":36345},[31676],[34,36347],{"className":36348,"style":32617},[31680],[34,36350,640],{"className":36351},[31685],[34,36353,36355,36383],{"className":36354},[31642],[34,36356,36358],{"className":36357},[31646],[31648,36359,36360],{"xmlns":31650},[31652,36361,36362,36380],{},[31655,36363,36364,36366,36368,36370,36372,36374,36376,36378],{},[31658,36365,31739],{},[31662,36367,456],{"stretchy":702},[31658,36369,31660],{},[31662,36371,31960],{"separator":758},[31658,36373,31759],{},[31662,36375,32226],{},[31749,36377,640],{},[31662,36379,2557],{"stretchy":702},[31665,36381,36382],{"encoding":31667},"f(n, m-1)",[34,36384,36386,36419],{"className":36385,"ariaHidden":758},[31672],[34,36387,36389,36392,36395,36398,36401,36404,36407,36410,36413,36416],{"className":36388},[31676],[34,36390],{"className":36391,"style":31774},[31680],[34,36393,31739],{"className":36394,"style":31778},[31685,31686],[34,36396,456],{"className":36397},[31782],[34,36399,31660],{"className":36400},[31685,31686],[34,36402,31960],{"className":36403},[32119],[34,36405],{"className":36406,"style":32123},[31845],[34,36408,31759],{"className":36409},[31685,31686],[34,36411],{"className":36412,"style":32604},[31845],[34,36414,32226],{"className":36415},[32024],[34,36417],{"className":36418,"style":32604},[31845],[34,36420,36422,36425,36428],{"className":36421},[31676],[34,36423],{"className":36424,"style":31774},[31680],[34,36426,640],{"className":36427},[31685],[34,36429,2557],{"className":36430},[31690],"。两者相加得 ",[34,36433,36435,36495],{"className":36434},[31642],[34,36436,36438],{"className":36437},[31646],[31648,36439,36440],{"xmlns":31650},[31652,36441,36442,36492],{},[31655,36443,36444,36446,36448,36450,36452,36454,36456,36458,36460,36462,36464,36466,36468,36470,36472,36474,36476,36478,36480,36482,36484,36486,36488,36490],{},[31658,36445,31739],{},[31662,36447,456],{"stretchy":702},[31658,36449,31660],{},[31662,36451,31960],{"separator":758},[31658,36453,31759],{},[31662,36455,2557],{"stretchy":702},[31662,36457,202],{},[31658,36459,31739],{},[31662,36461,456],{"stretchy":702},[31658,36463,31660],{},[31662,36465,31960],{"separator":758},[31658,36467,31759],{},[31662,36469,32226],{},[31749,36471,640],{},[31662,36473,2557],{"stretchy":702},[31662,36475,223],{},[31658,36477,31739],{},[31662,36479,456],{"stretchy":702},[31658,36481,31660],{},[31662,36483,32226],{},[31658,36485,31759],{},[31662,36487,31960],{"separator":758},[31658,36489,31759],{},[31662,36491,2557],{"stretchy":702},[31665,36493,36494],{"encoding":31667},"f(n,m) = f(n, m-1) + f(n-m, m)",[34,36496,36498,36534,36567,36588,36612],{"className":36497,"ariaHidden":758},[31672],[34,36499,36501,36504,36507,36510,36513,36516,36519,36522,36525,36528,36531],{"className":36500},[31676],[34,36502],{"className":36503,"style":31774},[31680],[34,36505,31739],{"className":36506,"style":31778},[31685,31686],[34,36508,456],{"className":36509},[31782],[34,36511,31660],{"className":36512},[31685,31686],[34,36514,31960],{"className":36515},[32119],[34,36517],{"className":36518,"style":32123},[31845],[34,36520,31759],{"className":36521},[31685,31686],[34,36523,2557],{"className":36524},[31690],[34,36526],{"className":36527,"style":31846},[31845],[34,36529,202],{"className":36530},[31850],[34,36532],{"className":36533,"style":31846},[31845],[34,36535,36537,36540,36543,36546,36549,36552,36555,36558,36561,36564],{"className":36536},[31676],[34,36538],{"className":36539,"style":31774},[31680],[34,36541,31739],{"className":36542,"style":31778},[31685,31686],[34,36544,456],{"className":36545},[31782],[34,36547,31660],{"className":36548},[31685,31686],[34,36550,31960],{"className":36551},[32119],[34,36553],{"className":36554,"style":32123},[31845],[34,36556,31759],{"className":36557},[31685,31686],[34,36559],{"className":36560,"style":32604},[31845],[34,36562,32226],{"className":36563},[32024],[34,36565],{"className":36566,"style":32604},[31845],[34,36568,36570,36573,36576,36579,36582,36585],{"className":36569},[31676],[34,36571],{"className":36572,"style":31774},[31680],[34,36574,640],{"className":36575},[31685],[34,36577,2557],{"className":36578},[31690],[34,36580],{"className":36581,"style":32604},[31845],[34,36583,223],{"className":36584},[32024],[34,36586],{"className":36587,"style":32604},[31845],[34,36589,36591,36594,36597,36600,36603,36606,36609],{"className":36590},[31676],[34,36592],{"className":36593,"style":31774},[31680],[34,36595,31739],{"className":36596,"style":31778},[31685,31686],[34,36598,456],{"className":36599},[31782],[34,36601,31660],{"className":36602},[31685,31686],[34,36604],{"className":36605,"style":32604},[31845],[34,36607,32226],{"className":36608},[32024],[34,36610],{"className":36611,"style":32604},[31845],[34,36613,36615,36618,36621,36624,36627,36630],{"className":36614},[31676],[34,36616],{"className":36617,"style":31774},[31680],[34,36619,31759],{"className":36620},[31685,31686],[34,36622,31960],{"className":36623},[32119],[34,36625],{"className":36626,"style":32123},[31845],[34,36628,31759],{"className":36629},[31685,31686],[34,36631,2557],{"className":36632},[31690],[108,36634,36635,2664,36638,347],{},[2627,36636,36637],{},"例子计算",[34,36639,36641,36736],{"className":36640},[31642],[34,36642,36644],{"className":36643},[31646],[31648,36645,36646],{"xmlns":31650},[31652,36647,36648,36733],{},[31655,36649,36650,36652,36654,36657,36659,36661,36663,36665,36667,36669,36671,36673,36675,36677,36679,36681,36683,36685,36687,36689,36691,36693,36695,36697,36699,36701,36703,36705,36707,36709,36711,36713,36715,36717,36719,36721,36724,36726,36728,36730],{},[31658,36651,31739],{},[31662,36653,456],{"stretchy":702},[31749,36655,36656],{},"6",[31662,36658,31960],{"separator":758},[31749,36660,15932],{},[31662,36662,2557],{"stretchy":702},[31662,36664,202],{},[31658,36666,31739],{},[31662,36668,456],{"stretchy":702},[31749,36670,36656],{},[31662,36672,31960],{"separator":758},[31749,36674,786],{},[31662,36676,2557],{"stretchy":702},[31662,36678,223],{},[31658,36680,31739],{},[31662,36682,456],{"stretchy":702},[31749,36684,768],{},[31662,36686,31960],{"separator":758},[31749,36688,15932],{},[31662,36690,2557],{"stretchy":702},[31662,36692,202],{},[31658,36694,31739],{},[31662,36696,456],{"stretchy":702},[31749,36698,36656],{},[31662,36700,31960],{"separator":758},[31749,36702,786],{},[31662,36704,2557],{"stretchy":702},[31662,36706,223],{},[31658,36708,31739],{},[31662,36710,456],{"stretchy":702},[31749,36712,768],{},[31662,36714,31960],{"separator":758},[31749,36716,768],{},[31662,36718,2557],{"stretchy":702},[31662,36720,202],{},[31749,36722,36723],{},"7",[31662,36725,223],{},[31749,36727,768],{},[31662,36729,202],{},[31749,36731,36732],{},"9",[31665,36734,36735],{"encoding":31667},"f(6,4) = f(6,3) + f(2,4) = f(6,3) + f(2,2) = 7 + 2 = 9",[34,36737,36739,36775,36811,36847,36883,36919,36937,36955],{"className":36738,"ariaHidden":758},[31672],[34,36740,36742,36745,36748,36751,36754,36757,36760,36763,36766,36769,36772],{"className":36741},[31676],[34,36743],{"className":36744,"style":31774},[31680],[34,36746,31739],{"className":36747,"style":31778},[31685,31686],[34,36749,456],{"className":36750},[31782],[34,36752,36656],{"className":36753},[31685],[34,36755,31960],{"className":36756},[32119],[34,36758],{"className":36759,"style":32123},[31845],[34,36761,15932],{"className":36762},[31685],[34,36764,2557],{"className":36765},[31690],[34,36767],{"className":36768,"style":31846},[31845],[34,36770,202],{"className":36771},[31850],[34,36773],{"className":36774,"style":31846},[31845],[34,36776,36778,36781,36784,36787,36790,36793,36796,36799,36802,36805,36808],{"className":36777},[31676],[34,36779],{"className":36780,"style":31774},[31680],[34,36782,31739],{"className":36783,"style":31778},[31685,31686],[34,36785,456],{"className":36786},[31782],[34,36788,36656],{"className":36789},[31685],[34,36791,31960],{"className":36792},[32119],[34,36794],{"className":36795,"style":32123},[31845],[34,36797,786],{"className":36798},[31685],[34,36800,2557],{"className":36801},[31690],[34,36803],{"className":36804,"style":32604},[31845],[34,36806,223],{"className":36807},[32024],[34,36809],{"className":36810,"style":32604},[31845],[34,36812,36814,36817,36820,36823,36826,36829,36832,36835,36838,36841,36844],{"className":36813},[31676],[34,36815],{"className":36816,"style":31774},[31680],[34,36818,31739],{"className":36819,"style":31778},[31685,31686],[34,36821,456],{"className":36822},[31782],[34,36824,768],{"className":36825},[31685],[34,36827,31960],{"className":36828},[32119],[34,36830],{"className":36831,"style":32123},[31845],[34,36833,15932],{"className":36834},[31685],[34,36836,2557],{"className":36837},[31690],[34,36839],{"className":36840,"style":31846},[31845],[34,36842,202],{"className":36843},[31850],[34,36845],{"className":36846,"style":31846},[31845],[34,36848,36850,36853,36856,36859,36862,36865,36868,36871,36874,36877,36880],{"className":36849},[31676],[34,36851],{"className":36852,"style":31774},[31680],[34,36854,31739],{"className":36855,"style":31778},[31685,31686],[34,36857,456],{"className":36858},[31782],[34,36860,36656],{"className":36861},[31685],[34,36863,31960],{"className":36864},[32119],[34,36866],{"className":36867,"style":32123},[31845],[34,36869,786],{"className":36870},[31685],[34,36872,2557],{"className":36873},[31690],[34,36875],{"className":36876,"style":32604},[31845],[34,36878,223],{"className":36879},[32024],[34,36881],{"className":36882,"style":32604},[31845],[34,36884,36886,36889,36892,36895,36898,36901,36904,36907,36910,36913,36916],{"className":36885},[31676],[34,36887],{"className":36888,"style":31774},[31680],[34,36890,31739],{"className":36891,"style":31778},[31685,31686],[34,36893,456],{"className":36894},[31782],[34,36896,768],{"className":36897},[31685],[34,36899,31960],{"className":36900},[32119],[34,36902],{"className":36903,"style":32123},[31845],[34,36905,768],{"className":36906},[31685],[34,36908,2557],{"className":36909},[31690],[34,36911],{"className":36912,"style":31846},[31845],[34,36914,202],{"className":36915},[31850],[34,36917],{"className":36918,"style":31846},[31845],[34,36920,36922,36925,36928,36931,36934],{"className":36921},[31676],[34,36923],{"className":36924,"style":35942},[31680],[34,36926,36723],{"className":36927},[31685],[34,36929],{"className":36930,"style":32604},[31845],[34,36932,223],{"className":36933},[32024],[34,36935],{"className":36936,"style":32604},[31845],[34,36938,36940,36943,36946,36949,36952],{"className":36939},[31676],[34,36941],{"className":36942,"style":32617},[31680],[34,36944,768],{"className":36945},[31685],[34,36947],{"className":36948,"style":31846},[31845],[34,36950,202],{"className":36951},[31850],[34,36953],{"className":36954,"style":31846},[31845],[34,36956,36958,36961],{"className":36957},[31676],[34,36959],{"className":36960,"style":32617},[31680],[34,36962,36732],{"className":36963},[31685],[32490,36965,36967],{"id":36966},"例题-6递归的简单选择排序与冒泡排序","例题 6：递归的简单选择排序与冒泡排序",[105,36969,36970,37264],{},[108,36971,36972,36975,36976,37036,37037,37081,37082,37125,37126,37208,37209,37212,37213,347],{},[2627,36973,36974],{},"简单选择排序推导","：在序列 ",[34,36977,36979,37003],{"className":36978},[31642],[34,36980,36982],{"className":36981},[31646],[31648,36983,36984],{"xmlns":31650},[31652,36985,36986,37000],{},[31655,36987,36988,36990,36992,36994,36996,36998],{},[31658,36989,111],{},[31662,36991,15972],{"stretchy":702},[31658,36993,31955],{},[31662,36995,33934],{},[31658,36997,31660],{},[31662,36999,32712],{"stretchy":702},[31665,37001,37002],{"encoding":31667},"a[i\\dots n]",[34,37004,37006],{"className":37005,"ariaHidden":758},[31672],[34,37007,37009,37012,37015,37018,37021,37024,37027,37030,37033],{"className":37008},[31676],[34,37010],{"className":37011,"style":31774},[31680],[34,37013,111],{"className":37014},[31685,31686],[34,37016,15972],{"className":37017},[31782],[34,37019,31955],{"className":37020},[31685,31686],[34,37022],{"className":37023,"style":32123},[31845],[34,37025,33934],{"className":37026},[34006],[34,37028],{"className":37029,"style":32123},[31845],[34,37031,31660],{"className":37032},[31685,31686],[34,37034,32712],{"className":37035},[31690]," 中找最小元素 ",[34,37038,37040,37060],{"className":37039},[31642],[34,37041,37043],{"className":37042},[31646],[31648,37044,37045],{"xmlns":31650},[31652,37046,37047,37057],{},[31655,37048,37049,37051,37053,37055],{},[31658,37050,111],{},[31662,37052,15972],{"stretchy":702},[31658,37054,33759],{},[31662,37056,32712],{"stretchy":702},[31665,37058,37059],{"encoding":31667},"a[k]",[34,37061,37063],{"className":37062,"ariaHidden":758},[31672],[34,37064,37066,37069,37072,37075,37078],{"className":37065},[31676],[34,37067],{"className":37068,"style":31774},[31680],[34,37070,111],{"className":37071},[31685,31686],[34,37073,15972],{"className":37074},[31782],[34,37076,33759],{"className":37077,"style":33774},[31685,31686],[34,37079,32712],{"className":37080},[31690],"，将其与 ",[34,37083,37085,37104],{"className":37084},[31642],[34,37086,37088],{"className":37087},[31646],[31648,37089,37090],{"xmlns":31650},[31652,37091,37092,37102],{},[31655,37093,37094,37096,37098,37100],{},[31658,37095,111],{},[31662,37097,15972],{"stretchy":702},[31658,37099,31955],{},[31662,37101,32712],{"stretchy":702},[31665,37103,33150],{"encoding":31667},[34,37105,37107],{"className":37106,"ariaHidden":758},[31672],[34,37108,37110,37113,37116,37119,37122],{"className":37109},[31676],[34,37111],{"className":37112,"style":31774},[31680],[34,37114,111],{"className":37115},[31685,31686],[34,37117,15972],{"className":37118},[31782],[34,37120,31955],{"className":37121},[31685,31686],[34,37123,32712],{"className":37124},[31690]," 交换。然后对剩下的 ",[34,37127,37129,37157],{"className":37128},[31642],[34,37130,37132],{"className":37131},[31646],[31648,37133,37134],{"xmlns":31650},[31652,37135,37136,37154],{},[31655,37137,37138,37140,37142,37144,37146,37148,37150,37152],{},[31658,37139,111],{},[31662,37141,15972],{"stretchy":702},[31658,37143,31955],{},[31662,37145,223],{},[31749,37147,640],{},[31662,37149,33934],{},[31658,37151,31660],{},[31662,37153,32712],{"stretchy":702},[31665,37155,37156],{"encoding":31667},"a[i+1\\dots n]",[34,37158,37160,37184],{"className":37159,"ariaHidden":758},[31672],[34,37161,37163,37166,37169,37172,37175,37178,37181],{"className":37162},[31676],[34,37164],{"className":37165,"style":31774},[31680],[34,37167,111],{"className":37168},[31685,31686],[34,37170,15972],{"className":37171},[31782],[34,37173,31955],{"className":37174},[31685,31686],[34,37176],{"className":37177,"style":32604},[31845],[34,37179,223],{"className":37180},[32024],[34,37182],{"className":37183,"style":32604},[31845],[34,37185,37187,37190,37193,37196,37199,37202,37205],{"className":37186},[31676],[34,37188],{"className":37189,"style":31774},[31680],[34,37191,640],{"className":37192},[31685],[34,37194],{"className":37195,"style":32123},[31845],[34,37197,33934],{"className":37198},[34006],[34,37200],{"className":37201,"style":32123},[31845],[34,37203,31660],{"className":37204},[31685,31686],[34,37206,32712],{"className":37207},[31690]," 递归调用 ",[31,37210,37211],{},"SelectSort(a, i+1, n)","。出口是 ",[34,37214,37216,37234],{"className":37215},[31642],[34,37217,37219],{"className":37218},[31646],[31648,37220,37221],{"xmlns":31650},[31652,37222,37223,37231],{},[31655,37224,37225,37227,37229],{},[31658,37226,31955],{},[31662,37228,202],{},[31658,37230,31660],{},[31665,37232,37233],{"encoding":31667},"i=n",[34,37235,37237,37255],{"className":37236,"ariaHidden":758},[31672],[34,37238,37240,37243,37246,37249,37252],{"className":37239},[31676],[34,37241],{"className":37242,"style":33104},[31680],[34,37244,31955],{"className":37245},[31685,31686],[34,37247],{"className":37248,"style":31846},[31845],[34,37250,202],{"className":37251},[31850],[34,37253],{"className":37254,"style":31846},[31845],[34,37256,37258,37261],{"className":37257},[31676],[34,37259],{"className":37260,"style":32527},[31680],[34,37262,31660],{"className":37263},[31685,31686],[108,37265,37266,37269,37270,37344,37345,37348],{},[2627,37267,37268],{},"冒泡排序推导","：将前 ",[34,37271,37273,37295],{"className":37272},[31642],[34,37274,37276],{"className":37275},[31646],[31648,37277,37278],{"xmlns":31650},[31652,37279,37280,37292],{},[31655,37281,37282,37284,37286,37288,37290],{},[31658,37283,31660],{},[31662,37285,32226],{},[31658,37287,31955],{},[31662,37289,223],{},[31749,37291,640],{},[31665,37293,37294],{"encoding":31667},"n-i+1",[34,37296,37298,37316,37335],{"className":37297,"ariaHidden":758},[31672],[34,37299,37301,37304,37307,37310,37313],{"className":37300},[31676],[34,37302],{"className":37303,"style":32597},[31680],[34,37305,31660],{"className":37306},[31685,31686],[34,37308],{"className":37309,"style":32604},[31845],[34,37311,32226],{"className":37312},[32024],[34,37314],{"className":37315,"style":32604},[31845],[34,37317,37319,37323,37326,37329,37332],{"className":37318},[31676],[34,37320],{"className":37321,"style":37322},[31680],"height:0.7429em;vertical-align:-0.0833em;",[34,37324,31955],{"className":37325},[31685,31686],[34,37327],{"className":37328,"style":32604},[31845],[34,37330,223],{"className":37331},[32024],[34,37333],{"className":37334,"style":32604},[31845],[34,37336,37338,37341],{"className":37337},[31676],[34,37339],{"className":37340,"style":32617},[31680],[34,37342,640],{"className":37343},[31685]," 个元素中的最大值通过相邻交换冒泡到最后。然后递归调用 ",[31,37346,37347],{},"BubbleSort(a, i+1, n)"," 对剩下的元素排序。",[32188,37350],{},[253,37352,37354],{"id":37353},"三-递归转化为非递归","三、 递归转化为非递归",[105,37356,37357,37367],{},[108,37358,37359,37362,37363,37366],{},[2627,37360,37361],{},"直接转化法","：直接用",[2627,37364,37365],{},"循环结构","的算法替代递归（不需要栈）。如阶乘的循环实现。",[108,37368,37369,37372,37373,37376],{},[2627,37370,37371],{},"间接转化法","：用",[2627,37374,37375],{},"栈","模拟系统底层运行过程，保存必须的信息。如树、图的遍历。",[32188,37378],{},[253,37380,37382],{"id":37381},"四-递归算法复杂度分析四大方法","四、 递归算法复杂度分析（四大方法）",[15,37384,37385,37386,37432],{},"递归算法的执行时间 ",[34,37387,37389,37410],{"className":37388},[31642],[34,37390,37392],{"className":37391},[31646],[31648,37393,37394],{"xmlns":31650},[31652,37395,37396,37407],{},[31655,37397,37398,37401,37403,37405],{},[31658,37399,37400],{},"T",[31662,37402,456],{"stretchy":702},[31658,37404,31660],{},[31662,37406,2557],{"stretchy":702},[31665,37408,37409],{"encoding":31667},"T(n)",[34,37411,37413],{"className":37412,"ariaHidden":758},[31672],[34,37414,37416,37419,37423,37426,37429],{"className":37415},[31676],[34,37417],{"className":37418,"style":31774},[31680],[34,37420,37400],{"className":37421,"style":37422},[31685,31686],"margin-right:0.1389em;",[34,37424,456],{"className":37425},[31782],[34,37427,31660],{"className":37428},[31685,31686],[34,37430,2557],{"className":37431},[31690]," 可用递归方程描述，常用以下四种方法求解：",[32490,37434,37436],{"id":37435},"_1-替换法","1. 替换法",[105,37438,37439,37445,38042],{},[108,37440,37441,37444],{},[2627,37442,37443],{},"原理","：将递归公式不断展开、代换子问题的规模，通过多项式整理推导解。",[108,37446,37447,2664,37450,37580,37581,37853,37854,347],{},[2627,37448,37449],{},"例题：汉诺塔",[34,37451,37453,37493],{"className":37452},[31642],[34,37454,37456],{"className":37455},[31646],[31648,37457,37458],{"xmlns":31650},[31652,37459,37460,37490],{},[31655,37461,37462,37464,37466,37468,37470,37472,37474,37476,37478,37480,37482,37484,37486,37488],{},[31658,37463,37400],{},[31662,37465,456],{"stretchy":702},[31658,37467,31660],{},[31662,37469,2557],{"stretchy":702},[31662,37471,202],{},[31749,37473,768],{},[31658,37475,37400],{},[31662,37477,456],{"stretchy":702},[31658,37479,31660],{},[31662,37481,32226],{},[31749,37483,640],{},[31662,37485,2557],{"stretchy":702},[31662,37487,223],{},[31749,37489,640],{},[31665,37491,37492],{"encoding":31667},"T(n) = 2T(n-1)+1",[34,37494,37496,37523,37550,37571],{"className":37495,"ariaHidden":758},[31672],[34,37497,37499,37502,37505,37508,37511,37514,37517,37520],{"className":37498},[31676],[34,37500],{"className":37501,"style":31774},[31680],[34,37503,37400],{"className":37504,"style":37422},[31685,31686],[34,37506,456],{"className":37507},[31782],[34,37509,31660],{"className":37510},[31685,31686],[34,37512,2557],{"className":37513},[31690],[34,37515],{"className":37516,"style":31846},[31845],[34,37518,202],{"className":37519},[31850],[34,37521],{"className":37522,"style":31846},[31845],[34,37524,37526,37529,37532,37535,37538,37541,37544,37547],{"className":37525},[31676],[34,37527],{"className":37528,"style":31774},[31680],[34,37530,768],{"className":37531},[31685],[34,37533,37400],{"className":37534,"style":37422},[31685,31686],[34,37536,456],{"className":37537},[31782],[34,37539,31660],{"className":37540},[31685,31686],[34,37542],{"className":37543,"style":32604},[31845],[34,37545,32226],{"className":37546},[32024],[34,37548],{"className":37549,"style":32604},[31845],[34,37551,37553,37556,37559,37562,37565,37568],{"className":37552},[31676],[34,37554],{"className":37555,"style":31774},[31680],[34,37557,640],{"className":37558},[31685],[34,37560,2557],{"className":37561},[31690],[34,37563],{"className":37564,"style":32604},[31845],[34,37566,223],{"className":37567},[32024],[34,37569],{"className":37570,"style":32604},[31845],[34,37572,37574,37577],{"className":37573},[31676],[34,37575],{"className":37576,"style":32617},[31680],[34,37578,640],{"className":37579},[31685],"。展开为 ",[34,37582,37584,37653],{"className":37583},[31642],[34,37585,37587],{"className":37586},[31646],[31648,37588,37589],{"xmlns":31650},[31652,37590,37591,37650],{},[31655,37592,37593,37595,37597,37599,37601,37603,37605,37607,37609,37611,37613,37615,37617,37619,37621,37623,37630,37632,37634,37636,37638,37640,37642,37644,37646,37648],{},[31749,37594,768],{},[31662,37596,456],{"stretchy":702},[31749,37598,768],{},[31658,37600,37400],{},[31662,37602,456],{"stretchy":702},[31658,37604,31660],{},[31662,37606,32226],{},[31749,37608,768],{},[31662,37610,2557],{"stretchy":702},[31662,37612,223],{},[31749,37614,640],{},[31662,37616,2557],{"stretchy":702},[31662,37618,223],{},[31749,37620,640],{},[31662,37622,202],{},[37624,37625,37626,37628],"msup",{},[31749,37627,768],{},[31749,37629,768],{},[31658,37631,37400],{},[31662,37633,456],{"stretchy":702},[31658,37635,31660],{},[31662,37637,32226],{},[31749,37639,768],{},[31662,37641,2557],{"stretchy":702},[31662,37643,223],{},[31749,37645,768],{},[31662,37647,223],{},[31749,37649,640],{},[31665,37651,37652],{"encoding":31667},"2(2T(n-2)+1)+1 = 2^2T(n-2)+2+1",[34,37654,37656,37689,37710,37731,37749,37805,37826,37844],{"className":37655,"ariaHidden":758},[31672],[34,37657,37659,37662,37665,37668,37671,37674,37677,37680,37683,37686],{"className":37658},[31676],[34,37660],{"className":37661,"style":31774},[31680],[34,37663,768],{"className":37664},[31685],[34,37666,456],{"className":37667},[31782],[34,37669,768],{"className":37670},[31685],[34,37672,37400],{"className":37673,"style":37422},[31685,31686],[34,37675,456],{"className":37676},[31782],[34,37678,31660],{"className":37679},[31685,31686],[34,37681],{"className":37682,"style":32604},[31845],[34,37684,32226],{"className":37685},[32024],[34,37687],{"className":37688,"style":32604},[31845],[34,37690,37692,37695,37698,37701,37704,37707],{"className":37691},[31676],[34,37693],{"className":37694,"style":31774},[31680],[34,37696,768],{"className":37697},[31685],[34,37699,2557],{"className":37700},[31690],[34,37702],{"className":37703,"style":32604},[31845],[34,37705,223],{"className":37706},[32024],[34,37708],{"className":37709,"style":32604},[31845],[34,37711,37713,37716,37719,37722,37725,37728],{"className":37712},[31676],[34,37714],{"className":37715,"style":31774},[31680],[34,37717,640],{"className":37718},[31685],[34,37720,2557],{"className":37721},[31690],[34,37723],{"className":37724,"style":32604},[31845],[34,37726,223],{"className":37727},[32024],[34,37729],{"className":37730,"style":32604},[31845],[34,37732,37734,37737,37740,37743,37746],{"className":37733},[31676],[34,37735],{"className":37736,"style":32617},[31680],[34,37738,640],{"className":37739},[31685],[34,37741],{"className":37742,"style":31846},[31845],[34,37744,202],{"className":37745},[31850],[34,37747],{"className":37748,"style":31846},[31845],[34,37750,37752,37756,37787,37790,37793,37796,37799,37802],{"className":37751},[31676],[34,37753],{"className":37754,"style":37755},[31680],"height:1.0641em;vertical-align:-0.25em;",[34,37757,37759,37762],{"className":37758},[31685],[34,37760,768],{"className":37761},[31685],[34,37763,37765],{"className":37764},[31792],[34,37766,37768],{"className":37767},[31796],[34,37769,37771],{"className":37770},[31801],[34,37772,37775],{"className":37773,"style":37774},[31805],"height:0.8141em;",[34,37776,37778,37781],{"style":37777},"top:-3.063em;margin-right:0.05em;",[34,37779],{"className":37780,"style":31814},[31813],[34,37782,37784],{"className":37783},[31818,31819,31820,31821],[34,37785,768],{"className":37786},[31685,31821],[34,37788,37400],{"className":37789,"style":37422},[31685,31686],[34,37791,456],{"className":37792},[31782],[34,37794,31660],{"className":37795},[31685,31686],[34,37797],{"className":37798,"style":32604},[31845],[34,37800,32226],{"className":37801},[32024],[34,37803],{"className":37804,"style":32604},[31845],[34,37806,37808,37811,37814,37817,37820,37823],{"className":37807},[31676],[34,37809],{"className":37810,"style":31774},[31680],[34,37812,768],{"className":37813},[31685],[34,37815,2557],{"className":37816},[31690],[34,37818],{"className":37819,"style":32604},[31845],[34,37821,223],{"className":37822},[32024],[34,37824],{"className":37825,"style":32604},[31845],[34,37827,37829,37832,37835,37838,37841],{"className":37828},[31676],[34,37830],{"className":37831,"style":35942},[31680],[34,37833,768],{"className":37834},[31685],[34,37836],{"className":37837,"style":32604},[31845],[34,37839,223],{"className":37840},[32024],[34,37842],{"className":37843,"style":32604},[31845],[34,37845,37847,37850],{"className":37846},[31676],[34,37848],{"className":37849,"style":32617},[31680],[34,37851,640],{"className":37852},[31685],"……最终化简为 ",[34,37855,37857,37904],{"className":37856},[31642],[34,37858,37860],{"className":37859},[31646],[31648,37861,37862],{"xmlns":31650},[31652,37863,37864,37901],{},[31655,37865,37866,37868,37870,37872,37874,37876,37882,37884,37886,37888,37891,37893,37899],{},[31658,37867,37400],{},[31662,37869,456],{"stretchy":702},[31658,37871,31660],{},[31662,37873,2557],{"stretchy":702},[31662,37875,202],{},[37624,37877,37878,37880],{},[31749,37879,768],{},[31658,37881,31660],{},[31662,37883,32226],{},[31749,37885,640],{},[31662,37887,202],{},[31658,37889,37890],{},"O",[31662,37892,456],{"stretchy":702},[37624,37894,37895,37897],{},[31749,37896,768],{},[31658,37898,31660],{},[31662,37900,2557],{"stretchy":702},[31665,37902,37903],{"encoding":31667},"T(n)=2^n-1 = O(2^n)",[34,37905,37907,37934,37980,37998],{"className":37906,"ariaHidden":758},[31672],[34,37908,37910,37913,37916,37919,37922,37925,37928,37931],{"className":37909},[31676],[34,37911],{"className":37912,"style":31774},[31680],[34,37914,37400],{"className":37915,"style":37422},[31685,31686],[34,37917,456],{"className":37918},[31782],[34,37920,31660],{"className":37921},[31685,31686],[34,37923,2557],{"className":37924},[31690],[34,37926],{"className":37927,"style":31846},[31845],[34,37929,202],{"className":37930},[31850],[34,37932],{"className":37933,"style":31846},[31845],[34,37935,37937,37941,37971,37974,37977],{"className":37936},[31676],[34,37938],{"className":37939,"style":37940},[31680],"height:0.7477em;vertical-align:-0.0833em;",[34,37942,37944,37947],{"className":37943},[31685],[34,37945,768],{"className":37946},[31685],[34,37948,37950],{"className":37949},[31792],[34,37951,37953],{"className":37952},[31796],[34,37954,37956],{"className":37955},[31801],[34,37957,37960],{"className":37958,"style":37959},[31805],"height:0.6644em;",[34,37961,37962,37965],{"style":37777},[34,37963],{"className":37964,"style":31814},[31813],[34,37966,37968],{"className":37967},[31818,31819,31820,31821],[34,37969,31660],{"className":37970},[31685,31686,31821],[34,37972],{"className":37973,"style":32604},[31845],[34,37975,32226],{"className":37976},[32024],[34,37978],{"className":37979,"style":32604},[31845],[34,37981,37983,37986,37989,37992,37995],{"className":37982},[31676],[34,37984],{"className":37985,"style":32617},[31680],[34,37987,640],{"className":37988},[31685],[34,37990],{"className":37991,"style":31846},[31845],[34,37993,202],{"className":37994},[31850],[34,37996],{"className":37997,"style":31846},[31845],[34,37999,38001,38004,38007,38010,38039],{"className":38000},[31676],[34,38002],{"className":38003,"style":31774},[31680],[34,38005,37890],{"className":38006,"style":34452},[31685,31686],[34,38008,456],{"className":38009},[31782],[34,38011,38013,38016],{"className":38012},[31685],[34,38014,768],{"className":38015},[31685],[34,38017,38019],{"className":38018},[31792],[34,38020,38022],{"className":38021},[31796],[34,38023,38025],{"className":38024},[31801],[34,38026,38028],{"className":38027,"style":37959},[31805],[34,38029,38030,38033],{"style":37777},[34,38031],{"className":38032,"style":31814},[31813],[34,38034,38036],{"className":38035},[31818,31819,31820,31821],[34,38037,31660],{"className":38038},[31685,31686,31821],[34,38040,2557],{"className":38041},[31690],[108,38043,38044,2664,38047,37580,38211,38404,38405,38433,38434,38516,38517,347],{},[2627,38045,38046],{},"例题：二路归并排序",[34,38048,38050,38098],{"className":38049},[31642],[34,38051,38053],{"className":38052},[31646],[31648,38054,38055],{"xmlns":31650},[31652,38056,38057,38095],{},[31655,38058,38059,38061,38063,38065,38067,38069,38071,38073,38075,38077,38081,38083,38085,38087,38093],{},[31658,38060,37400],{},[31662,38062,456],{"stretchy":702},[31658,38064,31660],{},[31662,38066,2557],{"stretchy":702},[31662,38068,202],{},[31749,38070,768],{},[31658,38072,37400],{},[31662,38074,456],{"stretchy":702},[31658,38076,31660],{},[31658,38078,38080],{"mathvariant":38079},"normal","\u002F",[31749,38082,768],{},[31662,38084,2557],{"stretchy":702},[31662,38086,223],{},[31743,38088,38089,38091],{},[31658,38090,31965],{},[31749,38092,768],{},[31658,38094,31660],{},[31665,38096,38097],{"encoding":31667},"T(n) = 2T(n\u002F2) + c_2n",[34,38099,38101,38128,38162],{"className":38100,"ariaHidden":758},[31672],[34,38102,38104,38107,38110,38113,38116,38119,38122,38125],{"className":38103},[31676],[34,38105],{"className":38106,"style":31774},[31680],[34,38108,37400],{"className":38109,"style":37422},[31685,31686],[34,38111,456],{"className":38112},[31782],[34,38114,31660],{"className":38115},[31685,31686],[34,38117,2557],{"className":38118},[31690],[34,38120],{"className":38121,"style":31846},[31845],[34,38123,202],{"className":38124},[31850],[34,38126],{"className":38127,"style":31846},[31845],[34,38129,38131,38134,38137,38140,38143,38146,38150,38153,38156,38159],{"className":38130},[31676],[34,38132],{"className":38133,"style":31774},[31680],[34,38135,768],{"className":38136},[31685],[34,38138,37400],{"className":38139,"style":37422},[31685,31686],[34,38141,456],{"className":38142},[31782],[34,38144,31660],{"className":38145},[31685,31686],[34,38147,38149],{"className":38148},[31685],"\u002F2",[34,38151,2557],{"className":38152},[31690],[34,38154],{"className":38155,"style":32604},[31845],[34,38157,223],{"className":38158},[32024],[34,38160],{"className":38161,"style":32604},[31845],[34,38163,38165,38168,38208],{"className":38164},[31676],[34,38166],{"className":38167,"style":31860},[31680],[34,38169,38171,38174],{"className":38170},[31685],[34,38172,31965],{"className":38173},[31685,31686],[34,38175,38177],{"className":38176},[31792],[34,38178,38180,38200],{"className":38179},[31796,31797],[34,38181,38183,38197],{"className":38182},[31801],[34,38184,38186],{"className":38185,"style":31806},[31805],[34,38187,38188,38191],{"style":31809},[34,38189],{"className":38190,"style":31814},[31813],[34,38192,38194],{"className":38193},[31818,31819,31820,31821],[34,38195,768],{"className":38196},[31685,31821],[34,38198,31829],{"className":38199},[31828],[34,38201,38203],{"className":38202},[31801],[34,38204,38206],{"className":38205,"style":31836},[31805],[34,38207],{},[34,38209,31660],{"className":38210},[31685,31686],[34,38212,38214,38260],{"className":38213},[31642],[34,38215,38217],{"className":38216},[31646],[31648,38218,38219],{"xmlns":31650},[31652,38220,38221,38257],{},[31655,38222,38223,38229,38231,38233,38235,38237,38243,38245,38247,38249,38255],{},[37624,38224,38225,38227],{},[31749,38226,768],{},[31749,38228,768],{},[31658,38230,37400],{},[31662,38232,456],{"stretchy":702},[31658,38234,31660],{},[31658,38236,38080],{"mathvariant":38079},[37624,38238,38239,38241],{},[31749,38240,768],{},[31749,38242,768],{},[31662,38244,2557],{"stretchy":702},[31662,38246,223],{},[31749,38248,768],{},[31743,38250,38251,38253],{},[31658,38252,31965],{},[31749,38254,768],{},[31658,38256,31660],{},[31665,38258,38259],{"encoding":31667},"2^2T(n\u002F2^2) + 2c_2n",[34,38261,38263,38351],{"className":38262,"ariaHidden":758},[31672],[34,38264,38266,38269,38298,38301,38304,38307,38310,38339,38342,38345,38348],{"className":38265},[31676],[34,38267],{"className":38268,"style":37755},[31680],[34,38270,38272,38275],{"className":38271},[31685],[34,38273,768],{"className":38274},[31685],[34,38276,38278],{"className":38277},[31792],[34,38279,38281],{"className":38280},[31796],[34,38282,38284],{"className":38283},[31801],[34,38285,38287],{"className":38286,"style":37774},[31805],[34,38288,38289,38292],{"style":37777},[34,38290],{"className":38291,"style":31814},[31813],[34,38293,38295],{"className":38294},[31818,31819,31820,31821],[34,38296,768],{"className":38297},[31685,31821],[34,38299,37400],{"className":38300,"style":37422},[31685,31686],[34,38302,456],{"className":38303},[31782],[34,38305,31660],{"className":38306},[31685,31686],[34,38308,38080],{"className":38309},[31685],[34,38311,38313,38316],{"className":38312},[31685],[34,38314,768],{"className":38315},[31685],[34,38317,38319],{"className":38318},[31792],[34,38320,38322],{"className":38321},[31796],[34,38323,38325],{"className":38324},[31801],[34,38326,38328],{"className":38327,"style":37774},[31805],[34,38329,38330,38333],{"style":37777},[34,38331],{"className":38332,"style":31814},[31813],[34,38334,38336],{"className":38335},[31818,31819,31820,31821],[34,38337,768],{"className":38338},[31685,31821],[34,38340,2557],{"className":38341},[31690],[34,38343],{"className":38344,"style":32604},[31845],[34,38346,223],{"className":38347},[32024],[34,38349],{"className":38350,"style":32604},[31845],[34,38352,38354,38358,38361,38401],{"className":38353},[31676],[34,38355],{"className":38356,"style":38357},[31680],"height:0.7944em;vertical-align:-0.15em;",[34,38359,768],{"className":38360},[31685],[34,38362,38364,38367],{"className":38363},[31685],[34,38365,31965],{"className":38366},[31685,31686],[34,38368,38370],{"className":38369},[31792],[34,38371,38373,38393],{"className":38372},[31796,31797],[34,38374,38376,38390],{"className":38375},[31801],[34,38377,38379],{"className":38378,"style":31806},[31805],[34,38380,38381,38384],{"style":31809},[34,38382],{"className":38383,"style":31814},[31813],[34,38385,38387],{"className":38386},[31818,31819,31820,31821],[34,38388,768],{"className":38389},[31685,31821],[34,38391,31829],{"className":38392},[31828],[34,38394,38396],{"className":38395},[31801],[34,38397,38399],{"className":38398,"style":31836},[31805],[34,38400],{},[34,38402,31660],{"className":38403},[31685,31686],"，推导 ",[34,38406,38408,38421],{"className":38407},[31642],[34,38409,38411],{"className":38410},[31646],[31648,38412,38413],{"xmlns":31650},[31652,38414,38415,38419],{},[31655,38416,38417],{},[31658,38418,33759],{},[31665,38420,33759],{"encoding":31667},[34,38422,38424],{"className":38423,"ariaHidden":758},[31672],[34,38425,38427,38430],{"className":38426},[31676],[34,38428],{"className":38429,"style":31681},[31680],[34,38431,33759],{"className":38432,"style":33774},[31685,31686]," 次后令 ",[34,38435,38437,38459],{"className":38436},[31642],[34,38438,38440],{"className":38439},[31646],[31648,38441,38442],{"xmlns":31650},[31652,38443,38444,38456],{},[31655,38445,38446,38448,38450],{},[31658,38447,31660],{},[31662,38449,202],{},[37624,38451,38452,38454],{},[31749,38453,768],{},[31658,38455,33759],{},[31665,38457,38458],{"encoding":31667},"n=2^k",[34,38460,38462,38480],{"className":38461,"ariaHidden":758},[31672],[34,38463,38465,38468,38471,38474,38477],{"className":38464},[31676],[34,38466],{"className":38467,"style":32527},[31680],[34,38469,31660],{"className":38470},[31685,31686],[34,38472],{"className":38473,"style":31846},[31845],[34,38475,202],{"className":38476},[31850],[34,38478],{"className":38479,"style":31846},[31845],[34,38481,38483,38487],{"className":38482},[31676],[34,38484],{"className":38485,"style":38486},[31680],"height:0.8491em;",[34,38488,38490,38493],{"className":38489},[31685],[34,38491,768],{"className":38492},[31685],[34,38494,38496],{"className":38495},[31792],[34,38497,38499],{"className":38498},[31796],[34,38500,38502],{"className":38501},[31801],[34,38503,38505],{"className":38504,"style":38486},[31805],[34,38506,38507,38510],{"style":37777},[34,38508],{"className":38509,"style":31814},[31813],[34,38511,38513],{"className":38512},[31818,31819,31820,31821],[34,38514,33759],{"className":38515,"style":33774},[31685,31686,31821],"，得 ",[34,38518,38520,38562],{"className":38519},[31642],[34,38521,38523],{"className":38522},[31646],[31648,38524,38525],{"xmlns":31650},[31652,38526,38527,38559],{},[31655,38528,38529,38531,38533,38535,38537,38539,38541,38543,38545,38555,38557],{},[31658,38530,37400],{},[31662,38532,456],{"stretchy":702},[31658,38534,31660],{},[31662,38536,2557],{"stretchy":702},[31662,38538,202],{},[31658,38540,37890],{},[31662,38542,456],{"stretchy":702},[31658,38544,31660],{},[31743,38546,38547,38553],{},[31655,38548,38549,38551],{},[31658,38550,885],{},[31662,38552,32682],{},[31749,38554,768],{},[31658,38556,31660],{},[31662,38558,2557],{"stretchy":702},[31665,38560,38561],{"encoding":31667},"T(n) = O(n\\log_2n)",[34,38563,38565,38592],{"className":38564,"ariaHidden":758},[31672],[34,38566,38568,38571,38574,38577,38580,38583,38586,38589],{"className":38567},[31676],[34,38569],{"className":38570,"style":31774},[31680],[34,38572,37400],{"className":38573,"style":37422},[31685,31686],[34,38575,456],{"className":38576},[31782],[34,38578,31660],{"className":38579},[31685,31686],[34,38581,2557],{"className":38582},[31690],[34,38584],{"className":38585,"style":31846},[31845],[34,38587,202],{"className":38588},[31850],[34,38590],{"className":38591,"style":31846},[31845],[34,38593,38595,38598,38601,38604,38607,38610,38657,38660,38663],{"className":38594},[31676],[34,38596],{"className":38597,"style":31774},[31680],[34,38599,37890],{"className":38600,"style":34452},[31685,31686],[34,38602,456],{"className":38603},[31782],[34,38605,31660],{"className":38606},[31685,31686],[34,38608],{"className":38609,"style":32123},[31845],[34,38611,38613,38620],{"className":38612},[32767],[34,38614,38616,38617],{"className":38615},[32767],"lo",[34,38618,31942],{"style":38619},"margin-right:0.0139em;",[34,38621,38623],{"className":38622},[31792],[34,38624,38626,38648],{"className":38625},[31796,31797],[34,38627,38629,38645],{"className":38628},[31801],[34,38630,38633],{"className":38631,"style":38632},[31805],"height:0.207em;",[34,38634,38636,38639],{"style":38635},"top:-2.4559em;margin-right:0.05em;",[34,38637],{"className":38638,"style":31814},[31813],[34,38640,38642],{"className":38641},[31818,31819,31820,31821],[34,38643,768],{"className":38644},[31685,31821],[34,38646,31829],{"className":38647},[31828],[34,38649,38651],{"className":38650},[31801],[34,38652,38655],{"className":38653,"style":38654},[31805],"height:0.2441em;",[34,38656],{},[34,38658],{"className":38659,"style":32123},[31845],[34,38661,31660],{"className":38662},[31685,31686],[34,38664,2557],{"className":38665},[31690],[32490,38667,38669],{"id":38668},"_2-特征方程法","2. 特征方程法",[105,38671,38672,38708,39257,39575,41146],{},[108,38673,38674,38676,38677,38707],{},[2627,38675,37443],{},"：用于解 ",[34,38678,38680,38694],{"className":38679},[31642],[34,38681,38683],{"className":38682},[31646],[31648,38684,38685],{"xmlns":31650},[31652,38686,38687,38692],{},[31655,38688,38689],{},[31658,38690,38691],{},"K",[31665,38693,38691],{"encoding":31667},[34,38695,38697],{"className":38696,"ariaHidden":758},[31672],[34,38698,38700,38703],{"className":38699},[31676],[34,38701],{"className":38702,"style":33218},[31680],[34,38704,38691],{"className":38705,"style":38706},[31685,31686],"margin-right:0.0715em;"," 阶常系数线性齐次\u002F非齐次递归方程。",[108,38709,38710,38713,38714,38742,38743,38816,38817,347],{},[2627,38711,38712],{},"根不相同","：若有 ",[34,38715,38717,38730],{"className":38716},[31642],[34,38718,38720],{"className":38719},[31646],[31648,38721,38722],{"xmlns":31650},[31652,38723,38724,38728],{},[31655,38725,38726],{},[31658,38727,33759],{},[31665,38729,33759],{"encoding":31667},[34,38731,38733],{"className":38732,"ariaHidden":758},[31672],[34,38734,38736,38739],{"className":38735},[31676],[34,38737],{"className":38738,"style":31681},[31680],[34,38740,33759],{"className":38741,"style":33774},[31685,31686]," 个不同根 ",[34,38744,38746,38765],{"className":38745},[31642],[34,38747,38749],{"className":38748},[31646],[31648,38750,38751],{"xmlns":31650},[31652,38752,38753,38762],{},[31655,38754,38755],{},[31743,38756,38757,38760],{},[31658,38758,38759],{},"q",[31658,38761,31955],{},[31665,38763,38764],{"encoding":31667},"q_i",[34,38766,38768],{"className":38767,"ariaHidden":758},[31672],[34,38769,38771,38775],{"className":38770},[31676],[34,38772],{"className":38773,"style":38774},[31680],"height:0.625em;vertical-align:-0.1944em;",[34,38776,38778,38781],{"className":38777},[31685],[34,38779,38759],{"className":38780,"style":32062},[31685,31686],[34,38782,38784],{"className":38783},[31792],[34,38785,38787,38808],{"className":38786},[31796,31797],[34,38788,38790,38805],{"className":38789},[31801],[34,38791,38793],{"className":38792,"style":32090},[31805],[34,38794,38796,38799],{"style":38795},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[34,38797],{"className":38798,"style":31814},[31813],[34,38800,38802],{"className":38801},[31818,31819,31820,31821],[34,38803,31955],{"className":38804},[31685,31686,31821],[34,38806,31829],{"className":38807},[31828],[34,38809,38811],{"className":38810},[31801],[34,38812,38814],{"className":38813,"style":31836},[31805],[34,38815],{},"，通解为 ",[34,38818,38820,38894],{"className":38819},[31642],[34,38821,38823],{"className":38822},[31646],[31648,38824,38825],{"xmlns":31650},[31652,38826,38827,38891],{},[31655,38828,38829,38831,38833,38835,38837,38839,38845,38854,38856,38862,38870,38872,38875,38877,38883],{},[31658,38830,37400],{},[31662,38832,456],{"stretchy":702},[31658,38834,31660],{},[31662,38836,2557],{"stretchy":702},[31662,38838,202],{},[31743,38840,38841,38843],{},[31658,38842,31965],{},[31749,38844,640],{},[38846,38847,38848,38850,38852],"msubsup",{},[31658,38849,38759],{},[31749,38851,640],{},[31658,38853,31660],{},[31662,38855,223],{},[31743,38857,38858,38860],{},[31658,38859,31965],{},[31749,38861,768],{},[38846,38863,38864,38866,38868],{},[31658,38865,38759],{},[31749,38867,768],{},[31658,38869,31660],{},[31662,38871,223],{},[31662,38873,38874],{},"⋯",[31662,38876,223],{},[31743,38878,38879,38881],{},[31658,38880,31965],{},[31658,38882,33759],{},[38846,38884,38885,38887,38889],{},[31658,38886,38759],{},[31658,38888,33759],{},[31658,38890,31660],{},[31665,38892,38893],{"encoding":31667},"T(n) = c_1q_1^n + c_2q_2^n + \\dots + c_kq_k^n",[34,38895,38897,38924,39033,39139,39157],{"className":38896,"ariaHidden":758},[31672],[34,38898,38900,38903,38906,38909,38912,38915,38918,38921],{"className":38899},[31676],[34,38901],{"className":38902,"style":31774},[31680],[34,38904,37400],{"className":38905,"style":37422},[31685,31686],[34,38907,456],{"className":38908},[31782],[34,38910,31660],{"className":38911},[31685,31686],[34,38913,2557],{"className":38914},[31690],[34,38916],{"className":38917,"style":31846},[31845],[34,38919,202],{"className":38920},[31850],[34,38922],{"className":38923,"style":31846},[31845],[34,38925,38927,38931,38971,39024,39027,39030],{"className":38926},[31676],[34,38928],{"className":38929,"style":38930},[31680],"height:0.9125em;vertical-align:-0.2481em;",[34,38932,38934,38937],{"className":38933},[31685],[34,38935,31965],{"className":38936},[31685,31686],[34,38938,38940],{"className":38939},[31792],[34,38941,38943,38963],{"className":38942},[31796,31797],[34,38944,38946,38960],{"className":38945},[31801],[34,38947,38949],{"className":38948,"style":31806},[31805],[34,38950,38951,38954],{"style":31809},[34,38952],{"className":38953,"style":31814},[31813],[34,38955,38957],{"className":38956},[31818,31819,31820,31821],[34,38958,640],{"className":38959},[31685,31821],[34,38961,31829],{"className":38962},[31828],[34,38964,38966],{"className":38965},[31801],[34,38967,38969],{"className":38968,"style":31836},[31805],[34,38970],{},[34,38972,38974,38977],{"className":38973},[31685],[34,38975,38759],{"className":38976,"style":32062},[31685,31686],[34,38978,38980],{"className":38979},[31792],[34,38981,38983,39015],{"className":38982},[31796,31797],[34,38984,38986,39012],{"className":38985},[31801],[34,38987,38989,39001],{"className":38988,"style":37959},[31805],[34,38990,38992,38995],{"style":38991},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[34,38993],{"className":38994,"style":31814},[31813],[34,38996,38998],{"className":38997},[31818,31819,31820,31821],[34,38999,640],{"className":39000},[31685,31821],[34,39002,39003,39006],{"style":37777},[34,39004],{"className":39005,"style":31814},[31813],[34,39007,39009],{"className":39008},[31818,31819,31820,31821],[34,39010,31660],{"className":39011},[31685,31686,31821],[34,39013,31829],{"className":39014},[31828],[34,39016,39018],{"className":39017},[31801],[34,39019,39022],{"className":39020,"style":39021},[31805],"height:0.2481em;",[34,39023],{},[34,39025],{"className":39026,"style":32604},[31845],[34,39028,223],{"className":39029},[32024],[34,39031],{"className":39032,"style":32604},[31845],[34,39034,39036,39039,39079,39130,39133,39136],{"className":39035},[31676],[34,39037],{"className":39038,"style":38930},[31680],[34,39040,39042,39045],{"className":39041},[31685],[34,39043,31965],{"className":39044},[31685,31686],[34,39046,39048],{"className":39047},[31792],[34,39049,39051,39071],{"className":39050},[31796,31797],[34,39052,39054,39068],{"className":39053},[31801],[34,39055,39057],{"className":39056,"style":31806},[31805],[34,39058,39059,39062],{"style":31809},[34,39060],{"className":39061,"style":31814},[31813],[34,39063,39065],{"className":39064},[31818,31819,31820,31821],[34,39066,768],{"className":39067},[31685,31821],[34,39069,31829],{"className":39070},[31828],[34,39072,39074],{"className":39073},[31801],[34,39075,39077],{"className":39076,"style":31836},[31805],[34,39078],{},[34,39080,39082,39085],{"className":39081},[31685],[34,39083,38759],{"className":39084,"style":32062},[31685,31686],[34,39086,39088],{"className":39087},[31792],[34,39089,39091,39122],{"className":39090},[31796,31797],[34,39092,39094,39119],{"className":39093},[31801],[34,39095,39097,39108],{"className":39096,"style":37959},[31805],[34,39098,39099,39102],{"style":38991},[34,39100],{"className":39101,"style":31814},[31813],[34,39103,39105],{"className":39104},[31818,31819,31820,31821],[34,39106,768],{"className":39107},[31685,31821],[34,39109,39110,39113],{"style":37777},[34,39111],{"className":39112,"style":31814},[31813],[34,39114,39116],{"className":39115},[31818,31819,31820,31821],[34,39117,31660],{"className":39118},[31685,31686,31821],[34,39120,31829],{"className":39121},[31828],[34,39123,39125],{"className":39124},[31801],[34,39126,39128],{"className":39127,"style":39021},[31805],[34,39129],{},[34,39131],{"className":39132,"style":32604},[31845],[34,39134,223],{"className":39135},[32024],[34,39137],{"className":39138,"style":32604},[31845],[34,39140,39142,39145,39148,39151,39154],{"className":39141},[31676],[34,39143],{"className":39144,"style":32597},[31680],[34,39146,38874],{"className":39147},[34006],[34,39149],{"className":39150,"style":32604},[31845],[34,39152,223],{"className":39153},[32024],[34,39155],{"className":39156,"style":32604},[31845],[34,39158,39160,39164,39204],{"className":39159},[31676],[34,39161],{"className":39162,"style":39163},[31680],"height:0.9475em;vertical-align:-0.2831em;",[34,39165,39167,39170],{"className":39166},[31685],[34,39168,31965],{"className":39169},[31685,31686],[34,39171,39173],{"className":39172},[31792],[34,39174,39176,39196],{"className":39175},[31796,31797],[34,39177,39179,39193],{"className":39178},[31801],[34,39180,39182],{"className":39181,"style":33833},[31805],[34,39183,39184,39187],{"style":31809},[34,39185],{"className":39186,"style":31814},[31813],[34,39188,39190],{"className":39189},[31818,31819,31820,31821],[34,39191,33759],{"className":39192,"style":33774},[31685,31686,31821],[34,39194,31829],{"className":39195},[31828],[34,39197,39199],{"className":39198},[31801],[34,39200,39202],{"className":39201,"style":31836},[31805],[34,39203],{},[34,39205,39207,39210],{"className":39206},[31685],[34,39208,38759],{"className":39209,"style":32062},[31685,31686],[34,39211,39213],{"className":39212},[31792],[34,39214,39216,39248],{"className":39215},[31796,31797],[34,39217,39219,39245],{"className":39218},[31801],[34,39220,39222,39234],{"className":39221,"style":37959},[31805],[34,39223,39225,39228],{"style":39224},"top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;",[34,39226],{"className":39227,"style":31814},[31813],[34,39229,39231],{"className":39230},[31818,31819,31820,31821],[34,39232,33759],{"className":39233,"style":33774},[31685,31686,31821],[34,39235,39236,39239],{"style":37777},[34,39237],{"className":39238,"style":31814},[31813],[34,39240,39242],{"className":39241},[31818,31819,31820,31821],[34,39243,31660],{"className":39244},[31685,31686,31821],[34,39246,31829],{"className":39247},[31828],[34,39249,39251],{"className":39250},[31801],[34,39252,39255],{"className":39253,"style":39254},[31805],"height:0.2831em;",[34,39256],{},[108,39258,39259,38713,39262,39290,39291,39319,39320,347],{},[2627,39260,39261],{},"有重根",[34,39263,39265,39278],{"className":39264},[31642],[34,39266,39268],{"className":39267},[31646],[31648,39269,39270],{"xmlns":31650},[31652,39271,39272,39276],{},[31655,39273,39274],{},[31658,39275,34395],{},[31665,39277,34395],{"encoding":31667},[34,39279,39281],{"className":39280,"ariaHidden":758},[31672],[34,39282,39284,39287],{"className":39283},[31676],[34,39285],{"className":39286,"style":32527},[31680],[34,39288,34395],{"className":39289,"style":34452},[31685,31686]," 个重根，通解会带有 ",[34,39292,39294,39307],{"className":39293},[31642],[34,39295,39297],{"className":39296},[31646],[31648,39298,39299],{"xmlns":31650},[31652,39300,39301,39305],{},[31655,39302,39303],{},[31658,39304,31660],{},[31665,39306,31660],{"encoding":31667},[34,39308,39310],{"className":39309,"ariaHidden":758},[31672],[34,39311,39313,39316],{"className":39312},[31676],[34,39314],{"className":39315,"style":32527},[31680],[34,39317,31660],{"className":39318},[31685,31686]," 的多项式系数，如 ",[34,39321,39323,39377],{"className":39322},[31642],[34,39324,39326],{"className":39325},[31646],[31648,39327,39328],{"xmlns":31650},[31652,39329,39330,39374],{},[31655,39331,39332,39334,39340,39342,39354,39356,39358,39360,39364,39366],{},[31662,39333,456],{"stretchy":702},[31743,39335,39336,39338],{},[31658,39337,31965],{},[31658,39339,31955],{},[31662,39341,223],{},[31743,39343,39344,39346],{},[31658,39345,31965],{},[31655,39347,39348,39350,39352],{},[31658,39349,31955],{},[31662,39351,223],{},[31749,39353,640],{},[31658,39355,31660],{},[31662,39357,223],{},[31662,39359,33934],{},[39361,39362,39363],"mtext",{}," ",[31662,39365,2557],{"stretchy":702},[38846,39367,39368,39370,39372],{},[31658,39369,38759],{},[31658,39371,31955],{},[31658,39373,31660],{},[31665,39375,39376],{"encoding":31667},"(c_i+c_{i+1}n+\\dots)q_i^n",[34,39378,39380,39438,39506],{"className":39379,"ariaHidden":758},[31672],[34,39381,39383,39386,39389,39429,39432,39435],{"className":39382},[31676],[34,39384],{"className":39385,"style":31774},[31680],[34,39387,456],{"className":39388},[31782],[34,39390,39392,39395],{"className":39391},[31685],[34,39393,31965],{"className":39394},[31685,31686],[34,39396,39398],{"className":39397},[31792],[34,39399,39401,39421],{"className":39400},[31796,31797],[34,39402,39404,39418],{"className":39403},[31801],[34,39405,39407],{"className":39406,"style":32090},[31805],[34,39408,39409,39412],{"style":31809},[34,39410],{"className":39411,"style":31814},[31813],[34,39413,39415],{"className":39414},[31818,31819,31820,31821],[34,39416,31955],{"className":39417},[31685,31686,31821],[34,39419,31829],{"className":39420},[31828],[34,39422,39424],{"className":39423},[31801],[34,39425,39427],{"className":39426,"style":31836},[31805],[34,39428],{},[34,39430],{"className":39431,"style":32604},[31845],[34,39433,223],{"className":39434},[32024],[34,39436],{"className":39437,"style":32604},[31845],[34,39439,39441,39445,39494,39497,39500,39503],{"className":39440},[31676],[34,39442],{"className":39443,"style":39444},[31680],"height:0.7917em;vertical-align:-0.2083em;",[34,39446,39448,39451],{"className":39447},[31685],[34,39449,31965],{"className":39450},[31685,31686],[34,39452,39454],{"className":39453},[31792],[34,39455,39457,39486],{"className":39456},[31796,31797],[34,39458,39460,39483],{"className":39459},[31801],[34,39461,39463],{"className":39462,"style":32090},[31805],[34,39464,39465,39468],{"style":31809},[34,39466],{"className":39467,"style":31814},[31813],[34,39469,39471],{"className":39470},[31818,31819,31820,31821],[34,39472,39474,39477,39480],{"className":39473},[31685,31821],[34,39475,31955],{"className":39476},[31685,31686,31821],[34,39478,223],{"className":39479},[32024,31821],[34,39481,640],{"className":39482},[31685,31821],[34,39484,31829],{"className":39485},[31828],[34,39487,39489],{"className":39488},[31801],[34,39490,39492],{"className":39491,"style":32037},[31805],[34,39493],{},[34,39495,31660],{"className":39496},[31685,31686],[34,39498],{"className":39499,"style":32604},[31845],[34,39501,223],{"className":39502},[32024],[34,39504],{"className":39505,"style":32604},[31845],[34,39507,39509,39513,39516,39519,39522],{"className":39508},[31676],[34,39510],{"className":39511,"style":39512},[31680],"height:1.0087em;vertical-align:-0.2587em;",[34,39514,33934],{"className":39515},[34006],[34,39517],{"className":39518,"style":32123},[31845],[34,39520,2557],{"className":39521},[31690],[34,39523,39525,39528],{"className":39524},[31685],[34,39526,38759],{"className":39527,"style":32062},[31685,31686],[34,39529,39531],{"className":39530},[31792],[34,39532,39534,39566],{"className":39533},[31796,31797],[34,39535,39537,39563],{"className":39536},[31801],[34,39538,39540,39552],{"className":39539,"style":37959},[31805],[34,39541,39543,39546],{"style":39542},"top:-2.4413em;margin-left:-0.0359em;margin-right:0.05em;",[34,39544],{"className":39545,"style":31814},[31813],[34,39547,39549],{"className":39548},[31818,31819,31820,31821],[34,39550,31955],{"className":39551},[31685,31686,31821],[34,39553,39554,39557],{"style":37777},[34,39555],{"className":39556,"style":31814},[31813],[34,39558,39560],{"className":39559},[31818,31819,31820,31821],[34,39561,31660],{"className":39562},[31685,31686,31821],[34,39564,31829],{"className":39565},[31828],[34,39567,39569],{"className":39568},[31801],[34,39570,39573],{"className":39571,"style":39572},[31805],"height:0.2587em;",[34,39574],{},[108,39576,39577,2664,39580,39817,39818,39984,39985],{},[2627,39578,39579],{},"例题 1",[34,39581,39583,39652],{"className":39582},[31642],[34,39584,39586],{"className":39585},[31646],[31648,39587,39588],{"xmlns":31650},[31652,39589,39590,39649],{},[31655,39591,39592,39594,39596,39598,39600,39602,39604,39606,39608,39610,39612,39614,39616,39618,39621,39623,39625,39627,39629,39631,39633,39635,39637,39639,39641,39643,39645,39647],{},[31658,39593,37400],{},[31662,39595,456],{"stretchy":702},[31658,39597,31660],{},[31662,39599,2557],{"stretchy":702},[31662,39601,202],{},[31749,39603,36656],{},[31658,39605,37400],{},[31662,39607,456],{"stretchy":702},[31658,39609,31660],{},[31662,39611,32226],{},[31749,39613,640],{},[31662,39615,2557],{"stretchy":702},[31662,39617,32226],{},[31749,39619,39620],{},"11",[31658,39622,37400],{},[31662,39624,456],{"stretchy":702},[31658,39626,31660],{},[31662,39628,32226],{},[31749,39630,768],{},[31662,39632,2557],{"stretchy":702},[31662,39634,223],{},[31749,39636,36656],{},[31658,39638,37400],{},[31662,39640,456],{"stretchy":702},[31658,39642,31660],{},[31662,39644,32226],{},[31749,39646,786],{},[31662,39648,2557],{"stretchy":702},[31665,39650,39651],{"encoding":31667},"T(n) = 6T(n-1) - 11T(n-2) + 6T(n-3)",[34,39653,39655,39682,39709,39730,39757,39778,39805],{"className":39654,"ariaHidden":758},[31672],[34,39656,39658,39661,39664,39667,39670,39673,39676,39679],{"className":39657},[31676],[34,39659],{"className":39660,"style":31774},[31680],[34,39662,37400],{"className":39663,"style":37422},[31685,31686],[34,39665,456],{"className":39666},[31782],[34,39668,31660],{"className":39669},[31685,31686],[34,39671,2557],{"className":39672},[31690],[34,39674],{"className":39675,"style":31846},[31845],[34,39677,202],{"className":39678},[31850],[34,39680],{"className":39681,"style":31846},[31845],[34,39683,39685,39688,39691,39694,39697,39700,39703,39706],{"className":39684},[31676],[34,39686],{"className":39687,"style":31774},[31680],[34,39689,36656],{"className":39690},[31685],[34,39692,37400],{"className":39693,"style":37422},[31685,31686],[34,39695,456],{"className":39696},[31782],[34,39698,31660],{"className":39699},[31685,31686],[34,39701],{"className":39702,"style":32604},[31845],[34,39704,32226],{"className":39705},[32024],[34,39707],{"className":39708,"style":32604},[31845],[34,39710,39712,39715,39718,39721,39724,39727],{"className":39711},[31676],[34,39713],{"className":39714,"style":31774},[31680],[34,39716,640],{"className":39717},[31685],[34,39719,2557],{"className":39720},[31690],[34,39722],{"className":39723,"style":32604},[31845],[34,39725,32226],{"className":39726},[32024],[34,39728],{"className":39729,"style":32604},[31845],[34,39731,39733,39736,39739,39742,39745,39748,39751,39754],{"className":39732},[31676],[34,39734],{"className":39735,"style":31774},[31680],[34,39737,39620],{"className":39738},[31685],[34,39740,37400],{"className":39741,"style":37422},[31685,31686],[34,39743,456],{"className":39744},[31782],[34,39746,31660],{"className":39747},[31685,31686],[34,39749],{"className":39750,"style":32604},[31845],[34,39752,32226],{"className":39753},[32024],[34,39755],{"className":39756,"style":32604},[31845],[34,39758,39760,39763,39766,39769,39772,39775],{"className":39759},[31676],[34,39761],{"className":39762,"style":31774},[31680],[34,39764,768],{"className":39765},[31685],[34,39767,2557],{"className":39768},[31690],[34,39770],{"className":39771,"style":32604},[31845],[34,39773,223],{"className":39774},[32024],[34,39776],{"className":39777,"style":32604},[31845],[34,39779,39781,39784,39787,39790,39793,39796,39799,39802],{"className":39780},[31676],[34,39782],{"className":39783,"style":31774},[31680],[34,39785,36656],{"className":39786},[31685],[34,39788,37400],{"className":39789,"style":37422},[31685,31686],[34,39791,456],{"className":39792},[31782],[34,39794,31660],{"className":39795},[31685,31686],[34,39797],{"className":39798,"style":32604},[31845],[34,39800,32226],{"className":39801},[32024],[34,39803],{"className":39804,"style":32604},[31845],[34,39806,39808,39811,39814],{"className":39807},[31676],[34,39809],{"className":39810,"style":31774},[31680],[34,39812,786],{"className":39813},[31685],[34,39815,2557],{"className":39816},[31690],"，初始 ",[34,39819,39821,39873],{"className":39820},[31642],[34,39822,39824],{"className":39823},[31646],[31648,39825,39826],{"xmlns":31650},[31652,39827,39828,39870],{},[31655,39829,39830,39832,39834,39836,39838,39840,39842,39844,39846,39848,39850,39852,39854,39856,39858,39860,39862,39864,39866,39868],{},[31658,39831,37400],{},[31662,39833,456],{"stretchy":702},[31749,39835,459],{},[31662,39837,2557],{"stretchy":702},[31662,39839,202],{},[31749,39841,459],{},[31662,39843,31960],{"separator":758},[31658,39845,37400],{},[31662,39847,456],{"stretchy":702},[31749,39849,640],{},[31662,39851,2557],{"stretchy":702},[31662,39853,202],{},[31749,39855,768],{},[31662,39857,31960],{"separator":758},[31658,39859,37400],{},[31662,39861,456],{"stretchy":702},[31749,39863,768],{},[31662,39865,2557],{"stretchy":702},[31662,39867,202],{},[31749,39869,3244],{},[31665,39871,39872],{"encoding":31667},"T(0)=0, T(1)=2, T(2)=10",[34,39874,39876,39903,39939,39975],{"className":39875,"ariaHidden":758},[31672],[34,39877,39879,39882,39885,39888,39891,39894,39897,39900],{"className":39878},[31676],[34,39880],{"className":39881,"style":31774},[31680],[34,39883,37400],{"className":39884,"style":37422},[31685,31686],[34,39886,456],{"className":39887},[31782],[34,39889,459],{"className":39890},[31685],[34,39892,2557],{"className":39893},[31690],[34,39895],{"className":39896,"style":31846},[31845],[34,39898,202],{"className":39899},[31850],[34,39901],{"className":39902,"style":31846},[31845],[34,39904,39906,39909,39912,39915,39918,39921,39924,39927,39930,39933,39936],{"className":39905},[31676],[34,39907],{"className":39908,"style":31774},[31680],[34,39910,459],{"className":39911},[31685],[34,39913,31960],{"className":39914},[32119],[34,39916],{"className":39917,"style":32123},[31845],[34,39919,37400],{"className":39920,"style":37422},[31685,31686],[34,39922,456],{"className":39923},[31782],[34,39925,640],{"className":39926},[31685],[34,39928,2557],{"className":39929},[31690],[34,39931],{"className":39932,"style":31846},[31845],[34,39934,202],{"className":39935},[31850],[34,39937],{"className":39938,"style":31846},[31845],[34,39940,39942,39945,39948,39951,39954,39957,39960,39963,39966,39969,39972],{"className":39941},[31676],[34,39943],{"className":39944,"style":31774},[31680],[34,39946,768],{"className":39947},[31685],[34,39949,31960],{"className":39950},[32119],[34,39952],{"className":39953,"style":32123},[31845],[34,39955,37400],{"className":39956,"style":37422},[31685,31686],[34,39958,456],{"className":39959},[31782],[34,39961,768],{"className":39962},[31685],[34,39964,2557],{"className":39965},[31690],[34,39967],{"className":39968,"style":31846},[31845],[34,39970,202],{"className":39971},[31850],[34,39973],{"className":39974,"style":31846},[31845],[34,39976,39978,39981],{"className":39977},[31676],[34,39979],{"className":39980,"style":32617},[31680],[34,39982,3244],{"className":39983},[31685],"。\n",[105,39986,39987,40424],{},[108,39988,39989,39990,40178,40179,347],{},"推导：特征方程 ",[34,39991,39993,40035],{"className":39992},[31642],[34,39994,39996],{"className":39995},[31646],[31648,39997,39998],{"xmlns":31650},[31652,39999,40000,40032],{},[31655,40001,40002,40008,40010,40012,40018,40020,40022,40024,40026,40028,40030],{},[37624,40003,40004,40006],{},[31658,40005,32936],{},[31749,40007,786],{},[31662,40009,32226],{},[31749,40011,36656],{},[37624,40013,40014,40016],{},[31658,40015,32936],{},[31749,40017,768],{},[31662,40019,223],{},[31749,40021,39620],{},[31658,40023,32936],{},[31662,40025,32226],{},[31749,40027,36656],{},[31662,40029,202],{},[31749,40031,459],{},[31665,40033,40034],{"encoding":31667},"x^3-6x^2+11x-6=0",[34,40036,40038,40083,40130,40151,40169],{"className":40037,"ariaHidden":758},[31672],[34,40039,40041,40045,40074,40077,40080],{"className":40040},[31676],[34,40042],{"className":40043,"style":40044},[31680],"height:0.8974em;vertical-align:-0.0833em;",[34,40046,40048,40051],{"className":40047},[31685],[34,40049,32936],{"className":40050},[31685,31686],[34,40052,40054],{"className":40053},[31792],[34,40055,40057],{"className":40056},[31796],[34,40058,40060],{"className":40059},[31801],[34,40061,40063],{"className":40062,"style":37774},[31805],[34,40064,40065,40068],{"style":37777},[34,40066],{"className":40067,"style":31814},[31813],[34,40069,40071],{"className":40070},[31818,31819,31820,31821],[34,40072,786],{"className":40073},[31685,31821],[34,40075],{"className":40076,"style":32604},[31845],[34,40078,32226],{"className":40079},[32024],[34,40081],{"className":40082,"style":32604},[31845],[34,40084,40086,40089,40092,40121,40124,40127],{"className":40085},[31676],[34,40087],{"className":40088,"style":40044},[31680],[34,40090,36656],{"className":40091},[31685],[34,40093,40095,40098],{"className":40094},[31685],[34,40096,32936],{"className":40097},[31685,31686],[34,40099,40101],{"className":40100},[31792],[34,40102,40104],{"className":40103},[31796],[34,40105,40107],{"className":40106},[31801],[34,40108,40110],{"className":40109,"style":37774},[31805],[34,40111,40112,40115],{"style":37777},[34,40113],{"className":40114,"style":31814},[31813],[34,40116,40118],{"className":40117},[31818,31819,31820,31821],[34,40119,768],{"className":40120},[31685,31821],[34,40122],{"className":40123,"style":32604},[31845],[34,40125,223],{"className":40126},[32024],[34,40128],{"className":40129,"style":32604},[31845],[34,40131,40133,40136,40139,40142,40145,40148],{"className":40132},[31676],[34,40134],{"className":40135,"style":35942},[31680],[34,40137,39620],{"className":40138},[31685],[34,40140,32936],{"className":40141},[31685,31686],[34,40143],{"className":40144,"style":32604},[31845],[34,40146,32226],{"className":40147},[32024],[34,40149],{"className":40150,"style":32604},[31845],[34,40152,40154,40157,40160,40163,40166],{"className":40153},[31676],[34,40155],{"className":40156,"style":32617},[31680],[34,40158,36656],{"className":40159},[31685],[34,40161],{"className":40162,"style":31846},[31845],[34,40164,202],{"className":40165},[31850],[34,40167],{"className":40168,"style":31846},[31845],[34,40170,40172,40175],{"className":40171},[31676],[34,40173],{"className":40174,"style":32617},[31680],[34,40176,459],{"className":40177},[31685],"，因式分解得根 ",[34,40180,40182,40228],{"className":40181},[31642],[34,40183,40185],{"className":40184},[31646],[31648,40186,40187],{"xmlns":31650},[31652,40188,40189,40225],{},[31655,40190,40191,40197,40199,40201,40203,40209,40211,40213,40215,40221,40223],{},[31743,40192,40193,40195],{},[31658,40194,38759],{},[31749,40196,640],{},[31662,40198,202],{},[31749,40200,640],{},[31662,40202,31960],{"separator":758},[31743,40204,40205,40207],{},[31658,40206,38759],{},[31749,40208,768],{},[31662,40210,202],{},[31749,40212,768],{},[31662,40214,31960],{"separator":758},[31743,40216,40217,40219],{},[31658,40218,38759],{},[31749,40220,786],{},[31662,40222,202],{},[31749,40224,786],{},[31665,40226,40227],{"encoding":31667},"q_1=1, q_2=2, q_3=3",[34,40229,40231,40286,40351,40415],{"className":40230,"ariaHidden":758},[31672],[34,40232,40234,40237,40277,40280,40283],{"className":40233},[31676],[34,40235],{"className":40236,"style":38774},[31680],[34,40238,40240,40243],{"className":40239},[31685],[34,40241,38759],{"className":40242,"style":32062},[31685,31686],[34,40244,40246],{"className":40245},[31792],[34,40247,40249,40269],{"className":40248},[31796,31797],[34,40250,40252,40266],{"className":40251},[31801],[34,40253,40255],{"className":40254,"style":31806},[31805],[34,40256,40257,40260],{"style":38795},[34,40258],{"className":40259,"style":31814},[31813],[34,40261,40263],{"className":40262},[31818,31819,31820,31821],[34,40264,640],{"className":40265},[31685,31821],[34,40267,31829],{"className":40268},[31828],[34,40270,40272],{"className":40271},[31801],[34,40273,40275],{"className":40274,"style":31836},[31805],[34,40276],{},[34,40278],{"className":40279,"style":31846},[31845],[34,40281,202],{"className":40282},[31850],[34,40284],{"className":40285,"style":31846},[31845],[34,40287,40289,40293,40296,40299,40302,40342,40345,40348],{"className":40288},[31676],[34,40290],{"className":40291,"style":40292},[31680],"height:0.8389em;vertical-align:-0.1944em;",[34,40294,640],{"className":40295},[31685],[34,40297,31960],{"className":40298},[32119],[34,40300],{"className":40301,"style":32123},[31845],[34,40303,40305,40308],{"className":40304},[31685],[34,40306,38759],{"className":40307,"style":32062},[31685,31686],[34,40309,40311],{"className":40310},[31792],[34,40312,40314,40334],{"className":40313},[31796,31797],[34,40315,40317,40331],{"className":40316},[31801],[34,40318,40320],{"className":40319,"style":31806},[31805],[34,40321,40322,40325],{"style":38795},[34,40323],{"className":40324,"style":31814},[31813],[34,40326,40328],{"className":40327},[31818,31819,31820,31821],[34,40329,768],{"className":40330},[31685,31821],[34,40332,31829],{"className":40333},[31828],[34,40335,40337],{"className":40336},[31801],[34,40338,40340],{"className":40339,"style":31836},[31805],[34,40341],{},[34,40343],{"className":40344,"style":31846},[31845],[34,40346,202],{"className":40347},[31850],[34,40349],{"className":40350,"style":31846},[31845],[34,40352,40354,40357,40360,40363,40366,40406,40409,40412],{"className":40353},[31676],[34,40355],{"className":40356,"style":40292},[31680],[34,40358,768],{"className":40359},[31685],[34,40361,31960],{"className":40362},[32119],[34,40364],{"className":40365,"style":32123},[31845],[34,40367,40369,40372],{"className":40368},[31685],[34,40370,38759],{"className":40371,"style":32062},[31685,31686],[34,40373,40375],{"className":40374},[31792],[34,40376,40378,40398],{"className":40377},[31796,31797],[34,40379,40381,40395],{"className":40380},[31801],[34,40382,40384],{"className":40383,"style":31806},[31805],[34,40385,40386,40389],{"style":38795},[34,40387],{"className":40388,"style":31814},[31813],[34,40390,40392],{"className":40391},[31818,31819,31820,31821],[34,40393,786],{"className":40394},[31685,31821],[34,40396,31829],{"className":40397},[31828],[34,40399,40401],{"className":40400},[31801],[34,40402,40404],{"className":40403,"style":31836},[31805],[34,40405],{},[34,40407],{"className":40408,"style":31846},[31845],[34,40410,202],{"className":40411},[31850],[34,40413],{"className":40414,"style":31846},[31845],[34,40416,40418,40421],{"className":40417},[31676],[34,40419],{"className":40420,"style":32617},[31680],[34,40422,786],{"className":40423},[31685],[108,40425,40426,40427,40732,40733,40982,40983,347],{},"通解为 ",[34,40428,40430,40486],{"className":40429},[31642],[34,40431,40433],{"className":40432},[31646],[31648,40434,40435],{"xmlns":31650},[31652,40436,40437,40483],{},[31655,40438,40439,40441,40443,40445,40447,40449,40455,40457,40463,40469,40471,40477],{},[31658,40440,37400],{},[31662,40442,456],{"stretchy":702},[31658,40444,31660],{},[31662,40446,2557],{"stretchy":702},[31662,40448,202],{},[31743,40450,40451,40453],{},[31658,40452,31965],{},[31749,40454,640],{},[31662,40456,223],{},[31743,40458,40459,40461],{},[31658,40460,31965],{},[31749,40462,768],{},[37624,40464,40465,40467],{},[31749,40466,768],{},[31658,40468,31660],{},[31662,40470,223],{},[31743,40472,40473,40475],{},[31658,40474,31965],{},[31749,40476,786],{},[37624,40478,40479,40481],{},[31749,40480,786],{},[31658,40482,31660],{},[31665,40484,40485],{"encoding":31667},"T(n) = c_1 + c_2 2^n + c_3 3^n",[34,40487,40489,40516,40572,40657],{"className":40488,"ariaHidden":758},[31672],[34,40490,40492,40495,40498,40501,40504,40507,40510,40513],{"className":40491},[31676],[34,40493],{"className":40494,"style":31774},[31680],[34,40496,37400],{"className":40497,"style":37422},[31685,31686],[34,40499,456],{"className":40500},[31782],[34,40502,31660],{"className":40503},[31685,31686],[34,40505,2557],{"className":40506},[31690],[34,40508],{"className":40509,"style":31846},[31845],[34,40511,202],{"className":40512},[31850],[34,40514],{"className":40515,"style":31846},[31845],[34,40517,40519,40523,40563,40566,40569],{"className":40518},[31676],[34,40520],{"className":40521,"style":40522},[31680],"height:0.7333em;vertical-align:-0.15em;",[34,40524,40526,40529],{"className":40525},[31685],[34,40527,31965],{"className":40528},[31685,31686],[34,40530,40532],{"className":40531},[31792],[34,40533,40535,40555],{"className":40534},[31796,31797],[34,40536,40538,40552],{"className":40537},[31801],[34,40539,40541],{"className":40540,"style":31806},[31805],[34,40542,40543,40546],{"style":31809},[34,40544],{"className":40545,"style":31814},[31813],[34,40547,40549],{"className":40548},[31818,31819,31820,31821],[34,40550,640],{"className":40551},[31685,31821],[34,40553,31829],{"className":40554},[31828],[34,40556,40558],{"className":40557},[31801],[34,40559,40561],{"className":40560,"style":31836},[31805],[34,40562],{},[34,40564],{"className":40565,"style":32604},[31845],[34,40567,223],{"className":40568},[32024],[34,40570],{"className":40571,"style":32604},[31845],[34,40573,40575,40579,40619,40648,40651,40654],{"className":40574},[31676],[34,40576],{"className":40577,"style":40578},[31680],"height:0.8144em;vertical-align:-0.15em;",[34,40580,40582,40585],{"className":40581},[31685],[34,40583,31965],{"className":40584},[31685,31686],[34,40586,40588],{"className":40587},[31792],[34,40589,40591,40611],{"className":40590},[31796,31797],[34,40592,40594,40608],{"className":40593},[31801],[34,40595,40597],{"className":40596,"style":31806},[31805],[34,40598,40599,40602],{"style":31809},[34,40600],{"className":40601,"style":31814},[31813],[34,40603,40605],{"className":40604},[31818,31819,31820,31821],[34,40606,768],{"className":40607},[31685,31821],[34,40609,31829],{"className":40610},[31828],[34,40612,40614],{"className":40613},[31801],[34,40615,40617],{"className":40616,"style":31836},[31805],[34,40618],{},[34,40620,40622,40625],{"className":40621},[31685],[34,40623,768],{"className":40624},[31685],[34,40626,40628],{"className":40627},[31792],[34,40629,40631],{"className":40630},[31796],[34,40632,40634],{"className":40633},[31801],[34,40635,40637],{"className":40636,"style":37959},[31805],[34,40638,40639,40642],{"style":37777},[34,40640],{"className":40641,"style":31814},[31813],[34,40643,40645],{"className":40644},[31818,31819,31820,31821],[34,40646,31660],{"className":40647},[31685,31686,31821],[34,40649],{"className":40650,"style":32604},[31845],[34,40652,223],{"className":40653},[32024],[34,40655],{"className":40656,"style":32604},[31845],[34,40658,40660,40663,40703],{"className":40659},[31676],[34,40661],{"className":40662,"style":40578},[31680],[34,40664,40666,40669],{"className":40665},[31685],[34,40667,31965],{"className":40668},[31685,31686],[34,40670,40672],{"className":40671},[31792],[34,40673,40675,40695],{"className":40674},[31796,31797],[34,40676,40678,40692],{"className":40677},[31801],[34,40679,40681],{"className":40680,"style":31806},[31805],[34,40682,40683,40686],{"style":31809},[34,40684],{"className":40685,"style":31814},[31813],[34,40687,40689],{"className":40688},[31818,31819,31820,31821],[34,40690,786],{"className":40691},[31685,31821],[34,40693,31829],{"className":40694},[31828],[34,40696,40698],{"className":40697},[31801],[34,40699,40701],{"className":40700,"style":31836},[31805],[34,40702],{},[34,40704,40706,40709],{"className":40705},[31685],[34,40707,786],{"className":40708},[31685],[34,40710,40712],{"className":40711},[31792],[34,40713,40715],{"className":40714},[31796],[34,40716,40718],{"className":40717},[31801],[34,40719,40721],{"className":40720,"style":37959},[31805],[34,40722,40723,40726],{"style":37777},[34,40724],{"className":40725,"style":31814},[31813],[34,40727,40729],{"className":40728},[31818,31819,31820,31821],[34,40730,31660],{"className":40731},[31685,31686,31821],"。代入初始条件解得 ",[34,40734,40736,40784],{"className":40735},[31642],[34,40737,40739],{"className":40738},[31646],[31648,40740,40741],{"xmlns":31650},[31652,40742,40743,40781],{},[31655,40744,40745,40751,40753,40755,40757,40763,40765,40767,40769,40771,40777,40779],{},[31743,40746,40747,40749],{},[31658,40748,31965],{},[31749,40750,640],{},[31662,40752,202],{},[31749,40754,459],{},[31662,40756,31960],{"separator":758},[31743,40758,40759,40761],{},[31658,40760,31965],{},[31749,40762,768],{},[31662,40764,202],{},[31662,40766,32226],{},[31749,40768,768],{},[31662,40770,31960],{"separator":758},[31743,40772,40773,40775],{},[31658,40774,31965],{},[31749,40776,786],{},[31662,40778,202],{},[31749,40780,768],{},[31665,40782,40783],{"encoding":31667},"c_1=0, c_2=-2, c_3=2",[34,40785,40787,40842,40906,40973],{"className":40786,"ariaHidden":758},[31672],[34,40788,40790,40793,40833,40836,40839],{"className":40789},[31676],[34,40791],{"className":40792,"style":31860},[31680],[34,40794,40796,40799],{"className":40795},[31685],[34,40797,31965],{"className":40798},[31685,31686],[34,40800,40802],{"className":40801},[31792],[34,40803,40805,40825],{"className":40804},[31796,31797],[34,40806,40808,40822],{"className":40807},[31801],[34,40809,40811],{"className":40810,"style":31806},[31805],[34,40812,40813,40816],{"style":31809},[34,40814],{"className":40815,"style":31814},[31813],[34,40817,40819],{"className":40818},[31818,31819,31820,31821],[34,40820,640],{"className":40821},[31685,31821],[34,40823,31829],{"className":40824},[31828],[34,40826,40828],{"className":40827},[31801],[34,40829,40831],{"className":40830,"style":31836},[31805],[34,40832],{},[34,40834],{"className":40835,"style":31846},[31845],[34,40837,202],{"className":40838},[31850],[34,40840],{"className":40841,"style":31846},[31845],[34,40843,40845,40848,40851,40854,40857,40897,40900,40903],{"className":40844},[31676],[34,40846],{"className":40847,"style":40292},[31680],[34,40849,459],{"className":40850},[31685],[34,40852,31960],{"className":40853},[32119],[34,40855],{"className":40856,"style":32123},[31845],[34,40858,40860,40863],{"className":40859},[31685],[34,40861,31965],{"className":40862},[31685,31686],[34,40864,40866],{"className":40865},[31792],[34,40867,40869,40889],{"className":40868},[31796,31797],[34,40870,40872,40886],{"className":40871},[31801],[34,40873,40875],{"className":40874,"style":31806},[31805],[34,40876,40877,40880],{"style":31809},[34,40878],{"className":40879,"style":31814},[31813],[34,40881,40883],{"className":40882},[31818,31819,31820,31821],[34,40884,768],{"className":40885},[31685,31821],[34,40887,31829],{"className":40888},[31828],[34,40890,40892],{"className":40891},[31801],[34,40893,40895],{"className":40894,"style":31836},[31805],[34,40896],{},[34,40898],{"className":40899,"style":31846},[31845],[34,40901,202],{"className":40902},[31850],[34,40904],{"className":40905,"style":31846},[31845],[34,40907,40909,40912,40915,40918,40921,40924,40964,40967,40970],{"className":40908},[31676],[34,40910],{"className":40911,"style":40292},[31680],[34,40913,32226],{"className":40914},[31685],[34,40916,768],{"className":40917},[31685],[34,40919,31960],{"className":40920},[32119],[34,40922],{"className":40923,"style":32123},[31845],[34,40925,40927,40930],{"className":40926},[31685],[34,40928,31965],{"className":40929},[31685,31686],[34,40931,40933],{"className":40932},[31792],[34,40934,40936,40956],{"className":40935},[31796,31797],[34,40937,40939,40953],{"className":40938},[31801],[34,40940,40942],{"className":40941,"style":31806},[31805],[34,40943,40944,40947],{"style":31809},[34,40945],{"className":40946,"style":31814},[31813],[34,40948,40950],{"className":40949},[31818,31819,31820,31821],[34,40951,786],{"className":40952},[31685,31821],[34,40954,31829],{"className":40955},[31828],[34,40957,40959],{"className":40958},[31801],[34,40960,40962],{"className":40961,"style":31836},[31805],[34,40963],{},[34,40965],{"className":40966,"style":31846},[31845],[34,40968,202],{"className":40969},[31850],[34,40971],{"className":40972,"style":31846},[31845],[34,40974,40976,40979],{"className":40975},[31676],[34,40977],{"className":40978,"style":32617},[31680],[34,40980,768],{"className":40981},[31685],"。最终解为 ",[34,40984,40986,41028],{"className":40985},[31642],[34,40987,40989],{"className":40988},[31646],[31648,40990,40991],{"xmlns":31650},[31652,40992,40993,41025],{},[31655,40994,40995,40997,40999,41001,41003,41005,41007,41009,41015,41017,41023],{},[31658,40996,37400],{},[31662,40998,456],{"stretchy":702},[31658,41000,31660],{},[31662,41002,2557],{"stretchy":702},[31662,41004,202],{},[31749,41006,768],{},[31662,41008,456],{"stretchy":702},[37624,41010,41011,41013],{},[31749,41012,786],{},[31658,41014,31660],{},[31662,41016,32226],{},[37624,41018,41019,41021],{},[31749,41020,768],{},[31658,41022,31660],{},[31662,41024,2557],{"stretchy":702},[31665,41026,41027],{"encoding":31667},"T(n) = 2(3^n - 2^n)",[34,41029,41031,41058,41108],{"className":41030,"ariaHidden":758},[31672],[34,41032,41034,41037,41040,41043,41046,41049,41052,41055],{"className":41033},[31676],[34,41035],{"className":41036,"style":31774},[31680],[34,41038,37400],{"className":41039,"style":37422},[31685,31686],[34,41041,456],{"className":41042},[31782],[34,41044,31660],{"className":41045},[31685,31686],[34,41047,2557],{"className":41048},[31690],[34,41050],{"className":41051,"style":31846},[31845],[34,41053,202],{"className":41054},[31850],[34,41056],{"className":41057,"style":31846},[31845],[34,41059,41061,41064,41067,41070,41099,41102,41105],{"className":41060},[31676],[34,41062],{"className":41063,"style":31774},[31680],[34,41065,768],{"className":41066},[31685],[34,41068,456],{"className":41069},[31782],[34,41071,41073,41076],{"className":41072},[31685],[34,41074,786],{"className":41075},[31685],[34,41077,41079],{"className":41078},[31792],[34,41080,41082],{"className":41081},[31796],[34,41083,41085],{"className":41084},[31801],[34,41086,41088],{"className":41087,"style":37959},[31805],[34,41089,41090,41093],{"style":37777},[34,41091],{"className":41092,"style":31814},[31813],[34,41094,41096],{"className":41095},[31818,31819,31820,31821],[34,41097,31660],{"className":41098},[31685,31686,31821],[34,41100],{"className":41101,"style":32604},[31845],[34,41103,32226],{"className":41104},[32024],[34,41106],{"className":41107,"style":32604},[31845],[34,41109,41111,41114,41143],{"className":41110},[31676],[34,41112],{"className":41113,"style":31774},[31680],[34,41115,41117,41120],{"className":41116},[31685],[34,41118,768],{"className":41119},[31685],[34,41121,41123],{"className":41122},[31792],[34,41124,41126],{"className":41125},[31796],[34,41127,41129],{"className":41128},[31801],[34,41130,41132],{"className":41131,"style":37959},[31805],[34,41133,41134,41137],{"style":37777},[34,41135],{"className":41136,"style":31814},[31813],[34,41138,41140],{"className":41139},[31818,31819,31820,31821],[34,41141,31660],{"className":41142},[31685,31686,31821],[34,41144,2557],{"className":41145},[31690],[108,41147,41148,2664,41151,41313,41314,41439,41440,41625,41626,347],{},[2627,41149,41150],{},"例题 2（Fibonacci数列）",[34,41152,41154,41202],{"className":41153},[31642],[34,41155,41157],{"className":41156},[31646],[31648,41158,41159],{"xmlns":31650},[31652,41160,41161,41199],{},[31655,41162,41163,41165,41167,41169,41171,41173,41175,41177,41179,41181,41183,41185,41187,41189,41191,41193,41195,41197],{},[31658,41164,31739],{},[31662,41166,456],{"stretchy":702},[31658,41168,31660],{},[31662,41170,2557],{"stretchy":702},[31662,41172,202],{},[31658,41174,31739],{},[31662,41176,456],{"stretchy":702},[31658,41178,31660],{},[31662,41180,32226],{},[31749,41182,640],{},[31662,41184,2557],{"stretchy":702},[31662,41186,223],{},[31658,41188,31739],{},[31662,41190,456],{"stretchy":702},[31658,41192,31660],{},[31662,41194,32226],{},[31749,41196,768],{},[31662,41198,2557],{"stretchy":702},[31665,41200,41201],{"encoding":31667},"f(n)=f(n-1)+f(n-2)",[34,41203,41205,41232,41256,41277,41301],{"className":41204,"ariaHidden":758},[31672],[34,41206,41208,41211,41214,41217,41220,41223,41226,41229],{"className":41207},[31676],[34,41209],{"className":41210,"style":31774},[31680],[34,41212,31739],{"className":41213,"style":31778},[31685,31686],[34,41215,456],{"className":41216},[31782],[34,41218,31660],{"className":41219},[31685,31686],[34,41221,2557],{"className":41222},[31690],[34,41224],{"className":41225,"style":31846},[31845],[34,41227,202],{"className":41228},[31850],[34,41230],{"className":41231,"style":31846},[31845],[34,41233,41235,41238,41241,41244,41247,41250,41253],{"className":41234},[31676],[34,41236],{"className":41237,"style":31774},[31680],[34,41239,31739],{"className":41240,"style":31778},[31685,31686],[34,41242,456],{"className":41243},[31782],[34,41245,31660],{"className":41246},[31685,31686],[34,41248],{"className":41249,"style":32604},[31845],[34,41251,32226],{"className":41252},[32024],[34,41254],{"className":41255,"style":32604},[31845],[34,41257,41259,41262,41265,41268,41271,41274],{"className":41258},[31676],[34,41260],{"className":41261,"style":31774},[31680],[34,41263,640],{"className":41264},[31685],[34,41266,2557],{"className":41267},[31690],[34,41269],{"className":41270,"style":32604},[31845],[34,41272,223],{"className":41273},[32024],[34,41275],{"className":41276,"style":32604},[31845],[34,41278,41280,41283,41286,41289,41292,41295,41298],{"className":41279},[31676],[34,41281],{"className":41282,"style":31774},[31680],[34,41284,31739],{"className":41285,"style":31778},[31685,31686],[34,41287,456],{"className":41288},[31782],[34,41290,31660],{"className":41291},[31685,31686],[34,41293],{"className":41294,"style":32604},[31845],[34,41296,32226],{"className":41297},[32024],[34,41299],{"className":41300,"style":32604},[31845],[34,41302,41304,41307,41310],{"className":41303},[31676],[34,41305],{"className":41306,"style":31774},[31680],[34,41308,768],{"className":41309},[31685],[34,41311,2557],{"className":41312},[31690],"。特征方程 ",[34,41315,41317,41347],{"className":41316},[31642],[34,41318,41320],{"className":41319},[31646],[31648,41321,41322],{"xmlns":31650},[31652,41323,41324,41344],{},[31655,41325,41326,41332,41334,41336,41338,41340,41342],{},[37624,41327,41328,41330],{},[31658,41329,32936],{},[31749,41331,768],{},[31662,41333,32226],{},[31658,41335,32936],{},[31662,41337,32226],{},[31749,41339,640],{},[31662,41341,202],{},[31749,41343,459],{},[31665,41345,41346],{"encoding":31667},"x^2-x-1=0",[34,41348,41350,41394,41412,41430],{"className":41349,"ariaHidden":758},[31672],[34,41351,41353,41356,41385,41388,41391],{"className":41352},[31676],[34,41354],{"className":41355,"style":40044},[31680],[34,41357,41359,41362],{"className":41358},[31685],[34,41360,32936],{"className":41361},[31685,31686],[34,41363,41365],{"className":41364},[31792],[34,41366,41368],{"className":41367},[31796],[34,41369,41371],{"className":41370},[31801],[34,41372,41374],{"className":41373,"style":37774},[31805],[34,41375,41376,41379],{"style":37777},[34,41377],{"className":41378,"style":31814},[31813],[34,41380,41382],{"className":41381},[31818,31819,31820,31821],[34,41383,768],{"className":41384},[31685,31821],[34,41386],{"className":41387,"style":32604},[31845],[34,41389,32226],{"className":41390},[32024],[34,41392],{"className":41393,"style":32604},[31845],[34,41395,41397,41400,41403,41406,41409],{"className":41396},[31676],[34,41398],{"className":41399,"style":32597},[31680],[34,41401,32936],{"className":41402},[31685,31686],[34,41404],{"className":41405,"style":32604},[31845],[34,41407,32226],{"className":41408},[32024],[34,41410],{"className":41411,"style":32604},[31845],[34,41413,41415,41418,41421,41424,41427],{"className":41414},[31676],[34,41416],{"className":41417,"style":32617},[31680],[34,41419,640],{"className":41420},[31685],[34,41422],{"className":41423,"style":31846},[31845],[34,41425,202],{"className":41426},[31850],[34,41428],{"className":41429,"style":31846},[31845],[34,41431,41433,41436],{"className":41432},[31676],[34,41434],{"className":41435,"style":32617},[31680],[34,41437,459],{"className":41438},[31685],"，解得根为 ",[34,41441,41443,41472],{"className":41442},[31642],[34,41444,41446],{"className":41445},[31646],[31648,41447,41448],{"xmlns":31650},[31652,41449,41450,41469],{},[31655,41451,41452],{},[41453,41454,41455,41467],"mfrac",{},[31655,41456,41457,41459,41462],{},[31749,41458,640],{},[31662,41460,41461],{},"±",[41463,41464,41465],"msqrt",{},[31749,41466,18136],{},[31749,41468,768],{},[31665,41470,41471],{"encoding":31667},"\\frac{1\\pm\\sqrt{5}}{2}",[34,41473,41475],{"className":41474,"ariaHidden":758},[31672],[34,41476,41478,41482],{"className":41477},[31676],[34,41479],{"className":41480,"style":41481},[31680],"height:1.383em;vertical-align:-0.345em;",[34,41483,41485,41489,41622],{"className":41484},[31685],[34,41486],{"className":41487},[31782,41488],"nulldelimiter",[34,41490,41492],{"className":41491},[41453],[34,41493,41495,41613],{"className":41494},[31796,31797],[34,41496,41498,41610],{"className":41497},[31801],[34,41499,41502,41518,41529],{"className":41500,"style":41501},[31805],"height:1.038em;",[34,41503,41505,41509],{"style":41504},"top:-2.655em;",[34,41506],{"className":41507,"style":41508},[31813],"height:3em;",[34,41510,41512],{"className":41511},[31818,31819,31820,31821],[34,41513,41515],{"className":41514},[31685,31821],[34,41516,768],{"className":41517},[31685,31821],[34,41519,41521,41524],{"style":41520},"top:-3.23em;",[34,41522],{"className":41523,"style":41508},[31813],[34,41525],{"className":41526,"style":41528},[41527],"frac-line","border-bottom-width:0.04em;",[34,41530,41532,41535],{"style":41531},"top:-3.399em;",[34,41533],{"className":41534,"style":41508},[31813],[34,41536,41538],{"className":41537},[31818,31819,31820,31821],[34,41539,41541,41544,41547],{"className":41540},[31685,31821],[34,41542,640],{"className":41543},[31685,31821],[34,41545,41461],{"className":41546},[32024,31821],[34,41548,41551],{"className":41549},[31685,41550,31821],"sqrt",[34,41552,41554,41601],{"className":41553},[31796,31797],[34,41555,41557,41598],{"className":41556},[31801],[34,41558,41561,41576],{"className":41559,"style":41560},[31805],"height:0.9128em;",[34,41562,41566,41569],{"className":41563,"style":41565},[41564],"svg-align","top:-3em;",[34,41567],{"className":41568,"style":41508},[31813],[34,41570,41573],{"className":41571,"style":41572},[31685,31821],"padding-left:0.833em;",[34,41574,18136],{"className":41575},[31685,31821],[34,41577,41579,41582],{"style":41578},"top:-2.8728em;",[34,41580],{"className":41581,"style":41508},[31813],[34,41583,41587],{"className":41584,"style":41586},[41585,31821],"hide-tail","min-width:0.853em;height:1.08em;",[41588,41589,41595],"svg",{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[20398,41596],{"d":41597},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[34,41599,31829],{"className":41600},[31828],[34,41602,41604],{"className":41603},[31801],[34,41605,41608],{"className":41606,"style":41607},[31805],"height:0.1272em;",[34,41609],{},[34,41611,31829],{"className":41612},[31828],[34,41614,41616],{"className":41615},[31801],[34,41617,41620],{"className":41618,"style":41619},[31805],"height:0.345em;",[34,41621],{},[34,41623],{"className":41624},[31690,41488],"，代入初始值得解 ",[34,41627,41629,41669],{"className":41628},[31642],[34,41630,41632],{"className":41631},[31646],[31648,41633,41634],{"xmlns":31650},[31652,41635,41636,41666],{},[31655,41637,41638,41640,41642,41644,41658,41664],{},[31658,41639,37890],{},[31662,41641,456],{"stretchy":702},[31662,41643,456],{"stretchy":702},[41453,41645,41646,41656],{},[31655,41647,41648,41650,41652],{},[31749,41649,640],{},[31662,41651,223],{},[41463,41653,41654],{},[31749,41655,18136],{},[31749,41657,768],{},[37624,41659,41660,41662],{},[31662,41661,2557],{"stretchy":702},[31658,41663,31660],{},[31662,41665,2557],{"stretchy":702},[31665,41667,41668],{"encoding":31667},"O((\\frac{1+\\sqrt{5}}{2})^n)",[34,41670,41672],{"className":41671,"ariaHidden":758},[31672],[34,41673,41675,41678,41681,41684,41802,41831],{"className":41674},[31676],[34,41676],{"className":41677,"style":41481},[31680],[34,41679,37890],{"className":41680,"style":34452},[31685,31686],[34,41682,1323],{"className":41683},[31782],[34,41685,41687,41690,41799],{"className":41686},[31685],[34,41688],{"className":41689},[31782,41488],[34,41691,41693],{"className":41692},[41453],[34,41694,41696,41791],{"className":41695},[31796,31797],[34,41697,41699,41788],{"className":41698},[31801],[34,41700,41702,41716,41724],{"className":41701,"style":41501},[31805],[34,41703,41704,41707],{"style":41504},[34,41705],{"className":41706,"style":41508},[31813],[34,41708,41710],{"className":41709},[31818,31819,31820,31821],[34,41711,41713],{"className":41712},[31685,31821],[34,41714,768],{"className":41715},[31685,31821],[34,41717,41718,41721],{"style":41520},[34,41719],{"className":41720,"style":41508},[31813],[34,41722],{"className":41723,"style":41528},[41527],[34,41725,41726,41729],{"style":41531},[34,41727],{"className":41728,"style":41508},[31813],[34,41730,41732],{"className":41731},[31818,31819,31820,31821],[34,41733,41735,41738,41741],{"className":41734},[31685,31821],[34,41736,640],{"className":41737},[31685,31821],[34,41739,223],{"className":41740},[32024,31821],[34,41742,41744],{"className":41743},[31685,41550,31821],[34,41745,41747,41780],{"className":41746},[31796,31797],[34,41748,41750,41777],{"className":41749},[31801],[34,41751,41753,41765],{"className":41752,"style":41560},[31805],[34,41754,41756,41759],{"className":41755,"style":41565},[41564],[34,41757],{"className":41758,"style":41508},[31813],[34,41760,41762],{"className":41761,"style":41572},[31685,31821],[34,41763,18136],{"className":41764},[31685,31821],[34,41766,41767,41770],{"style":41578},[34,41768],{"className":41769,"style":41508},[31813],[34,41771,41773],{"className":41772,"style":41586},[41585,31821],[41588,41774,41775],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,41776],{"d":41597},[34,41778,31829],{"className":41779},[31828],[34,41781,41783],{"className":41782},[31801],[34,41784,41786],{"className":41785,"style":41607},[31805],[34,41787],{},[34,41789,31829],{"className":41790},[31828],[34,41792,41794],{"className":41793},[31801],[34,41795,41797],{"className":41796,"style":41619},[31805],[34,41798],{},[34,41800],{"className":41801},[31690,41488],[34,41803,41805,41808],{"className":41804},[31690],[34,41806,2557],{"className":41807},[31690],[34,41809,41811],{"className":41810},[31792],[34,41812,41814],{"className":41813},[31796],[34,41815,41817],{"className":41816},[31801],[34,41818,41820],{"className":41819,"style":37959},[31805],[34,41821,41822,41825],{"style":37777},[34,41823],{"className":41824,"style":31814},[31813],[34,41826,41828],{"className":41827},[31818,31819,31820,31821],[34,41829,31660],{"className":41830},[31685,31686,31821],[34,41832,2557],{"className":41833},[31690],[32490,41835,41837],{"id":41836},"_3-递归树法","3. 递归树法",[105,41839,41840,41845,42504],{},[108,41841,41842,41844],{},[2627,41843,37443],{},"：展开递归方程构造递归树，把每一层的时间进行求和得出估计。",[108,41846,41847,2664,41849,41994,41995,42054,42055,42213,42214,42281,42282,347],{},[2627,41848,39579],{},[34,41850,41852,41896],{"className":41851},[31642],[34,41853,41855],{"className":41854},[31646],[31648,41856,41857],{"xmlns":31650},[31652,41858,41859,41893],{},[31655,41860,41861,41863,41865,41867,41869,41871,41873,41875,41877,41879,41881,41883,41885,41887],{},[31658,41862,37400],{},[31662,41864,456],{"stretchy":702},[31658,41866,31660],{},[31662,41868,2557],{"stretchy":702},[31662,41870,202],{},[31749,41872,768],{},[31658,41874,37400],{},[31662,41876,456],{"stretchy":702},[31658,41878,31660],{},[31658,41880,38080],{"mathvariant":38079},[31749,41882,768],{},[31662,41884,2557],{"stretchy":702},[31662,41886,223],{},[37624,41888,41889,41891],{},[31658,41890,31660],{},[31749,41892,768],{},[31665,41894,41895],{"encoding":31667},"T(n) = 2T(n\u002F2) + n^2",[34,41897,41899,41926,41959],{"className":41898,"ariaHidden":758},[31672],[34,41900,41902,41905,41908,41911,41914,41917,41920,41923],{"className":41901},[31676],[34,41903],{"className":41904,"style":31774},[31680],[34,41906,37400],{"className":41907,"style":37422},[31685,31686],[34,41909,456],{"className":41910},[31782],[34,41912,31660],{"className":41913},[31685,31686],[34,41915,2557],{"className":41916},[31690],[34,41918],{"className":41919,"style":31846},[31845],[34,41921,202],{"className":41922},[31850],[34,41924],{"className":41925,"style":31846},[31845],[34,41927,41929,41932,41935,41938,41941,41944,41947,41950,41953,41956],{"className":41928},[31676],[34,41930],{"className":41931,"style":31774},[31680],[34,41933,768],{"className":41934},[31685],[34,41936,37400],{"className":41937,"style":37422},[31685,31686],[34,41939,456],{"className":41940},[31782],[34,41942,31660],{"className":41943},[31685,31686],[34,41945,38149],{"className":41946},[31685],[34,41948,2557],{"className":41949},[31690],[34,41951],{"className":41952,"style":32604},[31845],[34,41954,223],{"className":41955},[32024],[34,41957],{"className":41958,"style":32604},[31845],[34,41960,41962,41965],{"className":41961},[31676],[34,41963],{"className":41964,"style":37774},[31680],[34,41966,41968,41971],{"className":41967},[31685],[34,41969,31660],{"className":41970},[31685,31686],[34,41972,41974],{"className":41973},[31792],[34,41975,41977],{"className":41976},[31796],[34,41978,41980],{"className":41979},[31801],[34,41981,41983],{"className":41982,"style":37774},[31805],[34,41984,41985,41988],{"style":37777},[34,41986],{"className":41987,"style":31814},[31813],[34,41989,41991],{"className":41990},[31818,31819,31820,31821],[34,41992,768],{"className":41993},[31685,31821],"。第一层代价 ",[34,41996,41998,42016],{"className":41997},[31642],[34,41999,42001],{"className":42000},[31646],[31648,42002,42003],{"xmlns":31650},[31652,42004,42005,42013],{},[31655,42006,42007],{},[37624,42008,42009,42011],{},[31658,42010,31660],{},[31749,42012,768],{},[31665,42014,42015],{"encoding":31667},"n^2",[34,42017,42019],{"className":42018,"ariaHidden":758},[31672],[34,42020,42022,42025],{"className":42021},[31676],[34,42023],{"className":42024,"style":37774},[31680],[34,42026,42028,42031],{"className":42027},[31685],[34,42029,31660],{"className":42030},[31685,31686],[34,42032,42034],{"className":42033},[31792],[34,42035,42037],{"className":42036},[31796],[34,42038,42040],{"className":42039},[31801],[34,42041,42043],{"className":42042,"style":37774},[31805],[34,42044,42045,42048],{"style":37777},[34,42046],{"className":42047,"style":31814},[31813],[34,42049,42051],{"className":42050},[31818,31819,31820,31821],[34,42052,768],{"className":42053},[31685,31821],"，第二层 ",[34,42056,42058,42101],{"className":42057},[31642],[34,42059,42061],{"className":42060},[31646],[31648,42062,42063],{"xmlns":31650},[31652,42064,42065,42098],{},[31655,42066,42067,42069,42072,42074,42076,42078,42080,42086,42088,42094,42096],{},[31749,42068,768],{},[31662,42070,42071],{},"×",[31662,42073,456],{"stretchy":702},[31658,42075,31660],{},[31658,42077,38080],{"mathvariant":38079},[31749,42079,768],{},[37624,42081,42082,42084],{},[31662,42083,2557],{"stretchy":702},[31749,42085,768],{},[31662,42087,202],{},[37624,42089,42090,42092],{},[31658,42091,31660],{},[31749,42093,768],{},[31658,42095,38080],{"mathvariant":38079},[31749,42097,768],{},[31665,42099,42100],{"encoding":31667},"2 \\times (n\u002F2)^2 = n^2\u002F2",[34,42102,42104,42122,42175],{"className":42103,"ariaHidden":758},[31672],[34,42105,42107,42110,42113,42116,42119],{"className":42106},[31676],[34,42108],{"className":42109,"style":35942},[31680],[34,42111,768],{"className":42112},[31685],[34,42114],{"className":42115,"style":32604},[31845],[34,42117,42071],{"className":42118},[32024],[34,42120],{"className":42121,"style":32604},[31845],[34,42123,42125,42128,42131,42134,42137,42166,42169,42172],{"className":42124},[31676],[34,42126],{"className":42127,"style":37755},[31680],[34,42129,456],{"className":42130},[31782],[34,42132,31660],{"className":42133},[31685,31686],[34,42135,38149],{"className":42136},[31685],[34,42138,42140,42143],{"className":42139},[31690],[34,42141,2557],{"className":42142},[31690],[34,42144,42146],{"className":42145},[31792],[34,42147,42149],{"className":42148},[31796],[34,42150,42152],{"className":42151},[31801],[34,42153,42155],{"className":42154,"style":37774},[31805],[34,42156,42157,42160],{"style":37777},[34,42158],{"className":42159,"style":31814},[31813],[34,42161,42163],{"className":42162},[31818,31819,31820,31821],[34,42164,768],{"className":42165},[31685,31821],[34,42167],{"className":42168,"style":31846},[31845],[34,42170,202],{"className":42171},[31850],[34,42173],{"className":42174,"style":31846},[31845],[34,42176,42178,42181,42210],{"className":42177},[31676],[34,42179],{"className":42180,"style":37755},[31680],[34,42182,42184,42187],{"className":42183},[31685],[34,42185,31660],{"className":42186},[31685,31686],[34,42188,42190],{"className":42189},[31792],[34,42191,42193],{"className":42192},[31796],[34,42194,42196],{"className":42195},[31801],[34,42197,42199],{"className":42198,"style":37774},[31805],[34,42200,42201,42204],{"style":37777},[34,42202],{"className":42203,"style":31814},[31813],[34,42205,42207],{"className":42206},[31818,31819,31820,31821],[34,42208,768],{"className":42209},[31685,31821],[34,42211,38149],{"className":42212},[31685],"，第三层 ",[34,42215,42217,42239],{"className":42216},[31642],[34,42218,42220],{"className":42219},[31646],[31648,42221,42222],{"xmlns":31650},[31652,42223,42224,42236],{},[31655,42225,42226,42232,42234],{},[37624,42227,42228,42230],{},[31658,42229,31660],{},[31749,42231,768],{},[31658,42233,38080],{"mathvariant":38079},[31749,42235,15932],{},[31665,42237,42238],{"encoding":31667},"n^2\u002F4",[34,42240,42242],{"className":42241,"ariaHidden":758},[31672],[34,42243,42245,42248,42277],{"className":42244},[31676],[34,42246],{"className":42247,"style":37755},[31680],[34,42249,42251,42254],{"className":42250},[31685],[34,42252,31660],{"className":42253},[31685,31686],[34,42255,42257],{"className":42256},[31792],[34,42258,42260],{"className":42259},[31796],[34,42261,42263],{"className":42262},[31801],[34,42264,42266],{"className":42265,"style":37774},[31805],[34,42267,42268,42271],{"style":37777},[34,42269],{"className":42270,"style":31814},[31813],[34,42272,42274],{"className":42273},[31818,31819,31820,31821],[34,42275,768],{"className":42276},[31685,31821],[34,42278,42280],{"className":42279},[31685],"\u002F4","。求和得 ",[34,42283,42285,42345],{"className":42284},[31642],[34,42286,42288],{"className":42287},[31646],[31648,42289,42290],{"xmlns":31650},[31652,42291,42292,42342],{},[31655,42293,42294,42300,42302,42304,42306,42308,42310,42312,42314,42316,42318,42320,42322,42324,42326,42328,42330,42332,42334,42340],{},[37624,42295,42296,42298],{},[31658,42297,31660],{},[31749,42299,768],{},[31662,42301,456],{"stretchy":702},[31749,42303,640],{},[31662,42305,223],{},[31749,42307,640],{},[31658,42309,38080],{"mathvariant":38079},[31749,42311,768],{},[31662,42313,223],{},[31749,42315,640],{},[31658,42317,38080],{"mathvariant":38079},[31749,42319,15932],{},[31662,42321,223],{},[31662,42323,33934],{},[39361,42325,39363],{},[31662,42327,2557],{"stretchy":702},[31662,42329,202],{},[31658,42331,37890],{},[31662,42333,456],{"stretchy":702},[37624,42335,42336,42338],{},[31658,42337,31660],{},[31749,42339,768],{},[31662,42341,2557],{"stretchy":702},[31665,42343,42344],{"encoding":31667},"n^2(1 + 1\u002F2 + 1\u002F4 + \\dots) = O(n^2)",[34,42346,42348,42398,42417,42436,42460],{"className":42347,"ariaHidden":758},[31672],[34,42349,42351,42354,42383,42386,42389,42392,42395],{"className":42350},[31676],[34,42352],{"className":42353,"style":37755},[31680],[34,42355,42357,42360],{"className":42356},[31685],[34,42358,31660],{"className":42359},[31685,31686],[34,42361,42363],{"className":42362},[31792],[34,42364,42366],{"className":42365},[31796],[34,42367,42369],{"className":42368},[31801],[34,42370,42372],{"className":42371,"style":37774},[31805],[34,42373,42374,42377],{"style":37777},[34,42375],{"className":42376,"style":31814},[31813],[34,42378,42380],{"className":42379},[31818,31819,31820,31821],[34,42381,768],{"className":42382},[31685,31821],[34,42384,456],{"className":42385},[31782],[34,42387,640],{"className":42388},[31685],[34,42390],{"className":42391,"style":32604},[31845],[34,42393,223],{"className":42394},[32024],[34,42396],{"className":42397,"style":32604},[31845],[34,42399,42401,42404,42408,42411,42414],{"className":42400},[31676],[34,42402],{"className":42403,"style":31774},[31680],[34,42405,42407],{"className":42406},[31685],"1\u002F2",[34,42409],{"className":42410,"style":32604},[31845],[34,42412,223],{"className":42413},[32024],[34,42415],{"className":42416,"style":32604},[31845],[34,42418,42420,42423,42427,42430,42433],{"className":42419},[31676],[34,42421],{"className":42422,"style":31774},[31680],[34,42424,42426],{"className":42425},[31685],"1\u002F4",[34,42428],{"className":42429,"style":32604},[31845],[34,42431,223],{"className":42432},[32024],[34,42434],{"className":42435,"style":32604},[31845],[34,42437,42439,42442,42445,42448,42451,42454,42457],{"className":42438},[31676],[34,42440],{"className":42441,"style":31774},[31680],[34,42443,33934],{"className":42444},[34006],[34,42446],{"className":42447,"style":32123},[31845],[34,42449,2557],{"className":42450},[31690],[34,42452],{"className":42453,"style":31846},[31845],[34,42455,202],{"className":42456},[31850],[34,42458],{"className":42459,"style":31846},[31845],[34,42461,42463,42466,42469,42472,42501],{"className":42462},[31676],[34,42464],{"className":42465,"style":37755},[31680],[34,42467,37890],{"className":42468,"style":34452},[31685,31686],[34,42470,456],{"className":42471},[31782],[34,42473,42475,42478],{"className":42474},[31685],[34,42476,31660],{"className":42477},[31685,31686],[34,42479,42481],{"className":42480},[31792],[34,42482,42484],{"className":42483},[31796],[34,42485,42487],{"className":42486},[31801],[34,42488,42490],{"className":42489,"style":37774},[31805],[34,42491,42492,42495],{"style":37777},[34,42493],{"className":42494,"style":31814},[31813],[34,42496,42498],{"className":42497},[31818,31819,31820,31821],[34,42499,768],{"className":42500},[31685,31821],[34,42502,2557],{"className":42503},[31690],[108,42505,42506,2664,42509,42669,42670,42698,42699,42796,42797,347],{},[2627,42507,42508],{},"例题 2（不平衡树）",[34,42510,42512,42566],{"className":42511},[31642],[34,42513,42515],{"className":42514},[31646],[31648,42516,42517],{"xmlns":31650},[31652,42518,42519,42563],{},[31655,42520,42521,42523,42525,42527,42529,42531,42533,42535,42537,42539,42541,42543,42545,42547,42549,42551,42553,42555,42557,42559,42561],{},[31658,42522,37400],{},[31662,42524,456],{"stretchy":702},[31658,42526,31660],{},[31662,42528,2557],{"stretchy":702},[31662,42530,202],{},[31658,42532,37400],{},[31662,42534,456],{"stretchy":702},[31658,42536,31660],{},[31658,42538,38080],{"mathvariant":38079},[31749,42540,786],{},[31662,42542,2557],{"stretchy":702},[31662,42544,223],{},[31658,42546,37400],{},[31662,42548,456],{"stretchy":702},[31749,42550,768],{},[31658,42552,31660],{},[31658,42554,38080],{"mathvariant":38079},[31749,42556,786],{},[31662,42558,2557],{"stretchy":702},[31662,42560,223],{},[31658,42562,31660],{},[31665,42564,42565],{"encoding":31667},"T(n) = T(n\u002F3) + T(2n\u002F3) + n",[34,42567,42569,42596,42627,42660],{"className":42568,"ariaHidden":758},[31672],[34,42570,42572,42575,42578,42581,42584,42587,42590,42593],{"className":42571},[31676],[34,42573],{"className":42574,"style":31774},[31680],[34,42576,37400],{"className":42577,"style":37422},[31685,31686],[34,42579,456],{"className":42580},[31782],[34,42582,31660],{"className":42583},[31685,31686],[34,42585,2557],{"className":42586},[31690],[34,42588],{"className":42589,"style":31846},[31845],[34,42591,202],{"className":42592},[31850],[34,42594],{"className":42595,"style":31846},[31845],[34,42597,42599,42602,42605,42608,42611,42615,42618,42621,42624],{"className":42598},[31676],[34,42600],{"className":42601,"style":31774},[31680],[34,42603,37400],{"className":42604,"style":37422},[31685,31686],[34,42606,456],{"className":42607},[31782],[34,42609,31660],{"className":42610},[31685,31686],[34,42612,42614],{"className":42613},[31685],"\u002F3",[34,42616,2557],{"className":42617},[31690],[34,42619],{"className":42620,"style":32604},[31845],[34,42622,223],{"className":42623},[32024],[34,42625],{"className":42626,"style":32604},[31845],[34,42628,42630,42633,42636,42639,42642,42645,42648,42651,42654,42657],{"className":42629},[31676],[34,42631],{"className":42632,"style":31774},[31680],[34,42634,37400],{"className":42635,"style":37422},[31685,31686],[34,42637,456],{"className":42638},[31782],[34,42640,768],{"className":42641},[31685],[34,42643,31660],{"className":42644},[31685,31686],[34,42646,42614],{"className":42647},[31685],[34,42649,2557],{"className":42650},[31690],[34,42652],{"className":42653,"style":32604},[31845],[34,42655,223],{"className":42656},[32024],[34,42658],{"className":42659,"style":32604},[31845],[34,42661,42663,42666],{"className":42662},[31676],[34,42664],{"className":42665,"style":32527},[31680],[34,42667,31660],{"className":42668},[31685,31686],"。每层的和均为 ",[34,42671,42673,42686],{"className":42672},[31642],[34,42674,42676],{"className":42675},[31646],[31648,42677,42678],{"xmlns":31650},[31652,42679,42680,42684],{},[31655,42681,42682],{},[31658,42683,31660],{},[31665,42685,31660],{"encoding":31667},[34,42687,42689],{"className":42688,"ariaHidden":758},[31672],[34,42690,42692,42695],{"className":42691},[31676],[34,42693],{"className":42694,"style":32527},[31680],[34,42696,31660],{"className":42697},[31685,31686],"。最长路径高度为 ",[34,42700,42702,42732],{"className":42701},[31642],[34,42703,42705],{"className":42704},[31646],[31648,42706,42707],{"xmlns":31650},[31652,42708,42709,42729],{},[31655,42710,42711,42727],{},[31743,42712,42713,42719],{},[31655,42714,42715,42717],{},[31658,42716,885],{},[31662,42718,32682],{},[31655,42720,42721,42723,42725],{},[31749,42722,786],{},[31658,42724,38080],{"mathvariant":38079},[31749,42726,768],{},[31658,42728,31660],{},[31665,42730,42731],{"encoding":31667},"\\log_{3\u002F2}n",[34,42733,42735],{"className":42734,"ariaHidden":758},[31672],[34,42736,42738,42742,42790,42793],{"className":42737},[31676],[34,42739],{"className":42740,"style":42741},[31680],"height:1.1136em;vertical-align:-0.4191em;",[34,42743,42745,42750],{"className":42744},[32767],[34,42746,38616,42748],{"className":42747},[32767],[34,42749,31942],{"style":38619},[34,42751,42753],{"className":42752},[31792],[34,42754,42756,42781],{"className":42755},[31796,31797],[34,42757,42759,42778],{"className":42758},[31801],[34,42760,42763],{"className":42761,"style":42762},[31805],"height:0.2809em;",[34,42764,42765,42768],{"style":38635},[34,42766],{"className":42767,"style":31814},[31813],[34,42769,42771],{"className":42770},[31818,31819,31820,31821],[34,42772,42774],{"className":42773},[31685,31821],[34,42775,42777],{"className":42776},[31685,31821],"3\u002F2",[34,42779,31829],{"className":42780},[31828],[34,42782,42784],{"className":42783},[31801],[34,42785,42788],{"className":42786,"style":42787},[31805],"height:0.4191em;",[34,42789],{},[34,42791],{"className":42792,"style":32123},[31845],[34,42794,31660],{"className":42795},[31685,31686],"。总和等于树高乘层和，即 ",[34,42798,42800,42860],{"className":42799},[31642],[34,42801,42803],{"className":42802},[31646],[31648,42804,42805],{"xmlns":31650},[31652,42806,42807,42857],{},[31655,42808,42809,42811,42813,42815,42831,42833,42835,42837,42839,42841,42843,42853,42855],{},[31658,42810,37890],{},[31662,42812,456],{"stretchy":702},[31658,42814,31660],{},[31743,42816,42817,42823],{},[31655,42818,42819,42821],{},[31658,42820,885],{},[31662,42822,32682],{},[31655,42824,42825,42827,42829],{},[31749,42826,786],{},[31658,42828,38080],{"mathvariant":38079},[31749,42830,768],{},[31658,42832,31660],{},[31662,42834,2557],{"stretchy":702},[31662,42836,202],{},[31658,42838,37890],{},[31662,42840,456],{"stretchy":702},[31658,42842,31660],{},[31743,42844,42845,42851],{},[31655,42846,42847,42849],{},[31658,42848,885],{},[31662,42850,32682],{},[31749,42852,768],{},[31658,42854,31660],{},[31662,42856,2557],{"stretchy":702},[31665,42858,42859],{"encoding":31667},"O(n \\log_{3\u002F2} n) = O(n\\log_2n)",[34,42861,42863,42945],{"className":42862,"ariaHidden":758},[31672],[34,42864,42866,42870,42873,42876,42879,42882,42927,42930,42933,42936,42939,42942],{"className":42865},[31676],[34,42867],{"className":42868,"style":42869},[31680],"height:1.1691em;vertical-align:-0.4191em;",[34,42871,37890],{"className":42872,"style":34452},[31685,31686],[34,42874,456],{"className":42875},[31782],[34,42877,31660],{"className":42878},[31685,31686],[34,42880],{"className":42881,"style":32123},[31845],[34,42883,42885,42890],{"className":42884},[32767],[34,42886,38616,42888],{"className":42887},[32767],[34,42889,31942],{"style":38619},[34,42891,42893],{"className":42892},[31792],[34,42894,42896,42919],{"className":42895},[31796,31797],[34,42897,42899,42916],{"className":42898},[31801],[34,42900,42902],{"className":42901,"style":42762},[31805],[34,42903,42904,42907],{"style":38635},[34,42905],{"className":42906,"style":31814},[31813],[34,42908,42910],{"className":42909},[31818,31819,31820,31821],[34,42911,42913],{"className":42912},[31685,31821],[34,42914,42777],{"className":42915},[31685,31821],[34,42917,31829],{"className":42918},[31828],[34,42920,42922],{"className":42921},[31801],[34,42923,42925],{"className":42924,"style":42787},[31805],[34,42926],{},[34,42928],{"className":42929,"style":32123},[31845],[34,42931,31660],{"className":42932},[31685,31686],[34,42934,2557],{"className":42935},[31690],[34,42937],{"className":42938,"style":31846},[31845],[34,42940,202],{"className":42941},[31850],[34,42943],{"className":42944,"style":31846},[31845],[34,42946,42948,42951,42954,42957,42960,42963,43005,43008,43011],{"className":42947},[31676],[34,42949],{"className":42950,"style":31774},[31680],[34,42952,37890],{"className":42953,"style":34452},[31685,31686],[34,42955,456],{"className":42956},[31782],[34,42958,31660],{"className":42959},[31685,31686],[34,42961],{"className":42962,"style":32123},[31845],[34,42964,42966,42971],{"className":42965},[32767],[34,42967,38616,42969],{"className":42968},[32767],[34,42970,31942],{"style":38619},[34,42972,42974],{"className":42973},[31792],[34,42975,42977,42997],{"className":42976},[31796,31797],[34,42978,42980,42994],{"className":42979},[31801],[34,42981,42983],{"className":42982,"style":38632},[31805],[34,42984,42985,42988],{"style":38635},[34,42986],{"className":42987,"style":31814},[31813],[34,42989,42991],{"className":42990},[31818,31819,31820,31821],[34,42992,768],{"className":42993},[31685,31821],[34,42995,31829],{"className":42996},[31828],[34,42998,43000],{"className":42999},[31801],[34,43001,43003],{"className":43002,"style":38654},[31805],[34,43004],{},[34,43006],{"className":43007,"style":32123},[31845],[34,43009,31660],{"className":43010},[31685,31686],[34,43012,2557],{"className":43013},[31690],[32490,43015,43017],{"id":43016},"_4-主方法-master-method","4. 主方法 (Master Method)",[105,43019,43020,43248,44254,44927],{},[108,43021,43022,43024,43025,43158,43159,31901],{},[2627,43023,37443],{},"：针对形式为 ",[34,43026,43028,43074],{"className":43027},[31642],[34,43029,43031],{"className":43030},[31646],[31648,43032,43033],{"xmlns":31650},[31652,43034,43035,43071],{},[31655,43036,43037,43039,43041,43043,43045,43047,43049,43051,43053,43055,43057,43059,43061,43063,43065,43067,43069],{},[31658,43038,37400],{},[31662,43040,456],{"stretchy":702},[31658,43042,31660],{},[31662,43044,2557],{"stretchy":702},[31662,43046,202],{},[31658,43048,111],{},[31658,43050,37400],{},[31662,43052,456],{"stretchy":702},[31658,43054,31660],{},[31658,43056,38080],{"mathvariant":38079},[31658,43058,1697],{},[31662,43060,2557],{"stretchy":702},[31662,43062,223],{},[31658,43064,31739],{},[31662,43066,456],{"stretchy":702},[31658,43068,31660],{},[31662,43070,2557],{"stretchy":702},[31665,43072,43073],{"encoding":31667},"T(n) = aT(n\u002Fb) + f(n)",[34,43075,43077,43104,43140],{"className":43076,"ariaHidden":758},[31672],[34,43078,43080,43083,43086,43089,43092,43095,43098,43101],{"className":43079},[31676],[34,43081],{"className":43082,"style":31774},[31680],[34,43084,37400],{"className":43085,"style":37422},[31685,31686],[34,43087,456],{"className":43088},[31782],[34,43090,31660],{"className":43091},[31685,31686],[34,43093,2557],{"className":43094},[31690],[34,43096],{"className":43097,"style":31846},[31845],[34,43099,202],{"className":43100},[31850],[34,43102],{"className":43103,"style":31846},[31845],[34,43105,43107,43110,43113,43116,43119,43122,43125,43128,43131,43134,43137],{"className":43106},[31676],[34,43108],{"className":43109,"style":31774},[31680],[34,43111,111],{"className":43112},[31685,31686],[34,43114,37400],{"className":43115,"style":37422},[31685,31686],[34,43117,456],{"className":43118},[31782],[34,43120,31660],{"className":43121},[31685,31686],[34,43123,38080],{"className":43124},[31685],[34,43126,1697],{"className":43127},[31685,31686],[34,43129,2557],{"className":43130},[31690],[34,43132],{"className":43133,"style":32604},[31845],[34,43135,223],{"className":43136},[32024],[34,43138],{"className":43139,"style":32604},[31845],[34,43141,43143,43146,43149,43152,43155],{"className":43142},[31676],[34,43144],{"className":43145,"style":31774},[31680],[34,43147,31739],{"className":43148,"style":31778},[31685,31686],[34,43150,456],{"className":43151},[31782],[34,43153,31660],{"className":43154},[31685,31686],[34,43156,2557],{"className":43157},[31690]," 的方程（",[34,43160,43162,43189],{"className":43161},[31642],[34,43163,43165],{"className":43164},[31646],[31648,43166,43167],{"xmlns":31650},[31652,43168,43169,43186],{},[31655,43170,43171,43173,43176,43178,43180,43182,43184],{},[31658,43172,111],{},[31662,43174,43175],{},"≥",[31749,43177,640],{},[31662,43179,31960],{"separator":758},[31658,43181,1697],{},[31662,43183,29622],{},[31749,43185,640],{},[31665,43187,43188],{"encoding":31667},"a\\ge1, b>1",[34,43190,43192,43211,43239],{"className":43191,"ariaHidden":758},[31672],[34,43193,43195,43199,43202,43205,43208],{"className":43194},[31676],[34,43196],{"className":43197,"style":43198},[31680],"height:0.7719em;vertical-align:-0.136em;",[34,43200,111],{"className":43201},[31685,31686],[34,43203],{"className":43204,"style":31846},[31845],[34,43206,43175],{"className":43207},[31850],[34,43209],{"className":43210,"style":31846},[31845],[34,43212,43214,43218,43221,43224,43227,43230,43233,43236],{"className":43213},[31676],[34,43215],{"className":43216,"style":43217},[31680],"height:0.8889em;vertical-align:-0.1944em;",[34,43219,640],{"className":43220},[31685],[34,43222,31960],{"className":43223},[32119],[34,43225],{"className":43226,"style":32123},[31845],[34,43228,1697],{"className":43229},[31685,31686],[34,43231],{"className":43232,"style":31846},[31845],[34,43234,29622],{"className":43235},[31850],[34,43237],{"className":43238,"style":31846},[31845],[34,43240,43242,43245],{"className":43241},[31676],[34,43243],{"className":43244,"style":32617},[31680],[34,43246,640],{"className":43247},[31685],[108,43249,43250,43253,43254,43298,43299,43434,43435],{},[2627,43251,43252],{},"三大情况","（通过比较 ",[34,43255,43257,43277],{"className":43256},[31642],[34,43258,43260],{"className":43259},[31646],[31648,43261,43262],{"xmlns":31650},[31652,43263,43264,43274],{},[31655,43265,43266,43268,43270,43272],{},[31658,43267,31739],{},[31662,43269,456],{"stretchy":702},[31658,43271,31660],{},[31662,43273,2557],{"stretchy":702},[31665,43275,43276],{"encoding":31667},"f(n)",[34,43278,43280],{"className":43279,"ariaHidden":758},[31672],[34,43281,43283,43286,43289,43292,43295],{"className":43282},[31676],[34,43284],{"className":43285,"style":31774},[31680],[34,43287,31739],{"className":43288,"style":31778},[31685,31686],[34,43290,456],{"className":43291},[31782],[34,43293,31660],{"className":43294},[31685,31686],[34,43296,2557],{"className":43297},[31690]," 与 ",[34,43300,43302,43332],{"className":43301},[31642],[34,43303,43305],{"className":43304},[31646],[31648,43306,43307],{"xmlns":31650},[31652,43308,43309,43329],{},[31655,43310,43311],{},[37624,43312,43313,43315],{},[31658,43314,31660],{},[31655,43316,43317,43327],{},[31743,43318,43319,43325],{},[31655,43320,43321,43323],{},[31658,43322,885],{},[31662,43324,32682],{},[31658,43326,1697],{},[31658,43328,111],{},[31665,43330,43331],{"encoding":31667},"n^{\\log_b a}",[34,43333,43335],{"className":43334,"ariaHidden":758},[31672],[34,43336,43338,43341],{"className":43337},[31676],[34,43339],{"className":43340,"style":38486},[31680],[34,43342,43344,43347],{"className":43343},[31685],[34,43345,31660],{"className":43346},[31685,31686],[34,43348,43350],{"className":43349},[31792],[34,43351,43353],{"className":43352},[31796],[34,43354,43356],{"className":43355},[31801],[34,43357,43359],{"className":43358,"style":38486},[31805],[34,43360,43361,43364],{"style":37777},[34,43362],{"className":43363,"style":31814},[31813],[34,43365,43367],{"className":43366},[31818,31819,31820,31821],[34,43368,43370,43427,43431],{"className":43369},[31685,31821],[34,43371,43373,43387],{"className":43372},[32767,31821],[34,43374,43376,43380,43384],{"className":43375},[32767,31821],[34,43377,43379],{"className":43378},[31821],"l",[34,43381,43383],{"className":43382},[31821],"o",[34,43385,31942],{"className":43386,"style":38619},[31821],[34,43388,43390],{"className":43389},[31792],[34,43391,43393,43418],{"className":43392},[31796,31797],[34,43394,43396,43415],{"className":43395},[31801],[34,43397,43400],{"className":43398,"style":43399},[31805],"height:0.2302em;",[34,43401,43403,43407],{"style":43402},"top:-2.2341em;margin-right:0.0714em;",[34,43404],{"className":43405,"style":43406},[31813],"height:2.5em;",[34,43408,43412],{"className":43409},[31818,43410,43411,31821],"reset-size3","size1",[34,43413,1697],{"className":43414},[31685,31686,31821],[34,43416,31829],{"className":43417},[31828],[34,43419,43421],{"className":43420},[31801],[34,43422,43425],{"className":43423,"style":43424},[31805],"height:0.2659em;",[34,43426],{},[34,43428],{"className":43429,"style":43430},[31845,31821],"margin-right:0.1952em;",[34,43432,111],{"className":43433},[31685,31686,31821],"）：\n",[2230,43436,43437,43745,44114],{},[108,43438,43439,43440,43565,43566,347],{},"若 ",[34,43441,43443,43472],{"className":43442},[31642],[34,43444,43446],{"className":43445},[31646],[31648,43447,43448],{"xmlns":31650},[31652,43449,43450,43470],{},[31655,43451,43452],{},[37624,43453,43454,43456],{},[31658,43455,31660],{},[31655,43457,43458,43468],{},[31743,43459,43460,43466],{},[31655,43461,43462,43464],{},[31658,43463,885],{},[31662,43465,32682],{},[31658,43467,1697],{},[31658,43469,111],{},[31665,43471,43331],{"encoding":31667},[34,43473,43475],{"className":43474,"ariaHidden":758},[31672],[34,43476,43478,43481],{"className":43477},[31676],[34,43479],{"className":43480,"style":38486},[31680],[34,43482,43484,43487],{"className":43483},[31685],[34,43485,31660],{"className":43486},[31685,31686],[34,43488,43490],{"className":43489},[31792],[34,43491,43493],{"className":43492},[31796],[34,43494,43496],{"className":43495},[31801],[34,43497,43499],{"className":43498,"style":38486},[31805],[34,43500,43501,43504],{"style":37777},[34,43502],{"className":43503,"style":31814},[31813],[34,43505,43507],{"className":43506},[31818,31819,31820,31821],[34,43508,43510,43559,43562],{"className":43509},[31685,31821],[34,43511,43513,43525],{"className":43512},[32767,31821],[34,43514,43516,43519,43522],{"className":43515},[32767,31821],[34,43517,43379],{"className":43518},[31821],[34,43520,43383],{"className":43521},[31821],[34,43523,31942],{"className":43524,"style":38619},[31821],[34,43526,43528],{"className":43527},[31792],[34,43529,43531,43551],{"className":43530},[31796,31797],[34,43532,43534,43548],{"className":43533},[31801],[34,43535,43537],{"className":43536,"style":43399},[31805],[34,43538,43539,43542],{"style":43402},[34,43540],{"className":43541,"style":43406},[31813],[34,43543,43545],{"className":43544},[31818,43410,43411,31821],[34,43546,1697],{"className":43547},[31685,31686,31821],[34,43549,31829],{"className":43550},[31828],[34,43552,43554],{"className":43553},[31801],[34,43555,43557],{"className":43556,"style":43424},[31805],[34,43558],{},[34,43560],{"className":43561,"style":43430},[31845,31821],[34,43563,111],{"className":43564},[31685,31686,31821]," 大（呈多项式大于），则 ",[34,43567,43569,43615],{"className":43568},[31642],[34,43570,43572],{"className":43571},[31646],[31648,43573,43574],{"xmlns":31650},[31652,43575,43576,43612],{},[31655,43577,43578,43580,43582,43584,43586,43588,43590,43592,43610],{},[31658,43579,37400],{},[31662,43581,456],{"stretchy":702},[31658,43583,31660],{},[31662,43585,2557],{"stretchy":702},[31662,43587,202],{},[31658,43589,37890],{},[31662,43591,456],{"stretchy":702},[37624,43593,43594,43596],{},[31658,43595,31660],{},[31655,43597,43598,43608],{},[31743,43599,43600,43606],{},[31655,43601,43602,43604],{},[31658,43603,885],{},[31662,43605,32682],{},[31658,43607,1697],{},[31658,43609,111],{},[31662,43611,2557],{"stretchy":702},[31665,43613,43614],{"encoding":31667},"T(n) = O(n^{\\log_b a})",[34,43616,43618,43645],{"className":43617,"ariaHidden":758},[31672],[34,43619,43621,43624,43627,43630,43633,43636,43639,43642],{"className":43620},[31676],[34,43622],{"className":43623,"style":31774},[31680],[34,43625,37400],{"className":43626,"style":37422},[31685,31686],[34,43628,456],{"className":43629},[31782],[34,43631,31660],{"className":43632},[31685,31686],[34,43634,2557],{"className":43635},[31690],[34,43637],{"className":43638,"style":31846},[31845],[34,43640,202],{"className":43641},[31850],[34,43643],{"className":43644,"style":31846},[31845],[34,43646,43648,43652,43655,43658,43742],{"className":43647},[31676],[34,43649],{"className":43650,"style":43651},[31680],"height:1.0991em;vertical-align:-0.25em;",[34,43653,37890],{"className":43654,"style":34452},[31685,31686],[34,43656,456],{"className":43657},[31782],[34,43659,43661,43664],{"className":43660},[31685],[34,43662,31660],{"className":43663},[31685,31686],[34,43665,43667],{"className":43666},[31792],[34,43668,43670],{"className":43669},[31796],[34,43671,43673],{"className":43672},[31801],[34,43674,43676],{"className":43675,"style":38486},[31805],[34,43677,43678,43681],{"style":37777},[34,43679],{"className":43680,"style":31814},[31813],[34,43682,43684],{"className":43683},[31818,31819,31820,31821],[34,43685,43687,43736,43739],{"className":43686},[31685,31821],[34,43688,43690,43702],{"className":43689},[32767,31821],[34,43691,43693,43696,43699],{"className":43692},[32767,31821],[34,43694,43379],{"className":43695},[31821],[34,43697,43383],{"className":43698},[31821],[34,43700,31942],{"className":43701,"style":38619},[31821],[34,43703,43705],{"className":43704},[31792],[34,43706,43708,43728],{"className":43707},[31796,31797],[34,43709,43711,43725],{"className":43710},[31801],[34,43712,43714],{"className":43713,"style":43399},[31805],[34,43715,43716,43719],{"style":43402},[34,43717],{"className":43718,"style":43406},[31813],[34,43720,43722],{"className":43721},[31818,43410,43411,31821],[34,43723,1697],{"className":43724},[31685,31686,31821],[34,43726,31829],{"className":43727},[31828],[34,43729,43731],{"className":43730},[31801],[34,43732,43734],{"className":43733,"style":43424},[31805],[34,43735],{},[34,43737],{"className":43738,"style":43430},[31845,31821],[34,43740,111],{"className":43741},[31685,31686,31821],[34,43743,2557],{"className":43744},[31690],[108,43746,43439,43747,43298,43872,43915,43916,347],{},[34,43748,43750,43779],{"className":43749},[31642],[34,43751,43753],{"className":43752},[31646],[31648,43754,43755],{"xmlns":31650},[31652,43756,43757,43777],{},[31655,43758,43759],{},[37624,43760,43761,43763],{},[31658,43762,31660],{},[31655,43764,43765,43775],{},[31743,43766,43767,43773],{},[31655,43768,43769,43771],{},[31658,43770,885],{},[31662,43772,32682],{},[31658,43774,1697],{},[31658,43776,111],{},[31665,43778,43331],{"encoding":31667},[34,43780,43782],{"className":43781,"ariaHidden":758},[31672],[34,43783,43785,43788],{"className":43784},[31676],[34,43786],{"className":43787,"style":38486},[31680],[34,43789,43791,43794],{"className":43790},[31685],[34,43792,31660],{"className":43793},[31685,31686],[34,43795,43797],{"className":43796},[31792],[34,43798,43800],{"className":43799},[31796],[34,43801,43803],{"className":43802},[31801],[34,43804,43806],{"className":43805,"style":38486},[31805],[34,43807,43808,43811],{"style":37777},[34,43809],{"className":43810,"style":31814},[31813],[34,43812,43814],{"className":43813},[31818,31819,31820,31821],[34,43815,43817,43866,43869],{"className":43816},[31685,31821],[34,43818,43820,43832],{"className":43819},[32767,31821],[34,43821,43823,43826,43829],{"className":43822},[32767,31821],[34,43824,43379],{"className":43825},[31821],[34,43827,43383],{"className":43828},[31821],[34,43830,31942],{"className":43831,"style":38619},[31821],[34,43833,43835],{"className":43834},[31792],[34,43836,43838,43858],{"className":43837},[31796,31797],[34,43839,43841,43855],{"className":43840},[31801],[34,43842,43844],{"className":43843,"style":43399},[31805],[34,43845,43846,43849],{"style":43402},[34,43847],{"className":43848,"style":43406},[31813],[34,43850,43852],{"className":43851},[31818,43410,43411,31821],[34,43853,1697],{"className":43854},[31685,31686,31821],[34,43856,31829],{"className":43857},[31828],[34,43859,43861],{"className":43860},[31801],[34,43862,43864],{"className":43863,"style":43424},[31805],[34,43865],{},[34,43867],{"className":43868,"style":43430},[31845,31821],[34,43870,111],{"className":43871},[31685,31686,31821],[34,43873,43875,43894],{"className":43874},[31642],[34,43876,43878],{"className":43877},[31646],[31648,43879,43880],{"xmlns":31650},[31652,43881,43882,43892],{},[31655,43883,43884,43886,43888,43890],{},[31658,43885,31739],{},[31662,43887,456],{"stretchy":702},[31658,43889,31660],{},[31662,43891,2557],{"stretchy":702},[31665,43893,43276],{"encoding":31667},[34,43895,43897],{"className":43896,"ariaHidden":758},[31672],[34,43898,43900,43903,43906,43909,43912],{"className":43899},[31676],[34,43901],{"className":43902,"style":31774},[31680],[34,43904,31739],{"className":43905,"style":31778},[31685,31686],[34,43907,456],{"className":43908},[31782],[34,43910,31660],{"className":43911},[31685,31686],[34,43913,2557],{"className":43914},[31690]," 一样大（同阶），则 ",[34,43917,43919,43971],{"className":43918},[31642],[34,43920,43922],{"className":43921},[31646],[31648,43923,43924],{"xmlns":31650},[31652,43925,43926,43968],{},[31655,43927,43928,43930,43932,43934,43936,43938,43940,43942,43960,43962,43964,43966],{},[31658,43929,37400],{},[31662,43931,456],{"stretchy":702},[31658,43933,31660],{},[31662,43935,2557],{"stretchy":702},[31662,43937,202],{},[31658,43939,37890],{},[31662,43941,456],{"stretchy":702},[37624,43943,43944,43946],{},[31658,43945,31660],{},[31655,43947,43948,43958],{},[31743,43949,43950,43956],{},[31655,43951,43952,43954],{},[31658,43953,885],{},[31662,43955,32682],{},[31658,43957,1697],{},[31658,43959,111],{},[31658,43961,885],{},[31662,43963,32682],{},[31658,43965,31660],{},[31662,43967,2557],{"stretchy":702},[31665,43969,43970],{"encoding":31667},"T(n) = O(n^{\\log_b a} \\log n)",[34,43972,43974,44001],{"className":43973,"ariaHidden":758},[31672],[34,43975,43977,43980,43983,43986,43989,43992,43995,43998],{"className":43976},[31676],[34,43978],{"className":43979,"style":31774},[31680],[34,43981,37400],{"className":43982,"style":37422},[31685,31686],[34,43984,456],{"className":43985},[31782],[34,43987,31660],{"className":43988},[31685,31686],[34,43990,2557],{"className":43991},[31690],[34,43993],{"className":43994,"style":31846},[31845],[34,43996,202],{"className":43997},[31850],[34,43999],{"className":44000,"style":31846},[31845],[34,44002,44004,44007,44010,44013,44097,44100,44105,44108,44111],{"className":44003},[31676],[34,44005],{"className":44006,"style":43651},[31680],[34,44008,37890],{"className":44009,"style":34452},[31685,31686],[34,44011,456],{"className":44012},[31782],[34,44014,44016,44019],{"className":44015},[31685],[34,44017,31660],{"className":44018},[31685,31686],[34,44020,44022],{"className":44021},[31792],[34,44023,44025],{"className":44024},[31796],[34,44026,44028],{"className":44027},[31801],[34,44029,44031],{"className":44030,"style":38486},[31805],[34,44032,44033,44036],{"style":37777},[34,44034],{"className":44035,"style":31814},[31813],[34,44037,44039],{"className":44038},[31818,31819,31820,31821],[34,44040,44042,44091,44094],{"className":44041},[31685,31821],[34,44043,44045,44057],{"className":44044},[32767,31821],[34,44046,44048,44051,44054],{"className":44047},[32767,31821],[34,44049,43379],{"className":44050},[31821],[34,44052,43383],{"className":44053},[31821],[34,44055,31942],{"className":44056,"style":38619},[31821],[34,44058,44060],{"className":44059},[31792],[34,44061,44063,44083],{"className":44062},[31796,31797],[34,44064,44066,44080],{"className":44065},[31801],[34,44067,44069],{"className":44068,"style":43399},[31805],[34,44070,44071,44074],{"style":43402},[34,44072],{"className":44073,"style":43406},[31813],[34,44075,44077],{"className":44076},[31818,43410,43411,31821],[34,44078,1697],{"className":44079},[31685,31686,31821],[34,44081,31829],{"className":44082},[31828],[34,44084,44086],{"className":44085},[31801],[34,44087,44089],{"className":44088,"style":43424},[31805],[34,44090],{},[34,44092],{"className":44093,"style":43430},[31845,31821],[34,44095,111],{"className":44096},[31685,31686,31821],[34,44098],{"className":44099,"style":32123},[31845],[34,44101,38616,44103],{"className":44102},[32767],[34,44104,31942],{"style":38619},[34,44106],{"className":44107,"style":32123},[31845],[34,44109,31660],{"className":44110},[31685,31686],[34,44112,2557],{"className":44113},[31690],[108,44115,43439,44116,44159,44160,347],{},[34,44117,44119,44138],{"className":44118},[31642],[34,44120,44122],{"className":44121},[31646],[31648,44123,44124],{"xmlns":31650},[31652,44125,44126,44136],{},[31655,44127,44128,44130,44132,44134],{},[31658,44129,31739],{},[31662,44131,456],{"stretchy":702},[31658,44133,31660],{},[31662,44135,2557],{"stretchy":702},[31665,44137,43276],{"encoding":31667},[34,44139,44141],{"className":44140,"ariaHidden":758},[31672],[34,44142,44144,44147,44150,44153,44156],{"className":44143},[31676],[34,44145],{"className":44146,"style":31774},[31680],[34,44148,31739],{"className":44149,"style":31778},[31685,31686],[34,44151,456],{"className":44152},[31782],[34,44154,31660],{"className":44155},[31685,31686],[34,44157,2557],{"className":44158},[31690]," 更大且满足正则条件，则 ",[34,44161,44163,44199],{"className":44162},[31642],[34,44164,44166],{"className":44165},[31646],[31648,44167,44168],{"xmlns":31650},[31652,44169,44170,44196],{},[31655,44171,44172,44174,44176,44178,44180,44182,44184,44186,44188,44190,44192,44194],{},[31658,44173,37400],{},[31662,44175,456],{"stretchy":702},[31658,44177,31660],{},[31662,44179,2557],{"stretchy":702},[31662,44181,202],{},[31658,44183,37890],{},[31662,44185,456],{"stretchy":702},[31658,44187,31739],{},[31662,44189,456],{"stretchy":702},[31658,44191,31660],{},[31662,44193,2557],{"stretchy":702},[31662,44195,2557],{"stretchy":702},[31665,44197,44198],{"encoding":31667},"T(n) = O(f(n))",[34,44200,44202,44229],{"className":44201,"ariaHidden":758},[31672],[34,44203,44205,44208,44211,44214,44217,44220,44223,44226],{"className":44204},[31676],[34,44206],{"className":44207,"style":31774},[31680],[34,44209,37400],{"className":44210,"style":37422},[31685,31686],[34,44212,456],{"className":44213},[31782],[34,44215,31660],{"className":44216},[31685,31686],[34,44218,2557],{"className":44219},[31690],[34,44221],{"className":44222,"style":31846},[31845],[34,44224,202],{"className":44225},[31850],[34,44227],{"className":44228,"style":31846},[31845],[34,44230,44232,44235,44238,44241,44244,44247,44250],{"className":44231},[31676],[34,44233],{"className":44234,"style":31774},[31680],[34,44236,37890],{"className":44237,"style":34452},[31685,31686],[34,44239,456],{"className":44240},[31782],[34,44242,31739],{"className":44243,"style":31778},[31685,31686],[34,44245,456],{"className":44246},[31782],[34,44248,31660],{"className":44249},[31685,31686],[34,44251,44253],{"className":44252},[31690],"))",[108,44255,44256,2664,44258,347,44373,44509,44510,44689,44690,44748,44749,44815,44816,347],{},[2627,44257,39579],{},[34,44259,44261,44301],{"className":44260},[31642],[34,44262,44264],{"className":44263},[31646],[31648,44265,44266],{"xmlns":31650},[31652,44267,44268,44298],{},[31655,44269,44270,44272,44274,44276,44278,44280,44282,44284,44286,44288,44290,44292,44294,44296],{},[31658,44271,37400],{},[31662,44273,456],{"stretchy":702},[31658,44275,31660],{},[31662,44277,2557],{"stretchy":702},[31662,44279,202],{},[31749,44281,15932],{},[31658,44283,37400],{},[31662,44285,456],{"stretchy":702},[31658,44287,31660],{},[31658,44289,38080],{"mathvariant":38079},[31749,44291,768],{},[31662,44293,2557],{"stretchy":702},[31662,44295,223],{},[31658,44297,31660],{},[31665,44299,44300],{"encoding":31667},"T(n) = 4T(n\u002F2) + n",[34,44302,44304,44331,44364],{"className":44303,"ariaHidden":758},[31672],[34,44305,44307,44310,44313,44316,44319,44322,44325,44328],{"className":44306},[31676],[34,44308],{"className":44309,"style":31774},[31680],[34,44311,37400],{"className":44312,"style":37422},[31685,31686],[34,44314,456],{"className":44315},[31782],[34,44317,31660],{"className":44318},[31685,31686],[34,44320,2557],{"className":44321},[31690],[34,44323],{"className":44324,"style":31846},[31845],[34,44326,202],{"className":44327},[31850],[34,44329],{"className":44330,"style":31846},[31845],[34,44332,44334,44337,44340,44343,44346,44349,44352,44355,44358,44361],{"className":44333},[31676],[34,44335],{"className":44336,"style":31774},[31680],[34,44338,15932],{"className":44339},[31685],[34,44341,37400],{"className":44342,"style":37422},[31685,31686],[34,44344,456],{"className":44345},[31782],[34,44347,31660],{"className":44348},[31685,31686],[34,44350,38149],{"className":44351},[31685],[34,44353,2557],{"className":44354},[31690],[34,44356],{"className":44357,"style":32604},[31845],[34,44359,223],{"className":44360},[32024],[34,44362],{"className":44363,"style":32604},[31845],[34,44365,44367,44370],{"className":44366},[31676],[34,44368],{"className":44369,"style":32527},[31680],[34,44371,31660],{"className":44372},[31685,31686],[34,44374,44376,44416],{"className":44375},[31642],[34,44377,44379],{"className":44378},[31646],[31648,44380,44381],{"xmlns":31650},[31652,44382,44383,44413],{},[31655,44384,44385,44387,44389,44391,44393,44395,44397,44399,44401,44403,44405,44407,44409,44411],{},[31658,44386,111],{},[31662,44388,202],{},[31749,44390,15932],{},[31662,44392,31960],{"separator":758},[31658,44394,1697],{},[31662,44396,202],{},[31749,44398,768],{},[31662,44400,31960],{"separator":758},[31658,44402,31739],{},[31662,44404,456],{"stretchy":702},[31658,44406,31660],{},[31662,44408,2557],{"stretchy":702},[31662,44410,202],{},[31658,44412,31660],{},[31665,44414,44415],{"encoding":31667},"a=4, b=2, f(n)=n",[34,44417,44419,44437,44464,44500],{"className":44418,"ariaHidden":758},[31672],[34,44420,44422,44425,44428,44431,44434],{"className":44421},[31676],[34,44423],{"className":44424,"style":32527},[31680],[34,44426,111],{"className":44427},[31685,31686],[34,44429],{"className":44430,"style":31846},[31845],[34,44432,202],{"className":44433},[31850],[34,44435],{"className":44436,"style":31846},[31845],[34,44438,44440,44443,44446,44449,44452,44455,44458,44461],{"className":44439},[31676],[34,44441],{"className":44442,"style":43217},[31680],[34,44444,15932],{"className":44445},[31685],[34,44447,31960],{"className":44448},[32119],[34,44450],{"className":44451,"style":32123},[31845],[34,44453,1697],{"className":44454},[31685,31686],[34,44456],{"className":44457,"style":31846},[31845],[34,44459,202],{"className":44460},[31850],[34,44462],{"className":44463,"style":31846},[31845],[34,44465,44467,44470,44473,44476,44479,44482,44485,44488,44491,44494,44497],{"className":44466},[31676],[34,44468],{"className":44469,"style":31774},[31680],[34,44471,768],{"className":44472},[31685],[34,44474,31960],{"className":44475},[32119],[34,44477],{"className":44478,"style":32123},[31845],[34,44480,31739],{"className":44481,"style":31778},[31685,31686],[34,44483,456],{"className":44484},[31782],[34,44486,31660],{"className":44487},[31685,31686],[34,44489,2557],{"className":44490},[31690],[34,44492],{"className":44493,"style":31846},[31845],[34,44495,202],{"className":44496},[31850],[34,44498],{"className":44499,"style":31846},[31845],[34,44501,44503,44506],{"className":44502},[31676],[34,44504],{"className":44505,"style":32527},[31680],[34,44507,31660],{"className":44508},[31685,31686],"。计算 ",[34,44511,44513,44551],{"className":44512},[31642],[34,44514,44516],{"className":44515},[31646],[31648,44517,44518],{"xmlns":31650},[31652,44519,44520,44548],{},[31655,44521,44522,44540,44542],{},[37624,44523,44524,44526],{},[31658,44525,31660],{},[31655,44527,44528,44538],{},[31743,44529,44530,44536],{},[31655,44531,44532,44534],{},[31658,44533,885],{},[31662,44535,32682],{},[31749,44537,768],{},[31749,44539,15932],{},[31662,44541,202],{},[37624,44543,44544,44546],{},[31658,44545,31660],{},[31749,44547,768],{},[31665,44549,44550],{"encoding":31667},"n^{\\log_2 4} = n^2",[34,44552,44554,44654],{"className":44553,"ariaHidden":758},[31672],[34,44555,44557,44560,44645,44648,44651],{"className":44556},[31676],[34,44558],{"className":44559,"style":38486},[31680],[34,44561,44563,44566],{"className":44562},[31685],[34,44564,31660],{"className":44565},[31685,31686],[34,44567,44569],{"className":44568},[31792],[34,44570,44572],{"className":44571},[31796],[34,44573,44575],{"className":44574},[31801],[34,44576,44578],{"className":44577,"style":38486},[31805],[34,44579,44580,44583],{"style":37777},[34,44581],{"className":44582,"style":31814},[31813],[34,44584,44586],{"className":44585},[31818,31819,31820,31821],[34,44587,44589,44639,44642],{"className":44588},[31685,31821],[34,44590,44592,44604],{"className":44591},[32767,31821],[34,44593,44595,44598,44601],{"className":44594},[32767,31821],[34,44596,43379],{"className":44597},[31821],[34,44599,43383],{"className":44600},[31821],[34,44602,31942],{"className":44603,"style":38619},[31821],[34,44605,44607],{"className":44606},[31792],[34,44608,44610,44631],{"className":44609},[31796,31797],[34,44611,44613,44628],{"className":44612},[31801],[34,44614,44617],{"className":44615,"style":44616},[31805],"height:0.1944em;",[34,44618,44619,44622],{"style":43402},[34,44620],{"className":44621,"style":43406},[31813],[34,44623,44625],{"className":44624},[31818,43410,43411,31821],[34,44626,768],{"className":44627},[31685,31821],[34,44629,31829],{"className":44630},[31828],[34,44632,44634],{"className":44633},[31801],[34,44635,44637],{"className":44636,"style":43424},[31805],[34,44638],{},[34,44640],{"className":44641,"style":43430},[31845,31821],[34,44643,15932],{"className":44644},[31685,31821],[34,44646],{"className":44647,"style":31846},[31845],[34,44649,202],{"className":44650},[31850],[34,44652],{"className":44653,"style":31846},[31845],[34,44655,44657,44660],{"className":44656},[31676],[34,44658],{"className":44659,"style":37774},[31680],[34,44661,44663,44666],{"className":44662},[31685],[34,44664,31660],{"className":44665},[31685,31686],[34,44667,44669],{"className":44668},[31792],[34,44670,44672],{"className":44671},[31796],[34,44673,44675],{"className":44674},[31801],[34,44676,44678],{"className":44677,"style":37774},[31805],[34,44679,44680,44683],{"style":37777},[34,44681],{"className":44682,"style":31814},[31813],[34,44684,44686],{"className":44685},[31818,31819,31820,31821],[34,44687,768],{"className":44688},[31685,31821],"。因为 ",[34,44691,44693,44710],{"className":44692},[31642],[34,44694,44696],{"className":44695},[31646],[31648,44697,44698],{"xmlns":31650},[31652,44699,44700,44708],{},[31655,44701,44702],{},[37624,44703,44704,44706],{},[31658,44705,31660],{},[31749,44707,768],{},[31665,44709,42015],{"encoding":31667},[34,44711,44713],{"className":44712,"ariaHidden":758},[31672],[34,44714,44716,44719],{"className":44715},[31676],[34,44717],{"className":44718,"style":37774},[31680],[34,44720,44722,44725],{"className":44721},[31685],[34,44723,31660],{"className":44724},[31685,31686],[34,44726,44728],{"className":44727},[31792],[34,44729,44731],{"className":44730},[31796],[34,44732,44734],{"className":44733},[31801],[34,44735,44737],{"className":44736,"style":37774},[31805],[34,44738,44739,44742],{"style":37777},[34,44740],{"className":44741,"style":31814},[31813],[34,44743,44745],{"className":44744},[31818,31819,31820,31821],[34,44746,768],{"className":44747},[31685,31821]," 比 ",[34,44750,44752,44776],{"className":44751},[31642],[34,44753,44755],{"className":44754},[31646],[31648,44756,44757],{"xmlns":31650},[31652,44758,44759,44773],{},[31655,44760,44761,44763,44765,44767,44769,44771],{},[31658,44762,31739],{},[31662,44764,456],{"stretchy":702},[31658,44766,31660],{},[31662,44768,2557],{"stretchy":702},[31662,44770,202],{},[31658,44772,31660],{},[31665,44774,44775],{"encoding":31667},"f(n)=n",[34,44777,44779,44806],{"className":44778,"ariaHidden":758},[31672],[34,44780,44782,44785,44788,44791,44794,44797,44800,44803],{"className":44781},[31676],[34,44783],{"className":44784,"style":31774},[31680],[34,44786,31739],{"className":44787,"style":31778},[31685,31686],[34,44789,456],{"className":44790},[31782],[34,44792,31660],{"className":44793},[31685,31686],[34,44795,2557],{"className":44796},[31690],[34,44798],{"className":44799,"style":31846},[31845],[34,44801,202],{"className":44802},[31850],[34,44804],{"className":44805,"style":31846},[31845],[34,44807,44809,44812],{"className":44808},[31676],[34,44810],{"className":44811,"style":32527},[31680],[34,44813,31660],{"className":44814},[31685,31686]," 大，属情况1，故 ",[34,44817,44819,44853],{"className":44818},[31642],[34,44820,44822],{"className":44821},[31646],[31648,44823,44824],{"xmlns":31650},[31652,44825,44826,44850],{},[31655,44827,44828,44830,44832,44834,44836,44838,44840,44842,44848],{},[31658,44829,37400],{},[31662,44831,456],{"stretchy":702},[31658,44833,31660],{},[31662,44835,2557],{"stretchy":702},[31662,44837,202],{},[31658,44839,37890],{},[31662,44841,456],{"stretchy":702},[37624,44843,44844,44846],{},[31658,44845,31660],{},[31749,44847,768],{},[31662,44849,2557],{"stretchy":702},[31665,44851,44852],{"encoding":31667},"T(n) = O(n^2)",[34,44854,44856,44883],{"className":44855,"ariaHidden":758},[31672],[34,44857,44859,44862,44865,44868,44871,44874,44877,44880],{"className":44858},[31676],[34,44860],{"className":44861,"style":31774},[31680],[34,44863,37400],{"className":44864,"style":37422},[31685,31686],[34,44866,456],{"className":44867},[31782],[34,44869,31660],{"className":44870},[31685,31686],[34,44872,2557],{"className":44873},[31690],[34,44875],{"className":44876,"style":31846},[31845],[34,44878,202],{"className":44879},[31850],[34,44881],{"className":44882,"style":31846},[31845],[34,44884,44886,44889,44892,44895,44924],{"className":44885},[31676],[34,44887],{"className":44888,"style":37755},[31680],[34,44890,37890],{"className":44891,"style":34452},[31685,31686],[34,44893,456],{"className":44894},[31782],[34,44896,44898,44901],{"className":44897},[31685],[34,44899,31660],{"className":44900},[31685,31686],[34,44902,44904],{"className":44903},[31792],[34,44905,44907],{"className":44906},[31796],[34,44908,44910],{"className":44909},[31801],[34,44911,44913],{"className":44912,"style":37774},[31805],[34,44914,44915,44918],{"style":37777},[34,44916],{"className":44917,"style":31814},[31813],[34,44919,44921],{"className":44920},[31818,31819,31820,31821],[34,44922,768],{"className":44923},[31685,31821],[34,44925,2557],{"className":44926},[31690],[108,44928,44929,2664,44932,347,45076,44509,45242,44689,45390,44748,45418,45514,45515,347],{},[2627,44930,44931],{},"例题 2",[34,44933,44935,44978],{"className":44934},[31642],[34,44936,44938],{"className":44937},[31646],[31648,44939,44940],{"xmlns":31650},[31652,44941,44942,44976],{},[31655,44943,44944,44946,44948,44950,44952,44954,44956,44958,44960,44962,44964,44966,44968,44970],{},[31658,44945,37400],{},[31662,44947,456],{"stretchy":702},[31658,44949,31660],{},[31662,44951,2557],{"stretchy":702},[31662,44953,202],{},[31749,44955,768],{},[31658,44957,37400],{},[31662,44959,456],{"stretchy":702},[31658,44961,31660],{},[31658,44963,38080],{"mathvariant":38079},[31749,44965,768],{},[31662,44967,2557],{"stretchy":702},[31662,44969,223],{},[37624,44971,44972,44974],{},[31658,44973,31660],{},[31749,44975,768],{},[31665,44977,41895],{"encoding":31667},[34,44979,44981,45008,45041],{"className":44980,"ariaHidden":758},[31672],[34,44982,44984,44987,44990,44993,44996,44999,45002,45005],{"className":44983},[31676],[34,44985],{"className":44986,"style":31774},[31680],[34,44988,37400],{"className":44989,"style":37422},[31685,31686],[34,44991,456],{"className":44992},[31782],[34,44994,31660],{"className":44995},[31685,31686],[34,44997,2557],{"className":44998},[31690],[34,45000],{"className":45001,"style":31846},[31845],[34,45003,202],{"className":45004},[31850],[34,45006],{"className":45007,"style":31846},[31845],[34,45009,45011,45014,45017,45020,45023,45026,45029,45032,45035,45038],{"className":45010},[31676],[34,45012],{"className":45013,"style":31774},[31680],[34,45015,768],{"className":45016},[31685],[34,45018,37400],{"className":45019,"style":37422},[31685,31686],[34,45021,456],{"className":45022},[31782],[34,45024,31660],{"className":45025},[31685,31686],[34,45027,38149],{"className":45028},[31685],[34,45030,2557],{"className":45031},[31690],[34,45033],{"className":45034,"style":32604},[31845],[34,45036,223],{"className":45037},[32024],[34,45039],{"className":45040,"style":32604},[31845],[34,45042,45044,45047],{"className":45043},[31676],[34,45045],{"className":45046,"style":37774},[31680],[34,45048,45050,45053],{"className":45049},[31685],[34,45051,31660],{"className":45052},[31685,31686],[34,45054,45056],{"className":45055},[31792],[34,45057,45059],{"className":45058},[31796],[34,45060,45062],{"className":45061},[31801],[34,45063,45065],{"className":45064,"style":37774},[31805],[34,45066,45067,45070],{"style":37777},[34,45068],{"className":45069,"style":31814},[31813],[34,45071,45073],{"className":45072},[31818,31819,31820,31821],[34,45074,768],{"className":45075},[31685,31821],[34,45077,45079,45123],{"className":45078},[31642],[34,45080,45082],{"className":45081},[31646],[31648,45083,45084],{"xmlns":31650},[31652,45085,45086,45120],{},[31655,45087,45088,45090,45092,45094,45096,45098,45100,45102,45104,45106,45108,45110,45112,45114],{},[31658,45089,111],{},[31662,45091,202],{},[31749,45093,768],{},[31662,45095,31960],{"separator":758},[31658,45097,1697],{},[31662,45099,202],{},[31749,45101,768],{},[31662,45103,31960],{"separator":758},[31658,45105,31739],{},[31662,45107,456],{"stretchy":702},[31658,45109,31660],{},[31662,45111,2557],{"stretchy":702},[31662,45113,202],{},[37624,45115,45116,45118],{},[31658,45117,31660],{},[31749,45119,768],{},[31665,45121,45122],{"encoding":31667},"a=2, b=2, f(n)=n^2",[34,45124,45126,45144,45171,45207],{"className":45125,"ariaHidden":758},[31672],[34,45127,45129,45132,45135,45138,45141],{"className":45128},[31676],[34,45130],{"className":45131,"style":32527},[31680],[34,45133,111],{"className":45134},[31685,31686],[34,45136],{"className":45137,"style":31846},[31845],[34,45139,202],{"className":45140},[31850],[34,45142],{"className":45143,"style":31846},[31845],[34,45145,45147,45150,45153,45156,45159,45162,45165,45168],{"className":45146},[31676],[34,45148],{"className":45149,"style":43217},[31680],[34,45151,768],{"className":45152},[31685],[34,45154,31960],{"className":45155},[32119],[34,45157],{"className":45158,"style":32123},[31845],[34,45160,1697],{"className":45161},[31685,31686],[34,45163],{"className":45164,"style":31846},[31845],[34,45166,202],{"className":45167},[31850],[34,45169],{"className":45170,"style":31846},[31845],[34,45172,45174,45177,45180,45183,45186,45189,45192,45195,45198,45201,45204],{"className":45173},[31676],[34,45175],{"className":45176,"style":31774},[31680],[34,45178,768],{"className":45179},[31685],[34,45181,31960],{"className":45182},[32119],[34,45184],{"className":45185,"style":32123},[31845],[34,45187,31739],{"className":45188,"style":31778},[31685,31686],[34,45190,456],{"className":45191},[31782],[34,45193,31660],{"className":45194},[31685,31686],[34,45196,2557],{"className":45197},[31690],[34,45199],{"className":45200,"style":31846},[31845],[34,45202,202],{"className":45203},[31850],[34,45205],{"className":45206,"style":31846},[31845],[34,45208,45210,45213],{"className":45209},[31676],[34,45211],{"className":45212,"style":37774},[31680],[34,45214,45216,45219],{"className":45215},[31685],[34,45217,31660],{"className":45218},[31685,31686],[34,45220,45222],{"className":45221},[31792],[34,45223,45225],{"className":45224},[31796],[34,45226,45228],{"className":45227},[31801],[34,45229,45231],{"className":45230,"style":37774},[31805],[34,45232,45233,45236],{"style":37777},[34,45234],{"className":45235,"style":31814},[31813],[34,45237,45239],{"className":45238},[31818,31819,31820,31821],[34,45240,768],{"className":45241},[31685,31821],[34,45243,45245,45279],{"className":45244},[31642],[34,45246,45248],{"className":45247},[31646],[31648,45249,45250],{"xmlns":31650},[31652,45251,45252,45276],{},[31655,45253,45254,45272,45274],{},[37624,45255,45256,45258],{},[31658,45257,31660],{},[31655,45259,45260,45270],{},[31743,45261,45262,45268],{},[31655,45263,45264,45266],{},[31658,45265,885],{},[31662,45267,32682],{},[31749,45269,768],{},[31749,45271,768],{},[31662,45273,202],{},[31658,45275,31660],{},[31665,45277,45278],{"encoding":31667},"n^{\\log_2 2} = n",[34,45280,45282,45381],{"className":45281,"ariaHidden":758},[31672],[34,45283,45285,45288,45372,45375,45378],{"className":45284},[31676],[34,45286],{"className":45287,"style":38486},[31680],[34,45289,45291,45294],{"className":45290},[31685],[34,45292,31660],{"className":45293},[31685,31686],[34,45295,45297],{"className":45296},[31792],[34,45298,45300],{"className":45299},[31796],[34,45301,45303],{"className":45302},[31801],[34,45304,45306],{"className":45305,"style":38486},[31805],[34,45307,45308,45311],{"style":37777},[34,45309],{"className":45310,"style":31814},[31813],[34,45312,45314],{"className":45313},[31818,31819,31820,31821],[34,45315,45317,45366,45369],{"className":45316},[31685,31821],[34,45318,45320,45332],{"className":45319},[32767,31821],[34,45321,45323,45326,45329],{"className":45322},[32767,31821],[34,45324,43379],{"className":45325},[31821],[34,45327,43383],{"className":45328},[31821],[34,45330,31942],{"className":45331,"style":38619},[31821],[34,45333,45335],{"className":45334},[31792],[34,45336,45338,45358],{"className":45337},[31796,31797],[34,45339,45341,45355],{"className":45340},[31801],[34,45342,45344],{"className":45343,"style":44616},[31805],[34,45345,45346,45349],{"style":43402},[34,45347],{"className":45348,"style":43406},[31813],[34,45350,45352],{"className":45351},[31818,43410,43411,31821],[34,45353,768],{"className":45354},[31685,31821],[34,45356,31829],{"className":45357},[31828],[34,45359,45361],{"className":45360},[31801],[34,45362,45364],{"className":45363,"style":43424},[31805],[34,45365],{},[34,45367],{"className":45368,"style":43430},[31845,31821],[34,45370,768],{"className":45371},[31685,31821],[34,45373],{"className":45374,"style":31846},[31845],[34,45376,202],{"className":45377},[31850],[34,45379],{"className":45380,"style":31846},[31845],[34,45382,45384,45387],{"className":45383},[31676],[34,45385],{"className":45386,"style":32527},[31680],[34,45388,31660],{"className":45389},[31685,31686],[34,45391,45393,45406],{"className":45392},[31642],[34,45394,45396],{"className":45395},[31646],[31648,45397,45398],{"xmlns":31650},[31652,45399,45400,45404],{},[31655,45401,45402],{},[31658,45403,31660],{},[31665,45405,31660],{"encoding":31667},[34,45407,45409],{"className":45408,"ariaHidden":758},[31672],[34,45410,45412,45415],{"className":45411},[31676],[34,45413],{"className":45414,"style":32527},[31680],[34,45416,31660],{"className":45417},[31685,31686],[34,45419,45421,45449],{"className":45420},[31642],[34,45422,45424],{"className":45423},[31646],[31648,45425,45426],{"xmlns":31650},[31652,45427,45428,45446],{},[31655,45429,45430,45432,45434,45436,45438,45440],{},[31658,45431,31739],{},[31662,45433,456],{"stretchy":702},[31658,45435,31660],{},[31662,45437,2557],{"stretchy":702},[31662,45439,202],{},[37624,45441,45442,45444],{},[31658,45443,31660],{},[31749,45445,768],{},[31665,45447,45448],{"encoding":31667},"f(n)=n^2",[34,45450,45452,45479],{"className":45451,"ariaHidden":758},[31672],[34,45453,45455,45458,45461,45464,45467,45470,45473,45476],{"className":45454},[31676],[34,45456],{"className":45457,"style":31774},[31680],[34,45459,31739],{"className":45460,"style":31778},[31685,31686],[34,45462,456],{"className":45463},[31782],[34,45465,31660],{"className":45466},[31685,31686],[34,45468,2557],{"className":45469},[31690],[34,45471],{"className":45472,"style":31846},[31845],[34,45474,202],{"className":45475},[31850],[34,45477],{"className":45478,"style":31846},[31845],[34,45480,45482,45485],{"className":45481},[31676],[34,45483],{"className":45484,"style":37774},[31680],[34,45486,45488,45491],{"className":45487},[31685],[34,45489,31660],{"className":45490},[31685,31686],[34,45492,45494],{"className":45493},[31792],[34,45495,45497],{"className":45496},[31796],[34,45498,45500],{"className":45499},[31801],[34,45501,45503],{"className":45502,"style":37774},[31805],[34,45504,45505,45508],{"style":37777},[34,45506],{"className":45507,"style":31814},[31813],[34,45509,45511],{"className":45510},[31818,31819,31820,31821],[34,45512,768],{"className":45513},[31685,31821]," 小，属情况3，故 ",[34,45516,45518,45551],{"className":45517},[31642],[34,45519,45521],{"className":45520},[31646],[31648,45522,45523],{"xmlns":31650},[31652,45524,45525,45549],{},[31655,45526,45527,45529,45531,45533,45535,45537,45539,45541,45547],{},[31658,45528,37400],{},[31662,45530,456],{"stretchy":702},[31658,45532,31660],{},[31662,45534,2557],{"stretchy":702},[31662,45536,202],{},[31658,45538,37890],{},[31662,45540,456],{"stretchy":702},[37624,45542,45543,45545],{},[31658,45544,31660],{},[31749,45546,768],{},[31662,45548,2557],{"stretchy":702},[31665,45550,44852],{"encoding":31667},[34,45552,45554,45581],{"className":45553,"ariaHidden":758},[31672],[34,45555,45557,45560,45563,45566,45569,45572,45575,45578],{"className":45556},[31676],[34,45558],{"className":45559,"style":31774},[31680],[34,45561,37400],{"className":45562,"style":37422},[31685,31686],[34,45564,456],{"className":45565},[31782],[34,45567,31660],{"className":45568},[31685,31686],[34,45570,2557],{"className":45571},[31690],[34,45573],{"className":45574,"style":31846},[31845],[34,45576,202],{"className":45577},[31850],[34,45579],{"className":45580,"style":31846},[31845],[34,45582,45584,45587,45590,45593,45622],{"className":45583},[31676],[34,45585],{"className":45586,"style":37755},[31680],[34,45588,37890],{"className":45589,"style":34452},[31685,31686],[34,45591,456],{"className":45592},[31782],[34,45594,45596,45599],{"className":45595},[31685],[34,45597,31660],{"className":45598},[31685,31686],[34,45600,45602],{"className":45601},[31792],[34,45603,45605],{"className":45604},[31796],[34,45606,45608],{"className":45607},[31801],[34,45609,45611],{"className":45610,"style":37774},[31805],[34,45612,45613,45616],{"style":37777},[34,45614],{"className":45615,"style":31814},[31813],[34,45617,45619],{"className":45618},[31818,31819,31820,31821],[34,45620,768],{"className":45621},[31685,31821],[34,45623,2557],{"className":45624},[31690],{"title":29,"searchDepth":43,"depth":43,"links":45626},[45627,45628,45629,45630],{"id":31564,"depth":49,"text":31565},{"id":32192,"depth":49,"text":32193},{"id":37353,"depth":49,"text":37354},{"id":37381,"depth":49,"text":37382},{},"\u002Fsource\u002Fcs\u002Falgo\u002Fnotes\u002Fchap2",{"title":31556,"description":29},"source\u002Fcs\u002Falgo\u002Fnotes\u002Fchap2","MIESxD1bSa-NqTsaslplHA_8WPeljDNRUC2XfGPNwCM",{"id":45637,"title":45638,"body":45639,"description":29,"extension":85,"meta":51536,"navigation":52,"path":51537,"seo":51538,"stem":51539,"__hash__":51540},"notes\u002Fsource\u002Fcs\u002Falgo\u002Fnotes\u002Fchap3.md","第三章：分治（Divide and Conquer）",{"type":8,"value":45640,"toc":51532},[45641,45644,45648,45653,45849,45854,45875,45880,45899,45901,45905,45909,46222,46226,46819,46823,47581,47585,48665,48669,50267,50271,51077,51081],[11,45642,45638],{"id":45643},"第三章分治divide-and-conquer",[253,45645,45647],{"id":45646},"一-分治法核心知识点","一、 分治法核心知识点",[15,45649,45650],{},[2627,45651,45652],{},"1. 分治法的设计思想",[105,45654,45655,45719,45729,45736],{},[108,45656,45657,45660,45661,45689,45690,45718],{},[2627,45658,45659],{},"核心理念","：“凡治众如治寡，分数是也”。对于一个规模为 ",[34,45662,45664,45677],{"className":45663},[31642],[34,45665,45667],{"className":45666},[31646],[31648,45668,45669],{"xmlns":31650},[31652,45670,45671,45675],{},[31655,45672,45673],{},[31658,45674,31660],{},[31665,45676,31660],{"encoding":31667},[34,45678,45680],{"className":45679,"ariaHidden":758},[31672],[34,45681,45683,45686],{"className":45682},[31676],[34,45684],{"className":45685,"style":32527},[31680],[34,45687,31660],{"className":45688},[31685,31686]," 的问题，若规模较小则直接解决；否则将其分解为 ",[34,45691,45693,45706],{"className":45692},[31642],[34,45694,45696],{"className":45695},[31646],[31648,45697,45698],{"xmlns":31650},[31652,45699,45700,45704],{},[31655,45701,45702],{},[31658,45703,33759],{},[31665,45705,33759],{"encoding":31667},[34,45707,45709],{"className":45708,"ariaHidden":758},[31672],[34,45710,45712,45715],{"className":45711},[31676],[34,45713],{"className":45714,"style":31681},[31680],[34,45716,33759],{"className":45717,"style":33774},[31685,31686]," 个规模较小的子问题。",[108,45720,45721,45722,45725,45726,347],{},"这些子问题必须",[2627,45723,45724],{},"互相独立","且与",[2627,45727,45728],{},"原问题形式相同",[108,45730,45731,45732,45735],{},"递归求解这些子问题，最后将各子问题的解",[2627,45733,45734],{},"合并","得到原问题的解。",[108,45737,45738,45741,45742,45793,45794,45845,45846,347],{},[2627,45739,45740],{},"二分法","：实践中发现，将问题分成大小相等的 ",[34,45743,45745,45763],{"className":45744},[31642],[34,45746,45748],{"className":45747},[31646],[31648,45749,45750],{"xmlns":31650},[31652,45751,45752,45760],{},[31655,45753,45754,45756,45758],{},[31658,45755,33759],{},[31662,45757,202],{},[31749,45759,768],{},[31665,45761,45762],{"encoding":31667},"k=2",[34,45764,45766,45784],{"className":45765,"ariaHidden":758},[31672],[34,45767,45769,45772,45775,45778,45781],{"className":45768},[31676],[34,45770],{"className":45771,"style":31681},[31680],[34,45773,33759],{"className":45774,"style":33774},[31685,31686],[34,45776],{"className":45777,"style":31846},[31845],[34,45779,202],{"className":45780},[31850],[34,45782],{"className":45783,"style":31846},[31845],[34,45785,45787,45790],{"className":45786},[31676],[34,45788],{"className":45789,"style":32617},[31680],[34,45791,768],{"className":45792},[31685]," 个子问题最为有效，这种平衡子问题的思想几乎总是优于子问题规模不等的做法。当 ",[34,45795,45797,45815],{"className":45796},[31642],[34,45798,45800],{"className":45799},[31646],[31648,45801,45802],{"xmlns":31650},[31652,45803,45804,45812],{},[31655,45805,45806,45808,45810],{},[31658,45807,33759],{},[31662,45809,202],{},[31749,45811,640],{},[31665,45813,45814],{"encoding":31667},"k=1",[34,45816,45818,45836],{"className":45817,"ariaHidden":758},[31672],[34,45819,45821,45824,45827,45830,45833],{"className":45820},[31676],[34,45822],{"className":45823,"style":31681},[31680],[34,45825,33759],{"className":45826,"style":33774},[31685,31686],[34,45828],{"className":45829,"style":31846},[31845],[34,45831,202],{"className":45832},[31850],[34,45834],{"className":45835,"style":31846},[31845],[34,45837,45839,45842],{"className":45838},[31676],[34,45840],{"className":45841,"style":32617},[31680],[34,45843,640],{"className":45844},[31685]," 时，这种策略被称为",[2627,45847,45848],{},"减治法",[15,45850,45851],{},[2627,45852,45853],{},"2. 适用分治法的特征",[105,45855,45856,45859,45862,45868],{},[108,45857,45858],{},"问题规模缩小到一定程度易于解决。",[108,45860,45861],{},"问题可分解为若干规模较小的相同子问题。",[108,45863,45864,45865,45867],{},"子问题的解可以",[2627,45866,45734],{},"为该问题的解。",[108,45869,45870,45871,45874],{},"分解出的各个子问题",[2627,45872,45873],{},"相互独立","（即子问题之间不包含公共的子问题）。",[15,45876,45877],{},[2627,45878,45879],{},"3. 求解过程（三大步骤）",[105,45881,45882,45888,45894],{},[108,45883,45884,45887],{},[2627,45885,45886],{},"分解","：将原问题分解为若干个规模较小、相互独立、与原问题形式相同的子问题。",[108,45889,45890,45893],{},[2627,45891,45892],{},"求解子问题","：若子问题规模较小而易于解决则直接求解，否则递归地求解各个子问题。",[108,45895,45896,45898],{},[2627,45897,45734],{},"：将各个子问题的解合并为原问题的解。",[32188,45900],{},[253,45902,45904],{"id":45903},"二-经典例题与推导过程精讲","二、 经典例题与推导过程精讲",[32490,45906,45908],{"id":45907},"例题-1快速排序-quick-sort","例题 1：快速排序 (Quick Sort)",[105,45910,45911,45946,46034],{},[108,45912,45913,45916,45917,45945],{},[2627,45914,45915],{},"题意描述","：对含有 ",[34,45918,45920,45933],{"className":45919},[31642],[34,45921,45923],{"className":45922},[31646],[31648,45924,45925],{"xmlns":31650},[31652,45926,45927,45931],{},[31655,45928,45929],{},[31658,45930,31660],{},[31665,45932,31660],{"encoding":31667},[34,45934,45936],{"className":45935,"ariaHidden":758},[31672],[34,45937,45939,45942],{"className":45938},[31676],[34,45940],{"className":45941,"style":32527},[31680],[34,45943,31660],{"className":45944},[31685,31686]," 个元素的序列进行排序。",[108,45947,45948,33183,45951],{},[2627,45949,45950],{},"算法设计与推导",[105,45952,45953,46019,46025],{},[108,45954,45955,45957,45958,45988,45989,46018],{},[2627,45956,45886],{},"：在序列中任取一个元素（通常取第一个）作为基准（pivot）。通过一趟扫描（Partition），将序列分割成两个子序列：左边所有元素 ",[34,45959,45961,45976],{"className":45960},[31642],[34,45962,45964],{"className":45963},[31646],[31648,45965,45966],{"xmlns":31650},[31652,45967,45968,45973],{},[31655,45969,45970],{},[31662,45971,45972],{},"≤",[31665,45974,45975],{"encoding":31667},"\\le",[34,45977,45979],{"className":45978,"ariaHidden":758},[31672],[34,45980,45982,45985],{"className":45981},[31676],[34,45983],{"className":45984,"style":43198},[31680],[34,45986,45972],{"className":45987},[31850]," 基准，右边所有元素 ",[34,45990,45992,46006],{"className":45991},[31642],[34,45993,45995],{"className":45994},[31646],[31648,45996,45997],{"xmlns":31650},[31652,45998,45999,46003],{},[31655,46000,46001],{},[31662,46002,43175],{},[31665,46004,46005],{"encoding":31667},"\\ge",[34,46007,46009],{"className":46008,"ariaHidden":758},[31672],[34,46010,46012,46015],{"className":46011},[31676],[34,46013],{"className":46014,"style":43198},[31680],[34,46016,43175],{"className":46017},[31850]," 基准，基准元素被放置在最终位置。",[108,46020,46021,46024],{},[2627,46022,46023],{},"求解","：对左右两个子序列分别递归地重复上述过程，直至子序列长度为0或1（此时子序列天然有序，直接返回）。",[108,46026,46027,46029,46030,46033],{},[2627,46028,45734],{},"：由于排序是",[2627,46031,46032],{},"就地进行","的（直接在原数组上交换元素），所以合并步骤不需要执行任何额外操作。",[108,46035,46036,46039,46040,46147,46148,347],{},[2627,46037,46038],{},"复杂度分析","：在最好和平均情况下，每次划分能将序列分成长度大致相等的两半，时间复杂度为 ",[34,46041,46043,46075],{"className":46042},[31642],[34,46044,46046],{"className":46045},[31646],[31648,46047,46048],{"xmlns":31650},[31652,46049,46050,46072],{},[31655,46051,46052,46054,46056,46058,46068,46070],{},[31658,46053,37890],{},[31662,46055,456],{"stretchy":702},[31658,46057,31660],{},[31743,46059,46060,46066],{},[31655,46061,46062,46064],{},[31658,46063,885],{},[31662,46065,32682],{},[31749,46067,768],{},[31658,46069,31660],{},[31662,46071,2557],{"stretchy":702},[31665,46073,46074],{"encoding":31667},"O(n \\log_2 n)",[34,46076,46078],{"className":46077,"ariaHidden":758},[31672],[34,46079,46081,46084,46087,46090,46093,46096,46138,46141,46144],{"className":46080},[31676],[34,46082],{"className":46083,"style":31774},[31680],[34,46085,37890],{"className":46086,"style":34452},[31685,31686],[34,46088,456],{"className":46089},[31782],[34,46091,31660],{"className":46092},[31685,31686],[34,46094],{"className":46095,"style":32123},[31845],[34,46097,46099,46104],{"className":46098},[32767],[34,46100,38616,46102],{"className":46101},[32767],[34,46103,31942],{"style":38619},[34,46105,46107],{"className":46106},[31792],[34,46108,46110,46130],{"className":46109},[31796,31797],[34,46111,46113,46127],{"className":46112},[31801],[34,46114,46116],{"className":46115,"style":38632},[31805],[34,46117,46118,46121],{"style":38635},[34,46119],{"className":46120,"style":31814},[31813],[34,46122,46124],{"className":46123},[31818,31819,31820,31821],[34,46125,768],{"className":46126},[31685,31821],[34,46128,31829],{"className":46129},[31828],[34,46131,46133],{"className":46132},[31801],[34,46134,46136],{"className":46135,"style":38654},[31805],[34,46137],{},[34,46139],{"className":46140,"style":32123},[31845],[34,46142,31660],{"className":46143},[31685,31686],[34,46145,2557],{"className":46146},[31690],"。最坏情况下（如原序列已经有序），每次划分极度不平衡，时间复杂度退化为 ",[34,46149,46151,46175],{"className":46150},[31642],[34,46152,46154],{"className":46153},[31646],[31648,46155,46156],{"xmlns":31650},[31652,46157,46158,46172],{},[31655,46159,46160,46162,46164,46170],{},[31658,46161,37890],{},[31662,46163,456],{"stretchy":702},[37624,46165,46166,46168],{},[31658,46167,31660],{},[31749,46169,768],{},[31662,46171,2557],{"stretchy":702},[31665,46173,46174],{"encoding":31667},"O(n^2)",[34,46176,46178],{"className":46177,"ariaHidden":758},[31672],[34,46179,46181,46184,46187,46190,46219],{"className":46180},[31676],[34,46182],{"className":46183,"style":37755},[31680],[34,46185,37890],{"className":46186,"style":34452},[31685,31686],[34,46188,456],{"className":46189},[31782],[34,46191,46193,46196],{"className":46192},[31685],[34,46194,31660],{"className":46195},[31685,31686],[34,46197,46199],{"className":46198},[31792],[34,46200,46202],{"className":46201},[31796],[34,46203,46205],{"className":46204},[31801],[34,46206,46208],{"className":46207,"style":37774},[31805],[34,46209,46210,46213],{"style":37777},[34,46211],{"className":46212,"style":31814},[31813],[34,46214,46216],{"className":46215},[31818,31819,31820,31821],[34,46217,768],{"className":46218},[31685,31821],[34,46220,2557],{"className":46221},[31690],[32490,46223,46225],{"id":46224},"例题-2查找最大和次大元素-最大和最小元素","例题 2：查找最大和次大元素 \u002F 最大和最小元素",[105,46227,46228,46233,46515],{},[108,46229,46230,46232],{},[2627,46231,45915],{},"：在一个给定的无序序列中，找出最大和次大的两个不同元素，或寻找最大和最小元素。",[108,46234,46235,33183,46238],{},[2627,46236,46237],{},"算法设计与推导（最大和次大元素）",[105,46239,46240,46259,46320,46349],{},[108,46241,46242,46245,46246,616,46249,46252,46253,616,46256,347],{},[2627,46243,46244],{},"边界情况","：若只有1个元素，",[31,46247,46248],{},"max1 = a[low]",[31,46250,46251],{},"max2 = -INF","；若有2个元素，",[31,46254,46255],{},"max1 = MAX(a[low], a[high])",[31,46257,46258],{},"max2 = MIN(a[low], a[high])",[108,46260,46261,46264,46265,46308,46309,46312,46313,46316,46317,347],{},[2627,46262,46263],{},"分解与求解","：若元素 ",[34,46266,46268,46284],{"className":46267},[31642],[34,46269,46271],{"className":46270},[31646],[31648,46272,46273],{"xmlns":31650},[31652,46274,46275,46281],{},[31655,46276,46277,46279],{},[31662,46278,29622],{},[31749,46280,768],{},[31665,46282,46283],{"encoding":31667},">2",[34,46285,46287,46299],{"className":46286,"ariaHidden":758},[31672],[34,46288,46290,46293,46296],{"className":46289},[31676],[34,46291],{"className":46292,"style":34812},[31680],[34,46294,29622],{"className":46295},[31850],[34,46297],{"className":46298,"style":31846},[31845],[34,46300,46302,46305],{"className":46301},[31676],[34,46303],{"className":46304,"style":32617},[31680],[34,46306,768],{"className":46307},[31685],"，将数组从中间 ",[31,46310,46311],{},"mid"," 一分为二，递归求出左区间的 ",[31,46314,46315],{},"(lmax1, lmax2)"," 和右区间的 ",[31,46318,46319],{},"(rmax1, rmax2)",[108,46321,46322,46324,46325,46328,46329,46332,46333,46336,46337,46340,46341,46344,46345,46348],{},[2627,46323,45734],{},"：如果 ",[31,46326,46327],{},"lmax1 > rmax1","，那么全局最大值 ",[31,46330,46331],{},"max1 = lmax1","，而次大值 ",[31,46334,46335],{},"max2"," 则是左区间的次大值 ",[31,46338,46339],{},"lmax2"," 和右区间最大值 ",[31,46342,46343],{},"rmax1"," 中的较大者（即 ",[31,46346,46347],{},"MAX(lmax2, rmax1)","）；反之亦然。",[108,46350,46351,46354,46355,46470,46471,347],{},[2627,46352,46353],{},"复杂度","：比较次数的递推式为 ",[34,46356,46358,46398],{"className":46357},[31642],[34,46359,46361],{"className":46360},[31646],[31648,46362,46363],{"xmlns":31650},[31652,46364,46365,46395],{},[31655,46366,46367,46369,46371,46373,46375,46377,46379,46381,46383,46385,46387,46389,46391,46393],{},[31658,46368,37400],{},[31662,46370,456],{"stretchy":702},[31658,46372,31660],{},[31662,46374,2557],{"stretchy":702},[31662,46376,202],{},[31749,46378,768],{},[31658,46380,37400],{},[31662,46382,456],{"stretchy":702},[31658,46384,31660],{},[31658,46386,38080],{"mathvariant":38079},[31749,46388,768],{},[31662,46390,2557],{"stretchy":702},[31662,46392,223],{},[31749,46394,640],{},[31665,46396,46397],{"encoding":31667},"T(n) = 2T(n\u002F2) + 1",[34,46399,46401,46428,46461],{"className":46400,"ariaHidden":758},[31672],[34,46402,46404,46407,46410,46413,46416,46419,46422,46425],{"className":46403},[31676],[34,46405],{"className":46406,"style":31774},[31680],[34,46408,37400],{"className":46409,"style":37422},[31685,31686],[34,46411,456],{"className":46412},[31782],[34,46414,31660],{"className":46415},[31685,31686],[34,46417,2557],{"className":46418},[31690],[34,46420],{"className":46421,"style":31846},[31845],[34,46423,202],{"className":46424},[31850],[34,46426],{"className":46427,"style":31846},[31845],[34,46429,46431,46434,46437,46440,46443,46446,46449,46452,46455,46458],{"className":46430},[31676],[34,46432],{"className":46433,"style":31774},[31680],[34,46435,768],{"className":46436},[31685],[34,46438,37400],{"className":46439,"style":37422},[31685,31686],[34,46441,456],{"className":46442},[31782],[34,46444,31660],{"className":46445},[31685,31686],[34,46447,38149],{"className":46448},[31685],[34,46450,2557],{"className":46451},[31690],[34,46453],{"className":46454,"style":32604},[31845],[34,46456,223],{"className":46457},[32024],[34,46459],{"className":46460,"style":32604},[31845],[34,46462,46464,46467],{"className":46463},[31676],[34,46465],{"className":46466,"style":32617},[31680],[34,46468,640],{"className":46469},[31685],"，时间复杂度为 ",[34,46472,46474,46494],{"className":46473},[31642],[34,46475,46477],{"className":46476},[31646],[31648,46478,46479],{"xmlns":31650},[31652,46480,46481,46491],{},[31655,46482,46483,46485,46487,46489],{},[31658,46484,37890],{},[31662,46486,456],{"stretchy":702},[31658,46488,31660],{},[31662,46490,2557],{"stretchy":702},[31665,46492,46493],{"encoding":31667},"O(n)",[34,46495,46497],{"className":46496,"ariaHidden":758},[31672],[34,46498,46500,46503,46506,46509,46512],{"className":46499},[31676],[34,46501],{"className":46502,"style":31774},[31680],[34,46504,37890],{"className":46505,"style":34452},[31685,31686],[34,46507,456],{"className":46508},[31782],[34,46510,31660],{"className":46511},[31685,31686],[34,46513,2557],{"className":46514},[31690],[108,46516,46517,33183,46520],{},[2627,46518,46519],{},"（最大和最小元素）的推导",[105,46521,46522,46532],{},[108,46523,46524,46525,46528,46529,347],{},"基本逻辑类似，返回左右区间合并的结果：",[31,46526,46527],{},"max = MAX(lmax, rmax)","，",[31,46530,46531],{},"min = MIN(lmin, rmin)",[108,46533,46534,46537,46538,46653,46654,347],{},[2627,46535,46536],{},"复杂度推导","：递推式为 ",[34,46539,46541,46581],{"className":46540},[31642],[34,46542,46544],{"className":46543},[31646],[31648,46545,46546],{"xmlns":31650},[31652,46547,46548,46578],{},[31655,46549,46550,46552,46554,46556,46558,46560,46562,46564,46566,46568,46570,46572,46574,46576],{},[31658,46551,37400],{},[31662,46553,456],{"stretchy":702},[31658,46555,31660],{},[31662,46557,2557],{"stretchy":702},[31662,46559,202],{},[31749,46561,768],{},[31658,46563,37400],{},[31662,46565,456],{"stretchy":702},[31658,46567,31660],{},[31658,46569,38080],{"mathvariant":38079},[31749,46571,768],{},[31662,46573,2557],{"stretchy":702},[31662,46575,223],{},[31749,46577,768],{},[31665,46579,46580],{"encoding":31667},"T(n) = 2T(n\u002F2) + 2",[34,46582,46584,46611,46644],{"className":46583,"ariaHidden":758},[31672],[34,46585,46587,46590,46593,46596,46599,46602,46605,46608],{"className":46586},[31676],[34,46588],{"className":46589,"style":31774},[31680],[34,46591,37400],{"className":46592,"style":37422},[31685,31686],[34,46594,456],{"className":46595},[31782],[34,46597,31660],{"className":46598},[31685,31686],[34,46600,2557],{"className":46601},[31690],[34,46603],{"className":46604,"style":31846},[31845],[34,46606,202],{"className":46607},[31850],[34,46609],{"className":46610,"style":31846},[31845],[34,46612,46614,46617,46620,46623,46626,46629,46632,46635,46638,46641],{"className":46613},[31676],[34,46615],{"className":46616,"style":31774},[31680],[34,46618,768],{"className":46619},[31685],[34,46621,37400],{"className":46622,"style":37422},[31685,31686],[34,46624,456],{"className":46625},[31782],[34,46627,31660],{"className":46628},[31685,31686],[34,46630,38149],{"className":46631},[31685],[34,46633,2557],{"className":46634},[31690],[34,46636],{"className":46637,"style":32604},[31845],[34,46639,223],{"className":46640},[32024],[34,46642],{"className":46643,"style":32604},[31845],[34,46645,46647,46650],{"className":46646},[31676],[34,46648],{"className":46649,"style":32617},[31680],[34,46651,768],{"className":46652},[31685],"，利用展开法推导可得 ",[34,46655,46657,46691],{"className":46656},[31642],[34,46658,46660],{"className":46659},[31646],[31648,46661,46662],{"xmlns":31650},[31652,46663,46664,46688],{},[31655,46665,46666,46668,46670,46672,46674,46676,46682,46684,46686],{},[31658,46667,37400],{},[31662,46669,456],{"stretchy":702},[31658,46671,31660],{},[31662,46673,2557],{"stretchy":702},[31662,46675,202],{},[41453,46677,46678,46680],{},[31749,46679,786],{},[31749,46681,768],{},[31658,46683,31660],{},[31662,46685,32226],{},[31749,46687,768],{},[31665,46689,46690],{"encoding":31667},"T(n) = \\frac{3}{2}n - 2",[34,46692,46694,46721,46810],{"className":46693,"ariaHidden":758},[31672],[34,46695,46697,46700,46703,46706,46709,46712,46715,46718],{"className":46696},[31676],[34,46698],{"className":46699,"style":31774},[31680],[34,46701,37400],{"className":46702,"style":37422},[31685,31686],[34,46704,456],{"className":46705},[31782],[34,46707,31660],{"className":46708},[31685,31686],[34,46710,2557],{"className":46711},[31690],[34,46713],{"className":46714,"style":31846},[31845],[34,46716,202],{"className":46717},[31850],[34,46719],{"className":46720,"style":31846},[31845],[34,46722,46724,46728,46798,46801,46804,46807],{"className":46723},[31676],[34,46725],{"className":46726,"style":46727},[31680],"height:1.1901em;vertical-align:-0.345em;",[34,46729,46731,46734,46795],{"className":46730},[31685],[34,46732],{"className":46733},[31782,41488],[34,46735,46737],{"className":46736},[41453],[34,46738,46740,46787],{"className":46739},[31796,31797],[34,46741,46743,46784],{"className":46742},[31801],[34,46744,46747,46761,46769],{"className":46745,"style":46746},[31805],"height:0.8451em;",[34,46748,46749,46752],{"style":41504},[34,46750],{"className":46751,"style":41508},[31813],[34,46753,46755],{"className":46754},[31818,31819,31820,31821],[34,46756,46758],{"className":46757},[31685,31821],[34,46759,768],{"className":46760},[31685,31821],[34,46762,46763,46766],{"style":41520},[34,46764],{"className":46765,"style":41508},[31813],[34,46767],{"className":46768,"style":41528},[41527],[34,46770,46772,46775],{"style":46771},"top:-3.394em;",[34,46773],{"className":46774,"style":41508},[31813],[34,46776,46778],{"className":46777},[31818,31819,31820,31821],[34,46779,46781],{"className":46780},[31685,31821],[34,46782,786],{"className":46783},[31685,31821],[34,46785,31829],{"className":46786},[31828],[34,46788,46790],{"className":46789},[31801],[34,46791,46793],{"className":46792,"style":41619},[31805],[34,46794],{},[34,46796],{"className":46797},[31690,41488],[34,46799,31660],{"className":46800},[31685,31686],[34,46802],{"className":46803,"style":32604},[31845],[34,46805,32226],{"className":46806},[32024],[34,46808],{"className":46809,"style":32604},[31845],[34,46811,46813,46816],{"className":46812},[31676],[34,46814],{"className":46815,"style":32617},[31680],[34,46817,768],{"className":46818},[31685],[32490,46820,46822],{"id":46821},"例题-3折半查找-binary-search","例题 3：折半查找 (Binary Search)",[105,46824,46825,46858,47365],{},[108,46826,46827,46829,46830,347],{},[2627,46828,45915],{},"：在一个有序序列中查找给定的关键字 ",[34,46831,46833,46846],{"className":46832},[31642],[34,46834,46836],{"className":46835},[31646],[31648,46837,46838],{"xmlns":31650},[31652,46839,46840,46844],{},[31655,46841,46842],{},[31658,46843,33759],{},[31665,46845,33759],{"encoding":31667},[34,46847,46849],{"className":46848,"ariaHidden":758},[31672],[34,46850,46852,46855],{"className":46851},[31676],[34,46853],{"className":46854,"style":31681},[31680],[34,46856,33759],{"className":46857,"style":33774},[31685,31686],[108,46859,46860,33183,46862],{},[2627,46861,45950],{},[105,46863,46864,46870,46949,47158],{},[108,46865,46866,46867,347],{},"确定区间中点 ",[31,46868,46869],{},"mid = (low + high) \u002F 2",[108,46871,43439,46872,46948],{},[34,46873,46875,46906],{"className":46874},[31642],[34,46876,46878],{"className":46877},[31646],[31648,46879,46880],{"xmlns":31650},[31652,46881,46882,46903],{},[31655,46883,46884,46886,46888,46890,46892,46894,46896,46898,46901],{},[31658,46885,33759],{},[31662,46887,202],{},[31662,46889,202],{},[31658,46891,111],{},[31662,46893,15972],{"stretchy":702},[31658,46895,31759],{},[31658,46897,31955],{},[31658,46899,46900],{},"d",[31662,46902,32712],{"stretchy":702},[31665,46904,46905],{"encoding":31667},"k == a[mid]",[34,46907,46909,46927],{"className":46908,"ariaHidden":758},[31672],[34,46910,46912,46915,46918,46921,46924],{"className":46911},[31676],[34,46913],{"className":46914,"style":31681},[31680],[34,46916,33759],{"className":46917,"style":33774},[31685,31686],[34,46919],{"className":46920,"style":31846},[31845],[34,46922,33376],{"className":46923},[31850],[34,46925],{"className":46926,"style":31846},[31845],[34,46928,46930,46933,46936,46939,46942,46945],{"className":46929},[31676],[34,46931],{"className":46932,"style":31774},[31680],[34,46934,111],{"className":46935},[31685,31686],[34,46937,15972],{"className":46938},[31782],[34,46940,31658],{"className":46941},[31685,31686],[34,46943,46900],{"className":46944},[31685,31686],[34,46946,32712],{"className":46947},[31690],"，查找成功，返回下标。",[108,46950,43439,46951,47025,47026,47054,47055,47157],{},[34,46952,46954,46982],{"className":46953},[31642],[34,46955,46957],{"className":46956},[31646],[31648,46958,46959],{"xmlns":31650},[31652,46960,46961,46979],{},[31655,46962,46963,46965,46967,46969,46971,46973,46975,46977],{},[31658,46964,33759],{},[31662,46966,932],{},[31658,46968,111],{},[31662,46970,15972],{"stretchy":702},[31658,46972,31759],{},[31658,46974,31955],{},[31658,46976,46900],{},[31662,46978,32712],{"stretchy":702},[31665,46980,46981],{"encoding":31667},"k \u003C a[mid]",[34,46983,46985,47004],{"className":46984,"ariaHidden":758},[31672],[34,46986,46988,46992,46995,46998,47001],{"className":46987},[31676],[34,46989],{"className":46990,"style":46991},[31680],"height:0.7335em;vertical-align:-0.0391em;",[34,46993,33759],{"className":46994,"style":33774},[31685,31686],[34,46996],{"className":46997,"style":31846},[31845],[34,46999,932],{"className":47000},[31850],[34,47002],{"className":47003,"style":31846},[31845],[34,47005,47007,47010,47013,47016,47019,47022],{"className":47006},[31676],[34,47008],{"className":47009,"style":31774},[31680],[34,47011,111],{"className":47012},[31685,31686],[34,47014,15972],{"className":47015},[31782],[34,47017,31658],{"className":47018},[31685,31686],[34,47020,46900],{"className":47021},[31685,31686],[34,47023,32712],{"className":47024},[31690],"，说明 ",[34,47027,47029,47042],{"className":47028},[31642],[34,47030,47032],{"className":47031},[31646],[31648,47033,47034],{"xmlns":31650},[31652,47035,47036,47040],{},[31655,47037,47038],{},[31658,47039,33759],{},[31665,47041,33759],{"encoding":31667},[34,47043,47045],{"className":47044,"ariaHidden":758},[31672],[34,47046,47048,47051],{"className":47047},[31676],[34,47049],{"className":47050,"style":31681},[31680],[34,47052,33759],{"className":47053,"style":33774},[31685,31686]," 必定位于左子表，新的查找区间更新为 ",[34,47056,47058,47095],{"className":47057},[31642],[34,47059,47061],{"className":47060},[31646],[31648,47062,47063],{"xmlns":31650},[31652,47064,47065,47092],{},[31655,47066,47067,47069,47071,47073,47075,47078,47080,47082,47084,47086,47088,47090],{},[31658,47068,111],{},[31662,47070,15972],{"stretchy":702},[31658,47072,43379],{},[31658,47074,43383],{},[31658,47076,47077],{},"w",[31662,47079,33934],{},[31658,47081,31759],{},[31658,47083,31955],{},[31658,47085,46900],{},[31662,47087,32226],{},[31749,47089,640],{},[31662,47091,32712],{"stretchy":702},[31665,47093,47094],{"encoding":31667},"a[low \\dots mid-1]",[34,47096,47098,47145],{"className":47097,"ariaHidden":758},[31672],[34,47099,47101,47104,47107,47110,47114,47117,47121,47124,47127,47130,47133,47136,47139,47142],{"className":47100},[31676],[34,47102],{"className":47103,"style":31774},[31680],[34,47105,111],{"className":47106},[31685,31686],[34,47108,15972],{"className":47109},[31782],[34,47111,43379],{"className":47112,"style":47113},[31685,31686],"margin-right:0.0197em;",[34,47115,43383],{"className":47116},[31685,31686],[34,47118,47077],{"className":47119,"style":47120},[31685,31686],"margin-right:0.0269em;",[34,47122],{"className":47123,"style":32123},[31845],[34,47125,33934],{"className":47126},[34006],[34,47128],{"className":47129,"style":32123},[31845],[34,47131,31658],{"className":47132},[31685,31686],[34,47134,46900],{"className":47135},[31685,31686],[34,47137],{"className":47138,"style":32604},[31845],[34,47140,32226],{"className":47141},[32024],[34,47143],{"className":47144,"style":32604},[31845],[34,47146,47148,47151,47154],{"className":47147},[31676],[34,47149],{"className":47150,"style":31774},[31680],[34,47152,640],{"className":47153},[31685],[34,47155,32712],{"className":47156},[31690],"，递归查找。",[108,47159,43439,47160,47025,47233,47261,47262,47157],{},[34,47161,47163,47191],{"className":47162},[31642],[34,47164,47166],{"className":47165},[31646],[31648,47167,47168],{"xmlns":31650},[31652,47169,47170,47188],{},[31655,47171,47172,47174,47176,47178,47180,47182,47184,47186],{},[31658,47173,33759],{},[31662,47175,29622],{},[31658,47177,111],{},[31662,47179,15972],{"stretchy":702},[31658,47181,31759],{},[31658,47183,31955],{},[31658,47185,46900],{},[31662,47187,32712],{"stretchy":702},[31665,47189,47190],{"encoding":31667},"k > a[mid]",[34,47192,47194,47212],{"className":47193,"ariaHidden":758},[31672],[34,47195,47197,47200,47203,47206,47209],{"className":47196},[31676],[34,47198],{"className":47199,"style":46991},[31680],[34,47201,33759],{"className":47202,"style":33774},[31685,31686],[34,47204],{"className":47205,"style":31846},[31845],[34,47207,29622],{"className":47208},[31850],[34,47210],{"className":47211,"style":31846},[31845],[34,47213,47215,47218,47221,47224,47227,47230],{"className":47214},[31676],[34,47216],{"className":47217,"style":31774},[31680],[34,47219,111],{"className":47220},[31685,31686],[34,47222,15972],{"className":47223},[31782],[34,47225,31658],{"className":47226},[31685,31686],[34,47228,46900],{"className":47229},[31685,31686],[34,47231,32712],{"className":47232},[31690],[34,47234,47236,47249],{"className":47235},[31642],[34,47237,47239],{"className":47238},[31646],[31648,47240,47241],{"xmlns":31650},[31652,47242,47243,47247],{},[31655,47244,47245],{},[31658,47246,33759],{},[31665,47248,33759],{"encoding":31667},[34,47250,47252],{"className":47251,"ariaHidden":758},[31672],[34,47253,47255,47258],{"className":47254},[31676],[34,47256],{"className":47257,"style":31681},[31680],[34,47259,33759],{"className":47260,"style":33774},[31685,31686]," 必定位于右子表，区间更新为 ",[34,47263,47265,47304],{"className":47264},[31642],[34,47266,47268],{"className":47267},[31646],[31648,47269,47270],{"xmlns":31650},[31652,47271,47272,47301],{},[31655,47273,47274,47276,47278,47280,47282,47284,47286,47288,47290,47293,47295,47297,47299],{},[31658,47275,111],{},[31662,47277,15972],{"stretchy":702},[31658,47279,31759],{},[31658,47281,31955],{},[31658,47283,46900],{},[31662,47285,223],{},[31749,47287,640],{},[31662,47289,33934],{},[31658,47291,47292],{},"h",[31658,47294,31955],{},[31658,47296,31942],{},[31658,47298,47292],{},[31662,47300,32712],{"stretchy":702},[31665,47302,47303],{"encoding":31667},"a[mid+1 \\dots high]",[34,47305,47307,47334],{"className":47306,"ariaHidden":758},[31672],[34,47308,47310,47313,47316,47319,47322,47325,47328,47331],{"className":47309},[31676],[34,47311],{"className":47312,"style":31774},[31680],[34,47314,111],{"className":47315},[31685,31686],[34,47317,15972],{"className":47318},[31782],[34,47320,31658],{"className":47321},[31685,31686],[34,47323,46900],{"className":47324},[31685,31686],[34,47326],{"className":47327,"style":32604},[31845],[34,47329,223],{"className":47330},[32024],[34,47332],{"className":47333,"style":32604},[31845],[34,47335,47337,47340,47343,47346,47349,47352,47356,47359,47362],{"className":47336},[31676],[34,47338],{"className":47339,"style":31774},[31680],[34,47341,640],{"className":47342},[31685],[34,47344],{"className":47345,"style":32123},[31845],[34,47347,33934],{"className":47348},[34006],[34,47350],{"className":47351,"style":32123},[31845],[34,47353,47355],{"className":47354},[31685,31686],"hi",[34,47357,31942],{"className":47358,"style":32062},[31685,31686],[34,47360,47292],{"className":47361},[31685,31686],[34,47363,32712],{"className":47364},[31690],[108,47366,47367,47369,47370,47481,47482,347],{},[2627,47368,46038],{},"：最坏情况下元素比较次数 ",[34,47371,47373,47412],{"className":47372},[31642],[34,47374,47376],{"className":47375},[31646],[31648,47377,47378],{"xmlns":31650},[31652,47379,47380,47409],{},[31655,47381,47382,47385,47387,47389,47391,47393,47395,47397,47399,47401,47403,47405,47407],{},[31658,47383,47384],{},"C",[31662,47386,456],{"stretchy":702},[31658,47388,31660],{},[31662,47390,2557],{"stretchy":702},[31662,47392,45972],{},[31749,47394,640],{},[31662,47396,223],{},[31658,47398,47384],{},[31662,47400,456],{"stretchy":702},[31658,47402,31660],{},[31658,47404,38080],{"mathvariant":38079},[31749,47406,768],{},[31662,47408,2557],{"stretchy":702},[31665,47410,47411],{"encoding":31667},"C(n) \\le 1 + C(n\u002F2)",[34,47413,47415,47442,47460],{"className":47414,"ariaHidden":758},[31672],[34,47416,47418,47421,47424,47427,47430,47433,47436,47439],{"className":47417},[31676],[34,47419],{"className":47420,"style":31774},[31680],[34,47422,47384],{"className":47423,"style":38706},[31685,31686],[34,47425,456],{"className":47426},[31782],[34,47428,31660],{"className":47429},[31685,31686],[34,47431,2557],{"className":47432},[31690],[34,47434],{"className":47435,"style":31846},[31845],[34,47437,45972],{"className":47438},[31850],[34,47440],{"className":47441,"style":31846},[31845],[34,47443,47445,47448,47451,47454,47457],{"className":47444},[31676],[34,47446],{"className":47447,"style":35942},[31680],[34,47449,640],{"className":47450},[31685],[34,47452],{"className":47453,"style":32604},[31845],[34,47455,223],{"className":47456},[32024],[34,47458],{"className":47459,"style":32604},[31845],[34,47461,47463,47466,47469,47472,47475,47478],{"className":47462},[31676],[34,47464],{"className":47465,"style":31774},[31680],[34,47467,47384],{"className":47468,"style":38706},[31685,31686],[34,47470,456],{"className":47471},[31782],[34,47473,31660],{"className":47474},[31685,31686],[34,47476,38149],{"className":47477},[31685],[34,47479,2557],{"className":47480},[31690],"，解得算法的时间复杂度为 ",[34,47483,47485,47515],{"className":47484},[31642],[34,47486,47488],{"className":47487},[31646],[31648,47489,47490],{"xmlns":31650},[31652,47491,47492,47512],{},[31655,47493,47494,47496,47498,47508,47510],{},[31658,47495,37890],{},[31662,47497,456],{"stretchy":702},[31743,47499,47500,47506],{},[31655,47501,47502,47504],{},[31658,47503,885],{},[31662,47505,32682],{},[31749,47507,768],{},[31658,47509,31660],{},[31662,47511,2557],{"stretchy":702},[31665,47513,47514],{"encoding":31667},"O(\\log_2 n)",[34,47516,47518],{"className":47517,"ariaHidden":758},[31672],[34,47519,47521,47524,47527,47530,47572,47575,47578],{"className":47520},[31676],[34,47522],{"className":47523,"style":31774},[31680],[34,47525,37890],{"className":47526,"style":34452},[31685,31686],[34,47528,456],{"className":47529},[31782],[34,47531,47533,47538],{"className":47532},[32767],[34,47534,38616,47536],{"className":47535},[32767],[34,47537,31942],{"style":38619},[34,47539,47541],{"className":47540},[31792],[34,47542,47544,47564],{"className":47543},[31796,31797],[34,47545,47547,47561],{"className":47546},[31801],[34,47548,47550],{"className":47549,"style":38632},[31805],[34,47551,47552,47555],{"style":38635},[34,47553],{"className":47554,"style":31814},[31813],[34,47556,47558],{"className":47557},[31818,31819,31820,31821],[34,47559,768],{"className":47560},[31685,31821],[34,47562,31829],{"className":47563},[31828],[34,47565,47567],{"className":47566},[31801],[34,47568,47570],{"className":47569,"style":38654},[31805],[34,47571],{},[34,47573],{"className":47574,"style":32123},[31845],[34,47576,31660],{"className":47577},[31685,31686],[34,47579,2557],{"className":47580},[31690],[32490,47582,47584],{"id":47583},"例题-4寻找一个序列中第-k-小元素","例题 4：寻找一个序列中第 k 小元素",[105,47586,47587,47735,48417],{},[108,47588,47589,47591,47592,47620,47621,47649,47650,47734],{},[2627,47590,45915],{},"：给定 ",[34,47593,47595,47608],{"className":47594},[31642],[34,47596,47598],{"className":47597},[31646],[31648,47599,47600],{"xmlns":31650},[31652,47601,47602,47606],{},[31655,47603,47604],{},[31658,47605,31660],{},[31665,47607,31660],{"encoding":31667},[34,47609,47611],{"className":47610,"ariaHidden":758},[31672],[34,47612,47614,47617],{"className":47613},[31676],[34,47615],{"className":47616,"style":32527},[31680],[34,47618,31660],{"className":47619},[31685,31686]," 个元素的无序序列，求其中第 ",[34,47622,47624,47637],{"className":47623},[31642],[34,47625,47627],{"className":47626},[31646],[31648,47628,47629],{"xmlns":31650},[31652,47630,47631,47635],{},[31655,47632,47633],{},[31658,47634,33759],{},[31665,47636,33759],{"encoding":31667},[34,47638,47640],{"className":47639,"ariaHidden":758},[31672],[34,47641,47643,47646],{"className":47642},[31676],[34,47644],{"className":47645,"style":31681},[31680],[34,47647,33759],{"className":47648,"style":33774},[31685,31686]," ",[34,47651,47653,47679],{"className":47652},[31642],[34,47654,47656],{"className":47655},[31646],[31648,47657,47658],{"xmlns":31650},[31652,47659,47660,47676],{},[31655,47661,47662,47664,47666,47668,47670,47672,47674],{},[31662,47663,456],{"stretchy":702},[31749,47665,640],{},[31662,47667,45972],{},[31658,47669,33759],{},[31662,47671,45972],{},[31658,47673,31660],{},[31662,47675,2557],{"stretchy":702},[31665,47677,47678],{"encoding":31667},"(1 \\le k \\le n)",[34,47680,47682,47703,47722],{"className":47681,"ariaHidden":758},[31672],[34,47683,47685,47688,47691,47694,47697,47700],{"className":47684},[31676],[34,47686],{"className":47687,"style":31774},[31680],[34,47689,456],{"className":47690},[31782],[34,47692,640],{"className":47693},[31685],[34,47695],{"className":47696,"style":31846},[31845],[34,47698,45972],{"className":47699},[31850],[34,47701],{"className":47702,"style":31846},[31845],[34,47704,47706,47710,47713,47716,47719],{"className":47705},[31676],[34,47707],{"className":47708,"style":47709},[31680],"height:0.8304em;vertical-align:-0.136em;",[34,47711,33759],{"className":47712,"style":33774},[31685,31686],[34,47714],{"className":47715,"style":31846},[31845],[34,47717,45972],{"className":47718},[31850],[34,47720],{"className":47721,"style":31846},[31845],[34,47723,47725,47728,47731],{"className":47724},[31676],[34,47726],{"className":47727,"style":31774},[31680],[34,47729,31660],{"className":47730},[31685,31686],[34,47732,2557],{"className":47733},[31690]," 小的元素。",[108,47736,47737,33183,47739],{},[2627,47738,45950],{},[105,47740,47741,47777,47885,48027,48206],{},[108,47742,47743,47744,47747,47748,47776],{},"借用快速排序的 ",[31,47745,47746],{},"Partition"," 划分思想，以第一个元素为基准，将其放入最终下标 ",[34,47749,47751,47764],{"className":47750},[31642],[34,47752,47754],{"className":47753},[31646],[31648,47755,47756],{"xmlns":31650},[31652,47757,47758,47762],{},[31655,47759,47760],{},[31658,47761,31955],{},[31665,47763,31955],{"encoding":31667},[34,47765,47767],{"className":47766,"ariaHidden":758},[31672],[34,47768,47770,47773],{"className":47769},[31676],[34,47771],{"className":47772,"style":33104},[31680],[34,47774,31955],{"className":47775},[31685,31686]," 处。",[108,47778,47779,47780,347],{},"计算基准元素是当前区间的第几个：",[34,47781,47783,47813],{"className":47782},[31642],[34,47784,47786],{"className":47785},[31646],[31648,47787,47788],{"xmlns":31650},[31652,47789,47790,47810],{},[31655,47791,47792,47794,47796,47798,47800,47802,47804,47806,47808],{},[31658,47793,43379],{},[31658,47795,33244],{},[31658,47797,31660],{},[31662,47799,202],{},[31658,47801,31955],{},[31662,47803,32226],{},[31658,47805,31747],{},[31662,47807,223],{},[31749,47809,640],{},[31665,47811,47812],{"encoding":31667},"len = i - s + 1",[34,47814,47816,47840,47858,47876],{"className":47815,"ariaHidden":758},[31672],[34,47817,47819,47822,47825,47828,47831,47834,47837],{"className":47818},[31676],[34,47820],{"className":47821,"style":31681},[31680],[34,47823,43379],{"className":47824,"style":47113},[31685,31686],[34,47826,33244],{"className":47827},[31685,31686],[34,47829,31660],{"className":47830},[31685,31686],[34,47832],{"className":47833,"style":31846},[31845],[34,47835,202],{"className":47836},[31850],[34,47838],{"className":47839,"style":31846},[31845],[34,47841,47843,47846,47849,47852,47855],{"className":47842},[31676],[34,47844],{"className":47845,"style":37322},[31680],[34,47847,31955],{"className":47848},[31685,31686],[34,47850],{"className":47851,"style":32604},[31845],[34,47853,32226],{"className":47854},[32024],[34,47856],{"className":47857,"style":32604},[31845],[34,47859,47861,47864,47867,47870,47873],{"className":47860},[31676],[34,47862],{"className":47863,"style":32597},[31680],[34,47865,31747],{"className":47866},[31685,31686],[34,47868],{"className":47869,"style":32604},[31845],[34,47871,223],{"className":47872},[32024],[34,47874],{"className":47875,"style":32604},[31845],[34,47877,47879,47882],{"className":47878},[31676],[34,47880],{"className":47881,"style":32617},[31680],[34,47883,640],{"className":47884},[31685],[108,47886,47887,47890,47891,47025,47954,47997,47998,48026],{},[2627,47888,47889],{},"情况1","：若 ",[34,47892,47894,47918],{"className":47893},[31642],[34,47895,47897],{"className":47896},[31646],[31648,47898,47899],{"xmlns":31650},[31652,47900,47901,47915],{},[31655,47902,47903,47905,47907,47909,47911,47913],{},[31658,47904,33759],{},[31662,47906,202],{},[31662,47908,202],{},[31658,47910,43379],{},[31658,47912,33244],{},[31658,47914,31660],{},[31665,47916,47917],{"encoding":31667},"k == len",[34,47919,47921,47939],{"className":47920,"ariaHidden":758},[31672],[34,47922,47924,47927,47930,47933,47936],{"className":47923},[31676],[34,47925],{"className":47926,"style":31681},[31680],[34,47928,33759],{"className":47929,"style":33774},[31685,31686],[34,47931],{"className":47932,"style":31846},[31845],[34,47934,33376],{"className":47935},[31850],[34,47937],{"className":47938,"style":31846},[31845],[34,47940,47942,47945,47948,47951],{"className":47941},[31676],[34,47943],{"className":47944,"style":31681},[31680],[34,47946,43379],{"className":47947,"style":47113},[31685,31686],[34,47949,33244],{"className":47950},[31685,31686],[34,47952,31660],{"className":47953},[31685,31686],[34,47955,47957,47976],{"className":47956},[31642],[34,47958,47960],{"className":47959},[31646],[31648,47961,47962],{"xmlns":31650},[31652,47963,47964,47974],{},[31655,47965,47966,47968,47970,47972],{},[31658,47967,111],{},[31662,47969,15972],{"stretchy":702},[31658,47971,31955],{},[31662,47973,32712],{"stretchy":702},[31665,47975,33150],{"encoding":31667},[34,47977,47979],{"className":47978,"ariaHidden":758},[31672],[34,47980,47982,47985,47988,47991,47994],{"className":47981},[31676],[34,47983],{"className":47984,"style":31774},[31680],[34,47986,111],{"className":47987},[31685,31686],[34,47989,15972],{"className":47990},[31782],[34,47992,31955],{"className":47993},[31685,31686],[34,47995,32712],{"className":47996},[31690]," 恰好就是第 ",[34,47999,48001,48014],{"className":48000},[31642],[34,48002,48004],{"className":48003},[31646],[31648,48005,48006],{"xmlns":31650},[31652,48007,48008,48012],{},[31655,48009,48010],{},[31658,48011,33759],{},[31665,48013,33759],{"encoding":31667},[34,48015,48017],{"className":48016,"ariaHidden":758},[31672],[34,48018,48020,48023],{"className":48019},[31676],[34,48021],{"className":48022,"style":31681},[31680],[34,48024,33759],{"className":48025,"style":33774},[31685,31686]," 小的元素，直接返回。",[108,48028,48029,47890,48032,48093,48094,48176,48177,48205],{},[2627,48030,48031],{},"情况2",[34,48033,48035,48057],{"className":48034},[31642],[34,48036,48038],{"className":48037},[31646],[31648,48039,48040],{"xmlns":31650},[31652,48041,48042,48054],{},[31655,48043,48044,48046,48048,48050,48052],{},[31658,48045,33759],{},[31662,48047,932],{},[31658,48049,43379],{},[31658,48051,33244],{},[31658,48053,31660],{},[31665,48055,48056],{"encoding":31667},"k \u003C len",[34,48058,48060,48078],{"className":48059,"ariaHidden":758},[31672],[34,48061,48063,48066,48069,48072,48075],{"className":48062},[31676],[34,48064],{"className":48065,"style":46991},[31680],[34,48067,33759],{"className":48068,"style":33774},[31685,31686],[34,48070],{"className":48071,"style":31846},[31845],[34,48073,932],{"className":48074},[31850],[34,48076],{"className":48077,"style":31846},[31845],[34,48079,48081,48084,48087,48090],{"className":48080},[31676],[34,48082],{"className":48083,"style":31681},[31680],[34,48085,43379],{"className":48086,"style":47113},[31685,31686],[34,48088,33244],{"className":48089},[31685,31686],[34,48091,31660],{"className":48092},[31685,31686],"，说明目标元素在左半部分，递归在 ",[34,48095,48097,48125],{"className":48096},[31642],[34,48098,48100],{"className":48099},[31646],[31648,48101,48102],{"xmlns":31650},[31652,48103,48104,48122],{},[31655,48105,48106,48108,48110,48112,48114,48116,48118,48120],{},[31658,48107,111],{},[31662,48109,15972],{"stretchy":702},[31658,48111,31747],{},[31662,48113,33934],{},[31658,48115,31955],{},[31662,48117,32226],{},[31749,48119,640],{},[31662,48121,32712],{"stretchy":702},[31665,48123,48124],{"encoding":31667},"a[s \\dots i-1]",[34,48126,48128,48164],{"className":48127,"ariaHidden":758},[31672],[34,48129,48131,48134,48137,48140,48143,48146,48149,48152,48155,48158,48161],{"className":48130},[31676],[34,48132],{"className":48133,"style":31774},[31680],[34,48135,111],{"className":48136},[31685,31686],[34,48138,15972],{"className":48139},[31782],[34,48141,31747],{"className":48142},[31685,31686],[34,48144],{"className":48145,"style":32123},[31845],[34,48147,33934],{"className":48148},[34006],[34,48150],{"className":48151,"style":32123},[31845],[34,48153,31955],{"className":48154},[31685,31686],[34,48156],{"className":48157,"style":32604},[31845],[34,48159,32226],{"className":48160},[32024],[34,48162],{"className":48163,"style":32604},[31845],[34,48165,48167,48170,48173],{"className":48166},[31676],[34,48168],{"className":48169,"style":31774},[31680],[34,48171,640],{"className":48172},[31685],[34,48174,32712],{"className":48175},[31690]," 中找第 ",[34,48178,48180,48193],{"className":48179},[31642],[34,48181,48183],{"className":48182},[31646],[31648,48184,48185],{"xmlns":31650},[31652,48186,48187,48191],{},[31655,48188,48189],{},[31658,48190,33759],{},[31665,48192,33759],{"encoding":31667},[34,48194,48196],{"className":48195,"ariaHidden":758},[31672],[34,48197,48199,48202],{"className":48198},[31676],[34,48200],{"className":48201,"style":31681},[31680],[34,48203,33759],{"className":48204,"style":33774},[31685,31686]," 小元素。",[108,48207,48208,47890,48211,48272,48273,48355,48356,47734],{},[2627,48209,48210],{},"情况3",[34,48212,48214,48236],{"className":48213},[31642],[34,48215,48217],{"className":48216},[31646],[31648,48218,48219],{"xmlns":31650},[31652,48220,48221,48233],{},[31655,48222,48223,48225,48227,48229,48231],{},[31658,48224,33759],{},[31662,48226,29622],{},[31658,48228,43379],{},[31658,48230,33244],{},[31658,48232,31660],{},[31665,48234,48235],{"encoding":31667},"k > len",[34,48237,48239,48257],{"className":48238,"ariaHidden":758},[31672],[34,48240,48242,48245,48248,48251,48254],{"className":48241},[31676],[34,48243],{"className":48244,"style":46991},[31680],[34,48246,33759],{"className":48247,"style":33774},[31685,31686],[34,48249],{"className":48250,"style":31846},[31845],[34,48252,29622],{"className":48253},[31850],[34,48255],{"className":48256,"style":31846},[31845],[34,48258,48260,48263,48266,48269],{"className":48259},[31676],[34,48261],{"className":48262,"style":31681},[31680],[34,48264,43379],{"className":48265,"style":47113},[31685,31686],[34,48267,33244],{"className":48268},[31685,31686],[34,48270,31660],{"className":48271},[31685,31686],"，说明目标在右半部分，递归在 ",[34,48274,48276,48304],{"className":48275},[31642],[34,48277,48279],{"className":48278},[31646],[31648,48280,48281],{"xmlns":31650},[31652,48282,48283,48301],{},[31655,48284,48285,48287,48289,48291,48293,48295,48297,48299],{},[31658,48286,111],{},[31662,48288,15972],{"stretchy":702},[31658,48290,31955],{},[31662,48292,223],{},[31749,48294,640],{},[31662,48296,33934],{},[31658,48298,33249],{},[31662,48300,32712],{"stretchy":702},[31665,48302,48303],{"encoding":31667},"a[i+1 \\dots t]",[34,48305,48307,48331],{"className":48306,"ariaHidden":758},[31672],[34,48308,48310,48313,48316,48319,48322,48325,48328],{"className":48309},[31676],[34,48311],{"className":48312,"style":31774},[31680],[34,48314,111],{"className":48315},[31685,31686],[34,48317,15972],{"className":48318},[31782],[34,48320,31955],{"className":48321},[31685,31686],[34,48323],{"className":48324,"style":32604},[31845],[34,48326,223],{"className":48327},[32024],[34,48329],{"className":48330,"style":32604},[31845],[34,48332,48334,48337,48340,48343,48346,48349,48352],{"className":48333},[31676],[34,48335],{"className":48336,"style":31774},[31680],[34,48338,640],{"className":48339},[31685],[34,48341],{"className":48342,"style":32123},[31845],[34,48344,33934],{"className":48345},[34006],[34,48347],{"className":48348,"style":32123},[31845],[34,48350,33249],{"className":48351},[31685,31686],[34,48353,32712],{"className":48354},[31690]," 中寻找第 ",[34,48357,48359,48381],{"className":48358},[31642],[34,48360,48362],{"className":48361},[31646],[31648,48363,48364],{"xmlns":31650},[31652,48365,48366,48378],{},[31655,48367,48368,48370,48372,48374,48376],{},[31658,48369,33759],{},[31662,48371,32226],{},[31658,48373,43379],{},[31658,48375,33244],{},[31658,48377,31660],{},[31665,48379,48380],{"encoding":31667},"k - len",[34,48382,48384,48402],{"className":48383,"ariaHidden":758},[31672],[34,48385,48387,48390,48393,48396,48399],{"className":48386},[31676],[34,48388],{"className":48389,"style":34299},[31680],[34,48391,33759],{"className":48392,"style":33774},[31685,31686],[34,48394],{"className":48395,"style":32604},[31845],[34,48397,32226],{"className":48398},[32024],[34,48400],{"className":48401,"style":32604},[31845],[34,48403,48405,48408,48411,48414],{"className":48404},[31676],[34,48406],{"className":48407,"style":31681},[31680],[34,48409,43379],{"className":48410,"style":47113},[31685,31686],[34,48412,33244],{"className":48413},[31685,31686],[34,48415,31660],{"className":48416},[31685,31686],[108,48418,48419,48421,48422,48547,48548,48591,48592,347],{},[2627,48420,46038],{},"：递推方程 ",[34,48423,48425,48469],{"className":48424},[31642],[34,48426,48428],{"className":48427},[31646],[31648,48429,48430],{"xmlns":31650},[31652,48431,48432,48466],{},[31655,48433,48434,48436,48438,48440,48442,48444,48446,48448,48450,48452,48454,48456,48458,48460,48462,48464],{},[31658,48435,37400],{},[31662,48437,456],{"stretchy":702},[31658,48439,31660],{},[31662,48441,2557],{"stretchy":702},[31662,48443,202],{},[31658,48445,37400],{},[31662,48447,456],{"stretchy":702},[31658,48449,31660],{},[31658,48451,38080],{"mathvariant":38079},[31749,48453,768],{},[31662,48455,2557],{"stretchy":702},[31662,48457,223],{},[31658,48459,37890],{},[31662,48461,456],{"stretchy":702},[31658,48463,31660],{},[31662,48465,2557],{"stretchy":702},[31665,48467,48468],{"encoding":31667},"T(n) = T(n\u002F2) + O(n)",[34,48470,48472,48499,48529],{"className":48471,"ariaHidden":758},[31672],[34,48473,48475,48478,48481,48484,48487,48490,48493,48496],{"className":48474},[31676],[34,48476],{"className":48477,"style":31774},[31680],[34,48479,37400],{"className":48480,"style":37422},[31685,31686],[34,48482,456],{"className":48483},[31782],[34,48485,31660],{"className":48486},[31685,31686],[34,48488,2557],{"className":48489},[31690],[34,48491],{"className":48492,"style":31846},[31845],[34,48494,202],{"className":48495},[31850],[34,48497],{"className":48498,"style":31846},[31845],[34,48500,48502,48505,48508,48511,48514,48517,48520,48523,48526],{"className":48501},[31676],[34,48503],{"className":48504,"style":31774},[31680],[34,48506,37400],{"className":48507,"style":37422},[31685,31686],[34,48509,456],{"className":48510},[31782],[34,48512,31660],{"className":48513},[31685,31686],[34,48515,38149],{"className":48516},[31685],[34,48518,2557],{"className":48519},[31690],[34,48521],{"className":48522,"style":32604},[31845],[34,48524,223],{"className":48525},[32024],[34,48527],{"className":48528,"style":32604},[31845],[34,48530,48532,48535,48538,48541,48544],{"className":48531},[31676],[34,48533],{"className":48534,"style":31774},[31680],[34,48536,37890],{"className":48537,"style":34452},[31685,31686],[34,48539,456],{"className":48540},[31782],[34,48542,31660],{"className":48543},[31685,31686],[34,48545,2557],{"className":48546},[31690],"。最好和平均情况（每次划分比较均匀）下，时间复杂度为 ",[34,48549,48551,48570],{"className":48550},[31642],[34,48552,48554],{"className":48553},[31646],[31648,48555,48556],{"xmlns":31650},[31652,48557,48558,48568],{},[31655,48559,48560,48562,48564,48566],{},[31658,48561,37890],{},[31662,48563,456],{"stretchy":702},[31658,48565,31660],{},[31662,48567,2557],{"stretchy":702},[31665,48569,46493],{"encoding":31667},[34,48571,48573],{"className":48572,"ariaHidden":758},[31672],[34,48574,48576,48579,48582,48585,48588],{"className":48575},[31676],[34,48577],{"className":48578,"style":31774},[31680],[34,48580,37890],{"className":48581,"style":34452},[31685,31686],[34,48583,456],{"className":48584},[31782],[34,48586,31660],{"className":48587},[31685,31686],[34,48589,2557],{"className":48590},[31690],"；最坏情况（每次划分都在边缘）时间复杂度退化为 ",[34,48593,48595,48618],{"className":48594},[31642],[34,48596,48598],{"className":48597},[31646],[31648,48599,48600],{"xmlns":31650},[31652,48601,48602,48616],{},[31655,48603,48604,48606,48608,48614],{},[31658,48605,37890],{},[31662,48607,456],{"stretchy":702},[37624,48609,48610,48612],{},[31658,48611,31660],{},[31749,48613,768],{},[31662,48615,2557],{"stretchy":702},[31665,48617,46174],{"encoding":31667},[34,48619,48621],{"className":48620,"ariaHidden":758},[31672],[34,48622,48624,48627,48630,48633,48662],{"className":48623},[31676],[34,48625],{"className":48626,"style":37755},[31680],[34,48628,37890],{"className":48629,"style":34452},[31685,31686],[34,48631,456],{"className":48632},[31782],[34,48634,48636,48639],{"className":48635},[31685],[34,48637,31660],{"className":48638},[31685,31686],[34,48640,48642],{"className":48641},[31792],[34,48643,48645],{"className":48644},[31796],[34,48646,48648],{"className":48647},[31801],[34,48649,48651],{"className":48650,"style":37774},[31805],[34,48652,48653,48656],{"style":37777},[34,48654],{"className":48655,"style":31814},[31813],[34,48657,48659],{"className":48658},[31818,31819,31820,31821],[34,48660,768],{"className":48661},[31685,31821],[34,48663,2557],{"className":48664},[31690],[32490,48666,48668],{"id":48667},"例题-5寻找两个等长有序序列的中位数重点","例题 5：寻找两个等长有序序列的中位数（重点）",[105,48670,48671,48762,50054],{},[108,48672,48673,48675,48676,48704,48705,258,48733,48761],{},[2627,48674,45915],{},"：给定两个长度均为 ",[34,48677,48679,48692],{"className":48678},[31642],[34,48680,48682],{"className":48681},[31646],[31648,48683,48684],{"xmlns":31650},[31652,48685,48686,48690],{},[31655,48687,48688],{},[31658,48689,31660],{},[31665,48691,31660],{"encoding":31667},[34,48693,48695],{"className":48694,"ariaHidden":758},[31672],[34,48696,48698,48701],{"className":48697},[31676],[34,48699],{"className":48700,"style":32527},[31680],[34,48702,31660],{"className":48703},[31685,31686]," 的升序序列 ",[34,48706,48708,48721],{"className":48707},[31642],[34,48709,48711],{"className":48710},[31646],[31648,48712,48713],{"xmlns":31650},[31652,48714,48715,48719],{},[31655,48716,48717],{},[31658,48718,111],{},[31665,48720,111],{"encoding":31667},[34,48722,48724],{"className":48723,"ariaHidden":758},[31672],[34,48725,48727,48730],{"className":48726},[31676],[34,48728],{"className":48729,"style":32527},[31680],[34,48731,111],{"className":48732},[31685,31686],[34,48734,48736,48749],{"className":48735},[31642],[34,48737,48739],{"className":48738},[31646],[31648,48740,48741],{"xmlns":31650},[31652,48742,48743,48747],{},[31655,48744,48745],{},[31658,48746,1697],{},[31665,48748,1697],{"encoding":31667},[34,48750,48752],{"className":48751,"ariaHidden":758},[31672],[34,48753,48755,48758],{"className":48754},[31676],[34,48756],{"className":48757,"style":31681},[31680],[34,48759,1697],{"className":48760},[31685,31686],"，求这两个序列合并后所有元素的中位数。",[108,48763,48764,33183,48766],{},[2627,48765,45950],{},[105,48767,48768,48930,49026,49737],{},[108,48769,48770,48773,48774,258,48802,48830,48831,258,48880,48929],{},[2627,48771,48772],{},"核心策略（舍弃法）","：分别求出 ",[34,48775,48777,48790],{"className":48776},[31642],[34,48778,48780],{"className":48779},[31646],[31648,48781,48782],{"xmlns":31650},[31652,48783,48784,48788],{},[31655,48785,48786],{},[31658,48787,111],{},[31665,48789,111],{"encoding":31667},[34,48791,48793],{"className":48792,"ariaHidden":758},[31672],[34,48794,48796,48799],{"className":48795},[31676],[34,48797],{"className":48798,"style":32527},[31680],[34,48800,111],{"className":48801},[31685,31686],[34,48803,48805,48818],{"className":48804},[31642],[34,48806,48808],{"className":48807},[31646],[31648,48809,48810],{"xmlns":31650},[31652,48811,48812,48816],{},[31655,48813,48814],{},[31658,48815,1697],{},[31665,48817,1697],{"encoding":31667},[34,48819,48821],{"className":48820,"ariaHidden":758},[31672],[34,48822,48824,48827],{"className":48823},[31676],[34,48825],{"className":48826,"style":31681},[31680],[34,48828,1697],{"className":48829},[31685,31686]," 当前区间的中位数 ",[34,48832,48834,48856],{"className":48833},[31642],[34,48835,48837],{"className":48836},[31646],[31648,48838,48839],{"xmlns":31650},[31652,48840,48841,48853],{},[31655,48842,48843,48845,48847,48849,48851],{},[31658,48844,111],{},[31662,48846,15972],{"stretchy":702},[31658,48848,31759],{},[31749,48850,640],{},[31662,48852,32712],{"stretchy":702},[31665,48854,48855],{"encoding":31667},"a[m1]",[34,48857,48859],{"className":48858,"ariaHidden":758},[31672],[34,48860,48862,48865,48868,48871,48874,48877],{"className":48861},[31676],[34,48863],{"className":48864,"style":31774},[31680],[34,48866,111],{"className":48867},[31685,31686],[34,48869,15972],{"className":48870},[31782],[34,48872,31759],{"className":48873},[31685,31686],[34,48875,640],{"className":48876},[31685],[34,48878,32712],{"className":48879},[31690],[34,48881,48883,48905],{"className":48882},[31642],[34,48884,48886],{"className":48885},[31646],[31648,48887,48888],{"xmlns":31650},[31652,48889,48890,48902],{},[31655,48891,48892,48894,48896,48898,48900],{},[31658,48893,1697],{},[31662,48895,15972],{"stretchy":702},[31658,48897,31759],{},[31749,48899,768],{},[31662,48901,32712],{"stretchy":702},[31665,48903,48904],{"encoding":31667},"b[m2]",[34,48906,48908],{"className":48907,"ariaHidden":758},[31672],[34,48909,48911,48914,48917,48920,48923,48926],{"className":48910},[31676],[34,48912],{"className":48913,"style":31774},[31680],[34,48915,1697],{"className":48916},[31685,31686],[34,48918,15972],{"className":48919},[31782],[34,48921,31759],{"className":48922},[31685,31686],[34,48924,768],{"className":48925},[31685],[34,48927,32712],{"className":48928},[31690],"。每次比较后，舍弃掉不可能包含中位数的一半元素，从而将问题规模减半。",[108,48931,43439,48932,49025],{},[34,48933,48935,48971],{"className":48934},[31642],[34,48936,48938],{"className":48937},[31646],[31648,48939,48940],{"xmlns":31650},[31652,48941,48942,48968],{},[31655,48943,48944,48946,48948,48950,48952,48954,48956,48958,48960,48962,48964,48966],{},[31658,48945,111],{},[31662,48947,15972],{"stretchy":702},[31658,48949,31759],{},[31749,48951,640],{},[31662,48953,32712],{"stretchy":702},[31662,48955,202],{},[31662,48957,202],{},[31658,48959,1697],{},[31662,48961,15972],{"stretchy":702},[31658,48963,31759],{},[31749,48965,768],{},[31662,48967,32712],{"stretchy":702},[31665,48969,48970],{"encoding":31667},"a[m1] == b[m2]",[34,48972,48974,49004],{"className":48973,"ariaHidden":758},[31672],[34,48975,48977,48980,48983,48986,48989,48992,48995,48998,49001],{"className":48976},[31676],[34,48978],{"className":48979,"style":31774},[31680],[34,48981,111],{"className":48982},[31685,31686],[34,48984,15972],{"className":48985},[31782],[34,48987,31759],{"className":48988},[31685,31686],[34,48990,640],{"className":48991},[31685],[34,48993,32712],{"className":48994},[31690],[34,48996],{"className":48997,"style":31846},[31845],[34,48999,33376],{"className":49000},[31850],[34,49002],{"className":49003,"style":31846},[31845],[34,49005,49007,49010,49013,49016,49019,49022],{"className":49006},[31676],[34,49008],{"className":49009,"style":31774},[31680],[34,49011,1697],{"className":49012},[31685,31686],[34,49014,15972],{"className":49015},[31782],[34,49017,31759],{"className":49018},[31685,31686],[34,49020,768],{"className":49021},[31685],[34,49023,32712],{"className":49024},[31690],"，则这个值就是全局中位数，直接返回。",[108,49027,49028,49121,49122,49150,49151,49179,49180,49208,49209,49237,49238],{},[2627,49029,43439,49030],{},[34,49031,49033,49067],{"className":49032},[31642],[34,49034,49036],{"className":49035},[31646],[31648,49037,49038],{"xmlns":31650},[31652,49039,49040,49064],{},[31655,49041,49042,49044,49046,49048,49050,49052,49054,49056,49058,49060,49062],{},[31658,49043,111],{},[31662,49045,15972],{"stretchy":702},[31658,49047,31759],{},[31749,49049,640],{},[31662,49051,32712],{"stretchy":702},[31662,49053,932],{},[31658,49055,1697],{},[31662,49057,15972],{"stretchy":702},[31658,49059,31759],{},[31749,49061,768],{},[31662,49063,32712],{"stretchy":702},[31665,49065,49066],{"encoding":31667},"a[m1] \u003C b[m2]",[34,49068,49070,49100],{"className":49069,"ariaHidden":758},[31672],[34,49071,49073,49076,49079,49082,49085,49088,49091,49094,49097],{"className":49072},[31676],[34,49074],{"className":49075,"style":31774},[31680],[34,49077,111],{"className":49078},[31685,31686],[34,49080,15972],{"className":49081},[31782],[34,49083,31759],{"className":49084},[31685,31686],[34,49086,640],{"className":49087},[31685],[34,49089,32712],{"className":49090},[31690],[34,49092],{"className":49093,"style":31846},[31845],[34,49095,932],{"className":49096},[31850],[34,49098],{"className":49099,"style":31846},[31845],[34,49101,49103,49106,49109,49112,49115,49118],{"className":49102},[31676],[34,49104],{"className":49105,"style":31774},[31680],[34,49107,1697],{"className":49108},[31685,31686],[34,49110,15972],{"className":49111},[31782],[34,49113,31759],{"className":49114},[31685,31686],[34,49116,768],{"className":49117},[31685],[34,49119,32712],{"className":49120},[31690],"：说明中位数一定不会出现在 ",[34,49123,49125,49138],{"className":49124},[31642],[34,49126,49128],{"className":49127},[31646],[31648,49129,49130],{"xmlns":31650},[31652,49131,49132,49136],{},[31655,49133,49134],{},[31658,49135,111],{},[31665,49137,111],{"encoding":31667},[34,49139,49141],{"className":49140,"ariaHidden":758},[31672],[34,49142,49144,49147],{"className":49143},[31676],[34,49145],{"className":49146,"style":32527},[31680],[34,49148,111],{"className":49149},[31685,31686]," 的前半部分（太小）和 ",[34,49152,49154,49167],{"className":49153},[31642],[34,49155,49157],{"className":49156},[31646],[31648,49158,49159],{"xmlns":31650},[31652,49160,49161,49165],{},[31655,49162,49163],{},[31658,49164,1697],{},[31665,49166,1697],{"encoding":31667},[34,49168,49170],{"className":49169,"ariaHidden":758},[31672],[34,49171,49173,49176],{"className":49172},[31676],[34,49174],{"className":49175,"style":31681},[31680],[34,49177,1697],{"className":49178},[31685,31686]," 的后半部分（太大）。此时舍弃 ",[34,49181,49183,49196],{"className":49182},[31642],[34,49184,49186],{"className":49185},[31646],[31648,49187,49188],{"xmlns":31650},[31652,49189,49190,49194],{},[31655,49191,49192],{},[31658,49193,111],{},[31665,49195,111],{"encoding":31667},[34,49197,49199],{"className":49198,"ariaHidden":758},[31672],[34,49200,49202,49205],{"className":49201},[31676],[34,49203],{"className":49204,"style":32527},[31680],[34,49206,111],{"className":49207},[31685,31686]," 的前半段和 ",[34,49210,49212,49225],{"className":49211},[31642],[34,49213,49215],{"className":49214},[31646],[31648,49216,49217],{"xmlns":31650},[31652,49218,49219,49223],{},[31655,49220,49221],{},[31658,49222,1697],{},[31665,49224,1697],{"encoding":31667},[34,49226,49228],{"className":49227,"ariaHidden":758},[31672],[34,49229,49231,49234],{"className":49230},[31676],[34,49232],{"className":49233,"style":31681},[31680],[34,49235,1697],{"className":49236},[31685,31686]," 的后半段。\n",[105,49239,49240,49245,49479],{},[108,49241,49242,2664],{},[2627,49243,49244],{},"奇偶数边界处理（极易错点）",[108,49246,49247,49248,49276,49277,49280,49281,49309,49310,258,49380,49408,49409,347],{},"如果当前元素个数 ",[34,49249,49251,49264],{"className":49250},[31642],[34,49252,49254],{"className":49253},[31646],[31648,49255,49256],{"xmlns":31650},[31652,49257,49258,49262],{},[31655,49259,49260],{},[31658,49261,31660],{},[31665,49263,31660],{"encoding":31667},[34,49265,49267],{"className":49266,"ariaHidden":758},[31672],[34,49268,49270,49273],{"className":49269},[31676],[34,49271],{"className":49272,"style":32527},[31680],[34,49274,31660],{"className":49275},[31685,31686]," 为",[2627,49278,49279],{},"奇数","：保留 ",[34,49282,49284,49297],{"className":49283},[31642],[34,49285,49287],{"className":49286},[31646],[31648,49288,49289],{"xmlns":31650},[31652,49290,49291,49295],{},[31655,49292,49293],{},[31658,49294,111],{},[31665,49296,111],{"encoding":31667},[34,49298,49300],{"className":49299,"ariaHidden":758},[31672],[34,49301,49303,49306],{"className":49302},[31676],[34,49304],{"className":49305,"style":32527},[31680],[34,49307,111],{"className":49308},[31685,31686]," 的后半部分 ",[34,49311,49313,49341],{"className":49312},[31642],[34,49314,49316],{"className":49315},[31646],[31648,49317,49318],{"xmlns":31650},[31652,49319,49320,49338],{},[31655,49321,49322,49324,49326,49328,49330,49332,49334,49336],{},[31658,49323,111],{},[31662,49325,15972],{"stretchy":702},[31658,49327,31759],{},[31749,49329,640],{},[31662,49331,33934],{},[31658,49333,33249],{},[31749,49335,640],{},[31662,49337,32712],{"stretchy":702},[31665,49339,49340],{"encoding":31667},"a[m1 \\dots t1]",[34,49342,49344],{"className":49343,"ariaHidden":758},[31672],[34,49345,49347,49350,49353,49356,49359,49362,49365,49368,49371,49374,49377],{"className":49346},[31676],[34,49348],{"className":49349,"style":31774},[31680],[34,49351,111],{"className":49352},[31685,31686],[34,49354,15972],{"className":49355},[31782],[34,49357,31759],{"className":49358},[31685,31686],[34,49360,640],{"className":49361},[31685],[34,49363],{"className":49364,"style":32123},[31845],[34,49366,33934],{"className":49367},[34006],[34,49369],{"className":49370,"style":32123},[31845],[34,49372,33249],{"className":49373},[31685,31686],[34,49375,640],{"className":49376},[31685],[34,49378,32712],{"className":49379},[31690],[34,49381,49383,49396],{"className":49382},[31642],[34,49384,49386],{"className":49385},[31646],[31648,49387,49388],{"xmlns":31650},[31652,49389,49390,49394],{},[31655,49391,49392],{},[31658,49393,1697],{},[31665,49395,1697],{"encoding":31667},[34,49397,49399],{"className":49398,"ariaHidden":758},[31672],[34,49400,49402,49405],{"className":49401},[31676],[34,49403],{"className":49404,"style":31681},[31680],[34,49406,1697],{"className":49407},[31685,31686]," 的前半部分 ",[34,49410,49412,49440],{"className":49411},[31642],[34,49413,49415],{"className":49414},[31646],[31648,49416,49417],{"xmlns":31650},[31652,49418,49419,49437],{},[31655,49420,49421,49423,49425,49427,49429,49431,49433,49435],{},[31658,49422,1697],{},[31662,49424,15972],{"stretchy":702},[31658,49426,31747],{},[31749,49428,768],{},[31662,49430,33934],{},[31658,49432,31759],{},[31749,49434,768],{},[31662,49436,32712],{"stretchy":702},[31665,49438,49439],{"encoding":31667},"b[s2 \\dots m2]",[34,49441,49443],{"className":49442,"ariaHidden":758},[31672],[34,49444,49446,49449,49452,49455,49458,49461,49464,49467,49470,49473,49476],{"className":49445},[31676],[34,49447],{"className":49448,"style":31774},[31680],[34,49450,1697],{"className":49451},[31685,31686],[34,49453,15972],{"className":49454},[31782],[34,49456,31747],{"className":49457},[31685,31686],[34,49459,768],{"className":49460},[31685],[34,49462],{"className":49463,"style":32123},[31845],[34,49465,33934],{"className":49466},[34006],[34,49468],{"className":49469,"style":32123},[31845],[34,49471,31759],{"className":49472},[31685,31686],[34,49474,768],{"className":49475},[31685],[34,49477,32712],{"className":49478},[31690],[108,49480,49247,49481,49276,49509,49512,49513,49516,49517,49545,49546,49638,49639,49667,49668,347],{},[34,49482,49484,49497],{"className":49483},[31642],[34,49485,49487],{"className":49486},[31646],[31648,49488,49489],{"xmlns":31650},[31652,49490,49491,49495],{},[31655,49492,49493],{},[31658,49494,31660],{},[31665,49496,31660],{"encoding":31667},[34,49498,49500],{"className":49499,"ariaHidden":758},[31672],[34,49501,49503,49506],{"className":49502},[31676],[34,49504],{"className":49505,"style":32527},[31680],[34,49507,31660],{"className":49508},[31685,31686],[2627,49510,49511],{},"偶数","：为了保证舍弃后两个序列",[2627,49514,49515],{},"仍然等长","，保留 ",[34,49518,49520,49533],{"className":49519},[31642],[34,49521,49523],{"className":49522},[31646],[31648,49524,49525],{"xmlns":31650},[31652,49526,49527,49531],{},[31655,49528,49529],{},[31658,49530,111],{},[31665,49532,111],{"encoding":31667},[34,49534,49536],{"className":49535,"ariaHidden":758},[31672],[34,49537,49539,49542],{"className":49538},[31676],[34,49540],{"className":49541,"style":32527},[31680],[34,49543,111],{"className":49544},[31685,31686]," 的后半部分必须是 ",[34,49547,49549,49581],{"className":49548},[31642],[34,49550,49552],{"className":49551},[31646],[31648,49553,49554],{"xmlns":31650},[31652,49555,49556,49578],{},[31655,49557,49558,49560,49562,49564,49566,49568,49570,49572,49574,49576],{},[31658,49559,111],{},[31662,49561,15972],{"stretchy":702},[31658,49563,31759],{},[31749,49565,640],{},[31662,49567,223],{},[31749,49569,640],{},[31662,49571,33934],{},[31658,49573,33249],{},[31749,49575,640],{},[31662,49577,32712],{"stretchy":702},[31665,49579,49580],{"encoding":31667},"a[m1+1 \\dots t1]",[34,49582,49584,49611],{"className":49583,"ariaHidden":758},[31672],[34,49585,49587,49590,49593,49596,49599,49602,49605,49608],{"className":49586},[31676],[34,49588],{"className":49589,"style":31774},[31680],[34,49591,111],{"className":49592},[31685,31686],[34,49594,15972],{"className":49595},[31782],[34,49597,31759],{"className":49598},[31685,31686],[34,49600,640],{"className":49601},[31685],[34,49603],{"className":49604,"style":32604},[31845],[34,49606,223],{"className":49607},[32024],[34,49609],{"className":49610,"style":32604},[31845],[34,49612,49614,49617,49620,49623,49626,49629,49632,49635],{"className":49613},[31676],[34,49615],{"className":49616,"style":31774},[31680],[34,49618,640],{"className":49619},[31685],[34,49621],{"className":49622,"style":32123},[31845],[34,49624,33934],{"className":49625},[34006],[34,49627],{"className":49628,"style":32123},[31845],[34,49630,33249],{"className":49631},[31685,31686],[34,49633,640],{"className":49634},[31685],[34,49636,32712],{"className":49637},[31690],"（连同较小的中间元素一并舍弃），保留 ",[34,49640,49642,49655],{"className":49641},[31642],[34,49643,49645],{"className":49644},[31646],[31648,49646,49647],{"xmlns":31650},[31652,49648,49649,49653],{},[31655,49650,49651],{},[31658,49652,1697],{},[31665,49654,1697],{"encoding":31667},[34,49656,49658],{"className":49657,"ariaHidden":758},[31672],[34,49659,49661,49664],{"className":49660},[31676],[34,49662],{"className":49663,"style":31681},[31680],[34,49665,1697],{"className":49666},[31685,31686]," 的前半部分为 ",[34,49669,49671,49698],{"className":49670},[31642],[34,49672,49674],{"className":49673},[31646],[31648,49675,49676],{"xmlns":31650},[31652,49677,49678,49696],{},[31655,49679,49680,49682,49684,49686,49688,49690,49692,49694],{},[31658,49681,1697],{},[31662,49683,15972],{"stretchy":702},[31658,49685,31747],{},[31749,49687,768],{},[31662,49689,33934],{},[31658,49691,31759],{},[31749,49693,768],{},[31662,49695,32712],{"stretchy":702},[31665,49697,49439],{"encoding":31667},[34,49699,49701],{"className":49700,"ariaHidden":758},[31672],[34,49702,49704,49707,49710,49713,49716,49719,49722,49725,49728,49731,49734],{"className":49703},[31676],[34,49705],{"className":49706,"style":31774},[31680],[34,49708,1697],{"className":49709},[31685,31686],[34,49711,15972],{"className":49712},[31782],[34,49714,31747],{"className":49715},[31685,31686],[34,49717,768],{"className":49718},[31685],[34,49720],{"className":49721,"style":32123},[31845],[34,49723,33934],{"className":49724},[34006],[34,49726],{"className":49727,"style":32123},[31845],[34,49729,31759],{"className":49730},[31685,31686],[34,49732,768],{"className":49733},[31685],[34,49735,32712],{"className":49736},[31690],[108,49738,49739,49832,49833,49861,49862,49890,49891,49961,49962,347],{},[2627,49740,43439,49741],{},[34,49742,49744,49778],{"className":49743},[31642],[34,49745,49747],{"className":49746},[31646],[31648,49748,49749],{"xmlns":31650},[31652,49750,49751,49775],{},[31655,49752,49753,49755,49757,49759,49761,49763,49765,49767,49769,49771,49773],{},[31658,49754,111],{},[31662,49756,15972],{"stretchy":702},[31658,49758,31759],{},[31749,49760,640],{},[31662,49762,32712],{"stretchy":702},[31662,49764,29622],{},[31658,49766,1697],{},[31662,49768,15972],{"stretchy":702},[31658,49770,31759],{},[31749,49772,768],{},[31662,49774,32712],{"stretchy":702},[31665,49776,49777],{"encoding":31667},"a[m1] > b[m2]",[34,49779,49781,49811],{"className":49780,"ariaHidden":758},[31672],[34,49782,49784,49787,49790,49793,49796,49799,49802,49805,49808],{"className":49783},[31676],[34,49785],{"className":49786,"style":31774},[31680],[34,49788,111],{"className":49789},[31685,31686],[34,49791,15972],{"className":49792},[31782],[34,49794,31759],{"className":49795},[31685,31686],[34,49797,640],{"className":49798},[31685],[34,49800,32712],{"className":49801},[31690],[34,49803],{"className":49804,"style":31846},[31845],[34,49806,29622],{"className":49807},[31850],[34,49809],{"className":49810,"style":31846},[31845],[34,49812,49814,49817,49820,49823,49826,49829],{"className":49813},[31676],[34,49815],{"className":49816,"style":31774},[31680],[34,49818,1697],{"className":49819},[31685,31686],[34,49821,15972],{"className":49822},[31782],[34,49824,31759],{"className":49825},[31685,31686],[34,49827,768],{"className":49828},[31685],[34,49830,32712],{"className":49831},[31690],"：舍弃 ",[34,49834,49836,49849],{"className":49835},[31642],[34,49837,49839],{"className":49838},[31646],[31648,49840,49841],{"xmlns":31650},[31652,49842,49843,49847],{},[31655,49844,49845],{},[31658,49846,111],{},[31665,49848,111],{"encoding":31667},[34,49850,49852],{"className":49851,"ariaHidden":758},[31672],[34,49853,49855,49858],{"className":49854},[31676],[34,49856],{"className":49857,"style":32527},[31680],[34,49859,111],{"className":49860},[31685,31686]," 的后半段和 ",[34,49863,49865,49878],{"className":49864},[31642],[34,49866,49868],{"className":49867},[31646],[31648,49869,49870],{"xmlns":31650},[31652,49871,49872,49876],{},[31655,49873,49874],{},[31658,49875,1697],{},[31665,49877,1697],{"encoding":31667},[34,49879,49881],{"className":49880,"ariaHidden":758},[31672],[34,49882,49884,49887],{"className":49883},[31676],[34,49885],{"className":49886,"style":31681},[31680],[34,49888,1697],{"className":49889},[31685,31686]," 的前半段。处理方式与上述对称：奇数保留 ",[34,49892,49894,49922],{"className":49893},[31642],[34,49895,49897],{"className":49896},[31646],[31648,49898,49899],{"xmlns":31650},[31652,49900,49901,49919],{},[31655,49902,49903,49905,49907,49909,49911,49913,49915,49917],{},[31658,49904,1697],{},[31662,49906,15972],{"stretchy":702},[31658,49908,31759],{},[31749,49910,768],{},[31662,49912,33934],{},[31658,49914,33249],{},[31749,49916,768],{},[31662,49918,32712],{"stretchy":702},[31665,49920,49921],{"encoding":31667},"b[m2 \\dots t2]",[34,49923,49925],{"className":49924,"ariaHidden":758},[31672],[34,49926,49928,49931,49934,49937,49940,49943,49946,49949,49952,49955,49958],{"className":49927},[31676],[34,49929],{"className":49930,"style":31774},[31680],[34,49932,1697],{"className":49933},[31685,31686],[34,49935,15972],{"className":49936},[31782],[34,49938,31759],{"className":49939},[31685,31686],[34,49941,768],{"className":49942},[31685],[34,49944],{"className":49945,"style":32123},[31845],[34,49947,33934],{"className":49948},[34006],[34,49950],{"className":49951,"style":32123},[31845],[34,49953,33249],{"className":49954},[31685,31686],[34,49956,768],{"className":49957},[31685],[34,49959,32712],{"className":49960},[31690],"，偶数保留 ",[34,49963,49965,49997],{"className":49964},[31642],[34,49966,49968],{"className":49967},[31646],[31648,49969,49970],{"xmlns":31650},[31652,49971,49972,49994],{},[31655,49973,49974,49976,49978,49980,49982,49984,49986,49988,49990,49992],{},[31658,49975,1697],{},[31662,49977,15972],{"stretchy":702},[31658,49979,31759],{},[31749,49981,768],{},[31662,49983,223],{},[31749,49985,640],{},[31662,49987,33934],{},[31658,49989,33249],{},[31749,49991,768],{},[31662,49993,32712],{"stretchy":702},[31665,49995,49996],{"encoding":31667},"b[m2+1 \\dots t2]",[34,49998,50000,50027],{"className":49999,"ariaHidden":758},[31672],[34,50001,50003,50006,50009,50012,50015,50018,50021,50024],{"className":50002},[31676],[34,50004],{"className":50005,"style":31774},[31680],[34,50007,1697],{"className":50008},[31685,31686],[34,50010,15972],{"className":50011},[31782],[34,50013,31759],{"className":50014},[31685,31686],[34,50016,768],{"className":50017},[31685],[34,50019],{"className":50020,"style":32604},[31845],[34,50022,223],{"className":50023},[32024],[34,50025],{"className":50026,"style":32604},[31845],[34,50028,50030,50033,50036,50039,50042,50045,50048,50051],{"className":50029},[31676],[34,50031],{"className":50032,"style":31774},[31680],[34,50034,640],{"className":50035},[31685],[34,50037],{"className":50038,"style":32123},[31845],[34,50040,33934],{"className":50041},[34006],[34,50043],{"className":50044,"style":32123},[31845],[34,50046,33249],{"className":50047},[31685,31686],[34,50049,768],{"className":50050},[31685],[34,50052,32712],{"className":50053},[31690],[108,50055,50056,50058,50059,46470,50169,347],{},[2627,50057,46038],{},"：每次将问题规模缩小一半，执行时间 ",[34,50060,50062,50100],{"className":50061},[31642],[34,50063,50065],{"className":50064},[31646],[31648,50066,50067],{"xmlns":31650},[31652,50068,50069,50097],{},[31655,50070,50071,50073,50075,50077,50079,50081,50083,50085,50087,50089,50091,50093,50095],{},[31658,50072,37400],{},[31662,50074,456],{"stretchy":702},[31658,50076,31660],{},[31662,50078,2557],{"stretchy":702},[31662,50080,202],{},[31658,50082,37400],{},[31662,50084,456],{"stretchy":702},[31658,50086,31660],{},[31658,50088,38080],{"mathvariant":38079},[31749,50090,768],{},[31662,50092,2557],{"stretchy":702},[31662,50094,223],{},[31749,50096,640],{},[31665,50098,50099],{"encoding":31667},"T(n) = T(n\u002F2) + 1",[34,50101,50103,50130,50160],{"className":50102,"ariaHidden":758},[31672],[34,50104,50106,50109,50112,50115,50118,50121,50124,50127],{"className":50105},[31676],[34,50107],{"className":50108,"style":31774},[31680],[34,50110,37400],{"className":50111,"style":37422},[31685,31686],[34,50113,456],{"className":50114},[31782],[34,50116,31660],{"className":50117},[31685,31686],[34,50119,2557],{"className":50120},[31690],[34,50122],{"className":50123,"style":31846},[31845],[34,50125,202],{"className":50126},[31850],[34,50128],{"className":50129,"style":31846},[31845],[34,50131,50133,50136,50139,50142,50145,50148,50151,50154,50157],{"className":50132},[31676],[34,50134],{"className":50135,"style":31774},[31680],[34,50137,37400],{"className":50138,"style":37422},[31685,31686],[34,50140,456],{"className":50141},[31782],[34,50143,31660],{"className":50144},[31685,31686],[34,50146,38149],{"className":50147},[31685],[34,50149,2557],{"className":50150},[31690],[34,50152],{"className":50153,"style":32604},[31845],[34,50155,223],{"className":50156},[32024],[34,50158],{"className":50159,"style":32604},[31845],[34,50161,50163,50166],{"className":50162},[31676],[34,50164],{"className":50165,"style":32617},[31680],[34,50167,640],{"className":50168},[31685],[34,50170,50172,50201],{"className":50171},[31642],[34,50173,50175],{"className":50174},[31646],[31648,50176,50177],{"xmlns":31650},[31652,50178,50179,50199],{},[31655,50180,50181,50183,50185,50195,50197],{},[31658,50182,37890],{},[31662,50184,456],{"stretchy":702},[31743,50186,50187,50193],{},[31655,50188,50189,50191],{},[31658,50190,885],{},[31662,50192,32682],{},[31749,50194,768],{},[31658,50196,31660],{},[31662,50198,2557],{"stretchy":702},[31665,50200,47514],{"encoding":31667},[34,50202,50204],{"className":50203,"ariaHidden":758},[31672],[34,50205,50207,50210,50213,50216,50258,50261,50264],{"className":50206},[31676],[34,50208],{"className":50209,"style":31774},[31680],[34,50211,37890],{"className":50212,"style":34452},[31685,31686],[34,50214,456],{"className":50215},[31782],[34,50217,50219,50224],{"className":50218},[32767],[34,50220,38616,50222],{"className":50221},[32767],[34,50223,31942],{"style":38619},[34,50225,50227],{"className":50226},[31792],[34,50228,50230,50250],{"className":50229},[31796,31797],[34,50231,50233,50247],{"className":50232},[31801],[34,50234,50236],{"className":50235,"style":38632},[31805],[34,50237,50238,50241],{"style":38635},[34,50239],{"className":50240,"style":31814},[31813],[34,50242,50244],{"className":50243},[31818,31819,31820,31821],[34,50245,768],{"className":50246},[31685,31821],[34,50248,31829],{"className":50249},[31828],[34,50251,50253],{"className":50252},[31801],[34,50254,50256],{"className":50255,"style":38654},[31805],[34,50257],{},[34,50259],{"className":50260,"style":32123},[31845],[34,50262,31660],{"className":50263},[31685,31686],[34,50265,2557],{"className":50266},[31690],[32490,50268,50270],{"id":50269},"例题-6求解最大连续子序列和问题","例题 6：求解最大连续子序列和问题",[105,50272,50273,50307,50806],{},[108,50274,50275,50277,50278,50306],{},[2627,50276,45915],{},"：给定含有 ",[34,50279,50281,50294],{"className":50280},[31642],[34,50282,50284],{"className":50283},[31646],[31648,50285,50286],{"xmlns":31650},[31652,50287,50288,50292],{},[31655,50289,50290],{},[31658,50291,31660],{},[31665,50293,31660],{"encoding":31667},[34,50295,50297],{"className":50296,"ariaHidden":758},[31672],[34,50298,50300,50303],{"className":50299},[31676],[34,50301],{"className":50302,"style":32527},[31680],[34,50304,31660],{"className":50305},[31685,31686]," 个整数的序列，求出其最大连续子序列的和。规定长度为0的子序列和为0（即如果全为负数则结果为0）。",[108,50308,50309,33183,50311],{},[2627,50310,45950],{},[105,50312,50313,50739,50753,50800],{},[108,50314,50315,50316,50413,50414,33183,50417],{},"取序列中间位置 ",[34,50317,50319,50353],{"className":50318},[31642],[34,50320,50322],{"className":50321},[31646],[31648,50323,50324],{"xmlns":31650},[31652,50325,50326,50350],{},[31655,50327,50328,50330,50332,50334,50336,50338,50340,50342,50344,50346,50348],{},[31658,50329,31759],{},[31658,50331,31955],{},[31658,50333,46900],{},[31662,50335,202],{},[31662,50337,456],{"stretchy":702},[31658,50339,31660],{},[31662,50341,32226],{},[31749,50343,640],{},[31662,50345,2557],{"stretchy":702},[31658,50347,38080],{"mathvariant":38079},[31749,50349,768],{},[31665,50351,50352],{"encoding":31667},"mid = (n-1)\u002F2",[34,50354,50356,50377,50398],{"className":50355,"ariaHidden":758},[31672],[34,50357,50359,50362,50365,50368,50371,50374],{"className":50358},[31676],[34,50360],{"className":50361,"style":31681},[31680],[34,50363,31658],{"className":50364},[31685,31686],[34,50366,46900],{"className":50367},[31685,31686],[34,50369],{"className":50370,"style":31846},[31845],[34,50372,202],{"className":50373},[31850],[34,50375],{"className":50376,"style":31846},[31845],[34,50378,50380,50383,50386,50389,50392,50395],{"className":50379},[31676],[34,50381],{"className":50382,"style":31774},[31680],[34,50384,456],{"className":50385},[31782],[34,50387,31660],{"className":50388},[31685,31686],[34,50390],{"className":50391,"style":32604},[31845],[34,50393,32226],{"className":50394},[32024],[34,50396],{"className":50397,"style":32604},[31845],[34,50399,50401,50404,50407,50410],{"className":50400},[31676],[34,50402],{"className":50403,"style":31774},[31680],[34,50405,640],{"className":50406},[31685],[34,50408,2557],{"className":50409},[31690],[34,50411,38149],{"className":50412},[31685],"。最大连续子序列可能存在于",[2627,50415,50416],{},"三个位置",[2230,50418,50419,50493,50609],{},[108,50420,50421,50422,47649,50425,50492],{},"完全落在",[2627,50423,50424],{},"左半部",[34,50426,50428,50456],{"className":50427},[31642],[34,50429,50431],{"className":50430},[31646],[31648,50432,50433],{"xmlns":31650},[31652,50434,50435,50453],{},[31655,50436,50437,50439,50441,50443,50445,50447,50449,50451],{},[31658,50438,111],{},[31662,50440,15972],{"stretchy":702},[31749,50442,459],{},[31662,50444,33934],{},[31658,50446,31759],{},[31658,50448,31955],{},[31658,50450,46900],{},[31662,50452,32712],{"stretchy":702},[31665,50454,50455],{"encoding":31667},"a[0 \\dots mid]",[34,50457,50459],{"className":50458,"ariaHidden":758},[31672],[34,50460,50462,50465,50468,50471,50474,50477,50480,50483,50486,50489],{"className":50461},[31676],[34,50463],{"className":50464,"style":31774},[31680],[34,50466,111],{"className":50467},[31685,31686],[34,50469,15972],{"className":50470},[31782],[34,50472,459],{"className":50473},[31685],[34,50475],{"className":50476,"style":32123},[31845],[34,50478,33934],{"className":50479},[34006],[34,50481],{"className":50482,"style":32123},[31845],[34,50484,31658],{"className":50485},[31685,31686],[34,50487,46900],{"className":50488},[31685,31686],[34,50490,32712],{"className":50491},[31690]," 中。",[108,50494,50421,50495,47649,50498,50492],{},[2627,50496,50497],{},"右半部",[34,50499,50501,50537],{"className":50500},[31642],[34,50502,50504],{"className":50503},[31646],[31648,50505,50506],{"xmlns":31650},[31652,50507,50508,50534],{},[31655,50509,50510,50512,50514,50516,50518,50520,50522,50524,50526,50528,50530,50532],{},[31658,50511,111],{},[31662,50513,15972],{"stretchy":702},[31658,50515,31759],{},[31658,50517,31955],{},[31658,50519,46900],{},[31662,50521,223],{},[31749,50523,640],{},[31662,50525,33934],{},[31658,50527,31660],{},[31662,50529,32226],{},[31749,50531,640],{},[31662,50533,32712],{"stretchy":702},[31665,50535,50536],{"encoding":31667},"a[mid+1 \\dots n-1]",[34,50538,50540,50567,50597],{"className":50539,"ariaHidden":758},[31672],[34,50541,50543,50546,50549,50552,50555,50558,50561,50564],{"className":50542},[31676],[34,50544],{"className":50545,"style":31774},[31680],[34,50547,111],{"className":50548},[31685,31686],[34,50550,15972],{"className":50551},[31782],[34,50553,31658],{"className":50554},[31685,31686],[34,50556,46900],{"className":50557},[31685,31686],[34,50559],{"className":50560,"style":32604},[31845],[34,50562,223],{"className":50563},[32024],[34,50565],{"className":50566,"style":32604},[31845],[34,50568,50570,50573,50576,50579,50582,50585,50588,50591,50594],{"className":50569},[31676],[34,50571],{"className":50572,"style":35942},[31680],[34,50574,640],{"className":50575},[31685],[34,50577],{"className":50578,"style":32123},[31845],[34,50580,33934],{"className":50581},[34006],[34,50583],{"className":50584,"style":32123},[31845],[34,50586,31660],{"className":50587},[31685,31686],[34,50589],{"className":50590,"style":32604},[31845],[34,50592,32226],{"className":50593},[32024],[34,50595],{"className":50596,"style":32604},[31845],[34,50598,50600,50603,50606],{"className":50599},[31676],[34,50601],{"className":50602,"style":31774},[31680],[34,50604,640],{"className":50605},[31685],[34,50607,32712],{"className":50608},[31690],[108,50610,50611,50614,50615,258,50666,347],{},[2627,50612,50613],{},"跨越序列中部","，包含 ",[34,50616,50618,50642],{"className":50617},[31642],[34,50619,50621],{"className":50620},[31646],[31648,50622,50623],{"xmlns":31650},[31652,50624,50625,50639],{},[31655,50626,50627,50629,50631,50633,50635,50637],{},[31658,50628,111],{},[31662,50630,15972],{"stretchy":702},[31658,50632,31759],{},[31658,50634,31955],{},[31658,50636,46900],{},[31662,50638,32712],{"stretchy":702},[31665,50640,50641],{"encoding":31667},"a[mid]",[34,50643,50645],{"className":50644,"ariaHidden":758},[31672],[34,50646,50648,50651,50654,50657,50660,50663],{"className":50647},[31676],[34,50649],{"className":50650,"style":31774},[31680],[34,50652,111],{"className":50653},[31685,31686],[34,50655,15972],{"className":50656},[31782],[34,50658,31658],{"className":50659},[31685,31686],[34,50661,46900],{"className":50662},[31685,31686],[34,50664,32712],{"className":50665},[31690],[34,50667,50669,50697],{"className":50668},[31642],[34,50670,50672],{"className":50671},[31646],[31648,50673,50674],{"xmlns":31650},[31652,50675,50676,50694],{},[31655,50677,50678,50680,50682,50684,50686,50688,50690,50692],{},[31658,50679,111],{},[31662,50681,15972],{"stretchy":702},[31658,50683,31759],{},[31658,50685,31955],{},[31658,50687,46900],{},[31662,50689,223],{},[31749,50691,640],{},[31662,50693,32712],{"stretchy":702},[31665,50695,50696],{"encoding":31667},"a[mid+1]",[34,50698,50700,50727],{"className":50699,"ariaHidden":758},[31672],[34,50701,50703,50706,50709,50712,50715,50718,50721,50724],{"className":50702},[31676],[34,50704],{"className":50705,"style":31774},[31680],[34,50707,111],{"className":50708},[31685,31686],[34,50710,15972],{"className":50711},[31782],[34,50713,31658],{"className":50714},[31685,31686],[34,50716,46900],{"className":50717},[31685,31686],[34,50719],{"className":50720,"style":32604},[31845],[34,50722,223],{"className":50723},[32024],[34,50725],{"className":50726,"style":32604},[31845],[34,50728,50730,50733,50736],{"className":50729},[31676],[34,50731],{"className":50732,"style":31774},[31680],[34,50734,640],{"className":50735},[31685],[34,50737,32712],{"className":50738},[31690],[108,50740,50741,50742,50745,50746,50749,50750,347],{},"对于位置 1 和位置 2：直接",[2627,50743,50744],{},"递归调用","，求出左半部最大和 ",[31,50747,50748],{},"maxLeftSum"," 以及右半部最大和 ",[31,50751,50752],{},"maxRightSum",[108,50754,50755,50756,50791,50792,50795,50796,50799],{},"对于位置 3：从 ",[34,50757,50759,50776],{"className":50758},[31642],[34,50760,50762],{"className":50761},[31646],[31648,50763,50764],{"xmlns":31650},[31652,50765,50766,50774],{},[31655,50767,50768,50770,50772],{},[31658,50769,31759],{},[31658,50771,31955],{},[31658,50773,46900],{},[31665,50775,46311],{"encoding":31667},[34,50777,50779],{"className":50778,"ariaHidden":758},[31672],[34,50780,50782,50785,50788],{"className":50781},[31676],[34,50783],{"className":50784,"style":31681},[31680],[34,50786,31658],{"className":50787},[31685,31686],[34,50789,46900],{"className":50790},[31685,31686]," 开始分别向左、向右连续扫描，分别求出左边延伸的最大和 ",[31,50793,50794],{},"maxLeftBorderSum"," 与右边延伸的最大和 ",[31,50797,50798],{},"maxRightBorderSum","。然后将两者相加。",[108,50801,50802,50803,347],{},"最终结果：取上述三种情况的最大值，即 ",[31,50804,50805],{},"max3(maxLeftSum, maxRightSum, maxLeftBorderSum + maxRightBorderSum)",[108,50807,50808,50810,50811,50854,50855,50970,50971,347],{},[2627,50809,46038],{},"：求跨越中部最大和需要 ",[34,50812,50814,50833],{"className":50813},[31642],[34,50815,50817],{"className":50816},[31646],[31648,50818,50819],{"xmlns":31650},[31652,50820,50821,50831],{},[31655,50822,50823,50825,50827,50829],{},[31658,50824,37890],{},[31662,50826,456],{"stretchy":702},[31658,50828,31660],{},[31662,50830,2557],{"stretchy":702},[31665,50832,46493],{"encoding":31667},[34,50834,50836],{"className":50835,"ariaHidden":758},[31672],[34,50837,50839,50842,50845,50848,50851],{"className":50838},[31676],[34,50840],{"className":50841,"style":31774},[31680],[34,50843,37890],{"className":50844,"style":34452},[31685,31686],[34,50846,456],{"className":50847},[31782],[34,50849,31660],{"className":50850},[31685,31686],[34,50852,2557],{"className":50853},[31690]," 次比较，递推式为 ",[34,50856,50858,50898],{"className":50857},[31642],[34,50859,50861],{"className":50860},[31646],[31648,50862,50863],{"xmlns":31650},[31652,50864,50865,50895],{},[31655,50866,50867,50869,50871,50873,50875,50877,50879,50881,50883,50885,50887,50889,50891,50893],{},[31658,50868,37400],{},[31662,50870,456],{"stretchy":702},[31658,50872,31660],{},[31662,50874,2557],{"stretchy":702},[31662,50876,202],{},[31749,50878,768],{},[31658,50880,37400],{},[31662,50882,456],{"stretchy":702},[31658,50884,31660],{},[31658,50886,38080],{"mathvariant":38079},[31749,50888,768],{},[31662,50890,2557],{"stretchy":702},[31662,50892,223],{},[31658,50894,31660],{},[31665,50896,50897],{"encoding":31667},"T(n) = 2T(n\u002F2) + n",[34,50899,50901,50928,50961],{"className":50900,"ariaHidden":758},[31672],[34,50902,50904,50907,50910,50913,50916,50919,50922,50925],{"className":50903},[31676],[34,50905],{"className":50906,"style":31774},[31680],[34,50908,37400],{"className":50909,"style":37422},[31685,31686],[34,50911,456],{"className":50912},[31782],[34,50914,31660],{"className":50915},[31685,31686],[34,50917,2557],{"className":50918},[31690],[34,50920],{"className":50921,"style":31846},[31845],[34,50923,202],{"className":50924},[31850],[34,50926],{"className":50927,"style":31846},[31845],[34,50929,50931,50934,50937,50940,50943,50946,50949,50952,50955,50958],{"className":50930},[31676],[34,50932],{"className":50933,"style":31774},[31680],[34,50935,768],{"className":50936},[31685],[34,50938,37400],{"className":50939,"style":37422},[31685,31686],[34,50941,456],{"className":50942},[31782],[34,50944,31660],{"className":50945},[31685,31686],[34,50947,38149],{"className":50948},[31685],[34,50950,2557],{"className":50951},[31690],[34,50953],{"className":50954,"style":32604},[31845],[34,50956,223],{"className":50957},[32024],[34,50959],{"className":50960,"style":32604},[31845],[34,50962,50964,50967],{"className":50963},[31676],[34,50965],{"className":50966,"style":32527},[31680],[34,50968,31660],{"className":50969},[31685,31686],"，根据主方法解得时间复杂度为 ",[34,50972,50974,51005],{"className":50973},[31642],[34,50975,50977],{"className":50976},[31646],[31648,50978,50979],{"xmlns":31650},[31652,50980,50981,51003],{},[31655,50982,50983,50985,50987,50989,50999,51001],{},[31658,50984,37890],{},[31662,50986,456],{"stretchy":702},[31658,50988,31660],{},[31743,50990,50991,50997],{},[31655,50992,50993,50995],{},[31658,50994,885],{},[31662,50996,32682],{},[31749,50998,768],{},[31658,51000,31660],{},[31662,51002,2557],{"stretchy":702},[31665,51004,46074],{"encoding":31667},[34,51006,51008],{"className":51007,"ariaHidden":758},[31672],[34,51009,51011,51014,51017,51020,51023,51026,51068,51071,51074],{"className":51010},[31676],[34,51012],{"className":51013,"style":31774},[31680],[34,51015,37890],{"className":51016,"style":34452},[31685,31686],[34,51018,456],{"className":51019},[31782],[34,51021,31660],{"className":51022},[31685,31686],[34,51024],{"className":51025,"style":32123},[31845],[34,51027,51029,51034],{"className":51028},[32767],[34,51030,38616,51032],{"className":51031},[32767],[34,51033,31942],{"style":38619},[34,51035,51037],{"className":51036},[31792],[34,51038,51040,51060],{"className":51039},[31796,31797],[34,51041,51043,51057],{"className":51042},[31801],[34,51044,51046],{"className":51045,"style":38632},[31805],[34,51047,51048,51051],{"style":38635},[34,51049],{"className":51050,"style":31814},[31813],[34,51052,51054],{"className":51053},[31818,31819,31820,31821],[34,51055,768],{"className":51056},[31685,31821],[34,51058,31829],{"className":51059},[31828],[34,51061,51063],{"className":51062},[31801],[34,51064,51066],{"className":51065,"style":38654},[31805],[34,51067],{},[34,51069],{"className":51070,"style":32123},[31845],[34,51072,31660],{"className":51073},[31685,31686],[34,51075,2557],{"className":51076},[31690],[32490,51078,51080],{"id":51079},"例题-7求解棋盘覆盖问题","例题 7：求解棋盘覆盖问题",[105,51082,51083,51201],{},[108,51084,51085,51087,51088,51200],{},[2627,51086,45915],{},"：在一个 ",[34,51089,51091,51117],{"className":51090},[31642],[34,51092,51094],{"className":51093},[31646],[31648,51095,51096],{"xmlns":31650},[31652,51097,51098,51114],{},[31655,51099,51100,51106,51108],{},[37624,51101,51102,51104],{},[31749,51103,768],{},[31658,51105,33759],{},[31662,51107,42071],{},[37624,51109,51110,51112],{},[31749,51111,768],{},[31658,51113,33759],{},[31665,51115,51116],{"encoding":31667},"2^k \\times 2^k",[34,51118,51120,51165],{"className":51119,"ariaHidden":758},[31672],[34,51121,51123,51127,51156,51159,51162],{"className":51122},[31676],[34,51124],{"className":51125,"style":51126},[31680],"height:0.9324em;vertical-align:-0.0833em;",[34,51128,51130,51133],{"className":51129},[31685],[34,51131,768],{"className":51132},[31685],[34,51134,51136],{"className":51135},[31792],[34,51137,51139],{"className":51138},[31796],[34,51140,51142],{"className":51141},[31801],[34,51143,51145],{"className":51144,"style":38486},[31805],[34,51146,51147,51150],{"style":37777},[34,51148],{"className":51149,"style":31814},[31813],[34,51151,51153],{"className":51152},[31818,31819,31820,31821],[34,51154,33759],{"className":51155,"style":33774},[31685,31686,31821],[34,51157],{"className":51158,"style":32604},[31845],[34,51160,42071],{"className":51161},[32024],[34,51163],{"className":51164,"style":32604},[31845],[34,51166,51168,51171],{"className":51167},[31676],[34,51169],{"className":51170,"style":38486},[31680],[34,51172,51174,51177],{"className":51173},[31685],[34,51175,768],{"className":51176},[31685],[34,51178,51180],{"className":51179},[31792],[34,51181,51183],{"className":51182},[31796],[34,51184,51186],{"className":51185},[31801],[34,51187,51189],{"className":51188,"style":38486},[31805],[34,51190,51191,51194],{"style":37777},[34,51192],{"className":51193,"style":31814},[31813],[34,51195,51197],{"className":51196},[31818,31819,31820,31821],[34,51198,33759],{"className":51199,"style":33774},[31685,31686,31821]," 的棋盘中，恰好有一个方格与其他不同（特殊方格）。用L型骨牌（占3个小方格）覆盖除特殊方格外所有方格，骨牌不重叠，求覆盖方法。",[108,51202,51203,33183,51205],{},[2627,51204,45950],{},[105,51206,51207,51369,51516,51526],{},[108,51208,51209,51210,51269,51270,51368],{},"由于棋盘大小是 ",[34,51211,51213,51231],{"className":51212},[31642],[34,51214,51216],{"className":51215},[31646],[31648,51217,51218],{"xmlns":31650},[31652,51219,51220,51228],{},[31655,51221,51222],{},[37624,51223,51224,51226],{},[31749,51225,15932],{},[31658,51227,33759],{},[31665,51229,51230],{"encoding":31667},"4^k",[34,51232,51234],{"className":51233,"ariaHidden":758},[31672],[34,51235,51237,51240],{"className":51236},[31676],[34,51238],{"className":51239,"style":38486},[31680],[34,51241,51243,51246],{"className":51242},[31685],[34,51244,15932],{"className":51245},[31685],[34,51247,51249],{"className":51248},[31792],[34,51250,51252],{"className":51251},[31796],[34,51253,51255],{"className":51254},[31801],[34,51256,51258],{"className":51257,"style":38486},[31805],[34,51259,51260,51263],{"style":37777},[34,51261],{"className":51262,"style":31814},[31813],[34,51264,51266],{"className":51265},[31818,31819,31820,31821],[34,51267,33759],{"className":51268,"style":33774},[31685,31686,31821],"，总共需要 ",[34,51271,51273,51303],{"className":51272},[31642],[34,51274,51276],{"className":51275},[31646],[31648,51277,51278],{"xmlns":31650},[31652,51279,51280,51300],{},[31655,51281,51282,51284,51290,51292,51294,51296,51298],{},[31662,51283,456],{"stretchy":702},[37624,51285,51286,51288],{},[31749,51287,15932],{},[31658,51289,33759],{},[31662,51291,32226],{},[31749,51293,640],{},[31662,51295,2557],{"stretchy":702},[31658,51297,38080],{"mathvariant":38079},[31749,51299,786],{},[31665,51301,51302],{"encoding":31667},"(4^k-1)\u002F3",[34,51304,51306,51353],{"className":51305,"ariaHidden":758},[31672],[34,51307,51309,51312,51315,51344,51347,51350],{"className":51308},[31676],[34,51310],{"className":51311,"style":43651},[31680],[34,51313,456],{"className":51314},[31782],[34,51316,51318,51321],{"className":51317},[31685],[34,51319,15932],{"className":51320},[31685],[34,51322,51324],{"className":51323},[31792],[34,51325,51327],{"className":51326},[31796],[34,51328,51330],{"className":51329},[31801],[34,51331,51333],{"className":51332,"style":38486},[31805],[34,51334,51335,51338],{"style":37777},[34,51336],{"className":51337,"style":31814},[31813],[34,51339,51341],{"className":51340},[31818,31819,31820,31821],[34,51342,33759],{"className":51343,"style":33774},[31685,31686,31821],[34,51345],{"className":51346,"style":32604},[31845],[34,51348,32226],{"className":51349},[32024],[34,51351],{"className":51352,"style":32604},[31845],[34,51354,51356,51359,51362,51365],{"className":51355},[31676],[34,51357],{"className":51358,"style":31774},[31680],[34,51360,640],{"className":51361},[31685],[34,51363,2557],{"className":51364},[31690],[34,51366,42614],{"className":51367},[31685]," 个L型骨牌。",[108,51370,51371,51373,51374,51515],{},[2627,51372,45886],{},"：将大棋盘十字划分为4个大小相同的 ",[34,51375,51377,51415],{"className":51376},[31642],[34,51378,51380],{"className":51379},[31646],[31648,51381,51382],{"xmlns":31650},[31652,51383,51384,51412],{},[31655,51385,51386,51398,51400],{},[37624,51387,51388,51390],{},[31749,51389,768],{},[31655,51391,51392,51394,51396],{},[31658,51393,33759],{},[31662,51395,32226],{},[31749,51397,640],{},[31662,51399,42071],{},[37624,51401,51402,51404],{},[31749,51403,768],{},[31655,51405,51406,51408,51410],{},[31658,51407,33759],{},[31662,51409,32226],{},[31749,51411,640],{},[31665,51413,51414],{"encoding":31667},"2^{k-1} \\times 2^{k-1}",[34,51416,51418,51471],{"className":51417,"ariaHidden":758},[31672],[34,51419,51421,51424,51462,51465,51468],{"className":51420},[31676],[34,51422],{"className":51423,"style":51126},[31680],[34,51425,51427,51430],{"className":51426},[31685],[34,51428,768],{"className":51429},[31685],[34,51431,51433],{"className":51432},[31792],[34,51434,51436],{"className":51435},[31796],[34,51437,51439],{"className":51438},[31801],[34,51440,51442],{"className":51441,"style":38486},[31805],[34,51443,51444,51447],{"style":37777},[34,51445],{"className":51446,"style":31814},[31813],[34,51448,51450],{"className":51449},[31818,31819,31820,31821],[34,51451,51453,51456,51459],{"className":51452},[31685,31821],[34,51454,33759],{"className":51455,"style":33774},[31685,31686,31821],[34,51457,32226],{"className":51458},[32024,31821],[34,51460,640],{"className":51461},[31685,31821],[34,51463],{"className":51464,"style":32604},[31845],[34,51466,42071],{"className":51467},[32024],[34,51469],{"className":51470,"style":32604},[31845],[34,51472,51474,51477],{"className":51473},[31676],[34,51475],{"className":51476,"style":38486},[31680],[34,51478,51480,51483],{"className":51479},[31685],[34,51481,768],{"className":51482},[31685],[34,51484,51486],{"className":51485},[31792],[34,51487,51489],{"className":51488},[31796],[34,51490,51492],{"className":51491},[31801],[34,51493,51495],{"className":51494,"style":38486},[31805],[34,51496,51497,51500],{"style":37777},[34,51498],{"className":51499,"style":31814},[31813],[34,51501,51503],{"className":51502},[31818,31819,31820,31821],[34,51504,51506,51509,51512],{"className":51505},[31685,31821],[34,51507,33759],{"className":51508,"style":33774},[31685,31686,31821],[34,51510,32226],{"className":51511},[32024,31821],[34,51513,640],{"className":51514},[31685,31821]," 的象限子棋盘。特殊方格必定落在其中一个象限中。",[108,51517,51518,51521,51522,51525],{},[2627,51519,51520],{},"骨牌放置与同构转化","：在棋盘正中央（4个象限的交界处），放置一个L型骨牌，去覆盖",[2627,51523,51524],{},"另外3个没有特殊方格的象限","的各一个角（即靠近中央的那个方格）。",[108,51527,51528,51531],{},[2627,51529,51530],{},"递归求解","：通过放置这个L型骨牌，这3个象限也被人为地制造出了一个“特殊方格”（即被骨牌覆盖的那个方格）。此时，4个子棋盘都各自拥有了1个“特殊方格”，完全转化为了与原问题形式相同的较小规模子问题，分别递归处理即可。",{"title":29,"searchDepth":43,"depth":43,"links":51533},[51534,51535],{"id":45646,"depth":49,"text":45647},{"id":45903,"depth":49,"text":45904},{},"\u002Fsource\u002Fcs\u002Falgo\u002Fnotes\u002Fchap3",{"title":45638,"description":29},"source\u002Fcs\u002Falgo\u002Fnotes\u002Fchap3","Nfw2PYWtXj02AYdP82DnNYFubJprGKyHO2UnLeIAZqo",{"id":51542,"title":51543,"body":51544,"description":29,"extension":85,"meta":58140,"navigation":52,"path":58141,"seo":58142,"stem":58143,"__hash__":58144},"notes\u002Fsource\u002Fcs\u002Falgo\u002Fnotes\u002Fchap4.md","第四章：动态规划（Dynamic Programming）",{"type":8,"value":51545,"toc":58136},[51546,51549,51553,51557,51564,51568,51588,51592,51595,51633,51637,51658,51660,51664,51668,52203,52264,52652,52656,53765,53769,53837,53841,54117,54121,55029,55033,55298,55302,56655,56659,57419,57423,57673,57677,57916,57920],[11,51547,51543],{"id":51548},"第四章动态规划dynamic-programming",[253,51550,51552],{"id":51551},"第一部分动态规划核心知识点","第一部分：动态规划核心知识点",[32490,51554,51556],{"id":51555},"_1-动态规划的基本概念","1. 动态规划的基本概念",[15,51558,51559,51560,51563],{},"动态规划是解决",[2627,51561,51562],{},"多阶段决策过程最优化","的一种数学方法。它将一个复杂的多阶段问题转换为一系列相互联系的单阶段子问题，逐个加以解决（填表法），从而避免了大量重复计算。",[32490,51565,51567],{"id":51566},"_2-适用动态规划求解问题的三大性质","2. 适用动态规划求解问题的三大性质",[105,51569,51570,51576,51582],{},[108,51571,51572,51575],{},[2627,51573,51574],{},"最优子结构（最基本条件）","：原问题的最优解包含其子问题的最优解。",[108,51577,51578,51581],{},[2627,51579,51580],{},"无后效性（马尔科夫性）","：某阶段状态一旦确定，就不受这个状态以后决策的影响。即当前状态以后的过程只与当前状态有关，与之前的状态无关。",[108,51583,51584,51587],{},[2627,51585,51586],{},"有重叠子问题","：子问题之间不独立，一个子问题在下一阶段决策中可能被多次用到。这也是动态规划优于分治法和穷举法的核心优势（通过填表保存结果，空间换时间）。",[32490,51589,51591],{"id":51590},"_3-动态规划求解的基本步骤","3. 动态规划求解的基本步骤",[15,51593,51594],{},"遇到实际问题时，通常遵循以下四步：",[2230,51596,51597,51603,51613,51627],{},[108,51598,51599,51602],{},[2627,51600,51601],{},"分析最优子结构","：分析问题的最优解特征，划分阶段。",[108,51604,51605,51608,51609,51612],{},[2627,51606,51607],{},"建立状态转移方程（递推公式）","：递归地定义最优解，确定动态规划数组（",[31,51610,51611],{},"dp","表）的含义。",[108,51614,51615,51618,51619,51622,51623,51626],{},[2627,51616,51617],{},"计算最优值","：通常采用",[2627,51620,51621],{},"自底向上","的填表方式计算，或者采用",[2627,51624,51625],{},"自顶向下带备忘录","的递归方式计算。",[108,51628,51629,51632],{},[2627,51630,51631],{},"构造最优解","：根据计算最优值时记录的信息，反向回溯（Traceback）构造出具体的最优解路径。",[32490,51634,51636],{"id":51635},"_4-动态规划与备忘录法的区别","4. 动态规划与备忘录法的区别",[105,51638,51639,51645],{},[108,51640,51641,51644],{},[2627,51642,51643],{},"相同点","：都用表格保存已解决的子问题答案，避免重复计算。",[108,51646,51647,51650,51651,51653,51654,51657],{},[2627,51648,51649],{},"不同点","：动态规划是",[2627,51652,51621],{},"递推；备忘录法是",[2627,51655,51656],{},"自顶向下","递归，控制结构与直接递归相同，但遇到已计算过的子问题直接查表返回。",[32188,51659],{},[253,51661,51663],{"id":51662},"第二部分经典例题与推导过程精讲","第二部分：经典例题与推导过程精讲",[32490,51665,51667],{"id":51666},"例题-1爬楼梯问题-斐波那契数列","例题 1：爬楼梯问题 \u002F 斐波那契数列",[105,51669,51670,51676,51815,51983,52120],{},[108,51671,51672,51675],{},[2627,51673,51674],{},"问题描述","：10级台阶，每次只能走1级或2级，求总走法数。",[108,51677,51678,51681,51682,51710,51711,51762,51763,51814],{},[2627,51679,51680],{},"推导过程","：到达第 ",[34,51683,51685,51698],{"className":51684},[31642],[34,51686,51688],{"className":51687},[31646],[31648,51689,51690],{"xmlns":31650},[31652,51691,51692,51696],{},[31655,51693,51694],{},[31658,51695,31955],{},[31665,51697,31955],{"encoding":31667},[34,51699,51701],{"className":51700,"ariaHidden":758},[31672],[34,51702,51704,51707],{"className":51703},[31676],[34,51705],{"className":51706,"style":33104},[31680],[34,51708,31955],{"className":51709},[31685,31686]," 级台阶只能从第 ",[34,51712,51714,51732],{"className":51713},[31642],[34,51715,51717],{"className":51716},[31646],[31648,51718,51719],{"xmlns":31650},[31652,51720,51721,51729],{},[31655,51722,51723,51725,51727],{},[31658,51724,31955],{},[31662,51726,32226],{},[31749,51728,640],{},[31665,51730,51731],{"encoding":31667},"i-1",[34,51733,51735,51753],{"className":51734,"ariaHidden":758},[31672],[34,51736,51738,51741,51744,51747,51750],{"className":51737},[31676],[34,51739],{"className":51740,"style":37322},[31680],[34,51742,31955],{"className":51743},[31685,31686],[34,51745],{"className":51746,"style":32604},[31845],[34,51748,32226],{"className":51749},[32024],[34,51751],{"className":51752,"style":32604},[31845],[34,51754,51756,51759],{"className":51755},[31676],[34,51757],{"className":51758,"style":32617},[31680],[34,51760,640],{"className":51761},[31685]," 级走1步，或者从第 ",[34,51764,51766,51784],{"className":51765},[31642],[34,51767,51769],{"className":51768},[31646],[31648,51770,51771],{"xmlns":31650},[31652,51772,51773,51781],{},[31655,51774,51775,51777,51779],{},[31658,51776,31955],{},[31662,51778,32226],{},[31749,51780,768],{},[31665,51782,51783],{"encoding":31667},"i-2",[34,51785,51787,51805],{"className":51786,"ariaHidden":758},[31672],[34,51788,51790,51793,51796,51799,51802],{"className":51789},[31676],[34,51791],{"className":51792,"style":37322},[31680],[34,51794,31955],{"className":51795},[31685,31686],[34,51797],{"className":51798,"style":32604},[31845],[34,51800,32226],{"className":51801},[32024],[34,51803],{"className":51804,"style":32604},[31845],[34,51806,51808,51811],{"className":51807},[31676],[34,51809],{"className":51810,"style":32617},[31680],[34,51812,768],{"className":51813},[31685]," 级走2步到达。",[108,51816,51817,2664,51820,347],{},[2627,51818,51819],{},"状态转移方程",[34,51821,51823,51872],{"className":51822},[31642],[34,51824,51826],{"className":51825},[31646],[31648,51827,51828],{"xmlns":31650},[31652,51829,51830,51869],{},[31655,51831,51832,51835,51837,51839,51841,51843,51845,51847,51849,51851,51853,51855,51857,51859,51861,51863,51865,51867],{},[31658,51833,51834],{},"F",[31662,51836,456],{"stretchy":702},[31658,51838,31660],{},[31662,51840,2557],{"stretchy":702},[31662,51842,202],{},[31658,51844,51834],{},[31662,51846,456],{"stretchy":702},[31658,51848,31660],{},[31662,51850,32226],{},[31749,51852,640],{},[31662,51854,2557],{"stretchy":702},[31662,51856,223],{},[31658,51858,51834],{},[31662,51860,456],{"stretchy":702},[31658,51862,31660],{},[31662,51864,32226],{},[31749,51866,768],{},[31662,51868,2557],{"stretchy":702},[31665,51870,51871],{"encoding":31667},"F(n) = F(n-1) + F(n-2)",[34,51873,51875,51902,51926,51947,51971],{"className":51874,"ariaHidden":758},[31672],[34,51876,51878,51881,51884,51887,51890,51893,51896,51899],{"className":51877},[31676],[34,51879],{"className":51880,"style":31774},[31680],[34,51882,51834],{"className":51883,"style":37422},[31685,31686],[34,51885,456],{"className":51886},[31782],[34,51888,31660],{"className":51889},[31685,31686],[34,51891,2557],{"className":51892},[31690],[34,51894],{"className":51895,"style":31846},[31845],[34,51897,202],{"className":51898},[31850],[34,51900],{"className":51901,"style":31846},[31845],[34,51903,51905,51908,51911,51914,51917,51920,51923],{"className":51904},[31676],[34,51906],{"className":51907,"style":31774},[31680],[34,51909,51834],{"className":51910,"style":37422},[31685,31686],[34,51912,456],{"className":51913},[31782],[34,51915,31660],{"className":51916},[31685,31686],[34,51918],{"className":51919,"style":32604},[31845],[34,51921,32226],{"className":51922},[32024],[34,51924],{"className":51925,"style":32604},[31845],[34,51927,51929,51932,51935,51938,51941,51944],{"className":51928},[31676],[34,51930],{"className":51931,"style":31774},[31680],[34,51933,640],{"className":51934},[31685],[34,51936,2557],{"className":51937},[31690],[34,51939],{"className":51940,"style":32604},[31845],[34,51942,223],{"className":51943},[32024],[34,51945],{"className":51946,"style":32604},[31845],[34,51948,51950,51953,51956,51959,51962,51965,51968],{"className":51949},[31676],[34,51951],{"className":51952,"style":31774},[31680],[34,51954,51834],{"className":51955,"style":37422},[31685,31686],[34,51957,456],{"className":51958},[31782],[34,51960,31660],{"className":51961},[31685,31686],[34,51963],{"className":51964,"style":32604},[31845],[34,51966,32226],{"className":51967},[32024],[34,51969],{"className":51970,"style":32604},[31845],[34,51972,51974,51977,51980],{"className":51973},[31676],[34,51975],{"className":51976,"style":31774},[31680],[34,51978,768],{"className":51979},[31685],[34,51981,2557],{"className":51982},[31690],[108,51984,51985,2664,51988,616,52054,347],{},[2627,51986,51987],{},"边界条件",[34,51989,51991,52015],{"className":51990},[31642],[34,51992,51994],{"className":51993},[31646],[31648,51995,51996],{"xmlns":31650},[31652,51997,51998,52012],{},[31655,51999,52000,52002,52004,52006,52008,52010],{},[31658,52001,51834],{},[31662,52003,456],{"stretchy":702},[31749,52005,640],{},[31662,52007,2557],{"stretchy":702},[31662,52009,202],{},[31749,52011,640],{},[31665,52013,52014],{"encoding":31667},"F(1)=1",[34,52016,52018,52045],{"className":52017,"ariaHidden":758},[31672],[34,52019,52021,52024,52027,52030,52033,52036,52039,52042],{"className":52020},[31676],[34,52022],{"className":52023,"style":31774},[31680],[34,52025,51834],{"className":52026,"style":37422},[31685,31686],[34,52028,456],{"className":52029},[31782],[34,52031,640],{"className":52032},[31685],[34,52034,2557],{"className":52035},[31690],[34,52037],{"className":52038,"style":31846},[31845],[34,52040,202],{"className":52041},[31850],[34,52043],{"className":52044,"style":31846},[31845],[34,52046,52048,52051],{"className":52047},[31676],[34,52049],{"className":52050,"style":32617},[31680],[34,52052,640],{"className":52053},[31685],[34,52055,52057,52081],{"className":52056},[31642],[34,52058,52060],{"className":52059},[31646],[31648,52061,52062],{"xmlns":31650},[31652,52063,52064,52078],{},[31655,52065,52066,52068,52070,52072,52074,52076],{},[31658,52067,51834],{},[31662,52069,456],{"stretchy":702},[31749,52071,768],{},[31662,52073,2557],{"stretchy":702},[31662,52075,202],{},[31749,52077,768],{},[31665,52079,52080],{"encoding":31667},"F(2)=2",[34,52082,52084,52111],{"className":52083,"ariaHidden":758},[31672],[34,52085,52087,52090,52093,52096,52099,52102,52105,52108],{"className":52086},[31676],[34,52088],{"className":52089,"style":31774},[31680],[34,52091,51834],{"className":52092,"style":37422},[31685,31686],[34,52094,456],{"className":52095},[31782],[34,52097,768],{"className":52098},[31685],[34,52100,2557],{"className":52101},[31690],[34,52103],{"className":52104,"style":31846},[31845],[34,52106,202],{"className":52107},[31850],[34,52109],{"className":52110,"style":31846},[31845],[34,52112,52114,52117],{"className":52113},[31676],[34,52115],{"className":52116,"style":32617},[31680],[34,52118,768],{"className":52119},[31685],[108,52121,52122,52124,52125,52127,52128,46528,52156,52159,52160,347],{},[2627,52123,46023],{},"：通过一维数组 ",[31,52126,51611],{}," 从 3 循环到 ",[34,52129,52131,52144],{"className":52130},[31642],[34,52132,52134],{"className":52133},[31646],[31648,52135,52136],{"xmlns":31650},[31652,52137,52138,52142],{},[31655,52139,52140],{},[31658,52141,31660],{},[31665,52143,31660],{"encoding":31667},[34,52145,52147],{"className":52146,"ariaHidden":758},[31672],[34,52148,52150,52153],{"className":52149},[31676],[34,52151],{"className":52152,"style":32527},[31680],[34,52154,31660],{"className":52155},[31685,31686],[31,52157,52158],{},"dp[i] = dp[i-1] + dp[i-2]","，时间复杂度 ",[34,52161,52163,52182],{"className":52162},[31642],[34,52164,52166],{"className":52165},[31646],[31648,52167,52168],{"xmlns":31650},[31652,52169,52170,52180],{},[31655,52171,52172,52174,52176,52178],{},[31658,52173,37890],{},[31662,52175,456],{"stretchy":702},[31658,52177,31660],{},[31662,52179,2557],{"stretchy":702},[31665,52181,46493],{"encoding":31667},[34,52183,52185],{"className":52184,"ariaHidden":758},[31672],[34,52186,52188,52191,52194,52197,52200],{"className":52187},[31676],[34,52189],{"className":52190,"style":31774},[31680],[34,52192,37890],{"className":52193,"style":34452},[31685,31686],[34,52195,456],{"className":52196},[31782],[34,52198,31660],{"className":52199},[31685,31686],[34,52201,2557],{"className":52202},[31690],[32490,52204,52206,52207],{"id":52205},"例题-2求解组合数-cnmcnmcnm","例题 2：求解组合数 ",[34,52208,52210,52234],{"className":52209},[31642],[34,52211,52213],{"className":52212},[31646],[31648,52214,52215],{"xmlns":31650},[31652,52216,52217,52231],{},[31655,52218,52219,52221,52223,52225,52227,52229],{},[31658,52220,47384],{},[31662,52222,456],{"stretchy":702},[31658,52224,31660],{},[31662,52226,31960],{"separator":758},[31658,52228,31759],{},[31662,52230,2557],{"stretchy":702},[31665,52232,52233],{"encoding":31667},"C(n,m)",[34,52235,52237],{"className":52236,"ariaHidden":758},[31672],[34,52238,52240,52243,52246,52249,52252,52255,52258,52261],{"className":52239},[31676],[34,52241],{"className":52242,"style":31774},[31680],[34,52244,47384],{"className":52245,"style":38706},[31685,31686],[34,52247,456],{"className":52248},[31782],[34,52250,31660],{"className":52251},[31685,31686],[34,52253,31960],{"className":52254},[32119],[34,52256],{"className":52257,"style":32123},[31845],[34,52259,31759],{"className":52260},[31685,31686],[34,52262,2557],{"className":52263},[31690],[105,52265,52266,52358,52557,52564,52574],{},[108,52267,52268,52270,52271,52357],{},[2627,52269,51674],{},"：求组合数 ",[34,52272,52274,52294],{"className":52273},[31642],[34,52275,52277],{"className":52276},[31646],[31648,52278,52279],{"xmlns":31650},[31652,52280,52281,52291],{},[31655,52282,52283],{},[38846,52284,52285,52287,52289],{},[31658,52286,47384],{},[31658,52288,31660],{},[31658,52290,31759],{},[31665,52292,52293],{"encoding":31667},"C_n^m",[34,52295,52297],{"className":52296,"ariaHidden":758},[31672],[34,52298,52300,52304],{"className":52299},[31676],[34,52301],{"className":52302,"style":52303},[31680],"height:0.9303em;vertical-align:-0.247em;",[34,52305,52307,52310],{"className":52306},[31685],[34,52308,47384],{"className":52309,"style":38706},[31685,31686],[34,52311,52313],{"className":52312},[31792],[34,52314,52316,52348],{"className":52315},[31796,31797],[34,52317,52319,52345],{"className":52318},[31801],[34,52320,52322,52334],{"className":52321,"style":37959},[31805],[34,52323,52325,52328],{"style":52324},"top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;",[34,52326],{"className":52327,"style":31814},[31813],[34,52329,52331],{"className":52330},[31818,31819,31820,31821],[34,52332,31660],{"className":52333},[31685,31686,31821],[34,52335,52336,52339],{"style":37777},[34,52337],{"className":52338,"style":31814},[31813],[34,52340,52342],{"className":52341},[31818,31819,31820,31821],[34,52343,31759],{"className":52344},[31685,31686,31821],[34,52346,31829],{"className":52347},[31828],[34,52349,52351],{"className":52350},[31801],[34,52352,52355],{"className":52353,"style":52354},[31805],"height:0.247em;",[34,52356],{},"，直接用阶乘公式易溢出且低效。",[108,52359,52360,52362,52363,52391,52392,52420,52421,52391,52449,52477,52478,52391,52506,52556],{},[2627,52361,51680],{},"：利用组合数学性质，从 ",[34,52364,52366,52379],{"className":52365},[31642],[34,52367,52369],{"className":52368},[31646],[31648,52370,52371],{"xmlns":31650},[31652,52372,52373,52377],{},[31655,52374,52375],{},[31658,52376,31660],{},[31665,52378,31660],{"encoding":31667},[34,52380,52382],{"className":52381,"ariaHidden":758},[31672],[34,52383,52385,52388],{"className":52384},[31676],[34,52386],{"className":52387,"style":32527},[31680],[34,52389,31660],{"className":52390},[31685,31686]," 个元素选 ",[34,52393,52395,52408],{"className":52394},[31642],[34,52396,52398],{"className":52397},[31646],[31648,52399,52400],{"xmlns":31650},[31652,52401,52402,52406],{},[31655,52403,52404],{},[31658,52405,31759],{},[31665,52407,31759],{"encoding":31667},[34,52409,52411],{"className":52410,"ariaHidden":758},[31672],[34,52412,52414,52417],{"className":52413},[31676],[34,52415],{"className":52416,"style":32527},[31680],[34,52418,31759],{"className":52419},[31685,31686]," 个，等于“不包含第 ",[34,52422,52424,52437],{"className":52423},[31642],[34,52425,52427],{"className":52426},[31646],[31648,52428,52429],{"xmlns":31650},[31652,52430,52431,52435],{},[31655,52432,52433],{},[31658,52434,31660],{},[31665,52436,31660],{"encoding":31667},[34,52438,52440],{"className":52439,"ariaHidden":758},[31672],[34,52441,52443,52446],{"className":52442},[31676],[34,52444],{"className":52445,"style":32527},[31680],[34,52447,31660],{"className":52448},[31685,31686],[34,52450,52452,52465],{"className":52451},[31642],[34,52453,52455],{"className":52454},[31646],[31648,52456,52457],{"xmlns":31650},[31652,52458,52459,52463],{},[31655,52460,52461],{},[31658,52462,31759],{},[31665,52464,31759],{"encoding":31667},[34,52466,52468],{"className":52467,"ariaHidden":758},[31672],[34,52469,52471,52474],{"className":52470},[31676],[34,52472],{"className":52473,"style":32527},[31680],[34,52475,31759],{"className":52476},[31685,31686]," 个”加上“包含第 ",[34,52479,52481,52494],{"className":52480},[31642],[34,52482,52484],{"className":52483},[31646],[31648,52485,52486],{"xmlns":31650},[31652,52487,52488,52492],{},[31655,52489,52490],{},[31658,52491,31660],{},[31665,52493,31660],{"encoding":31667},[34,52495,52497],{"className":52496,"ariaHidden":758},[31672],[34,52498,52500,52503],{"className":52499},[31676],[34,52501],{"className":52502,"style":32527},[31680],[34,52504,31660],{"className":52505},[31685,31686],[34,52507,52509,52526],{"className":52508},[31642],[34,52510,52512],{"className":52511},[31646],[31648,52513,52514],{"xmlns":31650},[31652,52515,52516,52524],{},[31655,52517,52518,52520,52522],{},[31658,52519,31759],{},[31662,52521,32226],{},[31749,52523,640],{},[31665,52525,36321],{"encoding":31667},[34,52527,52529,52547],{"className":52528,"ariaHidden":758},[31672],[34,52530,52532,52535,52538,52541,52544],{"className":52531},[31676],[34,52533],{"className":52534,"style":32597},[31680],[34,52536,31759],{"className":52537},[31685,31686],[34,52539],{"className":52540,"style":32604},[31845],[34,52542,32226],{"className":52543},[32024],[34,52545],{"className":52546,"style":32604},[31845],[34,52548,52550,52553],{"className":52549},[31676],[34,52551],{"className":52552,"style":32617},[31680],[34,52554,640],{"className":52555},[31685]," 个”。",[108,52558,52559,2664,52561,347],{},[2627,52560,51819],{},[31,52562,52563],{},"C[i][j] = C[i-1][j-1] + C[i-1][j]",[108,52565,52566,2664,52568,616,52571,347],{},[2627,52567,51987],{},[31,52569,52570],{},"C[i] = 1",[31,52572,52573],{},"C[i][i] = 1",[108,52575,52576,52578,52579,347],{},[2627,52577,46353],{},"：时间复杂度由递归的指数级降为动态规划填表的 ",[34,52580,52582,52605],{"className":52581},[31642],[34,52583,52585],{"className":52584},[31646],[31648,52586,52587],{"xmlns":31650},[31652,52588,52589,52603],{},[31655,52590,52591,52593,52595,52601],{},[31658,52592,37890],{},[31662,52594,456],{"stretchy":702},[37624,52596,52597,52599],{},[31658,52598,31660],{},[31749,52600,768],{},[31662,52602,2557],{"stretchy":702},[31665,52604,46174],{"encoding":31667},[34,52606,52608],{"className":52607,"ariaHidden":758},[31672],[34,52609,52611,52614,52617,52620,52649],{"className":52610},[31676],[34,52612],{"className":52613,"style":37755},[31680],[34,52615,37890],{"className":52616,"style":34452},[31685,31686],[34,52618,456],{"className":52619},[31782],[34,52621,52623,52626],{"className":52622},[31685],[34,52624,31660],{"className":52625},[31685,31686],[34,52627,52629],{"className":52628},[31792],[34,52630,52632],{"className":52631},[31796],[34,52633,52635],{"className":52634},[31801],[34,52636,52638],{"className":52637,"style":37774},[31805],[34,52639,52640,52643],{"style":37777},[34,52641],{"className":52642,"style":31814},[31813],[34,52644,52646],{"className":52645},[31818,31819,31820,31821],[34,52647,768],{"className":52648},[31685,31821],[34,52650,2557],{"className":52651},[31690],[32490,52653,52655],{"id":52654},"例题-3整数分解问题","例题 3：整数分解问题",[105,52657,52658,52719,53699,53756],{},[108,52659,52660,35203,52662,35232,52690,52718],{},[2627,52661,51674],{},[34,52663,52665,52678],{"className":52664},[31642],[34,52666,52668],{"className":52667},[31646],[31648,52669,52670],{"xmlns":31650},[31652,52671,52672,52676],{},[31655,52673,52674],{},[31658,52675,31660],{},[31665,52677,31660],{"encoding":31667},[34,52679,52681],{"className":52680,"ariaHidden":758},[31672],[34,52682,52684,52687],{"className":52683},[31676],[34,52685],{"className":52686,"style":32527},[31680],[34,52688,31660],{"className":52689},[31685,31686],[34,52691,52693,52706],{"className":52692},[31642],[34,52694,52696],{"className":52695},[31646],[31648,52697,52698],{"xmlns":31650},[31652,52699,52700,52704],{},[31655,52701,52702],{},[31658,52703,31759],{},[31665,52705,31759],{"encoding":31667},[34,52707,52709],{"className":52708,"ariaHidden":758},[31672],[34,52710,52712,52715],{"className":52711},[31676],[34,52713],{"className":52714,"style":32527},[31680],[34,52716,31759],{"className":52717},[31685,31686]," 的几种方案。",[108,52720,52721,34554,52723,52780,52781],{},[2627,52722,51680],{},[34,52724,52726,52750],{"className":52725},[31642],[34,52727,52729],{"className":52728},[31646],[31648,52730,52731],{"xmlns":31650},[31652,52732,52733,52747],{},[31655,52734,52735,52737,52739,52741,52743,52745],{},[31658,52736,31739],{},[31662,52738,456],{"stretchy":702},[31658,52740,31660],{},[31662,52742,31960],{"separator":758},[31658,52744,31759],{},[31662,52746,2557],{"stretchy":702},[31665,52748,52749],{"encoding":31667},"f(n,m)",[34,52751,52753],{"className":52752,"ariaHidden":758},[31672],[34,52754,52756,52759,52762,52765,52768,52771,52774,52777],{"className":52755},[31676],[34,52757],{"className":52758,"style":31774},[31680],[34,52760,31739],{"className":52761,"style":31778},[31685,31686],[34,52763,456],{"className":52764},[31782],[34,52766,31660],{"className":52767},[31685,31686],[34,52769,31960],{"className":52770},[32119],[34,52772],{"className":52773,"style":32123},[31845],[34,52775,31759],{"className":52776},[31685,31686],[34,52778,2557],{"className":52779},[31690]," 为方案数。\n",[105,52782,52783,52922,53293],{},[108,52784,35328,52785,52835,52836,52864,52865,347],{},[34,52786,52788,52805],{"className":52787},[31642],[34,52789,52791],{"className":52790},[31646],[31648,52792,52793],{"xmlns":31650},[31652,52794,52795,52803],{},[31655,52796,52797,52799,52801],{},[31658,52798,31660],{},[31662,52800,932],{},[31658,52802,31759],{},[31665,52804,35532],{"encoding":31667},[34,52806,52808,52826],{"className":52807,"ariaHidden":758},[31672],[34,52809,52811,52814,52817,52820,52823],{"className":52810},[31676],[34,52812],{"className":52813,"style":34812},[31680],[34,52815,31660],{"className":52816},[31685,31686],[34,52818],{"className":52819,"style":31846},[31845],[34,52821,932],{"className":52822},[31850],[34,52824],{"className":52825,"style":31846},[31845],[34,52827,52829,52832],{"className":52828},[31676],[34,52830],{"className":52831,"style":32527},[31680],[34,52833,31759],{"className":52834},[31685,31686]," 时，最大加数不可能超过 ",[34,52837,52839,52852],{"className":52838},[31642],[34,52840,52842],{"className":52841},[31646],[31648,52843,52844],{"xmlns":31650},[31652,52845,52846,52850],{},[31655,52847,52848],{},[31658,52849,31660],{},[31665,52851,31660],{"encoding":31667},[34,52853,52855],{"className":52854,"ariaHidden":758},[31672],[34,52856,52858,52861],{"className":52857},[31676],[34,52859],{"className":52860,"style":32527},[31680],[34,52862,31660],{"className":52863},[31685,31686],"，等价于 ",[34,52866,52868,52892],{"className":52867},[31642],[34,52869,52871],{"className":52870},[31646],[31648,52872,52873],{"xmlns":31650},[31652,52874,52875,52889],{},[31655,52876,52877,52879,52881,52883,52885,52887],{},[31658,52878,31739],{},[31662,52880,456],{"stretchy":702},[31658,52882,31660],{},[31662,52884,31960],{"separator":758},[31658,52886,31660],{},[31662,52888,2557],{"stretchy":702},[31665,52890,52891],{"encoding":31667},"f(n,n)",[34,52893,52895],{"className":52894,"ariaHidden":758},[31672],[34,52896,52898,52901,52904,52907,52910,52913,52916,52919],{"className":52897},[31676],[34,52899],{"className":52900,"style":31774},[31680],[34,52902,31739],{"className":52903,"style":31778},[31685,31686],[34,52905,456],{"className":52906},[31782],[34,52908,31660],{"className":52909},[31685,31686],[34,52911,31960],{"className":52912},[32119],[34,52914],{"className":52915,"style":32123},[31845],[34,52917,31660],{"className":52918},[31685,31686],[34,52920,2557],{"className":52921},[31690],[108,52923,35328,52924,52974,52975,53003,53004,53032,53033,53061,53062,53141,53142,347],{},[34,52925,52927,52944],{"className":52926},[31642],[34,52928,52930],{"className":52929},[31646],[31648,52931,52932],{"xmlns":31650},[31652,52933,52934,52942],{},[31655,52935,52936,52938,52940],{},[31658,52937,31660],{},[31662,52939,202],{},[31658,52941,31759],{},[31665,52943,35672],{"encoding":31667},[34,52945,52947,52965],{"className":52946,"ariaHidden":758},[31672],[34,52948,52950,52953,52956,52959,52962],{"className":52949},[31676],[34,52951],{"className":52952,"style":32527},[31680],[34,52954,31660],{"className":52955},[31685,31686],[34,52957],{"className":52958,"style":31846},[31845],[34,52960,202],{"className":52961},[31850],[34,52963],{"className":52964,"style":31846},[31845],[34,52966,52968,52971],{"className":52967},[31676],[34,52969],{"className":52970,"style":32527},[31680],[34,52972,31759],{"className":52973},[31685,31686]," 时，包含 ",[34,52976,52978,52991],{"className":52977},[31642],[34,52979,52981],{"className":52980},[31646],[31648,52982,52983],{"xmlns":31650},[31652,52984,52985,52989],{},[31655,52986,52987],{},[31658,52988,31759],{},[31665,52990,31759],{"encoding":31667},[34,52992,52994],{"className":52993,"ariaHidden":758},[31672],[34,52995,52997,53000],{"className":52996},[31676],[34,52998],{"className":52999,"style":32527},[31680],[34,53001,31759],{"className":53002},[31685,31686]," 的拆分只有 1 种（即 ",[34,53005,53007,53020],{"className":53006},[31642],[34,53008,53010],{"className":53009},[31646],[31648,53011,53012],{"xmlns":31650},[31652,53013,53014,53018],{},[31655,53015,53016],{},[31658,53017,31660],{},[31665,53019,31660],{"encoding":31667},[34,53021,53023],{"className":53022,"ariaHidden":758},[31672],[34,53024,53026,53029],{"className":53025},[31676],[34,53027],{"className":53028,"style":32527},[31680],[34,53030,31660],{"className":53031},[31685,31686]," 本身），不包含 ",[34,53034,53036,53049],{"className":53035},[31642],[34,53037,53039],{"className":53038},[31646],[31648,53040,53041],{"xmlns":31650},[31652,53042,53043,53047],{},[31655,53044,53045],{},[31658,53046,31759],{},[31665,53048,31759],{"encoding":31667},[34,53050,53052],{"className":53051,"ariaHidden":758},[31672],[34,53053,53055,53058],{"className":53054},[31676],[34,53056],{"className":53057,"style":32527},[31680],[34,53059,31759],{"className":53060},[31685,31686]," 的拆分为 ",[34,53063,53065,53093],{"className":53064},[31642],[34,53066,53068],{"className":53067},[31646],[31648,53069,53070],{"xmlns":31650},[31652,53071,53072,53090],{},[31655,53073,53074,53076,53078,53080,53082,53084,53086,53088],{},[31658,53075,31739],{},[31662,53077,456],{"stretchy":702},[31658,53079,31660],{},[31662,53081,31960],{"separator":758},[31658,53083,31759],{},[31662,53085,32226],{},[31749,53087,640],{},[31662,53089,2557],{"stretchy":702},[31665,53091,53092],{"encoding":31667},"f(n,m-1)",[34,53094,53096,53129],{"className":53095,"ariaHidden":758},[31672],[34,53097,53099,53102,53105,53108,53111,53114,53117,53120,53123,53126],{"className":53098},[31676],[34,53100],{"className":53101,"style":31774},[31680],[34,53103,31739],{"className":53104,"style":31778},[31685,31686],[34,53106,456],{"className":53107},[31782],[34,53109,31660],{"className":53110},[31685,31686],[34,53112,31960],{"className":53113},[32119],[34,53115],{"className":53116,"style":32123},[31845],[34,53118,31759],{"className":53119},[31685,31686],[34,53121],{"className":53122,"style":32604},[31845],[34,53124,32226],{"className":53125},[32024],[34,53127],{"className":53128,"style":32604},[31845],[34,53130,53132,53135,53138],{"className":53131},[31676],[34,53133],{"className":53134,"style":31774},[31680],[34,53136,640],{"className":53137},[31685],[34,53139,2557],{"className":53140},[31690],"，故 ",[34,53143,53145,53191],{"className":53144},[31642],[34,53146,53148],{"className":53147},[31646],[31648,53149,53150],{"xmlns":31650},[31652,53151,53152,53188],{},[31655,53153,53154,53156,53158,53160,53162,53164,53166,53168,53170,53172,53174,53176,53178,53180,53182,53184,53186],{},[31658,53155,31739],{},[31662,53157,456],{"stretchy":702},[31658,53159,31660],{},[31662,53161,31960],{"separator":758},[31658,53163,31759],{},[31662,53165,2557],{"stretchy":702},[31662,53167,202],{},[31658,53169,31739],{},[31662,53171,456],{"stretchy":702},[31658,53173,31660],{},[31662,53175,31960],{"separator":758},[31658,53177,31759],{},[31662,53179,32226],{},[31749,53181,640],{},[31662,53183,2557],{"stretchy":702},[31662,53185,223],{},[31749,53187,640],{},[31665,53189,53190],{"encoding":31667},"f(n,m) = f(n, m-1) + 1",[34,53192,53194,53230,53263,53284],{"className":53193,"ariaHidden":758},[31672],[34,53195,53197,53200,53203,53206,53209,53212,53215,53218,53221,53224,53227],{"className":53196},[31676],[34,53198],{"className":53199,"style":31774},[31680],[34,53201,31739],{"className":53202,"style":31778},[31685,31686],[34,53204,456],{"className":53205},[31782],[34,53207,31660],{"className":53208},[31685,31686],[34,53210,31960],{"className":53211},[32119],[34,53213],{"className":53214,"style":32123},[31845],[34,53216,31759],{"className":53217},[31685,31686],[34,53219,2557],{"className":53220},[31690],[34,53222],{"className":53223,"style":31846},[31845],[34,53225,202],{"className":53226},[31850],[34,53228],{"className":53229,"style":31846},[31845],[34,53231,53233,53236,53239,53242,53245,53248,53251,53254,53257,53260],{"className":53232},[31676],[34,53234],{"className":53235,"style":31774},[31680],[34,53237,31739],{"className":53238,"style":31778},[31685,31686],[34,53240,456],{"className":53241},[31782],[34,53243,31660],{"className":53244},[31685,31686],[34,53246,31960],{"className":53247},[32119],[34,53249],{"className":53250,"style":32123},[31845],[34,53252,31759],{"className":53253},[31685,31686],[34,53255],{"className":53256,"style":32604},[31845],[34,53258,32226],{"className":53259},[32024],[34,53261],{"className":53262,"style":32604},[31845],[34,53264,53266,53269,53272,53275,53278,53281],{"className":53265},[31676],[34,53267],{"className":53268,"style":31774},[31680],[34,53270,640],{"className":53271},[31685],[34,53273,2557],{"className":53274},[31690],[34,53276],{"className":53277,"style":32604},[31845],[34,53279,223],{"className":53280},[32024],[34,53282],{"className":53283,"style":32604},[31845],[34,53285,53287,53290],{"className":53286},[31676],[34,53288],{"className":53289,"style":32617},[31680],[34,53291,640],{"className":53292},[31685],[108,53294,35328,53295,53346,53347,47649,53350,53378,53379,53429,53430,53458,53459,53538,53539,47649,53542,53570,53571,53458,53621,31901],{},[34,53296,53298,53316],{"className":53297},[31642],[34,53299,53301],{"className":53300},[31646],[31648,53302,53303],{"xmlns":31650},[31652,53304,53305,53313],{},[31655,53306,53307,53309,53311],{},[31658,53308,31660],{},[31662,53310,29622],{},[31658,53312,31759],{},[31665,53314,53315],{"encoding":31667},"n>m",[34,53317,53319,53337],{"className":53318,"ariaHidden":758},[31672],[34,53320,53322,53325,53328,53331,53334],{"className":53321},[31676],[34,53323],{"className":53324,"style":34812},[31680],[34,53326,31660],{"className":53327},[31685,31686],[34,53329],{"className":53330,"style":31846},[31845],[34,53332,29622],{"className":53333},[31850],[34,53335],{"className":53336,"style":31846},[31845],[34,53338,53340,53343],{"className":53339},[31676],[34,53341],{"className":53342,"style":32527},[31680],[34,53344,31759],{"className":53345},[31685,31686]," 时，分为",[2627,53348,53349],{},"包含",[34,53351,53353,53366],{"className":53352},[31642],[34,53354,53356],{"className":53355},[31646],[31648,53357,53358],{"xmlns":31650},[31652,53359,53360,53364],{},[31655,53361,53362],{},[31658,53363,31759],{},[31665,53365,31759],{"encoding":31667},[34,53367,53369],{"className":53368,"ariaHidden":758},[31672],[34,53370,53372,53375],{"className":53371},[31676],[34,53373],{"className":53374,"style":32527},[31680],[34,53376,31759],{"className":53377},[31685,31686],"（将剩下的 ",[34,53380,53382,53399],{"className":53381},[31642],[34,53383,53385],{"className":53384},[31646],[31648,53386,53387],{"xmlns":31650},[31652,53388,53389,53397],{},[31655,53390,53391,53393,53395],{},[31658,53392,31660],{},[31662,53394,32226],{},[31658,53396,31759],{},[31665,53398,36128],{"encoding":31667},[34,53400,53402,53420],{"className":53401,"ariaHidden":758},[31672],[34,53403,53405,53408,53411,53414,53417],{"className":53404},[31676],[34,53406],{"className":53407,"style":32597},[31680],[34,53409,31660],{"className":53410},[31685,31686],[34,53412],{"className":53413,"style":32604},[31845],[34,53415,32226],{"className":53416},[32024],[34,53418],{"className":53419,"style":32604},[31845],[34,53421,53423,53426],{"className":53422},[31676],[34,53424],{"className":53425,"style":32527},[31680],[34,53427,31759],{"className":53428},[31685,31686]," 用最大不超过 ",[34,53431,53433,53446],{"className":53432},[31642],[34,53434,53436],{"className":53435},[31646],[31648,53437,53438],{"xmlns":31650},[31652,53439,53440,53444],{},[31655,53441,53442],{},[31658,53443,31759],{},[31665,53445,31759],{"encoding":31667},[34,53447,53449],{"className":53448,"ariaHidden":758},[31672],[34,53450,53452,53455],{"className":53451},[31676],[34,53453],{"className":53454,"style":32527},[31680],[34,53456,31759],{"className":53457},[31685,31686]," 的数拆分，即 ",[34,53460,53462,53490],{"className":53461},[31642],[34,53463,53465],{"className":53464},[31646],[31648,53466,53467],{"xmlns":31650},[31652,53468,53469,53487],{},[31655,53470,53471,53473,53475,53477,53479,53481,53483,53485],{},[31658,53472,31739],{},[31662,53474,456],{"stretchy":702},[31658,53476,31660],{},[31662,53478,32226],{},[31658,53480,31759],{},[31662,53482,31960],{"separator":758},[31658,53484,31759],{},[31662,53486,2557],{"stretchy":702},[31665,53488,53489],{"encoding":31667},"f(n-m,m)",[34,53491,53493,53517],{"className":53492,"ariaHidden":758},[31672],[34,53494,53496,53499,53502,53505,53508,53511,53514],{"className":53495},[31676],[34,53497],{"className":53498,"style":31774},[31680],[34,53500,31739],{"className":53501,"style":31778},[31685,31686],[34,53503,456],{"className":53504},[31782],[34,53506,31660],{"className":53507},[31685,31686],[34,53509],{"className":53510,"style":32604},[31845],[34,53512,32226],{"className":53513},[32024],[34,53515],{"className":53516,"style":32604},[31845],[34,53518,53520,53523,53526,53529,53532,53535],{"className":53519},[31676],[34,53521],{"className":53522,"style":31774},[31680],[34,53524,31759],{"className":53525},[31685,31686],[34,53527,31960],{"className":53528},[32119],[34,53530],{"className":53531,"style":32123},[31845],[34,53533,31759],{"className":53534},[31685,31686],[34,53536,2557],{"className":53537},[31690],"）和",[2627,53540,53541],{},"不包含",[34,53543,53545,53558],{"className":53544},[31642],[34,53546,53548],{"className":53547},[31646],[31648,53549,53550],{"xmlns":31650},[31652,53551,53552,53556],{},[31655,53553,53554],{},[31658,53555,31759],{},[31665,53557,31759],{"encoding":31667},[34,53559,53561],{"className":53560,"ariaHidden":758},[31672],[34,53562,53564,53567],{"className":53563},[31676],[34,53565],{"className":53566,"style":32527},[31680],[34,53568,31759],{"className":53569},[31685,31686],"（用最大不超过 ",[34,53572,53574,53591],{"className":53573},[31642],[34,53575,53577],{"className":53576},[31646],[31648,53578,53579],{"xmlns":31650},[31652,53580,53581,53589],{},[31655,53582,53583,53585,53587],{},[31658,53584,31759],{},[31662,53586,32226],{},[31749,53588,640],{},[31665,53590,36321],{"encoding":31667},[34,53592,53594,53612],{"className":53593,"ariaHidden":758},[31672],[34,53595,53597,53600,53603,53606,53609],{"className":53596},[31676],[34,53598],{"className":53599,"style":32597},[31680],[34,53601,31759],{"className":53602},[31685,31686],[34,53604],{"className":53605,"style":32604},[31845],[34,53607,32226],{"className":53608},[32024],[34,53610],{"className":53611,"style":32604},[31845],[34,53613,53615,53618],{"className":53614},[31676],[34,53616],{"className":53617,"style":32617},[31680],[34,53619,640],{"className":53620},[31685],[34,53622,53624,53651],{"className":53623},[31642],[34,53625,53627],{"className":53626},[31646],[31648,53628,53629],{"xmlns":31650},[31652,53630,53631,53649],{},[31655,53632,53633,53635,53637,53639,53641,53643,53645,53647],{},[31658,53634,31739],{},[31662,53636,456],{"stretchy":702},[31658,53638,31660],{},[31662,53640,31960],{"separator":758},[31658,53642,31759],{},[31662,53644,32226],{},[31749,53646,640],{},[31662,53648,2557],{"stretchy":702},[31665,53650,53092],{"encoding":31667},[34,53652,53654,53687],{"className":53653,"ariaHidden":758},[31672],[34,53655,53657,53660,53663,53666,53669,53672,53675,53678,53681,53684],{"className":53656},[31676],[34,53658],{"className":53659,"style":31774},[31680],[34,53661,31739],{"className":53662,"style":31778},[31685,31686],[34,53664,456],{"className":53665},[31782],[34,53667,31660],{"className":53668},[31685,31686],[34,53670,31960],{"className":53671},[32119],[34,53673],{"className":53674,"style":32123},[31845],[34,53676,31759],{"className":53677},[31685,31686],[34,53679],{"className":53680,"style":32604},[31845],[34,53682,32226],{"className":53683},[32024],[34,53685],{"className":53686,"style":32604},[31845],[34,53688,53690,53693,53696],{"className":53689},[31676],[34,53691],{"className":53692,"style":31774},[31680],[34,53694,640],{"className":53695},[31685],[34,53697,2557],{"className":53698},[31690],[108,53700,53701,33183,53703,1840,53706,2557],{},[2627,53702,51819],{},[31,53704,53705],{},"dp[n][m] = dp[n][m-1] + dp[n-m][m]",[34,53707,53709,53726],{"className":53708},[31642],[34,53710,53712],{"className":53711},[31646],[31648,53713,53714],{"xmlns":31650},[31652,53715,53716,53724],{},[31655,53717,53718,53720,53722],{},[31658,53719,31660],{},[31662,53721,29622],{},[31658,53723,31759],{},[31665,53725,53315],{"encoding":31667},[34,53727,53729,53747],{"className":53728,"ariaHidden":758},[31672],[34,53730,53732,53735,53738,53741,53744],{"className":53731},[31676],[34,53733],{"className":53734,"style":34812},[31680],[34,53736,31660],{"className":53737},[31685,31686],[34,53739],{"className":53740,"style":31846},[31845],[34,53742,29622],{"className":53743},[31850],[34,53745],{"className":53746,"style":31846},[31845],[34,53748,53750,53753],{"className":53749},[31676],[34,53751],{"className":53752,"style":32527},[31680],[34,53754,31759],{"className":53755},[31685,31686],[108,53757,53758,53761,53762,53764],{},[2627,53759,53760],{},"过程","：课件中详尽推导了 ",[31,53763,51611],{}," 的计算表，最终得到 7 种。",[32490,53766,53768],{"id":53767},"例题-4最大连续子序列和","例题 4：最大连续子序列和",[105,53770,53771,53776,53792,53805,53812],{},[108,53772,53773,53775],{},[2627,53774,51674],{},"：求序列中连续子数组的最大和。",[108,53777,53778,2664,53781,53784,53785,53787,53788,53791],{},[2627,53779,53780],{},"状态定义",[31,53782,53783],{},"dp[j]"," 表示以第 ",[31,53786,31968],{}," 个元素",[2627,53789,53790],{},"结尾","的最大连续子序列和。",[108,53793,53794,53796,53797,53800,53801,53804],{},[2627,53795,51680],{},"：如果前面的累加和 ",[31,53798,53799],{},"dp[j-1] > 0","，则加上当前元素有益；如果 ",[31,53802,53803],{},"dp[j-1] \u003C= 0","，则放弃前面的序列，从当前元素重新开始。",[108,53806,53807,2664,53809,347],{},[2627,53808,51819],{},[31,53810,53811],{},"dp[j] = max(dp[j-1] + a[j], a[j])",[108,53813,53814,53816,53817,53819,53820,53823,53824,53826,53827,53830,53831,53834,53835,347],{},[2627,53815,51631],{},"：找到 ",[31,53818,51611],{}," 数组中的最大值，其位置设为 ",[31,53821,53822],{},"maxj","。从 ",[31,53825,53822],{}," 往前找，直到遇到第一个 ",[31,53828,53829],{},"dp[k] \u003C= 0"," 的位置，序列即为 ",[31,53832,53833],{},"k+1"," 到 ",[31,53836,53822],{},[32490,53838,53840],{"id":53839},"例题-5最长递增子序列-lis-与合唱队形","例题 5：最长递增子序列 (LIS) 与合唱队形",[105,53842,53843,54010],{},[108,53844,53845,2664,53848,53851,53852,53854,53855,1840,53858,53909,53910,53913,53914,53916,53917,53919,53920],{},[2627,53846,53847],{},"最长递增子序列推导",[31,53849,53850],{},"dp[i]"," 表示以 ",[31,53853,33150],{}," 结尾的最长递增序列长度。检视前面的所有元素 ",[31,53856,53857],{},"a[j]",[34,53859,53861,53879],{"className":53860},[31642],[34,53862,53864],{"className":53863},[31646],[31648,53865,53866],{"xmlns":31650},[31652,53867,53868,53876],{},[31655,53869,53870,53872,53874],{},[31658,53871,31968],{},[31662,53873,932],{},[31658,53875,31955],{},[31665,53877,53878],{"encoding":31667},"j\u003Ci",[34,53880,53882,53900],{"className":53881,"ariaHidden":758},[31672],[34,53883,53885,53888,53891,53894,53897],{"className":53884},[31676],[34,53886],{"className":53887,"style":33123},[31680],[34,53889,31968],{"className":53890,"style":32153},[31685,31686],[34,53892],{"className":53893,"style":31846},[31845],[34,53895,932],{"className":53896},[31850],[34,53898],{"className":53899,"style":31846},[31845],[34,53901,53903,53906],{"className":53902},[31676],[34,53904],{"className":53905,"style":33104},[31680],[34,53907,31955],{"className":53908},[31685,31686],")，如果 ",[31,53911,53912],{},"a[i] > a[j]","，说明可以将 ",[31,53915,33150],{}," 接在 ",[31,53918,53857],{}," 后面。\n",[105,53921,53922],{},[108,53923,53924,2664,53926,53929,53930,347],{},[2627,53925,51819],{},[31,53927,53928],{},"dp[i] = max(dp[i], dp[j] + 1)","，前提是 ",[34,53931,53933,53962],{"className":53932},[31642],[34,53934,53936],{"className":53935},[31646],[31648,53937,53938],{"xmlns":31650},[31652,53939,53940,53960],{},[31655,53941,53942,53944,53946,53948,53950,53952,53954,53956,53958],{},[31658,53943,111],{},[31662,53945,15972],{"stretchy":702},[31658,53947,31955],{},[31662,53949,32712],{"stretchy":702},[31662,53951,29622],{},[31658,53953,111],{},[31662,53955,15972],{"stretchy":702},[31658,53957,31968],{},[31662,53959,32712],{"stretchy":702},[31665,53961,53912],{"encoding":31667},[34,53963,53965,53992],{"className":53964,"ariaHidden":758},[31672],[34,53966,53968,53971,53974,53977,53980,53983,53986,53989],{"className":53967},[31676],[34,53969],{"className":53970,"style":31774},[31680],[34,53972,111],{"className":53973},[31685,31686],[34,53975,15972],{"className":53976},[31782],[34,53978,31955],{"className":53979},[31685,31686],[34,53981,32712],{"className":53982},[31690],[34,53984],{"className":53985,"style":31846},[31845],[34,53987,29622],{"className":53988},[31850],[34,53990],{"className":53991,"style":31846},[31845],[34,53993,53995,53998,54001,54004,54007],{"className":53994},[31676],[34,53996],{"className":53997,"style":31774},[31680],[34,53999,111],{"className":54000},[31685,31686],[34,54002,15972],{"className":54003},[31782],[34,54005,31968],{"className":54006,"style":32153},[31685,31686],[34,54008,32712],{"className":54009},[31690],[108,54011,54012,54015,54016],{},[2627,54013,54014],{},"合唱队形拓展","：目标是找一个同学，其左边身高递增，右边身高递减，且总人数最多。\n",[105,54017,54018],{},[108,54019,54020,54023,54024,54027,54028,54031,54032,54035,54036,54039,54040,54043,54044,347],{},[2627,54021,54022],{},"解法","：原问题转化为求",[2627,54025,54026],{},"最长递增序列长度 (f1)"," 和",[2627,54029,54030],{},"最长递减序列长度 (f2)","。第一遍从左向右求 ",[31,54033,54034],{},"f1[i]","，第二遍从右向左求 ",[31,54037,54038],{},"f2[i]","。最终枚举所有人，取 ",[31,54041,54042],{},"max(f1[i] + f2[i] - 1)"," 即为最长合唱队形，时间复杂度 ",[34,54045,54047,54070],{"className":54046},[31642],[34,54048,54050],{"className":54049},[31646],[31648,54051,54052],{"xmlns":31650},[31652,54053,54054,54068],{},[31655,54055,54056,54058,54060,54066],{},[31658,54057,37890],{},[31662,54059,456],{"stretchy":702},[37624,54061,54062,54064],{},[31658,54063,31660],{},[31749,54065,768],{},[31662,54067,2557],{"stretchy":702},[31665,54069,46174],{"encoding":31667},[34,54071,54073],{"className":54072,"ariaHidden":758},[31672],[34,54074,54076,54079,54082,54085,54114],{"className":54075},[31676],[34,54077],{"className":54078,"style":37755},[31680],[34,54080,37890],{"className":54081,"style":34452},[31685,31686],[34,54083,456],{"className":54084},[31782],[34,54086,54088,54091],{"className":54087},[31685],[34,54089,31660],{"className":54090},[31685,31686],[34,54092,54094],{"className":54093},[31792],[34,54095,54097],{"className":54096},[31796],[34,54098,54100],{"className":54099},[31801],[34,54101,54103],{"className":54102,"style":37774},[31805],[34,54104,54105,54108],{"style":37777},[34,54106],{"className":54107,"style":31814},[31813],[34,54109,54111],{"className":54110},[31818,31819,31820,31821],[34,54112,768],{"className":54113},[31685,31821],[34,54115,2557],{"className":54116},[31690],[32490,54118,54120],{"id":54119},"例题-6最长公共子序列-lcs与最长公共子串","例题 6：最长公共子序列 (LCS)与最长公共子串",[105,54122,54123,54129,54284,54539,54847,54876],{},[108,54124,54125,54128],{},[2627,54126,54127],{},"LCS 问题描述","：求两个字符串 A 和 B 的最长不连续公共子序列。",[108,54130,54131,2664,54133,54136,54137,258,54210,54283],{},[2627,54132,53780],{},[31,54134,54135],{},"dp[i][j]"," 为子序列 ",[34,54138,54140,54168],{"className":54139},[31642],[34,54141,54143],{"className":54142},[31646],[31648,54144,54145],{"xmlns":31650},[31652,54146,54147,54165],{},[31655,54148,54149,54152,54154,54157,54159,54161,54163],{},[31658,54150,54151],{},"A",[31662,54153,15972],{"stretchy":702},[31749,54155,54156],{},"0..",[31658,54158,31955],{},[31662,54160,32226],{},[31749,54162,640],{},[31662,54164,32712],{"stretchy":702},[31665,54166,54167],{"encoding":31667},"A[0..i-1]",[34,54169,54171,54198],{"className":54170,"ariaHidden":758},[31672],[34,54172,54174,54177,54180,54183,54186,54189,54192,54195],{"className":54173},[31676],[34,54175],{"className":54176,"style":31774},[31680],[34,54178,54151],{"className":54179},[31685,31686],[34,54181,15972],{"className":54182},[31782],[34,54184,54156],{"className":54185},[31685],[34,54187,31955],{"className":54188},[31685,31686],[34,54190],{"className":54191,"style":32604},[31845],[34,54193,32226],{"className":54194},[32024],[34,54196],{"className":54197,"style":32604},[31845],[34,54199,54201,54204,54207],{"className":54200},[31676],[34,54202],{"className":54203,"style":31774},[31680],[34,54205,640],{"className":54206},[31685],[34,54208,32712],{"className":54209},[31690],[34,54211,54213,54240],{"className":54212},[31642],[34,54214,54216],{"className":54215},[31646],[31648,54217,54218],{"xmlns":31650},[31652,54219,54220,54237],{},[31655,54221,54222,54225,54227,54229,54231,54233,54235],{},[31658,54223,54224],{},"B",[31662,54226,15972],{"stretchy":702},[31749,54228,54156],{},[31658,54230,31968],{},[31662,54232,32226],{},[31749,54234,640],{},[31662,54236,32712],{"stretchy":702},[31665,54238,54239],{"encoding":31667},"B[0..j-1]",[34,54241,54243,54271],{"className":54242,"ariaHidden":758},[31672],[34,54244,54246,54249,54253,54256,54259,54262,54265,54268],{"className":54245},[31676],[34,54247],{"className":54248,"style":31774},[31680],[34,54250,54224],{"className":54251,"style":54252},[31685,31686],"margin-right:0.0502em;",[34,54254,15972],{"className":54255},[31782],[34,54257,54156],{"className":54258},[31685],[34,54260,31968],{"className":54261,"style":32153},[31685,31686],[34,54263],{"className":54264,"style":32604},[31845],[34,54266,32226],{"className":54267},[32024],[34,54269],{"className":54270,"style":32604},[31845],[34,54272,54274,54277,54280],{"className":54273},[31676],[34,54275],{"className":54276,"style":31774},[31680],[34,54278,640],{"className":54279},[31685],[34,54281,32712],{"className":54282},[31690]," 的最长公共子序列长度。",[108,54285,54286,33183,54288],{},[2627,54287,51680],{},[105,54289,54290,54421],{},[108,54291,54292,54293,54420],{},"若末尾字符相同（",[34,54294,54296,54336],{"className":54295},[31642],[34,54297,54299],{"className":54298},[31646],[31648,54300,54301],{"xmlns":31650},[31652,54302,54303,54333],{},[31655,54304,54305,54307,54309,54311,54313,54315,54317,54319,54321,54323,54325,54327,54329,54331],{},[31658,54306,54151],{},[31662,54308,15972],{"stretchy":702},[31658,54310,31955],{},[31662,54312,32226],{},[31749,54314,640],{},[31662,54316,32712],{"stretchy":702},[31662,54318,202],{},[31662,54320,202],{},[31658,54322,54224],{},[31662,54324,15972],{"stretchy":702},[31658,54326,31968],{},[31662,54328,32226],{},[31749,54330,640],{},[31662,54332,32712],{"stretchy":702},[31665,54334,54335],{"encoding":31667},"A[i-1] == B[j-1]",[34,54337,54339,54363,54384,54408],{"className":54338,"ariaHidden":758},[31672],[34,54340,54342,54345,54348,54351,54354,54357,54360],{"className":54341},[31676],[34,54343],{"className":54344,"style":31774},[31680],[34,54346,54151],{"className":54347},[31685,31686],[34,54349,15972],{"className":54350},[31782],[34,54352,31955],{"className":54353},[31685,31686],[34,54355],{"className":54356,"style":32604},[31845],[34,54358,32226],{"className":54359},[32024],[34,54361],{"className":54362,"style":32604},[31845],[34,54364,54366,54369,54372,54375,54378,54381],{"className":54365},[31676],[34,54367],{"className":54368,"style":31774},[31680],[34,54370,640],{"className":54371},[31685],[34,54373,32712],{"className":54374},[31690],[34,54376],{"className":54377,"style":31846},[31845],[34,54379,33376],{"className":54380},[31850],[34,54382],{"className":54383,"style":31846},[31845],[34,54385,54387,54390,54393,54396,54399,54402,54405],{"className":54386},[31676],[34,54388],{"className":54389,"style":31774},[31680],[34,54391,54224],{"className":54392,"style":54252},[31685,31686],[34,54394,15972],{"className":54395},[31782],[34,54397,31968],{"className":54398,"style":32153},[31685,31686],[34,54400],{"className":54401,"style":32604},[31845],[34,54403,32226],{"className":54404},[32024],[34,54406],{"className":54407,"style":32604},[31845],[34,54409,54411,54414,54417],{"className":54410},[31676],[34,54412],{"className":54413,"style":31774},[31680],[34,54415,640],{"className":54416},[31685],[34,54418,32712],{"className":54419},[31690],"），那么 LCS 长度必定等于它俩去掉末尾字符的 LCS 长度加 1。",[108,54422,54423,54424,54452,54453,54481,54482,43298,54510,54538],{},"若不同，LCS 只能来自于 ",[34,54425,54427,54440],{"className":54426},[31642],[34,54428,54430],{"className":54429},[31646],[31648,54431,54432],{"xmlns":31650},[31652,54433,54434,54438],{},[31655,54435,54436],{},[31658,54437,54151],{},[31665,54439,54151],{"encoding":31667},[34,54441,54443],{"className":54442,"ariaHidden":758},[31672],[34,54444,54446,54449],{"className":54445},[31676],[34,54447],{"className":54448,"style":33218},[31680],[34,54450,54151],{"className":54451},[31685,31686]," 去掉末尾字符与 ",[34,54454,54456,54469],{"className":54455},[31642],[34,54457,54459],{"className":54458},[31646],[31648,54460,54461],{"xmlns":31650},[31652,54462,54463,54467],{},[31655,54464,54465],{},[31658,54466,54224],{},[31665,54468,54224],{"encoding":31667},[34,54470,54472],{"className":54471,"ariaHidden":758},[31672],[34,54473,54475,54478],{"className":54474},[31676],[34,54476],{"className":54477,"style":33218},[31680],[34,54479,54224],{"className":54480,"style":54252},[31685,31686]," 的比较，或者 ",[34,54483,54485,54498],{"className":54484},[31642],[34,54486,54488],{"className":54487},[31646],[31648,54489,54490],{"xmlns":31650},[31652,54491,54492,54496],{},[31655,54493,54494],{},[31658,54495,54151],{},[31665,54497,54151],{"encoding":31667},[34,54499,54501],{"className":54500,"ariaHidden":758},[31672],[34,54502,54504,54507],{"className":54503},[31676],[34,54505],{"className":54506,"style":33218},[31680],[34,54508,54151],{"className":54509},[31685,31686],[34,54511,54513,54526],{"className":54512},[31642],[34,54514,54516],{"className":54515},[31646],[31648,54517,54518],{"xmlns":31650},[31652,54519,54520,54524],{},[31655,54521,54522],{},[31658,54523,54224],{},[31665,54525,54224],{"encoding":31667},[34,54527,54529],{"className":54528,"ariaHidden":758},[31672],[34,54530,54532,54535],{"className":54531},[31676],[34,54533],{"className":54534,"style":33218},[31680],[34,54536,54224],{"className":54537,"style":54252},[31685,31686]," 去掉末尾字符的比较，取二者最大值。",[108,54540,54541,33183,54543],{},[2627,54542,51819],{},[105,54544,54545,54676],{},[108,54546,54547,23100,54550,2557],{},[31,54548,54549],{},"dp[i][j] = dp[i-1][j-1] + 1",[34,54551,54553,54592],{"className":54552},[31642],[34,54554,54556],{"className":54555},[31646],[31648,54557,54558],{"xmlns":31650},[31652,54559,54560,54590],{},[31655,54561,54562,54564,54566,54568,54570,54572,54574,54576,54578,54580,54582,54584,54586,54588],{},[31658,54563,54151],{},[31662,54565,15972],{"stretchy":702},[31658,54567,31955],{},[31662,54569,32226],{},[31749,54571,640],{},[31662,54573,32712],{"stretchy":702},[31662,54575,202],{},[31662,54577,202],{},[31658,54579,54224],{},[31662,54581,15972],{"stretchy":702},[31658,54583,31968],{},[31662,54585,32226],{},[31749,54587,640],{},[31662,54589,32712],{"stretchy":702},[31665,54591,54335],{"encoding":31667},[34,54593,54595,54619,54640,54664],{"className":54594,"ariaHidden":758},[31672],[34,54596,54598,54601,54604,54607,54610,54613,54616],{"className":54597},[31676],[34,54599],{"className":54600,"style":31774},[31680],[34,54602,54151],{"className":54603},[31685,31686],[34,54605,15972],{"className":54606},[31782],[34,54608,31955],{"className":54609},[31685,31686],[34,54611],{"className":54612,"style":32604},[31845],[34,54614,32226],{"className":54615},[32024],[34,54617],{"className":54618,"style":32604},[31845],[34,54620,54622,54625,54628,54631,54634,54637],{"className":54621},[31676],[34,54623],{"className":54624,"style":31774},[31680],[34,54626,640],{"className":54627},[31685],[34,54629,32712],{"className":54630},[31690],[34,54632],{"className":54633,"style":31846},[31845],[34,54635,33376],{"className":54636},[31850],[34,54638],{"className":54639,"style":31846},[31845],[34,54641,54643,54646,54649,54652,54655,54658,54661],{"className":54642},[31676],[34,54644],{"className":54645,"style":31774},[31680],[34,54647,54224],{"className":54648,"style":54252},[31685,31686],[34,54650,15972],{"className":54651},[31782],[34,54653,31968],{"className":54654,"style":32153},[31685,31686],[34,54656],{"className":54657,"style":32604},[31845],[34,54659,32226],{"className":54660},[32024],[34,54662],{"className":54663,"style":32604},[31845],[34,54665,54667,54670,54673],{"className":54666},[31676],[34,54668],{"className":54669,"style":31774},[31680],[34,54671,640],{"className":54672},[31685],[34,54674,32712],{"className":54675},[31690],[108,54677,54678,1840,54681,2557],{},[31,54679,54680],{},"dp[i][j] = max(dp[i][j-1], dp[i-1][j])",[34,54682,54684,54723],{"className":54683},[31642],[34,54685,54687],{"className":54686},[31646],[31648,54688,54689],{"xmlns":31650},[31652,54690,54691,54720],{},[31655,54692,54693,54695,54697,54699,54701,54703,54705,54708,54710,54712,54714,54716,54718],{},[31658,54694,54151],{},[31662,54696,15972],{"stretchy":702},[31658,54698,31955],{},[31662,54700,32226],{},[31749,54702,640],{},[31662,54704,32712],{"stretchy":702},[31662,54706,54707],{"mathvariant":38079},"≠",[31658,54709,54224],{},[31662,54711,15972],{"stretchy":702},[31658,54713,31968],{},[31662,54715,32226],{},[31749,54717,640],{},[31662,54719,32712],{"stretchy":702},[31665,54721,54722],{"encoding":31667},"A[i-1] \\neq B[j-1]",[34,54724,54726,54750,54811,54835],{"className":54725,"ariaHidden":758},[31672],[34,54727,54729,54732,54735,54738,54741,54744,54747],{"className":54728},[31676],[34,54730],{"className":54731,"style":31774},[31680],[34,54733,54151],{"className":54734},[31685,31686],[34,54736,15972],{"className":54737},[31782],[34,54739,31955],{"className":54740},[31685,31686],[34,54742],{"className":54743,"style":32604},[31845],[34,54745,32226],{"className":54746},[32024],[34,54748],{"className":54749,"style":32604},[31845],[34,54751,54753,54756,54759,54762,54765,54808],{"className":54752},[31676],[34,54754],{"className":54755,"style":31774},[31680],[34,54757,640],{"className":54758},[31685],[34,54760,32712],{"className":54761},[31690],[34,54763],{"className":54764,"style":31846},[31845],[34,54766,54768,54801,54805],{"className":54767},[31850],[34,54769,54771],{"className":54770},[31850],[34,54772,54775],{"className":54773},[31685,54774],"vbox",[34,54776,54779],{"className":54777},[54778],"thinbox",[34,54780,54783,54786,54797],{"className":54781},[54782],"rlap",[34,54784],{"className":54785,"style":43217},[31680],[34,54787,54790],{"className":54788},[54789],"inner",[34,54791,54793],{"className":54792},[31685],[34,54794,54796],{"className":54795},[31850],"",[34,54798],{"className":54799},[54800],"fix",[34,54802],{"className":54803},[31845,54804],"nobreak",[34,54806,202],{"className":54807},[31850],[34,54809],{"className":54810,"style":31846},[31845],[34,54812,54814,54817,54820,54823,54826,54829,54832],{"className":54813},[31676],[34,54815],{"className":54816,"style":31774},[31680],[34,54818,54224],{"className":54819,"style":54252},[31685,31686],[34,54821,15972],{"className":54822},[31782],[34,54824,31968],{"className":54825,"style":32153},[31685,31686],[34,54827],{"className":54828,"style":32604},[31845],[34,54830,32226],{"className":54831},[32024],[34,54833],{"className":54834,"style":32604},[31845],[34,54836,54838,54841,54844],{"className":54837},[31676],[34,54839],{"className":54840,"style":31774},[31680],[34,54842,640],{"className":54843},[31685],[34,54845,32712],{"className":54846},[31690],[108,54848,54849,54852,54853,54856,54857,54860,54861,54864,54865,54868,54869,54872,54873,31901],{},[2627,54850,54851],{},"构造最优解（回溯）","：从 ",[31,54854,54855],{},"dp[m][n]"," 倒推。如果 ",[31,54858,54859],{},"dp[i][j] == dp[i-1][j]"," 则向上走（",[31,54862,54863],{},"i--","）；如果 ",[31,54866,54867],{},"dp[i][j] == dp[i][j-1]"," 则向左走（",[31,54870,54871],{},"j--","）；如果都不相等，说明该字符在 LCS 中，记录该字符并斜向左上走（",[31,54874,54875],{},"i--, j--",[108,54877,54878,54881,54882,39984,54885],{},[2627,54879,54880],{},"变体：最长公共子串","：子串要求",[2627,54883,54884],{},"必须连续",[105,54886,54887],{},[108,54888,54889,54890,55016,55017,36268,55019,55025,55026,55028],{},"状态转移方程：若 ",[34,54891,54893,54932],{"className":54892},[31642],[34,54894,54896],{"className":54895},[31646],[31648,54897,54898],{"xmlns":31650},[31652,54899,54900,54930],{},[31655,54901,54902,54904,54906,54908,54910,54912,54914,54916,54918,54920,54922,54924,54926,54928],{},[31658,54903,54151],{},[31662,54905,15972],{"stretchy":702},[31658,54907,31955],{},[31662,54909,32226],{},[31749,54911,640],{},[31662,54913,32712],{"stretchy":702},[31662,54915,202],{},[31662,54917,202],{},[31658,54919,54224],{},[31662,54921,15972],{"stretchy":702},[31658,54923,31968],{},[31662,54925,32226],{},[31749,54927,640],{},[31662,54929,32712],{"stretchy":702},[31665,54931,54335],{"encoding":31667},[34,54933,54935,54959,54980,55004],{"className":54934,"ariaHidden":758},[31672],[34,54936,54938,54941,54944,54947,54950,54953,54956],{"className":54937},[31676],[34,54939],{"className":54940,"style":31774},[31680],[34,54942,54151],{"className":54943},[31685,31686],[34,54945,15972],{"className":54946},[31782],[34,54948,31955],{"className":54949},[31685,31686],[34,54951],{"className":54952,"style":32604},[31845],[34,54954,32226],{"className":54955},[32024],[34,54957],{"className":54958,"style":32604},[31845],[34,54960,54962,54965,54968,54971,54974,54977],{"className":54961},[31676],[34,54963],{"className":54964,"style":31774},[31680],[34,54966,640],{"className":54967},[31685],[34,54969,32712],{"className":54970},[31690],[34,54972],{"className":54973,"style":31846},[31845],[34,54975,33376],{"className":54976},[31850],[34,54978],{"className":54979,"style":31846},[31845],[34,54981,54983,54986,54989,54992,54995,54998,55001],{"className":54982},[31676],[34,54984],{"className":54985,"style":31774},[31680],[34,54987,54224],{"className":54988,"style":54252},[31685,31686],[34,54990,15972],{"className":54991},[31782],[34,54993,31968],{"className":54994,"style":32153},[31685,31686],[34,54996],{"className":54997,"style":32604},[31845],[34,54999,32226],{"className":55000},[32024],[34,55002],{"className":55003,"style":32604},[31845],[34,55005,55007,55010,55013],{"className":55006},[31676],[34,55008],{"className":55009,"style":31774},[31680],[34,55011,640],{"className":55012},[31685],[34,55014,32712],{"className":55015},[31690],"，则 ",[31,55018,54549],{},[2627,55020,55021,55022],{},"否则 ",[31,55023,55024],{},"dp[i][j] = 0","（一旦断开，连续长度归零）。遍历时记录最大的 ",[31,55027,51611],{}," 值即可。",[32490,55030,55032],{"id":55031},"例题-7编辑距离-edit-distance","例题 7：编辑距离 (Edit Distance)",[105,55034,55035,55040,55189,55233,55240],{},[108,55036,55037,55039],{},[2627,55038,51674],{},"：将字符串 A 转换为 B 的最少操作次数（插入、删除、替换）。",[108,55041,55042,2664,55044,55046,55047,55117,55118,55188],{},[2627,55043,53780],{},[31,55045,54135],{}," 为将 ",[34,55048,55050,55075],{"className":55049},[31642],[34,55051,55053],{"className":55052},[31646],[31648,55054,55055],{"xmlns":31650},[31652,55056,55057,55073],{},[31655,55058,55059,55061,55063,55065,55067,55069,55071],{},[31658,55060,54151],{},[31662,55062,15972],{"stretchy":702},[31749,55064,54156],{},[31658,55066,31955],{},[31662,55068,32226],{},[31749,55070,640],{},[31662,55072,32712],{"stretchy":702},[31665,55074,54167],{"encoding":31667},[34,55076,55078,55105],{"className":55077,"ariaHidden":758},[31672],[34,55079,55081,55084,55087,55090,55093,55096,55099,55102],{"className":55080},[31676],[34,55082],{"className":55083,"style":31774},[31680],[34,55085,54151],{"className":55086},[31685,31686],[34,55088,15972],{"className":55089},[31782],[34,55091,54156],{"className":55092},[31685],[34,55094,31955],{"className":55095},[31685,31686],[34,55097],{"className":55098,"style":32604},[31845],[34,55100,32226],{"className":55101},[32024],[34,55103],{"className":55104,"style":32604},[31845],[34,55106,55108,55111,55114],{"className":55107},[31676],[34,55109],{"className":55110,"style":31774},[31680],[34,55112,640],{"className":55113},[31685],[34,55115,32712],{"className":55116},[31690]," 转换为 ",[34,55119,55121,55146],{"className":55120},[31642],[34,55122,55124],{"className":55123},[31646],[31648,55125,55126],{"xmlns":31650},[31652,55127,55128,55144],{},[31655,55129,55130,55132,55134,55136,55138,55140,55142],{},[31658,55131,54224],{},[31662,55133,15972],{"stretchy":702},[31749,55135,54156],{},[31658,55137,31968],{},[31662,55139,32226],{},[31749,55141,640],{},[31662,55143,32712],{"stretchy":702},[31665,55145,54239],{"encoding":31667},[34,55147,55149,55176],{"className":55148,"ariaHidden":758},[31672],[34,55150,55152,55155,55158,55161,55164,55167,55170,55173],{"className":55151},[31676],[34,55153],{"className":55154,"style":31774},[31680],[34,55156,54224],{"className":55157,"style":54252},[31685,31686],[34,55159,15972],{"className":55160},[31782],[34,55162,54156],{"className":55163},[31685],[34,55165,31968],{"className":55166,"style":32153},[31685,31686],[34,55168],{"className":55169,"style":32604},[31845],[34,55171,32226],{"className":55172},[32024],[34,55174],{"className":55175,"style":32604},[31845],[34,55177,55179,55182,55185],{"className":55178},[31676],[34,55180],{"className":55181,"style":31774},[31680],[34,55183,640],{"className":55184},[31685],[34,55186,32712],{"className":55187},[31690]," 的最少操作次数。",[108,55190,55191,33183,55193],{},[2627,55192,51680],{},[105,55194,55195,55201],{},[108,55196,55197,55198,347],{},"若当前字符相同：无需操作，代价同前缀，",[31,55199,55200],{},"dp[i][j] = dp[i-1][j-1]",[108,55202,55203,55204],{},"若字符不同，考虑三种操作的最小值：\n",[105,55205,55206,55215,55224],{},[108,55207,55208,55211,55212,347],{},[2627,55209,55210],{},"替换","：代价为 ",[31,55213,55214],{},"dp[i-1][j-1] + 1",[108,55216,55217,55220,55221,347],{},[2627,55218,55219],{},"插入","：相当于 A 不变，B 少了一个字符（因为插入的字符与 B 匹配了），代价为 ",[31,55222,55223],{},"dp[i][j-1] + 1",[108,55225,55226,55229,55230,347],{},[2627,55227,55228],{},"删除","：相当于 A 少了一个字符，B 不变，代价为 ",[31,55231,55232],{},"dp[i-1][j] + 1",[108,55234,55235,2664,55237,347],{},[2627,55236,51819],{},[31,55238,55239],{},"dp[i][j] = min(dp[i-1][j-1], dp[i][j-1], dp[i-1][j]) + 1",[108,55241,55242,2664,55244,55247,55248,55251,55252,347],{},[2627,55243,51987],{},[31,55245,55246],{},"dp[i] = i","（全部删除），",[31,55249,55250],{},"dp[j] = j","（全部插入）。复杂度 ",[34,55253,55255,55277],{"className":55254},[31642],[34,55256,55258],{"className":55257},[31646],[31648,55259,55260],{"xmlns":31650},[31652,55261,55262,55274],{},[31655,55263,55264,55266,55268,55270,55272],{},[31658,55265,37890],{},[31662,55267,456],{"stretchy":702},[31658,55269,31759],{},[31658,55271,31660],{},[31662,55273,2557],{"stretchy":702},[31665,55275,55276],{"encoding":31667},"O(mn)",[34,55278,55280],{"className":55279,"ariaHidden":758},[31672],[34,55281,55283,55286,55289,55292,55295],{"className":55282},[31676],[34,55284],{"className":55285,"style":31774},[31680],[34,55287,37890],{"className":55288,"style":34452},[31685,31686],[34,55290,456],{"className":55291},[31782],[34,55293,31749],{"className":55294},[31685,31686],[34,55296,2557],{"className":55297},[31690],[32490,55299,55301],{"id":55300},"例题-8矩阵连乘问题","例题 8：矩阵连乘问题",[105,55303,55304,55337,55569,56046,56526,56610],{},[108,55305,55306,47591,55308,55336],{},[2627,55307,51674],{},[34,55309,55311,55324],{"className":55310},[31642],[34,55312,55314],{"className":55313},[31646],[31648,55315,55316],{"xmlns":31650},[31652,55317,55318,55322],{},[31655,55319,55320],{},[31658,55321,31660],{},[31665,55323,31660],{"encoding":31667},[34,55325,55327],{"className":55326,"ariaHidden":758},[31672],[34,55328,55330,55333],{"className":55329},[31676],[34,55331],{"className":55332,"style":32527},[31680],[34,55334,31660],{"className":55335},[31685,31686]," 个矩阵，通过加括号确定乘法次序，使总数乘次数最少。",[108,55338,55339,2664,55341,1840,55344,55441,55442,55568],{},[2627,55340,53780],{},[31,55342,55343],{},"m[i][j]",[34,55345,55347,55373],{"className":55346},[31642],[34,55348,55350],{"className":55349},[31646],[31648,55351,55352],{"xmlns":31650},[31652,55353,55354,55370],{},[31655,55355,55356,55358,55360,55362,55364,55366,55368],{},[31749,55357,640],{},[31662,55359,45972],{},[31658,55361,31955],{},[31662,55363,45972],{},[31658,55365,31968],{},[31662,55367,45972],{},[31658,55369,31660],{},[31665,55371,55372],{"encoding":31667},"1 \\le i \\le j \\le n",[34,55374,55376,55395,55414,55432],{"className":55375,"ariaHidden":758},[31672],[34,55377,55379,55383,55386,55389,55392],{"className":55378},[31676],[34,55380],{"className":55381,"style":55382},[31680],"height:0.7804em;vertical-align:-0.136em;",[34,55384,640],{"className":55385},[31685],[34,55387],{"className":55388,"style":31846},[31845],[34,55390,45972],{"className":55391},[31850],[34,55393],{"className":55394,"style":31846},[31845],[34,55396,55398,55402,55405,55408,55411],{"className":55397},[31676],[34,55399],{"className":55400,"style":55401},[31680],"height:0.7955em;vertical-align:-0.136em;",[34,55403,31955],{"className":55404},[31685,31686],[34,55406],{"className":55407,"style":31846},[31845],[34,55409,45972],{"className":55410},[31850],[34,55412],{"className":55413,"style":31846},[31845],[34,55415,55417,55420,55423,55426,55429],{"className":55416},[31676],[34,55418],{"className":55419,"style":33123},[31680],[34,55421,31968],{"className":55422,"style":32153},[31685,31686],[34,55424],{"className":55425,"style":31846},[31845],[34,55427,45972],{"className":55428},[31850],[34,55430],{"className":55431,"style":31846},[31845],[34,55433,55435,55438],{"className":55434},[31676],[34,55436],{"className":55437,"style":32527},[31680],[34,55439,31660],{"className":55440},[31685,31686],") 为计算矩阵链 ",[34,55443,55445,55475],{"className":55444},[31642],[34,55446,55448],{"className":55447},[31646],[31648,55449,55450],{"xmlns":31650},[31652,55451,55452,55472],{},[31655,55453,55454,55460,55462,55464,55466],{},[31743,55455,55456,55458],{},[31658,55457,54151],{},[31658,55459,31955],{},[31658,55461,2547],{"mathvariant":38079},[31658,55463,2547],{"mathvariant":38079},[31658,55465,2547],{"mathvariant":38079},[31743,55467,55468,55470],{},[31658,55469,54151],{},[31658,55471,31968],{},[31665,55473,55474],{"encoding":31667},"A_i...A_j",[34,55476,55478],{"className":55477,"ariaHidden":758},[31672],[34,55479,55481,55485,55525,55528],{"className":55480},[31676],[34,55482],{"className":55483,"style":55484},[31680],"height:0.9694em;vertical-align:-0.2861em;",[34,55486,55488,55491],{"className":55487},[31685],[34,55489,54151],{"className":55490},[31685,31686],[34,55492,55494],{"className":55493},[31792],[34,55495,55497,55517],{"className":55496},[31796,31797],[34,55498,55500,55514],{"className":55499},[31801],[34,55501,55503],{"className":55502,"style":32090},[31805],[34,55504,55505,55508],{"style":31809},[34,55506],{"className":55507,"style":31814},[31813],[34,55509,55511],{"className":55510},[31818,31819,31820,31821],[34,55512,31955],{"className":55513},[31685,31686,31821],[34,55515,31829],{"className":55516},[31828],[34,55518,55520],{"className":55519},[31801],[34,55521,55523],{"className":55522,"style":31836},[31805],[34,55524],{},[34,55526,1167],{"className":55527},[31685],[34,55529,55531,55534],{"className":55530},[31685],[34,55532,54151],{"className":55533},[31685,31686],[34,55535,55537],{"className":55536},[31792],[34,55538,55540,55560],{"className":55539},[31796,31797],[34,55541,55543,55557],{"className":55542},[31801],[34,55544,55546],{"className":55545,"style":32090},[31805],[34,55547,55548,55551],{"style":31809},[34,55549],{"className":55550,"style":31814},[31813],[34,55552,55554],{"className":55553},[31818,31819,31820,31821],[34,55555,31968],{"className":55556,"style":32153},[31685,31686,31821],[34,55558,31829],{"className":55559},[31828],[34,55561,55563],{"className":55562},[31801],[34,55564,55566],{"className":55565,"style":32163},[31805],[34,55567],{}," 的最少数乘次数。",[108,55570,55571,55573,55574,55602,55603,55676,55677,55743,55744,55832,55833,31901],{},[2627,55572,51680],{},"：设在第 ",[34,55575,55577,55590],{"className":55576},[31642],[34,55578,55580],{"className":55579},[31646],[31648,55581,55582],{"xmlns":31650},[31652,55583,55584,55588],{},[31655,55585,55586],{},[31658,55587,33759],{},[31665,55589,33759],{"encoding":31667},[34,55591,55593],{"className":55592,"ariaHidden":758},[31672],[34,55594,55596,55599],{"className":55595},[31676],[34,55597],{"className":55598,"style":31681},[31680],[34,55600,33759],{"className":55601,"style":33774},[31685,31686]," 处断开（",[34,55604,55606,55628],{"className":55605},[31642],[34,55607,55609],{"className":55608},[31646],[31648,55610,55611],{"xmlns":31650},[31652,55612,55613,55625],{},[31655,55614,55615,55617,55619,55621,55623],{},[31658,55616,31955],{},[31662,55618,45972],{},[31658,55620,33759],{},[31662,55622,932],{},[31658,55624,31968],{},[31665,55626,55627],{"encoding":31667},"i \\le k \u003C j",[34,55629,55631,55649,55667],{"className":55630,"ariaHidden":758},[31672],[34,55632,55634,55637,55640,55643,55646],{"className":55633},[31676],[34,55635],{"className":55636,"style":55401},[31680],[34,55638,31955],{"className":55639},[31685,31686],[34,55641],{"className":55642,"style":31846},[31845],[34,55644,45972],{"className":55645},[31850],[34,55647],{"className":55648,"style":31846},[31845],[34,55650,55652,55655,55658,55661,55664],{"className":55651},[31676],[34,55653],{"className":55654,"style":46991},[31680],[34,55656,33759],{"className":55657,"style":33774},[31685,31686],[34,55659],{"className":55660,"style":31846},[31845],[34,55662,932],{"className":55663},[31850],[34,55665],{"className":55666,"style":31846},[31845],[34,55668,55670,55673],{"className":55669},[31676],[34,55671],{"className":55672,"style":33123},[31680],[34,55674,31968],{"className":55675,"style":32153},[31685,31686],"）。总计算量 = 前半段 ",[34,55678,55680,55704],{"className":55679},[31642],[34,55681,55683],{"className":55682},[31646],[31648,55684,55685],{"xmlns":31650},[31652,55686,55687,55701],{},[31655,55688,55689,55691,55693,55695,55697,55699],{},[31658,55690,54151],{},[31662,55692,15972],{"stretchy":702},[31658,55694,31955],{},[31662,55696,5598],{},[31658,55698,33759],{},[31662,55700,32712],{"stretchy":702},[31665,55702,55703],{"encoding":31667},"A[i:k]",[34,55705,55707,55731],{"className":55706,"ariaHidden":758},[31672],[34,55708,55710,55713,55716,55719,55722,55725,55728],{"className":55709},[31676],[34,55711],{"className":55712,"style":31774},[31680],[34,55714,54151],{"className":55715},[31685,31686],[34,55717,15972],{"className":55718},[31782],[34,55720,31955],{"className":55721},[31685,31686],[34,55723],{"className":55724,"style":31846},[31845],[34,55726,5598],{"className":55727},[31850],[34,55729],{"className":55730,"style":31846},[31845],[34,55732,55734,55737,55740],{"className":55733},[31676],[34,55735],{"className":55736,"style":31774},[31680],[34,55738,33759],{"className":55739,"style":33774},[31685,31686],[34,55741,32712],{"className":55742},[31690]," 的代价 + 后半段 ",[34,55745,55747,55775],{"className":55746},[31642],[34,55748,55750],{"className":55749},[31646],[31648,55751,55752],{"xmlns":31650},[31652,55753,55754,55772],{},[31655,55755,55756,55758,55760,55762,55764,55766,55768,55770],{},[31658,55757,54151],{},[31662,55759,15972],{"stretchy":702},[31658,55761,33759],{},[31662,55763,223],{},[31749,55765,640],{},[31662,55767,5598],{},[31658,55769,31968],{},[31662,55771,32712],{"stretchy":702},[31665,55773,55774],{"encoding":31667},"A[k+1:j]",[34,55776,55778,55802,55820],{"className":55777,"ariaHidden":758},[31672],[34,55779,55781,55784,55787,55790,55793,55796,55799],{"className":55780},[31676],[34,55782],{"className":55783,"style":31774},[31680],[34,55785,54151],{"className":55786},[31685,31686],[34,55788,15972],{"className":55789},[31782],[34,55791,33759],{"className":55792,"style":33774},[31685,31686],[34,55794],{"className":55795,"style":32604},[31845],[34,55797,223],{"className":55798},[32024],[34,55800],{"className":55801,"style":32604},[31845],[34,55803,55805,55808,55811,55814,55817],{"className":55804},[31676],[34,55806],{"className":55807,"style":32617},[31680],[34,55809,640],{"className":55810},[31685],[34,55812],{"className":55813,"style":31846},[31845],[34,55815,5598],{"className":55816},[31850],[34,55818],{"className":55819,"style":31846},[31845],[34,55821,55823,55826,55829],{"className":55822},[31676],[34,55824],{"className":55825,"style":31774},[31680],[34,55827,31968],{"className":55828,"style":32153},[31685,31686],[34,55830,32712],{"className":55831},[31690]," 的代价 + 两段结果矩阵相乘的代价（",[34,55834,55836,55876],{"className":55835},[31642],[34,55837,55839],{"className":55838},[31646],[31648,55840,55841],{"xmlns":31650},[31652,55842,55843,55873],{},[31655,55844,55845,55857,55859,55865,55867],{},[31743,55846,55847,55849],{},[31658,55848,15],{},[31655,55850,55851,55853,55855],{},[31658,55852,31955],{},[31662,55854,32226],{},[31749,55856,640],{},[31662,55858,42071],{},[31743,55860,55861,55863],{},[31658,55862,15],{},[31658,55864,33759],{},[31662,55866,42071],{},[31743,55868,55869,55871],{},[31658,55870,15],{},[31658,55872,31968],{},[31665,55874,55875],{"encoding":31667},"p_{i-1} \\times p_k \\times p_j",[34,55877,55879,55943,55999],{"className":55878,"ariaHidden":758},[31672],[34,55880,55882,55885,55934,55937,55940],{"className":55881},[31676],[34,55883],{"className":55884,"style":39444},[31680],[34,55886,55888,55891],{"className":55887},[31685],[34,55889,15],{"className":55890},[31685,31686],[34,55892,55894],{"className":55893},[31792],[34,55895,55897,55926],{"className":55896},[31796,31797],[34,55898,55900,55923],{"className":55899},[31801],[34,55901,55903],{"className":55902,"style":32090},[31805],[34,55904,55905,55908],{"style":31809},[34,55906],{"className":55907,"style":31814},[31813],[34,55909,55911],{"className":55910},[31818,31819,31820,31821],[34,55912,55914,55917,55920],{"className":55913},[31685,31821],[34,55915,31955],{"className":55916},[31685,31686,31821],[34,55918,32226],{"className":55919},[32024,31821],[34,55921,640],{"className":55922},[31685,31821],[34,55924,31829],{"className":55925},[31828],[34,55927,55929],{"className":55928},[31801],[34,55930,55932],{"className":55931,"style":32037},[31805],[34,55933],{},[34,55935],{"className":55936,"style":32604},[31845],[34,55938,42071],{"className":55939},[32024],[34,55941],{"className":55942,"style":32604},[31845],[34,55944,55946,55950,55990,55993,55996],{"className":55945},[31676],[34,55947],{"className":55948,"style":55949},[31680],"height:0.7778em;vertical-align:-0.1944em;",[34,55951,55953,55956],{"className":55952},[31685],[34,55954,15],{"className":55955},[31685,31686],[34,55957,55959],{"className":55958},[31792],[34,55960,55962,55982],{"className":55961},[31796,31797],[34,55963,55965,55979],{"className":55964},[31801],[34,55966,55968],{"className":55967,"style":33833},[31805],[34,55969,55970,55973],{"style":31809},[34,55971],{"className":55972,"style":31814},[31813],[34,55974,55976],{"className":55975},[31818,31819,31820,31821],[34,55977,33759],{"className":55978,"style":33774},[31685,31686,31821],[34,55980,31829],{"className":55981},[31828],[34,55983,55985],{"className":55984},[31801],[34,55986,55988],{"className":55987,"style":31836},[31805],[34,55989],{},[34,55991],{"className":55992,"style":32604},[31845],[34,55994,42071],{"className":55995},[32024],[34,55997],{"className":55998,"style":32604},[31845],[34,56000,56002,56006],{"className":56001},[31676],[34,56003],{"className":56004,"style":56005},[31680],"height:0.7167em;vertical-align:-0.2861em;",[34,56007,56009,56012],{"className":56008},[31685],[34,56010,15],{"className":56011},[31685,31686],[34,56013,56015],{"className":56014},[31792],[34,56016,56018,56038],{"className":56017},[31796,31797],[34,56019,56021,56035],{"className":56020},[31801],[34,56022,56024],{"className":56023,"style":32090},[31805],[34,56025,56026,56029],{"style":31809},[34,56027],{"className":56028,"style":31814},[31813],[34,56030,56032],{"className":56031},[31818,31819,31820,31821],[34,56033,31968],{"className":56034,"style":32153},[31685,31686,31821],[34,56036,31829],{"className":56037},[31828],[34,56039,56041],{"className":56040},[31801],[34,56042,56044],{"className":56043,"style":32163},[31805],[34,56045],{},[108,56047,56048,33183,56050],{},[2627,56049,51819],{},[34,56051,56053,56170],{"className":56052},[31642],[34,56054,56056],{"className":56055},[31646],[31648,56057,56058],{"xmlns":31650},[31652,56059,56060,56167],{},[31655,56061,56062,56064,56066,56068,56070,56072,56074,56076,56078,56099,56101,56103,56105,56107,56109,56111,56113,56115,56117,56119,56121,56123,56125,56127,56129,56131,56133,56135,56137,56149,56151,56157,56159,56165],{},[31658,56063,31759],{},[31662,56065,15972],{"stretchy":702},[31658,56067,31955],{},[31662,56069,32712],{"stretchy":702},[31662,56071,15972],{"stretchy":702},[31658,56073,31968],{},[31662,56075,32712],{"stretchy":702},[31662,56077,202],{},[31743,56079,56080,56087],{},[31655,56081,56082,56085],{},[31658,56083,56084],{},"min",[31662,56086,32682],{},[31655,56088,56089,56091,56093,56095,56097],{},[31658,56090,31955],{},[31662,56092,45972],{},[31658,56094,33759],{},[31662,56096,932],{},[31658,56098,31968],{},[31662,56100,32685],{"stretchy":702},[31658,56102,31759],{},[31662,56104,15972],{"stretchy":702},[31658,56106,31955],{},[31662,56108,32712],{"stretchy":702},[31662,56110,15972],{"stretchy":702},[31658,56112,33759],{},[31662,56114,32712],{"stretchy":702},[31662,56116,223],{},[31658,56118,31759],{},[31662,56120,15972],{"stretchy":702},[31658,56122,33759],{},[31662,56124,223],{},[31749,56126,640],{},[31662,56128,32712],{"stretchy":702},[31662,56130,15972],{"stretchy":702},[31658,56132,31968],{},[31662,56134,32712],{"stretchy":702},[31662,56136,223],{},[31743,56138,56139,56141],{},[31658,56140,15],{},[31655,56142,56143,56145,56147],{},[31658,56144,31955],{},[31662,56146,32226],{},[31749,56148,640],{},[31662,56150,42071],{},[31743,56152,56153,56155],{},[31658,56154,15],{},[31658,56156,33759],{},[31662,56158,42071],{},[31743,56160,56161,56163],{},[31658,56162,15],{},[31658,56164,31968],{},[31662,56166,32715],{"stretchy":702},[31665,56168,56169],{"encoding":31667},"m[i][j] = \\min_{i \\le k \u003C j} \\{m[i][k] + m[k+1][j] + p_{i-1} \\times p_k \\times p_j\\}",[34,56171,56173,56209,56304,56328,56358,56422,56477],{"className":56172,"ariaHidden":758},[31672],[34,56174,56176,56179,56182,56185,56188,56191,56194,56197,56200,56203,56206],{"className":56175},[31676],[34,56177],{"className":56178,"style":31774},[31680],[34,56180,31759],{"className":56181},[31685,31686],[34,56183,15972],{"className":56184},[31782],[34,56186,31955],{"className":56187},[31685,31686],[34,56189,32712],{"className":56190},[31690],[34,56192,15972],{"className":56193},[31782],[34,56195,31968],{"className":56196,"style":32153},[31685,31686],[34,56198,32712],{"className":56199},[31690],[34,56201],{"className":56202,"style":31846},[31845],[34,56204,202],{"className":56205},[31850],[34,56207],{"className":56208,"style":31846},[31845],[34,56210,56212,56215,56271,56274,56277,56280,56283,56286,56289,56292,56295,56298,56301],{"className":56211},[31676],[34,56213],{"className":56214,"style":32058},[31680],[34,56216,56218,56221],{"className":56217},[32767],[34,56219,56084],{"className":56220},[32767],[34,56222,56224],{"className":56223},[31792],[34,56225,56227,56263],{"className":56226},[31796,31797],[34,56228,56230,56260],{"className":56229},[31801],[34,56231,56233],{"className":56232,"style":33833},[31805],[34,56234,56236,56239],{"style":56235},"top:-2.55em;margin-right:0.05em;",[34,56237],{"className":56238,"style":31814},[31813],[34,56240,56242],{"className":56241},[31818,31819,31820,31821],[34,56243,56245,56248,56251,56254,56257],{"className":56244},[31685,31821],[34,56246,31955],{"className":56247},[31685,31686,31821],[34,56249,45972],{"className":56250},[31850,31821],[34,56252,33759],{"className":56253,"style":33774},[31685,31686,31821],[34,56255,932],{"className":56256},[31850,31821],[34,56258,31968],{"className":56259,"style":32153},[31685,31686,31821],[34,56261,31829],{"className":56262},[31828],[34,56264,56266],{"className":56265},[31801],[34,56267,56269],{"className":56268,"style":32163},[31805],[34,56270],{},[34,56272,32685],{"className":56273},[31782],[34,56275,31759],{"className":56276},[31685,31686],[34,56278,15972],{"className":56279},[31782],[34,56281,31955],{"className":56282},[31685,31686],[34,56284,32712],{"className":56285},[31690],[34,56287,15972],{"className":56288},[31782],[34,56290,33759],{"className":56291,"style":33774},[31685,31686],[34,56293,32712],{"className":56294},[31690],[34,56296],{"className":56297,"style":32604},[31845],[34,56299,223],{"className":56300},[32024],[34,56302],{"className":56303,"style":32604},[31845],[34,56305,56307,56310,56313,56316,56319,56322,56325],{"className":56306},[31676],[34,56308],{"className":56309,"style":31774},[31680],[34,56311,31759],{"className":56312},[31685,31686],[34,56314,15972],{"className":56315},[31782],[34,56317,33759],{"className":56318,"style":33774},[31685,31686],[34,56320],{"className":56321,"style":32604},[31845],[34,56323,223],{"className":56324},[32024],[34,56326],{"className":56327,"style":32604},[31845],[34,56329,56331,56334,56337,56340,56343,56346,56349,56352,56355],{"className":56330},[31676],[34,56332],{"className":56333,"style":31774},[31680],[34,56335,640],{"className":56336},[31685],[34,56338,32712],{"className":56339},[31690],[34,56341,15972],{"className":56342},[31782],[34,56344,31968],{"className":56345,"style":32153},[31685,31686],[34,56347,32712],{"className":56348},[31690],[34,56350],{"className":56351,"style":32604},[31845],[34,56353,223],{"className":56354},[32024],[34,56356],{"className":56357,"style":32604},[31845],[34,56359,56361,56364,56413,56416,56419],{"className":56360},[31676],[34,56362],{"className":56363,"style":39444},[31680],[34,56365,56367,56370],{"className":56366},[31685],[34,56368,15],{"className":56369},[31685,31686],[34,56371,56373],{"className":56372},[31792],[34,56374,56376,56405],{"className":56375},[31796,31797],[34,56377,56379,56402],{"className":56378},[31801],[34,56380,56382],{"className":56381,"style":32090},[31805],[34,56383,56384,56387],{"style":31809},[34,56385],{"className":56386,"style":31814},[31813],[34,56388,56390],{"className":56389},[31818,31819,31820,31821],[34,56391,56393,56396,56399],{"className":56392},[31685,31821],[34,56394,31955],{"className":56395},[31685,31686,31821],[34,56397,32226],{"className":56398},[32024,31821],[34,56400,640],{"className":56401},[31685,31821],[34,56403,31829],{"className":56404},[31828],[34,56406,56408],{"className":56407},[31801],[34,56409,56411],{"className":56410,"style":32037},[31805],[34,56412],{},[34,56414],{"className":56415,"style":32604},[31845],[34,56417,42071],{"className":56418},[32024],[34,56420],{"className":56421,"style":32604},[31845],[34,56423,56425,56428,56468,56471,56474],{"className":56424},[31676],[34,56426],{"className":56427,"style":55949},[31680],[34,56429,56431,56434],{"className":56430},[31685],[34,56432,15],{"className":56433},[31685,31686],[34,56435,56437],{"className":56436},[31792],[34,56438,56440,56460],{"className":56439},[31796,31797],[34,56441,56443,56457],{"className":56442},[31801],[34,56444,56446],{"className":56445,"style":33833},[31805],[34,56447,56448,56451],{"style":31809},[34,56449],{"className":56450,"style":31814},[31813],[34,56452,56454],{"className":56453},[31818,31819,31820,31821],[34,56455,33759],{"className":56456,"style":33774},[31685,31686,31821],[34,56458,31829],{"className":56459},[31828],[34,56461,56463],{"className":56462},[31801],[34,56464,56466],{"className":56465,"style":31836},[31805],[34,56467],{},[34,56469],{"className":56470,"style":32604},[31845],[34,56472,42071],{"className":56473},[32024],[34,56475],{"className":56476,"style":32604},[31845],[34,56478,56480,56483,56523],{"className":56479},[31676],[34,56481],{"className":56482,"style":32058},[31680],[34,56484,56486,56489],{"className":56485},[31685],[34,56487,15],{"className":56488},[31685,31686],[34,56490,56492],{"className":56491},[31792],[34,56493,56495,56515],{"className":56494},[31796,31797],[34,56496,56498,56512],{"className":56497},[31801],[34,56499,56501],{"className":56500,"style":32090},[31805],[34,56502,56503,56506],{"style":31809},[34,56504],{"className":56505,"style":31814},[31813],[34,56507,56509],{"className":56508},[31818,31819,31820,31821],[34,56510,31968],{"className":56511,"style":32153},[31685,31686,31821],[34,56513,31829],{"className":56514},[31828],[34,56516,56518],{"className":56517},[31801],[34,56519,56521],{"className":56520,"style":32163},[31805],[34,56522],{},[34,56524,32715],{"className":56525},[31690],[108,56527,56528,56531,56532,56535,56536,347],{},[2627,56529,56530],{},"求解顺序","：因为依赖更短的矩阵链，需",[2627,56533,56534],{},"按矩阵链的长度 (r) 自底向上计算","。例如先算所有长度为 2 的，再算长度为 3 的... 三重循环，时间复杂度 ",[34,56537,56539,56563],{"className":56538},[31642],[34,56540,56542],{"className":56541},[31646],[31648,56543,56544],{"xmlns":31650},[31652,56545,56546,56560],{},[31655,56547,56548,56550,56552,56558],{},[31658,56549,37890],{},[31662,56551,456],{"stretchy":702},[37624,56553,56554,56556],{},[31658,56555,31660],{},[31749,56557,786],{},[31662,56559,2557],{"stretchy":702},[31665,56561,56562],{"encoding":31667},"O(n^3)",[34,56564,56566],{"className":56565,"ariaHidden":758},[31672],[34,56567,56569,56572,56575,56578,56607],{"className":56568},[31676],[34,56570],{"className":56571,"style":37755},[31680],[34,56573,37890],{"className":56574,"style":34452},[31685,31686],[34,56576,456],{"className":56577},[31782],[34,56579,56581,56584],{"className":56580},[31685],[34,56582,31660],{"className":56583},[31685,31686],[34,56585,56587],{"className":56586},[31792],[34,56588,56590],{"className":56589},[31796],[34,56591,56593],{"className":56592},[31801],[34,56594,56596],{"className":56595,"style":37774},[31805],[34,56597,56598,56601],{"style":37777},[34,56599],{"className":56600,"style":31814},[31813],[34,56602,56604],{"className":56603},[31818,31819,31820,31821],[34,56605,786],{"className":56606},[31685,31821],[34,56608,2557],{"className":56609},[31690],[108,56611,56612,56614,56615,56618,56619,56647,56648,258,56651,56654],{},[2627,56613,51631],{},"：使用辅助数组 ",[31,56616,56617],{},"s[i][j]"," 记录取得最小值的断点 ",[34,56620,56622,56635],{"className":56621},[31642],[34,56623,56625],{"className":56624},[31646],[31648,56626,56627],{"xmlns":31650},[31652,56628,56629,56633],{},[31655,56630,56631],{},[31658,56632,33759],{},[31665,56634,33759],{"encoding":31667},[34,56636,56638],{"className":56637,"ariaHidden":758},[31672],[34,56639,56641,56644],{"className":56640},[31676],[34,56642],{"className":56643,"style":31681},[31680],[34,56645,33759],{"className":56646,"style":33774},[31685,31686],"。通过递归 ",[31,56649,56650],{},"traceback(s, i, s[i][j])",[31,56652,56653],{},"traceback(s, s[i][j]+1, j)"," 打印括号。",[32490,56656,56658],{"id":56657},"例题-901-背包问题","例题 9：0\u002F1 背包问题",[105,56660,56661,56868,56934,57275],{},[108,56662,56663,2664,56665,56693,56694,56765,56766,56837,56838,56867],{},[2627,56664,51674],{},[34,56666,56668,56681],{"className":56667},[31642],[34,56669,56671],{"className":56670},[31646],[31648,56672,56673],{"xmlns":31650},[31652,56674,56675,56679],{},[31655,56676,56677],{},[31658,56678,31660],{},[31665,56680,31660],{"encoding":31667},[34,56682,56684],{"className":56683,"ariaHidden":758},[31672],[34,56685,56687,56690],{"className":56686},[31676],[34,56688],{"className":56689,"style":32527},[31680],[34,56691,31660],{"className":56692},[31685,31686]," 个物品（重量 ",[34,56695,56697,56715],{"className":56696},[31642],[34,56698,56700],{"className":56699},[31646],[31648,56701,56702],{"xmlns":31650},[31652,56703,56704,56712],{},[31655,56705,56706],{},[31743,56707,56708,56710],{},[31658,56709,47077],{},[31658,56711,31955],{},[31665,56713,56714],{"encoding":31667},"w_i",[34,56716,56718],{"className":56717,"ariaHidden":758},[31672],[34,56719,56721,56724],{"className":56720},[31676],[34,56722],{"className":56723,"style":31860},[31680],[34,56725,56727,56730],{"className":56726},[31685],[34,56728,47077],{"className":56729,"style":47120},[31685,31686],[34,56731,56733],{"className":56732},[31792],[34,56734,56736,56757],{"className":56735},[31796,31797],[34,56737,56739,56754],{"className":56738},[31801],[34,56740,56742],{"className":56741,"style":32090},[31805],[34,56743,56745,56748],{"style":56744},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[34,56746],{"className":56747,"style":31814},[31813],[34,56749,56751],{"className":56750},[31818,31819,31820,31821],[34,56752,31955],{"className":56753},[31685,31686,31821],[34,56755,31829],{"className":56756},[31828],[34,56758,56760],{"className":56759},[31801],[34,56761,56763],{"className":56762,"style":31836},[31805],[34,56764],{},"，价值 ",[34,56767,56769,56788],{"className":56768},[31642],[34,56770,56772],{"className":56771},[31646],[31648,56773,56774],{"xmlns":31650},[31652,56775,56776,56785],{},[31655,56777,56778],{},[31743,56779,56780,56783],{},[31658,56781,56782],{},"v",[31658,56784,31955],{},[31665,56786,56787],{"encoding":31667},"v_i",[34,56789,56791],{"className":56790,"ariaHidden":758},[31672],[34,56792,56794,56797],{"className":56793},[31676],[34,56795],{"className":56796,"style":31860},[31680],[34,56798,56800,56803],{"className":56799},[31685],[34,56801,56782],{"className":56802,"style":32062},[31685,31686],[34,56804,56806],{"className":56805},[31792],[34,56807,56809,56829],{"className":56808},[31796,31797],[34,56810,56812,56826],{"className":56811},[31801],[34,56813,56815],{"className":56814,"style":32090},[31805],[34,56816,56817,56820],{"style":38795},[34,56818],{"className":56819,"style":31814},[31813],[34,56821,56823],{"className":56822},[31818,31819,31820,31821],[34,56824,31955],{"className":56825},[31685,31686,31821],[34,56827,31829],{"className":56828},[31828],[34,56830,56832],{"className":56831},[31801],[34,56833,56835],{"className":56834,"style":31836},[31805],[34,56836],{},"），背包容量 ",[34,56839,56841,56855],{"className":56840},[31642],[34,56842,56844],{"className":56843},[31646],[31648,56845,56846],{"xmlns":31650},[31652,56847,56848,56853],{},[31655,56849,56850],{},[31658,56851,56852],{},"W",[31665,56854,56852],{"encoding":31667},[34,56856,56858],{"className":56857,"ariaHidden":758},[31672],[34,56859,56861,56864],{"className":56860},[31676],[34,56862],{"className":56863,"style":33218},[31680],[34,56865,56852],{"className":56866,"style":37422},[31685,31686],"，每个物品选或不选，求最大价值。",[108,56869,56870,2664,56872,56875,56876,56904,56905,56933],{},[2627,56871,53780],{},[31,56873,56874],{},"dp[i][r]"," 表示已考虑前 ",[34,56877,56879,56892],{"className":56878},[31642],[34,56880,56882],{"className":56881},[31646],[31648,56883,56884],{"xmlns":31650},[31652,56885,56886,56890],{},[31655,56887,56888],{},[31658,56889,31955],{},[31665,56891,31955],{"encoding":31667},[34,56893,56895],{"className":56894,"ariaHidden":758},[31672],[34,56896,56898,56901],{"className":56897},[31676],[34,56899],{"className":56900,"style":33104},[31680],[34,56902,31955],{"className":56903},[31685,31686]," 个物品，背包剩余容量为 ",[34,56906,56908,56921],{"className":56907},[31642],[34,56909,56911],{"className":56910},[31646],[31648,56912,56913],{"xmlns":31650},[31652,56914,56915,56919],{},[31655,56916,56917],{},[31658,56918,34395],{},[31665,56920,34395],{"encoding":31667},[34,56922,56924],{"className":56923,"ariaHidden":758},[31672],[34,56925,56927,56930],{"className":56926},[31676],[34,56928],{"className":56929,"style":32527},[31680],[34,56931,34395],{"className":56932,"style":34452},[31685,31686]," 时的最大价值。",[108,56935,56936,33183,56938],{},[2627,56937,51680],{},[105,56939,56940,57038],{},[108,56941,2234,56942,57034,57035,347],{},[34,56943,56945,56967],{"className":56944},[31642],[34,56946,56948],{"className":56947},[31646],[31648,56949,56950],{"xmlns":31650},[31652,56951,56952,56964],{},[31655,56953,56954,56956,56958],{},[31658,56955,34395],{},[31662,56957,932],{},[31743,56959,56960,56962],{},[31658,56961,47077],{},[31658,56963,31955],{},[31665,56965,56966],{"encoding":31667},"r \u003C w_i",[34,56968,56970,56988],{"className":56969,"ariaHidden":758},[31672],[34,56971,56973,56976,56979,56982,56985],{"className":56972},[31676],[34,56974],{"className":56975,"style":34812},[31680],[34,56977,34395],{"className":56978,"style":34452},[31685,31686],[34,56980],{"className":56981,"style":31846},[31845],[34,56983,932],{"className":56984},[31850],[34,56986],{"className":56987,"style":31846},[31845],[34,56989,56991,56994],{"className":56990},[31676],[34,56992],{"className":56993,"style":31860},[31680],[34,56995,56997,57000],{"className":56996},[31685],[34,56998,47077],{"className":56999,"style":47120},[31685,31686],[34,57001,57003],{"className":57002},[31792],[34,57004,57006,57026],{"className":57005},[31796,31797],[34,57007,57009,57023],{"className":57008},[31801],[34,57010,57012],{"className":57011,"style":32090},[31805],[34,57013,57014,57017],{"style":56744},[34,57015],{"className":57016,"style":31814},[31813],[34,57018,57020],{"className":57019},[31818,31819,31820,31821],[34,57021,31955],{"className":57022},[31685,31686,31821],[34,57024,31829],{"className":57025},[31828],[34,57027,57029],{"className":57028},[31801],[34,57030,57032],{"className":57031,"style":31836},[31805],[34,57033],{},"，物品放不下，只能不选：",[31,57036,57037],{},"dp[i][r] = dp[i-1][r]",[108,57039,2234,57040,57132,57133,57202,57203,2664,57272,347],{},[34,57041,57043,57065],{"className":57042},[31642],[34,57044,57046],{"className":57045},[31646],[31648,57047,57048],{"xmlns":31650},[31652,57049,57050,57062],{},[31655,57051,57052,57054,57056],{},[31658,57053,34395],{},[31662,57055,43175],{},[31743,57057,57058,57060],{},[31658,57059,47077],{},[31658,57061,31955],{},[31665,57063,57064],{"encoding":31667},"r \\ge w_i",[34,57066,57068,57086],{"className":57067,"ariaHidden":758},[31672],[34,57069,57071,57074,57077,57080,57083],{"className":57070},[31676],[34,57072],{"className":57073,"style":43198},[31680],[34,57075,34395],{"className":57076,"style":34452},[31685,31686],[34,57078],{"className":57079,"style":31846},[31845],[34,57081,43175],{"className":57082},[31850],[34,57084],{"className":57085,"style":31846},[31845],[34,57087,57089,57092],{"className":57088},[31676],[34,57090],{"className":57091,"style":31860},[31680],[34,57093,57095,57098],{"className":57094},[31685],[34,57096,47077],{"className":57097,"style":47120},[31685,31686],[34,57099,57101],{"className":57100},[31792],[34,57102,57104,57124],{"className":57103},[31796,31797],[34,57105,57107,57121],{"className":57106},[31801],[34,57108,57110],{"className":57109,"style":32090},[31805],[34,57111,57112,57115],{"style":56744},[34,57113],{"className":57114,"style":31814},[31813],[34,57116,57118],{"className":57117},[31818,31819,31820,31821],[34,57119,31955],{"className":57120},[31685,31686,31821],[34,57122,31829],{"className":57123},[31828],[34,57125,57127],{"className":57126},[31801],[34,57128,57130],{"className":57129,"style":31836},[31805],[34,57131],{},"，取“不选”和“选”的最大值：“选”意味着腾出 ",[34,57134,57136,57153],{"className":57135},[31642],[34,57137,57139],{"className":57138},[31646],[31648,57140,57141],{"xmlns":31650},[31652,57142,57143,57151],{},[31655,57144,57145],{},[31743,57146,57147,57149],{},[31658,57148,47077],{},[31658,57150,31955],{},[31665,57152,56714],{"encoding":31667},[34,57154,57156],{"className":57155,"ariaHidden":758},[31672],[34,57157,57159,57162],{"className":57158},[31676],[34,57160],{"className":57161,"style":31860},[31680],[34,57163,57165,57168],{"className":57164},[31685],[34,57166,47077],{"className":57167,"style":47120},[31685,31686],[34,57169,57171],{"className":57170},[31792],[34,57172,57174,57194],{"className":57173},[31796,31797],[34,57175,57177,57191],{"className":57176},[31801],[34,57178,57180],{"className":57179,"style":32090},[31805],[34,57181,57182,57185],{"style":56744},[34,57183],{"className":57184,"style":31814},[31813],[34,57186,57188],{"className":57187},[31818,31819,31820,31821],[34,57189,31955],{"className":57190},[31685,31686,31821],[34,57192,31829],{"className":57193},[31828],[34,57195,57197],{"className":57196},[31801],[34,57198,57200],{"className":57199,"style":31836},[31805],[34,57201],{}," 的空间，价值增加 ",[34,57204,57206,57223],{"className":57205},[31642],[34,57207,57209],{"className":57208},[31646],[31648,57210,57211],{"xmlns":31650},[31652,57212,57213,57221],{},[31655,57214,57215],{},[31743,57216,57217,57219],{},[31658,57218,56782],{},[31658,57220,31955],{},[31665,57222,56787],{"encoding":31667},[34,57224,57226],{"className":57225,"ariaHidden":758},[31672],[34,57227,57229,57232],{"className":57228},[31676],[34,57230],{"className":57231,"style":31860},[31680],[34,57233,57235,57238],{"className":57234},[31685],[34,57236,56782],{"className":57237,"style":32062},[31685,31686],[34,57239,57241],{"className":57240},[31792],[34,57242,57244,57264],{"className":57243},[31796,31797],[34,57245,57247,57261],{"className":57246},[31801],[34,57248,57250],{"className":57249,"style":32090},[31805],[34,57251,57252,57255],{"style":38795},[34,57253],{"className":57254,"style":31814},[31813],[34,57256,57258],{"className":57257},[31818,31819,31820,31821],[34,57259,31955],{"className":57260},[31685,31686,31821],[34,57262,31829],{"className":57263},[31828],[34,57265,57267],{"className":57266},[31801],[34,57268,57270],{"className":57269,"style":31836},[31805],[34,57271],{},[31,57273,57274],{},"dp[i][r] = max(dp[i-1][r], dp[i-1][r-w_i] + v_i)",[108,57276,57277,54852,57280,57283,57284,57287,57288,57316,57317,57345,57346,57415,57416,57418],{},[2627,57278,57279],{},"构造解过程",[31,57281,57282],{},"dp[n][W]"," 开始倒推。如果 ",[31,57285,57286],{},"dp[i][r] != dp[i-1][r]","，说明物品 ",[34,57289,57291,57304],{"className":57290},[31642],[34,57292,57294],{"className":57293},[31646],[31648,57295,57296],{"xmlns":31650},[31652,57297,57298,57302],{},[31655,57299,57300],{},[31658,57301,31955],{},[31665,57303,31955],{"encoding":31667},[34,57305,57307],{"className":57306,"ariaHidden":758},[31672],[34,57308,57310,57313],{"className":57309},[31676],[34,57311],{"className":57312,"style":33104},[31680],[34,57314,31955],{"className":57315},[31685,31686]," 被选中了，将 ",[34,57318,57320,57333],{"className":57319},[31642],[34,57321,57323],{"className":57322},[31646],[31648,57324,57325],{"xmlns":31650},[31652,57326,57327,57331],{},[31655,57328,57329],{},[31658,57330,34395],{},[31665,57332,34395],{"encoding":31667},[34,57334,57336],{"className":57335,"ariaHidden":758},[31672],[34,57337,57339,57342],{"className":57338},[31676],[34,57340],{"className":57341,"style":32527},[31680],[34,57343,34395],{"className":57344,"style":34452},[31685,31686]," 减去 ",[34,57347,57349,57366],{"className":57348},[31642],[34,57350,57352],{"className":57351},[31646],[31648,57353,57354],{"xmlns":31650},[31652,57355,57356,57364],{},[31655,57357,57358],{},[31743,57359,57360,57362],{},[31658,57361,47077],{},[31658,57363,31955],{},[31665,57365,56714],{"encoding":31667},[34,57367,57369],{"className":57368,"ariaHidden":758},[31672],[34,57370,57372,57375],{"className":57371},[31676],[34,57373],{"className":57374,"style":31860},[31680],[34,57376,57378,57381],{"className":57377},[31685],[34,57379,47077],{"className":57380,"style":47120},[31685,31686],[34,57382,57384],{"className":57383},[31792],[34,57385,57387,57407],{"className":57386},[31796,31797],[34,57388,57390,57404],{"className":57389},[31801],[34,57391,57393],{"className":57392,"style":32090},[31805],[34,57394,57395,57398],{"style":56744},[34,57396],{"className":57397,"style":31814},[31813],[34,57399,57401],{"className":57400},[31818,31819,31820,31821],[34,57402,31955],{"className":57403},[31685,31686,31821],[34,57405,31829],{"className":57406},[31828],[34,57408,57410],{"className":57409},[31801],[34,57411,57413],{"className":57412,"style":31836},[31805],[34,57414],{},"，继续向上看 ",[31,57417,51731],{}," 层。",[32490,57420,57422],{"id":57421},"例题-10完全背包问题","例题 10：完全背包问题",[105,57424,57425,57430],{},[108,57426,57427,57429],{},[2627,57428,51674],{},"：与 0\u002F1 背包的区别在于，每种物品可以无限次重复选取。",[108,57431,57432,57434,57435,57463,57464,39984,57467,57470,57471,57499,57500,57503,57504,57532,57533,57536,57537,57539,57540,57543,57544,57623,57624,347],{},[2627,57433,51819],{},"：\n基本思路是枚举选 ",[34,57436,57438,57451],{"className":57437},[31642],[34,57439,57441],{"className":57440},[31646],[31648,57442,57443],{"xmlns":31650},[31652,57444,57445,57449],{},[31655,57446,57447],{},[31658,57448,33759],{},[31665,57450,33759],{"encoding":31667},[34,57452,57454],{"className":57453,"ariaHidden":758},[31672],[34,57455,57457,57460],{"className":57456},[31676],[34,57458],{"className":57459,"style":31681},[31680],[34,57461,33759],{"className":57462,"style":33774},[31685,31686]," 个：",[31,57465,57466],{},"dp[i][j] = max{dp[i-1][j - k*w[i]] + k*v[i]}",[2627,57468,57469],{},"算法改进","：由于选 ",[34,57472,57474,57487],{"className":57473},[31642],[34,57475,57477],{"className":57476},[31646],[31648,57478,57479],{"xmlns":31650},[31652,57480,57481,57485],{},[31655,57482,57483],{},[31658,57484,33759],{},[31665,57486,33759],{"encoding":31667},[34,57488,57490],{"className":57489,"ariaHidden":758},[31672],[34,57491,57493,57496],{"className":57492},[31676],[34,57494],{"className":57495,"style":31681},[31680],[34,57497,33759],{"className":57498,"style":33774},[31685,31686]," 个的情况已经在当前层 ",[31,57501,57502],{},"dp[i][j-w[i]]"," 中计算过了，因此可以去掉 ",[34,57505,57507,57520],{"className":57506},[31642],[34,57508,57510],{"className":57509},[31646],[31648,57511,57512],{"xmlns":31650},[31652,57513,57514,57518],{},[31655,57515,57516],{},[31658,57517,33759],{},[31665,57519,33759],{"encoding":31667},[34,57521,57523],{"className":57522,"ariaHidden":758},[31672],[34,57524,57526,57529],{"className":57525},[31676],[34,57527],{"className":57528,"style":31681},[31680],[34,57530,33759],{"className":57531,"style":33774},[31685,31686]," 的循环，直接优化为：\n",[31,57534,57535],{},"dp[i][j] = max(dp[i-1][j], dp[i][j-w[i]] + v[i])"," （注意第二项是 ",[31,57538,53850],{}," 而不是 ",[31,57541,57542],{},"dp[i-1]","）。\n时间复杂度从 ",[34,57545,57547,57573],{"className":57546},[31642],[34,57548,57550],{"className":57549},[31646],[31648,57551,57552],{"xmlns":31650},[31652,57553,57554,57570],{},[31655,57555,57556,57558,57560,57562,57568],{},[31658,57557,37890],{},[31662,57559,456],{"stretchy":702},[31658,57561,31660],{},[37624,57563,57564,57566],{},[31658,57565,47384],{},[31749,57567,768],{},[31662,57569,2557],{"stretchy":702},[31665,57571,57572],{"encoding":31667},"O(nC^2)",[34,57574,57576],{"className":57575,"ariaHidden":758},[31672],[34,57577,57579,57582,57585,57588,57591,57620],{"className":57578},[31676],[34,57580],{"className":57581,"style":37755},[31680],[34,57583,37890],{"className":57584,"style":34452},[31685,31686],[34,57586,456],{"className":57587},[31782],[34,57589,31660],{"className":57590},[31685,31686],[34,57592,57594,57597],{"className":57593},[31685],[34,57595,47384],{"className":57596,"style":38706},[31685,31686],[34,57598,57600],{"className":57599},[31792],[34,57601,57603],{"className":57602},[31796],[34,57604,57606],{"className":57605},[31801],[34,57607,57609],{"className":57608,"style":37774},[31805],[34,57610,57611,57614],{"style":37777},[34,57612],{"className":57613,"style":31814},[31813],[34,57615,57617],{"className":57616},[31818,31819,31820,31821],[34,57618,768],{"className":57619},[31685,31821],[34,57621,2557],{"className":57622},[31690]," 降为 ",[34,57625,57627,57649],{"className":57626},[31642],[34,57628,57630],{"className":57629},[31646],[31648,57631,57632],{"xmlns":31650},[31652,57633,57634,57646],{},[31655,57635,57636,57638,57640,57642,57644],{},[31658,57637,37890],{},[31662,57639,456],{"stretchy":702},[31658,57641,31660],{},[31658,57643,47384],{},[31662,57645,2557],{"stretchy":702},[31665,57647,57648],{"encoding":31667},"O(nC)",[34,57650,57652],{"className":57651,"ariaHidden":758},[31672],[34,57653,57655,57658,57661,57664,57667,57670],{"className":57654},[31676],[34,57656],{"className":57657,"style":31774},[31680],[34,57659,37890],{"className":57660,"style":34452},[31685,31686],[34,57662,456],{"className":57663},[31782],[34,57665,31660],{"className":57666},[31685,31686],[34,57668,47384],{"className":57669,"style":38706},[31685,31686],[34,57671,2557],{"className":57672},[31690],[32490,57674,57676],{"id":57675},"例题-11资源分配问题","例题 11：资源分配问题",[105,57678,57679,57684],{},[108,57680,57681,57683],{},[2627,57682,51674],{},"：将一定数量的资源（如 5 名员工）分配给若干使用者（如 3 个商店），使得总收益最大。",[108,57685,57686,2664,57688,57691,57692,46528,57765,57793,57794,57822,57823,57882,57883,57886,57887,57915],{},[2627,57687,51819],{},[31,57689,57690],{},"dp[i][j] = max(dp[i-1][j-k] + v[i][k])"," （其中 ",[34,57693,57695,57717],{"className":57694},[31642],[34,57696,57698],{"className":57697},[31646],[31648,57699,57700],{"xmlns":31650},[31652,57701,57702,57714],{},[31655,57703,57704,57706,57708,57710,57712],{},[31749,57705,459],{},[31662,57707,45972],{},[31658,57709,33759],{},[31662,57711,45972],{},[31658,57713,31968],{},[31665,57715,57716],{"encoding":31667},"0 \\le k \\le j",[34,57718,57720,57738,57756],{"className":57719,"ariaHidden":758},[31672],[34,57721,57723,57726,57729,57732,57735],{"className":57722},[31676],[34,57724],{"className":57725,"style":55382},[31680],[34,57727,459],{"className":57728},[31685],[34,57730],{"className":57731,"style":31846},[31845],[34,57733,45972],{"className":57734},[31850],[34,57736],{"className":57737,"style":31846},[31845],[34,57739,57741,57744,57747,57750,57753],{"className":57740},[31676],[34,57742],{"className":57743,"style":47709},[31680],[34,57745,33759],{"className":57746,"style":33774},[31685,31686],[34,57748],{"className":57749,"style":31846},[31845],[34,57751,45972],{"className":57752},[31850],[34,57754],{"className":57755,"style":31846},[31845],[34,57757,57759,57762],{"className":57758},[31676],[34,57760],{"className":57761,"style":33123},[31680],[34,57763,31968],{"className":57764,"style":32153},[31685,31686],[34,57766,57768,57781],{"className":57767},[31642],[34,57769,57771],{"className":57770},[31646],[31648,57772,57773],{"xmlns":31650},[31652,57774,57775,57779],{},[31655,57776,57777],{},[31658,57778,33759],{},[31665,57780,33759],{"encoding":31667},[34,57782,57784],{"className":57783,"ariaHidden":758},[31672],[34,57785,57787,57790],{"className":57786},[31676],[34,57788],{"className":57789,"style":31681},[31680],[34,57791,33759],{"className":57792,"style":33774},[31685,31686]," 表示分给当前第 ",[34,57795,57797,57810],{"className":57796},[31642],[34,57798,57800],{"className":57799},[31646],[31648,57801,57802],{"xmlns":31650},[31652,57803,57804,57808],{},[31655,57805,57806],{},[31658,57807,31955],{},[31665,57809,31955],{"encoding":31667},[34,57811,57813],{"className":57812,"ariaHidden":758},[31672],[34,57814,57816,57819],{"className":57815},[31676],[34,57817],{"className":57818,"style":33104},[31680],[34,57820,31955],{"className":57821},[31685,31686]," 个商店的资源数量，",[34,57824,57826,57852],{"className":57825},[31642],[34,57827,57829],{"className":57828},[31646],[31648,57830,57831],{"xmlns":31650},[31652,57832,57833,57849],{},[31655,57834,57835,57837,57839,57841,57843,57845,57847],{},[31658,57836,56782],{},[31662,57838,15972],{"stretchy":702},[31658,57840,31955],{},[31662,57842,32712],{"stretchy":702},[31662,57844,15972],{"stretchy":702},[31658,57846,33759],{},[31662,57848,32712],{"stretchy":702},[31665,57850,57851],{"encoding":31667},"v[i][k]",[34,57853,57855],{"className":57854,"ariaHidden":758},[31672],[34,57856,57858,57861,57864,57867,57870,57873,57876,57879],{"className":57857},[31676],[34,57859],{"className":57860,"style":31774},[31680],[34,57862,56782],{"className":57863,"style":32062},[31685,31686],[34,57865,15972],{"className":57866},[31782],[34,57868,31955],{"className":57869},[31685,31686],[34,57871,32712],{"className":57872},[31690],[34,57874,15972],{"className":57875},[31782],[34,57877,33759],{"className":57878,"style":33774},[31685,31686],[34,57880,32712],{"className":57881},[31690]," 是对应的收益表）。同时用 ",[31,57884,57885],{},"pnum[i][j]"," 记录取得最大值时的 ",[34,57888,57890,57903],{"className":57889},[31642],[34,57891,57893],{"className":57892},[31646],[31648,57894,57895],{"xmlns":31650},[31652,57896,57897,57901],{},[31655,57898,57899],{},[31658,57900,33759],{},[31665,57902,33759],{"encoding":31667},[34,57904,57906],{"className":57905,"ariaHidden":758},[31672],[34,57907,57909,57912],{"className":57908},[31676],[34,57910],{"className":57911,"style":31681},[31680],[34,57913,33759],{"className":57914,"style":33774},[31685,31686]," 供回溯使用。",[32490,57917,57919],{"id":57918},"例题-12长江一日游游艇租赁问题拓展","例题 12：长江一日游——游艇租赁问题（拓展）",[105,57921,57922,58099],{},[108,57923,57924,2664,57926,57954,57955,53834,57983,58011,58012,58069,58070,58098],{},[2627,57925,51674],{},[34,57927,57929,57942],{"className":57928},[31642],[34,57930,57932],{"className":57931},[31646],[31648,57933,57934],{"xmlns":31650},[31652,57935,57936,57940],{},[31655,57937,57938],{},[31658,57939,31660],{},[31665,57941,31660],{"encoding":31667},[34,57943,57945],{"className":57944,"ariaHidden":758},[31672],[34,57946,57948,57951],{"className":57947},[31676],[34,57949],{"className":57950,"style":32527},[31680],[34,57952,31660],{"className":57953},[31685,31686]," 个游艇出租站，从站点 ",[34,57956,57958,57971],{"className":57957},[31642],[34,57959,57961],{"className":57960},[31646],[31648,57962,57963],{"xmlns":31650},[31652,57964,57965,57969],{},[31655,57966,57967],{},[31658,57968,31955],{},[31665,57970,31955],{"encoding":31667},[34,57972,57974],{"className":57973,"ariaHidden":758},[31672],[34,57975,57977,57980],{"className":57976},[31676],[34,57978],{"className":57979,"style":33104},[31680],[34,57981,31955],{"className":57982},[31685,31686],[34,57984,57986,57999],{"className":57985},[31642],[34,57987,57989],{"className":57988},[31646],[31648,57990,57991],{"xmlns":31650},[31652,57992,57993,57997],{},[31655,57994,57995],{},[31658,57996,31968],{},[31665,57998,31968],{"encoding":31667},[34,58000,58002],{"className":58001,"ariaHidden":758},[31672],[34,58003,58005,58008],{"className":58004},[31676],[34,58006],{"className":58007,"style":33123},[31680],[34,58009,31968],{"className":58010,"style":32153},[31685,31686]," 租金为 ",[34,58013,58015,58039],{"className":58014},[31642],[34,58016,58018],{"className":58017},[31646],[31648,58019,58020],{"xmlns":31650},[31652,58021,58022,58036],{},[31655,58023,58024,58026,58028,58030,58032,58034],{},[31658,58025,34395],{},[31662,58027,456],{"stretchy":702},[31658,58029,31955],{},[31662,58031,31960],{"separator":758},[31658,58033,31968],{},[31662,58035,2557],{"stretchy":702},[31665,58037,58038],{"encoding":31667},"r(i,j)",[34,58040,58042],{"className":58041,"ariaHidden":758},[31672],[34,58043,58045,58048,58051,58054,58057,58060,58063,58066],{"className":58044},[31676],[34,58046],{"className":58047,"style":31774},[31680],[34,58049,34395],{"className":58050,"style":34452},[31685,31686],[34,58052,456],{"className":58053},[31782],[34,58055,31955],{"className":58056},[31685,31686],[34,58058,31960],{"className":58059},[32119],[34,58061],{"className":58062,"style":32123},[31845],[34,58064,31968],{"className":58065,"style":32153},[31685,31686],[34,58067,2557],{"className":58068},[31690],"，求从 1 到 ",[34,58071,58073,58086],{"className":58072},[31642],[34,58074,58076],{"className":58075},[31646],[31648,58077,58078],{"xmlns":31650},[31652,58079,58080,58084],{},[31655,58081,58082],{},[31658,58083,31660],{},[31665,58085,31660],{"encoding":31667},[34,58087,58089],{"className":58088,"ariaHidden":758},[31672],[34,58090,58092,58095],{"className":58091},[31676],[34,58093],{"className":58094,"style":32527},[31680],[34,58096,31660],{"className":58097},[31685,31686]," 最少租金。",[108,58100,58101,2664,58103,58106,58107,58135],{},[2627,58102,51819],{},[31,58104,58105],{},"m[i][j] = min_{i\u003Ck\u003Cj} \\{m[i][k] + m[k][j], r[i][j]\\}","。即直接到达，或者在中间某个站点 ",[34,58108,58110,58123],{"className":58109},[31642],[34,58111,58113],{"className":58112},[31646],[31648,58114,58115],{"xmlns":31650},[31652,58116,58117,58121],{},[31655,58118,58119],{},[31658,58120,33759],{},[31665,58122,33759],{"encoding":31667},[34,58124,58126],{"className":58125,"ariaHidden":758},[31672],[34,58127,58129,58132],{"className":58128},[31676],[34,58130],{"className":58131,"style":31681},[31680],[34,58133,33759],{"className":58134,"style":33774},[31685,31686]," 停靠的最小值。",{"title":29,"searchDepth":43,"depth":43,"links":58137},[58138,58139],{"id":51551,"depth":49,"text":51552},{"id":51662,"depth":49,"text":51663},{},"\u002Fsource\u002Fcs\u002Falgo\u002Fnotes\u002Fchap4",{"title":51543,"description":29},"source\u002Fcs\u002Falgo\u002Fnotes\u002Fchap4","pqaM2SjXbWvYPafNOn9u6guQCVshb2gyp-M6AIGZM1E",{"id":58146,"title":58147,"body":58148,"description":29,"extension":85,"meta":90805,"navigation":52,"path":90806,"seo":90807,"stem":90808,"__hash__":90809},"notes\u002Fsource\u002Fcs\u002Falgo\u002Fpastpapers\u002F2024-2025-se2.md","2024-2025 第二学期 算法设计与分析",{"type":8,"value":58149,"toc":90745},[58150,58153,58155,58159,58163,58166,58169,58175,58195,58200,58223,58225,58229,58232,58235,58239,58288,58292,58362,58364,58368,58371,58374,58378,58411,58415,58469,58471,58475,58478,58481,58486,58500,58504,58518,58520,58524,58527,58540,58543,58548,58581,58583,58586,59384,59387,59392,60493,60495,60498,61056,61059,61064,62756,62758,62761,63169,63172,63176,63601,63603,63607,63706,63709,63713,63931,63933,63937,64207,64210,64214,64453,64455,64459,64471,64474,64478,64488,64490,64494,64505,64508,64512,64549,64551,64555,65038,65041,65045,65354,65356,65360,65372,65375,65379,65406,65408,65412,65415,65418,65421,65433,65449,65451,65454,65457,65460,65465,65667,65669,65672,65732,65800,65803,65829,66071,66073,66076,66398,66401,66556,67288,67290,67293,68381,68384,68397,68734,68736,68740,68743,68746,68817,68886,68889,68894,69111,69116,70652,70654,70657,72360,72431,72434,72847,72919,82283,82347,82534,82569,83805,83807,83810,83987,83995,83998,84003,84116,85535,87642,87644,87648,87651,87654,87662,87665,87670,87742,87747,88115,88117,88120,88295,88481,88484,88487,88492,90512,90517,90743],[11,58151,58147],{"id":58152},"_2024-2025-第二学期-算法设计与分析",[32188,58154],{},[19,58156,58158],{"id":58157},"一简答题每题5分共20分","一、简答题（每题5分，共20分）",[253,58160,58162],{"id":58161},"_1-题目内容","1. 题目内容",[15,58164,58165],{},"简述分治法进行算法设计的三个基本步骤。",[253,58167,58168],{"id":58168},"答案与深度解析",[15,58170,58171,58174],{},[2627,58172,58173],{},"答案：","\n分治法（Divide and Conquer）的三个基本步骤为：",[2230,58176,58177,58183,58189],{},[108,58178,58179,58182],{},[2627,58180,58181],{},"分解（Divide）："," 将原问题分解为若干个规模较小、相互独立且与原问题形式相同的子问题。",[108,58184,58185,58188],{},[2627,58186,58187],{},"解决（Conquer）："," 若子问题规模较小且容易被解决，则直接求解；否则，递归地解决各个子问题。",[108,58190,58191,58194],{},[2627,58192,58193],{},"合并（Combine）："," 将各个子问题的解合并为原问题的解。",[15,58196,58197],{},[2627,58198,58199],{},"深度解析：",[105,58201,58202,58208,58217],{},[108,58203,58204,58207],{},[2627,58205,58206],{},"核心驱动力："," 面对庞大复杂的问题，计算机处理起来会非常耗时。分治法的底层逻辑是“降维打击”，通过不断缩小问题的规模，直到问题简单到可以直接得出答案（即递归的基线条件）。",[108,58209,58210,58213,58214,347],{},[2627,58211,58212],{},"关键点与风险："," 分治法成功的前提是“子问题必须相互独立”。如果子问题之间有重叠（即存在公共子子问题），分治法会进行大量重复计算，此时算法效率极低，应转而考虑",[2627,58215,58216],{},"动态规划",[108,58218,58219,58222],{},[2627,58220,58221],{},"应用实例："," 归并排序、快速排序、二分查找都严格遵循了这三个步骤。",[32188,58224],{},[253,58226,58228],{"id":58227},"_2-题目内容","2. 题目内容",[15,58230,58231],{},"简述回溯法与分支限界法的异同。",[253,58233,58168],{"id":58234},"答案与深度解析-1",[15,58236,58237],{},[2627,58238,58173],{},[105,58240,58241,58247],{},[108,58242,58243,58246],{},[2627,58244,58245],{},"相同点："," 两者都是在问题的状态空间树（解空间）上进行搜索的算法；都需要通过设计限界函数（剪枝函数）来避免无效搜索，提高算法效率。",[108,58248,58249,58252],{},[2627,58250,58251],{},"不同点：",[2230,58253,58254,58271],{},[108,58255,58256,58259,58260,58263,58264,347],{},[2627,58257,58258],{},"搜索方式不同："," 回溯法采用",[2627,58261,58262],{},"深度优先搜索（DFS）","；分支限界法采用",[2627,58265,58266,58267,58270],{},"广度优先搜索（BFS）",[2627,58268,58269],{},"或","最小耗费优先（优先队列）搜索",[108,58272,58273,58276,58277,58269,58280,58283,58284,58287],{},[2627,58274,58275],{},"求解目标不同："," 回溯法的目标通常是找出满足约束条件的",[2627,58278,58279],{},"所有解",[2627,58281,58282],{},"任一解","；分支限界法的目标则是找出满足约束条件的一个解，或者是某种意义下的",[2627,58285,58286],{},"最优解","（如最大值或最小值）。",[15,58289,58290],{},[2627,58291,58199],{},[105,58293,58294,58312],{},[108,58295,58296,58299,58300,58303,58304],{},[2627,58297,58298],{},"形象化理解："," * ",[2627,58301,58302],{},"回溯法","就像是在走迷宫，你顺着一条路一直走到黑（深度优先），遇到死胡同就退回上一个岔路口（回溯），尝试另一条路，直到把所有角落都走遍。\n",[105,58305,58306],{},[108,58307,58308,58311],{},[2627,58309,58310],{},"分支限界法","像是在雷达扫描，你站在起点，先把周围第一圈的岔路口都看一遍（广度优先），然后评估哪条路看起来最有可能通向宝藏（通过代价函数评估），接着优先去展开那条最有希望的路。",[108,58313,58314,58317,58318,58361],{},[2627,58315,58316],{},"空间复杂度差异："," 回溯法只需要保存当前从根节点到当前节点的路径，空间复杂度通常为 ",[34,58319,58321,58340],{"className":58320},[31642],[34,58322,58324],{"className":58323},[31646],[31648,58325,58326],{"xmlns":31650},[31652,58327,58328,58338],{},[31655,58329,58330,58332,58334,58336],{},[31658,58331,37890],{},[31662,58333,456],{"stretchy":702},[31658,58335,31660],{},[31662,58337,2557],{"stretchy":702},[31665,58339,46493],{"encoding":31667},[34,58341,58343],{"className":58342,"ariaHidden":758},[31672],[34,58344,58346,58349,58352,58355,58358],{"className":58345},[31676],[34,58347],{"className":58348,"style":31774},[31680],[34,58350,37890],{"className":58351,"style":34452},[31685,31686],[34,58353,456],{"className":58354},[31782],[34,58356,31660],{"className":58357},[31685,31686],[34,58359,2557],{"className":58360},[31690],"；而分支限界法需要维护一个活结点表（队列），在最坏情况下空间复杂度可能达到指数级。",[32188,58363],{},[253,58365,58367],{"id":58366},"_3-题目内容","3. 题目内容",[15,58369,58370],{},"简述拉斯维加斯算法与蒙特卡洛算法的异同。",[253,58372,58168],{"id":58373},"答案与深度解析-2",[15,58375,58376],{},[2627,58377,58173],{},[105,58379,58380,58385],{},[108,58381,58382,58384],{},[2627,58383,58245],{}," 两者都属于随机化算法，都在算法执行过程中引入了随机数来辅助决策，以期望获得更好的平均性能或绕过最坏情况。",[108,58386,58387,58389],{},[2627,58388,58251],{},[105,58390,58391,58401],{},[108,58392,58393,58396,58397,58400],{},[2627,58394,58395],{},"拉斯维加斯算法（Las Vegas）："," 只要算法返回了结果，这个结果就",[2627,58398,58399],{},"一定正确","。但它的执行时间是不确定的，有时可能找不到解并宣告失败。",[108,58402,58403,58406,58407,58410],{},[2627,58404,58405],{},"蒙特卡洛算法（Monte Carlo）："," 算法的执行时间是确定且通常较快的，它一定会给出一个答案，但这个答案",[2627,58408,58409],{},"不一定正确","，获得正确解的概率存在一个界限。",[15,58412,58413],{},[2627,58414,58199],{},[105,58416,58417,58459],{},[108,58418,58419,58422,58423],{},[2627,58420,58421],{},"如何记忆："," * 去“拉斯维加斯”赌场，要么赢钱（拿到绝对正确的解），要么输光离场（找不到解），但赌场绝对不会给你假钞（结果绝对正确）。\n",[105,58424,58425],{},[108,58426,58427,58428,58458],{},"“蒙特卡洛”常用于估算（例如用随机投点法估算圆周率 ",[34,58429,58431,58446],{"className":58430},[31642],[34,58432,58434],{"className":58433},[31646],[31648,58435,58436],{"xmlns":31650},[31652,58437,58438,58443],{},[31655,58439,58440],{},[31658,58441,58442],{},"π",[31665,58444,58445],{"encoding":31667},"\\pi",[34,58447,58449],{"className":58448,"ariaHidden":758},[31672],[34,58450,58452,58455],{"className":58451},[31676],[34,58453],{"className":58454,"style":32527},[31680],[34,58456,58442],{"className":58457,"style":32062},[31685,31686],"），投的点越多，估算越准确，但它始终是一个近似值（有概率犯错）。",[108,58460,58461,58464,58465,58468],{},[2627,58462,58463],{},"性能提升策略："," 蒙特卡洛算法可以通过",[2627,58466,58467],{},"反复多次执行","来无限逼近 100% 的正确率；而拉斯维加斯算法如果失败，只能重新运行，期望下一次运气好能找到解。",[32188,58470],{},[253,58472,58474],{"id":58473},"_4-题目内容","4. 题目内容",[15,58476,58477],{},"简述分支限界法中活结点表的组织方式。",[253,58479,58168],{"id":58480},"答案与深度解析-3",[15,58482,58483,58485],{},[2627,58484,58173],{},"\n分支限界法中，活结点表（用于存放已经被生成但尚未展开的节点）主要有两种组织方式：",[2230,58487,58488,58494],{},[108,58489,58490,58493],{},[2627,58491,58492],{},"队列式（FIFO）分支限界法："," 将活结点表组织成一个先进先出的队列。按照节点生成的顺序依次扩展。",[108,58495,58496,58499],{},[2627,58497,58498],{},"优先队列式分支限界法："," 将活结点表组织成一个优先队列（通常用堆来实现）。每个节点都有一个由代价函数计算出的优先级（如成本或价值），算法每次总是从活结点表中选取优先级最高（或代价最小）的节点作为下一个扩展的节点。",[15,58501,58502],{},[2627,58503,58199],{},[105,58505,58506,58512],{},[108,58507,58508,58511],{},[2627,58509,58510],{},"底层数据结构："," 队列式对应简单的数组队列或链式队列；优先队列式对应数据结构中的“大顶堆”或“小顶堆”。",[108,58513,58514,58517],{},[2627,58515,58516],{},"策略选择："," 在实际解决工程优化问题（如单源最短路径、旅行商问题）时，优先队列式分支限界法应用更广，因为它能更快地引导搜索过程逼近最优解，从而引发更多的“剪枝”，大幅缩减搜索空间。",[32188,58519],{},[19,58521,58523],{"id":58522},"二选择题每题1分共10分","二、选择题（每题1分，共10分）",[253,58525,58162],{"id":58526},"_1-题目内容-1",[15,58528,58529,58530,58533,58534,58536,58537,58539],{},"在算法设计中，要求算法对于错误的输入也有相应的处理即要求设计的算法具有（ ）。\nA. 确定性",[58531,58532],"br",{},"\nB. 健壮性",[58531,58535],{},"\nC. 可行性",[58531,58538],{},"\nD. 正确性",[253,58541,58168],{"id":58542},"答案与深度解析-4",[15,58544,58545],{},[2627,58546,58547],{},"答案：B",[105,58549,58550],{},[108,58551,58552,58555],{},[2627,58553,58554],{},"分析：",[105,58556,58557,58563,58569,58575],{},[108,58558,58559,58562],{},[2627,58560,58561],{},"确定性："," 算法的每一步都必须有明确的定义，无二义性。",[108,58564,58565,58568],{},[2627,58566,58567],{},"健壮性（Robustness）："," 也称鲁棒性，指当输入数据非法或出现异常情况时，算法能够适当地做出反应或进行处理，而不会产生莫名其妙的输出或直接崩溃。这正是题干描述的内容。",[108,58570,58571,58574],{},[2627,58572,58573],{},"可行性："," 算法中的操作必须足够基本，能够在有限次内完成。",[108,58576,58577,58580],{},[2627,58578,58579],{},"正确性："," 算法对合法的输入能产生满足要求的正确输出。",[32188,58582],{},[253,58584,58228],{"id":58585},"_2-题目内容-1",[15,58587,58588,58589,58707,58708,58782,58784,58785,58902,58903,58978,58980,58981,58707,59096,59194,59196,59197,58902,59310],{},"下述表达不正确的选项是（ ）。\nA. ",[34,58590,58592,58622],{"className":58591},[31642],[34,58593,58595],{"className":58594},[31646],[31648,58596,58597],{"xmlns":31650},[31652,58598,58599,58619],{},[31655,58600,58601,58607,58609,58611,58613],{},[37624,58602,58603,58605],{},[31658,58604,31660],{},[31749,58606,768],{},[31658,58608,38080],{"mathvariant":38079},[31749,58610,768],{},[31662,58612,223],{},[37624,58614,58615,58617],{},[31749,58616,768],{},[31658,58618,31660],{},[31665,58620,58621],{"encoding":31667},"n^2 \u002F 2 + 2^n",[34,58623,58625,58672],{"className":58624,"ariaHidden":758},[31672],[34,58626,58628,58631,58660,58663,58666,58669],{"className":58627},[31676],[34,58629],{"className":58630,"style":37755},[31680],[34,58632,58634,58637],{"className":58633},[31685],[34,58635,31660],{"className":58636},[31685,31686],[34,58638,58640],{"className":58639},[31792],[34,58641,58643],{"className":58642},[31796],[34,58644,58646],{"className":58645},[31801],[34,58647,58649],{"className":58648,"style":37774},[31805],[34,58650,58651,58654],{"style":37777},[34,58652],{"className":58653,"style":31814},[31813],[34,58655,58657],{"className":58656},[31818,31819,31820,31821],[34,58658,768],{"className":58659},[31685,31821],[34,58661,38149],{"className":58662},[31685],[34,58664],{"className":58665,"style":32604},[31845],[34,58667,223],{"className":58668},[32024],[34,58670],{"className":58671,"style":32604},[31845],[34,58673,58675,58678],{"className":58674},[31676],[34,58676],{"className":58677,"style":37959},[31680],[34,58679,58681,58684],{"className":58680},[31685],[34,58682,768],{"className":58683},[31685],[34,58685,58687],{"className":58686},[31792],[34,58688,58690],{"className":58689},[31796],[34,58691,58693],{"className":58692},[31801],[34,58694,58696],{"className":58695,"style":37959},[31805],[34,58697,58698,58701],{"style":37777},[34,58699],{"className":58700,"style":31814},[31813],[34,58702,58704],{"className":58703},[31818,31819,31820,31821],[34,58705,31660],{"className":58706},[31685,31686,31821]," 上界函数是 ",[34,58709,58711,58735],{"className":58710},[31642],[34,58712,58714],{"className":58713},[31646],[31648,58715,58716],{"xmlns":31650},[31652,58717,58718,58732],{},[31655,58719,58720,58722,58724,58730],{},[31658,58721,37890],{},[31662,58723,456],{"stretchy":702},[37624,58725,58726,58728],{},[31749,58727,768],{},[31658,58729,31660],{},[31662,58731,2557],{"stretchy":702},[31665,58733,58734],{"encoding":31667},"O(2^n)",[34,58736,58738],{"className":58737,"ariaHidden":758},[31672],[34,58739,58741,58744,58747,58750,58779],{"className":58740},[31676],[34,58742],{"className":58743,"style":31774},[31680],[34,58745,37890],{"className":58746,"style":34452},[31685,31686],[34,58748,456],{"className":58749},[31782],[34,58751,58753,58756],{"className":58752},[31685],[34,58754,768],{"className":58755},[31685],[34,58757,58759],{"className":58758},[31792],[34,58760,58762],{"className":58761},[31796],[34,58763,58765],{"className":58764},[31801],[34,58766,58768],{"className":58767,"style":37959},[31805],[34,58769,58770,58773],{"style":37777},[34,58771],{"className":58772,"style":31814},[31813],[34,58774,58776],{"className":58775},[31818,31819,31820,31821],[34,58777,31660],{"className":58778},[31685,31686,31821],[34,58780,2557],{"className":58781},[31690],[58531,58783],{},"\nB. ",[34,58786,58788,58817],{"className":58787},[31642],[34,58789,58791],{"className":58790},[31646],[31648,58792,58793],{"xmlns":31650},[31652,58794,58795,58815],{},[31655,58796,58797,58803,58805,58807,58809],{},[37624,58798,58799,58801],{},[31658,58800,31660],{},[31749,58802,768],{},[31658,58804,38080],{"mathvariant":38079},[31749,58806,768],{},[31662,58808,223],{},[37624,58810,58811,58813],{},[31749,58812,768],{},[31658,58814,31660],{},[31665,58816,58621],{"encoding":31667},[34,58818,58820,58867],{"className":58819,"ariaHidden":758},[31672],[34,58821,58823,58826,58855,58858,58861,58864],{"className":58822},[31676],[34,58824],{"className":58825,"style":37755},[31680],[34,58827,58829,58832],{"className":58828},[31685],[34,58830,31660],{"className":58831},[31685,31686],[34,58833,58835],{"className":58834},[31792],[34,58836,58838],{"className":58837},[31796],[34,58839,58841],{"className":58840},[31801],[34,58842,58844],{"className":58843,"style":37774},[31805],[34,58845,58846,58849],{"style":37777},[34,58847],{"className":58848,"style":31814},[31813],[34,58850,58852],{"className":58851},[31818,31819,31820,31821],[34,58853,768],{"className":58854},[31685,31821],[34,58856,38149],{"className":58857},[31685],[34,58859],{"className":58860,"style":32604},[31845],[34,58862,223],{"className":58863},[32024],[34,58865],{"className":58866,"style":32604},[31845],[34,58868,58870,58873],{"className":58869},[31676],[34,58871],{"className":58872,"style":37959},[31680],[34,58874,58876,58879],{"className":58875},[31685],[34,58877,768],{"className":58878},[31685],[34,58880,58882],{"className":58881},[31792],[34,58883,58885],{"className":58884},[31796],[34,58886,58888],{"className":58887},[31801],[34,58889,58891],{"className":58890,"style":37959},[31805],[34,58892,58893,58896],{"style":37777},[34,58894],{"className":58895,"style":31814},[31813],[34,58897,58899],{"className":58898},[31818,31819,31820,31821],[34,58900,31660],{"className":58901},[31685,31686,31821]," 的下界函数是 ",[34,58904,58906,58931],{"className":58905},[31642],[34,58907,58909],{"className":58908},[31646],[31648,58910,58911],{"xmlns":31650},[31652,58912,58913,58928],{},[31655,58914,58915,58918,58920,58926],{},[31658,58916,58917],{"mathvariant":38079},"Ω",[31662,58919,456],{"stretchy":702},[37624,58921,58922,58924],{},[31749,58923,768],{},[31658,58925,31660],{},[31662,58927,2557],{"stretchy":702},[31665,58929,58930],{"encoding":31667},"\\Omega(2^n)",[34,58932,58934],{"className":58933,"ariaHidden":758},[31672],[34,58935,58937,58940,58943,58946,58975],{"className":58936},[31676],[34,58938],{"className":58939,"style":31774},[31680],[34,58941,58917],{"className":58942},[31685],[34,58944,456],{"className":58945},[31782],[34,58947,58949,58952],{"className":58948},[31685],[34,58950,768],{"className":58951},[31685],[34,58953,58955],{"className":58954},[31792],[34,58956,58958],{"className":58957},[31796],[34,58959,58961],{"className":58960},[31801],[34,58962,58964],{"className":58963,"style":37959},[31805],[34,58965,58966,58969],{"style":37777},[34,58967],{"className":58968,"style":31814},[31813],[34,58970,58972],{"className":58971},[31818,31819,31820,31821],[34,58973,31660],{"className":58974},[31685,31686,31821],[34,58976,2557],{"className":58977},[31690],[58531,58979],{},"\nC. ",[34,58982,58984,59012],{"className":58983},[31642],[34,58985,58987],{"className":58986},[31646],[31648,58988,58989],{"xmlns":31650},[31652,58990,58991,59009],{},[31655,58992,58993,59003],{},[31743,58994,58995,59001],{},[31655,58996,58997,58999],{},[31658,58998,885],{},[31662,59000,32682],{},[31749,59002,768],{},[37624,59004,59005,59007],{},[31658,59006,31660],{},[31749,59008,786],{},[31665,59010,59011],{"encoding":31667},"\\log_2 n^3",[34,59013,59015],{"className":59014,"ariaHidden":758},[31672],[34,59016,59018,59022,59064,59067],{"className":59017},[31676],[34,59019],{"className":59020,"style":59021},[31680],"height:1.0582em;vertical-align:-0.2441em;",[34,59023,59025,59030],{"className":59024},[32767],[34,59026,38616,59028],{"className":59027},[32767],[34,59029,31942],{"style":38619},[34,59031,59033],{"className":59032},[31792],[34,59034,59036,59056],{"className":59035},[31796,31797],[34,59037,59039,59053],{"className":59038},[31801],[34,59040,59042],{"className":59041,"style":38632},[31805],[34,59043,59044,59047],{"style":38635},[34,59045],{"className":59046,"style":31814},[31813],[34,59048,59050],{"className":59049},[31818,31819,31820,31821],[34,59051,768],{"className":59052},[31685,31821],[34,59054,31829],{"className":59055},[31828],[34,59057,59059],{"className":59058},[31801],[34,59060,59062],{"className":59061,"style":38654},[31805],[34,59063],{},[34,59065],{"className":59066,"style":32123},[31845],[34,59068,59070,59073],{"className":59069},[31685],[34,59071,31660],{"className":59072},[31685,31686],[34,59074,59076],{"className":59075},[31792],[34,59077,59079],{"className":59078},[31796],[34,59080,59082],{"className":59081},[31801],[34,59083,59085],{"className":59084,"style":37774},[31805],[34,59086,59087,59090],{"style":37777},[34,59088],{"className":59089,"style":31814},[31813],[34,59091,59093],{"className":59092},[31818,31819,31820,31821],[34,59094,786],{"className":59095},[31685,31821],[34,59097,59099,59128],{"className":59098},[31642],[34,59100,59102],{"className":59101},[31646],[31648,59103,59104],{"xmlns":31650},[31652,59105,59106,59126],{},[31655,59107,59108,59110,59112,59122,59124],{},[31658,59109,37890],{},[31662,59111,456],{"stretchy":702},[31743,59113,59114,59120],{},[31655,59115,59116,59118],{},[31658,59117,885],{},[31662,59119,32682],{},[31749,59121,768],{},[31658,59123,31660],{},[31662,59125,2557],{"stretchy":702},[31665,59127,47514],{"encoding":31667},[34,59129,59131],{"className":59130,"ariaHidden":758},[31672],[34,59132,59134,59137,59140,59143,59185,59188,59191],{"className":59133},[31676],[34,59135],{"className":59136,"style":31774},[31680],[34,59138,37890],{"className":59139,"style":34452},[31685,31686],[34,59141,456],{"className":59142},[31782],[34,59144,59146,59151],{"className":59145},[32767],[34,59147,38616,59149],{"className":59148},[32767],[34,59150,31942],{"style":38619},[34,59152,59154],{"className":59153},[31792],[34,59155,59157,59177],{"className":59156},[31796,31797],[34,59158,59160,59174],{"className":59159},[31801],[34,59161,59163],{"className":59162,"style":38632},[31805],[34,59164,59165,59168],{"style":38635},[34,59166],{"className":59167,"style":31814},[31813],[34,59169,59171],{"className":59170},[31818,31819,31820,31821],[34,59172,768],{"className":59173},[31685,31821],[34,59175,31829],{"className":59176},[31828],[34,59178,59180],{"className":59179},[31801],[34,59181,59183],{"className":59182,"style":38654},[31805],[34,59184],{},[34,59186],{"className":59187,"style":32123},[31845],[34,59189,31660],{"className":59190},[31685,31686],[34,59192,2557],{"className":59193},[31690],[58531,59195],{},"\nD. ",[34,59198,59200,59227],{"className":59199},[31642],[34,59201,59203],{"className":59202},[31646],[31648,59204,59205],{"xmlns":31650},[31652,59206,59207,59225],{},[31655,59208,59209,59219],{},[31743,59210,59211,59217],{},[31655,59212,59213,59215],{},[31658,59214,885],{},[31662,59216,32682],{},[31749,59218,768],{},[37624,59220,59221,59223],{},[31658,59222,31660],{},[31749,59224,786],{},[31665,59226,59011],{"encoding":31667},[34,59228,59230],{"className":59229,"ariaHidden":758},[31672],[34,59231,59233,59236,59278,59281],{"className":59232},[31676],[34,59234],{"className":59235,"style":59021},[31680],[34,59237,59239,59244],{"className":59238},[32767],[34,59240,38616,59242],{"className":59241},[32767],[34,59243,31942],{"style":38619},[34,59245,59247],{"className":59246},[31792],[34,59248,59250,59270],{"className":59249},[31796,31797],[34,59251,59253,59267],{"className":59252},[31801],[34,59254,59256],{"className":59255,"style":38632},[31805],[34,59257,59258,59261],{"style":38635},[34,59259],{"className":59260,"style":31814},[31813],[34,59262,59264],{"className":59263},[31818,31819,31820,31821],[34,59265,768],{"className":59266},[31685,31821],[34,59268,31829],{"className":59269},[31828],[34,59271,59273],{"className":59272},[31801],[34,59274,59276],{"className":59275,"style":38654},[31805],[34,59277],{},[34,59279],{"className":59280,"style":32123},[31845],[34,59282,59284,59287],{"className":59283},[31685],[34,59285,31660],{"className":59286},[31685,31686],[34,59288,59290],{"className":59289},[31792],[34,59291,59293],{"className":59292},[31796],[34,59294,59296],{"className":59295},[31801],[34,59297,59299],{"className":59298,"style":37774},[31805],[34,59300,59301,59304],{"style":37777},[34,59302],{"className":59303,"style":31814},[31813],[34,59305,59307],{"className":59306},[31818,31819,31820,31821],[34,59308,786],{"className":59309},[31685,31821],[34,59311,59313,59337],{"className":59312},[31642],[34,59314,59316],{"className":59315},[31646],[31648,59317,59318],{"xmlns":31650},[31652,59319,59320,59334],{},[31655,59321,59322,59324,59326,59332],{},[31658,59323,58917],{"mathvariant":38079},[31662,59325,456],{"stretchy":702},[37624,59327,59328,59330],{},[31658,59329,31660],{},[31749,59331,786],{},[31662,59333,2557],{"stretchy":702},[31665,59335,59336],{"encoding":31667},"\\Omega(n^3)",[34,59338,59340],{"className":59339,"ariaHidden":758},[31672],[34,59341,59343,59346,59349,59352,59381],{"className":59342},[31676],[34,59344],{"className":59345,"style":37755},[31680],[34,59347,58917],{"className":59348},[31685],[34,59350,456],{"className":59351},[31782],[34,59353,59355,59358],{"className":59354},[31685],[34,59356,31660],{"className":59357},[31685,31686],[34,59359,59361],{"className":59360},[31792],[34,59362,59364],{"className":59363},[31796],[34,59365,59367],{"className":59366},[31801],[34,59368,59370],{"className":59369,"style":37774},[31805],[34,59371,59372,59375],{"style":37777},[34,59373],{"className":59374,"style":31814},[31813],[34,59376,59378],{"className":59377},[31818,31819,31820,31821],[34,59379,786],{"className":59380},[31685,31821],[34,59382,2557],{"className":59383},[31690],[253,59385,58168],{"id":59386},"答案与深度解析-5",[15,59388,59389],{},[2627,59390,59391],{},"答案：D",[105,59393,59394],{},[108,59395,59396,59398,59399,59427,59428],{},[2627,59397,58554],{}," 复杂度分析关注的是当问题规模 ",[34,59400,59402,59415],{"className":59401},[31642],[34,59403,59405],{"className":59404},[31646],[31648,59406,59407],{"xmlns":31650},[31652,59408,59409,59413],{},[31655,59410,59411],{},[31658,59412,31660],{},[31665,59414,31660],{"encoding":31667},[34,59416,59418],{"className":59417,"ariaHidden":758},[31672],[34,59419,59421,59424],{"className":59420},[31676],[34,59422],{"className":59423,"style":32527},[31680],[34,59425,31660],{"className":59426},[31685,31686]," 趋于无穷大时，函数的增长量级。我们在多项式相加时，只保留最高阶项。\n",[105,59429,59430,59847,60356],{},[108,59431,59432,59433,59550,59551,59610,59611,59669,59670,59728,59729,59757,59758,59787,59788,59846],{},"对于 ",[34,59434,59436,59465],{"className":59435},[31642],[34,59437,59439],{"className":59438},[31646],[31648,59440,59441],{"xmlns":31650},[31652,59442,59443,59463],{},[31655,59444,59445,59451,59453,59455,59457],{},[37624,59446,59447,59449],{},[31658,59448,31660],{},[31749,59450,768],{},[31658,59452,38080],{"mathvariant":38079},[31749,59454,768],{},[31662,59456,223],{},[37624,59458,59459,59461],{},[31749,59460,768],{},[31658,59462,31660],{},[31665,59464,58621],{"encoding":31667},[34,59466,59468,59515],{"className":59467,"ariaHidden":758},[31672],[34,59469,59471,59474,59503,59506,59509,59512],{"className":59470},[31676],[34,59472],{"className":59473,"style":37755},[31680],[34,59475,59477,59480],{"className":59476},[31685],[34,59478,31660],{"className":59479},[31685,31686],[34,59481,59483],{"className":59482},[31792],[34,59484,59486],{"className":59485},[31796],[34,59487,59489],{"className":59488},[31801],[34,59490,59492],{"className":59491,"style":37774},[31805],[34,59493,59494,59497],{"style":37777},[34,59495],{"className":59496,"style":31814},[31813],[34,59498,59500],{"className":59499},[31818,31819,31820,31821],[34,59501,768],{"className":59502},[31685,31821],[34,59504,38149],{"className":59505},[31685],[34,59507],{"className":59508,"style":32604},[31845],[34,59510,223],{"className":59511},[32024],[34,59513],{"className":59514,"style":32604},[31845],[34,59516,59518,59521],{"className":59517},[31676],[34,59519],{"className":59520,"style":37959},[31680],[34,59522,59524,59527],{"className":59523},[31685],[34,59525,768],{"className":59526},[31685],[34,59528,59530],{"className":59529},[31792],[34,59531,59533],{"className":59532},[31796],[34,59534,59536],{"className":59535},[31801],[34,59537,59539],{"className":59538,"style":37959},[31805],[34,59540,59541,59544],{"style":37777},[34,59542],{"className":59543,"style":31814},[31813],[34,59545,59547],{"className":59546},[31818,31819,31820,31821],[34,59548,31660],{"className":59549},[31685,31686,31821],"，指数级 ",[34,59552,59554,59572],{"className":59553},[31642],[34,59555,59557],{"className":59556},[31646],[31648,59558,59559],{"xmlns":31650},[31652,59560,59561,59569],{},[31655,59562,59563],{},[37624,59564,59565,59567],{},[31749,59566,768],{},[31658,59568,31660],{},[31665,59570,59571],{"encoding":31667},"2^n",[34,59573,59575],{"className":59574,"ariaHidden":758},[31672],[34,59576,59578,59581],{"className":59577},[31676],[34,59579],{"className":59580,"style":37959},[31680],[34,59582,59584,59587],{"className":59583},[31685],[34,59585,768],{"className":59586},[31685],[34,59588,59590],{"className":59589},[31792],[34,59591,59593],{"className":59592},[31796],[34,59594,59596],{"className":59595},[31801],[34,59597,59599],{"className":59598,"style":37959},[31805],[34,59600,59601,59604],{"style":37777},[34,59602],{"className":59603,"style":31814},[31813],[34,59605,59607],{"className":59606},[31818,31819,31820,31821],[34,59608,31660],{"className":59609},[31685,31686,31821]," 的增长速度远远快于多项式 ",[34,59612,59614,59631],{"className":59613},[31642],[34,59615,59617],{"className":59616},[31646],[31648,59618,59619],{"xmlns":31650},[31652,59620,59621,59629],{},[31655,59622,59623],{},[37624,59624,59625,59627],{},[31658,59626,31660],{},[31749,59628,768],{},[31665,59630,42015],{"encoding":31667},[34,59632,59634],{"className":59633,"ariaHidden":758},[31672],[34,59635,59637,59640],{"className":59636},[31676],[34,59638],{"className":59639,"style":37774},[31680],[34,59641,59643,59646],{"className":59642},[31685],[34,59644,31660],{"className":59645},[31685,31686],[34,59647,59649],{"className":59648},[31792],[34,59650,59652],{"className":59651},[31796],[34,59653,59655],{"className":59654},[31801],[34,59656,59658],{"className":59657,"style":37774},[31805],[34,59659,59660,59663],{"style":37777},[34,59661],{"className":59662,"style":31814},[31813],[34,59664,59666],{"className":59665},[31818,31819,31820,31821],[34,59667,768],{"className":59668},[31685,31821],"。因此最高阶项是 ",[34,59671,59673,59690],{"className":59672},[31642],[34,59674,59676],{"className":59675},[31646],[31648,59677,59678],{"xmlns":31650},[31652,59679,59680,59688],{},[31655,59681,59682],{},[37624,59683,59684,59686],{},[31749,59685,768],{},[31658,59687,31660],{},[31665,59689,59571],{"encoding":31667},[34,59691,59693],{"className":59692,"ariaHidden":758},[31672],[34,59694,59696,59699],{"className":59695},[31676],[34,59697],{"className":59698,"style":37959},[31680],[34,59700,59702,59705],{"className":59701},[31685],[34,59703,768],{"className":59704},[31685],[34,59706,59708],{"className":59707},[31792],[34,59709,59711],{"className":59710},[31796],[34,59712,59714],{"className":59713},[31801],[34,59715,59717],{"className":59716,"style":37959},[31805],[34,59718,59719,59722],{"style":37777},[34,59720],{"className":59721,"style":31814},[31813],[34,59723,59725],{"className":59724},[31818,31819,31820,31821],[34,59726,31660],{"className":59727},[31685,31686,31821],"。它的上界 ",[34,59730,59732,59745],{"className":59731},[31642],[34,59733,59735],{"className":59734},[31646],[31648,59736,59737],{"xmlns":31650},[31652,59738,59739,59743],{},[31655,59740,59741],{},[31658,59742,37890],{},[31665,59744,37890],{"encoding":31667},[34,59746,59748],{"className":59747,"ariaHidden":758},[31672],[34,59749,59751,59754],{"className":59750},[31676],[34,59752],{"className":59753,"style":33218},[31680],[34,59755,37890],{"className":59756,"style":34452},[31685,31686]," 和下界 ",[34,59759,59761,59775],{"className":59760},[31642],[34,59762,59764],{"className":59763},[31646],[31648,59765,59766],{"xmlns":31650},[31652,59767,59768,59772],{},[31655,59769,59770],{},[31658,59771,58917],{"mathvariant":38079},[31665,59773,59774],{"encoding":31667},"\\Omega",[34,59776,59778],{"className":59777,"ariaHidden":758},[31672],[34,59779,59781,59784],{"className":59780},[31676],[34,59782],{"className":59783,"style":33218},[31680],[34,59785,58917],{"className":59786},[31685]," 都可以是 ",[34,59789,59791,59808],{"className":59790},[31642],[34,59792,59794],{"className":59793},[31646],[31648,59795,59796],{"xmlns":31650},[31652,59797,59798,59806],{},[31655,59799,59800],{},[37624,59801,59802,59804],{},[31749,59803,768],{},[31658,59805,31660],{},[31665,59807,59571],{"encoding":31667},[34,59809,59811],{"className":59810,"ariaHidden":758},[31672],[34,59812,59814,59817],{"className":59813},[31676],[34,59815],{"className":59816,"style":37959},[31680],[34,59818,59820,59823],{"className":59819},[31685],[34,59821,768],{"className":59822},[31685],[34,59824,59826],{"className":59825},[31792],[34,59827,59829],{"className":59828},[31796],[34,59830,59832],{"className":59831},[31801],[34,59833,59835],{"className":59834,"style":37959},[31805],[34,59836,59837,59840],{"style":37777},[34,59838],{"className":59839,"style":31814},[31813],[34,59841,59843],{"className":59842},[31818,31819,31820,31821],[34,59844,31660],{"className":59845},[31685,31686,31821],"。A和B表述正确。",[108,59848,59432,59849,59962,59963,60056,60057,60157,60158,60256,60257,347],{},[34,59850,59852,59879],{"className":59851},[31642],[34,59853,59855],{"className":59854},[31646],[31648,59856,59857],{"xmlns":31650},[31652,59858,59859,59877],{},[31655,59860,59861,59871],{},[31743,59862,59863,59869],{},[31655,59864,59865,59867],{},[31658,59866,885],{},[31662,59868,32682],{},[31749,59870,768],{},[37624,59872,59873,59875],{},[31658,59874,31660],{},[31749,59876,786],{},[31665,59878,59011],{"encoding":31667},[34,59880,59882],{"className":59881,"ariaHidden":758},[31672],[34,59883,59885,59888,59930,59933],{"className":59884},[31676],[34,59886],{"className":59887,"style":59021},[31680],[34,59889,59891,59896],{"className":59890},[32767],[34,59892,38616,59894],{"className":59893},[32767],[34,59895,31942],{"style":38619},[34,59897,59899],{"className":59898},[31792],[34,59900,59902,59922],{"className":59901},[31796,31797],[34,59903,59905,59919],{"className":59904},[31801],[34,59906,59908],{"className":59907,"style":38632},[31805],[34,59909,59910,59913],{"style":38635},[34,59911],{"className":59912,"style":31814},[31813],[34,59914,59916],{"className":59915},[31818,31819,31820,31821],[34,59917,768],{"className":59918},[31685,31821],[34,59920,31829],{"className":59921},[31828],[34,59923,59925],{"className":59924},[31801],[34,59926,59928],{"className":59927,"style":38654},[31805],[34,59929],{},[34,59931],{"className":59932,"style":32123},[31845],[34,59934,59936,59939],{"className":59935},[31685],[34,59937,31660],{"className":59938},[31685,31686],[34,59940,59942],{"className":59941},[31792],[34,59943,59945],{"className":59944},[31796],[34,59946,59948],{"className":59947},[31801],[34,59949,59951],{"className":59950,"style":37774},[31805],[34,59952,59953,59956],{"style":37777},[34,59954],{"className":59955,"style":31814},[31813],[34,59957,59959],{"className":59958},[31818,31819,31820,31821],[34,59960,786],{"className":59961},[31685,31821],"，根据对数运算法则，它等于 ",[34,59964,59966,59992],{"className":59965},[31642],[34,59967,59969],{"className":59968},[31646],[31648,59970,59971],{"xmlns":31650},[31652,59972,59973,59989],{},[31655,59974,59975,59977,59987],{},[31749,59976,786],{},[31743,59978,59979,59985],{},[31655,59980,59981,59983],{},[31658,59982,885],{},[31662,59984,32682],{},[31749,59986,768],{},[31658,59988,31660],{},[31665,59990,59991],{"encoding":31667},"3 \\log_2 n",[34,59993,59995],{"className":59994,"ariaHidden":758},[31672],[34,59996,59998,60002,60005,60008,60050,60053],{"className":59997},[31676],[34,59999],{"className":60000,"style":60001},[31680],"height:0.9386em;vertical-align:-0.2441em;",[34,60003,786],{"className":60004},[31685],[34,60006],{"className":60007,"style":32123},[31845],[34,60009,60011,60016],{"className":60010},[32767],[34,60012,38616,60014],{"className":60013},[32767],[34,60015,31942],{"style":38619},[34,60017,60019],{"className":60018},[31792],[34,60020,60022,60042],{"className":60021},[31796,31797],[34,60023,60025,60039],{"className":60024},[31801],[34,60026,60028],{"className":60027,"style":38632},[31805],[34,60029,60030,60033],{"style":38635},[34,60031],{"className":60032,"style":31814},[31813],[34,60034,60036],{"className":60035},[31818,31819,31820,31821],[34,60037,768],{"className":60038},[31685,31821],[34,60040,31829],{"className":60041},[31828],[34,60043,60045],{"className":60044},[31801],[34,60046,60048],{"className":60047,"style":38654},[31805],[34,60049],{},[34,60051],{"className":60052,"style":32123},[31845],[34,60054,31660],{"className":60055},[31685,31686],"。常数系数 3 在量级分析中被忽略，因此它的确切量级是 ",[34,60058,60060,60091],{"className":60059},[31642],[34,60061,60063],{"className":60062},[31646],[31648,60064,60065],{"xmlns":31650},[31652,60066,60067,60088],{},[31655,60068,60069,60072,60074,60084,60086],{},[31658,60070,60071],{"mathvariant":38079},"Θ",[31662,60073,456],{"stretchy":702},[31743,60075,60076,60082],{},[31655,60077,60078,60080],{},[31658,60079,885],{},[31662,60081,32682],{},[31749,60083,768],{},[31658,60085,31660],{},[31662,60087,2557],{"stretchy":702},[31665,60089,60090],{"encoding":31667},"\\Theta(\\log_2 n)",[34,60092,60094],{"className":60093,"ariaHidden":758},[31672],[34,60095,60097,60100,60103,60106,60148,60151,60154],{"className":60096},[31676],[34,60098],{"className":60099,"style":31774},[31680],[34,60101,60071],{"className":60102},[31685],[34,60104,456],{"className":60105},[31782],[34,60107,60109,60114],{"className":60108},[32767],[34,60110,38616,60112],{"className":60111},[32767],[34,60113,31942],{"style":38619},[34,60115,60117],{"className":60116},[31792],[34,60118,60120,60140],{"className":60119},[31796,31797],[34,60121,60123,60137],{"className":60122},[31801],[34,60124,60126],{"className":60125,"style":38632},[31805],[34,60127,60128,60131],{"style":38635},[34,60129],{"className":60130,"style":31814},[31813],[34,60132,60134],{"className":60133},[31818,31819,31820,31821],[34,60135,768],{"className":60136},[31685,31821],[34,60138,31829],{"className":60139},[31828],[34,60141,60143],{"className":60142},[31801],[34,60144,60146],{"className":60145,"style":38654},[31805],[34,60147],{},[34,60149],{"className":60150,"style":32123},[31845],[34,60152,31660],{"className":60153},[31685,31686],[34,60155,2557],{"className":60156},[31690],"。这意味着它的上界是 ",[34,60159,60161,60190],{"className":60160},[31642],[34,60162,60164],{"className":60163},[31646],[31648,60165,60166],{"xmlns":31650},[31652,60167,60168,60188],{},[31655,60169,60170,60172,60174,60184,60186],{},[31658,60171,37890],{},[31662,60173,456],{"stretchy":702},[31743,60175,60176,60182],{},[31655,60177,60178,60180],{},[31658,60179,885],{},[31662,60181,32682],{},[31749,60183,768],{},[31658,60185,31660],{},[31662,60187,2557],{"stretchy":702},[31665,60189,47514],{"encoding":31667},[34,60191,60193],{"className":60192,"ariaHidden":758},[31672],[34,60194,60196,60199,60202,60205,60247,60250,60253],{"className":60195},[31676],[34,60197],{"className":60198,"style":31774},[31680],[34,60200,37890],{"className":60201,"style":34452},[31685,31686],[34,60203,456],{"className":60204},[31782],[34,60206,60208,60213],{"className":60207},[32767],[34,60209,38616,60211],{"className":60210},[32767],[34,60212,31942],{"style":38619},[34,60214,60216],{"className":60215},[31792],[34,60217,60219,60239],{"className":60218},[31796,31797],[34,60220,60222,60236],{"className":60221},[31801],[34,60223,60225],{"className":60224,"style":38632},[31805],[34,60226,60227,60230],{"style":38635},[34,60228],{"className":60229,"style":31814},[31813],[34,60231,60233],{"className":60232},[31818,31819,31820,31821],[34,60234,768],{"className":60235},[31685,31821],[34,60237,31829],{"className":60238},[31828],[34,60240,60242],{"className":60241},[31801],[34,60243,60245],{"className":60244,"style":38654},[31805],[34,60246],{},[34,60248],{"className":60249,"style":32123},[31845],[34,60251,31660],{"className":60252},[31685,31686],[34,60254,2557],{"className":60255},[31690],"，下界是 ",[34,60258,60260,60290],{"className":60259},[31642],[34,60261,60263],{"className":60262},[31646],[31648,60264,60265],{"xmlns":31650},[31652,60266,60267,60287],{},[31655,60268,60269,60271,60273,60283,60285],{},[31658,60270,58917],{"mathvariant":38079},[31662,60272,456],{"stretchy":702},[31743,60274,60275,60281],{},[31655,60276,60277,60279],{},[31658,60278,885],{},[31662,60280,32682],{},[31749,60282,768],{},[31658,60284,31660],{},[31662,60286,2557],{"stretchy":702},[31665,60288,60289],{"encoding":31667},"\\Omega(\\log_2 n)",[34,60291,60293],{"className":60292,"ariaHidden":758},[31672],[34,60294,60296,60299,60302,60305,60347,60350,60353],{"className":60295},[31676],[34,60297],{"className":60298,"style":31774},[31680],[34,60300,58917],{"className":60301},[31685],[34,60303,456],{"className":60304},[31782],[34,60306,60308,60313],{"className":60307},[32767],[34,60309,38616,60311],{"className":60310},[32767],[34,60312,31942],{"style":38619},[34,60314,60316],{"className":60315},[31792],[34,60317,60319,60339],{"className":60318},[31796,31797],[34,60320,60322,60336],{"className":60321},[31801],[34,60323,60325],{"className":60324,"style":38632},[31805],[34,60326,60327,60330],{"style":38635},[34,60328],{"className":60329,"style":31814},[31813],[34,60331,60333],{"className":60332},[31818,31819,31820,31821],[34,60334,768],{"className":60335},[31685,31821],[34,60337,31829],{"className":60338},[31828],[34,60340,60342],{"className":60341},[31801],[34,60343,60345],{"className":60344,"style":38654},[31805],[34,60346],{},[34,60348],{"className":60349,"style":32123},[31845],[34,60351,31660],{"className":60352},[31685,31686],[34,60354,2557],{"className":60355},[31690],[108,60357,60358,60359,60432,60433,60492],{},"选项 D 说下界是 ",[34,60360,60362,60385],{"className":60361},[31642],[34,60363,60365],{"className":60364},[31646],[31648,60366,60367],{"xmlns":31650},[31652,60368,60369,60383],{},[31655,60370,60371,60373,60375,60381],{},[31658,60372,58917],{"mathvariant":38079},[31662,60374,456],{"stretchy":702},[37624,60376,60377,60379],{},[31658,60378,31660],{},[31749,60380,786],{},[31662,60382,2557],{"stretchy":702},[31665,60384,59336],{"encoding":31667},[34,60386,60388],{"className":60387,"ariaHidden":758},[31672],[34,60389,60391,60394,60397,60400,60429],{"className":60390},[31676],[34,60392],{"className":60393,"style":37755},[31680],[34,60395,58917],{"className":60396},[31685],[34,60398,456],{"className":60399},[31782],[34,60401,60403,60406],{"className":60402},[31685],[34,60404,31660],{"className":60405},[31685,31686],[34,60407,60409],{"className":60408},[31792],[34,60410,60412],{"className":60411},[31796],[34,60413,60415],{"className":60414},[31801],[34,60416,60418],{"className":60417,"style":37774},[31805],[34,60419,60420,60423],{"style":37777},[34,60421],{"className":60422,"style":31814},[31813],[34,60424,60426],{"className":60425},[31818,31819,31820,31821],[34,60427,786],{"className":60428},[31685,31821],[34,60430,2557],{"className":60431},[31690],"，即认为该函数的增长速度大于等于 ",[34,60434,60436,60454],{"className":60435},[31642],[34,60437,60439],{"className":60438},[31646],[31648,60440,60441],{"xmlns":31650},[31652,60442,60443,60451],{},[31655,60444,60445],{},[37624,60446,60447,60449],{},[31658,60448,31660],{},[31749,60450,786],{},[31665,60452,60453],{"encoding":31667},"n^3",[34,60455,60457],{"className":60456,"ariaHidden":758},[31672],[34,60458,60460,60463],{"className":60459},[31676],[34,60461],{"className":60462,"style":37774},[31680],[34,60464,60466,60469],{"className":60465},[31685],[34,60467,31660],{"className":60468},[31685,31686],[34,60470,60472],{"className":60471},[31792],[34,60473,60475],{"className":60474},[31796],[34,60476,60478],{"className":60477},[31801],[34,60479,60481],{"className":60480,"style":37774},[31805],[34,60482,60483,60486],{"style":37777},[34,60484],{"className":60485,"style":31814},[31813],[34,60487,60489],{"className":60488},[31818,31819,31820,31821],[34,60490,786],{"className":60491},[31685,31821],"，这是极其荒谬的，对数函数的增长极其缓慢，远小于多项式函数。因此 D 错误。",[32188,60494],{},[253,60496,58367],{"id":60497},"_3-题目内容-1",[15,60499,60500,60501,60710,60711,60754,58784,60756,60829,58980,60831,60968,59196,60970],{},"某递归算法的时间复杂度公式为：",[34,60502,60504,60561],{"className":60503},[31642],[34,60505,60507],{"className":60506},[31646],[31648,60508,60509],{"xmlns":31650},[31652,60510,60511,60558],{},[31655,60512,60513,60515,60517,60519,60521,60523,60526,60528,60530,60532,60534,60536,60538,60540,60546,60556],{},[31658,60514,37400],{},[31662,60516,456],{"stretchy":702},[31658,60518,31660],{},[31662,60520,2557],{"stretchy":702},[31662,60522,202],{},[31749,60524,60525],{},"8",[31658,60527,37400],{},[31662,60529,456],{"stretchy":702},[31658,60531,31660],{},[31658,60533,38080],{"mathvariant":38079},[31749,60535,786],{},[31662,60537,2557],{"stretchy":702},[31662,60539,223],{},[37624,60541,60542,60544],{},[31658,60543,31660],{},[31749,60545,768],{},[31743,60547,60548,60554],{},[31655,60549,60550,60552],{},[31658,60551,885],{},[31662,60553,32682],{},[31749,60555,768],{},[31658,60557,31660],{},[31665,60559,60560],{"encoding":31667},"T(n)=8T(n\u002F3)+n^2\\log_2 n",[34,60562,60564,60591,60624],{"className":60563,"ariaHidden":758},[31672],[34,60565,60567,60570,60573,60576,60579,60582,60585,60588],{"className":60566},[31676],[34,60568],{"className":60569,"style":31774},[31680],[34,60571,37400],{"className":60572,"style":37422},[31685,31686],[34,60574,456],{"className":60575},[31782],[34,60577,31660],{"className":60578},[31685,31686],[34,60580,2557],{"className":60581},[31690],[34,60583],{"className":60584,"style":31846},[31845],[34,60586,202],{"className":60587},[31850],[34,60589],{"className":60590,"style":31846},[31845],[34,60592,60594,60597,60600,60603,60606,60609,60612,60615,60618,60621],{"className":60593},[31676],[34,60595],{"className":60596,"style":31774},[31680],[34,60598,60525],{"className":60599},[31685],[34,60601,37400],{"className":60602,"style":37422},[31685,31686],[34,60604,456],{"className":60605},[31782],[34,60607,31660],{"className":60608},[31685,31686],[34,60610,42614],{"className":60611},[31685],[34,60613,2557],{"className":60614},[31690],[34,60616],{"className":60617,"style":32604},[31845],[34,60619,223],{"className":60620},[32024],[34,60622],{"className":60623,"style":32604},[31845],[34,60625,60627,60630,60659,60662,60704,60707],{"className":60626},[31676],[34,60628],{"className":60629,"style":59021},[31680],[34,60631,60633,60636],{"className":60632},[31685],[34,60634,31660],{"className":60635},[31685,31686],[34,60637,60639],{"className":60638},[31792],[34,60640,60642],{"className":60641},[31796],[34,60643,60645],{"className":60644},[31801],[34,60646,60648],{"className":60647,"style":37774},[31805],[34,60649,60650,60653],{"style":37777},[34,60651],{"className":60652,"style":31814},[31813],[34,60654,60656],{"className":60655},[31818,31819,31820,31821],[34,60657,768],{"className":60658},[31685,31821],[34,60660],{"className":60661,"style":32123},[31845],[34,60663,60665,60670],{"className":60664},[32767],[34,60666,38616,60668],{"className":60667},[32767],[34,60669,31942],{"style":38619},[34,60671,60673],{"className":60672},[31792],[34,60674,60676,60696],{"className":60675},[31796,31797],[34,60677,60679,60693],{"className":60678},[31801],[34,60680,60682],{"className":60681,"style":38632},[31805],[34,60683,60684,60687],{"style":38635},[34,60685],{"className":60686,"style":31814},[31813],[34,60688,60690],{"className":60689},[31818,31819,31820,31821],[34,60691,768],{"className":60692},[31685,31821],[34,60694,31829],{"className":60695},[31828],[34,60697,60699],{"className":60698},[31801],[34,60700,60702],{"className":60701,"style":38654},[31805],[34,60703],{},[34,60705],{"className":60706,"style":32123},[31845],[34,60708,31660],{"className":60709},[31685,31686],"，则该算法的时间复杂度为（ ）。\nA. ",[34,60712,60714,60733],{"className":60713},[31642],[34,60715,60717],{"className":60716},[31646],[31648,60718,60719],{"xmlns":31650},[31652,60720,60721,60731],{},[31655,60722,60723,60725,60727,60729],{},[31658,60724,37890],{},[31662,60726,456],{"stretchy":702},[31658,60728,31660],{},[31662,60730,2557],{"stretchy":702},[31665,60732,46493],{"encoding":31667},[34,60734,60736],{"className":60735,"ariaHidden":758},[31672],[34,60737,60739,60742,60745,60748,60751],{"className":60738},[31676],[34,60740],{"className":60741,"style":31774},[31680],[34,60743,37890],{"className":60744,"style":34452},[31685,31686],[34,60746,456],{"className":60747},[31782],[34,60749,31660],{"className":60750},[31685,31686],[34,60752,2557],{"className":60753},[31690],[58531,60755],{},[34,60757,60759,60782],{"className":60758},[31642],[34,60760,60762],{"className":60761},[31646],[31648,60763,60764],{"xmlns":31650},[31652,60765,60766,60780],{},[31655,60767,60768,60770,60772,60778],{},[31658,60769,37890],{},[31662,60771,456],{"stretchy":702},[37624,60773,60774,60776],{},[31658,60775,31660],{},[31749,60777,768],{},[31662,60779,2557],{"stretchy":702},[31665,60781,46174],{"encoding":31667},[34,60783,60785],{"className":60784,"ariaHidden":758},[31672],[34,60786,60788,60791,60794,60797,60826],{"className":60787},[31676],[34,60789],{"className":60790,"style":37755},[31680],[34,60792,37890],{"className":60793,"style":34452},[31685,31686],[34,60795,456],{"className":60796},[31782],[34,60798,60800,60803],{"className":60799},[31685],[34,60801,31660],{"className":60802},[31685,31686],[34,60804,60806],{"className":60805},[31792],[34,60807,60809],{"className":60808},[31796],[34,60810,60812],{"className":60811},[31801],[34,60813,60815],{"className":60814,"style":37774},[31805],[34,60816,60817,60820],{"style":37777},[34,60818],{"className":60819,"style":31814},[31813],[34,60821,60823],{"className":60822},[31818,31819,31820,31821],[34,60824,768],{"className":60825},[31685,31821],[34,60827,2557],{"className":60828},[31690],[58531,60830],{},[34,60832,60834,60870],{"className":60833},[31642],[34,60835,60837],{"className":60836},[31646],[31648,60838,60839],{"xmlns":31650},[31652,60840,60841,60867],{},[31655,60842,60843,60845,60847,60853,60863,60865],{},[31658,60844,37890],{},[31662,60846,456],{"stretchy":702},[37624,60848,60849,60851],{},[31658,60850,31660],{},[31749,60852,768],{},[31743,60854,60855,60861],{},[31655,60856,60857,60859],{},[31658,60858,885],{},[31662,60860,32682],{},[31749,60862,768],{},[31658,60864,31660],{},[31662,60866,2557],{"stretchy":702},[31665,60868,60869],{"encoding":31667},"O(n^2 \\log_2 n)",[34,60871,60873],{"className":60872,"ariaHidden":758},[31672],[34,60874,60876,60879,60882,60885,60914,60917,60959,60962,60965],{"className":60875},[31676],[34,60877],{"className":60878,"style":37755},[31680],[34,60880,37890],{"className":60881,"style":34452},[31685,31686],[34,60883,456],{"className":60884},[31782],[34,60886,60888,60891],{"className":60887},[31685],[34,60889,31660],{"className":60890},[31685,31686],[34,60892,60894],{"className":60893},[31792],[34,60895,60897],{"className":60896},[31796],[34,60898,60900],{"className":60899},[31801],[34,60901,60903],{"className":60902,"style":37774},[31805],[34,60904,60905,60908],{"style":37777},[34,60906],{"className":60907,"style":31814},[31813],[34,60909,60911],{"className":60910},[31818,31819,31820,31821],[34,60912,768],{"className":60913},[31685,31821],[34,60915],{"className":60916,"style":32123},[31845],[34,60918,60920,60925],{"className":60919},[32767],[34,60921,38616,60923],{"className":60922},[32767],[34,60924,31942],{"style":38619},[34,60926,60928],{"className":60927},[31792],[34,60929,60931,60951],{"className":60930},[31796,31797],[34,60932,60934,60948],{"className":60933},[31801],[34,60935,60937],{"className":60936,"style":38632},[31805],[34,60938,60939,60942],{"style":38635},[34,60940],{"className":60941,"style":31814},[31813],[34,60943,60945],{"className":60944},[31818,31819,31820,31821],[34,60946,768],{"className":60947},[31685,31821],[34,60949,31829],{"className":60950},[31828],[34,60952,60954],{"className":60953},[31801],[34,60955,60957],{"className":60956,"style":38654},[31805],[34,60958],{},[34,60960],{"className":60961,"style":32123},[31845],[34,60963,31660],{"className":60964},[31685,31686],[34,60966,2557],{"className":60967},[31690],[58531,60969],{},[34,60971,60973,61003],{"className":60972},[31642],[34,60974,60976],{"className":60975},[31646],[31648,60977,60978],{"xmlns":31650},[31652,60979,60980,61000],{},[31655,60981,60982,60984,60986,60998],{},[31658,60983,37890],{},[31662,60985,456],{"stretchy":702},[37624,60987,60988,60990],{},[31658,60989,31660],{},[31655,60991,60992,60994,60996],{},[31749,60993,60525],{},[31658,60995,38080],{"mathvariant":38079},[31749,60997,786],{},[31662,60999,2557],{"stretchy":702},[31665,61001,61002],{"encoding":31667},"O(n^{8\u002F3})",[34,61004,61006],{"className":61005,"ariaHidden":758},[31672],[34,61007,61009,61013,61016,61019,61053],{"className":61008},[31676],[34,61010],{"className":61011,"style":61012},[31680],"height:1.138em;vertical-align:-0.25em;",[34,61014,37890],{"className":61015,"style":34452},[31685,31686],[34,61017,456],{"className":61018},[31782],[34,61020,61022,61025],{"className":61021},[31685],[34,61023,31660],{"className":61024},[31685,31686],[34,61026,61028],{"className":61027},[31792],[34,61029,61031],{"className":61030},[31796],[34,61032,61034],{"className":61033},[31801],[34,61035,61038],{"className":61036,"style":61037},[31805],"height:0.888em;",[34,61039,61040,61043],{"style":37777},[34,61041],{"className":61042,"style":31814},[31813],[34,61044,61046],{"className":61045},[31818,31819,31820,31821],[34,61047,61049],{"className":61048},[31685,31821],[34,61050,61052],{"className":61051},[31685,31821],"8\u002F3",[34,61054,2557],{"className":61055},[31690],[253,61057,58168],{"id":61058},"答案与深度解析-6",[15,61060,61061],{},[2627,61062,61063],{},"答案：C",[105,61065,61066],{},[108,61067,61068,61070,61071,61074,61075,61207,61208],{},[2627,61069,58554],{}," 此题考察求解递归式的终极武器——",[2627,61072,61073],{},"主定理（Master Theorem）","。\n主定理用于求解形式为 ",[34,61076,61078,61123],{"className":61077},[31642],[34,61079,61081],{"className":61080},[31646],[31648,61082,61083],{"xmlns":31650},[31652,61084,61085,61121],{},[31655,61086,61087,61089,61091,61093,61095,61097,61099,61101,61103,61105,61107,61109,61111,61113,61115,61117,61119],{},[31658,61088,37400],{},[31662,61090,456],{"stretchy":702},[31658,61092,31660],{},[31662,61094,2557],{"stretchy":702},[31662,61096,202],{},[31658,61098,111],{},[31658,61100,37400],{},[31662,61102,456],{"stretchy":702},[31658,61104,31660],{},[31658,61106,38080],{"mathvariant":38079},[31658,61108,1697],{},[31662,61110,2557],{"stretchy":702},[31662,61112,223],{},[31658,61114,31739],{},[31662,61116,456],{"stretchy":702},[31658,61118,31660],{},[31662,61120,2557],{"stretchy":702},[31665,61122,43073],{"encoding":31667},[34,61124,61126,61153,61189],{"className":61125,"ariaHidden":758},[31672],[34,61127,61129,61132,61135,61138,61141,61144,61147,61150],{"className":61128},[31676],[34,61130],{"className":61131,"style":31774},[31680],[34,61133,37400],{"className":61134,"style":37422},[31685,31686],[34,61136,456],{"className":61137},[31782],[34,61139,31660],{"className":61140},[31685,31686],[34,61142,2557],{"className":61143},[31690],[34,61145],{"className":61146,"style":31846},[31845],[34,61148,202],{"className":61149},[31850],[34,61151],{"className":61152,"style":31846},[31845],[34,61154,61156,61159,61162,61165,61168,61171,61174,61177,61180,61183,61186],{"className":61155},[31676],[34,61157],{"className":61158,"style":31774},[31680],[34,61160,111],{"className":61161},[31685,31686],[34,61163,37400],{"className":61164,"style":37422},[31685,31686],[34,61166,456],{"className":61167},[31782],[34,61169,31660],{"className":61170},[31685,31686],[34,61172,38080],{"className":61173},[31685],[34,61175,1697],{"className":61176},[31685,31686],[34,61178,2557],{"className":61179},[31690],[34,61181],{"className":61182,"style":32604},[31845],[34,61184,223],{"className":61185},[32024],[34,61187],{"className":61188,"style":32604},[31845],[34,61190,61192,61195,61198,61201,61204],{"className":61191},[31676],[34,61193],{"className":61194,"style":31774},[31680],[34,61196,31739],{"className":61197,"style":31778},[31685,31686],[34,61199,456],{"className":61200},[31782],[34,61202,31660],{"className":61203},[31685,31686],[34,61205,2557],{"className":61206},[31690]," 的递归方程。\n",[2230,61209,61210,61477,61984,62479,62530],{},[108,61211,61212,61213,61264,61265,61316,61317,61476],{},"提取参数：题干中 ",[34,61214,61216,61234],{"className":61215},[31642],[34,61217,61219],{"className":61218},[31646],[31648,61220,61221],{"xmlns":31650},[31652,61222,61223,61231],{},[31655,61224,61225,61227,61229],{},[31658,61226,111],{},[31662,61228,202],{},[31749,61230,60525],{},[31665,61232,61233],{"encoding":31667},"a = 8",[34,61235,61237,61255],{"className":61236,"ariaHidden":758},[31672],[34,61238,61240,61243,61246,61249,61252],{"className":61239},[31676],[34,61241],{"className":61242,"style":32527},[31680],[34,61244,111],{"className":61245},[31685,31686],[34,61247],{"className":61248,"style":31846},[31845],[34,61250,202],{"className":61251},[31850],[34,61253],{"className":61254,"style":31846},[31845],[34,61256,61258,61261],{"className":61257},[31676],[34,61259],{"className":61260,"style":32617},[31680],[34,61262,60525],{"className":61263},[31685],"（子问题个数），",[34,61266,61268,61286],{"className":61267},[31642],[34,61269,61271],{"className":61270},[31646],[31648,61272,61273],{"xmlns":31650},[31652,61274,61275,61283],{},[31655,61276,61277,61279,61281],{},[31658,61278,1697],{},[31662,61280,202],{},[31749,61282,786],{},[31665,61284,61285],{"encoding":31667},"b = 3",[34,61287,61289,61307],{"className":61288,"ariaHidden":758},[31672],[34,61290,61292,61295,61298,61301,61304],{"className":61291},[31676],[34,61293],{"className":61294,"style":31681},[31680],[34,61296,1697],{"className":61297},[31685,31686],[34,61299],{"className":61300,"style":31846},[31845],[34,61302,202],{"className":61303},[31850],[34,61305],{"className":61306,"style":31846},[31845],[34,61308,61310,61313],{"className":61309},[31676],[34,61311],{"className":61312,"style":32617},[31680],[34,61314,786],{"className":61315},[31685],"（问题规模缩小的比例），",[34,61318,61320,61360],{"className":61319},[31642],[34,61321,61323],{"className":61322},[31646],[31648,61324,61325],{"xmlns":31650},[31652,61326,61327,61357],{},[31655,61328,61329,61331,61333,61335,61337,61339,61345,61355],{},[31658,61330,31739],{},[31662,61332,456],{"stretchy":702},[31658,61334,31660],{},[31662,61336,2557],{"stretchy":702},[31662,61338,202],{},[37624,61340,61341,61343],{},[31658,61342,31660],{},[31749,61344,768],{},[31743,61346,61347,61353],{},[31655,61348,61349,61351],{},[31658,61350,885],{},[31662,61352,32682],{},[31749,61354,768],{},[31658,61356,31660],{},[31665,61358,61359],{"encoding":31667},"f(n) = n^2 \\log_2 n",[34,61361,61363,61390],{"className":61362,"ariaHidden":758},[31672],[34,61364,61366,61369,61372,61375,61378,61381,61384,61387],{"className":61365},[31676],[34,61367],{"className":61368,"style":31774},[31680],[34,61370,31739],{"className":61371,"style":31778},[31685,31686],[34,61373,456],{"className":61374},[31782],[34,61376,31660],{"className":61377},[31685,31686],[34,61379,2557],{"className":61380},[31690],[34,61382],{"className":61383,"style":31846},[31845],[34,61385,202],{"className":61386},[31850],[34,61388],{"className":61389,"style":31846},[31845],[34,61391,61393,61396,61425,61428,61470,61473],{"className":61392},[31676],[34,61394],{"className":61395,"style":59021},[31680],[34,61397,61399,61402],{"className":61398},[31685],[34,61400,31660],{"className":61401},[31685,31686],[34,61403,61405],{"className":61404},[31792],[34,61406,61408],{"className":61407},[31796],[34,61409,61411],{"className":61410},[31801],[34,61412,61414],{"className":61413,"style":37774},[31805],[34,61415,61416,61419],{"style":37777},[34,61417],{"className":61418,"style":31814},[31813],[34,61420,61422],{"className":61421},[31818,31819,31820,31821],[34,61423,768],{"className":61424},[31685,31821],[34,61426],{"className":61427,"style":32123},[31845],[34,61429,61431,61436],{"className":61430},[32767],[34,61432,38616,61434],{"className":61433},[32767],[34,61435,31942],{"style":38619},[34,61437,61439],{"className":61438},[31792],[34,61440,61442,61462],{"className":61441},[31796,31797],[34,61443,61445,61459],{"className":61444},[31801],[34,61446,61448],{"className":61447,"style":38632},[31805],[34,61449,61450,61453],{"style":38635},[34,61451],{"className":61452,"style":31814},[31813],[34,61454,61456],{"className":61455},[31818,31819,31820,31821],[34,61457,768],{"className":61458},[31685,31821],[34,61460,31829],{"className":61461},[31828],[34,61463,61465],{"className":61464},[31801],[34,61466,61468],{"className":61467,"style":38654},[31805],[34,61469],{},[34,61471],{"className":61472,"style":32123},[31845],[34,61474,31660],{"className":61475},[31685,31686],"（分解和合并的代价）。",[108,61478,61479,61480,61605,61606,44689,61732,13977,61836,61920,61921,347],{},"计算参照项：计算 ",[34,61481,61483,61512],{"className":61482},[31642],[34,61484,61486],{"className":61485},[31646],[31648,61487,61488],{"xmlns":31650},[31652,61489,61490,61510],{},[31655,61491,61492],{},[37624,61493,61494,61496],{},[31658,61495,31660],{},[31655,61497,61498,61508],{},[31743,61499,61500,61506],{},[31655,61501,61502,61504],{},[31658,61503,885],{},[31662,61505,32682],{},[31658,61507,1697],{},[31658,61509,111],{},[31665,61511,43331],{"encoding":31667},[34,61513,61515],{"className":61514,"ariaHidden":758},[31672],[34,61516,61518,61521],{"className":61517},[31676],[34,61519],{"className":61520,"style":38486},[31680],[34,61522,61524,61527],{"className":61523},[31685],[34,61525,31660],{"className":61526},[31685,31686],[34,61528,61530],{"className":61529},[31792],[34,61531,61533],{"className":61532},[31796],[34,61534,61536],{"className":61535},[31801],[34,61537,61539],{"className":61538,"style":38486},[31805],[34,61540,61541,61544],{"style":37777},[34,61542],{"className":61543,"style":31814},[31813],[34,61545,61547],{"className":61546},[31818,31819,31820,31821],[34,61548,61550,61599,61602],{"className":61549},[31685,31821],[34,61551,61553,61565],{"className":61552},[32767,31821],[34,61554,61556,61559,61562],{"className":61555},[32767,31821],[34,61557,43379],{"className":61558},[31821],[34,61560,43383],{"className":61561},[31821],[34,61563,31942],{"className":61564,"style":38619},[31821],[34,61566,61568],{"className":61567},[31792],[34,61569,61571,61591],{"className":61570},[31796,31797],[34,61572,61574,61588],{"className":61573},[31801],[34,61575,61577],{"className":61576,"style":43399},[31805],[34,61578,61579,61582],{"style":43402},[34,61580],{"className":61581,"style":43406},[31813],[34,61583,61585],{"className":61584},[31818,43410,43411,31821],[34,61586,1697],{"className":61587},[31685,31686,31821],[34,61589,31829],{"className":61590},[31828],[34,61592,61594],{"className":61593},[31801],[34,61595,61597],{"className":61596,"style":43424},[31805],[34,61598],{},[34,61600],{"className":61601,"style":43430},[31845,31821],[34,61603,111],{"className":61604},[31685,31686,31821],"，即 ",[34,61607,61609,61639],{"className":61608},[31642],[34,61610,61612],{"className":61611},[31646],[31648,61613,61614],{"xmlns":31650},[31652,61615,61616,61636],{},[31655,61617,61618],{},[37624,61619,61620,61622],{},[31658,61621,31660],{},[31655,61623,61624,61634],{},[31743,61625,61626,61632],{},[31655,61627,61628,61630],{},[31658,61629,885],{},[31662,61631,32682],{},[31749,61633,786],{},[31749,61635,60525],{},[31665,61637,61638],{"encoding":31667},"n^{\\log_3 8}",[34,61640,61642],{"className":61641,"ariaHidden":758},[31672],[34,61643,61645,61648],{"className":61644},[31676],[34,61646],{"className":61647,"style":38486},[31680],[34,61649,61651,61654],{"className":61650},[31685],[34,61652,31660],{"className":61653},[31685,31686],[34,61655,61657],{"className":61656},[31792],[34,61658,61660],{"className":61659},[31796],[34,61661,61663],{"className":61662},[31801],[34,61664,61666],{"className":61665,"style":38486},[31805],[34,61667,61668,61671],{"style":37777},[34,61669],{"className":61670,"style":31814},[31813],[34,61672,61674],{"className":61673},[31818,31819,31820,31821],[34,61675,61677,61726,61729],{"className":61676},[31685,31821],[34,61678,61680,61692],{"className":61679},[32767,31821],[34,61681,61683,61686,61689],{"className":61682},[32767,31821],[34,61684,43379],{"className":61685},[31821],[34,61687,43383],{"className":61688},[31821],[34,61690,31942],{"className":61691,"style":38619},[31821],[34,61693,61695],{"className":61694},[31792],[34,61696,61698,61718],{"className":61697},[31796,31797],[34,61699,61701,61715],{"className":61700},[31801],[34,61702,61704],{"className":61703,"style":44616},[31805],[34,61705,61706,61709],{"style":43402},[34,61707],{"className":61708,"style":43406},[31813],[34,61710,61712],{"className":61711},[31818,43410,43411,31821],[34,61713,786],{"className":61714},[31685,31821],[34,61716,31829],{"className":61717},[31828],[34,61719,61721],{"className":61720},[31801],[34,61722,61724],{"className":61723,"style":43424},[31805],[34,61725],{},[34,61727],{"className":61728,"style":43430},[31845,31821],[34,61730,60525],{"className":61731},[31685,31821],[34,61733,61735,61761],{"className":61734},[31642],[34,61736,61738],{"className":61737},[31646],[31648,61739,61740],{"xmlns":31650},[31652,61741,61742,61758],{},[31655,61743,61744,61750,61752,61754,61756],{},[37624,61745,61746,61748],{},[31749,61747,786],{},[31749,61749,768],{},[31662,61751,202],{},[31749,61753,36732],{},[31662,61755,29622],{},[31749,61757,60525],{},[31665,61759,61760],{"encoding":31667},"3^2 = 9 > 8",[34,61762,61764,61808,61827],{"className":61763,"ariaHidden":758},[31672],[34,61765,61767,61770,61799,61802,61805],{"className":61766},[31676],[34,61768],{"className":61769,"style":37774},[31680],[34,61771,61773,61776],{"className":61772},[31685],[34,61774,786],{"className":61775},[31685],[34,61777,61779],{"className":61778},[31792],[34,61780,61782],{"className":61781},[31796],[34,61783,61785],{"className":61784},[31801],[34,61786,61788],{"className":61787,"style":37774},[31805],[34,61789,61790,61793],{"style":37777},[34,61791],{"className":61792,"style":31814},[31813],[34,61794,61796],{"className":61795},[31818,31819,31820,31821],[34,61797,768],{"className":61798},[31685,31821],[34,61800],{"className":61801,"style":31846},[31845],[34,61803,202],{"className":61804},[31850],[34,61806],{"className":61807,"style":31846},[31845],[34,61809,61811,61815,61818,61821,61824],{"className":61810},[31676],[34,61812],{"className":61813,"style":61814},[31680],"height:0.6835em;vertical-align:-0.0391em;",[34,61816,36732],{"className":61817},[31685],[34,61819],{"className":61820,"style":31846},[31845],[34,61822,29622],{"className":61823},[31850],[34,61825],{"className":61826,"style":31846},[31845],[34,61828,61830,61833],{"className":61829},[31676],[34,61831],{"className":61832,"style":32617},[31680],[34,61834,60525],{"className":61835},[31685],[34,61837,61839,61863],{"className":61838},[31642],[34,61840,61842],{"className":61841},[31646],[31648,61843,61844],{"xmlns":31650},[31652,61845,61846,61860],{},[31655,61847,61848,61858],{},[31743,61849,61850,61856],{},[31655,61851,61852,61854],{},[31658,61853,885],{},[31662,61855,32682],{},[31749,61857,786],{},[31749,61859,60525],{},[31665,61861,61862],{"encoding":31667},"\\log_3 8",[34,61864,61866],{"className":61865,"ariaHidden":758},[31672],[34,61867,61869,61872,61914,61917],{"className":61868},[31676],[34,61870],{"className":61871,"style":60001},[31680],[34,61873,61875,61880],{"className":61874},[32767],[34,61876,38616,61878],{"className":61877},[32767],[34,61879,31942],{"style":38619},[34,61881,61883],{"className":61882},[31792],[34,61884,61886,61906],{"className":61885},[31796,31797],[34,61887,61889,61903],{"className":61888},[31801],[34,61890,61892],{"className":61891,"style":38632},[31805],[34,61893,61894,61897],{"style":38635},[34,61895],{"className":61896,"style":31814},[31813],[34,61898,61900],{"className":61899},[31818,31819,31820,31821],[34,61901,786],{"className":61902},[31685,31821],[34,61904,31829],{"className":61905},[31828],[34,61907,61909],{"className":61908},[31801],[34,61910,61912],{"className":61911,"style":38654},[31805],[34,61913],{},[34,61915],{"className":61916,"style":32123},[31845],[34,61918,60525],{"className":61919},[31685]," 的值略小于 2（约为 1.89）。即参照项约为 ",[34,61922,61924,61943],{"className":61923},[31642],[34,61925,61927],{"className":61926},[31646],[31648,61928,61929],{"xmlns":31650},[31652,61930,61931,61940],{},[31655,61932,61933],{},[37624,61934,61935,61937],{},[31658,61936,31660],{},[31749,61938,61939],{},"1.89",[31665,61941,61942],{"encoding":31667},"n^{1.89}",[34,61944,61946],{"className":61945,"ariaHidden":758},[31672],[34,61947,61949,61952],{"className":61948},[31676],[34,61950],{"className":61951,"style":37774},[31680],[34,61953,61955,61958],{"className":61954},[31685],[34,61956,31660],{"className":61957},[31685,31686],[34,61959,61961],{"className":61960},[31792],[34,61962,61964],{"className":61963},[31796],[34,61965,61967],{"className":61966},[31801],[34,61968,61970],{"className":61969,"style":37774},[31805],[34,61971,61972,61975],{"style":37777},[34,61973],{"className":61974,"style":31814},[31813],[34,61976,61978],{"className":61977},[31818,31819,31820,31821],[34,61979,61981],{"className":61980},[31685,31821],[34,61982,61939],{"className":61983},[31685,31821],[108,61985,61986,61987,62030,62031,43298,62189,62250,62251,62309,62310,62353,62354,347],{},"比较 ",[34,61988,61990,62009],{"className":61989},[31642],[34,61991,61993],{"className":61992},[31646],[31648,61994,61995],{"xmlns":31650},[31652,61996,61997,62007],{},[31655,61998,61999,62001,62003,62005],{},[31658,62000,31739],{},[31662,62002,456],{"stretchy":702},[31658,62004,31660],{},[31662,62006,2557],{"stretchy":702},[31665,62008,43276],{"encoding":31667},[34,62010,62012],{"className":62011,"ariaHidden":758},[31672],[34,62013,62015,62018,62021,62024,62027],{"className":62014},[31676],[34,62016],{"className":62017,"style":31774},[31680],[34,62019,31739],{"className":62020,"style":31778},[31685,31686],[34,62022,456],{"className":62023},[31782],[34,62025,31660],{"className":62026},[31685,31686],[34,62028,2557],{"className":62029},[31690]," 与参照项：我们将 ",[34,62032,62034,62073],{"className":62033},[31642],[34,62035,62037],{"className":62036},[31646],[31648,62038,62039],{"xmlns":31650},[31652,62040,62041,62071],{},[31655,62042,62043,62045,62047,62049,62051,62053,62059,62069],{},[31658,62044,31739],{},[31662,62046,456],{"stretchy":702},[31658,62048,31660],{},[31662,62050,2557],{"stretchy":702},[31662,62052,202],{},[37624,62054,62055,62057],{},[31658,62056,31660],{},[31749,62058,768],{},[31743,62060,62061,62067],{},[31655,62062,62063,62065],{},[31658,62064,885],{},[31662,62066,32682],{},[31749,62068,768],{},[31658,62070,31660],{},[31665,62072,61359],{"encoding":31667},[34,62074,62076,62103],{"className":62075,"ariaHidden":758},[31672],[34,62077,62079,62082,62085,62088,62091,62094,62097,62100],{"className":62078},[31676],[34,62080],{"className":62081,"style":31774},[31680],[34,62083,31739],{"className":62084,"style":31778},[31685,31686],[34,62086,456],{"className":62087},[31782],[34,62089,31660],{"className":62090},[31685,31686],[34,62092,2557],{"className":62093},[31690],[34,62095],{"className":62096,"style":31846},[31845],[34,62098,202],{"className":62099},[31850],[34,62101],{"className":62102,"style":31846},[31845],[34,62104,62106,62109,62138,62141,62183,62186],{"className":62105},[31676],[34,62107],{"className":62108,"style":59021},[31680],[34,62110,62112,62115],{"className":62111},[31685],[34,62113,31660],{"className":62114},[31685,31686],[34,62116,62118],{"className":62117},[31792],[34,62119,62121],{"className":62120},[31796],[34,62122,62124],{"className":62123},[31801],[34,62125,62127],{"className":62126,"style":37774},[31805],[34,62128,62129,62132],{"style":37777},[34,62130],{"className":62131,"style":31814},[31813],[34,62133,62135],{"className":62134},[31818,31819,31820,31821],[34,62136,768],{"className":62137},[31685,31821],[34,62139],{"className":62140,"style":32123},[31845],[34,62142,62144,62149],{"className":62143},[32767],[34,62145,38616,62147],{"className":62146},[32767],[34,62148,31942],{"style":38619},[34,62150,62152],{"className":62151},[31792],[34,62153,62155,62175],{"className":62154},[31796,31797],[34,62156,62158,62172],{"className":62157},[31801],[34,62159,62161],{"className":62160,"style":38632},[31805],[34,62162,62163,62166],{"style":38635},[34,62164],{"className":62165,"style":31814},[31813],[34,62167,62169],{"className":62168},[31818,31819,31820,31821],[34,62170,768],{"className":62171},[31685,31821],[34,62173,31829],{"className":62174},[31828],[34,62176,62178],{"className":62177},[31801],[34,62179,62181],{"className":62180,"style":38654},[31805],[34,62182],{},[34,62184],{"className":62185,"style":32123},[31845],[34,62187,31660],{"className":62188},[31685,31686],[34,62190,62192,62209],{"className":62191},[31642],[34,62193,62195],{"className":62194},[31646],[31648,62196,62197],{"xmlns":31650},[31652,62198,62199,62207],{},[31655,62200,62201],{},[37624,62202,62203,62205],{},[31658,62204,31660],{},[31749,62206,61939],{},[31665,62208,61942],{"encoding":31667},[34,62210,62212],{"className":62211,"ariaHidden":758},[31672],[34,62213,62215,62218],{"className":62214},[31676],[34,62216],{"className":62217,"style":37774},[31680],[34,62219,62221,62224],{"className":62220},[31685],[34,62222,31660],{"className":62223},[31685,31686],[34,62225,62227],{"className":62226},[31792],[34,62228,62230],{"className":62229},[31796],[34,62231,62233],{"className":62232},[31801],[34,62234,62236],{"className":62235,"style":37774},[31805],[34,62237,62238,62241],{"style":37777},[34,62239],{"className":62240,"style":31814},[31813],[34,62242,62244],{"className":62243},[31818,31819,31820,31821],[34,62245,62247],{"className":62246},[31685,31821],[34,62248,61939],{"className":62249},[31685,31821]," 比较。显然 ",[34,62252,62254,62271],{"className":62253},[31642],[34,62255,62257],{"className":62256},[31646],[31648,62258,62259],{"xmlns":31650},[31652,62260,62261,62269],{},[31655,62262,62263],{},[37624,62264,62265,62267],{},[31658,62266,31660],{},[31749,62268,768],{},[31665,62270,42015],{"encoding":31667},[34,62272,62274],{"className":62273,"ariaHidden":758},[31672],[34,62275,62277,62280],{"className":62276},[31676],[34,62278],{"className":62279,"style":37774},[31680],[34,62281,62283,62286],{"className":62282},[31685],[34,62284,31660],{"className":62285},[31685,31686],[34,62287,62289],{"className":62288},[31792],[34,62290,62292],{"className":62291},[31796],[34,62293,62295],{"className":62294},[31801],[34,62296,62298],{"className":62297,"style":37774},[31805],[34,62299,62300,62303],{"style":37777},[34,62301],{"className":62302,"style":31814},[31813],[34,62304,62306],{"className":62305},[31818,31819,31820,31821],[34,62307,768],{"className":62308},[31685,31821]," 的多项式阶数 2 大于 1.89。所以 ",[34,62311,62313,62332],{"className":62312},[31642],[34,62314,62316],{"className":62315},[31646],[31648,62317,62318],{"xmlns":31650},[31652,62319,62320,62330],{},[31655,62321,62322,62324,62326,62328],{},[31658,62323,31739],{},[31662,62325,456],{"stretchy":702},[31658,62327,31660],{},[31662,62329,2557],{"stretchy":702},[31665,62331,43276],{"encoding":31667},[34,62333,62335],{"className":62334,"ariaHidden":758},[31672],[34,62336,62338,62341,62344,62347,62350],{"className":62337},[31676],[34,62339],{"className":62340,"style":31774},[31680],[34,62342,31739],{"className":62343,"style":31778},[31685,31686],[34,62345,456],{"className":62346},[31782],[34,62348,31660],{"className":62349},[31685,31686],[34,62351,2557],{"className":62352},[31690]," 在多项式意义上大于 ",[34,62355,62357,62386],{"className":62356},[31642],[34,62358,62360],{"className":62359},[31646],[31648,62361,62362],{"xmlns":31650},[31652,62363,62364,62384],{},[31655,62365,62366],{},[37624,62367,62368,62370],{},[31658,62369,31660],{},[31655,62371,62372,62382],{},[31743,62373,62374,62380],{},[31655,62375,62376,62378],{},[31658,62377,885],{},[31662,62379,32682],{},[31658,62381,1697],{},[31658,62383,111],{},[31665,62385,43331],{"encoding":31667},[34,62387,62389],{"className":62388,"ariaHidden":758},[31672],[34,62390,62392,62395],{"className":62391},[31676],[34,62393],{"className":62394,"style":38486},[31680],[34,62396,62398,62401],{"className":62397},[31685],[34,62399,31660],{"className":62400},[31685,31686],[34,62402,62404],{"className":62403},[31792],[34,62405,62407],{"className":62406},[31796],[34,62408,62410],{"className":62409},[31801],[34,62411,62413],{"className":62412,"style":38486},[31805],[34,62414,62415,62418],{"style":37777},[34,62416],{"className":62417,"style":31814},[31813],[34,62419,62421],{"className":62420},[31818,31819,31820,31821],[34,62422,62424,62473,62476],{"className":62423},[31685,31821],[34,62425,62427,62439],{"className":62426},[32767,31821],[34,62428,62430,62433,62436],{"className":62429},[32767,31821],[34,62431,43379],{"className":62432},[31821],[34,62434,43383],{"className":62435},[31821],[34,62437,31942],{"className":62438,"style":38619},[31821],[34,62440,62442],{"className":62441},[31792],[34,62443,62445,62465],{"className":62444},[31796,31797],[34,62446,62448,62462],{"className":62447},[31801],[34,62449,62451],{"className":62450,"style":43399},[31805],[34,62452,62453,62456],{"style":43402},[34,62454],{"className":62455,"style":43406},[31813],[34,62457,62459],{"className":62458},[31818,43410,43411,31821],[34,62460,1697],{"className":62461},[31685,31686,31821],[34,62463,31829],{"className":62464},[31828],[34,62466,62468],{"className":62467},[31801],[34,62469,62471],{"className":62470,"style":43424},[31805],[34,62472],{},[34,62474],{"className":62475,"style":43430},[31845,31821],[34,62477,111],{"className":62478},[31685,31686,31821],[108,62480,62481,62482,62485,62486,62529],{},"匹配主定理情况：这属于主定理的",[2627,62483,62484],{},"第三种情况","（根节点的计算量占主导地位）。算法的时间复杂度由 ",[34,62487,62489,62508],{"className":62488},[31642],[34,62490,62492],{"className":62491},[31646],[31648,62493,62494],{"xmlns":31650},[31652,62495,62496,62506],{},[31655,62497,62498,62500,62502,62504],{},[31658,62499,31739],{},[31662,62501,456],{"stretchy":702},[31658,62503,31660],{},[31662,62505,2557],{"stretchy":702},[31665,62507,43276],{"encoding":31667},[34,62509,62511],{"className":62510,"ariaHidden":758},[31672],[34,62512,62514,62517,62520,62523,62526],{"className":62513},[31676],[34,62515],{"className":62516,"style":31774},[31680],[34,62518,31739],{"className":62519,"style":31778},[31685,31686],[34,62521,456],{"className":62522},[31782],[34,62524,31660],{"className":62525},[31685,31686],[34,62527,2557],{"className":62528},[31690]," 决定。",[108,62531,62532,62533,347],{},"结论：",[34,62534,62536,62598],{"className":62535},[31642],[34,62537,62539],{"className":62538},[31646],[31648,62540,62541],{"xmlns":31650},[31652,62542,62543,62595],{},[31655,62544,62545,62547,62549,62551,62553,62555,62557,62559,62561,62563,62565,62567,62569,62571,62573,62575,62581,62591,62593],{},[31658,62546,37400],{},[31662,62548,456],{"stretchy":702},[31658,62550,31660],{},[31662,62552,2557],{"stretchy":702},[31662,62554,202],{},[31658,62556,60071],{"mathvariant":38079},[31662,62558,456],{"stretchy":702},[31658,62560,31739],{},[31662,62562,456],{"stretchy":702},[31658,62564,31660],{},[31662,62566,2557],{"stretchy":702},[31662,62568,2557],{"stretchy":702},[31662,62570,202],{},[31658,62572,37890],{},[31662,62574,456],{"stretchy":702},[37624,62576,62577,62579],{},[31658,62578,31660],{},[31749,62580,768],{},[31743,62582,62583,62589],{},[31655,62584,62585,62587],{},[31658,62586,885],{},[31662,62588,32682],{},[31749,62590,768],{},[31658,62592,31660],{},[31662,62594,2557],{"stretchy":702},[31665,62596,62597],{"encoding":31667},"T(n) = \\Theta(f(n)) = O(n^2 \\log_2 n)",[34,62599,62601,62628,62661],{"className":62600,"ariaHidden":758},[31672],[34,62602,62604,62607,62610,62613,62616,62619,62622,62625],{"className":62603},[31676],[34,62605],{"className":62606,"style":31774},[31680],[34,62608,37400],{"className":62609,"style":37422},[31685,31686],[34,62611,456],{"className":62612},[31782],[34,62614,31660],{"className":62615},[31685,31686],[34,62617,2557],{"className":62618},[31690],[34,62620],{"className":62621,"style":31846},[31845],[34,62623,202],{"className":62624},[31850],[34,62626],{"className":62627,"style":31846},[31845],[34,62629,62631,62634,62637,62640,62643,62646,62649,62652,62655,62658],{"className":62630},[31676],[34,62632],{"className":62633,"style":31774},[31680],[34,62635,60071],{"className":62636},[31685],[34,62638,456],{"className":62639},[31782],[34,62641,31739],{"className":62642,"style":31778},[31685,31686],[34,62644,456],{"className":62645},[31782],[34,62647,31660],{"className":62648},[31685,31686],[34,62650,44253],{"className":62651},[31690],[34,62653],{"className":62654,"style":31846},[31845],[34,62656,202],{"className":62657},[31850],[34,62659],{"className":62660,"style":31846},[31845],[34,62662,62664,62667,62670,62673,62702,62705,62747,62750,62753],{"className":62663},[31676],[34,62665],{"className":62666,"style":37755},[31680],[34,62668,37890],{"className":62669,"style":34452},[31685,31686],[34,62671,456],{"className":62672},[31782],[34,62674,62676,62679],{"className":62675},[31685],[34,62677,31660],{"className":62678},[31685,31686],[34,62680,62682],{"className":62681},[31792],[34,62683,62685],{"className":62684},[31796],[34,62686,62688],{"className":62687},[31801],[34,62689,62691],{"className":62690,"style":37774},[31805],[34,62692,62693,62696],{"style":37777},[34,62694],{"className":62695,"style":31814},[31813],[34,62697,62699],{"className":62698},[31818,31819,31820,31821],[34,62700,768],{"className":62701},[31685,31821],[34,62703],{"className":62704,"style":32123},[31845],[34,62706,62708,62713],{"className":62707},[32767],[34,62709,38616,62711],{"className":62710},[32767],[34,62712,31942],{"style":38619},[34,62714,62716],{"className":62715},[31792],[34,62717,62719,62739],{"className":62718},[31796,31797],[34,62720,62722,62736],{"className":62721},[31801],[34,62723,62725],{"className":62724,"style":38632},[31805],[34,62726,62727,62730],{"style":38635},[34,62728],{"className":62729,"style":31814},[31813],[34,62731,62733],{"className":62732},[31818,31819,31820,31821],[34,62734,768],{"className":62735},[31685,31821],[34,62737,31829],{"className":62738},[31828],[34,62740,62742],{"className":62741},[31801],[34,62743,62745],{"className":62744,"style":38654},[31805],[34,62746],{},[34,62748],{"className":62749,"style":32123},[31845],[34,62751,31660],{"className":62752},[31685,31686],[34,62754,2557],{"className":62755},[31690],[32188,62757],{},[253,62759,58474],{"id":62760},"_4-题目内容-1",[15,62762,62763,62764,62874,62875,62941,58784,62943,63041,58980,63043,63101,59196,63103],{},"在棋盘覆盖问题中，关于 ",[34,62765,62767,62792],{"className":62766},[31642],[34,62768,62770],{"className":62769},[31646],[31648,62771,62772],{"xmlns":31650},[31652,62773,62774,62790],{},[31655,62775,62776,62782,62784],{},[37624,62777,62778,62780],{},[31749,62779,768],{},[31658,62781,33759],{},[31662,62783,42071],{},[37624,62785,62786,62788],{},[31749,62787,768],{},[31658,62789,33759],{},[31665,62791,51116],{"encoding":31667},[34,62793,62795,62839],{"className":62794,"ariaHidden":758},[31672],[34,62796,62798,62801,62830,62833,62836],{"className":62797},[31676],[34,62799],{"className":62800,"style":51126},[31680],[34,62802,62804,62807],{"className":62803},[31685],[34,62805,768],{"className":62806},[31685],[34,62808,62810],{"className":62809},[31792],[34,62811,62813],{"className":62812},[31796],[34,62814,62816],{"className":62815},[31801],[34,62817,62819],{"className":62818,"style":38486},[31805],[34,62820,62821,62824],{"style":37777},[34,62822],{"className":62823,"style":31814},[31813],[34,62825,62827],{"className":62826},[31818,31819,31820,31821],[34,62828,33759],{"className":62829,"style":33774},[31685,31686,31821],[34,62831],{"className":62832,"style":32604},[31845],[34,62834,42071],{"className":62835},[32024],[34,62837],{"className":62838,"style":32604},[31845],[34,62840,62842,62845],{"className":62841},[31676],[34,62843],{"className":62844,"style":38486},[31680],[34,62846,62848,62851],{"className":62847},[31685],[34,62849,768],{"className":62850},[31685],[34,62852,62854],{"className":62853},[31792],[34,62855,62857],{"className":62856},[31796],[34,62858,62860],{"className":62859},[31801],[34,62861,62863],{"className":62862,"style":38486},[31805],[34,62864,62865,62868],{"style":37777},[34,62866],{"className":62867,"style":31814},[31813],[34,62869,62871],{"className":62870},[31818,31819,31820,31821],[34,62872,33759],{"className":62873,"style":33774},[31685,31686,31821]," 的特殊棋盘（有一个特殊方块），所需的L型牌（可以覆盖3个方格）的个数是（ ）。\nA. ",[34,62876,62878,62900],{"className":62877},[31642],[34,62879,62881],{"className":62880},[31646],[31648,62882,62883],{"xmlns":31650},[31652,62884,62885,62897],{},[31655,62886,62887,62893,62895],{},[37624,62888,62889,62891],{},[31749,62890,15932],{},[31658,62892,33759],{},[31658,62894,38080],{"mathvariant":38079},[31749,62896,786],{},[31665,62898,62899],{"encoding":31667},"4^k \u002F 3",[34,62901,62903],{"className":62902,"ariaHidden":758},[31672],[34,62904,62906,62909,62938],{"className":62905},[31676],[34,62907],{"className":62908,"style":43651},[31680],[34,62910,62912,62915],{"className":62911},[31685],[34,62913,15932],{"className":62914},[31685],[34,62916,62918],{"className":62917},[31792],[34,62919,62921],{"className":62920},[31796],[34,62922,62924],{"className":62923},[31801],[34,62925,62927],{"className":62926,"style":38486},[31805],[34,62928,62929,62932],{"style":37777},[34,62930],{"className":62931,"style":31814},[31813],[34,62933,62935],{"className":62934},[31818,31819,31820,31821],[34,62936,33759],{"className":62937,"style":33774},[31685,31686,31821],[34,62939,42614],{"className":62940},[31685],[58531,62942],{},[34,62944,62946,62976],{"className":62945},[31642],[34,62947,62949],{"className":62948},[31646],[31648,62950,62951],{"xmlns":31650},[31652,62952,62953,62973],{},[31655,62954,62955,62957,62963,62965,62967,62969,62971],{},[31662,62956,456],{"stretchy":702},[37624,62958,62959,62961],{},[31749,62960,15932],{},[31658,62962,33759],{},[31662,62964,32226],{},[31749,62966,640],{},[31662,62968,2557],{"stretchy":702},[31658,62970,38080],{"mathvariant":38079},[31749,62972,786],{},[31665,62974,62975],{"encoding":31667},"(4^k - 1) \u002F 3",[34,62977,62979,63026],{"className":62978,"ariaHidden":758},[31672],[34,62980,62982,62985,62988,63017,63020,63023],{"className":62981},[31676],[34,62983],{"className":62984,"style":43651},[31680],[34,62986,456],{"className":62987},[31782],[34,62989,62991,62994],{"className":62990},[31685],[34,62992,15932],{"className":62993},[31685],[34,62995,62997],{"className":62996},[31792],[34,62998,63000],{"className":62999},[31796],[34,63001,63003],{"className":63002},[31801],[34,63004,63006],{"className":63005,"style":38486},[31805],[34,63007,63008,63011],{"style":37777},[34,63009],{"className":63010,"style":31814},[31813],[34,63012,63014],{"className":63013},[31818,31819,31820,31821],[34,63015,33759],{"className":63016,"style":33774},[31685,31686,31821],[34,63018],{"className":63019,"style":32604},[31845],[34,63021,32226],{"className":63022},[32024],[34,63024],{"className":63025,"style":32604},[31845],[34,63027,63029,63032,63035,63038],{"className":63028},[31676],[34,63030],{"className":63031,"style":31774},[31680],[34,63033,640],{"className":63034},[31685],[34,63036,2557],{"className":63037},[31690],[34,63039,42614],{"className":63040},[31685],[58531,63042],{},[34,63044,63046,63063],{"className":63045},[31642],[34,63047,63049],{"className":63048},[31646],[31648,63050,63051],{"xmlns":31650},[31652,63052,63053,63061],{},[31655,63054,63055],{},[37624,63056,63057,63059],{},[31749,63058,15932],{},[31658,63060,33759],{},[31665,63062,51230],{"encoding":31667},[34,63064,63066],{"className":63065,"ariaHidden":758},[31672],[34,63067,63069,63072],{"className":63068},[31676],[34,63070],{"className":63071,"style":38486},[31680],[34,63073,63075,63078],{"className":63074},[31685],[34,63076,15932],{"className":63077},[31685],[34,63079,63081],{"className":63080},[31792],[34,63082,63084],{"className":63083},[31796],[34,63085,63087],{"className":63086},[31801],[34,63088,63090],{"className":63089,"style":38486},[31805],[34,63091,63092,63095],{"style":37777},[34,63093],{"className":63094,"style":31814},[31813],[34,63096,63098],{"className":63097},[31818,31819,31820,31821],[34,63099,33759],{"className":63100,"style":33774},[31685,31686,31821],[58531,63102],{},[34,63104,63106,63128],{"className":63105},[31642],[34,63107,63109],{"className":63108},[31646],[31648,63110,63111],{"xmlns":31650},[31652,63112,63113,63125],{},[31655,63114,63115,63121,63123],{},[37624,63116,63117,63119],{},[31749,63118,768],{},[31658,63120,33759],{},[31658,63122,38080],{"mathvariant":38079},[31749,63124,786],{},[31665,63126,63127],{"encoding":31667},"2^k \u002F 3",[34,63129,63131],{"className":63130,"ariaHidden":758},[31672],[34,63132,63134,63137,63166],{"className":63133},[31676],[34,63135],{"className":63136,"style":43651},[31680],[34,63138,63140,63143],{"className":63139},[31685],[34,63141,768],{"className":63142},[31685],[34,63144,63146],{"className":63145},[31792],[34,63147,63149],{"className":63148},[31796],[34,63150,63152],{"className":63151},[31801],[34,63153,63155],{"className":63154,"style":38486},[31805],[34,63156,63157,63160],{"style":37777},[34,63158],{"className":63159,"style":31814},[31813],[34,63161,63163],{"className":63162},[31818,31819,31820,31821],[34,63164,33759],{"className":63165,"style":33774},[31685,31686,31821],[34,63167,42614],{"className":63168},[31685],[253,63170,58168],{"id":63171},"答案与深度解析-7",[15,63173,63174],{},[2627,63175,58547],{},[105,63177,63178],{},[108,63179,63180,63182,63183],{},[2627,63181,58554],{}," 这是一个纯粹的数学逻辑题，不需要死记硬背算法。\n",[2230,63184,63185,63411,63414,63498,63501],{},[108,63186,63187,63188,63247,63248,347],{},"棋盘的边长是 ",[34,63189,63191,63209],{"className":63190},[31642],[34,63192,63194],{"className":63193},[31646],[31648,63195,63196],{"xmlns":31650},[31652,63197,63198,63206],{},[31655,63199,63200],{},[37624,63201,63202,63204],{},[31749,63203,768],{},[31658,63205,33759],{},[31665,63207,63208],{"encoding":31667},"2^k",[34,63210,63212],{"className":63211,"ariaHidden":758},[31672],[34,63213,63215,63218],{"className":63214},[31676],[34,63216],{"className":63217,"style":38486},[31680],[34,63219,63221,63224],{"className":63220},[31685],[34,63222,768],{"className":63223},[31685],[34,63225,63227],{"className":63226},[31792],[34,63228,63230],{"className":63229},[31796],[34,63231,63233],{"className":63232},[31801],[34,63234,63236],{"className":63235,"style":38486},[31805],[34,63237,63238,63241],{"style":37777},[34,63239],{"className":63240,"style":31814},[31813],[34,63242,63244],{"className":63243},[31818,31819,31820,31821],[34,63245,33759],{"className":63246,"style":33774},[31685,31686,31821],"，因此总方格数是边长乘边长：",[34,63249,63251,63285],{"className":63250},[31642],[34,63252,63254],{"className":63253},[31646],[31648,63255,63256],{"xmlns":31650},[31652,63257,63258,63282],{},[31655,63259,63260,63266,63268,63274,63276],{},[37624,63261,63262,63264],{},[31749,63263,768],{},[31658,63265,33759],{},[31662,63267,42071],{},[37624,63269,63270,63272],{},[31749,63271,768],{},[31658,63273,33759],{},[31662,63275,202],{},[37624,63277,63278,63280],{},[31749,63279,15932],{},[31658,63281,33759],{},[31665,63283,63284],{"encoding":31667},"2^k \\times 2^k = 4^k",[34,63286,63288,63332,63376],{"className":63287,"ariaHidden":758},[31672],[34,63289,63291,63294,63323,63326,63329],{"className":63290},[31676],[34,63292],{"className":63293,"style":51126},[31680],[34,63295,63297,63300],{"className":63296},[31685],[34,63298,768],{"className":63299},[31685],[34,63301,63303],{"className":63302},[31792],[34,63304,63306],{"className":63305},[31796],[34,63307,63309],{"className":63308},[31801],[34,63310,63312],{"className":63311,"style":38486},[31805],[34,63313,63314,63317],{"style":37777},[34,63315],{"className":63316,"style":31814},[31813],[34,63318,63320],{"className":63319},[31818,31819,31820,31821],[34,63321,33759],{"className":63322,"style":33774},[31685,31686,31821],[34,63324],{"className":63325,"style":32604},[31845],[34,63327,42071],{"className":63328},[32024],[34,63330],{"className":63331,"style":32604},[31845],[34,63333,63335,63338,63367,63370,63373],{"className":63334},[31676],[34,63336],{"className":63337,"style":38486},[31680],[34,63339,63341,63344],{"className":63340},[31685],[34,63342,768],{"className":63343},[31685],[34,63345,63347],{"className":63346},[31792],[34,63348,63350],{"className":63349},[31796],[34,63351,63353],{"className":63352},[31801],[34,63354,63356],{"className":63355,"style":38486},[31805],[34,63357,63358,63361],{"style":37777},[34,63359],{"className":63360,"style":31814},[31813],[34,63362,63364],{"className":63363},[31818,31819,31820,31821],[34,63365,33759],{"className":63366,"style":33774},[31685,31686,31821],[34,63368],{"className":63369,"style":31846},[31845],[34,63371,202],{"className":63372},[31850],[34,63374],{"className":63375,"style":31846},[31845],[34,63377,63379,63382],{"className":63378},[31676],[34,63380],{"className":63381,"style":38486},[31680],[34,63383,63385,63388],{"className":63384},[31685],[34,63386,15932],{"className":63387},[31685],[34,63389,63391],{"className":63390},[31792],[34,63392,63394],{"className":63393},[31796],[34,63395,63397],{"className":63396},[31801],[34,63398,63400],{"className":63399,"style":38486},[31805],[34,63401,63402,63405],{"style":37777},[34,63403],{"className":63404,"style":31814},[31813],[34,63406,63408],{"className":63407},[31818,31819,31820,31821],[34,63409,33759],{"className":63410,"style":33774},[31685,31686,31821],[108,63412,63413],{},"题目说棋盘是“特殊棋盘”，意味着其中有 1 个方格是特殊方格（不需要被覆盖）。",[108,63415,63416,63417,347],{},"需要被L型牌覆盖的剩余方格总数为：",[34,63418,63420,63442],{"className":63419},[31642],[34,63421,63423],{"className":63422},[31646],[31648,63424,63425],{"xmlns":31650},[31652,63426,63427,63439],{},[31655,63428,63429,63435,63437],{},[37624,63430,63431,63433],{},[31749,63432,15932],{},[31658,63434,33759],{},[31662,63436,32226],{},[31749,63438,640],{},[31665,63440,63441],{"encoding":31667},"4^k - 1",[34,63443,63445,63489],{"className":63444,"ariaHidden":758},[31672],[34,63446,63448,63451,63480,63483,63486],{"className":63447},[31676],[34,63449],{"className":63450,"style":51126},[31680],[34,63452,63454,63457],{"className":63453},[31685],[34,63455,15932],{"className":63456},[31685],[34,63458,63460],{"className":63459},[31792],[34,63461,63463],{"className":63462},[31796],[34,63464,63466],{"className":63465},[31801],[34,63467,63469],{"className":63468,"style":38486},[31805],[34,63470,63471,63474],{"style":37777},[34,63472],{"className":63473,"style":31814},[31813],[34,63475,63477],{"className":63476},[31818,31819,31820,31821],[34,63478,33759],{"className":63479,"style":33774},[31685,31686,31821],[34,63481],{"className":63482,"style":32604},[31845],[34,63484,32226],{"className":63485},[32024],[34,63487],{"className":63488,"style":32604},[31845],[34,63490,63492,63495],{"className":63491},[31676],[34,63493],{"className":63494,"style":32617},[31680],[34,63496,640],{"className":63497},[31685],[108,63499,63500],{},"每一块L型骨牌刚好覆盖 3 个方格。",[108,63502,63503,63504,347],{},"因此所需骨牌数量为：",[34,63505,63507,63536],{"className":63506},[31642],[34,63508,63510],{"className":63509},[31646],[31648,63511,63512],{"xmlns":31650},[31652,63513,63514,63534],{},[31655,63515,63516,63518,63524,63526,63528,63530,63532],{},[31662,63517,456],{"stretchy":702},[37624,63519,63520,63522],{},[31749,63521,15932],{},[31658,63523,33759],{},[31662,63525,32226],{},[31749,63527,640],{},[31662,63529,2557],{"stretchy":702},[31658,63531,38080],{"mathvariant":38079},[31749,63533,786],{},[31665,63535,62975],{"encoding":31667},[34,63537,63539,63586],{"className":63538,"ariaHidden":758},[31672],[34,63540,63542,63545,63548,63577,63580,63583],{"className":63541},[31676],[34,63543],{"className":63544,"style":43651},[31680],[34,63546,456],{"className":63547},[31782],[34,63549,63551,63554],{"className":63550},[31685],[34,63552,15932],{"className":63553},[31685],[34,63555,63557],{"className":63556},[31792],[34,63558,63560],{"className":63559},[31796],[34,63561,63563],{"className":63562},[31801],[34,63564,63566],{"className":63565,"style":38486},[31805],[34,63567,63568,63571],{"style":37777},[34,63569],{"className":63570,"style":31814},[31813],[34,63572,63574],{"className":63573},[31818,31819,31820,31821],[34,63575,33759],{"className":63576,"style":33774},[31685,31686,31821],[34,63578],{"className":63579,"style":32604},[31845],[34,63581,32226],{"className":63582},[32024],[34,63584],{"className":63585,"style":32604},[31845],[34,63587,63589,63592,63595,63598],{"className":63588},[31676],[34,63590],{"className":63591,"style":31774},[31680],[34,63593,640],{"className":63594},[31685],[34,63596,2557],{"className":63597},[31690],[34,63599,42614],{"className":63600},[31685],[32188,63602],{},[253,63604,63606],{"id":63605},"_5-题目内容","5. 题目内容",[15,63608,63609,63610,63638,63639,63667,63668,63696,63697,63699,63700,63702,63703,63705],{},"在求解 ",[34,63611,63613,63626],{"className":63612},[31642],[34,63614,63616],{"className":63615},[31646],[31648,63617,63618],{"xmlns":31650},[31652,63619,63620,63624],{},[31655,63621,63622],{},[31658,63623,31660],{},[31665,63625,31660],{"encoding":31667},[34,63627,63629],{"className":63628,"ariaHidden":758},[31672],[34,63630,63632,63635],{"className":63631},[31676],[34,63633],{"className":63634,"style":32527},[31680],[34,63636,31660],{"className":63637},[31685,31686]," 个元素中第 ",[34,63640,63642,63655],{"className":63641},[31642],[34,63643,63645],{"className":63644},[31646],[31648,63646,63647],{"xmlns":31650},[31652,63648,63649,63653],{},[31655,63650,63651],{},[31658,63652,33759],{},[31665,63654,33759],{"encoding":31667},[34,63656,63658],{"className":63657,"ariaHidden":758},[31672],[34,63659,63661,63664],{"className":63660},[31676],[34,63662],{"className":63663,"style":31681},[31680],[34,63665,33759],{"className":63666,"style":33774},[31685,31686]," 小元素问题中，假设利用快速排序算法思想对 ",[34,63669,63671,63684],{"className":63670},[31642],[34,63672,63674],{"className":63673},[31646],[31648,63675,63676],{"xmlns":31650},[31652,63677,63678,63682],{},[31655,63679,63680],{},[31658,63681,31660],{},[31665,63683,31660],{"encoding":31667},[34,63685,63687],{"className":63686,"ariaHidden":758},[31672],[34,63688,63690,63693],{"className":63689},[31676],[34,63691],{"className":63692,"style":32527},[31680],[34,63694,31660],{"className":63695},[31685,31686]," 个元素进行划分，应如何选择划分基准（ ）。\nA. 随机选择一个元素作为划分基准",[58531,63698],{},"\nB. 取子序列的第一个元素作为划分基准",[58531,63701],{},"\nC. 用中位数的中位数方式寻觅划分基准",[58531,63704],{},"\nD. 以上皆可行，但不同方式，算法复杂度上界可能不同",[253,63707,58168],{"id":63708},"答案与深度解析-8",[15,63710,63711],{},[2627,63712,59391],{},[105,63714,63715],{},[108,63716,63717,63719,63720,63748,63749],{},[2627,63718,58554],{}," 求解第 ",[34,63721,63723,63736],{"className":63722},[31642],[34,63724,63726],{"className":63725},[31646],[31648,63727,63728],{"xmlns":31650},[31652,63729,63730,63734],{},[31655,63731,63732],{},[31658,63733,33759],{},[31665,63735,33759],{"encoding":31667},[34,63737,63739],{"className":63738,"ariaHidden":758},[31672],[34,63740,63742,63745],{"className":63741},[31676],[34,63743],{"className":63744,"style":31681},[31680],[34,63746,33759],{"className":63747,"style":33774},[31685,31686]," 小元素的经典算法是“快速选择算法”（Quickselect），它的核心也是划分（Partition）。\n",[105,63750,63751,63874,63928],{},[108,63752,63753,63756,63757,63800,63801,347],{},[2627,63754,63755],{},"取第一个元素（B）或随机取元素（A）："," 很容易实现，平均时间复杂度很好，是 ",[34,63758,63760,63779],{"className":63759},[31642],[34,63761,63763],{"className":63762},[31646],[31648,63764,63765],{"xmlns":31650},[31652,63766,63767,63777],{},[31655,63768,63769,63771,63773,63775],{},[31658,63770,37890],{},[31662,63772,456],{"stretchy":702},[31658,63774,31660],{},[31662,63776,2557],{"stretchy":702},[31665,63778,46493],{"encoding":31667},[34,63780,63782],{"className":63781,"ariaHidden":758},[31672],[34,63783,63785,63788,63791,63794,63797],{"className":63784},[31676],[34,63786],{"className":63787,"style":31774},[31680],[34,63789,37890],{"className":63790,"style":34452},[31685,31686],[34,63792,456],{"className":63793},[31782],[34,63795,31660],{"className":63796},[31685,31686],[34,63798,2557],{"className":63799},[31690],"。但如果是极端恶劣的数据（例如已经排好序的数组，且每次取第一个），最坏情况下的时间复杂度会退化到 ",[34,63802,63804,63827],{"className":63803},[31642],[34,63805,63807],{"className":63806},[31646],[31648,63808,63809],{"xmlns":31650},[31652,63810,63811,63825],{},[31655,63812,63813,63815,63817,63823],{},[31658,63814,37890],{},[31662,63816,456],{"stretchy":702},[37624,63818,63819,63821],{},[31658,63820,31660],{},[31749,63822,768],{},[31662,63824,2557],{"stretchy":702},[31665,63826,46174],{"encoding":31667},[34,63828,63830],{"className":63829,"ariaHidden":758},[31672],[34,63831,63833,63836,63839,63842,63871],{"className":63832},[31676],[34,63834],{"className":63835,"style":37755},[31680],[34,63837,37890],{"className":63838,"style":34452},[31685,31686],[34,63840,456],{"className":63841},[31782],[34,63843,63845,63848],{"className":63844},[31685],[34,63846,31660],{"className":63847},[31685,31686],[34,63849,63851],{"className":63850},[31792],[34,63852,63854],{"className":63853},[31796],[34,63855,63857],{"className":63856},[31801],[34,63858,63860],{"className":63859,"style":37774},[31805],[34,63861,63862,63865],{"style":37777},[34,63863],{"className":63864,"style":31814},[31813],[34,63866,63868],{"className":63867},[31818,31819,31820,31821],[34,63869,768],{"className":63870},[31685,31821],[34,63872,2557],{"className":63873},[31690],[108,63875,63876,63879,63880,63883,63884,63927],{},[2627,63877,63878],{},"中位数的中位数（C）："," 这是著名的 BFPRT 算法（线性时间选择算法）。它通过复杂的逻辑强制找到一个相对居中的元素作为基准。虽然常数项较大，但它能保证在",[2627,63881,63882],{},"最坏情况","下依然具有 ",[34,63885,63887,63906],{"className":63886},[31642],[34,63888,63890],{"className":63889},[31646],[31648,63891,63892],{"xmlns":31650},[31652,63893,63894,63904],{},[31655,63895,63896,63898,63900,63902],{},[31658,63897,37890],{},[31662,63899,456],{"stretchy":702},[31658,63901,31660],{},[31662,63903,2557],{"stretchy":702},[31665,63905,46493],{"encoding":31667},[34,63907,63909],{"className":63908,"ariaHidden":758},[31672],[34,63910,63912,63915,63918,63921,63924],{"className":63911},[31676],[34,63913],{"className":63914,"style":31774},[31680],[34,63916,37890],{"className":63917,"style":34452},[31685,31686],[34,63919,456],{"className":63920},[31782],[34,63922,31660],{"className":63923},[31685,31686],[34,63925,2557],{"className":63926},[31690]," 的时间复杂度上界。",[108,63929,63930],{},"因此，这三种方法在工程上和理论上都是可行的，只是它们的最坏时间复杂度上界存在差异。选项 D 描述最为客观全面。",[32188,63932],{},[253,63934,63936],{"id":63935},"_6-题目内容","6. 题目内容",[15,63938,63939,63940,64014,58784,64016,64074,58980,64076,64109,59196,64111],{},"用回溯法求解子集树问题，最坏情形下其解空间的叶结点数量为（ ）。\nA. ",[34,63941,63943,63967],{"className":63942},[31642],[34,63944,63946],{"className":63945},[31646],[31648,63947,63948],{"xmlns":31650},[31652,63949,63950,63964],{},[31655,63951,63952],{},[37624,63953,63954,63956],{},[31749,63955,768],{},[31655,63957,63958,63960,63962],{},[31658,63959,31660],{},[31662,63961,32226],{},[31749,63963,640],{},[31665,63965,63966],{"encoding":31667},"2^{n-1}",[34,63968,63970],{"className":63969,"ariaHidden":758},[31672],[34,63971,63973,63976],{"className":63972},[31676],[34,63974],{"className":63975,"style":37774},[31680],[34,63977,63979,63982],{"className":63978},[31685],[34,63980,768],{"className":63981},[31685],[34,63983,63985],{"className":63984},[31792],[34,63986,63988],{"className":63987},[31796],[34,63989,63991],{"className":63990},[31801],[34,63992,63994],{"className":63993,"style":37774},[31805],[34,63995,63996,63999],{"style":37777},[34,63997],{"className":63998,"style":31814},[31813],[34,64000,64002],{"className":64001},[31818,31819,31820,31821],[34,64003,64005,64008,64011],{"className":64004},[31685,31821],[34,64006,31660],{"className":64007},[31685,31686,31821],[34,64009,32226],{"className":64010},[32024,31821],[34,64012,640],{"className":64013},[31685,31821],[58531,64015],{},[34,64017,64019,64036],{"className":64018},[31642],[34,64020,64022],{"className":64021},[31646],[31648,64023,64024],{"xmlns":31650},[31652,64025,64026,64034],{},[31655,64027,64028],{},[37624,64029,64030,64032],{},[31749,64031,768],{},[31658,64033,31660],{},[31665,64035,59571],{"encoding":31667},[34,64037,64039],{"className":64038,"ariaHidden":758},[31672],[34,64040,64042,64045],{"className":64041},[31676],[34,64043],{"className":64044,"style":37959},[31680],[34,64046,64048,64051],{"className":64047},[31685],[34,64049,768],{"className":64050},[31685],[34,64052,64054],{"className":64053},[31792],[34,64055,64057],{"className":64056},[31796],[34,64058,64060],{"className":64059},[31801],[34,64061,64063],{"className":64062,"style":37959},[31805],[34,64064,64065,64068],{"style":37777},[34,64066],{"className":64067,"style":31814},[31813],[34,64069,64071],{"className":64070},[31818,31819,31820,31821],[34,64072,31660],{"className":64073},[31685,31686,31821],[58531,64075],{},[34,64077,64079,64094],{"className":64078},[31642],[34,64080,64082],{"className":64081},[31646],[31648,64083,64084],{"xmlns":31650},[31652,64085,64086,64092],{},[31655,64087,64088,64090],{},[31658,64089,31660],{},[31662,64091,5504],{"stretchy":702},[31665,64093,31668],{"encoding":31667},[34,64095,64097],{"className":64096,"ariaHidden":758},[31672],[34,64098,64100,64103,64106],{"className":64099},[31676],[34,64101],{"className":64102,"style":31681},[31680],[34,64104,31660],{"className":64105},[31685,31686],[34,64107,5504],{"className":64108},[31690],[58531,64110],{},[34,64112,64114,64142],{"className":64113},[31642],[34,64115,64117],{"className":64116},[31646],[31648,64118,64119],{"xmlns":31650},[31652,64120,64121,64139],{},[31655,64122,64123,64135,64137],{},[37624,64124,64125,64127],{},[31749,64126,768],{},[31655,64128,64129,64131,64133],{},[31658,64130,31660],{},[31662,64132,223],{},[31749,64134,640],{},[31662,64136,32226],{},[31749,64138,640],{},[31665,64140,64141],{"encoding":31667},"2^{n+1}-1",[34,64143,64145,64198],{"className":64144,"ariaHidden":758},[31672],[34,64146,64148,64151,64189,64192,64195],{"className":64147},[31676],[34,64149],{"className":64150,"style":40044},[31680],[34,64152,64154,64157],{"className":64153},[31685],[34,64155,768],{"className":64156},[31685],[34,64158,64160],{"className":64159},[31792],[34,64161,64163],{"className":64162},[31796],[34,64164,64166],{"className":64165},[31801],[34,64167,64169],{"className":64168,"style":37774},[31805],[34,64170,64171,64174],{"style":37777},[34,64172],{"className":64173,"style":31814},[31813],[34,64175,64177],{"className":64176},[31818,31819,31820,31821],[34,64178,64180,64183,64186],{"className":64179},[31685,31821],[34,64181,31660],{"className":64182},[31685,31686,31821],[34,64184,223],{"className":64185},[32024,31821],[34,64187,640],{"className":64188},[31685,31821],[34,64190],{"className":64191,"style":32604},[31845],[34,64193,32226],{"className":64194},[32024],[34,64196],{"className":64197,"style":32604},[31845],[34,64199,64201,64204],{"className":64200},[31676],[34,64202],{"className":64203,"style":32617},[31680],[34,64205,640],{"className":64206},[31685],[253,64208,58168],{"id":64209},"答案与深度解析-9",[15,64211,64212],{},[2627,64213,58547],{},[105,64215,64216],{},[108,64217,64218,64220],{},[2627,64219,58554],{},[105,64221,64222,64257,64289,64379],{},[108,64223,64224,64227,64228,64256],{},[2627,64225,64226],{},"子集树（Subset Tree）："," 当问题是从 ",[34,64229,64231,64244],{"className":64230},[31642],[34,64232,64234],{"className":64233},[31646],[31648,64235,64236],{"xmlns":31650},[31652,64237,64238,64242],{},[31655,64239,64240],{},[31658,64241,31660],{},[31665,64243,31660],{"encoding":31667},[34,64245,64247],{"className":64246,"ariaHidden":758},[31672],[34,64248,64250,64253],{"className":64249},[31676],[34,64251],{"className":64252,"style":32527},[31680],[34,64254,31660],{"className":64255},[31685,31686]," 个元素的集合中找出满足某种性质的子集时（例如 0\u002F1 背包问题、装载问题），每个元素只有两种状态：选（1）或不选（0）。",[108,64258,64259,64260,64288],{},"这相当于做了 ",[34,64261,64263,64276],{"className":64262},[31642],[34,64264,64266],{"className":64265},[31646],[31648,64267,64268],{"xmlns":31650},[31652,64269,64270,64274],{},[31655,64271,64272],{},[31658,64273,31660],{},[31665,64275,31660],{"encoding":31667},[34,64277,64279],{"className":64278,"ariaHidden":758},[31672],[34,64280,64282,64285],{"className":64281},[31676],[34,64283],{"className":64284,"style":32527},[31680],[34,64286,31660],{"className":64287},[31685,31686]," 次决策，每次决策有 2 个分支。",[108,64290,64291,64292,64320,64321,347],{},"画出一棵完整的状态树，第 1 层有 2 个节点，第 2 层有 4 个……到了第 ",[34,64293,64295,64308],{"className":64294},[31642],[34,64296,64298],{"className":64297},[31646],[31648,64299,64300],{"xmlns":31650},[31652,64301,64302,64306],{},[31655,64303,64304],{},[31658,64305,31660],{},[31665,64307,31660],{"encoding":31667},[34,64309,64311],{"className":64310,"ariaHidden":758},[31672],[34,64312,64314,64317],{"className":64313},[31676],[34,64315],{"className":64316,"style":32527},[31680],[34,64318,31660],{"className":64319},[31685,31686]," 层结束时，底部的叶子节点数量正好是所有可能的子集数量，即 ",[34,64322,64324,64341],{"className":64323},[31642],[34,64325,64327],{"className":64326},[31646],[31648,64328,64329],{"xmlns":31650},[31652,64330,64331,64339],{},[31655,64332,64333],{},[37624,64334,64335,64337],{},[31749,64336,768],{},[31658,64338,31660],{},[31665,64340,59571],{"encoding":31667},[34,64342,64344],{"className":64343,"ariaHidden":758},[31672],[34,64345,64347,64350],{"className":64346},[31676],[34,64348],{"className":64349,"style":37959},[31680],[34,64351,64353,64356],{"className":64352},[31685],[34,64354,768],{"className":64355},[31685],[34,64357,64359],{"className":64358},[31792],[34,64360,64362],{"className":64361},[31796],[34,64363,64365],{"className":64364},[31801],[34,64366,64368],{"className":64367,"style":37959},[31805],[34,64369,64370,64373],{"style":37777},[34,64371],{"className":64372,"style":31814},[31813],[34,64374,64376],{"className":64375},[31818,31819,31820,31821],[34,64377,31660],{"className":64378},[31685,31686,31821],[108,64380,64381,64382,64385,64386,64419,64420,347],{},"补充知识：如果题目问的是",[2627,64383,64384],{},"排列树","（例如旅行商问题、N皇后问题），所有可能的排列方式有 ",[34,64387,64389,64404],{"className":64388},[31642],[34,64390,64392],{"className":64391},[31646],[31648,64393,64394],{"xmlns":31650},[31652,64395,64396,64402],{},[31655,64397,64398,64400],{},[31658,64399,31660],{},[31662,64401,5504],{"stretchy":702},[31665,64403,31668],{"encoding":31667},[34,64405,64407],{"className":64406,"ariaHidden":758},[31672],[34,64408,64410,64413,64416],{"className":64409},[31676],[34,64411],{"className":64412,"style":31681},[31680],[34,64414,31660],{"className":64415},[31685,31686],[34,64417,5504],{"className":64418},[31690]," 种，此时叶节点数量为 ",[34,64421,64423,64438],{"className":64422},[31642],[34,64424,64426],{"className":64425},[31646],[31648,64427,64428],{"xmlns":31650},[31652,64429,64430,64436],{},[31655,64431,64432,64434],{},[31658,64433,31660],{},[31662,64435,5504],{"stretchy":702},[31665,64437,31668],{"encoding":31667},[34,64439,64441],{"className":64440,"ariaHidden":758},[31672],[34,64442,64444,64447,64450],{"className":64443},[31676],[34,64445],{"className":64446,"style":31681},[31680],[34,64448,31660],{"className":64449},[31685,31686],[34,64451,5504],{"className":64452},[31690],[32188,64454],{},[253,64456,64458],{"id":64457},"_7-题目内容","7. 题目内容",[15,64460,64461,64462,64464,64465,64467,64468,64470],{},"以深度优先方式对解空间进行搜索的算法是（ ）。\nA. 分支限界",[58531,64463],{},"\nB. 贪心算法",[58531,64466],{},"\nC. 动态规划",[58531,64469],{},"\nD. 回溯法",[253,64472,58168],{"id":64473},"答案与深度解析-10",[15,64475,64476],{},[2627,64477,59391],{},[105,64479,64480],{},[108,64481,64482,64484,64485,64487],{},[2627,64483,58554],{}," 这是一道概念识记题。正如简答题第 2 题所分析的：回溯法的核心引擎就是",[2627,64486,58262],{},"，再辅以剪枝函数。分支限界法是广度优先或优先队列。贪心和动态规划通常不直接描述为对树结构的穷举搜索。",[32188,64489],{},[253,64491,64493],{"id":64492},"_8-题目内容","8. 题目内容",[15,64495,64496,64497,64464,64499,64501,64502,64504],{},"备忘录算法是（ ）的变形。\nA. 分治法",[58531,64498],{},[58531,64500],{},"\nC. 动态规划法",[58531,64503],{},"\nD. 分支限界法",[253,64506,58168],{"id":64507},"答案与深度解析-11",[15,64509,64510],{},[2627,64511,61063],{},[105,64513,64514],{},[108,64515,64516,64518],{},[2627,64517,58554],{},[105,64519,64520,64523,64546],{},[108,64521,64522],{},"动态规划的核心痛点是“重叠子问题”，如果用普通递归，会引发指数级的重复计算。",[108,64524,64525,64526],{},"解决重复计算有两种路线：\n",[2230,64527,64528,64537],{},[108,64529,64530,64533,64534,64536],{},[2627,64531,64532],{},"自底向上："," 也就是狭义上的",[2627,64535,58216],{},"。先从最小的子问题开始算，用表格存起来，慢慢推导到大问题。",[108,64538,64539,64542,64543,347],{},[2627,64540,64541],{},"自顶向下："," 依然保持自然的递归写法，但是准备一个“备忘录”（通常是一个数组或哈希表）。每次递归计算出一个子问题的答案时，把它记在备忘录里；下次再遇到完全相同的子问题，直接查备忘录拿答案，不用再算一遍。这就是",[2627,64544,64545],{},"备忘录算法（Memoization）",[108,64547,64548],{},"因此，备忘录算法本质上是动态规划思想的“自顶向下”实现方式。",[32188,64550],{},[253,64552,64554],{"id":64553},"_9-题目内容","9. 题目内容",[15,64556,64557,64558,64586,64587,64615,64616,64644,64645,64714,64715,64758,58784,64760,64833,58980,64835,64942,59196,64944],{},"利用贪心算法求解 ",[34,64559,64561,64574],{"className":64560},[31642],[34,64562,64564],{"className":64563},[31646],[31648,64565,64566],{"xmlns":31650},[31652,64567,64568,64572],{},[31655,64569,64570],{},[31658,64571,31660],{},[31665,64573,31660],{"encoding":31667},[34,64575,64577],{"className":64576,"ariaHidden":758},[31672],[34,64578,64580,64583],{"className":64579},[31676],[34,64581],{"className":64582,"style":32527},[31680],[34,64584,31660],{"className":64585},[31685,31686]," 个集装箱最优装载问题（已知轮船载重量为 ",[34,64588,64590,64603],{"className":64589},[31642],[34,64591,64593],{"className":64592},[31646],[31648,64594,64595],{"xmlns":31650},[31652,64596,64597,64601],{},[31655,64598,64599],{},[31658,64600,56852],{},[31665,64602,56852],{"encoding":31667},[34,64604,64606],{"className":64605,"ariaHidden":758},[31672],[34,64607,64609,64612],{"className":64608},[31676],[34,64610],{"className":64611,"style":33218},[31680],[34,64613,56852],{"className":64614,"style":37422},[31685,31686],"，有 ",[34,64617,64619,64632],{"className":64618},[31642],[34,64620,64622],{"className":64621},[31646],[31648,64623,64624],{"xmlns":31650},[31652,64625,64626,64630],{},[31655,64627,64628],{},[31658,64629,31660],{},[31665,64631,31660],{"encoding":31667},[34,64633,64635],{"className":64634,"ariaHidden":758},[31672],[34,64636,64638,64641],{"className":64637},[31676],[34,64639],{"className":64640,"style":32527},[31680],[34,64642,31660],{"className":64643},[31685,31686]," 个集装箱重量分别为 ",[34,64646,64648,64665],{"className":64647},[31642],[34,64649,64651],{"className":64650},[31646],[31648,64652,64653],{"xmlns":31650},[31652,64654,64655,64663],{},[31655,64656,64657],{},[31743,64658,64659,64661],{},[31658,64660,47077],{},[31658,64662,31955],{},[31665,64664,56714],{"encoding":31667},[34,64666,64668],{"className":64667,"ariaHidden":758},[31672],[34,64669,64671,64674],{"className":64670},[31676],[34,64672],{"className":64673,"style":31860},[31680],[34,64675,64677,64680],{"className":64676},[31685],[34,64678,47077],{"className":64679,"style":47120},[31685,31686],[34,64681,64683],{"className":64682},[31792],[34,64684,64686,64706],{"className":64685},[31796,31797],[34,64687,64689,64703],{"className":64688},[31801],[34,64690,64692],{"className":64691,"style":32090},[31805],[34,64693,64694,64697],{"style":56744},[34,64695],{"className":64696,"style":31814},[31813],[34,64698,64700],{"className":64699},[31818,31819,31820,31821],[34,64701,31955],{"className":64702},[31685,31686,31821],[34,64704,31829],{"className":64705},[31828],[34,64707,64709],{"className":64708},[31801],[34,64710,64712],{"className":64711,"style":31836},[31805],[34,64713],{},"，设计使得轮船上能装载更多的集装箱）时间复杂度的是（ ）。\nA. ",[34,64716,64718,64737],{"className":64717},[31642],[34,64719,64721],{"className":64720},[31646],[31648,64722,64723],{"xmlns":31650},[31652,64724,64725,64735],{},[31655,64726,64727,64729,64731,64733],{},[31658,64728,37890],{},[31662,64730,456],{"stretchy":702},[31658,64732,31660],{},[31662,64734,2557],{"stretchy":702},[31665,64736,46493],{"encoding":31667},[34,64738,64740],{"className":64739,"ariaHidden":758},[31672],[34,64741,64743,64746,64749,64752,64755],{"className":64742},[31676],[34,64744],{"className":64745,"style":31774},[31680],[34,64747,37890],{"className":64748,"style":34452},[31685,31686],[34,64750,456],{"className":64751},[31782],[34,64753,31660],{"className":64754},[31685,31686],[34,64756,2557],{"className":64757},[31690],[58531,64759],{},[34,64761,64763,64786],{"className":64762},[31642],[34,64764,64766],{"className":64765},[31646],[31648,64767,64768],{"xmlns":31650},[31652,64769,64770,64784],{},[31655,64771,64772,64774,64776,64782],{},[31658,64773,37890],{},[31662,64775,456],{"stretchy":702},[37624,64777,64778,64780],{},[31658,64779,31660],{},[31749,64781,768],{},[31662,64783,2557],{"stretchy":702},[31665,64785,46174],{"encoding":31667},[34,64787,64789],{"className":64788,"ariaHidden":758},[31672],[34,64790,64792,64795,64798,64801,64830],{"className":64791},[31676],[34,64793],{"className":64794,"style":37755},[31680],[34,64796,37890],{"className":64797,"style":34452},[31685,31686],[34,64799,456],{"className":64800},[31782],[34,64802,64804,64807],{"className":64803},[31685],[34,64805,31660],{"className":64806},[31685,31686],[34,64808,64810],{"className":64809},[31792],[34,64811,64813],{"className":64812},[31796],[34,64814,64816],{"className":64815},[31801],[34,64817,64819],{"className":64818,"style":37774},[31805],[34,64820,64821,64824],{"style":37777},[34,64822],{"className":64823,"style":31814},[31813],[34,64825,64827],{"className":64826},[31818,31819,31820,31821],[34,64828,768],{"className":64829},[31685,31821],[34,64831,2557],{"className":64832},[31690],[58531,64834],{},[34,64836,64838,64870],{"className":64837},[31642],[34,64839,64841],{"className":64840},[31646],[31648,64842,64843],{"xmlns":31650},[31652,64844,64845,64867],{},[31655,64846,64847,64849,64851,64853,64863,64865],{},[31658,64848,37890],{},[31662,64850,456],{"stretchy":702},[31658,64852,31660],{},[31743,64854,64855,64861],{},[31655,64856,64857,64859],{},[31658,64858,885],{},[31662,64860,32682],{},[31749,64862,768],{},[31658,64864,31660],{},[31662,64866,2557],{"stretchy":702},[31665,64868,64869],{"encoding":31667},"O(n\\log_2 n)",[34,64871,64873],{"className":64872,"ariaHidden":758},[31672],[34,64874,64876,64879,64882,64885,64888,64891,64933,64936,64939],{"className":64875},[31676],[34,64877],{"className":64878,"style":31774},[31680],[34,64880,37890],{"className":64881,"style":34452},[31685,31686],[34,64883,456],{"className":64884},[31782],[34,64886,31660],{"className":64887},[31685,31686],[34,64889],{"className":64890,"style":32123},[31845],[34,64892,64894,64899],{"className":64893},[32767],[34,64895,38616,64897],{"className":64896},[32767],[34,64898,31942],{"style":38619},[34,64900,64902],{"className":64901},[31792],[34,64903,64905,64925],{"className":64904},[31796,31797],[34,64906,64908,64922],{"className":64907},[31801],[34,64909,64911],{"className":64910,"style":38632},[31805],[34,64912,64913,64916],{"style":38635},[34,64914],{"className":64915,"style":31814},[31813],[34,64917,64919],{"className":64918},[31818,31819,31820,31821],[34,64920,768],{"className":64921},[31685,31821],[34,64923,31829],{"className":64924},[31828],[34,64926,64928],{"className":64927},[31801],[34,64929,64931],{"className":64930,"style":38654},[31805],[34,64932],{},[34,64934],{"className":64935,"style":32123},[31845],[34,64937,31660],{"className":64938},[31685,31686],[34,64940,2557],{"className":64941},[31690],[58531,64943],{},[34,64945,64947,64969],{"className":64946},[31642],[34,64948,64950],{"className":64949},[31646],[31648,64951,64952],{"xmlns":31650},[31652,64953,64954,64966],{},[31655,64955,64956,64958,64960,64964],{},[31658,64957,37890],{},[31662,64959,456],{"stretchy":702},[41463,64961,64962],{},[31658,64963,31660],{},[31662,64965,2557],{"stretchy":702},[31665,64967,64968],{"encoding":31667},"O(\\sqrt{n})",[34,64970,64972],{"className":64971,"ariaHidden":758},[31672],[34,64973,64975,64979,64982,64985,65035],{"className":64974},[31676],[34,64976],{"className":64977,"style":64978},[31680],"height:1.0503em;vertical-align:-0.25em;",[34,64980,37890],{"className":64981,"style":34452},[31685,31686],[34,64983,456],{"className":64984},[31782],[34,64986,64988],{"className":64987},[31685,41550],[34,64989,64991,65026],{"className":64990},[31796,31797],[34,64992,64994,65023],{"className":64993},[31801],[34,64995,64998,65010],{"className":64996,"style":64997},[31805],"height:0.8003em;",[34,64999,65001,65004],{"className":65000,"style":41565},[41564],[34,65002],{"className":65003,"style":41508},[31813],[34,65005,65007],{"className":65006,"style":41572},[31685],[34,65008,31660],{"className":65009},[31685,31686],[34,65011,65013,65016],{"style":65012},"top:-2.7603em;",[34,65014],{"className":65015,"style":41508},[31813],[34,65017,65019],{"className":65018,"style":41586},[41585],[41588,65020,65021],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,65022],{"d":41597},[34,65024,31829],{"className":65025},[31828],[34,65027,65029],{"className":65028},[31801],[34,65030,65033],{"className":65031,"style":65032},[31805],"height:0.2397em;",[34,65034],{},[34,65036,2557],{"className":65037},[31690],[253,65039,58168],{"id":65040},"答案与深度解析-12",[15,65042,65043],{},[2627,65044,61063],{},[105,65046,65047],{},[108,65048,65049,65051,65052,65055,65056,65059,65060,65088,65089,65092,65093,65096,65097,65203,65204,65247,65248,347],{},[2627,65050,58554],{}," 1. ",[2627,65053,65054],{},"贪心策略："," 要想装载“最多”数量的集装箱，最直观且正确的策略就是：优先挑最轻的装。轻的占用的载重量少，留给后面集装箱的额度就多。\n2. ",[2627,65057,65058],{},"算法执行步骤："," 第一步，必须对 ",[34,65061,65063,65076],{"className":65062},[31642],[34,65064,65066],{"className":65065},[31646],[31648,65067,65068],{"xmlns":31650},[31652,65069,65070,65074],{},[31655,65071,65072],{},[31658,65073,31660],{},[31665,65075,31660],{"encoding":31667},[34,65077,65079],{"className":65078,"ariaHidden":758},[31672],[34,65080,65082,65085],{"className":65081},[31676],[34,65083],{"className":65084,"style":32527},[31680],[34,65086,31660],{"className":65087},[31685,31686]," 个集装箱的重量进行",[2627,65090,65091],{},"从小到大排序","。第二步，从轻到重依次遍历，只要没超重就装进去。\n3. ",[2627,65094,65095],{},"时间复杂度分析："," 最优的比较排序算法（如快速排序、归并排序）的时间复杂度为 ",[34,65098,65100,65131],{"className":65099},[31642],[34,65101,65103],{"className":65102},[31646],[31648,65104,65105],{"xmlns":31650},[31652,65106,65107,65129],{},[31655,65108,65109,65111,65113,65115,65125,65127],{},[31658,65110,37890],{},[31662,65112,456],{"stretchy":702},[31658,65114,31660],{},[31743,65116,65117,65123],{},[31655,65118,65119,65121],{},[31658,65120,885],{},[31662,65122,32682],{},[31749,65124,768],{},[31658,65126,31660],{},[31662,65128,2557],{"stretchy":702},[31665,65130,64869],{"encoding":31667},[34,65132,65134],{"className":65133,"ariaHidden":758},[31672],[34,65135,65137,65140,65143,65146,65149,65152,65194,65197,65200],{"className":65136},[31676],[34,65138],{"className":65139,"style":31774},[31680],[34,65141,37890],{"className":65142,"style":34452},[31685,31686],[34,65144,456],{"className":65145},[31782],[34,65147,31660],{"className":65148},[31685,31686],[34,65150],{"className":65151,"style":32123},[31845],[34,65153,65155,65160],{"className":65154},[32767],[34,65156,38616,65158],{"className":65157},[32767],[34,65159,31942],{"style":38619},[34,65161,65163],{"className":65162},[31792],[34,65164,65166,65186],{"className":65165},[31796,31797],[34,65167,65169,65183],{"className":65168},[31801],[34,65170,65172],{"className":65171,"style":38632},[31805],[34,65173,65174,65177],{"style":38635},[34,65175],{"className":65176,"style":31814},[31813],[34,65178,65180],{"className":65179},[31818,31819,31820,31821],[34,65181,768],{"className":65182},[31685,31821],[34,65184,31829],{"className":65185},[31828],[34,65187,65189],{"className":65188},[31801],[34,65190,65192],{"className":65191,"style":38654},[31805],[34,65193],{},[34,65195],{"className":65196,"style":32123},[31845],[34,65198,31660],{"className":65199},[31685,31686],[34,65201,2557],{"className":65202},[31690],"。第二步的遍历只需扫描一次，复杂度为 ",[34,65205,65207,65226],{"className":65206},[31642],[34,65208,65210],{"className":65209},[31646],[31648,65211,65212],{"xmlns":31650},[31652,65213,65214,65224],{},[31655,65215,65216,65218,65220,65222],{},[31658,65217,37890],{},[31662,65219,456],{"stretchy":702},[31658,65221,31660],{},[31662,65223,2557],{"stretchy":702},[31665,65225,46493],{"encoding":31667},[34,65227,65229],{"className":65228,"ariaHidden":758},[31672],[34,65230,65232,65235,65238,65241,65244],{"className":65231},[31676],[34,65233],{"className":65234,"style":31774},[31680],[34,65236,37890],{"className":65237,"style":34452},[31685,31686],[34,65239,456],{"className":65240},[31782],[34,65242,31660],{"className":65243},[31685,31686],[34,65245,2557],{"className":65246},[31690],"。两者相加，排序步骤主导了整体时间耗费，因此总时间复杂度为 ",[34,65249,65251,65282],{"className":65250},[31642],[34,65252,65254],{"className":65253},[31646],[31648,65255,65256],{"xmlns":31650},[31652,65257,65258,65280],{},[31655,65259,65260,65262,65264,65266,65276,65278],{},[31658,65261,37890],{},[31662,65263,456],{"stretchy":702},[31658,65265,31660],{},[31743,65267,65268,65274],{},[31655,65269,65270,65272],{},[31658,65271,885],{},[31662,65273,32682],{},[31749,65275,768],{},[31658,65277,31660],{},[31662,65279,2557],{"stretchy":702},[31665,65281,64869],{"encoding":31667},[34,65283,65285],{"className":65284,"ariaHidden":758},[31672],[34,65286,65288,65291,65294,65297,65300,65303,65345,65348,65351],{"className":65287},[31676],[34,65289],{"className":65290,"style":31774},[31680],[34,65292,37890],{"className":65293,"style":34452},[31685,31686],[34,65295,456],{"className":65296},[31782],[34,65298,31660],{"className":65299},[31685,31686],[34,65301],{"className":65302,"style":32123},[31845],[34,65304,65306,65311],{"className":65305},[32767],[34,65307,38616,65309],{"className":65308},[32767],[34,65310,31942],{"style":38619},[34,65312,65314],{"className":65313},[31792],[34,65315,65317,65337],{"className":65316},[31796,31797],[34,65318,65320,65334],{"className":65319},[31801],[34,65321,65323],{"className":65322,"style":38632},[31805],[34,65324,65325,65328],{"style":38635},[34,65326],{"className":65327,"style":31814},[31813],[34,65329,65331],{"className":65330},[31818,31819,31820,31821],[34,65332,768],{"className":65333},[31685,31821],[34,65335,31829],{"className":65336},[31828],[34,65338,65340],{"className":65339},[31801],[34,65341,65343],{"className":65342,"style":38654},[31805],[34,65344],{},[34,65346],{"className":65347,"style":32123},[31845],[34,65349,31660],{"className":65350},[31685,31686],[34,65352,2557],{"className":65353},[31690],[32188,65355],{},[253,65357,65359],{"id":65358},"_10-题目内容","10. 题目内容",[15,65361,65362,65363,65365,65366,65368,65369,65371],{},"调用一次下列随机算法进行问题求解时，可能成功也可能失败的是（ ）。\nA. 蒙特卡洛算法",[58531,65364],{},"\nB. 拉斯维加斯算法",[58531,65367],{},"\nC. 舍伍德算法",[58531,65370],{},"\nD. 数值概率算法",[253,65373,58168],{"id":65374},"答案与深度解析-13",[15,65376,65377],{},[2627,65378,58547],{},[105,65380,65381],{},[108,65382,65383,65385,65386],{},[2627,65384,58554],{}," 结合简答题第 3 题的知识：\n",[105,65387,65388,65394,65400],{},[108,65389,65390,65393],{},[2627,65391,65392],{},"拉斯维加斯算法","最大的特征就是：一旦它输出结果，结果绝对正确（100%）。但它不能保证在限定时间内一定能找到解，如果运气不好走入死胡同，它会返回“失败”（无解）。这完全契合题目描述的“可能成功也可能失败”。",[108,65395,65396,65399],{},[2627,65397,65398],{},"蒙特卡洛算法","必定会返回一个答案（一定成功运行），只是答案可能是错的。",[108,65401,65402,65405],{},[2627,65403,65404],{},"舍伍德算法","是为了消除最坏情况而引入随机性（例如随机选择快速排序的基准元素），它总能求得正确解，且必定成功。",[32188,65407],{},[19,65409,65411],{"id":65410},"三填空题每空2分共20分","三、填空题（每空2分，共20分）",[253,65413,58162],{"id":65414},"_1-题目内容-2",[15,65416,65417],{},"求解 0\u002F1 背包问题可采用动态规划法、回溯法和分支限界法，其中需要按照单位重量价值比进行排序以提高算法效率的是 _____，不需要进行排序的是 _____ 。",[253,65419,58168],{"id":65420},"答案与深度解析-14",[15,65422,65423,65425,65426,65429,65430],{},[2627,65424,58173],{},"\n第一空：",[2627,65427,65428],{},"回溯法和分支限界法","\n第二空：",[2627,65431,65432],{},"动态规划法",[105,65434,65435],{},[108,65436,65437,65439,65440],{},[2627,65438,58199],{}," * 为什么回溯和分支限界需要排序？因为这两种方法本质是在枚举所有组合。为了能尽早剪去不可能产生最优解的枝丫，我们需要一个强有力的“上界函数”。将物品按性价比（价值\u002F重量）降序排列，可以让我们用贪心思想迅速估算出一个“理想状态下的最大可能收益”，如果这个理想收益都比不过当前已经找到的解，直接砍掉这个搜索分支，大幅提升效率。\n",[105,65441,65442],{},[108,65443,65444,65445,65448],{},"为什么动态规划不需要？动态规划是基于状态转移方程 ",[31,65446,65447],{},"dp[i][j] = max(dp[i-1][j], dp[i-1][j-w[i]] + v[i])","，它通过穷尽表格所有容量状态来获得绝对最优解，物品出现的先后顺序不影响最终表格右下角的答案，因此排序纯属多此一举。",[32188,65450],{},[253,65452,58228],{"id":65453},"_2-题目内容-2",[15,65455,65456],{},"标识主元素的蒙特卡洛算法，获得正确解的概率为 0.5，则调用该算法 3 次可以将获得正确解的概率提高到 _____ 。",[253,65458,58168],{"id":65459},"答案与深度解析-15",[15,65461,65462],{},[2627,65463,65464],{},"答案：0.875 （或 87.5%、7\u002F8）",[105,65466,65467],{},[108,65468,65469,65471,65472],{},[2627,65470,58199],{}," 这是一个基础概率论计算题，需要逆向思维。\n",[105,65473,65474,65480,65490,65590],{},[108,65475,65476,65477,347],{},"算法运行 1 次，正确的概率是 0.5，",[2627,65478,65479],{},"错误的概率是 1 - 0.5 = 0.5",[108,65481,65482,65483,65486,65487,347],{},"“调用 3 次且最终拿到正确解”的含义是：只要这 3 次中",[2627,65484,65485],{},"有任意一次","是正确的，我们就成功了。它的反面是：",[2627,65488,65489],{},"3 次全部都算错了",[108,65491,65492,65493,347],{},"3 次全部算错的概率是：",[34,65494,65496,65524],{"className":65495},[31642],[34,65497,65499],{"className":65498},[31646],[31648,65500,65501],{"xmlns":31650},[31652,65502,65503,65521],{},[31655,65504,65505,65508,65510,65512,65514,65516,65518],{},[31749,65506,65507],{},"0.5",[31662,65509,42071],{},[31749,65511,65507],{},[31662,65513,42071],{},[31749,65515,65507],{},[31662,65517,202],{},[31749,65519,65520],{},"0.125",[31665,65522,65523],{"encoding":31667},"0.5 \\times 0.5 \\times 0.5 = 0.125",[34,65525,65527,65545,65563,65581],{"className":65526,"ariaHidden":758},[31672],[34,65528,65530,65533,65536,65539,65542],{"className":65529},[31676],[34,65531],{"className":65532,"style":35942},[31680],[34,65534,65507],{"className":65535},[31685],[34,65537],{"className":65538,"style":32604},[31845],[34,65540,42071],{"className":65541},[32024],[34,65543],{"className":65544,"style":32604},[31845],[34,65546,65548,65551,65554,65557,65560],{"className":65547},[31676],[34,65549],{"className":65550,"style":35942},[31680],[34,65552,65507],{"className":65553},[31685],[34,65555],{"className":65556,"style":32604},[31845],[34,65558,42071],{"className":65559},[32024],[34,65561],{"className":65562,"style":32604},[31845],[34,65564,65566,65569,65572,65575,65578],{"className":65565},[31676],[34,65567],{"className":65568,"style":32617},[31680],[34,65570,65507],{"className":65571},[31685],[34,65573],{"className":65574,"style":31846},[31845],[34,65576,202],{"className":65577},[31850],[34,65579],{"className":65580,"style":31846},[31845],[34,65582,65584,65587],{"className":65583},[31676],[34,65585],{"className":65586,"style":32617},[31680],[34,65588,65520],{"className":65589},[31685],[108,65591,65592,65593,347],{},"那么至少有一次算对（即最终获得正确解）的概率就是：",[34,65594,65596,65619],{"className":65595},[31642],[34,65597,65599],{"className":65598},[31646],[31648,65600,65601],{"xmlns":31650},[31652,65602,65603,65616],{},[31655,65604,65605,65607,65609,65611,65613],{},[31749,65606,640],{},[31662,65608,32226],{},[31749,65610,65520],{},[31662,65612,202],{},[31749,65614,65615],{},"0.875",[31665,65617,65618],{"encoding":31667},"1 - 0.125 = 0.875",[34,65620,65622,65640,65658],{"className":65621,"ariaHidden":758},[31672],[34,65623,65625,65628,65631,65634,65637],{"className":65624},[31676],[34,65626],{"className":65627,"style":35942},[31680],[34,65629,640],{"className":65630},[31685],[34,65632],{"className":65633,"style":32604},[31845],[34,65635,32226],{"className":65636},[32024],[34,65638],{"className":65639,"style":32604},[31845],[34,65641,65643,65646,65649,65652,65655],{"className":65642},[31676],[34,65644],{"className":65645,"style":32617},[31680],[34,65647,65520],{"className":65648},[31685],[34,65650],{"className":65651,"style":31846},[31845],[34,65653,202],{"className":65654},[31850],[34,65656],{"className":65657,"style":31846},[31845],[34,65659,65661,65664],{"className":65660},[31676],[34,65662],{"className":65663,"style":32617},[31680],[34,65665,65615],{"className":65666},[31685],[32188,65668],{},[253,65670,58367],{"id":65671},"_3-题目内容-2",[15,65673,65674,65702,65703,65731],{},[34,65675,65677,65690],{"className":65676},[31642],[34,65678,65680],{"className":65679},[31646],[31648,65681,65682],{"xmlns":31650},[31652,65683,65684,65688],{},[31655,65685,65686],{},[31658,65687,31660],{},[31665,65689,31660],{"encoding":31667},[34,65691,65693],{"className":65692,"ariaHidden":758},[31672],[34,65694,65696,65699],{"className":65695},[31676],[34,65697],{"className":65698,"style":32527},[31680],[34,65700,31660],{"className":65701},[31685,31686]," 皇后问题的满 ",[34,65704,65706,65719],{"className":65705},[31642],[34,65707,65709],{"className":65708},[31646],[31648,65710,65711],{"xmlns":31650},[31652,65712,65713,65717],{},[31655,65714,65715],{},[31658,65716,31660],{},[31665,65718,31660],{"encoding":31667},[34,65720,65722],{"className":65721,"ariaHidden":758},[31672],[34,65723,65725,65728],{"className":65724},[31676],[34,65726],{"className":65727,"style":32527},[31680],[34,65729,31660],{"className":65730},[31685,31686]," 叉树的求解算法的框架如下，补充完整该算法：",[24,65733,65736],{"className":65734,"code":65735,"language":31965,"meta":29,"style":29},"language-c shiki shiki-themes github-light github-dark","int sum = 0; \u002F\u002F sum 用于记录可行解个数\nvoid queen(int t) \u002F\u002F 放第 t 个皇后\n{\n if (t > n)\n _____(1)_____;\n else\n for (int j = 1; ____(2)____; j++)\n {\n ____(3)_____; \n if (place(t))\n ____(4)_____; \n }\n}\n",[31,65737,65738,65743,65748,65752,65757,65762,65767,65772,65777,65782,65787,65792,65796],{"__ignoreMap":29},[34,65739,65740],{"class":36,"line":37},[34,65741,65742],{},"int sum = 0; \u002F\u002F sum 用于记录可行解个数\n",[34,65744,65745],{"class":36,"line":43},[34,65746,65747],{},"void queen(int t) \u002F\u002F 放第 t 个皇后\n",[34,65749,65750],{"class":36,"line":49},[34,65751,20357],{},[34,65753,65754],{"class":36,"line":56},[34,65755,65756],{}," if (t > n)\n",[34,65758,65759],{"class":36,"line":62},[34,65760,65761],{}," _____(1)_____;\n",[34,65763,65764],{"class":36,"line":68},[34,65765,65766],{}," else\n",[34,65768,65769],{"class":36,"line":74},[34,65770,65771],{}," for (int j = 1; ____(2)____; j++)\n",[34,65773,65774],{"class":36,"line":1971},[34,65775,65776],{}," {\n",[34,65778,65779],{"class":36,"line":1976},[34,65780,65781],{}," ____(3)_____; \n",[34,65783,65784],{"class":36,"line":1985},[34,65785,65786],{}," if (place(t))\n",[34,65788,65789],{"class":36,"line":1990},[34,65790,65791],{}," ____(4)_____; \n",[34,65793,65794],{"class":36,"line":1999},[34,65795,26684],{},[34,65797,65798],{"class":36,"line":2022},[34,65799,77],{},[253,65801,58168],{"id":65802},"答案与深度解析-16",[15,65804,65805,65807,65808,65811,65812,65815,65816,65819,65820,65822,65823,65825,65826],{},[2627,65806,58173],{},"\n(1) ",[31,65809,65810],{},"sum++","\n(2) ",[31,65813,65814],{},"j \u003C= n","\n(3) ",[31,65817,65818],{},"x[t] = j"," （这里假设用于记录第 ",[31,65821,33249],{}," 个皇后所在的列号的全局数组名为 ",[31,65824,32936],{},"，这是极其通用的约定写法）\n(4) ",[31,65827,65828],{},"queen(t + 1)",[105,65830,65831],{},[108,65832,65833,65835,65836],{},[2627,65834,58199],{}," 这是经典的回溯法框架。我们一行一行从零梳理逻辑：\n",[105,65837,65838,65848,65967,66039,66061],{},[108,65839,65840,65841,65844,65845,65847],{},"函数 ",[31,65842,65843],{},"queen(t)"," 的意义是：去决定第 ",[31,65846,33249],{}," 行的皇后应该放在哪一列。",[108,65849,65850,65853,65854,65882,65883,65935,65936,65964,65965,347],{},[2627,65851,65852],{},"第(1)空："," 递归的终止条件。题目说一共有 ",[34,65855,65857,65870],{"className":65856},[31642],[34,65858,65860],{"className":65859},[31646],[31648,65861,65862],{"xmlns":31650},[31652,65863,65864,65868],{},[31655,65865,65866],{},[31658,65867,31660],{},[31665,65869,31660],{"encoding":31667},[34,65871,65873],{"className":65872,"ariaHidden":758},[31672],[34,65874,65876,65879],{"className":65875},[31676],[34,65877],{"className":65878,"style":32527},[31680],[34,65880,31660],{"className":65881},[31685,31686]," 个皇后，如果此时传入的 ",[34,65884,65886,65904],{"className":65885},[31642],[34,65887,65889],{"className":65888},[31646],[31648,65890,65891],{"xmlns":31650},[31652,65892,65893,65901],{},[31655,65894,65895,65897,65899],{},[31658,65896,33249],{},[31662,65898,29622],{},[31658,65900,31660],{},[31665,65902,65903],{"encoding":31667},"t > n",[34,65905,65907,65926],{"className":65906,"ariaHidden":758},[31672],[34,65908,65910,65914,65917,65920,65923],{"className":65909},[31676],[34,65911],{"className":65912,"style":65913},[31680],"height:0.6542em;vertical-align:-0.0391em;",[34,65915,33249],{"className":65916},[31685,31686],[34,65918],{"className":65919,"style":31846},[31845],[34,65921,29622],{"className":65922},[31850],[34,65924],{"className":65925,"style":31846},[31845],[34,65927,65929,65932],{"className":65928},[31676],[34,65930],{"className":65931,"style":32527},[31680],[34,65933,31660],{"className":65934},[31685,31686],"，说明第 1 到第 ",[34,65937,65939,65952],{"className":65938},[31642],[34,65940,65942],{"className":65941},[31646],[31648,65943,65944],{"xmlns":31650},[31652,65945,65946,65950],{},[31655,65947,65948],{},[31658,65949,31660],{},[31665,65951,31660],{"encoding":31667},[34,65953,65955],{"className":65954,"ariaHidden":758},[31672],[34,65956,65958,65961],{"className":65957},[31676],[34,65959],{"className":65960,"style":32527},[31680],[34,65962,31660],{"className":65963},[31685,31686]," 个皇后都已经成功放置在棋盘上了，这意味着我们找到了一个合法的可行解。此时当然要让解的计数器增加：",[31,65966,65810],{},[108,65968,65969,47649,65972,65974,65975,66003,66004,66006,66007,66035,66036,347],{},[2627,65970,65971],{},"第(2)空：",[31,65973,2101],{}," 块表示还在放置前 ",[34,65976,65978,65991],{"className":65977},[31642],[34,65979,65981],{"className":65980},[31646],[31648,65982,65983],{"xmlns":31650},[31652,65984,65985,65989],{},[31655,65986,65987],{},[31658,65988,31660],{},[31665,65990,31660],{"encoding":31667},[34,65992,65994],{"className":65993,"ariaHidden":758},[31672],[34,65995,65997,66000],{"className":65996},[31676],[34,65998],{"className":65999,"style":32527},[31680],[34,66001,31660],{"className":66002},[31685,31686]," 个皇后。要在第 ",[31,66005,33249],{}," 行放皇后，她能放在哪里呢？棋盘的列数是从第 1 列到第 ",[34,66008,66010,66023],{"className":66009},[31642],[34,66011,66013],{"className":66012},[31646],[31648,66014,66015],{"xmlns":31650},[31652,66016,66017,66021],{},[31655,66018,66019],{},[31658,66020,31660],{},[31665,66022,31660],{"encoding":31667},[34,66024,66026],{"className":66025,"ariaHidden":758},[31672],[34,66027,66029,66032],{"className":66028},[31676],[34,66030],{"className":66031,"style":32527},[31680],[34,66033,31660],{"className":66034},[31685,31686]," 列。所以我们要遍历所有的列尝试放置：",[31,66037,66038],{},"for (int j = 1; j \u003C= n; j++)",[108,66040,66041,66044,66045,66047,66048,66050,66051,66054,66055,66058,66059,347],{},[2627,66042,66043],{},"第(3)空："," 我们尝试把第 ",[31,66046,33249],{}," 个皇后放在第 ",[31,66049,31968],{}," 列。必须有一个数组把这个尝试记录下来，系统后面的 ",[31,66052,66053],{},"place(t)"," 函数才知道去哪里检查冲突。通常这个数组叫 ",[31,66056,66057],{},"x[]","。记录动作是 ",[31,66060,65818],{},[108,66062,66063,47649,66066,66068,66069,347],{},[2627,66064,66065],{},"第(4)空：",[31,66067,66053],{}," 是检查函数，如果刚才放的位置（不和前面几行的皇后同列、同对角线）是安全的（返回 true），那该怎么办？当然是继续去放下一行的皇后了，即递归调用：",[31,66070,65828],{},[32188,66072],{},[253,66074,58474],{"id":66075},"_4-题目内容-2",[15,66077,66078,66079,66135,66136,66164,66165,66193,66194,66244,66245,66331,66332,66397],{},"利用分治法求解正整数的划分数问题时，",[34,66080,66082,66105],{"className":66081},[31642],[34,66083,66085],{"className":66084},[31646],[31648,66086,66087],{"xmlns":31650},[31652,66088,66089,66103],{},[31655,66090,66091,66093,66095,66097,66099,66101],{},[31658,66092,31739],{},[31662,66094,456],{"stretchy":702},[31658,66096,31660],{},[31662,66098,31960],{"separator":758},[31658,66100,31759],{},[31662,66102,2557],{"stretchy":702},[31665,66104,52749],{"encoding":31667},[34,66106,66108],{"className":66107,"ariaHidden":758},[31672],[34,66109,66111,66114,66117,66120,66123,66126,66129,66132],{"className":66110},[31676],[34,66112],{"className":66113,"style":31774},[31680],[34,66115,31739],{"className":66116,"style":31778},[31685,31686],[34,66118,456],{"className":66119},[31782],[34,66121,31660],{"className":66122},[31685,31686],[34,66124,31960],{"className":66125},[32119],[34,66127],{"className":66128,"style":32123},[31845],[34,66130,31759],{"className":66131},[31685,31686],[34,66133,2557],{"className":66134},[31690]," 表示将正整数 ",[34,66137,66139,66152],{"className":66138},[31642],[34,66140,66142],{"className":66141},[31646],[31648,66143,66144],{"xmlns":31650},[31652,66145,66146,66150],{},[31655,66147,66148],{},[31658,66149,31660],{},[31665,66151,31660],{"encoding":31667},[34,66153,66155],{"className":66154,"ariaHidden":758},[31672],[34,66156,66158,66161],{"className":66157},[31676],[34,66159],{"className":66160,"style":32527},[31680],[34,66162,31660],{"className":66163},[31685,31686]," 划分成最大正整数不超过 ",[34,66166,66168,66181],{"className":66167},[31642],[34,66169,66171],{"className":66170},[31646],[31648,66172,66173],{"xmlns":31650},[31652,66174,66175,66179],{},[31655,66176,66177],{},[31658,66178,31759],{},[31665,66180,31759],{"encoding":31667},[34,66182,66184],{"className":66183,"ariaHidden":758},[31672],[34,66185,66187,66190],{"className":66186},[31676],[34,66188],{"className":66189,"style":32527},[31680],[34,66191,31759],{"className":66192},[31685,31686]," 的划分数，当 ",[34,66195,66197,66214],{"className":66196},[31642],[34,66198,66200],{"className":66199},[31646],[31648,66201,66202],{"xmlns":31650},[31652,66203,66204,66212],{},[31655,66205,66206,66208,66210],{},[31658,66207,31660],{},[31662,66209,29622],{},[31658,66211,31759],{},[31665,66213,53315],{"encoding":31667},[34,66215,66217,66235],{"className":66216,"ariaHidden":758},[31672],[34,66218,66220,66223,66226,66229,66232],{"className":66219},[31676],[34,66221],{"className":66222,"style":34812},[31680],[34,66224,31660],{"className":66225},[31685,31686],[34,66227],{"className":66228,"style":31846},[31845],[34,66230,29622],{"className":66231},[31850],[34,66233],{"className":66234,"style":31846},[31845],[34,66236,66238,66241],{"className":66237},[31676],[34,66239],{"className":66240,"style":32527},[31680],[34,66242,31759],{"className":66243},[31685,31686]," 时（",[34,66246,66248,66274],{"className":66247},[31642],[34,66249,66251],{"className":66250},[31646],[31648,66252,66253],{"xmlns":31650},[31652,66254,66255,66271],{},[31655,66256,66257,66259,66261,66263,66265,66267,66269],{},[31658,66258,31660],{},[31662,66260,29622],{},[31749,66262,459],{},[31662,66264,31960],{"separator":758},[31658,66266,31759],{},[31662,66268,29622],{},[31749,66270,459],{},[31665,66272,66273],{"encoding":31667},"n>0, m>0",[34,66275,66277,66295,66322],{"className":66276,"ariaHidden":758},[31672],[34,66278,66280,66283,66286,66289,66292],{"className":66279},[31676],[34,66281],{"className":66282,"style":34812},[31680],[34,66284,31660],{"className":66285},[31685,31686],[34,66287],{"className":66288,"style":31846},[31845],[34,66290,29622],{"className":66291},[31850],[34,66293],{"className":66294,"style":31846},[31845],[34,66296,66298,66301,66304,66307,66310,66313,66316,66319],{"className":66297},[31676],[34,66299],{"className":66300,"style":40292},[31680],[34,66302,459],{"className":66303},[31685],[34,66305,31960],{"className":66306},[32119],[34,66308],{"className":66309,"style":32123},[31845],[34,66311,31759],{"className":66312},[31685,31686],[34,66314],{"className":66315,"style":31846},[31845],[34,66317,29622],{"className":66318},[31850],[34,66320],{"className":66321,"style":31846},[31845],[34,66323,66325,66328],{"className":66324},[31676],[34,66326],{"className":66327,"style":32617},[31680],[34,66329,459],{"className":66330},[31685],"），",[34,66333,66335,66361],{"className":66334},[31642],[34,66336,66338],{"className":66337},[31646],[31648,66339,66340],{"xmlns":31650},[31652,66341,66342,66358],{},[31655,66343,66344,66346,66348,66350,66352,66354,66356],{},[31658,66345,31739],{},[31662,66347,456],{"stretchy":702},[31658,66349,31660],{},[31662,66351,31960],{"separator":758},[31658,66353,31759],{},[31662,66355,2557],{"stretchy":702},[31662,66357,202],{},[31665,66359,66360],{"encoding":31667},"f(n, m)=",[34,66362,66364],{"className":66363,"ariaHidden":758},[31672],[34,66365,66367,66370,66373,66376,66379,66382,66385,66388,66391,66394],{"className":66366},[31676],[34,66368],{"className":66369,"style":31774},[31680],[34,66371,31739],{"className":66372,"style":31778},[31685,31686],[34,66374,456],{"className":66375},[31782],[34,66377,31660],{"className":66378},[31685,31686],[34,66380,31960],{"className":66381},[32119],[34,66383],{"className":66384,"style":32123},[31845],[34,66386,31759],{"className":66387},[31685,31686],[34,66389,2557],{"className":66390},[31690],[34,66392],{"className":66393,"style":31846},[31845],[34,66395,202],{"className":66396},[31850]," _____",[253,66399,58168],{"id":66400},"答案与深度解析-17",[15,66402,66403],{},[2627,66404,58173,66405],{},[34,66406,66408,66454],{"className":66407},[31642],[34,66409,66411],{"className":66410},[31646],[31648,66412,66413],{"xmlns":31650},[31652,66414,66415,66451],{},[31655,66416,66417,66419,66421,66423,66425,66427,66429,66431,66433,66435,66437,66439,66441,66443,66445,66447,66449],{},[31658,66418,31739],{},[31662,66420,456],{"stretchy":702},[31658,66422,31660],{},[31662,66424,31960],{"separator":758},[31658,66426,31759],{},[31662,66428,32226],{},[31749,66430,640],{},[31662,66432,2557],{"stretchy":702},[31662,66434,223],{},[31658,66436,31739],{},[31662,66438,456],{"stretchy":702},[31658,66440,31660],{},[31662,66442,32226],{},[31658,66444,31759],{},[31662,66446,31960],{"separator":758},[31658,66448,31759],{},[31662,66450,2557],{"stretchy":702},[31665,66452,66453],{"encoding":31667},"f(n, m-1) + f(n-m, m)",[34,66455,66457,66490,66511,66535],{"className":66456,"ariaHidden":758},[31672],[34,66458,66460,66463,66466,66469,66472,66475,66478,66481,66484,66487],{"className":66459},[31676],[34,66461],{"className":66462,"style":31774},[31680],[34,66464,31739],{"className":66465,"style":31778},[31685,31686],[34,66467,456],{"className":66468},[31782],[34,66470,31660],{"className":66471},[31685,31686],[34,66473,31960],{"className":66474},[32119],[34,66476],{"className":66477,"style":32123},[31845],[34,66479,31759],{"className":66480},[31685,31686],[34,66482],{"className":66483,"style":32604},[31845],[34,66485,32226],{"className":66486},[32024],[34,66488],{"className":66489,"style":32604},[31845],[34,66491,66493,66496,66499,66502,66505,66508],{"className":66492},[31676],[34,66494],{"className":66495,"style":31774},[31680],[34,66497,640],{"className":66498},[31685],[34,66500,2557],{"className":66501},[31690],[34,66503],{"className":66504,"style":32604},[31845],[34,66506,223],{"className":66507},[32024],[34,66509],{"className":66510,"style":32604},[31845],[34,66512,66514,66517,66520,66523,66526,66529,66532],{"className":66513},[31676],[34,66515],{"className":66516,"style":31774},[31680],[34,66518,31739],{"className":66519,"style":31778},[31685,31686],[34,66521,456],{"className":66522},[31782],[34,66524,31660],{"className":66525},[31685,31686],[34,66527],{"className":66528,"style":32604},[31845],[34,66530,32226],{"className":66531},[32024],[34,66533],{"className":66534,"style":32604},[31845],[34,66536,66538,66541,66544,66547,66550,66553],{"className":66537},[31676],[34,66539],{"className":66540,"style":31774},[31680],[34,66542,31759],{"className":66543},[31685,31686],[34,66545,31960],{"className":66546},[32119],[34,66548],{"className":66549,"style":32123},[31845],[34,66551,31759],{"className":66552},[31685,31686],[34,66554,2557],{"className":66555},[31690],[105,66557,66558],{},[108,66559,66560,66562,66563,66566,66567,66595,66596,66624,66625,66676,66677,33183,66680],{},[2627,66561,58199],{}," 这是一个非常经典的递归数学模型。所谓划分，比如数字 5 划分且最大数不超过 3，可以是 ",[31,66564,66565],{},"3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1","。\n当要划分数字 ",[34,66568,66570,66583],{"className":66569},[31642],[34,66571,66573],{"className":66572},[31646],[31648,66574,66575],{"xmlns":31650},[31652,66576,66577,66581],{},[31655,66578,66579],{},[31658,66580,31660],{},[31665,66582,31660],{"encoding":31667},[34,66584,66586],{"className":66585,"ariaHidden":758},[31672],[34,66587,66589,66592],{"className":66588},[31676],[34,66590],{"className":66591,"style":32527},[31680],[34,66593,31660],{"className":66594},[31685,31686],"，且允许使用的最大数字是 ",[34,66597,66599,66612],{"className":66598},[31642],[34,66600,66602],{"className":66601},[31646],[31648,66603,66604],{"xmlns":31650},[31652,66605,66606,66610],{},[31655,66607,66608],{},[31658,66609,31759],{},[31665,66611,31759],{"encoding":31667},[34,66613,66615],{"className":66614,"ariaHidden":758},[31672],[34,66616,66618,66621],{"className":66617},[31676],[34,66619],{"className":66620,"style":32527},[31680],[34,66622,31759],{"className":66623},[31685,31686]," 时（前提 ",[34,66626,66628,66646],{"className":66627},[31642],[34,66629,66631],{"className":66630},[31646],[31648,66632,66633],{"xmlns":31650},[31652,66634,66635,66643],{},[31655,66636,66637,66639,66641],{},[31658,66638,31660],{},[31662,66640,29622],{},[31658,66642,31759],{},[31665,66644,66645],{"encoding":31667},"n > m",[34,66647,66649,66667],{"className":66648,"ariaHidden":758},[31672],[34,66650,66652,66655,66658,66661,66664],{"className":66651},[31676],[34,66653],{"className":66654,"style":34812},[31680],[34,66656,31660],{"className":66657},[31685,31686],[34,66659],{"className":66660,"style":31846},[31845],[34,66662,29622],{"className":66663},[31850],[34,66665],{"className":66666,"style":31846},[31845],[34,66668,66670,66673],{"className":66669},[31676],[34,66671],{"className":66672,"style":32527},[31680],[34,66674,31759],{"className":66675},[31685,31686],"），我们可以把所有可能的情况强制分为",[2627,66678,66679],{},"互不重叠的两大类",[105,66681,66682,66874,67135],{},[108,66683,66684,66715,66716,66744,66745,66795,66796,347],{},[2627,66685,66686,66687,347],{},"类别一：我们划分的过程中，打死都不用数字 ",[34,66688,66690,66703],{"className":66689},[31642],[34,66691,66693],{"className":66692},[31646],[31648,66694,66695],{"xmlns":31650},[31652,66696,66697,66701],{},[31655,66698,66699],{},[31658,66700,31759],{},[31665,66702,31759],{"encoding":31667},[34,66704,66706],{"className":66705,"ariaHidden":758},[31672],[34,66707,66709,66712],{"className":66708},[31676],[34,66710],{"className":66711,"style":32527},[31680],[34,66713,31759],{"className":66714},[31685,31686]," 既然不用 ",[34,66717,66719,66732],{"className":66718},[31642],[34,66720,66722],{"className":66721},[31646],[31648,66723,66724],{"xmlns":31650},[31652,66725,66726,66730],{},[31655,66727,66728],{},[31658,66729,31759],{},[31665,66731,31759],{"encoding":31667},[34,66733,66735],{"className":66734,"ariaHidden":758},[31672],[34,66736,66738,66741],{"className":66737},[31676],[34,66739],{"className":66740,"style":32527},[31680],[34,66742,31759],{"className":66743},[31685,31686],"，那我们能用的最大数字就降级成了 ",[34,66746,66748,66765],{"className":66747},[31642],[34,66749,66751],{"className":66750},[31646],[31648,66752,66753],{"xmlns":31650},[31652,66754,66755,66763],{},[31655,66756,66757,66759,66761],{},[31658,66758,31759],{},[31662,66760,32226],{},[31749,66762,640],{},[31665,66764,36321],{"encoding":31667},[34,66766,66768,66786],{"className":66767,"ariaHidden":758},[31672],[34,66769,66771,66774,66777,66780,66783],{"className":66770},[31676],[34,66772],{"className":66773,"style":32597},[31680],[34,66775,31759],{"className":66776},[31685,31686],[34,66778],{"className":66779,"style":32604},[31845],[34,66781,32226],{"className":66782},[32024],[34,66784],{"className":66785,"style":32604},[31845],[34,66787,66789,66792],{"className":66788},[31676],[34,66790],{"className":66791,"style":32617},[31680],[34,66793,640],{"className":66794},[31685],"。这种情况下，划分方式的数目等价于 ",[34,66797,66799,66826],{"className":66798},[31642],[34,66800,66802],{"className":66801},[31646],[31648,66803,66804],{"xmlns":31650},[31652,66805,66806,66824],{},[31655,66807,66808,66810,66812,66814,66816,66818,66820,66822],{},[31658,66809,31739],{},[31662,66811,456],{"stretchy":702},[31658,66813,31660],{},[31662,66815,31960],{"separator":758},[31658,66817,31759],{},[31662,66819,32226],{},[31749,66821,640],{},[31662,66823,2557],{"stretchy":702},[31665,66825,36382],{"encoding":31667},[34,66827,66829,66862],{"className":66828,"ariaHidden":758},[31672],[34,66830,66832,66835,66838,66841,66844,66847,66850,66853,66856,66859],{"className":66831},[31676],[34,66833],{"className":66834,"style":31774},[31680],[34,66836,31739],{"className":66837,"style":31778},[31685,31686],[34,66839,456],{"className":66840},[31782],[34,66842,31660],{"className":66843},[31685,31686],[34,66845,31960],{"className":66846},[32119],[34,66848],{"className":66849,"style":32123},[31845],[34,66851,31759],{"className":66852},[31685,31686],[34,66854],{"className":66855,"style":32604},[31845],[34,66857,32226],{"className":66858},[32024],[34,66860],{"className":66861,"style":32604},[31845],[34,66863,66865,66868,66871],{"className":66864},[31676],[34,66866],{"className":66867,"style":31774},[31680],[34,66869,640],{"className":66870},[31685],[34,66872,2557],{"className":66873},[31690],[108,66875,66876,66907,66908,66936,66937,66965,66966,67027,67028,67056,67057,347],{},[2627,66877,66878,66879,347],{},"类别二：我们划分的过程中，至少使用了一个数字 ",[34,66880,66882,66895],{"className":66881},[31642],[34,66883,66885],{"className":66884},[31646],[31648,66886,66887],{"xmlns":31650},[31652,66888,66889,66893],{},[31655,66890,66891],{},[31658,66892,31759],{},[31665,66894,31759],{"encoding":31667},[34,66896,66898],{"className":66897,"ariaHidden":758},[31672],[34,66899,66901,66904],{"className":66900},[31676],[34,66902],{"className":66903,"style":32527},[31680],[34,66905,31759],{"className":66906},[31685,31686]," 既然确定至少包含一个 ",[34,66909,66911,66924],{"className":66910},[31642],[34,66912,66914],{"className":66913},[31646],[31648,66915,66916],{"xmlns":31650},[31652,66917,66918,66922],{},[31655,66919,66920],{},[31658,66921,31759],{},[31665,66923,31759],{"encoding":31667},[34,66925,66927],{"className":66926,"ariaHidden":758},[31672],[34,66928,66930,66933],{"className":66929},[31676],[34,66931],{"className":66932,"style":32527},[31680],[34,66934,31759],{"className":66935},[31685,31686],"，那么把这个 ",[34,66938,66940,66953],{"className":66939},[31642],[34,66941,66943],{"className":66942},[31646],[31648,66944,66945],{"xmlns":31650},[31652,66946,66947,66951],{},[31655,66948,66949],{},[31658,66950,31759],{},[31665,66952,31759],{"encoding":31667},[34,66954,66956],{"className":66955,"ariaHidden":758},[31672],[34,66957,66959,66962],{"className":66958},[31676],[34,66960],{"className":66961,"style":32527},[31680],[34,66963,31759],{"className":66964},[31685,31686]," 先剥离出来，我们要面临的新问题就是如何把剩下的总数 ",[34,66967,66969,66991],{"className":66968},[31642],[34,66970,66972],{"className":66971},[31646],[31648,66973,66974],{"xmlns":31650},[31652,66975,66976,66988],{},[31655,66977,66978,66980,66982,66984,66986],{},[31662,66979,456],{"stretchy":702},[31658,66981,31660],{},[31662,66983,32226],{},[31658,66985,31759],{},[31662,66987,2557],{"stretchy":702},[31665,66989,66990],{"encoding":31667},"(n-m)",[34,66992,66994,67015],{"className":66993,"ariaHidden":758},[31672],[34,66995,66997,67000,67003,67006,67009,67012],{"className":66996},[31676],[34,66998],{"className":66999,"style":31774},[31680],[34,67001,456],{"className":67002},[31782],[34,67004,31660],{"className":67005},[31685,31686],[34,67007],{"className":67008,"style":32604},[31845],[34,67010,32226],{"className":67011},[32024],[34,67013],{"className":67014,"style":32604},[31845],[34,67016,67018,67021,67024],{"className":67017},[31676],[34,67019],{"className":67020,"style":31774},[31680],[34,67022,31759],{"className":67023},[31685,31686],[34,67025,2557],{"className":67026},[31690]," 继续进行划分。此时允许使用的最大数字依然是 ",[34,67029,67031,67044],{"className":67030},[31642],[34,67032,67034],{"className":67033},[31646],[31648,67035,67036],{"xmlns":31650},[31652,67037,67038,67042],{},[31655,67039,67040],{},[31658,67041,31759],{},[31665,67043,31759],{"encoding":31667},[34,67045,67047],{"className":67046,"ariaHidden":758},[31672],[34,67048,67050,67053],{"className":67049},[31676],[34,67051],{"className":67052,"style":32527},[31680],[34,67054,31759],{"className":67055},[31685,31686],"。这种情况下，划分数目等价于 ",[34,67058,67060,67087],{"className":67059},[31642],[34,67061,67063],{"className":67062},[31646],[31648,67064,67065],{"xmlns":31650},[31652,67066,67067,67085],{},[31655,67068,67069,67071,67073,67075,67077,67079,67081,67083],{},[31658,67070,31739],{},[31662,67072,456],{"stretchy":702},[31658,67074,31660],{},[31662,67076,32226],{},[31658,67078,31759],{},[31662,67080,31960],{"separator":758},[31658,67082,31759],{},[31662,67084,2557],{"stretchy":702},[31665,67086,36219],{"encoding":31667},[34,67088,67090,67114],{"className":67089,"ariaHidden":758},[31672],[34,67091,67093,67096,67099,67102,67105,67108,67111],{"className":67092},[31676],[34,67094],{"className":67095,"style":31774},[31680],[34,67097,31739],{"className":67098,"style":31778},[31685,31686],[34,67100,456],{"className":67101},[31782],[34,67103,31660],{"className":67104},[31685,31686],[34,67106],{"className":67107,"style":32604},[31845],[34,67109,32226],{"className":67110},[32024],[34,67112],{"className":67113,"style":32604},[31845],[34,67115,67117,67120,67123,67126,67129,67132],{"className":67116},[31676],[34,67118],{"className":67119,"style":31774},[31680],[34,67121,31759],{"className":67122},[31685,31686],[34,67124,31960],{"className":67125},[32119],[34,67127],{"className":67128,"style":32123},[31845],[34,67130,31759],{"className":67131},[31685,31686],[34,67133,2557],{"className":67134},[31690],[108,67136,67137,67138,347],{},"由于这两类覆盖了所有情况且互斥，运用加法原理，总数就是 ",[34,67139,67141,67186],{"className":67140},[31642],[34,67142,67144],{"className":67143},[31646],[31648,67145,67146],{"xmlns":31650},[31652,67147,67148,67184],{},[31655,67149,67150,67152,67154,67156,67158,67160,67162,67164,67166,67168,67170,67172,67174,67176,67178,67180,67182],{},[31658,67151,31739],{},[31662,67153,456],{"stretchy":702},[31658,67155,31660],{},[31662,67157,31960],{"separator":758},[31658,67159,31759],{},[31662,67161,32226],{},[31749,67163,640],{},[31662,67165,2557],{"stretchy":702},[31662,67167,223],{},[31658,67169,31739],{},[31662,67171,456],{"stretchy":702},[31658,67173,31660],{},[31662,67175,32226],{},[31658,67177,31759],{},[31662,67179,31960],{"separator":758},[31658,67181,31759],{},[31662,67183,2557],{"stretchy":702},[31665,67185,66453],{"encoding":31667},[34,67187,67189,67222,67243,67267],{"className":67188,"ariaHidden":758},[31672],[34,67190,67192,67195,67198,67201,67204,67207,67210,67213,67216,67219],{"className":67191},[31676],[34,67193],{"className":67194,"style":31774},[31680],[34,67196,31739],{"className":67197,"style":31778},[31685,31686],[34,67199,456],{"className":67200},[31782],[34,67202,31660],{"className":67203},[31685,31686],[34,67205,31960],{"className":67206},[32119],[34,67208],{"className":67209,"style":32123},[31845],[34,67211,31759],{"className":67212},[31685,31686],[34,67214],{"className":67215,"style":32604},[31845],[34,67217,32226],{"className":67218},[32024],[34,67220],{"className":67221,"style":32604},[31845],[34,67223,67225,67228,67231,67234,67237,67240],{"className":67224},[31676],[34,67226],{"className":67227,"style":31774},[31680],[34,67229,640],{"className":67230},[31685],[34,67232,2557],{"className":67233},[31690],[34,67235],{"className":67236,"style":32604},[31845],[34,67238,223],{"className":67239},[32024],[34,67241],{"className":67242,"style":32604},[31845],[34,67244,67246,67249,67252,67255,67258,67261,67264],{"className":67245},[31676],[34,67247],{"className":67248,"style":31774},[31680],[34,67250,31739],{"className":67251,"style":31778},[31685,31686],[34,67253,456],{"className":67254},[31782],[34,67256,31660],{"className":67257},[31685,31686],[34,67259],{"className":67260,"style":32604},[31845],[34,67262,32226],{"className":67263},[32024],[34,67265],{"className":67266,"style":32604},[31845],[34,67268,67270,67273,67276,67279,67282,67285],{"className":67269},[31676],[34,67271],{"className":67272,"style":31774},[31680],[34,67274,31759],{"className":67275},[31685,31686],[34,67277,31960],{"className":67278},[32119],[34,67280],{"className":67281,"style":32123},[31845],[34,67283,31759],{"className":67284},[31685,31686],[34,67286,2557],{"className":67287},[31690],[32188,67289],{},[253,67291,63606],{"id":67292},"_5-题目内容-1",[15,67294,67295,67296,67359,67360,67423,67424,67452,67453,67483,67484,67688,67689,67452,67717,67483,67746,67951,67952,68077,68078,68150,68151,68309,68310,66397],{},"利用动态规划法求解最长公共子序列问题时，设计动态规划数组 ",[34,67297,67299,67326],{"className":67298},[31642],[34,67300,67302],{"className":67301},[31646],[31648,67303,67304],{"xmlns":31650},[31652,67305,67306,67324],{},[31655,67307,67308,67310,67312,67314,67316,67318,67320,67322],{},[31658,67309,46900],{},[31658,67311,15],{},[31662,67313,15972],{"stretchy":702},[31658,67315,31759],{},[31662,67317,32712],{"stretchy":702},[31662,67319,15972],{"stretchy":702},[31658,67321,31660],{},[31662,67323,32712],{"stretchy":702},[31665,67325,54855],{"encoding":31667},[34,67327,67329],{"className":67328,"ariaHidden":758},[31672],[34,67330,67332,67335,67338,67341,67344,67347,67350,67353,67356],{"className":67331},[31676],[34,67333],{"className":67334,"style":31774},[31680],[34,67336,46900],{"className":67337},[31685,31686],[34,67339,15],{"className":67340},[31685,31686],[34,67342,15972],{"className":67343},[31782],[34,67345,31759],{"className":67346},[31685,31686],[34,67348,32712],{"className":67349},[31690],[34,67351,15972],{"className":67352},[31782],[34,67354,31660],{"className":67355},[31685,31686],[34,67357,32712],{"className":67358},[31690],"。其中 ",[34,67361,67363,67390],{"className":67362},[31642],[34,67364,67366],{"className":67365},[31646],[31648,67367,67368],{"xmlns":31650},[31652,67369,67370,67388],{},[31655,67371,67372,67374,67376,67378,67380,67382,67384,67386],{},[31658,67373,46900],{},[31658,67375,15],{},[31662,67377,15972],{"stretchy":702},[31658,67379,31955],{},[31662,67381,32712],{"stretchy":702},[31662,67383,15972],{"stretchy":702},[31658,67385,31968],{},[31662,67387,32712],{"stretchy":702},[31665,67389,54135],{"encoding":31667},[34,67391,67393],{"className":67392,"ariaHidden":758},[31672],[34,67394,67396,67399,67402,67405,67408,67411,67414,67417,67420],{"className":67395},[31676],[34,67397],{"className":67398,"style":31774},[31680],[34,67400,46900],{"className":67401},[31685,31686],[34,67403,15],{"className":67404},[31685,31686],[34,67406,15972],{"className":67407},[31782],[34,67409,31955],{"className":67410},[31685,31686],[34,67412,32712],{"className":67413},[31690],[34,67415,15972],{"className":67416},[31782],[34,67418,31968],{"className":67419,"style":32153},[31685,31686],[34,67421,32712],{"className":67422},[31690]," 为：长度为 ",[34,67425,67427,67440],{"className":67426},[31642],[34,67428,67430],{"className":67429},[31646],[31648,67431,67432],{"xmlns":31650},[31652,67433,67434,67438],{},[31655,67435,67436],{},[31658,67437,31955],{},[31665,67439,31955],{"encoding":31667},[34,67441,67443],{"className":67442,"ariaHidden":758},[31672],[34,67444,67446,67449],{"className":67445},[31676],[34,67447],{"className":67448,"style":33104},[31680],[34,67450,31955],{"className":67451},[31685,31686]," 的 ",[34,67454,67456,67470],{"className":67455},[31642],[34,67457,67459],{"className":67458},[31646],[31648,67460,67461],{"xmlns":31650},[31652,67462,67463,67468],{},[31655,67464,67465],{},[31658,67466,67467],{},"X",[31665,67469,67467],{"encoding":31667},[34,67471,67473],{"className":67472,"ariaHidden":758},[31672],[34,67474,67476,67479],{"className":67475},[31676],[34,67477],{"className":67478,"style":33218},[31680],[34,67480,67467],{"className":67481,"style":67482},[31685,31686],"margin-right:0.0785em;"," 子序列 ",[34,67485,67487,67529],{"className":67486},[31642],[34,67488,67490],{"className":67489},[31646],[31648,67491,67492],{"xmlns":31650},[31652,67493,67494,67526],{},[31655,67495,67496,67498,67504,67506,67512,67514,67516,67518,67524],{},[31662,67497,456],{"stretchy":702},[31743,67499,67500,67502],{},[31658,67501,32936],{},[31749,67503,459],{},[31662,67505,31960],{"separator":758},[31743,67507,67508,67510],{},[31658,67509,32936],{},[31749,67511,640],{},[31662,67513,31960],{"separator":758},[31662,67515,33934],{},[31662,67517,31960],{"separator":758},[31743,67519,67520,67522],{},[31658,67521,32936],{},[31658,67523,31955],{},[31662,67525,2557],{"stretchy":702},[31665,67527,67528],{"encoding":31667},"(x_0, x_1, \\dots, x_i)",[34,67530,67532],{"className":67531,"ariaHidden":758},[31672],[34,67533,67535,67538,67541,67581,67584,67587,67627,67630,67633,67636,67639,67642,67645,67685],{"className":67534},[31676],[34,67536],{"className":67537,"style":31774},[31680],[34,67539,456],{"className":67540},[31782],[34,67542,67544,67547],{"className":67543},[31685],[34,67545,32936],{"className":67546},[31685,31686],[34,67548,67550],{"className":67549},[31792],[34,67551,67553,67573],{"className":67552},[31796,31797],[34,67554,67556,67570],{"className":67555},[31801],[34,67557,67559],{"className":67558,"style":31806},[31805],[34,67560,67561,67564],{"style":31809},[34,67562],{"className":67563,"style":31814},[31813],[34,67565,67567],{"className":67566},[31818,31819,31820,31821],[34,67568,459],{"className":67569},[31685,31821],[34,67571,31829],{"className":67572},[31828],[34,67574,67576],{"className":67575},[31801],[34,67577,67579],{"className":67578,"style":31836},[31805],[34,67580],{},[34,67582,31960],{"className":67583},[32119],[34,67585],{"className":67586,"style":32123},[31845],[34,67588,67590,67593],{"className":67589},[31685],[34,67591,32936],{"className":67592},[31685,31686],[34,67594,67596],{"className":67595},[31792],[34,67597,67599,67619],{"className":67598},[31796,31797],[34,67600,67602,67616],{"className":67601},[31801],[34,67603,67605],{"className":67604,"style":31806},[31805],[34,67606,67607,67610],{"style":31809},[34,67608],{"className":67609,"style":31814},[31813],[34,67611,67613],{"className":67612},[31818,31819,31820,31821],[34,67614,640],{"className":67615},[31685,31821],[34,67617,31829],{"className":67618},[31828],[34,67620,67622],{"className":67621},[31801],[34,67623,67625],{"className":67624,"style":31836},[31805],[34,67626],{},[34,67628,31960],{"className":67629},[32119],[34,67631],{"className":67632,"style":32123},[31845],[34,67634,33934],{"className":67635},[34006],[34,67637],{"className":67638,"style":32123},[31845],[34,67640,31960],{"className":67641},[32119],[34,67643],{"className":67644,"style":32123},[31845],[34,67646,67648,67651],{"className":67647},[31685],[34,67649,32936],{"className":67650},[31685,31686],[34,67652,67654],{"className":67653},[31792],[34,67655,67657,67677],{"className":67656},[31796,31797],[34,67658,67660,67674],{"className":67659},[31801],[34,67661,67663],{"className":67662,"style":32090},[31805],[34,67664,67665,67668],{"style":31809},[34,67666],{"className":67667,"style":31814},[31813],[34,67669,67671],{"className":67670},[31818,31819,31820,31821],[34,67672,31955],{"className":67673},[31685,31686,31821],[34,67675,31829],{"className":67676},[31828],[34,67678,67680],{"className":67679},[31801],[34,67681,67683],{"className":67682,"style":31836},[31805],[34,67684],{},[34,67686,2557],{"className":67687},[31690]," 和长度为 ",[34,67690,67692,67705],{"className":67691},[31642],[34,67693,67695],{"className":67694},[31646],[31648,67696,67697],{"xmlns":31650},[31652,67698,67699,67703],{},[31655,67700,67701],{},[31658,67702,31968],{},[31665,67704,31968],{"encoding":31667},[34,67706,67708],{"className":67707,"ariaHidden":758},[31672],[34,67709,67711,67714],{"className":67710},[31676],[34,67712],{"className":67713,"style":33123},[31680],[34,67715,31968],{"className":67716,"style":32153},[31685,31686],[34,67718,67720,67734],{"className":67719},[31642],[34,67721,67723],{"className":67722},[31646],[31648,67724,67725],{"xmlns":31650},[31652,67726,67727,67732],{},[31655,67728,67729],{},[31658,67730,67731],{},"Y",[31665,67733,67731],{"encoding":31667},[34,67735,67737],{"className":67736,"ariaHidden":758},[31672],[34,67738,67740,67743],{"className":67739},[31676],[34,67741],{"className":67742,"style":33218},[31680],[34,67744,67731],{"className":67745,"style":32604},[31685,31686],[34,67747,67749,67792],{"className":67748},[31642],[34,67750,67752],{"className":67751},[31646],[31648,67753,67754],{"xmlns":31650},[31652,67755,67756,67789],{},[31655,67757,67758,67760,67767,67769,67775,67777,67779,67781,67787],{},[31662,67759,456],{"stretchy":702},[31743,67761,67762,67765],{},[31658,67763,67764],{},"y",[31749,67766,459],{},[31662,67768,31960],{"separator":758},[31743,67770,67771,67773],{},[31658,67772,67764],{},[31749,67774,640],{},[31662,67776,31960],{"separator":758},[31662,67778,33934],{},[31662,67780,31960],{"separator":758},[31743,67782,67783,67785],{},[31658,67784,67764],{},[31658,67786,31968],{},[31662,67788,2557],{"stretchy":702},[31665,67790,67791],{"encoding":31667},"(y_0, y_1, \\dots, y_j)",[34,67793,67795],{"className":67794,"ariaHidden":758},[31672],[34,67796,67798,67801,67804,67844,67847,67850,67890,67893,67896,67899,67902,67905,67908,67948],{"className":67797},[31676],[34,67799],{"className":67800,"style":32058},[31680],[34,67802,456],{"className":67803},[31782],[34,67805,67807,67810],{"className":67806},[31685],[34,67808,67764],{"className":67809,"style":32062},[31685,31686],[34,67811,67813],{"className":67812},[31792],[34,67814,67816,67836],{"className":67815},[31796,31797],[34,67817,67819,67833],{"className":67818},[31801],[34,67820,67822],{"className":67821,"style":31806},[31805],[34,67823,67824,67827],{"style":38795},[34,67825],{"className":67826,"style":31814},[31813],[34,67828,67830],{"className":67829},[31818,31819,31820,31821],[34,67831,459],{"className":67832},[31685,31821],[34,67834,31829],{"className":67835},[31828],[34,67837,67839],{"className":67838},[31801],[34,67840,67842],{"className":67841,"style":31836},[31805],[34,67843],{},[34,67845,31960],{"className":67846},[32119],[34,67848],{"className":67849,"style":32123},[31845],[34,67851,67853,67856],{"className":67852},[31685],[34,67854,67764],{"className":67855,"style":32062},[31685,31686],[34,67857,67859],{"className":67858},[31792],[34,67860,67862,67882],{"className":67861},[31796,31797],[34,67863,67865,67879],{"className":67864},[31801],[34,67866,67868],{"className":67867,"style":31806},[31805],[34,67869,67870,67873],{"style":38795},[34,67871],{"className":67872,"style":31814},[31813],[34,67874,67876],{"className":67875},[31818,31819,31820,31821],[34,67877,640],{"className":67878},[31685,31821],[34,67880,31829],{"className":67881},[31828],[34,67883,67885],{"className":67884},[31801],[34,67886,67888],{"className":67887,"style":31836},[31805],[34,67889],{},[34,67891,31960],{"className":67892},[32119],[34,67894],{"className":67895,"style":32123},[31845],[34,67897,33934],{"className":67898},[34006],[34,67900],{"className":67901,"style":32123},[31845],[34,67903,31960],{"className":67904},[32119],[34,67906],{"className":67907,"style":32123},[31845],[34,67909,67911,67914],{"className":67910},[31685],[34,67912,67764],{"className":67913,"style":32062},[31685,31686],[34,67915,67917],{"className":67916},[31792],[34,67918,67920,67940],{"className":67919},[31796,31797],[34,67921,67923,67937],{"className":67922},[31801],[34,67924,67926],{"className":67925,"style":32090},[31805],[34,67927,67928,67931],{"style":38795},[34,67929],{"className":67930,"style":31814},[31813],[34,67932,67934],{"className":67933},[31818,31819,31820,31821],[34,67935,31968],{"className":67936,"style":32153},[31685,31686,31821],[34,67938,31829],{"className":67939},[31828],[34,67941,67943],{"className":67942},[31801],[34,67944,67946],{"className":67945,"style":32163},[31805],[34,67947],{},[34,67949,2557],{"className":67950},[31690]," 的最长公共子序列的长度。若 ",[34,67953,67955,67993],{"className":67954},[31642],[34,67956,67958],{"className":67957},[31646],[31648,67959,67960],{"xmlns":31650},[31652,67961,67962,67990],{},[31655,67963,67964,67966,67968,67970,67972,67974,67976,67978,67980,67982,67984,67986,67988],{},[31658,67965,32936],{},[31662,67967,15972],{"stretchy":702},[31658,67969,31955],{},[31662,67971,32226],{},[31749,67973,640],{},[31662,67975,32712],{"stretchy":702},[31662,67977,202],{},[31658,67979,67764],{},[31662,67981,15972],{"stretchy":702},[31658,67983,31968],{},[31662,67985,32226],{},[31749,67987,640],{},[31662,67989,32712],{"stretchy":702},[31665,67991,67992],{"encoding":31667},"x[i-1] = y[j-1]",[34,67994,67996,68020,68041,68065],{"className":67995,"ariaHidden":758},[31672],[34,67997,67999,68002,68005,68008,68011,68014,68017],{"className":67998},[31676],[34,68000],{"className":68001,"style":31774},[31680],[34,68003,32936],{"className":68004},[31685,31686],[34,68006,15972],{"className":68007},[31782],[34,68009,31955],{"className":68010},[31685,31686],[34,68012],{"className":68013,"style":32604},[31845],[34,68015,32226],{"className":68016},[32024],[34,68018],{"className":68019,"style":32604},[31845],[34,68021,68023,68026,68029,68032,68035,68038],{"className":68022},[31676],[34,68024],{"className":68025,"style":31774},[31680],[34,68027,640],{"className":68028},[31685],[34,68030,32712],{"className":68031},[31690],[34,68033],{"className":68034,"style":31846},[31845],[34,68036,202],{"className":68037},[31850],[34,68039],{"className":68040,"style":31846},[31845],[34,68042,68044,68047,68050,68053,68056,68059,68062],{"className":68043},[31676],[34,68045],{"className":68046,"style":31774},[31680],[34,68048,67764],{"className":68049,"style":32062},[31685,31686],[34,68051,15972],{"className":68052},[31782],[34,68054,31968],{"className":68055,"style":32153},[31685,31686],[34,68057],{"className":68058,"style":32604},[31845],[34,68060,32226],{"className":68061},[32024],[34,68063],{"className":68064,"style":32604},[31845],[34,68066,68068,68071,68074],{"className":68067},[31676],[34,68069],{"className":68070,"style":31774},[31680],[34,68072,640],{"className":68073},[31685],[34,68075,32712],{"className":68076},[31690],"（当前两个字符相同）则 ",[34,68079,68081,68111],{"className":68080},[31642],[34,68082,68084],{"className":68083},[31646],[31648,68085,68086],{"xmlns":31650},[31652,68087,68088,68108],{},[31655,68089,68090,68092,68094,68096,68098,68100,68102,68104,68106],{},[31658,68091,46900],{},[31658,68093,15],{},[31662,68095,15972],{"stretchy":702},[31658,68097,31955],{},[31662,68099,32712],{"stretchy":702},[31662,68101,15972],{"stretchy":702},[31658,68103,31968],{},[31662,68105,32712],{"stretchy":702},[31662,68107,202],{},[31665,68109,68110],{"encoding":31667},"dp[i][j]=",[34,68112,68114],{"className":68113,"ariaHidden":758},[31672],[34,68115,68117,68120,68123,68126,68129,68132,68135,68138,68141,68144,68147],{"className":68116},[31676],[34,68118],{"className":68119,"style":31774},[31680],[34,68121,46900],{"className":68122},[31685,31686],[34,68124,15],{"className":68125},[31685,31686],[34,68127,15972],{"className":68128},[31782],[34,68130,31955],{"className":68131},[31685,31686],[34,68133,32712],{"className":68134},[31690],[34,68136,15972],{"className":68137},[31782],[34,68139,31968],{"className":68140,"style":32153},[31685,31686],[34,68142,32712],{"className":68143},[31690],[34,68145],{"className":68146,"style":31846},[31845],[34,68148,202],{"className":68149},[31850]," _____ ；若 ",[34,68152,68154,68192],{"className":68153},[31642],[34,68155,68157],{"className":68156},[31646],[31648,68158,68159],{"xmlns":31650},[31652,68160,68161,68189],{},[31655,68162,68163,68165,68167,68169,68171,68173,68175,68177,68179,68181,68183,68185,68187],{},[31658,68164,32936],{},[31662,68166,15972],{"stretchy":702},[31658,68168,31955],{},[31662,68170,32226],{},[31749,68172,640],{},[31662,68174,32712],{"stretchy":702},[31662,68176,54707],{"mathvariant":38079},[31658,68178,67764],{},[31662,68180,15972],{"stretchy":702},[31658,68182,31968],{},[31662,68184,32226],{},[31749,68186,640],{},[31662,68188,32712],{"stretchy":702},[31665,68190,68191],{"encoding":31667},"x[i-1] \\neq y[j-1]",[34,68193,68195,68219,68273,68297],{"className":68194,"ariaHidden":758},[31672],[34,68196,68198,68201,68204,68207,68210,68213,68216],{"className":68197},[31676],[34,68199],{"className":68200,"style":31774},[31680],[34,68202,32936],{"className":68203},[31685,31686],[34,68205,15972],{"className":68206},[31782],[34,68208,31955],{"className":68209},[31685,31686],[34,68211],{"className":68212,"style":32604},[31845],[34,68214,32226],{"className":68215},[32024],[34,68217],{"className":68218,"style":32604},[31845],[34,68220,68222,68225,68228,68231,68234,68270],{"className":68221},[31676],[34,68223],{"className":68224,"style":31774},[31680],[34,68226,640],{"className":68227},[31685],[34,68229,32712],{"className":68230},[31690],[34,68232],{"className":68233,"style":31846},[31845],[34,68235,68237,68264,68267],{"className":68236},[31850],[34,68238,68240],{"className":68239},[31850],[34,68241,68243],{"className":68242},[31685,54774],[34,68244,68246],{"className":68245},[54778],[34,68247,68249,68252,68261],{"className":68248},[54782],[34,68250],{"className":68251,"style":43217},[31680],[34,68253,68255],{"className":68254},[54789],[34,68256,68258],{"className":68257},[31685],[34,68259,54796],{"className":68260},[31850],[34,68262],{"className":68263},[54800],[34,68265],{"className":68266},[31845,54804],[34,68268,202],{"className":68269},[31850],[34,68271],{"className":68272,"style":31846},[31845],[34,68274,68276,68279,68282,68285,68288,68291,68294],{"className":68275},[31676],[34,68277],{"className":68278,"style":31774},[31680],[34,68280,67764],{"className":68281,"style":32062},[31685,31686],[34,68283,15972],{"className":68284},[31782],[34,68286,31968],{"className":68287,"style":32153},[31685,31686],[34,68289],{"className":68290,"style":32604},[31845],[34,68292,32226],{"className":68293},[32024],[34,68295],{"className":68296,"style":32604},[31845],[34,68298,68300,68303,68306],{"className":68299},[31676],[34,68301],{"className":68302,"style":31774},[31680],[34,68304,640],{"className":68305},[31685],[34,68307,32712],{"className":68308},[31690]," 则 ",[34,68311,68313,68342],{"className":68312},[31642],[34,68314,68316],{"className":68315},[31646],[31648,68317,68318],{"xmlns":31650},[31652,68319,68320,68340],{},[31655,68321,68322,68324,68326,68328,68330,68332,68334,68336,68338],{},[31658,68323,46900],{},[31658,68325,15],{},[31662,68327,15972],{"stretchy":702},[31658,68329,31955],{},[31662,68331,32712],{"stretchy":702},[31662,68333,15972],{"stretchy":702},[31658,68335,31968],{},[31662,68337,32712],{"stretchy":702},[31662,68339,202],{},[31665,68341,68110],{"encoding":31667},[34,68343,68345],{"className":68344,"ariaHidden":758},[31672],[34,68346,68348,68351,68354,68357,68360,68363,68366,68369,68372,68375,68378],{"className":68347},[31676],[34,68349],{"className":68350,"style":31774},[31680],[34,68352,46900],{"className":68353},[31685,31686],[34,68355,15],{"className":68356},[31685,31686],[34,68358,15972],{"className":68359},[31782],[34,68361,31955],{"className":68362},[31685,31686],[34,68364,32712],{"className":68365},[31690],[34,68367,15972],{"className":68368},[31782],[34,68370,31968],{"className":68371,"style":32153},[31685,31686],[34,68373,32712],{"className":68374},[31690],[34,68376],{"className":68377,"style":31846},[31845],[34,68379,202],{"className":68380},[31850],[253,68382,58168],{"id":68383},"答案与深度解析-18",[15,68385,68386,65425,68388,65429,68392],{},[2627,68387,58173],{},[2627,68389,68390],{},[31,68391,55214],{},[2627,68393,68394],{},[31,68395,68396],{},"max(dp[i-1][j], dp[i][j-1])",[105,68398,68399],{},[108,68400,68401,68403,68404],{},[2627,68402,58199],{}," 动态规划的核心在于推导“状态转移方程”。我们用白话进行逻辑推演：\n",[105,68405,68406,68414,68554],{},[108,68407,68408,68410,68411,31901],{},[31,68409,54135],{}," 就像是两个指针，分别指着字符串 X 的第 i 个字符，和字符串 Y 的第 j 个字符（注意题目给的暗示：C语言数组下标通常从0开始，所以第 i 个字符对应 ",[31,68412,68413],{},"x[i-1]",[108,68415,68416,68547,68548,68551,68552,347],{},[2627,68417,68418,68419,68546],{},"情景一：两边的末尾字符竟然一模一样（",[34,68420,68422,68462],{"className":68421},[31642],[34,68423,68425],{"className":68424},[31646],[31648,68426,68427],{"xmlns":31650},[31652,68428,68429,68459],{},[31655,68430,68431,68433,68435,68437,68439,68441,68443,68445,68447,68449,68451,68453,68455,68457],{},[31658,68432,32936],{},[31662,68434,15972],{"stretchy":702},[31658,68436,31955],{},[31662,68438,32226],{},[31749,68440,640],{},[31662,68442,32712],{"stretchy":702},[31662,68444,202],{},[31662,68446,202],{},[31658,68448,67764],{},[31662,68450,15972],{"stretchy":702},[31658,68452,31968],{},[31662,68454,32226],{},[31749,68456,640],{},[31662,68458,32712],{"stretchy":702},[31665,68460,68461],{"encoding":31667},"x[i-1] == y[j-1]",[34,68463,68465,68489,68510,68534],{"className":68464,"ariaHidden":758},[31672],[34,68466,68468,68471,68474,68477,68480,68483,68486],{"className":68467},[31676],[34,68469],{"className":68470,"style":31774},[31680],[34,68472,32936],{"className":68473},[31685,31686],[34,68475,15972],{"className":68476},[31782],[34,68478,31955],{"className":68479},[31685,31686],[34,68481],{"className":68482,"style":32604},[31845],[34,68484,32226],{"className":68485},[32024],[34,68487],{"className":68488,"style":32604},[31845],[34,68490,68492,68495,68498,68501,68504,68507],{"className":68491},[31676],[34,68493],{"className":68494,"style":31774},[31680],[34,68496,640],{"className":68497},[31685],[34,68499,32712],{"className":68500},[31690],[34,68502],{"className":68503,"style":31846},[31845],[34,68505,33376],{"className":68506},[31850],[34,68508],{"className":68509,"style":31846},[31845],[34,68511,68513,68516,68519,68522,68525,68528,68531],{"className":68512},[31676],[34,68514],{"className":68515,"style":31774},[31680],[34,68517,67764],{"className":68518,"style":32062},[31685,31686],[34,68520,15972],{"className":68521},[31782],[34,68523,31968],{"className":68524,"style":32153},[31685,31686],[34,68526],{"className":68527,"style":32604},[31845],[34,68529,32226],{"className":68530},[32024],[34,68532],{"className":68533,"style":32604},[31845],[34,68535,68537,68540,68543],{"className":68536},[31676],[34,68538],{"className":68539,"style":31774},[31680],[34,68541,640],{"className":68542},[31685],[34,68544,32712],{"className":68545},[31690],"）！"," 这简直是天上掉馅饼，这两个字符必定能成为公共子序列的最后一个字符。那么现在的长度，等于把这两个字符都扔掉后，前面剩余部分的公共长度再加上这1个字符。前面剩余部分的长度就是 ",[31,68549,68550],{},"dp[i-1][j-1]","。所以是 ",[31,68553,55214],{},[108,68555,68556,68716,68717],{},[2627,68557,68558,68559,31901],{},"情景二：两边的末尾字符不一样（",[34,68560,68562,68599],{"className":68561},[31642],[34,68563,68565],{"className":68564},[31646],[31648,68566,68567],{"xmlns":31650},[31652,68568,68569,68597],{},[31655,68570,68571,68573,68575,68577,68579,68581,68583,68585,68587,68589,68591,68593,68595],{},[31658,68572,32936],{},[31662,68574,15972],{"stretchy":702},[31658,68576,31955],{},[31662,68578,32226],{},[31749,68580,640],{},[31662,68582,32712],{"stretchy":702},[31662,68584,54707],{"mathvariant":38079},[31658,68586,67764],{},[31662,68588,15972],{"stretchy":702},[31658,68590,31968],{},[31662,68592,32226],{},[31749,68594,640],{},[31662,68596,32712],{"stretchy":702},[31665,68598,68191],{"encoding":31667},[34,68600,68602,68626,68680,68704],{"className":68601,"ariaHidden":758},[31672],[34,68603,68605,68608,68611,68614,68617,68620,68623],{"className":68604},[31676],[34,68606],{"className":68607,"style":31774},[31680],[34,68609,32936],{"className":68610},[31685,31686],[34,68612,15972],{"className":68613},[31782],[34,68615,31955],{"className":68616},[31685,31686],[34,68618],{"className":68619,"style":32604},[31845],[34,68621,32226],{"className":68622},[32024],[34,68624],{"className":68625,"style":32604},[31845],[34,68627,68629,68632,68635,68638,68641,68677],{"className":68628},[31676],[34,68630],{"className":68631,"style":31774},[31680],[34,68633,640],{"className":68634},[31685],[34,68636,32712],{"className":68637},[31690],[34,68639],{"className":68640,"style":31846},[31845],[34,68642,68644,68671,68674],{"className":68643},[31850],[34,68645,68647],{"className":68646},[31850],[34,68648,68650],{"className":68649},[31685,54774],[34,68651,68653],{"className":68652},[54778],[34,68654,68656,68659,68668],{"className":68655},[54782],[34,68657],{"className":68658,"style":43217},[31680],[34,68660,68662],{"className":68661},[54789],[34,68663,68665],{"className":68664},[31685],[34,68666,54796],{"className":68667},[31850],[34,68669],{"className":68670},[54800],[34,68672],{"className":68673},[31845,54804],[34,68675,202],{"className":68676},[31850],[34,68678],{"className":68679,"style":31846},[31845],[34,68681,68683,68686,68689,68692,68695,68698,68701],{"className":68682},[31676],[34,68684],{"className":68685,"style":31774},[31680],[34,68687,67764],{"className":68688,"style":32062},[31685,31686],[34,68690,15972],{"className":68691},[31782],[34,68693,31968],{"className":68694,"style":32153},[31685,31686],[34,68696],{"className":68697,"style":32604},[31845],[34,68699,32226],{"className":68700},[32024],[34,68702],{"className":68703,"style":32604},[31845],[34,68705,68707,68710,68713],{"className":68706},[31676],[34,68708],{"className":68709,"style":31774},[31680],[34,68711,640],{"className":68712},[31685],[34,68714,32712],{"className":68715},[31690]," 既然不一样，这两个字符绝对不可能同时出现在最长公共子序列的末尾。我们只能做取舍：\n",[105,68718,68719,68725],{},[108,68720,68721,68722,347],{},"要么抛弃 X 的末尾字符，看 X 的前 i-1 个字符和 Y 的前 j 个字符能凑出多长：",[31,68723,68724],{},"dp[i-1][j]",[108,68726,68727,68728,68731,68732,347],{},"要么抛弃 Y 的末尾字符，看 X 的前 i 个字符和 Y 的前 j-1 个字符能凑出多长：",[31,68729,68730],{},"dp[i][j-1]","。\n既然要求“最长”，我们就取这两种取舍方案中的较大值：",[31,68733,68396],{},[32188,68735],{},[19,68737,68739],{"id":68738},"四分析与构造题每题10分共30分","四、分析与构造题（每题10分，共30分）",[253,68741,58162],{"id":68742},"_1-题目内容-3",[15,68744,68745],{},"根据如下算法，回答问题。",[24,68747,68749],{"className":65734,"code":68748,"language":31965,"meta":29,"style":29},"void fun(int a[], int i, int n) \u002F\u002F 首次调用时: i为1\n{\n int j, k;\n if (i == n) return;\n else\n {\n k = i;\n for (j = i + 1; j \u003C= n; j++)\n if (a[j] \u003C a[k]) k = j;\n if (k != i)\n swap(a[i], a[k]);\n fun(a, i + 1, n);\n }\n}\n",[31,68750,68751,68756,68760,68765,68770,68774,68779,68784,68789,68794,68799,68804,68809,68813],{"__ignoreMap":29},[34,68752,68753],{"class":36,"line":37},[34,68754,68755],{},"void fun(int a[], int i, int n) \u002F\u002F 首次调用时: i为1\n",[34,68757,68758],{"class":36,"line":43},[34,68759,20357],{},[34,68761,68762],{"class":36,"line":49},[34,68763,68764],{}," int j, k;\n",[34,68766,68767],{"class":36,"line":56},[34,68768,68769],{}," if (i == n) return;\n",[34,68771,68772],{"class":36,"line":62},[34,68773,65766],{},[34,68775,68776],{"class":36,"line":68},[34,68777,68778],{}," {\n",[34,68780,68781],{"class":36,"line":74},[34,68782,68783],{}," k = i;\n",[34,68785,68786],{"class":36,"line":1971},[34,68787,68788],{}," for (j = i + 1; j \u003C= n; j++)\n",[34,68790,68791],{"class":36,"line":1976},[34,68792,68793],{}," if (a[j] \u003C a[k]) k = j;\n",[34,68795,68796],{"class":36,"line":1985},[34,68797,68798],{}," if (k != i)\n",[34,68800,68801],{"class":36,"line":1990},[34,68802,68803],{}," swap(a[i], a[k]);\n",[34,68805,68806],{"class":36,"line":1999},[34,68807,68808],{}," fun(a, i + 1, n);\n",[34,68810,68811],{"class":36,"line":2022},[34,68812,5540],{},[34,68814,68815],{"class":36,"line":2036},[34,68816,77],{},[2230,68818,68819,68883],{},[108,68820,43439,68821,68849,68850,68853,68854,68882],{},[34,68822,68824,68837],{"className":68823},[31642],[34,68825,68827],{"className":68826},[31646],[31648,68828,68829],{"xmlns":31650},[31652,68830,68831,68835],{},[31655,68832,68833],{},[31658,68834,111],{},[31665,68836,111],{"encoding":31667},[34,68838,68840],{"className":68839,"ariaHidden":758},[31672],[34,68841,68843,68846],{"className":68842},[31676],[34,68844],{"className":68845,"style":32527},[31680],[34,68847,111],{"className":68848},[31685,31686]," 数组中元素依次为 4, 9, 1, 3, 5, 6，分析并写出调用 ",[31,68851,68852],{},"fun(a,1,6)"," 后 ",[34,68855,68857,68870],{"className":68856},[31642],[34,68858,68860],{"className":68859},[31646],[31648,68861,68862],{"xmlns":31650},[31652,68863,68864,68868],{},[31655,68865,68866],{},[31658,68867,111],{},[31665,68869,111],{"encoding":31667},[34,68871,68873],{"className":68872,"ariaHidden":758},[31672],[34,68874,68876,68879],{"className":68875},[31676],[34,68877],{"className":68878,"style":32527},[31680],[34,68880,111],{"className":68881},[31685,31686]," 数组中元素值，并回答该算法完成的功能。（5分）",[108,68884,68885],{},"给出该算法时间复杂度递归方程（仅针对元素比较次数分析即可）并分析该算法的最坏时间复杂度。（5分）",[253,68887,58168],{"id":68888},"答案与深度解析-19",[15,68890,68891],{},[2627,68892,68893],{},"1) 数组变化及功能分析：",[105,68895,68896,69094,69102],{},[108,68897,68898,68901,68902,68904,68905,53834,68907,68909,68910,68913,68914],{},[2627,68899,68900],{},"模拟推理（从零推演）：","\n题目明确 ",[31,68903,111],{}," 的元素是 4, 9, 1, 3, 5, 6。从代码的界限 ",[31,68906,31955],{},[31,68908,31660],{},"（从 1 到 6），说明该题目将数组视为从索引 1 开始存数据的（即 ",[31,68911,68912],{},"a[1]=4, a[2]=9, ... a[6]=6","）。\n",[105,68915,68916,68959,68994,69022,69052,69081],{},[108,68917,68918,47649,68924,68927,68928,68931,68932,68935,68936,68939,68940,258,68943,68946,68947],{},[2627,68919,68920,68921,2664],{},"第一次调用 ",[31,68922,68923],{},"fun(a, 1, 6)",[31,68925,68926],{},"i=1","。内层循环从 ",[31,68929,68930],{},"j=2"," 到 6，寻找其中最小的元素。找到了 ",[31,68933,68934],{},"a[3]=1"," 是最小的，因此记录下标 ",[31,68937,68938],{},"k=3","。将 ",[31,68941,68942],{},"a[1]",[31,68944,68945],{},"a[3]"," 交换。\n",[105,68948,68949],{},[108,68950,68951,68952,68955,68956,347],{},"数组变为：",[2627,68953,68954],{},"1, 9, 4, 3, 5, 6","。随后递归调用 ",[31,68957,68958],{},"fun(a, 2, 6)",[108,68960,68961,47649,68966,68969,68970,68973,68974,68939,68977,258,68980,68946,68983],{},[2627,68962,68963,68964,2664],{},"第二次调用 ",[31,68965,68958],{},[31,68967,68968],{},"i=2","。此时数组有效区域是索引 2 到 6。寻找这部分最小元素。发现 ",[31,68971,68972],{},"a[4]=3"," 最小，记录 ",[31,68975,68976],{},"k=4",[31,68978,68979],{},"a[2]",[31,68981,68982],{},"a[4]",[105,68984,68985],{},[108,68986,68951,68987,68990,68991,347],{},[2627,68988,68989],{},"1, 3, 4, 9, 5, 6","。递归调用 ",[31,68992,68993],{},"fun(a, 3, 6)",[108,68995,68996,47649,69001,69004,69005,69008,69009,69011,69012],{},[2627,68997,68998,68999,2664],{},"第三次调用 ",[31,69000,68993],{},[31,69002,69003],{},"i=3","。寻找到 3..6 中最小的是 ",[31,69006,69007],{},"a[3]=4"," 本身，",[31,69010,68938],{},"，不交换。\n",[105,69013,69014],{},[108,69015,69016,69017,68990,69019,347],{},"数组保持：",[2627,69018,68989],{},[31,69020,69021],{},"fun(a, 4, 6)",[108,69023,69024,47649,69029,69032,69033,69036,69037,258,69039,39984,69042],{},[2627,69025,69026,69027,2664],{},"第四次调用 ",[31,69028,69021],{},[31,69030,69031],{},"i=4","。寻找 4..6 中最小元素，发现 ",[31,69034,69035],{},"a[5]=5"," 最小，交换 ",[31,69038,68982],{},[31,69040,69041],{},"a[5]",[105,69043,69044],{},[108,69045,68951,69046,68990,69049,347],{},[2627,69047,69048],{},"1, 3, 4, 5, 9, 6",[31,69050,69051],{},"fun(a, 5, 6)",[108,69053,69054,47649,69059,69062,69063,69036,69066,258,69068,39984,69071],{},[2627,69055,69056,69057,2664],{},"第五次调用 ",[31,69058,69051],{},[31,69060,69061],{},"i=5","。寻找 5..6 最小元素，发现 ",[31,69064,69065],{},"a[6]=6",[31,69067,69041],{},[31,69069,69070],{},"a[6]",[105,69072,69073],{},[108,69074,68951,69075,68990,69078,347],{},[2627,69076,69077],{},"1, 3, 4, 5, 6, 9",[31,69079,69080],{},"fun(a, 6, 6)",[108,69082,69083,47649,69088,69090,69091,69093],{},[2627,69084,69085,69086,2664],{},"第六次调用 ",[31,69087,69080],{},[31,69089,37233],{}," 满足界限条件，直接 ",[31,69092,1775],{},"，递归终止。",[108,69095,69096,69099,69100,347],{},[2627,69097,69098],{},"最终结果："," 经过算法处理，数组变为：",[2627,69101,69077],{},[108,69103,69104,69107,69108,347],{},[2627,69105,69106],{},"算法功能结论："," 这是一个递归实现的",[2627,69109,69110],{},"选择排序（从小到大排序）",[15,69112,69113],{},[2627,69114,69115],{},"2) 时间复杂度分析：",[105,69117,69118,69704,70571],{},[108,69119,69120,69123,69124,69167,69168,69196,69197,69225,69226],{},[2627,69121,69122],{},"建立递归方程："," 设 ",[34,69125,69127,69146],{"className":69126},[31642],[34,69128,69130],{"className":69129},[31646],[31648,69131,69132],{"xmlns":31650},[31652,69133,69134,69144],{},[31655,69135,69136,69138,69140,69142],{},[31658,69137,37400],{},[31662,69139,456],{"stretchy":702},[31658,69141,31660],{},[31662,69143,2557],{"stretchy":702},[31665,69145,37409],{"encoding":31667},[34,69147,69149],{"className":69148,"ariaHidden":758},[31672],[34,69150,69152,69155,69158,69161,69164],{"className":69151},[31676],[34,69153],{"className":69154,"style":31774},[31680],[34,69156,37400],{"className":69157,"style":37422},[31685,31686],[34,69159,456],{"className":69160},[31782],[34,69162,31660],{"className":69163},[31685,31686],[34,69165,2557],{"className":69166},[31690]," 为处理规模为 ",[34,69169,69171,69184],{"className":69170},[31642],[34,69172,69174],{"className":69173},[31646],[31648,69175,69176],{"xmlns":31650},[31652,69177,69178,69182],{},[31655,69179,69180],{},[31658,69181,31660],{},[31665,69183,31660],{"encoding":31667},[34,69185,69187],{"className":69186,"ariaHidden":758},[31672],[34,69188,69190,69193],{"className":69189},[31676],[34,69191],{"className":69192,"style":32527},[31680],[34,69194,31660],{"className":69195},[31685,31686]," 的数组（这里 ",[34,69198,69200,69213],{"className":69199},[31642],[34,69201,69203],{"className":69202},[31646],[31648,69204,69205],{"xmlns":31650},[31652,69206,69207,69211],{},[31655,69208,69209],{},[31658,69210,31660],{},[31665,69212,31660],{"encoding":31667},[34,69214,69216],{"className":69215,"ariaHidden":758},[31672],[34,69217,69219,69222],{"className":69218},[31676],[34,69220],{"className":69221,"style":32527},[31680],[34,69223,31660],{"className":69224},[31685,31686]," 代表待排序的剩余元素个数）所需要的比较次数。\n",[105,69227,69228,69301,69479],{},[108,69229,69230,69231,69234,69235,347],{},"当剩余 1 个元素时（即 ",[31,69232,69233],{},"i == n","），比较次数为 0，即 ",[34,69236,69238,69262],{"className":69237},[31642],[34,69239,69241],{"className":69240},[31646],[31648,69242,69243],{"xmlns":31650},[31652,69244,69245,69259],{},[31655,69246,69247,69249,69251,69253,69255,69257],{},[31658,69248,37400],{},[31662,69250,456],{"stretchy":702},[31749,69252,640],{},[31662,69254,2557],{"stretchy":702},[31662,69256,202],{},[31749,69258,459],{},[31665,69260,69261],{"encoding":31667},"T(1) = 0",[34,69263,69265,69292],{"className":69264,"ariaHidden":758},[31672],[34,69266,69268,69271,69274,69277,69280,69283,69286,69289],{"className":69267},[31676],[34,69269],{"className":69270,"style":31774},[31680],[34,69272,37400],{"className":69273,"style":37422},[31685,31686],[34,69275,456],{"className":69276},[31782],[34,69278,640],{"className":69279},[31685],[34,69281,2557],{"className":69282},[31690],[34,69284],{"className":69285,"style":31846},[31845],[34,69287,202],{"className":69288},[31850],[34,69290],{"className":69291,"style":31846},[31845],[34,69293,69295,69298],{"className":69294},[31676],[34,69296],{"className":69297,"style":32617},[31680],[34,69299,459],{"className":69300},[31685],[108,69302,69303,69304,69332,69333,69336,69337,69365,69366,69427,69428,69478],{},"对于规模为 ",[34,69305,69307,69320],{"className":69306},[31642],[34,69308,69310],{"className":69309},[31646],[31648,69311,69312],{"xmlns":31650},[31652,69313,69314,69318],{},[31655,69315,69316],{},[31658,69317,31660],{},[31665,69319,31660],{"encoding":31667},[34,69321,69323],{"className":69322,"ariaHidden":758},[31672],[34,69324,69326,69329],{"className":69325},[31676],[34,69327],{"className":69328,"style":32527},[31680],[34,69330,31660],{"className":69331},[31685,31686]," 的问题，我们要用 ",[31,69334,69335],{},"for"," 循环寻找当前 ",[34,69338,69340,69353],{"className":69339},[31642],[34,69341,69343],{"className":69342},[31646],[31648,69344,69345],{"xmlns":31650},[31652,69346,69347,69351],{},[31655,69348,69349],{},[31658,69350,31660],{},[31665,69352,31660],{"encoding":31667},[34,69354,69356],{"className":69355,"ariaHidden":758},[31672],[34,69357,69359,69362],{"className":69358},[31676],[34,69360],{"className":69361,"style":32527},[31680],[34,69363,31660],{"className":69364},[31685,31686]," 个元素中的最小值，需要进行 ",[34,69367,69369,69391],{"className":69368},[31642],[34,69370,69372],{"className":69371},[31646],[31648,69373,69374],{"xmlns":31650},[31652,69375,69376,69388],{},[31655,69377,69378,69380,69382,69384,69386],{},[31662,69379,456],{"stretchy":702},[31658,69381,31660],{},[31662,69383,32226],{},[31749,69385,640],{},[31662,69387,2557],{"stretchy":702},[31665,69389,69390],{"encoding":31667},"(n-1)",[34,69392,69394,69415],{"className":69393,"ariaHidden":758},[31672],[34,69395,69397,69400,69403,69406,69409,69412],{"className":69396},[31676],[34,69398],{"className":69399,"style":31774},[31680],[34,69401,456],{"className":69402},[31782],[34,69404,31660],{"className":69405},[31685,31686],[34,69407],{"className":69408,"style":32604},[31845],[34,69410,32226],{"className":69411},[32024],[34,69413],{"className":69414,"style":32604},[31845],[34,69416,69418,69421,69424],{"className":69417},[31676],[34,69419],{"className":69420,"style":31774},[31680],[34,69422,640],{"className":69423},[31685],[34,69425,2557],{"className":69426},[31690]," 次比较。然后针对剩下的 ",[34,69429,69431,69448],{"className":69430},[31642],[34,69432,69434],{"className":69433},[31646],[31648,69435,69436],{"xmlns":31650},[31652,69437,69438,69446],{},[31655,69439,69440,69442,69444],{},[31658,69441,31660],{},[31662,69443,32226],{},[31749,69445,640],{},[31665,69447,32587],{"encoding":31667},[34,69449,69451,69469],{"className":69450,"ariaHidden":758},[31672],[34,69452,69454,69457,69460,69463,69466],{"className":69453},[31676],[34,69455],{"className":69456,"style":32597},[31680],[34,69458,31660],{"className":69459},[31685,31686],[34,69461],{"className":69462,"style":32604},[31845],[34,69464,32226],{"className":69465},[32024],[34,69467],{"className":69468,"style":32604},[31845],[34,69470,69472,69475],{"className":69471},[31676],[34,69473],{"className":69474,"style":32617},[31680],[34,69476,640],{"className":69477},[31685]," 个元素进行递归。",[108,69480,69481,69482,69639],{},"由此得出严谨的递归方程：\n",[34,69483,69485,69531],{"className":69484},[31642],[34,69486,69488],{"className":69487},[31646],[31648,69489,69490],{"xmlns":31650},[31652,69491,69492,69528],{},[31655,69493,69494,69496,69498,69500,69502,69504,69506,69508,69510,69512,69514,69516,69518,69520,69522,69524,69526],{},[31658,69495,37400],{},[31662,69497,456],{"stretchy":702},[31658,69499,31660],{},[31662,69501,2557],{"stretchy":702},[31662,69503,202],{},[31658,69505,37400],{},[31662,69507,456],{"stretchy":702},[31658,69509,31660],{},[31662,69511,32226],{},[31749,69513,640],{},[31662,69515,2557],{"stretchy":702},[31662,69517,223],{},[31662,69519,456],{"stretchy":702},[31658,69521,31660],{},[31662,69523,32226],{},[31749,69525,640],{},[31662,69527,2557],{"stretchy":702},[31665,69529,69530],{"encoding":31667},"T(n) = T(n-1) + (n-1)",[34,69532,69534,69561,69585,69606,69627],{"className":69533,"ariaHidden":758},[31672],[34,69535,69537,69540,69543,69546,69549,69552,69555,69558],{"className":69536},[31676],[34,69538],{"className":69539,"style":31774},[31680],[34,69541,37400],{"className":69542,"style":37422},[31685,31686],[34,69544,456],{"className":69545},[31782],[34,69547,31660],{"className":69548},[31685,31686],[34,69550,2557],{"className":69551},[31690],[34,69553],{"className":69554,"style":31846},[31845],[34,69556,202],{"className":69557},[31850],[34,69559],{"className":69560,"style":31846},[31845],[34,69562,69564,69567,69570,69573,69576,69579,69582],{"className":69563},[31676],[34,69565],{"className":69566,"style":31774},[31680],[34,69568,37400],{"className":69569,"style":37422},[31685,31686],[34,69571,456],{"className":69572},[31782],[34,69574,31660],{"className":69575},[31685,31686],[34,69577],{"className":69578,"style":32604},[31845],[34,69580,32226],{"className":69581},[32024],[34,69583],{"className":69584,"style":32604},[31845],[34,69586,69588,69591,69594,69597,69600,69603],{"className":69587},[31676],[34,69589],{"className":69590,"style":31774},[31680],[34,69592,640],{"className":69593},[31685],[34,69595,2557],{"className":69596},[31690],[34,69598],{"className":69599,"style":32604},[31845],[34,69601,223],{"className":69602},[32024],[34,69604],{"className":69605,"style":32604},[31845],[34,69607,69609,69612,69615,69618,69621,69624],{"className":69608},[31676],[34,69610],{"className":69611,"style":31774},[31680],[34,69613,456],{"className":69614},[31782],[34,69616,31660],{"className":69617},[31685,31686],[34,69619],{"className":69620,"style":32604},[31845],[34,69622,32226],{"className":69623},[32024],[34,69625],{"className":69626,"style":32604},[31845],[34,69628,69630,69633,69636],{"className":69629},[31676],[34,69631],{"className":69632,"style":31774},[31680],[34,69634,640],{"className":69635},[31685],[34,69637,2557],{"className":69638},[31690],[34,69640,69642,69665],{"className":69641},[31642],[34,69643,69645],{"className":69644},[31646],[31648,69646,69647],{"xmlns":31650},[31652,69648,69649,69663],{},[31655,69650,69651,69653,69655,69657,69659,69661],{},[31658,69652,37400],{},[31662,69654,456],{"stretchy":702},[31749,69656,640],{},[31662,69658,2557],{"stretchy":702},[31662,69660,202],{},[31749,69662,459],{},[31665,69664,69261],{"encoding":31667},[34,69666,69668,69695],{"className":69667,"ariaHidden":758},[31672],[34,69669,69671,69674,69677,69680,69683,69686,69689,69692],{"className":69670},[31676],[34,69672],{"className":69673,"style":31774},[31680],[34,69675,37400],{"className":69676,"style":37422},[31685,31686],[34,69678,456],{"className":69679},[31782],[34,69681,640],{"className":69682},[31685],[34,69684,2557],{"className":69685},[31690],[34,69687],{"className":69688,"style":31846},[31845],[34,69690,202],{"className":69691},[31850],[34,69693],{"className":69694,"style":31846},[31845],[34,69696,69698,69701],{"className":69697},[31676],[34,69699],{"className":69700,"style":32617},[31680],[34,69702,459],{"className":69703},[31685],[108,69705,69706,69709,69710,69921,70186,70187,70442,70443,347],{},[2627,69707,69708],{},"求解与渐进复杂度：","\n不断展开这个方程：\n",[34,69711,69713,69771],{"className":69712},[31642],[34,69714,69716],{"className":69715},[31646],[31648,69717,69718],{"xmlns":31650},[31652,69719,69720,69768],{},[31655,69721,69722,69724,69726,69728,69730,69732,69734,69736,69738,69740,69742,69744,69746,69748,69750,69752,69754,69756,69758,69760,69762,69764,69766],{},[31658,69723,37400],{},[31662,69725,456],{"stretchy":702},[31658,69727,31660],{},[31662,69729,2557],{"stretchy":702},[31662,69731,202],{},[31658,69733,37400],{},[31662,69735,456],{"stretchy":702},[31658,69737,31660],{},[31662,69739,32226],{},[31749,69741,768],{},[31662,69743,2557],{"stretchy":702},[31662,69745,223],{},[31662,69747,456],{"stretchy":702},[31658,69749,31660],{},[31662,69751,32226],{},[31749,69753,768],{},[31662,69755,2557],{"stretchy":702},[31662,69757,223],{},[31662,69759,456],{"stretchy":702},[31658,69761,31660],{},[31662,69763,32226],{},[31749,69765,640],{},[31662,69767,2557],{"stretchy":702},[31665,69769,69770],{"encoding":31667},"T(n) = T(n-2) + (n-2) + (n-1)",[34,69772,69774,69801,69825,69846,69867,69888,69909],{"className":69773,"ariaHidden":758},[31672],[34,69775,69777,69780,69783,69786,69789,69792,69795,69798],{"className":69776},[31676],[34,69778],{"className":69779,"style":31774},[31680],[34,69781,37400],{"className":69782,"style":37422},[31685,31686],[34,69784,456],{"className":69785},[31782],[34,69787,31660],{"className":69788},[31685,31686],[34,69790,2557],{"className":69791},[31690],[34,69793],{"className":69794,"style":31846},[31845],[34,69796,202],{"className":69797},[31850],[34,69799],{"className":69800,"style":31846},[31845],[34,69802,69804,69807,69810,69813,69816,69819,69822],{"className":69803},[31676],[34,69805],{"className":69806,"style":31774},[31680],[34,69808,37400],{"className":69809,"style":37422},[31685,31686],[34,69811,456],{"className":69812},[31782],[34,69814,31660],{"className":69815},[31685,31686],[34,69817],{"className":69818,"style":32604},[31845],[34,69820,32226],{"className":69821},[32024],[34,69823],{"className":69824,"style":32604},[31845],[34,69826,69828,69831,69834,69837,69840,69843],{"className":69827},[31676],[34,69829],{"className":69830,"style":31774},[31680],[34,69832,768],{"className":69833},[31685],[34,69835,2557],{"className":69836},[31690],[34,69838],{"className":69839,"style":32604},[31845],[34,69841,223],{"className":69842},[32024],[34,69844],{"className":69845,"style":32604},[31845],[34,69847,69849,69852,69855,69858,69861,69864],{"className":69848},[31676],[34,69850],{"className":69851,"style":31774},[31680],[34,69853,456],{"className":69854},[31782],[34,69856,31660],{"className":69857},[31685,31686],[34,69859],{"className":69860,"style":32604},[31845],[34,69862,32226],{"className":69863},[32024],[34,69865],{"className":69866,"style":32604},[31845],[34,69868,69870,69873,69876,69879,69882,69885],{"className":69869},[31676],[34,69871],{"className":69872,"style":31774},[31680],[34,69874,768],{"className":69875},[31685],[34,69877,2557],{"className":69878},[31690],[34,69880],{"className":69881,"style":32604},[31845],[34,69883,223],{"className":69884},[32024],[34,69886],{"className":69887,"style":32604},[31845],[34,69889,69891,69894,69897,69900,69903,69906],{"className":69890},[31676],[34,69892],{"className":69893,"style":31774},[31680],[34,69895,456],{"className":69896},[31782],[34,69898,31660],{"className":69899},[31685,31686],[34,69901],{"className":69902,"style":32604},[31845],[34,69904,32226],{"className":69905},[32024],[34,69907],{"className":69908,"style":32604},[31845],[34,69910,69912,69915,69918],{"className":69911},[31676],[34,69913],{"className":69914,"style":31774},[31680],[34,69916,640],{"className":69917},[31685],[34,69919,2557],{"className":69920},[31690],[34,69922,69924,69994],{"className":69923},[31642],[34,69925,69927],{"className":69926},[31646],[31648,69928,69929],{"xmlns":31650},[31652,69930,69931,69991],{},[31655,69932,69933,69935,69937,69939,69941,69943,69945,69947,69949,69951,69953,69955,69957,69959,69961,69963,69965,69967,69969,69971,69973,69975,69977,69979,69981,69983,69985,69987,69989],{},[31658,69934,37400],{},[31662,69936,456],{"stretchy":702},[31658,69938,31660],{},[31662,69940,2557],{"stretchy":702},[31662,69942,202],{},[31658,69944,37400],{},[31662,69946,456],{"stretchy":702},[31658,69948,31660],{},[31662,69950,32226],{},[31749,69952,786],{},[31662,69954,2557],{"stretchy":702},[31662,69956,223],{},[31662,69958,456],{"stretchy":702},[31658,69960,31660],{},[31662,69962,32226],{},[31749,69964,786],{},[31662,69966,2557],{"stretchy":702},[31662,69968,223],{},[31662,69970,456],{"stretchy":702},[31658,69972,31660],{},[31662,69974,32226],{},[31749,69976,768],{},[31662,69978,2557],{"stretchy":702},[31662,69980,223],{},[31662,69982,456],{"stretchy":702},[31658,69984,31660],{},[31662,69986,32226],{},[31749,69988,640],{},[31662,69990,2557],{"stretchy":702},[31665,69992,69993],{"encoding":31667},"T(n) = T(n-3) + (n-3) + (n-2) + (n-1)",[34,69995,69997,70024,70048,70069,70090,70111,70132,70153,70174],{"className":69996,"ariaHidden":758},[31672],[34,69998,70000,70003,70006,70009,70012,70015,70018,70021],{"className":69999},[31676],[34,70001],{"className":70002,"style":31774},[31680],[34,70004,37400],{"className":70005,"style":37422},[31685,31686],[34,70007,456],{"className":70008},[31782],[34,70010,31660],{"className":70011},[31685,31686],[34,70013,2557],{"className":70014},[31690],[34,70016],{"className":70017,"style":31846},[31845],[34,70019,202],{"className":70020},[31850],[34,70022],{"className":70023,"style":31846},[31845],[34,70025,70027,70030,70033,70036,70039,70042,70045],{"className":70026},[31676],[34,70028],{"className":70029,"style":31774},[31680],[34,70031,37400],{"className":70032,"style":37422},[31685,31686],[34,70034,456],{"className":70035},[31782],[34,70037,31660],{"className":70038},[31685,31686],[34,70040],{"className":70041,"style":32604},[31845],[34,70043,32226],{"className":70044},[32024],[34,70046],{"className":70047,"style":32604},[31845],[34,70049,70051,70054,70057,70060,70063,70066],{"className":70050},[31676],[34,70052],{"className":70053,"style":31774},[31680],[34,70055,786],{"className":70056},[31685],[34,70058,2557],{"className":70059},[31690],[34,70061],{"className":70062,"style":32604},[31845],[34,70064,223],{"className":70065},[32024],[34,70067],{"className":70068,"style":32604},[31845],[34,70070,70072,70075,70078,70081,70084,70087],{"className":70071},[31676],[34,70073],{"className":70074,"style":31774},[31680],[34,70076,456],{"className":70077},[31782],[34,70079,31660],{"className":70080},[31685,31686],[34,70082],{"className":70083,"style":32604},[31845],[34,70085,32226],{"className":70086},[32024],[34,70088],{"className":70089,"style":32604},[31845],[34,70091,70093,70096,70099,70102,70105,70108],{"className":70092},[31676],[34,70094],{"className":70095,"style":31774},[31680],[34,70097,786],{"className":70098},[31685],[34,70100,2557],{"className":70101},[31690],[34,70103],{"className":70104,"style":32604},[31845],[34,70106,223],{"className":70107},[32024],[34,70109],{"className":70110,"style":32604},[31845],[34,70112,70114,70117,70120,70123,70126,70129],{"className":70113},[31676],[34,70115],{"className":70116,"style":31774},[31680],[34,70118,456],{"className":70119},[31782],[34,70121,31660],{"className":70122},[31685,31686],[34,70124],{"className":70125,"style":32604},[31845],[34,70127,32226],{"className":70128},[32024],[34,70130],{"className":70131,"style":32604},[31845],[34,70133,70135,70138,70141,70144,70147,70150],{"className":70134},[31676],[34,70136],{"className":70137,"style":31774},[31680],[34,70139,768],{"className":70140},[31685],[34,70142,2557],{"className":70143},[31690],[34,70145],{"className":70146,"style":32604},[31845],[34,70148,223],{"className":70149},[32024],[34,70151],{"className":70152,"style":32604},[31845],[34,70154,70156,70159,70162,70165,70168,70171],{"className":70155},[31676],[34,70157],{"className":70158,"style":31774},[31680],[34,70160,456],{"className":70161},[31782],[34,70163,31660],{"className":70164},[31685,31686],[34,70166],{"className":70167,"style":32604},[31845],[34,70169,32226],{"className":70170},[32024],[34,70172],{"className":70173,"style":32604},[31845],[34,70175,70177,70180,70183],{"className":70176},[31676],[34,70178],{"className":70179,"style":31774},[31680],[34,70181,640],{"className":70182},[31685],[34,70184,2557],{"className":70185},[31690],"\n...\n",[34,70188,70190,70256],{"className":70189},[31642],[34,70191,70193],{"className":70192},[31646],[31648,70194,70195],{"xmlns":31650},[31652,70196,70197,70253],{},[31655,70198,70199,70201,70203,70205,70207,70209,70211,70213,70215,70217,70219,70221,70223,70225,70227,70229,70231,70233,70235,70237,70239,70241,70243,70245,70247,70249,70251],{},[31658,70200,37400],{},[31662,70202,456],{"stretchy":702},[31658,70204,31660],{},[31662,70206,2557],{"stretchy":702},[31662,70208,202],{},[31658,70210,37400],{},[31662,70212,456],{"stretchy":702},[31749,70214,640],{},[31662,70216,2557],{"stretchy":702},[31662,70218,223],{},[31749,70220,640],{},[31662,70222,223],{},[31749,70224,768],{},[31662,70226,223],{},[31662,70228,38874],{},[31662,70230,223],{},[31662,70232,456],{"stretchy":702},[31658,70234,31660],{},[31662,70236,32226],{},[31749,70238,768],{},[31662,70240,2557],{"stretchy":702},[31662,70242,223],{},[31662,70244,456],{"stretchy":702},[31658,70246,31660],{},[31662,70248,32226],{},[31749,70250,640],{},[31662,70252,2557],{"stretchy":702},[31665,70254,70255],{"encoding":31667},"T(n) = T(1) + 1 + 2 + \\dots + (n-2) + (n-1)",[34,70257,70259,70286,70313,70331,70349,70367,70388,70409,70430],{"className":70258,"ariaHidden":758},[31672],[34,70260,70262,70265,70268,70271,70274,70277,70280,70283],{"className":70261},[31676],[34,70263],{"className":70264,"style":31774},[31680],[34,70266,37400],{"className":70267,"style":37422},[31685,31686],[34,70269,456],{"className":70270},[31782],[34,70272,31660],{"className":70273},[31685,31686],[34,70275,2557],{"className":70276},[31690],[34,70278],{"className":70279,"style":31846},[31845],[34,70281,202],{"className":70282},[31850],[34,70284],{"className":70285,"style":31846},[31845],[34,70287,70289,70292,70295,70298,70301,70304,70307,70310],{"className":70288},[31676],[34,70290],{"className":70291,"style":31774},[31680],[34,70293,37400],{"className":70294,"style":37422},[31685,31686],[34,70296,456],{"className":70297},[31782],[34,70299,640],{"className":70300},[31685],[34,70302,2557],{"className":70303},[31690],[34,70305],{"className":70306,"style":32604},[31845],[34,70308,223],{"className":70309},[32024],[34,70311],{"className":70312,"style":32604},[31845],[34,70314,70316,70319,70322,70325,70328],{"className":70315},[31676],[34,70317],{"className":70318,"style":35942},[31680],[34,70320,640],{"className":70321},[31685],[34,70323],{"className":70324,"style":32604},[31845],[34,70326,223],{"className":70327},[32024],[34,70329],{"className":70330,"style":32604},[31845],[34,70332,70334,70337,70340,70343,70346],{"className":70333},[31676],[34,70335],{"className":70336,"style":35942},[31680],[34,70338,768],{"className":70339},[31685],[34,70341],{"className":70342,"style":32604},[31845],[34,70344,223],{"className":70345},[32024],[34,70347],{"className":70348,"style":32604},[31845],[34,70350,70352,70355,70358,70361,70364],{"className":70351},[31676],[34,70353],{"className":70354,"style":32597},[31680],[34,70356,38874],{"className":70357},[34006],[34,70359],{"className":70360,"style":32604},[31845],[34,70362,223],{"className":70363},[32024],[34,70365],{"className":70366,"style":32604},[31845],[34,70368,70370,70373,70376,70379,70382,70385],{"className":70369},[31676],[34,70371],{"className":70372,"style":31774},[31680],[34,70374,456],{"className":70375},[31782],[34,70377,31660],{"className":70378},[31685,31686],[34,70380],{"className":70381,"style":32604},[31845],[34,70383,32226],{"className":70384},[32024],[34,70386],{"className":70387,"style":32604},[31845],[34,70389,70391,70394,70397,70400,70403,70406],{"className":70390},[31676],[34,70392],{"className":70393,"style":31774},[31680],[34,70395,768],{"className":70396},[31685],[34,70398,2557],{"className":70399},[31690],[34,70401],{"className":70402,"style":32604},[31845],[34,70404,223],{"className":70405},[32024],[34,70407],{"className":70408,"style":32604},[31845],[34,70410,70412,70415,70418,70421,70424,70427],{"className":70411},[31676],[34,70413],{"className":70414,"style":31774},[31680],[34,70416,456],{"className":70417},[31782],[34,70419,31660],{"className":70420},[31685,31686],[34,70422],{"className":70423,"style":32604},[31845],[34,70425,32226],{"className":70426},[32024],[34,70428],{"className":70429,"style":32604},[31845],[34,70431,70433,70436,70439],{"className":70432},[31676],[34,70434],{"className":70435,"style":31774},[31680],[34,70437,640],{"className":70438},[31685],[34,70440,2557],{"className":70441},[31690],"\n根据等差数列求和公式，总比较次数为 ",[34,70444,70446,70476],{"className":70445},[31642],[34,70447,70449],{"className":70448},[31646],[31648,70450,70451],{"xmlns":31650},[31652,70452,70453,70473],{},[31655,70454,70455],{},[41453,70456,70457,70471],{},[31655,70458,70459,70461,70463,70465,70467,70469],{},[31658,70460,31660],{},[31662,70462,456],{"stretchy":702},[31658,70464,31660],{},[31662,70466,32226],{},[31749,70468,640],{},[31662,70470,2557],{"stretchy":702},[31749,70472,768],{},[31665,70474,70475],{"encoding":31667},"\\frac{n(n-1)}{2}",[34,70477,70479],{"className":70478,"ariaHidden":758},[31672],[34,70480,70482,70486],{"className":70481},[31676],[34,70483],{"className":70484,"style":70485},[31680],"height:1.355em;vertical-align:-0.345em;",[34,70487,70489,70492,70568],{"className":70488},[31685],[34,70490],{"className":70491},[31782,41488],[34,70493,70495],{"className":70494},[41453],[34,70496,70498,70560],{"className":70497},[31796,31797],[34,70499,70501,70557],{"className":70500},[31801],[34,70502,70505,70519,70527],{"className":70503,"style":70504},[31805],"height:1.01em;",[34,70506,70507,70510],{"style":41504},[34,70508],{"className":70509,"style":41508},[31813],[34,70511,70513],{"className":70512},[31818,31819,31820,31821],[34,70514,70516],{"className":70515},[31685,31821],[34,70517,768],{"className":70518},[31685,31821],[34,70520,70521,70524],{"style":41520},[34,70522],{"className":70523,"style":41508},[31813],[34,70525],{"className":70526,"style":41528},[41527],[34,70528,70530,70533],{"style":70529},"top:-3.485em;",[34,70531],{"className":70532,"style":41508},[31813],[34,70534,70536],{"className":70535},[31818,31819,31820,31821],[34,70537,70539,70542,70545,70548,70551,70554],{"className":70538},[31685,31821],[34,70540,31660],{"className":70541},[31685,31686,31821],[34,70543,456],{"className":70544},[31782,31821],[34,70546,31660],{"className":70547},[31685,31686,31821],[34,70549,32226],{"className":70550},[32024,31821],[34,70552,640],{"className":70553},[31685,31821],[34,70555,2557],{"className":70556},[31690,31821],[34,70558,31829],{"className":70559},[31828],[34,70561,70563],{"className":70562},[31801],[34,70564,70566],{"className":70565,"style":41619},[31805],[34,70567],{},[34,70569],{"className":70570},[31690,41488],[108,70572,70573,70576,70577,347],{},[2627,70574,70575],{},"最坏时间复杂度："," 无论数组原本是否有序，这套寻找最小值的循环必须雷打不动地执行完。因此，最坏时间复杂度（实际上也是最好和平均情况）均为 ",[2627,70578,70579],{},[34,70580,70582,70605],{"className":70581},[31642],[34,70583,70585],{"className":70584},[31646],[31648,70586,70587],{"xmlns":31650},[31652,70588,70589,70603],{},[31655,70590,70591,70593,70595,70601],{},[31658,70592,37890],{},[31662,70594,456],{"stretchy":702},[37624,70596,70597,70599],{},[31658,70598,31660],{},[31749,70600,768],{},[31662,70602,2557],{"stretchy":702},[31665,70604,46174],{"encoding":31667},[34,70606,70608],{"className":70607,"ariaHidden":758},[31672],[34,70609,70611,70614,70617,70620,70649],{"className":70610},[31676],[34,70612],{"className":70613,"style":37755},[31680],[34,70615,37890],{"className":70616,"style":34452},[31685,31686],[34,70618,456],{"className":70619},[31782],[34,70621,70623,70626],{"className":70622},[31685],[34,70624,31660],{"className":70625},[31685,31686],[34,70627,70629],{"className":70628},[31792],[34,70630,70632],{"className":70631},[31796],[34,70633,70635],{"className":70634},[31801],[34,70636,70638],{"className":70637,"style":37774},[31805],[34,70639,70640,70643],{"style":37777},[34,70641],{"className":70642,"style":31814},[31813],[34,70644,70646],{"className":70645},[31818,31819,31820,31821],[34,70647,768],{"className":70648},[31685,31821],[34,70650,2557],{"className":70651},[31690],[32188,70653],{},[253,70655,58228],{"id":70656},"_2-题目内容-3",[15,70658,70659,70660,70688,70689,70884,70885,43298,70956,71042,71043,71806,71810,71811,2664],{},"给定 ",[34,70661,70663,70676],{"className":70662},[31642],[34,70664,70666],{"className":70665},[31646],[31648,70667,70668],{"xmlns":31650},[31652,70669,70670,70674],{},[31655,70671,70672],{},[31658,70673,31660],{},[31665,70675,31660],{"encoding":31667},[34,70677,70679],{"className":70678,"ariaHidden":758},[31672],[34,70680,70682,70685],{"className":70681},[31676],[34,70683],{"className":70684,"style":32527},[31680],[34,70686,31660],{"className":70687},[31685,31686]," 个矩阵 ",[34,70690,70692,70730],{"className":70691},[31642],[34,70693,70695],{"className":70694},[31646],[31648,70696,70697],{"xmlns":31650},[31652,70698,70699,70727],{},[31655,70700,70701,70707,70709,70715,70717,70719,70721],{},[31743,70702,70703,70705],{},[31658,70704,54151],{},[31749,70706,640],{},[31662,70708,31960],{"separator":758},[31743,70710,70711,70713],{},[31658,70712,54151],{},[31749,70714,768],{},[31662,70716,31960],{"separator":758},[31662,70718,33934],{},[31662,70720,31960],{"separator":758},[31743,70722,70723,70725],{},[31658,70724,54151],{},[31658,70726,31660],{},[31665,70728,70729],{"encoding":31667},"A_1, A_2, \\dots, A_n",[34,70731,70733],{"className":70732,"ariaHidden":758},[31672],[34,70734,70736,70740,70780,70783,70786,70826,70829,70832,70835,70838,70841,70844],{"className":70735},[31676],[34,70737],{"className":70738,"style":70739},[31680],"height:0.8778em;vertical-align:-0.1944em;",[34,70741,70743,70746],{"className":70742},[31685],[34,70744,54151],{"className":70745},[31685,31686],[34,70747,70749],{"className":70748},[31792],[34,70750,70752,70772],{"className":70751},[31796,31797],[34,70753,70755,70769],{"className":70754},[31801],[34,70756,70758],{"className":70757,"style":31806},[31805],[34,70759,70760,70763],{"style":31809},[34,70761],{"className":70762,"style":31814},[31813],[34,70764,70766],{"className":70765},[31818,31819,31820,31821],[34,70767,640],{"className":70768},[31685,31821],[34,70770,31829],{"className":70771},[31828],[34,70773,70775],{"className":70774},[31801],[34,70776,70778],{"className":70777,"style":31836},[31805],[34,70779],{},[34,70781,31960],{"className":70782},[32119],[34,70784],{"className":70785,"style":32123},[31845],[34,70787,70789,70792],{"className":70788},[31685],[34,70790,54151],{"className":70791},[31685,31686],[34,70793,70795],{"className":70794},[31792],[34,70796,70798,70818],{"className":70797},[31796,31797],[34,70799,70801,70815],{"className":70800},[31801],[34,70802,70804],{"className":70803,"style":31806},[31805],[34,70805,70806,70809],{"style":31809},[34,70807],{"className":70808,"style":31814},[31813],[34,70810,70812],{"className":70811},[31818,31819,31820,31821],[34,70813,768],{"className":70814},[31685,31821],[34,70816,31829],{"className":70817},[31828],[34,70819,70821],{"className":70820},[31801],[34,70822,70824],{"className":70823,"style":31836},[31805],[34,70825],{},[34,70827,31960],{"className":70828},[32119],[34,70830],{"className":70831,"style":32123},[31845],[34,70833,33934],{"className":70834},[34006],[34,70836],{"className":70837,"style":32123},[31845],[34,70839,31960],{"className":70840},[32119],[34,70842],{"className":70843,"style":32123},[31845],[34,70845,70847,70850],{"className":70846},[31685],[34,70848,54151],{"className":70849},[31685,31686],[34,70851,70853],{"className":70852},[31792],[34,70854,70856,70876],{"className":70855},[31796,31797],[34,70857,70859,70873],{"className":70858},[31801],[34,70860,70862],{"className":70861,"style":32362},[31805],[34,70863,70864,70867],{"style":31809},[34,70865],{"className":70866,"style":31814},[31813],[34,70868,70870],{"className":70869},[31818,31819,31820,31821],[34,70871,31660],{"className":70872},[31685,31686,31821],[34,70874,31829],{"className":70875},[31828],[34,70877,70879],{"className":70878},[31801],[34,70880,70882],{"className":70881,"style":31836},[31805],[34,70883],{},"，其中 ",[34,70886,70888,70906],{"className":70887},[31642],[34,70889,70891],{"className":70890},[31646],[31648,70892,70893],{"xmlns":31650},[31652,70894,70895,70903],{},[31655,70896,70897],{},[31743,70898,70899,70901],{},[31658,70900,54151],{},[31658,70902,31955],{},[31665,70904,70905],{"encoding":31667},"A_i",[34,70907,70909],{"className":70908,"ariaHidden":758},[31672],[34,70910,70912,70916],{"className":70911},[31676],[34,70913],{"className":70914,"style":70915},[31680],"height:0.8333em;vertical-align:-0.15em;",[34,70917,70919,70922],{"className":70918},[31685],[34,70920,54151],{"className":70921},[31685,31686],[34,70923,70925],{"className":70924},[31792],[34,70926,70928,70948],{"className":70927},[31796,31797],[34,70929,70931,70945],{"className":70930},[31801],[34,70932,70934],{"className":70933,"style":32090},[31805],[34,70935,70936,70939],{"style":31809},[34,70937],{"className":70938,"style":31814},[31813],[34,70940,70942],{"className":70941},[31818,31819,31820,31821],[34,70943,31955],{"className":70944},[31685,31686,31821],[34,70946,31829],{"className":70947},[31828],[34,70949,70951],{"className":70950},[31801],[34,70952,70954],{"className":70953,"style":31836},[31805],[34,70955],{},[34,70957,70959,70983],{"className":70958},[31642],[34,70960,70962],{"className":70961},[31646],[31648,70963,70964],{"xmlns":31650},[31652,70965,70966,70980],{},[31655,70967,70968],{},[31743,70969,70970,70972],{},[31658,70971,54151],{},[31655,70973,70974,70976,70978],{},[31658,70975,31955],{},[31662,70977,223],{},[31749,70979,640],{},[31665,70981,70982],{"encoding":31667},"A_{i+1}",[34,70984,70986],{"className":70985,"ariaHidden":758},[31672],[34,70987,70989,70993],{"className":70988},[31676],[34,70990],{"className":70991,"style":70992},[31680],"height:0.8917em;vertical-align:-0.2083em;",[34,70994,70996,70999],{"className":70995},[31685],[34,70997,54151],{"className":70998},[31685,31686],[34,71000,71002],{"className":71001},[31792],[34,71003,71005,71034],{"className":71004},[31796,31797],[34,71006,71008,71031],{"className":71007},[31801],[34,71009,71011],{"className":71010,"style":32090},[31805],[34,71012,71013,71016],{"style":31809},[34,71014],{"className":71015,"style":31814},[31813],[34,71017,71019],{"className":71018},[31818,31819,31820,31821],[34,71020,71022,71025,71028],{"className":71021},[31685,31821],[34,71023,31955],{"className":71024},[31685,31686,31821],[34,71026,223],{"className":71027},[32024,31821],[34,71029,640],{"className":71030},[31685,31821],[34,71032,31829],{"className":71033},[31828],[34,71035,71037],{"className":71036},[31801],[34,71038,71040],{"className":71039,"style":32037},[31805],[34,71041],{}," 保证是可乘的。不同的计算次序需要的乘法次数不同。用动态规划算法求解最小乘法次数，动态转移方程如下：\n",[34,71044,71046,71223],{"className":71045},[31642],[34,71047,71049],{"className":71048},[31646],[31648,71050,71051],{"xmlns":31650},[31652,71052,71053,71220],{},[31655,71054,71055,71057,71059,71061,71063,71065,71067,71069,71071,71073],{},[31658,71056,46900],{},[31658,71058,15],{},[31662,71060,15972],{"stretchy":702},[31658,71062,31955],{},[31662,71064,32712],{"stretchy":702},[31662,71066,15972],{"stretchy":702},[31658,71068,31968],{},[31662,71070,32712],{"stretchy":702},[31662,71072,202],{},[31655,71074,71075,71077],{},[31662,71076,32685],{"fence":758},[71078,71079,71083,71106],"mtable",{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},"0.36em","left left","1em",[71084,71085,71086,71094],"mtr",{},[71087,71088,71089],"mtd",{},[71090,71091,71092],"mstyle",{"scriptlevel":459,"displaystyle":702},[31749,71093,459],{},[71087,71095,71096],{},[71090,71097,71098],{"scriptlevel":459,"displaystyle":702},[31655,71099,71100,71102,71104],{},[31658,71101,31955],{},[31662,71103,202],{},[31658,71105,31968],{},[71084,71107,71108,71208],{},[71087,71109,71110],{},[71090,71111,71112],{"scriptlevel":459,"displaystyle":702},[71090,71113,71114,71135,71137,71139,71141,71143,71145,71147,71149,71151,71153,71155,71157,71159,71161,71163,71165,71167,71169,71171,71173,71175,71177,71189,71192,71198,71200,71206],{"scriptlevel":459,"displaystyle":758},[71115,71116,71117,71123],"munder",{},[31655,71118,71119,71121],{},[31658,71120,56084],{},[31662,71122,32682],{},[31655,71124,71125,71127,71129,71131,71133],{},[31658,71126,31955],{},[31662,71128,45972],{},[31658,71130,33759],{},[31662,71132,932],{},[31658,71134,31968],{},[31662,71136,32685],{"stretchy":702},[31658,71138,46900],{},[31658,71140,15],{},[31662,71142,15972],{"stretchy":702},[31658,71144,31955],{},[31662,71146,32712],{"stretchy":702},[31662,71148,15972],{"stretchy":702},[31658,71150,33759],{},[31662,71152,32712],{"stretchy":702},[31662,71154,223],{},[31658,71156,46900],{},[31658,71158,15],{},[31662,71160,15972],{"stretchy":702},[31658,71162,33759],{},[31662,71164,223],{},[31749,71166,640],{},[31662,71168,32712],{"stretchy":702},[31662,71170,15972],{"stretchy":702},[31658,71172,31968],{},[31662,71174,32712],{"stretchy":702},[31662,71176,223],{},[31743,71178,71179,71181],{},[31658,71180,15],{},[31655,71182,71183,71185,71187],{},[31658,71184,31955],{},[31662,71186,32226],{},[31749,71188,640],{},[31662,71190,71191],{},"⋅",[31743,71193,71194,71196],{},[31658,71195,15],{},[31658,71197,33759],{},[31662,71199,71191],{},[31743,71201,71202,71204],{},[31658,71203,15],{},[31658,71205,31968],{},[31662,71207,32715],{"stretchy":702},[71087,71209,71210],{},[71090,71211,71212],{"scriptlevel":459,"displaystyle":702},[31655,71213,71214,71216,71218],{},[31658,71215,31955],{},[31662,71217,932],{},[31658,71219,31968],{},[31665,71221,71222],{"encoding":31667},"dp[i][j] = \\begin{cases} 0 & i=j \\\\ \\displaystyle\\min_{i \\le k \u003C j}\\{dp[i][k]+dp[k+1][j]+ p_{i-1} \\cdot p_{k} \\cdot p_{j}\\} & i\u003Cj 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p_1=10, p_2=3, p_3=12, p_4=5, p_5=50, 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97],{"style":31809},[34,72195],{"className":72196,"style":31814},[31813],[34,72198,72200],{"className":72199},[31818,31819,31820,31821],[34,72201,15932],{"className":72202},[31685,31821],[34,72204,31829],{"className":72205},[31828],[34,72207,72209],{"className":72208},[31801],[34,72210,72212],{"className":72211,"style":31836},[31805],[34,72213],{},[34,72215],{"className":72216,"style":31846},[31845],[34,72218,202],{"className":72219},[31850],[34,72221],{"className":72222,"style":31846},[31845],[34,72224,72226,72229,72232,72235,72238,72278,72281,72284],{"className":72225},[31676],[34,72227],{"className":72228,"style":40292},[31680],[34,72230,18136],{"className":72231},[31685],[34,72233,31960],{"className":72234},[32119],[34,72236],{"className":72237,"style":32123},[31845],[34,72239,72241,72244],{"className":72240},[31685],[34,72242,15],{"className":72243},[31685,31686],[34,72245,72247],{"className":72246},[31792],[34,72248,72250,72270],{"className":72249},[31796,31797],[34,72251,72253,72267],{"className":72252},[31801],[34,72254,72256],{"className":72255,"style":31806},[31805],[34,72257,72258,72261],{"style":31809},[34,72259],{"className":72260,"style":31814},[31813],[34,72262,72264],{"className":72263},[31818,31819,31820,31821],[34,72265,18136],{"className":72266},[31685,31821],[34,72268,31829],{"className":72269},[31828],[34,72271,72273],{"className":72272},[31801],[34,72274,72276],{"className":72275,"style":31836},[31805],[34,72277],{},[34,72279],{"className":72280,"style":31846},[31845],[34,72282,202],{"className":72283},[31850],[34,72285],{"className":72286,"style":31846},[31845],[34,72288,72290,72293,72296,72299,72302,72342,72345,72348],{"className":72289},[31676],[34,72291],{"className":72292,"style":40292},[31680],[34,72294,5774],{"className":72295},[31685],[34,72297,31960],{"className":72298},[32119],[34,72300],{"className":72301,"style":32123},[31845],[34,72303,72305,72308],{"className":72304},[31685],[34,72306,15],{"className":72307},[31685,31686],[34,72309,72311],{"className":72310},[31792],[34,72312,72314,72334],{"className":72313},[31796,31797],[34,72315,72317,72331],{"className":72316},[31801],[34,72318,72320],{"className":72319,"style":31806},[31805],[34,72321,72322,72325],{"style":31809},[34,72323],{"className":72324,"style":31814},[31813],[34,72326,72328],{"className":72327},[31818,31819,31820,31821],[34,72329,36656],{"className":72330},[31685,31821],[34,72332,31829],{"className":72333},[31828],[34,72335,72337],{"className":72336},[31801],[34,72338,72340],{"className":72339,"style":31836},[31805],[34,72341],{},[34,72343],{"className":72344,"style":31846},[31845],[34,72346,202],{"className":72347},[31850],[34,72349],{"className":72350,"style":31846},[31845],[34,72352,72354,72357],{"className":72353},[31676],[34,72355],{"className":72356,"style":32617},[31680],[34,72358,36656],{"className":72359},[31685],[2230,72361,72362,72428],{},[108,72363,72364,72365,72398,72399,72427],{},"给出动态规划数组 ",[34,72366,72368,72383],{"className":72367},[31642],[34,72369,72371],{"className":72370},[31646],[31648,72372,72373],{"xmlns":31650},[31652,72374,72375,72381],{},[31655,72376,72377,72379],{},[31658,72378,46900],{},[31658,72380,15],{},[31665,72382,51611],{"encoding":31667},[34,72384,72386],{"className":72385,"ariaHidden":758},[31672],[34,72387,72389,72392,72395],{"className":72388},[31676],[34,72390],{"className":72391,"style":43217},[31680],[34,72393,46900],{"className":72394},[31685,31686],[34,72396,15],{"className":72397},[31685,31686]," 及矩阵连乘的断点数组 ",[34,72400,72402,72415],{"className":72401},[31642],[34,72403,72405],{"className":72404},[31646],[31648,72406,72407],{"xmlns":31650},[31652,72408,72409,72413],{},[31655,72410,72411],{},[31658,72412,31747],{},[31665,72414,31747],{"encoding":31667},[34,72416,72418],{"className":72417,"ariaHidden":758},[31672],[34,72419,72421,72424],{"className":72420},[31676],[34,72422],{"className":72423,"style":32527},[31680],[34,72425,31747],{"className":72426},[31685,31686],"。（8分）",[108,72429,72430],{},"写出最优的矩阵乘法运算顺序。（2分）",[253,72432,58168],{"id":72433},"答案与深度解析-20",[15,72435,72436,72437,72488,72489,72628,72629,72698,72699,347],{},"这道题是动态规划中经典的“矩阵连乘问题”，计算量大，我们需要严谨地列表步步为营。",[34,72438,72440,72458],{"className":72439},[31642],[34,72441,72443],{"className":72442},[31646],[31648,72444,72445],{"xmlns":31650},[31652,72446,72447,72455],{},[31655,72448,72449,72451,72453],{},[31658,72450,31660],{},[31662,72452,202],{},[31749,72454,36656],{},[31665,72456,72457],{"encoding":31667},"n=6",[34,72459,72461,72479],{"className":72460,"ariaHidden":758},[31672],[34,72462,72464,72467,72470,72473,72476],{"className":72463},[31676],[34,72465],{"className":72466,"style":32527},[31680],[34,72468,31660],{"className":72469},[31685,31686],[34,72471],{"className":72472,"style":31846},[31845],[34,72474,202],{"className":72475},[31850],[34,72477],{"className":72478,"style":31846},[31845],[34,72480,72482,72485],{"className":72481},[31676],[34,72483],{"className":72484,"style":32617},[31680],[34,72486,36656],{"className":72487},[31685],"，即 6 个矩阵连乘。维度向量 ",[34,72490,72492,72538],{"className":72491},[31642],[34,72493,72495],{"className":72494},[31646],[31648,72496,72497],{"xmlns":31650},[31652,72498,72499,72535],{},[31655,72500,72501,72503,72505,72507,72509,72511,72513,72515,72517,72519,72521,72523,72525,72527,72529,72531,72533],{},[31658,72502,15],{},[31662,72504,202],{},[31662,72506,15972],{"stretchy":702},[31749,72508,18136],{},[31662,72510,31960],{"separator":758},[31749,72512,3244],{},[31662,72514,31960],{"separator":758},[31749,72516,786],{},[31662,72518,31960],{"separator":758},[31749,72520,71869],{},[31662,72522,31960],{"separator":758},[31749,72524,18136],{},[31662,72526,31960],{"separator":758},[31749,72528,5774],{},[31662,72530,31960],{"separator":758},[31749,72532,36656],{},[31662,72534,32712],{"stretchy":702},[31665,72536,72537],{"encoding":31667},"p = [5, 10, 3, 12, 5, 50, 6]",[34,72539,72541,72559],{"className":72540,"ariaHidden":758},[31672],[34,72542,72544,72547,72550,72553,72556],{"className":72543},[31676],[34,72545],{"className":72546,"style":38774},[31680],[34,72548,15],{"className":72549},[31685,31686],[34,72551],{"className":72552,"style":31846},[31845],[34,72554,202],{"className":72555},[31850],[34,72557],{"className":72558,"style":31846},[31845],[34,72560,72562,72565,72568,72571,72574,72577,72580,72583,72586,72589,72592,72595,72598,72601,72604,72607,72610,72613,72616,72619,72622,72625],{"className":72561},[31676],[34,72563],{"className":72564,"style":31774},[31680],[34,72566,15972],{"className":72567},[31782],[34,72569,18136],{"className":72570},[31685],[34,72572,31960],{"className":72573},[32119],[34,72575],{"className":72576,"style":32123},[31845],[34,72578,3244],{"className":72579},[31685],[34,72581,31960],{"className":72582},[32119],[34,72584],{"className":72585,"style":32123},[31845],[34,72587,786],{"className":72588},[31685],[34,72590,31960],{"className":72591},[32119],[34,72593],{"className":72594,"style":32123},[31845],[34,72596,71869],{"className":72597},[31685],[34,72599,31960],{"className":72600},[32119],[34,72602],{"className":72603,"style":32123},[31845],[34,72605,18136],{"className":72606},[31685],[34,72608,31960],{"className":72609},[32119],[34,72611],{"className":72612,"style":32123},[31845],[34,72614,5774],{"className":72615},[31685],[34,72617,31960],{"className":72618},[32119],[34,72620],{"className":72621,"style":32123},[31845],[34,72623,36656],{"className":72624},[31685],[34,72626,32712],{"className":72627},[31690],"。矩阵 ",[34,72630,72632,72649],{"className":72631},[31642],[34,72633,72635],{"className":72634},[31646],[31648,72636,72637],{"xmlns":31650},[31652,72638,72639,72647],{},[31655,72640,72641],{},[31743,72642,72643,72645],{},[31658,72644,54151],{},[31658,72646,31955],{},[31665,72648,70905],{"encoding":31667},[34,72650,72652],{"className":72651,"ariaHidden":758},[31672],[34,72653,72655,72658],{"className":72654},[31676],[34,72656],{"className":72657,"style":70915},[31680],[34,72659,72661,72664],{"className":72660},[31685],[34,72662,54151],{"className":72663},[31685,31686],[34,72665,72667],{"className":72666},[31792],[34,72668,72670,72690],{"className":72669},[31796,31797],[34,72671,72673,72687],{"className":72672},[31801],[34,72674,72676],{"className":72675,"style":32090},[31805],[34,72677,72678,72681],{"style":31809},[34,72679],{"className":72680,"style":31814},[31813],[34,72682,72684],{"className":72683},[31818,31819,31820,31821],[34,72685,31955],{"className":72686},[31685,31686,31821],[34,72688,31829],{"className":72689},[31828],[34,72691,72693],{"className":72692},[31801],[34,72694,72696],{"className":72695,"style":31836},[31805],[34,72697],{}," 的维度是 ",[34,72700,72702,72734],{"className":72701},[31642],[34,72703,72705],{"className":72704},[31646],[31648,72706,72707],{"xmlns":31650},[31652,72708,72709,72731],{},[31655,72710,72711,72723,72725],{},[31743,72712,72713,72715],{},[31658,72714,15],{},[31655,72716,72717,72719,72721],{},[31658,72718,31955],{},[31662,72720,32226],{},[31749,72722,640],{},[31662,72724,42071],{},[31743,72726,72727,72729],{},[31658,72728,15],{},[31658,72730,31955],{},[31665,72732,72733],{"encoding":31667},"p_{i-1} \\times p_i",[34,72735,72737,72801],{"className":72736,"ariaHidden":758},[31672],[34,72738,72740,72743,72792,72795,72798],{"className":72739},[31676],[34,72741],{"className":72742,"style":39444},[31680],[34,72744,72746,72749],{"className":72745},[31685],[34,72747,15],{"className":72748},[31685,31686],[34,72750,72752],{"className":72751},[31792],[34,72753,72755,72784],{"className":72754},[31796,31797],[34,72756,72758,72781],{"className":72757},[31801],[34,72759,72761],{"className":72760,"style":32090},[31805],[34,72762,72763,72766],{"style":31809},[34,72764],{"className":72765,"style":31814},[31813],[34,72767,72769],{"className":72768},[31818,31819,31820,31821],[34,72770,72772,72775,72778],{"className":72771},[31685,31821],[34,72773,31955],{"className":72774},[31685,31686,31821],[34,72776,32226],{"className":72777},[32024,31821],[34,72779,640],{"className":72780},[31685,31821],[34,72782,31829],{"className":72783},[31828],[34,72785,72787],{"className":72786},[31801],[34,72788,72790],{"className":72789,"style":32037},[31805],[34,72791],{},[34,72793],{"className":72794,"style":32604},[31845],[34,72796,42071],{"className":72797},[32024],[34,72799],{"className":72800,"style":32604},[31845],[34,72802,72804,72807],{"className":72803},[31676],[34,72805],{"className":72806,"style":38774},[31680],[34,72808,72810,72813],{"className":72809},[31685],[34,72811,15],{"className":72812},[31685,31686],[34,72814,72816],{"className":72815},[31792],[34,72817,72819,72839],{"className":72818},[31796,31797],[34,72820,72822,72836],{"className":72821},[31801],[34,72823,72825],{"className":72824,"style":32090},[31805],[34,72826,72827,72830],{"style":31809},[34,72828],{"className":72829,"style":31814},[31813],[34,72831,72833],{"className":72832},[31818,31819,31820,31821],[34,72834,31955],{"className":72835},[31685,31686,31821],[34,72837,31829],{"className":72838},[31828],[34,72840,72842],{"className":72841},[31801],[34,72843,72845],{"className":72844,"style":31836},[31805],[34,72846],{},[15,72848,72849,72914,72915,72918],{},[2627,72850,72851,72852,72885,72886,2664],{},"1) 计算动态规划数组 ",[34,72853,72855,72870],{"className":72854},[31642],[34,72856,72858],{"className":72857},[31646],[31648,72859,72860],{"xmlns":31650},[31652,72861,72862,72868],{},[31655,72863,72864,72866],{},[31658,72865,46900],{},[31658,72867,15],{},[31665,72869,51611],{"encoding":31667},[34,72871,72873],{"className":72872,"ariaHidden":758},[31672],[34,72874,72876,72879,72882],{"className":72875},[31676],[34,72877],{"className":72878,"style":43217},[31680],[34,72880,46900],{"className":72881},[31685,31686],[34,72883,15],{"className":72884},[31685,31686]," 和断点数组 ",[34,72887,72889,72902],{"className":72888},[31642],[34,72890,72892],{"className":72891},[31646],[31648,72893,72894],{"xmlns":31650},[31652,72895,72896,72900],{},[31655,72897,72898],{},[31658,72899,31747],{},[31665,72901,31747],{"encoding":31667},[34,72903,72905],{"className":72904,"ariaHidden":758},[31672],[34,72906,72908,72911],{"className":72907},[31676],[34,72909],{"className":72910,"style":32527},[31680],[34,72912,31747],{"className":72913},[31685,31686],"\n填表的顺序是",[2627,72916,72917],{},"按照链长（长度为 L）从小到大","进行填写，这样我们在计算较长的链时，它所依赖的短链数据已经算好了。",[105,72920,72921,73150,75508,77753,79866,81296],{},[108,72922,72923,72926,72927],{},[2627,72924,72925],{},"长度 L = 1："," 即单一矩阵，不需要乘法。对角线全部为 0。\n",[34,72928,72930,73002],{"className":72929},[31642],[34,72931,72933],{"className":72932},[31646],[31648,72934,72935],{"xmlns":31650},[31652,72936,72937,72999],{},[31655,72938,72939,72941,72943,72945,72947,72949,72951,72953,72955,72957,72959,72961,72963,72965,72967,72969,72971,72973,72975,72977,72979,72981,72983,72985,72987,72989,72991,72993,72995,72997],{},[31658,72940,46900],{},[31658,72942,15],{},[31662,72944,15972],{"stretchy":702},[31749,72946,640],{},[31662,72948,32712],{"stretchy":702},[31662,72950,15972],{"stretchy":702},[31749,72952,640],{},[31662,72954,32712],{"stretchy":702},[31662,72956,202],{},[31658,72958,46900],{},[31658,72960,15],{},[31662,72962,15972],{"stretchy":702},[31749,72964,768],{},[31662,72966,32712],{"stretchy":702},[31662,72968,15972],{"stretchy":702},[31749,72970,768],{},[31662,72972,32712],{"stretchy":702},[31662,72974,202],{},[31662,72976,38874],{},[31662,72978,202],{},[31658,72980,46900],{},[31658,72982,15],{},[31662,72984,15972],{"stretchy":702},[31749,72986,36656],{},[31662,72988,32712],{"stretchy":702},[31662,72990,15972],{"stretchy":702},[31749,72992,36656],{},[31662,72994,32712],{"stretchy":702},[31662,72996,202],{},[31749,72998,459],{},[31665,73000,73001],{"encoding":31667},"dp[1][1] = dp[2][2] = \\dots = dp[6][6] = 0",[34,73003,73005,73044,73083,73102,73141],{"className":73004,"ariaHidden":758},[31672],[34,73006,73008,73011,73014,73017,73020,73023,73026,73029,73032,73035,73038,73041],{"className":73007},[31676],[34,73009],{"className":73010,"style":31774},[31680],[34,73012,46900],{"className":73013},[31685,31686],[34,73015,15],{"className":73016},[31685,31686],[34,73018,15972],{"className":73019},[31782],[34,73021,640],{"className":73022},[31685],[34,73024,32712],{"className":73025},[31690],[34,73027,15972],{"className":73028},[31782],[34,73030,640],{"className":73031},[31685],[34,73033,32712],{"className":73034},[31690],[34,73036],{"className":73037,"style":31846},[31845],[34,73039,202],{"className":73040},[31850],[34,73042],{"className":73043,"style":31846},[31845],[34,73045,73047,73050,73053,73056,73059,73062,73065,73068,73071,73074,73077,73080],{"className":73046},[31676],[34,73048],{"className":73049,"style":31774},[31680],[34,73051,46900],{"className":73052},[31685,31686],[34,73054,15],{"className":73055},[31685,31686],[34,73057,15972],{"className":73058},[31782],[34,73060,768],{"className":73061},[31685],[34,73063,32712],{"className":73064},[31690],[34,73066,15972],{"className":73067},[31782],[34,73069,768],{"className":73070},[31685],[34,73072,32712],{"className":73073},[31690],[34,73075],{"className":73076,"style":31846},[31845],[34,73078,202],{"className":73079},[31850],[34,73081],{"className":73082,"style":31846},[31845],[34,73084,73086,73090,73093,73096,73099],{"className":73085},[31676],[34,73087],{"className":73088,"style":73089},[31680],"height:0.3669em;",[34,73091,38874],{"className":73092},[34006],[34,73094],{"className":73095,"style":31846},[31845],[34,73097,202],{"className":73098},[31850],[34,73100],{"className":73101,"style":31846},[31845],[34,73103,73105,73108,73111,73114,73117,73120,73123,73126,73129,73132,73135,73138],{"className":73104},[31676],[34,73106],{"className":73107,"style":31774},[31680],[34,73109,46900],{"className":73110},[31685,31686],[34,73112,15],{"className":73113},[31685,31686],[34,73115,15972],{"className":73116},[31782],[34,73118,36656],{"className":73119},[31685],[34,73121,32712],{"className":73122},[31690],[34,73124,15972],{"className":73125},[31782],[34,73127,36656],{"className":73128},[31685],[34,73130,32712],{"className":73131},[31690],[34,73133],{"className":73134,"style":31846},[31845],[34,73136,202],{"className":73137},[31850],[34,73139],{"className":73140,"style":31846},[31845],[34,73142,73144,73147],{"className":73143},[31676],[34,73145],{"className":73146,"style":32617},[31680],[34,73148,459],{"className":73149},[31685],[108,73151,73152,73155,73156,73349,73350,73378,73379],{},[2627,73153,73154],{},"长度 L = 2："," 两个相邻矩阵相乘，公式为 ",[34,73157,73159,73201],{"className":73158},[31642],[34,73160,73162],{"className":73161},[31646],[31648,73163,73164],{"xmlns":31650},[31652,73165,73166,73198],{},[31655,73167,73168,73180,73186],{},[31743,73169,73170,73172],{},[31658,73171,15],{},[31655,73173,73174,73176,73178],{},[31658,73175,31955],{},[31662,73177,32226],{},[31749,73179,640],{},[31743,73181,73182,73184],{},[31658,73183,15],{},[31658,73185,31955],{},[31743,73187,73188,73190],{},[31658,73189,15],{},[31655,73191,73192,73194,73196],{},[31658,73193,31955],{},[31662,73195,223],{},[31749,73197,640],{},[31665,73199,73200],{"encoding":31667},"p_{i-1} p_i p_{i+1}",[34,73202,73204],{"className":73203,"ariaHidden":758},[31672],[34,73205,73207,73211,73260,73300],{"className":73206},[31676],[34,73208],{"className":73209,"style":73210},[31680],"height:0.6389em;vertical-align:-0.2083em;",[34,73212,73214,73217],{"className":73213},[31685],[34,73215,15],{"className":73216},[31685,31686],[34,73218,73220],{"className":73219},[31792],[34,73221,73223,73252],{"className":73222},[31796,31797],[34,73224,73226,73249],{"className":73225},[31801],[34,73227,73229],{"className":73228,"style":32090},[31805],[34,73230,73231,73234],{"style":31809},[34,73232],{"className":73233,"style":31814},[31813],[34,73235,73237],{"className":73236},[31818,31819,31820,31821],[34,73238,73240,73243,73246],{"className":73239},[31685,31821],[34,73241,31955],{"className":73242},[31685,31686,31821],[34,73244,32226],{"className":73245},[32024,31821],[34,73247,640],{"className":73248},[31685,31821],[34,73250,31829],{"className":73251},[31828],[34,73253,73255],{"className":73254},[31801],[34,73256,73258],{"className":73257,"style":32037},[31805],[34,73259],{},[34,73261,73263,73266],{"className":73262},[31685],[34,73264,15],{"className":73265},[31685,31686],[34,73267,73269],{"className":73268},[31792],[34,73270,73272,73292],{"className":73271},[31796,31797],[34,73273,73275,73289],{"className":73274},[31801],[34,73276,73278],{"className":73277,"style":32090},[31805],[34,73279,73280,73283],{"style":31809},[34,73281],{"className":73282,"style":31814},[31813],[34,73284,73286],{"className":73285},[31818,31819,31820,31821],[34,73287,31955],{"className":73288},[31685,31686,31821],[34,73290,31829],{"className":73291},[31828],[34,73293,73295],{"className":73294},[31801],[34,73296,73298],{"className":73297,"style":31836},[31805],[34,73299],{},[34,73301,73303,73306],{"className":73302},[31685],[34,73304,15],{"className":73305},[31685,31686],[34,73307,73309],{"className":73308},[31792],[34,73310,73312,73341],{"className":73311},[31796,31797],[34,73313,73315,73338],{"className":73314},[31801],[34,73316,73318],{"className":73317,"style":32090},[31805],[34,73319,73320,73323],{"style":31809},[34,73321],{"className":73322,"style":31814},[31813],[34,73324,73326],{"className":73325},[31818,31819,31820,31821],[34,73327,73329,73332,73335],{"className":73328},[31685,31821],[34,73330,31955],{"className":73331},[31685,31686,31821],[34,73333,223],{"className":73334},[32024,31821],[34,73336,640],{"className":73337},[31685,31821],[34,73339,31829],{"className":73340},[31828],[34,73342,73344],{"className":73343},[31801],[34,73345,73347],{"className":73346,"style":32037},[31805],[34,73348],{},"，没有别的断点选择，断点 ",[34,73351,73353,73366],{"className":73352},[31642],[34,73354,73356],{"className":73355},[31646],[31648,73357,73358],{"xmlns":31650},[31652,73359,73360,73364],{},[31655,73361,73362],{},[31658,73363,31747],{},[31665,73365,31747],{"encoding":31667},[34,73367,73369],{"className":73368,"ariaHidden":758},[31672],[34,73370,73372,73375],{"className":73371},[31676],[34,73373],{"className":73374,"style":32527},[31680],[34,73376,31747],{"className":73377},[31685,31686]," 即为左矩阵下标。",[105,73380,73381,73808,74233,74658,75083],{},[108,73382,73383,73726,73727],{},[34,73384,73386,73455],{"className":73385},[31642],[34,73387,73389],{"className":73388},[31646],[31648,73390,73391],{"xmlns":31650},[31652,73392,73393,73452],{},[31655,73394,73395,73397,73399,73401,73403,73405,73407,73409,73411,73413,73419,73421,73427,73429,73435,73437,73439,73441,73443,73445,73447,73449],{},[31658,73396,46900],{},[31658,73398,15],{},[31662,73400,15972],{"stretchy":702},[31749,73402,640],{},[31662,73404,32712],{"stretchy":702},[31662,73406,15972],{"stretchy":702},[31749,73408,768],{},[31662,73410,32712],{"stretchy":702},[31662,73412,202],{},[31743,73414,73415,73417],{},[31658,73416,15],{},[31749,73418,459],{},[31662,73420,71191],{},[31743,73422,73423,73425],{},[31658,73424,15],{},[31749,73426,640],{},[31662,73428,71191],{},[31743,73430,73431,73433],{},[31658,73432,15],{},[31749,73434,768],{},[31662,73436,202],{},[31749,73438,18136],{},[31662,73440,42071],{},[31749,73442,3244],{},[31662,73444,42071],{},[31749,73446,786],{},[31662,73448,202],{},[31749,73450,73451],{},"150",[31665,73453,73454],{"encoding":31667},"dp[1][2] = p_0 \\cdot p_1 \\cdot p_2 = 5 \\times 10 \\times 3 = 150",[34,73456,73458,73497,73553,73608,73663,73681,73699,73717],{"className":73457,"ariaHidden":758},[31672],[34,73459,73461,73464,73467,73470,73473,73476,73479,73482,73485,73488,73491,73494],{"className":73460},[31676],[34,73462],{"className":73463,"style":31774},[31680],[34,73465,46900],{"className":73466},[31685,31686],[34,73468,15],{"className":73469},[31685,31686],[34,73471,15972],{"className":73472},[31782],[34,73474,640],{"className":73475},[31685],[34,73477,32712],{"className":73478},[31690],[34,73480,15972],{"className":73481},[31782],[34,73483,768],{"className":73484},[31685],[34,73486,32712],{"className":73487},[31690],[34,73489],{"className":73490,"style":31846},[31845],[34,73492,202],{"className":73493},[31850],[34,73495],{"className":73496,"style":31846},[31845],[34,73498,73500,73504,73544,73547,73550],{"className":73499},[31676],[34,73501],{"className":73502,"style":73503},[31680],"height:0.6389em;vertical-align:-0.1944em;",[34,73505,73507,73510],{"className":73506},[31685],[34,73508,15],{"className":73509},[31685,31686],[34,73511,73513],{"className":73512},[31792],[34,73514,73516,73536],{"className":73515},[31796,31797],[34,73517,73519,73533],{"className":73518},[31801],[34,73520,73522],{"className":73521,"style":31806},[31805],[34,73523,73524,73527],{"style":31809},[34,73525],{"className":73526,"style":31814},[31813],[34,73528,73530],{"className":73529},[31818,31819,31820,31821],[34,73531,459],{"className":73532},[31685,31821],[34,73534,31829],{"className":73535},[31828],[34,73537,73539],{"className":73538},[31801],[34,73540,73542],{"className":73541,"style":31836},[31805],[34,73543],{},[34,73545],{"className":73546,"style":32604},[31845],[34,73548,71191],{"className":73549},[32024],[34,73551],{"className":73552,"style":32604},[31845],[34,73554,73556,73559,73599,73602,73605],{"className":73555},[31676],[34,73557],{"className":73558,"style":73503},[31680],[34,73560,73562,73565],{"className":73561},[31685],[34,73563,15],{"className":73564},[31685,31686],[34,73566,73568],{"className":73567},[31792],[34,73569,73571,73591],{"className":73570},[31796,31797],[34,73572,73574,73588],{"className":73573},[31801],[34,73575,73577],{"className":73576,"style":31806},[31805],[34,73578,73579,73582],{"style":31809},[34,73580],{"className":73581,"style":31814},[31813],[34,73583,73585],{"className":73584},[31818,31819,31820,31821],[34,73586,640],{"className":73587},[31685,31821],[34,73589,31829],{"className":73590},[31828],[34,73592,73594],{"className":73593},[31801],[34,73595,73597],{"className":73596,"style":31836},[31805],[34,73598],{},[34,73600],{"className":73601,"style":32604},[31845],[34,73603,71191],{"className":73604},[32024],[34,73606],{"className":73607,"style":32604},[31845],[34,73609,73611,73614,73654,73657,73660],{"className":73610},[31676],[34,73612],{"className":73613,"style":38774},[31680],[34,73615,73617,73620],{"className":73616},[31685],[34,73618,15],{"className":73619},[31685,31686],[34,73621,73623],{"className":73622},[31792],[34,73624,73626,73646],{"className":73625},[31796,31797],[34,73627,73629,73643],{"className":73628},[31801],[34,73630,73632],{"className":73631,"style":31806},[31805],[34,73633,73634,73637],{"style":31809},[34,73635],{"className":73636,"style":31814},[31813],[34,73638,73640],{"className":73639},[31818,31819,31820,31821],[34,73641,768],{"className":73642},[31685,31821],[34,73644,31829],{"className":73645},[31828],[34,73647,73649],{"className":73648},[31801],[34,73650,73652],{"className":73651,"style":31836},[31805],[34,73653],{},[34,73655],{"className":73656,"style":31846},[31845],[34,73658,202],{"className":73659},[31850],[34,73661],{"className":73662,"style":31846},[31845],[34,73664,73666,73669,73672,73675,73678],{"className":73665},[31676],[34,73667],{"className":73668,"style":35942},[31680],[34,73670,18136],{"className":73671},[31685],[34,73673],{"className":73674,"style":32604},[31845],[34,73676,42071],{"className":73677},[32024],[34,73679],{"className":73680,"style":32604},[31845],[34,73682,73684,73687,73690,73693,73696],{"className":73683},[31676],[34,73685],{"className":73686,"style":35942},[31680],[34,73688,3244],{"className":73689},[31685],[34,73691],{"className":73692,"style":32604},[31845],[34,73694,42071],{"className":73695},[32024],[34,73697],{"className":73698,"style":32604},[31845],[34,73700,73702,73705,73708,73711,73714],{"className":73701},[31676],[34,73703],{"className":73704,"style":32617},[31680],[34,73706,786],{"className":73707},[31685],[34,73709],{"className":73710,"style":31846},[31845],[34,73712,202],{"className":73713},[31850],[34,73715],{"className":73716,"style":31846},[31845],[34,73718,73720,73723],{"className":73719},[31676],[34,73721],{"className":73722,"style":32617},[31680],[34,73724,73451],{"className":73725},[31685],"，断点 ",[34,73728,73730,73760],{"className":73729},[31642],[34,73731,73733],{"className":73732},[31646],[31648,73734,73735],{"xmlns":31650},[31652,73736,73737,73757],{},[31655,73738,73739,73741,73743,73745,73747,73749,73751,73753,73755],{},[31658,73740,31747],{},[31662,73742,15972],{"stretchy":702},[31749,73744,640],{},[31662,73746,32712],{"stretchy":702},[31662,73748,15972],{"stretchy":702},[31749,73750,768],{},[31662,73752,32712],{"stretchy":702},[31662,73754,202],{},[31749,73756,640],{},[31665,73758,73759],{"encoding":31667},"s[1][2]=1",[34,73761,73763,73799],{"className":73762,"ariaHidden":758},[31672],[34,73764,73766,73769,73772,73775,73778,73781,73784,73787,73790,73793,73796],{"className":73765},[31676],[34,73767],{"className":73768,"style":31774},[31680],[34,73770,31747],{"className":73771},[31685,31686],[34,73773,15972],{"className":73774},[31782],[34,73776,640],{"className":73777},[31685],[34,73779,32712],{"className":73780},[31690],[34,73782,15972],{"className":73783},[31782],[34,73785,768],{"className":73786},[31685],[34,73788,32712],{"className":73789},[31690],[34,73791],{"className":73792,"style":31846},[31845],[34,73794,202],{"className":73795},[31850],[34,73797],{"className":73798,"style":31846},[31845],[34,73800,73802,73805],{"className":73801},[31676],[34,73803],{"className":73804,"style":32617},[31680],[34,73806,640],{"className":73807},[31685],[108,73809,73810,46528,74152],{},[34,73811,73813,73882],{"className":73812},[31642],[34,73814,73816],{"className":73815},[31646],[31648,73817,73818],{"xmlns":31650},[31652,73819,73820,73879],{},[31655,73821,73822,73824,73826,73828,73830,73832,73834,73836,73838,73840,73846,73848,73854,73856,73862,73864,73866,73868,73870,73872,73874,73876],{},[31658,73823,46900],{},[31658,73825,15],{},[31662,73827,15972],{"stretchy":702},[31749,73829,768],{},[31662,73831,32712],{"stretchy":702},[31662,73833,15972],{"stretchy":702},[31749,73835,786],{},[31662,73837,32712],{"stretchy":702},[31662,73839,202],{},[31743,73841,73842,73844],{},[31658,73843,15],{},[31749,73845,640],{},[31662,73847,71191],{},[31743,73849,73850,73852],{},[31658,73851,15],{},[31749,73853,768],{},[31662,73855,71191],{},[31743,73857,73858,73860],{},[31658,73859,15],{},[31749,73861,786],{},[31662,73863,202],{},[31749,73865,3244],{},[31662,73867,42071],{},[31749,73869,786],{},[31662,73871,42071],{},[31749,73873,71869],{},[31662,73875,202],{},[31749,73877,73878],{},"360",[31665,73880,73881],{"encoding":31667},"dp[2][3] = p_1 \\cdot p_2 \\cdot p_3 = 10 \\times 3 \\times 12 = 360",[34,73883,73885,73924,73979,74034,74089,74107,74125,74143],{"className":73884,"ariaHidden":758},[31672],[34,73886,73888,73891,73894,73897,73900,73903,73906,73909,73912,73915,73918,73921],{"className":73887},[31676],[34,73889],{"className":73890,"style":31774},[31680],[34,73892,46900],{"className":73893},[31685,31686],[34,73895,15],{"className":73896},[31685,31686],[34,73898,15972],{"className":73899},[31782],[34,73901,768],{"className":73902},[31685],[34,73904,32712],{"className":73905},[31690],[34,73907,15972],{"className":73908},[31782],[34,73910,786],{"className":73911},[31685],[34,73913,32712],{"className":73914},[31690],[34,73916],{"className":73917,"style":31846},[31845],[34,73919,202],{"className":73920},[31850],[34,73922],{"className":73923,"style":31846},[31845],[34,73925,73927,73930,73970,73973,73976],{"className":73926},[31676],[34,73928],{"className":73929,"style":73503},[31680],[34,73931,73933,73936],{"className":73932},[31685],[34,73934,15],{"className":73935},[31685,31686],[34,73937,73939],{"className":73938},[31792],[34,73940,73942,73962],{"className":73941},[31796,31797],[34,73943,73945,73959],{"className":73944},[31801],[34,73946,73948],{"className":73947,"style":31806},[31805],[34,73949,73950,73953],{"style":31809},[34,73951],{"className":73952,"style":31814},[31813],[34,73954,73956],{"className":73955},[31818,31819,31820,31821],[34,73957,640],{"className":73958},[31685,31821],[34,73960,31829],{"className":73961},[31828],[34,73963,73965],{"className":73964},[31801],[34,73966,73968],{"className":73967,"style":31836},[31805],[34,73969],{},[34,73971],{"className":73972,"style":32604},[31845],[34,73974,71191],{"className":73975},[32024],[34,73977],{"className":73978,"style":32604},[31845],[34,73980,73982,73985,74025,74028,74031],{"className":73981},[31676],[34,73983],{"className":73984,"style":73503},[31680],[34,73986,73988,73991],{"className":73987},[31685],[34,73989,15],{"className":73990},[31685,31686],[34,73992,73994],{"className":73993},[31792],[34,73995,73997,74017],{"className":73996},[31796,31797],[34,73998,74000,74014],{"className":73999},[31801],[34,74001,74003],{"className":74002,"style":31806},[31805],[34,74004,74005,74008],{"style":31809},[34,74006],{"className":74007,"style":31814},[31813],[34,74009,74011],{"className":74010},[31818,31819,31820,31821],[34,74012,768],{"className":74013},[31685,31821],[34,74015,31829],{"className":74016},[31828],[34,74018,74020],{"className":74019},[31801],[34,74021,74023],{"className":74022,"style":31836},[31805],[34,74024],{},[34,74026],{"className":74027,"style":32604},[31845],[34,74029,71191],{"className":74030},[32024],[34,74032],{"className":74033,"style":32604},[31845],[34,74035,74037,74040,74080,74083,74086],{"className":74036},[31676],[34,74038],{"className":74039,"style":38774},[31680],[34,74041,74043,74046],{"className":74042},[31685],[34,74044,15],{"className":74045},[31685,31686],[34,74047,74049],{"className":74048},[31792],[34,74050,74052,74072],{"className":74051},[31796,31797],[34,74053,74055,74069],{"className":74054},[31801],[34,74056,74058],{"className":74057,"style":31806},[31805],[34,74059,74060,74063],{"style":31809},[34,74061],{"className":74062,"style":31814},[31813],[34,74064,74066],{"className":74065},[31818,31819,31820,31821],[34,74067,786],{"className":74068},[31685,31821],[34,74070,31829],{"className":74071},[31828],[34,74073,74075],{"className":74074},[31801],[34,74076,74078],{"className":74077,"style":31836},[31805],[34,74079],{},[34,74081],{"className":74082,"style":31846},[31845],[34,74084,202],{"className":74085},[31850],[34,74087],{"className":74088,"style":31846},[31845],[34,74090,74092,74095,74098,74101,74104],{"className":74091},[31676],[34,74093],{"className":74094,"style":35942},[31680],[34,74096,3244],{"className":74097},[31685],[34,74099],{"className":74100,"style":32604},[31845],[34,74102,42071],{"className":74103},[32024],[34,74105],{"className":74106,"style":32604},[31845],[34,74108,74110,74113,74116,74119,74122],{"className":74109},[31676],[34,74111],{"className":74112,"style":35942},[31680],[34,74114,786],{"className":74115},[31685],[34,74117],{"className":74118,"style":32604},[31845],[34,74120,42071],{"className":74121},[32024],[34,74123],{"className":74124,"style":32604},[31845],[34,74126,74128,74131,74134,74137,74140],{"className":74127},[31676],[34,74129],{"className":74130,"style":32617},[31680],[34,74132,71869],{"className":74133},[31685],[34,74135],{"className":74136,"style":31846},[31845],[34,74138,202],{"className":74139},[31850],[34,74141],{"className":74142,"style":31846},[31845],[34,74144,74146,74149],{"className":74145},[31676],[34,74147],{"className":74148,"style":32617},[31680],[34,74150,73878],{"className":74151},[31685],[34,74153,74155,74185],{"className":74154},[31642],[34,74156,74158],{"className":74157},[31646],[31648,74159,74160],{"xmlns":31650},[31652,74161,74162,74182],{},[31655,74163,74164,74166,74168,74170,74172,74174,74176,74178,74180],{},[31658,74165,31747],{},[31662,74167,15972],{"stretchy":702},[31749,74169,768],{},[31662,74171,32712],{"stretchy":702},[31662,74173,15972],{"stretchy":702},[31749,74175,786],{},[31662,74177,32712],{"stretchy":702},[31662,74179,202],{},[31749,74181,768],{},[31665,74183,74184],{"encoding":31667},"s[2][3]=2",[34,74186,74188,74224],{"className":74187,"ariaHidden":758},[31672],[34,74189,74191,74194,74197,74200,74203,74206,74209,74212,74215,74218,74221],{"className":74190},[31676],[34,74192],{"className":74193,"style":31774},[31680],[34,74195,31747],{"className":74196},[31685,31686],[34,74198,15972],{"className":74199},[31782],[34,74201,768],{"className":74202},[31685],[34,74204,32712],{"className":74205},[31690],[34,74207,15972],{"className":74208},[31782],[34,74210,786],{"className":74211},[31685],[34,74213,32712],{"className":74214},[31690],[34,74216],{"className":74217,"style":31846},[31845],[34,74219,202],{"className":74220},[31850],[34,74222],{"className":74223,"style":31846},[31845],[34,74225,74227,74230],{"className":74226},[31676],[34,74228],{"className":74229,"style":32617},[31680],[34,74231,768],{"className":74232},[31685],[108,74234,74235,46528,74577],{},[34,74236,74238,74307],{"className":74237},[31642],[34,74239,74241],{"className":74240},[31646],[31648,74242,74243],{"xmlns":31650},[31652,74244,74245,74304],{},[31655,74246,74247,74249,74251,74253,74255,74257,74259,74261,74263,74265,74271,74273,74279,74281,74287,74289,74291,74293,74295,74297,74299,74301],{},[31658,74248,46900],{},[31658,74250,15],{},[31662,74252,15972],{"stretchy":702},[31749,74254,786],{},[31662,74256,32712],{"stretchy":702},[31662,74258,15972],{"stretchy":702},[31749,74260,15932],{},[31662,74262,32712],{"stretchy":702},[31662,74264,202],{},[31743,74266,74267,74269],{},[31658,74268,15],{},[31749,74270,768],{},[31662,74272,71191],{},[31743,74274,74275,74277],{},[31658,74276,15],{},[31749,74278,786],{},[31662,74280,71191],{},[31743,74282,74283,74285],{},[31658,74284,15],{},[31749,74286,15932],{},[31662,74288,202],{},[31749,74290,786],{},[31662,74292,42071],{},[31749,74294,71869],{},[31662,74296,42071],{},[31749,74298,18136],{},[31662,74300,202],{},[31749,74302,74303],{},"180",[31665,74305,74306],{"encoding":31667},"dp[3][4] = p_2 \\cdot p_3 \\cdot p_4 = 3 \\times 12 \\times 5 = 180",[34,74308,74310,74349,74404,74459,74514,74532,74550,74568],{"className":74309,"ariaHidden":758},[31672],[34,74311,74313,74316,74319,74322,74325,74328,74331,74334,74337,74340,74343,74346],{"className":74312},[31676],[34,74314],{"className":74315,"style":31774},[31680],[34,74317,46900],{"className":74318},[31685,31686],[34,74320,15],{"className":74321},[31685,31686],[34,74323,15972],{"className":74324},[31782],[34,74326,786],{"className":74327},[31685],[34,74329,32712],{"className":74330},[31690],[34,74332,15972],{"className":74333},[31782],[34,74335,15932],{"className":74336},[31685],[34,74338,32712],{"className":74339},[31690],[34,74341],{"className":74342,"style":31846},[31845],[34,74344,202],{"className":74345},[31850],[34,74347],{"className":74348,"style":31846},[31845],[34,74350,74352,74355,74395,74398,74401],{"className":74351},[31676],[34,74353],{"className":74354,"style":73503},[31680],[34,74356,74358,74361],{"className":74357},[31685],[34,74359,15],{"className":74360},[31685,31686],[34,74362,74364],{"className":74363},[31792],[34,74365,74367,74387],{"className":74366},[31796,31797],[34,74368,74370,74384],{"className":74369},[31801],[34,74371,74373],{"className":74372,"style":31806},[31805],[34,74374,74375,74378],{"style":31809},[34,74376],{"className":74377,"style":31814},[31813],[34,74379,74381],{"className":74380},[31818,31819,31820,31821],[34,74382,768],{"className":74383},[31685,31821],[34,74385,31829],{"className":74386},[31828],[34,74388,74390],{"className":74389},[31801],[34,74391,74393],{"className":74392,"style":31836},[31805],[34,74394],{},[34,74396],{"className":74397,"style":32604},[31845],[34,74399,71191],{"className":74400},[32024],[34,74402],{"className":74403,"style":32604},[31845],[34,74405,74407,74410,74450,74453,74456],{"className":74406},[31676],[34,74408],{"className":74409,"style":73503},[31680],[34,74411,74413,74416],{"className":74412},[31685],[34,74414,15],{"className":74415},[31685,31686],[34,74417,74419],{"className":74418},[31792],[34,74420,74422,74442],{"className":74421},[31796,31797],[34,74423,74425,74439],{"className":74424},[31801],[34,74426,74428],{"className":74427,"style":31806},[31805],[34,74429,74430,74433],{"style":31809},[34,74431],{"className":74432,"style":31814},[31813],[34,74434,74436],{"className":74435},[31818,31819,31820,31821],[34,74437,786],{"className":74438},[31685,31821],[34,74440,31829],{"className":74441},[31828],[34,74443,74445],{"className":74444},[31801],[34,74446,74448],{"className":74447,"style":31836},[31805],[34,74449],{},[34,74451],{"className":74452,"style":32604},[31845],[34,74454,71191],{"className":74455},[32024],[34,74457],{"className":74458,"style":32604},[31845],[34,74460,74462,74465,74505,74508,74511],{"className":74461},[31676],[34,74463],{"className":74464,"style":38774},[31680],[34,74466,74468,74471],{"className":74467},[31685],[34,74469,15],{"className":74470},[31685,31686],[34,74472,74474],{"className":74473},[31792],[34,74475,74477,74497],{"className":74476},[31796,31797],[34,74478,74480,74494],{"className":74479},[31801],[34,74481,74483],{"className":74482,"style":31806},[31805],[34,74484,74485,74488],{"style":31809},[34,74486],{"className":74487,"style":31814},[31813],[34,74489,74491],{"className":74490},[31818,31819,31820,31821],[34,74492,15932],{"className":74493},[31685,31821],[34,74495,31829],{"className":74496},[31828],[34,74498,74500],{"className":74499},[31801],[34,74501,74503],{"className":74502,"style":31836},[31805],[34,74504],{},[34,74506],{"className":74507,"style":31846},[31845],[34,74509,202],{"className":74510},[31850],[34,74512],{"className":74513,"style":31846},[31845],[34,74515,74517,74520,74523,74526,74529],{"className":74516},[31676],[34,74518],{"className":74519,"style":35942},[31680],[34,74521,786],{"className":74522},[31685],[34,74524],{"className":74525,"style":32604},[31845],[34,74527,42071],{"className":74528},[32024],[34,74530],{"className":74531,"style":32604},[31845],[34,74533,74535,74538,74541,74544,74547],{"className":74534},[31676],[34,74536],{"className":74537,"style":35942},[31680],[34,74539,71869],{"className":74540},[31685],[34,74542],{"className":74543,"style":32604},[31845],[34,74545,42071],{"className":74546},[32024],[34,74548],{"className":74549,"style":32604},[31845],[34,74551,74553,74556,74559,74562,74565],{"className":74552},[31676],[34,74554],{"className":74555,"style":32617},[31680],[34,74557,18136],{"className":74558},[31685],[34,74560],{"className":74561,"style":31846},[31845],[34,74563,202],{"className":74564},[31850],[34,74566],{"className":74567,"style":31846},[31845],[34,74569,74571,74574],{"className":74570},[31676],[34,74572],{"className":74573,"style":32617},[31680],[34,74575,74303],{"className":74576},[31685],[34,74578,74580,74610],{"className":74579},[31642],[34,74581,74583],{"className":74582},[31646],[31648,74584,74585],{"xmlns":31650},[31652,74586,74587,74607],{},[31655,74588,74589,74591,74593,74595,74597,74599,74601,74603,74605],{},[31658,74590,31747],{},[31662,74592,15972],{"stretchy":702},[31749,74594,786],{},[31662,74596,32712],{"stretchy":702},[31662,74598,15972],{"stretchy":702},[31749,74600,15932],{},[31662,74602,32712],{"stretchy":702},[31662,74604,202],{},[31749,74606,786],{},[31665,74608,74609],{"encoding":31667},"s[3][4]=3",[34,74611,74613,74649],{"className":74612,"ariaHidden":758},[31672],[34,74614,74616,74619,74622,74625,74628,74631,74634,74637,74640,74643,74646],{"className":74615},[31676],[34,74617],{"className":74618,"style":31774},[31680],[34,74620,31747],{"className":74621},[31685,31686],[34,74623,15972],{"className":74624},[31782],[34,74626,786],{"className":74627},[31685],[34,74629,32712],{"className":74630},[31690],[34,74632,15972],{"className":74633},[31782],[34,74635,15932],{"className":74636},[31685],[34,74638,32712],{"className":74639},[31690],[34,74641],{"className":74642,"style":31846},[31845],[34,74644,202],{"className":74645},[31850],[34,74647],{"className":74648,"style":31846},[31845],[34,74650,74652,74655],{"className":74651},[31676],[34,74653],{"className":74654,"style":32617},[31680],[34,74656,786],{"className":74657},[31685],[108,74659,74660,46528,75002],{},[34,74661,74663,74732],{"className":74662},[31642],[34,74664,74666],{"className":74665},[31646],[31648,74667,74668],{"xmlns":31650},[31652,74669,74670,74729],{},[31655,74671,74672,74674,74676,74678,74680,74682,74684,74686,74688,74690,74696,74698,74704,74706,74712,74714,74716,74718,74720,74722,74724,74726],{},[31658,74673,46900],{},[31658,74675,15],{},[31662,74677,15972],{"stretchy":702},[31749,74679,15932],{},[31662,74681,32712],{"stretchy":702},[31662,74683,15972],{"stretchy":702},[31749,74685,18136],{},[31662,74687,32712],{"stretchy":702},[31662,74689,202],{},[31743,74691,74692,74694],{},[31658,74693,15],{},[31749,74695,786],{},[31662,74697,71191],{},[31743,74699,74700,74702],{},[31658,74701,15],{},[31749,74703,15932],{},[31662,74705,71191],{},[31743,74707,74708,74710],{},[31658,74709,15],{},[31749,74711,18136],{},[31662,74713,202],{},[31749,74715,71869],{},[31662,74717,42071],{},[31749,74719,18136],{},[31662,74721,42071],{},[31749,74723,5774],{},[31662,74725,202],{},[31749,74727,74728],{},"3000",[31665,74730,74731],{"encoding":31667},"dp[4][5] = p_3 \\cdot p_4 \\cdot p_5 = 12 \\times 5 \\times 50 = 3000",[34,74733,74735,74774,74829,74884,74939,74957,74975,74993],{"className":74734,"ariaHidden":758},[31672],[34,74736,74738,74741,74744,74747,74750,74753,74756,74759,74762,74765,74768,74771],{"className":74737},[31676],[34,74739],{"className":74740,"style":31774},[31680],[34,74742,46900],{"className":74743},[31685,31686],[34,74745,15],{"className":74746},[31685,31686],[34,74748,15972],{"className":74749},[31782],[34,74751,15932],{"className":74752},[31685],[34,74754,32712],{"className":74755},[31690],[34,74757,15972],{"className":74758},[31782],[34,74760,18136],{"className":74761},[31685],[34,74763,32712],{"className":74764},[31690],[34,74766],{"className":74767,"style":31846},[31845],[34,74769,202],{"className":74770},[31850],[34,74772],{"className":74773,"style":31846},[31845],[34,74775,74777,74780,74820,74823,74826],{"className":74776},[31676],[34,74778],{"className":74779,"style":73503},[31680],[34,74781,74783,74786],{"className":74782},[31685],[34,74784,15],{"className":74785},[31685,31686],[34,74787,74789],{"className":74788},[31792],[34,74790,74792,74812],{"className":74791},[31796,31797],[34,74793,74795,74809],{"className":74794},[31801],[34,74796,74798],{"className":74797,"style":31806},[31805],[34,74799,74800,74803],{"style":31809},[34,74801],{"className":74802,"style":31814},[31813],[34,74804,74806],{"className":74805},[31818,31819,31820,31821],[34,74807,786],{"className":74808},[31685,31821],[34,74810,31829],{"className":74811},[31828],[34,74813,74815],{"className":74814},[31801],[34,74816,74818],{"className":74817,"style":31836},[31805],[34,74819],{},[34,74821],{"className":74822,"style":32604},[31845],[34,74824,71191],{"className":74825},[32024],[34,74827],{"className":74828,"style":32604},[31845],[34,74830,74832,74835,74875,74878,74881],{"className":74831},[31676],[34,74833],{"className":74834,"style":73503},[31680],[34,74836,74838,74841],{"className":74837},[31685],[34,74839,15],{"className":74840},[31685,31686],[34,74842,74844],{"className":74843},[31792],[34,74845,74847,74867],{"className":74846},[31796,31797],[34,74848,74850,74864],{"className":74849},[31801],[34,74851,74853],{"className":74852,"style":31806},[31805],[34,74854,74855,74858],{"style":31809},[34,74856],{"className":74857,"style":31814},[31813],[34,74859,74861],{"className":74860},[31818,31819,31820,31821],[34,74862,15932],{"className":74863},[31685,31821],[34,74865,31829],{"className":74866},[31828],[34,74868,74870],{"className":74869},[31801],[34,74871,74873],{"className":74872,"style":31836},[31805],[34,74874],{},[34,74876],{"className":74877,"style":32604},[31845],[34,74879,71191],{"className":74880},[32024],[34,74882],{"className":74883,"style":32604},[31845],[34,74885,74887,74890,74930,74933,74936],{"className":74886},[31676],[34,74888],{"className":74889,"style":38774},[31680],[34,74891,74893,74896],{"className":74892},[31685],[34,74894,15],{"className":74895},[31685,31686],[34,74897,74899],{"className":74898},[31792],[34,74900,74902,74922],{"className":74901},[31796,31797],[34,74903,74905,74919],{"className":74904},[31801],[34,74906,74908],{"className":74907,"style":31806},[31805],[34,74909,74910,74913],{"style":31809},[34,74911],{"className":74912,"style":31814},[31813],[34,74914,74916],{"className":74915},[31818,31819,31820,31821],[34,74917,18136],{"className":74918},[31685,31821],[34,74920,31829],{"className":74921},[31828],[34,74923,74925],{"className":74924},[31801],[34,74926,74928],{"className":74927,"style":31836},[31805],[34,74929],{},[34,74931],{"className":74932,"style":31846},[31845],[34,74934,202],{"className":74935},[31850],[34,74937],{"className":74938,"style":31846},[31845],[34,74940,74942,74945,74948,74951,74954],{"className":74941},[31676],[34,74943],{"className":74944,"style":35942},[31680],[34,74946,71869],{"className":74947},[31685],[34,74949],{"className":74950,"style":32604},[31845],[34,74952,42071],{"className":74953},[32024],[34,74955],{"className":74956,"style":32604},[31845],[34,74958,74960,74963,74966,74969,74972],{"className":74959},[31676],[34,74961],{"className":74962,"style":35942},[31680],[34,74964,18136],{"className":74965},[31685],[34,74967],{"className":74968,"style":32604},[31845],[34,74970,42071],{"className":74971},[32024],[34,74973],{"className":74974,"style":32604},[31845],[34,74976,74978,74981,74984,74987,74990],{"className":74977},[31676],[34,74979],{"className":74980,"style":32617},[31680],[34,74982,5774],{"className":74983},[31685],[34,74985],{"className":74986,"style":31846},[31845],[34,74988,202],{"className":74989},[31850],[34,74991],{"className":74992,"style":31846},[31845],[34,74994,74996,74999],{"className":74995},[31676],[34,74997],{"className":74998,"style":32617},[31680],[34,75000,74728],{"className":75001},[31685],[34,75003,75005,75035],{"className":75004},[31642],[34,75006,75008],{"className":75007},[31646],[31648,75009,75010],{"xmlns":31650},[31652,75011,75012,75032],{},[31655,75013,75014,75016,75018,75020,75022,75024,75026,75028,75030],{},[31658,75015,31747],{},[31662,75017,15972],{"stretchy":702},[31749,75019,15932],{},[31662,75021,32712],{"stretchy":702},[31662,75023,15972],{"stretchy":702},[31749,75025,18136],{},[31662,75027,32712],{"stretchy":702},[31662,75029,202],{},[31749,75031,15932],{},[31665,75033,75034],{"encoding":31667},"s[4][5]=4",[34,75036,75038,75074],{"className":75037,"ariaHidden":758},[31672],[34,75039,75041,75044,75047,75050,75053,75056,75059,75062,75065,75068,75071],{"className":75040},[31676],[34,75042],{"className":75043,"style":31774},[31680],[34,75045,31747],{"className":75046},[31685,31686],[34,75048,15972],{"className":75049},[31782],[34,75051,15932],{"className":75052},[31685],[34,75054,32712],{"className":75055},[31690],[34,75057,15972],{"className":75058},[31782],[34,75060,18136],{"className":75061},[31685],[34,75063,32712],{"className":75064},[31690],[34,75066],{"className":75067,"style":31846},[31845],[34,75069,202],{"className":75070},[31850],[34,75072],{"className":75073,"style":31846},[31845],[34,75075,75077,75080],{"className":75076},[31676],[34,75078],{"className":75079,"style":32617},[31680],[34,75081,15932],{"className":75082},[31685],[108,75084,75085,46528,75427],{},[34,75086,75088,75157],{"className":75087},[31642],[34,75089,75091],{"className":75090},[31646],[31648,75092,75093],{"xmlns":31650},[31652,75094,75095,75154],{},[31655,75096,75097,75099,75101,75103,75105,75107,75109,75111,75113,75115,75121,75123,75129,75131,75137,75139,75141,75143,75145,75147,75149,75151],{},[31658,75098,46900],{},[31658,75100,15],{},[31662,75102,15972],{"stretchy":702},[31749,75104,18136],{},[31662,75106,32712],{"stretchy":702},[31662,75108,15972],{"stretchy":702},[31749,75110,36656],{},[31662,75112,32712],{"stretchy":702},[31662,75114,202],{},[31743,75116,75117,75119],{},[31658,75118,15],{},[31749,75120,15932],{},[31662,75122,71191],{},[31743,75124,75125,75127],{},[31658,75126,15],{},[31749,75128,18136],{},[31662,75130,71191],{},[31743,75132,75133,75135],{},[31658,75134,15],{},[31749,75136,36656],{},[31662,75138,202],{},[31749,75140,18136],{},[31662,75142,42071],{},[31749,75144,5774],{},[31662,75146,42071],{},[31749,75148,36656],{},[31662,75150,202],{},[31749,75152,75153],{},"1500",[31665,75155,75156],{"encoding":31667},"dp[5][6] = p_4 \\cdot p_5 \\cdot p_6 = 5 \\times 50 \\times 6 = 1500",[34,75158,75160,75199,75254,75309,75364,75382,75400,75418],{"className":75159,"ariaHidden":758},[31672],[34,75161,75163,75166,75169,75172,75175,75178,75181,75184,75187,75190,75193,75196],{"className":75162},[31676],[34,75164],{"className":75165,"style":31774},[31680],[34,75167,46900],{"className":75168},[31685,31686],[34,75170,15],{"className":75171},[31685,31686],[34,75173,15972],{"className":75174},[31782],[34,75176,18136],{"className":75177},[31685],[34,75179,32712],{"className":75180},[31690],[34,75182,15972],{"className":75183},[31782],[34,75185,36656],{"className":75186},[31685],[34,75188,32712],{"className":75189},[31690],[34,75191],{"className":75192,"style":31846},[31845],[34,75194,202],{"className":75195},[31850],[34,75197],{"className":75198,"style":31846},[31845],[34,75200,75202,75205,75245,75248,75251],{"className":75201},[31676],[34,75203],{"className":75204,"style":73503},[31680],[34,75206,75208,75211],{"className":75207},[31685],[34,75209,15],{"className":75210},[31685,31686],[34,75212,75214],{"className":75213},[31792],[34,75215,75217,75237],{"className":75216},[31796,31797],[34,75218,75220,75234],{"className":75219},[31801],[34,75221,75223],{"className":75222,"style":31806},[31805],[34,75224,75225,75228],{"style":31809},[34,75226],{"className":75227,"style":31814},[31813],[34,75229,75231],{"className":75230},[31818,31819,31820,31821],[34,75232,15932],{"className":75233},[31685,31821],[34,75235,31829],{"className":75236},[31828],[34,75238,75240],{"className":75239},[31801],[34,75241,75243],{"className":75242,"style":31836},[31805],[34,75244],{},[34,75246],{"className":75247,"style":32604},[31845],[34,75249,71191],{"className":75250},[32024],[34,75252],{"className":75253,"style":32604},[31845],[34,75255,75257,75260,75300,75303,75306],{"className":75256},[31676],[34,75258],{"className":75259,"style":73503},[31680],[34,75261,75263,75266],{"className":75262},[31685],[34,75264,15],{"className":75265},[31685,31686],[34,75267,75269],{"className":75268},[31792],[34,75270,75272,75292],{"className":75271},[31796,31797],[34,75273,75275,75289],{"className":75274},[31801],[34,75276,75278],{"className":75277,"style":31806},[31805],[34,75279,75280,75283],{"style":31809},[34,75281],{"className":75282,"style":31814},[31813],[34,75284,75286],{"className":75285},[31818,31819,31820,31821],[34,75287,18136],{"className":75288},[31685,31821],[34,75290,31829],{"className":75291},[31828],[34,75293,75295],{"className":75294},[31801],[34,75296,75298],{"className":75297,"style":31836},[31805],[34,75299],{},[34,75301],{"className":75302,"style":32604},[31845],[34,75304,71191],{"className":75305},[32024],[34,75307],{"className":75308,"style":32604},[31845],[34,75310,75312,75315,75355,75358,75361],{"className":75311},[31676],[34,75313],{"className":75314,"style":38774},[31680],[34,75316,75318,75321],{"className":75317},[31685],[34,75319,15],{"className":75320},[31685,31686],[34,75322,75324],{"className":75323},[31792],[34,75325,75327,75347],{"className":75326},[31796,31797],[34,75328,75330,75344],{"className":75329},[31801],[34,75331,75333],{"className":75332,"style":31806},[31805],[34,75334,75335,75338],{"style":31809},[34,75336],{"className":75337,"style":31814},[31813],[34,75339,75341],{"className":75340},[31818,31819,31820,31821],[34,75342,36656],{"className":75343},[31685,31821],[34,75345,31829],{"className":75346},[31828],[34,75348,75350],{"className":75349},[31801],[34,75351,75353],{"className":75352,"style":31836},[31805],[34,75354],{},[34,75356],{"className":75357,"style":31846},[31845],[34,75359,202],{"className":75360},[31850],[34,75362],{"className":75363,"style":31846},[31845],[34,75365,75367,75370,75373,75376,75379],{"className":75366},[31676],[34,75368],{"className":75369,"style":35942},[31680],[34,75371,18136],{"className":75372},[31685],[34,75374],{"className":75375,"style":32604},[31845],[34,75377,42071],{"className":75378},[32024],[34,75380],{"className":75381,"style":32604},[31845],[34,75383,75385,75388,75391,75394,75397],{"className":75384},[31676],[34,75386],{"className":75387,"style":35942},[31680],[34,75389,5774],{"className":75390},[31685],[34,75392],{"className":75393,"style":32604},[31845],[34,75395,42071],{"className":75396},[32024],[34,75398],{"className":75399,"style":32604},[31845],[34,75401,75403,75406,75409,75412,75415],{"className":75402},[31676],[34,75404],{"className":75405,"style":32617},[31680],[34,75407,36656],{"className":75408},[31685],[34,75410],{"className":75411,"style":31846},[31845],[34,75413,202],{"className":75414},[31850],[34,75416],{"className":75417,"style":31846},[31845],[34,75419,75421,75424],{"className":75420},[31676],[34,75422],{"className":75423,"style":32617},[31680],[34,75425,75153],{"className":75426},[31685],[34,75428,75430,75460],{"className":75429},[31642],[34,75431,75433],{"className":75432},[31646],[31648,75434,75435],{"xmlns":31650},[31652,75436,75437,75457],{},[31655,75438,75439,75441,75443,75445,75447,75449,75451,75453,75455],{},[31658,75440,31747],{},[31662,75442,15972],{"stretchy":702},[31749,75444,18136],{},[31662,75446,32712],{"stretchy":702},[31662,75448,15972],{"stretchy":702},[31749,75450,36656],{},[31662,75452,32712],{"stretchy":702},[31662,75454,202],{},[31749,75456,18136],{},[31665,75458,75459],{"encoding":31667},"s[5][6]=5",[34,75461,75463,75499],{"className":75462,"ariaHidden":758},[31672],[34,75464,75466,75469,75472,75475,75478,75481,75484,75487,75490,75493,75496],{"className":75465},[31676],[34,75467],{"className":75468,"style":31774},[31680],[34,75470,31747],{"className":75471},[31685,31686],[34,75473,15972],{"className":75474},[31782],[34,75476,18136],{"className":75477},[31685],[34,75479,32712],{"className":75480},[31690],[34,75482,15972],{"className":75483},[31782],[34,75485,36656],{"className":75486},[31685],[34,75488,32712],{"className":75489},[31690],[34,75491],{"className":75492,"style":31846},[31845],[34,75494,202],{"className":75495},[31850],[34,75497],{"className":75498,"style":31846},[31845],[34,75500,75502,75505],{"className":75501},[31676],[34,75503],{"className":75504,"style":32617},[31680],[34,75506,18136],{"className":75507},[31685],[108,75509,75510,75513,75514,75542,75543],{},[2627,75511,75512],{},"长度 L = 3："," 试探断点 ",[34,75515,75517,75530],{"className":75516},[31642],[34,75518,75520],{"className":75519},[31646],[31648,75521,75522],{"xmlns":31650},[31652,75523,75524,75528],{},[31655,75525,75526],{},[31658,75527,33759],{},[31665,75529,33759],{"encoding":31667},[34,75531,75533],{"className":75532,"ariaHidden":758},[31672],[34,75534,75536,75539],{"className":75535},[31676],[34,75537],{"className":75538,"style":31681},[31680],[34,75540,33759],{"className":75541,"style":33774},[31685,31686]," 取哪一边使得代价最小。",[105,75544,75545,76282,76771,77262],{},[108,75546,75547,76198,76199,46528,76201,347],{},[34,75548,75550,75749],{"className":75549},[31642],[34,75551,75553],{"className":75552},[31646],[31648,75554,75555],{"xmlns":31650},[31652,75556,75557,75746],{},[31655,75558,75559,75561,75563,75565,75567,75569,75571,75573,75575,75577,75579,75581],{},[31658,75560,46900],{},[31658,75562,15],{},[31662,75564,15972],{"stretchy":702},[31749,75566,640],{},[31662,75568,32712],{"stretchy":702},[31662,75570,15972],{"stretchy":702},[31749,75572,786],{},[31662,75574,32712],{"stretchy":702},[31662,75576,202],{},[31658,75578,56084],{},[31662,75580,32682],{},[31655,75582,75583,75585],{},[31662,75584,32685],{"fence":758},[71078,75586,75587,75667],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,75588,75589],{},[71087,75590,75591],{},[71090,75592,75593],{"scriptlevel":459,"displaystyle":702},[31655,75594,75595,75597,75599,75601,75603,75605,75607,75609,75611,75613,75615,75617,75619,75621,75623,75625,75627,75629,75631,75633,75635,75637,75639,75641,75643,75645,75647,75649,75651,75653,75655,75657,75659,75662,75664],{},[31658,75596,33759],{},[31662,75598,202],{},[31749,75600,640],{},[31662,75602,5598],{},[31658,75604,46900],{},[31658,75606,15],{},[31662,75608,15972],{"stretchy":702},[31749,75610,640],{},[31662,75612,32712],{"stretchy":702},[31662,75614,15972],{"stretchy":702},[31749,75616,640],{},[31662,75618,32712],{"stretchy":702},[31662,75620,223],{},[31658,75622,46900],{},[31658,75624,15],{},[31662,75626,15972],{"stretchy":702},[31749,75628,768],{},[31662,75630,32712],{"stretchy":702},[31662,75632,15972],{"stretchy":702},[31749,75634,786],{},[31662,75636,32712],{"stretchy":702},[31662,75638,223],{},[31749,75640,18136],{},[31662,75642,42071],{},[31749,75644,3244],{},[31662,75646,42071],{},[31749,75648,71869],{},[31662,75650,202],{},[31749,75652,459],{},[31662,75654,223],{},[31749,75656,73878],{},[31662,75658,223],{},[31749,75660,75661],{},"600",[31662,75663,202],{},[31749,75665,75666],{},"960",[71084,75668,75669],{},[71087,75670,75671],{},[71090,75672,75673],{"scriptlevel":459,"displaystyle":702},[31655,75674,75675,75677,75679,75681,75683,75685,75687,75689,75691,75693,75695,75697,75699,75701,75703,75705,75707,75709,75711,75713,75715,75717,75719,75721,75723,75725,75727,75729,75731,75733,75735,75737,75739,75741,75743],{},[31658,75676,33759],{},[31662,75678,202],{},[31749,75680,768],{},[31662,75682,5598],{},[31658,75684,46900],{},[31658,75686,15],{},[31662,75688,15972],{"stretchy":702},[31749,75690,640],{},[31662,75692,32712],{"stretchy":702},[31662,75694,15972],{"stretchy":702},[31749,75696,768],{},[31662,75698,32712],{"stretchy":702},[31662,75700,223],{},[31658,75702,46900],{},[31658,75704,15],{},[31662,75706,15972],{"stretchy":702},[31749,75708,786],{},[31662,75710,32712],{"stretchy":702},[31662,75712,15972],{"stretchy":702},[31749,75714,786],{},[31662,75716,32712],{"stretchy":702},[31662,75718,223],{},[31749,75720,18136],{},[31662,75722,42071],{},[31749,75724,786],{},[31662,75726,42071],{},[31749,75728,71869],{},[31662,75730,202],{},[31749,75732,73451],{},[31662,75734,223],{},[31749,75736,459],{},[31662,75738,223],{},[31749,75740,74303],{},[31662,75742,202],{},[31749,75744,75745],{},"330",[31665,75747,75748],{"encoding":31667},"dp[1][3] = \\min \\begin{cases} k=1: dp[1][1] + dp[2][3] + 5\\times 10\\times 12 = 0 + 360 + 600 = 960 \\\\ k=2: dp[1][2] + dp[3][3] + 5\\times 3\\times 12 = 150 + 0 + 180 = 330 \\end{cases}",[34,75750,75752,75791],{"className":75751,"ariaHidden":758},[31672],[34,75753,75755,75758,75761,75764,75767,75770,75773,75776,75779,75782,75785,75788],{"className":75754},[31676],[34,75756],{"className":75757,"style":31774},[31680],[34,75759,46900],{"className":75760},[31685,31686],[34,75762,15],{"className":75763},[31685,31686],[34,75765,15972],{"className":75766},[31782],[34,75768,640],{"className":75769},[31685],[34,75771,32712],{"className":75772},[31690],[34,75774,15972],{"className":75775},[31782],[34,75777,786],{"className":75778},[31685],[34,75780,32712],{"className":75781},[31690],[34,75783],{"className":75784,"style":31846},[31845],[34,75786,202],{"className":75787},[31850],[34,75789],{"className":75790,"style":31846},[31845],[34,75792,75794,75798,75801,75804],{"className":75793},[31676],[34,75795],{"className":75796,"style":75797},[31680],"height:3em;vertical-align:-1.25em;",[34,75799,56084],{"className":75800},[32767],[34,75802],{"className":75803,"style":32123},[31845],[34,75805,75807,75816,76195],{"className":75806},[34006],[34,75808,75812],{"className":75809,"style":75811},[31782,75810],"delimcenter","top:0em;",[34,75813,32685],{"className":75814},[71281,75815],"size4",[34,75817,75819],{"className":75818},[31685],[34,75820,75822],{"className":75821},[71078],[34,75823,75825],{"className":75824},[71384],[34,75826,75828,76186],{"className":75827},[31796,31797],[34,75829,75831,76183],{"className":75830},[31801],[34,75832,75835,76009],{"className":75833,"style":75834},[31805],"height:1.69em;",[34,75836,75838,75841],{"style":75837},"top:-3.69em;",[34,75839],{"className":75840,"style":71401},[31813],[34,75842,75844,75847,75850,75853,75856,75859,75862,75865,75868,75871,75874,75877,75880,75883,75886,75889,75892,75895,75898,75901,75904,75907,75910,75913,75916,75919,75922,75925,75928,75931,75934,75937,75940,75943,75946,75949,75952,75955,75958,75961,75964,75967,75970,75973,75976,75979,75982,75985,75988,75991,75994,75997,76000,76003,76006],{"className":75843},[31685],[34,75845,33759],{"className":75846,"style":33774},[31685,31686],[34,75848],{"className":75849,"style":31846},[31845],[34,75851,202],{"className":75852},[31850],[34,75854],{"className":75855,"style":31846},[31845],[34,75857,640],{"className":75858},[31685],[34,75860],{"className":75861,"style":31846},[31845],[34,75863,5598],{"className":75864},[31850],[34,75866],{"className":75867,"style":31846},[31845],[34,75869,46900],{"className":75870},[31685,31686],[34,75872,15],{"className":75873},[31685,31686],[34,75875,15972],{"className":75876},[31782],[34,75878,640],{"className":75879},[31685],[34,75881,32712],{"className":75882},[31690],[34,75884,15972],{"className":75885},[31782],[34,75887,640],{"className":75888},[31685],[34,75890,32712],{"className":75891},[31690],[34,75893],{"className":75894,"style":32604},[31845],[34,75896,223],{"className":75897},[32024],[34,75899],{"className":75900,"style":32604},[31845],[34,75902,46900],{"className":75903},[31685,31686],[34,75905,15],{"className":75906},[31685,31686],[34,75908,15972],{"className":75909},[31782],[34,75911,768],{"className":75912},[31685],[34,75914,32712],{"className":75915},[31690],[34,75917,15972],{"className":75918},[31782],[34,75920,786],{"className":75921},[31685],[34,75923,32712],{"className":75924},[31690],[34,75926],{"className":75927,"style":32604},[31845],[34,75929,223],{"className":75930},[32024],[34,75932],{"className":75933,"style":32604},[31845],[34,75935,18136],{"className":75936},[31685],[34,75938],{"className":75939,"style":32604},[31845],[34,75941,42071],{"className":75942},[32024],[34,75944],{"className":75945,"style":32604},[31845],[34,75947,3244],{"className":75948},[31685],[34,75950],{"className":75951,"style":32604},[31845],[34,75953,42071],{"className":75954},[32024],[34,75956],{"className":75957,"style":32604},[31845],[34,75959,71869],{"className":75960},[31685],[34,75962],{"className":75963,"style":31846},[31845],[34,75965,202],{"className":75966},[31850],[34,75968],{"className":75969,"style":31846},[31845],[34,75971,459],{"className":75972},[31685],[34,75974],{"className":75975,"style":32604},[31845],[34,75977,223],{"className":75978},[32024],[34,75980],{"className":75981,"style":32604},[31845],[34,75983,73878],{"className":75984},[31685],[34,75986],{"className":75987,"style":32604},[31845],[34,75989,223],{"className":75990},[32024],[34,75992],{"className":75993,"style":32604},[31845],[34,75995,75661],{"className":75996},[31685],[34,75998],{"className":75999,"style":31846},[31845],[34,76001,202],{"className":76002},[31850],[34,76004],{"className":76005,"style":31846},[31845],[34,76007,75666],{"className":76008},[31685],[34,76010,76012,76015],{"style":76011},"top:-2.25em;",[34,76013],{"className":76014,"style":71401},[31813],[34,76016,76018,76021,76024,76027,76030,76033,76036,76039,76042,76045,76048,76051,76054,76057,76060,76063,76066,76069,76072,76075,76078,76081,76084,76087,76090,76093,76096,76099,76102,76105,76108,76111,76114,76117,76120,76123,7612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",[2627,76200,75745],{},[34,76202,76204,76234],{"className":76203},[31642],[34,76205,76207],{"className":76206},[31646],[31648,76208,76209],{"xmlns":31650},[31652,76210,76211,76231],{},[31655,76212,76213,76215,76217,76219,76221,76223,76225,76227,76229],{},[31658,76214,31747],{},[31662,76216,15972],{"stretchy":702},[31749,76218,640],{},[31662,76220,32712],{"stretchy":702},[31662,76222,15972],{"stretchy":702},[31749,76224,786],{},[31662,76226,32712],{"stretchy":702},[31662,76228,202],{},[31749,76230,768],{},[31665,76232,76233],{"encoding":31667},"s[1][3]=2",[34,76235,76237,76273],{"className":76236,"ariaHidden":758},[31672],[34,76238,76240,76243,76246,76249,76252,76255,76258,76261,76264,76267,76270],{"className":76239},[31676],[34,76241],{"className":76242,"style":31774},[31680],[34,76244,31747],{"className":76245},[31685,31686],[34,76247,15972],{"className":76248},[31782],[34,76250,640],{"className":76251},[31685],[34,76253,32712],{"className":76254},[31690],[34,76256,15972],{"className":76257},[31782],[34,76259,786],{"className":76260},[31685],[34,76262,32712],{"className":76263},[31690],[34,76265],{"className":76266,"style":31846},[31845],[34,76268,202],{"className":76269},[31850],[34,76271],{"className":76272,"style":31846},[31845],[34,76274,76276,76279],{"className":76275},[31676],[34,76277],{"className":76278,"style":32617},[31680],[34,76280,768],{"className":76281},[31685],[108,76283,76284,76198,76688,46528,76690,347],{},[34,76285,76287,76403],{"className":76286},[31642],[34,76288,76290],{"className":76289},[31646],[31648,76291,76292],{"xmlns":31650},[31652,76293,76294,76400],{},[31655,76295,76296,76298,76300,76302,76304,76306,76308,76310,76312,76314,76316,76318],{},[31658,76297,46900],{},[31658,76299,15],{},[31662,76301,15972],{"stretchy":702},[31749,76303,768],{},[31662,76305,32712],{"stretchy":702},[31662,76307,15972],{"stretchy":702},[31749,76309,15932],{},[31662,76311,32712],{"stretchy":702},[31662,76313,202],{},[31658,76315,56084],{},[31662,76317,32682],{},[31655,76319,76320,76322],{},[31662,76321,32685],{"fence":758},[71078,76323,76324,76362],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,76325,76326],{},[71087,76327,76328],{},[71090,76329,76330],{"scriptlevel":459,"displaystyle":702},[31655,76331,76332,76334,76336,76338,76340,76342,76344,76346,76348,76350,76352,76354,76356,76358,76360],{},[31658,76333,33759],{},[31662,76335,202],{},[31749,76337,768],{},[31662,76339,5598],{},[31749,76341,459],{},[31662,76343,223],{},[31749,76345,74303],{},[31662,76347,223],{},[31749,76349,3244],{},[31662,76351,42071],{},[31749,76353,786],{},[31662,76355,42071],{},[31749,76357,18136],{},[31662,76359,202],{},[31749,76361,75745],{},[71084,76363,76364],{},[71087,76365,76366],{},[71090,76367,76368],{"scriptlevel":459,"displaystyle":702},[31655,76369,76370,76372,76374,76376,76378,76380,76382,76384,76386,76388,76390,76392,76394,76396,76398],{},[31658,76371,33759],{},[31662,76373,202],{},[31749,76375,786],{},[31662,76377,5598],{},[31749,76379,73878],{},[31662,76381,223],{},[31749,76383,459],{},[31662,76385,223],{},[31749,76387,3244],{},[31662,76389,42071],{},[31749,76391,71869],{},[31662,76393,42071],{},[31749,76395,18136],{},[31662,76397,202],{},[31749,76399,75666],{},[31665,76401,76402],{"encoding":31667},"dp[2][4] = \\min \\begin{cases} k=2: 0 + 180 + 10\\times 3\\times 5 = 330 \\\\ k=3: 360 + 0 + 10\\times 12\\times 5 = 960 \\end{cases}",[34,76404,76406,76445],{"className":76405,"ariaHidden":758},[31672],[34,76407,76409,76412,76415,76418,76421,76424,76427,76430,76433,76436,76439,76442],{"className":76408},[31676],[34,76410],{"className":76411,"style":31774},[31680],[34,76413,46900],{"className":76414},[31685,31686],[34,76416,15],{"className":76417},[31685,31686],[34,76419,15972],{"className":76420},[31782],[34,76422,768],{"className":76423},[31685],[34,76425,32712],{"className":76426},[31690],[34,76428,15972],{"className":76429},[31782],[34,76431,15932],{"className":76432},[31685],[34,76434,32712],{"className":76435},[31690],[34,76437],{"className":76438,"style":31846},[31845],[34,76440,202],{"className":76441},[31850],[34,76443],{"className":76444,"style":31846},[31845],[34,76446,76448,76451,76454,76457],{"className":76447},[31676],[34,76449],{"className":76450,"style":75797},[31680],[34,76452,56084],{"className":76453},[32767],[34,76455],{"className":76456,"style":32123},[31845],[34,76458,76460,76466,76685],{"className":76459},[34006],[34,76461,76463],{"className":76462,"style":75811},[31782,75810],[34,76464,32685],{"className":76465},[71281,75815],[34,76467,76469],{"className":76468},[31685],[34,76470,76472],{"className":76471},[71078],[34,76473,76475],{"className":76474},[71384],[34,76476,76478,76677],{"className":76477},[31796,31797],[34,76479,76481,76674],{"className":76480},[31801],[34,76482,76484,76579],{"className":76483,"style":75834},[31805],[34,76485,76486,76489],{"style":75837},[34,76487],{"className":76488,"style":71401},[31813],[34,76490,76492,76495,76498,76501,76504,76507,76510,76513,76516,76519,76522,76525,76528,76531,76534,76537,76540,76543,76546,76549,76552,76555,76558,76561,76564,76567,76570,76573,76576],{"className":76491},[31685],[34,76493,33759],{"className":76494,"style":33774},[31685,31686],[34,76496],{"className":76497,"style":31846},[31845],[34,76499,202],{"className":76500},[31850],[34,76502],{"className":76503,"style":31846},[31845],[34,76505,768],{"className":76506},[31685],[34,76508],{"className":76509,"style":31846},[31845],[34,76511,5598],{"className":76512},[31850],[34,76514],{"className":76515,"style":31846},[31845],[34,76517,459],{"className":76518},[31685],[34,76520],{"className":76521,"style":32604},[31845],[34,76523,223],{"className":76524},[32024],[34,76526],{"className":76527,"style":32604},[31845],[34,76529,74303],{"className":76530},[31685],[34,76532],{"className":76533,"style":32604},[31845],[34,76535,223],{"className":76536},[32024],[34,76538],{"className":76539,"style":32604},[31845],[34,76541,3244],{"className":76542},[31685],[34,76544],{"className":76545,"style":32604},[31845],[34,76547,42071],{"className":76548},[32024],[34,76550],{"className":76551,"style":32604},[31845],[34,76553,786],{"className":76554},[31685],[34,76556],{"className":76557,"style":32604},[31845],[34,76559,42071],{"className":76560},[32024],[34,76562],{"className":76563,"style":32604},[31845],[34,76565,18136],{"className":76566},[31685],[34,76568],{"className":76569,"style":31846},[31845],[34,76571,202],{"className":76572},[31850],[34,76574],{"className":76575,"style":31846},[31845],[34,76577,75745],{"className":76578},[31685],[34,76580,76581,76584],{"style":76011},[34,76582],{"className":76583,"style":71401},[31813],[34,76585,76587,76590,76593,76596,76599,76602,76605,76608,76611,76614,76617,76620,76623,76626,76629,76632,76635,76638,76641,76644,76647,76650,76653,76656,76659,76662,76665,76668,76671],{"className":76586},[31685],[34,76588,33759],{"className":76589,"style":33774},[31685,31686],[34,76591],{"className":76592,"style":31846},[31845],[34,76594,202],{"className":76595},[31850],[34,76597],{"className":76598,"style":31846},[31845],[34,76600,786],{"className":76601},[31685],[34,76603],{"className":76604,"style":31846},[31845],[34,76606,5598],{"className":76607},[31850],[34,76609],{"className":76610,"style":31846},[31845],[34,76612,73878],{"className":76613},[31685],[34,76615],{"className":76616,"style":32604},[31845],[34,76618,223],{"className":76619},[32024],[34,76621],{"className":76622,"style":32604},[31845],[34,76624,459],{"className":76625},[31685],[34,76627],{"className":76628,"style":32604},[31845],[34,76630,223],{"className":76631},[32024],[34,76633],{"className":76634,"style":32604},[31845],[34,76636,3244],{"className":76637},[31685],[34,76639],{"className":76640,"style":32604},[31845],[34,76642,42071],{"className":76643},[32024],[34,76645],{"className":76646,"style":32604},[31845],[34,76648,71869],{"className":76649},[31685],[34,76651],{"className":76652,"style":32604},[31845],[34,76654,42071],{"className":76655},[32024],[34,76657],{"className":76658,"style":32604},[31845],[34,76660,18136],{"className":76661},[31685],[34,76663],{"className":76664,"style":31846},[31845],[34,76666,202],{"className":76667},[31850],[34,76669],{"className":76670,"style":31846},[31845],[34,76672,75666],{"className":76673},[31685],[34,76675,31829],{"className":76676},[31828],[34,76678,76680],{"className":76679},[31801],[34,76681,76683],{"className":76682,"style":76192},[31805],[34,76684],{},[34,76686],{"className":76687},[31690,41488],[2627,76689,75745],{},[34,76691,76693,76723],{"className":76692},[31642],[34,76694,76696],{"className":76695},[31646],[31648,76697,76698],{"xmlns":31650},[31652,76699,76700,76720],{},[31655,76701,76702,76704,76706,76708,76710,76712,76714,76716,76718],{},[31658,76703,31747],{},[31662,76705,15972],{"stretchy":702},[31749,76707,768],{},[31662,76709,32712],{"stretchy":702},[31662,76711,15972],{"stretchy":702},[31749,76713,15932],{},[31662,76715,32712],{"stretchy":702},[31662,76717,202],{},[31749,76719,768],{},[31665,76721,76722],{"encoding":31667},"s[2][4]=2",[34,76724,76726,76762],{"className":76725,"ariaHidden":758},[31672],[34,76727,76729,76732,76735,76738,76741,76744,76747,76750,76753,76756,76759],{"className":76728},[31676],[34,76730],{"className":76731,"style":31774},[31680],[34,76733,31747],{"className":76734},[31685,31686],[34,76736,15972],{"className":76737},[31782],[34,76739,768],{"className":76740},[31685],[34,76742,32712],{"className":76743},[31690],[34,76745,15972],{"className":76746},[31782],[34,76748,15932],{"className":76749},[31685],[34,76751,32712],{"className":76752},[31690],[34,76754],{"className":76755,"style":31846},[31845],[34,76757,202],{"className":76758},[31850],[34,76760],{"className":76761,"style":31846},[31845],[34,76763,76765,76768],{"className":76764},[31676],[34,76766],{"className":76767,"style":32617},[31680],[34,76769,768],{"className":76770},[31685],[108,76772,76773,76198,77179,46528,77181,347],{},[34,76774,76776,76894],{"className":76775},[31642],[34,76777,76779],{"className":76778},[31646],[31648,76780,76781],{"xmlns":31650},[31652,76782,76783,76891],{},[31655,76784,76785,76787,76789,76791,76793,76795,76797,76799,76801,76803,76805,76807],{},[31658,76786,46900],{},[31658,76788,15],{},[31662,76790,15972],{"stretchy":702},[31749,76792,786],{},[31662,76794,32712],{"stretchy":702},[31662,76796,15972],{"stretchy":702},[31749,76798,18136],{},[31662,76800,32712],{"stretchy":702},[31662,76802,202],{},[31658,76804,56084],{},[31662,76806,32682],{},[31655,76808,76809,76811],{},[31662,76810,32685],{"fence":758},[71078,76812,76813,76852],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,76814,76815],{},[71087,76816,76817],{},[71090,76818,76819],{"scriptlevel":459,"displaystyle":702},[31655,76820,76821,76823,76825,76827,76829,76831,76833,76835,76837,76839,76841,76843,76845,76847,76849],{},[31658,76822,33759],{},[31662,76824,202],{},[31749,76826,786],{},[31662,76828,5598],{},[31749,76830,459],{},[31662,76832,223],{},[31749,76834,74728],{},[31662,76836,223],{},[31749,76838,786],{},[31662,76840,42071],{},[31749,76842,71869],{},[31662,76844,42071],{},[31749,76846,5774],{},[31662,76848,202],{},[31749,76850,76851],{},"4800",[71084,76853,76854],{},[71087,76855,76856],{},[71090,76857,76858],{"scriptlevel":459,"displaystyle":702},[31655,76859,76860,76862,76864,76866,76868,76870,76872,76874,76876,76878,76880,76882,76884,76886,76888],{},[31658,76861,33759],{},[31662,76863,202],{},[31749,76865,15932],{},[31662,76867,5598],{},[31749,76869,74303],{},[31662,76871,223],{},[31749,76873,459],{},[31662,76875,223],{},[31749,76877,786],{},[31662,76879,42071],{},[31749,76881,18136],{},[31662,76883,42071],{},[31749,76885,5774],{},[31662,76887,202],{},[31749,76889,76890],{},"930",[31665,76892,76893],{"encoding":31667},"dp[3][5] = \\min \\begin{cases} k=3: 0 + 3000 + 3\\times 12\\times 50 = 4800 \\\\ k=4: 180 + 0 + 3\\times 5\\times 50 = 930 \\end{cases}",[34,76895,76897,76936],{"className":76896,"ariaHidden":758},[31672],[34,76898,76900,76903,76906,76909,76912,76915,76918,76921,76924,76927,76930,76933],{"className":76899},[31676],[34,76901],{"className":76902,"style":31774},[31680],[34,76904,46900],{"className":76905},[31685,31686],[34,76907,15],{"className":76908},[31685,31686],[34,76910,15972],{"className":76911},[31782],[34,76913,786],{"className":76914},[31685],[34,76916,32712],{"className":76917},[31690],[34,76919,15972],{"className":76920},[31782],[34,76922,18136],{"className":76923},[31685],[34,76925,32712],{"className":76926},[31690],[34,76928],{"className":76929,"style":31846},[31845],[34,76931,202],{"className":76932},[31850],[34,76934],{"className":76935,"style":31846},[31845],[34,76937,76939,76942,76945,76948],{"className":76938},[31676],[34,76940],{"className":76941,"style":75797},[31680],[34,76943,56084],{"className":76944},[32767],[34,76946],{"className":76947,"style":32123},[31845],[34,76949,76951,76957,77176],{"className":76950},[34006],[34,76952,76954],{"className":76953,"style":75811},[31782,75810],[34,76955,32685],{"className":76956},[71281,75815],[34,76958,76960],{"className":76959},[31685],[34,76961,76963],{"className":76962},[71078],[34,76964,76966],{"className":76965},[71384],[34,76967,76969,77168],{"className":76968},[31796,31797],[34,76970,76972,77165],{"className":76971},[31801],[34,76973,76975,77070],{"className":76974,"style":75834},[31805],[34,76976,76977,76980],{"style":75837},[34,76978],{"className":76979,"style":71401},[31813],[34,76981,76983,76986,76989,76992,76995,76998,77001,77004,77007,77010,77013,77016,77019,77022,77025,77028,77031,77034,77037,77040,77043,77046,77049,77052,77055,77058,77061,77064,77067],{"className":76982},[31685],[34,76984,33759],{"className":76985,"style":33774},[31685,31686],[34,76987],{"className":76988,"style":31846},[31845],[34,76990,202],{"className":76991},[31850],[34,76993],{"className":76994,"style":31846},[31845],[34,76996,786],{"className":76997},[31685],[34,76999],{"className":77000,"style":31846},[31845],[34,77002,5598],{"className":77003},[31850],[34,77005],{"className":77006,"style":31846},[31845],[34,77008,459],{"className":77009},[31685],[34,77011],{"className":77012,"style":32604},[31845],[34,77014,223],{"className":77015},[32024],[34,77017],{"className":77018,"style":32604},[31845],[34,77020,74728],{"className":77021},[31685],[34,77023],{"className":77024,"style":32604},[31845],[34,77026,223],{"className":77027},[32024],[34,77029],{"className":77030,"style":32604},[31845],[34,77032,786],{"className":77033},[31685],[34,77035],{"className":77036,"style":32604},[31845],[34,77038,42071],{"className":77039},[32024],[34,77041],{"className":77042,"style":32604},[31845],[34,77044,71869],{"className":77045},[31685],[34,77047],{"className":77048,"style":32604},[31845],[34,77050,42071],{"className":77051},[32024],[34,77053],{"className":77054,"style":32604},[31845],[34,77056,5774],{"className":77057},[31685],[34,77059],{"className":77060,"style":31846},[31845],[34,77062,202],{"className":77063},[31850],[34,77065],{"className":77066,"style":31846},[31845],[34,77068,76851],{"className":77069},[31685],[34,77071,77072,77075],{"style":76011},[34,77073],{"className":77074,"style":71401},[31813],[34,77076,77078,77081,77084,77087,77090,77093,77096,77099,77102,77105,77108,77111,77114,77117,77120,77123,77126,77129,77132,77135,77138,77141,77144,77147,77150,77153,77156,77159,77162],{"className":77077},[31685],[34,77079,33759],{"className":77080,"style":33774},[31685,31686],[34,77082],{"className":77083,"style":31846},[31845],[34,77085,202],{"className":77086},[31850],[34,77088],{"className":77089,"style":31846},[31845],[34,77091,15932],{"className":77092},[31685],[34,77094],{"className":77095,"style":31846},[31845],[34,77097,5598],{"className":77098},[31850],[34,77100],{"className":77101,"style":31846},[31845],[34,77103,74303],{"className":77104},[31685],[34,77106],{"className":77107,"style":32604},[31845],[34,77109,223],{"className":77110},[32024],[34,77112],{"className":77113,"style":32604},[31845],[34,77115,459],{"className":77116},[31685],[34,77118],{"className":77119,"style":32604},[31845],[34,77121,223],{"className":77122},[32024],[34,77124],{"className":77125,"style":32604},[31845],[34,77127,786],{"className":77128},[31685],[34,77130],{"className":77131,"style":32604},[31845],[34,77133,42071],{"className":77134},[32024],[34,77136],{"className":77137,"style":32604},[31845],[34,77139,18136],{"className":77140},[31685],[34,77142],{"className":77143,"style":32604},[31845],[34,77145,42071],{"className":77146},[32024],[34,77148],{"className":77149,"style":32604},[31845],[34,77151,5774],{"className":77152},[31685],[34,77154],{"className":77155,"style":31846},[31845],[34,77157,202],{"className":77158},[31850],[34,77160],{"className":77161,"style":31846},[31845],[34,77163,76890],{"className":77164},[31685],[34,77166,31829],{"className":77167},[31828],[34,77169,77171],{"className":77170},[31801],[34,77172,77174],{"className":77173,"style":76192},[31805],[34,77175],{},[34,77177],{"className":77178},[31690,41488],[2627,77180,76890],{},[34,77182,77184,77214],{"className":77183},[31642],[34,77185,77187],{"className":77186},[31646],[31648,77188,77189],{"xmlns":31650},[31652,77190,77191,77211],{},[31655,77192,77193,77195,77197,77199,77201,77203,77205,77207,77209],{},[31658,77194,31747],{},[31662,77196,15972],{"stretchy":702},[31749,77198,786],{},[31662,77200,32712],{"stretchy":702},[31662,77202,15972],{"stretchy":702},[31749,77204,18136],{},[31662,77206,32712],{"stretchy":702},[31662,77208,202],{},[31749,77210,15932],{},[31665,77212,77213],{"encoding":31667},"s[3][5]=4",[34,77215,77217,77253],{"className":77216,"ariaHidden":758},[31672],[34,77218,77220,77223,77226,77229,77232,77235,77238,77241,77244,77247,77250],{"className":77219},[31676],[34,77221],{"className":77222,"style":31774},[31680],[34,77224,31747],{"className":77225},[31685,31686],[34,77227,15972],{"className":77228},[31782],[34,77230,786],{"className":77231},[31685],[34,77233,32712],{"className":77234},[31690],[34,77236,15972],{"className":77237},[31782],[34,77239,18136],{"className":77240},[31685],[34,77242,32712],{"className":77243},[31690],[34,77245],{"className":77246,"style":31846},[31845],[34,77248,202],{"className":77249},[31850],[34,77251],{"className":77252,"style":31846},[31845],[34,77254,77256,77259],{"className":77255},[31676],[34,77257],{"className":77258,"style":32617},[31680],[34,77260,15932],{"className":77261},[31685],[108,77263,77264,76198,77670,46528,77672,347],{},[34,77265,77267,77385],{"className":77266},[31642],[34,77268,77270],{"className":77269},[31646],[31648,77271,77272],{"xmlns":31650},[31652,77273,77274,77382],{},[31655,77275,77276,77278,77280,77282,77284,77286,77288,77290,77292,77294,77296,77298],{},[31658,77277,46900],{},[31658,77279,15],{},[31662,77281,15972],{"stretchy":702},[31749,77283,15932],{},[31662,77285,32712],{"stretchy":702},[31662,77287,15972],{"stretchy":702},[31749,77289,36656],{},[31662,77291,32712],{"stretchy":702},[31662,77293,202],{},[31658,77295,56084],{},[31662,77297,32682],{},[31655,77299,77300,77302],{},[31662,77301,32685],{"fence":758},[71078,77303,77304,77343],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,77305,77306],{},[71087,77307,77308],{},[71090,77309,77310],{"scriptlevel":459,"displaystyle":702},[31655,77311,77312,77314,77316,77318,77320,77322,77324,77326,77328,77330,77332,77334,77336,77338,77340],{},[31658,77313,33759],{},[31662,77315,202],{},[31749,77317,15932],{},[31662,77319,5598],{},[31749,77321,459],{},[31662,77323,223],{},[31749,77325,75153],{},[31662,77327,223],{},[31749,77329,71869],{},[31662,77331,42071],{},[31749,77333,18136],{},[31662,77335,42071],{},[31749,77337,36656],{},[31662,77339,202],{},[31749,77341,77342],{},"1860",[71084,77344,77345],{},[71087,77346,77347],{},[71090,77348,77349],{"scriptlevel":459,"displaystyle":702},[31655,77350,77351,77353,77355,77357,77359,77361,77363,77365,77367,77369,77371,77373,77375,77377,77379],{},[31658,77352,33759],{},[31662,77354,202],{},[31749,77356,18136],{},[31662,77358,5598],{},[31749,77360,74728],{},[31662,77362,223],{},[31749,77364,459],{},[31662,77366,223],{},[31749,77368,71869],{},[31662,77370,42071],{},[31749,77372,5774],{},[31662,77374,42071],{},[31749,77376,36656],{},[31662,77378,202],{},[31749,77380,77381],{},"6600",[31665,77383,77384],{"encoding":31667},"dp[4][6] = \\min \\begin{cases} k=4: 0 + 1500 + 12\\times 5\\times 6 = 1860 \\\\ k=5: 3000 + 0 + 12\\times 50\\times 6 = 6600 \\end{cases}",[34,77386,77388,77427],{"className":77387,"ariaHidden":758},[31672],[34,77389,77391,77394,77397,77400,77403,77406,77409,77412,77415,77418,77421,77424],{"className":77390},[31676],[34,77392],{"className":77393,"style":31774},[31680],[34,77395,46900],{"className":77396},[31685,31686],[34,77398,15],{"className":77399},[31685,31686],[34,77401,15972],{"className":77402},[31782],[34,77404,15932],{"className":77405},[31685],[34,77407,32712],{"className":77408},[31690],[34,77410,15972],{"className":77411},[31782],[34,77413,36656],{"className":77414},[31685],[34,77416,32712],{"className":77417},[31690],[34,77419],{"className":77420,"style":31846},[31845],[34,77422,202],{"className":77423},[31850],[34,77425],{"className":77426,"style":31846},[31845],[34,77428,77430,77433,77436,77439],{"className":77429},[31676],[34,77431],{"className":77432,"style":75797},[31680],[34,77434,56084],{"className":77435},[32767],[34,77437],{"className":77438,"style":32123},[31845],[34,77440,77442,77448,77667],{"className":77441},[34006],[34,77443,77445],{"className":77444,"style":75811},[31782,75810],[34,77446,32685],{"className":77447},[71281,75815],[34,77449,77451],{"className":77450},[31685],[34,77452,77454],{"className":77453},[71078],[34,77455,77457],{"className":77456},[71384],[34,77458,77460,77659],{"className":77459},[31796,31797],[34,77461,77463,77656],{"className":77462},[31801],[34,77464,77466,77561],{"className":77465,"style":75834},[31805],[34,77467,77468,77471],{"style":75837},[34,77469],{"className":77470,"style":71401},[31813],[34,77472,77474,77477,77480,77483,77486,77489,77492,77495,77498,77501,77504,77507,77510,77513,77516,77519,77522,77525,77528,77531,77534,77537,77540,77543,77546,77549,77552,77555,77558],{"className":77473},[31685],[34,77475,33759],{"className":77476,"style":33774},[31685,31686],[34,77478],{"className":77479,"style":31846},[31845],[34,77481,202],{"className":77482},[31850],[34,77484],{"className":77485,"style":31846},[31845],[34,77487,15932],{"className":77488},[31685],[34,77490],{"className":77491,"style":31846},[31845],[34,77493,5598],{"className":77494},[31850],[34,77496],{"className":77497,"style":31846},[31845],[34,77499,459],{"className":77500},[31685],[34,77502],{"className":77503,"style":32604},[31845],[34,77505,223],{"className":77506},[32024],[34,77508],{"className":77509,"style":32604},[31845],[34,77511,75153],{"className":77512},[31685],[34,77514],{"className":77515,"style":32604},[31845],[34,77517,223],{"className":77518},[32024],[34,77520],{"className":77521,"style":32604},[31845],[34,77523,71869],{"className":77524},[31685],[34,77526],{"className":77527,"style":32604},[31845],[34,77529,42071],{"className":77530},[32024],[34,77532],{"className":77533,"style":32604},[31845],[34,77535,18136],{"className":77536},[31685],[34,77538],{"className":77539,"style":32604},[31845],[34,77541,42071],{"className":77542},[32024],[34,77544],{"className":77545,"style":32604},[31845],[34,77547,36656],{"className":77548},[31685],[34,77550],{"className":77551,"style":31846},[31845],[34,77553,202],{"className":77554},[31850],[34,77556],{"className":77557,"style":31846},[31845],[34,77559,77342],{"className":77560},[31685],[34,77562,77563,77566],{"style":76011},[34,77564],{"className":77565,"style":71401},[31813],[34,77567,77569,77572,77575,77578,77581,77584,77587,77590,77593,77596,77599,77602,77605,77608,77611,77614,77617,77620,77623,77626,77629,77632,77635,77638,77641,77644,77647,77650,77653],{"className":77568},[31685],[34,77570,33759],{"className":77571,"style":33774},[31685,31686],[34,77573],{"className":77574,"style":31846},[31845],[34,77576,202],{"className":77577},[31850],[34,77579],{"className":77580,"style":31846},[31845],[34,77582,18136],{"className":77583},[31685],[34,77585],{"className":77586,"style":31846},[31845],[34,77588,5598],{"className":77589},[31850],[34,77591],{"className":77592,"style":31846},[31845],[34,77594,74728],{"className":77595},[31685],[34,77597],{"className":77598,"style":32604},[31845],[34,77600,223],{"className":77601},[32024],[34,77603],{"className":77604,"style":32604},[31845],[34,77606,459],{"className":77607},[31685],[34,77609],{"className":77610,"style":32604},[31845],[34,77612,223],{"className":77613},[32024],[34,77615],{"className":77616,"style":32604},[31845],[34,77618,71869],{"className":77619},[31685],[34,77621],{"className":77622,"style":32604},[31845],[34,77624,42071],{"className":77625},[32024],[34,77627],{"className":77628,"style":32604},[31845],[34,77630,5774],{"className":77631},[31685],[34,77633],{"className":77634,"style":32604},[31845],[34,77636,42071],{"className":77637},[32024],[34,77639],{"className":77640,"style":32604},[31845],[34,77642,36656],{"className":77643},[31685],[34,77645],{"className":77646,"style":31846},[31845],[34,77648,202],{"className":77649},[31850],[34,77651],{"className":77652,"style":31846},[31845],[34,77654,77381],{"className":77655},[31685],[34,77657,31829],{"className":77658},[31828],[34,77660,77662],{"className":77661},[31801],[34,77663,77665],{"className":77664,"style":76192},[31805],[34,77666],{},[34,77668],{"className":77669},[31690,41488],[2627,77671,77342],{},[34,77673,77675,77705],{"className":77674},[31642],[34,77676,77678],{"className":77677},[31646],[31648,77679,77680],{"xmlns":31650},[31652,77681,77682,77702],{},[31655,77683,77684,77686,77688,77690,77692,77694,77696,77698,77700],{},[31658,77685,31747],{},[31662,77687,15972],{"stretchy":702},[31749,77689,15932],{},[31662,77691,32712],{"stretchy":702},[31662,77693,15972],{"stretchy":702},[31749,77695,36656],{},[31662,77697,32712],{"stretchy":702},[31662,77699,202],{},[31749,77701,15932],{},[31665,77703,77704],{"encoding":31667},"s[4][6]=4",[34,77706,77708,77744],{"className":77707,"ariaHidden":758},[31672],[34,77709,77711,77714,77717,77720,77723,77726,77729,77732,77735,77738,77741],{"className":77710},[31676],[34,77712],{"className":77713,"style":31774},[31680],[34,77715,31747],{"className":77716},[31685,31686],[34,77718,15972],{"className":77719},[31782],[34,77721,15932],{"className":77722},[31685],[34,77724,32712],{"className":77725},[31690],[34,77727,15972],{"className":77728},[31782],[34,77730,36656],{"className":77731},[31685],[34,77733,32712],{"className":77734},[31690],[34,77736],{"className":77737,"style":31846},[31845],[34,77739,202],{"className":77740},[31850],[34,77742],{"className":77743,"style":31846},[31845],[34,77745,77747,77750],{"className":77746},[31676],[34,77748],{"className":77749,"style":32617},[31680],[34,77751,15932],{"className":77752},[31685],[108,77754,77755,77758],{},[2627,77756,77757],{},"长度 L = 4：",[105,77759,77760,78472,79169],{},[108,77761,77762,76198,78389,46528,78391,347],{},[34,77763,77765,77922],{"className":77764},[31642],[34,77766,77768],{"className":77767},[31646],[31648,77769,77770],{"xmlns":31650},[31652,77771,77772,77919],{},[31655,77773,77774,77776,77778,77780,77782,77784,77786,77788,77790,77792,77794,77796],{},[31658,77775,46900],{},[31658,77777,15],{},[31662,77779,15972],{"stretchy":702},[31749,77781,640],{},[31662,77783,32712],{"stretchy":702},[31662,77785,15972],{"stretchy":702},[31749,77787,15932],{},[31662,77789,32712],{"stretchy":702},[31662,77791,202],{},[31658,77793,56084],{},[31662,77795,32682],{},[31655,77797,77798,77800],{},[31662,77799,32685],{"fence":758},[71078,77801,77802,77841,77880],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,77803,77804],{},[71087,77805,77806],{},[71090,77807,77808],{"scriptlevel":459,"displaystyle":702},[31655,77809,77810,77812,77814,77816,77818,77820,77822,77824,77826,77828,77830,77832,77834,77836,77838],{},[31658,77811,33759],{},[31662,77813,202],{},[31749,77815,640],{},[31662,77817,5598],{},[31749,77819,459],{},[31662,77821,223],{},[31749,77823,75745],{},[31662,77825,223],{},[31749,77827,18136],{},[31662,77829,42071],{},[31749,77831,3244],{},[31662,77833,42071],{},[31749,77835,18136],{},[31662,77837,202],{},[31749,77839,77840],{},"580",[71084,77842,77843],{},[71087,77844,77845],{},[71090,77846,77847],{"scriptlevel":459,"displaystyle":702},[31655,77848,77849,77851,77853,77855,77857,77859,77861,77863,77865,77867,77869,77871,77873,77875,77877],{},[31658,77850,33759],{},[31662,77852,202],{},[31749,77854,768],{},[31662,77856,5598],{},[31749,77858,73451],{},[31662,77860,223],{},[31749,77862,74303],{},[31662,77864,223],{},[31749,77866,18136],{},[31662,77868,42071],{},[31749,77870,786],{},[31662,77872,42071],{},[31749,77874,18136],{},[31662,77876,202],{},[31749,77878,77879],{},"405",[71084,77881,77882],{},[71087,77883,77884],{},[71090,77885,77886],{"scriptlevel":459,"displaystyle":702},[31655,77887,77888,77890,77892,77894,77896,77898,77900,77902,77904,77906,77908,77910,77912,77914,77916],{},[31658,77889,33759],{},[31662,77891,202],{},[31749,77893,786],{},[31662,77895,5598],{},[31749,77897,75745],{},[31662,77899,223],{},[31749,77901,459],{},[31662,77903,223],{},[31749,77905,18136],{},[31662,77907,42071],{},[31749,77909,71869],{},[31662,77911,42071],{},[31749,77913,18136],{},[31662,77915,202],{},[31749,77917,77918],{},"630",[31665,77920,77921],{"encoding":31667},"dp[1][4] = \\min \\begin{cases} k=1: 0 + 330 + 5\\times 10\\times 5 = 580 \\\\ k=2: 150 + 180 + 5\\times 3\\times 5 = 405 \\\\ k=3: 330 + 0 + 5\\times 12\\times 5 = 630 \\end{cases}",[34,77923,77925,77964],{"className":77924,"ariaHidden":758},[31672],[34,77926,77928,77931,77934,77937,77940,77943,77946,77949,77952,77955,77958,77961],{"className":77927},[31676],[34,77929],{"className":77930,"style":31774},[31680],[34,77932,46900],{"className":77933},[31685,31686],[34,77935,15],{"className":77936},[31685,31686],[34,77938,15972],{"className":77939},[31782],[34,77941,640],{"className":77942},[31685],[34,77944,32712],{"className":77945},[31690],[34,77947,15972],{"className":77948},[31782],[34,77950,15932],{"className":77951},[31685],[34,77953,32712],{"className":77954},[31690],[34,77956],{"className":77957,"style":31846},[31845],[34,77959,202],{"className":77960},[31850],[34,77962],{"className":77963,"style":31846},[31845],[34,77965,77967,77971,77974,77977],{"className":77966},[31676],[34,77968],{"className":77969,"style":77970},[31680],"height:4.32em;vertical-align:-1.91em;",[34,77972,56084],{"className":77973},[32767],[34,77975],{"className":77976,"style":32123},[31845],[34,77978,77980,78067,78386],{"className":77979},[34006],[34,77981,77983],{"className":77982},[31782],[34,77984,77986],{"className":77985},[71281,71282],[34,77987,77989,78058],{"className":77988},[31796,31797],[34,77990,77992,78055],{"className":77991},[31801],[34,77993,77996,78007,78023,78033,78044],{"className":77994,"style":77995},[31805],"height:2.35em;",[34,77997,77999,78002],{"style":77998},"top:-2.2em;",[34,78000],{"className":78001,"style":71299},[31813],[34,78003,78005],{"className":78004},[71303,71304],[34,78006,71307],{},[34,78008,78010,78013],{"style":78009},"top:-2.192em;",[34,78011],{"className":78012,"style":71299},[31813],[34,78014,78016],{"style":78015},"height:0.316em;width:0.8889em;",[41588,78017,78020],{"xmlns":41590,"width":71319,"height":78018,"style":71321,"viewBox":78019,"preserveAspectRatio":71323},"0.316em","0 0 888.89 316",[20398,78021],{"d":78022},"M384 0 H504 V316 H384z M384 0 H504 V316 H384z",[34,78024,78025,78028],{"style":71329},[34,78026],{"className":78027,"style":71299},[31813],[34,78029,78031],{"className":78030},[71303,71304],[34,78032,71338],{},[34,78034,78035,78038],{"style":71341},[34,78036],{"className":78037,"style":71299},[31813],[34,78039,78040],{"style":78015},[41588,78041,78042],{"xmlns":41590,"width":71319,"height":78018,"style":71321,"viewBox":78019,"preserveAspectRatio":71323},[20398,78043],{"d":78022},[34,78045,78047,78050],{"style":78046},"top:-4.6em;",[34,78048],{"className":78049,"style":71299},[31813],[34,78051,78053],{"className":78052},[71303,71304],[34,78054,71362],{},[34,78056,31829],{"className":78057},[31828],[34,78059,78061],{"className":78060},[31801],[34,78062,78065],{"className":78063,"style":78064},[31805],"height:1.85em;",[34,78066],{},[34,78068,78070],{"className":78069},[31685],[34,78071,78073],{"className":78072},[71078],[34,78074,78076],{"className":78075},[71384],[34,78077,78079,78377],{"className":78078},[31796,31797],[34,78080,78082,78374],{"className":78081},[31801],[34,78083,78086,78182,78278],{"className":78084,"style":78085},[31805],"height:2.41em;",[34,78087,78089,78092],{"style":78088},"top:-4.41em;",[34,78090],{"className":78091,"style":71401},[31813],[34,78093,78095,78098,78101,78104,78107,78110,78113,78116,78119,78122,78125,78128,78131,78134,78137,78140,78143,78146,78149,78152,78155,78158,78161,78164,78167,78170,78173,78176,78179],{"className":78094},[31685],[34,78096,33759],{"className":78097,"style":33774},[31685,31686],[34,78099],{"className":78100,"style":31846},[31845],[34,78102,202],{"className":78103},[31850],[34,78105],{"className":78106,"style":31846},[31845],[34,78108,640],{"className":78109},[31685],[34,78111],{"className":78112,"style":31846},[31845],[34,78114,5598],{"className":78115},[31850],[34,78117],{"className":78118,"style":31846},[31845],[34,78120,459],{"className":78121},[31685],[34,78123],{"className":78124,"style":32604},[31845],[34,78126,223],{"className":78127},[32024],[34,78129],{"className":78130,"style":32604},[31845],[34,78132,75745],{"className":78133},[31685],[34,78135],{"className":78136,"style":32604},[31845],[34,78138,223],{"className":78139},[32024],[34,78141],{"className":78142,"style":32604},[31845],[34,78144,18136],{"className":78145},[31685],[34,78147],{"className":78148,"style":32604},[31845],[34,78150,42071],{"className":78151},[32024],[34,78153],{"className":78154,"style":32604},[31845],[34,78156,3244],{"className":78157},[31685],[34,78159],{"className":78160,"style":32604},[31845],[34,78162,42071],{"className":78163},[32024],[34,78165],{"className":78166,"style":32604},[31845],[34,78168,18136],{"className":78169},[31685],[34,78171],{"className":78172,"style":31846},[31845],[34,78174,202],{"className":78175},[31850],[34,78177],{"className":78178,"style":31846},[31845],[34,78180,77840],{"className":78181},[31685],[34,78183,78185,78188],{"style":78184},"top:-2.97em;",[34,78186],{"className":78187,"style":71401},[31813],[34,78189,78191,78194,78197,78200,78203,78206,78209,78212,78215,78218,78221,78224,78227,78230,78233,78236,78239,78242,78245,78248,78251,78254,78257,78260,78263,78266,78269,78272,78275],{"className":78190},[31685],[34,78192,33759],{"className":78193,"style":33774},[31685,31686],[34,78195],{"className":78196,"style":31846},[31845],[34,78198,202],{"className":78199},[31850],[34,78201],{"className":78202,"style":31846},[31845],[34,78204,768],{"className":78205},[31685],[34,78207],{"className":78208,"style":31846},[31845],[34,78210,5598],{"className":78211},[31850],[34,78213],{"className":78214,"style":31846},[31845],[34,78216,73451],{"className":78217},[31685],[34,78219],{"className":78220,"style":32604},[31845],[34,78222,223],{"className":78223},[32024],[34,78225],{"className":78226,"style":32604},[31845],[34,78228,74303],{"className":78229},[31685],[34,78231],{"className":78232,"style":32604},[31845],[34,78234,223],{"className":78235},[32024],[34,78237],{"className":78238,"style":32604},[31845],[34,78240,18136],{"className":78241},[31685],[34,78243],{"className":78244,"style":32604},[31845],[34,78246,42071],{"className":78247},[32024],[34,78249],{"className":78250,"style":32604},[31845],[34,78252,786],{"className":78253},[31685],[34,78255],{"className":78256,"style":32604},[31845],[34,78258,42071],{"className":78259},[32024],[34,78261],{"className":78262,"style":32604},[31845],[34,78264,18136],{"className":78265},[31685],[34,78267],{"className":78268,"style":31846},[31845],[34,78270,202],{"className":78271},[31850],[34,78273],{"className":78274,"style":31846},[31845],[34,78276,77879],{"className":78277},[31685],[34,78279,78281,78284],{"style":78280},"top:-1.53em;",[34,78282],{"className":78283,"style":71401},[31813],[34,78285,78287,78290,78293,78296,78299,78302,78305,78308,78311,78314,78317,78320,78323,78326,78329,78332,78335,78338,78341,78344,78347,78350,78353,78356,78359,78362,78365,78368,78371],{"className":78286},[31685],[34,78288,33759],{"className":78289,"style":33774},[31685,316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0,78612,78614,78616,78618,78620,78622,78624,78626,78628],{},[31658,78601,33759],{},[31662,78603,202],{},[31749,78605,15932],{},[31662,78607,5598],{},[31749,78609,75745],{},[31662,78611,223],{},[31749,78613,459],{},[31662,78615,223],{},[31749,78617,3244],{},[31662,78619,42071],{},[31749,78621,18136],{},[31662,78623,42071],{},[31749,78625,5774],{},[31662,78627,202],{},[31749,78629,78630],{},"2830",[31665,78632,78633],{"encoding":31667},"dp[2][5] = \\min \\begin{cases} k=2: 0 + 930 + 10\\times 3\\times 50 = 2430 \\\\ k=3: 360 + 3000 + 10\\times 12\\times 50 = 9360 \\\\ k=4: 330 + 0 + 10\\times 5\\times 50 = 2830 \\end{cases}",[34,78635,78637,78676],{"className":78636,"ariaHidden":758},[31672],[34,78638,78640,78643,78646,78649,78652,78655,78658,78661,78664,78667,78670,78673],{"className":78639},[31676],[34,78641],{"className":78642,"style":31774},[31680],[34,78644,46900],{"className":78645},[31685,31686],[34,78647,15],{"className":78648},[31685,31686],[34,78650,15972],{"className":78651},[31782],[34,78653,768],{"className":78654},[31685],[34,78656,32712],{"className":78657},[31690],[34,78659,15972],{"className":78660},[31782],[34,78662,18136],{"className":78663},[31685],[34,78665,32712],{"className":78666},[31690],[34,78668],{"className":78669,"style":31846},[31845],[34,78671,202],{"className":78672},[31850],[34,78674],{"className":78675,"style":31846},[31845],[34,78677,78679,78682,78685,78688],{"className":78678},[31676],[34,78680],{"className":78681,"style":77970},[31680],[34,78683,56084],{"className":78684},[32767],[34,78686],{"className":78687,"style":32123},[31845],[34,78689,78691,78769,79083],{"className":78690},[34006],[34,78692,78694],{"className":78693},[31782],[34,78695,78697],{"className":78696},[71281,71282],[34,78698,78700,78761],{"className":78699},[31796,31797],[34,78701,78703,78758],{"className":78702},[31801],[34,78704,78706,78716,78727,78737,78748],{"className":78705,"style":77995},[31805],[34,78707,78708,78711],{"style":77998},[34,78709],{"className":78710,"style":71299},[31813],[34,78712,78714],{"className":78713},[71303,71304],[34,78715,71307],{},[34,78717,78718,78721],{"style":78009},[34,78719],{"className":78720,"style":71299},[31813],[34,78722,78723],{"style":78015},[41588,78724,78725],{"xmlns":41590,"width":71319,"height":78018,"style":71321,"viewBox":78019,"preserveAspectRatio":71323},[20398,78726],{"d":78022},[34,78728,78729,78732],{"style":71329},[34,78730],{"className":78731,"style":71299},[31813],[34,78733,78735],{"className":78734},[71303,71304],[34,78736,71338],{},[34,78738,78739,78742],{"style":71341},[34,78740],{"className":78741,"style":71299},[31813],[34,78743,78744],{"style":78015},[41588,78745,78746],{"xmlns":41590,"width":71319,"height":78018,"style":71321,"viewBox":78019,"preserveAspectRatio":71323},[20398,78747],{"d":78022},[34,78749,78750,78753],{"style":78046},[34,78751],{"className":78752,"style":71299},[31813],[34,78754,78756],{"className":78755},[71303,71304],[34,78757,71362],{},[34,78759,31829],{"className":78760},[31828],[34,78762,78764],{"className":78763},[31801],[34,78765,78767],{"className":78766,"style":78064},[31805],[34,78768],{},[34,78770,78772],{"className":78771},[31685],[34,78773,78775],{"className":78774},[71078],[34,78776,78778],{"className":78777},[71384],[34,78779,78781,79075],{"className":78780},[31796,31797],[34,78782,78784,79072],{"className":78783},[31801],[34,78785,78787,78882,78977],{"className":78786,"style":78085},[31805],[34,78788,78789,78792],{"style":78088},[34,78790],{"className":78791,"style":71401},[31813],[34,78793,78795,78798,78801,78804,78807,78810,78813,78816,78819,78822,78825,78828,78831,78834,78837,78840,78843,78846,78849,78852,78855,78858,78861,78864,78867,78870,78873,78876,78879],{"className":78794},[31685],[34,78796,33759],{"className":78797,"style":33774},[31685,31686],[34,78799],{"className":78800,"style":31846},[31845],[34,78802,202],{"className":78803},[31850],[34,78805],{"className":78806,"style":31846},[31845],[34,78808,768],{"className":78809},[31685],[34,78811],{"className":78812,"style":31846},[31845],[34,78814,5598],{"className":78815},[31850],[34,78817],{"className":78818,"style":31846},[31845],[34,78820,459],{"className":78821},[31685],[34,78823],{"className":78824,"style":32604},[31845],[34,78826,223],{"className":78827},[32024],[34,78829],{"className":78830,"style":32604},[31845],[34,78832,76890],{"className":78833},[31685],[34,78835],{"className":78836,"style":32604},[31845],[34,78838,223],{"className":78839},[32024],[34,78841],{"className":78842,"style":32604},[31845],[34,78844,3244],{"className":78845},[31685],[34,78847],{"className":78848,"style":32604},[31845],[34,78850,42071],{"className":78851},[32024],[34,78853],{"className":78854,"style":32604},[31845],[34,78856,786],{"className":78857},[31685],[34,78859],{"className":78860,"style":32604},[31845],[34,78862,42071],{"className":78863},[32024],[34,78865],{"className":78866,"style":32604},[31845],[34,78868,5774],{"className":78869},[31685],[34,78871],{"className":78872,"style":31846},[31845],[34,78874,202],{"className":78875},[31850],[34,78877],{"className":78878,"style":31846},[31845],[34,78880,78552],{"className":78881},[31685],[34,78883,78884,78887],{"style":78184},[34,78885],{"className":78886,"style":71401},[31813],[34,78888,78890,78893,78896,78899,78902,78905,78908,78911,78914,78917,78920,78923,78926,78929,78932,78935,78938,78941,78944,78947,78950,78953,78956,78959,78962,78965,78968,78971,78974],{"className":78889},[31685],[34,78891,33759],{"className":78892,"style":33774},[31685,31686],[34,78894],{"className":78895,"style":31846},[31845],[34,78897,202],{"c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= \\min \\begin{cases} k=3: 0 + 1860 + 3\\times 12\\times 6 = 2076 \\\\ k=4: 180 + 1500 + 3\\times 5\\times 6 = 1770 \\\\ k=5: 930 + 0 + 3\\times 50\\times 6 = 1830 \\end{cases}",[34,79332,79334,79373],{"className":79333,"ariaHidden":758},[31672],[34,79335,79337,79340,79343,79346,79349,79352,79355,79358,79361,79364,79367,79370],{"className":79336},[31676],[34,79338],{"className":79339,"style":31774},[31680],[34,79341,46900],{"className":79342},[31685,31686],[34,79344,15],{"className":79345},[31685,31686],[34,79347,15972],{"className":79348},[31782],[34,79350,786],{"className":79351},[31685],[34,79353,32712],{"className":79354},[31690],[34,79356,15972],{"className":79357},[31782],[34,79359,36656],{"className":79360},[31685],[34,79362,32712],{"className":79363},[31690],[34,79365],{"className":79366,"style":31846},[31845],[34,79368,202],{"className":79369},[31850],[34,79371],{"className":79372,"style":31846},[31845],[34,79374,79376,79379,79382,79385],{"className":79375},[31676],[34,79377],{"className":79378,"style":77970},[31680],[34,79380,56084],{"className":79381},[32767],[34,79383],{"className":79384,"style":32123},[31845],[34,79386,79388,79466,79780],{"className":79387},[34006],[34,79389,79391],{"className":79390},[31782],[34,79392,79394],{"className":79393},[71281,71282],[34,79395,79397,79458],{"className":79396},[31796,31797],[34,79398,79400,79455],{"className":79399},[31801],[34,79401,79403,79413,79424,79434,79445],{"className":79402,"style":77995},[31805],[34,79404,79405,79408],{"style":77998},[34,79406],{"className":79407,"style":71299},[31813],[34,79409,79411],{"className":79410},[71303,71304],[34,79412,71307],{},[34,79414,79415,79418],{"style":78009},[34,79416],{"className":79417,"style":71299},[31813],[34,79419,79420],{"style":78015},[41588,79421,79422],{"xmlns":41590,"width":71319,"height":78018,"style":71321,"viewBox":78019,"preserveAspectRatio":71323},[20398,79423],{"d":78022},[34,79425,79426,79429],{"style":71329},[34,79427],{"className":79428,"style":71299},[31813],[34,79430,79432],{"className":79431},[71303,71304],[34,79433,71338],{},[34,79435,79436,79439],{"style":71341},[34,79437],{"className":79438,"style":71299},[31813],[34,79440,79441],{"style":78015},[41588,79442,79443],{"xmlns":41590,"width":71319,"height":78018,"style":71321,"viewBox":78019,"preserveAspectRatio":71323},[20398,79444],{"d":78022},[34,79446,79447,79450],{"style":78046},[34,79448],{"className":79449,"style":71299},[31813],[34,79451,79453],{"className":79452},[71303,71304],[34,79454,71362],{},[34,79456,31829],{"className":79457},[31828],[34,79459,79461],{"className":79460},[31801],[34,79462,79464],{"className":79463,"style":78064},[31805],[34,79465],{},[34,79467,79469],{"className":79468},[31685],[34,79470,79472],{"className":79471},[71078],[34,79473,79475],{"className":79474},[71384],[34,79476,79478,79772],{"className":79477},[31796,31797],[34,79479,79481,79769],{"className":79480},[31801],[34,79482,79484,79579,79674],{"className":79483,"style":78085},[31805],[34,79485,79486,79489],{"style":78088},[34,79487],{"className":79488,"style":71401},[31813],[34,79490,79492,79495,79498,79501,79504,79507,79510,79513,79516,79519,79522,79525,79528,79531,79534,79537,79540,79543,79546,79549,79552,79555,79558,79561,79564,79567,79570,79573,79576],{"className":79491},[31685],[34,79493,33759],{"className":79494,"style":33774},[31685,31686],[34,79496],{"className":79497,"style":31846},[31845],[34,79499,202],{"className":79500},[31850],[34,79502],{"className":79503,"style":31846},[31845],[34,79505,786],{"className":79506},[31685],[34,79508],{"className":79509,"style":31846},[31845],[34,79511,5598],{"className":79512},[31850],[34,79514],{"className":79515,"style":31846},[31845],[34,79517,459],{"className":79518},[31685],[34,79520],{"className":79521,"style":32604},[31845],[34,79523,223],{"className":79524},[32024],[34,79526],{"className":79527,"style":32604},[31845],[34,79529,77342],{"className":79530},[31685],[34,79532],{"className":79533,"style":32604},[31845],[34,79535,223],{"className":79536},[32024],[34,79538],{"className":79539,"style":32604},[31845],[34,79541,786],{"className":79542},[31685],[34,79544],{"className":79545,"style":32604},[31845],[34,79547,42071],{"className":79548},[32024],[34,79550],{"className":79551,"style":32604},[31845],[34,79553,71869],{"className":79554},[31685],[34,79556],{"className":79557,"style":32604},[31845],[34,79559,42071],{"className":79560},[32024],[34,79562],{"className":79563,"style":32604},[31845],[34,79565,36656],{"className":79566},[31685],[34,79568],{"className":79569,"style":31846},[31845],[34,79571,202],{"className":79572},[31850],[34,79574],{"className":79575,"style":31846},[31845],[34,79577,79249],{"className":79578},[31685],[34,79580,79581,79584],{"style":78184},[34,79582],{"className":79583,"style":71401},[31813],[34,79585,79587,79590,79593,79596,79599,79602,79605,79608,79611,79614,79617,79620,79623,79626,79629,79632,79635,79638,79641,79644,79647,79650,79653,79656,79659,79662,79665,79668,79671],{"className":79586},[31685],[34,79588,33759],{"className":79589,"style":33774},[31685,31686],[34,79591],{"className":79592,"style":31846},[31845],[34,79594,202],{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L = 5：",[105,79872,79873,80592],{},[108,79874,79875,76198,80509,46528,80511,347],{},[34,79876,79878,80044],{"className":79877},[31642],[34,79879,79881],{"className":79880},[31646],[31648,79882,79883],{"xmlns":31650},[31652,79884,79885,80041],{},[31655,79886,79887,79889,79891,79893,79895,79897,79899,79901,79903,79905,79907,79909],{},[31658,79888,46900],{},[31658,79890,15],{},[31662,79892,15972],{"stretchy":702},[31749,79894,640],{},[31662,79896,32712],{"stretchy":702},[31662,79898,15972],{"stretchy":702},[31749,79900,18136],{},[31662,79902,32712],{"stretchy":702},[31662,79904,202],{},[31658,79906,56084],{},[31662,79908,32682],{},[31655,79910,79911,79913],{},[31662,79912,32685],{"fence":758},[71078,79914,79915,79947,79978,80009],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,79916,79917],{},[71087,79918,79919],{},[71090,79920,79921],{"scriptlevel":459,"displaystyle":702},[31655,79922,79923,79925,79927,79929,79931,79933,79935,79937,79939,79942,79944],{},[31658,79924,33759],{},[31662,79926,202],{},[31749,79928,640],{},[31662,79930,5598],{},[31749,79932,459],{},[31662,79934,223],{},[31749,79936,78552],{},[31662,79938,223],{},[31749,79940,79941],{},"2500",[31662,79943,202],{},[31749,79945,79946],{},"4930",[71084,79948,79949],{},[71087,79950,79951],{},[71090,79952,79953],{"scriptlevel":459,"displaystyle":702},[31655,79954,79955,79957,79959,79961,79963,79965,79967,79969,79971,79974,79976],{},[31658,79956,33759],{},[31662,79958,202],{},[31749,79960,768],{},[31662,79962,5598],{},[31749,79964,73451],{},[31662,79966,223],{},[31749,79968,76890],{},[31662,79970,223],{},[31749,79972,79973],{},"750",[31662,79975,202],{},[31749,79977,79327],{},[71084,79979,79980],{},[71087,79981,79982],{},[71090,79983,79984],{"scriptlevel":459,"displaystyle":702},[31655,79985,79986,79988,79990,79992,79994,79996,79998,80000,80002,80004,80006],{},[31658,79987,33759],{},[31662,79989,202],{},[31749,79991,786],{},[31662,79993,5598],{},[31749,79995,75745],{},[31662,79997,223],{},[31749,79999,74728],{},[31662,80001,223],{},[31749,80003,74728],{},[31662,80005,202],{},[31749,80007,80008],{},"6330",[71084,80010,80011],{},[71087,80012,80013],{},[71090,80014,80015],{"scriptlevel":459,"displaystyle":702},[31655,80016,80017,80019,80021,80023,80025,80027,80029,80031,80033,80036,80038],{},[31658,80018,33759],{},[31662,80020,202],{},[31749,80022,15932],{},[31662,80024,5598],{},[31749,80026,77879],{},[31662,80028,223],{},[31749,80030,459],{},[31662,80032,223],{},[31749,80034,80035],{},"1250",[31662,80037,202],{},[31749,80039,80040],{},"1655",[31665,80042,80043],{"encoding":31667},"dp[1][5] = \\min \\begin{cases} k=1: 0 + 2430 + 2500 = 4930 \\\\ k=2: 150 + 930 + 750 = 1830 \\\\ k=3: 330 + 3000 + 3000 = 6330 \\\\ k=4: 405 + 0 + 1250 = 1655 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= \\min \\begin{cases} k=2: 0 + 1770 + 180 = 1950 \\\\ k=3: 360 + 1860 + 720 = 2940 \\\\ k=4: 330 + 1500 + 300 = 2130 \\\\ k=5: 2430 + 0 + 3000 = 5430 \\end{cases}",[34,80763,80765,80804],{"className":80764,"ariaHidden":758},[31672],[34,80766,80768,80771,80774,80777,80780,80783,80786,80789,80792,80795,80798,80801],{"className":80767},[31676],[34,80769],{"className":80770,"style":31774},[31680],[34,80772,46900],{"className":80773},[31685,31686],[34,80775,15],{"className":80776},[31685,31686],[34,80778,15972],{"className":80779},[31782],[34,80781,768],{"className":80782},[31685],[34,80784,32712],{"className":80785},[31690],[34,80787,15972],{"className":80788},[31782],[34,80790,36656],{"className":80791},[31685],[34,80793,32712],{"className":80794},[31690],[34,80796],{"className":80797,"style":31846},[31845],[34,80799,202],{"className":80800},[31850],[34,80802],{"className":80803,"style":31846},[31845],[34,80805,80807,80810,80813,80816],{"className":80806},[31676],[34,80808],{"className":80809,"style":80092},[31680],[34,80811,56084],{"className":80812},[32767],[34,80814],{"className":80815,"style":32123},[31845],[34,80817,80819,80897,81210],{"className":80818},[34006],[34,80820,80822],{"className":80821},[31782],[34,80823,80825],{"className":80824},[71281,71282],[34,80826,80828,80889],{"className":80827},[31796,31797],[34,80829,80831,80886],{"className":80830},[31801],[34,80832,80834,80844,80855,80865,80876],{"className":80833,"style":80117},[31805],[34,80835,80836,80839],{"style":80120},[34,80837],{"className":80838,"style":71299},[31813],[34,80840,80842],{"className":80841},[71303,71304],[34,80843,71307],{},[34,80845,80846,80849],{"style":80131},[34,80847],{"className":80848,"style":71299},[31813],[34,80850,80851],{"style":80137},[41588,80852,80853],{"xmlns":41590,"width":71319,"height":80140,"style":71321,"viewBox":80141,"preserveAspectRatio":71323},[20398,80854],{"d":80144},[34,80856,80857,80860],{"style":71329},[34,80858],{"className":80859,"style":71299},[31813],[34,80861,80863],{"className":80862},[71303,71304],[34,80864,71338],{},[34,80866,80867,80870],{"style":71341},[34,80868],{"className":80869,"style":71299},[31813],[34,80871,80872],{"style":80137},[41588,80873,80874],{"xmlns":41590,"width":71319,"height":80140,"style":71321,"viewBox":80141,"preserveAspectRatio":71323},[20398,80875],{"d":80144},[34,80877,80878,80881],{"style":80168},[34,80879],{"className":80880,"style":71299},[31813],[34,80882,80884],{"className":80883},[71303,71304],[34,80885,71362],{},[34,80887,31829],{"className":80888},[31828],[34,80890,80892],{"className":80891},[31801],[34,80893,80895],{"className":80894,"style":80186},[31805],[34,80896],{},[34,80898,80900],{"className":80899},[31685],[34,80901,80903],{"className":80902},[71078],[34,80904,80906],{"className":80905},[71384],[34,80907,80909,81202],{"className":80908},[31796,31797],[34,80910,80912,81199],{"className":80911},[31801],[34,80913,80915,80986,81057,81128],{"className":80914,"style":80207},[31805],[34,80916,80917,80920],{"style":80210},[34,80918],{"className":80919,"style":71401},[31813],[34,80921,80923,80926,80929,80932,80935,80938,80941,80944,80947,80950,80953,80956,80959,80962,80965,80968,80971,80974,80977,80980,80983],{"className":80922},[31685],[34,80924,33759],{"className":80925,"style":33774},[31685,31686],[34,80927],{"className":80928,"style":31846},[31845],[34,80930,202],{"className":80931},[31850],[34,80933],{"className":80934,"style":31846},[31845],[34,80936,768],{"className":80937},[31685],[34,80939],{"className":80940,"style":31846},[31845],[34,80942,5598],{"className":80943},[31850],[34,80945],{"className":80946,"style":31846},[31845],[34,80948,459],{"className":80949},[31685],[34,80951],{"className":80952,"style":32604},[31845],[34,80954,223],{"className":80955},[32024],[34,80957],{"className":80958,"style":32604},[31845],[34,80960,79288],{"className":80961},[31685],[34,80963],{"className":80964,"style":32604},[31845],[34,80966,223],{"className":80967},[32024],[34,80969],{"className":80970,"style":32604},[31845],[34,80972,74303],{"className":80973},[31685],[34,80975],{"className":80976,"style":31846},[31845],[34,80978,202],{"className":80979},[31850],[34,80981],{"className":80982,"style":31846},[31845],[34,80984,80664],{"className":80985},[31685],[34,80987,80988,80991],{"style":75837},[34,80989],{"className":80990,"style":71401},[31813],[34,80992,80994,80997,81000,81003,81006,81009,81012,81015,81018,81021,81024,81027,81030,81033,81036,81039,81042,81045,81048,81051,81054],{"className":80993},[31685],[34,80995,33759],{"className":80996,"style":33774},[31685,31686],[34,80998],{"className":80999,"style":31846},[31845],[34,81001,202],{"className":81002},[31850],[34,81004],{"className":81005,"style":31846},[31845],[34,81007,786],{"className":81008},[31685],[34,81010],{"className":81011,"style":31846},[31845],[34,81013,5598],{"className":81014},[31850],[34,81016],{"className":81017,"style":31846},[31845],[34,81019,73878],{"className":81020},[31685],[34,81022],{"className":81023,"style":32604},[31845],[34,81025,223],{"className":81026},[32024],[34,81028],{"className":81029,"style":32604},[31845],[34,81031,773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L = 6（全局最终解）：",[105,81302,81303],{},[108,81304,81305,76198,82200,46528,82202,347],{},[34,81306,81308,81543],{"className":81307},[31642],[34,81309,81311],{"className":81310},[31646],[31648,81312,81313],{"xmlns":31650},[31652,81314,81315,81540],{},[31655,81316,81317,81319,81321,81323,81325,81327,81329,81331,81333,81335,81337,81339],{},[31658,81318,46900],{},[31658,81320,15],{},[31662,81322,15972],{"stretchy":702},[31749,81324,640],{},[31662,81326,32712],{"stretchy":702},[31662,81328,15972],{"stretchy":702},[31749,81330,36656],{},[31662,81332,32712],{"stretchy":702},[31662,81334,202],{},[31658,81336,56084],{},[31662,81338,32682],{},[31655,81340,81341,81343],{},[31662,81342,32685],{"fence":758},[71078,81344,81345,81384,81423,81462,81501],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,81346,81347],{},[71087,81348,81349],{},[71090,81350,81351],{"scriptlevel":459,"displaystyle":702},[31655,81352,81353,81355,81357,81359,81361,81363,81365,81367,81369,81371,81373,81375,81377,81379,81381],{},[31658,81354,33759],{},[31662,81356,202],{},[31749,81358,640],{},[31662,81360,5598],{},[31749,81362,459],{},[31662,81364,223],{},[31749,81366,80664],{},[31662,81368,223],{},[31749,81370,18136],{},[31662,81372,42071],{},[31749,81374,3244],{},[31662,81376,42071],{},[31749,81378,36656],{},[31662,81380,202],{},[31749,81382,81383],{},"2250",[71084,81385,81386],{},[71087,81387,81388],{},[71090,81389,81390],{"scriptlevel":459,"displaystyle":702},[31655,81391,81392,81394,81396,81398,81400,81402,81404,81406,81408,81410,81412,81414,81416,81418,81420],{},[31658,81393,33759],{},[31662,81395,202],{},[31749,81397,768],{},[31662,81399,5598],{},[31749,81401,73451],{},[31662,81403,223],{},[31749,81405,79288],{},[31662,81407,223],{},[31749,81409,18136],{},[31662,81411,42071],{},[31749,81413,786],{},[31662,81415,42071],{},[31749,81417,36656],{},[31662,81419,202],{},[31749,81421,81422],{},"2010",[71084,81424,81425],{},[71087,81426,81427],{},[71090,81428,81429],{"scriptlevel":459,"displaystyle":702},[31655,81430,81431,81433,81435,81437,81439,81441,81443,81445,81447,81449,81451,81453,81455,81457,81459],{},[31658,81432,33759],{},[31662,81434,202],{},[31749,81436,786],{},[31662,81438,5598],{},[31749,81440,75745],{},[31662,81442,223],{},[31749,81444,77342],{},[31662,81446,223],{},[31749,81448,18136],{},[31662,81450,42071],{},[31749,81452,71869],{},[31662,81454,42071],{},[31749,81456,36656],{},[31662,81458,202],{},[31749,81460,81461],{},"2550",[71084,81463,81464],{},[71087,81465,81466],{},[71090,81467,81468],{"scriptlevel":459,"displaystyle":702},[31655,81469,81470,81472,81474,81476,81478,81480,81482,81484,81486,81488,81490,81492,81494,81496,81498],{},[31658,81471,33759],{},[31662,81473,202],{},[31749,81475,15932],{},[31662,81477,5598],{},[31749,81479,77879],{},[31662,81481,223],{},[31749,81483,75153],{},[31662,81485,223],{},[31749,81487,18136],{},[31662,81489,42071],{},[31749,81491,18136],{},[31662,81493,42071],{},[31749,81495,36656],{},[31662,81497,202],{},[31749,81499,81500],{},"2055",[71084,81502,81503],{},[71087,81504,81505],{},[71090,81506,81507],{"scriptlevel":459,"displaystyle":702},[31655,81508,81509,81511,81513,81515,81517,81519,81521,81523,81525,81527,81529,81531,81533,81535,81537],{},[31658,81510,33759],{},[31662,81512,202],{},[31749,81514,18136],{},[31662,81516,5598],{},[31749,81518,80040],{},[31662,81520,223],{},[31749,81522,459],{},[31662,81524,223],{},[31749,81526,18136],{},[31662,81528,42071],{},[31749,81530,5774],{},[31662,81532,42071],{},[31749,81534,36656],{},[31662,81536,202],{},[31749,81538,81539],{},"3155",[31665,81541,81542],{"encoding":31667},"dp[1][6] = \\min \\begin{cases} k=1: 0 + 1950 + 5\\times 10\\times 6 = 2250 \\\\ k=2: 150 + 1770 + 5\\times 3\\times 6 = 2010 \\\\ k=3: 330 + 1860 + 5\\times 12\\times 6 = 2550 \\\\ k=4: 405 + 1500 + 5\\times 5\\times 6 = 2055 \\\\ k=5: 1655 + 0 + 5\\times 50\\times 6 = 3155 \\end{cases}",[34,81544,81546,81585],{"className":81545,"ariaHidden":758},[31672],[34,81547,81549,81552,81555,81558,81561,81564,81567,81570,81573,81576,81579,81582],{"className":81548},[31676],[34,81550],{"className":81551,"style":31774},[31680],[34,81553,46900],{"className":81554},[31685,31686],[34,81556,15],{"className":81557},[31685,31686],[34,81559,15972],{"className":81560},[31782],[34,81562,640],{"className":81563},[31685],[34,81565,32712],{"className":81566},[31690],[34,81568,15972],{"className":81569},[31782],[34,81571,36656],{"className":81572},[31685],[34,81574,32712],{"className":81575},[31690],[34,81577],{"className":81578,"style":31846},[31845],[34,81580,202],{"className":81581},[31850],[34,81583],{"className":81584,"style":31846},[31845],[34,81586,81588,81592,81595,81598],{"className":81587},[31676],[34,81589],{"className":81590,"style":81591},[31680],"height:7.2em;vertical-align:-3.35em;",[34,81593,56084],{"className":81594},[32767],[34,81596],{"className":81597,"style":32123},[31845],[34,81599,81601,81691,82197],{"className":81600},[34006],[34,81602,81604],{"className":81603},[31782],[34,81605,81607],{"className":81606},[71281,71282],[34,81608,81610,81682],{"className":81609},[31796,31797],[34,81611,81613,81679],{"className":81612},[31801],[34,81614,81617,81629,81645,81656,81668],{"className":81615,"style":81616},[31805],"height:3.85em;",[34,81618,81620,81624],{"style":81619},"top:-1.366em;",[34,81621],{"className":81622,"style":81623},[31813],"height:3.816em;",[34,81625,81627],{"className":81626},[71303,71304],[34,81628,71307],{},[34,81630,81632,81635],{"style":81631},"top:-1.358em;",[34,81633],{"className":81634,"style":81623},[31813],[34,81636,81638],{"style":81637},"height:1.816em;width:0.8889em;",[41588,81639,81642],{"xmlns":41590,"width":71319,"height":81640,"style":71321,"viewBox":81641,"preserveAspectRatio":71323},"1.816em","0 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(s=4)",[13858,82431,82432],{"align":82355},[2627,82433,82434],{},"2010 (s=2)",[13843,82436,82437,82441,82443,82445,82448,82450,82453],{},[13858,82438,82439],{"align":82355},[2627,82440,768],{},[13858,82442,2511],{"align":82355},[13858,82444,459],{"align":82355},[13858,82446,82447],{"align":82355},"360 (s=2)",[13858,82449,82423],{"align":82355},[13858,82451,82452],{"align":82355},"2430 (s=2)",[13858,82454,82455],{"align":82355},"1950 (s=2)",[13843,82457,82458,82462,82464,82466,82468,82471,82474],{},[13858,82459,82460],{"align":82355},[2627,82461,786],{},[13858,82463,2511],{"align":82355},[13858,82465,2511],{"align":82355},[13858,82467,459],{"align":82355},[13858,82469,82470],{"align":82355},"180 (s=3)",[13858,82472,82473],{"align":82355},"930 (s=4)",[13858,82475,82476],{"align":82355},"1770 (s=4)",[13843,82478,82479,82483,82485,82487,82489,82491,82494],{},[13858,82480,82481],{"align":82355},[2627,82482,15932],{},[13858,82484,2511],{"align":82355},[13858,82486,2511],{"align":82355},[13858,82488,2511],{"align":82355},[13858,82490,459],{"align":82355},[13858,82492,82493],{"align":82355},"3000 (s=4)",[13858,82495,82496],{"align":82355},"1860 (s=4)",[13843,82498,82499,82503,82505,82507,82509,82511,82513],{},[13858,82500,82501],{"align":82355},[2627,82502,18136],{},[13858,82504,2511],{"align":82355},[13858,82506,2511],{"align":82355},[13858,82508,2511],{"align":82355},[13858,82510,2511],{"align":82355},[13858,82512,459],{"align":82355},[13858,82514,82515],{"align":82355},"1500 (s=5)",[13843,82517,82518,82522,82524,82526,82528,82530,82532],{},[13858,82519,82520],{"align":82355},[2627,82521,36656],{},[13858,82523,2511],{"align":82355},[13858,82525,2511],{"align":82355},[13858,82527,2511],{"align":82355},[13858,82529,2511],{"align":82355},[13858,82531,2511],{"align":82355},[13858,82533,459],{"align":82355},[15,82535,82536,82539,82540,82568],{},[2627,82537,82538],{},"2) 最优的矩阵乘法运算顺序：","\n我们利用求出的断点数组 ",[34,82541,82543,82556],{"className":82542},[31642],[34,82544,82546],{"className":82545},[31646],[31648,82547,82548],{"xmlns":31650},[31652,82549,82550,82554],{},[31655,82551,82552],{},[31658,82553,31747],{},[31665,82555,31747],{"encoding":31667},[34,82557,82559],{"className":82558,"ariaHidden":758},[31672],[34,82560,82562,82565],{"className":82561},[31676],[34,82563],{"className":82564,"style":32527},[31680],[34,82566,31747],{"className":82567},[31685,31686],"，从大范围不断往里拆解。",[105,82570,82571,83000,83129,83466],{},[108,82572,82573,82574,82655,82656,258,82782,347],{},"全局范围是 1 到 6，查表得知 ",[34,82575,82577,82607],{"className":82576},[31642],[34,82578,82580],{"className":82579},[31646],[31648,82581,82582],{"xmlns":31650},[31652,82583,82584,82604],{},[31655,82585,82586,82588,82590,82592,82594,82596,82598,82600,82602],{},[31658,82587,31747],{},[31662,82589,15972],{"stretchy":702},[31749,82591,640],{},[31662,82593,32712],{"stretchy":702},[31662,82595,15972],{"stretchy":702},[31749,82597,36656],{},[31662,82599,32712],{"stretchy":702},[31662,82601,202],{},[31749,82603,768],{},[31665,82605,82606],{"encoding":31667},"s[1][6] = 2",[34,82608,82610,82646],{"className":82609,"ariaHidden":758},[31672],[34,82611,82613,82616,82619,82622,82625,82628,82631,82634,82637,82640,82643],{"className":82612},[31676],[34,82614],{"className":82615,"style":31774},[31680],[34,82617,31747],{"className":82618},[31685,31686],[34,82620,15972],{"className":82621},[31782],[34,82623,640],{"className":82624},[31685],[34,82626,32712],{"className":82627},[31690],[34,82629,15972],{"className":82630},[31782],[34,82632,36656],{"className":82633},[31685],[34,82635,32712],{"className":82636},[31690],[34,82638],{"className":82639,"style":31846},[31845],[34,82641,202],{"className":82642},[31850],[34,82644],{"className":82645,"style":31846},[31845],[34,82647,82649,82652],{"className":82648},[31676],[34,82650],{"className":82651,"style":32617},[31680],[34,82653,768],{"className":82654},[31685],"。说明第一刀切在第2个矩阵后。整体结构分为 ",[34,82657,82659,82687],{"className":82658},[31642],[34,82660,82662],{"className":82661},[31646],[31648,82663,82664],{"xmlns":31650},[31652,82665,82666,82684],{},[31655,82667,82668,82670,82676,82682],{},[31662,82669,456],{"stretchy":702},[31743,82671,82672,82674],{},[31658,82673,54151],{},[31749,82675,640],{},[31743,82677,82678,82680],{},[31658,82679,54151],{},[31749,82681,768],{},[31662,82683,2557],{"stretchy":702},[31665,82685,82686],{"encoding":31667},"(A_1 A_2)",[34,82688,82690],{"className":82689,"ariaHidden":758},[31672],[34,82691,82693,82696,82699,82739,82779],{"className":82692},[31676],[34,82694],{"className":82695,"style":31774},[31680],[34,82697,456],{"className":82698},[31782],[34,82700,82702,82705],{"className":82701},[31685],[34,82703,54151],{"className":82704},[31685,31686],[34,82706,82708],{"className":82707},[31792],[34,82709,82711,82731],{"className":82710},[31796,31797],[34,82712,82714,82728],{"className":82713},[31801],[34,82715,82717],{"className":82716,"style":31806},[31805],[34,82718,82719,82722],{"style":31809},[34,82720],{"className":82721,"style":31814},[31813],[34,82723,82725],{"className":82724},[31818,31819,31820,31821],[34,82726,640],{"className":82727},[31685,31821],[34,82729,31829],{"className":82730},[31828],[34,82732,82734],{"className":82733},[31801],[34,82735,82737],{"className":82736,"style":31836},[31805],[34,82738],{},[34,82740,82742,82745],{"className":82741},[31685],[34,82743,54151],{"className":82744},[31685,31686],[34,82746,82748],{"className":82747},[31792],[34,82749,82751,82771],{"className":82750},[31796,31797],[34,82752,82754,82768],{"className":82753},[31801],[34,82755,82757],{"className":82756,"style":31806},[31805],[34,82758,82759,82762],{"style":31809},[34,82760],{"className":82761,"style":31814},[31813],[34,82763,82765],{"className":82764},[31818,31819,31820,31821],[34,82766,768],{"className":82767},[31685,31821],[34,82769,31829],{"className":82770},[31828],[34,82772,82774],{"className":82773},[31801],[34,82775,82777],{"className":82776,"style":31836},[31805],[34,82778],{},[34,82780,2557],{"className":82781},[31690],[34,82783,82785,82825],{"className":82784},[31642],[34,82786,82788],{"className":82787},[31646],[31648,82789,82790],{"xmlns":31650},[31652,82791,82792,82822],{},[31655,82793,82794,82796,82802,82808,82814,82820],{},[31662,82795,456],{"stretchy":702},[31743,82797,82798,82800],{},[31658,82799,54151],{},[31749,82801,786],{},[31743,82803,82804,82806],{},[31658,82805,54151],{},[31749,82807,15932],{},[31743,82809,82810,82812],{},[31658,82811,54151],{},[31749,82813,18136],{},[31743,82815,82816,82818],{},[31658,82817,54151],{},[31749,82819,36656],{},[31662,82821,2557],{"stretchy":702},[31665,82823,82824],{"encoding":31667},"(A_3 A_4 A_5 A_6)",[34,82826,82828],{"className":82827,"ariaHidden":758},[31672],[34,82829,82831,82834,82837,82877,82917,82957,82997],{"className":82830},[31676],[34,82832],{"className":82833,"style":31774},[31680],[34,82835,456],{"className":82836},[31782],[34,82838,82840,82843],{"className":82839},[31685],[34,82841,54151],{"className":82842},[31685,31686],[34,82844,82846],{"className":82845},[31792],[34,82847,82849,82869],{"className":82848},[31796,31797],[34,82850,82852,82866],{"className":82851},[31801],[34,82853,82855],{"className":82854,"style":31806},[31805],[34,82856,82857,82860],{"style":31809},[34,82858],{"className":82859,"style":31814},[31813],[34,82861,82863],{"className":82862},[31818,31819,31820,31821],[34,82864,786],{"className":82865},[31685,31821],[34,82867,31829],{"className":82868},[31828],[34,82870,82872],{"className":82871},[31801],[34,82873,82875],{"className":82874,"style":31836},[31805],[34,82876],{},[34,82878,82880,82883],{"className":82879},[31685],[34,82881,54151],{"className":82882},[31685,31686],[34,82884,82886],{"className":82885},[31792],[34,82887,82889,82909],{"className":82888},[31796,31797],[34,82890,82892,82906],{"className":82891},[31801],[34,82893,82895],{"className":82894,"style":31806},[31805],[34,82896,82897,82900],{"style":31809},[34,82898],{"className":82899,"style":31814},[31813],[34,82901,82903],{"className":82902},[31818,31819,31820,31821],[34,82904,15932],{"className":82905},[31685,31821],[34,82907,31829],{"className":82908},[31828],[34,82910,82912],{"className":82911},[31801],[34,82913,82915],{"className":82914,"style":31836},[31805],[34,82916],{},[34,82918,82920,82923],{"className":82919},[31685],[34,82921,54151],{"className":82922},[31685,31686],[34,82924,82926],{"className":82925},[31792],[34,82927,82929,82949],{"className":82928},[31796,31797],[34,82930,82932,82946],{"className":82931},[31801],[34,82933,82935],{"className":82934,"style":31806},[31805],[34,82936,82937,82940],{"style":31809},[34,82938],{"className":82939,"style":31814},[31813],[34,82941,82943],{"className":82942},[31818,31819,31820,31821],[34,82944,18136],{"className":82945},[31685,31821],[34,82947,31829],{"className":82948},[31828],[34,82950,82952],{"className":82951},[31801],[34,82953,82955],{"className":82954,"style":31836},[31805],[34,82956],{},[34,82958,82960,82963],{"className":82959},[31685],[34,82961,54151],{"className":82962},[31685,31686],[34,82964,82966],{"className":82965},[31792],[34,82967,82969,82989],{"className":82968},[31796,31797],[34,82970,82972,82986],{"className":82971},[31801],[34,82973,82975],{"className":82974,"style":31806},[31805],[34,82976,82977,82980],{"style":31809},[34,82978],{"className":82979,"style":31814},[31813],[34,82981,82983],{"className":82982},[31818,31819,31820,31821],[34,82984,36656],{"className":82985},[31685,31821],[34,82987,31829],{"className":82988},[31828],[34,82990,82992],{"className":82991},[31801],[34,82993,82995],{"className":82994,"style":31836},[31805],[34,82996],{},[34,82998,2557],{"className":82999},[31690],[108,83001,83002,83003,83128],{},"对于左半边 ",[34,83004,83006,83033],{"className":83005},[31642],[34,83007,83009],{"className":83008},[31646],[31648,83010,83011],{"xmlns":31650},[31652,83012,83013,83031],{},[31655,83014,83015,83017,83023,83029],{},[31662,83016,456],{"stretchy":702},[31743,83018,83019,83021],{},[31658,83020,54151],{},[31749,83022,640],{},[31743,83024,83025,83027],{},[31658,83026,54151],{},[31749,83028,768],{},[31662,83030,2557],{"stretchy":702},[31665,83032,82686],{"encoding":31667},[34,83034,83036],{"className":83035,"ariaHidden":758},[31672],[34,83037,83039,83042,83045,83085,83125],{"className":83038},[31676],[34,83040],{"className":83041,"style":31774},[31680],[34,83043,456],{"className":83044},[31782],[34,83046,83048,83051],{"className":83047},[31685],[34,83049,54151],{"className":83050},[31685,31686],[34,83052,83054],{"className":83053},[31792],[34,83055,83057,83077],{"className":83056},[31796,31797],[34,83058,83060,83074],{"className":83059},[31801],[34,83061,83063],{"className":83062,"style":31806},[31805],[34,83064,83065,83068],{"style":31809},[34,83066],{"className":83067,"style":31814},[31813],[34,83069,83071],{"className":83070},[31818,31819,31820,31821],[34,83072,640],{"className":83073},[31685,31821],[34,83075,31829],{"className":83076},[31828],[34,83078,83080],{"className":83079},[31801],[34,83081,83083],{"className":83082,"style":31836},[31805],[34,83084],{},[34,83086,83088,83091],{"className":83087},[31685],[34,83089,54151],{"className":83090},[31685,31686],[34,83092,83094],{"className":83093},[31792],[34,83095,83097,83117],{"className":83096},[31796,31797],[34,83098,83100,83114],{"className":83099},[31801],[34,83101,83103],{"className":83102,"style":31806},[31805],[34,83104,83105,83108],{"style":31809},[34,83106],{"className":83107,"style":31814},[31813],[34,83109,83111],{"className":83110},[31818,31819,31820,31821],[34,83112,768],{"className":83113},[31685,31821],[34,83115,31829],{"className":83116},[31828],[34,83118,83120],{"className":83119},[31801],[34,83121,83123],{"className":83122,"style":31836},[31805],[34,83124],{},[34,83126,2557],{"className":83127},[31690],"，毫无疑问就是直接相乘。",[108,83130,83131,83132,83213,83214,258,83340,347],{},"对于右半边范围 3 到 6，查表得知 ",[34,83133,83135,83165],{"className":83134},[31642],[34,83136,83138],{"className":83137},[31646],[31648,83139,83140],{"xmlns":31650},[31652,83141,83142,83162],{},[31655,83143,83144,83146,83148,83150,83152,83154,83156,83158,83160],{},[31658,83145,31747],{},[31662,83147,15972],{"stretchy":702},[31749,83149,786],{},[31662,83151,32712],{"stretchy":702},[31662,83153,15972],{"stretchy":702},[31749,83155,36656],{},[31662,83157,32712],{"stretchy":702},[31662,83159,202],{},[31749,83161,15932],{},[31665,83163,83164],{"encoding":31667},"s[3][6] = 4",[34,83166,83168,83204],{"className":83167,"ariaHidden":758},[31672],[34,83169,83171,83174,83177,83180,83183,83186,83189,83192,83195,83198,83201],{"className":83170},[31676],[34,83172],{"className":83173,"style":31774},[31680],[34,83175,31747],{"className":83176},[31685,31686],[34,83178,15972],{"className":83179},[31782],[34,83181,786],{"className":83182},[31685],[34,83184,32712],{"className":83185},[31690],[34,83187,15972],{"className":83188},[31782],[34,83190,36656],{"className":83191},[31685],[34,83193,32712],{"className":83194},[31690],[34,83196],{"className":83197,"style":31846},[31845],[34,83199,202],{"className":83200},[31850],[34,83202],{"className":83203,"style":31846},[31845],[34,83205,83207,83210],{"className":83206},[31676],[34,83208],{"className":83209,"style":32617},[31680],[34,83211,15932],{"className":83212},[31685],"。说明这一刀切在第4个矩阵后。结构变为 ",[34,83215,83217,83245],{"className":83216},[31642],[34,83218,83220],{"className":83219},[31646],[31648,83221,83222],{"xmlns":31650},[31652,83223,83224,83242],{},[31655,83225,83226,83228,83234,83240],{},[31662,83227,456],{"stretchy":702},[31743,83229,83230,83232],{},[31658,83231,54151],{},[31749,83233,786],{},[31743,83235,83236,83238],{},[31658,83237,54151],{},[31749,83239,15932],{},[31662,83241,2557],{"stretchy":702},[31665,83243,83244],{"encoding":31667},"(A_3 A_4)",[34,83246,83248],{"className":83247,"ariaHidden":758},[31672],[34,83249,83251,83254,83257,83297,83337],{"className":83250},[31676],[34,83252],{"className":83253,"style":31774},[31680],[34,83255,456],{"className":83256},[31782],[34,83258,83260,83263],{"className":83259},[31685],[34,83261,54151],{"className":83262},[31685,31686],[34,83264,83266],{"className":83265},[31792],[34,83267,83269,83289],{"className":83268},[31796,31797],[34,83270,83272,83286],{"className":83271},[31801],[34,83273,83275],{"className":83274,"style":31806},[31805],[34,83276,83277,83280],{"style":31809},[34,83278],{"className":83279,"style":31814},[31813],[34,83281,83283],{"className":83282},[31818,31819,31820,31821],[34,83284,786],{"className":83285},[31685,31821],[34,83287,31829],{"className":83288},[31828],[34,83290,83292],{"className":83291},[31801],[34,83293,83295],{"className":83294,"style":31836},[31805],[34,83296],{},[34,83298,83300,83303],{"className":83299},[31685],[34,83301,54151],{"className":83302},[31685,31686],[34,83304,83306],{"className":83305},[31792],[34,83307,83309,83329],{"className":83308},[31796,31797],[34,83310,83312,83326],{"className":83311},[31801],[34,83313,83315],{"className":83314,"style":31806},[31805],[34,83316,83317,83320],{"style":31809},[34,83318],{"className":83319,"style":31814},[31813],[34,83321,83323],{"className":83322},[31818,31819,31820,31821],[34,83324,15932],{"className":83325},[31685,31821],[34,83327,31829],{"className":83328},[31828],[34,83330,83332],{"className":83331},[31801],[34,83333,83335],{"className":83334,"style":31836},[31805],[34,83336],{},[34,83338,2557],{"className":83339},[31690],[34,83341,83343,83371],{"className":83342},[31642],[34,83344,83346],{"className":83345},[31646],[31648,83347,83348],{"xmlns":31650},[31652,83349,83350,83368],{},[31655,83351,83352,83354,83360,83366],{},[31662,83353,456],{"stretchy":702},[31743,83355,83356,83358],{},[31658,83357,54151],{},[31749,83359,18136],{},[31743,83361,83362,83364],{},[31658,83363,54151],{},[31749,83365,36656],{},[31662,83367,2557],{"stretchy":702},[31665,83369,83370],{"encoding":31667},"(A_5 A_6)",[34,83372,83374],{"className":83373,"ariaHidden":758},[31672],[34,83375,83377,83380,83383,83423,83463],{"className":83376},[31676],[34,83378],{"className":83379,"style":31774},[31680],[34,83381,456],{"className":83382},[31782],[34,83384,83386,83389],{"className":83385},[31685],[34,83387,54151],{"className":83388},[31685,31686],[34,83390,83392],{"className":83391},[31792],[34,83393,83395,83415],{"className":83394},[31796,31797],[34,83396,83398,83412],{"className":83397},[31801],[34,83399,83401],{"className":83400,"style":31806},[31805],[34,83402,83403,83406],{"style":31809},[34,83404],{"className":83405,"style":31814},[31813],[34,83407,83409],{"className":83408},[31818,31819,31820,31821],[34,83410,18136],{"className":83411},[31685,31821],[34,83413,31829],{"className":83414},[31828],[34,83416,83418],{"className":83417},[31801],[34,83419,83421],{"className":83420,"style":31836},[31805],[34,83422],{},[34,83424,83426,83429],{"className":83425},[31685],[34,83427,54151],{"className":83428},[31685,31686],[34,83430,83432],{"className":83431},[31792],[34,83433,83435,83455],{"className":83434},[31796,31797],[34,83436,83438,83452],{"className":83437},[31801],[34,83439,83441],{"className":83440,"style":31806},[31805],[34,83442,83443,83446],{"style":31809},[34,83444],{"className":83445,"style":31814},[31813],[34,83447,83449],{"className":83448},[31818,31819,31820,31821],[34,83450,36656],{"className":83451},[31685,31821],[34,83453,31829],{"className":83454},[31828],[34,83456,83458],{"className":83457},[31801],[34,83459,83461],{"className":83460,"style":31836},[31805],[34,83462],{},[34,83464,2557],{"className":83465},[31690],[108,83467,83468,83469],{},"合并所有括号。最终的最优运算顺序（加括号方式）为：\n",[2627,83470,83471],{},[34,83472,83474,83538],{"className":83473},[31642],[34,83475,83477],{"className":83476},[31646],[31648,83478,83479],{"xmlns":31650},[31652,83480,83481,83535],{},[31655,83482,83483,83485,83491,83497,83499,83501,83503,83509,83515,83517,83519,83525,83531,83533],{},[31662,83484,456],{"stretchy":702},[31743,83486,83487,83489],{},[31658,83488,54151],{},[31749,83490,640],{},[31743,83492,83493,83495],{},[31658,83494,54151],{},[31749,83496,768],{},[31662,83498,2557],{"stretchy":702},[31662,83500,456],{"stretchy":702},[31662,83502,456],{"stretchy":702},[31743,83504,83505,83507],{},[31658,83506,54151],{},[31749,83508,786],{},[31743,83510,83511,83513],{},[31658,83512,54151],{},[31749,83514,15932],{},[31662,83516,2557],{"stretchy":702},[31662,83518,456],{"stretchy":702},[31743,83520,83521,83523],{},[31658,83522,54151],{},[31749,83524,18136],{},[31743,83526,83527,83529],{},[31658,83528,54151],{},[31749,83530,36656],{},[31662,83532,2557],{"stretchy":702},[31662,83534,2557],{"stretchy":702},[31665,83536,83537],{"encoding":31667},"(A_1 A_2)((A_3 A_4)(A_5 A_6))",[34,83539,83541],{"className":83540,"ariaHidden":758},[31672],[34,83542,83544,83547,83550,83590,83630,83633,83636,83676,83716,83719,83722,83762,83802],{"className":83543},[31676],[34,83545],{"className":83546,"style":31774},[31680],[34,83548,456],{"className":83549},[31782],[34,83551,83553,83556],{"className":83552},[31685],[34,83554,54151],{"className":83555},[31685,31686],[34,83557,83559],{"className":83558},[31792],[34,83560,83562,83582],{"className":83561},[31796,31797],[34,83563,83565,83579],{"className":83564},[31801],[34,83566,83568],{"className":83567,"style":31806},[31805],[34,83569,83570,83573],{"style":31809},[34,83571],{"className":83572,"style":31814},[31813],[34,83574,83576],{"className":83575},[31818,31819,31820,31821],[34,83577,640],{"className":83578},[31685,31821],[34,83580,31829],{"className":83581},[31828],[34,83583,83585],{"className":83584},[31801],[34,83586,83588],{"className":83587,"style":31836},[31805],[34,83589],{},[34,83591,83593,83596],{"className":83592},[31685],[34,83594,54151],{"className":83595},[31685,31686],[34,83597,83599],{"className":83598},[31792],[34,83600,83602,83622],{"className":83601},[31796,31797],[34,83603,83605,83619],{"className":83604},[31801],[34,83606,83608],{"className":83607,"style":31806},[31805],[34,83609,83610,83613],{"style":31809},[34,83611],{"className":83612,"style":31814},[31813],[34,83614,83616],{"className":83615},[31818,31819,31820,31821],[34,83617,768],{"className":83618},[31685,31821],[34,83620,31829],{"className":83621},[31828],[34,83623,83625],{"className":83624},[31801],[34,83626,83628],{"className":83627,"style":31836},[31805],[34,83629],{},[34,83631,2557],{"className":83632},[31690],[34,83634,1323],{"className":83635},[31782],[34,83637,83639,83642],{"className":83638},[31685],[34,83640,54151],{"className":83641},[31685,31686],[34,83643,83645],{"className":83644},[31792],[34,83646,83648,83668],{"className":83647},[31796,31797],[34,83649,83651,83665],{"className":83650},[31801],[34,83652,83654],{"className":83653,"style":31806},[31805],[34,83655,83656,83659],{"style":31809},[34,83657],{"className":83658,"style":31814},[31813],[34,83660,83662],{"className":83661},[31818,31819,31820,31821],[34,83663,786],{"className":83664},[31685,31821],[34,83666,31829],{"className":83667},[31828],[34,83669,83671],{"className":83670},[31801],[34,83672,83674],{"className":83673,"style":31836},[31805],[34,83675],{},[34,83677,83679,83682],{"className":83678},[31685],[34,83680,54151],{"className":83681},[31685,31686],[34,83683,83685],{"className":83684},[31792],[34,83686,83688,83708],{"className":83687},[31796,31797],[34,83689,83691,83705],{"className":83690},[31801],[34,83692,83694],{"className":83693,"style":31806},[31805],[34,83695,83696,83699],{"style":31809},[34,83697],{"className":83698,"style":31814},[31813],[34,83700,83702],{"className":83701},[31818,31819,31820,31821],[34,83703,15932],{"className":83704},[31685,31821],[34,83706,31829],{"className":83707},[31828],[34,83709,83711],{"className":83710},[31801],[34,83712,83714],{"className":83713,"style":31836},[31805],[34,83715],{},[34,83717,2557],{"className":83718},[31690],[34,83720,456],{"className":83721},[31782],[34,83723,83725,83728],{"className":83724},[31685],[34,83726,54151],{"className":83727},[31685,31686],[34,83729,83731],{"className":83730},[31792],[34,83732,83734,83754],{"className":83733},[31796,31797],[34,83735,83737,83751],{"className":83736},[31801],[34,83738,83740],{"className":83739,"style":31806},[31805],[34,83741,83742,83745],{"style":31809},[34,83743],{"className":83744,"style":31814},[31813],[34,83746,83748],{"className":83747},[31818,31819,31820,31821],[34,83749,18136],{"className":83750},[31685,31821],[34,83752,31829],{"className":83753},[31828],[34,83755,83757],{"className":83756},[31801],[34,83758,83760],{"className":83759,"style":31836},[31805],[34,83761],{},[34,83763,83765,83768],{"className":83764},[31685],[34,83766,54151],{"className":83767},[31685,31686],[34,83769,83771],{"className":83770},[31792],[34,83772,83774,83794],{"className":83773},[31796,31797],[34,83775,83777,83791],{"className":83776},[31801],[34,83778,83780],{"className":83779,"style":31806},[31805],[34,83781,83782,83785],{"style":31809},[34,83783],{"className":83784,"style":31814},[31813],[34,83786,83788],{"className":83787},[31818,31819,31820,31821],[34,83789,36656],{"className":83790},[31685,31821],[34,83792,31829],{"className":83793},[31828],[34,83795,83797],{"className":83796},[31801],[34,83798,83800],{"className":83799,"style":31836},[31805],[34,83801],{},[34,83803,44253],{"className":83804},[31690],[32188,83806],{},[253,83808,58367],{"id":83809},"_3-题目内容-3",[15,83811,83812,83813,83841,83842,83913,83914,31901],{},"在操作系统的单机作业调度问题中，",[34,83814,83816,83829],{"className":83815},[31642],[34,83817,83819],{"className":83818},[31646],[31648,83820,83821],{"xmlns":31650},[31652,83822,83823,83827],{},[31655,83824,83825],{},[31658,83826,31660],{},[31665,83828,31660],{"encoding":31667},[34,83830,83832],{"className":83831,"ariaHidden":758},[31672],[34,83833,83835,83838],{"className":83834},[31676],[34,83836],{"className":83837,"style":32527},[31680],[34,83839,31660],{"className":83840},[31685,31686]," 个作业同时提交给系统，每个作业需要 CPU 处理的时间分别为 ",[34,83843,83845,83863],{"className":83844},[31642],[34,83846,83848],{"className":83847},[31646],[31648,83849,83850],{"xmlns":31650},[31652,83851,83852,83860],{},[31655,83853,83854],{},[31743,83855,83856,83858],{},[31658,83857,33249],{},[31658,83859,31955],{},[31665,83861,83862],{"encoding":31667},"t_i",[34,83864,83866],{"className":83865,"ariaHidden":758},[31672],[34,83867,83869,83873],{"className":83868},[31676],[34,83870],{"className":83871,"style":83872},[31680],"height:0.7651em;vertical-align:-0.15em;",[34,83874,83876,83879],{"className":83875},[31685],[34,83877,33249],{"className":83878},[31685,31686],[34,83880,83882],{"className":83881},[31792],[34,83883,83885,83905],{"className":83884},[31796,31797],[34,83886,83888,83902],{"className":83887},[31801],[34,83889,83891],{"className":83890,"style":32090},[31805],[34,83892,83893,83896],{"style":31809},[34,83894],{"className":83895,"style":31814},[31813],[34,83897,83899],{"className":83898},[31818,31819,31820,31821],[34,83900,31955],{"className":83901},[31685,31686,31821],[34,83903,31829],{"className":83904},[31828],[34,83906,83908],{"className":83907},[31801],[34,83909,83911],{"className":83910,"style":31836},[31805],[34,83912],{},"（",[34,83915,83917,83939],{"className":83916},[31642],[34,83918,83920],{"className":83919},[31646],[31648,83921,83922],{"xmlns":31650},[31652,83923,83924,83936],{},[31655,83925,83926,83928,83930,83932,83934],{},[31749,83927,640],{},[31662,83929,45972],{},[31658,83931,31955],{},[31662,83933,45972],{},[31658,83935,31660],{},[31665,83937,83938],{"encoding":31667},"1 \\le i \\le n",[34,83940,83942,83960,83978],{"className":83941,"ariaHidden":758},[31672],[34,83943,83945,83948,83951,83954,83957],{"className":83944},[31676],[34,83946],{"className":83947,"style":55382},[31680],[34,83949,640],{"className":83950},[31685],[34,83952],{"className":83953,"style":31846},[31845],[34,83955,45972],{"className":83956},[31850],[34,83958],{"className":83959,"style":31846},[31845],[34,83961,83963,83966,83969,83972,83975],{"className":83962},[31676],[34,83964],{"className":83965,"style":55401},[31680],[34,83967,31955],{"className":83968},[31685,31686],[34,83970],{"className":83971,"style":31846},[31845],[34,83973,45972],{"className":83974},[31850],[34,83976],{"className":83977,"style":31846},[31845],[34,83979,83981,83984],{"className":83980},[31676],[34,83982],{"className":83983,"style":32527},[31680],[34,83985,31660],{"className":83986},[31685,31686],[2230,83988,83989,83992],{},[108,83990,83991],{},"请设计一种贪心策略对作业进行调度，使得所有作业的平均等待时间最短。（5分）",[108,83993,83994],{},"证明该贪心策略满足贪心选择和最优子结构性质。（5分）",[253,83996,58168],{"id":83997},"答案与深度解析-21",[15,83999,84000],{},[2627,84001,84002],{},"1) 贪心策略设计：",[105,84004,84005,84110],{},[108,84006,84007,84010,84011,84039,84040,84109],{},[2627,84008,84009],{},"策略（短作业优先，SJF）："," 将这 ",[34,84012,84014,84027],{"className":84013},[31642],[34,84015,84017],{"className":84016},[31646],[31648,84018,84019],{"xmlns":31650},[31652,84020,84021,84025],{},[31655,84022,84023],{},[31658,84024,31660],{},[31665,84026,31660],{"encoding":31667},[34,84028,84030],{"className":84029,"ariaHidden":758},[31672],[34,84031,84033,84036],{"className":84032},[31676],[34,84034],{"className":84035,"style":32527},[31680],[34,84037,31660],{"className":84038},[31685,31686]," 个作业按照它们所需 CPU 处理时间 ",[34,84041,84043,84060],{"className":84042},[31642],[34,84044,84046],{"className":84045},[31646],[31648,84047,84048],{"xmlns":31650},[31652,84049,84050,84058],{},[31655,84051,84052],{},[31743,84053,84054,84056],{},[31658,84055,33249],{},[31658,84057,31955],{},[31665,84059,83862],{"encoding":31667},[34,84061,84063],{"className":84062,"ariaHidden":758},[31672],[34,84064,84066,84069],{"className":84065},[31676],[34,84067],{"className":84068,"style":83872},[31680],[34,84070,84072,84075],{"className":84071},[31685],[34,84073,33249],{"className":84074},[31685,31686],[34,84076,84078],{"className":84077},[31792],[34,84079,84081,84101],{"className":84080},[31796,31797],[34,84082,84084,84098],{"className":84083},[31801],[34,84085,84087],{"className":84086,"style":32090},[31805],[34,84088,84089,84092],{"style":31809},[34,84090],{"className":84091,"style":31814},[31813],[34,84093,84095],{"className":84094},[31818,31819,31820,31821],[34,84096,31955],{"className":84097},[31685,31686,31821],[34,84099,31829],{"className":84100},[31828],[34,84102,84104],{"className":84103},[31801],[34,84105,84107],{"className":84106,"style":31836},[31805],[34,84108],{}," 进行**从小到大（升序）**排序，然后依次调度执行。",[108,84111,84112,84115],{},[2627,84113,84114],{},"零基础理解："," 想象去银行排队办业务，如果你办业务只需要 1 分钟，而前面站了一个需要办 1 小时的人，你是不是会觉得极其绝望（等待时间巨长）？为了让大家的总体体验最好（平均等待时间最短），银行正确的做法是让办事快的人先办。",[15,84117,84118,84121,84122,84193,84194,84264,84265,84398,84399,84907,84908,84977,84978,85028,85029,84977,85099,85150,85151,85505,85506,85534],{},[2627,84119,84120],{},"2) 数学严格证明：","\n在进行证明之前，我们要先梳理一个公式：\n假设调度顺序已经是 ",[34,84123,84125,84151],{"className":84124},[31642],[34,84126,84128],{"className":84127},[31646],[31648,84129,84130],{"xmlns":31650},[31652,84131,84132,84148],{},[31655,84133,84134,84136,84138,84140,84142,84144,84146],{},[31749,84135,640],{},[31662,84137,31960],{"separator":758},[31749,84139,768],{},[31662,84141,31960],{"separator":758},[31662,84143,33934],{},[31662,84145,31960],{"separator":758},[31658,84147,31660],{},[31665,84149,84150],{"encoding":31667},"1, 2, \\dots, n",[34,84152,84154],{"className":84153,"ariaHidden":758},[31672],[34,84155,84157,84160,84163,84166,84169,84172,84175,84178,84181,84184,84187,84190],{"className":84156},[31676],[34,84158],{"className":84159,"style":40292},[31680],[34,84161,640],{"className":84162},[31685],[34,84164,31960],{"className":84165},[32119],[34,84167],{"className":84168,"style":32123},[31845],[34,84170,768],{"className":84171},[31685],[34,84173,31960],{"className":84174},[32119],[34,84176],{"className":84177,"style":32123},[31845],[34,84179,33934],{"className":84180},[34006],[34,84182],{"className":84183,"style":32123},[31845],[34,84185,31960],{"className":84186},[32119],[34,84188],{"className":84189,"style":32123},[31845],[34,84191,31660],{"className":84192},[31685,31686],"。\n第 1 个作业无需等待，等待时间为 0。\n第 2 个作业需等待第 1 个作业执行完毕，等待时间为 ",[34,84195,84197,84215],{"className":84196},[31642],[34,84198,84200],{"className":84199},[31646],[31648,84201,84202],{"xmlns":31650},[31652,84203,84204,84212],{},[31655,84205,84206],{},[31743,84207,84208,84210],{},[31658,84209,33249],{},[31749,84211,640],{},[31665,84213,84214],{"encoding":31667},"t_1",[34,84216,84218],{"className":84217,"ariaHidden":758},[31672],[34,84219,84221,84224],{"className":84220},[31676],[34,84222],{"className":84223,"style":83872},[31680],[34,84225,84227,84230],{"className":84226},[31685],[34,84228,33249],{"className":84229},[31685,31686],[34,84231,84233],{"className":84232},[31792],[34,84234,84236,84256],{"className":84235},[31796,31797],[34,84237,84239,84253],{"className":84238},[31801],[34,84240,84242],{"className":84241,"style":31806},[31805],[34,84243,84244,84247],{"style":31809},[34,84245],{"className":84246,"style":31814},[31813],[34,84248,84250],{"className":84249},[31818,31819,31820,31821],[34,84251,640],{"className":84252},[31685,31821],[34,84254,31829],{"className":84255},[31828],[34,84257,84259],{"className":84258},[31801],[34,84260,84262],{"className":84261,"style":31836},[31805],[34,84263],{},"。\n第 3 个作业需等待前两个作业执行完，等待时间为 ",[34,84266,84268,84294],{"className":84267},[31642],[34,84269,84271],{"className":84270},[31646],[31648,84272,84273],{"xmlns":31650},[31652,84274,84275,84291],{},[31655,84276,84277,84283,84285],{},[31743,84278,84279,84281],{},[31658,84280,33249],{},[31749,84282,640],{},[31662,84284,223],{},[31743,84286,84287,84289],{},[31658,84288,33249],{},[31749,84290,768],{},[31665,84292,84293],{"encoding":31667},"t_1 + t_2",[34,84295,84297,84352],{"className":84296,"ariaHidden":758},[31672],[34,84298,84300,84303,84343,84346,84349],{"className":84299},[31676],[34,84301],{"className":84302,"style":83872},[31680],[34,84304,84306,84309],{"className":84305},[31685],[34,84307,33249],{"className":84308},[31685,31686],[34,84310,84312],{"className":84311},[31792],[34,84313,84315,84335],{"className":84314},[31796,31797],[34,84316,84318,84332],{"className":84317},[31801],[34,84319,84321],{"className":84320,"style":31806},[31805],[34,84322,84323,84326],{"style":31809},[34,84324],{"className":84325,"style":31814},[31813],[34,84327,84329],{"className":84328},[31818,31819,31820,31821],[34,84330,640],{"className":84331},[31685,31821],[34,84333,31829],{"className":84334},[31828],[34,84336,84338],{"className":84337},[31801],[34,84339,84341],{"className":84340,"style":31836},[31805],[34,84342],{},[34,84344],{"className":84345,"style":32604},[31845],[34,84347,223],{"className":84348},[32024],[34,84350],{"className":84351,"style":32604},[31845],[34,84353,84355,84358],{"className":84354},[31676],[34,84356],{"className":84357,"style":83872},[31680],[34,84359,84361,84364],{"className":84360},[31685],[34,84362,33249],{"className":84363},[31685,31686],[34,84365,84367],{"className":84366},[31792],[34,84368,84370,84390],{"className":84369},[31796,31797],[34,84371,84373,84387],{"className":84372},[31801],[34,84374,84376],{"className":84375,"style":31806},[31805],[34,84377,84378,84381],{"style":31809},[34,84379],{"className":84380,"style":31814},[31813],[34,84382,84384],{"className":84383},[31818,31819,31820,31821],[34,84385,768],{"className":84386},[31685,31821],[34,84388,31829],{"className":84389},[31828],[34,84391,84393],{"className":84392},[31801],[34,84394,84396],{"className":84395,"style":31836},[31805],[34,84397],{},"。\n...\n总等待时间 ",[34,84400,84402,84490],{"className":84401},[31642],[34,84403,84405],{"className":84404},[31646],[31648,84406,84407],{"xmlns":31650},[31652,84408,84409,84487],{},[31655,84410,84411,84413,84415,84417,84419,84425,84427,84429,84435,84437,84443,84445,84447,84449,84451,84453,84459,84461,84467,84469,84471,84473,84485],{},[31658,84412,37400],{},[31662,84414,202],{},[31749,84416,459],{},[31662,84418,223],{},[31743,84420,84421,84423],{},[31658,84422,33249],{},[31749,84424,640],{},[31662,84426,223],{},[31662,84428,456],{"stretchy":702},[31743,84430,84431,84433],{},[31658,84432,33249],{},[31749,84434,640],{},[31662,84436,223],{},[31743,84438,84439,84441],{},[31658,84440,33249],{},[31749,84442,768],{},[31662,84444,2557],{"stretchy":702},[31662,84446,223],{},[31662,84448,38874],{},[31662,84450,223],{},[31662,84452,456],{"stretchy":702},[31743,84454,84455,84457],{},[31658,84456,33249],{},[31749,84458,640],{},[31662,84460,223],{},[31743,84462,84463,84465],{},[31658,84464,33249],{},[31749,84466,768],{},[31662,84468,223],{},[31662,84470,38874],{},[31662,84472,223],{},[31743,84474,84475,84477],{},[31658,84476,33249],{},[31655,84478,84479,84481,84483],{},[31658,84480,31660],{},[31662,84482,32226],{},[31749,84484,640],{},[31662,84486,2557],{"stretchy":702},[31665,84488,84489],{"encoding":31667},"T = 0 + t_1 + (t_1 + t_2) + \\dots + (t_1 + t_2 + \\dots + t_{n-1})",[34,84491,84493,84511,84529,84584,84642,84700,84718,84776,84831,84849],{"className":84492,"ariaHidden":758},[31672],[34,84494,84496,84499,84502,84505,84508],{"className":84495},[31676],[34,84497],{"className":84498,"style":33218},[31680],[34,84500,37400],{"className":84501,"style":37422},[31685,31686],[34,84503],{"className":84504,"style":31846},[31845],[34,84506,202],{"className":84507},[31850],[34,84509],{"className":84510,"style":31846},[31845],[34,84512,84514,84517,84520,84523,84526],{"className":84513},[31676],[34,84515],{"className":84516,"style":35942},[31680],[34,84518,459],{"className":84519},[31685],[34,84521],{"className":84522,"style":32604},[31845],[34,84524,223],{"className":84525},[32024],[34,84527],{"className":84528,"style":32604},[31845],[34,84530,84532,84535,84575,84578,84581],{"className":84531},[31676],[34,84533],{"className":84534,"style":83872},[31680],[34,84536,84538,84541],{"className":84537},[31685],[34,84539,33249],{"className":84540},[31685,31686],[34,84542,84544],{"className":84543},[31792],[34,84545,84547,84567],{"className":84546},[31796,31797],[34,84548,84550,84564],{"className":84549},[31801],[34,84551,84553],{"className":84552,"style":31806},[31805],[34,84554,84555,84558],{"style":31809},[34,84556],{"className":84557,"style":31814},[31813],[34,84559,84561],{"className":84560},[31818,31819,31820,31821],[34,84562,640],{"className":84563},[31685,31821],[34,84565,31829],{"className":84566},[31828],[34,84568,84570],{"className":84569},[31801],[34,84571,84573],{"className":84572,"style":31836},[31805],[34,84574],{},[34,84576],{"className":84577,"style":32604},[31845],[34,84579,223],{"className":84580},[32024],[34,84582],{"className":84583,"style":32604},[31845],[34,84585,84587,84590,84593,84633,84636,84639],{"className":84586},[31676],[34,84588],{"className":84589,"style":31774},[31680],[34,84591,456],{"className":84592},[31782],[34,84594,84596,84599],{"className":84595},[31685],[34,84597,33249],{"className":84598},[31685,31686],[34,84600,84602],{"className":84601},[31792],[34,84603,84605,84625],{"className":84604},[31796,31797],[34,84606,84608,84622],{"className":84607},[31801],[34,84609,84611],{"className":84610,"style":31806},[31805],[34,84612,84613,84616],{"style":31809},[34,84614],{"className":84615,"style":31814},[31813],[34,84617,84619],{"className":84618},[31818,31819,31820,31821],[34,84620,640],{"className":84621},[31685,31821],[34,84623,31829],{"className":84624},[31828],[34,84626,84628],{"className":84627},[31801],[34,84629,84631],{"className":84630,"style":31836},[31805],[34,84632],{},[34,84634],{"className":84635,"style":32604},[31845],[34,84637,223],{"className":84638},[32024],[34,84640],{"className":84641,"style":32604},[31845],[34,84643,84645,84648,84688,84691,84694,84697],{"className":84644},[31676],[34,84646],{"className":84647,"style":31774},[31680],[34,84649,84651,84654],{"className":84650},[31685],[34,84652,33249],{"className":84653},[31685,31686],[34,84655,84657],{"className":84656},[31792],[34,84658,84660,84680],{"className":84659},[31796,31797],[34,84661,84663,84677],{"className":84662},[31801],[34,84664,84666],{"className":84665,"style":31806},[31805],[34,84667,84668,84671],{"style":31809},[34,84669],{"className":84670,"style":31814},[31813],[34,84672,84674],{"className":84673},[31818,31819,31820,31821],[34,84675,768],{"className":84676},[31685,31821],[34,84678,31829],{"className":84679},[31828],[34,84681,84683],{"className":84682},[31801],[34,84684,84686],{"className":84685,"style":31836},[31805],[34,84687],{},[34,84689,2557],{"className":84690},[31690],[34,84692],{"className":84693,"style":32604},[31845],[34,84695,223],{"className":84696},[32024],[34,84698],{"className":84699,"style":32604},[31845],[34,84701,84703,84706,84709,84712,84715],{"className":84702},[31676],[34,84704],{"className":84705,"style":32597},[31680],[34,84707,38874],{"className":84708},[34006],[34,84710],{"className":84711,"style":32604},[31845],[34,84713,223],{"className":84714},[32024],[34,84716],{"className":84717,"style":32604},[31845],[34,84719,84721,84724,84727,84767,84770,84773],{"className":84720},[31676],[34,84722],{"className":84723,"style":31774},[31680],[34,84725,456],{"className":84726},[31782],[34,84728,84730,84733],{"className":84729},[31685],[34,84731,33249],{"className":84732},[31685,31686],[34,84734,84736],{"className":84735},[31792],[34,84737,84739,84759],{"className":84738},[31796,31797],[34,84740,84742,84756],{"className":84741},[31801],[34,84743,84745],{"className":84744,"style":31806},[31805],[34,84746,84747,84750],{"style":31809},[34,84748],{"className":84749,"style":31814},[31813],[34,84751,84753],{"className":84752},[31818,31819,31820,31821],[34,84754,640],{"className":84755},[31685,31821],[34,84757,31829],{"className":84758},[31828],[34,84760,84762],{"className":84761},[31801],[34,84763,84765],{"className":84764,"style":31836},[31805],[34,84766],{},[34,84768],{"className":84769,"style":32604},[31845],[34,84771,223],{"className":84772},[32024],[34,8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",[34,84909,84911,84928],{"className":84910},[31642],[34,84912,84914],{"className":84913},[31646],[31648,84915,84916],{"xmlns":31650},[31652,84917,84918,84926],{},[31655,84919,84920],{},[31743,84921,84922,84924],{},[31658,84923,33249],{},[31749,84925,640],{},[31665,84927,84214],{"encoding":31667},[34,84929,84931],{"className":84930,"ariaHidden":758},[31672],[34,84932,84934,84937],{"className":84933},[31676],[34,84935],{"className":84936,"style":83872},[31680],[34,84938,84940,84943],{"className":84939},[31685],[34,84941,33249],{"className":84942},[31685,31686],[34,84944,84946],{"className":84945},[31792],[34,84947,84949,84969],{"className":84948},[31796,31797],[34,84950,84952,84966],{"className":84951},[31801],[34,84953,84955],{"className":84954,"style":31806},[31805],[34,84956,84957,84960],{"style":31809},[34,84958],{"className":84959,"style":31814},[31813],[34,84961,84963],{"className":84962},[31818,31819,31820,31821],[34,84964,640],{"className":84965},[31685,31821],[34,84967,31829],{"className":84968},[31828],[34,84970,84972],{"className":84971},[31801],[34,84973,84975],{"className":84974,"style":31836},[31805],[34,84976],{}," 累加了 ",[34,84979,84981,84998],{"className":84980},[31642],[34,84982,84984],{"className":84983},[31646],[31648,84985,84986],{"xmlns":31650},[31652,84987,84988,84996],{},[31655,84989,84990,84992,84994],{},[31658,84991,31660],{},[31662,84993,32226],{},[31749,84995,640],{},[31665,84997,32587],{"encoding":31667},[34,84999,85001,85019],{"className":85000,"ariaHidden":758},[31672],[34,85002,85004,85007,85010,85013,85016],{"className":85003},[31676],[34,85005],{"className":85006,"style":32597},[31680],[34,85008,31660],{"className":85009},[31685,31686],[34,85011],{"className":85012,"style":32604},[31845],[34,85014,32226],{"className":85015},[32024],[34,85017],{"className":85018,"style":32604},[31845],[34,85020,85022,85025],{"className":85021},[31676],[34,85023],{"className":85024,"style":32617},[31680],[34,85026,640],{"className":85027},[31685]," 次，",[34,85030,85032,85050],{"className":85031},[31642],[34,85033,85035],{"className":85034},[31646],[31648,85036,85037],{"xmlns":31650},[31652,85038,85039,85047],{},[31655,85040,85041],{},[31743,85042,85043,85045],{},[31658,85044,33249],{},[31749,85046,768],{},[31665,85048,85049],{"encoding":31667},"t_2",[34,85051,85053],{"className":85052,"ariaHidden":758},[31672],[34,85054,85056,85059],{"className":85055},[31676],[34,85057],{"className":85058,"style":83872},[31680],[34,85060,85062,85065],{"className":85061},[31685],[34,85063,33249],{"className":85064},[31685,31686],[34,85066,85068],{"className":85067},[31792],[34,85069,85071,85091],{"className":85070},[31796,31797],[34,85072,85074,85088],{"className":85073},[31801],[34,85075,85077],{"className":85076,"style":31806},[31805],[34,85078,85079,85082],{"style":31809},[34,85080],{"className":85081,"style":31814},[31813],[34,85083,85085],{"className":85084},[31818,31819,31820,31821],[34,85086,768],{"className":85087},[31685,31821],[34,85089,31829],{"className":85090},[31828],[34,85092,85094],{"className":85093},[31801],[34,85095,85097],{"className":85096,"style":31836},[31805],[34,85098],{},[34,85100,85102,85120],{"className":85101},[31642],[34,85103,85105],{"className":85104},[31646],[31648,85106,85107],{"xmlns":31650},[31652,85108,85109,85117],{},[31655,85110,85111,85113,85115],{},[31658,85112,31660],{},[31662,85114,32226],{},[31749,85116,768],{},[31665,85118,85119],{"encoding":31667},"n-2",[34,85121,85123,85141],{"className":85122,"ariaHidden":758},[31672],[34,85124,85126,85129,85132,85135,85138],{"className":85125},[31676],[34,85127],{"className":85128,"style":32597},[31680],[34,85130,31660],{"className":85131},[31685,31686],[34,85133],{"className":85134,"style":32604},[31845],[34,85136,32226],{"className":85137},[32024],[34,85139],{"className":85140,"style":32604},[31845],[34,85142,85144,85147],{"className":85143},[31676],[34,85145],{"className":85146,"style":32617},[31680],[34,85148,768],{"className":85149},[31685]," 次……\n所以，总等待时间 ",[2627,85152,85153],{},[34,85154,85156,85228],{"className":85155},[31642],[34,85157,85159],{"className":85158},[31646],[31648,85160,85161],{"xmlns":31650},[31652,85162,85163,85225],{},[31655,85164,85165,85167,85169,85171,85173,85175,85177,85179,85185,85187,85189,85191,85193,85195,85197,85203,85205,85207,85209,85211,85213],{},[31658,85166,37400],{},[31662,85168,202],{},[31662,85170,456],{"stretchy":702},[31658,85172,31660],{},[31662,85174,32226],{},[31749,85176,640],{},[31662,85178,2557],{"stretchy":702},[31743,85180,85181,85183],{},[31658,85182,33249],{},[31749,85184,640],{},[31662,85186,223],{},[31662,85188,456],{"stretchy":702},[31658,85190,31660],{},[31662,85192,32226],{},[31749,85194,768],{},[31662,85196,2557],{"stretchy":702},[31743,85198,85199,85201],{},[31658,85200,33249],{},[31749,85202,768],{},[31662,85204,223],{},[31662,85206,38874],{},[31662,85208,223],{},[31749,85210,640],{},[31662,85212,71191],{},[31743,85214,85215,85217],{},[31658,85216,33249],{},[31655,85218,85219,85221,85223],{},[31658,85220,31660],{},[31662,85222,32226],{},[31749,85224,640],{},[31665,85226,85227],{"encoding":31667},"T = (n-1)t_1 + (n-2)t_2 + \\dots + 1 \\cdot t_{n-1}",[34,85229,85231,85249,85270,85331,85352,85413,85431,85449],{"className":85230,"ariaHidden":758},[31672],[34,85232,85234,85237,85240,85243,85246],{"className":85233},[31676],[34,85235],{"className":85236,"style":33218},[31680],[34,85238,37400],{"className":85239,"style":37422},[31685,31686],[34,85241],{"className":85242,"style":31846},[31845],[34,85244,202],{"className":85245},[31850],[34,85247],{"className":85248,"style":31846},[31845],[34,85250,85252,85255,85258,85261,85264,85267],{"className":85251},[31676],[34,85253],{"className":85254,"style":31774},[31680],[34,85256,456],{"className":85257},[31782],[34,85259,31660],{"className":85260},[31685,31686],[34,85262],{"className":85263,"style":32604},[31845],[34,85265,32226],{"className":85266},[32024],[34,85268],{"className":85269,"style":32604},[31845],[34,85271,85273,85276,85279,85282,85322,85325,85328],{"className":85272},[31676],[34,85274],{"className":85275,"style":31774},[31680],[34,85277,640],{"className":85278},[31685],[34,85280,2557],{"className":85281},[31690],[34,85283,85285,85288],{"className":85284},[31685],[34,85286,33249],{"className":85287},[31685,31686],[34,85289,85291],{"className":85290},[31792],[34,85292,85294,85314],{"className":85293},[31796,31797],[34,85295,85297,85311],{"className":85296},[31801],[34,85298,85300],{"className":85299,"style":31806},[31805],[34,85301,85302,85305],{"style":31809},[34,85303],{"className":85304,"style":31814},[31813],[34,85306,85308],{"className":85307},[31818,31819,31820,31821],[34,85309,640],{"className":85310},[31685,31821],[34,85312,31829],{"className":85313},[31828],[34,85315,85317],{"className":85316},[31801],[34,85318,85320],{"className":85319,"style":31836},[31805],[34,85321],{},[34,85323],{"className":85324,"style":32604},[31845],[34,85326,223],{"className":85327},[32024],[34,85329],{"className":85330,"style":32604},[31845],[34,85332,85334,85337,85340,85343,85346,85349],{"className":85333},[31676],[34,85335],{"className":85336,"style":31774},[31680],[34,85338,456],{"className":85339},[31782],[34,85341,31660],{"className":85342},[31685,31686],[34,85344],{"className":85345,"style":32604},[31845],[34,85347,32226],{"className":85348},[32024],[34,85350],{"className":85351,"style":32604},[31845],[34,85353,85355,85358,85361,85364,85404,85407,85410],{"className":85354},[31676],[34,85356],{"className":85357,"style":31774},[31680],[34,85359,768],{"className":85360},[31685],[34,85362,2557],{"className":85363},[31690],[34,85365,85367,85370],{"className":85366},[31685],[34,85368,33249],{"className":85369},[31685,31686],[34,85371,85373],{"className":85372},[31792],[34,85374,85376,85396],{"className":85375},[31796,31797],[34,85377,85379,85393],{"className":85378},[31801],[34,85380,85382],{"className":85381,"style":31806},[31805],[34,85383,85384,85387],{"style":31809},[34,85385],{"className":85386,"style":31814},[31813],[34,85388,85390],{"className":85389},[31818,31819,31820,31821],[34,85391,768],{"className":85392},[31685,31821],[34,85394,31829],{"className":85395},[31828],[34,85397,85399],{"className":85398},[31801],[34,85400,85402],{"className":85401,"style":31836},[31805],[34,85403],{},[34,85405],{"className":85406,"style":32604},[31845],[34,85408,223],{"className":85409},[32024],[34,85411],{"className":85412,"style":32604},[31845],[34,85414,85416,85419,85422,85425,85428],{"className":85415},[31676],[34,85417],{"className":85418,"style":32597},[31680],[34,85420,38874],{"className":85421},[34006],[34,85423],{"className":85424,"style":32604},[31845],[34,85426,223],{"className":85427},[32024],[34,85429],{"className":85430,"style":32604},[31845],[34,85432,85434,85437,85440,85443,85446],{"className":85433},[31676],[34,85435],{"className":85436,"style":32617},[31680],[34,85438,640],{"className":85439},[31685],[34,85441],{"className":85442,"style":32604},[31845],[34,85444,71191],{"className":85445},[32024],[34,85447],{"className":85448,"style":32604},[31845],[34,85450,85452,85456],{"className":85451},[31676],[34,85453],{"className":85454,"style":85455},[31680],"height:0.8234em;vertical-align:-0.2083em;",[34,85457,85459,85462],{"className":85458},[31685],[34,85460,33249],{"className":85461},[31685,31686],[34,85463,85465],{"className":85464},[31792],[34,85466,85468,85497],{"className":85467},[31796,31797],[34,85469,85471,85494],{"className":85470},[31801],[34,85472,85474],{"className":85473,"style":31806},[31805],[34,85475,85476,85479],{"style":31809},[34,85477],{"className":85478,"style":31814},[31813],[34,85480,85482],{"className":85481},[31818,31819,31820,31821],[34,85483,85485,85488,85491],{"className":85484},[31685,31821],[34,85486,31660],{"className":85487},[31685,31686,31821],[34,85489,32226],{"className":85490},[32024,31821],[34,85492,640],{"className":85493},[31685,31821],[34,85495,31829],{"className":85496},[31828],[34,85498,85500],{"className":85499},[31801],[34,85501,85503],{"className":85502,"style":32037},[31805],[34,85504],{},"。\n要使得“平均等待时间最短”，就等同于使得“总等待时间 ",[34,85507,85509,85522],{"className":85508},[31642],[34,85510,85512],{"className":85511},[31646],[31648,85513,85514],{"xmlns":31650},[31652,85515,85516,85520],{},[31655,85517,85518],{},[31658,85519,37400],{},[31665,85521,37400],{"encoding":31667},[34,85523,85525],{"className":85524,"ariaHidden":758},[31672],[34,85526,85528,85531],{"className":85527},[31676],[34,85529],{"className":85530,"style":33218},[31680],[34,85532,37400],{"className":85533,"style":37422},[31685,31686]," 最小”。",[105,85536,85537,87010],{},[108,85538,85539,85542,85543,85571,85572,85600,85601,85629,85630,85658,85659,85793,85794,85822,85823,85884,85885,85946,85947,85975,85976,86004,86005,86621,86622,86674,86675,13977,86807,87009],{},[2627,85540,85541],{},"贪心选择性质证明（运用交换论证法）：","\n假设我们的短作业优先策略产生的升序排列不是最优的。这意味着存在某个最优调度方案，在这个方案中，必定存在“逆序对”——也就是说，存在排在前面的某个作业 ",[34,85544,85546,85559],{"className":85545},[31642],[34,85547,85549],{"className":85548},[31646],[31648,85550,85551],{"xmlns":31650},[31652,85552,85553,85557],{},[31655,85554,85555],{},[31658,85556,31955],{},[31665,85558,31955],{"encoding":31667},[34,85560,85562],{"className":85561,"ariaHidden":758},[31672],[34,85563,85565,85568],{"className":85564},[31676],[34,85566],{"className":85567,"style":33104},[31680],[34,85569,31955],{"className":85570},[31685,31686],"，其处理时间比排在后面的作业 ",[34,85573,85575,85588],{"className":85574},[31642],[34,85576,85578],{"className":85577},[31646],[31648,85579,85580],{"xmlns":31650},[31652,85581,85582,85586],{},[31655,85583,85584],{},[31658,85585,31968],{},[31665,85587,31968],{"encoding":31667},[34,85589,85591],{"className":85590,"ariaHidden":758},[31672],[34,85592,85594,85597],{"className":85593},[31676],[34,85595],{"className":85596,"style":33123},[31680],[34,85598,31968],{"className":85599,"style":32153},[31685,31686]," 更长（即存在某个排列次序使得靠前的位置 ",[34,85602,85604,85617],{"className":85603},[31642],[34,85605,85607],{"className":85606},[31646],[31648,85608,85609],{"xmlns":31650},[31652,85610,85611,85615],{},[31655,85612,85613],{},[31658,85614,32936],{},[31665,85616,32936],{"encoding":31667},[34,85618,85620],{"className":85619,"ariaHidden":758},[31672],[34,85621,85623,85626],{"className":85622},[31676],[34,85624],{"className":85625,"style":32527},[31680],[34,85627,32936],{"className":85628},[31685,31686]," 和靠后的位置 ",[34,85631,85633,85646],{"className":85632},[31642],[34,85634,85636],{"className":85635},[31646],[31648,85637,85638],{"xmlns":31650},[31652,85639,85640,85644],{},[31655,85641,85642],{},[31658,85643,67764],{},[31665,85645,67764],{"encoding":31667},[34,85647,85649],{"className":85648,"ariaHidden":758},[31672],[34,85650,85652,85655],{"className":85651},[31676],[34,85653],{"className":85654,"style":38774},[31680],[34,85656,67764],{"className":85657,"style":32062},[31685,31686]," 满足 ",[34,85660,85662,85688],{"className":85661},[31642],[34,85663,85665],{"className":85664},[31646],[31648,85666,85667],{"xmlns":31650},[31652,85668,85669,85685],{},[31655,85670,85671,85677,85679],{},[31743,85672,85673,85675],{},[31658,85674,33249],{},[31658,85676,32936],{},[31662,85678,29622],{},[31743,85680,85681,85683],{},[31658,85682,33249],{},[31658,85684,67764],{},[31665,85686,85687],{"encoding":31667},"t_x > t_y",[34,85689,85691,85746],{"className":85690,"ariaHidden":758},[31672],[34,85692,85694,85697,85737,85740,85743],{"className":85693},[31676],[34,85695],{"className":85696,"style":83872},[31680],[34,85698,85700,85703],{"className":85699},[31685],[34,85701,33249],{"className":85702},[31685,31686],[34,85704,85706],{"className":85705},[31792],[34,85707,85709,85729],{"className":85708},[31796,31797],[34,85710,85712,85726],{"className":85711},[31801],[34,85713,85715],{"className":85714,"style":32362},[31805],[34,85716,85717,85720],{"style":31809},[34,85718],{"className":85719,"style":31814},[31813],[34,85721,85723],{"className":85722},[31818,31819,31820,31821],[34,85724,32936],{"className":85725},[31685,31686,31821],[34,85727,31829],{"className":85728},[31828],[34,85730,85732],{"className":85731},[31801],[34,85733,85735],{"className":85734,"style":31836},[31805],[34,85736],{},[34,85738],{"className":85739,"style":31846},[31845],[34,85741,29622],{"className":85742},[31850],[34,85744],{"className":85745,"style":31846},[31845],[34,85747,85749,85753],{"className":85748},[31676],[34,85750],{"className":85751,"style":85752},[31680],"height:0.9012em;vertical-align:-0.2861em;",[34,85754,85756,85759],{"className":85755},[31685],[34,85757,33249],{"className":85758},[31685,31686],[34,85760,85762],{"className":85761},[31792],[34,85763,85765,85785],{"className":85764},[31796,31797],[34,85766,85768,85782],{"className":85767},[31801],[34,85769,85771],{"className":85770,"style":32362},[31805],[34,85772,85773,85776],{"style":31809},[34,85774],{"className":85775,"style":31814},[31813],[34,85777,85779],{"className":85778},[31818,31819,31820,31821],[34,85780,67764],{"className":85781,"style":32062},[31685,31686,31821],[34,85783,31829],{"className":85784},[31828],[34,85786,85788],{"className":85787},[31801],[34,85789,85791],{"className":85790,"style":32163},[31805],[34,85792],{},"）。\n在刚才推导的公式 ",[34,85795,85797,85810],{"className":85796},[31642],[34,85798,85800],{"className":85799},[31646],[31648,85801,85802],{"xmlns":31650},[31652,85803,85804,85808],{},[31655,85805,85806],{},[31658,85807,37400],{},[31665,85809,37400],{"encoding":31667},[34,85811,85813],{"className":85812,"ariaHidden":758},[31672],[34,85814,85816,85819],{"className":85815},[31676],[34,85817],{"className":85818,"style":33218},[31680],[34,85820,37400],{"className":85821,"style":37422},[31685,31686]," 中，越排在前面的作业，它乘以的权重系数越大（",[34,85824,85826,85848],{"className":85825},[31642],[34,85827,85829],{"className":85828},[31646],[31648,85830,85831],{"xmlns":31650},[31652,85832,85833,85845],{},[31655,85834,85835,85837,85839,85841,85843],{},[31662,85836,456],{"stretchy":702},[31658,85838,31660],{},[31662,85840,32226],{},[31658,85842,32936],{},[31662,85844,2557],{"stretchy":702},[31665,85846,85847],{"encoding":31667},"(n-x)",[34,85849,85851,85872],{"className":85850,"ariaHidden":758},[31672],[34,85852,85854,85857,85860,85863,85866,85869],{"className":85853},[31676],[34,85855],{"className":85856,"style":31774},[31680],[34,85858,456],{"className":85859},[31782],[34,85861,31660],{"className":85862},[31685,31686],[34,85864],{"className":85865,"style":32604},[31845],[34,85867,32226],{"className":85868},[32024],[34,85870],{"className":85871,"style":32604},[31845],[34,85873,85875,85878,85881],{"className":85874},[31676],[34,85876],{"className":85877,"style":31774},[31680],[34,85879,32936],{"className":85880},[31685,31686],[34,85882,2557],{"className":85883},[31690]," 大于 ",[34,85886,85888,85910],{"className":85887},[31642],[34,85889,85891],{"className":85890},[31646],[31648,85892,85893],{"xmlns":31650},[31652,85894,85895,85907],{},[31655,85896,85897,85899,85901,85903,85905],{},[31662,85898,456],{"stretchy":702},[31658,85900,31660],{},[31662,85902,32226],{},[31658,85904,67764],{},[31662,85906,2557],{"stretchy":702},[31665,85908,85909],{"encoding":31667},"(n-y)",[34,85911,85913,85934],{"className":85912,"ariaHidden":758},[31672],[34,85914,85916,85919,85922,85925,85928,85931],{"className":85915},[31676],[34,85917],{"className":85918,"style":31774},[31680],[34,85920,456],{"className":85921},[31782],[34,85923,31660],{"className":85924},[31685,31686],[34,85926],{"className":85927,"style":32604},[31845],[34,85929,32226],{"className":85930},[32024],[34,85932],{"className":85933,"style":32604},[31845],[34,85935,85937,85940,85943],{"className":85936},[31676],[34,85938],{"className":85939,"style":31774},[31680],[34,85941,67764],{"className":85942,"style":32062},[31685,31686],[34,85944,2557],{"className":85945},[31690],"）。\n如果我们将这个耗时较长的作业 ",[34,85948,85950,85963],{"className":85949},[31642],[34,85951,85953],{"className":85952},[31646],[31648,85954,85955],{"xmlns":31650},[31652,85956,85957,85961],{},[31655,85958,85959],{},[31658,85960,32936],{},[31665,85962,32936],{"encoding":31667},[34,85964,85966],{"className":85965,"ariaHidden":758},[31672],[34,85967,85969,85972],{"className":85968},[31676],[34,85970],{"className":85971,"style":32527},[31680],[34,85973,32936],{"className":85974},[31685,31686]," 与耗时较短的作业 ",[34,85977,85979,85992],{"className":85978},[31642],[34,85980,85982],{"className":85981},[31646],[31648,85983,85984],{"xmlns":31650},[31652,85985,85986,85990],{},[31655,85987,85988],{},[31658,85989,67764],{},[31665,85991,67764],{"encoding":31667},[34,85993,85995],{"className":85994,"ariaHidden":758},[31672],[34,85996,85998,86001],{"className":85997},[31676],[34,85999],{"className":86000,"style":38774},[31680],[34,86002,67764],{"className":86003,"style":32062},[31685,31686]," 交换位置。\n新的总耗时相比旧的总耗时，变化量为：",[34,86006,86008,86131],{"className":86007},[31642],[34,86009,86011],{"className":86010},[31646],[31648,86012,86013],{"xmlns":31650},[31652,86014,86015,86128],{},[31655,86016,86017,86020,86022,86024,86026,86028,86030,86032,86034,86040,86042,86044,86046,86048,86050,86052,86058,86060,86062,86064,86066,86068,86070,86072,86078,86080,86082,86084,86086,86088,86090,86096,86098,86100,86102,86104,86106,86108,86110,86112,86118,86120,86126],{},[31658,86018,86019],{"mathvariant":38079},"Δ",[31658,86021,37400],{},[31662,86023,202],{},[31662,86025,456],{"stretchy":702},[31658,86027,31660],{},[31662,86029,32226],{},[31658,86031,32936],{},[31662,86033,2557],{"stretchy":702},[31743,86035,86036,86038],{},[31658,86037,33249],{},[31658,86039,67764],{},[31662,86041,223],{},[31662,86043,456],{"stretchy":702},[31658,86045,31660],{},[31662,86047,32226],{},[31658,86049,67764],{},[31662,86051,2557],{"stretchy":702},[31743,86053,86054,86056],{},[31658,86055,33249],{},[31658,86057,32936],{},[31662,86059,32226],{},[31662,86061,15972],{"stretchy":702},[31662,86063,456],{"stretchy":702},[31658,86065,31660],{},[31662,86067,32226],{},[31658,86069,32936],{},[31662,86071,2557],{"stretchy":702},[31743,86073,86074,86076],{},[31658,86075,33249],{},[31658,86077,32936],{},[31662,86079,223],{},[31662,86081,456],{"stretchy":702},[31658,86083,31660],{},[31662,86085,32226],{},[31658,86087,67764],{},[31662,86089,2557],{"stretchy":702},[31743,86091,86092,86094],{},[31658,86093,33249],{},[31658,86095,67764],{},[31662,86097,32712],{"stretchy":702},[31662,86099,202],{},[31662,86101,456],{"stretchy":702},[31658,86103,67764],{},[31662,86105,32226],{},[31658,86107,32936],{},[31662,86109,2557],{"stretchy":702},[31662,86111,456],{"stretchy":702},[31743,86113,86114,86116],{},[31658,86115,33249],{},[31658,86117,32936],{},[31662,86119,32226],{},[31743,86121,86122,86124],{},[31658,86123,33249],{},[31658,86125,67764],{},[31662,86127,2557],{"stretchy":702},[31665,86129,86130],{"encoding":31667},"\\Delta T = (n-x)t_y + (n-y)t_x - [ (n-x)t_x + (n-y)t_y ] = (y-x)(t_x - t_y)",[34,86132,86134,86155,86176,86237,86258,86319,86341,86402,86423,86487,86508,86572],{"className":86133,"ariaHidden":758},[31672],[34,86135,86137,86140,86143,86146,86149,86152],{"className":86136},[31676],[34,86138],{"className":86139,"style":33218},[31680],[34,86141,86019],{"className":86142},[31685],[34,86144,37400],{"className":86145,"style":37422},[31685,31686],[34,86147],{"className":86148,"style":31846},[31845],[34,86150,202],{"className":86151},[31850],[34,86153],{"className":86154,"style":31846},[31845],[34,86156,86158,86161,86164,86167,86170,86173],{"className":86157},[31676],[34,86159],{"className":86160,"style":31774},[31680],[34,86162,456],{"className":86163},[31782],[34,86165,31660],{"className":86166},[31685,31686],[34,86168],{"className":86169,"style":32604},[31845],[34,86171,32226],{"className":86172},[32024],[34,86174],{"className":86175,"style":32604},[31845],[34,86177,86179,86182,86185,86188,86228,86231,86234],{"className":86178},[31676],[34,86180],{"className":86181,"style":32058},[31680],[34,86183,32936],{"className":86184},[31685,31686],[34,86186,2557],{"className":86187},[31690],[34,86189,86191,86194],{"className":86190},[31685],[34,86192,33249],{"className":86193},[31685,31686],[34,86195,86197],{"className":86196},[31792],[34,86198,86200,86220],{"className":86199},[31796,31797],[34,86201,86203,86217],{"className":86202},[31801],[34,86204,86206],{"className":86205,"style":32362},[31805],[34,86207,86208,86211],{"style":31809},[34,86209],{"className":86210,"style":31814},[31813],[34,86212,86214],{"className":86213},[31818,31819,31820,31821],[34,86215,67764],{"className":86216,"style":32062},[31685,31686,31821],[34,86218,31829],{"className":86219},[31828],[34,86221,86223],{"className":86222},[31801],[34,86224,86226],{"className":86225,"style":32163},[31805],[34,86227],{},[34,86229],{"className":86230,"style":32604},[31845],[34,86232,223],{"className":86233},[32024],[34,86235],{"className":86236,"style":32604},[31845],[34,86238,86240,86243,86246,86249,86252,86255],{"className":86239},[31676],[34,86241],{"className":86242,"style":31774},[31680],[34,86244,456],{"className":86245},[31782],[34,86247,31660],{"className":86248},[31685,31686],[34,86250],{"className":86251,"style":32604},[31845],[34,86253,32226],{"className":86254},[32024],[34,86256],{"className":86257,"style":32604},[31845],[34,86259,86261,86264,86267,86270,86310,86313,86316],{"className":86260},[31676],[34,86262],{"className":86263,"style":31774},[31680],[34,86265,67764],{"className":86266,"style":32062},[31685,31686],[34,86268,2557],{"className":86269},[31690],[34,86271,86273,86276],{"className":86272},[31685],[34,86274,33249],{"className":86275},[31685,31686],[34,86277,86279],{"className":86278},[31792],[34,86280,86282,86302],{"className":86281},[31796,31797],[34,86283,86285,86299],{"className":86284},[31801],[34,86286,86288],{"className":86287,"style":32362},[31805],[34,86289,86290,86293],{"style":31809},[34,86291],{"className":86292,"style":31814},[31813],[34,86294,86296],{"className":86295},[31818,31819,31820,31821],[34,86297,32936],{"className":86298},[31685,31686,31821],[34,86300,31829],{"className":86301},[31828],[34,86303,86305],{"className":86304},[31801],[34,86306,86308],{"className":86307,"style":31836},[31805],[34,86309],{},[34,86311],{"className":86312,"style":32604},[31845],[34,86314,32226],{"className":86315},[32024],[34,86317],{"className":86318,"style":32604},[31845],[34,86320,86322,86325,86329,86332,86335,86338],{"className":86321},[31676],[34,86323],{"className":86324,"style":31774},[31680],[34,86326,86328],{"className":86327},[31782],"[(",[34,86330,31660],{"className":86331},[31685,31686],[34,86333],{"className":86334,"style":32604},[31845],[34,86336,32226],{"className":86337},[32024],[34,86339],{"className":86340,"style":32604},[31845],[34,86342,86344,86347,86350,86353,86393,86396,86399],{"className":86343},[31676],[34,86345],{"className":86346,"style":31774},[31680],[34,86348,32936],{"className":86349},[31685,31686],[34,86351,2557],{"className":86352},[31690],[34,86354,86356,86359],{"className":86355},[31685],[34,86357,33249],{"className":86358},[31685,31686],[34,86360,86362],{"className":86361},[31792],[34,86363,86365,86385],{"className":86364},[31796,31797],[34,86366,86368,86382],{"className":86367},[31801],[34,86369,86371],{"className":86370,"style":32362},[31805],[34,86372,86373,86376],{"style":31809},[34,86374],{"className":86375,"style":31814},[31813],[34,86377,86379],{"className":86378},[31818,31819,31820,31821],[34,86380,32936],{"className":86381},[31685,31686,31821],[34,86383,31829],{"className":86384},[31828],[34,86386,86388],{"className":86387},[31801],[34,86389,86391],{"className":86390,"style":31836},[31805],[34,86392],{},[34,86394],{"className":86395,"style":32604},[31845],[34,86397,223],{"className":86398},[32024],[34,86400],{"className":86401,"style":32604},[31845],[34,86403,86405,86408,86411,86414,86417,86420],{"className":86404},[31676],[34,86406],{"className":86407,"style":31774},[31680],[34,86409,456],{"className":86410},[31782],[34,86412,31660],{"className":86413},[31685,31686],[34,86415],{"className":86416,"style":32604},[31845],[34,86418,32226],{"className":86419},[32024],[34,86421],{"className":86422,"style":32604},[31845],[34,86424,86426,86429,86432,86435,86475,86478,86481,86484],{"className":86425},[31676],[34,86427],{"className":86428,"style":32058},[31680],[34,86430,67764],{"className":86431,"style":32062},[31685,31686],[34,86433,2557],{"className":86434},[31690],[34,86436,86438,86441],{"className":86437},[31685],[34,86439,33249],{"className":86440},[31685,31686],[34,86442,86444],{"className":86443},[31792],[34,86445,86447,86467],{"className":86446},[31796,31797],[34,86448,86450,86464],{"className":86449},[31801],[34,86451,86453],{"className":86452,"style":32362},[31805],[34,86454,86455,86458],{"style":31809},[34,86456],{"className":86457,"style":31814},[31813],[34,86459,86461],{"className":86460},[31818,31819,31820,31821],[34,86462,67764],{"className":86463,"style":32062},[31685,31686,31821],[34,86465,31829],{"className":86466},[31828],[34,86468,86470],{"className":86469},[31801],[34,86471,86473],{"className":86472,"style":32163},[31805],[34,86474],{},[34,86476,32712],{"className":86477},[31690],[34,86479],{"className":86480,"style":31846},[31845],[34,86482,202],{"className":86483},[31850],[34,86485],{"className":86486,"style":31846},[31845],[34,86488,86490,86493,86496,86499,86502,86505],{"className":86489},[31676],[34,86491],{"className":86492,"style":31774},[31680],[34,86494,456],{"className":86495},[31782],[34,86497,67764],{"className":86498,"style":32062},[31685,31686],[34,86500],{"className":86501,"style":32604},[31845],[34,86503,32226],{"className":86504},[32024],[34,86506],{"className":86507,"style":32604},[31845],[34,86509,86511,86514,86517,86520,86523,86563,86566,86569],{"className":86510},[31676],[34,86512],{"className":86513,"style":31774},[31680],[34,86515,32936],{"className":86516},[31685,31686],[34,86518,2557],{"className":86519},[31690],[34,86521,456],{"className":86522},[31782],[34,86524,86526,86529],{"className":86525},[31685],[34,86527,33249],{"className":86528},[31685,31686],[34,86530,86532],{"className":86531},[31792],[34,86533,86535,86555],{"className":86534},[31796,31797],[34,86536,86538,86552],{"className":86537},[31801],[34,86539,86541],{"className":86540,"style":32362},[31805],[34,86542,86543,86546],{"style":31809},[34,86544],{"className":86545,"style":31814},[31813],[34,86547,86549],{"className":86548},[31818,31819,31820,31821],[34,86550,32936],{"className":86551},[31685,31686,31821],[34,86553,31829],{"className":86554},[31828],[34,86556,86558],{"className":86557},[31801],[34,86559,86561],{"className":86560,"style":31836},[31805],[34,86562],{},[34,86564],{"className":86565,"style":32604},[31845],[34,86567,32226],{"className":86568},[32024],[34,86570],{"className":86571,"style":32604},[31845],[34,86573,86575,86578,86618],{"className":86574},[31676],[34,86576],{"className":86577,"style":32058},[31680],[34,86579,86581,86584],{"className":86580},[31685],[34,86582,33249],{"className":86583},[31685,31686],[34,86585,86587],{"className":86586},[31792],[34,86588,86590,86610],{"className":86589},[31796,31797],[34,86591,86593,86607],{"className":86592},[31801],[34,86594,86596],{"className":86595,"style":32362},[31805],[34,86597,86598,86601],{"style":31809},[34,86599],{"className":86600,"style":31814},[31813],[34,86602,86604],{"className":86603},[31818,31819,31820,31821],[34,86605,67764],{"className":86606,"style":32062},[31685,31686,31821],[34,86608,31829],{"className":86609},[31828],[34,86611,86613],{"className":86612},[31801],[34,86614,86616],{"className":86615,"style":32163},[31805],[34,86617],{},[34,86619,2557],{"className":86620},[31690],"。\n因为 ",[34,86623,86625,86643],{"className":86624},[31642],[34,86626,86628],{"className":86627},[31646],[31648,86629,86630],{"xmlns":31650},[31652,86631,86632,86640],{},[31655,86633,86634,86636,86638],{},[31658,86635,67764],{},[31662,86637,29622],{},[31658,86639,32936],{},[31665,86641,86642],{"encoding":31667},"y > x",[34,86644,86646,86665],{"className":86645,"ariaHidden":758},[31672],[34,86647,86649,86653,86656,86659,86662],{"className":86648},[31676],[34,86650],{"className":86651,"style":86652},[31680],"height:0.7335em;vertical-align:-0.1944em;",[34,86654,67764],{"className":86655,"style":32062},[31685,31686],[34,86657],{"className":86658,"style":31846},[31845],[34,86660,29622],{"className":86661},[31850],[34,86663],{"className":86664,"style":31846},[31845],[34,86666,86668,86671],{"className":86667},[31676],[34,86669],{"className":86670,"style":32527},[31680],[34,86672,32936],{"className":86673},[31685,31686]," 且 ",[34,86676,86678,86703],{"className":86677},[31642],[34,86679,86681],{"className":86680},[31646],[31648,86682,86683],{"xmlns":31650},[31652,86684,86685,86701],{},[31655,86686,86687,86693,86695],{},[31743,86688,86689,86691],{},[31658,86690,33249],{},[31658,86692,32936],{},[31662,86694,29622],{},[31743,86696,86697,86699],{},[31658,86698,33249],{},[31658,86700,67764],{},[31665,86702,85687],{"encoding":31667},[34,86704,86706,86761],{"className":86705,"ariaHidden":758},[31672],[34,86707,86709,86712,86752,86755,86758],{"className":86708},[31676],[34,86710],{"className":86711,"style":83872},[31680],[34,86713,86715,86718],{"className":86714},[31685],[34,86716,33249],{"className":86717},[31685,31686],[34,86719,86721],{"className":86720},[31792],[34,86722,86724,86744],{"className":86723},[31796,31797],[34,86725,86727,86741],{"className":86726},[31801],[34,86728,86730],{"className":86729,"style":32362},[31805],[34,86731,86732,86735],{"style":31809},[34,86733],{"className":86734,"style":31814},[31813],[34,86736,86738],{"className":86737},[31818,31819,31820,31821],[34,86739,32936],{"className":86740},[31685,31686,31821],[34,86742,31829],{"className":86743},[31828],[34,86745,86747],{"className":86746},[31801],[34,86748,86750],{"className":86749,"style":31836},[31805],[34,86751],{},[34,86753],{"className":86754,"style":31846},[31845],[34,86756,29622],{"className":86757},[31850],[34,86759],{"className":86760,"style":31846},[31845],[34,86762,86764,86767],{"className":86763},[31676],[34,86765],{"className":86766,"style":85752},[31680],[34,86768,86770,86773],{"className":86769},[31685],[34,86771,33249],{"className":86772},[31685,31686],[34,86774,86776],{"className":86775},[31792],[34,86777,86779,86799],{"className":86778},[31796,31797],[34,86780,86782,86796],{"className":86781},[31801],[34,86783,86785],{"className":86784,"style":32362},[31805],[34,86786,86787,86790],{"style":31809},[34,86788],{"className":86789,"style":31814},[31813],[34,86791,86793],{"className":86792},[31818,31819,31820,31821],[34,86794,67764],{"className":86795,"style":32062},[31685,31686,31821],[34,86797,31829],{"className":86798},[31828],[34,86800,86802],{"className":86801},[31801],[34,86803,86805],{"className":86804,"style":32163},[31805],[34,86806],{},[34,86808,86810,86854],{"className":86809},[31642],[34,86811,86813],{"className":86812},[31646],[31648,86814,86815],{"xmlns":31650},[31652,86816,86817,86851],{},[31655,86818,86819,86821,86823,86825,86827,86829,86831,86837,86839,86845,86847,86849],{},[31662,86820,456],{"stretchy":702},[31658,86822,67764],{},[31662,86824,32226],{},[31658,86826,32936],{},[31662,86828,2557],{"stretchy":702},[31662,86830,456],{"stretchy":702},[31743,86832,86833,86835],{},[31658,86834,33249],{},[31658,86836,32936],{},[31662,86838,32226],{},[31743,86840,86841,86843],{},[31658,86842,33249],{},[31658,86844,67764],{},[31662,86846,2557],{"stretchy":702},[31662,86848,29622],{},[31749,86850,459],{},[31665,86852,86853],{"encoding":31667},"(y-x)(t_x - t_y) > 0",[34,86855,86857,86878,86942,87000],{"className":86856,"ariaHidden":758},[31672],[34,86858,86860,86863,86866,86869,86872,86875],{"className":86859},[31676],[34,86861],{"className":86862,"style":31774},[31680],[34,86864,456],{"className":86865},[31782],[34,86867,67764],{"className":86868,"style":32062},[31685,31686],[34,86870],{"className":86871,"style":32604},[31845],[34,86873,32226],{"className":86874},[32024],[34,86876],{"className":86877,"style":32604},[31845],[34,86879,86881,86884,86887,86890,86893,86933,86936,86939],{"className":86880},[31676],[34,86882],{"className":86883,"style":31774},[31680],[34,86885,32936],{"className":86886},[31685,31686],[34,86888,2557],{"className":86889},[31690],[34,86891,456],{"className":86892},[31782],[34,86894,86896,86899],{"className":86895},[31685],[34,86897,33249],{"className":86898},[31685,31686],[34,86900,86902],{"className":86901},[31792],[34,86903,86905,86925],{"className":86904},[31796,31797],[34,86906,86908,86922],{"className":86907},[31801],[34,86909,86911],{"className":86910,"style":32362},[31805],[34,86912,86913,86916],{"style":31809},[34,86914],{"className":86915,"style":31814},[31813],[34,86917,86919],{"className":86918},[31818,31819,31820,31821],[34,86920,32936],{"className":86921},[31685,31686,31821],[34,86923,31829],{"className":86924},[31828],[34,86926,86928],{"className":86927},[31801],[34,86929,86931],{"className":86930,"style":31836},[31805],[34,86932],{},[34,86934],{"className":86935,"style":32604},[31845],[34,86937,32226],{"className":86938},[32024],[34,86940],{"className":86941,"style":32604},[31845],[34,86943,86945,86948,86988,86991,86994,86997],{"className":86944},[31676],[34,86946],{"className":86947,"style":32058},[31680],[34,86949,86951,86954],{"className":86950},[31685],[34,86952,33249],{"className":86953},[31685,31686],[34,86955,86957],{"className":86956},[31792],[34,86958,86960,86980],{"className":86959},[31796,31797],[34,86961,86963,86977],{"className":86962},[31801],[34,86964,86966],{"className":86965,"style":32362},[31805],[34,86967,86968,86971],{"style":31809},[34,86969],{"className":86970,"style":31814},[31813],[34,86972,86974],{"className":86973},[31818,31819,31820,31821],[34,86975,67764],{"className":86976,"style":32062},[31685,31686,31821],[34,86978,31829],{"className":86979},[31828],[34,86981,86983],{"className":86982},[31801],[34,86984,86986],{"className":86985,"style":32163},[31805],[34,86987],{},[34,86989,2557],{"className":86990},[31690],[34,86992],{"className":86993,"style":31846},[31845],[34,86995,29622],{"className":86996},[31850],[34,86998],{"className":86999,"style":31846},[31845],[34,87001,87003,87006],{"className":87002},[31676],[34,87004],{"className":87005,"style":32617},[31680],[34,87007,459],{"className":87008},[31685],"。\n这意味着交换后总等待时间减小了。这与它原本是“最优方案”相矛盾。\n因此，按照时间升序排列（贪心选择）必定包含在最优解中。",[108,87011,87012,87015,87016,87066,87067,87510,87511,87561,87562,87612,87613,87641],{},[2627,87013,87014],{},"最优子结构性质证明：","\n假设我们按照贪心策略选出了耗时最短的作业 1。剩下的问题变成了：如何调度余下的 ",[34,87017,87019,87036],{"className":87018},[31642],[34,87020,87022],{"className":87021},[31646],[31648,87023,87024],{"xmlns":31650},[31652,87025,87026,87034],{},[31655,87027,87028,87030,87032],{},[31658,87029,31660],{},[31662,87031,32226],{},[31749,87033,640],{},[31665,87035,32587],{"encoding":31667},[34,87037,87039,87057],{"className":87038,"ariaHidden":758},[31672],[34,87040,87042,87045,87048,87051,87054],{"className":87041},[31676],[34,87043],{"className":87044,"style":32597},[31680],[34,87046,31660],{"className":87047},[31685,31686],[34,87049],{"className":87050,"style":32604},[31845],[34,87052,32226],{"className":87053},[32024],[34,87055],{"className":87056,"style":32604},[31845],[34,87058,87060,87063],{"className":87059},[31676],[34,87061],{"className":87062,"style":32617},[31680],[34,87064,640],{"className":87065},[31685]," 个作业使得总等待时间最小。\n根据公式推导 ",[34,87068,87070,87158],{"className":87069},[31642],[34,87071,87073],{"className":87072},[31646],[31648,87074,87075],{"xmlns":31650},[31652,87076,87077,87155],{},[31655,87078,87079,87081,87083,87085,87087,87089,87091,87093,87099,87101,87103,87105,87107,87109,87111,87113,87119,87121,87123,87125,87127,87129,87131,87137,87139,87141,87143,87145,87147,87153],{},[31658,87080,37400],{},[31662,87082,202],{},[31662,87084,456],{"stretchy":702},[31658,87086,31660],{},[31662,87088,32226],{},[31749,87090,640],{},[31662,87092,2557],{"stretchy":702},[31743,87094,87095,87097],{},[31658,87096,33249],{},[31749,87098,640],{},[31662,87100,223],{},[31662,87102,15972],{"stretchy":702},[31662,87104,456],{"stretchy":702},[31658,87106,31660],{},[31662,87108,32226],{},[31749,87110,768],{},[31662,87112,2557],{"stretchy":702},[31743,87114,87115,87117],{},[31658,87116,33249],{},[31749,87118,768],{},[31662,87120,223],{},[31662,87122,456],{"stretchy":702},[31658,87124,31660],{},[31662,87126,32226],{},[31749,87128,786],{},[31662,87130,2557],{"stretchy":702},[31743,87132,87133,87135],{},[31658,87134,33249],{},[31749,87136,786],{},[31662,87138,223],{},[31662,87140,38874],{},[31662,87142,223],{},[31749,87144,640],{},[31662,87146,71191],{},[31743,87148,87149,87151],{},[31658,87150,33249],{},[31658,87152,31660],{},[31662,87154,32712],{"stretchy":702},[31665,87156,87157],{"encoding":31667},"T = (n-1)t_1 + [ (n-2)t_2 + (n-3)t_3 + \\dots + 1 \\cdot t_n ]",[34,87159,87161,87179,87200,87261,87282,87343,87364,87425,87443,87461],{"className":87160,"ariaHidden":758},[31672],[34,87162,87164,87167,87170,87173,87176],{"className":87163},[31676],[34,87165],{"className":87166,"style":33218},[31680],[34,87168,37400],{"className":87169,"style":37422},[31685,31686],[34,87171],{"className":87172,"style":31846},[31845],[34,87174,202],{"className":87175},[31850],[34,87177],{"className":87178,"style":31846},[31845],[34,87180,87182,87185,87188,87191,87194,87197],{"className":87181},[31676],[34,87183],{"className":87184,"style":31774},[31680],[34,87186,456],{"className":87187},[31782],[34,87189,31660],{"className":87190},[31685,31686],[34,87192],{"className":87193,"style":32604},[31845],[34,87195,32226],{"className":87196},[32024],[34,87198],{"className":87199,"style":32604},[31845],[34,87201,87203,87206,87209,87212,87252,87255,87258],{"className":87202},[31676],[34,87204],{"className":87205,"style":31774},[31680],[34,87207,640],{"className":87208},[31685],[34,87210,2557],{"className":87211},[31690],[34,87213,87215,87218],{"className":87214},[31685],[34,87216,33249],{"className":87217},[31685,31686],[34,87219,87221],{"className":87220},[31792],[34,87222,87224,87244],{"className":87223},[31796,31797],[34,87225,87227,87241],{"className":87226},[31801],[34,87228,87230],{"className":87229,"style":31806},[31805],[34,87231,87232,87235],{"style":31809},[34,87233],{"className":87234,"style":31814},[31813],[34,87236,87238],{"className":87237},[31818,31819,31820,31821],[34,87239,640],{"className":87240},[31685,31821],[34,87242,31829],{"className":87243},[31828],[34,87245,87247],{"className":87246},[31801],[34,87248,87250],{"className":87249,"style":31836},[31805],[34,87251],{},[34,87253],{"className":87254,"style":32604},[31845],[34,87256,223],{"className":87257},[32024],[34,87259],{"className":87260,"style":32604},[31845],[34,87262,87264,87267,87270,87273,87276,87279],{"className":87263},[31676],[34,87265],{"className":87266,"style":31774},[31680],[34,87268,86328],{"className":87269},[31782],[34,87271,31660],{"className":87272},[31685,31686],[34,87274],{"className":87275,"style":32604},[31845],[34,87277,32226],{"className":87278},[32024],[34,87280],{"className":87281,"style":32604},[31845],[34,87283,87285,87288,87291,87294,87334,87337,87340],{"className":87284},[31676],[34,87286],{"className":87287,"style":31774},[31680],[34,87289,768],{"className":87290},[31685],[34,87292,2557],{"className":87293},[31690],[34,87295,87297,87300],{"className":87296},[31685],[34,87298,33249],{"className":87299},[31685,31686],[34,87301,87303],{"className":87302},[31792],[34,87304,87306,87326],{"className":87305},[31796,31797],[34,87307,87309,87323],{"className":87308},[31801],[34,87310,87312],{"className":87311,"style":31806},[31805],[34,87313,87314,87317],{"style":31809},[34,87315],{"className":87316,"style":31814},[31813],[34,87318,87320],{"className":87319},[31818,31819,31820,31821],[34,87321,768],{"className":87322},[31685,31821],[34,87324,31829],{"className":87325},[31828],[34,87327,87329],{"className":87328},[31801],[34,87330,87332],{"className":87331,"style":31836},[31805],[34,87333],{},[34,87335],{"className":87336,"style":32604},[31845],[34,87338,223],{"className":87339},[32024],[34,87341],{"className":87342,"style":32604},[31845],[34,87344,87346,87349,87352,87355,87358,87361],{"className":87345},[31676],[34,87347],{"className":87348,"style":31774},[31680],[34,87350,456],{"className":87351},[31782],[34,87353,31660],{"className":87354},[31685,31686],[34,87356],{"className":87357,"style":32604},[31845],[34,87359,32226],{"className":87360},[32024],[34,87362],{"className":87363,"style":32604},[31845],[34,87365,87367,87370,87373,87376,87416,87419,87422],{"className":87366},[31676],[34,87368],{"className":87369,"style":31774},[31680],[34,87371,786],{"className":87372},[31685],[34,87374,2557],{"className":87375},[31690],[34,87377,87379,87382],{"className":87378},[31685],[34,87380,33249],{"className":87381},[31685,31686],[34,87383,87385],{"className":87384},[31792],[34,87386,87388,87408],{"className":87387},[31796,31797],[34,87389,87391,87405],{"className":87390},[31801],[34,87392,87394],{"className":87393,"style":31806},[31805],[34,87395,87396,87399],{"style":31809},[34,87397],{"className":87398,"style":31814},[31813],[34,87400,87402],{"className":87401},[31818,31819,31820,31821],[34,87403,786],{"className":87404},[31685,31821],[34,87406,31829],{"className":87407},[31828],[34,87409,87411],{"className":87410},[31801],[34,87412,87414],{"className":87413,"style":31836},[31805],[34,87415],{},[34,87417],{"className":87418,"style":32604},[31845],[34,87420,223],{"className":87421},[32024],[34,87423],{"className":87424,"style":32604},[31845],[34,87426,87428,87431,87434,87437,87440],{"className":87427},[31676],[34,87429],{"className":87430,"style":32597},[31680],[34,87432,38874],{"className":87433},[34006],[34,87435],{"className":87436,"style":32604},[31845],[34,87438,223],{"className":87439},[32024],[34,8744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",[34,87512,87514,87531],{"className":87513},[31642],[34,87515,87517],{"className":87516},[31646],[31648,87518,87519],{"xmlns":31650},[31652,87520,87521,87529],{},[31655,87522,87523,87525,87527],{},[31658,87524,31660],{},[31662,87526,32226],{},[31749,87528,640],{},[31665,87530,32587],{"encoding":31667},[34,87532,87534,87552],{"className":87533,"ariaHidden":758},[31672],[34,87535,87537,87540,87543,87546,87549],{"className":87536},[31676],[34,87538],{"className":87539,"style":32597},[31680],[34,87541,31660],{"className":87542},[31685,31686],[34,87544],{"className":87545,"style":32604},[31845],[34,87547,32226],{"className":87548},[32024],[34,87550],{"className":87551,"style":32604},[31845],[34,87553,87555,87558],{"className":87554},[31676],[34,87556],{"className":87557,"style":32617},[31680],[34,87559,640],{"className":87560},[31685]," 个作业独立调度的总等待时间模型（只是常数项权重集体偏移）。\n如果这剩下的 ",[34,87563,87565,87582],{"className":87564},[31642],[34,87566,87568],{"className":87567},[31646],[31648,87569,87570],{"xmlns":31650},[31652,87571,87572,87580],{},[31655,87573,87574,87576,87578],{},[31658,87575,31660],{},[31662,87577,32226],{},[31749,87579,640],{},[31665,87581,32587],{"encoding":31667},[34,87583,87585,87603],{"className":87584,"ariaHidden":758},[31672],[34,87586,87588,87591,87594,87597,87600],{"className":87587},[31676],[34,87589],{"className":87590,"style":32597},[31680],[34,87592,31660],{"className":87593},[31685,31686],[34,87595],{"className":87596,"style":32604},[31845],[34,87598,32226],{"className":87599},[32024],[34,87601],{"className":87602,"style":32604},[31845],[34,87604,87606,87609],{"className":87605},[31676],[34,87607],{"className":87608,"style":32617},[31680],[34,87610,640],{"className":87611},[31685]," 个作业没有按照使方括号内总和最小的方式去排列，那么必定存在另一个排列能让方括号内的值更小，进而导致整体 ",[34,87614,87616,87629],{"className":87615},[31642],[34,87617,87619],{"className":87618},[31646],[31648,87620,87621],{"xmlns":31650},[31652,87622,87623,87627],{},[31655,87624,87625],{},[31658,87626,37400],{},[31665,87628,37400],{"encoding":31667},[34,87630,87632],{"className":87631,"ariaHidden":758},[31672],[34,87633,87635,87638],{"className":87634},[31676],[34,87636],{"className":87637,"style":33218},[31680],[34,87639,37400],{"className":87640,"style":37422},[31685,31686]," 更小，这与原问题的最优性矛盾。\n因此，原问题的最优解包含了子问题的最优解，满足最优子结构性质。",[32188,87643],{},[19,87645,87647],{"id":87646},"五算法设计每题10分共20分","五、算法设计（每题10分，共20分）",[253,87649,58162],{"id":87650},"_1-题目内容-4",[15,87652,87653],{},"设计递归算法求一个不带头节点的单链表长度。",[2230,87655,87656,87659],{},[108,87657,87658],{},"编写该算法。（5分）",[108,87660,87661],{},"分析所涉及的算法的时间和空间复杂度。（5分）",[253,87663,58168],{"id":87664},"答案与深度解析-22",[15,87666,87667],{},[2627,87668,87669],{},"1) 算法代码（C\u002FC++语言表示）：",[24,87671,87673],{"className":65734,"code":87672,"language":31965,"meta":29,"style":29},"\u002F\u002F 假设链表节点结构体定义如下：\n\u002F\u002F struct Node {\n\u002F\u002F int data;\n\u002F\u002F struct Node* next;\n\u002F\u002F };\n\nint getListLength(struct Node* head) {\n \u002F\u002F 基线条件：如果指针为空，说明走到了链表尽头（或本身是空链表），长度为0\n if (head == NULL) {\n return 0;\n }\n \u002F\u002F 递归思想：当前链表的总长度 = 当前这 1 个节点 + 后续链表的长度\n return 1 + getListLength(head->next);\n}\n",[31,87674,87675,87680,87685,87690,87695,87700,87704,87709,87714,87719,87724,87728,87733,87738],{"__ignoreMap":29},[34,87676,87677],{"class":36,"line":37},[34,87678,87679],{},"\u002F\u002F 假设链表节点结构体定义如下：\n",[34,87681,87682],{"class":36,"line":43},[34,87683,87684],{},"\u002F\u002F struct Node {\n",[34,87686,87687],{"class":36,"line":49},[34,87688,87689],{},"\u002F\u002F int data;\n",[34,87691,87692],{"class":36,"line":56},[34,87693,87694],{},"\u002F\u002F struct Node* next;\n",[34,87696,87697],{"class":36,"line":62},[34,87698,87699],{},"\u002F\u002F };\n",[34,87701,87702],{"class":36,"line":68},[34,87703,53],{"emptyLinePlaceholder":52},[34,87705,87706],{"class":36,"line":74},[34,87707,87708],{},"int getListLength(struct Node* head) {\n",[34,87710,87711],{"class":36,"line":1971},[34,87712,87713],{}," \u002F\u002F 基线条件：如果指针为空，说明走到了链表尽头（或本身是空链表），长度为0\n",[34,87715,87716],{"class":36,"line":1976},[34,87717,87718],{}," if (head == NULL) {\n",[34,87720,87721],{"class":36,"line":1985},[34,87722,87723],{}," return 0;\n",[34,87725,87726],{"class":36,"line":1990},[34,87727,5540],{},[34,87729,87730],{"class":36,"line":1999},[34,87731,87732],{}," \u002F\u002F 递归思想：当前链表的总长度 = 当前这 1 个节点 + 后续链表的长度\n",[34,87734,87735],{"class":36,"line":2022},[34,87736,87737],{}," return 1 + getListLength(head->next);\n",[34,87739,87740],{"class":36,"line":2036},[34,87741,77],{},[15,87743,87744],{},[2627,87745,87746],{},"2) 复杂度分析：",[105,87748,87749,87929],{},[108,87750,87751,87753,87754,87757,87758,87786,87787,87838,87839,87883,87884,347],{},[2627,87752,65095],{}," 该递归算法会从链表的首节点依次一直走到链表的末尾 ",[31,87755,87756],{},"NULL","。如果链表长度为 ",[34,87759,87761,87774],{"className":87760},[31642],[34,87762,87764],{"className":87763},[31646],[31648,87765,87766],{"xmlns":31650},[31652,87767,87768,87772],{},[31655,87769,87770],{},[31658,87771,31660],{},[31665,87773,31660],{"encoding":31667},[34,87775,87777],{"className":87776,"ariaHidden":758},[31672],[34,87778,87780,87783],{"className":87779},[31676],[34,87781],{"className":87782,"style":32527},[31680],[34,87784,31660],{"className":87785},[31685,31686],"，函数就会被调用执行 ",[34,87788,87790,87808],{"className":87789},[31642],[34,87791,87793],{"className":87792},[31646],[31648,87794,87795],{"xmlns":31650},[31652,87796,87797,87805],{},[31655,87798,87799,87801,87803],{},[31658,87800,31660],{},[31662,87802,223],{},[31749,87804,640],{},[31665,87806,87807],{"encoding":31667},"n+1",[34,87809,87811,87829],{"className":87810,"ariaHidden":758},[31672],[34,87812,87814,87817,87820,87823,87826],{"className":87813},[31676],[34,87815],{"className":87816,"style":32597},[31680],[34,87818,31660],{"className":87819},[31685,31686],[34,87821],{"className":87822,"style":32604},[31845],[34,87824,223],{"className":87825},[32024],[34,87827],{"className":87828,"style":32604},[31845],[34,87830,87832,87835],{"className":87831},[31676],[34,87833],{"className":87834,"style":32617},[31680],[34,87836,640],{"className":87837},[31685]," 次。每一次函数调用内部只做了简单的指针判空与加法运算，耗时为 ",[34,87840,87842,87862],{"className":87841},[31642],[34,87843,87845],{"className":87844},[31646],[31648,87846,87847],{"xmlns":31650},[31652,87848,87849,87859],{},[31655,87850,87851,87853,87855,87857],{},[31658,87852,37890],{},[31662,87854,456],{"stretchy":702},[31749,87856,640],{},[31662,87858,2557],{"stretchy":702},[31665,87860,87861],{"encoding":31667},"O(1)",[34,87863,87865],{"className":87864,"ariaHidden":758},[31672],[34,87866,87868,87871,87874,87877,87880],{"className":87867},[31676],[34,87869],{"className":87870,"style":31774},[31680],[34,87872,37890],{"className":87873,"style":34452},[31685,31686],[34,87875,456],{"className":87876},[31782],[34,87878,640],{"className":87879},[31685],[34,87881,2557],{"className":87882},[31690],"。因此总时间复杂度为 ",[2627,87885,87886],{},[34,87887,87889,87908],{"className":87888},[31642],[34,87890,87892],{"className":87891},[31646],[31648,87893,87894],{"xmlns":31650},[31652,87895,87896,87906],{},[31655,87897,87898,87900,87902,87904],{},[31658,87899,37890],{},[31662,87901,456],{"stretchy":702},[31658,87903,31660],{},[31662,87905,2557],{"stretchy":702},[31665,87907,46493],{"encoding":31667},[34,87909,87911],{"className":87910,"ariaHidden":758},[31672],[34,87912,87914,87917,87920,87923,87926],{"className":87913},[31676],[34,87915],{"className":87916,"style":31774},[31680],[34,87918,37890],{"className":87919,"style":34452},[31685,31686],[34,87921,456],{"className":87922},[31782],[34,87924,31660],{"className":87925},[31685,31686],[34,87927,2557],{"className":87928},[31690],[108,87930,87931,87934,87935,87938,87939,87967,87968,88018,88019,88069,88070,347],{},[2627,87932,87933],{},"空间复杂度分析："," 这一点极其容易被忽略。虽然代码里没有动态分配数组，但是",[2627,87936,87937],{},"系统执行递归是需要开辟“调用栈”的","。每次函数调用，系统要保存局部变量、返回地址等信息到栈帧中。链表长度为 ",[34,87940,87942,87955],{"className":87941},[31642],[34,87943,87945],{"className":87944},[31646],[31648,87946,87947],{"xmlns":31650},[31652,87948,87949,87953],{},[31655,87950,87951],{},[31658,87952,31660],{},[31665,87954,31660],{"encoding":31667},[34,87956,87958],{"className":87957,"ariaHidden":758},[31672],[34,87959,87961,87964],{"className":87960},[31676],[34,87962],{"className":87963,"style":32527},[31680],[34,87965,31660],{"className":87966},[31685,31686],"，递归就会深入 ",[34,87969,87971,87988],{"className":87970},[31642],[34,87972,87974],{"className":87973},[31646],[31648,87975,87976],{"xmlns":31650},[31652,87977,87978,87986],{},[31655,87979,87980,87982,87984],{},[31658,87981,31660],{},[31662,87983,223],{},[31749,87985,640],{},[31665,87987,87807],{"encoding":31667},[34,87989,87991,88009],{"className":87990,"ariaHidden":758},[31672],[34,87992,87994,87997,88000,88003,88006],{"className":87993},[31676],[34,87995],{"className":87996,"style":32597},[31680],[34,87998,31660],{"className":87999},[31685,31686],[34,88001],{"className":88002,"style":32604},[31845],[34,88004,223],{"className":88005},[32024],[34,88007],{"className":88008,"style":32604},[31845],[34,88010,88012,88015],{"className":88011},[31676],[34,88013],{"className":88014,"style":32617},[31680],[34,88016,640],{"className":88017},[31685]," 层，栈的最大深度就是 ",[34,88020,88022,88039],{"className":88021},[31642],[34,88023,88025],{"className":88024},[31646],[31648,88026,88027],{"xmlns":31650},[31652,88028,88029,88037],{},[31655,88030,88031,88033,88035],{},[31658,88032,31660],{},[31662,88034,223],{},[31749,88036,640],{},[31665,88038,87807],{"encoding":31667},[34,88040,88042,88060],{"className":88041,"ariaHidden":758},[31672],[34,88043,88045,88048,88051,88054,88057],{"className":88044},[31676],[34,88046],{"className":88047,"style":32597},[31680],[34,88049,31660],{"className":88050},[31685,31686],[34,88052],{"className":88053,"style":32604},[31845],[34,88055,223],{"className":88056},[32024],[34,88058],{"className":88059,"style":32604},[31845],[34,88061,88063,88066],{"className":88062},[31676],[34,88064],{"className":88065,"style":32617},[31680],[34,88067,640],{"className":88068},[31685],"。因此，空间复杂度同样为 ",[2627,88071,88072],{},[34,88073,88075,88094],{"className":88074},[31642],[34,88076,88078],{"className":88077},[31646],[31648,88079,88080],{"xmlns":31650},[31652,88081,88082,88092],{},[31655,88083,88084,88086,88088,88090],{},[31658,88085,37890],{},[31662,88087,456],{"stretchy":702},[31658,88089,31660],{},[31662,88091,2557],{"stretchy":702},[31665,88093,46493],{"encoding":31667},[34,88095,88097],{"className":88096,"ariaHidden":758},[31672],[34,88098,88100,88103,88106,88109,88112],{"className":88099},[31676],[34,88101],{"className":88102,"style":31774},[31680],[34,88104,37890],{"className":88105,"style":34452},[31685,31686],[34,88107,456],{"className":88108},[31782],[34,88110,31660],{"className":88111},[31685,31686],[34,88113,2557],{"className":88114},[31690],[32188,88116],{},[253,88118,58228],{"id":88119},"_2-题目内容-4",[15,88121,88122,88123,258,88151,88179,88180,258,88208,88236,88237,88265,88266,88294],{},"利用动态规划法设计算法求两个字符串 ",[34,88124,88126,88139],{"className":88125},[31642],[34,88127,88129],{"className":88128},[31646],[31648,88130,88131],{"xmlns":31650},[31652,88132,88133,88137],{},[31655,88134,88135],{},[31658,88136,54151],{},[31665,88138,54151],{"encoding":31667},[34,88140,88142],{"className":88141,"ariaHidden":758},[31672],[34,88143,88145,88148],{"className":88144},[31676],[34,88146],{"className":88147,"style":33218},[31680],[34,88149,54151],{"className":88150},[31685,31686],[34,88152,88154,88167],{"className":88153},[31642],[34,88155,88157],{"className":88156},[31646],[31648,88158,88159],{"xmlns":31650},[31652,88160,88161,88165],{},[31655,88162,88163],{},[31658,88164,54224],{},[31665,88166,54224],{"encoding":31667},[34,88168,88170],{"className":88169,"ariaHidden":758},[31672],[34,88171,88173,88176],{"className":88172},[31676],[34,88174],{"className":88175,"style":33218},[31680],[34,88177,54224],{"className":88178,"style":54252},[31685,31686]," 的编辑距离。设 ",[34,88181,88183,88196],{"className":88182},[31642],[34,88184,88186],{"className":88185},[31646],[31648,88187,88188],{"xmlns":31650},[31652,88189,88190,88194],{},[31655,88191,88192],{},[31658,88193,54151],{},[31665,88195,54151],{"encoding":31667},[34,88197,88199],{"className":88198,"ariaHidden":758},[31672],[34,88200,88202,88205],{"className":88201},[31676],[34,88203],{"className":88204,"style":33218},[31680],[34,88206,54151],{"className":88207},[31685,31686],[34,88209,88211,88224],{"className":88210},[31642],[34,88212,88214],{"className":88213},[31646],[31648,88215,88216],{"xmlns":31650},[31652,88217,88218,88222],{},[31655,88219,88220],{},[31658,88221,54224],{},[31665,88223,54224],{"encoding":31667},[34,88225,88227],{"className":88226,"ariaHidden":758},[31672],[34,88228,88230,88233],{"className":88229},[31676],[34,88231],{"className":88232,"style":33218},[31680],[34,88234,54224],{"className":88235,"style":54252},[31685,31686]," 是两个字符串。所谓的编辑距离即将字符串 ",[34,88238,88240,88253],{"className":88239},[31642],[34,88241,88243],{"className":88242},[31646],[31648,88244,88245],{"xmlns":31650},[31652,88246,88247,88251],{},[31655,88248,88249],{},[31658,88250,54151],{},[31665,88252,54151],{"encoding":31667},[34,88254,88256],{"className":88255,"ariaHidden":758},[31672],[34,88257,88259,88262],{"className":88258},[31676],[34,88260],{"className":88261,"style":33218},[31680],[34,88263,54151],{"className":88264},[31685,31686]," 转换成字符串 ",[34,88267,88269,88282],{"className":88268},[31642],[34,88270,88272],{"className":88271},[31646],[31648,88273,88274],{"xmlns":31650},[31652,88275,88276,88280],{},[31655,88277,88278],{},[31658,88279,54224],{},[31665,88281,54224],{"encoding":31667},[34,88283,88285],{"className":88284,"ariaHidden":758},[31672],[34,88286,88288,88291],{"className":88287},[31676],[34,88289],{"className":88290,"style":33218},[31680],[34,88292,54224],{"className":88293,"style":54252},[31685,31686]," 所经历的最少编辑操作次数。编辑操作共有 3 种：(1) 删除一个字符 delete；(2) 插入一个字符 insert；(3) 将一个字符替换另一个字符 replace。",[2230,88296,88297,88479],{},[108,88298,88299,88300,88363,88364,88392,88393,88421,88422,88392,88450,88478],{},"设计动态规划数组 ",[34,88301,88303,88330],{"className":88302},[31642],[34,88304,88306],{"className":88305},[31646],[31648,88307,88308],{"xmlns":31650},[31652,88309,88310,88328],{},[31655,88311,88312,88314,88316,88318,88320,88322,88324,88326],{},[31658,88313,46900],{},[31658,88315,15],{},[31662,88317,15972],{"stretchy":702},[31658,88319,31955],{},[31662,88321,32712],{"stretchy":702},[31662,88323,15972],{"stretchy":702},[31658,88325,31968],{},[31662,88327,32712],{"stretchy":702},[31665,88329,54135],{"encoding":31667},[34,88331,88333],{"className":88332,"ariaHidden":758},[31672],[34,88334,88336,88339,88342,88345,88348,88351,88354,88357,88360],{"className":88335},[31676],[34,88337],{"className":88338,"style":31774},[31680],[34,88340,46900],{"className":88341},[31685,31686],[34,88343,15],{"className":88344},[31685,31686],[34,88346,15972],{"className":88347},[31782],[34,88349,31955],{"className":88350},[31685,31686],[34,88352,32712],{"className":88353},[31690],[34,88355,15972],{"className":88356},[31782],[34,88358,31968],{"className":88359,"style":32153},[31685,31686],[34,88361,32712],{"className":88362},[31690],"（将长度为 ",[34,88365,88367,88380],{"className":88366},[31642],[34,88368,88370],{"className":88369},[31646],[31648,88371,88372],{"xmlns":31650},[31652,88373,88374,88378],{},[31655,88375,88376],{},[31658,88377,31955],{},[31665,88379,31955],{"encoding":31667},[34,88381,88383],{"className":88382,"ariaHidden":758},[31672],[34,88384,88386,88389],{"className":88385},[31676],[34,88387],{"className":88388,"style":33104},[31680],[34,88390,31955],{"className":88391},[31685,31686]," 的字符串 ",[34,88394,88396,88409],{"className":88395},[31642],[34,88397,88399],{"className":88398},[31646],[31648,88400,88401],{"xmlns":31650},[31652,88402,88403,88407],{},[31655,88404,88405],{},[31658,88406,54151],{},[31665,88408,54151],{"encoding":31667},[34,88410,88412],{"className":88411,"ariaHidden":758},[31672],[34,88413,88415,88418],{"className":88414},[31676],[34,88416],{"className":88417,"style":33218},[31680],[34,88419,54151],{"className":88420},[31685,31686]," 转换成长度为 ",[34,88423,88425,88438],{"className":88424},[31642],[34,88426,88428],{"className":88427},[31646],[31648,88429,88430],{"xmlns":31650},[31652,88431,88432,88436],{},[31655,88433,88434],{},[31658,88435,31968],{},[31665,88437,31968],{"encoding":31667},[34,88439,88441],{"className":88440,"ariaHidden":758},[31672],[34,88442,88444,88447],{"className":88443},[31676],[34,88445],{"className":88446,"style":33123},[31680],[34,88448,31968],{"className":88449,"style":32153},[31685,31686],[34,88451,88453,88466],{"className":88452},[31642],[34,88454,88456],{"className":88455},[31646],[31648,88457,88458],{"xmlns":31650},[31652,88459,88460,88464],{},[31655,88461,88462],{},[31658,88463,54224],{},[31665,88465,54224],{"encoding":31667},[34,88467,88469],{"className":88468,"ariaHidden":758},[31672],[34,88470,88472,88475],{"className":88471},[31676],[34,88473],{"className":88474,"style":33218},[31680],[34,88476,54224],{"className":88477,"style":54252},[31685,31686]," 的最少编辑操作次数）并给出其动态转移方程。（5分）",[108,88480,87658],{},[253,88482,58168],{"id":88483},"答案与深度解析-23",[15,88485,88486],{},"这是一道经典的“莱文斯坦距离（Levenshtein Distance）”算法设计题。我们一步步从基础进行设计：",[15,88488,88489],{},[2627,88490,88491],{},"1) DP 数组设计与动态转移方程推导：",[105,88493,88494,88617,89369,89611],{},[108,88495,88496,47649,88499,88501,88502,88530,88531,88559,88560,88530,88588,88616],{},[2627,88497,88498],{},"数组定义：",[31,88500,54135],{}," 表示将字符串 ",[34,88503,88505,88518],{"className":88504},[31642],[34,88506,88508],{"className":88507},[31646],[31648,88509,88510],{"xmlns":31650},[31652,88511,88512,88516],{},[31655,88513,88514],{},[31658,88515,54151],{},[31665,88517,54151],{"encoding":31667},[34,88519,88521],{"className":88520,"ariaHidden":758},[31672],[34,88522,88524,88527],{"className":88523},[31676],[34,88525],{"className":88526,"style":33218},[31680],[34,88528,54151],{"className":88529},[31685,31686]," 的前 ",[34,88532,88534,88547],{"className":88533},[31642],[34,88535,88537],{"className":88536},[31646],[31648,88538,88539],{"xmlns":31650},[31652,88540,88541,88545],{},[31655,88542,88543],{},[31658,88544,31955],{},[31665,88546,31955],{"encoding":31667},[34,88548,88550],{"className":88549,"ariaHidden":758},[31672],[34,88551,88553,88556],{"className":88552},[31676],[34,88554],{"className":88555,"style":33104},[31680],[34,88557,31955],{"className":88558},[31685,31686]," 个字符转换成字符串 ",[34,88561,88563,88576],{"className":88562},[31642],[34,88564,88566],{"className":88565},[31646],[31648,88567,88568],{"xmlns":31650},[31652,88569,88570,88574],{},[31655,88571,88572],{},[31658,88573,54224],{},[31665,88575,54224],{"encoding":31667},[34,88577,88579],{"className":88578,"ariaHidden":758},[31672],[34,88580,88582,88585],{"className":88581},[31676],[34,88583],{"className":88584,"style":33218},[31680],[34,88586,54224],{"className":88587,"style":54252},[31685,31686],[34,88589,88591,88604],{"className":88590},[31642],[34,88592,88594],{"className":88593},[31646],[31648,88595,88596],{"xmlns":31650},[31652,88597,88598,88602],{},[31655,88599,88600],{},[31658,88601,31968],{},[31665,88603,31968],{"encoding":31667},[34,88605,88607],{"className":88606,"ariaHidden":758},[31672],[34,88608,88610,88613],{"className":88609},[31676],[34,88611],{"className":88612,"style":33123},[31680],[34,88614,31968],{"className":88615,"style":32153},[31685,31686]," 个字符所需的最少编辑次数。",[108,88618,88619,88622,88623,88651,88652,88680,88681,88684,88685,88651,88713,88680,88741,43434,88744],{},[2627,88620,88621],{},"核心逻辑（从零理解为什么这么转移）：","\n比较 ",[34,88624,88626,88639],{"className":88625},[31642],[34,88627,88629],{"className":88628},[31646],[31648,88630,88631],{"xmlns":31650},[31652,88632,88633,88637],{},[31655,88634,88635],{},[31658,88636,54151],{},[31665,88638,54151],{"encoding":31667},[34,88640,88642],{"className":88641,"ariaHidden":758},[31672],[34,88643,88645,88648],{"className":88644},[31676],[34,88646],{"className":88647,"style":33218},[31680],[34,88649,54151],{"className":88650},[31685,31686]," 的第 ",[34,88653,88655,88668],{"className":88654},[31642],[34,88656,88658],{"className":88657},[31646],[31648,88659,88660],{"xmlns":31650},[31652,88661,88662,88666],{},[31655,88663,88664],{},[31658,88665,31955],{},[31665,88667,31955],{"encoding":31667},[34,88669,88671],{"className":88670,"ariaHidden":758},[31672],[34,88672,88674,88677],{"className":88673},[31676],[34,88675],{"className":88676,"style":33104},[31680],[34,88678,31955],{"className":88679},[31685,31686]," 个字符（",[31,88682,88683],{},"A[i-1]","）和 ",[34,88686,88688,88701],{"className":88687},[31642],[34,88689,88691],{"className":88690},[31646],[31648,88692,88693],{"xmlns":31650},[31652,88694,88695,88699],{},[31655,88696,88697],{},[31658,88698,54224],{},[31665,88700,54224],{"encoding":31667},[34,88702,88704],{"className":88703,"ariaHidden":758},[31672],[34,88705,88707,88710],{"className":88706},[31676],[34,88708],{"className":88709,"style":33218},[31680],[34,88711,54224],{"className":88712,"style":54252},[31685,31686],[34,88714,88716,88729],{"className":88715},[31642],[34,88717,88719],{"className":88718},[31646],[31648,88720,88721],{"xmlns":31650},[31652,88722,88723,88727],{},[31655,88724,88725],{},[31658,88726,31968],{},[31665,88728,31968],{"encoding":31667},[34,88730,88732],{"className":88731,"ariaHidden":758},[31672],[34,88733,88735,88738],{"className":88734},[31676],[34,88736],{"className":88737,"style":33123},[31680],[34,88739,31968],{"className":88740,"style":32153},[31685,31686],[31,88742,88743],{},"B[j-1]",[105,88745,88746,88756],{},[108,88747,88748,88753,88754],{},[2627,88749,88750,88751,82346],{},"情况一：两者相等（",[31,88752,54335],{},"。运气好，最后一个字符不用作任何处理。当前所需的编辑距离与它们都不包含各自最后一个字符时的距离一模一样。\n即：",[31,88755,55200],{},[108,88757,88758,88764,88765],{},[2627,88759,88760,88761,82346],{},"情况二：两者不等（",[31,88762,88763],{},"A[i-1] != B[j-1]","。既然最后一个字符匹配不上，我们就必须得动手实施三种操作之一，实施任意一次操作代价都为 1：\n",[105,88766,88767,88992,89164],{},[108,88768,88769,88772,88773,88801,88802,88830,88831,88530,88859,88909,88910,88530,88938,88989,88990,347],{},[2627,88770,88771],{},"如果执行“替换”："," 我们把 ",[34,88774,88776,88789],{"className":88775},[31642],[34,88777,88779],{"className":88778},[31646],[31648,88780,88781],{"xmlns":31650},[31652,88782,88783,88787],{},[31655,88784,88785],{},[31658,88786,54151],{},[31665,88788,54151],{"encoding":31667},[34,88790,88792],{"className":88791,"ariaHidden":758},[31672],[34,88793,88795,88798],{"className":88794},[31676],[34,88796],{"className":88797,"style":33218},[31680],[34,88799,54151],{"className":88800},[31685,31686]," 的最后一个字符强制替换成 ",[34,88803,88805,88818],{"className":88804},[31642],[34,88806,88808],{"className":88807},[31646],[31648,88809,88810],{"xmlns":31650},[31652,88811,88812,88816],{},[31655,88813,88814],{},[31658,88815,54224],{},[31665,88817,54224],{"encoding":31667},[34,88819,88821],{"className":88820,"ariaHidden":758},[31672],[34,88822,88824,88827],{"className":88823},[31676],[34,88825],{"className":88826,"style":33218},[31680],[34,88828,54224],{"className":88829,"style":54252},[31685,31686]," 的最后一个字符。替换完成后，这两者的末尾对齐了。那么之前的任务就是把 ",[34,88832,88834,88847],{"className":88833},[31642],[34,88835,88837],{"className":88836},[31646],[31648,88838,88839],{"xmlns":31650},[31652,88840,88841,88845],{},[31655,88842,88843],{},[31658,88844,54151],{},[31665,88846,54151],{"encoding":31667},[34,88848,88850],{"className":88849,"ariaHidden":758},[31672],[34,88851,88853,88856],{"className":88852},[31676],[34,88854],{"className":88855,"style":33218},[31680],[34,88857,54151],{"className":88858},[31685,31686],[34,88860,88862,88879],{"className":88861},[31642],[34,88863,88865],{"className":88864},[31646],[31648,88866,88867],{"xmlns":31650},[31652,88868,88869,88877],{},[31655,88870,88871,88873,88875],{},[31658,88872,31955],{},[31662,88874,32226],{},[31749,88876,640],{},[31665,88878,51731],{"encoding":31667},[34,88880,88882,88900],{"className":88881,"ariaHidden":758},[31672],[34,88883,88885,88888,88891,88894,88897],{"className":88884},[31676],[34,88886],{"className":88887,"style":37322},[31680],[34,88889,31955],{"className":88890},[31685,31686],[34,88892],{"className":88893,"style":32604},[31845],[34,88895,32226],{"className":88896},[32024],[34,88898],{"className":88899,"style":32604},[31845],[34,88901,88903,88906],{"className":88902},[31676],[34,88904],{"className":88905,"style":32617},[31680],[34,88907,640],{"className":88908},[31685]," 变成 ",[34,88911,88913,88926],{"className":88912},[31642],[34,88914,88916],{"className":88915},[31646],[31648,88917,88918],{"xmlns":31650},[31652,88919,88920,88924],{},[31655,88921,88922],{},[31658,88923,54224],{},[31665,88925,54224],{"encoding":31667},[34,88927,88929],{"className":88928,"ariaHidden":758},[31672],[34,88930,88932,88935],{"className":88931},[31676],[34,88933],{"className":88934,"style":33218},[31680],[34,88936,54224],{"className":88937,"style":54252},[31685,31686],[34,88939,88941,88959],{"className":88940},[31642],[34,88942,88944],{"className":88943},[31646],[31648,88945,88946],{"xmlns":31650},[31652,88947,88948,88956],{},[31655,88949,88950,88952,88954],{},[31658,88951,31968],{},[31662,88953,32226],{},[31749,88955,640],{},[31665,88957,88958],{"encoding":31667},"j-1",[34,88960,88962,88980],{"className":88961,"ariaHidden":758},[31672],[34,88963,88965,88968,88971,88974,88977],{"className":88964},[31676],[34,88966],{"className":88967,"style":33123},[31680],[34,88969,31968],{"className":88970,"style":32153},[31685,31686],[34,88972],{"className":88973,"style":32604},[31845],[34,88975,32226],{"className":88976},[32024],[34,88978],{"className":88979,"style":32604},[31845],[34,88981,88983,88986],{"className":88982},[31676],[34,88984],{"className":88985,"style":32617},[31680],[34,88987,640],{"className":88988},[31685],"。此时代价为：",[31,88991,55214],{},[108,88993,88994,88772,88997,89025,89026,88530,89054,89104,89105,88530,89133,89161,89162,347],{},[2627,88995,88996],{},"如果执行“删除”：",[34,88998,89000,89013],{"className":88999},[31642],[34,89001,89003],{"className":89002},[31646],[31648,89004,89005],{"xmlns":31650},[31652,89006,89007,89011],{},[31655,89008,89009],{},[31658,89010,54151],{},[31665,89012,54151],{"encoding":31667},[34,89014,89016],{"className":89015,"ariaHidden":758},[31672],[34,89017,89019,89022],{"className":89018},[31676],[34,89020],{"className":89021,"style":33218},[31680],[34,89023,54151],{"className":89024},[31685,31686]," 的最后一个字符删掉。这就指望 ",[34,89027,89029,89042],{"className":89028},[31642],[34,89030,89032],{"className":89031},[31646],[31648,89033,89034],{"xmlns":31650},[31652,89035,89036,89040],{},[31655,89037,89038],{},[31658,89039,54151],{},[31665,89041,54151],{"encoding":31667},[34,89043,89045],{"className":89044,"ariaHidden":758},[31672],[34,89046,89048,89051],{"className":89047},[31676],[34,89049],{"className":89050,"style":33218},[31680],[34,89052,54151],{"className":89053},[31685,31686],[34,89055,89057,89074],{"className":89056},[31642],[34,89058,89060],{"className":89059},[31646],[31648,89061,89062],{"xmlns":31650},[31652,89063,89064,89072],{},[31655,89065,89066,89068,89070],{},[31658,89067,31955],{},[31662,89069,32226],{},[31749,89071,640],{},[31665,89073,51731],{"encoding":31667},[34,89075,89077,89095],{"className":89076,"ariaHidden":758},[31672],[34,89078,89080,89083,89086,89089,89092],{"className":89079},[31676],[34,89081],{"className":89082,"style":37322},[31680],[34,89084,31955],{"className":89085},[31685,31686],[34,89087],{"className":89088,"style":32604},[31845],[34,89090,32226],{"className":89091},[32024],[34,89093],{"className":89094,"style":32604},[31845],[34,89096,89098,89101],{"className":89097},[31676],[34,89099],{"className":89100,"style":32617},[31680],[34,89102,640],{"className":89103},[31685]," 个字符自己能够变幻成 ",[34,89106,89108,89121],{"className":89107},[31642],[34,89109,89111],{"className":89110},[31646],[31648,89112,89113],{"xmlns":31650},[31652,89114,89115,89119],{},[31655,89116,89117],{},[31658,89118,54224],{},[31665,89120,54224],{"encoding":31667},[34,89122,89124],{"className":89123,"ariaHidden":758},[31672],[34,89125,89127,89130],{"className":89126},[31676],[34,89128],{"className":89129,"style":33218},[31680],[34,89131,54224],{"className":89132,"style":54252},[31685,31686],[34,89134,89136,89149],{"className":89135},[31642],[34,89137,89139],{"className":89138},[31646],[31648,89140,89141],{"xmlns":31650},[31652,89142,89143,89147],{},[31655,89144,89145],{},[31658,89146,31968],{},[31665,89148,31968],{"encoding":31667},[34,89150,89152],{"className":89151,"ariaHidden":758},[31672],[34,89153,89155,89158],{"className":89154},[31676],[34,89156],{"className":89157,"style":33123},[31680],[34,89159,31968],{"className":89160,"style":32153},[31685,31686]," 个字符。此时代价为：",[31,89163,55232],{},[108,89165,89166,89169,89170,89198,89199,89227,89228,88530,89256,89284,89285,88530,89313,89161,89363,89365,89366,347],{},[2627,89167,89168],{},"如果执行“插入”："," 我们在 ",[34,89171,89173,89186],{"className":89172},[31642],[34,89174,89176],{"className":89175},[31646],[31648,89177,89178],{"xmlns":31650},[31652,89179,89180,89184],{},[31655,89181,89182],{},[31658,89183,54151],{},[31665,89185,54151],{"encoding":31667},[34,89187,89189],{"className":89188,"ariaHidden":758},[31672],[34,89190,89192,89195],{"className":89191},[31676],[34,89193],{"className":89194,"style":33218},[31680],[34,89196,54151],{"className":89197},[31685,31686]," 的末尾无中生有地插入一个与 ",[34,89200,89202,89215],{"className":89201},[31642],[34,89203,89205],{"className":89204},[31646],[31648,89206,89207],{"xmlns":31650},[31652,89208,89209,89213],{},[31655,89210,89211],{},[31658,89212,54224],{},[31665,89214,54224],{"encoding":31667},[34,89216,89218],{"className":89217,"ariaHidden":758},[31672],[34,89219,89221,89224],{"className":89220},[31676],[34,89222],{"className":89223,"style":33218},[31680],[34,89225,54224],{"className":89226,"style":54252},[31685,31686]," 末尾一样的字符。末尾对齐后，我们需要将 ",[34,89229,89231,89244],{"className":89230},[31642],[34,89232,89234],{"className":89233},[31646],[31648,89235,89236],{"xmlns":31650},[31652,89237,89238,89242],{},[31655,89239,89240],{},[31658,89241,54151],{},[31665,89243,54151],{"encoding":31667},[34,89245,89247],{"className":89246,"ariaHidden":758},[31672],[34,89248,89250,89253],{"className":89249},[31676],[34,89251],{"className":89252,"style":33218},[31680],[34,89254,54151],{"className":89255},[31685,31686],[34,89257,89259,89272],{"className":89258},[31642],[34,89260,89262],{"className":89261},[31646],[31648,89263,89264],{"xmlns":31650},[31652,89265,89266,89270],{},[31655,89267,89268],{},[31658,89269,31955],{},[31665,89271,31955],{"encoding":31667},[34,89273,89275],{"className":89274,"ariaHidden":758},[31672],[34,89276,89278,89281],{"className":89277},[31676],[34,89279],{"className":89280,"style":33104},[31680],[34,89282,31955],{"className":89283},[31685,31686]," 个字符变幻成 ",[34,89286,89288,89301],{"className":89287},[31642],[34,89289,89291],{"className":89290},[31646],[31648,89292,89293],{"xmlns":31650},[31652,89294,89295,89299],{},[31655,89296,89297],{},[31658,89298,54224],{},[31665,89300,54224],{"encoding":31667},[34,89302,89304],{"className":89303,"ariaHidden":758},[31672],[34,89305,89307,89310],{"className":89306},[31676],[34,89308],{"className":89309,"style":33218},[31680],[34,89311,54224],{"className":89312,"style":54252},[31685,31686],[34,89314,89316,89333],{"className":89315},[31642],[34,89317,89319],{"className":89318},[31646],[31648,89320,89321],{"xmlns":31650},[31652,89322,89323,89331],{},[31655,89324,89325,89327,89329],{},[31658,89326,31968],{},[31662,89328,32226],{},[31749,89330,640],{},[31665,89332,88958],{"encoding":31667},[34,89334,89336,89354],{"className":89335,"ariaHidden":758},[31672],[34,89337,89339,89342,89345,89348,89351],{"className":89338},[31676],[34,89340],{"className":89341,"style":33123},[31680],[34,89343,31968],{"className":89344,"style":32153},[31685,31686],[34,89346],{"className":89347,"style":32604},[31845],[34,89349,32226],{"className":89350},[32024],[34,89352],{"className":89353,"style":32604},[31845],[34,89355,89357,89360],{"className":89356},[31676],[34,89358],{"className":89359,"style":32617},[31680],[34,89361,640],{"className":89362},[31685],[31,89364,55223],{},"。\n我们需要从中选择花费最小的一条路径：",[31,89367,89368],{},"min(替换代价, 删除代价, 插入代价)",[108,89370,89371,89374,89375,89403,89404,89432,89433,89461,89462,89490,89491,89494,89495,89403,89523,89432,89551,89579,89580,89490,89608,347],{},[2627,89372,89373],{},"边界条件：","\n如果 ",[34,89376,89378,89391],{"className":89377},[31642],[34,89379,89381],{"className":89380},[31646],[31648,89382,89383],{"xmlns":31650},[31652,89384,89385,89389],{},[31655,89386,89387],{},[31658,89388,54151],{},[31665,89390,54151],{"encoding":31667},[34,89392,89394],{"className":89393,"ariaHidden":758},[31672],[34,89395,89397,89400],{"className":89396},[31676],[34,89398],{"className":89399,"style":33218},[31680],[34,89401,54151],{"className":89402},[31685,31686]," 长度为 0，",[34,89405,89407,89420],{"className":89406},[31642],[34,89408,89410],{"className":89409},[31646],[31648,89411,89412],{"xmlns":31650},[31652,89413,89414,89418],{},[31655,89415,89416],{},[31658,89417,54224],{},[31665,89419,54224],{"encoding":31667},[34,89421,89423],{"className":89422,"ariaHidden":758},[31672],[34,89424,89426,89429],{"className":89425},[31676],[34,89427],{"className":89428,"style":33218},[31680],[34,89430,54224],{"className":89431,"style":54252},[31685,31686]," 长度为 ",[34,89434,89436,89449],{"className":89435},[31642],[34,89437,89439],{"className":89438},[31646],[31648,89440,89441],{"xmlns":31650},[31652,89442,89443,89447],{},[31655,89444,89445],{},[31658,89446,31968],{},[31665,89448,31968],{"encoding":31667},[34,89450,89452],{"className":89451,"ariaHidden":758},[31672],[34,89453,89455,89458],{"className":89454},[31676],[34,89456],{"className":89457,"style":33123},[31680],[34,89459,31968],{"className":89460,"style":32153},[31685,31686],"，要想转变过去，只能无脑插入 ",[34,89463,89465,89478],{"className":89464},[31642],[34,89466,89468],{"className":89467},[31646],[31648,89469,89470],{"xmlns":31650},[31652,89471,89472,89476],{},[31655,89473,89474],{},[31658,89475,31968],{},[31665,89477,31968],{"encoding":31667},[34,89479,89481],{"className":89480,"ariaHidden":758},[31672],[34,89482,89484,89487],{"className":89483},[31676],[34,89485],{"className":89486,"style":33123},[31680],[34,89488,31968],{"className":89489,"style":32153},[31685,31686]," 次。",[31,89492,89493],{},"dp[0][j] = j","。\n如果 ",[34,89496,89498,89511],{"className":89497},[31642],[34,89499,89501],{"className":89500},[31646],[31648,89502,89503],{"xmlns":31650},[31652,89504,89505,89509],{},[31655,89506,89507],{},[31658,89508,54224],{},[31665,89510,54224],{"encoding":31667},[34,89512,89514],{"className":89513,"ariaHidden":758},[31672],[34,89515,89517,89520],{"className":89516},[31676],[34,89518],{"className":89519,"style":33218},[31680],[34,89521,54224],{"className":89522,"style":54252},[31685,31686],[34,89524,89526,89539],{"className":89525},[31642],[34,89527,89529],{"className":89528},[31646],[31648,89530,89531],{"xmlns":31650},[31652,89532,89533,89537],{},[31655,89534,89535],{},[31658,89536,54151],{},[31665,89538,54151],{"encoding":31667},[34,89540,89542],{"className":89541,"ariaHidden":758},[31672],[34,89543,89545,89548],{"className":89544},[31676],[34,89546],{"className":89547,"style":33218},[31680],[34,89549,54151],{"className":89550},[31685,31686],[34,89552,89554,89567],{"className":89553},[31642],[34,89555,89557],{"className":89556},[31646],[31648,89558,89559],{"xmlns":31650},[31652,89560,89561,89565],{},[31655,89562,89563],{},[31658,89564,31955],{},[31665,89566,31955],{"encoding":31667},[34,89568,89570],{"className":89569,"ariaHidden":758},[31672],[34,89571,89573,89576],{"className":89572},[31676],[34,89574],{"className":89575,"style":33104},[31680],[34,89577,31955],{"className":89578},[31685,31686],"，要想转变过去，只能无脑删除 ",[34,89581,89583,89596],{"className":89582},[31642],[34,89584,89586],{"className":89585},[31646],[31648,89587,89588],{"xmlns":31650},[31652,89589,89590,89594],{},[31655,89591,89592],{},[31658,89593,31955],{},[31665,89595,31955],{"encoding":31667},[34,89597,89599],{"className":89598,"ariaHidden":758},[31672],[34,89600,89602,89605],{"className":89601},[31676],[34,89603],{"className":89604,"style":33104},[31680],[34,89606,31955],{"className":89607},[31685,31686],[31,89609,89610],{},"dp[i][0] = i",[108,89612,89613,89616],{},[2627,89614,89615],{},"严谨的动态转移方程：",[34,89617,89619,89879],{"className":89618},[31642],[34,89620,89622],{"className":89621},[31646],[31648,89623,89624],{"xmlns":31650},[31652,89625,89626,89876],{},[31655,89627,89628,89630,89632,89634,89636,89638,89640,89642,89644,89646],{},[31658,89629,46900],{},[31658,89631,15],{},[31662,89633,15972],{"stretchy":702},[31658,89635,31955],{},[31662,89637,32712],{"stretchy":702},[31662,89639,15972],{"stretchy":702},[31658,89641,31968],{},[31662,89643,32712],{"stretchy":702},[31662,89645,202],{},[31655,89647,89648,89650],{},[31662,89649,32685],{"fence":758},[71078,89651,89652,89672,89692,89756],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,89653,89654,89660],{},[71087,89655,89656],{},[71090,89657,89658],{"scriptlevel":459,"displaystyle":702},[31658,89659,31955],{},[71087,89661,89662],{},[71090,89663,89664],{"scriptlevel":459,"displaystyle":702},[31655,89665,89666,89668,89670],{},[31658,89667,31968],{},[31662,89669,202],{},[31749,89671,459],{},[71084,89673,89674,89680],{},[71087,89675,89676],{},[71090,89677,89678],{"scriptlevel":459,"displaystyle":702},[31658,89679,31968],{},[71087,89681,89682],{},[71090,89683,89684],{"scriptlevel":459,"displaystyle":702},[31655,89685,89686,89688,89690],{},[31658,89687,31955],{},[31662,89689,202],{},[31749,89691,459],{},[71084,89693,89694,89724],{},[71087,89695,89696],{},[71090,89697,89698],{"scriptlevel":459,"displaystyle":702},[31655,89699,89700,89702,89704,89706,89708,89710,89712,89714,89716,89718,89720,89722],{},[31658,89701,46900],{},[31658,89703,15],{},[31662,89705,15972],{"stretchy":702},[31658,89707,31955],{},[31662,89709,32226],{},[31749,89711,640],{},[31662,89713,32712],{"stretchy":702},[31662,89715,15972],{"stretchy":702},[31658,89717,31968],{},[31662,89719,32226],{},[31749,89721,640],{},[31662,89723,32712],{"stretchy":702},[71087,89725,89726],{},[71090,89727,89728],{"scriptlevel":459,"displaystyle":702},[31655,89729,89730,89732,89734,89736,89738,89740,89742,89744,89746,89748,89750,89752,89754],{},[31658,89731,54151],{},[31662,89733,15972],{"stretchy":702},[31658,89735,31955],{},[31662,89737,32226],{},[31749,89739,640],{},[31662,89741,32712],{"stretchy":702},[31662,89743,202],{},[31658,89745,54224],{},[31662,89747,15972],{"stretchy":702},[31658,89749,31968],{},[31662,89751,32226],{},[31749,89753,640],{},[31662,89755,32712],{"stretchy":702},[71084,89757,89758,89844],{},[71087,89759,89760],{},[71090,89761,89762],{"scriptlevel":459,"displaystyle":702},[31655,89763,89764,89766,89768,89770,89772,89774,89776,89778,89780,89782,89784,89786,89788,89790,89792,89794,89796,89798,89800,89802,89804,89806,89808,89810,89812,89814,89816,89818,89820,89822,89824,89826,89828,89830,89832,89834,89836,89838,89840,89842],{},[31658,89765,56084],{},[31662,89767,32682],{},[31662,89769,456],{"stretchy":702},[31658,89771,46900],{},[31658,89773,15],{},[31662,89775,15972],{"stretchy":702},[31658,89777,31955],{},[31662,89779,32226],{},[31749,89781,640],{},[31662,89783,32712],{"stretchy":702},[31662,89785,15972],{"stretchy":702},[31658,89787,31968],{},[31662,89789,32226],{},[31749,89791,640],{},[31662,89793,32712],{"stretchy":702},[31662,89795,31960],{"separator":758},[31658,89797,46900],{},[31658,89799,15],{},[31662,89801,15972],{"stretchy":702},[31658,89803,31955],{},[31662,89805,32226],{},[31749,89807,640],{},[31662,89809,32712],{"stretchy":702},[31662,89811,15972],{"stretchy":702},[31658,89813,31968],{},[31662,89815,32712],{"stretchy":702},[31662,89817,31960],{"separator":758},[31658,89819,46900],{},[31658,89821,15],{},[31662,89823,15972],{"stretchy":702},[31658,89825,31955],{},[31662,89827,32712],{"stretchy":702},[31662,89829,15972],{"stretchy":702},[31658,89831,31968],{},[31662,89833,32226],{},[31749,89835,640],{},[31662,89837,32712],{"stretchy":702},[31662,89839,2557],{"stretchy":702},[31662,89841,223],{},[31749,89843,640],{},[71087,89845,89846],{},[71090,89847,89848],{"scriptlevel":459,"displaystyle":702},[31655,89849,89850,89852,89854,89856,89858,89860,89862,89864,89866,89868,89870,89872,89874],{},[31658,89851,54151],{},[31662,89853,15972],{"stretchy":702},[31658,89855,31955],{},[31662,89857,32226],{},[31749,89859,640],{},[31662,89861,32712],{"stretchy":702},[31662,89863,54707],{"mathvariant":38079},[31658,89865,54224],{},[31662,89867,15972],{"stretchy":702},[31658,89869,31968],{},[31662,89871,32226],{},[31749,89873,640],{},[31662,89875,32712],{"stretchy":702},[31665,89877,89878],{"encoding":31667},"dp[i][j] = \\begin{cases} i & j = 0 \\\\ j & i = 0 \\\\ dp[i-1][j-1] & A[i-1] = B[j-1] \\\\ \\min(dp[i-1][j-1], dp[i-1][j], dp[i][j-1]) + 1 & A[i-1] \\neq B[j-1] \\end{cases}",[34,89880,89882,89921],{"className":89881,"ariaHidden":758},[31672],[34,89883,89885,89888,89891,89894,89897,89900,89903,89906,89909,89912,89915,89918],{"className":89884},[31676],[34,89886],{"className":89887,"style":31774},[31680],[34,89889,46900],{"className":89890},[31685,31686],[34,89892,15],{"className":89893},[31685,31686],[34,89895,15972],{"className":89896},[31782],[34,89898,31955],{"className":89899},[31685,31686],[34,89901,32712],{"className":89902},[31690],[34,89904,15972],{"className":89905},[31782],[34,89907,31968],{"className":89908,"style":32153},[31685,31686],[34,89910,32712],{"className":89911},[31690],[34,89913],{"className":89914,"style":31846},[31845],[34,89916,202],{"className":89917},[31850],[34,89919],{"className":89920,"style":31846},[31845],[34,89922,89924,89927],{"className":89923},[31676],[34,89925],{"className":89926,"style":80092},[31680],[34,89928,89930,90008,90509],{"className":89929},[34006],[34,89931,89933],{"className":89932},[31782],[34,89934,89936],{"className":89935},[71281,71282],[34,89937,89939,90000],{"className":89938},[31796,31797],[34,89940,89942,89997],{"className":89941},[31801],[34,89943,89945,89955,89966,89976,89987],{"className":89944,"style":80117},[31805],[34,89946,89947,89950],{"style":80120},[34,89948],{"className":89949,"style":71299},[31813],[34,89951,89953],{"className":89952},[71303,71304],[34,89954,71307],{},[34,89956,89957,89960],{"style":80131},[34,89958],{"className":89959,"style":71299},[31813],[34,89961,89962],{"style":80137},[41588,89963,89964],{"xmlns":41590,"width":71319,"height":80140,"style":71321,"viewBox":80141,"preserveAspectRatio":71323},[20398,89965],{"d":80144},[34,89967,89968,89971],{"style":71329},[34,89969],{"className":89970,"style":71299},[31813],[34,89972,89974],{"className":89973},[71303,71304],[34,89975,71338],{},[34,89977,89978,89981],{"style":71341},[34,89979],{"className":89980,"style":71299},[31813],[34,89982,89983],{"style":80137},[41588,89984,89985],{"xmlns":41590,"width":71319,"height":80140,"style":71321,"viewBox":80141,"preserveAspectRatio":71323},[20398,89986],{"d":80144},[34,89988,89989,89992],{"style":80168},[34,89990],{"className":89991,"style":71299},[31813],[34,89993,89995],{"className":89994},[71303,71304],[34,89996,71362],{},[34,89998,31829],{"className":89999},[31828],[34,90001,90003],{"className":90002},[31801],[34,90004,90006],{"className":90005,"style":80186},[31805],[34,90007],{},[34,90009,90011],{"className":90010},[31685],[34,90012,90014,90274,90277],{"className":90013},[71078],[34,90015,90017],{"className":90016},[71384],[34,90018,90020,90266],{"className":90019},[31796,31797],[34,90021,90023,90263],{"className":90022},[31801],[34,90024,90026,90037,90048,90104],{"className":90025,"style":80207},[31805],[34,90027,90028,90031],{"style":80210},[34,90029],{"className":90030,"style":71401},[31813],[34,90032,90034],{"className":90033},[31685],[34,90035,31955],{"className":90036},[31685,31686],[34,90038,90039,90042],{"style":75837},[34,90040],{"className":90041,"style":71401},[31813],[34,90043,90045],{"className":90044},[31685],[34,90046,31968],{"className":90047,"style":32153},[31685,31686],[34,90049,90050,90053],{"style":76011},[34,90051],{"className":90052,"style":71401},[31813],[34,90054,90056,90059,90062,90065,90068,90071,90074,90077,90080,90083,90086,90089,90092,90095,90098,90101],{"className":90055},[31685],[34,90057,46900],{"className":90058},[31685,31686],[34,90060,15],{"className":90061},[31685,31686],[34,90063,15972],{"className":90064},[31782],[34,90066,31955],{"className":90067},[31685,31686],[34,90069],{"className":90070,"style":32604},[31845],[34,90072,32226],{"className":90073},[32024],[34,90075],{"className":90076,"style":32604},[31845],[34,90078,640],{"className":90079},[31685],[34,90081,32712],{"className":90082},[31690],[34,90084,15972],{"className":90085},[31782],[34,90087,31968],{"className":90088,"style":32153},[31685,31686],[34,90090],{"className":90091,"style":32604},[31845],[34,90093,32226],{"className":90094},[32024],[34,90096],{"className":90097,"style":32604},[31845],[34,90099,640],{"className":90100},[31685],[34,90102,32712],{"className":90103},[31690],[34,90105,90106,90109],{"style":80424},[34,90107],{"className":90108,"style":71401},[31813],[34,90110,90112,90115,90118,90121,90124,90127,90130,90133,90136,90139,90142,90145,90148,90151,90154,90157,90160,90163,90166,90169,90172,90175,90178,90181,90184,90187,90190,90193,90196,90199,90202,90205,90208,90211,90214,90217,90220,90223,90226,90229,90232,90235,90238,90241,90244,90247,90251,90254,90257,90260],{"className":90111},[31685],[34,90113,56084],{"className":90114},[32767],[34,90116,456],{"className":90117},[31782],[34,90119,46900],{"className":90120},[31685,31686],[34,90122,15],{"className":90123},[31685,31686],[34,90125,15972],{"className":90126},[31782],[34,90128,31955],{"className":90129},[31685,31686],[34,90131],{"className":90132,"style":32604},[31845],[34,90134,32226],{"className":90135},[32024],[34,90137],{"className":90138,"style":32604},[31845],[34,90140,640],{"className":90141},[31685],[34,90143,32712],{"className":90144},[31690],[34,90146,15972],{"className":90147},[31782],[34,90149,31968],{"className":90150,"style":32153},[31685,31686],[34,90152],{"className":90153,"style":32604},[31845],[34,90155,32226],{"className":90156},[32024],[34,90158],{"className":90159,"style":32604},[31845],[34,90161,640],{"className":90162},[31685],[34,90164,32712],{"className":90165},[31690],[34,90167,31960],{"className":90168},[32119],[34,90170],{"className":90171,"style":32123},[31845],[34,90173,46900],{"className":90174},[31685,31686],[34,90176,15],{"className":90177},[31685,31686],[34,90179,15972],{"className":90180},[31782],[34,90182,31955],{"className":90183},[31685,31686],[34,90185],{"className":90186,"style":32604},[31845],[34,90188,32226],{"className":90189},[32024],[34,90191],{"className":90192,"style":32604},[31845],[34,90194,640],{"className":90195},[31685],[34,90197,32712],{"className":90198},[31690],[34,90200,15972],{"className":90201},[31782],[34,90203,31968],{"className":90204,"style":32153},[31685,31686],[34,90206,32712],{"className":90207},[31690],[34,90209,31960],{"className":90210},[32119],[34,90212],{"className":90213,"style":32123},[31845],[34,90215,46900],{"className":90216},[31685,31686],[34,90218,15],{"className":90219},[31685,31686],[34,90221,15972],{"className":90222},[31782],[34,90224,31955],{"className":90225},[31685,31686],[34,90227,32712],{"className":90228},[31690],[34,90230,15972],{"className":90231},[31782],[34,90233,31968],{"className":90234,"style":32153},[31685,31686],[34,90236],{"className":90237,"style":32604},[31845],[34,90239,32226],{"className":90240},[32024],[34,90242],{"className":90243,"style":32604},[31845],[34,90245,640],{"className":90246},[31685],[34,90248,90250],{"className":90249},[31690],"])",[34,90252],{"className":90253,"style":32604},[31845],[34,90255,223],{"className":90256},[32024],[34,90258],{"className":90259,"style":32604},[31845],[34,90261,640],{"className":90262},[31685],[34,90264,31829],{"className":90265},[31828],[34,90267,90269],{"className":90268},[31801],[34,90270,90272],{"className":90271,"style":80503},[31805],[34,90273],{},[34,90275],{"className":90276,"style":71733},[71732],[34,90278,90280],{"className":90279},[71384],[34,90281,90283,90501],{"className":90282},[31796,31797],[34,90284,90286,90498],{"className":90285},[31801],[34,90287,90289,90312,90335,90400],{"className":90288,"style":80207},[31805],[34,90290,90291,90294],{"style":80210},[34,90292],{"className":90293,"style":71401},[31813],[34,90295,90297,90300,90303,90306,90309],{"className":90296},[31685],[34,90298,31968],{"className":90299,"style":32153},[31685,31686],[34,90301],{"className":90302,"style":31846},[31845],[34,90304,202],{"className":90305},[31850],[34,90307],{"className":90308,"style":31846},[31845],[34,90310,459],{"className":90311},[31685],[34,90313,90314,90317],{"style":75837},[34,90315],{"className":90316,"style":71401},[31813],[34,90318,90320,90323,90326,90329,90332],{"className":90319},[31685],[34,90321,31955],{"className":90322},[31685,31686],[34,90324],{"className":90325,"style":31846},[31845],[34,90327,202],{"className":90328},[31850],[34,90330],{"className":90331,"style":31846},[31845],[34,90333,459],{"className":90334},[31685],[34,90336,90337,90340],{"style":76011},[34,90338],{"className":90339,"style":71401},[31813],[34,90341,90343,90346,90349,90352,90355,90358,90361,90364,90367,90370,90373,90376,90379,90382,90385,90388,90391,90394,90397],{"className":90342},[31685],[34,90344,54151],{"className":90345},[31685,31686],[34,90347,15972],{"className":90348},[31782],[34,90350,31955],{"className":90351},[31685,31686],[34,90353],{"className":90354,"style":32604},[31845],[34,90356,32226],{"className":90357},[32024],[34,90359],{"className":90360,"style":32604},[31845],[34,90362,640],{"className":90363},[31685],[34,90365,32712],{"className":90366},[31690],[34,90368],{"className":90369,"style":31846},[31845],[34,90371,202],{"className":90372},[31850],[34,90374],{"className":90375,"style":31846},[31845],[34,90377,54224],{"className":90378,"style":54252},[31685,31686],[34,90380,15972],{"className":90381},[31782],[34,90383,31968],{"className":90384,"style":32153},[31685,31686],[34,90386],{"className":90387,"style":32604},[31845],[34,90389,32226],{"className":90390},[32024],[34,90392],{"className":90393,"style":32604},[31845],[34,90395,640],{"className":90396},[31685],[34,90398,32712],{"className":90399},[31690],[34,90401,90402,90405],{"style":80424},[34,90403],{"className":90404,"style":71401},[31813],[34,90406,90408,90411,90414,90417,90420,90423,90426,90429,90432,90435,90471,90474,90477,90480,90483,90486,90489,90492,90495],{"className":90407},[31685],[34,90409,54151],{"className":90410},[31685,31686],[34,90412,15972],{"className":90413},[31782],[34,90415,31955],{"className":90416},[31685,31686],[34,90418],{"className":90419,"style":32604},[31845],[34,90421,32226],{"className":90422},[32024],[34,90424],{"className":90425,"style":32604},[31845],[34,90427,640],{"className":90428},[31685],[34,90430,32712],{"className":90431},[31690],[34,90433],{"className":90434,"style":31846},[31845],[34,90436,90438,90465,90468],{"className":90437},[31850],[34,90439,90441],{"className":90440},[31850],[34,90442,90444],{"className":90443},[31685,54774],[34,90445,90447],{"className":90446},[54778],[34,90448,90450,90453,90462],{"className":90449},[54782],[34,90451],{"className":90452,"style":43217},[31680],[34,90454,90456],{"className":90455},[54789],[34,90457,90459],{"className":90458},[31685],[34,90460,54796],{"className":90461},[31850],[34,90463],{"className":90464},[54800],[34,90466],{"className":90467},[31845,54804],[34,90469,202],{"className":90470},[31850],[34,90472],{"className":90473,"style":31846},[31845],[34,90475,54224],{"className":90476,"style":54252},[31685,31686],[34,90478,15972],{"className":90479},[31782],[34,90481,31968],{"className":90482,"style":32153},[31685,31686],[34,90484],{"className":90485,"style":32604},[31845],[34,90487,32226],{"className":90488},[32024],[34,90490],{"className":90491,"style":32604},[31845],[34,90493,640],{"className":90494},[31685],[34,90496,32712],{"className":90497},[31690],[34,90499,31829],{"className":90500},[31828],[34,90502,90504],{"className":90503},[31801],[34,90505,90507],{"className":90506,"style":80503},[31805],[34,90508],{},[34,90510],{"className":90511},[31690,41488],[15,90513,90514],{},[2627,90515,90516],{},"2) 算法代码（C\u002FC++实现）：",[24,90518,90520],{"className":65734,"code":90519,"language":31965,"meta":29,"style":29},"#include \u003Ciostream>\n#include \u003Cstring>\n#include \u003Cvector>\n#include \u003Calgorithm>\n\nusing namespace std;\n\n\u002F\u002F 获取三个数中最小值的辅助函数\nint getMin(int a, int b, int c) {\n return min(min(a, b), c);\n}\n\nint minDistance(string A, string B) {\n int m = A.length();\n int n = B.length();\n \n \u002F\u002F 创建二维动态规划数组，并初始化全为 0\n \u002F\u002F 注意数组维度是 (m+1) x (n+1)，因为我们要容纳前缀长度为0（空串）的情况\n vector\u003Cvector\u003Cint>> dp(m + 1, vector\u003Cint>(n + 1, 0));\n \n \u002F\u002F 初始化边界条件\n for (int i = 0; i \u003C= m; i++) {\n dp[i][0] = i; \u002F\u002F B为空串，A必须不断执行删除\n }\n for (int j = 0; j \u003C= n; j++) {\n dp[0][j] = j; \u002F\u002F A为空串，A必须不断执行插入\n }\n \n \u002F\u002F 使用状态转移方程逐行填表\n for (int i = 1; i \u003C= m; i++) {\n for (int j = 1; j \u003C= n; j++) {\n \u002F\u002F 注意：第 i 个字符在字符串中的索引是 i-1\n if (A[i - 1] == B[j - 1]) {\n dp[i][j] = dp[i - 1][j - 1]; \u002F\u002F 字符相同，无需操作代价\n } else {\n dp[i][j] = 1 + getMin(\n dp[i - 1][j - 1], \u002F\u002F 替换 (Replace)\n dp[i - 1][j], \u002F\u002F 删除 (Delete)\n dp[i][j - 1] \u002F\u002F 插入 (Insert)\n );\n }\n }\n }\n \n \u002F\u002F 表格右下角的值即为全局最小操作次数\n return dp[m][n];\n}\n",[31,90521,90522,90526,90531,90536,90541,90545,90549,90553,90558,90563,90568,90572,90576,90581,90586,90591,90596,90601,90606,90611,90615,90620,90625,90630,90634,90639,90644,90648,90652,90657,90662,90667,90672,90677,90682,90687,90692,90697,90702,90707,90712,90717,90721,90725,90729,90734,90739],{"__ignoreMap":29},[34,90523,90524],{"class":36,"line":37},[34,90525,40],{},[34,90527,90528],{"class":36,"line":43},[34,90529,90530],{},"#include \u003Cstring>\n",[34,90532,90533],{"class":36,"line":49},[34,90534,90535],{},"#include \u003Cvector>\n",[34,90537,90538],{"class":36,"line":56},[34,90539,90540],{},"#include \u003Calgorithm>\n",[34,90542,90543],{"class":36,"line":62},[34,90544,53],{"emptyLinePlaceholder":52},[34,90546,90547],{"class":36,"line":68},[34,90548,46],{},[34,90550,90551],{"class":36,"line":74},[34,90552,53],{"emptyLinePlaceholder":52},[34,90554,90555],{"class":36,"line":1971},[34,90556,90557],{},"\u002F\u002F 获取三个数中最小值的辅助函数\n",[34,90559,90560],{"class":36,"line":1976},[34,90561,90562],{},"int getMin(int a, int b, int c) {\n",[34,90564,90565],{"class":36,"line":1985},[34,90566,90567],{}," return min(min(a, b), c);\n",[34,90569,90570],{"class":36,"line":1990},[34,90571,77],{},[34,90573,90574],{"class":36,"line":1999},[34,90575,53],{"emptyLinePlaceholder":52},[34,90577,90578],{"class":36,"line":2022},[34,90579,90580],{},"int minDistance(string A, string B) {\n",[34,90582,90583],{"class":36,"line":2036},[34,90584,90585],{}," int m = A.length();\n",[34,90587,90588],{"class":36,"line":2603},[34,90589,90590],{}," int n = B.length();\n",[34,90592,90593],{"class":36,"line":5537},[34,90594,90595],{}," \n",[34,90597,90598],{"class":36,"line":5543},[34,90599,90600],{}," \u002F\u002F 创建二维动态规划数组，并初始化全为 0\n",[34,90602,90603],{"class":36,"line":5548},[34,90604,90605],{}," \u002F\u002F 注意数组维度是 (m+1) x (n+1)，因为我们要容纳前缀长度为0（空串）的情况\n",[34,90607,90608],{"class":36,"line":5564},[34,90609,90610],{}," vector\u003Cvector\u003Cint>> dp(m + 1, vector\u003Cint>(n + 1, 0));\n",[34,90612,90613],{"class":36,"line":5576},[34,90614,90595],{},[34,90616,90617],{"class":36,"line":5606},[34,90618,90619],{}," \u002F\u002F 初始化边界条件\n",[34,90621,90622],{"class":36,"line":5616},[34,90623,90624],{}," for (int i = 0; i \u003C= m; i++) {\n",[34,90626,90627],{"class":36,"line":5628},[34,90628,90629],{}," dp[i][0] = i; \u002F\u002F B为空串，A必须不断执行删除\n",[34,90631,90632],{"class":36,"line":5634},[34,90633,5540],{},[34,90635,90636],{"class":36,"line":6420},[34,90637,90638],{}," for (int j = 0; j \u003C= n; j++) {\n",[34,90640,90641],{"class":36,"line":11294},[34,90642,90643],{}," dp[0][j] = j; \u002F\u002F A为空串，A必须不断执行插入\n",[34,90645,90646],{"class":36,"line":11321},[34,90647,5540],{},[34,90649,90650],{"class":36,"line":11326},[34,90651,90595],{},[34,90653,90654],{"class":36,"line":11333},[34,90655,90656],{}," \u002F\u002F 使用状态转移方程逐行填表\n",[34,90658,90659],{"class":36,"line":11344},[34,90660,90661],{}," for (int i = 1; i \u003C= m; i++) {\n",[34,90663,90664],{"class":36,"line":11349},[34,90665,90666],{}," for (int j = 1; j \u003C= n; j++) {\n",[34,90668,90669],{"class":36,"line":11354},[34,90670,90671],{}," \u002F\u002F 注意：第 i 个字符在字符串中的索引是 i-1\n",[34,90673,90674],{"class":36,"line":11363},[34,90675,90676],{}," if (A[i - 1] == B[j - 1]) {\n",[34,90678,90679],{"class":36,"line":11368},[34,90680,90681],{}," dp[i][j] = dp[i - 1][j - 1]; \u002F\u002F 字符相同，无需操作代价\n",[34,90683,90684],{"class":36,"line":11377},[34,90685,90686],{}," } else {\n",[34,90688,90689],{"class":36,"line":11402},[34,90690,90691],{}," dp[i][j] = 1 + getMin(\n",[34,90693,90694],{"class":36,"line":11407},[34,90695,90696],{}," dp[i - 1][j - 1], \u002F\u002F 替换 (Replace)\n",[34,90698,90699],{"class":36,"line":11421},[34,90700,90701],{}," dp[i - 1][j], \u002F\u002F 删除 (Delete)\n",[34,90703,90704],{"class":36,"line":11426},[34,90705,90706],{}," dp[i][j - 1] \u002F\u002F 插入 (Insert)\n",[34,90708,90709],{"class":36,"line":11447},[34,90710,90711],{}," );\n",[34,90713,90714],{"class":36,"line":11452},[34,90715,90716],{}," }\n",[34,90718,90719],{"class":36,"line":11463},[34,90720,26684],{},[34,90722,90723],{"class":36,"line":11485},[34,90724,5540],{},[34,90726,90727],{"class":36,"line":11495},[34,90728,90595],{},[34,90730,90731],{"class":36,"line":11528},[34,90732,90733],{}," \u002F\u002F 表格右下角的值即为全局最小操作次数\n",[34,90735,90736],{"class":36,"line":11534},[34,90737,90738],{}," return dp[m][n];\n",[34,90740,90741],{"class":36,"line":11544},[34,90742,77],{},[79,90744,81],{},{"title":29,"searchDepth":43,"depth":43,"links":90746},[90747,90757,90779,90791,90799],{"id":58157,"depth":43,"text":58158,"children":90748},[90749,90750,90751,90752,90753,90754,90755,90756],{"id":58161,"depth":49,"text":58162},{"id":58168,"depth":49,"text":58168},{"id":58227,"depth":49,"text":58228},{"id":58234,"depth":49,"text":58168},{"id":58366,"depth":49,"text":58367},{"id":58373,"depth":49,"text":58168},{"id":58473,"depth":49,"text":58474},{"id":58480,"depth":49,"text":58168},{"id":58522,"depth":43,"text":58523,"children":90758},[90759,90760,90761,90762,90763,90764,90765,90766,90767,90768,90769,90770,90771,90772,90773,90774,90775,90776,90777,90778],{"id":58526,"depth":49,"text":58162},{"id":58542,"depth":49,"text":58168},{"id":58585,"depth":49,"text":58228},{"id":59386,"depth":49,"text":58168},{"id":60497,"depth":49,"text":58367},{"id":61058,"depth":49,"text":58168},{"id":62760,"depth":49,"text":58474},{"id":63171,"depth":49,"text":58168},{"id":63605,"depth":49,"text":63606},{"id":63708,"depth":49,"text":58168},{"id":63935,"depth":49,"text":63936},{"id":64209,"depth":49,"text":58168},{"id":64457,"depth":49,"text":64458},{"id":64473,"depth":49,"text":58168},{"id":64492,"depth":49,"text":64493},{"id":64507,"depth":49,"text":58168},{"id":64553,"depth":49,"text":64554},{"id":65040,"depth":49,"text":58168},{"id":65358,"depth":49,"text":65359},{"id":65374,"depth":49,"text":58168},{"id":65410,"depth":43,"text":65411,"children":90780},[90781,90782,90783,90784,90785,90786,90787,90788,90789,90790],{"id":65414,"depth":49,"text":58162},{"id":65420,"depth":49,"text":58168},{"id":65453,"depth":49,"text":58228},{"id":65459,"depth":49,"text":58168},{"id":65671,"depth":49,"text":58367},{"id":65802,"depth":49,"text":58168},{"id":66075,"depth":49,"text":58474},{"id":66400,"depth":49,"text":58168},{"id":67292,"depth":49,"text":63606},{"id":68383,"depth":49,"text":58168},{"id":68738,"depth":43,"text":68739,"children":90792},[90793,90794,90795,90796,90797,90798],{"id":68742,"depth":49,"text":58162},{"id":68888,"depth":49,"text":58168},{"id":70656,"depth":49,"text":58228},{"id":72433,"depth":49,"text":58168},{"id":83809,"depth":49,"text":58367},{"id":83997,"depth":49,"text":58168},{"id":87646,"depth":43,"text":87647,"children":90800},[90801,90802,90803,90804],{"id":87650,"depth":49,"text":58162},{"id":87664,"depth":49,"text":58168},{"id":88119,"depth":49,"text":58228},{"id":88483,"depth":49,"text":58168},{},"\u002Fsource\u002Fcs\u002Falgo\u002Fpastpapers\u002F2024-2025-se2",{"title":58147,"description":29},"source\u002Fcs\u002Falgo\u002Fpastpapers\u002F2024-2025-se2","SFPHjVtixKHENzt0ODswKs9ji6o8HqzxMKyRqUV-DsA",{"id":90811,"title":90812,"body":90813,"description":90820,"extension":85,"meta":91069,"navigation":52,"path":91070,"seo":91071,"stem":91072,"__hash__":91073},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002F00-复习索引.md","数据库补充习题 70+100 复习索引",{"type":8,"value":90814,"toc":91067},[90815,90818,90821,90841,90844,90858,90861,91034,91037],[11,90816,90812],{"id":90817},"数据库补充习题-70100-复习索引",[15,90819,90820],{},"整理依据：",[105,90822,90823,90832,90838],{},[108,90824,90825,90826,146,90829],{},"题面来源：",[31,90827,90828],{},"数据库系统-补充习题\u002F数据库70题.doc",[31,90830,90831],{},"数据库系统-补充习题\u002F数据库100题.doc",[108,90833,90834,90835],{},"第一优先参考：",[31,90836,90837],{},"数据库系统概论(第6版) (王珊,杜小勇,陈红) .pdf",[108,90839,90840],{},"SQL 代码：统一按 SQL Server \u002F T-SQL 写法整理",[15,90842,90843],{},"说明：",[105,90845,90846,90852,90855],{},[108,90847,90848,90851],{},[31,90849,90850],{},"数据库70题.doc"," 原文件存在跳号，例如综合题从 8 跳到 22、从 33 跳到 37、缺 53。这里保留原题号，不自行补号。",[108,90853,90854],{},"个别原题有明显笔误或表格数据不一致，我在对应答案中标注“题面笔误”或“按常见修正解答”。",[108,90856,90857],{},"教材第 6 版中相关重点章节：第 1 章绪论，第 2 章关系模型，第 3 章 SQL，第 4 章安全性，第 5 章完整性，第 6 章关系数据理论，第 7 章数据库设计，第 11 章数据库恢复技术，第 12 章并发控制。",[15,90859,90860],{},"每日复习安排：",[13837,90862,90863,90876],{},[13840,90864,90865],{},[13843,90866,90867,90870,90873],{},[13846,90868,90869],{},"天数",[13846,90871,90872],{},"文件",[13846,90874,90875],{},"内容",[13853,90877,90878,90891,90904,90917,90930,90943,90956,90969,90982,90995,91008,91021],{},[13843,90879,90880,90883,90888],{},[13858,90881,90882],{},"Day 01",[13858,90884,90885],{},[31,90886,90887],{},"Day01-数据库100题-选择01-20.md",[13858,90889,90890],{},"100 题单选 1-20",[13843,90892,90893,90896,90901],{},[13858,90894,90895],{},"Day 02",[13858,90897,90898],{},[31,90899,90900],{},"Day02-数据库100题-选择21-40.md",[13858,90902,90903],{},"100 题单选 21-40",[13843,90905,90906,90909,90914],{},[13858,90907,90908],{},"Day 03",[13858,90910,90911],{},[31,90912,90913],{},"Day03-数据库100题-选择41-50填空51-60.md",[13858,90915,90916],{},"100 题单选 41-50、填空 51-60",[13843,90918,90919,90922,90927],{},[13858,90920,90921],{},"Day 04",[13858,90923,90924],{},[31,90925,90926],{},"Day04-数据库100题-填空61-80.md",[13858,90928,90929],{},"100 题填空 61-80",[13843,90931,90932,90935,90940],{},[13858,90933,90934],{},"Day 05",[13858,90936,90937],{},[31,90938,90939],{},"Day05-数据库100题-填空81-100.md",[13858,90941,90942],{},"100 题填空 81-100",[13843,90944,90945,90948,90953],{},[13858,90946,90947],{},"Day 06",[13858,90949,90950],{},[31,90951,90952],{},"Day06-数据库70题-试题01-11.md",[13858,90954,90955],{},"70 题试题 1-11：视图、游标、SQL、范式",[13843,90957,90958,90961,90966],{},[13858,90959,90960],{},"Day 07",[13858,90962,90963],{},[31,90964,90965],{},"Day07-数据库70题-试题12-25.md",[13858,90967,90968],{},"70 题试题 12-25：保护、事务、恢复、完整性",[13843,90970,90971,90974,90979],{},[13858,90972,90973],{},"Day 08",[13858,90975,90976],{},[31,90977,90978],{},"Day08-数据库70题-综合01-08.md",[13858,90980,90981],{},"70 题综合 1-8：DBA、数据字典、关系代数",[13843,90983,90984,90987,90992],{},[13858,90985,90986],{},"Day 09",[13858,90988,90989],{},[31,90990,90991],{},"Day09-数据库70题-综合22-33.md",[13858,90993,90994],{},"70 题综合 22-33：关系演算、SQL、视图",[13843,90996,90997,91000,91005],{},[13858,90998,90999],{},"Day 10",[13858,91001,91002],{},[31,91003,91004],{},"Day10-数据库70题-综合37-47.md",[13858,91006,91007],{},"70 题综合 37-47：范式、候选码、分解",[13843,91009,91010,91013,91018],{},[13858,91011,91012],{},"Day 11",[13858,91014,91015],{},[31,91016,91017],{},"Day11-数据库70题-综合48-61.md",[13858,91019,91020],{},"70 题综合 48-61：E-R、完整性、事务并发",[13843,91022,91023,91026,91031],{},[13858,91024,91025],{},"Day 12",[13858,91027,91028],{},[31,91029,91030],{},"Day12-数据库70题-综合62-71.md",[13858,91032,91033],{},"70 题综合 62-71：转储、日志、恢复、并发控制",[15,91035,91036],{},"复习建议：",[2230,91038,91039,91042,91064],{},[108,91040,91041],{},"先看“答案”，再读“讲解”，最后遮住答案重新写一遍。",[108,91043,91044,91045,146,91048,146,91051,146,91054,146,91057,91060,91061,347],{},"SQL 题不要背 Oracle 旧写法，重点记 SQL Server 的 ",[31,91046,91047],{},"CREATE VIEW",[31,91049,91050],{},"UPDATE ... FROM",[31,91052,91053],{},"DROP INDEX index_name ON table_name",[31,91055,91056],{},"EXCEPT",[31,91058,91059],{},"INTERSECT","、双重 ",[31,91062,91063],{},"NOT EXISTS",[108,91065,91066],{},"范式题固定三步：先写函数依赖，再求候选码，最后判断部分依赖、传递依赖和 BCNF 条件。",{"title":29,"searchDepth":43,"depth":43,"links":91068},[],{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002F00",{"title":90812,"description":90820},"source\u002Fcs\u002Fdb\u002F100+70\u002F00-复习索引","XL5XCFJmg_ek8vdRBzeZCmksTvZ5gH4rTosRq7IqVOs",{"id":91075,"title":91076,"body":91077,"description":91084,"extension":85,"meta":91351,"navigation":52,"path":91352,"seo":91353,"stem":91354,"__hash__":91355},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay01-数据库100题-选择01-20.md","Day 01 - 数据库100题：单项选择 1-20",{"type":8,"value":91078,"toc":91329},[91079,91082,91085,91089,91092,91094,91097,91101,91104,91107,91110,91114,91117,91119,91122,91126,91129,91131,91134,91138,91141,91143,91146,91150,91153,91155,91158,91162,91165,91167,91170,91174,91177,91179,91182,91186,91189,91191,91194,91198,91201,91203,91206,91210,91213,91215,91218,91222,91225,91227,91230,91234,91237,91239,91242,91246,91249,91251,91254,91258,91261,91263,91266,91270,91273,91275,91278,91282,91285,91287,91290,91294,91297,91299,91305,91309,91312,91314,91317,91321,91324,91326],[11,91080,91076],{"id":91081},"day-01-数据库100题单项选择-1-20",[15,91083,91084],{},"参考重点：教材第 1 章数据库基本概念、第 2 章关系模型、第 3 章 SQL、第 11-12 章事务与并发恢复。",[19,91086,91088],{"id":91087},"_1-位于用户和操作系统之间的一层数据管理软件是","1. 位于用户和操作系统之间的一层数据管理软件是 。",[15,91090,91091],{},"A. DBS　B. DB　C. DBMS　D. MIS",[15,91093,61063],{},[15,91095,91096],{},"讲解：DBMS 是数据库管理系统，位于用户和操作系统之间，负责数据定义、组织存储、操纵、事务管理、恢复、安全和完整性控制。DB 是数据库本身，DBS 是由 DB、DBMS、应用系统、DBA 等组成的数据库系统。",[19,91098,91100],{"id":91099},"_2-数据库系统中的数据模型通常由-三部分组成","2. 数据库系统中的数据模型通常由 三部分组成。",[15,91102,91103],{},"A. 数据结构、数据操作和完整性约束\nB. 数据定义、数据操作和安全性约束\nC. 数据结构、数据管理和数据保护\nD. 数据定义、数据管理和运行控制",[15,91105,91106],{},"答案：A",[15,91108,91109],{},"讲解：教材把数据模型三要素归纳为数据结构、数据操作、完整性约束。数据结构描述静态特征，数据操作描述动态行为，完整性约束限制合法状态和合法变化。",[19,91111,91113],{"id":91112},"_3-codasyl-组织提出的-dbtg-报告中的数据模型是-的主要代表","3. CODASYL 组织提出的 DBTG 报告中的数据模型是 的主要代表。",[15,91115,91116],{},"A. 层次模型　B. 网状模型　C. 关系模型　D. 实体联系模型",[15,91118,58547],{},[15,91120,91121],{},"讲解：DBTG 报告是早期网状数据库模型的代表。层次模型典型代表是 IMS，关系模型由 E. F. Codd 提出，E-R 模型属于概念模型。",[19,91123,91125],{"id":91124},"_4-数据库的三级模式中数据的全局逻辑结构用-来描述","4. 数据库的三级模式中，数据的全局逻辑结构用 来描述。",[15,91127,91128],{},"A. 子模式　B. 用户模式　C. 模式　D. 存储模式",[15,91130,61063],{},[15,91132,91133],{},"讲解：三级模式为外模式、模式、内模式。模式也称概念模式，描述数据库中全体数据的全局逻辑结构；外模式面向用户视图；内模式描述物理存储。",[19,91135,91137],{"id":91136},"_5-用户涉及的逻辑结构用-描述","5. 用户涉及的逻辑结构用 描述。",[15,91139,91140],{},"A. 模式　B. 存储模式　C. 概念模式　D. 子模式",[15,91142,59391],{},[15,91144,91145],{},"讲解：子模式即外模式，是用户看到和使用的局部逻辑结构。一个数据库可以有多个外模式，以适应不同用户或应用。",[19,91147,91149],{"id":91148},"_6-数据库的并发控制完整性检查安全性检查等是对数据库的","6. 数据库的并发控制，完整性检查，安全性检查等是对数据库的 。",[15,91151,91152],{},"A. 设计　B. 保护　C. 操纵　D. 维护",[15,91154,58547],{},[15,91156,91157],{},"讲解：数据库保护通常包括安全性、完整性、并发控制和恢复。它们共同保证数据不被非法访问、不违反语义约束、不被并发破坏，并能在故障后恢复。",[19,91159,91161],{"id":91160},"_7-完成对数据库数据的查询与更新","7. 完成对数据库数据的查询与更新。",[15,91163,91164],{},"A. DCL　B. DDL　C. DML　D. DQL",[15,91166,61063],{},[15,91168,91169],{},"讲解：DML 是数据操纵语言，完成查询、插入、删除、修改等操作。DQL 常被看成 DML 中的查询部分；DDL 负责定义结构；DCL 负责授权等控制。",[19,91171,91173],{"id":91172},"_8-关系模型的程序员不需熟悉数据库的","8. 关系模型的程序员不需熟悉数据库的 。",[15,91175,91176],{},"A. 数据操作　B. 完整性约束条件　C. 存取路径　D. 数据定义",[15,91178,61063],{},[15,91180,91181],{},"讲解：关系模型具有较高的数据独立性，用户面向关系和 SQL 操作，不必关心底层索引、文件组织等存取路径。存取路径由 DBMS 和优化器决定。",[19,91183,91185],{"id":91184},"_9-dbms-提供-dml-实现对数据的操作可以独立交互使用的-dml-称为","9. DBMS 提供 DML 实现对数据的操作。可以独立交互使用的 DML 称为 。",[15,91187,91188],{},"A. 宿主型　B. 独立型　C. 自含型　D. 嵌入型",[15,91190,61063],{},[15,91192,91193],{},"讲解：自含型 DML 可以独立交互使用，例如直接在数据库客户端执行 SQL。宿主型或嵌入型 DML 则嵌入高级语言程序中使用。",[19,91195,91197],{"id":91196},"_10-dbms-提供-dml-实现对数据的操作嵌入高级语言中使用的-dml-称为","10. DBMS 提供 DML 实现对数据的操作。嵌入高级语言中使用的 DML 称为 。",[15,91199,91200],{},"A. 自主型　B. 自含型　C. 宿主型　D. 交互型",[15,91202,61063],{},[15,91204,91205],{},"讲解：嵌入 C、Java 等高级语言中的数据库操作语言称为宿主型 DML。高级语言是宿主语言，数据库语句嵌入其中。",[19,91207,91209],{"id":91208},"_11-下列运算中-不是关系代数的基本操作","11. 下列运算中 不是关系代数的基本操作。",[15,91211,91212],{},"A. 笛卡尔积　B. 自然联接　C. 投影　D. 并",[15,91214,58547],{},[15,91216,91217],{},"讲解：关系代数的基本运算通常包括选择、投影、并、差、笛卡尔积。自然连接可以由笛卡尔积、选择和投影组合得到，因此不是基本操作。",[19,91219,91221],{"id":91220},"_12-在关系中能惟一标识元组的属性集称为关系模型的","12. 在关系中能惟一标识元组的属性集称为关系模型的 。",[15,91223,91224],{},"A. 主键　B. 超键　C. 候选键　D. 外键",[15,91226,61063],{},[15,91228,91229],{},"讲解：候选键是能唯一标识元组且不含多余属性的最小属性集。主键是从候选键中选出的一个；超键能唯一标识元组，但可能包含冗余属性。",[19,91231,91233],{"id":91232},"_13-对关系-r-进行投影运算后得到关系-s则","13. 对关系 R 进行投影运算后，得到关系 S，则 。",[15,91235,91236],{},"A. 关系 R 的元组数等于关系 S 的元组数\nB. 关系 R 的元组数小于关系 S 的元组数\nC. 关系 R 的元组数大于关系 S 的元组数\nD. 关系 R 的元组数大于或等于关系 S 的元组数",[15,91238,59391],{},[15,91240,91241],{},"讲解：投影会减少列，并且关系是集合，重复元组要消除。因此投影后的元组数不可能多于原关系，可能相等，也可能更少。",[19,91243,91245],{"id":91244},"_14-关系-r-与关系-s-只有一个公共属性t1-是-r-与-s-等值联接的结果t2-是-r-与-s-等值自然联接的结果则","14. 关系 R 与关系 S 只有一个公共属性，T1 是 R 与 S 等值联接的结果，T2 是 R 与 S 等值自然联接的结果，则 。",[15,91247,91248],{},"A. T1 的属性个数等于 T2 的属性个数\nB. T1 的属性个数小于 T2 的属性个数\nC. T1 的属性个数大于 T2 的属性个数\nD. T2 的属性个数大于或等于 T2 的属性个数",[15,91250,61063],{},[15,91252,91253],{},"讲解：等值连接会保留两个关系中的公共属性列各一份；自然连接会把同名公共属性合并成一列。因此只有一个公共属性时，T1 比 T2 多一列。",[19,91255,91257],{"id":91256},"_15-在关系代数表达式优化策略中应尽可能早执行-操作","15. 在关系代数表达式优化策略中，应尽可能早执行 操作。",[15,91259,91260],{},"A. 投影　B. 联接　C. 选择　D. 笛卡尔积",[15,91262,61063],{},[15,91264,91265],{},"讲解：选择能尽早过滤元组，减少后续连接和笛卡尔积的中间结果规模。查询优化中常见规则是“选择下推”，其次是尽早投影。",[19,91267,91269],{"id":91268},"_16-关系数据库的查询语言是一种-语言","16. 关系数据库的查询语言是一种 语言。",[15,91271,91272],{},"A. 过程性　B. 非过程性　C. 第三代　D. 高级程序设计",[15,91274,58547],{},[15,91276,91277],{},"讲解：SQL 更接近非过程性语言，用户说明“要什么”，通常不必说明“怎样访问、怎样连接、使用哪个索引”。执行过程由 DBMS 优化器生成。",[19,91279,91281],{"id":91280},"_17-数据库中数据的正确性有效性和相容性称为_","17. 数据库中数据的正确性、有效性和相容性称为**_**。",[15,91283,91284],{},"A. 恢复　B. 并发控制　C. 完整性　D. 安全性",[15,91286,61063],{},[15,91288,91289],{},"讲解：完整性关注数据是否符合语义约束，例如主键非空且唯一、外键必须引用已有元组、成绩在 0-100 之间。安全性关注是否被未授权访问。",[19,91291,91293],{"id":91292},"_18-授权是-dbs-采用的","18. 授权是 DBS 采用的 。",[15,91295,91296],{},"A. 恢复措施　B. 并发控制措施　C. 完整性措施　D. 安全性措施",[15,91298,59391],{},[15,91300,91301,91302,347],{},"讲解：授权属于自主存取控制，用来规定用户能访问哪些数据对象、能执行哪些操作，如 ",[31,91303,91304],{},"GRANT SELECT ON ... TO ...",[19,91306,91308],{"id":91307},"_19-db-的转储属于-dbs-的","19. DB 的转储属于 DBS 的 。",[15,91310,91311],{},"A. 完整性措施　B. 一致性措施　C. 并发控制措施　D. 恢复措施",[15,91313,59391],{},[15,91315,91316],{},"讲解：转储就是备份数据库。发生介质故障时，可以用后援副本加日志恢复数据库，所以属于恢复技术。",[19,91318,91320],{"id":91319},"_20-事务的原子性是指","20. 事务的原子性是指 。",[15,91322,91323],{},"A. 事务中包括的所有操作要么都做，要么都不做\nB. 事务一旦提交，对数据库的改变就是永久的\nC. 一个事务内部的操作及使用的数据对并发的其他事务是隔离的\nD. 事务必须是使数据库从一个一致性状态变到另一个一致性状态",[15,91325,91106],{},[15,91327,91328],{},"讲解：原子性强调事务不可再分，提交则全部生效，失败则全部撤销。B 是持久性，C 是隔离性，D 是一致性。",{"title":29,"searchDepth":43,"depth":43,"links":91330},[91331,91332,91333,91334,91335,91336,91337,91338,91339,91340,91341,91342,91343,91344,91345,91346,91347,91348,91349,91350],{"id":91087,"depth":43,"text":91088},{"id":91099,"depth":43,"text":91100},{"id":91112,"depth":43,"text":91113},{"id":91124,"depth":43,"text":91125},{"id":91136,"depth":43,"text":91137},{"id":91148,"depth":43,"text":91149},{"id":91160,"depth":43,"text":91161},{"id":91172,"depth":43,"text":91173},{"id":91184,"depth":43,"text":91185},{"id":91196,"depth":43,"text":91197},{"id":91208,"depth":43,"text":91209},{"id":91220,"depth":43,"text":91221},{"id":91232,"depth":43,"text":91233},{"id":91244,"depth":43,"text":91245},{"id":91256,"depth":43,"text":91257},{"id":91268,"depth":43,"text":91269},{"id":91280,"depth":43,"text":91281},{"id":91292,"depth":43,"text":91293},{"id":91307,"depth":43,"text":91308},{"id":91319,"depth":43,"text":91320},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday01-100-01-20",{"title":91076,"description":91084},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay01-数据库100题-选择01-20","LDrNHeomN036_Bz1pdztvnbItpyx0tT9bZLahdUmN74",{"id":91357,"title":91358,"body":91359,"description":91366,"extension":85,"meta":91637,"navigation":52,"path":91638,"seo":91639,"stem":91640,"__hash__":91641},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay02-数据库100题-选择21-40.md","Day 02 - 数据库100题：单项选择 21-40",{"type":8,"value":91360,"toc":91615},[91361,91364,91367,91371,91374,91376,91379,91383,91386,91388,91391,91395,91398,91400,91403,91407,91409,91411,91414,91418,91421,91423,91426,91430,91433,91435,91438,91442,91445,91447,91450,91454,91457,91460,91463,91467,91470,91472,91475,91479,91482,91484,91487,91491,91494,91497,91499,91502,91506,91509,91511,91518,91522,91525,91527,91530,91534,91537,91539,91542,91546,91549,91551,91554,91558,91561,91563,91566,91570,91573,91576,91579,91583,91586,91588,91591,91595,91598,91600,91603,91607,91610,91612],[11,91362,91358],{"id":91363},"day-02-数据库100题单项选择-21-40",[15,91365,91366],{},"参考重点：教材第 1 章数据库系统特点，第 11 章恢复技术，第 12 章并发控制。",[19,91368,91370],{"id":91369},"_21-若一个事务执行成功则它的全部更新被提交若一个事务执行失败则-db-中被其更新过的数据恢复原状就像这些更新从未发生过这保持了数据库处于","21. 若一个事务执行成功，则它的全部更新被提交；若一个事务执行失败，则 DB 中被其更新过的数据恢复原状，就像这些更新从未发生过，这保持了数据库处于 。",[15,91372,91373],{},"A. 安全性状态　B. 一致性状态　C. 完整性状态　D. 可靠性状态",[15,91375,58547],{},[15,91377,91378],{},"讲解：事务要把数据库从一个一致性状态变到另一个一致性状态。失败事务通过回滚撤销影响，避免数据库停留在半完成状态。",[19,91380,91382],{"id":91381},"_22-事务的并发执行不会破坏-db-的完整性这个性质称为事务的","22. 事务的并发执行不会破坏 DB 的完整性，这个性质称为事务的 。",[15,91384,91385],{},"A. 原子性　B. 隔离性　C. 持久性　D. 一致性",[15,91387,58547],{},[15,91389,91390],{},"讲解：隔离性要求并发事务互不干扰，其结果等价于某个串行执行结果。这样并发执行才不会因交叉读写破坏数据库一致性。",[19,91392,91394],{"id":91393},"_23-事务的隔离性由-dbms-的","23. 事务的隔离性由 DBMS 的 。",[15,91396,91397],{},"A. 安全性子系统实现　B. 完整性子系统实现　C. 并发控制子系统实现　D. 恢复子系统实现",[15,91399,61063],{},[15,91401,91402],{},"讲解：隔离性依靠并发控制实现，典型机制包括封锁、时间戳、多版本并发控制等。",[19,91404,91406],{"id":91405},"_24-事务的持久性由-dbms-的","24. 事务的持久性由 DBMS 的 。",[15,91408,91397],{},[15,91410,59391],{},[15,91412,91413],{},"讲解：持久性要求事务一旦提交，其更新即使系统故障也不能丢失。日志、检查点、REDO 等恢复技术负责保证这一点。",[19,91415,91417],{"id":91416},"_25-如果事务-t1-获得了数据对象-d-上的排他锁那么","25. 如果事务 T1 获得了数据对象 D 上的排他锁，那么， 。",[15,91419,91420],{},"A. T1 对 D 只能读不能写\nB. T1 对 D 只能写不能读\nC. 根据 X 锁协议，另一事务 T2 对 D 既不能读也不能写\nD. 根据 S 锁协议，另一事务 T2 对 D 只能读不能写",[15,91422,61063],{},[15,91424,91425],{},"讲解：排他锁又称 X 锁或写锁。事务持有 X 锁时，其他事务不能再对同一对象加 S 锁或 X 锁，因此既不能读也不能写。",[19,91427,91429],{"id":91428},"_26-事务的执行次序称为","26. 事务的执行次序称为 。",[15,91431,91432],{},"A. 调度　B. 过程　C. 优先级　D. 步骤",[15,91434,91106],{},[15,91436,91437],{},"讲解：调度描述多个事务的读写操作交叉执行的顺序。判断并发执行是否正确，核心就是判断调度是否可串行化。",[19,91439,91441],{"id":91440},"_27-以下四个概念中可用来解决一个并发调度是否正确问题的是","27. 以下四个概念中，可用来解决“一个并发调度是否正确”问题的是 。",[15,91443,91444],{},"A. 串行调度　B. 并发事务的可串行化　C. 并发事务的可并行化　D. 并发事务的有效调度",[15,91446,58547],{},[15,91448,91449],{},"讲解：可串行化是并发调度正确性的准则。只要一个并发调度等价于某个串行调度，就认为它不会破坏事务隔离性。",[19,91451,91453],{"id":91452},"_28-所有事务都遵循两段锁协议与事务的并发调度是可串行化的二者关系是","28. “所有事务都遵循两段锁协议”与“事务的并发调度是可串行化的”，二者关系是？",[15,91455,91456],{},"A. 前者是后者的必要条件\nB. 后者是前者的充分条件\nC. 前者是后者的必然结果\nD. 后者是前者的充分必要条件",[15,91458,91459],{},"答案：C（按题项意图）",[15,91461,91462],{},"讲解：严格说，“所有事务遵循两段锁协议”是“冲突可串行化”的充分条件，不是必要条件。选项没有标准表述“前者是后者的充分条件”，所以通常按 C 理解为：遵循两段锁协议必然推出调度可串行化。",[19,91464,91466],{"id":91465},"_29-db-技术中脏数据是指","29. DB 技术中，“脏数据”是指 。",[15,91468,91469],{},"A. 未提交的数据\nB. 未提交的随后又被撤销的数据\nC. 违反访问权而写入 DB 的数据\nD. 输入时就有错的数据",[15,91471,58547],{},[15,91473,91474],{},"讲解：脏数据是事务读到了另一个尚未提交事务写入的数据，而该写入后来被回滚。读脏数据会导致事务基于不存在的结果继续计算。",[19,91476,91478],{"id":91477},"_30-数据库恢复的基本策略是数据冗余被转储的冗余数据包括","30. 数据库恢复的基本策略是数据冗余，被转储的冗余数据包括 。",[15,91480,91481],{},"A. 日志文件和数据库副本\nB. 应用程序和数据库副本\nC. 数据字典、日志文件和数据库副本\nD. 应用程序、数据字典、日志文件和数据库副本",[15,91483,91106],{},[15,91485,91486],{},"讲解：恢复依赖两类冗余：数据库后援副本和日志文件。副本用于恢复基线，日志用于重做已提交事务、撤销未提交事务。",[19,91488,91490],{"id":91489},"_31-设有两个事务-t1-和-t2它们的并发操作如下则正确的结论是","31. 设有两个事务 T1 和 T2，它们的并发操作如下，则正确的结论是 。",[15,91492,91493],{},"T1：读 S，S=S-2，写 S，随后 ROLLBACK 恢复 S 值。\nT2：在 T1 回滚前读 S，S=S-4。",[15,91495,91496],{},"A. 不存在问题　B. 丢失修改　C. 读了“脏”数据　D. 产生不一致分析",[15,91498,61063],{},[15,91500,91501],{},"讲解：T2 读到了 T1 尚未提交的写入值，而 T1 随后回滚，所以 T2 读到的是脏数据。避免方法是读前加 S 锁，或使用更高隔离级别。",[19,91503,91505],{"id":91504},"_32-sql-的-rollback-语句的主要作用是","32. SQL 的 ROLLBACK 语句的主要作用是 。",[15,91507,91508],{},"A. 终止程序　B. 中断程序　C. 事务提交　D. 事务回退",[15,91510,59391],{},[15,91512,91513,91514,91517],{},"讲解：",[31,91515,91516],{},"ROLLBACK"," 撤销当前事务中已经执行但尚未提交的更新，使数据库恢复到事务开始前的一致状态。",[19,91519,91521],{"id":91520},"_33-sql-中设置主键和外键的子句属于-dbs-的-范围","33. SQL 中设置主键和外键的子句属于 DBS 的 范围。",[15,91523,91524],{},"A. 完整性控制　B. 安全性控制　C. 并发控制控制　D. 恢复控制",[15,91526,91106],{},[15,91528,91529],{},"讲解：主键约束实现实体完整性，外键约束实现参照完整性，都属于完整性控制。",[19,91531,91533],{"id":91532},"_34-在数据管理技术的发展过程中数据独立性最高的是-阶段","34. 在数据管理技术的发展过程中，数据独立性最高的是 阶段。",[15,91535,91536],{},"A. 数据库系统　B. 文件系统　C. 人工管理　D. 数据项管理",[15,91538,91106],{},[15,91540,91541],{},"讲解：数据库系统通过三级模式和两级映像提高数据独立性。文件系统只有较弱的物理独立性，人工管理几乎没有独立性。",[19,91543,91545],{"id":91544},"_35-数据库系统与文件系统的主要区别是","35. 数据库系统与文件系统的主要区别是 。",[15,91547,91548],{},"A. 数据库系统复杂，而文件系统简单\nB. 文件系统不能解决数据冗余和数据独立性问题，而数据库系统可以解决\nC. 文件系统只能管理程序文件，而数据库系统能够管理各种类型的文件\nD. 文件系统管理的数据量较少，而数据库系统可以管理庞大的数据量",[15,91550,58547],{},[15,91552,91553],{},"讲解：本质区别在于数据库系统实现整体数据结构化、共享性强、冗余度低、数据独立性高，并由 DBMS 统一管理数据。",[19,91555,91557],{"id":91556},"_36-数据库的概念模型独立于","36. 数据库的概念模型独立于 。",[15,91559,91560],{},"A. 具体的机器和 DBMS　B. E-R 图　C. 信息世界　D. 现实世界",[15,91562,91106],{},[15,91564,91565],{},"讲解：概念模型面向信息世界，用来抽象现实世界，不依赖具体机器、存储结构或某个 DBMS。E-R 图是概念模型的常用表示方法。",[19,91567,91569],{"id":91568},"_37-数据库具有-最小的-和较高的","37. 数据库具有 ① 、最小的 ② 和较高的 ③ 。",[15,91571,91572],{},"① A. 程序结构化　B. 数据结构化　C. 程序标准化　D. 数据模块化\n② A. 冗余度　B. 存储量　C. 完整度　D. 有效性\n③ A. 程序与数据的可靠性　B. 程序与数据完整性　C. 程序与数据独立性　D. 程序与数据一致性",[15,91574,91575],{},"答案：①B；②A；③C",[15,91577,91578],{},"讲解：数据库系统阶段的典型特点是整体数据结构化、共享性高、冗余度低、数据独立性高，并具有统一的数据保护能力。",[19,91580,91582],{"id":91581},"_38-在数据库中存储的是","38. 在数据库中存储的是 。",[15,91584,91585],{},"A. 数据　B. 数据模型　C. 数据以及数据之间的联系　D. 信息",[15,91587,61063],{},[15,91589,91590],{},"讲解：数据库不仅保存数据本身，也通过数据模型保存和表达数据之间的联系。关系模型中，这些联系通常通过关系和外键体现。",[19,91592,91594],{"id":91593},"_39-数据库中数据的物理独立性是指","39. 数据库中，数据的物理独立性是指 。",[15,91596,91597],{},"A. 数据库与数据库管理系统的相互独立\nB. 用户程序与 DBMS 的相互独立\nC. 用户的应用程序与存储在磁盘上数据库中的数据是相互独立的\nD. 应用程序与数据库中数据的逻辑结构相互独立",[15,91599,61063],{},[15,91601,91602],{},"讲解：物理独立性指内模式改变，例如文件组织、索引、存储位置变化时，模式和应用程序尽量不变。逻辑独立性则是模式改变时外模式和应用程序尽量不变。",[19,91604,91606],{"id":91605},"_40-数据库的特点之一是数据的共享严格地讲这里的数据共享是指","40. 数据库的特点之一是数据的共享，严格地讲，这里的数据共享是指 。",[15,91608,91609],{},"A. 同一个应用中的多个程序共享一个数据集合\nB. 多个用户、同一种语言共享数据\nC. 多个用户共享一个数据文件\nD. 多种应用、多种语言、多个用户相互覆盖地使用数据集合",[15,91611,59391],{},[15,91613,91614],{},"讲解：数据库共享不是简单复制文件，而是在 DBMS 统一控制下共享同一份有组织的数据。这样可以减少冗余并提高一致性，但也需要安全、完整性和并发控制。",{"title":29,"searchDepth":43,"depth":43,"links":91616},[91617,91618,91619,91620,91621,91622,91623,91624,91625,91626,91627,91628,91629,91630,91631,91632,91633,91634,91635,91636],{"id":91369,"depth":43,"text":91370},{"id":91381,"depth":43,"text":91382},{"id":91393,"depth":43,"text":91394},{"id":91405,"depth":43,"text":91406},{"id":91416,"depth":43,"text":91417},{"id":91428,"depth":43,"text":91429},{"id":91440,"depth":43,"text":91441},{"id":91452,"depth":43,"text":91453},{"id":91465,"depth":43,"text":91466},{"id":91477,"depth":43,"text":91478},{"id":91489,"depth":43,"text":91490},{"id":91504,"depth":43,"text":91505},{"id":91520,"depth":43,"text":91521},{"id":91532,"depth":43,"text":91533},{"id":91544,"depth":43,"text":91545},{"id":91556,"depth":43,"text":91557},{"id":91568,"depth":43,"text":91569},{"id":91581,"depth":43,"text":91582},{"id":91593,"depth":43,"text":91594},{"id":91605,"depth":43,"text":91606},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday02-100-21-40",{"title":91358,"description":91366},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay02-数据库100题-选择21-40","MifkaBZkoVUl-X2c-hmeH1giBDTSLINvBn0TfwgKsaw",{"id":91643,"title":91644,"body":91645,"description":91652,"extension":85,"meta":91895,"navigation":52,"path":91896,"seo":91897,"stem":91898,"__hash__":91899},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay03-数据库100题-选择41-50填空51-60.md","Day 03 - 数据库100题：单选 41-50，填空 51-60",{"type":8,"value":91646,"toc":91873},[91647,91650,91653,91657,91660,91662,91665,91669,91672,91674,91677,91681,91684,91686,91689,91693,91696,91698,91701,91705,91708,91710,91713,91717,91720,91722,91725,91729,91732,91734,91737,91741,91744,91746,91749,91753,91756,91758,91761,91765,91768,91770,91773,91777,91780,91783,91787,91790,91793,91797,91800,91803,91807,91810,91813,91817,91820,91823,91827,91830,91833,91837,91840,91843,91847,91850,91853,91857,91860,91863,91867,91870],[11,91648,91644],{"id":91649},"day-03-数据库100题单选-41-50填空-51-60",[15,91651,91652],{},"参考重点：教材第 1 章数据库系统概述、三级模式、数据独立性。",[19,91654,91656],{"id":91655},"_41-数据库系统的核心是","41. 数据库系统的核心是 。",[15,91658,91659],{},"A. 数据库　B. 数据库管理系统　C. 数据模型　D. 软件工具",[15,91661,58547],{},[15,91663,91664],{},"讲解：DBMS 是数据库系统的核心软件，负责定义、组织、操纵、控制和维护数据库。数据库只是被管理的数据集合。",[19,91666,91668],{"id":91667},"_42-下述关于数据库的正确叙述是","42. 下述关于数据库的正确叙述是 。",[15,91670,91671],{},"A. 数据库中只存在数据项之间的联系\nB. 数据库的数据项之间和记录之间都存在联系\nC. 数据库数据项之间无联系，记录之间存在联系\nD. 数据库数据项之间和记录之间都不存在联系",[15,91673,58547],{},[15,91675,91676],{},"讲解：数据库中的数据按照某种数据模型组织，既包括数据本身，也包括数据之间的语义联系。",[19,91678,91680],{"id":91679},"_43-数据库db数据库系统dbs和数据库管理系统dbms三者之间的关系是","43. 数据库（DB）、数据库系统（DBS）和数据库管理系统（DBMS）三者之间的关系是 。",[15,91682,91683],{},"A. DBS 包括 DB 和 DBMS　B. DBMS 包括 DB 和 DBS\nC. DB 包括 DBS 和 DBMS　D. DBS 就是 DB，也就是 DBMS",[15,91685,91106],{},[15,91687,91688],{},"讲解：DBS 是引入数据库后的计算机系统，通常包括 DB、DBMS、应用系统、DBA 和相关硬件软件环境。",[19,91690,91692],{"id":91691},"_44-可以减少相同数据重复存储的现象","44. 可以减少相同数据重复存储的现象。",[15,91694,91695],{},"A. 记录　B. 字段　C. 文件　D. 数据库",[15,91697,59391],{},[15,91699,91700],{},"讲解：数据库通过统一组织和共享数据减少冗余。文件系统中不同应用往往各自保存相同数据，容易造成重复和不一致。",[19,91702,91704],{"id":91703},"_45-数据库管理系统的工作不包括","45. 数据库管理系统的工作不包括 。",[15,91706,91707],{},"A. 定义数据库\nB. 对已定义的数据库进行管理\nC. 为已定义的数据库提供操作系统\nD. 数据通信",[15,91709,61063],{},[15,91711,91712],{},"讲解：DBMS 是操作系统之上的基础软件，不负责提供操作系统。它可以定义数据库、管理数据库，并可能提供通信和数据转换功能。",[19,91714,91716],{"id":91715},"_46-数据库管理系统中用于定义和描述数据库逻辑结构的语言称为","46. 数据库管理系统中用于定义和描述数据库逻辑结构的语言称为 。",[15,91718,91719],{},"A. 数据库模式描述语言　B. 数据库子语言　C. 数据操纵语言　D. 数据结构语言",[15,91721,91106],{},[15,91723,91724],{},"讲解：这里的“数据库模式描述语言”就是用于定义数据库模式的 DDL。DDL 用来定义模式、表、视图、索引、完整性约束等；DML 用于查询和更新数据。",[19,91726,91728],{"id":91727},"_47-数据库系统的数据独立性是指","47. 数据库系统的数据独立性是指 。",[15,91730,91731],{},"A. 不会因为数据的变化而影响应用程序\nB. 不会因为系统数据存储结构与数据逻辑结构的变化而影响应用程序\nC. 不会因为存储策略的变化而影响存储结构\nD. 不会因为某些存储结构的变化而影响其他的存储结构",[15,91733,58547],{},[15,91735,91736],{},"讲解：数据独立性包括物理独立性和逻辑独立性。三级模式结构与两级映像是实现数据独立性的主要机制。",[19,91738,91740],{"id":91739},"_48-为使程序员编程时既可使用数据库语言又可使用常规的程序设计语言数据库系统需要把数据库语言嵌入到-中","48. 为使程序员编程时既可使用数据库语言又可使用常规的程序设计语言，数据库系统需要把数据库语言嵌入到 中。",[15,91742,91743],{},"A. 编译程序　B. 操作系统　C. 中间语言　D. 宿主语言",[15,91745,59391],{},[15,91747,91748],{},"讲解：嵌入式 SQL 把 SQL 语句嵌入 C、Java 等高级语言，由预编译器或驱动接口处理。",[19,91750,91752],{"id":91751},"_49-子模式是","49. 子模式是 。",[15,91754,91755],{},"A. 模式的副本　B. 模式的逻辑子集　C. 多个模式的集合　D. 以上三者都对",[15,91757,58547],{},[15,91759,91760],{},"讲解：子模式也称外模式，面向某一类用户或应用，是数据库全局模式的一个局部逻辑视图。",[19,91762,91764],{"id":91763},"_50-数据库三级模式体系结构的划分有利于保持数据库的","50. 数据库三级模式体系结构的划分，有利于保持数据库的 。",[15,91766,91767],{},"A. 数据独立性　B. 数据安全性　C. 结构规范化　D. 操作可行性",[15,91769,91106],{},[15,91771,91772],{},"讲解：外模式\u002F模式映像保证逻辑独立性，模式\u002F内模式映像保证物理独立性。",[19,91774,91776],{"id":91775},"_51-数据管理技术经历了-和三个阶段","51. 数据管理技术经历了 ①、②和③三个阶段。",[15,91778,91779],{},"答案：①人工管理；②文件系统；③数据库系统",[15,91781,91782],{},"讲解：人工管理阶段数据不保存、无共享、无独立性；文件系统阶段有文件管理但共享弱；数据库系统阶段由 DBMS 统一管理，整体结构化和共享性显著增强。",[19,91784,91786],{"id":91785},"_52-数据库是长期存储在计算机内有-的-的数据集合","52. 数据库是长期存储在计算机内、有 ① 的、② 的数据集合。",[15,91788,91789],{},"答案：①组织；②可共享",[15,91791,91792],{},"讲解：教材定义数据库为长期存储在计算机内、有组织、可共享的大量数据集合。它按数据模型组织、描述和存储。",[19,91794,91796],{"id":91795},"_53-数据库管理系统的主要功能有-数据库的运行管理和数据库的建立以及维护等-4-个方面","53. 数据库管理系统的主要功能有 ①、②、数据库的运行管理和数据库的建立以及维护等 4 个方面。",[15,91798,91799],{},"答案：①数据定义；②数据操纵",[15,91801,91802],{},"讲解：传统概括为数据定义、数据操纵、数据库运行管理、数据库建立和维护。第 6 版还把数据组织、存储和管理作为重要功能展开说明。",[19,91804,91806],{"id":91805},"_54-数据库语言包括-和-两大部分前者负责描述和定义数据库的各种特性后者用于说明对数据进行的各种操作","54. 数据库语言包括① 和② 两大部分，前者负责描述和定义数据库的各种特性，后者用于说明对数据进行的各种操作。",[15,91808,91809],{},"答案：①数据定义语言（DDL）；②数据操纵语言（DML）",[15,91811,91812],{},"讲解：DDL 定义结构和约束，DML 查询和更新数据。SQL 同时包含 DDL、DML、DCL 等功能。",[19,91814,91816],{"id":91815},"_55-开发管理和使用数据库的人员主要有和最终用户四类相关人员","55. 开发、管理和使用数据库的人员主要有①、②、③和最终用户四类相关人员。",[15,91818,91819],{},"答案：①数据库管理员（DBA）；②系统分析员和数据库设计人员；③应用程序员",[15,91821,91822],{},"讲解：DBA 负责运行管理；分析员和设计人员负责需求分析与数据库设计；应用程序员开发应用；最终用户通过应用或查询工具使用数据库。",[19,91824,91826],{"id":91825},"_56-由-负责全面管理和控制数据库系统","56. 由 负责全面管理和控制数据库系统。",[15,91828,91829],{},"答案：数据库管理员（DBA）",[15,91831,91832],{},"讲解：DBA 的职责包括模式定义、权限管理、完整性约束、备份恢复、性能监控和日常维护。",[19,91834,91836],{"id":91835},"_57-数据库系统与文件系统的本质区别在于","57. 数据库系统与文件系统的本质区别在于 。",[15,91838,91839],{},"答案：数据库系统实现了整体数据结构化",[15,91841,91842],{},"讲解：文件系统通常只做到记录内部有结构，文件之间缺乏统一联系；数据库系统面向整个组织描述数据及其联系。",[19,91844,91846],{"id":91845},"_58-数据独立性又可分为-与","58. 数据独立性又可分为① 与 ②。",[15,91848,91849],{},"答案：①物理独立性；②逻辑独立性",[15,91851,91852],{},"讲解：物理独立性关注存储结构改变不影响应用；逻辑独立性关注全局逻辑结构改变不影响外模式和应用。",[19,91854,91856],{"id":91855},"_59-当数据的物理存储改变了应用程序不变而由-dbms-处理这种改变这是指数据的","59. 当数据的物理存储改变了，应用程序不变，而由 DBMS 处理这种改变，这是指数据的 。",[15,91858,91859],{},"答案：物理独立性",[15,91861,91862],{},"讲解：例如增加索引、改变文件组织、调整存储位置，应用程序仍按原逻辑表访问数据。",[19,91864,91866],{"id":91865},"_60-数据模型质量的高低不会影响数据库性能的好坏这句话正确否","60. 数据模型质量的高低不会影响数据库性能的好坏，这句话正确否？",[15,91868,91869],{},"答案：不正确",[15,91871,91872],{},"讲解：数据模型直接影响冗余、完整性、查询复杂度、索引设计和事务更新成本。概念结构和逻辑结构设计不好，后续物理优化很难补救。",{"title":29,"searchDepth":43,"depth":43,"links":91874},[91875,91876,91877,91878,91879,91880,91881,91882,91883,91884,91885,91886,91887,91888,91889,91890,91891,91892,91893,91894],{"id":91655,"depth":43,"text":91656},{"id":91667,"depth":43,"text":91668},{"id":91679,"depth":43,"text":91680},{"id":91691,"depth":43,"text":91692},{"id":91703,"depth":43,"text":91704},{"id":91715,"depth":43,"text":91716},{"id":91727,"depth":43,"text":91728},{"id":91739,"depth":43,"text":91740},{"id":91751,"depth":43,"text":91752},{"id":91763,"depth":43,"text":91764},{"id":91775,"depth":43,"text":91776},{"id":91785,"depth":43,"text":91786},{"id":91795,"depth":43,"text":91796},{"id":91805,"depth":43,"text":91806},{"id":91815,"depth":43,"text":91816},{"id":91825,"depth":43,"text":91826},{"id":91835,"depth":43,"text":91836},{"id":91845,"depth":43,"text":91846},{"id":91855,"depth":43,"text":91856},{"id":91865,"depth":43,"text":91866},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday03-100-41-5051-60",{"title":91644,"description":91652},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay03-数据库100题-选择41-50填空51-60","iIdR5t0uNPIt78mHm2OGKkf_323NKT2QRR71XQ33kfE",{"id":91901,"title":91902,"body":91903,"description":91910,"extension":85,"meta":92178,"navigation":52,"path":92179,"seo":92180,"stem":92181,"__hash__":92182},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay04-数据库100题-填空61-80.md","Day 04 - 数据库100题：填空 61-80",{"type":8,"value":91904,"toc":92156},[91905,91908,91911,91915,91918,91921,91925,91928,91931,91935,91938,91941,91945,91947,91950,91954,91957,91960,91964,91967,91970,91974,91986,92003,92007,92010,92013,92017,92020,92023,92027,92030,92033,92037,92040,92055,92059,92062,92065,92069,92072,92075,92079,92082,92085,92089,92092,92095,92099,92102,92105,92109,92112,92115,92119,92122,92125,92129,92132,92135,92139,92142],[11,91906,91902],{"id":91907},"day-04-数据库100题填空-61-80",[15,91909,91910],{},"参考重点：教材第 1 章数据模型，第 2 章关系模型，第 3 章 SQL，第 4 章安全性，第 5 章完整性。",[19,91912,91914],{"id":91913},"_61-数据模型是由和三部分组成的","61. 数据模型是由①、②和③三部分组成的。",[15,91916,91917],{},"答案：①数据结构；②数据操作；③完整性约束",[15,91919,91920],{},"讲解：数据结构描述对象和联系，数据操作描述允许的查询与更新，完整性约束描述数据必须满足的语义规则。",[19,91922,91924],{"id":91923},"_62-非关系模型中数据结构的基本单位是","62. 非关系模型中数据结构的基本单位是 。",[15,91926,91927],{},"答案：记录",[15,91929,91930],{},"讲解：层次模型和网状模型以记录为基本单位，并用指针或层次结构表达记录之间的联系。关系模型的基本数据结构是关系。",[19,91932,91934],{"id":91933},"_63-层次模型中根结点以外的结点至多可有-个父结点","63. 层次模型中，根结点以外的结点至多可有 个父结点。",[15,91936,91937],{},"答案：一",[15,91939,91940],{},"讲解：层次模型的数据结构是一棵树，根结点无父结点，其他结点有且只有一个父结点。因此多对多联系需要额外处理。",[19,91942,91944],{"id":91943},"_64-关系模型是将数据之间的关系看成网络关系这句话是否正确","64. 关系模型是将数据之间的关系看成网络关系，这句话是否正确？",[15,91946,91869],{},[15,91948,91949],{},"讲解：关系模型用二维表形式的关系表示数据及联系，不是网状模型的网络结构。关系模型通过关系、主键和外键表达联系。",[19,91951,91953],{"id":91952},"_65-数据库体系结构按照和三级结构进行组织","65. 数据库体系结构按照①、②和③三级结构进行组织。",[15,91955,91956],{},"答案：①外模式；②模式；③内模式",[15,91958,91959],{},"讲解：外模式是用户视图，模式是全局逻辑结构，内模式是物理存储结构。三级模式支持两级映像和数据独立性。",[19,91961,91963],{"id":91962},"_66-数据冗余可能导致的问题有","66. 数据冗余可能导致的问题有①、②。",[15,91965,91966],{},"答案：①浪费存储空间；②数据不一致",[15,91968,91969],{},"讲解：同一事实重复存储后，更新时容易只改了某些副本，造成不一致；还可能引起插入、删除、修改异常。",[19,91971,91973],{"id":91972},"_67-一个关系模式的定义格式为","67. 一个关系模式的定义格式为 。",[15,91975,58173,91976,91979,91980,4374,91983],{},[31,91977,91978],{},"R(U, D, DOM, F)","，也常简写为 ",[31,91981,91982],{},"R(U)",[31,91984,91985],{},"关系名(属性1, 属性2, ...)",[15,91987,91988,91989,91991,91992,91995,91996,91999,92000,92002],{},"讲解：形式化定义中，",[31,91990,33350],{}," 是属性集，",[31,91993,91994],{},"D"," 是属性域，",[31,91997,91998],{},"DOM"," 是属性到域的映像，",[31,92001,51834],{}," 是数据依赖集合。做题时通常简写为关系名加属性列表。",[19,92004,92006],{"id":92005},"_68-一个关系模式的定义主要包括和关键字","68. 一个关系模式的定义主要包括①、②、③、④和关键字。",[15,92008,92009],{},"答案：①关系名；②属性名；③属性域；④属性到域的映像",[15,92011,92012],{},"讲解：关系模式要说明关系叫什么、有哪些属性、每个属性取值范围是什么，以及属性与域的对应关系。关键字用于唯一标识元组。",[19,92014,92016],{"id":92015},"_69-关系数据库中可命名的最小数据单位是","69. 关系数据库中可命名的最小数据单位是 。",[15,92018,92019],{},"答案：属性",[15,92021,92022],{},"讲解：关系中每一列是属性，属性名是关系数据库中可以命名的最小数据单位。单个单元格里的属性值通常不再单独命名。",[19,92024,92026],{"id":92025},"_70-在一个实体表示的信息中称-为关键字","70. 在一个实体表示的信息中，称 为关键字。",[15,92028,92029],{},"答案：能唯一标识实体的属性或属性组",[15,92031,92032],{},"讲解：例如学生实体中，学号通常能唯一标识一个学生，所以可作为关键字。关键字可以是单属性，也可以是属性组。",[19,92034,92036],{"id":92035},"_71-sql-语言的数据定义功能包括和","71. SQL 语言的数据定义功能包括①、②、③和④。",[15,92038,92039],{},"答案：①定义模式；②定义基本表；③定义视图；④定义索引",[15,92041,92042,92043,146,92046,146,92049,146,92051,92054],{},"讲解：对应 SQL Server 中常见语句为 ",[31,92044,92045],{},"CREATE SCHEMA",[31,92047,92048],{},"CREATE TABLE",[31,92050,91047],{},[31,92052,92053],{},"CREATE INDEX","。其中索引更多是物理存取优化对象。",[19,92056,92058],{"id":92057},"_72-视图是一个虚表它是从中导出的表在数据库中只存放视图的不存放视图的","72. 视图是一个虚表，它是从①中导出的表。在数据库中，只存放视图的②，不存放视图的③。",[15,92060,92061],{},"答案：①一个或多个基本表；②定义；③对应的数据",[15,92063,92064],{},"讲解：普通视图只保存查询定义，访问视图时由 DBMS 根据定义从基本表计算结果。物化视图或索引视图是特殊情况。",[19,92066,92068],{"id":92067},"_73-关系规范化的目的是","73. 关系规范化的目的是　　　　。",[15,92070,92071],{},"答案：消除不合适的数据依赖，降低冗余，避免插入、删除和修改异常",[15,92073,92074],{},"讲解：规范化通过分解关系模式，使函数依赖更合理。常见判断路线是 1NF、2NF、3NF、BCNF。",[19,92076,92078],{"id":92077},"_74-在关系-as-sn-d-和-bd-cn-nm-中a-的主键是-sb-的主键是-d则-d-在-a-中称为","74. 在关系 A(S, SN, D) 和 B(D, CN, NM) 中，A 的主键是 S，B 的主键是 D，则 D 在 A 中称为 　　　　。",[15,92080,92081],{},"答案：外键",[15,92083,92084],{},"讲解：A 中的属性 D 引用 B 的主键 D，因此 A.D 是外键。它体现 A 和 B 之间的参照完整性。",[19,92086,92088],{"id":92087},"_75-在一个关系-r-中若每个数据项都是不可再分割的那么-r-一定属于","75. 在一个关系 R 中，若每个数据项都是不可再分割的，那么 R 一定属于　　　　　。",[15,92090,92091],{},"答案：第一范式（1NF）",[15,92093,92094],{},"讲解：1NF 要求属性值是原子的、不可再分。更高范式还要继续考察函数依赖、候选码、部分依赖和传递依赖。",[19,92096,92098],{"id":92097},"_76-数据库保护包含数据的","76. 数据库保护包含数据的　　　　　。",[15,92100,92101],{},"答案：安全性、完整性、并发控制和恢复",[15,92103,92104],{},"讲解：安全性防非法访问，完整性防语义错误，并发控制防事务交叉破坏一致性，恢复防故障造成数据丢失或不一致。",[19,92106,92108],{"id":92107},"_77-保护数据安全性的一般方法是","77. 保护数据安全性的一般方法是　　　　　。",[15,92110,92111],{},"答案：存取控制",[15,92113,92114],{},"讲解：存取控制通过用户身份鉴别、授权、权限检查、角色、视图、审计等手段限制谁能访问哪些数据、执行哪些操作。",[19,92116,92118],{"id":92117},"_78-数据的安全性是指","78. 数据的安全性是指　　　　　。",[15,92120,92121],{},"答案：保护数据库，防止不合法使用造成数据泄露、更改或破坏",[15,92123,92124],{},"讲解：安全性强调“谁能看、谁能改”。完整性强调“数据本身是否符合规则”。",[19,92126,92128],{"id":92127},"_79-安全性控制的一般方法有和视图的保护五级安全措施","79. 安全性控制的一般方法有①、②、③、④和视图的保护五级安全措施。",[15,92130,92131],{},"答案：①用户标识和鉴别；②存取控制；③审计；④数据加密",[15,92133,92134],{},"讲解：第 6 版教材把安全性技术扩展到自主存取控制、强制存取控制、视图、审计、加密等。此题按常见五类填空。",[19,92136,92138],{"id":92137},"_80-存取权限包括两方面的内容一个是另一个是","80. 存取权限包括两方面的内容，一个是①，另一个是②。",[15,92140,92141],{},"答案：①数据对象；②操作类型",[15,92143,92144,92145,92148,92149,92152,92153,347],{},"讲解：授权要同时说明“对哪个对象”和“允许什么操作”。例如允许用户对表 ",[31,92146,92147],{},"Student"," 执行 ",[31,92150,92151],{},"SELECT","，但不允许 ",[31,92154,92155],{},"UPDATE",{"title":29,"searchDepth":43,"depth":43,"links":92157},[92158,92159,92160,92161,92162,92163,92164,92165,92166,92167,92168,92169,92170,92171,92172,92173,92174,92175,92176,92177],{"id":91913,"depth":43,"text":91914},{"id":91923,"depth":43,"text":91924},{"id":91933,"depth":43,"text":91934},{"id":91943,"depth":43,"text":91944},{"id":91952,"depth":43,"text":91953},{"id":91962,"depth":43,"text":91963},{"id":91972,"depth":43,"text":91973},{"id":92005,"depth":43,"text":92006},{"id":92015,"depth":43,"text":92016},{"id":92025,"depth":43,"text":92026},{"id":92035,"depth":43,"text":92036},{"id":92057,"depth":43,"text":92058},{"id":92067,"depth":43,"text":92068},{"id":92077,"depth":43,"text":92078},{"id":92087,"depth":43,"text":92088},{"id":92097,"depth":43,"text":92098},{"id":92107,"depth":43,"text":92108},{"id":92117,"depth":43,"text":92118},{"id":92127,"depth":43,"text":92128},{"id":92137,"depth":43,"text":92138},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday04-100-61-80",{"title":91902,"description":91910},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay04-数据库100题-填空61-80","JMijYSm9BfDc9S5CnMM6kXUiCdOz1O-AaDdbXSbAQEE",{"id":92184,"title":92185,"body":92186,"description":92193,"extension":85,"meta":92431,"navigation":52,"path":92432,"seo":92433,"stem":92434,"__hash__":92435},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay05-数据库100题-填空81-100.md","Day 05 - 数据库100题：填空 81-100",{"type":8,"value":92187,"toc":92409},[92188,92191,92194,92198,92201,92204,92208,92211,92214,92218,92221,92224,92228,92231,92234,92238,92241,92244,92248,92251,92254,92258,92261,92264,92268,92271,92274,92278,92281,92284,92288,92291,92294,92298,92301,92304,92308,92311,92314,92318,92321,92324,92328,92331,92334,92338,92341,92344,92348,92351,92354,92358,92361,92364,92368,92371,92377,92381,92384,92387,92391,92394],[11,92189,92185],{"id":92190},"day-05-数据库100题填空-81-100",[15,92192,92193],{},"参考重点：教材第 5 章完整性，第 11 章恢复技术，第 12 章并发控制。",[19,92195,92197],{"id":92196},"_81-是-dbms-的基本单位它是用户定义的一组逻辑一致的程序序列","81. 是 DBMS 的基本单位，它是用户定义的一组逻辑一致的程序序列。",[15,92199,92200],{},"答案：事务",[15,92202,92203],{},"讲解：事务是数据库恢复和并发控制的基本单位，具有 ACID 特性：原子性、一致性、隔离性、持久性。",[19,92205,92207],{"id":92206},"_82-dbms-的基本工作单位是事务它是用户定义的一组逻辑一致的程序序列并发控制的主要方法是-机制","82. DBMS 的基本工作单位是事务，它是用户定义的一组逻辑一致的程序序列；并发控制的主要方法是 机制。",[15,92209,92210],{},"答案：封锁",[15,92212,92213],{},"讲解：封锁机制通过 S 锁和 X 锁控制并发读写，使并发调度满足隔离性要求。SQL Server 中锁由引擎自动管理，也可用锁提示影响行为。",[19,92215,92217],{"id":92216},"_83-有两种基本类型的锁它们是和","83. 有两种基本类型的锁，它们是①和②。",[15,92219,92220],{},"答案：①共享锁（S 锁）；②排他锁（X 锁）",[15,92222,92223],{},"讲解：S 锁用于读，多个事务可以同时持有；X 锁用于写，与其他 S 锁和 X 锁都不相容。",[19,92225,92227],{"id":92226},"_84-如果数据库中只包含成功事务提交的结果就说数据库处于-状态","84. 如果数据库中只包含成功事务提交的结果，就说数据库处于 状态。",[15,92229,92230],{},"答案：一致性",[15,92232,92233],{},"讲解：数据库一致性要求所有完整性约束都被满足，且不含失败或未提交事务的残留结果。",[19,92235,92237],{"id":92236},"_85-对并发操作若不加以控制可能带来的不一致性有和和","85. 对并发操作若不加以控制，可能带来的不一致性有①和②和③。",[15,92239,92240],{},"答案：①丢失修改；②读脏数据；③不可重复读",[15,92242,92243],{},"讲解：丢失修改是两个事务覆盖彼此结果；读脏数据是读取了未提交且后来撤销的数据；不可重复读是同一事务两次读取同一数据得到不同结果。教材中有时也把不一致分析列为典型问题。",[19,92245,92247],{"id":92246},"_86-并发控制是对用户的-加以控制和协调","86. 并发控制是对用户的 加以控制和协调。",[15,92249,92250],{},"答案：并发操作",[15,92252,92253],{},"讲解：并发控制不是禁止并发，而是在保持隔离性和一致性的前提下提高吞吐量和资源利用率。",[19,92255,92257],{"id":92256},"_87-并发控制的主要方法是采用机制其类型有和两种","87. 并发控制的主要方法是采用①机制，其类型有②和③两种。",[15,92259,92260],{},"答案：①封锁；②共享锁；③排他锁",[15,92262,92263],{},"讲解：共享锁读读相容，排他锁用于更新。两段锁协议建立在这些锁类型之上，可保证冲突可串行化。",[19,92265,92267],{"id":92266},"_88-若事务-t-对数据对象-a-加了-s-锁则其他事务只能对数据-a-再加不能加直到事务-t-释放-a-上的锁","88. 若事务 T 对数据对象 A 加了 S 锁，则其他事务只能对数据 A 再加①，不能加②，直到事务 T 释放 A 上的锁。",[15,92269,92270],{},"答案：①S 锁；②X 锁",[15,92272,92273],{},"讲解：S 锁之间相容，所以其他事务仍可读；但 X 锁不相容，所以其他事务不能写。",[19,92275,92277],{"id":92276},"_89-若事务在运行过程中由于种种原因使事务未运行到正常终止点之前就被撤消这种情况称为","89. 若事务在运行过程中，由于种种原因，使事务未运行到正常终止点之前就被撤消，这种情况称为 。",[15,92279,92280],{},"答案：事务故障",[15,92282,92283],{},"讲解：事务故障只影响该事务本身，恢复时需要 UNDO 该事务已经执行的更新。",[19,92285,92287],{"id":92286},"_90-数据库恢复是将数据库从状态恢复到的过程","90. 数据库恢复是将数据库从①状态恢复到②的过程。",[15,92289,92290],{},"答案：①故障状态；②一致性状态",[15,92292,92293],{},"讲解：恢复技术的目标是保证事务的原子性和持久性，使数据库故障后仍能回到正确、一致的状态。",[19,92295,92297],{"id":92296},"_91-系统在运行过程中由于某种原因造成系统停止运行事务以非控制方式终止内存信息丢失而外存数据未破坏这种情况称为","91. 系统在运行过程中，由于某种原因造成系统停止运行，事务以非控制方式终止，内存信息丢失而外存数据未破坏，这种情况称为 。",[15,92299,92300],{},"答案：系统故障",[15,92302,92303],{},"讲解：系统故障也称软故障，如断电、操作系统崩溃。恢复时通常根据日志从检查点开始 REDO 已提交事务、UNDO 未提交事务。",[19,92305,92307],{"id":92306},"_92-系统在运行过程中由于某种硬件故障使存储在外存上的数据部分损失或全部损失这种情况称为","92. 系统在运行过程中，由于某种硬件故障，使存储在外存上的数据部分损失或全部损失，这种情况称为 。",[15,92309,92310],{},"答案：介质故障",[15,92312,92313],{},"讲解：介质故障也称硬故障，例如磁盘损坏。恢复时需要装入数据库后援副本，再利用日志重做备份以来已提交的事务。",[19,92315,92317],{"id":92316},"_93-数据库系统在运行过程中可能会发生故障故障主要有介质故障和四类","93. 数据库系统在运行过程中，可能会发生故障。故障主要有①、②、介质故障和③四类。",[15,92319,92320],{},"答案：①事务故障；②系统故障；③计算机病毒或恶意破坏",[15,92322,92323],{},"讲解：教材常把故障分为事务内部故障、系统故障、介质故障和计算机病毒等。不同故障的恢复粒度和责任主体不同。",[19,92325,92327],{"id":92326},"_94-数据库系统是利用存储在外存上其他地方的来重建被破坏的数据库它主要有两种和","94. 数据库系统是利用存储在外存上其他地方的①来重建被破坏的数据库。它主要有两种：②和③。",[15,92329,92330],{},"答案：①冗余数据；②数据转储；③登记日志文件",[15,92332,92333],{},"讲解：冗余是恢复的基础。转储提供数据库副本，日志记录事务更新过程，两者结合才能恢复到最近一致状态。",[19,92335,92337],{"id":92336},"_95-制作后援副本的过程称为它又分为和","95. 制作后援副本的过程称为①。它又分为②和③。",[15,92339,92340],{},"答案：①数据转储；②静态转储；③动态转储",[15,92342,92343],{},"讲解：静态转储期间数据库不能更新；动态转储允许事务并发执行，但必须配合日志保证副本可恢复。",[19,92345,92347],{"id":92346},"_96-事务故障系统故障的恢复是由完成的介质故障是由完成的","96. 事务故障、系统故障的恢复是由①完成的，介质故障是由②完成的。",[15,92349,92350],{},"答案：①系统自动；②数据库管理员配合系统工具",[15,92352,92353],{},"讲解：事务故障和系统故障通常由 DBMS 根据日志自动处理；介质故障需要先恢复备份介质，再应用日志，常需要 DBA 介入。",[19,92355,92357],{"id":92356},"_97-数据库的完整性是指数据的和","97. 数据库的完整性是指数据的①和②。",[15,92359,92360],{},"答案：①正确性；②相容性",[15,92362,92363],{},"讲解：更完整地说，完整性包括正确性、有效性和相容性。正确性强调数据值合法，有效性强调符合现实语义，相容性强调同一事实在不同位置不矛盾。",[19,92365,92367],{"id":92366},"_98-完整性约束是指和","98. 完整性约束是指①和②。",[15,92369,92370],{},"答案：①实体完整性约束；②参照完整性约束",[15,92372,92373,92374,347],{},"讲解：本题按两空常见答案填写。复习时还要记住第 6 版教材强调的第三类：用户定义完整性，例如 ",[31,92375,92376],{},"CHECK (分数 BETWEEN 0 AND 100)",[19,92378,92380],{"id":92379},"_99-为了保护数据库的实体完整性当用户程序对主码进行更新使主码值不惟一时dbms","99. 为了保护数据库的实体完整性，当用户程序对主码进行更新使主码值不惟一时，DBMS 。",[15,92382,92383],{},"答案：拒绝该操作",[15,92385,92386],{},"讲解：实体完整性要求主码唯一且非空。若插入或更新导致主码重复，DBMS 应进行违约处理，通常直接拒绝执行。",[19,92388,92390],{"id":92389},"_100-在数据库系统中对存取权限的定义称为","100. 在数据库系统中对存取权限的定义称为 。",[15,92392,92393],{},"答案：授权",[15,92395,92396,92397,92400,92401,92404,92405,92408],{},"讲解：授权定义用户或角色对数据对象的操作权限。SQL 中常用 ",[31,92398,92399],{},"GRANT"," 授权，",[31,92402,92403],{},"REVOKE"," 收回权限，SQL Server 还支持 ",[31,92406,92407],{},"DENY"," 显式拒绝。",{"title":29,"searchDepth":43,"depth":43,"links":92410},[92411,92412,92413,92414,92415,92416,92417,92418,92419,92420,92421,92422,92423,92424,92425,92426,92427,92428,92429,92430],{"id":92196,"depth":43,"text":92197},{"id":92206,"depth":43,"text":92207},{"id":92216,"depth":43,"text":92217},{"id":92226,"depth":43,"text":92227},{"id":92236,"depth":43,"text":92237},{"id":92246,"depth":43,"text":92247},{"id":92256,"depth":43,"text":92257},{"id":92266,"depth":43,"text":92267},{"id":92276,"depth":43,"text":92277},{"id":92286,"depth":43,"text":92287},{"id":92296,"depth":43,"text":92297},{"id":92306,"depth":43,"text":92307},{"id":92316,"depth":43,"text":92317},{"id":92326,"depth":43,"text":92327},{"id":92336,"depth":43,"text":92337},{"id":92346,"depth":43,"text":92347},{"id":92356,"depth":43,"text":92357},{"id":92366,"depth":43,"text":92367},{"id":92379,"depth":43,"text":92380},{"id":92389,"depth":43,"text":92390},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday05-100-81-100",{"title":92185,"description":92193},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay05-数据库100题-填空81-100","gq-tZQiWk2Sitd7InhIMJ1vGFrjn65aSqdu-ZjiTMIE",{"id":92437,"title":92438,"body":92439,"description":92446,"extension":85,"meta":93922,"navigation":52,"path":93923,"seo":93924,"stem":93925,"__hash__":93926},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay06-数据库70题-试题01-11.md","Day 06 - 数据库70题：试题 1-11",{"type":8,"value":92440,"toc":93887},[92441,92444,92447,92451,92453,92456,92473,92476,92480,92482,92489,92492,92496,92499,92525,92575,92582,92749,92769,92773,92798,92802,92822,92831,92835,92855,92859,92894,92908,92912,92942,92946,93015,93018,93022,93083,93086,93090,93155,93161,93165,93189,93193,93225,93242,93246,93280,93287,93298,93300,93348,93370,93381,93383,93421,93442,93450,93453,93455,93494,93504,93512,93514,93554,93557,93568,93574,93604,93611,93614,93625,93627,93698,93705,93713,93715,93782,93785,93793,93795,93878,93885],[11,92442,92438],{"id":92443},"day-06-数据库70题试题-1-11",[15,92445,92446],{},"参考重点：教材第 3 章 SQL、第 6 章关系数据理论。SQL 均按 SQL Server 写法。",[19,92448,92450],{"id":92449},"_1-简述在-sql-中使用视图的优点","1. 简述在 SQL 中使用视图的优点。",[15,92452,58173],{},[15,92454,92455],{},"视图的主要优点有：",[2230,92457,92458,92461,92464,92467,92470],{},[108,92459,92460],{},"简化用户操作，把复杂查询封装成一个虚表。",[108,92462,92463],{},"提高数据安全性，让用户只能看到授权范围内的列和行。",[108,92465,92466],{},"提供一定的逻辑独立性，基本表结构局部变化时可通过修改视图屏蔽影响。",[108,92468,92469],{},"使同一数据能按不同用户需求呈现不同外模式。",[108,92471,92472],{},"便于复用常用查询逻辑，统一统计口径。",[15,92474,92475],{},"讲解：视图不一定保存数据，普通视图保存的是查询定义。教材第 3 章把视图作为 SQL 的重要对象，重点强调“虚表”“简化查询”“安全控制”“逻辑独立性”。",[19,92477,92479],{"id":92478},"_2-简述游标的作用","2. 简述游标的作用。",[15,92481,58173],{},[15,92483,92484,92485,92488],{},"游标用于在过程化程序中逐行处理 SQL 查询结果集。它可以打开一个结果集，定位当前行，逐行 ",[31,92486,92487],{},"FETCH","，并可结合当前行进行更新或删除。",[15,92490,92491],{},"讲解：SQL 本身是面向集合的语言，但宿主语言或存储过程有时需要逐行处理，例如复杂业务校验、逐行调用过程。游标就是集合操作和逐行过程逻辑之间的桥。能用集合 SQL 解决时，SQL Server 中通常优先使用集合操作，游标只在确实需要逐行处理时使用。",[19,92493,92495],{"id":92494},"_3-对于下列四个关系模式试分别用-sql-语句表示各类查询","3. 对于下列四个关系模式，试分别用 SQL 语句表示各类查询。",[15,92497,92498],{},"题面：",[105,92500,92501,92507,92513,92519],{},[108,92502,92503,92504],{},"供应商关系：",[31,92505,92506],{},"SUPPLIERS(SNUMBER, SNAME, STATUS, SCITY)",[108,92508,92509,92510],{},"零件关系：",[31,92511,92512],{},"PARTS(PNUMBER, PNAME, COLOR, WEIGHT)",[108,92514,92515,92516],{},"工程项目关系：",[31,92517,92518],{},"JOBS(JNUMBER, JNAME, JCITY)",[108,92520,92521,92522],{},"供应关系：",[31,92523,92524],{},"SPJ(SNUMBER, PNUMBER, JNUMBER, QUANTITIES)",[15,92526,92527,92528,92531,92532,92535,92536,92539,92540,92543,92544,92547,92548,92551,92552,92555,92556,92559,92560,92563,92564,92567,92568,92543,92571,92574],{},"属性含义：",[31,92529,92530],{},"SNUMBER"," 供应商编号，",[31,92533,92534],{},"SNAME"," 供应商姓名，",[31,92537,92538],{},"STATUS"," 状态，",[31,92541,92542],{},"SCITY"," 城市，",[31,92545,92546],{},"PNUMBER"," 零件编号，",[31,92549,92550],{},"PNAME"," 零件名，",[31,92553,92554],{},"COLOR"," 颜色，",[31,92557,92558],{},"WEIGHT"," 重量，",[31,92561,92562],{},"JNUMBER"," 工程项目编号，",[31,92565,92566],{},"JNAME"," 工程项目名，",[31,92569,92570],{},"JCITY",[31,92572,92573],{},"QUANTITIES"," 数量。",[253,92576,92578,92579,92581],{"id":92577},"_31-用-create-table-建立上述四张表并指出主键和外键","3.1 用 ",[31,92580,92048],{}," 建立上述四张表，并指出主键和外键。",[24,92583,92587],{"className":92584,"code":92585,"language":92586,"meta":29,"style":29},"language-sql shiki shiki-themes github-light github-dark","CREATE TABLE dbo.Suppliers (\n SNumber varchar(20) NOT NULL,\n SName nvarchar(50) NOT NULL,\n [Status] int NULL,\n SCity nvarchar(50) NULL,\n CONSTRAINT PK_Suppliers PRIMARY KEY (SNumber)\n);\n\nCREATE TABLE dbo.Parts (\n PNumber varchar(20) NOT NULL,\n PName nvarchar(50) NOT NULL,\n Color nvarchar(20) NULL,\n [Weight] decimal(10, 2) NULL,\n CONSTRAINT PK_Parts PRIMARY KEY (PNumber)\n);\n\nCREATE TABLE dbo.Jobs (\n JNumber varchar(20) NOT NULL,\n JName nvarchar(50) NOT NULL,\n JCity nvarchar(50) NULL,\n CONSTRAINT PK_Jobs PRIMARY KEY (JNumber)\n);\n\nCREATE TABLE dbo.SPJ (\n SNumber varchar(20) NOT NULL,\n PNumber varchar(20) NOT NULL,\n JNumber varchar(20) NOT NULL,\n Quantities int NOT NULL,\n CONSTRAINT PK_SPJ PRIMARY KEY (SNumber, PNumber, JNumber),\n CONSTRAINT FK_SPJ_Suppliers FOREIGN KEY (SNumber) REFERENCES dbo.Suppliers(SNumber),\n CONSTRAINT FK_SPJ_Parts FOREIGN KEY (PNumber) REFERENCES dbo.Parts(PNumber),\n CONSTRAINT FK_SPJ_Jobs FOREIGN KEY (JNumber) REFERENCES dbo.Jobs(JNumber),\n CONSTRAINT CK_SPJ_Quantities CHECK (Quantities >= 0)\n);\n","sql",[31,92588,92589,92594,92599,92604,92609,92614,92619,92623,92627,92632,92637,92642,92647,92652,92657,92661,92665,92670,92675,92680,92685,92690,92694,92698,92703,92707,92711,92715,92720,92725,92730,92735,92740,92745],{"__ignoreMap":29},[34,92590,92591],{"class":36,"line":37},[34,92592,92593],{},"CREATE TABLE dbo.Suppliers (\n",[34,92595,92596],{"class":36,"line":43},[34,92597,92598],{}," SNumber varchar(20) NOT NULL,\n",[34,92600,92601],{"class":36,"line":49},[34,92602,92603],{}," SName nvarchar(50) NOT NULL,\n",[34,92605,92606],{"class":36,"line":56},[34,92607,92608],{}," [Status] int NULL,\n",[34,92610,92611],{"class":36,"line":62},[34,92612,92613],{}," SCity nvarchar(50) NULL,\n",[34,92615,92616],{"class":36,"line":68},[34,92617,92618],{}," CONSTRAINT PK_Suppliers PRIMARY KEY (SNumber)\n",[34,92620,92621],{"class":36,"line":74},[34,92622,462],{},[34,92624,92625],{"class":36,"line":1971},[34,92626,53],{"emptyLinePlaceholder":52},[34,92628,92629],{"class":36,"line":1976},[34,92630,92631],{},"CREATE TABLE dbo.Parts (\n",[34,92633,92634],{"class":36,"line":1985},[34,92635,92636],{}," PNumber varchar(20) NOT NULL,\n",[34,92638,92639],{"class":36,"line":1990},[34,92640,92641],{}," PName nvarchar(50) NOT NULL,\n",[34,92643,92644],{"class":36,"line":1999},[34,92645,92646],{}," Color nvarchar(20) NULL,\n",[34,92648,92649],{"class":36,"line":2022},[34,92650,92651],{}," [Weight] decimal(10, 2) NULL,\n",[34,92653,92654],{"class":36,"line":2036},[34,92655,92656],{}," CONSTRAINT PK_Parts PRIMARY KEY (PNumber)\n",[34,92658,92659],{"class":36,"line":2603},[34,92660,462],{},[34,92662,92663],{"class":36,"line":5537},[34,92664,53],{"emptyLinePlaceholder":52},[34,92666,92667],{"class":36,"line":5543},[34,92668,92669],{},"CREATE TABLE dbo.Jobs (\n",[34,92671,92672],{"class":36,"line":5548},[34,92673,92674],{}," JNumber varchar(20) NOT NULL,\n",[34,92676,92677],{"class":36,"line":5564},[34,92678,92679],{}," JName nvarchar(50) NOT NULL,\n",[34,92681,92682],{"class":36,"line":5576},[34,92683,92684],{}," JCity nvarchar(50) NULL,\n",[34,92686,92687],{"class":36,"line":5606},[34,92688,92689],{}," CONSTRAINT PK_Jobs PRIMARY KEY (JNumber)\n",[34,92691,92692],{"class":36,"line":5616},[34,92693,462],{},[34,92695,92696],{"class":36,"line":5628},[34,92697,53],{"emptyLinePlaceholder":52},[34,92699,92700],{"class":36,"line":5634},[34,92701,92702],{},"CREATE TABLE dbo.SPJ (\n",[34,92704,92705],{"class":36,"line":6420},[34,92706,92598],{},[34,92708,92709],{"class":36,"line":11294},[34,92710,92636],{},[34,92712,92713],{"class":36,"line":11321},[34,92714,92674],{},[34,92716,92717],{"class":36,"line":11326},[34,92718,92719],{}," Quantities int NOT NULL,\n",[34,92721,92722],{"class":36,"line":11333},[34,92723,92724],{}," CONSTRAINT PK_SPJ PRIMARY KEY (SNumber, PNumber, JNumber),\n",[34,92726,92727],{"class":36,"line":11344},[34,92728,92729],{}," CONSTRAINT FK_SPJ_Suppliers FOREIGN KEY (SNumber) REFERENCES dbo.Suppliers(SNumber),\n",[34,92731,92732],{"class":36,"line":11349},[34,92733,92734],{}," CONSTRAINT FK_SPJ_Parts FOREIGN KEY (PNumber) REFERENCES dbo.Parts(PNumber),\n",[34,92736,92737],{"class":36,"line":11354},[34,92738,92739],{}," CONSTRAINT FK_SPJ_Jobs FOREIGN KEY (JNumber) REFERENCES dbo.Jobs(JNumber),\n",[34,92741,92742],{"class":36,"line":11363},[34,92743,92744],{}," CONSTRAINT CK_SPJ_Quantities CHECK (Quantities >= 0)\n",[34,92746,92747],{"class":36,"line":11368},[34,92748,462],{},[15,92750,91513,92751,146,92754,146,92757,92760,92761,92764,92765,92768],{},[31,92752,92753],{},"Suppliers",[31,92755,92756],{},"Parts",[31,92758,92759],{},"Jobs"," 的主键分别是各自编号。",[31,92762,92763],{},"SPJ"," 表描述三者联系，常用 ",[31,92766,92767],{},"(SNumber, PNumber, JNumber)"," 作为组合主键，并分别引用三张实体表。",[253,92770,92772],{"id":92771},"_32-检索供应工程-j1-零件-p1-的供应商编号","3.2 检索供应工程 J1 零件 P1 的供应商编号。",[24,92774,92776],{"className":92584,"code":92775,"language":92586,"meta":29,"style":29},"SELECT SNumber\nFROM dbo.SPJ\nWHERE JNumber = 'J1'\n AND PNumber = 'P1';\n",[31,92777,92778,92783,92788,92793],{"__ignoreMap":29},[34,92779,92780],{"class":36,"line":37},[34,92781,92782],{},"SELECT SNumber\n",[34,92784,92785],{"class":36,"line":43},[34,92786,92787],{},"FROM dbo.SPJ\n",[34,92789,92790],{"class":36,"line":49},[34,92791,92792],{},"WHERE JNumber = 'J1'\n",[34,92794,92795],{"class":36,"line":56},[34,92796,92797],{}," AND PNumber = 'P1';\n",[253,92799,92801],{"id":92800},"_33-检索零件重量在-16-19-之间的零件的全部信息","3.3 检索零件重量在 16-19 之间的零件的全部信息。",[24,92803,92805],{"className":92584,"code":92804,"language":92586,"meta":29,"style":29},"SELECT *\nFROM dbo.Parts\nWHERE [Weight] BETWEEN 16 AND 19;\n",[31,92806,92807,92812,92817],{"__ignoreMap":29},[34,92808,92809],{"class":36,"line":37},[34,92810,92811],{},"SELECT *\n",[34,92813,92814],{"class":36,"line":43},[34,92815,92816],{},"FROM dbo.Parts\n",[34,92818,92819],{"class":36,"line":49},[34,92820,92821],{},"WHERE [Weight] BETWEEN 16 AND 19;\n",[15,92823,91513,92824,92827,92828,347],{},[31,92825,92826],{},"BETWEEN 16 AND 19"," 包含边界，相当于 ",[31,92829,92830],{},"Weight >= 16 AND Weight \u003C= 19",[253,92832,92834],{"id":92833},"_34-检索上海的供应商姓名","3.4 检索上海的供应商姓名。",[24,92836,92838],{"className":92584,"code":92837,"language":92586,"meta":29,"style":29},"SELECT SName\nFROM dbo.Suppliers\nWHERE SCity = N'上海';\n",[31,92839,92840,92845,92850],{"__ignoreMap":29},[34,92841,92842],{"class":36,"line":37},[34,92843,92844],{},"SELECT SName\n",[34,92846,92847],{"class":36,"line":43},[34,92848,92849],{},"FROM dbo.Suppliers\n",[34,92851,92852],{"class":36,"line":49},[34,92853,92854],{},"WHERE SCity = N'上海';\n",[253,92856,92858],{"id":92857},"_35-检索住在同一个城市的一对供应商编号","3.5 检索住在同一个城市的一对供应商编号。",[24,92860,92862],{"className":92584,"code":92861,"language":92586,"meta":29,"style":29},"SELECT s1.SNumber AS SNumber1,\n s2.SNumber AS SNumber2\nFROM dbo.Suppliers AS s1\nJOIN dbo.Suppliers AS s2\n ON s1.SCity = s2.SCity\n AND s1.SNumber \u003C s2.SNumber;\n",[31,92863,92864,92869,92874,92879,92884,92889],{"__ignoreMap":29},[34,92865,92866],{"class":36,"line":37},[34,92867,92868],{},"SELECT s1.SNumber AS SNumber1,\n",[34,92870,92871],{"class":36,"line":43},[34,92872,92873],{}," s2.SNumber AS SNumber2\n",[34,92875,92876],{"class":36,"line":49},[34,92877,92878],{},"FROM dbo.Suppliers AS s1\n",[34,92880,92881],{"class":36,"line":56},[34,92882,92883],{},"JOIN dbo.Suppliers AS s2\n",[34,92885,92886],{"class":36,"line":62},[34,92887,92888],{}," ON s1.SCity = s2.SCity\n",[34,92890,92891],{"class":36,"line":68},[34,92892,92893],{}," AND s1.SNumber \u003C s2.SNumber;\n",[15,92895,92896,92897,92900,92901,43298,92904,92907],{},"讲解：自连接用于比较同一张表中的不同行。",[31,92898,92899],{},"s1.SNumber \u003C s2.SNumber"," 避免自己和自己配对，也避免 ",[31,92902,92903],{},"(S1,S2)",[31,92905,92906],{},"(S2,S1)"," 重复。",[253,92909,92911],{"id":92910},"_36-检索供应商-s1-提供零件的工程项目名称","3.6 检索供应商 S1 提供零件的工程项目名称。",[24,92913,92915],{"className":92584,"code":92914,"language":92586,"meta":29,"style":29},"SELECT DISTINCT j.JName\nFROM dbo.SPJ AS spj\nJOIN dbo.Jobs AS j\n ON j.JNumber = spj.JNumber\nWHERE spj.SNumber = 'S1';\n",[31,92916,92917,92922,92927,92932,92937],{"__ignoreMap":29},[34,92918,92919],{"class":36,"line":37},[34,92920,92921],{},"SELECT DISTINCT j.JName\n",[34,92923,92924],{"class":36,"line":43},[34,92925,92926],{},"FROM dbo.SPJ AS spj\n",[34,92928,92929],{"class":36,"line":49},[34,92930,92931],{},"JOIN dbo.Jobs AS j\n",[34,92933,92934],{"class":36,"line":56},[34,92935,92936],{}," ON j.JNumber = spj.JNumber\n",[34,92938,92939],{"class":36,"line":62},[34,92940,92941],{},"WHERE spj.SNumber = 'S1';\n",[253,92943,92945],{"id":92944},"_37-求没有使用天津生产的红色零件的工程项目号","3.7 求没有使用天津生产的红色零件的工程项目号。",[24,92947,92949],{"className":92584,"code":92948,"language":92586,"meta":29,"style":29},"SELECT j.JNumber\nFROM dbo.Jobs AS j\nWHERE NOT EXISTS (\n SELECT 1\n FROM dbo.SPJ AS spj\n JOIN dbo.Suppliers AS s\n ON s.SNumber = spj.SNumber\n JOIN dbo.Parts AS p\n ON p.PNumber = spj.PNumber\n WHERE spj.JNumber = j.JNumber\n AND s.SCity = N'天津'\n AND p.Color = N'红色'\n);\n",[31,92950,92951,92956,92961,92966,92971,92976,92981,92986,92991,92996,93001,93006,93011],{"__ignoreMap":29},[34,92952,92953],{"class":36,"line":37},[34,92954,92955],{},"SELECT j.JNumber\n",[34,92957,92958],{"class":36,"line":43},[34,92959,92960],{},"FROM dbo.Jobs AS j\n",[34,92962,92963],{"class":36,"line":49},[34,92964,92965],{},"WHERE NOT EXISTS (\n",[34,92967,92968],{"class":36,"line":56},[34,92969,92970],{}," SELECT 1\n",[34,92972,92973],{"class":36,"line":62},[34,92974,92975],{}," FROM dbo.SPJ AS spj\n",[34,92977,92978],{"class":36,"line":68},[34,92979,92980],{}," JOIN dbo.Suppliers AS s\n",[34,92982,92983],{"class":36,"line":74},[34,92984,92985],{}," ON s.SNumber = spj.SNumber\n",[34,92987,92988],{"class":36,"line":1971},[34,92989,92990],{}," JOIN dbo.Parts AS p\n",[34,92992,92993],{"class":36,"line":1976},[34,92994,92995],{}," ON p.PNumber = spj.PNumber\n",[34,92997,92998],{"class":36,"line":1985},[34,92999,93000],{}," WHERE spj.JNumber = j.JNumber\n",[34,93002,93003],{"class":36,"line":1990},[34,93004,93005],{}," AND s.SCity = N'天津'\n",[34,93007,93008],{"class":36,"line":1999},[34,93009,93010],{}," AND p.Color = N'红色'\n",[34,93012,93013],{"class":36,"line":2022},[34,93014,462],{},[15,93016,93017],{},"讲解：题面中“天津生产”没有单独的零件产地属性，只能按供应商所在城市理解为“天津供应商供应的红色零件”。双重语义题优先写清假设。",[253,93019,93021],{"id":93020},"_38-求使用全部零件的工程项目名称","3.8 求使用全部零件的工程项目名称。",[24,93023,93025],{"className":92584,"code":93024,"language":92586,"meta":29,"style":29},"SELECT j.JName\nFROM dbo.Jobs AS j\nWHERE NOT EXISTS (\n SELECT 1\n FROM dbo.Parts AS p\n WHERE NOT EXISTS (\n SELECT 1\n FROM dbo.SPJ AS spj\n WHERE spj.JNumber = j.JNumber\n AND spj.PNumber = p.PNumber\n )\n);\n",[31,93026,93027,93032,93036,93040,93044,93049,93054,93059,93064,93069,93074,93079],{"__ignoreMap":29},[34,93028,93029],{"class":36,"line":37},[34,93030,93031],{},"SELECT j.JName\n",[34,93033,93034],{"class":36,"line":43},[34,93035,92960],{},[34,93037,93038],{"class":36,"line":49},[34,93039,92965],{},[34,93041,93042],{"class":36,"line":56},[34,93043,92970],{},[34,93045,93046],{"class":36,"line":62},[34,93047,93048],{}," FROM dbo.Parts AS p\n",[34,93050,93051],{"class":36,"line":68},[34,93052,93053],{}," WHERE NOT EXISTS (\n",[34,93055,93056],{"class":36,"line":74},[34,93057,93058],{}," SELECT 1\n",[34,93060,93061],{"class":36,"line":1971},[34,93062,93063],{}," FROM dbo.SPJ AS spj\n",[34,93065,93066],{"class":36,"line":1976},[34,93067,93068],{}," WHERE spj.JNumber = j.JNumber\n",[34,93070,93071],{"class":36,"line":1985},[34,93072,93073],{}," AND spj.PNumber = p.PNumber\n",[34,93075,93076],{"class":36,"line":1990},[34,93077,93078],{}," )\n",[34,93080,93081],{"class":36,"line":1999},[34,93082,462],{},[15,93084,93085],{},"讲解：这是关系除法的 SQL 写法。外层意思是“不存在某个零件，使得该工程没有使用它”。",[253,93087,93089],{"id":93088},"_39-求使用供应商-s1-所提供的全部零件的工程项目号","3.9 求使用供应商 S1 所提供的全部零件的工程项目号。",[24,93091,93093],{"className":92584,"code":93092,"language":92586,"meta":29,"style":29},"SELECT j.JNumber\nFROM dbo.Jobs AS j\nWHERE NOT EXISTS (\n SELECT 1\n FROM dbo.SPJ AS s1parts\n WHERE s1parts.SNumber = 'S1'\n AND NOT EXISTS (\n SELECT 1\n FROM dbo.SPJ AS spj\n WHERE spj.JNumber = j.JNumber\n AND spj.PNumber = s1parts.PNumber\n )\n);\n",[31,93094,93095,93099,93103,93107,93111,93116,93121,93126,93131,93136,93141,93146,93151],{"__ignoreMap":29},[34,93096,93097],{"class":36,"line":37},[34,93098,92955],{},[34,93100,93101],{"class":36,"line":43},[34,93102,92960],{},[34,93104,93105],{"class":36,"line":49},[34,93106,92965],{},[34,93108,93109],{"class":36,"line":56},[34,93110,92970],{},[34,93112,93113],{"class":36,"line":62},[34,93114,93115],{}," FROM dbo.SPJ AS s1parts\n",[34,93117,93118],{"class":36,"line":68},[34,93119,93120],{}," WHERE s1parts.SNumber = 'S1'\n",[34,93122,93123],{"class":36,"line":74},[34,93124,93125],{}," AND NOT EXISTS (\n",[34,93127,93128],{"class":36,"line":1971},[34,93129,93130],{}," SELECT 1\n",[34,93132,93133],{"class":36,"line":1976},[34,93134,93135],{}," FROM dbo.SPJ AS spj\n",[34,93137,93138],{"class":36,"line":1985},[34,93139,93140],{}," WHERE spj.JNumber = j.JNumber\n",[34,93142,93143],{"class":36,"line":1990},[34,93144,93145],{}," AND spj.PNumber = s1parts.PNumber\n",[34,93147,93148],{"class":36,"line":1999},[34,93149,93150],{}," )\n",[34,93152,93153],{"class":36,"line":2022},[34,93154,462],{},[15,93156,93157,93158,93160],{},"讲解：这里“全部零件”的范围不是 ",[31,93159,92756],{}," 表的所有零件，而是供应商 S1 曾经供应过的零件集合。",[253,93162,93164],{"id":93163},"_310-求由一个以上供应者提供的零件号","3.10 求由一个以上供应者提供的零件号。",[24,93166,93168],{"className":92584,"code":93167,"language":92586,"meta":29,"style":29},"SELECT PNumber\nFROM dbo.SPJ\nGROUP BY PNumber\nHAVING COUNT(DISTINCT SNumber) > 1;\n",[31,93169,93170,93175,93179,93184],{"__ignoreMap":29},[34,93171,93172],{"class":36,"line":37},[34,93173,93174],{},"SELECT PNumber\n",[34,93176,93177],{"class":36,"line":43},[34,93178,92787],{},[34,93180,93181],{"class":36,"line":49},[34,93182,93183],{},"GROUP BY PNumber\n",[34,93185,93186],{"class":36,"line":56},[34,93187,93188],{},"HAVING COUNT(DISTINCT SNumber) > 1;\n",[253,93190,93192],{"id":93191},"_311-将工程名为环保的所有供应数量提高-10","3.11 将工程名为“环保”的所有供应数量提高 10%。",[24,93194,93196],{"className":92584,"code":93195,"language":92586,"meta":29,"style":29},"UPDATE spj\nSET Quantities = CAST(ROUND(Quantities * 1.10, 0) AS int)\nFROM dbo.SPJ AS spj\nJOIN dbo.Jobs AS j\n ON j.JNumber = spj.JNumber\nWHERE j.JName = N'环保';\n",[31,93197,93198,93203,93208,93212,93216,93220],{"__ignoreMap":29},[34,93199,93200],{"class":36,"line":37},[34,93201,93202],{},"UPDATE spj\n",[34,93204,93205],{"class":36,"line":43},[34,93206,93207],{},"SET Quantities = CAST(ROUND(Quantities * 1.10, 0) AS int)\n",[34,93209,93210],{"class":36,"line":49},[34,93211,92926],{},[34,93213,93214],{"class":36,"line":56},[34,93215,92931],{},[34,93217,93218],{"class":36,"line":62},[34,93219,92936],{},[34,93221,93222],{"class":36,"line":68},[34,93223,93224],{},"WHERE j.JName = N'环保';\n",[15,93226,93227,93228,93231,93232,93235,93236,258,93239,347],{},"讲解：SQL Server 的多表更新常用 ",[31,93229,93230],{},"UPDATE 别名 SET ... FROM ... JOIN ...","。如果 ",[31,93233,93234],{},"Quantities"," 允许小数，可去掉 ",[31,93237,93238],{},"ROUND",[31,93240,93241],{},"CAST",[253,93243,93245],{"id":93244},"_312-建立一个视图包括每个工程项目使用零件的总数","3.12 建立一个视图，包括每个工程项目使用零件的总数。",[24,93247,93249],{"className":92584,"code":93248,"language":92586,"meta":29,"style":29},"CREATE VIEW dbo.ProjectPartTotals\nAS\nSELECT JNumber,\n SUM(Quantities) AS TotalQuantities\nFROM dbo.SPJ\nGROUP BY JNumber;\n",[31,93250,93251,93256,93261,93266,93271,93275],{"__ignoreMap":29},[34,93252,93253],{"class":36,"line":37},[34,93254,93255],{},"CREATE VIEW dbo.ProjectPartTotals\n",[34,93257,93258],{"class":36,"line":43},[34,93259,93260],{},"AS\n",[34,93262,93263],{"class":36,"line":49},[34,93264,93265],{},"SELECT JNumber,\n",[34,93267,93268],{"class":36,"line":56},[34,93269,93270],{}," SUM(Quantities) AS TotalQuantities\n",[34,93272,93273],{"class":36,"line":62},[34,93274,92787],{},[34,93276,93277],{"class":36,"line":68},[34,93278,93279],{},"GROUP BY JNumber;\n",[15,93281,93282,93283,93286],{},"讲解：视图中可以包含聚合查询。SQL Server 创建普通视图时不要在视图定义中写裸 ",[31,93284,93285],{},"ORDER BY","，排序应放在查询视图时完成。",[19,93288,93290,93291,46528,93294,93297],{"id":93289},"_4-设关系模式-ra-b-c-dfab-cd-a-d说明为什么-r-不是-2nf并分解成-2nf-模式集","4. 设关系模式 ",[31,93292,93293],{},"R(A, B, C, D)",[31,93295,93296],{},"F={AB -> CD, A -> D}","。说明为什么 R 不是 2NF，并分解成 2NF 模式集。",[15,93299,58173],{},[105,93301,93302,93308,93327,93339],{},[108,93303,93304,93305,347],{},"候选码：",[31,93306,93307],{},"AB",[108,93309,93310,93311,13977,93314,36268,93317,46528,93320,93323,93324,93326],{},"因为 ",[31,93312,93313],{},"AB -> CD",[31,93315,93316],{},"AB+ = ABCD",[31,93318,93319],{},"A+ = AD",[31,93321,93322],{},"B+ = B","，因此只有 ",[31,93325,93307],{}," 能推出全部属性。",[108,93328,93329,93330,93332,93333,61605,93335,93338],{},"非主属性 ",[31,93331,91994],{}," 部分依赖于候选码的一部分 ",[31,93334,54151],{},[31,93336,93337],{},"A -> D","，所以 R 不满足 2NF。",[108,93340,93341,93342,43298,93345,347],{},"分解为：",[31,93343,93344],{},"R1(A, D)",[31,93346,93347],{},"R2(A, B, C)",[15,93349,93350,93351,93354,93355,46528,93357,93354,93360,93363,93364,93366,93367,93369],{},"讲解：2NF 的核心是“非主属性不能部分依赖于候选码”。分解后，",[31,93352,93353],{},"R1"," 保存 ",[31,93356,93337],{},[31,93358,93359],{},"R2",[31,93361,93362],{},"AB -> C","，消除了 ",[31,93365,91994],{}," 对组合码 ",[31,93368,93307],{}," 的部分依赖。",[19,93371,93373,93374,46528,93377,93380],{"id":93372},"_5-设关系模式-ra-b-cfc-b-b-a说明为什么-r-不是-3nf并分解成-3nf-模式集","5. 设关系模式 ",[31,93375,93376],{},"R(A, B, C)",[31,93378,93379],{},"F={C -> B, B -> A}","。说明为什么 R 不是 3NF，并分解成 3NF 模式集。",[15,93382,58173],{},[105,93384,93385,93399,93405,93413],{},[108,93386,93304,93387,93389,93390,86674,93393,13977,93396,347],{},[31,93388,47384],{},"，因为 ",[31,93391,93392],{},"C -> B",[31,93394,93395],{},"B -> A",[31,93397,93398],{},"C+ = ABC",[108,93400,93401,93402,347],{},"存在传递依赖：",[31,93403,93404],{},"C -> B -> A",[108,93406,93407,93409,93410,93412],{},[31,93408,54224],{}," 不是超码，",[31,93411,54151],{}," 是非主属性，因此违反 3NF。",[108,93414,93341,93415,43298,93418,347],{},[31,93416,93417],{},"R1(C, B)",[31,93419,93420],{},"R2(B, A)",[15,93422,93423,93424,93427,93428,93430,93431,93433,93434,93436,93437,93409,93439,93441],{},"讲解：3NF 允许依赖 ",[31,93425,93426],{},"X -> A"," 的情况是：",[31,93429,67467],{}," 是超码，或 ",[31,93432,54151],{}," 是主属性。这里 ",[31,93435,93395],{}," 中 ",[31,93438,54224],{},[31,93440,54151],{}," 不是主属性，所以不满足 3NF。",[19,93443,93445,93446,93449],{"id":93444},"_6-设关系模式-rs-c-tname属性为学生学号课程号任课教师姓名每个学生可选多门课每门课只有一个教师任教","6. 设关系模式 ",[31,93447,93448],{},"R(S#, C#, TNAME)","，属性为学生学号、课程号、任课教师姓名。每个学生可选多门课，每门课只有一个教师任教。",[15,93451,93452],{},"要求：写出基本 FD 和关键码；说明为什么不是 2NF；举例说明冗余；分解成 2NF。",[15,93454,58173],{},[105,93456,93457,93467,93472,93482,93485],{},[108,93458,93459,93460,46528,93463,93466],{},"基本函数依赖：",[31,93461,93462],{},"C# -> TNAME",[31,93464,93465],{},"(S#, C#) -> TNAME"," 可由前者推出。",[108,93468,93304,93469,347],{},[31,93470,93471],{},"(S#, C#)",[108,93473,93474,93475,93478,93479,347],{},"不满足 2NF：",[31,93476,93477],{},"TNAME"," 是非主属性，却依赖于组合码的一部分 ",[31,93480,93481],{},"C#",[108,93483,93484],{},"冗余示例：很多学生选同一门课时，该课程教师姓名会重复存储多次。",[108,93486,93487,93488,43298,93491,347],{},"2NF 分解：",[31,93489,93490],{},"SC(S#, C#)",[31,93492,93493],{},"CT(C#, TNAME)",[15,93495,93496,93497,93500,93501,347],{},"讲解：选课事实和课程任课教师事实是两类信息，混在一个关系里会导致教师名重复。分解后，一个学生选一门课记录在 ",[31,93498,93499],{},"SC","，一门课对应一个教师记录在 ",[31,93502,93503],{},"CT",[19,93505,93507,93508,93511],{"id":93506},"_7-设关系模式-rs-tname-taddr属性分别表示学生学号指导教师姓名指导教师地址每个学生只有一个指导老师每个教师只有一个联系地址教师不允许同名同姓","7. 设关系模式 ",[31,93509,93510],{},"R(S#, TNAME, TADDR)","，属性分别表示学生学号、指导教师姓名、指导教师地址。每个学生只有一个指导老师；每个教师只有一个联系地址；教师不允许同名同姓。",[15,93513,58173],{},[105,93515,93516,93524,93529,93542,93545],{},[108,93517,93459,93518,46528,93521,347],{},[31,93519,93520],{},"S# -> TNAME",[31,93522,93523],{},"TNAME -> TADDR",[108,93525,93304,93526,347],{},[31,93527,93528],{},"S#",[108,93530,93531,93532,93535,93536,93409,93538,93541],{},"不满足 3NF：存在传递依赖 ",[31,93533,93534],{},"S# -> TNAME -> TADDR","，且 ",[31,93537,93477],{},[31,93539,93540],{},"TADDR"," 是非主属性。",[108,93543,93544],{},"冗余示例：同一导师指导多个学生时，导师地址重复出现多次。",[108,93546,93547,93548,43298,93551,347],{},"3NF 分解：",[31,93549,93550],{},"ST(S#, TNAME)",[31,93552,93553],{},"TA(TNAME, TADDR)",[15,93555,93556],{},"讲解：单属性候选码时不会有部分依赖，所以它至少满足 2NF；问题在于传递依赖，因此要分解到 3NF。",[19,93558,93560,93561,93563,93564,93567],{"id":93559},"_8-设关系模式-ra-b-c-d函数依赖集-fa-b-a-c写出-r-的关键码并说明理由","8. 设关系模式 ",[31,93562,93293],{},"，函数依赖集 ",[31,93565,93566],{},"F={A -> B, A -> C}","。写出 R 的关键码，并说明理由。",[15,93569,93570,93571,347],{},"答案：候选码是 ",[31,93572,93573],{},"AD",[15,93575,91513,93576,93578,93579,258,93581,93583,93584,36268,93586,93588,93589,347,93591,93535,93594,258,93597,93600,93601,93603],{},[31,93577,54151],{}," 能推出 ",[31,93580,54224],{},[31,93582,47384],{},"，但不能推出 ",[31,93585,91994],{},[31,93587,91994],{}," 不出现在任何函数依赖右部，所以任何候选码都必须包含 ",[31,93590,91994],{},[31,93592,93593],{},"AD+ = ABCD",[31,93595,93596],{},"A+",[31,93598,93599],{},"D+"," 都不能推出全部属性，所以 ",[31,93602,93573],{}," 是候选码。",[19,93605,93607,93608,347],{"id":93606},"_9-设某商业集团数据库中有关系模式-r商店编码-顾客编码-消费总额-顾客单位-地址-电话","9. 设某商业集团数据库中有关系模式 ",[31,93609,93610],{},"R(商店编码, 顾客编码, 消费总额, 顾客单位, 地址, 电话)",[15,93612,93613],{},"规定：",[2230,93615,93616,93619,93622],{},[108,93617,93618],{},"每个顾客在每个商店只有一个消费总额。",[108,93620,93621],{},"每个顾客只属于一个单位。",[108,93623,93624],{},"每个顾客单位只有一个地址、一个电话。",[15,93626,58173],{},[105,93628,93629,93649,93654,93664,93679],{},[108,93630,93631,93632],{},"基本函数依赖：\n",[105,93633,93634,93639,93644],{},[108,93635,93636],{},[31,93637,93638],{},"(商店编码, 顾客编码) -> 消费总额",[108,93640,93641],{},[31,93642,93643],{},"顾客编码 -> 顾客单位",[108,93645,93646],{},[31,93647,93648],{},"顾客单位 -> 地址, 电话",[108,93650,93304,93651,347],{},[31,93652,93653],{},"(商店编码, 顾客编码)",[108,93655,93474,93656,93659,93660,93663],{},[31,93657,93658],{},"顾客单位"," 部分依赖于组合码的一部分 ",[31,93661,93662],{},"顾客编码","，并且地址、电话也间接依赖于顾客编码。",[108,93665,93666,93667],{},"2NF 分解：\n",[105,93668,93669,93674],{},[108,93670,93671],{},[31,93672,93673],{},"R1(商店编码, 顾客编码, 消费总额)",[108,93675,93676],{},[31,93677,93678],{},"R2(顾客编码, 顾客单位, 地址, 电话)",[108,93680,93681,93682],{},"3NF 分解：\n",[105,93683,93684,93688,93693],{},[108,93685,93686],{},[31,93687,93673],{},[108,93689,93690],{},[31,93691,93692],{},"R2(顾客编码, 顾客单位)",[108,93694,93695],{},[31,93696,93697],{},"R3(顾客单位, 地址, 电话)",[15,93699,93700,93701,93704],{},"讲解：2NF 先消除组合码上的部分依赖，3NF 继续消除 ",[31,93702,93703],{},"顾客编码 -> 顾客单位 -> 地址, 电话"," 这种传递依赖。",[19,93706,93708,93709,93712],{"id":93707},"_10-设计一个记录各个球队队员每场比赛进球数的关系模式r队员编号-比赛场次-进球数-球队名-队长名每个队员只能属于一个球队每个球队只有一个队长","10. 设计一个记录各个球队队员每场比赛进球数的关系模式：",[31,93710,93711],{},"R(队员编号, 比赛场次, 进球数, 球队名, 队长名)","。每个队员只能属于一个球队，每个球队只有一个队长。",[15,93714,58173],{},[105,93716,93717,93736,93741,93750,93764],{},[108,93718,93631,93719],{},[105,93720,93721,93726,93731],{},[108,93722,93723],{},[31,93724,93725],{},"(队员编号, 比赛场次) -> 进球数",[108,93727,93728],{},[31,93729,93730],{},"队员编号 -> 球队名",[108,93732,93733],{},[31,93734,93735],{},"球队名 -> 队长名",[108,93737,93304,93738,347],{},[31,93739,93740],{},"(队员编号, 比赛场次)",[108,93742,93474,93743,93746,93747,347],{},[31,93744,93745],{},"球队名"," 部分依赖于组合码中的 ",[31,93748,93749],{},"队员编号",[108,93751,93666,93752],{},[105,93753,93754,93759],{},[108,93755,93756],{},[31,93757,93758],{},"R1(队员编号, 比赛场次, 进球数)",[108,93760,93761],{},[31,93762,93763],{},"R2(队员编号, 球队名, 队长名)",[108,93765,93681,93766],{},[105,93767,93768,93772,93777],{},[108,93769,93770],{},[31,93771,93758],{},[108,93773,93774],{},[31,93775,93776],{},"R2(队员编号, 球队名)",[108,93778,93779],{},[31,93780,93781],{},"R3(球队名, 队长名)",[15,93783,93784],{},"讲解：比赛进球数是“队员-比赛”的事实；队员所属球队是“队员”的事实；球队队长是“球队”的事实。分开存储后冗余更少，更新异常也更少。",[19,93786,93788,93789,93792],{"id":93787},"_11-设有关系模式-r职工名-项目名-工资-部门名-部门经理每个职工可参加多个项目各领一份工资每个项目只属于一个部门每个部门只有一个经理","11. 设有关系模式 ",[31,93790,93791],{},"R(职工名, 项目名, 工资, 部门名, 部门经理)","。每个职工可参加多个项目，各领一份工资；每个项目只属于一个部门；每个部门只有一个经理。",[15,93794,58173],{},[105,93796,93797,93816,93822,93837,93856],{},[108,93798,93631,93799],{},[105,93800,93801,93806,93811],{},[108,93802,93803],{},[31,93804,93805],{},"职工名 -> 工资",[108,93807,93808],{},[31,93809,93810],{},"项目名 -> 部门名",[108,93812,93813],{},[31,93814,93815],{},"部门名 -> 部门经理",[108,93817,93304,93818,93821],{},[31,93819,93820],{},"(职工名, 项目名)","，用来标识某职工参加某项目这一事实。",[108,93823,93474,93824,93827,93828,46528,93831,93827,93834,347],{},[31,93825,93826],{},"工资"," 部分依赖于 ",[31,93829,93830],{},"职工名",[31,93832,93833],{},"部门名",[31,93835,93836],{},"项目名",[108,93838,93666,93839],{},[105,93840,93841,93846,93851],{},[108,93842,93843],{},[31,93844,93845],{},"R1(职工名, 项目名)",[108,93847,93848],{},[31,93849,93850],{},"R2(职工名, 工资)",[108,93852,93853],{},[31,93854,93855],{},"R3(项目名, 部门名, 部门经理)",[108,93857,93681,93858],{},[105,93859,93860,93864,93868,93873],{},[108,93861,93862],{},[31,93863,93845],{},[108,93865,93866],{},[31,93867,93850],{},[108,93869,93870],{},[31,93871,93872],{},"R3(项目名, 部门名)",[108,93874,93875],{},[31,93876,93877],{},"R4(部门名, 部门经理)",[15,93879,93880,93881,93884],{},"讲解：题中“各领一份工资”按每个职工有一份工资理解。若某课程老师要求“工资随项目不同而不同”，则应把 ",[31,93882,93883],{},"(职工名, 项目名) -> 工资"," 写入依赖，并相应调整分解；但按题面常规语义，工资是职工属性。",[79,93886,81],{},{"title":29,"searchDepth":43,"depth":43,"links":93888},[93889,93890,93891,93906,93908,93910,93912,93914,93916,93918,93920],{"id":92449,"depth":43,"text":92450},{"id":92478,"depth":43,"text":92479},{"id":92494,"depth":43,"text":92495,"children":93892},[93893,93895,93896,93897,93898,93899,93900,93901,93902,93903,93904,93905],{"id":92577,"depth":49,"text":93894},"3.1 用 CREATE TABLE 建立上述四张表，并指出主键和外键。",{"id":92771,"depth":49,"text":92772},{"id":92800,"depth":49,"text":92801},{"id":92833,"depth":49,"text":92834},{"id":92857,"depth":49,"text":92858},{"id":92910,"depth":49,"text":92911},{"id":92944,"depth":49,"text":92945},{"id":93020,"depth":49,"text":93021},{"id":93088,"depth":49,"text":93089},{"id":93163,"depth":49,"text":93164},{"id":93191,"depth":49,"text":93192},{"id":93244,"depth":49,"text":93245},{"id":93289,"depth":43,"text":93907},"4. 设关系模式 R(A, B, C, D)，F={AB -> CD, A -> D}。说明为什么 R 不是 2NF，并分解成 2NF 模式集。",{"id":93372,"depth":43,"text":93909},"5. 设关系模式 R(A, B, C)，F={C -> B, B -> A}。说明为什么 R 不是 3NF，并分解成 3NF 模式集。",{"id":93444,"depth":43,"text":93911},"6. 设关系模式 R(S#, C#, TNAME)，属性为学生学号、课程号、任课教师姓名。每个学生可选多门课，每门课只有一个教师任教。",{"id":93506,"depth":43,"text":93913},"7. 设关系模式 R(S#, TNAME, TADDR)，属性分别表示学生学号、指导教师姓名、指导教师地址。每个学生只有一个指导老师；每个教师只有一个联系地址；教师不允许同名同姓。",{"id":93559,"depth":43,"text":93915},"8. 设关系模式 R(A, B, C, D)，函数依赖集 F={A -> B, A -> C}。写出 R 的关键码，并说明理由。",{"id":93606,"depth":43,"text":93917},"9. 设某商业集团数据库中有关系模式 R(商店编码, 顾客编码, 消费总额, 顾客单位, 地址, 电话)。",{"id":93707,"depth":43,"text":93919},"10. 设计一个记录各个球队队员每场比赛进球数的关系模式：R(队员编号, 比赛场次, 进球数, 球队名, 队长名)。每个队员只能属于一个球队，每个球队只有一个队长。",{"id":93787,"depth":43,"text":93921},"11. 设有关系模式 R(职工名, 项目名, 工资, 部门名, 部门经理)。每个职工可参加多个项目，各领一份工资；每个项目只属于一个部门；每个部门只有一个经理。",{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday06-70-01-11",{"title":92438,"description":92446},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay06-数据库70题-试题01-11","h-7ouEFAKK5R-RtQOKLxol1TeY3OxEl8pvIe3f5QlDU",{"id":93928,"title":93929,"body":93930,"description":93937,"extension":85,"meta":94378,"navigation":52,"path":94379,"seo":94380,"stem":94381,"__hash__":94382},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay07-数据库70题-试题12-25.md","Day 07 - 数据库70题：试题 12-25",{"type":8,"value":93931,"toc":94361},[93932,93935,93938,93942,93944,93947,93961,93964,93975,93977,93989,93992,94035,94043,94047,94049,94063,94071,94075,94077,94080,94094,94097,94101,94103,94106,94108,94122,94126,94128,94131,94139,94142,94146,94148,94151,94154,94158,94160,94163,94171,94174,94178,94180,94183,94205,94208,94212,94214,94225,94228,94232,94234,94237,94240,94244,94246,94249,94270,94272,94311,94314,94318,94320,94323,94334,94337,94341,94343,94346,94348,94359],[11,93933,93929],{"id":93934},"day-07-数据库70题试题-12-25",[15,93936,93937],{},"参考重点：教材第 4 章安全性、第 5 章完整性、第 11 章恢复技术、第 12 章并发控制。",[19,93939,93941],{"id":93940},"_12-数据库保护的四个方面各防止什么错误的发生","12. 数据库保护的四个方面各防止什么错误的发生？",[15,93943,58173],{},[15,93945,93946],{},"数据库保护包括安全性、完整性、并发控制和恢复四方面。",[105,93948,93949,93952,93955,93958],{},[108,93950,93951],{},"安全性：防止非法用户或越权用户访问、泄露、篡改、破坏数据。",[108,93953,93954],{},"完整性：防止不符合语义的数据进入数据库，例如主键重复、外键悬空、成绩超过范围。",[108,93956,93957],{},"并发控制：防止多个事务交叉执行造成丢失修改、读脏数据、不可重复读、不一致分析。",[108,93959,93960],{},"恢复：防止事务故障、系统故障、介质故障后数据库停留在错误或不一致状态。",[15,93962,93963],{},"讲解：可以把这四类保护记成“谁能动数据、数据是否合法、多事务怎么动、出故障怎么补救”。",[19,93965,93967,93968,93971,93972,93974],{"id":93966},"_13-试解释-commit-语句和-rollback-语句","13. 试解释 ",[31,93969,93970],{},"COMMIT"," 语句和 ",[31,93973,91516],{}," 语句。",[15,93976,58173],{},[105,93978,93979,93984],{},[108,93980,93981,93983],{},[31,93982,93970],{},"：提交事务，使事务对数据库所做的更新永久生效。",[108,93985,93986,93988],{},[31,93987,91516],{},"：回滚事务，撤销当前事务尚未提交的全部更新，使数据库回到事务开始前的一致状态。",[15,93990,93991],{},"SQL Server 示例：",[24,93993,93995],{"className":92584,"code":93994,"language":92586,"meta":29,"style":29},"BEGIN TRANSACTION;\n\nUPDATE dbo.Accounts\nSET Balance = Balance - 300\nWHERE AccountNo = 'A001';\n\nCOMMIT TRANSACTION;\n-- 如果出现错误，则执行 ROLLBACK TRANSACTION;\n",[31,93996,93997,94002,94006,94011,94016,94021,94025,94030],{"__ignoreMap":29},[34,93998,93999],{"class":36,"line":37},[34,94000,94001],{},"BEGIN TRANSACTION;\n",[34,94003,94004],{"class":36,"line":43},[34,94005,53],{"emptyLinePlaceholder":52},[34,94007,94008],{"class":36,"line":49},[34,94009,94010],{},"UPDATE dbo.Accounts\n",[34,94012,94013],{"class":36,"line":56},[34,94014,94015],{},"SET Balance = Balance - 300\n",[34,94017,94018],{"class":36,"line":62},[34,94019,94020],{},"WHERE AccountNo = 'A001';\n",[34,94022,94023],{"class":36,"line":68},[34,94024,53],{"emptyLinePlaceholder":52},[34,94026,94027],{"class":36,"line":74},[34,94028,94029],{},"COMMIT TRANSACTION;\n",[34,94031,94032],{"class":36,"line":1971},[34,94033,94034],{},"-- 如果出现错误，则执行 ROLLBACK TRANSACTION;\n",[15,94036,91513,94037,94039,94040,94042],{},[31,94038,93970],{}," 对应持久性，",[31,94041,91516],{}," 对应原子性。事务失败时，已做的局部更新不能留下。",[19,94044,94046],{"id":94045},"_14-试解释-undo-操作和-redo-操作","14. 试解释 UNDO 操作和 REDO 操作。",[15,94048,58173],{},[105,94050,94051,94057],{},[108,94052,94053,94056],{},[31,94054,94055],{},"UNDO","：撤销未提交或失败事务已经执行的更新，常利用日志中的旧值把数据改回去。",[108,94058,94059,94062],{},[31,94060,94061],{},"REDO","：重做已提交事务的更新，常利用日志中的新值把提交结果重新写入数据库。",[15,94064,94065,94066,94068,94069,347],{},"讲解：系统崩溃时，内存中已提交事务的结果可能尚未写到磁盘，所以要 ",[31,94067,94061],{},"；未提交事务的部分结果可能已经写到磁盘，所以要 ",[31,94070,94055],{},[19,94072,94074],{"id":94073},"_15-dbs-运行中一般有哪几类故障","15. DBS 运行中一般有哪几类故障？",[15,94076,58173],{},[15,94078,94079],{},"常见故障包括：",[2230,94081,94082,94085,94088,94091],{},[108,94083,94084],{},"事务故障：某个事务因逻辑错误、死锁回滚等提前终止。",[108,94086,94087],{},"系统故障：断电、操作系统崩溃等导致系统停止运行，但外存数据库未损坏。",[108,94089,94090],{},"介质故障：磁盘等外存介质损坏，导致数据库文件部分或全部丢失。",[108,94092,94093],{},"计算机病毒、恶意破坏或其他灾难性故障。",[15,94095,94096],{},"讲解：事务故障和系统故障通常可由 DBMS 自动恢复；介质故障需要后援副本和日志，往往需要 DBA 参与。",[19,94098,94100],{"id":94099},"_16-试叙述-db-的并发控制与恢复的关系","16. 试叙述 DB 的并发控制与恢复的关系。",[15,94102,58173],{},[15,94104,94105],{},"并发控制和恢复都是为了维护数据库一致性。并发控制在正常运行时保证多个事务隔离执行，恢复在故障后保证已提交事务不丢失、未提交事务不残留。",[15,94107,91513],{},[105,94109,94110,94113,94116,94119],{},[108,94111,94112],{},"并发控制解决“同时执行会不会互相干扰”。",[108,94114,94115],{},"恢复解决“执行中出故障后怎样回到一致状态”。",[108,94117,94118],{},"严格两段锁等协议能避免级联回滚，使恢复更简单。",[108,94120,94121],{},"日志记录必须反映事务更新顺序，才能正确 UNDO 和 REDO。",[19,94123,94125],{"id":94124},"_17-试叙述基本封锁类型的含义","17. 试叙述基本封锁类型的含义。",[15,94127,58173],{},[15,94129,94130],{},"基本封锁类型有共享锁和排他锁。",[105,94132,94133,94136],{},[108,94134,94135],{},"共享锁（S 锁）：事务对数据对象加 S 锁后，可以读该对象，其他事务也可以再加 S 锁读，但不能加 X 锁写。",[108,94137,94138],{},"排他锁（X 锁）：事务对数据对象加 X 锁后，可以读写该对象，其他事务不能再加 S 锁或 X 锁。",[15,94140,94141],{},"讲解：相容性表要记住：S-S 相容，S-X 不相容，X-X 不相容。",[19,94143,94145],{"id":94144},"_18-有些事务只要读数据为什么要加-s-锁","18. 有些事务只要读数据，为什么要加 S 锁？",[15,94147,58173],{},[15,94149,94150],{},"只读事务加 S 锁是为了防止读到正在被其他事务修改但尚未提交的数据，也防止读的过程中数据被改掉，从而保证隔离性。",[15,94152,94153],{},"讲解：如果读操作不加锁，可能发生读脏数据或不可重复读。S 锁允许多个读事务并发，但会阻止写事务同时修改同一对象。",[19,94155,94157],{"id":94156},"_19-试叙述两段锁协议","19. 试叙述两段锁协议。",[15,94159,58173],{},[15,94161,94162],{},"两段锁协议规定每个事务的封锁过程分成两个阶段：",[2230,94164,94165,94168],{},[108,94166,94167],{},"扩展阶段：事务可以申请并获得新的锁，但不能释放任何锁。",[108,94169,94170],{},"收缩阶段：事务可以释放锁，但不能再申请新的锁。",[15,94172,94173],{},"讲解：所有事务遵循两段锁协议时，所产生的并发调度是冲突可串行化的。两段锁协议不能保证不死锁；严格两段锁要求写锁通常保持到事务结束，可避免级联回滚。",[19,94175,94177],{"id":94176},"_20-列举基于日志的恢复方法的实现要点","20. 列举基于日志的恢复方法的实现要点。",[15,94179,58173],{},[15,94181,94182],{},"实现要点包括：",[2230,94184,94185,94188,94191,94194,94197],{},[108,94186,94187],{},"对每次更新登记日志，记录事务号、数据对象、旧值、新值等。",[108,94189,94190],{},"遵守先写日志原则（WAL）：更新数据页写入磁盘前，对应日志必须先写入稳定存储。",[108,94192,94193],{},"事务提交时必须先写入提交日志记录。",[108,94195,94196],{},"周期性建立检查点，减少恢复时扫描日志的范围。",[108,94198,94199,94200,94202,94203,347],{},"恢复时对已提交但可能未落盘的事务执行 ",[31,94201,94061],{},"，对未提交事务执行 ",[31,94204,94055],{},[15,94206,94207],{},"讲解：日志是恢复的“时间线”。没有日志，就无法知道哪些修改要重做、哪些修改要撤销。",[19,94209,94211],{"id":94210},"_21-列举三种解决死锁的常用方法","21. 列举三种解决死锁的常用方法。",[15,94213,58173],{},[2230,94215,94216,94219,94222],{},[108,94217,94218],{},"死锁预防：规定资源申请顺序，或采用时间戳策略，如 wait-die、wound-wait。",[108,94220,94221],{},"死锁检测与解除：构造等待图，发现环后回滚某个事务释放资源。",[108,94223,94224],{},"超时法：事务等待超过阈值就认为可能死锁，主动回滚。",[15,94226,94227],{},"讲解：预防法开销可能较大但避免死锁；检测法允许死锁发生再处理；超时法简单但可能误判。",[19,94229,94231],{"id":94230},"_22-什么叫转储转储有什么作用","22. 什么叫转储？转储有什么作用？",[15,94233,58173],{},[15,94235,94236],{},"转储是定期将数据库复制到后援存储介质上，形成数据库后援副本的过程。作用是在介质故障或严重破坏后，用后援副本和日志恢复数据库。",[15,94238,94239],{},"讲解：转储分静态转储和动态转储。静态转储期间不允许更新，简单但影响服务；动态转储允许并发事务执行，但必须配合日志。",[19,94241,94243],{"id":94242},"_23-简述-dbms-实现完整性的方法","23. 简述 DBMS 实现完整性的方法。",[15,94245,58173],{},[15,94247,94248],{},"DBMS 实现完整性通常包括三步：",[2230,94250,94251,94264,94267],{},[108,94252,94253,94254,146,94257,146,94260,94263],{},"定义完整性约束：主键、外键、",[31,94255,94256],{},"UNIQUE",[31,94258,94259],{},"NOT NULL",[31,94261,94262],{},"CHECK","、触发器等。",[108,94265,94266],{},"检查完整性约束：在插入、删除、修改时自动检查。",[108,94268,94269],{},"违约处理：拒绝操作，或按规则级联更新、级联删除、置空、置默认值。",[15,94271,93991],{},[24,94273,94275],{"className":92584,"code":94274,"language":92586,"meta":29,"style":29},"CREATE TABLE dbo.Score (\n Sno varchar(20) NOT NULL,\n Cno varchar(20) NOT NULL,\n Grade decimal(5, 2) NULL,\n CONSTRAINT PK_Score PRIMARY KEY (Sno, Cno),\n CONSTRAINT CK_Score_Grade CHECK (Grade BETWEEN 0 AND 100)\n);\n",[31,94276,94277,94282,94287,94292,94297,94302,94307],{"__ignoreMap":29},[34,94278,94279],{"class":36,"line":37},[34,94280,94281],{},"CREATE TABLE dbo.Score (\n",[34,94283,94284],{"class":36,"line":43},[34,94285,94286],{}," Sno varchar(20) NOT NULL,\n",[34,94288,94289],{"class":36,"line":49},[34,94290,94291],{}," Cno varchar(20) NOT NULL,\n",[34,94293,94294],{"class":36,"line":56},[34,94295,94296],{}," Grade decimal(5, 2) NULL,\n",[34,94298,94299],{"class":36,"line":62},[34,94300,94301],{}," CONSTRAINT PK_Score PRIMARY KEY (Sno, Cno),\n",[34,94303,94304],{"class":36,"line":68},[34,94305,94306],{}," CONSTRAINT CK_Score_Grade CHECK (Grade BETWEEN 0 AND 100)\n",[34,94308,94309],{"class":36,"line":74},[34,94310,462],{},[15,94312,94313],{},"讲解：完整性是“数据是否合理”的问题，不能只靠应用程序检查，应该尽量交给 DBMS 统一约束。",[19,94315,94317],{"id":94316},"_24-试叙述完整性分类","24. 试叙述完整性分类。",[15,94319,58173],{},[15,94321,94322],{},"常见分类有：",[105,94324,94325,94328,94331],{},[108,94326,94327],{},"按关系模型基本规则：实体完整性、参照完整性、用户定义完整性。",[108,94329,94330],{},"按约束对象：属性级约束、元组级约束、关系级约束。",[108,94332,94333],{},"按状态变化：静态约束和动态约束。",[15,94335,94336],{},"讲解：考试最常考的是前三类。实体完整性约束主键，参照完整性约束外键，用户定义完整性表达具体业务规则。",[19,94338,94340],{"id":94339},"_25-数据库安全性与完整性有什么区别","25. 数据库安全性与完整性有什么区别？",[15,94342,58173],{},[15,94344,94345],{},"安全性关注防止未授权访问和非法使用；完整性关注数据是否正确、有效、相容。",[15,94347,91513],{},[105,94349,94350,94353,94356],{},[108,94351,94352],{},"安全性问的是：“这个用户有没有资格看或改？”",[108,94354,94355],{},"完整性问的是：“这个数据值本身合不合法？”",[108,94357,94358],{},"例子：无权限用户查看成绩表是安全性问题；成绩写成 150 是完整性问题。",[79,94360,81],{},{"title":29,"searchDepth":43,"depth":43,"links":94362},[94363,94364,94366,94367,94368,94369,94370,94371,94372,94373,94374,94375,94376,94377],{"id":93940,"depth":43,"text":93941},{"id":93966,"depth":43,"text":94365},"13. 试解释 COMMIT 语句和 ROLLBACK 语句。",{"id":94045,"depth":43,"text":94046},{"id":94073,"depth":43,"text":94074},{"id":94099,"depth":43,"text":94100},{"id":94124,"depth":43,"text":94125},{"id":94144,"depth":43,"text":94145},{"id":94156,"depth":43,"text":94157},{"id":94176,"depth":43,"text":94177},{"id":94210,"depth":43,"text":94211},{"id":94230,"depth":43,"text":94231},{"id":94242,"depth":43,"text":94243},{"id":94316,"depth":43,"text":94317},{"id":94339,"depth":43,"text":94340},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday07-70-12-25",{"title":93929,"description":93937},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay07-数据库70题-试题12-25","qxRHW78EF0VNtRYH4Z_9fvfqAcVtn87i1BAnytod6Mw",{"id":94384,"title":94385,"body":94386,"description":94393,"extension":85,"meta":95036,"navigation":52,"path":95037,"seo":95038,"stem":95039,"__hash__":95040},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay08-数据库70题-综合01-08.md","Day 08 - 数据库70题：综合题 1-8",{"type":8,"value":94387,"toc":95014},[94388,94391,94394,94398,94400,94403,94423,94426,94430,94432,94435,94438,94455,94458,94462,94464,94467,94470,94487,94490,94494,94496,94507,94510,94513,94517,94519,94522,94524,94535,94556,94560,94562,94565,94567,94589,94601,94608,94612,94614,94631,94663,94666,94670,94673,94676,94731,94733,94781,94783,94850,94858,94862,94864,94869,94872,94876,94878,94883,94887,94889,94894,94900,94904,94906,94911,94914,94918,94920,94925,94933,94937,94939,94944,94947,94951,94953,94958,94961,94965,94967,94972,94975,94979,94981,94986,94990,94992,94997,95000,95004,95006,95011],[11,94389,94385],{"id":94390},"day-08-数据库70题综合题-1-8",[15,94392,94393],{},"参考重点：教材第 1 章数据库系统组成，第 2 章关系代数。",[19,94395,94397],{"id":94396},"_1-dba-的职责是什么","1. DBA 的职责是什么？",[15,94399,58173],{},[15,94401,94402],{},"DBA 即数据库管理员，主要职责包括：",[2230,94404,94405,94408,94411,94414,94417,94420],{},[108,94406,94407],{},"参与数据库设计，定义数据库模式、外模式和存储结构。",[108,94409,94410],{},"定义安全性规则，进行用户授权和权限管理。",[108,94412,94413],{},"定义完整性约束，保证数据正确、有效、相容。",[108,94415,94416],{},"制定备份和恢复策略，处理故障恢复。",[108,94418,94419],{},"监视数据库运行状态，进行性能调优、重组和空间管理。",[108,94421,94422],{},"维护数据字典和相关文档，协调不同用户和应用对数据库的使用。",[15,94424,94425],{},"讲解：DBA 不是普通程序员，而是数据库系统运行和数据资源管理的负责人。教材第 1 章把 DBA 列为数据库系统组成部分之一。",[19,94427,94429],{"id":94428},"_2-什么是数据字典数据字典包含哪些基本内容","2. 什么是数据字典？数据字典包含哪些基本内容？",[15,94431,58173],{},[15,94433,94434],{},"数据字典是 DBMS 维护的关于数据库中数据的描述信息，也称元数据仓库。",[15,94436,94437],{},"基本内容包括：",[105,94439,94440,94443,94446,94449,94452],{},[108,94441,94442],{},"数据库、模式、表、视图、索引等对象定义。",[108,94444,94445],{},"属性名、数据类型、长度、取值范围、默认值。",[108,94447,94448],{},"主键、外键、唯一、检查等完整性约束。",[108,94450,94451],{},"用户、角色、权限等安全信息。",[108,94453,94454],{},"存储结构、统计信息、访问路径等系统管理信息。",[15,94456,94457],{},"讲解：数据字典保存“关于数据的数据”。用户数据是业务内容，数据字典描述这些业务内容如何组织和受哪些规则约束。",[19,94459,94461],{"id":94460},"_3-叙述数据字典的主要任务和作用","3. 叙述数据字典的主要任务和作用。",[15,94463,58173],{},[15,94465,94466],{},"数据字典的主要任务是集中存储、维护和提供数据库元数据。",[15,94468,94469],{},"作用包括：",[2230,94471,94472,94475,94478,94481,94484],{},[108,94473,94474],{},"支持 DDL 执行和数据库对象管理。",[108,94476,94477],{},"支持查询优化器了解表结构、索引和统计信息。",[108,94479,94480],{},"支持完整性检查和安全性检查。",[108,94482,94483],{},"支持 DBA 进行监控、维护和文档管理。",[108,94485,94486],{},"为应用开发人员理解数据库结构提供依据。",[15,94488,94489],{},"讲解：如果没有数据字典，DBMS 就无法知道表有哪些列、列是什么类型、谁有权限访问、哪些约束需要检查。",[19,94491,94493],{"id":94492},"_4-叙述模型模式具体值三者之间的联系和区别","4. 叙述模型、模式、具体值三者之间的联系和区别。",[15,94495,58173],{},[105,94497,94498,94501,94504],{},[108,94499,94500],{},"数据模型：描述数据、联系、操作和约束的抽象工具，例如关系模型。",[108,94502,94503],{},"模式：用某种数据模型定义出来的数据库结构，是相对稳定的“型”。",[108,94505,94506],{},"具体值：某一时刻数据库中的实际数据，也称实例或当前值，是不断变化的“值”。",[15,94508,94509],{},"联系：数据模型规定能怎样建模；模式是在该模型下对某个数据库结构的定义；具体值必须符合模式及其约束。",[15,94511,94512],{},"讲解：类比编程：数据模型像语言规则，模式像类或表结构，具体值像运行时对象或表中的行。",[19,94514,94516],{"id":94515},"_5-叙述等值连接与自然连接的区别和联系","5. 叙述等值连接与自然连接的区别和联系。",[15,94518,58173],{},[15,94520,94521],{},"等值连接是在两个关系的属性之间用等号条件进行连接；自然连接是一种特殊等值连接，它自动比较两个关系中同名属性，并在结果中只保留一份同名属性列。",[15,94523,9971],{},[105,94525,94526,94529,94532],{},[108,94527,94528],{},"等值连接需要明确连接条件，可连接不同名属性。",[108,94530,94531],{},"自然连接按同名属性自动连接。",[108,94533,94534],{},"等值连接结果中通常保留两个被比较属性；自然连接会去掉重复同名属性。",[15,94536,94537,94538,43298,94541,94544,94545,94548,94549,94552,94553,347],{},"讲解：若 ",[31,94539,94540],{},"R(A,B)",[31,94542,94543],{},"S(B,C)"," 自然连接，结果属性是 ",[31,94546,94547],{},"(A,B,C)","；若做等值连接 ",[31,94550,94551],{},"R.B=S.B","，结果可能是 ",[31,94554,94555],{},"(R.A,R.B,S.B,S.C)",[19,94557,94559],{"id":94558},"_6-举例说明关系参照完整性的含义","6. 举例说明关系参照完整性的含义。",[15,94561,58173],{},[15,94563,94564],{},"参照完整性要求一个关系中的外键值，必须为空或等于被参照关系中某个主键值。",[15,94566,19553],{},[105,94568,94569,94578],{},[108,94570,94571,93436,94574,94577],{},[31,94572,94573],{},"Student(Sno, Sname)",[31,94575,94576],{},"Sno"," 是主键。",[108,94579,94580,93436,94583,94585,94586,347],{},[31,94581,94582],{},"SC(Sno, Cno, Grade)",[31,94584,94576],{}," 是外键，引用 ",[31,94587,94588],{},"Student(Sno)",[15,94590,94591,94592,94594,94595,94597,94598,94600],{},"则 ",[31,94593,93499],{}," 中每一条选课记录的 ",[31,94596,94576],{}," 必须在 ",[31,94599,92147],{}," 表中存在，不能出现一个不存在学生的选课记录。",[15,94602,94603,94604,94607],{},"讲解：参照完整性用于维护表之间联系的正确性。SQL Server 中用 ",[31,94605,94606],{},"FOREIGN KEY ... REFERENCES ..."," 定义。",[19,94609,94611],{"id":94610},"_7-将关系代数中的五种基本运算用元组关系演算表达式表示","7. 将关系代数中的五种基本运算用元组关系演算表达式表示。",[15,94613,58173],{},[15,94615,94616,94617,146,94620,94623,94624,146,94626,146,94629,347],{},"设关系谓词为 ",[31,94618,94619],{},"R(t)",[31,94621,94622],{},"S(t)","，元组变量为 ",[31,94625,33249],{},[31,94627,94628],{},"u",[31,94630,56782],{},[2230,94632,94633,94639,94645,94651,94657],{},[108,94634,94635,94636],{},"并：",[31,94637,94638],{},"R ∪ S = { t | R(t) OR S(t) }",[108,94640,94641,94642],{},"差：",[31,94643,94644],{},"R - S = { t | R(t) AND NOT S(t) }",[108,94646,94647,94648],{},"选择：",[31,94649,94650],{},"σF(R) = { t | R(t) AND F(t) }",[108,94652,94653,94654],{},"投影：",[31,94655,94656],{},"πA(R) = { u | EXISTS t (R(t) AND u[A] = t[A]) }",[108,94658,94659,94660],{},"笛卡尔积：",[31,94661,94662],{},"R × S = { v | EXISTS r EXISTS s (R(r) AND S(s) AND v = r ∘ s) }",[15,94664,94665],{},"讲解：关系代数是过程性更强的表达，关系演算是声明性表达。元组演算的核心是“用谓词描述结果元组应满足什么条件”。",[19,94667,94669],{"id":94668},"_8-有三个关系-sc-和-sc试用关系代数表达式表示查询","8. 有三个关系 S、C 和 SC，试用关系代数表达式表示查询。",[15,94671,94672],{},"题面关系：",[15,94674,94675],{},"S",[13837,94677,94678,94692],{},[13840,94679,94680],{},[13843,94681,94682,94684,94686,94689],{},[13846,94683,93528],{},[13846,94685,92534],{},[13846,94687,94688],{},"AGE",[13846,94690,94691],{},"SEX",[13853,94693,94694,94707,94720],{},[13843,94695,94696,94698,94701,94704],{},[13858,94697,640],{},[13858,94699,94700],{},"李强",[13858,94702,94703],{},"23",[13858,94705,94706],{},"男",[13843,94708,94709,94711,94714,94717],{},[13858,94710,768],{},[13858,94712,94713],{},"刘丽",[13858,94715,94716],{},"22",[13858,94718,94719],{},"女",[13843,94721,94722,94724,94727,94729],{},[13858,94723,18136],{},[13858,94725,94726],{},"张友",[13858,94728,94716],{},[13858,94730,94706],{},[15,94732,47384],{},[13837,94734,94735,94747],{},[13840,94736,94737],{},[13843,94738,94739,94741,94744],{},[13846,94740,93481],{},[13846,94742,94743],{},"CNAME",[13846,94745,94746],{},"TEACHER",[13853,94748,94749,94760,94771],{},[13843,94750,94751,94754,94757],{},[13858,94752,94753],{},"K1",[13858,94755,94756],{},"C语言",[13858,94758,94759],{},"王华",[13843,94761,94762,94765,94768],{},[13858,94763,94764],{},"K5",[13858,94766,94767],{},"数据库原理",[13858,94769,94770],{},"程军",[13843,94772,94773,94776,94779],{},[13858,94774,94775],{},"K8",[13858,94777,94778],{},"编译原理",[13858,94780,94770],{},[15,94782,93499],{},[13837,94784,94785,94796],{},[13840,94786,94787],{},[13843,94788,94789,94791,94793],{},[13846,94790,93528],{},[13846,94792,93481],{},[13846,94794,94795],{},"GRADE",[13853,94797,94798,94807,94816,94825,94833,94842],{},[13843,94799,94800,94802,94804],{},[13858,94801,640],{},[13858,94803,94753],{},[13858,94805,94806],{},"83",[13843,94808,94809,94811,94813],{},[13858,94810,768],{},[13858,94812,94753],{},[13858,94814,94815],{},"85",[13843,94817,94818,94820,94822],{},[13858,94819,18136],{},[13858,94821,94753],{},[13858,94823,94824],{},"92",[13843,94826,94827,94829,94831],{},[13858,94828,768],{},[13858,94830,94764],{},[13858,94832,854],{},[13843,94834,94835,94837,94839],{},[13858,94836,18136],{},[13858,94838,94764],{},[13858,94840,94841],{},"84",[13843,94843,94844,94846,94848],{},[13858,94845,18136],{},[13858,94847,94775],{},[13858,94849,27539],{},[15,94851,94852,94853,146,94855,94857],{},"为避免歧义，下面使用 ",[31,94854,92534],{},[31,94856,94743],{}," 作为姓名和课程名。",[253,94859,94861],{"id":94860},"_81-检索程军老师所授课程的课程号和课程名","8.1 检索“程军”老师所授课程的课程号和课程名。",[15,94863,58173],{},[15,94865,94866],{},[31,94867,94868],{},"πC#,CNAME(σTEACHER='程军'(C))",[15,94870,94871],{},"讲解：先选择教师为程军的课程，再投影课程号和课程名。",[253,94873,94875],{"id":94874},"_82-检索年龄大于-21-的男学生学号和姓名","8.2 检索年龄大于 21 的男学生学号和姓名。",[15,94877,58173],{},[15,94879,94880],{},[31,94881,94882],{},"πS#,SNAME(σAGE>21 AND SEX='男'(S))",[253,94884,94886],{"id":94885},"_83-检索至少选修程军老师所授全部课程的学生姓名","8.3 检索至少选修“程军”老师所授全部课程的学生姓名。",[15,94888,58173],{},[15,94890,94891],{},[31,94892,94893],{},"πSNAME(S ⋈ ((πS#,C#(SC)) ÷ (πC#(σTEACHER='程军'(C)))))",[15,94895,94896,94897,94899],{},"讲解：这是除法。被除数是学生选课对 ",[31,94898,93471],{},"，除数是程军老师讲授的全部课程号。结果是选了这些全部课程的学生号，再连接 S 取姓名。",[253,94901,94903],{"id":94902},"_84-检索李强同学不学课程的课程号","8.4 检索“李强”同学不学课程的课程号。",[15,94905,58173],{},[15,94907,94908],{},[31,94909,94910],{},"πC#(C) - πC#(σSNAME='李强'(S) ⋈ SC)",[15,94912,94913],{},"讲解：所有课程号减去李强已选课程号，就是李强未选课程。",[253,94915,94917],{"id":94916},"_85-检索至少选修两门课程的学生学号","8.5 检索至少选修两门课程的学生学号。",[15,94919,58173],{},[15,94921,94922],{},[31,94923,94924],{},"πSC1.S#(σSC1.S#=SC2.S# AND SC1.C#\u003C>SC2.C#(ρSC1(SC) × ρSC2(SC)))",[15,94926,94927,94928,94930,94931,347],{},"讲解：把 SC 改名成两份，找同一学生对应不同课程的配对。原题括号中写成 ",[31,94929,93481],{}," 应为 ",[31,94932,93528],{},[253,94934,94936],{"id":94935},"_86-检索全部学生都选修的课程的课程号和课程名","8.6 检索全部学生都选修的课程的课程号和课程名。",[15,94938,58173],{},[15,94940,94941],{},[31,94942,94943],{},"πC#,CNAME(C ⋈ ((πS#,C#(SC)) ÷ (πS#(S))))",[15,94945,94946],{},"讲解：除数是全部学生号，结果是被所有学生都选过的课程号。",[253,94948,94950],{"id":94949},"_87-检索选修课程包含程军老师所授课程之一的学生学号","8.7 检索选修课程包含“程军”老师所授课程之一的学生学号。",[15,94952,58173],{},[15,94954,94955],{},[31,94956,94957],{},"πS#(SC ⋈ πC#(σTEACHER='程军'(C)))",[15,94959,94960],{},"讲解：“之一”表示至少选一门，所以用连接即可，不用除法。",[253,94962,94964],{"id":94963},"_88-检索选修课程号为-k1-和-k5-的学生学号","8.8 检索选修课程号为 K1 和 K5 的学生学号。",[15,94966,58173],{},[15,94968,94969],{},[31,94970,94971],{},"πS#(σC#='K1'(SC)) ∩ πS#(σC#='K5'(SC))",[15,94973,94974],{},"讲解：同时满足两个课程条件，用交集表达最清楚。",[253,94976,94978],{"id":94977},"_89-检索选修全部课程的学生姓名","8.9 检索选修全部课程的学生姓名。",[15,94980,58173],{},[15,94982,94983],{},[31,94984,94985],{},"πSNAME(S ⋈ ((πS#,C#(SC)) ÷ (πC#(C))))",[253,94987,94989],{"id":94988},"_810-检索选修课程包含学号为-2-的学生所修课程的学生学号","8.10 检索选修课程包含学号为 2 的学生所修课程的学生学号。",[15,94991,58173],{},[15,94993,94994],{},[31,94995,94996],{},"πS#((πS#,C#(SC)) ÷ (πC#(σS#='2'(SC))))",[15,94998,94999],{},"讲解：结果学生的选课集合必须覆盖 2 号学生的选课集合。这也是除法题。",[253,95001,95003],{"id":95002},"_811-检索选修课程名为c语言的学生学号和姓名","8.11 检索选修课程名为“C语言”的学生学号和姓名。",[15,95005,58173],{},[15,95007,95008],{},[31,95009,95010],{},"πS#,SNAME(S ⋈ SC ⋈ σCNAME='C语言'(C))",[15,95012,95013],{},"讲解：先找到课程名为 C语言 的课程，再通过 SC 找选修学生，最后连接 S 取学号和姓名。",{"title":29,"searchDepth":43,"depth":43,"links":95015},[95016,95017,95018,95019,95020,95021,95022,95023],{"id":94396,"depth":43,"text":94397},{"id":94428,"depth":43,"text":94429},{"id":94460,"depth":43,"text":94461},{"id":94492,"depth":43,"text":94493},{"id":94515,"depth":43,"text":94516},{"id":94558,"depth":43,"text":94559},{"id":94610,"depth":43,"text":94611},{"id":94668,"depth":43,"text":94669,"children":95024},[95025,95026,95027,95028,95029,95030,95031,95032,95033,95034,95035],{"id":94860,"depth":49,"text":94861},{"id":94874,"depth":49,"text":94875},{"id":94885,"depth":49,"text":94886},{"id":94902,"depth":49,"text":94903},{"id":94916,"depth":49,"text":94917},{"id":94935,"depth":49,"text":94936},{"id":94949,"depth":49,"text":94950},{"id":94963,"depth":49,"text":94964},{"id":94977,"depth":49,"text":94978},{"id":94988,"depth":49,"text":94989},{"id":95002,"depth":49,"text":95003},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday08-70-01-08",{"title":94385,"description":94393},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay08-数据库70题-综合01-08","BNeeq6fXQGh2bCdwhvpKJLQ-Pi5pVeV3ho8QKGQEu1w",{"id":95042,"title":95043,"body":95044,"description":95051,"extension":85,"meta":97227,"navigation":52,"path":97228,"seo":97229,"stem":97230,"__hash__":97231},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay09-数据库70题-综合22-33.md","Day 09 - 数据库70题：综合题 22-33",{"type":8,"value":95045,"toc":97179},[95046,95049,95052,95056,95076,95080,95082,95087,95091,95093,95098,95102,95104,95109,95113,95115,95120,95124,95126,95131,95135,95137,95142,95146,95148,95153,95159,95163,95165,95170,95173,95177,95188,95192,95194,95199,95203,95205,95210,95214,95216,95347,95358,95362,95364,95366,95454,95456,95513,95515,95631,95638,95642,95687,95690,95750,95754,95789,95792,95810,95815,95819,95882,95885,95915,95918,95922,95926,95935,95939,95959,95963,95972,95982,95986,95988,95990,96044,96046,96088,96090,96118,96125,96169,96180,96184,96214,96221,96257,96261,96263,96265,96295,96297,96328,96332,96359,96363,96367,96399,96405,96409,96411,96413,96459,96461,96499,96502,96546,96548,96566,96573,96605,96608,96698,96713,96743,96746,96750,96752,96836,96838,96894,96897,96937,96940,96962,96966,96970,96990,96994,97034,97040,97044,97052,97055,97106,97112,97120,97174,97177],[11,95047,95043],{"id":95048},"day-09-数据库70题综合题-22-33",[15,95050,95051],{},"参考重点：教材第 2 章关系演算，第 3 章 SQL。SQL 均按 SQL Server 写法。",[19,95053,95055],{"id":95054},"_22-对于上题的关系-sc-和-sc试用元组演算表达式表示查询","22. 对于上题的关系 S、C 和 SC，试用元组演算表达式表示查询。",[15,95057,95058,95059,146,95062,146,95065,95068,95069,146,95072,95075],{},"使用谓词：",[31,95060,95061],{},"S(s)",[31,95063,95064],{},"C(c)",[31,95066,95067],{},"SC(sc)","，属性写作 ",[31,95070,95071],{},"s.S#",[31,95073,95074],{},"c.C#"," 等。",[253,95077,95079],{"id":95078},"_221-检索选修课程号为-k5-的学生学号和成绩","22.1 检索选修课程号为 K5 的学生学号和成绩。",[15,95081,58173],{},[15,95083,95084],{},[31,95085,95086],{},"{ \u003Cx, g> | EXISTS sc (SC(sc) AND sc.C#='K5' AND x=sc.S# AND g=sc.GRADE) }",[253,95088,95090],{"id":95089},"_222-检索选修课程号为-k8-的学生的学号和姓名","22.2 检索选修课程号为 K8 的学生的学号和姓名。",[15,95092,58173],{},[15,95094,95095],{},[31,95096,95097],{},"{ \u003Cx, n> | EXISTS s EXISTS sc (S(s) AND SC(sc) AND s.S#=sc.S# AND sc.C#='K8' AND x=s.S# AND n=s.SNAME) }",[253,95099,95101],{"id":95100},"_223-检索选修课程名为c语言的学生学号和姓名","22.3 检索选修课程名为“C语言”的学生学号和姓名。",[15,95103,58173],{},[15,95105,95106],{},[31,95107,95108],{},"{ \u003Cx, n> | EXISTS s EXISTS c EXISTS sc (S(s) AND C(c) AND SC(sc) AND c.CNAME='C语言' AND sc.C#=c.C# AND s.S#=sc.S# AND x=s.S# AND n=s.SNAME) }",[253,95110,95112],{"id":95111},"_224-检索选修课程号为-k1-或-k5-的学生学号","22.4 检索选修课程号为 K1 或 K5 的学生学号。",[15,95114,58173],{},[15,95116,95117],{},[31,95118,95119],{},"{ \u003Cx> | EXISTS sc (SC(sc) AND x=sc.S# AND (sc.C#='K1' OR sc.C#='K5')) }",[253,95121,95123],{"id":95122},"_225-检索选修课程号为-k1-和-k5-的学生学号","22.5 检索选修课程号为 K1 和 K5 的学生学号。",[15,95125,58173],{},[15,95127,95128],{},[31,95129,95130],{},"{ \u003Cx> | EXISTS sc1 EXISTS sc2 (SC(sc1) AND SC(sc2) AND sc1.S#=x AND sc2.S#=x AND sc1.C#='K1' AND sc2.C#='K5') }",[253,95132,95134],{"id":95133},"_226-检索不选修-k8-课程的学生姓名和年龄","22.6 检索不选修 K8 课程的学生姓名和年龄。",[15,95136,58173],{},[15,95138,95139],{},[31,95140,95141],{},"{ \u003Cn, a> | EXISTS s (S(s) AND n=s.SNAME AND a=s.AGE AND NOT EXISTS sc (SC(sc) AND sc.S#=s.S# AND sc.C#='K8')) }",[253,95143,95145],{"id":95144},"_227-检索选修全部课程的学生姓名","22.7 检索选修全部课程的学生姓名。",[15,95147,58173],{},[15,95149,95150],{},[31,95151,95152],{},"{ \u003Cn> | EXISTS s (S(s) AND n=s.SNAME AND FORALL c (C(c) -> EXISTS sc (SC(sc) AND sc.S#=s.S# AND sc.C#=c.C#))) }",[15,95154,91513,95155,95158],{},[31,95156,95157],{},"FORALL c"," 表示对每一门课程，都存在该学生的选课记录。这是关系除法在元组演算中的表达。",[253,95160,95162],{"id":95161},"_228-检索所选修课程包含-1-号学生所选课程的学生学号","22.8 检索所选修课程包含 1 号学生所选课程的学生学号。",[15,95164,58173],{},[15,95166,95167],{},[31,95168,95169],{},"{ \u003Cx> | EXISTS s (S(s) AND x=s.S# AND FORALL sc1 ((SC(sc1) AND sc1.S#='1') -> EXISTS sc2 (SC(sc2) AND sc2.S#=x AND sc2.C#=sc1.C#))) }",[15,95171,95172],{},"讲解：对 1 号学生选过的每一门课，结果学生 x 都必须也选过。",[19,95174,95176],{"id":95175},"_23-对于上题的关系-sc-和-sc试用域演算表达式表示查询","23. 对于上题的关系 S、C 和 SC，试用域演算表达式表示查询。",[15,95178,95058,95179,46528,95182,46528,95185,347],{},[31,95180,95181],{},"S(sno, sname, age, sex)",[31,95183,95184],{},"C(cno, cname, teacher)",[31,95186,95187],{},"SC(sno, cno, grade)",[253,95189,95191],{"id":95190},"_231-检索选修课程号为-k5-的学生学号和成绩","23.1 检索选修课程号为 K5 的学生学号和成绩。",[15,95193,58173],{},[15,95195,95196],{},[31,95197,95198],{},"{ \u003Csno, grade> | SC(sno, 'K5', grade) }",[253,95200,95202],{"id":95201},"_232-检索选修课程号为-k8-的学生学号和姓名","23.2 检索选修课程号为 K8 的学生学号和姓名。",[15,95204,58173],{},[15,95206,95207],{},[31,95208,95209],{},"{ \u003Csno, sname> | EXISTS age EXISTS sex EXISTS grade (S(sno, sname, age, sex) AND SC(sno, 'K8', grade)) }",[19,95211,95213],{"id":95212},"_25-叙述使用-sql-语言实现各种关系运算的方法","25. 叙述使用 SQL 语言实现各种关系运算的方法。",[15,95215,58173],{},[13837,95217,95218,95228],{},[13840,95219,95220],{},[13843,95221,95222,95225],{},[13846,95223,95224],{},"关系运算",[13846,95226,95227],{},"SQL Server 写法",[13853,95229,95230,95240,95250,95260,95269,95278,95288,95298,95309,95322],{},[13843,95231,95232,95235],{},[13858,95233,95234],{},"选择",[13858,95236,95237],{},[31,95238,95239],{},"WHERE",[13843,95241,95242,95245],{},[13858,95243,95244],{},"投影",[13858,95246,95247],{},[31,95248,95249],{},"SELECT 列名",[13843,95251,95252,95255],{},[13858,95253,95254],{},"并",[13858,95256,95257],{},[31,95258,95259],{},"UNION",[13843,95261,95262,95265],{},[13858,95263,95264],{},"差",[13858,95266,95267],{},[31,95268,91056],{},[13843,95270,95271,95274],{},[13858,95272,95273],{},"交",[13858,95275,95276],{},[31,95277,91059],{},[13843,95279,95280,95283],{},[13858,95281,95282],{},"笛卡尔积",[13858,95284,95285],{},[31,95286,95287],{},"CROSS JOIN",[13843,95289,95290,95293],{},[13858,95291,95292],{},"连接",[13858,95294,95295],{},[31,95296,95297],{},"JOIN ... ON ...",[13843,95299,95300,95303],{},[13858,95301,95302],{},"重命名",[13858,95304,95305,95306],{},"表别名、列别名，如 ",[31,95307,95308],{},"AS",[13843,95310,95311,95314],{},[13858,95312,95313],{},"除法",[13858,95315,95316,95317,4374,95319],{},"双重 ",[31,95318,91063],{},[31,95320,95321],{},"GROUP BY ... HAVING",[13843,95323,95324,95327],{},[13858,95325,95326],{},"聚集",[13858,95328,95329,146,95332,146,95335,146,95338,146,95341,43298,95344],{},[31,95330,95331],{},"COUNT",[31,95333,95334],{},"SUM",[31,95336,95337],{},"AVG",[31,95339,95340],{},"MAX",[31,95342,95343],{},"MIN",[31,95345,95346],{},"GROUP BY",[15,95348,95349,95350,95353,95354,95357],{},"讲解：SQL Server 没有教材中某些系统的 ",[31,95351,95352],{},"NATURAL JOIN","，自然连接要显式写 ",[31,95355,95356],{},"JOIN ... ON"," 并选择需要的列。",[19,95359,95361],{"id":95360},"_26-设有关系-abab试用-sql-写查询并给出结果","26. 设有关系 A、B、AB，试用 SQL 写查询并给出结果。",[15,95363,94672],{},[15,95365,54151],{},[13837,95367,95368,95384],{},[13840,95369,95370],{},[13843,95371,95372,95375,95378,95381],{},[13846,95373,95374],{},"A#",[13846,95376,95377],{},"ANAME",[13846,95379,95380],{},"WQTY",[13846,95382,95383],{},"CITY",[13853,95385,95386,95400,95414,95427,95440],{},[13843,95387,95388,95391,95394,95397],{},[13858,95389,95390],{},"101",[13858,95392,95393],{},"韶山商店",[13858,95395,95396],{},"15",[13858,95398,95399],{},"长沙",[13843,95401,95402,95405,95408,95411],{},[13858,95403,95404],{},"204",[13858,95406,95407],{},"前门百货商店",[13858,95409,95410],{},"89",[13858,95412,95413],{},"北京",[13843,95415,95416,95419,95422,95425],{},[13858,95417,95418],{},"256",[13858,95420,95421],{},"东风商店",[13858,95423,95424],{},"501",[13858,95426,95413],{},[13843,95428,95429,95432,95435,95438],{},[13858,95430,95431],{},"345",[13858,95433,95434],{},"铁道商店",[13858,95436,95437],{},"76",[13858,95439,95399],{},[13843,95441,95442,95445,95448,95451],{},[13858,95443,95444],{},"620",[13858,95446,95447],{},"第一百货公司",[13858,95449,95450],{},"413",[13858,95452,95453],{},"上海",[15,95455,54224],{},[13837,95457,95458,95471],{},[13840,95459,95460],{},[13843,95461,95462,95465,95468],{},[13846,95463,95464],{},"B#",[13846,95466,95467],{},"BNAME",[13846,95469,95470],{},"PRICE",[13853,95472,95473,95483,95493,95503],{},[13843,95474,95475,95477,95480],{},[13858,95476,640],{},[13858,95478,95479],{},"毛笔",[13858,95481,95482],{},"21",[13843,95484,95485,95487,95490],{},[13858,95486,768],{},[13858,95488,95489],{},"羽毛球",[13858,95491,95492],{},"784",[13843,95494,95495,95497,95500],{},[13858,95496,786],{},[13858,95498,95499],{},"收音机",[13858,95501,95502],{},"1325",[13843,95504,95505,95507,95510],{},[13858,95506,15932],{},[13858,95508,95509],{},"书包",[13858,95511,95512],{},"242",[15,95514,93307],{},[13837,95516,95517,95528],{},[13840,95518,95519],{},[13843,95520,95521,95523,95525],{},[13846,95522,95374],{},[13846,95524,95464],{},[13846,95526,95527],{},"QTY",[13853,95529,95530,95539,95548,95557,95566,95575,95585,95594,95603,95612,95621],{},[13843,95531,95532,95534,95536],{},[13858,95533,95390],{},[13858,95535,640],{},[13858,95537,95538],{},"105",[13843,95540,95541,95543,95545],{},[13858,95542,95390],{},[13858,95544,768],{},[13858,95546,95547],{},"42",[13843,95549,95550,95552,95554],{},[13858,95551,95390],{},[13858,95553,786],{},[13858,95555,95556],{},"25",[13843,95558,95559,95561,95563],{},[13858,95560,95390],{},[13858,95562,15932],{},[13858,95564,95565],{},"104",[13843,95567,95568,95570,95572],{},[13858,95569,95404],{},[13858,95571,786],{},[13858,95573,95574],{},"61",[13843,95576,95577,95580,95582],{},[13858,95578,95579],{},"206",[13858,95581,640],{},[13858,95583,95584],{},"241",[13843,95586,95587,95589,95591],{},[13858,95588,95579],{},[13858,95590,768],{},[13858,95592,95593],{},"91",[13843,95595,95596,95598,95600],{},[13858,95597,95431],{},[13858,95599,640],{},[13858,95601,95602],{},"141",[13843,95604,95605,95607,95609],{},[13858,95606,95431],{},[13858,95608,768],{},[13858,95610,95611],{},"18",[13843,95613,95614,95616,95618],{},[13858,95615,95431],{},[13858,95617,15932],{},[13858,95619,95620],{},"74",[13843,95622,95623,95626,95628],{},[13858,95624,95625],{},"602",[13858,95627,15932],{},[13858,95629,95630],{},"125",[15,95632,95633,95634,95637],{},"注意：原表中 ",[31,95635,95636],{},"AB.A#"," 出现 206 和 602，但 A 表没有这两个商店；题目第 3 问又说“代号为 256 的商店”。按常见教材题意，206 很可能是 256 的误写。下面 SQL 按题意写，结果说明会标注假设。",[253,95639,95641],{"id":95640},"_261-找出店员人数不超过-100-人或者在长沙的所有商店的代号和商品名","26.1 找出店员人数不超过 100 人或者在长沙的所有商店的代号和商品名。",[24,95643,95645],{"className":92584,"code":95644,"language":92586,"meta":29,"style":29},"SELECT DISTINCT a.[A#], b.BName\nFROM dbo.A AS a\nJOIN dbo.AB AS ab\n ON ab.[A#] = a.[A#]\nJOIN dbo.B AS b\n ON b.[B#] = ab.[B#]\nWHERE a.WQty \u003C= 100\n OR a.City = N'长沙';\n",[31,95646,95647,95652,95657,95662,95667,95672,95677,95682],{"__ignoreMap":29},[34,95648,95649],{"class":36,"line":37},[34,95650,95651],{},"SELECT DISTINCT a.[A#], b.BName\n",[34,95653,95654],{"class":36,"line":43},[34,95655,95656],{},"FROM dbo.A AS a\n",[34,95658,95659],{"class":36,"line":49},[34,95660,95661],{},"JOIN dbo.AB AS ab\n",[34,95663,95664],{"class":36,"line":56},[34,95665,95666],{}," ON ab.[A#] = a.[A#]\n",[34,95668,95669],{"class":36,"line":62},[34,95670,95671],{},"JOIN dbo.B AS b\n",[34,95673,95674],{"class":36,"line":68},[34,95675,95676],{}," ON b.[B#] = ab.[B#]\n",[34,95678,95679],{"class":36,"line":74},[34,95680,95681],{},"WHERE a.WQty \u003C= 100\n",[34,95683,95684],{"class":36,"line":1971},[34,95685,95686],{}," OR a.City = N'长沙';\n",[15,95688,95689],{},"按原表中能与 A 表连接上的记录，结果为：",[13837,95691,95692,95700],{},[13840,95693,95694],{},[13843,95695,95696,95698],{},[13846,95697,95374],{},[13846,95699,95467],{},[13853,95701,95702,95708,95714,95720,95726,95732,95738,95744],{},[13843,95703,95704,95706],{},[13858,95705,95390],{},[13858,95707,95479],{},[13843,95709,95710,95712],{},[13858,95711,95390],{},[13858,95713,95489],{},[13843,95715,95716,95718],{},[13858,95717,95390],{},[13858,95719,95499],{},[13843,95721,95722,95724],{},[13858,95723,95390],{},[13858,95725,95509],{},[13843,95727,95728,95730],{},[13858,95729,95404],{},[13858,95731,95499],{},[13843,95733,95734,95736],{},[13858,95735,95431],{},[13858,95737,95479],{},[13843,95739,95740,95742],{},[13858,95741,95431],{},[13858,95743,95489],{},[13843,95745,95746,95748],{},[13858,95747,95431],{},[13858,95749,95509],{},[253,95751,95753],{"id":95752},"_262-找出供应书包的商店名","26.2 找出供应书包的商店名。",[24,95755,95757],{"className":92584,"code":95756,"language":92586,"meta":29,"style":29},"SELECT DISTINCT a.AName\nFROM dbo.A AS a\nJOIN dbo.AB AS ab\n ON ab.[A#] = a.[A#]\nJOIN dbo.B AS b\n ON b.[B#] = ab.[B#]\nWHERE b.BName = N'书包';\n",[31,95758,95759,95764,95768,95772,95776,95780,95784],{"__ignoreMap":29},[34,95760,95761],{"class":36,"line":37},[34,95762,95763],{},"SELECT DISTINCT a.AName\n",[34,95765,95766],{"class":36,"line":43},[34,95767,95656],{},[34,95769,95770],{"class":36,"line":49},[34,95771,95661],{},[34,95773,95774],{"class":36,"line":56},[34,95775,95666],{},[34,95777,95778],{"class":36,"line":62},[34,95779,95671],{},[34,95781,95782],{"class":36,"line":68},[34,95783,95676],{},[34,95785,95786],{"class":36,"line":74},[34,95787,95788],{},"WHERE b.BName = N'书包';\n",[15,95790,95791],{},"结果：",[13837,95793,95794,95800],{},[13840,95795,95796],{},[13843,95797,95798],{},[13846,95799,95377],{},[13853,95801,95802,95806],{},[13843,95803,95804],{},[13858,95805,95393],{},[13843,95807,95808],{},[13858,95809,95434],{},[15,95811,91513,95812,95814],{},[31,95813,95625],{}," 在 A 表中没有对应商店，所以按内连接不会出现在结果中。",[253,95816,95818],{"id":95817},"_263-找出至少供应代号为-256-的商店所供应的全部商品的商店名和所在城市","26.3 找出至少供应代号为 256 的商店所供应的全部商品的商店名和所在城市。",[24,95820,95822],{"className":92584,"code":95821,"language":92586,"meta":29,"style":29},"SELECT a.AName, a.City\nFROM dbo.A AS a\nWHERE NOT EXISTS (\n SELECT 1\n FROM dbo.AB AS base_ab\n WHERE base_ab.[A#] = 256\n AND NOT EXISTS (\n SELECT 1\n FROM dbo.AB AS ab\n WHERE ab.[A#] = a.[A#]\n AND ab.[B#] = base_ab.[B#]\n )\n);\n",[31,95823,95824,95829,95833,95837,95841,95846,95851,95855,95859,95864,95869,95874,95878],{"__ignoreMap":29},[34,95825,95826],{"class":36,"line":37},[34,95827,95828],{},"SELECT a.AName, a.City\n",[34,95830,95831],{"class":36,"line":43},[34,95832,95656],{},[34,95834,95835],{"class":36,"line":49},[34,95836,92965],{},[34,95838,95839],{"class":36,"line":56},[34,95840,92970],{},[34,95842,95843],{"class":36,"line":62},[34,95844,95845],{}," FROM dbo.AB AS base_ab\n",[34,95847,95848],{"class":36,"line":68},[34,95849,95850],{}," WHERE base_ab.[A#] = 256\n",[34,95852,95853],{"class":36,"line":74},[34,95854,93125],{},[34,95856,95857],{"class":36,"line":1971},[34,95858,93130],{},[34,95860,95861],{"class":36,"line":1976},[34,95862,95863],{}," FROM dbo.AB AS ab\n",[34,95865,95866],{"class":36,"line":1985},[34,95867,95868],{}," WHERE ab.[A#] = a.[A#]\n",[34,95870,95871],{"class":36,"line":1990},[34,95872,95873],{}," AND ab.[B#] = base_ab.[B#]\n",[34,95875,95876],{"class":36,"line":1999},[34,95877,93150],{},[34,95879,95880],{"class":36,"line":2022},[34,95881,462],{},[15,95883,95884],{},"如果把原表中的 206 按题意修正为 256，则 256 供应商品 1 和 2，结果为：",[13837,95886,95887,95895],{},[13840,95888,95889],{},[13843,95890,95891,95893],{},[13846,95892,95377],{},[13846,95894,95383],{},[13853,95896,95897,95903,95909],{},[13843,95898,95899,95901],{},[13858,95900,95393],{},[13858,95902,95399],{},[13843,95904,95905,95907],{},[13858,95906,95421],{},[13858,95908,95413],{},[13843,95910,95911,95913],{},[13858,95912,95434],{},[13858,95914,95399],{},[15,95916,95917],{},"讲解：这是除法查询。逻辑是“不存在 256 供应的某个商品，而当前商店没有供应它”。",[19,95919,95921],{"id":95920},"_27-有图书登记表-tsbno-bc-bna-au-pub按要求用-sql-语句设计","27. 有图书登记表 TS(BNO, BC, BNA, AU, PUB)，按要求用 SQL 语句设计。",[253,95923,95925],{"id":95924},"_27a-按图书编号-bno-建立-ts-表的索引-its","27.a 按图书编号 BNO 建立 TS 表的索引 ITS。",[24,95927,95929],{"className":92584,"code":95928,"language":92586,"meta":29,"style":29},"CREATE INDEX ITS ON dbo.TS(BNO);\n",[31,95930,95931],{"__ignoreMap":29},[34,95932,95933],{"class":36,"line":37},[34,95934,95928],{},[253,95936,95938],{"id":95937},"_27b-查找按出版社统计其出版图书总数","27.b 查找按出版社统计其出版图书总数。",[24,95940,95942],{"className":92584,"code":95941,"language":92586,"meta":29,"style":29},"SELECT PUB, COUNT(*) AS BookCount\nFROM dbo.TS\nGROUP BY PUB;\n",[31,95943,95944,95949,95954],{"__ignoreMap":29},[34,95945,95946],{"class":36,"line":37},[34,95947,95948],{},"SELECT PUB, COUNT(*) AS BookCount\n",[34,95950,95951],{"class":36,"line":43},[34,95952,95953],{},"FROM dbo.TS\n",[34,95955,95956],{"class":36,"line":49},[34,95957,95958],{},"GROUP BY PUB;\n",[253,95960,95962],{"id":95961},"_27c-删除索引-its","27.c 删除索引 ITS。",[24,95964,95966],{"className":92584,"code":95965,"language":92586,"meta":29,"style":29},"DROP INDEX ITS ON dbo.TS;\n",[31,95967,95968],{"__ignoreMap":29},[34,95969,95970],{"class":36,"line":37},[34,95971,95965],{},[15,95973,95974,95975,95978,95979,347],{},"讲解：SQL Server 删除索引的语法是 ",[31,95976,95977],{},"DROP INDEX 索引名 ON 表名","，不同于 Oracle 的 ",[31,95980,95981],{},"DROP INDEX 索引名",[19,95983,95985],{"id":95984},"_28-已知三个关系-rs-和-t试用-sql-建立视图并分组求平均值","28. 已知三个关系 R、S 和 T，试用 SQL 建立视图并分组求平均值。",[15,95987,94672],{},[15,95989,34386],{},[13837,95991,95992,96002],{},[13840,95993,95994],{},[13843,95995,95996,95998,96000],{},[13846,95997,54151],{},[13846,95999,54224],{},[13846,96001,47384],{},[13853,96003,96004,96014,96024,96034],{},[13843,96005,96006,96009,96012],{},[13858,96007,96008],{},"a1",[13858,96010,96011],{},"b1",[13858,96013,3249],{},[13843,96015,96016,96019,96022],{},[13858,96017,96018],{},"a2",[13858,96020,96021],{},"b3",[13858,96023,94716],{},[13843,96025,96026,96029,96032],{},[13858,96027,96028],{},"a3",[13858,96030,96031],{},"b2",[13858,96033,95611],{},[13843,96035,96036,96039,96042],{},[13858,96037,96038],{},"a4",[13858,96040,96041],{},"b4",[13858,96043,71869],{},[15,96045,94675],{},[13837,96047,96048,96059],{},[13840,96049,96050],{},[13843,96051,96052,96054,96056],{},[13846,96053,54151],{},[13846,96055,91994],{},[13846,96057,96058],{},"E",[13853,96060,96061,96070,96079],{},[13843,96062,96063,96065,96068],{},[13858,96064,96008],{},[13858,96066,96067],{},"d1",[13858,96069,95396],{},[13843,96071,96072,96074,96077],{},[13858,96073,96018],{},[13858,96075,96076],{},"d2",[13858,96078,95611],{},[13843,96080,96081,96083,96085],{},[13858,96082,96008],{},[13858,96084,96076],{},[13858,96086,96087],{},"24",[15,96089,37400],{},[13837,96091,96092,96100],{},[13840,96093,96094],{},[13843,96095,96096,96098],{},[13846,96097,91994],{},[13846,96099,51834],{},[13853,96101,96102,96110],{},[13843,96103,96104,96107],{},[13858,96105,96106],{},"D2",[13858,96108,96109],{},"f2",[13843,96111,96112,96115],{},[13858,96113,96114],{},"d3",[13858,96116,96117],{},"f3",[253,96119,96121,96122,347],{"id":96120},"_281-将-rst-按关联属性建立视图-r-s-t","28.1 将 R、S、T 按关联属性建立视图 ",[31,96123,96124],{},"R-S-T",[24,96126,96128],{"className":92584,"code":96127,"language":92586,"meta":29,"style":29},"CREATE VIEW dbo.[R-S-T]\nAS\nSELECT r.A, r.B, r.C, s.D, s.E, t.F\nFROM dbo.R AS r\nJOIN dbo.S AS s\n ON s.A = r.A\nJOIN dbo.T AS t\n ON t.D = s.D;\n",[31,96129,96130,96135,96139,96144,96149,96154,96159,96164],{"__ignoreMap":29},[34,96131,96132],{"class":36,"line":37},[34,96133,96134],{},"CREATE VIEW dbo.[R-S-T]\n",[34,96136,96137],{"class":36,"line":43},[34,96138,93260],{},[34,96140,96141],{"class":36,"line":49},[34,96142,96143],{},"SELECT r.A, r.B, r.C, s.D, s.E, t.F\n",[34,96145,96146],{"class":36,"line":56},[34,96147,96148],{},"FROM dbo.R AS r\n",[34,96150,96151],{"class":36,"line":62},[34,96152,96153],{},"JOIN dbo.S AS s\n",[34,96155,96156],{"class":36,"line":68},[34,96157,96158],{}," ON s.A = r.A\n",[34,96160,96161],{"class":36,"line":74},[34,96162,96163],{},"JOIN dbo.T AS t\n",[34,96165,96166],{"class":36,"line":1971},[34,96167,96168],{}," ON t.D = s.D;\n",[15,96170,96171,96172,96174,96175,96177,96178,347],{},"讲解：题中 T 表第一行写作 ",[31,96173,96106],{},"，而 S 表为 ",[31,96176,96076],{},"。SQL Server 默认排序规则通常大小写不敏感，若数据库使用大小写敏感排序规则，需要统一为 ",[31,96179,96076],{},[253,96181,96183],{"id":96182},"_282-对视图按-a-分组求-c-和-e-的平均值","28.2 对视图按 A 分组，求 C 和 E 的平均值。",[24,96185,96187],{"className":92584,"code":96186,"language":92586,"meta":29,"style":29},"SELECT A,\n AVG(CAST(C AS decimal(10, 2))) AS AvgC,\n AVG(CAST(E AS decimal(10, 2))) AS AvgE\nFROM dbo.[R-S-T]\nGROUP BY A;\n",[31,96188,96189,96194,96199,96204,96209],{"__ignoreMap":29},[34,96190,96191],{"class":36,"line":37},[34,96192,96193],{},"SELECT A,\n",[34,96195,96196],{"class":36,"line":43},[34,96197,96198],{}," AVG(CAST(C AS decimal(10, 2))) AS AvgC,\n",[34,96200,96201],{"class":36,"line":49},[34,96202,96203],{}," AVG(CAST(E AS decimal(10, 2))) AS AvgE\n",[34,96205,96206],{"class":36,"line":56},[34,96207,96208],{},"FROM dbo.[R-S-T]\n",[34,96210,96211],{"class":36,"line":62},[34,96212,96213],{},"GROUP BY A;\n",[15,96215,96216,96217,96220],{},"在大小写不敏感且 ",[31,96218,96219],{},"D2=d2"," 的假设下，结果为：",[13837,96222,96223,96235],{},[13840,96224,96225],{},[13843,96226,96227,96229,96232],{},[13846,96228,54151],{},[13846,96230,96231],{},"AvgC",[13846,96233,96234],{},"AvgE",[13853,96236,96237,96247],{},[13843,96238,96239,96241,96244],{},[13858,96240,96008],{},[13858,96242,96243],{},"20.00",[13858,96245,96246],{},"24.00",[13843,96248,96249,96251,96254],{},[13858,96250,96018],{},[13858,96252,96253],{},"22.00",[13858,96255,96256],{},"18.00",[19,96258,96260],{"id":96259},"_29-设有关系-r-和-s试用-sql-实现查询和更新","29. 设有关系 R 和 S，试用 SQL 实现查询和更新。",[15,96262,94672],{},[15,96264,34386],{},[13837,96266,96267,96275],{},[13840,96268,96269],{},[13843,96270,96271,96273],{},[13846,96272,54151],{},[13846,96274,54224],{},[13853,96276,96277,96283,96289],{},[13843,96278,96279,96281],{},[13858,96280,96008],{},[13858,96282,96011],{},[13843,96284,96285,96287],{},[13858,96286,96018],{},[13858,96288,96031],{},[13843,96290,96291,96293],{},[13858,96292,96028],{},[13858,96294,96021],{},[15,96296,94675],{},[13837,96298,96299,96307],{},[13840,96300,96301],{},[13843,96302,96303,96305],{},[13846,96304,54151],{},[13846,96306,47384],{},[13853,96308,96309,96315,96321],{},[13843,96310,96311,96313],{},[13858,96312,96008],{},[13858,96314,5731],{},[13843,96316,96317,96319],{},[13858,96318,96018],{},[13858,96320,5774],{},[13843,96322,96323,96325],{},[13858,96324,96028],{},[13858,96326,96327],{},"55",[253,96329,96331],{"id":96330},"_291-查询属性-c-50-时r-中与之相关联的属性-b-之值","29.1 查询属性 C > 50 时，R 中与之相关联的属性 B 之值。",[24,96333,96335],{"className":92584,"code":96334,"language":92586,"meta":29,"style":29},"SELECT r.B\nFROM dbo.R AS r\nJOIN dbo.S AS s\n ON s.A = r.A\nWHERE s.C > 50;\n",[31,96336,96337,96342,96346,96350,96354],{"__ignoreMap":29},[34,96338,96339],{"class":36,"line":37},[34,96340,96341],{},"SELECT r.B\n",[34,96343,96344],{"class":36,"line":43},[34,96345,96148],{},[34,96347,96348],{"class":36,"line":49},[34,96349,96153],{},[34,96351,96352],{"class":36,"line":56},[34,96353,96158],{},[34,96355,96356],{"class":36,"line":62},[34,96357,96358],{},"WHERE s.C > 50;\n",[15,96360,95791,96361,347],{},[31,96362,96021],{},[253,96364,96366],{"id":96365},"_292-当属性-c-40-时将-r-中与之相关联的属性-b-值修改为-b4","29.2 当属性 C = 40 时，将 R 中与之相关联的属性 B 值修改为 B4。",[24,96368,96370],{"className":92584,"code":96369,"language":92586,"meta":29,"style":29},"UPDATE r\nSET B = 'B4'\nFROM dbo.R AS r\nJOIN dbo.S AS s\n ON s.A = r.A\nWHERE s.C = 40;\n",[31,96371,96372,96377,96382,96386,96390,96394],{"__ignoreMap":29},[34,96373,96374],{"class":36,"line":37},[34,96375,96376],{},"UPDATE r\n",[34,96378,96379],{"class":36,"line":43},[34,96380,96381],{},"SET B = 'B4'\n",[34,96383,96384],{"class":36,"line":49},[34,96385,96148],{},[34,96387,96388],{"class":36,"line":56},[34,96389,96153],{},[34,96391,96392],{"class":36,"line":62},[34,96393,96158],{},[34,96395,96396],{"class":36,"line":68},[34,96397,96398],{},"WHERE s.C = 40;\n",[15,96400,96401,96402,347],{},"讲解：SQL Server 的连接更新使用 ",[31,96403,96404],{},"UPDATE r SET ... FROM R AS r JOIN ...",[19,96406,96408],{"id":96407},"_30-已知关系-r-和-s执行给定-sql给出视图-h-内容和查询结果","30. 已知关系 R 和 S，执行给定 SQL，给出视图 H 内容和查询结果。",[15,96410,94672],{},[15,96412,34386],{},[13837,96414,96415,96425],{},[13840,96416,96417],{},[13843,96418,96419,96421,96423],{},[13846,96420,54151],{},[13846,96422,54224],{},[13846,96424,47384],{},[13853,96426,96427,96435,96443,96451],{},[13843,96428,96429,96431,96433],{},[13858,96430,96008],{},[13858,96432,96011],{},[13858,96434,3249],{},[13843,96436,96437,96439,96441],{},[13858,96438,96008],{},[13858,96440,96031],{},[13858,96442,94716],{},[13843,96444,96445,96447,96449],{},[13858,96446,96018],{},[13858,96448,96011],{},[13858,96450,95611],{},[13843,96452,96453,96455,96457],{},[13858,96454,96018],{},[13858,96456,96021],{},[13858,96458,96018],{},[15,96460,94675],{},[13837,96462,96463,96473],{},[13840,96464,96465],{},[13843,96466,96467,96469,96471],{},[13846,96468,54151],{},[13846,96470,91994],{},[13846,96472,96058],{},[13853,96474,96475,96483,96491],{},[13843,96476,96477,96479,96481],{},[13858,96478,96008],{},[13858,96480,96067],{},[13858,96482,95396],{},[13843,96484,96485,96487,96489],{},[13858,96486,96018],{},[13858,96488,96076],{},[13858,96490,95611],{},[13843,96492,96493,96495,96497],{},[13858,96494,96008],{},[13858,96496,96076],{},[13858,96498,96087],{},[15,96500,96501],{},"题面 SQL：",[24,96503,96505],{"className":92584,"code":96504,"language":92586,"meta":29,"style":29},"CREATE SQL VIEW H(A, B, C, D, E)\nAS SELECT A, B, R.C, D, E\nFROM R, S\nWHERE R.C = S.C;\n\nSELECT B, D, E\nFROM H\nWHERE C = 'C2';\n",[31,96506,96507,96512,96517,96522,96527,96531,96536,96541],{"__ignoreMap":29},[34,96508,96509],{"class":36,"line":37},[34,96510,96511],{},"CREATE SQL VIEW H(A, B, C, D, E)\n",[34,96513,96514],{"class":36,"line":43},[34,96515,96516],{},"AS SELECT A, B, R.C, D, E\n",[34,96518,96519],{"class":36,"line":49},[34,96520,96521],{},"FROM R, S\n",[34,96523,96524],{"class":36,"line":56},[34,96525,96526],{},"WHERE R.C = S.C;\n",[34,96528,96529],{"class":36,"line":62},[34,96530,53],{"emptyLinePlaceholder":52},[34,96532,96533],{"class":36,"line":68},[34,96534,96535],{},"SELECT B, D, E\n",[34,96537,96538],{"class":36,"line":74},[34,96539,96540],{},"FROM H\n",[34,96542,96543],{"class":36,"line":1971},[34,96544,96545],{},"WHERE C = 'C2';\n",[15,96547,58173],{},[15,96549,96550,96551,96553,96554,96557,96558,96561,96562,96565],{},"原题 SQL 有明显笔误：S 表没有 ",[31,96552,47384],{}," 列，因此 ",[31,96555,96556],{},"WHERE R.C = S.C"," 在题面关系上无法执行；并且题面 R.C 的值为 ",[31,96559,96560],{},"20, 22, 18, a2","，与查询条件 ",[31,96563,96564],{},"C='C2'"," 也不一致。",[15,96567,96568,96569,96572],{},"按最常见修正，把连接条件改为 ",[31,96570,96571],{},"R.A = S.A","，SQL Server 写法为：",[24,96574,96576],{"className":92584,"code":96575,"language":92586,"meta":29,"style":29},"CREATE VIEW dbo.H(A, B, C, D, E)\nAS\nSELECT r.A, r.B, r.C, s.D, s.E\nFROM dbo.R AS r\nJOIN dbo.S AS s\n ON s.A = r.A;\n",[31,96577,96578,96583,96587,96592,96596,96600],{"__ignoreMap":29},[34,96579,96580],{"class":36,"line":37},[34,96581,96582],{},"CREATE VIEW dbo.H(A, B, C, D, E)\n",[34,96584,96585],{"class":36,"line":43},[34,96586,93260],{},[34,96588,96589],{"class":36,"line":49},[34,96590,96591],{},"SELECT r.A, r.B, r.C, s.D, s.E\n",[34,96593,96594],{"class":36,"line":56},[34,96595,96148],{},[34,96597,96598],{"class":36,"line":62},[34,96599,96153],{},[34,96601,96602],{"class":36,"line":68},[34,96603,96604],{}," ON s.A = r.A;\n",[15,96606,96607],{},"则视图 H 为：",[13837,96609,96610,96624],{},[13840,96611,96612],{},[13843,96613,96614,96616,96618,96620,96622],{},[13846,96615,54151],{},[13846,96617,54224],{},[13846,96619,47384],{},[13846,96621,91994],{},[13846,96623,96058],{},[13853,96625,96626,96638,96650,96662,96674,96686],{},[13843,96627,96628,96630,96632,96634,96636],{},[13858,96629,96008],{},[13858,96631,96011],{},[13858,96633,3249],{},[13858,96635,96067],{},[13858,96637,95396],{},[13843,96639,96640,96642,96644,96646,96648],{},[13858,96641,96008],{},[13858,96643,96011],{},[13858,96645,3249],{},[13858,96647,96076],{},[13858,96649,96087],{},[13843,96651,96652,96654,96656,96658,96660],{},[13858,96653,96008],{},[13858,96655,96031],{},[13858,96657,94716],{},[13858,96659,96067],{},[13858,96661,95396],{},[13843,96663,96664,96666,96668,96670,96672],{},[13858,96665,96008],{},[13858,96667,96031],{},[13858,96669,94716],{},[13858,96671,96076],{},[13858,96673,96087],{},[13843,96675,96676,96678,96680,96682,96684],{},[13858,96677,96018],{},[13858,96679,96011],{},[13858,96681,95611],{},[13858,96683,96076],{},[13858,96685,95611],{},[13843,96687,96688,96690,96692,96694,96696],{},[13858,96689,96018],{},[13858,96691,96021],{},[13858,96693,96018],{},[13858,96695,96076],{},[13858,96697,95611],{},[15,96699,96700,96701,96704,96705,96708,96709,96712],{},"若仍按题面查询 ",[31,96702,96703],{},"WHERE C='C2'","，结果为空。若老师原意是 ",[31,96706,96707],{},"WHERE C=22"," 或原表 C 值应为 ",[31,96710,96711],{},"C2","，对应结果为：",[13837,96714,96715,96725],{},[13840,96716,96717],{},[13843,96718,96719,96721,96723],{},[13846,96720,54224],{},[13846,96722,91994],{},[13846,96724,96058],{},[13853,96726,96727,96735],{},[13843,96728,96729,96731,96733],{},[13858,96730,96031],{},[13858,96732,96067],{},[13858,96734,95396],{},[13843,96736,96737,96739,96741],{},[13858,96738,96031],{},[13858,96740,96076],{},[13858,96742,96087],{},[15,96744,96745],{},"讲解：复习时要会判断题面是否类型和属性一致。关系模式不含某属性时，SQL 不能执行；不能为了凑答案忽略这个错误。",[19,96747,96749],{"id":96748},"_31-已知关系-r按-a-分组求-c-的最大值和最小值并置于视图-rve-中再查询-a98-的记录","31. 已知关系 R，按 A 分组求 C 的最大值和最小值，并置于视图 RVE 中；再查询 A='98' 的记录。",[15,96751,94672],{},[13837,96753,96754,96764],{},[13840,96755,96756],{},[13843,96757,96758,96760,96762],{},[13846,96759,54151],{},[13846,96761,54224],{},[13846,96763,47384],{},[13853,96765,96766,96775,96783,96792,96801,96809,96818,96827],{},[13843,96767,96768,96771,96773],{},[13858,96769,96770],{},"97",[13858,96772,96011],{},[13858,96774,94841],{},[13843,96776,96777,96779,96781],{},[13858,96778,96770],{},[13858,96780,96031],{},[13858,96782,94824],{},[13843,96784,96785,96787,96789],{},[13858,96786,96770],{},[13858,96788,96021],{},[13858,96790,96791],{},"98",[13843,96793,96794,96796,96798],{},[13858,96795,96791],{},[13858,96797,96011],{},[13858,96799,96800],{},"72",[13843,96802,96803,96805,96807],{},[13858,96804,96791],{},[13858,96806,96031],{},[13858,96808,94841],{},[13843,96810,96811,96813,96815],{},[13858,96812,96791],{},[13858,96814,96021],{},[13858,96816,96817],{},"95",[13843,96819,96820,96822,96824],{},[13858,96821,29551],{},[13858,96823,96011],{},[13858,96825,96826],{},"88",[13843,96828,96829,96831,96833],{},[13858,96830,29551],{},[13858,96832,96031],{},[13858,96834,96835],{},"94",[15,96837,58173],{},[24,96839,96841],{"className":92584,"code":96840,"language":92586,"meta":29,"style":29},"CREATE VIEW dbo.RVE\nAS\nSELECT A,\n MAX(C) AS MaxC,\n MIN(C) AS MinC\nFROM dbo.R\nGROUP BY A;\n\nSELECT A, MaxC, MinC\nFROM dbo.RVE\nWHERE A = '98';\n",[31,96842,96843,96848,96852,96856,96861,96866,96871,96875,96879,96884,96889],{"__ignoreMap":29},[34,96844,96845],{"class":36,"line":37},[34,96846,96847],{},"CREATE VIEW dbo.RVE\n",[34,96849,96850],{"class":36,"line":43},[34,96851,93260],{},[34,96853,96854],{"class":36,"line":49},[34,96855,96193],{},[34,96857,96858],{"class":36,"line":56},[34,96859,96860],{}," MAX(C) AS MaxC,\n",[34,96862,96863],{"class":36,"line":62},[34,96864,96865],{}," MIN(C) AS MinC\n",[34,96867,96868],{"class":36,"line":68},[34,96869,96870],{},"FROM dbo.R\n",[34,96872,96873],{"class":36,"line":74},[34,96874,96213],{},[34,96876,96877],{"class":36,"line":1971},[34,96878,53],{"emptyLinePlaceholder":52},[34,96880,96881],{"class":36,"line":1976},[34,96882,96883],{},"SELECT A, MaxC, MinC\n",[34,96885,96886],{"class":36,"line":1985},[34,96887,96888],{},"FROM dbo.RVE\n",[34,96890,96891],{"class":36,"line":1990},[34,96892,96893],{},"WHERE A = '98';\n",[15,96895,96896],{},"视图 RVE：",[13837,96898,96899,96911],{},[13840,96900,96901],{},[13843,96902,96903,96905,96908],{},[13846,96904,54151],{},[13846,96906,96907],{},"MaxC",[13846,96909,96910],{},"MinC",[13853,96912,96913,96921,96929],{},[13843,96914,96915,96917,96919],{},[13858,96916,96770],{},[13858,96918,96791],{},[13858,96920,94841],{},[13843,96922,96923,96925,96927],{},[13858,96924,96791],{},[13858,96926,96817],{},[13858,96928,96800],{},[13843,96930,96931,96933,96935],{},[13858,96932,29551],{},[13858,96934,96835],{},[13858,96936,96826],{},[15,96938,96939],{},"查询结果：",[13837,96941,96942,96952],{},[13840,96943,96944],{},[13843,96945,96946,96948,96950],{},[13846,96947,54151],{},[13846,96949,96907],{},[13846,96951,96910],{},[13853,96953,96954],{},[13843,96955,96956,96958,96960],{},[13858,96957,96791],{},[13858,96959,96817],{},[13858,96961,96800],{},[19,96963,96965],{"id":96964},"_32-已知学生表-ssno-sn-sd-prov-和学生选课表-scsno-cn-gr用-sql-实现查询","32. 已知学生表 S(SNO, SN, SD, PROV) 和学生选课表 SC(SNO, CN, GR)，用 SQL 实现查询。",[253,96967,96969],{"id":96968},"_321-查询信息系的学生来自哪些省区","32.1 查询“信息系”的学生来自哪些省区。",[24,96971,96973],{"className":92584,"code":96972,"language":92586,"meta":29,"style":29},"SELECT DISTINCT PROV\nFROM dbo.S\nWHERE SD = N'信息系';\n",[31,96974,96975,96980,96985],{"__ignoreMap":29},[34,96976,96977],{"class":36,"line":37},[34,96978,96979],{},"SELECT DISTINCT PROV\n",[34,96981,96982],{"class":36,"line":43},[34,96983,96984],{},"FROM dbo.S\n",[34,96986,96987],{"class":36,"line":49},[34,96988,96989],{},"WHERE SD = N'信息系';\n",[253,96991,96993],{"id":96992},"_322-按分数降序排序输出英语系的学生选修了计算机课程的学生姓名和分数","32.2 按分数降序排序，输出“英语系”的学生选修了“计算机”课程的学生姓名和分数。",[24,96995,96997],{"className":92584,"code":96996,"language":92586,"meta":29,"style":29},"SELECT s.SN, sc.GR\nFROM dbo.S AS s\nJOIN dbo.SC AS sc\n ON sc.SNO = s.SNO\nWHERE s.SD = N'英语系'\n AND sc.CN = N'计算机'\nORDER BY sc.GR DESC;\n",[31,96998,96999,97004,97009,97014,97019,97024,97029],{"__ignoreMap":29},[34,97000,97001],{"class":36,"line":37},[34,97002,97003],{},"SELECT s.SN, sc.GR\n",[34,97005,97006],{"class":36,"line":43},[34,97007,97008],{},"FROM dbo.S AS s\n",[34,97010,97011],{"class":36,"line":49},[34,97012,97013],{},"JOIN dbo.SC AS sc\n",[34,97015,97016],{"class":36,"line":56},[34,97017,97018],{}," ON sc.SNO = s.SNO\n",[34,97020,97021],{"class":36,"line":62},[34,97022,97023],{},"WHERE s.SD = N'英语系'\n",[34,97025,97026],{"class":36,"line":68},[34,97027,97028],{}," AND sc.CN = N'计算机'\n",[34,97030,97031],{"class":36,"line":74},[34,97032,97033],{},"ORDER BY sc.GR DESC;\n",[15,97035,91513,97036,97039],{},[31,97037,97038],{},"DISTINCT"," 用于去重省区；排序只能保证最终查询结果顺序，不应依赖表内物理顺序。",[19,97041,97043],{"id":97042},"_33-设学生表-ssno-sn-和选课表-scsno-cno-cn-g完成视图和查询","33. 设学生表 S(SNO, SN) 和选课表 SC(SNO, CNO, CN, G)，完成视图和查询。",[253,97045,97047,97048,97051],{"id":97046},"_331-建立视图-v-sscsno-sn-cno-cn-g并按-cno-升序排序","33.1 建立视图 ",[31,97049,97050],{},"V-SSC(SNO, SN, CNO, CN, G)","，并按 CNO 升序排序。",[15,97053,97054],{},"SQL Server 写法：",[24,97056,97058],{"className":92584,"code":97057,"language":92586,"meta":29,"style":29},"CREATE VIEW dbo.[V-SSC]\nAS\nSELECT s.SNO, s.SN, sc.CNO, sc.CN, sc.G\nFROM dbo.S AS s\nJOIN dbo.SC AS sc\n ON sc.SNO = s.SNO;\n\nSELECT SNO, SN, CNO, CN, G\nFROM dbo.[V-SSC]\nORDER BY CNO ASC;\n",[31,97059,97060,97065,97069,97074,97078,97082,97087,97091,97096,97101],{"__ignoreMap":29},[34,97061,97062],{"class":36,"line":37},[34,97063,97064],{},"CREATE VIEW dbo.[V-SSC]\n",[34,97066,97067],{"class":36,"line":43},[34,97068,93260],{},[34,97070,97071],{"class":36,"line":49},[34,97072,97073],{},"SELECT s.SNO, s.SN, sc.CNO, sc.CN, sc.G\n",[34,97075,97076],{"class":36,"line":56},[34,97077,97008],{},[34,97079,97080],{"class":36,"line":62},[34,97081,97013],{},[34,97083,97084],{"class":36,"line":68},[34,97085,97086],{}," ON sc.SNO = s.SNO;\n",[34,97088,97089],{"class":36,"line":74},[34,97090,53],{"emptyLinePlaceholder":52},[34,97092,97093],{"class":36,"line":1971},[34,97094,97095],{},"SELECT SNO, SN, CNO, CN, G\n",[34,97097,97098],{"class":36,"line":1976},[34,97099,97100],{},"FROM dbo.[V-SSC]\n",[34,97102,97103],{"class":36,"line":1985},[34,97104,97105],{},"ORDER BY CNO ASC;\n",[15,97107,97108,97109,97111],{},"讲解：SQL Server 普通视图定义中不能为了显示顺序直接写 ",[31,97110,93285],{},"。视图只定义结果集合，排序应在查询视图时写。",[253,97113,97115,97116,97119],{"id":97114},"_332-从视图-v-ssc-上查询平均成绩在-90-分以上的-sncn-和-g","33.2 从视图 ",[31,97117,97118],{},"V-SSC"," 上查询平均成绩在 90 分以上的 SN、CN 和 G。",[24,97121,97123],{"className":92584,"code":97122,"language":92586,"meta":29,"style":29},"WITH StudentAvg AS (\n SELECT SNO\n FROM dbo.[V-SSC]\n GROUP BY SNO\n HAVING AVG(CAST(G AS decimal(5, 2))) >= 90\n)\nSELECT v.SN, v.CN, v.G\nFROM dbo.[V-SSC] AS v\nJOIN StudentAvg AS a\n ON a.SNO = v.SNO;\n",[31,97124,97125,97130,97135,97140,97145,97150,97154,97159,97164,97169],{"__ignoreMap":29},[34,97126,97127],{"class":36,"line":37},[34,97128,97129],{},"WITH StudentAvg AS (\n",[34,97131,97132],{"class":36,"line":43},[34,97133,97134],{}," SELECT SNO\n",[34,97136,97137],{"class":36,"line":49},[34,97138,97139],{}," FROM dbo.[V-SSC]\n",[34,97141,97142],{"class":36,"line":56},[34,97143,97144],{}," GROUP BY SNO\n",[34,97146,97147],{"class":36,"line":62},[34,97148,97149],{}," HAVING AVG(CAST(G AS decimal(5, 2))) >= 90\n",[34,97151,97152],{"class":36,"line":68},[34,97153,6111],{},[34,97155,97156],{"class":36,"line":74},[34,97157,97158],{},"SELECT v.SN, v.CN, v.G\n",[34,97160,97161],{"class":36,"line":1971},[34,97162,97163],{},"FROM dbo.[V-SSC] AS v\n",[34,97165,97166],{"class":36,"line":1976},[34,97167,97168],{},"JOIN StudentAvg AS a\n",[34,97170,97171],{"class":36,"line":1985},[34,97172,97173],{}," ON a.SNO = v.SNO;\n",[15,97175,97176],{},"讲解：题面说“平均成绩在 90 分以上”，通常按学生全部课程平均分理解。先按学生分组筛出平均分达标的学生，再返回这些学生的课程名和成绩。",[79,97178,81],{},{"title":29,"searchDepth":43,"depth":43,"links":97180},[97181,97191,97195,97196,97201,97206,97211,97215,97216,97217,97221],{"id":95054,"depth":43,"text":95055,"children":97182},[97183,97184,97185,97186,97187,97188,97189,97190],{"id":95078,"depth":49,"text":95079},{"id":95089,"depth":49,"text":95090},{"id":95100,"depth":49,"text":95101},{"id":95111,"depth":49,"text":95112},{"id":95122,"depth":49,"text":95123},{"id":95133,"depth":49,"text":95134},{"id":95144,"depth":49,"text":95145},{"id":95161,"depth":49,"text":95162},{"id":95175,"depth":43,"text":95176,"children":97192},[97193,97194],{"id":95190,"depth":49,"text":95191},{"id":95201,"depth":49,"text":95202},{"id":95212,"depth":43,"text":95213},{"id":95360,"depth":43,"text":95361,"children":97197},[97198,97199,97200],{"id":95640,"depth":49,"text":95641},{"id":95752,"depth":49,"text":95753},{"id":95817,"depth":49,"text":95818},{"id":95920,"depth":43,"text":95921,"children":97202},[97203,97204,97205],{"id":95924,"depth":49,"text":95925},{"id":95937,"depth":49,"text":95938},{"id":95961,"depth":49,"text":95962},{"id":95984,"depth":43,"text":95985,"children":97207},[97208,97210],{"id":96120,"depth":49,"text":97209},"28.1 将 R、S、T 按关联属性建立视图 R-S-T。",{"id":96182,"depth":49,"text":96183},{"id":96259,"depth":43,"text":96260,"children":97212},[97213,97214],{"id":96330,"depth":49,"text":96331},{"id":96365,"depth":49,"text":96366},{"id":96407,"depth":43,"text":96408},{"id":96748,"depth":43,"text":96749},{"id":96964,"depth":43,"text":96965,"children":97218},[97219,97220],{"id":96968,"depth":49,"text":96969},{"id":96992,"depth":49,"text":96993},{"id":97042,"depth":43,"text":97043,"children":97222},[97223,97225],{"id":97046,"depth":49,"text":97224},"33.1 建立视图 V-SSC(SNO, SN, CNO, CN, G)，并按 CNO 升序排序。",{"id":97114,"depth":49,"text":97226},"33.2 从视图 V-SSC 上查询平均成绩在 90 分以上的 SN、CN 和 G。",{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday09-70-22-33",{"title":95043,"description":95051},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay09-数据库70题-综合22-33","GQAoO4c7bveq6IcYc9nW5bYor_4SrrZ0UFhxvZ69WxY",{"id":97233,"title":97234,"body":97235,"description":97242,"extension":85,"meta":98724,"navigation":52,"path":98725,"seo":98726,"stem":98727,"__hash__":98728},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay10-数据库70题-综合37-47.md","Day 10 - 数据库70题：综合题 37-47",{"type":8,"value":97236,"toc":98680},[97237,97240,97243,97247,97250,97260,97263,97274,97283,97286,97311,97320,97322,97342,97351,97353,97364,97372,97374,97380,97390,97393,97408,97422,97427,97429,97454,97459,97469,97474,97495,97509,97517,97519,97541,97561,97570,97572,97602,97605,97622,97637,97649,97657,97659,97690,97697,97717,97720,97734,97738,97740,97757,97763,97767,97770,97787,97790,97801,97805,97807,97824,97836,97844,97846,97954,97956,97986,97988,98005,98008,98012,98015,98075,98077,98080,98083,98159,98162,98179,98182,98190,98192,98246,98248,98272,98275,98277,98289,98295,98303,98305,98398,98400,98406,98418,98422,98428,98430,98442,98445,98453,98455,98562,98582,98584,98621,98624,98635,98638,98650,98653,98669],[11,97238,97234],{"id":97239},"day-10-数据库70题综合题-37-47",[15,97241,97242],{},"参考重点：教材第 6 章关系数据理论。原文件中综合题 34-36 缺失，保留源文件跳号。",[19,97244,97246],{"id":97245},"_37-指出下列关系模式是第几范式并说明理由","37. 指出下列关系模式是第几范式，并说明理由。",[15,97248,97249],{},"默认所有关系都已满足 1NF，即属性值不可再分。",[253,97251,97253,97254,46528,97257],{"id":97252},"_371-rx-y-zfxy-z","37.1 ",[31,97255,97256],{},"R(X, Y, Z)",[31,97258,97259],{},"F={XY -> Z}",[15,97261,97262],{},"答案：BCNF。",[15,97264,97265,97266,97269,97270,97273],{},"讲解：候选码为 ",[31,97267,97268],{},"XY","。唯一非平凡依赖 ",[31,97271,97272],{},"XY -> Z"," 的左部是超码，所以满足 BCNF。",[253,97275,97277,97278,46528,97280],{"id":97276},"_372-rx-y-zfy-z-xz-y","37.2 ",[31,97279,97256],{},[31,97281,97282],{},"F={Y -> Z, XZ -> Y}",[15,97284,97285],{},"答案：3NF，但不满足 BCNF。",[15,97287,97288,97289,258,97291,13977,97294,146,97296,146,97298,97301,97302,93436,97305,97307,97308,97310],{},"讲解：候选码有 ",[31,97290,97268],{},[31,97292,97293],{},"XZ",[31,97295,67467],{},[31,97297,67731],{},[31,97299,97300],{},"Z"," 都是主属性。依赖 ",[31,97303,97304],{},"Y -> Z",[31,97306,67731],{}," 不是超码，违反 BCNF；但右部 ",[31,97309,97300],{}," 是主属性，所以满足 3NF。",[253,97312,97314,97315,46528,97317],{"id":97313},"_373-rx-y-zfx-y-y-x-x-yz","37.3 ",[31,97316,97256],{},[31,97318,97319],{},"F={X -> Y, Y -> X, X -> YZ}",[15,97321,97262],{},[15,97323,91513,97324,13977,97327,97329,97330,86674,97333,13977,97336,46528,97339,97341],{},[31,97325,97326],{},"X+ = XYZ",[31,97328,67467],{}," 是候选码；",[31,97331,97332],{},"Y -> X",[31,97334,97335],{},"X -> YZ",[31,97337,97338],{},"Y+ = XYZ",[31,97340,67731],{}," 也是候选码。非平凡依赖左部都是超码，因此满足 BCNF。",[253,97343,97345,97346,46528,97348],{"id":97344},"_374-rx-y-zfx-y-x-z","37.4 ",[31,97347,97256],{},[31,97349,97350],{},"F={X -> Y, X -> Z}",[15,97352,97262],{},[15,97354,91513,97355,97357,97358,97360,97361,97363],{},[31,97356,97326],{},"，候选码为 ",[31,97359,67467],{},"。所有非平凡依赖左部都是候选码 ",[31,97362,67467],{},"，满足 BCNF。",[253,97365,97367,97368,46528,97370],{"id":97366},"_375-rx-y-zfxy-z","37.5 ",[31,97369,97256],{},[31,97371,97259],{},[15,97373,97262],{},[15,97375,97376,97377,97379],{},"讲解：与 37.1 相同，候选码为 ",[31,97378,97268],{},"，决定因素是超码。",[253,97381,97383,97384,46528,97387],{"id":97382},"_376-rw-x-y-zfx-z-wx-y","37.6 ",[31,97385,97386],{},"R(W, X, Y, Z)",[31,97388,97389],{},"F={X -> Z, WX -> Y}",[15,97391,97392],{},"答案：1NF，不满足 2NF。",[15,97394,97265,97395,93389,97398,97401,97402,97404,97405,97407],{},[31,97396,97397],{},"WX",[31,97399,97400],{},"WX+ = WXYZ","。非主属性 ",[31,97403,97300],{}," 依赖于候选码的一部分 ",[31,97406,67467],{},"，存在部分依赖，所以不满足 2NF。",[19,97409,97411,97412,70884,97415,46528,97418,97421],{"id":97410},"_38-有关系模式-ru-f其中-ua-b-c-d-e-pfa-b-c-p-e-a-ce-d求-r-的所有候选关键字","38. 有关系模式 ",[31,97413,97414],{},"R(U, F)",[31,97416,97417],{},"U={A, B, C, D, E, P}",[31,97419,97420],{},"F={A -> B, C -> P, E -> A, CE -> D}","。求 R 的所有候选关键字。",[15,97423,58173,97424,347],{},[31,97425,97426],{},"CE",[15,97428,91513],{},[105,97430,97431,97439,97445],{},[108,97432,97433,97434,258,97436,97438],{},"没有出现在任何依赖右部的属性是 ",[31,97435,47384],{},[31,97437,96058],{},"，它们必须出现在每个候选码中。",[108,97440,97441,97444],{},[31,97442,97443],{},"(CE)+ = C E P A B D","，可推出全部属性。",[108,97446,97447,46528,97450,97453],{},[31,97448,97449],{},"C+ = CP",[31,97451,97452],{},"E+ = EAB","，都不能推出全部属性。",[15,97455,97456,97457,347],{},"因此唯一候选码是 ",[31,97458,97426],{},[19,97460,97462,97463,93563,97466,97421],{"id":97461},"_39-设有关系模式-rc-t-s-n-g函数依赖集-fc-t-cs-g-s-n求-r-的所有候选关键字","39. 设有关系模式 ",[31,97464,97465],{},"R(C, T, S, N, G)",[31,97467,97468],{},"F={C -> T, CS -> G, S -> N}",[15,97470,58173,97471,347],{},[31,97472,97473],{},"CS",[15,97475,97476,97477,258,97479,97481,97482,97485,97486,46528,97489,97492,97493,347],{},"讲解：没有出现在右部的属性是 ",[31,97478,47384],{},[31,97480,94675],{},"，所以候选码必须包含它们。",[31,97483,97484],{},"(CS)+ = C S T G N","，可以推出全部属性；",[31,97487,97488],{},"C+ = CT",[31,97490,97491],{},"S+ = SN"," 都不行。因此唯一候选码为 ",[31,97494,97473],{},[19,97496,97498,97499,46528,97502,46528,97505,97508],{"id":97497},"_40-设-rufuabcdfa-b-b-c-d-b把-r-分解成-bcnf-模式集","40. 设 ",[31,97500,97501],{},"R(U,F)",[31,97503,97504],{},"U={A,B,C,D}",[31,97506,97507],{},"F={A -> B, B -> C, D -> B}","，把 R 分解成 BCNF 模式集。",[253,97510,97512,97513,97516],{"id":97511},"_401-如果首先把-r-分解成-acd-bd求-f-在这两个模式上的投影","40.1 如果首先把 R 分解成 ",[31,97514,97515],{},"{ACD, BD}","，求 F 在这两个模式上的投影。",[15,97518,58173],{},[105,97520,97521,97533],{},[108,97522,2884,97523,97526,97527,46528,97530,347],{},[31,97524,97525],{},"ACD"," 上的投影：",[31,97528,97529],{},"A -> C",[31,97531,97532],{},"D -> C",[108,97534,2884,97535,97526,97538,347],{},[31,97536,97537],{},"BD",[31,97539,97540],{},"D -> B",[15,97542,91513,97543,97545,97546,97549,97550,97545,97552,97555,97556,97558,97559,347],{},[31,97544,97529],{}," 由 ",[31,97547,97548],{},"A -> B -> C"," 推出；",[31,97551,97532],{},[31,97553,97554],{},"D -> B -> C"," 推出。",[31,97557,97537],{}," 上只保留涉及 B、D 的依赖，得到 ",[31,97560,97540],{},[253,97562,97564,97565,258,97567,97569],{"id":97563},"_402-acd-和-bd-是-bcnf-吗如果不是进一步分解","40.2 ",[31,97566,97525],{},[31,97568,97537],{}," 是 BCNF 吗？如果不是，进一步分解。",[15,97571,58173],{},[105,97573,97574,97586],{},[108,97575,97576,97578,97579,93436,97581,11008,97583,97585],{},[31,97577,97537],{}," 是 BCNF，因为 ",[31,97580,97540],{},[31,97582,91994],{},[31,97584,97537],{}," 的候选码。",[108,97587,97588,97590,97591,97593,97594,97596,97597,146,97599,97601],{},[31,97589,97525],{}," 不是 BCNF。",[31,97592,97525],{}," 的候选码为 ",[31,97595,93573],{},"，而 ",[31,97598,97529],{},[31,97600,97532],{}," 的左部都不是超码。",[15,97603,97604],{},"一种 BCNF 分解为：",[105,97606,97607,97612,97617],{},[108,97608,97609],{},[31,97610,97611],{},"AC(A, C)",[108,97613,97614],{},[31,97615,97616],{},"AD(A, D)",[108,97618,97619],{},[31,97620,97621],{},"BD(B, D)",[15,97623,97624,97625,93436,97628,97630,97631,97633,97634,97636],{},"讲解：在 ",[31,97626,97627],{},"AC",[31,97629,54151],{}," 是码，满足 BCNF；",[31,97632,93573],{}," 上没有非平凡函数依赖，满足 BCNF；",[31,97635,97537],{}," 已满足 BCNF。",[19,97638,97640,97641,97644,97645,97648],{"id":97639},"_41-已知关系模式-rcity-st-zip-和函数依赖集-fcity-st-zip-zip-city找出-r-的两个候选关键字","41. 已知关系模式 ",[31,97642,97643],{},"R(CITY, ST, ZIP)"," 和函数依赖集 ",[31,97646,97647],{},"F={CITY, ST -> ZIP; ZIP -> CITY}","。找出 R 的两个候选关键字。",[15,97650,58173,97651,258,97654,347],{},[31,97652,97653],{},"(CITY, ST)",[31,97655,97656],{},"(ZIP, ST)",[15,97658,91513],{},[105,97660,97661,97666,97671],{},[108,97662,97663,347],{},[31,97664,97665],{},"(CITY, ST)+ = CITY, ST, ZIP",[108,97667,97668,347],{},[31,97669,97670],{},"(ZIP, ST)+ = ZIP, ST, CITY",[108,97672,97673,97674,97677,97678,97680,97681,97684,97685,4374,97687,97689],{},"单独 ",[31,97675,97676],{},"ZIP"," 只能推出 ",[31,97679,95383],{},"，不能推出 ",[31,97682,97683],{},"ST","；单独 ",[31,97686,95383],{},[31,97688,97683],{}," 也不能推出全部属性。",[15,97691,97692,97693,43298,97695,347],{},"因此两个候选码是 ",[31,97694,97653],{},[31,97696,97656],{},[19,97698,97700,97701,97703,97704,97706,97707,97710,97711,97714,97715,347],{"id":97699},"_42-教师管理数据库属性为学号-s课程号-c成绩-g任课教师-tn教师所在系-d","42. 教师管理数据库：属性为学号 ",[31,97702,93528],{},"、课程号 ",[31,97705,93481],{},"、成绩 ",[31,97708,97709],{},"G","、任课教师 ",[31,97712,97713],{},"TN","、教师所在系 ",[31,97716,91994],{},[15,97718,97719],{},"语义：",[2230,97721,97722,97725,97728,97731],{},[108,97723,97724],{},"学号和课程分别与其代表的学生和课程一一对应。",[108,97726,97727],{},"一个学生所修的每门课程都有一个成绩。",[108,97729,97730],{},"每门课程只有一位任课教师，但每位教师可以有多门课程。",[108,97732,97733],{},"教师没有重名，每个教师只属于一个系。",[253,97735,97737],{"id":97736},"_421-确定函数依赖集","42.1 确定函数依赖集。",[15,97739,58173],{},[105,97741,97742,97747,97752],{},[108,97743,97744],{},[31,97745,97746],{},"(S#, C#) -> G",[108,97748,97749],{},[31,97750,97751],{},"C# -> TN",[108,97753,97754],{},[31,97755,97756],{},"TN -> D",[15,97758,97759,97760,347],{},"由后两条还可推出 ",[31,97761,97762],{},"C# -> D",[253,97764,97766],{"id":97765},"_422-若用所有属性组成一个模式该关系模式为几范式并说明异常","42.2 若用所有属性组成一个模式，该关系模式为几范式？并说明异常。",[15,97768,97769],{},"答案：该模式为 1NF，不满足 2NF。",[15,97771,97772,97773,44689,97775,97777,97778,93659,97780,97782,97783,97786],{},"讲解：候选码是 ",[31,97774,93471],{},[31,97776,97751],{},"，非主属性 ",[31,97779,97713],{},[31,97781,93481],{},"，所以不满足 2NF；同时还有 ",[31,97784,97785],{},"C# -> TN -> D"," 的传递依赖。",[15,97788,97789],{},"异常示例：",[105,97791,97792,97795,97798],{},[108,97793,97794],{},"插入异常：一门新课程已有任课教师但还没有学生选课时，无法单独插入课程-教师信息。",[108,97796,97797],{},"删除异常：删除某课程最后一条选课记录，可能丢失该课程任课教师及教师所在系信息。",[108,97799,97800],{},"修改异常：教师所在系重复出现在很多选课记录中，修改时容易不一致。",[253,97802,97804],{"id":97803},"_423-分解为依赖保持无损连接的-3nf","42.3 分解为依赖保持、无损连接的 3NF。",[15,97806,58173],{},[105,97808,97809,97814,97819],{},[108,97810,97811],{},[31,97812,97813],{},"SC(S#, C#, G)",[108,97815,97816],{},[31,97817,97818],{},"CT(C#, TN)",[108,97820,97821],{},[31,97822,97823],{},"TD(TN, D)",[15,97825,91513,97826,97828,97829,97831,97832,97835],{},[31,97827,93499],{}," 保存选课成绩，",[31,97830,93503],{}," 保存课程教师，",[31,97833,97834],{},"TD"," 保存教师所在系。三个原始依赖都被保留，并且可通过外键连接恢复原关系。",[19,97837,97839,97840,97843],{"id":97838},"_43-给出关系-r工程号-材料号-数量-开工日期-完工日期-价格判断范式并分解","43. 给出关系 ",[31,97841,97842],{},"R(工程号, 材料号, 数量, 开工日期, 完工日期, 价格)","，判断范式并分解。",[15,97845,94672],{},[13837,97847,97848,97870],{},[13840,97849,97850],{},[13843,97851,97852,97855,97858,97861,97864,97867],{},[13846,97853,97854],{},"工程号",[13846,97856,97857],{},"材料号",[13846,97859,97860],{},"数量",[13846,97862,97863],{},"开工日期",[13846,97865,97866],{},"完工日期",[13846,97868,97869],{},"价格",[13853,97871,97872,97891,97906,97921,97938],{},[13843,97873,97874,97877,97880,97882,97885,97888],{},[13858,97875,97876],{},"P1",[13858,97878,97879],{},"I1",[13858,97881,15932],{},[13858,97883,97884],{},"9805",[13858,97886,97887],{},"9902",[13858,97889,97890],{},"250",[13843,97892,97893,97895,97898,97900,97902,97904],{},[13858,97894,97876],{},[13858,97896,97897],{},"I2",[13858,97899,36656],{},[13858,97901,97884],{},[13858,97903,97887],{},[13858,97905,25357],{},[13843,97907,97908,97910,97913,97915,97917,97919],{},[13858,97909,97876],{},[13858,97911,97912],{},"I3",[13858,97914,95396],{},[13858,97916,97884],{},[13858,97918,97887],{},[13858,97920,74303],{},[13843,97922,97923,97926,97928,97930,97933,97936],{},[13858,97924,97925],{},"P2",[13858,97927,97879],{},[13858,97929,36656],{},[13858,97931,97932],{},"9811",[13858,97934,97935],{},"9912",[13858,97937,97890],{},[13843,97939,97940,97942,97945,97947,97949,97951],{},[13858,97941,97925],{},[13858,97943,97944],{},"I4",[13858,97946,95611],{},[13858,97948,97932],{},[13858,97950,97935],{},[13858,97952,97953],{},"350",[15,97955,58173],{},[105,97957,97958,97963,97983],{},[108,97959,93304,97960,347],{},[31,97961,97962],{},"(工程号, 材料号)",[108,97964,97965,97966],{},"函数依赖：\n",[105,97967,97968,97973,97978],{},[108,97969,97970],{},[31,97971,97972],{},"工程号 -> 开工日期, 完工日期",[108,97974,97975],{},[31,97976,97977],{},"材料号 -> 价格",[108,97979,97980],{},[31,97981,97982],{},"(工程号, 材料号) -> 数量",[108,97984,97985],{},"该关系为 1NF，不满足 2NF，因为存在非主属性对组合码的部分依赖。",[15,97987,93341],{},[105,97989,97990,97995,98000],{},[108,97991,97992],{},[31,97993,97994],{},"工程(工程号, 开工日期, 完工日期)",[108,97996,97997],{},[31,97998,97999],{},"材料(材料号, 价格)",[108,98001,98002],{},[31,98003,98004],{},"用料(工程号, 材料号, 数量)",[15,98006,98007],{},"讲解：分解后，工程日期只存一次，材料价格只存一次，用料数量存放在工程和材料的联系表中，可避免更新、插入、删除异常。",[19,98009,98011],{"id":98010},"_44-判断给出的数据库集是否可直接作为关系数据库中的关系若不行改造为尽可能好的关系形式","44. 判断给出的数据库集是否可直接作为关系数据库中的关系；若不行，改造为尽可能好的关系形式。",[15,98013,98014],{},"题面表：",[13837,98016,98017,98030],{},[13840,98018,98019],{},[13843,98020,98021,98024,98027],{},[13846,98022,98023],{},"系名",[13846,98025,98026],{},"课程名",[13846,98028,98029],{},"教师名",[13853,98031,98032,98043,98054,98064],{},[13843,98033,98034,98037,98040],{},[13858,98035,98036],{},"计算机系",[13858,98038,98039],{},"DB",[13858,98041,98042],{},"李军，刘强",[13843,98044,98045,98048,98051],{},[13858,98046,98047],{},"机械系",[13858,98049,98050],{},"CAD",[13858,98052,98053],{},"金山，宋海",[13843,98055,98056,98059,98062],{},[13858,98057,98058],{},"造船系",[13858,98060,98061],{},"CAM",[13858,98063,94759],{},[13843,98065,98066,98069,98072],{},[13858,98067,98068],{},"自控系",[13858,98070,98071],{},"CTY",[13858,98073,98074],{},"张红，曾键",[15,98076,58173],{},[15,98078,98079],{},"不能直接作为关系数据库中的关系，因为“教师名”列中有多个教师名，属性值不是原子的，违反 1NF。",[15,98081,98082],{},"一种改造方式：",[13837,98084,98085,98095],{},[13840,98086,98087],{},[13843,98088,98089,98091,98093],{},[13846,98090,98023],{},[13846,98092,98026],{},[13846,98094,98029],{},[13853,98096,98097,98106,98115,98124,98133,98141,98150],{},[13843,98098,98099,98101,98103],{},[13858,98100,98036],{},[13858,98102,98039],{},[13858,98104,98105],{},"李军",[13843,98107,98108,98110,98112],{},[13858,98109,98036],{},[13858,98111,98039],{},[13858,98113,98114],{},"刘强",[13843,98116,98117,98119,98121],{},[13858,98118,98047],{},[13858,98120,98050],{},[13858,98122,98123],{},"金山",[13843,98125,98126,98128,98130],{},[13858,98127,98047],{},[13858,98129,98050],{},[13858,98131,98132],{},"宋海",[13843,98134,98135,98137,98139],{},[13858,98136,98058],{},[13858,98138,98061],{},[13858,98140,94759],{},[13843,98142,98143,98145,98147],{},[13858,98144,98068],{},[13858,98146,98071],{},[13858,98148,98149],{},"张红",[13843,98151,98152,98154,98156],{},[13858,98153,98068],{},[13858,98155,98071],{},[13858,98157,98158],{},"曾键",[15,98160,98161],{},"更规范的关系模式可设计为：",[105,98163,98164,98169,98174],{},[108,98165,98166],{},[31,98167,98168],{},"系(系名)",[108,98170,98171],{},[31,98172,98173],{},"课程(课程名, 系名)",[108,98175,98176],{},[31,98177,98178],{},"授课(课程名, 教师名)",[15,98180,98181],{},"讲解：先让每个属性值原子化，再根据业务语义继续分解，避免把多值属性塞进一个单元格。",[19,98183,98185,98186,98189],{"id":98184},"_45-设关系-r课程名-教师名-教师地址判断范式删除异常并分解","45. 设关系 ",[31,98187,98188],{},"R(课程名, 教师名, 教师地址)","，判断范式、删除异常，并分解。",[15,98191,94672],{},[13837,98193,98194,98205],{},[13840,98195,98196],{},[13843,98197,98198,98200,98202],{},[13846,98199,98026],{},[13846,98201,98029],{},[13846,98203,98204],{},"教师地址",[13853,98206,98207,98218,98227,98237],{},[13843,98208,98209,98212,98215],{},[13858,98210,98211],{},"C1",[13858,98213,98214],{},"马千里",[13858,98216,98217],{},"D1",[13843,98219,98220,98222,98225],{},[13858,98221,96711],{},[13858,98223,98224],{},"于得水",[13858,98226,98217],{},[13843,98228,98229,98232,98235],{},[13858,98230,98231],{},"C3",[13858,98233,98234],{},"余快",[13858,98236,96106],{},[13843,98238,98239,98242,98244],{},[13858,98240,98241],{},"C4",[13858,98243,98224],{},[13858,98245,98217],{},[15,98247,58173],{},[105,98249,98250,98253,98262,98266],{},[108,98251,98252],{},"假定一门课程只有一位教师，教师只有一个地址。",[108,98254,98255,98256,46528,98259,347],{},"函数依赖：",[31,98257,98258],{},"课程名 -> 教师名",[31,98260,98261],{},"教师名 -> 教师地址",[108,98263,93304,98264,347],{},[31,98265,98026],{},[108,98267,98268,98269,347],{},"该关系满足 2NF，但不满足 3NF，因为存在传递依赖 ",[31,98270,98271],{},"课程名 -> 教师名 -> 教师地址",[15,98273,98274],{},"删除异常：若删除于得水老师所教的最后一门课程，会同时丢失“于得水住址为 D1”的信息。",[15,98276,93341],{},[105,98278,98279,98284],{},[108,98280,98281],{},[31,98282,98283],{},"课程教师(课程名, 教师名)",[108,98285,98286],{},[31,98287,98288],{},"教师地址(教师名, 教师地址)",[15,98290,98291,98292,98294],{},"讲解：分解后，即使某教师暂时没有课程，也可以在 ",[31,98293,98204],{}," 中保存教师基本信息。",[19,98296,98298,98299,98302],{"id":98297},"_46-设关系-r职工号-职工名-年龄-性别-单位号-单位名判断是否属于-3nf若不是规范化为-3nf","46. 设关系 ",[31,98300,98301],{},"R(职工号, 职工名, 年龄, 性别, 单位号, 单位名)","，判断是否属于 3NF；若不是，规范化为 3NF。",[15,98304,94672],{},[13837,98306,98307,98328],{},[13840,98308,98309],{},[13843,98310,98311,98314,98316,98319,98322,98325],{},[13846,98312,98313],{},"职工号",[13846,98315,93830],{},[13846,98317,98318],{},"年龄",[13846,98320,98321],{},"性别",[13846,98323,98324],{},"单位号",[13846,98326,98327],{},"单位名",[13853,98329,98330,98348,98366,98383],{},[13843,98331,98332,98335,98338,98340,98342,98345],{},[13858,98333,98334],{},"E1",[13858,98336,98337],{},"ZHAO",[13858,98339,3249],{},[13858,98341,51834],{},[13858,98343,98344],{},"D3",[13858,98346,98347],{},"CCC",[13843,98349,98350,98353,98356,98358,98361,98363],{},[13858,98351,98352],{},"E2",[13858,98354,98355],{},"QIAN",[13858,98357,95556],{},[13858,98359,98360],{},"M",[13858,98362,98217],{},[13858,98364,98365],{},"AAA",[13843,98367,98368,98371,98374,98377,98379,98381],{},[13858,98369,98370],{},"E3",[13858,98372,98373],{},"SEN",[13858,98375,98376],{},"38",[13858,98378,98360],{},[13858,98380,98344],{},[13858,98382,98347],{},[13843,98384,98385,98387,98390,98392,98394,98396],{},[13858,98386,98370],{},[13858,98388,98389],{},"LI",[13858,98391,95556],{},[13858,98393,51834],{},[13858,98395,98344],{},[13858,98397,98347],{},[15,98399,58173],{},[15,98401,98402,98403,98405],{},"若按常规语义“职工号唯一标识职工”，题面数据中 ",[31,98404,98370],{}," 出现两次且姓名、年龄、性别不同，已经违反实体完整性。忽略该录入错误后，函数依赖可写为：",[105,98407,98408,98413],{},[108,98409,98410],{},[31,98411,98412],{},"职工号 -> 职工名, 年龄, 性别, 单位号",[108,98414,98415],{},[31,98416,98417],{},"单位号 -> 单位名",[15,98419,93304,98420,347],{},[31,98421,98313],{},[15,98423,98424,98425,347],{},"该关系不属于 3NF，属于 2NF。原因是候选码为单属性，不存在部分依赖；但存在传递依赖 ",[31,98426,98427],{},"职工号 -> 单位号 -> 单位名",[15,98429,93547],{},[105,98431,98432,98437],{},[108,98433,98434],{},[31,98435,98436],{},"职工(职工号, 职工名, 年龄, 性别, 单位号)",[108,98438,98439],{},[31,98440,98441],{},"单位(单位号, 单位名)",[15,98443,98444],{},"讲解：单位名是单位的属性，不应在每个职工元组中重复保存。",[19,98446,98448,98449,98452],{"id":98447},"_47-给出关系-scsno-cno-ctitle-iname-iloca-grade判断范式和异常并分解","47. 给出关系 ",[31,98450,98451],{},"SC(SNO, CNO, CTITLE, INAME, ILOCA, GRADE)","，判断范式和异常，并分解。",[15,98454,94672],{},[13837,98456,98457,98478],{},[13840,98458,98459],{},[13843,98460,98461,98464,98467,98470,98473,98476],{},[13846,98462,98463],{},"SNO",[13846,98465,98466],{},"CNO",[13846,98468,98469],{},"CTITLE",[13846,98471,98472],{},"INAME",[13846,98474,98475],{},"ILOCA",[13846,98477,94795],{},[13853,98479,98480,98498,98514,98530,98546],{},[13843,98481,98482,98485,98487,98490,98493,98495],{},[13858,98483,98484],{},"80152",[13858,98486,98211],{},[13858,98488,98489],{},"OS",[13858,98491,98492],{},"王平",[13858,98494,98217],{},[13858,98496,98497],{},"70",[13843,98499,98500,98503,98505,98507,98510,98512],{},[13858,98501,98502],{},"80153",[13858,98504,96711],{},[13858,98506,98039],{},[13858,98508,98509],{},"高升",[13858,98511,96106],{},[13858,98513,94815],{},[13843,98515,98516,98519,98521,98523,98525,98527],{},[13858,98517,98518],{},"80154",[13858,98520,98211],{},[13858,98522,98489],{},[13858,98524,98492],{},[13858,98526,98217],{},[13858,98528,98529],{},"86",[13843,98531,98532,98534,98536,98539,98542,98544],{},[13858,98533,98518],{},[13858,98535,98231],{},[13858,98537,98538],{},"AI",[13858,98540,98541],{},"杨杨",[13858,98543,98344],{},[13858,98545,96800],{},[13843,98547,98548,98551,98553,98556,98558,98560],{},[13858,98549,98550],{},"80155",[13858,98552,98241],{},[13858,98554,98555],{},"CL",[13858,98557,98509],{},[13858,98559,96106],{},[13858,98561,94824],{},[15,98563,92527,98564,98566,98567,98569,98570,98572,98573,98575,98576,98578,98579,98581],{},[31,98565,98463],{}," 学号，",[31,98568,98466],{}," 课程号，",[31,98571,98469],{}," 课程名，",[31,98574,98472],{}," 教师名，",[31,98577,98475],{}," 教师地址，",[31,98580,94795],{}," 成绩。",[15,98583,58173],{},[105,98585,98586,98591,98610],{},[108,98587,93304,98588,347],{},[31,98589,98590],{},"(SNO, CNO)",[108,98592,97965,98593],{},[105,98594,98595,98600,98605],{},[108,98596,98597],{},[31,98598,98599],{},"(SNO, CNO) -> GRADE",[108,98601,98602],{},[31,98603,98604],{},"CNO -> CTITLE, INAME",[108,98606,98607],{},[31,98608,98609],{},"INAME -> ILOCA",[108,98611,98612,98613,146,98615,146,98617,93746,98619,347],{},"该关系为 1NF，不满足 2NF，因为 ",[31,98614,98469],{},[31,98616,98472],{},[31,98618,98475],{},[31,98620,98466],{},[15,98622,98623],{},"异常：",[105,98625,98626,98629,98632],{},[108,98627,98628],{},"插入异常：新开课程还没有学生选课时，无法保存课程名、教师和教师地址。",[108,98630,98631],{},"删除异常：删除某课程最后一条选课记录时，会丢失课程和教师信息。",[108,98633,98634],{},"修改异常：教师地址重复保存，修改时可能不一致。",[15,98636,98637],{},"分解到 2NF：",[105,98639,98640,98645],{},[108,98641,98642],{},[31,98643,98644],{},"选课(SNO, CNO, GRADE)",[108,98646,98647],{},[31,98648,98649],{},"课程任课(CNO, CTITLE, INAME, ILOCA)",[15,98651,98652],{},"进一步分解到 3NF：",[105,98654,98655,98659,98664],{},[108,98656,98657],{},[31,98658,98644],{},[108,98660,98661],{},[31,98662,98663],{},"课程任课(CNO, CTITLE, INAME)",[108,98665,98666],{},[31,98667,98668],{},"教师(INAME, ILOCA)",[15,98670,98671,98672,98675,98676,98679],{},"讲解：只分解到 2NF 后，",[31,98673,98674],{},"课程任课"," 中仍有 ",[31,98677,98678],{},"CNO -> INAME -> ILOCA"," 的传递依赖。继续分解到 3NF 才能更好消除教师地址冗余。",{"title":29,"searchDepth":43,"depth":43,"links":98681},[98682,98696,98698,98700,98707,98709,98715,98717,98718,98720,98722],{"id":97245,"depth":43,"text":97246,"children":98683},[98684,98686,98688,98690,98692,98694],{"id":97252,"depth":49,"text":98685},"37.1 R(X, Y, Z)，F={XY -> Z}",{"id":97276,"depth":49,"text":98687},"37.2 R(X, Y, Z)，F={Y -> Z, XZ -> Y}",{"id":97313,"depth":49,"text":98689},"37.3 R(X, Y, Z)，F={X -> Y, Y -> X, X -> YZ}",{"id":97344,"depth":49,"text":98691},"37.4 R(X, Y, Z)，F={X -> Y, X -> Z}",{"id":97366,"depth":49,"text":98693},"37.5 R(X, Y, Z)，F={XY -> Z}",{"id":97382,"depth":49,"text":98695},"37.6 R(W, X, Y, Z)，F={X -> Z, WX -> Y}",{"id":97410,"depth":43,"text":98697},"38. 有关系模式 R(U, F)，其中 U={A, B, C, D, E, P}，F={A -> B, C -> P, E -> A, CE -> D}。求 R 的所有候选关键字。",{"id":97461,"depth":43,"text":98699},"39. 设有关系模式 R(C, T, S, N, G)，函数依赖集 F={C -> T, CS -> G, S -> N}。求 R 的所有候选关键字。",{"id":97497,"depth":43,"text":98701,"children":98702},"40. 设 R(U,F)，U={A,B,C,D}，F={A -> B, B -> C, D -> B}，把 R 分解成 BCNF 模式集。",[98703,98705],{"id":97511,"depth":49,"text":98704},"40.1 如果首先把 R 分解成 {ACD, BD}，求 F 在这两个模式上的投影。",{"id":97563,"depth":49,"text":98706},"40.2 ACD 和 BD 是 BCNF 吗？如果不是，进一步分解。",{"id":97639,"depth":43,"text":98708},"41. 已知关系模式 R(CITY, ST, ZIP) 和函数依赖集 F={CITY, ST -> ZIP; ZIP -> CITY}。找出 R 的两个候选关键字。",{"id":97699,"depth":43,"text":98710,"children":98711},"42. 教师管理数据库：属性为学号 S#、课程号 C#、成绩 G、任课教师 TN、教师所在系 D。",[98712,98713,98714],{"id":97736,"depth":49,"text":97737},{"id":97765,"depth":49,"text":97766},{"id":97803,"depth":49,"text":97804},{"id":97838,"depth":43,"text":98716},"43. 给出关系 R(工程号, 材料号, 数量, 开工日期, 完工日期, 价格)，判断范式并分解。",{"id":98010,"depth":43,"text":98011},{"id":98184,"depth":43,"text":98719},"45. 设关系 R(课程名, 教师名, 教师地址)，判断范式、删除异常，并分解。",{"id":98297,"depth":43,"text":98721},"46. 设关系 R(职工号, 职工名, 年龄, 性别, 单位号, 单位名)，判断是否属于 3NF；若不是，规范化为 3NF。",{"id":98447,"depth":43,"text":98723},"47. 给出关系 SC(SNO, CNO, CTITLE, INAME, ILOCA, GRADE)，判断范式和异常，并分解。",{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday10-70-37-47",{"title":97234,"description":97242},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay10-数据库70题-综合37-47","L7m0im9i_Y1VFrN20f4zbPtR_X5kQm_wC0iYqKSvicg",{"id":98730,"title":98731,"body":98732,"description":98739,"extension":85,"meta":99539,"navigation":52,"path":99540,"seo":99541,"stem":99542,"__hash__":99543},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay11-数据库70题-综合48-61.md","Day 11 - 数据库70题：综合题 48-61",{"type":8,"value":98733,"toc":99520},[98734,98737,98740,98744,98746,98760,98762,98776,98803,98817,98821,98827,98829,98832,98856,98870,98877,98880,98884,98886,98889,98951,98954,98989,98992,98996,98999,99013,99016,99020,99022,99150,99154,99156,99197,99201,99203,99243,99246,99250,99252,99255,99279,99281,99317,99320,99324,99326,99340,99343,99347,99349,99352,99355,99359,99361,99364,99378,99381,99402,99405,99409,99411,99414,99425,99428,99432,99434,99437,99440,99444,99446,99449,99452,99466,99469,99473,99475,99478,99486,99489,99493,99495,99498,99501,99515,99518],[11,98735,98731],{"id":98736},"day-11-数据库70题综合题-48-61",[15,98738,98739],{},"参考重点：教材第 7 章数据库设计，第 5 章完整性，第 11-12 章事务、恢复、并发控制。原文件缺综合题 53。",[19,98741,98743],{"id":98742},"_48-某大学实行学分制学生可根据自己情况选修课程每名学生可同时选修多门课程每门课程可由多位教师讲授每位教师可讲授多门课程不完整-e-r-图含学生课程教师三个实体","48. 某大学实行学分制，学生可根据自己情况选修课程。每名学生可同时选修多门课程，每门课程可由多位教师讲授；每位教师可讲授多门课程。不完整 E-R 图含学生、课程、教师三个实体。",[15,98745,5787],{},[2230,98747,98748,98751,98754,98757],{},[108,98749,98750],{},"指出学生与课程的联系类型，完善 E-R 图。",[108,98752,98753],{},"指出课程与教师的联系类型，完善 E-R 图。",[108,98755,98756],{},"若每名学生有一位教师指导，每一位教师指导多名学生，则学生与教师是何联系？",[108,98758,98759],{},"在原 E-R 图上补画教师与学生的联系。",[15,98761,58173],{},[2230,98763,98764,98767,98770,98773],{},[108,98765,98766],{},"学生与课程是多对多联系。一个学生可选多门课，一门课可被多名学生选。",[108,98768,98769],{},"课程与教师是多对多联系。一门课可由多位教师讲授，一位教师可讲多门课。",[108,98771,98772],{},"学生与教师是多对一联系，从教师到学生看是一对多：一名学生只有一位指导教师，一位教师可指导多名学生。",[108,98774,98775],{},"完整 E-R 可表示为：",[24,98777,98781],{"className":98778,"code":98779,"language":98780,"meta":29,"style":29},"language-mermaid shiki shiki-themes github-light github-dark","erDiagram\n 学生 }o--o{ 课程 : 选修\n 教师 }o--o{ 课程 : 讲授\n 教师 ||--o{ 学生 : 指导\n","mermaid",[31,98782,98783,98788,98793,98798],{"__ignoreMap":29},[34,98784,98785],{"class":36,"line":37},[34,98786,98787],{},"erDiagram\n",[34,98789,98790],{"class":36,"line":43},[34,98791,98792],{}," 学生 }o--o{ 课程 : 选修\n",[34,98794,98795],{"class":36,"line":49},[34,98796,98797],{}," 教师 }o--o{ 课程 : 讲授\n",[34,98799,98800],{"class":36,"line":56},[34,98801,98802],{}," 教师 ||--o{ 学生 : 指导\n",[15,98804,98805,98806,258,98809,98812,98813,98816],{},"讲解：E-R 联系类型判断只看业务约束。",[31,98807,98808],{},"选修",[31,98810,98811],{},"讲授"," 都是 M:N，转换为关系模式时一般需要单独的联系表；",[31,98814,98815],{},"指导"," 是 1:N，可在学生表中加入指导教师外键。",[19,98818,98820],{"id":98819},"_49-将图-56-的-e-r-图转换为关系模式菱形框中的属性自己确定","49. 将图 5.6 的 E-R 图转换为关系模式，菱形框中的属性自己确定。",[15,98822,98823,98824,347],{},"题面图示：实体“单位”含属性单位号、地址、电话；实体“职工”含属性职工号、姓名、性别、年龄、单位号；单位与职工之间有联系 ",[31,98825,98826],{},"D-3",[15,98828,58173],{},[15,98830,98831],{},"按常见语义，一个单位有多个职工，一个职工属于一个单位，是 1:N 联系。可转换为：",[105,98833,98834,98842],{},[108,98835,98836,98839,98840],{},[31,98837,98838],{},"单位(单位号, 地址, 电话)","，主键：",[31,98841,98324],{},[108,98843,98844,98839,98847,98849,98850,98852,98853],{},[31,98845,98846],{},"职工(职工号, 姓名, 性别, 年龄, 单位号)",[31,98848,98313],{},"，外键：",[31,98851,98324],{}," 引用 ",[31,98854,98855],{},"单位(单位号)",[15,98857,98858,98859,98861,98862,98865,98866,98869],{},"如果菱形联系 ",[31,98860,98826],{}," 有自定属性，例如 ",[31,98863,98864],{},"入职日期","，则该属性可并入 N 端的 ",[31,98867,98868],{},"职工"," 关系：",[105,98871,98872],{},[108,98873,98874],{},[31,98875,98876],{},"职工(职工号, 姓名, 性别, 年龄, 单位号, 入职日期)",[15,98878,98879],{},"讲解：1:N 联系转换为关系模式时，一般把 1 端主键加入 N 端作为外键；只有 M:N 联系才必须单独生成联系表。",[19,98881,98883],{"id":98882},"_50-设有商业销售记账数据库一个顾客顾客姓名单位电话号码可以买多种商品一种商品商品名称型号单价供应多个顾客试画出对应的-e-r-图","50. 设有商业销售记账数据库。一个顾客（顾客姓名，单位，电话号码）可以买多种商品，一种商品（商品名称，型号，单价）供应多个顾客。试画出对应的 E-R 图。",[15,98885,58173],{},[15,98887,98888],{},"顾客和商品是多对多联系，联系可命名为“购买”或“销售”，联系属性可按记账需要加入数量、日期、金额等。",[24,98890,98892],{"className":98778,"code":98891,"language":98780,"meta":29,"style":29},"erDiagram\n 顾客 }o--o{ 商品 : 购买\n 顾客 {\n string 顾客姓名\n string 单位\n string 电话号码\n }\n 商品 {\n string 商品名称\n string 型号\n decimal 单价\n }\n",[31,98893,98894,98898,98903,98908,98913,98918,98923,98927,98932,98937,98942,98947],{"__ignoreMap":29},[34,98895,98896],{"class":36,"line":37},[34,98897,98787],{},[34,98899,98900],{"class":36,"line":43},[34,98901,98902],{}," 顾客 }o--o{ 商品 : 购买\n",[34,98904,98905],{"class":36,"line":49},[34,98906,98907],{}," 顾客 {\n",[34,98909,98910],{"class":36,"line":56},[34,98911,98912],{}," string 顾客姓名\n",[34,98914,98915],{"class":36,"line":62},[34,98916,98917],{}," string 单位\n",[34,98919,98920],{"class":36,"line":68},[34,98921,98922],{}," string 电话号码\n",[34,98924,98925],{"class":36,"line":74},[34,98926,5540],{},[34,98928,98929],{"class":36,"line":1971},[34,98930,98931],{}," 商品 {\n",[34,98933,98934],{"class":36,"line":1976},[34,98935,98936],{}," string 商品名称\n",[34,98938,98939],{"class":36,"line":1985},[34,98940,98941],{}," string 型号\n",[34,98943,98944],{"class":36,"line":1990},[34,98945,98946],{}," decimal 单价\n",[34,98948,98949],{"class":36,"line":1999},[34,98950,5540],{},[15,98952,98953],{},"可转换为关系模式：",[105,98955,98956,98969,98981],{},[108,98957,98958,98961,98962,98965,98966],{},[31,98959,98960],{},"顾客(顾客姓名, 单位, 电话号码)","，主键可暂取 ",[31,98963,98964],{},"顾客姓名","，实际系统建议增加 ",[31,98967,98968],{},"顾客编号",[108,98970,98971,98974,98975,98965,98978],{},[31,98972,98973],{},"商品(商品名称, 型号, 单价)","，主键可取 ",[31,98976,98977],{},"(商品名称, 型号)",[31,98979,98980],{},"商品编号",[108,98982,98983,98974,98986],{},[31,98984,98985],{},"购买(顾客姓名, 商品名称, 型号, 数量, 购买日期)",[31,98987,98988],{},"(顾客姓名, 商品名称, 型号, 购买日期)",[15,98990,98991],{},"讲解：题面没有给出顾客编号和商品编号，考试可直接用自然属性作码；真实数据库设计中更建议使用稳定的代理键。",[19,98993,98995],{"id":98994},"_51-某医院病房计算机管理中需要如下信息并完成-e-r-图关系模型和候选码","51. 某医院病房计算机管理中需要如下信息，并完成 E-R 图、关系模型和候选码。",[15,98997,98998],{},"题面信息：",[105,99000,99001,99004,99007,99010],{},[108,99002,99003],{},"科室：科名，科地址，科电话，医生姓名",[108,99005,99006],{},"病房：病房号，床位号，所属科室名",[108,99008,99009],{},"医生：姓名，职称，所属科室名，年龄，工作证号",[108,99011,99012],{},"病人：病历号，姓名，诊断，主管医生，病房号",[15,99014,99015],{},"约束：一个科室有多个病房、多个医生；一个病房只能属于一个科室；一个医生只属于一个科室，但可负责多个病人的诊治；一个病人的主管医生只有一个。",[253,99017,99019],{"id":99018},"_511-e-r-图","51.1 E-R 图",[15,99021,58173],{},[24,99023,99025],{"className":98778,"code":99024,"language":98780,"meta":29,"style":29},"erDiagram\n 科室 ||--o{ 病房 : 拥有\n 科室 ||--o{ 医生 : 隶属\n 医生 ||--o{ 病人 : 主管\n 病房 ||--o{ 病人 : 收治\n\n 科室 {\n string 科名\n string 科地址\n string 科电话\n }\n 病房 {\n string 病房号\n string 床位号\n }\n 医生 {\n string 工作证号\n string 姓名\n string 职称\n int 年龄\n }\n 病人 {\n string 病历号\n string 姓名\n string 诊断\n }\n",[31,99026,99027,99031,99036,99041,99046,99051,99055,99060,99065,99070,99075,99079,99084,99089,99094,99098,99103,99108,99113,99118,99123,99127,99132,99137,99141,99146],{"__ignoreMap":29},[34,99028,99029],{"class":36,"line":37},[34,99030,98787],{},[34,99032,99033],{"class":36,"line":43},[34,99034,99035],{}," 科室 ||--o{ 病房 : 拥有\n",[34,99037,99038],{"class":36,"line":49},[34,99039,99040],{}," 科室 ||--o{ 医生 : 隶属\n",[34,99042,99043],{"class":36,"line":56},[34,99044,99045],{}," 医生 ||--o{ 病人 : 主管\n",[34,99047,99048],{"class":36,"line":62},[34,99049,99050],{}," 病房 ||--o{ 病人 : 收治\n",[34,99052,99053],{"class":36,"line":68},[34,99054,53],{"emptyLinePlaceholder":52},[34,99056,99057],{"class":36,"line":74},[34,99058,99059],{}," 科室 {\n",[34,99061,99062],{"class":36,"line":1971},[34,99063,99064],{}," string 科名\n",[34,99066,99067],{"class":36,"line":1976},[34,99068,99069],{}," string 科地址\n",[34,99071,99072],{"class":36,"line":1985},[34,99073,99074],{}," string 科电话\n",[34,99076,99077],{"class":36,"line":1990},[34,99078,5540],{},[34,99080,99081],{"class":36,"line":1999},[34,99082,99083],{}," 病房 {\n",[34,99085,99086],{"class":36,"line":2022},[34,99087,99088],{}," string 病房号\n",[34,99090,99091],{"class":36,"line":2036},[34,99092,99093],{}," string 床位号\n",[34,99095,99096],{"class":36,"line":2603},[34,99097,5540],{},[34,99099,99100],{"class":36,"line":5537},[34,99101,99102],{}," 医生 {\n",[34,99104,99105],{"class":36,"line":5543},[34,99106,99107],{}," string 工作证号\n",[34,99109,99110],{"class":36,"line":5548},[34,99111,99112],{}," string 姓名\n",[34,99114,99115],{"class":36,"line":5564},[34,99116,99117],{}," string 职称\n",[34,99119,99120],{"class":36,"line":5576},[34,99121,99122],{}," int 年龄\n",[34,99124,99125],{"class":36,"line":5606},[34,99126,5540],{},[34,99128,99129],{"class":36,"line":5616},[34,99130,99131],{}," 病人 {\n",[34,99133,99134],{"class":36,"line":5628},[34,99135,99136],{}," string 病历号\n",[34,99138,99139],{"class":36,"line":5634},[34,99140,99112],{},[34,99142,99143],{"class":36,"line":6420},[34,99144,99145],{}," string 诊断\n",[34,99147,99148],{"class":36,"line":11294},[34,99149,5540],{},[253,99151,99153],{"id":99152},"_512-转换为关系模型结构","51.2 转换为关系模型结构",[15,99155,58173],{},[105,99157,99158,99163,99172,99179,99191],{},[108,99159,99160,347],{},[31,99161,99162],{},"科室(科名, 科地址, 科电话)",[108,99164,99165,46528,99168,99171],{},[31,99166,99167],{},"医生(工作证号, 姓名, 职称, 年龄, 科名)",[31,99169,99170],{},"科名"," 为外键。",[108,99173,99174,46528,99177,99171],{},[31,99175,99176],{},"病房(病房号, 科名)",[31,99178,99170],{},[108,99180,99181,46528,99184,99187,99188,347],{},[31,99182,99183],{},"床位(病房号, 床位号)",[31,99185,99186],{},"病房号"," 为外键。若题目把“病房号+床位号”整体看作病房实体，也可写成 ",[31,99189,99190],{},"病房床位(病房号, 床位号, 科名)",[108,99192,99193,99196],{},[31,99194,99195],{},"病人(病历号, 姓名, 诊断, 主管医生工作证号, 病房号)","，主管医生和病房号为外键。",[253,99198,99200],{"id":99199},"_513-每个关系模式的候选码","51.3 每个关系模式的候选码",[15,99202,58173],{},[105,99204,99205,99212,99220,99227,99235],{},[108,99206,99207,2664,99210],{},[31,99208,99209],{},"科室",[31,99211,99170],{},[108,99213,99214,2664,99217],{},[31,99215,99216],{},"医生",[31,99218,99219],{},"工作证号",[108,99221,99222,2664,99225],{},[31,99223,99224],{},"病房",[31,99226,99186],{},[108,99228,99229,2664,99232],{},[31,99230,99231],{},"床位",[31,99233,99234],{},"(病房号, 床位号)",[108,99236,99237,2664,99240],{},[31,99238,99239],{},"病人",[31,99241,99242],{},"病历号",[15,99244,99245],{},"讲解：原题“科室”属性中列了“医生姓名”，但因为一个科室有多个医生，医生不应作为科室表中的多值属性，应拆成医生实体并用外键表示所属科室。",[19,99247,99249],{"id":99248},"_52-叙述数据库实现完整性检查的方法","52. 叙述数据库实现完整性检查的方法。",[15,99251,58173],{},[15,99253,99254],{},"DBMS 实现完整性检查一般包括：",[2230,99256,99257,99273,99276],{},[108,99258,99259,99260,146,99263,146,99266,146,99268,146,99270,99272],{},"提供完整性约束定义机制，如 ",[31,99261,99262],{},"PRIMARY KEY",[31,99264,99265],{},"FOREIGN KEY",[31,99267,94256],{},[31,99269,94262],{},[31,99271,94259],{},"、触发器。",[108,99274,99275],{},"在执行插入、删除、修改时自动检查约束。",[108,99277,99278],{},"发现违约时执行违约处理，如拒绝操作、级联更新、级联删除、置空、置默认值。",[15,99280,93991],{},[24,99282,99284],{"className":92584,"code":99283,"language":92586,"meta":29,"style":29},"CREATE TABLE dbo.Enrollment (\n Sno varchar(20) NOT NULL,\n Cno varchar(20) NOT NULL,\n Grade decimal(5, 2) NULL,\n CONSTRAINT PK_Enrollment PRIMARY KEY (Sno, Cno),\n CONSTRAINT CK_Enrollment_Grade CHECK (Grade BETWEEN 0 AND 100)\n);\n",[31,99285,99286,99291,99295,99299,99303,99308,99313],{"__ignoreMap":29},[34,99287,99288],{"class":36,"line":37},[34,99289,99290],{},"CREATE TABLE dbo.Enrollment (\n",[34,99292,99293],{"class":36,"line":43},[34,99294,94286],{},[34,99296,99297],{"class":36,"line":49},[34,99298,94291],{},[34,99300,99301],{"class":36,"line":56},[34,99302,94296],{},[34,99304,99305],{"class":36,"line":62},[34,99306,99307],{}," CONSTRAINT PK_Enrollment PRIMARY KEY (Sno, Cno),\n",[34,99309,99310],{"class":36,"line":68},[34,99311,99312],{}," CONSTRAINT CK_Enrollment_Grade CHECK (Grade BETWEEN 0 AND 100)\n",[34,99314,99315],{"class":36,"line":74},[34,99316,462],{},[15,99318,99319],{},"讲解：完整性检查应尽量靠 DBMS 统一执行，而不是分散在多个应用程序中。",[19,99321,99323],{"id":99322},"_54-事务中的提交和回滚是什么意思","54. 事务中的提交和回滚是什么意思？",[15,99325,58173],{},[105,99327,99328,99334],{},[108,99329,99330,99331,99333],{},"提交：事务成功结束，使用 ",[31,99332,93970],{}," 使所有更新永久生效。",[108,99335,99336,99337,99339],{},"回滚：事务失败或主动撤销，使用 ",[31,99338,91516],{}," 撤销所有未提交更新。",[15,99341,99342],{},"讲解：提交保证持久性，回滚保证原子性。一个事务不能只完成一半后把中间状态留在数据库里。",[19,99344,99346],{"id":99345},"_55-在数据库中为什么要有并发","55. 在数据库中为什么要有并发？",[15,99348,58173],{},[15,99350,99351],{},"数据库需要并发是因为多个用户和多个应用常常要同时访问数据库。并发可以提高系统吞吐量、改善资源利用率、缩短用户等待时间。",[15,99353,99354],{},"讲解：并发不是问题本身，未受控制的并发才是问题。DBMS 通过并发控制在效率和一致性之间取得平衡。",[19,99356,99358],{"id":99357},"_56-并发操作会产生几种不一致情况用什么方法避免各种不一致的情况","56. 并发操作会产生几种不一致情况？用什么方法避免各种不一致的情况？",[15,99360,58173],{},[15,99362,99363],{},"常见不一致情况：",[2230,99365,99366,99369,99372,99375],{},[108,99367,99368],{},"丢失修改：两个事务修改同一数据，后写覆盖先写。",[108,99370,99371],{},"读脏数据：读到其他事务尚未提交且后来回滚的数据。",[108,99373,99374],{},"不可重复读：同一事务两次读同一数据，中间被其他事务修改。",[108,99376,99377],{},"不一致分析：统计过程中部分数据被其他事务更新，得到混合状态结果。",[15,99379,99380],{},"避免方法：",[105,99382,99383,99386,99389],{},[108,99384,99385],{},"使用封锁机制，读加 S 锁、写加 X 锁。",[108,99387,99388],{},"遵循两段锁协议或严格两段锁协议。",[108,99390,99391,99392,146,99395,146,99398,99401],{},"使用合适隔离级别，如 SQL Server 的 ",[31,99393,99394],{},"READ COMMITTED",[31,99396,99397],{},"REPEATABLE READ",[31,99399,99400],{},"SERIALIZABLE","、快照隔离。",[15,99403,99404],{},"讲解：不同隔离级别防止的问题不同。考试中若未限定，写“封锁机制和两段锁协议”最稳。",[19,99406,99408],{"id":99407},"_57-叙述数据库中数据的一致性级别","57. 叙述数据库中数据的一致性级别。",[15,99410,58173],{},[15,99412,99413],{},"可按封锁协议理解一致性级别：",[105,99415,99416,99419,99422],{},[108,99417,99418],{},"一级封锁协议：事务修改数据前加 X 锁并保持到事务结束，可防止丢失修改。",[108,99420,99421],{},"二级封锁协议：在一级基础上，读数据前加 S 锁，读完即可释放，可防止读脏数据。",[108,99423,99424],{},"三级封锁协议：在一级基础上，读数据前加 S 锁并保持到事务结束，可防止不可重复读。",[15,99426,99427],{},"讲解：一致性级别越高，隔离性越强，但并发度越低。SQL 标准隔离级别与这些封锁协议有对应关系，但具体实现由 DBMS 决定。",[19,99429,99431],{"id":99430},"_58-叙述封锁的概念","58. 叙述封锁的概念。",[15,99433,58173],{},[15,99435,99436],{},"封锁是事务在访问某个数据对象前，先向 DBMS 请求对该对象加锁，以限制其他事务对同一对象的访问。访问结束或事务结束后释放锁。",[15,99438,99439],{},"讲解：封锁的目标是协调并发事务。常见锁类型是 S 锁和 X 锁，常见封锁粒度可以是行、页、表、数据库等。",[19,99441,99443],{"id":99442},"_59-叙述数据库中死锁产生的原因和解决死锁的方法","59. 叙述数据库中死锁产生的原因和解决死锁的方法。",[15,99445,58173],{},[15,99447,99448],{},"死锁产生原因：多个事务各自持有一部分锁，同时又等待对方释放锁，形成循环等待。",[15,99450,99451],{},"解决方法：",[2230,99453,99454,99457,99460,99463],{},[108,99455,99456],{},"死锁预防：规定统一加锁顺序，或用时间戳策略避免循环等待。",[108,99458,99459],{},"死锁检测：构造等待图，若出现环则说明死锁。",[108,99461,99462],{},"死锁解除：选择代价较小的事务回滚，释放其锁。",[108,99464,99465],{},"超时法：等待超过阈值后主动回滚。",[15,99467,99468],{},"讲解：两段锁协议可能产生死锁，因此实际 DBMS 通常需要死锁检测和牺牲事务回滚机制。",[19,99470,99472],{"id":99471},"_60-基本的封锁类型有几种试述它们的含义","60. 基本的封锁类型有几种？试述它们的含义。",[15,99474,58173],{},[15,99476,99477],{},"基本封锁类型有两种：",[105,99479,99480,99483],{},[108,99481,99482],{},"共享锁 S：用于读。多个事务可同时对同一对象加 S 锁。",[108,99484,99485],{},"排他锁 X：用于写。一个事务加 X 锁后，其他事务不能再加 S 锁或 X 锁。",[15,99487,99488],{},"讲解：记忆口诀：读读可并行，读写和写写互斥。",[19,99490,99492],{"id":99491},"_61-数据库中为什么要有恢复子系统它的功能是什么","61. 数据库中为什么要有恢复子系统？它的功能是什么？",[15,99494,58173],{},[15,99496,99497],{},"数据库需要恢复子系统，是因为事务故障、系统故障、介质故障等可能导致数据库不一致或数据丢失。恢复子系统负责在故障后把数据库恢复到正确一致状态。",[15,99499,99500],{},"功能包括：",[2230,99502,99503,99506,99509,99512],{},[108,99504,99505],{},"维护日志和检查点。",[108,99507,99508],{},"支持事务回滚，撤销失败事务。",[108,99510,99511],{},"系统重启后自动恢复，REDO 已提交事务、UNDO 未提交事务。",[108,99513,99514],{},"配合备份转储处理介质故障。",[15,99516,99517],{},"讲解：恢复子系统主要保证事务的原子性和持久性。",[79,99519,81],{},{"title":29,"searchDepth":43,"depth":43,"links":99521},[99522,99523,99524,99525,99530,99531,99532,99533,99534,99535,99536,99537,99538],{"id":98742,"depth":43,"text":98743},{"id":98819,"depth":43,"text":98820},{"id":98882,"depth":43,"text":98883},{"id":98994,"depth":43,"text":98995,"children":99526},[99527,99528,99529],{"id":99018,"depth":49,"text":99019},{"id":99152,"depth":49,"text":99153},{"id":99199,"depth":49,"text":99200},{"id":99248,"depth":43,"text":99249},{"id":99322,"depth":43,"text":99323},{"id":99345,"depth":43,"text":99346},{"id":99357,"depth":43,"text":99358},{"id":99407,"depth":43,"text":99408},{"id":99430,"depth":43,"text":99431},{"id":99442,"depth":43,"text":99443},{"id":99471,"depth":43,"text":99472},{"id":99491,"depth":43,"text":99492},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday11-70-48-61",{"title":98731,"description":98739},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay11-数据库70题-综合48-61","CbGuq7n12RgymuvwXBPEAq_BIxJysqtEZAuow3yQ1fc",{"id":99545,"title":99546,"body":99547,"description":99554,"extension":85,"meta":100409,"navigation":52,"path":100410,"seo":100411,"stem":100412,"__hash__":100413},"notes\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002FDay12-数据库70题-综合62-71.md","Day 12 - 数据库70题：综合题 62-71",{"type":8,"value":99548,"toc":100396},[99549,99552,99555,99559,99561,99564,99567,99571,99573,99576,99579,99583,99585,99588,99606,99609,99613,99615,99618,99632,99635,99639,99641,99643,99657,99660,99664,99666,99669,99672,99686,99689,99693,99695,99703,99705,99713,99716,99720,99722,99724,99772,99774,99803,99805,99871,99873,100014,100016,100056,100070,100072,100079,100153,100156,100160,100163,100219,100221,100241,100244,100321,100324,100385,100394],[11,99550,99546],{"id":99551},"day-12-数据库70题综合题-62-71",[15,99553,99554],{},"参考重点：教材第 5 章完整性，第 11 章恢复技术，第 12 章并发控制。SQL 均按 SQL Server 写法。",[19,99556,99558],{"id":99557},"_62-什么是数据库的转储转储的意义是什么","62. 什么是数据库的转储？转储的意义是什么？",[15,99560,58173],{},[15,99562,99563],{},"数据库转储是把数据库复制到后援存储介质上，形成后援副本的过程。意义是在数据库遭到破坏，尤其是发生介质故障时，可用后援副本和日志文件恢复数据库。",[15,99565,99566],{},"讲解：转储是恢复的基础。静态转储简单但影响业务；动态转储允许业务运行，但必须配合日志保证一致性。",[19,99568,99570],{"id":99569},"_63-为什么要建立日志文件","63. 为什么要建立日志文件？",[15,99572,58173],{},[15,99574,99575],{},"日志文件记录事务对数据库的更新操作，是事务故障和系统故障恢复的依据。通过日志可以撤销未提交事务、重做已提交事务，并配合检查点提高恢复效率。",[15,99577,99578],{},"讲解：日志通常记录事务标识、操作类型、数据对象、旧值、新值等。必须遵守先写日志原则：数据页写盘前，相应日志先写入稳定存储。",[19,99580,99582],{"id":99581},"_64-怎样进行系统故障的恢复","64. 怎样进行系统故障的恢复？",[15,99584,58173],{},[15,99586,99587],{},"系统故障恢复一般步骤：",[2230,99589,99590,99593,99598,99603],{},[108,99591,99592],{},"从最近检查点开始扫描日志。",[108,99594,99595,99596,347],{},"找出故障时已经提交但可能未完全写盘的事务，执行 ",[31,99597,94061],{},[108,99599,99600,99601,347],{},"找出故障时尚未提交的事务，执行 ",[31,99602,94055],{},[108,99604,99605],{},"恢复结束后数据库回到一致状态。",[15,99607,99608],{},"讲解：系统故障不破坏外存数据库文件，但会丢失内存缓冲区内容。因此既可能需要重做提交事务，也可能需要撤销未提交事务。",[19,99610,99612],{"id":99611},"_65-怎样进行介质故障的恢复","65. 怎样进行介质故障的恢复？",[15,99614,58173],{},[15,99616,99617],{},"介质故障恢复一般步骤：",[2230,99619,99620,99623,99626,99629],{},[108,99621,99622],{},"更换或修复故障介质。",[108,99624,99625],{},"装入最近一次数据库后援副本。",[108,99627,99628],{},"利用转储以来的日志文件，重做已经提交的事务。",[108,99630,99631],{},"必要时撤销未提交事务，使数据库达到一致状态。",[15,99633,99634],{},"讲解：介质故障会损坏外存数据，因此仅靠日志不够，必须先有数据库备份作为恢复起点。",[19,99636,99638],{"id":99637},"_66-数据库在运行过程可能产生的故障有哪几类","66. 数据库在运行过程可能产生的故障有哪几类？",[15,99640,58173],{},[15,99642,94079],{},[2230,99644,99645,99648,99651,99654],{},[108,99646,99647],{},"事务故障。",[108,99649,99650],{},"系统故障。",[108,99652,99653],{},"介质故障。",[108,99655,99656],{},"计算机病毒、恶意破坏或灾难性故障。",[15,99658,99659],{},"讲解：不同故障需要不同恢复策略。事务故障通常 UNDO 即可；系统故障需要 UNDO\u002FREDO；介质故障需要备份和日志。",[19,99661,99663],{"id":99662},"_67-叙述数据库的并发控制","67. 叙述数据库的并发控制。",[15,99665,58173],{},[15,99667,99668],{},"数据库并发控制是 DBMS 对多个事务同时访问同一数据的操作进行调度和协调，使并发执行结果等价于某个串行执行结果，从而保持数据库一致性。",[15,99670,99671],{},"常用方法：",[105,99673,99674,99677,99680,99683],{},[108,99675,99676],{},"封锁机制和两段锁协议。",[108,99678,99679],{},"时间戳排序。",[108,99681,99682],{},"多版本并发控制（MVCC）。",[108,99684,99685],{},"设置事务隔离级别。",[15,99687,99688],{},"讲解：并发控制的目标不是消灭并发，而是在高并发和正确性之间平衡。教材重点是封锁、封锁协议、可串行化和死锁处理。",[19,99690,99692],{"id":99691},"_68-在数据库保护中安全性和完整性有何区别","68. 在数据库保护中，安全性和完整性有何区别？",[15,99694,58173],{},[105,99696,99697,99700],{},[108,99698,99699],{},"安全性：保护数据库免受非法访问、非法使用、泄露或破坏。",[108,99701,99702],{},"完整性：保证数据库中的数据正确、有效、相容，符合语义约束。",[15,99704,19553],{},[105,99706,99707,99710],{},[108,99708,99709],{},"用户没有权限查看成绩表却试图查询，这是安全性问题。",[108,99711,99712],{},"成绩被写成 150 分，这是完整性问题。",[15,99714,99715],{},"讲解：安全性管“谁能做”，完整性管“做出来的数据是否合法”。",[19,99717,99719],{"id":99718},"_69-设数据库含-scd-三个关系表使用-sql-server-设置各个表操作之间的完整性约束","69. 设数据库含 S、C、D 三个关系表，使用 SQL Server 设置各个表操作之间的完整性约束。",[15,99721,94672],{},[15,99723,94675],{},[13837,99725,99726,99739],{},[13840,99727,99728],{},[13843,99729,99730,99733,99736],{},[13846,99731,99732],{},"学号",[13846,99734,99735],{},"姓名",[13846,99737,99738],{},"班号",[13853,99740,99741,99751,99762],{},[13843,99742,99743,99746,99748],{},[13858,99744,99745],{},"S2",[13858,99747,94759],{},[13858,99749,99750],{},"98001",[13843,99752,99753,99756,99759],{},[13858,99754,99755],{},"S6",[13858,99757,99758],{},"李英",[13858,99760,99761],{},"98002",[13843,99763,99764,99767,99769],{},[13858,99765,99766],{},"S12",[13858,99768,98114],{},[13858,99770,99771],{},"98003",[15,99773,47384],{},[13837,99775,99776,99786],{},[13840,99777,99778],{},[13843,99779,99780,99783],{},[13846,99781,99782],{},"课程号",[13846,99784,99785],{},"课程名称",[13853,99787,99788,99795],{},[13843,99789,99790,99792],{},[13858,99791,94753],{},[13858,99793,99794],{},"英语",[13843,99796,99797,99800],{},[13858,99798,99799],{},"K12",[13858,99801,99802],{},"高等数学",[15,99804,91994],{},[13837,99806,99807,99818],{},[13840,99808,99809],{},[13843,99810,99811,99813,99815],{},[13846,99812,99732],{},[13846,99814,99782],{},[13846,99816,99817],{},"分数",[13853,99819,99820,99828,99837,99845,99853,99862],{},[13843,99821,99822,99824,99826],{},[13858,99823,99745],{},[13858,99825,94753],{},[13858,99827,94841],{},[13843,99829,99830,99832,99834],{},[13858,99831,99755],{},[13858,99833,94753],{},[13858,99835,99836],{},"79",[13843,99838,99839,99841,99843],{},[13858,99840,99766],{},[13858,99842,94753],{},[13858,99844,96817],{},[13843,99846,99847,99849,99851],{},[13858,99848,99745],{},[13858,99850,99799],{},[13858,99852,95593],{},[13843,99854,99855,99857,99859],{},[13858,99856,99755],{},[13858,99858,99799],{},[13858,99860,99861],{},"82",[13843,99863,99864,99866,99868],{},[13858,99865,99766],{},[13858,99867,99799],{},[13858,99869,99870],{},"67",[15,99872,58173],{},[24,99874,99876],{"className":92584,"code":99875,"language":92586,"meta":29,"style":29},"CREATE TABLE dbo.S (\n 学号 varchar(20) NOT NULL,\n 姓名 nvarchar(50) NOT NULL,\n 班号 varchar(20) NOT NULL,\n CONSTRAINT PK_S PRIMARY KEY (学号)\n);\n\nCREATE TABLE dbo.C (\n 课程号 varchar(20) NOT NULL,\n 课程名称 nvarchar(100) NOT NULL,\n CONSTRAINT PK_C PRIMARY KEY (课程号),\n CONSTRAINT UQ_C_课程名称 UNIQUE (课程名称)\n);\n\nCREATE TABLE dbo.D (\n 学号 varchar(20) NOT NULL,\n 课程号 varchar(20) NOT NULL,\n 分数 decimal(5, 2) NULL,\n CONSTRAINT PK_D PRIMARY KEY (学号, 课程号),\n CONSTRAINT FK_D_S FOREIGN KEY (学号)\n REFERENCES dbo.S(学号)\n ON UPDATE CASCADE\n ON DELETE NO ACTION,\n CONSTRAINT FK_D_C FOREIGN KEY (课程号)\n REFERENCES dbo.C(课程号)\n ON UPDATE CASCADE\n ON DELETE NO ACTION,\n CONSTRAINT CK_D_分数 CHECK (分数 BETWEEN 0 AND 100)\n);\n",[31,99877,99878,99883,99888,99893,99898,99903,99907,99911,99916,99921,99926,99931,99936,99940,99944,99949,99953,99957,99962,99967,99972,99977,99982,99987,99992,99997,100001,100005,100010],{"__ignoreMap":29},[34,99879,99880],{"class":36,"line":37},[34,99881,99882],{},"CREATE TABLE dbo.S (\n",[34,99884,99885],{"class":36,"line":43},[34,99886,99887],{}," 学号 varchar(20) NOT NULL,\n",[34,99889,99890],{"class":36,"line":49},[34,99891,99892],{}," 姓名 nvarchar(50) NOT NULL,\n",[34,99894,99895],{"class":36,"line":56},[34,99896,99897],{}," 班号 varchar(20) NOT NULL,\n",[34,99899,99900],{"class":36,"line":62},[34,99901,99902],{}," CONSTRAINT PK_S PRIMARY KEY (学号)\n",[34,99904,99905],{"class":36,"line":68},[34,99906,462],{},[34,99908,99909],{"class":36,"line":74},[34,99910,53],{"emptyLinePlaceholder":52},[34,99912,99913],{"class":36,"line":1971},[34,99914,99915],{},"CREATE TABLE dbo.C (\n",[34,99917,99918],{"class":36,"line":1976},[34,99919,99920],{}," 课程号 varchar(20) NOT NULL,\n",[34,99922,99923],{"class":36,"line":1985},[34,99924,99925],{}," 课程名称 nvarchar(100) NOT NULL,\n",[34,99927,99928],{"class":36,"line":1990},[34,99929,99930],{}," CONSTRAINT PK_C PRIMARY KEY (课程号),\n",[34,99932,99933],{"class":36,"line":1999},[34,99934,99935],{}," CONSTRAINT UQ_C_课程名称 UNIQUE (课程名称)\n",[34,99937,99938],{"class":36,"line":2022},[34,99939,462],{},[34,99941,99942],{"class":36,"line":2036},[34,99943,53],{"emptyLinePlaceholder":52},[34,99945,99946],{"class":36,"line":2603},[34,99947,99948],{},"CREATE TABLE dbo.D (\n",[34,99950,99951],{"class":36,"line":5537},[34,99952,99887],{},[34,99954,99955],{"class":36,"line":5543},[34,99956,99920],{},[34,99958,99959],{"class":36,"line":5548},[34,99960,99961],{}," 分数 decimal(5, 2) NULL,\n",[34,99963,99964],{"class":36,"line":5564},[34,99965,99966],{}," CONSTRAINT PK_D PRIMARY KEY (学号, 课程号),\n",[34,99968,99969],{"class":36,"line":5576},[34,99970,99971],{}," CONSTRAINT FK_D_S FOREIGN KEY (学号)\n",[34,99973,99974],{"class":36,"line":5606},[34,99975,99976],{}," REFERENCES dbo.S(学号)\n",[34,99978,99979],{"class":36,"line":5616},[34,99980,99981],{}," ON UPDATE CASCADE\n",[34,99983,99984],{"class":36,"line":5628},[34,99985,99986],{}," ON DELETE NO ACTION,\n",[34,99988,99989],{"class":36,"line":5634},[34,99990,99991],{}," CONSTRAINT FK_D_C FOREIGN KEY (课程号)\n",[34,99993,99994],{"class":36,"line":6420},[34,99995,99996],{}," REFERENCES dbo.C(课程号)\n",[34,99998,99999],{"class":36,"line":11294},[34,100000,99981],{},[34,100002,100003],{"class":36,"line":11321},[34,100004,99986],{},[34,100006,100007],{"class":36,"line":11326},[34,100008,100009],{}," CONSTRAINT CK_D_分数 CHECK (分数 BETWEEN 0 AND 100)\n",[34,100011,100012],{"class":36,"line":11333},[34,100013,462],{},[15,100015,91513],{},[105,100017,100018,100027,100033,100042,100047],{},[108,100019,100020,258,100023,100026],{},[31,100021,100022],{},"S.学号",[31,100024,100025],{},"C.课程号"," 是实体完整性。",[108,100028,100029,100032],{},[31,100030,100031],{},"D(学号, 课程号)"," 是选课表组合主键，保证同一学生同一课程只有一条成绩。",[108,100034,100035,146,100038,100041],{},[31,100036,100037],{},"D.学号",[31,100039,100040],{},"D.课程号"," 是外键，保证参照完整性。",[108,100043,100044,100046],{},[31,100045,94262],{}," 保证用户定义完整性。",[108,100048,100049,100052,100053,347],{},[31,100050,100051],{},"ON DELETE NO ACTION"," 表示有选课记录时不能随便删除学生或课程；如果业务要求删除学生时同步删除成绩，可改成 ",[31,100054,100055],{},"ON DELETE CASCADE",[19,100057,100059,100060,46528,100063,46528,100066,100069],{"id":100058},"_70-设有三个事务-t1t2-和-t3动作分别为t1-aa2t2-aa2t3-aa2假定这三个事务允许并行操作讨论它们可能实施的调度","70. 设有三个事务 T1、T2 和 T3，动作分别为：",[31,100061,100062],{},"T1: A=A+2",[31,100064,100065],{},"T2: A=A*2",[31,100067,100068],{},"T3: A=A**2","。假定这三个事务允许并行操作，讨论它们可能实施的调度。",[15,100071,58173],{},[15,100073,100074,100075,100078],{},"若要求正确并发，调度结果应等价于某个串行调度。设初值为 ",[31,100076,100077],{},"A=a","，三个事务的 6 种串行顺序结果为：",[13837,100080,100081,100091],{},[13840,100082,100083],{},[13843,100084,100085,100088],{},[13846,100086,100087],{},"串行顺序",[13846,100089,100090],{},"最终结果",[13853,100092,100093,100103,100113,100123,100133,100143],{},[13843,100094,100095,100098],{},[13858,100096,100097],{},"T1 -> T2 -> T3",[13858,100099,100100],{},[31,100101,100102],{},"((a + 2) * 2)^2",[13843,100104,100105,100108],{},[13858,100106,100107],{},"T1 -> T3 -> T2",[13858,100109,100110],{},[31,100111,100112],{},"(a + 2)^2 * 2",[13843,100114,100115,100118],{},[13858,100116,100117],{},"T2 -> T1 -> T3",[13858,100119,100120],{},[31,100121,100122],{},"(2a + 2)^2",[13843,100124,100125,100128],{},[13858,100126,100127],{},"T2 -> T3 -> T1",[13858,100129,100130],{},[31,100131,100132],{},"(2a)^2 + 2",[13843,100134,100135,100138],{},[13858,100136,100137],{},"T3 -> T1 -> T2",[13858,100139,100140],{},[31,100141,100142],{},"(a^2 + 2) * 2",[13843,100144,100145,100148],{},[13858,100146,100147],{},"T3 -> T2 -> T1",[13858,100149,100150],{},[31,100151,100152],{},"2a^2 + 2",[15,100154,100155],{},"讲解：这三个操作都读写同一数据 A，彼此不交换顺序。例如先加 2 再乘 2，与先乘 2 再加 2 结果不同。因此正确调度必须可串行化。实际系统中可让每个事务对 A 加 X 锁并保持到提交，这样并发执行会退化为某个串行顺序，但结果正确。",[19,100157,100159],{"id":100158},"_71-假设存款余额-x1000-元甲事务取走-300-元乙事务取走-200-元执行时间如下如何实现这两个事务的并发控制","71. 假设存款余额 X=1000 元，甲事务取走 300 元，乙事务取走 200 元，执行时间如下。如何实现这两个事务的并发控制？",[15,100161,100162],{},"题面调度：",[13837,100164,100165,100178],{},[13840,100166,100167],{},[13843,100168,100169,100172,100175],{},[13846,100170,100171],{},"甲事务",[13846,100173,100174],{},"时间",[13846,100176,100177],{},"乙事务",[13853,100179,100180,100190,100199,100209],{},[13843,100181,100182,100185,100188],{},[13858,100183,100184],{},"读 X",[13858,100186,100187],{},"T1",[13858,100189],{},[13843,100191,100192,100194,100197],{},[13858,100193],{},[13858,100195,100196],{},"T2",[13858,100198,100184],{},[13843,100200,100201,100204,100207],{},[13858,100202,100203],{},"更新 X=X-300",[13858,100205,100206],{},"T3",[13858,100208],{},[13843,100210,100211,100213,100216],{},[13858,100212],{},[13858,100214,100215],{},"T4",[13858,100217,100218],{},"更新 X=X-200",[15,100220,58173],{},[15,100222,100223,100224,100227,100228,100231,100232,100235,100236,100238,100239,347],{},"该调度会产生丢失修改。甲、乙都读到 ",[31,100225,100226],{},"X=1000","，甲写回 ",[31,100229,100230],{},"700","，乙再按旧值写回 ",[31,100233,100234],{},"800","，最终余额可能为 ",[31,100237,100234],{},"，甲的取款被覆盖；正确结果应为 ",[31,100240,25024],{},[15,100242,100243],{},"可用排他锁或可串行化事务控制：",[24,100245,100247],{"className":92584,"code":100246,"language":92586,"meta":29,"style":29},"-- 甲事务\nBEGIN TRANSACTION;\n\nUPDATE dbo.Accounts\nSET Balance = Balance - 300\nWHERE AccountNo = 'X';\n\nCOMMIT TRANSACTION;\n\n-- 乙事务\nBEGIN TRANSACTION;\n\nUPDATE dbo.Accounts\nSET Balance = Balance - 200\nWHERE AccountNo = 'X';\n\nCOMMIT TRANSACTION;\n",[31,100248,100249,100254,100258,100262,100266,100270,100275,100279,100283,100287,100292,100296,100300,100304,100309,100313,100317],{"__ignoreMap":29},[34,100250,100251],{"class":36,"line":37},[34,100252,100253],{},"-- 甲事务\n",[34,100255,100256],{"class":36,"line":43},[34,100257,94001],{},[34,100259,100260],{"class":36,"line":49},[34,100261,53],{"emptyLinePlaceholder":52},[34,100263,100264],{"class":36,"line":56},[34,100265,94010],{},[34,100267,100268],{"class":36,"line":62},[34,100269,94015],{},[34,100271,100272],{"class":36,"line":68},[34,100273,100274],{},"WHERE AccountNo = 'X';\n",[34,100276,100277],{"class":36,"line":74},[34,100278,53],{"emptyLinePlaceholder":52},[34,100280,100281],{"class":36,"line":1971},[34,100282,94029],{},[34,100284,100285],{"class":36,"line":1976},[34,100286,53],{"emptyLinePlaceholder":52},[34,100288,100289],{"class":36,"line":1985},[34,100290,100291],{},"-- 乙事务\n",[34,100293,100294],{"class":36,"line":1990},[34,100295,94001],{},[34,100297,100298],{"class":36,"line":1999},[34,100299,53],{"emptyLinePlaceholder":52},[34,100301,100302],{"class":36,"line":2022},[34,100303,94010],{},[34,100305,100306],{"class":36,"line":2036},[34,100307,100308],{},"SET Balance = Balance - 200\n",[34,100310,100311],{"class":36,"line":2603},[34,100312,100274],{},[34,100314,100315],{"class":36,"line":5537},[34,100316,53],{"emptyLinePlaceholder":52},[34,100318,100319],{"class":36,"line":5543},[34,100320,94029],{},[15,100322,100323],{},"若必须先读再算，可在 SQL Server 中使用更新锁和保持锁：",[24,100325,100327],{"className":92584,"code":100326,"language":92586,"meta":29,"style":29},"BEGIN TRANSACTION;\n\nDECLARE @x decimal(12, 2);\n\nSELECT @x = Balance\nFROM dbo.Accounts WITH (UPDLOCK, HOLDLOCK)\nWHERE AccountNo = 'X';\n\nUPDATE dbo.Accounts\nSET Balance = @x - 300\nWHERE AccountNo = 'X';\n\nCOMMIT TRANSACTION;\n",[31,100328,100329,100333,100337,100342,100346,100351,100356,100360,100364,100368,100373,100377,100381],{"__ignoreMap":29},[34,100330,100331],{"class":36,"line":37},[34,100332,94001],{},[34,100334,100335],{"class":36,"line":43},[34,100336,53],{"emptyLinePlaceholder":52},[34,100338,100339],{"class":36,"line":49},[34,100340,100341],{},"DECLARE @x decimal(12, 2);\n",[34,100343,100344],{"class":36,"line":56},[34,100345,53],{"emptyLinePlaceholder":52},[34,100347,100348],{"class":36,"line":62},[34,100349,100350],{},"SELECT @x = Balance\n",[34,100352,100353],{"class":36,"line":68},[34,100354,100355],{},"FROM dbo.Accounts WITH (UPDLOCK, HOLDLOCK)\n",[34,100357,100358],{"class":36,"line":74},[34,100359,100274],{},[34,100361,100362],{"class":36,"line":1971},[34,100363,53],{"emptyLinePlaceholder":52},[34,100365,100366],{"class":36,"line":1976},[34,100367,94010],{},[34,100369,100370],{"class":36,"line":1985},[34,100371,100372],{},"SET Balance = @x - 300\n",[34,100374,100375],{"class":36,"line":1990},[34,100376,100274],{},[34,100378,100379],{"class":36,"line":1999},[34,100380,53],{"emptyLinePlaceholder":52},[34,100382,100383],{"class":36,"line":2022},[34,100384,94029],{},[15,100386,91513,100387,100390,100391,347],{},[31,100388,100389],{},"UPDLOCK, HOLDLOCK"," 使事务在读取余额时就锁住该行，并保持到事务结束，另一个事务必须等待。这样两个取款事务按某个串行顺序执行，最终余额为 ",[31,100392,100393],{},"1000 - 300 - 200 = 500",[79,100395,81],{},{"title":29,"searchDepth":43,"depth":43,"links":100397},[100398,100399,100400,100401,100402,100403,100404,100405,100406,100408],{"id":99557,"depth":43,"text":99558},{"id":99569,"depth":43,"text":99570},{"id":99581,"depth":43,"text":99582},{"id":99611,"depth":43,"text":99612},{"id":99637,"depth":43,"text":99638},{"id":99662,"depth":43,"text":99663},{"id":99691,"depth":43,"text":99692},{"id":99718,"depth":43,"text":99719},{"id":100058,"depth":43,"text":100407},"70. 设有三个事务 T1、T2 和 T3，动作分别为：T1: A=A+2，T2: A=A*2，T3: A=A**2。假定这三个事务允许并行操作，讨论它们可能实施的调度。",{"id":100158,"depth":43,"text":100159},{},"\u002Fsource\u002Fcs\u002Fdb\u002F100+70\u002Fday12-70-62-71",{"title":99546,"description":99554},"source\u002Fcs\u002Fdb\u002F100+70\u002FDay12-数据库70题-综合62-71","GeGDc6JLOkFxvzSukHx0Wvk32gJiKAN-WgQolTb2tdo",{"id":100415,"title":100416,"body":100417,"description":100434,"extension":85,"meta":100435,"navigation":52,"path":100436,"seo":100437,"stem":100438,"__hash__":100439},"notes\u002Fsource\u002Fcs\u002Fdb\u002Findex.md","Database Systems",{"type":8,"value":100418,"toc":100432},[100419,100429],[16016,100420,31533,100422,31533,100425],{"className":100421},[31532],[11,100423,100416],{"id":100424},"database-systems",[15,100426,100428],{"className":100427},[31540],"数据库系统",[31543,100430],{"path":100431,"unwrap-single-group":29},"\u002Fcs\u002Fdb\u002F",{"title":29,"searchDepth":43,"depth":43,"links":100433},[],"数据库系统学习笔记，包含数据库规范化（BCNF）、E-R图设计、SQL、关系代数等内容，附历年练习题。",{},"\u002Fsource\u002Fcs\u002Fdb",{"title":100416,"description":100434},"source\u002Fcs\u002Fdb\u002Findex","O09YIqwkFYK-DoDuZRL2xO6bIOXgiqqUtKpEwONAjEQ",{"id":100441,"title":100442,"body":100443,"description":101057,"extension":85,"meta":101058,"navigation":52,"path":101059,"seo":101060,"stem":101061,"__hash__":101062},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F01-基础概念与题型全文.md","基础概念与题型全文",{"type":8,"value":100444,"toc":101033},[100445,100447,100486,100489,100495,100498,100504,100507,100511,100514,100518,100540,100554,100557,100560,100564,100567,100581,100585,100588,100591,100593,100627,100630,100665,100668,100670,100722,100725,100757,100760,100764,100767,100770,100773,100776,100780,100783,100786,100789,100792,100796,100799,100802,100805,100809,100812,100815,100817,100821,100824,100839,100842,100858,100862,100865,100868,100870,100874,100877,100880,100883,100887,100890,100893,100896,100899,100902,100906,100909,100912,100916,100919,101007,101010,101031],[11,100446,100442],{"id":100442},[15,100448,100449,100450,146,100453,146,100456,146,100459,146,100462,146,100465,146,100468,146,100471,146,100474,146,100477,146,100480,146,100483,347],{},"覆盖图片：",[31,100451,100452],{},"FE1EB12F1021FF5AC008557D0D0FDC17.png",[31,100454,100455],{},"8087F11947A9E02D4BB95C6564592F0F.png",[31,100457,100458],{},"0B90006D47895530C9657C33567E7AF3.png",[31,100460,100461],{},"074F26CB7353DD83CAAA9B45A4BEF568.png",[31,100463,100464],{},"CA7860A16D7978E001F1205EE772B09B.png",[31,100466,100467],{},"7EFBDE17B0116BB9766F10AC8BFF9AE4.png",[31,100469,100470],{},"ce02642dbb6b3731ac80b12971821561_720.jpg",[31,100472,100473],{},"IMG_0751.JPG",[31,100475,100476],{},"IMG_0751 (1).JPG",[31,100478,100479],{},"IMG_0391.JPG",[31,100481,100482],{},"IMG_0391 (1).JPG",[31,100484,100485],{},"A82101FC9E13F4F2388DC3EEC2CFB6DB.png",[19,100487,100488],{"id":100488},"英文试卷题型结构",[24,100490,100493],{"className":100491,"code":100492,"language":2638,"meta":29},[2636],"I. Fill in the blanks. (20 * 1 points = 20 points)\nII. SQL. (8, 20 points)\nIII. Database programming. (2 * 10 points = 20 points)\nIV. Relational Database Design. (15 points)\nV. Query Optimization. (10 points)\nVI. Database Recovery. (10 points)\nVII. Concurrency Control. (5 points)\n",[31,100494,100492],{"__ignoreMap":29},[15,100496,100497],{},"另一页展示的结构：",[24,100499,100502],{"className":100500,"code":100501,"language":2638,"meta":29},[2636],"1. Explain the glossaries. (20 points)\n2. SQL. (20 points)\n3. Database programming. (20 points)\n4. Relational Database Design. (15 points)\n - Design an E-R diagram.\n - Design a set of relational schemas, and a list of constraints,\n including primary key and foreign key constraints.\n - Explain what normal form each relational schema satisfies, and why.\n5. Query Optimization. (10 points)\n - Give a relational algebra expression using Cartesian product\n equivalent to the query.\n - Draw a syntax tree for the above relational algebra expression.\n - Give a procedure for query optimization.\n - Give the optimized relational algebra expression.\n6. Database Recovery. (10 points)\n7. Concurrency Control. (5 points)\n - Add lock and unlock instructions to transactions T1 and T2.\n",[31,100503,100501],{"__ignoreMap":29},[19,100505,100506],{"id":100506},"术语解释",[253,100508,100510],{"id":100509},"relation","Relation",[15,100512,100513],{},"关系是域的笛卡尔积的子集。",[253,100515,100517],{"id":100516},"referential-integrity","Referential Integrity",[15,100519,100520,100521,100523,100524,100526,100527,100529,100530,100533,100534,100536,100537,100539],{},"参照完整性：若属性或属性组 ",[31,100522,51834],{}," 是基本关系 ",[31,100525,34386],{}," 的外码，它与基本关系 ",[31,100528,94675],{}," 的主码 ",[31,100531,100532],{},"Ks"," 相对应，则对于 ",[31,100535,34386],{}," 中每个元组在 ",[31,100538,51834],{}," 上的值必须满足：",[2230,100541,100542,100548],{},[108,100543,100544,100545,100547],{},"取空值，即 ",[31,100546,51834],{}," 的每个属性值均为空值；或",[108,100549,100550,100551,100553],{},"等于 ",[31,100552,94675],{}," 中某个元组的主码值。",[253,100555,100556],{"id":2310},"Role",[15,100558,100559],{},"角色是数据库中一组权限的集合。",[253,100561,100563],{"id":100562},"acid-properties","ACID Properties",[15,100565,100566],{},"事务的 ACID 特性：",[105,100568,100569,100572,100575,100578],{},[108,100570,100571],{},"Atomicity：原子性。",[108,100573,100574],{},"Consistency：一致性。",[108,100576,100577],{},"Isolation：隔离性。",[108,100579,100580],{},"Durability：持久性。",[253,100582,100584],{"id":100583},"dirty-data","Dirty Data",[15,100586,100587],{},"脏数据：假设事务 A 更新了一行数据，事务 B 读取了该行的新值；之后事务 A 回滚，则事务 B 读到的就是未提交且后来被撤销的数据，即脏数据。",[19,100589,100590],{"id":100590},"判断题与参考答案",[15,100592,92498],{},[2230,100594,100595,100598,100601,100604,100609,100612,100615,100618,100621,100624],{},[108,100596,100597],{},"一个基本表只能建立一个聚簇索引。",[108,100599,100600],{},"数据库的并发操作一定会出现异常，因此必须实行封锁机制。",[108,100602,100603],{},"突然停电属于介质故障。",[108,100605,100606,100608],{},[31,100607,87756],{}," 值代表了空字符串。",[108,100610,100611],{},"嵌入式 SQL 与主语言通信是靠 SQLCA 和变量来实现。",[108,100613,100614],{},"MAC 中主体和客体的密级通过对比来确定能否访问数据。",[108,100616,100617],{},"所有的视图都是可更新的。",[108,100619,100620],{},"关系的主码和外码中的属性均不能取空值。",[108,100622,100623],{},"“年龄限制在 16-25 岁之间”约束属于 DBMS 的安全性功能要求的实现。",[108,100625,100626],{},"查询优化时，连接运算应尽可能先做。",[15,100628,100629],{},"参考答案与理由：",[2230,100631,100632,100635,100638,100641,100647,100650,100653,100656,100659,100662],{},[108,100633,100634],{},"对。聚簇索引决定表中数据物理排列方式，一个表通常只能有一个。",[108,100636,100637],{},"错。并发操作可能出现异常，不是一定出现；封锁机制是为了预防并发异常。",[108,100639,100640],{},"错。突然停电通常属于系统故障，不是介质故障。",[108,100642,100643,100644,100646],{},"错。",[31,100645,87756],{}," 表示未知、缺失或不适用，不等同于空字符串。",[108,100648,100649],{},"对。嵌入式 SQL 通过 SQLCA 和主变量与宿主语言通信。",[108,100651,100652],{},"对。强制访问控制 MAC 依赖主体\u002F客体安全级别或密级比较。",[108,100654,100655],{},"错。复杂视图、含聚合\u002F分组\u002F连接等视图通常不可更新。",[108,100657,100658],{},"错。主码属性不能取空值；外码属性在允许时可以为空。",[108,100660,100661],{},"错。这是完整性约束，不是安全性功能。",[108,100663,100664],{},"错。查询优化通常尽量先做选择和投影，减少中间结果；连接顺序需优化，不能简单“尽可能先做”。",[19,100666,100667],{"id":100667},"填空题与参考答案",[15,100669,92498],{},[2230,100671,100672,100675,100678,100685,100688,100694,100701,100707,100714,100719],{},[108,100673,100674],{},"嵌入式 SQL 中，数据库工作单元与源程序工作单元之间的通信主要包括 SQLCA、____ 等。",[108,100676,100677],{},"为了避免磁盘介质故障，许多 DBMS 系统提供了 ____ 等功能用于数据库恢复。",[108,100679,100680,100681,100684],{},"数据模型通常由 _",[2627,100682,100683],{},"_、数据操作和 __"," 三部分组成。",[108,100686,100687],{},"避免活锁的基本方法是 ____。",[108,100689,100690,100691,347],{},"查询优化一般可分为 _",[2627,100692,100693],{},"_ 和 __",[108,100695,100696,100697,100700],{},"事务的 ACID 包括原子性、_",[2627,100698,100699],{},"_、__","、持续性。",[108,100702,100703,100704,347],{},"SQL Server 游标操作的步骤有：游标定义、游标打开、_",[2627,100705,100706],{},"_、关闭游标、__",[108,100708,100709,100710,100713],{},"基本封锁有两种：_",[2627,100711,100712],{},"_ 锁和 __"," 锁。",[108,100715,100716,100717,347],{},"数据库恢复中建立冗余数据最常用技术是 _",[2627,100718,100693],{},[108,100720,100721],{},"数据库建设是硬件、软件和 ____ 三方面的结合。",[15,100723,100724],{},"参考答案：",[2230,100726,100727,100730,100733,100736,100739,100742,100745,100748,100751,100754],{},[108,100728,100729],{},"主变量。",[108,100731,100732],{},"转储\u002F备份、日志。",[108,100734,100735],{},"数据结构；完整性约束。",[108,100737,100738],{},"先来先服务。",[108,100740,100741],{},"代数优化；物理优化。",[108,100743,100744],{},"一致性；隔离性。",[108,100746,100747],{},"推进游标\u002F读取游标；释放游标。",[108,100749,100750],{},"共享锁\u002FS 锁；排他锁\u002FX 锁。",[108,100752,100753],{},"数据转储；登记日志文件。",[108,100755,100756],{},"干件",[19,100758,100759],{"id":100759},"选择题与参考答案",[253,100761,100763],{"id":100762},"_1-规范化目的","1. 规范化目的",[15,100765,100766],{},"题面：以下四个叙述中，哪一个不是对关系模式进行规范化的主要目的？",[15,100768,100769],{},"A. 减少数据冗余\nB. 解决更新异常问题\nC. 加快查询速度\nD. 在模式设计上实现较好的语义表达\u002F单一化",[15,100771,100772],{},"答案：C。",[15,100774,100775],{},"方法：规范化主要解决冗余、插入异常、删除异常、更新异常和语义不清；性能提升不是主要目的，有时规范化还会增加连接成本。",[253,100777,100779],{"id":100778},"_2-教师只能访问自己课程成绩","2. 教师只能访问自己课程成绩",[15,100781,100782],{},"题面：对于学生、选课、课程三类关系，若要求“每个教师只能存取自己讲授课程的学生成绩”，应建立什么机制？",[15,100784,100785],{},"可见选项包括：A. 视图；C. 游标；D. 索引。",[15,100787,100788],{},"答案：A. 视图。",[15,100790,100791],{},"方法：按教师过滤课程与成绩，给教师授权访问对应视图，而不是直接访问基本表。",[253,100793,100795],{"id":100794},"_3-现实世界的特征对应数据世界的什么","3. 现实世界的“特征”对应数据世界的什么",[15,100797,100798],{},"题面：现实世界的“特征”对应于数据世界的（）。",[15,100800,100801],{},"A. 属性\nB. 联系\nC. 关系\u002F记录，原图局部不清\nD. 数据项",[15,100803,100804],{},"答案：A. 属性。",[253,100806,100808],{"id":100807},"_4-x-锁权限","4. X 锁权限",[15,100810,100811],{},"题面：如果事务 T 获得了数据项 Q 上的 X 锁，则 T 对 Q（）。",[15,100813,100814],{},"A. 只能读不能写\nB. 只能写不能读\nC. 既可读又可写\nD. 不能读不能写",[15,100816,100772],{},[253,100818,100820],{"id":100819},"_5-sql-空值操作中不正确的是","5. SQL 空值操作中不正确的是",[15,100822,100823],{},"题面：SQL 中，空值操作中不正确的是（）。",[15,100825,100826,100827,58784,100830,58980,100833,59196,100836],{},"A. ",[31,100828,100829],{},"NOT (SAGE IS NULL)",[31,100831,100832],{},"SAGE != NULL",[31,100834,100835],{},"SAGE IS NOT NULL",[31,100837,100838],{},"SAGE IS NULL",[15,100840,100841],{},"答案：B。",[15,100843,100844,100845,4374,100848,100851,100852,4374,100855,347],{},"方法：SQL 判断空值必须用 ",[31,100846,100847],{},"IS NULL",[31,100849,100850],{},"IS NOT NULL","，不能用 ",[31,100853,100854],{},"= NULL",[31,100856,100857],{},"!= NULL",[253,100859,100861],{"id":100860},"_6-取出某些列并消去重复元组","6. 取出某些列并消去重复元组",[15,100863,100864],{},"题面：取出关系中的某些列，并消去重复元组的关系代数运算称为（）。",[15,100866,100867],{},"A. 取列运算\nB. 投影运算\nC. 连接运算\nD. 选择运算",[15,100869,100841],{},[253,100871,100873],{"id":100872},"_7-全码关系模式最高可达到的范式","7. 全码关系模式最高可达到的范式",[15,100875,100876],{},"题面：全码关系模式最高可以达到的范式为（）。",[15,100878,100879],{},"A. 1NF\nB. 2NF\nC. 3NF\nD. BCNF",[15,100881,100882],{},"答案：D。",[253,100884,100886],{"id":100885},"_8-脏数据","8. 脏数据",[15,100888,100889],{},"题面：“脏数据”是指（）。",[15,100891,100892],{},"可见选项：",[15,100894,100895],{},"A. 未提交的数据\nB. 未提交的后又被撤销的数据\nC. 违反访问权而写入 DB 的数据\nD. 输入时就有错的数据",[15,100897,100898],{},"答案：B 更贴近题面；A 是概括性描述。",[15,100900,100901],{},"方法：脏读的关键是读取了其他事务未提交、并可能回滚的数据。",[253,100903,100905],{"id":100904},"_9-关系运算中耗时操作","9. 关系运算中耗时操作",[15,100907,100908],{},"题面：关系运算中花费时间可能很长的运算是（原图底部不完整）。",[15,100910,100911],{},"参考答案：连接或笛卡尔积。若题目强调查询优化中最应避免的大中间结果，通常选笛卡尔积。",[19,100913,100915],{"id":100914},"null完整性与补考名单例子","NULL、完整性与补考名单例子",[15,100917,100918],{},"可见课堂内容：",[24,100920,100922],{"className":92584,"code":100921,"language":92586,"meta":29,"style":29},"-- Dormno is NULL\n-- isNULL() \u002F setNull()\nSELECT *\nFROM student\nWHERE dormno IS NULL;\n\nSELECT *\nFROM student\nWHERE dormno IS NOT NULL;\n\n-- 下学期补考名单\nSELECT *\nFROM grade\nWHERE score \u003C 60 OR score IS NULL;\n\n-- 用户自定义完整性示例\nCHECK (sex = '男' OR sex = '女')\nCHECK (sage >= 1 AND sage \u003C= 150)\n",[31,100923,100924,100929,100934,100938,100943,100948,100952,100956,100960,100965,100969,100974,100978,100983,100988,100992,100997,101002],{"__ignoreMap":29},[34,100925,100926],{"class":36,"line":37},[34,100927,100928],{},"-- Dormno is NULL\n",[34,100930,100931],{"class":36,"line":43},[34,100932,100933],{},"-- isNULL() \u002F setNull()\n",[34,100935,100936],{"class":36,"line":49},[34,100937,92811],{},[34,100939,100940],{"class":36,"line":56},[34,100941,100942],{},"FROM student\n",[34,100944,100945],{"class":36,"line":62},[34,100946,100947],{},"WHERE dormno IS NULL;\n",[34,100949,100950],{"class":36,"line":68},[34,100951,53],{"emptyLinePlaceholder":52},[34,100953,100954],{"class":36,"line":74},[34,100955,92811],{},[34,100957,100958],{"class":36,"line":1971},[34,100959,100942],{},[34,100961,100962],{"class":36,"line":1976},[34,100963,100964],{},"WHERE dormno IS NOT NULL;\n",[34,100966,100967],{"class":36,"line":1985},[34,100968,53],{"emptyLinePlaceholder":52},[34,100970,100971],{"class":36,"line":1990},[34,100972,100973],{},"-- 下学期补考名单\n",[34,100975,100976],{"class":36,"line":1999},[34,100977,92811],{},[34,100979,100980],{"class":36,"line":2022},[34,100981,100982],{},"FROM grade\n",[34,100984,100985],{"class":36,"line":2036},[34,100986,100987],{},"WHERE score \u003C 60 OR score IS NULL;\n",[34,100989,100990],{"class":36,"line":2603},[34,100991,53],{"emptyLinePlaceholder":52},[34,100993,100994],{"class":36,"line":5537},[34,100995,100996],{},"-- 用户自定义完整性示例\n",[34,100998,100999],{"class":36,"line":5543},[34,101000,101001],{},"CHECK (sex = '男' OR sex = '女')\n",[34,101003,101004],{"class":36,"line":5548},[34,101005,101006],{},"CHECK (sage >= 1 AND sage \u003C= 150)\n",[15,101008,101009],{},"要点：",[105,101011,101012,101019,101025],{},[108,101013,101014,101015,101018],{},"参照完整性可配合 ",[31,101016,101017],{},"SET NULL"," 策略。",[108,101020,101021,101022,101024],{},"用户自定义完整性常用 ",[31,101023,94262],{}," 约束表达。",[108,101026,101027,101030],{},[31,101028,101029],{},"score IS NULL"," 与低于 60 分都可能进入补考名单，具体业务规则以教师说明为准。",[79,101032,81],{},{"title":29,"searchDepth":43,"depth":43,"links":101034},[101035,101036,101043,101044,101045,101056],{"id":100488,"depth":43,"text":100488},{"id":100506,"depth":43,"text":100506,"children":101037},[101038,101039,101040,101041,101042],{"id":100509,"depth":49,"text":100510},{"id":100516,"depth":49,"text":100517},{"id":2310,"depth":49,"text":100556},{"id":100562,"depth":49,"text":100563},{"id":100583,"depth":49,"text":100584},{"id":100590,"depth":43,"text":100590},{"id":100667,"depth":43,"text":100667},{"id":100759,"depth":43,"text":100759,"children":101046},[101047,101048,101049,101050,101051,101052,101053,101054,101055],{"id":100762,"depth":49,"text":100763},{"id":100778,"depth":49,"text":100779},{"id":100794,"depth":49,"text":100795},{"id":100807,"depth":49,"text":100808},{"id":100819,"depth":49,"text":100820},{"id":100860,"depth":49,"text":100861},{"id":100872,"depth":49,"text":100873},{"id":100885,"depth":49,"text":100886},{"id":100904,"depth":49,"text":100905},{"id":100914,"depth":43,"text":100915},"覆盖图片：FE1EB12F1021FF5AC008557D0D0FDC17.png、8087F11947A9E02D4BB95C6564592F0F.png、0B90006D47895530C9657C33567E7AF3.png、074F26CB7353DD83CAAA9B45A4BEF568.png、CA7860A16D7978E001F1205EE772B09B.png、7EFBDE17B0116BB9766F10AC8BFF9AE4.png、ce02642dbb6b3731ac80b12971821561_720.jpg、IMG_0751.JPG、IMG_0751 (1).JPG、IMG_0391.JPG、IMG_0391 (1).JPG、A82101FC9E13F4F2388DC3EEC2CFB6DB.png。",{},"\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F01",{"title":100442,"description":101057},"source\u002Fcs\u002Fdb\u002Fnotes\u002F01-基础概念与题型全文","PWAogBgG5fuBhIWs8XCON3DdnnChYJMm1BIRkFoXhig",{"id":101064,"title":101065,"body":101066,"description":102407,"extension":85,"meta":102408,"navigation":52,"path":102409,"seo":102410,"stem":102411,"__hash__":102412},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F02-SQL语句与数据库操作全文.md","SQL语句与数据库操作全文",{"type":8,"value":101067,"toc":102395},[101068,101071,101168,101172,101174,101180,101183,101189,101192,101256,101258,101448,101450,101469,101473,101475,101481,101483,101538,101541,101733,101739,101743,101746,101752,101755,101761,101854,101858,101861,101867,101870,101899,101902,101934,101938,101941,102003,102006,102035,102041,102073,102076,102108,102111,102114,102176,102178,102207,102210,102213,102251,102254,102262,102265,102268,102274,102277,102320,102323,102326,102328,102334,102337,102343,102347,102350,102365,102368,102374,102376,102393],[11,101069,101065],{"id":101070},"sql语句与数据库操作全文",[15,101072,100449,101073,146,101076,146,101078,146,101081,146,101083,146,101086,146,101089,146,101092,146,101095,146,101098,146,101101,146,101104,146,101107,146,101110,146,101113,146,101116,146,101118,146,101120,146,101123,146,101126,146,101129,146,101132,146,101135,146,101138,146,101141,146,101144,146,101147,146,101150,146,101153,146,101156,146,101159,146,101162,146,101165,347],{},[31,101074,101075],{},"0E164B889B39AE7BF5102E491F71E2D2.png",[31,101077,100485],{},[31,101079,101080],{},"1530EB82DD93E520018B5683CB9A9BD3.png",[31,101082,100461],{},[31,101084,101085],{},"635A1DB1DACFAAF5C23A0CCA49E90FF3.png",[31,101087,101088],{},"16CF0F453F7276996E66818C6C03E969.png",[31,101090,101091],{},"DD05FD2817A6A32159460B7DAEBBA6FC.png",[31,101093,101094],{},"324370E4515703A116DDF91B7DD677BD.png",[31,101096,101097],{},"D03F9CFE1ED63CF4B07E065C7CD338F3.png",[31,101099,101100],{},"IMG_0361.JPG",[31,101102,101103],{},"IMG_0361 (1).JPG",[31,101105,101106],{},"IMG_0362.JPG",[31,101108,101109],{},"IMG_0362 (1).JPG",[31,101111,101112],{},"IMG_0363.JPG",[31,101114,101115],{},"IMG_0363 (1).JPG",[31,101117,100479],{},[31,101119,100482],{},[31,101121,101122],{},"IMG_0459.JPG",[31,101124,101125],{},"IMG_0459 (1).JPG",[31,101127,101128],{},"IMG_0460.JPG",[31,101130,101131],{},"IMG_0460 (1).JPG",[31,101133,101134],{},"IMG_0477.JPG",[31,101136,101137],{},"IMG_0477 (1).JPG",[31,101139,101140],{},"IMG_0478.JPG",[31,101142,101143],{},"IMG_0478 (1).JPG",[31,101145,101146],{},"IMG_0479.JPG",[31,101148,101149],{},"IMG_0479 (1).JPG",[31,101151,101152],{},"IMG_0480.JPG",[31,101154,101155],{},"IMG_0480 (1).JPG",[31,101157,101158],{},"IMG_0481.JPG",[31,101160,101161],{},"IMG_0481 (1).JPG",[31,101163,101164],{},"IMG_0482.JPG",[31,101166,101167],{},"IMG_0482 (1).JPG",[19,101169,101171],{"id":101170},"公司数据库-sql-题","公司数据库 SQL 题",[15,101173,94672],{},[24,101175,101178],{"className":101176,"code":101177,"language":2638,"meta":29},[2636],"SaleDetail(SaleTime, SalemanNO, PNO, SaleNum)\nProduct(PNO, PName, Price)\nStock(PNO, StockNum, StockPlace)\n",[31,101179,101177],{"__ignoreMap":29},[15,101181,101182],{},"全库约束：",[24,101184,101187],{"className":101185,"code":101186,"language":2638,"meta":29},[2636],"The whole database satisfies cascading deletion is not allowed,\nbut cascading update is allowed.\n",[31,101188,101186],{"__ignoreMap":29},[15,101190,101191],{},"题目：",[2230,101193,101194,101201,101204,101210,101213,101216,101223,101238,101244],{},[108,101195,101196,101197,101200],{},"Create table ",[31,101198,101199],{},"SaleDetail"," with a list of constraints, including primary key and foreign key constraints.",[108,101202,101203],{},"Delete the sale details after Jan 1, 2022.",[108,101205,101206,101207,2547],{},"Insert a new product ",[31,101208,101209],{},"\u003C12345, Redmi Note12, 1699>",[108,101211,101212],{},"For each product, find the total sales of it in 2023, sort results by total sales in descending order.",[108,101214,101215],{},"Increase the price of each product by 10%.",[108,101217,101218,101219,101222],{},"Define a view ",[31,101220,101221],{},"v_warehouse1(PNO, PName, SaleTime, SaleNum)",". This view displays the sales information of the products stored in warehouse1.",[108,101224,101225,101226,101229,101230,101233,101234,101237],{},"Create an index ",[31,101227,101228],{},"idx_Pname"," on the ",[31,101231,101232],{},"Pname"," column in the ",[31,101235,101236],{},"Product"," table.",[108,101239,101240,101241,2547],{},"Create a new user ",[31,101242,101243],{},"Tom",[108,101245,101246,101247,101249,101250,101252,101253,101255],{},"Give user ",[31,101248,101243],{}," select authorization on the ",[31,101251,101236],{}," table, and let ",[31,101254,101243],{}," grant the permission to other security accounts.",[15,101257,100724],{},[24,101259,101261],{"className":92584,"code":101260,"language":92586,"meta":29,"style":29},"CREATE TABLE SaleDetail (\n SaleTime date,\n SalemanNO char(9),\n PNO char(9),\n SaleNum int,\n CONSTRAINT PK_SD PRIMARY KEY (SaleTime, SalemanNO, PNO),\n CONSTRAINT FK_SD_PNO FOREIGN KEY (PNO)\n REFERENCES Product(PNO)\n ON UPDATE CASCADE\n);\n\nDELETE FROM SaleDetail\nWHERE SaleTime > '2022-01-01';\n\nINSERT INTO Product(PNO, PName, Price)\nVALUES ('12345', 'Redmi Note12', 1699);\n\nSELECT PNO, SUM(SaleNum) AS TotalSales\nFROM SaleDetail\nWHERE YEAR(SaleTime) = 2023\nGROUP BY PNO\nORDER BY SUM(SaleNum) DESC;\n\nUPDATE Product\nSET Price = Price * 1.1;\n\nCREATE VIEW v_warehouse1(PNO, PName, SaleTime, SaleNum) AS\nSELECT sd.PNO, p.PName, sd.SaleTime, sd.SaleNum\nFROM SaleDetail AS sd\nJOIN Product AS p ON sd.PNO = p.PNO\nJOIN Stock AS s ON sd.PNO = s.PNO\nWHERE s.StockPlace = 'Warehouse1';\n\nCREATE INDEX idx_Pname\nON Product(PName);\n\nCREATE USER Tom;\n\nGRANT SELECT ON Product TO Tom WITH GRANT OPTION;\n",[31,101262,101263,101268,101273,101278,101283,101288,101293,101298,101303,101307,101311,101315,101320,101325,101329,101334,101339,101343,101348,101353,101358,101363,101368,101372,101377,101382,101386,101391,101396,101401,101406,101411,101416,101420,101425,101430,101434,101439,101443],{"__ignoreMap":29},[34,101264,101265],{"class":36,"line":37},[34,101266,101267],{},"CREATE TABLE SaleDetail (\n",[34,101269,101270],{"class":36,"line":43},[34,101271,101272],{}," SaleTime date,\n",[34,101274,101275],{"class":36,"line":49},[34,101276,101277],{}," SalemanNO char(9),\n",[34,101279,101280],{"class":36,"line":56},[34,101281,101282],{}," PNO char(9),\n",[34,101284,101285],{"class":36,"line":62},[34,101286,101287],{}," SaleNum int,\n",[34,101289,101290],{"class":36,"line":68},[34,101291,101292],{}," CONSTRAINT PK_SD PRIMARY KEY (SaleTime, SalemanNO, PNO),\n",[34,101294,101295],{"class":36,"line":74},[34,101296,101297],{}," CONSTRAINT FK_SD_PNO FOREIGN KEY (PNO)\n",[34,101299,101300],{"class":36,"line":1971},[34,101301,101302],{}," REFERENCES Product(PNO)\n",[34,101304,101305],{"class":36,"line":1976},[34,101306,99981],{},[34,101308,101309],{"class":36,"line":1985},[34,101310,462],{},[34,101312,101313],{"class":36,"line":1990},[34,101314,53],{"emptyLinePlaceholder":52},[34,101316,101317],{"class":36,"line":1999},[34,101318,101319],{},"DELETE FROM SaleDetail\n",[34,101321,101322],{"class":36,"line":2022},[34,101323,101324],{},"WHERE SaleTime > '2022-01-01';\n",[34,101326,101327],{"class":36,"line":2036},[34,101328,53],{"emptyLinePlaceholder":52},[34,101330,101331],{"class":36,"line":2603},[34,101332,101333],{},"INSERT INTO Product(PNO, PName, Price)\n",[34,101335,101336],{"class":36,"line":5537},[34,101337,101338],{},"VALUES ('12345', 'Redmi Note12', 1699);\n",[34,101340,101341],{"class":36,"line":5543},[34,101342,53],{"emptyLinePlaceholder":52},[34,101344,101345],{"class":36,"line":5548},[34,101346,101347],{},"SELECT PNO, SUM(SaleNum) AS TotalSales\n",[34,101349,101350],{"class":36,"line":5564},[34,101351,101352],{},"FROM SaleDetail\n",[34,101354,101355],{"class":36,"line":5576},[34,101356,101357],{},"WHERE YEAR(SaleTime) = 2023\n",[34,101359,101360],{"class":36,"line":5606},[34,101361,101362],{},"GROUP BY PNO\n",[34,101364,101365],{"class":36,"line":5616},[34,101366,101367],{},"ORDER BY SUM(SaleNum) DESC;\n",[34,101369,101370],{"class":36,"line":5628},[34,101371,53],{"emptyLinePlaceholder":52},[34,101373,101374],{"class":36,"line":5634},[34,101375,101376],{},"UPDATE Product\n",[34,101378,101379],{"class":36,"line":6420},[34,101380,101381],{},"SET Price = Price * 1.1;\n",[34,101383,101384],{"class":36,"line":11294},[34,101385,53],{"emptyLinePlaceholder":52},[34,101387,101388],{"class":36,"line":11321},[34,101389,101390],{},"CREATE VIEW v_warehouse1(PNO, PName, SaleTime, SaleNum) AS\n",[34,101392,101393],{"class":36,"line":11326},[34,101394,101395],{},"SELECT sd.PNO, p.PName, sd.SaleTime, sd.SaleNum\n",[34,101397,101398],{"class":36,"line":11333},[34,101399,101400],{},"FROM SaleDetail AS sd\n",[34,101402,101403],{"class":36,"line":11344},[34,101404,101405],{},"JOIN Product AS p ON sd.PNO = p.PNO\n",[34,101407,101408],{"class":36,"line":11349},[34,101409,101410],{},"JOIN Stock AS s ON sd.PNO = s.PNO\n",[34,101412,101413],{"class":36,"line":11354},[34,101414,101415],{},"WHERE s.StockPlace = 'Warehouse1';\n",[34,101417,101418],{"class":36,"line":11363},[34,101419,53],{"emptyLinePlaceholder":52},[34,101421,101422],{"class":36,"line":11368},[34,101423,101424],{},"CREATE INDEX idx_Pname\n",[34,101426,101427],{"class":36,"line":11377},[34,101428,101429],{},"ON Product(PName);\n",[34,101431,101432],{"class":36,"line":11402},[34,101433,53],{"emptyLinePlaceholder":52},[34,101435,101436],{"class":36,"line":11407},[34,101437,101438],{},"CREATE USER Tom;\n",[34,101440,101441],{"class":36,"line":11421},[34,101442,53],{"emptyLinePlaceholder":52},[34,101444,101445],{"class":36,"line":11426},[34,101446,101447],{},"GRANT SELECT ON Product TO Tom WITH GRANT OPTION;\n",[15,101449,8882],{},[105,101451,101452,101458,101463],{},[108,101453,101454,101455,101457],{},"原图中第 2 题写 “after Jan 1, 2022”，通常按 ",[31,101456,29622],{}," 处理。",[108,101459,101460,101461,347],{},"级联删除不允许，所以外键不写 ",[31,101462,100055],{},[108,101464,101465,101466,347],{},"级联更新允许，所以外键写 ",[31,101467,101468],{},"ON UPDATE CASCADE",[19,101470,101472],{"id":101471},"部门-职工关系-sql-综合题","部门-职工关系 SQL 综合题",[15,101474,94672],{},[24,101476,101479],{"className":101477,"code":101478,"language":2638,"meta":29},[2636],"Dept(dptno, dname, tele, mana)\nPe(pno, pname, sex, job, wage, dptno)\n\npno 职工号\npname 职工姓名\nsex 性别\njob 职务\nwage 工资\ndptno 所在部门号\ndname 部门名\nmana 负责人\n",[31,101480,101478],{"__ignoreMap":29},[15,101482,101191],{},[2230,101484,101485,101491,101502,101507,101518,101531],{},[108,101486,23611,101487,101490],{},[31,101488,101489],{},"Pe"," 表，要求“工资”在 1500-3500 元之间，要有主码、外码等完整性定义。",[108,101492,101493,101494,101497,101498,101501],{},"建立一个视图 ",[31,101495,101496],{},"V_DEPE","：统计每一个 ",[31,101499,101500],{},"mana"," 负责人管理的所有部门的员工工资之和，按部门分组。",[108,101503,2884,101504,101506],{},[31,101505,101489],{}," 表中，给部门名为“设计室”的部门员工，工资均提高 10%。",[108,101508,101509,101510,101513,101514,101517],{},"删除 ",[31,101511,101512],{},"Dept"," 表中部门号为 ",[31,101515,101516],{},"'Z01'"," 的部门信息，按置空原则实施。",[108,101519,2884,101520,101522,101523,101526,101527,101530],{},[31,101521,101489],{}," 表中增加元组：职工号 ",[31,101524,101525],{},"2015010","，姓名“王石”，性别“男”，归属于新部门 ",[31,101528,101529],{},"N01","、“综合室”，负责人“李明”。",[108,101532,101533,101534,101537],{},"编写带参数的存储过程 ",[31,101535,101536],{},"Sele_staff","：查询姓名第二个字为“武”的员工的部门名和负责人信息，并以标签格式输出“职工号 姓名 部门 负责人”。按 SQL Server 语法编写。",[15,101539,101540],{},"参考 SQL：",[24,101542,101544],{"className":92584,"code":101543,"language":92586,"meta":29,"style":29},"CREATE TABLE Dept (\n dptno char(10) PRIMARY KEY,\n dname varchar(20) NOT NULL,\n tele varchar(20),\n mana varchar(20)\n);\n\nCREATE TABLE Pe (\n pno char(10) PRIMARY KEY,\n pname varchar(20) NOT NULL,\n sex char(2),\n job varchar(20),\n wage int CHECK (wage BETWEEN 1500 AND 3500),\n dptno char(10) NULL,\n CONSTRAINT FK_Pe_Dept FOREIGN KEY (dptno)\n REFERENCES Dept(dptno)\n ON DELETE SET NULL\n);\n\nCREATE VIEW V_DEPE AS\nSELECT d.mana, d.dptno, d.dname, SUM(p.wage) AS TotalWage\nFROM Dept AS d\nJOIN Pe AS p ON d.dptno = p.dptno\nGROUP BY d.mana, d.dptno, d.dname;\n\nUPDATE p\nSET p.wage = p.wage * 1.1\nFROM Pe AS p\nJOIN Dept AS d ON p.dptno = d.dptno\nWHERE d.dname = N'设计室';\n\nDELETE FROM Dept\nWHERE dptno = 'Z01';\n\nINSERT INTO Dept(dptno, dname, mana)\nVALUES ('N01', N'综合室', N'李明');\n\nINSERT INTO Pe(pno, pname, sex, dptno)\nVALUES ('2015010', N'王石', N'男', 'N01');\n",[31,101545,101546,101551,101556,101561,101566,101571,101575,101579,101584,101589,101594,101599,101604,101609,101614,101619,101624,101629,101633,101637,101642,101647,101652,101657,101662,101666,101671,101676,101681,101686,101691,101695,101700,101705,101709,101714,101719,101723,101728],{"__ignoreMap":29},[34,101547,101548],{"class":36,"line":37},[34,101549,101550],{},"CREATE TABLE Dept (\n",[34,101552,101553],{"class":36,"line":43},[34,101554,101555],{}," dptno char(10) PRIMARY KEY,\n",[34,101557,101558],{"class":36,"line":49},[34,101559,101560],{}," dname varchar(20) NOT NULL,\n",[34,101562,101563],{"class":36,"line":56},[34,101564,101565],{}," tele varchar(20),\n",[34,101567,101568],{"class":36,"line":62},[34,101569,101570],{}," mana varchar(20)\n",[34,101572,101573],{"class":36,"line":68},[34,101574,462],{},[34,101576,101577],{"class":36,"line":74},[34,101578,53],{"emptyLinePlaceholder":52},[34,101580,101581],{"class":36,"line":1971},[34,101582,101583],{},"CREATE TABLE Pe (\n",[34,101585,101586],{"class":36,"line":1976},[34,101587,101588],{}," pno char(10) PRIMARY KEY,\n",[34,101590,101591],{"class":36,"line":1985},[34,101592,101593],{}," pname varchar(20) NOT NULL,\n",[34,101595,101596],{"class":36,"line":1990},[34,101597,101598],{}," sex char(2),\n",[34,101600,101601],{"class":36,"line":1999},[34,101602,101603],{}," job varchar(20),\n",[34,101605,101606],{"class":36,"line":2022},[34,101607,101608],{}," wage int CHECK (wage BETWEEN 1500 AND 3500),\n",[34,101610,101611],{"class":36,"line":2036},[34,101612,101613],{}," dptno char(10) NULL,\n",[34,101615,101616],{"class":36,"line":2603},[34,101617,101618],{}," CONSTRAINT FK_Pe_Dept FOREIGN KEY (dptno)\n",[34,101620,101621],{"class":36,"line":5537},[34,101622,101623],{}," REFERENCES Dept(dptno)\n",[34,101625,101626],{"class":36,"line":5543},[34,101627,101628],{}," ON DELETE SET NULL\n",[34,101630,101631],{"class":36,"line":5548},[34,101632,462],{},[34,101634,101635],{"class":36,"line":5564},[34,101636,53],{"emptyLinePlaceholder":52},[34,101638,101639],{"class":36,"line":5576},[34,101640,101641],{},"CREATE VIEW V_DEPE AS\n",[34,101643,101644],{"class":36,"line":5606},[34,101645,101646],{},"SELECT d.mana, d.dptno, d.dname, SUM(p.wage) AS TotalWage\n",[34,101648,101649],{"class":36,"line":5616},[34,101650,101651],{},"FROM Dept AS d\n",[34,101653,101654],{"class":36,"line":5628},[34,101655,101656],{},"JOIN Pe AS p ON d.dptno = p.dptno\n",[34,101658,101659],{"class":36,"line":5634},[34,101660,101661],{},"GROUP BY d.mana, d.dptno, d.dname;\n",[34,101663,101664],{"class":36,"line":6420},[34,101665,53],{"emptyLinePlaceholder":52},[34,101667,101668],{"class":36,"line":11294},[34,101669,101670],{},"UPDATE p\n",[34,101672,101673],{"class":36,"line":11321},[34,101674,101675],{},"SET p.wage = p.wage * 1.1\n",[34,101677,101678],{"class":36,"line":11326},[34,101679,101680],{},"FROM Pe AS p\n",[34,101682,101683],{"class":36,"line":11333},[34,101684,101685],{},"JOIN Dept AS d ON p.dptno = d.dptno\n",[34,101687,101688],{"class":36,"line":11344},[34,101689,101690],{},"WHERE d.dname = N'设计室';\n",[34,101692,101693],{"class":36,"line":11349},[34,101694,53],{"emptyLinePlaceholder":52},[34,101696,101697],{"class":36,"line":11354},[34,101698,101699],{},"DELETE FROM Dept\n",[34,101701,101702],{"class":36,"line":11363},[34,101703,101704],{},"WHERE dptno = 'Z01';\n",[34,101706,101707],{"class":36,"line":11368},[34,101708,53],{"emptyLinePlaceholder":52},[34,101710,101711],{"class":36,"line":11377},[34,101712,101713],{},"INSERT INTO Dept(dptno, dname, mana)\n",[34,101715,101716],{"class":36,"line":11402},[34,101717,101718],{},"VALUES ('N01', N'综合室', N'李明');\n",[34,101720,101721],{"class":36,"line":11407},[34,101722,53],{"emptyLinePlaceholder":52},[34,101724,101725],{"class":36,"line":11421},[34,101726,101727],{},"INSERT INTO Pe(pno, pname, sex, dptno)\n",[34,101729,101730],{"class":36,"line":11426},[34,101731,101732],{},"VALUES ('2015010', N'王石', N'男', 'N01');\n",[15,101734,101735,101736,347],{},"存储过程见 ",[31,101737,101738],{},"03-数据库编程全文.md",[19,101740,101742],{"id":101741},"供应商-零件-sql-题片段","供应商-零件 SQL 题片段",[15,101744,101745],{},"可见题目：",[24,101747,101750],{"className":101748,"code":101749,"language":2638,"meta":29},[2636],"3. 查询供应商 S1 供应的零件总数，统计字段命名为 Total_Qty。\n4. 查询供应数量大于 100 的 P2 零件的供应商名称、零件规格及供应数量。\n5. 查询没有供应零件 P3 的供应商信息。\n",[31,101751,101749],{"__ignoreMap":29},[15,101753,101754],{},"参考写法，表名按常见供应模式假设为：",[24,101756,101759],{"className":101757,"code":101758,"language":2638,"meta":29},[2636],"S(Sno, Sname, ...)\nP(Pno, Pname, Spec, ...)\nSP(Sno, Pno, Qty)\n",[31,101760,101758],{"__ignoreMap":29},[24,101762,101764],{"className":92584,"code":101763,"language":92586,"meta":29,"style":29},"SELECT SUM(Qty) AS Total_Qty\nFROM SP\nWHERE Sno = 'S1';\n\nSELECT s.Sname, p.Spec, sp.Qty\nFROM S AS s\nJOIN SP AS sp ON s.Sno = sp.Sno\nJOIN P AS p ON sp.Pno = p.Pno\nWHERE sp.Pno = 'P2'\n AND sp.Qty > 100;\n\nSELECT *\nFROM S AS s\nWHERE NOT EXISTS (\n SELECT 1\n FROM SP AS sp\n WHERE sp.Sno = s.Sno\n AND sp.Pno = 'P3'\n);\n",[31,101765,101766,101771,101776,101781,101785,101790,101795,101800,101805,101810,101815,101819,101823,101827,101831,101835,101840,101845,101850],{"__ignoreMap":29},[34,101767,101768],{"class":36,"line":37},[34,101769,101770],{},"SELECT SUM(Qty) AS Total_Qty\n",[34,101772,101773],{"class":36,"line":43},[34,101774,101775],{},"FROM SP\n",[34,101777,101778],{"class":36,"line":49},[34,101779,101780],{},"WHERE Sno = 'S1';\n",[34,101782,101783],{"class":36,"line":56},[34,101784,53],{"emptyLinePlaceholder":52},[34,101786,101787],{"class":36,"line":62},[34,101788,101789],{},"SELECT s.Sname, p.Spec, sp.Qty\n",[34,101791,101792],{"class":36,"line":68},[34,101793,101794],{},"FROM S AS s\n",[34,101796,101797],{"class":36,"line":74},[34,101798,101799],{},"JOIN SP AS sp ON s.Sno = sp.Sno\n",[34,101801,101802],{"class":36,"line":1971},[34,101803,101804],{},"JOIN P AS p ON sp.Pno = p.Pno\n",[34,101806,101807],{"class":36,"line":1976},[34,101808,101809],{},"WHERE sp.Pno = 'P2'\n",[34,101811,101812],{"class":36,"line":1985},[34,101813,101814],{}," AND sp.Qty > 100;\n",[34,101816,101817],{"class":36,"line":1990},[34,101818,53],{"emptyLinePlaceholder":52},[34,101820,101821],{"class":36,"line":1999},[34,101822,92811],{},[34,101824,101825],{"class":36,"line":2022},[34,101826,101794],{},[34,101828,101829],{"class":36,"line":2036},[34,101830,92965],{},[34,101832,101833],{"class":36,"line":2603},[34,101834,92970],{},[34,101836,101837],{"class":36,"line":5537},[34,101838,101839],{}," FROM SP AS sp\n",[34,101841,101842],{"class":36,"line":5543},[34,101843,101844],{}," WHERE sp.Sno = s.Sno\n",[34,101846,101847],{"class":36,"line":5548},[34,101848,101849],{}," AND sp.Pno = 'P3'\n",[34,101851,101852],{"class":36,"line":5564},[34,101853,462],{},[19,101855,101857],{"id":101856},"主键候选码与唯一约束","主键、候选码与唯一约束",[15,101859,101860],{},"课堂要点：",[24,101862,101865],{"className":101863,"code":101864,"language":2638,"meta":29},[2636],"一个数据表可以有多个候选码（not NULL unique），但只能有一个主码 primary key。\n实际应用中：确定的。\n全码：所有属性共同组成码。\n",[31,101866,101864],{"__ignoreMap":29},[15,101868,101869],{},"代码示例：",[24,101871,101873],{"className":92584,"code":101872,"language":92586,"meta":29,"style":29},"CREATE TABLE department (\n dno char(4) NOT NULL UNIQUE PRIMARY KEY,\n dname char(20) NOT NULL UNIQUE,\n head char(20)\n);\n",[31,101874,101875,101880,101885,101890,101895],{"__ignoreMap":29},[34,101876,101877],{"class":36,"line":37},[34,101878,101879],{},"CREATE TABLE department (\n",[34,101881,101882],{"class":36,"line":43},[34,101883,101884],{}," dno char(4) NOT NULL UNIQUE PRIMARY KEY,\n",[34,101886,101887],{"class":36,"line":49},[34,101888,101889],{}," dname char(20) NOT NULL UNIQUE,\n",[34,101891,101892],{"class":36,"line":56},[34,101893,101894],{}," head char(20)\n",[34,101896,101897],{"class":36,"line":62},[34,101898,462],{},[15,101900,101901],{},"如果要表达多个候选码但只选一个主码，可以写成：",[24,101903,101905],{"className":92584,"code":101904,"language":92586,"meta":29,"style":29},"CREATE TABLE department (\n dno char(4) NOT NULL,\n dname char(20) NOT NULL UNIQUE,\n head char(20),\n CONSTRAINT PK_department PRIMARY KEY (dno)\n);\n",[31,101906,101907,101911,101916,101920,101925,101930],{"__ignoreMap":29},[34,101908,101909],{"class":36,"line":37},[34,101910,101879],{},[34,101912,101913],{"class":36,"line":43},[34,101914,101915],{}," dno char(4) NOT NULL,\n",[34,101917,101918],{"class":36,"line":49},[34,101919,101889],{},[34,101921,101922],{"class":36,"line":56},[34,101923,101924],{}," head char(20),\n",[34,101926,101927],{"class":36,"line":62},[34,101928,101929],{}," CONSTRAINT PK_department PRIMARY KEY (dno)\n",[34,101931,101932],{"class":36,"line":68},[34,101933,462],{},[19,101935,101937],{"id":101936},"grade-表建表与联合主键","Grade 表建表与联合主键",[15,101939,101940],{},"课堂代码片段：",[24,101942,101944],{"className":92584,"code":101943,"language":92586,"meta":29,"style":29},"CREATE TABLE grade1 (\n sno char(6),\n cno char(2),\n score int,\n CONSTRAINT pk_grade1 PRIMARY KEY (sno, cno)\n);\n\nINSERT INTO grade1 VALUES ('990101', '01', 85);\nINSERT INTO grade1 VALUES ('990101', '03', 85);\nINSERT INTO grade1 VALUES ('990102', '01', 65);\n\nSELECT * FROM grade1;\n",[31,101945,101946,101951,101956,101961,101966,101971,101975,101979,101984,101989,101994,101998],{"__ignoreMap":29},[34,101947,101948],{"class":36,"line":37},[34,101949,101950],{},"CREATE TABLE grade1 (\n",[34,101952,101953],{"class":36,"line":43},[34,101954,101955],{}," sno char(6),\n",[34,101957,101958],{"class":36,"line":49},[34,101959,101960],{}," cno char(2),\n",[34,101962,101963],{"class":36,"line":56},[34,101964,101965],{}," score int,\n",[34,101967,101968],{"class":36,"line":62},[34,101969,101970],{}," CONSTRAINT pk_grade1 PRIMARY KEY (sno, cno)\n",[34,101972,101973],{"class":36,"line":68},[34,101974,462],{},[34,101976,101977],{"class":36,"line":74},[34,101978,53],{"emptyLinePlaceholder":52},[34,101980,101981],{"class":36,"line":1971},[34,101982,101983],{},"INSERT INTO grade1 VALUES ('990101', '01', 85);\n",[34,101985,101986],{"class":36,"line":1976},[34,101987,101988],{},"INSERT INTO grade1 VALUES ('990101', '03', 85);\n",[34,101990,101991],{"class":36,"line":1985},[34,101992,101993],{},"INSERT INTO grade1 VALUES ('990102', '01', 65);\n",[34,101995,101996],{"class":36,"line":1990},[34,101997,53],{"emptyLinePlaceholder":52},[34,101999,102000],{"class":36,"line":1999},[34,102001,102002],{},"SELECT * FROM grade1;\n",[15,102004,102005],{},"其他示例：",[24,102007,102009],{"className":92584,"code":102008,"language":92586,"meta":29,"style":29},"CREATE TABLE grade2 (\n sno char(6) PRIMARY KEY,\n cno char(2) PRIMARY KEY,\n score int\n);\n",[31,102010,102011,102016,102021,102026,102031],{"__ignoreMap":29},[34,102012,102013],{"class":36,"line":37},[34,102014,102015],{},"CREATE TABLE grade2 (\n",[34,102017,102018],{"class":36,"line":43},[34,102019,102020],{}," sno char(6) PRIMARY KEY,\n",[34,102022,102023],{"class":36,"line":49},[34,102024,102025],{}," cno char(2) PRIMARY KEY,\n",[34,102027,102028],{"class":36,"line":56},[34,102029,102030],{}," score int\n",[34,102032,102033],{"class":36,"line":62},[34,102034,462],{},[15,102036,102037,102038,102040],{},"上面这种写法在 SQL Server 中不推荐，因为一个表只能有一个 ",[31,102039,99262],{}," 约束；联合主键应写成表级约束：",[24,102042,102044],{"className":92584,"code":102043,"language":92586,"meta":29,"style":29},"CREATE TABLE grade2 (\n sno char(6) NOT NULL,\n cno char(2) NOT NULL,\n score int,\n PRIMARY KEY (sno, cno)\n);\n",[31,102045,102046,102050,102055,102060,102064,102069],{"__ignoreMap":29},[34,102047,102048],{"class":36,"line":37},[34,102049,102015],{},[34,102051,102052],{"class":36,"line":43},[34,102053,102054],{}," sno char(6) NOT NULL,\n",[34,102056,102057],{"class":36,"line":49},[34,102058,102059],{}," cno char(2) NOT NULL,\n",[34,102061,102062],{"class":36,"line":56},[34,102063,101965],{},[34,102065,102066],{"class":36,"line":62},[34,102067,102068],{}," PRIMARY KEY (sno, cno)\n",[34,102070,102071],{"class":36,"line":68},[34,102072,462],{},[15,102074,102075],{},"唯一约束示例：",[24,102077,102079],{"className":92584,"code":102078,"language":92586,"meta":29,"style":29},"CREATE TABLE grade3 (\n sno char(6) NOT NULL UNIQUE,\n cno char(2) NOT NULL UNIQUE,\n score int,\n PRIMARY KEY (sno, cno)\n);\n",[31,102080,102081,102086,102091,102096,102100,102104],{"__ignoreMap":29},[34,102082,102083],{"class":36,"line":37},[34,102084,102085],{},"CREATE TABLE grade3 (\n",[34,102087,102088],{"class":36,"line":43},[34,102089,102090],{}," sno char(6) NOT NULL UNIQUE,\n",[34,102092,102093],{"class":36,"line":49},[34,102094,102095],{}," cno char(2) NOT NULL UNIQUE,\n",[34,102097,102098],{"class":36,"line":56},[34,102099,101965],{},[34,102101,102102],{"class":36,"line":62},[34,102103,102068],{},[34,102105,102106],{"class":36,"line":68},[34,102107,462],{},[19,102109,102110],{"id":102110},"聚合与相关子查询",[15,102112,102113],{},"课堂可见 SQL：",[24,102115,102117],{"className":92584,"code":102116,"language":92586,"meta":29,"style":29},"SELECT AVG(score)\nFROM grade\nWHERE sno = '990101';\n\nSELECT sno, cno, score\nFROM grade AS g1\nWHERE score > (\n SELECT AVG(score)\n FROM grade AS g2\n WHERE g1.sno = g2.sno\n GROUP BY sno\n);\n",[31,102118,102119,102124,102128,102133,102137,102142,102147,102152,102157,102162,102167,102172],{"__ignoreMap":29},[34,102120,102121],{"class":36,"line":37},[34,102122,102123],{},"SELECT AVG(score)\n",[34,102125,102126],{"class":36,"line":43},[34,102127,100982],{},[34,102129,102130],{"class":36,"line":49},[34,102131,102132],{},"WHERE sno = '990101';\n",[34,102134,102135],{"class":36,"line":56},[34,102136,53],{"emptyLinePlaceholder":52},[34,102138,102139],{"class":36,"line":62},[34,102140,102141],{},"SELECT sno, cno, score\n",[34,102143,102144],{"class":36,"line":68},[34,102145,102146],{},"FROM grade AS g1\n",[34,102148,102149],{"class":36,"line":74},[34,102150,102151],{},"WHERE score > (\n",[34,102153,102154],{"class":36,"line":1971},[34,102155,102156],{}," SELECT AVG(score)\n",[34,102158,102159],{"class":36,"line":1976},[34,102160,102161],{}," FROM grade AS g2\n",[34,102163,102164],{"class":36,"line":1985},[34,102165,102166],{}," WHERE g1.sno = g2.sno\n",[34,102168,102169],{"class":36,"line":1990},[34,102170,102171],{}," GROUP BY sno\n",[34,102173,102174],{"class":36,"line":1999},[34,102175,462],{},[15,102177,101009],{},[105,102179,102180,102186,102192],{},[108,102181,102182,102185],{},[31,102183,102184],{},"AVG(score)"," 计算平均分。",[108,102187,102188,102191],{},[31,102189,102190],{},"GROUP BY sno"," 可按学号分组。",[108,102193,102194,102195,102198,102199,102202,102203,102206],{},"外层 ",[31,102196,102197],{},"g1"," 与内层 ",[31,102200,102201],{},"g2"," 通过 ",[31,102204,102205],{},"g1.sno = g2.sno"," 关联，属于相关子查询。",[19,102208,102209],{"id":102209},"事务删除与提交",[15,102211,102212],{},"课堂代码：",[24,102214,102216],{"className":92584,"code":102215,"language":92586,"meta":29,"style":29},"SELECT * FROM student WHERE sno = '990101';\nSELECT * FROM grade WHERE sno = '990101';\n\nBEGIN TRANSACTION;\nDELETE FROM grade WHERE sno = '990101';\nDELETE FROM student WHERE sno = '990101';\nCOMMIT;\n",[31,102217,102218,102223,102228,102232,102236,102241,102246],{"__ignoreMap":29},[34,102219,102220],{"class":36,"line":37},[34,102221,102222],{},"SELECT * FROM student WHERE sno = '990101';\n",[34,102224,102225],{"class":36,"line":43},[34,102226,102227],{},"SELECT * FROM grade WHERE sno = '990101';\n",[34,102229,102230],{"class":36,"line":49},[34,102231,53],{"emptyLinePlaceholder":52},[34,102233,102234],{"class":36,"line":56},[34,102235,94001],{},[34,102237,102238],{"class":36,"line":62},[34,102239,102240],{},"DELETE FROM grade WHERE sno = '990101';\n",[34,102242,102243],{"class":36,"line":68},[34,102244,102245],{},"DELETE FROM student WHERE sno = '990101';\n",[34,102247,102248],{"class":36,"line":74},[34,102249,102250],{},"COMMIT;\n",[15,102252,102253],{},"方法：",[105,102255,102256,102259],{},[108,102257,102258],{},"删除学生前先删除选课\u002F成绩等引用学生的子表记录，避免外键约束冲突。",[108,102260,102261],{},"使用事务包裹多条删除语句，保证要么全部成功，要么回滚。",[19,102263,102264],{"id":102264},"集合查询与相关子查询片段",[15,102266,102267],{},"可见内容：",[24,102269,102272],{"className":102270,"code":102271,"language":2638,"meta":29},[2636],"SELECT 语句的查询结果是元组的集合，所以对结果可进行集合操作。\n集合操作主要包括并操作 UNION、交操作 INTERSECT、差操作 EXCEPT。\n父查询和子查询均引用了 SC 表。\n用别名 SCX、SCY 将其区分。\n这是一个相关子查询。\n",[31,102273,102271],{"__ignoreMap":29},[15,102275,102276],{},"图中可见片段：",[24,102278,102280],{"className":92584,"code":102279,"language":92586,"meta":29,"style":29},"SELECT ...\nFROM SC AS SCX\nWHERE SCX.Sno = '20180002'\n AND ...;\n\nSELECT ...\nFROM SC AS SCY\nWHERE SCY.Cno = SCX.Cno;\n",[31,102281,102282,102287,102292,102297,102302,102306,102310,102315],{"__ignoreMap":29},[34,102283,102284],{"class":36,"line":37},[34,102285,102286],{},"SELECT ...\n",[34,102288,102289],{"class":36,"line":43},[34,102290,102291],{},"FROM SC AS SCX\n",[34,102293,102294],{"class":36,"line":49},[34,102295,102296],{},"WHERE SCX.Sno = '20180002'\n",[34,102298,102299],{"class":36,"line":56},[34,102300,102301],{}," AND ...;\n",[34,102303,102304],{"class":36,"line":62},[34,102305,53],{"emptyLinePlaceholder":52},[34,102307,102308],{"class":36,"line":68},[34,102309,102286],{},[34,102311,102312],{"class":36,"line":74},[34,102313,102314],{},"FROM SC AS SCY\n",[34,102316,102317],{"class":36,"line":1971},[34,102318,102319],{},"WHERE SCY.Cno = SCX.Cno;\n",[15,102321,102322],{},"照片局部被遮挡，完整 SQL 不清。语义是用相关子查询表达“和某学生选修课程有关”的集合条件。",[19,102324,102325],{"id":102325},"用户权限与角色",[15,102327,101860],{},[24,102329,102332],{"className":102330,"code":102331,"language":2638,"meta":29},[2636],"用户权限：\n1. 不是技术问题，是政策问题。\n2. 涉及对象：授权方、被授权方、数据对象、操作权限类型。\n3. 授权金字塔\u002F级联授权。\n4. 数据库权限：操作权限、管理权限。\n5. 权限组\u002F角色：简化授权。\n6. 构造实例：授权一定验证。前提：可授权用户、Public 权限。\n7. 无权：数据库操作、合法用户。\n8. Nopower：无权账号。\n9. Suser：系统角色 sysadmin(sa)，没有用户数据库授权。\n10. Muser：不是系统角色，仅是数据库用户。\n",[31,102333,102331],{"__ignoreMap":29},[15,102335,102336],{},"SQL Server 界面内容：",[24,102338,102341],{"className":102339,"code":102340,"language":2638,"meta":29},[2636],"用户：MUSER1\n服务器角色：public、sysadmin、securityadmin、serveradmin、setupadmin、\nprocessadmin、diskadmin、dbcreator、bulkadmin 等。\n用户映射、成员身份、安全对象、状态等属性页。\n",[31,102342,102340],{"__ignoreMap":29},[19,102344,102346],{"id":102345},"sql-server-对象访问错误","SQL Server 对象访问错误",[15,102348,102349],{},"可见命令：",[24,102351,102353],{"className":92584,"code":102352,"language":92586,"meta":29,"style":29},"SELECT * FROM student;\nSELECT * FROM dorm;\n",[31,102354,102355,102360],{"__ignoreMap":29},[34,102356,102357],{"class":36,"line":37},[34,102358,102359],{},"SELECT * FROM student;\n",[34,102361,102362],{"class":36,"line":43},[34,102363,102364],{},"SELECT * FROM dorm;\n",[15,102366,102367],{},"可见错误\u002F现象：",[24,102369,102372],{"className":102370,"code":102371,"language":2638,"meta":29},[2636],"无法访问 student DATA \u002F Object Explorer。\n拒绝了对对象 student 的访问。\n",[31,102373,102371],{"__ignoreMap":29},[15,102375,102253],{},[105,102377,102378,102381,102387],{},[108,102379,102380],{},"检查当前登录用户是否映射到对应数据库用户。",[108,102382,102383,102384,102386],{},"检查是否对表或视图授予了 ",[31,102385,92151],{}," 权限。",[108,102388,102389,102390,347],{},"检查数据库角色成员身份，如 ",[31,102391,102392],{},"db_datareader",[79,102394,81],{},{"title":29,"searchDepth":43,"depth":43,"links":102396},[102397,102398,102399,102400,102401,102402,102403,102404,102405,102406],{"id":101170,"depth":43,"text":101171},{"id":101471,"depth":43,"text":101472},{"id":101741,"depth":43,"text":101742},{"id":101856,"depth":43,"text":101857},{"id":101936,"depth":43,"text":101937},{"id":102110,"depth":43,"text":102110},{"id":102209,"depth":43,"text":102209},{"id":102264,"depth":43,"text":102264},{"id":102325,"depth":43,"text":102325},{"id":102345,"depth":43,"text":102346},"覆盖图片：0E164B889B39AE7BF5102E491F71E2D2.png、A82101FC9E13F4F2388DC3EEC2CFB6DB.png、1530EB82DD93E520018B5683CB9A9BD3.png、074F26CB7353DD83CAAA9B45A4BEF568.png、635A1DB1DACFAAF5C23A0CCA49E90FF3.png、16CF0F453F7276996E66818C6C03E969.png、DD05FD2817A6A32159460B7DAEBBA6FC.png、324370E4515703A116DDF91B7DD677BD.png、D03F9CFE1ED63CF4B07E065C7CD338F3.png、IMG_0361.JPG、IMG_0361 (1).JPG、IMG_0362.JPG、IMG_0362 (1).JPG、IMG_0363.JPG、IMG_0363 (1).JPG、IMG_0391.JPG、IMG_0391 (1).JPG、IMG_0459.JPG、IMG_0459 (1).JPG、IMG_0460.JPG、IMG_0460 (1).JPG、IMG_0477.JPG、IMG_0477 (1).JPG、IMG_0478.JPG、IMG_0478 (1).JPG、IMG_0479.JPG、IMG_0479 (1).JPG、IMG_0480.JPG、IMG_0480 (1).JPG、IMG_0481.JPG、IMG_0481 (1).JPG、IMG_0482.JPG、IMG_0482 (1).JPG。",{},"\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F02-sql",{"title":101065,"description":102407},"source\u002Fcs\u002Fdb\u002Fnotes\u002F02-SQL语句与数据库操作全文","yM0pUeG1YRO1dmBmHHy4YUIiAenLIP6HybjNXwsNCAA",{"id":102414,"title":102415,"body":102416,"description":103728,"extension":85,"meta":103729,"navigation":52,"path":103730,"seo":103731,"stem":103732,"__hash__":103733},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F03-数据库编程全文.md","数据库编程全文",{"type":8,"value":102417,"toc":103719},[102418,102420,102493,102497,102500,102506,102508,102532,102534,102561,102564,102566,102572,102575,102657,102660,102707,102710,102772,102775,102777,102783,102786,102877,102879,102940,102943,102945,103053,103056,103160,103163,103169,103173,103175,103181,103184,103397,103399,103420,103424,103426,103432,103434,103503,103506,103520,103523,103526,103676,103679,103717],[11,102419,102415],{"id":102415},[15,102421,100449,102422,146,102425,146,102427,146,102430,146,102433,146,102436,146,102439,146,102442,146,102445,146,102448,146,102451,146,102454,146,102457,146,102460,146,102463,146,102466,146,102469,146,102472,146,102475,146,102478,146,102481,146,102484,146,102487,146,102490,347],{},[31,102423,102424],{},"ED099ADE30A7E1EB3392D6470B8F236A.png",[31,102426,101075],{},[31,102428,102429],{},"C921AF903DCA13A80965ACDB7CEF8F9C.png",[31,102431,102432],{},"01BC0C9A088A35C1E0B81ED19A685DF9.png",[31,102434,102435],{},"CF05FAEBD7A4A45087B2C40104A2597B.png",[31,102437,102438],{},"2ee39b2d8b999926fff7500ba39d857e_720.jpg",[31,102440,102441],{},"315c533ac5d7e063a2d631ff068bbfb1_720.jpg",[31,102443,102444],{},"770d14317f811693903ab2941b92521b_720.jpg",[31,102446,102447],{},"ACCF76C5F58DF48DC5DBE5DDBF610E99.png",[31,102449,102450],{},"E961EEF841AF444B4CDFABFBBFB35179.png",[31,102452,102453],{},"600B1BF6B6F035E3068292F21E8B511C.png",[31,102455,102456],{},"2320E262911282DE3BB8D935D7356B5B.png",[31,102458,102459],{},"D89158FFD6D9DB14BB6336C1B0736D79.png",[31,102461,102462],{},"657B7FFC1D748C09FE75F0E6393BDF5F.png",[31,102464,102465],{},"IMG_0752.JPG",[31,102467,102468],{},"IMG_0752 (1).JPG",[31,102470,102471],{},"IMG_0753.JPG",[31,102473,102474],{},"IMG_0753 (1).JPG",[31,102476,102477],{},"IMG_0754.JPG",[31,102479,102480],{},"IMG_0754 (1).JPG",[31,102482,102483],{},"IMG_0755.JPG",[31,102485,102486],{},"IMG_0755 (1).JPG",[31,102488,102489],{},"IMG_0756.JPG",[31,102491,102492],{},"IMG_0756 (1).JPG",[19,102494,102496],{"id":102495},"datepart-函数","DATEPART 函数",[15,102498,102499],{},"题面\u002F课堂内容：",[24,102501,102504],{"className":102502,"code":102503,"language":2638,"meta":29},[2636],"SQL 函数 DATEPART\n日期\u002F时间函数，返回表示日期\u002F时间表达式指定部分的值的整数。\n格式：DATEPART(datepart, date-expression)\n",[31,102505,102503],{"__ignoreMap":29},[15,102507,30501],{},[24,102509,102511],{"className":92584,"code":102510,"language":92586,"meta":29,"style":29},"DECLARE @xs int, @xq int;\n\nSELECT @xs = DATEPART(hh, GETDATE());\nSELECT @xq = DATEPART(weekday, GETDATE());\n",[31,102512,102513,102518,102522,102527],{"__ignoreMap":29},[34,102514,102515],{"class":36,"line":37},[34,102516,102517],{},"DECLARE @xs int, @xq int;\n",[34,102519,102520],{"class":36,"line":43},[34,102521,53],{"emptyLinePlaceholder":52},[34,102523,102524],{"class":36,"line":49},[34,102525,102526],{},"SELECT @xs = DATEPART(hh, GETDATE());\n",[34,102528,102529],{"class":36,"line":56},[34,102530,102531],{},"SELECT @xq = DATEPART(weekday, GETDATE());\n",[15,102533,102253],{},[105,102535,102536,102542,102548],{},[108,102537,102538,102541],{},[31,102539,102540],{},"DATEPART(hh, GETDATE())"," 取当前小时。",[108,102543,102544,102547],{},[31,102545,102546],{},"DATEPART(weekday, GETDATE())"," 取当前星期几。",[108,102549,102550,102551,43298,102554,102557,102558,347],{},"SQL Server 默认 ",[31,102552,102553],{},"DATEPART(weekday, ...)",[31,102555,102556],{},"DATEFIRST"," 设置有关；课堂题中按周一到周五判断通常写 ",[31,102559,102560],{},"IN (2,3,4,5,6)",[19,102562,102563],{"id":102563},"公司数据库触发器题",[15,102565,92498],{},[24,102567,102570],{"className":102568,"code":102569,"language":2638,"meta":29},[2636],"Database programming\nConsider the company database of Question I.\n\n1. Write an SQL trigger to carry out the following action:\n The SaleDetail table is only permitted to be modified during business hours\n (9:00 AM - 5:00 PM) on weekdays (Monday-Friday).\n\n2. Create a stored procedure with parameters, give the SalemanNO of a staff,\n display the product names he sold.\n",[31,102571,102569],{"__ignoreMap":29},[15,102573,102574],{},"参考触发器：",[24,102576,102578],{"className":92584,"code":102577,"language":92586,"meta":29,"style":29},"CREATE TRIGGER trg_SaleDetail_business_hours\nON SaleDetail\nAFTER INSERT, UPDATE, DELETE\nAS\nBEGIN\n DECLARE @hour int, @weekday int;\n\n SELECT @hour = DATEPART(hh, GETDATE());\n SELECT @weekday = DATEPART(weekday, GETDATE());\n\n IF NOT (@hour BETWEEN 9 AND 17 AND @weekday IN (2, 3, 4, 5, 6))\n BEGIN\n PRINT 'Not business hours!';\n ROLLBACK;\n END\nEND;\n",[31,102579,102580,102585,102590,102595,102599,102604,102609,102613,102618,102623,102627,102632,102637,102642,102647,102652],{"__ignoreMap":29},[34,102581,102582],{"class":36,"line":37},[34,102583,102584],{},"CREATE TRIGGER trg_SaleDetail_business_hours\n",[34,102586,102587],{"class":36,"line":43},[34,102588,102589],{},"ON SaleDetail\n",[34,102591,102592],{"class":36,"line":49},[34,102593,102594],{},"AFTER INSERT, UPDATE, DELETE\n",[34,102596,102597],{"class":36,"line":56},[34,102598,93260],{},[34,102600,102601],{"class":36,"line":62},[34,102602,102603],{},"BEGIN\n",[34,102605,102606],{"class":36,"line":68},[34,102607,102608],{}," DECLARE @hour int, @weekday int;\n",[34,102610,102611],{"class":36,"line":74},[34,102612,53],{"emptyLinePlaceholder":52},[34,102614,102615],{"class":36,"line":1971},[34,102616,102617],{}," SELECT @hour = DATEPART(hh, GETDATE());\n",[34,102619,102620],{"class":36,"line":1976},[34,102621,102622],{}," SELECT @weekday = DATEPART(weekday, GETDATE());\n",[34,102624,102625],{"class":36,"line":1985},[34,102626,53],{"emptyLinePlaceholder":52},[34,102628,102629],{"class":36,"line":1990},[34,102630,102631],{}," IF NOT (@hour BETWEEN 9 AND 17 AND @weekday IN (2, 3, 4, 5, 6))\n",[34,102633,102634],{"class":36,"line":1999},[34,102635,102636],{}," BEGIN\n",[34,102638,102639],{"class":36,"line":2022},[34,102640,102641],{}," PRINT 'Not business hours!';\n",[34,102643,102644],{"class":36,"line":2036},[34,102645,102646],{}," ROLLBACK;\n",[34,102648,102649],{"class":36,"line":2603},[34,102650,102651],{}," END\n",[34,102653,102654],{"class":36,"line":5537},[34,102655,102656],{},"END;\n",[15,102658,102659],{},"原图可见代码片段：",[24,102661,102663],{"className":92584,"code":102662,"language":92586,"meta":29,"style":29},"BEGIN\n DECLARE @hour int, @weekday int;\n SELECT @hour = DATEPART(hh, GETDATE());\n SELECT @weekday = DATEPART(weekday, GETDATE());\n IF NOT (@hour BETWEEN 9 AND 17 AND @weekday IN (2,3,4,5,6))\n BEGIN\n PRINT 'Not business hours!';\n ROLLBACK;\n END\nEND\n",[31,102664,102665,102669,102673,102677,102681,102686,102690,102694,102698,102702],{"__ignoreMap":29},[34,102666,102667],{"class":36,"line":37},[34,102668,102603],{},[34,102670,102671],{"class":36,"line":43},[34,102672,102608],{},[34,102674,102675],{"class":36,"line":49},[34,102676,102617],{},[34,102678,102679],{"class":36,"line":56},[34,102680,102622],{},[34,102682,102683],{"class":36,"line":62},[34,102684,102685],{}," IF NOT (@hour BETWEEN 9 AND 17 AND @weekday IN (2,3,4,5,6))\n",[34,102687,102688],{"class":36,"line":68},[34,102689,102636],{},[34,102691,102692],{"class":36,"line":74},[34,102693,102641],{},[34,102695,102696],{"class":36,"line":1971},[34,102697,102646],{},[34,102699,102700],{"class":36,"line":1976},[34,102701,102651],{},[34,102703,102704],{"class":36,"line":1985},[34,102705,102706],{},"END\n",[15,102708,102709],{},"参考存储过程：",[24,102711,102713],{"className":92584,"code":102712,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE sp_sold_products\n @SalemanNO varchar(9)\nAS\nBEGIN\n SELECT PName\n FROM Product\n WHERE PNO IN (\n SELECT PNO\n FROM SaleDetail\n WHERE SalemanNO = @SalemanNO\n );\nEND;\n",[31,102714,102715,102720,102725,102729,102733,102738,102743,102748,102753,102758,102763,102768],{"__ignoreMap":29},[34,102716,102717],{"class":36,"line":37},[34,102718,102719],{},"CREATE PROCEDURE sp_sold_products\n",[34,102721,102722],{"class":36,"line":43},[34,102723,102724],{}," @SalemanNO varchar(9)\n",[34,102726,102727],{"class":36,"line":49},[34,102728,93260],{},[34,102730,102731],{"class":36,"line":56},[34,102732,102603],{},[34,102734,102735],{"class":36,"line":62},[34,102736,102737],{}," SELECT PName\n",[34,102739,102740],{"class":36,"line":68},[34,102741,102742],{}," FROM Product\n",[34,102744,102745],{"class":36,"line":74},[34,102746,102747],{}," WHERE PNO IN (\n",[34,102749,102750],{"class":36,"line":1971},[34,102751,102752],{}," SELECT PNO\n",[34,102754,102755],{"class":36,"line":1976},[34,102756,102757],{}," FROM SaleDetail\n",[34,102759,102760],{"class":36,"line":1985},[34,102761,102762],{}," WHERE SalemanNO = @SalemanNO\n",[34,102764,102765],{"class":36,"line":1990},[34,102766,102767],{}," );\n",[34,102769,102770],{"class":36,"line":1999},[34,102771,102656],{},[19,102773,102774],{"id":102774},"员工姓名输出参数存储过程",[15,102776,92498],{},[24,102778,102781],{"className":102779,"code":102780,"language":2638,"meta":29},[2636],"假设存在一个员工信息表 Employees(emp_id, emp_name, dept_id, salary)，\n编写一个 SQL 存储过程。\n\n存储过程名称：proc_get_employee_name\n输入参数：in emp_id INT（员工 ID）\n输出参数：out emp_name VARCHAR(50)（员工姓名）\n\n功能要求：\n若输入的员工 ID 存在，将该员工姓名赋值给输出参数；\n若输入的员工 ID 不存在或为 NULL，输出参数赋值为“无此员工”。\n",[31,102782,102780],{"__ignoreMap":29},[15,102784,102785],{},"参考写法，接近 MySQL\u002F通用存储过程语义：",[24,102787,102789],{"className":92584,"code":102788,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE proc_get_employee_name (\n IN in_emp_id INT,\n OUT out_emp_name VARCHAR(50)\n)\nBEGIN\n IF in_emp_id IS NULL THEN\n SET out_emp_name = '无此员工';\n ELSE\n SELECT emp_name\n INTO out_emp_name\n FROM Employees\n WHERE emp_id = in_emp_id;\n\n IF out_emp_name IS NULL THEN\n SET out_emp_name = '无此员工';\n END IF;\n END IF;\nEND;\n",[31,102790,102791,102796,102801,102806,102810,102814,102819,102824,102829,102834,102839,102844,102849,102853,102858,102863,102868,102873],{"__ignoreMap":29},[34,102792,102793],{"class":36,"line":37},[34,102794,102795],{},"CREATE PROCEDURE proc_get_employee_name (\n",[34,102797,102798],{"class":36,"line":43},[34,102799,102800],{}," IN in_emp_id INT,\n",[34,102802,102803],{"class":36,"line":49},[34,102804,102805],{}," OUT out_emp_name VARCHAR(50)\n",[34,102807,102808],{"class":36,"line":56},[34,102809,6111],{},[34,102811,102812],{"class":36,"line":62},[34,102813,102603],{},[34,102815,102816],{"class":36,"line":68},[34,102817,102818],{}," IF in_emp_id IS NULL THEN\n",[34,102820,102821],{"class":36,"line":74},[34,102822,102823],{}," SET out_emp_name = '无此员工';\n",[34,102825,102826],{"class":36,"line":1971},[34,102827,102828],{}," ELSE\n",[34,102830,102831],{"class":36,"line":1976},[34,102832,102833],{}," SELECT emp_name\n",[34,102835,102836],{"class":36,"line":1985},[34,102837,102838],{}," INTO out_emp_name\n",[34,102840,102841],{"class":36,"line":1990},[34,102842,102843],{}," FROM Employees\n",[34,102845,102846],{"class":36,"line":1999},[34,102847,102848],{}," WHERE emp_id = in_emp_id;\n",[34,102850,102851],{"class":36,"line":2022},[34,102852,53],{"emptyLinePlaceholder":52},[34,102854,102855],{"class":36,"line":2036},[34,102856,102857],{}," IF out_emp_name IS NULL THEN\n",[34,102859,102860],{"class":36,"line":2603},[34,102861,102862],{}," SET out_emp_name = '无此员工';\n",[34,102864,102865],{"class":36,"line":5537},[34,102866,102867],{}," END IF;\n",[34,102869,102870],{"class":36,"line":5543},[34,102871,102872],{}," END IF;\n",[34,102874,102875],{"class":36,"line":5548},[34,102876,102656],{},[15,102878,97054],{},[24,102880,102882],{"className":92584,"code":102881,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE proc_get_employee_name\n @in_emp_id int,\n @out_emp_name varchar(50) OUTPUT\nAS\nBEGIN\n SELECT @out_emp_name = emp_name\n FROM Employees\n WHERE emp_id = @in_emp_id;\n\n IF @in_emp_id IS NULL OR @out_emp_name IS NULL\n SET @out_emp_name = N'无此员工';\nEND;\n",[31,102883,102884,102889,102894,102899,102903,102907,102912,102917,102922,102926,102931,102936],{"__ignoreMap":29},[34,102885,102886],{"class":36,"line":37},[34,102887,102888],{},"CREATE PROCEDURE proc_get_employee_name\n",[34,102890,102891],{"class":36,"line":43},[34,102892,102893],{}," @in_emp_id int,\n",[34,102895,102896],{"class":36,"line":49},[34,102897,102898],{}," @out_emp_name varchar(50) OUTPUT\n",[34,102900,102901],{"class":36,"line":56},[34,102902,93260],{},[34,102904,102905],{"class":36,"line":62},[34,102906,102603],{},[34,102908,102909],{"class":36,"line":68},[34,102910,102911],{}," SELECT @out_emp_name = emp_name\n",[34,102913,102914],{"class":36,"line":74},[34,102915,102916],{}," FROM Employees\n",[34,102918,102919],{"class":36,"line":1971},[34,102920,102921],{}," WHERE emp_id = @in_emp_id;\n",[34,102923,102924],{"class":36,"line":1976},[34,102925,53],{"emptyLinePlaceholder":52},[34,102927,102928],{"class":36,"line":1985},[34,102929,102930],{}," IF @in_emp_id IS NULL OR @out_emp_name IS NULL\n",[34,102932,102933],{"class":36,"line":1990},[34,102934,102935],{}," SET @out_emp_name = N'无此员工';\n",[34,102937,102938],{"class":36,"line":1999},[34,102939,102656],{},[19,102941,102942],{"id":102942},"学生姓名输出参数存储过程",[15,102944,102212],{},[24,102946,102948],{"className":92584,"code":102947,"language":92586,"meta":29,"style":29},"IF EXISTS (\n SELECT name\n FROM sysobjects\n WHERE name = 'Sele_sname'\n AND type = 'p'\n)\nDROP PROCEDURE Sele_sname;\nGO\n\nCREATE PROCEDURE Sele_sname\n @in_sno char(6),\n @out_sname char(20) OUTPUT\nAS\nBEGIN\n SELECT @out_sname = sname\n FROM student\n WHERE sno = @in_sno;\n\n IF @out_sname IS NULL\n SET @out_sname = N'该学生不存在！';\nEND;\nGO\n",[31,102949,102950,102955,102960,102965,102970,102975,102979,102984,102989,102993,102998,103003,103008,103012,103016,103021,103026,103031,103035,103040,103045,103049],{"__ignoreMap":29},[34,102951,102952],{"class":36,"line":37},[34,102953,102954],{},"IF EXISTS (\n",[34,102956,102957],{"class":36,"line":43},[34,102958,102959],{}," SELECT name\n",[34,102961,102962],{"class":36,"line":49},[34,102963,102964],{}," FROM sysobjects\n",[34,102966,102967],{"class":36,"line":56},[34,102968,102969],{}," WHERE name = 'Sele_sname'\n",[34,102971,102972],{"class":36,"line":62},[34,102973,102974],{}," AND type = 'p'\n",[34,102976,102977],{"class":36,"line":68},[34,102978,6111],{},[34,102980,102981],{"class":36,"line":74},[34,102982,102983],{},"DROP PROCEDURE Sele_sname;\n",[34,102985,102986],{"class":36,"line":1971},[34,102987,102988],{},"GO\n",[34,102990,102991],{"class":36,"line":1976},[34,102992,53],{"emptyLinePlaceholder":52},[34,102994,102995],{"class":36,"line":1985},[34,102996,102997],{},"CREATE PROCEDURE Sele_sname\n",[34,102999,103000],{"class":36,"line":1990},[34,103001,103002],{}," @in_sno char(6),\n",[34,103004,103005],{"class":36,"line":1999},[34,103006,103007],{}," @out_sname char(20) OUTPUT\n",[34,103009,103010],{"class":36,"line":2022},[34,103011,93260],{},[34,103013,103014],{"class":36,"line":2036},[34,103015,102603],{},[34,103017,103018],{"class":36,"line":2603},[34,103019,103020],{}," SELECT @out_sname = sname\n",[34,103022,103023],{"class":36,"line":5537},[34,103024,103025],{}," FROM student\n",[34,103027,103028],{"class":36,"line":5543},[34,103029,103030],{}," WHERE sno = @in_sno;\n",[34,103032,103033],{"class":36,"line":5548},[34,103034,53],{"emptyLinePlaceholder":52},[34,103036,103037],{"class":36,"line":5564},[34,103038,103039],{}," IF @out_sname IS NULL\n",[34,103041,103042],{"class":36,"line":5576},[34,103043,103044],{}," SET @out_sname = N'该学生不存在！';\n",[34,103046,103047],{"class":36,"line":5606},[34,103048,102656],{},[34,103050,103051],{"class":36,"line":5616},[34,103052,102988],{},[15,103054,103055],{},"调用示例：",[24,103057,103059],{"className":92584,"code":103058,"language":92586,"meta":29,"style":29},"DECLARE @OUTPUT_sname char(20);\n\nEXEC Sele_sname\n @in_sno = '990101',\n @out_sname = @OUTPUT_sname OUTPUT;\n\nSELECT @OUTPUT_sname AS 学生姓名;\n\nDECLARE @OUTPUT_sname char(20);\n\nEXEC Sele_sname\n @in_sno = '990191',\n @out_sname = @OUTPUT_sname OUTPUT;\n\nSELECT @OUTPUT_sname AS 学生姓名;\n\nDECLARE @OUTPUT_sname char(20);\n\nEXEC Sele_sname\n @in_sno = NULL,\n @out_sname = @OUTPUT_sname OUTPUT;\n\nSELECT @OUTPUT_sname AS 学生姓名;\n",[31,103060,103061,103066,103070,103075,103080,103085,103089,103094,103098,103102,103106,103110,103115,103119,103123,103127,103131,103135,103139,103143,103148,103152,103156],{"__ignoreMap":29},[34,103062,103063],{"class":36,"line":37},[34,103064,103065],{},"DECLARE @OUTPUT_sname char(20);\n",[34,103067,103068],{"class":36,"line":43},[34,103069,53],{"emptyLinePlaceholder":52},[34,103071,103072],{"class":36,"line":49},[34,103073,103074],{},"EXEC Sele_sname\n",[34,103076,103077],{"class":36,"line":56},[34,103078,103079],{}," @in_sno = '990101',\n",[34,103081,103082],{"class":36,"line":62},[34,103083,103084],{}," @out_sname = @OUTPUT_sname OUTPUT;\n",[34,103086,103087],{"class":36,"line":68},[34,103088,53],{"emptyLinePlaceholder":52},[34,103090,103091],{"class":36,"line":74},[34,103092,103093],{},"SELECT @OUTPUT_sname AS 学生姓名;\n",[34,103095,103096],{"class":36,"line":1971},[34,103097,53],{"emptyLinePlaceholder":52},[34,103099,103100],{"class":36,"line":1976},[34,103101,103065],{},[34,103103,103104],{"class":36,"line":1985},[34,103105,53],{"emptyLinePlaceholder":52},[34,103107,103108],{"class":36,"line":1990},[34,103109,103074],{},[34,103111,103112],{"class":36,"line":1999},[34,103113,103114],{}," @in_sno = '990191',\n",[34,103116,103117],{"class":36,"line":2022},[34,103118,103084],{},[34,103120,103121],{"class":36,"line":2036},[34,103122,53],{"emptyLinePlaceholder":52},[34,103124,103125],{"class":36,"line":2603},[34,103126,103093],{},[34,103128,103129],{"class":36,"line":5537},[34,103130,53],{"emptyLinePlaceholder":52},[34,103132,103133],{"class":36,"line":5543},[34,103134,103065],{},[34,103136,103137],{"class":36,"line":5548},[34,103138,53],{"emptyLinePlaceholder":52},[34,103140,103141],{"class":36,"line":5564},[34,103142,103074],{},[34,103144,103145],{"class":36,"line":5576},[34,103146,103147],{}," @in_sno = NULL,\n",[34,103149,103150],{"class":36,"line":5606},[34,103151,103084],{},[34,103153,103154],{"class":36,"line":5616},[34,103155,53],{"emptyLinePlaceholder":52},[34,103157,103158],{"class":36,"line":5628},[34,103159,103093],{},[15,103161,103162],{},"可见执行结果：",[24,103164,103167],{"className":103165,"code":103166,"language":2638,"meta":29},[2636],"学生姓名\n该学生不存在！\n\n查询已成功执行\n",[31,103168,103166],{"__ignoreMap":29},[19,103170,103172],{"id":103171},"带参数存储过程-sele_staff","带参数存储过程 Sele_staff",[15,103174,92498],{},[24,103176,103179],{"className":103177,"code":103178,"language":2638,"meta":29},[2636],"编写带参数的存储过程 Sele_staff：\n查询姓名第二个字为“武”的员工的部门名和负责人信息，\n并以标签格式输出“职工号 姓名 部门 负责人”。\n按 SQL Server 语法编写。\n",[31,103180,103178],{"__ignoreMap":29},[15,103182,103183],{},"原图可见\u002F校正后的代码：",[24,103185,103187],{"className":92584,"code":103186,"language":92586,"meta":29,"style":29},"IF EXISTS (\n SELECT name\n FROM sysobjects\n WHERE name = 'Sele_staff'\n AND type = 'P'\n)\nDROP PROCEDURE Sele_staff;\nGO\n\nCREATE PROCEDURE Sele_staff\n @pe_name varchar(10)\nAS\nBEGIN\n DECLARE Cur1 CURSOR\n FOR\n SELECT p.pno, p.pname, d.dname, d.mana\n FROM Pe AS p\n JOIN Dept AS d ON p.dptno = d.dptno\n WHERE p.pname LIKE '_' + @pe_name + '%';\n\n OPEN Cur1;\n\n DECLARE @p_no varchar(10),\n @p_name varchar(10),\n @dp_name varchar(10),\n @dp_mana varchar(10);\n\n FETCH NEXT FROM Cur1\n INTO @p_no, @p_name, @dp_name, @dp_mana;\n\n WHILE @@FETCH_STATUS = 0\n BEGIN\n SELECT 职工号 = @p_no,\n 姓名 = @p_name,\n 部门 = @dp_name,\n 负责人 = @dp_mana;\n\n FETCH NEXT FROM Cur1\n INTO @p_no, @p_name, @dp_name, @dp_mana;\n END;\n\n CLOSE Cur1;\n DEALLOCATE Cur1;\nEND;\nGO\n",[31,103188,103189,103193,103197,103201,103206,103211,103215,103220,103224,103228,103233,103238,103242,103246,103251,103256,103261,103266,103271,103276,103280,103285,103289,103294,103299,103304,103309,103313,103318,103323,103327,103332,103336,103341,103346,103351,103356,103360,103365,103370,103375,103379,103384,103389,103393],{"__ignoreMap":29},[34,103190,103191],{"class":36,"line":37},[34,103192,102954],{},[34,103194,103195],{"class":36,"line":43},[34,103196,102959],{},[34,103198,103199],{"class":36,"line":49},[34,103200,102964],{},[34,103202,103203],{"class":36,"line":56},[34,103204,103205],{}," WHERE name = 'Sele_staff'\n",[34,103207,103208],{"class":36,"line":62},[34,103209,103210],{}," AND type = 'P'\n",[34,103212,103213],{"class":36,"line":68},[34,103214,6111],{},[34,103216,103217],{"class":36,"line":74},[34,103218,103219],{},"DROP PROCEDURE Sele_staff;\n",[34,103221,103222],{"class":36,"line":1971},[34,103223,102988],{},[34,103225,103226],{"class":36,"line":1976},[34,103227,53],{"emptyLinePlaceholder":52},[34,103229,103230],{"class":36,"line":1985},[34,103231,103232],{},"CREATE PROCEDURE Sele_staff\n",[34,103234,103235],{"class":36,"line":1990},[34,103236,103237],{}," @pe_name varchar(10)\n",[34,103239,103240],{"class":36,"line":1999},[34,103241,93260],{},[34,103243,103244],{"class":36,"line":2022},[34,103245,102603],{},[34,103247,103248],{"class":36,"line":2036},[34,103249,103250],{}," DECLARE Cur1 CURSOR\n",[34,103252,103253],{"class":36,"line":2603},[34,103254,103255],{}," FOR\n",[34,103257,103258],{"class":36,"line":5537},[34,103259,103260],{}," SELECT p.pno, p.pname, d.dname, d.mana\n",[34,103262,103263],{"class":36,"line":5543},[34,103264,103265],{}," FROM Pe AS p\n",[34,103267,103268],{"class":36,"line":5548},[34,103269,103270],{}," JOIN Dept AS d ON p.dptno = d.dptno\n",[34,103272,103273],{"class":36,"line":5564},[34,103274,103275],{}," WHERE p.pname LIKE '_' + @pe_name + '%';\n",[34,103277,103278],{"class":36,"line":5576},[34,103279,53],{"emptyLinePlaceholder":52},[34,103281,103282],{"class":36,"line":5606},[34,103283,103284],{}," OPEN Cur1;\n",[34,103286,103287],{"class":36,"line":5616},[34,103288,53],{"emptyLinePlaceholder":52},[34,103290,103291],{"class":36,"line":5628},[34,103292,103293],{}," DECLARE @p_no varchar(10),\n",[34,103295,103296],{"class":36,"line":5634},[34,103297,103298],{}," @p_name varchar(10),\n",[34,103300,103301],{"class":36,"line":6420},[34,103302,103303],{}," @dp_name varchar(10),\n",[34,103305,103306],{"class":36,"line":11294},[34,103307,103308],{}," @dp_mana varchar(10);\n",[34,103310,103311],{"class":36,"line":11321},[34,103312,53],{"emptyLinePlaceholder":52},[34,103314,103315],{"class":36,"line":11326},[34,103316,103317],{}," FETCH NEXT FROM Cur1\n",[34,103319,103320],{"class":36,"line":11333},[34,103321,103322],{}," INTO @p_no, @p_name, @dp_name, @dp_mana;\n",[34,103324,103325],{"class":36,"line":11344},[34,103326,53],{"emptyLinePlaceholder":52},[34,103328,103329],{"class":36,"line":11349},[34,103330,103331],{}," WHILE @@FETCH_STATUS = 0\n",[34,103333,103334],{"class":36,"line":11354},[34,103335,102636],{},[34,103337,103338],{"class":36,"line":11363},[34,103339,103340],{}," SELECT 职工号 = @p_no,\n",[34,103342,103343],{"class":36,"line":11368},[34,103344,103345],{}," 姓名 = @p_name,\n",[34,103347,103348],{"class":36,"line":11377},[34,103349,103350],{}," 部门 = @dp_name,\n",[34,103352,103353],{"class":36,"line":11402},[34,103354,103355],{}," 负责人 = @dp_mana;\n",[34,103357,103358],{"class":36,"line":11407},[34,103359,53],{"emptyLinePlaceholder":52},[34,103361,103362],{"class":36,"line":11421},[34,103363,103364],{}," FETCH NEXT FROM Cur1\n",[34,103366,103367],{"class":36,"line":11426},[34,103368,103369],{}," INTO @p_no, @p_name, @dp_name, @dp_mana;\n",[34,103371,103372],{"class":36,"line":11447},[34,103373,103374],{}," END;\n",[34,103376,103377],{"class":36,"line":11452},[34,103378,53],{"emptyLinePlaceholder":52},[34,103380,103381],{"class":36,"line":11463},[34,103382,103383],{}," CLOSE Cur1;\n",[34,103385,103386],{"class":36,"line":11485},[34,103387,103388],{}," DEALLOCATE Cur1;\n",[34,103390,103391],{"class":36,"line":11495},[34,103392,102656],{},[34,103394,103395],{"class":36,"line":11528},[34,103396,102988],{},[15,103398,90843],{},[105,103400,103401,103407],{},[108,103402,103403,103404,347],{},"“姓名第二个字为武”在 SQL Server 中可写 ",[31,103405,103406],{},"LIKE N'_武%'",[108,103408,103409,103410,43298,103413,103416,103417,347],{},"原图中有 ",[31,103411,103412],{},"Dname like '%' + @pe_name + '%'",[31,103414,103415],{},"sname like '%' + @stu_name + '%'"," 类似课堂示例；针对“第二个字”更准确的是 ",[31,103418,103419],{},"_武%",[19,103421,103423],{"id":103422},"sql-server-变量输出与流程控制","SQL Server 变量、输出与流程控制",[15,103425,101860],{},[24,103427,103430],{"className":103428,"code":103429,"language":2638,"meta":29},[2636],"3. 变量赋值：SET 用于标量赋值。\n4. 打印输出变量值。\n5. 流程控制语句：条件判断\u002F循环语句。\n6. 查询赋值：将查询结果赋值给变量的标准写法。\n7. 语句结束符。\n8. 变量定义。\n",[31,103431,103429],{"__ignoreMap":29},[15,103433,30501],{},[24,103435,103437],{"className":92584,"code":103436,"language":92586,"meta":29,"style":29},"SET @out_sname = N'该学生不存在';\n\nPRINT @msg;\n\nSELECT @name AS 学生姓名; -- 标签式格式化输出\n\nSELECT @变量 = 列名\nFROM 表\nWHERE 条件;\n\nIF \u003C条件>\nBEGIN\n 操作语句;\nEND;\n",[31,103438,103439,103444,103448,103453,103457,103462,103466,103471,103476,103481,103485,103490,103494,103499],{"__ignoreMap":29},[34,103440,103441],{"class":36,"line":37},[34,103442,103443],{},"SET @out_sname = N'该学生不存在';\n",[34,103445,103446],{"class":36,"line":43},[34,103447,53],{"emptyLinePlaceholder":52},[34,103449,103450],{"class":36,"line":49},[34,103451,103452],{},"PRINT @msg;\n",[34,103454,103455],{"class":36,"line":56},[34,103456,53],{"emptyLinePlaceholder":52},[34,103458,103459],{"class":36,"line":62},[34,103460,103461],{},"SELECT @name AS 学生姓名; -- 标签式格式化输出\n",[34,103463,103464],{"class":36,"line":68},[34,103465,53],{"emptyLinePlaceholder":52},[34,103467,103468],{"class":36,"line":74},[34,103469,103470],{},"SELECT @变量 = 列名\n",[34,103472,103473],{"class":36,"line":1971},[34,103474,103475],{},"FROM 表\n",[34,103477,103478],{"class":36,"line":1976},[34,103479,103480],{},"WHERE 条件;\n",[34,103482,103483],{"class":36,"line":1985},[34,103484,53],{"emptyLinePlaceholder":52},[34,103486,103487],{"class":36,"line":1990},[34,103488,103489],{},"IF \u003C条件>\n",[34,103491,103492],{"class":36,"line":1999},[34,103493,102603],{},[34,103495,103496],{"class":36,"line":2022},[34,103497,103498],{}," 操作语句;\n",[34,103500,103501],{"class":36,"line":2036},[34,103502,102656],{},[15,103504,103505],{},"可见代码：",[24,103507,103509],{"className":92584,"code":103508,"language":92586,"meta":29,"style":29},"SET @msg = STR(@@ROWCOUNT) + ' students updated by this statement';\nPRINT @msg;\n",[31,103510,103511,103516],{"__ignoreMap":29},[34,103512,103513],{"class":36,"line":37},[34,103514,103515],{},"SET @msg = STR(@@ROWCOUNT) + ' students updated by this statement';\n",[34,103517,103518],{"class":36,"line":43},[34,103519,103452],{},[19,103521,103522],{"id":103522},"游标定义与循环",[15,103524,103525],{},"课堂示例：",[24,103527,103529],{"className":92584,"code":103528,"language":92586,"meta":29,"style":29},"DECLARE @stuSno varchar(10),\n @stuName varchar(10),\n @stuSex varchar(10),\n @stuSage varchar(10);\n\nDROP PROCEDURE IF EXISTS Proc1;\n\nDECLARE Cur_Student2 CURSOR\nFOR\n SELECT sno, sname, sex, sage\n FROM student\n WHERE sname LIKE '%' + @stu_name + '%';\n\nOPEN Cur_Student2;\n\nFETCH NEXT FROM Cur_Student2\nINTO @stuSno, @stuName, @stuSex, @stuSage;\n\nWHILE @@FETCH_STATUS = 0\nBEGIN\n SELECT 学号 = @stuSno,\n 姓名 = @stuName,\n 性别 = @stuSex,\n 年龄 = @stuSage;\n\n FETCH NEXT FROM Cur_Student2\n INTO @stuSno, @stuName, @stuSex, @stuSage;\nEND;\n\nCLOSE Cur_Student2;\nDEALLOCATE Cur_Student2;\n",[31,103530,103531,103536,103541,103546,103551,103555,103560,103564,103569,103574,103579,103583,103588,103592,103597,103601,103606,103611,103615,103620,103624,103629,103634,103639,103644,103648,103653,103658,103662,103666,103671],{"__ignoreMap":29},[34,103532,103533],{"class":36,"line":37},[34,103534,103535],{},"DECLARE @stuSno varchar(10),\n",[34,103537,103538],{"class":36,"line":43},[34,103539,103540],{}," @stuName varchar(10),\n",[34,103542,103543],{"class":36,"line":49},[34,103544,103545],{}," @stuSex varchar(10),\n",[34,103547,103548],{"class":36,"line":56},[34,103549,103550],{}," @stuSage varchar(10);\n",[34,103552,103553],{"class":36,"line":62},[34,103554,53],{"emptyLinePlaceholder":52},[34,103556,103557],{"class":36,"line":68},[34,103558,103559],{},"DROP PROCEDURE IF EXISTS Proc1;\n",[34,103561,103562],{"class":36,"line":74},[34,103563,53],{"emptyLinePlaceholder":52},[34,103565,103566],{"class":36,"line":1971},[34,103567,103568],{},"DECLARE Cur_Student2 CURSOR\n",[34,103570,103571],{"class":36,"line":1976},[34,103572,103573],{},"FOR\n",[34,103575,103576],{"class":36,"line":1985},[34,103577,103578],{}," SELECT sno, sname, sex, sage\n",[34,103580,103581],{"class":36,"line":1990},[34,103582,103025],{},[34,103584,103585],{"class":36,"line":1999},[34,103586,103587],{}," WHERE sname LIKE '%' + @stu_name + '%';\n",[34,103589,103590],{"class":36,"line":2022},[34,103591,53],{"emptyLinePlaceholder":52},[34,103593,103594],{"class":36,"line":2036},[34,103595,103596],{},"OPEN Cur_Student2;\n",[34,103598,103599],{"class":36,"line":2603},[34,103600,53],{"emptyLinePlaceholder":52},[34,103602,103603],{"class":36,"line":5537},[34,103604,103605],{},"FETCH NEXT FROM Cur_Student2\n",[34,103607,103608],{"class":36,"line":5543},[34,103609,103610],{},"INTO @stuSno, @stuName, @stuSex, @stuSage;\n",[34,103612,103613],{"class":36,"line":5548},[34,103614,53],{"emptyLinePlaceholder":52},[34,103616,103617],{"class":36,"line":5564},[34,103618,103619],{},"WHILE @@FETCH_STATUS = 0\n",[34,103621,103622],{"class":36,"line":5576},[34,103623,102603],{},[34,103625,103626],{"class":36,"line":5606},[34,103627,103628],{}," SELECT 学号 = @stuSno,\n",[34,103630,103631],{"class":36,"line":5616},[34,103632,103633],{}," 姓名 = @stuName,\n",[34,103635,103636],{"class":36,"line":5628},[34,103637,103638],{}," 性别 = @stuSex,\n",[34,103640,103641],{"class":36,"line":5634},[34,103642,103643],{}," 年龄 = @stuSage;\n",[34,103645,103646],{"class":36,"line":6420},[34,103647,53],{"emptyLinePlaceholder":52},[34,103649,103650],{"class":36,"line":11294},[34,103651,103652],{}," FETCH NEXT FROM Cur_Student2\n",[34,103654,103655],{"class":36,"line":11321},[34,103656,103657],{}," INTO @stuSno, @stuName, @stuSex, @stuSage;\n",[34,103659,103660],{"class":36,"line":11326},[34,103661,102656],{},[34,103663,103664],{"class":36,"line":11333},[34,103665,53],{"emptyLinePlaceholder":52},[34,103667,103668],{"class":36,"line":11344},[34,103669,103670],{},"CLOSE Cur_Student2;\n",[34,103672,103673],{"class":36,"line":11349},[34,103674,103675],{},"DEALLOCATE Cur_Student2;\n",[15,103677,103678],{},"游标步骤：",[2230,103680,103681,103687,103693,103699,103705,103711],{},[108,103682,103683,103684],{},"定义游标：",[31,103685,103686],{},"DECLARE cursor_name CURSOR FOR SELECT ...",[108,103688,103689,103690],{},"打开游标：",[31,103691,103692],{},"OPEN cursor_name",[108,103694,103695,103696],{},"读取当前行：",[31,103697,103698],{},"FETCH NEXT FROM cursor_name INTO ...",[108,103700,103701,103702],{},"循环判断：",[31,103703,103704],{},"WHILE @@FETCH_STATUS = 0",[108,103706,103707,103708],{},"关闭游标：",[31,103709,103710],{},"CLOSE cursor_name",[108,103712,103713,103714],{},"释放游标：",[31,103715,103716],{},"DEALLOCATE cursor_name",[79,103718,81],{},{"title":29,"searchDepth":43,"depth":43,"links":103720},[103721,103722,103723,103724,103725,103726,103727],{"id":102495,"depth":43,"text":102496},{"id":102563,"depth":43,"text":102563},{"id":102774,"depth":43,"text":102774},{"id":102942,"depth":43,"text":102942},{"id":103171,"depth":43,"text":103172},{"id":103422,"depth":43,"text":103423},{"id":103522,"depth":43,"text":103522},"覆盖图片：ED099ADE30A7E1EB3392D6470B8F236A.png、0E164B889B39AE7BF5102E491F71E2D2.png、C921AF903DCA13A80965ACDB7CEF8F9C.png、01BC0C9A088A35C1E0B81ED19A685DF9.png、CF05FAEBD7A4A45087B2C40104A2597B.png、2ee39b2d8b999926fff7500ba39d857e_720.jpg、315c533ac5d7e063a2d631ff068bbfb1_720.jpg、770d14317f811693903ab2941b92521b_720.jpg、ACCF76C5F58DF48DC5DBE5DDBF610E99.png、E961EEF841AF444B4CDFABFBBFB35179.png、600B1BF6B6F035E3068292F21E8B511C.png、2320E262911282DE3BB8D935D7356B5B.png、D89158FFD6D9DB14BB6336C1B0736D79.png、657B7FFC1D748C09FE75F0E6393BDF5F.png、IMG_0752.JPG、IMG_0752 (1).JPG、IMG_0753.JPG、IMG_0753 (1).JPG、IMG_0754.JPG、IMG_0754 (1).JPG、IMG_0755.JPG、IMG_0755 (1).JPG、IMG_0756.JPG、IMG_0756 (1).JPG。",{},"\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F03",{"title":102415,"description":103728},"source\u002Fcs\u002Fdb\u002Fnotes\u002F03-数据库编程全文","duq7ALhaIjxcc2UuTynyit4_nrsHAnqpteKghvPUDb4",{"id":103735,"title":103736,"body":103737,"description":104270,"extension":85,"meta":104271,"navigation":52,"path":104272,"seo":104273,"stem":104274,"__hash__":104275},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F04-数据库设计与规范化全文.md","数据库设计与规范化全文",{"type":8,"value":103738,"toc":104259},[103739,103741,103809,103812,103815,103821,103824,103830,103833,103836,103842,103845,103848,103850,103856,103859,103884,103890,103949,103952,103954,103960,103963,103969,103971,103977,103980,103986,103989,103991,103997,104000,104006,104009,104015,104018,104024,104026,104040,104043,104045,104051,104053,104059,104062,104068,104070,104081,104084,104087,104093,104095,104112,104115,104121,104124,104178,104182,104185,104191,104194,104200,104203,104209,104212,104215,104217,104223,104226,104232,104235,104241,104243,104257],[11,103740,103736],{"id":103736},[15,103742,100449,103743,146,103746,146,103749,146,103752,146,103755,146,103758,146,103760,146,103762,146,103764,146,103767,146,103770,146,103773,146,103776,146,103779,146,103782,146,103785,146,103788,146,103791,146,103794,146,103797,146,103800,146,103803,146,103806,347],{},[31,103744,103745],{},"72a02a3e7087d67863ee7fbabfcab93a.png",[31,103747,103748],{},"fbe361cdc5283f542dcb76231701a884.png",[31,103750,103751],{},"a7c73797aa8be71a47f8305701a2a07e.png",[31,103753,103754],{},"6e76900d19df5e7c56b7e09cb36cc2e7.png",[31,103756,103757],{},"86567574E404D5112C0146C1244C013E.png",[31,103759,102453],{},[31,103761,101094],{},[31,103763,101097],{},[31,103765,103766],{},"482D900A31EF461C592934370DBBC2FE.png",[31,103768,103769],{},"IMG_0340.JPG",[31,103771,103772],{},"IMG_0340 (1).JPG",[31,103774,103775],{},"IMG_0342.JPG",[31,103777,103778],{},"IMG_0342 (1).JPG",[31,103780,103781],{},"IMG_0364.JPG",[31,103783,103784],{},"IMG_0364 (1).JPG",[31,103786,103787],{},"IMG_0381.JPG",[31,103789,103790],{},"IMG_0381 (1).JPG",[31,103792,103793],{},"IMG_0551.JPG",[31,103795,103796],{},"IMG_0551 (1).JPG",[31,103798,103799],{},"IMG_0553.JPG",[31,103801,103802],{},"IMG_0553 (1).JPG",[31,103804,103805],{},"IMG_0554.JPG",[31,103807,103808],{},"IMG_0554 (1).JPG",[19,103810,103811],{"id":103811},"规范化设计通用做题流程",[15,103813,103814],{},"课堂流程：",[24,103816,103819],{"className":103817,"code":103818,"language":2638,"meta":29},[2636],"规范化设计题目：\n书本上做题流程：\n\n1. 形式化：问题描述（自然语言）。\n2. 应用语义归纳：补充潜在语义\u002F默认语义。\n3. 形式化定义：R\u003CU, F>。\n 其中：\n U = { ... }\n F = { ... }\n 码：\n 非主属性集 = { ... }\n4. 说明范式，说明不合理性。\n5. 规范化推导（到 BCNF）。\n6. 规范化结果：\n R1(...), 码：..., 外码：...\n R2(...), 码：..., 外码：...\n\n经过以上规范化推导，问题已经设计到 BCNF，\n达到函数依赖范畴的最高级范式。\n",[31,103820,103818],{"__ignoreMap":29},[15,103822,103823],{},"考试项目设计模式：",[24,103825,103828],{"className":103826,"code":103827,"language":2638,"meta":29},[2636],"1. 形式化：问题描述（自然语言）。\n2. 应用语义归纳：补充潜在语义\u002F默认语义。\n3. 绘制 E-R 图。\n4. 转换关系模式。\n5. 说明范式，说明不合理性。\n6. 规范化推导（到 BCNF）。\n7. 给出设计结果：\n R1(...), 码：..., 外码：...\n R2(...), 码：..., 外码：...\n",[31,103829,103827],{"__ignoreMap":29},[19,103831,103832],{"id":103832},"黑板推导片段",[15,103834,103835],{},"原图局部为黑板板书，能辨认的内容：",[24,103837,103840],{"className":103838,"code":103839,"language":2638,"meta":29},[2636],"定义：R\u003CU, F>\n应用语义\u002F维表\u002F实例属于 1NF 关系。\n整理应用语义：挖掘\u002F补充潜在语义。\n关联连接条件。\n投影分解的依据。\n确定：码 = (SNO, CNO) 之类联合码。\nR ∈ 1NF 前提。\n规范化推导：非主属性集、部分依赖、传递依赖。\nR ∈ BCNF。\n\n示例关系：\nSNO -> SDEPT\nS(SNO, SNAME, SDEPT)\nSD(SDEPT, MNAME)\nC(CNO, CNAME)\nSC(SNO, CNO, GRADE)\n",[31,103841,103839],{"__ignoreMap":29},[15,103843,103844],{},"说明：黑板照片较远，部分字母和关系名不清。整体主题是从语义和函数依赖出发做投影分解，消除异常并逐步达到 BCNF。",[19,103846,103847],{"id":103847},"面向农业的电子商务系统规范化题",[15,103849,92498],{},[24,103851,103854],{"className":103852,"code":103853,"language":2638,"meta":29},[2636],"五、数据库规范化：（共15分）\n\n面向农业的电子商务系统部分需求概述：\n注册客户能看见农户基本信息、生产场地基本信息、\n生产的农产品及规格、价格、产能等信息，\n能查询往年生产统计信息：\n乡镇、镇长电话、农产品、总产能、平均价格。\n\n要求：\n1）用 E-R 图设计尽可能好的概念模型。（4分）\n2）将 E-R 图转换成关系模型。（3分）\n3）写出各个关系的函数依赖关系。（3分）\n4）写出各个关系的主码和外码。（2分）\n5）写出各个关系满足的范式级别。（2分）\n6）将规范级别低的关系转成高级别范式的关系。（1分）\n",[31,103855,103853],{"__ignoreMap":29},[15,103857,103858],{},"做题方法：",[2230,103860,103861,103864,103881],{},[108,103862,103863],{},"找实体：客户、农户、生产场地、农产品、规格、乡镇、年度生产统计。",[108,103865,103866,103867],{},"找联系：\n",[105,103868,103869,103872,103875,103878],{},[108,103870,103871],{},"客户查看农户\u002F产品信息。",[108,103873,103874],{},"农户拥有生产场地。",[108,103876,103877],{},"农户生产农产品。",[108,103879,103880],{},"乡镇统计农产品年度产能和平均价格。",[108,103882,103883],{},"关系模式示例：",[24,103885,103888],{"className":103886,"code":103887,"language":2638,"meta":29},[2636],"Customer(CustomerID, CustomerName, ...)\nTown(TownID, TownName, MayorPhone)\nFarmer(FarmerID, FarmerName, TownID, ...)\nProductionSite(SiteID, FarmerID, SiteName, Address, ...)\nProduct(ProductID, ProductName, Spec)\nFarmerProduct(FarmerID, ProductID, SiteID, Price, Capacity)\nAnnualStat(TownID, ProductID, Year, TotalCapacity, AvgPrice)\n",[31,103889,103887],{"__ignoreMap":29},[2230,103891,103892,103931],{"start":56},[108,103893,103894,103895],{},"主码\u002F外码示例：",[105,103896,103897,103903,103912,103920,103926],{},[108,103898,103899,103902],{},[31,103900,103901],{},"Town(TownID)"," 主码。",[108,103904,103905,103908,103909,347],{},[31,103906,103907],{},"Farmer.TownID"," 外码引用 ",[31,103910,103911],{},"Town",[108,103913,103914,103908,103917,347],{},[31,103915,103916],{},"ProductionSite.FarmerID",[31,103918,103919],{},"Farmer",[108,103921,103922,103925],{},[31,103923,103924],{},"FarmerProduct(FarmerID, ProductID, SiteID)"," 可作联合主码。",[108,103927,103928,103925],{},[31,103929,103930],{},"AnnualStat(TownID, ProductID, Year)",[108,103932,103933,103934],{},"范式说明：",[105,103935,103936,103939,103946],{},[108,103937,103938],{},"拆分后每个非主属性完全依赖本关系主码。",[108,103940,103941,103942,103945],{},"乡镇电话等不放入统计关系中，避免 ",[31,103943,103944],{},"TownID -> MayorPhone"," 造成传递依赖。",[108,103947,103948],{},"一般可说明达到 3NF\u002FBCNF，具体取决于写出的函数依赖。",[19,103950,103951],{"id":103951},"商品-销售数据库规范化题",[15,103953,92498],{},[24,103955,103958],{"className":103956,"code":103957,"language":2638,"meta":29},[2636],"五、数据库规范化：（共15分）\n\n设某商品-销售数据库中的信息有：\n员工号、员工名、工资、销售组名、销售组负责人、\n商品号、商品名\u002F商品价（原图局部作“商品价\u002F单价”）、销售日期、销售量、\n供应者号、供应者名、供应者地址。\n\n假定：\n1. 一个员工仅在一个销售组。\n2. 一个销售组可销售多种商品。\n3. 一种商品只能由一个组销售。\n4. 一种商品每天有一个销售量。\n5. 一个供应者可以供应多种商品。\n6. 一种商品可以多渠道供货。\n\n要求完成：\n1）绘制 E-R 图设计概念模型。（4分）\n2）将 E-R 图转换成关系模型。（2分）\n3）写出各个关系的函数依赖关系。（2分）\n4）写出各个关系的主码和外码。（5分）\n5）写出关系满足的范式级别，并转成高级别范式。（2分）\n",[31,103959,103957],{"__ignoreMap":29},[15,103961,103962],{},"参考实体与联系：",[24,103964,103967],{"className":103965,"code":103966,"language":2638,"meta":29},[2636],"实体：\nEmployee(员工)\nSalesGroup(销售组)\nProduct(商品)\nSupplier(供应者)\nSaleRecord(销售记录)\nSupply(供应联系)\n\n联系：\nEmployee 多对一 SalesGroup\nSalesGroup 一对多 Product\nProduct 一对多 SaleRecord\nSupplier 多对多 Product，通过 Supply 表\n",[31,103968,103966],{"__ignoreMap":29},[15,103970,103883],{},[24,103972,103975],{"className":103973,"code":103974,"language":2638,"meta":29},[2636],"SalesGroup(GroupID, GroupName, Manager)\nEmployee(EmpID, EmpName, Wage, GroupID)\nProduct(PID, PName, Price, GroupID)\nSaleRecord(PID, SaleDate, SaleQty)\nSupplier(SupplierID, SupplierName, SupplierAddress)\nSupply(SupplierID, PID)\n",[31,103976,103974],{"__ignoreMap":29},[15,103978,103979],{},"函数依赖示例：",[24,103981,103984],{"className":103982,"code":103983,"language":2638,"meta":29},[2636],"GroupID -> GroupName, Manager\nEmpID -> EmpName, Wage, GroupID\nPID -> PName, Price, GroupID\n(PID, SaleDate) -> SaleQty\nSupplierID -> SupplierName, SupplierAddress\n(SupplierID, PID) -> SupplierID, PID\n",[31,103985,103983],{"__ignoreMap":29},[19,103987,103988],{"id":103988},"高校教学管理信息系统设计题",[15,103990,92498],{},[24,103992,103995],{"className":103993,"code":103994,"language":2638,"meta":29},[2636],"五、数据库设计（共20分）\n\n某高校拟开发教学管理信息系统，用于管理学生、专业、课程、教师、学院\n等核心业务对象，系统需满足以下业务需求：\n\n学生与专业：\n每个学生主修一个专业，一个专业可招收多名学生；\n学生需选修多门课程，每门课程可被多名学生选修，\n需记录学生每门课程的选修成绩。\n\n专业与学院：\n每个学院可设置多个专业，每个专业隶属于一个学院。\n\n课程与学院：\n每门课程由一个学院开设，一个学院可开设多门课程。\n\n课程与教师：\n一门课程可由多名教师讲授，一名教师可讲授多门课程，\n需记录每门课程的授课教室与课时信息。\n\n教师与学院：\n每名教师隶属于一个学院，一个学院可拥有多名教师。\n\n题目：\n1. 根据上述业务需求，完成该系统的概念结构设计，绘制对应的 E-R 图。\n 要求标注所有实体型、联系型；标注每个联系的类型（1:1、1:n、m:n）；\n 标注联系的属性（如成绩、教室、课时）。（6分）\n2. 将该 E-R 图转换为关系数据库的关系模式。\n 要求标注每个关系模式的主键和外键。（7分）\n3. 指出上述关系模式分别为第几范式，为什么。（7分）\n",[31,103996,103994],{"__ignoreMap":29},[15,103998,103999],{},"文本化 E-R：",[24,104001,104004],{"className":104002,"code":104003,"language":2638,"meta":29},[2636],"学院 1 --- n 专业\n学院 1 --- n 课程\n学院 1 --- n 教师\n专业 1 --- n 学生\n学生 m --- n 课程，通过 选课 联系，联系属性：成绩\n教师 m --- n 课程，通过 授课 联系，联系属性：教室、课时\n",[31,104005,104003],{"__ignoreMap":29},[15,104007,104008],{},"关系模式：",[24,104010,104013],{"className":104011,"code":104012,"language":2638,"meta":29},[2636],"College(CollegeID, CollegeName)\nMajor(MajorID, MajorName, CollegeID)\nStudent(SID, SName, MajorID)\nCourse(CID, CName, CollegeID)\nTeacher(TID, TName, CollegeID)\nSC(SID, CID, Grade)\nTC(TID, CID, Classroom, Hours)\n",[31,104014,104012],{"__ignoreMap":29},[15,104016,104017],{},"主键与外键：",[24,104019,104022],{"className":104020,"code":104021,"language":2638,"meta":29},[2636],"College: PK(CollegeID)\nMajor: PK(MajorID), FK(CollegeID) -> College\nStudent: PK(SID), FK(MajorID) -> Major\nCourse: PK(CID), FK(CollegeID) -> College\nTeacher: PK(TID), FK(CollegeID) -> College\nSC: PK(SID, CID), FK(SID) -> Student, FK(CID) -> Course\nTC: PK(TID, CID), FK(TID) -> Teacher, FK(CID) -> Course\n",[31,104023,104021],{"__ignoreMap":29},[15,104025,103933],{},[105,104027,104028,104031,104034,104037],{},[108,104029,104030],{},"每个关系属性原子，满足 1NF。",[108,104032,104033],{},"非主属性完全依赖主键，满足 2NF。",[108,104035,104036],{},"非主属性不传递依赖主键，满足 3NF。",[108,104038,104039],{},"若每个函数依赖左部都是候选码，可进一步说明满足 BCNF。",[19,104041,104042],{"id":104042},"全球包裹递送公司设计题",[15,104044,92498],{},[24,104046,104049],{"className":104047,"code":104048,"language":2638,"meta":29},[2636],"Relational Database Design\n\nDesign a database for a worldwide package delivery company\n(e.g., EMS or SF Express).\n\nThe database must be able to keep track of customers who ship items\nand customers who receive items; some customers may do both.\n\nEach package must be identifiable and trackable, so the database\nmust be able to store the location of the package and its history\nof locations. Locations include trucks, planes, airports, and warehouses.\n\n1. Design an E-R diagram.\n2. Design a set of relational schemas, and a list of constraints,\n including primary key and foreign key constraints.\n3. Explain what normal form each relational schema satisfies, and why.\n",[31,104050,104048],{"__ignoreMap":29},[15,104052,103883],{},[24,104054,104057],{"className":104055,"code":104056,"language":2638,"meta":29},[2636],"Customer(CustomerID, Name, Phone, Address)\nPackage(PackageID, SenderID, ReceiverID, Weight, CreatedAt, CurrentLocationID)\nLocation(LocationID, LocationType, Description)\nTruck(LocationID, PlateNo)\nPlane(LocationID, FlightNo)\nAirport(LocationID, AirportCode, AirportName)\nWarehouse(LocationID, WarehouseName, Address)\nPackageLocationHistory(PackageID, LocationID, ArriveTime, LeaveTime)\n",[31,104058,104056],{"__ignoreMap":29},[15,104060,104061],{},"约束：",[24,104063,104066],{"className":104064,"code":104065,"language":2638,"meta":29},[2636],"Customer: PK(CustomerID)\nPackage: PK(PackageID),\n FK(SenderID) -> Customer(CustomerID),\n FK(ReceiverID) -> Customer(CustomerID),\n FK(CurrentLocationID) -> Location(LocationID)\nLocation: PK(LocationID)\nTruck\u002FPlane\u002FAirport\u002FWarehouse: PK\u002FFK(LocationID) -> Location(LocationID)\nPackageLocationHistory:\n PK(PackageID, LocationID, ArriveTime),\n FK(PackageID) -> Package(PackageID),\n FK(LocationID) -> Location(LocationID)\n",[31,104067,104065],{"__ignoreMap":29},[15,104069,103933],{},[105,104071,104072,104075,104078],{},[108,104073,104074],{},"将不同地点类型拆成子表，避免在一个大地点表中出现大量空值。",[108,104076,104077],{},"历史位置使用独立表，避免包裹表中重复存储多个历史地点。",[108,104079,104080],{},"各表非主属性依赖主键，通常可说明达到 3NF\u002FBCNF。",[19,104082,104083],{"id":104083},"学生与宿舍关系讨论",[15,104085,104086],{},"课堂问题清单：",[24,104088,104091],{"className":104089,"code":104090,"language":2638,"meta":29},[2636],"讨论：学生和宿舍的关系\n实践应用：\n1. 查询所有学生的住宿情况。\n2. 查询住宿学生的住宿情况。\n3. 查询学生的住宿的宿舍情况。\n4. 查询所有宿舍的学生住宿情况。\n5. 查询未住宿的宿舍情况。\n6. 查询所有宿舍的学生住宿情况。\n7. 查询所有学生和宿舍的住宿情况。\n8. 查询待分配宿舍的学生情况。\n9. 查询可分配宿舍的情况（假设一个宿舍最多 5 人）。\n10. 给出待分配学生的住宿的方案。\n",[31,104092,104090],{"__ignoreMap":29},[15,104094,102253],{},[105,104096,104097,104100,104103,104109],{},[108,104098,104099],{},"查询所有学生住宿情况：学生表左连接住宿\u002F宿舍表。",[108,104101,104102],{},"查询所有宿舍学生情况：宿舍表左连接住宿表和学生表。",[108,104104,104105,104106,104108],{},"同时保留未住宿学生和空宿舍：全外连接，或用 ",[31,104107,95259],{}," 合并左右外连接。",[108,104110,104111],{},"查询可分配宿舍：按宿舍分组统计人数，筛选人数小于 5。",[15,104113,104114],{},"示例模式：",[24,104116,104119],{"className":104117,"code":104118,"language":2638,"meta":29},[2636],"Student(Sno, Sname, ...)\nDorm(DormID, Building, RoomNo, Capacity)\nLive(Sno, DormID, BedNo)\n",[31,104120,104118],{"__ignoreMap":29},[15,104122,104123],{},"示例 SQL：",[24,104125,104127],{"className":92584,"code":104126,"language":92586,"meta":29,"style":29},"SELECT s.Sno, s.Sname, d.Building, d.RoomNo, l.BedNo\nFROM Student AS s\nLEFT JOIN Live AS l ON s.Sno = l.Sno\nLEFT JOIN Dorm AS d ON l.DormID = d.DormID;\n\nSELECT d.DormID, d.Building, d.RoomNo, COUNT(l.Sno) AS UsedBeds\nFROM Dorm AS d\nLEFT JOIN Live AS l ON d.DormID = l.DormID\nGROUP BY d.DormID, d.Building, d.RoomNo, d.Capacity\nHAVING COUNT(l.Sno) \u003C d.Capacity;\n",[31,104128,104129,104134,104139,104144,104149,104153,104158,104163,104168,104173],{"__ignoreMap":29},[34,104130,104131],{"class":36,"line":37},[34,104132,104133],{},"SELECT s.Sno, s.Sname, d.Building, d.RoomNo, l.BedNo\n",[34,104135,104136],{"class":36,"line":43},[34,104137,104138],{},"FROM Student AS s\n",[34,104140,104141],{"class":36,"line":49},[34,104142,104143],{},"LEFT JOIN Live AS l ON s.Sno = l.Sno\n",[34,104145,104146],{"class":36,"line":56},[34,104147,104148],{},"LEFT JOIN Dorm AS d ON l.DormID = d.DormID;\n",[34,104150,104151],{"class":36,"line":62},[34,104152,53],{"emptyLinePlaceholder":52},[34,104154,104155],{"class":36,"line":68},[34,104156,104157],{},"SELECT d.DormID, d.Building, d.RoomNo, COUNT(l.Sno) AS UsedBeds\n",[34,104159,104160],{"class":36,"line":74},[34,104161,104162],{},"FROM Dorm AS d\n",[34,104164,104165],{"class":36,"line":1971},[34,104166,104167],{},"LEFT JOIN Live AS l ON d.DormID = l.DormID\n",[34,104169,104170],{"class":36,"line":1976},[34,104171,104172],{},"GROUP BY d.DormID, d.Building, d.RoomNo, d.Capacity\n",[34,104174,104175],{"class":36,"line":1985},[34,104176,104177],{},"HAVING COUNT(l.Sno) \u003C d.Capacity;\n",[19,104179,104181],{"id":104180},"物资管理-e-r-示例","物资管理 E-R 示例",[15,104183,104184],{},"课堂图标题：",[24,104186,104189],{"className":104187,"code":104188,"language":2638,"meta":29},[2636],"例：物资管理\n实体及其属性 E-R 图\n实体及其联系 E-R 图\n",[31,104190,104188],{"__ignoreMap":29},[15,104192,104193],{},"可见实体\u002F属性\u002F联系：",[24,104195,104198],{"className":104196,"code":104197,"language":2638,"meta":29},[2636],"实体：供应商、仓库、职工、项目、零件。\n联系：供应、工作、领导、仓库\u002F项目\u002F零件相关供应联系。\n联系属性：供应量。\n属性可见：供应商号、仓库号、职工号、项目号、零件号等局部信息。\n",[31,104199,104197],{"__ignoreMap":29},[15,104201,104202],{},"文本化关系：",[24,104204,104207],{"className":104205,"code":104206,"language":2638,"meta":29},[2636],"供应商 --供应-- 零件\n供应商 --供应-- 项目\n仓库 --工作-- 职工\n职工 --领导-- 职工\n仓库\u002F项目\u002F零件之间存在库存或供应联系，联系属性含“供应量”\n",[31,104208,104206],{"__ignoreMap":29},[15,104210,104211],{},"原图部分被投影裁切，实体属性的具体名称未全部显示。",[19,104213,104214],{"id":104214},"数据库结构的数据访问",[15,104216,104184],{},[24,104218,104221],{"className":104219,"code":104220,"language":2638,"meta":29},[2636],"数据库结构的数据访问\n",[31,104222,104220],{"__ignoreMap":29},[15,104224,104225],{},"可见结构：",[24,104227,104230],{"className":104228,"code":104229,"language":2638,"meta":29},[2636],"应用程序 A\n应用程序 B\n对象的外模式\n程序工作区\nDBMS\n模式\n内模式\nOS\n数据库\nDB 的系统缓冲区\n",[31,104231,104229],{"__ignoreMap":29},[15,104233,104234],{},"文本化流程：",[24,104236,104239],{"className":104237,"code":104238,"language":2638,"meta":29},[2636],"应用程序 -> 外模式 -> 模式 -> 内模式 -> OS -> 数据库\n应用程序工作区 与 DBMS\u002F系统缓冲区之间进行数据交换。\n",[31,104240,104238],{"__ignoreMap":29},[15,104242,101009],{},[105,104244,104245,104248,104251,104254],{},[108,104246,104247],{},"外模式面向用户\u002F应用。",[108,104249,104250],{},"模式是数据库全局逻辑结构。",[108,104252,104253],{},"内模式描述物理存储。",[108,104255,104256],{},"DBMS 通过三级模式和两级映像实现数据独立性。",[79,104258,81],{},{"title":29,"searchDepth":43,"depth":43,"links":104260},[104261,104262,104263,104264,104265,104266,104267,104268,104269],{"id":103811,"depth":43,"text":103811},{"id":103832,"depth":43,"text":103832},{"id":103847,"depth":43,"text":103847},{"id":103951,"depth":43,"text":103951},{"id":103988,"depth":43,"text":103988},{"id":104042,"depth":43,"text":104042},{"id":104083,"depth":43,"text":104083},{"id":104180,"depth":43,"text":104181},{"id":104214,"depth":43,"text":104214},"覆盖图片：72a02a3e7087d67863ee7fbabfcab93a.png、fbe361cdc5283f542dcb76231701a884.png、a7c73797aa8be71a47f8305701a2a07e.png、6e76900d19df5e7c56b7e09cb36cc2e7.png、86567574E404D5112C0146C1244C013E.png、600B1BF6B6F035E3068292F21E8B511C.png、324370E4515703A116DDF91B7DD677BD.png、D03F9CFE1ED63CF4B07E065C7CD338F3.png、482D900A31EF461C592934370DBBC2FE.png、IMG_0340.JPG、IMG_0340 (1).JPG、IMG_0342.JPG、IMG_0342 (1).JPG、IMG_0364.JPG、IMG_0364 (1).JPG、IMG_0381.JPG、IMG_0381 (1).JPG、IMG_0551.JPG、IMG_0551 (1).JPG、IMG_0553.JPG、IMG_0553 (1).JPG、IMG_0554.JPG、IMG_0554 (1).JPG。",{},"\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F04",{"title":103736,"description":104270},"source\u002Fcs\u002Fdb\u002Fnotes\u002F04-数据库设计与规范化全文","PJvD55qBu6OJ8u9NCIFgCMPSsz396cX64g2Fvecidv4",{"id":104277,"title":104278,"body":104279,"description":104539,"extension":85,"meta":104540,"navigation":52,"path":104541,"seo":104542,"stem":104543,"__hash__":104544},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F05-查询优化恢复并发控制全文.md","查询优化、恢复、并发控制全文",{"type":8,"value":104280,"toc":104530},[104281,104284,104340,104343,104345,104351,104354,104360,104363,104377,104380,104386,104389,104395,104398,104401,104407,104410,104413,104419,104422,104428,104432,104435,104441,104444,104450,104452,104458,104461,104464,104466,104472,104475,104481,104484,104490,104493,104499,104503,104506,104512,104515,104521,104524],[11,104282,104278],{"id":104283},"查询优化恢复并发控制全文",[15,104285,100449,104286,146,104289,146,104292,146,104294,146,104296,146,104299,146,104302,146,104305,146,104308,146,104311,146,104314,146,104317,146,104320,146,104323,146,104326,146,104329,146,104332,146,104335,146,104338,347],{},[31,104287,104288],{},"546F40F2C52B5C38C4499ACAFD01FEDD.png",[31,104290,104291],{},"4B47BA6244099451BB27641110ABD42A.png",[31,104293,102432],{},[31,104295,102429],{},[31,104297,104298],{},"cab4c3c8b24470d4ddf52b9df6207c11.jpg",[31,104300,104301],{},"IMG_0689.JPG",[31,104303,104304],{},"IMG_0689 (1).JPG",[31,104306,104307],{},"IMG_0758.JPG",[31,104309,104310],{},"IMG_0758 (1).JPG",[31,104312,104313],{},"IMG_0759.JPG",[31,104315,104316],{},"IMG_0759 (1).JPG",[31,104318,104319],{},"C9D8CB7A918276A0773E0692137C82A9.png",[31,104321,104322],{},"45CF842DC915B004480C3FA46A6D8877.png",[31,104324,104325],{},"3616296FFDE186166B4E653F08EFFBEC.png",[31,104327,104328],{},"FFE74D06B1635830F1A4C3D7C083A176.png",[31,104330,104331],{},"7527539A3DBAAE0A5038034B956F2ECD.png",[31,104333,104334],{},"14BF3623B362548EEFD56F6081DAEC07.png",[31,104336,104337],{},"FC354D38D2E9BFC51DD56129E8A8BF83.png",[31,104339,100467],{},[19,104341,104342],{"id":104342},"查询优化题面",[15,104344,92498],{},[24,104346,104349],{"className":104347,"code":104348,"language":2638,"meta":29},[2636],"Query Optimization\n\nConsider the query:\n\nSELECT p.PNO, PName, Price, StockNum\nFROM Product p, Stock s\nWHERE p.PNO = s.PNO\n AND StockPlace = 'Warehouse1'\n AND Price > 4;\n\n1. Give a relational algebra expression using Cartesian product equivalent\n to the query.\n2. Draw a syntax tree for the above relational algebra expression and give\n a procedure for query optimization.\n3. Give the optimized relational algebra expression.\n",[31,104350,104348],{"__ignoreMap":29},[15,104352,104353],{},"笛卡尔积等价关系代数表达式：",[24,104355,104358],{"className":104356,"code":104357,"language":2638,"meta":29},[2636],"π_{p.PNO, PName, Price, StockNum}\n ( σ_{p.PNO = s.PNO ∧ StockPlace = 'Warehouse1' ∧ Price > 4}\n ( Product × Stock ) )\n",[31,104359,104357],{"__ignoreMap":29},[15,104361,104362],{},"优化思想：",[2230,104364,104365,104368,104371,104374],{},[108,104366,104367],{},"将选择条件分解。",[108,104369,104370],{},"将选择尽早下推到对应基本关系。",[108,104372,104373],{},"将笛卡尔积加选择转换为连接。",[108,104375,104376],{},"将投影尽早下推，只保留后续需要的列。",[15,104378,104379],{},"优化后的关系代数：",[24,104381,104384],{"className":104382,"code":104383,"language":2638,"meta":29},[2636],"π_{PNO, PName, Price, StockNum}\n(\n (π_{PNO, PName, Price}(σ_{Price > 4}(Product)))\n ⋈_{Product.PNO = Stock.PNO}\n (π_{PNO, StockNum}(σ_{StockPlace = 'Warehouse1'}(Stock)))\n)\n",[31,104385,104383],{"__ignoreMap":29},[15,104387,104388],{},"对应查询树，文本版：",[24,104390,104393],{"className":104391,"code":104392,"language":2638,"meta":29},[2636],"π PNO, PName, Price, StockNum\n└── ⋈ Product.PNO = Stock.PNO\n ├── π PNO, PName, Price\n │ └── σ Price > 4\n │ └── Product\n └── π PNO, StockNum\n └── σ StockPlace = 'Warehouse1'\n └── Stock\n",[31,104394,104392],{"__ignoreMap":29},[19,104396,104397],{"id":104397},"规范的查询优化步骤",[15,104399,104400],{},"课堂清单：",[24,104402,104405],{"className":104403,"code":104404,"language":2638,"meta":29},[2636],"规范的查询优化的步骤：\n1. 文字描述的应用问题场景。\n2. 给出其实现的 SQL 语句。\n3. 写出 SQL 语句对应的关系表达式。\n4. 绘制相应的查询树：project\u002Fselect\u002Fjoin。\n5. 绘制对应的关系代数语法树。\n6. 对其进行优化，给出优化依据的规则。\n7. 给出优化后的结果语法树。\n8. 给出对应的关系表达式。\n9. 编写查询执行策略\u002F计划书。\n",[31,104406,104404],{"__ignoreMap":29},[19,104408,104409],{"id":104409},"关系代数等价变换规则",[15,104411,104412],{},"课堂可见规则：",[24,104414,104417],{"className":104415,"code":104416,"language":2638,"meta":29},[2636],"关系代数等价变换规则\n关系代数表达式等价\n常用的等价变换规则：\n\n1. 连接、笛卡尔积交换律。\n2. 连接、笛卡尔积结合律。\n3. 选择的串接定律。\n4. 投影的串接定律。\n5. 选择与投影的交换律。\n6. 选择与笛卡尔积的交换律。\n7. 投影与笛卡尔积的交换。\n8. 选择与并的交换。\n9. 选择与差运算的交换。\n10. 投影与并的交换。\n\nE 是关系代数表达式，F 和 F_i 是连接\u002F选择条件。\n",[31,104418,104416],{"__ignoreMap":29},[15,104420,104421],{},"常用公式：",[24,104423,104426],{"className":104424,"code":104425,"language":2638,"meta":29},[2636],"R × S = S × R\n(R × S) × T = R × (S × T)\nR ⋈ S = S ⋈ R\n(R ⋈ S) ⋈ T = R ⋈ (S ⋈ T)\nσ_{F1 ∧ F2}(E) = σ_{F1}(σ_{F2}(E))\nπ_{A}(π_{A,B}(E)) = π_{A}(E)\nσ_F(R × S) 可在条件只涉及 R 或 S 时下推\nπ 可在保留连接属性和最终输出属性前提下下推\n",[31,104427,104425],{"__ignoreMap":29},[19,104429,104431],{"id":104430},"数据库恢复检查点恢复题","数据库恢复：检查点恢复题",[15,104433,104434],{},"题面可见：",[24,104436,104439],{"className":104437,"code":104438,"language":2638,"meta":29},[2636],"运用检查点恢复技术，填写以下策略和内容：\n图中划线、箭头、实心圆分别表示事务开始\u002F结束，事务未结束。\n\n对事务的处理策略：\nT1 事务：____\nT2 事务：____\nT3 事务：____\nT4 事务：____\n\n在基于检查点恢复过程中：\n开始：\nACTIVE-LIST = { ... } （2分）\n\n处理后，生成的事务队列：\nUNDO-LIST = { ... } （2分）\nREDO-LIST = { ... } （原图下部被遮挡）\n",[31,104440,104438],{"__ignoreMap":29},[15,104442,104443],{},"图形内容文本化：",[24,104445,104448],{"className":104446,"code":104447,"language":2638,"meta":29},[2636],"左竖线：检查点。\n右竖线：系统故障\u002F恢复时刻。\n多条事务时间线 T1、T2、T3、T4、T5、T6、T7、T8。\n\n可见状态：\n- 有些事务在检查点前开始，检查点后提交。\n- 有些事务在检查点前开始，到故障时仍未结束。\n- 有些事务在检查点后开始并在故障前提交。\n- 有些事务在检查点后开始，到故障时仍未结束。\n",[31,104449,104447],{"__ignoreMap":29},[15,104451,103858],{},[24,104453,104456],{"className":104454,"code":104455,"language":2638,"meta":29},[2636],"1. 检查点时正在执行的事务进入 ACTIVE-LIST。\n2. 从检查点之后的日志向后扫描：\n - 遇到 begin\u002Fstart，加入 UNDO-LIST。\n - 遇到 commit，若事务在 UNDO-LIST 中则移出，加入 REDO-LIST。\n3. 故障时未提交事务进入 UNDO-LIST。\n4. 故障前已提交事务进入 REDO-LIST。\n",[31,104457,104455],{"__ignoreMap":29},[15,104459,104460],{},"注意：照片中各事务编号笔迹较模糊，无法可靠确定每条事务线的编号，因此此处保留方法和可辨结构，不强行猜具体集合。",[19,104462,104463],{"id":104463},"并发控制题面",[15,104465,92498],{},[24,104467,104470],{"className":104468,"code":104469,"language":2638,"meta":29},[2636],"Concurrency Control:\nConsider the following two transactions:\n\nT1:\nread(A);\nread(B);\nif A = 0 then B := B + 1;\nwrite(B);\n\nT2:\nread(B);\nread(A);\nif B = 0 then A := A + 1;\nwrite(A);\n\nAdd lock and unlock instructions to transactions T1 and T2 so that\nthey observe the two-phase locking protocol.\nCan the execution of these transactions result in a deadlock?\n",[31,104471,104469],{"__ignoreMap":29},[15,104473,104474],{},"两阶段封锁协议参考答案：",[24,104476,104479],{"className":104477,"code":104478,"language":2638,"meta":29},[2636],"T1:\nSLock(A);\nSLock(B);\nread(A);\nread(B);\nif A = 0 then B := B + 1;\nXLock(B); -- 或一开始直接 XLock(B)\nwrite(B);\nUnlock(A);\nUnlock(B);\n\nT2:\nSLock(B);\nSLock(A);\nread(B);\nread(A);\nif B = 0 then A := A + 1;\nXLock(A); -- 或一开始直接 XLock(A)\nwrite(A);\nUnlock(B);\nUnlock(A);\n",[31,104480,104478],{"__ignoreMap":29},[15,104482,104483],{},"更严格且避免锁升级细节的写法：",[24,104485,104488],{"className":104486,"code":104487,"language":2638,"meta":29},[2636],"T1:\nSLock(A);\nXLock(B);\nread(A);\nread(B);\nif A = 0 then B := B + 1;\nwrite(B);\nUnlock(A);\nUnlock(B);\n\nT2:\nSLock(B);\nXLock(A);\nread(B);\nread(A);\nif B = 0 then A := A + 1;\nwrite(A);\nUnlock(B);\nUnlock(A);\n",[31,104489,104487],{"__ignoreMap":29},[15,104491,104492],{},"是否可能死锁：",[24,104494,104497],{"className":104495,"code":104496,"language":2638,"meta":29},[2636],"可能。\n例如 T1 先持有 A 的 S 锁并等待 B 的 X 锁；\nT2 先持有 B 的 S 锁并等待 A 的 X 锁；\n双方互相等待，可能形成死锁。\n",[31,104498,104496],{"__ignoreMap":29},[19,104500,104502],{"id":104501},"一级二级三级封锁协议课堂片段","一级、二级、三级封锁协议课堂片段",[15,104504,104505],{},"可见代码片段：",[24,104507,104510],{"className":104508,"code":104509,"language":2638,"meta":29},[2636],"VII. Concurrency Control\n\nT1【三级封锁协议】\nSLock(A)\nwhile ... GET\nXLock(B)\nwhile ... GET\nread(A);\nread(B);\nif A = 0 then B = B + 1;\nwrite(B);\nCOMMIT;\nUnlock(A);\nUnlock(B);\n\nT1【一级封锁协议】\nXLock(B)\nwhile ... GET\nread(A);\nread(B);\nif A = 0 then B = B + 1;\nwrite(B);\nUnlock(B);\n\nT1【二级封锁协议】\nSLock(A)\nwhile ...\nread(A);\nUnlock(A);\nXLock(B)\nwhile ... GET\nread(B);\nif A = 0 then B = B + 1;\nwrite(B);\nUnlock(B);\n",[31,104511,104509],{"__ignoreMap":29},[15,104513,104514],{},"课堂结论：",[24,104516,104519],{"className":104517,"code":104518,"language":2638,"meta":29},[2636],"可能会导致死锁。\n",[31,104520,104518],{"__ignoreMap":29},[19,104522,104523],{"id":104523},"封锁协议方法",[24,104525,104528],{"className":104526,"code":104527,"language":2638,"meta":29},[2636],"一级封锁协议：\n事务 T 修改数据 R 之前必须对 R 加 X 锁，直到事务结束才释放。\n可防止丢失修改。\n\n二级封锁协议：\n在一级基础上，读数据前必须加 S 锁，读完即可释放 S 锁。\n可防止读脏数据。\n\n三级封锁协议：\n在一级基础上，读数据前必须加 S 锁，并保持到事务结束。\n可防止不可重复读。\n\n两阶段封锁协议：\n事务分为加锁阶段和解锁阶段。\n一旦释放任何锁，就不能再申请新锁。\n可保证冲突可串行化，但可能发生死锁。\n",[31,104529,104527],{"__ignoreMap":29},{"title":29,"searchDepth":43,"depth":43,"links":104531},[104532,104533,104534,104535,104536,104537,104538],{"id":104342,"depth":43,"text":104342},{"id":104397,"depth":43,"text":104397},{"id":104409,"depth":43,"text":104409},{"id":104430,"depth":43,"text":104431},{"id":104463,"depth":43,"text":104463},{"id":104501,"depth":43,"text":104502},{"id":104523,"depth":43,"text":104523},"覆盖图片：546F40F2C52B5C38C4499ACAFD01FEDD.png、4B47BA6244099451BB27641110ABD42A.png、01BC0C9A088A35C1E0B81ED19A685DF9.png、C921AF903DCA13A80965ACDB7CEF8F9C.png、cab4c3c8b24470d4ddf52b9df6207c11.jpg、IMG_0689.JPG、IMG_0689 (1).JPG、IMG_0758.JPG、IMG_0758 (1).JPG、IMG_0759.JPG、IMG_0759 (1).JPG、C9D8CB7A918276A0773E0692137C82A9.png、45CF842DC915B004480C3FA46A6D8877.png、3616296FFDE186166B4E653F08EFFBEC.png、FFE74D06B1635830F1A4C3D7C083A176.png、7527539A3DBAAE0A5038034B956F2ECD.png、14BF3623B362548EEFD56F6081DAEC07.png、FC354D38D2E9BFC51DD56129E8A8BF83.png、7EFBDE17B0116BB9766F10AC8BFF9AE4.png。",{},"\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002F05",{"title":104278,"description":104539},"source\u002Fcs\u002Fdb\u002Fnotes\u002F05-查询优化恢复并发控制全文","Yk0BFzG3LTnibJu0Q0KsLYX14lAAmhpikCgbRq1bFNY",{"id":104546,"title":104547,"body":104548,"description":29,"extension":85,"meta":114741,"navigation":52,"path":114742,"seo":114743,"stem":114744,"__hash__":114745},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002Fnormalization.md","数据库规范化流程",{"type":8,"value":104549,"toc":114713},[104550,104552,104556,104832,104844,104846,104850,105361,105370,105372,105376,105380,105418,105487,105491,105494,105514,105518,105555,105559,105622,105625,105688,105692,105698,105702,105738,105740,105744,105747,105779,105781,105785,105788,107420,107422,107454,107486,108991,108993,108997,110097,110099,110103,110927,110929,110933,111010,111014,111828,111830,111833,111838,111902,111909,112396,112403,112405,112409,112559,112563,112675,112707,112816,112820,113129,113131,113135,113142,113145,113378,113409,113735,113738,114048,114050,114053],[11,104551,104547],{"id":104547},[19,104553,104555],{"id":104554},"流程一考试做题流程-项目设计模式","流程一：考试做题流程 \u002F 项目设计模式",[2230,104557,104558,104568,104578,104588,104598,104617,104627],{},[108,104559,104560,104563],{},[2627,104561,104562],{},"形式化：问题描述（自然语言）",[105,104564,104565],{},[108,104566,104567],{},"将需求方的口头要求或文字需求记录下来。",[108,104569,104570,104573],{},[2627,104571,104572],{},"应用语义归纳：补充潜在语义（默认）",[105,104574,104575],{},[108,104576,104577],{},"分析业务规则。例如：“一个员工只能属于一个部门”，这决定了数据之间的对应关系。",[108,104579,104580,104583],{},[2627,104581,104582],{},"绘制 E-R 图",[105,104584,104585],{},[108,104586,104587],{},"使用实体（框）、属性（圆）和联系（菱形）直观地表示数据结构。",[108,104589,104590,104593],{},[2627,104591,104592],{},"转换关系模式",[105,104594,104595],{},[108,104596,104597],{},"将 E-R 图转化为逻辑表格（关系模式）。",[108,104599,104600,104603],{},[2627,104601,104602],{},"说明范式，说明不合理性",[105,104604,104605],{},[108,104606,104607,104608,146,104611,58269,104614,347],{},"检查当前的表格属于第几范式（通常是 1NF 或 2NF），并指出存在的",[2627,104609,104610],{},"数据冗余",[2627,104612,104613],{},"插入异常",[2627,104615,104616],{},"删除异常",[108,104618,104619,104622],{},[2627,104620,104621],{},"规范化推导（到 BCNF）",[105,104623,104624],{},[108,104625,104626],{},"运用算法将表格拆分，逐步提升范式级别。",[108,104628,104629,104632],{},[2627,104630,104631],{},"给出设计结果：",[105,104633,104634,104733,104829],{},[108,104635,104636,616,104727,616,104730],{},[34,104637,104639,104665],{"className":104638},[31642],[34,104640,104642],{"className":104641},[31646],[31648,104643,104644],{"xmlns":31650},[31652,104645,104646,104662],{},[31655,104647,104648,104654,104656,104658,104660],{},[31743,104649,104650,104652],{},[31658,104651,34386],{},[31749,104653,640],{},[31662,104655,456],{"stretchy":702},[31662,104657,33934],{},[39361,104659,39363],{},[31662,104661,2557],{"stretchy":702},[31665,104663,104664],{"encoding":31667},"R_1 (\\dots)",[34,104666,104668],{"className":104667,"ariaHidden":758},[31672],[34,104669,104671,104674,104715,104718,104721,104724],{"className":104670},[31676],[34,104672],{"className":104673,"style":31774},[31680],[34,104675,104677,104680],{"className":104676},[31685],[34,104678,34386],{"className":104679,"style":34427},[31685,31686],[34,104681,104683],{"className":104682},[31792],[34,104684,104686,104707],{"className":104685},[31796,31797],[34,104687,104689,104704],{"className":104688},[31801],[34,104690,104692],{"className":104691,"style":31806},[31805],[34,104693,104695,104698],{"style":104694},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[34,104696],{"className":104697,"style":31814},[31813],[34,104699,104701],{"className":104700},[31818,31819,31820,31821],[34,104702,640],{"className":104703},[31685,31821],[34,104705,31829],{"className":104706},[31828],[34,104708,104710],{"className":104709},[31801],[34,104711,104713],{"className":104712,"style":31836},[31805],[34,104714],{},[34,104716,456],{"className":104717},[31782],[34,104719,33934],{"className":104720},[34006],[34,104722],{"className":104723,"style":32123},[31845],[34,104725,2557],{"className":104726},[31690],[2627,104728,104729],{},"码：",[2627,104731,104732],{},"外码：",[108,104734,104735,616,104825,616,104827],{},[34,104736,104738,104764],{"className":104737},[31642],[34,104739,104741],{"className":104740},[31646],[31648,104742,104743],{"xmlns":31650},[31652,104744,104745,104761],{},[31655,104746,104747,104753,104755,104757,104759],{},[31743,104748,104749,104751],{},[31658,104750,34386],{},[31749,104752,768],{},[31662,104754,456],{"stretchy":702},[31662,104756,33934],{},[39361,104758,39363],{},[31662,104760,2557],{"stretchy":702},[31665,104762,104763],{"encoding":31667},"R_2 (\\dots)",[34,104765,104767],{"className":104766,"ariaHidden":758},[31672],[34,104768,104770,104773,104813,104816,104819,104822],{"className":104769},[31676],[34,104771],{"className":104772,"style":31774},[31680],[34,104774,104776,104779],{"className":104775},[31685],[34,104777,34386],{"className":104778,"style":34427},[31685,31686],[34,104780,104782],{"className":104781},[31792],[34,104783,104785,104805],{"className":104784},[31796,31797],[34,104786,104788,104802],{"className":104787},[31801],[34,104789,104791],{"className":104790,"style":31806},[31805],[34,104792,104793,104796],{"style":104694},[34,104794],{"className":104795,"style":31814},[31813],[34,104797,104799],{"className":104798},[31818,31819,31820,31821],[34,104800,768],{"className":104801},[31685,31821],[34,104803,31829],{"className":104804},[31828],[34,104806,104808],{"className":104807},[31801],[34,104809,104811],{"className":104810,"style":31836},[31805],[34,104812],{},[34,104814,456],{"className":104815},[31782],[34,104817,33934],{"className":104818},[34006],[34,104820],{"className":104821,"style":32123},[31845],[34,104823,2557],{"className":104824},[31690],[2627,104826,104729],{},[2627,104828,104732],{},[108,104830,104831],{},"……",[104833,104834,104835],"blockquote",{},[15,104836,104837,104839,104840,104843],{},[2627,104838,62532],{}," 经过以上规范化推导，问题已经设计到 ",[2627,104841,104842],{},"BCNF (Boyce-Codd Normal Form)","，达到函数依赖范畴的最高级范式。",[32188,104845],{},[19,104847,104849],{"id":104848},"流程二书本上做题流程","流程二：书本上做题流程",[2230,104851,104852,104856,104860,105154,105158,105162],{},[108,104853,104854],{},[2627,104855,104562],{},[108,104857,104858],{},[2627,104859,104572],{},[108,104861,104862,104938],{},[2627,104863,104864,104865],{},"形式化定义：",[34,104866,104868,104892],{"className":104867},[31642],[34,104869,104871],{"className":104870},[31646],[31648,104872,104873],{"xmlns":31650},[31652,104874,104875,104889],{},[31655,104876,104877,104879,104881,104883,104885,104887],{},[31658,104878,34386],{},[31662,104880,932],{},[31658,104882,33350],{},[31662,104884,31960],{"separator":758},[31658,104886,51834],{},[31662,104888,29622],{},[31665,104890,104891],{"encoding":31667},"R \u003CU, F>",[34,104893,104895,104914],{"className":104894,"ariaHidden":758},[31672],[34,104896,104898,104902,104905,104908,104911],{"className":104897},[31676],[34,104899],{"className":104900,"style":104901},[31680],"height:0.7224em;vertical-align:-0.0391em;",[34,104903,34386],{"className":104904,"style":34427},[31685,31686],[34,104906],{"className":104907,"style":31846},[31845],[34,104909,932],{"className":104910},[31850],[34,104912],{"className":104913,"style":31846},[31845],[34,104915,104917,104920,104923,104926,104929,104932,104935],{"className":104916},[31676],[34,104918],{"className":104919,"style":70739},[31680],[34,104921,33350],{"className":104922,"style":33389},[31685,31686],[34,104924,31960],{"className":104925},[32119],[34,104927],{"className":104928,"style":32123},[31845],[34,104930,51834],{"className":104931,"style":37422},[31685,31686],[34,104933],{"className":104934,"style":31846},[31845],[34,104936,29622],{"className":104937},[31850],[105,104939,104940],{},[108,104941,104942,104945],{},[2627,104943,104944],{},"其中：",[105,104946,104947,105016,105085,105090],{},[108,104948,104949,105015],{},[34,104950,104952,104976],{"className":104951},[31642],[34,104953,104955],{"className":104954},[31646],[31648,104956,104957],{"xmlns":31650},[31652,104958,104959,104973],{},[31655,104960,104961,104963,104965,104967,104969,104971],{},[31658,104962,33350],{},[31662,104964,202],{},[31662,104966,32685],{"stretchy":702},[31662,104968,33934],{},[39361,104970,39363],{},[31662,104972,32715],{"stretchy":702},[31665,104974,104975],{"encoding":31667},"U = \\{ \\dots \\}",[34,104977,104979,104997],{"className":104978,"ariaHidden":758},[31672],[34,104980,104982,104985,104988,104991,104994],{"className":104981},[31676],[34,104983],{"className":104984,"style":33218},[31680],[34,104986,33350],{"className":104987,"style":33389},[31685,31686],[34,104989],{"className":104990,"style":31846},[31845],[34,104992,202],{"className":104993},[31850],[34,104995],{"className":104996,"style":31846},[31845],[34,104998,105000,105003,105006,105009,105012],{"className":104999},[31676],[34,105001],{"className":105002,"style":31774},[31680],[34,105004,32685],{"className":105005},[31782],[34,105007,33934],{"className":105008},[34006],[34,105010],{"className":105011,"style":32123},[31845],[34,105013,32715],{"className":105014},[31690]," （属性全集，即表里所有的列）",[108,105017,105018,105084],{},[34,105019,105021,105045],{"className":105020},[31642],[34,105022,105024],{"className":105023},[31646],[31648,105025,105026],{"xmlns":31650},[31652,105027,105028,105042],{},[31655,105029,105030,105032,105034,105036,105038,105040],{},[31658,105031,51834],{},[31662,105033,202],{},[31662,105035,32685],{"stretchy":702},[31662,105037,33934],{},[39361,105039,39363],{},[31662,105041,32715],{"stretchy":702},[31665,105043,105044],{"encoding":31667},"F = \\{ \\dots \\}",[34,105046,105048,105066],{"className":105047,"ariaHidden":758},[31672],[34,105049,105051,105054,105057,105060,105063],{"className":105050},[31676],[34,105052],{"className":105053,"style":33218},[31680],[34,105055,51834],{"className":105056,"style":37422},[31685,31686],[34,105058],{"className":105059,"style":31846},[31845],[34,105061,202],{"className":105062},[31850],[34,105064],{"className":105065,"style":31846},[31845],[34,105067,105069,105072,105075,105078,105081],{"className":105068},[31676],[34,105070],{"className":105071,"style":31774},[31680],[34,105073,32685],{"className":105074},[31782],[34,105076,33934],{"className":105077},[34006],[34,105079],{"className":105080,"style":32123},[31845],[34,105082,32715],{"className":105083},[31690]," （函数依赖集，即列与列之间的决定关系）",[108,105086,105087,105089],{},[2627,105088,104729],{}," （Primary Key，能唯一标识一行的最小属性组）",[108,105091,105092,105153],{},[2627,105093,105094,105095],{},"非主属性集 ",[34,105096,105098,105120],{"className":105097},[31642],[34,105099,105101],{"className":105100},[31646],[31648,105102,105103],{"xmlns":31650},[31652,105104,105105,105117],{},[31655,105106,105107,105109,105111,105113,105115],{},[31662,105108,202],{},[31662,105110,32685],{"stretchy":702},[31662,105112,33934],{},[39361,105114,39363],{},[31662,105116,32715],{"stretchy":702},[31665,105118,105119],{"encoding":31667},"= \\{ \\dots \\}",[34,105121,105123,105135],{"className":105122,"ariaHidden":758},[31672],[34,105124,105126,105129,105132],{"className":105125},[31676],[34,105127],{"className":105128,"style":73089},[31680],[34,105130,202],{"className":105131},[31850],[34,105133],{"className":105134,"style":31846},[31845],[34,105136,105138,105141,105144,105147,105150],{"className":105137},[31676],[34,105139],{"className":105140,"style":31774},[31680],[34,105142,32685],{"className":105143},[31782],[34,105145,33934],{"className":105146},[34006],[34,105148],{"className":105149,"style":32123},[31845],[34,105151,32715],{"className":105152},[31690]," （不包含在任何候选码中的属性）",[108,105155,105156],{},[2627,105157,104602],{},[108,105159,105160],{},[2627,105161,104621],{},[108,105163,105164,105167],{},[2627,105165,105166],{},"规范化结果：",[105,105168,105169,105264,105359],{},[108,105170,105171,616,105260,616,105262],{},[34,105172,105174,105199],{"className":105173},[31642],[34,105175,105177],{"className":105176},[31646],[31648,105178,105179],{"xmlns":31650},[31652,105180,105181,105197],{},[31655,105182,105183,105189,105191,105193,105195],{},[31743,105184,105185,105187],{},[31658,105186,34386],{},[31749,105188,640],{},[31662,105190,456],{"stretchy":702},[31662,105192,33934],{},[39361,105194,39363],{},[31662,105196,2557],{"stretchy":702},[31665,105198,104664],{"encoding":31667},[34,105200,105202],{"className":105201,"ariaHidden":758},[31672],[34,105203,105205,105208,105248,105251,105254,105257],{"className":105204},[31676],[34,105206],{"className":105207,"style":31774},[31680],[34,105209,105211,105214],{"className":105210},[31685],[34,105212,34386],{"className":105213,"style":34427},[31685,31686],[34,105215,105217],{"className":105216},[31792],[34,105218,105220,105240],{"className":105219},[31796,31797],[34,105221,105223,105237],{"className":105222},[31801],[34,105224,105226],{"className":105225,"style":31806},[31805],[34,105227,105228,105231],{"style":104694},[34,105229],{"className":105230,"style":31814},[31813],[34,105232,105234],{"className":105233},[31818,31819,31820,31821],[34,105235,640],{"className":105236},[31685,31821],[34,105238,31829],{"className":105239},[31828],[34,105241,105243],{"className":105242},[31801],[34,105244,105246],{"className":105245,"style":31836},[31805],[34,105247],{},[34,105249,456],{"className":105250},[31782],[34,105252,33934],{"className":105253},[34006],[34,105255],{"className":105256,"style":32123},[31845],[34,105258,2557],{"className":105259},[31690],[2627,105261,104729],{},[2627,105263,104732],{},[108,105265,105266,616,105355,616,105357],{},[34,105267,105269,105294],{"className":105268},[31642],[34,105270,105272],{"className":105271},[31646],[31648,105273,105274],{"xmlns":31650},[31652,105275,105276,105292],{},[31655,105277,105278,105284,105286,105288,105290],{},[31743,105279,105280,105282],{},[31658,105281,34386],{},[31749,105283,768],{},[31662,105285,456],{"stretchy":702},[31662,105287,33934],{},[39361,105289,39363],{},[31662,105291,2557],{"stretchy":702},[31665,105293,104763],{"encoding":31667},[34,105295,105297],{"className":105296,"ariaHidden":758},[31672],[34,105298,105300,105303,105343,105346,105349,105352],{"className":105299},[31676],[34,105301],{"className":105302,"style":31774},[31680],[34,105304,105306,105309],{"className":105305},[31685],[34,105307,34386],{"className":105308,"style":34427},[31685,31686],[34,105310,105312],{"className":105311},[31792],[34,105313,105315,105335],{"className":105314},[31796,31797],[34,105316,105318,105332],{"className":105317},[31801],[34,105319,105321],{"className":105320,"style":31806},[31805],[34,105322,105323,105326],{"style":104694},[34,105324],{"className":105325,"style":31814},[31813],[34,105327,105329],{"className":105328},[31818,31819,31820,31821],[34,105330,768],{"className":105331},[31685,31821],[34,105333,31829],{"className":105334},[31828],[34,105336,105338],{"className":105337},[31801],[34,105339,105341],{"className":105340,"style":31836},[31805],[34,105342],{},[34,105344,456],{"className":105345},[31782],[34,105347,33934],{"className":105348},[34006],[34,105350],{"className":105351,"style":32123},[31845],[34,105353,2557],{"className":105354},[31690],[2627,105356,104729],{},[2627,105358,104732],{},[108,105360,104831],{},[104833,105362,105363],{},[15,105364,105365,104839,105367,104843],{},[2627,105366,62532],{},[2627,105368,105369],{},"BCNF",[32188,105371],{},[19,105373,105375],{"id":105374},"深度解析这些步骤是什么意思","深度解析：这些步骤是什么意思？",[253,105377,105379],{"id":105378},"_1-为什么要从语义开始","1. 为什么要从“语义”开始？",[15,105381,105382,105383,105386,105387,347],{},"数据库设计不是直接画表。",[2627,105384,105385],{},"语义","决定了",[2627,105388,105389,105390,2557],{},"函数依赖 (",[34,105391,105393,105406],{"className":105392},[31642],[34,105394,105396],{"className":105395},[31646],[31648,105397,105398],{"xmlns":31650},[31652,105399,105400,105404],{},[31655,105401,105402],{},[31658,105403,51834],{},[31665,105405,51834],{"encoding":31667},[34,105407,105409],{"className":105408,"ariaHidden":758},[31672],[34,105410,105412,105415],{"className":105411},[31676],[34,105413],{"className":105414,"style":33218},[31680],[34,105416,51834],{"className":105417,"style":37422},[31685,31686],[105,105419,105420,105484],{},[108,105421,105422,105424,105425,347],{},[71807,105423,19553],{}," 如果业务规定“身份证号可以确定姓名”，那么就存在函数依赖：",[34,105426,105428,105447],{"className":105427},[31642],[34,105429,105431],{"className":105430},[31646],[31648,105432,105433],{"xmlns":31650},[31652,105434,105435,105444],{},[31655,105436,105437,105440,105442],{},[39361,105438,105439],{},"身份证号",[31662,105441,33239],{},[39361,105443,99735],{},[31665,105445,105446],{"encoding":31667},"\\text{身份证号} \\rightarrow \\text{姓名}",[34,105448,105450,105472],{"className":105449,"ariaHidden":758},[31672],[34,105451,105453,105456,105463,105466,105469],{"className":105452},[31676],[34,105454],{"className":105455,"style":33218},[31680],[34,105457,105459],{"className":105458},[31685,2638],[34,105460,105439],{"className":105461},[31685,105462],"cjk_fallback",[34,105464],{"className":105465,"style":31846},[31845],[34,105467,33239],{"className":105468},[31850],[34,105470],{"className":105471,"style":31846},[31845],[34,105473,105475,105478],{"className":105474},[31676],[34,105476],{"className":105477,"style":33218},[31680],[34,105479,105481],{"className":105480},[31685,2638],[34,105482,99735],{"className":105483},[31685,105462],[108,105485,105486],{},"如果你不清楚业务逻辑（语义），你就无法确定谁是“码”（主键），也就无法进行后续推导。",[253,105488,105490],{"id":105489},"_2-什么是不合理性","2. 什么是“不合理性”？",[15,105492,105493],{},"当一个表的设计不好时（范式过低），会出现以下问题：",[105,105495,105496,105502,105508],{},[108,105497,105498,105501],{},[2627,105499,105500],{},"数据冗余："," 同一个部门的名字在每一个员工行都存一遍，浪费空间。",[108,105503,105504,105507],{},[2627,105505,105506],{},"插入异常："," 如果一个部门刚成立还没员工，由于主键（员工号）不能为空，你可能无法把这个部门存进数据库。",[108,105509,105510,105513],{},[2627,105511,105512],{},"删除异常："," 如果把最后一个员工删了，结果连同这个部门的信息也一起消失了。",[253,105515,105517],{"id":105516},"_3-为什么目标是-bcnf","3. 为什么目标是 BCNF？",[105,105519,105520,105526,105535,105543,105549],{},[108,105521,105522,105525],{},[2627,105523,105524],{},"1NF (第一范式)："," 属性不可再分（列不能再拆）。",[108,105527,105528,105531,105532,347],{},[2627,105529,105530],{},"2NF (第二范式)："," 消除了非主属性对码的",[2627,105533,105534],{},"部分依赖",[108,105536,105537,105531,105540,347],{},[2627,105538,105539],{},"3NF (第三范式)：",[2627,105541,105542],{},"传递依赖",[108,105544,105545,105548],{},[2627,105546,105547],{},"BCNF (BC范式)："," 在 3NF 基础上进一步消除主属性对码的部分或传递依赖。",[108,105550,105551,105554],{},[2627,105552,105553],{},"意义："," 到达 BCNF 意味着在处理“一个变量决定另一个变量”这种逻辑关系时，你的数据库已经达到了物理结构上的“最干净”状态。",[253,105556,105558],{"id":105557},"_4-转换流程的差异","4. 转换流程的差异",[105,105560,105561,105571],{},[108,105562,105563,105566,105567,105570],{},[2627,105564,105565],{},"实战（流程一）"," 强调 ",[2627,105568,105569],{},"E-R 图","，因为它是连接人类思维和计算机逻辑的桥梁。",[108,105572,105573,105566,105576,105621],{},[2627,105574,105575],{},"理论（流程二）",[2627,105577,105578,105579,2557],{},"数学定义 (",[34,105580,105582,105600],{"className":105581},[31642],[34,105583,105585],{"className":105584},[31646],[31648,105586,105587],{"xmlns":31650},[31652,105588,105589,105597],{},[31655,105590,105591,105593,105595],{},[31658,105592,33350],{},[31662,105594,31960],{"separator":758},[31658,105596,51834],{},[31665,105598,105599],{"encoding":31667},"U, F",[34,105601,105603],{"className":105602,"ariaHidden":758},[31672],[34,105604,105606,105609,105612,105615,105618],{"className":105605},[31676],[34,105607],{"className":105608,"style":70739},[31680],[34,105610,33350],{"className":105611,"style":33389},[31685,31686],[34,105613,31960],{"className":105614},[32119],[34,105616],{"className":105617,"style":32123},[31845],[34,105619,51834],{"className":105620,"style":37422},[31685,31686],"，它是为了方便使用算法（如模式分解算法）进行严谨的证明。",[253,105623,105624],{"id":105624},"建议",[15,105626,105627,105628,105658,105659,105687],{},"在实际操作中，请务必先写出所有的 ",[2627,105629,105389,105630,2557],{},[34,105631,105633,105646],{"className":105632},[31642],[34,105634,105636],{"className":105635},[31646],[31648,105637,105638],{"xmlns":31650},[31652,105639,105640,105644],{},[31655,105641,105642],{},[31658,105643,51834],{},[31665,105645,51834],{"encoding":31667},[34,105647,105649],{"className":105648,"ariaHidden":758},[31672],[34,105650,105652,105655],{"className":105651},[31676],[34,105653],{"className":105654,"style":33218},[31680],[34,105656,51834],{"className":105657,"style":37422},[31685,31686],"。这是所有推导的根基。如果 ",[34,105660,105662,105675],{"className":105661},[31642],[34,105663,105665],{"className":105664},[31646],[31648,105666,105667],{"xmlns":31650},[31652,105668,105669,105673],{},[31655,105670,105671],{},[31658,105672,51834],{},[31665,105674,51834],{"encoding":31667},[34,105676,105678],{"className":105677,"ariaHidden":758},[31672],[34,105679,105681,105684],{"className":105680},[31676],[34,105682],{"className":105683,"style":33218},[31680],[34,105685,51834],{"className":105686,"style":37422},[31685,31686]," 写错了，后面的范式判断和拆表（规范化推导）都会全盘皆错。",[19,105689,105691],{"id":105690},"例题-week-8-hw","例题 （Week 8. HW）",[15,105693,105694,105697],{},[2627,105695,105696],{},"面向农业的电子商务系统部分需求概述：","\n注册客户能看见农户基本信息、生产场地基本信息、生产的农产品及规格、价格、产能等信息，能查询往年生产统计信息（乡镇，镇长电话，农产品、总产能、平均价格）；",[253,105699,105701],{"id":105700},"答题要求","答题要求：",[2230,105703,105704,105710,105716,105721,105727,105732],{},[108,105705,105706,105709],{},[2627,105707,105708],{},"用 E-R 图设计尽可能好的概念模型；","（4 分）",[108,105711,105712,105715],{},[2627,105713,105714],{},"将 E-R 图转换成关系模型；","（3 分）",[108,105717,105718,105715],{},[2627,105719,105720],{},"写出各个关系的函数依赖关系；",[108,105722,105723,105726],{},[2627,105724,105725],{},"写出各个关系的主码和外码；","（2 分）",[108,105728,105729,105726],{},[2627,105730,105731],{},"写出各个关系满足的范式级别；",[108,105733,105734,105737],{},[2627,105735,105736],{},"将规范级别低的关系转成高级别范式的关系。","（1 分）",[32188,105739],{},[253,105741,105743],{"id":105742},"_1-需求剖析与概念模型设计-e-r-图","1. 需求剖析与概念模型设计 (E-R 图)",[15,105745,105746],{},"在设计前，我们首先识别核心实体及其逻辑关联：",[105,105748,105749,105755,105761,105767,105773],{},[108,105750,105751,105754],{},[2627,105752,105753],{},"乡镇 (Township)","：作为地理管理单元，拥有唯一的镇长联系方式。",[108,105756,105757,105760],{},[2627,105758,105759],{},"农户 (Farmer)","：属于某个乡镇，拥有基本信息。",[108,105762,105763,105766],{},[2627,105764,105765],{},"生产场地 (Site)","：由农户经营，是农产品的产出地。",[108,105768,105769,105772],{},[2627,105770,105771],{},"农产品 (Product)","：挂载于场地之下，具有规格、单价、产能。",[108,105774,105775,105778],{},[2627,105776,105777],{},"生产统计 (Statistics)","：这是一个派生或聚合信息，通常涉及“乡镇、年份、产品”三个维度的关联。",[32188,105780],{},[253,105782,105784],{"id":105783},"_2-关系模型转换","2. 关系模型转换",[15,105786,105787],{},"基于 E-R 图，转换得到的逻辑关系模式如下（初始状态）：",[2230,105789,105790,106020,106288,106542,106918],{},[108,105791,105792,47649,105795],{},[2627,105793,105794],{},"乡镇表",[34,105796,105798,105866],{"className":105797},[31642],[34,105799,105801],{"className":105800},[31646],[31648,105802,105803],{"xmlns":31650},[31652,105804,105805,105863],{},[31655,105806,105807,105813,105815,105838,105840,105842,105844,105846,105848,105850,105853,105855,105857,105859,105861],{},[31743,105808,105809,105811],{},[31658,105810,34386],{},[31749,105812,640],{},[31662,105814,456],{"stretchy":702},[71115,105816,105817,105835],{"accentunder":758},[31655,105818,105819,105821,105823,105825,105827,105829,105831,105833],{},[31658,105820,37400],{},[31658,105822,43383],{},[31658,105824,47077],{},[31658,105826,31660],{},[31658,105828,33347],{},[31658,105830,111],{},[31658,105832,31759],{},[31658,105834,33244],{},[31662,105836,105837],{"stretchy":758},"‾",[31662,105839,31960],{"separator":758},[31658,105841,98360],{},[31658,105843,111],{},[31658,105845,67764],{},[31658,105847,43383],{},[31658,105849,34395],{},[31658,105851,105852],{},"P",[31658,105854,47292],{},[31658,105856,43383],{},[31658,105858,31660],{},[31658,105860,33244],{},[31662,105862,2557],{"stretchy":702},[31665,105864,105865],{"encoding":31667},"R_1(\\underline{TownName}, MayorPhone)",[34,105867,105869],{"className":105868,"ariaHidden":758},[31672],[34,105870,105872,105875,105915,105918,105983,105986,105989,105992,105995,105998,106002,106005,106008,106011,106014,106017],{"className":105871},[31676],[34,105873],{"className":105874,"style":31774},[31680],[34,105876,105878,105881],{"className":105877},[31685],[34,105879,34386],{"className":105880,"style":34427},[31685,31686],[34,105882,105884],{"className":105883},[31792],[34,105885,105887,105907],{"className":105886},[31796,31797],[34,105888,105890,105904],{"className":105889},[31801],[34,105891,105893],{"className":105892,"style":31806},[31805],[34,105894,105895,105898],{"style":104694},[34,105896],{"className":105897,"style":31814},[31813],[34,105899,105901],{"className":105900},[31818,31819,31820,31821],[34,105902,640],{"className":105903},[31685,31821],[34,105905,31829],{"className":105906},[31828],[34,105908,105910],{"className":105909},[31801],[34,105911,105913],{"className":105912,"style":31836},[31805],[34,105914],{},[34,105916,456],{"className":105917},[31782],[34,105919,105922],{"className":105920},[31685,105921],"underline",[34,105923,105925,105974],{"className":105924},[31796,31797],[34,105926,105928,105971],{"className":105927},[31801],[34,105929,105931,105941],{"className":105930,"style":33218},[31805],[34,105932,105934,105937],{"style":105933},"top:-2.84em;",[34,105935],{"className":105936,"style":41508},[31813],[34,105938],{"className":105939,"style":41528},[105940],"underline-line",[34,105942,105943,105946],{"style":41565},[34,105944],{"className":105945,"style":41508},[31813],[34,105947,105949,105952,105955,105958,105961,105964,105968],{"className":105948},[31685],[34,105950,37400],{"className":105951,"style":37422},[31685,31686],[34,105953,43383],{"className":105954},[31685,31686],[34,105956,47077],{"className":105957,"style":47120},[31685,31686],[34,105959,31660],{"className":105960},[31685,31686],[34,105962,33347],{"className":105963,"style":33389},[31685,31686],[34,105965,105967],{"className":105966},[31685,31686],"am",[34,105969,33244],{"className":105970},[31685,31686],[34,105972,31829],{"className":105973},[31828],[34,105975,105977],{"className":105976},[31801],[34,105978,105981],{"className":105979,"style":105980},[31805],"height:0.2em;",[34,105982],{},[34,105984,31960],{"className":105985},[32119],[34,105987],{"className":105988,"style":32123},[31845],[34,105990,98360],{"className":105991,"style":33389},[31685,31686],[34,105993,111],{"className":105994},[31685,31686],[34,105996,67764],{"className":105997,"style":32062},[31685,31686],[34,105999,106001],{"className":106000,"style":34452},[31685,31686],"or",[34,106003,105852],{"className":106004,"style":37422},[31685,31686],[34,106006,47292],{"className":106007},[31685,31686],[34,106009,43383],{"className":106010},[31685,31686],[34,106012,31660],{"className":106013},[31685,31686],[34,106015,33244],{"className":106016},[31685,31686],[34,106018,2557],{"className":106019},[31690],[108,106021,106022,47649,106025],{},[2627,106023,106024],{},"农户表",[34,106026,106028,106113],{"className":106027},[31642],[34,106029,106031],{"className":106030},[31646],[31648,106032,106033],{"xmlns":31650},[31652,106034,106035,106110],{},[31655,106036,106037,106043,106045,106068,106070,106072,106074,106076,106078,106080,106082,106084,106086,106088,106090,106092,106094,106096,106098,106100,106102,106104,106106,106108],{},[31743,106038,106039,106041],{},[31658,106040,34386],{},[31749,106042,768],{},[31662,106044,456],{"stretchy":702},[71115,106046,106047,106066],{"accentunder":758},[31655,106048,106049,106051,106053,106055,106057,106059,106061,106064],{},[31658,106050,51834],{},[31658,106052,111],{},[31658,106054,34395],{},[31658,106056,31759],{},[31658,106058,33244],{},[31658,106060,34395],{},[31658,106062,106063],{},"I",[31658,106065,91994],{},[31662,106067,105837],{"stretchy":758},[31662,106069,31960],{"separator":758},[31658,106071,51834],{},[31658,106073,111],{},[31658,106075,34395],{},[31658,106077,31759],{},[31658,106079,33244],{},[31658,106081,34395],{},[31658,106083,106063],{},[31658,106085,31660],{},[31658,106087,31739],{},[31658,106089,43383],{},[31662,106091,31960],{"separator":758},[31658,106093,37400],{},[31658,106095,43383],{},[31658,106097,47077],{},[31658,106099,31660],{},[31658,106101,33347],{},[31658,106103,111],{},[31658,106105,31759],{},[31658,106107,33244],{},[31662,106109,2557],{"stretchy":702},[31665,106111,106112],{"encoding":31667},"R_2(\\underline{FarmerID}, FarmerInfo, 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AvgPrice)",[34,107073,107075],{"className":107074,"ariaHidden":758},[31672],[34,107076,107078,107082,107122,107125,107238,107241,107244,107247,107250,107253,107256,107259,107262,107265,107268,107271,107274,107277,107280,107283,107286,107289,107292,107295,107298,107301,107304,107307,107310,107313,107316,107319,107322,107325,107328,107331,107334,107337,107340,107343],{"className":107077},[31676],[34,107079],{"className":107080,"style":107081},[31680],"height:1.1444em;vertical-align:-0.3944em;",[34,107083,107085,107088],{"className":107084},[31685],[34,107086,34386],{"className":107087,"style":34427},[31685,31686],[34,107089,107091],{"className":107090},[31792],[34,107092,107094,107114],{"className":107093},[31796,31797],[34,107095,107097,107111],{"className":107096},[31801],[34,107098,107100],{"className":107099,"style":31806},[31805],[34,107101,107102,107105],{"style":104694},[34,107103],{"className":107104,"style":31814},[31813],[34,107106,107108],{"className":107107},[31818,31819,31820,31821],[34,107109,18136],{"className":107110},[31685,31821],[34,107112,31829],{"className":107113},[31828],[34,107115,107117],{"className":107116},[31801],[34,107118,107120],{"className":107119,"style":31836},[31805],[34,107121],{},[34,107123,456],{"className":107124},[31782],[34,107126,107128],{"className":107127},[31685,105921],[34,107129,107131,107229],{"className":107130},[31796,31797],[34,107132,107134,107226],{"className":107133},[31801],[34,107135,107137,107146],{"className":107136,"style":31681},[31805],[34,107138,107140,107143],{"style":107139},"top:-2.6456em;",[34,107141],{"className":107142,"style":41508},[31813],[34,107144],{"className":107145,"style":41528},[105940],[34,107147,107148,107151],{"style":41565},[34,107149],{"className":107150,"style":41508},[31813],[34,107152,107154,107157,107160,107163,107166,107169,107172,107175,107178,107181,107184,107187,107190,107193,107196,107199,107202,107205,107208,107211,107214,107217,107220,107223],{"className":107153},[31685],[34,107155,37400],{"className":107156,"style":37422},[31685,31686],[34,107158,43383],{"className":107159},[31685,31686],[34,107161,47077],{"className":107162,"style":47120},[31685,31686],[34,107164,31660],{"className":107165},[31685,31686],[34,107167,33347],{"className":107168,"style":33389},[31685,31686],[34,107170,105967],{"className":107171},[31685,31686],[34,107173,33244],{"className":107174},[31685,31686],[34,107176,31960],{"className":107177},[32119],[34,107179],{"className":107180,"style":32123},[31845],[34,107182,105852],{"className":107183,"style":37422},[31685,31686],[34,107185,34395],{"className":107186,"style":34452},[31685,31686],[34,107188,43383],{"className":107189},[31685,31686],[34,107191,46900],{"className":107192},[31685,31686],[34,107194,94628],{"className":107195},[31685,31686],[34,107197,31965],{"className":107198},[31685,31686],[34,107200,106819],{"className":107201,"style":33389},[31685,31686],[34,107203,105967],{"className":107204},[31685,31686],[34,107206,33244],{"className":107207},[31685,31686],[34,107209,31960],{"className":107210},[32119],[34,107212],{"className":107213,"style":32123},[31845],[34,107215,67731],{"className":107216,"style":32604},[31685,31686],[34,107218,33244],{"className":107219},[31685,31686],[34,107221,111],{"className":107222},[31685,31686],[34,107224,34395],{"className":107225,"style":34452},[31685,31686],[34,107227,31829],{"className":107228},[31828],[34,107230,107232],{"className":107231},[31801],[34,107233,107236],{"className":107234,"style":107235},[31805],"height:0.3944em;",[34,107237],{},[34,107239,31960],{"className":107240},[32119],[34,107242],{"className":107243,"style":32123},[31845],[34,107245,98360],{"className":107246,"style":33389},[31685,31686],[34,107248,111],{"className":107249},[31685,31686],[34,107251,67764],{"className":107252,"style":32062},[31685,31686],[34,107254,106001],{"className":107255,"style":34452},[31685,31686],[34,107257,105852],{"className":107258,"style":37422},[31685,31686],[34,107260,47292],{"className":107261},[31685,31686],[34,107263,43383],{"className":107264},[31685,31686],[34,107266,31660],{"className":107267},[31685,31686],[34,107269,33244],{"className":107270},[31685,31686],[34,107272,31960],{"className":107273},[32119],[34,107275],{"className":107276,"style":32123},[31845],[34,107278,37400],{"className":107279,"style":37422},[31685,31686],[34,107281,43383],{"className":107282},[31685,31686],[34,107284,33249],{"className":107285},[31685,31686],[34,107287,111],{"className":107288},[31685,31686],[34,107290,43379],{"className":107291,"style":47113},[31685,31686],[34,107293,47384],{"className":107294,"style":38706},[31685,31686],[34,107296,111],{"className":107297},[31685,31686],[34,107299,15],{"className":107300},[31685,31686],[34,107302,111],{"className":107303},[31685,31686],[34,107305,31965],{"className":107306},[31685,31686],[34,107308,31955],{"className":107309},[31685,31686],[34,107311,33249],{"className":107312},[31685,31686],[34,107314,67764],{"className":107315,"style":32062},[31685,31686],[34,107317,31960],{"className":107318},[32119],[34,107320],{"className":107321,"style":32123},[31845],[34,107323,54151],{"className":107324},[31685,31686],[34,107326,56782],{"className":107327,"style":32062},[31685,31686],[34,107329,31942],{"className":107330,"style":32062},[31685,31686],[34,107332,105852],{"className":107333,"style":37422},[31685,31686],[34,107335,34395],{"className":107336,"style":34452},[31685,31686],[34,107338,31955],{"className":107339},[31685,31686],[34,107341,106860],{"className":107342},[31685,31686],[34,107344,2557],{"className":107345},[31690],[71807,107347,107348,107349,107419],{},"> 注：为了符合题目后续“规范化”的要求，我们在 ",[34,107350,107352,107370],{"className":107351},[31642],[34,107353,107355],{"className":107354},[31646],[31648,107356,107357],{"xmlns":31650},[31652,107358,107359,107367],{},[31655,107360,107361],{},[31743,107362,107363,107365],{},[31658,107364,34386],{},[31749,107366,18136],{},[31665,107368,107369],{"encoding":31667},"R_5",[34,107371,107373],{"className":107372,"ariaHidden":758},[31672],[34,107374,107376,107379],{"className":107375},[31676],[34,107377],{"className":107378,"style":70915},[31680],[34,107380,107382,107385],{"className":107381},[31685],[34,107383,34386],{"className":107384,"style":34427},[31685,31686],[34,107386,107388],{"className":107387},[31792],[34,107389,107391,107411],{"className":107390},[31796,31797],[34,107392,107394,107408],{"className":107393},[31801],[34,107395,107397],{"className":107396,"style":31806},[31805],[34,107398,107399,107402],{"style":104694},[34,107400],{"className":107401,"style":31814},[31813],[34,107403,107405],{"className":107404},[31818,31819,31820,31821],[34,107406,18136],{"className":107407},[31685,31821],[34,107409,31829],{"className":107410},[31828],[34,107412,107414],{"className":107413},[31801],[34,107415,107417],{"className":107416,"style":31836},[31805],[34,107418],{}," 中故意引入了冗余。",[32188,107421],{},[253,107423,107425,107426,2557],{"id":107424},"_3-函数依赖关系分析-fff","3. 函数依赖关系分析 (",[34,107427,107429,107442],{"className":107428},[31642],[34,107430,107432],{"className":107431},[31646],[31648,107433,107434],{"xmlns":31650},[31652,107435,107436,107440],{},[31655,107437,107438],{},[31658,107439,51834],{},[31665,107441,51834],{"encoding":31667},[34,107443,107445],{"className":107444,"ariaHidden":758},[31672],[34,107446,107448,107451],{"className":107447},[31676],[34,107449],{"className":107450,"style":33218},[31680],[34,107452,51834],{"className":107453,"style":37422},[31685,31686],[15,107455,107456,107457,107485],{},"针对上述关系，其函数依赖集 ",[34,107458,107460,107473],{"className":107459},[31642],[34,107461,107463],{"className":107462},[31646],[31648,107464,107465],{"xmlns":31650},[31652,107466,107467,107471],{},[31655,107468,107469],{},[31658,107470,51834],{},[31665,107472,51834],{"encoding":31667},[34,107474,107476],{"className":107475,"ariaHidden":758},[31672],[34,107477,107479,107482],{"className":107478},[31676],[34,107480],{"className":107481,"style":33218},[31680],[34,107483,51834],{"className":107484,"style":37422},[31685,31686]," 如下：",[105,107487,107488,107685,107927,108155,108503],{},[108,107489,59432,107490,2664,107560],{},[34,107491,107493,107511],{"className":107492},[31642],[34,107494,107496],{"className":107495},[31646],[31648,107497,107498],{"xmlns":31650},[31652,107499,107500,107508],{},[31655,107501,107502],{},[31743,107503,107504,107506],{},[31658,107505,34386],{},[31749,107507,640],{},[31665,107509,107510],{"encoding":31667},"R_1",[34,107512,107514],{"className":107513,"ariaHidden":758},[31672],[34,107515,107517,107520],{"className":107516},[31676],[34,107518],{"className":107519,"style":70915},[31680],[34,107521,107523,107526],{"className":107522},[31685],[34,107524,34386],{"className":107525,"style":34427},[31685,31686],[34,107527,107529],{"className":107528},[31792],[34,107530,107532,107552],{"className":107531},[31796,31797],[34,107533,107535,107549],{"className":107534},[31801],[34,107536,107538],{"className":107537,"style":31806},[31805],[34,107539,107540,107543],{"style":104694},[34,107541],{"className":107542,"style":31814},[31813],[34,107544,107546],{"className":107545},[31818,31819,31820,31821],[34,107547,640],{"className":107548},[31685,31821],[34,107550,31829],{"className":107551},[31828],[34,107553,107555],{"className":107554},[31801],[34,107556,107558],{"className":107557,"style":31836},[31805],[34,107559],{},[34,107561,107563,107613],{"className":107562},[31642],[34,107564,107566],{"className":107565},[31646],[31648,107567,107568],{"xmlns":31650},[31652,107569,107570,107610],{},[31655,107571,107572,107574,107576,107578,107580,107582,107584,107586,107588,107590,107592,107594,107596,107598,107600,107602,107604,107606,107608],{},[31658,107573,37400],{},[31658,107575,43383],{},[31658,107577,47077],{},[31658,107579,31660],{},[31658,107581,33347],{},[31658,107583,111],{},[31658,107585,31759],{},[31658,107587,33244],{},[31662,107589,33239],{},[31658,107591,98360],{},[31658,107593,111],{},[31658,107595,67764],{},[31658,107597,43383],{},[31658,107599,34395],{},[31658,107601,105852],{},[31658,107603,47292],{},[31658,107605,43383],{},[31658,107607,31660],{},[31658,107609,33244],{},[31665,107611,107612],{"encoding":31667},"TownName \\rightarrow MayorPhone",[34,107614,107616,107652],{"className":107615,"ariaHidden":758},[31672],[34,107617,107619,107622,107625,107628,107631,107634,107637,107640,107643,107646,107649],{"className":107618},[31676],[34,107620],{"className":107621,"style":33218},[31680],[34,107623,37400],{"className":107624,"style":37422},[31685,31686],[34,107626,43383],{"className":107627},[31685,31686],[34,107629,47077],{"className":107630,"style":47120},[31685,31686],[34,107632,31660],{"className":107633},[31685,31686],[34,107635,33347],{"className":107636,"style":33389},[31685,31686],[34,107638,105967],{"className":107639},[31685,31686],[34,107641,33244],{"className":107642},[31685,31686],[34,107644],{"className":107645,"style":31846},[31845],[34,107647,33239],{"className":107648},[31850],[34,107650],{"className":107651,"style":31846},[31845],[34,107653,107655,107658,107661,107664,107667,107670,107673,107676,107679,107682],{"className":107654},[31676],[34,107656],{"className":107657,"style":43217},[31680],[34,107659,98360],{"className":107660,"style":33389},[31685,31686],[34,107662,111],{"className":107663},[31685,31686],[34,107665,67764],{"className":107666,"style":32062},[31685,31686],[34,107668,106001],{"className":107669,"style":34452},[31685,31686],[34,107671,105852],{"className":107672,"style":37422},[31685,31686],[34,107674,47292],{"className":107675},[31685,31686],[34,107677,43383],{"className":107678},[31685,31686],[34,107680,31660],{"className":107681},[31685,31686],[34,107683,33244],{"className":107684},[31685,31686],[108,107686,59432,107687,2664,107757],{},[34,107688,107690,107708],{"className":107689},[31642],[34,107691,107693],{"className":107692},[31646],[31648,107694,107695],{"xmlns":31650},[31652,107696,107697,107705],{},[31655,107698,107699],{},[31743,107700,107701,107703],{},[31658,107702,34386],{},[31749,107704,768],{},[31665,107706,107707],{"encoding":31667},"R_2",[34,107709,107711],{"className":107710,"ariaHidden":758},[31672],[34,107712,107714,107717],{"className":107713},[31676],[34,107715],{"className":107716,"style":70915},[31680],[34,107718,107720,107723],{"className":107719},[31685],[34,107721,34386],{"className":107722,"style":34427},[31685,31686],[34,107724,107726],{"className":107725},[31792],[34,107727,107729,107749],{"className":107728},[31796,31797],[34,107730,107732,107746],{"className":107731},[31801],[34,107733,107735],{"className":107734,"style":31806},[31805],[34,107736,107737,107740],{"style":104694},[34,107738],{"className":107739,"style":31814},[31813],[34,107741,107743],{"className":107742},[31818,31819,31820,31821],[34,107744,768],{"className":107745},[31685,31821],[34,107747,31829],{"className":107748},[31828],[34,107750,107752],{"className":107751},[31801],[34,107753,107755],{"className":107754,"style":31836},[31805],[34,107756],{},[34,107758,107760,107828],{"className":107759},[31642],[34,107761,107763],{"className":107762},[31646],[31648,107764,107765],{"xmlns":31650},[31652,107766,107767,107825],{},[31655,107768,107769,107771,107773,107775,107777,107779,107781,107783,107785,107787,107789,107791,107793,107795,107797,107799,107801,107803,107805,107807,107809,107811,107813,107815,107817,107819,107821,107823],{},[31658,107770,51834],{},[31658,107772,111],{},[31658,107774,34395],{},[31658,107776,31759],{},[31658,107778,33244],{},[31658,107780,34395],{},[31658,107782,106063],{},[31658,107784,91994],{},[31662,107786,33239],{},[31658,107788,51834],{},[31658,107790,111],{},[31658,107792,34395],{},[31658,107794,31759],{},[31658,107796,33244],{},[31658,107798,34395],{},[31658,107800,106063],{},[31658,107802,31660],{},[31658,107804,31739],{},[31658,107806,43383],{},[31662,107808,31960],{"separator":758},[31658,107810,37400],{},[31658,107812,43383],{},[31658,107814,47077],{},[31658,107816,31660],{},[31658,107818,33347],{},[31658,107820,111],{},[31658,107822,31759],{},[31658,107824,33244],{},[31665,107826,107827],{"encoding":31667},"FarmerID \\rightarrow FarmerInfo, TownName",[34,107829,107831,107867],{"className":107830,"ariaHidden":758},[31672],[34,107832,107834,107837,107840,107843,107846,107849,107852,107855,107858,107861,107864],{"className":107833},[31676],[34,107835],{"className":107836,"style":33218},[31680],[34,107838,51834],{"className":107839,"style":37422},[31685,31686],[34,107841,111],{"className":107842},[31685,31686],[34,107844,34395],{"className":107845,"style":34452},[31685,31686],[34,107847,31759],{"className":107848},[31685,31686],[34,107850,34599],{"className":107851,"style":34452},[31685,31686],[34,107853,106063],{"className":107854,"style":67482},[31685,31686],[34,107856,91994],{"className":107857,"style":34452},[31685,31686],[34,107859],{"className":107860,"style":31846},[31845],[34,107862,33239],{"className":107863},[31850],[34,107865],{"className":107866,"style":31846},[31845],[34,107868,107870,107873,107876,107879,107882,107885,107888,107891,107894,107897,107900,107903,107906,107909,107912,107915,107918,107921,107924],{"className":107869},[31676],[34,107871],{"className":107872,"style":43217},[31680],[34,107874,51834],{"className":107875,"style":37422},[31685,31686],[34,107877,111],{"className":107878},[31685,31686],[34,107880,34395],{"className":107881,"style":34452},[31685,31686],[34,107883,31759],{"className":107884},[31685,31686],[34,107886,34599],{"className":107887,"style":34452},[31685,31686],[34,107889,106063],{"className":107890,"style":67482},[31685,31686],[34,107892,31660],{"className":107893},[31685,31686],[34,107895,31739],{"className":107896,"style":31778},[31685,31686],[34,107898,43383],{"className":107899},[31685,31686],[34,107901,31960],{"className":107902},[32119],[34,107904],{"className":107905,"style":32123},[31845],[34,107907,37400],{"className":107908,"style":37422},[31685,31686],[34,107910,43383],{"className":107911},[31685,31686],[34,107913,47077],{"className":107914,"style":47120},[31685,31686],[34,107916,31660],{"className":107917},[31685,31686],[34,107919,33347],{"className":107920,"style":33389},[31685,31686],[34,107922,105967],{"className":107923},[31685,31686],[34,107925,33244],{"className":107926},[31685,31686],[108,107928,59432,107929,2664,107999],{},[34,107930,107932,107950],{"className":107931},[31642],[34,107933,107935],{"className":107934},[31646],[31648,107936,107937],{"xmlns":31650},[31652,107938,107939,107947],{},[31655,107940,107941],{},[31743,107942,107943,107945],{},[31658,107944,34386],{},[31749,107946,786],{},[31665,107948,107949],{"encoding":31667},"R_3",[34,107951,107953],{"className":107952,"ariaHidden":758},[31672],[34,107954,107956,107959],{"className":107955},[31676],[34,107957],{"className":107958,"style":70915},[31680],[34,107960,107962,107965],{"className":107961},[31685],[34,107963,34386],{"className":107964,"style":34427},[31685,31686],[34,107966,107968],{"className":107967},[31792],[34,107969,107971,107991],{"className":107970},[31796,31797],[34,107972,107974,107988],{"className":107973},[31801],[34,107975,107977],{"className":107976,"style":31806},[31805],[34,107978,107979,107982],{"style":104694},[34,107980],{"className":107981,"style":31814},[31813],[34,107983,107985],{"className":107984},[31818,31819,31820,31821],[34,107986,786],{"className":107987},[31685,31821],[34,107989,31829],{"className":107990},[31828],[34,107992,107994],{"className":107993},[31801],[34,107995,107997],{"className":107996,"style":31836},[31805],[34,107998],{},[34,108000,108002,108062],{"className":108001},[31642],[34,108003,108005],{"className":108004},[31646],[31648,108006,108007],{"xmlns":31650},[31652,108008,108009,108059],{},[31655,108010,108011,108013,108015,108017,108019,108021,108023,108025,108027,108029,108031,108033,108035,108037,108039,108041,108043,108045,108047,108049,108051,108053,108055,108057],{},[31658,108012,94675],{},[31658,108014,31955],{},[31658,108016,33249],{},[31658,108018,33244],{},[31658,108020,106063],{},[31658,108022,91994],{},[31662,108024,33239],{},[31658,108026,94675],{},[31658,108028,31955],{},[31658,108030,33249],{},[31658,108032,33244],{},[31658,108034,106063],{},[31658,108036,31660],{},[31658,108038,31739],{},[31658,108040,43383],{},[31662,108042,31960],{"separator":758},[31658,108044,51834],{},[31658,108046,111],{},[31658,108048,34395],{},[31658,108050,31759],{},[31658,108052,33244],{},[31658,108054,34395],{},[31658,108056,106063],{},[31658,108058,91994],{},[31665,108060,108061],{"encoding":31667},"SiteID \\rightarrow SiteInfo, FarmerID",[34,108063,108065,108098],{"className":108064,"ariaHidden":758},[31672],[34,108066,108068,108071,108074,108077,108080,108083,108086,108089,108092,108095],{"className":108067},[31676],[34,108069],{"className":108070,"style":33218},[31680],[34,108072,94675],{"className":108073,"style":106455},[31685,31686],[34,108075,31955],{"className":108076},[31685,31686],[34,108078,33249],{"className":108079},[31685,31686],[34,108081,33244],{"className":108082},[31685,31686],[34,108084,106063],{"className":108085,"style":67482},[31685,31686],[34,108087,91994],{"className":108088,"style":34452},[31685,31686],[34,108090],{"className":108091,"style":31846},[31845],[34,108093,33239],{"className":108094},[31850],[34,108096],{"className":108097,"style":31846},[31845],[34,108099,108101,108104,108107,108110,108113,108116,108119,108122,108125,108128,108131,108134,108137,108140,108143,108146,108149,108152],{"className":108100},[31676],[34,108102],{"className":108103,"style":43217},[31680],[34,108105,94675],{"className":108106,"style":106455},[31685,31686],[34,108108,31955],{"className":108109},[31685,31686],[34,108111,33249],{"className":108112},[31685,31686],[34,108114,33244],{"className":108115},[31685,31686],[34,108117,106063],{"className":108118,"style":67482},[31685,31686],[34,108120,31660],{"className":108121},[31685,31686],[34,108123,31739],{"className":108124,"style":31778},[31685,31686],[34,108126,43383],{"className":108127},[31685,31686],[34,108129,31960],{"className":108130},[32119],[34,108132],{"className":108133,"style":32123},[31845],[34,108135,51834],{"className":108136,"style":37422},[31685,31686],[34,108138,111],{"className":108139},[31685,31686],[34,108141,34395],{"className":108142,"style":34452},[31685,31686],[34,108144,31759],{"className":108145},[31685,31686],[34,108147,34599],{"className":108148,"style":34452},[31685,31686],[34,108150,106063],{"className":108151,"style":67482},[31685,31686],[34,108153,91994],{"className":108154,"style":34452},[31685,31686],[108,108156,59432,108157,2664,108227],{},[34,108158,108160,108178],{"className":108159},[31642],[34,108161,108163],{"className":108162},[31646],[31648,108164,108165],{"xmlns":31650},[31652,108166,108167,108175],{},[31655,108168,108169],{},[31743,108170,108171,108173],{},[31658,108172,34386],{},[31749,108174,15932],{},[31665,108176,108177],{"encoding":31667},"R_4",[34,108179,108181],{"className":108180,"ariaHidden":758},[31672],[34,108182,108184,108187],{"className":108183},[31676],[34,108185],{"className":108186,"style":70915},[31680],[34,108188,108190,108193],{"className":108189},[31685],[34,108191,34386],{"className":108192,"style":34427},[31685,31686],[34,108194,108196],{"className":108195},[31792],[34,108197,108199,108219],{"className":108198},[31796,31797],[34,108200,108202,108216],{"className":108201},[31801],[34,108203,108205],{"className":108204,"style":31806},[31805],[34,108206,108207,108210],{"style":104694},[34,108208],{"className":108209,"style":31814},[31813],[34,108211,108213],{"className":108212},[31818,31819,31820,31821],[34,108214,15932],{"className":108215},[31685,31821],[34,108217,31829],{"className":108218},[31828],[34,108220,108222],{"className":108221},[31801],[34,108223,108225],{"className":108224,"style":31836},[31805],[34,108226],{},[34,108228,108230,108338],{"className":108229},[31642],[34,108231,108233],{"className":108232},[31646],[31648,108234,108235],{"xmlns":31650},[31652,108236,108237,108335],{},[31655,108238,108239,108241,108243,108245,108247,108249,108251,108253,108255,108257,108259,108261,108263,108265,108267,108269,108271,108273,108275,108277,108279,108281,108283,108285,108287,108289,108291,108293,108295,108297,108299,108301,108303,108305,108307,108309,108311,108313,108315,108317,108319,108321,108323,108325,108327,108329,108331,108333],{},[31658,108240,105852],{},[31658,108242,34395],{},[31658,108244,43383],{},[31658,108246,46900],{},[31658,108248,94628],{},[31658,108250,31965],{},[31658,108252,33249],{},[31658,108254,106063],{},[31658,108256,91994],{},[31662,108258,33239],{},[31658,108260,105852],{},[31658,108262,34395],{},[31658,108264,43383],{},[31658,108266,46900],{},[31658,108268,94628],{},[31658,108270,31965],{},[31658,108272,33249],{},[31658,108274,33347],{},[31658,108276,111],{},[31658,108278,31759],{},[31658,108280,33244],{},[31662,108282,31960],{"separator":758},[31658,108284,94675],{},[31658,108286,15],{},[31658,108288,33244],{},[31658,108290,31965],{},[31662,108292,31960],{"separator":758},[31658,108294,105852],{},[31658,108296,34395],{},[31658,108298,31955],{},[31658,108300,31965],{},[31658,108302,33244],{},[31662,108304,31960],{"separator":758},[31658,108306,47384],{},[31658,108308,111],{},[31658,108310,15],{},[31658,108312,111],{},[31658,108314,31965],{},[31658,108316,31955],{},[31658,108318,33249],{},[31658,108320,67764],{},[31662,108322,31960],{"separator":758},[31658,108324,94675],{},[31658,108326,31955],{},[31658,108328,33249],{},[31658,108330,33244],{},[31658,108332,106063],{},[31658,108334,91994],{},[31665,108336,108337],{"encoding":31667},"ProductID \\rightarrow ProductName, Spec, Price, Capacity, SiteID",[34,108339,108341,108383],{"className":108340,"ariaHidden":758},[31672],[34,108342,108344,108347,108350,108353,108356,108359,108362,108365,108368,108371,108374,108377,108380],{"className":108343},[31676],[34,108345],{"className":108346,"style":31681},[31680],[34,108348,105852],{"className":108349,"style":37422},[31685,31686],[34,108351,34395],{"className":108352,"style":34452},[31685,31686],[34,108354,43383],{"className":108355},[31685,31686],[34,108357,46900],{"className":108358},[31685,31686],[34,108360,94628],{"className":108361},[31685,31686],[34,108363,31965],{"className":108364},[31685,31686],[34,108366,33249],{"className":108367},[31685,31686],[34,108369,106063],{"className":108370,"style":67482},[31685,31686],[34,108372,91994],{"className":108373,"style":34452},[31685,31686],[34,108375],{"className":108376,"style":31846},[31845],[34,108378,33239],{"className":108379},[31850],[34,108381],{"className":108382,"style":31846},[31845],[34,108384,108386,108389,108392,108395,108398,108401,108404,108407,108410,108413,108416,108419,108422,108425,108428,108431,108434,108437,108440,108443,108446,108449,108452,108455,108458,108461,108464,108467,108470,108473,108476,108479,108482,108485,108488,108491,108494,108497,108500],{"className":108385},[31676],[34,108387],{"className":108388,"style":43217},[31680],[34,108390,105852],{"className":108391,"style":37422},[31685,31686],[34,108393,34395],{"className":108394,"style":34452},[31685,31686],[34,108396,43383],{"className":108397},[31685,31686],[34,108399,46900],{"className":108400},[31685,31686],[34,108402,94628],{"className":108403},[31685,31686],[34,108405,31965],{"className":108406},[31685,31686],[34,108408,106819],{"className":108409,"style":33389},[31685,31686],[34,108411,105967],{"className":108412},[31685,31686],[34,108414,33244],{"className":108415},[31685,31686],[34,108417,31960],{"className":108418},[32119],[34,108420],{"className":108421,"style":32123},[31845],[34,108423,94675],{"className":108424,"style":106455},[31685,31686],[34,108426,15],{"className":108427},[31685,31686],[34,108429,106841],{"className":108430},[31685,31686],[34,108432,31960],{"className":108433},[32119],[34,108435],{"className":108436,"style":32123},[31845],[34,108438,105852],{"className":108439,"style":37422},[31685,31686],[34,108441,34395],{"className":108442,"style":34452},[31685,31686],[34,108444,31955],{"className":108445},[31685,31686],[34,108447,106860],{"className":108448},[31685,31686],[34,108450,31960],{"className":108451},[32119],[34,108453],{"className":108454,"style":32123},[31845],[34,108456,47384],{"className":108457,"style":38706},[31685,31686],[34,108459,111],{"className":108460},[31685,31686],[34,108462,15],{"className":108463},[31685,31686],[34,108465,111],{"className":108466},[31685,31686],[34,108468,31965],{"className":108469},[31685,31686],[34,108471,31955],{"className":108472},[31685,31686],[34,108474,33249],{"className":108475},[31685,31686],[34,108477,67764],{"className":108478,"style":32062},[31685,31686],[34,108480,31960],{"className":108481},[32119],[34,108483],{"className":108484,"style":32123},[31845],[34,108486,94675],{"className":108487,"style":106455},[31685,31686],[34,108489,31955],{"className":108490},[31685,31686],[34,108492,33249],{"className":108493},[31685,31686],[34,108495,33244],{"className":108496},[31685,31686],[34,108498,106063],{"className":108499,"style":67482},[31685,31686],[34,108501,91994],{"className":108502,"style":34452},[31685,31686],[108,108504,59432,108505,33183,108574],{},[34,108506,108508,108525],{"className":108507},[31642],[34,108509,108511],{"className":108510},[31646],[31648,108512,108513],{"xmlns":31650},[31652,108514,108515,108523],{},[31655,108516,108517],{},[31743,108518,108519,108521],{},[31658,108520,34386],{},[31749,108522,18136],{},[31665,108524,107369],{"encoding":31667},[34,108526,108528],{"className":108527,"ariaHidden":758},[31672],[34,108529,108531,108534],{"className":108530},[31676],[34,108532],{"className":108533,"style":70915},[31680],[34,108535,108537,108540],{"className":108536},[31685],[34,108538,34386],{"className":108539,"style":34427},[31685,31686],[34,108541,108543],{"className":108542},[31792],[34,108544,108546,108566],{"className":108545},[31796,31797],[34,108547,108549,108563],{"className":108548},[31801],[34,108550,108552],{"className":108551,"style":31806},[31805],[34,108553,108554,108557],{"style":104694},[34,108555],{"className":108556,"style":31814},[31813],[34,108558,108560],{"className":108559},[31818,31819,31820,31821],[34,108561,18136],{"className":108562},[31685,31821],[34,108564,31829],{"className":108565},[31828],[34,108567,108569],{"className":108568},[31801],[34,108570,108572],{"className":108571,"style":31836},[31805],[34,108573],{},[2230,108575,108576,108861],{},[108,108577,108578],{},[34,108579,108581,108693],{"className":108580},[31642],[34,108582,108584],{"className":108583},[31646],[31648,108585,108586],{"xmlns":31650},[31652,108587,108588,108690],{},[31655,108589,108590,108592,108594,108596,108598,108600,108602,108604,108606,108608,108610,108612,108614,108616,108618,108620,108622,108624,108626,108628,108630,108632,108634,108636,108638,108640,108642,108644,108646,108648,108650,108652,108654,108656,108658,108660,108662,108664,108666,108668,108670,108672,108674,108676,108678,108680,108682,108684,108686,108688],{},[31662,108591,456],{"stretchy":702},[31658,108593,37400],{},[31658,108595,43383],{},[31658,108597,47077],{},[31658,108599,31660],{},[31658,108601,33347],{},[31658,108603,111],{},[31658,108605,31759],{},[31658,108607,33244],{},[31662,108609,31960],{"separator":758},[31658,108611,105852],{},[31658,108613,34395],{},[31658,108615,43383],{},[31658,108617,46900],{},[31658,108619,94628],{},[31658,108621,31965],{},[31658,108623,33249],{},[31658,108625,33347],{},[31658,108627,111],{},[31658,108629,31759],{},[31658,108631,33244],{},[31662,108633,31960],{"separator":758},[31658,108635,67731],{},[31658,108637,33244],{},[31658,108639,111],{},[31658,108641,34395],{},[31662,108643,2557],{"stretchy":702},[31662,108645,33239],{},[31658,108647,37400],{},[31658,108649,43383],{},[31658,108651,33249],{},[31658,108653,111],{},[31658,108655,43379],{},[31658,108657,47384],{},[31658,108659,111],{},[31658,108661,15],{},[31658,108663,111],{},[31658,108665,31965],{},[31658,108667,31955],{},[31658,108669,33249],{},[31658,108671,67764],{},[31662,108673,31960],{"separator":758},[31658,108675,54151],{},[31658,108677,56782],{},[31658,108679,31942],{},[31658,108681,105852],{},[31658,108683,34395],{},[31658,108685,31955],{},[31658,108687,31965],{},[31658,108689,33244],{},[31665,108691,108692],{"encoding":31667},"(TownName, ProductName, Year) \\rightarrow TotalCapacity, AvgPrice",[34,108694,108696,108789],{"className":108695,"ariaHidden":758},[31672],[34,108697,108699,108702,108705,108708,108711,108714,108717,108720,108723,108726,108729,108732,108735,108738,108741,108744,108747,108750,108753,108756,108759,108762,108765,108768,108771,108774,108777,108780,108783,108786],{"className":108698},[31676],[34,108700],{"className":108701,"style":31774},[31680],[34,108703,456],{"className":108704},[31782],[34,108706,37400],{"className":108707,"style":37422},[31685,31686],[34,108709,43383],{"className":108710},[31685,31686],[34,108712,47077],{"className":108713,"style":47120},[31685,31686],[34,108715,31660],{"className":108716},[31685,31686],[34,108718,33347],{"className":108719,"style":33389},[31685,31686],[34,108721,105967],{"className":108722},[31685,31686],[34,108724,33244],{"className":108725},[31685,31686],[34,108727,31960],{"className":108728},[32119],[34,108730],{"className":108731,"style":32123},[31845],[34,108733,105852],{"className":108734,"style":37422},[31685,31686],[34,108736,34395],{"className":108737,"style":34452},[31685,31686],[34,108739,43383],{"className":108740},[31685,31686],[34,108742,46900],{"className":108743},[31685,31686],[34,108745,94628],{"className":108746},[31685,31686],[34,108748,31965],{"className":108749},[31685,31686],[34,108751,106819],{"className":108752,"style":33389},[31685,31686],[34,108754,105967],{"className":108755},[31685,31686],[34,108757,33244],{"className":108758},[31685,31686],[34,108760,31960],{"className":108761},[32119],[34,108763],{"className":108764,"style":32123},[31845],[34,108766,67731],{"className":108767,"style":32604},[31685,31686],[34,108769,33244],{"className":108770},[31685,31686],[34,108772,111],{"className":108773},[31685,31686],[34,108775,34395],{"className":108776,"style":34452},[31685,31686],[34,108778,2557],{"className":108779},[31690],[34,108781],{"className":108782,"style":31846},[31845],[34,108784,33239],{"className":108785},[31850],[34,108787],{"className":108788,"style":31846},[31845],[34,108790,108792,108795,108798,108801,108804,108807,108810,108813,108816,108819,108822,108825,108828,108831,108834,108837,108840,108843,108846,108849,108852,108855,108858],{"className":108791},[31676],[34,108793],{"className":108794,"style":43217},[31680],[34,108796,37400],{"className":108797,"style":37422},[31685,31686],[34,108799,43383],{"className":108800},[31685,31686],[34,108802,33249],{"className":108803},[31685,31686],[34,108805,111],{"className":108806},[31685,31686],[34,108808,43379],{"className":108809,"style":47113},[31685,31686],[34,108811,47384],{"className":108812,"style":38706},[31685,31686],[34,108814,111],{"className":108815},[31685,31686],[34,108817,15],{"className":108818},[31685,31686],[34,108820,111],{"className":108821},[31685,31686],[34,108823,31965],{"className":108824},[31685,31686],[34,108826,31955],{"className":108827},[31685,31686],[34,108829,33249],{"className":108830},[31685,31686],[34,108832,67764],{"className":108833,"style":32062},[31685,31686],[34,108835,31960],{"className":108836},[32119],[34,108838],{"className":108839,"style":32123},[31845],[34,108841,54151],{"className":108842},[31685,31686],[34,108844,56782],{"className":108845,"style":32062},[31685,31686],[34,108847,31942],{"className":108848,"style":32062},[31685,31686],[34,108850,105852],{"className":108851,"style":37422},[31685,31686],[34,108853,34395],{"className":108854,"style":34452},[31685,31686],[34,108856,31955],{"className":108857},[31685,31686],[34,108859,106860],{"className":108860},[31685,31686],[108,108862,108863,108987,108988,82346],{},[34,108864,108866,108915],{"className":108865},[31642],[34,108867,108869],{"className":108868},[31646],[31648,108870,108871],{"xmlns":31650},[31652,108872,108873,108913],{},[31655,108874,108875,108877,108879,108881,108883,108885,108887,108889,108891,108893,108895,108897,108899,108901,108903,108905,108907,108909,108911],{},[31658,108876,37400],{},[31658,108878,43383],{},[31658,108880,47077],{},[31658,108882,31660],{},[31658,108884,33347],{},[31658,108886,111],{},[31658,108888,31759],{},[31658,108890,33244],{},[31662,108892,33239],{},[31658,108894,98360],{},[31658,108896,111],{},[31658,108898,67764],{},[31658,108900,43383],{},[31658,108902,34395],{},[31658,108904,105852],{},[31658,108906,47292],{},[31658,108908,43383],{},[31658,108910,31660],{},[31658,108912,33244],{},[31665,108914,107612],{"encoding":31667},[34,108916,108918,108954],{"className":108917,"ariaHidden":758},[31672],[34,108919,108921,108924,108927,108930,108933,108936,108939,108942,108945,108948,108951],{"className":108920},[31676],[34,108922],{"className":108923,"style":33218},[31680],[34,108925,37400],{"className":108926,"style":37422},[31685,31686],[34,108928,43383],{"className":108929},[31685,31686],[34,108931,47077],{"className":108932,"style":47120},[31685,31686],[34,108934,31660],{"className":108935},[31685,31686],[34,108937,33347],{"className":108938,"style":33389},[31685,31686],[34,108940,105967],{"className":108941},[31685,31686],[34,108943,33244],{"className":108944},[31685,31686],[34,108946],{"className":108947,"style":31846},[31845],[34,108949,33239],{"className":108950},[31850],[34,108952],{"className":108953,"style":31846},[31845],[34,108955,108957,108960,108963,108966,108969,108972,108975,108978,108981,108984],{"className":108956},[31676],[34,108958],{"className":108959,"style":43217},[31680],[34,108961,98360],{"className":108962,"style":33389},[31685,31686],[34,108964,111],{"className":108965},[31685,31686],[34,108967,67764],{"className":108968,"style":32062},[31685,31686],[34,108970,106001],{"className":108971,"style":34452},[31685,31686],[34,108973,105852],{"className":108974,"style":37422},[31685,31686],[34,108976,47292],{"className":108977},[31685,31686],[34,108979,43383],{"className":108980},[31685,31686],[34,108982,31660],{"className":108983},[31685,31686],[34,108985,33244],{"className":108986},[31685,31686]," （",[2627,108989,108990],{},"关键：部分函数依赖",[32188,108992],{},[253,108994,108996],{"id":108995},"_4-主码与外码标识","4. 主码与外码标识",[13837,108998,108999,109012],{},[13840,109000,109001],{},[13843,109002,109003,109006,109009],{},[13846,109004,109005],{"align":82355},"关系模式",[13846,109007,109008],{"align":82355},"主码 (Primary Key)",[13846,109010,109011],{"align":82355},"外码 (Foreign 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ProductName",[34,110035,110037],{"className":110036,"ariaHidden":758},[31672],[34,110038,110040,110043,110046,110049,110052,110055,110058,110061,110064,110067,110070,110073,110076,110079,110082,110085,110088,110091,110094],{"className":110039},[31676],[34,110041],{"className":110042,"style":43217},[31680],[34,110044,37400],{"className":110045,"style":37422},[31685,31686],[34,110047,43383],{"className":110048},[31685,31686],[34,110050,47077],{"className":110051,"style":47120},[31685,31686],[34,110053,31660],{"className":110054},[31685,31686],[34,110056,33347],{"className":110057,"style":33389},[31685,31686],[34,110059,105967],{"className":110060},[31685,31686],[34,110062,33244],{"className":110063},[31685,31686],[34,110065,31960],{"className":110066},[32119],[34,110068],{"className":110069,"style":32123},[31845],[34,110071,105852],{"className":110072,"style":37422},[31685,31686],[34,110074,34395],{"className":110075,"style":34452},[31685,31686],[34,110077,43383],{"className":110078},[31685,31686],[34,110080,46900],{"className":110081},[31685,31686],[34,110083,94628],{"className":110084},[31685,31686],[34,110086,31965],{"className":110087},[31685,31686],[34,110089,106819],{"className":110090,"style":33389},[31685,31686],[34,110092,105967],{"className":110093},[31685,31686],[34,110095,33244],{"className":110096},[31685,31686],[32188,110098],{},[253,110100,110102],{"id":110101},"_5-范式级别判定与不合理性说明","5. 范式级别判定与不合理性说明",[105,110104,110105,110345],{},[108,110106,110107,110341,110342,110344],{},[2627,110108,110109],{},[34,110110,110112,110154],{"className":110111},[31642],[34,110113,110115],{"className":110114},[31646],[31648,110116,110117],{"xmlns":31650},[31652,110118,110119,110151],{},[31655,110120,110121,110127,110129,110135,110137,110143,110145],{},[31743,110122,110123,110125],{},[31658,110124,34386],{},[31749,110126,640],{},[31662,110128,31960],{"separator":758},[31743,110130,110131,110133],{},[31658,110132,34386],{},[31749,110134,768],{},[31662,110136,31960],{"separator":758},[31743,110138,110139,110141],{},[31658,110140,34386],{},[31749,110142,786],{},[31662,110144,31960],{"separator":758},[31743,110146,110147,110149],{},[31658,110148,34386],{},[31749,110150,15932],{},[31665,110152,110153],{"encoding":31667},"R_1, R_2, R_3, R_4",[34,110155,110157],{"className":110156,"ariaHidden":758},[31672],[34,110158,110160,110163,110203,110206,110209,110249,110252,110255,110295,110298,110301],{"className":110159},[31676],[34,110161],{"className":110162,"style":70739},[31680],[34,110164,110166,110169],{"className":110165},[31685],[34,110167,34386],{"className":110168,"style":34427},[31685,31686],[34,110170,110172],{"className":110171},[31792],[34,110173,110175,110195],{"className":110174},[31796,31797],[34,110176,110178,110192],{"className":110177},[31801],[34,110179,110181],{"className":110180,"style":31806},[31805],[34,110182,110183,110186],{"style":104694},[34,110184],{"className":110185,"style":31814},[31813],[34,110187,110189],{"className":110188},[31818,31819,31820,31821],[34,110190,640],{"className":110191},[31685,31821],[34,110193,31829],{"className":110194},[31828],[34,110196,110198],{"className":110197},[31801],[34,110199,110201],{"className":110200,"style":31836},[31805],[34,110202],{},[34,110204,31960],{"className":110205},[32119],[34,110207],{"className":110208,"style":32123},[31845],[34,110210,110212,110215],{"className":110211},[31685],[34,110213,34386],{"className":110214,"style":34427},[31685,31686],[34,110216,110218],{"className":110217},[31792],[34,110219,110221,110241],{"className":110220},[31796,31797],[34,110222,110224,110238],{"className":110223},[31801],[34,110225,110227],{"className":110226,"style":31806},[31805],[34,110228,110229,110232],{"style":104694},[34,110230],{"className":110231,"style":31814},[31813],[34,110233,110235],{"className":110234},[31818,31819,31820,31821],[34,110236,768],{"className":110237},[31685,31821],[34,110239,31829],{"className":110240},[31828],[34,110242,110244],{"className":110243},[31801],[34,110245,110247],{"className":110246,"style":31836},[31805],[34,110248],{},[34,110250,31960],{"className":110251},[32119],[34,110253],{"className":110254,"style":32123},[31845],[34,110256,110258,110261],{"className":110257},[31685],[34,110259,34386],{"className":110260,"style":34427},[31685,31686],[34,110262,110264],{"className":110263},[31792],[34,110265,110267,110287],{"className":110266},[31796,31797],[34,110268,110270,110284],{"className":110269},[31801],[34,110271,110273],{"className":110272,"style":31806},[31805],[34,110274,110275,110278],{"style":104694},[34,110276],{"className":110277,"style":31814},[31813],[34,110279,110281],{"className":110280},[31818,31819,31820,31821],[34,110282,786],{"className":110283},[31685,31821],[34,110285,31829],{"className":110286},[31828],[34,110288,110290],{"className":110289},[31801],[34,110291,110293],{"className":110292,"style":31836},[31805],[34,110294],{},[34,110296,31960],{"className":110297},[32119],[34,110299],{"className":110300,"style":32123},[31845],[34,110302,110304,110307],{"className":110303},[31685],[34,110305,34386],{"className":110306,"style":34427},[31685,31686],[34,110308,110310],{"className":110309},[31792],[34,110311,110313,110333],{"className":110312},[31796,31797],[34,110314,110316,110330],{"className":110315},[31801],[34,110317,110319],{"className":110318,"style":31806},[31805],[34,110320,110321,110324],{"style":104694},[34,110322],{"className":110323,"style":31814},[31813],[34,110325,110327],{"className":110326},[31818,31819,31820,31821],[34,110328,15932],{"className":110329},[31685,31821],[34,110331,31829],{"className":110332},[31828],[34,110334,110336],{"className":110335},[31801],[34,110337,110339],{"className":110338,"style":31836},[31805],[34,110340],{},"：均满足 ",[2627,110343,105369],{},"。非主属性对码既不存在部分依赖，也不存在传递依赖。",[108,110346,110347,110419,110420,110423,110424,39984,110427],{},[2627,110348,110349,110418],{},[34,110350,110352,110369],{"className":110351},[31642],[34,110353,110355],{"className":110354},[31646],[31648,110356,110357],{"xmlns":31650},[31652,110358,110359,110367],{},[31655,110360,110361],{},[31743,110362,110363,110365],{},[31658,110364,34386],{},[31749,110366,18136],{},[31665,110368,107369],{"encoding":31667},[34,110370,110372],{"className":110371,"ariaHidden":758},[31672],[34,110373,110375,110378],{"className":110374},[31676],[34,110376],{"className":110377,"style":70915},[31680],[34,110379,110381,110384],{"className":110380},[31685],[34,110382,34386],{"className":110383,"style":34427},[31685,31686],[34,110385,110387],{"className":110386},[31792],[34,110388,110390,110410],{"className":110389},[31796,31797],[34,110391,110393,110407],{"className":110392},[31801],[34,110394,110396],{"className":110395,"style":31806},[31805],[34,110397,110398,110401],{"style":104694},[34,110399],{"className":110400,"style":31814},[31813],[34,110402,110404],{"className":110403},[31818,31819,31820,31821],[34,110405,18136],{"className":110406},[31685,31821],[34,110408,31829],{"className":110409},[31828],[34,110411,110413],{"className":110412},[31801],[34,110414,110416],{"className":110415,"style":31836},[31805],[34,110417],{}," (往年统计表)","：仅满足 ",[2627,110421,110422],{},"1NF","，不满足 ",[2627,110425,110426],{},"2NF",[105,110428,110429,110921],{},[108,110430,110431,2664,110434,110503,110504,110659,110660,110784,110785,110856,110857,110917,110918,347],{},[2627,110432,110433],{},"判定理由",[34,110435,110437,110454],{"className":110436},[31642],[34,110438,110440],{"className":110439},[31646],[31648,110441,110442],{"xmlns":31650},[31652,110443,110444,110452],{},[31655,110445,110446],{},[31743,110447,110448,110450],{},[31658,110449,34386],{},[31749,110451,18136],{},[31665,110453,107369],{"encoding":31667},[34,110455,110457],{"className":110456,"ariaHidden":758},[31672],[34,110458,110460,110463],{"className":110459},[31676],[34,110461],{"className":110462,"style":70915},[31680],[34,110464,110466,110469],{"className":110465},[31685],[34,110467,34386],{"className":110468,"style":34427},[31685,31686],[34,110470,110472],{"className":110471},[31792],[34,110473,110475,110495],{"className":110474},[31796,31797],[34,110476,110478,110492],{"className":110477},[31801],[34,110479,110481],{"className":110480,"style":31806},[31805],[34,110482,110483,110486],{"style":104694},[34,110484],{"className":110485,"style":31814},[31813],[34,110487,110489],{"className":110488},[31818,31819,31820,31821],[34,110490,18136],{"className":110491},[31685,31821],[34,110493,31829],{"className":110494},[31828],[34,110496,110498],{"className":110497},[31801],[34,110499,110501],{"className":110500,"style":31836},[31805],[34,110502],{}," 的码为 ",[34,110505,110507,110572],{"className":110506},[31642],[34,110508,110510],{"className":110509},[31646],[31648,110511,110512],{"xmlns":31650},[31652,110513,110514,110570],{},[31655,110515,110516,110518,110520,110522,110524,110526,110528,110530,110532,110534,110536,110538,110540,110542,110544,110546,110548,110550,110552,110554,110556,110558,110560,110562,110564,110566,110568],{},[31662,110517,456],{"stretchy":702},[31658,110519,37400],{},[31658,110521,43383],{},[31658,110523,47077],{},[31658,110525,31660],{},[31658,110527,33347],{},[31658,110529,111],{},[31658,110531,31759],{},[31658,110533,33244],{},[31662,110535,31960],{"separator":758},[31658,110537,105852],{},[31658,110539,34395],{},[31658,110541,43383],{},[31658,110543,46900],{},[31658,110545,94628],{},[31658,110547,31965],{},[31658,110549,33249],{},[31658,110551,33347],{},[31658,110553,111],{},[31658,110555,31759],{},[31658,110557,33244],{},[31662,110559,31960],{"separator":758},[31658,110561,67731],{},[31658,110563,33244],{},[31658,110565,111],{},[31658,110567,34395],{},[31662,110569,2557],{"stretchy":702},[31665,110571,109889],{"encoding":31667},[34,110573,110575],{"className":110574,"ariaHidden":758},[31672],[34,110576,110578,110581,110584,110587,110590,110593,110596,110599,110602,110605,110608,110611,110614,110617,110620,110623,110626,110629,110632,110635,110638,110641,110644,110647,110650,110653,110656],{"className":110577},[31676],[34,110579],{"className":110580,"style":31774},[31680],[34,110582,456],{"className":110583},[31782],[34,110585,37400],{"className":110586,"style":37422},[31685,31686],[34,110588,43383],{"className":110589},[31685,31686],[34,110591,47077],{"className":110592,"style":47120},[31685,31686],[34,110594,31660],{"className":110595},[31685,31686],[34,110597,33347],{"className":110598,"style":33389},[31685,31686],[34,110600,105967],{"className":110601},[31685,31686],[34,110603,33244],{"className":110604},[31685,31686],[34,110606,31960],{"className":110607},[32119],[34,110609],{"className":110610,"style":32123},[31845],[34,110612,105852],{"className":110613,"style":37422},[31685,31686],[34,110615,34395],{"className":110616,"style":34452},[31685,31686],[34,110618,43383],{"className":110619},[31685,31686],[34,110621,46900],{"className":110622},[31685,31686],[34,110624,94628],{"className":110625},[31685,31686],[34,110627,31965],{"className":110628},[31685,31686],[34,110630,106819],{"className":110631,"style":33389},[31685,31686],[34,110633,105967],{"className":110634},[31685,31686],[34,110636,33244],{"className":110637},[31685,31686],[34,110639,31960],{"className":110640},[32119],[34,110642],{"className":110643,"style":32123},[31845],[34,110645,67731],{"className":110646,"style":32604},[31685,31686],[34,110648,33244],{"className":110649},[31685,31686],[34,110651,111],{"className":110652},[31685,31686],[34,110654,34395],{"className":110655,"style":34452},[31685,31686],[34,110657,2557],{"className":110658},[31690],"。然而，存在函数依赖 ",[34,110661,110663,110712],{"className":110662},[31642],[34,110664,110666],{"className":110665},[31646],[31648,110667,110668],{"xmlns":31650},[31652,110669,110670,110710],{},[31655,110671,110672,110674,110676,110678,110680,110682,110684,110686,110688,110690,110692,110694,110696,110698,110700,110702,110704,110706,110708],{},[31658,110673,37400],{},[31658,110675,43383],{},[31658,110677,47077],{},[31658,110679,31660],{},[31658,110681,33347],{},[31658,110683,111],{},[31658,110685,31759],{},[31658,110687,33244],{},[31662,110689,33239],{},[31658,110691,98360],{},[31658,110693,111],{},[31658,110695,67764],{},[31658,110697,43383],{},[31658,110699,34395],{},[31658,110701,105852],{},[31658,110703,47292],{},[31658,110705,43383],{},[31658,110707,31660],{},[31658,110709,33244],{},[31665,110711,107612],{"encoding":31667},[34,110713,110715,110751],{"className":110714,"ariaHidden":758},[31672],[34,110716,110718,110721,110724,110727,110730,110733,110736,110739,110742,110745,110748],{"className":110717},[31676],[34,110719],{"className":110720,"style":33218},[31680],[34,110722,37400],{"className":110723,"style":37422},[31685,31686],[34,110725,43383],{"className":110726},[31685,31686],[34,110728,47077],{"className":110729,"style":47120},[31685,31686],[34,110731,31660],{"className":110732},[31685,31686],[34,110734,33347],{"className":110735,"style":33389},[31685,31686],[34,110737,105967],{"className":110738},[31685,31686],[34,110740,33244],{"className":110741},[31685,31686],[34,110743],{"className":110744,"style":31846},[31845],[34,110746,33239],{"className":110747},[31850],[34,110749],{"className":110750,"style":31846},[31845],[34,110752,110754,110757,110760,110763,110766,110769,110772,110775,110778,110781],{"className":110753},[31676],[34,110755],{"className":110756,"style":43217},[31680],[34,110758,98360],{"className":110759,"style":33389},[31685,31686],[34,110761,111],{"className":110762},[31685,31686],[34,110764,67764],{"className":110765,"style":32062},[31685,31686],[34,110767,106001],{"className":110768,"style":34452},[31685,31686],[34,110770,105852],{"className":110771,"style":37422},[31685,31686],[34,110773,47292],{"className":110774},[31685,31686],[34,110776,43383],{"className":110777},[31685,31686],[34,110779,31660],{"className":110780},[31685,31686],[34,110782,33244],{"className":110783},[31685,31686],"。这意味着非主属性 ",[34,110786,110788,110820],{"className":110787},[31642],[34,110789,110791],{"className":110790},[31646],[31648,110792,110793],{"xmlns":31650},[31652,110794,110795,110817],{},[31655,110796,110797,110799,110801,110803,110805,110807,110809,110811,110813,110815],{},[31658,110798,98360],{},[31658,110800,111],{},[31658,110802,67764],{},[31658,110804,43383],{},[31658,110806,34395],{},[31658,110808,105852],{},[31658,110810,47292],{},[31658,110812,43383],{},[31658,110814,31660],{},[31658,110816,33244],{},[31665,110818,110819],{"encoding":31667},"MayorPhone",[34,110821,110823],{"className":110822,"ariaHidden":758},[31672],[34,110824,110826,110829,110832,110835,110838,110841,110844,110847,110850,110853],{"className":110825},[31676],[34,110827],{"className":110828,"style":43217},[31680],[34,110830,98360],{"className":110831,"style":33389},[31685,31686],[34,110833,111],{"className":110834},[31685,31686],[34,110836,67764],{"className":110837,"style":32062},[31685,31686],[34,110839,106001],{"className":110840,"style":34452},[31685,31686],[34,110842,105852],{"className":110843,"style":37422},[31685,31686],[34,110845,47292],{"className":110846},[31685,31686],[34,110848,43383],{"className":110849},[31685,31686],[34,110851,31660],{"className":110852},[31685,31686],[34,110854,33244],{"className":110855},[31685,31686]," 只依赖于码的一部分（",[34,110858,110860,110887],{"className":110859},[31642],[34,110861,110863],{"className":110862},[31646],[31648,110864,110865],{"xmlns":31650},[31652,110866,110867,110885],{},[31655,110868,110869,110871,110873,110875,110877,110879,110881,110883],{},[31658,110870,37400],{},[31658,110872,43383],{},[31658,110874,47077],{},[31658,110876,31660],{},[31658,110878,33347],{},[31658,110880,111],{},[31658,110882,31759],{},[31658,110884,33244],{},[31665,110886,109120],{"encoding":31667},[34,110888,110890],{"className":110889,"ariaHidden":758},[31672],[34,110891,110893,110896,110899,110902,110905,110908,110911,110914],{"className":110892},[31676],[34,110894],{"className":110895,"style":33218},[31680],[34,110897,37400],{"className":110898,"style":37422},[31685,31686],[34,110900,43383],{"className":110901},[31685,31686],[34,110903,47077],{"className":110904,"style":47120},[31685,31686],[34,110906,31660],{"className":110907},[31685,31686],[34,110909,33347],{"className":110910,"style":33389},[31685,31686],[34,110912,105967],{"className":110913},[31685,31686],[34,110915,33244],{"className":110916},[31685,31686],"），即存在",[2627,110919,110920],{},"部分函数依赖",[108,110922,110923,110926],{},[2627,110924,110925],{},"不合理性","：如果一个镇有 100 种农产品，镇长的电话就会重复存储 100 次（数据冗余）；如果镇长换了，需要修改大量记录（更新异常）。",[32188,110928],{},[253,110930,110932],{"id":110931},"_6-规范化推导向高级别范式转换","6. 规范化推导：向高级别范式转换",[15,110934,110935,110936,111005,111006,111009],{},"针对低规范化级别的 ",[34,110937,110939,110956],{"className":110938},[31642],[34,110940,110942],{"className":110941},[31646],[31648,110943,110944],{"xmlns":31650},[31652,110945,110946,110954],{},[31655,110947,110948],{},[31743,110949,110950,110952],{},[31658,110951,34386],{},[31749,110953,18136],{},[31665,110955,107369],{"encoding":31667},[34,110957,110959],{"className":110958,"ariaHidden":758},[31672],[34,110960,110962,110965],{"className":110961},[31676],[34,110963],{"className":110964,"style":70915},[31680],[34,110966,110968,110971],{"className":110967},[31685],[34,110969,34386],{"className":110970,"style":34427},[31685,31686],[34,110972,110974],{"className":110973},[31792],[34,110975,110977,110997],{"className":110976},[31796,31797],[34,110978,110980,110994],{"className":110979},[31801],[34,110981,110983],{"className":110982,"style":31806},[31805],[34,110984,110985,110988],{"style":104694},[34,110986],{"className":110987,"style":31814},[31813],[34,110989,110991],{"className":110990},[31818,31819,31820,31821],[34,110992,18136],{"className":110993},[31685,31821],[34,110995,31829],{"className":110996},[31828],[34,110998,111000],{"className":110999},[31801],[34,111001,111003],{"className":111002,"style":31836},[31805],[34,111004],{}," 进行",[2627,111007,111008],{},"投影分解","，消除部分函数依赖：",[32490,111011,111013],{"id":111012},"分解结果","分解结果：",[2230,111015,111016,111428],{},[108,111017,111018,2664,111095,111414],{},[2627,111019,111020,111094],{},[34,111021,111023,111042],{"className":111022},[31642],[34,111024,111026],{"className":111025},[31646],[31648,111027,111028],{"xmlns":31650},[31652,111029,111030,111039],{},[31655,111031,111032],{},[31743,111033,111034,111036],{},[31658,111035,34386],{},[31749,111037,111038],{},"51",[31665,111040,111041],{"encoding":31667},"R_{51}",[34,111043,111045],{"className":111044,"ariaHidden":758},[31672],[34,111046,111048,111051],{"className":111047},[31676],[34,111049],{"className":111050,"style":70915},[31680],[34,111052,111054,111057],{"className":111053},[31685],[34,111055,34386],{"className":111056,"style":34427},[31685,31686],[34,111058,111060],{"className":111059},[31792],[34,111061,111063,111086],{"className":111062},[31796,31797],[34,111064,111066,111083],{"className":111065},[31801],[34,111067,111069],{"className":111068,"style":31806},[31805],[34,111070,111071,111074],{"style":104694},[34,111072],{"className":111073,"style":31814},[31813],[34,111075,111077],{"className":111076},[31818,31819,31820,31821],[34,111078,111080],{"className":111079},[31685,31821],[34,111081,111038],{"className":111082},[31685,31821],[34,111084,31829],{"className":111085},[31828],[34,111087,111089],{"className":111088},[31801],[34,111090,111092],{"className":111091,"style":31836},[31805],[34,111093],{}," (统计核心表)",[34,111096,111098,111216],{"className":111097},[31642],[34,111099,111101],{"className":111100},[31646],[31648,111102,111103],{"xmlns":31650},[31652,111104,111105,111213],{},[31655,111106,111107,111109,111165,111167,111169,111171,111173,111175,111177,111179,111181,111183,111185,111187,111189,111191,111193,111195,111197,111199,111201,111203,111205,111207,111209,111211],{},[31662,111108,456],{"stretchy":702},[71115,111110,111111,111163],{"accentunder":758},[31655,111112,111113,111115,111117,111119,111121,111123,111125,111127,111129,111131,111133,111135,111137,111139,111141,111143,111145,111147,111149,111151,111153,111155,111157,111159,111161],{},[31658,111114,37400],{},[31658,111116,43383],{},[31658,111118,47077],{},[31658,111120,31660],{},[31658,111122,33347],{},[31658,111124,111],{},[31658,111126,31759],{},[31658,111128,33244],{},[31662,111130,31960],{"separator":758},[31658,111132,105852],{},[31658,111134,34395],{},[31658,111136,43383],{},[31658,111138,46900],{},[31658,111140,94628],{},[31658,111142,31965],{},[31658,111144,33249],{},[31658,111146,33347],{},[31658,111148,111],{},[31658,111150,31759],{},[31658,111152,33244],{},[31662,111154,31960],{"separator":758},[31658,111156,67731],{},[31658,111158,33244],{},[31658,111160,111],{},[31658,111162,34395],{},[31662,111164,105837],{"stretchy":758},[31662,111166,31960],{"separator":758},[31658,111168,37400],{},[31658,111170,43383],{},[31658,111172,33249],{},[31658,111174,111],{},[31658,111176,43379],{},[31658,111178,47384],{},[31658,111180,111],{},[31658,111182,15],{},[31658,111184,111],{},[31658,111186,31965],{},[31658,111188,31955],{},[31658,111190,33249],{},[31658,111192,67764],{},[31662,111194,31960],{"separator":758},[31658,111196,54151],{},[31658,111198,56782],{},[31658,111200,31942],{},[31658,111202,105852],{},[31658,111204,34395],{},[31658,111206,31955],{},[31658,111208,31965],{},[31658,111210,33244],{},[31662,111212,2557],{"stretchy":702},[31665,111214,111215],{"encoding":31667},"(\\underline{TownName, ProductName, Year}, TotalCapacity, AvgPrice)",[34,111217,111219],{"className":111218,"ariaHidden":758},[31672],[34,111220,111222,111225,111228,111339,111342,111345,111348,111351,111354,111357,111360,111363,111366,111369,111372,111375,111378,111381,111384,111387,111390,111393,111396,111399,111402,111405,111408,111411],{"className":111221},[31676],[34,111223],{"className":111224,"style":107081},[31680],[34,111226,456],{"className":111227},[31782],[34,111229,111231],{"className":111230},[31685,105921],[34,111232,111234,111331],{"className":111233},[31796,31797],[34,111235,111237,111328],{"className":111236},[31801],[34,111238,111240,111248],{"className":111239,"style":31681},[31805],[34,111241,111242,111245],{"style":107139},[34,111243],{"className":111244,"style":41508},[31813],[34,111246],{"className":111247,"style":41528},[105940],[34,111249,111250,111253],{"style":41565},[34,111251],{"className":111252,"style":41508},[31813],[34,111254,111256,111259,111262,111265,111268,111271,111274,111277,111280,111283,111286,111289,111292,111295,111298,111301,111304,111307,111310,111313,111316,111319,111322,111325],{"className":111255},[31685],[34,111257,37400],{"className":111258,"style":37422},[31685,31686],[34,111260,43383],{"className":111261},[31685,31686],[34,111263,47077],{"className":111264,"style":47120},[31685,31686],[34,111266,31660],{"className":111267},[31685,31686],[34,111269,33347],{"className":111270,"style":33389},[31685,31686],[34,111272,105967],{"className":111273},[31685,31686],[34,111275,33244],{"className":111276},[31685,31686],[34,111278,31960],{"className":111279},[32119],[34,111281],{"className":111282,"style":32123},[31845],[34,111284,105852],{"className":111285,"style":37422},[31685,31686],[34,111287,34395],{"className":111288,"style":34452},[31685,31686],[34,111290,43383],{"className":111291},[31685,31686],[34,111293,46900],{"className":111294},[31685,31686],[34,111296,94628],{"className":111297},[31685,31686],[34,111299,31965],{"className":111300},[31685,31686],[34,111302,106819],{"className":111303,"style":33389},[31685,31686],[34,111305,105967],{"className":111306},[31685,31686],[34,111308,33244],{"className":111309},[31685,31686],[34,111311,31960],{"className":111312},[32119],[34,111314],{"className":111315,"style":32123},[31845],[34,111317,67731],{"className":111318,"style":32604},[31685,31686],[34,111320,33244],{"className":111321},[31685,31686],[34,111323,111],{"className":111324},[31685,31686],[34,111326,34395],{"className":111327,"style":34452},[31685,31686],[34,111329,31829],{"className":111330},[31828],[34,111332,111334],{"className":111333},[31801],[34,111335,111337],{"className":111336,"style":107235},[31805],[34,111338],{},[34,111340,31960],{"className":111341},[32119],[34,111343],{"className":111344,"style":32123},[31845],[34,111346,37400],{"className":111347,"style":37422},[31685,31686],[34,111349,43383],{"className":111350},[31685,31686],[34,111352,33249],{"className":111353},[31685,31686],[34,111355,111],{"className":111356},[31685,31686],[34,111358,43379],{"className":111359,"style":47113},[31685,31686],[34,111361,47384],{"className":111362,"style":38706},[31685,31686],[34,111364,111],{"className":111365},[31685,31686],[34,111367,15],{"className":111368},[31685,31686],[34,111370,111],{"className":111371},[31685,31686],[34,111373,31965],{"className":111374},[31685,31686],[34,111376,31955],{"className":111377},[31685,31686],[34,111379,33249],{"className":111380},[31685,31686],[34,111382,67764],{"className":111383,"style":32062},[31685,31686],[34,111385,31960],{"className":111386},[32119],[34,111388],{"className":111389,"style":32123},[31845],[34,111391,54151],{"className":111392},[31685,31686],[34,111394,56782],{"className":111395,"style":32062},[31685,31686],[34,111397,31942],{"className":111398,"style":32062},[31685,31686],[34,111400,105852],{"className":111401,"style":37422},[31685,31686],[34,111403,34395],{"className":111404,"style":34452},[31685,31686],[34,111406,31955],{"className":111407},[31685,31686],[34,111409,106860],{"className":111410},[31685,31686],[34,111412,2557],{"className":111413},[31690],[105,111415,111416,111422],{},[108,111417,111418,111421],{},[2627,111419,111420],{},"范式级别","：BCNF",[108,111423,111424,111427],{},[2627,111425,111426],{},"说明","：所有非主属性完全函数依赖于码。",[108,111429,111430,2664,111507,111678],{},[2627,111431,111432,111506],{},[34,111433,111435,111454],{"className":111434},[31642],[34,111436,111438],{"className":111437},[31646],[31648,111439,111440],{"xmlns":31650},[31652,111441,111442,111451],{},[31655,111443,111444],{},[31743,111445,111446,111448],{},[31658,111447,34386],{},[31749,111449,111450],{},"52",[31665,111452,111453],{"encoding":31667},"R_{52}",[34,111455,111457],{"className":111456,"ariaHidden":758},[31672],[34,111458,111460,111463],{"className":111459},[31676],[34,111461],{"className":111462,"style":70915},[31680],[34,111464,111466,111469],{"className":111465},[31685],[34,111467,34386],{"className":111468,"style":34427},[31685,31686],[34,111470,111472],{"className":111471},[31792],[34,111473,111475,111498],{"className":111474},[31796,31797],[34,111476,111478,111495],{"className":111477},[31801],[34,111479,111481],{"className":111480,"style":31806},[31805],[34,111482,111483,111486],{"style":104694},[34,111484],{"className":111485,"style":31814},[31813],[34,111487,111489],{"className":111488},[31818,31819,31820,31821],[34,111490,111492],{"className":111491},[31685,31821],[34,111493,111450],{"className":111494},[31685,31821],[34,111496,31829],{"className":111497},[31828],[34,111499,111501],{"className":111500},[31801],[34,111502,111504],{"className":111503,"style":31836},[31805],[34,111505],{}," (镇长联系表)",[34,111508,111510,111570],{"className":111509},[31642],[34,111511,111513],{"className":111512},[31646],[31648,111514,111515],{"xmlns":31650},[31652,111516,111517,111567],{},[31655,111518,111519,111521,111543,111545,111547,111549,111551,111553,111555,111557,111559,111561,111563,111565],{},[31662,111520,456],{"stretchy":702},[71115,111522,111523,111541],{"accentunder":758},[31655,111524,111525,111527,111529,111531,111533,111535,111537,111539],{},[31658,111526,37400],{},[31658,111528,43383],{},[31658,111530,47077],{},[31658,111532,31660],{},[31658,111534,33347],{},[31658,111536,111],{},[31658,111538,31759],{},[31658,111540,33244],{},[31662,111542,105837],{"stretchy":758},[31662,111544,31960],{"separator":758},[31658,111546,98360],{},[31658,111548,111],{},[31658,111550,67764],{},[31658,111552,43383],{},[31658,111554,34395],{},[31658,111556,105852],{},[31658,111558,47292],{},[31658,111560,43383],{},[31658,111562,31660],{},[31658,111564,33244],{},[31662,111566,2557],{"stretchy":702},[31665,111568,111569],{"encoding":31667},"(\\underline{TownName}, MayorPhone)",[34,111571,111573],{"className":111572,"ariaHidden":758},[31672],[34,111574,111576,111579,111582,111642,111645,111648,111651,111654,111657,111660,111663,111666,111669,111672,111675],{"className":111575},[31676],[34,111577],{"className":111578,"style":31774},[31680],[34,111580,456],{"className":111581},[31782],[34,111583,111585],{"className":111584},[31685,105921],[34,111586,111588,111634],{"className":111587},[31796,31797],[34,111589,111591,111631],{"className":111590},[31801],[34,111592,111594,111602],{"className":111593,"style":33218},[31805],[34,111595,111596,111599],{"style":105933},[34,111597],{"className":111598,"style":41508},[31813],[34,111600],{"className":111601,"style":41528},[105940],[34,111603,111604,111607],{"style":41565},[34,111605],{"className":111606,"style":41508},[31813],[34,111608,111610,111613,111616,111619,111622,111625,111628],{"className":111609},[31685],[34,111611,37400],{"className":111612,"style":37422},[31685,31686],[34,111614,43383],{"className":111615},[31685,31686],[34,111617,47077],{"className":111618,"style":47120},[31685,31686],[34,111620,31660],{"className":111621},[31685,31686],[34,111623,33347],{"className":111624,"style":33389},[31685,31686],[34,111626,105967],{"className":111627},[31685,31686],[34,111629,33244],{"className":111630},[31685,31686],[34,111632,31829],{"className":111633},[31828],[34,111635,111637],{"className":111636},[31801],[34,111638,111640],{"className":111639,"style":105980},[31805],[34,111641],{},[34,111643,31960],{"className":111644},[32119],[34,111646],{"className":111647,"style":32123},[31845],[34,111649,98360],{"className":111650,"style":33389},[31685,31686],[34,111652,111],{"className":111653},[31685,31686],[34,111655,67764],{"className":111656,"style":32062},[31685,31686],[34,111658,106001],{"className":111659,"style":34452},[31685,31686],[34,111661,105852],{"className":111662,"style":37422},[31685,31686],[34,111664,47292],{"className":111665},[31685,31686],[34,111667,43383],{"className":111668},[31685,31686],[34,111670,31660],{"className":111671},[31685,31686],[34,111673,33244],{"className":111674},[31685,31686],[34,111676,2557],{"className":111677},[31690],[105,111679,111680,111684],{},[108,111681,111682,111421],{},[2627,111683,111420],{},[108,111685,111686,111688,111689,111758,111759,347],{},[2627,111687,111426],{},"：由于该表与 ",[34,111690,111692,111709],{"className":111691},[31642],[34,111693,111695],{"className":111694},[31646],[31648,111696,111697],{"xmlns":31650},[31652,111698,111699,111707],{},[31655,111700,111701],{},[31743,111702,111703,111705],{},[31658,111704,34386],{},[31749,111706,640],{},[31665,111708,107510],{"encoding":31667},[34,111710,111712],{"className":111711,"ariaHidden":758},[31672],[34,111713,111715,111718],{"className":111714},[31676],[34,111716],{"className":111717,"style":70915},[31680],[34,111719,111721,111724],{"className":111720},[31685],[34,111722,34386],{"className":111723,"style":34427},[31685,31686],[34,111725,111727],{"className":111726},[31792],[34,111728,111730,111750],{"className":111729},[31796,31797],[34,111731,111733,111747],{"className":111732},[31801],[34,111734,111736],{"className":111735,"style":31806},[31805],[34,111737,111738,111741],{"style":104694},[34,111739],{"className":111740,"style":31814},[31813],[34,111742,111744],{"className":111743},[31818,31819,31820,31821],[34,111745,640],{"className":111746},[31685,31821],[34,111748,31829],{"className":111749},[31828],[34,111751,111753],{"className":111752},[31801],[34,111754,111756],{"className":111755,"style":31836},[31805],[34,111757],{}," 结构一致，实际开发中应直接合并入 ",[34,111760,111762,111779],{"className":111761},[31642],[34,111763,111765],{"className":111764},[31646],[31648,111766,111767],{"xmlns":31650},[31652,111768,111769,111777],{},[31655,111770,111771],{},[31743,111772,111773,111775],{},[31658,111774,34386],{},[31749,111776,640],{},[31665,111778,107510],{"encoding":31667},[34,111780,111782],{"className":111781,"ariaHidden":758},[31672],[34,111783,111785,111788],{"className":111784},[31676],[34,111786],{"className":111787,"style":70915},[31680],[34,111789,111791,111794],{"className":111790},[31685],[34,111792,34386],{"className":111793,"style":34427},[31685,31686],[34,111795,111797],{"className":111796},[31792],[34,111798,111800,111820],{"className":111799},[31796,31797],[34,111801,111803,111817],{"className":111802},[31801],[34,111804,111806],{"className":111805,"style":31806},[31805],[34,111807,111808,111811],{"style":104694},[34,111809],{"className":111810,"style":31814},[31813],[34,111812,111814],{"className":111813},[31818,31819,31820,31821],[34,111815,640],{"className":111816},[31685,31821],[34,111818,31829],{"className":111819},[31828],[34,111821,111823],{"className":111822},[31801],[34,111824,111826],{"className":111825,"style":31836},[31805],[34,111827],{},[32188,111829],{},[253,111831,111832],{"id":111832},"结论与建议",[15,111834,111835,111836,104843],{},"经过以上规范化推导，问题已经设计到 ",[2627,111837,104842],{},[19,111839,111841,111842,111845,111846,111901],{"id":111840},"设计一个不是传递依赖的-xyzx-y-zxyz-关系","设计一个",[2627,111843,111844],{},"不是传递依赖","的 ",[34,111847,111849,111871],{"className":111848},[31642],[34,111850,111852],{"className":111851},[31646],[31648,111853,111854],{"xmlns":31650},[31652,111855,111856,111868],{},[31655,111857,111858,111860,111862,111864,111866],{},[31658,111859,67467],{},[31662,111861,31960],{"separator":758},[31658,111863,67731],{},[31662,111865,31960],{"separator":758},[31658,111867,97300],{},[31665,111869,111870],{"encoding":31667},"X, Y, Z",[34,111872,111874],{"className":111873,"ariaHidden":758},[31672],[34,111875,111877,111880,111883,111886,111889,111892,111895,111898],{"className":111876},[31676],[34,111878],{"className":111879,"style":70739},[31680],[34,111881,67467],{"className":111882,"style":67482},[31685,31686],[34,111884,31960],{"className":111885},[32119],[34,111887],{"className":111888,"style":32123},[31845],[34,111890,67731],{"className":111891,"style":32604},[31685,31686],[34,111893,31960],{"className":111894},[32119],[34,111896],{"className":111897,"style":32123},[31845],[34,111899,97300],{"className":111900,"style":38706},[31685,31686]," 关系",[15,111903,111904,111905,111908],{},"根据数据库权威教材（如 ",[71807,111906,111907],{},"Database System Concepts"," by Silberschatz），传递依赖的定义是：",[104833,111910,111911,112104],{},[15,111912,111913,111914,111942,111943,258,111994,112045,112046,112074,112075,112103],{},"在关系模式 ",[34,111915,111917,111930],{"className":111916},[31642],[34,111918,111920],{"className":111919},[31646],[31648,111921,111922],{"xmlns":31650},[31652,111923,111924,111928],{},[31655,111925,111926],{},[31658,111927,34386],{},[31665,111929,34386],{"encoding":31667},[34,111931,111933],{"className":111932,"ariaHidden":758},[31672],[34,111934,111936,111939],{"className":111935},[31676],[34,111937],{"className":111938,"style":33218},[31680],[34,111940,34386],{"className":111941,"style":34427},[31685,31686]," 中，如果存在函数依赖 ",[34,111944,111946,111964],{"className":111945},[31642],[34,111947,111949],{"className":111948},[31646],[31648,111950,111951],{"xmlns":31650},[31652,111952,111953,111961],{},[31655,111954,111955,111957,111959],{},[31658,111956,67467],{},[31662,111958,33239],{},[31658,111960,67731],{},[31665,111962,111963],{"encoding":31667},"X \\to Y",[34,111965,111967,111985],{"className":111966,"ariaHidden":758},[31672],[34,111968,111970,111973,111976,111979,111982],{"className":111969},[31676],[34,111971],{"className":111972,"style":33218},[31680],[34,111974,67467],{"className":111975,"style":67482},[31685,31686],[34,111977],{"className":111978,"style":31846},[31845],[34,111980,33239],{"className":111981},[31850],[34,111983],{"className":111984,"style":31846},[31845],[34,111986,111988,111991],{"className":111987},[31676],[34,111989],{"className":111990,"style":33218},[31680],[34,111992,67731],{"className":111993,"style":32604},[31685,31686],[34,111995,111997,112015],{"className":111996},[31642],[34,111998,112000],{"className":111999},[31646],[31648,112001,112002],{"xmlns":31650},[31652,112003,112004,112012],{},[31655,112005,112006,112008,112010],{},[31658,112007,67731],{},[31662,112009,33239],{},[31658,112011,97300],{},[31665,112013,112014],{"encoding":31667},"Y \\to Z",[34,112016,112018,112036],{"className":112017,"ariaHidden":758},[31672],[34,112019,112021,112024,112027,112030,112033],{"className":112020},[31676],[34,112022],{"className":112023,"style":33218},[31680],[34,112025,67731],{"className":112026,"style":32604},[31685,31686],[34,112028],{"className":112029,"style":31846},[31845],[34,112031,33239],{"className":112032},[31850],[34,112034],{"className":112035,"style":31846},[31845],[34,112037,112039,112042],{"className":112038},[31676],[34,112040],{"className":112041,"style":33218},[31680],[34,112043,97300],{"className":112044,"style":38706},[31685,31686],"，且满足以下条件，则称 ",[34,112047,112049,112062],{"className":112048},[31642],[34,112050,112052],{"className":112051},[31646],[31648,112053,112054],{"xmlns":31650},[31652,112055,112056,112060],{},[31655,112057,112058],{},[31658,112059,97300],{},[31665,112061,97300],{"encoding":31667},[34,112063,112065],{"className":112064,"ariaHidden":758},[31672],[34,112066,112068,112071],{"className":112067},[31676],[34,112069],{"className":112070,"style":33218},[31680],[34,112072,97300],{"className":112073,"style":38706},[31685,31686]," 对 ",[34,112076,112078,112091],{"className":112077},[31642],[34,112079,112081],{"className":112080},[31646],[31648,112082,112083],{"xmlns":31650},[31652,112084,112085,112089],{},[31655,112086,112087],{},[31658,112088,67467],{},[31665,112090,67467],{"encoding":31667},[34,112092,112094],{"className":112093,"ariaHidden":758},[31672],[34,112095,112097,112100],{"className":112096},[31676],[34,112098],{"className":112099,"style":33218},[31680],[34,112101,67467],{"className":112102,"style":67482},[31685,31686]," 传递函数依赖：",[2230,112105,112106,112248],{},[108,112107,112108,33775,112190,112218,112219,112247],{},[34,112109,112111,112130],{"className":112110},[31642],[34,112112,112114],{"className":112113},[31646],[31648,112115,112116],{"xmlns":31650},[31652,112117,112118,112127],{},[31655,112119,112120,112122,112125],{},[31658,112121,67731],{},[31662,112123,112124],{},"↛",[31658,112126,67467],{},[31665,112128,112129],{"encoding":31667},"Y \\not\\to X",[34,112131,112133,112181],{"className":112132,"ariaHidden":758},[31672],[34,112134,112136,112139,112142,112145,112172,112175,112178],{"className":112135},[31676],[34,112137],{"className":112138,"style":43217},[31680],[34,112140,67731],{"className":112141,"style":32604},[31685,31686],[34,112143],{"className":112144,"style":31846},[31845],[34,112146,112148],{"className":112147},[31850],[34,112149,112151],{"className":112150},[31685,54774],[34,112152,112154],{"className":112153},[54778],[34,112155,112157,112160,112169],{"className":112156},[54782],[34,112158],{"className":112159,"style":43217},[31680],[34,112161,112163],{"className":112162},[54789],[34,112164,112166],{"className":112165},[31685],[34,112167,54796],{"className":112168},[31850],[34,112170],{"className":112171},[54800],[34,112173],{"className":112174},[31845,54804],[34,112176,33239],{"className":112177},[31850],[34,112179],{"className":112180,"style":31846},[31845],[34,112182,112184,112187],{"className":112183},[31676],[34,112185],{"className":112186,"style":33218},[31680],[34,112188,67467],{"className":112189,"style":67482},[31685,31686],[34,112191,112193,112206],{"className":112192},[31642],[34,112194,112196],{"className":112195},[31646],[31648,112197,112198],{"xmlns":31650},[31652,112199,112200,112204],{},[31655,112201,112202],{},[31658,112203,67731],{},[31665,112205,67731],{"encoding":31667},[34,112207,112209],{"className":112208,"ariaHidden":758},[31672],[34,112210,112212,112215],{"className":112211},[31676],[34,112213],{"className":112214,"style":33218},[31680],[34,112216,67731],{"className":112217,"style":32604},[31685,31686]," 不决定 ",[34,112220,112222,112235],{"className":112221},[31642],[34,112223,112225],{"className":112224},[31646],[31648,112226,112227],{"xmlns":31650},[31652,112228,112229,112233],{},[31655,112230,112231],{},[31658,112232,67467],{},[31665,112234,67467],{"encoding":31667},[34,112236,112238],{"className":112237,"ariaHidden":758},[31672],[34,112239,112241,112244],{"className":112240},[31676],[34,112242],{"className":112243,"style":33218},[31680],[34,112245,67467],{"className":112246,"style":67482},[31685,31686],"，两者不是等价的码）。",[108,112249,112250,33775,112338,112366,112367,112395],{},[34,112251,112253,112272],{"className":112252},[31642],[34,112254,112256],{"className":112255},[31646],[31648,112257,112258],{"xmlns":31650},[31652,112259,112260,112269],{},[31655,112261,112262,112264,112267],{},[31658,112263,97300],{},[31662,112265,112266],{"mathvariant":38079},"∉",[31658,112268,67731],{},[31665,112270,112271],{"encoding":31667},"Z \\notin Y",[34,112273,112275,112329],{"className":112274,"ariaHidden":758},[31672],[34,112276,112278,112281,112284,112287,112326],{"className":112277},[31676],[34,112279],{"className":112280,"style":31774},[31680],[34,112282,97300],{"className":112283,"style":38706},[31685,31686],[34,112285],{"className":112286,"style":31846},[31845],[34,112288,112290,112297],{"className":112289},[31850],[34,112291,112293],{"className":112292},[31685],[34,112294,112296],{"className":112295},[31850],"∈",[34,112298,112300],{"className":112299},[31685,54774],[34,112301,112303],{"className":112302},[54778],[34,112304,112307,112310,112323],{"className":112305},[112306],"llap",[34,112308],{"className":112309,"style":31774},[31680],[34,112311,112313],{"className":112312},[54789],[34,112314,112316,112319],{"className":112315},[31685],[34,112317,38080],{"className":112318},[31685],[34,112320],{"className":112321,"style":112322},[31845],"margin-right:0.0556em;",[34,112324],{"className":112325},[54800],[34,112327],{"className":112328,"style":31846},[31845],[34,112330,112332,112335],{"className":112331},[31676],[34,112333],{"className":112334,"style":33218},[31680],[34,112336,67731],{"className":112337,"style":32604},[31685,31686],[34,112339,112341,112354],{"className":112340},[31642],[34,112342,112344],{"className":112343},[31646],[31648,112345,112346],{"xmlns":31650},[31652,112347,112348,112352],{},[31655,112349,112350],{},[31658,112351,97300],{},[31665,112353,97300],{"encoding":31667},[34,112355,112357],{"className":112356,"ariaHidden":758},[31672],[34,112358,112360,112363],{"className":112359},[31676],[34,112361],{"className":112362,"style":33218},[31680],[34,112364,97300],{"className":112365,"style":38706},[31685,31686]," 不是 ",[34,112368,112370,112383],{"className":112369},[31642],[34,112371,112373],{"className":112372},[31646],[31648,112374,112375],{"xmlns":31650},[31652,112376,112377,112381],{},[31655,112378,112379],{},[31658,112380,67731],{},[31665,112382,67731],{"encoding":31667},[34,112384,112386],{"className":112385,"ariaHidden":758},[31672],[34,112387,112389,112392],{"className":112388},[31676],[34,112390],{"className":112391,"style":33218},[31680],[34,112393,67731],{"className":112394,"style":32604},[31685,31686]," 的子集）。",[15,112397,112398,112399,112402],{},"要设计一个",[2627,112400,112401],{},"非传递依赖","，我们可以从以下两个逻辑角度出发：",[32188,112404],{},[253,112406,112408],{"id":112407},"方案一分支直接依赖-branching-direct-dependency","方案一：分支直接依赖 (Branching Direct Dependency)",[15,112410,112411,112412,112415,112416,112444,112445,258,112473,112501,112502,258,112530,112558],{},"这是最常见的符合 ",[2627,112413,112414],{},"3NF (Third Normal Form)"," 的设计。在这种设计中，",[34,112417,112419,112432],{"className":112418},[31642],[34,112420,112422],{"className":112421},[31646],[31648,112423,112424],{"xmlns":31650},[31652,112425,112426,112430],{},[31655,112427,112428],{},[31658,112429,67467],{},[31665,112431,67467],{"encoding":31667},[34,112433,112435],{"className":112434,"ariaHidden":758},[31672],[34,112436,112438,112441],{"className":112437},[31676],[34,112439],{"className":112440,"style":33218},[31680],[34,112442,67467],{"className":112443,"style":67482},[31685,31686]," 同时决定 ",[34,112446,112448,112461],{"className":112447},[31642],[34,112449,112451],{"className":112450},[31646],[31648,112452,112453],{"xmlns":31650},[31652,112454,112455,112459],{},[31655,112456,112457],{},[31658,112458,67731],{},[31665,112460,67731],{"encoding":31667},[34,112462,112464],{"className":112463,"ariaHidden":758},[31672],[34,112465,112467,112470],{"className":112466},[31676],[34,112468],{"className":112469,"style":33218},[31680],[34,112471,67731],{"className":112472,"style":32604},[31685,31686],[34,112474,112476,112489],{"className":112475},[31642],[34,112477,112479],{"className":112478},[31646],[31648,112480,112481],{"xmlns":31650},[31652,112482,112483,112487],{},[31655,112484,112485],{},[31658,112486,97300],{},[31665,112488,97300],{"encoding":31667},[34,112490,112492],{"className":112491,"ariaHidden":758},[31672],[34,112493,112495,112498],{"className":112494},[31676],[34,112496],{"className":112497,"style":33218},[31680],[34,112499,97300],{"className":112500,"style":38706},[31685,31686],"，但 ",[34,112503,112505,112518],{"className":112504},[31642],[34,112506,112508],{"className":112507},[31646],[31648,112509,112510],{"xmlns":31650},[31652,112511,112512,112516],{},[31655,112513,112514],{},[31658,112515,67731],{},[31665,112517,67731],{"encoding":31667},[34,112519,112521],{"className":112520,"ariaHidden":758},[31672],[34,112522,112524,112527],{"className":112523},[31676],[34,112525],{"className":112526,"style":33218},[31680],[34,112528,67731],{"className":112529,"style":32604},[31685,31686],[34,112531,112533,112546],{"className":112532},[31642],[34,112534,112536],{"className":112535},[31646],[31648,112537,112538],{"xmlns":31650},[31652,112539,112540,112544],{},[31655,112541,112542],{},[31658,112543,97300],{},[31665,112545,97300],{"encoding":31667},[34,112547,112549],{"className":112548,"ariaHidden":758},[31672],[34,112550,112552,112555],{"className":112551},[31676],[34,112553],{"className":112554,"style":33218},[31680],[34,112556,97300],{"className":112557,"style":38706},[31685,31686]," 之间没有任何函数依赖关系。",[32490,112560,112562],{"id":112561},"_1-变量定义","1. 变量定义",[105,112564,112565,112602,112639],{},[108,112566,112567,2664,112598,112601],{},[2627,112568,112569,112597],{},[34,112570,112572,112585],{"className":112571},[31642],[34,112573,112575],{"className":112574},[31646],[31648,112576,112577],{"xmlns":31650},[31652,112578,112579,112583],{},[31655,112580,112581],{},[31658,112582,67467],{},[31665,112584,67467],{"encoding":31667},[34,112586,112588],{"className":112587,"ariaHidden":758},[31672],[34,112589,112591,112594],{"className":112590},[31676],[34,112592],{"className":112593,"style":33218},[31680],[34,112595,67467],{"className":112596,"style":67482},[31685,31686]," (主码)",[31,112599,112600],{},"EmployeeID"," (员工编号)",[108,112603,112604,2664,112635,112638],{},[2627,112605,112606,112634],{},[34,112607,112609,112622],{"className":112608},[31642],[34,112610,112612],{"className":112611},[31646],[31648,112613,112614],{"xmlns":31650},[31652,112615,112616,112620],{},[31655,112617,112618],{},[31658,112619,67731],{},[31665,112621,67731],{"encoding":31667},[34,112623,112625],{"className":112624,"ariaHidden":758},[31672],[34,112626,112628,112631],{"className":112627},[31676],[34,112629],{"className":112630,"style":33218},[31680],[34,112632,67731],{"className":112633,"style":32604},[31685,31686]," (非主属性)",[31,112636,112637],{},"Name"," (姓名)",[108,112640,112641,2664,112671,112674],{},[2627,112642,112643,112634],{},[34,112644,112646,112659],{"className":112645},[31642],[34,112647,112649],{"className":112648},[31646],[31648,112650,112651],{"xmlns":31650},[31652,112652,112653,112657],{},[31655,112654,112655],{},[31658,112656,97300],{},[31665,112658,97300],{"encoding":31667},[34,112660,112662],{"className":112661,"ariaHidden":758},[31672],[34,112663,112665,112668],{"className":112664},[31676],[34,112666],{"className":112667,"style":33218},[31680],[34,112669,97300],{"className":112670,"style":38706},[31685,31686],[31,112672,112673],{},"Salary"," (薪水)",[32490,112676,112678,112679],{"id":112677},"_2-函数依赖集-fff","2. 函数依赖集 ",[34,112680,112682,112695],{"className":112681},[31642],[34,112683,112685],{"className":112684},[31646],[31648,112686,112687],{"xmlns":31650},[31652,112688,112689,112693],{},[31655,112690,112691],{},[31658,112692,51834],{},[31665,112694,51834],{"encoding":31667},[34,112696,112698],{"className":112697,"ariaHidden":758},[31672],[34,112699,112701,112704],{"className":112700},[31676],[34,112702],{"className":112703,"style":33218},[31680],[34,112705,51834],{"className":112706,"style":37422},[31685,31686],[105,112708,112709,112762],{},[108,112710,112711,112761],{},[34,112712,112714,112731],{"className":112713},[31642],[34,112715,112717],{"className":112716},[31646],[31648,112718,112719],{"xmlns":31650},[31652,112720,112721,112729],{},[31655,112722,112723,112725,112727],{},[31658,112724,67467],{},[31662,112726,33239],{},[31658,112728,67731],{},[31665,112730,111963],{"encoding":31667},[34,112732,112734,112752],{"className":112733,"ariaHidden":758},[31672],[34,112735,112737,112740,112743,112746,112749],{"className":112736},[31676],[34,112738],{"className":112739,"style":33218},[31680],[34,112741,67467],{"className":112742,"style":67482},[31685,31686],[34,112744],{"className":112745,"style":31846},[31845],[34,112747,33239],{"className":112748},[31850],[34,112750],{"className":112751,"style":31846},[31845],[34,112753,112755,112758],{"className":112754},[31676],[34,112756],{"className":112757,"style":33218},[31680],[34,112759,67731],{"className":112760,"style":32604},[31685,31686]," (员工编号决定姓名)",[108,112763,112764,112815],{},[34,112765,112767,112785],{"className":112766},[31642],[34,112768,112770],{"className":112769},[31646],[31648,112771,112772],{"xmlns":31650},[31652,112773,112774,112782],{},[31655,112775,112776,112778,112780],{},[31658,112777,67467],{},[31662,112779,33239],{},[31658,112781,97300],{},[31665,112783,112784],{"encoding":31667},"X \\to Z",[34,112786,112788,112806],{"className":112787,"ariaHidden":758},[31672],[34,112789,112791,112794,112797,112800,112803],{"className":112790},[31676],[34,112792],{"className":112793,"style":33218},[31680],[34,112795,67467],{"className":112796,"style":67482},[31685,31686],[34,112798],{"className":112799,"style":31846},[31845],[34,112801,33239],{"className":112802},[31850],[34,112804],{"className":112805,"style":31846},[31845],[34,112807,112809,112812],{"className":112808},[31676],[34,112810],{"className":112811,"style":33218},[31680],[34,112813,97300],{"className":112814,"style":38706},[31685,31686]," (员工编号决定薪水)",[32490,112817,112819],{"id":112818},"_3-为什么不是传递依赖","3. 为什么不是传递依赖？",[105,112821,112822,112959,113020],{},[108,112823,112824,112825,112875,112876,347],{},"虽然 ",[34,112826,112828,112845],{"className":112827},[31642],[34,112829,112831],{"className":112830},[31646],[31648,112832,112833],{"xmlns":31650},[31652,112834,112835,112843],{},[31655,112836,112837,112839,112841],{},[31658,112838,67467],{},[31662,112840,33239],{},[31658,112842,67731],{},[31665,112844,111963],{"encoding":31667},[34,112846,112848,112866],{"className":112847,"ariaHidden":758},[31672],[34,112849,112851,112854,112857,112860,112863],{"className":112850},[31676],[34,112852],{"className":112853,"style":33218},[31680],[34,112855,67467],{"className":112856,"style":67482},[31685,31686],[34,112858],{"className":112859,"style":31846},[31845],[34,112861,33239],{"className":112862},[31850],[34,112864],{"className":112865,"style":31846},[31845],[34,112867,112869,112872],{"className":112868},[31676],[34,112870],{"className":112871,"style":33218},[31680],[34,112873,67731],{"className":112874,"style":32604},[31685,31686]," 成立，但 ",[2627,112877,112878],{},[34,112879,112881,112899],{"className":112880},[31642],[34,112882,112884],{"className":112883},[31646],[31648,112885,112886],{"xmlns":31650},[31652,112887,112888,112896],{},[31655,112889,112890,112892,112894],{},[31658,112891,67731],{},[31662,112893,112124],{},[31658,112895,97300],{},[31665,112897,112898],{"encoding":31667},"Y \\not\\to Z",[34,112900,112902,112950],{"className":112901,"ariaHidden":758},[31672],[34,112903,112905,112908,112911,112914,112941,112944,112947],{"className":112904},[31676],[34,112906],{"className":112907,"style":43217},[31680],[34,112909,67731],{"className":112910,"style":32604},[31685,31686],[34,112912],{"className":112913,"style":31846},[31845],[34,112915,112917],{"className":112916},[31850],[34,112918,112920],{"className":112919},[31685,54774],[34,112921,112923],{"className":112922},[54778],[34,112924,112926,112929,112938],{"className":112925},[54782],[34,112927],{"className":112928,"style":43217},[31680],[34,112930,112932],{"className":112931},[54789],[34,112933,112935],{"className":112934},[31685],[34,112936,54796],{"className":112937},[31850],[34,112939],{"className":112940},[54800],[34,112942],{"className":112943},[31845,54804],[34,112945,33239],{"className":112946},[31850],[34,112948],{"className":112949,"style":31846},[31845],[34,112951,112953,112956],{"className":112952},[31676],[34,112954],{"className":112955,"style":33218},[31680],[34,112957,97300],{"className":112958,"style":38706},[31685,31686],[108,112960,112961,112962,112990,112991,113019],{},"在应用语义中，姓名（",[34,112963,112965,112978],{"className":112964},[31642],[34,112966,112968],{"className":112967},[31646],[31648,112969,112970],{"xmlns":31650},[31652,112971,112972,112976],{},[31655,112973,112974],{},[31658,112975,67731],{},[31665,112977,67731],{"encoding":31667},[34,112979,112981],{"className":112980,"ariaHidden":758},[31672],[34,112982,112984,112987],{"className":112983},[31676],[34,112985],{"className":112986,"style":33218},[31680],[34,112988,67731],{"className":112989,"style":32604},[31685,31686],"）无法唯一决定薪水（",[34,112992,112994,113007],{"className":112993},[31642],[34,112995,112997],{"className":112996},[31646],[31648,112998,112999],{"xmlns":31650},[31652,113000,113001,113005],{},[31655,113002,113003],{},[31658,113004,97300],{},[31665,113006,97300],{"encoding":31667},[34,113008,113010],{"className":113009,"ariaHidden":758},[31672],[34,113011,113013,113016],{"className":113012},[31676],[34,113014],{"className":113015,"style":33218},[31680],[34,113017,97300],{"className":113018,"style":38706},[31685,31686],"），因为可能存在同名的人且薪水不同。",[108,113021,113022,113023,113073,113074,113124,113125,113128],{},"由于不存在 ",[34,113024,113026,113043],{"className":113025},[31642],[34,113027,113029],{"className":113028},[31646],[31648,113030,113031],{"xmlns":31650},[31652,113032,113033,113041],{},[31655,113034,113035,113037,113039],{},[31658,113036,67731],{},[31662,113038,33239],{},[31658,113040,97300],{},[31665,113042,112014],{"encoding":31667},[34,113044,113046,113064],{"className":113045,"ariaHidden":758},[31672],[34,113047,113049,113052,113055,113058,113061],{"className":113048},[31676],[34,113050],{"className":113051,"style":33218},[31680],[34,113053,67731],{"className":113054,"style":32604},[31685,31686],[34,113056],{"className":113057,"style":31846},[31845],[34,113059,33239],{"className":113060},[31850],[34,113062],{"className":113063,"style":31846},[31845],[34,113065,113067,113070],{"className":113066},[31676],[34,113068],{"className":113069,"style":33218},[31680],[34,113071,97300],{"className":113072,"style":38706},[31685,31686]," 这个中间环节，因此 ",[34,113075,113077,113094],{"className":113076},[31642],[34,113078,113080],{"className":113079},[31646],[31648,113081,113082],{"xmlns":31650},[31652,113083,113084,113092],{},[31655,113085,113086,113088,113090],{},[31658,113087,67467],{},[31662,113089,33239],{},[31658,113091,97300],{},[31665,113093,112784],{"encoding":31667},[34,113095,113097,113115],{"className":113096,"ariaHidden":758},[31672],[34,113098,113100,113103,113106,113109,113112],{"className":113099},[31676],[34,113101],{"className":113102,"style":33218},[31680],[34,113104,67467],{"className":113105,"style":67482},[31685,31686],[34,113107],{"className":113108,"style":31846},[31845],[34,113110,33239],{"className":113111},[31850],[34,113113],{"className":113114,"style":31846},[31845],[34,113116,113118,113121],{"className":113117},[31676],[34,113119],{"className":113120,"style":33218},[31680],[34,113122,97300],{"className":113123,"style":38706},[31685,31686]," 是",[2627,113126,113127],{},"直接函数依赖","，而非传递依赖。",[32188,113130],{},[253,113132,113134],{"id":113133},"方案二部分函数依赖-partial-functional-dependency","方案二：部分函数依赖 (Partial Functional Dependency)",[15,113136,113137,113138,113141],{},"这种设计通常出现在 ",[2627,113139,113140],{},"2NF (Second Normal Form)"," 以下的模式中。它违反了 2NF，但从定义上看，它并不构成 3NF 所针对的“传递依赖”。",[32490,113143,112562],{"id":113144},"_1-变量定义-1",[105,113146,113147,113306,113342],{},[108,113148,113149,2664,113180,113305],{},[2627,113150,113151,113179],{},[34,113152,113154,113167],{"className":113153},[31642],[34,113155,113157],{"className":113156},[31646],[31648,113158,113159],{"xmlns":31650},[31652,113160,113161,113165],{},[31655,113162,113163],{},[31658,113164,67467],{},[31665,113166,67467],{"encoding":31667},[34,113168,113170],{"className":113169,"ariaHidden":758},[31672],[34,113171,113173,113176],{"className":113172},[31676],[34,113174],{"className":113175,"style":33218},[31680],[34,113177,67467],{"className":113178,"style":67482},[31685,31686]," (复合主码)",[34,113181,113183,113235],{"className":113182},[31642],[34,113184,113186],{"className":113185},[31646],[31648,113187,113188],{"xmlns":31650},[31652,113189,113190,113232],{},[31655,113191,113192,113194,113196,113198,113200,113202,113204,113206,113208,113210,113212,113214,113216,113218,113220,113222,113224,113226,113228,113230],{},[31662,113193,32685],{"stretchy":702},[31658,113195,94675],{},[31658,113197,33249],{},[31658,113199,94628],{},[31658,113201,46900],{},[31658,113203,33244],{},[31658,113205,31660],{},[31658,113207,33249],{},[31658,113209,106063],{},[31658,113211,91994],{},[31662,113213,31960],{"separator":758},[31658,113215,47384],{},[31658,113217,43383],{},[31658,113219,94628],{},[31658,113221,34395],{},[31658,113223,31747],{},[31658,113225,33244],{},[31658,113227,106063],{},[31658,113229,91994],{},[31662,113231,32715],{"stretchy":702},[31665,113233,113234],{"encoding":31667},"\\{StudentID, CourseID\\}",[34,113236,113238],{"className":113237,"ariaHidden":758},[31672],[34,113239,113241,113244,113247,113250,113253,113256,113259,113262,113265,113268,113271,113274,113277,113280,113283,113286,113289,113292,113296,113299,113302],{"className":113240},[31676],[34,113242],{"className":113243,"style":31774},[31680],[34,113245,32685],{"className":113246},[31782],[34,113248,94675],{"className":113249,"style":106455},[31685,31686],[34,113251,33249],{"className":113252},[31685,31686],[34,113254,94628],{"className":113255},[31685,31686],[34,113257,46900],{"className":113258},[31685,31686],[34,113260,33244],{"className":113261},[31685,31686],[34,113263,31660],{"className":113264},[31685,31686],[34,113266,33249],{"className":113267},[31685,31686],[34,113269,106063],{"className":113270,"style":67482},[31685,31686],[34,113272,91994],{"className":113273,"style":34452},[31685,31686],[34,113275,31960],{"className":113276},[32119],[34,113278],{"className":113279,"style":32123},[31845],[34,113281,47384],{"className":113282,"style":38706},[31685,31686],[34,113284,43383],{"className":113285},[31685,31686],[34,113287,94628],{"className":113288},[31685,31686],[34,113290,34395],{"className":113291,"style":34452},[31685,31686],[34,113293,113295],{"className":113294},[31685,31686],"se",[34,113297,106063],{"className":113298,"style":67482},[31685,31686],[34,113300,91994],{"className":113301,"style":34452},[31685,31686],[34,113303,32715],{"className":113304},[31690]," (学号, 课程号)",[108,113307,113308,2664,113338,113341],{},[2627,113309,113310,112634],{},[34,113311,113313,113326],{"className":113312},[31642],[34,113314,113316],{"className":113315},[31646],[31648,113317,113318],{"xmlns":31650},[31652,113319,113320,113324],{},[31655,113321,113322],{},[31658,113323,67731],{},[31665,113325,67731],{"encoding":31667},[34,113327,113329],{"className":113328,"ariaHidden":758},[31672],[34,113330,113332,113335],{"className":113331},[31676],[34,113333],{"className":113334,"style":33218},[31680],[34,113336,67731],{"className":113337,"style":32604},[31685,31686],[31,113339,113340],{},"StudentName"," (学生姓名)",[108,113343,113344,2664,113374,113377],{},[2627,113345,113346,112634],{},[34,113347,113349,113362],{"className":113348},[31642],[34,113350,113352],{"className":113351},[31646],[31648,113353,113354],{"xmlns":31650},[31652,113355,113356,113360],{},[31655,113357,113358],{},[31658,113359,97300],{},[31665,113361,97300],{"encoding":31667},[34,113363,113365],{"className":113364,"ariaHidden":758},[31672],[34,113366,113368,113371],{"className":113367},[31676],[34,113369],{"className":113370,"style":33218},[31680],[34,113372,97300],{"className":113373,"style":38706},[31685,31686],[31,113375,113376],{},"Grade"," (成绩)",[32490,113379,112678,113381],{"id":113380},"_2-函数依赖集-fff-1",[34,113382,113384,113397],{"className":113383},[31642],[34,113385,113387],{"className":113386},[31646],[31648,113388,113389],{"xmlns":31650},[31652,113390,113391,113395],{},[31655,113392,113393],{},[31658,113394,51834],{},[31665,113396,51834],{"encoding":31667},[34,113398,113400],{"className":113399,"ariaHidden":758},[31672],[34,113401,113403,113406],{"className":113402},[31676],[34,113404],{"className":113405,"style":33218},[31680],[34,113407,51834],{"className":113408,"style":37422},[31685,31686],[105,113410,113411,113631],{},[108,113412,113413,113463,113464,113630],{},[34,113414,113416,113433],{"className":113415},[31642],[34,113417,113419],{"className":113418},[31646],[31648,113420,113421],{"xmlns":31650},[31652,113422,113423,113431],{},[31655,113424,113425,113427,113429],{},[31658,113426,67467],{},[31662,113428,33239],{},[31658,113430,97300],{},[31665,113432,112784],{"encoding":31667},[34,113434,113436,113454],{"className":113435,"ariaHidden":758},[31672],[34,113437,113439,113442,113445,113448,113451],{"className":113438},[31676],[34,113440],{"className":113441,"style":33218},[31680],[34,113443,67467],{"className":113444,"style":67482},[31685,31686],[34,113446],{"className":113447,"style":31846},[31845],[34,113449,33239],{"className":113450},[31850],[34,113452],{"className":113453,"style":31846},[31845],[34,113455,113457,113460],{"className":113456},[31676],[34,113458],{"className":113459,"style":33218},[31680],[34,113461,97300],{"className":113462,"style":38706},[31685,31686],"：即 ",[34,113465,113467,113531],{"className":113466},[31642],[34,113468,113470],{"className":113469},[31646],[31648,113471,113472],{"xmlns":31650},[31652,113473,113474,113528],{},[31655,113475,113476,113478,113480,113482,113484,113486,113488,113490,113492,113494,113496,113498,113500,113502,113504,113506,113508,113510,113512,113514,113516,113518,113520,113522,113524,113526],{},[31662,113477,456],{"stretchy":702},[31658,113479,94675],{},[31658,113481,33249],{},[31658,113483,94628],{},[31658,113485,46900],{},[31658,113487,33244],{},[31658,113489,31660],{},[31658,113491,33249],{},[31658,113493,106063],{},[31658,113495,91994],{},[31662,113497,31960],{"separator":758},[31658,113499,47384],{},[31658,113501,43383],{},[31658,113503,94628],{},[31658,113505,34395],{},[31658,113507,31747],{},[31658,113509,33244],{},[31658,113511,106063],{},[31658,113513,91994],{},[31662,113515,2557],{"stretchy":702},[31662,113517,33239],{},[31658,113519,97709],{},[31658,113521,34395],{},[31658,113523,111],{},[31658,113525,46900],{},[31658,113527,33244],{},[31665,113529,113530],{"encoding":31667},"(StudentID, CourseID) \\to Grade",[34,113532,113534,113609],{"className":113533,"ariaHidden":758},[31672],[34,113535,113537,113540,113543,113546,113549,113552,113555,113558,113561,113564,113567,113570,113573,113576,113579,113582,113585,113588,113591,113594,113597,113600,113603,113606],{"className":113536},[31676],[34,113538],{"className":113539,"style":31774},[31680],[34,113541,456],{"className":113542},[31782],[34,113544,94675],{"className":113545,"style":106455},[31685,31686],[34,113547,33249],{"className":113548},[31685,31686],[34,113550,94628],{"className":113551},[31685,31686],[34,113553,46900],{"className":113554},[31685,31686],[34,113556,33244],{"className":113557},[31685,31686],[34,113559,31660],{"className":113560},[31685,31686],[34,113562,33249],{"className":113563},[31685,31686],[34,113565,106063],{"className":113566,"style":67482},[31685,31686],[34,113568,91994],{"className":113569,"style":34452},[31685,31686],[34,113571,31960],{"className":113572},[32119],[34,113574],{"className":113575,"style":32123},[31845],[34,113577,47384],{"className":113578,"style":38706},[31685,31686],[34,113580,43383],{"className":113581},[31685,31686],[34,113583,94628],{"className":113584},[31685,31686],[34,113586,34395],{"className":113587,"style":34452},[31685,31686],[34,113589,113295],{"className":113590},[31685,31686],[34,113592,106063],{"className":113593,"style":67482},[31685,31686],[34,113595,91994],{"className":113596,"style":34452},[31685,31686],[34,113598,2557],{"className":113599},[31690],[34,113601],{"className":113602,"style":31846},[31845],[34,113604,33239],{"className":113605},[31850],[34,113607],{"className":113608,"style":31846},[31845],[34,113610,113612,113615,113618,113621,113624,113627],{"className":113611},[31676],[34,113613],{"className":113614,"style":31681},[31680],[34,113616,97709],{"className":113617},[31685,31686],[34,113619,34395],{"className":113620,"style":34452},[31685,31686],[34,113622,111],{"className":113623},[31685,31686],[34,113625,46900],{"className":113626},[31685,31686],[34,113628,33244],{"className":113629},[31685,31686],"（完全函数依赖）",[108,113632,113633,113734],{},[34,113634,113636,113674],{"className":113635},[31642],[34,113637,113639],{"className":113638},[31646],[31648,113640,113641],{"xmlns":31650},[31652,113642,113643,113671],{},[31655,113644,113645,113647,113649,113651,113653,113655,113657,113659,113661,113663,113665,113667,113669],{},[31662,113646,456],{"stretchy":702},[31658,113648,94675],{},[31658,113650,33249],{},[31658,113652,94628],{},[31658,113654,46900],{},[31658,113656,33244],{},[31658,113658,31660],{},[31658,113660,33249],{},[31658,113662,106063],{},[31658,113664,91994],{},[31662,113666,2557],{"stretchy":702},[31662,113668,33239],{},[31658,113670,67731],{},[31665,113672,113673],{"encoding":31667},"(StudentID) \\to Y",[34,113675,113677,113725],{"className":113676,"ariaHidden":758},[31672],[34,113678,113680,113683,113686,113689,113692,113695,113698,113701,113704,113707,113710,113713,113716,113719,113722],{"className":113679},[31676],[34,113681],{"className":113682,"style":31774},[31680],[34,113684,456],{"className":113685},[31782],[34,113687,94675],{"className":113688,"style":106455},[31685,31686],[34,113690,33249],{"className":113691},[31685,31686],[34,113693,94628],{"className":113694},[31685,31686],[34,113696,46900],{"className":113697},[31685,31686],[34,113699,33244],{"className":113700},[31685,31686],[34,113702,31660],{"className":113703},[31685,31686],[34,113705,33249],{"className":113706},[31685,31686],[34,113708,106063],{"className":113709,"style":67482},[31685,31686],[34,113711,91994],{"className":113712,"style":34452},[31685,31686],[34,113714,2557],{"className":113715},[31690],[34,113717],{"className":113718,"style":31846},[31845],[34,113720,33239],{"className":113721},[31850],[34,113723],{"className":113724,"style":31846},[31845],[34,113726,113728,113731],{"className":113727},[31676],[34,113729],{"className":113730,"style":33218},[31680],[34,113732,67731],{"className":113733,"style":32604},[31685,31686],"：即学号决定姓名（部分函数依赖）",[32490,113736,112819],{"id":113737},"_3-为什么不是传递依赖-1",[105,113739,113740,113843,113979,114039],{},[108,113741,113742,113743,86674,113793,347],{},"要构成传递依赖，必须满足 ",[34,113744,113746,113763],{"className":113745},[31642],[34,113747,113749],{"className":113748},[31646],[31648,113750,113751],{"xmlns":31650},[31652,113752,113753,113761],{},[31655,113754,113755,113757,113759],{},[31658,113756,67467],{},[31662,113758,33239],{},[31658,113760,67731],{},[31665,113762,111963],{"encoding":31667},[34,113764,113766,113784],{"className":113765,"ariaHidden":758},[31672],[34,113767,113769,113772,113775,113778,113781],{"className":113768},[31676],[34,113770],{"className":113771,"style":33218},[31680],[34,113773,67467],{"className":113774,"style":67482},[31685,31686],[34,113776],{"className":113777,"style":31846},[31845],[34,113779,33239],{"className":113780},[31850],[34,113782],{"className":113783,"style":31846},[31845],[34,113785,113787,113790],{"className":113786},[31676],[34,113788],{"className":113789,"style":33218},[31680],[34,113791,67731],{"className":113792,"style":32604},[31685,31686],[34,113794,113796,113813],{"className":113795},[31642],[34,113797,113799],{"className":113798},[31646],[31648,113800,113801],{"xmlns":31650},[31652,113802,113803,113811],{},[31655,113804,113805,113807,113809],{},[31658,113806,67731],{},[31662,113808,33239],{},[31658,113810,97300],{},[31665,113812,112014],{"encoding":31667},[34,113814,113816,113834],{"className":113815,"ariaHidden":758},[31672],[34,113817,113819,113822,113825,113828,113831],{"className":113818},[31676],[34,113820],{"className":113821,"style":33218},[31680],[34,113823,67731],{"className":113824,"style":32604},[31685,31686],[34,113826],{"className":113827,"style":31846},[31845],[34,113829,33239],{"className":113830},[31850],[34,113832],{"className":113833,"style":31846},[31845],[34,113835,113837,113840],{"className":113836},[31676],[34,113838],{"className":113839,"style":33218},[31680],[34,113841,97300],{"className":113842,"style":38706},[31685,31686],[108,113844,113845,113846,113896,113897,347],{},"在这个例子中，虽然 ",[34,113847,113849,113866],{"className":113848},[31642],[34,113850,113852],{"className":113851},[31646],[31648,113853,113854],{"xmlns":31650},[31652,113855,113856,113864],{},[31655,113857,113858,113860,113862],{},[31658,113859,67467],{},[31662,113861,33239],{},[31658,113863,67731],{},[31665,113865,111963],{"encoding":31667},[34,113867,113869,113887],{"className":113868,"ariaHidden":758},[31672],[34,113870,113872,113875,113878,113881,113884],{"className":113871},[31676],[34,113873],{"className":113874,"style":33218},[31680],[34,113876,67467],{"className":113877,"style":67482},[31685,31686],[34,113879],{"className":113880,"style":31846},[31845],[34,113882,33239],{"className":113883},[31850],[34,113885],{"className":113886,"style":31846},[31845],[34,113888,113890,113893],{"className":113889},[31676],[34,113891],{"className":113892,"style":33218},[31680],[34,113894,67731],{"className":113895,"style":32604},[31685,31686]," 成立（通过部分依赖），但 ",[2627,113898,113899],{},[34,113900,113902,113919],{"className":113901},[31642],[34,113903,113905],{"className":113904},[31646],[31648,113906,113907],{"xmlns":31650},[31652,113908,113909,113917],{},[31655,113910,113911,113913,113915],{},[31658,113912,67731],{},[31662,113914,112124],{},[31658,113916,97300],{},[31665,113918,112898],{"encoding":31667},[34,113920,113922,113970],{"className":113921,"ariaHidden":758},[31672],[34,113923,113925,113928,113931,113934,113961,113964,113967],{"className":113924},[31676],[34,113926],{"className":113927,"style":43217},[31680],[34,113929,67731],{"className":113930,"style":32604},[31685,31686],[34,113932],{"className":113933,"style":31846},[31845],[34,113935,113937],{"className":113936},[31850],[34,113938,113940],{"className":113939},[31685,54774],[34,113941,113943],{"className":113942},[54778],[34,113944,113946,113949,113958],{"className":113945},[54782],[34,113947],{"className":113948,"style":43217},[31680],[34,113950,113952],{"className":113951},[54789],[34,113953,113955],{"className":113954},[31685],[34,113956,54796],{"className":113957},[31850],[34,113959],{"className":113960},[54800],[34,113962],{"className":113963},[31845,54804],[34,113965,33239],{"className":113966},[31850],[34,113968],{"className":113969,"style":31846},[31845],[34,113971,113973,113976],{"className":113972},[31676],[34,113974],{"className":113975,"style":33218},[31680],[34,113977,97300],{"className":113978,"style":38706},[31685,31686],[108,113980,113981,113982,114010,114011,31901],{},"学生姓名（",[34,113983,113985,113998],{"className":113984},[31642],[34,113986,113988],{"className":113987},[31646],[31648,113989,113990],{"xmlns":31650},[31652,113991,113992,113996],{},[31655,113993,113994],{},[31658,113995,67731],{},[31665,113997,67731],{"encoding":31667},[34,113999,114001],{"className":114000,"ariaHidden":758},[31672],[34,114002,114004,114007],{"className":114003},[31676],[34,114005],{"className":114006,"style":33218},[31680],[34,114008,67731],{"className":114009,"style":32604},[31685,31686],"）无法决定某门课的成绩（",[34,114012,114014,114027],{"className":114013},[31642],[34,114015,114017],{"className":114016},[31646],[31648,114018,114019],{"xmlns":31650},[31652,114020,114021,114025],{},[31655,114022,114023],{},[31658,114024,97300],{},[31665,114026,97300],{"encoding":31667},[34,114028,114030],{"className":114029,"ariaHidden":758},[31672],[34,114031,114033,114036],{"className":114032},[31676],[34,114034],{"className":114035,"style":33218},[31680],[34,114037,97300],{"className":114038,"style":38706},[31685,31686],[108,114040,114041,114044,114045,114047],{},[2627,114042,114043],{},"结论","：该关系模式存在",[2627,114046,105534],{},"（违反 2NF），但不存在传递依赖。",[32188,114049],{},[253,114051,114052],{"id":114052},"对比分析",[13837,114054,114055,114071],{},[13840,114056,114057],{},[13843,114058,114059,114062,114065,114068],{},[13846,114060,114061],{"align":82355},"类型",[13846,114063,114064],{"align":82355},"函数依赖链",[13846,114066,114067],{"align":82355},"范式状态",[13846,114069,114070],{"align":82355},"核心特征",[13853,114072,114073,114324,114447],{},[13843,114074,114075,114080,114233,114236],{},[13858,114076,114077],{"align":82355},[2627,114078,114079],{},"直接依赖",[13858,114081,114082],{"align":82355},[34,114083,114085,114119],{"className":114084},[31642],[34,114086,114088],{"className":114087},[31646],[31648,114089,114090],{"xmlns":31650},[31652,114091,114092,114116],{},[31655,114093,114094,114096,114098,114100,114102,114104,114106,114108,114110,114112,114114],{},[31658,114095,67467],{},[31662,114097,33239],{},[31658,114099,67731],{},[31662,114101,31960],{"separator":758},[31658,114103,67467],{},[31662,114105,33239],{},[31658,114107,97300],{},[31662,114109,31960],{"separator":758},[31658,114111,67731],{},[31662,114113,112124],{},[31658,114115,97300],{},[31665,114117,114118],{"encoding":31667},"X \\to Y, X \\to Z, Y \\not\\to Z",[34,114120,114122,114140,114167,114224],{"className":114121,"ariaHidden":758},[31672],[34,114123,114125,114128,114131,114134,114137],{"className":114124},[31676],[34,114126],{"className":114127,"style":33218},[31680],[34,114129,67467],{"className":114130,"style":67482},[31685,31686],[34,114132],{"className":114133,"style":31846},[31845],[34,114135,33239],{"className":114136},[31850],[34,114138],{"className":114139,"style":31846},[31845],[34,114141,114143,114146,114149,114152,114155,114158,114161,114164],{"className":114142},[31676],[34,114144],{"className":114145,"style":70739},[31680],[34,114147,67731],{"className":114148,"style":32604},[31685,31686],[34,114150,31960],{"className":114151},[32119],[34,114153],{"className":114154,"style":32123},[31845],[34,114156,67467],{"className":114157,"style":67482},[31685,31686],[34,114159],{"className":114160,"style":31846},[31845],[34,114162,33239],{"className":114163},[31850],[34,114165],{"className":114166,"style":31846},[31845],[34,114168,114170,114173,114176,114179,114182,114185,114188,114215,114218,114221],{"className":114169},[31676],[34,114171],{"className":114172,"style":43217},[31680],[34,114174,97300],{"className":114175,"style":38706},[31685,31686],[34,114177,31960],{"className":114178},[32119],[34,114180],{"className":114181,"style":32123},[31845],[34,114183,67731],{"className":114184,"style":32604},[31685,31686],[34,114186],{"className":114187,"style":31846},[31845],[34,114189,114191],{"className":114190},[31850],[34,114192,114194],{"className":114193},[31685,54774],[34,114195,114197],{"className":114196},[54778],[34,114198,114200,114203,114212],{"className":114199},[54782],[34,114201],{"className":114202,"style":43217},[31680],[34,114204,114206],{"className":114205},[54789],[34,114207,114209],{"className":114208},[31685],[34,114210,54796],{"className":114211},[31850],[34,114213],{"className":114214},[54800],[34,114216],{"className":114217},[31845,54804],[34,114219,33239],{"className":114220},[31850],[34,114222],{"className":114223,"style":31846},[31845],[34,114225,114227,114230],{"className":114226},[31676],[34,114228],{"className":114229,"style":33218},[31680],[34,114231,97300],{"className":114232,"style":38706},[31685,31686],[13858,114234,114235],{"align":82355},"满足 3NF",[13858,114237,114238,258,114266,114294,114295,114323],{"align":82355},[34,114239,114241,114254],{"className":114240},[31642],[34,114242,114244],{"className":114243},[31646],[31648,114245,114246],{"xmlns":31650},[31652,114247,114248,114252],{},[31655,114249,114250],{},[31658,114251,67731],{},[31665,114253,67731],{"encoding":31667},[34,114255,114257],{"className":114256,"ariaHidden":758},[31672],[34,114258,114260,114263],{"className":114259},[31676],[34,114261],{"className":114262,"style":33218},[31680],[34,114264,67731],{"className":114265,"style":32604},[31685,31686],[34,114267,114269,114282],{"className":114268},[31642],[34,114270,114272],{"className":114271},[31646],[31648,114273,114274],{"xmlns":31650},[31652,114275,114276,114280],{},[31655,114277,114278],{},[31658,114279,97300],{},[31665,114281,97300],{"encoding":31667},[34,114283,114285],{"className":114284,"ariaHidden":758},[31672],[34,114286,114288,114291],{"className":114287},[31676],[34,114289],{"className":114290,"style":33218},[31680],[34,114292,97300],{"className":114293,"style":38706},[31685,31686]," 互不干扰，都是 ",[34,114296,114298,114311],{"className":114297},[31642],[34,114299,114301],{"className":114300},[31646],[31648,114302,114303],{"xmlns":31650},[31652,114304,114305,114309],{},[31655,114306,114307],{},[31658,114308,67467],{},[31665,114310,67467],{"encoding":31667},[34,114312,114314],{"className":114313,"ariaHidden":758},[31672],[34,114315,114317,114320],{"className":114316},[31676],[34,114318],{"className":114319,"style":33218},[31680],[34,114321,67467],{"className":114322,"style":67482},[31685,31686]," 的直接描述。",[13843,114325,114326,114330,114441,114444],{},[13858,114327,114328],{"align":82355},[2627,114329,105534],{},[13858,114331,114332],{"align":82355},[34,114333,114335,114369],{"className":114334},[31642],[34,114336,114338],{"className":114337},[31646],[31648,114339,114340],{"xmlns":31650},[31652,114341,114342,114366],{},[31655,114343,114344,114346,114348,114350,114352,114354,114356,114358,114360,114362,114364],{},[31662,114345,456],{"stretchy":702},[31658,114347,54151],{},[31662,114349,31960],{"separator":758},[31658,114351,54224],{},[31662,114353,2557],{"stretchy":702},[31662,114355,33239],{},[31658,114357,97300],{},[31662,114359,31960],{"separator":758},[31658,114361,54151],{},[31662,114363,33239],{},[31658,114365,67731],{},[31665,114367,114368],{"encoding":31667},"(A, B) \\to Z, A \\to Y",[34,114370,114372,114405,114432],{"className":114371,"ariaHidden":758},[31672],[34,114373,114375,114378,114381,114384,114387,114390,114393,114396,114399,114402],{"className":114374},[31676],[34,114376],{"className":114377,"style":31774},[31680],[34,114379,456],{"className":114380},[31782],[34,114382,54151],{"className":114383},[31685,31686],[34,114385,31960],{"className":114386},[32119],[34,114388],{"className":114389,"style":32123},[31845],[34,114391,54224],{"className":114392,"style":54252},[31685,31686],[34,114394,2557],{"className":114395},[31690],[34,114397],{"className":114398,"style":31846},[31845],[34,114400,33239],{"className":114401},[31850],[34,114403],{"className":114404,"style":31846},[31845],[34,114406,114408,114411,114414,114417,114420,114423,114426,114429],{"className":114407},[31676],[34,114409],{"className":114410,"style":70739},[31680],[34,114412,97300],{"className":114413,"style":38706},[31685,31686],[34,114415,31960],{"className":114416},[32119],[34,114418],{"className":114419,"style":32123},[31845],[34,114421,54151],{"className":114422},[31685,31686],[34,114424],{"className":114425,"style":31846},[31845],[34,114427,33239],{"className":114428},[31850],[34,114430],{"className":114431,"style":31846},[31845],[34,114433,114435,114438],{"className":114434},[31676],[34,114436],{"className":114437,"style":33218},[31680],[34,114439,67731],{"className":114440,"style":32604},[31685,31686],[13858,114442,114443],{"align":82355},"违反 2NF",[13858,114445,114446],{"align":82355},"非主属性只依赖于复合码的一部分。",[13843,114448,114449,114453,114622,114625],{},[13858,114450,114451],{"align":82355},[2627,114452,105542],{},[13858,114454,114455,114541,114542,2557],{"align":82355},[34,114456,114458,114484],{"className":114457},[31642],[34,114459,114461],{"className":114460},[31646],[31648,114462,114463],{"xmlns":31650},[31652,114464,114465,114481],{},[31655,114466,114467,114469,114471,114473,114475,114477,114479],{},[31658,114468,67467],{},[31662,114470,33239],{},[31658,114472,67731],{},[31662,114474,31960],{"separator":758},[31658,114476,67731],{},[31662,114478,33239],{},[31658,114480,97300],{},[31665,114482,114483],{"encoding":31667},"X \\to Y, Y \\to Z",[34,114485,114487,114505,114532],{"className":114486,"ariaHidden":758},[31672],[34,114488,114490,114493,114496,114499,114502],{"className":114489},[31676],[34,114491],{"className":114492,"style":33218},[31680],[34,114494,67467],{"className":114495,"style":67482},[31685,31686],[34,114497],{"className":114498,"style":31846},[31845],[34,114500,33239],{"className":114501},[31850],[34,114503],{"className":114504,"style":31846},[31845],[34,114506,114508,114511,114514,114517,114520,114523,114526,114529],{"className":114507},[31676],[34,114509],{"className":114510,"style":70739},[31680],[34,114512,67731],{"className":114513,"style":32604},[31685,31686],[34,114515,31960],{"className":114516},[32119],[34,114518],{"className":114519,"style":32123},[31845],[34,114521,67731],{"className":114522,"style":32604},[31685,31686],[34,114524],{"className":114525,"style":31846},[31845],[34,114527,33239],{"className":114528},[31850],[34,114530],{"className":114531,"style":31846},[31845],[34,114533,114535,114538],{"className":114534},[31676],[34,114536],{"className":114537,"style":33218},[31680],[34,114539,97300],{"className":114540,"style":38706},[31685,31686]," (且 ",[34,114543,114545,114562],{"className":114544},[31642],[34,114546,114548],{"className":114547},[31646],[31648,114549,114550],{"xmlns":31650},[31652,114551,114552,114560],{},[31655,114553,114554,114556,114558],{},[31658,114555,67731],{},[31662,114557,112124],{},[31658,114559,67467],{},[31665,114561,112129],{"encoding":31667},[34,114563,114565,114613],{"className":114564,"ariaHidden":758},[31672],[34,114566,114568,114571,114574,114577,114604,114607,114610],{"className":114567},[31676],[34,114569],{"className":114570,"style":43217},[31680],[34,114572,67731],{"className":114573,"style":32604},[31685,31686],[34,114575],{"className":114576,"style":31846},[31845],[34,114578,114580],{"className":114579},[31850],[34,114581,114583],{"className":114582},[31685,54774],[34,114584,114586],{"className":114585},[54778],[34,114587,114589,114592,114601],{"className":114588},[54782],[34,114590],{"className":114591,"style":43217},[31680],[34,114593,114595],{"className":114594},[54789],[34,114596,114598],{"className":114597},[31685],[34,114599,54796],{"className":114600},[31850],[34,114602],{"className":114603},[54800],[34,114605],{"className":114606},[31845,54804],[34,114608,33239],{"className":114609},[31850],[34,114611],{"className":114612,"style":31846},[31845],[34,114614,114616,114619],{"className":114615},[31676],[34,114617],{"className":114618,"style":33218},[31680],[34,114620,67467],{"className":114621,"style":67482},[31685,31686],[13858,114623,114624],{"align":82355},"违反 3NF",[13858,114626,114627,114655,114656,114684,114685,347],{"align":82355},[34,114628,114630,114643],{"className":114629},[31642],[34,114631,114633],{"className":114632},[31646],[31648,114634,114635],{"xmlns":31650},[31652,114636,114637,114641],{},[31655,114638,114639],{},[31658,114640,97300],{},[31665,114642,97300],{"encoding":31667},[34,114644,114646],{"className":114645,"ariaHidden":758},[31672],[34,114647,114649,114652],{"className":114648},[31676],[34,114650],{"className":114651,"style":33218},[31680],[34,114653,97300],{"className":114654,"style":38706},[31685,31686]," 通过中间人 ",[34,114657,114659,114672],{"className":114658},[31642],[34,114660,114662],{"className":114661},[31646],[31648,114663,114664],{"xmlns":31650},[31652,114665,114666,114670],{},[31655,114667,114668],{},[31658,114669,67731],{},[31665,114671,67731],{"encoding":31667},[34,114673,114675],{"className":114674,"ariaHidden":758},[31672],[34,114676,114678,114681],{"className":114677},[31676],[34,114679],{"className":114680,"style":33218},[31680],[34,114682,67731],{"className":114683,"style":32604},[31685,31686]," 间接依赖于 ",[34,114686,114688,114701],{"className":114687},[31642],[34,114689,114691],{"className":114690},[31646],[31648,114692,114693],{"xmlns":31650},[31652,114694,114695,114699],{},[31655,114696,114697],{},[31658,114698,67467],{},[31665,114700,67467],{"encoding":31667},[34,114702,114704],{"className":114703,"ariaHidden":758},[31672],[34,114705,114707,114710],{"className":114706},[31676],[34,114708],{"className":114709,"style":33218},[31680],[34,114711,67467],{"className":114712,"style":67482},[31685,31686],{"title":29,"searchDepth":43,"depth":43,"links":114714},[114715,114716,114717,114724,114735],{"id":104554,"depth":43,"text":104555},{"id":104848,"depth":43,"text":104849},{"id":105374,"depth":43,"text":105375,"children":114718},[114719,114720,114721,114722,114723],{"id":105378,"depth":49,"text":105379},{"id":105489,"depth":49,"text":105490},{"id":105516,"depth":49,"text":105517},{"id":105557,"depth":49,"text":105558},{"id":105624,"depth":49,"text":105624},{"id":105690,"depth":43,"text":105691,"children":114725},[114726,114727,114728,114729,114731,114732,114733,114734],{"id":105700,"depth":49,"text":105701},{"id":105742,"depth":49,"text":105743},{"id":105783,"depth":49,"text":105784},{"id":107424,"depth":49,"text":114730},"3. 函数依赖关系分析 (FFF)",{"id":108995,"depth":49,"text":108996},{"id":110101,"depth":49,"text":110102},{"id":110931,"depth":49,"text":110932},{"id":111832,"depth":49,"text":111832},{"id":111840,"depth":43,"text":114736,"children":114737},"设计一个不是传递依赖的 X,Y,ZX, Y, ZX,Y,Z 关系",[114738,114739,114740],{"id":112407,"depth":49,"text":112408},{"id":113133,"depth":49,"text":113134},{"id":114052,"depth":49,"text":114052},{},"\u002Fsource\u002Fcs\u002Fdb\u002Fnotes\u002Fnormalization",{"title":104547,"description":29},"source\u002Fcs\u002Fdb\u002Fnotes\u002Fnormalization","5BImyTMmdQ8Shz3Hur8c7UyKfQ-uN08tRAA8FQaowq8",{"id":114747,"title":114748,"body":114749,"description":118532,"extension":85,"meta":118533,"navigation":52,"path":118534,"seo":118535,"stem":118536,"__hash__":118537},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fpastpapers\u002F25-26cs.md","2025-2026-2 计科数据库系统期末",{"type":8,"value":114750,"toc":118480},[114751,114755,114780,114826,114828,114832,115089,115091,115095,115136,115138,115142,115153,115155,115159,115162,115165,115168,115171,115174,115182,115185,115196,115198,115202,115205,115222,115224,115235,115237,115241,115244,115247,115250,115253,115256,115259,115262,115264,115267,115270,115274,115276,115287,115289,115292,115296,115298,115309,115311,115314,115318,115320,115331,115333,115336,115340,115342,115353,115355,115388,115392,115394,115405,115407,115410,115414,115416,115427,115429,115432,115436,115438,115449,115451,115454,115458,115460,115471,115473,115476,115480,115482,115493,115495,115498,115502,115504,115515,115517,115520,115524,115526,115537,115539,115542,115546,115548,115559,115561,115564,115568,115570,115581,115583,115586,115590,115592,115603,115605,115608,115612,115614,115622,115624,115627,115629,115632,115635,115644,115647,115650,115653,115660,115663,115666,115669,115676,115679,115682,115685,115693,115696,115699,115702,115710,115713,115716,115718,115721,115724,115726,115729,115732,116286,116289,116291,116293,116316,116351,116354,116357,116359,116361,116364,116410,116412,116415,116418,116420,116422,116424,116427,116429,116431,116434,116749,116810,116813,116815,116817,116819,117327,117330,117333,117335,117337,117340,117359,117362,117365,117367,117369,117371,117429,117432,117435,117437,117439,117441,117493,117496,117498,117501,117504,117506,117518,117521,117523,117525,117528,117545,117548,117607,117610,117619,117627,117636,117643,117651,117659,117662,117665,117667,117669,117674,117677,117680,117685,117688,117691,117694,117699,117702,117705,117708,117713,117716,117719,117721,117726,117729,117732,117734,117739,117742,117745,117748,117751,117756,117759,117762,117765,117767,117770,117773,117775,117777,117780,117803,117806,118251,118254,118256,118259,118262,118264,118266,118268,118270,118272,118274,118276,118278,118281,118421,118424,118438,118440,118478],[11,114752,114754],{"id":114753},"西北大学-2025-2026-学年第-2-学期本科考试出题专用纸","西北大学 2025 —— 2026 学年第 2 学期本科考试出题专用纸",[15,114756,114757,114760,114761,114764,114765,114767,114768,114771,114772,114775,114776,114779],{},[2627,114758,114759],{},"考试科目："," 数据库系统\n",[2627,114762,114763],{},"本卷为："," 闭卷\n",[2627,114766,114763],{}," A 卷\n",[2627,114769,114770],{},"印数："," 110\n",[2627,114773,114774],{},"出题院系："," 计算机学院\n",[2627,114777,114778],{},"出题日期："," 2026 年 4 月 20 日",[13837,114781,114782,114807],{},[13840,114783,114784],{},[13843,114785,114786,114789,114792,114795,114798,114801,114804],{},[13846,114787,114788],{},"题号",[13846,114790,114791],{},"一",[13846,114793,114794],{},"二",[13846,114796,114797],{},"三",[13846,114799,114800],{},"四",[13846,114802,114803],{},"五",[13846,114805,114806],{},"六",[13853,114808,114809],{},[13843,114810,114811,114814,114816,114818,114820,114822,114824],{},[13858,114812,114813],{},"得分",[13858,114815],{},[13858,114817],{},[13858,114819],{},[13858,114821],{},[13858,114823],{},[13858,114825],{},[32188,114827],{},[19,114829,114831],{"id":114830},"一选择题1-分题-15-题-15-分","一、选择题（1 分\u002F题 * 15 题 = 15 分）",[2230,114833,114834,114851,114868,114885,114902,114919,114936,114953,114970,114987,115004,115021,115038,115055,115072],{},[108,114835,114836,114837,114839,114840,114842,114843,114845,114846,114848,114849],{},"数据库（DB）、数据库管理系统（DBMS）和数据库系统（DBS）三者之间的关系是（ ）",[58531,114838],{},"A. DB 包含 DBMS 和 DBS",[58531,114841],{},"B. DBS 包含 DB 和 DBMS",[58531,114844],{},"C. DBMS 包含 DB 和 DBS",[58531,114847],{},"D. DBS 就是 DBMS",[58531,114850],{},[108,114852,114853,114854,114856,114857,114859,114860,114862,114863,114865,114866],{},"数据独立性是指（ ）",[58531,114855],{},"A. 数据之间相互独立",[58531,114858],{},"B. 应用程序与数据库的数据结构相互独立",[58531,114861],{},"C. 数据存储与数据逻辑相互独立",[58531,114864],{},"D. 数据与操作系统相互独立",[58531,114867],{},[108,114869,114870,114871,114873,114874,114876,114877,114879,114880,114882,114883],{},"关系数据库中，元组对应传统文件中的（ ）",[58531,114872],{},"A. 记录",[58531,114875],{},"B. 字段",[58531,114878],{},"C. 文件",[58531,114881],{},"D. 数据项",[58531,114884],{},[108,114886,114887,114888,114890,114891,114893,114894,114896,114897,114899,114900],{},"关系代数中，从关系中取出满足条件的元组的操作称为（ ）",[58531,114889],{},"A. 投影",[58531,114892],{},"B. 选择",[58531,114895],{},"C. 连接",[58531,114898],{},"D. 除",[58531,114901],{},[108,114903,114904,114905,114907,114908,114910,114911,114913,114914,114916,114917],{},"SQL 语言中，用于删除表结构的命令是（ ）",[58531,114906],{},"A. DELETE",[58531,114909],{},"B. DROP",[58531,114912],{},"C. CLEAR",[58531,114915],{},"D. REMOVE",[58531,114918],{},[108,114920,114921,114922,114924,114925,114927,114928,114930,114931,114933,114934],{},"视图创建完成后，数据字典中存放的是（ ）",[58531,114923],{},"A. 查询语句",[58531,114926],{},"B. 查询结果",[58531,114929],{},"C. 视图所包含的数据",[58531,114932],{},"D. 视图所依赖的基本表",[58531,114935],{},[108,114937,114938,114939,114941,114942,114944,114945,114947,114948,114950,114951],{},"数据库触发器的主要作用是（ ）。",[58531,114940],{},"A. 优化查询执行计划",[58531,114943],{},"B. 提高数据插入速度",[58531,114946],{},"C. 实现用户权限管理",[58531,114949],{},"D. 自动维护数据完整性规则",[58531,114952],{},[108,114954,114955,114956,114958,114959,114961,114962,114964,114965,114967,114968],{},"将 E-R 图转换为关系模型，属于数据库设计的（ ）",[58531,114957],{},"A. 需求分析",[58531,114960],{},"B. 概念设计",[58531,114963],{},"C. 逻辑设计",[58531,114966],{},"D. 物理设计",[58531,114969],{},[108,114971,114972,114973,114975,114976,114978,114979,114981,114982,114984,114985],{},"若一个关系模式 R 属于 1NF，且它的每一个非主属性都完全函数依赖于主键，则 R 属于（ ）",[58531,114974],{},"A. 1NF",[58531,114977],{},"B. 2NF",[58531,114980],{},"C. 3NF",[58531,114983],{},"D. BCNF",[58531,114986],{},[108,114988,114989,114990,114992,114993,114995,114996,114998,114999,115001,115002],{},"在数据库中，关于游标的描述，错误的是（ ）",[58531,114991],{},"A. 游标可以对查询结果集进行逐行处理",[58531,114994],{},"B. 使用游标通常需要经历声明、打开、读取、关闭、释放几个步骤",[58531,114997],{},"C. 游标常用于存储过程中处理复杂的逐行逻辑",[58531,115000],{},"D. 游标能提高大数据量批量操作的执行效率",[58531,115003],{},[108,115005,115006,115007,115009,115010,115012,115013,115015,115016,115018,115019],{},"数据库中，数据的正确性、有效性和相容性称为（ ）",[58531,115008],{},"A. 安全性",[58531,115011],{},"B. 完整性",[58531,115014],{},"C. 并发控制",[58531,115017],{},"D. 恢复",[58531,115020],{},[108,115022,115023,115024,115026,115027,115029,115030,115032,115033,115035,115036],{},"事务的原子性是指（ ）",[58531,115025],{},"A. 事务要么全做，要么全不做",[58531,115028],{},"B. 事务执行后数据库永远一致",[58531,115031],{},"C. 事务之间相互隔离",[58531,115034],{},"D. 事务提交后结果永久有效",[58531,115037],{},[108,115039,115040,115041,115043,115044,115046,115047,115049,115050,115052,115053],{},"数据库中，“读脏数据”是由（ ）引起的",[58531,115042],{},"A. 事务未提交就被其他事务读取",[58531,115045],{},"B. 不可重复读",[58531,115048],{},"C. 幻影读",[58531,115051],{},"D. 死锁",[58531,115054],{},[108,115056,115057,115058,115060,115061,115063,115064,115066,115067,115069,115070],{},"解决并发操作带来的数据不一致问题，普遍采用（ ）",[58531,115059],{},"A. 封锁机制",[58531,115062],{},"B. 存取控制",[58531,115065],{},"C. 日志文件",[58531,115068],{},"D. 备份恢复",[58531,115071],{},[108,115073,115074,115075,115077,115078,115080,115081,115083,115084,115086,115087],{},"在关系代数优化中，下列策略不正确的是（ ）",[58531,115076],{},"A. 尽可能早地执行选择运算",[58531,115079],{},"\nB. 尽可能早地执行投影运算",[58531,115082],{},"\nC. 把选择与笛卡尔积结合转换为等值连接",[58531,115085],{},"\nD. 先做笛卡尔积，再做选择运算",[58531,115088],{},[32188,115090],{},[19,115092,115094],{"id":115093},"二填空题2-分空-10-空-20-分","二、填空题（2 分\u002F空 * 10 空 = 20 分）",[2230,115096,115097,115105,115112,115120,115128],{},[108,115098,115099,115100,258,115103,347],{},"建立冗余数据的方式有 ",[2627,115101,115102],{},"__",[2627,115104,115102],{},[108,115106,115107,115108,258,115110,347],{},"数据转储的方法有 ",[2627,115109,115102],{},[2627,115111,115102],{},[108,115113,115114,115115,258,115117,347],{},"查询优化包含两类，有 ",[2627,115116,115102],{},[2627,115118,115119],{},"__*",[108,115121,115122,115123,115125,115126,347],{},"游标是系统为用户开设的一个 ",[2627,115124,115102],{},"，用来存放 SQL 语句的 ",[2627,115127,115102],{},[108,115129,115130,115131,146,115133,115135],{},"数据库工作单元与源程序工作单元之间的通信方式有 ",[2627,115132,115102],{},[2627,115134,115102],{}," 和游标。",[32188,115137],{},[19,115139,115141],{"id":115140},"三简答题5-分题-3-题-15-分","三、简答题（5 分\u002F题 * 3 题 = 15 分）",[2230,115143,115144,115147,115150],{},[108,115145,115146],{},"将 1NF 的函数关系逐步分解为 BCNF，说明每一步消除了什么类型的函数依赖。",[108,115148,115149],{},"解释共享锁和排他锁的含义，并给出其相容矩阵。",[108,115151,115152],{},"简述数据库故障的主要分类及各类故障的成因与恢复方式。",[32188,115154],{},[19,115156,115158],{"id":115157},"四关系代数及-sql-语句4-分题-5-题-20-分","四、关系代数及 SQL 语句（4 分\u002F题 * 5 题 = 20 分）",[15,115160,115161],{},"有如下三个关系：",[15,115163,115164],{},"供应商表 Supplier（供应商编号 Sno，供应商名称 Sname，所在城市 City）",[15,115166,115167],{},"零件表 Part（零件编号 Pno，零件名称 Pname，零件规格 Spec）",[15,115169,115170],{},"供应表 Supply（供应商编号 Sno，零件编号 Pno，供应数量 Qty）",[15,115172,115173],{},"写出下列两个问题的关系代数表达式：",[2230,115175,115176,115179],{},[108,115177,115178],{},"查询所有位于“西安”的供应商名称和供应商编号。",[108,115180,115181],{},"查询供应了 P1 零件的供应商名称、所在城市及供应数量。",[15,115183,115184],{},"用 SQL 语句回答下列三个问题：",[2230,115186,115187,115190,115193],{"start":49},[108,115188,115189],{},"查询供应商 S1 供应的零件总数（统计字段命名为 Total_Qty）。",[108,115191,115192],{},"查询供应数量大于 100 的 P2 零件的供应商名称、零件规格及供应数量。",[108,115194,115195],{},"查询没有供应零件 P3 的供应商信息。",[32188,115197],{},[19,115199,115201],{"id":115200},"五数据库设计共-20-分","五、数据库设计（共 20 分）",[15,115203,115204],{},"某高校拟开发教学管理信息系统，用于管理学生、专业、课程、教师、学院等核心业务对象，系统需满足以下业务需求：",[105,115206,115207,115210,115213,115216,115219],{},[108,115208,115209],{},"学生与专业：每个学生主修一个专业，一个专业可招收多名学生；学生需选修多门课程，每门课程可被多名学生选修，需记录学生每门课程的选修成绩。",[108,115211,115212],{},"专业与学院：每个学院可设置多个专业，每个专业仅隶属于一个学院。",[108,115214,115215],{},"课程与学院：每门课程由一个学院开设，一个学院可开设多门课程。",[108,115217,115218],{},"课程与教师：一门课程可由多名教师讲授，一名教师可讲授多门课程，需记录每门课程的授课教室与课时信息。",[108,115220,115221],{},"教师与学院：每名教师隶属于一个学院，一个学院可拥有多名教师。",[15,115223,101191],{},[2230,115225,115226,115229,115232],{},[108,115227,115228],{},"请根据上述业务需求，完成该系统的概念结构设计，绘制对应的 E-R 图，要求标注所有实体型、联系型；标注每个联系的类型（1:1、1:n、m:n）；标注联系的属性（如成绩、教室、课时）。（6 分）",[108,115230,115231],{},"请将该 ER 图转换为关系数据库的关系模式，要求标注每个关系模式的主键和外键。（7 分）",[108,115233,115234],{},"指出上述关系模式分别为第几范式，为什么。（7 分）",[32188,115236],{},[19,115238,115240],{"id":115239},"六sql-编程共-10-分","六、SQL 编程（共 10 分）",[15,115242,115243],{},"假设存在一个员工信息表 Employees（emp_id, emp_name, dept_id, salary），请编写一个 SQL 存储过程，要求如下：",[15,115245,115246],{},"存储过程名称：proc_get_employee_name;",[15,115248,115249],{},"输入参数：in_emp_id INT（员工 ID）；",[15,115251,115252],{},"输出参数：out_emp_name VARCHAR(50)（员工姓名）；",[15,115254,115255],{},"功能要求：",[15,115257,115258],{},"若输入的员工 ID 存在，将该员工的姓名赋值给输出参数；",[15,115260,115261],{},"若输入的员工 ID 不存在或为 NULL，输出参数赋值为“无此员工”。",[32188,115263],{},[11,115265,115266],{"id":115266},"数据库系统概论题目答案与解析",[19,115268,114831],{"id":115269},"一选择题1-分题-15-题-15-分-1",[253,115271,115273],{"id":115272},"_1-题目","1. 题目",[15,115275,114836],{},[15,115277,114839,115278,115280,115281,115283,115284,115286],{},[58531,115279],{},"\nB. DBS 包含 DB 和 DBMS",[58531,115282],{},"\nC. DBMS 包含 DB 和 DBS",[58531,115285],{},"\nD. DBS 就是 DBMS",[15,115288,58547],{},[15,115290,115291],{},"解析：\n数据库系统 DBS 通常由数据库 DB、数据库管理系统 DBMS、应用程序和数据库管理员等组成。因此 DBS 包含 DB 和 DBMS。",[253,115293,115295],{"id":115294},"_2-题目","2. 题目",[15,115297,114853],{},[15,115299,114856,115300,115302,115303,115305,115306,115308],{},[58531,115301],{},"\nB. 应用程序与数据库的数据结构相互独立",[58531,115304],{},"\nC. 数据存储与数据逻辑相互独立",[58531,115307],{},"\nD. 数据与操作系统相互独立",[15,115310,58547],{},[15,115312,115313],{},"解析：\n数据独立性是指应用程序与数据库中数据的逻辑结构、物理存储结构之间具有相对独立性。教材口径中包括逻辑独立性和物理独立性。",[253,115315,115317],{"id":115316},"_3-题目","3. 题目",[15,115319,114870],{},[15,115321,114873,115322,115324,115325,115327,115328,115330],{},[58531,115323],{},"\nB. 字段",[58531,115326],{},"\nC. 文件",[58531,115329],{},"\nD. 数据项",[15,115332,91106],{},[15,115334,115335],{},"解析：\n关系中的一个元组对应传统文件中的一条记录；属性对应字段。",[253,115337,115339],{"id":115338},"_4-题目","4. 题目",[15,115341,114887],{},[15,115343,114890,115344,115346,115347,115349,115350,115352],{},[58531,115345],{},"\nB. 选择",[58531,115348],{},"\nC. 连接",[58531,115351],{},"\nD. 除",[15,115354,58547],{},[15,115356,115357,115358,347],{},"解析：\n选择运算是从关系中选出满足给定条件的元组，记作 ",[34,115359,115361,115376],{"className":115360},[31642],[34,115362,115364],{"className":115363},[31646],[31648,115365,115366],{"xmlns":31650},[31652,115367,115368,115373],{},[31655,115369,115370],{},[31658,115371,115372],{},"σ",[31665,115374,115375],{"encoding":31667},"\\sigma",[34,115377,115379],{"className":115378,"ariaHidden":758},[31672],[34,115380,115382,115385],{"className":115381},[31676],[34,115383],{"className":115384,"style":32527},[31680],[34,115386,115372],{"className":115387,"style":32062},[31685,31686],[253,115389,115391],{"id":115390},"_5-题目","5. 题目",[15,115393,114904],{},[15,115395,114907,115396,115398,115399,115401,115402,115404],{},[58531,115397],{},"\nB. DROP",[58531,115400],{},"\nC. CLEAR",[58531,115403],{},"\nD. REMOVE",[15,115406,58547],{},[15,115408,115409],{},"解析：\nDROP TABLE 用于删除基本表的定义及其数据；DELETE 只删除表中的数据，不删除表结构。",[253,115411,115413],{"id":115412},"_6-题目","6. 题目",[15,115415,114921],{},[15,115417,114924,115418,115420,115421,115423,115424,115426],{},[58531,115419],{},"\nB. 查询结果",[58531,115422],{},"\nC. 视图所包含的数据",[58531,115425],{},"\nD. 视图所依赖的基本表",[15,115428,91106],{},[15,115430,115431],{},"解析：\n视图是虚表。数据库中只保存视图的定义，即查询语句，不保存视图对应的查询结果。",[253,115433,115435],{"id":115434},"_7-题目","7. 题目",[15,115437,114938],{},[15,115439,114941,115440,115442,115443,115445,115446,115448],{},[58531,115441],{},"\nB. 提高数据插入速度",[58531,115444],{},"\nC. 实现用户权限管理",[58531,115447],{},"\nD. 自动维护数据完整性规则",[15,115450,59391],{},[15,115452,115453],{},"解析：\n触发器可在插入、删除、修改数据时自动执行，常用于实现复杂的完整性约束和业务规则。",[253,115455,115457],{"id":115456},"_8-题目","8. 题目",[15,115459,114955],{},[15,115461,114958,115462,115464,115465,115467,115468,115470],{},[58531,115463],{},"\nB. 概念设计",[58531,115466],{},"\nC. 逻辑设计",[58531,115469],{},"\nD. 物理设计",[15,115472,61063],{},[15,115474,115475],{},"解析：\nE-R 图属于概念结构设计成果，将 E-R 图转换为关系模式属于逻辑结构设计。",[253,115477,115479],{"id":115478},"_9-题目","9. 题目",[15,115481,114972],{},[15,115483,114975,115484,115486,115487,115489,115490,115492],{},[58531,115485],{},"\nB. 2NF",[58531,115488],{},"\nC. 3NF",[58531,115491],{},"\nD. BCNF",[15,115494,58547],{},[15,115496,115497],{},"解析：\n2NF 的要求是在 1NF 基础上，每一个非主属性都完全函数依赖于任何一个候选码，消除非主属性对码的部分函数依赖。",[253,115499,115501],{"id":115500},"_10-题目","10. 题目",[15,115503,114989],{},[15,115505,114992,115506,115508,115509,115511,115512,115514],{},[58531,115507],{},"\nB. 使用游标通常需要经历声明、打开、读取、关闭、释放几个步骤",[58531,115510],{},"\nC. 游标常用于存储过程中处理复杂的逐行逻辑",[58531,115513],{},"\nD. 游标能提高大数据量批量操作的执行效率",[15,115516,59391],{},[15,115518,115519],{},"解析：\n游标适合逐行处理复杂逻辑，但通常效率低于集合操作。大数据量批量操作一般不应优先使用游标。",[253,115521,115523],{"id":115522},"_11-题目","11. 题目",[15,115525,115006],{},[15,115527,115009,115528,115530,115531,115533,115534,115536],{},[58531,115529],{},"\nB. 完整性",[58531,115532],{},"\nC. 并发控制",[58531,115535],{},"\nD. 恢复",[15,115538,58547],{},[15,115540,115541],{},"解析：\n数据库完整性是指数据的正确性、有效性和相容性。",[253,115543,115545],{"id":115544},"_12-题目","12. 题目",[15,115547,115023],{},[15,115549,115026,115550,115552,115553,115555,115556,115558],{},[58531,115551],{},"\nB. 事务执行后数据库永远一致",[58531,115554],{},"\nC. 事务之间相互隔离",[58531,115557],{},"\nD. 事务提交后结果永久有效",[15,115560,91106],{},[15,115562,115563],{},"解析：\n原子性要求事务中的操作要么全部完成，要么全部不做。",[253,115565,115567],{"id":115566},"_13-题目","13. 题目",[15,115569,115040],{},[15,115571,115043,115572,115574,115575,115577,115578,115580],{},[58531,115573],{},"\nB. 不可重复读",[58531,115576],{},"\nC. 幻影读",[58531,115579],{},"\nD. 死锁",[15,115582,91106],{},[15,115584,115585],{},"解析：\n读脏数据是指一个事务读取了另一个事务尚未提交的数据。",[253,115587,115589],{"id":115588},"_14-题目","14. 题目",[15,115591,115057],{},[15,115593,115060,115594,115596,115597,115599,115600,115602],{},[58531,115595],{},"\nB. 存取控制",[58531,115598],{},"\nC. 日志文件",[58531,115601],{},"\nD. 备份恢复",[15,115604,91106],{},[15,115606,115607],{},"解析：\n并发控制中普遍采用封锁机制来保证事务调度的正确性，防止数据不一致。",[253,115609,115611],{"id":115610},"_15-题目","15. 题目",[15,115613,115074],{},[15,115615,115077,115616,115080,115618,115083,115620,115086],{},[58531,115617],{},[58531,115619],{},[58531,115621],{},[15,115623,59391],{},[15,115625,115626],{},"解析：\n关系代数优化通常应尽早执行选择和投影，减少中间结果规模。先做笛卡尔积会产生大量无用中间结果，因此不正确。",[32188,115628],{},[19,115630,115094],{"id":115631},"二填空题2-分空-10-空-20-分-1",[253,115633,115273],{"id":115634},"_1-题目-1",[15,115636,115637,115638,115641,115642,347],{},"建立冗余数据的方式有 *",[2627,115639,115640],{},"*__**"," 和 *",[2627,115643,115640],{},[15,115645,115646],{},"答案：\n数据转储；登记日志文件",[15,115648,115649],{},"解析：\n数据库恢复的基本原理是利用冗余数据。建立冗余数据最常用的技术是数据转储和登记日志文件。",[253,115651,115295],{"id":115652},"_2-题目-1",[15,115654,115655,115656,115641,115658,347],{},"数据转储的方法有 *",[2627,115657,115640],{},[2627,115659,115640],{},[15,115661,115662],{},"答案：\n静态转储；动态转储",[15,115664,115665],{},"解析：\n按转储期间是否允许事务运行，可分为静态转储和动态转储。补充：若按转储范围划分，也可分为海量转储和增量转储。",[253,115667,115317],{"id":115668},"_3-题目-1",[15,115670,115671,115672,115641,115674,347],{},"查询优化包含两类，有 *",[2627,115673,115640],{},[2627,115675,115640],{},[15,115677,115678],{},"答案：\n代数优化；物理优化",[15,115680,115681],{},"解析：\n查询优化通常包括代数优化和物理优化。代数优化主要改变查询语句中操作的次序和组合；物理优化主要选择合适的存取路径和执行算法。",[253,115683,115339],{"id":115684},"_4-题目-1",[15,115686,115687,115688,115690,115691,347],{},"游标是系统为用户开设的一个 *",[2627,115689,115640],{},"，用来存放 SQL 语句的 *",[2627,115692,115640],{},[15,115694,115695],{},"答案：\n数据缓冲区；执行结果",[15,115697,115698],{},"解析：\n游标用于处理 SQL 查询结果集。它相当于系统为用户开设的数据缓冲区，用来存放 SQL 语句的执行结果。",[253,115700,115391],{"id":115701},"_5-题目-1",[15,115703,115704,115705,115707,115708,115135],{},"数据库工作单元与源程序工作单元之间的通信方式有 *",[2627,115706,115640],{},"、*",[2627,115709,115640],{},[15,115711,115712],{},"答案：\nSQL 通信区；主变量",[15,115714,115715],{},"解析：\n嵌入式 SQL 中，数据库工作单元和源程序工作单元之间主要通过 SQL 通信区、主变量和游标进行通信。",[32188,115717],{},[19,115719,115141],{"id":115720},"三简答题5-分题-3-题-15-分-1",[253,115722,115273],{"id":115723},"_1-题目-2",[15,115725,115146],{},[15,115727,115728],{},"答案与解析：",[15,115730,115731],{},"从 1NF 到 BCNF 的规范化过程如下：",[2230,115733,115734,115939,116166],{},[108,115735,115736,115737,115739,115740,115742,115743,115745,115746,93535,115798,115849,115851,115852,115880,115881,115909,115910,115938],{},"1NF → 2NF",[58531,115738],{},"要求：\n在 1NF 的基础上，每一个非主属性都完全函数依赖于任何一个候选码。",[58531,115741],{},"消除的问题：\n消除非主属性对码的部分函数依赖。",[58531,115744],{},"即若存在：\n",[34,115747,115749,115768],{"className":115748},[31642],[34,115750,115752],{"className":115751},[31646],[31648,115753,115754],{"xmlns":31650},[31652,115755,115756,115765],{},[31655,115757,115758,115760,115763],{},[31658,115759,67467],{},[31662,115761,115762],{},"⊂",[31658,115764,38691],{},[31665,115766,115767],{"encoding":31667},"X \\subset K",[34,115769,115771,115789],{"className":115770,"ariaHidden":758},[31672],[34,115772,115774,115777,115780,115783,115786],{"className":115773},[31676],[34,115775],{"className":115776,"style":104901},[31680],[34,115778,67467],{"className":115779,"style":67482},[31685,31686],[34,115781],{"className":115782,"style":31846},[31845],[34,115784,115762],{"className":115785},[31850],[34,115787],{"className":115788,"style":31846},[31845],[34,115790,115792,115795],{"className":115791},[31676],[34,115793],{"className":115794,"style":33218},[31680],[34,115796,38691],{"className":115797,"style":38706},[31685,31686],[34,115799,115801,115819],{"className":115800},[31642],[34,115802,115804],{"className":115803},[31646],[31648,115805,115806],{"xmlns":31650},[31652,115807,115808,115816],{},[31655,115809,115810,115812,115814],{},[31658,115811,67467],{},[31662,115813,33239],{},[31658,115815,54151],{},[31665,115817,115818],{"encoding":31667},"X \\to A",[34,115820,115822,115840],{"className":115821,"ariaHidden":758},[31672],[34,115823,115825,115828,115831,115834,115837],{"className":115824},[31676],[34,115826],{"className":115827,"style":33218},[31680],[34,115829,67467],{"className":115830,"style":67482},[31685,31686],[34,115832],{"className":115833,"style":31846},[31845],[34,115835,33239],{"className":115836},[31850],[34,115838],{"className":115839,"style":31846},[31845],[34,115841,115843,115846],{"className":115842},[31676],[34,115844],{"className":115845,"style":33218},[31680],[34,115847,54151],{"className":115848},[31685,31686],[58531,115850],{},"其中 ",[34,115853,115855,115868],{"className":115854},[31642],[34,115856,115858],{"className":115857},[31646],[31648,115859,115860],{"xmlns":31650},[31652,115861,115862,115866],{},[31655,115863,115864],{},[31658,115865,38691],{},[31665,115867,38691],{"encoding":31667},[34,115869,115871],{"className":115870,"ariaHidden":758},[31672],[34,115872,115874,115877],{"className":115873},[31676],[34,115875],{"className":115876,"style":33218},[31680],[34,115878,38691],{"className":115879,"style":38706},[31685,31686]," 是候选码，",[34,115882,115884,115897],{"className":115883},[31642],[34,115885,115887],{"className":115886},[31646],[31648,115888,115889],{"xmlns":31650},[31652,115890,115891,115895],{},[31655,115892,115893],{},[31658,115894,54151],{},[31665,115896,54151],{"encoding":31667},[34,115898,115900],{"className":115899,"ariaHidden":758},[31672],[34,115901,115903,115906],{"className":115902},[31676],[34,115904],{"className":115905,"style":33218},[31680],[34,115907,54151],{"className":115908},[31685,31686]," 是非主属性，则说明 ",[34,115911,115913,115926],{"className":115912},[31642],[34,115914,115916],{"className":115915},[31646],[31648,115917,115918],{"xmlns":31650},[31652,115919,115920,115924],{},[31655,115921,115922],{},[31658,115923,54151],{},[31665,115925,54151],{"encoding":31667},[34,115927,115929],{"className":115928,"ariaHidden":758},[31672],[34,115930,115932,115935],{"className":115931},[31676],[34,115933],{"className":115934,"style":33218},[31680],[34,115936,54151],{"className":115937},[31685,31686]," 部分依赖于码，需要分解。",[108,115940,115941,115942,115944,115945,115947,115948,115745,115950,46528,116001,93535,116051,116079,116080,116108,116109,116111,116112,116163,116165],{},"2NF → 3NF",[58531,115943],{},"要求：\n在 2NF 的基础上，每一个非主属性都不传递依赖于候选码。",[58531,115946],{},"消除的问题：\n消除非主属性对码的传递函数依赖。",[58531,115949],{},[34,115951,115953,115971],{"className":115952},[31642],[34,115954,115956],{"className":115955},[31646],[31648,115957,115958],{"xmlns":31650},[31652,115959,115960,115968],{},[31655,115961,115962,115964,115966],{},[31658,115963,38691],{},[31662,115965,33239],{},[31658,115967,67467],{},[31665,115969,115970],{"encoding":31667},"K \\to X",[34,115972,115974,115992],{"className":115973,"ariaHidden":758},[31672],[34,115975,115977,115980,115983,115986,115989],{"className":115976},[31676],[34,115978],{"className":115979,"style":33218},[31680],[34,115981,38691],{"className":115982,"style":38706},[31685,31686],[34,115984],{"className":115985,"style":31846},[31845],[34,115987,33239],{"className":115988},[31850],[34,115990],{"className":115991,"style":31846},[31845],[34,115993,115995,115998],{"className":115994},[31676],[34,115996],{"className":115997,"style":33218},[31680],[34,115999,67467],{"className":116000,"style":67482},[31685,31686],[34,116002,116004,116021],{"className":116003},[31642],[34,116005,116007],{"className":116006},[31646],[31648,116008,116009],{"xmlns":31650},[31652,116010,116011,116019],{},[31655,116012,116013,116015,116017],{},[31658,116014,67467],{},[31662,116016,33239],{},[31658,116018,54151],{},[31665,116020,115818],{"encoding":31667},[34,116022,116024,116042],{"className":116023,"ariaHidden":758},[31672],[34,116025,116027,116030,116033,116036,116039],{"className":116026},[31676],[34,116028],{"className":116029,"style":33218},[31680],[34,116031,67467],{"className":116032,"style":67482},[31685,31686],[34,116034],{"className":116035,"style":31846},[31845],[34,116037,33239],{"className":116038},[31850],[34,116040],{"className":116041,"style":31846},[31845],[34,116043,116045,116048],{"className":116044},[31676],[34,116046],{"className":116047,"style":33218},[31680],[34,116049,54151],{"className":116050},[31685,31686],[34,116052,116054,116067],{"className":116053},[31642],[34,116055,116057],{"className":116056},[31646],[31648,116058,116059],{"xmlns":31650},[31652,116060,116061,116065],{},[31655,116062,116063],{},[31658,116064,67467],{},[31665,116066,67467],{"encoding":31667},[34,116068,116070],{"className":116069,"ariaHidden":758},[31672],[34,116071,116073,116076],{"className":116072},[31676],[34,116074],{"className":116075,"style":33218},[31680],[34,116077,67467],{"className":116078,"style":67482},[31685,31686]," 不是候选码，",[34,116081,116083,116096],{"className":116082},[31642],[34,116084,116086],{"className":116085},[31646],[31648,116087,116088],{"xmlns":31650},[31652,116089,116090,116094],{},[31655,116091,116092],{},[31658,116093,54151],{},[31665,116095,54151],{"encoding":31667},[34,116097,116099],{"className":116098,"ariaHidden":758},[31672],[34,116100,116102,116105],{"className":116101},[31676],[34,116103],{"className":116104,"style":33218},[31680],[34,116106,54151],{"className":116107},[31685,31686]," 是非主属性",[58531,116110],{},"则有：\n",[34,116113,116115,116133],{"className":116114},[31642],[34,116116,116118],{"className":116117},[31646],[31648,116119,116120],{"xmlns":31650},[31652,116121,116122,116130],{},[31655,116123,116124,116126,116128],{},[31658,116125,38691],{},[31662,116127,33239],{},[31658,116129,54151],{},[31665,116131,116132],{"encoding":31667},"K \\to A",[34,116134,116136,116154],{"className":116135,"ariaHidden":758},[31672],[34,116137,116139,116142,116145,116148,116151],{"className":116138},[31676],[34,116140],{"className":116141,"style":33218},[31680],[34,116143,38691],{"className":116144,"style":38706},[31685,31686],[34,116146],{"className":116147,"style":31846},[31845],[34,116149,33239],{"className":116150},[31850],[34,116152],{"className":116153,"style":31846},[31845],[34,116155,116157,116160],{"className":116156},[31676],[34,116158],{"className":116159,"style":33218},[31680],[34,116161,54151],{"className":116162},[31685,31686],[58531,116164],{},"这种依赖属于传递依赖，需要分解。",[108,116167,116168,116169,116171,116172,46528,116222,116250,116251,116279,116280,116282,116283,116285],{},"3NF → BCNF",[58531,116170],{},"要求：\n对于关系模式中的每一个非平凡函数依赖 ",[34,116173,116175,116192],{"className":116174},[31642],[34,116176,116178],{"className":116177},[31646],[31648,116179,116180],{"xmlns":31650},[31652,116181,116182,116190],{},[31655,116183,116184,116186,116188],{},[31658,116185,67467],{},[31662,116187,33239],{},[31658,116189,67731],{},[31665,116191,111963],{"encoding":31667},[34,116193,116195,116213],{"className":116194,"ariaHidden":758},[31672],[34,116196,116198,116201,116204,116207,116210],{"className":116197},[31676],[34,116199],{"className":116200,"style":33218},[31680],[34,116202,67467],{"className":116203,"style":67482},[31685,31686],[34,116205],{"className":116206,"style":31846},[31845],[34,116208,33239],{"className":116209},[31850],[34,116211],{"className":116212,"style":31846},[31845],[34,116214,116216,116219],{"className":116215},[31676],[34,116217],{"className":116218,"style":33218},[31680],[34,116220,67731],{"className":116221,"style":32604},[31685,31686],[34,116223,116225,116238],{"className":116224},[31642],[34,116226,116228],{"className":116227},[31646],[31648,116229,116230],{"xmlns":31650},[31652,116231,116232,116236],{},[31655,116233,116234],{},[31658,116235,67467],{},[31665,116237,67467],{"encoding":31667},[34,116239,116241],{"className":116240,"ariaHidden":758},[31672],[34,116242,116244,116247],{"className":116243},[31676],[34,116245],{"className":116246,"style":33218},[31680],[34,116248,67467],{"className":116249,"style":67482},[31685,31686]," 必须包含候选码，即 ",[34,116252,116254,116267],{"className":116253},[31642],[34,116255,116257],{"className":116256},[31646],[31648,116258,116259],{"xmlns":31650},[31652,116260,116261,116265],{},[31655,116262,116263],{},[31658,116264,67467],{},[31665,116266,67467],{"encoding":31667},[34,116268,116270],{"className":116269,"ariaHidden":758},[31672],[34,116271,116273,116276],{"className":116272},[31676],[34,116274],{"className":116275,"style":33218},[31680],[34,116277,67467],{"className":116278,"style":67482},[31685,31686]," 必须是超码。",[58531,116281],{},"消除的问题：\n消除所有决定因素不是候选码的函数依赖。",[58531,116284],{},"换句话说，BCNF 比 3NF 更严格，不仅要求非主属性不能部分依赖、传递依赖于码，还要求任何函数依赖的决定因素都必须是码。",[253,116287,115295],{"id":116288},"_2-题目-2",[15,116290,115149],{},[15,116292,115728],{},[2230,116294,116295,116304,116313],{},[108,116296,116297,116298,116300,116301,116303],{},"共享锁，简称 S 锁",[58531,116299],{},"若事务 T 对数据对象 A 加上 S 锁，则事务 T 可以读取 A，但不能修改 A。",[58531,116302],{},"其他事务可以再对 A 加 S 锁，但不能对 A 加 X 锁。",[108,116305,116306,116307,116309,116310,116312],{},"排他锁，简称 X 锁",[58531,116308],{},"若事务 T 对数据对象 A 加上 X 锁，则事务 T 可以读取和修改 A。",[58531,116311],{},"其他事务不能再对 A 加 S 锁，也不能对 A 加 X 锁。",[108,116314,116315],{},"相容矩阵",[13837,116317,116318,116331],{},[13840,116319,116320],{},[13843,116321,116322,116325,116328],{},[13846,116323,116324],{},"已加锁 \\ 申请锁",[13846,116326,116327],{},"S 锁",[13846,116329,116330],{},"X 锁",[13853,116332,116333,116343],{},[13843,116334,116335,116337,116340],{},[13858,116336,116327],{},[13858,116338,116339],{},"相容",[13858,116341,116342],{},"不相容",[13843,116344,116345,116347,116349],{},[13858,116346,116330],{},[13858,116348,116342],{},[13858,116350,116342],{},[15,116352,116353],{},"解析：\n共享锁之间相容，因为多个事务可以同时读同一个数据对象；排他锁与任何锁都不相容，因为修改数据时必须保证独占访问。",[253,116355,115317],{"id":116356},"_3-题目-2",[15,116358,115152],{},[15,116360,115728],{},[15,116362,116363],{},"数据库故障主要包括事务故障、系统故障、介质故障和计算机病毒等。",[2230,116365,116366,116375,116392,116401],{},[108,116367,116368,116369,116371,116372,116374],{},"事务故障",[58531,116370],{},"成因：\n某个事务在运行过程中因输入错误、运算溢出、违反完整性约束、死锁等原因不能继续执行。",[58531,116373],{},"恢复方式：\n撤销该事务已经对数据库所做的修改，即执行 UNDO，使数据库恢复到事务开始前的状态。",[108,116376,116377,116378,116380,116381,116383,116384],{},"系统故障",[58531,116379],{},"成因：\n操作系统故障、DBMS 故障、停电、硬件瞬时故障等，导致系统停止运行，但磁盘数据一般没有破坏。",[58531,116382],{},"恢复方式：\n系统重新启动后，根据日志文件：",[105,116385,116386,116389],{},[108,116387,116388],{},"撤销故障发生时尚未提交的事务；",[108,116390,116391],{},"重做故障发生前已经提交但其结果可能尚未写入数据库的事务。",[108,116393,116394,116395,116397,116398,116400],{},"介质故障",[58531,116396],{},"成因：\n磁盘损坏、磁头碰撞、强磁干扰等导致数据库或日志文件部分或全部丢失。",[58531,116399],{},"恢复方式：\n利用最近一次数据转储副本恢复数据库，然后利用日志文件重做转储以后已经提交的事务。",[108,116402,116403,116404,116406,116407,116409],{},"计算机病毒",[58531,116405],{},"成因：\n恶意程序破坏数据库文件或日志文件。",[58531,116408],{},"恢复方式：\n先清除病毒，再利用备份副本和日志文件进行恢复。实际系统中还需要加强安全控制和备份管理。",[32188,116411],{},[19,116413,115158],{"id":116414},"四关系代数及-sql-语句4-分题-5-题-20-分-1",[15,116416,116417],{},"已知关系：",[15,116419,115164],{},[15,116421,115167],{},[15,116423,115170],{},[253,116425,115273],{"id":116426},"_1-题目-3",[15,116428,115178],{},[15,116430,58173],{},[15,116432,116433],{},"关系代数表达式：",[15,116435,116436],{},[34,116437,116439,116531],{"className":116438},[31642],[34,116440,116442],{"className":116441},[31646],[31648,116443,116444],{"xmlns":31650},[31652,116445,116446,116528],{},[31655,116447,116448,116472,116474,116506,116508,116510,116512,116514,116516,116518,116520,116522,116524,116526],{},[31743,116449,116450,116452],{},[31658,116451,58442],{},[31655,116453,116454,116456,116458,116460,116462,116464,116466,116468,116470],{},[31658,116455,94675],{},[31658,116457,31660],{},[31658,116459,111],{},[31658,116461,31759],{},[31658,116463,33244],{},[31662,116465,31960],{"separator":758},[31658,116467,94675],{},[31658,116469,31660],{},[31658,116471,43383],{},[31662,116473,456],{"stretchy":702},[31743,116475,116476,116478],{},[31658,116477,115372],{},[31655,116479,116480,116482,116484,116486,116488,116496,116499],{},[31658,116481,47384],{},[31658,116483,31955],{},[31658,116485,33249],{},[31658,116487,67764],{},[37624,116489,116490,116492],{},[31662,116491,202],{},[31662,116493,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},"0em","′",[39361,116497,116498],{},"西",[37624,116500,116501,116504],{},[39361,116502,116503],{},"安",[31662,116505,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,116507,456],{"stretchy":702},[31658,116509,94675],{},[31658,116511,94628],{},[31658,116513,15],{},[31658,116515,15],{},[31658,116517,43379],{},[31658,116519,31955],{},[31658,116521,33244],{},[31658,116523,34395],{},[31662,116525,2557],{"stretchy":702},[31662,116527,2557],{"stretchy":702},[31665,116529,116530],{"encoding":31667},"\\pi_{Sname, Sno}(\\sigma_{City='西安'}(Supplier))",[34,116532,116534],{"className":116533,"ariaHidden":758},[31672],[34,116535,116537,116540,116603,116606,116727,116730,116733,116736,116740,116743,116746],{"className":116536},[31676],[34,116538],{"className":116539,"style":32058},[31680],[34,116541,116543,116546],{"className":116542},[31685],[34,116544,58442],{"className":116545,"style":32062},[31685,31686],[34,116547,116549],{"className":116548},[31792],[34,116550,116552,116595],{"className":116551},[31796,31797],[34,116553,116555,116592],{"className":116554},[31801],[34,116556,116559],{"className":116557,"style":116558},[31805],"height:0.3283em;",[34,116560,116561,116564],{"style":38795},[34,116562],{"className":116563,"style":31814},[31813],[34,116565,116567],{"className":116566},[31818,31819,31820,31821],[34,116568,116570,116573,116577,116580,116583,116586,116589],{"className":116569},[31685,31821],[34,116571,94675],{"className":116572,"style":106455},[31685,31686,31821],[34,116574,116576],{"className":116575},[31685,31686,31821],"nam",[34,116578,33244],{"className":116579},[31685,31686,31821],[34,116581,31960],{"className":116582},[32119,31821],[34,116584,94675],{"className":116585,"style":106455},[31685,31686,31821],[34,116587,31660],{"className":116588},[31685,31686,31821],[34,116590,43383],{"className":116591},[31685,31686,31821],[34,116593,31829],{"className":116594},[31828],[34,116596,116598],{"className":116597},[31801],[34,116599,116601],{"className":116600,"style":32163},[31805],[34,116602],{},[34,116604,456],{"className":116605},[31782],[34,116607,116609,116612],{"className":116608},[31685],[34,116610,115372],{"className":116611,"style":32062},[31685,31686],[34,116613,116615],{"className":116614},[31792],[34,116616,116618,116719],{"className":116617},[31796,31797],[34,116619,116621,116716],{"className":116620},[31801],[34,116622,116624],{"className":116623,"style":116558},[31805],[34,116625,116626,116629],{"style":38795},[34,116627],{"className":116628,"style":31814},[31813],[34,116630,116632],{"className":116631},[31818,31819,31820,31821],[34,116633,116635,116638,116641,116644,116647,116681,116684],{"className":116634},[31685,31821],[34,116636,47384],{"className":116637,"style":38706},[31685,31686,31821],[34,116639,31955],{"className":116640},[31685,31686,31821],[34,116642,33249],{"className":116643},[31685,31686,31821],[34,116645,67764],{"className":116646,"style":32062},[31685,31686,31821],[34,116648,116650,116653],{"className":116649},[31850,31821],[34,116651,202],{"className":116652},[31850,31821],[34,116654,116656],{"className":116655},[31792],[34,116657,116659],{"className":116658},[31796],[34,116660,116662],{"className":116661},[31801],[34,116663,116666],{"className":116664,"style":116665},[31805],"height:0.6828em;",[34,116667,116669,116672],{"style":116668},"top:-2.786em;margin-right:0.0714em;",[34,116670],{"className":116671,"style":43406},[31813],[34,116673,116675],{"className":116674},[31818,43410,43411,31821],[34,116676,116678],{"className":116677},[31685,31821],[34,116679,116495],{"className":116680},[31685,31821],[34,116682,116498],{"className":116683},[31685,105462,31821],[34,116685,116687,116690],{"className":116686},[31685,31821],[34,116688,116503],{"className":116689},[31685,105462,31821],[34,116691,116693],{"className":116692},[31792],[34,116694,116696],{"className":116695},[31796],[34,116697,116699],{"className":116698},[31801],[34,116700,116702],{"className":116701,"style":116665},[31805],[34,116703,116704,116707],{"style":116668},[34,116705],{"className":116706,"style":43406},[31813],[34,116708,116710],{"className":116709},[31818,43410,43411,31821],[34,116711,116713],{"className":116712},[31685,31821],[34,116714,116495],{"className":116715},[31685,31821],[34,116717,31829],{"className":116718},[31828],[34,116720,116722],{"className":116721},[31801],[34,116723,116725],{"className":116724,"style":32163},[31805],[34,116726],{},[34,116728,456],{"className":116729},[31782],[34,116731,94675],{"className":116732,"style":106455},[31685,31686],[34,116734,94628],{"className":116735},[31685,31686],[34,116737,116739],{"className":116738,"style":47113},[31685,31686],"ppl",[34,116741,31955],{"className":116742},[31685,31686],[34,116744,34599],{"className":116745,"style":34452},[31685,31686],[34,116747,44253],{"className":116748},[31690],[15,116750,116751,116752,116780,116781,116809],{},"解析：\n先用选择运算 ",[34,116753,116755,116768],{"className":116754},[31642],[34,116756,116758],{"className":116757},[31646],[31648,116759,116760],{"xmlns":31650},[31652,116761,116762,116766],{},[31655,116763,116764],{},[31658,116765,115372],{},[31665,116767,115375],{"encoding":31667},[34,116769,116771],{"className":116770,"ariaHidden":758},[31672],[34,116772,116774,116777],{"className":116773},[31676],[34,116775],{"className":116776,"style":32527},[31680],[34,116778,115372],{"className":116779,"style":32062},[31685,31686]," 选出 City 为“西安”的供应商，再用投影运算 ",[34,116782,116784,116797],{"className":116783},[31642],[34,116785,116787],{"className":116786},[31646],[31648,116788,116789],{"xmlns":31650},[31652,116790,116791,116795],{},[31655,116792,116793],{},[31658,116794,58442],{},[31665,116796,58445],{"encoding":31667},[34,116798,116800],{"className":116799,"ariaHidden":758},[31672],[34,116801,116803,116806],{"className":116802},[31676],[34,116804],{"className":116805,"style":32527},[31680],[34,116807,58442],{"className":116808,"style":32062},[31685,31686]," 取出供应商名称和供应商编号。",[253,116811,115295],{"id":116812},"_2-题目-3",[15,116814,115181],{},[15,116816,58173],{},[15,116818,116433],{},[15,116820,116821],{},[34,116822,116824,116986],{"className":116823},[31642],[34,116825,116827],{"className":116826},[31646],[31648,116828,116829],{"xmlns":31650},[31652,116830,116831,116983],{},[31655,116832,116833,116868,116870,116896,116898,116900,116902,116904,116906,116908,116910,116912,116914,116967,116969,116971,116973,116975,116977,116979,116981],{},[31743,116834,116835,116837],{},[31658,116836,58442],{},[31655,116838,116839,116841,116843,116845,116847,116849,116851,116853,116855,116857,116859,116861,116864,116866],{},[31658,116840,94675],{},[31658,116842,31660],{},[31658,116844,111],{},[31658,116846,31759],{},[31658,116848,33244],{},[31662,116850,31960],{"separator":758},[31658,116852,47384],{},[31658,116854,31955],{},[31658,116856,33249],{},[31658,116858,67764],{},[31662,116860,31960],{"separator":758},[31658,116862,116863],{},"Q",[31658,116865,33249],{},[31658,116867,67764],{},[31662,116869,456],{"stretchy":702},[31743,116871,116872,116874],{},[31658,116873,115372],{},[31655,116875,116876,116878,116880,116882,116888,116890],{},[31658,116877,105852],{},[31658,116879,31660],{},[31658,116881,43383],{},[37624,116883,116884,116886],{},[31662,116885,202],{},[31662,116887,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,116889,105852],{},[37624,116891,116892,116894],{},[31749,116893,640],{},[31662,116895,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,116897,456],{"stretchy":702},[31658,116899,94675],{},[31658,116901,94628],{},[31658,116903,15],{},[31658,116905,15],{},[31658,116907,43379],{},[31658,116909,31955],{},[31658,116911,33244],{},[31658,116913,34395],{},[31743,116915,116916,116919],{},[31662,116917,116918],{},"⋈",[31655,116920,116921,116923,116925,116927,116929,116931,116933,116935,116937,116939,116941,116943,116945,116947,116949,116951,116953,116955,116957,116959,116961,116963,116965],{},[31658,116922,94675],{},[31658,116924,94628],{},[31658,116926,15],{},[31658,116928,15],{},[31658,116930,43379],{},[31658,116932,31955],{},[31658,116934,33244],{},[31658,116936,34395],{},[31658,116938,2547],{"mathvariant":38079},[31658,116940,94675],{},[31658,116942,31660],{},[31658,116944,43383],{},[31662,116946,202],{},[31658,116948,94675],{},[31658,116950,94628],{},[31658,116952,15],{},[31658,116954,15],{},[31658,116956,43379],{},[31658,116958,67764],{},[31658,116960,2547],{"mathvariant":38079},[31658,116962,94675],{},[31658,116964,31660],{},[31658,116966,43383],{},[31658,116968,94675],{},[31658,116970,94628],{},[31658,116972,15],{},[31658,116974,15],{},[31658,116976,43379],{},[31658,116978,67764],{},[31662,116980,2557],{"stretchy":702},[31662,116982,2557],{"stretchy":702},[31665,116984,116985],{"encoding":31667},"\\pi_{Sname, City, Qty}(\\sigma_{Pno='P1'}(Supplier \\bowtie_{Supplier.Sno=Supply.Sno} Supply))",[34,116987,116989,117306],{"className":116988,"ariaHidden":758},[31672],[34,116990,116992,116995,117069,117072,117188,117191,117194,117197,117200,117203,117206,117209,117303],{"className":116991},[31676],[34,116993],{"className":116994,"style":32058},[31680],[34,116996,116998,117001],{"className":116997},[31685],[34,116999,58442],{"className":117000,"style":32062},[31685,31686],[34,117002,117004],{"className":117003},[31792],[34,117005,117007,117061],{"className":117006},[31796,31797],[34,117008,117010,117058],{"className":117009},[31801],[34,117011,117013],{"className":117012,"style":116558},[31805],[34,117014,117015,117018],{"style":38795},[34,117016],{"className":117017,"style":31814},[31813],[34,117019,117021],{"className":117020},[31818,31819,31820,31821],[34,117022,117024,117027,117030,117033,117036,117039,117042,117045,117048,117051,117055],{"className":117023},[31685,31821],[34,117025,94675],{"className":117026,"style":106455},[31685,31686,31821],[34,117028,116576],{"className":117029},[31685,31686,31821],[34,117031,33244],{"className":117032},[31685,31686,31821],[34,117034,31960],{"className":117035},[32119,31821],[34,117037,47384],{"className":117038,"style":38706},[31685,31686,31821],[34,117040,31955],{"className":117041},[31685,31686,31821],[34,117043,33249],{"className":117044},[31685,31686,31821],[34,117046,67764],{"className":117047,"style":32062},[31685,31686,31821],[34,117049,31960],{"className":117050},[32119,31821],[34,117052,117054],{"className":117053},[31685,31686,31821],"Qt",[34,117056,67764],{"className":117057,"style":32062},[31685,31686,31821],[34,117059,31829],{"className":117060},[31828],[34,117062,117064],{"className":117063},[31801],[34,117065,117067],{"className":117066,"style":32163},[31805],[34,117068],{},[34,117070,456],{"className":117071},[31782],[34,117073,117075,117078],{"className":117074},[31685],[34,117076,115372],{"className":117077,"style":32062},[31685,31686],[34,117079,117081],{"className":117080},[31792],[34,117082,117084,117180],{"className":117083},[31796,31797],[34,117085,117087,117177],{"className":117086},[31801],[34,117088,117090],{"className":117089,"style":116558},[31805],[34,117091,117092,117095],{"style":38795},[34,117093],{"className":117094,"style":31814},[31813],[34,117096,117098],{"className":117097},[31818,31819,31820,31821],[34,117099,117101,117104,117107,117110,117142,117145],{"className":117100},[31685,31821],[34,117102,105852],{"className":117103,"style":37422},[31685,31686,31821],[34,117105,31660],{"className":117106},[31685,31686,31821],[34,117108,43383],{"className":117109},[31685,31686,31821],[34,117111,117113,117116],{"className":117112},[31850,31821],[34,117114,202],{"className":117115},[31850,31821],[34,117117,117119],{"className":117118},[31792],[34,117120,117122],{"className":117121},[31796],[34,117123,117125],{"className":117124},[31801],[34,117126,117128],{"className":117127,"style":116665},[31805],[34,117129,117130,117133],{"style":116668},[34,117131],{"className":117132,"style":43406},[31813],[34,117134,117136],{"className":117135},[31818,43410,43411,31821],[34,117137,117139],{"className":117138},[31685,31821],[34,117140,116495],{"className":117141},[31685,31821],[34,117143,105852],{"className":117144,"style":37422},[31685,31686,31821],[34,117146,117148,117151],{"className":117147},[31685,31821],[34,117149,640],{"className":117150},[31685,31821],[34,117152,117154],{"className":117153},[31792],[34,117155,117157],{"className":117156},[31796],[34,117158,117160],{"className":117159},[31801],[34,117161,117163],{"className":117162,"style":116665},[31805],[34,117164,117165,117168],{"style":116668},[34,117166],{"className":117167,"style":43406},[31813],[34,117169,117171],{"className":117170},[31818,43410,43411,31821],[34,117172,117174],{"className":117173},[31685,31821],[34,117175,116495],{"className":117176},[31685,31821],[34,117178,31829],{"className":117179},[31828],[34,117181,117183],{"className":117182},[31801],[34,117184,117186],{"className":117185,"style":31836},[31805],[34,117187],{},[34,117189,456],{"className":117190},[31782],[34,117192,94675],{"className":117193,"style":106455},[31685,31686],[34,117195,94628],{"className":117196},[31685,31686],[34,117198,116739],{"className":117199,"style":47113},[31685,31686],[34,117201,31955],{"className":117202},[31685,31686],[34,117204,34599],{"className":117205,"style":34452},[31685,31686],[34,117207],{"className":117208,"style":31846},[31845],[34,117210,117212,117215],{"className":117211},[31850],[34,117213,116918],{"className":117214},[31850],[34,117216,117218],{"className":117217},[31792],[34,117219,117221,117295],{"className":117220},[31796,31797],[34,117222,117224,117292],{"className":117223},[31801],[34,117225,117227],{"className":117226,"style":33833},[31805],[34,117228,117229,117232],{"style":31809},[34,117230],{"className":117231,"style":31814},[31813],[34,117233,117235],{"className":117234},[31818,31819,31820,31821],[34,117236,117238,117241,117244,117247,117250,117253,117256,117259,117262,117265,117268,117271,117274,117277,117280,117283,117286,117289],{"className":117237},[31685,31821],[34,117239,94675],{"className":117240,"style":106455},[31685,31686,31821],[34,117242,94628],{"className":117243},[31685,31686,31821],[34,117245,116739],{"className":117246,"style":47113},[31685,31686,31821],[34,117248,31955],{"className":117249},[31685,31686,31821],[34,117251,34599],{"className":117252,"style":34452},[31685,31686,31821],[34,117254,2547],{"className":117255},[31685,31821],[34,117257,94675],{"className":117258,"style":106455},[31685,31686,31821],[34,117260,31660],{"className":117261},[31685,31686,31821],[34,117263,43383],{"className":117264},[31685,31686,31821],[34,117266,202],{"className":117267},[31850,31821],[34,117269,94675],{"className":117270,"style":106455},[31685,31686,31821],[34,117272,94628],{"className":117273},[31685,31686,31821],[34,117275,116739],{"className":117276,"style":47113},[31685,31686,31821],[34,117278,67764],{"className":117279,"style":32062},[31685,31686,31821],[34,117281,2547],{"className":117282},[31685,31821],[34,117284,94675],{"className":117285,"style":106455},[31685,31686,31821],[34,117287,31660],{"className":117288},[31685,31686,31821],[34,117290,43383],{"className":117291},[31685,31686,31821],[34,117293,31829],{"className":117294},[31828],[34,117296,117298],{"className":117297},[31801],[34,117299,117301],{"className":117300,"style":32163},[31805],[34,117302],{},[34,117304],{"className":117305,"style":31846},[31845],[34,117307,117309,117312,117315,117318,117321,117324],{"className":117308},[31676],[34,117310],{"className":117311,"style":31774},[31680],[34,117313,94675],{"className":117314,"style":106455},[31685,31686],[34,117316,94628],{"className":117317},[31685,31686],[34,117319,116739],{"className":117320,"style":47113},[31685,31686],[34,117322,67764],{"className":117323,"style":32062},[31685,31686],[34,117325,44253],{"className":117326},[31690],[15,117328,117329],{},"解析：\n先将 Supplier 与 Supply 按供应商编号连接，再选择 Pno 为 P1 的元组，最后投影出供应商名称、所在城市和供应数量。",[253,117331,115317],{"id":117332},"_3-题目-3",[15,117334,115189],{},[15,117336,58173],{},[15,117338,117339],{},"SQL Server 语句：",[24,117341,117343],{"className":92584,"code":117342,"language":92586,"meta":29,"style":29},"SELECT ISNULL(SUM(Qty), 0) AS Total_Qty\nFROM Supply\nWHERE Sno = 'S1';\n",[31,117344,117345,117350,117355],{"__ignoreMap":29},[34,117346,117347],{"class":36,"line":37},[34,117348,117349],{},"SELECT ISNULL(SUM(Qty), 0) AS Total_Qty\n",[34,117351,117352],{"class":36,"line":43},[34,117353,117354],{},"FROM Supply\n",[34,117356,117357],{"class":36,"line":49},[34,117358,101780],{},[15,117360,117361],{},"解析：\nSUM(Qty) 用于统计供应数量总和。使用 ISNULL 是为了避免 S1 没有供应记录时返回 NULL。",[253,117363,115339],{"id":117364},"_4-题目-2",[15,117366,115192],{},[15,117368,58173],{},[15,117370,117339],{},[24,117372,117374],{"className":92584,"code":117373,"language":92586,"meta":29,"style":29},"SELECT\n s.Sname,\n p.Spec,\n sp.Qty\nFROM Supplier AS s\nJOIN Supply AS sp\n ON s.Sno = sp.Sno\nJOIN Part AS p\n ON sp.Pno = p.Pno\nWHERE sp.Pno = 'P2'\n AND sp.Qty > 100;\n",[31,117375,117376,117381,117386,117391,117396,117401,117406,117411,117416,117421,117425],{"__ignoreMap":29},[34,117377,117378],{"class":36,"line":37},[34,117379,117380],{},"SELECT\n",[34,117382,117383],{"class":36,"line":43},[34,117384,117385],{}," s.Sname,\n",[34,117387,117388],{"class":36,"line":49},[34,117389,117390],{}," p.Spec,\n",[34,117392,117393],{"class":36,"line":56},[34,117394,117395],{}," sp.Qty\n",[34,117397,117398],{"class":36,"line":62},[34,117399,117400],{},"FROM Supplier AS s\n",[34,117402,117403],{"class":36,"line":68},[34,117404,117405],{},"JOIN Supply AS sp\n",[34,117407,117408],{"class":36,"line":74},[34,117409,117410],{}," ON s.Sno = sp.Sno\n",[34,117412,117413],{"class":36,"line":1971},[34,117414,117415],{},"JOIN Part AS p\n",[34,117417,117418],{"class":36,"line":1976},[34,117419,117420],{}," ON sp.Pno = p.Pno\n",[34,117422,117423],{"class":36,"line":1985},[34,117424,101809],{},[34,117426,117427],{"class":36,"line":1990},[34,117428,101814],{},[15,117430,117431],{},"解析：\n该查询需要同时使用供应商表、零件表和供应表。通过 Sno 连接供应商和供应关系，通过 Pno 连接零件和供应关系。",[253,117433,115391],{"id":117434},"_5-题目-2",[15,117436,115195],{},[15,117438,58173],{},[15,117440,117339],{},[24,117442,117444],{"className":92584,"code":117443,"language":92586,"meta":29,"style":29},"SELECT\n s.Sno,\n s.Sname,\n s.City\nFROM Supplier AS s\nWHERE NOT EXISTS (\n SELECT 1\n FROM Supply AS sp\n WHERE sp.Sno = s.Sno\n AND sp.Pno = 'P3'\n);\n",[31,117445,117446,117450,117455,117459,117464,117468,117472,117476,117481,117485,117489],{"__ignoreMap":29},[34,117447,117448],{"class":36,"line":37},[34,117449,117380],{},[34,117451,117452],{"class":36,"line":43},[34,117453,117454],{}," s.Sno,\n",[34,117456,117457],{"class":36,"line":49},[34,117458,117385],{},[34,117460,117461],{"class":36,"line":56},[34,117462,117463],{}," s.City\n",[34,117465,117466],{"class":36,"line":62},[34,117467,117400],{},[34,117469,117470],{"class":36,"line":68},[34,117471,92965],{},[34,117473,117474],{"class":36,"line":74},[34,117475,92970],{},[34,117477,117478],{"class":36,"line":1971},[34,117479,117480],{}," FROM Supply AS sp\n",[34,117482,117483],{"class":36,"line":1976},[34,117484,101844],{},[34,117486,117487],{"class":36,"line":1985},[34,117488,101849],{},[34,117490,117491],{"class":36,"line":1990},[34,117492,462],{},[15,117494,117495],{},"解析：\nNOT EXISTS 表示不存在对应记录。该语句查询在 Supply 表中不存在供应 P3 记录的供应商。",[32188,117497],{},[19,117499,115201],{"id":117500},"五数据库设计共-20-分-1",[15,117502,117503],{},"题目背景：",[15,117505,115204],{},[105,117507,117508,117510,117512,117514,117516],{},[108,117509,115209],{},[108,117511,115212],{},[108,117513,115215],{},[108,117515,115218],{},[108,117517,115221],{},[253,117519,115273],{"id":117520},"_1-题目-4",[15,117522,115228],{},[15,117524,58173],{},[15,117526,117527],{},"实体型：",[2230,117529,117530,117533,117536,117539,117542],{},[108,117531,117532],{},"学生：Student",[108,117534,117535],{},"专业：Major",[108,117537,117538],{},"课程：Course",[108,117540,117541],{},"教师：Teacher",[108,117543,117544],{},"学院：College",[15,117546,117547],{},"联系型：",[2230,117549,117550,117559,117571,117580,117588,117596],{},[108,117551,117552,117553,117555,117556,117558],{},"学生 —— 主修 —— 专业",[58531,117554],{},"联系类型：n:1",[58531,117557],{},"含义：每个学生主修一个专业，一个专业可招收多名学生。",[108,117560,117561,117562,117564,117565,117567,117568,117570],{},"学生 —— 选修 —— 课程",[58531,117563],{},"联系类型：m:n",[58531,117566],{},"联系属性：成绩 Grade",[58531,117569],{},"含义：一个学生可选修多门课程，一门课程可被多名学生选修。",[108,117572,117573,117574,117576,117577,117579],{},"学院 —— 设置 —— 专业",[58531,117575],{},"联系类型：1:n",[58531,117578],{},"含义：一个学院可设置多个专业，一个专业只隶属于一个学院。",[108,117581,117582,117583,117576,117585,117587],{},"学院 —— 开设 —— 课程",[58531,117584],{},[58531,117586],{},"含义：一个学院可开设多门课程，一门课程只由一个学院开设。",[108,117589,117590,117591,117576,117593,117595],{},"学院 —— 拥有\u002F隶属 —— 教师",[58531,117592],{},[58531,117594],{},"含义：一个学院可拥有多名教师，一名教师只隶属于一个学院。",[108,117597,117598,117599,117564,117601,117603,117604,117606],{},"教师 —— 讲授 —— 课程",[58531,117600],{},[58531,117602],{},"联系属性：教室 Classroom，课时 Hours",[58531,117605],{},"含义：一名教师可讲授多门课程，一门课程可由多名教师讲授。",[15,117608,117609],{},"E-R 图文字表示：",[15,117611,117612,117615,117616],{},[34,117613,117614],{},"学院 College"," 1 —— 设置 —— n ",[34,117617,117618],{},"专业 Major",[15,117620,117621,117623,117624],{},[34,117622,117618],{}," 1 —— 主修 —— n ",[34,117625,117626],{},"学生 Student",[15,117628,117629,117631,117632,117635],{},[34,117630,117626],{}," m —— 选修 —— n ",[34,117633,117634],{},"课程 Course","\n|\n成绩 Grade",[15,117637,117638,117640,117641],{},[34,117639,117614],{}," 1 —— 开设 —— n ",[34,117642,117634],{},[15,117644,117645,117647,117648],{},[34,117646,117614],{}," 1 —— 拥有\u002F隶属 —— n ",[34,117649,117650],{},"教师 Teacher",[15,117652,117653,117655,117656,117658],{},[34,117654,117650],{}," m —— 讲授 —— n ",[34,117657,117634],{},"\n|\n教室 Classroom，课时 Hours",[15,117660,117661],{},"解析：\nE-R 图的核心是识别实体型和联系型。1:n 联系通常可以通过在 n 端加入外键实现；m:n 联系通常需要单独转换为一个关系模式。",[253,117663,115295],{"id":117664},"_2-题目-4",[15,117666,115231],{},[15,117668,58173],{},[2230,117670,117671],{},[108,117672,117673],{},"学院表 College",[15,117675,117676],{},"College(\nCollegeID,\nCollegeName\n)",[15,117678,117679],{},"主键：\nCollegeID",[2230,117681,117682],{"start":43},[108,117683,117684],{},"专业表 Major",[15,117686,117687],{},"Major(\nMajorID,\nMajorName,\nCollegeID\n)",[15,117689,117690],{},"主键：\nMajorID",[15,117692,117693],{},"外键：\nCollegeID REFERENCES College(CollegeID)",[2230,117695,117696],{"start":49},[108,117697,117698],{},"学生表 Student",[15,117700,117701],{},"Student(\nStudentID,\nStudentName,\nMajorID\n)",[15,117703,117704],{},"主键：\nStudentID",[15,117706,117707],{},"外键：\nMajorID REFERENCES Major(MajorID)",[2230,117709,117710],{"start":56},[108,117711,117712],{},"课程表 Course",[15,117714,117715],{},"Course(\nCourseID,\nCourseName,\nCollegeID\n)",[15,117717,117718],{},"主键：\nCourseID",[15,117720,117693],{},[2230,117722,117723],{"start":62},[108,117724,117725],{},"教师表 Teacher",[15,117727,117728],{},"Teacher(\nTeacherID,\nTeacherName,\nCollegeID\n)",[15,117730,117731],{},"主键：\nTeacherID",[15,117733,117693],{},[2230,117735,117736],{"start":68},[108,117737,117738],{},"选修表 SC",[15,117740,117741],{},"SC(\nStudentID,\nCourseID,\nGrade\n)",[15,117743,117744],{},"主键：\n(StudentID, CourseID)",[15,117746,117747],{},"外键：\nStudentID REFERENCES Student(StudentID)",[15,117749,117750],{},"外键：\nCourseID REFERENCES Course(CourseID)",[2230,117752,117753],{"start":74},[108,117754,117755],{},"授课表 Teaching",[15,117757,117758],{},"Teaching(\nTeacherID,\nCourseID,\nClassroom,\nHours\n)",[15,117760,117761],{},"主键：\n(TeacherID, CourseID)",[15,117763,117764],{},"外键：\nTeacherID REFERENCES Teacher(TeacherID)",[15,117766,117750],{},[15,117768,117769],{},"解析：\n1:n 联系可以在 n 端加入外键，例如 Student 中加入 MajorID。\nm:n 联系需要单独转换为关系模式，例如 SC 和 Teaching。\n联系的属性应放入由该联系转换成的关系模式中，因此 Grade 放入 SC，Classroom 和 Hours 放入 Teaching。",[253,117771,115317],{"id":117772},"_3-题目-4",[15,117774,115234],{},[15,117776,58173],{},[15,117778,117779],{},"在以下基本函数依赖假设下：",[2230,117781,117782,117785,117788,117791,117794,117797,117800],{},[108,117783,117784],{},"CollegeID → CollegeName",[108,117786,117787],{},"MajorID → MajorName, CollegeID",[108,117789,117790],{},"StudentID → StudentName, MajorID",[108,117792,117793],{},"CourseID → CourseName, CollegeID",[108,117795,117796],{},"TeacherID → TeacherName, CollegeID",[108,117798,117799],{},"(StudentID, CourseID) → Grade",[108,117801,117802],{},"(TeacherID, CourseID) → Classroom, Hours",[15,117804,117805],{},"各关系模式范式如下：",[2230,117807,117808,117817,117825,117833,117841,117849,118027],{},[108,117809,117810,117811,117813,117814,117816],{},"College(CollegeID, CollegeName)",[58531,117812],{},"范式：BCNF",[58531,117815],{},"原因：\n其唯一非平凡函数依赖为 CollegeID → CollegeName，决定因素 CollegeID 是候选码。",[108,117818,117819,117820,117813,117822,117824],{},"Major(MajorID, MajorName, CollegeID)",[58531,117821],{},[58531,117823],{},"原因：\nMajorID 是主键，MajorName 和 CollegeID 都完全函数依赖于 MajorID，不存在部分依赖和传递依赖。",[108,117826,117827,117828,117813,117830,117832],{},"Student(StudentID, StudentName, MajorID)",[58531,117829],{},[58531,117831],{},"原因：\nStudentID 是主键，StudentName 和 MajorID 都完全函数依赖于 StudentID，且决定因素是候选码。",[108,117834,117835,117836,117813,117838,117840],{},"Course(CourseID, CourseName, CollegeID)",[58531,117837],{},[58531,117839],{},"原因：\nCourseID 是主键，CourseName 和 CollegeID 都完全函数依赖于 CourseID，不存在非主属性对码的部分依赖和传递依赖。",[108,117842,117843,117844,117813,117846,117848],{},"Teacher(TeacherID, TeacherName, CollegeID)",[58531,117845],{},[58531,117847],{},"原因：\nTeacherID 是主键，TeacherName 和 CollegeID 都完全函数依赖于 TeacherID，决定因素是候选码。",[108,117850,117851,117852,117813,117854,117856,117857,117859,118024,118026],{},"SC(StudentID, CourseID, Grade)",[58531,117853],{},[58531,117855],{},"原因：\n主键为 (StudentID, CourseID)，Grade 表示某学生选修某课程的成绩，完全函数依赖于整个主键：",[58531,117858],{},[34,117860,117862,117925],{"className":117861},[31642],[34,117863,117865],{"className":117864},[31646],[31648,117866,117867],{"xmlns":31650},[31652,117868,117869,117923],{},[31655,117870,117871,117873,117875,117877,117879,117881,117883,117885,117887,117889,117891,117893,117895,117897,117899,117901,117903,117905,117907,117909,117911,117913,117915,117917,117919,117921],{},[31662,117872,456],{"stretchy":702},[31658,117874,94675],{},[31658,117876,33249],{},[31658,117878,94628],{},[31658,117880,46900],{},[31658,117882,33244],{},[31658,117884,31660],{},[31658,117886,33249],{},[31658,117888,106063],{},[31658,117890,91994],{},[31662,117892,31960],{"separator":758},[31658,117894,47384],{},[31658,117896,43383],{},[31658,117898,94628],{},[31658,117900,34395],{},[31658,117902,31747],{},[31658,117904,33244],{},[31658,117906,106063],{},[31658,117908,91994],{},[31662,117910,2557],{"stretchy":702},[31662,117912,33239],{},[31658,117914,97709],{},[31658,117916,34395],{},[31658,117918,111],{},[31658,117920,46900],{},[31658,117922,33244],{},[31665,117924,113530],{"encoding":31667},[34,117926,117928,118003],{"className":117927,"ariaHidden":758},[31672],[34,117929,117931,117934,117937,117940,117943,117946,117949,117952,117955,117958,117961,117964,117967,117970,117973,117976,117979,117982,117985,117988,117991,117994,117997,118000],{"className":117930},[31676],[34,117932],{"className":117933,"style":31774},[31680],[34,117935,456],{"className":117936},[31782],[34,117938,94675],{"className":117939,"style":106455},[31685,31686],[34,117941,33249],{"className":117942},[31685,31686],[34,117944,94628],{"className":117945},[31685,31686],[34,117947,46900],{"className":117948},[31685,31686],[34,117950,33244],{"className":117951},[31685,31686],[34,117953,31660],{"className":117954},[31685,31686],[34,117956,33249],{"className":117957},[31685,31686],[34,117959,106063],{"className":117960,"style":67482},[31685,31686],[34,117962,91994],{"className":117963,"style":34452},[31685,31686],[34,117965,31960],{"className":117966},[32119],[34,117968],{"className":117969,"style":32123},[31845],[34,117971,47384],{"className":117972,"style":38706},[31685,31686],[34,117974,43383],{"className":117975},[31685,31686],[34,117977,94628],{"className":117978},[31685,31686],[34,117980,34395],{"className":117981,"style":34452},[31685,31686],[34,117983,113295],{"className":117984},[31685,31686],[34,117986,106063],{"className":117987,"style":67482},[31685,31686],[34,117989,91994],{"className":117990,"style":34452},[31685,31686],[34,117992,2557],{"className":117993},[31690],[34,117995],{"className":117996,"style":31846},[31845],[34,117998,33239],{"className":117999},[31850],[34,118001],{"className":118002,"style":31846},[31845],[34,118004,118006,118009,118012,118015,118018,118021],{"className":118005},[31676],[34,118007],{"className":118008,"style":31681},[31680],[34,118010,97709],{"className":118011},[31685,31686],[34,118013,34395],{"className":118014,"style":34452},[31685,31686],[34,118016,111],{"className":118017},[31685,31686],[34,118019,46900],{"className":118020},[31685,31686],[34,118022,33244],{"className":118023},[31685,31686],[58531,118025],{},"Grade 不依赖于 StudentID 或 CourseID 的某一部分，因此不存在部分函数依赖，也不存在传递函数依赖。",[108,118028,118029,118030,117813,118032,118034,118035,118037,118248,118250],{},"Teaching(TeacherID, CourseID, Classroom, Hours)",[58531,118031],{},[58531,118033],{},"原因：\n主键为 (TeacherID, CourseID)，Classroom 和 Hours 表示某教师讲授某课程时的教室和课时信息，完全函数依赖于整个主键：",[58531,118036],{},[34,118038,118040,118125],{"className":118039},[31642],[34,118041,118043],{"className":118042},[31646],[31648,118044,118045],{"xmlns":31650},[31652,118046,118047,118122],{},[31655,118048,118049,118051,118053,118055,118057,118059,118061,118063,118065,118067,118069,118071,118073,118075,118077,118079,118081,118083,118085,118087,118089,118091,118093,118095,118097,118099,118101,118103,118105,118107,118109,118111,118114,118116,118118,118120],{},[31662,118050,456],{"stretchy":702},[31658,118052,37400],{},[31658,118054,33244],{},[31658,118056,111],{},[31658,118058,31965],{},[31658,118060,47292],{},[31658,118062,33244],{},[31658,118064,34395],{},[31658,118066,106063],{},[31658,118068,91994],{},[31662,118070,31960],{"separator":758},[31658,118072,47384],{},[31658,118074,43383],{},[31658,118076,94628],{},[31658,118078,34395],{},[31658,118080,31747],{},[31658,118082,33244],{},[31658,118084,106063],{},[31658,118086,91994],{},[31662,118088,2557],{"stretchy":702},[31662,118090,33239],{},[31658,118092,47384],{},[31658,118094,43379],{},[31658,118096,111],{},[31658,118098,31747],{},[31658,118100,31747],{},[31658,118102,34395],{},[31658,118104,43383],{},[31658,118106,43383],{},[31658,118108,31759],{},[31662,118110,31960],{"separator":758},[31658,118112,118113],{},"H",[31658,118115,43383],{},[31658,118117,94628],{},[31658,118119,34395],{},[31658,118121,31747],{},[31665,118123,118124],{"encoding":31667},"(TeacherID, CourseID) \\to Classroom, Hours",[34,118126,118128,118200],{"className":118127,"ariaHidden":758},[31672],[34,118129,118131,118134,118137,118140,118143,118146,118149,118152,118155,118158,118161,118164,118167,118170,118173,118176,118179,118182,118185,118188,118191,118194,118197],{"className":118130},[31676],[34,118132],{"className":118133,"style":31774},[31680],[34,118135,456],{"className":118136},[31782],[34,118138,37400],{"className":118139,"style":37422},[31685,31686],[34,118141,33244],{"className":118142},[31685,31686],[34,118144,111],{"className":118145},[31685,31686],[34,118147,31965],{"className":118148},[31685,31686],[34,118150,47292],{"className":118151},[31685,31686],[34,118153,34599],{"className":118154,"style":34452},[31685,31686],[34,118156,106063],{"className":118157,"style":67482},[31685,31686],[34,118159,91994],{"className":118160,"style":34452},[31685,31686],[34,118162,31960],{"className":118163},[32119],[34,118165],{"className":118166,"style":32123},[31845],[34,118168,47384],{"className":118169,"style":38706},[31685,31686],[34,118171,43383],{"className":118172},[31685,31686],[34,118174,94628],{"className":118175},[31685,31686],[34,118177,34395],{"className":118178,"style":34452},[31685,31686],[34,118180,113295],{"className":118181},[31685,31686],[34,118183,106063],{"className":118184,"style":67482},[31685,31686],[34,118186,91994],{"className":118187,"style":34452},[31685,31686],[34,118189,2557],{"className":118190},[31690],[34,118192],{"className":118193,"style":31846},[31845],[34,118195,33239],{"className":118196},[31850],[34,118198],{"className":118199,"style":31846},[31845],[34,118201,118203,118206,118209,118212,118215,118219,118223,118226,118229,118232,118236,118239,118242,118245],{"className":118202},[31676],[34,118204],{"className":118205,"style":43217},[31680],[34,118207,47384],{"className":118208,"style":38706},[31685,31686],[34,118210,43379],{"className":118211,"style":47113},[31685,31686],[34,118213,111],{"className":118214},[31685,31686],[34,118216,118218],{"className":118217,"style":34452},[31685,31686],"ssr",[34,118220,118222],{"className":118221},[31685,31686],"oo",[34,118224,31759],{"className":118225},[31685,31686],[34,118227,31960],{"className":118228},[32119],[34,118230],{"className":118231,"style":32123},[31845],[34,118233,118113],{"className":118234,"style":118235},[31685,31686],"margin-right:0.0813em;",[34,118237,43383],{"className":118238},[31685,31686],[34,118240,94628],{"className":118241},[31685,31686],[34,118243,34395],{"className":118244,"style":34452},[31685,31686],[34,118246,31747],{"className":118247},[31685,31686],[58531,118249],{},"因此不存在部分依赖和传递依赖，且决定因素是候选码。",[15,118252,118253],{},"总结：\n上述关系模式在属性原子性的基础上，非主属性均完全依赖于候选码，且不存在非主属性对候选码的传递依赖；进一步看，每个非平凡函数依赖的决定因素都是候选码，因此均可认为达到 BCNF。",[32188,118255],{},[19,118257,115240],{"id":118258},"六sql-编程共-10-分-1",[253,118260,118261],{"id":118261},"题目",[15,118263,115243],{},[15,118265,115246],{},[15,118267,115249],{},[15,118269,115252],{},[15,118271,115255],{},[15,118273,115258],{},[15,118275,115261],{},[15,118277,58173],{},[15,118279,118280],{},"SQL Server 存储过程：",[24,118282,118284],{"className":92584,"code":118283,"language":92586,"meta":29,"style":29},"IF OBJECT_ID(N'dbo.proc_get_employee_name', N'P') IS NOT NULL\nBEGIN\nDROP PROCEDURE dbo.proc_get_employee_name;\nEND;\nGO\n\nCREATE PROCEDURE dbo.proc_get_employee_name\n@in_emp_id INT,\n@out_emp_name VARCHAR(50) OUTPUT\nAS\nBEGIN\nSET NOCOUNT ON;\n\n IF @in_emp_id IS NULL\n OR NOT EXISTS (\n SELECT 1\n FROM dbo.Employees\n WHERE emp_id = @in_emp_id\n )\n BEGIN\n SET @out_emp_name = '无此员工';\n RETURN;\n END;\n\n SELECT @out_emp_name = emp_name\n FROM dbo.Employees\n WHERE emp_id = @in_emp_id;\n\nEND;\nGO\n",[31,118285,118286,118291,118295,118300,118304,118308,118312,118317,118322,118327,118331,118335,118340,118344,118349,118354,118359,118364,118369,118374,118378,118383,118388,118392,118396,118400,118405,118409,118413,118417],{"__ignoreMap":29},[34,118287,118288],{"class":36,"line":37},[34,118289,118290],{},"IF OBJECT_ID(N'dbo.proc_get_employee_name', N'P') IS NOT NULL\n",[34,118292,118293],{"class":36,"line":43},[34,118294,102603],{},[34,118296,118297],{"class":36,"line":49},[34,118298,118299],{},"DROP PROCEDURE dbo.proc_get_employee_name;\n",[34,118301,118302],{"class":36,"line":56},[34,118303,102656],{},[34,118305,118306],{"class":36,"line":62},[34,118307,102988],{},[34,118309,118310],{"class":36,"line":68},[34,118311,53],{"emptyLinePlaceholder":52},[34,118313,118314],{"class":36,"line":74},[34,118315,118316],{},"CREATE PROCEDURE dbo.proc_get_employee_name\n",[34,118318,118319],{"class":36,"line":1971},[34,118320,118321],{},"@in_emp_id INT,\n",[34,118323,118324],{"class":36,"line":1976},[34,118325,118326],{},"@out_emp_name VARCHAR(50) OUTPUT\n",[34,118328,118329],{"class":36,"line":1985},[34,118330,93260],{},[34,118332,118333],{"class":36,"line":1990},[34,118334,102603],{},[34,118336,118337],{"class":36,"line":1999},[34,118338,118339],{},"SET NOCOUNT ON;\n",[34,118341,118342],{"class":36,"line":2022},[34,118343,53],{"emptyLinePlaceholder":52},[34,118345,118346],{"class":36,"line":2036},[34,118347,118348],{}," IF @in_emp_id IS NULL\n",[34,118350,118351],{"class":36,"line":2603},[34,118352,118353],{}," OR NOT EXISTS (\n",[34,118355,118356],{"class":36,"line":5537},[34,118357,118358],{}," SELECT 1\n",[34,118360,118361],{"class":36,"line":5543},[34,118362,118363],{}," FROM dbo.Employees\n",[34,118365,118366],{"class":36,"line":5548},[34,118367,118368],{}," WHERE emp_id = @in_emp_id\n",[34,118370,118371],{"class":36,"line":5564},[34,118372,118373],{}," )\n",[34,118375,118376],{"class":36,"line":5576},[34,118377,102636],{},[34,118379,118380],{"class":36,"line":5606},[34,118381,118382],{}," SET @out_emp_name = '无此员工';\n",[34,118384,118385],{"class":36,"line":5616},[34,118386,118387],{}," RETURN;\n",[34,118389,118390],{"class":36,"line":5628},[34,118391,103374],{},[34,118393,118394],{"class":36,"line":5634},[34,118395,53],{"emptyLinePlaceholder":52},[34,118397,118398],{"class":36,"line":6420},[34,118399,102911],{},[34,118401,118402],{"class":36,"line":11294},[34,118403,118404],{}," FROM dbo.Employees\n",[34,118406,118407],{"class":36,"line":11321},[34,118408,102921],{},[34,118410,118411],{"class":36,"line":11326},[34,118412,53],{"emptyLinePlaceholder":52},[34,118414,118415],{"class":36,"line":11333},[34,118416,102656],{},[34,118418,118419],{"class":36,"line":11344},[34,118420,102988],{},[15,118422,118423],{},"解析：",[2230,118425,118426,118429,118432,118435],{},[108,118427,118428],{},"IF OBJECT_ID(...) IS NOT NULL 用于判断存储过程是否已经存在，若存在则先删除，避免重复创建时报错。",[108,118430,118431],{},"@in_emp_id 是输入参数，@out_emp_name 是输出参数。",[108,118433,118434],{},"如果 @in_emp_id 为 NULL，或者 Employees 表中不存在该员工编号，则将输出参数赋值为“无此员工”。",[108,118436,118437],{},"如果员工存在，则通过 SELECT @out_emp_name = emp_name 将员工姓名赋值给输出参数。",[15,118439,103055],{},[24,118441,118443],{"className":92584,"code":118442,"language":92586,"meta":29,"style":29},"DECLARE @name VARCHAR(50);\n\nEXEC dbo.proc_get_employee_name\n@in_emp_id = 1,\n@out_emp_name = @name OUTPUT;\n\nSELECT @name AS emp_name;\n",[31,118444,118445,118450,118454,118459,118464,118469,118473],{"__ignoreMap":29},[34,118446,118447],{"class":36,"line":37},[34,118448,118449],{},"DECLARE @name VARCHAR(50);\n",[34,118451,118452],{"class":36,"line":43},[34,118453,53],{"emptyLinePlaceholder":52},[34,118455,118456],{"class":36,"line":49},[34,118457,118458],{},"EXEC dbo.proc_get_employee_name\n",[34,118460,118461],{"class":36,"line":56},[34,118462,118463],{},"@in_emp_id = 1,\n",[34,118465,118466],{"class":36,"line":62},[34,118467,118468],{},"@out_emp_name = @name OUTPUT;\n",[34,118470,118471],{"class":36,"line":68},[34,118472,53],{"emptyLinePlaceholder":52},[34,118474,118475],{"class":36,"line":74},[34,118476,118477],{},"SELECT @name AS emp_name;\n",[79,118479,81],{},{"title":29,"searchDepth":43,"depth":43,"links":118481},[118482,118483,118484,118485,118486,118487,118488,118505,118512,118517,118524,118529],{"id":114830,"depth":43,"text":114831},{"id":115093,"depth":43,"text":115094},{"id":115140,"depth":43,"text":115141},{"id":115157,"depth":43,"text":115158},{"id":115200,"depth":43,"text":115201},{"id":115239,"depth":43,"text":115240},{"id":115269,"depth":43,"text":114831,"children":118489},[118490,118491,118492,118493,118494,118495,118496,118497,118498,118499,118500,118501,118502,118503,118504],{"id":115272,"depth":49,"text":115273},{"id":115294,"depth":49,"text":115295},{"id":115316,"depth":49,"text":115317},{"id":115338,"depth":49,"text":115339},{"id":115390,"depth":49,"text":115391},{"id":115412,"depth":49,"text":115413},{"id":115434,"depth":49,"text":115435},{"id":115456,"depth":49,"text":115457},{"id":115478,"depth":49,"text":115479},{"id":115500,"depth":49,"text":115501},{"id":115522,"depth":49,"text":115523},{"id":115544,"depth":49,"text":115545},{"id":115566,"depth":49,"text":115567},{"id":115588,"depth":49,"text":115589},{"id":115610,"depth":49,"text":115611},{"id":115631,"depth":43,"text":115094,"children":118506},[118507,118508,118509,118510,118511],{"id":115634,"depth":49,"text":115273},{"id":115652,"depth":49,"text":115295},{"id":115668,"depth":49,"text":115317},{"id":115684,"depth":49,"text":115339},{"id":115701,"depth":49,"text":115391},{"id":115720,"depth":43,"text":115141,"children":118513},[118514,118515,118516],{"id":115723,"depth":49,"text":115273},{"id":116288,"depth":49,"text":115295},{"id":116356,"depth":49,"text":115317},{"id":116414,"depth":43,"text":115158,"children":118518},[118519,118520,118521,118522,118523],{"id":116426,"depth":49,"text":115273},{"id":116812,"depth":49,"text":115295},{"id":117332,"depth":49,"text":115317},{"id":117364,"depth":49,"text":115339},{"id":117434,"depth":49,"text":115391},{"id":117500,"depth":43,"text":115201,"children":118525},[118526,118527,118528],{"id":117520,"depth":49,"text":115273},{"id":117664,"depth":49,"text":115295},{"id":117772,"depth":49,"text":115317},{"id":118258,"depth":43,"text":115240,"children":118530},[118531],{"id":118261,"depth":49,"text":118261},"考试科目： 数据库系统\n本卷为： 闭卷\n本卷为： A 卷\n印数： 110\n出题院系： 计算机学院\n出题日期： 2026 年 4 月 20 日",{},"\u002Fsource\u002Fcs\u002Fdb\u002Fpastpapers\u002F25-26cs",{"title":114748,"description":118532},"source\u002Fcs\u002Fdb\u002Fpastpapers\u002F25-26cs","3KSCoVir438NJRqcmQBuLYB8F8XKHUT6FDs4kpgY1W4",{"id":118539,"title":118540,"body":118541,"description":29,"extension":85,"meta":119847,"navigation":52,"path":119848,"seo":119849,"stem":119850,"__hash__":119851},"notes\u002Fsource\u002Fcs\u002Fdb\u002Fpastpapers\u002Fdb-pastpaper-1.md","2024-2025-2 计科数据库系统期末",{"type":8,"value":118542,"toc":119839},[118543,118547,118551,118557,118560,118563,118566,118569,118604,118610,118616,118629,118635,118638,118655,118661,118678,118684,118700,118706,118733,118739,118756,118769,118787,118793,118810,118816,118833,118839,118866,118872,118885,118891,118999,119005,119018,119024,119037,119039,119043,119048,119060,119065,119077,119086,119098,119104,119116,119128,119140,119142,119146,119151,119183,119188,119214,119219,119262,119267,119303,119305,119309,119314,119325,119330,119353,119358,119375,119380,119401,119406,119428,119433,119454,119459,119480,119485,119503,119505,119509,119514,119519,119561,119566,119599,119604,119616,119621,119732,119734,119738,119743,119748],[11,118544,118546],{"id":118545},"数据库系统-试题","数据库系统 试题",[253,118548,118550],{"id":118549},"一-选择题1分题-15-15分","一、 选择题（1分\u002F题 * 15 = 15分）",[15,118552,118553,118556],{},[2627,118554,118555],{},"题目 1："," 数据库系统中的数据模型通常由()三部分组成。",[15,118558,118559],{},"A. 数据结构、数据操作和完整性约束",[15,118561,118562],{},"B. 数据定义、数据操作和安全性约束",[15,118564,118565],{},"C. 数据结构、数据管理和数据保护",[15,118567,118568],{},"D. 数据定义、数据管理和运行控制",[105,118570,118571,118578],{},[108,118572,118573,2664,118576],{},[2627,118574,118575],{},"答案",[2627,118577,54151],{},[108,118579,118580,118583,118584],{},[2627,118581,118582],{},"解析","：数据模型是数据库系统的核心概念，用于描述数据、数据之间的联系以及数据的语义。它必须包含三个基本要素：\n",[2230,118585,118586,118592,118598],{},[108,118587,118588,118591],{},[2627,118589,118590],{},"数据结构","：描述系统的静态特性（例如，关系模型中的二维表结构）。",[108,118593,118594,118597],{},[2627,118595,118596],{},"数据操作","：描述系统的动态特性，即允许执行的操作（如查询、插入、删除、修改）。",[108,118599,118600,118603],{},[2627,118601,118602],{},"完整性约束","：保证数据的正确性和有效性（例如，主键不能为空，外键必须存在等）。其他选项混淆了数据库管理系统（DBMS）的功能（如数据管理、安全控制）与数据模型本身的组成。",[15,118605,118606,118609],{},[2627,118607,118608],{},"题目 2："," 数据库系统的核心是( )。",[15,118611,118612,118613],{},"A. 数据库\nB. 数据库管理系统\nC. 数据模型\nD. 软件工具\n",[71807,118614,118615],{},"(注：原卷此处排版缺失A选项，通常A为“数据库”)",[105,118617,118618,118624],{},[108,118619,118620,2664,118622],{},[2627,118621,118575],{},[2627,118623,54224],{},[108,118625,118626,118628],{},[2627,118627,118582],{},"：数据库系统（DBS）是一个大概念，包含数据库（DB）、数据库管理系统（DBMS）、应用程序和数据库管理员（DBA）。其中，**数据库管理系统（DBMS）**是负责建立、使用和维护数据库的系统软件，它统一管理和控制数据，是整个系统的核心与大脑。",[15,118630,118631,118634],{},[2627,118632,118633],{},"题目 3："," 下列四项中,不属于数据库系统的特点的是()",[15,118636,118637],{},"A. 数据结构化\nB. 数据由DBMS统一管理\nC. 数据冗余度大\nD. 数据独立性高",[105,118639,118640,118646],{},[108,118641,118642,2664,118644],{},[2627,118643,118575],{},[2627,118645,47384],{},[108,118647,118648,118650,118651,118654],{},[2627,118649,118582],{},"：文件系统由于各自维护自己的文件，会导致同一数据多处存放（冗余度大）。而数据库系统的首要目的之一就是",[2627,118652,118653],{},"减少数据冗余","，实现数据共享。因此，“数据冗余度大”是文件系统的缺点，绝不是数据库系统的特点。",[15,118656,118657,118660],{},[2627,118658,118659],{},"题目 4："," E-R 图是数据库设计的工具之一,它适用于建立数据库的( )。\nA. 概念模型 B. 逻辑模型 C. 结构模型 D. 物理模型",[105,118662,118663,118669],{},[108,118664,118665,2664,118667],{},[2627,118666,118575],{},[2627,118668,54151],{},[108,118670,118671,118673,118674,118677],{},[2627,118672,118582],{},"：E-R图（实体-联系图）是用于需求分析后，将现实世界中的业务逻辑转化为计算机能理解的模型的第一步。它独立于任何具体的DBMS，完全从业务概念出发，因此对应的是",[2627,118675,118676],{},"概念模型","。逻辑模型是将E-R图转化为具体的表结构；物理模型则涉及数据在磁盘上的具体存储。",[15,118679,118680,118683],{},[2627,118681,118682],{},"题目 5："," 在关系数据库设计中,设计关系模式是数据库设计中()阶段的任务。\nA. 逻辑设计 B. 物理设计 C. 需求分析 D. 概念设计",[105,118685,118686,118692],{},[108,118687,118688,2664,118690],{},[2627,118689,118575],{},[2627,118691,54151],{},[108,118693,118694,118696,118697,347],{},[2627,118695,118582],{},"：设计关系模式（即确定有哪些表，表中有哪些列，主外键是什么）是将概念阶段的E-R图转化为特定关系型数据库（如MySQL, SQL Server）支持的数据模型的过程。这一步属于",[2627,118698,118699],{},"逻辑结构设计",[15,118701,118702,118705],{},[2627,118703,118704],{},"题目 6："," 数据的物理独立性是指( )\nA. 数据库与数据库管理系统相互独立\nB. 用户程序与数据库管理系统相互独立\nC. 用户的应用程序与存储在磁盘上数据库中的数据是相互独立的\nD. 应用程序与数据库中数据的逻辑结构是相互独立的",[105,118707,118708,118714],{},[108,118709,118710,2664,118712],{},[2627,118711,118575],{},[2627,118713,47384],{},[108,118715,118716,118718,118719],{},[2627,118717,118582],{},"： 数据库具有“三级模式和两级映像”。\n",[105,118720,118721,118727],{},[108,118722,118723,118726],{},[2627,118724,118725],{},"物理独立性","：底层数据在磁盘上的存储位置、存储格式改变时，上层应用程序不需要修改。",[108,118728,118729,118732],{},[2627,118730,118731],{},"逻辑独立性","（对应选项D）：数据库的整体表结构（逻辑层面）增加或修改时，上层应用程序不需要修改。题目问的是物理独立性，故选C。",[15,118734,118735,118738],{},[2627,118736,118737],{},"题目 7："," 要保证数据库的数据独立性,需要修改的是( )\nA. 三层模式之间的两种映射 B. 模式与内模式 C. 模式与外模式 D. 三层模式",[105,118740,118741,118747],{},[108,118742,118743,2664,118745],{},[2627,118744,118575],{},[2627,118746,54151],{},[108,118748,118749,118751,118752,118755],{},[2627,118750,118582],{},"：数据的独立性完全依靠DBMS在三级模式（外模式、模式、内模式）之间提供的",[2627,118753,118754],{},"两种映射","（外模式\u002F模式映射、模式\u002F内模式映射）来保证。当底层发生变化时，DBA只需要修改这两种映射关系，就能保证上层不受影响。",[15,118757,118758,118761,118762,118765,118766],{},[2627,118759,118760],{},"题目 8："," “授权”和“收回权限”是DBS采用的( )措施。\n",[71807,118763,118764],{},"(注：原卷缺失文字，合理推断为“收回权限”即GRANT和REVOKE)","\nA. 安全性 B. 完整性 C. 并发控制 D. 数据恢复\n",[71807,118767,118768],{},"(注：排版错乱，A为安全性，推测B为完整性等)",[105,118770,118771,118778],{},[108,118772,118773,2664,118775],{},[2627,118774,118575],{},[2627,118776,118777],{},"A（安全性）",[108,118779,118780,118782,118783,118786],{},[2627,118781,118582],{},"：授权（GRANT）指的是允许特定用户查看或修改数据的权限，这是为了防止恶意用户非法访问数据，属于",[2627,118784,118785],{},"安全性","控制范畴。完整性则是防止合法用户输入错误的数据。",[15,118788,118789,118792],{},[2627,118790,118791],{},"题目 9："," DBMS能实现对数据库中数据的查询、插入、修改和删除等操作,这种功能称为()\nA. 数据定义功能 B. 数据管理功能 C. 数据操纵功能 D. 数据控制功能",[105,118794,118795,118801],{},[108,118796,118797,2664,118799],{},[2627,118798,118575],{},[2627,118800,47384],{},[108,118802,118803,118805,118806,118809],{},[2627,118804,118582],{},"：在SQL语言中，操作数据的动作（增删改查：INSERT, DELETE, UPDATE, SELECT）统称为",[2627,118807,118808],{},"数据操纵语言（DML - Data Manipulation Language）","。数据定义功能（DDL）是用来建表、删表的。",[15,118811,118812,118815],{},[2627,118813,118814],{},"题目 10："," 有一名为“列车运营”实体,含有:车次、日期、实际发车时间、实际抵达时间、情况摘要等属性,该实体主键是( )\nA. 车次 B. 日期 C. 车次+日期 D. 车次+情况摘要",[105,118817,118818,118824],{},[108,118819,118820,2664,118822],{},[2627,118821,118575],{},[2627,118823,47384],{},[108,118825,118826,118828,118829,118832],{},[2627,118827,118582],{},"：主键的作用是唯一标识一条记录。一趟列车（如G123）每天都会发车，因此单靠“车次”无法唯一确定一条具体的运营记录。单靠“日期”更不行，因为每天有很多车次。只有联合“",[2627,118830,118831],{},"车次+日期","”，才能唯一确定特定一天、特定一趟列车的运营情况。",[15,118834,118835,118838],{},[2627,118836,118837],{},"题目 11："," 学校数据库中有学生和宿舍两个关系:学生(学号,姓名)和宿舍(楼名,房间号,床位号,学号)。假设有的学生不住宿,床位也可能空闲,如果要列出所有学生住宿和宿舍分配的情况,包括没有住宿的学生和空闲的床位,则应执行( )\nA. 全外连接 B. 左外连接 C. 右外连接 D. 自然连接",[105,118840,118841,118847],{},[108,118842,118843,2664,118845],{},[2627,118844,118575],{},[2627,118846,54151],{},[108,118848,118849,118851,118852],{},[2627,118850,118582],{},"： 连接查询中，自然连接只会保留匹配的行。\n",[105,118853,118854,118857,118860],{},[108,118855,118856],{},"左外连接：保留左表所有行，右表没有的补空。",[108,118858,118859],{},"右外连接：保留右表所有行，左表没有的补空。",[108,118861,118862,118865],{},[2627,118863,118864],{},"全外连接（Full Outer Join）","：左右两表的所有行都保留。题目要求既要保留“没有住宿的学生（左表未匹配）”，又要保留“空闲的床位（右表未匹配）”，因此必须使用全外连接。",[15,118867,118868,118871],{},[2627,118869,118870],{},"题目 12："," 关系规范化中的插入操作异常是指()\nA. 不该删除的数据被删除 B. 不该插入的数据被插入 C. 应该删除的数据未被刪除 D. 应该插入的数据未被插入",[105,118873,118874,118880],{},[108,118875,118876,2664,118878],{},[2627,118877,118575],{},[2627,118879,91994],{},[108,118881,118882,118884],{},[2627,118883,118582],{},"：插入异常是指由于表结构设计不合理（如存在部分依赖），导致某些本身存在且应该被录入的信息，因为缺少主键的一部分而无法被插入到数据库中。",[15,118886,118887,118890],{},[2627,118888,118889],{},"题目 13："," 设 D1={张三,李四}, D2={数学,语文}, D3={优,良} 则笛卡尔积D1×D2×D3的基是()\nA. 3 B. 6 C. 8 D. 9",[105,118892,118893,118899],{},[108,118894,118895,2664,118897],{},[2627,118896,118575],{},[2627,118898,47384],{},[108,118900,118901,118903,118904,347],{},[2627,118902,118582],{},"：笛卡尔积是指将各个集合中的元素进行所有可能的排列组合。其结果的基（也就是组合后的记录总数）等于各个集合元素个数的乘积。由于这三个集合都包含2个元素，计算过程为：",[34,118905,118907,118933],{"className":118906},[31642],[34,118908,118910],{"className":118909},[31646],[31648,118911,118912],{"xmlns":31650},[31652,118913,118914,118930],{},[31655,118915,118916,118918,118920,118922,118924,118926,118928],{},[31749,118917,768],{},[31662,118919,42071],{},[31749,118921,768],{},[31662,118923,42071],{},[31749,118925,768],{},[31662,118927,202],{},[31749,118929,60525],{},[31665,118931,118932],{"encoding":31667},"2 \\times 2 \\times 2 = 8",[34,118934,118936,118954,118972,118990],{"className":118935,"ariaHidden":758},[31672],[34,118937,118939,118942,118945,118948,118951],{"className":118938},[31676],[34,118940],{"className":118941,"style":35942},[31680],[34,118943,768],{"className":118944},[31685],[34,118946],{"className":118947,"style":32604},[31845],[34,118949,42071],{"className":118950},[32024],[34,118952],{"className":118953,"style":32604},[31845],[34,118955,118957,118960,118963,118966,118969],{"className":118956},[31676],[34,118958],{"className":118959,"style":35942},[31680],[34,118961,768],{"className":118962},[31685],[34,118964],{"className":118965,"style":32604},[31845],[34,118967,42071],{"className":118968},[32024],[34,118970],{"className":118971,"style":32604},[31845],[34,118973,118975,118978,118981,118984,118987],{"className":118974},[31676],[34,118976],{"className":118977,"style":32617},[31680],[34,118979,768],{"className":118980},[31685],[34,118982],{"className":118983,"style":31846},[31845],[34,118985,202],{"className":118986},[31850],[34,118988],{"className":118989,"style":31846},[31845],[34,118991,118993,118996],{"className":118992},[31676],[34,118994],{"className":118995,"style":32617},[31680],[34,118997,60525],{"className":118998},[31685],[15,119000,119001,119004],{},[2627,119002,119003],{},"题目 14："," 在视图上不能完成的操作是( )。\nA. 更新视图 B. 查询 C. 在视图上定义新的基本表 D. 在视图上定义新视图",[105,119006,119007,119013],{},[108,119008,119009,2664,119011],{},[2627,119010,118575],{},[2627,119012,47384],{},[108,119014,119015,119017],{},[2627,119016,118582],{},"：视图被称为“虚表”，它自身不存储数据，其数据来源于背后的基本表。因此，你不能在一个虚拟的表上再去建立物理存储的“基本表”。",[15,119019,119020,119023],{},[2627,119021,119022],{},"题目 15："," 在数据库中,产生数据不一致的根本原因是()。\nA. 数据存储量太大 B. 没有严格保护数据 C. 未对数据进行完整性控制 D. 数据冗余",[105,119025,119026,119032],{},[108,119027,119028,2664,119030],{},[2627,119029,118575],{},[2627,119031,91994],{},[108,119033,119034,119036],{},[2627,119035,118582],{},"：如果同一个数据在数据库中存了多份（数据冗余），当需要修改这个数据时，如果只修改了其中一份而忘记修改另一份，就会导致同一数据的不同副本出现差异。这是数据不一致的根本诱因。",[32188,119038],{},[253,119040,119042],{"id":119041},"二-填空题1分空-10空-10分","二、 填空题（1分\u002F空 * 10空 = 10分）",[15,119044,119045],{},[2627,119046,119047],{},"1. 数据库与文件系统的根本区别是",[105,119049,119050,119055],{},[108,119051,119052,119054],{},[2627,119053,118575],{},"：数据的结构化",[108,119056,119057,119059],{},[2627,119058,118582],{},"：文件系统内部可能有结构，但整体是无结构的；而数据库中的数据整体上是高度结构化的（具有模式）。",[15,119061,119062],{},[2627,119063,119064],{},"2. 关系R(A,B,C)和S(A, D, E)中,R的主键是(A,B,C),S是的主键是A,则A在R中称为",[105,119066,119067,119072],{},[108,119068,119069,119071],{},[2627,119070,118575],{},"：外键（或外部码）",[108,119073,119074,119076],{},[2627,119075,118582],{},"：关系R中包含了关系S的主键A，因此A在R中作为参照另一张表的桥梁，被称为外键。",[15,119078,119079,119082,119083],{},[2627,119080,119081],{},"3. SQL 是一种综合性的语言,是","*、*",[2627,119084,119085],{},"、数据控制语言的组合。",[105,119087,119088,119093],{},[108,119089,119090,119092],{},[2627,119091,118575],{},"：数据定义语言（DDL）、数据操纵语言（DML）",[108,119094,119095,119097],{},[2627,119096,118582],{},"：SQL语言分为三大部分：定义表结构的DDL，增删改查数据的DML，以及进行权限管理的数据控制语言（DCL）。",[15,119099,119100,119103],{},[2627,119101,119102],{},"4. 关系模型的基本数据结构是","* ,其数据库存储时的基本组织方式***。**",[105,119105,119106,119111],{},[108,119107,119108,119110],{},[2627,119109,118575],{},"：关系（或二维表）、文件",[108,119112,119113,119115],{},[2627,119114,118582],{},"：在用户看来，关系模型就是一张张的二维表；而在操作系统的底层存储上，数据库通常是以文件的形式组织在磁盘上的。",[15,119117,119118,119121,119124,119125],{},[2627,119119,119120],{},"5. 关系的完整性包含",[71807,119122,119123],{},"和","**。**\n",[71807,119126,119127],{},"(注：标准理论为三类，原卷留有两个空)",[105,119129,119130,119135],{},[108,119131,119132,119134],{},[2627,119133,118575],{},"：实体完整性、参照完整性（或用户定义的完整性）",[108,119136,119137,119139],{},[2627,119138,118582],{},"：实体完整性要求主键不为空；参照完整性要求外键必须在主表中存在；用户定义的完整性则是针对特定业务规则的约束。填前两者即可。",[32188,119141],{},[253,119143,119145],{"id":119144},"三-简答题5分题-4题-20分","三、 简答题（5分\u002F题 * 4题 = 20分）",[15,119147,119148],{},[2627,119149,119150],{},"1. 试述使用数据库系统的好处。",[105,119152,119153],{},[108,119154,119155,33183,119157],{},[2627,119156,118575],{},[2230,119158,119159,119165,119171,119177],{},[108,119160,119161,119164],{},[2627,119162,119163],{},"实现数据共享，减少数据冗余","：多个应用程序可以共用同一个数据库，避免重复存储。",[108,119166,119167,119170],{},[2627,119168,119169],{},"具有高度的数据独立性","：底层数据结构的改变不会严重影响上层应用程序的使用。",[108,119172,119173,119176],{},[2627,119174,119175],{},"提供统一的数据控制机制","：由DBMS统一保证数据的完整性（不符合规则的数据无法写入）、安全性（无权限无法访问）、并发控制（多人同时修改不出错）以及数据恢复（崩溃后可恢复）。",[108,119178,119179,119182],{},[2627,119180,119181],{},"数据整体结构化","：能够清晰表达复杂事物及其之间的联系。",[15,119184,119185],{},[2627,119186,119187],{},"2. 试述视图的优点。",[105,119189,119190],{},[108,119191,119192,33183,119194],{},[2627,119193,118575],{},[2230,119195,119196,119202,119208],{},[108,119197,119198,119201],{},[2627,119199,119200],{},"简化用户操作","：可以将多张表复杂的连接查询逻辑封装在视图中，用户直接查询视图即可。",[108,119203,119204,119207],{},[2627,119205,119206],{},"增强数据安全性","：可以通过视图向不同用户展示不同的数据列，隐藏敏感字段。",[108,119209,119210,119213],{},[2627,119211,119212],{},"提供逻辑数据独立性","：当底层物理表的结构发生变化时，可以通过修改视图的定义，使得对外暴露的视图结构保持不变，从而不用修改应用程序。",[15,119215,119216],{},[2627,119217,119218],{},"3. 请描述数据库设计的步骤,并对每个步骤进行描述。",[105,119220,119221],{},[108,119222,119223,33183,119225],{},[2627,119224,118575],{},[2230,119226,119227,119233,119239,119244,119250,119256],{},[108,119228,119229,119232],{},[2627,119230,119231],{},"需求分析","：调查和分析用户的业务活动和数据需求，明确系统要“做什么”。",[108,119234,119235,119238],{},[2627,119236,119237],{},"概念结构设计","：将需求转化为抽象的、独立于底层系统的E-R图。",[108,119240,119241,119243],{},[2627,119242,118699],{},"：将E-R图转化为特定DBMS支持的数据模型（如关系模式），并进行规范化优化。",[108,119245,119246,119249],{},[2627,119247,119248],{},"物理结构设计","：为逻辑数据模型选取最适合应用环境的物理存储结构和存取方法（如建立索引）。",[108,119251,119252,119255],{},[2627,119253,119254],{},"数据库实施","：编写SQL语句建库建表，组织数据入库，并编写调试应用程序。",[108,119257,119258,119261],{},[2627,119259,119260],{},"数据库运行与维护","：系统投入运行后，进行长期的监控、备份、性能优化及必要的结构调整。",[15,119263,119264],{},[2627,119265,119266],{},"4. 请简述关系的完整性与安全性的不同。",[105,119268,119269],{},[108,119270,119271,33183,119273],{},[2627,119272,118575],{},[105,119274,119275,119292],{},[108,119276,119277,119280,119281,119284,119285,119288,119289,347],{},[2627,119278,119279],{},"完整性","：主要防范的是",[2627,119282,119283],{},"合法用户","由于误操作等原因录入了",[2627,119286,119287],{},"不正确、不符合语义规则的数据","（例如年龄输入为负数，或关联了不存在的外键）。核心目标是保证数据的",[2627,119290,119291],{},"准确性和有效性",[108,119293,119294,119280,119296,119299,119300,347],{},[2627,119295,118785],{},[2627,119297,119298],{},"非法用户","或恶意攻击对数据的窃取、破坏或未经授权的篡改。核心目标是保护数据",[2627,119301,119302],{},"不被越权访问",[32188,119304],{},[253,119306,119308],{"id":119307},"四-sql语句共20分","四、 SQL语句（共20分）",[15,119310,119311,2664],{},[2627,119312,119313],{},"前提关系说明",[105,119315,119316,119319,119322],{},[108,119317,119318],{},"学生 Student(学号 Sno, 学生姓名 Sname, 性别 Sgender, 出生日期 Birthday, 系 Sdept)",[108,119320,119321],{},"课程 Course(课程号 Cno, 课程名 Cname, 先行课 Cpno, 学分 Ccredit)",[108,119323,119324],{},"学生选课 SC(学号 Sno, 课程号 Cno, 分数 Grade)",[15,119326,119327],{},[2627,119328,119329],{},"1. 在Student表中新增一个Email列,其数据类型为varchar(30)。(2分)",[105,119331,119332,119340],{},[108,119333,119334,2664,119337],{},[2627,119335,119336],{},"SQL语句",[31,119338,119339],{},"ALTER TABLE Student ADD Email varchar(30);",[108,119341,119342,119344,119345,119348,119349,119352],{},[2627,119343,118582],{},"：修改已有表的结构必须使用 ",[31,119346,119347],{},"ALTER TABLE","，新增列使用 ",[31,119350,119351],{},"ADD"," 关键字。",[15,119354,119355],{},[2627,119356,119357],{},"2. 为出生日期 Birthday 列创建索引。(2分)",[105,119359,119360,119367],{},[108,119361,119362,2664,119364],{},[2627,119363,119336],{},[31,119365,119366],{},"CREATE INDEX idx_birthday ON Student(Birthday);",[108,119368,119369,119371,119372,119374],{},[2627,119370,118582],{},"：创建索引使用 ",[31,119373,92053],{}," 语法，后跟自定义的索引名称（如idx_birthday），并指定表名和要建立索引的列名。",[15,119376,119377],{},[2627,119378,119379],{},"3. 在Student表中查询性别为“男”且系别为“计算机科学与技术”的学生数据。(2分)",[105,119381,119382,119389],{},[108,119383,119384,2664,119386],{},[2627,119385,119336],{},[31,119387,119388],{},"SELECT * FROM Student WHERE Sgender='男' AND Sdept='计算机科学与技术';",[108,119390,119391,119393,119394,119396,119397,119400],{},[2627,119392,118582],{},"：这题考察基础的单表条件查询。使用 ",[31,119395,95239],{}," 子句过滤，多个条件需同时满足时使用逻辑运算符 ",[31,119398,119399],{},"AND"," 连接。",[15,119402,119403],{},[2627,119404,119405],{},"4. 在Student表中查询邮箱域名为“@stumail.nwu.edu.cn”的学生数据。(2分)",[105,119407,119408,119415],{},[108,119409,119410,2664,119412],{},[2627,119411,119336],{},[31,119413,119414],{},"SELECT * FROM Student WHERE Email LIKE '%@stumail.nwu.edu.cn';",[108,119416,119417,119419,119420,119423,119424,119427],{},[2627,119418,118582],{},"：由于域名是在邮箱字符串的结尾，我们需要使用模糊匹配。",[31,119421,119422],{},"LIKE"," 配合通配符 ",[31,119425,119426],{},"%"," 表示匹配任意长度的任意字符序列。",[15,119429,119430],{},[2627,119431,119432],{},"5. 找出Student表中年龄最大和年龄最小的学生出生日期。(4分)",[105,119434,119435,119442],{},[108,119436,119437,2664,119439],{},[2627,119438,119336],{},[31,119440,119441],{},"SELECT MIN(Birthday), MAX(Birthday) FROM Student;",[108,119443,119444,119446,119447,258,119450,119453],{},[2627,119445,118582],{},"：年龄最大意味着出生日期最早（值最小），年龄最小意味着出生日期最晚（值最大）。使用聚合函数 ",[31,119448,119449],{},"MIN()",[31,119451,119452],{},"MAX()"," 即可。",[15,119455,119456],{},[2627,119457,119458],{},"6. 分系别统计Student表中的学生人数。(4分)",[105,119460,119461,119468],{},[108,119462,119463,2664,119465],{},[2627,119464,119336],{},[31,119466,119467],{},"SELECT Sdept, COUNT(*) FROM Student GROUP BY Sdept;",[108,119469,119470,119472,119473,119476,119477,347],{},[2627,119471,118582],{},"：“分系别”意味着需要以“系（Sdept）”为依据进行分组，故使用 ",[31,119474,119475],{},"GROUP BY Sdept","；统计人数使用聚合函数 ",[31,119478,119479],{},"COUNT(*)",[15,119481,119482],{},[2627,119483,119484],{},"7. 将学生的学号及其平均成绩定义为一个试图。(4分)",[105,119486,119487,119494],{},[108,119488,119489,2664,119491],{},[2627,119490,119336],{},[31,119492,119493],{},"CREATE VIEW Stu_AvgGrade AS SELECT Sno, AVG(Grade) FROM SC GROUP BY Sno;",[108,119495,119496,119498,119499,119502],{},[2627,119497,118582],{},"：创建视图使用 ",[31,119500,119501],{},"CREATE VIEW 视图名 AS","，后面紧跟对应的查询语句。本题需按学号分组查询SC表中的平均成绩。",[32188,119504],{},[253,119506,119508],{"id":119507},"五-数据库设计共20分","五、 数据库设计（共20分）",[15,119510,119511,119513],{},[2627,119512,117503],{},"\n设有关系模式 R(读者编号, 姓名, 单位编号, 单位名称, 图书号, 书名, 借阅日期, 还书日期)，用来表示读者借阅图书信息。\n规则：每个读者属于一个单位；借阅呈多对多关系；每个读者每天每本书最多借一次。",[15,119515,119516],{},[2627,119517,119518],{},"1. 根据上述条件,试画出反应R中关系的E-R图(不必画实体的属性)。(6分)",[105,119520,119521],{},[108,119522,119523,119526,119527],{},[2627,119524,119525],{},"答案解析","：由于系统限制纯文本输出，可以这样画：\n",[105,119528,119529,119535],{},[108,119530,119531,119534],{},[2627,119532,119533],{},"三个实体（用矩形框表示）","：读者、单位、图书。",[108,119536,119537,33183,119540],{},[2627,119538,119539],{},"两个联系（用菱形框表示）",[105,119541,119542,119551],{},[108,119543,119544,119547,119548,347],{},[2627,119545,119546],{},"所属","（连接“读者”和“单位”）：由于每个读者只属于一个单位，而一个单位有多个读者，连线标记为 ",[2627,119549,119550],{},"N : 1",[108,119552,119553,119556,119557,119560],{},[2627,119554,119555],{},"借阅","（连接“读者”和“图书”）：多对多关系，连线标记为 ",[2627,119558,119559],{},"M : N","。需要注意的是，由于“每天借一次”的规定，借阅联系需要附带属性：借阅日期、还书日期（连在借阅菱形上）。",[15,119562,119563],{},[2627,119564,119565],{},"2. R最高属于第几范式,为什么?(5分)",[105,119567,119568,119576],{},[108,119569,119570,119572,119573,347],{},[2627,119571,118575],{},"：最高属于",[2627,119574,119575],{},"第一范式（1NF）",[108,119577,119578,119580,119581,119584,119585,33183,119587],{},[2627,119579,118582],{},"：\n首先明确主键：因为一个读者可以多次借阅同一本书，要唯一确认一条借阅记录，必须依赖 ",[2627,119582,119583],{},"(读者编号, 图书号, 借阅日期)"," 这三个字段共同构成复合主键。\n产生问题的原因在于",[2627,119586,110920],{},[105,119588,119589,119592],{},[108,119590,119591],{},"“姓名”、“单位编号” 只依赖于主键中的“读者编号”，与“图书号”、“借阅日期”无关。",[108,119593,119594,119595,119598],{},"“书名” 只依赖于主键中的“图书号”。\n这种非主属性（如姓名）依赖于主键的",[2627,119596,119597],{},"一部分","的情况，违反了第二范式（2NF）的要求（2NF要求所有非主属性必须完全依赖于整个主键）。因此，它仅仅满足最基础的列不可分的1NF。",[15,119600,119601],{},[2627,119602,119603],{},"3. 举例说明R在何种情况下会发生删除异常。(5分)",[105,119605,119606],{},[108,119607,119608,119611,119612,119615],{},[2627,119609,119610],{},"答案与解析","：\n假设读者“张三”（单位编号1，单位名称“信息学院”）归还了所有他借过的图书，并且我们需要将该借阅历史从关系R中删除。由于R不仅包含了借阅记录，还混杂了张三的基本信息，当删除张三所有借阅记录的那一刻，",[2627,119613,119614],{},"张三的姓名、所在单位的信息也会随着借阅记录一起被彻底从数据库中抹除","。这就叫删除异常（不该丢失的数据丢失了）。",[15,119617,119618],{},[2627,119619,119620],{},"4. 请将R规范到3NF,并标注每种关系中的主码。(4分)",[105,119622,119623,119658],{},[108,119624,119625,119627,119628],{},[2627,119626,118575],{},"：需要将其拆分为四张表，以消除部分依赖和传递依赖（带下划线的为主码）：\n",[2230,119629,119630,119637,119644,119651],{},[108,119631,119632,119633,119636],{},"读者 (",[94628,119634,119635],{},"读者编号",", 姓名, 单位编号)",[108,119638,119639,119640,119643],{},"单位 (",[94628,119641,119642],{},"单位编号",", 单位名称)",[108,119645,119646,119647,119650],{},"图书 (",[94628,119648,119649],{},"图书号",", 书名)",[108,119652,119653,119654,119657],{},"借阅 (",[94628,119655,119656],{},"读者编号, 图书号, 借阅日期",", 还书日期)",[108,119659,119660,33183,119662],{},[2627,119661,118582],{},[105,119663,119664,119667,119670],{},[108,119665,119666],{},"将图书信息独立成表，消除了“书名”对主键的部分依赖。",[108,119668,119669],{},"将借阅历史独立成表，由原复合主键和业务相关属性构成。",[108,119671,119672,119673,119702,119703,119731],{},"将读者自身信息独立成表，进一步发现，在原表逻辑中“读者编号 ",[34,119674,119676,119690],{"className":119675},[31642],[34,119677,119679],{"className":119678},[31646],[31648,119680,119681],{"xmlns":31650},[31652,119682,119683,119687],{},[31655,119684,119685],{},[31662,119686,33239],{},[31665,119688,119689],{"encoding":31667},"\\rightarrow",[34,119691,119693],{"className":119692,"ariaHidden":758},[31672],[34,119694,119696,119699],{"className":119695},[31676],[34,119697],{"className":119698,"style":73089},[31680],[34,119700,33239],{"className":119701},[31850]," 单位编号 ",[34,119704,119706,119719],{"className":119705},[31642],[34,119707,119709],{"className":119708},[31646],[31648,119710,119711],{"xmlns":31650},[31652,119712,119713,119717],{},[31655,119714,119715],{},[31662,119716,33239],{},[31665,119718,119689],{"encoding":31667},[34,119720,119722],{"className":119721,"ariaHidden":758},[31672],[34,119723,119725,119728],{"className":119724},[31676],[34,119726],{"className":119727,"style":73089},[31680],[34,119729,33239],{"className":119730},[31850]," 单位名称”，存在传递依赖（违反3NF）。因此必须把单位信息单独拆离出来，使系统达到第三范式（3NF）。",[32188,119733],{},[253,119735,119737],{"id":119736},"六-嵌入式sql填空15分","六、 嵌入式SQL填空（15分）",[15,119739,119740,119742],{},[2627,119741,117503],{}," 这是一段将SQL语句嵌入到C语言中执行的代码。核心逻辑是通过游标遍历查询结果，并交互式地询问用户是否要修改当前记录的年龄。",[15,119744,119745,2664],{},[2627,119746,119747],{},"代码填空与解析",[105,119749,119750,119765,119787,119802,119820],{},[108,119751,119752,2664,119755,119758],{},[2627,119753,119754],{},"空 (1)",[31,119756,119757],{},"EXEC SQL INCLUDE SQLCA;",[105,119759,119760],{},[108,119761,119762,119764],{},[2627,119763,118582],{},"：在C语言主程序的声明部分，必须首先引入SQL通信区（SQLCA）。它是DBMS与宿主语言程序之间交流状态信息（例如操作是否成功、返回错误码等）的数据结构。",[108,119766,119767,2664,119770,119773],{},[2627,119768,119769],{},"空 (2)",[31,119771,119772],{},"EXEC SQL DECLARE SX CURSOR FOR",[105,119774,119775],{},[108,119776,119777,119779,119780,119783,119784,347],{},[2627,119778,118582],{},"：在执行连接数据库的代码后，代码上方有一个 ",[31,119781,119782],{},"SELECT Stno, Sname... FROM Student"," 的查询语句。在嵌入式SQL中，对于返回多行结果的查询，必须声明一个游标（Cursor）来逐行处理。该空的作用就是声明游标 ",[31,119785,119786],{},"SX",[108,119788,119789,2664,119792,119795],{},[2627,119790,119791],{},"空 (3)",[31,119793,119794],{},"EXEC SQL OPEN SX;",[105,119796,119797],{},[108,119798,119799,119801],{},[2627,119800,118582],{},"：注释提示“打开游标SX”。在使用游标之前，必须用OPEN命令打开它，此时DBMS才真正执行查询并将结果集准备好。",[108,119803,119804,2664,119807,119810],{},[2627,119805,119806],{},"空 (4)",[31,119808,119809],{},"EXEC SQL FETCH SX INTO :HSno, :HSname, :HSsex, :HSage;",[105,119811,119812],{},[108,119813,119814,119816,119817,119819],{},[2627,119815,118582],{},"：进入循环后，注释提示“推进游标,将当前数据放入主变量”。FETCH操作用于将游标当前指向的某一条记录提取出来，并赋值给C语言中定义的变量（主变量前需加冒号 ",[31,119818,5598],{}," 标识）。",[108,119821,119822,2664,119825,119828],{},[2627,119823,119824],{},"空 (5)",[31,119826,119827],{},"EXEC SQL UPDATE Student",[105,119829,119830],{},[108,119831,119832,119834,119835,119838],{},[2627,119833,118582],{},"：注释提示“嵌入式 SQL 更新”。紧随其后的代码是 ",[31,119836,119837],{},"SET Sage=:NEWAGE WHERE CURRENT OF SX;","，这属于“定位更新”（修改游标当前指向的行）。因此这里需要补全UPDATE的基础语法，即指定要更新的表名。",{"title":29,"searchDepth":43,"depth":43,"links":119840},[119841,119842,119843,119844,119845,119846],{"id":118549,"depth":49,"text":118550},{"id":119041,"depth":49,"text":119042},{"id":119144,"depth":49,"text":119145},{"id":119307,"depth":49,"text":119308},{"id":119507,"depth":49,"text":119508},{"id":119736,"depth":49,"text":119737},{},"\u002Fsource\u002Fcs\u002Fdb\u002Fpastpapers\u002Fdb-pastpaper-1",{"title":118540,"description":29},"source\u002Fcs\u002Fdb\u002Fpastpapers\u002Fdb-pastpaper-1","TE4XIlRN0s_V3UsFQMM5_mSZFZkVroeweQx96rS1Bgc",{"id":119853,"title":119854,"body":119855,"description":120062,"extension":85,"meta":120063,"navigation":52,"path":120064,"seo":120065,"stem":120066,"__hash__":120067},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F00-总览与考试题型地图.md","数据库系统完整讲义总览与考试题型地图",{"type":8,"value":119856,"toc":120053},[119857,119859,119870,119874,119877,119880,119906,119910,119913,119936,119940,119943,119993,119997,120000,120003,120029,120033,120036],[11,119858,119854],{"id":119854},[15,119860,119861,119862,119865,119866,119869],{},"这套讲义按本课程近年考试和课堂投影整理，不按教材目录机械复述，而是按“考试会怎么问、答题时怎么写、为什么这样写”组织。主要参考资料包括：王珊等《数据库系统概论》第 6 版、2020-2024 历年期末题、2025-2026 新上传 ",[31,119863,119864],{},"new-pic"," 投影\u002F板书、",[31,119867,119868],{},"pic整理文本"," 课堂材料、补充习题 70+100 题。",[19,119871,119873],{"id":119872},"一近年题型结构","一、近年题型结构",[253,119875,119876],{"id":119876},"中文卷常见结构",[15,119878,119879],{},"常见分布为：",[105,119881,119882,119885,119888,119891,119894,119897,119900,119903],{},[108,119883,119884],{},"选择题：数据库基础概念、关系模型、SQL 语法、完整性\u002F安全性、事务、封锁、恢复。",[108,119886,119887],{},"填空题：教材中的精确定义、三层模式、数据模型三要素、SQL 组成、完整性分类、恢复\u002F并发术语。",[108,119889,119890],{},"简答题：DBMS 功能、数据库系统优点、视图优点、完整性与安全性区别、事务 ACID、死锁、恢复策略。",[108,119892,119893],{},"SQL 题：建表、约束、索引、视图、查询、聚合、嵌套查询、删除\u002F更新、授权。",[108,119895,119896],{},"数据库设计题：E-R 图、E-R 转关系模式、主码外码、函数依赖、范式判断、异常、分解到 3NF\u002FBCNF。",[108,119898,119899],{},"数据库编程题：存储过程、触发器、游标、嵌入式 SQL、事务控制。",[108,119901,119902],{},"查询优化题：写关系代数、画原始查询树\u002F语法树、做等价变换、画优化树、写优化后的关系代数。",[108,119904,119905],{},"恢复与并发题：检查点 UNDO\u002FREDO、事务调度、封锁协议、两阶段封锁、死锁判断。",[253,119907,119909],{"id":119908},"new-pic-最新样题结构","new-pic 最新样题结构",[15,119911,119912],{},"新图里出现了更明确的专题权重：",[105,119914,119915,119918,119921,119924,119927,119930,119933],{},[108,119916,119917],{},"Explain the glossaries \u002F Fill in the blanks：关系、参照完整性、角色、ACID、脏数据等术语。",[108,119919,119920],{},"SQL：SaleDetail\u002FProduct\u002FStock 这类公司数据库，要求建表、级联更新、禁止级联删除、删除某日期后销售明细、插入产品、统计 2023 销售额、涨价、建视图、建索引、建用户并授权。",[108,119922,119923],{},"Database programming：营业时间触发器、带参数存储过程、游标循环、输出参数。",[108,119925,119926],{},"Relational Database Design：E-R 图、关系模式、主码外码、范式说明。",[108,119928,119929],{},"Query Optimization：用笛卡尔积写等价关系代数，画原始树，给优化过程，写优化表达式。",[108,119931,119932],{},"Database Recovery：检查点恢复，判断哪些事务 UNDO、哪些 REDO。",[108,119934,119935],{},"Concurrency Control：给事务加锁\u002F解锁，满足两阶段封锁协议，并判断是否可能死锁。",[19,119937,119939],{"id":119938},"二本讲义文件怎么用","二、本讲义文件怎么用",[15,119941,119942],{},"建议按下面顺序复习：",[2230,119944,119945,119951,119957,119963,119969,119975,119981,119987],{},[108,119946,119947,119950],{},[31,119948,119949],{},"01-数据库基础与关系模型.md","：把选择填空的根打牢。",[108,119952,119953,119956],{},[31,119954,119955],{},"02-SQL与关系代数.md","：覆盖 SQL 大题和小题。",[108,119958,119959,119962],{},[31,119960,119961],{},"03-数据库编程专项.md","：集中练存储过程、触发器、游标、嵌入式 SQL。",[108,119964,119965,119968],{},[31,119966,119967],{},"04-数据库设计ER与规范化.md","：设计题和范式题是大分值核心。",[108,119970,119971,119974],{},[31,119972,119973],{},"05-查询优化专项.md","：解决“画几棵树、怎么画”的问题。",[108,119976,119977,119980],{},[31,119978,119979],{},"06-事务并发控制与恢复.md","：检查点、UNDO\u002FREDO、封锁协议、死锁。",[108,119982,119983,119986],{},[31,119984,119985],{},"07-填空选择速记清单.md","：考前背诵小知识点。",[108,119988,119989,119992],{},[31,119990,119991],{},"08-大题答题模板.md","：考场直接套用的结构化模板。",[19,119994,119996],{"id":119995},"三最重要的得分策略","三、最重要的得分策略",[15,119998,119999],{},"大题不要只写结果，要写“依据”。数据库设计题要写实体、联系、主码、外码、范式理由；查询优化题要写原始表达式、原始树、优化规则、优化树；恢复题要写判断规则；编程题要写参数、条件判断、异常\u002F不存在处理、游标关闭释放或事务提交回滚。",[15,120001,120002],{},"填空题不要凭感觉写同义词。教材里很多术语有固定表述，例如：",[105,120004,120005,120008,120011,120014,120017,120020,120023,120026],{},[108,120006,120007],{},"数据模型三要素：数据结构、数据操作、完整性约束。",[108,120009,120010],{},"数据库系统核心：DBMS。",[108,120012,120013],{},"数据库与文件系统本质区别：数据结构化。",[108,120015,120016],{},"数据库系统特点：数据结构化、共享性高\u002F冗余低、独立性高、由 DBMS 统一管理。",[108,120018,120019],{},"关系完整性：实体完整性、参照完整性、用户自定义完整性。",[108,120021,120022],{},"恢复冗余技术：数据转储、登记日志文件。",[108,120024,120025],{},"基本封锁：共享锁 S、排他锁 X。",[108,120027,120028],{},"SQL 综合语言：DDL、DML、DCL，有时教材也把查询语言 DQL 单列。",[19,120030,120032],{"id":120031},"四考场时间分配","四、考场时间分配",[15,120034,120035],{},"如果是 120 分钟闭卷，建议：",[105,120037,120038,120041,120044,120047,120050],{},[108,120039,120040],{},"选择填空：20-25 分钟。不会的先标记，后面凭大题知识回填。",[108,120042,120043],{},"SQL：20 分钟。先写能确定的基本语句，复杂嵌套最后补。",[108,120045,120046],{},"设计\u002F规范化：25-30 分钟。先列实体和函数依赖，防止画图卡太久。",[108,120048,120049],{},"编程：15-20 分钟。先写结构：参数、声明、判断、核心 SQL、提交\u002F关闭。",[108,120051,120052],{},"优化\u002F恢复\u002F并发：20 分钟。严格按模板写，树可以画简洁文本结构。",{"title":29,"searchDepth":43,"depth":43,"links":120054},[120055,120059,120060,120061],{"id":119872,"depth":43,"text":119873,"children":120056},[120057,120058],{"id":119876,"depth":49,"text":119876},{"id":119908,"depth":49,"text":119909},{"id":119938,"depth":43,"text":119939},{"id":119995,"depth":43,"text":119996},{"id":120031,"depth":43,"text":120032},"这套讲义按本课程近年考试和课堂投影整理，不按教材目录机械复述，而是按“考试会怎么问、答题时怎么写、为什么这样写”组织。主要参考资料包括：王珊等《数据库系统概论》第 6 版、2020-2024 历年期末题、2025-2026 新上传 new-pic 投影\u002F板书、pic整理文本 课堂材料、补充习题 70+100 题。",{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F00",{"title":119854,"description":120062},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F00-总览与考试题型地图","1yNhwVPIeXJ5k7jE4TPV9L99XZpmfwnXO0TwOdAEi2k",{"id":120069,"title":120070,"body":120071,"description":120078,"extension":85,"meta":120491,"navigation":52,"path":120492,"seo":120493,"stem":120494,"__hash__":120495},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F01-数据库基础与关系模型.md","01 数据库基础与关系模型",{"type":8,"value":120072,"toc":120478},[120073,120076,120079,120083,120087,120098,120110,120113,120116,120130,120133,120137,120140,120151,120154,120168,120171,120175,120178,120189,120192,120200,120203,120206,120209,120213,120216,120219,120264,120267,120287,120291,120294,120305,120307,120332,120335,120343,120346,120350,120353,120359,120365,120371,120377,120383,120389,120392,120397,120401,120404,120419,120422,120437,120440,120443,120457,120460,120464,120467,120470,120473,120476],[11,120074,120070],{"id":120075},"_01-数据库基础与关系模型",[15,120077,120078],{},"这一章主要对应选择题、填空题、简答题，也会支撑后面的设计题和恢复并发题。",[19,120080,120082],{"id":120081},"一数据库系统基本概念","一、数据库系统基本概念",[253,120084,120086],{"id":120085},"dbdbmsdbs","DB、DBMS、DBS",[105,120088,120089,120092,120095],{},[108,120090,120091],{},"DB：Database，数据库，是长期存储在计算机内、有组织、可共享的数据集合。",[108,120093,120094],{},"DBMS：Database Management System，数据库管理系统，是位于用户和操作系统之间的数据管理软件。",[108,120096,120097],{},"DBS：Database System，数据库系统，通常包括数据库、DBMS、应用程序、数据库管理员 DBA、用户等。",[15,120099,120100,120101,120104,120105,146,120107,347],{},"关系：",[31,120102,120103],{},"DBS 包括 DB 和 DBMS","。考试经常问“三者关系”和“数据库系统核心”，答案分别是 ",[31,120106,120103],{},[31,120108,120109],{},"DBMS",[253,120111,120112],{"id":120112},"数据库系统的特点",[15,120114,120115],{},"数据库系统相对文件系统的主要优点：",[105,120117,120118,120121,120124,120127],{},[108,120119,120120],{},"数据结构化：这是数据库系统与文件系统的本质区别。",[108,120122,120123],{},"数据共享性高，冗余度低，易扩充。",[108,120125,120126],{},"数据独立性高：应用程序尽量不受数据逻辑结构和物理存储变化影响。",[108,120128,120129],{},"数据由 DBMS 统一管理和控制：包括安全性、完整性、并发控制、恢复。",[15,120131,120132],{},"常见陷阱：“数据冗余度大”不是数据库系统特点，而是文件系统常见问题。",[19,120134,120136],{"id":120135},"二数据模型","二、数据模型",[15,120138,120139],{},"数据模型通常由三部分组成：",[2230,120141,120142,120145,120148],{},[108,120143,120144],{},"数据结构：描述数据对象及对象之间的联系，是静态特征。",[108,120146,120147],{},"数据操作：描述允许对数据做什么，如查询、插入、删除、修改，是动态特征。",[108,120149,120150],{},"完整性约束：描述数据必须满足的语义规则。",[15,120152,120153],{},"常见数据模型：",[105,120155,120156,120159,120162,120165],{},[108,120157,120158],{},"层次模型：树结构，一个子结点至多一个父结点。",[108,120160,120161],{},"网状模型：图结构，CODASYL\u002FDBTG 是代表。",[108,120163,120164],{},"关系模型：用二维表\u002F关系表示数据和联系。",[108,120166,120167],{},"E-R 模型：概念模型，用实体、属性、联系描述现实世界。",[15,120169,120170],{},"概念模型独立于具体机器和 DBMS；E-R 图用于概念结构设计，不是逻辑设计或物理设计。",[19,120172,120174],{"id":120173},"三三级模式与两级映像","三、三级模式与两级映像",[15,120176,120177],{},"数据库系统常用三级模式结构：",[105,120179,120180,120183,120186],{},[108,120181,120182],{},"外模式：用户或应用程序看到的局部逻辑结构，也叫子模式、用户模式。",[108,120184,120185],{},"模式：数据库的全局逻辑结构，也叫概念模式。",[108,120187,120188],{},"内模式：数据在物理存储层面的描述，也叫存储模式。",[15,120190,120191],{},"两级映像：",[105,120193,120194,120197],{},[108,120195,120196],{},"外模式\u002F模式映像：保证逻辑独立性。",[108,120198,120199],{},"模式\u002F内模式映像：保证物理独立性。",[15,120201,120202],{},"物理独立性：数据的物理存储位置、存储结构、存取路径改变时，应用程序不需要修改。",[15,120204,120205],{},"逻辑独立性：数据库的全局逻辑结构改变时，外模式和应用程序尽量不需要修改。",[15,120207,120208],{},"考试问“要保证数据独立性，需要修改什么”，答：三级模式之间的两种映像。",[19,120210,120212],{"id":120211},"四关系模型基本术语","四、关系模型基本术语",[15,120214,120215],{},"关系模型的基本数据结构是关系。关系可以理解为二维表，但要满足关系模型规则。",[15,120217,120218],{},"常用术语：",[105,120220,120221,120224,120234,120237,120240,120243,120246,120249,120252,120255,120258,120261],{},[108,120222,120223],{},"域：一组具有相同数据类型的值的集合。",[108,120225,120226,120227,120230,120231,347],{},"笛卡尔积：多个域所有可能组合的集合。若 ",[31,120228,120229],{},"|D1|=2, |D2|=2, |D3|=2","，则基数为 ",[31,120232,120233],{},"2*2*2=8",[108,120235,120236],{},"关系：域的笛卡尔积的子集。",[108,120238,120239],{},"元组：关系中的一行。",[108,120241,120242],{},"属性：关系中的一列；现实世界的“特征”对应数据世界的属性。",[108,120244,120245],{},"码\u002F键：能唯一标识元组的属性组。",[108,120247,120248],{},"超码：能唯一标识元组的属性组，可包含多余属性。",[108,120250,120251],{},"候选码：最小超码，不含多余属性。",[108,120253,120254],{},"主码：从候选码中选定的一个。",[108,120256,120257],{},"主属性：包含在任一候选码中的属性。",[108,120259,120260],{},"非主属性：不包含在任何候选码中的属性。",[108,120262,120263],{},"外码：一个关系中的属性或属性组，引用另一个关系的主码。",[15,120265,120266],{},"关系的性质：",[105,120268,120269,120272,120275,120278,120281,120284],{},[108,120270,120271],{},"列是同质的，每列来自同一域。",[108,120273,120274],{},"不同列可以来自同一域，但属性名必须不同。",[108,120276,120277],{},"列的顺序可交换。",[108,120279,120280],{},"行的顺序可交换。",[108,120282,120283],{},"任意两个元组不能完全相同。",[108,120285,120286],{},"每个分量必须是不可再分的数据项，这是 1NF 的要求。",[19,120288,120290],{"id":120289},"五关系完整性","五、关系完整性",[15,120292,120293],{},"关系模型有三类完整性：",[2230,120295,120296,120299,120302],{},[108,120297,120298],{},"实体完整性：主码属性不能为空，且主码值唯一。",[108,120300,120301],{},"参照完整性：外码值要么为空，要么等于被参照关系中某个主码值。",[108,120303,120304],{},"用户自定义完整性：由具体业务定义，如年龄范围、性别取值、工资下限。",[15,120306,19553],{},[24,120308,120310],{"className":92584,"code":120309,"language":92586,"meta":29,"style":29},"Sno char(9) PRIMARY KEY\nFOREIGN KEY (Sno) REFERENCES Student(Sno)\nCHECK (Age BETWEEN 18 AND 55)\nCHECK (Sex IN ('男', '女'))\n",[31,120311,120312,120317,120322,120327],{"__ignoreMap":29},[34,120313,120314],{"class":36,"line":37},[34,120315,120316],{},"Sno char(9) PRIMARY KEY\n",[34,120318,120319],{"class":36,"line":43},[34,120320,120321],{},"FOREIGN KEY (Sno) REFERENCES Student(Sno)\n",[34,120323,120324],{"class":36,"line":49},[34,120325,120326],{},"CHECK (Age BETWEEN 18 AND 55)\n",[34,120328,120329],{"class":36,"line":56},[34,120330,120331],{},"CHECK (Sex IN ('男', '女'))\n",[15,120333,120334],{},"安全性与完整性区别：",[105,120336,120337,120340],{},[108,120338,120339],{},"安全性：防止非法用户非法访问，关键词是授权、口令、角色、视图保护、审计。",[108,120341,120342],{},"完整性：防止合法用户输入或产生不符合语义的数据，关键词是主键、外键、CHECK、触发器。",[15,120344,120345],{},"“年龄限制 16-25 岁”是完整性，不是安全性。“授权”和“撤权”是安全性。",[19,120347,120349],{"id":120348},"六关系代数基础","六、关系代数基础",[15,120351,120352],{},"五种基本运算常见说法：并、差、笛卡尔积、选择、投影。交、连接、自然连接、除法可由基本运算导出。",[15,120354,120355,120356,120358],{},"选择 ",[31,120357,115372],{},"：按条件选行。",[24,120360,120363],{"className":120361,"code":120362,"language":2638,"meta":29},[2636],"σ_{Age > 20}(Student)\n",[31,120364,120362],{"__ignoreMap":29},[15,120366,120367,120368,120370],{},"投影 ",[31,120369,58442],{},"：选列，并消去重复元组。",[24,120372,120375],{"className":120373,"code":120374,"language":2638,"meta":29},[2636],"π_{Sno, Sname}(Student)\n",[31,120376,120374],{"__ignoreMap":29},[15,120378,120379,120380,120382],{},"连接 ",[31,120381,116918],{},"：按条件把两个关系相关元组拼接。",[24,120384,120387],{"className":120385,"code":120386,"language":2638,"meta":29},[2636],"Student ⋈_{Student.Sno = SC.Sno} SC\n",[31,120388,120386],{"__ignoreMap":29},[15,120390,120391],{},"自然连接：在公共属性上做等值连接，并去掉重复公共属性列。",[15,120393,120394,120395,347],{},"除法：表达“全部”“至少包含全部”这类查询，例如“选修了所有课程的学生”。SQL 中通常用双重 ",[31,120396,91063],{},[19,120398,120400],{"id":120399},"七null-与视图","七、NULL 与视图",[15,120402,120403],{},"NULL 表示未知、不存在或不适用，不等于 0、空格或空字符串。判断 NULL 必须写：",[24,120405,120407],{"className":92584,"code":120406,"language":92586,"meta":29,"style":29},"WHERE Sage IS NULL\nWHERE Sage IS NOT NULL\n",[31,120408,120409,120414],{"__ignoreMap":29},[34,120410,120411],{"class":36,"line":37},[34,120412,120413],{},"WHERE Sage IS NULL\n",[34,120415,120416],{"class":36,"line":43},[34,120417,120418],{},"WHERE Sage IS NOT NULL\n",[15,120420,120421],{},"不能写：",[24,120423,120425],{"className":92584,"code":120424,"language":92586,"meta":29,"style":29},"WHERE Sage = NULL\nWHERE Sage != NULL\n",[31,120426,120427,120432],{"__ignoreMap":29},[34,120428,120429],{"class":36,"line":37},[34,120430,120431],{},"WHERE Sage = NULL\n",[34,120433,120434],{"class":36,"line":43},[34,120435,120436],{},"WHERE Sage != NULL\n",[15,120438,120439],{},"视图是从一个或多个基本表导出的虚表。数据库中只保存视图定义，不保存视图对应的数据。",[15,120441,120442],{},"视图优点：",[105,120444,120445,120448,120451,120454],{},[108,120446,120447],{},"简化用户查询。",[108,120449,120450],{},"使用户以不同角度看同一数据。",[108,120452,120453],{},"提供一定逻辑独立性。",[108,120455,120456],{},"可以作为安全保护手段，只授权访问过滤后的视图。",[15,120458,120459],{},"视图可查询，也可在视图上再定义视图；但不能在视图上定义新的基本表。并非所有视图都可更新，含聚合、分组、去重、多表复杂连接的视图通常不可更新。",[19,120461,120463],{"id":120462},"八索引","八、索引",[15,120465,120466],{},"索引能提高查询效率，因为它为查找条件建立了有序或散列式的辅助访问路径，避免全表扫描。",[15,120468,120469],{},"但索引不是越多越好。插入、删除、更新时索引也要维护，会增加开销。",[15,120471,120472],{},"聚簇索引决定数据物理排列，一个基本表通常只能有一个聚簇索引。非聚簇索引可以有多个。",[15,120474,120475],{},"关系代数优化中应尽早做选择和投影；不是尽早做连接，更不是尽早做笛卡尔积。",[79,120477,81],{},{"title":29,"searchDepth":43,"depth":43,"links":120479},[120480,120484,120485,120486,120487,120488,120489,120490],{"id":120081,"depth":43,"text":120082,"children":120481},[120482,120483],{"id":120085,"depth":49,"text":120086},{"id":120112,"depth":49,"text":120112},{"id":120135,"depth":43,"text":120136},{"id":120173,"depth":43,"text":120174},{"id":120211,"depth":43,"text":120212},{"id":120289,"depth":43,"text":120290},{"id":120348,"depth":43,"text":120349},{"id":120399,"depth":43,"text":120400},{"id":120462,"depth":43,"text":120463},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F01",{"title":120070,"description":120078},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F01-数据库基础与关系模型","WMuhuCHLzncE60TS91ap4aobrhoJj7JZ_6_bOBE6lvY",{"id":120497,"title":120498,"body":120499,"description":120506,"extension":85,"meta":121720,"navigation":52,"path":121721,"seo":121722,"stem":121723,"__hash__":121724},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F02-SQL与关系代数.md","02 SQL 与关系代数",{"type":8,"value":120500,"toc":121698},[120501,120504,120507,120511,120514,120548,120559,120563,120566,120618,120622,120625,120659,120670,120673,120676,120699,120702,120708,120719,120770,120773,120793,120798,120802,120805,120820,120823,120847,120850,120865,120868,120877,120880,120899,120905,120930,120934,120937,120956,120959,120978,120981,121000,121004,121013,121016,121060,121065,121101,121105,121142,121146,121156,121159,121217,121220,121224,121227,121251,121254,121269,121272,121291,121297,121301,121305,121307,121312,121315,121365,121368,121381,121384,121397,121400,121425,121428,121442,121445,121474,121478,121483,121532,121535,121560,121563,121589,121592,121605,121610,121614,121616,121622,121624,121630,121633,121639,121642,121696],[11,120502,120498],{"id":120503},"_02-sql-与关系代数",[15,120505,120506],{},"SQL 大题通常不难，但容易因为约束、外键动作、聚合分组、嵌套查询、视图写法丢分。本章按考试写法整理。",[19,120508,120510],{"id":120509},"一sql-语言分类","一、SQL 语言分类",[15,120512,120513],{},"SQL 是综合性语言，常见分类：",[105,120515,120516,120528,120541],{},[108,120517,120518,120519,146,120522,146,120525,347],{},"DDL：数据定义语言，",[31,120520,120521],{},"CREATE",[31,120523,120524],{},"ALTER",[31,120526,120527],{},"DROP",[108,120529,120530,120531,146,120533,146,120536,146,120539,347],{},"DML：数据操纵语言，",[31,120532,92151],{},[31,120534,120535],{},"INSERT",[31,120537,120538],{},"DELETE",[31,120540,92155],{},[108,120542,120543,120544,146,120546,347],{},"DCL：数据控制语言，",[31,120545,92399],{},[31,120547,92403],{},[15,120549,120550,120551,120553,120554,146,120556,120558],{},"有些教材也把查询 ",[31,120552,92151],{}," 称为 DQL，把 ",[31,120555,93970],{},[31,120557,91516],{}," 称为事务控制 TCL。填空题若按教材“三类”问，通常答 DDL、DML、DCL。",[19,120560,120562],{"id":120561},"二建表与完整性约束","二、建表与完整性约束",[253,120564,120565],{"id":120565},"基本模板",[24,120567,120569],{"className":92584,"code":120568,"language":92586,"meta":29,"style":29},"CREATE TABLE 表名 (\n 列名 数据类型 [列级约束],\n 列名 数据类型 [列级约束],\n CONSTRAINT 约束名 PRIMARY KEY (列1, 列2),\n CONSTRAINT 约束名 FOREIGN KEY (外码列)\n REFERENCES 被参照表(主码列)\n ON DELETE SET NULL\n ON UPDATE CASCADE,\n CONSTRAINT 约束名 CHECK (条件)\n);\n",[31,120570,120571,120576,120581,120585,120590,120595,120600,120604,120609,120614],{"__ignoreMap":29},[34,120572,120573],{"class":36,"line":37},[34,120574,120575],{},"CREATE TABLE 表名 (\n",[34,120577,120578],{"class":36,"line":43},[34,120579,120580],{}," 列名 数据类型 [列级约束],\n",[34,120582,120583],{"class":36,"line":49},[34,120584,120580],{},[34,120586,120587],{"class":36,"line":56},[34,120588,120589],{}," CONSTRAINT 约束名 PRIMARY KEY (列1, 列2),\n",[34,120591,120592],{"class":36,"line":62},[34,120593,120594],{}," CONSTRAINT 约束名 FOREIGN KEY (外码列)\n",[34,120596,120597],{"class":36,"line":68},[34,120598,120599],{}," REFERENCES 被参照表(主码列)\n",[34,120601,120602],{"class":36,"line":74},[34,120603,101628],{},[34,120605,120606],{"class":36,"line":1971},[34,120607,120608],{}," ON UPDATE CASCADE,\n",[34,120610,120611],{"class":36,"line":1976},[34,120612,120613],{}," CONSTRAINT 约束名 CHECK (条件)\n",[34,120615,120616],{"class":36,"line":1985},[34,120617,462],{},[253,120619,120621],{"id":120620},"主键唯一非空","主键、唯一、非空",[15,120623,120624],{},"一个表只能有一个主键约束，但主键可以由多列组成。",[24,120626,120628],{"className":92584,"code":120627,"language":92586,"meta":29,"style":29},"CREATE TABLE SC (\n Sno char(9) NOT NULL,\n Cno char(4) NOT NULL,\n Grade int,\n CONSTRAINT PK_SC PRIMARY KEY (Sno, Cno)\n);\n",[31,120629,120630,120635,120640,120645,120650,120655],{"__ignoreMap":29},[34,120631,120632],{"class":36,"line":37},[34,120633,120634],{},"CREATE TABLE SC (\n",[34,120636,120637],{"class":36,"line":43},[34,120638,120639],{}," Sno char(9) NOT NULL,\n",[34,120641,120642],{"class":36,"line":49},[34,120643,120644],{}," Cno char(4) NOT NULL,\n",[34,120646,120647],{"class":36,"line":56},[34,120648,120649],{}," Grade int,\n",[34,120651,120652],{"class":36,"line":62},[34,120653,120654],{}," CONSTRAINT PK_SC PRIMARY KEY (Sno, Cno)\n",[34,120656,120657],{"class":36,"line":68},[34,120658,462],{},[15,120660,120661,120663,120664,120667,120668,347],{},[31,120662,94256],{}," 表示候选码性质；",[31,120665,120666],{},"NOT NULL UNIQUE"," 可以唯一标识元组，但一个表只能选一个作为 ",[31,120669,99262],{},[253,120671,120672],{"id":120672},"外键动作",[15,120674,120675],{},"外键常见动作：",[105,120677,120678,120683,120689,120694],{},[108,120679,120680,120682],{},[31,120681,100055],{},"：删除父表记录时，级联删除子表记录。",[108,120684,120685,120688],{},[31,120686,120687],{},"ON DELETE SET NULL","：删除父表记录时，把子表外键置空。",[108,120690,120691,120693],{},[31,120692,100051],{}," 或不写：若有子表引用，则拒绝删除。",[108,120695,120696,120698],{},[31,120697,101468],{},"：父表主码更新时，子表外键跟着更新。",[15,120700,120701],{},"考题里三种说法要分清：",[24,120703,120706],{"className":120704,"code":120705,"language":2638,"meta":29},[2636],"级联删除：ON DELETE CASCADE\n按置空原则删除：ON DELETE SET NULL\n不允许级联删除：不要写 ON DELETE CASCADE，通常保持 NO ACTION\u002FRESTRICT\n",[31,120707,120705],{"__ignoreMap":29},[15,120709,120710,120711,120714,120715,120718],{},"如果父表 ",[31,120712,120713],{},"Dept(dptno)"," 被子表 ",[31,120716,120717],{},"Pe(dptno)"," 引用：",[24,120720,120722],{"className":92584,"code":120721,"language":92586,"meta":29,"style":29},"-- 删除部门时，自动删除该部门员工记录，叫级联删除。\nFOREIGN KEY (dptno) REFERENCES Dept(dptno)\n ON DELETE CASCADE\n\n-- 删除部门时，员工仍保留，但 dptno 变 NULL，叫置空。\nFOREIGN KEY (dptno) REFERENCES Dept(dptno)\n ON DELETE SET NULL\n\n-- 删除部门时，如果还有员工引用该部门，则拒绝删除。\nFOREIGN KEY (dptno) REFERENCES Dept(dptno)\n",[31,120723,120724,120729,120734,120739,120743,120748,120752,120757,120761,120766],{"__ignoreMap":29},[34,120725,120726],{"class":36,"line":37},[34,120727,120728],{},"-- 删除部门时，自动删除该部门员工记录，叫级联删除。\n",[34,120730,120731],{"class":36,"line":43},[34,120732,120733],{},"FOREIGN KEY (dptno) REFERENCES Dept(dptno)\n",[34,120735,120736],{"class":36,"line":49},[34,120737,120738],{}," ON DELETE CASCADE\n",[34,120740,120741],{"class":36,"line":56},[34,120742,53],{"emptyLinePlaceholder":52},[34,120744,120745],{"class":36,"line":62},[34,120746,120747],{},"-- 删除部门时，员工仍保留，但 dptno 变 NULL，叫置空。\n",[34,120749,120750],{"class":36,"line":68},[34,120751,120733],{},[34,120753,120754],{"class":36,"line":74},[34,120755,120756],{}," ON DELETE SET NULL\n",[34,120758,120759],{"class":36,"line":1971},[34,120760,53],{"emptyLinePlaceholder":52},[34,120762,120763],{"class":36,"line":1976},[34,120764,120765],{},"-- 删除部门时，如果还有员工引用该部门，则拒绝删除。\n",[34,120767,120768],{"class":36,"line":1985},[34,120769,120733],{},[15,120771,120772],{},"new-pic 中公司数据库说明“cascading deletion is not allowed, but cascading update is allowed”，建表时应写：",[24,120774,120776],{"className":92584,"code":120775,"language":92586,"meta":29,"style":29},"CONSTRAINT FK_SD_PNO FOREIGN KEY (PNO)\n REFERENCES Product(PNO)\n ON UPDATE CASCADE\n",[31,120777,120778,120783,120788],{"__ignoreMap":29},[34,120779,120780],{"class":36,"line":37},[34,120781,120782],{},"CONSTRAINT FK_SD_PNO FOREIGN KEY (PNO)\n",[34,120784,120785],{"class":36,"line":43},[34,120786,120787],{}," REFERENCES Product(PNO)\n",[34,120789,120790],{"class":36,"line":49},[34,120791,120792],{}," ON UPDATE CASCADE\n",[15,120794,120795,120796,347],{},"不要写 ",[31,120797,100055],{},[19,120799,120801],{"id":120800},"三基础查询模板","三、基础查询模板",[15,120803,120804],{},"选择列：",[24,120806,120808],{"className":92584,"code":120807,"language":92586,"meta":29,"style":29},"SELECT Sno, Sname\nFROM Student;\n",[31,120809,120810,120815],{"__ignoreMap":29},[34,120811,120812],{"class":36,"line":37},[34,120813,120814],{},"SELECT Sno, Sname\n",[34,120816,120817],{"class":36,"line":43},[34,120818,120819],{},"FROM Student;\n",[15,120821,120822],{},"条件查询：",[24,120824,120826],{"className":92584,"code":120825,"language":92586,"meta":29,"style":29},"SELECT *\nFROM Student\nWHERE Sgender = N'男'\n AND Sdept = N'计算机科学与技术';\n",[31,120827,120828,120832,120837,120842],{"__ignoreMap":29},[34,120829,120830],{"class":36,"line":37},[34,120831,92811],{},[34,120833,120834],{"class":36,"line":43},[34,120835,120836],{},"FROM Student\n",[34,120838,120839],{"class":36,"line":49},[34,120840,120841],{},"WHERE Sgender = N'男'\n",[34,120843,120844],{"class":36,"line":56},[34,120845,120846],{}," AND Sdept = N'计算机科学与技术';\n",[15,120848,120849],{},"模糊匹配：",[24,120851,120853],{"className":92584,"code":120852,"language":92586,"meta":29,"style":29},"WHERE Email LIKE '%@stumail.nwu.edu.cn'\nWHERE Sname LIKE N'_武%' -- 第二个字为“武”\n",[31,120854,120855,120860],{"__ignoreMap":29},[34,120856,120857],{"class":36,"line":37},[34,120858,120859],{},"WHERE Email LIKE '%@stumail.nwu.edu.cn'\n",[34,120861,120862],{"class":36,"line":43},[34,120863,120864],{},"WHERE Sname LIKE N'_武%' -- 第二个字为“武”\n",[15,120866,120867],{},"排序：",[24,120869,120871],{"className":92584,"code":120870,"language":92586,"meta":29,"style":29},"ORDER BY Qty DESC, Pno ASC\n",[31,120872,120873],{"__ignoreMap":29},[34,120874,120875],{"class":36,"line":37},[34,120876,120870],{},[15,120878,120879],{},"聚合：",[24,120881,120883],{"className":92584,"code":120882,"language":92586,"meta":29,"style":29},"SELECT Sdept, COUNT(*) AS StudentCount\nFROM Student\nGROUP BY Sdept;\n",[31,120884,120885,120890,120894],{"__ignoreMap":29},[34,120886,120887],{"class":36,"line":37},[34,120888,120889],{},"SELECT Sdept, COUNT(*) AS StudentCount\n",[34,120891,120892],{"class":36,"line":43},[34,120893,120836],{},[34,120895,120896],{"class":36,"line":49},[34,120897,120898],{},"GROUP BY Sdept;\n",[15,120900,120901,120902,2664],{},"分组后筛选用 ",[31,120903,120904],{},"HAVING",[24,120906,120908],{"className":92584,"code":120907,"language":92586,"meta":29,"style":29},"SELECT Sno, AVG(Grade) AS AvgGrade\nFROM SC\nGROUP BY Sno\nHAVING AVG(Grade) >= 90;\n",[31,120909,120910,120915,120920,120925],{"__ignoreMap":29},[34,120911,120912],{"class":36,"line":37},[34,120913,120914],{},"SELECT Sno, AVG(Grade) AS AvgGrade\n",[34,120916,120917],{"class":36,"line":43},[34,120918,120919],{},"FROM SC\n",[34,120921,120922],{"class":36,"line":49},[34,120923,120924],{},"GROUP BY Sno\n",[34,120926,120927],{"class":36,"line":56},[34,120928,120929],{},"HAVING AVG(Grade) >= 90;\n",[19,120931,120933],{"id":120932},"四连接查询","四、连接查询",[15,120935,120936],{},"内连接：",[24,120938,120940],{"className":92584,"code":120939,"language":92586,"meta":29,"style":29},"SELECT s.Sno, s.Sname, sc.Cno, sc.Grade\nFROM Student AS s\nJOIN SC AS sc ON s.Sno = sc.Sno;\n",[31,120941,120942,120947,120951],{"__ignoreMap":29},[34,120943,120944],{"class":36,"line":37},[34,120945,120946],{},"SELECT s.Sno, s.Sname, sc.Cno, sc.Grade\n",[34,120948,120949],{"class":36,"line":43},[34,120950,104138],{},[34,120952,120953],{"class":36,"line":49},[34,120954,120955],{},"JOIN SC AS sc ON s.Sno = sc.Sno;\n",[15,120957,120958],{},"左外连接：保留左表全部行。",[24,120960,120962],{"className":92584,"code":120961,"language":92586,"meta":29,"style":29},"SELECT s.Sno, s.Sname, d.RoomNo\nFROM Student AS s\nLEFT JOIN DormAssign AS d ON s.Sno = d.Sno;\n",[31,120963,120964,120969,120973],{"__ignoreMap":29},[34,120965,120966],{"class":36,"line":37},[34,120967,120968],{},"SELECT s.Sno, s.Sname, d.RoomNo\n",[34,120970,120971],{"class":36,"line":43},[34,120972,104138],{},[34,120974,120975],{"class":36,"line":49},[34,120976,120977],{},"LEFT JOIN DormAssign AS d ON s.Sno = d.Sno;\n",[15,120979,120980],{},"全外连接：左右两边未匹配行都保留。题目说“包括没有住宿的学生和空闲床位”，应使用全外连接。",[24,120982,120984],{"className":92584,"code":120983,"language":92586,"meta":29,"style":29},"SELECT s.Sno, s.Sname, b.Building, b.RoomNo, b.BedNo\nFROM Student AS s\nFULL OUTER JOIN Bed AS b ON s.Sno = b.Sno;\n",[31,120985,120986,120991,120995],{"__ignoreMap":29},[34,120987,120988],{"class":36,"line":37},[34,120989,120990],{},"SELECT s.Sno, s.Sname, b.Building, b.RoomNo, b.BedNo\n",[34,120992,120993],{"class":36,"line":43},[34,120994,104138],{},[34,120996,120997],{"class":36,"line":49},[34,120998,120999],{},"FULL OUTER JOIN Bed AS b ON s.Sno = b.Sno;\n",[19,121001,121003],{"id":121002},"五嵌套查询与全部问题","五、嵌套查询与“全部”问题",[253,121005,121007,43298,121010],{"id":121006},"in-与-exists",[31,121008,121009],{},"IN",[31,121011,121012],{},"EXISTS",[15,121014,121015],{},"查询参加“法国拉力赛”的车手编号和姓名：",[24,121017,121019],{"className":92584,"code":121018,"language":92586,"meta":29,"style":29},"SELECT Dno, Dname\nFROM D\nWHERE Dno IN (\n SELECT DC.Dno\n FROM DC\n JOIN C ON DC.Cno = C.Cno\n WHERE C.Cname = N'法国拉力赛'\n);\n",[31,121020,121021,121026,121031,121036,121041,121046,121051,121056],{"__ignoreMap":29},[34,121022,121023],{"class":36,"line":37},[34,121024,121025],{},"SELECT Dno, Dname\n",[34,121027,121028],{"class":36,"line":43},[34,121029,121030],{},"FROM D\n",[34,121032,121033],{"class":36,"line":49},[34,121034,121035],{},"WHERE Dno IN (\n",[34,121037,121038],{"class":36,"line":56},[34,121039,121040],{}," SELECT DC.Dno\n",[34,121042,121043],{"class":36,"line":62},[34,121044,121045],{}," FROM DC\n",[34,121047,121048],{"class":36,"line":68},[34,121049,121050],{}," JOIN C ON DC.Cno = C.Cno\n",[34,121052,121053],{"class":36,"line":74},[34,121054,121055],{}," WHERE C.Cname = N'法国拉力赛'\n",[34,121057,121058],{"class":36,"line":1971},[34,121059,462],{},[15,121061,121062,121064],{},[31,121063,121012],{}," 判断子查询是否非空：",[24,121066,121068],{"className":92584,"code":121067,"language":92586,"meta":29,"style":29},"SELECT *\nFROM S AS s\nWHERE EXISTS (\n SELECT 1\n FROM SPJ AS spj\n WHERE spj.Sno = s.Sno\n);\n",[31,121069,121070,121074,121078,121083,121087,121092,121097],{"__ignoreMap":29},[34,121071,121072],{"class":36,"line":37},[34,121073,92811],{},[34,121075,121076],{"class":36,"line":43},[34,121077,101794],{},[34,121079,121080],{"class":36,"line":49},[34,121081,121082],{},"WHERE EXISTS (\n",[34,121084,121085],{"class":36,"line":56},[34,121086,92970],{},[34,121088,121089],{"class":36,"line":62},[34,121090,121091],{}," FROM SPJ AS spj\n",[34,121093,121094],{"class":36,"line":68},[34,121095,121096],{}," WHERE spj.Sno = s.Sno\n",[34,121098,121099],{"class":36,"line":74},[34,121100,462],{},[253,121102,121104],{"id":121103},"没有供应-p3-的供应商","没有供应 P3 的供应商",[24,121106,121108],{"className":92584,"code":121107,"language":92586,"meta":29,"style":29},"SELECT *\nFROM S AS s\nWHERE NOT EXISTS (\n SELECT 1\n FROM SP AS sp\n WHERE sp.Sno = s.Sno\n AND sp.Pno = 'P3'\n);\n",[31,121109,121110,121114,121118,121122,121126,121130,121134,121138],{"__ignoreMap":29},[34,121111,121112],{"class":36,"line":37},[34,121113,92811],{},[34,121115,121116],{"class":36,"line":43},[34,121117,101794],{},[34,121119,121120],{"class":36,"line":49},[34,121121,92965],{},[34,121123,121124],{"class":36,"line":56},[34,121125,92970],{},[34,121127,121128],{"class":36,"line":62},[34,121129,101839],{},[34,121131,121132],{"class":36,"line":68},[34,121133,101844],{},[34,121135,121136],{"class":36,"line":74},[34,121137,101849],{},[34,121139,121140],{"class":36,"line":1971},[34,121141,462],{},[253,121143,121145],{"id":121144},"选修全部课程的双重-not-exists","“选修全部课程”的双重 NOT EXISTS",[15,121147,121148,121149,616,121151,616,121154,347],{},"模式：",[31,121150,94573],{},[31,121152,121153],{},"Course(Cno)",[31,121155,94582],{},[15,121157,121158],{},"语义：“不存在一门课程，使得该学生没有选修它”。",[24,121160,121162],{"className":92584,"code":121161,"language":92586,"meta":29,"style":29},"SELECT s.Sno, s.Sname\nFROM Student AS s\nWHERE NOT EXISTS (\n SELECT 1\n FROM Course AS c\n WHERE NOT EXISTS (\n SELECT 1\n FROM SC AS sc\n WHERE sc.Sno = s.Sno\n AND sc.Cno = c.Cno\n )\n);\n",[31,121163,121164,121169,121173,121177,121181,121186,121190,121194,121199,121204,121209,121213],{"__ignoreMap":29},[34,121165,121166],{"class":36,"line":37},[34,121167,121168],{},"SELECT s.Sno, s.Sname\n",[34,121170,121171],{"class":36,"line":43},[34,121172,104138],{},[34,121174,121175],{"class":36,"line":49},[34,121176,92965],{},[34,121178,121179],{"class":36,"line":56},[34,121180,92970],{},[34,121182,121183],{"class":36,"line":62},[34,121184,121185],{}," FROM Course AS c\n",[34,121187,121188],{"class":36,"line":68},[34,121189,93053],{},[34,121191,121192],{"class":36,"line":74},[34,121193,93058],{},[34,121195,121196],{"class":36,"line":1971},[34,121197,121198],{}," FROM SC AS sc\n",[34,121200,121201],{"class":36,"line":1976},[34,121202,121203],{}," WHERE sc.Sno = s.Sno\n",[34,121205,121206],{"class":36,"line":1985},[34,121207,121208],{}," AND sc.Cno = c.Cno\n",[34,121210,121211],{"class":36,"line":1990},[34,121212,93078],{},[34,121214,121215],{"class":36,"line":1999},[34,121216,462],{},[15,121218,121219],{},"这是除法查询的 SQL 典型写法。",[19,121221,121223],{"id":121222},"六视图索引授权","六、视图、索引、授权",[15,121225,121226],{},"创建视图：",[24,121228,121230],{"className":92584,"code":121229,"language":92586,"meta":29,"style":29},"CREATE VIEW V_AvgGrade(Sno, AvgGrade) AS\nSELECT Sno, AVG(Grade)\nFROM SC\nGROUP BY Sno;\n",[31,121231,121232,121237,121242,121246],{"__ignoreMap":29},[34,121233,121234],{"class":36,"line":37},[34,121235,121236],{},"CREATE VIEW V_AvgGrade(Sno, AvgGrade) AS\n",[34,121238,121239],{"class":36,"line":43},[34,121240,121241],{},"SELECT Sno, AVG(Grade)\n",[34,121243,121244],{"class":36,"line":49},[34,121245,120919],{},[34,121247,121248],{"class":36,"line":56},[34,121249,121250],{},"GROUP BY Sno;\n",[15,121252,121253],{},"创建索引：",[24,121255,121257],{"className":92584,"code":121256,"language":92586,"meta":29,"style":29},"CREATE INDEX idx_Birthday ON Student(Birthday);\nCREATE INDEX idx_Pname ON Product(PName);\n",[31,121258,121259,121264],{"__ignoreMap":29},[34,121260,121261],{"class":36,"line":37},[34,121262,121263],{},"CREATE INDEX idx_Birthday ON Student(Birthday);\n",[34,121265,121266],{"class":36,"line":43},[34,121267,121268],{},"CREATE INDEX idx_Pname ON Product(PName);\n",[15,121270,121271],{},"授权：",[24,121273,121275],{"className":92584,"code":121274,"language":92586,"meta":29,"style":29},"GRANT SELECT ON Product TO Tom;\nGRANT SELECT ON Product TO Tom WITH GRANT OPTION;\nREVOKE SELECT ON Product FROM Tom;\n",[31,121276,121277,121282,121286],{"__ignoreMap":29},[34,121278,121279],{"class":36,"line":37},[34,121280,121281],{},"GRANT SELECT ON Product TO Tom;\n",[34,121283,121284],{"class":36,"line":43},[34,121285,101447],{},[34,121287,121288],{"class":36,"line":49},[34,121289,121290],{},"REVOKE SELECT ON Product FROM Tom;\n",[15,121292,121293,121296],{},[31,121294,121295],{},"WITH GRANT OPTION"," 表示被授权者可以把权限再授予他人。",[19,121298,121300],{"id":121299},"七常考-sql-题型示例","七、常考 SQL 题型示例",[253,121302,121304],{"id":121303},"saledetailproductstock","SaleDetail\u002FProduct\u002FStock",[15,121306,120100],{},[24,121308,121310],{"className":121309,"code":101177,"language":2638,"meta":29},[2636],[31,121311,101177],{"__ignoreMap":29},[15,121313,121314],{},"建销售明细表：",[24,121316,121318],{"className":92584,"code":121317,"language":92586,"meta":29,"style":29},"CREATE TABLE SaleDetail (\n SaleTime date NOT NULL,\n SalemanNO char(9) NOT NULL,\n PNO char(9) NOT NULL,\n SaleNum int,\n CONSTRAINT PK_SaleDetail PRIMARY KEY (SaleTime, SalemanNO, PNO),\n CONSTRAINT FK_SaleDetail_Product FOREIGN KEY (PNO)\n REFERENCES Product(PNO)\n ON UPDATE CASCADE\n);\n",[31,121319,121320,121324,121329,121334,121339,121343,121348,121353,121357,121361],{"__ignoreMap":29},[34,121321,121322],{"class":36,"line":37},[34,121323,101267],{},[34,121325,121326],{"class":36,"line":43},[34,121327,121328],{}," SaleTime date NOT NULL,\n",[34,121330,121331],{"class":36,"line":49},[34,121332,121333],{}," SalemanNO char(9) NOT NULL,\n",[34,121335,121336],{"class":36,"line":56},[34,121337,121338],{}," PNO char(9) NOT NULL,\n",[34,121340,121341],{"class":36,"line":62},[34,121342,101287],{},[34,121344,121345],{"class":36,"line":68},[34,121346,121347],{}," CONSTRAINT PK_SaleDetail PRIMARY KEY (SaleTime, SalemanNO, PNO),\n",[34,121349,121350],{"class":36,"line":74},[34,121351,121352],{}," CONSTRAINT FK_SaleDetail_Product FOREIGN KEY (PNO)\n",[34,121354,121355],{"class":36,"line":1971},[34,121356,101302],{},[34,121358,121359],{"class":36,"line":1976},[34,121360,99981],{},[34,121362,121363],{"class":36,"line":1985},[34,121364,462],{},[15,121366,121367],{},"删除 2022-01-01 之后的销售明细：",[24,121369,121371],{"className":92584,"code":121370,"language":92586,"meta":29,"style":29},"DELETE FROM SaleDetail\nWHERE SaleTime > '2022-01-01';\n",[31,121372,121373,121377],{"__ignoreMap":29},[34,121374,121375],{"class":36,"line":37},[34,121376,101319],{},[34,121378,121379],{"class":36,"line":43},[34,121380,101324],{},[15,121382,121383],{},"插入产品：",[24,121385,121387],{"className":92584,"code":121386,"language":92586,"meta":29,"style":29},"INSERT INTO Product(PNO, PName, Price)\nVALUES ('12345', 'Redmi Note12', 1699);\n",[31,121388,121389,121393],{"__ignoreMap":29},[34,121390,121391],{"class":36,"line":37},[34,121392,101333],{},[34,121394,121395],{"class":36,"line":43},[34,121396,101338],{},[15,121398,121399],{},"统计每个产品 2023 年销售总量并降序：",[24,121401,121403],{"className":92584,"code":121402,"language":92586,"meta":29,"style":29},"SELECT PNO, SUM(SaleNum) AS TotalSales\nFROM SaleDetail\nWHERE YEAR(SaleTime) = 2023\nGROUP BY PNO\nORDER BY SUM(SaleNum) DESC;\n",[31,121404,121405,121409,121413,121417,121421],{"__ignoreMap":29},[34,121406,121407],{"class":36,"line":37},[34,121408,101347],{},[34,121410,121411],{"class":36,"line":43},[34,121412,101352],{},[34,121414,121415],{"class":36,"line":49},[34,121416,101357],{},[34,121418,121419],{"class":36,"line":56},[34,121420,101362],{},[34,121422,121423],{"class":36,"line":62},[34,121424,101367],{},[15,121426,121427],{},"每个产品涨价 10%：",[24,121429,121431],{"className":92584,"code":121430,"language":92586,"meta":29,"style":29},"UPDATE Product\nSET Price = Price * 1.10;\n",[31,121432,121433,121437],{"__ignoreMap":29},[34,121434,121435],{"class":36,"line":37},[34,121436,101376],{},[34,121438,121439],{"class":36,"line":43},[34,121440,121441],{},"SET Price = Price * 1.10;\n",[15,121443,121444],{},"Warehouse1 销售信息视图：",[24,121446,121448],{"className":92584,"code":121447,"language":92586,"meta":29,"style":29},"CREATE VIEW v_warehouse1(PNO, PName, SaleTime, SaleNum) AS\nSELECT sd.PNO, p.PName, sd.SaleTime, sd.SaleNum\nFROM SaleDetail AS sd\nJOIN Product AS p ON sd.PNO = p.PNO\nJOIN Stock AS s ON sd.PNO = s.PNO\nWHERE s.StockPlace = 'Warehouse1';\n",[31,121449,121450,121454,121458,121462,121466,121470],{"__ignoreMap":29},[34,121451,121452],{"class":36,"line":37},[34,121453,101390],{},[34,121455,121456],{"class":36,"line":43},[34,121457,101395],{},[34,121459,121460],{"class":36,"line":49},[34,121461,101400],{},[34,121463,121464],{"class":36,"line":56},[34,121465,101405],{},[34,121467,121468],{"class":36,"line":62},[34,121469,101410],{},[34,121471,121472],{"class":36,"line":68},[34,121473,101415],{},[253,121475,121477],{"id":121476},"deptpe-部门职工题","Dept\u002FPe 部门职工题",[15,121479,23611,121480,121482],{},[31,121481,101489],{}," 表，工资 1500-3500，有主码外码：",[24,121484,121486],{"className":92584,"code":121485,"language":92586,"meta":29,"style":29},"CREATE TABLE Pe (\n pno char(10) PRIMARY KEY,\n pname varchar(20) NOT NULL,\n sex char(2),\n job varchar(20),\n wage int CHECK (wage BETWEEN 1500 AND 3500),\n dptno char(10) NULL,\n CONSTRAINT FK_Pe_Dept FOREIGN KEY (dptno)\n REFERENCES Dept(dptno)\n ON DELETE SET NULL\n);\n",[31,121487,121488,121492,121496,121500,121504,121508,121512,121516,121520,121524,121528],{"__ignoreMap":29},[34,121489,121490],{"class":36,"line":37},[34,121491,101583],{},[34,121493,121494],{"class":36,"line":43},[34,121495,101588],{},[34,121497,121498],{"class":36,"line":49},[34,121499,101593],{},[34,121501,121502],{"class":36,"line":56},[34,121503,101598],{},[34,121505,121506],{"class":36,"line":62},[34,121507,101603],{},[34,121509,121510],{"class":36,"line":68},[34,121511,101608],{},[34,121513,121514],{"class":36,"line":74},[34,121515,101613],{},[34,121517,121518],{"class":36,"line":1971},[34,121519,101618],{},[34,121521,121522],{"class":36,"line":1976},[34,121523,101623],{},[34,121525,121526],{"class":36,"line":1985},[34,121527,101628],{},[34,121529,121530],{"class":36,"line":1990},[34,121531,462],{},[15,121533,121534],{},"负责人、部门工资总和视图：",[24,121536,121538],{"className":92584,"code":121537,"language":92586,"meta":29,"style":29},"CREATE VIEW V_DEPE AS\nSELECT d.mana, d.dptno, d.dname, SUM(p.wage) AS TotalWage\nFROM Dept AS d\nJOIN Pe AS p ON d.dptno = p.dptno\nGROUP BY d.mana, d.dptno, d.dname;\n",[31,121539,121540,121544,121548,121552,121556],{"__ignoreMap":29},[34,121541,121542],{"class":36,"line":37},[34,121543,101641],{},[34,121545,121546],{"class":36,"line":43},[34,121547,101646],{},[34,121549,121550],{"class":36,"line":49},[34,121551,101651],{},[34,121553,121554],{"class":36,"line":56},[34,121555,101656],{},[34,121557,121558],{"class":36,"line":62},[34,121559,101661],{},[15,121561,121562],{},"给设计室员工涨工资：",[24,121564,121566],{"className":92584,"code":121565,"language":92586,"meta":29,"style":29},"UPDATE p\nSET p.wage = p.wage * 1.10\nFROM Pe AS p\nJOIN Dept AS d ON p.dptno = d.dptno\nWHERE d.dname = N'设计室';\n",[31,121567,121568,121572,121577,121581,121585],{"__ignoreMap":29},[34,121569,121570],{"class":36,"line":37},[34,121571,101670],{},[34,121573,121574],{"class":36,"line":43},[34,121575,121576],{},"SET p.wage = p.wage * 1.10\n",[34,121578,121579],{"class":36,"line":49},[34,121580,101680],{},[34,121582,121583],{"class":36,"line":56},[34,121584,101685],{},[34,121586,121587],{"class":36,"line":62},[34,121588,101690],{},[15,121590,121591],{},"按置空原则删除部门：",[24,121593,121595],{"className":92584,"code":121594,"language":92586,"meta":29,"style":29},"DELETE FROM Dept\nWHERE dptno = 'Z01';\n",[31,121596,121597,121601],{"__ignoreMap":29},[34,121598,121599],{"class":36,"line":37},[34,121600,101699],{},[34,121602,121603],{"class":36,"line":43},[34,121604,101704],{},[15,121606,121607,121608,347],{},"前提是外键定义了 ",[31,121609,120687],{},[19,121611,121613],{"id":121612},"八关系代数与-sql-对照","八、关系代数与 SQL 对照",[15,121615,94647],{},[24,121617,121620],{"className":121618,"code":121619,"language":2638,"meta":29},[2636],"σ_{条件}(R)\nSELECT * FROM R WHERE 条件;\n",[31,121621,121619],{"__ignoreMap":29},[15,121623,94653],{},[24,121625,121628],{"className":121626,"code":121627,"language":2638,"meta":29},[2636],"π_{A,B}(R)\nSELECT DISTINCT A, B FROM R;\n",[31,121629,121627],{"__ignoreMap":29},[15,121631,121632],{},"连接：",[24,121634,121637],{"className":121635,"code":121636,"language":2638,"meta":29},[2636],"R ⋈_{R.A=S.A} S\nSELECT * FROM R JOIN S ON R.A = S.A;\n",[31,121638,121636],{"__ignoreMap":29},[15,121640,121641],{},"并、交、差：",[24,121643,121645],{"className":92584,"code":121644,"language":92586,"meta":29,"style":29},"SELECT Sno FROM A\nUNION\nSELECT Sno FROM B;\n\nSELECT Sno FROM A\nINTERSECT\nSELECT Sno FROM B;\n\nSELECT Sno FROM A\nEXCEPT\nSELECT Sno FROM B;\n",[31,121646,121647,121652,121657,121662,121666,121670,121675,121679,121683,121687,121692],{"__ignoreMap":29},[34,121648,121649],{"class":36,"line":37},[34,121650,121651],{},"SELECT Sno FROM A\n",[34,121653,121654],{"class":36,"line":43},[34,121655,121656],{},"UNION\n",[34,121658,121659],{"class":36,"line":49},[34,121660,121661],{},"SELECT Sno FROM B;\n",[34,121663,121664],{"class":36,"line":56},[34,121665,53],{"emptyLinePlaceholder":52},[34,121667,121668],{"class":36,"line":62},[34,121669,121651],{},[34,121671,121672],{"class":36,"line":68},[34,121673,121674],{},"INTERSECT\n",[34,121676,121677],{"class":36,"line":74},[34,121678,121661],{},[34,121680,121681],{"class":36,"line":1971},[34,121682,53],{"emptyLinePlaceholder":52},[34,121684,121685],{"class":36,"line":1976},[34,121686,121651],{},[34,121688,121689],{"class":36,"line":1985},[34,121690,121691],{},"EXCEPT\n",[34,121693,121694],{"class":36,"line":1990},[34,121695,121661],{},[79,121697,81],{},{"title":29,"searchDepth":43,"depth":43,"links":121699},[121700,121701,121706,121707,121708,121714,121715,121719],{"id":120509,"depth":43,"text":120510},{"id":120561,"depth":43,"text":120562,"children":121702},[121703,121704,121705],{"id":120565,"depth":49,"text":120565},{"id":120620,"depth":49,"text":120621},{"id":120672,"depth":49,"text":120672},{"id":120800,"depth":43,"text":120801},{"id":120932,"depth":43,"text":120933},{"id":121002,"depth":43,"text":121003,"children":121709},[121710,121712,121713],{"id":121006,"depth":49,"text":121711},"IN 与 EXISTS",{"id":121103,"depth":49,"text":121104},{"id":121144,"depth":49,"text":121145},{"id":121222,"depth":43,"text":121223},{"id":121299,"depth":43,"text":121300,"children":121716},[121717,121718],{"id":121303,"depth":49,"text":121304},{"id":121476,"depth":49,"text":121477},{"id":121612,"depth":43,"text":121613},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F02-sql",{"title":120498,"description":120506},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F02-SQL与关系代数","SL_snUp50TdMeedPObH3G31mIq8g5PkXT5VJyY4OifY",{"id":121726,"title":121727,"body":121728,"description":121735,"extension":85,"meta":123278,"navigation":52,"path":123279,"seo":123280,"stem":123281,"__hash__":123282},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F03-数据库编程专项.md","03 数据库编程专项：存储过程、触发器、游标、嵌入式 SQL",{"type":8,"value":121729,"toc":123259},[121730,121733,121736,121740,121743,121746,121769,121773,121776,121821,121824,121878,121881,121888,121890,121991,122000,122078,122081,122085,122088,122101,122263,122266,122269,122373,122377,122380,122383,122403,122405,122549,122551,122572,122576,122579,122777,122781,122788,122806,122809,122831,122834,122846,122849,122854,122928,122940,122944,122947,122950,123070,123076,123090,123094,123097,123100,123117,123120,123190,123193,123257],[11,121731,121727],{"id":121732},"_03-数据库编程专项存储过程触发器游标嵌入式-sql",[15,121734,121735],{},"数据库编程题是本课程大题高频点。new-pic 和历年题都出现过：带参数存储过程、输出参数、游标、触发器、嵌入式 SQL 填空、事务错误处理。",[19,121737,121739],{"id":121738},"一存储过程总思路","一、存储过程总思路",[15,121741,121742],{},"存储过程是一组预先编译保存在数据库中的 SQL 语句和流程控制语句。它适合完成经常调用、逻辑较复杂、需要参数的数据库操作。",[15,121744,121745],{},"答题结构：",[2230,121747,121748,121751,121757,121760,121763,121766],{},[108,121749,121750],{},"如有要求，先删除同名过程。",[108,121752,121753,121756],{},[31,121754,121755],{},"CREATE PROCEDURE"," 声明过程名和参数。",[108,121758,121759],{},"声明局部变量。",[108,121761,121762],{},"写核心 SQL。",[108,121764,121765],{},"按题意处理不存在、NULL、库存不足等情况。",[108,121767,121768],{},"必要时写调用验证。",[19,121770,121772],{"id":121771},"二不带参数与带参数存储过程","二、不带参数与带参数存储过程",[253,121774,121775],{"id":121775},"不带参数",[24,121777,121779],{"className":92584,"code":121778,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE ListAllStudents\nAS\nBEGIN\n SELECT Sno, Sname, Sdept\n FROM Student;\nEND;\nGO\n\nEXEC ListAllStudents;\n",[31,121780,121781,121786,121790,121794,121799,121804,121808,121812,121816],{"__ignoreMap":29},[34,121782,121783],{"class":36,"line":37},[34,121784,121785],{},"CREATE PROCEDURE ListAllStudents\n",[34,121787,121788],{"class":36,"line":43},[34,121789,93260],{},[34,121791,121792],{"class":36,"line":49},[34,121793,102603],{},[34,121795,121796],{"class":36,"line":56},[34,121797,121798],{}," SELECT Sno, Sname, Sdept\n",[34,121800,121801],{"class":36,"line":62},[34,121802,121803],{}," FROM Student;\n",[34,121805,121806],{"class":36,"line":68},[34,121807,102656],{},[34,121809,121810],{"class":36,"line":74},[34,121811,102988],{},[34,121813,121814],{"class":36,"line":1971},[34,121815,53],{"emptyLinePlaceholder":52},[34,121817,121818],{"class":36,"line":1976},[34,121819,121820],{},"EXEC ListAllStudents;\n",[253,121822,121823],{"id":121823},"带输入参数",[24,121825,121827],{"className":92584,"code":121826,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE ListStudentsByDept\n @dept varchar(20)\nAS\nBEGIN\n SELECT Sno, Sname, Sdept\n FROM Student\n WHERE Sdept = @dept;\nEND;\nGO\n\nEXEC ListStudentsByDept @dept = N'计算机科学与技术';\n",[31,121828,121829,121834,121839,121843,121847,121851,121856,121861,121865,121869,121873],{"__ignoreMap":29},[34,121830,121831],{"class":36,"line":37},[34,121832,121833],{},"CREATE PROCEDURE ListStudentsByDept\n",[34,121835,121836],{"class":36,"line":43},[34,121837,121838],{}," @dept varchar(20)\n",[34,121840,121841],{"class":36,"line":49},[34,121842,93260],{},[34,121844,121845],{"class":36,"line":56},[34,121846,102603],{},[34,121848,121849],{"class":36,"line":62},[34,121850,121798],{},[34,121852,121853],{"class":36,"line":68},[34,121854,121855],{}," FROM Student\n",[34,121857,121858],{"class":36,"line":74},[34,121859,121860],{}," WHERE Sdept = @dept;\n",[34,121862,121863],{"class":36,"line":1971},[34,121864,102656],{},[34,121866,121867],{"class":36,"line":1976},[34,121868,102988],{},[34,121870,121871],{"class":36,"line":1985},[34,121872,53],{"emptyLinePlaceholder":52},[34,121874,121875],{"class":36,"line":1990},[34,121876,121877],{},"EXEC ListStudentsByDept @dept = N'计算机科学与技术';\n",[253,121879,121880],{"id":121880},"带输出参数",[15,121882,121883,121884,121887],{},"题型：",[31,121885,121886],{},"Employees(emp_id, emp_name, dept_id, salary)","，输入员工号，输出员工姓名；若不存在或输入为 NULL，输出“无此员工”。",[15,121889,97054],{},[24,121891,121893],{"className":92584,"code":121892,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE proc_get_employee_name\n @in_emp_id int,\n @out_emp_name varchar(50) OUTPUT\nAS\nBEGIN\n IF @in_emp_id IS NULL\n BEGIN\n SET @out_emp_name = N'无此员工';\n RETURN;\n END;\n\n SELECT @out_emp_name = emp_name\n FROM Employees\n WHERE emp_id = @in_emp_id;\n\n IF @out_emp_name IS NULL\n SET @out_emp_name = N'无此员工';\nEND;\nGO\n\nDECLARE @name varchar(50);\nEXEC proc_get_employee_name 1001, @name OUTPUT;\nSELECT @name AS 员工姓名;\n",[31,121894,121895,121899,121903,121907,121911,121915,121919,121923,121927,121931,121935,121939,121943,121947,121951,121955,121960,121964,121968,121972,121976,121981,121986],{"__ignoreMap":29},[34,121896,121897],{"class":36,"line":37},[34,121898,102888],{},[34,121900,121901],{"class":36,"line":43},[34,121902,102893],{},[34,121904,121905],{"class":36,"line":49},[34,121906,102898],{},[34,121908,121909],{"class":36,"line":56},[34,121910,93260],{},[34,121912,121913],{"class":36,"line":62},[34,121914,102603],{},[34,121916,121917],{"class":36,"line":68},[34,121918,118348],{},[34,121920,121921],{"class":36,"line":74},[34,121922,102636],{},[34,121924,121925],{"class":36,"line":1971},[34,121926,102935],{},[34,121928,121929],{"class":36,"line":1976},[34,121930,118387],{},[34,121932,121933],{"class":36,"line":1985},[34,121934,103374],{},[34,121936,121937],{"class":36,"line":1990},[34,121938,53],{"emptyLinePlaceholder":52},[34,121940,121941],{"class":36,"line":1999},[34,121942,102911],{},[34,121944,121945],{"class":36,"line":2022},[34,121946,102916],{},[34,121948,121949],{"class":36,"line":2036},[34,121950,102921],{},[34,121952,121953],{"class":36,"line":2603},[34,121954,53],{"emptyLinePlaceholder":52},[34,121956,121957],{"class":36,"line":5537},[34,121958,121959],{}," IF @out_emp_name IS NULL\n",[34,121961,121962],{"class":36,"line":5543},[34,121963,102935],{},[34,121965,121966],{"class":36,"line":5548},[34,121967,102656],{},[34,121969,121970],{"class":36,"line":5564},[34,121971,102988],{},[34,121973,121974],{"class":36,"line":5576},[34,121975,53],{"emptyLinePlaceholder":52},[34,121977,121978],{"class":36,"line":5606},[34,121979,121980],{},"DECLARE @name varchar(50);\n",[34,121982,121983],{"class":36,"line":5616},[34,121984,121985],{},"EXEC proc_get_employee_name 1001, @name OUTPUT;\n",[34,121987,121988],{"class":36,"line":5628},[34,121989,121990],{},"SELECT @name AS 员工姓名;\n",[15,121992,121993,121994,146,121996,121999],{},"MySQL\u002F通用过程题若明确写 ",[31,121995,121009],{},[31,121997,121998],{},"OUT","，可写：",[24,122001,122003],{"className":92584,"code":122002,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE proc_get_employee_name(\n IN in_emp_id INT,\n OUT out_emp_name VARCHAR(50)\n)\nBEGIN\n IF in_emp_id IS NULL THEN\n SET out_emp_name = '无此员工';\n ELSE\n SELECT emp_name\n INTO out_emp_name\n FROM Employees\n WHERE emp_id = in_emp_id;\n\n IF out_emp_name IS NULL THEN\n SET out_emp_name = '无此员工';\n END IF;\n END IF;\nEND;\n",[31,122004,122005,122010,122014,122018,122022,122026,122030,122034,122038,122042,122046,122050,122054,122058,122062,122066,122070,122074],{"__ignoreMap":29},[34,122006,122007],{"class":36,"line":37},[34,122008,122009],{},"CREATE PROCEDURE proc_get_employee_name(\n",[34,122011,122012],{"class":36,"line":43},[34,122013,102800],{},[34,122015,122016],{"class":36,"line":49},[34,122017,102805],{},[34,122019,122020],{"class":36,"line":56},[34,122021,6111],{},[34,122023,122024],{"class":36,"line":62},[34,122025,102603],{},[34,122027,122028],{"class":36,"line":68},[34,122029,102818],{},[34,122031,122032],{"class":36,"line":74},[34,122033,102823],{},[34,122035,122036],{"class":36,"line":1971},[34,122037,102828],{},[34,122039,122040],{"class":36,"line":1976},[34,122041,102833],{},[34,122043,122044],{"class":36,"line":1985},[34,122045,102838],{},[34,122047,122048],{"class":36,"line":1990},[34,122049,102843],{},[34,122051,122052],{"class":36,"line":1999},[34,122053,102848],{},[34,122055,122056],{"class":36,"line":2022},[34,122057,53],{"emptyLinePlaceholder":52},[34,122059,122060],{"class":36,"line":2036},[34,122061,102857],{},[34,122063,122064],{"class":36,"line":2603},[34,122065,102862],{},[34,122067,122068],{"class":36,"line":5537},[34,122069,102867],{},[34,122071,122072],{"class":36,"line":5543},[34,122073,102872],{},[34,122075,122076],{"class":36,"line":5548},[34,122077,102656],{},[15,122079,122080],{},"考试若没有指定 DBMS，本课程课堂多按 SQL Server\u002FT-SQL 写。",[19,122082,122084],{"id":122083},"三库存出库入库存储过程","三、库存出库\u002F入库存储过程",[253,122086,122087],{"id":122087},"出库题模板",[15,122089,121883,122090,122093,122094,122097,122098,347],{},[31,122091,122092],{},"BOOK(BID, BNAME, QTY)","，出库时若库存足够则扣减并通知余量，否则提示库存不足。参数：书号 ",[31,122095,122096],{},"@bid","、出库数量 ",[31,122099,122100],{},"@s",[24,122102,122104],{"className":92584,"code":122103,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE BookOut\n @bid char(10),\n @s int\nAS\nBEGIN\n DECLARE @qty int;\n\n SELECT @qty = QTY\n FROM BOOK\n WHERE BID = @bid;\n\n IF @qty IS NULL\n BEGIN\n PRINT N'无此图书';\n RETURN;\n END;\n\n IF @qty >= @s\n BEGIN\n UPDATE BOOK\n SET QTY = QTY - @s\n WHERE BID = @bid;\n\n SELECT BID, BNAME, QTY AS 库存余量\n FROM BOOK\n WHERE BID = @bid;\n END\n ELSE\n BEGIN\n PRINT N'库存不足';\n END\nEND;\nGO\n\nEXEC BookOut @bid = 'B001', @s = 10;\n",[31,122105,122106,122111,122116,122121,122125,122129,122134,122138,122143,122148,122153,122157,122162,122166,122171,122175,122179,122183,122188,122192,122197,122202,122207,122211,122216,122221,122225,122229,122233,122237,122242,122246,122250,122254,122258],{"__ignoreMap":29},[34,122107,122108],{"class":36,"line":37},[34,122109,122110],{},"CREATE PROCEDURE BookOut\n",[34,122112,122113],{"class":36,"line":43},[34,122114,122115],{}," @bid char(10),\n",[34,122117,122118],{"class":36,"line":49},[34,122119,122120],{}," @s int\n",[34,122122,122123],{"class":36,"line":56},[34,122124,93260],{},[34,122126,122127],{"class":36,"line":62},[34,122128,102603],{},[34,122130,122131],{"class":36,"line":68},[34,122132,122133],{}," DECLARE @qty int;\n",[34,122135,122136],{"class":36,"line":74},[34,122137,53],{"emptyLinePlaceholder":52},[34,122139,122140],{"class":36,"line":1971},[34,122141,122142],{}," SELECT @qty = QTY\n",[34,122144,122145],{"class":36,"line":1976},[34,122146,122147],{}," FROM BOOK\n",[34,122149,122150],{"class":36,"line":1985},[34,122151,122152],{}," WHERE BID = @bid;\n",[34,122154,122155],{"class":36,"line":1990},[34,122156,53],{"emptyLinePlaceholder":52},[34,122158,122159],{"class":36,"line":1999},[34,122160,122161],{}," IF @qty IS NULL\n",[34,122163,122164],{"class":36,"line":2022},[34,122165,102636],{},[34,122167,122168],{"class":36,"line":2036},[34,122169,122170],{}," PRINT N'无此图书';\n",[34,122172,122173],{"class":36,"line":2603},[34,122174,118387],{},[34,122176,122177],{"class":36,"line":5537},[34,122178,103374],{},[34,122180,122181],{"class":36,"line":5543},[34,122182,53],{"emptyLinePlaceholder":52},[34,122184,122185],{"class":36,"line":5548},[34,122186,122187],{}," IF @qty >= @s\n",[34,122189,122190],{"class":36,"line":5564},[34,122191,102636],{},[34,122193,122194],{"class":36,"line":5576},[34,122195,122196],{}," UPDATE BOOK\n",[34,122198,122199],{"class":36,"line":5606},[34,122200,122201],{}," SET QTY = QTY - @s\n",[34,122203,122204],{"class":36,"line":5616},[34,122205,122206],{}," WHERE BID = @bid;\n",[34,122208,122209],{"class":36,"line":5628},[34,122210,53],{"emptyLinePlaceholder":52},[34,122212,122213],{"class":36,"line":5634},[34,122214,122215],{}," SELECT BID, BNAME, QTY AS 库存余量\n",[34,122217,122218],{"class":36,"line":6420},[34,122219,122220],{}," FROM BOOK\n",[34,122222,122223],{"class":36,"line":11294},[34,122224,122206],{},[34,122226,122227],{"class":36,"line":11321},[34,122228,102651],{},[34,122230,122231],{"class":36,"line":11326},[34,122232,102828],{},[34,122234,122235],{"class":36,"line":11333},[34,122236,102636],{},[34,122238,122239],{"class":36,"line":11344},[34,122240,122241],{}," PRINT N'库存不足';\n",[34,122243,122244],{"class":36,"line":11349},[34,122245,102651],{},[34,122247,122248],{"class":36,"line":11354},[34,122249,102656],{},[34,122251,122252],{"class":36,"line":11363},[34,122253,102988],{},[34,122255,122256],{"class":36,"line":11368},[34,122257,53],{"emptyLinePlaceholder":52},[34,122259,122260],{"class":36,"line":11377},[34,122261,122262],{},"EXEC BookOut @bid = 'B001', @s = 10;\n",[253,122264,122265],{"id":122265},"入库题模板",[15,122267,122268],{},"若存在则增加库存；不存在则插入新书，书名暂为空，并提示“新书入库”。",[24,122270,122272],{"className":92584,"code":122271,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE BookIn\n @bid char(10),\n @s int\nAS\nBEGIN\n IF EXISTS (SELECT 1 FROM BOOK WHERE BID = @bid)\n BEGIN\n UPDATE BOOK\n SET QTY = QTY + @s\n WHERE BID = @bid;\n\n SELECT BID, BNAME, QTY AS 最新库存\n FROM BOOK\n WHERE BID = @bid;\n END\n ELSE\n BEGIN\n INSERT INTO BOOK(BID, BNAME, QTY)\n VALUES (@bid, NULL, @s);\n PRINT N'新书入库';\n END\nEND;\nGO\n",[31,122273,122274,122279,122283,122287,122291,122295,122300,122304,122308,122313,122317,122321,122326,122330,122334,122338,122342,122346,122351,122356,122361,122365,122369],{"__ignoreMap":29},[34,122275,122276],{"class":36,"line":37},[34,122277,122278],{},"CREATE PROCEDURE BookIn\n",[34,122280,122281],{"class":36,"line":43},[34,122282,122115],{},[34,122284,122285],{"class":36,"line":49},[34,122286,122120],{},[34,122288,122289],{"class":36,"line":56},[34,122290,93260],{},[34,122292,122293],{"class":36,"line":62},[34,122294,102603],{},[34,122296,122297],{"class":36,"line":68},[34,122298,122299],{}," IF EXISTS (SELECT 1 FROM BOOK WHERE BID = @bid)\n",[34,122301,122302],{"class":36,"line":74},[34,122303,102636],{},[34,122305,122306],{"class":36,"line":1971},[34,122307,122196],{},[34,122309,122310],{"class":36,"line":1976},[34,122311,122312],{}," SET QTY = QTY + @s\n",[34,122314,122315],{"class":36,"line":1985},[34,122316,122206],{},[34,122318,122319],{"class":36,"line":1990},[34,122320,53],{"emptyLinePlaceholder":52},[34,122322,122323],{"class":36,"line":1999},[34,122324,122325],{}," SELECT BID, BNAME, QTY AS 最新库存\n",[34,122327,122328],{"class":36,"line":2022},[34,122329,122220],{},[34,122331,122332],{"class":36,"line":2036},[34,122333,122206],{},[34,122335,122336],{"class":36,"line":2603},[34,122337,102651],{},[34,122339,122340],{"class":36,"line":5537},[34,122341,102828],{},[34,122343,122344],{"class":36,"line":5543},[34,122345,102636],{},[34,122347,122348],{"class":36,"line":5548},[34,122349,122350],{}," INSERT INTO BOOK(BID, BNAME, QTY)\n",[34,122352,122353],{"class":36,"line":5564},[34,122354,122355],{}," VALUES (@bid, NULL, @s);\n",[34,122357,122358],{"class":36,"line":5576},[34,122359,122360],{}," PRINT N'新书入库';\n",[34,122362,122363],{"class":36,"line":5606},[34,122364,102651],{},[34,122366,122367],{"class":36,"line":5616},[34,122368,102656],{},[34,122370,122371],{"class":36,"line":5628},[34,122372,102988],{},[19,122374,122376],{"id":122375},"四游标","四、游标",[15,122378,122379],{},"游标用于协调 SQL 的集合处理方式与宿主语言\u002F过程语言的单记录处理方式。选择填空常问：游标是用于逐行处理查询结果集的机制。",[15,122381,122382],{},"SQL Server 游标步骤：",[2230,122384,122385,122388,122391,122394,122397,122400],{},[108,122386,122387],{},"定义游标。",[108,122389,122390],{},"打开游标。",[108,122392,122393],{},"读取\u002F推进游标。",[108,122395,122396],{},"循环处理。",[108,122398,122399],{},"关闭游标。",[108,122401,122402],{},"释放游标。",[15,122404,1589],{},[24,122406,122408],{"className":92584,"code":122407,"language":92586,"meta":29,"style":29},"DECLARE Cur1 CURSOR\nFOR\n SELECT p.pno, p.pname, d.dname, d.mana\n FROM Pe AS p\n JOIN Dept AS d ON p.dptno = d.dptno\n WHERE p.pname LIKE N'_武%';\n\nOPEN Cur1;\n\nDECLARE @pno varchar(10),\n @pname varchar(20),\n @dname varchar(20),\n @mana varchar(20);\n\nFETCH NEXT FROM Cur1\nINTO @pno, @pname, @dname, @mana;\n\nWHILE @@FETCH_STATUS = 0\nBEGIN\n SELECT 职工号 = @pno,\n 姓名 = @pname,\n 部门 = @dname,\n 负责人 = @mana;\n\n FETCH NEXT FROM Cur1\n INTO @pno, @pname, @dname, @mana;\nEND;\n\nCLOSE Cur1;\nDEALLOCATE Cur1;\n",[31,122409,122410,122415,122419,122424,122429,122434,122439,122443,122448,122452,122457,122462,122467,122472,122476,122481,122486,122490,122494,122498,122503,122508,122513,122518,122522,122526,122531,122535,122539,122544],{"__ignoreMap":29},[34,122411,122412],{"class":36,"line":37},[34,122413,122414],{},"DECLARE Cur1 CURSOR\n",[34,122416,122417],{"class":36,"line":43},[34,122418,103573],{},[34,122420,122421],{"class":36,"line":49},[34,122422,122423],{}," SELECT p.pno, p.pname, d.dname, d.mana\n",[34,122425,122426],{"class":36,"line":56},[34,122427,122428],{}," FROM Pe AS p\n",[34,122430,122431],{"class":36,"line":62},[34,122432,122433],{}," JOIN Dept AS d ON p.dptno = d.dptno\n",[34,122435,122436],{"class":36,"line":68},[34,122437,122438],{}," WHERE p.pname LIKE N'_武%';\n",[34,122440,122441],{"class":36,"line":74},[34,122442,53],{"emptyLinePlaceholder":52},[34,122444,122445],{"class":36,"line":1971},[34,122446,122447],{},"OPEN Cur1;\n",[34,122449,122450],{"class":36,"line":1976},[34,122451,53],{"emptyLinePlaceholder":52},[34,122453,122454],{"class":36,"line":1985},[34,122455,122456],{},"DECLARE @pno varchar(10),\n",[34,122458,122459],{"class":36,"line":1990},[34,122460,122461],{}," @pname varchar(20),\n",[34,122463,122464],{"class":36,"line":1999},[34,122465,122466],{}," @dname varchar(20),\n",[34,122468,122469],{"class":36,"line":2022},[34,122470,122471],{}," @mana varchar(20);\n",[34,122473,122474],{"class":36,"line":2036},[34,122475,53],{"emptyLinePlaceholder":52},[34,122477,122478],{"class":36,"line":2603},[34,122479,122480],{},"FETCH NEXT FROM Cur1\n",[34,122482,122483],{"class":36,"line":5537},[34,122484,122485],{},"INTO @pno, @pname, @dname, @mana;\n",[34,122487,122488],{"class":36,"line":5543},[34,122489,53],{"emptyLinePlaceholder":52},[34,122491,122492],{"class":36,"line":5548},[34,122493,103619],{},[34,122495,122496],{"class":36,"line":5564},[34,122497,102603],{},[34,122499,122500],{"class":36,"line":5576},[34,122501,122502],{}," SELECT 职工号 = @pno,\n",[34,122504,122505],{"class":36,"line":5606},[34,122506,122507],{}," 姓名 = @pname,\n",[34,122509,122510],{"class":36,"line":5616},[34,122511,122512],{}," 部门 = @dname,\n",[34,122514,122515],{"class":36,"line":5628},[34,122516,122517],{}," 负责人 = @mana;\n",[34,122519,122520],{"class":36,"line":5634},[34,122521,53],{"emptyLinePlaceholder":52},[34,122523,122524],{"class":36,"line":6420},[34,122525,103317],{},[34,122527,122528],{"class":36,"line":11294},[34,122529,122530],{}," INTO @pno, @pname, @dname, @mana;\n",[34,122532,122533],{"class":36,"line":11321},[34,122534,102656],{},[34,122536,122537],{"class":36,"line":11326},[34,122538,53],{"emptyLinePlaceholder":52},[34,122540,122541],{"class":36,"line":11333},[34,122542,122543],{},"CLOSE Cur1;\n",[34,122545,122546],{"class":36,"line":11344},[34,122547,122548],{},"DEALLOCATE Cur1;\n",[15,122550,8882],{},[105,122552,122553,122559,122565],{},[108,122554,122555,122558],{},[31,122556,122557],{},"FETCH NEXT FROM 游标 INTO 变量列表"," 的变量数量和查询列数量必须一致。",[108,122560,122561,122562,347],{},"循环条件是 ",[31,122563,122564],{},"@@FETCH_STATUS = 0",[108,122566,122567,122568,122571],{},"关闭后还要 ",[31,122569,122570],{},"DEALLOCATE"," 释放。",[19,122573,122575],{"id":122574},"五带参数存储过程加游标","五、带参数存储过程加游标",[15,122577,122578],{},"题型：查询姓名第二个字为“武”的员工的部门名和负责人，并以标签格式输出。",[24,122580,122582],{"className":92584,"code":122581,"language":92586,"meta":29,"style":29},"IF EXISTS (\n SELECT name\n FROM sysobjects\n WHERE name = 'Sele_staff'\n AND type = 'P'\n)\nDROP PROCEDURE Sele_staff;\nGO\n\nCREATE PROCEDURE Sele_staff\n @second_char nchar(1)\nAS\nBEGIN\n DECLARE Cur1 CURSOR\n FOR\n SELECT p.pno, p.pname, d.dname, d.mana\n FROM Pe AS p\n JOIN Dept AS d ON p.dptno = d.dptno\n WHERE p.pname LIKE N'_' + @second_char + N'%';\n\n OPEN Cur1;\n\n DECLARE @pno varchar(10),\n @pname nvarchar(20),\n @dname nvarchar(20),\n @mana nvarchar(20);\n\n FETCH NEXT FROM Cur1 INTO @pno, @pname, @dname, @mana;\n\n WHILE @@FETCH_STATUS = 0\n BEGIN\n SELECT 职工号 = @pno,\n 姓名 = @pname,\n 部门 = @dname,\n 负责人 = @mana;\n\n FETCH NEXT FROM Cur1 INTO @pno, @pname, @dname, @mana;\n END;\n\n CLOSE Cur1;\n DEALLOCATE Cur1;\nEND;\nGO\n\nEXEC Sele_staff N'武';\n",[31,122583,122584,122588,122592,122596,122600,122604,122608,122612,122616,122620,122624,122629,122633,122637,122641,122645,122649,122653,122657,122662,122666,122670,122674,122679,122684,122689,122694,122698,122703,122707,122711,122715,122720,122725,122730,122735,122739,122744,122748,122752,122756,122760,122764,122768,122772],{"__ignoreMap":29},[34,122585,122586],{"class":36,"line":37},[34,122587,102954],{},[34,122589,122590],{"class":36,"line":43},[34,122591,102959],{},[34,122593,122594],{"class":36,"line":49},[34,122595,102964],{},[34,122597,122598],{"class":36,"line":56},[34,122599,103205],{},[34,122601,122602],{"class":36,"line":62},[34,122603,103210],{},[34,122605,122606],{"class":36,"line":68},[34,122607,6111],{},[34,122609,122610],{"class":36,"line":74},[34,122611,103219],{},[34,122613,122614],{"class":36,"line":1971},[34,122615,102988],{},[34,122617,122618],{"class":36,"line":1976},[34,122619,53],{"emptyLinePlaceholder":52},[34,122621,122622],{"class":36,"line":1985},[34,122623,103232],{},[34,122625,122626],{"class":36,"line":1990},[34,122627,122628],{}," @second_char nchar(1)\n",[34,122630,122631],{"class":36,"line":1999},[34,122632,93260],{},[34,122634,122635],{"class":36,"line":2022},[34,122636,102603],{},[34,122638,122639],{"class":36,"line":2036},[34,122640,103250],{},[34,122642,122643],{"class":36,"line":2603},[34,122644,103255],{},[34,122646,122647],{"class":36,"line":5537},[34,122648,103260],{},[34,122650,122651],{"class":36,"line":5543},[34,122652,103265],{},[34,122654,122655],{"class":36,"line":5548},[34,122656,103270],{},[34,122658,122659],{"class":36,"line":5564},[34,122660,122661],{}," WHERE p.pname LIKE N'_' + @second_char + N'%';\n",[34,122663,122664],{"class":36,"line":5576},[34,122665,53],{"emptyLinePlaceholder":52},[34,122667,122668],{"class":36,"line":5606},[34,122669,103284],{},[34,122671,122672],{"class":36,"line":5616},[34,122673,53],{"emptyLinePlaceholder":52},[34,122675,122676],{"class":36,"line":5628},[34,122677,122678],{}," DECLARE @pno varchar(10),\n",[34,122680,122681],{"class":36,"line":5634},[34,122682,122683],{}," @pname nvarchar(20),\n",[34,122685,122686],{"class":36,"line":6420},[34,122687,122688],{}," @dname nvarchar(20),\n",[34,122690,122691],{"class":36,"line":11294},[34,122692,122693],{}," @mana nvarchar(20);\n",[34,122695,122696],{"class":36,"line":11321},[34,122697,53],{"emptyLinePlaceholder":52},[34,122699,122700],{"class":36,"line":11326},[34,122701,122702],{}," FETCH NEXT FROM Cur1 INTO @pno, @pname, @dname, @mana;\n",[34,122704,122705],{"class":36,"line":11333},[34,122706,53],{"emptyLinePlaceholder":52},[34,122708,122709],{"class":36,"line":11344},[34,122710,103331],{},[34,122712,122713],{"class":36,"line":11349},[34,122714,102636],{},[34,122716,122717],{"class":36,"line":11354},[34,122718,122719],{}," SELECT 职工号 = @pno,\n",[34,122721,122722],{"class":36,"line":11363},[34,122723,122724],{}," 姓名 = @pname,\n",[34,122726,122727],{"class":36,"line":11368},[34,122728,122729],{}," 部门 = @dname,\n",[34,122731,122732],{"class":36,"line":11377},[34,122733,122734],{}," 负责人 = @mana;\n",[34,122736,122737],{"class":36,"line":11402},[34,122738,53],{"emptyLinePlaceholder":52},[34,122740,122741],{"class":36,"line":11407},[34,122742,122743],{}," FETCH NEXT FROM Cur1 INTO @pno, @pname, @dname, @mana;\n",[34,122745,122746],{"class":36,"line":11421},[34,122747,103374],{},[34,122749,122750],{"class":36,"line":11426},[34,122751,53],{"emptyLinePlaceholder":52},[34,122753,122754],{"class":36,"line":11447},[34,122755,103383],{},[34,122757,122758],{"class":36,"line":11452},[34,122759,103388],{},[34,122761,122762],{"class":36,"line":11463},[34,122763,102656],{},[34,122765,122766],{"class":36,"line":11485},[34,122767,102988],{},[34,122769,122770],{"class":36,"line":11495},[34,122771,53],{"emptyLinePlaceholder":52},[34,122773,122774],{"class":36,"line":11528},[34,122775,122776],{},"EXEC Sele_staff N'武';\n",[19,122778,122780],{"id":122779},"六触发器","六、触发器",[15,122782,122783,122784,122787],{},"触发器是一种特殊存储过程，由事件自动触发，不能通过普通 ",[31,122785,122786],{},"EXEC"," 调用。它常用于复杂完整性约束、审计、联动维护。",[15,122789,122790,122791,122794,122795,122798,122799,258,122802,122805],{},"触发器通常“不带用户手工传入的参数”。它不是像存储过程那样 ",[31,122792,122793],{},"EXEC proc @x=..."," 调用，而是在表发生 ",[31,122796,122797],{},"INSERT\u002FUPDATE\u002FDELETE"," 时由 DBMS 自动执行。触发器要获取本次被修改的数据，主要靠 ",[31,122800,122801],{},"inserted",[31,122803,122804],{},"deleted"," 这两张逻辑表。",[15,122807,122808],{},"常见触发时机：",[105,122810,122811,122816,122821,122826],{},[108,122812,122813],{},[31,122814,122815],{},"AFTER INSERT",[108,122817,122818],{},[31,122819,122820],{},"AFTER UPDATE",[108,122822,122823],{},[31,122824,122825],{},"AFTER DELETE",[108,122827,122828],{},[31,122829,122830],{},"INSTEAD OF INSERT\u002FUPDATE\u002FDELETE",[15,122832,122833],{},"SQL Server 触发器中常用临时表：",[105,122835,122836,122841],{},[108,122837,122838,122840],{},[31,122839,122801],{},"：插入后的新行；更新时也保存更新后的新行。",[108,122842,122843,122845],{},[31,122844,122804],{},"：删除前的旧行；更新时也保存更新前的旧行。",[253,122847,122848],{"id":122848},"营业时间触发器",[15,122850,121883,122851,122853],{},[31,122852,101199],{}," 只能在工作日 9:00-17:00 修改。",[24,122855,122857],{"className":92584,"code":122856,"language":92586,"meta":29,"style":29},"CREATE TRIGGER trg_SaleDetail_business_hours\nON SaleDetail\nAFTER INSERT, UPDATE, DELETE\nAS\nBEGIN\n DECLARE @hour int, @weekday int;\n\n SELECT @hour = DATEPART(hh, GETDATE());\n SELECT @weekday = DATEPART(weekday, GETDATE());\n\n IF NOT (@hour BETWEEN 9 AND 17 AND @weekday IN (2, 3, 4, 5, 6))\n BEGIN\n PRINT N'Not business hours!';\n ROLLBACK;\n END;\nEND;\nGO\n",[31,122858,122859,122863,122867,122871,122875,122879,122883,122887,122891,122895,122899,122903,122907,122912,122916,122920,122924],{"__ignoreMap":29},[34,122860,122861],{"class":36,"line":37},[34,122862,102584],{},[34,122864,122865],{"class":36,"line":43},[34,122866,102589],{},[34,122868,122869],{"class":36,"line":49},[34,122870,102594],{},[34,122872,122873],{"class":36,"line":56},[34,122874,93260],{},[34,122876,122877],{"class":36,"line":62},[34,122878,102603],{},[34,122880,122881],{"class":36,"line":68},[34,122882,102608],{},[34,122884,122885],{"class":36,"line":74},[34,122886,53],{"emptyLinePlaceholder":52},[34,122888,122889],{"class":36,"line":1971},[34,122890,102617],{},[34,122892,122893],{"class":36,"line":1976},[34,122894,102622],{},[34,122896,122897],{"class":36,"line":1985},[34,122898,53],{"emptyLinePlaceholder":52},[34,122900,122901],{"class":36,"line":1990},[34,122902,102631],{},[34,122904,122905],{"class":36,"line":1999},[34,122906,102636],{},[34,122908,122909],{"class":36,"line":2022},[34,122910,122911],{}," PRINT N'Not business hours!';\n",[34,122913,122914],{"class":36,"line":2036},[34,122915,102646],{},[34,122917,122918],{"class":36,"line":2603},[34,122919,103374],{},[34,122921,122922],{"class":36,"line":5537},[34,122923,102656],{},[34,122925,122926],{"class":36,"line":5543},[34,122927,102988],{},[15,122929,122930,122932,122933,122935,122936,122939],{},[31,122931,102546],{}," 的结果受 ",[31,122934,102556],{}," 影响。课堂题一般默认周一到周五为 2-6。若想更稳，可先 ",[31,122937,122938],{},"SET DATEFIRST 7","，但考场通常不用展开。",[19,122941,122943],{"id":122942},"七事务与错误处理","七、事务与错误处理",[15,122945,122946],{},"多步删除\u002F修改必须使用事务，保证全部成功才提交，任何一步失败要回滚。",[15,122948,122949],{},"例：删除学生时先删除成绩，再删除学生。",[24,122951,122953],{"className":92584,"code":122952,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE DeleteStudent\n @sno char(9)\nAS\nBEGIN\n BEGIN TRANSACTION;\n\n DELETE FROM Grade WHERE Sno = @sno;\n IF @@ERROR \u003C> 0\n BEGIN\n ROLLBACK TRANSACTION;\n PRINT N'删除成绩失败，事务已回滚';\n RETURN;\n END;\n\n DELETE FROM Student WHERE Sno = @sno;\n IF @@ERROR \u003C> 0\n BEGIN\n ROLLBACK TRANSACTION;\n PRINT N'删除学生失败，事务已回滚';\n RETURN;\n END;\n\n COMMIT TRANSACTION;\n PRINT N'学生删除成功，事务已提交';\nEND;\nGO\n",[31,122954,122955,122960,122965,122969,122973,122978,122982,122987,122992,122996,123001,123006,123010,123014,123018,123023,123027,123031,123035,123040,123044,123048,123052,123057,123062,123066],{"__ignoreMap":29},[34,122956,122957],{"class":36,"line":37},[34,122958,122959],{},"CREATE PROCEDURE DeleteStudent\n",[34,122961,122962],{"class":36,"line":43},[34,122963,122964],{}," @sno char(9)\n",[34,122966,122967],{"class":36,"line":49},[34,122968,93260],{},[34,122970,122971],{"class":36,"line":56},[34,122972,102603],{},[34,122974,122975],{"class":36,"line":62},[34,122976,122977],{}," BEGIN TRANSACTION;\n",[34,122979,122980],{"class":36,"line":68},[34,122981,53],{"emptyLinePlaceholder":52},[34,122983,122984],{"class":36,"line":74},[34,122985,122986],{}," DELETE FROM Grade WHERE Sno = @sno;\n",[34,122988,122989],{"class":36,"line":1971},[34,122990,122991],{}," IF @@ERROR \u003C> 0\n",[34,122993,122994],{"class":36,"line":1976},[34,122995,102636],{},[34,122997,122998],{"class":36,"line":1985},[34,122999,123000],{}," ROLLBACK TRANSACTION;\n",[34,123002,123003],{"class":36,"line":1990},[34,123004,123005],{}," PRINT N'删除成绩失败，事务已回滚';\n",[34,123007,123008],{"class":36,"line":1999},[34,123009,118387],{},[34,123011,123012],{"class":36,"line":2022},[34,123013,103374],{},[34,123015,123016],{"class":36,"line":2036},[34,123017,53],{"emptyLinePlaceholder":52},[34,123019,123020],{"class":36,"line":2603},[34,123021,123022],{}," DELETE FROM Student WHERE Sno = @sno;\n",[34,123024,123025],{"class":36,"line":5537},[34,123026,122991],{},[34,123028,123029],{"class":36,"line":5543},[34,123030,102636],{},[34,123032,123033],{"class":36,"line":5548},[34,123034,123000],{},[34,123036,123037],{"class":36,"line":5564},[34,123038,123039],{}," PRINT N'删除学生失败，事务已回滚';\n",[34,123041,123042],{"class":36,"line":5576},[34,123043,118387],{},[34,123045,123046],{"class":36,"line":5606},[34,123047,103374],{},[34,123049,123050],{"class":36,"line":5616},[34,123051,53],{"emptyLinePlaceholder":52},[34,123053,123054],{"class":36,"line":5628},[34,123055,123056],{}," COMMIT TRANSACTION;\n",[34,123058,123059],{"class":36,"line":5634},[34,123060,123061],{}," PRINT N'学生删除成功，事务已提交';\n",[34,123063,123064],{"class":36,"line":6420},[34,123065,102656],{},[34,123067,123068],{"class":36,"line":11294},[34,123069,102988],{},[15,123071,123072,123075],{},[31,123073,123074],{},"@@ERROR"," 的规则：",[105,123077,123078,123081,123084,123087],{},[108,123079,123080],{},"每条语句执行后必须立刻判断。",[108,123082,123083],{},"不能等多条语句执行完再判断，因为会被后续语句覆盖。",[108,123085,123086],{},"出错必须回滚。",[108,123088,123089],{},"出错必须退出过程，避免继续执行破坏一致性。",[19,123091,123093],{"id":123092},"八嵌入式-sql","八、嵌入式 SQL",[15,123095,123096],{},"嵌入式 SQL 是把 SQL 语句嵌入 C 等宿主语言中使用。它解决的问题是：SQL 一次处理一组记录，宿主语言通常一次处理一条记录，两者通过主变量、SQLCA、游标协调。",[15,123098,123099],{},"核心术语：",[105,123101,123102,123108,123114],{},[108,123103,123104,123105,347],{},"主变量：宿主语言变量，SQL 中使用时前面加冒号，如 ",[31,123106,123107],{},":deptname",[108,123109,123110,123111,347],{},"SQLCA：SQL 通信区，保存 SQL 执行状态，如 ",[31,123112,123113],{},"sqlca.sqlcode",[108,123115,123116],{},"游标：逐行处理查询结果集。",[15,123118,123119],{},"典型填空：",[24,123121,123123],{"className":65734,"code":123122,"language":31965,"meta":29,"style":29},"EXEC SQL INCLUDE SQLCA; \u002F* 定义 SQL 通信区 *\u002F\n\nEXEC SQL DECLARE SX CURSOR FOR\nSELECT Sno, Sname, Ssex, Sage\nFROM Student\nWHERE Sdept = :deptname;\n\nEXEC SQL OPEN SX;\n\nEXEC SQL FETCH SX INTO :HSno, :HSname, :HSsex, :HSage;\n\nEXEC SQL UPDATE Student\nSET Sage = :NEWAGE\nWHERE CURRENT OF SX;\n",[31,123124,123125,123130,123134,123139,123144,123148,123153,123157,123162,123166,123171,123175,123180,123185],{"__ignoreMap":29},[34,123126,123127],{"class":36,"line":37},[34,123128,123129],{},"EXEC SQL INCLUDE SQLCA; \u002F* 定义 SQL 通信区 *\u002F\n",[34,123131,123132],{"class":36,"line":43},[34,123133,53],{"emptyLinePlaceholder":52},[34,123135,123136],{"class":36,"line":49},[34,123137,123138],{},"EXEC SQL DECLARE SX CURSOR FOR\n",[34,123140,123141],{"class":36,"line":56},[34,123142,123143],{},"SELECT Sno, Sname, Ssex, Sage\n",[34,123145,123146],{"class":36,"line":62},[34,123147,120836],{},[34,123149,123150],{"class":36,"line":68},[34,123151,123152],{},"WHERE Sdept = :deptname;\n",[34,123154,123155],{"class":36,"line":74},[34,123156,53],{"emptyLinePlaceholder":52},[34,123158,123159],{"class":36,"line":1971},[34,123160,123161],{},"EXEC SQL OPEN SX;\n",[34,123163,123164],{"class":36,"line":1976},[34,123165,53],{"emptyLinePlaceholder":52},[34,123167,123168],{"class":36,"line":1985},[34,123169,123170],{},"EXEC SQL FETCH SX INTO :HSno, :HSname, :HSsex, :HSage;\n",[34,123172,123173],{"class":36,"line":1990},[34,123174,53],{"emptyLinePlaceholder":52},[34,123176,123177],{"class":36,"line":1999},[34,123178,123179],{},"EXEC SQL UPDATE Student\n",[34,123181,123182],{"class":36,"line":2022},[34,123183,123184],{},"SET Sage = :NEWAGE\n",[34,123186,123187],{"class":36,"line":2036},[34,123188,123189],{},"WHERE CURRENT OF SX;\n",[15,123191,123192],{},"完整流程：",[2230,123194,123195,123204,123209,123215,123221,123227,123233,123240,123246],{},[108,123196,123197,258,123200,123203],{},[31,123198,123199],{},"EXEC SQL BEGIN DECLARE SECTION;",[31,123201,123202],{},"END DECLARE SECTION;"," 声明主变量。",[108,123205,123206,123208],{},[31,123207,119757],{}," 引入通信区。",[108,123210,123211,123214],{},[31,123212,123213],{},"EXEC SQL CONNECT ..."," 连接数据库。",[108,123216,123217,123220],{},[31,123218,123219],{},"DECLARE cursor CURSOR FOR SELECT ..."," 定义游标。",[108,123222,123223,123226],{},[31,123224,123225],{},"OPEN cursor"," 打开游标。",[108,123228,123229,123230,347],{},"循环 ",[31,123231,123232],{},"FETCH cursor INTO :变量",[108,123234,123235,123236,123239],{},"判断 ",[31,123237,123238],{},"sqlca.sqlcode != 0"," 退出。",[108,123241,123242,123243,347],{},"如需更新当前行，用 ",[31,123244,123245],{},"WHERE CURRENT OF cursor",[108,123247,123248,46528,123251,46528,123254,347],{},[31,123249,123250],{},"CLOSE cursor",[31,123252,123253],{},"COMMIT WORK",[31,123255,123256],{},"DISCONNECT",[79,123258,81],{},{"title":29,"searchDepth":43,"depth":43,"links":123260},[123261,123262,123267,123271,123272,123273,123276,123277],{"id":121738,"depth":43,"text":121739},{"id":121771,"depth":43,"text":121772,"children":123263},[123264,123265,123266],{"id":121775,"depth":49,"text":121775},{"id":121823,"depth":49,"text":121823},{"id":121880,"depth":49,"text":121880},{"id":122083,"depth":43,"text":122084,"children":123268},[123269,123270],{"id":122087,"depth":49,"text":122087},{"id":122265,"depth":49,"text":122265},{"id":122375,"depth":43,"text":122376},{"id":122574,"depth":43,"text":122575},{"id":122779,"depth":43,"text":122780,"children":123274},[123275],{"id":122848,"depth":49,"text":122848},{"id":122942,"depth":43,"text":122943},{"id":123092,"depth":43,"text":123093},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F03",{"title":121727,"description":121735},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F03-数据库编程专项","IKDsMRs7lNgEtLPT8v75B10XgannBBFiRlWFQY1NS04",{"id":123284,"title":123285,"body":123286,"description":123293,"extension":85,"meta":123857,"navigation":52,"path":123858,"seo":123859,"stem":123860,"__hash__":123861},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F04-数据库设计ER与规范化.md","04 数据库设计、E-R 图与规范化",{"type":8,"value":123287,"toc":123825},[123288,123291,123294,123298,123301,123321,123324,123328,123331,123334,123337,123340,123343,123357,123360,123363,123374,123377,123380,123383,123394,123398,123401,123404,123410,123414,123417,123421,123424,123430,123434,123437,123443,123446,123449,123455,123459,123466,123469,123483,123486,123492,123496,123499,123502,123505,123508,123511,123513,123519,123529,123533,123536,123538,123544,123547,123553,123556,123562,123565,123569,123572,123586,123591,123594,123598,123621,123625,123628,123631,123651,123653,123658,123661,123666,123669,123672,123675,123681,123683,123697,123699,123705,123707,123713,123715,123721,123723,123729,123732,123735,123741,123743,123760,123762,123767,123770,123776,123780,123783,123789,123791,123809,123813,123816,123822],[11,123289,123285],{"id":123290},"_04-数据库设计e-r-图与规范化",[15,123292,123293],{},"数据库设计题通常 15-20 分，是最能拉开差距的大题。核心不是画得漂亮，而是语义识别准确、联系类型正确、关系模式主外码明确、范式理由能自圆其说。",[19,123295,123297],{"id":123296},"一数据库设计步骤","一、数据库设计步骤",[15,123299,123300],{},"教材常见步骤：",[2230,123302,123303,123306,123309,123312,123315,123318],{},[108,123304,123305],{},"需求分析：调查用户需求，形成数据字典、数据流图、需求说明。",[108,123307,123308],{},"概念结构设计：用 E-R 图描述实体、属性、联系，独立于具体 DBMS。",[108,123310,123311],{},"逻辑结构设计：把 E-R 图转换成关系模式，确定主码外码，并进行规范化。",[108,123313,123314],{},"物理结构设计：选择存储结构、索引、存取路径、分区等。",[108,123316,123317],{},"数据库实施：建库、建表、装载数据、编写应用。",[108,123319,123320],{},"运行和维护：监控、调优、备份恢复、安全管理。",[15,123322,123323],{},"题目问“设计关系模式属于哪一阶段”，答：逻辑结构设计。",[19,123325,123327],{"id":123326},"二e-r-图怎么画","二、E-R 图怎么画",[253,123329,123330],{"id":123330},"找实体",[15,123332,123333],{},"实体通常是名词，能独立存在并需要保存信息。例如学生、课程、教师、学院、图书、读者、供应商、零件、工程。",[15,123335,123336],{},"不要把所有名词都当实体。若只是某实体的特征，如姓名、年龄、地址、电话，通常是属性。",[253,123338,123339],{"id":123339},"找联系",[15,123341,123342],{},"联系通常来自动词或业务规则：",[105,123344,123345,123348,123351,123354],{},[108,123346,123347],{},"学生主修专业：学生 n:1 专业。",[108,123349,123350],{},"学生选修课程：学生 m:n 课程，联系属性是成绩。",[108,123352,123353],{},"教师讲授课程：教师 m:n 课程，联系属性是教室、课时。",[108,123355,123356],{},"读者借阅图书：读者 m:n 图书，联系属性是借阅日期、还书日期。",[253,123358,123359],{"id":123359},"判断联系类型",[15,123361,123362],{},"题目中的“一”“多”就是线索：",[105,123364,123365,123368,123371],{},[108,123366,123367],{},"“每个学生主修一个专业，一个专业可招收多名学生”：学生 n:1 专业。",[108,123369,123370],{},"“每个学院可设置多个专业，每个专业属于一个学院”：学院 1:n 专业。",[108,123372,123373],{},"“每个学生可选多门课，每门课可被多名学生选修”：学生 m:n 课程。",[253,123375,123376],{"id":123376},"联系属性",[15,123378,123379],{},"若某属性不属于单个实体，而属于一次联系，就放在联系上。",[15,123381,123382],{},"例：",[105,123384,123385,123388,123391],{},[108,123386,123387],{},"选课成绩属于“学生选修课程”这次联系，不属于学生，也不属于课程。",[108,123389,123390],{},"授课教室、课时属于“教师讲授课程”这次联系。",[108,123392,123393],{},"借阅日期、还书日期属于“读者借阅图书”这次联系。",[19,123395,123397],{"id":123396},"三e-r-转关系模式规则","三、E-R 转关系模式规则",[253,123399,123400],{"id":123400},"实体转表",[15,123402,123403],{},"每个实体型转换为一个关系模式，实体的属性成为表的属性，实体码成为主码。",[24,123405,123408],{"className":123406,"code":123407,"language":2638,"meta":29},[2636],"Student(SID, SName, MajorID)\nPK: SID\n",[31,123409,123407],{"__ignoreMap":29},[253,123411,123413],{"id":123412},"_11-联系","1:1 联系",[15,123415,123416],{},"可把任一方主码加入另一方作外码，也可单独建联系表。若联系有属性，通常单独建表更清晰。",[253,123418,123420],{"id":123419},"_1n-联系","1:n 联系",[15,123422,123423],{},"把 1 方主码加入 n 方作为外码。若联系有属性，也一并放入 n 方，除非联系本身需要独立记录。",[24,123425,123428],{"className":123426,"code":123427,"language":2638,"meta":29},[2636],"College(CollegeID, CollegeName)\nMajor(MajorID, MajorName, CollegeID)\nFK Major.CollegeID -> College\n",[31,123429,123427],{"__ignoreMap":29},[253,123431,123433],{"id":123432},"mn-联系","m:n 联系",[15,123435,123436],{},"必须单独转换为联系表，主码通常是双方主码组合，联系属性也放入该表。",[24,123438,123441],{"className":123439,"code":123440,"language":2638,"meta":29},[2636],"SC(SID, CID, Grade)\nPK: (SID, CID)\nFK: SID -> Student, CID -> Course\n",[31,123442,123440],{"__ignoreMap":29},[253,123444,123445],{"id":123445},"多元联系",[15,123447,123448],{},"如供应商-零件-工程三元供应关系，单独建联系表，主码通常是各实体主码组合。",[24,123450,123453],{"className":123451,"code":123452,"language":2638,"meta":29},[2636],"SPJ(Sno, Pno, Jno, Qty)\nPK: (Sno, Pno, Jno)\n",[31,123454,123452],{"__ignoreMap":29},[19,123456,123458],{"id":123457},"四函数依赖与码","四、函数依赖与码",[15,123460,123461,123462,123465],{},"函数依赖 ",[31,123463,123464],{},"X -> Y"," 的含义：在关系模式 R 的任一可能关系中，若两个元组 X 值相等，则 Y 值也相等。",[15,123467,123468],{},"候选码求法：",[2230,123470,123471,123474,123477,123480],{},[108,123472,123473],{},"先找从未出现在任何依赖右部的属性，这些属性必须在候选码中。",[108,123475,123476],{},"求闭包，看是否能推出全部属性。",[108,123478,123479],{},"若不能，补充属性继续求闭包。",[108,123481,123482],{},"删除多余属性，得到最小属性组。",[15,123484,123485],{},"闭包例：",[24,123487,123490],{"className":123488,"code":123489,"language":2638,"meta":29},[2636],"U = {A,B,C,D,E}\nF = {AB -> C, B -> D, C -> E, EC -> B, AC -> B}\n\n(AC)+:\n初始 {A,C}\nC -> E，得 {A,C,E}\nEC -> B，得 {A,B,C,E}\nB -> D，得 {A,B,C,D,E}\n所以 AC 是超码。若 A+ 或 C+ 不能推出全部，则 AC 是候选码。\n",[31,123491,123489],{"__ignoreMap":29},[19,123493,123495],{"id":123494},"五范式判断","五、范式判断",[253,123497,110422],{"id":123498},"_1nf",[15,123500,123501],{},"每个属性都是不可再分的数据项。关系数据库中的规范关系至少应满足 1NF。",[253,123503,110426],{"id":123504},"_2nf",[15,123506,123507],{},"在 1NF 基础上，每个非主属性完全函数依赖于候选码，不能部分依赖候选码的一部分。",[15,123509,123510],{},"只有候选码是组合码时，才需要特别检查部分依赖。",[15,123512,123382],{},[24,123514,123517],{"className":123515,"code":123516,"language":2638,"meta":29},[2636],"R(Sno, Cno, Grade, Cname)\n候选码：(Sno, Cno)\n(Sno, Cno) -> Grade\nCno -> Cname\n",[31,123518,123516],{"__ignoreMap":29},[15,123520,123521,123524,123525,123528],{},[31,123522,123523],{},"Cname"," 只依赖组合码的一部分 ",[31,123526,123527],{},"Cno","，存在部分依赖，不满足 2NF。",[253,123530,123532],{"id":123531},"_3nf","3NF",[15,123534,123535],{},"在 2NF 基础上，非主属性不能传递依赖于候选码。",[15,123537,123382],{},[24,123539,123542],{"className":123540,"code":123541,"language":2638,"meta":29},[2636],"Student(Sno, Sname, DeptNo, DeptName)\nSno -> DeptNo\nDeptNo -> DeptName\n所以 Sno -> DeptName 是传递依赖\n",[31,123543,123541],{"__ignoreMap":29},[15,123545,123546],{},"应拆为：",[24,123548,123551],{"className":123549,"code":123550,"language":2638,"meta":29},[2636],"Student(Sno, Sname, DeptNo)\nDept(DeptNo, DeptName)\n",[31,123552,123550],{"__ignoreMap":29},[253,123554,105369],{"id":123555},"bcnf",[15,123557,123558,123559,123561],{},"对每个非平凡函数依赖 ",[31,123560,123464],{},"，X 都必须是超码。BCNF 比 3NF 更严格。",[15,123563,123564],{},"全码关系模式通常可达到 BCNF，因为所有属性都是主属性，常见异常较少；但严格判断仍应看所有函数依赖左部是否为超码。",[19,123566,123568],{"id":123567},"六操作异常","六、操作异常",[15,123570,123571],{},"不合理关系模式常见异常：",[105,123573,123574,123577,123580,123583],{},[108,123575,123576],{},"插入异常：应该插入的数据因缺少主码一部分而插不进去。",[108,123578,123579],{},"删除异常：删除某条记录时，把不该删除的信息也删掉。",[108,123581,123582],{},"更新异常：同一事实重复存储，修改时必须多处更新，否则不一致。",[108,123584,123585],{},"数据冗余：相同信息反复存储。",[15,123587,123382,123588,347],{},[31,123589,123590],{},"R(读者编号, 姓名, 单位编号, 单位名称, 图书号, 书名, 借阅日期, 还书日期)",[15,123592,123593],{},"若某单位只有一个读者，且删除该读者最后一条借阅记录，则单位名称也随记录消失，这就是删除异常。",[19,123595,123597],{"id":123596},"七3nf-分解通用流程","七、3NF 分解通用流程",[2230,123599,123600,123603,123606,123609,123612,123615,123618],{},[108,123601,123602],{},"写全集 U。",[108,123604,123605],{},"写基本函数依赖集 F。",[108,123607,123608],{},"求候选码。",[108,123610,123611],{},"判断最高范式。",[108,123613,123614],{},"按依赖拆表。",[108,123616,123617],{},"每张表标主码、外码。",[108,123619,123620],{},"说明分解后消除了什么依赖和异常。",[19,123622,123624],{"id":123623},"八典型设计题","八、典型设计题",[253,123626,123627],{"id":123627},"高校教学管理系统",[15,123629,123630],{},"业务：",[105,123632,123633,123636,123639,123642,123645,123648],{},[108,123634,123635],{},"学院 1:n 专业。",[108,123637,123638],{},"学院 1:n 课程。",[108,123640,123641],{},"学院 1:n 教师。",[108,123643,123644],{},"专业 1:n 学生。",[108,123646,123647],{},"学生 m:n 课程，联系属性：成绩。",[108,123649,123650],{},"教师 m:n 课程，联系属性：教室、课时。",[15,123652,104008],{},[24,123654,123656],{"className":123655,"code":104012,"language":2638,"meta":29},[2636],[31,123657,104012],{"__ignoreMap":29},[15,123659,123660],{},"主外码：",[24,123662,123664],{"className":123663,"code":104021,"language":2638,"meta":29},[2636],[31,123665,104021],{"__ignoreMap":29},[15,123667,123668],{},"范式说明：各表属性原子；非主属性完全依赖主码；无非主属性对主码的传递依赖，一般可说明达到 3NF。若进一步假定编号唯一决定其余属性，且无其他非平凡依赖，则达到 BCNF。",[253,123670,123671],{"id":123671},"读者借阅图书",[15,123673,123674],{},"原关系：",[24,123676,123679],{"className":123677,"code":123678,"language":2638,"meta":29},[2636],"R(读者编号, 姓名, 单位编号, 单位名称, 图书号, 书名, 借阅日期, 还书日期)\n",[31,123680,123678],{"__ignoreMap":29},[15,123682,97719],{},[105,123684,123685,123688,123691,123694],{},[108,123686,123687],{},"每个读者属于一个单位。",[108,123689,123690],{},"每个读者可借多本书，每本书可被多名读者借。",[108,123692,123693],{},"每个读者可对同一本书多次借阅。",[108,123695,123696],{},"每个读者每天每本书最多借一次。",[15,123698,98255],{},[24,123700,123703],{"className":123701,"code":123702,"language":2638,"meta":29},[2636],"读者编号 -> 姓名, 单位编号\n单位编号 -> 单位名称\n图书号 -> 书名\n(读者编号, 图书号, 借阅日期) -> 还书日期\n",[31,123704,123702],{"__ignoreMap":29},[15,123706,93304],{},[24,123708,123711],{"className":123709,"code":123710,"language":2638,"meta":29},[2636],"(读者编号, 图书号, 借阅日期)\n",[31,123712,123710],{"__ignoreMap":29},[15,123714,93547],{},[24,123716,123719],{"className":123717,"code":123718,"language":2638,"meta":29},[2636],"Reader(读者编号, 姓名, 单位编号)\nUnit(单位编号, 单位名称)\nBook(图书号, 书名)\nBorrow(读者编号, 图书号, 借阅日期, 还书日期)\n",[31,123720,123718],{"__ignoreMap":29},[15,123722,123660],{},[24,123724,123727],{"className":123725,"code":123726,"language":2638,"meta":29},[2636],"Reader: PK(读者编号), FK(单位编号) -> Unit\nUnit: PK(单位编号)\nBook: PK(图书号)\nBorrow: PK(读者编号, 图书号, 借阅日期),\n FK(读者编号) -> Reader,\n FK(图书号) -> Book\n",[31,123728,123726],{"__ignoreMap":29},[253,123730,123731],{"id":123731},"商品-销售数据库",[15,123733,123734],{},"实体：",[24,123736,123739],{"className":123737,"code":123738,"language":2638,"meta":29},[2636],"Employee(员工)\nSalesGroup(销售组)\nProduct(商品)\nSupplier(供应者)\nSaleRecord(销售记录)\n",[31,123740,123738],{"__ignoreMap":29},[15,123742,97719],{},[105,123744,123745,123748,123751,123754,123757],{},[108,123746,123747],{},"员工只在一个销售组。",[108,123749,123750],{},"销售组可销售多种商品。",[108,123752,123753],{},"一种商品只能由一个组销售。",[108,123755,123756],{},"一种商品每天有一个销售量。",[108,123758,123759],{},"供应者和商品是多对多。",[15,123761,120100],{},[24,123763,123765],{"className":123764,"code":103974,"language":2638,"meta":29},[2636],[31,123766,103974],{"__ignoreMap":29},[15,123768,123769],{},"依赖：",[24,123771,123774],{"className":123772,"code":123773,"language":2638,"meta":29},[2636],"GroupID -> GroupName, Manager\nEmpID -> EmpName, Wage, GroupID\nPID -> PName, Price, GroupID\n(PID, SaleDate) -> SaleQty\nSupplierID -> SupplierName, SupplierAddress\n",[31,123775,123773],{"__ignoreMap":29},[253,123777,123779],{"id":123778},"互联网教育项目","“互联网+教育”项目",[15,123781,123782],{},"可设计实体：",[24,123784,123787],{"className":123785,"code":123786,"language":2638,"meta":29},[2636],"Teacher(TeacherID, TeacherName)\nCourse(CourseID, CourseName)\nMember(MemberID, MemberName, Balance)\nTeach(TeacherID, CourseID)\nEnroll(MemberID, CourseID)\nLearningRecord(MemberID, CourseID, StartTime, EndTime, Cost)\nRecharge(MemberID, RechargeTime, Amount)\nTeacherIncome(TeacherID, CourseID, IncomeDate, Amount)\nCompanyIncome(IncomeID, Source, Amount, IncomeDate)\n",[31,123788,123786],{"__ignoreMap":29},[15,123790,90843],{},[105,123792,123793,123799,123806],{},[108,123794,123795,123796,347],{},"教师与课程多对多，用 ",[31,123797,123798],{},"Teach",[108,123800,123801,123802,123805],{},"会员与课程多对多，用 ",[31,123803,123804],{},"Enroll"," 或学习记录。",[108,123807,123808],{},"充值、学习消费、教师分成、公司收益都属于独立业务事件，最好单独成表，避免把多次事件塞成重复列。",[19,123810,123812],{"id":123811},"九答题时画图可以用文字结构代替吗","九、答题时画图可以用文字结构代替吗",[15,123814,123815],{},"正式试卷要求画 E-R 图时，最好画矩形、菱形、椭圆，并标 1:n\u002Fm:n。若时间紧，可以用清晰的文本 E-R 辅助：",[24,123817,123820],{"className":123818,"code":123819,"language":2638,"meta":29},[2636],"学院 1 --- n 专业\n专业 1 --- n 学生\n学生 m --- n 课程 [成绩]\n教师 m --- n 课程 [教室, 课时]\n学院 1 --- n 教师\n学院 1 --- n 课程\n",[31,123821,123819],{"__ignoreMap":29},[15,123823,123824],{},"再写“矩形为实体，方括号为联系属性”，通常能保住主要分数。",{"title":29,"searchDepth":43,"depth":43,"links":123826},[123827,123828,123834,123841,123842,123848,123849,123850,123856],{"id":123296,"depth":43,"text":123297},{"id":123326,"depth":43,"text":123327,"children":123829},[123830,123831,123832,123833],{"id":123330,"depth":49,"text":123330},{"id":123339,"depth":49,"text":123339},{"id":123359,"depth":49,"text":123359},{"id":123376,"depth":49,"text":123376},{"id":123396,"depth":43,"text":123397,"children":123835},[123836,123837,123838,123839,123840],{"id":123400,"depth":49,"text":123400},{"id":123412,"depth":49,"text":123413},{"id":123419,"depth":49,"text":123420},{"id":123432,"depth":49,"text":123433},{"id":123445,"depth":49,"text":123445},{"id":123457,"depth":43,"text":123458},{"id":123494,"depth":43,"text":123495,"children":123843},[123844,123845,123846,123847],{"id":123498,"depth":49,"text":110422},{"id":123504,"depth":49,"text":110426},{"id":123531,"depth":49,"text":123532},{"id":123555,"depth":49,"text":105369},{"id":123567,"depth":43,"text":123568},{"id":123596,"depth":43,"text":123597},{"id":123623,"depth":43,"text":123624,"children":123851},[123852,123853,123854,123855],{"id":123627,"depth":49,"text":123627},{"id":123671,"depth":49,"text":123671},{"id":123731,"depth":49,"text":123731},{"id":123778,"depth":49,"text":123779},{"id":123811,"depth":43,"text":123812},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F04-er",{"title":123285,"description":123293},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F04-数据库设计ER与规范化","4txUCZaNZcRxAcczW1EMNQ04srOxLoCw5UaLibMqKH8",{"id":123863,"title":123864,"body":123865,"description":123872,"extension":85,"meta":124276,"navigation":52,"path":124277,"seo":124278,"stem":124279,"__hash__":124280},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F05-查询优化专项.md","05 查询优化专项：关系代数、查询树、优化树",{"type":8,"value":123866,"toc":124261},[123867,123870,123873,123877,123880,123891,123894,123911,123914,123918,123921,123944,123948,123951,123981,123985,123988,123994,123998,124004,124007,124011,124014,124020,124023,124029,124032,124038,124041,124055,124059,124064,124068,124074,124078,124081,124102,124105,124111,124114,124120,124123,124129,124132,124138,124141,124145,124148,124154,124157,124163,124166,124172,124175,124181,124184,124190,124193,124196,124202,124205,124208,124214,124218,124249,124253,124259],[11,123868,123864],{"id":123869},"_05-查询优化专项关系代数查询树优化树",[15,123871,123872],{},"查询优化题经常让人慌，因为题目会说“画语法树”“画优化树”“给优化过程”。其实套路非常固定。",[19,123874,123876],{"id":123875},"一考试到底要画几棵树","一、考试到底要画几棵树",[15,123878,123879],{},"看题目措辞：",[105,123881,123882,123885,123888],{},[108,123883,123884],{},"“画出原始语法树及优化后的语法树”：画 2 棵树，原始树 + 优化树。",[108,123886,123887],{},"“写关系代数表达式，画语法树，并给优化后的表达式”：至少画 1 棵原始树；为了得满分，最好再画 1 棵优化树。",[108,123889,123890],{},"“Draw a syntax tree ... and give a procedure for query optimization”：画原始树，然后列优化步骤；若时间允许补优化树。",[15,123892,123893],{},"推荐考场固定输出：",[2230,123895,123896,123899,123902,123905,123908],{},[108,123897,123898],{},"SQL 对应的原始关系代数表达式，通常用笛卡尔积。",[108,123900,123901],{},"原始查询树。",[108,123903,123904],{},"优化规则。",[108,123906,123907],{},"优化后的查询树。",[108,123909,123910],{},"优化后的关系代数表达式。",[15,123912,123913],{},"这样无论题目要求 1 棵还是 2 棵树，都能覆盖。",[19,123915,123917],{"id":123916},"二基本优化规则","二、基本优化规则",[15,123919,123920],{},"最核心的规则：",[2230,123922,123923,123929,123932,123935,123938,123941],{},[108,123924,123925,123926],{},"选择条件分解：",[31,123927,123928],{},"σ_{F1 ∧ F2}(E) = σ_{F1}(σ_{F2}(E))",[108,123930,123931],{},"选择下推：只涉及某个关系的选择尽早下推到该关系。",[108,123933,123934],{},"投影下推：只保留最终输出属性、连接属性、选择属性。",[108,123936,123937],{},"笛卡尔积 + 连接条件改写为连接。",[108,123939,123940],{},"先做选择、投影，减少中间结果；避免大笛卡尔积。",[108,123942,123943],{},"连接顺序要让中间结果尽量小。",[19,123945,123947],{"id":123946},"三标准例题","三、标准例题",[15,123949,123950],{},"SQL：",[24,123952,123954],{"className":92584,"code":123953,"language":92586,"meta":29,"style":29},"SELECT p.PNO, PName, Price, StockNum\nFROM Product AS p, Stock AS s\nWHERE p.PNO = s.PNO\n AND StockPlace = 'Warehouse1'\n AND Price > 4;\n",[31,123955,123956,123961,123966,123971,123976],{"__ignoreMap":29},[34,123957,123958],{"class":36,"line":37},[34,123959,123960],{},"SELECT p.PNO, PName, Price, StockNum\n",[34,123962,123963],{"class":36,"line":43},[34,123964,123965],{},"FROM Product AS p, Stock AS s\n",[34,123967,123968],{"class":36,"line":49},[34,123969,123970],{},"WHERE p.PNO = s.PNO\n",[34,123972,123973],{"class":36,"line":56},[34,123974,123975],{}," AND StockPlace = 'Warehouse1'\n",[34,123977,123978],{"class":36,"line":62},[34,123979,123980],{}," AND Price > 4;\n",[253,123982,123984],{"id":123983},"_1-原始关系代数","1. 原始关系代数",[15,123986,123987],{},"题目要求“using Cartesian product equivalent to the query”，所以先写笛卡尔积：",[24,123989,123992],{"className":123990,"code":123991,"language":2638,"meta":29},[2636],"π_{p.PNO, PName, Price, StockNum}\n(\n σ_{p.PNO = s.PNO ∧ StockPlace = 'Warehouse1' ∧ Price > 4}\n (Product × Stock)\n)\n",[31,123993,123991],{"__ignoreMap":29},[253,123995,123997],{"id":123996},"_2-原始查询树","2. 原始查询树",[24,123999,124002],{"className":124000,"code":124001,"language":2638,"meta":29},[2636],"π p.PNO, PName, Price, StockNum\n└── σ p.PNO=s.PNO ∧ StockPlace='Warehouse1' ∧ Price>4\n └── ×\n ├── Product\n └── Stock\n",[31,124003,124001],{"__ignoreMap":29},[15,124005,124006],{},"这棵树的特点：先做笛卡尔积，产生巨大中间结果，再筛选，效率差。",[253,124008,124010],{"id":124009},"_3-优化过程","3. 优化过程",[15,124012,124013],{},"把选择条件拆开：",[24,124015,124018],{"className":124016,"code":124017,"language":2638,"meta":29},[2636],"p.PNO = s.PNO\nStockPlace = 'Warehouse1'\nPrice > 4\n",[31,124019,124017],{"__ignoreMap":29},[15,124021,124022],{},"下推单表选择：",[24,124024,124027],{"className":124025,"code":124026,"language":2638,"meta":29},[2636],"σ_{Price > 4}(Product)\nσ_{StockPlace = 'Warehouse1'}(Stock)\n",[31,124028,124026],{"__ignoreMap":29},[15,124030,124031],{},"把笛卡尔积加连接条件改为连接：",[24,124033,124036],{"className":124034,"code":124035,"language":2638,"meta":29},[2636],"σ_{p.PNO=s.PNO}(Product × Stock)\n=> Product ⋈_{p.PNO=s.PNO} Stock\n",[31,124037,124035],{"__ignoreMap":29},[15,124039,124040],{},"投影下推：",[105,124042,124043,124049],{},[108,124044,124045,124046],{},"Product 侧保留：",[31,124047,124048],{},"PNO, PName, Price",[108,124050,124051,124052],{},"Stock 侧保留：",[31,124053,124054],{},"PNO, StockNum",[253,124056,124058],{"id":124057},"_4-优化后的查询树","4. 优化后的查询树",[24,124060,124062],{"className":124061,"code":104392,"language":2638,"meta":29},[2636],[31,124063,104392],{"__ignoreMap":29},[253,124065,124067],{"id":124066},"_5-优化后的关系代数","5. 优化后的关系代数",[24,124069,124072],{"className":124070,"code":124071,"language":2638,"meta":29},[2636],"π_{PNO, PName, Price, StockNum}\n(\n π_{PNO, PName, Price}(σ_{Price > 4}(Product))\n ⋈_{Product.PNO = Stock.PNO}\n π_{PNO, StockNum}(σ_{StockPlace = 'Warehouse1'}(Stock))\n)\n",[31,124073,124071],{"__ignoreMap":29},[19,124075,124077],{"id":124076},"四画树规范","四、画树规范",[15,124079,124080],{},"树从上到下：",[105,124082,124083,124088,124099],{},[108,124084,124085,124086,347],{},"根节点一般是最终投影 ",[31,124087,58442],{},[108,124089,124090,124091,124093,124094,124096,124097,347],{},"中间是选择 ",[31,124092,115372],{},"、连接 ",[31,124095,116918],{},"、笛卡尔积 ",[31,124098,42071],{},[108,124100,124101],{},"叶子是基本表。",[15,124103,124104],{},"原始树常见结构：",[24,124106,124109],{"className":124107,"code":124108,"language":2638,"meta":29},[2636],"π\n└── σ\n └── ×\n ├── R\n └── S\n",[31,124110,124108],{"__ignoreMap":29},[15,124112,124113],{},"优化树常见结构：",[24,124115,124118],{"className":124116,"code":124117,"language":2638,"meta":29},[2636],"π\n└── ⋈\n ├── π\n │ └── σ\n │ └── R\n └── π\n └── σ\n └── S\n",[31,124119,124117],{"__ignoreMap":29},[15,124121,124122],{},"如果有三个表，原始树：",[24,124124,124127],{"className":124125,"code":124126,"language":2638,"meta":29},[2636],"π\n└── σ\n └── ×\n ├── ×\n │ ├── R\n │ └── S\n └── T\n",[31,124128,124126],{"__ignoreMap":29},[15,124130,124131],{},"优化树：",[24,124133,124136],{"className":124134,"code":124135,"language":2638,"meta":29},[2636],"π\n└── ⋈\n ├── ⋈\n │ ├── σ\u002Fπ R\n │ └── σ\u002Fπ S\n └── σ\u002Fπ T\n",[31,124137,124135],{"__ignoreMap":29},[15,124139,124140],{},"连接顺序按条件和选择率决定；考试没给统计信息时，优先把选择条件最强的关系先过滤，再连接。",[19,124142,124144],{"id":124143},"五关系代数等价变换常用公式","五、关系代数等价变换常用公式",[15,124146,124147],{},"交换律：",[24,124149,124152],{"className":124150,"code":124151,"language":2638,"meta":29},[2636],"R × S = S × R\nR ⋈ S = S ⋈ R\n",[31,124153,124151],{"__ignoreMap":29},[15,124155,124156],{},"结合律：",[24,124158,124161],{"className":124159,"code":124160,"language":2638,"meta":29},[2636],"(R × S) × T = R × (S × T)\n(R ⋈ S) ⋈ T = R ⋈ (S ⋈ T)\n",[31,124162,124160],{"__ignoreMap":29},[15,124164,124165],{},"选择串接：",[24,124167,124170],{"className":124168,"code":124169,"language":2638,"meta":29},[2636],"σ_{F1 ∧ F2}(E) = σ_{F1}(σ_{F2}(E))\n",[31,124171,124169],{"__ignoreMap":29},[15,124173,124174],{},"投影串接：",[24,124176,124179],{"className":124177,"code":124178,"language":2638,"meta":29},[2636],"π_A(π_{A,B}(E)) = π_A(E)\n",[31,124180,124178],{"__ignoreMap":29},[15,124182,124183],{},"选择与投影交换：",[24,124185,124188],{"className":124186,"code":124187,"language":2638,"meta":29},[2636],"π_A(σ_F(E)) = π_A(σ_F(π_{A ∪ attrs(F)}(E)))\n",[31,124189,124187],{"__ignoreMap":29},[15,124191,124192],{},"意思是：投影下推时不能把选择条件需要的属性提前删掉。",[15,124194,124195],{},"选择与笛卡尔积交换：",[24,124197,124200],{"className":124198,"code":124199,"language":2638,"meta":29},[2636],"σ_F(R × S) = σ_F(R) × S\n",[31,124201,124199],{"__ignoreMap":29},[15,124203,124204],{},"前提是 F 只涉及 R 的属性；若只涉及 S 就下推到 S。",[15,124206,124207],{},"连接改写：",[24,124209,124212],{"className":124210,"code":124211,"language":2638,"meta":29},[2636],"σ_{R.a=S.a}(R × S) = R ⋈_{R.a=S.a} S\n",[31,124213,124211],{"__ignoreMap":29},[19,124215,124217],{"id":124216},"六常见扣分点","六、常见扣分点",[105,124219,124220,124223,124226,124233,124236,124246],{},[108,124221,124222],{},"一开始就写连接表达式，题目明明要求“using Cartesian product”。应先写笛卡尔积原始表达式，再优化为连接。",[108,124224,124225],{},"原始树和优化树画成一样。",[108,124227,124228,124229,124232],{},"投影下推时删掉连接属性，如 Stock 侧删掉 ",[31,124230,124231],{},"PNO","，导致后面无法连接。",[108,124234,124235],{},"选择条件没有拆分，无法下推。",[108,124237,2892,124238,124241,124242,124245],{},[31,124239,124240],{},"Price > 4"," 下推到 Stock，把 ",[31,124243,124244],{},"StockPlace"," 下推到 Product，表搞反。",[108,124247,124248],{},"优化表达式最终输出列与 SQL 不一致。",[19,124250,124252],{"id":124251},"七答题模板","七、答题模板",[24,124254,124257],{"className":124255,"code":124256,"language":2638,"meta":29},[2636],"原始关系代数：\nπ_{输出属性}(σ_{连接条件 ∧ 单表条件1 ∧ 单表条件2}(R × S))\n\n原始树：\nπ\n└── σ\n └── ×\n ├── R\n └── S\n\n优化规则：\n1. 分解选择条件。\n2. 将只涉及单个关系的选择下推。\n3. 将笛卡尔积和连接选择转换为连接。\n4. 投影下推，保留输出属性和连接属性。\n\n优化树：\nπ\n└── ⋈\n ├── π\n │ └── σ\n │ └── R\n └── π\n └── σ\n └── S\n\n优化关系代数：\nπ_{输出属性}((π(...σ...(R))) ⋈ 条件 (π(...σ...(S))))\n",[31,124258,124256],{"__ignoreMap":29},[79,124260,81],{},{"title":29,"searchDepth":43,"depth":43,"links":124262},[124263,124264,124265,124272,124273,124274,124275],{"id":123875,"depth":43,"text":123876},{"id":123916,"depth":43,"text":123917},{"id":123946,"depth":43,"text":123947,"children":124266},[124267,124268,124269,124270,124271],{"id":123983,"depth":49,"text":123984},{"id":123996,"depth":49,"text":123997},{"id":124009,"depth":49,"text":124010},{"id":124057,"depth":49,"text":124058},{"id":124066,"depth":49,"text":124067},{"id":124076,"depth":43,"text":124077},{"id":124143,"depth":43,"text":124144},{"id":124216,"depth":43,"text":124217},{"id":124251,"depth":43,"text":124252},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F05",{"title":123864,"description":123872},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F05-查询优化专项","TZh4l_7YYYqfOflI8BC6ZUgpMEkUgLqRJtf4y_Yk_Ok",{"id":124282,"title":124283,"body":124284,"description":124291,"extension":85,"meta":124835,"navigation":52,"path":124836,"seo":124837,"stem":124838,"__hash__":124839},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F06-事务并发控制与恢复.md","06 事务、并发控制与数据库恢复",{"type":8,"value":124285,"toc":124821},[124286,124289,124292,124296,124299,124302,124316,124319,124330,124334,124337,124351,124354,124358,124361,124369,124372,124378,124381,124385,124388,124396,124399,124407,124410,124418,124421,124429,124431,124439,124443,124445,124451,124454,124459,124462,124465,124479,124482,124486,124489,124503,124506,124510,124513,124521,124524,124538,124541,124555,124558,124562,124565,124568,124571,124582,124585,124593,124596,124604,124608,124611,124614,124625,124628,124681,124684,124690,124693,124704,124708,124714,124717,124721,124726,124731,124734,124812,124816,124819],[11,124287,124283],{"id":124288},"_06-事务并发控制与数据库恢复",[15,124290,124291],{},"这一章对应选择填空、简答、检查点恢复大题、封锁协议大题。",[19,124293,124295],{"id":124294},"一事务与-acid","一、事务与 ACID",[15,124297,124298],{},"事务是用户定义的一个数据库操作序列，是 DBMS 的基本工作单位。事务要么全部成功，要么全部失败。",[15,124300,124301],{},"ACID：",[105,124303,124304,124307,124310,124313],{},[108,124305,124306],{},"原子性 Atomicity：事务中所有操作要么都做，要么都不做。",[108,124308,124309],{},"一致性 Consistency：事务使数据库从一个一致状态转变到另一个一致状态。",[108,124311,124312],{},"隔离性 Isolation：并发事务之间相互隔离，不应互相干扰。",[108,124314,124315],{},"持久性 Durability：事务一旦提交，对数据库的改变就是永久的。",[15,124317,124318],{},"实现子系统：",[105,124320,124321,124324,124327],{},[108,124322,124323],{},"原子性、持久性主要由恢复子系统保证。",[108,124325,124326],{},"隔离性主要由并发控制子系统保证。",[108,124328,124329],{},"一致性由完整性约束、并发控制和恢复共同保证。",[19,124331,124333],{"id":124332},"二并发操作异常","二、并发操作异常",[15,124335,124336],{},"不加控制的并发可能产生：",[2230,124338,124339,124342,124345,124348],{},[108,124340,124341],{},"丢失修改：两个事务都修改同一数据，后写覆盖前写。",[108,124343,124344],{},"读脏数据：一个事务读取了另一个事务未提交且后来回滚的数据。",[108,124346,124347],{},"不可重复读：同一事务两次读同一数据，中间被其他事务修改并提交，结果不同。",[108,124349,124350],{},"幻读：同一事务按条件两次查询，第二次出现了其他事务新插入的满足条件的行。",[15,124352,124353],{},"“脏数据”严格说是未提交后又被撤销的数据。",[19,124355,124357],{"id":124356},"三锁","三、锁",[15,124359,124360],{},"基本封锁类型：",[105,124362,124363,124366],{},[108,124364,124365],{},"S 锁，共享锁，读锁。事务对数据加 S 锁后，可以读，不能写；其他事务可再加 S 锁，不能加 X 锁。",[108,124367,124368],{},"X 锁，排他锁，写锁。事务对数据加 X 锁后，可以读写；其他事务不能再加 S 锁或 X 锁。",[15,124370,124371],{},"兼容矩阵：",[24,124373,124376],{"className":124374,"code":124375,"language":2638,"meta":29},[2636]," 已有 S 已有 X\n请求 S 兼容 不兼容\n请求 X 不兼容 不兼容\n",[31,124377,124375],{"__ignoreMap":29},[15,124379,124380],{},"如果事务 T 获得数据项 Q 上的 X 锁，则 T 对 Q 既可读又可写；其他事务既不能读也不能写。",[19,124382,124384],{"id":124383},"四封锁协议","四、封锁协议",[15,124386,124387],{},"一级封锁协议：",[105,124389,124390,124393],{},[108,124391,124392],{},"修改数据前必须加 X 锁，事务结束释放。",[108,124394,124395],{},"可防止丢失修改。",[15,124397,124398],{},"二级封锁协议：",[105,124400,124401,124404],{},[108,124402,124403],{},"在一级基础上，读数据前必须加 S 锁，读完即可释放。",[108,124405,124406],{},"可防止读脏数据。",[15,124408,124409],{},"三级封锁协议：",[105,124411,124412,124415],{},[108,124413,124414],{},"在一级基础上，读数据前必须加 S 锁，并保持到事务结束。",[108,124416,124417],{},"可防止不可重复读。",[15,124419,124420],{},"两阶段封锁协议 2PL：",[105,124422,124423,124426],{},[108,124424,124425],{},"加锁阶段：可以申请锁，不能释放锁。",[108,124427,124428],{},"解锁阶段：一旦释放任何锁，就不能再申请新锁。",[15,124430,62532],{},[105,124432,124433,124436],{},[108,124434,124435],{},"遵守两阶段封锁协议可保证冲突可串行化。",[108,124437,124438],{},"两阶段封锁协议不能保证不死锁，反而可能死锁。",[19,124440,124442],{"id":124441},"五两阶段封锁例题","五、两阶段封锁例题",[15,124444,101191],{},[24,124446,124449],{"className":124447,"code":124448,"language":2638,"meta":29},[2636],"T1:\nread(A);\nread(B);\nif A = 0 then B := B + 1;\nwrite(B);\n\nT2:\nread(B);\nread(A);\nif B = 0 then A := A + 1;\nwrite(A);\n",[31,124450,124448],{"__ignoreMap":29},[15,124452,124453],{},"一种满足 2PL 的写法：",[24,124455,124457],{"className":124456,"code":104487,"language":2638,"meta":29},[2636],[31,124458,104487],{"__ignoreMap":29},[15,124460,124461],{},"是否可能死锁：可能。若 T1 持有 A 的 S 锁等待 B 的 X 锁，T2 持有 B 的 S 锁等待 A 的 X 锁，则互相等待。",[15,124463,124464],{},"避免死锁的常用方法：",[105,124466,124467,124470,124473,124476],{},[108,124468,124469],{},"一次封锁法：事务开始前一次申请所有锁。",[108,124471,124472],{},"顺序封锁法：所有事务按统一顺序申请锁。",[108,124474,124475],{},"死锁检测与解除：构造等待图，发现环后撤销某事务。",[108,124477,124478],{},"超时法：等待超过阈值就回滚。",[15,124480,124481],{},"避免活锁：先来先服务。",[19,124483,124485],{"id":124484},"六数据库故障类型","六、数据库故障类型",[15,124487,124488],{},"常见四类故障：",[2230,124490,124491,124494,124497,124500],{},[108,124492,124493],{},"事务故障：事务内部错误、违反约束、死锁被撤销等。",[108,124495,124496],{},"系统故障：断电、操作系统崩溃，内存丢失，外存数据库未物理损坏。",[108,124498,124499],{},"介质故障：磁盘损坏，外存数据部分或全部丢失。",[108,124501,124502],{},"计算机病毒或人为破坏等。",[15,124504,124505],{},"突然停电通常归为系统故障，不是介质故障。",[19,124507,124509],{"id":124508},"七恢复的基本原理","七、恢复的基本原理",[15,124511,124512],{},"恢复的核心是冗余。最常用冗余技术：",[105,124514,124515,124518],{},[108,124516,124517],{},"数据转储：制作数据库后援副本。",[108,124519,124520],{},"登记日志文件：记录事务对数据库的更新。",[15,124522,124523],{},"转储类型：",[105,124525,124526,124529,124532,124535],{},[108,124527,124528],{},"静态转储：转储期间不允许事务运行。",[108,124530,124531],{},"动态转储：转储期间允许事务运行，必须配合日志。",[108,124533,124534],{},"海量转储：每次转储全部数据库。",[108,124536,124537],{},"增量转储：只转储上次转储后变化的数据。",[15,124539,124540],{},"日志记录通常包括：",[105,124542,124543,124546,124549,124552],{},[108,124544,124545],{},"事务开始记录。",[108,124547,124548],{},"更新记录：事务标识、数据项、旧值、新值。",[108,124550,124551],{},"提交记录。",[108,124553,124554],{},"回滚记录。",[15,124556,124557],{},"日志原则：先写日志，再写数据库。即 WAL。",[19,124559,124561],{"id":124560},"八undo-与-redo","八、UNDO 与 REDO",[15,124563,124564],{},"UNDO：撤销未提交事务的影响，用旧值恢复。",[15,124566,124567],{},"REDO：重做已提交事务的影响，用新值重做。",[15,124569,124570],{},"判断规则：",[105,124572,124573,124576,124579],{},[108,124574,124575],{},"故障时未提交的事务：UNDO。",[108,124577,124578],{},"故障前已经提交但其结果可能尚未写入数据库的事务：REDO。",[108,124580,124581],{},"检查点前已提交且已写入稳定数据库的事务：通常不用 REDO。",[15,124583,124584],{},"系统故障恢复通常：",[2230,124586,124587,124590],{},[108,124588,124589],{},"UNDO 未完成事务。",[108,124591,124592],{},"REDO 已提交但可能未落盘事务。",[15,124594,124595],{},"介质故障恢复：",[2230,124597,124598,124601],{},[108,124599,124600],{},"装入最近的后援副本。",[108,124602,124603],{},"利用日志 REDO 自副本以来已提交事务。",[19,124605,124607],{"id":124606},"九检查点恢复","九、检查点恢复",[15,124609,124610],{},"检查点的作用：减少恢复时需要扫描的日志范围。",[15,124612,124613],{},"检查点记录通常包含：",[105,124615,124616,124622],{},[108,124617,124618,124619,347],{},"检查点时刻正在执行的事务列表 ",[31,124620,124621],{},"ACTIVE-LIST",[108,124623,124624],{},"这些事务最近日志记录地址等信息。",[15,124626,124627],{},"基于检查点恢复的一般算法：",[2230,124629,124630,124633,124640,124666],{},[108,124631,124632],{},"从最后一个检查点开始。",[108,124634,124635,124636,124639],{},"把检查点时正在运行的事务放入 ",[31,124637,124638],{},"UNDO-LIST"," 或活动集合。",[108,124641,124642,124643],{},"向后扫描日志到故障点：\n",[105,124644,124645,124654],{},[108,124646,124647,124648,124651,124652,347],{},"遇到 ",[31,124649,124650],{},"START T","：把 T 加入 ",[31,124653,124638],{},[108,124655,124647,124656,124659,124660,124662,124663,347],{},[31,124657,124658],{},"COMMIT T","：把 T 从 ",[31,124661,124638],{}," 移出，加入 ",[31,124664,124665],{},"REDO-LIST",[108,124667,124668,124669],{},"扫描结束后：\n",[105,124670,124671,124676],{},[108,124672,124673,124675],{},[31,124674,124638],{}," 中的是故障时未提交事务，要 UNDO。",[108,124677,124678,124680],{},[31,124679,124665],{}," 中的是故障前已提交事务，要 REDO。",[15,124682,124683],{},"考试图形判断：",[24,124685,124688],{"className":124686,"code":124687,"language":2638,"meta":29},[2636],"时间轴： 检查点 CP ---------------- 故障 CRASH\n\nT1: ----commit---- before CP 不处理\nT2: ----start---- CP ----commit---- REDO\nT3: ----start---- CP --------crash UNDO\nT4: CP ----start--commit REDO\nT5: CP ----start----crash UNDO\n",[31,124689,124687],{"__ignoreMap":29},[15,124691,124692],{},"一句话：",[105,124694,124695,124698,124701],{},[108,124696,124697],{},"故障时没结束：UNDO。",[108,124699,124700],{},"故障前已提交且可能受检查点影响：REDO。",[108,124702,124703],{},"检查点前已提交：一般不用管。",[19,124705,124707],{"id":124706},"十检查点题答题模板","十、检查点题答题模板",[24,124709,124712],{"className":124710,"code":124711,"language":2638,"meta":29},[2636],"1. 先找检查点时正在运行的事务，写 ACTIVE-LIST = {...}。\n2. 从检查点向故障点扫描：\n 遇到开始但未提交，加入 UNDO-LIST；\n 遇到提交，移出 UNDO-LIST，加入 REDO-LIST。\n3. 故障点处仍未提交的事务执行 UNDO。\n4. 故障点前已经提交但在检查点之后相关的事务执行 REDO。\n",[31,124713,124711],{"__ignoreMap":29},[15,124715,124716],{},"如果图上事务线看不清，也要把规则写清楚，再根据可辨认编号填集合。",[19,124718,124720],{"id":124719},"十一commit-与-rollback","十一、COMMIT 与 ROLLBACK",[15,124722,124723,124725],{},[31,124724,93970],{},"：提交事务，使事务对数据库的更新永久生效。",[15,124727,124728,124730],{},[31,124729,91516],{},"：回滚事务，撤销事务已经做过但未提交的更新。",[15,124732,124733],{},"SQL Server 事务模板：",[24,124735,124737],{"className":92584,"code":124736,"language":92586,"meta":29,"style":29},"BEGIN TRANSACTION;\n\nUPDATE Account SET Balance = Balance - 300 WHERE Ano = 'A';\nIF @@ERROR \u003C> 0\nBEGIN\n ROLLBACK TRANSACTION;\n RETURN;\nEND;\n\nUPDATE Account SET Balance = Balance + 300 WHERE Ano = 'B';\nIF @@ERROR \u003C> 0\nBEGIN\n ROLLBACK TRANSACTION;\n RETURN;\nEND;\n\nCOMMIT TRANSACTION;\n",[31,124738,124739,124743,124747,124752,124757,124761,124766,124771,124775,124779,124784,124788,124792,124796,124800,124804,124808],{"__ignoreMap":29},[34,124740,124741],{"class":36,"line":37},[34,124742,94001],{},[34,124744,124745],{"class":36,"line":43},[34,124746,53],{"emptyLinePlaceholder":52},[34,124748,124749],{"class":36,"line":49},[34,124750,124751],{},"UPDATE Account SET Balance = Balance - 300 WHERE Ano = 'A';\n",[34,124753,124754],{"class":36,"line":56},[34,124755,124756],{},"IF @@ERROR \u003C> 0\n",[34,124758,124759],{"class":36,"line":62},[34,124760,102603],{},[34,124762,124763],{"class":36,"line":68},[34,124764,124765],{}," ROLLBACK TRANSACTION;\n",[34,124767,124768],{"class":36,"line":74},[34,124769,124770],{}," RETURN;\n",[34,124772,124773],{"class":36,"line":1971},[34,124774,102656],{},[34,124776,124777],{"class":36,"line":1976},[34,124778,53],{"emptyLinePlaceholder":52},[34,124780,124781],{"class":36,"line":1985},[34,124782,124783],{},"UPDATE Account SET Balance = Balance + 300 WHERE Ano = 'B';\n",[34,124785,124786],{"class":36,"line":1990},[34,124787,124756],{},[34,124789,124790],{"class":36,"line":1999},[34,124791,102603],{},[34,124793,124794],{"class":36,"line":2022},[34,124795,124765],{},[34,124797,124798],{"class":36,"line":2036},[34,124799,124770],{},[34,124801,124802],{"class":36,"line":2603},[34,124803,102656],{},[34,124805,124806],{"class":36,"line":5537},[34,124807,53],{"emptyLinePlaceholder":52},[34,124809,124810],{"class":36,"line":5543},[34,124811,94029],{},[19,124813,124815],{"id":124814},"十二并发控制与恢复的关系","十二、并发控制与恢复的关系",[15,124817,124818],{},"并发控制保证事务隔离执行，防止并发造成不一致；恢复保证故障后数据库回到一致状态。两者都服务于事务正确性。并发控制处理“同时运行”的冲突，恢复处理“失败之后”的修复。",[79,124820,81],{},{"title":29,"searchDepth":43,"depth":43,"links":124822},[124823,124824,124825,124826,124827,124828,124829,124830,124831,124832,124833,124834],{"id":124294,"depth":43,"text":124295},{"id":124332,"depth":43,"text":124333},{"id":124356,"depth":43,"text":124357},{"id":124383,"depth":43,"text":124384},{"id":124441,"depth":43,"text":124442},{"id":124484,"depth":43,"text":124485},{"id":124508,"depth":43,"text":124509},{"id":124560,"depth":43,"text":124561},{"id":124606,"depth":43,"text":124607},{"id":124706,"depth":43,"text":124707},{"id":124719,"depth":43,"text":124720},{"id":124814,"depth":43,"text":124815},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F06",{"title":124283,"description":124291},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F06-事务并发控制与恢复","jirbXB_fcfKjFSBr62EmSdyl4EX2sj1Psm_elTvbgxM",{"id":124841,"title":124842,"body":124843,"description":124850,"extension":85,"meta":125404,"navigation":52,"path":125405,"seo":125406,"stem":125407,"__hash__":125408},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F07-填空选择速记清单.md","07 填空选择速记清单",{"type":8,"value":124844,"toc":125386},[124845,124848,124851,124855,124880,124884,124898,124902,124905,124925,124928,124932,124951,124955,124975,124979,125007,125011,125029,125033,125056,125060,125083,125087,125179,125183,125203,125207,125239,125243,125280,125284,125322,125326,125351,125355],[11,124846,124842],{"id":124847},"_07-填空选择速记清单",[15,124849,124850],{},"这一章专门背小题。很多内容来自教材固定表述、补充习题和新图判断填空。",[19,124852,124854],{"id":124853},"一数据库基础","一、数据库基础",[105,124856,124857,124860,124862,124865,124868,124871,124874,124877],{},[108,124858,124859],{},"位于用户和操作系统之间的数据管理软件：DBMS。",[108,124861,120010],{},[108,124863,124864],{},"数据库是长期存储在计算机内、有组织、可共享的数据集合。",[108,124866,124867],{},"数据库系统与文件系统的本质区别：数据结构化。",[108,124869,124870],{},"数据库系统特点：数据结构化、共享性高、冗余度低、独立性高、由 DBMS 统一管理。",[108,124872,124873],{},"数据管理技术阶段：人工管理、文件系统、数据库系统。",[108,124875,124876],{},"数据库中存储的是数据以及数据之间的联系。",[108,124878,124879],{},"数据库建设是硬件、软件、干件三方面结合。这里“干件”指人、管理、制度等因素，教材\u002F旧题常这样写。",[19,124881,124883],{"id":124882},"二dbdbmsdbs","二、DB、DBMS、DBS",[105,124885,124886,124889,124892,124895],{},[108,124887,124888],{},"DB：数据库。",[108,124890,124891],{},"DBMS：数据库管理系统。",[108,124893,124894],{},"DBS：数据库系统。",[108,124896,124897],{},"三者关系：DBS 包括 DB 和 DBMS。",[19,124899,124901],{"id":124900},"三dbms-功能","三、DBMS 功能",[15,124903,124904],{},"DBMS 主要功能：",[105,124906,124907,124910,124913,124916,124919,124922],{},[108,124908,124909],{},"数据定义。",[108,124911,124912],{},"数据组织、存储和管理。",[108,124914,124915],{},"数据操纵。",[108,124917,124918],{},"数据库运行管理。",[108,124920,124921],{},"数据库建立和维护。",[108,124923,124924],{},"通信接口等。",[15,124926,124927],{},"若题目问四大功能，常答：数据定义、数据操纵、数据库运行管理、数据库建立和维护。",[19,124929,124931],{"id":124930},"四数据模型","四、数据模型",[105,124933,124934,124936,124939,124942,124945,124948],{},[108,124935,120007],{},[108,124937,124938],{},"层次模型：树。",[108,124940,124941],{},"网状模型：图，CODASYL\u002FDBTG。",[108,124943,124944],{},"关系模型：关系\u002F二维表。",[108,124946,124947],{},"E-R 图用于概念结构设计。",[108,124949,124950],{},"现实世界的事物对应实体；特征对应属性；事物之间的关联对应联系。",[19,124952,124954],{"id":124953},"五三级模式","五、三级模式",[105,124956,124957,124960,124963,124966,124969,124972],{},[108,124958,124959],{},"外模式：单个用户使用的数据视图描述，也叫子模式\u002F用户模式。",[108,124961,124962],{},"模式：全局逻辑结构，也叫概念模式。",[108,124964,124965],{},"内模式：物理存储结构，也叫存储模式。",[108,124967,124968],{},"两级映像：外模式\u002F模式映像，模式\u002F内模式映像。",[108,124970,124971],{},"物理独立性由模式\u002F内模式映像保证。",[108,124973,124974],{},"逻辑独立性由外模式\u002F模式映像保证。",[19,124976,124978],{"id":124977},"六关系模型","六、关系模型",[105,124980,124981,124983,124986,124989,124992,124995,124998,125001,125004],{},[108,124982,120236],{},[108,124984,124985],{},"关系数据库中可命名的最小数据单位：属性。",[108,124987,124988],{},"关系中行、列顺序均可交换。",[108,124990,124991],{},"关系中每个属性必须来自同一域。",[108,124993,124994],{},"关系中不允许多值属性。",[108,124996,124997],{},"1NF：每个分量不可再分。",[108,124999,125000],{},"主码属性不能取空值。",[108,125002,125003],{},"外码在允许时可以取空值。",[108,125005,125006],{},"全码关系模式最高通常可达到 BCNF。",[19,125008,125010],{"id":125009},"七码","七、码",[105,125012,125013,125016,125019,125022,125025,125027],{},[108,125014,125015],{},"超码：能唯一标识元组的属性组。",[108,125017,125018],{},"候选码：不含多余属性的超码。",[108,125020,125021],{},"主码：选定的候选码。",[108,125023,125024],{},"外码：本关系属性引用另一关系主码。",[108,125026,120257],{},[108,125028,120260],{},[19,125030,125032],{"id":125031},"八关系完整性","八、关系完整性",[105,125034,125035,125038,125041,125044,125047],{},[108,125036,125037],{},"实体完整性：主码唯一且非空。",[108,125039,125040],{},"参照完整性：外码要么为空，要么等于被参照关系中某个主码值。",[108,125042,125043],{},"用户自定义完整性：业务规则，如工资范围、年龄范围、性别取值。",[108,125045,125046],{},"“年龄 16-25 岁”属于完整性。",[108,125048,125049,146,125051,146,125053,125055],{},[31,125050,99262],{},[31,125052,99265],{},[31,125054,94262],{}," 属于完整性控制。",[19,125057,125059],{"id":125058},"九安全性","九、安全性",[105,125061,125062,125065,125068,125071,125074,125077,125080],{},[108,125063,125064],{},"数据安全性：防止非法用户非法使用数据库。",[108,125066,125067],{},"授权和撤权属于安全性措施。",[108,125069,125070],{},"常见安全措施：用户标识与鉴别、存取控制、视图、审计、数据加密。",[108,125072,125073],{},"存取权限包括：数据对象、操作类型。",[108,125075,125076],{},"角色：一组权限的集合，可以简化授权工作量。",[108,125078,125079],{},"自主存取控制 DAC：用户可按权限访问对象，授权者可转授。",[108,125081,125082],{},"强制存取控制 MAC：主体和客体按安全级别\u002F密级比较。",[19,125084,125086],{"id":125085},"十sql","十、SQL",[105,125088,125089,125098,125109,125116,125122,125128,125134,125140,125146,125153,125162,125168,125173,125176],{},[108,125090,125091,125092,146,125094,146,125096,347],{},"DDL：",[31,125093,120521],{},[31,125095,120524],{},[31,125097,120527],{},[108,125099,125100,125101,146,125103,146,125105,146,125107,347],{},"DML：",[31,125102,92151],{},[31,125104,120535],{},[31,125106,120538],{},[31,125108,92155],{},[108,125110,125111,125112,146,125114,347],{},"DCL：",[31,125113,92399],{},[31,125115,92403],{},[108,125117,125118,125121],{},[31,125119,125120],{},"DELETE FROM SC","：删除 SC 表中的全部记录，不删除表结构。",[108,125123,125124,125127],{},[31,125125,125126],{},"DROP TABLE SC","：删除基本表。",[108,125129,125130,125133],{},[31,125131,125132],{},"ALTER TABLE Student ADD Email varchar(30)","：增加列。",[108,125135,125136,125137,347],{},"建索引：",[31,125138,125139],{},"CREATE INDEX idx ON T(col)",[108,125141,125142,125143,347],{},"删除索引 SQL Server：",[31,125144,125145],{},"DROP INDEX idx ON T",[108,125147,125148,125149,146,125151,347],{},"空值判断：",[31,125150,100847],{},[31,125152,100850],{},[108,125154,125155,125157,125158,125161],{},[31,125156,119479],{}," 统计包含 NULL 的行；",[31,125159,125160],{},"COUNT(col)"," 不统计 col 为 NULL 的行。",[108,125163,125164,125167],{},[31,125165,125166],{},"> ALL","：大于子查询结果中的所有值。",[108,125169,125170,125172],{},[31,125171,121012],{},"：测试集合是否非空。",[108,125174,125175],{},"视图是虚表，只保存定义，不保存数据。",[108,125177,125178],{},"在视图上不能定义新的基本表。",[19,125180,125182],{"id":125181},"十一关系代数","十一、关系代数",[105,125184,125185,125188,125191,125194,125197,125200],{},[108,125186,125187],{},"基本运算：并、差、笛卡尔积、选择、投影。",[108,125189,125190],{},"选择：选行。",[108,125192,125193],{},"投影：选列并消去重复元组。",[108,125195,125196],{},"自然连接：公共属性等值连接并去掉重复列。",[108,125198,125199],{},"查询优化应尽早做选择和投影。",[108,125201,125202],{},"最容易产生巨大中间结果的是笛卡尔积和连接。",[19,125204,125206],{"id":125205},"十二规范化","十二、规范化",[105,125208,125209,125212,125215,125218,125221,125224,125227,125230,125233],{},[108,125210,125211],{},"规范化目的：减少冗余，消除插入、删除、更新异常，使语义更清晰。",[108,125213,125214],{},"“加快查询速度”不是规范化主要目的。",[108,125216,125217],{},"插入异常：应该插入的数据插不进去。",[108,125219,125220],{},"删除异常：不该删除的数据被连带删除。",[108,125222,125223],{},"更新异常：同一事实多处存储导致修改困难和不一致。",[108,125225,125226],{},"2NF：消除非主属性对码的部分依赖。",[108,125228,125229],{},"3NF：消除非主属性对码的传递依赖。",[108,125231,125232],{},"BCNF：每个非平凡函数依赖左部都是超码。",[108,125234,125235,125236,347],{},"两个函数依赖集 F 和 G 等价的充要条件：",[31,125237,125238],{},"F+ = G+",[19,125240,125242],{"id":125241},"十三事务与并发","十三、事务与并发",[105,125244,125245,125248,125251,125254,125257,125260,125263,125266,125269,125272,125274,125277],{},[108,125246,125247],{},"事务：用户定义的一组逻辑一致的程序序列。",[108,125249,125250],{},"ACID：原子性、一致性、隔离性、持久性。",[108,125252,125253],{},"事务执行次序：调度。",[108,125255,125256],{},"并发调度正确性标准：可串行化。",[108,125258,125259],{},"并发控制主要方法：封锁机制。",[108,125261,125262],{},"基本锁：S 锁、X 锁。",[108,125264,125265],{},"S 锁：可读不可写，其他事务可再加 S 锁。",[108,125267,125268],{},"X 锁：可读可写，其他事务不能读写。",[108,125270,125271],{},"两阶段封锁协议可保证冲突可串行化，但可能死锁。",[108,125273,124481],{},[108,125275,125276],{},"死锁原因：多个事务互相等待对方持有的资源。",[108,125278,125279],{},"死锁解决：预防、检测解除、超时、一次封锁、顺序封锁。",[19,125281,125283],{"id":125282},"十四恢复","十四、恢复",[105,125285,125286,125289,125292,125295,125298,125301,125304,125307,125310,125313,125316,125319],{},[108,125287,125288],{},"恢复：把数据库从错误状态恢复到某个已知正确状态。",[108,125290,125291],{},"恢复基本策略：冗余。",[108,125293,125294],{},"冗余技术：数据转储、登记日志文件。",[108,125296,125297],{},"转储：制作后援副本。",[108,125299,125300],{},"日志文件作用：记录事务更新，用于 UNDO\u002FREDO。",[108,125302,125303],{},"WAL：先写日志，后写数据库。",[108,125305,125306],{},"事务故障恢复：UNDO。",[108,125308,125309],{},"系统故障恢复：UNDO 未提交事务，REDO 已提交但可能未写入数据库的事务。",[108,125311,125312],{},"介质故障恢复：装入后援副本，再 REDO 日志中已提交事务。",[108,125314,125315],{},"检查点作用：减少恢复扫描日志范围。",[108,125317,125318],{},"突然停电：系统故障。",[108,125320,125321],{},"磁盘损坏：介质故障。",[19,125323,125325],{"id":125324},"十五嵌入式-sql-与游标","十五、嵌入式 SQL 与游标",[105,125327,125328,125331,125336,125339,125342,125345],{},[108,125329,125330],{},"嵌入式 SQL 与主语言通信：SQLCA 和主变量。",[108,125332,125333,125334,347],{},"主变量在 SQL 中前面加冒号，如 ",[31,125335,123107],{},[108,125337,125338],{},"SQLCA：SQL 通信区。",[108,125340,125341],{},"游标作用：协调 SQL 集合处理方式与主语言单记录处理方式。",[108,125343,125344],{},"SQL Server 游标步骤：定义、打开、读取\u002F推进、关闭、释放。",[108,125346,125347,125348,347],{},"嵌入式 SQL 更新当前游标行：",[31,125349,125350],{},"WHERE CURRENT OF cursor_name",[19,125352,125354],{"id":125353},"十六触发器","十六、触发器",[105,125356,125357,125360,125367,125375,125378],{},[108,125358,125359],{},"触发器是特殊存储过程。",[108,125361,125362,125363,125366],{},"触发器自动触发，不通过 ",[31,125364,125365],{},"EXECUTE"," 调用。",[108,125368,125369,125370,146,125372,125374],{},"触发器一般不带用户手工传入参数；它通过 ",[31,125371,122801],{},[31,125373,122804],{}," 获取本次变更数据。",[108,125376,125377],{},"触发器必须定义在特定表或视图上。",[108,125379,125380,125381,258,125383,125385],{},"SQL Server 触发器使用 ",[31,125382,122801],{},[31,125384,122804],{}," 两张临时表。",{"title":29,"searchDepth":43,"depth":43,"links":125387},[125388,125389,125390,125391,125392,125393,125394,125395,125396,125397,125398,125399,125400,125401,125402,125403],{"id":124853,"depth":43,"text":124854},{"id":124882,"depth":43,"text":124883},{"id":124900,"depth":43,"text":124901},{"id":124930,"depth":43,"text":124931},{"id":124953,"depth":43,"text":124954},{"id":124977,"depth":43,"text":124978},{"id":125009,"depth":43,"text":125010},{"id":125031,"depth":43,"text":125032},{"id":125058,"depth":43,"text":125059},{"id":125085,"depth":43,"text":125086},{"id":125181,"depth":43,"text":125182},{"id":125205,"depth":43,"text":125206},{"id":125241,"depth":43,"text":125242},{"id":125282,"depth":43,"text":125283},{"id":125324,"depth":43,"text":125325},{"id":125353,"depth":43,"text":125354},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F07",{"title":124842,"description":124850},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F07-填空选择速记清单","8L1Maxxvl8qWQjqvv6O3Ad0UQS1aA-yp7TT4lehfq3U",{"id":125410,"title":125411,"body":125412,"description":125419,"extension":85,"meta":126352,"navigation":52,"path":126353,"seo":126354,"stem":126355,"__hash__":126356},"notes\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F08-大题答题模板.md","08 大题答题模板",{"type":8,"value":125413,"toc":126321},[125414,125417,125420,125424,125427,125483,125486,125521,125524,125548,125552,125588,125592,125646,125650,125654,125721,125724,125789,125792,125854,125858,125940,125944,125947,126018,126022,126098,126102,126164,126167,126194,126198,126201,126207,126211,126217,126220,126226,126230,126236,126240,126246,126249,126260,126264,126270,126272,126278,126282,126285,126291,126295,126301,126304,126310,126313,126319],[11,125415,125411],{"id":125416},"_08-大题答题模板",[15,125418,125419],{},"这一章是考场版。遇到类似题，照模板填表名、属性、条件即可。",[19,125421,125423],{"id":125422},"一sql-大题模板","一、SQL 大题模板",[253,125425,125426],{"id":125426},"建表",[24,125428,125430],{"className":92584,"code":125429,"language":92586,"meta":29,"style":29},"CREATE TABLE 表名 (\n A char(10) NOT NULL,\n B varchar(20),\n C int CHECK (C BETWEEN 下限 AND 上限),\n D char(10) NULL,\n CONSTRAINT PK_表名 PRIMARY KEY (A),\n CONSTRAINT FK_表名_父表 FOREIGN KEY (D)\n REFERENCES 父表(D)\n ON DELETE SET NULL\n ON UPDATE CASCADE\n);\n",[31,125431,125432,125436,125441,125446,125451,125456,125461,125466,125471,125475,125479],{"__ignoreMap":29},[34,125433,125434],{"class":36,"line":37},[34,125435,120575],{},[34,125437,125438],{"class":36,"line":43},[34,125439,125440],{}," A char(10) NOT NULL,\n",[34,125442,125443],{"class":36,"line":49},[34,125444,125445],{}," B varchar(20),\n",[34,125447,125448],{"class":36,"line":56},[34,125449,125450],{}," C int CHECK (C BETWEEN 下限 AND 上限),\n",[34,125452,125453],{"class":36,"line":62},[34,125454,125455],{}," D char(10) NULL,\n",[34,125457,125458],{"class":36,"line":68},[34,125459,125460],{}," CONSTRAINT PK_表名 PRIMARY KEY (A),\n",[34,125462,125463],{"class":36,"line":74},[34,125464,125465],{}," CONSTRAINT FK_表名_父表 FOREIGN KEY (D)\n",[34,125467,125468],{"class":36,"line":1971},[34,125469,125470],{}," REFERENCES 父表(D)\n",[34,125472,125473],{"class":36,"line":1976},[34,125474,101628],{},[34,125476,125477],{"class":36,"line":1985},[34,125478,99981],{},[34,125480,125481],{"class":36,"line":1990},[34,125482,462],{},[253,125484,125485],{"id":125485},"视图",[24,125487,125489],{"className":92584,"code":125488,"language":92586,"meta":29,"style":29},"CREATE VIEW 视图名(列1, 列2, 统计列) AS\nSELECT 分组列1, 分组列2, SUM(数值列)\nFROM ...\nJOIN ... ON ...\nWHERE ...\nGROUP BY 分组列1, 分组列2;\n",[31,125490,125491,125496,125501,125506,125511,125516],{"__ignoreMap":29},[34,125492,125493],{"class":36,"line":37},[34,125494,125495],{},"CREATE VIEW 视图名(列1, 列2, 统计列) AS\n",[34,125497,125498],{"class":36,"line":43},[34,125499,125500],{},"SELECT 分组列1, 分组列2, SUM(数值列)\n",[34,125502,125503],{"class":36,"line":49},[34,125504,125505],{},"FROM ...\n",[34,125507,125508],{"class":36,"line":56},[34,125509,125510],{},"JOIN ... ON ...\n",[34,125512,125513],{"class":36,"line":62},[34,125514,125515],{},"WHERE ...\n",[34,125517,125518],{"class":36,"line":68},[34,125519,125520],{},"GROUP BY 分组列1, 分组列2;\n",[253,125522,125523],{"id":125523},"更新关联表",[24,125525,125526],{"className":92584,"code":121565,"language":92586,"meta":29,"style":29},[31,125527,125528,125532,125536,125540,125544],{"__ignoreMap":29},[34,125529,125530],{"class":36,"line":37},[34,125531,101670],{},[34,125533,125534],{"class":36,"line":43},[34,125535,121576],{},[34,125537,125538],{"class":36,"line":49},[34,125539,101680],{},[34,125541,125542],{"class":36,"line":56},[34,125543,101685],{},[34,125545,125546],{"class":36,"line":62},[34,125547,101690],{},[253,125549,125551],{"id":125550},"没有查询","“没有”查询",[24,125553,125554],{"className":92584,"code":121107,"language":92586,"meta":29,"style":29},[31,125555,125556,125560,125564,125568,125572,125576,125580,125584],{"__ignoreMap":29},[34,125557,125558],{"class":36,"line":37},[34,125559,92811],{},[34,125561,125562],{"class":36,"line":43},[34,125563,101794],{},[34,125565,125566],{"class":36,"line":49},[34,125567,92965],{},[34,125569,125570],{"class":36,"line":56},[34,125571,92970],{},[34,125573,125574],{"class":36,"line":62},[34,125575,101839],{},[34,125577,125578],{"class":36,"line":68},[34,125579,101844],{},[34,125581,125582],{"class":36,"line":74},[34,125583,101849],{},[34,125585,125586],{"class":36,"line":1971},[34,125587,462],{},[253,125589,125591],{"id":125590},"全部查询","“全部”查询",[24,125593,125595],{"className":92584,"code":125594,"language":92586,"meta":29,"style":29},"SELECT s.Sno\nFROM Student AS s\nWHERE NOT EXISTS (\n SELECT 1\n FROM Course AS c\n WHERE NOT EXISTS (\n SELECT 1\n FROM SC AS sc\n WHERE sc.Sno = s.Sno\n AND sc.Cno = c.Cno\n )\n);\n",[31,125596,125597,125602,125606,125610,125614,125618,125622,125626,125630,125634,125638,125642],{"__ignoreMap":29},[34,125598,125599],{"class":36,"line":37},[34,125600,125601],{},"SELECT s.Sno\n",[34,125603,125604],{"class":36,"line":43},[34,125605,104138],{},[34,125607,125608],{"class":36,"line":49},[34,125609,92965],{},[34,125611,125612],{"class":36,"line":56},[34,125613,92970],{},[34,125615,125616],{"class":36,"line":62},[34,125617,121185],{},[34,125619,125620],{"class":36,"line":68},[34,125621,93053],{},[34,125623,125624],{"class":36,"line":74},[34,125625,93058],{},[34,125627,125628],{"class":36,"line":1971},[34,125629,121198],{},[34,125631,125632],{"class":36,"line":1976},[34,125633,121203],{},[34,125635,125636],{"class":36,"line":1985},[34,125637,121208],{},[34,125639,125640],{"class":36,"line":1990},[34,125641,93078],{},[34,125643,125644],{"class":36,"line":1999},[34,125645,462],{},[19,125647,125649],{"id":125648},"二存储过程模板","二、存储过程模板",[253,125651,125653],{"id":125652},"输入参数-条件判断","输入参数 + 条件判断",[24,125655,125657],{"className":92584,"code":125656,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE 过程名\n @参数1 int,\n @参数2 int\nAS\nBEGIN\n IF @参数1 IS NULL\n BEGIN\n PRINT N'参数错误';\n RETURN;\n END;\n\n -- 核心 SQL\nEND;\nGO\n",[31,125658,125659,125664,125669,125674,125678,125682,125687,125691,125696,125700,125704,125708,125713,125717],{"__ignoreMap":29},[34,125660,125661],{"class":36,"line":37},[34,125662,125663],{},"CREATE PROCEDURE 过程名\n",[34,125665,125666],{"class":36,"line":43},[34,125667,125668],{}," @参数1 int,\n",[34,125670,125671],{"class":36,"line":49},[34,125672,125673],{}," @参数2 int\n",[34,125675,125676],{"class":36,"line":56},[34,125677,93260],{},[34,125679,125680],{"class":36,"line":62},[34,125681,102603],{},[34,125683,125684],{"class":36,"line":68},[34,125685,125686],{}," IF @参数1 IS NULL\n",[34,125688,125689],{"class":36,"line":74},[34,125690,102636],{},[34,125692,125693],{"class":36,"line":1971},[34,125694,125695],{}," PRINT N'参数错误';\n",[34,125697,125698],{"class":36,"line":1976},[34,125699,118387],{},[34,125701,125702],{"class":36,"line":1985},[34,125703,103374],{},[34,125705,125706],{"class":36,"line":1990},[34,125707,53],{"emptyLinePlaceholder":52},[34,125709,125710],{"class":36,"line":1999},[34,125711,125712],{}," -- 核心 SQL\n",[34,125714,125715],{"class":36,"line":2022},[34,125716,102656],{},[34,125718,125719],{"class":36,"line":2036},[34,125720,102988],{},[253,125722,125723],{"id":125723},"输出参数",[24,125725,125727],{"className":92584,"code":125726,"language":92586,"meta":29,"style":29},"CREATE PROCEDURE proc_get_name\n @id int,\n @name varchar(50) OUTPUT\nAS\nBEGIN\n SELECT @name = Name\n FROM T\n WHERE ID = @id;\n\n IF @id IS NULL OR @name IS NULL\n SET @name = N'无此记录';\nEND;\nGO\n",[31,125728,125729,125734,125739,125744,125748,125752,125757,125762,125767,125771,125776,125781,125785],{"__ignoreMap":29},[34,125730,125731],{"class":36,"line":37},[34,125732,125733],{},"CREATE PROCEDURE proc_get_name\n",[34,125735,125736],{"class":36,"line":43},[34,125737,125738],{}," @id int,\n",[34,125740,125741],{"class":36,"line":49},[34,125742,125743],{}," @name varchar(50) OUTPUT\n",[34,125745,125746],{"class":36,"line":56},[34,125747,93260],{},[34,125749,125750],{"class":36,"line":62},[34,125751,102603],{},[34,125753,125754],{"class":36,"line":68},[34,125755,125756],{}," SELECT @name = Name\n",[34,125758,125759],{"class":36,"line":74},[34,125760,125761],{}," FROM T\n",[34,125763,125764],{"class":36,"line":1971},[34,125765,125766],{}," WHERE ID = @id;\n",[34,125768,125769],{"class":36,"line":1976},[34,125770,53],{"emptyLinePlaceholder":52},[34,125772,125773],{"class":36,"line":1985},[34,125774,125775],{}," IF @id IS NULL OR @name IS NULL\n",[34,125777,125778],{"class":36,"line":1990},[34,125779,125780],{}," SET @name = N'无此记录';\n",[34,125782,125783],{"class":36,"line":1999},[34,125784,102656],{},[34,125786,125787],{"class":36,"line":2022},[34,125788,102988],{},[253,125790,125791],{"id":125791},"库存判断",[24,125793,125795],{"className":92584,"code":125794,"language":92586,"meta":29,"style":29},"DECLARE @qty int;\nSELECT @qty = QTY FROM BOOK WHERE BID = @bid;\n\nIF @qty IS NULL\n PRINT N'无此图书';\nELSE IF @qty >= @s\nBEGIN\n UPDATE BOOK SET QTY = QTY - @s WHERE BID = @bid;\n SELECT QTY FROM BOOK WHERE BID = @bid;\nEND\nELSE\n PRINT N'库存不足';\n",[31,125796,125797,125802,125807,125811,125816,125821,125826,125830,125835,125840,125844,125849],{"__ignoreMap":29},[34,125798,125799],{"class":36,"line":37},[34,125800,125801],{},"DECLARE @qty int;\n",[34,125803,125804],{"class":36,"line":43},[34,125805,125806],{},"SELECT @qty = QTY FROM BOOK WHERE BID = @bid;\n",[34,125808,125809],{"class":36,"line":49},[34,125810,53],{"emptyLinePlaceholder":52},[34,125812,125813],{"class":36,"line":56},[34,125814,125815],{},"IF @qty IS NULL\n",[34,125817,125818],{"class":36,"line":62},[34,125819,125820],{}," PRINT N'无此图书';\n",[34,125822,125823],{"class":36,"line":68},[34,125824,125825],{},"ELSE IF @qty >= @s\n",[34,125827,125828],{"class":36,"line":74},[34,125829,102603],{},[34,125831,125832],{"class":36,"line":1971},[34,125833,125834],{}," UPDATE BOOK SET QTY = QTY - @s WHERE BID = @bid;\n",[34,125836,125837],{"class":36,"line":1976},[34,125838,125839],{}," SELECT QTY FROM BOOK WHERE BID = @bid;\n",[34,125841,125842],{"class":36,"line":1985},[34,125843,102706],{},[34,125845,125846],{"class":36,"line":1990},[34,125847,125848],{},"ELSE\n",[34,125850,125851],{"class":36,"line":1999},[34,125852,125853],{}," PRINT N'库存不足';\n",[19,125855,125857],{"id":125856},"三游标模板","三、游标模板",[24,125859,125861],{"className":92584,"code":125860,"language":92586,"meta":29,"style":29},"DECLARE Cur CURSOR\nFOR SELECT 列1, 列2 FROM 表 WHERE 条件;\n\nOPEN Cur;\n\nDECLARE @v1 数据类型, @v2 数据类型;\n\nFETCH NEXT FROM Cur INTO @v1, @v2;\n\nWHILE @@FETCH_STATUS = 0\nBEGIN\n SELECT 标签1 = @v1, 标签2 = @v2;\n FETCH NEXT FROM Cur INTO @v1, @v2;\nEND;\n\nCLOSE Cur;\nDEALLOCATE Cur;\n",[31,125862,125863,125868,125873,125877,125882,125886,125891,125895,125900,125904,125908,125912,125917,125922,125926,125930,125935],{"__ignoreMap":29},[34,125864,125865],{"class":36,"line":37},[34,125866,125867],{},"DECLARE Cur CURSOR\n",[34,125869,125870],{"class":36,"line":43},[34,125871,125872],{},"FOR SELECT 列1, 列2 FROM 表 WHERE 条件;\n",[34,125874,125875],{"class":36,"line":49},[34,125876,53],{"emptyLinePlaceholder":52},[34,125878,125879],{"class":36,"line":56},[34,125880,125881],{},"OPEN Cur;\n",[34,125883,125884],{"class":36,"line":62},[34,125885,53],{"emptyLinePlaceholder":52},[34,125887,125888],{"class":36,"line":68},[34,125889,125890],{},"DECLARE @v1 数据类型, @v2 数据类型;\n",[34,125892,125893],{"class":36,"line":74},[34,125894,53],{"emptyLinePlaceholder":52},[34,125896,125897],{"class":36,"line":1971},[34,125898,125899],{},"FETCH NEXT FROM Cur INTO @v1, @v2;\n",[34,125901,125902],{"class":36,"line":1976},[34,125903,53],{"emptyLinePlaceholder":52},[34,125905,125906],{"class":36,"line":1985},[34,125907,103619],{},[34,125909,125910],{"class":36,"line":1990},[34,125911,102603],{},[34,125913,125914],{"class":36,"line":1999},[34,125915,125916],{}," SELECT 标签1 = @v1, 标签2 = @v2;\n",[34,125918,125919],{"class":36,"line":2022},[34,125920,125921],{}," FETCH NEXT FROM Cur INTO @v1, @v2;\n",[34,125923,125924],{"class":36,"line":2036},[34,125925,102656],{},[34,125927,125928],{"class":36,"line":2603},[34,125929,53],{"emptyLinePlaceholder":52},[34,125931,125932],{"class":36,"line":5537},[34,125933,125934],{},"CLOSE Cur;\n",[34,125936,125937],{"class":36,"line":5543},[34,125938,125939],{},"DEALLOCATE Cur;\n",[19,125941,125943],{"id":125942},"四触发器模板","四、触发器模板",[253,125945,125946],{"id":125946},"营业时间限制",[24,125948,125950],{"className":92584,"code":125949,"language":92586,"meta":29,"style":29},"CREATE TRIGGER trg_table_time\nON 表名\nAFTER INSERT, UPDATE, DELETE\nAS\nBEGIN\n DECLARE @hour int, @weekday int;\n SELECT @hour = DATEPART(hh, GETDATE());\n SELECT @weekday = DATEPART(weekday, GETDATE());\n\n IF NOT (@hour BETWEEN 9 AND 17 AND @weekday IN (2,3,4,5,6))\n BEGIN\n PRINT N'Not business hours!';\n ROLLBACK;\n END;\nEND;\nGO\n",[31,125951,125952,125957,125962,125966,125970,125974,125978,125982,125986,125990,125994,125998,126002,126006,126010,126014],{"__ignoreMap":29},[34,125953,125954],{"class":36,"line":37},[34,125955,125956],{},"CREATE TRIGGER trg_table_time\n",[34,125958,125959],{"class":36,"line":43},[34,125960,125961],{},"ON 表名\n",[34,125963,125964],{"class":36,"line":49},[34,125965,102594],{},[34,125967,125968],{"class":36,"line":56},[34,125969,93260],{},[34,125971,125972],{"class":36,"line":62},[34,125973,102603],{},[34,125975,125976],{"class":36,"line":68},[34,125977,102608],{},[34,125979,125980],{"class":36,"line":74},[34,125981,102617],{},[34,125983,125984],{"class":36,"line":1971},[34,125985,102622],{},[34,125987,125988],{"class":36,"line":1976},[34,125989,53],{"emptyLinePlaceholder":52},[34,125991,125992],{"class":36,"line":1985},[34,125993,102685],{},[34,125995,125996],{"class":36,"line":1990},[34,125997,102636],{},[34,125999,126000],{"class":36,"line":1999},[34,126001,122911],{},[34,126003,126004],{"class":36,"line":2022},[34,126005,102646],{},[34,126007,126008],{"class":36,"line":2036},[34,126009,103374],{},[34,126011,126012],{"class":36,"line":2603},[34,126013,102656],{},[34,126015,126016],{"class":36,"line":5537},[34,126017,102988],{},[253,126019,126021],{"id":126020},"用-inserteddeleted","用 inserted\u002Fdeleted",[24,126023,126025],{"className":92584,"code":126024,"language":92586,"meta":29,"style":29},"CREATE TRIGGER trg_check_update\nON T\nAFTER UPDATE\nAS\nBEGIN\n IF EXISTS (\n SELECT 1\n FROM inserted\n WHERE 某列 \u003C 0\n )\n BEGIN\n PRINT N'违反约束';\n ROLLBACK;\n END;\nEND;\nGO\n",[31,126026,126027,126032,126037,126042,126046,126050,126055,126059,126064,126069,126073,126077,126082,126086,126090,126094],{"__ignoreMap":29},[34,126028,126029],{"class":36,"line":37},[34,126030,126031],{},"CREATE TRIGGER trg_check_update\n",[34,126033,126034],{"class":36,"line":43},[34,126035,126036],{},"ON T\n",[34,126038,126039],{"class":36,"line":49},[34,126040,126041],{},"AFTER UPDATE\n",[34,126043,126044],{"class":36,"line":56},[34,126045,93260],{},[34,126047,126048],{"class":36,"line":62},[34,126049,102603],{},[34,126051,126052],{"class":36,"line":68},[34,126053,126054],{}," IF EXISTS (\n",[34,126056,126057],{"class":36,"line":74},[34,126058,93058],{},[34,126060,126061],{"class":36,"line":1971},[34,126062,126063],{}," FROM inserted\n",[34,126065,126066],{"class":36,"line":1976},[34,126067,126068],{}," WHERE 某列 \u003C 0\n",[34,126070,126071],{"class":36,"line":1985},[34,126072,93078],{},[34,126074,126075],{"class":36,"line":1990},[34,126076,102636],{},[34,126078,126079],{"class":36,"line":1999},[34,126080,126081],{}," PRINT N'违反约束';\n",[34,126083,126084],{"class":36,"line":2022},[34,126085,102646],{},[34,126087,126088],{"class":36,"line":2036},[34,126089,103374],{},[34,126091,126092],{"class":36,"line":2603},[34,126093,102656],{},[34,126095,126096],{"class":36,"line":5537},[34,126097,102988],{},[19,126099,126101],{"id":126100},"五嵌入式-sql-填空模板","五、嵌入式 SQL 填空模板",[24,126103,126105],{"className":65734,"code":126104,"language":31965,"meta":29,"style":29},"EXEC SQL INCLUDE SQLCA;\n\nEXEC SQL DECLARE SX CURSOR FOR\nSELECT Sno, Sname, Ssex, Sage\nFROM Student\nWHERE Sdept = :deptname;\n\nEXEC SQL OPEN SX;\n\nEXEC SQL FETCH SX INTO :HSno, :HSname, :HSsex, :HSage;\n\nEXEC SQL UPDATE Student\nSET Sage = :NEWAGE\nWHERE CURRENT OF SX;\n",[31,126106,126107,126112,126116,126120,126124,126128,126132,126136,126140,126144,126148,126152,126156,126160],{"__ignoreMap":29},[34,126108,126109],{"class":36,"line":37},[34,126110,126111],{},"EXEC SQL INCLUDE SQLCA;\n",[34,126113,126114],{"class":36,"line":43},[34,126115,53],{"emptyLinePlaceholder":52},[34,126117,126118],{"class":36,"line":49},[34,126119,123138],{},[34,126121,126122],{"class":36,"line":56},[34,126123,123143],{},[34,126125,126126],{"class":36,"line":62},[34,126127,120836],{},[34,126129,126130],{"class":36,"line":68},[34,126131,123152],{},[34,126133,126134],{"class":36,"line":74},[34,126135,53],{"emptyLinePlaceholder":52},[34,126137,126138],{"class":36,"line":1971},[34,126139,123161],{},[34,126141,126142],{"class":36,"line":1976},[34,126143,53],{"emptyLinePlaceholder":52},[34,126145,126146],{"class":36,"line":1985},[34,126147,123170],{},[34,126149,126150],{"class":36,"line":1990},[34,126151,53],{"emptyLinePlaceholder":52},[34,126153,126154],{"class":36,"line":1999},[34,126155,123179],{},[34,126157,126158],{"class":36,"line":2022},[34,126159,123184],{},[34,126161,126162],{"class":36,"line":2036},[34,126163,123189],{},[15,126165,126166],{},"填空记忆：",[105,126168,126169,126174,126178,126182,126188],{},[108,126170,126171,126172],{},"定义 SQL 通信区：",[31,126173,119757],{},[108,126175,103683,126176],{},[31,126177,119772],{},[108,126179,103689,126180],{},[31,126181,119794],{},[108,126183,126184,126185],{},"推进游标：",[31,126186,126187],{},"EXEC SQL FETCH SX INTO :...;",[108,126189,126190,126191],{},"当前游标行更新：",[31,126192,126193],{},"EXEC SQL UPDATE 表名 SET ... WHERE CURRENT OF SX;",[19,126195,126197],{"id":126196},"六e-r-设计模板","六、E-R 设计模板",[15,126199,126200],{},"答题顺序：",[24,126202,126205],{"className":126203,"code":126204,"language":2638,"meta":29},[2636],"1. 实体：\n E1(...), E2(...), E3(...)\n\n2. 联系和联系类型：\n E1 1 --- n E2\n E2 m --- n E3 [联系属性]\n\n3. 关系模式：\n R1(属性...), PK: ..., FK: ...\n R2(属性...), PK: ..., FK: ...\n\n4. 范式说明：\n 各属性原子，满足 1NF；\n 非主属性完全依赖主码，满足 2NF；\n 无非主属性传递依赖主码，满足 3NF；\n 若每个非平凡函数依赖左部都是候选码，则满足 BCNF。\n",[31,126206,126204],{"__ignoreMap":29},[19,126208,126210],{"id":126209},"七规范化模板","七、规范化模板",[24,126212,126215],{"className":126213,"code":126214,"language":2638,"meta":29},[2636],"U = {所有属性}\nF = {\n A -> B,\n C -> D,\n (A, C) -> E\n}\n\n候选码：\n先求 (A,C)+ = ...\n因为 (A,C)+ = U，且 A+、C+ 均不能推出 U，\n所以候选码为 (A,C)。\n\n范式判断：\nR 满足 1NF。\n由于存在 C -> D，D 为非主属性，C 是候选码 (A,C) 的真子集，\n所以存在部分依赖，不满足 2NF。\n\n分解：\nR1(C, D), PK(C)\nR2(A, C, E, ...), PK(A,C)\n",[31,126216,126214],{"__ignoreMap":29},[15,126218,126219],{},"若是传递依赖：",[24,126221,126224],{"className":126222,"code":126223,"language":2638,"meta":29},[2636],"A -> B, B -> C，因此 A -> C 为传递依赖。\n分解为 R1(A,B), R2(B,C)。\n",[31,126225,126223],{"__ignoreMap":29},[19,126227,126229],{"id":126228},"八查询优化模板","八、查询优化模板",[24,126231,126234],{"className":126232,"code":126233,"language":2638,"meta":29},[2636],"原始表达式：\nπ_{输出列}(σ_{连接条件 ∧ 单表条件1 ∧ 单表条件2}(R × S))\n\n原始树：\nπ 输出列\n└── σ 全部条件\n └── ×\n ├── R\n └── S\n\n优化规则：\n1. 分解选择条件。\n2. 单表选择下推。\n3. 笛卡尔积 + 连接条件改为连接。\n4. 投影下推，只保留输出属性和连接属性。\n\n优化树：\nπ 输出列\n└── ⋈ 连接条件\n ├── π R需要列\n │ └── σ R条件\n │ └── R\n └── π S需要列\n └── σ S条件\n └── S\n\n优化表达式：\nπ_{输出列}(π(...σ...(R)) ⋈_{条件} π(...σ...(S)))\n",[31,126235,126233],{"__ignoreMap":29},[19,126237,126239],{"id":126238},"九检查点恢复模板","九、检查点恢复模板",[24,126241,126244],{"className":126242,"code":126243,"language":2638,"meta":29},[2636],"检查点时正在执行的事务：\nACTIVE-LIST = {...}\n\n从检查点向故障点扫描日志：\n遇到 START Ti：Ti 加入 UNDO-LIST。\n遇到 COMMIT Ti：Ti 从 UNDO-LIST 移出，加入 REDO-LIST。\n\n故障时：\nUNDO-LIST = {...} 这些事务未提交，要撤销。\nREDO-LIST = {...} 这些事务已提交，要重做。\n",[31,126245,126243],{"__ignoreMap":29},[15,126247,126248],{},"判断口诀：",[105,126250,126251,126254,126257],{},[108,126252,126253],{},"故障前已提交：REDO。",[108,126255,126256],{},"故障时未提交：UNDO。",[108,126258,126259],{},"检查点前已提交：通常不处理。",[19,126261,126263],{"id":126262},"十两阶段封锁模板","十、两阶段封锁模板",[24,126265,126268],{"className":126266,"code":126267,"language":2638,"meta":29},[2636],"T1:\nSLock(读对象);\nXLock(写对象);\nread(...);\nwrite(...);\nUnlock(...);\nUnlock(...);\n\nT2:\nSLock(读对象);\nXLock(写对象);\nread(...);\nwrite(...);\nUnlock(...);\nUnlock(...);\n",[31,126269,126267],{"__ignoreMap":29},[15,126271,90843],{},[24,126273,126276],{"className":126274,"code":126275,"language":2638,"meta":29},[2636],"上述调度中，每个事务在释放任何锁之前已经申请完所需锁，\n释放锁之后不再申请新锁，因此满足两阶段封锁协议。\n但如果两个事务按相反顺序申请锁，可能互相等待，因此可能死锁。\n",[31,126277,126275],{"__ignoreMap":29},[19,126279,126281],{"id":126280},"十一简答题万能结构","十一、简答题万能结构",[253,126283,126284],{"id":126284},"视图优点",[24,126286,126289],{"className":126287,"code":126288,"language":2638,"meta":29},[2636],"视图能够简化查询，使用户以不同角度观察数据；\n能够提供一定逻辑独立性；\n能够通过只授权访问视图来保护数据安全；\n也有利于屏蔽复杂连接和敏感字段。\n",[31,126290,126288],{"__ignoreMap":29},[253,126292,126294],{"id":126293},"dbms-功能","DBMS 功能",[24,126296,126299],{"className":126297,"code":126298,"language":2638,"meta":29},[2636],"DBMS 提供数据定义、数据组织存储和管理、数据操纵、\n数据库运行管理、数据库建立和维护、通信接口等功能。\n其中运行管理包括安全性、完整性、并发控制和恢复。\n",[31,126300,126298],{"__ignoreMap":29},[253,126302,126303],{"id":126303},"完整性与安全性区别",[24,126305,126308],{"className":126306,"code":126307,"language":2638,"meta":29},[2636],"安全性是防止非法用户非法访问数据库；\n完整性是防止合法用户对数据库加入不符合语义的数据。\n安全性通过用户标识、授权、视图、审计等实现；\n完整性通过主码、外码、CHECK、触发器等实现。\n",[31,126309,126307],{"__ignoreMap":29},[253,126311,126312],{"id":126312},"恢复基本原理",[24,126314,126317],{"className":126315,"code":126316,"language":2638,"meta":29},[2636],"数据库恢复利用冗余数据把数据库从错误状态恢复到一致状态。\n冗余数据主要包括后援副本和日志文件。\n事务故障通常 UNDO，系统故障对未提交事务 UNDO、对已提交但可能未落盘事务 REDO，\n介质故障先装入后援副本再利用日志 REDO。\n",[31,126318,126316],{"__ignoreMap":29},[79,126320,81],{},{"title":29,"searchDepth":43,"depth":43,"links":126322},[126323,126330,126335,126336,126340,126341,126342,126343,126344,126345,126346],{"id":125422,"depth":43,"text":125423,"children":126324},[126325,126326,126327,126328,126329],{"id":125426,"depth":49,"text":125426},{"id":125485,"depth":49,"text":125485},{"id":125523,"depth":49,"text":125523},{"id":125550,"depth":49,"text":125551},{"id":125590,"depth":49,"text":125591},{"id":125648,"depth":43,"text":125649,"children":126331},[126332,126333,126334],{"id":125652,"depth":49,"text":125653},{"id":125723,"depth":49,"text":125723},{"id":125791,"depth":49,"text":125791},{"id":125856,"depth":43,"text":125857},{"id":125942,"depth":43,"text":125943,"children":126337},[126338,126339],{"id":125946,"depth":49,"text":125946},{"id":126020,"depth":49,"text":126021},{"id":126100,"depth":43,"text":126101},{"id":126196,"depth":43,"text":126197},{"id":126209,"depth":43,"text":126210},{"id":126228,"depth":43,"text":126229},{"id":126238,"depth":43,"text":126239},{"id":126262,"depth":43,"text":126263},{"id":126280,"depth":43,"text":126281,"children":126347},[126348,126349,126350,126351],{"id":126284,"depth":49,"text":126284},{"id":126293,"depth":49,"text":126294},{"id":126303,"depth":49,"text":126303},{"id":126312,"depth":49,"text":126312},{},"\u002Fsource\u002Fcs\u002Fdb\u002Ftutorial\u002F08",{"title":125411,"description":125419},"source\u002Fcs\u002Fdb\u002Ftutorial\u002F08-大题答题模板","EOp1Q4KfzGUxU8_3tfmiknQdWLmoAs2wJD8tXE-i-fI",{"id":126358,"title":126359,"body":126360,"description":29,"extension":85,"meta":126367,"navigation":52,"path":126368,"seo":126369,"stem":126370,"__hash__":126371},"notes\u002Fsource\u002Fcs\u002Fdm\u002Findex.md","Discrete Math",{"type":8,"value":126361,"toc":126365},[126362],[31543,126363],{"path":126364,"unwrap-single-group":29},"\u002Fcs\u002Fdm\u002F",{"title":29,"searchDepth":43,"depth":43,"links":126366},[],{},"\u002Fsource\u002Fcs\u002Fdm",{"title":126359,"description":29},"source\u002Fcs\u002Fdm\u002Findex","fs_5jdzJhu0lnyjO66DhGdVN47OtRdkD2HAXI72o4dE",{"id":126373,"title":126374,"body":126375,"description":126381,"extension":85,"meta":161994,"navigation":52,"path":161995,"seo":161996,"stem":161997,"__hash__":161998},"notes\u002Fsource\u002Fcs\u002Fdm\u002Fnote\u002Fchap12.md","第十二章总线索",{"type":8,"value":126376,"toc":161878},[126377,126379,126382,126485,126488,126505,126508,126602,126604,126724,126727,126816,126818,126822,126826,126829,126918,126920,127021,127027,127033,127035,127039,127043,127046,127121,127198,127201,127249,127251,127305,127365,127367,127371,127374,127435,127512,127514,127578,127581,127635,127637,127641,127645,127654,127656,127864,127867,127869,127873,127878,127881,127884,127886,127890,127978,128054,128128,128130,128134,128140,128142,128219,128278,128280,128284,128290,128292,128515,128670,128672,128676,128682,128685,128762,128765,128767,128771,128777,128779,129001,129171,129173,129177,129183,129186,129188,129192,129196,129202,129205,129262,129264,129268,129274,129276,129280,129286,129289,129291,129295,129301,129411,129414,129416,129420,129426,129458,129532,129535,129727,129758,129966,129969,130094,130096,130100,130104,130139,130186,130218,130220,130321,130324,130326,130330,130333,130336,130368,130447,130450,130482,130559,130562,130743,130745,130749,130752,130798,130832,130834,130880,130914,130916,130920,130983,130985,130993,130995,130999,131002,131109,131315,131318,131323,131325,131328,131330,131334,131337,131480,131483,131486,131488,131577,131580,131582,131586,131589,131830,131832,131995,132091,132093,132097,132189,132309,132312,132432,132527,132529,132533,132535,132622,132625,132712,132809,132811,132816,132818,132822,132825,132912,132981,133061,133064,133069,133071,133075,133077,133163,133169,133249,133251,133255,133345,133399,133463,133550,133552,133556,133587,133641,133706,133793,133795,133799,133830,133884,133886,133890,133894,133897,133902,133904,133908,133998,134067,134099,134102,134207,134210,134314,134317,134466,134468,134472,134476,134572,134575,134714,134807,134895,134897,134901,134904,134924,134926,134930,134932,134968,134975,134978,134980,134994,134996,135000,135003,135005,135009,135011,135087,135090,135473,135475,135814,135817,135908,135910,135914,135917,136207,136210,136386,136388,136392,136398,136401,136403,136407,136417,136420,136509,136512,136514,136518,136521,136657,136666,136668,136672,136678,136680,136684,136690,136692,136696,136785,136788,136790,136794,136853,136857,136860,136969,136972,137084,137086,137090,137093,137184,137187,137189,137193,137197,137258,137289,137292,137323,137326,137329,137382,137384,137386,137390,137393,137396,137398,137402,137404,137435,137438,137469,137471,137502,137504,137535,137538,137569,137571,137602,137605,137607,137611,137739,137881,137884,137938,138102,138247,138412,138555,138696,138838,138840,138844,138847,139154,139157,139159,139163,139167,139198,139232,139234,139238,139241,139247,139250,139298,139329,139398,139400,139404,139413,139416,139522,139553,139556,139561,139563,139567,139573,139575,139681,139712,139714,139718,139721,139725,139770,139823,139825,139876,139882,139884,139889,139891,139895,139939,139989,139991,140041,140046,140048,140052,140058,140060,140064,140196,140199,140201,140212,140214,140218,140222,140228,140230,140234,140240,140242,140246,140252,140254,140258,140261,140272,140274,140278,140281,140284,140298,140300,140304,140307,140396,140398,140622,140846,140849,140903,140906,140909,140911,140915,140918,141089,141091,141745,141747,141755,141757,141761,141775,141777,141781,141784,141787,142653,142656,142801,142803,142807,142810,142899,142901,143124,143127,143181,143183,143185,143189,143192,143360,143362,143464,143635,143638,143740,143742,143746,143748,143849,144018,144020,144024,144055,144066,144068,144153,144155,144159,144161,144164,144442,144474,144477,144752,144783,144786,145090,145092,145096,145099,145130,145133,145196,145242,145484,145486,145539,145711,145713,145716,145718,145857,145860,146005,146007,146011,146014,146019,146022,146054,146166,146168,146704,146706,146710,146712,146814,146816,146952,146954,146956,147058,147061,147063,147067,147149,147152,147397,147399,147570,147573,147747,147750,147752,147756,147788,147873,147876,147878,147979,147982,148087,148089,148161,148164,148236,148239,148241,148245,148248,148250,148362,148364,148576,148579,148681,148683,148736,148738,148841,148843,148847,148850,148856,148858,148862,148865,148868,148879,148882,148993,148995,148999,149002,149011,149043,149090,149093,149113,149115,149119,149122,149125,149323,149325,149628,149630,149634,149637,149726,149729,149731,149735,149738,149821,149824,149922,149983,149985,149989,150061,150065,150175,150178,150799,150801,150805,150935,150937,150941,151083,151381,151384,151682,151684,151688,151748,151750,151754,151757,151774,151776,151780,151783,151872,151875,151877,151881,151884,152074,152076,152184,152320,152323,152453,152455,152508,152510,152642,152644,152648,152651,153167,153169,153287,153319,153321,153325,153328,153424,153427,153430,153507,153510,153512,153516,153546,153548,153654,153656,153660,153664,153667,153670,153868,153871,153874,153876,153880,153886,153889,153894,153896,153900,154126,154401,154403,154408,154410,154414,154417,154421,154424,154441,154443,154447,154520,154613,154686,154688,154803,154806,155123,155264,155266,155270,155342,155435,155508,155511,155687,155689,156006,156147,156149,156153,156156,156250,156256,156259,156261,156265,156268,156271,156274,156276,156279,158218,158220,158223,158227,158230,158250,158253,158342,158431,158433,158437,158439,158447,158449,158460,158462,158466,158469,158472,158528,158531,158534,158537,158590,158592,158596,158599,158758,158761,158764,158766,158770,158773,158793,158796,158799,158801,158805,158808,158919,158922,159049,159051,159055,159058,159111,159114,159117,159170,159173,159175,159179,159182,159352,159355,159474,159476,159480,159483,159724,159727,159729,159733,159736,159824,159827,159915,159917,159920,159924,159960,159963,160052,160054,160058,160090,160092,160096,160185,160274,160363,160365,160369,160413,160415,160419,160446,160448,160451,160454,160543,160632,160721,160810,160899,160902,161092,161181,161309,161644,161772,161872],[11,126378,126374],{"id":126374},[15,126380,126381],{},"图论研究的是：",[34,126383,126386],{"className":126384},[126385],"katex-display",[34,126387,126389,126415],{"className":126388},[31642],[34,126390,126392],{"className":126391},[31646],[31648,126393,126395],{"xmlns":31650,"display":126394},"block",[31652,126396,126397,126412],{},[31655,126398,126399],{},[126400,126401,126403],"menclose",{"notation":126402},"box",[71090,126404,126405],{"scriptlevel":459,"displaystyle":702},[71090,126406,126407],{"scriptlevel":459,"displaystyle":702},[71090,126408,126409],{"scriptlevel":459,"displaystyle":758},[39361,126410,126411],{},"对象之间的连接关系",[31665,126413,126414],{"encoding":31667},"\\boxed{\\text{对象之间的连接关系}}",[34,126416,126418],{"className":126417,"ariaHidden":758},[31672],[34,126419,126421,126425],{"className":126420},[31676],[34,126422],{"className":126423,"style":126424},[31680],"height:1.3633em;vertical-align:-0.34em;",[34,126426,126428],{"className":126427},[31685],[34,126429,126431,126476],{"className":126430},[31796,31797],[34,126432,126434,126473],{"className":126433},[31801],[34,126435,126438,126461],{"className":126436,"style":126437},[31805],"height:1.0233em;",[34,126439,126441,126445],{"style":126440},"top:-3.3633em;",[34,126442],{"className":126443,"style":126444},[31813],"height:3.3633em;",[34,126446,126449],{"className":126447},[126448],"boxpad",[34,126450,126452],{"className":126451},[31685],[34,126453,126455],{"className":126454},[31685],[34,126456,126458],{"className":126457},[31685,2638],[34,126459,126411],{"className":126460},[31685,105462],[34,126462,126464,126467],{"style":126463},"top:-3.0233em;",[34,126465],{"className":126466,"style":126444},[31813],[34,126468],{"className":126469,"style":126472},[126470,126471],"stretchy","fbox","height:1.3633em;border-style:solid;border-width:0.04em;",[34,126474,31829],{"className":126475},[31828],[34,126477,126479],{"className":126478},[31801],[34,126480,126483],{"className":126481,"style":126482},[31805],"height:0.34em;",[34,126484],{},[15,126486,126487],{},"比如：",[105,126489,126490,126493,126496,126499,126502],{},[108,126491,126492],{},"城市之间有没有路；",[108,126494,126495],{},"人和人之间是否认识；",[108,126497,126498],{},"程序是否等待某个资源；",[108,126500,126501],{},"任务之间谁必须先完成；",[108,126503,126504],{},"两个点之间最短路线是多少。",[15,126506,126507],{},"图的基本结构就是：",[34,126509,126511],{"className":126510},[126385],[34,126512,126514,126548],{"className":126513},[31642],[34,126515,126517],{"className":126516},[31646],[31648,126518,126519],{"xmlns":31650,"display":126394},[31652,126520,126521,126545],{},[31655,126522,126523,126525,126527,126530,126533,126535,126537,126539,126542],{},[31658,126524,97709],{},[31662,126526,202],{},[31662,126528,126529],{"stretchy":702},"⟨",[31658,126531,126532],{},"V",[31662,126534,31960],{"separator":758},[31658,126536,96058],{},[31662,126538,31960],{"separator":758},[31658,126540,126541],{},"φ",[31662,126543,126544],{"stretchy":702},"⟩",[31665,126546,126547],{"encoding":31667},"G=\\langle V,E,\\varphi\\rangle",[34,126549,126551,126569],{"className":126550,"ariaHidden":758},[31672],[34,126552,126554,126557,126560,126563,126566],{"className":126553},[31676],[34,126555],{"className":126556,"style":33218},[31680],[34,126558,97709],{"className":126559},[31685,31686],[34,126561],{"className":126562,"style":31846},[31845],[34,126564,202],{"className":126565},[31850],[34,126567],{"className":126568,"style":31846},[31845],[34,126570,126572,126575,126578,126581,126584,126587,126590,126593,126596,126599],{"className":126571},[31676],[34,126573],{"className":126574,"style":31774},[31680],[34,126576,126529],{"className":126577},[31782],[34,126579,126532],{"className":126580,"style":32604},[31685,31686],[34,126582,31960],{"className":126583},[32119],[34,126585],{"className":126586,"style":32123},[31845],[34,126588,96058],{"className":126589,"style":106455},[31685,31686],[34,126591,31960],{"className":126592},[32119],[34,126594],{"className":126595,"style":32123},[31845],[34,126597,126541],{"className":126598},[31685,31686],[34,126600,126544],{"className":126601},[31690],[15,126603,104944],{},[13837,126605,126606,126616],{},[13840,126607,126608],{},[13843,126609,126610,126613],{},[13846,126611,126612],{},"符号",[13846,126614,126615],{},"含义",[13853,126617,126618,126653,126688],{},[13843,126619,126620,126650],{},[13858,126621,126622],{},[34,126623,126625,126638],{"className":126624},[31642],[34,126626,126628],{"className":126627},[31646],[31648,126629,126630],{"xmlns":31650},[31652,126631,126632,126636],{},[31655,126633,126634],{},[31658,126635,126532],{},[31665,126637,126532],{"encoding":31667},[34,126639,126641],{"className":126640,"ariaHidden":758},[31672],[34,126642,126644,126647],{"className":126643},[31676],[34,126645],{"className":126646,"style":33218},[31680],[34,126648,126532],{"className":126649,"style":32604},[31685,31686],[13858,126651,126652],{},"顶点集",[13843,126654,126655,126685],{},[13858,126656,126657],{},[34,126658,126660,126673],{"className":126659},[31642],[34,126661,126663],{"className":126662},[31646],[31648,126664,126665],{"xmlns":31650},[31652,126666,126667,126671],{},[31655,126668,126669],{},[31658,126670,96058],{},[31665,126672,96058],{"encoding":31667},[34,126674,126676],{"className":126675,"ariaHidden":758},[31672],[34,126677,126679,126682],{"className":126678},[31676],[34,126680],{"className":126681,"style":33218},[31680],[34,126683,96058],{"className":126684,"style":106455},[31685,31686],[13858,126686,126687],{},"边集",[13843,126689,126690,126721],{},[13858,126691,126692],{},[34,126693,126695,126709],{"className":126694},[31642],[34,126696,126698],{"className":126697},[31646],[31648,126699,126700],{"xmlns":31650},[31652,126701,126702,126706],{},[31655,126703,126704],{},[31658,126705,126541],{},[31665,126707,126708],{"encoding":31667},"\\varphi",[34,126710,126712],{"className":126711,"ariaHidden":758},[31672],[34,126713,126715,126718],{"className":126714},[31676],[34,126716],{"className":126717,"style":38774},[31680],[34,126719,126541],{"className":126720},[31685,31686],[13858,126722,126723],{},"关联函数，说明每条边连接哪些顶点",[15,126725,126726],{},"最直观一句话：",[34,126728,126730],{"className":126729},[126385],[34,126731,126733,126756],{"className":126732},[31642],[34,126734,126736],{"className":126735},[31646],[31648,126737,126738],{"xmlns":31650,"display":126394},[31652,126739,126740,126753],{},[31655,126741,126742],{},[126400,126743,126744],{"notation":126402},[71090,126745,126746],{"scriptlevel":459,"displaystyle":702},[71090,126747,126748],{"scriptlevel":459,"displaystyle":702},[71090,126749,126750],{"scriptlevel":459,"displaystyle":758},[39361,126751,126752],{},"顶点表示对象，边表示对象之间的关系",[31665,126754,126755],{"encoding":31667},"\\boxed{\\text{顶点表示对象，边表示对象之间的关系}}",[34,126757,126759],{"className":126758,"ariaHidden":758},[31672],[34,126760,126762,126765],{"className":126761},[31676],[34,126763],{"className":126764,"style":126424},[31680],[34,126766,126768],{"className":126767},[31685],[34,126769,126771,126808],{"className":126770},[31796,31797],[34,126772,126774,126805],{"className":126773},[31801],[34,126775,126777,126797],{"className":126776,"style":126437},[31805],[34,126778,126779,126782],{"style":126440},[34,126780],{"className":126781,"style":126444},[31813],[34,126783,126785],{"className":126784},[126448],[34,126786,126788],{"className":126787},[31685],[34,126789,126791],{"className":126790},[31685],[34,126792,126794],{"className":126793},[31685,2638],[34,126795,126752],{"className":126796},[31685,105462],[34,126798,126799,126802],{"style":126463},[34,126800],{"className":126801,"style":126444},[31813],[34,126803],{"className":126804,"style":126472},[126470,126471],[34,126806,31829],{"className":126807},[31828],[34,126809,126811],{"className":126810},[31801],[34,126812,126814],{"className":126813,"style":126482},[31805],[34,126815],{},[32188,126817],{},[11,126819,126821],{"id":126820},"_121-图与子图","12.1 图与子图",[19,126823,126825],{"id":126824},"一图的定义","一、图的定义",[15,126827,126828],{},"图通常记作：",[34,126830,126832],{"className":126831},[126385],[34,126833,126835,126864],{"className":126834},[31642],[34,126836,126838],{"className":126837},[31646],[31648,126839,126840],{"xmlns":31650,"display":126394},[31652,126841,126842,126862],{},[31655,126843,126844,126846,126848,126850,126852,126854,126856,126858,126860],{},[31658,126845,97709],{},[31662,126847,202],{},[31662,126849,126529],{"stretchy":702},[31658,126851,126532],{},[31662,126853,31960],{"separator":758},[31658,126855,96058],{},[31662,126857,31960],{"separator":758},[31658,126859,126541],{},[31662,126861,126544],{"stretchy":702},[31665,126863,126547],{"encoding":31667},[34,126865,126867,126885],{"className":126866,"ariaHidden":758},[31672],[34,126868,126870,126873,126876,126879,126882],{"className":126869},[31676],[34,126871],{"className":126872,"style":33218},[31680],[34,126874,97709],{"className":126875},[31685,31686],[34,126877],{"className":126878,"style":31846},[31845],[34,126880,202],{"className":126881},[31850],[34,126883],{"className":126884,"style":31846},[31845],[34,126886,126888,126891,126894,126897,126900,126903,126906,126909,126912,126915],{"className":126887},[31676],[34,126889],{"className":126890,"style":31774},[31680],[34,126892,126529],{"className":126893},[31782],[34,126895,126532],{"className":126896,"style":32604},[31685,31686],[34,126898,31960],{"className":126899},[32119],[34,126901],{"className":126902,"style":32123},[31845],[34,126904,96058],{"className":126905,"style":106455},[31685,31686],[34,126907,31960],{"className":126908},[32119],[34,126910],{"className":126911,"style":32123},[31845],[34,126913,126541],{"className":126914},[31685,31686],[34,126916,126544],{"className":126917},[31690],[15,126919,104944],{},[105,126921,126922,126956,126990],{},[108,126923,126924,126952,126953,36268],{},[34,126925,126927,126940],{"className":126926},[31642],[34,126928,126930],{"className":126929},[31646],[31648,126931,126932],{"xmlns":31650},[31652,126933,126934,126938],{},[31655,126935,126936],{},[31658,126937,126532],{},[31665,126939,126532],{"encoding":31667},[34,126941,126943],{"className":126942,"ariaHidden":758},[31672],[34,126944,126946,126949],{"className":126945},[31676],[34,126947],{"className":126948,"style":33218},[31680],[34,126950,126532],{"className":126951,"style":32604},[31685,31686]," 是非空有限集合，元素叫",[2627,126954,126955],{},"顶点",[108,126957,126958,126986,126987,36268],{},[34,126959,126961,126974],{"className":126960},[31642],[34,126962,126964],{"className":126963},[31646],[31648,126965,126966],{"xmlns":31650},[31652,126967,126968,126972],{},[31655,126969,126970],{},[31658,126971,96058],{},[31665,126973,96058],{"encoding":31667},[34,126975,126977],{"className":126976,"ariaHidden":758},[31672],[34,126978,126980,126983],{"className":126979},[31676],[34,126981],{"className":126982,"style":33218},[31680],[34,126984,96058],{"className":126985,"style":106455},[31685,31686]," 是有限集合，元素叫",[2627,126988,126989],{},"边",[108,126991,126992,127020],{},[34,126993,126995,127008],{"className":126994},[31642],[34,126996,126998],{"className":126997},[31646],[31648,126999,127000],{"xmlns":31650},[31652,127001,127002,127006],{},[31655,127003,127004],{},[31658,127005,126541],{},[31665,127007,126708],{"encoding":31667},[34,127009,127011],{"className":127010,"ariaHidden":758},[31672],[34,127012,127014,127017],{"className":127013},[31676],[34,127015],{"className":127016,"style":38774},[31680],[34,127018,126541],{"className":127019},[31685,31686]," 把每条边对应到某个顶点对。",[15,127022,127023,127024,347],{},"如果边没有方向，叫",[2627,127025,127026],{},"无向图",[15,127028,127029,127030,347],{},"如果边有方向，叫",[2627,127031,127032],{},"有向图",[32188,127034],{},[19,127036,127038],{"id":127037},"二无向边与有向边","二、无向边与有向边",[19,127040,127042],{"id":127041},"_1-无向边","1. 无向边",[15,127044,127045],{},"无向边连接两个顶点，不分起点终点。",[15,127047,127048,127049,127077,127078,127120],{},"如果边 ",[34,127050,127052,127065],{"className":127051},[31642],[34,127053,127055],{"className":127054},[31646],[31648,127056,127057],{"xmlns":31650},[31652,127058,127059,127063],{},[31655,127060,127061],{},[31658,127062,33244],{},[31665,127064,33244],{"encoding":31667},[34,127066,127068],{"className":127067,"ariaHidden":758},[31672],[34,127069,127071,127074],{"className":127070},[31676],[34,127072],{"className":127073,"style":32527},[31680],[34,127075,33244],{"className":127076},[31685,31686]," 连接 ",[34,127079,127081,127099],{"className":127080},[31642],[34,127082,127084],{"className":127083},[31646],[31648,127085,127086],{"xmlns":31650},[31652,127087,127088,127096],{},[31655,127089,127090,127092,127094],{},[31658,127091,94628],{},[31662,127093,31960],{"separator":758},[31658,127095,56782],{},[31665,127097,127098],{"encoding":31667},"u,v",[34,127100,127102],{"className":127101,"ariaHidden":758},[31672],[34,127103,127105,127108,127111,127114,127117],{"className":127104},[31676],[34,127106],{"className":127107,"style":38774},[31680],[34,127109,94628],{"className":127110},[31685,31686],[34,127112,31960],{"className":127113},[32119],[34,127115],{"className":127116,"style":32123},[31845],[34,127118,56782],{"className":127119,"style":32062},[31685,31686],"，可以写作：",[34,127122,127124],{"className":127123},[126385],[34,127125,127127,127153],{"className":127126},[31642],[34,127128,127130],{"className":127129},[31646],[31648,127131,127132],{"xmlns":31650,"display":126394},[31652,127133,127134,127150],{},[31655,127135,127136,127138,127140,127142,127144,127146,127148],{},[31658,127137,33244],{},[31662,127139,202],{},[31662,127141,456],{"stretchy":702},[31658,127143,94628],{},[31662,127145,31960],{"separator":758},[31658,127147,56782],{},[31662,127149,2557],{"stretchy":702},[31665,127151,127152],{"encoding":31667},"e=(u,v)",[34,127154,127156,127174],{"className":127155,"ariaHidden":758},[31672],[34,127157,127159,127162,127165,127168,127171],{"className":127158},[31676],[34,127160],{"className":127161,"style":32527},[31680],[34,127163,33244],{"className":127164},[31685,31686],[34,127166],{"className":127167,"style":31846},[31845],[34,127169,202],{"className":127170},[31850],[34,127172],{"className":127173,"style":31846},[31845],[34,127175,127177,127180,127183,127186,127189,127192,127195],{"className":127176},[31676],[34,127178],{"className":127179,"style":31774},[31680],[34,127181,456],{"className":127182},[31782],[34,127184,94628],{"className":127185},[31685,31686],[34,127187,31960],{"className":127188},[32119],[34,127190],{"className":127191,"style":32123},[31845],[34,127193,56782],{"className":127194,"style":32062},[31685,31686],[34,127196,2557],{"className":127197},[31690],[15,127199,127200],{},"或者理解成无序对：",[34,127202,127204],{"className":127203},[126385],[34,127205,127207,127225],{"className":127206},[31642],[34,127208,127210],{"className":127209},[31646],[31648,127211,127212],{"xmlns":31650,"display":126394},[31652,127213,127214,127222],{},[31655,127215,127216,127218,127220],{},[31658,127217,94628],{},[31662,127219,31960],{"separator":758},[31658,127221,56782],{},[31665,127223,127224],{"encoding":31667},"{u,v}",[34,127226,127228],{"className":127227,"ariaHidden":758},[31672],[34,127229,127231,127234],{"className":127230},[31676],[34,127232],{"className":127233,"style":38774},[31680],[34,127235,127237,127240,127243,127246],{"className":127236},[31685],[34,127238,94628],{"className":127239},[31685,31686],[34,127241,31960],{"className":127242},[32119],[34,127244],{"className":127245,"style":32123},[31845],[34,127247,56782],{"className":127248,"style":32062},[31685,31686],[15,127250,2632],{},[34,127252,127254],{"className":127253},[126385],[34,127255,127257,127275],{"className":127256},[31642],[34,127258,127260],{"className":127259},[31646],[31648,127261,127262],{"xmlns":31650,"display":126394},[31652,127263,127264,127272],{},[31655,127265,127266,127268,127270],{},[31658,127267,94628],{},[31662,127269,32226],{},[31658,127271,56782],{},[31665,127273,127274],{"encoding":31667},"u-v",[34,127276,127278,127296],{"className":127277,"ariaHidden":758},[31672],[34,127279,127281,127284,127287,127290,127293],{"className":127280},[31676],[34,127282],{"className":127283,"style":32597},[31680],[34,127285,94628],{"className":127286},[31685,31686],[34,127288],{"className":127289,"style":32604},[31845],[34,127291,32226],{"className":127292},[32024],[34,127294],{"className":127295,"style":32604},[31845],[34,127297,127299,127302],{"className":127298},[31676],[34,127300],{"className":127301,"style":32527},[31680],[34,127303,56782],{"className":127304,"style":32062},[31685,31686],[15,127306,127307,127308,258,127336,127364],{},"表示 ",[34,127309,127311,127324],{"className":127310},[31642],[34,127312,127314],{"className":127313},[31646],[31648,127315,127316],{"xmlns":31650},[31652,127317,127318,127322],{},[31655,127319,127320],{},[31658,127321,94628],{},[31665,127323,94628],{"encoding":31667},[34,127325,127327],{"className":127326,"ariaHidden":758},[31672],[34,127328,127330,127333],{"className":127329},[31676],[34,127331],{"className":127332,"style":32527},[31680],[34,127334,94628],{"className":127335},[31685,31686],[34,127337,127339,127352],{"className":127338},[31642],[34,127340,127342],{"className":127341},[31646],[31648,127343,127344],{"xmlns":31650},[31652,127345,127346,127350],{},[31655,127347,127348],{},[31658,127349,56782],{},[31665,127351,56782],{"encoding":31667},[34,127353,127355],{"className":127354,"ariaHidden":758},[31672],[34,127356,127358,127361],{"className":127357},[31676],[34,127359],{"className":127360,"style":32527},[31680],[34,127362,56782],{"className":127363,"style":32062},[31685,31686]," 相连。",[32188,127366],{},[19,127368,127370],{"id":127369},"_2-有向边","2. 有向边",[15,127372,127373],{},"有向边有方向，有起点和终点。",[15,127375,127376,127377,127405,127406,127434],{},"如果边从 ",[34,127378,127380,127393],{"className":127379},[31642],[34,127381,127383],{"className":127382},[31646],[31648,127384,127385],{"xmlns":31650},[31652,127386,127387,127391],{},[31655,127388,127389],{},[31658,127390,94628],{},[31665,127392,94628],{"encoding":31667},[34,127394,127396],{"className":127395,"ariaHidden":758},[31672],[34,127397,127399,127402],{"className":127398},[31676],[34,127400],{"className":127401,"style":32527},[31680],[34,127403,94628],{"className":127404},[31685,31686]," 指向 ",[34,127407,127409,127422],{"className":127408},[31642],[34,127410,127412],{"className":127411},[31646],[31648,127413,127414],{"xmlns":31650},[31652,127415,127416,127420],{},[31655,127417,127418],{},[31658,127419,56782],{},[31665,127421,56782],{"encoding":31667},[34,127423,127425],{"className":127424,"ariaHidden":758},[31672],[34,127426,127428,127431],{"className":127427},[31676],[34,127429],{"className":127430,"style":32527},[31680],[34,127432,56782],{"className":127433,"style":32062},[31685,31686],"，记作：",[34,127436,127438],{"className":127437},[126385],[34,127439,127441,127467],{"className":127440},[31642],[34,127442,127444],{"className":127443},[31646],[31648,127445,127446],{"xmlns":31650,"display":126394},[31652,127447,127448,127464],{},[31655,127449,127450,127452,127454,127456,127458,127460,127462],{},[31658,127451,33244],{},[31662,127453,202],{},[31662,127455,126529],{"stretchy":702},[31658,127457,94628],{},[31662,127459,31960],{"separator":758},[31658,127461,56782],{},[31662,127463,126544],{"stretchy":702},[31665,127465,127466],{"encoding":31667},"e=\\langle u,v\\rangle",[34,127468,127470,127488],{"className":127469,"ariaHidden":758},[31672],[34,127471,127473,127476,127479,127482,127485],{"className":127472},[31676],[34,127474],{"className":127475,"style":32527},[31680],[34,127477,33244],{"className":127478},[31685,31686],[34,127480],{"className":127481,"style":31846},[31845],[34,127483,202],{"className":127484},[31850],[34,127486],{"className":127487,"style":31846},[31845],[34,127489,127491,127494,127497,127500,127503,127506,127509],{"className":127490},[31676],[34,127492],{"className":127493,"style":31774},[31680],[34,127495,126529],{"className":127496},[31782],[34,127498,94628],{"className":127499},[31685,31686],[34,127501,31960],{"className":127502},[32119],[34,127504],{"className":127505,"style":32123},[31845],[34,127507,56782],{"className":127508,"style":32062},[31685,31686],[34,127510,126544],{"className":127511},[31690],[15,127513,104944],{},[105,127515,127516,127547],{},[108,127517,127518,127546],{},[34,127519,127521,127534],{"className":127520},[31642],[34,127522,127524],{"className":127523},[31646],[31648,127525,127526],{"xmlns":31650},[31652,127527,127528,127532],{},[31655,127529,127530],{},[31658,127531,94628],{},[31665,127533,94628],{"encoding":31667},[34,127535,127537],{"className":127536,"ariaHidden":758},[31672],[34,127538,127540,127543],{"className":127539},[31676],[34,127541],{"className":127542,"style":32527},[31680],[34,127544,94628],{"className":127545},[31685,31686],"：起点；",[108,127548,127549,127577],{},[34,127550,127552,127565],{"className":127551},[31642],[34,127553,127555],{"className":127554},[31646],[31648,127556,127557],{"xmlns":31650},[31652,127558,127559,127563],{},[31655,127560,127561],{},[31658,127562,56782],{},[31665,127564,56782],{"encoding":31667},[34,127566,127568],{"className":127567,"ariaHidden":758},[31672],[34,127569,127571,127574],{"className":127570},[31676],[34,127572],{"className":127573,"style":32527},[31680],[34,127575,56782],{"className":127576,"style":32062},[31685,31686],"：终点。",[15,127579,127580],{},"画图时用箭头表示：",[34,127582,127584],{"className":127583},[126385],[34,127585,127587,127605],{"className":127586},[31642],[34,127588,127590],{"className":127589},[31646],[31648,127591,127592],{"xmlns":31650,"display":126394},[31652,127593,127594,127602],{},[31655,127595,127596,127598,127600],{},[31658,127597,94628],{},[31662,127599,33239],{},[31658,127601,56782],{},[31665,127603,127604],{"encoding":31667},"u\\to v",[34,127606,127608,127626],{"className":127607,"ariaHidden":758},[31672],[34,127609,127611,127614,127617,127620,127623],{"className":127610},[31676],[34,127612],{"className":127613,"style":32527},[31680],[34,127615,94628],{"className":127616},[31685,31686],[34,127618],{"className":127619,"style":31846},[31845],[34,127621,33239],{"className":127622},[31850],[34,127624],{"className":127625,"style":31846},[31845],[34,127627,127629,127632],{"className":127628},[31676],[34,127630],{"className":127631,"style":32527},[31680],[34,127633,56782],{"className":127634,"style":32062},[31685,31686],[32188,127636],{},[11,127638,127640],{"id":127639},"三图的基本术语","三、图的基本术语",[19,127642,127644],{"id":127643},"_1-顶点","1. 顶点",[15,127646,127647,127648,127650,127651,347],{},"图中的点叫",[2627,127649,126955],{},"，也叫",[2627,127652,127653],{},"结点",[15,127655,2632],{},[34,127657,127659],{"className":127658},[126385],[34,127660,127662,127702],{"className":127661},[31642],[34,127663,127665],{"className":127664},[31646],[31648,127666,127667],{"xmlns":31650,"display":126394},[31652,127668,127669,127699],{},[31655,127670,127671,127673,127675],{},[31658,127672,126532],{},[31662,127674,202],{},[31655,127676,127677,127683,127685,127691,127693],{},[31743,127678,127679,127681],{},[31658,127680,56782],{},[31749,127682,640],{},[31662,127684,31960],{"separator":758},[31743,127686,127687,127689],{},[31658,127688,56782],{},[31749,127690,768],{},[31662,127692,31960],{"separator":758},[31743,127694,127695,127697],{},[31658,127696,56782],{},[31749,127698,786],{},[31665,127700,127701],{"encoding":31667},"V={v_1,v_2,v_3}",[34,127703,127705,127723],{"className":127704,"ariaHidden":758},[31672],[34,127706,127708,127711,127714,127717,127720],{"className":127707},[31676],[34,127709],{"className":127710,"style":33218},[31680],[34,127712,126532],{"className":127713,"style":32604},[31685,31686],[34,127715],{"className":127716,"style":31846},[31845],[34,127718,202],{"className":127719},[31850],[34,127721],{"className":127722,"style":31846},[31845],[34,127724,127726,127729],{"className":127725},[31676],[34,127727],{"className":127728,"style":38774},[31680],[34,127730,127732,127772,127775,127778,127818,127821,127824],{"className":127731},[31685],[34,127733,127735,127738],{"className":127734},[31685],[34,127736,56782],{"className":127737,"style":32062},[31685,31686],[34,127739,127741],{"className":127740},[31792],[34,127742,127744,127764],{"className":127743},[31796,31797],[34,127745,127747,127761],{"className":127746},[31801],[34,127748,127750],{"className":127749,"style":31806},[31805],[34,127751,127752,127755],{"style":38795},[34,127753],{"className":127754,"style":31814},[31813],[34,127756,127758],{"className":127757},[31818,31819,31820,31821],[34,127759,640],{"className":127760},[31685,31821],[34,127762,31829],{"className":127763},[31828],[34,127765,127767],{"className":127766},[31801],[34,127768,127770],{"className":127769,"style":31836},[31805],[34,127771],{},[34,127773,31960],{"className":127774},[32119],[34,127776],{"className":127777,"style":32123},[31845],[34,127779,127781,127784],{"className":127780},[31685],[34,127782,56782],{"className":127783,"style":32062},[31685,31686],[34,127785,127787],{"className":127786},[31792],[34,127788,127790,127810],{"className":127789},[31796,31797],[34,127791,127793,127807],{"className":127792},[31801],[34,127794,127796],{"className":127795,"style":31806},[31805],[34,127797,127798,127801],{"style":38795},[34,127799],{"className":127800,"style":31814},[31813],[34,127802,127804],{"className":127803},[31818,31819,31820,31821],[34,127805,768],{"className":127806},[31685,31821],[34,127808,31829],{"className":127809},[31828],[34,127811,127813],{"className":127812},[31801],[34,127814,127816],{"className":127815,"style":31836},[31805],[34,127817],{},[34,127819,31960],{"className":127820},[32119],[34,127822],{"className":127823,"style":32123},[31845],[34,127825,127827,127830],{"className":127826},[31685],[34,127828,56782],{"className":127829,"style":32062},[31685,31686],[34,127831,127833],{"className":127832},[31792],[34,127834,127836,127856],{"className":127835},[31796,31797],[34,127837,127839,127853],{"className":127838},[31801],[34,127840,127842],{"className":127841,"style":31806},[31805],[34,127843,127844,127847],{"style":38795},[34,127845],{"className":127846,"style":31814},[31813],[34,127848,127850],{"className":127849},[31818,31819,31820,31821],[34,127851,786],{"className":127852},[31685,31821],[34,127854,31829],{"className":127855},[31828],[34,127857,127859],{"className":127858},[31801],[34,127860,127862],{"className":127861,"style":31836},[31805],[34,127863],{},[15,127865,127866],{},"表示图有三个顶点。",[32188,127868],{},[19,127870,127872],{"id":127871},"_2-边","2. 边",[15,127874,127875,127876,347],{},"图中连接顶点的线叫",[2627,127877,126989],{},[15,127879,127880],{},"无向图的边没有方向。",[15,127882,127883],{},"有向图的边有方向。",[32188,127885],{},[19,127887,127889],{"id":127888},"_3-关联","3. 关联",[15,127891,127048,127892,127920,127921,258,127949,127977],{},[34,127893,127895,127908],{"className":127894},[31642],[34,127896,127898],{"className":127897},[31646],[31648,127899,127900],{"xmlns":31650},[31652,127901,127902,127906],{},[31655,127903,127904],{},[31658,127905,33244],{},[31665,127907,33244],{"encoding":31667},[34,127909,127911],{"className":127910,"ariaHidden":758},[31672],[34,127912,127914,127917],{"className":127913},[31676],[34,127915],{"className":127916,"style":32527},[31680],[34,127918,33244],{"className":127919},[31685,31686]," 连接顶点 ",[34,127922,127924,127937],{"className":127923},[31642],[34,127925,127927],{"className":127926},[31646],[31648,127928,127929],{"xmlns":31650},[31652,127930,127931,127935],{},[31655,127932,127933],{},[31658,127934,94628],{},[31665,127936,94628],{"encoding":31667},[34,127938,127940],{"className":127939,"ariaHidden":758},[31672],[34,127941,127943,127946],{"className":127942},[31676],[34,127944],{"className":127945,"style":32527},[31680],[34,127947,94628],{"className":127948},[31685,31686],[34,127950,127952,127965],{"className":127951},[31642],[34,127953,127955],{"className":127954},[31646],[31648,127956,127957],{"xmlns":31650},[31652,127958,127959,127963],{},[31655,127960,127961],{},[31658,127962,56782],{},[31665,127964,56782],{"encoding":31667},[34,127966,127968],{"className":127967,"ariaHidden":758},[31672],[34,127969,127971,127974],{"className":127970},[31676],[34,127972],{"className":127973,"style":32527},[31680],[34,127975,56782],{"className":127976,"style":32062},[31685,31686],"，则称：",[104833,127979,127980],{},[15,127981,127982,127983,128011,128012,128053],{},"边 ",[34,127984,127986,127999],{"className":127985},[31642],[34,127987,127989],{"className":127988},[31646],[31648,127990,127991],{"xmlns":31650},[31652,127992,127993,127997],{},[31655,127994,127995],{},[31658,127996,33244],{},[31665,127998,33244],{"encoding":31667},[34,128000,128002],{"className":128001,"ariaHidden":758},[31672],[34,128003,128005,128008],{"className":128004},[31676],[34,128006],{"className":128007,"style":32527},[31680],[34,128009,33244],{"className":128010},[31685,31686]," 与顶点 ",[34,128013,128015,128032],{"className":128014},[31642],[34,128016,128018],{"className":128017},[31646],[31648,128019,128020],{"xmlns":31650},[31652,128021,128022,128030],{},[31655,128023,128024,128026,128028],{},[31658,128025,94628],{},[31662,128027,31960],{"separator":758},[31658,128029,56782],{},[31665,128031,127098],{"encoding":31667},[34,128033,128035],{"className":128034,"ariaHidden":758},[31672],[34,128036,128038,128041,128044,128047,128050],{"className":128037},[31676],[34,128039],{"className":128040,"style":38774},[31680],[34,128042,94628],{"className":128043},[31685,31686],[34,128045,31960],{"className":128046},[32119],[34,128048],{"className":128049,"style":32123},[31845],[34,128051,56782],{"className":128052,"style":32062},[31685,31686]," 关联。",[15,128055,128056,128057,128098,128099,128127],{},"也说 ",[34,128058,128060,128077],{"className":128059},[31642],[34,128061,128063],{"className":128062},[31646],[31648,128064,128065],{"xmlns":31650},[31652,128066,128067,128075],{},[31655,128068,128069,128071,128073],{},[31658,128070,94628],{},[31662,128072,31960],{"separator":758},[31658,128074,56782],{},[31665,128076,127098],{"encoding":31667},[34,128078,128080],{"className":128079,"ariaHidden":758},[31672],[34,128081,128083,128086,128089,128092,128095],{"className":128082},[31676],[34,128084],{"className":128085,"style":38774},[31680],[34,128087,94628],{"className":128088},[31685,31686],[34,128090,31960],{"className":128091},[32119],[34,128093],{"className":128094,"style":32123},[31845],[34,128096,56782],{"className":128097,"style":32062},[31685,31686]," 是边 ",[34,128100,128102,128115],{"className":128101},[31642],[34,128103,128105],{"className":128104},[31646],[31648,128106,128107],{"xmlns":31650},[31652,128108,128109,128113],{},[31655,128110,128111],{},[31658,128112,33244],{},[31665,128114,33244],{"encoding":31667},[34,128116,128118],{"className":128117,"ariaHidden":758},[31672],[34,128119,128121,128124],{"className":128120},[31676],[34,128122],{"className":128123,"style":32527},[31680],[34,128125,33244],{"className":128126},[31685,31686]," 的端点。",[32188,128129],{},[19,128131,128133],{"id":128132},"_4-邻接顶点","4. 邻接顶点",[15,128135,128136,128137,347],{},"如果两个顶点之间有边相连，就称它们是",[2627,128138,128139],{},"邻接顶点",[15,128141,2632],{},[34,128143,128145],{"className":128144},[126385],[34,128146,128148,128174],{"className":128147},[31642],[34,128149,128151],{"className":128150},[31646],[31648,128152,128153],{"xmlns":31650,"display":126394},[31652,128154,128155,128171],{},[31655,128156,128157,128159,128161,128163,128165,128167,128169],{},[31662,128158,456],{"stretchy":702},[31658,128160,94628],{},[31662,128162,31960],{"separator":758},[31658,128164,56782],{},[31662,128166,2557],{"stretchy":702},[31662,128168,112296],{},[31658,128170,96058],{},[31665,128172,128173],{"encoding":31667},"(u,v)\\in E",[34,128175,128177,128210],{"className":128176,"ariaHidden":758},[31672],[34,128178,128180,128183,128186,128189,128192,128195,128198,128201,128204,128207],{"className":128179},[31676],[34,128181],{"className":128182,"style":31774},[31680],[34,128184,456],{"className":128185},[31782],[34,128187,94628],{"className":128188},[31685,31686],[34,128190,31960],{"className":128191},[32119],[34,128193],{"className":128194,"style":32123},[31845],[34,128196,56782],{"className":128197,"style":32062},[31685,31686],[34,128199,2557],{"className":128200},[31690],[34,128202],{"className":128203,"style":31846},[31845],[34,128205,112296],{"className":128206},[31850],[34,128208],{"className":128209,"style":31846},[31845],[34,128211,128213,128216],{"className":128212},[31676],[34,128214],{"className":128215,"style":33218},[31680],[34,128217,96058],{"className":128218,"style":106455},[31685,31686],[15,128220,94591,128221,258,128249,128277],{},[34,128222,128224,128237],{"className":128223},[31642],[34,128225,128227],{"className":128226},[31646],[31648,128228,128229],{"xmlns":31650},[31652,128230,128231,128235],{},[31655,128232,128233],{},[31658,128234,94628],{},[31665,128236,94628],{"encoding":31667},[34,128238,128240],{"className":128239,"ariaHidden":758},[31672],[34,128241,128243,128246],{"className":128242},[31676],[34,128244],{"className":128245,"style":32527},[31680],[34,128247,94628],{"className":128248},[31685,31686],[34,128250,128252,128265],{"className":128251},[31642],[34,128253,128255],{"className":128254},[31646],[31648,128256,128257],{"xmlns":31650},[31652,128258,128259,128263],{},[31655,128260,128261],{},[31658,128262,56782],{},[31665,128264,56782],{"encoding":31667},[34,128266,128268],{"className":128267,"ariaHidden":758},[31672],[34,128269,128271,128274],{"className":128270},[31676],[34,128272],{"className":128273,"style":32527},[31680],[34,128275,56782],{"className":128276,"style":32062},[31685,31686]," 邻接。",[32188,128279],{},[19,128281,128283],{"id":128282},"_5-邻接边","5. 邻接边",[15,128285,128286,128287,347],{},"如果两条边有公共端点，则称它们是",[2627,128288,128289],{},"邻接边",[15,128291,126487],{},[34,128293,128295],{"className":128294},[126385],[34,128296,128298,128350],{"className":128297},[31642],[34,128299,128301],{"className":128300},[31646],[31648,128302,128303],{"xmlns":31650,"display":126394},[31652,128304,128305,128347],{},[31655,128306,128307,128313,128315,128317,128319,128321,128323,128325,128327,128329,128335,128337,128339,128341,128343,128345],{},[31743,128308,128309,128311],{},[31658,128310,33244],{},[31749,128312,640],{},[31662,128314,202],{},[31662,128316,456],{"stretchy":702},[31658,128318,94628],{},[31662,128320,31960],{"separator":758},[31658,128322,56782],{},[31662,128324,2557],{"stretchy":702},[31662,128326,31960],{"separator":758},[31845,128328],{"width":71082},[31743,128330,128331,128333],{},[31658,128332,33244],{},[31749,128334,768],{},[31662,128336,202],{},[31662,128338,456],{"stretchy":702},[31658,128340,56782],{},[31662,128342,31960],{"separator":758},[31658,128344,47077],{},[31662,128346,2557],{"stretchy":702},[31665,128348,128349],{"encoding":31667},"e_1=(u,v),\\quad e_2=(v,w)",[34,128351,128353,128408,128491],{"className":128352,"ariaHidden":758},[31672],[34,128354,128356,128359,128399,128402,128405],{"className":128355},[31676],[34,128357],{"className":128358,"style":31860},[31680],[34,128360,128362,128365],{"className":128361},[31685],[34,128363,33244],{"className":128364},[31685,31686],[34,128366,128368],{"className":128367},[31792],[34,128369,128371,128391],{"className":128370},[31796,31797],[34,128372,128374,128388],{"className":128373},[31801],[34,128375,128377],{"className":128376,"style":31806},[31805],[34,128378,128379,128382],{"style":31809},[34,128380],{"className":128381,"style":31814},[31813],[34,128383,128385],{"className":128384},[31818,31819,31820,31821],[34,128386,640],{"className":128387},[31685,31821],[34,128389,31829],{"className":128390},[31828],[34,128392,128394],{"className":128393},[31801],[34,128395,128397],{"className":128396,"style":31836},[31805],[34,128398],{},[34,128400],{"className":128401,"style":31846},[31845],[34,128403,202],{"className":128404},[31850],[34,128406],{"className":128407,"style":31846},[31845],[34,128409,128411,128414,128417,128420,128423,128426,128429,128432,128435,128439,128442,128482,128485,128488],{"className":128410},[31676],[34,128412],{"className":128413,"style":31774},[31680],[34,128415,456],{"className":128416},[31782],[34,128418,94628],{"className":128419},[31685,31686],[34,128421,31960],{"className":128422},[32119],[34,128424],{"className":128425,"style":32123},[31845],[34,128427,56782],{"className":128428,"style":32062},[31685,31686],[34,128430,2557],{"className":128431},[31690],[34,128433,31960],{"className":128434},[32119],[34,128436],{"className":128437,"style":128438},[31845],"margin-right:1em;",[34,128440],{"className":128441,"style":32123},[31845],[34,128443,128445,128448],{"className":128444},[31685],[34,128446,33244],{"className":128447},[31685,31686],[34,128449,128451],{"className":128450},[31792],[34,128452,128454,128474],{"className":128453},[31796,31797],[34,128455,128457,128471],{"className":128456},[31801],[34,128458,128460],{"className":128459,"style":31806},[31805],[34,128461,128462,128465],{"style":31809},[34,128463],{"className":128464,"style":31814},[31813],[34,128466,128468],{"className":128467},[31818,31819,31820,31821],[34,128469,768],{"className":128470},[31685,31821],[34,128472,31829],{"className":128473},[31828],[34,128475,128477],{"className":128476},[31801],[34,128478,128480],{"className":128479,"style":31836},[31805],[34,128481],{},[34,128483],{"className":128484,"style":31846},[31845],[34,128486,202],{"className":128487},[31850],[34,128489],{"className":128490,"style":31846},[31845],[34,128492,128494,128497,128500,128503,128506,128509,128512],{"className":128493},[31676],[34,128495],{"className":128496,"style":31774},[31680],[34,128498,456],{"className":128499},[31782],[34,128501,56782],{"className":128502,"style":32062},[31685,31686],[34,128504,31960],{"className":128505},[32119],[34,128507],{"className":128508,"style":32123},[31845],[34,128510,47077],{"className":128511,"style":47120},[31685,31686],[34,128513,2557],{"className":128514},[31690],[15,128516,128517,128518,13977,128546,128277],{},"它们都经过 ",[34,128519,128521,128534],{"className":128520},[31642],[34,128522,128524],{"className":128523},[31646],[31648,128525,128526],{"xmlns":31650},[31652,128527,128528,128532],{},[31655,128529,128530],{},[31658,128531,56782],{},[31665,128533,56782],{"encoding":31667},[34,128535,128537],{"className":128536,"ariaHidden":758},[31672],[34,128538,128540,128543],{"className":128539},[31676],[34,128541],{"className":128542,"style":32527},[31680],[34,128544,56782],{"className":128545,"style":32062},[31685,31686],[34,128547,128549,128575],{"className":128548},[31642],[34,128550,128552],{"className":128551},[31646],[31648,128553,128554],{"xmlns":31650},[31652,128555,128556,128572],{},[31655,128557,128558,128564,128566],{},[31743,128559,128560,128562],{},[31658,128561,33244],{},[31749,128563,640],{},[31662,128565,31960],{"separator":758},[31743,128567,128568,128570],{},[31658,128569,33244],{},[31749,128571,768],{},[31665,128573,128574],{"encoding":31667},"e_1,e_2",[34,128576,128578],{"className":128577,"ariaHidden":758},[31672],[34,128579,128581,128584,128624,128627,128630],{"className":128580},[31676],[34,128582],{"className":128583,"style":38774},[31680],[34,128585,128587,128590],{"className":128586},[31685],[34,128588,33244],{"className":128589},[31685,31686],[34,128591,128593],{"className":128592},[31792],[34,128594,128596,128616],{"className":128595},[31796,31797],[34,128597,128599,128613],{"className":128598},[31801],[34,128600,128602],{"className":128601,"style":31806},[31805],[34,128603,128604,128607],{"style":31809},[34,128605],{"className":128606,"style":31814},[31813],[34,128608,128610],{"className":128609},[31818,31819,31820,31821],[34,128611,640],{"className":128612},[31685,31821],[34,128614,31829],{"className":128615},[31828],[34,128617,128619],{"className":128618},[31801],[34,128620,128622],{"className":128621,"style":31836},[31805],[34,128623],{},[34,128625,31960],{"className":128626},[32119],[34,128628],{"className":128629,"style":32123},[31845],[34,128631,128633,128636],{"className":128632},[31685],[34,128634,33244],{"className":128635},[31685,31686],[34,128637,128639],{"className":128638},[31792],[34,128640,128642,128662],{"className":128641},[31796,31797],[34,128643,128645,128659],{"className":128644},[31801],[34,128646,128648],{"className":128647,"style":31806},[31805],[34,128649,128650,128653],{"style":31809},[34,128651],{"className":128652,"style":31814},[31813],[34,128654,128656],{"className":128655},[31818,31819,31820,31821],[34,128657,768],{"className":128658},[31685,31821],[34,128660,31829],{"className":128661},[31828],[34,128663,128665],{"className":128664},[31801],[34,128666,128668],{"className":128667,"style":31836},[31805],[34,128669],{},[32188,128671],{},[19,128673,128675],{"id":128674},"_6-自环","6. 自环",[15,128677,128678,128679,347],{},"一条边的两个端点相同，叫",[2627,128680,128681],{},"自环",[15,128683,128684],{},"形式：",[34,128686,128688],{"className":128687},[126385],[34,128689,128691,128717],{"className":128690},[31642],[34,128692,128694],{"className":128693},[31646],[31648,128695,128696],{"xmlns":31650,"display":126394},[31652,128697,128698,128714],{},[31655,128699,128700,128702,128704,128706,128708,128710,128712],{},[31658,128701,33244],{},[31662,128703,202],{},[31662,128705,456],{"stretchy":702},[31658,128707,56782],{},[31662,128709,31960],{"separator":758},[31658,128711,56782],{},[31662,128713,2557],{"stretchy":702},[31665,128715,128716],{"encoding":31667},"e=(v,v)",[34,128718,128720,128738],{"className":128719,"ariaHidden":758},[31672],[34,128721,128723,128726,128729,128732,128735],{"className":128722},[31676],[34,128724],{"className":128725,"style":32527},[31680],[34,128727,33244],{"className":128728},[31685,31686],[34,128730],{"className":128731,"style":31846},[31845],[34,128733,202],{"className":128734},[31850],[34,128736],{"className":128737,"style":31846},[31845],[34,128739,128741,128744,128747,128750,128753,128756,128759],{"className":128740},[31676],[34,128742],{"className":128743,"style":31774},[31680],[34,128745,456],{"className":128746},[31782],[34,128748,56782],{"className":128749,"style":32062},[31685,31686],[34,128751,31960],{"className":128752},[32119],[34,128754],{"className":128755,"style":32123},[31845],[34,128757,56782],{"className":128758,"style":32062},[31685,31686],[34,128760,2557],{"className":128761},[31690],[15,128763,128764],{},"画图时就是顶点上绕一圈的边。",[32188,128766],{},[19,128768,128770],{"id":128769},"_7-重边","7. 重边",[15,128772,128773,128774,347],{},"如果两个顶点之间有多条边相连，这些边叫",[2627,128775,128776],{},"重边",[15,128778,2632],{},[34,128780,128782],{"className":128781},[126385],[34,128783,128785,128837],{"className":128784},[31642],[34,128786,128788],{"className":128787},[31646],[31648,128789,128790],{"xmlns":31650,"display":126394},[31652,128791,128792,128834],{},[31655,128793,128794,128800,128802,128804,128806,128808,128810,128812,128814,128816,128822,128824,128826,128828,128830,128832],{},[31743,128795,128796,128798],{},[31658,128797,33244],{},[31749,128799,640],{},[31662,128801,202],{},[31662,128803,456],{"stretchy":702},[31658,128805,94628],{},[31662,128807,31960],{"separator":758},[31658,128809,56782],{},[31662,128811,2557],{"stretchy":702},[31662,128813,31960],{"separator":758},[31845,128815],{"width":71082},[31743,128817,128818,128820],{},[31658,128819,33244],{},[31749,128821,768],{},[31662,128823,202],{},[31662,128825,456],{"stretchy":702},[31658,128827,94628],{},[31662,128829,31960],{"separator":758},[31658,128831,56782],{},[31662,128833,2557],{"stretchy":702},[31665,128835,128836],{"encoding":31667},"e_1=(u,v),\\quad e_2=(u,v)",[34,128838,128840,128895,128977],{"className":128839,"ariaHidden":758},[31672],[34,128841,128843,128846,128886,128889,128892],{"className":128842},[31676],[34,128844],{"className":128845,"style":31860},[31680],[34,128847,128849,128852],{"className":128848},[31685],[34,128850,33244],{"className":128851},[31685,31686],[34,128853,128855],{"className":128854},[31792],[34,128856,128858,128878],{"className":128857},[31796,31797],[34,128859,128861,128875],{"className":128860},[31801],[34,128862,128864],{"className":128863,"style":31806},[31805],[34,128865,128866,128869],{"style":31809},[34,128867],{"className":128868,"style":31814},[31813],[34,128870,128872],{"className":128871},[31818,31819,31820,31821],[34,128873,640],{"className":128874},[31685,31821],[34,128876,31829],{"className":128877},[31828],[34,128879,128881],{"className":128880},[31801],[34,128882,128884],{"className":128883,"style":31836},[31805],[34,128885],{},[34,128887],{"className":128888,"style":31846},[31845],[34,128890,202],{"className":128891},[31850],[34,128893],{"className":128894,"style":31846},[31845],[34,128896,128898,128901,128904,128907,128910,128913,128916,128919,128922,128925,128928,128968,128971,128974],{"className":128897},[31676],[34,128899],{"className":128900,"style":31774},[31680],[34,128902,456],{"className":128903},[31782],[34,128905,94628],{"className":128906},[31685,31686],[34,128908,31960],{"className":128909},[32119],[34,128911],{"className":128912,"style":32123},[31845],[34,128914,56782],{"className":128915,"style":32062},[31685,31686],[34,128917,2557],{"className":128918},[31690],[34,128920,31960],{"className":128921},[32119],[34,128923],{"className":128924,"style":128438},[31845],[34,128926],{"className":128927,"style":32123},[31845],[34,128929,128931,128934],{"className":128930},[31685],[34,128932,33244],{"className":128933},[31685,31686],[34,128935,128937],{"className":128936},[31792],[34,128938,128940,128960],{"className":128939},[31796,31797],[34,128941,128943,128957],{"className":128942},[31801],[34,128944,128946],{"className":128945,"style":31806},[31805],[34,128947,128948,128951],{"style":31809},[34,128949],{"className":128950,"style":31814},[31813],[34,128952,128954],{"className":128953},[31818,31819,31820,31821],[34,128955,768],{"className":128956},[31685,31821],[34,128958,31829],{"className":128959},[31828],[34,128961,128963],{"className":128962},[31801],[34,128964,128966],{"className":128965,"style":31836},[31805],[34,128967],{},[34,128969],{"className":128970,"style":31846},[31845],[34,128972,202],{"className":128973},[31850],[34,128975],{"className":128976,"style":31846},[31845],[34,128978,128980,128983,128986,128989,128992,128995,128998],{"className":128979},[31676],[34,128981],{"className":128982,"style":31774},[31680],[34,128984,456],{"className":128985},[31782],[34,128987,94628],{"className":128988},[31685,31686],[34,128990,31960],{"className":128991},[32119],[34,128993],{"className":128994,"style":32123},[31845],[34,128996,56782],{"className":128997,"style":32062},[31685,31686],[34,128999,2557],{"className":129000},[31690],[15,129002,129003,129004,129170],{},"且 ",[34,129005,129007,129033],{"className":129006},[31642],[34,129008,129010],{"className":129009},[31646],[31648,129011,129012],{"xmlns":31650},[31652,129013,129014,129030],{},[31655,129015,129016,129022,129024],{},[31743,129017,129018,129020],{},[31658,129019,33244],{},[31749,129021,640],{},[31662,129023,54707],{"mathvariant":38079},[31743,129025,129026,129028],{},[31658,129027,33244],{},[31749,129029,768],{},[31665,129031,129032],{"encoding":31667},"e_1\\neq e_2",[34,129034,129036,129124],{"className":129035,"ariaHidden":758},[31672],[34,129037,129039,129042,129082,129085,129121],{"className":129038},[31676],[34,129040],{"className":129041,"style":43217},[31680],[34,129043,129045,129048],{"className":129044},[31685],[34,129046,33244],{"className":129047},[31685,31686],[34,129049,129051],{"className":129050},[31792],[34,129052,129054,129074],{"className":129053},[31796,31797],[34,129055,129057,129071],{"className":129056},[31801],[34,129058,129060],{"className":129059,"style":31806},[31805],[34,129061,129062,129065],{"style":31809},[34,129063],{"className":129064,"style":31814},[31813],[34,129066,129068],{"className":129067},[31818,31819,31820,31821],[34,129069,640],{"className":129070},[31685,31821],[34,129072,31829],{"className":129073},[31828],[34,129075,129077],{"className":129076},[31801],[34,129078,129080],{"className":129079,"style":31836},[31805],[34,129081],{},[34,129083],{"className":129084,"style":31846},[31845],[34,129086,129088,129115,129118],{"className":129087},[31850],[34,129089,129091],{"className":129090},[31850],[34,129092,129094],{"className":129093},[31685,54774],[34,129095,129097],{"className":129096},[54778],[34,129098,129100,129103,129112],{"className":129099},[54782],[34,129101],{"className":129102,"style":43217},[31680],[34,129104,129106],{"className":129105},[54789],[34,129107,129109],{"className":129108},[31685],[34,129110,54796],{"className":129111},[31850],[34,129113],{"className":129114},[54800],[34,129116],{"className":129117},[31845,54804],[34,129119,202],{"className":129120},[31850],[34,129122],{"className":129123,"style":31846},[31845],[34,129125,129127,129130],{"className":129126},[31676],[34,129128],{"className":129129,"style":31860},[31680],[34,129131,129133,129136],{"className":129132},[31685],[34,129134,33244],{"className":129135},[31685,31686],[34,129137,129139],{"className":129138},[31792],[34,129140,129142,129162],{"className":129141},[31796,31797],[34,129143,129145,129159],{"className":129144},[31801],[34,129146,129148],{"className":129147,"style":31806},[31805],[34,129149,129150,129153],{"style":31809},[34,129151],{"className":129152,"style":31814},[31813],[34,129154,129156],{"className":129155},[31818,31819,31820,31821],[34,129157,768],{"className":129158},[31685,31821],[34,129160,31829],{"className":129161},[31828],[34,129163,129165],{"className":129164},[31801],[34,129166,129168],{"className":129167,"style":31836},[31805],[34,129169],{},"，它们就是重边。",[32188,129172],{},[19,129174,129176],{"id":129175},"_8-孤立点","8. 孤立点",[15,129178,129179,129180,347],{},"如果某个顶点没有任何边与它关联，则称它是",[2627,129181,129182],{},"孤立点",[15,129184,129185],{},"孤立点的度为 0。",[32188,129187],{},[11,129189,129191],{"id":129190},"四图的分类","四、图的分类",[19,129193,129195],{"id":129194},"_1-零图","1. 零图",[15,129197,129198,129199,347],{},"只含一些孤立点、没有边的图，叫",[2627,129200,129201],{},"零图",[15,129203,129204],{},"也就是：",[34,129206,129208],{"className":129207},[126385],[34,129209,129211,129230],{"className":129210},[31642],[34,129212,129214],{"className":129213},[31646],[31648,129215,129216],{"xmlns":31650,"display":126394},[31652,129217,129218,129227],{},[31655,129219,129220,129222,129224],{},[31658,129221,96058],{},[31662,129223,202],{},[31658,129225,129226],{"mathvariant":38079},"∅",[31665,129228,129229],{"encoding":31667},"E=\\varnothing",[34,129231,129233,129251],{"className":129232,"ariaHidden":758},[31672],[34,129234,129236,129239,129242,129245,129248],{"className":129235},[31676],[34,129237],{"className":129238,"style":33218},[31680],[34,129240,96058],{"className":129241,"style":106455},[31685,31686],[34,129243],{"className":129244,"style":31846},[31845],[34,129246,202],{"className":129247},[31850],[34,129249],{"className":129250,"style":31846},[31845],[34,129252,129254,129258],{"className":129253},[31676],[34,129255],{"className":129256,"style":129257},[31680],"height:0.6633em;vertical-align:-0.0817em;",[34,129259,129226],{"className":129260},[31685,129261],"amsrm",[32188,129263],{},[19,129265,129267],{"id":129266},"_2-平凡图","2. 平凡图",[15,129269,129270,129271,347],{},"只有一个孤立顶点的图，叫",[2627,129272,129273],{},"平凡图",[32188,129275],{},[19,129277,129279],{"id":129278},"_3-多重图","3. 多重图",[15,129281,129282,129283,347],{},"含有重边的图叫",[2627,129284,129285],{},"多重图",[15,129287,129288],{},"教材里含重边的图称为多重边图。",[32188,129290],{},[19,129292,129294],{"id":129293},"_4-简单图","4. 简单图",[15,129296,129297,129298,347],{},"不含自环，也不含重边的图，叫",[2627,129299,129300],{},"简单图",[34,129302,129304],{"className":129303},[126385],[34,129305,129307,129330],{"className":129306},[31642],[34,129308,129310],{"className":129309},[31646],[31648,129311,129312],{"xmlns":31650,"display":126394},[31652,129313,129314,129327],{},[31655,129315,129316],{},[126400,129317,129318],{"notation":126402},[71090,129319,129320],{"scriptlevel":459,"displaystyle":702},[71090,129321,129322],{"scriptlevel":459,"displaystyle":702},[71090,129323,129324],{"scriptlevel":459,"displaystyle":758},[39361,129325,129326],{},"简单图 = 无自环 + 无重边",[31665,129328,129329],{"encoding":31667},"\\boxed{\\text{简单图 = 无自环 + 无重边}}",[34,129331,129333],{"className":129332,"ariaHidden":758},[31672],[34,129334,129336,129340],{"className":129335},[31676],[34,129337],{"className":129338,"style":129339},[31680],"height:1.4467em;vertical-align:-0.4233em;",[34,129341,129343],{"className":129342},[31685],[34,129344,129346,129402],{"className":129345},[31796,31797],[34,129347,129349,129399],{"className":129348},[31801],[34,129350,129352,129390],{"className":129351,"style":126437},[31805],[34,129353,129355,129359],{"style":129354},"top:-3.4467em;",[34,129356],{"className":129357,"style":129358},[31813],"height:3.4467em;",[34,129360,129362],{"className":129361},[126448],[34,129363,129365],{"className":129364},[31685],[34,129366,129368],{"className":129367},[31685],[34,129369,129371,129374,129378,129382,129386],{"className":129370},[31685,2638],[34,129372,129300],{"className":129373},[31685,105462],[34,129375,129377],{"className":129376},[31685]," = ",[34,129379,129381],{"className":129380},[31685,105462],"无自环",[34,129383,129385],{"className":129384},[31685]," + ",[34,129387,129389],{"className":129388},[31685,105462],"无重边",[34,129391,129392,129395],{"style":126463},[34,129393],{"className":129394,"style":129358},[31813],[34,129396],{"className":129397,"style":129398},[126470,126471],"height:1.4467em;border-style:solid;border-width:0.04em;",[34,129400,31829],{"className":129401},[31828],[34,129403,129405],{"className":129404},[31801],[34,129406,129409],{"className":129407,"style":129408},[31805],"height:0.4233em;",[34,129410],{},[15,129412,129413],{},"这是后面讨论很多定理时最常用的图。",[32188,129415],{},[19,129417,129419],{"id":129418},"_5-完全图","5. 完全图",[15,129421,129422,129423,347],{},"如果一个简单图中任意两个不同顶点之间都有边相连，则叫",[2627,129424,129425],{},"完全图",[15,129427,129428,129429,129457],{},"含 ",[34,129430,129432,129445],{"className":129431},[31642],[34,129433,129435],{"className":129434},[31646],[31648,129436,129437],{"xmlns":31650},[31652,129438,129439,129443],{},[31655,129440,129441],{},[31658,129442,31660],{},[31665,129444,31660],{"encoding":31667},[34,129446,129448],{"className":129447,"ariaHidden":758},[31672],[34,129449,129451,129454],{"className":129450},[31676],[34,129452],{"className":129453,"style":32527},[31680],[34,129455,31660],{"className":129456},[31685,31686]," 个顶点的无向完全图记作：",[34,129459,129461],{"className":129460},[126385],[34,129462,129464,129482],{"className":129463},[31642],[34,129465,129467],{"className":129466},[31646],[31648,129468,129469],{"xmlns":31650,"display":126394},[31652,129470,129471,129479],{},[31655,129472,129473],{},[31743,129474,129475,129477],{},[31658,129476,38691],{},[31658,129478,31660],{},[31665,129480,129481],{"encoding":31667},"K_n",[34,129483,129485],{"className":129484,"ariaHidden":758},[31672],[34,129486,129488,129491],{"className":129487},[31676],[34,129489],{"className":129490,"style":70915},[31680],[34,129492,129494,129497],{"className":129493},[31685],[34,129495,38691],{"className":129496,"style":38706},[31685,31686],[34,129498,129500],{"className":129499},[31792],[34,129501,129503,129524],{"className":129502},[31796,31797],[34,129504,129506,129521],{"className":129505},[31801],[34,129507,129509],{"className":129508,"style":32362},[31805],[34,129510,129512,129515],{"style":129511},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[34,129513],{"className":129514,"style":31814},[31813],[34,129516,129518],{"className":129517},[31818,31819,31820,31821],[34,129519,31660],{"className":129520},[31685,31686,31821],[34,129522,31829],{"className":129523},[31828],[34,129525,129527],{"className":129526},[31801],[34,129528,129530],{"className":129529,"style":31836},[31805],[34,129531],{},[15,129533,129534],{},"无向完全图的边数是：",[34,129536,129538],{"className":129537},[126385],[34,129539,129541,129579],{"className":129540},[31642],[34,129542,129544],{"className":129543},[31646],[31648,129545,129546],{"xmlns":31650,"display":126394},[31652,129547,129548,129576],{},[31655,129549,129550],{},[126400,129551,129552],{"notation":126402},[71090,129553,129554],{"scriptlevel":459,"displaystyle":702},[71090,129555,129556],{"scriptlevel":459,"displaystyle":702},[71090,129557,129558],{"scriptlevel":459,"displaystyle":758},[41453,129559,129560,129574],{},[31655,129561,129562,129564,129566,129568,129570,129572],{},[31658,129563,31660],{},[31662,129565,456],{"stretchy":702},[31658,129567,31660],{},[31662,129569,32226],{},[31749,129571,640],{},[31662,129573,2557],{"stretchy":702},[31749,129575,768],{},[31665,129577,129578],{"encoding":31667},"\\boxed{\\frac{n(n-1)}{2}}",[34,129580,129582],{"className":129581,"ariaHidden":758},[31672],[34,129583,129585,129589],{"className":129584},[31676],[34,129586],{"className":129587,"style":129588},[31680],"height:2.793em;vertical-align:-1.026em;",[34,129590,129592],{"className":129591},[31685],[34,129593,129595,129718],{"className":129594},[31796,31797],[34,129596,129598,129715],{"className":129597},[31801],[34,129599,129602,129705],{"className":129600,"style":129601},[31805],"height:1.767em;",[34,129603,129605,129609],{"style":129604},"top:-4.793em;",[34,129606],{"className":129607,"style":129608},[31813],"height:4.793em;",[34,129610,129612],{"className":129611},[126448],[34,129613,129615],{"className":129614},[31685],[34,129616,129618],{"className":129617},[31685],[34,129619,129621,129624,129702],{"className":129620},[31685],[34,129622],{"className":129623},[31782,41488],[34,129625,129627],{"className":129626},[41453],[34,129628,129630,129693],{"className":129629},[31796,31797],[34,129631,129633,129690],{"className":129632},[31801],[34,129634,129637,129649,129657],{"className":129635,"style":129636},[31805],"height:1.427em;",[34,129638,129640,129643],{"style":129639},"top:-2.314em;",[34,129641],{"className":129642,"style":41508},[31813],[34,129644,129646],{"className":129645},[31685],[34,129647,768],{"className":129648},[31685],[34,129650,129651,129654],{"style":41520},[34,129652],{"className":129653,"style":41508},[31813],[34,129655],{"className":129656,"style":41528},[41527],[34,129658,129660,129663],{"style":129659},"top:-3.677em;",[34,129661],{"className":129662,"style":41508},[31813],[34,129664,129666,129669,129672,129675,129678,129681,129684,129687],{"className":129665},[31685],[34,129667,31660],{"className":129668},[31685,31686],[34,129670,456],{"className":129671},[31782],[34,129673,31660],{"className":129674},[31685,31686],[34,129676],{"className":129677,"style":32604},[31845],[34,129679,32226],{"className":129680},[32024],[34,129682],{"className":129683,"style":32604},[31845],[34,129685,640],{"className":129686},[31685],[34,129688,2557],{"className":129689},[31690],[34,129691,31829],{"className":129692},[31828],[34,129694,129696],{"className":129695},[31801],[34,129697,129700],{"className":129698,"style":129699},[31805],"height:0.686em;",[34,129701],{},[34,129703],{"className":129704},[31690,41488],[34,129706,129708,129711],{"style":129707},"top:-3.767em;",[34,129709],{"className":129710,"style":129608},[31813],[34,129712],{"className":129713,"style":129714},[126470,126471],"height:2.793em;border-style:solid;border-width:0.04em;",[34,129716,31829],{"className":129717},[31828],[34,129719,129721],{"className":129720},[31801],[34,129722,129725],{"className":129723,"style":129724},[31805],"height:1.026em;",[34,129726],{},[15,129728,93310,129729,129757],{},[34,129730,129732,129745],{"className":129731},[31642],[34,129733,129735],{"className":129734},[31646],[31648,129736,129737],{"xmlns":31650},[31652,129738,129739,129743],{},[31655,129740,129741],{},[31658,129742,31660],{},[31665,129744,31660],{"encoding":31667},[34,129746,129748],{"className":129747,"ariaHidden":758},[31672],[34,129749,129751,129754],{"className":129750},[31676],[34,129752],{"className":129753,"style":32527},[31680],[34,129755,31660],{"className":129756},[31685,31686]," 个顶点中任意选两个顶点连一条边：",[34,129759,129761],{"className":129760},[126385],[34,129762,129764,129804],{"className":129763},[31642],[34,129765,129767],{"className":129766},[31646],[31648,129768,129769],{"xmlns":31650,"display":126394},[31652,129770,129771,129801],{},[31655,129772,129773,129781,129783],{},[38846,129774,129775,129777,129779],{},[31658,129776,47384],{},[31658,129778,31660],{},[31749,129780,768],{},[31662,129782,202],{},[41453,129784,129785,129799],{},[31655,129786,129787,129789,129791,129793,129795,129797],{},[31658,129788,31660],{},[31662,129790,456],{"stretchy":702},[31658,129792,31660],{},[31662,129794,32226],{},[31749,129796,640],{},[31662,129798,2557],{"stretchy":702},[31749,129800,768],{},[31665,129802,129803],{"encoding":31667},"C_n^2=\\frac{n(n-1)}{2}",[34,129805,129807,129876],{"className":129806,"ariaHidden":758},[31672],[34,129808,129810,129814,129867,129870,129873],{"className":129809},[31676],[34,129811],{"className":129812,"style":129813},[31680],"height:1.1111em;vertical-align:-0.247em;",[34,129815,129817,129820],{"className":129816},[31685],[34,129818,47384],{"className":129819,"style":38706},[31685,31686],[34,129821,129823],{"className":129822},[31792],[34,129824,129826,129859],{"className":129825},[31796,31797],[34,129827,129829,129856],{"className":129828},[31801],[34,129830,129833,129844],{"className":129831,"style":129832},[31805],"height:0.8641em;",[34,129834,129835,129838],{"style":52324},[34,129836],{"className":129837,"style":31814},[31813],[34,129839,129841],{"className":129840},[31818,31819,31820,31821],[34,129842,31660],{"className":129843},[31685,31686,31821],[34,129845,129847,129850],{"style":129846},"top:-3.113em;margin-right:0.05em;",[34,129848],{"className":129849,"style":31814},[31813],[34,129851,129853],{"className":129852},[31818,31819,31820,31821],[34,129854,768],{"className":129855},[31685,31821],[34,129857,31829],{"className":129858},[31828],[34,129860,129862],{"className":129861},[31801],[34,129863,129865],{"className":129864,"style":52354},[31805],[34,129866],{},[34,129868],{"className":129869,"style":31846},[31845],[34,129871,202],{"className":129872},[31850],[34,129874],{"className":129875,"style":31846},[31845],[34,129877,129879,129883],{"className":129878},[31676],[34,129880],{"className":129881,"style":129882},[31680],"height:2.113em;vertical-align:-0.686em;",[34,129884,129886,129889,129963],{"className":129885},[31685],[34,129887],{"className":129888},[31782,41488],[34,129890,129892],{"className":129891},[41453],[34,129893,129895,129955],{"className":129894},[31796,31797],[34,129896,129898,129952],{"className":129897},[31801],[34,129899,129901,129912,129920],{"className":129900,"style":129636},[31805],[34,129902,129903,129906],{"style":129639},[34,129904],{"className":129905,"style":41508},[31813],[34,129907,129909],{"className":129908},[31685],[34,129910,768],{"className":129911},[31685],[34,129913,129914,129917],{"style":41520},[34,129915],{"className":129916,"style":41508},[31813],[34,129918],{"className":129919,"style":41528},[41527],[34,129921,129922,129925],{"style":129659},[34,129923],{"className":129924,"style":41508},[31813],[34,129926,129928,129931,129934,129937,129940,129943,129946,129949],{"className":129927},[31685],[34,129929,31660],{"className":129930},[31685,31686],[34,129932,456],{"className":129933},[31782],[34,129935,31660],{"className":129936},[31685,31686],[34,129938],{"className":129939,"style":32604},[31845],[34,129941,32226],{"className":129942},[32024],[34,129944],{"className":129945,"style":32604},[31845],[34,129947,640],{"className":129948},[31685],[34,129950,2557],{"className":129951},[31690],[34,129953,31829],{"className":129954},[31828],[34,129956,129958],{"className":129957},[31801],[34,129959,129961],{"className":129960,"style":129699},[31805],[34,129962],{},[34,129964],{"className":129965},[31690,41488],[15,129967,129968],{},"如果是有向完全图，每一对不同顶点之间有两个相反方向的有向边，则边数是：",[34,129970,129972],{"className":129971},[126385],[34,129973,129975,130009],{"className":129974},[31642],[34,129976,129978],{"className":129977},[31646],[31648,129979,129980],{"xmlns":31650,"display":126394},[31652,129981,129982,130006],{},[31655,129983,129984],{},[126400,129985,129986],{"notation":126402},[71090,129987,129988],{"scriptlevel":459,"displaystyle":702},[71090,129989,129990],{"scriptlevel":459,"displaystyle":702},[71090,129991,129992],{"scriptlevel":459,"displaystyle":758},[31655,129993,129994,129996,129998,130000,130002,130004],{},[31658,129995,31660],{},[31662,129997,456],{"stretchy":702},[31658,129999,31660],{},[31662,130001,32226],{},[31749,130003,640],{},[31662,130005,2557],{"stretchy":702},[31665,130007,130008],{"encoding":31667},"\\boxed{n(n-1)}",[34,130010,130012],{"className":130011,"ariaHidden":758},[31672],[34,130013,130015,130019],{"className":130014},[31676],[34,130016],{"className":130017,"style":130018},[31680],"height:1.68em;vertical-align:-0.59em;",[34,130020,130022],{"className":130021},[31685],[34,130023,130025,130085],{"className":130024},[31796,31797],[34,130026,130028,130082],{"className":130027},[31801],[34,130029,130032,130072],{"className":130030,"style":130031},[31805],"height:1.09em;",[34,130033,130035,130039],{"style":130034},"top:-3.68em;",[34,130036],{"className":130037,"style":130038},[31813],"height:3.68em;",[34,130040,130042],{"className":130041},[126448],[34,130043,130045],{"className":130044},[31685],[34,130046,130048,130051,130054,130057,130060,130063,130066,130069],{"className":130047},[31685],[34,130049,31660],{"className":130050},[31685,31686],[34,130052,456],{"className":130053},[31782],[34,130055,31660],{"className":130056},[31685,31686],[34,130058],{"className":130059,"style":32604},[31845],[34,130061,32226],{"className":130062},[32024],[34,130064],{"className":130065,"style":32604},[31845],[34,130067,640],{"className":130068},[31685],[34,130070,2557],{"className":130071},[31690],[34,130073,130075,130078],{"style":130074},"top:-3.09em;",[34,130076],{"className":130077,"style":130038},[31813],[34,130079],{"className":130080,"style":130081},[126470,126471],"height:1.68em;border-style:solid;border-width:0.04em;",[34,130083,31829],{"className":130084},[31828],[34,130086,130088],{"className":130087},[31801],[34,130089,130092],{"className":130090,"style":130091},[31805],"height:0.59em;",[34,130093],{},[32188,130095],{},[11,130097,130099],{"id":130098},"五顶点的度","五、顶点的度",[19,130101,130103],{"id":130102},"_1-无向图中的度","1. 无向图中的度",[15,130105,130106,130107,130135,130136,127434],{},"在无向图中，一个顶点 ",[34,130108,130110,130123],{"className":130109},[31642],[34,130111,130113],{"className":130112},[31646],[31648,130114,130115],{"xmlns":31650},[31652,130116,130117,130121],{},[31655,130118,130119],{},[31658,130120,56782],{},[31665,130122,56782],{"encoding":31667},[34,130124,130126],{"className":130125,"ariaHidden":758},[31672],[34,130127,130129,130132],{"className":130128},[31676],[34,130130],{"className":130131,"style":32527},[31680],[34,130133,56782],{"className":130134,"style":32062},[31685,31686]," 的",[2627,130137,130138],{},"度",[34,130140,130142],{"className":130141},[126385],[34,130143,130145,130165],{"className":130144},[31642],[34,130146,130148],{"className":130147},[31646],[31648,130149,130150],{"xmlns":31650,"display":126394},[31652,130151,130152,130162],{},[31655,130153,130154,130156,130158,130160],{},[31658,130155,46900],{},[31662,130157,456],{"stretchy":702},[31658,130159,56782],{},[31662,130161,2557],{"stretchy":702},[31665,130163,130164],{"encoding":31667},"d(v)",[34,130166,130168],{"className":130167,"ariaHidden":758},[31672],[34,130169,130171,130174,130177,130180,130183],{"className":130170},[31676],[34,130172],{"className":130173,"style":31774},[31680],[34,130175,46900],{"className":130176},[31685,31686],[34,130178,456],{"className":130179},[31782],[34,130181,56782],{"className":130182,"style":32062},[31685,31686],[34,130184,2557],{"className":130185},[31690],[15,130187,130188,130189,130217],{},"表示与 ",[34,130190,130192,130205],{"className":130191},[31642],[34,130193,130195],{"className":130194},[31646],[31648,130196,130197],{"xmlns":31650},[31652,130198,130199,130203],{},[31655,130200,130201],{},[31658,130202,56782],{},[31665,130204,56782],{"encoding":31667},[34,130206,130208],{"className":130207,"ariaHidden":758},[31672],[34,130209,130211,130214],{"className":130210},[31676],[34,130212],{"className":130213,"style":32527},[31680],[34,130215,56782],{"className":130216,"style":32062},[31685,31686]," 关联的边数。",[15,130219,8882],{},[34,130221,130223],{"className":130222},[126385],[34,130224,130226,130253],{"className":130225},[31642],[34,130227,130229],{"className":130228},[31646],[31648,130230,130231],{"xmlns":31650,"display":126394},[31652,130232,130233,130250],{},[31655,130234,130235],{},[126400,130236,130237],{"notation":126402},[71090,130238,130239],{"scriptlevel":459,"displaystyle":702},[71090,130240,130241],{"scriptlevel":459,"displaystyle":702},[71090,130242,130243],{"scriptlevel":459,"displaystyle":758},[31655,130244,130245,130248],{},[39361,130246,130247],{},"自环对度的贡献是 ",[31749,130249,768],{},[31665,130251,130252],{"encoding":31667},"\\boxed{\\text{自环对度的贡献是 }2}",[34,130254,130256],{"className":130255,"ariaHidden":758},[31672],[34,130257,130259,130262],{"className":130258},[31676],[34,130260],{"className":130261,"style":126424},[31680],[34,130263,130265],{"className":130264},[31685],[34,130266,130268,130313],{"className":130267},[31796,31797],[34,130269,130271,130310],{"className":130270},[31801],[34,130272,130274,130302],{"className":130273,"style":126437},[31805],[34,130275,130276,130279],{"style":126440},[34,130277],{"className":130278,"style":126444},[31813],[34,130280,130282],{"className":130281},[126448],[34,130283,130285],{"className":130284},[31685],[34,130286,130288,130299],{"className":130287},[31685],[34,130289,130291,130295],{"className":130290},[31685,2638],[34,130292,130294],{"className":130293},[31685,105462],"自环对度的贡献是",[34,130296,130298],{"className":130297},[31685]," ",[34,130300,768],{"className":130301},[31685],[34,130303,130304,130307],{"style":126463},[34,130305],{"className":130306,"style":126444},[31813],[34,130308],{"className":130309,"style":126472},[126470,126471],[34,130311,31829],{"className":130312},[31828],[34,130314,130316],{"className":130315},[31801],[34,130317,130319],{"className":130318,"style":126482},[31805],[34,130320],{},[15,130322,130323],{},"因为自环的两个端点都在同一个顶点上。",[32188,130325],{},[19,130327,130329],{"id":130328},"_2-有向图中的出度和入度","2. 有向图中的出度和入度",[15,130331,130332],{},"在有向图中：",[253,130334,130335],{"id":130335},"出度",[15,130337,130338,130339,130367],{},"从顶点 ",[34,130340,130342,130355],{"className":130341},[31642],[34,130343,130345],{"className":130344},[31646],[31648,130346,130347],{"xmlns":31650},[31652,130348,130349,130353],{},[31655,130350,130351],{},[31658,130352,56782],{},[31665,130354,56782],{"encoding":31667},[34,130356,130358],{"className":130357,"ariaHidden":758},[31672],[34,130359,130361,130364],{"className":130360},[31676],[34,130362],{"className":130363,"style":32527},[31680],[34,130365,56782],{"className":130366,"style":32062},[31685,31686]," 发出的边数叫出度，记作：",[34,130369,130371],{"className":130370},[126385],[34,130372,130374,130398],{"className":130373},[31642],[34,130375,130377],{"className":130376},[31646],[31648,130378,130379],{"xmlns":31650,"display":126394},[31652,130380,130381,130395],{},[31655,130382,130383,130389,130391,130393],{},[37624,130384,130385,130387],{},[31658,130386,46900],{},[31662,130388,223],{},[31662,130390,456],{"stretchy":702},[31658,130392,56782],{},[31662,130394,2557],{"stretchy":702},[31665,130396,130397],{"encoding":31667},"d^+(v)",[34,130399,130401],{"className":130400,"ariaHidden":758},[31672],[34,130402,130404,130408,130438,130441,130444],{"className":130403},[31676],[34,130405],{"className":130406,"style":130407},[31680],"height:1.0713em;vertical-align:-0.25em;",[34,130409,130411,130414],{"className":130410},[31685],[34,130412,46900],{"className":130413},[31685,31686],[34,130415,130417],{"className":130416},[31792],[34,130418,130420],{"className":130419},[31796],[34,130421,130423],{"className":130422},[31801],[34,130424,130427],{"className":130425,"style":130426},[31805],"height:0.8213em;",[34,130428,130429,130432],{"style":129846},[34,130430],{"className":130431,"style":31814},[31813],[34,130433,130435],{"className":130434},[31818,31819,31820,31821],[34,130436,223],{"className":130437},[32024,31821],[34,130439,456],{"className":130440},[31782],[34,130442,56782],{"className":130443,"style":32062},[31685,31686],[34,130445,2557],{"className":130446},[31690],[253,130448,130449],{"id":130449},"入度",[15,130451,130452,130453,130481],{},"进入顶点 ",[34,130454,130456,130469],{"className":130455},[31642],[34,130457,130459],{"className":130458},[31646],[31648,130460,130461],{"xmlns":31650},[31652,130462,130463,130467],{},[31655,130464,130465],{},[31658,130466,56782],{},[31665,130468,56782],{"encoding":31667},[34,130470,130472],{"className":130471,"ariaHidden":758},[31672],[34,130473,130475,130478],{"className":130474},[31676],[34,130476],{"className":130477,"style":32527},[31680],[34,130479,56782],{"className":130480,"style":32062},[31685,31686]," 的边数叫入度，记作：",[34,130483,130485],{"className":130484},[126385],[34,130486,130488,130512],{"className":130487},[31642],[34,130489,130491],{"className":130490},[31646],[31648,130492,130493],{"xmlns":31650,"display":126394},[31652,130494,130495,130509],{},[31655,130496,130497,130503,130505,130507],{},[37624,130498,130499,130501],{},[31658,130500,46900],{},[31662,130502,32226],{},[31662,130504,456],{"stretchy":702},[31658,130506,56782],{},[31662,130508,2557],{"stretchy":702},[31665,130510,130511],{"encoding":31667},"d^-(v)",[34,130513,130515],{"className":130514,"ariaHidden":758},[31672],[34,130516,130518,130521,130550,130553,130556],{"className":130517},[31676],[34,130519],{"className":130520,"style":130407},[31680],[34,130522,130524,130527],{"className":130523},[31685],[34,130525,46900],{"className":130526},[31685,31686],[34,130528,130530],{"className":130529},[31792],[34,130531,130533],{"className":130532},[31796],[34,130534,130536],{"className":130535},[31801],[34,130537,130539],{"className":130538,"style":130426},[31805],[34,130540,130541,130544],{"style":129846},[34,130542],{"className":130543,"style":31814},[31813],[34,130545,130547],{"className":130546},[31818,31819,31820,31821],[34,130548,32226],{"className":130549},[32024,31821],[34,130551,456],{"className":130552},[31782],[34,130554,56782],{"className":130555,"style":32062},[31685,31686],[34,130557,2557],{"className":130558},[31690],[253,130560,130561],{"id":130561},"总度",[34,130563,130565],{"className":130564},[126385],[34,130566,130568,130616],{"className":130567},[31642],[34,130569,130571],{"className":130570},[31646],[31648,130572,130573],{"xmlns":31650,"display":126394},[31652,130574,130575,130613],{},[31655,130576,130577,130579,130581,130583,130585,130587,130593,130595,130597,130599,130601,130607,130609,130611],{},[31658,130578,46900],{},[31662,130580,456],{"stretchy":702},[31658,130582,56782],{},[31662,130584,2557],{"stretchy":702},[31662,130586,202],{},[37624,130588,130589,130591],{},[31658,130590,46900],{},[31662,130592,223],{},[31662,130594,456],{"stretchy":702},[31658,130596,56782],{},[31662,130598,2557],{"stretchy":702},[31662,130600,223],{},[37624,130602,130603,130605],{},[31658,130604,46900],{},[31662,130606,32226],{},[31662,130608,456],{"stretchy":702},[31658,130610,56782],{},[31662,130612,2557],{"stretchy":702},[31665,130614,130615],{"encoding":31667},"d(v)=d^+(v)+d^-(v)",[34,130617,130619,130646,130699],{"className":130618,"ariaHidden":758},[31672],[34,130620,130622,130625,130628,130631,130634,130637,130640,130643],{"className":130621},[31676],[34,130623],{"className":130624,"style":31774},[31680],[34,130626,46900],{"className":130627},[31685,31686],[34,130629,456],{"className":130630},[31782],[34,130632,56782],{"className":130633,"style":32062},[31685,31686],[34,130635,2557],{"className":130636},[31690],[34,130638],{"className":130639,"style":31846},[31845],[34,130641,202],{"className":130642},[31850],[34,130644],{"className":130645,"style":31846},[31845],[34,130647,130649,130652,130681,130684,130687,130690,130693,130696],{"className":130648},[31676],[34,130650],{"className":130651,"style":130407},[31680],[34,130653,130655,130658],{"className":130654},[31685],[34,130656,46900],{"className":130657},[31685,31686],[34,130659,130661],{"className":130660},[31792],[34,130662,130664],{"className":130663},[31796],[34,130665,130667],{"className":130666},[31801],[34,130668,130670],{"className":130669,"style":130426},[31805],[34,130671,130672,130675],{"style":129846},[34,130673],{"className":130674,"style":31814},[31813],[34,130676,130678],{"className":130677},[31818,31819,31820,31821],[34,130679,223],{"className":130680},[32024,31821],[34,130682,456],{"className":130683},[31782],[34,130685,56782],{"className":130686,"style":32062},[31685,31686],[34,130688,2557],{"className":130689},[31690],[34,130691],{"className":130692,"style":32604},[31845],[34,130694,223],{"className":130695},[32024],[34,130697],{"className":130698,"style":32604},[31845],[34,130700,130702,130705,130734,130737,130740],{"className":130701},[31676],[34,130703],{"className":130704,"style":130407},[31680],[34,130706,130708,130711],{"className":130707},[31685],[34,130709,46900],{"className":130710},[31685,31686],[34,130712,130714],{"className":130713},[31792],[34,130715,130717],{"className":130716},[31796],[34,130718,130720],{"className":130719},[31801],[34,130721,130723],{"className":130722,"style":130426},[31805],[34,130724,130725,130728],{"style":129846},[34,130726],{"className":130727,"style":31814},[31813],[34,130729,130731],{"className":130730},[31818,31819,31820,31821],[34,130732,32226],{"className":130733},[32024,31821],[34,130735,456],{"className":130736},[31782],[34,130738,56782],{"className":130739,"style":32062},[31685,31686],[34,130741,2557],{"className":130742},[31690],[32188,130744],{},[19,130746,130748],{"id":130747},"_3-奇顶点和偶顶点","3. 奇顶点和偶顶点",[15,130750,130751],{},"如果：",[34,130753,130755],{"className":130754},[126385],[34,130756,130758,130777],{"className":130757},[31642],[34,130759,130761],{"className":130760},[31646],[31648,130762,130763],{"xmlns":31650,"display":126394},[31652,130764,130765,130775],{},[31655,130766,130767,130769,130771,130773],{},[31658,130768,46900],{},[31662,130770,456],{"stretchy":702},[31658,130772,56782],{},[31662,130774,2557],{"stretchy":702},[31665,130776,130164],{"encoding":31667},[34,130778,130780],{"className":130779,"ariaHidden":758},[31672],[34,130781,130783,130786,130789,130792,130795],{"className":130782},[31676],[34,130784],{"className":130785,"style":31774},[31680],[34,130787,46900],{"className":130788},[31685,31686],[34,130790,456],{"className":130791},[31782],[34,130793,56782],{"className":130794,"style":32062},[31685,31686],[34,130796,2557],{"className":130797},[31690],[15,130799,130800,130801,113124,130829,347],{},"是奇数，则 ",[34,130802,130804,130817],{"className":130803},[31642],[34,130805,130807],{"className":130806},[31646],[31648,130808,130809],{"xmlns":31650},[31652,130810,130811,130815],{},[31655,130812,130813],{},[31658,130814,56782],{},[31665,130816,56782],{"encoding":31667},[34,130818,130820],{"className":130819,"ariaHidden":758},[31672],[34,130821,130823,130826],{"className":130822},[31676],[34,130824],{"className":130825,"style":32527},[31680],[34,130827,56782],{"className":130828,"style":32062},[31685,31686],[2627,130830,130831],{},"奇顶点",[15,130833,130751],{},[34,130835,130837],{"className":130836},[126385],[34,130838,130840,130859],{"className":130839},[31642],[34,130841,130843],{"className":130842},[31646],[31648,130844,130845],{"xmlns":31650,"display":126394},[31652,130846,130847,130857],{},[31655,130848,130849,130851,130853,130855],{},[31658,130850,46900],{},[31662,130852,456],{"stretchy":702},[31658,130854,56782],{},[31662,130856,2557],{"stretchy":702},[31665,130858,130164],{"encoding":31667},[34,130860,130862],{"className":130861,"ariaHidden":758},[31672],[34,130863,130865,130868,130871,130874,130877],{"className":130864},[31676],[34,130866],{"className":130867,"style":31774},[31680],[34,130869,46900],{"className":130870},[31685,31686],[34,130872,456],{"className":130873},[31782],[34,130875,56782],{"className":130876,"style":32062},[31685,31686],[34,130878,2557],{"className":130879},[31690],[15,130881,130882,130883,113124,130911,347],{},"是偶数，则 ",[34,130884,130886,130899],{"className":130885},[31642],[34,130887,130889],{"className":130888},[31646],[31648,130890,130891],{"xmlns":31650},[31652,130892,130893,130897],{},[31655,130894,130895],{},[31658,130896,56782],{},[31665,130898,56782],{"encoding":31667},[34,130900,130902],{"className":130901,"ariaHidden":758},[31672],[34,130903,130905,130908],{"className":130904},[31676],[34,130906],{"className":130907,"style":32527},[31680],[34,130909,56782],{"className":130910,"style":32062},[31685,31686],[2627,130912,130913],{},"偶顶点",[32188,130915],{},[19,130917,130919],{"id":130918},"_4-正则图","4. 正则图",[15,130921,130922,130923,130951,130952,347],{},"如果无向图中所有顶点的度都等于 ",[34,130924,130926,130939],{"className":130925},[31642],[34,130927,130929],{"className":130928},[31646],[31648,130930,130931],{"xmlns":31650},[31652,130932,130933,130937],{},[31655,130934,130935],{},[31658,130936,33759],{},[31665,130938,33759],{"encoding":31667},[34,130940,130942],{"className":130941,"ariaHidden":758},[31672],[34,130943,130945,130948],{"className":130944},[31676],[34,130946],{"className":130947,"style":31681},[31680],[34,130949,33759],{"className":130950,"style":33774},[31685,31686],"，则称它是 ",[2627,130953,130954,130982],{},[34,130955,130957,130970],{"className":130956},[31642],[34,130958,130960],{"className":130959},[31646],[31648,130961,130962],{"xmlns":31650},[31652,130963,130964,130968],{},[31655,130965,130966],{},[31658,130967,33759],{},[31665,130969,33759],{"encoding":31667},[34,130971,130973],{"className":130972,"ariaHidden":758},[31672],[34,130974,130976,130979],{"className":130975},[31676],[34,130977],{"className":130978,"style":31681},[31680],[34,130980,33759],{"className":130981,"style":33774},[31685,31686]," 正则图",[15,130984,2632],{},[105,130986,130987,130990],{},[108,130988,130989],{},"所有顶点度都是 2：2 正则图；",[108,130991,130992],{},"所有顶点度都是 3：3 正则图。",[32188,130994],{},[11,130996,130998],{"id":130997},"六握手定理","六、握手定理",[15,131000,131001],{},"这是本章最重要公式之一。",[15,131003,131004,131005,131079,131080,131108],{},"设无向图 ",[34,131006,131008,131034],{"className":131007},[31642],[34,131009,131011],{"className":131010},[31646],[31648,131012,131013],{"xmlns":31650},[31652,131014,131015,131031],{},[31655,131016,131017,131019,131021,131023,131025,131027,131029],{},[31658,131018,97709],{},[31662,131020,202],{},[31662,131022,126529],{"stretchy":702},[31658,131024,126532],{},[31662,131026,31960],{"separator":758},[31658,131028,96058],{},[31662,131030,126544],{"stretchy":702},[31665,131032,131033],{"encoding":31667},"G=\\langle V,E\\rangle",[34,131035,131037,131055],{"className":131036,"ariaHidden":758},[31672],[34,131038,131040,131043,131046,131049,131052],{"className":131039},[31676],[34,131041],{"className":131042,"style":33218},[31680],[34,131044,97709],{"className":131045},[31685,31686],[34,131047],{"className":131048,"style":31846},[31845],[34,131050,202],{"className":131051},[31850],[34,131053],{"className":131054,"style":31846},[31845],[34,131056,131058,131061,131064,131067,131070,131073,131076],{"className":131057},[31676],[34,131059],{"className":131060,"style":31774},[31680],[34,131062,126529],{"className":131063},[31782],[34,131065,126532],{"className":131066,"style":32604},[31685,31686],[34,131068,31960],{"className":131069},[32119],[34,131071],{"className":131072,"style":32123},[31845],[34,131074,96058],{"className":131075,"style":106455},[31685,31686],[34,131077,126544],{"className":131078},[31690],"，边数为 ",[34,131081,131083,131096],{"className":131082},[31642],[34,131084,131086],{"className":131085},[31646],[31648,131087,131088],{"xmlns":31650},[31652,131089,131090,131094],{},[31655,131091,131092],{},[31658,131093,31759],{},[31665,131095,31759],{"encoding":31667},[34,131097,131099],{"className":131098,"ariaHidden":758},[31672],[34,131100,131102,131105],{"className":131101},[31676],[34,131103],{"className":131104,"style":32527},[31680],[34,131106,31759],{"className":131107},[31685,31686],"，则：",[34,131110,131112],{"className":131111},[126385],[34,131113,131115,131164],{"className":131114},[31642],[34,131116,131118],{"className":131117},[31646],[31648,131119,131120],{"xmlns":31650,"display":126394},[31652,131121,131122,131161],{},[31655,131123,131124],{},[126400,131125,131126],{"notation":126402},[71090,131127,131128],{"scriptlevel":459,"displaystyle":702},[71090,131129,131130],{"scriptlevel":459,"displaystyle":702},[71090,131131,131132],{"scriptlevel":459,"displaystyle":758},[31655,131133,131134,131147,131149,131151,131153,131155,131157,131159],{},[71115,131135,131136,131139],{},[31662,131137,131138],{},"∑",[31655,131140,131141,131143,131145],{},[31658,131142,56782],{},[31662,131144,112296],{},[31658,131146,126532],{},[31658,131148,46900],{},[31662,131150,456],{"stretchy":702},[31658,131152,56782],{},[31662,131154,2557],{"stretchy":702},[31662,131156,202],{},[31749,131158,768],{},[31658,131160,31759],{},[31665,131162,131163],{"encoding":31667},"\\boxed{\\sum_{v\\in V}d(v)=2m}",[34,131165,131167],{"className":131166,"ariaHidden":758},[31672],[34,131168,131170,131174],{"className":131169},[31676],[34,131171],{"className":131172,"style":131173},[31680],"height:3.0517em;vertical-align:-1.6617em;",[34,131175,131177],{"className":131176},[31685],[34,131178,131180,131306],{"className":131179},[31796,31797],[34,131181,131183,131303],{"className":131182},[31801],[34,131184,131187,131293],{"className":131185,"style":131186},[31805],"height:1.39em;",[34,131188,131190,131194],{"style":131189},"top:-5.0517em;",[34,131191],{"className":131192,"style":131193},[31813],"height:5.0517em;",[34,131195,131197],{"className":131196},[126448],[34,131198,131200],{"className":131199},[31685],[34,131201,131203,131263,131266,131269,131272,131275,131278,131281,131284,131287,131290],{"className":131202},[31685],[34,131204,131206],{"className":131205},[32767,71420],[34,131207,131209,131254],{"className":131208},[31796,31797],[34,131210,131212,131251],{"className":131211},[31801],[34,131213,131216,131238],{"className":131214,"style":131215},[31805],"height:1.05em;",[34,131217,131219,131223],{"style":131218},"top:-1.8557em;margin-left:0em;",[34,131220],{"className":131221,"style":131222},[31813],"height:3.05em;",[34,131224,131226],{"className":131225},[31818,31819,31820,31821],[34,131227,131229,131232,131235],{"className":131228},[31685,31821],[34,131230,56782],{"className":131231,"style":32062},[31685,31686,31821],[34,131233,112296],{"className":131234},[31850,31821],[34,131236,126532],{"className":131237,"style":32604},[31685,31686,31821],[34,131239,131241,131244],{"style":131240},"top:-3.05em;",[34,131242],{"className":131243,"style":131222},[31813],[34,131245,131246],{},[34,131247,131138],{"className":131248},[32767,131249,131250],"op-symbol","large-op",[34,131252,31829],{"className":131253},[31828],[34,131255,131257],{"className":131256},[31801],[34,131258,131261],{"className":131259,"style":131260},[31805],"height:1.3217em;",[34,131262],{},[34,131264],{"className":131265,"style":32123},[31845],[34,131267,46900],{"className":131268},[31685,31686],[34,131270,456],{"className":131271},[31782],[34,131273,56782],{"className":131274,"style":32062},[31685,31686],[34,131276,2557],{"className":131277},[31690],[34,131279],{"className":131280,"style":31846},[31845],[34,131282,202],{"className":131283},[31850],[34,131285],{"className":131286,"style":31846},[31845],[34,131288,768],{"className":131289},[31685],[34,131291,31759],{"className":131292},[31685,31686],[34,131294,131296,131299],{"style":131295},"top:-3.39em;",[34,131297],{"className":131298,"style":131193},[31813],[34,131300],{"className":131301,"style":131302},[126470,126471],"height:3.0517em;border-style:solid;border-width:0.04em;",[34,131304,31829],{"className":131305},[31828],[34,131307,131309],{"className":131308},[31801],[34,131310,131313],{"className":131311,"style":131312},[31805],"height:1.6617em;",[34,131314],{},[15,131316,131317],{},"意思是：",[104833,131319,131320],{},[15,131321,131322],{},"所有顶点度数之和 = 边数的 2 倍。",[15,131324,12275],{},[15,131326,131327],{},"因为每条普通边贡献两个端点，自环也贡献 2。",[32188,131329],{},[19,131331,131333],{"id":131332},"推论奇度顶点个数一定为偶数","推论：奇度顶点个数一定为偶数",[15,131335,131336],{},"因为：",[34,131338,131340],{"className":131339},[126385],[34,131341,131343,131381],{"className":131342},[31642],[34,131344,131346],{"className":131345},[31646],[31648,131347,131348],{"xmlns":31650,"display":126394},[31652,131349,131350,131378],{},[31655,131351,131352,131364,131366,131368,131370,131372,131374,131376],{},[71115,131353,131354,131356],{},[31662,131355,131138],{},[31655,131357,131358,131360,131362],{},[31658,131359,56782],{},[31662,131361,112296],{},[31658,131363,126532],{},[31658,131365,46900],{},[31662,131367,456],{"stretchy":702},[31658,131369,56782],{},[31662,131371,2557],{"stretchy":702},[31662,131373,202],{},[31749,131375,768],{},[31658,131377,31759],{},[31665,131379,131380],{"encoding":31667},"\\sum_{v\\in V}d(v)=2m",[34,131382,131384,131468],{"className":131383,"ariaHidden":758},[31672],[34,131385,131387,131391,131444,131447,131450,131453,131456,131459,131462,131465],{"className":131386},[31676],[34,131388],{"className":131389,"style":131390},[31680],"height:2.3717em;vertical-align:-1.3217em;",[34,131392,131394],{"className":131393},[32767,71420],[34,131395,131397,131436],{"className":131396},[31796,31797],[34,131398,131400,131433],{"className":131399},[31801],[34,131401,131403,131423],{"className":131402,"style":131215},[31805],[34,131404,131405,131408],{"style":131218},[34,131406],{"className":131407,"style":131222},[31813],[34,131409,131411],{"className":131410},[31818,31819,31820,31821],[34,131412,131414,131417,131420],{"className":131413},[31685,31821],[34,131415,56782],{"className":131416,"style":32062},[31685,31686,31821],[34,131418,112296],{"className":131419},[31850,31821],[34,131421,126532],{"className":131422,"style":32604},[31685,31686,31821],[34,131424,131425,131428],{"style":131240},[34,131426],{"className":131427,"style":131222},[31813],[34,131429,131430],{},[34,131431,131138],{"className":131432},[32767,131249,131250],[34,131434,31829],{"className":131435},[31828],[34,131437,131439],{"className":131438},[31801],[34,131440,131442],{"className":131441,"style":131260},[31805],[34,131443],{},[34,131445],{"className":131446,"style":32123},[31845],[34,131448,46900],{"className":131449},[31685,31686],[34,131451,456],{"className":131452},[31782],[34,131454,56782],{"className":131455,"style":32062},[31685,31686],[34,131457,2557],{"className":131458},[31690],[34,131460],{"className":131461,"style":31846},[31845],[34,131463,202],{"className":131464},[31850],[34,131466],{"className":131467,"style":31846},[31845],[34,131469,131471,131474,131477],{"className":131470},[31676],[34,131472],{"className":131473,"style":32617},[31680],[34,131475,768],{"className":131476},[31685],[34,131478,31759],{"className":131479},[31685,31686],[15,131481,131482],{},"一定是偶数。",[15,131484,131485],{},"偶数度顶点加起来还是偶数，所以奇数度顶点的个数必须是偶数。",[15,131487,62532],{},[34,131489,131491],{"className":131490},[126385],[34,131492,131494,131517],{"className":131493},[31642],[34,131495,131497],{"className":131496},[31646],[31648,131498,131499],{"xmlns":31650,"display":126394},[31652,131500,131501,131514],{},[31655,131502,131503],{},[126400,131504,131505],{"notation":126402},[71090,131506,131507],{"scriptlevel":459,"displaystyle":702},[71090,131508,131509],{"scriptlevel":459,"displaystyle":702},[71090,131510,131511],{"scriptlevel":459,"displaystyle":758},[39361,131512,131513],{},"任意无向图中，奇顶点个数为偶数",[31665,131515,131516],{"encoding":31667},"\\boxed{\\text{任意无向图中，奇顶点个数为偶数}}",[34,131518,131520],{"className":131519,"ariaHidden":758},[31672],[34,131521,131523,131526],{"className":131522},[31676],[34,131524],{"className":131525,"style":126424},[31680],[34,131527,131529],{"className":131528},[31685],[34,131530,131532,131569],{"className":131531},[31796,31797],[34,131533,131535,131566],{"className":131534},[31801],[34,131536,131538,131558],{"className":131537,"style":126437},[31805],[34,131539,131540,131543],{"style":126440},[34,131541],{"className":131542,"style":126444},[31813],[34,131544,131546],{"className":131545},[126448],[34,131547,131549],{"className":131548},[31685],[34,131550,131552],{"className":131551},[31685],[34,131553,131555],{"className":131554},[31685,2638],[34,131556,131513],{"className":131557},[31685,105462],[34,131559,131560,131563],{"style":126463},[34,131561],{"className":131562,"style":126444},[31813],[34,131564],{"className":131565,"style":126472},[126470,126471],[34,131567,31829],{"className":131568},[31828],[34,131570,131572],{"className":131571},[31801],[34,131573,131575],{"className":131574,"style":126482},[31805],[34,131576],{},[15,131578,131579],{},"这条非常常考。",[32188,131581],{},[11,131583,131585],{"id":131584},"七子图","七、子图",[15,131587,131588],{},"设：",[34,131590,131592],{"className":131591},[126385],[34,131593,131595,131651],{"className":131594},[31642],[34,131596,131598],{"className":131597},[31646],[31648,131599,131600],{"xmlns":31650,"display":126394},[31652,131601,131602,131648],{},[31655,131603,131604,131606,131608,131610,131612,131614,131616,131618,131620,131622,131628,131630,131632,131638,131640,131646],{},[31658,131605,97709],{},[31662,131607,202],{},[31662,131609,126529],{"stretchy":702},[31658,131611,126532],{},[31662,131613,31960],{"separator":758},[31658,131615,96058],{},[31662,131617,126544],{"stretchy":702},[31662,131619,31960],{"separator":758},[31845,131621],{"width":71082},[37624,131623,131624,131626],{},[31658,131625,97709],{},[31662,131627,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,131629,202],{},[31662,131631,126529],{"stretchy":702},[37624,131633,131634,131636],{},[31658,131635,126532],{},[31662,131637,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,131639,31960],{"separator":758},[37624,131641,131642,131644],{},[31658,131643,96058],{},[31662,131645,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,131647,126544],{"stretchy":702},[31665,131649,131650],{"encoding":31667},"G=\\langle V,E\\rangle,\\quad G'=\\langle V',E'\\rangle",[34,131652,131654,131672,131748],{"className":131653,"ariaHidden":758},[31672],[34,131655,131657,131660,131663,131666,131669],{"className":131656},[31676],[34,131658],{"className":131659,"style":33218},[31680],[34,131661,97709],{"className":131662},[31685,31686],[34,131664],{"className":131665,"style":31846},[31845],[34,131667,202],{"className":131668},[31850],[34,131670],{"className":131671,"style":31846},[31845],[34,131673,131675,131679,131682,131685,131688,131691,131694,131697,131700,131703,131706,131739,131742,131745],{"className":131674},[31676],[34,131676],{"className":131677,"style":131678},[31680],"height:1.0519em;vertical-align:-0.25em;",[34,131680,126529],{"className":131681},[31782],[34,131683,126532],{"className":131684,"style":32604},[31685,31686],[34,131686,31960],{"className":131687},[32119],[34,131689],{"className":131690,"style":32123},[31845],[34,131692,96058],{"className":131693,"style":106455},[31685,31686],[34,131695,126544],{"className":131696},[31690],[34,131698,31960],{"className":131699},[32119],[34,131701],{"className":131702,"style":128438},[31845],[34,131704],{"className":131705,"style":32123},[31845],[34,131707,131709,131712],{"className":131708},[31685],[34,131710,97709],{"className":131711},[31685,31686],[34,131713,131715],{"className":131714},[31792],[34,131716,131718],{"className":131717},[31796],[34,131719,131721],{"className":131720},[31801],[34,131722,131725],{"className":131723,"style":131724},[31805],"height:0.8019em;",[34,131726,131727,131730],{"style":129846},[34,131728],{"className":131729,"style":31814},[31813],[34,131731,131733],{"className":131732},[31818,31819,31820,31821],[34,131734,131736],{"className":131735},[31685,31821],[34,131737,116495],{"className":131738},[31685,31821],[34,131740],{"className":131741,"style":31846},[31845],[34,131743,202],{"className":131744},[31850],[34,131746],{"className":131747,"style":31846},[31845],[34,131749,131751,131754,131757,131789,131792,131795,131827],{"className":131750},[31676],[34,131752],{"className":131753,"style":131678},[31680],[34,131755,126529],{"className":131756},[31782],[34,131758,131760,131763],{"className":131759},[31685],[34,131761,126532],{"className":131762,"style":32604},[31685,31686],[34,131764,131766],{"className":131765},[31792],[34,131767,131769],{"className":131768},[31796],[34,131770,131772],{"className":131771},[31801],[34,131773,131775],{"className":131774,"style":131724},[31805],[34,131776,131777,131780],{"style":129846},[34,131778],{"className":131779,"style":31814},[31813],[34,131781,131783],{"className":131782},[31818,31819,31820,31821],[34,131784,131786],{"className":131785},[31685,31821],[34,131787,116495],{"className":131788},[31685,31821],[34,131790,31960],{"className":131791},[32119],[34,131793],{"className":131794,"style":32123},[31845],[34,131796,131798,131801],{"className":131797},[31685],[34,131799,96058],{"className":131800,"style":106455},[31685,31686],[34,131802,131804],{"className":131803},[31792],[34,131805,131807],{"className":131806},[31796],[34,131808,131810],{"className":131809},[31801],[34,131811,131813],{"className":131812,"style":131724},[31805],[34,131814,131815,131818],{"style":129846},[34,131816],{"className":131817,"style":31814},[31813],[34,131819,131821],{"className":131820},[31818,31819,31820,31821],[34,131822,131824],{"className":131823},[31685,31821],[34,131825,116495],{"className":131826},[31685,31821],[34,131828,126544],{"className":131829},[31690],[15,131831,130751],{},[34,131833,131835],{"className":131834},[126385],[34,131836,131838,131875],{"className":131837},[31642],[34,131839,131841],{"className":131840},[31646],[31648,131842,131843],{"xmlns":31650,"display":126394},[31652,131844,131845,131872],{},[31655,131846,131847,131853,131856,131858,131860,131862,131868,131870],{},[37624,131848,131849,131851],{},[31658,131850,126532],{},[31662,131852,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,131854,131855],{},"⊆",[31658,131857,126532],{},[31662,131859,31960],{"separator":758},[31845,131861],{"width":71082},[37624,131863,131864,131866],{},[31658,131865,96058],{},[31662,131867,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,131869,131855],{},[31658,131871,96058],{},[31665,131873,131874],{"encoding":31667},"V'\\subseteq V,\\quad E'\\subseteq E",[34,131876,131878,131926,131986],{"className":131877,"ariaHidden":758},[31672],[34,131879,131881,131885,131917,131920,131923],{"className":131880},[31676],[34,131882],{"className":131883,"style":131884},[31680],"height:0.9379em;vertical-align:-0.136em;",[34,131886,131888,131891],{"className":131887},[31685],[34,131889,126532],{"className":131890,"style":32604},[31685,31686],[34,131892,131894],{"className":131893},[31792],[34,131895,131897],{"className":131896},[31796],[34,131898,131900],{"className":131899},[31801],[34,131901,131903],{"className":131902,"style":131724},[31805],[34,131904,131905,131908],{"style":129846},[34,131906],{"className":131907,"style":31814},[31813],[34,131909,131911],{"className":131910},[31818,31819,31820,31821],[34,131912,131914],{"className":131913},[31685,31821],[34,131915,116495],{"className":131916},[31685,31821],[34,131918],{"className":131919,"style":31846},[31845],[34,131921,131855],{"className":131922},[31850],[34,131924],{"className":131925,"style":31846},[31845],[34,131927,131929,131933,131936,131939,131942,131945,131977,131980,131983],{"className":131928},[31676],[34,131930],{"className":131931,"style":131932},[31680],"height:0.9963em;vertical-align:-0.1944em;",[34,131934,126532],{"className":131935,"style":32604},[31685,31686],[34,131937,31960],{"className":131938},[32119],[34,131940],{"className":131941,"style":128438},[31845],[34,131943],{"className":131944,"style":32123},[31845],[34,131946,131948,131951],{"className":131947},[31685],[34,131949,96058],{"className":131950,"style":106455},[31685,31686],[34,131952,131954],{"className":131953},[31792],[34,131955,131957],{"className":131956},[31796],[34,131958,131960],{"className":131959},[31801],[34,131961,131963],{"className":131962,"style":131724},[31805],[34,131964,131965,131968],{"style":129846},[34,131966],{"className":131967,"style":31814},[31813],[34,131969,131971],{"className":131970},[31818,31819,31820,31821],[34,131972,131974],{"className":131973},[31685,31821],[34,131975,116495],{"className":131976},[31685,31821],[34,131978],{"className":131979,"style":31846},[31845],[34,131981,131855],{"className":131982},[31850],[34,131984],{"className":131985,"style":31846},[31845],[34,131987,131989,131992],{"className":131988},[31676],[34,131990],{"className":131991,"style":33218},[31680],[34,131993,96058],{"className":131994,"style":106455},[31685,31686],[15,131996,94591,131997,11008,132060,130135,132088,347],{},[34,131998,132000,132018],{"className":131999},[31642],[34,132001,132003],{"className":132002},[31646],[31648,132004,132005],{"xmlns":31650},[31652,132006,132007,132015],{},[31655,132008,132009],{},[37624,132010,132011,132013],{},[31658,132012,97709],{},[31662,132014,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,132016,132017],{"encoding":31667},"G'",[34,132019,132021],{"className":132020,"ariaHidden":758},[31672],[34,132022,132024,132028],{"className":132023},[31676],[34,132025],{"className":132026,"style":132027},[31680],"height:0.7519em;",[34,132029,132031,132034],{"className":132030},[31685],[34,132032,97709],{"className":132033},[31685,31686],[34,132035,132037],{"className":132036},[31792],[34,132038,132040],{"className":132039},[31796],[34,132041,132043],{"className":132042},[31801],[34,132044,132046],{"className":132045,"style":132027},[31805],[34,132047,132048,132051],{"style":37777},[34,132049],{"className":132050,"style":31814},[31813],[34,132052,132054],{"className":132053},[31818,31819,31820,31821],[34,132055,132057],{"className":132056},[31685,31821],[34,132058,116495],{"className":132059},[31685,31821],[34,132061,132063,132076],{"className":132062},[31642],[34,132064,132066],{"className":132065},[31646],[31648,132067,132068],{"xmlns":31650},[31652,132069,132070,132074],{},[31655,132071,132072],{},[31658,132073,97709],{},[31665,132075,97709],{"encoding":31667},[34,132077,132079],{"className":132078,"ariaHidden":758},[31672],[34,132080,132082,132085],{"className":132081},[31676],[34,132083],{"className":132084,"style":33218},[31680],[34,132086,97709],{"className":132087},[31685,31686],[2627,132089,132090],{},"子图",[32188,132092],{},[19,132094,132096],{"id":132095},"_1-真子图","1. 真子图",[15,132098,2234,132099,11008,132160,132188],{},[34,132100,132102,132119],{"className":132101},[31642],[34,132103,132105],{"className":132104},[31646],[31648,132106,132107],{"xmlns":31650},[31652,132108,132109,132117],{},[31655,132110,132111],{},[37624,132112,132113,132115],{},[31658,132114,97709],{},[31662,132116,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,132118,132017],{"encoding":31667},[34,132120,132122],{"className":132121,"ariaHidden":758},[31672],[34,132123,132125,132128],{"className":132124},[31676],[34,132126],{"className":132127,"style":132027},[31680],[34,132129,132131,132134],{"className":132130},[31685],[34,132132,97709],{"className":132133},[31685,31686],[34,132135,132137],{"className":132136},[31792],[34,132138,132140],{"className":132139},[31796],[34,132141,132143],{"className":132142},[31801],[34,132144,132146],{"className":132145,"style":132027},[31805],[34,132147,132148,132151],{"style":37777},[34,132149],{"className":132150,"style":31814},[31813],[34,132152,132154],{"className":132153},[31818,31819,31820,31821],[34,132155,132157],{"className":132156},[31685,31821],[34,132158,116495],{"className":132159},[31685,31821],[34,132161,132163,132176],{"className":132162},[31642],[34,132164,132166],{"className":132165},[31646],[31648,132167,132168],{"xmlns":31650},[31652,132169,132170,132174],{},[31655,132171,132172],{},[31658,132173,97709],{},[31665,132175,97709],{"encoding":31667},[34,132177,132179],{"className":132178,"ariaHidden":758},[31672],[34,132180,132182,132185],{"className":132181},[31676],[34,132183],{"className":132184,"style":33218},[31680],[34,132186,97709],{"className":132187},[31685,31686]," 的子图，但：",[34,132190,132192],{"className":132191},[126385],[34,132193,132195,132217],{"className":132194},[31642],[34,132196,132198],{"className":132197},[31646],[31648,132199,132200],{"xmlns":31650,"display":126394},[31652,132201,132202,132214],{},[31655,132203,132204,132210,132212],{},[37624,132205,132206,132208],{},[31658,132207,126532],{},[31662,132209,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,132211,54707],{"mathvariant":38079},[31658,132213,126532],{},[31665,132215,132216],{"encoding":31667},"V'\\neq V",[34,132218,132220,132300],{"className":132219,"ariaHidden":758},[31672],[34,132221,132223,132226,132258,132261,132297],{"className":132222},[31676],[34,132224],{"className":132225,"style":131932},[31680],[34,132227,132229,132232],{"className":132228},[31685],[34,132230,126532],{"className":132231,"style":32604},[31685,31686],[34,132233,132235],{"className":132234},[31792],[34,132236,132238],{"className":132237},[31796],[34,132239,132241],{"className":132240},[31801],[34,132242,132244],{"className":132243,"style":131724},[31805],[34,132245,132246,132249],{"style":129846},[34,132247],{"className":132248,"style":31814},[31813],[34,132250,132252],{"className":132251},[31818,31819,31820,31821],[34,132253,132255],{"className":132254},[31685,31821],[34,132256,116495],{"className":132257},[31685,31821],[34,132259],{"className":132260,"style":31846},[31845],[34,132262,132264,132291,132294],{"className":132263},[31850],[34,132265,132267],{"className":132266},[31850],[34,132268,132270],{"className":132269},[31685,54774],[34,132271,132273],{"className":132272},[54778],[34,132274,132276,132279,132288],{"className":132275},[54782],[34,132277],{"className":132278,"style":43217},[31680],[34,132280,132282],{"className":132281},[54789],[34,132283,132285],{"className":132284},[31685],[34,132286,54796],{"className":132287},[31850],[34,132289],{"className":132290},[54800],[34,132292],{"className":132293},[31845,54804],[34,132295,202],{"className":132296},[31850],[34,132298],{"className":132299,"style":31846},[31845],[34,132301,132303,132306],{"className":132302},[31676],[34,132304],{"className":132305,"style":33218},[31680],[34,132307,126532],{"className":132308,"style":32604},[31685,31686],[15,132310,132311],{},"或：",[34,132313,132315],{"className":132314},[126385],[34,132316,132318,132340],{"className":132317},[31642],[34,132319,132321],{"className":132320},[31646],[31648,132322,132323],{"xmlns":31650,"display":126394},[31652,132324,132325,132337],{},[31655,132326,132327,132333,132335],{},[37624,132328,132329,132331],{},[31658,132330,96058],{},[31662,132332,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,132334,54707],{"mathvariant":38079},[31658,132336,96058],{},[31665,132338,132339],{"encoding":31667},"E'\\neq E",[34,132341,132343,132423],{"className":132342,"ariaHidden":758},[31672],[34,132344,132346,132349,132381,132384,132420],{"className":132345},[31676],[34,132347],{"className":132348,"style":131932},[31680],[34,132350,132352,132355],{"className":132351},[31685],[34,132353,96058],{"className":132354,"style":106455},[31685,31686],[34,132356,132358],{"className":132357},[31792],[34,132359,132361],{"className":132360},[31796],[34,132362,132364],{"className":132363},[31801],[34,132365,132367],{"className":132366,"style":131724},[31805],[34,132368,132369,132372],{"style":129846},[34,132370],{"className":132371,"style":31814},[31813],[34,132373,132375],{"className":132374},[31818,31819,31820,31821],[34,132376,132378],{"className":132377},[31685,31821],[34,132379,116495],{"className":132380},[31685,31821],[34,132382],{"className":132383,"style":31846},[31845],[34,132385,132387,132414,132417],{"className":132386},[31850],[34,132388,132390],{"className":132389},[31850],[34,132391,132393],{"className":132392},[31685,54774],[34,132394,132396],{"className":132395},[54778],[34,132397,132399,132402,132411],{"className":132398},[54782],[34,132400],{"className":132401,"style":43217},[31680],[34,132403,132405],{"className":132404},[54789],[34,132406,132408],{"className":132407},[31685],[34,132409,54796],{"className":132410},[31850],[34,132412],{"className":132413},[54800],[34,132415],{"className":132416},[31845,54804],[34,132418,202],{"className":132419},[31850],[34,132421],{"className":132422,"style":31846},[31845],[34,132424,132426,132429],{"className":132425},[31676],[34,132427],{"className":132428,"style":33218},[31680],[34,132430,96058],{"className":132431,"style":106455},[31685,31686],[15,132433,132434,132435,11008,132496,130135,132524,347],{},"则称 ",[34,132436,132438,132455],{"className":132437},[31642],[34,132439,132441],{"className":132440},[31646],[31648,132442,132443],{"xmlns":31650},[31652,132444,132445,132453],{},[31655,132446,132447],{},[37624,132448,132449,132451],{},[31658,132450,97709],{},[31662,132452,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,132454,132017],{"encoding":31667},[34,132456,132458],{"className":132457,"ariaHidden":758},[31672],[34,132459,132461,132464],{"className":132460},[31676],[34,132462],{"className":132463,"style":132027},[31680],[34,132465,132467,132470],{"className":132466},[31685],[34,132468,97709],{"className":132469},[31685,31686],[34,132471,132473],{"className":132472},[31792],[34,132474,132476],{"className":132475},[31796],[34,132477,132479],{"className":132478},[31801],[34,132480,132482],{"className":132481,"style":132027},[31805],[34,132483,132484,132487],{"style":37777},[34,132485],{"className":132486,"style":31814},[31813],[34,132488,132490],{"className":132489},[31818,31819,31820,31821],[34,132491,132493],{"className":132492},[31685,31821],[34,132494,116495],{"className":132495},[31685,31821],[34,132497,132499,132512],{"className":132498},[31642],[34,132500,132502],{"className":132501},[31646],[31648,132503,132504],{"xmlns":31650},[31652,132505,132506,132510],{},[31655,132507,132508],{},[31658,132509,97709],{},[31665,132511,97709],{"encoding":31667},[34,132513,132515],{"className":132514,"ariaHidden":758},[31672],[34,132516,132518,132521],{"className":132517},[31676],[34,132519],{"className":132520,"style":33218},[31680],[34,132522,97709],{"className":132523},[31685,31686],[2627,132525,132526],{},"真子图",[32188,132528],{},[19,132530,132532],{"id":132531},"_2-生成子图-支撑子图","2. 生成子图 \u002F 支撑子图",[15,132534,130751],{},[34,132536,132538],{"className":132537},[126385],[34,132539,132541,132563],{"className":132540},[31642],[34,132542,132544],{"className":132543},[31646],[31648,132545,132546],{"xmlns":31650,"display":126394},[31652,132547,132548,132560],{},[31655,132549,132550,132556,132558],{},[37624,132551,132552,132554],{},[31658,132553,126532],{},[31662,132555,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,132557,202],{},[31658,132559,126532],{},[31665,132561,132562],{"encoding":31667},"V'=V",[34,132564,132566,132613],{"className":132565,"ariaHidden":758},[31672],[34,132567,132569,132572,132604,132607,132610],{"className":132568},[31676],[34,132570],{"className":132571,"style":131724},[31680],[34,132573,132575,132578],{"className":132574},[31685],[34,132576,126532],{"className":132577,"style":32604},[31685,31686],[34,132579,132581],{"className":132580},[31792],[34,132582,132584],{"className":132583},[31796],[34,132585,132587],{"className":132586},[31801],[34,132588,132590],{"className":132589,"style":131724},[31805],[34,132591,132592,132595],{"style":129846},[34,132593],{"className":132594,"style":31814},[31813],[34,132596,132598],{"className":132597},[31818,31819,31820,31821],[34,132599,132601],{"className":132600},[31685,31821],[34,132602,116495],{"className":132603},[31685,31821],[34,132605],{"className":132606,"style":31846},[31845],[34,132608,202],{"className":132609},[31850],[34,132611],{"className":132612,"style":31846},[31845],[34,132614,132616,132619],{"className":132615},[31676],[34,132617],{"className":132618,"style":33218},[31680],[34,132620,126532],{"className":132621,"style":32604},[31685,31686],[15,132623,132624],{},"但：",[34,132626,132628],{"className":132627},[126385],[34,132629,132631,132653],{"className":132630},[31642],[34,132632,132634],{"className":132633},[31646],[31648,132635,132636],{"xmlns":31650,"display":126394},[31652,132637,132638,132650],{},[31655,132639,132640,132646,132648],{},[37624,132641,132642,132644],{},[31658,132643,96058],{},[31662,132645,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,132647,131855],{},[31658,132649,96058],{},[31665,132651,132652],{"encoding":31667},"E'\\subseteq E",[34,132654,132656,132703],{"className":132655,"ariaHidden":758},[31672],[34,132657,132659,132662,132694,132697,132700],{"className":132658},[31676],[34,132660],{"className":132661,"style":131884},[31680],[34,132663,132665,132668],{"className":132664},[31685],[34,132666,96058],{"className":132667,"style":106455},[31685,31686],[34,132669,132671],{"className":132670},[31792],[34,132672,132674],{"className":132673},[31796],[34,132675,132677],{"className":132676},[31801],[34,132678,132680],{"className":132679,"style":131724},[31805],[34,132681,132682,132685],{"style":129846},[34,132683],{"className":132684,"style":31814},[31813],[34,132686,132688],{"className":132687},[31818,31819,31820,31821],[34,132689,132691],{"className":132690},[31685,31821],[34,132692,116495],{"className":132693},[31685,31821],[34,132695],{"className":132696,"style":31846},[31845],[34,132698,131855],{"className":132699},[31850],[34,132701],{"className":132702,"style":31846},[31845],[34,132704,132706,132709],{"className":132705},[31676],[34,132707],{"className":132708,"style":33218},[31680],[34,132710,96058],{"className":132711,"style":106455},[31685,31686],[15,132713,132434,132714,11008,132775,130135,132803,127650,132806,347],{},[34,132715,132717,132734],{"className":132716},[31642],[34,132718,132720],{"className":132719},[31646],[31648,132721,132722],{"xmlns":31650},[31652,132723,132724,132732],{},[31655,132725,132726],{},[37624,132727,132728,132730],{},[31658,132729,97709],{},[31662,132731,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,132733,132017],{"encoding":31667},[34,132735,132737],{"className":132736,"ariaHidden":758},[31672],[34,132738,132740,132743],{"className":132739},[31676],[34,132741],{"className":132742,"style":132027},[31680],[34,132744,132746,132749],{"className":132745},[31685],[34,132747,97709],{"className":132748},[31685,31686],[34,132750,132752],{"className":132751},[31792],[34,132753,132755],{"className":132754},[31796],[34,132756,132758],{"className":132757},[31801],[34,132759,132761],{"className":132760,"style":132027},[31805],[34,132762,132763,132766],{"style":37777},[34,132764],{"className":132765,"style":31814},[31813],[34,132767,132769],{"className":132768},[31818,31819,31820,31821],[34,132770,132772],{"className":132771},[31685,31821],[34,132773,116495],{"className":132774},[31685,31821],[34,132776,132778,132791],{"className":132777},[31642],[34,132779,132781],{"className":132780},[31646],[31648,132782,132783],{"xmlns":31650},[31652,132784,132785,132789],{},[31655,132786,132787],{},[31658,132788,97709],{},[31665,132790,97709],{"encoding":31667},[34,132792,132794],{"className":132793,"ariaHidden":758},[31672],[34,132795,132797,132800],{"className":132796},[31676],[34,132798],{"className":132799,"style":33218},[31680],[34,132801,97709],{"className":132802},[31685,31686],[2627,132804,132805],{},"生成子图",[2627,132807,132808],{},"支撑子图",[15,132810,131317],{},[104833,132812,132813],{},[15,132814,132815],{},"顶点全保留，只删一些边。",[32188,132817],{},[19,132819,132821],{"id":132820},"_3-顶点导出子图","3. 顶点导出子图",[15,132823,132824],{},"如果取：",[34,132826,132828],{"className":132827},[126385],[34,132829,132831,132853],{"className":132830},[31642],[34,132832,132834],{"className":132833},[31646],[31648,132835,132836],{"xmlns":31650,"display":126394},[31652,132837,132838,132850],{},[31655,132839,132840,132846,132848],{},[37624,132841,132842,132844],{},[31658,132843,126532],{},[31662,132845,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,132847,131855],{},[31658,132849,126532],{},[31665,132851,132852],{"encoding":31667},"V'\\subseteq V",[34,132854,132856,132903],{"className":132855,"ariaHidden":758},[31672],[34,132857,132859,132862,132894,132897,132900],{"className":132858},[31676],[34,132860],{"className":132861,"style":131884},[31680],[34,132863,132865,132868],{"className":132864},[31685],[34,132866,126532],{"className":132867,"style":32604},[31685,31686],[34,132869,132871],{"className":132870},[31792],[34,132872,132874],{"className":132873},[31796],[34,132875,132877],{"className":132876},[31801],[34,132878,132880],{"className":132879,"style":131724},[31805],[34,132881,132882,132885],{"style":129846},[34,132883],{"className":132884,"style":31814},[31813],[34,132886,132888],{"className":132887},[31818,31819,31820,31821],[34,132889,132891],{"className":132890},[31685,31821],[34,132892,116495],{"className":132893},[31685,31821],[34,132895],{"className":132896,"style":31846},[31845],[34,132898,131855],{"className":132899},[31850],[34,132901],{"className":132902,"style":31846},[31845],[34,132904,132906,132909],{"className":132905},[31676],[34,132907],{"className":132908,"style":33218},[31680],[34,132910,126532],{"className":132911,"style":32604},[31685,31686],[15,132913,132914,132915,132977,132978,127434],{},"并保留 ",[34,132916,132918,132936],{"className":132917},[31642],[34,132919,132921],{"className":132920},[31646],[31648,132922,132923],{"xmlns":31650},[31652,132924,132925,132933],{},[31655,132926,132927],{},[37624,132928,132929,132931],{},[31658,132930,126532],{},[31662,132932,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,132934,132935],{"encoding":31667},"V'",[34,132937,132939],{"className":132938,"ariaHidden":758},[31672],[34,132940,132942,132945],{"className":132941},[31676],[34,132943],{"className":132944,"style":132027},[31680],[34,132946,132948,132951],{"className":132947},[31685],[34,132949,126532],{"className":132950,"style":32604},[31685,31686],[34,132952,132954],{"className":132953},[31792],[34,132955,132957],{"className":132956},[31796],[34,132958,132960],{"className":132959},[31801],[34,132961,132963],{"className":132962,"style":132027},[31805],[34,132964,132965,132968],{"style":37777},[34,132966],{"className":132967,"style":31814},[31813],[34,132969,132971],{"className":132970},[31818,31819,31820,31821],[34,132972,132974],{"className":132973},[31685,31821],[34,132975,116495],{"className":132976},[31685,31821]," 中顶点之间在原图中所有的边，则得到的子图叫",[2627,132979,132980],{},"顶点导出子图",[34,132982,132984],{"className":132983},[126385],[34,132985,132987,133011],{"className":132986},[31642],[34,132988,132990],{"className":132989},[31646],[31648,132991,132992],{"xmlns":31650,"display":126394},[31652,132993,132994,133008],{},[31655,132995,132996,132998,133000,133006],{},[31658,132997,97709],{},[31662,132999,15972],{"stretchy":702},[37624,133001,133002,133004],{},[31658,133003,126532],{},[31662,133005,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,133007,32712],{"stretchy":702},[31665,133009,133010],{"encoding":31667},"G[V']",[34,133012,133014],{"className":133013,"ariaHidden":758},[31672],[34,133015,133017,133020,133023,133026,133058],{"className":133016},[31676],[34,133018],{"className":133019,"style":131678},[31680],[34,133021,97709],{"className":133022},[31685,31686],[34,133024,15972],{"className":133025},[31782],[34,133027,133029,133032],{"className":133028},[31685],[34,133030,126532],{"className":133031,"style":32604},[31685,31686],[34,133033,133035],{"className":133034},[31792],[34,133036,133038],{"className":133037},[31796],[34,133039,133041],{"className":133040},[31801],[34,133042,133044],{"className":133043,"style":131724},[31805],[34,133045,133046,133049],{"style":129846},[34,133047],{"className":133048,"style":31814},[31813],[34,133050,133052],{"className":133051},[31818,31819,31820,31821],[34,133053,133055],{"className":133054},[31685,31821],[34,133056,116495],{"className":133057},[31685,31821],[34,133059,32712],{"className":133060},[31690],[15,133062,133063],{},"通俗说：",[104833,133065,133066],{},[15,133067,133068],{},"选一些顶点，把这些顶点之间原来有的边全部保留。",[32188,133070],{},[19,133072,133074],{"id":133073},"_4-边导出子图","4. 边导出子图",[15,133076,132824],{},[34,133078,133080],{"className":133079},[126385],[34,133081,133083,133104],{"className":133082},[31642],[34,133084,133086],{"className":133085},[31646],[31648,133087,133088],{"xmlns":31650,"display":126394},[31652,133089,133090,133102],{},[31655,133091,133092,133098,133100],{},[37624,133093,133094,133096],{},[31658,133095,96058],{},[31662,133097,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,133099,131855],{},[31658,133101,96058],{},[31665,133103,132652],{"encoding":31667},[34,133105,133107,133154],{"className":133106,"ariaHidden":758},[31672],[34,133108,133110,133113,133145,133148,133151],{"className":133109},[31676],[34,133111],{"className":133112,"style":131884},[31680],[34,133114,133116,133119],{"className":133115},[31685],[34,133117,96058],{"className":133118,"style":106455},[31685,31686],[34,133120,133122],{"className":133121},[31792],[34,133123,133125],{"className":133124},[31796],[34,133126,133128],{"className":133127},[31801],[34,133129,133131],{"className":133130,"style":131724},[31805],[34,133132,133133,133136],{"style":129846},[34,133134],{"className":133135,"style":31814},[31813],[34,133137,133139],{"className":133138},[31818,31819,31820,31821],[34,133140,133142],{"className":133141},[31685,31821],[34,133143,116495],{"className":133144},[31685,31821],[34,133146],{"className":133147,"style":31846},[31845],[34,133149,131855],{"className":133150},[31850],[34,133152],{"className":133153,"style":31846},[31845],[34,133155,133157,133160],{"className":133156},[31676],[34,133158],{"className":133159,"style":33218},[31680],[34,133161,96058],{"className":133162,"style":106455},[31685,31686],[15,133164,133165,133166,127434],{},"并把这些边及其端点组成子图，则叫",[2627,133167,133168],{},"边导出子图",[34,133170,133172],{"className":133171},[126385],[34,133173,133175,133199],{"className":133174},[31642],[34,133176,133178],{"className":133177},[31646],[31648,133179,133180],{"xmlns":31650,"display":126394},[31652,133181,133182,133196],{},[31655,133183,133184,133186,133188,133194],{},[31658,133185,97709],{},[31662,133187,15972],{"stretchy":702},[37624,133189,133190,133192],{},[31658,133191,96058],{},[31662,133193,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,133195,32712],{"stretchy":702},[31665,133197,133198],{"encoding":31667},"G[E']",[34,133200,133202],{"className":133201,"ariaHidden":758},[31672],[34,133203,133205,133208,133211,133214,133246],{"className":133204},[31676],[34,133206],{"className":133207,"style":131678},[31680],[34,133209,97709],{"className":133210},[31685,31686],[34,133212,15972],{"className":133213},[31782],[34,133215,133217,133220],{"className":133216},[31685],[34,133218,96058],{"className":133219,"style":106455},[31685,31686],[34,133221,133223],{"className":133222},[31792],[34,133224,133226],{"className":133225},[31796],[34,133227,133229],{"className":133228},[31801],[34,133230,133232],{"className":133231,"style":131724},[31805],[34,133233,133234,133237],{"style":129846},[34,133235],{"className":133236,"style":31814},[31813],[34,133238,133240],{"className":133239},[31818,31819,31820,31821],[34,133241,133243],{"className":133242},[31685,31821],[34,133244,116495],{"className":133245},[31685,31821],[34,133247,32712],{"className":133248},[31690],[32188,133250],{},[19,133252,133254],{"id":133253},"_5-删除顶点","5. 删除顶点",[15,133256,133257,133258,133286,133287,133315,133316,133344],{},"从图 ",[34,133259,133261,133274],{"className":133260},[31642],[34,133262,133264],{"className":133263},[31646],[31648,133265,133266],{"xmlns":31650},[31652,133267,133268,133272],{},[31655,133269,133270],{},[31658,133271,97709],{},[31665,133273,97709],{"encoding":31667},[34,133275,133277],{"className":133276,"ariaHidden":758},[31672],[34,133278,133280,133283],{"className":133279},[31676],[34,133281],{"className":133282,"style":33218},[31680],[34,133284,97709],{"className":133285},[31685,31686]," 中删去顶点 ",[34,133288,133290,133303],{"className":133289},[31642],[34,133291,133293],{"className":133292},[31646],[31648,133294,133295],{"xmlns":31650},[31652,133296,133297,133301],{},[31655,133298,133299],{},[31658,133300,56782],{},[31665,133302,56782],{"encoding":31667},[34,133304,133306],{"className":133305,"ariaHidden":758},[31672],[34,133307,133309,133312],{"className":133308},[31676],[34,133310],{"className":133311,"style":32527},[31680],[34,133313,56782],{"className":133314,"style":32062},[31685,31686],"，同时删去与 ",[34,133317,133319,133332],{"className":133318},[31642],[34,133320,133322],{"className":133321},[31646],[31648,133323,133324],{"xmlns":31650},[31652,133325,133326,133330],{},[31655,133327,133328],{},[31658,133329,56782],{},[31665,133331,56782],{"encoding":31667},[34,133333,133335],{"className":133334,"ariaHidden":758},[31672],[34,133336,133338,133341],{"className":133337},[31676],[34,133339],{"className":133340,"style":32527},[31680],[34,133342,56782],{"className":133343,"style":32062},[31685,31686]," 关联的所有边，记作：",[34,133346,133348],{"className":133347},[126385],[34,133349,133351,133369],{"className":133350},[31642],[34,133352,133354],{"className":133353},[31646],[31648,133355,133356],{"xmlns":31650,"display":126394},[31652,133357,133358,133366],{},[31655,133359,133360,133362,133364],{},[31658,133361,97709],{},[31662,133363,32226],{},[31658,133365,56782],{},[31665,133367,133368],{"encoding":31667},"G-v",[34,133370,133372,133390],{"className":133371,"ariaHidden":758},[31672],[34,133373,133375,133378,133381,133384,133387],{"className":133374},[31676],[34,133376],{"className":133377,"style":34944},[31680],[34,133379,97709],{"className":133380},[31685,31686],[34,133382],{"className":133383,"style":32604},[31845],[34,133385,32226],{"className":133386},[32024],[34,133388],{"className":133389,"style":32604},[31845],[34,133391,133393,133396],{"className":133392},[31676],[34,133394],{"className":133395,"style":32527},[31680],[34,133397,56782],{"className":133398,"style":32062},[31685,31686],[15,133400,133401,133402,127434],{},"如果删去顶点集合 ",[34,133403,133405,133422],{"className":133404},[31642],[34,133406,133408],{"className":133407},[31646],[31648,133409,133410],{"xmlns":31650},[31652,133411,133412,133420],{},[31655,133413,133414],{},[37624,133415,133416,133418],{},[31658,133417,126532],{},[31662,133419,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,133421,132935],{"encoding":31667},[34,133423,133425],{"className":133424,"ariaHidden":758},[31672],[34,133426,133428,133431],{"className":133427},[31676],[34,133429],{"className":133430,"style":132027},[31680],[34,133432,133434,133437],{"className":133433},[31685],[34,133435,126532],{"className":133436,"style":32604},[31685,31686],[34,133438,133440],{"className":133439},[31792],[34,133441,133443],{"className":133442},[31796],[34,133444,133446],{"className":133445},[31801],[34,133447,133449],{"className":133448,"style":132027},[31805],[34,133450,133451,133454],{"style":37777},[34,133452],{"className":133453,"style":31814},[31813],[34,133455,133457],{"className":133456},[31818,31819,31820,31821],[34,133458,133460],{"className":133459},[31685,31821],[34,133461,116495],{"className":133462},[31685,31821],[34,133464,133466],{"className":133465},[126385],[34,133467,133469,133491],{"className":133468},[31642],[34,133470,133472],{"className":133471},[31646],[31648,133473,133474],{"xmlns":31650,"display":126394},[31652,133475,133476,133488],{},[31655,133477,133478,133480,133482],{},[31658,133479,97709],{},[31662,133481,32226],{},[37624,133483,133484,133486],{},[31658,133485,126532],{},[31662,133487,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,133489,133490],{"encoding":31667},"G-V'",[34,133492,133494,133512],{"className":133493,"ariaHidden":758},[31672],[34,133495,133497,133500,133503,133506,133509],{"className":133496},[31676],[34,133498],{"className":133499,"style":34944},[31680],[34,133501,97709],{"className":133502},[31685,31686],[34,133504],{"className":133505,"style":32604},[31845],[34,133507,32226],{"className":133508},[32024],[34,133510],{"className":133511,"style":32604},[31845],[34,133513,133515,133518],{"className":133514},[31676],[34,133516],{"className":133517,"style":131724},[31680],[34,133519,133521,133524],{"className":133520},[31685],[34,133522,126532],{"className":133523,"style":32604},[31685,31686],[34,133525,133527],{"className":133526},[31792],[34,133528,133530],{"className":133529},[31796],[34,133531,133533],{"className":133532},[31801],[34,133534,133536],{"className":133535,"style":131724},[31805],[34,133537,133538,133541],{"style":129846},[34,133539],{"className":133540,"style":31814},[31813],[34,133542,133544],{"className":133543},[31818,31819,31820,31821],[34,133545,133547],{"className":133546},[31685,31821],[34,133548,116495],{"className":133549},[31685,31821],[32188,133551],{},[19,133553,133555],{"id":133554},"_6-删除边","6. 删除边",[15,133557,133558,133559,127434],{},"从图中删去边 ",[34,133560,133562,133575],{"className":133561},[31642],[34,133563,133565],{"className":133564},[31646],[31648,133566,133567],{"xmlns":31650},[31652,133568,133569,133573],{},[31655,133570,133571],{},[31658,133572,33244],{},[31665,133574,33244],{"encoding":31667},[34,133576,133578],{"className":133577,"ariaHidden":758},[31672],[34,133579,133581,133584],{"className":133580},[31676],[34,133582],{"className":133583,"style":32527},[31680],[34,133585,33244],{"className":133586},[31685,31686],[34,133588,133590],{"className":133589},[126385],[34,133591,133593,133611],{"className":133592},[31642],[34,133594,133596],{"className":133595},[31646],[31648,133597,133598],{"xmlns":31650,"display":126394},[31652,133599,133600,133608],{},[31655,133601,133602,133604,133606],{},[31658,133603,97709],{},[31662,133605,32226],{},[31658,133607,33244],{},[31665,133609,133610],{"encoding":31667},"G-e",[34,133612,133614,133632],{"className":133613,"ariaHidden":758},[31672],[34,133615,133617,133620,133623,133626,133629],{"className":133616},[31676],[34,133618],{"className":133619,"style":34944},[31680],[34,133621,97709],{"className":133622},[31685,31686],[34,133624],{"className":133625,"style":32604},[31845],[34,133627,32226],{"className":133628},[32024],[34,133630],{"className":133631,"style":32604},[31845],[34,133633,133635,133638],{"className":133634},[31676],[34,133636],{"className":133637,"style":32527},[31680],[34,133639,33244],{"className":133640},[31685,31686],[15,133642,133643,133644,127434],{},"从图中删去边集 ",[34,133645,133647,133665],{"className":133646},[31642],[34,133648,133650],{"className":133649},[31646],[31648,133651,133652],{"xmlns":31650},[31652,133653,133654,133662],{},[31655,133655,133656],{},[37624,133657,133658,133660],{},[31658,133659,96058],{},[31662,133661,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,133663,133664],{"encoding":31667},"E'",[34,133666,133668],{"className":133667,"ariaHidden":758},[31672],[34,133669,133671,133674],{"className":133670},[31676],[34,133672],{"className":133673,"style":132027},[31680],[34,133675,133677,133680],{"className":133676},[31685],[34,133678,96058],{"className":133679,"style":106455},[31685,31686],[34,133681,133683],{"className":133682},[31792],[34,133684,133686],{"className":133685},[31796],[34,133687,133689],{"className":133688},[31801],[34,133690,133692],{"className":133691,"style":132027},[31805],[34,133693,133694,133697],{"style":37777},[34,133695],{"className":133696,"style":31814},[31813],[34,133698,133700],{"className":133699},[31818,31819,31820,31821],[34,133701,133703],{"className":133702},[31685,31821],[34,133704,116495],{"className":133705},[31685,31821],[34,133707,133709],{"className":133708},[126385],[34,133710,133712,133734],{"className":133711},[31642],[34,133713,133715],{"className":133714},[31646],[31648,133716,133717],{"xmlns":31650,"display":126394},[31652,133718,133719,133731],{},[31655,133720,133721,133723,133725],{},[31658,133722,97709],{},[31662,133724,32226],{},[37624,133726,133727,133729],{},[31658,133728,96058],{},[31662,133730,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,133732,133733],{"encoding":31667},"G-E'",[34,133735,133737,133755],{"className":133736,"ariaHidden":758},[31672],[34,133738,133740,133743,133746,133749,133752],{"className":133739},[31676],[34,133741],{"className":133742,"style":34944},[31680],[34,133744,97709],{"className":133745},[31685,31686],[34,133747],{"className":133748,"style":32604},[31845],[34,133750,32226],{"className":133751},[32024],[34,133753],{"className":133754,"style":32604},[31845],[34,133756,133758,133761],{"className":133757},[31676],[34,133759],{"className":133760,"style":131724},[31680],[34,133762,133764,133767],{"className":133763},[31685],[34,133765,96058],{"className":133766,"style":106455},[31685,31686],[34,133768,133770],{"className":133769},[31792],[34,133771,133773],{"className":133772},[31796],[34,133774,133776],{"className":133775},[31801],[34,133777,133779],{"className":133778,"style":131724},[31805],[34,133780,133781,133784],{"style":129846},[34,133782],{"className":133783,"style":31814},[31813],[34,133785,133787],{"className":133786},[31818,31819,31820,31821],[34,133788,133790],{"className":133789},[31685,31821],[34,133791,116495],{"className":133792},[31685,31821],[32188,133794],{},[19,133796,133798],{"id":133797},"_7-加边","7. 加边",[15,133800,133801,133802,127434],{},"如果在图中加入边 ",[34,133803,133805,133818],{"className":133804},[31642],[34,133806,133808],{"className":133807},[31646],[31648,133809,133810],{"xmlns":31650},[31652,133811,133812,133816],{},[31655,133813,133814],{},[31658,133815,33244],{},[31665,133817,33244],{"encoding":31667},[34,133819,133821],{"className":133820,"ariaHidden":758},[31672],[34,133822,133824,133827],{"className":133823},[31676],[34,133825],{"className":133826,"style":32527},[31680],[34,133828,33244],{"className":133829},[31685,31686],[34,133831,133833],{"className":133832},[126385],[34,133834,133836,133854],{"className":133835},[31642],[34,133837,133839],{"className":133838},[31646],[31648,133840,133841],{"xmlns":31650,"display":126394},[31652,133842,133843,133851],{},[31655,133844,133845,133847,133849],{},[31658,133846,97709],{},[31662,133848,223],{},[31658,133850,33244],{},[31665,133852,133853],{"encoding":31667},"G+e",[34,133855,133857,133875],{"className":133856,"ariaHidden":758},[31672],[34,133858,133860,133863,133866,133869,133872],{"className":133859},[31676],[34,133861],{"className":133862,"style":34944},[31680],[34,133864,97709],{"className":133865},[31685,31686],[34,133867],{"className":133868,"style":32604},[31845],[34,133870,223],{"className":133871},[32024],[34,133873],{"className":133874,"style":32604},[31845],[34,133876,133878,133881],{"className":133877},[31676],[34,133879],{"className":133880,"style":32527},[31680],[34,133882,33244],{"className":133883},[31685,31686],[32188,133885],{},[11,133887,133889],{"id":133888},"八补图","八、补图",[19,133891,133893],{"id":133892},"_1-一般补图思想","1. 一般补图思想",[15,133895,133896],{},"补图的意思是：",[104833,133898,133899],{},[15,133900,133901],{},"原图没有的边，在补图里有；原图有的边，在补图里没有。",[32188,133903],{},[19,133905,133907],{"id":133906},"_2-简单图的补图","2. 简单图的补图",[15,133909,133910,133911,133939,133940,133968,133969,133997],{},"设 ",[34,133912,133914,133927],{"className":133913},[31642],[34,133915,133917],{"className":133916},[31646],[31648,133918,133919],{"xmlns":31650},[31652,133920,133921,133925],{},[31655,133922,133923],{},[31658,133924,97709],{},[31665,133926,97709],{"encoding":31667},[34,133928,133930],{"className":133929,"ariaHidden":758},[31672],[34,133931,133933,133936],{"className":133932},[31676],[34,133934],{"className":133935,"style":33218},[31680],[34,133937,97709],{"className":133938},[31685,31686]," 是含 ",[34,133941,133943,133956],{"className":133942},[31642],[34,133944,133946],{"className":133945},[31646],[31648,133947,133948],{"xmlns":31650},[31652,133949,133950,133954],{},[31655,133951,133952],{},[31658,133953,31660],{},[31665,133955,31660],{"encoding":31667},[34,133957,133959],{"className":133958,"ariaHidden":758},[31672],[34,133960,133962,133965],{"className":133961},[31676],[34,133963],{"className":133964,"style":32527},[31680],[34,133966,31660],{"className":133967},[31685,31686]," 个顶点的简单图，则 ",[34,133970,133972,133985],{"className":133971},[31642],[34,133973,133975],{"className":133974},[31646],[31648,133976,133977],{"xmlns":31650},[31652,133978,133979,133983],{},[31655,133980,133981],{},[31658,133982,97709],{},[31665,133984,97709],{"encoding":31667},[34,133986,133988],{"className":133987,"ariaHidden":758},[31672],[34,133989,133991,133994],{"className":133990},[31676],[34,133992],{"className":133993,"style":33218},[31680],[34,133995,97709],{"className":133996},[31685,31686]," 的补图记作：",[34,133999,134001],{"className":134000},[126385],[34,134002,134004,134023],{"className":134003},[31642],[34,134005,134007],{"className":134006},[31646],[31648,134008,134009],{"xmlns":31650,"display":126394},[31652,134010,134011,134020],{},[31655,134012,134013],{},[134014,134015,134016,134018],"mover",{"accent":758},[31658,134017,97709],{},[31662,134019,105837],{"stretchy":758},[31665,134021,134022],{"encoding":31667},"\\overline G",[34,134024,134026],{"className":134025,"ariaHidden":758},[31672],[34,134027,134029,134033],{"className":134028},[31676],[34,134030],{"className":134031,"style":134032},[31680],"height:0.8833em;",[34,134034,134037],{"className":134035},[31685,134036],"overline",[34,134038,134040],{"className":134039},[31796],[34,134041,134043],{"className":134042},[31801],[34,134044,134046,134057],{"className":134045,"style":134032},[31805],[34,134047,134048,134051],{"style":41565},[34,134049],{"className":134050,"style":41508},[31813],[34,134052,134054],{"className":134053},[31685],[34,134055,97709],{"className":134056},[31685,31686],[34,134058,134060,134063],{"style":134059},"top:-3.8033em;",[34,134061],{"className":134062,"style":41508},[31813],[34,134064],{"className":134065,"style":41528},[134066],"overline-line",[15,134068,134069,134070,134098],{},"它的顶点集和 ",[34,134071,134073,134086],{"className":134072},[31642],[34,134074,134076],{"className":134075},[31646],[31648,134077,134078],{"xmlns":31650},[31652,134079,134080,134084],{},[31655,134081,134082],{},[31658,134083,97709],{},[31665,134085,97709],{"encoding":31667},[34,134087,134089],{"className":134088,"ariaHidden":758},[31672],[34,134090,134092,134095],{"className":134091},[31676],[34,134093],{"className":134094,"style":33218},[31680],[34,134096,97709],{"className":134097},[31685,31686]," 相同。",[15,134100,134101],{},"并且：",[34,134103,134105],{"className":134104},[126385],[34,134106,134108,134138],{"className":134107},[31642],[34,134109,134111],{"className":134110},[31646],[31648,134112,134113],{"xmlns":31650,"display":126394},[31652,134114,134115,134135],{},[31655,134116,134117,134119,134121,134123,134125,134127,134133],{},[31658,134118,94628],{},[31658,134120,56782],{},[31662,134122,112296],{},[31658,134124,96058],{},[31662,134126,456],{"stretchy":702},[134014,134128,134129,134131],{"accent":758},[31658,134130,97709],{},[31662,134132,105837],{"stretchy":758},[31662,134134,2557],{"stretchy":702},[31665,134136,134137],{"encoding":31667},"uv\\in E(\\overline G)",[34,134139,134141,134160],{"className":134140,"ariaHidden":758},[31672],[34,134142,134144,134147,134151,134154,134157],{"className":134143},[31676],[34,134145],{"className":134146,"style":34812},[31680],[34,134148,134150],{"className":134149,"style":32062},[31685,31686],"uv",[34,134152],{"className":134153,"style":31846},[31845],[34,134155,112296],{"className":134156},[31850],[34,134158],{"className":134159,"style":31846},[31845],[34,134161,134163,134167,134170,134173,134204],{"className":134162},[31676],[34,134164],{"className":134165,"style":134166},[31680],"height:1.1333em;vertical-align:-0.25em;",[34,134168,96058],{"className":134169,"style":106455},[31685,31686],[34,134171,456],{"className":134172},[31782],[34,134174,134176],{"className":134175},[31685,134036],[34,134177,134179],{"className":134178},[31796],[34,134180,134182],{"className":134181},[31801],[34,134183,134185,134196],{"className":134184,"style":134032},[31805],[34,134186,134187,134190],{"style":41565},[34,134188],{"className":134189,"style":41508},[31813],[34,134191,134193],{"className":134192},[31685],[34,134194,97709],{"className":134195},[31685,31686],[34,134197,134198,134201],{"style":134059},[34,134199],{"className":134200,"style":41508},[31813],[34,134202],{"className":134203,"style":41528},[134066],[34,134205,2557],{"className":134206},[31690],[15,134208,134209],{},"当且仅当：",[34,134211,134213],{"className":134212},[126385],[34,134214,134216,134242],{"className":134215},[31642],[34,134217,134219],{"className":134218},[31646],[31648,134220,134221],{"xmlns":31650,"display":126394},[31652,134222,134223,134239],{},[31655,134224,134225,134227,134229,134231,134233,134235,134237],{},[31658,134226,94628],{},[31658,134228,56782],{},[31662,134230,112266],{"mathvariant":38079},[31658,134232,96058],{},[31662,134234,456],{"stretchy":702},[31658,134236,97709],{},[31662,134238,2557],{"stretchy":702},[31665,134240,134241],{"encoding":31667},"uv\\notin E(G)",[34,134243,134245,134296],{"className":134244,"ariaHidden":758},[31672],[34,134246,134248,134251,134254,134257,134293],{"className":134247},[31676],[34,134249],{"className":134250,"style":31774},[31680],[34,134252,134150],{"className":134253,"style":32062},[31685,31686],[34,134255],{"className":134256,"style":31846},[31845],[34,134258,134260,134266],{"className":134259},[31850],[34,134261,134263],{"className":134262},[31685],[34,134264,112296],{"className":134265},[31850],[34,134267,134269],{"className":134268},[31685,54774],[34,134270,134272],{"className":134271},[54778],[34,134273,134275,134278,134290],{"className":134274},[112306],[34,134276],{"className":134277,"style":31774},[31680],[34,134279,134281],{"className":134280},[54789],[34,134282,134284,134287],{"className":134283},[31685],[34,134285,38080],{"className":134286},[31685],[34,134288],{"className":134289,"style":112322},[31845],[34,134291],{"className":134292},[54800],[34,134294],{"className":134295,"style":31846},[31845],[34,134297,134299,134302,134305,134308,134311],{"className":134298},[31676],[34,134300],{"className":134301,"style":31774},[31680],[34,134303,96058],{"className":134304,"style":106455},[31685,31686],[34,134306,456],{"className":134307},[31782],[34,134309,97709],{"className":134310},[31685,31686],[34,134312,2557],{"className":134313},[31690],[15,134315,134316],{},"也就是说：",[34,134318,134320],{"className":134319},[126385],[34,134321,134323,134353],{"className":134322},[31642],[34,134324,134326],{"className":134325},[31646],[31648,134327,134328],{"xmlns":31650,"display":126394},[31652,134329,134330,134350],{},[31655,134331,134332,134334,134336,134342,134344],{},[31658,134333,97709],{},[31662,134335,223],{},[134014,134337,134338,134340],{"accent":758},[31658,134339,97709],{},[31662,134341,105837],{"stretchy":758},[31662,134343,202],{},[31743,134345,134346,134348],{},[31658,134347,38691],{},[31658,134349,31660],{},[31665,134351,134352],{"encoding":31667},"G+\\overline G=K_n",[34,134354,134356,134374,134420],{"className":134355,"ariaHidden":758},[31672],[34,134357,134359,134362,134365,134368,134371],{"className":134358},[31676],[34,134360],{"className":134361,"style":34944},[31680],[34,134363,97709],{"className":134364},[31685,31686],[34,134366],{"className":134367,"style":32604},[31845],[34,134369,223],{"className":134370},[32024],[34,134372],{"className":134373,"style":32604},[31845],[34,134375,134377,134380,134411,134414,134417],{"className":134376},[31676],[34,134378],{"className":134379,"style":134032},[31680],[34,134381,134383],{"className":134382},[31685,134036],[34,134384,134386],{"className":134385},[31796],[34,134387,134389],{"className":134388},[31801],[34,134390,134392,134403],{"className":134391,"style":134032},[31805],[34,134393,134394,134397],{"style":41565},[34,134395],{"className":134396,"style":41508},[31813],[34,134398,134400],{"className":134399},[31685],[34,134401,97709],{"className":134402},[31685,31686],[34,134404,134405,134408],{"style":134059},[34,134406],{"className":134407,"style":41508},[31813],[34,134409],{"className":134410,"style":41528},[134066],[34,134412],{"className":134413,"style":31846},[31845],[34,134415,202],{"className":134416},[31850],[34,134418],{"className":134419,"style":31846},[31845],[34,134421,134423,134426],{"className":134422},[31676],[34,134424],{"className":134425,"style":70915},[31680],[34,134427,134429,134432],{"className":134428},[31685],[34,134430,38691],{"className":134431,"style":38706},[31685,31686],[34,134433,134435],{"className":134434},[31792],[34,134436,134438,134458],{"className":134437},[31796,31797],[34,134439,134441,134455],{"className":134440},[31801],[34,134442,134444],{"className":134443,"style":32362},[31805],[34,134445,134446,134449],{"style":129511},[34,134447],{"className":134448,"style":31814},[31813],[34,134450,134452],{"className":134451},[31818,31819,31820,31821],[34,134453,31660],{"className":134454},[31685,31686,31821],[34,134456,31829],{"className":134457},[31828],[34,134459,134461],{"className":134460},[31801],[34,134462,134464],{"className":134463,"style":31836},[31805],[34,134465],{},[32188,134467],{},[11,134469,134471],{"id":134470},"九图的同构","九、图的同构",[19,134473,134475],{"id":134474},"_1-定义","1. 定义",[15,134477,134478,134479,258,134507,134568,134569,347],{},"两个图 ",[34,134480,134482,134495],{"className":134481},[31642],[34,134483,134485],{"className":134484},[31646],[31648,134486,134487],{"xmlns":31650},[31652,134488,134489,134493],{},[31655,134490,134491],{},[31658,134492,97709],{},[31665,134494,97709],{"encoding":31667},[34,134496,134498],{"className":134497,"ariaHidden":758},[31672],[34,134499,134501,134504],{"className":134500},[31676],[34,134502],{"className":134503,"style":33218},[31680],[34,134505,97709],{"className":134506},[31685,31686],[34,134508,134510,134527],{"className":134509},[31642],[34,134511,134513],{"className":134512},[31646],[31648,134514,134515],{"xmlns":31650},[31652,134516,134517,134525],{},[31655,134518,134519],{},[37624,134520,134521,134523],{},[31658,134522,97709],{},[31662,134524,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,134526,132017],{"encoding":31667},[34,134528,134530],{"className":134529,"ariaHidden":758},[31672],[34,134531,134533,134536],{"className":134532},[31676],[34,134534],{"className":134535,"style":132027},[31680],[34,134537,134539,134542],{"className":134538},[31685],[34,134540,97709],{"className":134541},[31685,31686],[34,134543,134545],{"className":134544},[31792],[34,134546,134548],{"className":134547},[31796],[34,134549,134551],{"className":134550},[31801],[34,134552,134554],{"className":134553,"style":132027},[31805],[34,134555,134556,134559],{"style":37777},[34,134557],{"className":134558,"style":31814},[31813],[34,134560,134562],{"className":134561},[31818,31819,31820,31821],[34,134563,134565],{"className":134564},[31685,31821],[34,134566,116495],{"className":134567},[31685,31821]," 如果只是“画法不同”，但结构完全一样，就叫",[2627,134570,134571],{},"同构",[15,134573,134574],{},"严格说，如果存在一个顶点双射：",[34,134576,134578],{"className":134577},[126385],[34,134579,134581,134619],{"className":134580},[31642],[34,134582,134584],{"className":134583},[31646],[31648,134585,134586],{"xmlns":31650,"display":126394},[31652,134587,134588,134616],{},[31655,134589,134590,134592,134594,134596,134598,134600,134602,134604,134606,134608,134614],{},[31658,134591,31739],{},[31662,134593,5598],{},[31658,134595,126532],{},[31662,134597,456],{"stretchy":702},[31658,134599,97709],{},[31662,134601,2557],{"stretchy":702},[31662,134603,33239],{},[31658,134605,126532],{},[31662,134607,456],{"stretchy":702},[37624,134609,134610,134612],{},[31658,134611,97709],{},[31662,134613,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,134615,2557],{"stretchy":702},[31665,134617,134618],{"encoding":31667},"f:V(G)\\to V(G')",[34,134620,134622,134640,134667],{"className":134621,"ariaHidden":758},[31672],[34,134623,134625,134628,134631,134634,134637],{"className":134624},[31676],[34,134626],{"className":134627,"style":43217},[31680],[34,134629,31739],{"className":134630,"style":31778},[31685,31686],[34,134632],{"className":134633,"style":31846},[31845],[34,134635,5598],{"className":134636},[31850],[34,134638],{"className":134639,"style":31846},[31845],[34,134641,134643,134646,134649,134652,134655,134658,134661,134664],{"className":134642},[31676],[34,134644],{"className":134645,"style":31774},[31680],[34,134647,126532],{"className":134648,"style":32604},[31685,31686],[34,134650,456],{"className":134651},[31782],[34,134653,97709],{"className":134654},[31685,31686],[34,134656,2557],{"className":134657},[31690],[34,134659],{"className":134660,"style":31846},[31845],[34,134662,33239],{"className":134663},[31850],[34,134665],{"className":134666,"style":31846},[31845],[34,134668,134670,134673,134676,134679,134711],{"className":134669},[31676],[34,134671],{"className":134672,"style":131678},[31680],[34,134674,126532],{"className":134675,"style":32604},[31685,31686],[34,134677,456],{"className":134678},[31782],[34,134680,134682,134685],{"className":134681},[31685],[34,134683,97709],{"className":134684},[31685,31686],[34,134686,134688],{"className":134687},[31792],[34,134689,134691],{"className":134690},[31796],[34,134692,134694],{"className":134693},[31801],[34,134695,134697],{"className":134696,"style":131724},[31805],[34,134698,134699,134702],{"style":129846},[34,134700],{"className":134701,"style":31814},[31813],[34,134703,134705],{"className":134704},[31818,31819,31820,31821],[34,134706,134708],{"className":134707},[31685,31821],[34,134709,116495],{"className":134710},[31685,31821],[34,134712,2557],{"className":134713},[31690],[15,134715,134716,134717,43298,134745,134806],{},"并且保持邻接关系，则称 ",[34,134718,134720,134733],{"className":134719},[31642],[34,134721,134723],{"className":134722},[31646],[31648,134724,134725],{"xmlns":31650},[31652,134726,134727,134731],{},[31655,134728,134729],{},[31658,134730,97709],{},[31665,134732,97709],{"encoding":31667},[34,134734,134736],{"className":134735,"ariaHidden":758},[31672],[34,134737,134739,134742],{"className":134738},[31676],[34,134740],{"className":134741,"style":33218},[31680],[34,134743,97709],{"className":134744},[31685,31686],[34,134746,134748,134765],{"className":134747},[31642],[34,134749,134751],{"className":134750},[31646],[31648,134752,134753],{"xmlns":31650},[31652,134754,134755,134763],{},[31655,134756,134757],{},[37624,134758,134759,134761],{},[31658,134760,97709],{},[31662,134762,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,134764,132017],{"encoding":31667},[34,134766,134768],{"className":134767,"ariaHidden":758},[31672],[34,134769,134771,134774],{"className":134770},[31676],[34,134772],{"className":134773,"style":132027},[31680],[34,134775,134777,134780],{"className":134776},[31685],[34,134778,97709],{"className":134779},[31685,31686],[34,134781,134783],{"className":134782},[31792],[34,134784,134786],{"className":134785},[31796],[34,134787,134789],{"className":134788},[31801],[34,134790,134792],{"className":134791,"style":132027},[31805],[34,134793,134794,134797],{"style":37777},[34,134795],{"className":134796,"style":31814},[31813],[34,134798,134800],{"className":134799},[31818,31819,31820,31821],[34,134801,134803],{"className":134802},[31685,31821],[34,134804,116495],{"className":134805},[31685,31821]," 同构，记作：",[34,134808,134810],{"className":134809},[126385],[34,134811,134813,134836],{"className":134812},[31642],[34,134814,134816],{"className":134815},[31646],[31648,134817,134818],{"xmlns":31650,"display":126394},[31652,134819,134820,134833],{},[31655,134821,134822,134824,134827],{},[31658,134823,97709],{},[31662,134825,134826],{},"≅",[37624,134828,134829,134831],{},[31658,134830,97709],{},[31662,134832,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,134834,134835],{"encoding":31667},"G\\cong G'",[34,134837,134839,134857],{"className":134838,"ariaHidden":758},[31672],[34,134840,134842,134845,134848,134851,134854],{"className":134841},[31676],[34,134843],{"className":134844,"style":33218},[31680],[34,134846,97709],{"className":134847},[31685,31686],[34,134849],{"className":134850,"style":31846},[31845],[34,134852,134826],{"className":134853},[31850],[34,134855],{"className":134856,"style":31846},[31845],[34,134858,134860,134863],{"className":134859},[31676],[34,134861],{"className":134862,"style":131724},[31680],[34,134864,134866,134869],{"className":134865},[31685],[34,134867,97709],{"className":134868},[31685,31686],[34,134870,134872],{"className":134871},[31792],[34,134873,134875],{"className":134874},[31796],[34,134876,134878],{"className":134877},[31801],[34,134879,134881],{"className":134880,"style":131724},[31805],[34,134882,134883,134886],{"style":129846},[34,134884],{"className":134885,"style":31814},[31813],[34,134887,134889],{"className":134888},[31818,31819,31820,31821],[34,134890,134892],{"className":134891},[31685,31821],[34,134893,116495],{"className":134894},[31685,31821],[32188,134896],{},[19,134898,134900],{"id":134899},"_2-图同构要保持什么","2. 图同构要保持什么",[15,134902,134903],{},"如果两个图同构，必须保持：",[2230,134905,134906,134909,134912,134915,134918,134921],{},[108,134907,134908],{},"顶点数相同；",[108,134910,134911],{},"边数相同；",[108,134913,134914],{},"度数序列相同；",[108,134916,134917],{},"邻接关系相同；",[108,134919,134920],{},"有向图还要保持方向；",[108,134922,134923],{},"自环、重边数量也要对应。",[32188,134925],{},[19,134927,134929],{"id":134928},"_3-注意这些只是必要条件不一定充分","3. 注意：这些只是必要条件，不一定充分",[15,134931,134316],{},[34,134933,134935],{"className":134934},[126385],[34,134936,134938,134953],{"className":134937},[31642],[34,134939,134941],{"className":134940},[31646],[31648,134942,134943],{"xmlns":31650,"display":126394},[31652,134944,134945,134950],{},[31655,134946,134947],{},[39361,134948,134949],{},"顶点数相同、边数相同、度数序列相同",[31665,134951,134952],{"encoding":31667},"\\text{顶点数相同、边数相同、度数序列相同}",[34,134954,134956],{"className":134955,"ariaHidden":758},[31672],[34,134957,134959,134962],{"className":134958},[31676],[34,134960],{"className":134961,"style":33218},[31680],[34,134963,134965],{"className":134964},[31685,2638],[34,134966,134949],{"className":134967},[31685,105462],[15,134969,134970,134971,134974],{},"只能说明它们",[2627,134972,134973],{},"可能同构","，不能保证一定同构。",[15,134976,134977],{},"判断不同构时，常用方法是找不变量不同。",[15,134979,2632],{},[105,134981,134982,134985,134988,134991],{},[108,134983,134984],{},"一个有孤立点，另一个没有；",[108,134986,134987],{},"一个有三角形，另一个没有；",[108,134989,134990],{},"一个有自环，另一个没有；",[108,134992,134993],{},"一个连通，另一个不连通。",[32188,134995],{},[11,134997,134999],{"id":134998},"_122-路径与循环","12.2 路径与循环",[15,135001,135002],{},"这一节研究“能不能从一个顶点走到另一个顶点”。",[32188,135004],{},[11,135006,135008],{"id":135007},"一路径","一、路径",[15,135010,131588],{},[34,135012,135014],{"className":135013},[126385],[34,135015,135017,135042],{"className":135016},[31642],[34,135018,135020],{"className":135019},[31646],[31648,135021,135022],{"xmlns":31650,"display":126394},[31652,135023,135024,135040],{},[31655,135025,135026,135028,135030,135032,135034,135036,135038],{},[31658,135027,97709],{},[31662,135029,202],{},[31662,135031,126529],{"stretchy":702},[31658,135033,126532],{},[31662,135035,31960],{"separator":758},[31658,135037,96058],{},[31662,135039,126544],{"stretchy":702},[31665,135041,131033],{"encoding":31667},[34,135043,135045,135063],{"className":135044,"ariaHidden":758},[31672],[34,135046,135048,135051,135054,135057,135060],{"className":135047},[31676],[34,135049],{"className":135050,"style":33218},[31680],[34,135052,97709],{"className":135053},[31685,31686],[34,135055],{"className":135056,"style":31846},[31845],[34,135058,202],{"className":135059},[31850],[34,135061],{"className":135062,"style":31846},[31845],[34,135064,135066,135069,135072,135075,135078,135081,135084],{"className":135065},[31676],[34,135067],{"className":135068,"style":31774},[31680],[34,135070,126529],{"className":135071},[31782],[34,135073,126532],{"className":135074,"style":32604},[31685,31686],[34,135076,31960],{"className":135077},[32119],[34,135079],{"className":135080,"style":32123},[31845],[34,135082,96058],{"className":135083,"style":106455},[31685,31686],[34,135085,126544],{"className":135086},[31690],[15,135088,135089],{},"一个路径可以写成：",[34,135091,135093],{"className":135092},[126385],[34,135094,135096,135157],{"className":135095},[31642],[34,135097,135099],{"className":135098},[31646],[31648,135100,135101],{"xmlns":31650,"display":126394},[31652,135102,135103,135154],{},[31655,135104,135105,135108,135110,135116,135122,135128,135134,135140,135142,135148],{},[31658,135106,135107],{},"μ",[31662,135109,202],{},[31743,135111,135112,135114],{},[31658,135113,56782],{},[31749,135115,459],{},[31743,135117,135118,135120],{},[31658,135119,33244],{},[31749,135121,640],{},[31743,135123,135124,135126],{},[31658,135125,56782],{},[31749,135127,640],{},[31743,135129,135130,135132],{},[31658,135131,33244],{},[31749,135133,768],{},[31743,135135,135136,135138],{},[31658,135137,56782],{},[31749,135139,768],{},[31662,135141,38874],{},[31743,135143,135144,135146],{},[31658,135145,33244],{},[31658,135147,33759],{},[31743,135149,135150,135152],{},[31658,135151,56782],{},[31658,135153,33759],{},[31665,135155,135156],{"encoding":31667},"\\mu=v_0e_1v_1e_2v_2\\cdots e_kv_k",[34,135158,135160,135178],{"className":135159,"ariaHidden":758},[31672],[34,135161,135163,135166,135169,135172,135175],{"className":135162},[31676],[34,135164],{"className":135165,"style":38774},[31680],[34,135167,135107],{"className":135168},[31685,31686],[34,135170],{"className":135171,"style":31846},[31845],[34,135173,202],{"className":135174},[31850],[34,135176],{"className":135177,"style":31846},[31845],[34,135179,135181,135184,135224,135264,135304,135344,135384,135387,135390,135393,135433],{"className":135180},[31676],[34,135182],{"className":135183,"style":31860},[31680],[34,135185,135187,135190],{"className":135186},[31685],[34,135188,56782],{"className":135189,"style":32062},[31685,31686],[34,135191,135193],{"className":135192},[31792],[34,135194,135196,135216],{"className":135195},[31796,31797],[34,135197,135199,135213],{"className":135198},[31801],[34,135200,135202],{"className":135201,"style":31806},[31805],[34,135203,135204,135207],{"style":38795},[34,135205],{"className":135206,"style":31814},[31813],[34,135208,135210],{"className":135209},[31818,31819,31820,31821],[34,135211,459],{"className":135212},[31685,31821],[34,135214,31829],{"className":135215},[31828],[34,135217,135219],{"className":135218},[31801],[34,135220,135222],{"className":135221,"style":31836},[31805],[34,135223],{},[34,135225,135227,135230],{"className":135226},[31685],[34,135228,33244],{"className":135229},[31685,31686],[34,135231,135233],{"className":135232},[31792],[34,135234,135236,135256],{"className":135235},[31796,31797],[34,135237,135239,135253],{"className":135238},[31801],[34,135240,135242],{"className":135241,"style":31806},[31805],[34,135243,135244,135247],{"style":31809},[34,135245],{"className":135246,"style":31814},[31813],[34,135248,135250],{"className":135249},[31818,31819,31820,31821],[34,135251,640],{"className":135252},[31685,31821],[34,135254,31829],{"className":135255},[31828],[34,135257,135259],{"className":135258},[31801],[34,135260,135262],{"className":135261,"style":31836},[31805],[34,135263],{},[34,135265,135267,135270],{"className":135266},[31685],[34,135268,56782],{"className":135269,"style":32062},[31685,31686],[34,135271,135273],{"className":135272},[31792],[34,135274,135276,135296],{"className":135275},[31796,31797],[34,135277,135279,135293],{"className":135278},[31801],[34,135280,135282],{"className":135281,"style":31806},[31805],[34,135283,135284,135287],{"style":38795},[34,135285],{"className":135286,"style":31814},[31813],[34,135288,135290],{"className":135289},[31818,31819,31820,31821],[34,135291,640],{"className":135292},[31685,31821],[34,135294,31829],{"className":135295},[31828],[34,135297,135299],{"className":135298},[31801],[34,135300,135302],{"className":135301,"style":31836},[31805],[34,135303],{},[34,135305,135307,135310],{"className":135306},[31685],[34,135308,33244],{"className":135309},[31685,31686],[34,135311,135313],{"className":135312},[31792],[34,135314,135316,135336],{"className":135315},[31796,31797],[34,135317,135319,135333],{"className":135318},[31801],[34,135320,135322],{"className":135321,"style":31806},[31805],[34,135323,135324,135327],{"style":31809},[34,135325],{"className":135326,"style":31814},[31813],[34,135328,135330],{"className":135329},[31818,31819,31820,31821],[34,135331,768],{"className":135332},[31685,31821],[34,135334,31829],{"className":135335},[31828],[34,135337,135339],{"className":135338},[31801],[34,135340,135342],{"className":135341,"style":31836},[31805],[34,135343],{},[34,135345,135347,135350],{"className":135346},[31685],[34,135348,56782],{"className":135349,"style":32062},[31685,31686],[34,135351,135353],{"className":135352},[31792],[34,135354,135356,135376],{"className":135355},[31796,31797],[34,135357,135359,135373],{"className":135358},[31801],[34,135360,135362],{"className":135361,"style":31806},[31805],[34,135363,135364,135367],{"style":38795},[34,135365],{"className":135366,"style":31814},[31813],[34,135368,135370],{"className":135369},[31818,31819,31820,31821],[34,135371,768],{"className":135372},[31685,31821],[34,135374,31829],{"className":135375},[31828],[34,135377,135379],{"className":135378},[31801],[34,135380,135382],{"className":135381,"style":31836},[31805],[34,135383],{},[34,135385],{"className":135386,"style":32123},[31845],[34,135388,38874],{"className":135389},[34006],[34,135391],{"className":135392,"style":32123},[31845],[34,135394,135396,135399],{"className":135395},[31685],[34,135397,33244],{"className":135398},[31685,31686],[34,135400,135402],{"className":135401},[31792],[34,135403,135405,135425],{"className":135404},[31796,31797],[34,135406,135408,135422],{"className":135407},[31801],[34,135409,135411],{"className":135410,"style":33833},[31805],[34,135412,135413,135416],{"style":31809},[34,135414],{"className":135415,"style":31814},[31813],[34,135417,135419],{"className":135418},[31818,31819,31820,31821],[34,135420,33759],{"className":135421,"style":33774},[31685,31686,31821],[34,135423,31829],{"className":135424},[31828],[34,135426,135428],{"className":135427},[31801],[34,135429,135431],{"className":135430,"style":31836},[31805],[34,135432],{},[34,135434,135436,135439],{"className":135435},[31685],[34,135437,56782],{"className":135438,"style":32062},[31685,31686],[34,135440,135442],{"className":135441},[31792],[34,135443,135445,135465],{"className":135444},[31796,31797],[34,135446,135448,135462],{"className":135447},[31801],[34,135449,135451],{"className":135450,"style":33833},[31805],[34,135452,135453,135456],{"style":38795},[34,135454],{"className":135455,"style":31814},[31813],[34,135457,135459],{"className":135458},[31818,31819,31820,31821],[34,135460,33759],{"className":135461,"style":33774},[31685,31686,31821],[34,135463,31829],{"className":135464},[31828],[34,135466,135468],{"className":135467},[31801],[34,135469,135471],{"className":135470,"style":31836},[31805],[34,135472],{},[15,135474,131317],{},[15,135476,135477,135478,135548,135549,135743,135744,347],{},"从 ",[34,135479,135481,135499],{"className":135480},[31642],[34,135482,135484],{"className":135483},[31646],[31648,135485,135486],{"xmlns":31650},[31652,135487,135488,135496],{},[31655,135489,135490],{},[31743,135491,135492,135494],{},[31658,135493,56782],{},[31749,135495,459],{},[31665,135497,135498],{"encoding":31667},"v_0",[34,135500,135502],{"className":135501,"ariaHidden":758},[31672],[34,135503,135505,135508],{"className":135504},[31676],[34,135506],{"className":135507,"style":31860},[31680],[34,135509,135511,135514],{"className":135510},[31685],[34,135512,56782],{"className":135513,"style":32062},[31685,31686],[34,135515,135517],{"className":135516},[31792],[34,135518,135520,135540],{"className":135519},[31796,31797],[34,135521,135523,135537],{"className":135522},[31801],[34,135524,135526],{"className":135525,"style":31806},[31805],[34,135527,135528,135531],{"style":38795},[34,135529],{"className":135530,"style":31814},[31813],[34,135532,135534],{"className":135533},[31818,31819,31820,31821],[34,135535,459],{"className":135536},[31685,31821],[34,135538,31829],{"className":135539},[31828],[34,135541,135543],{"className":135542},[31801],[34,135544,135546],{"className":135545,"style":31836},[31805],[34,135547],{}," 出发，经过边 ",[34,135550,135552,135590],{"className":135551},[31642],[34,135553,135555],{"className":135554},[31646],[31648,135556,135557],{"xmlns":31650},[31652,135558,135559,135587],{},[31655,135560,135561,135567,135569,135575,135577,135579,135581],{},[31743,135562,135563,135565],{},[31658,135564,33244],{},[31749,135566,640],{},[31662,135568,31960],{"separator":758},[31743,135570,135571,135573],{},[31658,135572,33244],{},[31749,135574,768],{},[31662,135576,31960],{"separator":758},[31662,135578,33934],{},[31662,135580,31960],{"separator":758},[31743,135582,135583,135585],{},[31658,135584,33244],{},[31658,135586,33759],{},[31665,135588,135589],{"encoding":31667},"e_1,e_2,\\dots,e_k",[34,135591,135593],{"className":135592,"ariaHidden":758},[31672],[34,135594,135596,135599,135639,135642,135645,135685,135688,135691,135694,135697,135700,135703],{"className":135595},[31676],[34,135597],{"className":135598,"style":38774},[31680],[34,135600,135602,135605],{"className":135601},[31685],[34,135603,33244],{"className":135604},[31685,31686],[34,135606,135608],{"className":135607},[31792],[34,135609,135611,135631],{"className":135610},[31796,31797],[34,135612,135614,135628],{"className":135613},[31801],[34,135615,135617],{"className":135616,"style":31806},[31805],[34,135618,135619,135622],{"style":31809},[34,135620],{"className":135621,"style":31814},[31813],[34,135623,135625],{"className":135624},[31818,31819,31820,31821],[34,135626,640],{"className":135627},[31685,31821],[34,135629,31829],{"className":135630},[31828],[34,135632,135634],{"className":135633},[31801],[34,135635,135637],{"className":135636,"style":31836},[31805],[34,135638],{},[34,135640,31960],{"className":135641},[32119],[34,135643],{"className":135644,"style":32123},[31845],[34,135646,135648,135651],{"className":135647},[31685],[34,135649,33244],{"className":135650},[31685,31686],[34,135652,135654],{"className":135653},[31792],[34,135655,135657,135677],{"className":135656},[31796,31797],[34,135658,135660,135674],{"className":135659},[31801],[34,135661,135663],{"className":135662,"style":31806},[31805],[34,135664,135665,135668],{"style":31809},[34,135666],{"className":135667,"style":31814},[31813],[34,135669,135671],{"className":135670},[31818,31819,31820,31821],[34,135672,768],{"className":135673},[31685,31821],[34,135675,31829],{"className":135676},[31828],[34,135678,135680],{"className":135679},[31801],[34,135681,135683],{"className":135682,"style":31836},[31805],[34,135684],{},[34,135686,31960],{"className":135687},[32119],[34,135689],{"className":135690,"style":32123},[31845],[34,135692,33934],{"className":135693},[34006],[34,135695],{"className":135696,"style":32123},[31845],[34,135698,31960],{"className":135699},[32119],[34,135701],{"className":135702,"style":32123},[31845],[34,135704,135706,135709],{"className":135705},[31685],[34,135707,33244],{"className":135708},[31685,31686],[34,135710,135712],{"className":135711},[31792],[34,135713,135715,135735],{"className":135714},[31796,31797],[34,135716,135718,135732],{"className":135717},[31801],[34,135719,135721],{"className":135720,"style":33833},[31805],[34,135722,135723,135726],{"style":31809},[34,135724],{"className":135725,"style":31814},[31813],[34,135727,135729],{"className":135728},[31818,31819,31820,31821],[34,135730,33759],{"className":135731,"style":33774},[31685,31686,31821],[34,135733,31829],{"className":135734},[31828],[34,135736,135738],{"className":135737},[31801],[34,135739,135741],{"className":135740,"style":31836},[31805],[34,135742],{},"，最后到达 ",[34,135745,135747,135765],{"className":135746},[31642],[34,135748,135750],{"className":135749},[31646],[31648,135751,135752],{"xmlns":31650},[31652,135753,135754,135762],{},[31655,135755,135756],{},[31743,135757,135758,135760],{},[31658,135759,56782],{},[31658,135761,33759],{},[31665,135763,135764],{"encoding":31667},"v_k",[34,135766,135768],{"className":135767,"ariaHidden":758},[31672],[34,135769,135771,135774],{"className":135770},[31676],[34,135772],{"className":135773,"style":31860},[31680],[34,135775,135777,135780],{"className":135776},[31685],[34,135778,56782],{"className":135779,"style":32062},[31685,31686],[34,135781,135783],{"className":135782},[31792],[34,135784,135786,135806],{"className":135785},[31796,31797],[34,135787,135789,135803],{"className":135788},[31801],[34,135790,135792],{"className":135791,"style":33833},[31805],[34,135793,135794,135797],{"style":38795},[34,135795],{"className":135796,"style":31814},[31813],[34,135798,135800],{"className":135799},[31818,31819,31820,31821],[34,135801,33759],{"className":135802,"style":33774},[31685,31686,31821],[34,135804,31829],{"className":135805},[31828],[34,135807,135809],{"className":135808},[31801],[34,135810,135812],{"className":135811,"style":31836},[31805],[34,135813],{},[15,135815,135816],{},"路径长度是边数：",[34,135818,135820],{"className":135819},[126385],[34,135821,135823,135845],{"className":135822},[31642],[34,135824,135826],{"className":135825},[31646],[31648,135827,135828],{"xmlns":31650,"display":126394},[31652,135829,135830,135842],{},[31655,135831,135832],{},[126400,135833,135834],{"notation":126402},[71090,135835,135836],{"scriptlevel":459,"displaystyle":702},[71090,135837,135838],{"scriptlevel":459,"displaystyle":702},[71090,135839,135840],{"scriptlevel":459,"displaystyle":758},[31658,135841,33759],{},[31665,135843,135844],{"encoding":31667},"\\boxed{k}",[34,135846,135848],{"className":135847,"ariaHidden":758},[31672],[34,135849,135851,135855],{"className":135850},[31676],[34,135852],{"className":135853,"style":135854},[31680],"height:1.3744em;vertical-align:-0.34em;",[34,135856,135858],{"className":135857},[31685],[34,135859,135861,135900],{"className":135860},[31796,31797],[34,135862,135864,135897],{"className":135863},[31801],[34,135865,135868,135887],{"className":135866,"style":135867},[31805],"height:1.0344em;",[34,135869,135871,135875],{"style":135870},"top:-3.3744em;",[34,135872],{"className":135873,"style":135874},[31813],"height:3.3744em;",[34,135876,135878],{"className":135877},[126448],[34,135879,135881],{"className":135880},[31685],[34,135882,135884],{"className":135883},[31685],[34,135885,33759],{"className":135886,"style":33774},[31685,31686],[34,135888,135890,135893],{"style":135889},"top:-3.0344em;",[34,135891],{"className":135892,"style":135874},[31813],[34,135894],{"className":135895,"style":135896},[126470,126471],"height:1.3744em;border-style:solid;border-width:0.04em;",[34,135898,31829],{"className":135899},[31828],[34,135901,135903],{"className":135902},[31801],[34,135904,135906],{"className":135905,"style":126482},[31805],[34,135907],{},[32188,135909],{},[19,135911,135913],{"id":135912},"_1-路径的起点和终点","1. 路径的起点和终点",[15,135915,135916],{},"在：",[34,135918,135920],{"className":135919},[126385],[34,135921,135923,135971],{"className":135922},[31642],[34,135924,135926],{"className":135925},[31646],[31648,135927,135928],{"xmlns":31650,"display":126394},[31652,135929,135930,135968],{},[31655,135931,135932,135934,135936,135942,135948,135954,135956,135962],{},[31658,135933,135107],{},[31662,135935,202],{},[31743,135937,135938,135940],{},[31658,135939,56782],{},[31749,135941,459],{},[31743,135943,135944,135946],{},[31658,135945,33244],{},[31749,135947,640],{},[31743,135949,135950,135952],{},[31658,135951,56782],{},[31749,135953,640],{},[31662,135955,38874],{},[31743,135957,135958,135960],{},[31658,135959,33244],{},[31658,135961,33759],{},[31743,135963,135964,135966],{},[31658,135965,56782],{},[31658,135967,33759],{},[31665,135969,135970],{"encoding":31667},"\\mu=v_0e_1v_1\\cdots e_kv_k",[34,135972,135974,135992],{"className":135973,"ariaHidden":758},[31672],[34,135975,135977,135980,135983,135986,135989],{"className":135976},[31676],[34,135978],{"className":135979,"style":38774},[31680],[34,135981,135107],{"className":135982},[31685,31686],[34,135984],{"className":135985,"style":31846},[31845],[34,135987,202],{"className":135988},[31850],[34,135990],{"className":135991,"style":31846},[31845],[34,135993,135995,135998,136038,136078,136118,136121,136124,136127,136167],{"className":135994},[31676],[34,135996],{"className":135997,"style":31860},[31680],[34,135999,136001,136004],{"className":136000},[31685],[34,136002,56782],{"className":136003,"style":32062},[31685,31686],[34,136005,136007],{"className":136006},[31792],[34,136008,136010,136030],{"className":136009},[31796,31797],[34,136011,136013,136027],{"className":136012},[31801],[34,136014,136016],{"className":136015,"style":31806},[31805],[34,136017,136018,136021],{"style":38795},[34,136019],{"className":136020,"style":31814},[31813],[34,136022,136024],{"className":136023},[31818,31819,31820,31821],[34,136025,459],{"className":136026},[31685,31821],[34,136028,31829],{"className":136029},[31828],[34,136031,136033],{"className":136032},[31801],[34,136034,136036],{"className":136035,"style":31836},[31805],[34,136037],{},[34,136039,136041,136044],{"className":136040},[31685],[34,136042,33244],{"className":136043},[31685,31686],[34,136045,136047],{"className":136046},[31792],[34,136048,136050,136070],{"className":136049},[31796,31797],[34,136051,136053,136067],{"className":136052},[31801],[34,136054,136056],{"className":136055,"style":31806},[31805],[34,136057,136058,136061],{"style":31809},[34,136059],{"className":136060,"style":31814},[31813],[34,136062,136064],{"className":136063},[31818,31819,31820,31821],[34,136065,640],{"className":136066},[31685,31821],[34,136068,31829],{"className":136069},[31828],[34,136071,136073],{"className":136072},[31801],[34,136074,136076],{"className":136075,"style":31836},[31805],[34,136077],{},[34,136079,136081,136084],{"className":136080},[31685],[34,136082,56782],{"className":136083,"style":32062},[31685,31686],[34,136085,136087],{"className":136086},[31792],[34,136088,136090,136110],{"className":136089},[31796,31797],[34,136091,136093,136107],{"className":136092},[31801],[34,136094,136096],{"className":136095,"style":31806},[31805],[34,136097,136098,136101],{"style":38795},[34,136099],{"className":136100,"style":31814},[31813],[34,136102,136104],{"className":136103},[31818,31819,31820,31821],[34,136105,640],{"className":136106},[31685,31821],[34,136108,31829],{"className":136109},[31828],[34,136111,136113],{"className":136112},[31801],[34,136114,136116],{"className":136115,"style":31836},[31805],[34,136117],{},[34,136119],{"className":136120,"style":32123},[31845],[34,136122,38874],{"className":136123},[34006],[34,136125],{"className":136126,"style":32123},[31845],[34,136128,136130,136133],{"className":136129},[31685],[34,136131,33244],{"className":136132},[31685,31686],[34,136134,136136],{"className":136135},[31792],[34,136137,136139,136159],{"className":136138},[31796,31797],[34,136140,136142,136156],{"className":136141},[31801],[34,136143,136145],{"className":136144,"style":33833},[31805],[34,136146,136147,136150],{"style":31809},[34,136148],{"className":136149,"style":31814},[31813],[34,136151,136153],{"className":136152},[31818,31819,31820,31821],[34,136154,33759],{"className":136155,"style":33774},[31685,31686,31821],[34,136157,31829],{"className":136158},[31828],[34,136160,136162],{"className":136161},[31801],[34,136163,136165],{"className":136164,"style":31836},[31805],[34,136166],{},[34,136168,136170,136173],{"className":136169},[31685],[34,136171,56782],{"className":136172,"style":32062},[31685,31686],[34,136174,136176],{"className":136175},[31792],[34,136177,136179,136199],{"className":136178},[31796,31797],[34,136180,136182,136196],{"className":136181},[31801],[34,136183,136185],{"className":136184,"style":33833},[31805],[34,136186,136187,136190],{"style":38795},[34,136188],{"className":136189,"style":31814},[31813],[34,136191,136193],{"className":136192},[31818,31819,31820,31821],[34,136194,33759],{"className":136195,"style":33774},[31685,31686,31821],[34,136197,31829],{"className":136198},[31828],[34,136200,136202],{"className":136201},[31801],[34,136203,136205],{"className":136204,"style":31836},[31805],[34,136206],{},[15,136208,136209],{},"中：",[105,136211,136212,136283,136355],{},[108,136213,136214,127546],{},[34,136215,136217,136234],{"className":136216},[31642],[34,136218,136220],{"className":136219},[31646],[31648,136221,136222],{"xmlns":31650},[31652,136223,136224,136232],{},[31655,136225,136226],{},[31743,136227,136228,136230],{},[31658,136229,56782],{},[31749,136231,459],{},[31665,136233,135498],{"encoding":31667},[34,136235,136237],{"className":136236,"ariaHidden":758},[31672],[34,136238,136240,136243],{"className":136239},[31676],[34,136241],{"className":136242,"style":31860},[31680],[34,136244,136246,136249],{"className":136245},[31685],[34,136247,56782],{"className":136248,"style":32062},[31685,31686],[34,136250,136252],{"className":136251},[31792],[34,136253,136255,136275],{"className":136254},[31796,31797],[34,136256,136258,136272],{"className":136257},[31801],[34,136259,136261],{"className":136260,"style":31806},[31805],[34,136262,136263,136266],{"style":38795},[34,136264],{"className":136265,"style":31814},[31813],[34,136267,136269],{"className":136268},[31818,31819,31820,31821],[34,136270,459],{"className":136271},[31685,31821],[34,136273,31829],{"className":136274},[31828],[34,136276,136278],{"className":136277},[31801],[34,136279,136281],{"className":136280,"style":31836},[31805],[34,136282],{},[108,136284,136285,136354],{},[34,136286,136288,136305],{"className":136287},[31642],[34,136289,136291],{"className":136290},[31646],[31648,136292,136293],{"xmlns":31650},[31652,136294,136295,136303],{},[31655,136296,136297],{},[31743,136298,136299,136301],{},[31658,136300,56782],{},[31658,136302,33759],{},[31665,136304,135764],{"encoding":31667},[34,136306,136308],{"className":136307,"ariaHidden":758},[31672],[34,136309,136311,136314],{"className":136310},[31676],[34,136312],{"className":136313,"style":31860},[31680],[34,136315,136317,136320],{"className":136316},[31685],[34,136318,56782],{"className":136319,"style":32062},[31685,31686],[34,136321,136323],{"className":136322},[31792],[34,136324,136326,136346],{"className":136325},[31796,31797],[34,136327,136329,136343],{"className":136328},[31801],[34,136330,136332],{"className":136331,"style":33833},[31805],[34,136333,136334,136337],{"style":38795},[34,136335],{"className":136336,"style":31814},[31813],[34,136338,136340],{"className":136339},[31818,31819,31820,31821],[34,136341,33759],{"className":136342,"style":33774},[31685,31686,31821],[34,136344,31829],{"className":136345},[31828],[34,136347,136349],{"className":136348},[31801],[34,136350,136352],{"className":136351,"style":31836},[31805],[34,136353],{},"：终点；",[108,136356,136357,136385],{},[34,136358,136360,136373],{"className":136359},[31642],[34,136361,136363],{"className":136362},[31646],[31648,136364,136365],{"xmlns":31650},[31652,136366,136367,136371],{},[31655,136368,136369],{},[31658,136370,33759],{},[31665,136372,33759],{"encoding":31667},[34,136374,136376],{"className":136375,"ariaHidden":758},[31672],[34,136377,136379,136382],{"className":136378},[31676],[34,136380],{"className":136381,"style":31681},[31680],[34,136383,33759],{"className":136384,"style":33774},[31685,31686],"：路径长度。",[32188,136387],{},[19,136389,136391],{"id":136390},"_2-简单路径","2. 简单路径",[15,136393,136394,136395,347],{},"如果路径中的边互不相同，则称为",[2627,136396,136397],{},"简单路径",[15,136399,136400],{},"教材中“简单路径”重点看边不重复。",[32188,136402],{},[19,136404,136406],{"id":136405},"_3-基本路径-通路","3. 基本路径 \u002F 通路",[15,136408,136409,136410,136413,136414,347],{},"如果路径中的顶点互不相同，则称为",[2627,136411,136412],{},"基本路径","，也常叫",[2627,136415,136416],{},"通路",[15,136418,136419],{},"顶点不重复自然边也不会重复，所以：",[34,136421,136423],{"className":136422},[126385],[34,136424,136426,136449],{"className":136425},[31642],[34,136427,136429],{"className":136428},[31646],[31648,136430,136431],{"xmlns":31650,"display":126394},[31652,136432,136433,136446],{},[31655,136434,136435],{},[126400,136436,136437],{"notation":126402},[71090,136438,136439],{"scriptlevel":459,"displaystyle":702},[71090,136440,136441],{"scriptlevel":459,"displaystyle":702},[71090,136442,136443],{"scriptlevel":459,"displaystyle":758},[39361,136444,136445],{},"基本路径一定是简单路径",[31665,136447,136448],{"encoding":31667},"\\boxed{\\text{基本路径一定是简单路径}}",[34,136450,136452],{"className":136451,"ariaHidden":758},[31672],[34,136453,136455,136458],{"className":136454},[31676],[34,136456],{"className":136457,"style":126424},[31680],[34,136459,136461],{"className":136460},[31685],[34,136462,136464,136501],{"className":136463},[31796,31797],[34,136465,136467,136498],{"className":136466},[31801],[34,136468,136470,136490],{"className":136469,"style":126437},[31805],[34,136471,136472,136475],{"style":126440},[34,136473],{"className":136474,"style":126444},[31813],[34,136476,136478],{"className":136477},[126448],[34,136479,136481],{"className":136480},[31685],[34,136482,136484],{"className":136483},[31685],[34,136485,136487],{"className":136486},[31685,2638],[34,136488,136445],{"className":136489},[31685,105462],[34,136491,136492,136495],{"style":126463},[34,136493],{"className":136494,"style":126444},[31813],[34,136496],{"className":136497,"style":126472},[126470,126471],[34,136499,31829],{"className":136500},[31828],[34,136502,136504],{"className":136503},[31801],[34,136505,136507],{"className":136506,"style":126482},[31805],[34,136508],{},[15,136510,136511],{},"但简单路径不一定是基本路径。",[32188,136513],{},[11,136515,136517],{"id":136516},"二循环-回路","二、循环 \u002F 回路",[15,136519,136520],{},"如果路径的起点和终点相同：",[34,136522,136524],{"className":136523},[126385],[34,136525,136527,136553],{"className":136526},[31642],[34,136528,136530],{"className":136529},[31646],[31648,136531,136532],{"xmlns":31650,"display":126394},[31652,136533,136534,136550],{},[31655,136535,136536,136542,136544],{},[31743,136537,136538,136540],{},[31658,136539,56782],{},[31749,136541,459],{},[31662,136543,202],{},[31743,136545,136546,136548],{},[31658,136547,56782],{},[31658,136549,33759],{},[31665,136551,136552],{"encoding":31667},"v_0=v_k",[34,136554,136556,136611],{"className":136555,"ariaHidden":758},[31672],[34,136557,136559,136562,136602,136605,136608],{"className":136558},[31676],[34,136560],{"className":136561,"style":31860},[31680],[34,136563,136565,136568],{"className":136564},[31685],[34,136566,56782],{"className":136567,"style":32062},[31685,31686],[34,136569,136571],{"className":136570},[31792],[34,136572,136574,136594],{"className":136573},[31796,31797],[34,136575,136577,136591],{"className":136576},[31801],[34,136578,136580],{"className":136579,"style":31806},[31805],[34,136581,136582,136585],{"style":38795},[34,136583],{"className":136584,"style":31814},[31813],[34,136586,136588],{"className":136587},[31818,31819,31820,31821],[34,136589,459],{"className":136590},[31685,31821],[34,136592,31829],{"className":136593},[31828],[34,136595,136597],{"className":136596},[31801],[34,136598,136600],{"className":136599,"style":31836},[31805],[34,136601],{},[34,136603],{"className":136604,"style":31846},[31845],[34,136606,202],{"className":136607},[31850],[34,136609],{"className":136610,"style":31846},[31845],[34,136612,136614,136617],{"className":136613},[31676],[34,136615],{"className":136616,"style":31860},[31680],[34,136618,136620,136623],{"className":136619},[31685],[34,136621,56782],{"className":136622,"style":32062},[31685,31686],[34,136624,136626],{"className":136625},[31792],[34,136627,136629,136649],{"className":136628},[31796,31797],[34,136630,136632,136646],{"className":136631},[31801],[34,136633,136635],{"className":136634,"style":33833},[31805],[34,136636,136637,136640],{"style":38795},[34,136638],{"className":136639,"style":31814},[31813],[34,136641,136643],{"className":136642},[31818,31819,31820,31821],[34,136644,33759],{"className":136645,"style":33774},[31685,31686,31821],[34,136647,31829],{"className":136648},[31828],[34,136650,136652],{"className":136651},[31801],[34,136653,136655],{"className":136654,"style":31836},[31805],[34,136656],{},[15,136658,136659,136660,136413,136663,347],{},"则称为",[2627,136661,136662],{},"循环",[2627,136664,136665],{},"回路",[32188,136667],{},[19,136669,136671],{"id":136670},"_1-简单循环","1. 简单循环",[15,136673,136674,136675,347],{},"如果循环中的边互不相同，则称为",[2627,136676,136677],{},"简单循环",[32188,136679],{},[19,136681,136683],{"id":136682},"_2-基本循环","2. 基本循环",[15,136685,136686,136687,347],{},"如果除了起点和终点相同之外，其余顶点互不相同，则称为",[2627,136688,136689],{},"基本循环",[32188,136691],{},[19,136693,136695],{"id":136694},"_3-基本循环与简单循环关系","3. 基本循环与简单循环关系",[34,136697,136699],{"className":136698},[126385],[34,136700,136702,136725],{"className":136701},[31642],[34,136703,136705],{"className":136704},[31646],[31648,136706,136707],{"xmlns":31650,"display":126394},[31652,136708,136709,136722],{},[31655,136710,136711],{},[126400,136712,136713],{"notation":126402},[71090,136714,136715],{"scriptlevel":459,"displaystyle":702},[71090,136716,136717],{"scriptlevel":459,"displaystyle":702},[71090,136718,136719],{"scriptlevel":459,"displaystyle":758},[39361,136720,136721],{},"基本循环一定是简单循环",[31665,136723,136724],{"encoding":31667},"\\boxed{\\text{基本循环一定是简单循环}}",[34,136726,136728],{"className":136727,"ariaHidden":758},[31672],[34,136729,136731,136734],{"className":136730},[31676],[34,136732],{"className":136733,"style":126424},[31680],[34,136735,136737],{"className":136736},[31685],[34,136738,136740,136777],{"className":136739},[31796,31797],[34,136741,136743,136774],{"className":136742},[31801],[34,136744,136746,136766],{"className":136745,"style":126437},[31805],[34,136747,136748,136751],{"style":126440},[34,136749],{"className":136750,"style":126444},[31813],[34,136752,136754],{"className":136753},[126448],[34,136755,136757],{"className":136756},[31685],[34,136758,136760],{"className":136759},[31685],[34,136761,136763],{"className":136762},[31685,2638],[34,136764,136721],{"className":136765},[31685,105462],[34,136767,136768,136771],{"style":126463},[34,136769],{"className":136770,"style":126444},[31813],[34,136772],{"className":136773,"style":126472},[126470,126471],[34,136775,31829],{"className":136776},[31828],[34,136778,136780],{"className":136779},[31801],[34,136781,136783],{"className":136782,"style":126482},[31805],[34,136784],{},[15,136786,136787],{},"但简单循环不一定是基本循环。",[32188,136789],{},[11,136791,136793],{"id":136792},"三路径长度定理","三、路径长度定理",[15,136795,133910,136796,133939,136824,136852],{},[34,136797,136799,136812],{"className":136798},[31642],[34,136800,136802],{"className":136801},[31646],[31648,136803,136804],{"xmlns":31650},[31652,136805,136806,136810],{},[31655,136807,136808],{},[31658,136809,97709],{},[31665,136811,97709],{"encoding":31667},[34,136813,136815],{"className":136814,"ariaHidden":758},[31672],[34,136816,136818,136821],{"className":136817},[31676],[34,136819],{"className":136820,"style":33218},[31680],[34,136822,97709],{"className":136823},[31685,31686],[34,136825,136827,136840],{"className":136826},[31642],[34,136828,136830],{"className":136829},[31646],[31648,136831,136832],{"xmlns":31650},[31652,136833,136834,136838],{},[31655,136835,136836],{},[31658,136837,31660],{},[31665,136839,31660],{"encoding":31667},[34,136841,136843],{"className":136842,"ariaHidden":758},[31672],[34,136844,136846,136849],{"className":136845},[31676],[34,136847],{"className":136848,"style":32527},[31680],[34,136850,31660],{"className":136851},[31685,31686]," 个顶点的有向图。",[19,136854,136856],{"id":136855},"_1-基本路径长度","1. 基本路径长度",[15,136858,136859],{},"任意基本路径的长度最多为：",[34,136861,136863],{"className":136862},[126385],[34,136864,136866,136894],{"className":136865},[31642],[34,136867,136869],{"className":136868},[31646],[31648,136870,136871],{"xmlns":31650,"display":126394},[31652,136872,136873,136891],{},[31655,136874,136875],{},[126400,136876,136877],{"notation":126402},[71090,136878,136879],{"scriptlevel":459,"displaystyle":702},[71090,136880,136881],{"scriptlevel":459,"displaystyle":702},[71090,136882,136883],{"scriptlevel":459,"displaystyle":758},[31655,136884,136885,136887,136889],{},[31658,136886,31660],{},[31662,136888,32226],{},[31749,136890,640],{},[31665,136892,136893],{"encoding":31667},"\\boxed{n-1}",[34,136895,136897],{"className":136896,"ariaHidden":758},[31672],[34,136898,136900,136904],{"className":136899},[31676],[34,136901],{"className":136902,"style":136903},[31680],"height:1.4078em;vertical-align:-0.4233em;",[34,136905,136907],{"className":136906},[31685],[34,136908,136910,136961],{"className":136909},[31796,31797],[34,136911,136913,136958],{"className":136912},[31801],[34,136914,136917,136948],{"className":136915,"style":136916},[31805],"height:0.9844em;",[34,136918,136920,136924],{"style":136919},"top:-3.4078em;",[34,136921],{"className":136922,"style":136923},[31813],"height:3.4078em;",[34,136925,136927],{"className":136926},[126448],[34,136928,136930],{"className":136929},[31685],[34,136931,136933,136936,136939,136942,136945],{"className":136932},[31685],[34,136934,31660],{"className":136935},[31685,31686],[34,136937],{"className":136938,"style":32604},[31845],[34,136940,32226],{"className":136941},[32024],[34,136943],{"className":136944,"style":32604},[31845],[34,136946,640],{"className":136947},[31685],[34,136949,136951,136954],{"style":136950},"top:-2.9844em;",[34,136952],{"className":136953,"style":136923},[31813],[34,136955],{"className":136956,"style":136957},[126470,126471],"height:1.4078em;border-style:solid;border-width:0.04em;",[34,136959,31829],{"className":136960},[31828],[34,136962,136964],{"className":136963},[31801],[34,136965,136967],{"className":136966,"style":129408},[31805],[34,136968],{},[15,136970,136971],{},"因为基本路径不能重复顶点。",[15,136973,136974,136975,137003,137004,137032,137033,137083],{},"如果有 ",[34,136976,136978,136991],{"className":136977},[31642],[34,136979,136981],{"className":136980},[31646],[31648,136982,136983],{"xmlns":31650},[31652,136984,136985,136989],{},[31655,136986,136987],{},[31658,136988,31660],{},[31665,136990,31660],{"encoding":31667},[34,136992,136994],{"className":136993,"ariaHidden":758},[31672],[34,136995,136997,137000],{"className":136996},[31676],[34,136998],{"className":136999,"style":32527},[31680],[34,137001,31660],{"className":137002},[31685,31686]," 个顶点，最多经过 ",[34,137005,137007,137020],{"className":137006},[31642],[34,137008,137010],{"className":137009},[31646],[31648,137011,137012],{"xmlns":31650},[31652,137013,137014,137018],{},[31655,137015,137016],{},[31658,137017,31660],{},[31665,137019,31660],{"encoding":31667},[34,137021,137023],{"className":137022,"ariaHidden":758},[31672],[34,137024,137026,137029],{"className":137025},[31676],[34,137027],{"className":137028,"style":32527},[31680],[34,137030,31660],{"className":137031},[31685,31686]," 个顶点，因此最多有 ",[34,137034,137036,137053],{"className":137035},[31642],[34,137037,137039],{"className":137038},[31646],[31648,137040,137041],{"xmlns":31650},[31652,137042,137043,137051],{},[31655,137044,137045,137047,137049],{},[31658,137046,31660],{},[31662,137048,32226],{},[31749,137050,640],{},[31665,137052,32587],{"encoding":31667},[34,137054,137056,137074],{"className":137055,"ariaHidden":758},[31672],[34,137057,137059,137062,137065,137068,137071],{"className":137058},[31676],[34,137060],{"className":137061,"style":32597},[31680],[34,137063,31660],{"className":137064},[31685,31686],[34,137066],{"className":137067,"style":32604},[31845],[34,137069,32226],{"className":137070},[32024],[34,137072],{"className":137073,"style":32604},[31845],[34,137075,137077,137080],{"className":137076},[31676],[34,137078],{"className":137079,"style":32617},[31680],[34,137081,640],{"className":137082},[31685]," 条边。",[32188,137085],{},[19,137087,137089],{"id":137088},"_2-基本循环长度","2. 基本循环长度",[15,137091,137092],{},"任意基本循环的长度最多为：",[34,137094,137096],{"className":137095},[126385],[34,137097,137099,137121],{"className":137098},[31642],[34,137100,137102],{"className":137101},[31646],[31648,137103,137104],{"xmlns":31650,"display":126394},[31652,137105,137106,137118],{},[31655,137107,137108],{},[126400,137109,137110],{"notation":126402},[71090,137111,137112],{"scriptlevel":459,"displaystyle":702},[71090,137113,137114],{"scriptlevel":459,"displaystyle":702},[71090,137115,137116],{"scriptlevel":459,"displaystyle":758},[31658,137117,31660],{},[31665,137119,137120],{"encoding":31667},"\\boxed{n}",[34,137122,137124],{"className":137123,"ariaHidden":758},[31672],[34,137125,137127,137131],{"className":137126},[31676],[34,137128],{"className":137129,"style":137130},[31680],"height:1.1106em;vertical-align:-0.34em;",[34,137132,137134],{"className":137133},[31685],[34,137135,137137,137176],{"className":137136},[31796,31797],[34,137138,137140,137173],{"className":137139},[31801],[34,137141,137144,137163],{"className":137142,"style":137143},[31805],"height:0.7706em;",[34,137145,137147,137151],{"style":137146},"top:-3.1106em;",[34,137148],{"className":137149,"style":137150},[31813],"height:3.1106em;",[34,137152,137154],{"className":137153},[126448],[34,137155,137157],{"className":137156},[31685],[34,137158,137160],{"className":137159},[31685],[34,137161,31660],{"className":137162},[31685,31686],[34,137164,137166,137169],{"style":137165},"top:-2.7706em;",[34,137167],{"className":137168,"style":137150},[31813],[34,137170],{"className":137171,"style":137172},[126470,126471],"height:1.1106em;border-style:solid;border-width:0.04em;",[34,137174,31829],{"className":137175},[31828],[34,137177,137179],{"className":137178},[31801],[34,137180,137182],{"className":137181,"style":126482},[31805],[34,137183],{},[15,137185,137186],{},"因为基本循环除了起点终点相同外，其他顶点不重复。",[32188,137188],{},[11,137190,137192],{"id":137191},"四可达性","四、可达性",[19,137194,137196],{"id":137195},"_1-可达","1. 可达",[15,137198,137199,137200,137228,137229,137257],{},"在有向图中，如果从顶点 ",[34,137201,137203,137216],{"className":137202},[31642],[34,137204,137206],{"className":137205},[31646],[31648,137207,137208],{"xmlns":31650},[31652,137209,137210,137214],{},[31655,137211,137212],{},[31658,137213,94628],{},[31665,137215,94628],{"encoding":31667},[34,137217,137219],{"className":137218,"ariaHidden":758},[31672],[34,137220,137222,137225],{"className":137221},[31676],[34,137223],{"className":137224,"style":32527},[31680],[34,137226,94628],{"className":137227},[31685,31686]," 到顶点 ",[34,137230,137232,137245],{"className":137231},[31642],[34,137233,137235],{"className":137234},[31646],[31648,137236,137237],{"xmlns":31650},[31652,137238,137239,137243],{},[31655,137240,137241],{},[31658,137242,56782],{},[31665,137244,56782],{"encoding":31667},[34,137246,137248],{"className":137247,"ariaHidden":758},[31672],[34,137249,137251,137254],{"className":137250},[31676],[34,137252],{"className":137253,"style":32527},[31680],[34,137255,56782],{"className":137256,"style":32062},[31685,31686]," 存在路径，则称：",[34,137259,137261],{"className":137260},[126385],[34,137262,137264,137277],{"className":137263},[31642],[34,137265,137267],{"className":137266},[31646],[31648,137268,137269],{"xmlns":31650,"display":126394},[31652,137270,137271,137275],{},[31655,137272,137273],{},[31658,137274,56782],{},[31665,137276,56782],{"encoding":31667},[34,137278,137280],{"className":137279,"ariaHidden":758},[31672],[34,137281,137283,137286],{"className":137282},[31676],[34,137284],{"className":137285,"style":32527},[31680],[34,137287,56782],{"className":137288,"style":32062},[31685,31686],[15,137290,137291],{},"从：",[34,137293,137295],{"className":137294},[126385],[34,137296,137298,137311],{"className":137297},[31642],[34,137299,137301],{"className":137300},[31646],[31648,137302,137303],{"xmlns":31650,"display":126394},[31652,137304,137305,137309],{},[31655,137306,137307],{},[31658,137308,94628],{},[31665,137310,94628],{"encoding":31667},[34,137312,137314],{"className":137313,"ariaHidden":758},[31672],[34,137315,137317,137320],{"className":137316},[31676],[34,137318],{"className":137319,"style":32527},[31680],[34,137321,94628],{"className":137322},[31685,31686],[15,137324,137325],{},"可达。",[15,137327,137328],{},"也可以说：",[34,137330,137332],{"className":137331},[126385],[34,137333,137335,137352],{"className":137334},[31642],[34,137336,137338],{"className":137337},[31646],[31648,137339,137340],{"xmlns":31650,"display":126394},[31652,137341,137342,137350],{},[31655,137343,137344,137346,137348],{},[31658,137345,94628],{},[31662,137347,33239],{},[31658,137349,56782],{},[31665,137351,127604],{"encoding":31667},[34,137353,137355,137373],{"className":137354,"ariaHidden":758},[31672],[34,137356,137358,137361,137364,137367,137370],{"className":137357},[31676],[34,137359],{"className":137360,"style":32527},[31680],[34,137362,94628],{"className":137363},[31685,31686],[34,137365],{"className":137366,"style":31846},[31845],[34,137368,33239],{"className":137369},[31850],[34,137371],{"className":137372,"style":31846},[31845],[34,137374,137376,137379],{"className":137375},[31676],[34,137377],{"className":137378,"style":32527},[31680],[34,137380,56782],{"className":137381,"style":32062},[31685,31686],[15,137383,137325],{},[32188,137385],{},[19,137387,137389],{"id":137388},"_2-顶点到自身可达","2. 顶点到自身可达",[15,137391,137392],{},"规定任意顶点到自己都是可达的。",[15,137394,137395],{},"所以可达关系具有自反性。",[32188,137397],{},[19,137399,137401],{"id":137400},"_3-可达关系具有传递性","3. 可达关系具有传递性",[15,137403,130751],{},[34,137405,137407],{"className":137406},[126385],[34,137408,137410,137423],{"className":137409},[31642],[34,137411,137413],{"className":137412},[31646],[31648,137414,137415],{"xmlns":31650,"display":126394},[31652,137416,137417,137421],{},[31655,137418,137419],{},[31658,137420,94628],{},[31665,137422,94628],{"encoding":31667},[34,137424,137426],{"className":137425,"ariaHidden":758},[31672],[34,137427,137429,137432],{"className":137428},[31676],[34,137430],{"className":137431,"style":32527},[31680],[34,137433,94628],{"className":137434},[31685,31686],[15,137436,137437],{},"可达：",[34,137439,137441],{"className":137440},[126385],[34,137442,137444,137457],{"className":137443},[31642],[34,137445,137447],{"className":137446},[31646],[31648,137448,137449],{"xmlns":31650,"display":126394},[31652,137450,137451,137455],{},[31655,137452,137453],{},[31658,137454,56782],{},[31665,137456,56782],{"encoding":31667},[34,137458,137460],{"className":137459,"ariaHidden":758},[31672],[34,137461,137463,137466],{"className":137462},[31676],[34,137464],{"className":137465,"style":32527},[31680],[34,137467,56782],{"className":137468,"style":32062},[31685,31686],[15,137470,134101],{},[34,137472,137474],{"className":137473},[126385],[34,137475,137477,137490],{"className":137476},[31642],[34,137478,137480],{"className":137479},[31646],[31648,137481,137482],{"xmlns":31650,"display":126394},[31652,137483,137484,137488],{},[31655,137485,137486],{},[31658,137487,56782],{},[31665,137489,56782],{"encoding":31667},[34,137491,137493],{"className":137492,"ariaHidden":758},[31672],[34,137494,137496,137499],{"className":137495},[31676],[34,137497],{"className":137498,"style":32527},[31680],[34,137500,56782],{"className":137501,"style":32062},[31685,31686],[15,137503,137437],{},[34,137505,137507],{"className":137506},[126385],[34,137508,137510,137523],{"className":137509},[31642],[34,137511,137513],{"className":137512},[31646],[31648,137514,137515],{"xmlns":31650,"display":126394},[31652,137516,137517,137521],{},[31655,137518,137519],{},[31658,137520,47077],{},[31665,137522,47077],{"encoding":31667},[34,137524,137526],{"className":137525,"ariaHidden":758},[31672],[34,137527,137529,137532],{"className":137528},[31676],[34,137530],{"className":137531,"style":32527},[31680],[34,137533,47077],{"className":137534,"style":47120},[31685,31686],[15,137536,137537],{},"则：",[34,137539,137541],{"className":137540},[126385],[34,137542,137544,137557],{"className":137543},[31642],[34,137545,137547],{"className":137546},[31646],[31648,137548,137549],{"xmlns":31650,"display":126394},[31652,137550,137551,137555],{},[31655,137552,137553],{},[31658,137554,94628],{},[31665,137556,94628],{"encoding":31667},[34,137558,137560],{"className":137559,"ariaHidden":758},[31672],[34,137561,137563,137566],{"className":137562},[31676],[34,137564],{"className":137565,"style":32527},[31680],[34,137567,94628],{"className":137568},[31685,31686],[15,137570,137437],{},[34,137572,137574],{"className":137573},[126385],[34,137575,137577,137590],{"className":137576},[31642],[34,137578,137580],{"className":137579},[31646],[31648,137581,137582],{"xmlns":31650,"display":126394},[31652,137583,137584,137588],{},[31655,137585,137586],{},[31658,137587,47077],{},[31665,137589,47077],{"encoding":31667},[34,137591,137593],{"className":137592,"ariaHidden":758},[31672],[34,137594,137596,137599],{"className":137595},[31676],[34,137597],{"className":137598,"style":32527},[31680],[34,137600,47077],{"className":137601,"style":47120},[31685,31686],[15,137603,137604],{},"所以可达关系是自反、传递的。",[32188,137606],{},[11,137608,137610],{"id":137609},"五距离","五、距离",[15,137612,137613,137614,137738],{},"在有向图中，两个顶点 ",[34,137615,137617,137643],{"className":137616},[31642],[34,137618,137620],{"className":137619},[31646],[31648,137621,137622],{"xmlns":31650},[31652,137623,137624,137640],{},[31655,137625,137626,137632,137634],{},[31743,137627,137628,137630],{},[31658,137629,56782],{},[31658,137631,31955],{},[31662,137633,31960],{"separator":758},[31743,137635,137636,137638],{},[31658,137637,56782],{},[31658,137639,31968],{},[31665,137641,137642],{"encoding":31667},"v_i,v_j",[34,137644,137646],{"className":137645,"ariaHidden":758},[31672],[34,137647,137649,137652,137692,137695,137698],{"className":137648},[31676],[34,137650],{"className":137651,"style":56005},[31680],[34,137653,137655,137658],{"className":137654},[31685],[34,137656,56782],{"className":137657,"style":32062},[31685,31686],[34,137659,137661],{"className":137660},[31792],[34,137662,137664,137684],{"className":137663},[31796,31797],[34,137665,137667,137681],{"className":137666},[31801],[34,137668,137670],{"className":137669,"style":32090},[31805],[34,137671,137672,137675],{"style":38795},[34,137673],{"className":137674,"style":31814},[31813],[34,137676,137678],{"className":137677},[31818,31819,31820,31821],[34,137679,31955],{"className":137680},[31685,31686,31821],[34,137682,31829],{"className":137683},[31828],[34,137685,137687],{"className":137686},[31801],[34,137688,137690],{"className":137689,"style":31836},[31805],[34,137691],{},[34,137693,31960],{"className":137694},[32119],[34,137696],{"className":137697,"style":32123},[31845],[34,137699,137701,137704],{"className":137700},[31685],[34,137702,56782],{"className":137703,"style":32062},[31685,31686],[34,137705,137707],{"className":137706},[31792],[34,137708,137710,137730],{"className":137709},[31796,31797],[34,137711,137713,137727],{"className":137712},[31801],[34,137714,137716],{"className":137715,"style":32090},[31805],[34,137717,137718,137721],{"style":38795},[34,137719],{"className":137720,"style":31814},[31813],[34,137722,137724],{"className":137723},[31818,31819,31820,31821],[34,137725,31968],{"className":137726,"style":32153},[31685,31686,31821],[34,137728,31829],{"className":137729},[31828],[34,137731,137733],{"className":137732},[31801],[34,137734,137736],{"className":137735,"style":32163},[31805],[34,137737],{}," 的距离记作：",[34,137740,137742],{"className":137741},[126385],[34,137743,137745,137777],{"className":137744},[31642],[34,137746,137748],{"className":137747},[31646],[31648,137749,137750],{"xmlns":31650,"display":126394},[31652,137751,137752,137774],{},[31655,137753,137754,137756,137758,137764,137766,137772],{},[31658,137755,46900],{},[31662,137757,15972],{"stretchy":702},[31743,137759,137760,137762],{},[31658,137761,56782],{},[31658,137763,31955],{},[31662,137765,31960],{"separator":758},[31743,137767,137768,137770],{},[31658,137769,56782],{},[31658,137771,31968],{},[31662,137773,32712],{"stretchy":702},[31665,137775,137776],{"encoding":31667},"d[v_i,v_j]",[34,137778,137780],{"className":137779,"ariaHidden":758},[31672],[34,137781,137783,137786,137789,137792,137832,137835,137838,137878],{"className":137782},[31676],[34,137784],{"className":137785,"style":32058},[31680],[34,137787,46900],{"className":137788},[31685,31686],[34,137790,15972],{"className":137791},[31782],[34,137793,137795,137798],{"className":137794},[31685],[34,137796,56782],{"className":137797,"style":32062},[31685,31686],[34,137799,137801],{"className":137800},[31792],[34,137802,137804,137824],{"className":137803},[31796,31797],[34,137805,137807,137821],{"className":137806},[31801],[34,137808,137810],{"className":137809,"style":32090},[31805],[34,137811,137812,137815],{"style":38795},[34,137813],{"className":137814,"style":31814},[31813],[34,137816,137818],{"className":137817},[31818,31819,31820,31821],[34,137819,31955],{"className":137820},[31685,31686,31821],[34,137822,31829],{"className":137823},[31828],[34,137825,137827],{"className":137826},[31801],[34,137828,137830],{"className":137829,"style":31836},[31805],[34,137831],{},[34,137833,31960],{"className":137834},[32119],[34,137836],{"className":137837,"style":32123},[31845],[34,137839,137841,137844],{"className":137840},[31685],[34,137842,56782],{"className":137843,"style":32062},[31685,31686],[34,137845,137847],{"className":137846},[31792],[34,137848,137850,137870],{"className":137849},[31796,31797],[34,137851,137853,137867],{"className":137852},[31801],[34,137854,137856],{"className":137855,"style":32090},[31805],[34,137857,137858,137861],{"style":38795},[34,137859],{"className":137860,"style":31814},[31813],[34,137862,137864],{"className":137863},[31818,31819,31820,31821],[34,137865,31968],{"className":137866,"style":32153},[31685,31686,31821],[34,137868,31829],{"className":137869},[31828],[34,137871,137873],{"className":137872},[31801],[34,137874,137876],{"className":137875,"style":32163},[31805],[34,137877],{},[34,137879,32712],{"className":137880},[31690],[15,137882,137883],{},"定义如下：",[2230,137885,137886],{},[108,137887,2234,137888,131108],{},[34,137889,137891,137908],{"className":137890},[31642],[34,137892,137894],{"className":137893},[31646],[31648,137895,137896],{"xmlns":31650},[31652,137897,137898,137906],{},[31655,137899,137900,137902,137904],{},[31658,137901,31955],{},[31662,137903,202],{},[31658,137905,31968],{},[31665,137907,33094],{"encoding":31667},[34,137909,137911,137929],{"className":137910,"ariaHidden":758},[31672],[34,137912,137914,137917,137920,137923,137926],{"className":137913},[31676],[34,137915],{"className":137916,"style":33104},[31680],[34,137918,31955],{"className":137919},[31685,31686],[34,137921],{"className":137922,"style":31846},[31845],[34,137924,202],{"className":137925},[31850],[34,137927],{"className":137928,"style":31846},[31845],[34,137930,137932,137935],{"className":137931},[31676],[34,137933],{"className":137934,"style":33123},[31680],[34,137936,31968],{"className":137937,"style":32153},[31685,31686],[34,137939,137941],{"className":137940},[126385],[34,137942,137944,137980],{"className":137943},[31642],[34,137945,137947],{"className":137946},[31646],[31648,137948,137949],{"xmlns":31650,"display":126394},[31652,137950,137951,137977],{},[31655,137952,137953,137955,137957,137963,137965,137971,137973,137975],{},[31658,137954,46900],{},[31662,137956,15972],{"stretchy":702},[31743,137958,137959,137961],{},[31658,137960,56782],{},[31658,137962,31955],{},[31662,137964,31960],{"separator":758},[31743,137966,137967,137969],{},[31658,137968,56782],{},[31658,137970,31968],{},[31662,137972,32712],{"stretchy":702},[31662,137974,202],{},[31749,137976,459],{},[31665,137978,137979],{"encoding":31667},"d[v_i,v_j]=0",[34,137981,137983,138093],{"className":137982,"ariaHidden":758},[31672],[34,137984,137986,137989,137992,137995,138035,138038,138041,138081,138084,138087,138090],{"className":137985},[31676],[34,137987],{"className":137988,"style":32058},[31680],[34,137990,46900],{"className":137991},[31685,31686],[34,137993,15972],{"className":137994},[31782],[34,137996,137998,138001],{"className":137997},[31685],[34,137999,56782],{"className":138000,"style":32062},[31685,31686],[34,138002,138004],{"className":138003},[31792],[34,138005,138007,138027],{"className":138006},[31796,31797],[34,138008,138010,138024],{"className":138009},[31801],[34,138011,138013],{"className":138012,"style":32090},[31805],[34,138014,138015,138018],{"style":38795},[34,138016],{"className":138017,"style":31814},[31813],[34,138019,138021],{"className":138020},[31818,31819,31820,31821],[34,138022,31955],{"className":138023},[31685,31686,31821],[34,138025,31829],{"className":138026},[31828],[34,138028,138030],{"className":138029},[31801],[34,138031,138033],{"className":138032,"style":31836},[31805],[34,138034],{},[34,138036,31960],{"className":138037},[32119],[34,138039],{"className":138040,"style":32123},[31845],[34,138042,138044,138047],{"className":138043},[31685],[34,138045,56782],{"className":138046,"style":32062},[31685,31686],[34,138048,138050],{"className":138049},[31792],[34,138051,138053,138073],{"className":138052},[31796,31797],[34,138054,138056,138070],{"className":138055},[31801],[34,138057,138059],{"className":138058,"style":32090},[31805],[34,138060,138061,138064],{"style":38795},[34,138062],{"className":138063,"style":31814},[31813],[34,138065,138067],{"className":138066},[31818,31819,31820,31821],[34,138068,31968],{"className":138069,"style":32153},[31685,31686,31821],[34,138071,31829],{"className":138072},[31828],[34,138074,138076],{"className":138075},[31801],[34,138077,138079],{"className":138078,"style":32163},[31805],[34,138080],{},[34,138082,32712],{"className":138083},[31690],[34,138085],{"className":138086,"style":31846},[31845],[34,138088,202],{"className":138089},[31850],[34,138091],{"className":138092,"style":31846},[31845],[34,138094,138096,138099],{"className":138095},[31676],[34,138097],{"className":138098,"style":32617},[31680],[34,138100,459],{"className":138101},[31685],[2230,138103,138104],{"start":43},[108,138105,2234,138106,138176,138177,138246],{},[34,138107,138109,138127],{"className":138108},[31642],[34,138110,138112],{"className":138111},[31646],[31648,138113,138114],{"xmlns":31650},[31652,138115,138116,138124],{},[31655,138117,138118],{},[31743,138119,138120,138122],{},[31658,138121,56782],{},[31658,138123,31968],{},[31665,138125,138126],{"encoding":31667},"v_j",[34,138128,138130],{"className":138129,"ariaHidden":758},[31672],[34,138131,138133,138136],{"className":138132},[31676],[34,138134],{"className":138135,"style":56005},[31680],[34,138137,138139,138142],{"className":138138},[31685],[34,138140,56782],{"className":138141,"style":32062},[31685,31686],[34,138143,138145],{"className":138144},[31792],[34,138146,138148,138168],{"className":138147},[31796,31797],[34,138149,138151,138165],{"className":138150},[31801],[34,138152,138154],{"className":138153,"style":32090},[31805],[34,138155,138156,138159],{"style":38795},[34,138157],{"className":138158,"style":31814},[31813],[34,138160,138162],{"className":138161},[31818,31819,31820,31821],[34,138163,31968],{"className":138164,"style":32153},[31685,31686,31821],[34,138166,31829],{"className":138167},[31828],[34,138169,138171],{"className":138170},[31801],[34,138172,138174],{"className":138173,"style":32163},[31805],[34,138175],{}," 从 ",[34,138178,138180,138197],{"className":138179},[31642],[34,138181,138183],{"className":138182},[31646],[31648,138184,138185],{"xmlns":31650},[31652,138186,138187,138195],{},[31655,138188,138189],{},[31743,138190,138191,138193],{},[31658,138192,56782],{},[31658,138194,31955],{},[31665,138196,56787],{"encoding":31667},[34,138198,138200],{"className":138199,"ariaHidden":758},[31672],[34,138201,138203,138206],{"className":138202},[31676],[34,138204],{"className":138205,"style":31860},[31680],[34,138207,138209,138212],{"className":138208},[31685],[34,138210,56782],{"className":138211,"style":32062},[31685,31686],[34,138213,138215],{"className":138214},[31792],[34,138216,138218,138238],{"className":138217},[31796,31797],[34,138219,138221,138235],{"className":138220},[31801],[34,138222,138224],{"className":138223,"style":32090},[31805],[34,138225,138226,138229],{"style":38795},[34,138227],{"className":138228,"style":31814},[31813],[34,138230,138232],{"className":138231},[31818,31819,31820,31821],[34,138233,31955],{"className":138234},[31685,31686,31821],[34,138236,31829],{"className":138237},[31828],[34,138239,138241],{"className":138240},[31801],[34,138242,138244],{"className":138243,"style":31836},[31805],[34,138245],{}," 不可达，则：",[34,138248,138250],{"className":138249},[126385],[34,138251,138253,138290],{"className":138252},[31642],[34,138254,138256],{"className":138255},[31646],[31648,138257,138258],{"xmlns":31650,"display":126394},[31652,138259,138260,138287],{},[31655,138261,138262,138264,138266,138272,138274,138280,138282,138284],{},[31658,138263,46900],{},[31662,138265,15972],{"stretchy":702},[31743,138267,138268,138270],{},[31658,138269,56782],{},[31658,138271,31955],{},[31662,138273,31960],{"separator":758},[31743,138275,138276,138278],{},[31658,138277,56782],{},[31658,138279,31968],{},[31662,138281,32712],{"stretchy":702},[31662,138283,202],{},[31658,138285,138286],{"mathvariant":38079},"∞",[31665,138288,138289],{"encoding":31667},"d[v_i,v_j]=\\infty",[34,138291,138293,138403],{"className":138292,"ariaHidden":758},[31672],[34,138294,138296,138299,138302,138305,138345,138348,138351,138391,138394,138397,138400],{"className":138295},[31676],[34,138297],{"className":138298,"style":32058},[31680],[34,138300,46900],{"className":138301},[31685,31686],[34,138303,15972],{"className":138304},[31782],[34,138306,138308,138311],{"className":138307},[31685],[34,138309,56782],{"className":138310,"style":32062},[31685,31686],[34,138312,138314],{"className":138313},[31792],[34,138315,138317,138337],{"className":138316},[31796,31797],[34,138318,138320,138334],{"className":138319},[31801],[34,138321,138323],{"className":138322,"style":32090},[31805],[34,138324,138325,138328],{"style":38795},[34,138326],{"className":138327,"style":31814},[31813],[34,138329,138331],{"className":138330},[31818,31819,31820,31821],[34,138332,31955],{"className":138333},[31685,31686,31821],[34,138335,31829],{"className":138336},[31828],[34,138338,138340],{"className":138339},[31801],[34,138341,138343],{"className":138342,"style":31836},[31805],[34,138344],{},[34,138346,31960],{"className":138347},[32119],[34,138349],{"className":138350,"style":32123},[31845],[34,138352,138354,138357],{"className":138353},[31685],[34,138355,56782],{"className":138356,"style":32062},[31685,31686],[34,138358,138360],{"className":138359},[31792],[34,138361,138363,138383],{"className":138362},[31796,31797],[34,138364,138366,138380],{"className":138365},[31801],[34,138367,138369],{"className":138368,"style":32090},[31805],[34,138370,138371,138374],{"style":38795},[34,138372],{"className":138373,"style":31814},[31813],[34,138375,138377],{"className":138376},[31818,31819,31820,31821],[34,138378,31968],{"className":138379,"style":32153},[31685,31686,31821],[34,138381,31829],{"className":138382},[31828],[34,138384,138386],{"className":138385},[31801],[34,138387,138389],{"className":138388,"style":32163},[31805],[34,138390],{},[34,138392,32712],{"className":138393},[31690],[34,138395],{"className":138396,"style":31846},[31845],[34,138398,202],{"className":138399},[31850],[34,138401],{"className":138402,"style":31846},[31845],[34,138404,138406,138409],{"className":138405},[31676],[34,138407],{"className":138408,"style":32527},[31680],[34,138410,138286],{"className":138411},[31685],[2230,138413,138414],{"start":49},[108,138415,2234,138416,138176,138485,138554],{},[34,138417,138419,138436],{"className":138418},[31642],[34,138420,138422],{"className":138421},[31646],[31648,138423,138424],{"xmlns":31650},[31652,138425,138426,138434],{},[31655,138427,138428],{},[31743,138429,138430,138432],{},[31658,138431,56782],{},[31658,138433,31968],{},[31665,138435,138126],{"encoding":31667},[34,138437,138439],{"className":138438,"ariaHidden":758},[31672],[34,138440,138442,138445],{"className":138441},[31676],[34,138443],{"className":138444,"style":56005},[31680],[34,138446,138448,138451],{"className":138447},[31685],[34,138449,56782],{"className":138450,"style":32062},[31685,31686],[34,138452,138454],{"className":138453},[31792],[34,138455,138457,138477],{"className":138456},[31796,31797],[34,138458,138460,138474],{"className":138459},[31801],[34,138461,138463],{"className":138462,"style":32090},[31805],[34,138464,138465,138468],{"style":38795},[34,138466],{"className":138467,"style":31814},[31813],[34,138469,138471],{"className":138470},[31818,31819,31820,31821],[34,138472,31968],{"className":138473,"style":32153},[31685,31686,31821],[34,138475,31829],{"className":138476},[31828],[34,138478,138480],{"className":138479},[31801],[34,138481,138483],{"className":138482,"style":32163},[31805],[34,138484],{},[34,138486,138488,138505],{"className":138487},[31642],[34,138489,138491],{"className":138490},[31646],[31648,138492,138493],{"xmlns":31650},[31652,138494,138495,138503],{},[31655,138496,138497],{},[31743,138498,138499,138501],{},[31658,138500,56782],{},[31658,138502,31955],{},[31665,138504,56787],{"encoding":31667},[34,138506,138508],{"className":138507,"ariaHidden":758},[31672],[34,138509,138511,138514],{"className":138510},[31676],[34,138512],{"className":138513,"style":31860},[31680],[34,138515,138517,138520],{"className":138516},[31685],[34,138518,56782],{"className":138519,"style":32062},[31685,31686],[34,138521,138523],{"className":138522},[31792],[34,138524,138526,138546],{"className":138525},[31796,31797],[34,138527,138529,138543],{"className":138528},[31801],[34,138530,138532],{"className":138531,"style":32090},[31805],[34,138533,138534,138537],{"style":38795},[34,138535],{"className":138536,"style":31814},[31813],[34,138538,138540],{"className":138539},[31818,31819,31820,31821],[34,138541,31955],{"className":138542},[31685,31686,31821],[34,138544,31829],{"className":138545},[31828],[34,138547,138549],{"className":138548},[31801],[34,138550,138552],{"className":138551,"style":31836},[31805],[34,138553],{}," 可达，则：",[34,138556,138558],{"className":138557},[126385],[34,138559,138561,138592],{"className":138560},[31642],[34,138562,138564],{"className":138563},[31646],[31648,138565,138566],{"xmlns":31650,"display":126394},[31652,138567,138568,138590],{},[31655,138569,138570,138572,138574,138580,138582,138588],{},[31658,138571,46900],{},[31662,138573,15972],{"stretchy":702},[31743,138575,138576,138578],{},[31658,138577,56782],{},[31658,138579,31955],{},[31662,138581,31960],{"separator":758},[31743,138583,138584,138586],{},[31658,138585,56782],{},[31658,138587,31968],{},[31662,138589,32712],{"stretchy":702},[31665,138591,137776],{"encoding":31667},[34,138593,138595],{"className":138594,"ariaHidden":758},[31672],[34,138596,138598,138601,138604,138607,138647,138650,138653,138693],{"className":138597},[31676],[34,138599],{"className":138600,"style":32058},[31680],[34,138602,46900],{"className":138603},[31685,31686],[34,138605,15972],{"className":138606},[31782],[34,138608,138610,138613],{"className":138609},[31685],[34,138611,56782],{"className":138612,"style":32062},[31685,31686],[34,138614,138616],{"className":138615},[31792],[34,138617,138619,138639],{"className":138618},[31796,31797],[34,138620,138622,138636],{"className":138621},[31801],[34,138623,138625],{"className":138624,"style":32090},[31805],[34,138626,138627,138630],{"style":38795},[34,138628],{"className":138629,"style":31814},[31813],[34,138631,138633],{"className":138632},[31818,31819,31820,31821],[34,138634,31955],{"className":138635},[31685,31686,31821],[34,138637,31829],{"className":138638},[31828],[34,138640,138642],{"className":138641},[31801],[34,138643,138645],{"className":138644,"style":31836},[31805],[34,138646],{},[34,138648,31960],{"className":138649},[32119],[34,138651],{"className":138652,"style":32123},[31845],[34,138654,138656,138659],{"className":138655},[31685],[34,138657,56782],{"className":138658,"style":32062},[31685,31686],[34,138660,138662],{"className":138661},[31792],[34,138663,138665,138685],{"className":138664},[31796,31797],[34,138666,138668,138682],{"className":138667},[31801],[34,138669,138671],{"className":138670,"style":32090},[31805],[34,138672,138673,138676],{"style":38795},[34,138674],{"className":138675,"style":31814},[31813],[34,138677,138679],{"className":138678},[31818,31819,31820,31821],[34,138680,31968],{"className":138681,"style":32153},[31685,31686,31821],[34,138683,31829],{"className":138684},[31828],[34,138686,138688],{"className":138687},[31801],[34,138689,138691],{"className":138690,"style":32163},[31805],[34,138692],{},[34,138694,32712],{"className":138695},[31690],[15,138697,138698,138699,53834,138768,138837],{},"等于从 ",[34,138700,138702,138719],{"className":138701},[31642],[34,138703,138705],{"className":138704},[31646],[31648,138706,138707],{"xmlns":31650},[31652,138708,138709,138717],{},[31655,138710,138711],{},[31743,138712,138713,138715],{},[31658,138714,56782],{},[31658,138716,31955],{},[31665,138718,56787],{"encoding":31667},[34,138720,138722],{"className":138721,"ariaHidden":758},[31672],[34,138723,138725,138728],{"className":138724},[31676],[34,138726],{"className":138727,"style":31860},[31680],[34,138729,138731,138734],{"className":138730},[31685],[34,138732,56782],{"className":138733,"style":32062},[31685,31686],[34,138735,138737],{"className":138736},[31792],[34,138738,138740,138760],{"className":138739},[31796,31797],[34,138741,138743,138757],{"className":138742},[31801],[34,138744,138746],{"className":138745,"style":32090},[31805],[34,138747,138748,138751],{"style":38795},[34,138749],{"className":138750,"style":31814},[31813],[34,138752,138754],{"className":138753},[31818,31819,31820,31821],[34,138755,31955],{"className":138756},[31685,31686,31821],[34,138758,31829],{"className":138759},[31828],[34,138761,138763],{"className":138762},[31801],[34,138764,138766],{"className":138765,"style":31836},[31805],[34,138767],{},[34,138769,138771,138788],{"className":138770},[31642],[34,138772,138774],{"className":138773},[31646],[31648,138775,138776],{"xmlns":31650},[31652,138777,138778,138786],{},[31655,138779,138780],{},[31743,138781,138782,138784],{},[31658,138783,56782],{},[31658,138785,31968],{},[31665,138787,138126],{"encoding":31667},[34,138789,138791],{"className":138790,"ariaHidden":758},[31672],[34,138792,138794,138797],{"className":138793},[31676],[34,138795],{"className":138796,"style":56005},[31680],[34,138798,138800,138803],{"className":138799},[31685],[34,138801,56782],{"className":138802,"style":32062},[31685,31686],[34,138804,138806],{"className":138805},[31792],[34,138807,138809,138829],{"className":138808},[31796,31797],[34,138810,138812,138826],{"className":138811},[31801],[34,138813,138815],{"className":138814,"style":32090},[31805],[34,138816,138817,138820],{"style":38795},[34,138818],{"className":138819,"style":31814},[31813],[34,138821,138823],{"className":138822},[31818,31819,31820,31821],[34,138824,31968],{"className":138825,"style":32153},[31685,31686,31821],[34,138827,31829],{"className":138828},[31828],[34,138830,138832],{"className":138831},[31801],[34,138833,138835],{"className":138834,"style":32163},[31805],[34,138836],{}," 的最短路径长度。",[32188,138839],{},[19,138841,138843],{"id":138842},"注意有向图距离不一定对称","注意：有向图距离不一定对称",[15,138845,138846],{},"一般：",[34,138848,138850],{"className":138849},[126385],[34,138851,138853,138907],{"className":138852},[31642],[34,138854,138856],{"className":138855},[31646],[31648,138857,138858],{"xmlns":31650,"display":126394},[31652,138859,138860,138904],{},[31655,138861,138862,138864,138866,138872,138874,138880,138882,138884,138886,138888,138894,138896,138902],{},[31658,138863,46900],{},[31662,138865,15972],{"stretchy":702},[31743,138867,138868,138870],{},[31658,138869,56782],{},[31658,138871,31955],{},[31662,138873,31960],{"separator":758},[31743,138875,138876,138878],{},[31658,138877,56782],{},[31658,138879,31968],{},[31662,138881,32712],{"stretchy":702},[31662,138883,54707],{"mathvariant":38079},[31658,138885,46900],{},[31662,138887,15972],{"stretchy":702},[31743,138889,138890,138892],{},[31658,138891,56782],{},[31658,138893,31968],{},[31662,138895,31960],{"separator":758},[31743,138897,138898,138900],{},[31658,138899,56782],{},[31658,138901,31955],{},[31662,138903,32712],{"stretchy":702},[31665,138905,138906],{"encoding":31667},"d[v_i,v_j]\\neq d[v_j,v_i]",[34,138908,138910,139053],{"className":138909,"ariaHidden":758},[31672],[34,138911,138913,138916,138919,138922,138962,138965,138968,139008,139011,139014,139050],{"className":138912},[31676],[34,138914],{"className":138915,"style":32058},[31680],[34,138917,46900],{"className":138918},[31685,31686],[34,138920,15972],{"className":138921},[31782],[34,138923,138925,138928],{"className":138924},[31685],[34,138926,56782],{"className":138927,"style":32062},[31685,31686],[34,138929,138931],{"className":138930},[31792],[34,138932,138934,138954],{"className":138933},[31796,31797],[34,138935,138937,138951],{"className":138936},[31801],[34,138938,138940],{"className":138939,"style":32090},[31805],[34,138941,138942,138945],{"style":38795},[34,138943],{"className":138944,"style":31814},[31813],[34,138946,138948],{"className":138947},[31818,31819,31820,31821],[34,138949,31955],{"className":138950},[31685,31686,31821],[34,138952,31829],{"className":138953},[31828],[34,138955,138957],{"className":138956},[31801],[34,138958,138960],{"className":138959,"style":31836},[31805],[34,138961],{},[34,138963,31960],{"className":138964},[32119],[34,138966],{"className":138967,"style":32123},[31845],[34,138969,138971,138974],{"className":138970},[31685],[34,138972,56782],{"className":138973,"style":32062},[31685,31686],[34,138975,138977],{"className":138976},[31792],[34,138978,138980,139000],{"className":138979},[31796,31797],[34,138981,138983,138997],{"className":138982},[31801],[34,138984,138986],{"className":138985,"style":32090},[31805],[34,138987,138988,138991],{"style":38795},[34,138989],{"className":138990,"style":31814},[31813],[34,138992,138994],{"className":138993},[31818,31819,31820,31821],[34,138995,31968],{"className":138996,"style":32153},[31685,31686,31821],[34,138998,31829],{"className":138999},[31828],[34,139001,139003],{"className":139002},[31801],[34,139004,139006],{"className":139005,"style":32163},[31805],[34,139007],{},[34,139009,32712],{"className":139010},[31690],[34,139012],{"className":139013,"style":31846},[31845],[34,139015,139017,139044,139047],{"className":139016},[31850],[34,139018,139020],{"className":139019},[31850],[34,139021,139023],{"className":139022},[31685,54774],[34,139024,139026],{"className":139025},[54778],[34,139027,139029,139032,139041],{"className":139028},[54782],[34,139030],{"className":139031,"style":43217},[31680],[34,139033,139035],{"className":139034},[54789],[34,139036,139038],{"className":139037},[31685],[34,139039,54796],{"className":139040},[31850],[34,139042],{"className":139043},[54800],[34,139045],{"className":139046},[31845,54804],[34,139048,202],{"className":139049},[31850],[34,139051],{"className":139052,"style":31846},[31845],[34,139054,139056,139059,139062,139065,139105,139108,139111,139151],{"className":139055},[31676],[34,139057],{"className":139058,"style":32058},[31680],[34,139060,46900],{"className":139061},[31685,31686],[34,139063,15972],{"className":139064},[31782],[34,139066,139068,139071],{"className":139067},[31685],[34,139069,56782],{"className":139070,"style":32062},[31685,31686],[34,139072,139074],{"className":139073},[31792],[34,139075,139077,139097],{"className":139076},[31796,31797],[34,139078,139080,139094],{"className":139079},[31801],[34,139081,139083],{"className":139082,"style":32090},[31805],[34,139084,139085,139088],{"style":38795},[34,139086],{"className":139087,"style":31814},[31813],[34,139089,139091],{"className":139090},[31818,31819,31820,31821],[34,139092,31968],{"className":139093,"style":32153},[31685,31686,31821],[34,139095,31829],{"className":139096},[31828],[34,139098,139100],{"className":139099},[31801],[34,139101,139103],{"className":139102,"style":32163},[31805],[34,139104],{},[34,139106,31960],{"className":139107},[32119],[34,139109],{"className":139110,"style":32123},[31845],[34,139112,139114,139117],{"className":139113},[31685],[34,139115,56782],{"className":139116,"style":32062},[31685,31686],[34,139118,139120],{"className":139119},[31792],[34,139121,139123,139143],{"className":139122},[31796,31797],[34,139124,139126,139140],{"className":139125},[31801],[34,139127,139129],{"className":139128,"style":32090},[31805],[34,139130,139131,139134],{"style":38795},[34,139132],{"className":139133,"style":31814},[31813],[34,139135,139137],{"className":139136},[31818,31819,31820,31821],[34,139138,31955],{"className":139139},[31685,31686,31821],[34,139141,31829],{"className":139142},[31828],[34,139144,139146],{"className":139145},[31801],[34,139147,139149],{"className":139148,"style":31836},[31805],[34,139150],{},[34,139152,32712],{"className":139153},[31690],[15,139155,139156],{},"因为有向边有方向。",[32188,139158],{},[11,139160,139162],{"id":139161},"六无向图的连通性","六、无向图的连通性",[19,139164,139166],{"id":139165},"_1-连通图","1. 连通图",[15,139168,133910,139169,139197],{},[34,139170,139172,139185],{"className":139171},[31642],[34,139173,139175],{"className":139174},[31646],[31648,139176,139177],{"xmlns":31650},[31652,139178,139179,139183],{},[31655,139180,139181],{},[31658,139182,97709],{},[31665,139184,97709],{"encoding":31667},[34,139186,139188],{"className":139187,"ariaHidden":758},[31672],[34,139189,139191,139194],{"className":139190},[31676],[34,139192],{"className":139193,"style":33218},[31680],[34,139195,97709],{"className":139196},[31685,31686]," 是无向图。",[15,139199,139200,139201,113124,139229,347],{},"如果任意两个顶点之间都有路径相连，则称 ",[34,139202,139204,139217],{"className":139203},[31642],[34,139205,139207],{"className":139206},[31646],[31648,139208,139209],{"xmlns":31650},[31652,139210,139211,139215],{},[31655,139212,139213],{},[31658,139214,97709],{},[31665,139216,97709],{"encoding":31667},[34,139218,139220],{"className":139219,"ariaHidden":758},[31672],[34,139221,139223,139226],{"className":139222},[31676],[34,139224],{"className":139225,"style":33218},[31680],[34,139227,97709],{"className":139228},[31685,31686],[2627,139230,139231],{},"连通图",[32188,139233],{},[19,139235,139237],{"id":139236},"_2-连通分支","2. 连通分支",[15,139239,139240],{},"如果无向图不连通，可以分成若干个极大连通子图。",[15,139242,139243,139244,347],{},"这些极大连通子图叫",[2627,139245,139246],{},"连通分支",[15,139248,139249],{},"连通分支个数记作：",[34,139251,139253],{"className":139252},[126385],[34,139254,139256,139277],{"className":139255},[31642],[34,139257,139259],{"className":139258},[31646],[31648,139260,139261],{"xmlns":31650,"display":126394},[31652,139262,139263,139274],{},[31655,139264,139265,139268,139270,139272],{},[31658,139266,139267],{},"ω",[31662,139269,456],{"stretchy":702},[31658,139271,97709],{},[31662,139273,2557],{"stretchy":702},[31665,139275,139276],{"encoding":31667},"\\omega(G)",[34,139278,139280],{"className":139279,"ariaHidden":758},[31672],[34,139281,139283,139286,139289,139292,139295],{"className":139282},[31676],[34,139284],{"className":139285,"style":31774},[31680],[34,139287,139267],{"className":139288,"style":32062},[31685,31686],[34,139290,456],{"className":139291},[31782],[34,139293,97709],{"className":139294},[31685,31686],[34,139296,2557],{"className":139297},[31690],[15,139299,2234,139300,139328],{},[34,139301,139303,139316],{"className":139302},[31642],[34,139304,139306],{"className":139305},[31646],[31648,139307,139308],{"xmlns":31650},[31652,139309,139310,139314],{},[31655,139311,139312],{},[31658,139313,97709],{},[31665,139315,97709],{"encoding":31667},[34,139317,139319],{"className":139318,"ariaHidden":758},[31672],[34,139320,139322,139325],{"className":139321},[31676],[34,139323],{"className":139324,"style":33218},[31680],[34,139326,97709],{"className":139327},[31685,31686]," 是连通图，则：",[34,139330,139332],{"className":139331},[126385],[34,139333,139335,139359],{"className":139334},[31642],[34,139336,139338],{"className":139337},[31646],[31648,139339,139340],{"xmlns":31650,"display":126394},[31652,139341,139342,139356],{},[31655,139343,139344,139346,139348,139350,139352,139354],{},[31658,139345,139267],{},[31662,139347,456],{"stretchy":702},[31658,139349,97709],{},[31662,139351,2557],{"stretchy":702},[31662,139353,202],{},[31749,139355,640],{},[31665,139357,139358],{"encoding":31667},"\\omega(G)=1",[34,139360,139362,139389],{"className":139361,"ariaHidden":758},[31672],[34,139363,139365,139368,139371,139374,139377,139380,139383,139386],{"className":139364},[31676],[34,139366],{"className":139367,"style":31774},[31680],[34,139369,139267],{"className":139370,"style":32062},[31685,31686],[34,139372,456],{"className":139373},[31782],[34,139375,97709],{"className":139376},[31685,31686],[34,139378,2557],{"className":139379},[31690],[34,139381],{"className":139382,"style":31846},[31845],[34,139384,202],{"className":139385},[31850],[34,139387],{"className":139388,"style":31846},[31845],[34,139390,139392,139395],{"className":139391},[31676],[34,139393],{"className":139394,"style":32617},[31680],[34,139396,640],{"className":139397},[31685],[32188,139399],{},[19,139401,139403],{"id":139402},"_3-割边-桥","3. 割边 \u002F 桥",[15,139405,139406,139407,127650,139410,347],{},"如果删去某条边后，图的连通分支数增加，则这条边叫",[2627,139408,139409],{},"割边",[2627,139411,139412],{},"桥",[15,139414,139415],{},"即：",[34,139417,139419],{"className":139418},[126385],[34,139420,139422,139456],{"className":139421},[31642],[34,139423,139425],{"className":139424},[31646],[31648,139426,139427],{"xmlns":31650,"display":126394},[31652,139428,139429,139453],{},[31655,139430,139431,139433,139435,139437,139439,139441,139443,139445,139447,139449,139451],{},[31658,139432,139267],{},[31662,139434,456],{"stretchy":702},[31658,139436,97709],{},[31662,139438,32226],{},[31658,139440,33244],{},[31662,139442,2557],{"stretchy":702},[31662,139444,29622],{},[31658,139446,139267],{},[31662,139448,456],{"stretchy":702},[31658,139450,97709],{},[31662,139452,2557],{"stretchy":702},[31665,139454,139455],{"encoding":31667},"\\omega(G-e)>\\omega(G)",[34,139457,139459,139483,139504],{"className":139458,"ariaHidden":758},[31672],[34,139460,139462,139465,139468,139471,139474,139477,139480],{"className":139461},[31676],[34,139463],{"className":139464,"style":31774},[31680],[34,139466,139267],{"className":139467,"style":32062},[31685,31686],[34,139469,456],{"className":139470},[31782],[34,139472,97709],{"className":139473},[31685,31686],[34,139475],{"className":139476,"style":32604},[31845],[34,139478,32226],{"className":139479},[32024],[34,139481],{"className":139482,"style":32604},[31845],[34,139484,139486,139489,139492,139495,139498,139501],{"className":139485},[31676],[34,139487],{"className":139488,"style":31774},[31680],[34,139490,33244],{"className":139491},[31685,31686],[34,139493,2557],{"className":139494},[31690],[34,139496],{"className":139497,"style":31846},[31845],[34,139499,29622],{"className":139500},[31850],[34,139502],{"className":139503,"style":31846},[31845],[34,139505,139507,139510,139513,139516,139519],{"className":139506},[31676],[34,139508],{"className":139509,"style":31774},[31680],[34,139511,139267],{"className":139512,"style":32062},[31685,31686],[34,139514,456],{"className":139515},[31782],[34,139517,97709],{"className":139518},[31685,31686],[34,139520,2557],{"className":139521},[31690],[15,139523,94591,139524,139552],{},[34,139525,139527,139540],{"className":139526},[31642],[34,139528,139530],{"className":139529},[31646],[31648,139531,139532],{"xmlns":31650},[31652,139533,139534,139538],{},[31655,139535,139536],{},[31658,139537,33244],{},[31665,139539,33244],{"encoding":31667},[34,139541,139543],{"className":139542,"ariaHidden":758},[31672],[34,139544,139546,139549],{"className":139545},[31676],[34,139547],{"className":139548,"style":32527},[31680],[34,139550,33244],{"className":139551},[31685,31686]," 是割边。",[15,139554,139555],{},"直观理解：",[104833,139557,139558],{},[15,139559,139560],{},"一删这条边，图就更散了。",[32188,139562],{},[19,139564,139566],{"id":139565},"_4-割点","4. 割点",[15,139568,139569,139570,347],{},"如果删去某个顶点以及与它关联的边后，图的连通分支数增加，则这个顶点叫",[2627,139571,139572],{},"割点",[15,139574,139415],{},[34,139576,139578],{"className":139577},[126385],[34,139579,139581,139615],{"className":139580},[31642],[34,139582,139584],{"className":139583},[31646],[31648,139585,139586],{"xmlns":31650,"display":126394},[31652,139587,139588,139612],{},[31655,139589,139590,139592,139594,139596,139598,139600,139602,139604,139606,139608,139610],{},[31658,139591,139267],{},[31662,139593,456],{"stretchy":702},[31658,139595,97709],{},[31662,139597,32226],{},[31658,139599,56782],{},[31662,139601,2557],{"stretchy":702},[31662,139603,29622],{},[31658,139605,139267],{},[31662,139607,456],{"stretchy":702},[31658,139609,97709],{},[31662,139611,2557],{"stretchy":702},[31665,139613,139614],{"encoding":31667},"\\omega(G-v)>\\omega(G)",[34,139616,139618,139642,139663],{"className":139617,"ariaHidden":758},[31672],[34,139619,139621,139624,139627,139630,139633,139636,139639],{"className":139620},[31676],[34,139622],{"className":139623,"style":31774},[31680],[34,139625,139267],{"className":139626,"style":32062},[31685,31686],[34,139628,456],{"className":139629},[31782],[34,139631,97709],{"className":139632},[31685,31686],[34,139634],{"className":139635,"style":32604},[31845],[34,139637,32226],{"className":139638},[32024],[34,139640],{"className":139641,"style":32604},[31845],[34,139643,139645,139648,139651,139654,139657,139660],{"className":139644},[31676],[34,139646],{"className":139647,"style":31774},[31680],[34,139649,56782],{"className":139650,"style":32062},[31685,31686],[34,139652,2557],{"className":139653},[31690],[34,139655],{"className":139656,"style":31846},[31845],[34,139658,29622],{"className":139659},[31850],[34,139661],{"className":139662,"style":31846},[31845],[34,139664,139666,139669,139672,139675,139678],{"className":139665},[31676],[34,139667],{"className":139668,"style":31774},[31680],[34,139670,139267],{"className":139671,"style":32062},[31685,31686],[34,139673,456],{"className":139674},[31782],[34,139676,97709],{"className":139677},[31685,31686],[34,139679,2557],{"className":139680},[31690],[15,139682,94591,139683,139711],{},[34,139684,139686,139699],{"className":139685},[31642],[34,139687,139689],{"className":139688},[31646],[31648,139690,139691],{"xmlns":31650},[31652,139692,139693,139697],{},[31655,139694,139695],{},[31658,139696,56782],{},[31665,139698,56782],{"encoding":31667},[34,139700,139702],{"className":139701,"ariaHidden":758},[31672],[34,139703,139705,139708],{"className":139704},[31676],[34,139706],{"className":139707,"style":32527},[31680],[34,139709,56782],{"className":139710,"style":32062},[31685,31686]," 是割点。",[32188,139713],{},[11,139715,139717],{"id":139716},"七有向图的连通性","七、有向图的连通性",[15,139719,139720],{},"有向图比无向图复杂，教材分三种：",[19,139722,139724],{"id":139723},"_1-强连通","1. 强连通",[15,139726,139727,139728,139769],{},"如果任意两个顶点 ",[34,139729,139731,139748],{"className":139730},[31642],[34,139732,139734],{"className":139733},[31646],[31648,139735,139736],{"xmlns":31650},[31652,139737,139738,139746],{},[31655,139739,139740,139742,139744],{},[31658,139741,94628],{},[31662,139743,31960],{"separator":758},[31658,139745,56782],{},[31665,139747,127098],{"encoding":31667},[34,139749,139751],{"className":139750,"ariaHidden":758},[31672],[34,139752,139754,139757,139760,139763,139766],{"className":139753},[31676],[34,139755],{"className":139756,"style":38774},[31680],[34,139758,94628],{"className":139759},[31685,31686],[34,139761,31960],{"className":139762},[32119],[34,139764],{"className":139765,"style":32123},[31845],[34,139767,56782],{"className":139768,"style":32062},[31685,31686],"，都有：",[34,139771,139773],{"className":139772},[126385],[34,139774,139776,139795],{"className":139775},[31642],[34,139777,139779],{"className":139778},[31646],[31648,139780,139781],{"xmlns":31650,"display":126394},[31652,139782,139783,139792],{},[31655,139784,139785,139787,139790],{},[31658,139786,94628],{},[39361,139788,139789],{}," 可达 ",[31658,139791,56782],{},[31665,139793,139794],{"encoding":31667},"u\\text{ 可达 }v",[34,139796,139798],{"className":139797,"ariaHidden":758},[31672],[34,139799,139801,139804,139807,139820],{"className":139800},[31676],[34,139802],{"className":139803,"style":33218},[31680],[34,139805,94628],{"className":139806},[31685,31686],[34,139808,139810,139813,139817],{"className":139809},[31685,2638],[34,139811,130298],{"className":139812},[31685],[34,139814,139816],{"className":139815},[31685,105462],"可达",[34,139818,130298],{"className":139819},[31685],[34,139821,56782],{"className":139822,"style":32062},[31685,31686],[15,139824,134101],{},[34,139826,139828],{"className":139827},[126385],[34,139829,139831,139849],{"className":139830},[31642],[34,139832,139834],{"className":139833},[31646],[31648,139835,139836],{"xmlns":31650,"display":126394},[31652,139837,139838,139846],{},[31655,139839,139840,139842,139844],{},[31658,139841,56782],{},[39361,139843,139789],{},[31658,139845,94628],{},[31665,139847,139848],{"encoding":31667},"v\\text{ 可达 }u",[34,139850,139852],{"className":139851,"ariaHidden":758},[31672],[34,139853,139855,139858,139861,139873],{"className":139854},[31676],[34,139856],{"className":139857,"style":33218},[31680],[34,139859,56782],{"className":139860,"style":32062},[31685,31686],[34,139862,139864,139867,139870],{"className":139863},[31685,2638],[34,139865,130298],{"className":139866},[31685],[34,139868,139816],{"className":139869},[31685,105462],[34,139871,130298],{"className":139872},[31685],[34,139874,94628],{"className":139875},[31685,31686],[15,139877,139878,139879,347],{},"则称有向图是",[2627,139880,139881],{},"强连通图",[15,139883,133063],{},[104833,139885,139886],{},[15,139887,139888],{},"任意两个点之间都能双向到达。",[32188,139890],{},[19,139892,139894],{"id":139893},"_2-单向连通","2. 单向连通",[15,139896,139727,139897,139938],{},[34,139898,139900,139917],{"className":139899},[31642],[34,139901,139903],{"className":139902},[31646],[31648,139904,139905],{"xmlns":31650},[31652,139906,139907,139915],{},[31655,139908,139909,139911,139913],{},[31658,139910,94628],{},[31662,139912,31960],{"separator":758},[31658,139914,56782],{},[31665,139916,127098],{"encoding":31667},[34,139918,139920],{"className":139919,"ariaHidden":758},[31672],[34,139921,139923,139926,139929,139932,139935],{"className":139922},[31676],[34,139924],{"className":139925,"style":38774},[31680],[34,139927,94628],{"className":139928},[31685,31686],[34,139930,31960],{"className":139931},[32119],[34,139933],{"className":139934,"style":32123},[31845],[34,139936,56782],{"className":139937,"style":32062},[31685,31686],"，至少有一个方向可达：",[34,139940,139942],{"className":139941},[126385],[34,139943,139945,139962],{"className":139944},[31642],[34,139946,139948],{"className":139947},[31646],[31648,139949,139950],{"xmlns":31650,"display":126394},[31652,139951,139952,139960],{},[31655,139953,139954,139956,139958],{},[31658,139955,94628],{},[39361,139957,139789],{},[31658,139959,56782],{},[31665,139961,139794],{"encoding":31667},[34,139963,139965],{"className":139964,"ariaHidden":758},[31672],[34,139966,139968,139971,139974,139986],{"className":139967},[31676],[34,139969],{"className":139970,"style":33218},[31680],[34,139972,94628],{"className":139973},[31685,31686],[34,139975,139977,139980,139983],{"className":139976},[31685,2638],[34,139978,130298],{"className":139979},[31685],[34,139981,139816],{"className":139982},[31685,105462],[34,139984,130298],{"className":139985},[31685],[34,139987,56782],{"className":139988,"style":32062},[31685,31686],[15,139990,132311],{},[34,139992,139994],{"className":139993},[126385],[34,139995,139997,140014],{"className":139996},[31642],[34,139998,140000],{"className":139999},[31646],[31648,140001,140002],{"xmlns":31650,"display":126394},[31652,140003,140004,140012],{},[31655,140005,140006,140008,140010],{},[31658,140007,56782],{},[39361,140009,139789],{},[31658,140011,94628],{},[31665,140013,139848],{"encoding":31667},[34,140015,140017],{"className":140016,"ariaHidden":758},[31672],[34,140018,140020,140023,140026,140038],{"className":140019},[31676],[34,140021],{"className":140022,"style":33218},[31680],[34,140024,56782],{"className":140025,"style":32062},[31685,31686],[34,140027,140029,140032,140035],{"className":140028},[31685,2638],[34,140030,130298],{"className":140031},[31685],[34,140033,139816],{"className":140034},[31685,105462],[34,140036,130298],{"className":140037},[31685],[34,140039,94628],{"className":140040},[31685,31686],[15,140042,136659,140043,347],{},[2627,140044,140045],{},"单向连通图",[32188,140047],{},[19,140049,140051],{"id":140050},"_3-弱连通","3. 弱连通",[15,140053,140054,140055,347],{},"如果把有向图中所有边的方向去掉后，得到的无向图是连通图，则称原有向图是",[2627,140056,140057],{},"弱连通图",[32188,140059],{},[19,140061,140063],{"id":140062},"_4-三者关系","4. 三者关系",[34,140065,140067],{"className":140066},[126385],[34,140068,140070,140106],{"className":140069},[31642],[34,140071,140073],{"className":140072},[31646],[31648,140074,140075],{"xmlns":31650,"display":126394},[31652,140076,140077,140103],{},[31655,140078,140079],{},[126400,140080,140081],{"notation":126402},[71090,140082,140083],{"scriptlevel":459,"displaystyle":702},[71090,140084,140085],{"scriptlevel":459,"displaystyle":702},[71090,140086,140087],{"scriptlevel":459,"displaystyle":758},[31655,140088,140089,140092,140095,140098,140100],{},[39361,140090,140091],{},"强连通",[31662,140093,140094],{},"⇒",[39361,140096,140097],{},"单向连通",[31662,140099,140094],{},[39361,140101,140102],{},"弱连通",[31665,140104,140105],{"encoding":31667},"\\boxed{\\text{强连通} \\Rightarrow \\text{单向连通} \\Rightarrow \\text{弱连通}}",[34,140107,140109],{"className":140108,"ariaHidden":758},[31672],[34,140110,140112,140115],{"className":140111},[31676],[34,140113],{"className":140114,"style":126424},[31680],[34,140116,140118],{"className":140117},[31685],[34,140119,140121,140188],{"className":140120},[31796,31797],[34,140122,140124,140185],{"className":140123},[31801],[34,140125,140127,140177],{"className":140126,"style":126437},[31805],[34,140128,140129,140132],{"style":126440},[34,140130],{"className":140131,"style":126444},[31813],[34,140133,140135],{"className":140134},[126448],[34,140136,140138],{"className":140137},[31685],[34,140139,140141,140147,140150,140153,140156,140162,140165,140168,140171],{"className":140140},[31685],[34,140142,140144],{"className":140143},[31685,2638],[34,140145,140091],{"className":140146},[31685,105462],[34,140148],{"className":140149,"style":31846},[31845],[34,140151,140094],{"className":140152},[31850],[34,140154],{"className":140155,"style":31846},[31845],[34,140157,140159],{"className":140158},[31685,2638],[34,140160,140097],{"className":140161},[31685,105462],[34,140163],{"className":140164,"style":31846},[31845],[34,140166,140094],{"className":140167},[31850],[34,140169],{"className":140170,"style":31846},[31845],[34,140172,140174],{"className":140173},[31685,2638],[34,140175,140102],{"className":140176},[31685,105462],[34,140178,140179,140182],{"style":126463},[34,140180],{"className":140181,"style":126444},[31813],[34,140183],{"className":140184,"style":126472},[126470,126471],[34,140186,31829],{"className":140187},[31828],[34,140189,140191],{"className":140190},[31801],[34,140192,140194],{"className":140193,"style":126482},[31805],[34,140195],{},[15,140197,140198],{},"反过来一般不成立。",[15,140200,134316],{},[105,140202,140203,140206,140209],{},[108,140204,140205],{},"强连通最强；",[108,140207,140208],{},"单向连通居中；",[108,140210,140211],{},"弱连通最弱。",[32188,140213],{},[11,140215,140217],{"id":140216},"八极大连通子图","八、极大连通子图",[19,140219,140221],{"id":140220},"_1-强分图","1. 强分图",[15,140223,140224,140225,347],{},"有向图中的极大强连通子图叫",[2627,140226,140227],{},"强分图",[32188,140229],{},[19,140231,140233],{"id":140232},"_2-单向分图","2. 单向分图",[15,140235,140236,140237,347],{},"有向图中的极大单向连通子图叫",[2627,140238,140239],{},"单向分图",[32188,140241],{},[19,140243,140245],{"id":140244},"_3-弱分图","3. 弱分图",[15,140247,140248,140249,347],{},"有向图中的极大弱连通子图叫",[2627,140250,140251],{},"弱分图",[32188,140253],{},[19,140255,140257],{"id":140256},"_4-常见性质","4. 常见性质",[15,140259,140260],{},"对于有向图：",[105,140262,140263,140266,140269],{},[108,140264,140265],{},"每个顶点恰好属于一个强分图；",[108,140267,140268],{},"每个顶点和每条边至少属于一个单向分图；",[108,140270,140271],{},"每个顶点和每条边恰好属于一个弱分图。",[32188,140273],{},[11,140275,140277],{"id":140276},"_123-图的矩阵表示","12.3 图的矩阵表示",[15,140279,140280],{},"这一节很重要，因为它把图变成矩阵，方便计算机处理。",[15,140282,140283],{},"主要有：",[2230,140285,140286,140289,140292,140295],{},[108,140287,140288],{},"关联矩阵；",[108,140290,140291],{},"邻接矩阵；",[108,140293,140294],{},"可达矩阵；",[108,140296,140297],{},"距离矩阵。",[32188,140299],{},[11,140301,140303],{"id":140302},"一关联矩阵","一、关联矩阵",[15,140305,140306],{},"关联矩阵描述的是：",[34,140308,140310],{"className":140309},[126385],[34,140311,140313,140336],{"className":140312},[31642],[34,140314,140316],{"className":140315},[31646],[31648,140317,140318],{"xmlns":31650,"display":126394},[31652,140319,140320,140333],{},[31655,140321,140322],{},[126400,140323,140324],{"notation":126402},[71090,140325,140326],{"scriptlevel":459,"displaystyle":702},[71090,140327,140328],{"scriptlevel":459,"displaystyle":702},[71090,140329,140330],{"scriptlevel":459,"displaystyle":758},[39361,140331,140332],{},"顶点和边之间的关系",[31665,140334,140335],{"encoding":31667},"\\boxed{\\text{顶点和边之间的关系}}",[34,140337,140339],{"className":140338,"ariaHidden":758},[31672],[34,140340,140342,140345],{"className":140341},[31676],[34,140343],{"className":140344,"style":126424},[31680],[34,140346,140348],{"className":140347},[31685],[34,140349,140351,140388],{"className":140350},[31796,31797],[34,140352,140354,140385],{"className":140353},[31801],[34,140355,140357,140377],{"className":140356,"style":126437},[31805],[34,140358,140359,140362],{"style":126440},[34,140360],{"className":140361,"style":126444},[31813],[34,140363,140365],{"className":140364},[126448],[34,140366,140368],{"className":140367},[31685],[34,140369,140371],{"className":140370},[31685],[34,140372,140374],{"className":140373},[31685,2638],[34,140375,140332],{"className":140376},[31685,105462],[34,140378,140379,140382],{"style":126463},[34,140380],{"className":140381,"style":126444},[31813],[34,140383],{"className":140384,"style":126472},[126470,126471],[34,140386,31829],{"className":140387},[31828],[34,140389,140391],{"className":140390},[31801],[34,140392,140394],{"className":140393,"style":126482},[31805],[34,140395],{},[15,140397,131588],{},[34,140399,140401],{"className":140400},[126385],[34,140402,140404,140448],{"className":140403},[31642],[34,140405,140407],{"className":140406},[31646],[31648,140408,140409],{"xmlns":31650,"display":126394},[31652,140410,140411,140445],{},[31655,140412,140413,140415,140417],{},[31658,140414,126532],{},[31662,140416,202],{},[31655,140418,140419,140425,140427,140433,140435,140437,140439],{},[31743,140420,140421,140423],{},[31658,140422,56782],{},[31749,140424,640],{},[31662,140426,31960],{"separator":758},[31743,140428,140429,140431],{},[31658,140430,56782],{},[31749,140432,768],{},[31662,140434,31960],{"separator":758},[31662,140436,33934],{},[31662,140438,31960],{"separator":758},[31743,140440,140441,140443],{},[31658,140442,56782],{},[31658,140444,31660],{},[31665,140446,140447],{"encoding":31667},"V={v_1,v_2,\\dots,v_n}",[34,140449,140451,140469],{"className":140450,"ariaHidden":758},[31672],[34,140452,140454,140457,140460,140463,140466],{"className":140453},[31676],[34,140455],{"className":140456,"style":33218},[31680],[34,140458,126532],{"className":140459,"style":32604},[31685,31686],[34,140461],{"className":140462,"style":31846},[31845],[34,140464,202],{"className":140465},[31850],[34,140467],{"className":140468,"style":31846},[31845],[34,140470,140472,140475],{"className":140471},[31676],[34,140473],{"className":140474,"style":38774},[31680],[34,140476,140478,140518,140521,140524,140564,140567,140570,140573,140576,140579,140582],{"className":140477},[31685],[34,140479,140481,140484],{"className":140480},[31685],[34,140482,56782],{"className":140483,"style":32062},[31685,31686],[34,140485,140487],{"className":140486},[31792],[34,140488,140490,140510],{"className":140489},[31796,31797],[34,140491,140493,140507],{"className":140492},[31801],[34,140494,140496],{"className":140495,"style":31806},[31805],[34,140497,140498,140501],{"style":38795},[34,140499],{"className":140500,"style":31814},[31813],[34,140502,140504],{"className":140503},[31818,31819,31820,31821],[34,140505,640],{"className":140506},[31685,31821],[34,140508,31829],{"className":140509},[31828],[34,140511,140513],{"className":140512},[31801],[34,140514,140516],{"className":140515,"style":31836},[31805],[34,140517],{},[34,140519,31960],{"className":140520},[32119],[34,140522],{"className":140523,"style":32123},[31845],[34,140525,140527,140530],{"className":140526},[31685],[34,140528,56782],{"className":140529,"style":32062},[31685,31686],[34,140531,140533],{"className":140532},[31792],[34,140534,140536,140556],{"className":140535},[31796,31797],[34,140537,140539,140553],{"className":140538},[31801],[34,140540,140542],{"className":140541,"style":31806},[31805],[34,140543,140544,140547],{"style":38795},[34,140545],{"className":140546,"style":31814},[31813],[34,140548,140550],{"className":140549},[31818,31819,31820,31821],[34,140551,768],{"className":140552},[31685,31821],[34,140554,31829],{"className":140555},[31828],[34,140557,140559],{"className":140558},[31801],[34,140560,140562],{"className":140561,"style":31836},[31805],[34,140563],{},[34,140565,31960],{"className":140566},[32119],[34,140568],{"className":140569,"style":32123},[31845],[34,140571,33934],{"className":140572},[34006],[34,140574],{"className":140575,"style":32123},[31845],[34,140577,31960],{"className":140578},[32119],[34,140580],{"className":140581,"style":32123},[31845],[34,140583,140585,140588],{"className":140584},[31685],[34,140586,56782],{"className":140587,"style":32062},[31685,31686],[34,140589,140591],{"className":140590},[31792],[34,140592,140594,140614],{"className":140593},[31796,31797],[34,140595,140597,140611],{"className":140596},[31801],[34,140598,140600],{"className":140599,"style":32362},[31805],[34,140601,140602,140605],{"style":38795},[34,140603],{"className":140604,"style":31814},[31813],[34,140606,140608],{"className":140607},[31818,31819,31820,31821],[34,140609,31660],{"className":140610},[31685,31686,31821],[34,140612,31829],{"className":140613},[31828],[34,140615,140617],{"className":140616},[31801],[34,140618,140620],{"className":140619,"style":31836},[31805],[34,140621],{},[34,140623,140625],{"className":140624},[126385],[34,140626,140628,140672],{"className":140627},[31642],[34,140629,140631],{"className":140630},[31646],[31648,140632,140633],{"xmlns":31650,"display":126394},[31652,140634,140635,140669],{},[31655,140636,140637,140639,140641],{},[31658,140638,96058],{},[31662,140640,202],{},[31655,140642,140643,140649,140651,140657,140659,140661,140663],{},[31743,140644,140645,140647],{},[31658,140646,33244],{},[31749,140648,640],{},[31662,140650,31960],{"separator":758},[31743,140652,140653,140655],{},[31658,140654,33244],{},[31749,140656,768],{},[31662,140658,31960],{"separator":758},[31662,140660,33934],{},[31662,140662,31960],{"separator":758},[31743,140664,140665,140667],{},[31658,140666,33244],{},[31658,140668,31759],{},[31665,140670,140671],{"encoding":31667},"E={e_1,e_2,\\dots,e_m}",[34,140673,140675,140693],{"className":140674,"ariaHidden":758},[31672],[34,140676,140678,140681,140684,140687,140690],{"className":140677},[31676],[34,140679],{"className":140680,"style":33218},[31680],[34,140682,96058],{"className":140683,"style":106455},[31685,31686],[34,140685],{"className":140686,"style":31846},[31845],[34,140688,202],{"className":140689},[31850],[34,140691],{"className":140692,"style":31846},[31845],[34,140694,140696,140699],{"className":140695},[31676],[34,140697],{"className":140698,"style":38774},[31680],[34,140700,140702,140742,140745,140748,140788,140791,140794,140797,140800,140803,140806],{"className":140701},[31685],[34,140703,140705,140708],{"className":140704},[31685],[34,140706,33244],{"className":140707},[31685,31686],[34,140709,140711],{"className":140710},[31792],[34,140712,140714,140734],{"className":140713},[31796,31797],[34,140715,140717,140731],{"className":140716},[31801],[34,140718,140720],{"className":140719,"style":31806},[31805],[34,140721,140722,140725],{"style":31809},[34,140723],{"className":140724,"style":31814},[31813],[34,140726,140728],{"className":140727},[31818,31819,31820,31821],[34,140729,640],{"className":140730},[31685,31821],[34,140732,31829],{"className":140733},[31828],[34,140735,140737],{"className":140736},[31801],[34,140738,140740],{"className":140739,"style":31836},[31805],[34,140741],{},[34,140743,31960],{"className":140744},[32119],[34,140746],{"className":140747,"style":32123},[31845],[34,140749,140751,140754],{"className":140750},[31685],[34,140752,33244],{"className":140753},[31685,31686],[34,140755,140757],{"className":140756},[31792],[34,140758,140760,140780],{"className":140759},[31796,31797],[34,140761,140763,140777],{"className":140762},[31801],[34,140764,140766],{"className":140765,"style":31806},[31805],[34,140767,140768,140771],{"style":31809},[34,140769],{"className":140770,"style":31814},[31813],[34,140772,140774],{"className":140773},[31818,31819,31820,31821],[34,140775,768],{"className":140776},[31685,31821],[34,140778,31829],{"className":140779},[31828],[34,140781,140783],{"className":140782},[31801],[34,140784,140786],{"className":140785,"style":31836},[31805],[34,140787],{},[34,140789,31960],{"className":140790},[32119],[34,140792],{"className":140793,"style":32123},[31845],[34,140795,33934],{"className":140796},[34006],[34,140798],{"className":140799,"style":32123},[31845],[34,140801,31960],{"className":140802},[32119],[34,140804],{"className":140805,"style":32123},[31845],[34,140807,140809,140812],{"className":140808},[31685],[34,140810,33244],{"className":140811},[31685,31686],[34,140813,140815],{"className":140814},[31792],[34,140816,140818,140838],{"className":140817},[31796,31797],[34,140819,140821,140835],{"className":140820},[31801],[34,140822,140824],{"className":140823,"style":32362},[31805],[34,140825,140826,140829],{"style":31809},[34,140827],{"className":140828,"style":31814},[31813],[34,140830,140832],{"className":140831},[31818,31819,31820,31821],[34,140833,31759],{"className":140834},[31685,31686,31821],[34,140836,31829],{"className":140837},[31828],[34,140839,140841],{"className":140840},[31801],[34,140842,140844],{"className":140843,"style":31836},[31805],[34,140845],{},[15,140847,140848],{},"关联矩阵是一个：",[34,140850,140852],{"className":140851},[126385],[34,140853,140855,140873],{"className":140854},[31642],[34,140856,140858],{"className":140857},[31646],[31648,140859,140860],{"xmlns":31650,"display":126394},[31652,140861,140862,140870],{},[31655,140863,140864,140866,140868],{},[31658,140865,31660],{},[31662,140867,42071],{},[31658,140869,31759],{},[31665,140871,140872],{"encoding":31667},"n\\times m",[34,140874,140876,140894],{"className":140875,"ariaHidden":758},[31672],[34,140877,140879,140882,140885,140888,140891],{"className":140878},[31676],[34,140880],{"className":140881,"style":32597},[31680],[34,140883,31660],{"className":140884},[31685,31686],[34,140886],{"className":140887,"style":32604},[31845],[34,140889,42071],{"className":140890},[32024],[34,140892],{"className":140893,"style":32604},[31845],[34,140895,140897,140900],{"className":140896},[31676],[34,140898],{"className":140899,"style":32527},[31680],[34,140901,31759],{"className":140902},[31685,31686],[15,140904,140905],{},"矩阵。",[15,140907,140908],{},"行对应顶点，列对应边。",[32188,140910],{},[19,140912,140914],{"id":140913},"_1-无向图关联矩阵","1. 无向图关联矩阵",[15,140916,140917],{},"无向图中：",[34,140919,140921],{"className":140920},[126385],[34,140922,140924,140964],{"className":140923},[31642],[34,140925,140927],{"className":140926},[31646],[31648,140928,140929],{"xmlns":31650,"display":126394},[31652,140930,140931,140961],{},[31655,140932,140933,140935,140937,140939,140949],{},[31658,140934,51834],{},[31662,140936,202],{},[31662,140938,456],{"stretchy":702},[31743,140940,140941,140943],{},[31658,140942,31739],{},[31655,140944,140945,140947],{},[31658,140946,31955],{},[31658,140948,31968],{},[31743,140950,140951,140953],{},[31662,140952,2557],{"stretchy":702},[31655,140954,140955,140957,140959],{},[31658,140956,31660],{},[31662,140958,42071],{},[31658,140960,31759],{},[31665,140962,140963],{"encoding":31667},"F=(f_{ij})_{n\\times m}",[34,140965,140967,140985],{"className":140966,"ariaHidden":758},[31672],[34,140968,140970,140973,140976,140979,140982],{"className":140969},[31676],[34,140971],{"className":140972,"style":33218},[31680],[34,140974,51834],{"className":140975,"style":37422},[31685,31686],[34,140977],{"className":140978,"style":31846},[31845],[34,140980,202],{"className":140981},[31850],[34,140983],{"className":140984,"style":31846},[31845],[34,140986,140988,140991,140994,141039],{"className":140987},[31676],[34,140989],{"className":140990,"style":32058},[31680],[34,140992,456],{"className":140993},[31782],[34,140995,140997,141000],{"className":140996},[31685],[34,140998,31739],{"className":140999,"style":31778},[31685,31686],[34,141001,141003],{"className":141002},[31792],[34,141004,141006,141031],{"className":141005},[31796,31797],[34,141007,141009,141028],{"className":141008},[31801],[34,141010,141012],{"className":141011,"style":32090},[31805],[34,141013,141015,141018],{"style":141014},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[34,141016],{"className":141017,"style":31814},[31813],[34,141019,141021],{"className":141020},[31818,31819,31820,31821],[34,141022,141024],{"className":141023},[31685,31821],[34,141025,141027],{"className":141026,"style":32153},[31685,31686,31821],"ij",[34,141029,31829],{"className":141030},[31828],[34,141032,141034],{"className":141033},[31801],[34,141035,141037],{"className":141036,"style":32163},[31805],[34,141038],{},[34,141040,141042,141045],{"className":141041},[31690],[34,141043,2557],{"className":141044},[31690],[34,141046,141048],{"className":141047},[31792],[34,141049,141051,141081],{"className":141050},[31796,31797],[34,141052,141054,141078],{"className":141053},[31801],[34,141055,141058],{"className":141056,"style":141057},[31805],"height:0.2583em;",[34,141059,141060,141063],{"style":31809},[34,141061],{"className":141062,"style":31814},[31813],[34,141064,141066],{"className":141065},[31818,31819,31820,31821],[34,141067,141069,141072,141075],{"className":141068},[31685,31821],[34,141070,31660],{"className":141071},[31685,31686,31821],[34,141073,42071],{"className":141074},[32024,31821],[34,141076,31759],{"className":141077},[31685,31686,31821],[34,141079,31829],{"className":141080},[31828],[34,141082,141084],{"className":141083},[31801],[34,141085,141087],{"className":141086,"style":32037},[31805],[34,141088],{},[15,141090,104944],{},[34,141092,141094],{"className":141093},[126385],[34,141095,141097,141214],{"className":141096},[31642],[34,141098,141100],{"className":141099},[31646],[31648,141101,141102],{"xmlns":31650,"display":126394},[31652,141103,141104,141211],{},[31655,141105,141106,141116,141118],{},[31743,141107,141108,141110],{},[31658,141109,31739],{},[31655,141111,141112,141114],{},[31658,141113,31955],{},[31658,141115,31968],{},[31662,141117,202],{},[31655,141119,141120,141122],{},[31662,141121,32685],{"fence":758},[71078,141123,141124],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,141125,141126,141136,141163,141190],{},[71087,141127,141128],{},[71090,141129,141130],{"scriptlevel":459,"displaystyle":702},[31655,141131,141132,141134],{},[31749,141133,459],{},[31662,141135,31960],{"separator":758},[71087,141137,141138],{},[71090,141139,141140],{"scriptlevel":459,"displaystyle":702},[31655,141141,141142,141148,141151,141157,141159,141161],{},[31743,141143,141144,141146],{},[31658,141145,33244],{},[31658,141147,31968],{},[39361,141149,141150],{}," 不关联 ",[31743,141152,141153,141155],{},[31658,141154,56782],{},[31658,141156,31955],{},[39361,141158,130298],{},[31749,141160,640],{},[31662,141162,31960],{"separator":758},[71087,141164,141165],{},[71090,141166,141167],{"scriptlevel":459,"displaystyle":702},[31655,141168,141169,141175,141178,141184,141186,141188],{},[31743,141170,141171,141173],{},[31658,141172,33244],{},[31658,141174,31968],{},[39361,141176,141177],{}," 是非自环且关联 ",[31743,141179,141180,141182],{},[31658,141181,56782],{},[31658,141183,31955],{},[39361,141185,130298],{},[31749,141187,768],{},[31662,141189,31960],{"separator":758},[71087,141191,141192],{},[71090,141193,141194],{"scriptlevel":459,"displaystyle":702},[31655,141195,141196,141202,141205],{},[31743,141197,141198,141200],{},[31658,141199,33244],{},[31658,141201,31968],{},[39361,141203,141204],{}," 是自环且关联 ",[31743,141206,141207,141209],{},[31658,141208,56782],{},[31658,141210,31955],{},[31665,141212,141213],{"encoding":31667},"f_{ij}=\n\\begin{cases}\n0,& e_j\\text{ 不关联 }v_i\\\n1,& e_j\\text{ 是非自环且关联 }v_i\\\n2,& e_j\\text{ 是自环且关联 }v_i\n\\end{cases}",[34,141215,141217,141276],{"className":141216,"ariaHidden":758},[31672],[34,141218,141220,141224,141267,141270,141273],{"className":141219},[31676],[34,141221],{"className":141222,"style":141223},[31680],"height:0.9805em;vertical-align:-0.2861em;",[34,141225,141227,141230],{"className":141226},[31685],[34,141228,31739],{"className":141229,"style":31778},[31685,31686],[34,141231,141233],{"className":141232},[31792],[34,141234,141236,141259],{"className":141235},[31796,31797],[34,141237,141239,141256],{"className":141238},[31801],[34,141240,141242],{"className":141241,"style":32090},[31805],[34,141243,141244,141247],{"style":141014},[34,141245],{"className":141246,"style":31814},[31813],[34,141248,141250],{"className":141249},[31818,31819,31820,31821],[34,141251,141253],{"className":141252},[31685,31821],[34,141254,141027],{"className":141255,"style":32153},[31685,31686,31821],[34,141257,31829],{"className":141258},[31828],[34,141260,141262],{"className":141261},[31801],[34,141263,141265],{"className":141264,"style":32163},[31805],[34,141266],{},[34,141268],{"className":141269,"style":31846},[31845],[34,141271,202],{"className":141272},[31850],[34,141274],{"className":141275,"style":31846},[31845],[34,141277,141279,141283],{"className":141278},[31676],[34,141280],{"className":141281,"style":141282},[31680],"height:1.8em;vertical-align:-0.65em;",[34,141284,141286,141293,141742],{"className":141285},[34006],[34,141287,141289],{"className":141288,"style":75811},[31782,75810],[34,141290,32685],{"className":141291},[71281,141292],"size2",[34,141294,141296],{"className":141295},[31685],[34,141297,141299,141338,141341,141474,141478,141612,141615,141618],{"className":141298},[71078],[34,141300,141302],{"className":141301},[71384],[34,141303,141305,141329],{"className":141304},[31796,31797],[34,141306,141308,141326],{"className":141307},[31801],[34,141309,141312],{"className":141310,"style":141311},[31805],"height:0.97em;",[34,141313,141314,141317],{"style":78184},[34,141315],{"className":141316,"style":71401},[31813],[34,141318,141320,141323],{"className":141319},[31685],[34,141321,459],{"className":141322},[31685],[34,141324,31960],{"className":141325},[32119],[34,141327,31829],{"className":141328},[31828],[34,141330,141332],{"className":141331},[31801],[34,141333,141336],{"className":141334,"style":141335},[31805],"height:0.47em;",[34,141337],{},[34,141339],{"className":141340,"style":71733},[71732],[34,141342,141344],{"className":141343},[71384],[34,141345,141347,141466],{"className":141346},[31796,31797],[34,141348,141350,141463],{"className":141349},[31801],[34,141351,141353],{"className":141352,"style":141311},[31805],[34,141354,141355,141358],{"style":78184},[34,141356],{"className":141357,"style":71401},[31813],[34,141359,141361,141401,141414,141454,141457,141460],{"className":141360},[31685],[34,141362,141364,141367],{"className":141363},[31685],[34,141365,33244],{"className":141366},[31685,31686],[34,141368,141370],{"className":141369},[31792],[34,141371,141373,141393],{"className":141372},[31796,31797],[34,141374,141376,141390],{"className":141375},[31801],[34,141377,141379],{"className":141378,"style":32090},[31805],[34,141380,141381,141384],{"style":31809},[34,141382],{"className":141383,"style":31814},[31813],[34,141385,141387],{"className":141386},[31818,31819,31820,31821],[34,141388,31968],{"className":141389,"style":32153},[31685,31686,31821],[34,141391,31829],{"className":141392},[31828],[34,141394,141396],{"className":141395},[31801],[34,141397,141399],{"className":141398,"style":32163},[31805],[34,141400],{},[34,141402,141404,141407,141411],{"className":141403},[31685,2638],[34,141405,130298],{"className":141406},[31685],[34,141408,141410],{"className":141409},[31685,105462],"不关联",[34,141412,130298],{"className":141413},[31685],[34,141415,141417,141420],{"className":141416},[31685],[34,141418,56782],{"className":141419,"style":32062},[31685,31686],[34,141421,141423],{"className":141422},[31792],[34,141424,141426,141446],{"className":141425},[31796,31797],[34,141427,141429,141443],{"className":141428},[31801],[34,141430,141432],{"className":141431,"style":32090},[31805],[34,141433,141434,141437],{"style":38795},[34,141435],{"className":141436,"style":31814},[31813],[34,141438,141440],{"className":141439},[31818,31819,31820,31821],[34,141441,31955],{"className":141442},[31685,31686,31821],[34,141444,31829],{"className":141445},[31828],[34,141447,141449],{"className":141448},[31801],[34,141450,141452],{"className":141451,"style":31836},[31805],[34,141453],{},[34,141455,130298],{"className":141456},[31845],[34,141458,640],{"className":141459},[31685],[34,141461,31960],{"className":141462},[32119],[34,141464,31829],{"className":141465},[31828],[34,141467,141469],{"className":141468},[31801],[34,141470,141472],{"className":141471,"style":141335},[31805],[34,141473],{},[34,141475],{"className":141476,"style":141477},[71732],"width:0.5em;",[34,141479,141482],{"className":141480},[141481],"col-align-c",[34,141483,141485,141604],{"className":141484},[31796,31797],[34,141486,141488,141601],{"className":141487},[31801],[34,141489,141491],{"className":141490,"style":141311},[31805],[34,141492,141493,141496],{"style":78184},[34,141494],{"className":141495,"style":71401},[31813],[34,141497,141499,141539,141552,141592,141595,141598],{"className":141498},[31685],[34,141500,141502,141505],{"className":141501},[31685],[34,141503,33244],{"className":141504},[31685,31686],[34,141506,141508],{"className":141507},[31792],[34,141509,141511,141531],{"className":141510},[31796,31797],[34,141512,141514,141528],{"className":141513},[31801],[34,141515,141517],{"className":141516,"style":32090},[31805],[34,141518,141519,141522],{"style":31809},[34,141520],{"className":141521,"style":31814},[31813],[34,141523,141525],{"className":141524},[31818,31819,31820,31821],[34,141526,31968],{"className":141527,"style":32153},[31685,31686,31821],[34,141529,31829],{"className":141530},[31828],[34,141532,141534],{"className":141533},[31801],[34,141535,141537],{"className":141536,"style":32163},[31805],[34,141538],{},[34,141540,141542,141545,141549],{"className":141541},[31685,2638],[34,141543,130298],{"className":141544},[31685],[34,141546,141548],{"className":141547},[31685,105462],"是非自环且关联",[34,141550,130298],{"className":141551},[31685],[34,141553,141555,141558],{"className":141554},[31685],[34,141556,56782],{"className":141557,"style":32062},[31685,31686],[34,141559,141561],{"className":141560},[31792],[34,141562,141564,141584],{"className":141563},[31796,31797],[34,141565,141567,141581],{"className":141566},[31801],[34,141568,141570],{"className":141569,"style":32090},[31805],[34,141571,141572,141575],{"style":38795},[34,141573],{"className":141574,"style":31814},[31813],[34,141576,141578],{"className":141577},[31818,31819,31820,31821],[34,141579,31955],{"className":141580},[31685,31686,31821],[34,141582,31829],{"className":141583},[31828],[34,141585,141587],{"className":141586},[31801],[34,141588,141590],{"className":141589,"style":31836},[31805],[34,141591],{},[34,141593,130298],{"className":141594},[31845],[34,141596,768],{"className":141597},[31685],[34,141599,31960],{"className":141600},[32119],[34,141602,31829],{"className":141603},[31828],[34,141605,141607],{"className":141606},[31801],[34,141608,141610],{"className":141609,"style":141335},[31805],[34,141611],{},[34,141613],{"className":141614,"style":141477},[71732],[34,141616],{"className":141617,"style":141477},[71732],[34,141619,141621],{"className":141620},[141481],[34,141622,141624,141734],{"className":141623},[31796,31797],[34,141625,141627,141731],{"className":141626},[31801],[34,141628,141630],{"className":141629,"style":141311},[31805],[34,141631,141632,141635],{"style":78184},[34,141633],{"className":141634,"style":71401},[31813],[34,141636,141638,141678,141691],{"className":141637},[31685],[34,141639,141641,141644],{"className":141640},[31685],[34,141642,33244],{"className":141643},[31685,31686],[34,141645,141647],{"className":141646},[31792],[34,141648,141650,141670],{"className":141649},[31796,31797],[34,141651,141653,141667],{"className":141652},[31801],[34,141654,141656],{"className":141655,"style":32090},[31805],[34,141657,141658,141661],{"style":31809},[34,141659],{"className":141660,"style":31814},[31813],[34,141662,141664],{"className":141663},[31818,31819,31820,31821],[34,141665,31968],{"className":141666,"style":32153},[31685,31686,31821],[34,141668,31829],{"className":141669},[31828],[34,141671,141673],{"className":141672},[31801],[34,141674,141676],{"className":141675,"style":32163},[31805],[34,141677],{},[34,141679,141681,141684,141688],{"className":141680},[31685,2638],[34,141682,130298],{"className":141683},[31685],[34,141685,141687],{"className":141686},[31685,105462],"是自环且关联",[34,141689,130298],{"className":141690},[31685],[34,141692,141694,141697],{"className":141693},[31685],[34,141695,56782],{"className":141696,"style":32062},[31685,31686],[34,141698,141700],{"className":141699},[31792],[34,141701,141703,141723],{"className":141702},[31796,31797],[34,141704,141706,141720],{"className":141705},[31801],[34,141707,141709],{"className":141708,"style":32090},[31805],[34,141710,141711,141714],{"style":38795},[34,141712],{"className":141713,"style":31814},[31813],[34,141715,141717],{"className":141716},[31818,31819,31820,31821],[34,141718,31955],{"className":141719},[31685,31686,31821],[34,141721,31829],{"className":141722},[31828],[34,141724,141726],{"className":141725},[31801],[34,141727,141729],{"className":141728,"style":31836},[31805],[34,141730],{},[34,141732,31829],{"className":141733},[31828],[34,141735,141737],{"className":141736},[31801],[34,141738,141740],{"className":141739,"style":141335},[31805],[34,141741],{},[34,141743],{"className":141744},[31690,41488],[15,141746,22058],{},[105,141748,141749,141752],{},[108,141750,141751],{},"普通边连接两个顶点，所以对应两行各有一个 1；",[108,141753,141754],{},"自环在一个顶点上贡献 2。",[32188,141756],{},[19,141758,141760],{"id":141759},"_2-无向图关联矩阵性质","2. 无向图关联矩阵性质",[2230,141762,141763,141766,141769,141772],{},[108,141764,141765],{},"每一列元素和为 2；",[108,141767,141768],{},"每一行元素和等于对应顶点的度；",[108,141770,141771],{},"全 0 行对应孤立点；",[108,141773,141774],{},"重边对应相同的列。",[32188,141776],{},[19,141778,141780],{"id":141779},"_3-有向图关联矩阵","3. 有向图关联矩阵",[15,141782,141783],{},"有向图中，关联矩阵要体现方向。",[15,141785,141786],{},"一般按教材定义：",[34,141788,141790],{"className":141789},[126385],[34,141791,141793,141944],{"className":141792},[31642],[34,141794,141796],{"className":141795},[31646],[31648,141797,141798],{"xmlns":31650,"display":126394},[31652,141799,141800,141941],{},[31655,141801,141802,141812,141814],{},[31743,141803,141804,141806],{},[31658,141805,31739],{},[31655,141807,141808,141810],{},[31658,141809,31955],{},[31658,141811,31968],{},[31662,141813,202],{},[31655,141815,141816,141818],{},[31662,141817,32685],{"fence":758},[71078,141819,141820],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,141821,141822,141832,141858,141888,141917],{},[71087,141823,141824],{},[71090,141825,141826],{"scriptlevel":459,"displaystyle":702},[31655,141827,141828,141830],{},[31749,141829,459],{},[31662,141831,31960],{"separator":758},[71087,141833,141834],{},[71090,141835,141836],{"scriptlevel":459,"displaystyle":702},[31655,141837,141838,141844,141846,141852,141854,141856],{},[31743,141839,141840,141842],{},[31658,141841,33244],{},[31658,141843,31968],{},[39361,141845,141150],{},[31743,141847,141848,141850],{},[31658,141849,56782],{},[31658,141851,31955],{},[39361,141853,130298],{},[31749,141855,640],{},[31662,141857,31960],{"separator":758},[71087,141859,141860],{},[71090,141861,141862],{"scriptlevel":459,"displaystyle":702},[31655,141863,141864,141870,141873,141879,141882,141884,141886],{},[31743,141865,141866,141868],{},[31658,141867,33244],{},[31658,141869,31968],{},[39361,141871,141872],{}," 以 ",[31743,141874,141875,141877],{},[31658,141876,56782],{},[31658,141878,31955],{},[39361,141880,141881],{}," 为起点 ",[31662,141883,32226],{},[31749,141885,640],{},[31662,141887,31960],{"separator":758},[71087,141889,141890],{},[71090,141891,141892],{"scriptlevel":459,"displaystyle":702},[31655,141893,141894,141900,141902,141908,141911,141913,141915],{},[31743,141895,141896,141898],{},[31658,141897,33244],{},[31658,141899,31968],{},[39361,141901,141872],{},[31743,141903,141904,141906],{},[31658,141905,56782],{},[31658,141907,31955],{},[39361,141909,141910],{}," 为终点 ",[31662,141912,32226],{},[31749,141914,768],{},[31662,141916,31960],{"separator":758},[71087,141918,141919],{},[71090,141920,141921],{"scriptlevel":459,"displaystyle":702},[31655,141922,141923,141929,141932,141938],{},[31743,141924,141925,141927],{},[31658,141926,33244],{},[31658,141928,31968],{},[39361,141930,141931],{}," 是关联 ",[31743,141933,141934,141936],{},[31658,141935,56782],{},[31658,141937,31955],{},[39361,141939,141940],{}," 的自环",[31665,141942,141943],{"encoding":31667},"f_{ij}=\n\\begin{cases}\n0,& e_j\\text{ 不关联 }v_i\\\n1,& e_j\\text{ 以 }v_i\\text{ 为起点}\\\n-1,& e_j\\text{ 以 }v_i\\text{ 为终点}\\\n-2,& e_j\\text{ 是关联 }v_i\\text{ 的自环}\n\\end{cases}",[34,141945,141947,142005],{"className":141946,"ariaHidden":758},[31672],[34,141948,141950,141953,141996,141999,142002],{"className":141949},[31676],[34,141951],{"className":141952,"style":141223},[31680],[34,141954,141956,141959],{"className":141955},[31685],[34,141957,31739],{"className":141958,"style":31778},[31685,31686],[34,141960,141962],{"className":141961},[31792],[34,141963,141965,141988],{"className":141964},[31796,31797],[34,141966,141968,141985],{"className":141967},[31801],[34,141969,141971],{"className":141970,"style":32090},[31805],[34,141972,141973,141976],{"style":141014},[34,141974],{"className":141975,"style":31814},[31813],[34,141977,141979],{"className":141978},[31818,31819,31820,31821],[34,141980,141982],{"className":141981},[31685,31821],[34,141983,141027],{"className":141984,"style":32153},[31685,31686,31821],[34,141986,31829],{"className":141987},[31828],[34,141989,141991],{"className":141990},[31801],[34,141992,141994],{"className":141993,"style":32163},[31805],[34,141995],{},[34,141997],{"className":141998,"style":31846},[31845],[34,142000,202],{"className":142001},[31850],[34,142003],{"className":142004,"style":31846},[31845],[34,142006,142008,142011],{"className":142007},[31676],[34,142009],{"className":142010,"style":141282},[31680],[34,142012,142014,142020,142650],{"className":142013},[34006],[34,142015,142017],{"className":142016,"style":75811},[31782,75810],[34,142018,32685],{"className":142019},[71281,141292],[34,142021,142023],{"className":142022},[31685],[34,142024,142026,142063,142066,142198,142201,142353,142356,142359,142510,142513,142516],{"className":142025},[71078],[34,142027,142029],{"className":142028},[71384],[34,142030,142032,142055],{"className":142031},[31796,31797],[34,142033,142035,142052],{"className":142034},[31801],[34,142036,142038],{"className":142037,"style":141311},[31805],[34,142039,142040,142043],{"style":78184},[34,142041],{"className":142042,"style":71401},[31813],[34,142044,142046,142049],{"className":142045},[31685],[34,142047,459],{"className":142048},[31685],[34,142050,31960],{"className":142051},[32119],[34,142053,31829],{"className":142054},[31828],[34,142056,142058],{"className":142057},[31801],[34,142059,142061],{"className":142060,"style":141335},[31805],[34,142062],{},[34,142064],{"className":142065,"style":71733},[71732],[34,142067,142069],{"className":142068},[71384],[34,142070,142072,142190],{"className":142071},[31796,31797],[34,142073,142075,142187],{"className":142074},[31801],[34,142076,142078],{"className":142077,"style":141311},[31805],[34,142079,142080,142083],{"style":78184},[34,142081],{"className":142082,"style":71401},[31813],[34,142084,142086,142126,142138,142178,142181,142184],{"className":142085},[31685],[34,142087,142089,142092],{"className":142088},[31685],[34,142090,33244],{"className":142091},[31685,31686],[34,142093,142095],{"className":142094},[31792],[34,142096,142098,142118],{"className":142097},[31796,31797],[34,142099,142101,142115],{"className":142100},[31801],[34,142102,142104],{"className":142103,"style":32090},[31805],[34,142105,142106,142109],{"style":31809},[34,142107],{"className":142108,"style":31814},[31813],[34,142110,142112],{"className":142111},[31818,31819,31820,31821],[34,142113,31968],{"className":142114,"style":32153},[31685,31686,31821],[34,142116,31829],{"className":142117},[31828],[34,142119,142121],{"className":142120},[31801],[34,142122,142124],{"className":142123,"style":32163},[31805],[34,142125],{},[34,142127,142129,142132,142135],{"className":142128},[31685,2638],[34,142130,130298],{"className":142131},[31685],[34,142133,141410],{"className":142134},[31685,105462],[34,142136,130298],{"className":142137},[31685],[34,142139,142141,142144],{"className":142140},[31685],[34,142142,56782],{"className":142143,"style":32062},[31685,31686],[34,142145,142147],{"className":142146},[31792],[34,142148,142150,142170],{"className":142149},[31796,31797],[34,142151,142153,142167],{"className":142152},[31801],[34,142154,142156],{"className":142155,"style":32090},[31805],[34,142157,142158,142161],{"style":38795},[34,142159],{"className":142160,"style":31814},[31813],[34,142162,142164],{"className":142163},[31818,31819,31820,31821],[34,142165,31955],{"className":142166},[31685,31686,31821],[34,142168,31829],{"className":142169},[31828],[34,142171,142173],{"className":142172},[31801],[34,142174,142176],{"className":142175,"style":31836},[31805],[34,142177],{},[34,142179,130298],{"className":142180},[31845],[34,142182,640],{"className":142183},[31685],[34,142185,31960],{"className":142186},[32119],[34,142188,31829],{"className":142189},[31828],[34,142191,142193],{"className":142192},[31801],[34,142194,142196],{"className":142195,"style":141335},[31805],[34,142197],{},[34,142199],{"className":142200,"style":141477},[71732],[34,142202,142204],{"className":142203},[141481],[34,142205,142207,142345],{"className":142206},[31796,31797],[34,142208,142210,142342],{"className":142209},[31801],[34,142211,142213],{"className":142212,"style":141311},[31805],[34,142214,142215,142218],{"style":78184},[34,142216],{"className":142217,"style":71401},[31813],[34,142219,142221,142261,142274,142314,142324,142327,142330,142333,142336,142339],{"className":142220},[31685],[34,142222,142224,142227],{"className":142223},[31685],[34,142225,33244],{"className":142226},[31685,31686],[34,142228,142230],{"className":142229},[31792],[34,142231,142233,142253],{"className":142232},[31796,31797],[34,142234,142236,142250],{"className":142235},[31801],[34,142237,142239],{"className":142238,"style":32090},[31805],[34,142240,142241,142244],{"style":31809},[34,142242],{"className":142243,"style":31814},[31813],[34,142245,142247],{"className":142246},[31818,31819,31820,31821],[34,142248,31968],{"className":142249,"style":32153},[31685,31686,31821],[34,142251,31829],{"className":142252},[31828],[34,142254,142256],{"className":142255},[31801],[34,142257,142259],{"className":142258,"style":32163},[31805],[34,142260],{},[34,142262,142264,142267,142271],{"className":142263},[31685,2638],[34,142265,130298],{"className":142266},[31685],[34,142268,142270],{"className":142269},[31685,105462],"以",[34,142272,130298],{"className":142273},[31685],[34,142275,142277,142280],{"className":142276},[31685],[34,142278,56782],{"className":142279,"style":32062},[31685,31686],[34,142281,142283],{"className":142282},[31792],[34,142284,142286,142306],{"className":142285},[31796,31797],[34,142287,142289,142303],{"className":142288},[31801],[34,142290,142292],{"className":142291,"style":32090},[31805],[34,142293,142294,142297],{"style":38795},[34,142295],{"className":142296,"style":31814},[31813],[34,142298,142300],{"className":142299},[31818,31819,31820,31821],[34,142301,31955],{"className":142302},[31685,31686,31821],[34,142304,31829],{"className":142305},[31828],[34,142307,142309],{"className":142308},[31801],[34,142310,142312],{"className":142311,"style":31836},[31805],[34,142313],{},[34,142315,142317,142320],{"className":142316},[31685,2638],[34,142318,130298],{"className":142319},[31685],[34,142321,142323],{"className":142322},[31685,105462],"为起点",[34,142325,130298],{"className":142326},[31845],[34,142328],{"className":142329,"style":32604},[31845],[34,142331,32226],{"className":142332},[32024],[34,142334],{"className":142335,"style":32604},[31845],[34,142337,640],{"className":142338},[31685],[34,142340,31960],{"className":142341},[32119],[34,142343,31829],{"className":142344},[31828],[34,142346,142348],{"className":142347},[31801],[34,142349,142351],{"className":142350,"style":141335},[31805],[34,142352],{},[34,142354],{"className":142355,"style":141477},[71732],[34,142357],{"className":142358,"style":141477},[71732],[34,142360,142362],{"className":142361},[141481],[34,142363,142365,142502],{"className":142364},[31796,31797],[34,142366,142368,142499],{"className":142367},[31801],[34,142369,142371],{"className":142370,"style":141311},[31805],[34,142372,142373,142376],{"style":78184},[34,142374],{"className":142375,"style":71401},[31813],[34,142377,142379,142419,142431,142471,142481,142484,142487,142490,142493,142496],{"className":142378},[31685],[34,142380,142382,142385],{"className":142381},[31685],[34,142383,33244],{"className":142384},[31685,31686],[34,142386,142388],{"className":142387},[31792],[34,142389,142391,142411],{"className":142390},[31796,31797],[34,142392,142394,142408],{"className":142393},[31801],[34,142395,142397],{"className":142396,"style":32090},[31805],[34,142398,142399,142402],{"style":31809},[34,142400],{"className":142401,"style":31814},[31813],[34,142403,142405],{"className":142404},[31818,31819,31820,31821],[34,142406,31968],{"className":142407,"style":32153},[31685,31686,31821],[34,142409,31829],{"className":142410},[31828],[34,142412,142414],{"className":142413},[31801],[34,142415,142417],{"className":142416,"style":32163},[31805],[34,142418],{},[34,142420,142422,142425,142428],{"className":142421},[31685,2638],[34,142423,130298],{"className":142424},[31685],[34,142426,142270],{"className":142427},[31685,105462],[34,142429,130298],{"className":142430},[31685],[34,142432,142434,142437],{"className":142433},[31685],[34,142435,56782],{"className":142436,"style":32062},[31685,31686],[34,142438,142440],{"className":142439},[31792],[34,142441,142443,142463],{"className":142442},[31796,31797],[34,142444,142446,142460],{"className":142445},[31801],[34,142447,142449],{"className":142448,"style":32090},[31805],[34,142450,142451,142454],{"style":38795},[34,142452],{"className":142453,"style":31814},[31813],[34,142455,142457],{"className":142456},[31818,31819,31820,31821],[34,142458,31955],{"className":142459},[31685,31686,31821],[34,142461,31829],{"className":142462},[31828],[34,142464,142466],{"className":142465},[31801],[34,142467,142469],{"className":142468,"style":31836},[31805],[34,142470],{},[34,142472,142474,142477],{"className":142473},[31685,2638],[34,142475,130298],{"className":142476},[31685],[34,142478,142480],{"className":142479},[31685,105462],"为终点",[34,142482,130298],{"className":142483},[31845],[34,142485],{"className":142486,"style":32604},[31845],[34,142488,32226],{"className":142489},[32024],[34,142491],{"className":142492,"style":32604},[31845],[34,142494,768],{"className":142495},[31685],[34,142497,31960],{"className":142498},[32119],[34,142500,31829],{"className":142501},[31828],[34,142503,142505],{"className":142504},[31801],[34,142506,142508],{"className":142507,"style":141335},[31805],[34,142509],{},[34,142511],{"className":142512,"style":141477},[71732],[34,142514],{"className":142515,"style":141477},[71732],[34,142517,142519],{"className":142518},[141481],[34,142520,142522,142642],{"className":142521},[31796,31797],[34,142523,142525,142639],{"className":142524},[31801],[34,142526,142528],{"className":142527,"style":141311},[31805],[34,142529,142530,142533],{"style":78184},[34,142531],{"className":142532,"style":71401},[31813],[34,142534,142536,142576,142589,142629],{"className":142535},[31685],[34,142537,142539,142542],{"className":142538},[31685],[34,142540,33244],{"className":142541},[31685,31686],[34,142543,142545],{"className":142544},[31792],[34,142546,142548,142568],{"className":142547},[31796,31797],[34,142549,142551,142565],{"className":142550},[31801],[34,142552,142554],{"className":142553,"style":32090},[31805],[34,142555,142556,142559],{"style":31809},[34,142557],{"className":142558,"style":31814},[31813],[34,142560,142562],{"className":142561},[31818,31819,31820,31821],[34,142563,31968],{"className":142564,"style":32153},[31685,31686,31821],[34,142566,31829],{"className":142567},[31828],[34,142569,142571],{"className":142570},[31801],[34,142572,142574],{"className":142573,"style":32163},[31805],[34,142575],{},[34,142577,142579,142582,142586],{"className":142578},[31685,2638],[34,142580,130298],{"className":142581},[31685],[34,142583,142585],{"className":142584},[31685,105462],"是关联",[34,142587,130298],{"className":142588},[31685],[34,142590,142592,142595],{"className":142591},[31685],[34,142593,56782],{"className":142594,"style":32062},[31685,31686],[34,142596,142598],{"className":142597},[31792],[34,142599,142601,142621],{"className":142600},[31796,31797],[34,142602,142604,142618],{"className":142603},[31801],[34,142605,142607],{"className":142606,"style":32090},[31805],[34,142608,142609,142612],{"style":38795},[34,142610],{"className":142611,"style":31814},[31813],[34,142613,142615],{"className":142614},[31818,31819,31820,31821],[34,142616,31955],{"className":142617},[31685,31686,31821],[34,142619,31829],{"className":142620},[31828],[34,142622,142624],{"className":142623},[31801],[34,142625,142627],{"className":142626,"style":31836},[31805],[34,142628],{},[34,142630,142632,142635],{"className":142631},[31685,2638],[34,142633,130298],{"className":142634},[31685],[34,142636,142638],{"className":142637},[31685,105462],"的自环",[34,142640,31829],{"className":142641},[31828],[34,142643,142645],{"className":142644},[31801],[34,142646,142648],{"className":142647,"style":141335},[31805],[34,142649],{},[34,142651],{"className":142652},[31690,41488],[15,142654,142655],{},"不同教材对有向自环的矩阵记法可能略有差异，但普通有向边最重要的是：",[34,142657,142659],{"className":142658},[126385],[34,142660,142662,142698],{"className":142661},[31642],[34,142663,142665],{"className":142664},[31646],[31648,142666,142667],{"xmlns":31650,"display":126394},[31652,142668,142669,142695],{},[31655,142670,142671],{},[126400,142672,142673],{"notation":126402},[71090,142674,142675],{"scriptlevel":459,"displaystyle":702},[71090,142676,142677],{"scriptlevel":459,"displaystyle":702},[71090,142678,142679],{"scriptlevel":459,"displaystyle":758},[31655,142680,142681,142684,142686,142688,142691,142693],{},[39361,142682,142683],{},"起点记 ",[31749,142685,640],{},[31662,142687,31960],{"separator":758},[39361,142689,142690],{}," 终点记 ",[31662,142692,32226],{},[31749,142694,640],{},[31665,142696,142697],{"encoding":31667},"\\boxed{\\text{起点记 }1,\\text{ 终点记 }-1}",[34,142699,142701],{"className":142700,"ariaHidden":758},[31672],[34,142702,142704,142708],{"className":142703},[31676],[34,142705],{"className":142706,"style":142707},[31680],"height:1.5578em;vertical-align:-0.5344em;",[34,142709,142711],{"className":142710},[31685],[34,142712,142714,142792],{"className":142713},[31796,31797],[34,142715,142717,142789],{"className":142716},[31801],[34,142718,142720,142780],{"className":142719,"style":126437},[31805],[34,142721,142723,142727],{"style":142722},"top:-3.5578em;",[34,142724],{"className":142725,"style":142726},[31813],"height:3.5578em;",[34,142728,142730],{"className":142729},[126448],[34,142731,142733],{"className":142732},[31685],[34,142734,142736,142746,142749,142752,142755,142768,142771,142774,142777],{"className":142735},[31685],[34,142737,142739,142743],{"className":142738},[31685,2638],[34,142740,142742],{"className":142741},[31685,105462],"起点记",[34,142744,130298],{"className":142745},[31685],[34,142747,640],{"className":142748},[31685],[34,142750,31960],{"className":142751},[32119],[34,142753],{"className":142754,"style":32123},[31845],[34,142756,142758,142761,142765],{"className":142757},[31685,2638],[34,142759,130298],{"className":142760},[31685],[34,142762,142764],{"className":142763},[31685,105462],"终点记",[34,142766,130298],{"className":142767},[31685],[34,142769],{"className":142770,"style":32604},[31845],[34,142772,32226],{"className":142773},[32024],[34,142775],{"className":142776,"style":32604},[31845],[34,142778,640],{"className":142779},[31685],[34,142781,142782,142785],{"style":126463},[34,142783],{"className":142784,"style":142726},[31813],[34,142786],{"className":142787,"style":142788},[126470,126471],"height:1.5578em;border-style:solid;border-width:0.04em;",[34,142790,31829],{"className":142791},[31828],[34,142793,142795],{"className":142794},[31801],[34,142796,142799],{"className":142797,"style":142798},[31805],"height:0.5344em;",[34,142800],{},[32188,142802],{},[11,142804,142806],{"id":142805},"二邻接矩阵","二、邻接矩阵",[15,142808,142809],{},"邻接矩阵描述的是：",[34,142811,142813],{"className":142812},[126385],[34,142814,142816,142839],{"className":142815},[31642],[34,142817,142819],{"className":142818},[31646],[31648,142820,142821],{"xmlns":31650,"display":126394},[31652,142822,142823,142836],{},[31655,142824,142825],{},[126400,142826,142827],{"notation":126402},[71090,142828,142829],{"scriptlevel":459,"displaystyle":702},[71090,142830,142831],{"scriptlevel":459,"displaystyle":702},[71090,142832,142833],{"scriptlevel":459,"displaystyle":758},[39361,142834,142835],{},"顶点和顶点之间的连接关系",[31665,142837,142838],{"encoding":31667},"\\boxed{\\text{顶点和顶点之间的连接关系}}",[34,142840,142842],{"className":142841,"ariaHidden":758},[31672],[34,142843,142845,142848],{"className":142844},[31676],[34,142846],{"className":142847,"style":126424},[31680],[34,142849,142851],{"className":142850},[31685],[34,142852,142854,142891],{"className":142853},[31796,31797],[34,142855,142857,142888],{"className":142856},[31801],[34,142858,142860,142880],{"className":142859,"style":126437},[31805],[34,142861,142862,142865],{"style":126440},[34,142863],{"className":142864,"style":126444},[31813],[34,142866,142868],{"className":142867},[126448],[34,142869,142871],{"className":142870},[31685],[34,142872,142874],{"className":142873},[31685],[34,142875,142877],{"className":142876},[31685,2638],[34,142878,142835],{"className":142879},[31685,105462],[34,142881,142882,142885],{"style":126463},[34,142883],{"className":142884,"style":126444},[31813],[34,142886],{"className":142887,"style":126472},[126470,126471],[34,142889,31829],{"className":142890},[31828],[34,142892,142894],{"className":142893},[31801],[34,142895,142897],{"className":142896,"style":126482},[31805],[34,142898],{},[15,142900,131588],{},[34,142902,142904],{"className":142903},[126385],[34,142905,142907,142950],{"className":142906},[31642],[34,142908,142910],{"className":142909},[31646],[31648,142911,142912],{"xmlns":31650,"display":126394},[31652,142913,142914,142948],{},[31655,142915,142916,142918,142920],{},[31658,142917,126532],{},[31662,142919,202],{},[31655,142921,142922,142928,142930,142936,142938,142940,142942],{},[31743,142923,142924,142926],{},[31658,142925,56782],{},[31749,142927,640],{},[31662,142929,31960],{"separator":758},[31743,142931,142932,142934],{},[31658,142933,56782],{},[31749,142935,768],{},[31662,142937,31960],{"separator":758},[31662,142939,33934],{},[31662,142941,31960],{"separator":758},[31743,142943,142944,142946],{},[31658,142945,56782],{},[31658,142947,31660],{},[31665,142949,140447],{"encoding":31667},[34,142951,142953,142971],{"className":142952,"ariaHidden":758},[31672],[34,142954,142956,142959,142962,142965,142968],{"className":142955},[31676],[34,142957],{"className":142958,"style":33218},[31680],[34,142960,126532],{"className":142961,"style":32604},[31685,31686],[34,142963],{"className":142964,"style":31846},[31845],[34,142966,202],{"className":142967},[31850],[34,142969],{"className":142970,"style":31846},[31845],[34,142972,142974,142977],{"className":142973},[31676],[34,142975],{"className":142976,"style":38774},[31680],[34,142978,142980,143020,143023,143026,143066,143069,143072,143075,143078,143081,143084],{"className":142979},[31685],[34,142981,142983,142986],{"className":142982},[31685],[34,142984,56782],{"className":142985,"style":32062},[31685,31686],[34,142987,142989],{"className":142988},[31792],[34,142990,142992,143012],{"className":142991},[31796,31797],[34,142993,142995,143009],{"className":142994},[31801],[34,142996,142998],{"className":142997,"style":31806},[31805],[34,142999,143000,143003],{"style":38795},[34,143001],{"className":143002,"style":31814},[31813],[34,143004,143006],{"className":143005},[31818,31819,31820,31821],[34,143007,640],{"className":143008},[31685,31821],[34,143010,31829],{"className":143011},[31828],[34,143013,143015],{"className":143014},[31801],[34,143016,143018],{"className":143017,"style":31836},[31805],[34,143019],{},[34,143021,31960],{"className":143022},[32119],[34,143024],{"className":143025,"style":32123},[31845],[34,143027,143029,143032],{"className":143028},[31685],[34,143030,56782],{"className":143031,"style":32062},[31685,31686],[34,143033,143035],{"className":143034},[31792],[34,143036,143038,143058],{"className":143037},[31796,31797],[34,143039,143041,143055],{"className":143040},[31801],[34,143042,143044],{"className":143043,"style":31806},[31805],[34,143045,143046,143049],{"style":38795},[34,143047],{"className":143048,"style":31814},[31813],[34,143050,143052],{"className":143051},[31818,31819,31820,31821],[34,143053,768],{"className":143054},[31685,31821],[34,143056,31829],{"className":143057},[31828],[34,143059,143061],{"className":143060},[31801],[34,143062,143064],{"className":143063,"style":31836},[31805],[34,143065],{},[34,143067,31960],{"className":143068},[32119],[34,143070],{"className":143071,"style":32123},[31845],[34,143073,33934],{"className":143074},[34006],[34,143076],{"className":143077,"style":32123},[31845],[34,143079,31960],{"className":143080},[32119],[34,143082],{"className":143083,"style":32123},[31845],[34,143085,143087,143090],{"className":143086},[31685],[34,143088,56782],{"className":143089,"style":32062},[31685,31686],[34,143091,143093],{"className":143092},[31792],[34,143094,143096,143116],{"className":143095},[31796,31797],[34,143097,143099,143113],{"className":143098},[31801],[34,143100,143102],{"className":143101,"style":32362},[31805],[34,143103,143104,143107],{"style":38795},[34,143105],{"className":143106,"style":31814},[31813],[34,143108,143110],{"className":143109},[31818,31819,31820,31821],[34,143111,31660],{"className":143112},[31685,31686,31821],[34,143114,31829],{"className":143115},[31828],[34,143117,143119],{"className":143118},[31801],[34,143120,143122],{"className":143121,"style":31836},[31805],[34,143123],{},[15,143125,143126],{},"邻接矩阵是：",[34,143128,143130],{"className":143129},[126385],[34,143131,143133,143151],{"className":143132},[31642],[34,143134,143136],{"className":143135},[31646],[31648,143137,143138],{"xmlns":31650,"display":126394},[31652,143139,143140,143148],{},[31655,143141,143142,143144,143146],{},[31658,143143,31660],{},[31662,143145,42071],{},[31658,143147,31660],{},[31665,143149,143150],{"encoding":31667},"n\\times n",[34,143152,143154,143172],{"className":143153,"ariaHidden":758},[31672],[34,143155,143157,143160,143163,143166,143169],{"className":143156},[31676],[34,143158],{"className":143159,"style":32597},[31680],[34,143161,31660],{"className":143162},[31685,31686],[34,143164],{"className":143165,"style":32604},[31845],[34,143167,42071],{"className":143168},[32024],[34,143170],{"className":143171,"style":32604},[31845],[34,143173,143175,143178],{"className":143174},[31676],[34,143176],{"className":143177,"style":32527},[31680],[34,143179,31660],{"className":143180},[31685,31686],[15,143182,140905],{},[32188,143184],{},[19,143186,143188],{"id":143187},"_1-有向图邻接矩阵","1. 有向图邻接矩阵",[15,143190,143191],{},"有向图中：",[34,143193,143195],{"className":143194},[126385],[34,143196,143198,143238],{"className":143197},[31642],[34,143199,143201],{"className":143200},[31646],[31648,143202,143203],{"xmlns":31650,"display":126394},[31652,143204,143205,143235],{},[31655,143206,143207,143209,143211,143213,143223],{},[31658,143208,54151],{},[31662,143210,202],{},[31662,143212,456],{"stretchy":702},[31743,143214,143215,143217],{},[31658,143216,111],{},[31655,143218,143219,143221],{},[31658,143220,31955],{},[31658,143222,31968],{},[31743,143224,143225,143227],{},[31662,143226,2557],{"stretchy":702},[31655,143228,143229,143231,143233],{},[31658,143230,31660],{},[31662,143232,42071],{},[31658,143234,31660],{},[31665,143236,143237],{"encoding":31667},"A=(a_{ij})_{n\\times n}",[34,143239,143241,143259],{"className":143240,"ariaHidden":758},[31672],[34,143242,143244,143247,143250,143253,143256],{"className":143243},[31676],[34,143245],{"className":143246,"style":33218},[31680],[34,143248,54151],{"className":143249},[31685,31686],[34,143251],{"className":143252,"style":31846},[31845],[34,143254,202],{"className":143255},[31850],[34,143257],{"className":143258,"style":31846},[31845],[34,143260,143262,143265,143268,143311],{"className":143261},[31676],[34,143263],{"className":143264,"style":32058},[31680],[34,143266,456],{"className":143267},[31782],[34,143269,143271,143274],{"className":143270},[31685],[34,143272,111],{"className":143273},[31685,31686],[34,143275,143277],{"className":143276},[31792],[34,143278,143280,143303],{"className":143279},[31796,31797],[34,143281,143283,143300],{"className":143282},[31801],[34,143284,143286],{"className":143285,"style":32090},[31805],[34,143287,143288,143291],{"style":31809},[34,143289],{"className":143290,"style":31814},[31813],[34,143292,143294],{"className":143293},[31818,31819,31820,31821],[34,143295,143297],{"className":143296},[31685,31821],[34,143298,141027],{"className":143299,"style":32153},[31685,31686,31821],[34,143301,31829],{"className":143302},[31828],[34,143304,143306],{"className":143305},[31801],[34,143307,143309],{"className":143308,"style":32163},[31805],[34,143310],{},[34,143312,143314,143317],{"className":143313},[31690],[34,143315,2557],{"className":143316},[31690],[34,143318,143320],{"className":143319},[31792],[34,143321,143323,143352],{"className":143322},[31796,31797],[34,143324,143326,143349],{"className":143325},[31801],[34,143327,143329],{"className":143328,"style":141057},[31805],[34,143330,143331,143334],{"style":31809},[34,143332],{"className":143333,"style":31814},[31813],[34,143335,143337],{"className":143336},[31818,31819,31820,31821],[34,143338,143340,143343,143346],{"className":143339},[31685,31821],[34,143341,31660],{"className":143342},[31685,31686,31821],[34,143344,42071],{"className":143345},[32024,31821],[34,143347,31660],{"className":143348},[31685,31686,31821],[34,143350,31829],{"className":143351},[31828],[34,143353,143355],{"className":143354},[31801],[34,143356,143358],{"className":143357,"style":32037},[31805],[34,143359],{},[15,143361,104944],{},[34,143363,143365],{"className":143364},[126385],[34,143366,143368,143394],{"className":143367},[31642],[34,143369,143371],{"className":143370},[31646],[31648,143372,143373],{"xmlns":31650,"display":126394},[31652,143374,143375,143391],{},[31655,143376,143377,143387,143389],{},[31743,143378,143379,143381],{},[31658,143380,111],{},[31655,143382,143383,143385],{},[31658,143384,31955],{},[31658,143386,31968],{},[31662,143388,202],{},[31658,143390,33759],{},[31665,143392,143393],{"encoding":31667},"a_{ij}=k",[34,143395,143397,143455],{"className":143396,"ariaHidden":758},[31672],[34,143398,143400,143403,143446,143449,143452],{"className":143399},[31676],[34,143401],{"className":143402,"style":56005},[31680],[34,143404,143406,143409],{"className":143405},[31685],[34,143407,111],{"className":143408},[31685,31686],[34,143410,143412],{"className":143411},[31792],[34,143413,143415,143438],{"className":143414},[31796,31797],[34,143416,143418,143435],{"className":143417},[31801],[34,143419,143421],{"className":143420,"style":32090},[31805],[34,143422,143423,143426],{"style":31809},[34,143424],{"className":143425,"style":31814},[31813],[34,143427,143429],{"className":143428},[31818,31819,31820,31821],[34,143430,143432],{"className":143431},[31685,31821],[34,143433,141027],{"className":143434,"style":32153},[31685,31686,31821],[34,143436,31829],{"className":143437},[31828],[34,143439,143441],{"className":143440},[31801],[34,143442,143444],{"className":143443,"style":32163},[31805],[34,143445],{},[34,143447],{"className":143448,"style":31846},[31845],[34,143450,202],{"className":143451},[31850],[34,143453],{"className":143454,"style":31846},[31845],[34,143456,143458,143461],{"className":143457},[31676],[34,143459],{"className":143460,"style":31681},[31680],[34,143462,33759],{"className":143463,"style":33774},[31685,31686],[15,143465,143466,143467,53834,143536,143605,143606,143634],{},"表示从 ",[34,143468,143470,143487],{"className":143469},[31642],[34,143471,143473],{"className":143472},[31646],[31648,143474,143475],{"xmlns":31650},[31652,143476,143477,143485],{},[31655,143478,143479],{},[31743,143480,143481,143483],{},[31658,143482,56782],{},[31658,143484,31955],{},[31665,143486,56787],{"encoding":31667},[34,143488,143490],{"className":143489,"ariaHidden":758},[31672],[34,143491,143493,143496],{"className":143492},[31676],[34,143494],{"className":143495,"style":31860},[31680],[34,143497,143499,143502],{"className":143498},[31685],[34,143500,56782],{"className":143501,"style":32062},[31685,31686],[34,143503,143505],{"className":143504},[31792],[34,143506,143508,143528],{"className":143507},[31796,31797],[34,143509,143511,143525],{"className":143510},[31801],[34,143512,143514],{"className":143513,"style":32090},[31805],[34,143515,143516,143519],{"style":38795},[34,143517],{"className":143518,"style":31814},[31813],[34,143520,143522],{"className":143521},[31818,31819,31820,31821],[34,143523,31955],{"className":143524},[31685,31686,31821],[34,143526,31829],{"className":143527},[31828],[34,143529,143531],{"className":143530},[31801],[34,143532,143534],{"className":143533,"style":31836},[31805],[34,143535],{},[34,143537,143539,143556],{"className":143538},[31642],[34,143540,143542],{"className":143541},[31646],[31648,143543,143544],{"xmlns":31650},[31652,143545,143546,143554],{},[31655,143547,143548],{},[31743,143549,143550,143552],{},[31658,143551,56782],{},[31658,143553,31968],{},[31665,143555,138126],{"encoding":31667},[34,143557,143559],{"className":143558,"ariaHidden":758},[31672],[34,143560,143562,143565],{"className":143561},[31676],[34,143563],{"className":143564,"style":56005},[31680],[34,143566,143568,143571],{"className":143567},[31685],[34,143569,56782],{"className":143570,"style":32062},[31685,31686],[34,143572,143574],{"className":143573},[31792],[34,143575,143577,143597],{"className":143576},[31796,31797],[34,143578,143580,143594],{"className":143579},[31801],[34,143581,143583],{"className":143582,"style":32090},[31805],[34,143584,143585,143588],{"style":38795},[34,143586],{"className":143587,"style":31814},[31813],[34,143589,143591],{"className":143590},[31818,31819,31820,31821],[34,143592,31968],{"className":143593,"style":32153},[31685,31686,31821],[34,143595,31829],{"className":143596},[31828],[34,143598,143600],{"className":143599},[31801],[34,143601,143603],{"className":143602,"style":32163},[31805],[34,143604],{}," 的有向边有 ",[34,143607,143609,143622],{"className":143608},[31642],[34,143610,143612],{"className":143611},[31646],[31648,143613,143614],{"xmlns":31650},[31652,143615,143616,143620],{},[31655,143617,143618],{},[31658,143619,33759],{},[31665,143621,33759],{"encoding":31667},[34,143623,143625],{"className":143624,"ariaHidden":758},[31672],[34,143626,143628,143631],{"className":143627},[31676],[34,143629],{"className":143630,"style":31681},[31680],[34,143632,33759],{"className":143633,"style":33774},[31685,31686]," 条。",[15,143636,143637],{},"如果没有边，则：",[34,143639,143641],{"className":143640},[126385],[34,143642,143644,143670],{"className":143643},[31642],[34,143645,143647],{"className":143646},[31646],[31648,143648,143649],{"xmlns":31650,"display":126394},[31652,143650,143651,143667],{},[31655,143652,143653,143663,143665],{},[31743,143654,143655,143657],{},[31658,143656,111],{},[31655,143658,143659,143661],{},[31658,143660,31955],{},[31658,143662,31968],{},[31662,143664,202],{},[31749,143666,459],{},[31665,143668,143669],{"encoding":31667},"a_{ij}=0",[34,143671,143673,143731],{"className":143672,"ariaHidden":758},[31672],[34,143674,143676,143679,143722,143725,143728],{"className":143675},[31676],[34,143677],{"className":143678,"style":56005},[31680],[34,143680,143682,143685],{"className":143681},[31685],[34,143683,111],{"className":143684},[31685,31686],[34,143686,143688],{"className":143687},[31792],[34,143689,143691,143714],{"className":143690},[31796,31797],[34,143692,143694,143711],{"className":143693},[31801],[34,143695,143697],{"className":143696,"style":32090},[31805],[34,143698,143699,143702],{"style":31809},[34,143700],{"className":143701,"style":31814},[31813],[34,143703,143705],{"className":143704},[31818,31819,31820,31821],[34,143706,143708],{"className":143707},[31685,31821],[34,143709,141027],{"className":143710,"style":32153},[31685,31686,31821],[34,143712,31829],{"className":143713},[31828],[34,143715,143717],{"className":143716},[31801],[34,143718,143720],{"className":143719,"style":32163},[31805],[34,143721],{},[34,143723],{"className":143724,"style":31846},[31845],[34,143726,202],{"className":143727},[31850],[34,143729],{"className":143730,"style":31846},[31845],[34,143732,143734,143737],{"className":143733},[31676],[34,143735],{"className":143736,"style":32617},[31680],[34,143738,459],{"className":143739},[31685],[32188,143741],{},[19,143743,143745],{"id":143744},"_2-无向图邻接矩阵","2. 无向图邻接矩阵",[15,143747,140917],{},[34,143749,143751],{"className":143750},[126385],[34,143752,143754,143779],{"className":143753},[31642],[34,143755,143757],{"className":143756},[31646],[31648,143758,143759],{"xmlns":31650,"display":126394},[31652,143760,143761,143777],{},[31655,143762,143763,143773,143775],{},[31743,143764,143765,143767],{},[31658,143766,111],{},[31655,143768,143769,143771],{},[31658,143770,31955],{},[31658,143772,31968],{},[31662,143774,202],{},[31658,143776,33759],{},[31665,143778,143393],{"encoding":31667},[34,143780,143782,143840],{"className":143781,"ariaHidden":758},[31672],[34,143783,143785,143788,143831,143834,143837],{"className":143784},[31676],[34,143786],{"className":143787,"style":56005},[31680],[34,143789,143791,143794],{"className":143790},[31685],[34,143792,111],{"className":143793},[31685,31686],[34,143795,143797],{"className":143796},[31792],[34,143798,143800,143823],{"className":143799},[31796,31797],[34,143801,143803,143820],{"className":143802},[31801],[34,143804,143806],{"className":143805,"style":32090},[31805],[34,143807,143808,143811],{"style":31809},[34,143809],{"className":143810,"style":31814},[31813],[34,143812,143814],{"className":143813},[31818,31819,31820,31821],[34,143815,143817],{"className":143816},[31685,31821],[34,143818,141027],{"className":143819,"style":32153},[31685,31686,31821],[34,143821,31829],{"className":143822},[31828],[34,143824,143826],{"className":143825},[31801],[34,143827,143829],{"className":143828,"style":32163},[31805],[34,143830],{},[34,143832],{"className":143833,"style":31846},[31845],[34,143835,202],{"className":143836},[31850],[34,143838],{"className":143839,"style":31846},[31845],[34,143841,143843,143846],{"className":143842},[31676],[34,143844],{"className":143845,"style":31681},[31680],[34,143847,33759],{"className":143848,"style":33774},[31685,31686],[15,143850,127307,143851,258,143920,143989,143990,137083],{},[34,143852,143854,143871],{"className":143853},[31642],[34,143855,143857],{"className":143856},[31646],[31648,143858,143859],{"xmlns":31650},[31652,143860,143861,143869],{},[31655,143862,143863],{},[31743,143864,143865,143867],{},[31658,143866,56782],{},[31658,143868,31955],{},[31665,143870,56787],{"encoding":31667},[34,143872,143874],{"className":143873,"ariaHidden":758},[31672],[34,143875,143877,143880],{"className":143876},[31676],[34,143878],{"className":143879,"style":31860},[31680],[34,143881,143883,143886],{"className":143882},[31685],[34,143884,56782],{"className":143885,"style":32062},[31685,31686],[34,143887,143889],{"className":143888},[31792],[34,143890,143892,143912],{"className":143891},[31796,31797],[34,143893,143895,143909],{"className":143894},[31801],[34,143896,143898],{"className":143897,"style":32090},[31805],[34,143899,143900,143903],{"style":38795},[34,143901],{"className":143902,"style":31814},[31813],[34,143904,143906],{"className":143905},[31818,31819,31820,31821],[34,143907,31955],{"className":143908},[31685,31686,31821],[34,143910,31829],{"className":143911},[31828],[34,143913,143915],{"className":143914},[31801],[34,143916,143918],{"className":143917,"style":31836},[31805],[34,143919],{},[34,143921,143923,143940],{"className":143922},[31642],[34,143924,143926],{"className":143925},[31646],[31648,143927,143928],{"xmlns":31650},[31652,143929,143930,143938],{},[31655,143931,143932],{},[31743,143933,143934,143936],{},[31658,143935,56782],{},[31658,143937,31968],{},[31665,143939,138126],{"encoding":31667},[34,143941,143943],{"className":143942,"ariaHidden":758},[31672],[34,143944,143946,143949],{"className":143945},[31676],[34,143947],{"className":143948,"style":56005},[31680],[34,143950,143952,143955],{"className":143951},[31685],[34,143953,56782],{"className":143954,"style":32062},[31685,31686],[34,143956,143958],{"className":143957},[31792],[34,143959,143961,143981],{"className":143960},[31796,31797],[34,143962,143964,143978],{"className":143963},[31801],[34,143965,143967],{"className":143966,"style":32090},[31805],[34,143968,143969,143972],{"style":38795},[34,143970],{"className":143971,"style":31814},[31813],[34,143973,143975],{"className":143974},[31818,31819,31820,31821],[34,143976,31968],{"className":143977,"style":32153},[31685,31686,31821],[34,143979,31829],{"className":143980},[31828],[34,143982,143984],{"className":143983},[31801],[34,143985,143987],{"className":143986,"style":32163},[31805],[34,143988],{}," 之间有 ",[34,143991,143993,144006],{"className":143992},[31642],[34,143994,143996],{"className":143995},[31646],[31648,143997,143998],{"xmlns":31650},[31652,143999,144000,144004],{},[31655,144001,144002],{},[31658,144003,33759],{},[31665,144005,33759],{"encoding":31667},[34,144007,144009],{"className":144008,"ariaHidden":758},[31672],[34,144010,144012,144015],{"className":144011},[31676],[34,144013],{"className":144014,"style":31681},[31680],[34,144016,33759],{"className":144017,"style":33774},[31685,31686],[32188,144019],{},[19,144021,144023],{"id":144022},"_3-简单图邻接矩阵特点","3. 简单图邻接矩阵特点",[15,144025,2234,144026,144054],{},[34,144027,144029,144042],{"className":144028},[31642],[34,144030,144032],{"className":144031},[31646],[31648,144033,144034],{"xmlns":31650},[31652,144035,144036,144040],{},[31655,144037,144038],{},[31658,144039,97709],{},[31665,144041,97709],{"encoding":31667},[34,144043,144045],{"className":144044,"ariaHidden":758},[31672],[34,144046,144048,144051],{"className":144047},[31676],[34,144049],{"className":144050,"style":33218},[31680],[34,144052,97709],{"className":144053},[31685,31686]," 是简单图，则：",[2230,144056,144057,144060,144063],{},[108,144058,144059],{},"主对角线元素全为 0；",[108,144061,144062],{},"每个元素只能是 0 或 1；",[108,144064,144065],{},"无向简单图的邻接矩阵是对称矩阵。",[15,144067,139415],{},[34,144069,144071],{"className":144070},[126385],[34,144072,144074,144096],{"className":144073},[31642],[34,144075,144077],{"className":144076},[31646],[31648,144078,144079],{"xmlns":31650,"display":126394},[31652,144080,144081,144093],{},[31655,144082,144083,144085,144087],{},[31658,144084,54151],{},[31662,144086,202],{},[37624,144088,144089,144091],{},[31658,144090,54151],{},[31658,144092,37400],{},[31665,144094,144095],{"encoding":31667},"A=A^T",[34,144097,144099,144117],{"className":144098,"ariaHidden":758},[31672],[34,144100,144102,144105,144108,144111,144114],{"className":144101},[31676],[34,144103],{"className":144104,"style":33218},[31680],[34,144106,54151],{"className":144107},[31685,31686],[34,144109],{"className":144110,"style":31846},[31845],[34,144112,202],{"className":144113},[31850],[34,144115],{"className":144116,"style":31846},[31845],[34,144118,144120,144124],{"className":144119},[31676],[34,144121],{"className":144122,"style":144123},[31680],"height:0.8913em;",[34,144125,144127,144130],{"className":144126},[31685],[34,144128,54151],{"className":144129},[31685,31686],[34,144131,144133],{"className":144132},[31792],[34,144134,144136],{"className":144135},[31796],[34,144137,144139],{"className":144138},[31801],[34,144140,144142],{"className":144141,"style":144123},[31805],[34,144143,144144,144147],{"style":129846},[34,144145],{"className":144146,"style":31814},[31813],[34,144148,144150],{"className":144149},[31818,31819,31820,31821],[34,144151,37400],{"className":144152,"style":37422},[31685,31686,31821],[32188,144154],{},[19,144156,144158],{"id":144157},"_4-用邻接矩阵求度","4. 用邻接矩阵求度",[15,144160,140260],{},[253,144162,130335],{"id":144163},"出度-1",[34,144165,144167],{"className":144166},[126385],[34,144168,144170,144225],{"className":144169},[31642],[34,144171,144173],{"className":144172},[31646],[31648,144174,144175],{"xmlns":31650,"display":126394},[31652,144176,144177,144222],{},[31655,144178,144179,144185,144187,144193,144195,144197,144212],{},[37624,144180,144181,144183],{},[31658,144182,46900],{},[31662,144184,223],{},[31662,144186,456],{"stretchy":702},[31743,144188,144189,144191],{},[31658,144190,56782],{},[31658,144192,31955],{},[31662,144194,2557],{"stretchy":702},[31662,144196,202],{},[144198,144199,144200,144202,144210],"munderover",{},[31662,144201,131138],{},[31655,144203,144204,144206,144208],{},[31658,144205,31968],{},[31662,144207,202],{},[31749,144209,640],{},[31658,144211,31660],{},[31743,144213,144214,144216],{},[31658,144215,111],{},[31655,144217,144218,144220],{},[31658,144219,31955],{},[31658,144221,31968],{},[31665,144223,144224],{"encoding":31667},"d^+(v_i)=\\sum_{j=1}^{n}a_{ij}",[34,144226,144228,144318],{"className":144227,"ariaHidden":758},[31672],[34,144229,144231,144234,144263,144266,144306,144309,144312,144315],{"className":144230},[31676],[34,144232],{"className":144233,"style":130407},[31680],[34,144235,144237,144240],{"className":144236},[31685],[34,144238,46900],{"className":144239},[31685,31686],[34,144241,144243],{"className":144242},[31792],[34,144244,144246],{"className":144245},[31796],[34,144247,144249],{"className":144248},[31801],[34,144250,144252],{"className":144251,"style":130426},[31805],[34,144253,144254,144257],{"style":129846},[34,144255],{"className":144256,"style":31814},[31813],[34,144258,144260],{"className":144259},[31818,31819,31820,31821],[34,144261,223],{"className":144262},[32024,31821],[34,144264,456],{"className":144265},[31782],[34,144267,144269,144272],{"className":144268},[31685],[34,144270,56782],{"className":144271,"style":32062},[31685,31686],[34,144273,144275],{"className":144274},[31792],[34,144276,144278,144298],{"className":144277},[31796,31797],[34,144279,144281,144295],{"className":144280},[31801],[34,144282,144284],{"className":144283,"style":32090},[31805],[34,144285,144286,144289],{"style":38795},[34,144287],{"className":144288,"style":31814},[31813],[34,144290,144292],{"className":144291},[31818,31819,31820,31821],[34,144293,31955],{"className":144294},[31685,31686,31821],[34,144296,31829],{"className":144297},[31828],[34,144299,144301],{"className":144300},[31801],[34,144302,144304],{"className":144303,"style":31836},[31805],[34,144305],{},[34,144307,2557],{"className":144308},[31690],[34,144310],{"className":144311,"style":31846},[31845],[34,144313,202],{"className":144314},[31850],[34,144316],{"className":144317,"style":31846},[31845],[34,144319,144321,144325,144396,144399],{"className":144320},[31676],[34,144322],{"className":144323,"style":144324},[31680],"height:3.0652em;vertical-align:-1.4138em;",[34,144326,144328],{"className":144327},[32767,71420],[34,144329,144331,144387],{"className":144330},[31796,31797],[34,144332,144334,144384],{"className":144333},[31801],[34,144335,144338,144359,144369],{"className":144336,"style":144337},[31805],"height:1.6514em;",[34,144339,144341,144344],{"style":144340},"top:-1.8723em;margin-left:0em;",[34,144342],{"className":144343,"style":131222},[31813],[34,144345,144347],{"className":144346},[31818,31819,31820,31821],[34,144348,144350,144353,144356],{"className":144349},[31685,31821],[34,144351,31968],{"className":144352,"style":32153},[31685,31686,31821],[34,144354,202],{"className":144355},[31850,31821],[34,144357,640],{"className":144358},[31685,31821],[34,144360,144361,144364],{"style":131240},[34,144362],{"className":144363,"style":131222},[31813],[34,144365,144366],{},[34,144367,131138],{"className":144368},[32767,131249,131250],[34,144370,144372,144375],{"style":144371},"top:-4.3em;margin-left:0em;",[34,144373],{"className":144374,"style":131222},[31813],[34,144376,144378],{"className":144377},[31818,31819,31820,31821],[34,144379,144381],{"className":144380},[31685,31821],[34,144382,31660],{"className":144383},[31685,31686,31821],[34,144385,31829],{"className":144386},[31828],[34,144388,144390],{"className":144389},[31801],[34,144391,144394],{"className":144392,"style":144393},[31805],"height:1.4138em;",[34,144395],{},[34,144397],{"className":144398,"style":32123},[31845],[34,144400,144402,144405],{"className":144401},[31685],[34,144403,111],{"className":144404},[31685,31686],[34,144406,144408],{"className":144407},[31792],[34,144409,144411,144434],{"className":144410},[31796,31797],[34,144412,144414,144431],{"className":144413},[31801],[34,144415,144417],{"className":144416,"style":32090},[31805],[34,144418,144419,144422],{"style":31809},[34,144420],{"className":144421,"style":31814},[31813],[34,144423,144425],{"className":144424},[31818,31819,31820,31821],[34,144426,144428],{"className":144427},[31685,31821],[34,144429,141027],{"className":144430,"style":32153},[31685,31686,31821],[34,144432,31829],{"className":144433},[31828],[34,144435,144437],{"className":144436},[31801],[34,144438,144440],{"className":144439,"style":32163},[31805],[34,144441],{},[15,144443,144444,144445,144473],{},"也就是第 ",[34,144446,144448,144461],{"className":144447},[31642],[34,144449,144451],{"className":144450},[31646],[31648,144452,144453],{"xmlns":31650},[31652,144454,144455,144459],{},[31655,144456,144457],{},[31658,144458,31955],{},[31665,144460,31955],{"encoding":31667},[34,144462,144464],{"className":144463,"ariaHidden":758},[31672],[34,144465,144467,144470],{"className":144466},[31676],[34,144468],{"className":144469,"style":33104},[31680],[34,144471,31955],{"className":144472},[31685,31686]," 行元素之和。",[253,144475,130449],{"id":144476},"入度-1",[34,144478,144480],{"className":144479},[126385],[34,144481,144483,144537],{"className":144482},[31642],[34,144484,144486],{"className":144485},[31646],[31648,144487,144488],{"xmlns":31650,"display":126394},[31652,144489,144490,144534],{},[31655,144491,144492,144498,144500,144506,144508,144510,144524],{},[37624,144493,144494,144496],{},[31658,144495,46900],{},[31662,144497,32226],{},[31662,144499,456],{"stretchy":702},[31743,144501,144502,144504],{},[31658,144503,56782],{},[31658,144505,31955],{},[31662,144507,2557],{"stretchy":702},[31662,144509,202],{},[144198,144511,144512,144514,144522],{},[31662,144513,131138],{},[31655,144515,144516,144518,144520],{},[31658,144517,31968],{},[31662,144519,202],{},[31749,144521,640],{},[31658,144523,31660],{},[31743,144525,144526,144528],{},[31658,144527,111],{},[31655,144529,144530,144532],{},[31658,144531,31968],{},[31658,144533,31955],{},[31665,144535,144536],{"encoding":31667},"d^-(v_i)=\\sum_{j=1}^{n}a_{ji}",[34,144538,144540,144630],{"className":144539,"ariaHidden":758},[31672],[34,144541,144543,144546,144575,144578,144618,144621,144624,144627],{"className":144542},[31676],[34,144544],{"className":144545,"style":130407},[31680],[34,144547,144549,144552],{"className":144548},[31685],[34,144550,46900],{"className":144551},[31685,31686],[34,144553,144555],{"className":144554},[31792],[34,144556,144558],{"className":144557},[31796],[34,144559,144561],{"className":144560},[31801],[34,144562,144564],{"className":144563,"style":130426},[31805],[34,144565,144566,144569],{"style":129846},[34,144567],{"className":144568,"style":31814},[31813],[34,144570,144572],{"className":144571},[31818,31819,31820,31821],[34,144573,32226],{"className":144574},[32024,31821],[34,144576,456],{"className":144577},[31782],[34,144579,144581,144584],{"className":144580},[31685],[34,144582,56782],{"className":144583,"style":32062},[31685,31686],[34,144585,144587],{"className":144586},[31792],[34,144588,144590,144610],{"className":144589},[31796,31797],[34,144591,144593,144607],{"className":144592},[31801],[34,144594,144596],{"className":144595,"style":32090},[31805],[34,144597,144598,144601],{"style":38795},[34,144599],{"className":144600,"style":31814},[31813],[34,144602,144604],{"className":144603},[31818,31819,31820,31821],[34,144605,31955],{"className":144606},[31685,31686,31821],[34,144608,31829],{"className":144609},[31828],[34,144611,144613],{"className":144612},[31801],[34,144614,144616],{"className":144615,"style":31836},[31805],[34,144617],{},[34,144619,2557],{"className":144620},[31690],[34,144622],{"className":144623,"style":31846},[31845],[34,144625,202],{"className":144626},[31850],[34,144628],{"className":144629,"style":31846},[31845],[34,144631,144633,144636,144703,144706],{"className":144632},[31676],[34,144634],{"className":144635,"style":144324},[31680],[34,144637,144639],{"className":144638},[32767,71420],[34,144640,144642,144695],{"className":144641},[31796,31797],[34,144643,144645,144692],{"className":144644},[31801],[34,144646,144648,144668,144678],{"className":144647,"style":144337},[31805],[34,144649,144650,144653],{"style":144340},[34,144651],{"className":144652,"style":131222},[31813],[34,144654,144656],{"className":144655},[31818,31819,31820,31821],[34,144657,144659,144662,144665],{"className":144658},[31685,31821],[34,144660,31968],{"className":144661,"style":32153},[31685,31686,31821],[34,144663,202],{"className":144664},[31850,31821],[34,144666,640],{"className":144667},[31685,31821],[34,144669,144670,144673],{"style":131240},[34,144671],{"className":144672,"style":131222},[31813],[34,144674,144675],{},[34,144676,131138],{"className":144677},[32767,131249,131250],[34,144679,144680,144683],{"style":144371},[34,144681],{"className":144682,"style":131222},[31813],[34,144684,144686],{"className":144685},[31818,31819,31820,31821],[34,144687,144689],{"className":144688},[31685,31821],[34,144690,31660],{"className":144691},[31685,31686,31821],[34,144693,31829],{"className":144694},[31828],[34,144696,144698],{"className":144697},[31801],[34,144699,144701],{"className":144700,"style":144393},[31805],[34,144702],{},[34,144704],{"className":144705,"style":32123},[31845],[34,144707,144709,144712],{"className":144708},[31685],[34,144710,111],{"className":144711},[31685,31686],[34,144713,144715],{"className":144714},[31792],[34,144716,144718,144744],{"className":144717},[31796,31797],[34,144719,144721,144741],{"className":144720},[31801],[34,144722,144724],{"className":144723,"style":32090},[31805],[34,144725,144726,144729],{"style":31809},[34,144727],{"className":144728,"style":31814},[31813],[34,144730,144732],{"className":144731},[31818,31819,31820,31821],[34,144733,144735,144738],{"className":144734},[31685,31821],[34,144736,31968],{"className":144737,"style":32153},[31685,31686,31821],[34,144739,31955],{"className":144740},[31685,31686,31821],[34,144742,31829],{"className":144743},[31828],[34,144745,144747],{"className":144746},[31801],[34,144748,144750],{"className":144749,"style":32163},[31805],[34,144751],{},[15,144753,144444,144754,144782],{},[34,144755,144757,144770],{"className":144756},[31642],[34,144758,144760],{"className":144759},[31646],[31648,144761,144762],{"xmlns":31650},[31652,144763,144764,144768],{},[31655,144765,144766],{},[31658,144767,31955],{},[31665,144769,31955],{"encoding":31667},[34,144771,144773],{"className":144772,"ariaHidden":758},[31672],[34,144774,144776,144779],{"className":144775},[31676],[34,144777],{"className":144778,"style":33104},[31680],[34,144780,31955],{"className":144781},[31685,31686]," 列元素之和。",[253,144784,130561],{"id":144785},"总度-1",[34,144787,144789],{"className":144788},[126385],[34,144790,144792,144852],{"className":144791},[31642],[34,144793,144795],{"className":144794},[31646],[31648,144796,144797],{"xmlns":31650,"display":126394},[31652,144798,144799,144849],{},[31655,144800,144801,144803,144805,144811,144813,144815,144821,144823,144829,144831,144833,144839,144841,144847],{},[31658,144802,46900],{},[31662,144804,456],{"stretchy":702},[31743,144806,144807,144809],{},[31658,144808,56782],{},[31658,144810,31955],{},[31662,144812,2557],{"stretchy":702},[31662,144814,202],{},[37624,144816,144817,144819],{},[31658,144818,46900],{},[31662,144820,223],{},[31662,144822,456],{"stretchy":702},[31743,144824,144825,144827],{},[31658,144826,56782],{},[31658,144828,31955],{},[31662,144830,2557],{"stretchy":702},[31662,144832,223],{},[37624,144834,144835,144837],{},[31658,144836,46900],{},[31662,144838,32226],{},[31662,144840,456],{"stretchy":702},[31743,144842,144843,144845],{},[31658,144844,56782],{},[31658,144846,31955],{},[31662,144848,2557],{"stretchy":702},[31665,144850,144851],{"encoding":31667},"d(v_i)=d^+(v_i)+d^-(v_i)",[34,144853,144855,144919,145009],{"className":144854,"ariaHidden":758},[31672],[34,144856,144858,144861,144864,144867,144907,144910,144913,144916],{"className":144857},[31676],[34,144859],{"className":144860,"style":31774},[31680],[34,144862,46900],{"className":144863},[31685,31686],[34,144865,456],{"className":144866},[31782],[34,144868,144870,144873],{"className":144869},[31685],[34,144871,56782],{"className":144872,"style":32062},[31685,31686],[34,144874,144876],{"className":144875},[31792],[34,144877,144879,144899],{"className":144878},[31796,31797],[34,144880,144882,144896],{"className":144881},[31801],[34,144883,144885],{"className":144884,"style":32090},[31805],[34,144886,144887,144890],{"style":38795},[34,144888],{"className":144889,"style":31814},[31813],[34,144891,144893],{"className":144892},[31818,31819,31820,31821],[34,144894,31955],{"className":144895},[31685,31686,31821],[34,144897,31829],{"className":144898},[31828],[34,144900,144902],{"className":144901},[31801],[34,144903,144905],{"className":144904,"style":31836},[31805],[34,144906],{},[34,144908,2557],{"className":144909},[31690],[34,144911],{"className":144912,"style":31846},[31845],[34,144914,202],{"className":144915},[31850],[34,144917],{"className":144918,"style":31846},[31845],[34,144920,144922,144925,144954,144957,144997,145000,145003,145006],{"className":144921},[31676],[34,144923],{"className":144924,"style":130407},[31680],[34,144926,144928,144931],{"className":144927},[31685],[34,144929,46900],{"className":144930},[31685,31686],[34,144932,144934],{"className":144933},[31792],[34,144935,144937],{"className":144936},[31796],[34,144938,144940],{"className":144939},[31801],[34,144941,144943],{"className":144942,"style":130426},[31805],[34,144944,144945,144948],{"style":129846},[34,144946],{"className":144947,"style":31814},[31813],[34,144949,144951],{"className":144950},[31818,31819,31820,31821],[34,144952,223],{"className":144953},[32024,31821],[34,144955,456],{"className":144956},[31782],[34,144958,144960,144963],{"className":144959},[31685],[34,144961,56782],{"className":144962,"style":32062},[31685,31686],[34,144964,144966],{"className":144965},[31792],[34,144967,144969,144989],{"className":144968},[31796,31797],[34,144970,144972,144986],{"className":144971},[31801],[34,144973,144975],{"className":144974,"style":32090},[31805],[34,144976,144977,144980],{"style":38795},[34,144978],{"className":144979,"style":31814},[31813],[34,144981,144983],{"className":144982},[31818,31819,31820,31821],[34,144984,31955],{"className":144985},[31685,31686,31821],[34,144987,31829],{"className":144988},[31828],[34,144990,144992],{"className":144991},[31801],[34,144993,144995],{"className":144994,"style":31836},[31805],[34,144996],{},[34,144998,2557],{"className":144999},[31690],[34,145001],{"className":145002,"style":32604},[31845],[34,145004,223],{"className":145005},[32024],[34,145007],{"className":145008,"style":32604},[31845],[34,145010,145012,145015,145044,145047,145087],{"className":145011},[31676],[34,145013],{"className":145014,"style":130407},[31680],[34,145016,145018,145021],{"className":145017},[31685],[34,145019,46900],{"className":145020},[31685,31686],[34,145022,145024],{"className":145023},[31792],[34,145025,145027],{"className":145026},[31796],[34,145028,145030],{"className":145029},[31801],[34,145031,145033],{"className":145032,"style":130426},[31805],[34,145034,145035,145038],{"style":129846},[34,145036],{"className":145037,"style":31814},[31813],[34,145039,145041],{"className":145040},[31818,31819,31820,31821],[34,145042,32226],{"className":145043},[32024,31821],[34,145045,456],{"className":145046},[31782],[34,145048,145050,145053],{"className":145049},[31685],[34,145051,56782],{"className":145052,"style":32062},[31685,31686],[34,145054,145056],{"className":145055},[31792],[34,145057,145059,145079],{"className":145058},[31796,31797],[34,145060,145062,145076],{"className":145061},[31801],[34,145063,145065],{"className":145064,"style":32090},[31805],[34,145066,145067,145070],{"style":38795},[34,145068],{"className":145069,"style":31814},[31813],[34,145071,145073],{"className":145072},[31818,31819,31820,31821],[34,145074,31955],{"className":145075},[31685,31686,31821],[34,145077,31829],{"className":145078},[31828],[34,145080,145082],{"className":145081},[31801],[34,145083,145085],{"className":145084,"style":31836},[31805],[34,145086],{},[34,145088,2557],{"className":145089},[31690],[32188,145091],{},[11,145093,145095],{"id":145094},"三邻接矩阵的幂","三、邻接矩阵的幂",[15,145097,145098],{},"这是很重要的计算结论。",[15,145100,133910,145101,145129],{},[34,145102,145104,145117],{"className":145103},[31642],[34,145105,145107],{"className":145106},[31646],[31648,145108,145109],{"xmlns":31650},[31652,145110,145111,145115],{},[31655,145112,145113],{},[31658,145114,54151],{},[31665,145116,54151],{"encoding":31667},[34,145118,145120],{"className":145119,"ariaHidden":758},[31672],[34,145121,145123,145126],{"className":145122},[31676],[34,145124],{"className":145125,"style":33218},[31680],[34,145127,54151],{"className":145128},[31685,31686]," 是无重边有向图的邻接矩阵。",[15,145131,145132],{},"那么：",[34,145134,145136],{"className":145135},[126385],[34,145137,145139,145157],{"className":145138},[31642],[34,145140,145142],{"className":145141},[31646],[31648,145143,145144],{"xmlns":31650,"display":126394},[31652,145145,145146,145154],{},[31655,145147,145148],{},[37624,145149,145150,145152],{},[31658,145151,54151],{},[31658,145153,31759],{},[31665,145155,145156],{"encoding":31667},"A^m",[34,145158,145160],{"className":145159,"ariaHidden":758},[31672],[34,145161,145163,145167],{"className":145162},[31676],[34,145164],{"className":145165,"style":145166},[31680],"height:0.7144em;",[34,145168,145170,145173],{"className":145169},[31685],[34,145171,54151],{"className":145172},[31685,31686],[34,145174,145176],{"className":145175},[31792],[34,145177,145179],{"className":145178},[31796],[34,145180,145182],{"className":145181},[31801],[34,145183,145185],{"className":145184,"style":145166},[31805],[34,145186,145187,145190],{"style":129846},[34,145188],{"className":145189,"style":31814},[31813],[34,145191,145193],{"className":145192},[31818,31819,31820,31821],[34,145194,31759],{"className":145195},[31685,31686,31821],[15,145197,145198,145199,145241],{},"中第 ",[34,145200,145202,145220],{"className":145201},[31642],[34,145203,145205],{"className":145204},[31646],[31648,145206,145207],{"xmlns":31650},[31652,145208,145209,145217],{},[31655,145210,145211,145213,145215],{},[31658,145212,31955],{},[31662,145214,31960],{"separator":758},[31658,145216,31968],{},[31665,145218,145219],{"encoding":31667},"i,j",[34,145221,145223],{"className":145222,"ariaHidden":758},[31672],[34,145224,145226,145229,145232,145235,145238],{"className":145225},[31676],[34,145227],{"className":145228,"style":33123},[31680],[34,145230,31955],{"className":145231},[31685,31686],[34,145233,31960],{"className":145234},[32119],[34,145236],{"className":145237,"style":32123},[31845],[34,145239,31968],{"className":145240,"style":32153},[31685,31686]," 个元素表示：",[34,145243,145245],{"className":145244},[126385],[34,145246,145248,145296],{"className":145247},[31642],[34,145249,145251],{"className":145250},[31646],[31648,145252,145253],{"xmlns":31650,"display":126394},[31652,145254,145255,145293],{},[31655,145256,145257],{},[126400,145258,145259],{"notation":126402},[71090,145260,145261],{"scriptlevel":459,"displaystyle":702},[71090,145262,145263],{"scriptlevel":459,"displaystyle":702},[71090,145264,145265],{"scriptlevel":459,"displaystyle":758},[31655,145266,145267,145270,145276,145279,145285,145288,145290],{},[39361,145268,145269],{},"从 ",[31743,145271,145272,145274],{},[31658,145273,56782],{},[31658,145275,31955],{},[39361,145277,145278],{}," 到 ",[31743,145280,145281,145283],{},[31658,145282,56782],{},[31658,145284,31968],{},[39361,145286,145287],{}," 的长度为 ",[31658,145289,31759],{},[39361,145291,145292],{}," 的路径条数",[31665,145294,145295],{"encoding":31667},"\\boxed{\\text{从 }v_i\\text{ 到 }v_j\\text{ 的长度为 }m\\text{ 的路径条数}}",[34,145297,145299],{"className":145298,"ariaHidden":758},[31672],[34,145300,145302,145306],{"className":145301},[31676],[34,145303],{"className":145304,"style":145305},[31680],"height:1.6494em;vertical-align:-0.6261em;",[34,145307,145309],{"className":145308},[31685],[34,145310,145312,145475],{"className":145311},[31796,31797],[34,145313,145315,145472],{"className":145314},[31801],[34,145316,145318,145463],{"className":145317,"style":126437},[31805],[34,145319,145321,145325],{"style":145320},"top:-3.6494em;",[34,145322],{"className":145323,"style":145324},[31813],"height:3.6494em;",[34,145326,145328],{"className":145327},[126448],[34,145329,145331],{"className":145330},[31685],[34,145332,145334,145344,145384,145397,145437,145450,145453],{"className":145333},[31685],[34,145335,145337,145341],{"className":145336},[31685,2638],[34,145338,145340],{"className":145339},[31685,105462],"从",[34,145342,130298],{"className":145343},[31685],[34,145345,145347,145350],{"className":145346},[31685],[34,145348,56782],{"className":145349,"style":32062},[31685,31686],[34,145351,145353],{"className":145352},[31792],[34,145354,145356,145376],{"className":145355},[31796,31797],[34,145357,145359,145373],{"className":145358},[31801],[34,145360,145362],{"className":145361,"style":32090},[31805],[34,145363,145364,145367],{"style":38795},[34,145365],{"className":145366,"style":31814},[31813],[34,145368,145370],{"className":145369},[31818,31819,31820,31821],[34,145371,31955],{"className":145372},[31685,31686,31821],[34,145374,31829],{"className":145375},[31828],[34,145377,145379],{"className":145378},[31801],[34,145380,145382],{"className":145381,"style":31836},[31805],[34,145383],{},[34,145385,145387,145390,145394],{"className":145386},[31685,2638],[34,145388,130298],{"className":145389},[31685],[34,145391,145393],{"className":145392},[31685,105462],"到",[34,145395,130298],{"className":145396},[31685],[34,145398,145400,145403],{"className":145399},[31685],[34,145401,56782],{"className":145402,"style":32062},[31685,31686],[34,145404,145406],{"className":145405},[31792],[34,145407,145409,145429],{"className":145408},[31796,31797],[34,145410,145412,145426],{"className":145411},[31801],[34,145413,145415],{"className":145414,"style":32090},[31805],[34,145416,145417,145420],{"style":38795},[34,145418],{"className":145419,"style":31814},[31813],[34,145421,145423],{"className":145422},[31818,31819,31820,31821],[34,145424,31968],{"className":145425,"style":32153},[31685,31686,31821],[34,145427,31829],{"className":145428},[31828],[34,145430,145432],{"className":145431},[31801],[34,145433,145435],{"className":145434,"style":32163},[31805],[34,145436],{},[34,145438,145440,145443,145447],{"className":145439},[31685,2638],[34,145441,130298],{"className":145442},[31685],[34,145444,145446],{"className":145445},[31685,105462],"的长度为",[34,145448,130298],{"className":145449},[31685],[34,145451,31759],{"className":145452},[31685,31686],[34,145454,145456,145459],{"className":145455},[31685,2638],[34,145457,130298],{"className":145458},[31685],[34,145460,145462],{"className":145461},[31685,105462],"的路径条数",[34,145464,145465,145468],{"style":126463},[34,145466],{"className":145467,"style":145324},[31813],[34,145469],{"className":145470,"style":145471},[126470,126471],"height:1.6494em;border-style:solid;border-width:0.04em;",[34,145473,31829],{"className":145474},[31828],[34,145476,145478],{"className":145477},[31801],[34,145479,145482],{"className":145480,"style":145481},[31805],"height:0.6261em;",[34,145483],{},[15,145485,130751],{},[34,145487,145489],{"className":145488},[126385],[34,145490,145492,145509],{"className":145491},[31642],[34,145493,145495],{"className":145494},[31646],[31648,145496,145497],{"xmlns":31650,"display":126394},[31652,145498,145499,145507],{},[31655,145500,145501,145503,145505],{},[31658,145502,31955],{},[31662,145504,202],{},[31658,145506,31968],{},[31665,145508,33094],{"encoding":31667},[34,145510,145512,145530],{"className":145511,"ariaHidden":758},[31672],[34,145513,145515,145518,145521,145524,145527],{"className":145514},[31676],[34,145516],{"className":145517,"style":33104},[31680],[34,145519,31955],{"className":145520},[31685,31686],[34,145522],{"className":145523,"style":31846},[31845],[34,145525,202],{"className":145526},[31850],[34,145528],{"className":145529,"style":31846},[31845],[34,145531,145533,145536],{"className":145532},[31676],[34,145534],{"className":145535,"style":33123},[31680],[34,145537,31968],{"className":145538,"style":32153},[31685,31686],[15,145540,145541,145542,145611,145612,145681,145682,145710],{},"则表示从 ",[34,145543,145545,145562],{"className":145544},[31642],[34,145546,145548],{"className":145547},[31646],[31648,145549,145550],{"xmlns":31650},[31652,145551,145552,145560],{},[31655,145553,145554],{},[31743,145555,145556,145558],{},[31658,145557,56782],{},[31658,145559,31955],{},[31665,145561,56787],{"encoding":31667},[34,145563,145565],{"className":145564,"ariaHidden":758},[31672],[34,145566,145568,145571],{"className":145567},[31676],[34,145569],{"className":145570,"style":31860},[31680],[34,145572,145574,145577],{"className":145573},[31685],[34,145575,56782],{"className":145576,"style":32062},[31685,31686],[34,145578,145580],{"className":145579},[31792],[34,145581,145583,145603],{"className":145582},[31796,31797],[34,145584,145586,145600],{"className":145585},[31801],[34,145587,145589],{"className":145588,"style":32090},[31805],[34,145590,145591,145594],{"style":38795},[34,145592],{"className":145593,"style":31814},[31813],[34,145595,145597],{"className":145596},[31818,31819,31820,31821],[34,145598,31955],{"className":145599},[31685,31686,31821],[34,145601,31829],{"className":145602},[31828],[34,145604,145606],{"className":145605},[31801],[34,145607,145609],{"className":145608,"style":31836},[31805],[34,145610],{}," 出发又回到 ",[34,145613,145615,145632],{"className":145614},[31642],[34,145616,145618],{"className":145617},[31646],[31648,145619,145620],{"xmlns":31650},[31652,145621,145622,145630],{},[31655,145623,145624],{},[31743,145625,145626,145628],{},[31658,145627,56782],{},[31658,145629,31955],{},[31665,145631,56787],{"encoding":31667},[34,145633,145635],{"className":145634,"ariaHidden":758},[31672],[34,145636,145638,145641],{"className":145637},[31676],[34,145639],{"className":145640,"style":31860},[31680],[34,145642,145644,145647],{"className":145643},[31685],[34,145645,56782],{"className":145646,"style":32062},[31685,31686],[34,145648,145650],{"className":145649},[31792],[34,145651,145653,145673],{"className":145652},[31796,31797],[34,145654,145656,145670],{"className":145655},[31801],[34,145657,145659],{"className":145658,"style":32090},[31805],[34,145660,145661,145664],{"style":38795},[34,145662],{"className":145663,"style":31814},[31813],[34,145665,145667],{"className":145666},[31818,31819,31820,31821],[34,145668,31955],{"className":145669},[31685,31686,31821],[34,145671,31829],{"className":145672},[31828],[34,145674,145676],{"className":145675},[31801],[34,145677,145679],{"className":145678,"style":31836},[31805],[34,145680],{}," 的长度为 ",[34,145683,145685,145698],{"className":145684},[31642],[34,145686,145688],{"className":145687},[31646],[31648,145689,145690],{"xmlns":31650},[31652,145691,145692,145696],{},[31655,145693,145694],{},[31658,145695,31759],{},[31665,145697,31759],{"encoding":31667},[34,145699,145701],{"className":145700,"ariaHidden":758},[31672],[34,145702,145704,145707],{"className":145703},[31676],[34,145705],{"className":145706,"style":32527},[31680],[34,145708,31759],{"className":145709},[31685,31686]," 的循环条数。",[32188,145712],{},[19,145714,145715],{"id":145715},"例子理解",[15,145717,130751],{},[34,145719,145721],{"className":145720},[126385],[34,145722,145724,145754],{"className":145723},[31642],[34,145725,145727],{"className":145726},[31646],[31648,145728,145729],{"xmlns":31650,"display":126394},[31652,145730,145731,145751],{},[31655,145732,145733,145735,145741,145747,145749],{},[31662,145734,456],{"stretchy":702},[37624,145736,145737,145739],{},[31658,145738,54151],{},[31749,145740,786],{},[31743,145742,145743,145745],{},[31662,145744,2557],{"stretchy":702},[31749,145746,96087],{},[31662,145748,202],{},[31749,145750,18136],{},[31665,145752,145753],{"encoding":31667},"(A^3)_{24}=5",[34,145755,145757,145848],{"className":145756,"ariaHidden":758},[31672],[34,145758,145760,145764,145767,145796,145839,145842,145845],{"className":145759},[31676],[34,145761],{"className":145762,"style":145763},[31680],"height:1.1141em;vertical-align:-0.25em;",[34,145765,456],{"className":145766},[31782],[34,145768,145770,145773],{"className":145769},[31685],[34,145771,54151],{"className":145772},[31685,31686],[34,145774,145776],{"className":145775},[31792],[34,145777,145779],{"className":145778},[31796],[34,145780,145782],{"className":145781},[31801],[34,145783,145785],{"className":145784,"style":129832},[31805],[34,145786,145787,145790],{"style":129846},[34,145788],{"className":145789,"style":31814},[31813],[34,145791,145793],{"className":145792},[31818,31819,31820,31821],[34,145794,786],{"className":145795},[31685,31821],[34,145797,145799,145802],{"className":145798},[31690],[34,145800,2557],{"className":145801},[31690],[34,145803,145805],{"className":145804},[31792],[34,145806,145808,145831],{"className":145807},[31796,31797],[34,145809,145811,145828],{"className":145810},[31801],[34,145812,145814],{"className":145813,"style":31806},[31805],[34,145815,145816,145819],{"style":31809},[34,145817],{"className":145818,"style":31814},[31813],[34,145820,145822],{"className":145821},[31818,31819,31820,31821],[34,145823,145825],{"className":145824},[31685,31821],[34,145826,96087],{"className":145827},[31685,31821],[34,145829,31829],{"className":145830},[31828],[34,145832,145834],{"className":145833},[31801],[34,145835,145837],{"className":145836,"style":31836},[31805],[34,145838],{},[34,145840],{"className":145841,"style":31846},[31845],[34,145843,202],{"className":145844},[31850],[34,145846],{"className":145847,"style":31846},[31845],[34,145849,145851,145854],{"className":145850},[31676],[34,145852],{"className":145853,"style":32617},[31680],[34,145855,18136],{"className":145856},[31685],[15,145858,145859],{},"表示：",[104833,145861,145862],{},[15,145863,135477,145864,53834,145934,146004],{},[34,145865,145867,145885],{"className":145866},[31642],[34,145868,145870],{"className":145869},[31646],[31648,145871,145872],{"xmlns":31650},[31652,145873,145874,145882],{},[31655,145875,145876],{},[31743,145877,145878,145880],{},[31658,145879,56782],{},[31749,145881,768],{},[31665,145883,145884],{"encoding":31667},"v_2",[34,145886,145888],{"className":145887,"ariaHidden":758},[31672],[34,145889,145891,145894],{"className":145890},[31676],[34,145892],{"className":145893,"style":31860},[31680],[34,145895,145897,145900],{"className":145896},[31685],[34,145898,56782],{"className":145899,"style":32062},[31685,31686],[34,145901,145903],{"className":145902},[31792],[34,145904,145906,145926],{"className":145905},[31796,31797],[34,145907,145909,145923],{"className":145908},[31801],[34,145910,145912],{"className":145911,"style":31806},[31805],[34,145913,145914,145917],{"style":38795},[34,145915],{"className":145916,"style":31814},[31813],[34,145918,145920],{"className":145919},[31818,31819,31820,31821],[34,145921,768],{"className":145922},[31685,31821],[34,145924,31829],{"className":145925},[31828],[34,145927,145929],{"className":145928},[31801],[34,145930,145932],{"className":145931,"style":31836},[31805],[34,145933],{},[34,145935,145937,145955],{"className":145936},[31642],[34,145938,145940],{"className":145939},[31646],[31648,145941,145942],{"xmlns":31650},[31652,145943,145944,145952],{},[31655,145945,145946],{},[31743,145947,145948,145950],{},[31658,145949,56782],{},[31749,145951,15932],{},[31665,145953,145954],{"encoding":31667},"v_4",[34,145956,145958],{"className":145957,"ariaHidden":758},[31672],[34,145959,145961,145964],{"className":145960},[31676],[34,145962],{"className":145963,"style":31860},[31680],[34,145965,145967,145970],{"className":145966},[31685],[34,145968,56782],{"className":145969,"style":32062},[31685,31686],[34,145971,145973],{"className":145972},[31792],[34,145974,145976,145996],{"className":145975},[31796,31797],[34,145977,145979,145993],{"className":145978},[31801],[34,145980,145982],{"className":145981,"style":31806},[31805],[34,145983,145984,145987],{"style":38795},[34,145985],{"className":145986,"style":31814},[31813],[34,145988,145990],{"className":145989},[31818,31819,31820,31821],[34,145991,15932],{"className":145992},[31685,31821],[34,145994,31829],{"className":145995},[31828],[34,145997,145999],{"className":145998},[31801],[34,146000,146002],{"className":146001,"style":31836},[31805],[34,146003],{}," 的长度为 3 的路径有 5 条。",[32188,146006],{},[11,146008,146010],{"id":146009},"四可达矩阵","四、可达矩阵",[15,146012,146013],{},"可达矩阵只关心：",[104833,146015,146016],{},[15,146017,146018],{},"能不能到达。",[15,146020,146021],{},"不关心有几条路径，也不关心路径长度。",[15,146023,146024,146025,146053],{},"设有向图 ",[34,146026,146028,146041],{"className":146027},[31642],[34,146029,146031],{"className":146030},[31646],[31648,146032,146033],{"xmlns":31650},[31652,146034,146035,146039],{},[31655,146036,146037],{},[31658,146038,97709],{},[31665,146040,97709],{"encoding":31667},[34,146042,146044],{"className":146043,"ariaHidden":758},[31672],[34,146045,146047,146050],{"className":146046},[31676],[34,146048],{"className":146049,"style":33218},[31680],[34,146051,97709],{"className":146052},[31685,31686]," 的可达矩阵为：",[34,146055,146057],{"className":146056},[126385],[34,146058,146060,146090],{"className":146059},[31642],[34,146061,146063],{"className":146062},[31646],[31648,146064,146065],{"xmlns":31650,"display":126394},[31652,146066,146067,146087],{},[31655,146068,146069,146071,146073,146075,146085],{},[31658,146070,105852],{},[31662,146072,202],{},[31662,146074,456],{"stretchy":702},[31743,146076,146077,146079],{},[31658,146078,15],{},[31655,146080,146081,146083],{},[31658,146082,31955],{},[31658,146084,31968],{},[31662,146086,2557],{"stretchy":702},[31665,146088,146089],{"encoding":31667},"P=(p_{ij})",[34,146091,146093,146111],{"className":146092,"ariaHidden":758},[31672],[34,146094,146096,146099,146102,146105,146108],{"className":146095},[31676],[34,146097],{"className":146098,"style":33218},[31680],[34,146100,105852],{"className":146101,"style":37422},[31685,31686],[34,146103],{"className":146104,"style":31846},[31845],[34,146106,202],{"className":146107},[31850],[34,146109],{"className":146110,"style":31846},[31845],[34,146112,146114,146117,146120,146163],{"className":146113},[31676],[34,146115],{"className":146116,"style":32058},[31680],[34,146118,456],{"className":146119},[31782],[34,146121,146123,146126],{"className":146122},[31685],[34,146124,15],{"className":146125},[31685,31686],[34,146127,146129],{"className":146128},[31792],[34,146130,146132,146155],{"className":146131},[31796,31797],[34,146133,146135,146152],{"className":146134},[31801],[34,146136,146138],{"className":146137,"style":32090},[31805],[34,146139,146140,146143],{"style":31809},[34,146141],{"className":146142,"style":31814},[31813],[34,146144,146146],{"className":146145},[31818,31819,31820,31821],[34,146147,146149],{"className":146148},[31685,31821],[34,146150,141027],{"className":146151,"style":32153},[31685,31686,31821],[34,146153,31829],{"className":146154},[31828],[34,146156,146158],{"className":146157},[31801],[34,146159,146161],{"className":146160,"style":32163},[31805],[34,146162],{},[34,146164,2557],{"className":146165},[31690],[15,146167,104944],{},[34,146169,146171],{"className":146170},[126385],[34,146172,146174,146275],{"className":146173},[31642],[34,146175,146177],{"className":146176},[31646],[31648,146178,146179],{"xmlns":31650,"display":126394},[31652,146180,146181,146272],{},[31655,146182,146183,146193,146195],{},[31743,146184,146185,146187],{},[31658,146186,15],{},[31655,146188,146189,146191],{},[31658,146190,31955],{},[31658,146192,31968],{},[31662,146194,202],{},[31655,146196,146197,146199],{},[31662,146198,32685],{"fence":758},[71078,146200,146201],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,146202,146203,146213,146249],{},[71087,146204,146205],{},[71090,146206,146207],{"scriptlevel":459,"displaystyle":702},[31655,146208,146209,146211],{},[31749,146210,640],{},[31662,146212,31960],{"separator":758},[71087,146214,146215],{},[71090,146216,146217],{"scriptlevel":459,"displaystyle":702},[31655,146218,146219,146221,146223,146225,146228,146234,146237,146243,146245,146247],{},[31658,146220,31955],{},[31662,146222,202],{},[31658,146224,31968],{},[39361,146226,146227],{}," 或 ",[31743,146229,146230,146232],{},[31658,146231,56782],{},[31658,146233,31968],{},[39361,146235,146236],{}," 从 ",[31743,146238,146239,146241],{},[31658,146240,56782],{},[31658,146242,31955],{},[39361,146244,139789],{},[31749,146246,459],{},[31662,146248,31960],{"separator":758},[71087,146250,146251],{},[71090,146252,146253],{"scriptlevel":459,"displaystyle":702},[31655,146254,146255,146261,146263,146269],{},[31743,146256,146257,146259],{},[31658,146258,56782],{},[31658,146260,31968],{},[39361,146262,146236],{},[31743,146264,146265,146267],{},[31658,146266,56782],{},[31658,146268,31955],{},[39361,146270,146271],{}," 不可达",[31665,146273,146274],{"encoding":31667},"p_{ij}=\n\\begin{cases}\n1,& i=j\\text{ 或 }v_j\\text{ 从 }v_i\\text{ 可达}\\\n0,& v_j\\text{ 从 }v_i\\text{ 不可达}\n\\end{cases}",[34,146276,146278,146336],{"className":146277,"ariaHidden":758},[31672],[34,146279,146281,146284,146327,146330,146333],{"className":146280},[31676],[34,146282],{"className":146283,"style":56005},[31680],[34,146285,146287,146290],{"className":146286},[31685],[34,146288,15],{"className":146289},[31685,31686],[34,146291,146293],{"className":146292},[31792],[34,146294,146296,146319],{"className":146295},[31796,31797],[34,146297,146299,146316],{"className":146298},[31801],[34,146300,146302],{"className":146301,"style":32090},[31805],[34,146303,146304,146307],{"style":31809},[34,146305],{"className":146306,"style":31814},[31813],[34,146308,146310],{"className":146309},[31818,31819,31820,31821],[34,146311,146313],{"className":146312},[31685,31821],[34,146314,141027],{"className":146315,"style":32153},[31685,31686,31821],[34,146317,31829],{"className":146318},[31828],[34,146320,146322],{"className":146321},[31801],[34,146323,146325],{"className":146324,"style":32163},[31805],[34,146326],{},[34,146328],{"className":146329,"style":31846},[31845],[34,146331,202],{"className":146332},[31850],[34,146334],{"className":146335,"style":31846},[31845],[34,146337,146339,146342],{"className":146338},[31676],[34,146340],{"className":146341,"style":141282},[31680],[34,146343,146345,146351,146701],{"className":146344},[34006],[34,146346,146348],{"className":146347,"style":75811},[31782,75810],[34,146349,32685],{"className":146350},[71281,141292],[34,146352,146354],{"className":146353},[31685],[34,146355,146357,146394,146397,146565,146568],{"className":146356},[71078],[34,146358,146360],{"className":146359},[71384],[34,146361,146363,146386],{"className":146362},[31796,31797],[34,146364,146366,146383],{"className":146365},[31801],[34,146367,146369],{"className":146368,"style":141311},[31805],[34,146370,146371,146374],{"style":78184},[34,146372],{"className":146373,"style":71401},[31813],[34,146375,146377,146380],{"className":146376},[31685],[34,146378,640],{"className":146379},[31685],[34,146381,31960],{"className":146382},[32119],[34,146384,31829],{"className":146385},[31828],[34,146387,146389],{"className":146388},[31801],[34,146390,146392],{"className":146391,"style":141335},[31805],[34,146393],{},[34,146395],{"className":146396,"style":71733},[71732],[34,146398,146400],{"className":146399},[71384],[34,146401,146403,146557],{"className":146402},[31796,31797],[34,146404,146406,146554],{"className":146405},[31801],[34,146407,146409],{"className":146408,"style":141311},[31805],[34,146410,146411,146414],{"style":78184},[34,146412],{"className":146413,"style":71401},[31813],[34,146415,146417,146420,146423,146426,146429,146432,146444,146484,146496,146536,146545,146548,146551],{"className":146416},[31685],[34,146418,31955],{"className":146419},[31685,31686],[34,146421],{"className":146422,"style":31846},[31845],[34,146424,202],{"className":146425},[31850],[34,146427],{"className":146428,"style":31846},[31845],[34,146430,31968],{"className":146431,"style":32153},[31685,31686],[34,146433,146435,146438,146441],{"className":146434},[31685,2638],[34,146436,130298],{"className":146437},[31685],[34,146439,58269],{"className":146440},[31685,105462],[34,146442,130298],{"className":146443},[31685],[34,146445,146447,146450],{"className":146446},[31685],[34,146448,56782],{"className":146449,"style":32062},[31685,31686],[34,146451,146453],{"className":146452},[31792],[34,146454,146456,146476],{"className":146455},[31796,31797],[34,146457,146459,146473],{"className":146458},[31801],[34,146460,146462],{"className":146461,"style":32090},[31805],[34,146463,146464,146467],{"style":38795},[34,146465],{"className":146466,"style":31814},[31813],[34,146468,146470],{"className":146469},[31818,31819,31820,31821],[34,146471,31968],{"className":146472,"style":32153},[31685,31686,31821],[34,146474,31829],{"className":146475},[31828],[34,146477,146479],{"className":146478},[31801],[34,146480,146482],{"className":146481,"style":32163},[31805],[34,146483],{},[34,146485,146487,146490,146493],{"className":146486},[31685,2638],[34,146488,130298],{"className":146489},[31685],[34,146491,145340],{"className":146492},[31685,105462],[34,146494,130298],{"className":146495},[31685],[34,146497,146499,146502],{"className":146498},[31685],[34,146500,56782],{"className":146501,"style":32062},[31685,31686],[34,146503,146505],{"className":146504},[31792],[34,146506,146508,146528],{"className":146507},[31796,31797],[34,146509,146511,146525],{"className":146510},[31801],[34,146512,146514],{"className":146513,"style":32090},[31805],[34,146515,146516,146519],{"style":38795},[34,146517],{"className":146518,"style":31814},[31813],[34,146520,146522],{"className":146521},[31818,31819,31820,31821],[34,146523,31955],{"className":146524},[31685,31686,31821],[34,146526,31829],{"className":146527},[31828],[34,146529,146531],{"className":146530},[31801],[34,146532,146534],{"className":146533,"style":31836},[31805],[34,146535],{},[34,146537,146539,146542],{"className":146538},[31685,2638],[34,146540,130298],{"className":146541},[31685],[34,146543,139816],{"className":146544},[31685,105462],[34,146546,130298],{"className":146547},[31845],[34,146549,459],{"className":146550},[31685],[34,146552,31960],{"className":146553},[32119],[34,146555,31829],{"className":146556},[31828],[34,146558,146560],{"className":146559},[31801],[34,146561,146563],{"className":146562,"style":141335},[31805],[34,146564],{},[34,146566],{"className":146567,"style":141477},[71732],[34,146569,146571],{"className":146570},[141481],[34,146572,146574,146693],{"className":146573},[31796,31797],[34,146575,146577,146690],{"className":146576},[31801],[34,146578,146580],{"className":146579,"style":141311},[31805],[34,146581,146582,146585],{"style":78184},[34,146583],{"className":146584,"style":71401},[31813],[34,146586,146588,146628,146640,146680],{"className":146587},[31685],[34,146589,146591,146594],{"className":146590},[31685],[34,146592,56782],{"className":146593,"style":32062},[31685,31686],[34,146595,146597],{"className":146596},[31792],[34,146598,146600,146620],{"className":146599},[31796,31797],[34,146601,146603,146617],{"className":146602},[31801],[34,146604,146606],{"className":146605,"style":32090},[31805],[34,146607,146608,146611],{"style":38795},[34,146609],{"className":146610,"style":31814},[31813],[34,146612,146614],{"className":146613},[31818,31819,31820,31821],[34,146615,31968],{"className":146616,"style":32153},[31685,31686,31821],[34,146618,31829],{"className":146619},[31828],[34,146621,146623],{"className":146622},[31801],[34,146624,146626],{"className":146625,"style":32163},[31805],[34,146627],{},[34,146629,146631,146634,146637],{"className":146630},[31685,2638],[34,146632,130298],{"className":146633},[31685],[34,146635,145340],{"className":146636},[31685,105462],[34,146638,130298],{"className":146639},[31685],[34,146641,146643,146646],{"className":146642},[31685],[34,146644,56782],{"className":146645,"style":32062},[31685,31686],[34,146647,146649],{"className":146648},[31792],[34,146650,146652,146672],{"className":146651},[31796,31797],[34,146653,146655,146669],{"className":146654},[31801],[34,146656,146658],{"className":146657,"style":32090},[31805],[34,146659,146660,146663],{"style":38795},[34,146661],{"className":146662,"style":31814},[31813],[34,146664,146666],{"className":146665},[31818,31819,31820,31821],[34,146667,31955],{"className":146668},[31685,31686,31821],[34,146670,31829],{"className":146671},[31828],[34,146673,146675],{"className":146674},[31801],[34,146676,146678],{"className":146677,"style":31836},[31805],[34,146679],{},[34,146681,146683,146686],{"className":146682},[31685,2638],[34,146684,130298],{"className":146685},[31685],[34,146687,146689],{"className":146688},[31685,105462],"不可达",[34,146691,31829],{"className":146692},[31828],[34,146694,146696],{"className":146695},[31801],[34,146697,146699],{"className":146698,"style":141335},[31805],[34,146700],{},[34,146702],{"className":146703},[31690,41488],[32188,146705],{},[19,146707,146709],{"id":146708},"_1-可达矩阵的含义","1. 可达矩阵的含义",[15,146711,130751],{},[34,146713,146715],{"className":146714},[126385],[34,146716,146718,146744],{"className":146717},[31642],[34,146719,146721],{"className":146720},[31646],[31648,146722,146723],{"xmlns":31650,"display":126394},[31652,146724,146725,146741],{},[31655,146726,146727,146737,146739],{},[31743,146728,146729,146731],{},[31658,146730,15],{},[31655,146732,146733,146735],{},[31658,146734,31955],{},[31658,146736,31968],{},[31662,146738,202],{},[31749,146740,640],{},[31665,146742,146743],{"encoding":31667},"p_{ij}=1",[34,146745,146747,146805],{"className":146746,"ariaHidden":758},[31672],[34,146748,146750,146753,146796,146799,146802],{"className":146749},[31676],[34,146751],{"className":146752,"style":56005},[31680],[34,146754,146756,146759],{"className":146755},[31685],[34,146757,15],{"className":146758},[31685,31686],[34,146760,146762],{"className":146761},[31792],[34,146763,146765,146788],{"className":146764},[31796,31797],[34,146766,146768,146785],{"className":146767},[31801],[34,146769,146771],{"className":146770,"style":32090},[31805],[34,146772,146773,146776],{"style":31809},[34,146774],{"className":146775,"style":31814},[31813],[34,146777,146779],{"className":146778},[31818,31819,31820,31821],[34,146780,146782],{"className":146781},[31685,31821],[34,146783,141027],{"className":146784,"style":32153},[31685,31686,31821],[34,146786,31829],{"className":146787},[31828],[34,146789,146791],{"className":146790},[31801],[34,146792,146794],{"className":146793,"style":32163},[31805],[34,146795],{},[34,146797],{"className":146798,"style":31846},[31845],[34,146800,202],{"className":146801},[31850],[34,146803],{"className":146804,"style":31846},[31845],[34,146806,146808,146811],{"className":146807},[31676],[34,146809],{"className":146810,"style":32617},[31680],[34,146812,640],{"className":146813},[31685],[15,146815,145859],{},[34,146817,146819],{"className":146818},[126385],[34,146820,146822,146848],{"className":146821},[31642],[34,146823,146825],{"className":146824},[31646],[31648,146826,146827],{"xmlns":31650,"display":126394},[31652,146828,146829,146845],{},[31655,146830,146831,146837,146839],{},[31743,146832,146833,146835],{},[31658,146834,56782],{},[31658,146836,31955],{},[31662,146838,33239],{},[31743,146840,146841,146843],{},[31658,146842,56782],{},[31658,146844,31968],{},[31665,146846,146847],{"encoding":31667},"v_i\\to v_j",[34,146849,146851,146906],{"className":146850,"ariaHidden":758},[31672],[34,146852,146854,146857,146897,146900,146903],{"className":146853},[31676],[34,146855],{"className":146856,"style":31860},[31680],[34,146858,146860,146863],{"className":146859},[31685],[34,146861,56782],{"className":146862,"style":32062},[31685,31686],[34,146864,146866],{"className":146865},[31792],[34,146867,146869,146889],{"className":146868},[31796,31797],[34,146870,146872,146886],{"className":146871},[31801],[34,146873,146875],{"className":146874,"style":32090},[31805],[34,146876,146877,146880],{"style":38795},[34,146878],{"className":146879,"style":31814},[31813],[34,146881,146883],{"className":146882},[31818,31819,31820,31821],[34,146884,31955],{"className":146885},[31685,31686,31821],[34,146887,31829],{"className":146888},[31828],[34,146890,146892],{"className":146891},[31801],[34,146893,146895],{"className":146894,"style":31836},[31805],[34,146896],{},[34,146898],{"className":146899,"style":31846},[31845],[34,146901,33239],{"className":146902},[31850],[34,146904],{"className":146905,"style":31846},[31845],[34,146907,146909,146912],{"className":146908},[31676],[34,146910],{"className":146911,"style":56005},[31680],[34,146913,146915,146918],{"className":146914},[31685],[34,146916,56782],{"className":146917,"style":32062},[31685,31686],[34,146919,146921],{"className":146920},[31792],[34,146922,146924,146944],{"className":146923},[31796,31797],[34,146925,146927,146941],{"className":146926},[31801],[34,146928,146930],{"className":146929,"style":32090},[31805],[34,146931,146932,146935],{"style":38795},[34,146933],{"className":146934,"style":31814},[31813],[34,146936,146938],{"className":146937},[31818,31819,31820,31821],[34,146939,31968],{"className":146940,"style":32153},[31685,31686,31821],[34,146942,31829],{"className":146943},[31828],[34,146945,146947],{"className":146946},[31801],[34,146948,146950],{"className":146949,"style":32163},[31805],[34,146951],{},[15,146953,137325],{},[15,146955,130751],{},[34,146957,146959],{"className":146958},[126385],[34,146960,146962,146988],{"className":146961},[31642],[34,146963,146965],{"className":146964},[31646],[31648,146966,146967],{"xmlns":31650,"display":126394},[31652,146968,146969,146985],{},[31655,146970,146971,146981,146983],{},[31743,146972,146973,146975],{},[31658,146974,15],{},[31655,146976,146977,146979],{},[31658,146978,31955],{},[31658,146980,31968],{},[31662,146982,202],{},[31749,146984,459],{},[31665,146986,146987],{"encoding":31667},"p_{ij}=0",[34,146989,146991,147049],{"className":146990,"ariaHidden":758},[31672],[34,146992,146994,146997,147040,147043,147046],{"className":146993},[31676],[34,146995],{"className":146996,"style":56005},[31680],[34,146998,147000,147003],{"className":146999},[31685],[34,147001,15],{"className":147002},[31685,31686],[34,147004,147006],{"className":147005},[31792],[34,147007,147009,147032],{"className":147008},[31796,31797],[34,147010,147012,147029],{"className":147011},[31801],[34,147013,147015],{"className":147014,"style":32090},[31805],[34,147016,147017,147020],{"style":31809},[34,147018],{"className":147019,"style":31814},[31813],[34,147021,147023],{"className":147022},[31818,31819,31820,31821],[34,147024,147026],{"className":147025},[31685,31821],[34,147027,141027],{"className":147028,"style":32153},[31685,31686,31821],[34,147030,31829],{"className":147031},[31828],[34,147033,147035],{"className":147034},[31801],[34,147036,147038],{"className":147037,"style":32163},[31805],[34,147039],{},[34,147041],{"className":147042,"style":31846},[31845],[34,147044,202],{"className":147045},[31850],[34,147047],{"className":147048,"style":31846},[31845],[34,147050,147052,147055],{"className":147051},[31676],[34,147053],{"className":147054,"style":32617},[31680],[34,147056,459],{"className":147057},[31685],[15,147059,147060],{},"表示不可达。",[32188,147062],{},[19,147064,147066],{"id":147065},"_2-用邻接矩阵求可达矩阵","2. 用邻接矩阵求可达矩阵",[15,147068,136974,147069,147097,147098,147148],{},[34,147070,147072,147085],{"className":147071},[31642],[34,147073,147075],{"className":147074},[31646],[31648,147076,147077],{"xmlns":31650},[31652,147078,147079,147083],{},[31655,147080,147081],{},[31658,147082,31660],{},[31665,147084,31660],{"encoding":31667},[34,147086,147088],{"className":147087,"ariaHidden":758},[31672],[34,147089,147091,147094],{"className":147090},[31676],[34,147092],{"className":147093,"style":32527},[31680],[34,147095,31660],{"className":147096},[31685,31686]," 个顶点，则只需考虑长度不超过 ",[34,147099,147101,147118],{"className":147100},[31642],[34,147102,147104],{"className":147103},[31646],[31648,147105,147106],{"xmlns":31650},[31652,147107,147108,147116],{},[31655,147109,147110,147112,147114],{},[31658,147111,31660],{},[31662,147113,32226],{},[31749,147115,640],{},[31665,147117,32587],{"encoding":31667},[34,147119,147121,147139],{"className":147120,"ariaHidden":758},[31672],[34,147122,147124,147127,147130,147133,147136],{"className":147123},[31676],[34,147125],{"className":147126,"style":32597},[31680],[34,147128,31660],{"className":147129},[31685,31686],[34,147131],{"className":147132,"style":32604},[31845],[34,147134,32226],{"className":147135},[32024],[34,147137],{"className":147138,"style":32604},[31845],[34,147140,147142,147145],{"className":147141},[31676],[34,147143],{"className":147144,"style":32617},[31680],[34,147146,640],{"className":147147},[31685]," 的基本路径。",[15,147150,147151],{},"所以可以用布尔矩阵运算：",[34,147153,147155],{"className":147154},[126385],[34,147156,147158,147217],{"className":147157},[31642],[34,147159,147161],{"className":147160},[31646],[31648,147162,147163],{"xmlns":31650,"display":126394},[31652,147164,147165,147214],{},[31655,147166,147167,147169,147171,147173,147176,147178,147180,147192,147194,147196,147198],{},[31658,147168,105852],{},[31662,147170,202],{},[31658,147172,106063],{},[31662,147174,147175],{},"∨",[31658,147177,54151],{},[31662,147179,147175],{},[37624,147181,147182,147184],{},[31658,147183,54151],{},[31655,147185,147186,147188,147190],{},[31662,147187,456],{"stretchy":702},[31749,147189,768],{},[31662,147191,2557],{"stretchy":702},[31662,147193,147175],{},[31662,147195,38874],{},[31662,147197,147175],{},[37624,147199,147200,147202],{},[31658,147201,54151],{},[31655,147203,147204,147206,147208,147210,147212],{},[31662,147205,456],{"stretchy":702},[31658,147207,31660],{},[31662,147209,32226],{},[31749,147211,640],{},[31662,147213,2557],{"stretchy":702},[31665,147215,147216],{"encoding":31667},"P=I\\lor A\\lor A^{(2)}\\lor\\cdots\\lor A^{(n-1)}",[34,147218,147220,147238,147256,147274,147328,147347],{"className":147219,"ariaHidden":758},[31672],[34,147221,147223,147226,147229,147232,147235],{"className":147222},[31676],[34,147224],{"className":147225,"style":33218},[31680],[34,147227,105852],{"className":147228,"style":37422},[31685,31686],[34,147230],{"className":147231,"style":31846},[31845],[34,147233,202],{"className":147234},[31850],[34,147236],{"className":147237,"style":31846},[31845],[34,147239,147241,147244,147247,147250,147253],{"className":147240},[31676],[34,147242],{"className":147243,"style":33218},[31680],[34,147245,106063],{"className":147246,"style":67482},[31685,31686],[34,147248],{"className":147249,"style":32604},[31845],[34,147251,147175],{"className":147252},[32024],[34,147254],{"className":147255,"style":32604},[31845],[34,147257,147259,147262,147265,147268,147271],{"className":147258},[31676],[34,147260],{"className":147261,"style":33218},[31680],[34,147263,54151],{"className":147264},[31685,31686],[34,147266],{"className":147267,"style":32604},[31845],[34,147269,147175],{"className":147270},[32024],[34,147272],{"className":147273,"style":32604},[31845],[34,147275,147277,147281,147319,147322,147325],{"className":147276},[31676],[34,147278],{"className":147279,"style":147280},[31680],"height:0.938em;",[34,147282,147284,147287],{"className":147283},[31685],[34,147285,54151],{"className":147286},[31685,31686],[34,147288,147290],{"className":147289},[31792],[34,147291,147293],{"className":147292},[31796],[34,147294,147296],{"className":147295},[31801],[34,147297,147299],{"className":147298,"style":147280},[31805],[34,147300,147301,147304],{"style":129846},[34,147302],{"className":147303,"style":31814},[31813],[34,147305,147307],{"className":147306},[31818,31819,31820,31821],[34,147308,147310,147313,147316],{"className":147309},[31685,31821],[34,147311,456],{"className":147312},[31782,31821],[34,147314,768],{"className":147315},[31685,31821],[34,147317,2557],{"className":147318},[31690,31821],[34,147320],{"className":147321,"style":32604},[31845],[34,147323,147175],{"className":147324},[32024],[34,147326],{"className":147327,"style":32604},[31845],[34,147329,147331,147335,147338,147341,147344],{"className":147330},[31676],[34,147332],{"className":147333,"style":147334},[31680],"height:0.5556em;",[34,147336,38874],{"className":147337},[34006],[34,147339],{"className":147340,"style":32604},[31845],[34,147342,147175],{"className":147343},[32024],[34,147345],{"className":147346,"style":32604},[31845],[34,147348,147350,147353],{"className":147349},[31676],[34,147351],{"className":147352,"style":147280},[31680],[34,147354,147356,147359],{"className":147355},[31685],[34,147357,54151],{"className":147358},[31685,31686],[34,147360,147362],{"className":147361},[31792],[34,147363,147365],{"className":147364},[31796],[34,147366,147368],{"className":147367},[31801],[34,147369,147371],{"className":147370,"style":147280},[31805],[34,147372,147373,147376],{"style":129846},[34,147374],{"className":147375,"style":31814},[31813],[34,147377,147379],{"className":147378},[31818,31819,31820,31821],[34,147380,147382,147385,147388,147391,147394],{"className":147381},[31685,31821],[34,147383,456],{"className":147384},[31782,31821],[34,147386,31660],{"className":147387},[31685,31686,31821],[34,147389,32226],{"className":147390},[32024,31821],[34,147392,640],{"className":147393},[31685,31821],[34,147395,2557],{"className":147396},[31690,31821],[15,147398,104944],{},[105,147400,147401,147432,147538],{},[108,147402,147403,147431],{},[34,147404,147406,147419],{"className":147405},[31642],[34,147407,147409],{"className":147408},[31646],[31648,147410,147411],{"xmlns":31650},[31652,147412,147413,147417],{},[31655,147414,147415],{},[31658,147416,106063],{},[31665,147418,106063],{"encoding":31667},[34,147420,147422],{"className":147421,"ariaHidden":758},[31672],[34,147423,147425,147428],{"className":147424},[31676],[34,147426],{"className":147427,"style":33218},[31680],[34,147429,106063],{"className":147430,"style":67482},[31685,31686],"：单位矩阵，表示自己到自己可达；",[108,147433,147434,147508,147509,147537],{},[34,147435,147437,147461],{"className":147436},[31642],[34,147438,147440],{"className":147439},[31646],[31648,147441,147442],{"xmlns":31650},[31652,147443,147444,147458],{},[31655,147445,147446],{},[37624,147447,147448,147450],{},[31658,147449,54151],{},[31655,147451,147452,147454,147456],{},[31662,147453,456],{"stretchy":702},[31658,147455,33759],{},[31662,147457,2557],{"stretchy":702},[31665,147459,147460],{"encoding":31667},"A^{(k)}",[34,147462,147464],{"className":147463,"ariaHidden":758},[31672],[34,147465,147467,147470],{"className":147466},[31676],[34,147468],{"className":147469,"style":61037},[31680],[34,147471,147473,147476],{"className":147472},[31685],[34,147474,54151],{"className":147475},[31685,31686],[34,147477,147479],{"className":147478},[31792],[34,147480,147482],{"className":147481},[31796],[34,147483,147485],{"className":147484},[31801],[34,147486,147488],{"className":147487,"style":61037},[31805],[34,147489,147490,147493],{"style":37777},[34,147491],{"className":147492,"style":31814},[31813],[34,147494,147496],{"className":147495},[31818,31819,31820,31821],[34,147497,147499,147502,147505],{"className":147498},[31685,31821],[34,147500,456],{"className":147501},[31782,31821],[34,147503,33759],{"className":147504,"style":33774},[31685,31686,31821],[34,147506,2557],{"className":147507},[31690,31821],"：布尔意义下的 ",[34,147510,147512,147525],{"className":147511},[31642],[34,147513,147515],{"className":147514},[31646],[31648,147516,147517],{"xmlns":31650},[31652,147518,147519,147523],{},[31655,147520,147521],{},[31658,147522,33759],{},[31665,147524,33759],{"encoding":31667},[34,147526,147528],{"className":147527,"ariaHidden":758},[31672],[34,147529,147531,147534],{"className":147530},[31676],[34,147532],{"className":147533,"style":31681},[31680],[34,147535,33759],{"className":147536,"style":33774},[31685,31686]," 次幂；",[108,147539,147540,147569],{},[34,147541,147543,147557],{"className":147542},[31642],[34,147544,147546],{"className":147545},[31646],[31648,147547,147548],{"xmlns":31650},[31652,147549,147550,147554],{},[31655,147551,147552],{},[31662,147553,147175],{},[31665,147555,147556],{"encoding":31667},"\\lor",[34,147558,147560],{"className":147559,"ariaHidden":758},[31672],[34,147561,147563,147566],{"className":147562},[31676],[34,147564],{"className":147565,"style":147334},[31680],[34,147567,147175],{"className":147568},[31685],"：布尔或。",[15,147571,147572],{},"也可以先求普通幂：",[34,147574,147576],{"className":147575},[126385],[34,147577,147579,147619],{"className":147578},[31642],[34,147580,147582],{"className":147581},[31646],[31648,147583,147584],{"xmlns":31650,"display":126394},[31652,147585,147586,147616],{},[31655,147587,147588,147590,147592,147598,147600,147602,147604],{},[31658,147589,54151],{},[31662,147591,223],{},[37624,147593,147594,147596],{},[31658,147595,54151],{},[31749,147597,768],{},[31662,147599,223],{},[31662,147601,38874],{},[31662,147603,223],{},[37624,147605,147606,147608],{},[31658,147607,54151],{},[31655,147609,147610,147612,147614],{},[31658,147611,31660],{},[31662,147613,32226],{},[31749,147615,640],{},[31665,147617,147618],{"encoding":31667},"A+A^2+\\cdots+A^{n-1}",[34,147620,147622,147640,147685,147703],{"className":147621,"ariaHidden":758},[31672],[34,147623,147625,147628,147631,147634,147637],{"className":147624},[31676],[34,147626],{"className":147627,"style":34944},[31680],[34,147629,54151],{"className":147630},[31685,31686],[34,147632],{"className":147633,"style":32604},[31845],[34,147635,223],{"className":147636},[32024],[34,147638],{"className":147639,"style":32604},[31845],[34,147641,147643,147647,147676,147679,147682],{"className":147642},[31676],[34,147644],{"className":147645,"style":147646},[31680],"height:0.9474em;vertical-align:-0.0833em;",[34,147648,147650,147653],{"className":147649},[31685],[34,147651,54151],{"className":147652},[31685,31686],[34,147654,147656],{"className":147655},[31792],[34,147657,147659],{"className":147658},[31796],[34,147660,147662],{"className":147661},[31801],[34,147663,147665],{"className":147664,"style":129832},[31805],[34,147666,147667,147670],{"style":129846},[34,147668],{"className":147669,"style":31814},[31813],[34,147671,147673],{"className":147672},[31818,31819,31820,31821],[34,147674,768],{"className":147675},[31685,31821],[34,147677],{"className":147678,"style":32604},[31845],[34,147680,223],{"className":147681},[32024],[34,147683],{"className":147684,"style":32604},[31845],[34,147686,147688,147691,147694,147697,147700],{"className":147687},[31676],[34,147689],{"className":147690,"style":32597},[31680],[34,147692,38874],{"className":147693},[34006],[34,147695],{"className":147696,"style":32604},[31845],[34,147698,223],{"className":147699},[32024],[34,147701],{"className":147702,"style":32604},[31845],[34,147704,147706,147709],{"className":147705},[31676],[34,147707],{"className":147708,"style":129832},[31680],[34,147710,147712,147715],{"className":147711},[31685],[34,147713,54151],{"className":147714},[31685,31686],[34,147716,147718],{"className":147717},[31792],[34,147719,147721],{"className":147720},[31796],[34,147722,147724],{"className":147723},[31801],[34,147725,147727],{"className":147726,"style":129832},[31805],[34,147728,147729,147732],{"style":129846},[34,147730],{"className":147731,"style":31814},[31813],[34,147733,147735],{"className":147734},[31818,31819,31820,31821],[34,147736,147738,147741,147744],{"className":147737},[31685,31821],[34,147739,31660],{"className":147740},[31685,31686,31821],[34,147742,32226],{"className":147743},[32024,31821],[34,147745,640],{"className":147746},[31685,31821],[15,147748,147749],{},"再把非零元素改成 1。",[32188,147751],{},[19,147753,147755],{"id":147754},"_3-强分图与可达矩阵","3. 强分图与可达矩阵",[15,147757,147758,147759,147787],{},"如果有可达矩阵 ",[34,147760,147762,147775],{"className":147761},[31642],[34,147763,147765],{"className":147764},[31646],[31648,147766,147767],{"xmlns":31650},[31652,147768,147769,147773],{},[31655,147770,147771],{},[31658,147772,105852],{},[31665,147774,105852],{"encoding":31667},[34,147776,147778],{"className":147777,"ariaHidden":758},[31672],[34,147779,147781,147784],{"className":147780},[31676],[34,147782],{"className":147783,"style":33218},[31680],[34,147785,105852],{"className":147786,"style":37422},[31685,31686],"，那么：",[34,147789,147791],{"className":147790},[126385],[34,147792,147794,147817],{"className":147793},[31642],[34,147795,147797],{"className":147796},[31646],[31648,147798,147799],{"xmlns":31650,"display":126394},[31652,147800,147801,147814],{},[31655,147802,147803,147805,147808],{},[31658,147804,105852],{},[31662,147806,147807],{},"∧",[37624,147809,147810,147812],{},[31658,147811,105852],{},[31658,147813,37400],{},[31665,147815,147816],{"encoding":31667},"P\\land P^T",[34,147818,147820,147838],{"className":147819,"ariaHidden":758},[31672],[34,147821,147823,147826,147829,147832,147835],{"className":147822},[31676],[34,147824],{"className":147825,"style":33218},[31680],[34,147827,105852],{"className":147828,"style":37422},[31685,31686],[34,147830],{"className":147831,"style":32604},[31845],[34,147833,147807],{"className":147834},[32024],[34,147836],{"className":147837,"style":32604},[31845],[34,147839,147841,147844],{"className":147840},[31676],[34,147842],{"className":147843,"style":144123},[31680],[34,147845,147847,147850],{"className":147846},[31685],[34,147848,105852],{"className":147849,"style":37422},[31685,31686],[34,147851,147853],{"className":147852},[31792],[34,147854,147856],{"className":147855},[31796],[34,147857,147859],{"className":147858},[31801],[34,147860,147862],{"className":147861,"style":144123},[31805],[34,147863,147864,147867],{"style":129846},[34,147865],{"className":147866,"style":31814},[31813],[34,147868,147870],{"className":147869},[31818,31819,31820,31821],[34,147871,37400],{"className":147872,"style":37422},[31685,31686,31821],[15,147874,147875],{},"可以表示互相可达关系。",[15,147877,130751],{},[34,147879,147881],{"className":147880},[126385],[34,147882,147884,147909],{"className":147883},[31642],[34,147885,147887],{"className":147886},[31646],[31648,147888,147889],{"xmlns":31650,"display":126394},[31652,147890,147891,147907],{},[31655,147892,147893,147903,147905],{},[31743,147894,147895,147897],{},[31658,147896,15],{},[31655,147898,147899,147901],{},[31658,147900,31955],{},[31658,147902,31968],{},[31662,147904,202],{},[31749,147906,640],{},[31665,147908,146743],{"encoding":31667},[34,147910,147912,147970],{"className":147911,"ariaHidden":758},[31672],[34,147913,147915,147918,147961,147964,147967],{"className":147914},[31676],[34,147916],{"className":147917,"style":56005},[31680],[34,147919,147921,147924],{"className":147920},[31685],[34,147922,15],{"className":147923},[31685,31686],[34,147925,147927],{"className":147926},[31792],[34,147928,147930,147953],{"className":147929},[31796,31797],[34,147931,147933,147950],{"className":147932},[31801],[34,147934,147936],{"className":147935,"style":32090},[31805],[34,147937,147938,147941],{"style":31809},[34,147939],{"className":147940,"style":31814},[31813],[34,147942,147944],{"className":147943},[31818,31819,31820,31821],[34,147945,147947],{"className":147946},[31685,31821],[34,147948,141027],{"className":147949,"style":32153},[31685,31686,31821],[34,147951,31829],{"className":147952},[31828],[34,147954,147956],{"className":147955},[31801],[34,147957,147959],{"className":147958,"style":32163},[31805],[34,147960],{},[34,147962],{"className":147963,"style":31846},[31845],[34,147965,202],{"className":147966},[31850],[34,147968],{"className":147969,"style":31846},[31845],[34,147971,147973,147976],{"className":147972},[31676],[34,147974],{"className":147975,"style":32617},[31680],[34,147977,640],{"className":147978},[31685],[15,147980,147981],{},"且：",[34,147983,147985],{"className":147984},[126385],[34,147986,147988,148014],{"className":147987},[31642],[34,147989,147991],{"className":147990},[31646],[31648,147992,147993],{"xmlns":31650,"display":126394},[31652,147994,147995,148011],{},[31655,147996,147997,148007,148009],{},[31743,147998,147999,148001],{},[31658,148000,15],{},[31655,148002,148003,148005],{},[31658,148004,31968],{},[31658,148006,31955],{},[31662,148008,202],{},[31749,148010,640],{},[31665,148012,148013],{"encoding":31667},"p_{ji}=1",[34,148015,148017,148078],{"className":148016,"ariaHidden":758},[31672],[34,148018,148020,148023,148069,148072,148075],{"className":148019},[31676],[34,148021],{"className":148022,"style":56005},[31680],[34,148024,148026,148029],{"className":148025},[31685],[34,148027,15],{"className":148028},[31685,31686],[34,148030,148032],{"className":148031},[31792],[34,148033,148035,148061],{"className":148034},[31796,31797],[34,148036,148038,148058],{"className":148037},[31801],[34,148039,148041],{"className":148040,"style":32090},[31805],[34,148042,148043,148046],{"style":31809},[34,148044],{"className":148045,"style":31814},[31813],[34,148047,148049],{"className":148048},[31818,31819,31820,31821],[34,148050,148052,148055],{"className":148051},[31685,31821],[34,148053,31968],{"className":148054,"style":32153},[31685,31686,31821],[34,148056,31955],{"className":148057},[31685,31686,31821],[34,148059,31829],{"className":148060},[31828],[34,148062,148064],{"className":148063},[31801],[34,148065,148067],{"className":148066,"style":32163},[31805],[34,148068],{},[34,148070],{"className":148071,"style":31846},[31845],[34,148073,202],{"className":148074},[31850],[34,148076],{"className":148077,"style":31846},[31845],[34,148079,148081,148084],{"className":148080},[31676],[34,148082],{"className":148083,"style":32617},[31680],[34,148085,640],{"className":148086},[31685],[15,148088,90843],{},[34,148090,148092],{"className":148091},[126385],[34,148093,148095,148112],{"className":148094},[31642],[34,148096,148098],{"className":148097},[31646],[31648,148099,148100],{"xmlns":31650,"display":126394},[31652,148101,148102,148110],{},[31655,148103,148104],{},[31743,148105,148106,148108],{},[31658,148107,56782],{},[31658,148109,31955],{},[31665,148111,56787],{"encoding":31667},[34,148113,148115],{"className":148114,"ariaHidden":758},[31672],[34,148116,148118,148121],{"className":148117},[31676],[34,148119],{"className":148120,"style":31860},[31680],[34,148122,148124,148127],{"className":148123},[31685],[34,148125,56782],{"className":148126,"style":32062},[31685,31686],[34,148128,148130],{"className":148129},[31792],[34,148131,148133,148153],{"className":148132},[31796,31797],[34,148134,148136,148150],{"className":148135},[31801],[34,148137,148139],{"className":148138,"style":32090},[31805],[34,148140,148141,148144],{"style":38795},[34,148142],{"className":148143,"style":31814},[31813],[34,148145,148147],{"className":148146},[31818,31819,31820,31821],[34,148148,31955],{"className":148149},[31685,31686,31821],[34,148151,31829],{"className":148152},[31828],[34,148154,148156],{"className":148155},[31801],[34,148157,148159],{"className":148158,"style":31836},[31805],[34,148160],{},[15,148162,148163],{},"和：",[34,148165,148167],{"className":148166},[126385],[34,148168,148170,148187],{"className":148169},[31642],[34,148171,148173],{"className":148172},[31646],[31648,148174,148175],{"xmlns":31650,"display":126394},[31652,148176,148177,148185],{},[31655,148178,148179],{},[31743,148180,148181,148183],{},[31658,148182,56782],{},[31658,148184,31968],{},[31665,148186,138126],{"encoding":31667},[34,148188,148190],{"className":148189,"ariaHidden":758},[31672],[34,148191,148193,148196],{"className":148192},[31676],[34,148194],{"className":148195,"style":56005},[31680],[34,148197,148199,148202],{"className":148198},[31685],[34,148200,56782],{"className":148201,"style":32062},[31685,31686],[34,148203,148205],{"className":148204},[31792],[34,148206,148208,148228],{"className":148207},[31796,31797],[34,148209,148211,148225],{"className":148210},[31801],[34,148212,148214],{"className":148213,"style":32090},[31805],[34,148215,148216,148219],{"style":38795},[34,148217],{"className":148218,"style":31814},[31813],[34,148220,148222],{"className":148221},[31818,31819,31820,31821],[34,148223,31968],{"className":148224,"style":32153},[31685,31686,31821],[34,148226,31829],{"className":148227},[31828],[34,148229,148231],{"className":148230},[31801],[34,148232,148234],{"className":148233,"style":32163},[31805],[34,148235],{},[15,148237,148238],{},"互相可达，可能属于同一个强分图。",[32188,148240],{},[11,148242,148244],{"id":148243},"五距离矩阵","五、距离矩阵",[15,148246,148247],{},"距离矩阵记录的是两个顶点之间最短距离。",[15,148249,131588],{},[34,148251,148253],{"className":148252},[126385],[34,148254,148256,148286],{"className":148255},[31642],[34,148257,148259],{"className":148258},[31646],[31648,148260,148261],{"xmlns":31650,"display":126394},[31652,148262,148263,148283],{},[31655,148264,148265,148267,148269,148271,148281],{},[31658,148266,91994],{},[31662,148268,202],{},[31662,148270,456],{"stretchy":702},[31743,148272,148273,148275],{},[31658,148274,46900],{},[31655,148276,148277,148279],{},[31658,148278,31955],{},[31658,148280,31968],{},[31662,148282,2557],{"stretchy":702},[31665,148284,148285],{"encoding":31667},"D=(d_{ij})",[34,148287,148289,148307],{"className":148288,"ariaHidden":758},[31672],[34,148290,148292,148295,148298,148301,148304],{"className":148291},[31676],[34,148293],{"className":148294,"style":33218},[31680],[34,148296,91994],{"className":148297,"style":34452},[31685,31686],[34,148299],{"className":148300,"style":31846},[31845],[34,148302,202],{"className":148303},[31850],[34,148305],{"className":148306,"style":31846},[31845],[34,148308,148310,148313,148316,148359],{"className":148309},[31676],[34,148311],{"className":148312,"style":32058},[31680],[34,148314,456],{"className":148315},[31782],[34,148317,148319,148322],{"className":148318},[31685],[34,148320,46900],{"className":148321},[31685,31686],[34,148323,148325],{"className":148324},[31792],[34,148326,148328,148351],{"className":148327},[31796,31797],[34,148329,148331,148348],{"className":148330},[31801],[34,148332,148334],{"className":148333,"style":32090},[31805],[34,148335,148336,148339],{"style":31809},[34,148337],{"className":148338,"style":31814},[31813],[34,148340,148342],{"className":148341},[31818,31819,31820,31821],[34,148343,148345],{"className":148344},[31685,31821],[34,148346,141027],{"className":148347,"style":32153},[31685,31686,31821],[34,148349,31829],{"className":148350},[31828],[34,148352,148354],{"className":148353},[31801],[34,148355,148357],{"className":148356,"style":32163},[31805],[34,148358],{},[34,148360,2557],{"className":148361},[31690],[15,148363,104944],{},[34,148365,148367],{"className":148366},[126385],[34,148368,148370,148414],{"className":148369},[31642],[34,148371,148373],{"className":148372},[31646],[31648,148374,148375],{"xmlns":31650,"display":126394},[31652,148376,148377,148411],{},[31655,148378,148379,148389,148391,148393,148395,148401,148403,148409],{},[31743,148380,148381,148383],{},[31658,148382,46900],{},[31655,148384,148385,148387],{},[31658,148386,31955],{},[31658,148388,31968],{},[31662,148390,202],{},[31658,148392,46900],{},[31662,148394,15972],{"stretchy":702},[31743,148396,148397,148399],{},[31658,148398,56782],{},[31658,148400,31955],{},[31662,148402,31960],{"separator":758},[31743,148404,148405,148407],{},[31658,148406,56782],{},[31658,148408,31968],{},[31662,148410,32712],{"stretchy":702},[31665,148412,148413],{"encoding":31667},"d_{ij}=d[v_i,v_j]",[34,148415,148417,148475],{"className":148416,"ariaHidden":758},[31672],[34,148418,148420,148423,148466,148469,148472],{"className":148419},[31676],[34,148421],{"className":148422,"style":141223},[31680],[34,148424,148426,148429],{"className":148425},[31685],[34,148427,46900],{"className":148428},[31685,31686],[34,148430,148432],{"className":148431},[31792],[34,148433,148435,148458],{"className":148434},[31796,31797],[34,148436,148438,148455],{"className":148437},[31801],[34,148439,148441],{"className":148440,"style":32090},[31805],[34,148442,148443,148446],{"style":31809},[34,148444],{"className":148445,"style":31814},[31813],[34,148447,148449],{"className":148448},[31818,31819,31820,31821],[34,148450,148452],{"className":148451},[31685,31821],[34,148453,141027],{"className":148454,"style":32153},[31685,31686,31821],[34,148456,31829],{"className":148457},[31828],[34,148459,148461],{"className":148460},[31801],[34,148462,148464],{"className":148463,"style":32163},[31805],[34,148465],{},[34,148467],{"className":148468,"style":31846},[31845],[34,148470,202],{"className":148471},[31850],[34,148473],{"className":148474,"style":31846},[31845],[34,148476,148478,148481,148484,148487,148527,148530,148533,148573],{"className":148477},[31676],[34,148479],{"className":148480,"style":32058},[31680],[34,148482,46900],{"className":148483},[31685,31686],[34,148485,15972],{"className":148486},[31782],[34,148488,148490,148493],{"className":148489},[31685],[34,148491,56782],{"className":148492,"style":32062},[31685,31686],[34,148494,148496],{"className":148495},[31792],[34,148497,148499,148519],{"className":148498},[31796,31797],[34,148500,148502,148516],{"className":148501},[31801],[34,148503,148505],{"className":148504,"style":32090},[31805],[34,148506,148507,148510],{"style":38795},[34,148508],{"className":148509,"style":31814},[31813],[34,148511,148513],{"className":148512},[31818,31819,31820,31821],[34,148514,31955],{"className":148515},[31685,31686,31821],[34,148517,31829],{"className":148518},[31828],[34,148520,148522],{"className":148521},[31801],[34,148523,148525],{"className":148524,"style":31836},[31805],[34,148526],{},[34,148528,31960],{"className":148529},[32119],[34,148531],{"className":148532,"style":32123},[31845],[34,148534,148536,148539],{"className":148535},[31685],[34,148537,56782],{"className":148538,"style":32062},[31685,31686],[34,148540,148542],{"className":148541},[31792],[34,148543,148545,148565],{"className":148544},[31796,31797],[34,148546,148548,148562],{"className":148547},[31801],[34,148549,148551],{"className":148550,"style":32090},[31805],[34,148552,148553,148556],{"style":38795},[34,148554],{"className":148555,"style":31814},[31813],[34,148557,148559],{"className":148558},[31818,31819,31820,31821],[34,148560,31968],{"className":148561,"style":32153},[31685,31686,31821],[34,148563,31829],{"className":148564},[31828],[34,148566,148568],{"className":148567},[31801],[34,148569,148571],{"className":148570,"style":32163},[31805],[34,148572],{},[34,148574,32712],{"className":148575},[31690],[15,148577,148578],{},"如果不可达，则：",[34,148580,148582],{"className":148581},[126385],[34,148583,148585,148611],{"className":148584},[31642],[34,148586,148588],{"className":148587},[31646],[31648,148589,148590],{"xmlns":31650,"display":126394},[31652,148591,148592,148608],{},[31655,148593,148594,148604,148606],{},[31743,148595,148596,148598],{},[31658,148597,46900],{},[31655,148599,148600,148602],{},[31658,148601,31955],{},[31658,148603,31968],{},[31662,148605,202],{},[31658,148607,138286],{"mathvariant":38079},[31665,148609,148610],{"encoding":31667},"d_{ij}=\\infty",[34,148612,148614,148672],{"className":148613,"ariaHidden":758},[31672],[34,148615,148617,148620,148663,148666,148669],{"className":148616},[31676],[34,148618],{"className":148619,"style":141223},[31680],[34,148621,148623,148626],{"className":148622},[31685],[34,148624,46900],{"className":148625},[31685,31686],[34,148627,148629],{"className":148628},[31792],[34,148630,148632,148655],{"className":148631},[31796,31797],[34,148633,148635,148652],{"className":148634},[31801],[34,148636,148638],{"className":148637,"style":32090},[31805],[34,148639,148640,148643],{"style":31809},[34,148641],{"className":148642,"style":31814},[31813],[34,148644,148646],{"className":148645},[31818,31819,31820,31821],[34,148647,148649],{"className":148648},[31685,31821],[34,148650,141027],{"className":148651,"style":32153},[31685,31686,31821],[34,148653,31829],{"className":148654},[31828],[34,148656,148658],{"className":148657},[31801],[34,148659,148661],{"className":148660,"style":32163},[31805],[34,148662],{},[34,148664],{"className":148665,"style":31846},[31845],[34,148667,202],{"className":148668},[31850],[34,148670],{"className":148671,"style":31846},[31845],[34,148673,148675,148678],{"className":148674},[31676],[34,148676],{"className":148677,"style":32527},[31680],[34,148679,138286],{"className":148680},[31685],[15,148682,130751],{},[34,148684,148686],{"className":148685},[126385],[34,148687,148689,148706],{"className":148688},[31642],[34,148690,148692],{"className":148691},[31646],[31648,148693,148694],{"xmlns":31650,"display":126394},[31652,148695,148696,148704],{},[31655,148697,148698,148700,148702],{},[31658,148699,31955],{},[31662,148701,202],{},[31658,148703,31968],{},[31665,148705,33094],{"encoding":31667},[34,148707,148709,148727],{"className":148708,"ariaHidden":758},[31672],[34,148710,148712,148715,148718,148721,148724],{"className":148711},[31676],[34,148713],{"className":148714,"style":33104},[31680],[34,148716,31955],{"className":148717},[31685,31686],[34,148719],{"className":148720,"style":31846},[31845],[34,148722,202],{"className":148723},[31850],[34,148725],{"className":148726,"style":31846},[31845],[34,148728,148730,148733],{"className":148729},[31676],[34,148731],{"className":148732,"style":33123},[31680],[34,148734,31968],{"className":148735,"style":32153},[31685,31686],[15,148737,137537],{},[34,148739,148741],{"className":148740},[126385],[34,148742,148744,148770],{"className":148743},[31642],[34,148745,148747],{"className":148746},[31646],[31648,148748,148749],{"xmlns":31650,"display":126394},[31652,148750,148751,148767],{},[31655,148752,148753,148763,148765],{},[31743,148754,148755,148757],{},[31658,148756,46900],{},[31655,148758,148759,148761],{},[31658,148760,31955],{},[31658,148762,31955],{},[31662,148764,202],{},[31749,148766,459],{},[31665,148768,148769],{"encoding":31667},"d_{ii}=0",[34,148771,148773,148832],{"className":148772,"ariaHidden":758},[31672],[34,148774,148776,148779,148823,148826,148829],{"className":148775},[31676],[34,148777],{"className":148778,"style":33814},[31680],[34,148780,148782,148785],{"className":148781},[31685],[34,148783,46900],{"className":148784},[31685,31686],[34,148786,148788],{"className":148787},[31792],[34,148789,148791,148815],{"className":148790},[31796,31797],[34,148792,148794,148812],{"className":148793},[31801],[34,148795,148797],{"className":148796,"style":32090},[31805],[34,148798,148799,148802],{"style":31809},[34,148800],{"className":148801,"style":31814},[31813],[34,148803,148805],{"className":148804},[31818,31819,31820,31821],[34,148806,148808],{"className":148807},[31685,31821],[34,148809,148811],{"className":148810},[31685,31686,31821],"ii",[34,148813,31829],{"className":148814},[31828],[34,148816,148818],{"className":148817},[31801],[34,148819,148821],{"className":148820,"style":31836},[31805],[34,148822],{},[34,148824],{"className":148825,"style":31846},[31845],[34,148827,202],{"className":148828},[31850],[34,148830],{"className":148831,"style":31846},[31845],[34,148833,148835,148838],{"className":148834},[31676],[34,148836],{"className":148837,"style":32617},[31680],[34,148839,459],{"className":148840},[31685],[32188,148842],{},[11,148844,148846],{"id":148845},"_124-应用举例","12.4 应用举例",[15,148848,148849],{},"这一节主要讲图论应用。",[15,148851,148852,148853,347],{},"教材里有几类：",[2627,148854,148855],{},"均分问题、最短路径问题、最优路径问题、关键路径问题",[32188,148857],{},[11,148859,148861],{"id":148860},"一均分问题","一、均分问题",[15,148863,148864],{},"这种题一般把“状态”看成顶点，把“一步操作”看成边。",[15,148866,148867],{},"比如三个容器倒水问题：",[105,148869,148870,148873,148876],{},[108,148871,148872],{},"每一种水量分布是一个状态；",[108,148874,148875],{},"从一种状态倒水到另一种状态，就是一条边；",[108,148877,148878],{},"问能否均分，就是问图中是否存在从初始状态到目标状态的路径。",[15,148880,148881],{},"核心思想：",[34,148883,148885],{"className":148884},[126385],[34,148886,148888,148911],{"className":148887},[31642],[34,148889,148891],{"className":148890},[31646],[31648,148892,148893],{"xmlns":31650,"display":126394},[31652,148894,148895,148908],{},[31655,148896,148897],{},[126400,148898,148899],{"notation":126402},[71090,148900,148901],{"scriptlevel":459,"displaystyle":702},[71090,148902,148903],{"scriptlevel":459,"displaystyle":702},[71090,148904,148905],{"scriptlevel":459,"displaystyle":758},[39361,148906,148907],{},"状态 = 顶点，操作 = 边，求解 = 找路径",[31665,148909,148910],{"encoding":31667},"\\boxed{\\text{状态 = 顶点，操作 = 边，求解 = 找路径}}",[34,148912,148914],{"className":148913,"ariaHidden":758},[31672],[34,148915,148917,148920],{"className":148916},[31676],[34,148918],{"className":148919,"style":126424},[31680],[34,148921,148923],{"className":148922},[31685],[34,148924,148926,148985],{"className":148925},[31796,31797],[34,148927,148929,148982],{"className":148928},[31801],[34,148930,148932,148974],{"className":148931,"style":126437},[31805],[34,148933,148934,148937],{"style":126440},[34,148935],{"className":148936,"style":126444},[31813],[34,148938,148940],{"className":148939},[126448],[34,148941,148943],{"className":148942},[31685],[34,148944,148946],{"className":148945},[31685],[34,148947,148949,148953,148956,148960,148963,148967,148970],{"className":148948},[31685,2638],[34,148950,148952],{"className":148951},[31685,105462],"状态",[34,148954,129377],{"className":148955},[31685],[34,148957,148959],{"className":148958},[31685,105462],"顶点，操作",[34,148961,129377],{"className":148962},[31685],[34,148964,148966],{"className":148965},[31685,105462],"边，求解",[34,148968,129377],{"className":148969},[31685],[34,148971,148973],{"className":148972},[31685,105462],"找路径",[34,148975,148976,148979],{"style":126463},[34,148977],{"className":148978,"style":126444},[31813],[34,148980],{"className":148981,"style":126472},[126470,126471],[34,148983,31829],{"className":148984},[31828],[34,148986,148988],{"className":148987},[31801],[34,148989,148991],{"className":148990,"style":126482},[31805],[34,148992],{},[32188,148994],{},[11,148996,148998],{"id":148997},"二赋权图","二、赋权图",[19,149000,134475],{"id":149001},"_1-定义-1",[15,149003,149004,149005,127650,149008,347],{},"如果图的每条边都赋有一个数值，称为",[2627,149006,149007],{},"赋权图",[2627,149009,149010],{},"带权图",[15,149012,149013,149014,149042],{},"记边 ",[34,149015,149017,149030],{"className":149016},[31642],[34,149018,149020],{"className":149019},[31646],[31648,149021,149022],{"xmlns":31650},[31652,149023,149024,149028],{},[31655,149025,149026],{},[31658,149027,33244],{},[31665,149029,33244],{"encoding":31667},[34,149031,149033],{"className":149032,"ariaHidden":758},[31672],[34,149034,149036,149039],{"className":149035},[31676],[34,149037],{"className":149038,"style":32527},[31680],[34,149040,33244],{"className":149041},[31685,31686]," 的权为：",[34,149044,149046],{"className":149045},[126385],[34,149047,149049,149069],{"className":149048},[31642],[34,149050,149052],{"className":149051},[31646],[31648,149053,149054],{"xmlns":31650,"display":126394},[31652,149055,149056,149066],{},[31655,149057,149058,149060,149062,149064],{},[31658,149059,56852],{},[31662,149061,456],{"stretchy":702},[31658,149063,33244],{},[31662,149065,2557],{"stretchy":702},[31665,149067,149068],{"encoding":31667},"W(e)",[34,149070,149072],{"className":149071,"ariaHidden":758},[31672],[34,149073,149075,149078,149081,149084,149087],{"className":149074},[31676],[34,149076],{"className":149077,"style":31774},[31680],[34,149079,56852],{"className":149080,"style":37422},[31685,31686],[34,149082,456],{"className":149083},[31782],[34,149085,33244],{"className":149086},[31685,31686],[34,149088,2557],{"className":149089},[31690],[15,149091,149092],{},"权可以表示：",[105,149094,149095,149098,149101,149104,149107,149110],{},[108,149096,149097],{},"距离；",[108,149099,149100],{},"时间；",[108,149102,149103],{},"费用；",[108,149105,149106],{},"风险；",[108,149108,149109],{},"容量；",[108,149111,149112],{},"损耗。",[32188,149114],{},[19,149116,149118],{"id":149117},"_2-路径长度","2. 路径长度",[15,149120,149121],{},"在赋权图中，一条路径的长度不是边数，而是路径上所有边权之和。",[15,149123,149124],{},"如果路径：",[34,149126,149128],{"className":149127},[126385],[34,149129,149131,149167],{"className":149130},[31642],[34,149132,149134],{"className":149133},[31646],[31648,149135,149136],{"xmlns":31650,"display":126394},[31652,149137,149138,149164],{},[31655,149139,149140,149142,149144,149150,149156,149158],{},[31658,149141,135107],{},[31662,149143,202],{},[31743,149145,149146,149148],{},[31658,149147,33244],{},[31749,149149,640],{},[31743,149151,149152,149154],{},[31658,149153,33244],{},[31749,149155,768],{},[31662,149157,38874],{},[31743,149159,149160,149162],{},[31658,149161,33244],{},[31658,149163,33759],{},[31665,149165,149166],{"encoding":31667},"\\mu=e_1e_2\\cdots e_k",[34,149168,149170,149188],{"className":149169,"ariaHidden":758},[31672],[34,149171,149173,149176,149179,149182,149185],{"className":149172},[31676],[34,149174],{"className":149175,"style":38774},[31680],[34,149177,135107],{"className":149178},[31685,31686],[34,149180],{"className":149181,"style":31846},[31845],[34,149183,202],{"className":149184},[31850],[34,149186],{"className":149187,"style":31846},[31845],[34,149189,149191,149194,149234,149274,149277,149280,149283],{"className":149190},[31676],[34,149192],{"className":149193,"style":31860},[31680],[34,149195,149197,149200],{"className":149196},[31685],[34,149198,33244],{"className":149199},[31685,31686],[34,149201,149203],{"className":149202},[31792],[34,149204,149206,149226],{"className":149205},[31796,31797],[34,149207,149209,149223],{"className":149208},[31801],[34,149210,149212],{"className":149211,"style":31806},[31805],[34,149213,149214,149217],{"style":31809},[34,149215],{"className":149216,"style":31814},[31813],[34,149218,149220],{"className":149219},[31818,31819,31820,31821],[34,149221,640],{"className":149222},[31685,31821],[34,149224,31829],{"className":149225},[31828],[34,149227,149229],{"className":149228},[31801],[34,149230,149232],{"className":149231,"style":31836},[31805],[34,149233],{},[34,149235,149237,149240],{"className":149236},[31685],[34,149238,33244],{"className":149239},[31685,31686],[34,149241,149243],{"className":149242},[31792],[34,149244,149246,149266],{"className":149245},[31796,31797],[34,149247,149249,149263],{"className":149248},[31801],[34,149250,149252],{"className":149251,"style":31806},[31805],[34,149253,149254,149257],{"style":31809},[34,149255],{"className":149256,"style":31814},[31813],[34,149258,149260],{"className":149259},[31818,31819,31820,31821],[34,149261,768],{"className":149262},[31685,31821],[34,149264,31829],{"className":149265},[31828],[34,149267,149269],{"className":149268},[31801],[34,149270,149272],{"className":149271,"style":31836},[31805],[34,149273],{},[34,149275],{"className":149276,"style":32123},[31845],[34,149278,38874],{"className":149279},[34006],[34,149281],{"className":149282,"style":32123},[31845],[34,149284,149286,149289],{"className":149285},[31685],[34,149287,33244],{"className":149288},[31685,31686],[34,149290,149292],{"className":149291},[31792],[34,149293,149295,149315],{"className":149294},[31796,31797],[34,149296,149298,149312],{"className":149297},[31801],[34,149299,149301],{"className":149300,"style":33833},[31805],[34,149302,149303,149306],{"style":31809},[34,149304],{"className":149305,"style":31814},[31813],[34,149307,149309],{"className":149308},[31818,31819,31820,31821],[34,149310,33759],{"className":149311,"style":33774},[31685,31686,31821],[34,149313,31829],{"className":149314},[31828],[34,149316,149318],{"className":149317},[31801],[34,149319,149321],{"className":149320,"style":31836},[31805],[34,149322],{},[15,149324,137537],{},[34,149326,149328],{"className":149327},[126385],[34,149329,149331,149397],{"className":149330},[31642],[34,149332,149334],{"className":149333},[31646],[31648,149335,149336],{"xmlns":31650,"display":126394},[31652,149337,149338,149394],{},[31655,149339,149340,149342,149344,149346,149348,149350,149352,149354,149360,149362,149364,149366,149368,149374,149376,149378,149380,149382,149384,149386,149392],{},[31658,149341,43379],{},[31662,149343,456],{"stretchy":702},[31658,149345,135107],{},[31662,149347,2557],{"stretchy":702},[31662,149349,202],{},[31658,149351,56852],{},[31662,149353,456],{"stretchy":702},[31743,149355,149356,149358],{},[31658,149357,33244],{},[31749,149359,640],{},[31662,149361,2557],{"stretchy":702},[31662,149363,223],{},[31658,149365,56852],{},[31662,149367,456],{"stretchy":702},[31743,149369,149370,149372],{},[31658,149371,33244],{},[31749,149373,768],{},[31662,149375,2557],{"stretchy":702},[31662,149377,223],{},[31662,149379,38874],{},[31662,149381,223],{},[31658,149383,56852],{},[31662,149385,456],{"stretchy":702},[31743,149387,149388,149390],{},[31658,149389,33244],{},[31658,149391,33759],{},[31662,149393,2557],{"stretchy":702},[31665,149395,149396],{"encoding":31667},"l(\\mu)=W(e_1)+W(e_2)+\\cdots+W(e_k)",[34,149398,149400,149427,149491,149555,149573],{"className":149399,"ariaHidden":758},[31672],[34,149401,149403,149406,149409,149412,149415,149418,149421,149424],{"className":149402},[31676],[34,149404],{"className":149405,"style":31774},[31680],[34,149407,43379],{"className":149408,"style":47113},[31685,31686],[34,149410,456],{"className":149411},[31782],[34,149413,135107],{"className":149414},[31685,31686],[34,149416,2557],{"className":149417},[31690],[34,149419],{"className":149420,"style":31846},[31845],[34,149422,202],{"className":149423},[31850],[34,149425],{"className":149426,"style":31846},[31845],[34,149428,149430,149433,149436,149439,149479,149482,149485,149488],{"className":149429},[31676],[34,149431],{"className":149432,"style":31774},[31680],[34,149434,56852],{"className":149435,"style":37422},[31685,31686],[34,149437,456],{"className":149438},[31782],[34,149440,149442,149445],{"className":149441},[31685],[34,149443,33244],{"className":149444},[31685,31686],[34,149446,149448],{"className":149447},[31792],[34,149449,149451,149471],{"className":149450},[31796,31797],[34,149452,149454,149468],{"className":149453},[31801],[34,149455,149457],{"className":149456,"style":31806},[31805],[34,149458,149459,149462],{"style":31809},[34,149460],{"className":149461,"style":31814},[31813],[34,149463,149465],{"className":149464},[31818,31819,31820,31821],[34,149466,640],{"className":149467},[31685,31821],[34,149469,31829],{"className":149470},[31828],[34,149472,149474],{"className":149473},[31801],[34,149475,149477],{"className":149476,"style":31836},[31805],[34,149478],{},[34,149480,2557],{"className":149481},[31690],[34,149483],{"className":149484,"style":32604},[31845],[34,149486,223],{"className":149487},[32024],[34,149489],{"className":149490,"style":32604},[31845],[34,149492,149494,149497,149500,149503,149543,149546,149549,149552],{"className":149493},[31676],[34,149495],{"className":149496,"style":31774},[31680],[34,149498,56852],{"className":149499,"style":37422},[31685,31686],[34,149501,456],{"className":149502},[31782],[34,149504,149506,149509],{"className":149505},[31685],[34,149507,33244],{"className":149508},[31685,31686],[34,149510,149512],{"className":149511},[31792],[34,149513,149515,149535],{"className":149514},[31796,31797],[34,149516,149518,149532],{"className":149517},[31801],[34,149519,149521],{"className":149520,"style":31806},[31805],[34,149522,149523,149526],{"style":31809},[34,149524],{"className":149525,"style":31814},[31813],[34,149527,149529],{"className":149528},[31818,31819,31820,31821],[34,149530,768],{"className":149531},[31685,31821],[34,149533,31829],{"className":149534},[31828],[34,149536,149538],{"className":149537},[31801],[34,149539,149541],{"className":149540,"style":31836},[31805],[34,149542],{},[34,149544,2557],{"className":149545},[31690],[34,149547],{"className":149548,"style":32604},[31845],[34,149550,223],{"className":149551},[32024],[34,149553],{"className":149554,"style":32604},[31845],[34,149556,149558,149561,149564,149567,149570],{"className":149557},[31676],[34,149559],{"className":149560,"style":32597},[31680],[34,149562,38874],{"className":149563},[34006],[34,149565],{"className":149566,"style":32604},[31845],[34,149568,223],{"className":149569},[32024],[34,149571],{"className":149572,"style":32604},[31845],[34,149574,149576,149579,149582,149585,149625],{"className":149575},[31676],[34,149577],{"className":149578,"style":31774},[31680],[34,149580,56852],{"className":149581,"style":37422},[31685,31686],[34,149583,456],{"className":149584},[31782],[34,149586,149588,149591],{"className":149587},[31685],[34,149589,33244],{"className":149590},[31685,31686],[34,149592,149594],{"className":149593},[31792],[34,149595,149597,149617],{"className":149596},[31796,31797],[34,149598,149600,149614],{"className":149599},[31801],[34,149601,149603],{"className":149602,"style":33833},[31805],[34,149604,149605,149608],{"style":31809},[34,149606],{"className":149607,"style":31814},[31813],[34,149609,149611],{"className":149610},[31818,31819,31820,31821],[34,149612,33759],{"className":149613,"style":33774},[31685,31686,31821],[34,149615,31829],{"className":149616},[31828],[34,149618,149620],{"className":149619},[31801],[34,149621,149623],{"className":149622,"style":31836},[31805],[34,149624],{},[34,149626,2557],{"className":149627},[31690],[32188,149629],{},[11,149631,149633],{"id":149632},"三dijkstra-算法","三、Dijkstra 算法",[15,149635,149636],{},"Dijkstra 算法用于求：",[34,149638,149640],{"className":149639},[126385],[34,149641,149643,149666],{"className":149642},[31642],[34,149644,149646],{"className":149645},[31646],[31648,149647,149648],{"xmlns":31650,"display":126394},[31652,149649,149650,149663],{},[31655,149651,149652],{},[126400,149653,149654],{"notation":126402},[71090,149655,149656],{"scriptlevel":459,"displaystyle":702},[71090,149657,149658],{"scriptlevel":459,"displaystyle":702},[71090,149659,149660],{"scriptlevel":459,"displaystyle":758},[39361,149661,149662],{},"从一个源点到其他顶点的最短路径",[31665,149664,149665],{"encoding":31667},"\\boxed{\\text{从一个源点到其他顶点的最短路径}}",[34,149667,149669],{"className":149668,"ariaHidden":758},[31672],[34,149670,149672,149675],{"className":149671},[31676],[34,149673],{"className":149674,"style":126424},[31680],[34,149676,149678],{"className":149677},[31685],[34,149679,149681,149718],{"className":149680},[31796,31797],[34,149682,149684,149715],{"className":149683},[31801],[34,149685,149687,149707],{"className":149686,"style":126437},[31805],[34,149688,149689,149692],{"style":126440},[34,149690],{"className":149691,"style":126444},[31813],[34,149693,149695],{"className":149694},[126448],[34,149696,149698],{"className":149697},[31685],[34,149699,149701],{"className":149700},[31685],[34,149702,149704],{"className":149703},[31685,2638],[34,149705,149662],{"className":149706},[31685,105462],[34,149708,149709,149712],{"style":126463},[34,149710],{"className":149711,"style":126444},[31813],[34,149713],{"className":149714,"style":126472},[126470,126471],[34,149716,31829],{"className":149717},[31828],[34,149719,149721],{"className":149720},[31801],[34,149722,149724],{"className":149723,"style":126482},[31805],[34,149725],{},[15,149727,149728],{},"通常要求边权非负。",[32188,149730],{},[19,149732,149734],{"id":149733},"_1-算法思想","1. 算法思想",[15,149736,149737],{},"把顶点分成两类：",[13837,149739,149740,149749],{},[13840,149741,149742],{},[13843,149743,149744,149747],{},[13846,149745,149746],{},"集合",[13846,149748,126615],{},[13853,149750,149751,149786],{},[13843,149752,149753,149783],{},[13858,149754,149755],{},[34,149756,149758,149771],{"className":149757},[31642],[34,149759,149761],{"className":149760},[31646],[31648,149762,149763],{"xmlns":31650},[31652,149764,149765,149769],{},[31655,149766,149767],{},[31658,149768,105852],{},[31665,149770,105852],{"encoding":31667},[34,149772,149774],{"className":149773,"ariaHidden":758},[31672],[34,149775,149777,149780],{"className":149776},[31676],[34,149778],{"className":149779,"style":33218},[31680],[34,149781,105852],{"className":149782,"style":37422},[31685,31686],[13858,149784,149785],{},"已确定最短路径的顶点",[13843,149787,149788,149818],{},[13858,149789,149790],{},[34,149791,149793,149806],{"className":149792},[31642],[34,149794,149796],{"className":149795},[31646],[31648,149797,149798],{"xmlns":31650},[31652,149799,149800,149804],{},[31655,149801,149802],{},[31658,149803,37400],{},[31665,149805,37400],{"encoding":31667},[34,149807,149809],{"className":149808,"ariaHidden":758},[31672],[34,149810,149812,149815],{"className":149811},[31676],[34,149813],{"className":149814,"style":33218},[31680],[34,149816,37400],{"className":149817,"style":37422},[31685,31686],[13858,149819,149820],{},"暂时未确定的顶点",[15,149822,149823],{},"初始时：",[34,149825,149827],{"className":149826},[126385],[34,149828,149830,149852],{"className":149829},[31642],[34,149831,149833],{"className":149832},[31646],[31648,149834,149835],{"xmlns":31650,"display":126394},[31652,149836,149837,149849],{},[31655,149838,149839,149841,149843],{},[31658,149840,105852],{},[31662,149842,202],{},[31743,149844,149845,149847],{},[31658,149846,56782],{},[31749,149848,459],{},[31665,149850,149851],{"encoding":31667},"P={v_0}",[34,149853,149855,149873],{"className":149854,"ariaHidden":758},[31672],[34,149856,149858,149861,149864,149867,149870],{"className":149857},[31676],[34,149859],{"className":149860,"style":33218},[31680],[34,149862,105852],{"className":149863,"style":37422},[31685,31686],[34,149865],{"className":149866,"style":31846},[31845],[34,149868,202],{"className":149869},[31850],[34,149871],{"className":149872,"style":31846},[31845],[34,149874,149876,149879],{"className":149875},[31676],[34,149877],{"className":149878,"style":31860},[31680],[34,149880,149882],{"className":149881},[31685],[34,149883,149885,149888],{"className":149884},[31685],[34,149886,56782],{"className":149887,"style":32062},[31685,31686],[34,149889,149891],{"className":149890},[31792],[34,149892,149894,149914],{"className":149893},[31796,31797],[34,149895,149897,149911],{"className":149896},[31801],[34,149898,149900],{"className":149899,"style":31806},[31805],[34,149901,149902,149905],{"style":38795},[34,149903],{"className":149904,"style":31814},[31813],[34,149906,149908],{"className":149907},[31818,31819,31820,31821],[34,149909,459],{"className":149910},[31685,31821],[34,149912,31829],{"className":149913},[31828],[34,149915,149917],{"className":149916},[31801],[34,149918,149920],{"className":149919,"style":31836},[31805],[34,149921],{},[15,149923,149924,149925,149953,149954,149982],{},"然后不断从 ",[34,149926,149928,149941],{"className":149927},[31642],[34,149929,149931],{"className":149930},[31646],[31648,149932,149933],{"xmlns":31650},[31652,149934,149935,149939],{},[31655,149936,149937],{},[31658,149938,37400],{},[31665,149940,37400],{"encoding":31667},[34,149942,149944],{"className":149943,"ariaHidden":758},[31672],[34,149945,149947,149950],{"className":149946},[31676],[34,149948],{"className":149949,"style":33218},[31680],[34,149951,37400],{"className":149952,"style":37422},[31685,31686]," 中选出当前距离最小的顶点，加入 ",[34,149955,149957,149970],{"className":149956},[31642],[34,149958,149960],{"className":149959},[31646],[31648,149961,149962],{"xmlns":31650},[31652,149963,149964,149968],{},[31655,149965,149966],{},[31658,149967,105852],{},[31665,149969,105852],{"encoding":31667},[34,149971,149973],{"className":149972,"ariaHidden":758},[31672],[34,149974,149976,149979],{"className":149975},[31676],[34,149977],{"className":149978,"style":33218},[31680],[34,149980,105852],{"className":149981,"style":37422},[31685,31686],"，并更新它的邻接点距离。",[32188,149984],{},[19,149986,149988],{"id":149987},"_2-算法步骤","2. 算法步骤",[15,149990,149991,149992,347],{},"设源点为 ",[34,149993,149995,150012],{"className":149994},[31642],[34,149996,149998],{"className":149997},[31646],[31648,149999,150000],{"xmlns":31650},[31652,150001,150002,150010],{},[31655,150003,150004],{},[31743,150005,150006,150008],{},[31658,150007,56782],{},[31749,150009,459],{},[31665,150011,135498],{"encoding":31667},[34,150013,150015],{"className":150014,"ariaHidden":758},[31672],[34,150016,150018,150021],{"className":150017},[31676],[34,150019],{"className":150020,"style":31860},[31680],[34,150022,150024,150027],{"className":150023},[31685],[34,150025,56782],{"className":150026,"style":32062},[31685,31686],[34,150028,150030],{"className":150029},[31792],[34,150031,150033,150053],{"className":150032},[31796,31797],[34,150034,150036,150050],{"className":150035},[31801],[34,150037,150039],{"className":150038,"style":31806},[31805],[34,150040,150041,150044],{"style":38795},[34,150042],{"className":150043,"style":31814},[31813],[34,150045,150047],{"className":150046},[31818,31819,31820,31821],[34,150048,459],{"className":150049},[31685,31821],[34,150051,31829],{"className":150052},[31828],[34,150054,150056],{"className":150055},[31801],[34,150057,150059],{"className":150058,"style":31836},[31805],[34,150060],{},[253,150062,150064],{"id":150063},"第一步初始化","第一步：初始化",[34,150066,150068],{"className":150067},[126385],[34,150069,150071,150099],{"className":150070},[31642],[34,150072,150074],{"className":150073},[31646],[31648,150075,150076],{"xmlns":31650,"display":126394},[31652,150077,150078,150096],{},[31655,150079,150080,150082,150084,150090,150092,150094],{},[31658,150081,46900],{},[31662,150083,456],{"stretchy":702},[31743,150085,150086,150088],{},[31658,150087,56782],{},[31749,150089,459],{},[31662,150091,2557],{"stretchy":702},[31662,150093,202],{},[31749,150095,459],{},[31665,150097,150098],{"encoding":31667},"d(v_0)=0",[34,150100,150102,150166],{"className":150101,"ariaHidden":758},[31672],[34,150103,150105,150108,150111,150114,150154,150157,150160,150163],{"className":150104},[31676],[34,150106],{"className":150107,"style":31774},[31680],[34,150109,46900],{"className":150110},[31685,31686],[34,150112,456],{"className":150113},[31782],[34,150115,150117,150120],{"className":150116},[31685],[34,150118,56782],{"className":150119,"style":32062},[31685,31686],[34,150121,150123],{"className":150122},[31792],[34,150124,150126,150146],{"className":150125},[31796,31797],[34,150127,150129,150143],{"className":150128},[31801],[34,150130,150132],{"className":150131,"style":31806},[31805],[34,150133,150134,150137],{"style":38795},[34,150135],{"className":150136,"style":31814},[31813],[34,150138,150140],{"className":150139},[31818,31819,31820,31821],[34,150141,459],{"className":150142},[31685,31821],[34,150144,31829],{"className":150145},[31828],[34,150147,150149],{"className":150148},[31801],[34,150150,150152],{"className":150151,"style":31836},[31805],[34,150153],{},[34,150155,2557],{"className":150156},[31690],[34,150158],{"className":150159,"style":31846},[31845],[34,150161,202],{"className":150162},[31850],[34,150164],{"className":150165,"style":31846},[31845],[34,150167,150169,150172],{"className":150168},[31676],[34,150170],{"className":150171,"style":32617},[31680],[34,150173,459],{"className":150174},[31685],[15,150176,150177],{},"对于其他顶点：",[34,150179,150181],{"className":150180},[126385],[34,150182,150184,150297],{"className":150183},[31642],[34,150185,150187],{"className":150186},[31646],[31648,150188,150189],{"xmlns":31650,"display":126394},[31652,150190,150191,150294],{},[31655,150192,150193,150195,150197,150203,150205,150207],{},[31658,150194,46900],{},[31662,150196,456],{"stretchy":702},[31743,150198,150199,150201],{},[31658,150200,56782],{},[31658,150202,31955],{},[31662,150204,2557],{"stretchy":702},[31662,150206,202],{},[31655,150208,150209,150211],{},[31662,150210,32685],{"fence":758},[71078,150212,150213],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,150214,150215,150243,150271],{},[71087,150216,150217],{},[71090,150218,150219],{"scriptlevel":459,"displaystyle":702},[31655,150220,150221,150223,150225,150231,150233,150239,150241],{},[31658,150222,56852],{},[31662,150224,456],{"stretchy":702},[31743,150226,150227,150229],{},[31658,150228,56782],{},[31749,150230,459],{},[31662,150232,31960],{"separator":758},[31743,150234,150235,150237],{},[31658,150236,56782],{},[31658,150238,31955],{},[31662,150240,2557],{"stretchy":702},[31662,150242,31960],{"separator":758},[71087,150244,150245],{},[71090,150246,150247],{"scriptlevel":459,"displaystyle":702},[31655,150248,150249,150255,150258,150264,150267,150269],{},[31743,150250,150251,150253],{},[31658,150252,56782],{},[31749,150254,459],{},[39361,150256,150257],{}," 与 ",[31743,150259,150260,150262],{},[31658,150261,56782],{},[31658,150263,31955],{},[39361,150265,150266],{}," 有边 ",[31658,150268,138286],{"mathvariant":38079},[31662,150270,31960],{"separator":758},[71087,150272,150273],{},[71090,150274,150275],{"scriptlevel":459,"displaystyle":702},[31655,150276,150277,150283,150285,150291],{},[31743,150278,150279,150281],{},[31658,150280,56782],{},[31749,150282,459],{},[39361,150284,150257],{},[31743,150286,150287,150289],{},[31658,150288,56782],{},[31658,150290,31955],{},[39361,150292,150293],{}," 无边",[31665,150295,150296],{"encoding":31667},"d(v_i)=\n\\begin{cases}\nW(v_0,v_i),& v_0\\text{ 与 }v_i\\text{ 有边}\\\n\\infty,& v_0\\text{ 与 }v_i\\text{ 无边}\n\\end{cases}",[34,150298,150300,150364],{"className":150299,"ariaHidden":758},[31672],[34,150301,150303,150306,150309,150312,150352,150355,150358,150361],{"className":150302},[31676],[34,150304],{"className":150305,"style":31774},[31680],[34,150307,46900],{"className":150308},[31685,31686],[34,150310,456],{"className":150311},[31782],[34,150313,150315,150318],{"className":150314},[31685],[34,150316,56782],{"className":150317,"style":32062},[31685,31686],[34,150319,150321],{"className":150320},[31792],[34,150322,150324,150344],{"className":150323},[31796,31797],[34,150325,150327,150341],{"className":150326},[31801],[34,150328,150330],{"className":150329,"style":32090},[31805],[34,150331,150332,150335],{"style":38795},[34,150333],{"className":150334,"style":31814},[31813],[34,150336,150338],{"className":150337},[31818,31819,31820,31821],[34,150339,31955],{"className":150340},[31685,31686,31821],[34,150342,31829],{"className":150343},[31828],[34,150345,150347],{"className":150346},[31801],[34,150348,150350],{"className":150349,"style":31836},[31805],[34,150351],{},[34,150353,2557],{"className":150354},[31690],[34,150356],{"className":150357,"style":31846},[31845],[34,150359,202],{"className":150360},[31850],[34,150362],{"className":150363,"style":31846},[31845],[34,150365,150367,150370],{"className":150366},[31676],[34,150368],{"className":150369,"style":141282},[31680],[34,150371,150373,150379,150796],{"className":150372},[34006],[34,150374,150376],{"className":150375,"style":75811},[31782,75810],[34,150377,32685],{"className":150378},[71281,141292],[34,150380,150382],{"className":150381},[31685],[34,150383,150385,150514,150517,150660,150663],{"className":150384},[71078],[34,150386,150388],{"className":150387},[71384],[34,150389,150391,150506],{"className":150390},[31796,31797],[34,150392,150394,150503],{"className":150393},[31801],[34,150395,150397],{"className":150396,"style":141311},[31805],[34,150398,150399,150402],{"style":78184},[34,150400],{"className":150401,"style":71401},[31813],[34,150403,150405,150408,150411,150451,150454,150457,150497,150500],{"className":150404},[31685],[34,150406,56852],{"className":150407,"style":37422},[31685,31686],[34,150409,456],{"className":150410},[31782],[34,150412,150414,150417],{"className":150413},[31685],[34,150415,56782],{"className":150416,"style":32062},[31685,31686],[34,150418,150420],{"className":150419},[31792],[34,150421,150423,150443],{"className":150422},[31796,31797],[34,150424,150426,150440],{"className":150425},[31801],[34,150427,150429],{"className":150428,"style":31806},[31805],[34,150430,150431,150434],{"style":38795},[34,150432],{"className":150433,"style":31814},[31813],[34,150435,150437],{"className":150436},[31818,31819,31820,31821],[34,150438,459],{"className":150439},[31685,31821],[34,150441,31829],{"className":150442},[31828],[34,150444,150446],{"className":150445},[31801],[34,150447,150449],{"className":150448,"style":31836},[31805],[34,150450],{},[34,150452,31960],{"className":150453},[32119],[34,150455],{"className":150456,"style":32123},[31845],[34,150458,150460,150463],{"className":150459},[31685],[34,150461,56782],{"className":150462,"style":32062},[31685,31686],[34,150464,150466],{"className":150465},[31792],[34,150467,150469,150489],{"className":150468},[31796,31797],[34,150470,150472,150486],{"className":150471},[31801],[34,150473,150475],{"className":150474,"style":32090},[31805],[34,150476,150477,150480],{"style":38795},[34,150478],{"className":150479,"style":31814},[31813],[34,150481,150483],{"className":150482},[31818,31819,31820,31821],[34,150484,31955],{"className":150485},[31685,31686,31821],[34,150487,31829],{"className":150488},[31828],[34,150490,150492],{"className":150491},[31801],[34,150493,150495],{"className":150494,"style":31836},[31805],[34,150496],{},[34,150498,2557],{"className":150499},[31690],[34,150501,31960],{"className":150502},[32119],[34,150504,31829],{"className":150505},[31828],[34,150507,150509],{"className":150508},[31801],[34,150510,150512],{"className":150511,"style":141335},[31805],[34,150513],{},[34,150515],{"className":150516,"style":71733},[71732],[34,150518,150520],{"className":150519},[71384],[34,150521,150523,150652],{"className":150522},[31796,31797],[34,150524,150526,150649],{"className":150525},[31801],[34,150527,150529],{"className":150528,"style":141311},[31805],[34,150530,150531,150534],{"style":78184},[34,150532],{"className":150533,"style":71401},[31813],[34,150535,150537,150577,150590,150630,150640,150643,150646],{"className":150536},[31685],[34,150538,150540,150543],{"className":150539},[31685],[34,150541,56782],{"className":150542,"style":32062},[31685,31686],[34,150544,150546],{"className":150545},[31792],[34,150547,150549,150569],{"className":150548},[31796,31797],[34,150550,150552,150566],{"className":150551},[31801],[34,150553,150555],{"className":150554,"style":31806},[31805],[34,150556,150557,150560],{"style":38795},[34,150558],{"className":150559,"style":31814},[31813],[34,150561,150563],{"className":150562},[31818,31819,31820,31821],[34,150564,459],{"className":150565},[31685,31821],[34,150567,31829],{"className":150568},[31828],[34,150570,150572],{"className":150571},[31801],[34,150573,150575],{"className":150574,"style":31836},[31805],[34,150576],{},[34,150578,150580,150583,150587],{"className":150579},[31685,2638],[34,150581,130298],{"className":150582},[31685],[34,150584,150586],{"className":150585},[31685,105462],"与",[34,150588,130298],{"className":150589},[31685],[34,150591,150593,150596],{"className":150592},[31685],[34,150594,56782],{"className":150595,"style":32062},[31685,31686],[34,150597,150599],{"className":150598},[31792],[34,150600,150602,150622],{"className":150601},[31796,31797],[34,150603,150605,150619],{"className":150604},[31801],[34,150606,150608],{"className":150607,"style":32090},[31805],[34,150609,150610,150613],{"style":38795},[34,150611],{"className":150612,"style":31814},[31813],[34,150614,150616],{"className":150615},[31818,31819,31820,31821],[34,150617,31955],{"className":150618},[31685,31686,31821],[34,150620,31829],{"className":150621},[31828],[34,150623,150625],{"className":150624},[31801],[34,150626,150628],{"className":150627,"style":31836},[31805],[34,150629],{},[34,150631,150633,150636],{"className":150632},[31685,2638],[34,150634,130298],{"className":150635},[31685],[34,150637,150639],{"className":150638},[31685,105462],"有边",[34,150641,130298],{"className":150642},[31845],[34,150644,138286],{"className":150645},[31685],[34,150647,31960],{"className":150648},[32119],[34,150650,31829],{"className":150651},[31828],[34,150653,150655],{"className":150654},[31801],[34,150656,150658],{"className":150657,"style":141335},[31805],[34,150659],{},[34,150661],{"className":150662,"style":141477},[71732],[34,150664,150666],{"className":150665},[141481],[34,150667,150669,150788],{"className":150668},[31796,31797],[34,150670,150672,150785],{"className":150671},[31801],[34,150673,150675],{"className":150674,"style":141311},[31805],[34,150676,150677,150680],{"style":78184},[34,150678],{"className":150679,"style":71401},[31813],[34,150681,150683,150723,150735,150775],{"className":150682},[31685],[34,150684,150686,150689],{"className":150685},[31685],[34,150687,56782],{"className":150688,"style":32062},[31685,31686],[34,150690,150692],{"className":150691},[31792],[34,150693,150695,150715],{"className":150694},[31796,31797],[34,150696,150698,150712],{"className":150697},[31801],[34,150699,150701],{"className":150700,"style":31806},[31805],[34,150702,150703,150706],{"style":38795},[34,150704],{"className":150705,"style":31814},[31813],[34,150707,150709],{"className":150708},[31818,31819,31820,31821],[34,150710,459],{"className":150711},[31685,31821],[34,150713,31829],{"className":150714},[31828],[34,150716,150718],{"className":150717},[31801],[34,150719,150721],{"className":150720,"style":31836},[31805],[34,150722],{},[34,150724,150726,150729,150732],{"className":150725},[31685,2638],[34,150727,130298],{"className":150728},[31685],[34,150730,150586],{"className":150731},[31685,105462],[34,150733,130298],{"className":150734},[31685],[34,150736,150738,150741],{"className":150737},[31685],[34,150739,56782],{"className":150740,"style":32062},[31685,31686],[34,150742,150744],{"className":150743},[31792],[34,150745,150747,150767],{"className":150746},[31796,31797],[34,150748,150750,150764],{"className":150749},[31801],[34,150751,150753],{"className":150752,"style":32090},[31805],[34,150754,150755,150758],{"style":38795},[34,150756],{"className":150757,"style":31814},[31813],[34,150759,150761],{"className":150760},[31818,31819,31820,31821],[34,150762,31955],{"className":150763},[31685,31686,31821],[34,150765,31829],{"className":150766},[31828],[34,150768,150770],{"className":150769},[31801],[34,150771,150773],{"className":150772,"style":31836},[31805],[34,150774],{},[34,150776,150778,150781],{"className":150777},[31685,2638],[34,150779,130298],{"className":150780},[31685],[34,150782,150784],{"className":150783},[31685,105462],"无边",[34,150786,31829],{"className":150787},[31828],[34,150789,150791],{"className":150790},[31801],[34,150792,150794],{"className":150793,"style":141335},[31805],[34,150795],{},[34,150797],{"className":150798},[31690,41488],[32188,150800],{},[253,150802,150804],{"id":150803},"第二步选最小","第二步：选最小",[15,150806,150807,150808,150836,150837,150906,150907,347],{},"从未确定顶点中选 ",[34,150809,150811,150824],{"className":150810},[31642],[34,150812,150814],{"className":150813},[31646],[31648,150815,150816],{"xmlns":31650},[31652,150817,150818,150822],{},[31655,150819,150820],{},[31658,150821,46900],{},[31665,150823,46900],{"encoding":31667},[34,150825,150827],{"className":150826,"ariaHidden":758},[31672],[34,150828,150830,150833],{"className":150829},[31676],[34,150831],{"className":150832,"style":31681},[31680],[34,150834,46900],{"className":150835},[31685,31686]," 值最小的顶点 ",[34,150838,150840,150857],{"className":150839},[31642],[34,150841,150843],{"className":150842},[31646],[31648,150844,150845],{"xmlns":31650},[31652,150846,150847,150855],{},[31655,150848,150849],{},[31743,150850,150851,150853],{},[31658,150852,56782],{},[31658,150854,33759],{},[31665,150856,135764],{"encoding":31667},[34,150858,150860],{"className":150859,"ariaHidden":758},[31672],[34,150861,150863,150866],{"className":150862},[31676],[34,150864],{"className":150865,"style":31860},[31680],[34,150867,150869,150872],{"className":150868},[31685],[34,150870,56782],{"className":150871,"style":32062},[31685,31686],[34,150873,150875],{"className":150874},[31792],[34,150876,150878,150898],{"className":150877},[31796,31797],[34,150879,150881,150895],{"className":150880},[31801],[34,150882,150884],{"className":150883,"style":33833},[31805],[34,150885,150886,150889],{"style":38795},[34,150887],{"className":150888,"style":31814},[31813],[34,150890,150892],{"className":150891},[31818,31819,31820,31821],[34,150893,33759],{"className":150894,"style":33774},[31685,31686,31821],[34,150896,31829],{"className":150897},[31828],[34,150899,150901],{"className":150900},[31801],[34,150902,150904],{"className":150903,"style":31836},[31805],[34,150905],{},"，加入 ",[34,150908,150910,150923],{"className":150909},[31642],[34,150911,150913],{"className":150912},[31646],[31648,150914,150915],{"xmlns":31650},[31652,150916,150917,150921],{},[31655,150918,150919],{},[31658,150920,105852],{},[31665,150922,105852],{"encoding":31667},[34,150924,150926],{"className":150925,"ariaHidden":758},[31672],[34,150927,150929,150932],{"className":150928},[31676],[34,150930],{"className":150931,"style":33218},[31680],[34,150933,105852],{"className":150934,"style":37422},[31685,31686],[32188,150936],{},[253,150938,150940],{"id":150939},"第三步更新","第三步：更新",[15,150942,3101,150943,151012,151013,151082],{},[34,150944,150946,150963],{"className":150945},[31642],[34,150947,150949],{"className":150948},[31646],[31648,150950,150951],{"xmlns":31650},[31652,150952,150953,150961],{},[31655,150954,150955],{},[31743,150956,150957,150959],{},[31658,150958,56782],{},[31658,150960,33759],{},[31665,150962,135764],{"encoding":31667},[34,150964,150966],{"className":150965,"ariaHidden":758},[31672],[34,150967,150969,150972],{"className":150968},[31676],[34,150970],{"className":150971,"style":31860},[31680],[34,150973,150975,150978],{"className":150974},[31685],[34,150976,56782],{"className":150977,"style":32062},[31685,31686],[34,150979,150981],{"className":150980},[31792],[34,150982,150984,151004],{"className":150983},[31796,31797],[34,150985,150987,151001],{"className":150986},[31801],[34,150988,150990],{"className":150989,"style":33833},[31805],[34,150991,150992,150995],{"style":38795},[34,150993],{"className":150994,"style":31814},[31813],[34,150996,150998],{"className":150997},[31818,31819,31820,31821],[34,150999,33759],{"className":151000,"style":33774},[31685,31686,31821],[34,151002,31829],{"className":151003},[31828],[34,151005,151007],{"className":151006},[31801],[34,151008,151010],{"className":151009,"style":31836},[31805],[34,151011],{}," 的邻接顶点 ",[34,151014,151016,151033],{"className":151015},[31642],[34,151017,151019],{"className":151018},[31646],[31648,151020,151021],{"xmlns":31650},[31652,151022,151023,151031],{},[31655,151024,151025],{},[31743,151026,151027,151029],{},[31658,151028,56782],{},[31658,151030,31955],{},[31665,151032,56787],{"encoding":31667},[34,151034,151036],{"className":151035,"ariaHidden":758},[31672],[34,151037,151039,151042],{"className":151038},[31676],[34,151040],{"className":151041,"style":31860},[31680],[34,151043,151045,151048],{"className":151044},[31685],[34,151046,56782],{"className":151047,"style":32062},[31685,31686],[34,151049,151051],{"className":151050},[31792],[34,151052,151054,151074],{"className":151053},[31796,31797],[34,151055,151057,151071],{"className":151056},[31801],[34,151058,151060],{"className":151059,"style":32090},[31805],[34,151061,151062,151065],{"style":38795},[34,151063],{"className":151064,"style":31814},[31813],[34,151066,151068],{"className":151067},[31818,31819,31820,31821],[34,151069,31955],{"className":151070},[31685,31686,31821],[34,151072,31829],{"className":151073},[31828],[34,151075,151077],{"className":151076},[31801],[34,151078,151080],{"className":151079,"style":31836},[31805],[34,151081],{},"，若：",[34,151084,151086],{"className":151085},[126385],[34,151087,151089,151149],{"className":151088},[31642],[34,151090,151092],{"className":151091},[31646],[31648,151093,151094],{"xmlns":31650,"display":126394},[31652,151095,151096,151146],{},[31655,151097,151098,151100,151102,151108,151110,151112,151114,151116,151122,151124,151130,151132,151134,151136,151138,151144],{},[31658,151099,46900],{},[31662,151101,456],{"stretchy":702},[31743,151103,151104,151106],{},[31658,151105,56782],{},[31658,151107,33759],{},[31662,151109,2557],{"stretchy":702},[31662,151111,223],{},[31658,151113,56852],{},[31662,151115,456],{"stretchy":702},[31743,151117,151118,151120],{},[31658,151119,56782],{},[31658,151121,33759],{},[31662,151123,31960],{"separator":758},[31743,151125,151126,151128],{},[31658,151127,56782],{},[31658,151129,31955],{},[31662,151131,2557],{"stretchy":702},[31662,151133,932],{},[31658,151135,46900],{},[31662,151137,456],{"stretchy":702},[31743,151139,151140,151142],{},[31658,151141,56782],{},[31658,151143,31955],{},[31662,151145,2557],{"stretchy":702},[31665,151147,151148],{"encoding":31667},"d(v_k)+W(v_k,v_i)\u003Cd(v_i)",[34,151150,151152,151216,151326],{"className":151151,"ariaHidden":758},[31672],[34,151153,151155,151158,151161,151164,151204,151207,151210,151213],{"className":151154},[31676],[34,151156],{"className":151157,"style":31774},[31680],[34,151159,46900],{"className":151160},[31685,31686],[34,151162,456],{"className":151163},[31782],[34,151165,151167,151170],{"className":151166},[31685],[34,151168,56782],{"className":151169,"style":32062},[31685,31686],[34,151171,151173],{"className":151172},[31792],[34,151174,151176,151196],{"className":151175},[31796,31797],[34,151177,151179,151193],{"className":151178},[31801],[34,151180,151182],{"className":151181,"style":33833},[31805],[34,151183,151184,151187],{"style":38795},[34,151185],{"className":151186,"style":31814},[31813],[34,151188,151190],{"className":151189},[31818,31819,31820,31821],[34,151191,33759],{"className":151192,"style":33774},[31685,31686,31821],[34,151194,31829],{"className":151195},[31828],[34,151197,151199],{"className":151198},[31801],[34,151200,151202],{"className":151201,"style":31836},[31805],[34,151203],{},[34,151205,2557],{"className":151206},[31690],[34,151208],{"className":151209,"style":32604},[31845],[34,151211,223],{"className":151212},[32024],[34,151214],{"className":151215,"style":32604},[31845],[34,151217,151219,151222,151225,151228,151268,151271,151274,151314,151317,151320,151323],{"className":151218},[31676],[34,151220],{"className":151221,"style":31774},[31680],[34,151223,56852],{"className":151224,"style":37422},[31685,31686],[34,151226,456],{"className":151227},[31782],[34,151229,151231,151234],{"className":151230},[31685],[34,151232,56782],{"className":151233,"style":32062},[31685,31686],[34,151235,151237],{"className":151236},[31792],[34,151238,151240,151260],{"className":151239},[31796,31797],[34,151241,151243,151257],{"className":151242},[31801],[34,151244,151246],{"className":151245,"style":33833},[31805],[34,151247,151248,151251],{"style":38795},[34,151249],{"className":151250,"style":31814},[31813],[34,151252,151254],{"className":151253},[31818,31819,31820,31821],[34,151255,33759],{"className":151256,"style":33774},[31685,31686,31821],[34,151258,31829],{"className":151259},[31828],[34,151261,151263],{"className":151262},[31801],[34,151264,151266],{"className":151265,"style":31836},[31805],[34,151267],{},[34,151269,31960],{"className":151270},[32119],[34,151272],{"className":151273,"style":32123},[31845],[34,151275,151277,151280],{"className":151276},[31685],[34,151278,56782],{"className":151279,"style":32062},[31685,31686],[34,151281,151283],{"className":151282},[31792],[34,151284,151286,151306],{"className":151285},[31796,31797],[34,151287,151289,151303],{"className":151288},[31801],[34,151290,151292],{"className":151291,"style":32090},[31805],[34,151293,151294,151297],{"style":38795},[34,151295],{"className":151296,"style":31814},[31813],[34,151298,151300],{"className":151299},[31818,31819,31820,31821],[34,151301,31955],{"className":151302},[31685,31686,31821],[34,151304,31829],{"className":151305},[31828],[34,151307,151309],{"className":151308},[31801],[34,151310,151312],{"className":151311,"style":31836},[31805],[34,151313],{},[34,151315,2557],{"className":151316},[31690],[34,151318],{"className":151319,"style":31846},[31845],[34,151321,932],{"className":151322},[31850],[34,151324],{"className":151325,"style":31846},[31845],[34,151327,151329,151332,151335,151338,151378],{"className":151328},[31676],[34,151330],{"className":151331,"style":31774},[31680],[34,151333,46900],{"className":151334},[31685,31686],[34,151336,456],{"className":151337},[31782],[34,151339,151341,151344],{"className":151340},[31685],[34,151342,56782],{"className":151343,"style":32062},[31685,31686],[34,151345,151347],{"className":151346},[31792],[34,151348,151350,151370],{"className":151349},[31796,31797],[34,151351,151353,151367],{"className":151352},[31801],[34,151354,151356],{"className":151355,"style":32090},[31805],[34,151357,151358,151361],{"style":38795},[34,151359],{"className":151360,"style":31814},[31813],[34,151362,151364],{"className":151363},[31818,31819,31820,31821],[34,151365,31955],{"className":151366},[31685,31686,31821],[34,151368,31829],{"className":151369},[31828],[34,151371,151373],{"className":151372},[31801],[34,151374,151376],{"className":151375,"style":31836},[31805],[34,151377],{},[34,151379,2557],{"className":151380},[31690],[15,151382,151383],{},"则更新：",[34,151385,151387],{"className":151386},[126385],[34,151388,151390,151450],{"className":151389},[31642],[34,151391,151393],{"className":151392},[31646],[31648,151394,151395],{"xmlns":31650,"display":126394},[31652,151396,151397,151447],{},[31655,151398,151399,151401,151403,151409,151411,151413,151415,151417,151423,151425,151427,151429,151431,151437,151439,151445],{},[31658,151400,46900],{},[31662,151402,456],{"stretchy":702},[31743,151404,151405,151407],{},[31658,151406,56782],{},[31658,151408,31955],{},[31662,151410,2557],{"stretchy":702},[31662,151412,202],{},[31658,151414,46900],{},[31662,151416,456],{"stretchy":702},[31743,151418,151419,151421],{},[31658,151420,56782],{},[31658,151422,33759],{},[31662,151424,2557],{"stretchy":702},[31662,151426,223],{},[31658,151428,56852],{},[31662,151430,456],{"stretchy":702},[31743,151432,151433,151435],{},[31658,151434,56782],{},[31658,151436,33759],{},[31662,151438,31960],{"separator":758},[31743,151440,151441,151443],{},[31658,151442,56782],{},[31658,151444,31955],{},[31662,151446,2557],{"stretchy":702},[31665,151448,151449],{"encoding":31667},"d(v_i)=d(v_k)+W(v_k,v_i)",[34,151451,151453,151517,151581],{"className":151452,"ariaHidden":758},[31672],[34,151454,151456,151459,151462,151465,151505,151508,151511,151514],{"className":151455},[31676],[34,151457],{"className":151458,"style":31774},[31680],[34,151460,46900],{"className":151461},[31685,31686],[34,151463,456],{"className":151464},[31782],[34,151466,151468,151471],{"className":151467},[31685],[34,151469,56782],{"className":151470,"style":32062},[31685,31686],[34,151472,151474],{"className":151473},[31792],[34,151475,151477,151497],{"className":151476},[31796,31797],[34,151478,151480,151494],{"className":151479},[31801],[34,151481,151483],{"className":151482,"style":32090},[31805],[34,151484,151485,151488],{"style":38795},[34,151486],{"className":151487,"style":31814},[31813],[34,151489,151491],{"className":151490},[31818,31819,31820,31821],[34,151492,31955],{"className":151493},[31685,31686,31821],[34,151495,31829],{"className":151496},[31828],[34,151498,151500],{"className":151499},[31801],[34,151501,151503],{"className":151502,"style":31836},[31805],[34,151504],{},[34,151506,2557],{"className":151507},[31690],[34,151509],{"className":151510,"style":31846},[31845],[34,151512,202],{"className":151513},[31850],[34,151515],{"className":151516,"style":31846},[31845],[34,151518,151520,151523,151526,151529,151569,151572,151575,151578],{"className":151519},[31676],[34,151521],{"className":151522,"style":31774},[31680],[34,151524,46900],{"className":151525},[31685,31686],[34,151527,456],{"className":151528},[31782],[34,151530,151532,151535],{"className":151531},[31685],[34,151533,56782],{"className":151534,"style":32062},[31685,31686],[34,151536,151538],{"className":151537},[31792],[34,151539,151541,151561],{"className":151540},[31796,31797],[34,151542,151544,151558],{"className":151543},[31801],[34,151545,151547],{"className":151546,"style":33833},[31805],[34,151548,151549,151552],{"style":38795},[34,151550],{"className":151551,"style":31814},[31813],[34,151553,151555],{"className":151554},[31818,31819,31820,31821],[34,151556,33759],{"className":151557,"style":33774},[31685,31686,31821],[34,151559,31829],{"className":151560},[31828],[34,151562,151564],{"className":151563},[31801],[34,151565,151567],{"className":151566,"style":31836},[31805],[34,151568],{},[34,151570,2557],{"className":151571},[31690],[34,151573],{"className":151574,"style":32604},[31845],[34,151576,223],{"className":151577},[32024],[34,151579],{"className":151580,"style":32604},[31845],[34,151582,151584,151587,151590,151593,151633,151636,151639,151679],{"className":151583},[31676],[34,151585],{"className":151586,"style":31774},[31680],[34,151588,56852],{"className":151589,"style":37422},[31685,31686],[34,151591,456],{"className":151592},[31782],[34,151594,151596,151599],{"className":151595},[31685],[34,151597,56782],{"className":151598,"style":32062},[31685,31686],[34,151600,151602],{"className":151601},[31792],[34,151603,151605,151625],{"className":151604},[31796,31797],[34,151606,151608,151622],{"className":151607},[31801],[34,151609,151611],{"className":151610,"style":33833},[31805],[34,151612,151613,151616],{"style":38795},[34,151614],{"className":151615,"style":31814},[31813],[34,151617,151619],{"className":151618},[31818,31819,31820,31821],[34,151620,33759],{"className":151621,"style":33774},[31685,31686,31821],[34,151623,31829],{"className":151624},[31828],[34,151626,151628],{"className":151627},[31801],[34,151629,151631],{"className":151630,"style":31836},[31805],[34,151632],{},[34,151634,31960],{"className":151635},[32119],[34,151637],{"className":151638,"style":32123},[31845],[34,151640,151642,151645],{"className":151641},[31685],[34,151643,56782],{"className":151644,"style":32062},[31685,31686],[34,151646,151648],{"className":151647},[31792],[34,151649,151651,151671],{"className":151650},[31796,31797],[34,151652,151654,151668],{"className":151653},[31801],[34,151655,151657],{"className":151656,"style":32090},[31805],[34,151658,151659,151662],{"style":38795},[34,151660],{"className":151661,"style":31814},[31813],[34,151663,151665],{"className":151664},[31818,31819,31820,31821],[34,151666,31955],{"className":151667},[31685,31686,31821],[34,151669,31829],{"className":151670},[31828],[34,151672,151674],{"className":151673},[31801],[34,151675,151677],{"className":151676,"style":31836},[31805],[34,151678],{},[34,151680,2557],{"className":151681},[31690],[32188,151683],{},[253,151685,151687],{"id":151686},"第四步重复","第四步：重复",[15,151689,151690,151691,151719,151720,347],{},"一直重复，直到目标点被加入 ",[34,151692,151694,151707],{"className":151693},[31642],[34,151695,151697],{"className":151696},[31646],[31648,151698,151699],{"xmlns":31650},[31652,151700,151701,151705],{},[31655,151702,151703],{},[31658,151704,105852],{},[31665,151706,105852],{"encoding":31667},[34,151708,151710],{"className":151709,"ariaHidden":758},[31672],[34,151711,151713,151716],{"className":151712},[31676],[34,151714],{"className":151715,"style":33218},[31680],[34,151717,105852],{"className":151718,"style":37422},[31685,31686],"，或者所有顶点都加入 ",[34,151721,151723,151736],{"className":151722},[31642],[34,151724,151726],{"className":151725},[31646],[31648,151727,151728],{"xmlns":31650},[31652,151729,151730,151734],{},[31655,151731,151732],{},[31658,151733,105852],{},[31665,151735,105852],{"encoding":31667},[34,151737,151739],{"className":151738,"ariaHidden":758},[31672],[34,151740,151742,151745],{"className":151741},[31676],[34,151743],{"className":151744,"style":33218},[31680],[34,151746,105852],{"className":151747,"style":37422},[31685,31686],[32188,151749],{},[19,151751,151753],{"id":151752},"_3-适用场景","3. 适用场景",[15,151755,151756],{},"Dijkstra 适合：",[105,151758,151759,151762,151765,151768,151771],{},[108,151760,151761],{},"单源最短路径；",[108,151763,151764],{},"非负权图；",[108,151766,151767],{},"地图导航；",[108,151769,151770],{},"网络路由；",[108,151772,151773],{},"成本最小问题。",[32188,151775],{},[11,151777,151779],{"id":151778},"四floyd-算法","四、Floyd 算法",[15,151781,151782],{},"Floyd 算法用于求：",[34,151784,151786],{"className":151785},[126385],[34,151787,151789,151812],{"className":151788},[31642],[34,151790,151792],{"className":151791},[31646],[31648,151793,151794],{"xmlns":31650,"display":126394},[31652,151795,151796,151809],{},[31655,151797,151798],{},[126400,151799,151800],{"notation":126402},[71090,151801,151802],{"scriptlevel":459,"displaystyle":702},[71090,151803,151804],{"scriptlevel":459,"displaystyle":702},[71090,151805,151806],{"scriptlevel":459,"displaystyle":758},[39361,151807,151808],{},"任意两点之间的最短路径",[31665,151810,151811],{"encoding":31667},"\\boxed{\\text{任意两点之间的最短路径}}",[34,151813,151815],{"className":151814,"ariaHidden":758},[31672],[34,151816,151818,151821],{"className":151817},[31676],[34,151819],{"className":151820,"style":126424},[31680],[34,151822,151824],{"className":151823},[31685],[34,151825,151827,151864],{"className":151826},[31796,31797],[34,151828,151830,151861],{"className":151829},[31801],[34,151831,151833,151853],{"className":151832,"style":126437},[31805],[34,151834,151835,151838],{"style":126440},[34,151836],{"className":151837,"style":126444},[31813],[34,151839,151841],{"className":151840},[126448],[34,151842,151844],{"className":151843},[31685],[34,151845,151847],{"className":151846},[31685],[34,151848,151850],{"className":151849},[31685,2638],[34,151851,151808],{"className":151852},[31685,105462],[34,151854,151855,151858],{"style":126463},[34,151856],{"className":151857,"style":126444},[31813],[34,151859],{"className":151860,"style":126472},[126470,126471],[34,151862,31829],{"className":151863},[31828],[34,151865,151867],{"className":151866},[31801],[34,151868,151870],{"className":151869,"style":126482},[31805],[34,151871],{},[15,151873,151874],{},"也叫全源最短路径算法。",[32188,151876],{},[19,151878,151880],{"id":151879},"_1-初始矩阵","1. 初始矩阵",[15,151882,151883],{},"设初始距离矩阵为：",[34,151885,151887],{"className":151886},[126385],[34,151888,151890,151938],{"className":151889},[31642],[34,151891,151893],{"className":151892},[31646],[31648,151894,151895],{"xmlns":31650,"display":126394},[31652,151896,151897,151935],{},[31655,151898,151899,151911,151913,151915,151933],{},[37624,151900,151901,151903],{},[31658,151902,91994],{},[31655,151904,151905,151907,151909],{},[31662,151906,456],{"stretchy":702},[31749,151908,459],{},[31662,151910,2557],{"stretchy":702},[31662,151912,202],{},[31662,151914,456],{"stretchy":702},[38846,151916,151917,151919,151925],{},[31658,151918,46900],{},[31655,151920,151921,151923],{},[31658,151922,31955],{},[31658,151924,31968],{},[31655,151926,151927,151929,151931],{},[31662,151928,456],{"stretchy":702},[31749,151930,459],{},[31662,151932,2557],{"stretchy":702},[31662,151934,2557],{"stretchy":702},[31665,151936,151937],{"encoding":31667},"D^{(0)}=(d_{ij}^{(0)})",[34,151939,151941,151994],{"className":151940,"ariaHidden":758},[31672],[34,151942,151944,151947,151985,151988,151991],{"className":151943},[31676],[34,151945],{"className":151946,"style":147280},[31680],[34,151948,151950,151953],{"className":151949},[31685],[34,151951,91994],{"className":151952,"style":34452},[31685,31686],[34,151954,151956],{"className":151955},[31792],[34,151957,151959],{"className":151958},[31796],[34,151960,151962],{"className":151961},[31801],[34,151963,151965],{"className":151964,"style":147280},[31805],[34,151966,151967,151970],{"style":129846},[34,151968],{"className":151969,"style":31814},[31813],[34,151971,151973],{"className":151972},[31818,31819,31820,31821],[34,151974,151976,151979,151982],{"className":151975},[31685,31821],[34,151977,456],{"className":151978},[31782,31821],[34,151980,459],{"className":151981},[31685,31821],[34,151983,2557],{"className":151984},[31690,31821],[34,151986],{"className":151987,"style":31846},[31845],[34,151989,202],{"className":151990},[31850],[34,151992],{"className":151993,"style":31846},[31845],[34,151995,151997,152001,152004,152071],{"className":151996},[31676],[34,151998],{"className":151999,"style":152000},[31680],"height:1.4578em;vertical-align:-0.413em;",[34,152002,456],{"className":152003},[31782],[34,152005,152007,152010],{"className":152006},[31685],[34,152008,46900],{"className":152009},[31685,31686],[34,152011,152013],{"className":152012},[31792],[34,152014,152016,152062],{"className":152015},[31796,31797],[34,152017,152019,152059],{"className":152018},[31801],[34,152020,152023,152038],{"className":152021,"style":152022},[31805],"height:1.0448em;",[34,152024,152026,152029],{"style":152025},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[34,152027],{"className":152028,"style":31814},[31813],[34,152030,152032],{"className":152031},[31818,31819,31820,31821],[34,152033,152035],{"className":152034},[31685,31821],[34,152036,141027],{"className":152037,"style":32153},[31685,31686,31821],[34,152039,152041,152044],{"style":152040},"top:-3.2198em;margin-right:0.05em;",[34,152042],{"className":152043,"style":31814},[31813],[34,152045,152047],{"className":152046},[31818,31819,31820,31821],[34,152048,152050,152053,152056],{"className":152049},[31685,31821],[34,152051,456],{"className":152052},[31782,31821],[34,152054,459],{"className":152055},[31685,31821],[34,152057,2557],{"className":152058},[31690,31821],[34,152060,31829],{"className":152061},[31828],[34,152063,152065],{"className":152064},[31801],[34,152066,152069],{"className":152067,"style":152068},[31805],"height:0.413em;",[34,152070],{},[34,152072,2557],{"className":152073},[31690],[15,152075,104944],{},[34,152077,152079],{"className":152078},[126385],[34,152080,152082,152112],{"className":152081},[31642],[34,152083,152085],{"className":152084},[31646],[31648,152086,152087],{"xmlns":31650,"display":126394},[31652,152088,152089,152109],{},[31655,152090,152091],{},[38846,152092,152093,152095,152101],{},[31658,152094,46900],{},[31655,152096,152097,152099],{},[31658,152098,31955],{},[31658,152100,31968],{},[31655,152102,152103,152105,152107],{},[31662,152104,456],{"stretchy":702},[31749,152106,459],{},[31662,152108,2557],{"stretchy":702},[31665,152110,152111],{"encoding":31667},"d_{ij}^{(0)}",[34,152113,152115],{"className":152114,"ariaHidden":758},[31672],[34,152116,152118,152121],{"className":152117},[31676],[34,152119],{"className":152120,"style":152000},[31680],[34,152122,152124,152127],{"className":152123},[31685],[34,152125,46900],{"className":152126},[31685,31686],[34,152128,152130],{"className":152129},[31792],[34,152131,152133,152176],{"className":152132},[31796,31797],[34,152134,152136,152173],{"className":152135},[31801],[34,152137,152139,152153],{"className":152138,"style":152022},[31805],[34,152140,152141,152144],{"style":152025},[34,152142],{"className":152143,"style":31814},[31813],[34,152145,152147],{"className":152146},[31818,31819,31820,31821],[34,152148,152150],{"className":152149},[31685,31821],[34,152151,141027],{"className":152152,"style":32153},[31685,31686,31821],[34,152154,152155,152158],{"style":152040},[34,152156],{"className":152157,"style":31814},[31813],[34,152159,152161],{"className":152160},[31818,31819,31820,31821],[34,152162,152164,152167,152170],{"className":152163},[31685,31821],[34,152165,456],{"className":152166},[31782,31821],[34,152168,459],{"className":152169},[31685,31821],[34,152171,2557],{"className":152172},[31690,31821],[34,152174,31829],{"className":152175},[31828],[34,152177,152179],{"className":152178},[31801],[34,152180,152182],{"className":152181,"style":152068},[31805],[34,152183],{},[15,152185,152186,152187,152319],{},"表示边 ",[34,152188,152190,152215],{"className":152189},[31642],[34,152191,152193],{"className":152192},[31646],[31648,152194,152195],{"xmlns":31650},[31652,152196,152197,152213],{},[31655,152198,152199,152205,152207],{},[31743,152200,152201,152203],{},[31658,152202,56782],{},[31658,152204,31955],{},[31662,152206,33239],{},[31743,152208,152209,152211],{},[31658,152210,56782],{},[31658,152212,31968],{},[31665,152214,146847],{"encoding":31667},[34,152216,152218,152273],{"className":152217,"ariaHidden":758},[31672],[34,152219,152221,152224,152264,152267,152270],{"className":152220},[31676],[34,152222],{"className":152223,"style":31860},[31680],[34,152225,152227,152230],{"className":152226},[31685],[34,152228,56782],{"className":152229,"style":32062},[31685,31686],[34,152231,152233],{"className":152232},[31792],[34,152234,152236,152256],{"className":152235},[31796,31797],[34,152237,152239,152253],{"className":152238},[31801],[34,152240,152242],{"className":152241,"style":32090},[31805],[34,152243,152244,152247],{"style":38795},[34,152245],{"className":152246,"style":31814},[31813],[34,152248,152250],{"className":152249},[31818,31819,31820,31821],[34,152251,31955],{"className":152252},[31685,31686,31821],[34,152254,31829],{"className":152255},[31828],[34,152257,152259],{"className":152258},[31801],[34,152260,152262],{"className":152261,"style":31836},[31805],[34,152263],{},[34,152265],{"className":152266,"style":31846},[31845],[34,152268,33239],{"className":152269},[31850],[34,152271],{"className":152272,"style":31846},[31845],[34,152274,152276,152279],{"className":152275},[31676],[34,152277],{"className":152278,"style":56005},[31680],[34,152280,152282,152285],{"className":152281},[31685],[34,152283,56782],{"className":152284,"style":32062},[31685,31686],[34,152286,152288],{"className":152287},[31792],[34,152289,152291,152311],{"className":152290},[31796,31797],[34,152292,152294,152308],{"className":152293},[31801],[34,152295,152297],{"className":152296,"style":32090},[31805],[34,152298,152299,152302],{"style":38795},[34,152300],{"className":152301,"style":31814},[31813],[34,152303,152305],{"className":152304},[31818,31819,31820,31821],[34,152306,31968],{"className":152307,"style":32153},[31685,31686,31821],[34,152309,31829],{"className":152310},[31828],[34,152312,152314],{"className":152313},[31801],[34,152315,152317],{"className":152316,"style":32163},[31805],[34,152318],{}," 的权。",[15,152321,152322],{},"如果没有边：",[34,152324,152326],{"className":152325},[126385],[34,152327,152329,152363],{"className":152328},[31642],[34,152330,152332],{"className":152331},[31646],[31648,152333,152334],{"xmlns":31650,"display":126394},[31652,152335,152336,152360],{},[31655,152337,152338,152356,152358],{},[38846,152339,152340,152342,152348],{},[31658,152341,46900],{},[31655,152343,152344,152346],{},[31658,152345,31955],{},[31658,152347,31968],{},[31655,152349,152350,152352,152354],{},[31662,152351,456],{"stretchy":702},[31749,152353,459],{},[31662,152355,2557],{"stretchy":702},[31662,152357,202],{},[31658,152359,138286],{"mathvariant":38079},[31665,152361,152362],{"encoding":31667},"d_{ij}^{(0)}=\\infty",[34,152364,152366,152444],{"className":152365,"ariaHidden":758},[31672],[34,152367,152369,152372,152435,152438,152441],{"className":152368},[31676],[34,152370],{"className":152371,"style":152000},[31680],[34,152373,152375,152378],{"className":152374},[31685],[34,152376,46900],{"className":152377},[31685,31686],[34,152379,152381],{"className":152380},[31792],[34,152382,152384,152427],{"className":152383},[31796,31797],[34,152385,152387,152424],{"className":152386},[31801],[34,152388,152390,152404],{"className":152389,"style":152022},[31805],[34,152391,152392,152395],{"style":152025},[34,152393],{"className":152394,"style":31814},[31813],[34,152396,152398],{"className":152397},[31818,31819,31820,31821],[34,152399,152401],{"className":152400},[31685,31821],[34,152402,141027],{"className":152403,"style":32153},[31685,31686,31821],[34,152405,152406,152409],{"style":152040},[34,152407],{"className":152408,"style":31814},[31813],[34,152410,152412],{"className":152411},[31818,31819,31820,31821],[34,152413,152415,152418,152421],{"className":152414},[31685,31821],[34,152416,456],{"className":152417},[31782,31821],[34,152419,459],{"className":152420},[31685,31821],[34,152422,2557],{"className":152423},[31690,31821],[34,152425,31829],{"className":152426},[31828],[34,152428,152430],{"className":152429},[31801],[34,152431,152433],{"className":152432,"style":152068},[31805],[34,152434],{},[34,152436],{"className":152437,"style":31846},[31845],[34,152439,202],{"className":152440},[31850],[34,152442],{"className":152443,"style":31846},[31845],[34,152445,152447,152450],{"className":152446},[31676],[34,152448],{"className":152449,"style":32527},[31680],[34,152451,138286],{"className":152452},[31685],[15,152454,130751],{},[34,152456,152458],{"className":152457},[126385],[34,152459,152461,152478],{"className":152460},[31642],[34,152462,152464],{"className":152463},[31646],[31648,152465,152466],{"xmlns":31650,"display":126394},[31652,152467,152468,152476],{},[31655,152469,152470,152472,152474],{},[31658,152471,31955],{},[31662,152473,202],{},[31658,152475,31968],{},[31665,152477,33094],{"encoding":31667},[34,152479,152481,152499],{"className":152480,"ariaHidden":758},[31672],[34,152482,152484,152487,152490,152493,152496],{"className":152483},[31676],[34,152485],{"className":152486,"style":33104},[31680],[34,152488,31955],{"className":152489},[31685,31686],[34,152491],{"className":152492,"style":31846},[31845],[34,152494,202],{"className":152495},[31850],[34,152497],{"className":152498,"style":31846},[31845],[34,152500,152502,152505],{"className":152501},[31676],[34,152503],{"className":152504,"style":33123},[31680],[34,152506,31968],{"className":152507,"style":32153},[31685,31686],[15,152509,137537],{},[34,152511,152513],{"className":152512},[126385],[34,152514,152516,152550],{"className":152515},[31642],[34,152517,152519],{"className":152518},[31646],[31648,152520,152521],{"xmlns":31650,"display":126394},[31652,152522,152523,152547],{},[31655,152524,152525,152543,152545],{},[38846,152526,152527,152529,152535],{},[31658,152528,46900],{},[31655,152530,152531,152533],{},[31658,152532,31955],{},[31658,152534,31955],{},[31655,152536,152537,152539,152541],{},[31662,152538,456],{"stretchy":702},[31749,152540,459],{},[31662,152542,2557],{"stretchy":702},[31662,152544,202],{},[31749,152546,459],{},[31665,152548,152549],{"encoding":31667},"d_{ii}^{(0)}=0",[34,152551,152553,152633],{"className":152552,"ariaHidden":758},[31672],[34,152554,152556,152560,152624,152627,152630],{"className":152555},[31676],[34,152557],{"className":152558,"style":152559},[31680],"height:1.3217em;vertical-align:-0.2769em;",[34,152561,152563,152566],{"className":152562},[31685],[34,152564,46900],{"className":152565},[31685,31686],[34,152567,152569],{"className":152568},[31792],[34,152570,152572,152615],{"className":152571},[31796,31797],[34,152573,152575,152612],{"className":152574},[31801],[34,152576,152578,152592],{"className":152577,"style":152022},[31805],[34,152579,152580,152583],{"style":152025},[34,152581],{"className":152582,"style":31814},[31813],[34,152584,152586],{"className":152585},[31818,31819,31820,31821],[34,152587,152589],{"className":152588},[31685,31821],[34,152590,148811],{"className":152591},[31685,31686,31821],[34,152593,152594,152597],{"style":152040},[34,152595],{"className":152596,"style":31814},[31813],[34,152598,152600],{"className":152599},[31818,31819,31820,31821],[34,152601,152603,152606,152609],{"className":152602},[31685,31821],[34,152604,456],{"className":152605},[31782,31821],[34,152607,459],{"className":152608},[31685,31821],[34,152610,2557],{"className":152611},[31690,31821],[34,152613,31829],{"className":152614},[31828],[34,152616,152618],{"className":152617},[31801],[34,152619,152622],{"className":152620,"style":152621},[31805],"height:0.2769em;",[34,152623],{},[34,152625],{"className":152626,"style":31846},[31845],[34,152628,202],{"className":152629},[31850],[34,152631],{"className":152632,"style":31846},[31845],[34,152634,152636,152639],{"className":152635},[31676],[34,152637],{"className":152638,"style":32617},[31680],[34,152640,459],{"className":152641},[31685],[32188,152643],{},[19,152645,152647],{"id":152646},"_2-核心更新公式","2. 核心更新公式",[15,152649,152650],{},"Floyd 算法的核心是：",[34,152652,152654],{"className":152653},[126385],[34,152655,152657,152781],{"className":152656},[31642],[34,152658,152660],{"className":152659},[31646],[31648,152661,152662],{"xmlns":31650,"display":126394},[31652,152663,152664,152778],{},[31655,152665,152666],{},[126400,152667,152668],{"notation":126402},[71090,152669,152670],{"scriptlevel":459,"displaystyle":702},[71090,152671,152672],{"scriptlevel":459,"displaystyle":702},[71090,152673,152674],{"scriptlevel":459,"displaystyle":758},[31655,152675,152676,152694,152696,152698,152700],{},[38846,152677,152678,152680,152686],{},[31658,152679,46900],{},[31655,152681,152682,152684],{},[31658,152683,31955],{},[31658,152685,31968],{},[31655,152687,152688,152690,152692],{},[31662,152689,456],{"stretchy":702},[31658,152691,33759],{},[31662,152693,2557],{"stretchy":702},[31662,152695,202],{},[31658,152697,56084],{},[31662,152699,32682],{},[31655,152701,152702,152704,152726,152728,152730,152752,152754,152776],{},[31662,152703,456],{"fence":758},[38846,152705,152706,152708,152714],{},[31658,152707,46900],{},[31655,152709,152710,152712],{},[31658,152711,31955],{},[31658,152713,31968],{},[31655,152715,152716,152718,152720,152722,152724],{},[31662,152717,456],{"stretchy":702},[31658,152719,33759],{},[31662,152721,32226],{},[31749,152723,640],{},[31662,152725,2557],{"stretchy":702},[31662,152727,31960],{"separator":758},[39361,152729,130298],{},[38846,152731,152732,152734,152740],{},[31658,152733,46900],{},[31655,152735,152736,152738],{},[31658,152737,31955],{},[31658,152739,33759],{},[31655,152741,152742,152744,152746,152748,152750],{},[31662,152743,456],{"stretchy":702},[31658,152745,33759],{},[31662,152747,32226],{},[31749,152749,640],{},[31662,152751,2557],{"stretchy":702},[31662,152753,223],{},[38846,152755,152756,152758,152764],{},[31658,152757,46900],{},[31655,152759,152760,152762],{},[31658,152761,33759],{},[31658,152763,31968],{},[31655,152765,152766,152768,152770,152772,152774],{},[31662,152767,456],{"stretchy":702},[31658,152769,33759],{},[31662,152771,32226],{},[31749,152773,640],{},[31662,152775,2557],{"stretchy":702},[31662,152777,2557],{"fence":758},[31665,152779,152780],{"encoding":31667},"\\boxed{\nd_{ij}^{(k)}\n=\n\\min\\left(d_{ij}^{(k-1)},\\ d_{ik}^{(k-1)}+d_{kj}^{(k-1)}\\right)\n}",[34,152782,152784],{"className":152783,"ariaHidden":758},[31672],[34,152785,152787,152791],{"className":152786},[31676],[34,152788],{"className":152789,"style":152790},[31680],"height:2.48em;vertical-align:-0.99em;",[34,152792,152794],{"className":152793},[31685],[34,152795,152797,153158],{"className":152796},[31796,31797],[34,152798,152800,153155],{"className":152799},[31801],[34,152801,152804,153145],{"className":152802,"style":152803},[31805],"height:1.49em;",[34,152805,152807,152811],{"style":152806},"top:-4.48em;",[34,152808],{"className":152809,"style":152810},[31813],"height:4.48em;",[34,152812,152814],{"className":152813},[126448],[34,152815,152817],{"className":152816},[31685],[34,152818,152820,152883,152886,152889,152892,152895,152898],{"className":152819},[31685],[34,152821,152823,152826],{"className":152822},[31685],[34,152824,46900],{"className":152825},[31685,31686],[34,152827,152829],{"className":152828},[31792],[34,152830,152832,152875],{"className":152831},[31796,31797],[34,152833,152835,152872],{"className":152834},[31801],[34,152836,152838,152852],{"className":152837,"style":152022},[31805],[34,152839,152840,152843],{"style":152025},[34,152841],{"className":152842,"style":31814},[31813],[34,152844,152846],{"className":152845},[31818,31819,31820,31821],[34,152847,152849],{"className":152848},[31685,31821],[34,152850,141027],{"className":152851,"style":32153},[31685,31686,31821],[34,152853,152854,152857],{"style":152040},[34,152855],{"className":152856,"style":31814},[31813],[34,152858,152860],{"className":152859},[31818,31819,31820,31821],[34,152861,152863,152866,152869],{"className":152862},[31685,31821],[34,152864,456],{"className":152865},[31782,31821],[34,152867,33759],{"className":152868,"style":33774},[31685,31686,31821],[34,152870,2557],{"className":152871},[31690,31821],[34,152873,31829],{"className":152874},[31828],[34,152876,152878],{"className":152877},[31801],[34,152879,152881],{"className":152880,"style":152068},[31805],[34,152882],{},[34,152884],{"className":152885,"style":31846},[31845],[34,152887,202],{"className":152888},[31850],[34,152890],{"className":152891,"style":31846},[31845],[34,152893,56084],{"className":152894},[32767],[34,152896],{"className":152897,"style":32123},[31845],[34,152899,152901,152907,152976,152979,152982,152985,153057,153060,153063,153066,153139],{"className":152900},[34006],[34,152902,152904],{"className":152903,"style":75811},[31782,75810],[34,152905,456],{"className":152906},[71281,141292],[34,152908,152910,152913],{"className":152909},[31685],[34,152911,46900],{"className":152912},[31685,31686],[34,152914,152916],{"className":152915},[31792],[34,152917,152919,152968],{"className":152918},[31796,31797],[34,152920,152922,152965],{"className":152921},[31801],[34,152923,152925,152939],{"className":152924,"style":152022},[31805],[34,152926,152927,152930],{"style":152025},[34,152928],{"className":152929,"style":31814},[31813],[34,152931,152933],{"className":152932},[31818,31819,31820,31821],[34,152934,152936],{"className":152935},[31685,31821],[34,152937,141027],{"className":152938,"style":32153},[31685,31686,31821],[34,152940,152941,152944],{"style":152040},[34,152942],{"className":152943,"style":31814},[31813],[34,152945,152947],{"className":152946},[31818,31819,31820,31821],[34,152948,152950,152953,152956,152959,152962],{"className":152949},[31685,31821],[34,152951,456],{"className":152952},[31782,31821],[34,152954,33759],{"className":152955,"style":33774},[31685,31686,31821],[34,152957,32226],{"className":152958},[32024,31821],[34,152960,640],{"className":152961},[31685,31821],[34,152963,2557],{"className":152964},[31690,31821],[34,152966,31829],{"className":152967},[31828],[34,152969,152971],{"className":152970},[31801],[34,152972,152974],{"className":152973,"style":152068},[31805],[34,152975],{},[34,152977,31960],{"className":152978},[32119],[34,152980,130298],{"className":152981},[31845],[34,152983],{"className":152984,"style":32123},[31845],[34,152986,152988,152991],{"className":152987},[31685],[34,152989,46900],{"className":152990},[31685,31686],[34,152992,152994],{"className":152993},[31792],[34,152995,152997,153048],{"className":152996},[31796,31797],[34,152998,153000,153045],{"className":152999},[31801],[34,153001,153003,153019],{"className":153002,"style":152022},[31805],[34,153004,153006,153009],{"style":153005},"top:-2.3987em;margin-left:0em;margin-right:0.05em;",[34,153007],{"className":153008,"style":31814},[31813],[34,153010,153012],{"className":153011},[31818,31819,31820,31821],[34,153013,153015],{"className":153014},[31685,31821],[34,153016,153018],{"className":153017,"style":33774},[31685,31686,31821],"ik",[34,153020,153021,153024],{"style":152040},[34,153022],{"className":153023,"style":31814},[31813],[34,153025,153027],{"className":153026},[31818,31819,31820,31821],[34,153028,153030,153033,153036,153039,153042],{"className":153029},[31685,31821],[34,153031,456],{"className":153032},[31782,31821],[34,153034,33759],{"className":153035,"style":33774},[31685,31686,31821],[34,153037,32226],{"className":153038},[32024,31821],[34,153040,640],{"className":153041},[31685,31821],[34,153043,2557],{"className":153044},[31690,31821],[34,153046,31829],{"className":153047},[31828],[34,153049,153051],{"className":153050},[31801],[34,153052,153055],{"className":153053,"style":153054},[31805],"height:0.3013em;",[34,153056],{},[34,153058],{"className":153059,"style":32604},[31845],[34,153061,223],{"className":153062},[32024],[34,153064],{"className":153065,"style":32604},[31845],[34,153067,153069,153072],{"className":153068},[31685],[34,153070,46900],{"className":153071},[31685,31686],[34,153073,153075],{"className":153074},[31792],[34,153076,153078,153130],{"className":153077},[31796,31797],[34,153079,153081,153127],{"className":153080},[31801],[34,153082,153084,153101],{"className":153083,"style":152022},[31805],[34,153085,153086,153089],{"style":153005},[34,153087],{"className":153088,"style":31814},[31813],[34,153090,153092],{"className":153091},[31818,31819,31820,31821],[34,153093,153095,153098],{"className":153094},[31685,31821],[34,153096,33759],{"className":153097,"style":33774},[31685,31686,31821],[34,153099,31968],{"className":153100,"style":32153},[31685,31686,31821],[34,153102,153103,153106],{"style":152040},[34,153104],{"className":153105,"style":31814},[31813],[34,153107,153109],{"className":153108},[31818,31819,31820,31821],[34,153110,153112,153115,153118,153121,153124],{"className":153111},[31685,31821],[34,153113,456],{"className":153114},[31782,31821],[34,153116,33759],{"className":153117,"style":33774},[31685,31686,31821],[34,153119,32226],{"className":153120},[32024,31821],[34,153122,640],{"className":153123},[31685,31821],[34,153125,2557],{"className":153126},[31690,31821],[34,153128,31829],{"className":153129},[31828],[34,153131,153133],{"className":153132},[31801],[34,153134,153137],{"className":153135,"style":153136},[31805],"height:0.4374em;",[34,153138],{},[34,153140,153142],{"className":153141,"style":75811},[31690,75810],[34,153143,2557],{"className":153144},[71281,141292],[34,153146,153148,153151],{"style":153147},"top:-3.49em;",[34,153149],{"className":153150,"style":152810},[31813],[34,153152],{"className":153153,"style":153154},[126470,126471],"height:2.48em;border-style:solid;border-width:0.04em;",[34,153156,31829],{"className":153157},[31828],[34,153159,153161],{"className":153160},[31801],[34,153162,153165],{"className":153163,"style":153164},[31805],"height:0.99em;",[34,153166],{},[15,153168,131317],{},[104833,153170,153171],{},[15,153172,135477,153173,53834,153201,153229,153230,153258,153259,347],{},[34,153174,153176,153189],{"className":153175},[31642],[34,153177,153179],{"className":153178},[31646],[31648,153180,153181],{"xmlns":31650},[31652,153182,153183,153187],{},[31655,153184,153185],{},[31658,153186,31955],{},[31665,153188,31955],{"encoding":31667},[34,153190,153192],{"className":153191,"ariaHidden":758},[31672],[34,153193,153195,153198],{"className":153194},[31676],[34,153196],{"className":153197,"style":33104},[31680],[34,153199,31955],{"className":153200},[31685,31686],[34,153202,153204,153217],{"className":153203},[31642],[34,153205,153207],{"className":153206},[31646],[31648,153208,153209],{"xmlns":31650},[31652,153210,153211,153215],{},[31655,153212,153213],{},[31658,153214,31968],{},[31665,153216,31968],{"encoding":31667},[34,153218,153220],{"className":153219,"ariaHidden":758},[31672],[34,153221,153223,153226],{"className":153222},[31676],[34,153224],{"className":153225,"style":33123},[31680],[34,153227,31968],{"className":153228,"style":32153},[31685,31686]," 的最短路，要么不经过 ",[34,153231,153233,153246],{"className":153232},[31642],[34,153234,153236],{"className":153235},[31646],[31648,153237,153238],{"xmlns":31650},[31652,153239,153240,153244],{},[31655,153241,153242],{},[31658,153243,33759],{},[31665,153245,33759],{"encoding":31667},[34,153247,153249],{"className":153248,"ariaHidden":758},[31672],[34,153250,153252,153255],{"className":153251},[31676],[34,153253],{"className":153254,"style":31681},[31680],[34,153256,33759],{"className":153257,"style":33774},[31685,31686],"，要么经过 ",[34,153260,153262,153275],{"className":153261},[31642],[34,153263,153265],{"className":153264},[31646],[31648,153266,153267],{"xmlns":31650},[31652,153268,153269,153273],{},[31655,153270,153271],{},[31658,153272,33759],{},[31665,153274,33759],{"encoding":31667},[34,153276,153278],{"className":153277,"ariaHidden":758},[31672],[34,153279,153281,153284],{"className":153280},[31676],[34,153282],{"className":153283,"style":31681},[31680],[34,153285,33759],{"className":153286,"style":33774},[31685,31686],[15,153288,153289,153290,153318],{},"如果经过 ",[34,153291,153293,153306],{"className":153292},[31642],[34,153294,153296],{"className":153295},[31646],[31648,153297,153298],{"xmlns":31650},[31652,153299,153300,153304],{},[31655,153301,153302],{},[31658,153303,33759],{},[31665,153305,33759],{"encoding":31667},[34,153307,153309],{"className":153308,"ariaHidden":758},[31672],[34,153310,153312,153315],{"className":153311},[31676],[34,153313],{"className":153314,"style":31681},[31680],[34,153316,33759],{"className":153317,"style":33774},[31685,31686]," 更短，就更新。",[32188,153320],{},[19,153322,153324],{"id":153323},"_3-算法过程","3. 算法过程",[15,153326,153327],{},"依次令：",[34,153329,153331],{"className":153330},[126385],[34,153332,153334,153364],{"className":153333},[31642],[34,153335,153337],{"className":153336},[31646],[31648,153338,153339],{"xmlns":31650,"display":126394},[31652,153340,153341,153361],{},[31655,153342,153343,153345,153347,153349,153351,153353,153355,153357,153359],{},[31658,153344,33759],{},[31662,153346,202],{},[31749,153348,640],{},[31662,153350,31960],{"separator":758},[31749,153352,768],{},[31662,153354,31960],{"separator":758},[31662,153356,33934],{},[31662,153358,31960],{"separator":758},[31658,153360,31660],{},[31665,153362,153363],{"encoding":31667},"k=1,2,\\dots,n",[34,153365,153367,153385],{"className":153366,"ariaHidden":758},[31672],[34,153368,153370,153373,153376,153379,153382],{"className":153369},[31676],[34,153371],{"className":153372,"style":31681},[31680],[34,153374,33759],{"className":153375,"style":33774},[31685,31686],[34,153377],{"className":153378,"style":31846},[31845],[34,153380,202],{"className":153381},[31850],[34,153383],{"className":153384,"style":31846},[31845],[34,153386,153388,153391,153394,153397,153400,153403,153406,153409,153412,153415,153418,153421],{"className":153387},[31676],[34,153389],{"className":153390,"style":40292},[31680],[34,153392,640],{"className":153393},[31685],[34,153395,31960],{"className":153396},[32119],[34,153398],{"className":153399,"style":32123},[31845],[34,153401,768],{"className":153402},[31685],[34,153404,31960],{"className":153405},[32119],[34,153407],{"className":153408,"style":32123},[31845],[34,153410,33934],{"className":153411},[34006],[34,153413],{"className":153414,"style":32123},[31845],[34,153416,31960],{"className":153417},[32119],[34,153419],{"className":153420,"style":32123},[31845],[34,153422,31660],{"className":153423},[31685,31686],[15,153425,153426],{},"逐步允许路径经过更多中间顶点。",[15,153428,153429],{},"最终：",[34,153431,153433],{"className":153432},[126385],[34,153434,153436,153460],{"className":153435},[31642],[34,153437,153439],{"className":153438},[31646],[31648,153440,153441],{"xmlns":31650,"display":126394},[31652,153442,153443,153457],{},[31655,153444,153445],{},[37624,153446,153447,153449],{},[31658,153448,91994],{},[31655,153450,153451,153453,153455],{},[31662,153452,456],{"stretchy":702},[31658,153454,31660],{},[31662,153456,2557],{"stretchy":702},[31665,153458,153459],{"encoding":31667},"D^{(n)}",[34,153461,153463],{"className":153462,"ariaHidden":758},[31672],[34,153464,153466,153469],{"className":153465},[31676],[34,153467],{"className":153468,"style":147280},[31680],[34,153470,153472,153475],{"className":153471},[31685],[34,153473,91994],{"className":153474,"style":34452},[31685,31686],[34,153476,153478],{"className":153477},[31792],[34,153479,153481],{"className":153480},[31796],[34,153482,153484],{"className":153483},[31801],[34,153485,153487],{"className":153486,"style":147280},[31805],[34,153488,153489,153492],{"style":129846},[34,153490],{"className":153491,"style":31814},[31813],[34,153493,153495],{"className":153494},[31818,31819,31820,31821],[34,153496,153498,153501,153504],{"className":153497},[31685,31821],[34,153499,456],{"className":153500},[31782,31821],[34,153502,31660],{"className":153503},[31685,31686,31821],[34,153505,2557],{"className":153506},[31690,31821],[15,153508,153509],{},"就是所有顶点对之间的最短距离矩阵。",[32188,153511],{},[19,153513,153515],{"id":153514},"_4-dijkstra-与-floyd-区别","4. Dijkstra 与 Floyd 区别",[13837,153517,153518,153528],{},[13840,153519,153520],{},[13843,153521,153522,153525],{},[13846,153523,153524],{},"算法",[13846,153526,153527],{},"作用",[13853,153529,153530,153538],{},[13843,153531,153532,153535],{},[13858,153533,153534],{},"Dijkstra",[13858,153536,153537],{},"求一个源点到其他点的最短路径",[13843,153539,153540,153543],{},[13858,153541,153542],{},"Floyd",[13858,153544,153545],{},"求任意两点之间的最短路径",[15,153547,124692],{},[34,153549,153551],{"className":153550},[126385],[34,153552,153554,153577],{"className":153553},[31642],[34,153555,153557],{"className":153556},[31646],[31648,153558,153559],{"xmlns":31650,"display":126394},[31652,153560,153561,153574],{},[31655,153562,153563],{},[126400,153564,153565],{"notation":126402},[71090,153566,153567],{"scriptlevel":459,"displaystyle":702},[71090,153568,153569],{"scriptlevel":459,"displaystyle":702},[71090,153570,153571],{"scriptlevel":459,"displaystyle":758},[39361,153572,153573],{},"Dijkstra 是单源，Floyd 是全源",[31665,153575,153576],{"encoding":31667},"\\boxed{\\text{Dijkstra 是单源，Floyd 是全源}}",[34,153578,153580],{"className":153579,"ariaHidden":758},[31672],[34,153581,153583,153587],{"className":153582},[31676],[34,153584],{"className":153585,"style":153586},[31680],"height:1.5689em;vertical-align:-0.5344em;",[34,153588,153590],{"className":153589},[31685],[34,153591,153593,153646],{"className":153592},[31796,31797],[34,153594,153596,153643],{"className":153595},[31801],[34,153597,153599,153634],{"className":153598,"style":135867},[31805],[34,153600,153602,153606],{"style":153601},"top:-3.5689em;",[34,153603],{"className":153604,"style":153605},[31813],"height:3.5689em;",[34,153607,153609],{"className":153608},[126448],[34,153610,153612],{"className":153611},[31685],[34,153613,153615],{"className":153614},[31685],[34,153616,153618,153622,153626,153630],{"className":153617},[31685,2638],[34,153619,153621],{"className":153620},[31685],"Dijkstra ",[34,153623,153625],{"className":153624},[31685,105462],"是单源，",[34,153627,153629],{"className":153628},[31685],"Floyd ",[34,153631,153633],{"className":153632},[31685,105462],"是全源",[34,153635,153636,153639],{"style":135889},[34,153637],{"className":153638,"style":153605},[31813],[34,153640],{"className":153641,"style":153642},[126470,126471],"height:1.5689em;border-style:solid;border-width:0.04em;",[34,153644,31829],{"className":153645},[31828],[34,153647,153649],{"className":153648},[31801],[34,153650,153652],{"className":153651,"style":142798},[31805],[34,153653],{},[32188,153655],{},[11,153657,153659],{"id":153658},"五决策图与最优路径","五、决策图与最优路径",[19,153661,153663],{"id":153662},"_1-决策图","1. 决策图",[15,153665,153666],{},"决策图是一类有向赋权图。",[15,153668,153669],{},"顶点分层：",[34,153671,153673],{"className":153672},[126385],[34,153674,153676,153714],{"className":153675},[31642],[34,153677,153679],{"className":153678},[31646],[31648,153680,153681],{"xmlns":31650,"display":126394},[31652,153682,153683,153711],{},[31655,153684,153685,153691,153693,153699,153701,153703,153705],{},[31743,153686,153687,153689],{},[31658,153688,126532],{},[31749,153690,640],{},[31662,153692,31960],{"separator":758},[31743,153694,153695,153697],{},[31658,153696,126532],{},[31749,153698,768],{},[31662,153700,31960],{"separator":758},[31662,153702,33934],{},[31662,153704,31960],{"separator":758},[31743,153706,153707,153709],{},[31658,153708,126532],{},[31658,153710,31747],{},[31665,153712,153713],{"encoding":31667},"V_1,V_2,\\dots,V_s",[34,153715,153717],{"className":153716,"ariaHidden":758},[31672],[34,153718,153720,153723,153764,153767,153770,153810,153813,153816,153819,153822,153825,153828],{"className":153719},[31676],[34,153721],{"className":153722,"style":70739},[31680],[34,153724,153726,153729],{"className":153725},[31685],[34,153727,126532],{"className":153728,"style":32604},[31685,31686],[34,153730,153732],{"className":153731},[31792],[34,153733,153735,153756],{"className":153734},[31796,31797],[34,153736,153738,153753],{"className":153737},[31801],[34,153739,153741],{"className":153740,"style":31806},[31805],[34,153742,153744,153747],{"style":153743},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[34,153745],{"className":153746,"style":31814},[31813],[34,153748,153750],{"className":153749},[31818,31819,31820,31821],[34,153751,640],{"className":153752},[31685,31821],[34,153754,31829],{"className":153755},[31828],[34,153757,153759],{"className":153758},[31801],[34,153760,153762],{"className":153761,"style":31836},[31805],[34,153763],{},[34,153765,31960],{"className":153766},[32119],[34,153768],{"className":153769,"style":32123},[31845],[34,153771,153773,153776],{"className":153772},[31685],[34,153774,126532],{"className":153775,"style":32604},[31685,31686],[34,153777,153779],{"className":153778},[31792],[34,153780,153782,153802],{"className":153781},[31796,31797],[34,153783,153785,153799],{"className":153784},[31801],[34,153786,153788],{"className":153787,"style":31806},[31805],[34,153789,153790,153793],{"style":153743},[34,153791],{"className":153792,"style":31814},[31813],[34,153794,153796],{"className":153795},[31818,31819,31820,31821],[34,153797,768],{"className":153798},[31685,31821],[34,153800,31829],{"className":153801},[31828],[34,153803,153805],{"className":153804},[31801],[34,153806,153808],{"className":153807,"style":31836},[31805],[34,153809],{},[34,153811,31960],{"className":153812},[32119],[34,153814],{"className":153815,"style":32123},[31845],[34,153817,33934],{"className":153818},[34006],[34,153820],{"className":153821,"style":32123},[31845],[34,153823,31960],{"className":153824},[32119],[34,153826],{"className":153827,"style":32123},[31845],[34,153829,153831,153834],{"className":153830},[31685],[34,153832,126532],{"className":153833,"style":32604},[31685,31686],[34,153835,153837],{"className":153836},[31792],[34,153838,153840,153860],{"className":153839},[31796,31797],[34,153841,153843,153857],{"className":153842},[31801],[34,153844,153846],{"className":153845,"style":32362},[31805],[34,153847,153848,153851],{"style":153743},[34,153849],{"className":153850,"style":31814},[31813],[34,153852,153854],{"className":153853},[31818,31819,31820,31821],[34,153855,31747],{"className":153856},[31685,31686,31821],[34,153858,31829],{"className":153859},[31828],[34,153861,153863],{"className":153862},[31801],[34,153864,153866],{"className":153865,"style":31836},[31805],[34,153867],{},[15,153869,153870],{},"边只从前一层指向后一层。",[15,153872,153873],{},"这种图常用来表示多阶段决策问题。",[32188,153875],{},[19,153877,153879],{"id":153878},"_2-最优路径","2. 最优路径",[15,153881,153882,153883,347],{},"从初始顶点到终止顶点的路径中，权值总和最小的路径叫",[2627,153884,153885],{},"最优路径",[15,153887,153888],{},"求法通常是动态规划思想：",[104833,153890,153891],{},[15,153892,153893],{},"从后往前，逐层计算每个顶点到终点的最优值。",[32188,153895],{},[19,153897,153899],{"id":153898},"_3-核心思想","3. 核心思想",[15,153901,153902,153903,153931,153932,131108],{},"如果从 ",[34,153904,153906,153919],{"className":153905},[31642],[34,153907,153909],{"className":153908},[31646],[31648,153910,153911],{"xmlns":31650},[31652,153912,153913,153917],{},[31655,153914,153915],{},[31658,153916,56782],{},[31665,153918,56782],{"encoding":31667},[34,153920,153922],{"className":153921,"ariaHidden":758},[31672],[34,153923,153925,153928],{"className":153924},[31676],[34,153926],{"className":153927,"style":32527},[31680],[34,153929,56782],{"className":153930,"style":32062},[31685,31686]," 出发，可以走到若干后继顶点 ",[34,153933,153935,153973],{"className":153934},[31642],[34,153936,153938],{"className":153937},[31646],[31648,153939,153940],{"xmlns":31650},[31652,153941,153942,153970],{},[31655,153943,153944,153950,153952,153958,153960,153962,153964],{},[31743,153945,153946,153948],{},[31658,153947,94628],{},[31749,153949,640],{},[31662,153951,31960],{"separator":758},[31743,153953,153954,153956],{},[31658,153955,94628],{},[31749,153957,768],{},[31662,153959,31960],{"separator":758},[31662,153961,33934],{},[31662,153963,31960],{"separator":758},[31743,153965,153966,153968],{},[31658,153967,94628],{},[31658,153969,33759],{},[31665,153971,153972],{"encoding":31667},"u_1,u_2,\\dots,u_k",[34,153974,153976],{"className":153975,"ariaHidden":758},[31672],[34,153977,153979,153982,154022,154025,154028,154068,154071,154074,154077,154080,154083,154086],{"className":153978},[31676],[34,153980],{"className":153981,"style":38774},[31680],[34,153983,153985,153988],{"className":153984},[31685],[34,153986,94628],{"className":153987},[31685,31686],[34,153989,153991],{"className":153990},[31792],[34,153992,153994,154014],{"className":153993},[31796,31797],[34,153995,153997,154011],{"className":153996},[31801],[34,153998,154000],{"className":153999,"style":31806},[31805],[34,154001,154002,154005],{"style":31809},[34,154003],{"className":154004,"style":31814},[31813],[34,154006,154008],{"className":154007},[31818,31819,31820,31821],[34,154009,640],{"className":154010},[31685,31821],[34,154012,31829],{"className":154013},[31828],[34,154015,154017],{"className":154016},[31801],[34,154018,154020],{"className":154019,"style":31836},[31805],[34,154021],{},[34,154023,31960],{"className":154024},[32119],[34,154026],{"className":154027,"style":32123},[31845],[34,154029,154031,154034],{"className":154030},[31685],[34,154032,94628],{"className":154033},[31685,31686],[34,154035,154037],{"className":154036},[31792],[34,154038,154040,154060],{"className":154039},[31796,31797],[34,154041,154043,154057],{"className":154042},[31801],[34,154044,154046],{"className":154045,"style":31806},[31805],[34,154047,154048,154051],{"style":31809},[34,154049],{"className":154050,"style":31814},[31813],[34,154052,154054],{"className":154053},[31818,31819,31820,31821],[34,154055,768],{"className":154056},[31685,31821],[34,154058,31829],{"className":154059},[31828],[34,154061,154063],{"className":154062},[31801],[34,154064,154066],{"className":154065,"style":31836},[31805],[34,154067],{},[34,154069,31960],{"className":154070},[32119],[34,154072],{"className":154073,"style":32123},[31845],[34,154075,33934],{"className":154076},[34006],[34,154078],{"className":154079,"style":32123},[31845],[34,154081,31960],{"className":154082},[32119],[34,154084],{"className":154085,"style":32123},[31845],[34,154087,154089,154092],{"className":154088},[31685],[34,154090,94628],{"className":154091},[31685,31686],[34,154093,154095],{"className":154094},[31792],[34,154096,154098,154118],{"className":154097},[31796,31797],[34,154099,154101,154115],{"className":154100},[31801],[34,154102,154104],{"className":154103,"style":33833},[31805],[34,154105,154106,154109],{"style":31809},[34,154107],{"className":154108,"style":31814},[31813],[34,154110,154112],{"className":154111},[31818,31819,31820,31821],[34,154113,33759],{"className":154114,"style":33774},[31685,31686,31821],[34,154116,31829],{"className":154117},[31828],[34,154119,154121],{"className":154120},[31801],[34,154122,154124],{"className":154123,"style":31836},[31805],[34,154125],{},[34,154127,154129],{"className":154128},[126385],[34,154130,154132,154196],{"className":154131},[31642],[34,154133,154135],{"className":154134},[31646],[31648,154136,154137],{"xmlns":31650,"display":126394},[31652,154138,154139,154193],{},[31655,154140,154141,154143,154145,154147,154149,154151,154161],{},[31658,154142,51834],{},[31662,154144,456],{"stretchy":702},[31658,154146,56782],{},[31662,154148,2557],{"stretchy":702},[31662,154150,202],{},[71115,154152,154153,154159],{},[31655,154154,154155,154157],{},[31658,154156,56084],{},[31662,154158,32682],{},[31658,154160,31955],{},[31655,154162,154163,154165,154167,154169,154171,154177,154179,154181,154183,154185,154191],{},[31658,154164,56852],{},[31662,154166,456],{"stretchy":702},[31658,154168,56782],{},[31662,154170,31960],{"separator":758},[31743,154172,154173,154175],{},[31658,154174,94628],{},[31658,154176,31955],{},[31662,154178,2557],{"stretchy":702},[31662,154180,223],{},[31658,154182,51834],{},[31662,154184,456],{"stretchy":702},[31743,154186,154187,154189],{},[31658,154188,94628],{},[31658,154190,31955],{},[31662,154192,2557],{"stretchy":702},[31665,154194,154195],{"encoding":31667},"F(v)=\\min_i{W(v,u_i)+F(u_i)}",[34,154197,154199,154226],{"className":154198,"ariaHidden":758},[31672],[34,154200,154202,154205,154208,154211,154214,154217,154220,154223],{"className":154201},[31676],[34,154203],{"className":154204,"style":31774},[31680],[34,154206,51834],{"className":154207,"style":37422},[31685,31686],[34,154209,456],{"className":154210},[31782],[34,154212,56782],{"className":154213,"style":32062},[31685,31686],[34,154215,2557],{"className":154216},[31690],[34,154218],{"className":154219,"style":31846},[31845],[34,154221,202],{"className":154222},[31850],[34,154224],{"className":154225,"style":31846},[31845],[34,154227,154229,154233,154279,154282],{"className":154228},[31676],[34,154230],{"className":154231,"style":154232},[31680],"height:1.4777em;vertical-align:-0.7277em;",[34,154234,154236],{"className":154235},[32767,71420],[34,154237,154239,154270],{"className":154238},[31796,31797],[34,154240,154242,154267],{"className":154241},[31801],[34,154243,154245,154257],{"className":154244,"style":71430},[31805],[34,154246,154248,154251],{"style":154247},"top:-2.3723em;margin-left:0em;",[34,154249],{"className":154250,"style":41508},[31813],[34,154252,154254],{"className":154253},[31818,31819,31820,31821],[34,154255,31955],{"className":154256},[31685,31686,31821],[34,154258,154259,154262],{"style":41565},[34,154260],{"className":154261,"style":41508},[31813],[34,154263,154264],{},[34,154265,56084],{"className":154266},[32767],[34,154268,31829],{"className":154269},[31828],[34,154271,154273],{"className":154272},[31801],[34,154274,154277],{"className":154275,"style":154276},[31805],"height:0.7277em;",[34,154278],{},[34,154280],{"className":154281,"style":32123},[31845],[34,154283,154285,154288,154291,154294,154297,154300,154340,154343,154346,154349,154352,154355,154358,154398],{"className":154284},[31685],[34,154286,56852],{"className":154287,"style":37422},[31685,31686],[34,154289,456],{"className":154290},[31782],[34,154292,56782],{"className":154293,"style":32062},[31685,31686],[34,154295,31960],{"className":154296},[32119],[34,154298],{"className":154299,"style":32123},[31845],[34,154301,154303,154306],{"className":154302},[31685],[34,154304,94628],{"className":154305},[31685,31686],[34,154307,154309],{"className":154308},[31792],[34,154310,154312,154332],{"className":154311},[31796,31797],[34,154313,154315,154329],{"className":154314},[31801],[34,154316,154318],{"className":154317,"style":32090},[31805],[34,154319,154320,154323],{"style":31809},[34,154321],{"className":154322,"style":31814},[31813],[34,154324,154326],{"className":154325},[31818,31819,31820,31821],[34,154327,31955],{"className":154328},[31685,31686,31821],[34,154330,31829],{"className":154331},[31828],[34,154333,154335],{"className":154334},[31801],[34,154336,154338],{"className":154337,"style":31836},[31805],[34,154339],{},[34,154341,2557],{"className":154342},[31690],[34,154344],{"className":154345,"style":32604},[31845],[34,154347,223],{"className":154348},[32024],[34,154350],{"className":154351,"style":32604},[31845],[34,154353,51834],{"className":154354,"style":37422},[31685,31686],[34,154356,456],{"className":154357},[31782],[34,154359,154361,154364],{"className":154360},[31685],[34,154362,94628],{"className":154363},[31685,31686],[34,154365,154367],{"className":154366},[31792],[34,154368,154370,154390],{"className":154369},[31796,31797],[34,154371,154373,154387],{"className":154372},[31801],[34,154374,154376],{"className":154375,"style":32090},[31805],[34,154377,154378,154381],{"style":31809},[34,154379],{"className":154380,"style":31814},[31813],[34,154382,154384],{"className":154383},[31818,31819,31820,31821],[34,154385,31955],{"className":154386},[31685,31686,31821],[34,154388,31829],{"className":154389},[31828],[34,154391,154393],{"className":154392},[31801],[34,154394,154396],{"className":154395,"style":31836},[31805],[34,154397],{},[34,154399,2557],{"className":154400},[31690],[15,154402,131317],{},[104833,154404,154405],{},[15,154406,154407],{},"从当前点出发的最优值 = 走一条边的代价 + 后面最优值 的最小值。",[32188,154409],{},[11,154411,154413],{"id":154412},"六pert-图与关键路径","六、PERT 图与关键路径",[15,154415,154416],{},"PERT 图用于工程项目管理。",[19,154418,154420],{"id":154419},"_1-pert-图的特点","1. PERT 图的特点",[15,154422,154423],{},"PERT 图是一种有向无环图。",[105,154425,154426,154429,154432,154435,154438],{},[108,154427,154428],{},"顶点表示事件；",[108,154430,154431],{},"边表示任务；",[108,154433,154434],{},"边权表示完成任务所需时间；",[108,154436,154437],{},"有一个源点；",[108,154439,154440],{},"有一个汇点。",[32188,154442],{},[19,154444,154446],{"id":154445},"_2-最早完成时间","2. 最早完成时间",[15,154448,154449,154450,154519],{},"顶点 ",[34,154451,154453,154470],{"className":154452},[31642],[34,154454,154456],{"className":154455},[31646],[31648,154457,154458],{"xmlns":31650},[31652,154459,154460,154468],{},[31655,154461,154462],{},[31743,154463,154464,154466],{},[31658,154465,56782],{},[31658,154467,31955],{},[31665,154469,56787],{"encoding":31667},[34,154471,154473],{"className":154472,"ariaHidden":758},[31672],[34,154474,154476,154479],{"className":154475},[31676],[34,154477],{"className":154478,"style":31860},[31680],[34,154480,154482,154485],{"className":154481},[31685],[34,154483,56782],{"className":154484,"style":32062},[31685,31686],[34,154486,154488],{"className":154487},[31792],[34,154489,154491,154511],{"className":154490},[31796,31797],[34,154492,154494,154508],{"className":154493},[31801],[34,154495,154497],{"className":154496,"style":32090},[31805],[34,154498,154499,154502],{"style":38795},[34,154500],{"className":154501,"style":31814},[31813],[34,154503,154505],{"className":154504},[31818,31819,31820,31821],[34,154506,31955],{"className":154507},[31685,31686,31821],[34,154509,31829],{"className":154510},[31828],[34,154512,154514],{"className":154513},[31801],[34,154515,154517],{"className":154516,"style":31836},[31805],[34,154518],{}," 的最早完成时间记作：",[34,154521,154523],{"className":154522},[126385],[34,154524,154526,154552],{"className":154525},[31642],[34,154527,154529],{"className":154528},[31646],[31648,154530,154531],{"xmlns":31650,"display":126394},[31652,154532,154533,154549],{},[31655,154534,154535,154537,154539,154541,154547],{},[31658,154536,37400],{},[31658,154538,96058],{},[31662,154540,456],{"stretchy":702},[31743,154542,154543,154545],{},[31658,154544,56782],{},[31658,154546,31955],{},[31662,154548,2557],{"stretchy":702},[31665,154550,154551],{"encoding":31667},"TE(v_i)",[34,154553,154555],{"className":154554,"ariaHidden":758},[31672],[34,154556,154558,154561,154564,154567,154570,154610],{"className":154557},[31676],[34,154559],{"className":154560,"style":31774},[31680],[34,154562,37400],{"className":154563,"style":37422},[31685,31686],[34,154565,96058],{"className":154566,"style":106455},[31685,31686],[34,154568,456],{"className":154569},[31782],[34,154571,154573,154576],{"className":154572},[31685],[34,154574,56782],{"className":154575,"style":32062},[31685,31686],[34,154577,154579],{"className":154578},[31792],[34,154580,154582,154602],{"className":154581},[31796,31797],[34,154583,154585,154599],{"className":154584},[31801],[34,154586,154588],{"className":154587,"style":32090},[31805],[34,154589,154590,154593],{"style":38795},[34,154591],{"className":154592,"style":31814},[31813],[34,154594,154596],{"className":154595},[31818,31819,31820,31821],[34,154597,31955],{"className":154598},[31685,31686,31821],[34,154600,31829],{"className":154601},[31828],[34,154603,154605],{"className":154604},[31801],[34,154606,154608],{"className":154607,"style":31836},[31805],[34,154609],{},[34,154611,2557],{"className":154612},[31690],[15,154614,154615,154616,154685],{},"表示在不违反先后顺序的情况下，事件 ",[34,154617,154619,154636],{"className":154618},[31642],[34,154620,154622],{"className":154621},[31646],[31648,154623,154624],{"xmlns":31650},[31652,154625,154626,154634],{},[31655,154627,154628],{},[31743,154629,154630,154632],{},[31658,154631,56782],{},[31658,154633,31955],{},[31665,154635,56787],{"encoding":31667},[34,154637,154639],{"className":154638,"ariaHidden":758},[31672],[34,154640,154642,154645],{"className":154641},[31676],[34,154643],{"className":154644,"style":31860},[31680],[34,154646,154648,154651],{"className":154647},[31685],[34,154649,56782],{"className":154650,"style":32062},[31685,31686],[34,154652,154654],{"className":154653},[31792],[34,154655,154657,154677],{"className":154656},[31796,31797],[34,154658,154660,154674],{"className":154659},[31801],[34,154661,154663],{"className":154662,"style":32090},[31805],[34,154664,154665,154668],{"style":38795},[34,154666],{"className":154667,"style":31814},[31813],[34,154669,154671],{"className":154670},[31818,31819,31820,31821],[34,154672,31955],{"className":154673},[31685,31686,31821],[34,154675,31829],{"className":154676},[31828],[34,154678,154680],{"className":154679},[31801],[34,154681,154683],{"className":154682,"style":31836},[31805],[34,154684],{}," 最早什么时候能发生。",[15,154687,19690],{},[34,154689,154691],{"className":154690},[126385],[34,154692,154694,154724],{"className":154693},[31642],[34,154695,154697],{"className":154696},[31646],[31648,154698,154699],{"xmlns":31650,"display":126394},[31652,154700,154701,154721],{},[31655,154702,154703,154705,154707,154709,154715,154717,154719],{},[31658,154704,37400],{},[31658,154706,96058],{},[31662,154708,456],{"stretchy":702},[31743,154710,154711,154713],{},[31658,154712,56782],{},[31749,154714,640],{},[31662,154716,2557],{"stretchy":702},[31662,154718,202],{},[31749,154720,459],{},[31665,154722,154723],{"encoding":31667},"TE(v_1)=0",[34,154725,154727,154794],{"className":154726,"ariaHidden":758},[31672],[34,154728,154730,154733,154736,154739,154742,154782,154785,154788,154791],{"className":154729},[31676],[34,154731],{"className":154732,"style":31774},[31680],[34,154734,37400],{"className":154735,"style":37422},[31685,31686],[34,154737,96058],{"className":154738,"style":106455},[31685,31686],[34,154740,456],{"className":154741},[31782],[34,154743,154745,154748],{"className":154744},[31685],[34,154746,56782],{"className":154747,"style":32062},[31685,31686],[34,154749,154751],{"className":154750},[31792],[34,154752,154754,154774],{"className":154753},[31796,31797],[34,154755,154757,154771],{"className":154756},[31801],[34,154758,154760],{"className":154759,"style":31806},[31805],[34,154761,154762,154765],{"style":38795},[34,154763],{"className":154764,"style":31814},[31813],[34,154766,154768],{"className":154767},[31818,31819,31820,31821],[34,154769,640],{"className":154770},[31685,31821],[34,154772,31829],{"className":154773},[31828],[34,154775,154777],{"className":154776},[31801],[34,154778,154780],{"className":154779,"style":31836},[31805],[34,154781],{},[34,154783,2557],{"className":154784},[31690],[34,154786],{"className":154787,"style":31846},[31845],[34,154789,202],{"className":154790},[31850],[34,154792],{"className":154793,"style":31846},[31845],[34,154795,154797,154800],{"className":154796},[31676],[34,154798],{"className":154799,"style":32617},[31680],[34,154801,459],{"className":154802},[31685],[15,154804,154805],{},"对其他点：",[34,154807,154809],{"className":154808},[126385],[34,154810,154812,154882],{"className":154811},[31642],[34,154813,154815],{"className":154814},[31646],[31648,154816,154817],{"xmlns":31650,"display":126394},[31652,154818,154819,154879],{},[31655,154820,154821,154823,154825,154827,154833,154835,154837,154839,154841],{},[31658,154822,37400],{},[31658,154824,96058],{},[31662,154826,456],{"stretchy":702},[31743,154828,154829,154831],{},[31658,154830,56782],{},[31658,154832,31955],{},[31662,154834,2557],{"stretchy":702},[31662,154836,202],{},[31658,154838,32679],{},[31662,154840,32682],{},[31655,154842,154843,154845,154847,154849,154855,154857,154859,154861,154863,154869,154871,154877],{},[31658,154844,37400],{},[31658,154846,96058],{},[31662,154848,456],{"stretchy":702},[31743,154850,154851,154853],{},[31658,154852,56782],{},[31658,154854,31968],{},[31662,154856,2557],{"stretchy":702},[31662,154858,223],{},[31658,154860,56852],{},[31662,154862,456],{"stretchy":702},[31743,154864,154865,154867],{},[31658,154866,56782],{},[31658,154868,31968],{},[31662,154870,31960],{"separator":758},[31743,154872,154873,154875],{},[31658,154874,56782],{},[31658,154876,31955],{},[31662,154878,2557],{"stretchy":702},[31665,154880,154881],{"encoding":31667},"TE(v_i)=\\max{TE(v_j)+W(v_j,v_i)}",[34,154883,154885,154952],{"className":154884,"ariaHidden":758},[31672],[34,154886,154888,154891,154894,154897,154900,154940,154943,154946,154949],{"className":154887},[31676],[34,154889],{"className":154890,"style":31774},[31680],[34,154892,37400],{"className":154893,"style":37422},[31685,31686],[34,154895,96058],{"className":154896,"style":106455},[31685,31686],[34,154898,456],{"className":154899},[31782],[34,154901,154903,154906],{"className":154902},[31685],[34,154904,56782],{"className":154905,"style":32062},[31685,31686],[34,154907,154909],{"className":154908},[31792],[34,154910,154912,154932],{"className":154911},[31796,31797],[34,154913,154915,154929],{"className":154914},[31801],[34,154916,154918],{"className":154917,"style":32090},[31805],[34,154919,154920,154923],{"style":38795},[34,154921],{"className":154922,"style":31814},[31813],[34,154924,154926],{"className":154925},[31818,31819,31820,31821],[34,154927,31955],{"className":154928},[31685,31686,31821],[34,154930,31829],{"className":154931},[31828],[34,154933,154935],{"className":154934},[31801],[34,154936,154938],{"className":154937,"style":31836},[31805],[34,154939],{},[34,154941,2557],{"className":154942},[31690],[34,154944],{"className":154945,"style":31846},[31845],[34,154947,202],{"className":154948},[31850],[34,154950],{"className":154951,"style":31846},[31845],[34,154953,154955,154958,154961,154964],{"className":154954},[31676],[34,154956],{"className":154957,"style":32058},[31680],[34,154959,32679],{"className":154960},[32767],[34,154962],{"className":154963,"style":32123},[31845],[34,154965,154967,154970,154973,154976,155016,155019,155022,155025,155028,155031,155034,155074,155077,155080,155120],{"className":154966},[31685],[34,154968,37400],{"className":154969,"style":37422},[31685,31686],[34,154971,96058],{"className":154972,"style":106455},[31685,31686],[34,154974,456],{"className":154975},[31782],[34,154977,154979,154982],{"className":154978},[31685],[34,154980,56782],{"className":154981,"style":32062},[31685,31686],[34,154983,154985],{"className":154984},[31792],[34,154986,154988,155008],{"className":154987},[31796,31797],[34,154989,154991,155005],{"className":154990},[31801],[34,154992,154994],{"className":154993,"style":32090},[31805],[34,154995,154996,154999],{"style":38795},[34,154997],{"className":154998,"style":31814},[31813],[34,155000,155002],{"className":155001},[31818,31819,31820,31821],[34,155003,31968],{"className":155004,"style":32153},[31685,31686,31821],[34,155006,31829],{"className":155007},[31828],[34,155009,155011],{"className":155010},[31801],[34,155012,155014],{"className":155013,"style":32163},[31805],[34,155015],{},[34,155017,2557],{"className":155018},[31690],[34,155020],{"className":155021,"style":32604},[31845],[34,155023,223],{"className":155024},[32024],[34,155026],{"className":155027,"style":32604},[31845],[34,155029,56852],{"className":155030,"style":37422},[31685,31686],[34,155032,456],{"className":155033},[31782],[34,155035,155037,155040],{"className":155036},[31685],[34,155038,56782],{"className":155039,"style":32062},[31685,31686],[34,155041,155043],{"className":155042},[31792],[34,155044,155046,155066],{"className":155045},[31796,31797],[34,155047,155049,155063],{"className":155048},[31801],[34,155050,155052],{"className":155051,"style":32090},[31805],[34,155053,155054,155057],{"style":38795},[34,155055],{"className":155056,"style":31814},[31813],[34,155058,155060],{"className":155059},[31818,31819,31820,31821],[34,155061,31968],{"className":155062,"style":32153},[31685,31686,31821],[34,155064,31829],{"className":155065},[31828],[34,155067,155069],{"className":155068},[31801],[34,155070,155072],{"className":155071,"style":32163},[31805],[34,155073],{},[34,155075,31960],{"className":155076},[32119],[34,155078],{"className":155079,"style":32123},[31845],[34,155081,155083,155086],{"className":155082},[31685],[34,155084,56782],{"className":155085,"style":32062},[31685,31686],[34,155087,155089],{"className":155088},[31792],[34,155090,155092,155112],{"className":155091},[31796,31797],[34,155093,155095,155109],{"className":155094},[31801],[34,155096,155098],{"className":155097,"style":32090},[31805],[34,155099,155100,155103],{"style":38795},[34,155101],{"className":155102,"style":31814},[31813],[34,155104,155106],{"className":155105},[31818,31819,31820,31821],[34,155107,31955],{"className":155108},[31685,31686,31821],[34,155110,31829],{"className":155111},[31828],[34,155113,155115],{"className":155114},[31801],[34,155116,155118],{"className":155117,"style":31836},[31805],[34,155119],{},[34,155121,2557],{"className":155122},[31690],[15,155124,115851,155125,11008,155194,155263],{},[34,155126,155128,155145],{"className":155127},[31642],[34,155129,155131],{"className":155130},[31646],[31648,155132,155133],{"xmlns":31650},[31652,155134,155135,155143],{},[31655,155136,155137],{},[31743,155138,155139,155141],{},[31658,155140,56782],{},[31658,155142,31968],{},[31665,155144,138126],{"encoding":31667},[34,155146,155148],{"className":155147,"ariaHidden":758},[31672],[34,155149,155151,155154],{"className":155150},[31676],[34,155152],{"className":155153,"style":56005},[31680],[34,155155,155157,155160],{"className":155156},[31685],[34,155158,56782],{"className":155159,"style":32062},[31685,31686],[34,155161,155163],{"className":155162},[31792],[34,155164,155166,155186],{"className":155165},[31796,31797],[34,155167,155169,155183],{"className":155168},[31801],[34,155170,155172],{"className":155171,"style":32090},[31805],[34,155173,155174,155177],{"style":38795},[34,155175],{"className":155176,"style":31814},[31813],[34,155178,155180],{"className":155179},[31818,31819,31820,31821],[34,155181,31968],{"className":155182,"style":32153},[31685,31686,31821],[34,155184,31829],{"className":155185},[31828],[34,155187,155189],{"className":155188},[31801],[34,155190,155192],{"className":155191,"style":32163},[31805],[34,155193],{},[34,155195,155197,155214],{"className":155196},[31642],[34,155198,155200],{"className":155199},[31646],[31648,155201,155202],{"xmlns":31650},[31652,155203,155204,155212],{},[31655,155205,155206],{},[31743,155207,155208,155210],{},[31658,155209,56782],{},[31658,155211,31955],{},[31665,155213,56787],{"encoding":31667},[34,155215,155217],{"className":155216,"ariaHidden":758},[31672],[34,155218,155220,155223],{"className":155219},[31676],[34,155221],{"className":155222,"style":31860},[31680],[34,155224,155226,155229],{"className":155225},[31685],[34,155227,56782],{"className":155228,"style":32062},[31685,31686],[34,155230,155232],{"className":155231},[31792],[34,155233,155235,155255],{"className":155234},[31796,31797],[34,155236,155238,155252],{"className":155237},[31801],[34,155239,155241],{"className":155240,"style":32090},[31805],[34,155242,155243,155246],{"style":38795},[34,155244],{"className":155245,"style":31814},[31813],[34,155247,155249],{"className":155248},[31818,31819,31820,31821],[34,155250,31955],{"className":155251},[31685,31686,31821],[34,155253,31829],{"className":155254},[31828],[34,155256,155258],{"className":155257},[31801],[34,155259,155261],{"className":155260,"style":31836},[31805],[34,155262],{}," 的前驱顶点。",[32188,155265],{},[19,155267,155269],{"id":155268},"_3-最迟完成时间","3. 最迟完成时间",[15,155271,154449,155272,155341],{},[34,155273,155275,155292],{"className":155274},[31642],[34,155276,155278],{"className":155277},[31646],[31648,155279,155280],{"xmlns":31650},[31652,155281,155282,155290],{},[31655,155283,155284],{},[31743,155285,155286,155288],{},[31658,155287,56782],{},[31658,155289,31955],{},[31665,155291,56787],{"encoding":31667},[34,155293,155295],{"className":155294,"ariaHidden":758},[31672],[34,155296,155298,155301],{"className":155297},[31676],[34,155299],{"className":155300,"style":31860},[31680],[34,155302,155304,155307],{"className":155303},[31685],[34,155305,56782],{"className":155306,"style":32062},[31685,31686],[34,155308,155310],{"className":155309},[31792],[34,155311,155313,155333],{"className":155312},[31796,31797],[34,155314,155316,155330],{"className":155315},[31801],[34,155317,155319],{"className":155318,"style":32090},[31805],[34,155320,155321,155324],{"style":38795},[34,155322],{"className":155323,"style":31814},[31813],[34,155325,155327],{"className":155326},[31818,31819,31820,31821],[34,155328,31955],{"className":155329},[31685,31686,31821],[34,155331,31829],{"className":155332},[31828],[34,155334,155336],{"className":155335},[31801],[34,155337,155339],{"className":155338,"style":31836},[31805],[34,155340],{}," 的最迟完成时间记作：",[34,155343,155345],{"className":155344},[126385],[34,155346,155348,155374],{"className":155347},[31642],[34,155349,155351],{"className":155350},[31646],[31648,155352,155353],{"xmlns":31650,"display":126394},[31652,155354,155355,155371],{},[31655,155356,155357,155359,155361,155363,155369],{},[31658,155358,37400],{},[31658,155360,33206],{},[31662,155362,456],{"stretchy":702},[31743,155364,155365,155367],{},[31658,155366,56782],{},[31658,155368,31955],{},[31662,155370,2557],{"stretchy":702},[31665,155372,155373],{"encoding":31667},"TL(v_i)",[34,155375,155377],{"className":155376,"ariaHidden":758},[31672],[34,155378,155380,155383,155386,155389,155392,155432],{"className":155379},[31676],[34,155381],{"className":155382,"style":31774},[31680],[34,155384,37400],{"className":155385,"style":37422},[31685,31686],[34,155387,33206],{"className":155388},[31685,31686],[34,155390,456],{"className":155391},[31782],[34,155393,155395,155398],{"className":155394},[31685],[34,155396,56782],{"className":155397,"style":32062},[31685,31686],[34,155399,155401],{"className":155400},[31792],[34,155402,155404,155424],{"className":155403},[31796,31797],[34,155405,155407,155421],{"className":155406},[31801],[34,155408,155410],{"className":155409,"style":32090},[31805],[34,155411,155412,155415],{"style":38795},[34,155413],{"className":155414,"style":31814},[31813],[34,155416,155418],{"className":155417},[31818,31819,31820,31821],[34,155419,31955],{"className":155420},[31685,31686,31821],[34,155422,31829],{"className":155423},[31828],[34,155425,155427],{"className":155426},[31801],[34,155428,155430],{"className":155429,"style":31836},[31805],[34,155431],{},[34,155433,2557],{"className":155434},[31690],[15,155436,155437,155438,155507],{},"表示在不延误总工期的情况下，事件 ",[34,155439,155441,155458],{"className":155440},[31642],[34,155442,155444],{"className":155443},[31646],[31648,155445,155446],{"xmlns":31650},[31652,155447,155448,155456],{},[31655,155449,155450],{},[31743,155451,155452,155454],{},[31658,155453,56782],{},[31658,155455,31955],{},[31665,155457,56787],{"encoding":31667},[34,155459,155461],{"className":155460,"ariaHidden":758},[31672],[34,155462,155464,155467],{"className":155463},[31676],[34,155465],{"className":155466,"style":31860},[31680],[34,155468,155470,155473],{"className":155469},[31685],[34,155471,56782],{"className":155472,"style":32062},[31685,31686],[34,155474,155476],{"className":155475},[31792],[34,155477,155479,155499],{"className":155478},[31796,31797],[34,155480,155482,155496],{"className":155481},[31801],[34,155483,155485],{"className":155484,"style":32090},[31805],[34,155486,155487,155490],{"style":38795},[34,155488],{"className":155489,"style":31814},[31813],[34,155491,155493],{"className":155492},[31818,31819,31820,31821],[34,155494,31955],{"className":155495},[31685,31686,31821],[34,155497,31829],{"className":155498},[31828],[34,155500,155502],{"className":155501},[31801],[34,155503,155505],{"className":155504,"style":31836},[31805],[34,155506],{}," 最晚什么时候必须发生。",[15,155509,155510],{},"终点：",[34,155512,155514],{"className":155513},[126385],[34,155515,155517,155559],{"className":155516},[31642],[34,155518,155520],{"className":155519},[31646],[31648,155521,155522],{"xmlns":31650,"display":126394},[31652,155523,155524,155556],{},[31655,155525,155526,155528,155530,155532,155538,155540,155542,155544,155546,155548,155554],{},[31658,155527,37400],{},[31658,155529,33206],{},[31662,155531,456],{"stretchy":702},[31743,155533,155534,155536],{},[31658,155535,56782],{},[31658,155537,31660],{},[31662,155539,2557],{"stretchy":702},[31662,155541,202],{},[31658,155543,37400],{},[31658,155545,96058],{},[31662,155547,456],{"stretchy":702},[31743,155549,155550,155552],{},[31658,155551,56782],{},[31658,155553,31660],{},[31662,155555,2557],{"stretchy":702},[31665,155557,155558],{"encoding":31667},"TL(v_n)=TE(v_n)",[34,155560,155562,155629],{"className":155561,"ariaHidden":758},[31672],[34,155563,155565,155568,155571,155574,155577,155617,155620,155623,155626],{"className":155564},[31676],[34,155566],{"className":155567,"style":31774},[31680],[34,155569,37400],{"className":155570,"style":37422},[31685,31686],[34,155572,33206],{"className":155573},[31685,31686],[34,155575,456],{"className":155576},[31782],[34,155578,155580,155583],{"className":155579},[31685],[34,155581,56782],{"className":155582,"style":32062},[31685,31686],[34,155584,155586],{"className":155585},[31792],[34,155587,155589,155609],{"className":155588},[31796,31797],[34,155590,155592,155606],{"className":155591},[31801],[34,155593,155595],{"className":155594,"style":32362},[31805],[34,155596,155597,155600],{"style":38795},[34,155598],{"className":155599,"style":31814},[31813],[34,155601,155603],{"className":155602},[31818,31819,31820,31821],[34,155604,31660],{"className":155605},[31685,31686,31821],[34,155607,31829],{"className":155608},[31828],[34,155610,155612],{"className":155611},[31801],[34,155613,155615],{"className":155614,"style":31836},[31805],[34,155616],{},[34,155618,2557],{"className":155619},[31690],[34,155621],{"className":155622,"style":31846},[31845],[34,155624,202],{"className":155625},[31850],[34,155627],{"className":155628,"style":31846},[31845],[34,155630,155632,155635,155638,155641,155644,155684],{"className":155631},[31676],[34,155633],{"className":155634,"style":31774},[31680],[34,155636,37400],{"className":155637,"style":37422},[31685,31686],[34,155639,96058],{"className":155640,"style":106455},[31685,31686],[34,155642,456],{"className":155643},[31782],[34,155645,155647,155650],{"className":155646},[31685],[34,155648,56782],{"className":155649,"style":32062},[31685,31686],[34,155651,155653],{"className":155652},[31792],[34,155654,155656,155676],{"className":155655},[31796,31797],[34,155657,155659,155673],{"className":155658},[31801],[34,155660,155662],{"className":155661,"style":32362},[31805],[34,155663,155664,155667],{"style":38795},[34,155665],{"className":155666,"style":31814},[31813],[34,155668,155670],{"className":155669},[31818,31819,31820,31821],[34,155671,31660],{"className":155672},[31685,31686,31821],[34,155674,31829],{"className":155675},[31828],[34,155677,155679],{"className":155678},[31801],[34,155680,155682],{"className":155681,"style":31836},[31805],[34,155683],{},[34,155685,2557],{"className":155686},[31690],[15,155688,154805],{},[34,155690,155692],{"className":155691},[126385],[34,155693,155695,155765],{"className":155694},[31642],[34,155696,155698],{"className":155697},[31646],[31648,155699,155700],{"xmlns":31650,"display":126394},[31652,155701,155702,155762],{},[31655,155703,155704,155706,155708,155710,155716,155718,155720,155722,155724],{},[31658,155705,37400],{},[31658,155707,33206],{},[31662,155709,456],{"stretchy":702},[31743,155711,155712,155714],{},[31658,155713,56782],{},[31658,155715,31955],{},[31662,155717,2557],{"stretchy":702},[31662,155719,202],{},[31658,155721,56084],{},[31662,155723,32682],{},[31655,155725,155726,155728,155730,155732,155738,155740,155742,155744,155746,155752,155754,155760],{},[31658,155727,37400],{},[31658,155729,33206],{},[31662,155731,456],{"stretchy":702},[31743,155733,155734,155736],{},[31658,155735,56782],{},[31658,155737,31968],{},[31662,155739,2557],{"stretchy":702},[31662,155741,32226],{},[31658,155743,56852],{},[31662,155745,456],{"stretchy":702},[31743,155747,155748,155750],{},[31658,155749,56782],{},[31658,155751,31955],{},[31662,155753,31960],{"separator":758},[31743,155755,155756,155758],{},[31658,155757,56782],{},[31658,155759,31968],{},[31662,155761,2557],{"stretchy":702},[31665,155763,155764],{"encoding":31667},"TL(v_i)=\\min{TL(v_j)-W(v_i,v_j)}",[34,155766,155768,155835],{"className":155767,"ariaHidden":758},[31672],[34,155769,155771,155774,155777,155780,155783,155823,155826,155829,155832],{"className":155770},[31676],[34,155772],{"className":155773,"style":31774},[31680],[34,155775,37400],{"className":155776,"style":37422},[31685,31686],[34,155778,33206],{"className":155779},[31685,31686],[34,155781,456],{"className":155782},[31782],[34,155784,155786,155789],{"className":155785},[31685],[34,155787,56782],{"className":155788,"style":32062},[31685,31686],[34,155790,155792],{"className":155791},[31792],[34,155793,155795,155815],{"className":155794},[31796,31797],[34,155796,155798,155812],{"className":155797},[31801],[34,155799,155801],{"className":155800,"style":32090},[31805],[34,155802,155803,155806],{"style":38795},[34,155804],{"className":155805,"style":31814},[31813],[34,155807,155809],{"className":155808},[31818,31819,31820,31821],[34,155810,31955],{"className":155811},[31685,31686,31821],[34,155813,31829],{"className":155814},[31828],[34,155816,155818],{"className":155817},[31801],[34,155819,155821],{"className":155820,"style":31836},[31805],[34,155822],{},[34,155824,2557],{"className":155825},[31690],[34,155827],{"className":155828,"style":31846},[31845],[34,155830,202],{"className":155831},[31850],[34,155833],{"className":155834,"style":31846},[31845],[34,155836,155838,155841,155844,155847],{"className":155837},[31676],[34,155839],{"className":155840,"style":32058},[31680],[34,155842,56084],{"className":155843},[32767],[34,155845],{"className":155846,"style":32123},[31845],[34,155848,155850,155853,155856,155859,155899,155902,155905,155908,155911,155914,155917,155957,155960,155963,156003],{"className":155849},[31685],[34,155851,37400],{"className":155852,"style":37422},[31685,31686],[34,155854,33206],{"className":155855},[31685,31686],[34,155857,456],{"className":155858},[31782],[34,155860,155862,155865],{"className":155861},[31685],[34,155863,56782],{"className":155864,"style":32062},[31685,31686],[34,155866,155868],{"className":155867},[31792],[34,155869,155871,155891],{"className":155870},[31796,31797],[34,155872,155874,155888],{"className":155873},[31801],[34,155875,155877],{"className":155876,"style":32090},[31805],[34,155878,155879,155882],{"style":38795},[34,155880],{"className":155881,"style":31814},[31813],[34,155883,155885],{"className":155884},[31818,31819,31820,31821],[34,155886,31968],{"className":155887,"style":32153},[31685,31686,31821],[34,155889,31829],{"className":155890},[31828],[34,155892,155894],{"className":155893},[31801],[34,155895,155897],{"className":155896,"style":32163},[31805],[34,155898],{},[34,155900,2557],{"className":155901},[31690],[34,155903],{"className":155904,"style":32604},[31845],[34,155906,32226],{"className":155907},[32024],[34,155909],{"className":155910,"style":32604},[31845],[34,155912,56852],{"className":155913,"style":37422},[31685,31686],[34,155915,456],{"className":155916},[31782],[34,155918,155920,155923],{"className":155919},[31685],[34,155921,56782],{"className":155922,"style":32062},[31685,31686],[34,155924,155926],{"className":155925},[31792],[34,155927,155929,155949],{"className":155928},[31796,31797],[34,155930,155932,155946],{"className":155931},[31801],[34,155933,155935],{"className":155934,"style":32090},[31805],[34,155936,155937,155940],{"style":38795},[34,155938],{"className":155939,"style":31814},[31813],[34,155941,155943],{"className":155942},[31818,31819,31820,31821],[34,155944,31955],{"className":155945},[31685,31686,31821],[34,155947,31829],{"className":155948},[31828],[34,155950,155952],{"className":155951},[31801],[34,155953,155955],{"className":155954,"style":31836},[31805],[34,155956],{},[34,155958,31960],{"className":155959},[32119],[34,155961],{"className":155962,"style":32123},[31845],[34,155964,155966,155969],{"className":155965},[31685],[34,155967,56782],{"className":155968,"style":32062},[31685,31686],[34,155970,155972],{"className":155971},[31792],[34,155973,155975,155995],{"className":155974},[31796,31797],[34,155976,155978,155992],{"className":155977},[31801],[34,155979,155981],{"className":155980,"style":32090},[31805],[34,155982,155983,155986],{"style":38795},[34,155984],{"className":155985,"style":31814},[31813],[34,155987,155989],{"className":155988},[31818,31819,31820,31821],[34,155990,31968],{"className":155991,"style":32153},[31685,31686,31821],[34,155993,31829],{"className":155994},[31828],[34,155996,155998],{"className":155997},[31801],[34,155999,156001],{"className":156000,"style":32163},[31805],[34,156002],{},[34,156004,2557],{"className":156005},[31690],[15,156007,115851,156008,11008,156077,156146],{},[34,156009,156011,156028],{"className":156010},[31642],[34,156012,156014],{"className":156013},[31646],[31648,156015,156016],{"xmlns":31650},[31652,156017,156018,156026],{},[31655,156019,156020],{},[31743,156021,156022,156024],{},[31658,156023,56782],{},[31658,156025,31968],{},[31665,156027,138126],{"encoding":31667},[34,156029,156031],{"className":156030,"ariaHidden":758},[31672],[34,156032,156034,156037],{"className":156033},[31676],[34,156035],{"className":156036,"style":56005},[31680],[34,156038,156040,156043],{"className":156039},[31685],[34,156041,56782],{"className":156042,"style":32062},[31685,31686],[34,156044,156046],{"className":156045},[31792],[34,156047,156049,156069],{"className":156048},[31796,31797],[34,156050,156052,156066],{"className":156051},[31801],[34,156053,156055],{"className":156054,"style":32090},[31805],[34,156056,156057,156060],{"style":38795},[34,156058],{"className":156059,"style":31814},[31813],[34,156061,156063],{"className":156062},[31818,31819,31820,31821],[34,156064,31968],{"className":156065,"style":32153},[31685,31686,31821],[34,156067,31829],{"className":156068},[31828],[34,156070,156072],{"className":156071},[31801],[34,156073,156075],{"className":156074,"style":32163},[31805],[34,156076],{},[34,156078,156080,156097],{"className":156079},[31642],[34,156081,156083],{"className":156082},[31646],[31648,156084,156085],{"xmlns":31650},[31652,156086,156087,156095],{},[31655,156088,156089],{},[31743,156090,156091,156093],{},[31658,156092,56782],{},[31658,156094,31955],{},[31665,156096,56787],{"encoding":31667},[34,156098,156100],{"className":156099,"ariaHidden":758},[31672],[34,156101,156103,156106],{"className":156102},[31676],[34,156104],{"className":156105,"style":31860},[31680],[34,156107,156109,156112],{"className":156108},[31685],[34,156110,56782],{"className":156111,"style":32062},[31685,31686],[34,156113,156115],{"className":156114},[31792],[34,156116,156118,156138],{"className":156117},[31796,31797],[34,156119,156121,156135],{"className":156120},[31801],[34,156122,156124],{"className":156123,"style":32090},[31805],[34,156125,156126,156129],{"style":38795},[34,156127],{"className":156128,"style":31814},[31813],[34,156130,156132],{"className":156131},[31818,31819,31820,31821],[34,156133,31955],{"className":156134},[31685,31686,31821],[34,156136,31829],{"className":156137},[31828],[34,156139,156141],{"className":156140},[31801],[34,156142,156144],{"className":156143,"style":31836},[31805],[34,156145],{}," 的后继顶点。",[32188,156148],{},[19,156150,156152],{"id":156151},"_4-关键路径","4. 关键路径",[15,156154,156155],{},"如果某条路径上的顶点都满足：",[34,156157,156159],{"className":156158},[126385],[34,156160,156162,156196],{"className":156161},[31642],[34,156163,156165],{"className":156164},[31646],[31648,156166,156167],{"xmlns":31650,"display":126394},[31652,156168,156169,156193],{},[31655,156170,156171,156173,156175,156177,156179,156181,156183,156185,156187,156189,156191],{},[31658,156172,37400],{},[31658,156174,96058],{},[31662,156176,456],{"stretchy":702},[31658,156178,56782],{},[31662,156180,2557],{"stretchy":702},[31662,156182,202],{},[31658,156184,37400],{},[31658,156186,33206],{},[31662,156188,456],{"stretchy":702},[31658,156190,56782],{},[31662,156192,2557],{"stretchy":702},[31665,156194,156195],{"encoding":31667},"TE(v)=TL(v)",[34,156197,156199,156229],{"className":156198,"ariaHidden":758},[31672],[34,156200,156202,156205,156208,156211,156214,156217,156220,156223,156226],{"className":156201},[31676],[34,156203],{"className":156204,"style":31774},[31680],[34,156206,37400],{"className":156207,"style":37422},[31685,31686],[34,156209,96058],{"className":156210,"style":106455},[31685,31686],[34,156212,456],{"className":156213},[31782],[34,156215,56782],{"className":156216,"style":32062},[31685,31686],[34,156218,2557],{"className":156219},[31690],[34,156221],{"className":156222,"style":31846},[31845],[34,156224,202],{"className":156225},[31850],[34,156227],{"className":156228,"style":31846},[31845],[34,156230,156232,156235,156238,156241,156244,156247],{"className":156231},[31676],[34,156233],{"className":156234,"style":31774},[31680],[34,156236,37400],{"className":156237,"style":37422},[31685,31686],[34,156239,33206],{"className":156240},[31685,31686],[34,156242,456],{"className":156243},[31782],[34,156245,56782],{"className":156246,"style":32062},[31685,31686],[34,156248,2557],{"className":156249},[31690],[15,156251,156252,156253,347],{},"则这条路径可能是",[2627,156254,156255],{},"关键路径",[15,156257,156258],{},"关键路径的长度就是整个工程的最短完成时间。",[32188,156260],{},[19,156262,156264],{"id":156263},"_5-关键路径的意义","5. 关键路径的意义",[15,156266,156267],{},"关键路径上的任务不能延误。",[15,156269,156270],{},"只要关键路径上任一任务延误，整个项目就会延误。",[15,156272,156273],{},"非关键路径上的任务通常有一定机动时间。",[32188,156275],{},[11,156277,156278],{"id":156278},"第十二章核心公式总结",[13837,156280,156281,156291],{},[13840,156282,156283],{},[13843,156284,156285,156288],{},[13846,156286,156287],{},"知识点",[13846,156289,156290],{},"公式 \u002F 结论",[13853,156292,156293,156424,156497,156644,156652,156709,156744,156990,157234,157388,157633,158081],{},[13843,156294,156295,156298],{},[13858,156296,156297],{},"无向完全图边数",[13858,156299,156300],{},[34,156301,156303,156332],{"className":156302},[31642],[34,156304,156306],{"className":156305},[31646],[31648,156307,156308],{"xmlns":31650},[31652,156309,156310,156330],{},[31655,156311,156312],{},[41453,156313,156314,156328],{},[31655,156315,156316,156318,156320,156322,156324,156326],{},[31658,156317,31660],{},[31662,156319,456],{"stretchy":702},[31658,156321,31660],{},[31662,156323,32226],{},[31749,156325,640],{},[31662,156327,2557],{"stretchy":702},[31749,156329,768],{},[31665,156331,70475],{"encoding":31667},[34,156333,156335],{"className":156334,"ariaHidden":758},[31672],[34,156336,156338,156341],{"className":156337},[31676],[34,156339],{"className":156340,"style":70485},[31680],[34,156342,156344,156347,156421],{"className":156343},[31685],[34,156345],{"className":156346},[31782,41488],[34,156348,156350],{"className":156349},[41453],[34,156351,156353,156413],{"className":156352},[31796,31797],[34,156354,156356,156410],{"className":156355},[31801],[34,156357,156359,156373,156381],{"className":156358,"style":70504},[31805],[34,156360,156361,156364],{"style":41504},[34,156362],{"className":156363,"style":41508},[31813],[34,156365,156367],{"className":156366},[31818,31819,31820,31821],[34,156368,156370],{"className":156369},[31685,31821],[34,156371,768],{"className":156372},[31685,31821],[34,156374,156375,156378],{"style":41520},[34,156376],{"className":156377,"style":41508},[31813],[34,156379],{"className":156380,"style":41528},[41527],[34,156382,156383,156386],{"style":70529},[34,156384],{"className":156385,"style":41508},[31813],[34,156387,156389],{"className":156388},[31818,31819,31820,31821],[34,156390,156392,156395,156398,156401,156404,156407],{"className":156391},[31685,31821],[34,156393,31660],{"className":156394},[31685,31686,31821],[34,156396,456],{"className":156397},[31782,31821],[34,156399,31660],{"className":156400},[31685,31686,31821],[34,156402,32226],{"className":156403},[32024,31821],[34,156405,640],{"className":156406},[31685,31821],[34,156408,2557],{"className":156409},[31690,31821],[34,156411,31829],{"className":156412},[31828],[34,156414,156416],{"className":156415},[31801],[34,156417,156419],{"className":156418,"style":41619},[31805],[34,156420],{},[34,156422],{"className":156423},[31690,41488],[13843,156425,156426,156429],{},[13858,156427,156428],{},"有向完全图边数",[13858,156430,156431],{},[34,156432,156434,156458],{"className":156433},[31642],[34,156435,156437],{"className":156436},[31646],[31648,156438,156439],{"xmlns":31650},[31652,156440,156441,156455],{},[31655,156442,156443,156445,156447,156449,156451,156453],{},[31658,156444,31660],{},[31662,156446,456],{"stretchy":702},[31658,156448,31660],{},[31662,156450,32226],{},[31749,156452,640],{},[31662,156454,2557],{"stretchy":702},[31665,156456,156457],{"encoding":31667},"n(n-1)",[34,156459,156461,156485],{"className":156460,"ariaHidden":758},[31672],[34,156462,156464,156467,156470,156473,156476,156479,156482],{"className":156463},[31676],[34,156465],{"className":156466,"style":31774},[31680],[34,156468,31660],{"className":156469},[31685,31686],[34,156471,456],{"className":156472},[31782],[34,156474,31660],{"className":156475},[31685,31686],[34,156477],{"className":156478,"style":32604},[31845],[34,156480,32226],{"className":156481},[32024],[34,156483],{"className":156484,"style":32604},[31845],[34,156486,156488,156491,156494],{"className":156487},[31676],[34,156489],{"className":156490,"style":31774},[31680],[34,156492,640],{"className":156493},[31685],[34,156495,2557],{"className":156496},[31690],[13843,156498,156499,156502],{},[13858,156500,156501],{},"握手定理",[13858,156503,156504],{},[34,156505,156507,156544],{"className":156506},[31642],[34,156508,156510],{"className":156509},[31646],[31648,156511,156512],{"xmlns":31650},[31652,156513,156514,156542],{},[31655,156515,156516,156528,156530,156532,156534,156536,156538,156540],{},[31743,156517,156518,156520],{},[31662,156519,131138],{},[31655,156521,156522,156524,156526],{},[31658,156523,56782],{},[31662,156525,112296],{},[31658,156527,126532],{},[31658,156529,46900],{},[31662,156531,456],{"stretchy":702},[31658,156533,56782],{},[31662,156535,2557],{"stretchy":702},[31662,156537,202],{},[31749,156539,768],{},[31658,156541,31759],{},[31665,156543,131380],{"encoding":31667},[34,156545,156547,156632],{"className":156546,"ariaHidden":758},[31672],[34,156548,156550,156554,156608,156611,156614,156617,156620,156623,156626,156629],{"className":156549},[31676],[34,156551],{"className":156552,"style":156553},[31680],"height:1.0771em;vertical-align:-0.3271em;",[34,156555,156557,156562],{"className":156556},[32767],[34,156558,131138],{"className":156559,"style":156561},[32767,131249,156560],"small-op","position:relative;top:0em;",[34,156563,156565],{"className":156564},[31792],[34,156566,156568,156599],{"className":156567},[31796,31797],[34,156569,156571,156596],{"className":156570},[31801],[34,156572,156575],{"className":156573,"style":156574},[31805],"height:0.1786em;",[34,156576,156578,156581],{"style":156577},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[34,156579],{"className":156580,"style":31814},[31813],[34,156582,156584],{"className":156583},[31818,31819,31820,31821],[34,156585,156587,156590,156593],{"className":156586},[31685,31821],[34,156588,56782],{"className":156589,"style":32062},[31685,31686,31821],[34,156591,112296],{"className":156592},[31850,31821],[34,156594,126532],{"className":156595,"style":32604},[31685,31686,31821],[34,156597,31829],{"className":156598},[31828],[34,156600,156602],{"className":156601},[31801],[34,156603,156606],{"className":156604,"style":156605},[31805],"height:0.3271em;",[34,156607],{},[34,156609],{"className":156610,"style":32123},[31845],[34,156612,46900],{"className":156613},[31685,31686],[34,156615,456],{"className":156616},[31782],[34,156618,56782],{"className":156619,"style":32062},[31685,31686],[34,156621,2557],{"className":156622},[31690],[34,156624],{"className":156625,"style":31846},[31845],[34,156627,202],{"className":156628},[31850],[34,156630],{"className":156631,"style":31846},[31845],[34,156633,156635,156638,156641],{"className":156634},[31676],[34,156636],{"className":156637,"style":32617},[31680],[34,156639,768],{"className":156640},[31685],[34,156642,31759],{"className":156643},[31685,31686],[13843,156645,156646,156649],{},[13858,156647,156648],{},"奇顶点个数",[13858,156650,156651],{},"一定是偶数",[13843,156653,156654,156657],{},[13858,156655,156656],{},"基本路径最大长度",[13858,156658,156659],{},[34,156660,156662,156679],{"className":156661},[31642],[34,156663,156665],{"className":156664},[31646],[31648,156666,156667],{"xmlns":31650},[31652,156668,156669,156677],{},[31655,156670,156671,156673,156675],{},[31658,156672,31660],{},[31662,156674,32226],{},[31749,156676,640],{},[31665,156678,32587],{"encoding":31667},[34,156680,156682,156700],{"className":156681,"ariaHidden":758},[31672],[34,156683,156685,156688,156691,156694,156697],{"className":156684},[31676],[34,156686],{"className":156687,"style":32597},[31680],[34,156689,31660],{"className":156690},[31685,31686],[34,156692],{"className":156693,"style":32604},[31845],[34,156695,32226],{"className":156696},[32024],[34,156698],{"className":156699,"style":32604},[31845],[34,156701,156703,156706],{"className":156702},[31676],[34,156704],{"className":156705,"style":32617},[31680],[34,156707,640],{"className":156708},[31685],[13843,156710,156711,156714],{},[13858,156712,156713],{},"基本循环最大长度",[13858,156715,156716],{},[34,156717,156719,156732],{"className":156718},[31642],[34,156720,156722],{"className":156721},[31646],[31648,156723,156724],{"xmlns":31650},[31652,156725,156726,156730],{},[31655,156727,156728],{},[31658,156729,31660],{},[31665,156731,31660],{"encoding":31667},[34,156733,156735],{"className":156734,"ariaHidden":758},[31672],[34,156736,156738,156741],{"className":156737},[31676],[34,156739],{"className":156740,"style":32527},[31680],[34,156742,31660],{"className":156743},[31685,31686],[13843,156745,156746,156749],{},[13858,156747,156748],{},"有向图出度",[13858,156750,156751],{},[34,156752,156754,156800],{"className":156753},[31642],[34,156755,156757],{"className":156756},[31646],[31648,156758,156759],{"xmlns":31650},[31652,156760,156761,156797],{},[31655,156762,156763,156769,156771,156777,156779,156781,156787],{},[37624,156764,156765,156767],{},[31658,156766,46900],{},[31662,156768,223],{},[31662,156770,456],{"stretchy":702},[31743,156772,156773,156775],{},[31658,156774,56782],{},[31658,156776,31955],{},[31662,156778,2557],{"stretchy":702},[31662,156780,202],{},[31743,156782,156783,156785],{},[31662,156784,131138],{},[31658,156786,31968],{},[31743,156788,156789,156791],{},[31658,156790,111],{},[31655,156792,156793,156795],{},[31658,156794,31955],{},[31658,156796,31968],{},[31665,156798,156799],{"encoding":31667},"d^+(v_i)=\\sum_j a_{ij}",[34,156801,156803,156895],{"className":156802,"ariaHidden":758},[31672],[34,156804,156806,156810,156840,156843,156883,156886,156889,156892],{"className":156805},[31676],[34,156807],{"className":156808,"style":156809},[31680],"height:1.0213em;vertical-align:-0.25em;",[34,156811,156813,156816],{"className":156812},[31685],[34,156814,46900],{"className":156815},[31685,31686],[34,156817,156819],{"className":156818},[31792],[34,156820,156822],{"className":156821},[31796],[34,156823,156825],{"className":156824},[31801],[34,156826,156829],{"className":156827,"style":156828},[31805],"height:0.7713em;",[34,156830,156831,156834],{"style":37777},[34,156832],{"className":156833,"style":31814},[31813],[34,156835,156837],{"className":156836},[31818,31819,31820,31821],[34,156838,223],{"className":156839},[32024,31821],[34,156841,456],{"className":156842},[31782],[34,156844,156846,156849],{"className":156845},[31685],[34,156847,56782],{"className":156848,"style":32062},[31685,31686],[34,156850,156852],{"className":156851},[31792],[34,156853,156855,156875],{"className":156854},[31796,31797],[34,156856,156858,156872],{"className":156857},[31801],[34,156859,156861],{"className":156860,"style":32090},[31805],[34,156862,156863,156866],{"style":38795},[34,156864],{"className":156865,"style":31814},[31813],[34,156867,156869],{"className":156868},[31818,31819,31820,31821],[34,156870,31955],{"className":156871},[31685,31686,31821],[34,156873,31829],{"className":156874},[31828],[34,156876,156878],{"className":156877},[31801],[34,156879,156881],{"className":156880,"style":31836},[31805],[34,156882],{},[34,156884,2557],{"className":156885},[31690],[34,156887],{"className":156888,"style":31846},[31845],[34,156890,202],{"className":156891},[31850],[34,156893],{"className":156894,"style":31846},[31845],[34,156896,156898,156902,156944,156947],{"className":156897},[31676],[34,156899],{"className":156900,"style":156901},[31680],"height:1.1858em;vertical-align:-0.4358em;",[34,156903,156905,156908],{"className":156904},[32767],[34,156906,131138],{"className":156907,"style":156561},[32767,131249,156560],[34,156909,156911],{"className":156910},[31792],[34,156912,156914,156935],{"className":156913},[31796,31797],[34,156915,156917,156932],{"className":156916},[31801],[34,156918,156921],{"className":156919,"style":156920},[31805],"height:0.162em;",[34,156922,156923,156926],{"style":156577},[34,156924],{"className":156925,"style":31814},[31813],[34,156927,156929],{"className":156928},[31818,31819,31820,31821],[34,156930,31968],{"className":156931,"style":32153},[31685,31686,31821],[34,156933,31829],{"className":156934},[31828],[34,156936,156938],{"className":156937},[31801],[34,156939,156942],{"className":156940,"style":156941},[31805],"height:0.4358em;",[34,156943],{},[34,156945],{"className":156946,"style":32123},[31845],[34,156948,156950,156953],{"className":156949},[31685],[34,156951,111],{"className":156952},[31685,31686],[34,156954,156956],{"className":156955},[31792],[34,156957,156959,156982],{"className":156958},[31796,31797],[34,156960,156962,156979],{"className":156961},[31801],[34,156963,156965],{"className":156964,"style":32090},[31805],[34,156966,156967,156970],{"style":31809},[34,156968],{"className":156969,"style":31814},[31813],[34,156971,156973],{"className":156972},[31818,31819,31820,31821],[34,156974,156976],{"className":156975},[31685,31821],[34,156977,141027],{"className":156978,"style":32153},[31685,31686,31821],[34,156980,31829],{"className":156981},[31828],[34,156983,156985],{"className":156984},[31801],[34,156986,156988],{"className":156987,"style":32163},[31805],[34,156989],{},[13843,156991,156992,156995],{},[13858,156993,156994],{},"有向图入度",[13858,156996,156997],{},[34,156998,157000,157046],{"className":156999},[31642],[34,157001,157003],{"className":157002},[31646],[31648,157004,157005],{"xmlns":31650},[31652,157006,157007,157043],{},[31655,157008,157009,157015,157017,157023,157025,157027,157033],{},[37624,157010,157011,157013],{},[31658,157012,46900],{},[31662,157014,32226],{},[31662,157016,456],{"stretchy":702},[31743,157018,157019,157021],{},[31658,157020,56782],{},[31658,157022,31955],{},[31662,157024,2557],{"stretchy":702},[31662,157026,202],{},[31743,157028,157029,157031],{},[31662,157030,131138],{},[31658,157032,31968],{},[31743,157034,157035,157037],{},[31658,157036,111],{},[31655,157038,157039,157041],{},[31658,157040,31968],{},[31658,157042,31955],{},[31665,157044,157045],{"encoding":31667},"d^-(v_i)=\\sum_j a_{ji}",[34,157047,157049,157139],{"className":157048,"ariaHidden":758},[31672],[34,157050,157052,157055,157084,157087,157127,157130,157133,157136],{"className":157051},[31676],[34,157053],{"className":157054,"style":156809},[31680],[34,157056,157058,157061],{"className":157057},[31685],[34,157059,46900],{"className":157060},[31685,31686],[34,157062,157064],{"className":157063},[31792],[34,157065,157067],{"className":157066},[31796],[34,157068,157070],{"className":157069},[31801],[34,157071,157073],{"className":157072,"style":156828},[31805],[34,157074,157075,157078],{"style":37777},[34,157076],{"className":157077,"style":31814},[31813],[34,157079,157081],{"className":157080},[31818,31819,31820,31821],[34,157082,32226],{"className":157083},[32024,31821],[34,157085,456],{"className":157086},[31782],[34,157088,157090,157093],{"className":157089},[31685],[34,157091,56782],{"className":157092,"style":32062},[31685,31686],[34,157094,157096],{"className":157095},[31792],[34,157097,157099,157119],{"className":157098},[31796,31797],[34,157100,157102,157116],{"className":157101},[31801],[34,157103,157105],{"className":157104,"style":32090},[31805],[34,157106,157107,157110],{"style":38795},[34,157108],{"className":157109,"style":31814},[31813],[34,157111,157113],{"className":157112},[31818,31819,31820,31821],[34,157114,31955],{"className":157115},[31685,31686,31821],[34,157117,31829],{"className":157118},[31828],[34,157120,157122],{"className":157121},[31801],[34,157123,157125],{"className":157124,"style":31836},[31805],[34,157126],{},[34,157128,2557],{"className":157129},[31690],[34,157131],{"className":157132,"style":31846},[31845],[34,157134,202],{"className":157135},[31850],[34,157137],{"className":157138,"style":31846},[31845],[34,157140,157142,157145,157185,157188],{"className":157141},[31676],[34,157143],{"className":157144,"style":156901},[31680],[34,157146,157148,157151],{"className":157147},[32767],[34,157149,131138],{"className":157150,"style":156561},[32767,131249,156560],[34,157152,157154],{"className":157153},[31792],[34,157155,157157,157177],{"className":157156},[31796,31797],[34,157158,157160,157174],{"className":157159},[31801],[34,157161,157163],{"className":157162,"style":156920},[31805],[34,157164,157165,157168],{"style":156577},[34,157166],{"className":157167,"style":31814},[31813],[34,157169,157171],{"className":157170},[31818,31819,31820,31821],[34,157172,31968],{"className":157173,"style":32153},[31685,31686,31821],[34,157175,31829],{"className":157176},[31828],[34,157178,157180],{"className":157179},[31801],[34,157181,157183],{"className":157182,"style":156941},[31805],[34,157184],{},[34,157186],{"className":157187,"style":32123},[31845],[34,157189,157191,157194],{"className":157190},[31685],[34,157192,111],{"className":157193},[31685,31686],[34,157195,157197],{"className":157196},[31792],[34,157198,157200,157226],{"className":157199},[31796,31797],[34,157201,157203,157223],{"className":157202},[31801],[34,157204,157206],{"className":157205,"style":32090},[31805],[34,157207,157208,157211],{"style":31809},[34,157209],{"className":157210,"style":31814},[31813],[34,157212,157214],{"className":157213},[31818,31819,31820,31821],[34,157215,157217,157220],{"className":157216},[31685,31821],[34,157218,31968],{"className":157219,"style":32153},[31685,31686,31821],[34,157221,31955],{"className":157222},[31685,31686,31821],[34,157224,31829],{"className":157225},[31828],[34,157227,157229],{"className":157228},[31801],[34,157230,157232],{"className":157231,"style":32163},[31805],[34,157233],{},[13843,157235,157236,157239],{},[13858,157237,157238],{},"邻接矩阵幂",[13858,157240,157241,157358,157359,157387],{},[34,157242,157244,157274],{"className":157243},[31642],[34,157245,157247],{"className":157246},[31646],[31648,157248,157249],{"xmlns":31650},[31652,157250,157251,157271],{},[31655,157252,157253,157255,157261],{},[31662,157254,456],{"stretchy":702},[37624,157256,157257,157259],{},[31658,157258,54151],{},[31658,157260,31759],{},[31743,157262,157263,157265],{},[31662,157264,2557],{"stretchy":702},[31655,157266,157267,157269],{},[31658,157268,31955],{},[31658,157270,31968],{},[31665,157272,157273],{"encoding":31667},"(A^m)_{ij}",[34,157275,157277],{"className":157276,"ariaHidden":758},[31672],[34,157278,157280,157283,157286,157315],{"className":157279},[31676],[34,157281],{"className":157282,"style":32058},[31680],[34,157284,456],{"className":157285},[31782],[34,157287,157289,157292],{"className":157288},[31685],[34,157290,54151],{"className":157291},[31685,31686],[34,157293,157295],{"className":157294},[31792],[34,157296,157298],{"className":157297},[31796],[34,157299,157301],{"className":157300},[31801],[34,157302,157304],{"className":157303,"style":37959},[31805],[34,157305,157306,157309],{"style":37777},[34,157307],{"className":157308,"style":31814},[31813],[34,157310,157312],{"className":157311},[31818,31819,31820,31821],[34,157313,31759],{"className":157314},[31685,31686,31821],[34,157316,157318,157321],{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表示长度为 ",[34,157360,157362,157375],{"className":157361},[31642],[34,157363,157365],{"className":157364},[31646],[31648,157366,157367],{"xmlns":31650},[31652,157368,157369,157373],{},[31655,157370,157371],{},[31658,157372,31759],{},[31665,157374,31759],{"encoding":31667},[34,157376,157378],{"className":157377,"ariaHidden":758},[31672],[34,157379,157381,157384],{"className":157380},[31676],[34,157382],{"className":157383,"style":32527},[31680],[34,157385,31759],{"className":157386},[31685,31686]," 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更新",[13858,157639,157640],{},[34,157641,157643,157753],{"className":157642},[31642],[34,157644,157646],{"className":157645},[31646],[31648,157647,157648],{"xmlns":31650},[31652,157649,157650,157750],{},[31655,157651,157652,157670,157672,157674,157676,157678,157700,157702,157724,157726,157748],{},[38846,157653,157654,157656,157662],{},[31658,157655,46900],{},[31655,157657,157658,157660],{},[31658,157659,31955],{},[31658,157661,31968],{},[31655,157663,157664,157666,157668],{},[31662,157665,456],{"stretchy":702},[31658,157667,33759],{},[31662,157669,2557],{"stretchy":702},[31662,157671,202],{},[31658,157673,56084],{},[31662,157675,32682],{},[31662,157677,456],{"stretchy":702},[38846,157679,157680,157682,157688],{},[31658,157681,46900],{},[31655,157683,157684,157686],{},[31658,157685,31955],{},[31658,157687,31968],{},[31655,157689,157690,157692,157694,157696,157698],{},[31662,157691,456],{"stretchy":702},[31658,157693,33759],{},[31662,157695,32226],{},[31749,157697,640],{},[31662,157699,2557],{"stretchy":702},[31662,157701,31960],{"separator":758},[38846,157703,157704,157706,157712],{},[31658,157705,46900],{},[31655,157707,157708,157710],{},[31658,157709,31955],{},[31658,157711,33759],{},[31655,157713,157714,157716,157718,157720,157722],{},[31662,157715,456],{"stretchy":702},[31658,157717,33759],{},[31662,157719,32226],{},[31749,157721,640],{},[31662,157723,2557],{"stretchy":702},[31662,157725,223],{},[38846,157727,157728,157730,157736],{},[31658,157729,46900],{},[31655,157731,157732,157734],{},[31658,157733,33759],{},[31658,157735,31968],{},[31655,157737,157738,157740,157742,157744,157746],{},[31662,157739,456],{"stretchy":702},[31658,157741,33759],{},[31662,157743,32226],{},[31749,157745,640],{},[31662,157747,2557],{"stretchy":702},[31662,157749,2557],{"stretchy":702},[31665,157751,157752],{"encoding":31667},"d_{ij}^{(k)}=\\min(d_{ij}^{(k-1)},d_{ik}^{(k-1)}+d_{kj}^{(k-1)})",[34,157754,157756,157834,157999],{"className":157755,"ariaHidden":758},[31672],[34,157757,157759,157762,157825,157828,157831],{"className":157758},[31676],[34,157760],{"className":157761,"style":152000},[31680],[34,157763,157765,157768],{"className":157764},[31685],[34,157766,46900],{"className":157767},[31685,31686],[34,157769,157771],{"className":157770},[31792],[34,157772,157774,157817],{"className":157773},[31796,31797],[34,157775,157777,157814],{"className":157776},[31801],[34,157778,157780,157794],{"className":157779,"style":152022},[31805],[34,157781,157782,157785],{"style":152025},[34,157783],{"className":157784,"style":31814},[31813],[34,157786,157788],{"className":157787},[31818,31819,31820,31821],[34,157789,157791],{"className":157790},[31685,31821],[34,157792,141027],{"className":157793,"style":32153},[31685,31686,31821],[34,157795,157796,157799],{"style":152040},[34,157797],{"className":157798,"style":31814},[31813],[34,157800,157802],{"className":157801},[31818,31819,31820,31821],[34,157803,157805,157808,157811],{"className":157804},[31685,31821],[34,157806,456],{"className":157807},[31782,31821],[34,157809,33759],{"className":157810,"style":33774},[31685,31686,31821],[34,157812,2557],{"className":157813},[31690,31821],[34,157815,31829],{"className":157816},[31828],[34,157818,157820],{"className":157819},[31801],[34,157821,157823],{"className":157822,"style":152068},[31805],[34,157824],{},[34,157826],{"className":157827,"style":31846},[31845],[34,157829,202],{"className":157830},[31850],[34,157832],{"className":157833,"style":31846},[31845],[34,157835,157837,157840,157843,157846,157915,157918,157921,157990,157993,157996],{"className":157836},[31676],[34,157838],{"className":157839,"style":152000},[31680],[34,157841,56084],{"className":157842},[32767],[34,157844,456],{"className":157845},[31782],[34,157847,157849,157852],{"className":157848},[31685],[34,157850,46900],{"className":157851},[31685,31686],[34,157853,157855],{"className":157854},[31792],[34,157856,157858,157907],{"className":157857},[31796,31797],[34,157859,157861,157904],{"className":157860},[31801],[34,157862,157864,157878],{"className":157863,"style":152022},[31805],[34,157865,157866,157869],{"style":152025},[34,157867],{"className":157868,"style":31814},[31813],[34,157870,157872],{"className":157871},[31818,31819,31820,31821],[34,157873,157875],{"className":157874},[31685,31821],[34,157876,141027],{"className":157877,"style":32153},[31685,31686,31821],[34,157879,157880,157883],{"style":152040},[34,157881],{"className":157882,"style":31814},[31813],[34,157884,157886],{"className":157885},[31818,31819,31820,31821],[34,157887,157889,157892,157895,157898,157901],{"className":157888},[31685,31821],[34,157890,456],{"className":157891},[31782,31821],[34,157893,33759],{"className":157894,"style":33774},[31685,31686,31821],[34,157896,32226],{"className":157897},[32024,31821],[34,157899,640],{"className":157900},[31685,31821],[34,157902,2557],{"className":157903},[31690,31821],[34,157905,31829],{"className":157906},[31828],[34,157908,157910],{"className":157909},[31801],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W(e_i)",[34,158128,158130,158157],{"className":158129,"ariaHidden":758},[31672],[34,158131,158133,158136,158139,158142,158145,158148,158151,158154],{"className":158132},[31676],[34,158134],{"className":158135,"style":31774},[31680],[34,158137,43379],{"className":158138,"style":47113},[31685,31686],[34,158140,456],{"className":158141},[31782],[34,158143,135107],{"className":158144},[31685,31686],[34,158146,2557],{"className":158147},[31690],[34,158149],{"className":158150,"style":31846},[31845],[34,158152,202],{"className":158153},[31850],[34,158155],{"className":158156,"style":31846},[31845],[34,158158,158160,158163,158166,158169,158172,158175,158215],{"className":158159},[31676],[34,158161],{"className":158162,"style":31774},[31680],[34,158164,131138],{"className":158165,"style":156561},[32767,131249,156560],[34,158167],{"className":158168,"style":32123},[31845],[34,158170,56852],{"className":158171,"style":37422},[31685,31686],[34,158173,456],{"className":158174},[31782],[34,158176,158178,158181],{"className":158177},[31685],[34,158179,33244],{"className":158180},[31685,31686],[34,158182,158184],{"className":158183},[31792],[34,158185,158187,158207],{"className":158186},[31796,31797],[34,158188,158190,158204],{"className":158189},[31801],[34,158191,158193],{"className":158192,"style":32090},[31805],[34,158194,158195,158198],{"style":31809},[34,158196],{"className":158197,"style":31814},[31813],[34,158199,158201],{"className":158200},[31818,31819,31820,31821],[34,158202,31955],{"className":158203},[31685,31686,31821],[34,158205,31829],{"className":158206},[31828],[34,158208,158210],{"className":158209},[31801],[34,158211,158213],{"className":158212,"style":31836},[31805],[34,158214],{},[34,158216,2557],{"className":158217},[31690],[32188,158219],{},[11,158221,158222],{"id":158222},"第十二章常考题型",[19,158224,158226],{"id":158225},"题型一判断图的类型","题型一：判断图的类型",[15,158228,158229],{},"常问：",[105,158231,158232,158235,158238,158241,158244,158247],{},[108,158233,158234],{},"是无向图还是有向图？",[108,158236,158237],{},"是简单图还是多重图？",[108,158239,158240],{},"有没有自环？",[108,158242,158243],{},"有没有重边？",[108,158245,158246],{},"是不是完全图？",[108,158248,158249],{},"是不是正则图？",[15,158251,158252],{},"口诀：",[34,158254,158256],{"className":158255},[126385],[34,158257,158259,158282],{"className":158258},[31642],[34,158260,158262],{"className":158261},[31646],[31648,158263,158264],{"xmlns":31650,"display":126394},[31652,158265,158266,158279],{},[31655,158267,158268],{},[126400,158269,158270],{"notation":126402},[71090,158271,158272],{"scriptlevel":459,"displaystyle":702},[71090,158273,158274],{"scriptlevel":459,"displaystyle":702},[71090,158275,158276],{"scriptlevel":459,"displaystyle":758},[39361,158277,158278],{},"简单图：无自环、无重边",[31665,158280,158281],{"encoding":31667},"\\boxed{\\text{简单图：无自环、无重边}}",[34,158283,158285],{"className":158284,"ariaHidden":758},[31672],[34,158286,158288,158291],{"className":158287},[31676],[34,158289],{"className":158290,"style":126424},[31680],[34,158292,158294],{"className":158293},[31685],[34,158295,158297,158334],{"className":158296},[31796,31797],[34,158298,158300,158331],{"className":158299},[31801],[34,158301,158303,158323],{"className":158302,"style":126437},[31805],[34,158304,158305,158308],{"style":126440},[34,158306],{"className":158307,"style":126444},[31813],[34,158309,158311],{"className":158310},[126448],[34,158312,158314],{"className":158313},[31685],[34,158315,158317],{"className":158316},[31685],[34,158318,158320],{"className":158319},[31685,2638],[34,158321,158278],{"className":158322},[31685,105462],[34,158324,158325,158328],{"style":126463},[34,158326],{"className":158327,"style":126444},[31813],[34,158329],{"className":158330,"style":126472},[126470,126471],[34,158332,31829],{"className":158333},[31828],[34,158335,158337],{"className":158336},[31801],[34,158338,158340],{"className":158339,"style":126482},[31805],[34,158341],{},[34,158343,158345],{"className":158344},[126385],[34,158346,158348,158371],{"className":158347},[31642],[34,158349,158351],{"className":158350},[31646],[31648,158352,158353],{"xmlns":31650,"display":126394},[31652,158354,158355,158368],{},[31655,158356,158357],{},[126400,158358,158359],{"notation":126402},[71090,158360,158361],{"scriptlevel":459,"displaystyle":702},[71090,158362,158363],{"scriptlevel":459,"displaystyle":702},[71090,158364,158365],{"scriptlevel":459,"displaystyle":758},[39361,158366,158367],{},"完全图：任意两点都有边",[31665,158369,158370],{"encoding":31667},"\\boxed{\\text{完全图：任意两点都有边}}",[34,158372,158374],{"className":158373,"ariaHidden":758},[31672],[34,158375,158377,158380],{"className":158376},[31676],[34,158378],{"className":158379,"style":126424},[31680],[34,158381,158383],{"className":158382},[31685],[34,158384,158386,158423],{"className":158385},[31796,31797],[34,158387,158389,158420],{"className":158388},[31801],[34,158390,158392,158412],{"className":158391,"style":126437},[31805],[34,158393,158394,158397],{"style":126440},[34,158395],{"className":158396,"style":126444},[31813],[34,158398,158400],{"className":158399},[126448],[34,158401,158403],{"className":158402},[31685],[34,158404,158406],{"className":158405},[31685],[34,158407,158409],{"className":158408},[31685,2638],[34,158410,158367],{"className":158411},[31685,105462],[34,158413,158414,158417],{"style":126463},[34,158415],{"className":158416,"style":126444},[31813],[34,158418],{"className":158419,"style":126472},[126470,126471],[34,158421,31829],{"className":158422},[31828],[34,158424,158426],{"className":158425},[31801],[34,158427,158429],{"className":158428,"style":126482},[31805],[34,158430],{},[32188,158432],{},[19,158434,158436],{"id":158435},"题型二求顶点的度","题型二：求顶点的度",[15,158438,140917],{},[105,158440,158441,158444],{},[108,158442,158443],{},"普通边贡献 1；",[108,158445,158446],{},"自环贡献 2。",[15,158448,143191],{},[105,158450,158451,158454,158457],{},[108,158452,158453],{},"出边算出度；",[108,158455,158456],{},"入边算入度；",[108,158458,158459],{},"总度 = 出度 + 入度。",[32188,158461],{},[19,158463,158465],{"id":158464},"题型三用握手定理判断图是否存在","题型三：用握手定理判断图是否存在",[15,158467,158468],{},"给出度数序列，问是否可能构成无向图。",[15,158470,158471],{},"先看：",[34,158473,158475],{"className":158474},[126385],[34,158476,158478,158500],{"className":158477},[31642],[34,158479,158481],{"className":158480},[31646],[31648,158482,158483],{"xmlns":31650,"display":126394},[31652,158484,158485,158497],{},[31655,158486,158487,158489,158491,158493,158495],{},[31662,158488,131138],{},[31658,158490,46900],{},[31662,158492,456],{"stretchy":702},[31658,158494,56782],{},[31662,158496,2557],{"stretchy":702},[31665,158498,158499],{"encoding":31667},"\\sum d(v)",[34,158501,158503],{"className":158502,"ariaHidden":758},[31672],[34,158504,158506,158510,158513,158516,158519,158522,158525],{"className":158505},[31676],[34,158507],{"className":158508,"style":158509},[31680],"height:1.6em;vertical-align:-0.55em;",[34,158511,131138],{"className":158512,"style":156561},[32767,131249,131250],[34,158514],{"className":158515,"style":32123},[31845],[34,158517,46900],{"className":158518},[31685,31686],[34,158520,456],{"className":158521},[31782],[34,158523,56782],{"className":158524,"style":32062},[31685,31686],[34,158526,2557],{"className":158527},[31690],[15,158529,158530],{},"是否为偶数。",[15,158532,158533],{},"如果是奇数，必不可能。",[15,158535,158536],{},"还要看是否违反简单图条件，比如某个顶点度数超过：",[34,158538,158540],{"className":158539},[126385],[34,158541,158543,158560],{"className":158542},[31642],[34,158544,158546],{"className":158545},[31646],[31648,158547,158548],{"xmlns":31650,"display":126394},[31652,158549,158550,158558],{},[31655,158551,158552,158554,158556],{},[31658,158553,31660],{},[31662,158555,32226],{},[31749,158557,640],{},[31665,158559,32587],{"encoding":31667},[34,158561,158563,158581],{"className":158562,"ariaHidden":758},[31672],[34,158564,158566,158569,158572,158575,158578],{"className":158565},[31676],[34,158567],{"className":158568,"style":32597},[31680],[34,158570,31660],{"className":158571},[31685,31686],[34,158573],{"className":158574,"style":32604},[31845],[34,158576,32226],{"className":158577},[32024],[34,158579],{"className":158580,"style":32604},[31845],[34,158582,158584,158587],{"className":158583},[31676],[34,158585],{"className":158586,"style":32617},[31680],[34,158588,640],{"className":158589},[31685],[32188,158591],{},[19,158593,158595],{"id":158594},"题型四判断子图","题型四：判断子图",[15,158597,158598],{},"要看：",[34,158600,158602],{"className":158601},[126385],[34,158603,158605,158640],{"className":158604},[31642],[34,158606,158608],{"className":158607},[31646],[31648,158609,158610],{"xmlns":31650,"display":126394},[31652,158611,158612,158638],{},[31655,158613,158614,158620,158622,158624,158626,158628,158634,158636],{},[37624,158615,158616,158618],{},[31658,158617,126532],{},[31662,158619,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,158621,131855],{},[31658,158623,126532],{},[31662,158625,31960],{"separator":758},[31845,158627],{"width":71082},[37624,158629,158630,158632],{},[31658,158631,96058],{},[31662,158633,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,158635,131855],{},[31658,158637,96058],{},[31665,158639,131874],{"encoding":31667},[34,158641,158643,158690,158749],{"className":158642,"ariaHidden":758},[31672],[34,158644,158646,158649,158681,158684,158687],{"className":158645},[31676],[34,158647],{"className":158648,"style":131884},[31680],[34,158650,158652,158655],{"className":158651},[31685],[34,158653,126532],{"className":158654,"style":32604},[31685,31686],[34,158656,158658],{"className":158657},[31792],[34,158659,158661],{"className":158660},[31796],[34,158662,158664],{"className":158663},[31801],[34,158665,158667],{"className":158666,"style":131724},[31805],[34,158668,158669,158672],{"style":129846},[34,158670],{"className":158671,"style":31814},[31813],[34,158673,158675],{"className":158674},[31818,31819,31820,31821],[34,158676,158678],{"className":158677},[31685,31821],[34,158679,116495],{"className":158680},[31685,31821],[34,158682],{"className":158683,"style":31846},[31845],[34,158685,131855],{"className":158686},[31850],[34,158688],{"className":158689,"style":31846},[31845],[34,158691,158693,158696,158699,158702,158705,158708,158740,158743,158746],{"className":158692},[31676],[34,158694],{"className":158695,"style":131932},[31680],[34,158697,126532],{"className":158698,"style":32604},[31685,31686],[34,158700,31960],{"className":158701},[32119],[34,158703],{"className":158704,"style":128438},[31845],[34,158706],{"className":158707,"style":32123},[31845],[34,158709,158711,158714],{"className":158710},[31685],[34,158712,96058],{"className":158713,"style":106455},[31685,31686],[34,158715,158717],{"className":158716},[31792],[34,158718,158720],{"className":158719},[31796],[34,158721,158723],{"className":158722},[31801],[34,158724,158726],{"className":158725,"style":131724},[31805],[34,158727,158728,158731],{"style":129846},[34,158729],{"className":158730,"style":31814},[31813],[34,158732,158734],{"className":158733},[31818,31819,31820,31821],[34,158735,158737],{"className":158736},[31685,31821],[34,158738,116495],{"className":158739},[31685,31821],[34,158741],{"className":158742,"style":31846},[31845],[34,158744,131855],{"className":158745},[31850],[34,158747],{"className":158748,"style":31846},[31845],[34,158750,158752,158755],{"className":158751},[31676],[34,158753],{"className":158754,"style":33218},[31680],[34,158756,96058],{"className":158757,"style":106455},[31685,31686],[15,158759,158760],{},"生成子图看顶点是否全部保留。",[15,158762,158763],{},"导出子图看是否保留了对应的所有边。",[32188,158765],{},[19,158767,158769],{"id":158768},"题型五判断图同构","题型五：判断图同构",[15,158771,158772],{},"先查：",[2230,158774,158775,158778,158781,158784,158787,158790],{},[108,158776,158777],{},"顶点数；",[108,158779,158780],{},"边数；",[108,158782,158783],{},"度数序列；",[108,158785,158786],{},"是否连通；",[108,158788,158789],{},"是否有自环、重边；",[108,158791,158792],{},"是否有三角形、孤立点、割点等结构。",[15,158794,158795],{},"如果这些不同，肯定不同构。",[15,158797,158798],{},"如果都相同，还要进一步找顶点对应关系。",[32188,158800],{},[19,158802,158804],{"id":158803},"题型六判断连通性","题型六：判断连通性",[15,158806,158807],{},"无向图：",[34,158809,158811],{"className":158810},[126385],[34,158812,158814,158844],{"className":158813},[31642],[34,158815,158817],{"className":158816},[31646],[31648,158818,158819],{"xmlns":31650,"display":126394},[31652,158820,158821,158841],{},[31655,158822,158823],{},[126400,158824,158825],{"notation":126402},[71090,158826,158827],{"scriptlevel":459,"displaystyle":702},[71090,158828,158829],{"scriptlevel":459,"displaystyle":702},[71090,158830,158831],{"scriptlevel":459,"displaystyle":758},[31655,158832,158833,158836,158838],{},[39361,158834,158835],{},"任意两点有路径",[31662,158837,140094],{},[39361,158839,158840],{},"连通",[31665,158842,158843],{"encoding":31667},"\\boxed{\\text{任意两点有路径} \\Rightarrow \\text{连通}}",[34,158845,158847],{"className":158846,"ariaHidden":758},[31672],[34,158848,158850,158853],{"className":158849},[31676],[34,158851],{"className":158852,"style":126424},[31680],[34,158854,158856],{"className":158855},[31685],[34,158857,158859,158911],{"className":158858},[31796,31797],[34,158860,158862,158908],{"className":158861},[31801],[34,158863,158865,158900],{"className":158864,"style":126437},[31805],[34,158866,158867,158870],{"style":126440},[34,158868],{"className":158869,"style":126444},[31813],[34,158871,158873],{"className":158872},[126448],[34,158874,158876],{"className":158875},[31685],[34,158877,158879,158885,158888,158891,158894],{"className":158878},[31685],[34,158880,158882],{"className":158881},[31685,2638],[34,158883,158835],{"className":158884},[31685,105462],[34,158886],{"className":158887,"style":31846},[31845],[34,158889,140094],{"className":158890},[31850],[34,158892],{"className":158893,"style":31846},[31845],[34,158895,158897],{"className":158896},[31685,2638],[34,158898,158840],{"className":158899},[31685,105462],[34,158901,158902,158905],{"style":126463},[34,158903],{"className":158904,"style":126444},[31813],[34,158906],{"className":158907,"style":126472},[126470,126471],[34,158909,31829],{"className":158910},[31828],[34,158912,158914],{"className":158913},[31801],[34,158915,158917],{"className":158916,"style":126482},[31805],[34,158918],{},[15,158920,158921],{},"有向图：",[34,158923,158925],{"className":158924},[126385],[34,158926,158928,158959],{"className":158927},[31642],[34,158929,158931],{"className":158930},[31646],[31648,158932,158933],{"xmlns":31650,"display":126394},[31652,158934,158935,158957],{},[31655,158936,158937],{},[126400,158938,158939],{"notation":126402},[71090,158940,158941],{"scriptlevel":459,"displaystyle":702},[71090,158942,158943],{"scriptlevel":459,"displaystyle":702},[71090,158944,158945],{"scriptlevel":459,"displaystyle":758},[31655,158946,158947,158949,158951,158953,158955],{},[39361,158948,140091],{},[31662,158950,140094],{},[39361,158952,140097],{},[31662,158954,140094],{},[39361,158956,140102],{},[31665,158958,140105],{"encoding":31667},[34,158960,158962],{"className":158961,"ariaHidden":758},[31672],[34,158963,158965,158968],{"className":158964},[31676],[34,158966],{"className":158967,"style":126424},[31680],[34,158969,158971],{"className":158970},[31685],[34,158972,158974,159041],{"className":158973},[31796,31797],[34,158975,158977,159038],{"className":158976},[31801],[34,158978,158980,159030],{"className":158979,"style":126437},[31805],[34,158981,158982,158985],{"style":126440},[34,158983],{"className":158984,"style":126444},[31813],[34,158986,158988],{"className":158987},[126448],[34,158989,158991],{"className":158990},[31685],[34,158992,158994,159000,159003,159006,159009,159015,159018,159021,159024],{"className":158993},[31685],[34,158995,158997],{"className":158996},[31685,2638],[34,158998,140091],{"className":158999},[31685,105462],[34,159001],{"className":159002,"style":31846},[31845],[34,159004,140094],{"className":159005},[31850],[34,159007],{"className":159008,"style":31846},[31845],[34,159010,159012],{"className":159011},[31685,2638],[34,159013,140097],{"className":159014},[31685,105462],[34,159016],{"className":159017,"style":31846},[31845],[34,159019,140094],{"className":159020},[31850],[34,159022],{"className":159023,"style":31846},[31845],[34,159025,159027],{"className":159026},[31685,2638],[34,159028,140102],{"className":159029},[31685,105462],[34,159031,159032,159035],{"style":126463},[34,159033],{"className":159034,"style":126444},[31813],[34,159036],{"className":159037,"style":126472},[126470,126471],[34,159039,31829],{"className":159040},[31828],[34,159042,159044],{"className":159043},[31801],[34,159045,159047],{"className":159046,"style":126482},[31805],[34,159048],{},[32188,159050],{},[19,159052,159054],{"id":159053},"题型七求邻接矩阵关联矩阵","题型七：求邻接矩阵、关联矩阵",[15,159056,159057],{},"邻接矩阵：",[34,159059,159061],{"className":159060},[126385],[34,159062,159064,159081],{"className":159063},[31642],[34,159065,159067],{"className":159066},[31646],[31648,159068,159069],{"xmlns":31650,"display":126394},[31652,159070,159071,159079],{},[31655,159072,159073,159075,159077],{},[31658,159074,31660],{},[31662,159076,42071],{},[31658,159078,31660],{},[31665,159080,143150],{"encoding":31667},[34,159082,159084,159102],{"className":159083,"ariaHidden":758},[31672],[34,159085,159087,159090,159093,159096,159099],{"className":159086},[31676],[34,159088],{"className":159089,"style":32597},[31680],[34,159091,31660],{"className":159092},[31685,31686],[34,159094],{"className":159095,"style":32604},[31845],[34,159097,42071],{"className":159098},[32024],[34,159100],{"className":159101,"style":32604},[31845],[34,159103,159105,159108],{"className":159104},[31676],[34,159106],{"className":159107,"style":32527},[31680],[34,159109,31660],{"className":159110},[31685,31686],[15,159112,159113],{},"表示顶点和顶点关系。",[15,159115,159116],{},"关联矩阵：",[34,159118,159120],{"className":159119},[126385],[34,159121,159123,159140],{"className":159122},[31642],[34,159124,159126],{"className":159125},[31646],[31648,159127,159128],{"xmlns":31650,"display":126394},[31652,159129,159130,159138],{},[31655,159131,159132,159134,159136],{},[31658,159133,31660],{},[31662,159135,42071],{},[31658,159137,31759],{},[31665,159139,140872],{"encoding":31667},[34,159141,159143,159161],{"className":159142,"ariaHidden":758},[31672],[34,159144,159146,159149,159152,159155,159158],{"className":159145},[31676],[34,159147],{"className":159148,"style":32597},[31680],[34,159150,31660],{"className":159151},[31685,31686],[34,159153],{"className":159154,"style":32604},[31845],[34,159156,42071],{"className":159157},[32024],[34,159159],{"className":159160,"style":32604},[31845],[34,159162,159164,159167],{"className":159163},[31676],[34,159165],{"className":159166,"style":32527},[31680],[34,159168,31759],{"className":159169},[31685,31686],[15,159171,159172],{},"表示顶点和边关系。",[32188,159174],{},[19,159176,159178],{"id":159177},"题型八用矩阵求路径条数","题型八：用矩阵求路径条数",[15,159180,159181],{},"如果题目问：",[15,159183,159184,159185,53834,159254,89432,159323,159351],{},"“从 ",[34,159186,159188,159205],{"className":159187},[31642],[34,159189,159191],{"className":159190},[31646],[31648,159192,159193],{"xmlns":31650},[31652,159194,159195,159203],{},[31655,159196,159197],{},[31743,159198,159199,159201],{},[31658,159200,56782],{},[31658,159202,31955],{},[31665,159204,56787],{"encoding":31667},[34,159206,159208],{"className":159207,"ariaHidden":758},[31672],[34,159209,159211,159214],{"className":159210},[31676],[34,159212],{"className":159213,"style":31860},[31680],[34,159215,159217,159220],{"className":159216},[31685],[34,159218,56782],{"className":159219,"style":32062},[31685,31686],[34,159221,159223],{"className":159222},[31792],[34,159224,159226,159246],{"className":159225},[31796,31797],[34,159227,159229,159243],{"className":159228},[31801],[34,159230,159232],{"className":159231,"style":32090},[31805],[34,159233,159234,159237],{"style":38795},[34,159235],{"className":159236,"style":31814},[31813],[34,159238,159240],{"className":159239},[31818,31819,31820,31821],[34,159241,31955],{"className":159242},[31685,31686,31821],[34,159244,31829],{"className":159245},[31828],[34,159247,159249],{"className":159248},[31801],[34,159250,159252],{"className":159251,"style":31836},[31805],[34,159253],{},[34,159255,159257,159274],{"className":159256},[31642],[34,159258,159260],{"className":159259},[31646],[31648,159261,159262],{"xmlns":31650},[31652,159263,159264,159272],{},[31655,159265,159266],{},[31743,159267,159268,159270],{},[31658,159269,56782],{},[31658,159271,31968],{},[31665,159273,138126],{"encoding":31667},[34,159275,159277],{"className":159276,"ariaHidden":758},[31672],[34,159278,159280,159283],{"className":159279},[31676],[34,159281],{"className":159282,"style":56005},[31680],[34,159284,159286,159289],{"className":159285},[31685],[34,159287,56782],{"className":159288,"style":32062},[31685,31686],[34,159290,159292],{"className":159291},[31792],[34,159293,159295,159315],{"className":159294},[31796,31797],[34,159296,159298,159312],{"className":159297},[31801],[34,159299,159301],{"className":159300,"style":32090},[31805],[34,159302,159303,159306],{"style":38795},[34,159304],{"className":159305,"style":31814},[31813],[34,159307,159309],{"className":159308},[31818,31819,31820,31821],[34,159310,31968],{"className":159311,"style":32153},[31685,31686,31821],[34,159313,31829],{"className":159314},[31828],[34,159316,159318],{"className":159317},[31801],[34,159319,159321],{"className":159320,"style":32163},[31805],[34,159322],{},[34,159324,159326,159339],{"className":159325},[31642],[34,159327,159329],{"className":159328},[31646],[31648,159330,159331],{"xmlns":31650},[31652,159332,159333,159337],{},[31655,159334,159335],{},[31658,159336,31759],{},[31665,159338,31759],{"encoding":31667},[34,159340,159342],{"className":159341,"ariaHidden":758},[31672],[34,159343,159345,159348],{"className":159344},[31676],[34,159346],{"className":159347,"style":32527},[31680],[34,159349,31759],{"className":159350},[31685,31686]," 的路径有几条？”",[15,159353,159354],{},"就看：",[34,159356,159358],{"className":159357},[126385],[34,159359,159361,159390],{"className":159360},[31642],[34,159362,159364],{"className":159363},[31646],[31648,159365,159366],{"xmlns":31650,"display":126394},[31652,159367,159368,159388],{},[31655,159369,159370,159372,159378],{},[31662,159371,456],{"stretchy":702},[37624,159373,159374,159376],{},[31658,159375,54151],{},[31658,159377,31759],{},[31743,159379,159380,159382],{},[31662,159381,2557],{"stretchy":702},[31655,159383,159384,159386],{},[31658,159385,31955],{},[31658,159387,31968],{},[31665,159389,157273],{"encoding":31667},[34,159391,159393],{"className":159392,"ariaHidden":758},[31672],[34,159394,159396,159399,159402,159431],{"className":159395},[31676],[34,159397],{"className":159398,"style":32058},[31680],[34,159400,456],{"className":159401},[31782],[34,159403,159405,159408],{"className":159404},[31685],[34,159406,54151],{"className":159407},[31685,31686],[34,159409,159411],{"className":159410},[31792],[34,159412,159414],{"className":159413},[31796],[34,159415,159417],{"className":159416},[31801],[34,159418,159420],{"className":159419,"style":145166},[31805],[34,159421,159422,159425],{"style":129846},[34,159423],{"className":159424,"style":31814},[31813],[34,159426,159428],{"className":159427},[31818,31819,31820,31821],[34,159429,31759],{"className":159430},[31685,31686,31821],[34,159432,159434,159437],{"className":159433},[31690],[34,159435,2557],{"className":159436},[31690],[34,159438,159440],{"className":159439},[31792],[34,159441,159443,159466],{"className":159442},[31796,31797],[34,159444,159446,159463],{"className":159445},[31801],[34,159447,159449],{"className":159448,"style":32090},[31805],[34,159450,159451,159454],{"style":31809},[34,159452],{"className":159453,"style":31814},[31813],[34,159455,159457],{"className":159456},[31818,31819,31820,31821],[34,159458,159460],{"className":159459},[31685,31821],[34,159461,141027],{"className":159462,"style":32153},[31685,31686,31821],[34,159464,31829],{"className":159465},[31828],[34,159467,159469],{"className":159468},[31801],[34,159470,159472],{"className":159471,"style":32163},[31805],[34,159473],{},[32188,159475],{},[19,159477,159479],{"id":159478},"题型九求可达矩阵","题型九：求可达矩阵",[15,159481,159482],{},"用：",[34,159484,159486],{"className":159485},[126385],[34,159487,159489,159546],{"className":159488},[31642],[34,159490,159492],{"className":159491},[31646],[31648,159493,159494],{"xmlns":31650,"display":126394},[31652,159495,159496,159544],{},[31655,159497,159498,159500,159502,159504,159506,159508,159510,159522,159524,159526,159528],{},[31658,159499,105852],{},[31662,159501,202],{},[31658,159503,106063],{},[31662,159505,147175],{},[31658,159507,54151],{},[31662,159509,147175],{},[37624,159511,159512,159514],{},[31658,159513,54151],{},[31655,159515,159516,159518,159520],{},[31662,159517,456],{"stretchy":702},[31749,159519,768],{},[31662,159521,2557],{"stretchy":702},[31662,159523,147175],{},[31662,159525,38874],{},[31662,159527,147175],{},[37624,159529,159530,159532],{},[31658,159531,54151],{},[31655,159533,159534,159536,159538,159540,159542],{},[31662,159535,456],{"stretchy":702},[31658,159537,31660],{},[31662,159539,32226],{},[31749,159541,640],{},[31662,159543,2557],{"stretchy":702},[31665,159545,147216],{"encoding":31667},[34,159547,159549,159567,159585,159603,159656,159674],{"className":159548,"ariaHidden":758},[31672],[34,159550,159552,159555,159558,159561,159564],{"className":159551},[31676],[34,159553],{"className":159554,"style":33218},[31680],[34,159556,105852],{"className":159557,"style":37422},[31685,31686],[34,159559],{"className":159560,"style":31846},[31845],[34,159562,202],{"className":159563},[31850],[34,159565],{"className":159566,"style":31846},[31845],[34,159568,159570,159573,159576,159579,159582],{"className":159569},[31676],[34,159571],{"className":159572,"style":33218},[31680],[34,159574,106063],{"className":159575,"style":67482},[31685,31686],[34,159577],{"className":159578,"style":32604},[31845],[34,159580,147175],{"className":159581},[32024],[34,159583],{"className":159584,"style":32604},[31845],[34,159586,159588,159591,159594,159597,159600],{"className":159587},[31676],[34,159589],{"className":159590,"style":33218},[31680],[34,159592,54151],{"className":159593},[31685,31686],[34,159595],{"className":159596,"style":32604},[31845],[34,159598,147175],{"className":159599},[32024],[34,159601],{"className":159602,"style":32604},[31845],[34,159604,159606,159609,159647,159650,159653],{"className":159605},[31676],[34,159607],{"className":159608,"style":147280},[31680],[34,159610,159612,159615],{"className":159611},[31685],[34,159613,54151],{"className":159614},[31685,31686],[34,159616,159618],{"className":159617},[31792],[34,159619,159621],{"className":159620},[31796],[34,159622,159624],{"className":159623},[31801],[34,159625,159627],{"className":159626,"style":147280},[31805],[34,159628,159629,159632],{"style":129846},[34,159630],{"className":159631,"style":31814},[31813],[34,159633,159635],{"className":159634},[31818,31819,31820,31821],[34,159636,159638,159641,159644],{"className":159637},[31685,31821],[34,159639,456],{"className":159640},[31782,31821],[34,159642,768],{"className":159643},[31685,31821],[34,159645,2557],{"className":159646},[31690,31821],[34,159648],{"className":159649,"style":32604},[31845],[34,159651,147175],{"className":159652},[32024],[34,159654],{"className":159655,"style":32604},[31845],[34,159657,159659,159662,159665,159668,159671],{"className":159658},[31676],[34,159660],{"className":159661,"style":147334},[31680],[34,159663,38874],{"className":159664},[34006],[34,159666],{"className":159667,"style":32604},[31845],[34,159669,147175],{"className":159670},[32024],[34,159672],{"className":159673,"style":32604},[31845],[34,159675,159677,159680],{"className":159676},[31676],[34,159678],{"className":159679,"style":147280},[31680],[34,159681,159683,159686],{"className":159682},[31685],[34,159684,54151],{"className":159685},[31685,31686],[34,159687,159689],{"className":159688},[31792],[34,159690,159692],{"className":159691},[31796],[34,159693,159695],{"className":159694},[31801],[34,159696,159698],{"className":159697,"style":147280},[31805],[34,159699,159700,159703],{"style":129846},[34,159701],{"className":159702,"style":31814},[31813],[34,159704,159706],{"className":159705},[31818,31819,31820,31821],[34,159707,159709,159712,159715,159718,159721],{"className":159708},[31685,31821],[34,159710,456],{"className":159711},[31782,31821],[34,159713,31660],{"className":159714},[31685,31686,31821],[34,159716,32226],{"className":159717},[32024,31821],[34,159719,640],{"className":159720},[31685,31821],[34,159722,2557],{"className":159723},[31690,31821],[15,159725,159726],{},"或用 Warshall 类方法。",[32188,159728],{},[19,159730,159732],{"id":159731},"题型十最短路径","题型十：最短路径",[15,159734,159735],{},"单源最短路径：",[34,159737,159739],{"className":159738},[126385],[34,159740,159742,159764],{"className":159741},[31642],[34,159743,159745],{"className":159744},[31646],[31648,159746,159747],{"xmlns":31650,"display":126394},[31652,159748,159749,159761],{},[31655,159750,159751],{},[126400,159752,159753],{"notation":126402},[71090,159754,159755],{"scriptlevel":459,"displaystyle":702},[71090,159756,159757],{"scriptlevel":459,"displaystyle":702},[71090,159758,159759],{"scriptlevel":459,"displaystyle":758},[39361,159760,153534],{},[31665,159762,159763],{"encoding":31667},"\\boxed{\\text{Dijkstra}}",[34,159765,159767],{"className":159766,"ariaHidden":758},[31672],[34,159768,159770,159773],{"className":159769},[31676],[34,159771],{"className":159772,"style":153586},[31680],[34,159774,159776],{"className":159775},[31685],[34,159777,159779,159816],{"className":159778},[31796,31797],[34,159780,159782,159813],{"className":159781},[31801],[34,159783,159785,159805],{"className":159784,"style":135867},[31805],[34,159786,159787,159790],{"style":153601},[34,159788],{"className":159789,"style":153605},[31813],[34,159791,159793],{"className":159792},[126448],[34,159794,159796],{"className":159795},[31685],[34,159797,159799],{"className":159798},[31685],[34,159800,159802],{"className":159801},[31685,2638],[34,159803,153534],{"className":159804},[31685],[34,159806,159807,159810],{"style":135889},[34,159808],{"className":159809,"style":153605},[31813],[34,159811],{"className":159812,"style":153642},[126470,126471],[34,159814,31829],{"className":159815},[31828],[34,159817,159819],{"className":159818},[31801],[34,159820,159822],{"className":159821,"style":142798},[31805],[34,159823],{},[15,159825,159826],{},"任意两点最短路径：",[34,159828,159830],{"className":159829},[126385],[34,159831,159833,159855],{"className":159832},[31642],[34,159834,159836],{"className":159835},[31646],[31648,159837,159838],{"xmlns":31650,"display":126394},[31652,159839,159840,159852],{},[31655,159841,159842],{},[126400,159843,159844],{"notation":126402},[71090,159845,159846],{"scriptlevel":459,"displaystyle":702},[71090,159847,159848],{"scriptlevel":459,"displaystyle":702},[71090,159849,159850],{"scriptlevel":459,"displaystyle":758},[39361,159851,153542],{},[31665,159853,159854],{"encoding":31667},"\\boxed{\\text{Floyd}}",[34,159856,159858],{"className":159857,"ariaHidden":758},[31672],[34,159859,159861,159864],{"className":159860},[31676],[34,159862],{"className":159863,"style":153586},[31680],[34,159865,159867],{"className":159866},[31685],[34,159868,159870,159907],{"className":159869},[31796,31797],[34,159871,159873,159904],{"className":159872},[31801],[34,159874,159876,159896],{"className":159875,"style":135867},[31805],[34,159877,159878,159881],{"style":153601},[34,159879],{"className":159880,"style":153605},[31813],[34,159882,159884],{"className":159883},[126448],[34,159885,159887],{"className":159886},[31685],[34,159888,159890],{"className":159889},[31685],[34,159891,159893],{"className":159892},[31685,2638],[34,159894,153542],{"className":159895},[31685],[34,159897,159898,159901],{"style":135889},[34,159899],{"className":159900,"style":153605},[31813],[34,159902],{"className":159903,"style":153642},[126470,126471],[34,159905,31829],{"className":159906},[31828],[34,159908,159910],{"className":159909},[31801],[34,159911,159913],{"className":159912,"style":142798},[31805],[34,159914],{},[32188,159916],{},[11,159918,159919],{"id":159919},"最容易混的点",[19,159921,159923],{"id":159922},"_1-路径简单路径基本路径","1. 路径、简单路径、基本路径",[13837,159925,159926,159936],{},[13840,159927,159928],{},[13843,159929,159930,159933],{},[13846,159931,159932],{},"名称",[13846,159934,159935],{},"要求",[13853,159937,159938,159946,159953],{},[13843,159939,159940,159943],{},[13858,159941,159942],{},"路径",[13858,159944,159945],{},"可以重复边、重复点",[13843,159947,159948,159950],{},[13858,159949,136397],{},[13858,159951,159952],{},"边不重复",[13843,159954,159955,159957],{},[13858,159956,136412],{},[13858,159958,159959],{},"顶点不重复",[15,159961,159962],{},"所以：",[34,159964,159966],{"className":159965},[126385],[34,159967,159969,159992],{"className":159968},[31642],[34,159970,159972],{"className":159971},[31646],[31648,159973,159974],{"xmlns":31650,"display":126394},[31652,159975,159976,159989],{},[31655,159977,159978],{},[126400,159979,159980],{"notation":126402},[71090,159981,159982],{"scriptlevel":459,"displaystyle":702},[71090,159983,159984],{"scriptlevel":459,"displaystyle":702},[71090,159985,159986],{"scriptlevel":459,"displaystyle":758},[39361,159987,159988],{},"基本路径一定是简单路径，反之不一定",[31665,159990,159991],{"encoding":31667},"\\boxed{\\text{基本路径一定是简单路径，反之不一定}}",[34,159993,159995],{"className":159994,"ariaHidden":758},[31672],[34,159996,159998,160001],{"className":159997},[31676],[34,159999],{"className":160000,"style":126424},[31680],[34,160002,160004],{"className":160003},[31685],[34,160005,160007,160044],{"className":160006},[31796,31797],[34,160008,160010,160041],{"className":160009},[31801],[34,160011,160013,160033],{"className":160012,"style":126437},[31805],[34,160014,160015,160018],{"style":126440},[34,160016],{"className":160017,"style":126444},[31813],[34,160019,160021],{"className":160020},[126448],[34,160022,160024],{"className":160023},[31685],[34,160025,160027],{"className":160026},[31685],[34,160028,160030],{"className":160029},[31685,2638],[34,160031,159988],{"className":160032},[31685,105462],[34,160034,160035,160038],{"style":126463},[34,160036],{"className":160037,"style":126444},[31813],[34,160039],{"className":160040,"style":126472},[126470,126471],[34,160042,31829],{"className":160043},[31828],[34,160045,160047],{"className":160046},[31801],[34,160048,160050],{"className":160049,"style":126482},[31805],[34,160051],{},[32188,160053],{},[19,160055,160057],{"id":160056},"_2-循环简单循环基本循环","2. 循环、简单循环、基本循环",[13837,160059,160060,160068],{},[13840,160061,160062],{},[13843,160063,160064,160066],{},[13846,160065,159932],{},[13846,160067,159935],{},[13853,160069,160070,160077,160083],{},[13843,160071,160072,160074],{},[13858,160073,136662],{},[13858,160075,160076],{},"起点终点相同",[13843,160078,160079,160081],{},[13858,160080,136677],{},[13858,160082,159952],{},[13843,160084,160085,160087],{},[13858,160086,136689],{},[13858,160088,160089],{},"除起点终点外，顶点不重复",[32188,160091],{},[19,160093,160095],{"id":160094},"_3-强连通单向连通弱连通","3. 强连通、单向连通、弱连通",[34,160097,160099],{"className":160098},[126385],[34,160100,160102,160125],{"className":160101},[31642],[34,160103,160105],{"className":160104},[31646],[31648,160106,160107],{"xmlns":31650,"display":126394},[31652,160108,160109,160122],{},[31655,160110,160111],{},[126400,160112,160113],{"notation":126402},[71090,160114,160115],{"scriptlevel":459,"displaystyle":702},[71090,160116,160117],{"scriptlevel":459,"displaystyle":702},[71090,160118,160119],{"scriptlevel":459,"displaystyle":758},[39361,160120,160121],{},"强：双向可达",[31665,160123,160124],{"encoding":31667},"\\boxed{\\text{强：双向可达}}",[34,160126,160128],{"className":160127,"ariaHidden":758},[31672],[34,160129,160131,160134],{"className":160130},[31676],[34,160132],{"className":160133,"style":126424},[31680],[34,160135,160137],{"className":160136},[31685],[34,160138,160140,160177],{"className":160139},[31796,31797],[34,160141,160143,160174],{"className":160142},[31801],[34,160144,160146,160166],{"className":160145,"style":126437},[31805],[34,160147,160148,160151],{"style":126440},[34,160149],{"className":160150,"style":126444},[31813],[34,160152,160154],{"className":160153},[126448],[34,160155,160157],{"className":160156},[31685],[34,160158,160160],{"className":160159},[31685],[34,160161,160163],{"className":160162},[31685,2638],[34,160164,160121],{"className":160165},[31685,105462],[34,160167,160168,160171],{"style":126463},[34,160169],{"className":160170,"style":126444},[31813],[34,160172],{"className":160173,"style":126472},[126470,126471],[34,160175,31829],{"className":160176},[31828],[34,160178,160180],{"className":160179},[31801],[34,160181,160183],{"className":160182,"style":126482},[31805],[34,160184],{},[34,160186,160188],{"className":160187},[126385],[34,160189,160191,160214],{"className":160190},[31642],[34,160192,160194],{"className":160193},[31646],[31648,160195,160196],{"xmlns":31650,"display":126394},[31652,160197,160198,160211],{},[31655,160199,160200],{},[126400,160201,160202],{"notation":126402},[71090,160203,160204],{"scriptlevel":459,"displaystyle":702},[71090,160205,160206],{"scriptlevel":459,"displaystyle":702},[71090,160207,160208],{"scriptlevel":459,"displaystyle":758},[39361,160209,160210],{},"单向：至少一个方向可达",[31665,160212,160213],{"encoding":31667},"\\boxed{\\text{单向：至少一个方向可达}}",[34,160215,160217],{"className":160216,"ariaHidden":758},[31672],[34,160218,160220,160223],{"className":160219},[31676],[34,160221],{"className":160222,"style":126424},[31680],[34,160224,160226],{"className":160225},[31685],[34,160227,160229,160266],{"className":160228},[31796,31797],[34,160230,160232,160263],{"className":160231},[31801],[34,160233,160235,160255],{"className":160234,"style":126437},[31805],[34,160236,160237,160240],{"style":126440},[34,160238],{"className":160239,"style":126444},[31813],[34,160241,160243],{"className":160242},[126448],[34,160244,160246],{"className":160245},[31685],[34,160247,160249],{"className":160248},[31685],[34,160250,160252],{"className":160251},[31685,2638],[34,160253,160210],{"className":160254},[31685,105462],[34,160256,160257,160260],{"style":126463},[34,160258],{"className":160259,"style":126444},[31813],[34,160261],{"className":160262,"style":126472},[126470,126471],[34,160264,31829],{"className":160265},[31828],[34,160267,160269],{"className":160268},[31801],[34,160270,160272],{"className":160271,"style":126482},[31805],[34,160273],{},[34,160275,160277],{"className":160276},[126385],[34,160278,160280,160303],{"className":160279},[31642],[34,160281,160283],{"className":160282},[31646],[31648,160284,160285],{"xmlns":31650,"display":126394},[31652,160286,160287,160300],{},[31655,160288,160289],{},[126400,160290,160291],{"notation":126402},[71090,160292,160293],{"scriptlevel":459,"displaystyle":702},[71090,160294,160295],{"scriptlevel":459,"displaystyle":702},[71090,160296,160297],{"scriptlevel":459,"displaystyle":758},[39361,160298,160299],{},"弱：去掉方向后连通",[31665,160301,160302],{"encoding":31667},"\\boxed{\\text{弱：去掉方向后连通}}",[34,160304,160306],{"className":160305,"ariaHidden":758},[31672],[34,160307,160309,160312],{"className":160308},[31676],[34,160310],{"className":160311,"style":126424},[31680],[34,160313,160315],{"className":160314},[31685],[34,160316,160318,160355],{"className":160317},[31796,31797],[34,160319,160321,160352],{"className":160320},[31801],[34,160322,160324,160344],{"className":160323,"style":126437},[31805],[34,160325,160326,160329],{"style":126440},[34,160327],{"className":160328,"style":126444},[31813],[34,160330,160332],{"className":160331},[126448],[34,160333,160335],{"className":160334},[31685],[34,160336,160338],{"className":160337},[31685],[34,160339,160341],{"className":160340},[31685,2638],[34,160342,160299],{"className":160343},[31685,105462],[34,160345,160346,160349],{"style":126463},[34,160347],{"className":160348,"style":126444},[31813],[34,160350],{"className":160351,"style":126472},[126470,126471],[34,160353,31829],{"className":160354},[31828],[34,160356,160358],{"className":160357},[31801],[34,160359,160361],{"className":160360,"style":126482},[31805],[34,160362],{},[32188,160364],{},[19,160366,160368],{"id":160367},"_4-关联矩阵和邻接矩阵","4. 关联矩阵和邻接矩阵",[13837,160370,160371,160387],{},[13840,160372,160373],{},[13843,160374,160375,160378,160381,160384],{},[13846,160376,160377],{},"矩阵",[13846,160379,160380],{},"行",[13846,160382,160383],{},"列",[13846,160385,160386],{},"表示",[13853,160388,160389,160401],{},[13843,160390,160391,160394,160396,160398],{},[13858,160392,160393],{},"关联矩阵",[13858,160395,126955],{},[13858,160397,126989],{},[13858,160399,160400],{},"顶点与边是否关联",[13843,160402,160403,160406,160408,160410],{},[13858,160404,160405],{},"邻接矩阵",[13858,160407,126955],{},[13858,160409,126955],{},[13858,160411,160412],{},"顶点之间是否有边",[32188,160414],{},[19,160416,160418],{"id":160417},"_5-dijkstra-和-floyd","5. Dijkstra 和 Floyd",[13837,160420,160421,160430],{},[13840,160422,160423],{},[13843,160424,160425,160427],{},[13846,160426,153524],{},[13846,160428,160429],{},"用途",[13853,160431,160432,160439],{},[13843,160433,160434,160436],{},[13858,160435,153534],{},[13858,160437,160438],{},"一个点到其他点",[13843,160440,160441,160443],{},[13858,160442,153542],{},[13858,160444,160445],{},"所有点到所有点",[32188,160447],{},[11,160449,160450],{"id":160450},"第十二章一句话总结",[15,160452,160453],{},"第十二章就是图论入门，核心主线是：",[34,160455,160457],{"className":160456},[126385],[34,160458,160460,160483],{"className":160459},[31642],[34,160461,160463],{"className":160462},[31646],[31648,160464,160465],{"xmlns":31650,"display":126394},[31652,160466,160467,160480],{},[31655,160468,160469],{},[126400,160470,160471],{"notation":126402},[71090,160472,160473],{"scriptlevel":459,"displaystyle":702},[71090,160474,160475],{"scriptlevel":459,"displaystyle":702},[71090,160476,160477],{"scriptlevel":459,"displaystyle":758},[39361,160478,160479],{},"图是什么",[31665,160481,160482],{"encoding":31667},"\\boxed{\\text{图是什么}}",[34,160484,160486],{"className":160485,"ariaHidden":758},[31672],[34,160487,160489,160492],{"className":160488},[31676],[34,160490],{"className":160491,"style":126424},[31680],[34,160493,160495],{"className":160494},[31685],[34,160496,160498,160535],{"className":160497},[31796,31797],[34,160499,160501,160532],{"className":160500},[31801],[34,160502,160504,160524],{"className":160503,"style":126437},[31805],[34,160505,160506,160509],{"style":126440},[34,160507],{"className":160508,"style":126444},[31813],[34,160510,160512],{"className":160511},[126448],[34,160513,160515],{"className":160514},[31685],[34,160516,160518],{"className":160517},[31685],[34,160519,160521],{"className":160520},[31685,2638],[34,160522,160479],{"className":160523},[31685,105462],[34,160525,160526,160529],{"style":126463},[34,160527],{"className":160528,"style":126444},[31813],[34,160530],{"className":160531,"style":126472},[126470,126471],[34,160533,31829],{"className":160534},[31828],[34,160536,160538],{"className":160537},[31801],[34,160539,160541],{"className":160540,"style":126482},[31805],[34,160542],{},[34,160544,160546],{"className":160545},[126385],[34,160547,160549,160572],{"className":160548},[31642],[34,160550,160552],{"className":160551},[31646],[31648,160553,160554],{"xmlns":31650,"display":126394},[31652,160555,160556,160569],{},[31655,160557,160558],{},[126400,160559,160560],{"notation":126402},[71090,160561,160562],{"scriptlevel":459,"displaystyle":702},[71090,160563,160564],{"scriptlevel":459,"displaystyle":702},[71090,160565,160566],{"scriptlevel":459,"displaystyle":758},[39361,160567,160568],{},"图有哪些基本结构",[31665,160570,160571],{"encoding":31667},"\\boxed{\\text{图有哪些基本结构}}",[34,160573,160575],{"className":160574,"ariaHidden":758},[31672],[34,160576,160578,160581],{"className":160577},[31676],[34,160579],{"className":160580,"style":126424},[31680],[34,160582,160584],{"className":160583},[31685],[34,160585,160587,160624],{"className":160586},[31796,31797],[34,160588,160590,160621],{"className":160589},[31801],[34,160591,160593,160613],{"className":160592,"style":126437},[31805],[34,160594,160595,160598],{"style":126440},[34,160596],{"className":160597,"style":126444},[31813],[34,160599,160601],{"className":160600},[126448],[34,160602,160604],{"className":160603},[31685],[34,160605,160607],{"className":160606},[31685],[34,160608,160610],{"className":160609},[31685,2638],[34,160611,160568],{"className":160612},[31685,105462],[34,160614,160615,160618],{"style":126463},[34,160616],{"className":160617,"style":126444},[31813],[34,160619],{"className":160620,"style":126472},[126470,126471],[34,160622,31829],{"className":160623},[31828],[34,160625,160627],{"className":160626},[31801],[34,160628,160630],{"className":160629,"style":126482},[31805],[34,160631],{},[34,160633,160635],{"className":160634},[126385],[34,160636,160638,160661],{"className":160637},[31642],[34,160639,160641],{"className":160640},[31646],[31648,160642,160643],{"xmlns":31650,"display":126394},[31652,160644,160645,160658],{},[31655,160646,160647],{},[126400,160648,160649],{"notation":126402},[71090,160650,160651],{"scriptlevel":459,"displaystyle":702},[71090,160652,160653],{"scriptlevel":459,"displaystyle":702},[71090,160654,160655],{"scriptlevel":459,"displaystyle":758},[39361,160656,160657],{},"图中能不能走通",[31665,160659,160660],{"encoding":31667},"\\boxed{\\text{图中能不能走通}}",[34,160662,160664],{"className":160663,"ariaHidden":758},[31672],[34,160665,160667,160670],{"className":160666},[31676],[34,160668],{"className":160669,"style":126424},[31680],[34,160671,160673],{"className":160672},[31685],[34,160674,160676,160713],{"className":160675},[31796,31797],[34,160677,160679,160710],{"className":160678},[31801],[34,160680,160682,160702],{"className":160681,"style":126437},[31805],[34,160683,160684,160687],{"style":126440},[34,160685],{"className":160686,"style":126444},[31813],[34,160688,160690],{"className":160689},[126448],[34,160691,160693],{"className":160692},[31685],[34,160694,160696],{"className":160695},[31685],[34,160697,160699],{"className":160698},[31685,2638],[34,160700,160657],{"className":160701},[31685,105462],[34,160703,160704,160707],{"style":126463},[34,160705],{"className":160706,"style":126444},[31813],[34,160708],{"className":160709,"style":126472},[126470,126471],[34,160711,31829],{"className":160712},[31828],[34,160714,160716],{"className":160715},[31801],[34,160717,160719],{"className":160718,"style":126482},[31805],[34,160720],{},[34,160722,160724],{"className":160723},[126385],[34,160725,160727,160750],{"className":160726},[31642],[34,160728,160730],{"className":160729},[31646],[31648,160731,160732],{"xmlns":31650,"display":126394},[31652,160733,160734,160747],{},[31655,160735,160736],{},[126400,160737,160738],{"notation":126402},[71090,160739,160740],{"scriptlevel":459,"displaystyle":702},[71090,160741,160742],{"scriptlevel":459,"displaystyle":702},[71090,160743,160744],{"scriptlevel":459,"displaystyle":758},[39361,160745,160746],{},"如何用矩阵表示图",[31665,160748,160749],{"encoding":31667},"\\boxed{\\text{如何用矩阵表示图}}",[34,160751,160753],{"className":160752,"ariaHidden":758},[31672],[34,160754,160756,160759],{"className":160755},[31676],[34,160757],{"className":160758,"style":126424},[31680],[34,160760,160762],{"className":160761},[31685],[34,160763,160765,160802],{"className":160764},[31796,31797],[34,160766,160768,160799],{"className":160767},[31801],[34,160769,160771,160791],{"className":160770,"style":126437},[31805],[34,160772,160773,160776],{"style":126440},[34,160774],{"className":160775,"style":126444},[31813],[34,160777,160779],{"className":160778},[126448],[34,160780,160782],{"className":160781},[31685],[34,160783,160785],{"className":160784},[31685],[34,160786,160788],{"className":160787},[31685,2638],[34,160789,160746],{"className":160790},[31685,105462],[34,160792,160793,160796],{"style":126463},[34,160794],{"className":160795,"style":126444},[31813],[34,160797],{"className":160798,"style":126472},[126470,126471],[34,160800,31829],{"className":160801},[31828],[34,160803,160805],{"className":160804},[31801],[34,160806,160808],{"className":160807,"style":126482},[31805],[34,160809],{},[34,160811,160813],{"className":160812},[126385],[34,160814,160816,160839],{"className":160815},[31642],[34,160817,160819],{"className":160818},[31646],[31648,160820,160821],{"xmlns":31650,"display":126394},[31652,160822,160823,160836],{},[31655,160824,160825],{},[126400,160826,160827],{"notation":126402},[71090,160828,160829],{"scriptlevel":459,"displaystyle":702},[71090,160830,160831],{"scriptlevel":459,"displaystyle":702},[71090,160832,160833],{"scriptlevel":459,"displaystyle":758},[39361,160834,160835],{},"如何解决最短路径和关键路径问题",[31665,160837,160838],{"encoding":31667},"\\boxed{\\text{如何解决最短路径和关键路径问题}}",[34,160840,160842],{"className":160841,"ariaHidden":758},[31672],[34,160843,160845,160848],{"className":160844},[31676],[34,160846],{"className":160847,"style":126424},[31680],[34,160849,160851],{"className":160850},[31685],[34,160852,160854,160891],{"className":160853},[31796,31797],[34,160855,160857,160888],{"className":160856},[31801],[34,160858,160860,160880],{"className":160859,"style":126437},[31805],[34,160861,160862,160865],{"style":126440},[34,160863],{"className":160864,"style":126444},[31813],[34,160866,160868],{"className":160867},[126448],[34,160869,160871],{"className":160870},[31685],[34,160872,160874],{"className":160873},[31685],[34,160875,160877],{"className":160876},[31685,2638],[34,160878,160835],{"className":160879},[31685,105462],[34,160881,160882,160885],{"style":126463},[34,160883],{"className":160884,"style":126444},[31813],[34,160886],{"className":160887,"style":126472},[126470,126471],[34,160889,31829],{"className":160890},[31828],[34,160892,160894],{"className":160893},[31801],[34,160895,160897],{"className":160896,"style":126482},[31805],[34,160898],{},[15,160900,160901],{},"最该背的是：",[34,160903,160905],{"className":160904},[126385],[34,160906,160908,160955],{"className":160907},[31642],[34,160909,160911],{"className":160910},[31646],[31648,160912,160913],{"xmlns":31650,"display":126394},[31652,160914,160915,160953],{},[31655,160916,160917],{},[126400,160918,160919],{"notation":126402},[71090,160920,160921],{"scriptlevel":459,"displaystyle":702},[71090,160922,160923],{"scriptlevel":459,"displaystyle":702},[71090,160924,160925],{"scriptlevel":459,"displaystyle":758},[31655,160926,160927,160939,160941,160943,160945,160947,160949,160951],{},[71115,160928,160929,160931],{},[31662,160930,131138],{},[31655,160932,160933,160935,160937],{},[31658,160934,56782],{},[31662,160936,112296],{},[31658,160938,126532],{},[31658,160940,46900],{},[31662,160942,456],{"stretchy":702},[31658,160944,56782],{},[31662,160946,2557],{"stretchy":702},[31662,160948,202],{},[31749,160950,768],{},[31658,160952,31759],{},[31665,160954,131163],{"encoding":31667},[34,160956,160958],{"className":160957,"ariaHidden":758},[31672],[34,160959,160961,160964],{"className":160960},[31676],[34,160962],{"className":160963,"style":131173},[31680],[34,160965,160967],{"className":160966},[31685],[34,160968,160970,161084],{"className":160969},[31796,31797],[34,160971,160973,161081],{"className":160972},[31801],[34,160974,160976,161073],{"className":160975,"style":131186},[31805],[34,160977,160978,160981],{"style":131189},[34,160979],{"className":160980,"style":131193},[31813],[34,160982,160984],{"className":160983},[126448],[34,160985,160987],{"className":160986},[31685],[34,160988,160990,161043,161046,161049,161052,161055,161058,161061,161064,161067,161070],{"className":160989},[31685],[34,160991,160993],{"className":160992},[32767,71420],[34,160994,160996,161035],{"className":160995},[31796,31797],[34,160997,160999,161032],{"className":160998},[31801],[34,161000,161002,161022],{"className":161001,"style":131215},[31805],[34,161003,161004,161007],{"style":131218},[34,161005],{"className":161006,"style":131222},[31813],[34,161008,161010],{"className":161009},[31818,31819,31820,31821],[34,161011,161013,161016,161019],{"className":161012},[31685,31821],[34,161014,56782],{"className":161015,"style":32062},[31685,31686,31821],[34,161017,112296],{"className":161018},[31850,31821],[34,161020,126532],{"className":161021,"style":32604},[31685,31686,31821],[34,161023,161024,161027],{"style":131240},[34,161025],{"className":161026,"style":131222},[31813],[34,161028,161029],{},[34,161030,131138],{"className":161031},[32767,131249,131250],[34,161033,31829],{"className":161034},[31828],[34,161036,161038],{"className":161037},[31801],[34,161039,161041],{"className":161040,"style":131260},[31805],[34,161042],{},[34,161044],{"className":161045,"style":32123},[31845],[34,161047,46900],{"className":161048},[31685,31686],[34,161050,456],{"className":161051},[31782],[34,161053,56782],{"className":161054,"style":32062},[31685,31686],[34,161056,2557],{"className":161057},[31690],[34,161059],{"className":161060,"style":31846},[31845],[34,161062,202],{"className":161063},[31850],[34,161065],{"className":161066,"style":31846},[31845],[34,161068,768],{"className":161069},[31685],[34,161071,31759],{"className":161072},[31685,31686],[34,161074,161075,161078],{"style":131295},[34,161076],{"className":161077,"style":131193},[31813],[34,161079],{"className":161080,"style":131302},[126470,126471],[34,161082,31829],{"className":161083},[31828],[34,161085,161087],{"className":161086},[31801],[34,161088,161090],{"className":161089,"style":131312},[31805],[34,161091],{},[34,161093,161095],{"className":161094},[126385],[34,161096,161098,161121],{"className":161097},[31642],[34,161099,161101],{"className":161100},[31646],[31648,161102,161103],{"xmlns":31650,"display":126394},[31652,161104,161105,161118],{},[31655,161106,161107],{},[126400,161108,161109],{"notation":126402},[71090,161110,161111],{"scriptlevel":459,"displaystyle":702},[71090,161112,161113],{"scriptlevel":459,"displaystyle":702},[71090,161114,161115],{"scriptlevel":459,"displaystyle":758},[39361,161116,161117],{},"奇度顶点个数一定为偶数",[31665,161119,161120],{"encoding":31667},"\\boxed{\\text{奇度顶点个数一定为偶数}}",[34,161122,161124],{"className":161123,"ariaHidden":758},[31672],[34,161125,161127,161130],{"className":161126},[31676],[34,161128],{"className":161129,"style":126424},[31680],[34,161131,161133],{"className":161132},[31685],[34,161134,161136,161173],{"className":161135},[31796,31797],[34,161137,161139,161170],{"className":161138},[31801],[34,161140,161142,161162],{"className":161141,"style":126437},[31805],[34,161143,161144,161147],{"style":126440},[34,161145],{"className":161146,"style":126444},[31813],[34,161148,161150],{"className":161149},[126448],[34,161151,161153],{"className":161152},[31685],[34,161154,161156],{"className":161155},[31685],[34,161157,161159],{"className":161158},[31685,2638],[34,161160,161117],{"className":161161},[31685,105462],[34,161163,161164,161167],{"style":126463},[34,161165],{"className":161166,"style":126444},[31813],[34,161168],{"className":161169,"style":126472},[126470,126471],[34,161171,31829],{"className":161172},[31828],[34,161174,161176],{"className":161175},[31801],[34,161177,161179],{"className":161178,"style":126482},[31805],[34,161180],{},[34,161182,161184],{"className":161183},[126385],[34,161185,161187,161220],{"className":161186},[31642],[34,161188,161190],{"className":161189},[31646],[31648,161191,161192],{"xmlns":31650,"display":126394},[31652,161193,161194,161217],{},[31655,161195,161196],{},[126400,161197,161198],{"notation":126402},[71090,161199,161200],{"scriptlevel":459,"displaystyle":702},[71090,161201,161202],{"scriptlevel":459,"displaystyle":702},[71090,161203,161204],{"scriptlevel":459,"displaystyle":758},[31655,161205,161206,161209,161211,161213,161215],{},[39361,161207,161208],{},"基本路径长度",[31662,161210,45972],{},[31658,161212,31660],{},[31662,161214,32226],{},[31749,161216,640],{},[31665,161218,161219],{"encoding":31667},"\\boxed{\\text{基本路径长度}\\leq n-1}",[34,161221,161223],{"className":161222,"ariaHidden":758},[31672],[34,161224,161226,161230],{"className":161225},[31676],[34,161227],{"className":161228,"style":161229},[31680],"height:1.4993em;vertical-align:-0.476em;",[34,161231,161233],{"className":161232},[31685],[34,161234,161236,161300],{"className":161235},[31796,31797],[34,161237,161239,161297],{"className":161238},[31801],[34,161240,161242,161288],{"className":161241,"style":126437},[31805],[34,161243,161245,161249],{"style":161244},"top:-3.4993em;",[34,161246],{"className":161247,"style":161248},[31813],"height:3.4993em;",[34,161250,161252],{"className":161251},[126448],[34,161253,161255],{"className":161254},[31685],[34,161256,161258,161264,161267,161270,161273,161276,161279,161282,161285],{"className":161257},[31685],[34,161259,161261],{"className":161260},[31685,2638],[34,161262,161208],{"className":161263},[31685,105462],[34,161265],{"className":161266,"style":31846},[31845],[34,161268,45972],{"className":161269},[31850],[34,161271],{"className":161272,"style":31846},[31845],[34,161274,31660],{"className":161275},[31685,31686],[34,161277],{"className":161278,"style":32604},[31845],[34,161280,32226],{"className":161281},[32024],[34,161283],{"className":161284,"style":32604},[31845],[34,161286,640],{"className":161287},[31685],[34,161289,161290,161293],{"style":126463},[34,161291],{"className":161292,"style":161248},[31813],[34,161294],{"className":161295,"style":161296},[126470,126471],"height:1.4993em;border-style:solid;border-width:0.04em;",[34,161298,31829],{"className":161299},[31828],[34,161301,161303],{"className":161302},[31801],[34,161304,161307],{"className":161305,"style":161306},[31805],"height:0.476em;",[34,161308],{},[34,161310,161312],{"className":161311},[126385],[34,161313,161315,161380],{"className":161314},[31642],[34,161316,161318],{"className":161317},[31646],[31648,161319,161320],{"xmlns":31650,"display":126394},[31652,161321,161322,161377],{},[31655,161323,161324],{},[126400,161325,161326],{"notation":126402},[71090,161327,161328],{"scriptlevel":459,"displaystyle":702},[71090,161329,161330],{"scriptlevel":459,"displaystyle":702},[71090,161331,161332],{"scriptlevel":459,"displaystyle":758},[31655,161333,161334,161336,161342,161352,161355,161361,161363,161369,161372,161374],{},[31662,161335,456],{"stretchy":702},[37624,161337,161338,161340],{},[31658,161339,54151],{},[31658,161341,31759],{},[31743,161343,161344,161346],{},[31662,161345,2557],{"stretchy":702},[31655,161347,161348,161350],{},[31658,161349,31955],{},[31658,161351,31968],{},[39361,161353,161354],{}," 表示从 ",[31743,161356,161357,161359],{},[31658,161358,56782],{},[31658,161360,31955],{},[39361,161362,145278],{},[31743,161364,161365,161367],{},[31658,161366,56782],{},[31658,161368,31968],{},[39361,161370,161371],{}," 长度为 ",[31658,161373,31759],{},[39361,161375,161376],{}," 的路径数",[31665,161378,161379],{"encoding":31667},"\\boxed{(A^m)_{ij}\\text{ 表示从 }v_i\\text{ 到 }v_j\\text{ 长度为 }m\\text{ 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求任意两点最短路}}",[34,161802,161804],{"className":161803,"ariaHidden":758},[31672],[34,161805,161807,161810],{"className":161806},[31676],[34,161808],{"className":161809,"style":153586},[31680],[34,161811,161813],{"className":161812},[31685],[34,161814,161816,161864],{"className":161815},[31796,31797],[34,161817,161819,161861],{"className":161818},[31801],[34,161820,161822,161853],{"className":161821,"style":135867},[31805],[34,161823,161824,161827],{"style":153601},[34,161825],{"className":161826,"style":153605},[31813],[34,161828,161830],{"className":161829},[126448],[34,161831,161833],{"className":161832},[31685],[34,161834,161836],{"className":161835},[31685],[34,161837,161839,161842,161846,161849],{"className":161838},[31685,2638],[34,161840,153621],{"className":161841},[31685],[34,161843,161845],{"className":161844},[31685,105462],"求单源最短路，",[34,161847,153629],{"className":161848},[31685],[34,161850,161852],{"className":161851},[31685,105462],"求任意两点最短路",[34,161854,161855,161858],{"style":135889},[34,161856],{"className":161857,"style":153605},[31813],[34,161859],{"className":161860,"style":153642},[126470,126471],[34,161862,31829],{"className":161863},[31828],[34,161865,161867],{"className":161866},[31801],[34,161868,161870],{"className":161869,"style":142798},[31805],[34,161871],{},[15,161873,161874,161875],{},"这章概念很多，但逻辑很清楚：",[2627,161876,161877],{},"先认识图，再研究走法，再用矩阵计算，最后拿去解路径问题。",{"title":29,"searchDepth":43,"depth":43,"links":161879},[161880,161881,161882,161883,161884,161885,161886,161887,161888,161889,161890,161891,161892,161893,161894,161895,161896,161897,161898,161903,161904,161905,161906,161907,161908,161909,161910,161911,161912,161913,161914,161915,161916,161917,161918,161919,161920,161921,161922,161923,161924,161925,161926,161927,161928,161929,161930,161931,161932,161933,161934,161935,161936,161937,161938,161939,161940,161941,161942,161943,161944,161945,161946,161947,161948,161953,161954,161955,161956,161957,161958,161959,161960,161966,161967,161968,161969,161970,161971,161972,161973,161974,161975,161976,161977,161978,161979,161980,161981,161982,161983,161984,161985,161986,161987,161988,161989,161990,161991,161992,161993],{"id":126824,"depth":43,"text":126825},{"id":127037,"depth":43,"text":127038},{"id":127041,"depth":43,"text":127042},{"id":127369,"depth":43,"text":127370},{"id":127643,"depth":43,"text":127644},{"id":127871,"depth":43,"text":127872},{"id":127888,"depth":43,"text":127889},{"id":128132,"depth":43,"text":128133},{"id":128282,"depth":43,"text":128283},{"id":128674,"depth":43,"text":128675},{"id":128769,"depth":43,"text":128770},{"id":129175,"depth":43,"text":129176},{"id":129194,"depth":43,"text":129195},{"id":129266,"depth":43,"text":129267},{"id":129278,"depth":43,"text":129279},{"id":129293,"depth":43,"text":129294},{"id":129418,"depth":43,"text":129419},{"id":130102,"depth":43,"text":130103},{"id":130328,"depth":43,"text":130329,"children":161899},[161900,161901,161902],{"id":130335,"depth":49,"text":130335},{"id":130449,"depth":49,"text":130449},{"id":130561,"depth":49,"text":130561},{"id":130747,"depth":43,"text":130748},{"id":130918,"depth":43,"text":130919},{"id":131332,"depth":43,"text":131333},{"id":132095,"depth":43,"text":132096},{"id":132531,"depth":43,"text":132532},{"id":132820,"depth":43,"text":132821},{"id":133073,"depth":43,"text":133074},{"id":133253,"depth":43,"text":133254},{"id":133554,"depth":43,"text":133555},{"id":133797,"depth":43,"text":133798},{"id":133892,"depth":43,"text":133893},{"id":133906,"depth":43,"text":133907},{"id":134474,"depth":43,"text":134475},{"id":134899,"depth":43,"text":134900},{"id":134928,"depth":43,"text":134929},{"id":135912,"depth":43,"text":135913},{"id":136390,"depth":43,"text":136391},{"id":136405,"depth":43,"text":136406},{"id":136670,"depth":43,"text":136671},{"id":136682,"depth":43,"text":136683},{"id":136694,"depth":43,"text":136695},{"id":136855,"depth":43,"text":136856},{"id":137088,"depth":43,"text":137089},{"id":137195,"depth":43,"text":137196},{"id":137388,"depth":43,"text":137389},{"id":137400,"depth":43,"text":137401},{"id":138842,"depth":43,"text":138843},{"id":139165,"depth":43,"text":139166},{"id":139236,"depth":43,"text":139237},{"id":139402,"depth":43,"text":139403},{"id":139565,"depth":43,"text":139566},{"id":139723,"depth":43,"text":139724},{"id":139893,"depth":43,"text":139894},{"id":140050,"depth":43,"text":140051},{"id":140062,"depth":43,"text":140063},{"id":140220,"depth":43,"text":140221},{"id":140232,"depth":43,"text":140233},{"id":140244,"depth":43,"text":140245},{"id":140256,"depth":43,"text":140257},{"id":140913,"depth":43,"text":140914},{"id":141759,"depth":43,"text":141760},{"id":141779,"depth":43,"text":141780},{"id":143187,"depth":43,"text":143188},{"id":143744,"depth":43,"text":143745},{"id":144022,"depth":43,"text":144023},{"id":144157,"depth":43,"text":144158,"children":161949},[161950,161951,161952],{"id":144163,"depth":49,"text":130335},{"id":144476,"depth":49,"text":130449},{"id":144785,"depth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Science",{"type":8,"value":162003,"toc":162018},[162004,162014],[16016,162005,31533,162007,31533,162010],{"className":162006},[31532],[11,162008,162001],{"id":162009},"computer-science",[15,162011,162013],{"className":162012},[31540],"計算機科学のノート・過去問・宿題をまとめています。",[31543,162015],{"path":162016,"variant":162017},"\u002Fcs\u002F","courses",{"title":29,"searchDepth":43,"depth":43,"links":162019},[],"计算机科学课程笔记汇总，包含操作系统、算法设计与分析、数据库系统、软件工程等科目的课程内容、作业与历年考题。",{"layout":162022,"sidebar":162023},"doc",false,"\u002Fsource\u002Fcs",{"title":162001,"description":162020},"source\u002Fcs\u002Findex","Y1hGUwBJxPSTxFD9-gGIA1OQOBm4wf-DP7ru9Aenzsc",{"id":162029,"title":162030,"body":162031,"description":29,"extension":85,"meta":175216,"navigation":52,"path":175217,"seo":175218,"stem":175219,"__hash__":175220},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap11-hw.md","Chap. 11 HW",{"type":8,"value":162032,"toc":175167},[162033,162037,162041,162044,162047,162050,162053,162056,162059,162062,162064,162067,162070,162142,162145,162147,162151,162154,162157,162395,162398,162480,162562,162644,162725,162807,162889,162971,163053,163135,163138,163368,163370,163373,163403,163405,163409,163412,163415,163418,163421,163423,163426,163479,163482,163485,163488,163541,163544,163597,163600,163653,163656,163709,163712,163715,163718,163771,163774,163827,163830,163883,163886,163939,163942,164171,164173,164252,164331,164411,164490,164569,164649,164728,164808,164888,164890,165120,165122,165125,165155,165157,165161,165164,165167,165170,165173,165176,165295,165298,165301,165304,165533,165535,165615,165693,165771,165849,165928,166006,166084,166162,166240,166242,166472,166474,166477,166507,166509,166512,166515,166518,166548,166551,166802,166805,167057,167060,167090,167092,167096,167099,167129,167132,167162,167165,167195,167198,167228,167230,167234,167237,167240,167242,167286,167289,167292,167324,167327,167359,167361,167364,167367,167398,167401,167441,167444,167540,167542,167584,167586,167589,167592,167638,167641,167644,167704,167707,167776,167779,167848,167851,167933,167935,167977,167979,167983,167986,168122,168125,168275,168277,168280,168313,168316,168318,168321,168324,168327,168408,168411,168517,168520,168695,168698,168730,168732,168736,168739,168771,168773,168777,168780,168783,168786,168789,168792,168795,168798,168801,168803,168806,168808,168811,168814,168819,168822,168825,168828,168898,168968,168971,169038,169041,169108,169111,169224,169226,169231,169234,169237,169240,169354,169357,169471,169474,169586,169589,169724,169726,169730,169733,169846,169849,169901,169903,169907,169910,169978,169981,170031,170033,170037,170040,170043,170046,170049,170051,170054,170057,170060,170141,170144,170185,170188,170229,170232,170338,170340,170343,170383,170386,170427,170430,170433,170435,170439,170442,170445,170448,170451,170555,170558,170561,170601,170604,170644,170647,170752,170755,170860,170863,170904,170907,170910,171014,171017,171112,171115,171211,171214,171320,171322,171326,171329,171370,171372,171376,171378,171381,171384,171424,171427,171467,171470,171593,171596,171599,171602,171604,171607,171610,171613,171616,171619,171622,171625,171628,171631,171634,171637,171640,171643,171646,171649,171652,171655,171658,171706,171709,171712,171715,171717,171720,171723,171766,171806,171809,171904,171907,171910,171951,171954,171957,171997,172000,172040,172043,172085,172088,172194,172197,172303,172306,172412,172415,172418,172458,172461,172501,172504,172544,172547,172743,172746,172850,172853,172958,172960,172964,172967,172969,172972,173013,173015,173019,173022,173062,173065,173067,173070,173110,173112,173116,173119,173122,173124,173127,173130,173133,173163,173165,173195,173198,173229,173232,173263,173266,173297,173300,173302,173305,173308,173311,173381,173383,173459,173462,173538,173541,173543,173546,173549,173552,173612,173615,173645,173647,173649,173652,173655,173658,173706,173709,173767,173770,173834,173836,173866,173869,173872,173875,173877,173954,173957,174032,174035,174037,174040,174043,174046,174092,174095,174153,174156,174220,174222,174252,174255,174258,174261,174359,174361,174364,174440,174442,174445,174513,174515,174518,174521,174524,174554,174556,174558,174562,174564,174594,174596,174626,174628,174658,174661,174664,174666,174670,174672,174702,174704,174734,174736,174766,174769,174799,174801,174805,174808,174840,174842,174846,174849,174916,174918,174968,174970,174974,174977,175017,175020,175022,175025,175065,175067,175071,175073,175103,175105,175135,175137],[11,162034,162036],{"id":162035},"os-chap11-hw","OS Chap.11 HW",[19,162038,162040],{"id":162039},"_1-磁盘调度算法-fifosstfscan-计算磁头移动磁道数","1. 磁盘调度算法 FIFO、SSTF、SCAN 计算磁头移动磁道数",[253,162042,162043],{"id":162043},"题目内容",[15,162045,162046],{},"Assume a disk with 200 tracks and that the disk request queue has random requests in it. The requested tracks, in the order received by the disk scheduler, are 27, 129, 110, 186, 147, 41, 10, 64, 120. Assume that the disk head is initially positioned over track 100 and is moving in the direction of decreasing track number. Using disk scheduling algorithms FIFO, SSFT, SCAN to calculate the number of tracks traversed.",[15,162048,162049],{},"中文意思：",[15,162051,162052],{},"假设一块磁盘有 200 个磁道，磁道号通常可理解为 0 到 199。磁盘请求队列中的请求磁道号按到达顺序为：",[15,162054,162055],{},"27, 129, 110, 186, 147, 41, 10, 64, 120",[15,162057,162058],{},"磁头初始位置在磁道 100，并且正在向磁道号减小的方向移动。分别使用 FIFO、SSTF、SCAN 算法，计算磁头总共移动了多少个磁道。",[15,162060,162061],{},"注意：题目里的 SSFT 应该是 SSTF，即 Shortest Seek Time First，最短寻道时间优先。",[32188,162063],{},[253,162065,162066],{"id":162066},"解题核心",[15,162068,162069],{},"磁头移动距离的计算公式是：",[15,162071,162072],{},[34,162073,162075,162100],{"className":162074},[31642],[34,162076,162078],{"className":162077},[31646],[31648,162079,162080],{"xmlns":31650},[31652,162081,162082,162097],{},[31655,162083,162084,162087,162090,162092,162095],{},[31658,162085,162086],{"mathvariant":38079},"∣",[39361,162088,162089],{},"当前磁道",[31662,162091,32226],{},[39361,162093,162094],{},"下一个访问磁道",[31658,162096,162086],{"mathvariant":38079},[31665,162098,162099],{"encoding":31667},"|\\text{当前磁道} - \\text{下一个访问磁道}|",[34,162101,162103,162127],{"className":162102,"ariaHidden":758},[31672],[34,162104,162106,162109,162112,162118,162121,162124],{"className":162105},[31676],[34,162107],{"className":162108,"style":31774},[31680],[34,162110,162086],{"className":162111},[31685],[34,162113,162115],{"className":162114},[31685,2638],[34,162116,162089],{"className":162117},[31685,105462],[34,162119],{"className":162120,"style":32604},[31845],[34,162122,32226],{"className":162123},[32024],[34,162125],{"className":162126,"style":32604},[31845],[34,162128,162130,162133,162139],{"className":162129},[31676],[34,162131],{"className":162132,"style":31774},[31680],[34,162134,162136],{"className":162135},[31685,2638],[34,162137,162094],{"className":162138},[31685,105462],[34,162140,162086],{"className":162141},[31685],[15,162143,162144],{},"总移动磁道数就是每一次移动距离之和。",[32188,162146],{},[253,162148,162150],{"id":162149},"一fifo-算法","一、FIFO 算法",[15,162152,162153],{},"FIFO 即 First-In First-Out，先来先服务。",[15,162155,162156],{},"也就是说，请求按原来的到达顺序处理：",[15,162158,162159],{},[34,162160,162162,162221],{"className":162161},[31642],[34,162163,162165],{"className":162164},[31646],[31648,162166,162167],{"xmlns":31650},[31652,162168,162169,162218],{},[31655,162170,162171,162174,162176,162179,162181,162184,162186,162189,162191,162194,162196,162199,162201,162204,162206,162208,162210,162213,162215],{},[31749,162172,162173],{},"100",[31662,162175,33239],{},[31749,162177,162178],{},"27",[31662,162180,33239],{},[31749,162182,162183],{},"129",[31662,162185,33239],{},[31749,162187,162188],{},"110",[31662,162190,33239],{},[31749,162192,162193],{},"186",[31662,162195,33239],{},[31749,162197,162198],{},"147",[31662,162200,33239],{},[31749,162202,162203],{},"41",[31662,162205,33239],{},[31749,162207,3244],{},[31662,162209,33239],{},[31749,162211,162212],{},"64",[31662,162214,33239],{},[31749,162216,162217],{},"120",[31665,162219,162220],{"encoding":31667},"100 \\rightarrow 27 \\rightarrow 129 \\rightarrow 110 \\rightarrow 186 \\rightarrow 147 \\rightarrow 41 \\rightarrow 10 \\rightarrow 64 \\rightarrow 120",[34,162222,162224,162242,162260,162278,162296,162314,162332,162350,162368,162386],{"className":162223,"ariaHidden":758},[31672],[34,162225,162227,162230,162233,162236,162239],{"className":162226},[31676],[34,162228],{"className":162229,"style":32617},[31680],[34,162231,162173],{"className":162232},[31685],[34,162234],{"className":162235,"style":31846},[31845],[34,162237,33239],{"className":162238},[31850],[34,162240],{"className":162241,"style":31846},[31845],[34,162243,162245,162248,162251,162254,162257],{"className":162244},[31676],[34,162246],{"className":162247,"style":32617},[31680],[34,162249,162178],{"className":162250},[31685],[34,162252],{"className":162253,"style":31846},[31845],[34,162255,33239],{"className":162256},[31850],[34,162258],{"className":162259,"style":31846},[31845],[34,162261,162263,162266,162269,162272,162275],{"className":162262},[31676],[34,162264],{"className":162265,"style":32617},[31680],[34,162267,162183],{"className":162268},[31685],[34,162270],{"className":162271,"style":31846},[31845],[34,162273,33239],{"className":162274},[31850],[34,162276],{"className":162277,"style":31846},[31845],[34,162279,162281,162284,162287,162290,162293],{"className":162280},[31676],[34,162282],{"className":162283,"style":32617},[31680],[34,162285,162188],{"className":162286},[31685],[34,162288],{"className":162289,"style":31846},[31845],[34,162291,33239],{"className":162292},[31850],[34,162294],{"className":162295,"style":31846},[31845],[34,162297,162299,162302,162305,162308,162311],{"className":162298},[31676],[34,162300],{"className":162301,"style":32617},[31680],[34,162303,162193],{"className":162304},[31685],[34,162306],{"className":162307,"style":31846},[31845],[34,162309,33239],{"className":162310},[31850],[34,162312],{"className":162313,"style":31846},[31845],[34,162315,162317,162320,162323,162326,162329],{"className":162316},[31676],[34,162318],{"className":162319,"style":32617},[31680],[34,162321,162198],{"className":162322},[31685],[34,162324],{"className":162325,"style":31846},[31845],[34,162327,33239],{"className":162328},[31850],[34,162330],{"className":162331,"style":31846},[31845],[34,162333,162335,162338,162341,162344,162347],{"className":162334},[31676],[34,162336],{"className":162337,"style":32617},[31680],[34,162339,162203],{"className":162340},[31685],[34,162342],{"className":162343,"style":31846},[31845],[34,162345,33239],{"className":162346},[31850],[34,162348],{"className":162349,"style":31846},[31845],[34,162351,162353,162356,162359,162362,162365],{"className":162352},[31676],[34,162354],{"className":162355,"style":32617},[31680],[34,162357,3244],{"className":162358},[31685],[34,162360],{"className":162361,"style":31846},[31845],[34,162363,33239],{"className":162364},[31850],[34,162366],{"className":162367,"style":31846},[31845],[34,162369,162371,162374,162377,162380,162383],{"className":162370},[31676],[34,162372],{"className":162373,"style":32617},[31680],[34,162375,162212],{"className":162376},[31685],[34,162378],{"className":162379,"style":31846},[31845],[34,162381,33239],{"className":162382},[31850],[34,162384],{"className":162385,"style":31846},[31845],[34,162387,162389,162392],{"className":162388},[31676],[34,162390],{"className":162391,"style":32617},[31680],[34,162393,162217],{"className":162394},[31685],[15,162396,162397],{},"逐步计算：",[15,162399,162400],{},[34,162401,162403,162430],{"className":162402},[31642],[34,162404,162406],{"className":162405},[31646],[31648,162407,162408],{"xmlns":31650},[31652,162409,162410,162427],{},[31655,162411,162412,162414,162416,162418,162420,162422,162424],{},[31658,162413,162086],{"mathvariant":38079},[31749,162415,162173],{},[31662,162417,32226],{},[31749,162419,162178],{},[31658,162421,162086],{"mathvariant":38079},[31662,162423,202],{},[31749,162425,162426],{},"73",[31665,162428,162429],{"encoding":31667},"|100 - 27| = 73",[34,162431,162433,162452,162471],{"className":162432,"ariaHidden":758},[31672],[34,162434,162436,162439,162443,162446,162449],{"className":162435},[31676],[34,162437],{"className":162438,"style":31774},[31680],[34,162440,162442],{"className":162441},[31685],"∣100",[34,162444],{"className":162445,"style":32604},[31845],[34,162447,32226],{"className":162448},[32024],[34,162450],{"className":162451,"style":32604},[31845],[34,162453,162455,162458,162462,162465,162468],{"className":162454},[31676],[34,162456],{"className":162457,"style":31774},[31680],[34,162459,162461],{"className":162460},[31685],"27∣",[34,162463],{"className":162464,"style":31846},[31845],[34,162466,202],{"className":162467},[31850],[34,162469],{"className":162470,"style":31846},[31845],[34,162472,162474,162477],{"className":162473},[31676],[34,162475],{"className":162476,"style":32617},[31680],[34,162478,162426],{"className":162479},[31685],[15,162481,162482],{},[34,162483,162485,162512],{"className":162484},[31642],[34,162486,162488],{"className":162487},[31646],[31648,162489,162490],{"xmlns":31650},[31652,162491,162492,162509],{},[31655,162493,162494,162496,162498,162500,162502,162504,162506],{},[31658,162495,162086],{"mathvariant":38079},[31749,162497,162178],{},[31662,162499,32226],{},[31749,162501,162183],{},[31658,162503,162086],{"mathvariant":38079},[31662,162505,202],{},[31749,162507,162508],{},"102",[31665,162510,162511],{"encoding":31667},"|27 - 129| = 102",[34,162513,162515,162534,162553],{"className":162514,"ariaHidden":758},[31672],[34,162516,162518,162521,162525,162528,162531],{"className":162517},[31676],[34,162519],{"className":162520,"style":31774},[31680],[34,162522,162524],{"className":162523},[31685],"∣27",[34,162526],{"className":162527,"style":32604},[31845],[34,162529,32226],{"className":162530},[32024],[34,162532],{"className":162533,"style":32604},[31845],[34,162535,162537,162540,162544,162547,162550],{"className":162536},[31676],[34,162538],{"className":162539,"style":31774},[31680],[34,162541,162543],{"className":162542},[31685],"129∣",[34,162545],{"className":162546,"style":31846},[31845],[34,162548,202],{"className":162549},[31850],[34,162551],{"className":162552,"style":31846},[31845],[34,162554,162556,162559],{"className":162555},[31676],[34,162557],{"className":162558,"style":32617},[31680],[34,162560,162508],{"className":162561},[31685],[15,162563,162564],{},[34,162565,162567,162594],{"className":162566},[31642],[34,162568,162570],{"className":162569},[31646],[31648,162571,162572],{"xmlns":31650},[31652,162573,162574,162591],{},[31655,162575,162576,162578,162580,162582,162584,162586,162588],{},[31658,162577,162086],{"mathvariant":38079},[31749,162579,162183],{},[31662,162581,32226],{},[31749,162583,162188],{},[31658,162585,162086],{"mathvariant":38079},[31662,162587,202],{},[31749,162589,162590],{},"19",[31665,162592,162593],{"encoding":31667},"|129 - 110| = 19",[34,162595,162597,162616,162635],{"className":162596,"ariaHidden":758},[31672],[34,162598,162600,162603,162607,162610,162613],{"className":162599},[31676],[34,162601],{"className":162602,"style":31774},[31680],[34,162604,162606],{"className":162605},[31685],"∣129",[34,162608],{"className":162609,"style":32604},[31845],[34,162611,32226],{"className":162612},[32024],[34,162614],{"className":162615,"style":32604},[31845],[34,162617,162619,162622,162626,162629,162632],{"className":162618},[31676],[34,162620],{"className":162621,"style":31774},[31680],[34,162623,162625],{"className":162624},[31685],"110∣",[34,162627],{"className":162628,"style":31846},[31845],[34,162630,202],{"className":162631},[31850],[34,162633],{"className":162634,"style":31846},[31845],[34,162636,162638,162641],{"className":162637},[31676],[34,162639],{"className":162640,"style":32617},[31680],[34,162642,162590],{"className":162643},[31685],[15,162645,162646],{},[34,162647,162649,162675],{"className":162648},[31642],[34,162650,162652],{"className":162651},[31646],[31648,162653,162654],{"xmlns":31650},[31652,162655,162656,162672],{},[31655,162657,162658,162660,162662,162664,162666,162668,162670],{},[31658,162659,162086],{"mathvariant":38079},[31749,162661,162188],{},[31662,162663,32226],{},[31749,162665,162193],{},[31658,162667,162086],{"mathvariant":38079},[31662,162669,202],{},[31749,162671,95437],{},[31665,162673,162674],{"encoding":31667},"|110 - 186| = 76",[34,162676,162678,162697,162716],{"className":162677,"ariaHidden":758},[31672],[34,162679,162681,162684,162688,162691,162694],{"className":162680},[31676],[34,162682],{"className":162683,"style":31774},[31680],[34,162685,162687],{"className":162686},[31685],"∣110",[34,162689],{"className":162690,"style":32604},[31845],[34,162692,32226],{"className":162693},[32024],[34,162695],{"className":162696,"style":32604},[31845],[34,162698,162700,162703,162707,162710,162713],{"className":162699},[31676],[34,162701],{"className":162702,"style":31774},[31680],[34,162704,162706],{"className":162705},[31685],"186∣",[34,162708],{"className":162709,"style":31846},[31845],[34,162711,202],{"className":162712},[31850],[34,162714],{"className":162715,"style":31846},[31845],[34,162717,162719,162722],{"className":162718},[31676],[34,162720],{"className":162721,"style":32617},[31680],[34,162723,95437],{"className":162724},[31685],[15,162726,162727],{},[34,162728,162730,162757],{"className":162729},[31642],[34,162731,162733],{"className":162732},[31646],[31648,162734,162735],{"xmlns":31650},[31652,162736,162737,162754],{},[31655,162738,162739,162741,162743,162745,162747,162749,162751],{},[31658,162740,162086],{"mathvariant":38079},[31749,162742,162193],{},[31662,162744,32226],{},[31749,162746,162198],{},[31658,162748,162086],{"mathvariant":38079},[31662,162750,202],{},[31749,162752,162753],{},"39",[31665,162755,162756],{"encoding":31667},"|186 - 147| = 39",[34,162758,162760,162779,162798],{"className":162759,"ariaHidden":758},[31672],[34,162761,162763,162766,162770,162773,162776],{"className":162762},[31676],[34,162764],{"className":162765,"style":31774},[31680],[34,162767,162769],{"className":162768},[31685],"∣186",[34,162771],{"className":162772,"style":32604},[31845],[34,162774,32226],{"className":162775},[32024],[34,162777],{"className":162778,"style":32604},[31845],[34,162780,162782,162785,162789,162792,162795],{"className":162781},[31676],[34,162783],{"className":162784,"style":31774},[31680],[34,162786,162788],{"className":162787},[31685],"147∣",[34,162790],{"className":162791,"style":31846},[31845],[34,162793,202],{"className":162794},[31850],[34,162796],{"className":162797,"style":31846},[31845],[34,162799,162801,162804],{"className":162800},[31676],[34,162802],{"className":162803,"style":32617},[31680],[34,162805,162753],{"className":162806},[31685],[15,162808,162809],{},[34,162810,162812,162839],{"className":162811},[31642],[34,162813,162815],{"className":162814},[31646],[31648,162816,162817],{"xmlns":31650},[31652,162818,162819,162836],{},[31655,162820,162821,162823,162825,162827,162829,162831,162833],{},[31658,162822,162086],{"mathvariant":38079},[31749,162824,162198],{},[31662,162826,32226],{},[31749,162828,162203],{},[31658,162830,162086],{"mathvariant":38079},[31662,162832,202],{},[31749,162834,162835],{},"106",[31665,162837,162838],{"encoding":31667},"|147 - 41| = 106",[34,162840,162842,162861,162880],{"className":162841,"ariaHidden":758},[31672],[34,162843,162845,162848,162852,162855,162858],{"className":162844},[31676],[34,162846],{"className":162847,"style":31774},[31680],[34,162849,162851],{"className":162850},[31685],"∣147",[34,162853],{"className":162854,"style":32604},[31845],[34,162856,32226],{"className":162857},[32024],[34,162859],{"className":162860,"style":32604},[31845],[34,162862,162864,162867,162871,162874,162877],{"className":162863},[31676],[34,162865],{"className":162866,"style":31774},[31680],[34,162868,162870],{"className":162869},[31685],"41∣",[34,162872],{"className":162873,"style":31846},[31845],[34,162875,202],{"className":162876},[31850],[34,162878],{"className":162879,"style":31846},[31845],[34,162881,162883,162886],{"className":162882},[31676],[34,162884],{"className":162885,"style":32617},[31680],[34,162887,162835],{"className":162888},[31685],[15,162890,162891],{},[34,162892,162894,162921],{"className":162893},[31642],[34,162895,162897],{"className":162896},[31646],[31648,162898,162899],{"xmlns":31650},[31652,162900,162901,162918],{},[31655,162902,162903,162905,162907,162909,162911,162913,162915],{},[31658,162904,162086],{"mathvariant":38079},[31749,162906,162203],{},[31662,162908,32226],{},[31749,162910,3244],{},[31658,162912,162086],{"mathvariant":38079},[31662,162914,202],{},[31749,162916,162917],{},"31",[31665,162919,162920],{"encoding":31667},"|41 - 10| = 31",[34,162922,162924,162943,162962],{"className":162923,"ariaHidden":758},[31672],[34,162925,162927,162930,162934,162937,162940],{"className":162926},[31676],[34,162928],{"className":162929,"style":31774},[31680],[34,162931,162933],{"className":162932},[31685],"∣41",[34,162935],{"className":162936,"style":32604},[31845],[34,162938,32226],{"className":162939},[32024],[34,162941],{"className":162942,"style":32604},[31845],[34,162944,162946,162949,162953,162956,162959],{"className":162945},[31676],[34,162947],{"className":162948,"style":31774},[31680],[34,162950,162952],{"className":162951},[31685],"10∣",[34,162954],{"className":162955,"style":31846},[31845],[34,162957,202],{"className":162958},[31850],[34,162960],{"className":162961,"style":31846},[31845],[34,162963,162965,162968],{"className":162964},[31676],[34,162966],{"className":162967,"style":32617},[31680],[34,162969,162917],{"className":162970},[31685],[15,162972,162973],{},[34,162974,162976,163003],{"className":162975},[31642],[34,162977,162979],{"className":162978},[31646],[31648,162980,162981],{"xmlns":31650},[31652,162982,162983,163000],{},[31655,162984,162985,162987,162989,162991,162993,162995,162997],{},[31658,162986,162086],{"mathvariant":38079},[31749,162988,3244],{},[31662,162990,32226],{},[31749,162992,162212],{},[31658,162994,162086],{"mathvariant":38079},[31662,162996,202],{},[31749,162998,162999],{},"54",[31665,163001,163002],{"encoding":31667},"|10 - 64| = 54",[34,163004,163006,163025,163044],{"className":163005,"ariaHidden":758},[31672],[34,163007,163009,163012,163016,163019,163022],{"className":163008},[31676],[34,163010],{"className":163011,"style":31774},[31680],[34,163013,163015],{"className":163014},[31685],"∣10",[34,163017],{"className":163018,"style":32604},[31845],[34,163020,32226],{"className":163021},[32024],[34,163023],{"className":163024,"style":32604},[31845],[34,163026,163028,163031,163035,163038,163041],{"className":163027},[31676],[34,163029],{"className":163030,"style":31774},[31680],[34,163032,163034],{"className":163033},[31685],"64∣",[34,163036],{"className":163037,"style":31846},[31845],[34,163039,202],{"className":163040},[31850],[34,163042],{"className":163043,"style":31846},[31845],[34,163045,163047,163050],{"className":163046},[31676],[34,163048],{"className":163049,"style":32617},[31680],[34,163051,162999],{"className":163052},[31685],[15,163054,163055],{},[34,163056,163058,163085],{"className":163057},[31642],[34,163059,163061],{"className":163060},[31646],[31648,163062,163063],{"xmlns":31650},[31652,163064,163065,163082],{},[31655,163066,163067,163069,163071,163073,163075,163077,163079],{},[31658,163068,162086],{"mathvariant":38079},[31749,163070,162212],{},[31662,163072,32226],{},[31749,163074,162217],{},[31658,163076,162086],{"mathvariant":38079},[31662,163078,202],{},[31749,163080,163081],{},"56",[31665,163083,163084],{"encoding":31667},"|64 - 120| = 56",[34,163086,163088,163107,163126],{"className":163087,"ariaHidden":758},[31672],[34,163089,163091,163094,163098,163101,163104],{"className":163090},[31676],[34,163092],{"className":163093,"style":31774},[31680],[34,163095,163097],{"className":163096},[31685],"∣64",[34,163099],{"className":163100,"style":32604},[31845],[34,163102,32226],{"className":163103},[32024],[34,163105],{"className":163106,"style":32604},[31845],[34,163108,163110,163113,163117,163120,163123],{"className":163109},[31676],[34,163111],{"className":163112,"style":31774},[31680],[34,163114,163116],{"className":163115},[31685],"120∣",[34,163118],{"className":163119,"style":31846},[31845],[34,163121,202],{"className":163122},[31850],[34,163124],{"className":163125,"style":31846},[31845],[34,163127,163129,163132],{"className":163128},[31676],[34,163130],{"className":163131,"style":32617},[31680],[34,163133,163081],{"className":163134},[31685],[15,163136,163137],{},"总移动磁道数：",[15,163139,163140],{},[34,163141,163143,163194],{"className":163142},[31642],[34,163144,163146],{"className":163145},[31646],[31648,163147,163148],{"xmlns":31650},[31652,163149,163150,163191],{},[31655,163151,163152,163154,163156,163158,163160,163162,163164,163166,163168,163170,163172,163174,163176,163178,163180,163182,163184,163186,163188],{},[31749,163153,162426],{},[31662,163155,223],{},[31749,163157,162508],{},[31662,163159,223],{},[31749,163161,162590],{},[31662,163163,223],{},[31749,163165,95437],{},[31662,163167,223],{},[31749,163169,162753],{},[31662,163171,223],{},[31749,163173,162835],{},[31662,163175,223],{},[31749,163177,162917],{},[31662,163179,223],{},[31749,163181,162999],{},[31662,163183,223],{},[31749,163185,163081],{},[31662,163187,202],{},[31749,163189,163190],{},"556",[31665,163192,163193],{"encoding":31667},"73 + 102 + 19 + 76 + 39 + 106 + 31 + 54 + 56 = 556",[34,163195,163197,163215,163233,163251,163269,163287,163305,163323,163341,163359],{"className":163196,"ariaHidden":758},[31672],[34,163198,163200,163203,163206,163209,163212],{"className":163199},[31676],[34,163201],{"className":163202,"style":35942},[31680],[34,163204,162426],{"className":163205},[31685],[34,163207],{"className":163208,"style":32604},[31845],[34,163210,223],{"className":163211},[32024],[34,163213],{"className":163214,"style":32604},[31845],[34,163216,163218,163221,163224,163227,163230],{"className":163217},[31676],[34,163219],{"className":163220,"style":35942},[31680],[34,163222,162508],{"className":163223},[31685],[34,163225],{"className":163226,"style":32604},[31845],[34,163228,223],{"className":163229},[32024],[34,163231],{"className":163232,"style":32604},[31845],[34,163234,163236,163239,163242,163245,163248],{"className":163235},[31676],[34,163237],{"className":163238,"style":35942},[31680],[34,163240,162590],{"className":163241},[31685],[34,163243],{"className":163244,"style":32604},[31845],[34,163246,223],{"className":163247},[32024],[34,163249],{"className":163250,"style":32604},[31845],[34,163252,163254,163257,163260,163263,163266],{"className":163253},[31676],[34,163255],{"className":163256,"style":35942},[31680],[34,163258,95437],{"className":163259},[31685],[34,163261],{"className":163262,"style":32604},[31845],[34,163264,223],{"className":163265},[32024],[34,163267],{"className":163268,"style":32604},[31845],[34,163270,163272,163275,163278,163281,163284],{"className":163271},[31676],[34,163273],{"className":163274,"style":35942},[31680],[34,163276,162753],{"className":163277},[31685],[34,163279],{"className":163280,"style":32604},[31845],[34,163282,223],{"className":163283},[32024],[34,163285],{"className":163286,"style":32604},[31845],[34,163288,163290,163293,163296,163299,163302],{"className":163289},[31676],[34,163291],{"className":163292,"style":35942},[31680],[34,163294,162835],{"className":163295},[31685],[34,163297],{"className":163298,"style":32604},[31845],[34,163300,223],{"className":163301},[32024],[34,163303],{"className":163304,"style":32604},[31845],[34,163306,163308,163311,163314,163317,163320],{"className":163307},[31676],[34,163309],{"className":163310,"style":35942},[31680],[34,163312,162917],{"className":163313},[31685],[34,163315],{"className":163316,"style":32604},[31845],[34,163318,223],{"className":163319},[32024],[34,163321],{"className":163322,"style":32604},[31845],[34,163324,163326,163329,163332,163335,163338],{"className":163325},[31676],[34,163327],{"className":163328,"style":35942},[31680],[34,163330,162999],{"className":163331},[31685],[34,163333],{"className":163334,"style":32604},[31845],[34,163336,223],{"className":163337},[32024],[34,163339],{"className":163340,"style":32604},[31845],[34,163342,163344,163347,163350,163353,163356],{"className":163343},[31676],[34,163345],{"className":163346,"style":32617},[31680],[34,163348,163081],{"className":163349},[31685],[34,163351],{"className":163352,"style":31846},[31845],[34,163354,202],{"className":163355},[31850],[34,163357],{"className":163358,"style":31846},[31845],[34,163360,163362,163365],{"className":163361},[31676],[34,163363],{"className":163364,"style":32617},[31680],[34,163366,163190],{"className":163367},[31685],[15,163369,159962],{},[15,163371,163372],{},"FIFO 总移动磁道数为：",[15,163374,163375],{},[34,163376,163378,163391],{"className":163377},[31642],[34,163379,163381],{"className":163380},[31646],[31648,163382,163383],{"xmlns":31650},[31652,163384,163385,163389],{},[31655,163386,163387],{},[31749,163388,163190],{},[31665,163390,163190],{"encoding":31667},[34,163392,163394],{"className":163393,"ariaHidden":758},[31672],[34,163395,163397,163400],{"className":163396},[31676],[34,163398],{"className":163399,"style":32617},[31680],[34,163401,163190],{"className":163402},[31685],[32188,163404],{},[253,163406,163408],{"id":163407},"二sstf-算法","二、SSTF 算法",[15,163410,163411],{},"SSTF 即 Shortest Seek Time First，最短寻道时间优先。",[15,163413,163414],{},"每次都选择距离当前磁头位置最近的请求。",[15,163416,163417],{},"初始位置是 100。",[15,163419,163420],{},"请求队列：",[15,163422,162055],{},[15,163424,163425],{},"从 100 出发，离 100 最近的是 110：",[15,163427,163428],{},[34,163429,163431,163449],{"className":163430},[31642],[34,163432,163434],{"className":163433},[31646],[31648,163435,163436],{"xmlns":31650},[31652,163437,163438,163446],{},[31655,163439,163440,163442,163444],{},[31749,163441,162173],{},[31662,163443,33239],{},[31749,163445,162188],{},[31665,163447,163448],{"encoding":31667},"100 \\rightarrow 110",[34,163450,163452,163470],{"className":163451,"ariaHidden":758},[31672],[34,163453,163455,163458,163461,163464,163467],{"className":163454},[31676],[34,163456],{"className":163457,"style":32617},[31680],[34,163459,162173],{"className":163460},[31685],[34,163462],{"className":163463,"style":31846},[31845],[34,163465,33239],{"className":163466},[31850],[34,163468],{"className":163469,"style":31846},[31845],[34,163471,163473,163476],{"className":163472},[31676],[34,163474],{"className":163475,"style":32617},[31680],[34,163477,162188],{"className":163478},[31685],[15,163480,163481],{},"剩余：",[15,163483,163484],{},"27, 129, 186, 147, 41, 10, 64, 120",[15,163486,163487],{},"从 110 出发，最近的是 120：",[15,163489,163490],{},[34,163491,163493,163511],{"className":163492},[31642],[34,163494,163496],{"className":163495},[31646],[31648,163497,163498],{"xmlns":31650},[31652,163499,163500,163508],{},[31655,163501,163502,163504,163506],{},[31749,163503,162188],{},[31662,163505,33239],{},[31749,163507,162217],{},[31665,163509,163510],{"encoding":31667},"110 \\rightarrow 120",[34,163512,163514,163532],{"className":163513,"ariaHidden":758},[31672],[34,163515,163517,163520,163523,163526,163529],{"className":163516},[31676],[34,163518],{"className":163519,"style":32617},[31680],[34,163521,162188],{"className":163522},[31685],[34,163524],{"className":163525,"style":31846},[31845],[34,163527,33239],{"className":163528},[31850],[34,163530],{"className":163531,"style":31846},[31845],[34,163533,163535,163538],{"className":163534},[31676],[34,163536],{"className":163537,"style":32617},[31680],[34,163539,162217],{"className":163540},[31685],[15,163542,163543],{},"从 120 出发，最近的是 129：",[15,163545,163546],{},[34,163547,163549,163567],{"className":163548},[31642],[34,163550,163552],{"className":163551},[31646],[31648,163553,163554],{"xmlns":31650},[31652,163555,163556,163564],{},[31655,163557,163558,163560,163562],{},[31749,163559,162217],{},[31662,163561,33239],{},[31749,163563,162183],{},[31665,163565,163566],{"encoding":31667},"120 \\rightarrow 129",[34,163568,163570,163588],{"className":163569,"ariaHidden":758},[31672],[34,163571,163573,163576,163579,163582,163585],{"className":163572},[31676],[34,163574],{"className":163575,"style":32617},[31680],[34,163577,162217],{"className":163578},[31685],[34,163580],{"className":163581,"style":31846},[31845],[34,163583,33239],{"className":163584},[31850],[34,163586],{"className":163587,"style":31846},[31845],[34,163589,163591,163594],{"className":163590},[31676],[34,163592],{"className":163593,"style":32617},[31680],[34,163595,162183],{"className":163596},[31685],[15,163598,163599],{},"从 129 出发，最近的是 147：",[15,163601,163602],{},[34,163603,163605,163623],{"className":163604},[31642],[34,163606,163608],{"className":163607},[31646],[31648,163609,163610],{"xmlns":31650},[31652,163611,163612,163620],{},[31655,163613,163614,163616,163618],{},[31749,163615,162183],{},[31662,163617,33239],{},[31749,163619,162198],{},[31665,163621,163622],{"encoding":31667},"129 \\rightarrow 147",[34,163624,163626,163644],{"className":163625,"ariaHidden":758},[31672],[34,163627,163629,163632,163635,163638,163641],{"className":163628},[31676],[34,163630],{"className":163631,"style":32617},[31680],[34,163633,162183],{"className":163634},[31685],[34,163636],{"className":163637,"style":31846},[31845],[34,163639,33239],{"className":163640},[31850],[34,163642],{"className":163643,"style":31846},[31845],[34,163645,163647,163650],{"className":163646},[31676],[34,163648],{"className":163649,"style":32617},[31680],[34,163651,162198],{"className":163652},[31685],[15,163654,163655],{},"从 147 出发，最近的是 186：",[15,163657,163658],{},[34,163659,163661,163679],{"className":163660},[31642],[34,163662,163664],{"className":163663},[31646],[31648,163665,163666],{"xmlns":31650},[31652,163667,163668,163676],{},[31655,163669,163670,163672,163674],{},[31749,163671,162198],{},[31662,163673,33239],{},[31749,163675,162193],{},[31665,163677,163678],{"encoding":31667},"147 \\rightarrow 186",[34,163680,163682,163700],{"className":163681,"ariaHidden":758},[31672],[34,163683,163685,163688,163691,163694,163697],{"className":163684},[31676],[34,163686],{"className":163687,"style":32617},[31680],[34,163689,162198],{"className":163690},[31685],[34,163692],{"className":163693,"style":31846},[31845],[34,163695,33239],{"className":163696},[31850],[34,163698],{"className":163699,"style":31846},[31845],[34,163701,163703,163706],{"className":163702},[31676],[34,163704],{"className":163705,"style":32617},[31680],[34,163707,162193],{"className":163708},[31685],[15,163710,163711],{},"此时剩余请求都在较小磁道方向：",[15,163713,163714],{},"27, 41, 10, 64",[15,163716,163717],{},"从 186 出发，最近的是 64：",[15,163719,163720],{},[34,163721,163723,163741],{"className":163722},[31642],[34,163724,163726],{"className":163725},[31646],[31648,163727,163728],{"xmlns":31650},[31652,163729,163730,163738],{},[31655,163731,163732,163734,163736],{},[31749,163733,162193],{},[31662,163735,33239],{},[31749,163737,162212],{},[31665,163739,163740],{"encoding":31667},"186 \\rightarrow 64",[34,163742,163744,163762],{"className":163743,"ariaHidden":758},[31672],[34,163745,163747,163750,163753,163756,163759],{"className":163746},[31676],[34,163748],{"className":163749,"style":32617},[31680],[34,163751,162193],{"className":163752},[31685],[34,163754],{"className":163755,"style":31846},[31845],[34,163757,33239],{"className":163758},[31850],[34,163760],{"className":163761,"style":31846},[31845],[34,163763,163765,163768],{"className":163764},[31676],[34,163766],{"className":163767,"style":32617},[31680],[34,163769,162212],{"className":163770},[31685],[15,163772,163773],{},"从 64 出发，最近的是 41：",[15,163775,163776],{},[34,163777,163779,163797],{"className":163778},[31642],[34,163780,163782],{"className":163781},[31646],[31648,163783,163784],{"xmlns":31650},[31652,163785,163786,163794],{},[31655,163787,163788,163790,163792],{},[31749,163789,162212],{},[31662,163791,33239],{},[31749,163793,162203],{},[31665,163795,163796],{"encoding":31667},"64 \\rightarrow 41",[34,163798,163800,163818],{"className":163799,"ariaHidden":758},[31672],[34,163801,163803,163806,163809,163812,163815],{"className":163802},[31676],[34,163804],{"className":163805,"style":32617},[31680],[34,163807,162212],{"className":163808},[31685],[34,163810],{"className":163811,"style":31846},[31845],[34,163813,33239],{"className":163814},[31850],[34,163816],{"className":163817,"style":31846},[31845],[34,163819,163821,163824],{"className":163820},[31676],[34,163822],{"className":163823,"style":32617},[31680],[34,163825,162203],{"className":163826},[31685],[15,163828,163829],{},"从 41 出发，最近的是 27：",[15,163831,163832],{},[34,163833,163835,163853],{"className":163834},[31642],[34,163836,163838],{"className":163837},[31646],[31648,163839,163840],{"xmlns":31650},[31652,163841,163842,163850],{},[31655,163843,163844,163846,163848],{},[31749,163845,162203],{},[31662,163847,33239],{},[31749,163849,162178],{},[31665,163851,163852],{"encoding":31667},"41 \\rightarrow 27",[34,163854,163856,163874],{"className":163855,"ariaHidden":758},[31672],[34,163857,163859,163862,163865,163868,163871],{"className":163858},[31676],[34,163860],{"className":163861,"style":32617},[31680],[34,163863,162203],{"className":163864},[31685],[34,163866],{"className":163867,"style":31846},[31845],[34,163869,33239],{"className":163870},[31850],[34,163872],{"className":163873,"style":31846},[31845],[34,163875,163877,163880],{"className":163876},[31676],[34,163878],{"className":163879,"style":32617},[31680],[34,163881,162178],{"className":163882},[31685],[15,163884,163885],{},"最后：",[15,163887,163888],{},[34,163889,163891,163909],{"className":163890},[31642],[34,163892,163894],{"className":163893},[31646],[31648,163895,163896],{"xmlns":31650},[31652,163897,163898,163906],{},[31655,163899,163900,163902,163904],{},[31749,163901,162178],{},[31662,163903,33239],{},[31749,163905,3244],{},[31665,163907,163908],{"encoding":31667},"27 \\rightarrow 10",[34,163910,163912,163930],{"className":163911,"ariaHidden":758},[31672],[34,163913,163915,163918,163921,163924,163927],{"className":163914},[31676],[34,163916],{"className":163917,"style":32617},[31680],[34,163919,162178],{"className":163920},[31685],[34,163922],{"className":163923,"style":31846},[31845],[34,163925,33239],{"className":163926},[31850],[34,163928],{"className":163929,"style":31846},[31845],[34,163931,163933,163936],{"className":163932},[31676],[34,163934],{"className":163935,"style":32617},[31680],[34,163937,3244],{"className":163938},[31685],[15,163940,163941],{},"所以 SSTF 的访问顺序为：",[15,163943,163944],{},[34,163945,163947,163997],{"className":163946},[31642],[34,163948,163950],{"className":163949},[31646],[31648,163951,163952],{"xmlns":31650},[31652,163953,163954,163994],{},[31655,163955,163956,163958,163960,163962,163964,163966,163968,163970,163972,163974,163976,163978,163980,163982,163984,163986,163988,163990,163992],{},[31749,163957,162173],{},[31662,163959,33239],{},[31749,163961,162188],{},[31662,163963,33239],{},[31749,163965,162217],{},[31662,163967,33239],{},[31749,163969,162183],{},[31662,163971,33239],{},[31749,163973,162198],{},[31662,163975,33239],{},[31749,163977,162193],{},[31662,163979,33239],{},[31749,163981,162212],{},[31662,163983,33239],{},[31749,163985,162203],{},[31662,163987,33239],{},[31749,163989,162178],{},[31662,163991,33239],{},[31749,163993,3244],{},[31665,163995,163996],{"encoding":31667},"100 \\rightarrow 110 \\rightarrow 120 \\rightarrow 129 \\rightarrow 147 \\rightarrow 186 \\rightarrow 64 \\rightarrow 41 \\rightarrow 27 \\rightarrow 10",[34,163998,164000,164018,164036,164054,164072,164090,164108,164126,164144,164162],{"className":163999,"ariaHidden":758},[31672],[34,164001,164003,164006,164009,164012,164015],{"className":164002},[31676],[34,164004],{"className":164005,"style":32617},[31680],[34,164007,162173],{"className":164008},[31685],[34,164010],{"className":164011,"style":31846},[31845],[34,164013,33239],{"className":164014},[31850],[34,164016],{"className":164017,"style":31846},[31845],[34,164019,164021,164024,164027,164030,164033],{"className":164020},[31676],[34,164022],{"className":164023,"style":32617},[31680],[34,164025,162188],{"className":164026},[31685],[34,164028],{"className":164029,"style":31846},[31845],[34,164031,33239],{"className":164032},[31850],[34,164034],{"className":164035,"style":31846},[31845],[34,164037,164039,164042,164045,164048,164051],{"className":164038},[31676],[34,164040],{"className":164041,"style":32617},[31680],[34,164043,162217],{"className":164044},[31685],[34,164046],{"className":164047,"style":31846},[31845],[34,164049,33239],{"className":164050},[31850],[34,164052],{"className":164053,"style":31846},[31845],[34,164055,164057,164060,164063,164066,164069],{"className":164056},[31676],[34,164058],{"className":164059,"style":32617},[31680],[34,164061,162183],{"className":164062},[31685],[34,164064],{"className":164065,"style":31846},[31845],[34,164067,33239],{"className":164068},[31850],[34,164070],{"className":164071,"style":31846},[31845],[34,164073,164075,164078,164081,164084,164087],{"className":164074},[31676],[34,164076],{"className":164077,"style":32617},[31680],[34,164079,162198],{"className":164080},[31685],[34,164082],{"className":164083,"style":31846},[31845],[34,164085,33239],{"className":164086},[31850],[34,164088],{"className":164089,"style":31846},[31845],[34,164091,164093,164096,164099,164102,164105],{"className":164092},[31676],[34,164094],{"className":164095,"style":32617},[31680],[34,164097,162193],{"className":164098},[31685],[34,164100],{"className":164101,"style":31846},[31845],[34,164103,33239],{"className":164104},[31850],[34,164106],{"className":164107,"style":31846},[31845],[34,164109,164111,164114,164117,164120,164123],{"className":164110},[31676],[34,164112],{"className":164113,"style":32617},[31680],[34,164115,162212],{"className":164116},[31685],[34,164118],{"className":164119,"style":31846},[31845],[34,164121,33239],{"className":164122},[31850],[34,164124],{"className":164125,"style":31846},[31845],[34,164127,164129,164132,164135,164138,164141],{"className":164128},[31676],[34,164130],{"className":164131,"style":32617},[31680],[34,164133,162203],{"className":164134},[31685],[34,164136],{"className":164137,"style":31846},[31845],[34,164139,33239],{"className":164140},[31850],[34,164142],{"className":164143,"style":31846},[31845],[34,164145,164147,164150,164153,164156,164159],{"className":164146},[31676],[34,164148],{"className":164149,"style":32617},[31680],[34,164151,162178],{"className":164152},[31685],[34,164154],{"className":164155,"style":31846},[31845],[34,164157,33239],{"className":164158},[31850],[34,164160],{"className":164161,"style":31846},[31845],[34,164163,164165,164168],{"className":164164},[31676],[34,164166],{"className":164167,"style":32617},[31680],[34,164169,3244],{"className":164170},[31685],[15,164172,162397],{},[15,164174,164175],{},[34,164176,164178,164204],{"className":164177},[31642],[34,164179,164181],{"className":164180},[31646],[31648,164182,164183],{"xmlns":31650},[31652,164184,164185,164201],{},[31655,164186,164187,164189,164191,164193,164195,164197,164199],{},[31658,164188,162086],{"mathvariant":38079},[31749,164190,162173],{},[31662,164192,32226],{},[31749,164194,162188],{},[31658,164196,162086],{"mathvariant":38079},[31662,164198,202],{},[31749,164200,3244],{},[31665,164202,164203],{"encoding":31667},"|100 - 110| = 10",[34,164205,164207,164225,164243],{"className":164206,"ariaHidden":758},[31672],[34,164208,164210,164213,164216,164219,164222],{"className":164209},[31676],[34,164211],{"className":164212,"style":31774},[31680],[34,164214,162442],{"className":164215},[31685],[34,164217],{"className":164218,"style":32604},[31845],[34,164220,32226],{"className":164221},[32024],[34,164223],{"className":164224,"style":32604},[31845],[34,164226,164228,164231,164234,164237,164240],{"className":164227},[31676],[34,164229],{"className":164230,"style":31774},[31680],[34,164232,162625],{"className":164233},[31685],[34,164235],{"className":164236,"style":31846},[31845],[34,164238,202],{"className":164239},[31850],[34,164241],{"className":164242,"style":31846},[31845],[34,164244,164246,164249],{"className":164245},[31676],[34,164247],{"className":164248,"style":32617},[31680],[34,164250,3244],{"className":164251},[31685],[15,164253,164254],{},[34,164255,164257,164283],{"className":164256},[31642],[34,164258,164260],{"className":164259},[31646],[31648,164261,164262],{"xmlns":31650},[31652,164263,164264,164280],{},[31655,164265,164266,164268,164270,164272,164274,164276,164278],{},[31658,164267,162086],{"mathvariant":38079},[31749,164269,162188],{},[31662,164271,32226],{},[31749,164273,162217],{},[31658,164275,162086],{"mathvariant":38079},[31662,164277,202],{},[31749,164279,3244],{},[31665,164281,164282],{"encoding":31667},"|110 - 120| = 10",[34,164284,164286,164304,164322],{"className":164285,"ariaHidden":758},[31672],[34,164287,164289,164292,164295,164298,164301],{"className":164288},[31676],[34,164290],{"className":164291,"style":31774},[31680],[34,164293,162687],{"className":164294},[31685],[34,164296],{"className":164297,"style":32604},[31845],[34,164299,32226],{"className":164300},[32024],[34,164302],{"className":164303,"style":32604},[31845],[34,164305,164307,164310,164313,164316,164319],{"className":164306},[31676],[34,164308],{"className":164309,"style":31774},[31680],[34,164311,163116],{"className":164312},[31685],[34,164314],{"className":164315,"style":31846},[31845],[34,164317,202],{"className":164318},[31850],[34,164320],{"className":164321,"style":31846},[31845],[34,164323,164325,164328],{"className":164324},[31676],[34,164326],{"className":164327,"style":32617},[31680],[34,164329,3244],{"className":164330},[31685],[15,164332,164333],{},[34,164334,164336,164362],{"className":164335},[31642],[34,164337,164339],{"className":164338},[31646],[31648,164340,164341],{"xmlns":31650},[31652,164342,164343,164359],{},[31655,164344,164345,164347,164349,164351,164353,164355,164357],{},[31658,164346,162086],{"mathvariant":38079},[31749,164348,162217],{},[31662,164350,32226],{},[31749,164352,162183],{},[31658,164354,162086],{"mathvariant":38079},[31662,164356,202],{},[31749,164358,36732],{},[31665,164360,164361],{"encoding":31667},"|120 - 129| = 9",[34,164363,164365,164384,164402],{"className":164364,"ariaHidden":758},[31672],[34,164366,164368,164371,164375,164378,164381],{"className":164367},[31676],[34,164369],{"className":164370,"style":31774},[31680],[34,164372,164374],{"className":164373},[31685],"∣120",[34,164376],{"className":164377,"style":32604},[31845],[34,164379,32226],{"className":164380},[32024],[34,164382],{"className":164383,"style":32604},[31845],[34,164385,164387,164390,164393,164396,164399],{"className":164386},[31676],[34,164388],{"className":164389,"style":31774},[31680],[34,164391,162543],{"className":164392},[31685],[34,164394],{"className":164395,"style":31846},[31845],[34,164397,202],{"className":164398},[31850],[34,164400],{"className":164401,"style":31846},[31845],[34,164403,164405,164408],{"className":164404},[31676],[34,164406],{"className":164407,"style":32617},[31680],[34,164409,36732],{"className":164410},[31685],[15,164412,164413],{},[34,164414,164416,164442],{"className":164415},[31642],[34,164417,164419],{"className":164418},[31646],[31648,164420,164421],{"xmlns":31650},[31652,164422,164423,164439],{},[31655,164424,164425,164427,164429,164431,164433,164435,164437],{},[31658,164426,162086],{"mathvariant":38079},[31749,164428,162183],{},[31662,164430,32226],{},[31749,164432,162198],{},[31658,164434,162086],{"mathvariant":38079},[31662,164436,202],{},[31749,164438,95611],{},[31665,164440,164441],{"encoding":31667},"|129 - 147| = 18",[34,164443,164445,164463,164481],{"className":164444,"ariaHidden":758},[31672],[34,164446,164448,164451,164454,164457,164460],{"className":164447},[31676],[34,164449],{"className":164450,"style":31774},[31680],[34,164452,162606],{"className":164453},[31685],[34,164455],{"className":164456,"style":32604},[31845],[34,164458,32226],{"className":164459},[32024],[34,164461],{"className":164462,"style":32604},[31845],[34,164464,164466,164469,164472,164475,164478],{"className":164465},[31676],[34,164467],{"className":164468,"style":31774},[31680],[34,164470,162788],{"className":164471},[31685],[34,164473],{"className":164474,"style":31846},[31845],[34,164476,202],{"className":164477},[31850],[34,164479],{"className":164480,"style":31846},[31845],[34,164482,164484,164487],{"className":164483},[31676],[34,164485],{"className":164486,"style":32617},[31680],[34,164488,95611],{"className":164489},[31685],[15,164491,164492],{},[34,164493,164495,164521],{"className":164494},[31642],[34,164496,164498],{"className":164497},[31646],[31648,164499,164500],{"xmlns":31650},[31652,164501,164502,164518],{},[31655,164503,164504,164506,164508,164510,164512,164514,164516],{},[31658,164505,162086],{"mathvariant":38079},[31749,164507,162198],{},[31662,164509,32226],{},[31749,164511,162193],{},[31658,164513,162086],{"mathvariant":38079},[31662,164515,202],{},[31749,164517,162753],{},[31665,164519,164520],{"encoding":31667},"|147 - 186| = 39",[34,164522,164524,164542,164560],{"className":164523,"ariaHidden":758},[31672],[34,164525,164527,164530,164533,164536,164539],{"className":164526},[31676],[34,164528],{"className":164529,"style":31774},[31680],[34,164531,162851],{"className":164532},[31685],[34,164534],{"className":164535,"style":32604},[31845],[34,164537,32226],{"className":164538},[32024],[34,164540],{"className":164541,"style":32604},[31845],[34,164543,164545,164548,164551,164554,164557],{"className":164544},[31676],[34,164546],{"className":164547,"style":31774},[31680],[34,164549,162706],{"className":164550},[31685],[34,164552],{"className":164553,"style":31846},[31845],[34,164555,202],{"className":164556},[31850],[34,164558],{"className":164559,"style":31846},[31845],[34,164561,164563,164566],{"className":164562},[31676],[34,164564],{"className":164565,"style":32617},[31680],[34,164567,162753],{"className":164568},[31685],[15,164570,164571],{},[34,164572,164574,164601],{"className":164573},[31642],[34,164575,164577],{"className":164576},[31646],[31648,164578,164579],{"xmlns":31650},[31652,164580,164581,164598],{},[31655,164582,164583,164585,164587,164589,164591,164593,164595],{},[31658,164584,162086],{"mathvariant":38079},[31749,164586,162193],{},[31662,164588,32226],{},[31749,164590,162212],{},[31658,164592,162086],{"mathvariant":38079},[31662,164594,202],{},[31749,164596,164597],{},"122",[31665,164599,164600],{"encoding":31667},"|186 - 64| = 122",[34,164602,164604,164622,164640],{"className":164603,"ariaHidden":758},[31672],[34,164605,164607,164610,164613,164616,164619],{"className":164606},[31676],[34,164608],{"className":164609,"style":31774},[31680],[34,164611,162769],{"className":164612},[31685],[34,164614],{"className":164615,"style":32604},[31845],[34,164617,32226],{"className":164618},[32024],[34,164620],{"className":164621,"style":32604},[31845],[34,164623,164625,164628,164631,164634,164637],{"className":164624},[31676],[34,164626],{"className":164627,"style":31774},[31680],[34,164629,163034],{"className":164630},[31685],[34,164632],{"className":164633,"style":31846},[31845],[34,164635,202],{"className":164636},[31850],[34,164638],{"className":164639,"style":31846},[31845],[34,164641,164643,164646],{"className":164642},[31676],[34,164644],{"className":164645,"style":32617},[31680],[34,164647,164597],{"className":164648},[31685],[15,164650,164651],{},[34,164652,164654,164680],{"className":164653},[31642],[34,164655,164657],{"className":164656},[31646],[31648,164658,164659],{"xmlns":31650},[31652,164660,164661,164677],{},[31655,164662,164663,164665,164667,164669,164671,164673,164675],{},[31658,164664,162086],{"mathvariant":38079},[31749,164666,162212],{},[31662,164668,32226],{},[31749,164670,162203],{},[31658,164672,162086],{"mathvariant":38079},[31662,164674,202],{},[31749,164676,94703],{},[31665,164678,164679],{"encoding":31667},"|64 - 41| = 23",[34,164681,164683,164701,164719],{"className":164682,"ariaHidden":758},[31672],[34,164684,164686,164689,164692,164695,164698],{"className":164685},[31676],[34,164687],{"className":164688,"style":31774},[31680],[34,164690,163097],{"className":164691},[31685],[34,164693],{"className":164694,"style":32604},[31845],[34,164696,32226],{"className":164697},[32024],[34,164699],{"className":164700,"style":32604},[31845],[34,164702,164704,164707,164710,164713,164716],{"className":164703},[31676],[34,164705],{"className":164706,"style":31774},[31680],[34,164708,162870],{"className":164709},[31685],[34,164711],{"className":164712,"style":31846},[31845],[34,164714,202],{"className":164715},[31850],[34,164717],{"className":164718,"style":31846},[31845],[34,164720,164722,164725],{"className":164721},[31676],[34,164723],{"className":164724,"style":32617},[31680],[34,164726,94703],{"className":164727},[31685],[15,164729,164730],{},[34,164731,164733,164760],{"className":164732},[31642],[34,164734,164736],{"className":164735},[31646],[31648,164737,164738],{"xmlns":31650},[31652,164739,164740,164757],{},[31655,164741,164742,164744,164746,164748,164750,164752,164754],{},[31658,164743,162086],{"mathvariant":38079},[31749,164745,162203],{},[31662,164747,32226],{},[31749,164749,162178],{},[31658,164751,162086],{"mathvariant":38079},[31662,164753,202],{},[31749,164755,164756],{},"14",[31665,164758,164759],{"encoding":31667},"|41 - 27| = 14",[34,164761,164763,164781,164799],{"className":164762,"ariaHidden":758},[31672],[34,164764,164766,164769,164772,164775,164778],{"className":164765},[31676],[34,164767],{"className":164768,"style":31774},[31680],[34,164770,162933],{"className":164771},[31685],[34,164773],{"className":164774,"style":32604},[31845],[34,164776,32226],{"className":164777},[32024],[34,164779],{"className":164780,"style":32604},[31845],[34,164782,164784,164787,164790,164793,164796],{"className":164783},[31676],[34,164785],{"className":164786,"style":31774},[31680],[34,164788,162461],{"className":164789},[31685],[34,164791],{"className":164792,"style":31846},[31845],[34,164794,202],{"className":164795},[31850],[34,164797],{"className":164798,"style":31846},[31845],[34,164800,164802,164805],{"className":164801},[31676],[34,164803],{"className":164804,"style":32617},[31680],[34,164806,164756],{"className":164807},[31685],[15,164809,164810],{},[34,164811,164813,164840],{"className":164812},[31642],[34,164814,164816],{"className":164815},[31646],[31648,164817,164818],{"xmlns":31650},[31652,164819,164820,164837],{},[31655,164821,164822,164824,164826,164828,164830,164832,164834],{},[31658,164823,162086],{"mathvariant":38079},[31749,164825,162178],{},[31662,164827,32226],{},[31749,164829,3244],{},[31658,164831,162086],{"mathvariant":38079},[31662,164833,202],{},[31749,164835,164836],{},"17",[31665,164838,164839],{"encoding":31667},"|27 - 10| = 17",[34,164841,164843,164861,164879],{"className":164842,"ariaHidden":758},[31672],[34,164844,164846,164849,164852,164855,164858],{"className":164845},[31676],[34,164847],{"className":164848,"style":31774},[31680],[34,164850,162524],{"className":164851},[31685],[34,164853],{"className":164854,"style":32604},[31845],[34,164856,32226],{"className":164857},[32024],[34,164859],{"className":164860,"style":32604},[31845],[34,164862,164864,164867,164870,164873,164876],{"className":164863},[31676],[34,164865],{"className":164866,"style":31774},[31680],[34,164868,162952],{"className":164869},[31685],[34,164871],{"className":164872,"style":31846},[31845],[34,164874,202],{"className":164875},[31850],[34,164877],{"className":164878,"style":31846},[31845],[34,164880,164882,164885],{"className":164881},[31676],[34,164883],{"className":164884,"style":32617},[31680],[34,164886,164836],{"className":164887},[31685],[15,164889,163137],{},[15,164891,164892],{},[34,164893,164895,164946],{"className":164894},[31642],[34,164896,164898],{"className":164897},[31646],[31648,164899,164900],{"xmlns":31650},[31652,164901,164902,164943],{},[31655,164903,164904,164906,164908,164910,164912,164914,164916,164918,164920,164922,164924,164926,164928,164930,164932,164934,164936,164938,164940],{},[31749,164905,3244],{},[31662,164907,223],{},[31749,164909,3244],{},[31662,164911,223],{},[31749,164913,36732],{},[31662,164915,223],{},[31749,164917,95611],{},[31662,164919,223],{},[31749,164921,162753],{},[31662,164923,223],{},[31749,164925,164597],{},[31662,164927,223],{},[31749,164929,94703],{},[31662,164931,223],{},[31749,164933,164756],{},[31662,164935,223],{},[31749,164937,164836],{},[31662,164939,202],{},[31749,164941,164942],{},"262",[31665,164944,164945],{"encoding":31667},"10 + 10 + 9 + 18 + 39 + 122 + 23 + 14 + 17 = 262",[34,164947,164949,164967,164985,165003,165021,165039,165057,165075,165093,165111],{"className":164948,"ariaHidden":758},[31672],[34,164950,164952,164955,164958,164961,164964],{"className":164951},[31676],[34,164953],{"className":164954,"style":35942},[31680],[34,164956,3244],{"className":164957},[31685],[34,164959],{"className":164960,"style":32604},[31845],[34,164962,223],{"className":164963},[32024],[34,164965],{"className":164966,"style":32604},[31845],[34,164968,164970,164973,164976,164979,164982],{"className":164969},[31676],[34,164971],{"className":164972,"style":35942},[31680],[34,164974,3244],{"className":164975},[31685],[34,164977],{"className":164978,"style":32604},[31845],[34,164980,223],{"className":164981},[32024],[34,164983],{"className":164984,"style":32604},[31845],[34,164986,164988,164991,164994,164997,165000],{"className":164987},[31676],[34,164989],{"className":164990,"style":35942},[31680],[34,164992,36732],{"className":164993},[31685],[34,164995],{"className":164996,"style":32604},[31845],[34,164998,223],{"className":164999},[32024],[34,165001],{"className":165002,"style":32604},[31845],[34,165004,165006,165009,165012,165015,165018],{"className":165005},[31676],[34,165007],{"className":165008,"style":35942},[31680],[34,165010,95611],{"className":165011},[31685],[34,165013],{"className":165014,"style":32604},[31845],[34,165016,223],{"className":165017},[32024],[34,165019],{"className":165020,"style":32604},[31845],[34,165022,165024,165027,165030,165033,165036],{"className":165023},[31676],[34,165025],{"className":165026,"style":35942},[31680],[34,165028,162753],{"className":165029},[31685],[34,165031],{"className":165032,"style":32604},[31845],[34,165034,223],{"className":165035},[32024],[34,165037],{"className":165038,"style":32604},[31845],[34,165040,165042,165045,165048,165051,165054],{"className":165041},[31676],[34,165043],{"className":165044,"style":35942},[31680],[34,165046,164597],{"className":165047},[31685],[34,165049],{"className":165050,"style":32604},[31845],[34,165052,223],{"className":165053},[32024],[34,165055],{"className":165056,"style":32604},[31845],[34,165058,165060,165063,165066,165069,165072],{"className":165059},[31676],[34,165061],{"className":165062,"style":35942},[31680],[34,165064,94703],{"className":165065},[31685],[34,165067],{"className":165068,"style":32604},[31845],[34,165070,223],{"className":165071},[32024],[34,165073],{"className":165074,"style":32604},[31845],[34,165076,165078,165081,165084,165087,165090],{"className":165077},[31676],[34,165079],{"className":165080,"style":35942},[31680],[34,165082,164756],{"className":165083},[31685],[34,165085],{"className":165086,"style":32604},[31845],[34,165088,223],{"className":165089},[32024],[34,165091],{"className":165092,"style":32604},[31845],[34,165094,165096,165099,165102,165105,165108],{"className":165095},[31676],[34,165097],{"className":165098,"style":32617},[31680],[34,165100,164836],{"className":165101},[31685],[34,165103],{"className":165104,"style":31846},[31845],[34,165106,202],{"className":165107},[31850],[34,165109],{"className":165110,"style":31846},[31845],[34,165112,165114,165117],{"className":165113},[31676],[34,165115],{"className":165116,"style":32617},[31680],[34,165118,164942],{"className":165119},[31685],[15,165121,159962],{},[15,165123,165124],{},"SSTF 总移动磁道数为：",[15,165126,165127],{},[34,165128,165130,165143],{"className":165129},[31642],[34,165131,165133],{"className":165132},[31646],[31648,165134,165135],{"xmlns":31650},[31652,165136,165137,165141],{},[31655,165138,165139],{},[31749,165140,164942],{},[31665,165142,164942],{"encoding":31667},[34,165144,165146],{"className":165145,"ariaHidden":758},[31672],[34,165147,165149,165152],{"className":165148},[31676],[34,165150],{"className":165151,"style":32617},[31680],[34,165153,164942],{"className":165154},[31685],[32188,165156],{},[253,165158,165160],{"id":165159},"三scan-算法","三、SCAN 算法",[15,165162,165163],{},"SCAN 又叫电梯算法。",[15,165165,165166],{},"题目说磁头初始方向是磁道号减小的方向，所以先处理小于 100 的请求，再反向处理大于 100 的请求。",[15,165168,165169],{},"小于 100 的请求有：",[15,165171,165172],{},"64, 41, 27, 10",[15,165174,165175],{},"按磁头减小方向访问：",[15,165177,165178],{},[34,165179,165181,165211],{"className":165180},[31642],[34,165182,165184],{"className":165183},[31646],[31648,165185,165186],{"xmlns":31650},[31652,165187,165188,165208],{},[31655,165189,165190,165192,165194,165196,165198,165200,165202,165204,165206],{},[31749,165191,162173],{},[31662,165193,33239],{},[31749,165195,162212],{},[31662,165197,33239],{},[31749,165199,162203],{},[31662,165201,33239],{},[31749,165203,162178],{},[31662,165205,33239],{},[31749,165207,3244],{},[31665,165209,165210],{"encoding":31667},"100 \\rightarrow 64 \\rightarrow 41 \\rightarrow 27 \\rightarrow 10",[34,165212,165214,165232,165250,165268,165286],{"className":165213,"ariaHidden":758},[31672],[34,165215,165217,165220,165223,165226,165229],{"className":165216},[31676],[34,165218],{"className":165219,"style":32617},[31680],[34,165221,162173],{"className":165222},[31685],[34,165224],{"className":165225,"style":31846},[31845],[34,165227,33239],{"className":165228},[31850],[34,165230],{"className":165231,"style":31846},[31845],[34,165233,165235,165238,165241,165244,165247],{"className":165234},[31676],[34,165236],{"className":165237,"style":32617},[31680],[34,165239,162212],{"className":165240},[31685],[34,165242],{"className":165243,"style":31846},[31845],[34,165245,33239],{"className":165246},[31850],[34,165248],{"className":165249,"style":31846},[31845],[34,165251,165253,165256,165259,165262,165265],{"className":165252},[31676],[34,165254],{"className":165255,"style":32617},[31680],[34,165257,162203],{"className":165258},[31685],[34,165260],{"className":165261,"style":31846},[31845],[34,165263,33239],{"className":165264},[31850],[34,165266],{"className":165267,"style":31846},[31845],[34,165269,165271,165274,165277,165280,165283],{"className":165270},[31676],[34,165272],{"className":165273,"style":32617},[31680],[34,165275,162178],{"className":165276},[31685],[34,165278],{"className":165279,"style":31846},[31845],[34,165281,33239],{"className":165282},[31850],[34,165284],{"className":165285,"style":31846},[31845],[34,165287,165289,165292],{"className":165288},[31676],[34,165290],{"className":165291,"style":32617},[31680],[34,165293,3244],{"className":165294},[31685],[15,165296,165297],{},"然后反向访问大于 100 的请求：",[15,165299,165300],{},"110, 120, 129, 147, 186",[15,165302,165303],{},"所以访问顺序为：",[15,165305,165306],{},[34,165307,165309,165359],{"className":165308},[31642],[34,165310,165312],{"className":165311},[31646],[31648,165313,165314],{"xmlns":31650},[31652,165315,165316,165356],{},[31655,165317,165318,165320,165322,165324,165326,165328,165330,165332,165334,165336,165338,165340,165342,165344,165346,165348,165350,165352,165354],{},[31749,165319,162173],{},[31662,165321,33239],{},[31749,165323,162212],{},[31662,165325,33239],{},[31749,165327,162203],{},[31662,165329,33239],{},[31749,165331,162178],{},[31662,165333,33239],{},[31749,165335,3244],{},[31662,165337,33239],{},[31749,165339,162188],{},[31662,165341,33239],{},[31749,165343,162217],{},[31662,165345,33239],{},[31749,165347,162183],{},[31662,165349,33239],{},[31749,165351,162198],{},[31662,165353,33239],{},[31749,165355,162193],{},[31665,165357,165358],{"encoding":31667},"100 \\rightarrow 64 \\rightarrow 41 \\rightarrow 27 \\rightarrow 10 \\rightarrow 110 \\rightarrow 120 \\rightarrow 129 \\rightarrow 147 \\rightarrow 186",[34,165360,165362,165380,165398,165416,165434,165452,165470,165488,165506,165524],{"className":165361,"ariaHidden":758},[31672],[34,165363,165365,165368,165371,165374,165377],{"className":165364},[31676],[34,165366],{"className":165367,"style":32617},[31680],[34,165369,162173],{"className":165370},[31685],[34,165372],{"className":165373,"style":31846},[31845],[34,165375,33239],{"className":165376},[31850],[34,165378],{"className":165379,"style":31846},[31845],[34,165381,165383,165386,165389,165392,165395],{"className":165382},[31676],[34,165384],{"className":165385,"style":32617},[31680],[34,165387,162212],{"className":165388},[31685],[34,165390],{"className":165391,"style":31846},[31845],[34,165393,33239],{"className":165394},[31850],[34,165396],{"className":165397,"style":31846},[31845],[34,165399,165401,165404,165407,165410,165413],{"className":165400},[31676],[34,165402],{"className":165403,"style":32617},[31680],[34,165405,162203],{"className":165406},[31685],[34,165408],{"className":165409,"style":31846},[31845],[34,165411,33239],{"className":165412},[31850],[34,165414],{"className":165415,"style":31846},[31845],[34,165417,165419,165422,165425,165428,165431],{"className":165418},[31676],[34,165420],{"className":165421,"style":32617},[31680],[34,165423,162178],{"className":165424},[31685],[34,165426],{"className":165427,"style":31846},[31845],[34,165429,33239],{"className":165430},[31850],[34,165432],{"className":165433,"style":31846},[31845],[34,165435,165437,165440,165443,165446,165449],{"className":165436},[31676],[34,165438],{"className":165439,"style":32617},[31680],[34,165441,3244],{"className":165442},[31685],[34,165444],{"className":165445,"style":31846},[31845],[34,165447,33239],{"className":165448},[31850],[34,165450],{"className":165451,"style":31846},[31845],[34,165453,165455,165458,165461,165464,165467],{"className":165454},[31676],[34,165456],{"className":165457,"style":32617},[31680],[34,165459,162188],{"className":165460},[31685],[34,165462],{"className":165463,"style":31846},[31845],[34,165465,33239],{"className":165466},[31850],[34,165468],{"className":165469,"style":31846},[31845],[34,165471,165473,165476,165479,165482,165485],{"className":165472},[31676],[34,165474],{"className":165475,"style":32617},[31680],[34,165477,162217],{"className":165478},[31685],[34,165480],{"className":165481,"style":31846},[31845],[34,165483,33239],{"className":165484},[31850],[34,165486],{"className":165487,"style":31846},[31845],[34,165489,165491,165494,165497,165500,165503],{"className":165490},[31676],[34,165492],{"className":165493,"style":32617},[31680],[34,165495,162183],{"className":165496},[31685],[34,165498],{"className":165499,"style":31846},[31845],[34,165501,33239],{"className":165502},[31850],[34,165504],{"className":165505,"style":31846},[31845],[34,165507,165509,165512,165515,165518,165521],{"className":165508},[31676],[34,165510],{"className":165511,"style":32617},[31680],[34,165513,162198],{"className":165514},[31685],[34,165516],{"className":165517,"style":31846},[31845],[34,165519,33239],{"className":165520},[31850],[34,165522],{"className":165523,"style":31846},[31845],[34,165525,165527,165530],{"className":165526},[31676],[34,165528],{"className":165529,"style":32617},[31680],[34,165531,162193],{"className":165532},[31685],[15,165534,162397],{},[15,165536,165537],{},[34,165538,165540,165567],{"className":165539},[31642],[34,165541,165543],{"className":165542},[31646],[31648,165544,165545],{"xmlns":31650},[31652,165546,165547,165564],{},[31655,165548,165549,165551,165553,165555,165557,165559,165561],{},[31658,165550,162086],{"mathvariant":38079},[31749,165552,162173],{},[31662,165554,32226],{},[31749,165556,162212],{},[31658,165558,162086],{"mathvariant":38079},[31662,165560,202],{},[31749,165562,165563],{},"36",[31665,165565,165566],{"encoding":31667},"|100 - 64| = 36",[34,165568,165570,165588,165606],{"className":165569,"ariaHidden":758},[31672],[34,165571,165573,165576,165579,165582,165585],{"className":165572},[31676],[34,165574],{"className":165575,"style":31774},[31680],[34,165577,162442],{"className":165578},[31685],[34,165580],{"className":165581,"style":32604},[31845],[34,165583,32226],{"className":165584},[32024],[34,165586],{"className":165587,"style":32604},[31845],[34,165589,165591,165594,165597,165600,165603],{"className":165590},[31676],[34,165592],{"className":165593,"style":31774},[31680],[34,165595,163034],{"className":165596},[31685],[34,165598],{"className":165599,"style":31846},[31845],[34,165601,202],{"className":165602},[31850],[34,165604],{"className":165605,"style":31846},[31845],[34,165607,165609,165612],{"className":165608},[31676],[34,165610],{"className":165611,"style":32617},[31680],[34,165613,165563],{"className":165614},[31685],[15,165616,165617],{},[34,165618,165620,165645],{"className":165619},[31642],[34,165621,165623],{"className":165622},[31646],[31648,165624,165625],{"xmlns":31650},[31652,165626,165627,165643],{},[31655,165628,165629,165631,165633,165635,165637,165639,165641],{},[31658,165630,162086],{"mathvariant":38079},[31749,165632,162212],{},[31662,165634,32226],{},[31749,165636,162203],{},[31658,165638,162086],{"mathvariant":38079},[31662,165640,202],{},[31749,165642,94703],{},[31665,165644,164679],{"encoding":31667},[34,165646,165648,165666,165684],{"className":165647,"ariaHidden":758},[31672],[34,165649,165651,165654,165657,165660,165663],{"className":165650},[31676],[34,165652],{"className":165653,"style":31774},[31680],[34,165655,163097],{"className":165656},[31685],[34,165658],{"className":165659,"style":32604},[31845],[34,165661,32226],{"className":165662},[32024],[34,165664],{"className":165665,"style":32604},[31845],[34,165667,165669,165672,165675,165678,165681],{"className":165668},[31676],[34,165670],{"className":165671,"style":31774},[31680],[34,165673,162870],{"className":165674},[31685],[34,165676],{"className":165677,"style":31846},[31845],[34,165679,202],{"className":165680},[31850],[34,165682],{"className":165683,"style":31846},[31845],[34,165685,165687,165690],{"className":165686},[31676],[34,165688],{"className":165689,"style":32617},[31680],[34,165691,94703],{"className":165692},[31685],[15,165694,165695],{},[34,165696,165698,165723],{"className":165697},[31642],[34,165699,165701],{"className":165700},[31646],[31648,165702,165703],{"xmlns":31650},[31652,165704,165705,165721],{},[31655,165706,165707,165709,165711,165713,165715,165717,165719],{},[31658,165708,162086],{"mathvariant":38079},[31749,165710,162203],{},[31662,165712,32226],{},[31749,165714,162178],{},[31658,165716,162086],{"mathvariant":38079},[31662,165718,202],{},[31749,165720,164756],{},[31665,165722,164759],{"encoding":31667},[34,165724,165726,165744,165762],{"className":165725,"ariaHidden":758},[31672],[34,165727,165729,165732,165735,165738,165741],{"className":165728},[31676],[34,165730],{"className":165731,"style":31774},[31680],[34,165733,162933],{"className":165734},[31685],[34,165736],{"className":165737,"style":32604},[31845],[34,165739,32226],{"className":165740},[32024],[34,165742],{"className":165743,"style":32604},[31845],[34,165745,165747,165750,165753,165756,165759],{"className":165746},[31676],[34,165748],{"className":165749,"style":31774},[31680],[34,165751,162461],{"className":165752},[31685],[34,165754],{"className":165755,"style":31846},[31845],[34,165757,202],{"className":165758},[31850],[34,165760],{"className":165761,"style":31846},[31845],[34,165763,165765,165768],{"className":165764},[31676],[34,165766],{"className":165767,"style":32617},[31680],[34,165769,164756],{"className":165770},[31685],[15,165772,165773],{},[34,165774,165776,165801],{"className":165775},[31642],[34,165777,165779],{"className":165778},[31646],[31648,165780,165781],{"xmlns":31650},[31652,165782,165783,165799],{},[31655,165784,165785,165787,165789,165791,165793,165795,165797],{},[31658,165786,162086],{"mathvariant":38079},[31749,165788,162178],{},[31662,165790,32226],{},[31749,165792,3244],{},[31658,165794,162086],{"mathvariant":38079},[31662,165796,202],{},[31749,165798,164836],{},[31665,165800,164839],{"encoding":31667},[34,165802,165804,165822,165840],{"className":165803,"ariaHidden":758},[31672],[34,165805,165807,165810,165813,165816,165819],{"className":165806},[31676],[34,165808],{"className":165809,"style":31774},[31680],[34,165811,162524],{"className":165812},[31685],[34,165814],{"className":165815,"style":32604},[31845],[34,165817,32226],{"className":165818},[32024],[34,165820],{"className":165821,"style":32604},[31845],[34,165823,165825,165828,165831,165834,165837],{"className":165824},[31676],[34,165826],{"className":165827,"style":31774},[31680],[34,165829,162952],{"className":165830},[31685],[34,165832],{"className":165833,"style":31846},[31845],[34,165835,202],{"className":165836},[31850],[34,165838],{"className":165839,"style":31846},[31845],[34,165841,165843,165846],{"className":165842},[31676],[34,165844],{"className":165845,"style":32617},[31680],[34,165847,164836],{"className":165848},[31685],[15,165850,165851],{},[34,165852,165854,165880],{"className":165853},[31642],[34,165855,165857],{"className":165856},[31646],[31648,165858,165859],{"xmlns":31650},[31652,165860,165861,165877],{},[31655,165862,165863,165865,165867,165869,165871,165873,165875],{},[31658,165864,162086],{"mathvariant":38079},[31749,165866,3244],{},[31662,165868,32226],{},[31749,165870,162188],{},[31658,165872,162086],{"mathvariant":38079},[31662,165874,202],{},[31749,165876,162173],{},[31665,165878,165879],{"encoding":31667},"|10 - 110| = 100",[34,165881,165883,165901,165919],{"className":165882,"ariaHidden":758},[31672],[34,165884,165886,165889,165892,165895,165898],{"className":165885},[31676],[34,165887],{"className":165888,"style":31774},[31680],[34,165890,163015],{"className":165891},[31685],[34,165893],{"className":165894,"style":32604},[31845],[34,165896,32226],{"className":165897},[32024],[34,165899],{"className":165900,"style":32604},[31845],[34,165902,165904,165907,165910,165913,165916],{"className":165903},[31676],[34,165905],{"className":165906,"style":31774},[31680],[34,165908,162625],{"className":165909},[31685],[34,165911],{"className":165912,"style":31846},[31845],[34,165914,202],{"className":165915},[31850],[34,165917],{"className":165918,"style":31846},[31845],[34,165920,165922,165925],{"className":165921},[31676],[34,165923],{"className":165924,"style":32617},[31680],[34,165926,162173],{"className":165927},[31685],[15,165929,165930],{},[34,165931,165933,165958],{"className":165932},[31642],[34,165934,165936],{"className":165935},[31646],[31648,165937,165938],{"xmlns":31650},[31652,165939,165940,165956],{},[31655,165941,165942,165944,165946,165948,165950,165952,165954],{},[31658,165943,162086],{"mathvariant":38079},[31749,165945,162188],{},[31662,165947,32226],{},[31749,165949,162217],{},[31658,165951,162086],{"mathvariant":38079},[31662,165953,202],{},[31749,165955,3244],{},[31665,165957,164282],{"encoding":31667},[34,165959,165961,165979,165997],{"className":165960,"ariaHidden":758},[31672],[34,165962,165964,165967,165970,165973,165976],{"className":165963},[31676],[34,165965],{"className":165966,"style":31774},[31680],[34,165968,162687],{"className":165969},[31685],[34,165971],{"className":165972,"style":32604},[31845],[34,165974,32226],{"className":165975},[32024],[34,165977],{"className":165978,"style":32604},[31845],[34,165980,165982,165985,165988,165991,165994],{"className":165981},[31676],[34,165983],{"className":165984,"style":31774},[31680],[34,165986,163116],{"className":165987},[31685],[34,165989],{"className":165990,"style":31846},[31845],[34,165992,202],{"className":165993},[31850],[34,165995],{"className":165996,"style":31846},[31845],[34,165998,166000,166003],{"className":165999},[31676],[34,166001],{"className":166002,"style":32617},[31680],[34,166004,3244],{"className":166005},[31685],[15,166007,166008],{},[34,166009,166011,166036],{"className":166010},[31642],[34,166012,166014],{"className":166013},[31646],[31648,166015,166016],{"xmlns":31650},[31652,166017,166018,166034],{},[31655,166019,166020,166022,166024,166026,166028,166030,166032],{},[31658,166021,162086],{"mathvariant":38079},[31749,166023,162217],{},[31662,166025,32226],{},[31749,166027,162183],{},[31658,166029,162086],{"mathvariant":38079},[31662,166031,202],{},[31749,166033,36732],{},[31665,166035,164361],{"encoding":31667},[34,166037,166039,166057,166075],{"className":166038,"ariaHidden":758},[31672],[34,166040,166042,166045,166048,166051,166054],{"className":166041},[31676],[34,166043],{"className":166044,"style":31774},[31680],[34,166046,164374],{"className":166047},[31685],[34,166049],{"className":166050,"style":32604},[31845],[34,166052,32226],{"className":166053},[32024],[34,166055],{"className":166056,"style":32604},[31845],[34,166058,166060,166063,166066,166069,166072],{"className":166059},[31676],[34,166061],{"className":166062,"style":31774},[31680],[34,166064,162543],{"className":166065},[31685],[34,166067],{"className":166068,"style":31846},[31845],[34,166070,202],{"className":166071},[31850],[34,166073],{"className":166074,"style":31846},[31845],[34,166076,166078,166081],{"className":166077},[31676],[34,166079],{"className":166080,"style":32617},[31680],[34,166082,36732],{"className":166083},[31685],[15,166085,166086],{},[34,166087,166089,166114],{"className":166088},[31642],[34,166090,166092],{"className":166091},[31646],[31648,166093,166094],{"xmlns":31650},[31652,166095,166096,166112],{},[31655,166097,166098,166100,166102,166104,166106,166108,166110],{},[31658,166099,162086],{"mathvariant":38079},[31749,166101,162183],{},[31662,166103,32226],{},[31749,166105,162198],{},[31658,166107,162086],{"mathvariant":38079},[31662,166109,202],{},[31749,166111,95611],{},[31665,166113,164441],{"encoding":31667},[34,166115,166117,166135,166153],{"className":166116,"ariaHidden":758},[31672],[34,166118,166120,166123,166126,166129,166132],{"className":166119},[31676],[34,166121],{"className":166122,"style":31774},[31680],[34,166124,162606],{"className":166125},[31685],[34,166127],{"className":166128,"style":32604},[31845],[34,166130,32226],{"className":166131},[32024],[34,166133],{"className":166134,"style":32604},[31845],[34,166136,166138,166141,166144,166147,166150],{"className":166137},[31676],[34,166139],{"className":166140,"style":31774},[31680],[34,166142,162788],{"className":166143},[31685],[34,166145],{"className":166146,"style":31846},[31845],[34,166148,202],{"className":166149},[31850],[34,166151],{"className":166152,"style":31846},[31845],[34,166154,166156,166159],{"className":166155},[31676],[34,166157],{"className":166158,"style":32617},[31680],[34,166160,95611],{"className":166161},[31685],[15,166163,166164],{},[34,166165,166167,166192],{"className":166166},[31642],[34,166168,166170],{"className":166169},[31646],[31648,166171,166172],{"xmlns":31650},[31652,166173,166174,166190],{},[31655,166175,166176,166178,166180,166182,166184,166186,166188],{},[31658,166177,162086],{"mathvariant":38079},[31749,166179,162198],{},[31662,166181,32226],{},[31749,166183,162193],{},[31658,166185,162086],{"mathvariant":38079},[31662,166187,202],{},[31749,166189,162753],{},[31665,166191,164520],{"encoding":31667},[34,166193,166195,166213,166231],{"className":166194,"ariaHidden":758},[31672],[34,166196,166198,166201,166204,166207,166210],{"className":166197},[31676],[34,166199],{"className":166200,"style":31774},[31680],[34,166202,162851],{"className":166203},[31685],[34,166205],{"className":166206,"style":32604},[31845],[34,166208,32226],{"className":166209},[32024],[34,166211],{"className":166212,"style":32604},[31845],[34,166214,166216,166219,166222,166225,166228],{"className":166215},[31676],[34,166217],{"className":166218,"style":31774},[31680],[34,166220,162706],{"className":166221},[31685],[34,166223],{"className":166224,"style":31846},[31845],[34,166226,202],{"className":166227},[31850],[34,166229],{"className":166230,"style":31846},[31845],[34,166232,166234,166237],{"className":166233},[31676],[34,166235],{"className":166236,"style":32617},[31680],[34,166238,162753],{"className":166239},[31685],[15,166241,163137],{},[15,166243,166244],{},[34,166245,166247,166298],{"className":166246},[31642],[34,166248,166250],{"className":166249},[31646],[31648,166251,166252],{"xmlns":31650},[31652,166253,166254,166295],{},[31655,166255,166256,166258,166260,166262,166264,166266,166268,166270,166272,166274,166276,166278,166280,166282,166284,166286,166288,166290,166292],{},[31749,166257,165563],{},[31662,166259,223],{},[31749,166261,94703],{},[31662,166263,223],{},[31749,166265,164756],{},[31662,166267,223],{},[31749,166269,164836],{},[31662,166271,223],{},[31749,166273,162173],{},[31662,166275,223],{},[31749,166277,3244],{},[31662,166279,223],{},[31749,166281,36732],{},[31662,166283,223],{},[31749,166285,95611],{},[31662,166287,223],{},[31749,166289,162753],{},[31662,166291,202],{},[31749,166293,166294],{},"266",[31665,166296,166297],{"encoding":31667},"36 + 23 + 14 + 17 + 100 + 10 + 9 + 18 + 39 = 266",[34,166299,166301,166319,166337,166355,166373,166391,166409,166427,166445,166463],{"className":166300,"ariaHidden":758},[31672],[34,166302,166304,166307,166310,166313,166316],{"className":166303},[31676],[34,166305],{"className":166306,"style":35942},[31680],[34,166308,165563],{"className":166309},[31685],[34,166311],{"className":166312,"style":32604},[31845],[34,166314,223],{"className":166315},[32024],[34,166317],{"className":166318,"style":32604},[31845],[34,166320,166322,166325,166328,166331,166334],{"className":166321},[31676],[34,166323],{"className":166324,"style":35942},[31680],[34,166326,94703],{"className":166327},[31685],[34,166329],{"className":166330,"style":32604},[31845],[34,166332,223],{"className":166333},[32024],[34,166335],{"className":166336,"style":32604},[31845],[34,166338,166340,166343,166346,166349,166352],{"className":166339},[31676],[34,166341],{"className":166342,"style":35942},[31680],[34,166344,164756],{"className":166345},[31685],[34,166347],{"className":166348,"style":32604},[31845],[34,166350,223],{"className":166351},[32024],[34,166353],{"className":166354,"style":32604},[31845],[34,166356,166358,166361,166364,166367,166370],{"className":166357},[31676],[34,166359],{"className":166360,"style":35942},[31680],[34,166362,164836],{"className":166363},[31685],[34,166365],{"className":166366,"style":32604},[31845],[34,166368,223],{"className":166369},[32024],[34,166371],{"className":166372,"style":32604},[31845],[34,166374,166376,166379,166382,166385,166388],{"className":166375},[31676],[34,166377],{"className":166378,"style":35942},[31680],[34,166380,162173],{"className":166381},[31685],[34,166383],{"className":166384,"style":32604},[31845],[34,166386,223],{"className":166387},[32024],[34,166389],{"className":166390,"style":32604},[31845],[34,166392,166394,166397,166400,166403,166406],{"className":166393},[31676],[34,166395],{"className":166396,"style":35942},[31680],[34,166398,3244],{"className":166399},[31685],[34,166401],{"className":166402,"style":32604},[31845],[34,166404,223],{"className":166405},[32024],[34,166407],{"className":166408,"style":32604},[31845],[34,166410,166412,166415,166418,166421,166424],{"className":166411},[31676],[34,166413],{"className":166414,"style":35942},[31680],[34,166416,36732],{"className":166417},[31685],[34,166419],{"className":166420,"style":32604},[31845],[34,166422,223],{"className":166423},[32024],[34,166425],{"className":166426,"style":32604},[31845],[34,166428,166430,166433,166436,166439,166442],{"className":166429},[31676],[34,166431],{"className":166432,"style":35942},[31680],[34,166434,95611],{"className":166435},[31685],[34,166437],{"className":166438,"style":32604},[31845],[34,166440,223],{"className":166441},[32024],[34,166443],{"className":166444,"style":32604},[31845],[34,166446,166448,166451,166454,166457,166460],{"className":166447},[31676],[34,166449],{"className":166450,"style":32617},[31680],[34,166452,162753],{"className":166453},[31685],[34,166455],{"className":166456,"style":31846},[31845],[34,166458,202],{"className":166459},[31850],[34,166461],{"className":166462,"style":31846},[31845],[34,166464,166466,166469],{"className":166465},[31676],[34,166467],{"className":166468,"style":32617},[31680],[34,166470,166294],{"className":166471},[31685],[15,166473,159962],{},[15,166475,166476],{},"SCAN 总移动磁道数为：",[15,166478,166479],{},[34,166480,166482,166495],{"className":166481},[31642],[34,166483,166485],{"className":166484},[31646],[31648,166486,166487],{"xmlns":31650},[31652,166488,166489,166493],{},[31655,166490,166491],{},[31749,166492,166294],{},[31665,166494,166294],{"encoding":31667},[34,166496,166498],{"className":166497,"ariaHidden":758},[31672],[34,166499,166501,166504],{"className":166500},[31676],[34,166502],{"className":166503,"style":32617},[31680],[34,166505,166294],{"className":166506},[31685],[32188,166508],{},[253,166510,166511],{"id":166511},"补充说明",[15,166513,166514],{},"按 Stallings 表 11.2 的 SCAN 算法示例，磁头到达当前方向上最后一个请求后就反向，不一定额外移动到磁盘最边界。",[15,166516,166517],{},"所以本题 SCAN 取：",[15,166519,166520],{},[34,166521,166523,166536],{"className":166522},[31642],[34,166524,166526],{"className":166525},[31646],[31648,166527,166528],{"xmlns":31650},[31652,166529,166530,166534],{},[31655,166531,166532],{},[31749,166533,166294],{},[31665,166535,166294],{"encoding":31667},[34,166537,166539],{"className":166538,"ariaHidden":758},[31672],[34,166540,166542,166545],{"className":166541},[31676],[34,166543],{"className":166544,"style":32617},[31680],[34,166546,166294],{"className":166547},[31685],[15,166549,166550],{},"如果你的老师要求严格 SCAN 必须先走到磁道 0 再反向，则访问过程为：",[15,166552,166553],{},[34,166554,166556,166610],{"className":166555},[31642],[34,166557,166559],{"className":166558},[31646],[31648,166560,166561],{"xmlns":31650},[31652,166562,166563,166607],{},[31655,166564,166565,166567,166569,166571,166573,166575,166577,166579,166581,166583,166585,166587,166589,166591,166593,166595,166597,166599,166601,166603,166605],{},[31749,166566,162173],{},[31662,166568,33239],{},[31749,166570,162212],{},[31662,166572,33239],{},[31749,166574,162203],{},[31662,166576,33239],{},[31749,166578,162178],{},[31662,166580,33239],{},[31749,166582,3244],{},[31662,166584,33239],{},[31749,166586,459],{},[31662,166588,33239],{},[31749,166590,162188],{},[31662,166592,33239],{},[31749,166594,162217],{},[31662,166596,33239],{},[31749,166598,162183],{},[31662,166600,33239],{},[31749,166602,162198],{},[31662,166604,33239],{},[31749,166606,162193],{},[31665,166608,166609],{"encoding":31667},"100 \\rightarrow 64 \\rightarrow 41 \\rightarrow 27 \\rightarrow 10 \\rightarrow 0 \\rightarrow 110 \\rightarrow 120 \\rightarrow 129 \\rightarrow 147 \\rightarrow 186",[34,166611,166613,166631,166649,166667,166685,166703,166721,166739,166757,166775,166793],{"className":166612,"ariaHidden":758},[31672],[34,166614,166616,166619,166622,166625,166628],{"className":166615},[31676],[34,166617],{"className":166618,"style":32617},[31680],[34,166620,162173],{"className":166621},[31685],[34,166623],{"className":166624,"style":31846},[31845],[34,166626,33239],{"className":166627},[31850],[34,166629],{"className":166630,"style":31846},[31845],[34,166632,166634,166637,166640,166643,166646],{"className":166633},[31676],[34,166635],{"className":166636,"style":32617},[31680],[34,166638,162212],{"className":166639},[31685],[34,166641],{"className":166642,"style":31846},[31845],[34,166644,33239],{"className":166645},[31850],[34,166647],{"className":166648,"style":31846},[31845],[34,166650,166652,166655,166658,166661,166664],{"className":166651},[31676],[34,166653],{"className":166654,"style":32617},[31680],[34,166656,162203],{"className":166657},[31685],[34,166659],{"className":166660,"style":31846},[31845],[34,166662,33239],{"className":166663},[31850],[34,166665],{"className":166666,"style":31846},[31845],[34,166668,166670,166673,166676,166679,166682],{"className":166669},[31676],[34,166671],{"className":166672,"style":32617},[31680],[34,166674,162178],{"className":166675},[31685],[34,166677],{"className":166678,"style":31846},[31845],[34,166680,33239],{"className":166681},[31850],[34,166683],{"className":166684,"style":31846},[31845],[34,166686,166688,166691,166694,166697,166700],{"className":166687},[31676],[34,166689],{"className":166690,"style":32617},[31680],[34,166692,3244],{"className":166693},[31685],[34,166695],{"className":166696,"style":31846},[31845],[34,166698,33239],{"className":166699},[31850],[34,166701],{"className":166702,"style":31846},[31845],[34,166704,166706,166709,166712,166715,166718],{"className":166705},[31676],[34,166707],{"className":166708,"style":32617},[31680],[34,166710,459],{"className":166711},[31685],[34,166713],{"className":166714,"style":31846},[31845],[34,166716,33239],{"className":166717},[31850],[34,166719],{"className":166720,"style":31846},[31845],[34,166722,166724,166727,166730,166733,166736],{"className":166723},[31676],[34,166725],{"className":166726,"style":32617},[31680],[34,166728,162188],{"className":166729},[31685],[34,166731],{"className":166732,"style":31846},[31845],[34,166734,33239],{"className":166735},[31850],[34,166737],{"className":166738,"style":31846},[31845],[34,166740,166742,166745,166748,166751,166754],{"className":166741},[31676],[34,166743],{"className":166744,"style":32617},[31680],[34,166746,162217],{"className":166747},[31685],[34,166749],{"className":166750,"style":31846},[31845],[34,166752,33239],{"className":166753},[31850],[34,166755],{"className":166756,"style":31846},[31845],[34,166758,166760,166763,166766,166769,166772],{"className":166759},[31676],[34,166761],{"className":166762,"style":32617},[31680],[34,166764,162183],{"className":166765},[31685],[34,166767],{"className":166768,"style":31846},[31845],[34,166770,33239],{"className":166771},[31850],[34,166773],{"className":166774,"style":31846},[31845],[34,166776,166778,166781,166784,166787,166790],{"className":166777},[31676],[34,166779],{"className":166780,"style":32617},[31680],[34,166782,162198],{"className":166783},[31685],[34,166785],{"className":166786,"style":31846},[31845],[34,166788,33239],{"className":166789},[31850],[34,166791],{"className":166792,"style":31846},[31845],[34,166794,166796,166799],{"className":166795},[31676],[34,166797],{"className":166798,"style":32617},[31680],[34,166800,162193],{"className":166801},[31685],[15,166803,166804],{},"总移动磁道数为：",[15,166806,166807],{},[34,166808,166810,166865],{"className":166809},[31642],[34,166811,166813],{"className":166812},[31646],[31648,166814,166815],{"xmlns":31650},[31652,166816,166817,166862],{},[31655,166818,166819,166821,166823,166825,166827,166829,166831,166833,166835,166837,166839,166841,166843,166845,166847,166849,166851,166853,166855,166857,166859],{},[31749,166820,165563],{},[31662,166822,223],{},[31749,166824,94703],{},[31662,166826,223],{},[31749,166828,164756],{},[31662,166830,223],{},[31749,166832,164836],{},[31662,166834,223],{},[31749,166836,3244],{},[31662,166838,223],{},[31749,166840,162188],{},[31662,166842,223],{},[31749,166844,3244],{},[31662,166846,223],{},[31749,166848,36732],{},[31662,166850,223],{},[31749,166852,95611],{},[31662,166854,223],{},[31749,166856,162753],{},[31662,166858,202],{},[31749,166860,166861],{},"286",[31665,166863,166864],{"encoding":31667},"36 + 23 + 14 + 17 + 10 + 110 + 10 + 9 + 18 + 39 = 286",[34,166866,166868,166886,166904,166922,166940,166958,166976,166994,167012,167030,167048],{"className":166867,"ariaHidden":758},[31672],[34,166869,166871,166874,166877,166880,166883],{"className":166870},[31676],[34,166872],{"className":166873,"style":35942},[31680],[34,166875,165563],{"className":166876},[31685],[34,166878],{"className":166879,"style":32604},[31845],[34,166881,223],{"className":166882},[32024],[34,166884],{"className":166885,"style":32604},[31845],[34,166887,166889,166892,166895,166898,166901],{"className":166888},[31676],[34,166890],{"className":166891,"style":35942},[31680],[34,166893,94703],{"className":166894},[31685],[34,166896],{"className":166897,"style":32604},[31845],[34,166899,223],{"className":166900},[32024],[34,166902],{"className":166903,"style":32604},[31845],[34,166905,166907,166910,166913,166916,166919],{"className":166906},[31676],[34,166908],{"className":166909,"style":35942},[31680],[34,166911,164756],{"className":166912},[31685],[34,166914],{"className":166915,"style":32604},[31845],[34,166917,223],{"className":166918},[32024],[34,166920],{"className":166921,"style":32604},[31845],[34,166923,166925,166928,166931,166934,166937],{"className":166924},[31676],[34,166926],{"className":166927,"style":35942},[31680],[34,166929,164836],{"className":166930},[31685],[34,166932],{"className":166933,"style":32604},[31845],[34,166935,223],{"className":166936},[32024],[34,166938],{"className":166939,"style":32604},[31845],[34,166941,166943,166946,166949,166952,166955],{"className":166942},[31676],[34,166944],{"className":166945,"style":35942},[31680],[34,166947,3244],{"className":166948},[31685],[34,166950],{"className":166951,"style":32604},[31845],[34,166953,223],{"className":166954},[32024],[34,166956],{"className":166957,"style":32604},[31845],[34,166959,166961,166964,166967,166970,166973],{"className":166960},[31676],[34,166962],{"className":166963,"style":35942},[31680],[34,166965,162188],{"className":166966},[31685],[34,166968],{"className":166969,"style":32604},[31845],[34,166971,223],{"className":166972},[32024],[34,166974],{"className":166975,"style":32604},[31845],[34,166977,166979,166982,166985,166988,166991],{"className":166978},[31676],[34,166980],{"className":166981,"style":35942},[31680],[34,166983,3244],{"className":166984},[31685],[34,166986],{"className":166987,"style":32604},[31845],[34,166989,223],{"className":166990},[32024],[34,166992],{"className":166993,"style":32604},[31845],[34,166995,166997,167000,167003,167006,167009],{"className":166996},[31676],[34,166998],{"className":166999,"style":35942},[31680],[34,167001,36732],{"className":167002},[31685],[34,167004],{"className":167005,"style":32604},[31845],[34,167007,223],{"className":167008},[32024],[34,167010],{"className":167011,"style":32604},[31845],[34,167013,167015,167018,167021,167024,167027],{"className":167014},[31676],[34,167016],{"className":167017,"style":35942},[31680],[34,167019,95611],{"className":167020},[31685],[34,167022],{"className":167023,"style":32604},[31845],[34,167025,223],{"className":167026},[32024],[34,167028],{"className":167029,"style":32604},[31845],[34,167031,167033,167036,167039,167042,167045],{"className":167032},[31676],[34,167034],{"className":167035,"style":32617},[31680],[34,167037,162753],{"className":167038},[31685],[34,167040],{"className":167041,"style":31846},[31845],[34,167043,202],{"className":167044},[31850],[34,167046],{"className":167047,"style":31846},[31845],[34,167049,167051,167054],{"className":167050},[31676],[34,167052],{"className":167053,"style":32617},[31680],[34,167055,166861],{"className":167056},[31685],[15,167058,167059],{},"考试按 Stallings 的表格风格，通常写：",[15,167061,167062],{},[34,167063,167065,167078],{"className":167064},[31642],[34,167066,167068],{"className":167067},[31646],[31648,167069,167070],{"xmlns":31650},[31652,167071,167072,167076],{},[31655,167073,167074],{},[31749,167075,166294],{},[31665,167077,166294],{"encoding":31667},[34,167079,167081],{"className":167080,"ariaHidden":758},[31672],[34,167082,167084,167087],{"className":167083},[31676],[34,167085],{"className":167086,"style":32617},[31680],[34,167088,166294],{"className":167089},[31685],[32188,167091],{},[253,167093,167095],{"id":167094},"第-1-题最终答案","第 1 题最终答案",[15,167097,167098],{},"FIFO：",[15,167100,167101],{},[34,167102,167104,167117],{"className":167103},[31642],[34,167105,167107],{"className":167106},[31646],[31648,167108,167109],{"xmlns":31650},[31652,167110,167111,167115],{},[31655,167112,167113],{},[31749,167114,163190],{},[31665,167116,163190],{"encoding":31667},[34,167118,167120],{"className":167119,"ariaHidden":758},[31672],[34,167121,167123,167126],{"className":167122},[31676],[34,167124],{"className":167125,"style":32617},[31680],[34,167127,163190],{"className":167128},[31685],[15,167130,167131],{},"SSTF：",[15,167133,167134],{},[34,167135,167137,167150],{"className":167136},[31642],[34,167138,167140],{"className":167139},[31646],[31648,167141,167142],{"xmlns":31650},[31652,167143,167144,167148],{},[31655,167145,167146],{},[31749,167147,164942],{},[31665,167149,164942],{"encoding":31667},[34,167151,167153],{"className":167152,"ariaHidden":758},[31672],[34,167154,167156,167159],{"className":167155},[31676],[34,167157],{"className":167158,"style":32617},[31680],[34,167160,164942],{"className":167161},[31685],[15,167163,167164],{},"SCAN：",[15,167166,167167],{},[34,167168,167170,167183],{"className":167169},[31642],[34,167171,167173],{"className":167172},[31646],[31648,167174,167175],{"xmlns":31650},[31652,167176,167177,167181],{},[31655,167178,167179],{},[31749,167180,166294],{},[31665,167182,166294],{"encoding":31667},[34,167184,167186],{"className":167185,"ariaHidden":758},[31672],[34,167187,167189,167192],{"className":167188},[31676],[34,167190],{"className":167191,"style":32617},[31680],[34,167193,166294],{"className":167194},[31685],[15,167196,167197],{},"如果严格要求 SCAN 到磁盘边界 0：",[15,167199,167200],{},[34,167201,167203,167216],{"className":167202},[31642],[34,167204,167206],{"className":167205},[31646],[31648,167207,167208],{"xmlns":31650},[31652,167209,167210,167214],{},[31655,167211,167212],{},[31749,167213,166861],{},[31665,167215,166861],{"encoding":31667},[34,167217,167219],{"className":167218,"ariaHidden":758},[31672],[34,167220,167222,167225],{"className":167221},[31676],[34,167223],{"className":167224,"style":32617},[31680],[34,167226,166861],{"className":167227},[31685],[32188,167229],{},[19,167231,167233],{"id":167232},"_2-中断驱动-io-中-cpu-花费多少时间处理中断","2. 中断驱动 I\u002FO 中 CPU 花费多少时间处理中断",[253,167235,162043],{"id":167236},"题目内容-1",[15,167238,167239],{},"Assume that the disk rotates at 360 rpm. A processor reads one sector from the disk using interrupt-driven I\u002FO, with one interrupt per byte. If it takes 2.5 µs to process each interrupt, what percentage of the time will the processor spend handling I\u002FO (disregard seek time)?",[15,167241,162049],{},[15,167243,167244,167245,167285],{},"假设磁盘转速为 360 rpm。处理器使用中断驱动 I\u002FO 从磁盘读取一个扇区，每读取 1 字节产生一次中断。如果处理一次中断需要 ",[34,167246,167248,167267],{"className":167247},[31642],[34,167249,167251],{"className":167250},[31646],[31648,167252,167253],{"xmlns":31650},[31652,167254,167255,167264],{},[31655,167256,167257,167260,167262],{},[31749,167258,167259],{},"2.5",[31658,167261,135107],{},[31658,167263,31747],{},[31665,167265,167266],{"encoding":31667},"2.5\\mu s",[34,167268,167270],{"className":167269,"ariaHidden":758},[31672],[34,167271,167273,167276,167279,167282],{"className":167272},[31676],[34,167274],{"className":167275,"style":40292},[31680],[34,167277,167259],{"className":167278},[31685],[34,167280,135107],{"className":167281},[31685,31686],[34,167283,31747],{"className":167284},[31685,31686],"，问处理器有百分之多少的时间用于处理中断？忽略寻道时间。",[15,167287,167288],{},"本题使用第 5 题给出的磁盘信息：",[15,167290,167291],{},"每个扇区大小：",[15,167293,167294,167323],{},[34,167295,167297,167311],{"className":167296},[31642],[34,167298,167300],{"className":167299},[31646],[31648,167301,167302],{"xmlns":31650},[31652,167303,167304,167309],{},[31655,167305,167306],{},[31749,167307,167308],{},"512",[31665,167310,167308],{"encoding":31667},[34,167312,167314],{"className":167313,"ariaHidden":758},[31672],[34,167315,167317,167320],{"className":167316},[31676],[34,167318],{"className":167319,"style":32617},[31680],[34,167321,167308],{"className":167322},[31685]," 字节",[15,167325,167326],{},"每个磁道：",[15,167328,167329,167358],{},[34,167330,167332,167346],{"className":167331},[31642],[34,167333,167335],{"className":167334},[31646],[31648,167336,167337],{"xmlns":31650},[31652,167338,167339,167344],{},[31655,167340,167341],{},[31749,167342,167343],{},"96",[31665,167345,167343],{"encoding":31667},[34,167347,167349],{"className":167348,"ariaHidden":758},[31672],[34,167350,167352,167355],{"className":167351},[31676],[34,167353],{"className":167354,"style":32617},[31680],[34,167356,167343],{"className":167357},[31685]," 个扇区",[32188,167360],{},[253,167362,162066],{"id":167363},"解题核心-1",[15,167365,167366],{},"每字节一次中断，所以读一个扇区会产生：",[15,167368,167369,167397],{},[34,167370,167372,167385],{"className":167371},[31642],[34,167373,167375],{"className":167374},[31646],[31648,167376,167377],{"xmlns":31650},[31652,167378,167379,167383],{},[31655,167380,167381],{},[31749,167382,167308],{},[31665,167384,167308],{"encoding":31667},[34,167386,167388],{"className":167387,"ariaHidden":758},[31672],[34,167389,167391,167394],{"className":167390},[31676],[34,167392],{"className":167393,"style":32617},[31680],[34,167395,167308],{"className":167396},[31685]," 次中断",[15,167399,167400],{},"每次中断处理时间：",[15,167402,167403],{},[34,167404,167406,167423],{"className":167405},[31642],[34,167407,167409],{"className":167408},[31646],[31648,167410,167411],{"xmlns":31650},[31652,167412,167413,167421],{},[31655,167414,167415,167417,167419],{},[31749,167416,167259],{},[31658,167418,135107],{},[31658,167420,31747],{},[31665,167422,167266],{"encoding":31667},[34,167424,167426],{"className":167425,"ariaHidden":758},[31672],[34,167427,167429,167432,167435,167438],{"className":167428},[31676],[34,167430],{"className":167431,"style":40292},[31680],[34,167433,167259],{"className":167434},[31685],[34,167436,135107],{"className":167437},[31685,31686],[34,167439,31747],{"className":167440},[31685,31686],[15,167442,167443],{},"所以处理中断总时间：",[15,167445,167446],{},[34,167447,167449,167480],{"className":167448},[31642],[34,167450,167452],{"className":167451},[31646],[31648,167453,167454],{"xmlns":31650},[31652,167455,167456,167477],{},[31655,167457,167458,167460,167462,167464,167466,167468,167470,167473,167475],{},[31749,167459,167308],{},[31662,167461,42071],{},[31749,167463,167259],{},[31658,167465,135107],{},[31658,167467,31747],{},[31662,167469,202],{},[31749,167471,167472],{},"1280",[31658,167474,135107],{},[31658,167476,31747],{},[31665,167478,167479],{"encoding":31667},"512 \\times 2.5\\mu s = 1280\\mu s",[34,167481,167483,167501,167525],{"className":167482,"ariaHidden":758},[31672],[34,167484,167486,167489,167492,167495,167498],{"className":167485},[31676],[34,167487],{"className":167488,"style":35942},[31680],[34,167490,167308],{"className":167491},[31685],[34,167493],{"className":167494,"style":32604},[31845],[34,167496,42071],{"className":167497},[32024],[34,167499],{"className":167500,"style":32604},[31845],[34,167502,167504,167507,167510,167513,167516,167519,167522],{"className":167503},[31676],[34,167505],{"className":167506,"style":40292},[31680],[34,167508,167259],{"className":167509},[31685],[34,167511,135107],{"className":167512},[31685,31686],[34,167514,31747],{"className":167515},[31685,31686],[34,167517],{"className":167518,"style":31846},[31845],[34,167520,202],{"className":167521},[31850],[34,167523],{"className":167524,"style":31846},[31845],[34,167526,167528,167531,167534,167537],{"className":167527},[31676],[34,167529],{"className":167530,"style":40292},[31680],[34,167532,167472],{"className":167533},[31685],[34,167535,135107],{"className":167536},[31685,31686],[34,167538,31747],{"className":167539},[31685,31686],[15,167541,139415],{},[15,167543,167544],{},[34,167545,167547,167566],{"className":167546},[31642],[34,167548,167550],{"className":167549},[31646],[31648,167551,167552],{"xmlns":31650},[31652,167553,167554,167563],{},[31655,167555,167556,167559,167561],{},[31749,167557,167558],{},"1.28",[31658,167560,31759],{},[31658,167562,31747],{},[31665,167564,167565],{"encoding":31667},"1.28ms",[34,167567,167569],{"className":167568,"ariaHidden":758},[31672],[34,167570,167572,167575,167578,167581],{"className":167571},[31676],[34,167573],{"className":167574,"style":32617},[31680],[34,167576,167558],{"className":167577},[31685],[34,167579,31759],{"className":167580},[31685,31686],[34,167582,31747],{"className":167583},[31685,31686],[32188,167585],{},[253,167587,167588],{"id":167588},"计算磁盘传输一个扇区的时间",[15,167590,167591],{},"磁盘转速：",[15,167593,167594],{},[34,167595,167597,167617],{"className":167596},[31642],[34,167598,167600],{"className":167599},[31646],[31648,167601,167602],{"xmlns":31650},[31652,167603,167604,167614],{},[31655,167605,167606,167608,167610,167612],{},[31749,167607,73878],{},[31658,167609,34395],{},[31658,167611,15],{},[31658,167613,31759],{},[31665,167615,167616],{"encoding":31667},"360rpm",[34,167618,167620],{"className":167619,"ariaHidden":758},[31672],[34,167621,167623,167626,167629,167632,167635],{"className":167622},[31676],[34,167624],{"className":167625,"style":40292},[31680],[34,167627,73878],{"className":167628},[31685],[34,167630,34395],{"className":167631,"style":34452},[31685,31686],[34,167633,15],{"className":167634},[31685,31686],[34,167636,31759],{"className":167637},[31685,31686],[15,167639,167640],{},"意思是每分钟转 360 圈。",[15,167642,167643],{},"每秒转数：",[15,167645,167646,167703],{},[34,167647,167649,167672],{"className":167648},[31642],[34,167650,167652],{"className":167651},[31646],[31648,167653,167654],{"xmlns":31650},[31652,167655,167656,167669],{},[31655,167657,167658,167660,167662,167665,167667],{},[31749,167659,73878],{},[31658,167661,38080],{"mathvariant":38079},[31749,167663,167664],{},"60",[31662,167666,202],{},[31749,167668,36656],{},[31665,167670,167671],{"encoding":31667},"360 \u002F 60 = 6",[34,167673,167675,167694],{"className":167674,"ariaHidden":758},[31672],[34,167676,167678,167681,167685,167688,167691],{"className":167677},[31676],[34,167679],{"className":167680,"style":31774},[31680],[34,167682,167684],{"className":167683},[31685],"360\u002F60",[34,167686],{"className":167687,"style":31846},[31845],[34,167689,202],{"className":167690},[31850],[34,167692],{"className":167693,"style":31846},[31845],[34,167695,167697,167700],{"className":167696},[31676],[34,167698],{"className":167699,"style":32617},[31680],[34,167701,36656],{"className":167702},[31685]," 转\u002F秒",[15,167705,167706],{},"所以一圈时间：",[15,167708,167709],{},[34,167710,167712,167739],{"className":167711},[31642],[34,167713,167715],{"className":167714},[31646],[31648,167716,167717],{"xmlns":31650},[31652,167718,167719,167736],{},[31655,167720,167721,167723,167725,167727,167729,167731,167734],{},[31749,167722,640],{},[31658,167724,38080],{"mathvariant":38079},[31749,167726,36656],{},[31658,167728,31747],{},[31662,167730,202],{},[31749,167732,167733],{},"0.1667",[31658,167735,31747],{},[31665,167737,167738],{"encoding":31667},"1 \u002F 6s = 0.1667s",[34,167740,167742,167764],{"className":167741,"ariaHidden":758},[31672],[34,167743,167745,167748,167752,167755,167758,167761],{"className":167744},[31676],[34,167746],{"className":167747,"style":31774},[31680],[34,167749,167751],{"className":167750},[31685],"1\u002F6",[34,167753,31747],{"className":167754},[31685,31686],[34,167756],{"className":167757,"style":31846},[31845],[34,167759,202],{"className":167760},[31850],[34,167762],{"className":167763,"style":31846},[31845],[34,167765,167767,167770,167773],{"className":167766},[31676],[34,167768],{"className":167769,"style":32617},[31680],[34,167771,167733],{"className":167772},[31685],[34,167774,31747],{"className":167775},[31685,31686],[15,167777,167778],{},"换成毫秒：",[15,167780,167781],{},[34,167782,167784,167809],{"className":167783},[31642],[34,167785,167787],{"className":167786},[31646],[31648,167788,167789],{"xmlns":31650},[31652,167790,167791,167806],{},[31655,167792,167793,167795,167797,167799,167802,167804],{},[31749,167794,167733],{},[31658,167796,31747],{},[31662,167798,202],{},[31749,167800,167801],{},"166.7",[31658,167803,31759],{},[31658,167805,31747],{},[31665,167807,167808],{"encoding":31667},"0.1667s = 166.7ms",[34,167810,167812,167833],{"className":167811,"ariaHidden":758},[31672],[34,167813,167815,167818,167821,167824,167827,167830],{"className":167814},[31676],[34,167816],{"className":167817,"style":32617},[31680],[34,167819,167733],{"className":167820},[31685],[34,167822,31747],{"className":167823},[31685,31686],[34,167825],{"className":167826,"style":31846},[31845],[34,167828,202],{"className":167829},[31850],[34,167831],{"className":167832,"style":31846},[31845],[34,167834,167836,167839,167842,167845],{"className":167835},[31676],[34,167837],{"className":167838,"style":32617},[31680],[34,167840,167801],{"className":167841},[31685],[34,167843,31759],{"className":167844},[31685,31686],[34,167846,31747],{"className":167847},[31685,31686],[15,167849,167850],{},"每个磁道有 96 个扇区，所以一个扇区经过磁头的时间是：",[15,167852,167853],{},[34,167854,167856,167887],{"className":167855},[31642],[34,167857,167859],{"className":167858},[31646],[31648,167860,167861],{"xmlns":31650},[31652,167862,167863,167884],{},[31655,167864,167865,167867,167869,167871,167873,167875,167877,167880,167882],{},[31749,167866,167801],{},[31658,167868,31759],{},[31658,167870,31747],{},[31658,167872,38080],{"mathvariant":38079},[31749,167874,167343],{},[31662,167876,202],{},[31749,167878,167879],{},"1.736",[31658,167881,31759],{},[31658,167883,31747],{},[31665,167885,167886],{"encoding":31667},"166.7ms \u002F 96 = 1.736ms",[34,167888,167890,167918],{"className":167889,"ariaHidden":758},[31672],[34,167891,167893,167896,167899,167902,167905,167909,167912,167915],{"className":167892},[31676],[34,167894],{"className":167895,"style":31774},[31680],[34,167897,167801],{"className":167898},[31685],[34,167900,31759],{"className":167901},[31685,31686],[34,167903,31747],{"className":167904},[31685,31686],[34,167906,167908],{"className":167907},[31685],"\u002F96",[34,167910],{"className":167911,"style":31846},[31845],[34,167913,202],{"className":167914},[31850],[34,167916],{"className":167917,"style":31846},[31845],[34,167919,167921,167924,167927,167930],{"className":167920},[31676],[34,167922],{"className":167923,"style":32617},[31680],[34,167925,167879],{"className":167926},[31685],[34,167928,31759],{"className":167929},[31685,31686],[34,167931,31747],{"className":167932},[31685,31686],[15,167934,139415],{},[15,167936,167937],{},[34,167938,167940,167959],{"className":167939},[31642],[34,167941,167943],{"className":167942},[31646],[31648,167944,167945],{"xmlns":31650},[31652,167946,167947,167956],{},[31655,167948,167949,167952,167954],{},[31749,167950,167951],{},"1736",[31658,167953,135107],{},[31658,167955,31747],{},[31665,167957,167958],{"encoding":31667},"1736\\mu s",[34,167960,167962],{"className":167961,"ariaHidden":758},[31672],[34,167963,167965,167968,167971,167974],{"className":167964},[31676],[34,167966],{"className":167967,"style":40292},[31680],[34,167969,167951],{"className":167970},[31685],[34,167972,135107],{"className":167973},[31685,31686],[34,167975,31747],{"className":167976},[31685,31686],[32188,167978],{},[253,167980,167982],{"id":167981},"计算-cpu-用于处理中断的比例","计算 CPU 用于处理中断的比例",[15,167984,167985],{},"比例为：",[15,167987,167988],{},[34,167989,167991,168017],{"className":167990},[31642],[34,167992,167994],{"className":167993},[31646],[31648,167995,167996],{"xmlns":31650},[31652,167997,167998,168014],{},[31655,167999,168000,168008,168010,168012],{},[41453,168001,168002,168005],{},[39361,168003,168004],{},"中断处理总时间",[39361,168006,168007],{},"一个扇区传输时间",[31662,168009,42071],{},[31749,168011,162173],{},[31658,168013,119426],{"mathvariant":38079},[31665,168015,168016],{"encoding":31667},"\\frac{\\text{中断处理总时间}}{\\text{一个扇区传输时间}} \\times 100\\%",[34,168018,168020,168111],{"className":168019,"ariaHidden":758},[31672],[34,168021,168023,168027,168102,168105,168108],{"className":168022},[31676],[34,168024],{"className":168025,"style":168026},[31680],"height:1.2173em;vertical-align:-0.345em;",[34,168028,168030,168033,168099],{"className":168029},[31685],[34,168031],{"className":168032},[31782,41488],[34,168034,168036],{"className":168035},[41453],[34,168037,168039,168091],{"className":168038},[31796,31797],[34,168040,168042,168088],{"className":168041},[31801],[34,168043,168046,168063,168071],{"className":168044,"style":168045},[31805],"height:0.8723em;",[34,168047,168048,168051],{"style":41504},[34,168049],{"className":168050,"style":41508},[31813],[34,168052,168054],{"className":168053},[31818,31819,31820,31821],[34,168055,168057],{"className":168056},[31685,31821],[34,168058,168060],{"className":168059},[31685,2638,31821],[34,168061,168007],{"className":168062},[31685,105462,31821],[34,168064,168065,168068],{"style":41520},[34,168066],{"className":168067,"style":41508},[31813],[34,168069],{"className":168070,"style":41528},[41527],[34,168072,168073,168076],{"style":46771},[34,168074],{"className":168075,"style":41508},[31813],[34,168077,168079],{"className":168078},[31818,31819,31820,31821],[34,168080,168082],{"className":168081},[31685,31821],[34,168083,168085],{"className":168084},[31685,2638,31821],[34,168086,168004],{"className":168087},[31685,105462,31821],[34,168089,31829],{"className":168090},[31828],[34,168092,168094],{"className":168093},[31801],[34,168095,168097],{"className":168096,"style":41619},[31805],[34,168098],{},[34,168100],{"className":168101},[31690,41488],[34,168103],{"className":168104,"style":32604},[31845],[34,168106,42071],{"className":168107},[32024],[34,168109],{"className":168110,"style":32604},[31845],[34,168112,168114,168118],{"className":168113},[31676],[34,168115],{"className":168116,"style":168117},[31680],"height:0.8056em;vertical-align:-0.0556em;",[34,168119,168121],{"className":168120},[31685],"100%",[15,168123,168124],{},"代入：",[15,168126,168127],{},[34,168128,168130,168161],{"className":168129},[31642],[34,168131,168133],{"className":168132},[31646],[31648,168134,168135],{"xmlns":31650},[31652,168136,168137,168158],{},[31655,168138,168139,168145,168147,168149,168151,168153,168156],{},[41453,168140,168141,168143],{},[31749,168142,167472],{},[31749,168144,167951],{},[31662,168146,42071],{},[31749,168148,162173],{},[31658,168150,119426],{"mathvariant":38079},[31662,168152,202],{},[31749,168154,168155],{},"73.7",[31658,168157,119426],{"mathvariant":38079},[31665,168159,168160],{"encoding":31667},"\\frac{1280}{1736} \\times 100\\% = 73.7\\%",[34,168162,168164,168247,168265],{"className":168163,"ariaHidden":758},[31672],[34,168165,168167,168170,168238,168241,168244],{"className":168166},[31676],[34,168168],{"className":168169,"style":46727},[31680],[34,168171,168173,168176,168235],{"className":168172},[31685],[34,168174],{"className":168175},[31782,41488],[34,168177,168179],{"className":168178},[41453],[34,168180,168182,168227],{"className":168181},[31796,31797],[34,168183,168185,168224],{"className":168184},[31801],[34,168186,168188,168202,168210],{"className":168187,"style":46746},[31805],[34,168189,168190,168193],{"style":41504},[34,168191],{"className":168192,"style":41508},[31813],[34,168194,168196],{"className":168195},[31818,31819,31820,31821],[34,168197,168199],{"className":168198},[31685,31821],[34,168200,167951],{"className":168201},[31685,31821],[34,168203,168204,168207],{"style":41520},[34,168205],{"className":168206,"style":41508},[31813],[34,168208],{"className":168209,"style":41528},[41527],[34,168211,168212,168215],{"style":46771},[34,168213],{"className":168214,"style":41508},[31813],[34,168216,168218],{"className":168217},[31818,31819,31820,31821],[34,168219,168221],{"className":168220},[31685,31821],[34,168222,167472],{"className":168223},[31685,31821],[34,168225,31829],{"className":168226},[31828],[34,168228,168230],{"className":168229},[31801],[34,168231,168233],{"className":168232,"style":41619},[31805],[34,168234],{},[34,168236],{"className":168237},[31690,41488],[34,168239],{"className":168240,"style":32604},[31845],[34,168242,42071],{"className":168243},[32024],[34,168245],{"className":168246,"style":32604},[31845],[34,168248,168250,168253,168256,168259,168262],{"className":168249},[31676],[34,168251],{"className":168252,"style":168117},[31680],[34,168254,168121],{"className":168255},[31685],[34,168257],{"className":168258,"style":31846},[31845],[34,168260,202],{"className":168261},[31850],[34,168263],{"className":168264,"style":31846},[31845],[34,168266,168268,168271],{"className":168267},[31676],[34,168269],{"className":168270,"style":168117},[31680],[34,168272,168274],{"className":168273},[31685],"73.7%",[15,168276,159962],{},[15,168278,168279],{},"CPU 大约有：",[15,168281,168282],{},[34,168283,168285,168301],{"className":168284},[31642],[34,168286,168288],{"className":168287},[31646],[31648,168289,168290],{"xmlns":31650},[31652,168291,168292,168298],{},[31655,168293,168294,168296],{},[31749,168295,168155],{},[31658,168297,119426],{"mathvariant":38079},[31665,168299,168300],{"encoding":31667},"73.7\\%",[34,168302,168304],{"className":168303,"ariaHidden":758},[31672],[34,168305,168307,168310],{"className":168306},[31676],[34,168308],{"className":168309,"style":168117},[31680],[34,168311,168274],{"className":168312},[31685],[15,168314,168315],{},"的时间花在处理中断上。",[32188,168317],{},[253,168319,166511],{"id":168320},"补充说明-1",[15,168322,168323],{},"这个题通常只把“扇区传输时间”作为分母，不把平均旋转等待时间算进去。",[15,168325,168326],{},"如果老师把“随机读一个扇区”理解成还要包括平均旋转延迟，则平均旋转延迟为半圈：",[15,168328,168329],{},[34,168330,168332,168363],{"className":168331},[31642],[34,168333,168335],{"className":168334},[31646],[31648,168336,168337],{"xmlns":31650},[31652,168338,168339,168360],{},[31655,168340,168341,168343,168345,168347,168349,168351,168353,168356,168358],{},[31749,168342,167801],{},[31658,168344,31759],{},[31658,168346,31747],{},[31658,168348,38080],{"mathvariant":38079},[31749,168350,768],{},[31662,168352,202],{},[31749,168354,168355],{},"83.3",[31658,168357,31759],{},[31658,168359,31747],{},[31665,168361,168362],{"encoding":31667},"166.7ms \u002F 2 = 83.3ms",[34,168364,168366,168393],{"className":168365,"ariaHidden":758},[31672],[34,168367,168369,168372,168375,168378,168381,168384,168387,168390],{"className":168368},[31676],[34,168370],{"className":168371,"style":31774},[31680],[34,168373,167801],{"className":168374},[31685],[34,168376,31759],{"className":168377},[31685,31686],[34,168379,31747],{"className":168380},[31685,31686],[34,168382,38149],{"className":168383},[31685],[34,168385],{"className":168386,"style":31846},[31845],[34,168388,202],{"className":168389},[31850],[34,168391],{"className":168392,"style":31846},[31845],[34,168394,168396,168399,168402,168405],{"className":168395},[31676],[34,168397],{"className":168398,"style":32617},[31680],[34,168400,168355],{"className":168401},[31685],[34,168403,31759],{"className":168404},[31685,31686],[34,168406,31747],{"className":168407},[31685,31686],[15,168409,168410],{},"总时间约为：",[15,168412,168413],{},[34,168414,168416,168451],{"className":168415},[31642],[34,168417,168419],{"className":168418},[31646],[31648,168420,168421],{"xmlns":31650},[31652,168422,168423,168448],{},[31655,168424,168425,168427,168429,168431,168433,168435,168437,168439,168441,168444,168446],{},[31749,168426,168355],{},[31658,168428,31759],{},[31658,168430,31747],{},[31662,168432,223],{},[31749,168434,167879],{},[31658,168436,31759],{},[31658,168438,31747],{},[31662,168440,202],{},[31749,168442,168443],{},"85.036",[31658,168445,31759],{},[31658,168447,31747],{},[31665,168449,168450],{"encoding":31667},"83.3ms + 1.736ms = 85.036ms",[34,168452,168454,168478,168502],{"className":168453,"ariaHidden":758},[31672],[34,168455,168457,168460,168463,168466,168469,168472,168475],{"className":168456},[31676],[34,168458],{"className":168459,"style":35942},[31680],[34,168461,168355],{"className":168462},[31685],[34,168464,31759],{"className":168465},[31685,31686],[34,168467,31747],{"className":168468},[31685,31686],[34,168470],{"className":168471,"style":32604},[31845],[34,168473,223],{"className":168474},[32024],[34,168476],{"className":168477,"style":32604},[31845],[34,168479,168481,168484,168487,168490,168493,168496,168499],{"className":168480},[31676],[34,168482],{"className":168483,"style":32617},[31680],[34,168485,167879],{"className":168486},[31685],[34,168488,31759],{"className":168489},[31685,31686],[34,168491,31747],{"className":168492},[31685,31686],[34,168494],{"className":168495,"style":31846},[31845],[34,168497,202],{"className":168498},[31850],[34,168500],{"className":168501,"style":31846},[31845],[34,168503,168505,168508,168511,168514],{"className":168504},[31676],[34,168506],{"className":168507,"style":32617},[31680],[34,168509,168443],{"className":168510},[31685],[34,168512,31759],{"className":168513},[31685,31686],[34,168515,31747],{"className":168516},[31685,31686],[15,168518,168519],{},"此时比例为：",[15,168521,168522],{},[34,168523,168525,168569],{"className":168524},[31642],[34,168526,168528],{"className":168527},[31646],[31648,168529,168530],{"xmlns":31650},[31652,168531,168532,168566],{},[31655,168533,168534,168552,168554,168556,168558,168561,168564],{},[41453,168535,168536,168544],{},[31655,168537,168538,168540,168542],{},[31749,168539,167558],{},[31658,168541,31759],{},[31658,168543,31747],{},[31655,168545,168546,168548,168550],{},[31749,168547,168443],{},[31658,168549,31759],{},[31658,168551,31747],{},[31662,168553,42071],{},[31749,168555,162173],{},[31658,168557,119426],{"mathvariant":38079},[31662,168559,168560],{},"≈",[31749,168562,168563],{},"1.50",[31658,168565,119426],{"mathvariant":38079},[31665,168567,168568],{"encoding":31667},"\\frac{1.28ms}{85.036ms} \\times 100\\% \\approx 1.50\\%",[34,168570,168572,168667,168685],{"className":168571,"ariaHidden":758},[31672],[34,168573,168575,168578,168658,168661,168664],{"className":168574},[31676],[34,168576],{"className":168577,"style":46727},[31680],[34,168579,168581,168584,168655],{"className":168580},[31685],[34,168582],{"className":168583},[31782,41488],[34,168585,168587],{"className":168586},[41453],[34,168588,168590,168647],{"className":168589},[31796,31797],[34,168591,168593,168644],{"className":168592},[31801],[34,168594,168596,168616,168624],{"className":168595,"style":46746},[31805],[34,168597,168598,168601],{"style":41504},[34,168599],{"className":168600,"style":41508},[31813],[34,168602,168604],{"className":168603},[31818,31819,31820,31821],[34,168605,168607,168610,168613],{"className":168606},[31685,31821],[34,168608,168443],{"className":168609},[31685,31821],[34,168611,31759],{"className":168612},[31685,31686,31821],[34,168614,31747],{"className":168615},[31685,31686,31821],[34,168617,168618,168621],{"style":41520},[34,168619],{"className":168620,"style":41508},[31813],[34,168622],{"className":168623,"style":41528},[41527],[34,168625,168626,168629],{"style":46771},[34,168627],{"className":168628,"style":41508},[31813],[34,168630,168632],{"className":168631},[31818,31819,31820,31821],[34,168633,168635,168638,168641],{"className":168634},[31685,31821],[34,168636,167558],{"className":168637},[31685,31821],[34,168639,31759],{"className":168640},[31685,31686,31821],[34,168642,31747],{"className":168643},[31685,31686,31821],[34,168645,31829],{"className":168646},[31828],[34,168648,168650],{"className":168649},[31801],[34,168651,168653],{"className":168652,"style":41619},[31805],[34,168654],{},[34,168656],{"className":168657},[31690,41488],[34,168659],{"className":168660,"style":32604},[31845],[34,168662,42071],{"className":168663},[32024],[34,168665],{"className":168666,"style":32604},[31845],[34,168668,168670,168673,168676,168679,168682],{"className":168669},[31676],[34,168671],{"className":168672,"style":168117},[31680],[34,168674,168121],{"className":168675},[31685],[34,168677],{"className":168678,"style":31846},[31845],[34,168680,168560],{"className":168681},[31850],[34,168683],{"className":168684,"style":31846},[31845],[34,168686,168688,168691],{"className":168687},[31676],[34,168689],{"className":168690,"style":168117},[31680],[34,168692,168694],{"className":168693},[31685],"1.50%",[15,168696,168697],{},"但 Stallings 这类题通常考的是 I\u002FO 传输期间中断开销，所以主要答案写：",[15,168699,168700],{},[34,168701,168703,168718],{"className":168702},[31642],[34,168704,168706],{"className":168705},[31646],[31648,168707,168708],{"xmlns":31650},[31652,168709,168710,168716],{},[31655,168711,168712,168714],{},[31749,168713,168155],{},[31658,168715,119426],{"mathvariant":38079},[31665,168717,168300],{"encoding":31667},[34,168719,168721],{"className":168720,"ariaHidden":758},[31672],[34,168722,168724,168727],{"className":168723},[31676],[34,168725],{"className":168726,"style":168117},[31680],[34,168728,168274],{"className":168729},[31685],[32188,168731],{},[253,168733,168735],{"id":168734},"第-2-题最终答案","第 2 题最终答案",[15,168737,168738],{},"CPU 处理中断的时间比例约为：",[15,168740,168741],{},[34,168742,168744,168759],{"className":168743},[31642],[34,168745,168747],{"className":168746},[31646],[31648,168748,168749],{"xmlns":31650},[31652,168750,168751,168757],{},[31655,168752,168753,168755],{},[31749,168754,168155],{},[31658,168756,119426],{"mathvariant":38079},[31665,168758,168300],{"encoding":31667},[34,168760,168762],{"className":168761,"ariaHidden":758},[31672],[34,168763,168765,168768],{"className":168764},[31676],[34,168766],{"className":168767,"style":168117},[31680],[34,168769,168274],{"className":168770},[31685],[32188,168772],{},[19,168774,168776],{"id":168775},"_3-计算最大聚合-io-传输速率","3. 计算最大聚合 I\u002FO 传输速率",[253,168778,162043],{"id":168779},"题目内容-2",[15,168781,168782],{},"A 32-bit computer has two selector channels and one multiplexor channel. Each selector channel supports two magnetic disk and two magnetic tape units. The multiplexor channel has two line printers, two card readers, and ten VDT terminals connected to it. Assume the following transfer rates:",[15,168784,168785],{},"Disk drive: 800 Kbytes\u002Fs",[15,168787,168788],{},"Magnetic tape drive: 200 Kbytes\u002Fs",[15,168790,168791],{},"Line printer: 6.6 Kbytes\u002Fs",[15,168793,168794],{},"Card reader: 1.2 Kbytes\u002Fs",[15,168796,168797],{},"VDT: 1 Kbytes\u002Fs",[15,168799,168800],{},"Estimate the maximum aggregate I\u002FO transfer rate in this system.",[15,168802,162049],{},[15,168804,168805],{},"一台 32 位计算机有两个选择通道和一个多路复用通道。每个选择通道连接两个磁盘驱动器和两个磁带机。多路复用通道连接两个行式打印机、两个读卡机和十个 VDT 终端。给出各设备传输速率，求系统最大总 I\u002FO 传输速率。",[32188,168807],{},[253,168809,162066],{"id":168810},"解题核心-2",[15,168812,168813],{},"这里要区分两种通道：",[2230,168815,168816],{},[108,168817,168818],{},"选择通道 selector channel",[15,168820,168821],{},"选择通道通常用于高速设备，但一次只能服务一个设备。",[15,168823,168824],{},"所以一个选择通道的最大传输速率不是把所有设备加起来，而是取它能服务的最快设备速率。",[15,168826,168827],{},"每个选择通道连接：",[15,168829,168830,168831],{},"2 个磁盘，每个 ",[34,168832,168834,168864],{"className":168833},[31642],[34,168835,168837],{"className":168836},[31646],[31648,168838,168839],{"xmlns":31650},[31652,168840,168841,168861],{},[31655,168842,168843,168845,168847,168849,168851,168853,168855,168857,168859],{},[31749,168844,100234],{},[31658,168846,38691],{},[31658,168848,1697],{},[31658,168850,67764],{},[31658,168852,33249],{},[31658,168854,33244],{},[31658,168856,31747],{},[31658,168858,38080],{"mathvariant":38079},[31658,168860,31747],{},[31665,168862,168863],{"encoding":31667},"800Kbytes\u002Fs",[34,168865,168867],{"className":168866,"ariaHidden":758},[31672],[34,168868,168870,168873,168876,168879,168882,168885,168888,168892,168895],{"className":168869},[31676],[34,168871],{"className":168872,"style":31774},[31680],[34,168874,100234],{"className":168875},[31685],[34,168877,38691],{"className":168878,"style":38706},[31685,31686],[34,168880,1697],{"className":168881},[31685,31686],[34,168883,67764],{"className":168884,"style":32062},[31685,31686],[34,168886,33249],{"className":168887},[31685,31686],[34,168889,168891],{"className":168890},[31685,31686],"es",[34,168893,38080],{"className":168894},[31685],[34,168896,31747],{"className":168897},[31685,31686],[15,168899,168900,168901],{},"2 个磁带机，每个 ",[34,168902,168904,168935],{"className":168903},[31642],[34,168905,168907],{"className":168906},[31646],[31648,168908,168909],{"xmlns":31650},[31652,168910,168911,168932],{},[31655,168912,168913,168916,168918,168920,168922,168924,168926,168928,168930],{},[31749,168914,168915],{},"200",[31658,168917,38691],{},[31658,168919,1697],{},[31658,168921,67764],{},[31658,168923,33249],{},[31658,168925,33244],{},[31658,168927,31747],{},[31658,168929,38080],{"mathvariant":38079},[31658,168931,31747],{},[31665,168933,168934],{"encoding":31667},"200Kbytes\u002Fs",[34,168936,168938],{"className":168937,"ariaHidden":758},[31672],[34,168939,168941,168944,168947,168950,168953,168956,168959,168962,168965],{"className":168940},[31676],[34,168942],{"className":168943,"style":31774},[31680],[34,168945,168915],{"className":168946},[31685],[34,168948,38691],{"className":168949,"style":38706},[31685,31686],[34,168951,1697],{"className":168952},[31685,31686],[34,168954,67764],{"className":168955,"style":32062},[31685,31686],[34,168957,33249],{"className":168958},[31685,31686],[34,168960,168891],{"className":168961},[31685,31686],[34,168963,38080],{"className":168964},[31685],[34,168966,31747],{"className":168967},[31685,31686],[15,168969,168970],{},"最快设备是磁盘：",[15,168972,168973],{},[34,168974,168976,169005],{"className":168975},[31642],[34,168977,168979],{"className":168978},[31646],[31648,168980,168981],{"xmlns":31650},[31652,168982,168983,169003],{},[31655,168984,168985,168987,168989,168991,168993,168995,168997,168999,169001],{},[31749,168986,100234],{},[31658,168988,38691],{},[31658,168990,1697],{},[31658,168992,67764],{},[31658,168994,33249],{},[31658,168996,33244],{},[31658,168998,31747],{},[31658,169000,38080],{"mathvariant":38079},[31658,169002,31747],{},[31665,169004,168863],{"encoding":31667},[34,169006,169008],{"className":169007,"ariaHidden":758},[31672],[34,169009,169011,169014,169017,169020,169023,169026,169029,169032,169035],{"className":169010},[31676],[34,169012],{"className":169013,"style":31774},[31680],[34,169015,100234],{"className":169016},[31685],[34,169018,38691],{"className":169019,"style":38706},[31685,31686],[34,169021,1697],{"className":169022},[31685,31686],[34,169024,67764],{"className":169025,"style":32062},[31685,31686],[34,169027,33249],{"className":169028},[31685,31686],[34,169030,168891],{"className":169031},[31685,31686],[34,169033,38080],{"className":169034},[31685],[34,169036,31747],{"className":169037},[31685,31686],[15,169039,169040],{},"所以每个选择通道最大传输速率为：",[15,169042,169043],{},[34,169044,169046,169075],{"className":169045},[31642],[34,169047,169049],{"className":169048},[31646],[31648,169050,169051],{"xmlns":31650},[31652,169052,169053,169073],{},[31655,169054,169055,169057,169059,169061,169063,169065,169067,169069,169071],{},[31749,169056,100234],{},[31658,169058,38691],{},[31658,169060,1697],{},[31658,169062,67764],{},[31658,169064,33249],{},[31658,169066,33244],{},[31658,169068,31747],{},[31658,169070,38080],{"mathvariant":38079},[31658,169072,31747],{},[31665,169074,168863],{"encoding":31667},[34,169076,169078],{"className":169077,"ariaHidden":758},[31672],[34,169079,169081,169084,169087,169090,169093,169096,169099,169102,169105],{"className":169080},[31676],[34,169082],{"className":169083,"style":31774},[31680],[34,169085,100234],{"className":169086},[31685],[34,169088,38691],{"className":169089,"style":38706},[31685,31686],[34,169091,1697],{"className":169092},[31685,31686],[34,169094,67764],{"className":169095,"style":32062},[31685,31686],[34,169097,33249],{"className":169098},[31685,31686],[34,169100,168891],{"className":169101},[31685,31686],[34,169103,38080],{"className":169104},[31685],[34,169106,31747],{"className":169107},[31685,31686],[15,169109,169110],{},"有两个选择通道：",[15,169112,169113],{},[34,169114,169116,169155],{"className":169115},[31642],[34,169117,169119],{"className":169118},[31646],[31648,169120,169121],{"xmlns":31650},[31652,169122,169123,169152],{},[31655,169124,169125,169127,169129,169131,169133,169136,169138,169140,169142,169144,169146,169148,169150],{},[31749,169126,768],{},[31662,169128,42071],{},[31749,169130,100234],{},[31662,169132,202],{},[31749,169134,169135],{},"1600",[31658,169137,38691],{},[31658,169139,1697],{},[31658,169141,67764],{},[31658,169143,33249],{},[31658,169145,33244],{},[31658,169147,31747],{},[31658,169149,38080],{"mathvariant":38079},[31658,169151,31747],{},[31665,169153,169154],{"encoding":31667},"2 \\times 800 = 1600Kbytes\u002Fs",[34,169156,169158,169176,169194],{"className":169157,"ariaHidden":758},[31672],[34,169159,169161,169164,169167,169170,169173],{"className":169160},[31676],[34,169162],{"className":169163,"style":35942},[31680],[34,169165,768],{"className":169166},[31685],[34,169168],{"className":169169,"style":32604},[31845],[34,169171,42071],{"className":169172},[32024],[34,169174],{"className":169175,"style":32604},[31845],[34,169177,169179,169182,169185,169188,169191],{"className":169178},[31676],[34,169180],{"className":169181,"style":32617},[31680],[34,169183,100234],{"className":169184},[31685],[34,169186],{"className":169187,"style":31846},[31845],[34,169189,202],{"className":169190},[31850],[34,169192],{"className":169193,"style":31846},[31845],[34,169195,169197,169200,169203,169206,169209,169212,169215,169218,169221],{"className":169196},[31676],[34,169198],{"className":169199,"style":31774},[31680],[34,169201,169135],{"className":169202},[31685],[34,169204,38691],{"className":169205,"style":38706},[31685,31686],[34,169207,1697],{"className":169208},[31685,31686],[34,169210,67764],{"className":169211,"style":32062},[31685,31686],[34,169213,33249],{"className":169214},[31685,31686],[34,169216,168891],{"className":169217},[31685,31686],[34,169219,38080],{"className":169220},[31685],[34,169222,31747],{"className":169223},[31685,31686],[32188,169225],{},[2230,169227,169228],{"start":43},[108,169229,169230],{},"多路复用通道 multiplexor channel",[15,169232,169233],{},"多路复用通道用于低速设备，可以让多个低速设备交错传输。",[15,169235,169236],{},"所以它的最大速率可以把连接设备速率相加。",[15,169238,169239],{},"两个行式打印机：",[15,169241,169242],{},[34,169243,169245,169285],{"className":169244},[31642],[34,169246,169248],{"className":169247},[31646],[31648,169249,169250],{"xmlns":31650},[31652,169251,169252,169282],{},[31655,169253,169254,169256,169258,169261,169263,169266,169268,169270,169272,169274,169276,169278,169280],{},[31749,169255,768],{},[31662,169257,42071],{},[31749,169259,169260],{},"6.6",[31662,169262,202],{},[31749,169264,169265],{},"13.2",[31658,169267,38691],{},[31658,169269,1697],{},[31658,169271,67764],{},[31658,169273,33249],{},[31658,169275,33244],{},[31658,169277,31747],{},[31658,169279,38080],{"mathvariant":38079},[31658,169281,31747],{},[31665,169283,169284],{"encoding":31667},"2 \\times 6.6 = 13.2Kbytes\u002Fs",[34,169286,169288,169306,169324],{"className":169287,"ariaHidden":758},[31672],[34,169289,169291,169294,169297,169300,169303],{"className":169290},[31676],[34,169292],{"className":169293,"style":35942},[31680],[34,169295,768],{"className":169296},[31685],[34,169298],{"className":169299,"style":32604},[31845],[34,169301,42071],{"className":169302},[32024],[34,169304],{"className":169305,"style":32604},[31845],[34,169307,169309,169312,169315,169318,169321],{"className":169308},[31676],[34,169310],{"className":169311,"style":32617},[31680],[34,169313,169260],{"className":169314},[31685],[34,169316],{"className":169317,"style":31846},[31845],[34,169319,202],{"className":169320},[31850],[34,169322],{"className":169323,"style":31846},[31845],[34,169325,169327,169330,169333,169336,169339,169342,169345,169348,169351],{"className":169326},[31676],[34,169328],{"className":169329,"style":31774},[31680],[34,169331,169265],{"className":169332},[31685],[34,169334,38691],{"className":169335,"style":38706},[31685,31686],[34,169337,1697],{"className":169338},[31685,31686],[34,169340,67764],{"className":169341,"style":32062},[31685,31686],[34,169343,33249],{"className":169344},[31685,31686],[34,169346,168891],{"className":169347},[31685,31686],[34,169349,38080],{"className":169350},[31685],[34,169352,31747],{"className":169353},[31685,31686],[15,169355,169356],{},"两个读卡机：",[15,169358,169359],{},[34,169360,169362,169402],{"className":169361},[31642],[34,169363,169365],{"className":169364},[31646],[31648,169366,169367],{"xmlns":31650},[31652,169368,169369,169399],{},[31655,169370,169371,169373,169375,169378,169380,169383,169385,169387,169389,169391,169393,169395,169397],{},[31749,169372,768],{},[31662,169374,42071],{},[31749,169376,169377],{},"1.2",[31662,169379,202],{},[31749,169381,169382],{},"2.4",[31658,169384,38691],{},[31658,169386,1697],{},[31658,169388,67764],{},[31658,169390,33249],{},[31658,169392,33244],{},[31658,169394,31747],{},[31658,169396,38080],{"mathvariant":38079},[31658,169398,31747],{},[31665,169400,169401],{"encoding":31667},"2 \\times 1.2 = 2.4Kbytes\u002Fs",[34,169403,169405,169423,169441],{"className":169404,"ariaHidden":758},[31672],[34,169406,169408,169411,169414,169417,169420],{"className":169407},[31676],[34,169409],{"className":169410,"style":35942},[31680],[34,169412,768],{"className":169413},[31685],[34,169415],{"className":169416,"style":32604},[31845],[34,169418,42071],{"className":169419},[32024],[34,169421],{"className":169422,"style":32604},[31845],[34,169424,169426,169429,169432,169435,169438],{"className":169425},[31676],[34,169427],{"className":169428,"style":32617},[31680],[34,169430,169377],{"className":169431},[31685],[34,169433],{"className":169434,"style":31846},[31845],[34,169436,202],{"className":169437},[31850],[34,169439],{"className":169440,"style":31846},[31845],[34,169442,169444,169447,169450,169453,169456,169459,169462,169465,169468],{"className":169443},[31676],[34,169445],{"className":169446,"style":31774},[31680],[34,169448,169382],{"className":169449},[31685],[34,169451,38691],{"className":169452,"style":38706},[31685,31686],[34,169454,1697],{"className":169455},[31685,31686],[34,169457,67764],{"className":169458,"style":32062},[31685,31686],[34,169460,33249],{"className":169461},[31685,31686],[34,169463,168891],{"className":169464},[31685,31686],[34,169466,38080],{"className":169467},[31685],[34,169469,31747],{"className":169470},[31685,31686],[15,169472,169473],{},"十个 VDT 终端：",[15,169475,169476],{},[34,169477,169479,169517],{"className":169478},[31642],[34,169480,169482],{"className":169481},[31646],[31648,169483,169484],{"xmlns":31650},[31652,169485,169486,169514],{},[31655,169487,169488,169490,169492,169494,169496,169498,169500,169502,169504,169506,169508,169510,169512],{},[31749,169489,3244],{},[31662,169491,42071],{},[31749,169493,640],{},[31662,169495,202],{},[31749,169497,3244],{},[31658,169499,38691],{},[31658,169501,1697],{},[31658,169503,67764],{},[31658,169505,33249],{},[31658,169507,33244],{},[31658,169509,31747],{},[31658,169511,38080],{"mathvariant":38079},[31658,169513,31747],{},[31665,169515,169516],{"encoding":31667},"10 \\times 1 = 10Kbytes\u002Fs",[34,169518,169520,169538,169556],{"className":169519,"ariaHidden":758},[31672],[34,169521,169523,169526,169529,169532,169535],{"className":169522},[31676],[34,169524],{"className":169525,"style":35942},[31680],[34,169527,3244],{"className":169528},[31685],[34,169530],{"className":169531,"style":32604},[31845],[34,169533,42071],{"className":169534},[32024],[34,169536],{"className":169537,"style":32604},[31845],[34,169539,169541,169544,169547,169550,169553],{"className":169540},[31676],[34,169542],{"className":169543,"style":32617},[31680],[34,169545,640],{"className":169546},[31685],[34,169548],{"className":169549,"style":31846},[31845],[34,169551,202],{"className":169552},[31850],[34,169554],{"className":169555,"style":31846},[31845],[34,169557,169559,169562,169565,169568,169571,169574,169577,169580,169583],{"className":169558},[31676],[34,169560],{"className":169561,"style":31774},[31680],[34,169563,3244],{"className":169564},[31685],[34,169566,38691],{"className":169567,"style":38706},[31685,31686],[34,169569,1697],{"className":169570},[31685,31686],[34,169572,67764],{"className":169573,"style":32062},[31685,31686],[34,169575,33249],{"className":169576},[31685,31686],[34,169578,168891],{"className":169579},[31685,31686],[34,169581,38080],{"className":169582},[31685],[34,169584,31747],{"className":169585},[31685,31686],[15,169587,169588],{},"多路复用通道总速率：",[15,169590,169591],{},[34,169592,169594,169637],{"className":169593},[31642],[34,169595,169597],{"className":169596},[31646],[31648,169598,169599],{"xmlns":31650},[31652,169600,169601,169634],{},[31655,169602,169603,169605,169607,169609,169611,169613,169615,169618,169620,169622,169624,169626,169628,169630,169632],{},[31749,169604,169265],{},[31662,169606,223],{},[31749,169608,169382],{},[31662,169610,223],{},[31749,169612,3244],{},[31662,169614,202],{},[31749,169616,169617],{},"25.6",[31658,169619,38691],{},[31658,169621,1697],{},[31658,169623,67764],{},[31658,169625,33249],{},[31658,169627,33244],{},[31658,169629,31747],{},[31658,169631,38080],{"mathvariant":38079},[31658,169633,31747],{},[31665,169635,169636],{"encoding":31667},"13.2 + 2.4 + 10 = 25.6Kbytes\u002Fs",[34,169638,169640,169658,169676,169694],{"className":169639,"ariaHidden":758},[31672],[34,169641,169643,169646,169649,169652,169655],{"className":169642},[31676],[34,169644],{"className":169645,"style":35942},[31680],[34,169647,169265],{"className":169648},[31685],[34,169650],{"className":169651,"style":32604},[31845],[34,169653,223],{"className":169654},[32024],[34,169656],{"className":169657,"style":32604},[31845],[34,169659,169661,169664,169667,169670,169673],{"className":169660},[31676],[34,169662],{"className":169663,"style":35942},[31680],[34,169665,169382],{"className":169666},[31685],[34,169668],{"className":169669,"style":32604},[31845],[34,169671,223],{"className":169672},[32024],[34,169674],{"className":169675,"style":32604},[31845],[34,169677,169679,169682,169685,169688,169691],{"className":169678},[31676],[34,169680],{"className":169681,"style":32617},[31680],[34,169683,3244],{"className":169684},[31685],[34,169686],{"className":169687,"style":31846},[31845],[34,169689,202],{"className":169690},[31850],[34,169692],{"className":169693,"style":31846},[31845],[34,169695,169697,169700,169703,169706,169709,169712,169715,169718,169721],{"className":169696},[31676],[34,169698],{"className":169699,"style":31774},[31680],[34,169701,169617],{"className":169702},[31685],[34,169704,38691],{"className":169705,"style":38706},[31685,31686],[34,169707,1697],{"className":169708},[31685,31686],[34,169710,67764],{"className":169711,"style":32062},[31685,31686],[34,169713,33249],{"className":169714},[31685,31686],[34,169716,168891],{"className":169717},[31685,31686],[34,169719,38080],{"className":169720},[31685],[34,169722,31747],{"className":169723},[31685,31686],[32188,169725],{},[253,169727,169729],{"id":169728},"系统最大聚合-io-传输速率","系统最大聚合 I\u002FO 传输速率",[15,169731,169732],{},"总速率为：",[15,169734,169735],{},[34,169736,169738,169777],{"className":169737},[31642],[34,169739,169741],{"className":169740},[31646],[31648,169742,169743],{"xmlns":31650},[31652,169744,169745,169774],{},[31655,169746,169747,169749,169751,169753,169755,169758,169760,169762,169764,169766,169768,169770,169772],{},[31749,169748,169135],{},[31662,169750,223],{},[31749,169752,169617],{},[31662,169754,202],{},[31749,169756,169757],{},"1625.6",[31658,169759,38691],{},[31658,169761,1697],{},[31658,169763,67764],{},[31658,169765,33249],{},[31658,169767,33244],{},[31658,169769,31747],{},[31658,169771,38080],{"mathvariant":38079},[31658,169773,31747],{},[31665,169775,169776],{"encoding":31667},"1600 + 25.6 = 1625.6Kbytes\u002Fs",[34,169778,169780,169798,169816],{"className":169779,"ariaHidden":758},[31672],[34,169781,169783,169786,169789,169792,169795],{"className":169782},[31676],[34,169784],{"className":169785,"style":35942},[31680],[34,169787,169135],{"className":169788},[31685],[34,169790],{"className":169791,"style":32604},[31845],[34,169793,223],{"className":169794},[32024],[34,169796],{"className":169797,"style":32604},[31845],[34,169799,169801,169804,169807,169810,169813],{"className":169800},[31676],[34,169802],{"className":169803,"style":32617},[31680],[34,169805,169617],{"className":169806},[31685],[34,169808],{"className":169809,"style":31846},[31845],[34,169811,202],{"className":169812},[31850],[34,169814],{"className":169815,"style":31846},[31845],[34,169817,169819,169822,169825,169828,169831,169834,169837,169840,169843],{"className":169818},[31676],[34,169820],{"className":169821,"style":31774},[31680],[34,169823,169757],{"className":169824},[31685],[34,169826,38691],{"className":169827,"style":38706},[31685,31686],[34,169829,1697],{"className":169830},[31685,31686],[34,169832,67764],{"className":169833,"style":32062},[31685,31686],[34,169835,33249],{"className":169836},[31685,31686],[34,169838,168891],{"className":169839},[31685,31686],[34,169841,38080],{"className":169842},[31685],[34,169844,31747],{"className":169845},[31685,31686],[15,169847,169848],{},"换成 MB\u002Fs 约为：",[15,169850,169851],{},[34,169852,169854,169877],{"className":169853},[31642],[34,169855,169857],{"className":169856},[31646],[31648,169858,169859],{"xmlns":31650},[31652,169860,169861,169874],{},[31655,169862,169863,169866,169868,169870,169872],{},[31749,169864,169865],{},"1.626",[31658,169867,98360],{},[31658,169869,54224],{},[31658,169871,38080],{"mathvariant":38079},[31658,169873,31747],{},[31665,169875,169876],{"encoding":31667},"1.626MB\u002Fs",[34,169878,169880],{"className":169879,"ariaHidden":758},[31672],[34,169881,169883,169886,169889,169892,169895,169898],{"className":169882},[31676],[34,169884],{"className":169885,"style":31774},[31680],[34,169887,169865],{"className":169888},[31685],[34,169890,98360],{"className":169891,"style":33389},[31685,31686],[34,169893,54224],{"className":169894,"style":54252},[31685,31686],[34,169896,38080],{"className":169897},[31685],[34,169899,31747],{"className":169900},[31685,31686],[32188,169902],{},[253,169904,169906],{"id":169905},"第-3-题最终答案","第 3 题最终答案",[15,169908,169909],{},"最大聚合 I\u002FO 传输速率为：",[15,169911,169912],{},[34,169913,169915,169945],{"className":169914},[31642],[34,169916,169918],{"className":169917},[31646],[31648,169919,169920],{"xmlns":31650},[31652,169921,169922,169942],{},[31655,169923,169924,169926,169928,169930,169932,169934,169936,169938,169940],{},[31749,169925,169757],{},[31658,169927,38691],{},[31658,169929,1697],{},[31658,169931,67764],{},[31658,169933,33249],{},[31658,169935,33244],{},[31658,169937,31747],{},[31658,169939,38080],{"mathvariant":38079},[31658,169941,31747],{},[31665,169943,169944],{"encoding":31667},"1625.6Kbytes\u002Fs",[34,169946,169948],{"className":169947,"ariaHidden":758},[31672],[34,169949,169951,169954,169957,169960,169963,169966,169969,169972,169975],{"className":169950},[31676],[34,169952],{"className":169953,"style":31774},[31680],[34,169955,169757],{"className":169956},[31685],[34,169958,38691],{"className":169959,"style":38706},[31685,31686],[34,169961,1697],{"className":169962},[31685,31686],[34,169964,67764],{"className":169965,"style":32062},[31685,31686],[34,169967,33249],{"className":169968},[31685,31686],[34,169970,168891],{"className":169971},[31685,31686],[34,169973,38080],{"className":169974},[31685],[34,169976,31747],{"className":169977},[31685,31686],[15,169979,169980],{},"约等于：",[15,169982,169983],{},[34,169984,169986,170007],{"className":169985},[31642],[34,169987,169989],{"className":169988},[31646],[31648,169990,169991],{"xmlns":31650},[31652,169992,169993,170005],{},[31655,169994,169995,169997,169999,170001,170003],{},[31749,169996,169865],{},[31658,169998,98360],{},[31658,170000,54224],{},[31658,170002,38080],{"mathvariant":38079},[31658,170004,31747],{},[31665,170006,169876],{"encoding":31667},[34,170008,170010],{"className":170009,"ariaHidden":758},[31672],[34,170011,170013,170016,170019,170022,170025,170028],{"className":170012},[31676],[34,170014],{"className":170015,"style":31774},[31680],[34,170017,169865],{"className":170018},[31685],[34,170020,98360],{"className":170021,"style":33389},[31685,31686],[34,170023,54224],{"className":170024,"style":54252},[31685,31686],[34,170026,38080],{"className":170027},[31685],[34,170029,31747],{"className":170030},[31685,31686],[32188,170032],{},[19,170034,170036],{"id":170035},"_4-磁盘记录顺序存放与优化存放","4. 磁盘记录顺序存放与优化存放",[253,170038,162043],{"id":170039},"题目内容-3",[15,170041,170042],{},"假设有 8 个记录 A、B、C、D、E、F、G、H 存放在磁盘里，每个磁道有 8 个扇区，正好可以存放 8 个记录。假设磁盘旋转速度为 20ms\u002F转，处理程序每读出一个记录后，用 2ms 的时间进行处理，请问：",[15,170044,170045],{},"a. 当记录 A、B、C、D、E、F、G、H 按顺序放在磁道上时，顺序处理这 8 个记录花费的总时间是多少？假设启动时的位置正好在 A 扇区的起点。",[15,170047,170048],{},"b. 如何采取优化方法，使处理这些记录所花费的总时间最短？并求出该最短时间。",[32188,170050],{},[253,170052,162066],{"id":170053},"解题核心-3",[15,170055,170056],{},"一圈有 8 个扇区，一圈时间为 20ms。",[15,170058,170059],{},"所以每个扇区经过磁头的时间是：",[15,170061,170062],{},[34,170063,170065,170095],{"className":170064},[31642],[34,170066,170068],{"className":170067},[31646],[31648,170069,170070],{"xmlns":31650},[31652,170071,170072,170092],{},[31655,170073,170074,170076,170078,170080,170082,170084,170086,170088,170090],{},[31749,170075,3249],{},[31658,170077,31759],{},[31658,170079,31747],{},[31658,170081,38080],{"mathvariant":38079},[31749,170083,60525],{},[31662,170085,202],{},[31749,170087,167259],{},[31658,170089,31759],{},[31658,170091,31747],{},[31665,170093,170094],{"encoding":31667},"20ms \u002F 8 = 2.5ms",[34,170096,170098,170126],{"className":170097,"ariaHidden":758},[31672],[34,170099,170101,170104,170107,170110,170113,170117,170120,170123],{"className":170100},[31676],[34,170102],{"className":170103,"style":31774},[31680],[34,170105,3249],{"className":170106},[31685],[34,170108,31759],{"className":170109},[31685,31686],[34,170111,31747],{"className":170112},[31685,31686],[34,170114,170116],{"className":170115},[31685],"\u002F8",[34,170118],{"className":170119,"style":31846},[31845],[34,170121,202],{"className":170122},[31850],[34,170124],{"className":170125,"style":31846},[31845],[34,170127,170129,170132,170135,170138],{"className":170128},[31676],[34,170130],{"className":170131,"style":32617},[31680],[34,170133,167259],{"className":170134},[31685],[34,170136,31759],{"className":170137},[31685,31686],[34,170139,31747],{"className":170140},[31685,31686],[15,170142,170143],{},"读取一个记录需要经过一个扇区，所以读取一个记录的时间为：",[15,170145,170146],{},[34,170147,170149,170167],{"className":170148},[31642],[34,170150,170152],{"className":170151},[31646],[31648,170153,170154],{"xmlns":31650},[31652,170155,170156,170164],{},[31655,170157,170158,170160,170162],{},[31749,170159,167259],{},[31658,170161,31759],{},[31658,170163,31747],{},[31665,170165,170166],{"encoding":31667},"2.5ms",[34,170168,170170],{"className":170169,"ariaHidden":758},[31672],[34,170171,170173,170176,170179,170182],{"className":170172},[31676],[34,170174],{"className":170175,"style":32617},[31680],[34,170177,167259],{"className":170178},[31685],[34,170180,31759],{"className":170181},[31685,31686],[34,170183,31747],{"className":170184},[31685,31686],[15,170186,170187],{},"读完一个记录后，CPU 还要处理：",[15,170189,170190],{},[34,170191,170193,170211],{"className":170192},[31642],[34,170194,170196],{"className":170195},[31646],[31648,170197,170198],{"xmlns":31650},[31652,170199,170200,170208],{},[31655,170201,170202,170204,170206],{},[31749,170203,768],{},[31658,170205,31759],{},[31658,170207,31747],{},[31665,170209,170210],{"encoding":31667},"2ms",[34,170212,170214],{"className":170213,"ariaHidden":758},[31672],[34,170215,170217,170220,170223,170226],{"className":170216},[31676],[34,170218],{"className":170219,"style":32617},[31680],[34,170221,768],{"className":170222},[31685],[34,170224,31759],{"className":170225},[31685,31686],[34,170227,31747],{"className":170228},[31685,31686],[15,170230,170231],{},"因此处理一个记录之后，下一次能准备好继续读的时间为：",[15,170233,170234],{},[34,170235,170237,170272],{"className":170236},[31642],[34,170238,170240],{"className":170239},[31646],[31648,170241,170242],{"xmlns":31650},[31652,170243,170244,170269],{},[31655,170245,170246,170248,170250,170252,170254,170256,170258,170260,170262,170265,170267],{},[31749,170247,167259],{},[31658,170249,31759],{},[31658,170251,31747],{},[31662,170253,223],{},[31749,170255,768],{},[31658,170257,31759],{},[31658,170259,31747],{},[31662,170261,202],{},[31749,170263,170264],{},"4.5",[31658,170266,31759],{},[31658,170268,31747],{},[31665,170270,170271],{"encoding":31667},"2.5ms + 2ms = 4.5ms",[34,170273,170275,170299,170323],{"className":170274,"ariaHidden":758},[31672],[34,170276,170278,170281,170284,170287,170290,170293,170296],{"className":170277},[31676],[34,170279],{"className":170280,"style":35942},[31680],[34,170282,167259],{"className":170283},[31685],[34,170285,31759],{"className":170286},[31685,31686],[34,170288,31747],{"className":170289},[31685,31686],[34,170291],{"className":170292,"style":32604},[31845],[34,170294,223],{"className":170295},[32024],[34,170297],{"className":170298,"style":32604},[31845],[34,170300,170302,170305,170308,170311,170314,170317,170320],{"className":170301},[31676],[34,170303],{"className":170304,"style":32617},[31680],[34,170306,768],{"className":170307},[31685],[34,170309,31759],{"className":170310},[31685,31686],[34,170312,31747],{"className":170313},[31685,31686],[34,170315],{"className":170316,"style":31846},[31845],[34,170318,202],{"className":170319},[31850],[34,170321],{"className":170322,"style":31846},[31845],[34,170324,170326,170329,170332,170335],{"className":170325},[31676],[34,170327],{"className":170328,"style":32617},[31680],[34,170330,170264],{"className":170331},[31685],[34,170333,31759],{"className":170334},[31685,31686],[34,170336,31747],{"className":170337},[31685,31686],[15,170339,8882],{},[15,170341,170342],{},"相邻扇区之间只隔：",[15,170344,170345],{},[34,170346,170348,170365],{"className":170347},[31642],[34,170349,170351],{"className":170350},[31646],[31648,170352,170353],{"xmlns":31650},[31652,170354,170355,170363],{},[31655,170356,170357,170359,170361],{},[31749,170358,167259],{},[31658,170360,31759],{},[31658,170362,31747],{},[31665,170364,170166],{"encoding":31667},[34,170366,170368],{"className":170367,"ariaHidden":758},[31672],[34,170369,170371,170374,170377,170380],{"className":170370},[31676],[34,170372],{"className":170373,"style":32617},[31680],[34,170375,167259],{"className":170376},[31685],[34,170378,31759],{"className":170379},[31685,31686],[34,170381,31747],{"className":170382},[31685,31686],[15,170384,170385],{},"但是读完并处理完一个记录需要：",[15,170387,170388],{},[34,170389,170391,170409],{"className":170390},[31642],[34,170392,170394],{"className":170393},[31646],[31648,170395,170396],{"xmlns":31650},[31652,170397,170398,170406],{},[31655,170399,170400,170402,170404],{},[31749,170401,170264],{},[31658,170403,31759],{},[31658,170405,31747],{},[31665,170407,170408],{"encoding":31667},"4.5ms",[34,170410,170412],{"className":170411,"ariaHidden":758},[31672],[34,170413,170415,170418,170421,170424],{"className":170414},[31676],[34,170416],{"className":170417,"style":32617},[31680],[34,170419,170264],{"className":170420},[31685],[34,170422,31759],{"className":170423},[31685,31686],[34,170425,31747],{"className":170426},[31685,31686],[15,170428,170429],{},"所以如果记录按 A、B、C、D、E、F、G、H 连续存放，读完并处理 A 之后，B 已经从磁头下转过去了，必须等 B 下一圈再回来。",[15,170431,170432],{},"这就是磁盘记录顺序存放导致效率低的原因。",[32188,170434],{},[253,170436,170438],{"id":170437},"a-顺序存放时的总时间","a. 顺序存放时的总时间",[15,170440,170441],{},"记录按顺序放置：",[15,170443,170444],{},"A B C D E F G H",[15,170446,170447],{},"启动时正好在 A 扇区起点。",[15,170449,170450],{},"读取并处理 A 需要：",[15,170452,170453],{},[34,170454,170456,170489],{"className":170455},[31642],[34,170457,170459],{"className":170458},[31646],[31648,170460,170461],{"xmlns":31650},[31652,170462,170463,170487],{},[31655,170464,170465,170467,170469,170471,170473,170475,170477,170479,170481,170483,170485],{},[31749,170466,167259],{},[31658,170468,31759],{},[31658,170470,31747],{},[31662,170472,223],{},[31749,170474,768],{},[31658,170476,31759],{},[31658,170478,31747],{},[31662,170480,202],{},[31749,170482,170264],{},[31658,170484,31759],{},[31658,170486,31747],{},[31665,170488,170271],{"encoding":31667},[34,170490,170492,170516,170540],{"className":170491,"ariaHidden":758},[31672],[34,170493,170495,170498,170501,170504,170507,170510,170513],{"className":170494},[31676],[34,170496],{"className":170497,"style":35942},[31680],[34,170499,167259],{"className":170500},[31685],[34,170502,31759],{"className":170503},[31685,31686],[34,170505,31747],{"className":170506},[31685,31686],[34,170508],{"className":170509,"style":32604},[31845],[34,170511,223],{"className":170512},[32024],[34,170514],{"className":170515,"style":32604},[31845],[34,170517,170519,170522,170525,170528,170531,170534,170537],{"className":170518},[31676],[34,170520],{"className":170521,"style":32617},[31680],[34,170523,768],{"className":170524},[31685],[34,170526,31759],{"className":170527},[31685,31686],[34,170529,31747],{"className":170530},[31685,31686],[34,170532],{"className":170533,"style":31846},[31845],[34,170535,202],{"className":170536},[31850],[34,170538],{"className":170539,"style":31846},[31845],[34,170541,170543,170546,170549,170552],{"className":170542},[31676],[34,170544],{"className":170545,"style":32617},[31680],[34,170547,170264],{"className":170548},[31685],[34,170550,31759],{"className":170551},[31685,31686],[34,170553,31747],{"className":170554},[31685,31686],[15,170556,170557],{},"但处理完 A 时，B 的扇区起点早就过去了，所以要等待 B 下一圈回来。",[15,170559,170560],{},"B 的起点在 A 起点之后：",[15,170562,170563],{},[34,170564,170566,170583],{"className":170565},[31642],[34,170567,170569],{"className":170568},[31646],[31648,170570,170571],{"xmlns":31650},[31652,170572,170573,170581],{},[31655,170574,170575,170577,170579],{},[31749,170576,167259],{},[31658,170578,31759],{},[31658,170580,31747],{},[31665,170582,170166],{"encoding":31667},[34,170584,170586],{"className":170585,"ariaHidden":758},[31672],[34,170587,170589,170592,170595,170598],{"className":170588},[31676],[34,170590],{"className":170591,"style":32617},[31680],[34,170593,167259],{"className":170594},[31685],[34,170596,31759],{"className":170597},[31685,31686],[34,170599,31747],{"className":170600},[31685,31686],[15,170602,170603],{},"处理完 A 的时间是：",[15,170605,170606],{},[34,170607,170609,170626],{"className":170608},[31642],[34,170610,170612],{"className":170611},[31646],[31648,170613,170614],{"xmlns":31650},[31652,170615,170616,170624],{},[31655,170617,170618,170620,170622],{},[31749,170619,170264],{},[31658,170621,31759],{},[31658,170623,31747],{},[31665,170625,170408],{"encoding":31667},[34,170627,170629],{"className":170628,"ariaHidden":758},[31672],[34,170630,170632,170635,170638,170641],{"className":170631},[31676],[34,170633],{"className":170634,"style":32617},[31680],[34,170636,170264],{"className":170637},[31685],[34,170639,31759],{"className":170640},[31685,31686],[34,170642,31747],{"className":170643},[31685,31686],[15,170645,170646],{},"已经错过 B 起点：",[15,170648,170649],{},[34,170650,170652,170686],{"className":170651},[31642],[34,170653,170655],{"className":170654},[31646],[31648,170656,170657],{"xmlns":31650},[31652,170658,170659,170683],{},[31655,170660,170661,170663,170665,170667,170669,170671,170673,170675,170677,170679,170681],{},[31749,170662,170264],{},[31658,170664,31759],{},[31658,170666,31747],{},[31662,170668,32226],{},[31749,170670,167259],{},[31658,170672,31759],{},[31658,170674,31747],{},[31662,170676,202],{},[31749,170678,768],{},[31658,170680,31759],{},[31658,170682,31747],{},[31665,170684,170685],{"encoding":31667},"4.5ms - 2.5ms = 2ms",[34,170687,170689,170713,170737],{"className":170688,"ariaHidden":758},[31672],[34,170690,170692,170695,170698,170701,170704,170707,170710],{"className":170691},[31676],[34,170693],{"className":170694,"style":35942},[31680],[34,170696,170264],{"className":170697},[31685],[34,170699,31759],{"className":170700},[31685,31686],[34,170702,31747],{"className":170703},[31685,31686],[34,170705],{"className":170706,"style":32604},[31845],[34,170708,32226],{"className":170709},[32024],[34,170711],{"className":170712,"style":32604},[31845],[34,170714,170716,170719,170722,170725,170728,170731,170734],{"className":170715},[31676],[34,170717],{"className":170718,"style":32617},[31680],[34,170720,167259],{"className":170721},[31685],[34,170723,31759],{"className":170724},[31685,31686],[34,170726,31747],{"className":170727},[31685,31686],[34,170729],{"className":170730,"style":31846},[31845],[34,170732,202],{"className":170733},[31850],[34,170735],{"className":170736,"style":31846},[31845],[34,170738,170740,170743,170746,170749],{"className":170739},[31676],[34,170741],{"className":170742,"style":32617},[31680],[34,170744,768],{"className":170745},[31685],[34,170747,31759],{"className":170748},[31685,31686],[34,170750,31747],{"className":170751},[31685,31686],[15,170753,170754],{},"所以需要等到下一圈的 B：",[15,170756,170757],{},[34,170758,170760,170794],{"className":170759},[31642],[34,170761,170763],{"className":170762},[31646],[31648,170764,170765],{"xmlns":31650},[31652,170766,170767,170791],{},[31655,170768,170769,170771,170773,170775,170777,170779,170781,170783,170785,170787,170789],{},[31749,170770,3249],{},[31658,170772,31759],{},[31658,170774,31747],{},[31662,170776,32226],{},[31749,170778,768],{},[31658,170780,31759],{},[31658,170782,31747],{},[31662,170784,202],{},[31749,170786,95611],{},[31658,170788,31759],{},[31658,170790,31747],{},[31665,170792,170793],{"encoding":31667},"20ms - 2ms = 18ms",[34,170795,170797,170821,170845],{"className":170796,"ariaHidden":758},[31672],[34,170798,170800,170803,170806,170809,170812,170815,170818],{"className":170799},[31676],[34,170801],{"className":170802,"style":35942},[31680],[34,170804,3249],{"className":170805},[31685],[34,170807,31759],{"className":170808},[31685,31686],[34,170810,31747],{"className":170811},[31685,31686],[34,170813],{"className":170814,"style":32604},[31845],[34,170816,32226],{"className":170817},[32024],[34,170819],{"className":170820,"style":32604},[31845],[34,170822,170824,170827,170830,170833,170836,170839,170842],{"className":170823},[31676],[34,170825],{"className":170826,"style":32617},[31680],[34,170828,768],{"className":170829},[31685],[34,170831,31759],{"className":170832},[31685,31686],[34,170834,31747],{"className":170835},[31685,31686],[34,170837],{"className":170838,"style":31846},[31845],[34,170840,202],{"className":170841},[31850],[34,170843],{"className":170844,"style":31846},[31845],[34,170846,170848,170851,170854,170857],{"className":170847},[31676],[34,170849],{"className":170850,"style":32617},[31680],[34,170852,95611],{"className":170853},[31685],[34,170855,31759],{"className":170856},[31685,31686],[34,170858,31747],{"className":170859},[31685,31686],[15,170861,170862],{},"也就是说，每两个相邻逻辑记录之间都会多等：",[15,170864,170865],{},[34,170866,170868,170886],{"className":170867},[31642],[34,170869,170871],{"className":170870},[31646],[31648,170872,170873],{"xmlns":31650},[31652,170874,170875,170883],{},[31655,170876,170877,170879,170881],{},[31749,170878,95611],{},[31658,170880,31759],{},[31658,170882,31747],{},[31665,170884,170885],{"encoding":31667},"18ms",[34,170887,170889],{"className":170888,"ariaHidden":758},[31672],[34,170890,170892,170895,170898,170901],{"className":170891},[31676],[34,170893],{"className":170894,"style":32617},[31680],[34,170896,95611],{"className":170897},[31685],[34,170899,31759],{"className":170900},[31685,31686],[34,170902,31747],{"className":170903},[31685,31686],[15,170905,170906],{},"8 个记录总共有 7 次等待。",[15,170908,170909],{},"每个记录的读取加处理时间：",[15,170911,170912],{},[34,170913,170915,170948],{"className":170914},[31642],[34,170916,170918],{"className":170917},[31646],[31648,170919,170920],{"xmlns":31650},[31652,170921,170922,170946],{},[31655,170923,170924,170926,170928,170930,170932,170934,170936,170938,170940,170942,170944],{},[31749,170925,167259],{},[31658,170927,31759],{},[31658,170929,31747],{},[31662,170931,223],{},[31749,170933,768],{},[31658,170935,31759],{},[31658,170937,31747],{},[31662,170939,202],{},[31749,170941,170264],{},[31658,170943,31759],{},[31658,170945,31747],{},[31665,170947,170271],{"encoding":31667},[34,170949,170951,170975,170999],{"className":170950,"ariaHidden":758},[31672],[34,170952,170954,170957,170960,170963,170966,170969,170972],{"className":170953},[31676],[34,170955],{"className":170956,"style":35942},[31680],[34,170958,167259],{"className":170959},[31685],[34,170961,31759],{"className":170962},[31685,31686],[34,170964,31747],{"className":170965},[31685,31686],[34,170967],{"className":170968,"style":32604},[31845],[34,170970,223],{"className":170971},[32024],[34,170973],{"className":170974,"style":32604},[31845],[34,170976,170978,170981,170984,170987,170990,170993,170996],{"className":170977},[31676],[34,170979],{"className":170980,"style":32617},[31680],[34,170982,768],{"className":170983},[31685],[34,170985,31759],{"className":170986},[31685,31686],[34,170988,31747],{"className":170989},[31685,31686],[34,170991],{"className":170992,"style":31846},[31845],[34,170994,202],{"className":170995},[31850],[34,170997],{"className":170998,"style":31846},[31845],[34,171000,171002,171005,171008,171011],{"className":171001},[31676],[34,171003],{"className":171004,"style":32617},[31680],[34,171006,170264],{"className":171007},[31685],[34,171009,31759],{"className":171010},[31685,31686],[34,171012,31747],{"className":171013},[31685,31686],[15,171015,171016],{},"8 个记录共：",[15,171018,171019],{},[34,171020,171022,171052],{"className":171021},[31642],[34,171023,171025],{"className":171024},[31646],[31648,171026,171027],{"xmlns":31650},[31652,171028,171029,171049],{},[31655,171030,171031,171033,171035,171037,171039,171041,171043,171045,171047],{},[31749,171032,60525],{},[31662,171034,42071],{},[31749,171036,170264],{},[31658,171038,31759],{},[31658,171040,31747],{},[31662,171042,202],{},[31749,171044,165563],{},[31658,171046,31759],{},[31658,171048,31747],{},[31665,171050,171051],{"encoding":31667},"8 \\times 4.5ms = 36ms",[34,171053,171055,171073,171097],{"className":171054,"ariaHidden":758},[31672],[34,171056,171058,171061,171064,171067,171070],{"className":171057},[31676],[34,171059],{"className":171060,"style":35942},[31680],[34,171062,60525],{"className":171063},[31685],[34,171065],{"className":171066,"style":32604},[31845],[34,171068,42071],{"className":171069},[32024],[34,171071],{"className":171072,"style":32604},[31845],[34,171074,171076,171079,171082,171085,171088,171091,171094],{"className":171075},[31676],[34,171077],{"className":171078,"style":32617},[31680],[34,171080,170264],{"className":171081},[31685],[34,171083,31759],{"className":171084},[31685,31686],[34,171086,31747],{"className":171087},[31685,31686],[34,171089],{"className":171090,"style":31846},[31845],[34,171092,202],{"className":171093},[31850],[34,171095],{"className":171096,"style":31846},[31845],[34,171098,171100,171103,171106,171109],{"className":171099},[31676],[34,171101],{"className":171102,"style":32617},[31680],[34,171104,165563],{"className":171105},[31685],[34,171107,31759],{"className":171108},[31685,31686],[34,171110,31747],{"className":171111},[31685,31686],[15,171113,171114],{},"等待时间共：",[15,171116,171117],{},[34,171118,171120,171151],{"className":171119},[31642],[34,171121,171123],{"className":171122},[31646],[31648,171124,171125],{"xmlns":31650},[31652,171126,171127,171148],{},[31655,171128,171129,171131,171133,171135,171137,171139,171141,171144,171146],{},[31749,171130,36723],{},[31662,171132,42071],{},[31749,171134,95611],{},[31658,171136,31759],{},[31658,171138,31747],{},[31662,171140,202],{},[31749,171142,171143],{},"126",[31658,171145,31759],{},[31658,171147,31747],{},[31665,171149,171150],{"encoding":31667},"7 \\times 18ms = 126ms",[34,171152,171154,171172,171196],{"className":171153,"ariaHidden":758},[31672],[34,171155,171157,171160,171163,171166,171169],{"className":171156},[31676],[34,171158],{"className":171159,"style":35942},[31680],[34,171161,36723],{"className":171162},[31685],[34,171164],{"className":171165,"style":32604},[31845],[34,171167,42071],{"className":171168},[32024],[34,171170],{"className":171171,"style":32604},[31845],[34,171173,171175,171178,171181,171184,171187,171190,171193],{"className":171174},[31676],[34,171176],{"className":171177,"style":32617},[31680],[34,171179,95611],{"className":171180},[31685],[34,171182,31759],{"className":171183},[31685,31686],[34,171185,31747],{"className":171186},[31685,31686],[34,171188],{"className":171189,"style":31846},[31845],[34,171191,202],{"className":171192},[31850],[34,171194],{"className":171195,"style":31846},[31845],[34,171197,171199,171202,171205,171208],{"className":171198},[31676],[34,171200],{"className":171201,"style":32617},[31680],[34,171203,171143],{"className":171204},[31685],[34,171206,31759],{"className":171207},[31685,31686],[34,171209,31747],{"className":171210},[31685,31686],[15,171212,171213],{},"所以总时间为：",[15,171215,171216],{},[34,171217,171219,171254],{"className":171218},[31642],[34,171220,171222],{"className":171221},[31646],[31648,171223,171224],{"xmlns":31650},[31652,171225,171226,171251],{},[31655,171227,171228,171230,171232,171234,171236,171238,171240,171242,171244,171247,171249],{},[31749,171229,165563],{},[31658,171231,31759],{},[31658,171233,31747],{},[31662,171235,223],{},[31749,171237,171143],{},[31658,171239,31759],{},[31658,171241,31747],{},[31662,171243,202],{},[31749,171245,171246],{},"162",[31658,171248,31759],{},[31658,171250,31747],{},[31665,171252,171253],{"encoding":31667},"36ms + 126ms = 162ms",[34,171255,171257,171281,171305],{"className":171256,"ariaHidden":758},[31672],[34,171258,171260,171263,171266,171269,171272,171275,171278],{"className":171259},[31676],[34,171261],{"className":171262,"style":35942},[31680],[34,171264,165563],{"className":171265},[31685],[34,171267,31759],{"className":171268},[31685,31686],[34,171270,31747],{"className":171271},[31685,31686],[34,171273],{"className":171274,"style":32604},[31845],[34,171276,223],{"className":171277},[32024],[34,171279],{"className":171280,"style":32604},[31845],[34,171282,171284,171287,171290,171293,171296,171299,171302],{"className":171283},[31676],[34,171285],{"className":171286,"style":32617},[31680],[34,171288,171143],{"className":171289},[31685],[34,171291,31759],{"className":171292},[31685,31686],[34,171294,31747],{"className":171295},[31685,31686],[34,171297],{"className":171298,"style":31846},[31845],[34,171300,202],{"className":171301},[31850],[34,171303],{"className":171304,"style":31846},[31845],[34,171306,171308,171311,171314,171317],{"className":171307},[31676],[34,171309],{"className":171310,"style":32617},[31680],[34,171312,171246],{"className":171313},[31685],[34,171315,31759],{"className":171316},[31685,31686],[34,171318,31747],{"className":171319},[31685,31686],[32188,171321],{},[253,171323,171325],{"id":171324},"a-的答案","a 的答案",[15,171327,171328],{},"顺序存放时，总时间为：",[15,171330,171331],{},[34,171332,171334,171352],{"className":171333},[31642],[34,171335,171337],{"className":171336},[31646],[31648,171338,171339],{"xmlns":31650},[31652,171340,171341,171349],{},[31655,171342,171343,171345,171347],{},[31749,171344,171246],{},[31658,171346,31759],{},[31658,171348,31747],{},[31665,171350,171351],{"encoding":31667},"162ms",[34,171353,171355],{"className":171354,"ariaHidden":758},[31672],[34,171356,171358,171361,171364,171367],{"className":171357},[31676],[34,171359],{"className":171360,"style":32617},[31680],[34,171362,171246],{"className":171363},[31685],[34,171365,31759],{"className":171366},[31685,31686],[34,171368,31747],{"className":171369},[31685,31686],[32188,171371],{},[253,171373,171375],{"id":171374},"b-优化方法","b. 优化方法",[15,171377,104362],{},[15,171379,171380],{},"不要把逻辑上相邻的记录放在物理上相邻的扇区，而是进行交错存放，也叫磁盘交错 interleaving。",[15,171382,171383],{},"因为读完并处理一个记录需要：",[15,171385,171386],{},[34,171387,171389,171406],{"className":171388},[31642],[34,171390,171392],{"className":171391},[31646],[31648,171393,171394],{"xmlns":31650},[31652,171395,171396,171404],{},[31655,171397,171398,171400,171402],{},[31749,171399,170264],{},[31658,171401,31759],{},[31658,171403,31747],{},[31665,171405,170408],{"encoding":31667},[34,171407,171409],{"className":171408,"ariaHidden":758},[31672],[34,171410,171412,171415,171418,171421],{"className":171411},[31676],[34,171413],{"className":171414,"style":32617},[31680],[34,171416,170264],{"className":171417},[31685],[34,171419,31759],{"className":171420},[31685,31686],[34,171422,31747],{"className":171423},[31685,31686],[15,171425,171426],{},"而一个扇区时间是：",[15,171428,171429],{},[34,171430,171432,171449],{"className":171431},[31642],[34,171433,171435],{"className":171434},[31646],[31648,171436,171437],{"xmlns":31650},[31652,171438,171439,171447],{},[31655,171440,171441,171443,171445],{},[31749,171442,167259],{},[31658,171444,31759],{},[31658,171446,31747],{},[31665,171448,170166],{"encoding":31667},[34,171450,171452],{"className":171451,"ariaHidden":758},[31672],[34,171453,171455,171458,171461,171464],{"className":171454},[31676],[34,171456],{"className":171457,"style":32617},[31680],[34,171459,167259],{"className":171460},[31685],[34,171462,31759],{"className":171463},[31685,31686],[34,171465,31747],{"className":171466},[31685,31686],[15,171468,171469],{},"所以读完并处理后，磁盘已经转过：",[15,171471,171472],{},[34,171473,171475,171498],{"className":171474},[31642],[34,171476,171478],{"className":171477},[31646],[31648,171479,171480],{"xmlns":31650},[31652,171481,171482,171495],{},[31655,171483,171484,171490,171492],{},[41453,171485,171486,171488],{},[31749,171487,170264],{},[31749,171489,167259],{},[31662,171491,202],{},[31749,171493,171494],{},"1.8",[31665,171496,171497],{"encoding":31667},"\\frac{4.5}{2.5} = 1.8",[34,171499,171501,171584],{"className":171500,"ariaHidden":758},[31672],[34,171502,171504,171507,171575,171578,171581],{"className":171503},[31676],[34,171505],{"className":171506,"style":46727},[31680],[34,171508,171510,171513,171572],{"className":171509},[31685],[34,171511],{"className":171512},[31782,41488],[34,171514,171516],{"className":171515},[41453],[34,171517,171519,171564],{"className":171518},[31796,31797],[34,171520,171522,171561],{"className":171521},[31801],[34,171523,171525,171539,171547],{"className":171524,"style":46746},[31805],[34,171526,171527,171530],{"style":41504},[34,171528],{"className":171529,"style":41508},[31813],[34,171531,171533],{"className":171532},[31818,31819,31820,31821],[34,171534,171536],{"className":171535},[31685,31821],[34,171537,167259],{"className":171538},[31685,31821],[34,171540,171541,171544],{"style":41520},[34,171542],{"className":171543,"style":41508},[31813],[34,171545],{"className":171546,"style":41528},[41527],[34,171548,171549,171552],{"style":46771},[34,171550],{"className":171551,"style":41508},[31813],[34,171553,171555],{"className":171554},[31818,31819,31820,31821],[34,171556,171558],{"className":171557},[31685,31821],[34,171559,170264],{"className":171560},[31685,31821],[34,171562,31829],{"className":171563},[31828],[34,171565,171567],{"className":171566},[31801],[34,171568,171570],{"className":171569,"style":41619},[31805],[34,171571],{},[34,171573],{"className":171574},[31690,41488],[34,171576],{"className":171577,"style":31846},[31845],[34,171579,202],{"className":171580},[31850],[34,171582],{"className":171583,"style":31846},[31845],[34,171585,171587,171590],{"className":171586},[31676],[34,171588],{"className":171589,"style":32617},[31680],[34,171591,171494],{"className":171592},[31685],[15,171594,171595],{},"个扇区。",[15,171597,171598],{},"下一次读应该安排在大约 2 个扇区之后。",[15,171600,171601],{},"因此，逻辑上的下一个记录最好放在当前记录后面第 2 个扇区的位置。",[32188,171603],{},[253,171605,171606],{"id":171606},"构造最优存放方式",[15,171608,171609],{},"按照磁盘旋转方向，把 8 个扇区编号为：",[15,171611,171612],{},"0, 1, 2, 3, 4, 5, 6, 7",[15,171614,171615],{},"让 A 放在 0 号扇区。",[15,171617,171618],{},"为了让下一个逻辑记录在 2 个扇区之后出现，可以安排：",[15,171620,171621],{},"A 在 0 号扇区",[15,171623,171624],{},"B 在 2 号扇区",[15,171626,171627],{},"C 在 4 号扇区",[15,171629,171630],{},"D 在 6 号扇区",[15,171632,171633],{},"继续往后时，因为 0 号扇区已经被 A 占了，所以 E 不能放回 0 号扇区，只能放在下一个合适的位置 1 号扇区。",[15,171635,171636],{},"于是可以安排：",[15,171638,171639],{},"E 在 1 号扇区",[15,171641,171642],{},"F 在 3 号扇区",[15,171644,171645],{},"G 在 5 号扇区",[15,171647,171648],{},"H 在 7 号扇区",[15,171650,171651],{},"所以沿磁盘旋转方向的物理存放顺序为：",[15,171653,171654],{},"A E B F C G D H",[15,171656,171657],{},"也可以写成扇区表：",[13837,171659,171660,171683],{},[13840,171661,171662],{},[13843,171663,171664,171667,171669,171671,171673,171675,171677,171679,171681],{},[13846,171665,171666],{},"扇区号",[13846,171668,459],{},[13846,171670,640],{},[13846,171672,768],{},[13846,171674,786],{},[13846,171676,15932],{},[13846,171678,18136],{},[13846,171680,36656],{},[13846,171682,36723],{},[13853,171684,171685],{},[13843,171686,171687,171690,171692,171694,171696,171698,171700,171702,171704],{},[13858,171688,171689],{},"记录",[13858,171691,54151],{},[13858,171693,96058],{},[13858,171695,54224],{},[13858,171697,51834],{},[13858,171699,47384],{},[13858,171701,97709],{},[13858,171703,91994],{},[13858,171705,118113],{},[15,171707,171708],{},"这样逻辑处理顺序仍然是：",[15,171710,171711],{},"A, B, C, D, E, F, G, H",[15,171713,171714],{},"只是它们在磁盘上的物理位置被交错安排了。",[32188,171716],{},[253,171718,171719],{"id":171719},"优化后的时间计算",[15,171721,171722],{},"访问 A：",[15,171724,171725,171726,171765],{},"A 从 ",[34,171727,171729,171747],{"className":171728},[31642],[34,171730,171732],{"className":171731},[31646],[31648,171733,171734],{"xmlns":31650},[31652,171735,171736,171744],{},[31655,171737,171738,171740,171742],{},[31749,171739,459],{},[31658,171741,31759],{},[31658,171743,31747],{},[31665,171745,171746],{"encoding":31667},"0ms",[34,171748,171750],{"className":171749,"ariaHidden":758},[31672],[34,171751,171753,171756,171759,171762],{"className":171752},[31676],[34,171754],{"className":171755,"style":32617},[31680],[34,171757,459],{"className":171758},[31685],[34,171760,31759],{"className":171761},[31685,31686],[34,171763,31747],{"className":171764},[31685,31686]," 开始读，读完并处理完需要：",[15,171767,171768],{},[34,171769,171771,171788],{"className":171770},[31642],[34,171772,171774],{"className":171773},[31646],[31648,171775,171776],{"xmlns":31650},[31652,171777,171778,171786],{},[31655,171779,171780,171782,171784],{},[31749,171781,170264],{},[31658,171783,31759],{},[31658,171785,31747],{},[31665,171787,170408],{"encoding":31667},[34,171789,171791],{"className":171790,"ariaHidden":758},[31672],[34,171792,171794,171797,171800,171803],{"className":171793},[31676],[34,171795],{"className":171796,"style":32617},[31680],[34,171798,170264],{"className":171799},[31685],[34,171801,31759],{"className":171802},[31685,31686],[34,171804,31747],{"className":171805},[31685,31686],[15,171807,171808],{},"B 在 2 号扇区，开始时间是：",[15,171810,171811],{},[34,171812,171814,171844],{"className":171813},[31642],[34,171815,171817],{"className":171816},[31646],[31648,171818,171819],{"xmlns":31650},[31652,171820,171821,171841],{},[31655,171822,171823,171825,171827,171829,171831,171833,171835,171837,171839],{},[31749,171824,768],{},[31662,171826,42071],{},[31749,171828,167259],{},[31658,171830,31759],{},[31658,171832,31747],{},[31662,171834,202],{},[31749,171836,18136],{},[31658,171838,31759],{},[31658,171840,31747],{},[31665,171842,171843],{"encoding":31667},"2 \\times 2.5ms = 5ms",[34,171845,171847,171865,171889],{"className":171846,"ariaHidden":758},[31672],[34,171848,171850,171853,171856,171859,171862],{"className":171849},[31676],[34,171851],{"className":171852,"style":35942},[31680],[34,171854,768],{"className":171855},[31685],[34,171857],{"className":171858,"style":32604},[31845],[34,171860,42071],{"className":171861},[32024],[34,171863],{"className":171864,"style":32604},[31845],[34,171866,171868,171871,171874,171877,171880,171883,171886],{"className":171867},[31676],[34,171869],{"className":171870,"style":32617},[31680],[34,171872,167259],{"className":171873},[31685],[34,171875,31759],{"className":171876},[31685,31686],[34,171878,31747],{"className":171879},[31685,31686],[34,171881],{"className":171882,"style":31846},[31845],[34,171884,202],{"className":171885},[31850],[34,171887],{"className":171888,"style":31846},[31845],[34,171890,171892,171895,171898,171901],{"className":171891},[31676],[34,171893],{"className":171894,"style":32617},[31680],[34,171896,18136],{"className":171897},[31685],[34,171899,31759],{"className":171900},[31685,31686],[34,171902,31747],{"className":171903},[31685,31686],[15,171905,171906],{},"所以刚好赶得上 B。",[15,171908,171909],{},"A 到 B 的起点间隔：",[15,171911,171912],{},[34,171913,171915,171933],{"className":171914},[31642],[34,171916,171918],{"className":171917},[31646],[31648,171919,171920],{"xmlns":31650},[31652,171921,171922,171930],{},[31655,171923,171924,171926,171928],{},[31749,171925,18136],{},[31658,171927,31759],{},[31658,171929,31747],{},[31665,171931,171932],{"encoding":31667},"5ms",[34,171934,171936],{"className":171935,"ariaHidden":758},[31672],[34,171937,171939,171942,171945,171948],{"className":171938},[31676],[34,171940],{"className":171941,"style":32617},[31680],[34,171943,18136],{"className":171944},[31685],[34,171946,31759],{"className":171947},[31685,31686],[34,171949,31747],{"className":171950},[31685,31686],[15,171952,171953],{},"同理：",[15,171955,171956],{},"B 到 C：",[15,171958,171959],{},[34,171960,171962,171979],{"className":171961},[31642],[34,171963,171965],{"className":171964},[31646],[31648,171966,171967],{"xmlns":31650},[31652,171968,171969,171977],{},[31655,171970,171971,171973,171975],{},[31749,171972,18136],{},[31658,171974,31759],{},[31658,171976,31747],{},[31665,171978,171932],{"encoding":31667},[34,171980,171982],{"className":171981,"ariaHidden":758},[31672],[34,171983,171985,171988,171991,171994],{"className":171984},[31676],[34,171986],{"className":171987,"style":32617},[31680],[34,171989,18136],{"className":171990},[31685],[34,171992,31759],{"className":171993},[31685,31686],[34,171995,31747],{"className":171996},[31685,31686],[15,171998,171999],{},"C 到 D：",[15,172001,172002],{},[34,172003,172005,172022],{"className":172004},[31642],[34,172006,172008],{"className":172007},[31646],[31648,172009,172010],{"xmlns":31650},[31652,172011,172012,172020],{},[31655,172013,172014,172016,172018],{},[31749,172015,18136],{},[31658,172017,31759],{},[31658,172019,31747],{},[31665,172021,171932],{"encoding":31667},[34,172023,172025],{"className":172024,"ariaHidden":758},[31672],[34,172026,172028,172031,172034,172037],{"className":172027},[31676],[34,172029],{"className":172030,"style":32617},[31680],[34,172032,18136],{"className":172033},[31685],[34,172035,31759],{"className":172036},[31685,31686],[34,172038,31747],{"className":172039},[31685,31686],[15,172041,172042],{},"D 到 E：",[15,172044,172045,172046],{},"D 在 6 号扇区，起点是 ",[34,172047,172049,172067],{"className":172048},[31642],[34,172050,172052],{"className":172051},[31646],[31648,172053,172054],{"xmlns":31650},[31652,172055,172056,172064],{},[31655,172057,172058,172060,172062],{},[31749,172059,95396],{},[31658,172061,31759],{},[31658,172063,31747],{},[31665,172065,172066],{"encoding":31667},"15ms",[34,172068,172070],{"className":172069,"ariaHidden":758},[31672],[34,172071,172073,172076,172079,172082],{"className":172072},[31676],[34,172074],{"className":172075,"style":32617},[31680],[34,172077,95396],{"className":172078},[31685],[34,172080,31759],{"className":172081},[31685,31686],[34,172083,31747],{"className":172084},[31685,31686],[15,172086,172087],{},"读完并处理完 D 到：",[15,172089,172090],{},[34,172091,172093,172128],{"className":172092},[31642],[34,172094,172096],{"className":172095},[31646],[31648,172097,172098],{"xmlns":31650},[31652,172099,172100,172125],{},[31655,172101,172102,172104,172106,172108,172110,172112,172114,172116,172118,172121,172123],{},[31749,172103,95396],{},[31658,172105,31759],{},[31658,172107,31747],{},[31662,172109,223],{},[31749,172111,170264],{},[31658,172113,31759],{},[31658,172115,31747],{},[31662,172117,202],{},[31749,172119,172120],{},"19.5",[31658,172122,31759],{},[31658,172124,31747],{},[31665,172126,172127],{"encoding":31667},"15ms + 4.5ms = 19.5ms",[34,172129,172131,172155,172179],{"className":172130,"ariaHidden":758},[31672],[34,172132,172134,172137,172140,172143,172146,172149,172152],{"className":172133},[31676],[34,172135],{"className":172136,"style":35942},[31680],[34,172138,95396],{"className":172139},[31685],[34,172141,31759],{"className":172142},[31685,31686],[34,172144,31747],{"className":172145},[31685,31686],[34,172147],{"className":172148,"style":32604},[31845],[34,172150,223],{"className":172151},[32024],[34,172153],{"className":172154,"style":32604},[31845],[34,172156,172158,172161,172164,172167,172170,172173,172176],{"className":172157},[31676],[34,172159],{"className":172160,"style":32617},[31680],[34,172162,170264],{"className":172163},[31685],[34,172165,31759],{"className":172166},[31685,31686],[34,172168,31747],{"className":172169},[31685,31686],[34,172171],{"className":172172,"style":31846},[31845],[34,172174,202],{"className":172175},[31850],[34,172177],{"className":172178,"style":31846},[31845],[34,172180,172182,172185,172188,172191],{"className":172181},[31676],[34,172183],{"className":172184,"style":32617},[31680],[34,172186,172120],{"className":172187},[31685],[34,172189,31759],{"className":172190},[31685,31686],[34,172192,31747],{"className":172193},[31685,31686],[15,172195,172196],{},"E 在 1 号扇区，下一次出现时间是：",[15,172198,172199],{},[34,172200,172202,172237],{"className":172201},[31642],[34,172203,172205],{"className":172204},[31646],[31648,172206,172207],{"xmlns":31650},[31652,172208,172209,172234],{},[31655,172210,172211,172213,172215,172217,172219,172221,172223,172225,172227,172230,172232],{},[31749,172212,3249],{},[31658,172214,31759],{},[31658,172216,31747],{},[31662,172218,223],{},[31749,172220,167259],{},[31658,172222,31759],{},[31658,172224,31747],{},[31662,172226,202],{},[31749,172228,172229],{},"22.5",[31658,172231,31759],{},[31658,172233,31747],{},[31665,172235,172236],{"encoding":31667},"20ms + 2.5ms = 22.5ms",[34,172238,172240,172264,172288],{"className":172239,"ariaHidden":758},[31672],[34,172241,172243,172246,172249,172252,172255,172258,172261],{"className":172242},[31676],[34,172244],{"className":172245,"style":35942},[31680],[34,172247,3249],{"className":172248},[31685],[34,172250,31759],{"className":172251},[31685,31686],[34,172253,31747],{"className":172254},[31685,31686],[34,172256],{"className":172257,"style":32604},[31845],[34,172259,223],{"className":172260},[32024],[34,172262],{"className":172263,"style":32604},[31845],[34,172265,172267,172270,172273,172276,172279,172282,172285],{"className":172266},[31676],[34,172268],{"className":172269,"style":32617},[31680],[34,172271,167259],{"className":172272},[31685],[34,172274,31759],{"className":172275},[31685,31686],[34,172277,31747],{"className":172278},[31685,31686],[34,172280],{"className":172281,"style":31846},[31845],[34,172283,202],{"className":172284},[31850],[34,172286],{"className":172287,"style":31846},[31845],[34,172289,172291,172294,172297,172300],{"className":172290},[31676],[34,172292],{"className":172293,"style":32617},[31680],[34,172295,172229],{"className":172296},[31685],[34,172298,31759],{"className":172299},[31685,31686],[34,172301,31747],{"className":172302},[31685,31686],[15,172304,172305],{},"所以 D 到 E 的起点间隔：",[15,172307,172308],{},[34,172309,172311,172346],{"className":172310},[31642],[34,172312,172314],{"className":172313},[31646],[31648,172315,172316],{"xmlns":31650},[31652,172317,172318,172343],{},[31655,172319,172320,172322,172324,172326,172328,172330,172332,172334,172336,172339,172341],{},[31749,172321,172229],{},[31658,172323,31759],{},[31658,172325,31747],{},[31662,172327,32226],{},[31749,172329,95396],{},[31658,172331,31759],{},[31658,172333,31747],{},[31662,172335,202],{},[31749,172337,172338],{},"7.5",[31658,172340,31759],{},[31658,172342,31747],{},[31665,172344,172345],{"encoding":31667},"22.5ms - 15ms = 7.5ms",[34,172347,172349,172373,172397],{"className":172348,"ariaHidden":758},[31672],[34,172350,172352,172355,172358,172361,172364,172367,172370],{"className":172351},[31676],[34,172353],{"className":172354,"style":35942},[31680],[34,172356,172229],{"className":172357},[31685],[34,172359,31759],{"className":172360},[31685,31686],[34,172362,31747],{"className":172363},[31685,31686],[34,172365],{"className":172366,"style":32604},[31845],[34,172368,32226],{"className":172369},[32024],[34,172371],{"className":172372,"style":32604},[31845],[34,172374,172376,172379,172382,172385,172388,172391,172394],{"className":172375},[31676],[34,172377],{"className":172378,"style":32617},[31680],[34,172380,95396],{"className":172381},[31685],[34,172383,31759],{"className":172384},[31685,31686],[34,172386,31747],{"className":172387},[31685,31686],[34,172389],{"className":172390,"style":31846},[31845],[34,172392,202],{"className":172393},[31850],[34,172395],{"className":172396,"style":31846},[31845],[34,172398,172400,172403,172406,172409],{"className":172399},[31676],[34,172401],{"className":172402,"style":32617},[31680],[34,172404,172338],{"className":172405},[31685],[34,172407,31759],{"className":172408},[31685,31686],[34,172410,31747],{"className":172411},[31685,31686],[15,172413,172414],{},"然后：",[15,172416,172417],{},"E 到 F：",[15,172419,172420],{},[34,172421,172423,172440],{"className":172422},[31642],[34,172424,172426],{"className":172425},[31646],[31648,172427,172428],{"xmlns":31650},[31652,172429,172430,172438],{},[31655,172431,172432,172434,172436],{},[31749,172433,18136],{},[31658,172435,31759],{},[31658,172437,31747],{},[31665,172439,171932],{"encoding":31667},[34,172441,172443],{"className":172442,"ariaHidden":758},[31672],[34,172444,172446,172449,172452,172455],{"className":172445},[31676],[34,172447],{"className":172448,"style":32617},[31680],[34,172450,18136],{"className":172451},[31685],[34,172453,31759],{"className":172454},[31685,31686],[34,172456,31747],{"className":172457},[31685,31686],[15,172459,172460],{},"F 到 G：",[15,172462,172463],{},[34,172464,172466,172483],{"className":172465},[31642],[34,172467,172469],{"className":172468},[31646],[31648,172470,172471],{"xmlns":31650},[31652,172472,172473,172481],{},[31655,172474,172475,172477,172479],{},[31749,172476,18136],{},[31658,172478,31759],{},[31658,172480,31747],{},[31665,172482,171932],{"encoding":31667},[34,172484,172486],{"className":172485,"ariaHidden":758},[31672],[34,172487,172489,172492,172495,172498],{"className":172488},[31676],[34,172490],{"className":172491,"style":32617},[31680],[34,172493,18136],{"className":172494},[31685],[34,172496,31759],{"className":172497},[31685,31686],[34,172499,31747],{"className":172500},[31685,31686],[15,172502,172503],{},"G 到 H：",[15,172505,172506],{},[34,172507,172509,172526],{"className":172508},[31642],[34,172510,172512],{"className":172511},[31646],[31648,172513,172514],{"xmlns":31650},[31652,172515,172516,172524],{},[31655,172517,172518,172520,172522],{},[31749,172519,18136],{},[31658,172521,31759],{},[31658,172523,31747],{},[31665,172525,171932],{"encoding":31667},[34,172527,172529],{"className":172528,"ariaHidden":758},[31672],[34,172530,172532,172535,172538,172541],{"className":172531},[31676],[34,172533],{"className":172534,"style":32617},[31680],[34,172536,18136],{"className":172537},[31685],[34,172539,31759],{"className":172540},[31685,31686],[34,172542,31747],{"className":172543},[31685,31686],[15,172545,172546],{},"所以从 A 起点到 H 起点的间隔总和为：",[15,172548,172549],{},[34,172550,172552,172599],{"className":172551},[31642],[34,172553,172555],{"className":172554},[31646],[31648,172556,172557],{"xmlns":31650},[31652,172558,172559,172596],{},[31655,172560,172561,172563,172565,172567,172569,172571,172573,172575,172577,172579,172581,172583,172585,172587,172589,172592,172594],{},[31749,172562,18136],{},[31662,172564,223],{},[31749,172566,18136],{},[31662,172568,223],{},[31749,172570,18136],{},[31662,172572,223],{},[31749,172574,172338],{},[31662,172576,223],{},[31749,172578,18136],{},[31662,172580,223],{},[31749,172582,18136],{},[31662,172584,223],{},[31749,172586,18136],{},[31662,172588,202],{},[31749,172590,172591],{},"37.5",[31658,172593,31759],{},[31658,172595,31747],{},[31665,172597,172598],{"encoding":31667},"5 + 5 + 5 + 7.5 + 5 + 5 + 5 = 37.5ms",[34,172600,172602,172620,172638,172656,172674,172692,172710,172728],{"className":172601,"ariaHidden":758},[31672],[34,172603,172605,172608,172611,172614,172617],{"className":172604},[31676],[34,172606],{"className":172607,"style":35942},[31680],[34,172609,18136],{"className":172610},[31685],[34,172612],{"className":172613,"style":32604},[31845],[34,172615,223],{"className":172616},[32024],[34,172618],{"className":172619,"style":32604},[31845],[34,172621,172623,172626,172629,172632,172635],{"className":172622},[31676],[34,172624],{"className":172625,"style":35942},[31680],[34,172627,18136],{"className":172628},[31685],[34,172630],{"className":172631,"style":32604},[31845],[34,172633,223],{"className":172634},[32024],[34,172636],{"className":172637,"style":32604},[31845],[34,172639,172641,172644,172647,172650,172653],{"className":172640},[31676],[34,172642],{"className":172643,"style":35942},[31680],[34,172645,18136],{"className":172646},[31685],[34,172648],{"className":172649,"style":32604},[31845],[34,172651,223],{"className":172652},[32024],[34,172654],{"className":172655,"style":32604},[31845],[34,172657,172659,172662,172665,172668,172671],{"className":172658},[31676],[34,172660],{"className":172661,"style":35942},[31680],[34,172663,172338],{"className":172664},[31685],[34,172666],{"className":172667,"style":32604},[31845],[34,172669,223],{"className":172670},[32024],[34,172672],{"className":172673,"style":32604},[31845],[34,172675,172677,172680,172683,172686,172689],{"className":172676},[31676],[34,172678],{"className":172679,"style":35942},[31680],[34,172681,18136],{"className":172682},[31685],[34,172684],{"className":172685,"style":32604},[31845],[34,172687,223],{"className":172688},[32024],[34,172690],{"className":172691,"style":32604},[31845],[34,172693,172695,172698,172701,172704,172707],{"className":172694},[31676],[34,172696],{"className":172697,"style":35942},[31680],[34,172699,18136],{"className":172700},[31685],[34,172702],{"className":172703,"style":32604},[31845],[34,172705,223],{"className":172706},[32024],[34,172708],{"className":172709,"style":32604},[31845],[34,172711,172713,172716,172719,172722,172725],{"className":172712},[31676],[34,172714],{"className":172715,"style":32617},[31680],[34,172717,18136],{"className":172718},[31685],[34,172720],{"className":172721,"style":31846},[31845],[34,172723,202],{"className":172724},[31850],[34,172726],{"className":172727,"style":31846},[31845],[34,172729,172731,172734,172737,172740],{"className":172730},[31676],[34,172732],{"className":172733,"style":32617},[31680],[34,172735,172591],{"className":172736},[31685],[34,172738,31759],{"className":172739},[31685,31686],[34,172741,31747],{"className":172742},[31685,31686],[15,172744,172745],{},"H 本身还需要读取并处理：",[15,172747,172748],{},[34,172749,172751,172784],{"className":172750},[31642],[34,172752,172754],{"className":172753},[31646],[31648,172755,172756],{"xmlns":31650},[31652,172757,172758,172782],{},[31655,172759,172760,172762,172764,172766,172768,172770,172772,172774,172776,172778,172780],{},[31749,172761,167259],{},[31658,172763,31759],{},[31658,172765,31747],{},[31662,172767,223],{},[31749,172769,768],{},[31658,172771,31759],{},[31658,172773,31747],{},[31662,172775,202],{},[31749,172777,170264],{},[31658,172779,31759],{},[31658,172781,31747],{},[31665,172783,170271],{"encoding":31667},[34,172785,172787,172811,172835],{"className":172786,"ariaHidden":758},[31672],[34,172788,172790,172793,172796,172799,172802,172805,172808],{"className":172789},[31676],[34,172791],{"className":172792,"style":35942},[31680],[34,172794,167259],{"className":172795},[31685],[34,172797,31759],{"className":172798},[31685,31686],[34,172800,31747],{"className":172801},[31685,31686],[34,172803],{"className":172804,"style":32604},[31845],[34,172806,223],{"className":172807},[32024],[34,172809],{"className":172810,"style":32604},[31845],[34,172812,172814,172817,172820,172823,172826,172829,172832],{"className":172813},[31676],[34,172815],{"className":172816,"style":32617},[31680],[34,172818,768],{"className":172819},[31685],[34,172821,31759],{"className":172822},[31685,31686],[34,172824,31747],{"className":172825},[31685,31686],[34,172827],{"className":172828,"style":31846},[31845],[34,172830,202],{"className":172831},[31850],[34,172833],{"className":172834,"style":31846},[31845],[34,172836,172838,172841,172844,172847],{"className":172837},[31676],[34,172839],{"className":172840,"style":32617},[31680],[34,172842,170264],{"className":172843},[31685],[34,172845,31759],{"className":172846},[31685,31686],[34,172848,31747],{"className":172849},[31685,31686],[15,172851,172852],{},"所以总时间：",[15,172854,172855],{},[34,172856,172858,172892],{"className":172857},[31642],[34,172859,172861],{"className":172860},[31646],[31648,172862,172863],{"xmlns":31650},[31652,172864,172865,172889],{},[31655,172866,172867,172869,172871,172873,172875,172877,172879,172881,172883,172885,172887],{},[31749,172868,172591],{},[31658,172870,31759],{},[31658,172872,31747],{},[31662,172874,223],{},[31749,172876,170264],{},[31658,172878,31759],{},[31658,172880,31747],{},[31662,172882,202],{},[31749,172884,95547],{},[31658,172886,31759],{},[31658,172888,31747],{},[31665,172890,172891],{"encoding":31667},"37.5ms + 4.5ms = 42ms",[34,172893,172895,172919,172943],{"className":172894,"ariaHidden":758},[31672],[34,172896,172898,172901,172904,172907,172910,172913,172916],{"className":172897},[31676],[34,172899],{"className":172900,"style":35942},[31680],[34,172902,172591],{"className":172903},[31685],[34,172905,31759],{"className":172906},[31685,31686],[34,172908,31747],{"className":172909},[31685,31686],[34,172911],{"className":172912,"style":32604},[31845],[34,172914,223],{"className":172915},[32024],[34,172917],{"className":172918,"style":32604},[31845],[34,172920,172922,172925,172928,172931,172934,172937,172940],{"className":172921},[31676],[34,172923],{"className":172924,"style":32617},[31680],[34,172926,170264],{"className":172927},[31685],[34,172929,31759],{"className":172930},[31685,31686],[34,172932,31747],{"className":172933},[31685,31686],[34,172935],{"className":172936,"style":31846},[31845],[34,172938,202],{"className":172939},[31850],[34,172941],{"className":172942,"style":31846},[31845],[34,172944,172946,172949,172952,172955],{"className":172945},[31676],[34,172947],{"className":172948,"style":32617},[31680],[34,172950,95547],{"className":172951},[31685],[34,172953,31759],{"className":172954},[31685,31686],[34,172956,31747],{"className":172957},[31685,31686],[32188,172959],{},[253,172961,172963],{"id":172962},"b-的答案","b 的答案",[15,172965,172966],{},"最优物理存放顺序可以为：",[15,172968,171654],{},[15,172970,172971],{},"顺序处理 A、B、C、D、E、F、G、H 的最短时间为：",[15,172973,172974],{},[34,172975,172977,172995],{"className":172976},[31642],[34,172978,172980],{"className":172979},[31646],[31648,172981,172982],{"xmlns":31650},[31652,172983,172984,172992],{},[31655,172985,172986,172988,172990],{},[31749,172987,95547],{},[31658,172989,31759],{},[31658,172991,31747],{},[31665,172993,172994],{"encoding":31667},"42ms",[34,172996,172998],{"className":172997,"ariaHidden":758},[31672],[34,172999,173001,173004,173007,173010],{"className":173000},[31676],[34,173002],{"className":173003,"style":32617},[31680],[34,173005,95547],{"className":173006},[31685],[34,173008,31759],{"className":173009},[31685,31686],[34,173011,31747],{"className":173012},[31685,31686],[32188,173014],{},[253,173016,173018],{"id":173017},"第-4-题最终答案","第 4 题最终答案",[15,173020,173021],{},"a. 顺序存放时：",[15,173023,173024],{},[34,173025,173027,173044],{"className":173026},[31642],[34,173028,173030],{"className":173029},[31646],[31648,173031,173032],{"xmlns":31650},[31652,173033,173034,173042],{},[31655,173035,173036,173038,173040],{},[31749,173037,171246],{},[31658,173039,31759],{},[31658,173041,31747],{},[31665,173043,171351],{"encoding":31667},[34,173045,173047],{"className":173046,"ariaHidden":758},[31672],[34,173048,173050,173053,173056,173059],{"className":173049},[31676],[34,173051],{"className":173052,"style":32617},[31680],[34,173054,171246],{"className":173055},[31685],[34,173057,31759],{"className":173058},[31685,31686],[34,173060,31747],{"className":173061},[31685,31686],[15,173063,173064],{},"b. 采用交错存放，例如：",[15,173066,171654],{},[15,173068,173069],{},"最短处理时间：",[15,173071,173072],{},[34,173073,173075,173092],{"className":173074},[31642],[34,173076,173078],{"className":173077},[31646],[31648,173079,173080],{"xmlns":31650},[31652,173081,173082,173090],{},[31655,173083,173084,173086,173088],{},[31749,173085,95547],{},[31658,173087,31759],{},[31658,173089,31747],{},[31665,173091,172994],{"encoding":31667},[34,173093,173095],{"className":173094,"ariaHidden":758},[31672],[34,173096,173098,173101,173104,173107],{"className":173097},[31676],[34,173099],{"className":173100,"style":32617},[31680],[34,173102,95547],{"className":173103},[31685],[34,173105,31759],{"className":173106},[31685,31686],[34,173108,31747],{"className":173109},[31685,31686],[32188,173111],{},[19,173113,173115],{"id":173114},"_5-计算存储-300000-条逻辑记录需要多少磁盘空间","5. 计算存储 300000 条逻辑记录需要多少磁盘空间",[253,173117,162043],{"id":173118},"题目内容-4",[15,173120,173121],{},"Calculate how much disk space (in sectors, tracks, and surfaces) will be required to store 300,000 120-byte logical records if the disk is fixed-sector with 512 bytes\u002Fsector, with 96 sectors\u002Ftrack, 110 tracks per surface, and 8 usable surfaces. Ignore any file header record(s) and track indexes, and assume that records cannot span two sectors.",[15,173123,162049],{},[15,173125,173126],{},"计算存储 300000 条逻辑记录需要多少磁盘空间，要求用扇区数、磁道数、盘面数表示。",[15,173128,173129],{},"已知：",[15,173131,173132],{},"每条逻辑记录大小：",[15,173134,173135,167323],{},[34,173136,173138,173151],{"className":173137},[31642],[34,173139,173141],{"className":173140},[31646],[31648,173142,173143],{"xmlns":31650},[31652,173144,173145,173149],{},[31655,173146,173147],{},[31749,173148,162217],{},[31665,173150,162217],{"encoding":31667},[34,173152,173154],{"className":173153,"ariaHidden":758},[31672],[34,173155,173157,173160],{"className":173156},[31676],[34,173158],{"className":173159,"style":32617},[31680],[34,173161,162217],{"className":173162},[31685],[15,173164,167291],{},[15,173166,173167,167323],{},[34,173168,173170,173183],{"className":173169},[31642],[34,173171,173173],{"className":173172},[31646],[31648,173174,173175],{"xmlns":31650},[31652,173176,173177,173181],{},[31655,173178,173179],{},[31749,173180,167308],{},[31665,173182,167308],{"encoding":31667},[34,173184,173186],{"className":173185,"ariaHidden":758},[31672],[34,173187,173189,173192],{"className":173188},[31676],[34,173190],{"className":173191,"style":32617},[31680],[34,173193,167308],{"className":173194},[31685],[15,173196,173197],{},"每磁道扇区数：",[15,173199,173200,173228],{},[34,173201,173203,173216],{"className":173202},[31642],[34,173204,173206],{"className":173205},[31646],[31648,173207,173208],{"xmlns":31650},[31652,173209,173210,173214],{},[31655,173211,173212],{},[31749,173213,167343],{},[31665,173215,167343],{"encoding":31667},[34,173217,173219],{"className":173218,"ariaHidden":758},[31672],[34,173220,173222,173225],{"className":173221},[31676],[34,173223],{"className":173224,"style":32617},[31680],[34,173226,167343],{"className":173227},[31685]," 个扇区\u002F磁道",[15,173230,173231],{},"每盘面磁道数：",[15,173233,173234,173262],{},[34,173235,173237,173250],{"className":173236},[31642],[34,173238,173240],{"className":173239},[31646],[31648,173241,173242],{"xmlns":31650},[31652,173243,173244,173248],{},[31655,173245,173246],{},[31749,173247,162188],{},[31665,173249,162188],{"encoding":31667},[34,173251,173253],{"className":173252,"ariaHidden":758},[31672],[34,173254,173256,173259],{"className":173255},[31676],[34,173257],{"className":173258,"style":32617},[31680],[34,173260,162188],{"className":173261},[31685]," 个磁道\u002F盘面",[15,173264,173265],{},"可用盘面数：",[15,173267,173268,173296],{},[34,173269,173271,173284],{"className":173270},[31642],[34,173272,173274],{"className":173273},[31646],[31648,173275,173276],{"xmlns":31650},[31652,173277,173278,173282],{},[31655,173279,173280],{},[31749,173281,60525],{},[31665,173283,60525],{"encoding":31667},[34,173285,173287],{"className":173286,"ariaHidden":758},[31672],[34,173288,173290,173293],{"className":173289},[31676],[34,173291],{"className":173292,"style":32617},[31680],[34,173294,60525],{"className":173295},[31685]," 个",[15,173298,173299],{},"忽略文件头和磁道索引，并且假设一条记录不能跨越两个扇区。",[32188,173301],{},[253,173303,162066],{"id":173304},"解题核心-4",[15,173306,173307],{},"因为一条记录不能跨扇区，所以一个扇区里能放多少条记录，要用向下取整。",[15,173309,173310],{},"每扇区可放记录数：",[15,173312,173313],{},[34,173314,173316,173344],{"className":173315},[31642],[34,173317,173319],{"className":173318},[31646],[31648,173320,173321],{"xmlns":31650},[31652,173322,173323,173341],{},[31655,173324,173325,173328,173330,173332,173334,173337,173339],{},[31662,173326,173327],{"stretchy":702},"⌊",[31749,173329,167308],{},[31658,173331,38080],{"mathvariant":38079},[31749,173333,162217],{},[31662,173335,173336],{"stretchy":702},"⌋",[31662,173338,202],{},[31749,173340,15932],{},[31665,173342,173343],{"encoding":31667},"\\lfloor 512 \u002F 120 \\rfloor = 4",[34,173345,173347,173372],{"className":173346,"ariaHidden":758},[31672],[34,173348,173350,173353,173356,173360,173363,173366,173369],{"className":173349},[31676],[34,173351],{"className":173352,"style":31774},[31680],[34,173354,173327],{"className":173355},[31782],[34,173357,173359],{"className":173358},[31685],"512\u002F120",[34,173361,173336],{"className":173362},[31690],[34,173364],{"className":173365,"style":31846},[31845],[34,173367,202],{"className":173368},[31850],[34,173370],{"className":173371,"style":31846},[31845],[34,173373,173375,173378],{"className":173374},[31676],[34,173376],{"className":173377,"style":32617},[31680],[34,173379,15932],{"className":173380},[31685],[15,173382,131336],{},[15,173384,173385],{},[34,173386,173388,173411],{"className":173387},[31642],[34,173389,173391],{"className":173390},[31646],[31648,173392,173393],{"xmlns":31650},[31652,173394,173395,173408],{},[31655,173396,173397,173399,173401,173403,173405],{},[31749,173398,15932],{},[31662,173400,42071],{},[31749,173402,162217],{},[31662,173404,202],{},[31749,173406,173407],{},"480",[31665,173409,173410],{"encoding":31667},"4 \\times 120 = 480",[34,173412,173414,173432,173450],{"className":173413,"ariaHidden":758},[31672],[34,173415,173417,173420,173423,173426,173429],{"className":173416},[31676],[34,173418],{"className":173419,"style":35942},[31680],[34,173421,15932],{"className":173422},[31685],[34,173424],{"className":173425,"style":32604},[31845],[34,173427,42071],{"className":173428},[32024],[34,173430],{"className":173431,"style":32604},[31845],[34,173433,173435,173438,173441,173444,173447],{"className":173434},[31676],[34,173436],{"className":173437,"style":32617},[31680],[34,173439,162217],{"className":173440},[31685],[34,173442],{"className":173443,"style":31846},[31845],[34,173445,202],{"className":173446},[31850],[34,173448],{"className":173449,"style":31846},[31845],[34,173451,173453,173456],{"className":173452},[31676],[34,173454],{"className":173455,"style":32617},[31680],[34,173457,173407],{"className":173458},[31685],[15,173460,173461],{},"一个扇区剩余：",[15,173463,173464],{},[34,173465,173467,173490],{"className":173466},[31642],[34,173468,173470],{"className":173469},[31646],[31648,173471,173472],{"xmlns":31650},[31652,173473,173474,173487],{},[31655,173475,173476,173478,173480,173482,173484],{},[31749,173477,167308],{},[31662,173479,32226],{},[31749,173481,173407],{},[31662,173483,202],{},[31749,173485,173486],{},"32",[31665,173488,173489],{"encoding":31667},"512 - 480 = 32",[34,173491,173493,173511,173529],{"className":173492,"ariaHidden":758},[31672],[34,173494,173496,173499,173502,173505,173508],{"className":173495},[31676],[34,173497],{"className":173498,"style":35942},[31680],[34,173500,167308],{"className":173501},[31685],[34,173503],{"className":173504,"style":32604},[31845],[34,173506,32226],{"className":173507},[32024],[34,173509],{"className":173510,"style":32604},[31845],[34,173512,173514,173517,173520,173523,173526],{"className":173513},[31676],[34,173515],{"className":173516,"style":32617},[31680],[34,173518,173407],{"className":173519},[31685],[34,173521],{"className":173522,"style":31846},[31845],[34,173524,202],{"className":173525},[31850],[34,173527],{"className":173528,"style":31846},[31845],[34,173530,173532,173535],{"className":173531},[31676],[34,173533],{"className":173534,"style":32617},[31680],[34,173536,173486],{"className":173537},[31685],[15,173539,173540],{},"字节无法再放一条完整记录，所以浪费掉。",[32188,173542],{},[253,173544,173545],{"id":173545},"计算需要的扇区数",[15,173547,173548],{},"共有 300000 条记录，每个扇区放 4 条。",[15,173550,173551],{},"需要扇区数：",[15,173553,173554],{},[34,173555,173557,173581],{"className":173556},[31642],[34,173558,173560],{"className":173559},[31646],[31648,173561,173562],{"xmlns":31650},[31652,173563,173564,173578],{},[31655,173565,173566,173569,173571,173573,173575],{},[31749,173567,173568],{},"300000",[31658,173570,38080],{"mathvariant":38079},[31749,173572,15932],{},[31662,173574,202],{},[31749,173576,173577],{},"75000",[31665,173579,173580],{"encoding":31667},"300000 \u002F 4 = 75000",[34,173582,173584,173603],{"className":173583,"ariaHidden":758},[31672],[34,173585,173587,173590,173594,173597,173600],{"className":173586},[31676],[34,173588],{"className":173589,"style":31774},[31680],[34,173591,173593],{"className":173592},[31685],"300000\u002F4",[34,173595],{"className":173596,"style":31846},[31845],[34,173598,202],{"className":173599},[31850],[34,173601],{"className":173602,"style":31846},[31845],[34,173604,173606,173609],{"className":173605},[31676],[34,173607],{"className":173608,"style":32617},[31680],[34,173610,173577],{"className":173611},[31685],[15,173613,173614],{},"所以需要：",[15,173616,173617],{},[34,173618,173620,173633],{"className":173619},[31642],[34,173621,173623],{"className":173622},[31646],[31648,173624,173625],{"xmlns":31650},[31652,173626,173627,173631],{},[31655,173628,173629],{},[31749,173630,173577],{},[31665,173632,173577],{"encoding":31667},[34,173634,173636],{"className":173635,"ariaHidden":758},[31672],[34,173637,173639,173642],{"className":173638},[31676],[34,173640],{"className":173641,"style":32617},[31680],[34,173643,173577],{"className":173644},[31685],[15,173646,171595],{},[32188,173648],{},[253,173650,173651],{"id":173651},"计算需要的磁道数",[15,173653,173654],{},"每个磁道有 96 个扇区。",[15,173656,173657],{},"需要磁道数：",[15,173659,173660],{},[34,173661,173663,173687],{"className":173662},[31642],[34,173664,173666],{"className":173665},[31646],[31648,173667,173668],{"xmlns":31650},[31652,173669,173670,173684],{},[31655,173671,173672,173675,173677,173679,173681],{},[31662,173673,173674],{"stretchy":702},"⌈",[31749,173676,173577],{},[31658,173678,38080],{"mathvariant":38079},[31749,173680,167343],{},[31662,173682,173683],{"stretchy":702},"⌉",[31665,173685,173686],{"encoding":31667},"\\lceil 75000 \u002F 96 \\rceil",[34,173688,173690],{"className":173689,"ariaHidden":758},[31672],[34,173691,173693,173696,173699,173703],{"className":173692},[31676],[34,173694],{"className":173695,"style":31774},[31680],[34,173697,173674],{"className":173698},[31782],[34,173700,173702],{"className":173701},[31685],"75000\u002F96",[34,173704,173683],{"className":173705},[31690],[15,173707,173708],{},"先计算：",[15,173710,173711],{},[34,173712,173714,173737],{"className":173713},[31642],[34,173715,173717],{"className":173716},[31646],[31648,173718,173719],{"xmlns":31650},[31652,173720,173721,173734],{},[31655,173722,173723,173725,173727,173729,173731],{},[31749,173724,173577],{},[31658,173726,38080],{"mathvariant":38079},[31749,173728,167343],{},[31662,173730,202],{},[31749,173732,173733],{},"781.25",[31665,173735,173736],{"encoding":31667},"75000 \u002F 96 = 781.25",[34,173738,173740,173758],{"className":173739,"ariaHidden":758},[31672],[34,173741,173743,173746,173749,173752,173755],{"className":173742},[31676],[34,173744],{"className":173745,"style":31774},[31680],[34,173747,173702],{"className":173748},[31685],[34,173750],{"className":173751,"style":31846},[31845],[34,173753,202],{"className":173754},[31850],[34,173756],{"className":173757,"style":31846},[31845],[34,173759,173761,173764],{"className":173760},[31676],[34,173762],{"className":173763,"style":32617},[31680],[34,173765,173733],{"className":173766},[31685],[15,173768,173769],{},"因为磁道数必须是整数，所以向上取整：",[15,173771,173772],{},[34,173773,173775,173798],{"className":173774},[31642],[34,173776,173778],{"className":173777},[31646],[31648,173779,173780],{"xmlns":31650},[31652,173781,173782,173795],{},[31655,173783,173784,173786,173788,173790,173792],{},[31662,173785,173674],{"stretchy":702},[31749,173787,173733],{},[31662,173789,173683],{"stretchy":702},[31662,173791,202],{},[31749,173793,173794],{},"782",[31665,173796,173797],{"encoding":31667},"\\lceil 781.25 \\rceil = 782",[34,173799,173801,173825],{"className":173800,"ariaHidden":758},[31672],[34,173802,173804,173807,173810,173813,173816,173819,173822],{"className":173803},[31676],[34,173805],{"className":173806,"style":31774},[31680],[34,173808,173674],{"className":173809},[31782],[34,173811,173733],{"className":173812},[31685],[34,173814,173683],{"className":173815},[31690],[34,173817],{"className":173818,"style":31846},[31845],[34,173820,202],{"className":173821},[31850],[34,173823],{"className":173824,"style":31846},[31845],[34,173826,173828,173831],{"className":173827},[31676],[34,173829],{"className":173830,"style":32617},[31680],[34,173832,173794],{"className":173833},[31685],[15,173835,173614],{},[15,173837,173838],{},[34,173839,173841,173854],{"className":173840},[31642],[34,173842,173844],{"className":173843},[31646],[31648,173845,173846],{"xmlns":31650},[31652,173847,173848,173852],{},[31655,173849,173850],{},[31749,173851,173794],{},[31665,173853,173794],{"encoding":31667},[34,173855,173857],{"className":173856,"ariaHidden":758},[31672],[34,173858,173860,173863],{"className":173859},[31676],[34,173861],{"className":173862,"style":32617},[31680],[34,173864,173794],{"className":173865},[31685],[15,173867,173868],{},"个磁道。",[15,173870,173871],{},"也可以理解为：",[15,173873,173874],{},"781 个完整磁道加 24 个扇区。",[15,173876,131336],{},[15,173878,173879],{},[34,173880,173882,173906],{"className":173881},[31642],[34,173883,173885],{"className":173884},[31646],[31648,173886,173887],{"xmlns":31650},[31652,173888,173889,173903],{},[31655,173890,173891,173894,173896,173898,173900],{},[31749,173892,173893],{},"781",[31662,173895,42071],{},[31749,173897,167343],{},[31662,173899,202],{},[31749,173901,173902],{},"74976",[31665,173904,173905],{"encoding":31667},"781 \\times 96 = 74976",[34,173907,173909,173927,173945],{"className":173908,"ariaHidden":758},[31672],[34,173910,173912,173915,173918,173921,173924],{"className":173911},[31676],[34,173913],{"className":173914,"style":35942},[31680],[34,173916,173893],{"className":173917},[31685],[34,173919],{"className":173920,"style":32604},[31845],[34,173922,42071],{"className":173923},[32024],[34,173925],{"className":173926,"style":32604},[31845],[34,173928,173930,173933,173936,173939,173942],{"className":173929},[31676],[34,173931],{"className":173932,"style":32617},[31680],[34,173934,167343],{"className":173935},[31685],[34,173937],{"className":173938,"style":31846},[31845],[34,173940,202],{"className":173941},[31850],[34,173943],{"className":173944,"style":31846},[31845],[34,173946,173948,173951],{"className":173947},[31676],[34,173949],{"className":173950,"style":32617},[31680],[34,173952,173902],{"className":173953},[31685],[15,173955,173956],{},"剩余扇区：",[15,173958,173959],{},[34,173960,173962,173984],{"className":173961},[31642],[34,173963,173965],{"className":173964},[31646],[31648,173966,173967],{"xmlns":31650},[31652,173968,173969,173981],{},[31655,173970,173971,173973,173975,173977,173979],{},[31749,173972,173577],{},[31662,173974,32226],{},[31749,173976,173902],{},[31662,173978,202],{},[31749,173980,96087],{},[31665,173982,173983],{"encoding":31667},"75000 - 74976 = 24",[34,173985,173987,174005,174023],{"className":173986,"ariaHidden":758},[31672],[34,173988,173990,173993,173996,173999,174002],{"className":173989},[31676],[34,173991],{"className":173992,"style":35942},[31680],[34,173994,173577],{"className":173995},[31685],[34,173997],{"className":173998,"style":32604},[31845],[34,174000,32226],{"className":174001},[32024],[34,174003],{"className":174004,"style":32604},[31845],[34,174006,174008,174011,174014,174017,174020],{"className":174007},[31676],[34,174009],{"className":174010,"style":32617},[31680],[34,174012,173902],{"className":174013},[31685],[34,174015],{"className":174016,"style":31846},[31845],[34,174018,202],{"className":174019},[31850],[34,174021],{"className":174022,"style":31846},[31845],[34,174024,174026,174029],{"className":174025},[31676],[34,174027],{"className":174028,"style":32617},[31680],[34,174030,96087],{"className":174031},[31685],[15,174033,174034],{},"所以需要再占用第 782 个磁道的一部分。",[32188,174036],{},[253,174038,174039],{"id":174039},"计算需要的盘面数",[15,174041,174042],{},"每个盘面有 110 个磁道。",[15,174044,174045],{},"需要盘面数：",[15,174047,174048],{},[34,174049,174051,174073],{"className":174050},[31642],[34,174052,174054],{"className":174053},[31646],[31648,174055,174056],{"xmlns":31650},[31652,174057,174058,174070],{},[31655,174059,174060,174062,174064,174066,174068],{},[31662,174061,173674],{"stretchy":702},[31749,174063,173794],{},[31658,174065,38080],{"mathvariant":38079},[31749,174067,162188],{},[31662,174069,173683],{"stretchy":702},[31665,174071,174072],{"encoding":31667},"\\lceil 782 \u002F 110 \\rceil",[34,174074,174076],{"className":174075,"ariaHidden":758},[31672],[34,174077,174079,174082,174085,174089],{"className":174078},[31676],[34,174080],{"className":174081,"style":31774},[31680],[34,174083,173674],{"className":174084},[31782],[34,174086,174088],{"className":174087},[31685],"782\u002F110",[34,174090,173683],{"className":174091},[31690],[15,174093,174094],{},"计算：",[15,174096,174097],{},[34,174098,174100,174123],{"className":174099},[31642],[34,174101,174103],{"className":174102},[31646],[31648,174104,174105],{"xmlns":31650},[31652,174106,174107,174120],{},[31655,174108,174109,174111,174113,174115,174117],{},[31749,174110,173794],{},[31658,174112,38080],{"mathvariant":38079},[31749,174114,162188],{},[31662,174116,202],{},[31749,174118,174119],{},"7.109...",[31665,174121,174122],{"encoding":31667},"782 \u002F 110 = 7.109...",[34,174124,174126,174144],{"className":174125,"ariaHidden":758},[31672],[34,174127,174129,174132,174135,174138,174141],{"className":174128},[31676],[34,174130],{"className":174131,"style":31774},[31680],[34,174133,174088],{"className":174134},[31685],[34,174136],{"className":174137,"style":31846},[31845],[34,174139,202],{"className":174140},[31850],[34,174142],{"className":174143,"style":31846},[31845],[34,174145,174147,174150],{"className":174146},[31676],[34,174148],{"className":174149,"style":32617},[31680],[34,174151,174119],{"className":174152},[31685],[15,174154,174155],{},"向上取整：",[15,174157,174158],{},[34,174159,174161,174184],{"className":174160},[31642],[34,174162,174164],{"className":174163},[31646],[31648,174165,174166],{"xmlns":31650},[31652,174167,174168,174181],{},[31655,174169,174170,174172,174175,174177,174179],{},[31662,174171,173674],{"stretchy":702},[31749,174173,174174],{},"7.109",[31662,174176,173683],{"stretchy":702},[31662,174178,202],{},[31749,174180,60525],{},[31665,174182,174183],{"encoding":31667},"\\lceil 7.109 \\rceil = 8",[34,174185,174187,174211],{"className":174186,"ariaHidden":758},[31672],[34,174188,174190,174193,174196,174199,174202,174205,174208],{"className":174189},[31676],[34,174191],{"className":174192,"style":31774},[31680],[34,174194,173674],{"className":174195},[31782],[34,174197,174174],{"className":174198},[31685],[34,174200,173683],{"className":174201},[31690],[34,174203],{"className":174204,"style":31846},[31845],[34,174206,202],{"className":174207},[31850],[34,174209],{"className":174210,"style":31846},[31845],[34,174212,174214,174217],{"className":174213},[31676],[34,174215],{"className":174216,"style":32617},[31680],[34,174218,60525],{"className":174219},[31685],[15,174221,173614],{},[15,174223,174224],{},[34,174225,174227,174240],{"className":174226},[31642],[34,174228,174230],{"className":174229},[31646],[31648,174231,174232],{"xmlns":31650},[31652,174233,174234,174238],{},[31655,174235,174236],{},[31749,174237,60525],{},[31665,174239,60525],{"encoding":31667},[34,174241,174243],{"className":174242,"ariaHidden":758},[31672],[34,174244,174246,174249],{"className":174245},[31676],[34,174247],{"className":174248,"style":32617},[31680],[34,174250,60525],{"className":174251},[31685],[15,174253,174254],{},"个盘面。",[15,174256,174257],{},"更细一点说：",[15,174259,174260],{},"7 个完整盘面可以提供：",[15,174262,174263],{},[34,174264,174266,174293],{"className":174265},[31642],[34,174267,174269],{"className":174268},[31646],[31648,174270,174271],{"xmlns":31650},[31652,174272,174273,174290],{},[31655,174274,174275,174277,174279,174281,174283,174285,174287],{},[31749,174276,36723],{},[31662,174278,42071],{},[31749,174280,162188],{},[31662,174282,42071],{},[31749,174284,167343],{},[31662,174286,202],{},[31749,174288,174289],{},"73920",[31665,174291,174292],{"encoding":31667},"7 \\times 110 \\times 96 = 73920",[34,174294,174296,174314,174332,174350],{"className":174295,"ariaHidden":758},[31672],[34,174297,174299,174302,174305,174308,174311],{"className":174298},[31676],[34,174300],{"className":174301,"style":35942},[31680],[34,174303,36723],{"className":174304},[31685],[34,174306],{"className":174307,"style":32604},[31845],[34,174309,42071],{"className":174310},[32024],[34,174312],{"className":174313,"style":32604},[31845],[34,174315,174317,174320,174323,174326,174329],{"className":174316},[31676],[34,174318],{"className":174319,"style":35942},[31680],[34,174321,162188],{"className":174322},[31685],[34,174324],{"className":174325,"style":32604},[31845],[34,174327,42071],{"className":174328},[32024],[34,174330],{"className":174331,"style":32604},[31845],[34,174333,174335,174338,174341,174344,174347],{"className":174334},[31676],[34,174336],{"className":174337,"style":32617},[31680],[34,174339,167343],{"className":174340},[31685],[34,174342],{"className":174343,"style":31846},[31845],[34,174345,202],{"className":174346},[31850],[34,174348],{"className":174349,"style":31846},[31845],[34,174351,174353,174356],{"className":174352},[31676],[34,174354],{"className":174355,"style":32617},[31680],[34,174357,174289],{"className":174358},[31685],[15,174360,171595],{},[15,174362,174363],{},"还剩：",[15,174365,174366],{},[34,174367,174369,174392],{"className":174368},[31642],[34,174370,174372],{"className":174371},[31646],[31648,174373,174374],{"xmlns":31650},[31652,174375,174376,174389],{},[31655,174377,174378,174380,174382,174384,174386],{},[31749,174379,173577],{},[31662,174381,32226],{},[31749,174383,174289],{},[31662,174385,202],{},[31749,174387,174388],{},"1080",[31665,174390,174391],{"encoding":31667},"75000 - 73920 = 1080",[34,174393,174395,174413,174431],{"className":174394,"ariaHidden":758},[31672],[34,174396,174398,174401,174404,174407,174410],{"className":174397},[31676],[34,174399],{"className":174400,"style":35942},[31680],[34,174402,173577],{"className":174403},[31685],[34,174405],{"className":174406,"style":32604},[31845],[34,174408,32226],{"className":174409},[32024],[34,174411],{"className":174412,"style":32604},[31845],[34,174414,174416,174419,174422,174425,174428],{"className":174415},[31676],[34,174417],{"className":174418,"style":32617},[31680],[34,174420,174289],{"className":174421},[31685],[34,174423],{"className":174424,"style":31846},[31845],[34,174426,202],{"className":174427},[31850],[34,174429],{"className":174430,"style":31846},[31845],[34,174432,174434,174437],{"className":174433},[31676],[34,174435],{"className":174436,"style":32617},[31680],[34,174438,174388],{"className":174439},[31685],[15,174441,171595],{},[15,174443,174444],{},"这些扇区需要：",[15,174446,174447],{},[34,174448,174450,174476],{"className":174449},[31642],[34,174451,174453],{"className":174452},[31646],[31648,174454,174455],{"xmlns":31650},[31652,174456,174457,174473],{},[31655,174458,174459,174461,174463,174465,174467,174469,174471],{},[31662,174460,173674],{"stretchy":702},[31749,174462,174388],{},[31658,174464,38080],{"mathvariant":38079},[31749,174466,167343],{},[31662,174468,173683],{"stretchy":702},[31662,174470,202],{},[31749,174472,71869],{},[31665,174474,174475],{"encoding":31667},"\\lceil 1080 \u002F 96 \\rceil = 12",[34,174477,174479,174504],{"className":174478,"ariaHidden":758},[31672],[34,174480,174482,174485,174488,174492,174495,174498,174501],{"className":174481},[31676],[34,174483],{"className":174484,"style":31774},[31680],[34,174486,173674],{"className":174487},[31782],[34,174489,174491],{"className":174490},[31685],"1080\u002F96",[34,174493,173683],{"className":174494},[31690],[34,174496],{"className":174497,"style":31846},[31845],[34,174499,202],{"className":174500},[31850],[34,174502],{"className":174503,"style":31846},[31845],[34,174505,174507,174510],{"className":174506},[31676],[34,174508],{"className":174509,"style":32617},[31680],[34,174511,71869],{"className":174512},[31685],[15,174514,173868],{},[15,174516,174517],{},"所以实际占用为：",[15,174519,174520],{},"7 个完整盘面 + 第 8 个盘面的 12 个磁道",[15,174522,174523],{},"因此按整数盘面数计算，需要：",[15,174525,174526],{},[34,174527,174529,174542],{"className":174528},[31642],[34,174530,174532],{"className":174531},[31646],[31648,174533,174534],{"xmlns":31650},[31652,174535,174536,174540],{},[31655,174537,174538],{},[31749,174539,60525],{},[31665,174541,60525],{"encoding":31667},[34,174543,174545],{"className":174544,"ariaHidden":758},[31672],[34,174546,174548,174551],{"className":174547},[31676],[34,174549],{"className":174550,"style":32617},[31680],[34,174552,60525],{"className":174553},[31685],[15,174555,174254],{},[32188,174557],{},[253,174559,174561],{"id":174560},"第-5-题最终答案","第 5 题最终答案",[15,174563,173551],{},[15,174565,174566],{},[34,174567,174569,174582],{"className":174568},[31642],[34,174570,174572],{"className":174571},[31646],[31648,174573,174574],{"xmlns":31650},[31652,174575,174576,174580],{},[31655,174577,174578],{},[31749,174579,173577],{},[31665,174581,173577],{"encoding":31667},[34,174583,174585],{"className":174584,"ariaHidden":758},[31672],[34,174586,174588,174591],{"className":174587},[31676],[34,174589],{"className":174590,"style":32617},[31680],[34,174592,173577],{"className":174593},[31685],[15,174595,173657],{},[15,174597,174598],{},[34,174599,174601,174614],{"className":174600},[31642],[34,174602,174604],{"className":174603},[31646],[31648,174605,174606],{"xmlns":31650},[31652,174607,174608,174612],{},[31655,174609,174610],{},[31749,174611,173794],{},[31665,174613,173794],{"encoding":31667},[34,174615,174617],{"className":174616,"ariaHidden":758},[31672],[34,174618,174620,174623],{"className":174619},[31676],[34,174621],{"className":174622,"style":32617},[31680],[34,174624,173794],{"className":174625},[31685],[15,174627,174045],{},[15,174629,174630],{},[34,174631,174633,174646],{"className":174632},[31642],[34,174634,174636],{"className":174635},[31646],[31648,174637,174638],{"xmlns":31650},[31652,174639,174640,174644],{},[31655,174641,174642],{},[31749,174643,60525],{},[31665,174645,60525],{"encoding":31667},[34,174647,174649],{"className":174648,"ariaHidden":758},[31672],[34,174650,174652,174655],{"className":174651},[31676],[34,174653],{"className":174654,"style":32617},[31680],[34,174656,60525],{"className":174657},[31685],[15,174659,174660],{},"更精确地说：",[15,174662,174663],{},"占用 7 个完整盘面，再占用第 8 个盘面的 12 个磁道。",[32188,174665],{},[19,174667,174669],{"id":174668},"第-1-题","第 1 题",[15,174671,167098],{},[15,174673,174674],{},[34,174675,174677,174690],{"className":174676},[31642],[34,174678,174680],{"className":174679},[31646],[31648,174681,174682],{"xmlns":31650},[31652,174683,174684,174688],{},[31655,174685,174686],{},[31749,174687,163190],{},[31665,174689,163190],{"encoding":31667},[34,174691,174693],{"className":174692,"ariaHidden":758},[31672],[34,174694,174696,174699],{"className":174695},[31676],[34,174697],{"className":174698,"style":32617},[31680],[34,174700,163190],{"className":174701},[31685],[15,174703,167131],{},[15,174705,174706],{},[34,174707,174709,174722],{"className":174708},[31642],[34,174710,174712],{"className":174711},[31646],[31648,174713,174714],{"xmlns":31650},[31652,174715,174716,174720],{},[31655,174717,174718],{},[31749,174719,164942],{},[31665,174721,164942],{"encoding":31667},[34,174723,174725],{"className":174724,"ariaHidden":758},[31672],[34,174726,174728,174731],{"className":174727},[31676],[34,174729],{"className":174730,"style":32617},[31680],[34,174732,164942],{"className":174733},[31685],[15,174735,167164],{},[15,174737,174738],{},[34,174739,174741,174754],{"className":174740},[31642],[34,174742,174744],{"className":174743},[31646],[31648,174745,174746],{"xmlns":31650},[31652,174747,174748,174752],{},[31655,174749,174750],{},[31749,174751,166294],{},[31665,174753,166294],{"encoding":31667},[34,174755,174757],{"className":174756,"ariaHidden":758},[31672],[34,174758,174760,174763],{"className":174759},[31676],[34,174761],{"className":174762,"style":32617},[31680],[34,174764,166294],{"className":174765},[31685],[15,174767,174768],{},"如果严格要求 SCAN 必须走到磁道 0 再反向：",[15,174770,174771],{},[34,174772,174774,174787],{"className":174773},[31642],[34,174775,174777],{"className":174776},[31646],[31648,174778,174779],{"xmlns":31650},[31652,174780,174781,174785],{},[31655,174782,174783],{},[31749,174784,166861],{},[31665,174786,166861],{"encoding":31667},[34,174788,174790],{"className":174789,"ariaHidden":758},[31672],[34,174791,174793,174796],{"className":174792},[31676],[34,174794],{"className":174795,"style":32617},[31680],[34,174797,166861],{"className":174798},[31685],[32188,174800],{},[19,174802,174804],{"id":174803},"第-2-题","第 2 题",[15,174806,174807],{},"CPU 用于处理中断的时间比例约为：",[15,174809,174810],{},[34,174811,174813,174828],{"className":174812},[31642],[34,174814,174816],{"className":174815},[31646],[31648,174817,174818],{"xmlns":31650},[31652,174819,174820,174826],{},[31655,174821,174822,174824],{},[31749,174823,168155],{},[31658,174825,119426],{"mathvariant":38079},[31665,174827,168300],{"encoding":31667},[34,174829,174831],{"className":174830,"ariaHidden":758},[31672],[34,174832,174834,174837],{"className":174833},[31676],[34,174835],{"className":174836,"style":168117},[31680],[34,174838,168274],{"className":174839},[31685],[32188,174841],{},[19,174843,174845],{"id":174844},"第-3-题","第 3 题",[15,174847,174848],{},"最大聚合 I\u002FO 传输速率：",[15,174850,174851],{},[34,174852,174854,174883],{"className":174853},[31642],[34,174855,174857],{"className":174856},[31646],[31648,174858,174859],{"xmlns":31650},[31652,174860,174861,174881],{},[31655,174862,174863,174865,174867,174869,174871,174873,174875,174877,174879],{},[31749,174864,169757],{},[31658,174866,38691],{},[31658,174868,1697],{},[31658,174870,67764],{},[31658,174872,33249],{},[31658,174874,33244],{},[31658,174876,31747],{},[31658,174878,38080],{"mathvariant":38079},[31658,174880,31747],{},[31665,174882,169944],{"encoding":31667},[34,174884,174886],{"className":174885,"ariaHidden":758},[31672],[34,174887,174889,174892,174895,174898,174901,174904,174907,174910,174913],{"className":174888},[31676],[34,174890],{"className":174891,"style":31774},[31680],[34,174893,169757],{"className":174894},[31685],[34,174896,38691],{"className":174897,"style":38706},[31685,31686],[34,174899,1697],{"className":174900},[31685,31686],[34,174902,67764],{"className":174903,"style":32062},[31685,31686],[34,174905,33249],{"className":174906},[31685,31686],[34,174908,168891],{"className":174909},[31685,31686],[34,174911,38080],{"className":174912},[31685],[34,174914,31747],{"className":174915},[31685,31686],[15,174917,169980],{},[15,174919,174920],{},[34,174921,174923,174944],{"className":174922},[31642],[34,174924,174926],{"className":174925},[31646],[31648,174927,174928],{"xmlns":31650},[31652,174929,174930,174942],{},[31655,174931,174932,174934,174936,174938,174940],{},[31749,174933,169865],{},[31658,174935,98360],{},[31658,174937,54224],{},[31658,174939,38080],{"mathvariant":38079},[31658,174941,31747],{},[31665,174943,169876],{"encoding":31667},[34,174945,174947],{"className":174946,"ariaHidden":758},[31672],[34,174948,174950,174953,174956,174959,174962,174965],{"className":174949},[31676],[34,174951],{"className":174952,"style":31774},[31680],[34,174954,169865],{"className":174955},[31685],[34,174957,98360],{"className":174958,"style":33389},[31685,31686],[34,174960,54224],{"className":174961,"style":54252},[31685,31686],[34,174963,38080],{"className":174964},[31685],[34,174966,31747],{"className":174967},[31685,31686],[32188,174969],{},[19,174971,174973],{"id":174972},"第-4-题","第 4 题",[15,174975,174976],{},"顺序存放：",[15,174978,174979],{},[34,174980,174982,174999],{"className":174981},[31642],[34,174983,174985],{"className":174984},[31646],[31648,174986,174987],{"xmlns":31650},[31652,174988,174989,174997],{},[31655,174990,174991,174993,174995],{},[31749,174992,171246],{},[31658,174994,31759],{},[31658,174996,31747],{},[31665,174998,171351],{"encoding":31667},[34,175000,175002],{"className":175001,"ariaHidden":758},[31672],[34,175003,175005,175008,175011,175014],{"className":175004},[31676],[34,175006],{"className":175007,"style":32617},[31680],[34,175009,171246],{"className":175010},[31685],[34,175012,31759],{"className":175013},[31685,31686],[34,175015,31747],{"className":175016},[31685,31686],[15,175018,175019],{},"优化交错存放，例如：",[15,175021,171654],{},[15,175023,175024],{},"最短时间：",[15,175026,175027],{},[34,175028,175030,175047],{"className":175029},[31642],[34,175031,175033],{"className":175032},[31646],[31648,175034,175035],{"xmlns":31650},[31652,175036,175037,175045],{},[31655,175038,175039,175041,175043],{},[31749,175040,95547],{},[31658,175042,31759],{},[31658,175044,31747],{},[31665,175046,172994],{"encoding":31667},[34,175048,175050],{"className":175049,"ariaHidden":758},[31672],[34,175051,175053,175056,175059,175062],{"className":175052},[31676],[34,175054],{"className":175055,"style":32617},[31680],[34,175057,95547],{"className":175058},[31685],[34,175060,31759],{"className":175061},[31685,31686],[34,175063,31747],{"className":175064},[31685,31686],[32188,175066],{},[19,175068,175070],{"id":175069},"第-5-题","第 5 题",[15,175072,173551],{},[15,175074,175075],{},[34,175076,175078,175091],{"className":175077},[31642],[34,175079,175081],{"className":175080},[31646],[31648,175082,175083],{"xmlns":31650},[31652,175084,175085,175089],{},[31655,175086,175087],{},[31749,175088,173577],{},[31665,175090,173577],{"encoding":31667},[34,175092,175094],{"className":175093,"ariaHidden":758},[31672],[34,175095,175097,175100],{"className":175096},[31676],[34,175098],{"className":175099,"style":32617},[31680],[34,175101,173577],{"className":175102},[31685],[15,175104,173657],{},[15,175106,175107],{},[34,175108,175110,175123],{"className":175109},[31642],[34,175111,175113],{"className":175112},[31646],[31648,175114,175115],{"xmlns":31650},[31652,175116,175117,175121],{},[31655,175118,175119],{},[31749,175120,173794],{},[31665,175122,173794],{"encoding":31667},[34,175124,175126],{"className":175125,"ariaHidden":758},[31672],[34,175127,175129,175132],{"className":175128},[31676],[34,175130],{"className":175131,"style":32617},[31680],[34,175133,173794],{"className":175134},[31685],[15,175136,174045],{},[15,175138,175139],{},[34,175140,175142,175155],{"className":175141},[31642],[34,175143,175145],{"className":175144},[31646],[31648,175146,175147],{"xmlns":31650},[31652,175148,175149,175153],{},[31655,175150,175151],{},[31749,175152,60525],{},[31665,175154,60525],{"encoding":31667},[34,175156,175158],{"className":175157,"ariaHidden":758},[31672],[34,175159,175161,175164],{"className":175160},[31676],[34,175162],{"className":175163,"style":32617},[31680],[34,175165,60525],{"className":175166},[31685],{"title":29,"searchDepth":43,"depth":43,"links":175168},[175169,175178,175186,175192,175203,175211,175212,175213,175214,175215],{"id":162039,"depth":43,"text":162040,"children":175170},[175171,175172,175173,175174,175175,175176,175177],{"id":162043,"depth":49,"text":162043},{"id":162066,"depth":49,"text":162066},{"id":162149,"depth":49,"text":162150},{"id":163407,"depth":49,"text":163408},{"id":165159,"depth":49,"text":165160},{"id":166511,"depth":49,"text":166511},{"id":167094,"depth":49,"text":167095},{"id":167232,"depth":43,"text":167233,"children":175179},[175180,175181,175182,175183,175184,175185],{"id":167236,"depth":49,"text":162043},{"id":167363,"depth":49,"text":162066},{"id":167588,"depth":49,"text":167588},{"id":167981,"depth":49,"text":167982},{"id":168320,"depth":49,"text":166511},{"id":168734,"depth":49,"text":168735},{"id":168775,"depth":43,"text":168776,"children":175187},[175188,175189,175190,175191],{"id":168779,"depth":49,"text":162043},{"id":168810,"depth":49,"text":162066},{"id":169728,"depth":49,"text":169729},{"id":169905,"depth":49,"text":169906},{"id":170035,"depth":43,"text":170036,"children":175193},[175194,175195,175196,175197,175198,175199,175200,175201,175202],{"id":170039,"depth":49,"text":162043},{"id":170053,"depth":49,"text":162066},{"id":170437,"depth":49,"text":170438},{"id":171324,"depth":49,"text":171325},{"id":171374,"depth":49,"text":171375},{"id":171606,"depth":49,"text":171606},{"id":171719,"depth":49,"text":171719},{"id":172962,"depth":49,"text":172963},{"id":173017,"depth":49,"text":173018},{"id":173114,"depth":43,"text":173115,"children":175204},[175205,175206,175207,175208,175209,175210],{"id":173118,"depth":49,"text":162043},{"id":173304,"depth":49,"text":162066},{"id":173545,"depth":49,"text":173545},{"id":173651,"depth":49,"text":173651},{"id":174039,"depth":49,"text":174039},{"id":174560,"depth":49,"text":174561},{"id":174668,"depth":43,"text":174669},{"id":174803,"depth":43,"text":174804},{"id":174844,"depth":43,"text":174845},{"id":174972,"depth":43,"text":174973},{"id":175069,"depth":43,"text":175070},{"order":1990},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap11-hw",{"title":162030,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap11-hw","gPTFv75I2zzOLhy6xar9iQtjjtiM8yk5dtzWDThMD3U",{"id":175222,"title":175223,"body":175224,"description":29,"extension":85,"meta":193837,"navigation":52,"path":193838,"seo":193839,"stem":193840,"__hash__":193841},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap12-hw.md","Chap. 12 HW",{"type":8,"value":175225,"toc":193773},[175226,175230,175234,175236,175241,175253,175258,175270,175275,175287,175298,175300,175303,175305,175309,175311,175355,175358,175361,175364,175394,175397,175427,175429,175433,175435,175486,175489,175527,175557,175594,175596,175667,175670,175673,175710,175713,175749,175779,175816,175846,175848,175919,175990,176060,176063,176093,176096,176126,176128,176164,176234,176270,176273,176303,176305,176335,176338,176374,176404,176440,176470,176609,176611,176669,176727,176730,176760,176762,176766,176768,176812,176815,176818,176866,176868,176898,176901,176931,176933,176937,176939,176980,176983,177037,177040,177116,177119,177121,177204,177207,177278,177281,177366,177369,177438,177440,177508,177511,177551,177553,177557,177560,177563,177566,177569,177572,177575,177578,177580,177584,177587,177589,177672,177675,177677,177688,177749,177824,177827,177838,177840,177844,177847,177849,177969,177972,177974,177985,177987,178030,178032,178036,178039,178042,178125,178128,178130,178144,178146,178157,178160,178162,178166,178169,178172,178175,178178,178181,178184,178187,178190,178193,178195,178198,178202,178205,178208,178213,178216,178219,178222,178225,178228,178230,178234,178237,178242,178245,178247,178258,178263,178266,178269,178271,178279,178281,178289,178294,178297,178299,178307,178312,178315,178318,178323,178326,178328,178405,178407,178412,178414,178422,178424,178428,178430,178433,178436,178439,178441,178444,178446,178449,178452,178455,178458,178461,178551,178554,178557,178559,178562,178564,178568,178600,178603,178633,178636,178639,178669,178671,178674,178747,178778,178780,178783,178786,178816,178819,178822,178852,178854,178857,178887,178889,178892,178922,178924,178956,179032,179034,179066,179230,179233,179316,179319,179424,179426,179551,179554,179680,179683,179838,179841,180046,180048,180052,180055,180058,180090,180165,180168,180171,180320,180322,180483,180486,180635,180638,180794,180796,180799,180831,180913,180916,181014,181017,181047,181050,181217,181220,181361,181363,181365,181368,181556,181559,181562,181722,181725,181866,181868,182014,182017,182180,182182,182186,182188,182191,182193,182231,182268,182305,182342,182447,182449,182452,182455,182563,182566,182571,182624,182629,182683,182688,182742,182747,182872,182874,182911,182913,183014,183016,183097,183099,183180,183182,183332,183334,183427,183429,183466,183468,183569,183571,183652,183654,183735,183737,183887,183889,183977,183979,184016,184019,184101,184103,184184,184187,184189,184220,184222,184303,184305,184386,184388,184538,184540,184628,184630,184632,184667,184754,184789,184876,184911,184998,185000,185004,185006,185009,185011,185014,185016,185019,185049,185052,185127,185130,185171,185174,185216,185219,185278,185281,185333,185336,185386,185388,185392,185395,185398,185401,185404,185472,185475,185544,185547,185633,185636,185692,185695,185771,185774,185862,185864,185868,185871,185874,185877,185967,185970,186020,186023,186114,186117,186147,186150,186153,186183,186186,186189,186274,186276,186389,186392,186456,186459,186523,186525,186567,186570,186648,186650,186746,186749,186840,186842,186938,186940,186944,186947,186950,186999,187002,187075,187078,187179,187182,187262,187265,187361,187363,187365,187368,187451,187454,187597,187600,187695,187697,187701,187703,187706,187708,187711,187713,187716,187746,187749,187779,187782,187812,187815,187845,187848,187884,187887,187954,187957,188024,188026,188030,188033,188068,188071,188107,188110,188168,188171,188201,188204,188206,188210,188213,188216,188219,188249,188251,188254,188349,188351,188355,188358,188361,188364,188394,188397,188400,188403,188478,188480,188575,188577,188581,188584,188587,188590,188593,188596,188599,188675,188677,188772,188774,188778,188781,188900,188902,189000,189003,189098,189101,189182,189185,189259,189261,189263,189266,189362,189364,189455,189457,189558,189560,189564,189567,189570,189572,189575,189577,189580,189582,189585,189652,189655,189690,189693,189730,189732,189736,189739,189742,189788,189790,189866,189869,189899,189902,189905,189981,189983,189985,189989,189991,190026,190029,190064,190067,190123,190125,190155,190157,190159,190163,190166,190169,190172,190175,190178,190181,190184,190187,190241,190244,190274,190276,190279,190333,190335,190432,190434,190464,190467,190519,190522,190525,190528,190627,190629,190631,190634,190637,190640,190693,190696,190750,190753,190783,190786,190789,190885,190887,190891,190894,190929,190932,190962,190965,191061,191063,191148,191150,191153,191155,191158,191255,191258,191364,191367,191464,191467,191563,191566,191654,191657,191745,191748,191836,191838,191934,191936,191938,191940,192127,192129,192131,192291,192293,192434,192436,192582,192584,192586,192621,192708,192743,192830,192865,192952,192954,192956,192958,193041,193044,193186,193188,193283,193285,193287,193290,193385,193387,193477,193479,193579,193581,193585,193588,193684,193686],[11,175227,175229],{"id":175228},"os-chap12-hw","OS Chap.12 HW",[19,175231,175233],{"id":175232},"_1-选择题与简答题","1. 选择题与简答题",[253,175235,162043],{"id":162043},[2230,175237,175238],{},[108,175239,175240],{},"UNIX中的文件系统采用（ ）。",[15,175242,175243,175244,175246,175247,175249,175250,175252],{},"A 网状文件",[58531,175245],{},"\nB 记录式文件",[58531,175248],{},"\nC 索引文件",[58531,175251],{},"\nD 流式文件",[2230,175254,175255],{"start":43},[108,175256,175257],{},"设文件F1当前引用计数值为1，先建立F1的符号链接文件F2，再建立F1的硬链接文件F3，然后删除F1。此时，F2和F3的引用计数值分别是( )",[15,175259,175260,175261,175263,175264,175266,175267,175269],{},"A 0，1",[58531,175262],{},"\nB 1，2",[58531,175265],{},"\nC 1，1",[58531,175268],{},"\nD 2，1",[2230,175271,175272],{"start":49},[108,175273,175274],{},"对磁盘而言，输入输出操作的信息传送单位为 ( ) 。",[15,175276,175277,175278,175280,175281,175283,175284,175286],{},"A. 字符",[58531,175279],{},"\nB. 字",[58531,175282],{},"\nC. 块",[58531,175285],{},"\nD. 文件",[2230,175288,175289,175292,175295],{"start":56},[108,175290,175291],{},"计算机系统中，屏幕显示分辨率为640×480，若要存储一屏256彩色的图像，需要( )KB存储空间。",[108,175293,175294],{},"什么是文件的物理结构？有哪几种组织方式？",[108,175296,175297],{},"什么是文件的逻辑结构？有哪几种组织方式？",[32188,175299],{},[253,175301,175302],{"id":175302},"答案和讲解",[32188,175304],{},[253,175306,175308],{"id":175307},"_11-unix中的文件系统采用什么","1.1 UNIX中的文件系统采用什么？",[15,175310,58173],{},[15,175312,175313],{},[34,175314,175316,175333],{"className":175315},[31642],[34,175317,175319],{"className":175318},[31646],[31648,175320,175321],{"xmlns":31650},[31652,175322,175323,175330],{},[31655,175324,175325,175327],{},[31658,175326,91994],{},[39361,175328,175329],{}," 流式文件",[31665,175331,175332],{"encoding":31667},"D \\text{ 流式文件}",[34,175334,175336],{"className":175335,"ariaHidden":758},[31672],[34,175337,175339,175342,175345],{"className":175338},[31676],[34,175340],{"className":175341,"style":33218},[31680],[34,175343,91994],{"className":175344,"style":34452},[31685,31686],[34,175346,175348,175351],{"className":175347},[31685,2638],[34,175349,130298],{"className":175350},[31685],[34,175352,175354],{"className":175353},[31685,105462],"流式文件",[15,175356,175357],{},"UNIX\u002FLinux 普通文件一般被看成连续的字节流，也就是 byte stream。",[15,175359,175360],{},"它不要求文件内部必须是一条条记录，也不要求文件内部必须有复杂的树形、网状结构。",[15,175362,175363],{},"所以 UNIX 文件的逻辑结构通常说成：",[15,175365,175366],{},[34,175367,175369,175382],{"className":175368},[31642],[34,175370,175372],{"className":175371},[31646],[31648,175373,175374],{"xmlns":31650},[31652,175375,175376,175380],{},[31655,175377,175378],{},[39361,175379,175354],{},[31665,175381,175354],{"encoding":31667},[34,175383,175385],{"className":175384,"ariaHidden":758},[31672],[34,175386,175388,175391],{"className":175387},[31676],[34,175389],{"className":175390,"style":33218},[31680],[34,175392,175354],{"className":175393},[31685,105462],[15,175395,175396],{},"因此答案选：",[15,175398,175399],{},[34,175400,175402,175415],{"className":175401},[31642],[34,175403,175405],{"className":175404},[31646],[31648,175406,175407],{"xmlns":31650},[31652,175408,175409,175413],{},[31655,175410,175411],{},[31658,175412,91994],{},[31665,175414,91994],{"encoding":31667},[34,175416,175418],{"className":175417,"ariaHidden":758},[31672],[34,175419,175421,175424],{"className":175420},[31676],[34,175422],{"className":175423,"style":33218},[31680],[34,175425,91994],{"className":175426,"style":34452},[31685,31686],[32188,175428],{},[253,175430,175432],{"id":175431},"_12-符号链接硬链接与引用计数","1.2 符号链接、硬链接与引用计数",[15,175434,58173],{},[15,175436,175437],{},[34,175438,175440,175462],{"className":175439},[31642],[34,175441,175443],{"className":175442},[31646],[31648,175444,175445],{"xmlns":31650},[31652,175446,175447,175459],{},[31655,175448,175449,175451,175453,175455,175457],{},[31658,175450,47384],{},[39361,175452,130298],{},[31749,175454,640],{},[39361,175456,46528],{},[31749,175458,640],{},[31665,175460,175461],{"encoding":31667},"C \\ 1，1",[34,175463,175465],{"className":175464,"ariaHidden":758},[31672],[34,175466,175468,175471,175474,175477,175480,175483],{"className":175467},[31676],[34,175469],{"className":175470,"style":33218},[31680],[34,175472,47384],{"className":175473,"style":38706},[31685,31686],[34,175475,130298],{"className":175476},[31845],[34,175478,640],{"className":175479},[31685],[34,175481,46528],{"className":175482},[31685,105462],[34,175484,640],{"className":175485},[31685],[15,175487,175488],{},"初始条件：",[15,175490,175491,175492,175526],{},"文件 ",[34,175493,175495,175511],{"className":175494},[31642],[34,175496,175498],{"className":175497},[31646],[31648,175499,175500],{"xmlns":31650},[31652,175501,175502,175508],{},[31655,175503,175504,175506],{},[31658,175505,51834],{},[31749,175507,640],{},[31665,175509,175510],{"encoding":31667},"F1",[34,175512,175514],{"className":175513,"ariaHidden":758},[31672],[34,175515,175517,175520,175523],{"className":175516},[31676],[34,175518],{"className":175519,"style":33218},[31680],[34,175521,51834],{"className":175522,"style":37422},[31685,31686],[34,175524,640],{"className":175525},[31685]," 当前引用计数为：",[15,175528,175529],{},[34,175530,175532,175545],{"className":175531},[31642],[34,175533,175535],{"className":175534},[31646],[31648,175536,175537],{"xmlns":31650},[31652,175538,175539,175543],{},[31655,175540,175541],{},[31749,175542,640],{},[31665,175544,640],{"encoding":31667},[34,175546,175548],{"className":175547,"ariaHidden":758},[31672],[34,175549,175551,175554],{"className":175550},[31676],[34,175552],{"className":175553,"style":32617},[31680],[34,175555,640],{"className":175556},[31685],[15,175558,175559,175560,175593],{},"这说明目前只有目录项 ",[34,175561,175563,175578],{"className":175562},[31642],[34,175564,175566],{"className":175565},[31646],[31648,175567,175568],{"xmlns":31650},[31652,175569,175570,175576],{},[31655,175571,175572,175574],{},[31658,175573,51834],{},[31749,175575,640],{},[31665,175577,175510],{"encoding":31667},[34,175579,175581],{"className":175580,"ariaHidden":758},[31672],[34,175582,175584,175587,175590],{"className":175583},[31676],[34,175585],{"className":175586,"style":33218},[31680],[34,175588,51834],{"className":175589,"style":37422},[31685,31686],[34,175591,640],{"className":175592},[31685]," 指向该文件的 inode。",[32188,175595],{},[15,175597,175598,175599,175632,175633,347],{},"第一步：建立 ",[34,175600,175602,175617],{"className":175601},[31642],[34,175603,175605],{"className":175604},[31646],[31648,175606,175607],{"xmlns":31650},[31652,175608,175609,175615],{},[31655,175610,175611,175613],{},[31658,175612,51834],{},[31749,175614,640],{},[31665,175616,175510],{"encoding":31667},[34,175618,175620],{"className":175619,"ariaHidden":758},[31672],[34,175621,175623,175626,175629],{"className":175622},[31676],[34,175624],{"className":175625,"style":33218},[31680],[34,175627,51834],{"className":175628,"style":37422},[31685,31686],[34,175630,640],{"className":175631},[31685]," 的符号链接文件 ",[34,175634,175636,175652],{"className":175635},[31642],[34,175637,175639],{"className":175638},[31646],[31648,175640,175641],{"xmlns":31650},[31652,175642,175643,175649],{},[31655,175644,175645,175647],{},[31658,175646,51834],{},[31749,175648,768],{},[31665,175650,175651],{"encoding":31667},"F2",[34,175653,175655],{"className":175654,"ariaHidden":758},[31672],[34,175656,175658,175661,175664],{"className":175657},[31676],[34,175659],{"className":175660,"style":33218},[31680],[34,175662,51834],{"className":175663,"style":37422},[31685,31686],[34,175665,768],{"className":175666},[31685],[15,175668,175669],{},"符号链接，也叫软链接。",[15,175671,175672],{},"它本质上是一个新的特殊文件，文件内容保存的是目标路径名，例如保存“F1”这个路径。",[15,175674,175675,175676,175709],{},"所以建立符号链接不会让 ",[34,175677,175679,175694],{"className":175678},[31642],[34,175680,175682],{"className":175681},[31646],[31648,175683,175684],{"xmlns":31650},[31652,175685,175686,175692],{},[31655,175687,175688,175690],{},[31658,175689,51834],{},[31749,175691,640],{},[31665,175693,175510],{"encoding":31667},[34,175695,175697],{"className":175696,"ariaHidden":758},[31672],[34,175698,175700,175703,175706],{"className":175699},[31676],[34,175701],{"className":175702,"style":33218},[31680],[34,175704,51834],{"className":175705,"style":37422},[31685,31686],[34,175707,640],{"className":175708},[31685]," 的 inode 引用计数增加。",[15,175711,175712],{},"此时：",[15,175714,175715,175748],{},[34,175716,175718,175733],{"className":175717},[31642],[34,175719,175721],{"className":175720},[31646],[31648,175722,175723],{"xmlns":31650},[31652,175724,175725,175731],{},[31655,175726,175727,175729],{},[31658,175728,51834],{},[31749,175730,640],{},[31665,175732,175510],{"encoding":31667},[34,175734,175736],{"className":175735,"ariaHidden":758},[31672],[34,175737,175739,175742,175745],{"className":175738},[31676],[34,175740],{"className":175741,"style":33218},[31680],[34,175743,51834],{"className":175744,"style":37422},[31685,31686],[34,175746,640],{"className":175747},[31685]," 的引用计数仍为：",[15,175750,175751],{},[34,175752,175754,175767],{"className":175753},[31642],[34,175755,175757],{"className":175756},[31646],[31648,175758,175759],{"xmlns":31650},[31652,175760,175761,175765],{},[31655,175762,175763],{},[31749,175764,640],{},[31665,175766,640],{"encoding":31667},[34,175768,175770],{"className":175769,"ariaHidden":758},[31672],[34,175771,175773,175776],{"className":175772},[31676],[34,175774],{"className":175775,"style":32617},[31680],[34,175777,640],{"className":175778},[31685],[15,175780,175781,175782,175815],{},"而 ",[34,175783,175785,175800],{"className":175784},[31642],[34,175786,175788],{"className":175787},[31646],[31648,175789,175790],{"xmlns":31650},[31652,175791,175792,175798],{},[31655,175793,175794,175796],{},[31658,175795,51834],{},[31749,175797,768],{},[31665,175799,175651],{"encoding":31667},[34,175801,175803],{"className":175802,"ariaHidden":758},[31672],[34,175804,175806,175809,175812],{"className":175805},[31676],[34,175807],{"className":175808,"style":33218},[31680],[34,175810,51834],{"className":175811,"style":37422},[31685,31686],[34,175813,768],{"className":175814},[31685]," 自己作为一个独立文件，也有自己的 inode，自己的引用计数为：",[15,175817,175818],{},[34,175819,175821,175834],{"className":175820},[31642],[34,175822,175824],{"className":175823},[31646],[31648,175825,175826],{"xmlns":31650},[31652,175827,175828,175832],{},[31655,175829,175830],{},[31749,175831,640],{},[31665,175833,640],{"encoding":31667},[34,175835,175837],{"className":175836,"ariaHidden":758},[31672],[34,175838,175840,175843],{"className":175839},[31676],[34,175841],{"className":175842,"style":32617},[31680],[34,175844,640],{"className":175845},[31685],[32188,175847],{},[15,175849,175850,175851,175884,175885,347],{},"第二步：建立 ",[34,175852,175854,175869],{"className":175853},[31642],[34,175855,175857],{"className":175856},[31646],[31648,175858,175859],{"xmlns":31650},[31652,175860,175861,175867],{},[31655,175862,175863,175865],{},[31658,175864,51834],{},[31749,175866,640],{},[31665,175868,175510],{"encoding":31667},[34,175870,175872],{"className":175871,"ariaHidden":758},[31672],[34,175873,175875,175878,175881],{"className":175874},[31676],[34,175876],{"className":175877,"style":33218},[31680],[34,175879,51834],{"className":175880,"style":37422},[31685,31686],[34,175882,640],{"className":175883},[31685]," 的硬链接文件 ",[34,175886,175888,175904],{"className":175887},[31642],[34,175889,175891],{"className":175890},[31646],[31648,175892,175893],{"xmlns":31650},[31652,175894,175895,175901],{},[31655,175896,175897,175899],{},[31658,175898,51834],{},[31749,175900,786],{},[31665,175902,175903],{"encoding":31667},"F3",[34,175905,175907],{"className":175906,"ariaHidden":758},[31672],[34,175908,175910,175913,175916],{"className":175909},[31676],[34,175911],{"className":175912,"style":33218},[31680],[34,175914,51834],{"className":175915,"style":37422},[31685,31686],[34,175917,786],{"className":175918},[31685],[15,175920,175921,175922,175955,175956,175989],{},"硬链接会让新的目录项 ",[34,175923,175925,175940],{"className":175924},[31642],[34,175926,175928],{"className":175927},[31646],[31648,175929,175930],{"xmlns":31650},[31652,175931,175932,175938],{},[31655,175933,175934,175936],{},[31658,175935,51834],{},[31749,175937,786],{},[31665,175939,175903],{"encoding":31667},[34,175941,175943],{"className":175942,"ariaHidden":758},[31672],[34,175944,175946,175949,175952],{"className":175945},[31676],[34,175947],{"className":175948,"style":33218},[31680],[34,175950,51834],{"className":175951,"style":37422},[31685,31686],[34,175953,786],{"className":175954},[31685]," 直接指向 ",[34,175957,175959,175974],{"className":175958},[31642],[34,175960,175962],{"className":175961},[31646],[31648,175963,175964],{"xmlns":31650},[31652,175965,175966,175972],{},[31655,175967,175968,175970],{},[31658,175969,51834],{},[31749,175971,640],{},[31665,175973,175510],{"encoding":31667},[34,175975,175977],{"className":175976,"ariaHidden":758},[31672],[34,175978,175980,175983,175986],{"className":175979},[31676],[34,175981],{"className":175982,"style":33218},[31680],[34,175984,51834],{"className":175985,"style":37422},[31685,31686],[34,175987,640],{"className":175988},[31685]," 原来的 inode。",[15,175991,175992,175993,258,176026,176059],{},"所以 ",[34,175994,175996,176011],{"className":175995},[31642],[34,175997,175999],{"className":175998},[31646],[31648,176000,176001],{"xmlns":31650},[31652,176002,176003,176009],{},[31655,176004,176005,176007],{},[31658,176006,51834],{},[31749,176008,640],{},[31665,176010,175510],{"encoding":31667},[34,176012,176014],{"className":176013,"ariaHidden":758},[31672],[34,176015,176017,176020,176023],{"className":176016},[31676],[34,176018],{"className":176019,"style":33218},[31680],[34,176021,51834],{"className":176022,"style":37422},[31685,31686],[34,176024,640],{"className":176025},[31685],[34,176027,176029,176044],{"className":176028},[31642],[34,176030,176032],{"className":176031},[31646],[31648,176033,176034],{"xmlns":31650},[31652,176035,176036,176042],{},[31655,176037,176038,176040],{},[31658,176039,51834],{},[31749,176041,786],{},[31665,176043,175903],{"encoding":31667},[34,176045,176047],{"className":176046,"ariaHidden":758},[31672],[34,176048,176050,176053,176056],{"className":176049},[31676],[34,176051],{"className":176052,"style":33218},[31680],[34,176054,51834],{"className":176055,"style":37422},[31685,31686],[34,176057,786],{"className":176058},[31685]," 实际上是同一个文件实体的两个名字。",[15,176061,176062],{},"此时原文件 inode 的引用计数从：",[15,176064,176065],{},[34,176066,176068,176081],{"className":176067},[31642],[34,176069,176071],{"className":176070},[31646],[31648,176072,176073],{"xmlns":31650},[31652,176074,176075,176079],{},[31655,176076,176077],{},[31749,176078,640],{},[31665,176080,640],{"encoding":31667},[34,176082,176084],{"className":176083,"ariaHidden":758},[31672],[34,176085,176087,176090],{"className":176086},[31676],[34,176088],{"className":176089,"style":32617},[31680],[34,176091,640],{"className":176092},[31685],[15,176094,176095],{},"变成：",[15,176097,176098],{},[34,176099,176101,176114],{"className":176100},[31642],[34,176102,176104],{"className":176103},[31646],[31648,176105,176106],{"xmlns":31650},[31652,176107,176108,176112],{},[31655,176109,176110],{},[31749,176111,768],{},[31665,176113,768],{"encoding":31667},[34,176115,176117],{"className":176116,"ariaHidden":758},[31672],[34,176118,176120,176123],{"className":176119},[31676],[34,176121],{"className":176122,"style":32617},[31680],[34,176124,768],{"className":176125},[31685],[32188,176127],{},[15,176129,176130,176131,347],{},"第三步：删除 ",[34,176132,176134,176149],{"className":176133},[31642],[34,176135,176137],{"className":176136},[31646],[31648,176138,176139],{"xmlns":31650},[31652,176140,176141,176147],{},[31655,176142,176143,176145],{},[31658,176144,51834],{},[31749,176146,640],{},[31665,176148,175510],{"encoding":31667},[34,176150,176152],{"className":176151,"ariaHidden":758},[31672],[34,176153,176155,176158,176161],{"className":176154},[31676],[34,176156],{"className":176157,"style":33218},[31680],[34,176159,51834],{"className":176160,"style":37422},[31685,31686],[34,176162,640],{"className":176163},[31685],[15,176165,101509,176166,176199,176200,176233],{},[34,176167,176169,176184],{"className":176168},[31642],[34,176170,176172],{"className":176171},[31646],[31648,176173,176174],{"xmlns":31650},[31652,176175,176176,176182],{},[31655,176177,176178,176180],{},[31658,176179,51834],{},[31749,176181,640],{},[31665,176183,175510],{"encoding":31667},[34,176185,176187],{"className":176186,"ariaHidden":758},[31672],[34,176188,176190,176193,176196],{"className":176189},[31676],[34,176191],{"className":176192,"style":33218},[31680],[34,176194,51834],{"className":176195,"style":37422},[31685,31686],[34,176197,640],{"className":176198},[31685]," 只是删除目录项 ",[34,176201,176203,176218],{"className":176202},[31642],[34,176204,176206],{"className":176205},[31646],[31648,176207,176208],{"xmlns":31650},[31652,176209,176210,176216],{},[31655,176211,176212,176214],{},[31658,176213,51834],{},[31749,176215,640],{},[31665,176217,175510],{"encoding":31667},[34,176219,176221],{"className":176220,"ariaHidden":758},[31672],[34,176222,176224,176227,176230],{"className":176223},[31676],[34,176225],{"className":176226,"style":33218},[31680],[34,176228,51834],{"className":176229,"style":37422},[31685,31686],[34,176231,640],{"className":176232},[31685],"，并不是立刻删除文件内容。",[15,176235,93310,176236,176269],{},[34,176237,176239,176254],{"className":176238},[31642],[34,176240,176242],{"className":176241},[31646],[31648,176243,176244],{"xmlns":31650},[31652,176245,176246,176252],{},[31655,176247,176248,176250],{},[31658,176249,51834],{},[31749,176251,786],{},[31665,176253,175903],{"encoding":31667},[34,176255,176257],{"className":176256,"ariaHidden":758},[31672],[34,176258,176260,176263,176266],{"className":176259},[31676],[34,176261],{"className":176262,"style":33218},[31680],[34,176264,51834],{"className":176265,"style":37422},[31685,31686],[34,176267,786],{"className":176268},[31685]," 仍然指向原来的 inode。",[15,176271,176272],{},"所以原 inode 的引用计数从：",[15,176274,176275],{},[34,176276,176278,176291],{"className":176277},[31642],[34,176279,176281],{"className":176280},[31646],[31648,176282,176283],{"xmlns":31650},[31652,176284,176285,176289],{},[31655,176286,176287],{},[31749,176288,768],{},[31665,176290,768],{"encoding":31667},[34,176292,176294],{"className":176293,"ariaHidden":758},[31672],[34,176295,176297,176300],{"className":176296},[31676],[34,176298],{"className":176299,"style":32617},[31680],[34,176301,768],{"className":176302},[31685],[15,176304,176095],{},[15,176306,176307],{},[34,176308,176310,176323],{"className":176309},[31642],[34,176311,176313],{"className":176312},[31646],[31648,176314,176315],{"xmlns":31650},[31652,176316,176317,176321],{},[31655,176318,176319],{},[31749,176320,640],{},[31665,176322,640],{"encoding":31667},[34,176324,176326],{"className":176325,"ariaHidden":758},[31672],[34,176327,176329,176332],{"className":176328},[31676],[34,176330],{"className":176331,"style":32617},[31680],[34,176333,640],{"className":176334},[31685],[15,176336,176337],{},"因此：",[15,176339,176340,176373],{},[34,176341,176343,176358],{"className":176342},[31642],[34,176344,176346],{"className":176345},[31646],[31648,176347,176348],{"xmlns":31650},[31652,176349,176350,176356],{},[31655,176351,176352,176354],{},[31658,176353,51834],{},[31749,176355,786],{},[31665,176357,175903],{"encoding":31667},[34,176359,176361],{"className":176360,"ariaHidden":758},[31672],[34,176362,176364,176367,176370],{"className":176363},[31676],[34,176365],{"className":176366,"style":33218},[31680],[34,176368,51834],{"className":176369,"style":37422},[31685,31686],[34,176371,786],{"className":176372},[31685]," 对应的原文件 inode 引用计数为：",[15,176375,176376],{},[34,176377,176379,176392],{"className":176378},[31642],[34,176380,176382],{"className":176381},[31646],[31648,176383,176384],{"xmlns":31650},[31652,176385,176386,176390],{},[31655,176387,176388],{},[31749,176389,640],{},[31665,176391,640],{"encoding":31667},[34,176393,176395],{"className":176394,"ariaHidden":758},[31672],[34,176396,176398,176401],{"className":176397},[31676],[34,176399],{"className":176400,"style":32617},[31680],[34,176402,640],{"className":176403},[31685],[15,176405,175781,176406,176439],{},[34,176407,176409,176424],{"className":176408},[31642],[34,176410,176412],{"className":176411},[31646],[31648,176413,176414],{"xmlns":31650},[31652,176415,176416,176422],{},[31655,176417,176418,176420],{},[31658,176419,51834],{},[31749,176421,768],{},[31665,176423,175651],{"encoding":31667},[34,176425,176427],{"className":176426,"ariaHidden":758},[31672],[34,176428,176430,176433,176436],{"className":176429},[31676],[34,176431],{"className":176432,"style":33218},[31680],[34,176434,51834],{"className":176435,"style":37422},[31685,31686],[34,176437,768],{"className":176438},[31685]," 是符号链接文件，它本身没有被删除，所以它自己的引用计数仍为：",[15,176441,176442],{},[34,176443,176445,176458],{"className":176444},[31642],[34,176446,176448],{"className":176447},[31646],[31648,176449,176450],{"xmlns":31650},[31652,176451,176452,176456],{},[31655,176453,176454],{},[31749,176455,640],{},[31665,176457,640],{"encoding":31667},[34,176459,176461],{"className":176460,"ariaHidden":758},[31672],[34,176462,176464,176467],{"className":176463},[31676],[34,176465],{"className":176466,"style":32617},[31680],[34,176468,640],{"className":176469},[31685],[15,176471,176472,176473,176506,176507,176540,176541,176574,176575,176608],{},"但是要注意：",[34,176474,176476,176491],{"className":176475},[31642],[34,176477,176479],{"className":176478},[31646],[31648,176480,176481],{"xmlns":31650},[31652,176482,176483,176489],{},[31655,176484,176485,176487],{},[31658,176486,51834],{},[31749,176488,768],{},[31665,176490,175651],{"encoding":31667},[34,176492,176494],{"className":176493,"ariaHidden":758},[31672],[34,176495,176497,176500,176503],{"className":176496},[31676],[34,176498],{"className":176499,"style":33218},[31680],[34,176501,51834],{"className":176502,"style":37422},[31685,31686],[34,176504,768],{"className":176505},[31685]," 保存的是路径 ",[34,176508,176510,176525],{"className":176509},[31642],[34,176511,176513],{"className":176512},[31646],[31648,176514,176515],{"xmlns":31650},[31652,176516,176517,176523],{},[31655,176518,176519,176521],{},[31658,176520,51834],{},[31749,176522,640],{},[31665,176524,175510],{"encoding":31667},[34,176526,176528],{"className":176527,"ariaHidden":758},[31672],[34,176529,176531,176534,176537],{"className":176530},[31676],[34,176532],{"className":176533,"style":33218},[31680],[34,176535,51834],{"className":176536,"style":37422},[31685,31686],[34,176538,640],{"className":176539},[31685],"，现在 ",[34,176542,176544,176559],{"className":176543},[31642],[34,176545,176547],{"className":176546},[31646],[31648,176548,176549],{"xmlns":31650},[31652,176550,176551,176557],{},[31655,176552,176553,176555],{},[31658,176554,51834],{},[31749,176556,640],{},[31665,176558,175510],{"encoding":31667},[34,176560,176562],{"className":176561,"ariaHidden":758},[31672],[34,176563,176565,176568,176571],{"className":176564},[31676],[34,176566],{"className":176567,"style":33218},[31680],[34,176569,51834],{"className":176570,"style":37422},[31685,31686],[34,176572,640],{"className":176573},[31685]," 被删了，所以 ",[34,176576,176578,176593],{"className":176577},[31642],[34,176579,176581],{"className":176580},[31646],[31648,176582,176583],{"xmlns":31650},[31652,176584,176585,176591],{},[31655,176586,176587,176589],{},[31658,176588,51834],{},[31749,176590,768],{},[31665,176592,175651],{"encoding":31667},[34,176594,176596],{"className":176595,"ariaHidden":758},[31672],[34,176597,176599,176602,176605],{"className":176598},[31676],[34,176600],{"className":176601,"style":33218},[31680],[34,176603,51834],{"className":176604,"style":37422},[31685,31686],[34,176606,768],{"className":176607},[31685]," 会变成悬空符号链接。",[15,176610,153429],{},[15,176612,176613],{},[34,176614,176616,176636],{"className":176615},[31642],[34,176617,176619],{"className":176618},[31646],[31648,176620,176621],{"xmlns":31650},[31652,176622,176623,176633],{},[31655,176624,176625,176627,176629,176631],{},[31658,176626,51834],{},[31749,176628,768],{},[31662,176630,202],{},[31749,176632,640],{},[31665,176634,176635],{"encoding":31667},"F2 = 1",[34,176637,176639,176660],{"className":176638,"ariaHidden":758},[31672],[34,176640,176642,176645,176648,176651,176654,176657],{"className":176641},[31676],[34,176643],{"className":176644,"style":33218},[31680],[34,176646,51834],{"className":176647,"style":37422},[31685,31686],[34,176649,768],{"className":176650},[31685],[34,176652],{"className":176653,"style":31846},[31845],[34,176655,202],{"className":176656},[31850],[34,176658],{"className":176659,"style":31846},[31845],[34,176661,176663,176666],{"className":176662},[31676],[34,176664],{"className":176665,"style":32617},[31680],[34,176667,640],{"className":176668},[31685],[15,176670,176671],{},[34,176672,176674,176694],{"className":176673},[31642],[34,176675,176677],{"className":176676},[31646],[31648,176678,176679],{"xmlns":31650},[31652,176680,176681,176691],{},[31655,176682,176683,176685,176687,176689],{},[31658,176684,51834],{},[31749,176686,786],{},[31662,176688,202],{},[31749,176690,640],{},[31665,176692,176693],{"encoding":31667},"F3 = 1",[34,176695,176697,176718],{"className":176696,"ariaHidden":758},[31672],[34,176698,176700,176703,176706,176709,176712,176715],{"className":176699},[31676],[34,176701],{"className":176702,"style":33218},[31680],[34,176704,51834],{"className":176705,"style":37422},[31685,31686],[34,176707,786],{"className":176708},[31685],[34,176710],{"className":176711,"style":31846},[31845],[34,176713,202],{"className":176714},[31850],[34,176716],{"className":176717,"style":31846},[31845],[34,176719,176721,176724],{"className":176720},[31676],[34,176722],{"className":176723,"style":32617},[31680],[34,176725,640],{"className":176726},[31685],[15,176728,176729],{},"答案选：",[15,176731,176732],{},[34,176733,176735,176748],{"className":176734},[31642],[34,176736,176738],{"className":176737},[31646],[31648,176739,176740],{"xmlns":31650},[31652,176741,176742,176746],{},[31655,176743,176744],{},[31658,176745,47384],{},[31665,176747,47384],{"encoding":31667},[34,176749,176751],{"className":176750,"ariaHidden":758},[31672],[34,176752,176754,176757],{"className":176753},[31676],[34,176755],{"className":176756,"style":33218},[31680],[34,176758,47384],{"className":176759,"style":38706},[31685,31686],[32188,176761],{},[253,176763,176765],{"id":176764},"_13-磁盘输入输出操作的信息传送单位","1.3 磁盘输入输出操作的信息传送单位",[15,176767,58173],{},[15,176769,176770],{},[34,176771,176773,176790],{"className":176772},[31642],[34,176774,176776],{"className":176775},[31646],[31648,176777,176778],{"xmlns":31650},[31652,176779,176780,176787],{},[31655,176781,176782,176784],{},[31658,176783,47384],{},[39361,176785,176786],{}," 块",[31665,176788,176789],{"encoding":31667},"C \\text{ 块}",[34,176791,176793],{"className":176792,"ariaHidden":758},[31672],[34,176794,176796,176799,176802],{"className":176795},[31676],[34,176797],{"className":176798,"style":33218},[31680],[34,176800,47384],{"className":176801,"style":38706},[31685,31686],[34,176803,176805,176808],{"className":176804},[31685,2638],[34,176806,130298],{"className":176807},[31685],[34,176809,176811],{"className":176810},[31685,105462],"块",[15,176813,176814],{},"磁盘 I\u002FO 通常不是按字符传输，也不是按字传输，而是按块传输。",[15,176816,176817],{},"一个磁盘块通常包含多个字节，文件系统读写磁盘时，基本单位是：",[15,176819,176820],{},[34,176821,176823,176844],{"className":176822},[31642],[34,176824,176826],{"className":176825},[31646],[31648,176827,176828],{"xmlns":31650},[31652,176829,176830,176842],{},[31655,176831,176832,176834,176836,176838,176840],{},[31658,176833,1697],{},[31658,176835,43379],{},[31658,176837,43383],{},[31658,176839,31965],{},[31658,176841,33759],{},[31665,176843,126394],{"encoding":31667},[34,176845,176847],{"className":176846,"ariaHidden":758},[31672],[34,176848,176850,176853,176856,176859,176863],{"className":176849},[31676],[34,176851],{"className":176852,"style":31681},[31680],[34,176854,1697],{"className":176855},[31685,31686],[34,176857,43379],{"className":176858,"style":47113},[31685,31686],[34,176860,176862],{"className":176861},[31685,31686],"oc",[34,176864,33759],{"className":176865,"style":33774},[31685,31686],[15,176867,129204],{},[15,176869,176870],{},[34,176871,176873,176886],{"className":176872},[31642],[34,176874,176876],{"className":176875},[31646],[31648,176877,176878],{"xmlns":31650},[31652,176879,176880,176884],{},[31655,176881,176882],{},[39361,176883,176811],{},[31665,176885,176811],{"encoding":31667},[34,176887,176889],{"className":176888,"ariaHidden":758},[31672],[34,176890,176892,176895],{"className":176891},[31676],[34,176893],{"className":176894,"style":33218},[31680],[34,176896,176811],{"className":176897},[31685,105462],[15,176899,176900],{},"所以答案选：",[15,176902,176903],{},[34,176904,176906,176919],{"className":176905},[31642],[34,176907,176909],{"className":176908},[31646],[31648,176910,176911],{"xmlns":31650},[31652,176912,176913,176917],{},[31655,176914,176915],{},[31658,176916,47384],{},[31665,176918,47384],{"encoding":31667},[34,176920,176922],{"className":176921,"ariaHidden":758},[31672],[34,176923,176925,176928],{"className":176924},[31676],[34,176926],{"className":176927,"style":33218},[31680],[34,176929,47384],{"className":176930,"style":38706},[31685,31686],[32188,176932],{},[253,176934,176936],{"id":176935},"_14-640480256色图像需要多少kb","1.4 640×480，256色图像需要多少KB？",[15,176938,58173],{},[15,176940,176941],{},[34,176942,176944,176962],{"className":176943},[31642],[34,176945,176947],{"className":176946},[31646],[31648,176948,176949],{"xmlns":31650},[31652,176950,176951,176959],{},[31655,176952,176953,176955,176957],{},[31749,176954,25357],{},[31658,176956,38691],{},[31658,176958,54224],{},[31665,176960,176961],{"encoding":31667},"300KB",[34,176963,176965],{"className":176964,"ariaHidden":758},[31672],[34,176966,176968,176971,176974,176977],{"className":176967},[31676],[34,176969],{"className":176970,"style":33218},[31680],[34,176972,25357],{"className":176973},[31685],[34,176975,38691],{"className":176976,"style":38706},[31685,31686],[34,176978,54224],{"className":176979,"style":54252},[31685,31686],[15,176981,176982],{},"屏幕分辨率为：",[15,176984,176985],{},[34,176986,176988,177007],{"className":176987},[31642],[34,176989,176991],{"className":176990},[31646],[31648,176992,176993],{"xmlns":31650},[31652,176994,176995,177004],{},[31655,176996,176997,177000,177002],{},[31749,176998,176999],{},"640",[31662,177001,42071],{},[31749,177003,173407],{},[31665,177005,177006],{"encoding":31667},"640 \\times 480",[34,177008,177010,177028],{"className":177009,"ariaHidden":758},[31672],[34,177011,177013,177016,177019,177022,177025],{"className":177012},[31676],[34,177014],{"className":177015,"style":35942},[31680],[34,177017,176999],{"className":177018},[31685],[34,177020],{"className":177021,"style":32604},[31845],[34,177023,42071],{"className":177024},[32024],[34,177026],{"className":177027,"style":32604},[31845],[34,177029,177031,177034],{"className":177030},[31676],[34,177032],{"className":177033,"style":32617},[31680],[34,177035,173407],{"className":177036},[31685],[15,177038,177039],{},"所以总像素数为：",[15,177041,177042],{},[34,177043,177045,177068],{"className":177044},[31642],[34,177046,177048],{"className":177047},[31646],[31648,177049,177050],{"xmlns":31650},[31652,177051,177052,177065],{},[31655,177053,177054,177056,177058,177060,177062],{},[31749,177055,176999],{},[31662,177057,42071],{},[31749,177059,173407],{},[31662,177061,202],{},[31749,177063,177064],{},"307200",[31665,177066,177067],{"encoding":31667},"640 \\times 480 = 307200",[34,177069,177071,177089,177107],{"className":177070,"ariaHidden":758},[31672],[34,177072,177074,177077,177080,177083,177086],{"className":177073},[31676],[34,177075],{"className":177076,"style":35942},[31680],[34,177078,176999],{"className":177079},[31685],[34,177081],{"className":177082,"style":32604},[31845],[34,177084,42071],{"className":177085},[32024],[34,177087],{"className":177088,"style":32604},[31845],[34,177090,177092,177095,177098,177101,177104],{"className":177091},[31676],[34,177093],{"className":177094,"style":32617},[31680],[34,177096,173407],{"className":177097},[31685],[34,177099],{"className":177100,"style":31846},[31845],[34,177102,202],{"className":177103},[31850],[34,177105],{"className":177106,"style":31846},[31845],[34,177108,177110,177113],{"className":177109},[31676],[34,177111],{"className":177112,"style":32617},[31680],[34,177114,177064],{"className":177115},[31685],[15,177117,177118],{},"256 色表示每个像素有 256 种可能颜色。",[15,177120,131336],{},[15,177122,177123],{},[34,177124,177126,177148],{"className":177125},[31642],[34,177127,177129],{"className":177128},[31646],[31648,177130,177131],{"xmlns":31650},[31652,177132,177133,177145],{},[31655,177134,177135,177137,177139],{},[31749,177136,95418],{},[31662,177138,202],{},[37624,177140,177141,177143],{},[31749,177142,768],{},[31749,177144,60525],{},[31665,177146,177147],{"encoding":31667},"256 = 2^8",[34,177149,177151,177169],{"className":177150,"ariaHidden":758},[31672],[34,177152,177154,177157,177160,177163,177166],{"className":177153},[31676],[34,177155],{"className":177156,"style":32617},[31680],[34,177158,95418],{"className":177159},[31685],[34,177161],{"className":177162,"style":31846},[31845],[34,177164,202],{"className":177165},[31850],[34,177167],{"className":177168,"style":31846},[31845],[34,177170,177172,177175],{"className":177171},[31676],[34,177173],{"className":177174,"style":37774},[31680],[34,177176,177178,177181],{"className":177177},[31685],[34,177179,768],{"className":177180},[31685],[34,177182,177184],{"className":177183},[31792],[34,177185,177187],{"className":177186},[31796],[34,177188,177190],{"className":177189},[31801],[34,177191,177193],{"className":177192,"style":37774},[31805],[34,177194,177195,177198],{"style":37777},[34,177196],{"className":177197,"style":31814},[31813],[34,177199,177201],{"className":177200},[31818,31819,31820,31821],[34,177202,60525],{"className":177203},[31685,31821],[15,177205,177206],{},"所以每个像素需要：",[15,177208,177209],{},[34,177210,177212,177238],{"className":177211},[31642],[34,177213,177215],{"className":177214},[31646],[31648,177216,177217],{"xmlns":31650},[31652,177218,177219,177235],{},[31655,177220,177221,177223,177225,177227,177229,177231,177233],{},[31749,177222,60525],{},[31658,177224,1697],{},[31658,177226,31955],{},[31658,177228,33249],{},[31662,177230,202],{},[31749,177232,640],{},[31658,177234,54224],{},[31665,177236,177237],{"encoding":31667},"8bit = 1B",[34,177239,177241,177266],{"className":177240,"ariaHidden":758},[31672],[34,177242,177244,177247,177250,177254,177257,177260,177263],{"className":177243},[31676],[34,177245],{"className":177246,"style":31681},[31680],[34,177248,60525],{"className":177249},[31685],[34,177251,177253],{"className":177252},[31685,31686],"bi",[34,177255,33249],{"className":177256},[31685,31686],[34,177258],{"className":177259,"style":31846},[31845],[34,177261,202],{"className":177262},[31850],[34,177264],{"className":177265,"style":31846},[31845],[34,177267,177269,177272,177275],{"className":177268},[31676],[34,177270],{"className":177271,"style":33218},[31680],[34,177273,640],{"className":177274},[31685],[34,177276,54224],{"className":177277,"style":54252},[31685,31686],[15,177279,177280],{},"总存储空间为：",[15,177282,177283],{},[34,177284,177286,177312],{"className":177285},[31642],[34,177287,177289],{"className":177288},[31646],[31648,177290,177291],{"xmlns":31650},[31652,177292,177293,177309],{},[31655,177294,177295,177297,177299,177301,177303,177305,177307],{},[31749,177296,177064],{},[31662,177298,42071],{},[31749,177300,640],{},[31658,177302,54224],{},[31662,177304,202],{},[31749,177306,177064],{},[31658,177308,54224],{},[31665,177310,177311],{"encoding":31667},"307200 \\times 1B = 307200B",[34,177313,177315,177333,177354],{"className":177314,"ariaHidden":758},[31672],[34,177316,177318,177321,177324,177327,177330],{"className":177317},[31676],[34,177319],{"className":177320,"style":35942},[31680],[34,177322,177064],{"className":177323},[31685],[34,177325],{"className":177326,"style":32604},[31845],[34,177328,42071],{"className":177329},[32024],[34,177331],{"className":177332,"style":32604},[31845],[34,177334,177336,177339,177342,177345,177348,177351],{"className":177335},[31676],[34,177337],{"className":177338,"style":33218},[31680],[34,177340,640],{"className":177341},[31685],[34,177343,54224],{"className":177344,"style":54252},[31685,31686],[34,177346],{"className":177347,"style":31846},[31845],[34,177349,202],{"className":177350},[31850],[34,177352],{"className":177353,"style":31846},[31845],[34,177355,177357,177360,177363],{"className":177356},[31676],[34,177358],{"className":177359,"style":33218},[31680],[34,177361,177064],{"className":177362},[31685],[34,177364,54224],{"className":177365,"style":54252},[31685,31686],[15,177367,177368],{},"换算成 KB：",[15,177370,177371],{},[34,177372,177374,177399],{"className":177373},[31642],[34,177375,177377],{"className":177376},[31646],[31648,177378,177379],{"xmlns":31650},[31652,177380,177381,177396],{},[31655,177382,177383,177385,177387,177389,177391,177394],{},[31749,177384,640],{},[31658,177386,38691],{},[31658,177388,54224],{},[31662,177390,202],{},[31749,177392,177393],{},"1024",[31658,177395,54224],{},[31665,177397,177398],{"encoding":31667},"1KB = 1024B",[34,177400,177402,177426],{"className":177401,"ariaHidden":758},[31672],[34,177403,177405,177408,177411,177414,177417,177420,177423],{"className":177404},[31676],[34,177406],{"className":177407,"style":33218},[31680],[34,177409,640],{"className":177410},[31685],[34,177412,38691],{"className":177413,"style":38706},[31685,31686],[34,177415,54224],{"className":177416,"style":54252},[31685,31686],[34,177418],{"className":177419,"style":31846},[31845],[34,177421,202],{"className":177422},[31850],[34,177424],{"className":177425,"style":31846},[31845],[34,177427,177429,177432,177435],{"className":177428},[31676],[34,177430],{"className":177431,"style":33218},[31680],[34,177433,177393],{"className":177434},[31685],[34,177436,54224],{"className":177437,"style":54252},[31685,31686],[15,177439,159962],{},[15,177441,177442],{},[34,177443,177445,177471],{"className":177444},[31642],[34,177446,177448],{"className":177447},[31646],[31648,177449,177450],{"xmlns":31650},[31652,177451,177452,177468],{},[31655,177453,177454,177456,177458,177460,177462,177464,177466],{},[31749,177455,177064],{},[31658,177457,38080],{"mathvariant":38079},[31749,177459,177393],{},[31662,177461,202],{},[31749,177463,25357],{},[31658,177465,38691],{},[31658,177467,54224],{},[31665,177469,177470],{"encoding":31667},"307200 \u002F 1024 = 300KB",[34,177472,177474,177493],{"className":177473,"ariaHidden":758},[31672],[34,177475,177477,177480,177484,177487,177490],{"className":177476},[31676],[34,177478],{"className":177479,"style":31774},[31680],[34,177481,177483],{"className":177482},[31685],"307200\u002F1024",[34,177485],{"className":177486,"style":31846},[31845],[34,177488,202],{"className":177489},[31850],[34,177491],{"className":177492,"style":31846},[31845],[34,177494,177496,177499,177502,177505],{"className":177495},[31676],[34,177497],{"className":177498,"style":33218},[31680],[34,177500,25357],{"className":177501},[31685],[34,177503,38691],{"className":177504,"style":38706},[31685,31686],[34,177506,54224],{"className":177507,"style":54252},[31685,31686],[15,177509,177510],{},"最终答案：",[15,177512,177513],{},[34,177514,177516,177533],{"className":177515},[31642],[34,177517,177519],{"className":177518},[31646],[31648,177520,177521],{"xmlns":31650},[31652,177522,177523,177531],{},[31655,177524,177525,177527,177529],{},[31749,177526,25357],{},[31658,177528,38691],{},[31658,177530,54224],{},[31665,177532,176961],{"encoding":31667},[34,177534,177536],{"className":177535,"ariaHidden":758},[31672],[34,177537,177539,177542,177545,177548],{"className":177538},[31676],[34,177540],{"className":177541,"style":33218},[31680],[34,177543,25357],{"className":177544},[31685],[34,177546,38691],{"className":177547,"style":38706},[31685,31686],[34,177549,54224],{"className":177550,"style":54252},[31685,31686],[32188,177552],{},[253,177554,177556],{"id":177555},"_15-什么是文件的物理结构有哪几种组织方式","1.5 什么是文件的物理结构？有哪几种组织方式？",[15,177558,177559],{},"文件的物理结构是指文件在外存，也就是磁盘上的实际存放方式。",[15,177561,177562],{},"换句话说，它关心的是：",[15,177564,177565],{},"文件被分成哪些磁盘块？",[15,177567,177568],{},"这些磁盘块在磁盘上如何分布？",[15,177570,177571],{},"操作系统如何记录这些块的位置？",[15,177573,177574],{},"Stallings 第 12 章在 secondary storage management 中讲到，文件在二级存储上由一组 blocks 组成，操作系统或文件管理系统负责给文件分配这些块。",[15,177576,177577],{},"常见的文件物理组织方式主要有三种：",[32188,177579],{},[32490,177581,177583],{"id":177582},"_1-连续分配-contiguous-allocation","1. 连续分配 Contiguous Allocation",[15,177585,177586],{},"文件占用一组连续的磁盘块。",[15,177588,2632],{},[15,177590,177591],{},[34,177592,177594,177624],{"className":177593},[31642],[34,177595,177597],{"className":177596},[31646],[31648,177598,177599],{"xmlns":31650},[31652,177600,177601,177621],{},[31655,177602,177603,177605,177607,177609,177611,177613,177615,177617,177619],{},[31749,177604,3249],{},[31662,177606,31960],{"separator":758},[31749,177608,95482],{},[31662,177610,31960],{"separator":758},[31749,177612,94716],{},[31662,177614,31960],{"separator":758},[31749,177616,94703],{},[31662,177618,31960],{"separator":758},[31749,177620,96087],{},[31665,177622,177623],{"encoding":31667},"20, 21, 22, 23, 24",[34,177625,177627],{"className":177626,"ariaHidden":758},[31672],[34,177628,177630,177633,177636,177639,177642,177645,177648,177651,177654,177657,177660,177663,177666,177669],{"className":177629},[31676],[34,177631],{"className":177632,"style":40292},[31680],[34,177634,3249],{"className":177635},[31685],[34,177637,31960],{"className":177638},[32119],[34,177640],{"className":177641,"style":32123},[31845],[34,177643,95482],{"className":177644},[31685],[34,177646,31960],{"className":177647},[32119],[34,177649],{"className":177650,"style":32123},[31845],[34,177652,94716],{"className":177653},[31685],[34,177655,31960],{"className":177656},[32119],[34,177658],{"className":177659,"style":32123},[31845],[34,177661,94703],{"className":177662},[31685],[34,177664,31960],{"className":177665},[32119],[34,177667],{"className":177668,"style":32123},[31845],[34,177670,96087],{"className":177671},[31685],[15,177673,177674],{},"这些块连续存放同一个文件。",[15,177676,14543],{},[2230,177678,177679,177682,177685],{},[108,177680,177681],{},"顺序访问速度快。",[108,177683,177684],{},"随机访问也方便。",[108,177686,177687],{},"文件分配表只需要记录起始块号和文件长度。",[15,177689,177690,177691,177719,177720,177748],{},"如果文件从块 ",[34,177692,177694,177707],{"className":177693},[31642],[34,177695,177697],{"className":177696},[31646],[31648,177698,177699],{"xmlns":31650},[31652,177700,177701,177705],{},[31655,177702,177703],{},[31658,177704,1697],{},[31665,177706,1697],{"encoding":31667},[34,177708,177710],{"className":177709,"ariaHidden":758},[31672],[34,177711,177713,177716],{"className":177712},[31676],[34,177714],{"className":177715,"style":31681},[31680],[34,177717,1697],{"className":177718},[31685,31686]," 开始，要访问第 ",[34,177721,177723,177736],{"className":177722},[31642],[34,177724,177726],{"className":177725},[31646],[31648,177727,177728],{"xmlns":31650},[31652,177729,177730,177734],{},[31655,177731,177732],{},[31658,177733,31955],{},[31665,177735,31955],{"encoding":31667},[34,177737,177739],{"className":177738,"ariaHidden":758},[31672],[34,177740,177742,177745],{"className":177741},[31676],[34,177743],{"className":177744,"style":33104},[31680],[34,177746,31955],{"className":177747},[31685,31686]," 块，则物理位置可以直接算：",[15,177750,177751],{},[34,177752,177754,177776],{"className":177753},[31642],[34,177755,177757],{"className":177756},[31646],[31648,177758,177759],{"xmlns":31650},[31652,177760,177761,177773],{},[31655,177762,177763,177765,177767,177769,177771],{},[31658,177764,1697],{},[31662,177766,223],{},[31658,177768,31955],{},[31662,177770,32226],{},[31749,177772,640],{},[31665,177774,177775],{"encoding":31667},"b + i - 1",[34,177777,177779,177797,177815],{"className":177778,"ariaHidden":758},[31672],[34,177780,177782,177785,177788,177791,177794],{"className":177781},[31676],[34,177783],{"className":177784,"style":34299},[31680],[34,177786,1697],{"className":177787},[31685,31686],[34,177789],{"className":177790,"style":32604},[31845],[34,177792,223],{"className":177793},[32024],[34,177795],{"className":177796,"style":32604},[31845],[34,177798,177800,177803,177806,177809,177812],{"className":177799},[31676],[34,177801],{"className":177802,"style":37322},[31680],[34,177804,31955],{"className":177805},[31685,31686],[34,177807],{"className":177808,"style":32604},[31845],[34,177810,32226],{"className":177811},[32024],[34,177813],{"className":177814,"style":32604},[31845],[34,177816,177818,177821],{"className":177817},[31676],[34,177819],{"className":177820,"style":32617},[31680],[34,177822,640],{"className":177823},[31685],[15,177825,177826],{},"缺点：",[2230,177828,177829,177832,177835],{},[108,177830,177831],{},"容易产生外部碎片。",[108,177833,177834],{},"文件增长困难。",[108,177836,177837],{},"创建文件时可能需要提前知道文件最大长度。",[32188,177839],{},[32490,177841,177843],{"id":177842},"_2-链式分配-chained-allocation","2. 链式分配 Chained Allocation",[15,177845,177846],{},"文件由多个磁盘块组成，每个块中保存指向下一个块的指针。",[15,177848,2632],{},[15,177850,177851],{},[34,177852,177854,177885],{"className":177853},[31642],[34,177855,177857],{"className":177856},[31646],[31648,177858,177859],{"xmlns":31650},[31652,177860,177861,177882],{},[31655,177862,177863,177865,177867,177870,177872,177874,177876,177878,177880],{},[31749,177864,36732],{},[31662,177866,33239],{},[31749,177868,177869],{},"16",[31662,177871,33239],{},[31749,177873,640],{},[31662,177875,33239],{},[31749,177877,95556],{},[31662,177879,33239],{},[31749,177881,36723],{},[31665,177883,177884],{"encoding":31667},"9 \\rightarrow 16 \\rightarrow 1 \\rightarrow 25 \\rightarrow 7",[34,177886,177888,177906,177924,177942,177960],{"className":177887,"ariaHidden":758},[31672],[34,177889,177891,177894,177897,177900,177903],{"className":177890},[31676],[34,177892],{"className":177893,"style":32617},[31680],[34,177895,36732],{"className":177896},[31685],[34,177898],{"className":177899,"style":31846},[31845],[34,177901,33239],{"className":177902},[31850],[34,177904],{"className":177905,"style":31846},[31845],[34,177907,177909,177912,177915,177918,177921],{"className":177908},[31676],[34,177910],{"className":177911,"style":32617},[31680],[34,177913,177869],{"className":177914},[31685],[34,177916],{"className":177917,"style":31846},[31845],[34,177919,33239],{"className":177920},[31850],[34,177922],{"className":177923,"style":31846},[31845],[34,177925,177927,177930,177933,177936,177939],{"className":177926},[31676],[34,177928],{"className":177929,"style":32617},[31680],[34,177931,640],{"className":177932},[31685],[34,177934],{"className":177935,"style":31846},[31845],[34,177937,33239],{"className":177938},[31850],[34,177940],{"className":177941,"style":31846},[31845],[34,177943,177945,177948,177951,177954,177957],{"className":177944},[31676],[34,177946],{"className":177947,"style":32617},[31680],[34,177949,95556],{"className":177950},[31685],[34,177952],{"className":177953,"style":31846},[31845],[34,177955,33239],{"className":177956},[31850],[34,177958],{"className":177959,"style":31846},[31845],[34,177961,177963,177966],{"className":177962},[31676],[34,177964],{"className":177965,"style":32617},[31680],[34,177967,36723],{"className":177968},[31685],[15,177970,177971],{},"这些块不必连续。",[15,177973,14543],{},[2230,177975,177976,177979,177982],{},[108,177977,177978],{},"不要求连续空间。",[108,177980,177981],{},"文件增长比较方便。",[108,177983,177984],{},"不容易产生外部碎片。",[15,177986,177826],{},[2230,177988,177989,177992,178024,178027],{},[108,177990,177991],{},"随机访问效率低。",[108,177993,177994,177995,178023],{},"要访问第 ",[34,177996,177998,178011],{"className":177997},[31642],[34,177999,178001],{"className":178000},[31646],[31648,178002,178003],{"xmlns":31650},[31652,178004,178005,178009],{},[31655,178006,178007],{},[31658,178008,31955],{},[31665,178010,31955],{"encoding":31667},[34,178012,178014],{"className":178013,"ariaHidden":758},[31672],[34,178015,178017,178020],{"className":178016},[31676],[34,178018],{"className":178019,"style":33104},[31680],[34,178021,31955],{"className":178022},[31685,31686]," 块，必须从头沿链找。",[108,178025,178026],{},"指针损坏可能导致后续块丢失。",[108,178028,178029],{},"不符合局部性，连续读文件时可能在磁盘上跳来跳去。",[32188,178031],{},[32490,178033,178035],{"id":178034},"_3-索引分配-indexed-allocation","3. 索引分配 Indexed Allocation",[15,178037,178038],{},"为每个文件建立一个索引块，索引块中保存该文件所有数据块的地址。",[15,178040,178041],{},"例如索引块中保存：",[15,178043,178044],{},[34,178045,178047,178077],{"className":178046},[31642],[34,178048,178050],{"className":178049},[31646],[31648,178051,178052],{"xmlns":31650},[31652,178053,178054,178074],{},[31655,178055,178056,178058,178060,178062,178064,178066,178068,178070,178072],{},[31749,178057,36723],{},[31662,178059,31960],{"separator":758},[31749,178061,3249],{},[31662,178063,31960],{"separator":758},[31749,178065,786],{},[31662,178067,31960],{"separator":758},[31749,178069,96826],{},[31662,178071,31960],{"separator":758},[31749,178073,95396],{},[31665,178075,178076],{"encoding":31667},"7, 20, 3, 88, 15",[34,178078,178080],{"className":178079,"ariaHidden":758},[31672],[34,178081,178083,178086,178089,178092,178095,178098,178101,178104,178107,178110,178113,178116,178119,178122],{"className":178082},[31676],[34,178084],{"className":178085,"style":40292},[31680],[34,178087,36723],{"className":178088},[31685],[34,178090,31960],{"className":178091},[32119],[34,178093],{"className":178094,"style":32123},[31845],[34,178096,3249],{"className":178097},[31685],[34,178099,31960],{"className":178100},[32119],[34,178102],{"className":178103,"style":32123},[31845],[34,178105,786],{"className":178106},[31685],[34,178108,31960],{"className":178109},[32119],[34,178111],{"className":178112,"style":32123},[31845],[34,178114,96826],{"className":178115},[31685],[34,178117,31960],{"className":178118},[32119],[34,178120],{"className":178121,"style":32123},[31845],[34,178123,95396],{"className":178124},[31685],[15,178126,178127],{},"表示文件数据分别存放在这些磁盘块中。",[15,178129,14543],{},[2230,178131,178132,178135,178138,178141],{},[108,178133,178134],{},"支持直接访问。",[108,178136,178137],{},"支持顺序访问。",[108,178139,178140],{},"不要求文件块连续。",[108,178142,178143],{},"相比链式分配，随机访问效率更高。",[15,178145,177826],{},[2230,178147,178148,178151,178154],{},[108,178149,178150],{},"索引块本身需要额外空间。",[108,178152,178153],{},"小文件也可能有索引开销。",[108,178155,178156],{},"大文件可能需要多级索引。",[15,178158,178159],{},"Stallings 认为索引分配解决了连续分配和链式分配的许多问题，因此是很常见的文件分配方式。",[32188,178161],{},[253,178163,178165],{"id":178164},"_16-什么是文件的逻辑结构有哪几种组织方式","1.6 什么是文件的逻辑结构？有哪几种组织方式？",[15,178167,178168],{},"文件的逻辑结构是指从用户或应用程序角度看到的文件内部组织方式。",[15,178170,178171],{},"它关心的是：",[15,178173,178174],{},"文件中的数据在逻辑上如何排列？",[15,178176,178177],{},"是字节流？",[15,178179,178180],{},"是一条条记录？",[15,178182,178183],{},"是否有索引？",[15,178185,178186],{},"是否按关键字顺序排列？",[15,178188,178189],{},"Stallings 将 file organization 定义为由访问方式决定的记录逻辑结构。",[15,178191,178192],{},"常见逻辑结构可以分两套说法。",[32188,178194],{},[32490,178196,178197],{"id":178197},"国内教材常见分类",[2230,178199,178200],{},[108,178201,175354],{},[15,178203,178204],{},"文件被看作连续的字节流。",[15,178206,178207],{},"UNIX\u002FLinux 普通文件就是典型例子。",[2230,178209,178210],{"start":43},[108,178211,178212],{},"记录式文件",[15,178214,178215],{},"文件由一条条记录组成。",[15,178217,178218],{},"例如学生信息文件、工资表文件、成绩表文件。",[15,178220,178221],{},"记录式文件还可以分为：",[15,178223,178224],{},"定长记录文件",[15,178226,178227],{},"变长记录文件",[32188,178229],{},[32490,178231,178233],{"id":178232},"stallings-第-12-章常见分类","Stallings 第 12 章常见分类",[15,178235,178236],{},"Stallings 讲了五种基本 file organization：",[2230,178238,178239],{},[108,178240,178241],{},"堆文件 Pile",[15,178243,178244],{},"数据按照到达顺序直接加入文件。",[15,178246,4160],{},[105,178248,178249,178252,178255],{},[108,178250,178251],{},"结构最简单；",[108,178253,178254],{},"适合收集数据；",[108,178256,178257],{},"查找效率较低。",[2230,178259,178260],{"start":43},[108,178261,178262],{},"顺序文件 Sequential File",[15,178264,178265],{},"记录按某个关键字顺序排列。",[15,178267,178268],{},"例如按学号递增存放学生记录。",[15,178270,14543],{},[105,178272,178273,178276],{},[108,178274,178275],{},"顺序处理效率高；",[108,178277,178278],{},"适合批处理。",[15,178280,177826],{},[105,178282,178283,178286],{},[108,178284,178285],{},"插入和删除不方便；",[108,178287,178288],{},"随机查找效率不够高。",[2230,178290,178291],{"start":49},[108,178292,178293],{},"索引顺序文件 Indexed Sequential File",[15,178295,178296],{},"既保留顺序文件的顺序结构，又建立索引。",[15,178298,14543],{},[105,178300,178301,178304],{},[108,178302,178303],{},"可以顺序访问；",[108,178305,178306],{},"也可以通过索引较快查找。",[2230,178308,178309],{"start":56},[108,178310,178311],{},"索引文件 Indexed File",[15,178313,178314],{},"通过索引来访问记录。",[15,178316,178317],{},"适合需要频繁随机查找的场景。",[2230,178319,178320],{"start":62},[108,178321,178322],{},"直接文件或散列文件 Direct \u002F Hashed File",[15,178324,178325],{},"通过散列函数根据关键字直接计算记录存放位置。",[15,178327,2632],{},[15,178329,178330],{},[34,178331,178333,178362],{"className":178332},[31642],[34,178334,178336],{"className":178335},[31646],[31648,178337,178338],{"xmlns":31650},[31652,178339,178340,178359],{},[31655,178341,178342,178345,178347,178349,178351,178353,178355,178357],{},[39361,178343,178344],{},"位置",[31662,178346,202],{},[31658,178348,47292],{},[31662,178350,456],{"stretchy":702},[31658,178352,33759],{},[31658,178354,33244],{},[31658,178356,67764],{},[31662,178358,2557],{"stretchy":702},[31665,178360,178361],{"encoding":31667},"位置 = h(key)",[34,178363,178365,178383],{"className":178364,"ariaHidden":758},[31672],[34,178366,178368,178371,178374,178377,178380],{"className":178367},[31676],[34,178369],{"className":178370,"style":33218},[31680],[34,178372,178344],{"className":178373},[31685,105462],[34,178375],{"className":178376,"style":31846},[31845],[34,178378,202],{"className":178379},[31850],[34,178381],{"className":178382,"style":31846},[31845],[34,178384,178386,178389,178392,178395,178398,178402],{"className":178385},[31676],[34,178387],{"className":178388,"style":31774},[31680],[34,178390,47292],{"className":178391},[31685,31686],[34,178393,456],{"className":178394},[31782],[34,178396,33759],{"className":178397,"style":33774},[31685,31686],[34,178399,178401],{"className":178400,"style":32062},[31685,31686],"ey",[34,178403,2557],{"className":178404},[31690],[15,178406,14543],{},[105,178408,178409],{},[108,178410,178411],{},"按关键字访问速度很快。",[15,178413,177826],{},[105,178415,178416,178419],{},[108,178417,178418],{},"可能有冲突；",[108,178420,178421],{},"不适合顺序遍历。",[32188,178423],{},[19,178425,178427],{"id":178426},"_2-文件增长时按-1248-块分配空间","2. 文件增长时按 1、2、4、8... 块分配空间",[253,178429,162043],{"id":167236},[15,178431,178432],{},"One scheme to avoid the problem of pre-allocation versus waste or lack of contiguity is to allocate portions of increasing size as the file grows. For example, begin with a portion size of one block, and double the portion size for each allocation. Consider a file of n records with a blocking factor of F, and suppose that a simple one-level index is used as a file allocation table.",[15,178434,178435],{},"a. Give an upper limit on the number of entries in the file allocation table as a function of F and n.",[15,178437,178438],{},"b. What is the maximum amount of the allocated file space that is unused at any time?",[15,178440,162049],{},[15,178442,178443],{},"为了避免预分配造成浪费，或者文件块不连续的问题，可以随着文件增长，按越来越大的 portion 分配空间。",[15,178445,2632],{},[15,178447,178448],{},"第一次分配 1 块；",[15,178450,178451],{},"第二次分配 2 块；",[15,178453,178454],{},"第三次分配 4 块；",[15,178456,178457],{},"第四次分配 8 块；",[15,178459,178460],{},"之后每次翻倍。",[15,178462,178463,178464,178492,178493,178521,178522,178550],{},"设文件有 ",[34,178465,178467,178480],{"className":178466},[31642],[34,178468,178470],{"className":178469},[31646],[31648,178471,178472],{"xmlns":31650},[31652,178473,178474,178478],{},[31655,178475,178476],{},[31658,178477,31660],{},[31665,178479,31660],{"encoding":31667},[34,178481,178483],{"className":178482,"ariaHidden":758},[31672],[34,178484,178486,178489],{"className":178485},[31676],[34,178487],{"className":178488,"style":32527},[31680],[34,178490,31660],{"className":178491},[31685,31686]," 条记录，阻塞因子为 ",[34,178494,178496,178509],{"className":178495},[31642],[34,178497,178499],{"className":178498},[31646],[31648,178500,178501],{"xmlns":31650},[31652,178502,178503,178507],{},[31655,178504,178505],{},[31658,178506,51834],{},[31665,178508,51834],{"encoding":31667},[34,178510,178512],{"className":178511,"ariaHidden":758},[31672],[34,178513,178515,178518],{"className":178514},[31676],[34,178516],{"className":178517,"style":33218},[31680],[34,178519,51834],{"className":178520,"style":37422},[31685,31686],"，即每个磁盘块可以存放 ",[34,178523,178525,178538],{"className":178524},[31642],[34,178526,178528],{"className":178527},[31646],[31648,178529,178530],{"xmlns":31650},[31652,178531,178532,178536],{},[31655,178533,178534],{},[31658,178535,51834],{},[31665,178537,51834],{"encoding":31667},[34,178539,178541],{"className":178540,"ariaHidden":758},[31672],[34,178542,178544,178547],{"className":178543},[31676],[34,178545],{"className":178546,"style":33218},[31680],[34,178548,51834],{"className":178549,"style":37422},[31685,31686]," 条记录。假设使用简单的一级索引作为文件分配表。",[15,178552,178553],{},"a. 求文件分配表中的索引项数量上限。",[15,178555,178556],{},"b. 任意时刻已经分配但未使用的文件空间最大是多少？",[32188,178558],{},[253,178560,175302],{"id":178561},"答案和讲解-1",[32188,178563],{},[253,178565,178567],{"id":178566},"_2a-文件分配表项数上限","2.a 文件分配表项数上限",[15,178569,178570,178571,178599],{},"阻塞因子 ",[34,178572,178574,178587],{"className":178573},[31642],[34,178575,178577],{"className":178576},[31646],[31648,178578,178579],{"xmlns":31650},[31652,178580,178581,178585],{},[31655,178582,178583],{},[31658,178584,51834],{},[31665,178586,51834],{"encoding":31667},[34,178588,178590],{"className":178589,"ariaHidden":758},[31672],[34,178591,178593,178596],{"className":178592},[31676],[34,178594],{"className":178595,"style":33218},[31680],[34,178597,51834],{"className":178598,"style":37422},[31685,31686]," 表示：",[15,178601,178602],{},"每个磁盘块可以存放：",[15,178604,178605],{},[34,178606,178608,178621],{"className":178607},[31642],[34,178609,178611],{"className":178610},[31646],[31648,178612,178613],{"xmlns":31650},[31652,178614,178615,178619],{},[31655,178616,178617],{},[31658,178618,51834],{},[31665,178620,51834],{"encoding":31667},[34,178622,178624],{"className":178623,"ariaHidden":758},[31672],[34,178625,178627,178630],{"className":178626},[31676],[34,178628],{"className":178629,"style":33218},[31680],[34,178631,51834],{"className":178632,"style":37422},[31685,31686],[15,178634,178635],{},"条记录。",[15,178637,178638],{},"文件共有：",[15,178640,178641],{},[34,178642,178644,178657],{"className":178643},[31642],[34,178645,178647],{"className":178646},[31646],[31648,178648,178649],{"xmlns":31650},[31652,178650,178651,178655],{},[31655,178652,178653],{},[31658,178654,31660],{},[31665,178656,31660],{"encoding":31667},[34,178658,178660],{"className":178659,"ariaHidden":758},[31672],[34,178661,178663,178666],{"className":178662},[31676],[34,178664],{"className":178665,"style":32527},[31680],[34,178667,31660],{"className":178668},[31685,31686],[15,178670,178635],{},[15,178672,178673],{},"所以需要的数据块数为：",[15,178675,178676],{},[34,178677,178679,178705],{"className":178678},[31642],[34,178680,178682],{"className":178681},[31646],[31648,178683,178684],{"xmlns":31650},[31652,178685,178686,178702],{},[31655,178687,178688,178690,178692,178694,178696,178698,178700],{},[31658,178689,54224],{},[31662,178691,202],{},[31662,178693,173674],{"stretchy":702},[31658,178695,31660],{},[31658,178697,38080],{"mathvariant":38079},[31658,178699,51834],{},[31662,178701,173683],{"stretchy":702},[31665,178703,178704],{"encoding":31667},"B = \\lceil n \u002F F \\rceil",[34,178706,178708,178726],{"className":178707,"ariaHidden":758},[31672],[34,178709,178711,178714,178717,178720,178723],{"className":178710},[31676],[34,178712],{"className":178713,"style":33218},[31680],[34,178715,54224],{"className":178716,"style":54252},[31685,31686],[34,178718],{"className":178719,"style":31846},[31845],[34,178721,202],{"className":178722},[31850],[34,178724],{"className":178725,"style":31846},[31845],[34,178727,178729,178732,178735,178738,178741,178744],{"className":178728},[31676],[34,178730],{"className":178731,"style":31774},[31680],[34,178733,173674],{"className":178734},[31782],[34,178736,31660],{"className":178737},[31685,31686],[34,178739,38080],{"className":178740},[31685],[34,178742,51834],{"className":178743,"style":37422},[31685,31686],[34,178745,173683],{"className":178746},[31690],[15,178748,25496,178749,178777],{},[34,178750,178752,178765],{"className":178751},[31642],[34,178753,178755],{"className":178754},[31646],[31648,178756,178757],{"xmlns":31650},[31652,178758,178759,178763],{},[31655,178760,178761],{},[31658,178762,54224],{},[31665,178764,54224],{"encoding":31667},[34,178766,178768],{"className":178767,"ariaHidden":758},[31672],[34,178769,178771,178774],{"className":178770},[31676],[34,178772],{"className":178773,"style":33218},[31680],[34,178775,54224],{"className":178776,"style":54252},[31685,31686]," 表示实际至少需要多少个磁盘块。",[32188,178779],{},[15,178781,178782],{},"按题目规则，文件空间分配为：",[15,178784,178785],{},"第一次：",[15,178787,178788],{},[34,178789,178791,178804],{"className":178790},[31642],[34,178792,178794],{"className":178793},[31646],[31648,178795,178796],{"xmlns":31650},[31652,178797,178798,178802],{},[31655,178799,178800],{},[31749,178801,640],{},[31665,178803,640],{"encoding":31667},[34,178805,178807],{"className":178806,"ariaHidden":758},[31672],[34,178808,178810,178813],{"className":178809},[31676],[34,178811],{"className":178812,"style":32617},[31680],[34,178814,640],{"className":178815},[31685],[15,178817,178818],{},"块。",[15,178820,178821],{},"第二次：",[15,178823,178824],{},[34,178825,178827,178840],{"className":178826},[31642],[34,178828,178830],{"className":178829},[31646],[31648,178831,178832],{"xmlns":31650},[31652,178833,178834,178838],{},[31655,178835,178836],{},[31749,178837,768],{},[31665,178839,768],{"encoding":31667},[34,178841,178843],{"className":178842,"ariaHidden":758},[31672],[34,178844,178846,178849],{"className":178845},[31676],[34,178847],{"className":178848,"style":32617},[31680],[34,178850,768],{"className":178851},[31685],[15,178853,178818],{},[15,178855,178856],{},"第三次：",[15,178858,178859],{},[34,178860,178862,178875],{"className":178861},[31642],[34,178863,178865],{"className":178864},[31646],[31648,178866,178867],{"xmlns":31650},[31652,178868,178869,178873],{},[31655,178870,178871],{},[31749,178872,15932],{},[31665,178874,15932],{"encoding":31667},[34,178876,178878],{"className":178877,"ariaHidden":758},[31672],[34,178879,178881,178884],{"className":178880},[31676],[34,178882],{"className":178883,"style":32617},[31680],[34,178885,15932],{"className":178886},[31685],[15,178888,178818],{},[15,178890,178891],{},"第四次：",[15,178893,178894],{},[34,178895,178897,178910],{"className":178896},[31642],[34,178898,178900],{"className":178899},[31646],[31648,178901,178902],{"xmlns":31650},[31652,178903,178904,178908],{},[31655,178905,178906],{},[31749,178907,60525],{},[31665,178909,60525],{"encoding":31667},[34,178911,178913],{"className":178912,"ariaHidden":758},[31672],[34,178914,178916,178919],{"className":178915},[31676],[34,178917],{"className":178918,"style":32617},[31680],[34,178920,60525],{"className":178921},[31685],[15,178923,178818],{},[15,178925,178926,178927,178955],{},"第 ",[34,178928,178930,178943],{"className":178929},[31642],[34,178931,178933],{"className":178932},[31646],[31648,178934,178935],{"xmlns":31650},[31652,178936,178937,178941],{},[31655,178938,178939],{},[31658,178940,33759],{},[31665,178942,33759],{"encoding":31667},[34,178944,178946],{"className":178945,"ariaHidden":758},[31672],[34,178947,178949,178952],{"className":178948},[31676],[34,178950],{"className":178951,"style":31681},[31680],[34,178953,33759],{"className":178954,"style":33774},[31685,31686]," 次：",[15,178957,178958],{},[34,178959,178961,178985],{"className":178960},[31642],[34,178962,178964],{"className":178963},[31646],[31648,178965,178966],{"xmlns":31650},[31652,178967,178968,178982],{},[31655,178969,178970],{},[37624,178971,178972,178974],{},[31749,178973,768],{},[31655,178975,178976,178978,178980],{},[31658,178977,33759],{},[31662,178979,32226],{},[31749,178981,640],{},[31665,178983,178984],{"encoding":31667},"2^{k-1}",[34,178986,178988],{"className":178987,"ariaHidden":758},[31672],[34,178989,178991,178994],{"className":178990},[31676],[34,178992],{"className":178993,"style":38486},[31680],[34,178995,178997,179000],{"className":178996},[31685],[34,178998,768],{"className":178999},[31685],[34,179001,179003],{"className":179002},[31792],[34,179004,179006],{"className":179005},[31796],[34,179007,179009],{"className":179008},[31801],[34,179010,179012],{"className":179011,"style":38486},[31805],[34,179013,179014,179017],{"style":37777},[34,179015],{"className":179016,"style":31814},[31813],[34,179018,179020],{"className":179019},[31818,31819,31820,31821],[34,179021,179023,179026,179029],{"className":179022},[31685,31821],[34,179024,33759],{"className":179025,"style":33774},[31685,31686,31821],[34,179027,32226],{"className":179028},[32024,31821],[34,179030,640],{"className":179031},[31685,31821],[15,179033,178818],{},[15,179035,179036,179037,179065],{},"前 ",[34,179038,179040,179053],{"className":179039},[31642],[34,179041,179043],{"className":179042},[31646],[31648,179044,179045],{"xmlns":31650},[31652,179046,179047,179051],{},[31655,179048,179049],{},[31658,179050,33759],{},[31665,179052,33759],{"encoding":31667},[34,179054,179056],{"className":179055,"ariaHidden":758},[31672],[34,179057,179059,179062],{"className":179058},[31676],[34,179060],{"className":179061,"style":31681},[31680],[34,179063,33759],{"className":179064,"style":33774},[31685,31686]," 次分配的总块数为：",[15,179067,179068],{},[34,179069,179071,179111],{"className":179070},[31642],[34,179072,179074],{"className":179073},[31646],[31648,179075,179076],{"xmlns":31650},[31652,179077,179078,179108],{},[31655,179079,179080,179082,179084,179086,179088,179090,179092,179094,179096],{},[31749,179081,640],{},[31662,179083,223],{},[31749,179085,768],{},[31662,179087,223],{},[31749,179089,15932],{},[31662,179091,223],{},[31662,179093,38874],{},[31662,179095,223],{},[37624,179097,179098,179100],{},[31749,179099,768],{},[31655,179101,179102,179104,179106],{},[31658,179103,33759],{},[31662,179105,32226],{},[31749,179107,640],{},[31665,179109,179110],{"encoding":31667},"1 + 2 + 4 + \\cdots + 2^{k-1}",[34,179112,179114,179132,179150,179168,179186],{"className":179113,"ariaHidden":758},[31672],[34,179115,179117,179120,179123,179126,179129],{"className":179116},[31676],[34,179118],{"className":179119,"style":35942},[31680],[34,179121,640],{"className":179122},[31685],[34,179124],{"className":179125,"style":32604},[31845],[34,179127,223],{"className":179128},[32024],[34,179130],{"className":179131,"style":32604},[31845],[34,179133,179135,179138,179141,179144,179147],{"className":179134},[31676],[34,179136],{"className":179137,"style":35942},[31680],[34,179139,768],{"className":179140},[31685],[34,179142],{"className":179143,"style":32604},[31845],[34,179145,223],{"className":179146},[32024],[34,179148],{"className":179149,"style":32604},[31845],[34,179151,179153,179156,179159,179162,179165],{"className":179152},[31676],[34,179154],{"className":179155,"style":35942},[31680],[34,179157,15932],{"className":179158},[31685],[34,179160],{"className":179161,"style":32604},[31845],[34,179163,223],{"className":179164},[32024],[34,179166],{"className":179167,"style":32604},[31845],[34,179169,179171,179174,179177,179180,179183],{"className":179170},[31676],[34,179172],{"className":179173,"style":32597},[31680],[34,179175,38874],{"className":179176},[34006],[34,179178],{"className":179179,"style":32604},[31845],[34,179181,223],{"className":179182},[32024],[34,179184],{"className":179185,"style":32604},[31845],[34,179187,179189,179192],{"className":179188},[31676],[34,179190],{"className":179191,"style":38486},[31680],[34,179193,179195,179198],{"className":179194},[31685],[34,179196,768],{"className":179197},[31685],[34,179199,179201],{"className":179200},[31792],[34,179202,179204],{"className":179203},[31796],[34,179205,179207],{"className":179206},[31801],[34,179208,179210],{"className":179209,"style":38486},[31805],[34,179211,179212,179215],{"style":37777},[34,179213],{"className":179214,"style":31814},[31813],[34,179216,179218],{"className":179217},[31818,31819,31820,31821],[34,179219,179221,179224,179227],{"className":179220},[31685,31821],[34,179222,33759],{"className":179223,"style":33774},[31685,31686,31821],[34,179225,32226],{"className":179226},[32024,31821],[34,179228,640],{"className":179229},[31685,31821],[15,179231,179232],{},"这是等比数列，和为：",[15,179234,179235],{},[34,179236,179238,179260],{"className":179237},[31642],[34,179239,179241],{"className":179240},[31646],[31648,179242,179243],{"xmlns":31650},[31652,179244,179245,179257],{},[31655,179246,179247,179253,179255],{},[37624,179248,179249,179251],{},[31749,179250,768],{},[31658,179252,33759],{},[31662,179254,32226],{},[31749,179256,640],{},[31665,179258,179259],{"encoding":31667},"2^k - 1",[34,179261,179263,179307],{"className":179262,"ariaHidden":758},[31672],[34,179264,179266,179269,179298,179301,179304],{"className":179265},[31676],[34,179267],{"className":179268,"style":51126},[31680],[34,179270,179272,179275],{"className":179271},[31685],[34,179273,768],{"className":179274},[31685],[34,179276,179278],{"className":179277},[31792],[34,179279,179281],{"className":179280},[31796],[34,179282,179284],{"className":179283},[31801],[34,179285,179287],{"className":179286,"style":38486},[31805],[34,179288,179289,179292],{"style":37777},[34,179290],{"className":179291,"style":31814},[31813],[34,179293,179295],{"className":179294},[31818,31819,31820,31821],[34,179296,33759],{"className":179297,"style":33774},[31685,31686,31821],[34,179299],{"className":179300,"style":32604},[31845],[34,179302,32226],{"className":179303},[32024],[34,179305],{"className":179306,"style":32604},[31845],[34,179308,179310,179313],{"className":179309},[31676],[34,179311],{"className":179312,"style":32617},[31680],[34,179314,640],{"className":179315},[31685],[15,179317,179318],{},"为了容纳文件，必须满足：",[15,179320,179321],{},[34,179322,179324,179350],{"className":179323},[31642],[34,179325,179327],{"className":179326},[31646],[31648,179328,179329],{"xmlns":31650},[31652,179330,179331,179347],{},[31655,179332,179333,179339,179341,179343,179345],{},[37624,179334,179335,179337],{},[31749,179336,768],{},[31658,179338,33759],{},[31662,179340,32226],{},[31749,179342,640],{},[31662,179344,43175],{},[31658,179346,54224],{},[31665,179348,179349],{"encoding":31667},"2^k - 1 \\ge B",[34,179351,179353,179397,179415],{"className":179352,"ariaHidden":758},[31672],[34,179354,179356,179359,179388,179391,179394],{"className":179355},[31676],[34,179357],{"className":179358,"style":51126},[31680],[34,179360,179362,179365],{"className":179361},[31685],[34,179363,768],{"className":179364},[31685],[34,179366,179368],{"className":179367},[31792],[34,179369,179371],{"className":179370},[31796],[34,179372,179374],{"className":179373},[31801],[34,179375,179377],{"className":179376,"style":38486},[31805],[34,179378,179379,179382],{"style":37777},[34,179380],{"className":179381,"style":31814},[31813],[34,179383,179385],{"className":179384},[31818,31819,31820,31821],[34,179386,33759],{"className":179387,"style":33774},[31685,31686,31821],[34,179389],{"className":179390,"style":32604},[31845],[34,179392,32226],{"className":179393},[32024],[34,179395],{"className":179396,"style":32604},[31845],[34,179398,179400,179403,179406,179409,179412],{"className":179399},[31676],[34,179401],{"className":179402,"style":55382},[31680],[34,179404,640],{"className":179405},[31685],[34,179407],{"className":179408,"style":31846},[31845],[34,179410,43175],{"className":179411},[31850],[34,179413],{"className":179414,"style":31846},[31845],[34,179416,179418,179421],{"className":179417},[31676],[34,179419],{"className":179420,"style":33218},[31680],[34,179422,54224],{"className":179423,"style":54252},[31685,31686],[15,179425,129204],{},[15,179427,179428],{},[34,179429,179431,179465],{"className":179430},[31642],[34,179432,179434],{"className":179433},[31646],[31648,179435,179436],{"xmlns":31650},[31652,179437,179438,179462],{},[31655,179439,179440,179446,179448,179450,179452,179454,179456,179458,179460],{},[37624,179441,179442,179444],{},[31749,179443,768],{},[31658,179445,33759],{},[31662,179447,32226],{},[31749,179449,640],{},[31662,179451,43175],{},[31662,179453,173674],{"stretchy":702},[31658,179455,31660],{},[31658,179457,38080],{"mathvariant":38079},[31658,179459,51834],{},[31662,179461,173683],{"stretchy":702},[31665,179463,179464],{"encoding":31667},"2^k - 1 \\ge \\lceil n \u002F F \\rceil",[34,179466,179468,179512,179530],{"className":179467,"ariaHidden":758},[31672],[34,179469,179471,179474,179503,179506,179509],{"className":179470},[31676],[34,179472],{"className":179473,"style":51126},[31680],[34,179475,179477,179480],{"className":179476},[31685],[34,179478,768],{"className":179479},[31685],[34,179481,179483],{"className":179482},[31792],[34,179484,179486],{"className":179485},[31796],[34,179487,179489],{"className":179488},[31801],[34,179490,179492],{"className":179491,"style":38486},[31805],[34,179493,179494,179497],{"style":37777},[34,179495],{"className":179496,"style":31814},[31813],[34,179498,179500],{"className":179499},[31818,31819,31820,31821],[34,179501,33759],{"className":179502,"style":33774},[31685,31686,31821],[34,179504],{"className":179505,"style":32604},[31845],[34,179507,32226],{"className":179508},[32024],[34,179510],{"className":179511,"style":32604},[31845],[34,179513,179515,179518,179521,179524,179527],{"className":179514},[31676],[34,179516],{"className":179517,"style":55382},[31680],[34,179519,640],{"className":179520},[31685],[34,179522],{"className":179523,"style":31846},[31845],[34,179525,43175],{"className":179526},[31850],[34,179528],{"className":179529,"style":31846},[31845],[34,179531,179533,179536,179539,179542,179545,179548],{"className":179532},[31676],[34,179534],{"className":179535,"style":31774},[31680],[34,179537,173674],{"className":179538},[31782],[34,179540,31660],{"className":179541},[31685,31686],[34,179543,38080],{"className":179544},[31685],[34,179546,51834],{"className":179547,"style":37422},[31685,31686],[34,179549,173683],{"className":179550},[31690],[15,179552,179553],{},"移项：",[15,179555,179556],{},[34,179557,179559,179593],{"className":179558},[31642],[34,179560,179562],{"className":179561},[31646],[31648,179563,179564],{"xmlns":31650},[31652,179565,179566,179590],{},[31655,179567,179568,179574,179576,179578,179580,179582,179584,179586,179588],{},[37624,179569,179570,179572],{},[31749,179571,768],{},[31658,179573,33759],{},[31662,179575,43175],{},[31662,179577,173674],{"stretchy":702},[31658,179579,31660],{},[31658,179581,38080],{"mathvariant":38079},[31658,179583,51834],{},[31662,179585,173683],{"stretchy":702},[31662,179587,223],{},[31749,179589,640],{},[31665,179591,179592],{"encoding":31667},"2^k \\ge \\lceil n \u002F F \\rceil + 1",[34,179594,179596,179641,179671],{"className":179595,"ariaHidden":758},[31672],[34,179597,179599,179603,179632,179635,179638],{"className":179598},[31676],[34,179600],{"className":179601,"style":179602},[31680],"height:0.9851em;vertical-align:-0.136em;",[34,179604,179606,179609],{"className":179605},[31685],[34,179607,768],{"className":179608},[31685],[34,179610,179612],{"className":179611},[31792],[34,179613,179615],{"className":179614},[31796],[34,179616,179618],{"className":179617},[31801],[34,179619,179621],{"className":179620,"style":38486},[31805],[34,179622,179623,179626],{"style":37777},[34,179624],{"className":179625,"style":31814},[31813],[34,179627,179629],{"className":179628},[31818,31819,31820,31821],[34,179630,33759],{"className":179631,"style":33774},[31685,31686,31821],[34,179633],{"className":179634,"style":31846},[31845],[34,179636,43175],{"className":179637},[31850],[34,179639],{"className":179640,"style":31846},[31845],[34,179642,179644,179647,179650,179653,179656,179659,179662,179665,179668],{"className":179643},[31676],[34,179645],{"className":179646,"style":31774},[31680],[34,179648,173674],{"className":179649},[31782],[34,179651,31660],{"className":179652},[31685,31686],[34,179654,38080],{"className":179655},[31685],[34,179657,51834],{"className":179658,"style":37422},[31685,31686],[34,179660,173683],{"className":179661},[31690],[34,179663],{"className":179664,"style":32604},[31845],[34,179666,223],{"className":179667},[32024],[34,179669],{"className":179670,"style":32604},[31845],[34,179672,179674,179677],{"className":179673},[31676],[34,179675],{"className":179676,"style":32617},[31680],[34,179678,640],{"className":179679},[31685],[15,179681,179682],{},"两边取以 2 为底的对数：",[15,179684,179685],{},[34,179686,179688,179732],{"className":179687},[31642],[34,179689,179691],{"className":179690},[31646],[31648,179692,179693],{"xmlns":31650},[31652,179694,179695,179729],{},[31655,179696,179697,179699,179701,179711,179713,179715,179717,179719,179721,179723,179725,179727],{},[31658,179698,33759],{},[31662,179700,43175],{},[31743,179702,179703,179709],{},[31655,179704,179705,179707],{},[31658,179706,885],{},[31662,179708,32682],{},[31749,179710,768],{},[31662,179712,456],{"stretchy":702},[31662,179714,173674],{"stretchy":702},[31658,179716,31660],{},[31658,179718,38080],{"mathvariant":38079},[31658,179720,51834],{},[31662,179722,173683],{"stretchy":702},[31662,179724,223],{},[31749,179726,640],{},[31662,179728,2557],{"stretchy":702},[31665,179730,179731],{"encoding":31667},"k \\ge \\log_2(\\lceil n \u002F F \\rceil + 1)",[34,179733,179735,179753,179826],{"className":179734,"ariaHidden":758},[31672],[34,179736,179738,179741,179744,179747,179750],{"className":179737},[31676],[34,179739],{"className":179740,"style":47709},[31680],[34,179742,33759],{"className":179743,"style":33774},[31685,31686],[34,179745],{"className":179746,"style":31846},[31845],[34,179748,43175],{"className":179749},[31850],[34,179751],{"className":179752,"style":31846},[31845],[34,179754,179756,179759,179801,179805,179808,179811,179814,179817,179820,179823],{"className":179755},[31676],[34,179757],{"className":179758,"style":31774},[31680],[34,179760,179762,179767],{"className":179761},[32767],[34,179763,38616,179765],{"className":179764},[32767],[34,179766,31942],{"style":38619},[34,179768,179770],{"className":179769},[31792],[34,179771,179773,179793],{"className":179772},[31796,31797],[34,179774,179776,179790],{"className":179775},[31801],[34,179777,179779],{"className":179778,"style":38632},[31805],[34,179780,179781,179784],{"style":38635},[34,179782],{"className":179783,"style":31814},[31813],[34,179785,179787],{"className":179786},[31818,31819,31820,31821],[34,179788,768],{"className":179789},[31685,31821],[34,179791,31829],{"className":179792},[31828],[34,179794,179796],{"className":179795},[31801],[34,179797,179799],{"className":179798,"style":38654},[31805],[34,179800],{},[34,179802,179804],{"className":179803},[31782],"(⌈",[34,179806,31660],{"className":179807},[31685,31686],[34,179809,38080],{"className":179810},[31685],[34,179812,51834],{"className":179813,"style":37422},[31685,31686],[34,179815,173683],{"className":179816},[31690],[34,179818],{"className":179819,"style":32604},[31845],[34,179821,223],{"className":179822},[32024],[34,179824],{"className":179825,"style":32604},[31845],[34,179827,179829,179832,179835],{"className":179828},[31676],[34,179830],{"className":179831,"style":31774},[31680],[34,179833,640],{"className":179834},[31685],[34,179836,2557],{"className":179837},[31690],[15,179839,179840],{},"所以文件分配表项数上限为：",[15,179842,179843],{},[34,179844,179846,179904],{"className":179845},[31642],[34,179847,179849],{"className":179848},[31646],[31648,179850,179851],{"xmlns":31650},[31652,179852,179853,179901],{},[31655,179854,179855],{},[126400,179856,179857],{"notation":126402},[71090,179858,179859],{"scriptlevel":459,"displaystyle":702},[71090,179860,179861],{"scriptlevel":459,"displaystyle":702},[71090,179862,179863],{"scriptlevel":459,"displaystyle":758},[31655,179864,179865,179867,179869,179871,179881,179883,179885,179887,179889,179891,179893,179895,179897,179899],{},[31658,179866,33759],{},[31662,179868,202],{},[31662,179870,173674],{"stretchy":702},[31743,179872,179873,179879],{},[31655,179874,179875,179877],{},[31658,179876,885],{},[31662,179878,32682],{},[31749,179880,768],{},[31662,179882,456],{"stretchy":702},[31662,179884,173674],{"stretchy":702},[31658,179886,31660],{},[31658,179888,38080],{"mathvariant":38079},[31658,179890,51834],{},[31662,179892,173683],{"stretchy":702},[31662,179894,223],{},[31749,179896,640],{},[31662,179898,2557],{"stretchy":702},[31662,179900,173683],{"stretchy":702},[31665,179902,179903],{"encoding":31667},"\\boxed{k = \\lceil \\log_2(\\lceil n \u002F F \\rceil + 1) \\rceil}",[34,179905,179907],{"className":179906,"ariaHidden":758},[31672],[34,179908,179910,179913],{"className":179909},[31676],[34,179911],{"className":179912,"style":130018},[31680],[34,179914,179916],{"className":179915},[31685],[34,179917,179919,180038],{"className":179918},[31796,31797],[34,179920,179922,180035],{"className":179921},[31801],[34,179923,179925,180027],{"className":179924,"style":130031},[31805],[34,179926,179927,179930],{"style":130034},[34,179928],{"className":179929,"style":130038},[31813],[34,179931,179933],{"className":179932},[126448],[34,179934,179936],{"className":179935},[31685],[34,179937,179939,179942,179945,179948,179951,179954,179996,179999,180002,180005,180008,180011,180014,180017,180020,180023],{"className":179938},[31685],[34,179940,33759],{"className":179941,"style":33774},[31685,31686],[34,179943],{"className":179944,"style":31846},[31845],[34,179946,202],{"className":179947},[31850],[34,179949],{"className":179950,"style":31846},[31845],[34,179952,173674],{"className":179953},[31782],[34,179955,179957,179962],{"className":179956},[32767],[34,179958,38616,179960],{"className":179959},[32767],[34,179961,31942],{"style":38619},[34,179963,179965],{"className":179964},[31792],[34,179966,179968,179988],{"className":179967},[31796,31797],[34,179969,179971,179985],{"className":179970},[31801],[34,179972,179974],{"className":179973,"style":38632},[31805],[34,179975,179976,179979],{"style":38635},[34,179977],{"className":179978,"style":31814},[31813],[34,179980,179982],{"className":179981},[31818,31819,31820,31821],[34,179983,768],{"className":179984},[31685,31821],[34,179986,31829],{"className":179987},[31828],[34,179989,179991],{"className":179990},[31801],[34,179992,179994],{"className":179993,"style":38654},[31805],[34,179995],{},[34,179997,179804],{"className":179998},[31782],[34,180000,31660],{"className":180001},[31685,31686],[34,180003,38080],{"className":180004},[31685],[34,180006,51834],{"className":180007,"style":37422},[31685,31686],[34,180009,173683],{"className":180010},[31690],[34,180012],{"className":180013,"style":32604},[31845],[34,180015,223],{"className":180016},[32024],[34,180018],{"className":180019,"style":32604},[31845],[34,180021,640],{"className":180022},[31685],[34,180024,180026],{"className":180025},[31690],")⌉",[34,180028,180029,180032],{"style":130074},[34,180030],{"className":180031,"style":130038},[31813],[34,180033],{"className":180034,"style":130081},[126470,126471],[34,180036,31829],{"className":180037},[31828],[34,180039,180041],{"className":180040},[31801],[34,180042,180044],{"className":180043,"style":130091},[31805],[34,180045],{},[32188,180047],{},[253,180049,180051],{"id":180050},"_2b-最大未使用空间","2.b 最大未使用空间",[15,180053,180054],{},"已经分配但未使用的空间主要出现在刚刚分配一个新 portion 的时候。",[15,180056,180057],{},"因为每次新分配的 portion 都比前一次大一倍，所以最大的浪费发生在最后一次，也就是最大 portion 刚分配出来的时候。",[15,180059,180060,180061,180089],{},"如果文件共用了 ",[34,180062,180064,180077],{"className":180063},[31642],[34,180065,180067],{"className":180066},[31646],[31648,180068,180069],{"xmlns":31650},[31652,180070,180071,180075],{},[31655,180072,180073],{},[31658,180074,33759],{},[31665,180076,33759],{"encoding":31667},[34,180078,180080],{"className":180079,"ariaHidden":758},[31672],[34,180081,180083,180086],{"className":180082},[31676],[34,180084],{"className":180085,"style":31681},[31680],[34,180087,33759],{"className":180088,"style":33774},[31685,31686]," 个索引项，那么最后一次分配的 portion 大小是：",[15,180091,180092],{},[34,180093,180095,180118],{"className":180094},[31642],[34,180096,180098],{"className":180097},[31646],[31648,180099,180100],{"xmlns":31650},[31652,180101,180102,180116],{},[31655,180103,180104],{},[37624,180105,180106,180108],{},[31749,180107,768],{},[31655,180109,180110,180112,180114],{},[31658,180111,33759],{},[31662,180113,32226],{},[31749,180115,640],{},[31665,180117,178984],{"encoding":31667},[34,180119,180121],{"className":180120,"ariaHidden":758},[31672],[34,180122,180124,180127],{"className":180123},[31676],[34,180125],{"className":180126,"style":38486},[31680],[34,180128,180130,180133],{"className":180129},[31685],[34,180131,768],{"className":180132},[31685],[34,180134,180136],{"className":180135},[31792],[34,180137,180139],{"className":180138},[31796],[34,180140,180142],{"className":180141},[31801],[34,180143,180145],{"className":180144,"style":38486},[31805],[34,180146,180147,180150],{"style":37777},[34,180148],{"className":180149,"style":31814},[31813],[34,180151,180153],{"className":180152},[31818,31819,31820,31821],[34,180154,180156,180159,180162],{"className":180155},[31685,31821],[34,180157,33759],{"className":180158,"style":33774},[31685,31686,31821],[34,180160,32226],{"className":180161},[32024,31821],[34,180163,640],{"className":180164},[31685,31821],[15,180166,180167],{},"个块。",[15,180169,180170],{},"所以按块数计，最大未使用空间不超过：",[15,180172,180173],{},[34,180174,180176,180212],{"className":180175},[31642],[34,180177,180179],{"className":180178},[31646],[31648,180180,180181],{"xmlns":31650},[31652,180182,180183,180209],{},[31655,180184,180185],{},[126400,180186,180187],{"notation":126402},[71090,180188,180189],{"scriptlevel":459,"displaystyle":702},[71090,180190,180191],{"scriptlevel":459,"displaystyle":702},[71090,180192,180193],{"scriptlevel":459,"displaystyle":758},[31655,180194,180195,180207],{},[37624,180196,180197,180199],{},[31749,180198,768],{},[31655,180200,180201,180203,180205],{},[31658,180202,33759],{},[31662,180204,32226],{},[31749,180206,640],{},[39361,180208,176786],{},[31665,180210,180211],{"encoding":31667},"\\boxed{2^{k-1} \\text{ 块}}",[34,180213,180215],{"className":180214,"ariaHidden":758},[31672],[34,180216,180218,180222],{"className":180217},[31676],[34,180219],{"className":180220,"style":180221},[31680],"height:1.5791em;vertical-align:-0.34em;",[34,180223,180225],{"className":180224},[31685],[34,180226,180228,180312],{"className":180227},[31796,31797],[34,180229,180231,180309],{"className":180230},[31801],[34,180232,180235,180299],{"className":180233,"style":180234},[31805],"height:1.2391em;",[34,180236,180238,180242],{"style":180237},"top:-3.5791em;",[34,180239],{"className":180240,"style":180241},[31813],"height:3.5791em;",[34,180243,180245],{"className":180244},[126448],[34,180246,180248],{"className":180247},[31685],[34,180249,180251,180290],{"className":180250},[31685],[34,180252,180254,180257],{"className":180253},[31685],[34,180255,768],{"className":180256},[31685],[34,180258,180260],{"className":180259},[31792],[34,180261,180263],{"className":180262},[31796],[34,180264,180266],{"className":180265},[31801],[34,180267,180270],{"className":180268,"style":180269},[31805],"height:0.8991em;",[34,180271,180272,180275],{"style":129846},[34,180273],{"className":180274,"style":31814},[31813],[34,180276,180278],{"className":180277},[31818,31819,31820,31821],[34,180279,180281,180284,180287],{"className":180280},[31685,31821],[34,180282,33759],{"className":180283,"style":33774},[31685,31686,31821],[34,180285,32226],{"className":180286},[32024,31821],[34,180288,640],{"className":180289},[31685,31821],[34,180291,180293,180296],{"className":180292},[31685,2638],[34,180294,130298],{"className":180295},[31685],[34,180297,176811],{"className":180298},[31685,105462],[34,180300,180302,180305],{"style":180301},"top:-3.2391em;",[34,180303],{"className":180304,"style":180241},[31813],[34,180306],{"className":180307,"style":180308},[126470,126471],"height:1.5791em;border-style:solid;border-width:0.04em;",[34,180310,31829],{"className":180311},[31828],[34,180313,180315],{"className":180314},[31801],[34,180316,180318],{"className":180317,"style":126482},[31805],[34,180319],{},[15,180321,104944],{},[15,180323,180324],{},[34,180325,180327,180375],{"className":180326},[31642],[34,180328,180330],{"className":180329},[31646],[31648,180331,180332],{"xmlns":31650},[31652,180333,180334,180372],{},[31655,180335,180336,180338,180340,180342,180352,180354,180356,180358,180360,180362,180364,180366,180368,180370],{},[31658,180337,33759],{},[31662,180339,202],{},[31662,180341,173674],{"stretchy":702},[31743,180343,180344,180350],{},[31655,180345,180346,180348],{},[31658,180347,885],{},[31662,180349,32682],{},[31749,180351,768],{},[31662,180353,456],{"stretchy":702},[31662,180355,173674],{"stretchy":702},[31658,180357,31660],{},[31658,180359,38080],{"mathvariant":38079},[31658,180361,51834],{},[31662,180363,173683],{"stretchy":702},[31662,180365,223],{},[31749,180367,640],{},[31662,180369,2557],{"stretchy":702},[31662,180371,173683],{"stretchy":702},[31665,180373,180374],{"encoding":31667},"k = \\lceil \\log_2(\\lceil n \u002F F \\rceil + 1) \\rceil",[34,180376,180378,180396,180471],{"className":180377,"ariaHidden":758},[31672],[34,180379,180381,180384,180387,180390,180393],{"className":180380},[31676],[34,180382],{"className":180383,"style":31681},[31680],[34,180385,33759],{"className":180386,"style":33774},[31685,31686],[34,180388],{"className":180389,"style":31846},[31845],[34,180391,202],{"className":180392},[31850],[34,180394],{"className":180395,"style":31846},[31845],[34,180397,180399,180402,180405,180447,180450,180453,180456,180459,180462,180465,180468],{"className":180398},[31676],[34,180400],{"className":180401,"style":31774},[31680],[34,180403,173674],{"className":180404},[31782],[34,180406,180408,180413],{"className":180407},[32767],[34,180409,38616,180411],{"className":180410},[32767],[34,180412,31942],{"style":38619},[34,180414,180416],{"className":180415},[31792],[34,180417,180419,180439],{"className":180418},[31796,31797],[34,180420,180422,180436],{"className":180421},[31801],[34,180423,180425],{"className":180424,"style":38632},[31805],[34,180426,180427,180430],{"style":38635},[34,180428],{"className":180429,"style":31814},[31813],[34,180431,180433],{"className":180432},[31818,31819,31820,31821],[34,180434,768],{"className":180435},[31685,31821],[34,180437,31829],{"className":180438},[31828],[34,180440,180442],{"className":180441},[31801],[34,180443,180445],{"className":180444,"style":38654},[31805],[34,180446],{},[34,180448,179804],{"className":180449},[31782],[34,180451,31660],{"className":180452},[31685,31686],[34,180454,38080],{"className":180455},[31685],[34,180457,51834],{"className":180458,"style":37422},[31685,31686],[34,180460,173683],{"className":180461},[31690],[34,180463],{"className":180464,"style":32604},[31845],[34,180466,223],{"className":180467},[32024],[34,180469],{"className":180470,"style":32604},[31845],[34,180472,180474,180477,180480],{"className":180473},[31676],[34,180475],{"className":180476,"style":31774},[31680],[34,180478,640],{"className":180479},[31685],[34,180481,180026],{"className":180482},[31690],[15,180484,180485],{},"如果换成记录容量：",[15,180487,180488],{},[34,180489,180491,180530],{"className":180490},[31642],[34,180492,180494],{"className":180493},[31646],[31648,180495,180496],{"xmlns":31650},[31652,180497,180498,180527],{},[31655,180499,180500],{},[126400,180501,180502],{"notation":126402},[71090,180503,180504],{"scriptlevel":459,"displaystyle":702},[71090,180505,180506],{"scriptlevel":459,"displaystyle":702},[71090,180507,180508],{"scriptlevel":459,"displaystyle":758},[31655,180509,180510,180522,180524],{},[37624,180511,180512,180514],{},[31749,180513,768],{},[31655,180515,180516,180518,180520],{},[31658,180517,33759],{},[31662,180519,32226],{},[31749,180521,640],{},[31658,180523,51834],{},[39361,180525,180526],{}," 条记录容量",[31665,180528,180529],{"encoding":31667},"\\boxed{2^{k-1}F \\text{ 条记录容量}}",[34,180531,180533],{"className":180532,"ariaHidden":758},[31672],[34,180534,180536,180539],{"className":180535},[31676],[34,180537],{"className":180538,"style":180221},[31680],[34,180540,180542],{"className":180541},[31685],[34,180543,180545,180627],{"className":180544},[31796,31797],[34,180546,180548,180624],{"className":180547},[31801],[34,180549,180551,180616],{"className":180550,"style":180234},[31805],[34,180552,180553,180556],{"style":180237},[34,180554],{"className":180555,"style":180241},[31813],[34,180557,180559],{"className":180558},[126448],[34,180560,180562],{"className":180561},[31685],[34,180563,180565,180603,180606],{"className":180564},[31685],[34,180566,180568,180571],{"className":180567},[31685],[34,180569,768],{"className":180570},[31685],[34,180572,180574],{"className":180573},[31792],[34,180575,180577],{"className":180576},[31796],[34,180578,180580],{"className":180579},[31801],[34,180581,180583],{"className":180582,"style":180269},[31805],[34,180584,180585,180588],{"style":129846},[34,180586],{"className":180587,"style":31814},[31813],[34,180589,180591],{"className":180590},[31818,31819,31820,31821],[34,180592,180594,180597,180600],{"className":180593},[31685,31821],[34,180595,33759],{"className":180596,"style":33774},[31685,31686,31821],[34,180598,32226],{"className":180599},[32024,31821],[34,180601,640],{"className":180602},[31685,31821],[34,180604,51834],{"className":180605,"style":37422},[31685,31686],[34,180607,180609,180612],{"className":180608},[31685,2638],[34,180610,130298],{"className":180611},[31685],[34,180613,180615],{"className":180614},[31685,105462],"条记录容量",[34,180617,180618,180621],{"style":180301},[34,180619],{"className":180620,"style":180241},[31813],[34,180622],{"className":180623,"style":180308},[126470,126471],[34,180625,31829],{"className":180626},[31828],[34,180628,180630],{"className":180629},[31801],[34,180631,180633],{"className":180632,"style":126482},[31805],[34,180634],{},[15,180636,180637],{},"如果认为“刚分配后马上写入一条记录”，那么最大未使用记录槽位约为：",[15,180639,180640],{},[34,180641,180643,180683],{"className":180642},[31642],[34,180644,180646],{"className":180645},[31646],[31648,180647,180648],{"xmlns":31650},[31652,180649,180650,180680],{},[31655,180651,180652],{},[126400,180653,180654],{"notation":126402},[71090,180655,180656],{"scriptlevel":459,"displaystyle":702},[71090,180657,180658],{"scriptlevel":459,"displaystyle":702},[71090,180659,180660],{"scriptlevel":459,"displaystyle":758},[31655,180661,180662,180674,180676,180678],{},[37624,180663,180664,180666],{},[31749,180665,768],{},[31655,180667,180668,180670,180672],{},[31658,180669,33759],{},[31662,180671,32226],{},[31749,180673,640],{},[31658,180675,51834],{},[31662,180677,32226],{},[31749,180679,640],{},[31665,180681,180682],{"encoding":31667},"\\boxed{2^{k-1}F - 1}",[34,180684,180686],{"className":180685,"ariaHidden":758},[31672],[34,180687,180689,180693],{"className":180688},[31676],[34,180690],{"className":180691,"style":180692},[31680],"height:1.6624em;vertical-align:-0.4233em;",[34,180694,180696],{"className":180695},[31685],[34,180697,180699,180786],{"className":180698},[31796,31797],[34,180700,180702,180783],{"className":180701},[31801],[34,180703,180705,180774],{"className":180704,"style":180234},[31805],[34,180706,180708,180712],{"style":180707},"top:-3.6624em;",[34,180709],{"className":180710,"style":180711},[31813],"height:3.6624em;",[34,180713,180715],{"className":180714},[126448],[34,180716,180718],{"className":180717},[31685],[34,180719,180721,180759,180762,180765,180768,180771],{"className":180720},[31685],[34,180722,180724,180727],{"className":180723},[31685],[34,180725,768],{"className":180726},[31685],[34,180728,180730],{"className":180729},[31792],[34,180731,180733],{"className":180732},[31796],[34,180734,180736],{"className":180735},[31801],[34,180737,180739],{"className":180738,"style":180269},[31805],[34,180740,180741,180744],{"style":129846},[34,180742],{"className":180743,"style":31814},[31813],[34,180745,180747],{"className":180746},[31818,31819,31820,31821],[34,180748,180750,180753,180756],{"className":180749},[31685,31821],[34,180751,33759],{"className":180752,"style":33774},[31685,31686,31821],[34,180754,32226],{"className":180755},[32024,31821],[34,180757,640],{"className":180758},[31685,31821],[34,180760,51834],{"className":180761,"style":37422},[31685,31686],[34,180763],{"className":180764,"style":32604},[31845],[34,180766,32226],{"className":180767},[32024],[34,180769],{"className":180770,"style":32604},[31845],[34,180772,640],{"className":180773},[31685],[34,180775,180776,180779],{"style":180301},[34,180777],{"className":180778,"style":180711},[31813],[34,180780],{"className":180781,"style":180782},[126470,126471],"height:1.6624em;border-style:solid;border-width:0.04em;",[34,180784,31829],{"className":180785},[31828],[34,180787,180789],{"className":180788},[31801],[34,180790,180792],{"className":180791,"style":129408},[31805],[34,180793],{},[32188,180795],{},[253,180797,180798],{"id":180798},"另一种更精确的最终文件浪费",[15,180800,180801,180802,180830],{},"如果题目问的是“最终文件有 ",[34,180803,180805,180818],{"className":180804},[31642],[34,180806,180808],{"className":180807},[31646],[31648,180809,180810],{"xmlns":31650},[31652,180811,180812,180816],{},[31655,180813,180814],{},[31658,180815,31660],{},[31665,180817,31660],{"encoding":31667},[34,180819,180821],{"className":180820,"ariaHidden":758},[31672],[34,180822,180824,180827],{"className":180823},[31676],[34,180825],{"className":180826,"style":32527},[31680],[34,180828,31660],{"className":180829},[31685,31686]," 条记录时实际浪费多少”，则总分配块数为：",[15,180832,180833],{},[34,180834,180836,180857],{"className":180835},[31642],[34,180837,180839],{"className":180838},[31646],[31648,180840,180841],{"xmlns":31650},[31652,180842,180843,180855],{},[31655,180844,180845,180851,180853],{},[37624,180846,180847,180849],{},[31749,180848,768],{},[31658,180850,33759],{},[31662,180852,32226],{},[31749,180854,640],{},[31665,180856,179259],{"encoding":31667},[34,180858,180860,180904],{"className":180859,"ariaHidden":758},[31672],[34,180861,180863,180866,180895,180898,180901],{"className":180862},[31676],[34,180864],{"className":180865,"style":51126},[31680],[34,180867,180869,180872],{"className":180868},[31685],[34,180870,768],{"className":180871},[31685],[34,180873,180875],{"className":180874},[31792],[34,180876,180878],{"className":180877},[31796],[34,180879,180881],{"className":180880},[31801],[34,180882,180884],{"className":180883,"style":38486},[31805],[34,180885,180886,180889],{"style":37777},[34,180887],{"className":180888,"style":31814},[31813],[34,180890,180892],{"className":180891},[31818,31819,31820,31821],[34,180893,33759],{"className":180894,"style":33774},[31685,31686,31821],[34,180896],{"className":180897,"style":32604},[31845],[34,180899,32226],{"className":180900},[32024],[34,180902],{"className":180903,"style":32604},[31845],[34,180905,180907,180910],{"className":180906},[31676],[34,180908],{"className":180909,"style":32617},[31680],[34,180911,640],{"className":180912},[31685],[15,180914,180915],{},"总分配记录容量为：",[15,180917,180918],{},[34,180919,180921,180949],{"className":180920},[31642],[34,180922,180924],{"className":180923},[31646],[31648,180925,180926],{"xmlns":31650},[31652,180927,180928,180946],{},[31655,180929,180930,180932,180934,180940,180942,180944],{},[31658,180931,51834],{},[31662,180933,456],{"stretchy":702},[37624,180935,180936,180938],{},[31749,180937,768],{},[31658,180939,33759],{},[31662,180941,32226],{},[31749,180943,640],{},[31662,180945,2557],{"stretchy":702},[31665,180947,180948],{"encoding":31667},"F(2^k - 1)",[34,180950,180952,181002],{"className":180951,"ariaHidden":758},[31672],[34,180953,180955,180958,180961,180964,180993,180996,180999],{"className":180954},[31676],[34,180956],{"className":180957,"style":43651},[31680],[34,180959,51834],{"className":180960,"style":37422},[31685,31686],[34,180962,456],{"className":180963},[31782],[34,180965,180967,180970],{"className":180966},[31685],[34,180968,768],{"className":180969},[31685],[34,180971,180973],{"className":180972},[31792],[34,180974,180976],{"className":180975},[31796],[34,180977,180979],{"className":180978},[31801],[34,180980,180982],{"className":180981,"style":38486},[31805],[34,180983,180984,180987],{"style":37777},[34,180985],{"className":180986,"style":31814},[31813],[34,180988,180990],{"className":180989},[31818,31819,31820,31821],[34,180991,33759],{"className":180992,"style":33774},[31685,31686,31821],[34,180994],{"className":180995,"style":32604},[31845],[34,180997,32226],{"className":180998},[32024],[34,181000],{"className":181001,"style":32604},[31845],[34,181003,181005,181008,181011],{"className":181004},[31676],[34,181006],{"className":181007,"style":31774},[31680],[34,181009,640],{"className":181010},[31685],[34,181012,2557],{"className":181013},[31690],[15,181015,181016],{},"实际使用记录数为：",[15,181018,181019],{},[34,181020,181022,181035],{"className":181021},[31642],[34,181023,181025],{"className":181024},[31646],[31648,181026,181027],{"xmlns":31650},[31652,181028,181029,181033],{},[31655,181030,181031],{},[31658,181032,31660],{},[31665,181034,31660],{"encoding":31667},[34,181036,181038],{"className":181037,"ariaHidden":758},[31672],[34,181039,181041,181044],{"className":181040},[31676],[34,181042],{"className":181043,"style":32527},[31680],[34,181045,31660],{"className":181046},[31685,31686],[15,181048,181049],{},"所以最终未使用记录槽位为：",[15,181051,181052],{},[34,181053,181055,181097],{"className":181054},[31642],[34,181056,181058],{"className":181057},[31646],[31648,181059,181060],{"xmlns":31650},[31652,181061,181062,181094],{},[31655,181063,181064],{},[126400,181065,181066],{"notation":126402},[71090,181067,181068],{"scriptlevel":459,"displaystyle":702},[71090,181069,181070],{"scriptlevel":459,"displaystyle":702},[71090,181071,181072],{"scriptlevel":459,"displaystyle":758},[31655,181073,181074,181076,181078,181084,181086,181088,181090,181092],{},[31658,181075,51834],{},[31662,181077,456],{"stretchy":702},[37624,181079,181080,181082],{},[31749,181081,768],{},[31658,181083,33759],{},[31662,181085,32226],{},[31749,181087,640],{},[31662,181089,2557],{"stretchy":702},[31662,181091,32226],{},[31658,181093,31660],{},[31665,181095,181096],{"encoding":31667},"\\boxed{F(2^k - 1) - n}",[34,181098,181100],{"className":181099,"ariaHidden":758},[31672],[34,181101,181103,181107],{"className":181102},[31676],[34,181104],{"className":181105,"style":181106},[31680],"height:1.8291em;vertical-align:-0.59em;",[34,181108,181110],{"className":181109},[31685],[34,181111,181113,181209],{"className":181112},[31796,31797],[34,181114,181116,181206],{"className":181115},[31801],[34,181117,181119,181197],{"className":181118,"style":180234},[31805],[34,181120,181122,181126],{"style":181121},"top:-3.8291em;",[34,181123],{"className":181124,"style":181125},[31813],"height:3.8291em;",[34,181127,181129],{"className":181128},[126448],[34,181130,181132],{"className":181131},[31685],[34,181133,181135,181138,181141,181170,181173,181176,181179,181182,181185,181188,181191,181194],{"className":181134},[31685],[34,181136,51834],{"className":181137,"style":37422},[31685,31686],[34,181139,456],{"className":181140},[31782],[34,181142,181144,181147],{"className":181143},[31685],[34,181145,768],{"className":181146},[31685],[34,181148,181150],{"className":181149},[31792],[34,181151,181153],{"className":181152},[31796],[34,181154,181156],{"className":181155},[31801],[34,181157,181159],{"className":181158,"style":180269},[31805],[34,181160,181161,181164],{"style":129846},[34,181162],{"className":181163,"style":31814},[31813],[34,181165,181167],{"className":181166},[31818,31819,31820,31821],[34,181168,33759],{"className":181169,"style":33774},[31685,31686,31821],[34,181171],{"className":181172,"style":32604},[31845],[34,181174,32226],{"className":181175},[32024],[34,181177],{"className":181178,"style":32604},[31845],[34,181180,640],{"className":181181},[31685],[34,181183,2557],{"className":181184},[31690],[34,181186],{"className":181187,"style":32604},[31845],[34,181189,32226],{"className":181190},[32024],[34,181192],{"className":181193,"style":32604},[31845],[34,181195,31660],{"className":181196},[31685,31686],[34,181198,181199,181202],{"style":180301},[34,181200],{"className":181201,"style":181125},[31813],[34,181203],{"className":181204,"style":181205},[126470,126471],"height:1.8291em;border-style:solid;border-width:0.04em;",[34,181207,31829],{"className":181208},[31828],[34,181210,181212],{"className":181211},[31801],[34,181213,181215],{"className":181214,"style":130091},[31805],[34,181216],{},[15,181218,181219],{},"但是本题问的是“at any time”的最大未使用空间，所以更常用答案是：",[15,181221,181222],{},[34,181223,181225,181260],{"className":181224},[31642],[34,181226,181228],{"className":181227},[31646],[31648,181229,181230],{"xmlns":31650},[31652,181231,181232,181258],{},[31655,181233,181234],{},[126400,181235,181236],{"notation":126402},[71090,181237,181238],{"scriptlevel":459,"displaystyle":702},[71090,181239,181240],{"scriptlevel":459,"displaystyle":702},[71090,181241,181242],{"scriptlevel":459,"displaystyle":758},[31655,181243,181244,181256],{},[37624,181245,181246,181248],{},[31749,181247,768],{},[31655,181249,181250,181252,181254],{},[31658,181251,33759],{},[31662,181253,32226],{},[31749,181255,640],{},[39361,181257,176786],{},[31665,181259,180211],{"encoding":31667},[34,181261,181263],{"className":181262,"ariaHidden":758},[31672],[34,181264,181266,181269],{"className":181265},[31676],[34,181267],{"className":181268,"style":180221},[31680],[34,181270,181272],{"className":181271},[31685],[34,181273,181275,181353],{"className":181274},[31796,31797],[34,181276,181278,181350],{"className":181277},[31801],[34,181279,181281,181342],{"className":181280,"style":180234},[31805],[34,181282,181283,181286],{"style":180237},[34,181284],{"className":181285,"style":180241},[31813],[34,181287,181289],{"className":181288},[126448],[34,181290,181292],{"className":181291},[31685],[34,181293,181295,181333],{"className":181294},[31685],[34,181296,181298,181301],{"className":181297},[31685],[34,181299,768],{"className":181300},[31685],[34,181302,181304],{"className":181303},[31792],[34,181305,181307],{"className":181306},[31796],[34,181308,181310],{"className":181309},[31801],[34,181311,181313],{"className":181312,"style":180269},[31805],[34,181314,181315,181318],{"style":129846},[34,181316],{"className":181317,"style":31814},[31813],[34,181319,181321],{"className":181320},[31818,31819,31820,31821],[34,181322,181324,181327,181330],{"className":181323},[31685,31821],[34,181325,33759],{"className":181326,"style":33774},[31685,31686,31821],[34,181328,32226],{"className":181329},[32024,31821],[34,181331,640],{"className":181332},[31685,31821],[34,181334,181336,181339],{"className":181335},[31685,2638],[34,181337,130298],{"className":181338},[31685],[34,181340,176811],{"className":181341},[31685,105462],[34,181343,181344,181347],{"style":180301},[34,181345],{"className":181346,"style":180241},[31813],[34,181348],{"className":181349,"style":180308},[126470,126471],[34,181351,31829],{"className":181352},[31828],[34,181354,181356],{"className":181355},[31801],[34,181357,181359],{"className":181358,"style":126482},[31805],[34,181360],{},[32188,181362],{},[253,181364,168735],{"id":168734},[15,181366,181367],{},"a.",[15,181369,181370],{},[34,181371,181373,181427],{"className":181372},[31642],[34,181374,181376],{"className":181375},[31646],[31648,181377,181378],{"xmlns":31650},[31652,181379,181380,181424],{},[31655,181381,181382],{},[126400,181383,181384],{"notation":126402},[71090,181385,181386],{"scriptlevel":459,"displaystyle":702},[71090,181387,181388],{"scriptlevel":459,"displaystyle":702},[71090,181389,181390],{"scriptlevel":459,"displaystyle":758},[31655,181391,181392,181394,181404,181406,181408,181410,181412,181414,181416,181418,181420,181422],{},[31662,181393,173674],{"stretchy":702},[31743,181395,181396,181402],{},[31655,181397,181398,181400],{},[31658,181399,885],{},[31662,181401,32682],{},[31749,181403,768],{},[31662,181405,456],{"stretchy":702},[31662,181407,173674],{"stretchy":702},[31658,181409,31660],{},[31658,181411,38080],{"mathvariant":38079},[31658,181413,51834],{},[31662,181415,173683],{"stretchy":702},[31662,181417,223],{},[31749,181419,640],{},[31662,181421,2557],{"stretchy":702},[31662,181423,173683],{"stretchy":702},[31665,181425,181426],{"encoding":31667},"\\boxed{\\lceil \\log_2(\\lceil n \u002F F \\rceil + 1) \\rceil}",[34,181428,181430],{"className":181429,"ariaHidden":758},[31672],[34,181431,181433,181436],{"className":181432},[31676],[34,181434],{"className":181435,"style":130018},[31680],[34,181437,181439],{"className":181438},[31685],[34,181440,181442,181548],{"className":181441},[31796,31797],[34,181443,181445,181545],{"className":181444},[31801],[34,181446,181448,181537],{"className":181447,"style":130031},[31805],[34,181449,181450,181453],{"style":130034},[34,181451],{"className":181452,"style":130038},[31813],[34,181454,181456],{"className":181455},[126448],[34,181457,181459],{"className":181458},[31685],[34,181460,181462,181465,181507,181510,181513,181516,181519,181522,181525,181528,181531,181534],{"className":181461},[31685],[34,181463,173674],{"className":181464},[31782],[34,181466,181468,181473],{"className":181467},[32767],[34,181469,38616,181471],{"className":181470},[32767],[34,181472,31942],{"style":38619},[34,181474,181476],{"className":181475},[31792],[34,181477,181479,181499],{"className":181478},[31796,31797],[34,181480,181482,181496],{"className":181481},[31801],[34,181483,181485],{"className":181484,"style":38632},[31805],[34,181486,181487,181490],{"style":38635},[34,181488],{"className":181489,"style":31814},[31813],[34,181491,181493],{"className":181492},[31818,31819,31820,31821],[34,181494,768],{"className":181495},[31685,31821],[34,181497,31829],{"className":181498},[31828],[34,181500,181502],{"className":181501},[31801],[34,181503,181505],{"className":181504,"style":38654},[31805],[34,181506],{},[34,181508,179804],{"className":181509},[31782],[34,181511,31660],{"className":181512},[31685,31686],[34,181514,38080],{"className":181515},[31685],[34,181517,51834],{"className":181518,"style":37422},[31685,31686],[34,181520,173683],{"className":181521},[31690],[34,181523],{"className":181524,"style":32604},[31845],[34,181526,223],{"className":181527},[32024],[34,181529],{"className":181530,"style":32604},[31845],[34,181532,640],{"className":181533},[31685],[34,181535,180026],{"className":181536},[31690],[34,181538,181539,181542],{"style":130074},[34,181540],{"className":181541,"style":130038},[31813],[34,181543],{"className":181544,"style":130081},[126470,126471],[34,181546,31829],{"className":181547},[31828],[34,181549,181551],{"className":181550},[31801],[34,181552,181554],{"className":181553,"style":130091},[31805],[34,181555],{},[15,181557,181558],{},"b.",[15,181560,181561],{},"令：",[15,181563,181564],{},[34,181565,181567,181614],{"className":181566},[31642],[34,181568,181570],{"className":181569},[31646],[31648,181571,181572],{"xmlns":31650},[31652,181573,181574,181612],{},[31655,181575,181576,181578,181580,181582,181592,181594,181596,181598,181600,181602,181604,181606,181608,181610],{},[31658,181577,33759],{},[31662,181579,202],{},[31662,181581,173674],{"stretchy":702},[31743,181583,181584,181590],{},[31655,181585,181586,181588],{},[31658,181587,885],{},[31662,181589,32682],{},[31749,181591,768],{},[31662,181593,456],{"stretchy":702},[31662,181595,173674],{"stretchy":702},[31658,181597,31660],{},[31658,181599,38080],{"mathvariant":38079},[31658,181601,51834],{},[31662,181603,173683],{"stretchy":702},[31662,181605,223],{},[31749,181607,640],{},[31662,181609,2557],{"stretchy":702},[31662,181611,173683],{"stretchy":702},[31665,181613,180374],{"encoding":31667},[34,181615,181617,181635,181710],{"className":181616,"ariaHidden":758},[31672],[34,181618,181620,181623,181626,181629,181632],{"className":181619},[31676],[34,181621],{"className":181622,"style":31681},[31680],[34,181624,33759],{"className":181625,"style":33774},[31685,31686],[34,181627],{"className":181628,"style":31846},[31845],[34,181630,202],{"className":181631},[31850],[34,181633],{"className":181634,"style":31846},[31845],[34,181636,181638,181641,181644,181686,181689,181692,181695,181698,181701,181704,181707],{"className":181637},[31676],[34,181639],{"className":181640,"style":31774},[31680],[34,181642,173674],{"className":181643},[31782],[34,181645,181647,181652],{"className":181646},[32767],[34,181648,38616,181650],{"className":181649},[32767],[34,181651,31942],{"style":38619},[34,181653,181655],{"className":181654},[31792],[34,181656,181658,181678],{"className":181657},[31796,31797],[34,181659,181661,181675],{"className":181660},[31801],[34,181662,181664],{"className":181663,"style":38632},[31805],[34,181665,181666,181669],{"style":38635},[34,181667],{"className":181668,"style":31814},[31813],[34,181670,181672],{"className":181671},[31818,31819,31820,31821],[34,181673,768],{"className":181674},[31685,31821],[34,181676,31829],{"className":181677},[31828],[34,181679,181681],{"className":181680},[31801],[34,181682,181684],{"className":181683,"style":38654},[31805],[34,181685],{},[34,181687,179804],{"className":181688},[31782],[34,181690,31660],{"className":181691},[31685,31686],[34,181693,38080],{"className":181694},[31685],[34,181696,51834],{"className":181697,"style":37422},[31685,31686],[34,181699,173683],{"className":181700},[31690],[34,181702],{"className":181703,"style":32604},[31845],[34,181705,223],{"className":181706},[32024],[34,181708],{"className":181709,"style":32604},[31845],[34,181711,181713,181716,181719],{"className":181712},[31676],[34,181714],{"className":181715,"style":31774},[31680],[34,181717,640],{"className":181718},[31685],[34,181720,180026],{"className":181721},[31690],[15,181723,181724],{},"最大未使用空间为：",[15,181726,181727],{},[34,181728,181730,181765],{"className":181729},[31642],[34,181731,181733],{"className":181732},[31646],[31648,181734,181735],{"xmlns":31650},[31652,181736,181737,181763],{},[31655,181738,181739],{},[126400,181740,181741],{"notation":126402},[71090,181742,181743],{"scriptlevel":459,"displaystyle":702},[71090,181744,181745],{"scriptlevel":459,"displaystyle":702},[71090,181746,181747],{"scriptlevel":459,"displaystyle":758},[31655,181748,181749,181761],{},[37624,181750,181751,181753],{},[31749,181752,768],{},[31655,181754,181755,181757,181759],{},[31658,181756,33759],{},[31662,181758,32226],{},[31749,181760,640],{},[39361,181762,176786],{},[31665,181764,180211],{"encoding":31667},[34,181766,181768],{"className":181767,"ariaHidden":758},[31672],[34,181769,181771,181774],{"className":181770},[31676],[34,181772],{"className":181773,"style":180221},[31680],[34,181775,181777],{"className":181776},[31685],[34,181778,181780,181858],{"className":181779},[31796,31797],[34,181781,181783,181855],{"className":181782},[31801],[34,181784,181786,181847],{"className":181785,"style":180234},[31805],[34,181787,181788,181791],{"style":180237},[34,181789],{"className":181790,"style":180241},[31813],[34,181792,181794],{"className":181793},[126448],[34,181795,181797],{"className":181796},[31685],[34,181798,181800,181838],{"className":181799},[31685],[34,181801,181803,181806],{"className":181802},[31685],[34,181804,768],{"className":181805},[31685],[34,181807,181809],{"className":181808},[31792],[34,181810,181812],{"className":181811},[31796],[34,181813,181815],{"className":181814},[31801],[34,181816,181818],{"className":181817,"style":180269},[31805],[34,181819,181820,181823],{"style":129846},[34,181821],{"className":181822,"style":31814},[31813],[34,181824,181826],{"className":181825},[31818,31819,31820,31821],[34,181827,181829,181832,181835],{"className":181828},[31685,31821],[34,181830,33759],{"className":181831,"style":33774},[31685,31686,31821],[34,181833,32226],{"className":181834},[32024,31821],[34,181836,640],{"className":181837},[31685,31821],[34,181839,181841,181844],{"className":181840},[31685,2638],[34,181842,130298],{"className":181843},[31685],[34,181845,176811],{"className":181846},[31685,105462],[34,181848,181849,181852],{"style":180301},[34,181850],{"className":181851,"style":180241},[31813],[34,181853],{"className":181854,"style":180308},[126470,126471],[34,181856,31829],{"className":181857},[31828],[34,181859,181861],{"className":181860},[31801],[34,181862,181864],{"className":181863,"style":126482},[31805],[34,181865],{},[15,181867,132311],{},[15,181869,181870],{},[34,181871,181873,181910],{"className":181872},[31642],[34,181874,181876],{"className":181875},[31646],[31648,181877,181878],{"xmlns":31650},[31652,181879,181880,181908],{},[31655,181881,181882],{},[126400,181883,181884],{"notation":126402},[71090,181885,181886],{"scriptlevel":459,"displaystyle":702},[71090,181887,181888],{"scriptlevel":459,"displaystyle":702},[71090,181889,181890],{"scriptlevel":459,"displaystyle":758},[31655,181891,181892,181904,181906],{},[37624,181893,181894,181896],{},[31749,181895,768],{},[31655,181897,181898,181900,181902],{},[31658,181899,33759],{},[31662,181901,32226],{},[31749,181903,640],{},[31658,181905,51834],{},[39361,181907,180526],{},[31665,181909,180529],{"encoding":31667},[34,181911,181913],{"className":181912,"ariaHidden":758},[31672],[34,181914,181916,181919],{"className":181915},[31676],[34,181917],{"className":181918,"style":180221},[31680],[34,181920,181922],{"className":181921},[31685],[34,181923,181925,182006],{"className":181924},[31796,31797],[34,181926,181928,182003],{"className":181927},[31801],[34,181929,181931,181995],{"className":181930,"style":180234},[31805],[34,181932,181933,181936],{"style":180237},[34,181934],{"className":181935,"style":180241},[31813],[34,181937,181939],{"className":181938},[126448],[34,181940,181942],{"className":181941},[31685],[34,181943,181945,181983,181986],{"className":181944},[31685],[34,181946,181948,181951],{"className":181947},[31685],[34,181949,768],{"className":181950},[31685],[34,181952,181954],{"className":181953},[31792],[34,181955,181957],{"className":181956},[31796],[34,181958,181960],{"className":181959},[31801],[34,181961,181963],{"className":181962,"style":180269},[31805],[34,181964,181965,181968],{"style":129846},[34,181966],{"className":181967,"style":31814},[31813],[34,181969,181971],{"className":181970},[31818,31819,31820,31821],[34,181972,181974,181977,181980],{"className":181973},[31685,31821],[34,181975,33759],{"className":181976,"style":33774},[31685,31686,31821],[34,181978,32226],{"className":181979},[32024,31821],[34,181981,640],{"className":181982},[31685,31821],[34,181984,51834],{"className":181985,"style":37422},[31685,31686],[34,181987,181989,181992],{"className":181988},[31685,2638],[34,181990,130298],{"className":181991},[31685],[34,181993,180615],{"className":181994},[31685,105462],[34,181996,181997,182000],{"style":180301},[34,181998],{"className":181999,"style":180241},[31813],[34,182001],{"className":182002,"style":180308},[126470,126471],[34,182004,31829],{"className":182005},[31828],[34,182007,182009],{"className":182008},[31801],[34,182010,182012],{"className":182011,"style":126482},[31805],[34,182013],{},[15,182015,182016],{},"若刚写入触发分配的那条记录后再统计，则为：",[15,182018,182019],{},[34,182020,182022,182064],{"className":182021},[31642],[34,182023,182025],{"className":182024},[31646],[31648,182026,182027],{"xmlns":31650},[31652,182028,182029,182061],{},[31655,182030,182031],{},[126400,182032,182033],{"notation":126402},[71090,182034,182035],{"scriptlevel":459,"displaystyle":702},[71090,182036,182037],{"scriptlevel":459,"displaystyle":702},[71090,182038,182039],{"scriptlevel":459,"displaystyle":758},[31655,182040,182041,182053,182055,182057,182059],{},[37624,182042,182043,182045],{},[31749,182044,768],{},[31655,182046,182047,182049,182051],{},[31658,182048,33759],{},[31662,182050,32226],{},[31749,182052,640],{},[31658,182054,51834],{},[31662,182056,32226],{},[31749,182058,640],{},[39361,182060,180526],{},[31665,182062,182063],{"encoding":31667},"\\boxed{2^{k-1}F - 1 \\text{ 条记录容量}}",[34,182065,182067],{"className":182066,"ariaHidden":758},[31672],[34,182068,182070,182073],{"className":182069},[31676],[34,182071],{"className":182072,"style":180692},[31680],[34,182074,182076],{"className":182075},[31685],[34,182077,182079,182172],{"className":182078},[31796,31797],[34,182080,182082,182169],{"className":182081},[31801],[34,182083,182085,182161],{"className":182084,"style":180234},[31805],[34,182086,182087,182090],{"style":180707},[34,182088],{"className":182089,"style":180711},[31813],[34,182091,182093],{"className":182092},[126448],[34,182094,182096],{"className":182095},[31685],[34,182097,182099,182137,182140,182143,182146,182149,182152],{"className":182098},[31685],[34,182100,182102,182105],{"className":182101},[31685],[34,182103,768],{"className":182104},[31685],[34,182106,182108],{"className":182107},[31792],[34,182109,182111],{"className":182110},[31796],[34,182112,182114],{"className":182113},[31801],[34,182115,182117],{"className":182116,"style":180269},[31805],[34,182118,182119,182122],{"style":129846},[34,182120],{"className":182121,"style":31814},[31813],[34,182123,182125],{"className":182124},[31818,31819,31820,31821],[34,182126,182128,182131,182134],{"className":182127},[31685,31821],[34,182129,33759],{"className":182130,"style":33774},[31685,31686,31821],[34,182132,32226],{"className":182133},[32024,31821],[34,182135,640],{"className":182136},[31685,31821],[34,182138,51834],{"className":182139,"style":37422},[31685,31686],[34,182141],{"className":182142,"style":32604},[31845],[34,182144,32226],{"className":182145},[32024],[34,182147],{"className":182148,"style":32604},[31845],[34,182150,640],{"className":182151},[31685],[34,182153,182155,182158],{"className":182154},[31685,2638],[34,182156,130298],{"className":182157},[31685],[34,182159,180615],{"className":182160},[31685,105462],[34,182162,182163,182166],{"style":180301},[34,182164],{"className":182165,"style":180711},[31813],[34,182167],{"className":182168,"style":180782},[126470,126471],[34,182170,31829],{"className":182171},[31828],[34,182173,182175],{"className":182174},[31801],[34,182176,182178],{"className":182177,"style":129408},[31805],[34,182179],{},[32188,182181],{},[19,182183,182185],{"id":182184},"_3-计算最后一个块未装满造成的空间浪费百分比","3. 计算最后一个块未装满造成的空间浪费百分比",[253,182187,162043],{"id":168779},[15,182189,182190],{},"Ignoring overhead for directories and file descriptors, consider a file system in which files are stored in blocks of 16K bytes. For each of the following file sizes, calculate the percentage of wasted file space due to incomplete filling of the last block: 41,600 bytes; 640,000 bytes; 4.064,000 bytes.",[15,182192,162049],{},[15,182194,182195,182196,182230],{},"忽略目录和文件描述符开销，假设文件系统以 ",[34,182197,182199,182215],{"className":182198},[31642],[34,182200,182202],{"className":182201},[31646],[31648,182203,182204],{"xmlns":31650},[31652,182205,182206,182212],{},[31655,182207,182208,182210],{},[31749,182209,177869],{},[31658,182211,38691],{},[31665,182213,182214],{"encoding":31667},"16K",[34,182216,182218],{"className":182217,"ariaHidden":758},[31672],[34,182219,182221,182224,182227],{"className":182220},[31676],[34,182222],{"className":182223,"style":33218},[31680],[34,182225,177869],{"className":182226},[31685],[34,182228,38691],{"className":182229,"style":38706},[31685,31686]," 字节为一个块存储文件。对于下面三种文件大小，计算由于最后一个块未装满而造成的空间浪费百分比：",[15,182232,182233],{},[34,182234,182236,182253],{"className":182235},[31642],[34,182237,182239],{"className":182238},[31646],[31648,182240,182241],{"xmlns":31650},[31652,182242,182243,182250],{},[31655,182244,182245,182248],{},[31749,182246,182247],{},"41600",[31658,182249,54224],{},[31665,182251,182252],{"encoding":31667},"41600B",[34,182254,182256],{"className":182255,"ariaHidden":758},[31672],[34,182257,182259,182262,182265],{"className":182258},[31676],[34,182260],{"className":182261,"style":33218},[31680],[34,182263,182247],{"className":182264},[31685],[34,182266,54224],{"className":182267,"style":54252},[31685,31686],[15,182269,182270],{},[34,182271,182273,182290],{"className":182272},[31642],[34,182274,182276],{"className":182275},[31646],[31648,182277,182278],{"xmlns":31650},[31652,182279,182280,182287],{},[31655,182281,182282,182285],{},[31749,182283,182284],{},"640000",[31658,182286,54224],{},[31665,182288,182289],{"encoding":31667},"640000B",[34,182291,182293],{"className":182292,"ariaHidden":758},[31672],[34,182294,182296,182299,182302],{"className":182295},[31676],[34,182297],{"className":182298,"style":33218},[31680],[34,182300,182284],{"className":182301},[31685],[34,182303,54224],{"className":182304,"style":54252},[31685,31686],[15,182306,182307],{},[34,182308,182310,182327],{"className":182309},[31642],[34,182311,182313],{"className":182312},[31646],[31648,182314,182315],{"xmlns":31650},[31652,182316,182317,182324],{},[31655,182318,182319,182322],{},[31749,182320,182321],{},"4064000",[31658,182323,54224],{},[31665,182325,182326],{"encoding":31667},"4064000B",[34,182328,182330],{"className":182329,"ariaHidden":758},[31672],[34,182331,182333,182336,182339],{"className":182332},[31676],[34,182334],{"className":182335,"style":33218},[31680],[34,182337,182321],{"className":182338},[31685],[34,182340,54224],{"className":182341,"style":54252},[31685,31686],[15,182343,182344,182345,182389,182390,182446],{},"注意：题目中的 ",[34,182346,182348,182368],{"className":182347},[31642],[34,182349,182351],{"className":182350},[31646],[31648,182352,182353],{"xmlns":31650},[31652,182354,182355,182365],{},[31655,182356,182357,182360,182362],{},[31749,182358,182359],{},"4.064",[31662,182361,31960],{"separator":758},[31749,182363,182364],{},"000",[31665,182366,182367],{"encoding":31667},"4.064,000",[34,182369,182371],{"className":182370,"ariaHidden":758},[31672],[34,182372,182374,182377,182380,182383,182386],{"className":182373},[31676],[34,182375],{"className":182376,"style":40292},[31680],[34,182378,182359],{"className":182379},[31685],[34,182381,31960],{"className":182382},[32119],[34,182384],{"className":182385,"style":32123},[31845],[34,182387,182364],{"className":182388},[31685]," 通常应理解为 ",[34,182391,182393,182416],{"className":182392},[31642],[34,182394,182396],{"className":182395},[31646],[31648,182397,182398],{"xmlns":31650},[31652,182399,182400,182413],{},[31655,182401,182402,182404,182406,182409,182411],{},[31749,182403,15932],{},[31662,182405,31960],{"separator":758},[31749,182407,182408],{},"064",[31662,182410,31960],{"separator":758},[31749,182412,182364],{},[31665,182414,182415],{"encoding":31667},"4,064,000",[34,182417,182419],{"className":182418,"ariaHidden":758},[31672],[34,182420,182422,182425,182428,182431,182434,182437,182440,182443],{"className":182421},[31676],[34,182423],{"className":182424,"style":40292},[31680],[34,182426,15932],{"className":182427},[31685],[34,182429,31960],{"className":182430},[32119],[34,182432],{"className":182433,"style":32123},[31845],[34,182435,182408],{"className":182436},[31685],[34,182438,31960],{"className":182439},[32119],[34,182441],{"className":182442,"style":32123},[31845],[34,182444,182364],{"className":182445},[31685]," 字节。",[32188,182448],{},[253,182450,175302],{"id":182451},"答案和讲解-2",[15,182453,182454],{},"块大小为：",[15,182456,182457],{},[34,182458,182460,182491],{"className":182459},[31642],[34,182461,182463],{"className":182462},[31646],[31648,182464,182465],{"xmlns":31650},[31652,182466,182467,182488],{},[31655,182468,182469,182471,182473,182475,182477,182479,182481,182483,182486],{},[31749,182470,177869],{},[31658,182472,38691],{},[31662,182474,202],{},[31749,182476,177869],{},[31662,182478,42071],{},[31749,182480,177393],{},[31662,182482,202],{},[31749,182484,182485],{},"16384",[31658,182487,54224],{},[31665,182489,182490],{"encoding":31667},"16K = 16 \\times 1024 = 16384B",[34,182492,182494,182515,182533,182551],{"className":182493,"ariaHidden":758},[31672],[34,182495,182497,182500,182503,182506,182509,182512],{"className":182496},[31676],[34,182498],{"className":182499,"style":33218},[31680],[34,182501,177869],{"className":182502},[31685],[34,182504,38691],{"className":182505,"style":38706},[31685,31686],[34,182507],{"className":182508,"style":31846},[31845],[34,182510,202],{"className":182511},[31850],[34,182513],{"className":182514,"style":31846},[31845],[34,182516,182518,182521,182524,182527,182530],{"className":182517},[31676],[34,182519],{"className":182520,"style":35942},[31680],[34,182522,177869],{"className":182523},[31685],[34,182525],{"className":182526,"style":32604},[31845],[34,182528,42071],{"className":182529},[32024],[34,182531],{"className":182532,"style":32604},[31845],[34,182534,182536,182539,182542,182545,182548],{"className":182535},[31676],[34,182537],{"className":182538,"style":32617},[31680],[34,182540,177393],{"className":182541},[31685],[34,182543],{"className":182544,"style":31846},[31845],[34,182546,202],{"className":182547},[31850],[34,182549],{"className":182550,"style":31846},[31845],[34,182552,182554,182557,182560],{"className":182553},[31676],[34,182555],{"className":182556,"style":33218},[31680],[34,182558,182485],{"className":182559},[31685],[34,182561,54224],{"className":182562,"style":54252},[31685,31686],[15,182564,182565],{},"计算步骤：",[2230,182567,182568],{},[108,182569,182570],{},"需要块数：",[15,182572,182573],{},[34,182574,182576,182600],{"className":182575},[31642],[34,182577,182579],{"className":182578},[31646],[31648,182580,182581],{"xmlns":31650},[31652,182582,182583,182597],{},[31655,182584,182585,182587,182590,182592,182595],{},[31662,182586,173674],{"stretchy":702},[39361,182588,182589],{},"文件大小",[31658,182591,38080],{"mathvariant":38079},[39361,182593,182594],{},"块大小",[31662,182596,173683],{"stretchy":702},[31665,182598,182599],{"encoding":31667},"\\lceil 文件大小 \u002F 块大小 \\rceil",[34,182601,182603],{"className":182602,"ariaHidden":758},[31672],[34,182604,182606,182609,182612,182615,182618,182621],{"className":182605},[31676],[34,182607],{"className":182608,"style":31774},[31680],[34,182610,173674],{"className":182611},[31782],[34,182613,182589],{"className":182614},[31685,105462],[34,182616,38080],{"className":182617},[31685],[34,182619,182594],{"className":182620},[31685,105462],[34,182622,173683],{"className":182623},[31690],[2230,182625,182626],{"start":43},[108,182627,182628],{},"实际分配空间：",[15,182630,182631],{},[34,182632,182634,182653],{"className":182633},[31642],[34,182635,182637],{"className":182636},[31646],[31648,182638,182639],{"xmlns":31650},[31652,182640,182641,182650],{},[31655,182642,182643,182646,182648],{},[39361,182644,182645],{},"需要块数",[31662,182647,42071],{},[39361,182649,182594],{},[31665,182651,182652],{"encoding":31667},"需要块数 \\times 块大小",[34,182654,182656,182674],{"className":182655,"ariaHidden":758},[31672],[34,182657,182659,182662,182665,182668,182671],{"className":182658},[31676],[34,182660],{"className":182661,"style":34944},[31680],[34,182663,182645],{"className":182664},[31685,105462],[34,182666],{"className":182667,"style":32604},[31845],[34,182669,42071],{"className":182670},[32024],[34,182672],{"className":182673,"style":32604},[31845],[34,182675,182677,182680],{"className":182676},[31676],[34,182678],{"className":182679,"style":33218},[31680],[34,182681,182594],{"className":182682},[31685,105462],[2230,182684,182685],{"start":49},[108,182686,182687],{},"浪费空间：",[15,182689,182690],{},[34,182691,182693,182712],{"className":182692},[31642],[34,182694,182696],{"className":182695},[31646],[31648,182697,182698],{"xmlns":31650},[31652,182699,182700,182709],{},[31655,182701,182702,182705,182707],{},[39361,182703,182704],{},"实际分配空间",[31662,182706,32226],{},[39361,182708,182589],{},[31665,182710,182711],{"encoding":31667},"实际分配空间 - 文件大小",[34,182713,182715,182733],{"className":182714,"ariaHidden":758},[31672],[34,182716,182718,182721,182724,182727,182730],{"className":182717},[31676],[34,182719],{"className":182720,"style":34944},[31680],[34,182722,182704],{"className":182723},[31685,105462],[34,182725],{"className":182726,"style":32604},[31845],[34,182728,32226],{"className":182729},[32024],[34,182731],{"className":182732,"style":32604},[31845],[34,182734,182736,182739],{"className":182735},[31676],[34,182737],{"className":182738,"style":33218},[31680],[34,182740,182589],{"className":182741},[31685,105462],[2230,182743,182744],{"start":56},[108,182745,182746],{},"浪费百分比：",[15,182748,182749],{},[34,182750,182752,182777],{"className":182751},[31642],[34,182753,182755],{"className":182754},[31646],[31648,182756,182757],{"xmlns":31650},[31652,182758,182759,182774],{},[31655,182760,182761,182768,182770,182772],{},[41453,182762,182763,182766],{},[39361,182764,182765],{},"浪费空间",[39361,182767,182704],{},[31662,182769,42071],{},[31749,182771,162173],{},[31658,182773,119426],{"mathvariant":38079},[31665,182775,182776],{"encoding":31667},"\\frac{浪费空间}{实际分配空间} \\times 100\\%",[34,182778,182780,182863],{"className":182779,"ariaHidden":758},[31672],[34,182781,182783,182786,182854,182857,182860],{"className":182782},[31676],[34,182784],{"className":182785,"style":168026},[31680],[34,182787,182789,182792,182851],{"className":182788},[31685],[34,182790],{"className":182791},[31782,41488],[34,182793,182795],{"className":182794},[41453],[34,182796,182798,182843],{"className":182797},[31796,31797],[34,182799,182801,182840],{"className":182800},[31801],[34,182802,182804,182818,182826],{"className":182803,"style":168045},[31805],[34,182805,182806,182809],{"style":41504},[34,182807],{"className":182808,"style":41508},[31813],[34,182810,182812],{"className":182811},[31818,31819,31820,31821],[34,182813,182815],{"className":182814},[31685,31821],[34,182816,182704],{"className":182817},[31685,105462,31821],[34,182819,182820,182823],{"style":41520},[34,182821],{"className":182822,"style":41508},[31813],[34,182824],{"className":182825,"style":41528},[41527],[34,182827,182828,182831],{"style":46771},[34,182829],{"className":182830,"style":41508},[31813],[34,182832,182834],{"className":182833},[31818,31819,31820,31821],[34,182835,182837],{"className":182836},[31685,31821],[34,182838,182765],{"className":182839},[31685,105462,31821],[34,182841,31829],{"className":182842},[31828],[34,182844,182846],{"className":182845},[31801],[34,182847,182849],{"className":182848,"style":41619},[31805],[34,182850],{},[34,182852],{"className":182853},[31690,41488],[34,182855],{"className":182856,"style":32604},[31845],[34,182858,42071],{"className":182859},[32024],[34,182861],{"className":182862,"style":32604},[31845],[34,182864,182866,182869],{"className":182865},[31676],[34,182867],{"className":182868,"style":168117},[31680],[34,182870,168121],{"className":182871},[31685],[32188,182873],{},[253,182875,182877,182878],{"id":182876},"_31-文件大小为-41600b41600b41600b","3.1 文件大小为 ",[34,182879,182881,182896],{"className":182880},[31642],[34,182882,182884],{"className":182883},[31646],[31648,182885,182886],{"xmlns":31650},[31652,182887,182888,182894],{},[31655,182889,182890,182892],{},[31749,182891,182247],{},[31658,182893,54224],{},[31665,182895,182252],{"encoding":31667},[34,182897,182899],{"className":182898,"ariaHidden":758},[31672],[34,182900,182902,182905,182908],{"className":182901},[31676],[34,182903],{"className":182904,"style":33218},[31680],[34,182906,182247],{"className":182907},[31685],[34,182909,54224],{"className":182910,"style":54252},[31685,31686],[15,182912,182570],{},[15,182914,182915],{},[34,182916,182918,182953],{"className":182917},[31642],[34,182919,182921],{"className":182920},[31646],[31648,182922,182923],{"xmlns":31650},[31652,182924,182925,182950],{},[31655,182926,182927,182929,182931,182933,182935,182937,182939,182941,182944,182946,182948],{},[31662,182928,173674],{"stretchy":702},[31749,182930,182247],{},[31658,182932,38080],{"mathvariant":38079},[31749,182934,182485],{},[31662,182936,173683],{"stretchy":702},[31662,182938,202],{},[31662,182940,173674],{"stretchy":702},[31749,182942,182943],{},"2.539",[31662,182945,173683],{"stretchy":702},[31662,182947,202],{},[31749,182949,786],{},[31665,182951,182952],{"encoding":31667},"\\lceil 41600 \u002F 16384 \\rceil = \\lceil 2.539 \\rceil = 3",[34,182954,182956,182981,183005],{"className":182955,"ariaHidden":758},[31672],[34,182957,182959,182962,182965,182969,182972,182975,182978],{"className":182958},[31676],[34,182960],{"className":182961,"style":31774},[31680],[34,182963,173674],{"className":182964},[31782],[34,182966,182968],{"className":182967},[31685],"41600\u002F16384",[34,182970,173683],{"className":182971},[31690],[34,182973],{"className":182974,"style":31846},[31845],[34,182976,202],{"className":182977},[31850],[34,182979],{"className":182980,"style":31846},[31845],[34,182982,182984,182987,182990,182993,182996,182999,183002],{"className":182983},[31676],[34,182985],{"className":182986,"style":31774},[31680],[34,182988,173674],{"className":182989},[31782],[34,182991,182943],{"className":182992},[31685],[34,182994,173683],{"className":182995},[31690],[34,182997],{"className":182998,"style":31846},[31845],[34,183000,202],{"className":183001},[31850],[34,183003],{"className":183004,"style":31846},[31845],[34,183006,183008,183011],{"className":183007},[31676],[34,183009],{"className":183010,"style":32617},[31680],[34,183012,786],{"className":183013},[31685],[15,183015,182628],{},[15,183017,183018],{},[34,183019,183021,183046],{"className":183020},[31642],[34,183022,183024],{"className":183023},[31646],[31648,183025,183026],{"xmlns":31650},[31652,183027,183028,183043],{},[31655,183029,183030,183032,183034,183036,183038,183041],{},[31749,183031,786],{},[31662,183033,42071],{},[31749,183035,182485],{},[31662,183037,202],{},[31749,183039,183040],{},"49152",[31658,183042,54224],{},[31665,183044,183045],{"encoding":31667},"3 \\times 16384 = 49152B",[34,183047,183049,183067,183085],{"className":183048,"ariaHidden":758},[31672],[34,183050,183052,183055,183058,183061,183064],{"className":183051},[31676],[34,183053],{"className":183054,"style":35942},[31680],[34,183056,786],{"className":183057},[31685],[34,183059],{"className":183060,"style":32604},[31845],[34,183062,42071],{"className":183063},[32024],[34,183065],{"className":183066,"style":32604},[31845],[34,183068,183070,183073,183076,183079,183082],{"className":183069},[31676],[34,183071],{"className":183072,"style":32617},[31680],[34,183074,182485],{"className":183075},[31685],[34,183077],{"className":183078,"style":31846},[31845],[34,183080,202],{"className":183081},[31850],[34,183083],{"className":183084,"style":31846},[31845],[34,183086,183088,183091,183094],{"className":183087},[31676],[34,183089],{"className":183090,"style":33218},[31680],[34,183092,183040],{"className":183093},[31685],[34,183095,54224],{"className":183096,"style":54252},[31685,31686],[15,183098,182687],{},[15,183100,183101],{},[34,183102,183104,183129],{"className":183103},[31642],[34,183105,183107],{"className":183106},[31646],[31648,183108,183109],{"xmlns":31650},[31652,183110,183111,183126],{},[31655,183112,183113,183115,183117,183119,183121,183124],{},[31749,183114,183040],{},[31662,183116,32226],{},[31749,183118,182247],{},[31662,183120,202],{},[31749,183122,183123],{},"7552",[31658,183125,54224],{},[31665,183127,183128],{"encoding":31667},"49152 - 41600 = 7552B",[34,183130,183132,183150,183168],{"className":183131,"ariaHidden":758},[31672],[34,183133,183135,183138,183141,183144,183147],{"className":183134},[31676],[34,183136],{"className":183137,"style":35942},[31680],[34,183139,183040],{"className":183140},[31685],[34,183142],{"className":183143,"style":32604},[31845],[34,183145,32226],{"className":183146},[32024],[34,183148],{"className":183149,"style":32604},[31845],[34,183151,183153,183156,183159,183162,183165],{"className":183152},[31676],[34,183154],{"className":183155,"style":32617},[31680],[34,183157,182247],{"className":183158},[31685],[34,183160],{"className":183161,"style":31846},[31845],[34,183163,202],{"className":183164},[31850],[34,183166],{"className":183167,"style":31846},[31845],[34,183169,183171,183174,183177],{"className":183170},[31676],[34,183172],{"className":183173,"style":33218},[31680],[34,183175,183123],{"className":183176},[31685],[34,183178,54224],{"className":183179,"style":54252},[31685,31686],[15,183181,182746],{},[15,183183,183184],{},[34,183185,183187,183218],{"className":183186},[31642],[34,183188,183190],{"className":183189},[31646],[31648,183191,183192],{"xmlns":31650},[31652,183193,183194,183215],{},[31655,183195,183196,183202,183204,183206,183208,183210,183213],{},[41453,183197,183198,183200],{},[31749,183199,183123],{},[31749,183201,183040],{},[31662,183203,42071],{},[31749,183205,162173],{},[31658,183207,119426],{"mathvariant":38079},[31662,183209,168560],{},[31749,183211,183212],{},"15.36",[31658,183214,119426],{"mathvariant":38079},[31665,183216,183217],{"encoding":31667},"\\frac{7552}{49152} \\times 100\\% \\approx 15.36\\%",[34,183219,183221,183304,183322],{"className":183220,"ariaHidden":758},[31672],[34,183222,183224,183227,183295,183298,183301],{"className":183223},[31676],[34,183225],{"className":183226,"style":46727},[31680],[34,183228,183230,183233,183292],{"className":183229},[31685],[34,183231],{"className":183232},[31782,41488],[34,183234,183236],{"className":183235},[41453],[34,183237,183239,183284],{"className":183238},[31796,31797],[34,183240,183242,183281],{"className":183241},[31801],[34,183243,183245,183259,183267],{"className":183244,"style":46746},[31805],[34,183246,183247,183250],{"style":41504},[34,183248],{"className":183249,"style":41508},[31813],[34,183251,183253],{"className":183252},[31818,31819,31820,31821],[34,183254,183256],{"className":183255},[31685,31821],[34,183257,183040],{"className":183258},[31685,31821],[34,183260,183261,183264],{"style":41520},[34,183262],{"className":183263,"style":41508},[31813],[34,183265],{"className":183266,"style":41528},[41527],[34,183268,183269,183272],{"style":46771},[34,183270],{"className":183271,"style":41508},[31813],[34,183273,183275],{"className":183274},[31818,31819,31820,31821],[34,183276,183278],{"className":183277},[31685,31821],[34,183279,183123],{"className":183280},[31685,31821],[34,183282,31829],{"className":183283},[31828],[34,183285,183287],{"className":183286},[31801],[34,183288,183290],{"className":183289,"style":41619},[31805],[34,183291],{},[34,183293],{"className":183294},[31690,41488],[34,183296],{"className":183297,"style":32604},[31845],[34,183299,42071],{"className":183300},[32024],[34,183302],{"className":183303,"style":32604},[31845],[34,183305,183307,183310,183313,183316,183319],{"className":183306},[31676],[34,183308],{"className":183309,"style":168117},[31680],[34,183311,168121],{"className":183312},[31685],[34,183314],{"className":183315,"style":31846},[31845],[34,183317,168560],{"className":183318},[31850],[34,183320],{"className":183321,"style":31846},[31845],[34,183323,183325,183328],{"className":183324},[31676],[34,183326],{"className":183327,"style":168117},[31680],[34,183329,183331],{"className":183330},[31685],"15.36%",[15,183333,58173],{},[15,183335,183336],{},[34,183337,183339,183365],{"className":183338},[31642],[34,183340,183342],{"className":183341},[31646],[31648,183343,183344],{"xmlns":31650},[31652,183345,183346,183362],{},[31655,183347,183348],{},[126400,183349,183350],{"notation":126402},[71090,183351,183352],{"scriptlevel":459,"displaystyle":702},[71090,183353,183354],{"scriptlevel":459,"displaystyle":702},[71090,183355,183356],{"scriptlevel":459,"displaystyle":758},[31655,183357,183358,183360],{},[31749,183359,183212],{},[31658,183361,119426],{"mathvariant":38079},[31665,183363,183364],{"encoding":31667},"\\boxed{15.36\\%}",[34,183366,183368],{"className":183367,"ariaHidden":758},[31672],[34,183369,183371,183375],{"className":183370},[31676],[34,183372],{"className":183373,"style":183374},[31680],"height:1.4856em;vertical-align:-0.3956em;",[34,183376,183378],{"className":183377},[31685],[34,183379,183381,183418],{"className":183380},[31796,31797],[34,183382,183384,183415],{"className":183383},[31801],[34,183385,183387,183406],{"className":183386,"style":130031},[31805],[34,183388,183390,183394],{"style":183389},"top:-3.4856em;",[34,183391],{"className":183392,"style":183393},[31813],"height:3.4856em;",[34,183395,183397],{"className":183396},[126448],[34,183398,183400],{"className":183399},[31685],[34,183401,183403],{"className":183402},[31685],[34,183404,183331],{"className":183405},[31685],[34,183407,183408,183411],{"style":130074},[34,183409],{"className":183410,"style":183393},[31813],[34,183412],{"className":183413,"style":183414},[126470,126471],"height:1.4856em;border-style:solid;border-width:0.04em;",[34,183416,31829],{"className":183417},[31828],[34,183419,183421],{"className":183420},[31801],[34,183422,183425],{"className":183423,"style":183424},[31805],"height:0.3956em;",[34,183426],{},[32188,183428],{},[253,183430,183432,183433],{"id":183431},"_32-文件大小为-640000b640000b640000b","3.2 文件大小为 ",[34,183434,183436,183451],{"className":183435},[31642],[34,183437,183439],{"className":183438},[31646],[31648,183440,183441],{"xmlns":31650},[31652,183442,183443,183449],{},[31655,183444,183445,183447],{},[31749,183446,182284],{},[31658,183448,54224],{},[31665,183450,182289],{"encoding":31667},[34,183452,183454],{"className":183453,"ariaHidden":758},[31672],[34,183455,183457,183460,183463],{"className":183456},[31676],[34,183458],{"className":183459,"style":33218},[31680],[34,183461,182284],{"className":183462},[31685],[34,183464,54224],{"className":183465,"style":54252},[31685,31686],[15,183467,182570],{},[15,183469,183470],{},[34,183471,183473,183508],{"className":183472},[31642],[34,183474,183476],{"className":183475},[31646],[31648,183477,183478],{"xmlns":31650},[31652,183479,183480,183505],{},[31655,183481,183482,183484,183486,183488,183490,183492,183494,183496,183499,183501,183503],{},[31662,183483,173674],{"stretchy":702},[31749,183485,182284],{},[31658,183487,38080],{"mathvariant":38079},[31749,183489,182485],{},[31662,183491,173683],{"stretchy":702},[31662,183493,202],{},[31662,183495,173674],{"stretchy":702},[31749,183497,183498],{},"39.0625",[31662,183500,173683],{"stretchy":702},[31662,183502,202],{},[31749,183504,5731],{},[31665,183506,183507],{"encoding":31667},"\\lceil 640000 \u002F 16384 \\rceil = \\lceil 39.0625 \\rceil = 40",[34,183509,183511,183536,183560],{"className":183510,"ariaHidden":758},[31672],[34,183512,183514,183517,183520,183524,183527,183530,183533],{"className":183513},[31676],[34,183515],{"className":183516,"style":31774},[31680],[34,183518,173674],{"className":183519},[31782],[34,183521,183523],{"className":183522},[31685],"640000\u002F16384",[34,183525,173683],{"className":183526},[31690],[34,183528],{"className":183529,"style":31846},[31845],[34,183531,202],{"className":183532},[31850],[34,183534],{"className":183535,"style":31846},[31845],[34,183537,183539,183542,183545,183548,183551,183554,183557],{"className":183538},[31676],[34,183540],{"className":183541,"style":31774},[31680],[34,183543,173674],{"className":183544},[31782],[34,183546,183498],{"className":183547},[31685],[34,183549,173683],{"className":183550},[31690],[34,183552],{"className":183553,"style":31846},[31845],[34,183555,202],{"className":183556},[31850],[34,183558],{"className":183559,"style":31846},[31845],[34,183561,183563,183566],{"className":183562},[31676],[34,183564],{"className":183565,"style":32617},[31680],[34,183567,5731],{"className":183568},[31685],[15,183570,182628],{},[15,183572,183573],{},[34,183574,183576,183601],{"className":183575},[31642],[34,183577,183579],{"className":183578},[31646],[31648,183580,183581],{"xmlns":31650},[31652,183582,183583,183598],{},[31655,183584,183585,183587,183589,183591,183593,183596],{},[31749,183586,5731],{},[31662,183588,42071],{},[31749,183590,182485],{},[31662,183592,202],{},[31749,183594,183595],{},"655360",[31658,183597,54224],{},[31665,183599,183600],{"encoding":31667},"40 \\times 16384 = 655360B",[34,183602,183604,183622,183640],{"className":183603,"ariaHidden":758},[31672],[34,183605,183607,183610,183613,183616,183619],{"className":183606},[31676],[34,183608],{"className":183609,"style":35942},[31680],[34,183611,5731],{"className":183612},[31685],[34,183614],{"className":183615,"style":32604},[31845],[34,183617,42071],{"className":183618},[32024],[34,183620],{"className":183621,"style":32604},[31845],[34,183623,183625,183628,183631,183634,183637],{"className":183624},[31676],[34,183626],{"className":183627,"style":32617},[31680],[34,183629,182485],{"className":183630},[31685],[34,183632],{"className":183633,"style":31846},[31845],[34,183635,202],{"className":183636},[31850],[34,183638],{"className":183639,"style":31846},[31845],[34,183641,183643,183646,183649],{"className":183642},[31676],[34,183644],{"className":183645,"style":33218},[31680],[34,183647,183595],{"className":183648},[31685],[34,183650,54224],{"className":183651,"style":54252},[31685,31686],[15,183653,182687],{},[15,183655,183656],{},[34,183657,183659,183684],{"className":183658},[31642],[34,183660,183662],{"className":183661},[31646],[31648,183663,183664],{"xmlns":31650},[31652,183665,183666,183681],{},[31655,183667,183668,183670,183672,183674,183676,183679],{},[31749,183669,183595],{},[31662,183671,32226],{},[31749,183673,182284],{},[31662,183675,202],{},[31749,183677,183678],{},"15360",[31658,183680,54224],{},[31665,183682,183683],{"encoding":31667},"655360 - 640000 = 15360B",[34,183685,183687,183705,183723],{"className":183686,"ariaHidden":758},[31672],[34,183688,183690,183693,183696,183699,183702],{"className":183689},[31676],[34,183691],{"className":183692,"style":35942},[31680],[34,183694,183595],{"className":183695},[31685],[34,183697],{"className":183698,"style":32604},[31845],[34,183700,32226],{"className":183701},[32024],[34,183703],{"className":183704,"style":32604},[31845],[34,183706,183708,183711,183714,183717,183720],{"className":183707},[31676],[34,183709],{"className":183710,"style":32617},[31680],[34,183712,182284],{"className":183713},[31685],[34,183715],{"className":183716,"style":31846},[31845],[34,183718,202],{"className":183719},[31850],[34,183721],{"className":183722,"style":31846},[31845],[34,183724,183726,183729,183732],{"className":183725},[31676],[34,183727],{"className":183728,"style":33218},[31680],[34,183730,183678],{"className":183731},[31685],[34,183733,54224],{"className":183734,"style":54252},[31685,31686],[15,183736,182746],{},[15,183738,183739],{},[34,183740,183742,183773],{"className":183741},[31642],[34,183743,183745],{"className":183744},[31646],[31648,183746,183747],{"xmlns":31650},[31652,183748,183749,183770],{},[31655,183750,183751,183757,183759,183761,183763,183765,183768],{},[41453,183752,183753,183755],{},[31749,183754,183678],{},[31749,183756,183595],{},[31662,183758,42071],{},[31749,183760,162173],{},[31658,183762,119426],{"mathvariant":38079},[31662,183764,168560],{},[31749,183766,183767],{},"2.34",[31658,183769,119426],{"mathvariant":38079},[31665,183771,183772],{"encoding":31667},"\\frac{15360}{655360} \\times 100\\% \\approx 2.34\\%",[34,183774,183776,183859,183877],{"className":183775,"ariaHidden":758},[31672],[34,183777,183779,183782,183850,183853,183856],{"className":183778},[31676],[34,183780],{"className":183781,"style":46727},[31680],[34,183783,183785,183788,183847],{"className":183784},[31685],[34,183786],{"className":183787},[31782,41488],[34,183789,183791],{"className":183790},[41453],[34,183792,183794,183839],{"className":183793},[31796,31797],[34,183795,183797,183836],{"className":183796},[31801],[34,183798,183800,183814,183822],{"className":183799,"style":46746},[31805],[34,183801,183802,183805],{"style":41504},[34,183803],{"className":183804,"style":41508},[31813],[34,183806,183808],{"className":183807},[31818,31819,31820,31821],[34,183809,183811],{"className":183810},[31685,31821],[34,183812,183595],{"className":183813},[31685,31821],[34,183815,183816,183819],{"style":41520},[34,183817],{"className":183818,"style":41508},[31813],[34,183820],{"className":183821,"style":41528},[41527],[34,183823,183824,183827],{"style":46771},[34,183825],{"className":183826,"style":41508},[31813],[34,183828,183830],{"className":183829},[31818,31819,31820,31821],[34,183831,183833],{"className":183832},[31685,31821],[34,183834,183678],{"className":183835},[31685,31821],[34,183837,31829],{"className":183838},[31828],[34,183840,183842],{"className":183841},[31801],[34,183843,183845],{"className":183844,"style":41619},[31805],[34,183846],{},[34,183848],{"className":183849},[31690,41488],[34,183851],{"className":183852,"style":32604},[31845],[34,183854,42071],{"className":183855},[32024],[34,183857],{"className":183858,"style":32604},[31845],[34,183860,183862,183865,183868,183871,183874],{"className":183861},[31676],[34,183863],{"className":183864,"style":168117},[31680],[34,183866,168121],{"className":183867},[31685],[34,183869],{"className":183870,"style":31846},[31845],[34,183872,168560],{"className":183873},[31850],[34,183875],{"className":183876,"style":31846},[31845],[34,183878,183880,183883],{"className":183879},[31676],[34,183881],{"className":183882,"style":168117},[31680],[34,183884,183886],{"className":183885},[31685],"2.34%",[15,183888,58173],{},[15,183890,183891],{},[34,183892,183894,183920],{"className":183893},[31642],[34,183895,183897],{"className":183896},[31646],[31648,183898,183899],{"xmlns":31650},[31652,183900,183901,183917],{},[31655,183902,183903],{},[126400,183904,183905],{"notation":126402},[71090,183906,183907],{"scriptlevel":459,"displaystyle":702},[71090,183908,183909],{"scriptlevel":459,"displaystyle":702},[71090,183910,183911],{"scriptlevel":459,"displaystyle":758},[31655,183912,183913,183915],{},[31749,183914,183767],{},[31658,183916,119426],{"mathvariant":38079},[31665,183918,183919],{"encoding":31667},"\\boxed{2.34\\%}",[34,183921,183923],{"className":183922,"ariaHidden":758},[31672],[34,183924,183926,183929],{"className":183925},[31676],[34,183927],{"className":183928,"style":183374},[31680],[34,183930,183932],{"className":183931},[31685],[34,183933,183935,183969],{"className":183934},[31796,31797],[34,183936,183938,183966],{"className":183937},[31801],[34,183939,183941,183958],{"className":183940,"style":130031},[31805],[34,183942,183943,183946],{"style":183389},[34,183944],{"className":183945,"style":183393},[31813],[34,183947,183949],{"className":183948},[126448],[34,183950,183952],{"className":183951},[31685],[34,183953,183955],{"className":183954},[31685],[34,183956,183886],{"className":183957},[31685],[34,183959,183960,183963],{"style":130074},[34,183961],{"className":183962,"style":183393},[31813],[34,183964],{"className":183965,"style":183414},[126470,126471],[34,183967,31829],{"className":183968},[31828],[34,183970,183972],{"className":183971},[31801],[34,183973,183975],{"className":183974,"style":183424},[31805],[34,183976],{},[32188,183978],{},[253,183980,183982,183983],{"id":183981},"_33-文件大小为-4064000b4064000b4064000b","3.3 文件大小为 ",[34,183984,183986,184001],{"className":183985},[31642],[34,183987,183989],{"className":183988},[31646],[31648,183990,183991],{"xmlns":31650},[31652,183992,183993,183999],{},[31655,183994,183995,183997],{},[31749,183996,182321],{},[31658,183998,54224],{},[31665,184000,182326],{"encoding":31667},[34,184002,184004],{"className":184003,"ariaHidden":758},[31672],[34,184005,184007,184010,184013],{"className":184006},[31676],[34,184008],{"className":184009,"style":33218},[31680],[34,184011,182321],{"className":184012},[31685],[34,184014,54224],{"className":184015,"style":54252},[31685,31686],[15,184017,184018],{},"先算：",[15,184020,184021],{},[34,184022,184024,184050],{"className":184023},[31642],[34,184025,184027],{"className":184026},[31646],[31648,184028,184029],{"xmlns":31650},[31652,184030,184031,184047],{},[31655,184032,184033,184035,184037,184040,184042,184045],{},[31749,184034,182485],{},[31662,184036,42071],{},[31749,184038,184039],{},"248",[31662,184041,202],{},[31749,184043,184044],{},"4063232",[31658,184046,54224],{},[31665,184048,184049],{"encoding":31667},"16384 \\times 248 = 4063232B",[34,184051,184053,184071,184089],{"className":184052,"ariaHidden":758},[31672],[34,184054,184056,184059,184062,184065,184068],{"className":184055},[31676],[34,184057],{"className":184058,"style":35942},[31680],[34,184060,182485],{"className":184061},[31685],[34,184063],{"className":184064,"style":32604},[31845],[34,184066,42071],{"className":184067},[32024],[34,184069],{"className":184070,"style":32604},[31845],[34,184072,184074,184077,184080,184083,184086],{"className":184073},[31676],[34,184075],{"className":184076,"style":32617},[31680],[34,184078,184039],{"className":184079},[31685],[34,184081],{"className":184082,"style":31846},[31845],[34,184084,202],{"className":184085},[31850],[34,184087],{"className":184088,"style":31846},[31845],[34,184090,184092,184095,184098],{"className":184091},[31676],[34,184093],{"className":184094,"style":33218},[31680],[34,184096,184044],{"className":184097},[31685],[34,184099,54224],{"className":184100,"style":54252},[31685,31686],[15,184102,174363],{},[15,184104,184105],{},[34,184106,184108,184133],{"className":184107},[31642],[34,184109,184111],{"className":184110},[31646],[31648,184112,184113],{"xmlns":31650},[31652,184114,184115,184130],{},[31655,184116,184117,184119,184121,184123,184125,184128],{},[31749,184118,182321],{},[31662,184120,32226],{},[31749,184122,184044],{},[31662,184124,202],{},[31749,184126,184127],{},"768",[31658,184129,54224],{},[31665,184131,184132],{"encoding":31667},"4064000 - 4063232 = 768B",[34,184134,184136,184154,184172],{"className":184135,"ariaHidden":758},[31672],[34,184137,184139,184142,184145,184148,184151],{"className":184138},[31676],[34,184140],{"className":184141,"style":35942},[31680],[34,184143,182321],{"className":184144},[31685],[34,184146],{"className":184147,"style":32604},[31845],[34,184149,32226],{"className":184150},[32024],[34,184152],{"className":184153,"style":32604},[31845],[34,184155,184157,184160,184163,184166,184169],{"className":184156},[31676],[34,184158],{"className":184159,"style":32617},[31680],[34,184161,184044],{"className":184162},[31685],[34,184164],{"className":184165,"style":31846},[31845],[34,184167,202],{"className":184168},[31850],[34,184170],{"className":184171,"style":31846},[31845],[34,184173,184175,184178,184181],{"className":184174},[31676],[34,184176],{"className":184177,"style":33218},[31680],[34,184179,184127],{"className":184180},[31685],[34,184182,54224],{"className":184183,"style":54252},[31685,31686],[15,184185,184186],{},"所以需要再多一个块。",[15,184188,182570],{},[15,184190,184191],{},[34,184192,184194,184208],{"className":184193},[31642],[34,184195,184197],{"className":184196},[31646],[31648,184198,184199],{"xmlns":31650},[31652,184200,184201,184206],{},[31655,184202,184203],{},[31749,184204,184205],{},"249",[31665,184207,184205],{"encoding":31667},[34,184209,184211],{"className":184210,"ariaHidden":758},[31672],[34,184212,184214,184217],{"className":184213},[31676],[34,184215],{"className":184216,"style":32617},[31680],[34,184218,184205],{"className":184219},[31685],[15,184221,182628],{},[15,184223,184224],{},[34,184225,184227,184252],{"className":184226},[31642],[34,184228,184230],{"className":184229},[31646],[31648,184231,184232],{"xmlns":31650},[31652,184233,184234,184249],{},[31655,184235,184236,184238,184240,184242,184244,184247],{},[31749,184237,184205],{},[31662,184239,42071],{},[31749,184241,182485],{},[31662,184243,202],{},[31749,184245,184246],{},"4079616",[31658,184248,54224],{},[31665,184250,184251],{"encoding":31667},"249 \\times 16384 = 4079616B",[34,184253,184255,184273,184291],{"className":184254,"ariaHidden":758},[31672],[34,184256,184258,184261,184264,184267,184270],{"className":184257},[31676],[34,184259],{"className":184260,"style":35942},[31680],[34,184262,184205],{"className":184263},[31685],[34,184265],{"className":184266,"style":32604},[31845],[34,184268,42071],{"className":184269},[32024],[34,184271],{"className":184272,"style":32604},[31845],[34,184274,184276,184279,184282,184285,184288],{"className":184275},[31676],[34,184277],{"className":184278,"style":32617},[31680],[34,184280,182485],{"className":184281},[31685],[34,184283],{"className":184284,"style":31846},[31845],[34,184286,202],{"className":184287},[31850],[34,184289],{"className":184290,"style":31846},[31845],[34,184292,184294,184297,184300],{"className":184293},[31676],[34,184295],{"className":184296,"style":33218},[31680],[34,184298,184246],{"className":184299},[31685],[34,184301,54224],{"className":184302,"style":54252},[31685,31686],[15,184304,182687],{},[15,184306,184307],{},[34,184308,184310,184335],{"className":184309},[31642],[34,184311,184313],{"className":184312},[31646],[31648,184314,184315],{"xmlns":31650},[31652,184316,184317,184332],{},[31655,184318,184319,184321,184323,184325,184327,184330],{},[31749,184320,184246],{},[31662,184322,32226],{},[31749,184324,182321],{},[31662,184326,202],{},[31749,184328,184329],{},"15616",[31658,184331,54224],{},[31665,184333,184334],{"encoding":31667},"4079616 - 4064000 = 15616B",[34,184336,184338,184356,184374],{"className":184337,"ariaHidden":758},[31672],[34,184339,184341,184344,184347,184350,184353],{"className":184340},[31676],[34,184342],{"className":184343,"style":35942},[31680],[34,184345,184246],{"className":184346},[31685],[34,184348],{"className":184349,"style":32604},[31845],[34,184351,32226],{"className":184352},[32024],[34,184354],{"className":184355,"style":32604},[31845],[34,184357,184359,184362,184365,184368,184371],{"className":184358},[31676],[34,184360],{"className":184361,"style":32617},[31680],[34,184363,182321],{"className":184364},[31685],[34,184366],{"className":184367,"style":31846},[31845],[34,184369,202],{"className":184370},[31850],[34,184372],{"className":184373,"style":31846},[31845],[34,184375,184377,184380,184383],{"className":184376},[31676],[34,184378],{"className":184379,"style":33218},[31680],[34,184381,184329],{"className":184382},[31685],[34,184384,54224],{"className":184385,"style":54252},[31685,31686],[15,184387,182746],{},[15,184389,184390],{},[34,184391,184393,184424],{"className":184392},[31642],[34,184394,184396],{"className":184395},[31646],[31648,184397,184398],{"xmlns":31650},[31652,184399,184400,184421],{},[31655,184401,184402,184408,184410,184412,184414,184416,184419],{},[41453,184403,184404,184406],{},[31749,184405,184329],{},[31749,184407,184246],{},[31662,184409,42071],{},[31749,184411,162173],{},[31658,184413,119426],{"mathvariant":38079},[31662,184415,168560],{},[31749,184417,184418],{},"0.383",[31658,184420,119426],{"mathvariant":38079},[31665,184422,184423],{"encoding":31667},"\\frac{15616}{4079616} \\times 100\\% \\approx 0.383\\%",[34,184425,184427,184510,184528],{"className":184426,"ariaHidden":758},[31672],[34,184428,184430,184433,184501,184504,184507],{"className":184429},[31676],[34,184431],{"className":184432,"style":46727},[31680],[34,184434,184436,184439,184498],{"className":184435},[31685],[34,184437],{"className":184438},[31782,41488],[34,184440,184442],{"className":184441},[41453],[34,184443,184445,184490],{"className":184444},[31796,31797],[34,184446,184448,184487],{"className":184447},[31801],[34,184449,184451,184465,184473],{"className":184450,"style":46746},[31805],[34,184452,184453,184456],{"style":41504},[34,184454],{"className":184455,"style":41508},[31813],[34,184457,184459],{"className":184458},[31818,31819,31820,31821],[34,184460,184462],{"className":184461},[31685,31821],[34,184463,184246],{"className":184464},[31685,31821],[34,184466,184467,184470],{"style":41520},[34,184468],{"className":184469,"style":41508},[31813],[34,184471],{"className":184472,"style":41528},[41527],[34,184474,184475,184478],{"style":46771},[34,184476],{"className":184477,"style":41508},[31813],[34,184479,184481],{"className":184480},[31818,31819,31820,31821],[34,184482,184484],{"className":184483},[31685,31821],[34,184485,184329],{"className":184486},[31685,31821],[34,184488,31829],{"className":184489},[31828],[34,184491,184493],{"className":184492},[31801],[34,184494,184496],{"className":184495,"style":41619},[31805],[34,184497],{},[34,184499],{"className":184500},[31690,41488],[34,184502],{"className":184503,"style":32604},[31845],[34,184505,42071],{"className":184506},[32024],[34,184508],{"className":184509,"style":32604},[31845],[34,184511,184513,184516,184519,184522,184525],{"className":184512},[31676],[34,184514],{"className":184515,"style":168117},[31680],[34,184517,168121],{"className":184518},[31685],[34,184520],{"className":184521,"style":31846},[31845],[34,184523,168560],{"className":184524},[31850],[34,184526],{"className":184527,"style":31846},[31845],[34,184529,184531,184534],{"className":184530},[31676],[34,184532],{"className":184533,"style":168117},[31680],[34,184535,184537],{"className":184536},[31685],"0.383%",[15,184539,58173],{},[15,184541,184542],{},[34,184543,184545,184571],{"className":184544},[31642],[34,184546,184548],{"className":184547},[31646],[31648,184549,184550],{"xmlns":31650},[31652,184551,184552,184568],{},[31655,184553,184554],{},[126400,184555,184556],{"notation":126402},[71090,184557,184558],{"scriptlevel":459,"displaystyle":702},[71090,184559,184560],{"scriptlevel":459,"displaystyle":702},[71090,184561,184562],{"scriptlevel":459,"displaystyle":758},[31655,184563,184564,184566],{},[31749,184565,184418],{},[31658,184567,119426],{"mathvariant":38079},[31665,184569,184570],{"encoding":31667},"\\boxed{0.383\\%}",[34,184572,184574],{"className":184573,"ariaHidden":758},[31672],[34,184575,184577,184580],{"className":184576},[31676],[34,184578],{"className":184579,"style":183374},[31680],[34,184581,184583],{"className":184582},[31685],[34,184584,184586,184620],{"className":184585},[31796,31797],[34,184587,184589,184617],{"className":184588},[31801],[34,184590,184592,184609],{"className":184591,"style":130031},[31805],[34,184593,184594,184597],{"style":183389},[34,184595],{"className":184596,"style":183393},[31813],[34,184598,184600],{"className":184599},[126448],[34,184601,184603],{"className":184602},[31685],[34,184604,184606],{"className":184605},[31685],[34,184607,184537],{"className":184608},[31685],[34,184610,184611,184614],{"style":130074},[34,184612],{"className":184613,"style":183393},[31813],[34,184615],{"className":184616,"style":183414},[126470,126471],[34,184618,31829],{"className":184619},[31828],[34,184621,184623],{"className":184622},[31801],[34,184624,184626],{"className":184625,"style":183424},[31805],[34,184627],{},[32188,184629],{},[253,184631,169906],{"id":169905},[15,184633,184634,2664],{},[34,184635,184637,184652],{"className":184636},[31642],[34,184638,184640],{"className":184639},[31646],[31648,184641,184642],{"xmlns":31650},[31652,184643,184644,184650],{},[31655,184645,184646,184648],{},[31749,184647,182247],{},[31658,184649,54224],{},[31665,184651,182252],{"encoding":31667},[34,184653,184655],{"className":184654,"ariaHidden":758},[31672],[34,184656,184658,184661,184664],{"className":184657},[31676],[34,184659],{"className":184660,"style":33218},[31680],[34,184662,182247],{"className":184663},[31685],[34,184665,54224],{"className":184666,"style":54252},[31685,31686],[15,184668,184669],{},[34,184670,184672,184697],{"className":184671},[31642],[34,184673,184675],{"className":184674},[31646],[31648,184676,184677],{"xmlns":31650},[31652,184678,184679,184695],{},[31655,184680,184681],{},[126400,184682,184683],{"notation":126402},[71090,184684,184685],{"scriptlevel":459,"displaystyle":702},[71090,184686,184687],{"scriptlevel":459,"displaystyle":702},[71090,184688,184689],{"scriptlevel":459,"displaystyle":758},[31655,184690,184691,184693],{},[31749,184692,183212],{},[31658,184694,119426],{"mathvariant":38079},[31665,184696,183364],{"encoding":31667},[34,184698,184700],{"className":184699,"ariaHidden":758},[31672],[34,184701,184703,184706],{"className":184702},[31676],[34,184704],{"className":184705,"style":183374},[31680],[34,184707,184709],{"className":184708},[31685],[34,184710,184712,184746],{"className":184711},[31796,31797],[34,184713,184715,184743],{"className":184714},[31801],[34,184716,184718,184735],{"className":184717,"style":130031},[31805],[34,184719,184720,184723],{"style":183389},[34,184721],{"className":184722,"style":183393},[31813],[34,184724,184726],{"className":184725},[126448],[34,184727,184729],{"className":184728},[31685],[34,184730,184732],{"className":184731},[31685],[34,184733,183331],{"className":184734},[31685],[34,184736,184737,184740],{"style":130074},[34,184738],{"className":184739,"style":183393},[31813],[34,184741],{"className":184742,"style":183414},[126470,126471],[34,184744,31829],{"className":184745},[31828],[34,184747,184749],{"className":184748},[31801],[34,184750,184752],{"className":184751,"style":183424},[31805],[34,184753],{},[15,184755,184756,2664],{},[34,184757,184759,184774],{"className":184758},[31642],[34,184760,184762],{"className":184761},[31646],[31648,184763,184764],{"xmlns":31650},[31652,184765,184766,184772],{},[31655,184767,184768,184770],{},[31749,184769,182284],{},[31658,184771,54224],{},[31665,184773,182289],{"encoding":31667},[34,184775,184777],{"className":184776,"ariaHidden":758},[31672],[34,184778,184780,184783,184786],{"className":184779},[31676],[34,184781],{"className":184782,"style":33218},[31680],[34,184784,182284],{"className":184785},[31685],[34,184787,54224],{"className":184788,"style":54252},[31685,31686],[15,184790,184791],{},[34,184792,184794,184819],{"className":184793},[31642],[34,184795,184797],{"className":184796},[31646],[31648,184798,184799],{"xmlns":31650},[31652,184800,184801,184817],{},[31655,184802,184803],{},[126400,184804,184805],{"notation":126402},[71090,184806,184807],{"scriptlevel":459,"displaystyle":702},[71090,184808,184809],{"scriptlevel":459,"displaystyle":702},[71090,184810,184811],{"scriptlevel":459,"displaystyle":758},[31655,184812,184813,184815],{},[31749,184814,183767],{},[31658,184816,119426],{"mathvariant":38079},[31665,184818,183919],{"encoding":31667},[34,184820,184822],{"className":184821,"ariaHidden":758},[31672],[34,184823,184825,184828],{"className":184824},[31676],[34,184826],{"className":184827,"style":183374},[31680],[34,184829,184831],{"className":184830},[31685],[34,184832,184834,184868],{"className":184833},[31796,31797],[34,184835,184837,184865],{"className":184836},[31801],[34,184838,184840,184857],{"className":184839,"style":130031},[31805],[34,184841,184842,184845],{"style":183389},[34,184843],{"className":184844,"style":183393},[31813],[34,184846,184848],{"className":184847},[126448],[34,184849,184851],{"className":184850},[31685],[34,184852,184854],{"className":184853},[31685],[34,184855,183886],{"className":184856},[31685],[34,184858,184859,184862],{"style":130074},[34,184860],{"className":184861,"style":183393},[31813],[34,184863],{"className":184864,"style":183414},[126470,126471],[34,184866,31829],{"className":184867},[31828],[34,184869,184871],{"className":184870},[31801],[34,184872,184874],{"className":184873,"style":183424},[31805],[34,184875],{},[15,184877,184878,2664],{},[34,184879,184881,184896],{"className":184880},[31642],[34,184882,184884],{"className":184883},[31646],[31648,184885,184886],{"xmlns":31650},[31652,184887,184888,184894],{},[31655,184889,184890,184892],{},[31749,184891,182321],{},[31658,184893,54224],{},[31665,184895,182326],{"encoding":31667},[34,184897,184899],{"className":184898,"ariaHidden":758},[31672],[34,184900,184902,184905,184908],{"className":184901},[31676],[34,184903],{"className":184904,"style":33218},[31680],[34,184906,182321],{"className":184907},[31685],[34,184909,54224],{"className":184910,"style":54252},[31685,31686],[15,184912,184913],{},[34,184914,184916,184941],{"className":184915},[31642],[34,184917,184919],{"className":184918},[31646],[31648,184920,184921],{"xmlns":31650},[31652,184922,184923,184939],{},[31655,184924,184925],{},[126400,184926,184927],{"notation":126402},[71090,184928,184929],{"scriptlevel":459,"displaystyle":702},[71090,184930,184931],{"scriptlevel":459,"displaystyle":702},[71090,184932,184933],{"scriptlevel":459,"displaystyle":758},[31655,184934,184935,184937],{},[31749,184936,184418],{},[31658,184938,119426],{"mathvariant":38079},[31665,184940,184570],{"encoding":31667},[34,184942,184944],{"className":184943,"ariaHidden":758},[31672],[34,184945,184947,184950],{"className":184946},[31676],[34,184948],{"className":184949,"style":183374},[31680],[34,184951,184953],{"className":184952},[31685],[34,184954,184956,184990],{"className":184955},[31796,31797],[34,184957,184959,184987],{"className":184958},[31801],[34,184960,184962,184979],{"className":184961,"style":130031},[31805],[34,184963,184964,184967],{"style":183389},[34,184965],{"className":184966,"style":183393},[31813],[34,184968,184970],{"className":184969},[126448],[34,184971,184973],{"className":184972},[31685],[34,184974,184976],{"className":184975},[31685],[34,184977,184537],{"className":184978},[31685],[34,184980,184981,184984],{"style":130074},[34,184982],{"className":184983,"style":183393},[31813],[34,184985],{"className":184986,"style":183414},[126470,126471],[34,184988,31829],{"className":184989},[31828],[34,184991,184993],{"className":184992},[31801],[34,184994,184996],{"className":184995,"style":183424},[31805],[34,184997],{},[32188,184999],{},[19,185001,185003],{"id":185002},"_4-磁盘组柱面数总存储量平均等待时间","4. 磁盘组柱面数、总存储量、平均等待时间",[253,185005,162043],{"id":170039},[15,185007,185008],{},"某磁盘组有6片盘片，每片有两个记录面，存储区域内径为22cm，外径为33cm，道存储密度为40道\u002Fcm，内层位存储密度为400b\u002Fcm，转速为3000r\u002Fmin（转\u002F分），问共有多少柱面？盘组总存储量为多少？平均等待时间为多少？",[32188,185010],{},[253,185012,175302],{"id":185013},"答案和讲解-3",[15,185015,173129],{},[15,185017,185018],{},"盘片数：",[15,185020,185021],{},[34,185022,185024,185037],{"className":185023},[31642],[34,185025,185027],{"className":185026},[31646],[31648,185028,185029],{"xmlns":31650},[31652,185030,185031,185035],{},[31655,185032,185033],{},[31749,185034,36656],{},[31665,185036,36656],{"encoding":31667},[34,185038,185040],{"className":185039,"ariaHidden":758},[31672],[34,185041,185043,185046],{"className":185042},[31676],[34,185044],{"className":185045,"style":32617},[31680],[34,185047,36656],{"className":185048},[31685],[15,185050,185051],{},"每片有两个记录面，所以记录面总数为：",[15,185053,185054],{},[34,185055,185057,185079],{"className":185056},[31642],[34,185058,185060],{"className":185059},[31646],[31648,185061,185062],{"xmlns":31650},[31652,185063,185064,185076],{},[31655,185065,185066,185068,185070,185072,185074],{},[31749,185067,36656],{},[31662,185069,42071],{},[31749,185071,768],{},[31662,185073,202],{},[31749,185075,71869],{},[31665,185077,185078],{"encoding":31667},"6 \\times 2 = 12",[34,185080,185082,185100,185118],{"className":185081,"ariaHidden":758},[31672],[34,185083,185085,185088,185091,185094,185097],{"className":185084},[31676],[34,185086],{"className":185087,"style":35942},[31680],[34,185089,36656],{"className":185090},[31685],[34,185092],{"className":185093,"style":32604},[31845],[34,185095,42071],{"className":185096},[32024],[34,185098],{"className":185099,"style":32604},[31845],[34,185101,185103,185106,185109,185112,185115],{"className":185102},[31676],[34,185104],{"className":185105,"style":32617},[31680],[34,185107,768],{"className":185108},[31685],[34,185110],{"className":185111,"style":31846},[31845],[34,185113,202],{"className":185114},[31850],[34,185116],{"className":185117,"style":31846},[31845],[34,185119,185121,185124],{"className":185120},[31676],[34,185122],{"className":185123,"style":32617},[31680],[34,185125,71869],{"className":185126},[31685],[15,185128,185129],{},"存储区域内径：",[15,185131,185132],{},[34,185133,185135,185153],{"className":185134},[31642],[34,185136,185138],{"className":185137},[31646],[31648,185139,185140],{"xmlns":31650},[31652,185141,185142,185150],{},[31655,185143,185144,185146,185148],{},[31749,185145,94716],{},[31658,185147,31965],{},[31658,185149,31759],{},[31665,185151,185152],{"encoding":31667},"22cm",[34,185154,185156],{"className":185155,"ariaHidden":758},[31672],[34,185157,185159,185162,185165,185168],{"className":185158},[31676],[34,185160],{"className":185161,"style":32617},[31680],[34,185163,94716],{"className":185164},[31685],[34,185166,31965],{"className":185167},[31685,31686],[34,185169,31759],{"className":185170},[31685,31686],[15,185172,185173],{},"存储区域外径：",[15,185175,185176],{},[34,185177,185179,185198],{"className":185178},[31642],[34,185180,185182],{"className":185181},[31646],[31648,185183,185184],{"xmlns":31650},[31652,185185,185186,185195],{},[31655,185187,185188,185191,185193],{},[31749,185189,185190],{},"33",[31658,185192,31965],{},[31658,185194,31759],{},[31665,185196,185197],{"encoding":31667},"33cm",[34,185199,185201],{"className":185200,"ariaHidden":758},[31672],[34,185202,185204,185207,185210,185213],{"className":185203},[31676],[34,185205],{"className":185206,"style":32617},[31680],[34,185208,185190],{"className":185209},[31685],[34,185211,31965],{"className":185212},[31685,31686],[34,185214,31759],{"className":185215},[31685,31686],[15,185217,185218],{},"道存储密度：",[15,185220,185221],{},[34,185222,185224,185247],{"className":185223},[31642],[34,185225,185227],{"className":185226},[31646],[31648,185228,185229],{"xmlns":31650},[31652,185230,185231,185244],{},[31655,185232,185233,185235,185238,185240,185242],{},[31749,185234,5731],{},[39361,185236,185237],{}," 道",[31658,185239,38080],{"mathvariant":38079},[31658,185241,31965],{},[31658,185243,31759],{},[31665,185245,185246],{"encoding":31667},"40 \\text{ 道}\u002Fcm",[34,185248,185250],{"className":185249,"ariaHidden":758},[31672],[34,185251,185253,185256,185259,185269,185272,185275],{"className":185252},[31676],[34,185254],{"className":185255,"style":31774},[31680],[34,185257,5731],{"className":185258},[31685],[34,185260,185262,185265],{"className":185261},[31685,2638],[34,185263,130298],{"className":185264},[31685],[34,185266,185268],{"className":185267},[31685,105462],"道",[34,185270,38080],{"className":185271},[31685],[34,185273,31965],{"className":185274},[31685,31686],[34,185276,31759],{"className":185277},[31685,31686],[15,185279,185280],{},"内层位存储密度：",[15,185282,185283],{},[34,185284,185286,185309],{"className":185285},[31642],[34,185287,185289],{"className":185288},[31646],[31648,185290,185291],{"xmlns":31650},[31652,185292,185293,185306],{},[31655,185294,185295,185298,185300,185302,185304],{},[31749,185296,185297],{},"400",[31658,185299,1697],{},[31658,185301,38080],{"mathvariant":38079},[31658,185303,31965],{},[31658,185305,31759],{},[31665,185307,185308],{"encoding":31667},"400b\u002Fcm",[34,185310,185312],{"className":185311,"ariaHidden":758},[31672],[34,185313,185315,185318,185321,185324,185327,185330],{"className":185314},[31676],[34,185316],{"className":185317,"style":31774},[31680],[34,185319,185297],{"className":185320},[31685],[34,185322,1697],{"className":185323},[31685,31686],[34,185325,38080],{"className":185326},[31685],[34,185328,31965],{"className":185329},[31685,31686],[34,185331,31759],{"className":185332},[31685,31686],[15,185334,185335],{},"转速：",[15,185337,185338],{},[34,185339,185341,185365],{"className":185340},[31642],[34,185342,185344],{"className":185343},[31646],[31648,185345,185346],{"xmlns":31650},[31652,185347,185348,185362],{},[31655,185349,185350,185352,185354,185356,185358,185360],{},[31749,185351,74728],{},[31658,185353,34395],{},[31658,185355,38080],{"mathvariant":38079},[31658,185357,31759],{},[31658,185359,31955],{},[31658,185361,31660],{},[31665,185363,185364],{"encoding":31667},"3000r\u002Fmin",[34,185366,185368],{"className":185367,"ariaHidden":758},[31672],[34,185369,185371,185374,185377,185380,185383],{"className":185370},[31676],[34,185372],{"className":185373,"style":31774},[31680],[34,185375,74728],{"className":185376},[31685],[34,185378,34395],{"className":185379,"style":34452},[31685,31686],[34,185381,38080],{"className":185382},[31685],[34,185384,56084],{"className":185385},[31685,31686],[32188,185387],{},[253,185389,185391],{"id":185390},"_41-共有多少柱面","4.1 共有多少柱面？",[15,185393,185394],{},"柱面数等于每个记录面上的磁道数。",[15,185396,185397],{},"注意题目给的是内径和外径，不是半径。",[15,185399,185400],{},"所以先求半径：",[15,185402,185403],{},"内半径：",[15,185405,185406],{},[34,185407,185409,185435],{"className":185408},[31642],[34,185410,185412],{"className":185411},[31646],[31648,185413,185414],{"xmlns":31650},[31652,185415,185416,185432],{},[31655,185417,185418,185420,185422,185424,185426,185428,185430],{},[31749,185419,94716],{},[31658,185421,38080],{"mathvariant":38079},[31749,185423,768],{},[31662,185425,202],{},[31749,185427,39620],{},[31658,185429,31965],{},[31658,185431,31759],{},[31665,185433,185434],{"encoding":31667},"22 \u002F 2 = 11cm",[34,185436,185438,185457],{"className":185437,"ariaHidden":758},[31672],[34,185439,185441,185444,185448,185451,185454],{"className":185440},[31676],[34,185442],{"className":185443,"style":31774},[31680],[34,185445,185447],{"className":185446},[31685],"22\u002F2",[34,185449],{"className":185450,"style":31846},[31845],[34,185452,202],{"className":185453},[31850],[34,185455],{"className":185456,"style":31846},[31845],[34,185458,185460,185463,185466,185469],{"className":185459},[31676],[34,185461],{"className":185462,"style":32617},[31680],[34,185464,39620],{"className":185465},[31685],[34,185467,31965],{"className":185468},[31685,31686],[34,185470,31759],{"className":185471},[31685,31686],[15,185473,185474],{},"外半径：",[15,185476,185477],{},[34,185478,185480,185507],{"className":185479},[31642],[34,185481,185483],{"className":185482},[31646],[31648,185484,185485],{"xmlns":31650},[31652,185486,185487,185504],{},[31655,185488,185489,185491,185493,185495,185497,185500,185502],{},[31749,185490,185190],{},[31658,185492,38080],{"mathvariant":38079},[31749,185494,768],{},[31662,185496,202],{},[31749,185498,185499],{},"16.5",[31658,185501,31965],{},[31658,185503,31759],{},[31665,185505,185506],{"encoding":31667},"33 \u002F 2 = 16.5cm",[34,185508,185510,185529],{"className":185509,"ariaHidden":758},[31672],[34,185511,185513,185516,185520,185523,185526],{"className":185512},[31676],[34,185514],{"className":185515,"style":31774},[31680],[34,185517,185519],{"className":185518},[31685],"33\u002F2",[34,185521],{"className":185522,"style":31846},[31845],[34,185524,202],{"className":185525},[31850],[34,185527],{"className":185528,"style":31846},[31845],[34,185530,185532,185535,185538,185541],{"className":185531},[31676],[34,185533],{"className":185534,"style":32617},[31680],[34,185536,185499],{"className":185537},[31685],[34,185539,31965],{"className":185540},[31685,31686],[34,185542,31759],{"className":185543},[31685,31686],[15,185545,185546],{},"存储区域宽度：",[15,185548,185549],{},[34,185550,185552,185579],{"className":185551},[31642],[34,185553,185555],{"className":185554},[31646],[31648,185556,185557],{"xmlns":31650},[31652,185558,185559,185576],{},[31655,185560,185561,185563,185565,185567,185569,185572,185574],{},[31749,185562,185499],{},[31662,185564,32226],{},[31749,185566,39620],{},[31662,185568,202],{},[31749,185570,185571],{},"5.5",[31658,185573,31965],{},[31658,185575,31759],{},[31665,185577,185578],{"encoding":31667},"16.5 - 11 = 5.5cm",[34,185580,185582,185600,185618],{"className":185581,"ariaHidden":758},[31672],[34,185583,185585,185588,185591,185594,185597],{"className":185584},[31676],[34,185586],{"className":185587,"style":35942},[31680],[34,185589,185499],{"className":185590},[31685],[34,185592],{"className":185593,"style":32604},[31845],[34,185595,32226],{"className":185596},[32024],[34,185598],{"className":185599,"style":32604},[31845],[34,185601,185603,185606,185609,185612,185615],{"className":185602},[31676],[34,185604],{"className":185605,"style":32617},[31680],[34,185607,39620],{"className":185608},[31685],[34,185610],{"className":185611,"style":31846},[31845],[34,185613,202],{"className":185614},[31850],[34,185616],{"className":185617,"style":31846},[31845],[34,185619,185621,185624,185627,185630],{"className":185620},[31676],[34,185622],{"className":185623,"style":32617},[31680],[34,185625,185571],{"className":185626},[31685],[34,185628,31965],{"className":185629},[31685,31686],[34,185631,31759],{"className":185632},[31685,31686],[15,185634,185635],{},"道密度为：",[15,185637,185638],{},[34,185639,185641,185662],{"className":185640},[31642],[34,185642,185644],{"className":185643},[31646],[31648,185645,185646],{"xmlns":31650},[31652,185647,185648,185660],{},[31655,185649,185650,185652,185654,185656,185658],{},[31749,185651,5731],{},[39361,185653,185237],{},[31658,185655,38080],{"mathvariant":38079},[31658,185657,31965],{},[31658,185659,31759],{},[31665,185661,185246],{"encoding":31667},[34,185663,185665],{"className":185664,"ariaHidden":758},[31672],[34,185666,185668,185671,185674,185683,185686,185689],{"className":185667},[31676],[34,185669],{"className":185670,"style":31774},[31680],[34,185672,5731],{"className":185673},[31685],[34,185675,185677,185680],{"className":185676},[31685,2638],[34,185678,130298],{"className":185679},[31685],[34,185681,185268],{"className":185682},[31685,105462],[34,185684,38080],{"className":185685},[31685],[34,185687,31965],{"className":185688},[31685,31686],[34,185690,31759],{"className":185691},[31685,31686],[15,185693,185694],{},"所以每个记录面上的磁道数为：",[15,185696,185697],{},[34,185698,185700,185723],{"className":185699},[31642],[34,185701,185703],{"className":185702},[31646],[31648,185704,185705],{"xmlns":31650},[31652,185706,185707,185720],{},[31655,185708,185709,185711,185713,185715,185717],{},[31749,185710,185571],{},[31662,185712,42071],{},[31749,185714,5731],{},[31662,185716,202],{},[31749,185718,185719],{},"220",[31665,185721,185722],{"encoding":31667},"5.5 \\times 40 = 220",[34,185724,185726,185744,185762],{"className":185725,"ariaHidden":758},[31672],[34,185727,185729,185732,185735,185738,185741],{"className":185728},[31676],[34,185730],{"className":185731,"style":35942},[31680],[34,185733,185571],{"className":185734},[31685],[34,185736],{"className":185737,"style":32604},[31845],[34,185739,42071],{"className":185740},[32024],[34,185742],{"className":185743,"style":32604},[31845],[34,185745,185747,185750,185753,185756,185759],{"className":185746},[31676],[34,185748],{"className":185749,"style":32617},[31680],[34,185751,5731],{"className":185752},[31685],[34,185754],{"className":185755,"style":31846},[31845],[34,185757,202],{"className":185758},[31850],[34,185760],{"className":185761,"style":31846},[31845],[34,185763,185765,185768],{"className":185764},[31676],[34,185766],{"className":185767,"style":32617},[31680],[34,185769,185719],{"className":185770},[31685],[15,185772,185773],{},"因此柱面数为：",[15,185775,185776],{},[34,185777,185779,185801],{"className":185778},[31642],[34,185780,185782],{"className":185781},[31646],[31648,185783,185784],{"xmlns":31650},[31652,185785,185786,185798],{},[31655,185787,185788],{},[126400,185789,185790],{"notation":126402},[71090,185791,185792],{"scriptlevel":459,"displaystyle":702},[71090,185793,185794],{"scriptlevel":459,"displaystyle":702},[71090,185795,185796],{"scriptlevel":459,"displaystyle":758},[31749,185797,185719],{},[31665,185799,185800],{"encoding":31667},"\\boxed{220}",[34,185802,185804],{"className":185803,"ariaHidden":758},[31672],[34,185805,185807,185811],{"className":185806},[31676],[34,185808],{"className":185809,"style":185810},[31680],"height:1.3244em;vertical-align:-0.34em;",[34,185812,185814],{"className":185813},[31685],[34,185815,185817,185854],{"className":185816},[31796,31797],[34,185818,185820,185851],{"className":185819},[31801],[34,185821,185823,185842],{"className":185822,"style":136916},[31805],[34,185824,185826,185830],{"style":185825},"top:-3.3244em;",[34,185827],{"className":185828,"style":185829},[31813],"height:3.3244em;",[34,185831,185833],{"className":185832},[126448],[34,185834,185836],{"className":185835},[31685],[34,185837,185839],{"className":185838},[31685],[34,185840,185719],{"className":185841},[31685],[34,185843,185844,185847],{"style":136950},[34,185845],{"className":185846,"style":185829},[31813],[34,185848],{"className":185849,"style":185850},[126470,126471],"height:1.3244em;border-style:solid;border-width:0.04em;",[34,185852,31829],{"className":185853},[31828],[34,185855,185857],{"className":185856},[31801],[34,185858,185860],{"className":185859,"style":126482},[31805],[34,185861],{},[32188,185863],{},[253,185865,185867],{"id":185866},"_42-盘组总存储量","4.2 盘组总存储量",[15,185869,185870],{},"题目给的是内层位存储密度。",[15,185872,185873],{},"一般这种题按内层磁道容量作为每条磁道容量来算，因为外层磁道虽然周长更长，但通常为了保持统一格式，按内层容量确定每道容量。",[15,185875,185876],{},"内层磁道周长为：",[15,185878,185879],{},[34,185880,185882,185910],{"className":185881},[31642],[34,185883,185885],{"className":185884},[31646],[31648,185886,185887],{"xmlns":31650},[31652,185888,185889,185907],{},[31655,185890,185891,185893,185895,185897,185899,185901,185903,185905],{},[31658,185892,58442],{},[31662,185894,42071],{},[31749,185896,94716],{},[31662,185898,202],{},[31749,185900,94716],{},[31658,185902,58442],{},[31658,185904,31965],{},[31658,185906,31759],{},[31665,185908,185909],{"encoding":31667},"\\pi \\times 22 = 22\\pi cm",[34,185911,185913,185931,185949],{"className":185912,"ariaHidden":758},[31672],[34,185914,185916,185919,185922,185925,185928],{"className":185915},[31676],[34,185917],{"className":185918,"style":32597},[31680],[34,185920,58442],{"className":185921,"style":32062},[31685,31686],[34,185923],{"className":185924,"style":32604},[31845],[34,185926,42071],{"className":185927},[32024],[34,185929],{"className":185930,"style":32604},[31845],[34,185932,185934,185937,185940,185943,185946],{"className":185933},[31676],[34,185935],{"className":185936,"style":32617},[31680],[34,185938,94716],{"className":185939},[31685],[34,185941],{"className":185942,"style":31846},[31845],[34,185944,202],{"className":185945},[31850],[34,185947],{"className":185948,"style":31846},[31845],[34,185950,185952,185955,185958,185961,185964],{"className":185951},[31676],[34,185953],{"className":185954,"style":32617},[31680],[34,185956,94716],{"className":185957},[31685],[34,185959,58442],{"className":185960,"style":32062},[31685,31686],[34,185962,31965],{"className":185963},[31685,31686],[34,185965,31759],{"className":185966},[31685,31686],[15,185968,185969],{},"内层位密度为：",[15,185971,185972],{},[34,185973,185975,185996],{"className":185974},[31642],[34,185976,185978],{"className":185977},[31646],[31648,185979,185980],{"xmlns":31650},[31652,185981,185982,185994],{},[31655,185983,185984,185986,185988,185990,185992],{},[31749,185985,185297],{},[31658,185987,1697],{},[31658,185989,38080],{"mathvariant":38079},[31658,185991,31965],{},[31658,185993,31759],{},[31665,185995,185308],{"encoding":31667},[34,185997,185999],{"className":185998,"ariaHidden":758},[31672],[34,186000,186002,186005,186008,186011,186014,186017],{"className":186001},[31676],[34,186003],{"className":186004,"style":31774},[31680],[34,186006,185297],{"className":186007},[31685],[34,186009,1697],{"className":186010},[31685,31686],[34,186012,38080],{"className":186013},[31685],[34,186015,31965],{"className":186016},[31685,31686],[34,186018,31759],{"className":186019},[31685,31686],[15,186021,186022],{},"所以每条磁道可存储位数为：",[15,186024,186025],{},[34,186026,186028,186057],{"className":186027},[31642],[34,186029,186031],{"className":186030},[31646],[31648,186032,186033],{"xmlns":31650},[31652,186034,186035,186054],{},[31655,186036,186037,186039,186041,186043,186045,186047,186050,186052],{},[31749,186038,94716],{},[31658,186040,58442],{},[31662,186042,42071],{},[31749,186044,185297],{},[31662,186046,202],{},[31749,186048,186049],{},"8800",[31658,186051,58442],{},[31658,186053,1697],{},[31665,186055,186056],{"encoding":31667},"22\\pi \\times 400 = 8800\\pi b",[34,186058,186060,186081,186099],{"className":186059,"ariaHidden":758},[31672],[34,186061,186063,186066,186069,186072,186075,186078],{"className":186062},[31676],[34,186064],{"className":186065,"style":35942},[31680],[34,186067,94716],{"className":186068},[31685],[34,186070,58442],{"className":186071,"style":32062},[31685,31686],[34,186073],{"className":186074,"style":32604},[31845],[34,186076,42071],{"className":186077},[32024],[34,186079],{"className":186080,"style":32604},[31845],[34,186082,186084,186087,186090,186093,186096],{"className":186083},[31676],[34,186085],{"className":186086,"style":32617},[31680],[34,186088,185297],{"className":186089},[31685],[34,186091],{"className":186092,"style":31846},[31845],[34,186094,202],{"className":186095},[31850],[34,186097],{"className":186098,"style":31846},[31845],[34,186100,186102,186105,186108,186111],{"className":186101},[31676],[34,186103],{"className":186104,"style":31681},[31680],[34,186106,186049],{"className":186107},[31685],[34,186109,58442],{"className":186110,"style":32062},[31685,31686],[34,186112,1697],{"className":186113},[31685,31686],[15,186115,186116],{},"每个记录面有：",[15,186118,186119],{},[34,186120,186122,186135],{"className":186121},[31642],[34,186123,186125],{"className":186124},[31646],[31648,186126,186127],{"xmlns":31650},[31652,186128,186129,186133],{},[31655,186130,186131],{},[31749,186132,185719],{},[31665,186134,185719],{"encoding":31667},[34,186136,186138],{"className":186137,"ariaHidden":758},[31672],[34,186139,186141,186144],{"className":186140},[31676],[34,186142],{"className":186143,"style":32617},[31680],[34,186145,185719],{"className":186146},[31685],[15,186148,186149],{},"条磁道。",[15,186151,186152],{},"共有：",[15,186154,186155],{},[34,186156,186158,186171],{"className":186157},[31642],[34,186159,186161],{"className":186160},[31646],[31648,186162,186163],{"xmlns":31650},[31652,186164,186165,186169],{},[31655,186166,186167],{},[31749,186168,71869],{},[31665,186170,71869],{"encoding":31667},[34,186172,186174],{"className":186173,"ariaHidden":758},[31672],[34,186175,186177,186180],{"className":186176},[31676],[34,186178],{"className":186179,"style":32617},[31680],[34,186181,71869],{"className":186182},[31685],[15,186184,186185],{},"个记录面。",[15,186187,186188],{},"所以盘组总存储量为：",[15,186190,186191],{},[34,186192,186194,186220],{"className":186193},[31642],[34,186195,186197],{"className":186196},[31646],[31648,186198,186199],{"xmlns":31650},[31652,186200,186201,186217],{},[31655,186202,186203,186205,186207,186209,186211,186213,186215],{},[31749,186204,185719],{},[31662,186206,42071],{},[31749,186208,71869],{},[31662,186210,42071],{},[31749,186212,186049],{},[31658,186214,58442],{},[31658,186216,1697],{},[31665,186218,186219],{"encoding":31667},"220 \\times 12 \\times 8800\\pi b",[34,186221,186223,186241,186259],{"className":186222,"ariaHidden":758},[31672],[34,186224,186226,186229,186232,186235,186238],{"className":186225},[31676],[34,186227],{"className":186228,"style":35942},[31680],[34,186230,185719],{"className":186231},[31685],[34,186233],{"className":186234,"style":32604},[31845],[34,186236,42071],{"className":186237},[32024],[34,186239],{"className":186240,"style":32604},[31845],[34,186242,186244,186247,186250,186253,186256],{"className":186243},[31676],[34,186245],{"className":186246,"style":35942},[31680],[34,186248,71869],{"className":186249},[31685],[34,186251],{"className":186252,"style":32604},[31845],[34,186254,42071],{"className":186255},[32024],[34,186257],{"className":186258,"style":32604},[31845],[34,186260,186262,186265,186268,186271],{"className":186261},[31676],[34,186263],{"className":186264,"style":31681},[31680],[34,186266,186049],{"className":186267},[31685],[34,186269,58442],{"className":186270,"style":32062},[31685,31686],[34,186272,1697],{"className":186273},[31685,31686],[15,186275,174094],{},[15,186277,186278],{},[34,186279,186281,186314],{"className":186280},[31642],[34,186282,186284],{"className":186283},[31646],[31648,186285,186286],{"xmlns":31650},[31652,186287,186288,186311],{},[31655,186289,186290,186292,186294,186296,186298,186300,186302,186304,186307,186309],{},[31749,186291,185719],{},[31662,186293,42071],{},[31749,186295,71869],{},[31662,186297,42071],{},[31749,186299,186049],{},[31658,186301,58442],{},[31662,186303,202],{},[31749,186305,186306],{},"23232000",[31658,186308,58442],{},[31658,186310,1697],{},[31665,186312,186313],{"encoding":31667},"220 \\times 12 \\times 8800\\pi = 23232000\\pi b",[34,186315,186317,186335,186353,186374],{"className":186316,"ariaHidden":758},[31672],[34,186318,186320,186323,186326,186329,186332],{"className":186319},[31676],[34,186321],{"className":186322,"style":35942},[31680],[34,186324,185719],{"className":186325},[31685],[34,186327],{"className":186328,"style":32604},[31845],[34,186330,42071],{"className":186331},[32024],[34,186333],{"className":186334,"style":32604},[31845],[34,186336,186338,186341,186344,186347,186350],{"className":186337},[31676],[34,186339],{"className":186340,"style":35942},[31680],[34,186342,71869],{"className":186343},[31685],[34,186345],{"className":186346,"style":32604},[31845],[34,186348,42071],{"className":186349},[32024],[34,186351],{"className":186352,"style":32604},[31845],[34,186354,186356,186359,186362,186365,186368,186371],{"className":186355},[31676],[34,186357],{"className":186358,"style":32617},[31680],[34,186360,186049],{"className":186361},[31685],[34,186363,58442],{"className":186364,"style":32062},[31685,31686],[34,186366],{"className":186367,"style":31846},[31845],[34,186369,202],{"className":186370},[31850],[34,186372],{"className":186373,"style":31846},[31845],[34,186375,186377,186380,186383,186386],{"className":186376},[31676],[34,186378],{"className":186379,"style":31681},[31680],[34,186381,186306],{"className":186382},[31685],[34,186384,58442],{"className":186385,"style":32062},[31685,31686],[34,186387,1697],{"className":186388},[31685,31686],[15,186390,186391],{},"近似为：",[15,186393,186394],{},[34,186395,186397,186420],{"className":186396},[31642],[34,186398,186400],{"className":186399},[31646],[31648,186401,186402],{"xmlns":31650},[31652,186403,186404,186417],{},[31655,186405,186406,186408,186410,186412,186415],{},[31749,186407,186306],{},[31658,186409,58442],{},[31662,186411,168560],{},[31749,186413,186414],{},"72985480",[31658,186416,1697],{},[31665,186418,186419],{"encoding":31667},"23232000\\pi \\approx 72985480b",[34,186421,186423,186444],{"className":186422,"ariaHidden":758},[31672],[34,186424,186426,186429,186432,186435,186438,186441],{"className":186425},[31676],[34,186427],{"className":186428,"style":32617},[31680],[34,186430,186306],{"className":186431},[31685],[34,186433,58442],{"className":186434,"style":32062},[31685,31686],[34,186436],{"className":186437,"style":31846},[31845],[34,186439,168560],{"className":186440},[31850],[34,186442],{"className":186443,"style":31846},[31845],[34,186445,186447,186450,186453],{"className":186446},[31676],[34,186448],{"className":186449,"style":31681},[31680],[34,186451,186414],{"className":186452},[31685],[34,186454,1697],{"className":186455},[31685,31686],[15,186457,186458],{},"换算为字节：",[15,186460,186461],{},[34,186462,186464,186489],{"className":186463},[31642],[34,186465,186467],{"className":186466},[31646],[31648,186468,186469],{"xmlns":31650},[31652,186470,186471,186486],{},[31655,186472,186473,186475,186477,186479,186481,186484],{},[31749,186474,186414],{},[31658,186476,38080],{"mathvariant":38079},[31749,186478,60525],{},[31662,186480,168560],{},[31749,186482,186483],{},"9123185",[31658,186485,54224],{},[31665,186487,186488],{"encoding":31667},"72985480 \u002F 8 \\approx 9123185B",[34,186490,186492,186511],{"className":186491,"ariaHidden":758},[31672],[34,186493,186495,186498,186502,186505,186508],{"className":186494},[31676],[34,186496],{"className":186497,"style":31774},[31680],[34,186499,186501],{"className":186500},[31685],"72985480\u002F8",[34,186503],{"className":186504,"style":31846},[31845],[34,186506,168560],{"className":186507},[31850],[34,186509],{"className":186510,"style":31846},[31845],[34,186512,186514,186517,186520],{"className":186513},[31676],[34,186515],{"className":186516,"style":33218},[31680],[34,186518,186483],{"className":186519},[31685],[34,186521,54224],{"className":186522,"style":54252},[31685,31686],[15,186524,169980],{},[15,186526,186527],{},[34,186528,186530,186549],{"className":186529},[31642],[34,186531,186533],{"className":186532},[31646],[31648,186534,186535],{"xmlns":31650},[31652,186536,186537,186546],{},[31655,186538,186539,186542,186544],{},[31749,186540,186541],{},"9.12",[31658,186543,98360],{},[31658,186545,54224],{},[31665,186547,186548],{"encoding":31667},"9.12MB",[34,186550,186552],{"className":186551,"ariaHidden":758},[31672],[34,186553,186555,186558,186561,186564],{"className":186554},[31676],[34,186556],{"className":186557,"style":33218},[31680],[34,186559,186541],{"className":186560},[31685],[34,186562,98360],{"className":186563,"style":33389},[31685,31686],[34,186565,54224],{"className":186566,"style":54252},[31685,31686],[15,186568,186569],{},"如果按二进制 MiB 计算：",[15,186571,186572],{},[34,186573,186575,186608],{"className":186574},[31642],[34,186576,186578],{"className":186577},[31646],[31648,186579,186580],{"xmlns":31650},[31652,186581,186582,186605],{},[31655,186583,186584,186586,186588,186590,186592,186594,186596,186599,186601,186603],{},[31749,186585,186483],{},[31658,186587,38080],{"mathvariant":38079},[31749,186589,177393],{},[31658,186591,38080],{"mathvariant":38079},[31749,186593,177393],{},[31662,186595,168560],{},[31749,186597,186598],{},"8.70",[31658,186600,98360],{},[31658,186602,31955],{},[31658,186604,54224],{},[31665,186606,186607],{"encoding":31667},"9123185 \u002F 1024 \u002F 1024 \\approx 8.70MiB",[34,186609,186611,186630],{"className":186610,"ariaHidden":758},[31672],[34,186612,186614,186617,186621,186624,186627],{"className":186613},[31676],[34,186615],{"className":186616,"style":31774},[31680],[34,186618,186620],{"className":186619},[31685],"9123185\u002F1024\u002F1024",[34,186622],{"className":186623,"style":31846},[31845],[34,186625,168560],{"className":186626},[31850],[34,186628],{"className":186629,"style":31846},[31845],[34,186631,186633,186636,186639,186642,186645],{"className":186632},[31676],[34,186634],{"className":186635,"style":33218},[31680],[34,186637,186598],{"className":186638},[31685],[34,186640,98360],{"className":186641,"style":33389},[31685,31686],[34,186643,31955],{"className":186644},[31685,31686],[34,186646,54224],{"className":186647,"style":54252},[31685,31686],[15,186649,186188],{},[15,186651,186652],{},[34,186653,186655,186683],{"className":186654},[31642],[34,186656,186658],{"className":186657},[31646],[31648,186659,186660],{"xmlns":31650},[31652,186661,186662,186680],{},[31655,186663,186664],{},[126400,186665,186666],{"notation":126402},[71090,186667,186668],{"scriptlevel":459,"displaystyle":702},[71090,186669,186670],{"scriptlevel":459,"displaystyle":702},[71090,186671,186672],{"scriptlevel":459,"displaystyle":758},[31655,186673,186674,186676,186678],{},[31749,186675,186306],{},[31658,186677,58442],{},[31658,186679,1697],{},[31665,186681,186682],{"encoding":31667},"\\boxed{23232000\\pi b}",[34,186684,186686],{"className":186685,"ariaHidden":758},[31672],[34,186687,186689,186692],{"className":186688},[31676],[34,186690],{"className":186691,"style":135854},[31680],[34,186693,186695],{"className":186694},[31685],[34,186696,186698,186738],{"className":186697},[31796,31797],[34,186699,186701,186735],{"className":186700},[31801],[34,186702,186704,186727],{"className":186703,"style":135867},[31805],[34,186705,186706,186709],{"style":135870},[34,186707],{"className":186708,"style":135874},[31813],[34,186710,186712],{"className":186711},[126448],[34,186713,186715],{"className":186714},[31685],[34,186716,186718,186721,186724],{"className":186717},[31685],[34,186719,186306],{"className":186720},[31685],[34,186722,58442],{"className":186723,"style":32062},[31685,31686],[34,186725,1697],{"className":186726},[31685,31686],[34,186728,186729,186732],{"style":135889},[34,186730],{"className":186731,"style":135874},[31813],[34,186733],{"className":186734,"style":135896},[126470,126471],[34,186736,31829],{"className":186737},[31828],[34,186739,186741],{"className":186740},[31801],[34,186742,186744],{"className":186743,"style":126482},[31805],[34,186745],{},[15,186747,186748],{},"约为：",[15,186750,186751],{},[34,186752,186754,186780],{"className":186753},[31642],[34,186755,186757],{"className":186756},[31646],[31648,186758,186759],{"xmlns":31650},[31652,186760,186761,186777],{},[31655,186762,186763],{},[126400,186764,186765],{"notation":126402},[71090,186766,186767],{"scriptlevel":459,"displaystyle":702},[71090,186768,186769],{"scriptlevel":459,"displaystyle":702},[71090,186770,186771],{"scriptlevel":459,"displaystyle":758},[31655,186772,186773,186775],{},[31749,186774,186414],{},[31658,186776,1697],{},[31665,186778,186779],{"encoding":31667},"\\boxed{72985480b}",[34,186781,186783],{"className":186782,"ariaHidden":758},[31672],[34,186784,186786,186789],{"className":186785},[31676],[34,186787],{"className":186788,"style":135854},[31680],[34,186790,186792],{"className":186791},[31685],[34,186793,186795,186832],{"className":186794},[31796,31797],[34,186796,186798,186829],{"className":186797},[31801],[34,186799,186801,186821],{"className":186800,"style":135867},[31805],[34,186802,186803,186806],{"style":135870},[34,186804],{"className":186805,"style":135874},[31813],[34,186807,186809],{"className":186808},[126448],[34,186810,186812],{"className":186811},[31685],[34,186813,186815,186818],{"className":186814},[31685],[34,186816,186414],{"className":186817},[31685],[34,186819,1697],{"className":186820},[31685,31686],[34,186822,186823,186826],{"style":135889},[34,186824],{"className":186825,"style":135874},[31813],[34,186827],{"className":186828,"style":135896},[126470,126471],[34,186830,31829],{"className":186831},[31828],[34,186833,186835],{"className":186834},[31801],[34,186836,186838],{"className":186837,"style":126482},[31805],[34,186839],{},[15,186841,186748],{},[15,186843,186844],{},[34,186845,186847,186875],{"className":186846},[31642],[34,186848,186850],{"className":186849},[31646],[31648,186851,186852],{"xmlns":31650},[31652,186853,186854,186872],{},[31655,186855,186856],{},[126400,186857,186858],{"notation":126402},[71090,186859,186860],{"scriptlevel":459,"displaystyle":702},[71090,186861,186862],{"scriptlevel":459,"displaystyle":702},[71090,186863,186864],{"scriptlevel":459,"displaystyle":758},[31655,186865,186866,186868,186870],{},[31749,186867,186541],{},[31658,186869,98360],{},[31658,186871,54224],{},[31665,186873,186874],{"encoding":31667},"\\boxed{9.12MB}",[34,186876,186878],{"className":186877,"ariaHidden":758},[31672],[34,186879,186881,186884],{"className":186880},[31676],[34,186882],{"className":186883,"style":126424},[31680],[34,186885,186887],{"className":186886},[31685],[34,186888,186890,186930],{"className":186889},[31796,31797],[34,186891,186893,186927],{"className":186892},[31801],[34,186894,186896,186919],{"className":186895,"style":126437},[31805],[34,186897,186898,186901],{"style":126440},[34,186899],{"className":186900,"style":126444},[31813],[34,186902,186904],{"className":186903},[126448],[34,186905,186907],{"className":186906},[31685],[34,186908,186910,186913,186916],{"className":186909},[31685],[34,186911,186541],{"className":186912},[31685],[34,186914,98360],{"className":186915,"style":33389},[31685,31686],[34,186917,54224],{"className":186918,"style":54252},[31685,31686],[34,186920,186921,186924],{"style":126463},[34,186922],{"className":186923,"style":126444},[31813],[34,186925],{"className":186926,"style":126472},[126470,126471],[34,186928,31829],{"className":186929},[31828],[34,186931,186933],{"className":186932},[31801],[34,186934,186936],{"className":186935,"style":126482},[31805],[34,186937],{},[32188,186939],{},[253,186941,186943],{"id":186942},"_43-平均等待时间","4.3 平均等待时间",[15,186945,186946],{},"这里的平均等待时间通常指平均旋转等待时间，也叫平均旋转延迟。",[15,186948,186949],{},"转速为：",[15,186951,186952],{},[34,186953,186955,186978],{"className":186954},[31642],[34,186956,186958],{"className":186957},[31646],[31648,186959,186960],{"xmlns":31650},[31652,186961,186962,186976],{},[31655,186963,186964,186966,186968,186970,186972,186974],{},[31749,186965,74728],{},[31658,186967,34395],{},[31658,186969,38080],{"mathvariant":38079},[31658,186971,31759],{},[31658,186973,31955],{},[31658,186975,31660],{},[31665,186977,185364],{"encoding":31667},[34,186979,186981],{"className":186980,"ariaHidden":758},[31672],[34,186982,186984,186987,186990,186993,186996],{"className":186983},[31676],[34,186985],{"className":186986,"style":31774},[31680],[34,186988,74728],{"className":186989},[31685],[34,186991,34395],{"className":186992,"style":34452},[31685,31686],[34,186994,38080],{"className":186995},[31685],[34,186997,56084],{"className":186998},[31685,31686],[15,187000,187001],{},"换成每秒转数：",[15,187003,187004],{},[34,187005,187007,187035],{"className":187006},[31642],[34,187008,187010],{"className":187009},[31646],[31648,187011,187012],{"xmlns":31650},[31652,187013,187014,187032],{},[31655,187015,187016,187018,187020,187022,187024,187026,187028,187030],{},[31749,187017,74728],{},[31658,187019,38080],{"mathvariant":38079},[31749,187021,167664],{},[31662,187023,202],{},[31749,187025,5774],{},[31658,187027,34395],{},[31658,187029,38080],{"mathvariant":38079},[31658,187031,31747],{},[31665,187033,187034],{"encoding":31667},"3000 \u002F 60 = 50r\u002Fs",[34,187036,187038,187057],{"className":187037,"ariaHidden":758},[31672],[34,187039,187041,187044,187048,187051,187054],{"className":187040},[31676],[34,187042],{"className":187043,"style":31774},[31680],[34,187045,187047],{"className":187046},[31685],"3000\u002F60",[34,187049],{"className":187050,"style":31846},[31845],[34,187052,202],{"className":187053},[31850],[34,187055],{"className":187056,"style":31846},[31845],[34,187058,187060,187063,187066,187069,187072],{"className":187059},[31676],[34,187061],{"className":187062,"style":31774},[31680],[34,187064,5774],{"className":187065},[31685],[34,187067,34395],{"className":187068,"style":34452},[31685,31686],[34,187070,38080],{"className":187071},[31685],[34,187073,31747],{"className":187074},[31685,31686],[15,187076,187077],{},"所以一圈时间为：",[15,187079,187080],{},[34,187081,187083,187118],{"className":187082},[31642],[34,187084,187086],{"className":187085},[31646],[31648,187087,187088],{"xmlns":31650},[31652,187089,187090,187115],{},[31655,187091,187092,187094,187096,187098,187100,187102,187105,187107,187109,187111,187113],{},[31749,187093,640],{},[31658,187095,38080],{"mathvariant":38079},[31749,187097,5774],{},[31658,187099,31747],{},[31662,187101,202],{},[31749,187103,187104],{},"0.02",[31658,187106,31747],{},[31662,187108,202],{},[31749,187110,3249],{},[31658,187112,31759],{},[31658,187114,31747],{},[31665,187116,187117],{"encoding":31667},"1 \u002F 50s = 0.02s = 20ms",[34,187119,187121,187143,187164],{"className":187120,"ariaHidden":758},[31672],[34,187122,187124,187127,187131,187134,187137,187140],{"className":187123},[31676],[34,187125],{"className":187126,"style":31774},[31680],[34,187128,187130],{"className":187129},[31685],"1\u002F50",[34,187132,31747],{"className":187133},[31685,31686],[34,187135],{"className":187136,"style":31846},[31845],[34,187138,202],{"className":187139},[31850],[34,187141],{"className":187142,"style":31846},[31845],[34,187144,187146,187149,187152,187155,187158,187161],{"className":187145},[31676],[34,187147],{"className":187148,"style":32617},[31680],[34,187150,187104],{"className":187151},[31685],[34,187153,31747],{"className":187154},[31685,31686],[34,187156],{"className":187157,"style":31846},[31845],[34,187159,202],{"className":187160},[31850],[34,187162],{"className":187163,"style":31846},[31845],[34,187165,187167,187170,187173,187176],{"className":187166},[31676],[34,187168],{"className":187169,"style":32617},[31680],[34,187171,3249],{"className":187172},[31685],[34,187174,31759],{"className":187175},[31685,31686],[34,187177,31747],{"className":187178},[31685,31686],[15,187180,187181],{},"平均等待时间为半圈时间：",[15,187183,187184],{},[34,187185,187187,187217],{"className":187186},[31642],[34,187188,187190],{"className":187189},[31646],[31648,187191,187192],{"xmlns":31650},[31652,187193,187194,187214],{},[31655,187195,187196,187198,187200,187202,187204,187206,187208,187210,187212],{},[31749,187197,3249],{},[31658,187199,31759],{},[31658,187201,31747],{},[31658,187203,38080],{"mathvariant":38079},[31749,187205,768],{},[31662,187207,202],{},[31749,187209,3244],{},[31658,187211,31759],{},[31658,187213,31747],{},[31665,187215,187216],{"encoding":31667},"20ms \u002F 2 = 10ms",[34,187218,187220,187247],{"className":187219,"ariaHidden":758},[31672],[34,187221,187223,187226,187229,187232,187235,187238,187241,187244],{"className":187222},[31676],[34,187224],{"className":187225,"style":31774},[31680],[34,187227,3249],{"className":187228},[31685],[34,187230,31759],{"className":187231},[31685,31686],[34,187233,31747],{"className":187234},[31685,31686],[34,187236,38149],{"className":187237},[31685],[34,187239],{"className":187240,"style":31846},[31845],[34,187242,202],{"className":187243},[31850],[34,187245],{"className":187246,"style":31846},[31845],[34,187248,187250,187253,187256,187259],{"className":187249},[31676],[34,187251],{"className":187252,"style":32617},[31680],[34,187254,3244],{"className":187255},[31685],[34,187257,31759],{"className":187258},[31685,31686],[34,187260,31747],{"className":187261},[31685,31686],[15,187263,187264],{},"所以平均等待时间为：",[15,187266,187267],{},[34,187268,187270,187298],{"className":187269},[31642],[34,187271,187273],{"className":187272},[31646],[31648,187274,187275],{"xmlns":31650},[31652,187276,187277,187295],{},[31655,187278,187279],{},[126400,187280,187281],{"notation":126402},[71090,187282,187283],{"scriptlevel":459,"displaystyle":702},[71090,187284,187285],{"scriptlevel":459,"displaystyle":702},[71090,187286,187287],{"scriptlevel":459,"displaystyle":758},[31655,187288,187289,187291,187293],{},[31749,187290,3244],{},[31658,187292,31759],{},[31658,187294,31747],{},[31665,187296,187297],{"encoding":31667},"\\boxed{10ms}",[34,187299,187301],{"className":187300,"ariaHidden":758},[31672],[34,187302,187304,187307],{"className":187303},[31676],[34,187305],{"className":187306,"style":185810},[31680],[34,187308,187310],{"className":187309},[31685],[34,187311,187313,187353],{"className":187312},[31796,31797],[34,187314,187316,187350],{"className":187315},[31801],[34,187317,187319,187342],{"className":187318,"style":136916},[31805],[34,187320,187321,187324],{"style":185825},[34,187322],{"className":187323,"style":185829},[31813],[34,187325,187327],{"className":187326},[126448],[34,187328,187330],{"className":187329},[31685],[34,187331,187333,187336,187339],{"className":187332},[31685],[34,187334,3244],{"className":187335},[31685],[34,187337,31759],{"className":187338},[31685,31686],[34,187340,31747],{"className":187341},[31685,31686],[34,187343,187344,187347],{"style":136950},[34,187345],{"className":187346,"style":185829},[31813],[34,187348],{"className":187349,"style":185850},[126470,126471],[34,187351,31829],{"className":187352},[31828],[34,187354,187356],{"className":187355},[31801],[34,187357,187359],{"className":187358,"style":126482},[31805],[34,187360],{},[32188,187362],{},[253,187364,173018],{"id":173017},[15,187366,187367],{},"柱面数：",[15,187369,187370],{},[34,187371,187373,187394],{"className":187372},[31642],[34,187374,187376],{"className":187375},[31646],[31648,187377,187378],{"xmlns":31650},[31652,187379,187380,187392],{},[31655,187381,187382],{},[126400,187383,187384],{"notation":126402},[71090,187385,187386],{"scriptlevel":459,"displaystyle":702},[71090,187387,187388],{"scriptlevel":459,"displaystyle":702},[71090,187389,187390],{"scriptlevel":459,"displaystyle":758},[31749,187391,185719],{},[31665,187393,185800],{"encoding":31667},[34,187395,187397],{"className":187396,"ariaHidden":758},[31672],[34,187398,187400,187403],{"className":187399},[31676],[34,187401],{"className":187402,"style":185810},[31680],[34,187404,187406],{"className":187405},[31685],[34,187407,187409,187443],{"className":187408},[31796,31797],[34,187410,187412,187440],{"className":187411},[31801],[34,187413,187415,187432],{"className":187414,"style":136916},[31805],[34,187416,187417,187420],{"style":185825},[34,187418],{"className":187419,"style":185829},[31813],[34,187421,187423],{"className":187422},[126448],[34,187424,187426],{"className":187425},[31685],[34,187427,187429],{"className":187428},[31685],[34,187430,185719],{"className":187431},[31685],[34,187433,187434,187437],{"style":136950},[34,187435],{"className":187436,"style":185829},[31813],[34,187438],{"className":187439,"style":185850},[126470,126471],[34,187441,31829],{"className":187442},[31828],[34,187444,187446],{"className":187445},[31801],[34,187447,187449],{"className":187448,"style":126482},[31805],[34,187450],{},[15,187452,187453],{},"盘组总存储量：",[15,187455,187456],{},[34,187457,187459,187501],{"className":187458},[31642],[34,187460,187462],{"className":187461},[31646],[31648,187463,187464],{"xmlns":31650},[31652,187465,187466,187498],{},[31655,187467,187468],{},[126400,187469,187470],{"notation":126402},[71090,187471,187472],{"scriptlevel":459,"displaystyle":702},[71090,187473,187474],{"scriptlevel":459,"displaystyle":702},[71090,187475,187476],{"scriptlevel":459,"displaystyle":758},[31655,187477,187478,187480,187482,187484,187486,187488,187490,187492,187494,187496],{},[31749,187479,186306],{},[31658,187481,58442],{},[31658,187483,1697],{},[31662,187485,168560],{},[31749,187487,186414],{},[31658,187489,1697],{},[31662,187491,168560],{},[31749,187493,186541],{},[31658,187495,98360],{},[31658,187497,54224],{},[31665,187499,187500],{"encoding":31667},"\\boxed{23232000\\pi b \\approx 72985480b \\approx 9.12MB}",[34,187502,187504],{"className":187503,"ariaHidden":758},[31672],[34,187505,187507,187510],{"className":187506},[31676],[34,187508],{"className":187509,"style":135854},[31680],[34,187511,187513],{"className":187512},[31685],[34,187514,187516,187589],{"className":187515},[31796,31797],[34,187517,187519,187586],{"className":187518},[31801],[34,187520,187522,187578],{"className":187521,"style":135867},[31805],[34,187523,187524,187527],{"style":135870},[34,187525],{"className":187526,"style":135874},[31813],[34,187528,187530],{"className":187529},[126448],[34,187531,187533],{"className":187532},[31685],[34,187534,187536,187539,187542,187545,187548,187551,187554,187557,187560,187563,187566,187569,187572,187575],{"className":187535},[31685],[34,187537,186306],{"className":187538},[31685],[34,187540,58442],{"className":187541,"style":32062},[31685,31686],[34,187543,1697],{"className":187544},[31685,31686],[34,187546],{"className":187547,"style":31846},[31845],[34,187549,168560],{"className":187550},[31850],[34,187552],{"className":187553,"style":31846},[31845],[34,187555,186414],{"className":187556},[31685],[34,187558,1697],{"className":187559},[31685,31686],[34,187561],{"className":187562,"style":31846},[31845],[34,187564,168560],{"className":187565},[31850],[34,187567],{"className":187568,"style":31846},[31845],[34,187570,186541],{"className":187571},[31685],[34,187573,98360],{"className":187574,"style":33389},[31685,31686],[34,187576,54224],{"className":187577,"style":54252},[31685,31686],[34,187579,187580,187583],{"style":135889},[34,187581],{"className":187582,"style":135874},[31813],[34,187584],{"className":187585,"style":135896},[126470,126471],[34,187587,31829],{"className":187588},[31828],[34,187590,187592],{"className":187591},[31801],[34,187593,187595],{"className":187594,"style":126482},[31805],[34,187596],{},[15,187598,187599],{},"平均等待时间：",[15,187601,187602],{},[34,187603,187605,187632],{"className":187604},[31642],[34,187606,187608],{"className":187607},[31646],[31648,187609,187610],{"xmlns":31650},[31652,187611,187612,187630],{},[31655,187613,187614],{},[126400,187615,187616],{"notation":126402},[71090,187617,187618],{"scriptlevel":459,"displaystyle":702},[71090,187619,187620],{"scriptlevel":459,"displaystyle":702},[71090,187621,187622],{"scriptlevel":459,"displaystyle":758},[31655,187623,187624,187626,187628],{},[31749,187625,3244],{},[31658,187627,31759],{},[31658,187629,31747],{},[31665,187631,187297],{"encoding":31667},[34,187633,187635],{"className":187634,"ariaHidden":758},[31672],[34,187636,187638,187641],{"className":187637},[31676],[34,187639],{"className":187640,"style":185810},[31680],[34,187642,187644],{"className":187643},[31685],[34,187645,187647,187687],{"className":187646},[31796,31797],[34,187648,187650,187684],{"className":187649},[31801],[34,187651,187653,187676],{"className":187652,"style":136916},[31805],[34,187654,187655,187658],{"style":185825},[34,187656],{"className":187657,"style":185829},[31813],[34,187659,187661],{"className":187660},[126448],[34,187662,187664],{"className":187663},[31685],[34,187665,187667,187670,187673],{"className":187666},[31685],[34,187668,3244],{"className":187669},[31685],[34,187671,31759],{"className":187672},[31685,31686],[34,187674,31747],{"className":187675},[31685,31686],[34,187677,187678,187681],{"style":136950},[34,187679],{"className":187680,"style":185829},[31813],[34,187682],{"className":187683,"style":185850},[126470,126471],[34,187685,31829],{"className":187686},[31828],[34,187688,187690],{"className":187689},[31801],[34,187691,187693],{"className":187692,"style":126482},[31805],[34,187694],{},[32188,187696],{},[19,187698,187700],{"id":187699},"_5-多级索引文件最大长度","5. 多级索引文件最大长度",[253,187702,162043],{"id":173118},[15,187704,187705],{},"设文件索引节点中有7个地址项，其中4个为直接地址索引，2个是一级间接地址索引，1个是二级间接地址索引，地址项大小为4B，若磁盘索引块和磁盘数据块大小均为1KB，求可表示的单个文件的最大长度。",[32188,187707],{},[253,187709,175302],{"id":187710},"答案和讲解-4",[15,187712,173129],{},[15,187714,187715],{},"地址项总数：",[15,187717,187718],{},[34,187719,187721,187734],{"className":187720},[31642],[34,187722,187724],{"className":187723},[31646],[31648,187725,187726],{"xmlns":31650},[31652,187727,187728,187732],{},[31655,187729,187730],{},[31749,187731,36723],{},[31665,187733,36723],{"encoding":31667},[34,187735,187737],{"className":187736,"ariaHidden":758},[31672],[34,187738,187740,187743],{"className":187739},[31676],[34,187741],{"className":187742,"style":32617},[31680],[34,187744,36723],{"className":187745},[31685],[15,187747,187748],{},"直接地址索引：",[15,187750,187751],{},[34,187752,187754,187767],{"className":187753},[31642],[34,187755,187757],{"className":187756},[31646],[31648,187758,187759],{"xmlns":31650},[31652,187760,187761,187765],{},[31655,187762,187763],{},[31749,187764,15932],{},[31665,187766,15932],{"encoding":31667},[34,187768,187770],{"className":187769,"ariaHidden":758},[31672],[34,187771,187773,187776],{"className":187772},[31676],[34,187774],{"className":187775,"style":32617},[31680],[34,187777,15932],{"className":187778},[31685],[15,187780,187781],{},"一级间接地址索引：",[15,187783,187784],{},[34,187785,187787,187800],{"className":187786},[31642],[34,187788,187790],{"className":187789},[31646],[31648,187791,187792],{"xmlns":31650},[31652,187793,187794,187798],{},[31655,187795,187796],{},[31749,187797,768],{},[31665,187799,768],{"encoding":31667},[34,187801,187803],{"className":187802,"ariaHidden":758},[31672],[34,187804,187806,187809],{"className":187805},[31676],[34,187807],{"className":187808,"style":32617},[31680],[34,187810,768],{"className":187811},[31685],[15,187813,187814],{},"二级间接地址索引：",[15,187816,187817],{},[34,187818,187820,187833],{"className":187819},[31642],[34,187821,187823],{"className":187822},[31646],[31648,187824,187825],{"xmlns":31650},[31652,187826,187827,187831],{},[31655,187828,187829],{},[31749,187830,640],{},[31665,187832,640],{"encoding":31667},[34,187834,187836],{"className":187835,"ariaHidden":758},[31672],[34,187837,187839,187842],{"className":187838},[31676],[34,187840],{"className":187841,"style":32617},[31680],[34,187843,640],{"className":187844},[31685],[15,187846,187847],{},"地址项大小：",[15,187849,187850],{},[34,187851,187853,187869],{"className":187852},[31642],[34,187854,187856],{"className":187855},[31646],[31648,187857,187858],{"xmlns":31650},[31652,187859,187860,187866],{},[31655,187861,187862,187864],{},[31749,187863,15932],{},[31658,187865,54224],{},[31665,187867,187868],{"encoding":31667},"4B",[34,187870,187872],{"className":187871,"ariaHidden":758},[31672],[34,187873,187875,187878,187881],{"className":187874},[31676],[34,187876],{"className":187877,"style":33218},[31680],[34,187879,15932],{"className":187880},[31685],[34,187882,54224],{"className":187883,"style":54252},[31685,31686],[15,187885,187886],{},"索引块大小：",[15,187888,187889],{},[34,187890,187892,187915],{"className":187891},[31642],[34,187893,187895],{"className":187894},[31646],[31648,187896,187897],{"xmlns":31650},[31652,187898,187899,187913],{},[31655,187900,187901,187903,187905,187907,187909,187911],{},[31749,187902,640],{},[31658,187904,38691],{},[31658,187906,54224],{},[31662,187908,202],{},[31749,187910,177393],{},[31658,187912,54224],{},[31665,187914,177398],{"encoding":31667},[34,187916,187918,187942],{"className":187917,"ariaHidden":758},[31672],[34,187919,187921,187924,187927,187930,187933,187936,187939],{"className":187920},[31676],[34,187922],{"className":187923,"style":33218},[31680],[34,187925,640],{"className":187926},[31685],[34,187928,38691],{"className":187929,"style":38706},[31685,31686],[34,187931,54224],{"className":187932,"style":54252},[31685,31686],[34,187934],{"className":187935,"style":31846},[31845],[34,187937,202],{"className":187938},[31850],[34,187940],{"className":187941,"style":31846},[31845],[34,187943,187945,187948,187951],{"className":187944},[31676],[34,187946],{"className":187947,"style":33218},[31680],[34,187949,177393],{"className":187950},[31685],[34,187952,54224],{"className":187953,"style":54252},[31685,31686],[15,187955,187956],{},"数据块大小：",[15,187958,187959],{},[34,187960,187962,187985],{"className":187961},[31642],[34,187963,187965],{"className":187964},[31646],[31648,187966,187967],{"xmlns":31650},[31652,187968,187969,187983],{},[31655,187970,187971,187973,187975,187977,187979,187981],{},[31749,187972,640],{},[31658,187974,38691],{},[31658,187976,54224],{},[31662,187978,202],{},[31749,187980,177393],{},[31658,187982,54224],{},[31665,187984,177398],{"encoding":31667},[34,187986,187988,188012],{"className":187987,"ariaHidden":758},[31672],[34,187989,187991,187994,187997,188000,188003,188006,188009],{"className":187990},[31676],[34,187992],{"className":187993,"style":33218},[31680],[34,187995,640],{"className":187996},[31685],[34,187998,38691],{"className":187999,"style":38706},[31685,31686],[34,188001,54224],{"className":188002,"style":54252},[31685,31686],[34,188004],{"className":188005,"style":31846},[31845],[34,188007,202],{"className":188008},[31850],[34,188010],{"className":188011,"style":31846},[31845],[34,188013,188015,188018,188021],{"className":188014},[31676],[34,188016],{"className":188017,"style":33218},[31680],[34,188019,177393],{"className":188020},[31685],[34,188022,54224],{"className":188023,"style":54252},[31685,31686],[32188,188025],{},[253,188027,188029],{"id":188028},"_51-一个索引块能存放多少个地址项","5.1 一个索引块能存放多少个地址项？",[15,188031,188032],{},"每个地址项大小为：",[15,188034,188035],{},[34,188036,188038,188053],{"className":188037},[31642],[34,188039,188041],{"className":188040},[31646],[31648,188042,188043],{"xmlns":31650},[31652,188044,188045,188051],{},[31655,188046,188047,188049],{},[31749,188048,15932],{},[31658,188050,54224],{},[31665,188052,187868],{"encoding":31667},[34,188054,188056],{"className":188055,"ariaHidden":758},[31672],[34,188057,188059,188062,188065],{"className":188058},[31676],[34,188060],{"className":188061,"style":33218},[31680],[34,188063,15932],{"className":188064},[31685],[34,188066,54224],{"className":188067,"style":54252},[31685,31686],[15,188069,188070],{},"一个索引块大小为：",[15,188072,188073],{},[34,188074,188076,188092],{"className":188075},[31642],[34,188077,188079],{"className":188078},[31646],[31648,188080,188081],{"xmlns":31650},[31652,188082,188083,188089],{},[31655,188084,188085,188087],{},[31749,188086,177393],{},[31658,188088,54224],{},[31665,188090,188091],{"encoding":31667},"1024B",[34,188093,188095],{"className":188094,"ariaHidden":758},[31672],[34,188096,188098,188101,188104],{"className":188097},[31676],[34,188099],{"className":188100,"style":33218},[31680],[34,188102,177393],{"className":188103},[31685],[34,188105,54224],{"className":188106,"style":54252},[31685,31686],[15,188108,188109],{},"所以一个索引块能存放的地址项数量为：",[15,188111,188112],{},[34,188113,188115,188137],{"className":188114},[31642],[34,188116,188118],{"className":188117},[31646],[31648,188119,188120],{"xmlns":31650},[31652,188121,188122,188134],{},[31655,188123,188124,188126,188128,188130,188132],{},[31749,188125,177393],{},[31658,188127,38080],{"mathvariant":38079},[31749,188129,15932],{},[31662,188131,202],{},[31749,188133,95418],{},[31665,188135,188136],{"encoding":31667},"1024 \u002F 4 = 256",[34,188138,188140,188159],{"className":188139,"ariaHidden":758},[31672],[34,188141,188143,188146,188150,188153,188156],{"className":188142},[31676],[34,188144],{"className":188145,"style":31774},[31680],[34,188147,188149],{"className":188148},[31685],"1024\u002F4",[34,188151],{"className":188152,"style":31846},[31845],[34,188154,202],{"className":188155},[31850],[34,188157],{"className":188158,"style":31846},[31845],[34,188160,188162,188165],{"className":188161},[31676],[34,188163],{"className":188164,"style":32617},[31680],[34,188166,95418],{"className":188167},[31685],[15,188169,188170],{},"也就是说，一个一级间接索引块可以指向：",[15,188172,188173],{},[34,188174,188176,188189],{"className":188175},[31642],[34,188177,188179],{"className":188178},[31646],[31648,188180,188181],{"xmlns":31650},[31652,188182,188183,188187],{},[31655,188184,188185],{},[31749,188186,95418],{},[31665,188188,95418],{"encoding":31667},[34,188190,188192],{"className":188191,"ariaHidden":758},[31672],[34,188193,188195,188198],{"className":188194},[31676],[34,188196],{"className":188197,"style":32617},[31680],[34,188199,95418],{"className":188200},[31685],[15,188202,188203],{},"个数据块。",[32188,188205],{},[253,188207,188209],{"id":188208},"_52-直接地址索引能表示多少数据","5.2 直接地址索引能表示多少数据？",[15,188211,188212],{},"有 4 个直接地址项。",[15,188214,188215],{},"每个直接地址项直接指向 1 个数据块。",[15,188217,188218],{},"所以直接地址索引可以表示：",[15,188220,188221],{},[34,188222,188224,188237],{"className":188223},[31642],[34,188225,188227],{"className":188226},[31646],[31648,188228,188229],{"xmlns":31650},[31652,188230,188231,188235],{},[31655,188232,188233],{},[31749,188234,15932],{},[31665,188236,15932],{"encoding":31667},[34,188238,188240],{"className":188239,"ariaHidden":758},[31672],[34,188241,188243,188246],{"className":188242},[31676],[34,188244],{"className":188245,"style":32617},[31680],[34,188247,15932],{"className":188248},[31685],[15,188250,188203],{},[15,188252,188253],{},"容量为：",[15,188255,188256],{},[34,188257,188259,188289],{"className":188258},[31642],[34,188260,188262],{"className":188261},[31646],[31648,188263,188264],{"xmlns":31650},[31652,188265,188266,188286],{},[31655,188267,188268,188270,188272,188274,188276,188278,188280,188282,188284],{},[31749,188269,15932],{},[31662,188271,42071],{},[31749,188273,640],{},[31658,188275,38691],{},[31658,188277,54224],{},[31662,188279,202],{},[31749,188281,15932],{},[31658,188283,38691],{},[31658,188285,54224],{},[31665,188287,188288],{"encoding":31667},"4 \\times 1KB = 4KB",[34,188290,188292,188310,188334],{"className":188291,"ariaHidden":758},[31672],[34,188293,188295,188298,188301,188304,188307],{"className":188294},[31676],[34,188296],{"className":188297,"style":35942},[31680],[34,188299,15932],{"className":188300},[31685],[34,188302],{"className":188303,"style":32604},[31845],[34,188305,42071],{"className":188306},[32024],[34,188308],{"className":188309,"style":32604},[31845],[34,188311,188313,188316,188319,188322,188325,188328,188331],{"className":188312},[31676],[34,188314],{"className":188315,"style":33218},[31680],[34,188317,640],{"className":188318},[31685],[34,188320,38691],{"className":188321,"style":38706},[31685,31686],[34,188323,54224],{"className":188324,"style":54252},[31685,31686],[34,188326],{"className":188327,"style":31846},[31845],[34,188329,202],{"className":188330},[31850],[34,188332],{"className":188333,"style":31846},[31845],[34,188335,188337,188340,188343,188346],{"className":188336},[31676],[34,188338],{"className":188339,"style":33218},[31680],[34,188341,15932],{"className":188342},[31685],[34,188344,38691],{"className":188345,"style":38706},[31685,31686],[34,188347,54224],{"className":188348,"style":54252},[31685,31686],[32188,188350],{},[253,188352,188354],{"id":188353},"_53-一级间接地址索引能表示多少数据","5.3 一级间接地址索引能表示多少数据？",[15,188356,188357],{},"有 2 个一级间接地址项。",[15,188359,188360],{},"每个一级间接地址项指向一个索引块。",[15,188362,188363],{},"每个索引块中有：",[15,188365,188366],{},[34,188367,188369,188382],{"className":188368},[31642],[34,188370,188372],{"className":188371},[31646],[31648,188373,188374],{"xmlns":31650},[31652,188375,188376,188380],{},[31655,188377,188378],{},[31749,188379,95418],{},[31665,188381,95418],{"encoding":31667},[34,188383,188385],{"className":188384,"ariaHidden":758},[31672],[34,188386,188388,188391],{"className":188387},[31676],[34,188389],{"className":188390,"style":32617},[31680],[34,188392,95418],{"className":188393},[31685],[15,188395,188396],{},"个地址项。",[15,188398,188399],{},"每个地址项指向一个数据块。",[15,188401,188402],{},"所以一级间接索引可以表示的数据块数为：",[15,188404,188405],{},[34,188406,188408,188430],{"className":188407},[31642],[34,188409,188411],{"className":188410},[31646],[31648,188412,188413],{"xmlns":31650},[31652,188414,188415,188427],{},[31655,188416,188417,188419,188421,188423,188425],{},[31749,188418,768],{},[31662,188420,42071],{},[31749,188422,95418],{},[31662,188424,202],{},[31749,188426,167308],{},[31665,188428,188429],{"encoding":31667},"2 \\times 256 = 512",[34,188431,188433,188451,188469],{"className":188432,"ariaHidden":758},[31672],[34,188434,188436,188439,188442,188445,188448],{"className":188435},[31676],[34,188437],{"className":188438,"style":35942},[31680],[34,188440,768],{"className":188441},[31685],[34,188443],{"className":188444,"style":32604},[31845],[34,188446,42071],{"className":188447},[32024],[34,188449],{"className":188450,"style":32604},[31845],[34,188452,188454,188457,188460,188463,188466],{"className":188453},[31676],[34,188455],{"className":188456,"style":32617},[31680],[34,188458,95418],{"className":188459},[31685],[34,188461],{"className":188462,"style":31846},[31845],[34,188464,202],{"className":188465},[31850],[34,188467],{"className":188468,"style":31846},[31845],[34,188470,188472,188475],{"className":188471},[31676],[34,188473],{"className":188474,"style":32617},[31680],[34,188476,167308],{"className":188477},[31685],[15,188479,188253],{},[15,188481,188482],{},[34,188483,188485,188515],{"className":188484},[31642],[34,188486,188488],{"className":188487},[31646],[31648,188489,188490],{"xmlns":31650},[31652,188491,188492,188512],{},[31655,188493,188494,188496,188498,188500,188502,188504,188506,188508,188510],{},[31749,188495,167308],{},[31662,188497,42071],{},[31749,188499,640],{},[31658,188501,38691],{},[31658,188503,54224],{},[31662,188505,202],{},[31749,188507,167308],{},[31658,188509,38691],{},[31658,188511,54224],{},[31665,188513,188514],{"encoding":31667},"512 \\times 1KB = 512KB",[34,188516,188518,188536,188560],{"className":188517,"ariaHidden":758},[31672],[34,188519,188521,188524,188527,188530,188533],{"className":188520},[31676],[34,188522],{"className":188523,"style":35942},[31680],[34,188525,167308],{"className":188526},[31685],[34,188528],{"className":188529,"style":32604},[31845],[34,188531,42071],{"className":188532},[32024],[34,188534],{"className":188535,"style":32604},[31845],[34,188537,188539,188542,188545,188548,188551,188554,188557],{"className":188538},[31676],[34,188540],{"className":188541,"style":33218},[31680],[34,188543,640],{"className":188544},[31685],[34,188546,38691],{"className":188547,"style":38706},[31685,31686],[34,188549,54224],{"className":188550,"style":54252},[31685,31686],[34,188552],{"className":188553,"style":31846},[31845],[34,188555,202],{"className":188556},[31850],[34,188558],{"className":188559,"style":31846},[31845],[34,188561,188563,188566,188569,188572],{"className":188562},[31676],[34,188564],{"className":188565,"style":33218},[31680],[34,188567,167308],{"className":188568},[31685],[34,188570,38691],{"className":188571,"style":38706},[31685,31686],[34,188573,54224],{"className":188574,"style":54252},[31685,31686],[32188,188576],{},[253,188578,188580],{"id":188579},"_54-二级间接地址索引能表示多少数据","5.4 二级间接地址索引能表示多少数据？",[15,188582,188583],{},"二级间接结构是：",[15,188585,188586],{},"inode 中的二级间接地址项",[15,188588,188589],{},"指向一个一级索引块",[15,188591,188592],{},"这个一级索引块中的每个地址项又指向一个二级索引块",[15,188594,188595],{},"每个二级索引块再指向数据块",[15,188597,188598],{},"所以一个二级间接地址项可以表示的数据块数为：",[15,188600,188601],{},[34,188602,188604,188627],{"className":188603},[31642],[34,188605,188607],{"className":188606},[31646],[31648,188608,188609],{"xmlns":31650},[31652,188610,188611,188624],{},[31655,188612,188613,188615,188617,188619,188621],{},[31749,188614,95418],{},[31662,188616,42071],{},[31749,188618,95418],{},[31662,188620,202],{},[31749,188622,188623],{},"65536",[31665,188625,188626],{"encoding":31667},"256 \\times 256 = 65536",[34,188628,188630,188648,188666],{"className":188629,"ariaHidden":758},[31672],[34,188631,188633,188636,188639,188642,188645],{"className":188632},[31676],[34,188634],{"className":188635,"style":35942},[31680],[34,188637,95418],{"className":188638},[31685],[34,188640],{"className":188641,"style":32604},[31845],[34,188643,42071],{"className":188644},[32024],[34,188646],{"className":188647,"style":32604},[31845],[34,188649,188651,188654,188657,188660,188663],{"className":188650},[31676],[34,188652],{"className":188653,"style":32617},[31680],[34,188655,95418],{"className":188656},[31685],[34,188658],{"className":188659,"style":31846},[31845],[34,188661,202],{"className":188662},[31850],[34,188664],{"className":188665,"style":31846},[31845],[34,188667,188669,188672],{"className":188668},[31676],[34,188670],{"className":188671,"style":32617},[31680],[34,188673,188623],{"className":188674},[31685],[15,188676,188253],{},[15,188678,188679],{},[34,188680,188682,188712],{"className":188681},[31642],[34,188683,188685],{"className":188684},[31646],[31648,188686,188687],{"xmlns":31650},[31652,188688,188689,188709],{},[31655,188690,188691,188693,188695,188697,188699,188701,188703,188705,188707],{},[31749,188692,188623],{},[31662,188694,42071],{},[31749,188696,640],{},[31658,188698,38691],{},[31658,188700,54224],{},[31662,188702,202],{},[31749,188704,188623],{},[31658,188706,38691],{},[31658,188708,54224],{},[31665,188710,188711],{"encoding":31667},"65536 \\times 1KB = 65536KB",[34,188713,188715,188733,188757],{"className":188714,"ariaHidden":758},[31672],[34,188716,188718,188721,188724,188727,188730],{"className":188717},[31676],[34,188719],{"className":188720,"style":35942},[31680],[34,188722,188623],{"className":188723},[31685],[34,188725],{"className":188726,"style":32604},[31845],[34,188728,42071],{"className":188729},[32024],[34,188731],{"className":188732,"style":32604},[31845],[34,188734,188736,188739,188742,188745,188748,188751,188754],{"className":188735},[31676],[34,188737],{"className":188738,"style":33218},[31680],[34,188740,640],{"className":188741},[31685],[34,188743,38691],{"className":188744,"style":38706},[31685,31686],[34,188746,54224],{"className":188747,"style":54252},[31685,31686],[34,188749],{"className":188750,"style":31846},[31845],[34,188752,202],{"className":188753},[31850],[34,188755],{"className":188756,"style":31846},[31845],[34,188758,188760,188763,188766,188769],{"className":188759},[31676],[34,188761],{"className":188762,"style":33218},[31680],[34,188764,188623],{"className":188765},[31685],[34,188767,38691],{"className":188768,"style":38706},[31685,31686],[34,188770,54224],{"className":188771,"style":54252},[31685,31686],[32188,188773],{},[253,188775,188777],{"id":188776},"_55-单个文件最大长度","5.5 单个文件最大长度",[15,188779,188780],{},"总数据块数为：",[15,188782,188783],{},[34,188784,188786,188816],{"className":188785},[31642],[34,188787,188789],{"className":188788},[31646],[31648,188790,188791],{"xmlns":31650},[31652,188792,188793,188813],{},[31655,188794,188795,188797,188799,188801,188803,188805,188807,188809,188811],{},[31749,188796,15932],{},[31662,188798,223],{},[31749,188800,768],{},[31662,188802,42071],{},[31749,188804,95418],{},[31662,188806,223],{},[31749,188808,95418],{},[31662,188810,42071],{},[31749,188812,95418],{},[31665,188814,188815],{"encoding":31667},"4 + 2 \\times 256 + 256 \\times 256",[34,188817,188819,188837,188855,188873,188891],{"className":188818,"ariaHidden":758},[31672],[34,188820,188822,188825,188828,188831,188834],{"className":188821},[31676],[34,188823],{"className":188824,"style":35942},[31680],[34,188826,15932],{"className":188827},[31685],[34,188829],{"className":188830,"style":32604},[31845],[34,188832,223],{"className":188833},[32024],[34,188835],{"className":188836,"style":32604},[31845],[34,188838,188840,188843,188846,188849,188852],{"className":188839},[31676],[34,188841],{"className":188842,"style":35942},[31680],[34,188844,768],{"className":188845},[31685],[34,188847],{"className":188848,"style":32604},[31845],[34,188850,42071],{"className":188851},[32024],[34,188853],{"className":188854,"style":32604},[31845],[34,188856,188858,188861,188864,188867,188870],{"className":188857},[31676],[34,188859],{"className":188860,"style":35942},[31680],[34,188862,95418],{"className":188863},[31685],[34,188865],{"className":188866,"style":32604},[31845],[34,188868,223],{"className":188869},[32024],[34,188871],{"className":188872,"style":32604},[31845],[34,188874,188876,188879,188882,188885,188888],{"className":188875},[31676],[34,188877],{"className":188878,"style":35942},[31680],[34,188880,95418],{"className":188881},[31685],[34,188883],{"className":188884,"style":32604},[31845],[34,188886,42071],{"className":188887},[32024],[34,188889],{"className":188890,"style":32604},[31845],[34,188892,188894,188897],{"className":188893},[31676],[34,188895],{"className":188896,"style":32617},[31680],[34,188898,95418],{"className":188899},[31685],[15,188901,174094],{},[15,188903,188904],{},[34,188905,188907,188934],{"className":188906},[31642],[34,188908,188910],{"className":188909},[31646],[31648,188911,188912],{"xmlns":31650},[31652,188913,188914,188931],{},[31655,188915,188916,188918,188920,188922,188924,188926,188928],{},[31749,188917,15932],{},[31662,188919,223],{},[31749,188921,167308],{},[31662,188923,223],{},[31749,188925,188623],{},[31662,188927,202],{},[31749,188929,188930],{},"66052",[31665,188932,188933],{"encoding":31667},"4 + 512 + 65536 = 66052",[34,188935,188937,188955,188973,188991],{"className":188936,"ariaHidden":758},[31672],[34,188938,188940,188943,188946,188949,188952],{"className":188939},[31676],[34,188941],{"className":188942,"style":35942},[31680],[34,188944,15932],{"className":188945},[31685],[34,188947],{"className":188948,"style":32604},[31845],[34,188950,223],{"className":188951},[32024],[34,188953],{"className":188954,"style":32604},[31845],[34,188956,188958,188961,188964,188967,188970],{"className":188957},[31676],[34,188959],{"className":188960,"style":35942},[31680],[34,188962,167308],{"className":188963},[31685],[34,188965],{"className":188966,"style":32604},[31845],[34,188968,223],{"className":188969},[32024],[34,188971],{"className":188972,"style":32604},[31845],[34,188974,188976,188979,188982,188985,188988],{"className":188975},[31676],[34,188977],{"className":188978,"style":32617},[31680],[34,188980,188623],{"className":188981},[31685],[34,188983],{"className":188984,"style":31846},[31845],[34,188986,202],{"className":188987},[31850],[34,188989],{"className":188990,"style":31846},[31845],[34,188992,188994,188997],{"className":188993},[31676],[34,188995],{"className":188996,"style":32617},[31680],[34,188998,188930],{"className":188999},[31685],[15,189001,189002],{},"所以最大文件长度为：",[15,189004,189005],{},[34,189006,189008,189038],{"className":189007},[31642],[34,189009,189011],{"className":189010},[31646],[31648,189012,189013],{"xmlns":31650},[31652,189014,189015,189035],{},[31655,189016,189017,189019,189021,189023,189025,189027,189029,189031,189033],{},[31749,189018,188930],{},[31662,189020,42071],{},[31749,189022,640],{},[31658,189024,38691],{},[31658,189026,54224],{},[31662,189028,202],{},[31749,189030,188930],{},[31658,189032,38691],{},[31658,189034,54224],{},[31665,189036,189037],{"encoding":31667},"66052 \\times 1KB = 66052KB",[34,189039,189041,189059,189083],{"className":189040,"ariaHidden":758},[31672],[34,189042,189044,189047,189050,189053,189056],{"className":189043},[31676],[34,189045],{"className":189046,"style":35942},[31680],[34,189048,188930],{"className":189049},[31685],[34,189051],{"className":189052,"style":32604},[31845],[34,189054,42071],{"className":189055},[32024],[34,189057],{"className":189058,"style":32604},[31845],[34,189060,189062,189065,189068,189071,189074,189077,189080],{"className":189061},[31676],[34,189063],{"className":189064,"style":33218},[31680],[34,189066,640],{"className":189067},[31685],[34,189069,38691],{"className":189070,"style":38706},[31685,31686],[34,189072,54224],{"className":189073,"style":54252},[31685,31686],[34,189075],{"className":189076,"style":31846},[31845],[34,189078,202],{"className":189079},[31850],[34,189081],{"className":189082,"style":31846},[31845],[34,189084,189086,189089,189092,189095],{"className":189085},[31676],[34,189087],{"className":189088,"style":33218},[31680],[34,189090,188930],{"className":189091},[31685],[34,189093,38691],{"className":189094,"style":38706},[31685,31686],[34,189096,54224],{"className":189097,"style":54252},[31685,31686],[15,189099,189100],{},"换算成字节：",[15,189102,189103],{},[34,189104,189106,189131],{"className":189105},[31642],[34,189107,189109],{"className":189108},[31646],[31648,189110,189111],{"xmlns":31650},[31652,189112,189113,189128],{},[31655,189114,189115,189117,189119,189121,189123,189126],{},[31749,189116,188930],{},[31662,189118,42071],{},[31749,189120,177393],{},[31662,189122,202],{},[31749,189124,189125],{},"67637248",[31658,189127,54224],{},[31665,189129,189130],{"encoding":31667},"66052 \\times 1024 = 67637248B",[34,189132,189134,189152,189170],{"className":189133,"ariaHidden":758},[31672],[34,189135,189137,189140,189143,189146,189149],{"className":189136},[31676],[34,189138],{"className":189139,"style":35942},[31680],[34,189141,188930],{"className":189142},[31685],[34,189144],{"className":189145,"style":32604},[31845],[34,189147,42071],{"className":189148},[32024],[34,189150],{"className":189151,"style":32604},[31845],[34,189153,189155,189158,189161,189164,189167],{"className":189154},[31676],[34,189156],{"className":189157,"style":32617},[31680],[34,189159,177393],{"className":189160},[31685],[34,189162],{"className":189163,"style":31846},[31845],[34,189165,202],{"className":189166},[31850],[34,189168],{"className":189169,"style":31846},[31845],[34,189171,189173,189176,189179],{"className":189172},[31676],[34,189174],{"className":189175,"style":33218},[31680],[34,189177,189125],{"className":189178},[31685],[34,189180,54224],{"className":189181,"style":54252},[31685,31686],[15,189183,189184],{},"换算成 MiB：",[15,189186,189187],{},[34,189188,189190,189219],{"className":189189},[31642],[34,189191,189193],{"className":189192},[31646],[31648,189194,189195],{"xmlns":31650},[31652,189196,189197,189216],{},[31655,189198,189199,189201,189203,189205,189207,189210,189212,189214],{},[31749,189200,188930],{},[31658,189202,38080],{"mathvariant":38079},[31749,189204,177393],{},[31662,189206,168560],{},[31749,189208,189209],{},"64.50",[31658,189211,98360],{},[31658,189213,31955],{},[31658,189215,54224],{},[31665,189217,189218],{"encoding":31667},"66052 \u002F 1024 \\approx 64.50MiB",[34,189220,189222,189241],{"className":189221,"ariaHidden":758},[31672],[34,189223,189225,189228,189232,189235,189238],{"className":189224},[31676],[34,189226],{"className":189227,"style":31774},[31680],[34,189229,189231],{"className":189230},[31685],"66052\u002F1024",[34,189233],{"className":189234,"style":31846},[31845],[34,189236,168560],{"className":189237},[31850],[34,189239],{"className":189240,"style":31846},[31845],[34,189242,189244,189247,189250,189253,189256],{"className":189243},[31676],[34,189245],{"className":189246,"style":33218},[31680],[34,189248,189209],{"className":189249},[31685],[34,189251,98360],{"className":189252,"style":33389},[31685,31686],[34,189254,31955],{"className":189255},[31685,31686],[34,189257,54224],{"className":189258,"style":54252},[31685,31686],[32188,189260],{},[253,189262,174561],{"id":174560},[15,189264,189265],{},"可表示的单个文件最大长度为：",[15,189267,189268],{},[34,189269,189271,189299],{"className":189270},[31642],[34,189272,189274],{"className":189273},[31646],[31648,189275,189276],{"xmlns":31650},[31652,189277,189278,189296],{},[31655,189279,189280],{},[126400,189281,189282],{"notation":126402},[71090,189283,189284],{"scriptlevel":459,"displaystyle":702},[71090,189285,189286],{"scriptlevel":459,"displaystyle":702},[71090,189287,189288],{"scriptlevel":459,"displaystyle":758},[31655,189289,189290,189292,189294],{},[31749,189291,188930],{},[31658,189293,38691],{},[31658,189295,54224],{},[31665,189297,189298],{"encoding":31667},"\\boxed{66052KB}",[34,189300,189302],{"className":189301,"ariaHidden":758},[31672],[34,189303,189305,189308],{"className":189304},[31676],[34,189306],{"className":189307,"style":126424},[31680],[34,189309,189311],{"className":189310},[31685],[34,189312,189314,189354],{"className":189313},[31796,31797],[34,189315,189317,189351],{"className":189316},[31801],[34,189318,189320,189343],{"className":189319,"style":126437},[31805],[34,189321,189322,189325],{"style":126440},[34,189323],{"className":189324,"style":126444},[31813],[34,189326,189328],{"className":189327},[126448],[34,189329,189331],{"className":189330},[31685],[34,189332,189334,189337,189340],{"className":189333},[31685],[34,189335,188930],{"className":189336},[31685],[34,189338,38691],{"className":189339,"style":38706},[31685,31686],[34,189341,54224],{"className":189342,"style":54252},[31685,31686],[34,189344,189345,189348],{"style":126463},[34,189346],{"className":189347,"style":126444},[31813],[34,189349],{"className":189350,"style":126472},[126470,126471],[34,189352,31829],{"className":189353},[31828],[34,189355,189357],{"className":189356},[31801],[34,189358,189360],{"className":189359,"style":126482},[31805],[34,189361],{},[15,189363,129204],{},[15,189365,189366],{},[34,189367,189369,189395],{"className":189368},[31642],[34,189370,189372],{"className":189371},[31646],[31648,189373,189374],{"xmlns":31650},[31652,189375,189376,189392],{},[31655,189377,189378],{},[126400,189379,189380],{"notation":126402},[71090,189381,189382],{"scriptlevel":459,"displaystyle":702},[71090,189383,189384],{"scriptlevel":459,"displaystyle":702},[71090,189385,189386],{"scriptlevel":459,"displaystyle":758},[31655,189387,189388,189390],{},[31749,189389,189125],{},[31658,189391,54224],{},[31665,189393,189394],{"encoding":31667},"\\boxed{67637248B}",[34,189396,189398],{"className":189397,"ariaHidden":758},[31672],[34,189399,189401,189404],{"className":189400},[31676],[34,189402],{"className":189403,"style":126424},[31680],[34,189405,189407],{"className":189406},[31685],[34,189408,189410,189447],{"className":189409},[31796,31797],[34,189411,189413,189444],{"className":189412},[31801],[34,189414,189416,189436],{"className":189415,"style":126437},[31805],[34,189417,189418,189421],{"style":126440},[34,189419],{"className":189420,"style":126444},[31813],[34,189422,189424],{"className":189423},[126448],[34,189425,189427],{"className":189426},[31685],[34,189428,189430,189433],{"className":189429},[31685],[34,189431,189125],{"className":189432},[31685],[34,189434,54224],{"className":189435,"style":54252},[31685,31686],[34,189437,189438,189441],{"style":126463},[34,189439],{"className":189440,"style":126444},[31813],[34,189442],{"className":189443,"style":126472},[126470,126471],[34,189445,31829],{"className":189446},[31828],[34,189448,189450],{"className":189449},[31801],[34,189451,189453],{"className":189452,"style":126482},[31805],[34,189454],{},[15,189456,169980],{},[15,189458,189459],{},[34,189460,189462,189492],{"className":189461},[31642],[34,189463,189465],{"className":189464},[31646],[31648,189466,189467],{"xmlns":31650},[31652,189468,189469,189489],{},[31655,189470,189471],{},[126400,189472,189473],{"notation":126402},[71090,189474,189475],{"scriptlevel":459,"displaystyle":702},[71090,189476,189477],{"scriptlevel":459,"displaystyle":702},[71090,189478,189479],{"scriptlevel":459,"displaystyle":758},[31655,189480,189481,189483,189485,189487],{},[31749,189482,189209],{},[31658,189484,98360],{},[31658,189486,31955],{},[31658,189488,54224],{},[31665,189490,189491],{"encoding":31667},"\\boxed{64.50MiB}",[34,189493,189495],{"className":189494,"ariaHidden":758},[31672],[34,189496,189498,189501],{"className":189497},[31676],[34,189499],{"className":189500,"style":126424},[31680],[34,189502,189504],{"className":189503},[31685],[34,189505,189507,189550],{"className":189506},[31796,31797],[34,189508,189510,189547],{"className":189509},[31801],[34,189511,189513,189539],{"className":189512,"style":126437},[31805],[34,189514,189515,189518],{"style":126440},[34,189516],{"className":189517,"style":126444},[31813],[34,189519,189521],{"className":189520},[126448],[34,189522,189524],{"className":189523},[31685],[34,189525,189527,189530,189533,189536],{"className":189526},[31685],[34,189528,189209],{"className":189529},[31685],[34,189531,98360],{"className":189532,"style":33389},[31685,31686],[34,189534,31955],{"className":189535},[31685,31686],[34,189537,54224],{"className":189538,"style":54252},[31685,31686],[34,189540,189541,189544],{"style":126463},[34,189542],{"className":189543,"style":126444},[31813],[34,189545],{"className":189546,"style":126472},[126470,126471],[34,189548,31829],{"className":189549},[31828],[34,189551,189553],{"className":189552},[31801],[34,189554,189556],{"className":189555,"style":126482},[31805],[34,189557],{},[32188,189559],{},[19,189561,189563],{"id":189562},"_6-unixlinux-文件访问偏移-263168-字节处数据需要几次间接","6. UNIX\u002FLinux 文件访问偏移 263168 字节处数据需要几次间接？",[253,189565,162043],{"id":189566},"题目内容-5",[15,189568,189569],{},"一个UNIX\u002FLinux文件，如果一个盘块的大小为1KB，每个盘块占4个字节，那么，若进程欲访问偏移为263168字节处的数据，需经过几次间接？",[15,189571,8882],{},[15,189573,189574],{},"题目中的“每个盘块占4个字节”应理解为“每个盘块号\u002F地址项占4个字节”。",[32188,189576],{},[253,189578,175302],{"id":189579},"答案和讲解-5",[15,189581,173129],{},[15,189583,189584],{},"盘块大小：",[15,189586,189587],{},[34,189588,189590,189613],{"className":189589},[31642],[34,189591,189593],{"className":189592},[31646],[31648,189594,189595],{"xmlns":31650},[31652,189596,189597,189611],{},[31655,189598,189599,189601,189603,189605,189607,189609],{},[31749,189600,640],{},[31658,189602,38691],{},[31658,189604,54224],{},[31662,189606,202],{},[31749,189608,177393],{},[31658,189610,54224],{},[31665,189612,177398],{"encoding":31667},[34,189614,189616,189640],{"className":189615,"ariaHidden":758},[31672],[34,189617,189619,189622,189625,189628,189631,189634,189637],{"className":189618},[31676],[34,189620],{"className":189621,"style":33218},[31680],[34,189623,640],{"className":189624},[31685],[34,189626,38691],{"className":189627,"style":38706},[31685,31686],[34,189629,54224],{"className":189630,"style":54252},[31685,31686],[34,189632],{"className":189633,"style":31846},[31845],[34,189635,202],{"className":189636},[31850],[34,189638],{"className":189639,"style":31846},[31845],[34,189641,189643,189646,189649],{"className":189642},[31676],[34,189644],{"className":189645,"style":33218},[31680],[34,189647,177393],{"className":189648},[31685],[34,189650,54224],{"className":189651,"style":54252},[31685,31686],[15,189653,189654],{},"每个盘块号，也就是每个地址项大小：",[15,189656,189657],{},[34,189658,189660,189675],{"className":189659},[31642],[34,189661,189663],{"className":189662},[31646],[31648,189664,189665],{"xmlns":31650},[31652,189666,189667,189673],{},[31655,189668,189669,189671],{},[31749,189670,15932],{},[31658,189672,54224],{},[31665,189674,187868],{"encoding":31667},[34,189676,189678],{"className":189677,"ariaHidden":758},[31672],[34,189679,189681,189684,189687],{"className":189680},[31676],[34,189682],{"className":189683,"style":33218},[31680],[34,189685,15932],{"className":189686},[31685],[34,189688,54224],{"className":189689,"style":54252},[31685,31686],[15,189691,189692],{},"目标偏移量：",[15,189694,189695],{},[34,189696,189698,189715],{"className":189697},[31642],[34,189699,189701],{"className":189700},[31646],[31648,189702,189703],{"xmlns":31650},[31652,189704,189705,189712],{},[31655,189706,189707,189710],{},[31749,189708,189709],{},"263168",[31658,189711,54224],{},[31665,189713,189714],{"encoding":31667},"263168B",[34,189716,189718],{"className":189717,"ariaHidden":758},[31672],[34,189719,189721,189724,189727],{"className":189720},[31676],[34,189722],{"className":189723,"style":33218},[31680],[34,189725,189709],{"className":189726},[31685],[34,189728,54224],{"className":189729,"style":54252},[31685,31686],[32188,189731],{},[253,189733,189735],{"id":189734},"_61-先求该偏移位于第几个逻辑块","6.1 先求该偏移位于第几个逻辑块",[15,189737,189738],{},"逻辑块号从 0 开始编号。",[15,189740,189741],{},"目标逻辑块号为：",[15,189743,189744],{},[34,189745,189747,189769],{"className":189746},[31642],[34,189748,189750],{"className":189749},[31646],[31648,189751,189752],{"xmlns":31650},[31652,189753,189754,189766],{},[31655,189755,189756,189758,189760,189762,189764],{},[31662,189757,173327],{"stretchy":702},[31749,189759,189709],{},[31658,189761,38080],{"mathvariant":38079},[31749,189763,177393],{},[31662,189765,173336],{"stretchy":702},[31665,189767,189768],{"encoding":31667},"\\lfloor 263168 \u002F 1024 \\rfloor",[34,189770,189772],{"className":189771,"ariaHidden":758},[31672],[34,189773,189775,189778,189781,189785],{"className":189774},[31676],[34,189776],{"className":189777,"style":31774},[31680],[34,189779,173327],{"className":189780},[31782],[34,189782,189784],{"className":189783},[31685],"263168\u002F1024",[34,189786,173336],{"className":189787},[31690],[15,189789,131336],{},[15,189791,189792],{},[34,189793,189795,189818],{"className":189794},[31642],[34,189796,189798],{"className":189797},[31646],[31648,189799,189800],{"xmlns":31650},[31652,189801,189802,189815],{},[31655,189803,189804,189806,189808,189811,189813],{},[31749,189805,177393],{},[31662,189807,42071],{},[31749,189809,189810],{},"257",[31662,189812,202],{},[31749,189814,189709],{},[31665,189816,189817],{"encoding":31667},"1024 \\times 257 = 263168",[34,189819,189821,189839,189857],{"className":189820,"ariaHidden":758},[31672],[34,189822,189824,189827,189830,189833,189836],{"className":189823},[31676],[34,189825],{"className":189826,"style":35942},[31680],[34,189828,177393],{"className":189829},[31685],[34,189831],{"className":189832,"style":32604},[31845],[34,189834,42071],{"className":189835},[32024],[34,189837],{"className":189838,"style":32604},[31845],[34,189840,189842,189845,189848,189851,189854],{"className":189841},[31676],[34,189843],{"className":189844,"style":32617},[31680],[34,189846,189810],{"className":189847},[31685],[34,189849],{"className":189850,"style":31846},[31845],[34,189852,202],{"className":189853},[31850],[34,189855],{"className":189856,"style":31846},[31845],[34,189858,189860,189863],{"className":189859},[31676],[34,189861],{"className":189862,"style":32617},[31680],[34,189864,189709],{"className":189865},[31685],[15,189867,189868],{},"所以该偏移正好位于逻辑块：",[15,189870,189871],{},[34,189872,189874,189887],{"className":189873},[31642],[34,189875,189877],{"className":189876},[31646],[31648,189878,189879],{"xmlns":31650},[31652,189880,189881,189885],{},[31655,189882,189883],{},[31749,189884,189810],{},[31665,189886,189810],{"encoding":31667},[34,189888,189890],{"className":189889,"ariaHidden":758},[31672],[34,189891,189893,189896],{"className":189892},[31676],[34,189894],{"className":189895,"style":32617},[31680],[34,189897,189810],{"className":189898},[31685],[15,189900,189901],{},"的起始位置。",[15,189903,189904],{},"如果从 1 开始数，它就是第：",[15,189906,189907],{},[34,189908,189910,189933],{"className":189909},[31642],[34,189911,189913],{"className":189912},[31646],[31648,189914,189915],{"xmlns":31650},[31652,189916,189917,189930],{},[31655,189918,189919,189921,189923,189925,189927],{},[31749,189920,189810],{},[31662,189922,223],{},[31749,189924,640],{},[31662,189926,202],{},[31749,189928,189929],{},"258",[31665,189931,189932],{"encoding":31667},"257 + 1 = 258",[34,189934,189936,189954,189972],{"className":189935,"ariaHidden":758},[31672],[34,189937,189939,189942,189945,189948,189951],{"className":189938},[31676],[34,189940],{"className":189941,"style":35942},[31680],[34,189943,189810],{"className":189944},[31685],[34,189946],{"className":189947,"style":32604},[31845],[34,189949,223],{"className":189950},[32024],[34,189952],{"className":189953,"style":32604},[31845],[34,189955,189957,189960,189963,189966,189969],{"className":189956},[31676],[34,189958],{"className":189959,"style":32617},[31680],[34,189961,640],{"className":189962},[31685],[34,189964],{"className":189965,"style":31846},[31845],[34,189967,202],{"className":189968},[31850],[34,189970],{"className":189971,"style":31846},[31845],[34,189973,189975,189978],{"className":189974},[31676],[34,189976],{"className":189977,"style":32617},[31680],[34,189979,189929],{"className":189980},[31685],[15,189982,188203],{},[32188,189984],{},[253,189986,189988],{"id":189987},"_62-一个间接索引块可以存多少个块号","6.2 一个间接索引块可以存多少个块号？",[15,189990,188070],{},[15,189992,189993],{},[34,189994,189996,190011],{"className":189995},[31642],[34,189997,189999],{"className":189998},[31646],[31648,190000,190001],{"xmlns":31650},[31652,190002,190003,190009],{},[31655,190004,190005,190007],{},[31749,190006,177393],{},[31658,190008,54224],{},[31665,190010,188091],{"encoding":31667},[34,190012,190014],{"className":190013,"ariaHidden":758},[31672],[34,190015,190017,190020,190023],{"className":190016},[31676],[34,190018],{"className":190019,"style":33218},[31680],[34,190021,177393],{"className":190022},[31685],[34,190024,54224],{"className":190025,"style":54252},[31685,31686],[15,190027,190028],{},"每个块号大小为：",[15,190030,190031],{},[34,190032,190034,190049],{"className":190033},[31642],[34,190035,190037],{"className":190036},[31646],[31648,190038,190039],{"xmlns":31650},[31652,190040,190041,190047],{},[31655,190042,190043,190045],{},[31749,190044,15932],{},[31658,190046,54224],{},[31665,190048,187868],{"encoding":31667},[34,190050,190052],{"className":190051,"ariaHidden":758},[31672],[34,190053,190055,190058,190061],{"className":190054},[31676],[34,190056],{"className":190057,"style":33218},[31680],[34,190059,15932],{"className":190060},[31685],[34,190062,54224],{"className":190063,"style":54252},[31685,31686],[15,190065,190066],{},"所以一个索引块可以存放块号数：",[15,190068,190069],{},[34,190070,190072,190093],{"className":190071},[31642],[34,190073,190075],{"className":190074},[31646],[31648,190076,190077],{"xmlns":31650},[31652,190078,190079,190091],{},[31655,190080,190081,190083,190085,190087,190089],{},[31749,190082,177393],{},[31658,190084,38080],{"mathvariant":38079},[31749,190086,15932],{},[31662,190088,202],{},[31749,190090,95418],{},[31665,190092,188136],{"encoding":31667},[34,190094,190096,190114],{"className":190095,"ariaHidden":758},[31672],[34,190097,190099,190102,190105,190108,190111],{"className":190098},[31676],[34,190100],{"className":190101,"style":31774},[31680],[34,190103,188149],{"className":190104},[31685],[34,190106],{"className":190107,"style":31846},[31845],[34,190109,202],{"className":190110},[31850],[34,190112],{"className":190113,"style":31846},[31845],[34,190115,190117,190120],{"className":190116},[31676],[34,190118],{"className":190119,"style":32617},[31680],[34,190121,95418],{"className":190122},[31685],[15,190124,188170],{},[15,190126,190127],{},[34,190128,190130,190143],{"className":190129},[31642],[34,190131,190133],{"className":190132},[31646],[31648,190134,190135],{"xmlns":31650},[31652,190136,190137,190141],{},[31655,190138,190139],{},[31749,190140,95418],{},[31665,190142,95418],{"encoding":31667},[34,190144,190146],{"className":190145,"ariaHidden":758},[31672],[34,190147,190149,190152],{"className":190148},[31676],[34,190150],{"className":190151,"style":32617},[31680],[34,190153,95418],{"className":190154},[31685],[15,190156,188203],{},[32188,190158],{},[253,190160,190162],{"id":190161},"_63-判断需要几次间接","6.3 判断需要几次间接",[15,190164,190165],{},"UNIX\u002FLinux inode 通常使用：",[15,190167,190168],{},"直接地址项",[15,190170,190171],{},"一级间接地址项",[15,190173,190174],{},"二级间接地址项",[15,190176,190177],{},"三级间接地址项",[15,190179,190180],{},"Stallings 中 UNIX inode 也采用 direct pointers 加 single、double、triple indirect pointers 的方式。",[15,190182,190183],{},"常见 UNIX\u002FLinux inode 有 12 个直接块地址。",[15,190185,190186],{},"直接块覆盖逻辑块：",[15,190188,190189],{},[34,190190,190192,190211],{"className":190191},[31642],[34,190193,190195],{"className":190194},[31646],[31648,190196,190197],{"xmlns":31650},[31652,190198,190199,190208],{},[31655,190200,190201,190203,190206],{},[31749,190202,459],{},[31662,190204,190205],{},"∼",[31749,190207,39620],{},[31665,190209,190210],{"encoding":31667},"0 \\sim 11",[34,190212,190214,190232],{"className":190213,"ariaHidden":758},[31672],[34,190215,190217,190220,190223,190226,190229],{"className":190216},[31676],[34,190218],{"className":190219,"style":32617},[31680],[34,190221,459],{"className":190222},[31685],[34,190224],{"className":190225,"style":31846},[31845],[34,190227,190205],{"className":190228},[31850],[34,190230],{"className":190231,"style":31846},[31845],[34,190233,190235,190238],{"className":190234},[31676],[34,190236],{"className":190237,"style":32617},[31680],[34,190239,39620],{"className":190240},[31685],[15,190242,190243],{},"一级间接块可以再覆盖：",[15,190245,190246],{},[34,190247,190249,190262],{"className":190248},[31642],[34,190250,190252],{"className":190251},[31646],[31648,190253,190254],{"xmlns":31650},[31652,190255,190256,190260],{},[31655,190257,190258],{},[31749,190259,95418],{},[31665,190261,95418],{"encoding":31667},[34,190263,190265],{"className":190264,"ariaHidden":758},[31672],[34,190266,190268,190271],{"className":190267},[31676],[34,190269],{"className":190270,"style":32617},[31680],[34,190272,95418],{"className":190273},[31685],[15,190275,188203],{},[15,190277,190278],{},"所以一级间接覆盖逻辑块：",[15,190280,190281],{},[34,190282,190284,190303],{"className":190283},[31642],[34,190285,190287],{"className":190286},[31646],[31648,190288,190289],{"xmlns":31650},[31652,190290,190291,190300],{},[31655,190292,190293,190295,190297],{},[31749,190294,71869],{},[31662,190296,190205],{},[31749,190298,190299],{},"267",[31665,190301,190302],{"encoding":31667},"12 \\sim 267",[34,190304,190306,190324],{"className":190305,"ariaHidden":758},[31672],[34,190307,190309,190312,190315,190318,190321],{"className":190308},[31676],[34,190310],{"className":190311,"style":32617},[31680],[34,190313,71869],{"className":190314},[31685],[34,190316],{"className":190317,"style":31846},[31845],[34,190319,190205],{"className":190320},[31850],[34,190322],{"className":190323,"style":31846},[31845],[34,190325,190327,190330],{"className":190326},[31676],[34,190328],{"className":190329,"style":32617},[31680],[34,190331,190299],{"className":190332},[31685],[15,190334,131336],{},[15,190336,190337],{},[34,190338,190340,190366],{"className":190339},[31642],[34,190341,190343],{"className":190342},[31646],[31648,190344,190345],{"xmlns":31650},[31652,190346,190347,190363],{},[31655,190348,190349,190351,190353,190355,190357,190359,190361],{},[31749,190350,71869],{},[31662,190352,223],{},[31749,190354,95418],{},[31662,190356,32226],{},[31749,190358,640],{},[31662,190360,202],{},[31749,190362,190299],{},[31665,190364,190365],{"encoding":31667},"12 + 256 - 1 = 267",[34,190367,190369,190387,190405,190423],{"className":190368,"ariaHidden":758},[31672],[34,190370,190372,190375,190378,190381,190384],{"className":190371},[31676],[34,190373],{"className":190374,"style":35942},[31680],[34,190376,71869],{"className":190377},[31685],[34,190379],{"className":190380,"style":32604},[31845],[34,190382,223],{"className":190383},[32024],[34,190385],{"className":190386,"style":32604},[31845],[34,190388,190390,190393,190396,190399,190402],{"className":190389},[31676],[34,190391],{"className":190392,"style":35942},[31680],[34,190394,95418],{"className":190395},[31685],[34,190397],{"className":190398,"style":32604},[31845],[34,190400,32226],{"className":190401},[32024],[34,190403],{"className":190404,"style":32604},[31845],[34,190406,190408,190411,190414,190417,190420],{"className":190407},[31676],[34,190409],{"className":190410,"style":32617},[31680],[34,190412,640],{"className":190413},[31685],[34,190415],{"className":190416,"style":31846},[31845],[34,190418,202],{"className":190419},[31850],[34,190421],{"className":190422,"style":31846},[31845],[34,190424,190426,190429],{"className":190425},[31676],[34,190427],{"className":190428,"style":32617},[31680],[34,190430,190299],{"className":190431},[31685],[15,190433,189741],{},[15,190435,190436],{},[34,190437,190439,190452],{"className":190438},[31642],[34,190440,190442],{"className":190441},[31646],[31648,190443,190444],{"xmlns":31650},[31652,190445,190446,190450],{},[31655,190447,190448],{},[31749,190449,189810],{},[31665,190451,189810],{"encoding":31667},[34,190453,190455],{"className":190454,"ariaHidden":758},[31672],[34,190456,190458,190461],{"className":190457},[31676],[34,190459],{"className":190460,"style":32617},[31680],[34,190462,189810],{"className":190463},[31685],[15,190465,190466],{},"它落在：",[15,190468,190469],{},[34,190470,190472,190489],{"className":190471},[31642],[34,190473,190475],{"className":190474},[31646],[31648,190476,190477],{"xmlns":31650},[31652,190478,190479,190487],{},[31655,190480,190481,190483,190485],{},[31749,190482,71869],{},[31662,190484,190205],{},[31749,190486,190299],{},[31665,190488,190302],{"encoding":31667},[34,190490,190492,190510],{"className":190491,"ariaHidden":758},[31672],[34,190493,190495,190498,190501,190504,190507],{"className":190494},[31676],[34,190496],{"className":190497,"style":32617},[31680],[34,190499,71869],{"className":190500},[31685],[34,190502],{"className":190503,"style":31846},[31845],[34,190505,190205],{"className":190506},[31850],[34,190508],{"className":190509,"style":31846},[31845],[34,190511,190513,190516],{"className":190512},[31676],[34,190514],{"className":190515,"style":32617},[31680],[34,190517,190299],{"className":190518},[31685],[15,190520,190521],{},"范围内。",[15,190523,190524],{},"因此它只需要通过一级间接索引访问。",[15,190526,190527],{},"所以需要经过：",[15,190529,190530],{},[34,190531,190533,190560],{"className":190532},[31642],[34,190534,190536],{"className":190535},[31646],[31648,190537,190538],{"xmlns":31650},[31652,190539,190540,190557],{},[31655,190541,190542],{},[126400,190543,190544],{"notation":126402},[71090,190545,190546],{"scriptlevel":459,"displaystyle":702},[71090,190547,190548],{"scriptlevel":459,"displaystyle":702},[71090,190549,190550],{"scriptlevel":459,"displaystyle":758},[31655,190551,190552,190554],{},[31749,190553,640],{},[39361,190555,190556],{}," 次间接",[31665,190558,190559],{"encoding":31667},"\\boxed{1 \\text{ 次间接}}",[34,190561,190563],{"className":190562,"ariaHidden":758},[31672],[34,190564,190566,190569],{"className":190565},[31676],[34,190567],{"className":190568,"style":126424},[31680],[34,190570,190572],{"className":190571},[31685],[34,190573,190575,190619],{"className":190574},[31796,31797],[34,190576,190578,190616],{"className":190577},[31801],[34,190579,190581,190608],{"className":190580,"style":126437},[31805],[34,190582,190583,190586],{"style":126440},[34,190584],{"className":190585,"style":126444},[31813],[34,190587,190589],{"className":190588},[126448],[34,190590,190592],{"className":190591},[31685],[34,190593,190595,190598],{"className":190594},[31685],[34,190596,640],{"className":190597},[31685],[34,190599,190601,190604],{"className":190600},[31685,2638],[34,190602,130298],{"className":190603},[31685],[34,190605,190607],{"className":190606},[31685,105462],"次间接",[34,190609,190610,190613],{"style":126463},[34,190611],{"className":190612,"style":126444},[31813],[34,190614],{"className":190615,"style":126472},[126470,126471],[34,190617,31829],{"className":190618},[31828],[34,190620,190622],{"className":190621},[31801],[34,190623,190625],{"className":190624,"style":126482},[31805],[34,190626],{},[32188,190628],{},[253,190630,166511],{"id":166511},[15,190632,190633],{},"即使某些传统 UNIX 使用 10 个直接块地址，结论仍然一样。",[15,190635,190636],{},"如果有 10 个直接块：",[15,190638,190639],{},"直接块覆盖：",[15,190641,190642],{},[34,190643,190645,190663],{"className":190644},[31642],[34,190646,190648],{"className":190647},[31646],[31648,190649,190650],{"xmlns":31650},[31652,190651,190652,190660],{},[31655,190653,190654,190656,190658],{},[31749,190655,459],{},[31662,190657,190205],{},[31749,190659,36732],{},[31665,190661,190662],{"encoding":31667},"0 \\sim 9",[34,190664,190666,190684],{"className":190665,"ariaHidden":758},[31672],[34,190667,190669,190672,190675,190678,190681],{"className":190668},[31676],[34,190670],{"className":190671,"style":32617},[31680],[34,190673,459],{"className":190674},[31685],[34,190676],{"className":190677,"style":31846},[31845],[34,190679,190205],{"className":190680},[31850],[34,190682],{"className":190683,"style":31846},[31845],[34,190685,190687,190690],{"className":190686},[31676],[34,190688],{"className":190689,"style":32617},[31680],[34,190691,36732],{"className":190692},[31685],[15,190694,190695],{},"一级间接覆盖：",[15,190697,190698],{},[34,190699,190701,190720],{"className":190700},[31642],[34,190702,190704],{"className":190703},[31646],[31648,190705,190706],{"xmlns":31650},[31652,190707,190708,190717],{},[31655,190709,190710,190712,190714],{},[31749,190711,3244],{},[31662,190713,190205],{},[31749,190715,190716],{},"265",[31665,190718,190719],{"encoding":31667},"10 \\sim 265",[34,190721,190723,190741],{"className":190722,"ariaHidden":758},[31672],[34,190724,190726,190729,190732,190735,190738],{"className":190725},[31676],[34,190727],{"className":190728,"style":32617},[31680],[34,190730,3244],{"className":190731},[31685],[34,190733],{"className":190734,"style":31846},[31845],[34,190736,190205],{"className":190737},[31850],[34,190739],{"className":190740,"style":31846},[31845],[34,190742,190744,190747],{"className":190743},[31676],[34,190745],{"className":190746,"style":32617},[31680],[34,190748,190716],{"className":190749},[31685],[15,190751,190752],{},"目标逻辑块：",[15,190754,190755],{},[34,190756,190758,190771],{"className":190757},[31642],[34,190759,190761],{"className":190760},[31646],[31648,190762,190763],{"xmlns":31650},[31652,190764,190765,190769],{},[31655,190766,190767],{},[31749,190768,189810],{},[31665,190770,189810],{"encoding":31667},[34,190772,190774],{"className":190773,"ariaHidden":758},[31672],[34,190775,190777,190780],{"className":190776},[31676],[34,190778],{"className":190779,"style":32617},[31680],[34,190781,189810],{"className":190782},[31685],[15,190784,190785],{},"仍然在一级间接范围内。",[15,190787,190788],{},"所以答案依然是：",[15,190790,190791],{},[34,190792,190794,190819],{"className":190793},[31642],[34,190795,190797],{"className":190796},[31646],[31648,190798,190799],{"xmlns":31650},[31652,190800,190801,190817],{},[31655,190802,190803],{},[126400,190804,190805],{"notation":126402},[71090,190806,190807],{"scriptlevel":459,"displaystyle":702},[71090,190808,190809],{"scriptlevel":459,"displaystyle":702},[71090,190810,190811],{"scriptlevel":459,"displaystyle":758},[31655,190812,190813,190815],{},[31749,190814,640],{},[39361,190816,190556],{},[31665,190818,190559],{"encoding":31667},[34,190820,190822],{"className":190821,"ariaHidden":758},[31672],[34,190823,190825,190828],{"className":190824},[31676],[34,190826],{"className":190827,"style":126424},[31680],[34,190829,190831],{"className":190830},[31685],[34,190832,190834,190877],{"className":190833},[31796,31797],[34,190835,190837,190874],{"className":190836},[31801],[34,190838,190840,190866],{"className":190839,"style":126437},[31805],[34,190841,190842,190845],{"style":126440},[34,190843],{"className":190844,"style":126444},[31813],[34,190846,190848],{"className":190847},[126448],[34,190849,190851],{"className":190850},[31685],[34,190852,190854,190857],{"className":190853},[31685],[34,190855,640],{"className":190856},[31685],[34,190858,190860,190863],{"className":190859},[31685,2638],[34,190861,130298],{"className":190862},[31685],[34,190864,190607],{"className":190865},[31685,105462],[34,190867,190868,190871],{"style":126463},[34,190869],{"className":190870,"style":126444},[31813],[34,190872],{"className":190873,"style":126472},[126470,126471],[34,190875,31829],{"className":190876},[31828],[34,190878,190880],{"className":190879},[31801],[34,190881,190883],{"className":190882,"style":126482},[31805],[34,190884],{},[32188,190886],{},[253,190888,190890],{"id":190889},"第-6-题最终答案","第 6 题最终答案",[15,190892,190893],{},"访问偏移：",[15,190895,190896],{},[34,190897,190899,190914],{"className":190898},[31642],[34,190900,190902],{"className":190901},[31646],[31648,190903,190904],{"xmlns":31650},[31652,190905,190906,190912],{},[31655,190907,190908,190910],{},[31749,190909,189709],{},[31658,190911,54224],{},[31665,190913,189714],{"encoding":31667},[34,190915,190917],{"className":190916,"ariaHidden":758},[31672],[34,190918,190920,190923,190926],{"className":190919},[31676],[34,190921],{"className":190922,"style":33218},[31680],[34,190924,189709],{"className":190925},[31685],[34,190927,54224],{"className":190928,"style":54252},[31685,31686],[15,190930,190931],{},"对应逻辑块号：",[15,190933,190934],{},[34,190935,190937,190950],{"className":190936},[31642],[34,190938,190940],{"className":190939},[31646],[31648,190941,190942],{"xmlns":31650},[31652,190943,190944,190948],{},[31655,190945,190946],{},[31749,190947,189810],{},[31665,190949,189810],{"encoding":31667},[34,190951,190953],{"className":190952,"ariaHidden":758},[31672],[34,190954,190956,190959],{"className":190955},[31676],[34,190957],{"className":190958,"style":32617},[31680],[34,190960,189810],{"className":190961},[31685],[15,190963,190964],{},"需要经过：",[15,190966,190967],{},[34,190968,190970,190995],{"className":190969},[31642],[34,190971,190973],{"className":190972},[31646],[31648,190974,190975],{"xmlns":31650},[31652,190976,190977,190993],{},[31655,190978,190979],{},[126400,190980,190981],{"notation":126402},[71090,190982,190983],{"scriptlevel":459,"displaystyle":702},[71090,190984,190985],{"scriptlevel":459,"displaystyle":702},[71090,190986,190987],{"scriptlevel":459,"displaystyle":758},[31655,190988,190989,190991],{},[31749,190990,640],{},[39361,190992,190556],{},[31665,190994,190559],{"encoding":31667},[34,190996,190998],{"className":190997,"ariaHidden":758},[31672],[34,190999,191001,191004],{"className":191000},[31676],[34,191002],{"className":191003,"style":126424},[31680],[34,191005,191007],{"className":191006},[31685],[34,191008,191010,191053],{"className":191009},[31796,31797],[34,191011,191013,191050],{"className":191012},[31801],[34,191014,191016,191042],{"className":191015,"style":126437},[31805],[34,191017,191018,191021],{"style":126440},[34,191019],{"className":191020,"style":126444},[31813],[34,191022,191024],{"className":191023},[126448],[34,191025,191027],{"className":191026},[31685],[34,191028,191030,191033],{"className":191029},[31685],[34,191031,640],{"className":191032},[31685],[34,191034,191036,191039],{"className":191035},[31685,2638],[34,191037,130298],{"className":191038},[31685],[34,191040,190607],{"className":191041},[31685,105462],[34,191043,191044,191047],{"style":126463},[34,191045],{"className":191046,"style":126444},[31813],[34,191048],{"className":191049,"style":126472},[126470,126471],[34,191051,31829],{"className":191052},[31828],[34,191054,191056],{"className":191055},[31801],[34,191057,191059],{"className":191058,"style":126482},[31805],[34,191060],{},[15,191062,129204],{},[15,191064,191065],{},[34,191066,191068,191091],{"className":191067},[31642],[34,191069,191071],{"className":191070},[31646],[31648,191072,191073],{"xmlns":31650},[31652,191074,191075,191088],{},[31655,191076,191077],{},[126400,191078,191079],{"notation":126402},[71090,191080,191081],{"scriptlevel":459,"displaystyle":702},[71090,191082,191083],{"scriptlevel":459,"displaystyle":702},[71090,191084,191085],{"scriptlevel":459,"displaystyle":758},[39361,191086,191087],{},"一级间接索引",[31665,191089,191090],{"encoding":31667},"\\boxed{一级间接索引}",[34,191092,191094],{"className":191093,"ariaHidden":758},[31672],[34,191095,191097,191100],{"className":191096},[31676],[34,191098],{"className":191099,"style":126424},[31680],[34,191101,191103],{"className":191102},[31685],[34,191104,191106,191140],{"className":191105},[31796,31797],[34,191107,191109,191137],{"className":191108},[31801],[34,191110,191112,191129],{"className":191111,"style":126437},[31805],[34,191113,191114,191117],{"style":126440},[34,191115],{"className":191116,"style":126444},[31813],[34,191118,191120],{"className":191119},[126448],[34,191121,191123],{"className":191122},[31685],[34,191124,191126],{"className":191125},[31685],[34,191127,191087],{"className":191128},[31685,105462],[34,191130,191131,191134],{"style":126463},[34,191132],{"className":191133,"style":126444},[31813],[34,191135],{"className":191136,"style":126472},[126470,126471],[34,191138,31829],{"className":191139},[31828],[34,191141,191143],{"className":191142},[31801],[34,191144,191146],{"className":191145,"style":126482},[31805],[34,191147],{},[32188,191149],{},[11,191151,191152],{"id":191152},"汇总答案",[19,191154,174669],{"id":174668},[15,191156,191157],{},"UNIX 文件系统采用：",[15,191159,191160],{},[34,191161,191163,191189],{"className":191162},[31642],[34,191164,191166],{"className":191165},[31646],[31648,191167,191168],{"xmlns":31650},[31652,191169,191170,191186],{},[31655,191171,191172],{},[126400,191173,191174],{"notation":126402},[71090,191175,191176],{"scriptlevel":459,"displaystyle":702},[71090,191177,191178],{"scriptlevel":459,"displaystyle":702},[71090,191179,191180],{"scriptlevel":459,"displaystyle":758},[31655,191181,191182,191184],{},[31658,191183,91994],{},[39361,191185,175329],{},[31665,191187,191188],{"encoding":31667},"\\boxed{D \\text{ 流式文件}}",[34,191190,191192],{"className":191191,"ariaHidden":758},[31672],[34,191193,191195,191198],{"className":191194},[31676],[34,191196],{"className":191197,"style":126424},[31680],[34,191199,191201],{"className":191200},[31685],[34,191202,191204,191247],{"className":191203},[31796,31797],[34,191205,191207,191244],{"className":191206},[31801],[34,191208,191210,191236],{"className":191209,"style":126437},[31805],[34,191211,191212,191215],{"style":126440},[34,191213],{"className":191214,"style":126444},[31813],[34,191216,191218],{"className":191217},[126448],[34,191219,191221],{"className":191220},[31685],[34,191222,191224,191227],{"className":191223},[31685],[34,191225,91994],{"className":191226,"style":34452},[31685,31686],[34,191228,191230,191233],{"className":191229},[31685,2638],[34,191231,130298],{"className":191232},[31685],[34,191234,175354],{"className":191235},[31685,105462],[34,191237,191238,191241],{"style":126463},[34,191239],{"className":191240,"style":126444},[31813],[34,191242],{"className":191243,"style":126472},[126470,126471],[34,191245,31829],{"className":191246},[31828],[34,191248,191250],{"className":191249},[31801],[34,191251,191253],{"className":191252,"style":126482},[31805],[34,191254],{},[15,191256,191257],{},"F2 和 F3 的引用计数分别为：",[15,191259,191260],{},[34,191261,191263,191295],{"className":191262},[31642],[34,191264,191266],{"className":191265},[31646],[31648,191267,191268],{"xmlns":31650},[31652,191269,191270,191292],{},[31655,191271,191272],{},[126400,191273,191274],{"notation":126402},[71090,191275,191276],{"scriptlevel":459,"displaystyle":702},[71090,191277,191278],{"scriptlevel":459,"displaystyle":702},[71090,191279,191280],{"scriptlevel":459,"displaystyle":758},[31655,191281,191282,191284,191286,191288,191290],{},[31658,191283,47384],{},[39361,191285,130298],{},[31749,191287,640],{},[39361,191289,46528],{},[31749,191291,640],{},[31665,191293,191294],{"encoding":31667},"\\boxed{C \\ 1，1}",[34,191296,191298],{"className":191297,"ariaHidden":758},[31672],[34,191299,191301,191304],{"className":191300},[31676],[34,191302],{"className":191303,"style":126424},[31680],[34,191305,191307],{"className":191306},[31685],[34,191308,191310,191356],{"className":191309},[31796,31797],[34,191311,191313,191353],{"className":191312},[31801],[34,191314,191316,191345],{"className":191315,"style":126437},[31805],[34,191317,191318,191321],{"style":126440},[34,191319],{"className":191320,"style":126444},[31813],[34,191322,191324],{"className":191323},[126448],[34,191325,191327],{"className":191326},[31685],[34,191328,191330,191333,191336,191339,191342],{"className":191329},[31685],[34,191331,47384],{"className":191332,"style":38706},[31685,31686],[34,191334,130298],{"className":191335},[31845],[34,191337,640],{"className":191338},[31685],[34,191340,46528],{"className":191341},[31685,105462],[34,191343,640],{"className":191344},[31685],[34,191346,191347,191350],{"style":126463},[34,191348],{"className":191349,"style":126444},[31813],[34,191351],{"className":191352,"style":126472},[126470,126471],[34,191354,31829],{"className":191355},[31828],[34,191357,191359],{"className":191358},[31801],[34,191360,191362],{"className":191361,"style":126482},[31805],[34,191363],{},[15,191365,191366],{},"磁盘 I\u002FO 信息传送单位：",[15,191368,191369],{},[34,191370,191372,191398],{"className":191371},[31642],[34,191373,191375],{"className":191374},[31646],[31648,191376,191377],{"xmlns":31650},[31652,191378,191379,191395],{},[31655,191380,191381],{},[126400,191382,191383],{"notation":126402},[71090,191384,191385],{"scriptlevel":459,"displaystyle":702},[71090,191386,191387],{"scriptlevel":459,"displaystyle":702},[71090,191388,191389],{"scriptlevel":459,"displaystyle":758},[31655,191390,191391,191393],{},[31658,191392,47384],{},[39361,191394,176786],{},[31665,191396,191397],{"encoding":31667},"\\boxed{C \\text{ 块}}",[34,191399,191401],{"className":191400,"ariaHidden":758},[31672],[34,191402,191404,191407],{"className":191403},[31676],[34,191405],{"className":191406,"style":126424},[31680],[34,191408,191410],{"className":191409},[31685],[34,191411,191413,191456],{"className":191412},[31796,31797],[34,191414,191416,191453],{"className":191415},[31801],[34,191417,191419,191445],{"className":191418,"style":126437},[31805],[34,191420,191421,191424],{"style":126440},[34,191422],{"className":191423,"style":126444},[31813],[34,191425,191427],{"className":191426},[126448],[34,191428,191430],{"className":191429},[31685],[34,191431,191433,191436],{"className":191432},[31685],[34,191434,47384],{"className":191435,"style":38706},[31685,31686],[34,191437,191439,191442],{"className":191438},[31685,2638],[34,191440,130298],{"className":191441},[31685],[34,191443,176811],{"className":191444},[31685,105462],[34,191446,191447,191450],{"style":126463},[34,191448],{"className":191449,"style":126444},[31813],[34,191451],{"className":191452,"style":126472},[126470,126471],[34,191454,31829],{"className":191455},[31828],[34,191457,191459],{"className":191458},[31801],[34,191460,191462],{"className":191461,"style":126482},[31805],[34,191463],{},[15,191465,191466],{},"一屏 256 色图像存储空间：",[15,191468,191469],{},[34,191470,191472,191500],{"className":191471},[31642],[34,191473,191475],{"className":191474},[31646],[31648,191476,191477],{"xmlns":31650},[31652,191478,191479,191497],{},[31655,191480,191481],{},[126400,191482,191483],{"notation":126402},[71090,191484,191485],{"scriptlevel":459,"displaystyle":702},[71090,191486,191487],{"scriptlevel":459,"displaystyle":702},[71090,191488,191489],{"scriptlevel":459,"displaystyle":758},[31655,191490,191491,191493,191495],{},[31749,191492,25357],{},[31658,191494,38691],{},[31658,191496,54224],{},[31665,191498,191499],{"encoding":31667},"\\boxed{300KB}",[34,191501,191503],{"className":191502,"ariaHidden":758},[31672],[34,191504,191506,191509],{"className":191505},[31676],[34,191507],{"className":191508,"style":126424},[31680],[34,191510,191512],{"className":191511},[31685],[34,191513,191515,191555],{"className":191514},[31796,31797],[34,191516,191518,191552],{"className":191517},[31801],[34,191519,191521,191544],{"className":191520,"style":126437},[31805],[34,191522,191523,191526],{"style":126440},[34,191524],{"className":191525,"style":126444},[31813],[34,191527,191529],{"className":191528},[126448],[34,191530,191532],{"className":191531},[31685],[34,191533,191535,191538,191541],{"className":191534},[31685],[34,191536,25357],{"className":191537},[31685],[34,191539,38691],{"className":191540,"style":38706},[31685,31686],[34,191542,54224],{"className":191543,"style":54252},[31685,31686],[34,191545,191546,191549],{"style":126463},[34,191547],{"className":191548,"style":126444},[31813],[34,191550],{"className":191551,"style":126472},[126470,126471],[34,191553,31829],{"className":191554},[31828],[34,191556,191558],{"className":191557},[31801],[34,191559,191561],{"className":191560,"style":126482},[31805],[34,191562],{},[15,191564,191565],{},"文件物理结构：",[15,191567,191568],{},[34,191569,191571,191594],{"className":191570},[31642],[34,191572,191574],{"className":191573},[31646],[31648,191575,191576],{"xmlns":31650},[31652,191577,191578,191591],{},[31655,191579,191580],{},[126400,191581,191582],{"notation":126402},[71090,191583,191584],{"scriptlevel":459,"displaystyle":702},[71090,191585,191586],{"scriptlevel":459,"displaystyle":702},[71090,191587,191588],{"scriptlevel":459,"displaystyle":758},[39361,191589,191590],{},"文件在磁盘上的实际存放方式",[31665,191592,191593],{"encoding":31667},"\\boxed{\\text{文件在磁盘上的实际存放方式}}",[34,191595,191597],{"className":191596,"ariaHidden":758},[31672],[34,191598,191600,191603],{"className":191599},[31676],[34,191601],{"className":191602,"style":126424},[31680],[34,191604,191606],{"className":191605},[31685],[34,191607,191609,191646],{"className":191608},[31796,31797],[34,191610,191612,191643],{"className":191611},[31801],[34,191613,191615,191635],{"className":191614,"style":126437},[31805],[34,191616,191617,191620],{"style":126440},[34,191618],{"className":191619,"style":126444},[31813],[34,191621,191623],{"className":191622},[126448],[34,191624,191626],{"className":191625},[31685],[34,191627,191629],{"className":191628},[31685],[34,191630,191632],{"className":191631},[31685,2638],[34,191633,191590],{"className":191634},[31685,105462],[34,191636,191637,191640],{"style":126463},[34,191638],{"className":191639,"style":126444},[31813],[34,191641],{"className":191642,"style":126472},[126470,126471],[34,191644,31829],{"className":191645},[31828],[34,191647,191649],{"className":191648},[31801],[34,191650,191652],{"className":191651,"style":126482},[31805],[34,191653],{},[15,191655,191656],{},"常见方式：",[15,191658,191659],{},[34,191660,191662,191685],{"className":191661},[31642],[34,191663,191665],{"className":191664},[31646],[31648,191666,191667],{"xmlns":31650},[31652,191668,191669,191682],{},[31655,191670,191671],{},[126400,191672,191673],{"notation":126402},[71090,191674,191675],{"scriptlevel":459,"displaystyle":702},[71090,191676,191677],{"scriptlevel":459,"displaystyle":702},[71090,191678,191679],{"scriptlevel":459,"displaystyle":758},[39361,191680,191681],{},"连续分配、链式分配、索引分配",[31665,191683,191684],{"encoding":31667},"\\boxed{\\text{连续分配、链式分配、索引分配}}",[34,191686,191688],{"className":191687,"ariaHidden":758},[31672],[34,191689,191691,191694],{"className":191690},[31676],[34,191692],{"className":191693,"style":126424},[31680],[34,191695,191697],{"className":191696},[31685],[34,191698,191700,191737],{"className":191699},[31796,31797],[34,191701,191703,191734],{"className":191702},[31801],[34,191704,191706,191726],{"className":191705,"style":126437},[31805],[34,191707,191708,191711],{"style":126440},[34,191709],{"className":191710,"style":126444},[31813],[34,191712,191714],{"className":191713},[126448],[34,191715,191717],{"className":191716},[31685],[34,191718,191720],{"className":191719},[31685],[34,191721,191723],{"className":191722},[31685,2638],[34,191724,191681],{"className":191725},[31685,105462],[34,191727,191728,191731],{"style":126463},[34,191729],{"className":191730,"style":126444},[31813],[34,191732],{"className":191733,"style":12647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style":54252},[31685,31686],[34,193367,193368,193371],{"style":126463},[34,193369],{"className":193370,"style":126444},[31813],[34,193372],{"className":193373,"style":126472},[126470,126471],[34,193375,31829],{"className":193376},[31828],[34,193378,193380],{"className":193379},[31801],[34,193381,193383],{"className":193382,"style":126482},[31805],[34,193384],{},[15,193386,129204],{},[15,193388,193389],{},[34,193390,193392,193417],{"className":193391},[31642],[34,193393,193395],{"className":193394},[31646],[31648,193396,193397],{"xmlns":31650},[31652,193398,193399,193415],{},[31655,193400,193401],{},[126400,193402,193403],{"notation":126402},[71090,193404,193405],{"scriptlevel":459,"displaystyle":702},[71090,193406,193407],{"scriptlevel":459,"displaystyle":702},[71090,193408,193409],{"scriptlevel":459,"displaystyle":758},[31655,193410,193411,193413],{},[31749,193412,189125],{},[31658,193414,54224],{},[31665,193416,189394],{"encoding":31667},[34,193418,193420],{"className":193419,"ariaHidden":758},[31672],[34,193421,193423,193426],{"className":193422},[31676],[34,193424],{"className":193425,"style":126424},[31680],[34,193427,193429],{"className":193428},[31685],[34,193430,193432,193469],{"className":193431},[31796,31797],[34,193433,193435,193466],{"className":193434},[31801],[34,193436,193438,193458],{"className":193437,"style":126437},[31805],[34,193439,193440,193443],{"style":126440},[34,193441],{"className":193442,"style":126444},[31813],[34,193444,193446],{"className":193445},[126448],[34,193447,193449],{"className":193448},[31685],[34,193450,193452,193455],{"className":193451},[31685],[34,193453,189125],{"className":193454},[31685],[34,193456,54224],{"className":193457,"style":54252},[31685,31686],[34,193459,193460,193463],{"style":126463},[34,193461],{"className":193462,"style":126444},[31813],[34,193464],{"className":193465,"style":126472},[126470,126471],[34,193467,31829],{"className":193468},[31828],[34,193470,193472],{"className":193471},[31801],[34,193473,193475],{"className":193474,"style":126482},[31805],[34,193476],{},[15,193478,186748],{},[15,193480,193481],{},[34,193482,193484,193513],{"className":193483},[31642],[34,193485,193487],{"className":193486},[31646],[31648,193488,193489],{"xmlns":31650},[31652,193490,193491,193511],{},[31655,193492,193493],{},[126400,193494,193495],{"notation":126402},[71090,193496,193497],{"scriptlevel":459,"displaystyle":702},[71090,193498,193499],{"scriptlevel":459,"displaystyle":702},[71090,193500,193501],{"scriptlevel":459,"displaystyle":758},[31655,193502,193503,193505,193507,193509],{},[31749,193504,189209],{},[31658,193506,98360],{},[31658,193508,31955],{},[31658,193510,54224],{},[31665,193512,189491],{"encoding":31667},[34,193514,193516],{"className":193515,"ariaHidden":758},[31672],[34,193517,193519,193522],{"className":193518},[31676],[34,193520],{"className":193521,"style":126424},[31680],[34,193523,193525],{"className":193524},[31685],[34,193526,193528,193571],{"className":193527},[31796,31797],[34,193529,193531,193568],{"className":193530},[31801],[34,193532,193534,193560],{"className":193533,"style":126437},[31805],[34,193535,193536,193539],{"style":126440},[34,193537],{"className":193538,"style":126444},[31813],[34,193540,193542],{"className":193541},[126448],[34,193543,193545],{"className":193544},[31685],[34,193546,193548,193551,193554,193557],{"className":193547},[31685],[34,193549,189209],{"className":193550},[31685],[34,193552,98360],{"className":193553,"style":33389},[31685,31686],[34,193555,31955],{"className":193556},[31685,31686],[34,193558,54224],{"className":193559,"style":54252},[31685,31686],[34,193561,193562,193565],{"style":126463},[34,193563],{"className":193564,"style":126444},[31813],[34,193566],{"className":193567,"style":126472},[126470,126471],[34,193569,31829],{"className":193570},[31828],[34,193572,193574],{"className":193573},[31801],[34,193575,193577],{"className":193576,"style":126482},[31805],[34,193578],{},[32188,193580],{},[19,193582,193584],{"id":193583},"第-6-题","第 6 题",[15,193586,193587],{},"需要：",[15,193589,193590],{},[34,193591,193593,193618],{"className":193592},[31642],[34,193594,193596],{"className":193595},[31646],[31648,193597,193598],{"xmlns":31650},[31652,193599,193600,193616],{},[31655,193601,193602],{},[126400,193603,193604],{"notation":126402},[71090,193605,193606],{"scriptlevel":459,"displaystyle":702},[71090,193607,193608],{"scriptlevel":459,"displaystyle":702},[71090,193609,193610],{"scriptlevel":459,"displaystyle":758},[31655,193611,193612,193614],{},[31749,193613,640],{},[39361,193615,190556],{},[31665,193617,190559],{"encoding":31667},[34,193619,193621],{"className":193620,"ariaHidden":758},[31672],[34,193622,193624,193627],{"className":193623},[31676],[34,193625],{"className":193626,"style":126424},[31680],[34,193628,193630],{"className":193629},[31685],[34,193631,193633,193676],{"className":193632},[31796,31797],[34,193634,193636,193673],{"className":193635},[31801],[34,193637,193639,193665],{"className":193638,"style":126437},[31805],[34,193640,193641,193644],{"style":126440},[34,193642],{"className":193643,"style":126444},[31813],[34,193645,193647],{"className":193646},[126448],[34,193648,193650],{"className":193649},[31685],[34,193651,193653,193656],{"className":193652},[31685],[34,193654,640],{"className":193655},[31685],[34,193657,193659,193662],{"className":193658},[31685,2638],[34,193660,130298],{"className":193661},[31685],[34,193663,190607],{"className":193664},[31685,105462],[34,193666,193667,193670],{"style":126463},[34,193668],{"className":193669,"style":126444},[31813],[34,193671],{"className":193672,"style":126472},[126470,126471],[34,193674,31829],{"className":193675},[31828],[34,193677,193679],{"className":193678},[31801],[34,193680,193682],{"className":193681,"style":126482},[31805],[34,193683],{},[15,193685,139415],{},[15,193687,193688],{},[34,193689,193691,193713],{"className":193690},[31642],[34,193692,193694],{"className":193693},[31646],[31648,193695,193696],{"xmlns":31650},[31652,193697,193698,193710],{},[31655,193699,193700],{},[126400,193701,193702],{"notation":126402},[71090,193703,193704],{"scriptlevel":459,"displaystyle":702},[71090,193705,193706],{"scriptlevel":459,"displaystyle":702},[71090,193707,193708],{"scriptlevel":459,"displaystyle":758},[39361,193709,191087],{},[31665,193711,193712],{"encoding":31667},"\\boxed{\\text{一级间接索引}}",[34,193714,193716],{"className":193715,"ariaHidden":758},[31672],[34,193717,193719,193722],{"className":193718},[31676],[34,193720],{"className":193721,"style":126424},[31680],[34,193723,193725],{"className":193724},[31685],[34,193726,193728,193765],{"className":193727},[31796,31797],[34,193729,193731,193762],{"className":193730},[31801],[34,193732,193734,193754],{"className":193733,"style":126437},[31805],[34,193735,193736,193739],{"style":126440},[34,193737],{"className":193738,"style":126444},[31813],[34,193740,193742],{"className":193741},[126448],[34,193743,193745],{"className":193744},[31685],[34,193746,193748],{"className":193747},[31685],[34,193749,193751],{"className":193750},[31685,2638],[34,193752,191087],{"className":193753},[31685,105462],[34,193755,193756,193759],{"style":126463},[34,193757],{"className":193758,"style":126444},[31813],[34,193760],{"className":193761,"style":126472},[126470,126471],[34,193763,31829],{"className":193764},[31828],[34,193766,193768],{"className":193767},[31801],[34,193769,193771],{"className":193770,"style":126482},[31805],[34,193772],{},{"title":29,"searchDepth":43,"depth":43,"links":193774},[193775,193785,193793,193804,193812,193822,193831,193832,193833,193834,193835,193836],{"id":175232,"depth":43,"text":175233,"children":193776},[193777,193778,193779,193780,193781,193782,193783,193784],{"id":162043,"depth":49,"text":162043},{"id":175302,"depth":49,"text":175302},{"id":175307,"depth":49,"text":175308},{"id":175431,"depth":49,"text":175432},{"id":176764,"depth":49,"text":176765},{"id":176935,"depth":49,"text":176936},{"id":177555,"depth":49,"text":177556},{"id":178164,"depth":49,"text":178165},{"id":178426,"depth":43,"text":178427,"children":193786},[193787,193788,193789,193790,193791,193792],{"id":167236,"depth":49,"text":162043},{"id":178561,"depth":49,"text":175302},{"id":178566,"depth":49,"text":178567},{"id":180050,"depth":49,"text":180051},{"id":180798,"depth":49,"text":180798},{"id":168734,"depth":49,"text":168735},{"id":182184,"depth":43,"text":182185,"children":193794},[193795,193796,193797,193799,193801,193803],{"id":168779,"depth":49,"text":162043},{"id":182451,"depth":49,"text":175302},{"id":182876,"depth":49,"text":193798},"3.1 文件大小为 41600B41600B41600B",{"id":183431,"depth":49,"text":193800},"3.2 文件大小为 640000B640000B640000B",{"id":183981,"depth":49,"text":193802},"3.3 文件大小为 4064000B4064000B4064000B",{"id":169905,"depth":49,"text":169906},{"id":185002,"depth":43,"text":185003,"children":193805},[193806,193807,193808,193809,193810,193811],{"id":170039,"depth":49,"text":162043},{"id":185013,"depth":49,"text":175302},{"id":185390,"depth":49,"text":185391},{"id":185866,"depth":49,"text":185867},{"id":186942,"depth":49,"text":186943},{"id":173017,"depth":49,"text":173018},{"id":187699,"depth":43,"text":187700,"children":193813},[193814,193815,193816,193817,193818,193819,193820,193821],{"id":173118,"depth":49,"text":162043},{"id":187710,"depth":49,"text":175302},{"id":188028,"depth":49,"text":188029},{"id":188208,"depth":49,"text":188209},{"id":188353,"depth":49,"text":188354},{"id":188579,"depth":49,"text":188580},{"id":188776,"depth":49,"text":188777},{"id":174560,"depth":49,"text":174561},{"id":189562,"depth":43,"text":189563,"children":193823},[193824,193825,193826,193827,193828,193829,193830],{"id":189566,"depth":49,"text":162043},{"id":189579,"depth":49,"text":175302},{"id":189734,"depth":49,"text":189735},{"id":189987,"depth":49,"text":189988},{"id":190161,"depth":49,"text":190162},{"id":166511,"depth":49,"text":166511},{"id":190889,"depth":49,"text":190890},{"id":174668,"depth":43,"text":174669},{"id":174803,"depth":43,"text":174804},{"id":174844,"depth":43,"text":174845},{"id":174972,"depth":43,"text":174973},{"id":175069,"depth":43,"text":175070},{"id":193583,"depth":43,"text":193584},{"order":1999},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap12-hw",{"title":175223,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap12-hw","1cIyfro2WQzIpTxvBuXLsXkkVTUL_wIdpMG-zEqS_8I",{"id":193843,"title":193844,"body":193845,"description":29,"extension":85,"meta":194799,"navigation":52,"path":194800,"seo":194801,"stem":194802,"__hash__":194803},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap2-hw.md","Chap. 2 HW",{"type":8,"value":193846,"toc":194788},[193847,193851,193855,193906,193910,193976,193980,193991,193996,194000,194005,194037,194041,194188,194717,194721,194726,194730,194733,194774,194778],[11,193848,193850],{"id":193849},"os-chap2-hw","OS Chap.2 HW",[19,193852,193854],{"id":193853},"q1-definition-objectives-functions-and-features-of-the-operating-system","Q1. Definition, Objectives, Functions, and Features of the Operating System",[105,193856,193857,193863,193881,193900],{},[108,193858,193859,193862],{},[2627,193860,193861],{},"Definition:"," An operating system (OS) is a program that controls the execution of application programs and acts as an interface between applications and the computer hardware.",[108,193864,193865,193868,193869,193872,193873,193876,193877,193880],{},[2627,193866,193867],{},"Objectives:"," The three main objectives of an OS are ",[2627,193870,193871],{},"convenience"," (making the computer easier to use), ",[2627,193874,193875],{},"efficiency"," (allowing the computer's resources to be used in an efficient manner), and ",[2627,193878,193879],{},"ability to evolve"," (permitting the development and introduction of new system functions without interfering with existing services).",[108,193882,193883,193886],{},[2627,193884,193885],{},"Functions:",[105,193887,193888,193894],{},[108,193889,193890,193893],{},[2627,193891,193892],{},"User\u002FComputer Interface:"," The OS masks hardware details and provides services for program development, program execution, uniform access to I\u002FO devices, controlled access to files, system access (protection and conflict resolution), error detection\u002Fresponse, and accounting.",[108,193895,193896,193899],{},[2627,193897,193898],{},"Resource Manager:"," The OS is responsible for managing the computer's resources, including allocating main memory, determining which I\u002FO devices can be used by executing programs, and deciding how much processor time is devoted to specific user programs.",[108,193901,193902,193905],{},[2627,193903,193904],{},"Features:"," Modern OS features include instruction set architectures, application programming interfaces (APIs), dual-mode execution (user mode vs. kernel mode) to protect memory and prevent unauthorized use of privileged instructions, and interrupt\u002Ftrap handling to switch execution contexts.",[19,193907,193909],{"id":193908},"q2-main-features-of-different-os-types","Q2. Main Features of Different OS Types",[105,193911,193912,193926,193940,193966],{},[108,193913,193914,193917,193918,193921,193922,193925],{},[2627,193915,193916],{},"Batch Processing OS:"," The central feature is a software program called a ",[2627,193919,193920],{},"monitor",". Users submit jobs offline, which are batched together sequentially and handled by the monitor. The monitor automatically loads and executes the next program in the batch once the current one finishes. ",[71807,193923,193924],{},"Multiprogrammed batch systems"," enhance this by keeping multiple jobs in memory simultaneously; when one job waits for I\u002FO, the processor switches to another job, highly improving efficiency and CPU utilization.",[108,193927,193928,193931,193932,193935,193936,193939],{},[2627,193929,193930],{},"Time-Sharing OS:"," Designed to handle multiple interactive jobs simultaneously. The OS uses ",[2627,193933,193934],{},"multiprogramming and time slicing"," (allocating a short burst or \"quantum\" of computation to each process) so that multiple users can access the system through terminals concurrently. Its principal objective is to ",[2627,193937,193938],{},"minimize response time"," rather than strictly maximizing processor use.",[108,193941,193942,193945,193946,193949,193950,193953,193954,193957,193958,193961,193962,193965],{},[2627,193943,193944],{},"Real-Time OS:"," Designed for applications where correctness depends not just on logical results, but on the ",[2627,193947,193948],{},"time at which the results are produced",". Features include ",[2627,193951,193952],{},"determinism"," (performing operations at fixed times), fast ",[2627,193955,193956],{},"responsiveness"," to external interrupts, extensive ",[2627,193959,193960],{},"user control"," over task priorities, strict preemptive scheduling for meeting deadlines (hard vs. soft tasks), and ",[2627,193963,193964],{},"fail-soft operation"," (the ability to fail in a way that preserves maximum capability).",[108,193967,193968,193971,193972,193975],{},[2627,193969,193970],{},"Distributed OS:"," Operates across a cluster of separate computers (multicomputer system) but provides the ",[2627,193973,193974],{},"illusion of a single main memory space and secondary memory space",", along with unified access facilities like a distributed file system.",[19,193977,193979],{"id":193978},"q3-user-behavior-and-core-mindset","Q3. User Behavior and Core Mindset",[15,193981,193982,193983,193986,193987,193990],{},"The provided sources focus heavily on the technical and architectural internals of operating systems rather than the psychological or behavioral analysis of users. However, the sources do note that in interactive systems (like time-sharing), human reaction time is \"relatively slow,\" allowing many users to share a system without noticeable interference. Furthermore, users expect ",[2627,193984,193985],{},"predictability"," (variations in response time are distracting) and rely on the OS to provide a ",[2627,193988,193989],{},"convenient interface"," that abstracts away complex hardware details.",[15,193992,193993],{},[71807,193994,193995],{},"(Note: A deep psychological or behavioral analysis of the \"core mindset\" falls outside the scope of the provided sources and would require independent verification).",[19,193997,193999],{"id":193998},"q4-cpu-time-spent-on-clock-interrupts","Q4. CPU Time Spent on Clock Interrupts",[15,194001,194002],{},[71807,194003,194004],{},"(Note: The following calculation uses the system parameters you provided, which require external mathematical application not explicitly found in the sources).",[105,194006,194007,194016,194022,194028],{},[108,194008,194009,194012,194013],{},[2627,194010,194011],{},"Time per interrupt sequence:"," 1ms (handle interrupt) + 2ms (process scheduling) + 1ms (allocate CPU) = ",[2627,194014,194015],{},"4ms total per clock interrupt.",[108,194017,194018,194021],{},[2627,194019,194020],{},"Total interrupts per second:"," 100.",[108,194023,194024,194027],{},[2627,194025,194026],{},"Total time spent per second:"," 100 interrupts × 4ms = 400ms.",[108,194029,194030,194033,194034,2547],{},[2627,194031,194032],{},"Percentage of CPU time:"," 400ms out of 1000ms (1 second) = ",[2627,194035,194036],{},"40% of the CPU's time",[19,194038,194040],{"id":194039},"q5-increase-in-cpu-utilization-with-added-memory","Q5. Increase in CPU Utilization with Added Memory",[15,194042,194043],{},[71807,194044,194045,194046,194129,194130,194158,194159,194187],{},"(Note: This uses the standard probabilistic CPU utilization formula ",[34,194047,194049,194071],{"className":194048},[31642],[34,194050,194052],{"className":194051},[31646],[31648,194053,194054],{"xmlns":31650},[31652,194055,194056,194068],{},[31655,194057,194058,194060,194062],{},[31749,194059,640],{},[31662,194061,32226],{},[37624,194063,194064,194066],{},[31658,194065,15],{},[31658,194067,33347],{},[31665,194069,194070],{"encoding":31667},"1 - p^N",[34,194072,194074,194092],{"className":194073,"ariaHidden":758},[31672],[34,194075,194077,194080,194083,194086,194089],{"className":194076},[31676],[34,194078],{"className":194079,"style":35942},[31680],[34,194081,640],{"className":194082},[31685],[34,194084],{"className":194085,"style":32604},[31845],[34,194087,32226],{"className":194088},[32024],[34,194090],{"className":194091,"style":32604},[31845],[34,194093,194095,194099],{"className":194094},[31676],[34,194096],{"className":194097,"style":194098},[31680],"height:1.0358em;vertical-align:-0.1944em;",[34,194100,194102,194105],{"className":194101},[31685],[34,194103,15],{"className":194104},[31685,31686],[34,194106,194108],{"className":194107},[31792],[34,194109,194111],{"className":194110},[31796],[34,194112,194114],{"className":194113},[31801],[34,194115,194118],{"className":194116,"style":194117},[31805],"height:0.8413em;",[34,194119,194120,194123],{"style":37777},[34,194121],{"className":194122,"style":31814},[31813],[34,194124,194126],{"className":194125},[31818,31819,31820,31821],[34,194127,33347],{"className":194128,"style":33389},[31685,31686,31821]," where ",[34,194131,194133,194146],{"className":194132},[31642],[34,194134,194136],{"className":194135},[31646],[31648,194137,194138],{"xmlns":31650},[31652,194139,194140,194144],{},[31655,194141,194142],{},[31658,194143,15],{},[31665,194145,15],{"encoding":31667},[34,194147,194149],{"className":194148,"ariaHidden":758},[31672],[34,194150,194152,194155],{"className":194151},[31676],[34,194153],{"className":194154,"style":38774},[31680],[34,194156,15],{"className":194157},[31685,31686]," is the I\u002FO wait probability and ",[34,194160,194162,194175],{"className":194161},[31642],[34,194163,194165],{"className":194164},[31646],[31648,194166,194167],{"xmlns":31650},[31652,194168,194169,194173],{},[31655,194170,194171],{},[31658,194172,33347],{},[31665,194174,33347],{"encoding":31667},[34,194176,194178],{"className":194177,"ariaHidden":758},[31672],[34,194179,194181,194184],{"className":194180},[31676],[34,194182],{"className":194183,"style":33218},[31680],[34,194185,33347],{"className":194186,"style":33389},[31685,31686]," is the number of processes. This specific formula and problem are not derived from the text).",[105,194189,194190,194453,194708],{},[108,194191,194192,194195],{},[2627,194193,194194],{},"Initial State:",[105,194196,194197,194200,194203,194293],{},[108,194198,194199],{},"Available memory = 1MB (1024KB).",[108,194201,194202],{},"User space = 1024KB - 200KB (OS) = 824KB.",[108,194204,194205,194206,194234,194235,194292],{},"Number of processes (",[34,194207,194209,194222],{"className":194208},[31642],[34,194210,194212],{"className":194211},[31646],[31648,194213,194214],{"xmlns":31650},[31652,194215,194216,194220],{},[31655,194217,194218],{},[31658,194219,33347],{},[31665,194221,33347],{"encoding":31667},[34,194223,194225],{"className":194224,"ariaHidden":758},[31672],[34,194226,194228,194231],{"className":194227},[31676],[34,194229],{"className":194230,"style":33218},[31680],[34,194232,33347],{"className":194233,"style":33389},[31685,31686],") = ",[34,194236,194238,194261],{"className":194237},[31642],[34,194239,194241],{"className":194240},[31646],[31648,194242,194243],{"xmlns":31650},[31652,194244,194245,194258],{},[31655,194246,194247,194250,194252,194254,194256],{},[31749,194248,194249],{},"824",[31658,194251,38080],{"mathvariant":38079},[31749,194253,168915],{},[31662,194255,202],{},[31749,194257,15932],{},[31665,194259,194260],{"encoding":31667},"824 \u002F 200 = 4",[34,194262,194264,194283],{"className":194263,"ariaHidden":758},[31672],[34,194265,194267,194270,194274,194277,194280],{"className":194266},[31676],[34,194268],{"className":194269,"style":31774},[31680],[34,194271,194273],{"className":194272},[31685],"824\u002F200",[34,194275],{"className":194276,"style":31846},[31845],[34,194278,202],{"className":194279},[31850],[34,194281],{"className":194282,"style":31846},[31845],[34,194284,194286,194289],{"className":194285},[31676],[34,194287],{"className":194288,"style":32617},[31680],[34,194290,15932],{"className":194291},[31685]," complete processes.",[108,194294,194295,194296,1840,194450,26192],{},"CPU Utilization = ",[34,194297,194299,194336],{"className":194298},[31642],[34,194300,194302],{"className":194301},[31646],[31648,194303,194304],{"xmlns":31650},[31652,194305,194306,194333],{},[31655,194307,194308,194310,194312,194319,194321,194323,194325,194328,194330],{},[31749,194309,640],{},[31662,194311,32226],{},[37624,194313,194314,194317],{},[31749,194315,194316],{},"0.80",[31749,194318,15932],{},[31662,194320,202],{},[31749,194322,640],{},[31662,194324,32226],{},[31749,194326,194327],{},"0.4096",[31662,194329,202],{},[31749,194331,194332],{},"0.5904",[31665,194334,194335],{"encoding":31667},"1 - 0.80^4 = 1 - 0.4096 = 0.5904",[34,194337,194339,194357,194405,194423,194441],{"className":194338,"ariaHidden":758},[31672],[34,194340,194342,194345,194348,194351,194354],{"className":194341},[31676],[34,194343],{"className":194344,"style":35942},[31680],[34,194346,640],{"className":194347},[31685],[34,194349],{"className":194350,"style":32604},[31845],[34,194352,32226],{"className":194353},[32024],[34,194355],{"className":194356,"style":32604},[31845],[34,194358,194360,194363,194367,194396,194399,194402],{"className":194359},[31676],[34,194361],{"className":194362,"style":37774},[31680],[34,194364,194366],{"className":194365},[31685],"0.8",[34,194368,194370,194373],{"className":194369},[31685],[34,194371,459],{"className":194372},[31685],[34,194374,194376],{"className":194375},[31792],[34,194377,194379],{"className":194378},[31796],[34,194380,194382],{"className":194381},[31801],[34,194383,194385],{"className":194384,"style":37774},[31805],[34,194386,194387,194390],{"style":37777},[34,194388],{"className":194389,"style":31814},[31813],[34,194391,194393],{"className":194392},[31818,31819,31820,31821],[34,194394,15932],{"className":194395},[31685,31821],[34,194397],{"className":194398,"style":31846},[31845],[34,194400,202],{"className":194401},[31850],[34,194403],{"className":194404,"style":31846},[31845],[34,194406,194408,194411,194414,194417,194420],{"className":194407},[31676],[34,194409],{"className":194410,"style":35942},[31680],[34,194412,640],{"className":194413},[31685],[34,194415],{"className":194416,"style":32604},[31845],[34,194418,32226],{"className":194419},[32024],[34,194421],{"className":194422,"style":32604},[31845],[34,194424,194426,194429,194432,194435,194438],{"className":194425},[31676],[34,194427],{"className":194428,"style":32617},[31680],[34,194430,194327],{"className":194431},[31685],[34,194433],{"className":194434,"style":31846},[31845],[34,194436,202],{"className":194437},[31850],[34,194439],{"className":194440,"style":31846},[31845],[34,194442,194444,194447],{"className":194443},[31676],[34,194445],{"className":194446,"style":32617},[31680],[34,194448,194332],{"className":194449},[31685],[2627,194451,194452],{},"59.04%",[108,194454,194455,194458],{},[2627,194456,194457],{},"After adding 1MB (Total 2MB \u002F 2048KB):",[105,194459,194460,194463,194550],{},[108,194461,194462],{},"User space = 2048KB - 200KB (OS) = 1848KB.",[108,194464,194205,194465,194234,194493,194292],{},[34,194466,194468,194481],{"className":194467},[31642],[34,194469,194471],{"className":194470},[31646],[31648,194472,194473],{"xmlns":31650},[31652,194474,194475,194479],{},[31655,194476,194477],{},[31658,194478,33347],{},[31665,194480,33347],{"encoding":31667},[34,194482,194484],{"className":194483,"ariaHidden":758},[31672],[34,194485,194487,194490],{"className":194486},[31676],[34,194488],{"className":194489,"style":33218},[31680],[34,194491,33347],{"className":194492,"style":33389},[31685,31686],[34,194494,194496,194519],{"className":194495},[31642],[34,194497,194499],{"className":194498},[31646],[31648,194500,194501],{"xmlns":31650},[31652,194502,194503,194516],{},[31655,194504,194505,194508,194510,194512,194514],{},[31749,194506,194507],{},"1848",[31658,194509,38080],{"mathvariant":38079},[31749,194511,168915],{},[31662,194513,202],{},[31749,194515,36732],{},[31665,194517,194518],{"encoding":31667},"1848 \u002F 200 = 9",[34,194520,194522,194541],{"className":194521,"ariaHidden":758},[31672],[34,194523,194525,194528,194532,194535,194538],{"className":194524},[31676],[34,194526],{"className":194527,"style":31774},[31680],[34,194529,194531],{"className":194530},[31685],"1848\u002F200",[34,194533],{"className":194534,"style":31846},[31845],[34,194536,202],{"className":194537},[31850],[34,194539],{"className":194540,"style":31846},[31845],[34,194542,194544,194547],{"className":194543},[31676],[34,194545],{"className":194546,"style":32617},[31680],[34,194548,36732],{"className":194549},[31685],[108,194551,194552,194553,1840,194705,26192],{},"New CPU Utilization = ",[34,194554,194556,194592],{"className":194555},[31642],[34,194557,194559],{"className":194558},[31646],[31648,194560,194561],{"xmlns":31650},[31652,194562,194563,194589],{},[31655,194564,194565,194567,194569,194575,194577,194579,194581,194584,194586],{},[31749,194566,640],{},[31662,194568,32226],{},[37624,194570,194571,194573],{},[31749,194572,194316],{},[31749,194574,36732],{},[31662,194576,202],{},[31749,194578,640],{},[31662,194580,32226],{},[31749,194582,194583],{},"0.1342",[31662,194585,202],{},[31749,194587,194588],{},"0.8658",[31665,194590,194591],{"encoding":31667},"1 - 0.80^9 = 1 - 0.1342 = 0.8658",[34,194593,194595,194613,194660,194678,194696],{"className":194594,"ariaHidden":758},[31672],[34,194596,194598,194601,194604,194607,194610],{"className":194597},[31676],[34,194599],{"className":194600,"style":35942},[31680],[34,194602,640],{"className":194603},[31685],[34,194605],{"className":194606,"style":32604},[31845],[34,194608,32226],{"className":194609},[32024],[34,194611],{"className":194612,"style":32604},[31845],[34,194614,194616,194619,194622,194651,194654,194657],{"className":194615},[31676],[34,194617],{"className":194618,"style":37774},[31680],[34,194620,194366],{"className":194621},[31685],[34,194623,194625,194628],{"className":194624},[31685],[34,194626,459],{"className":194627},[31685],[34,194629,194631],{"className":194630},[31792],[34,194632,194634],{"className":194633},[31796],[34,194635,194637],{"className":194636},[31801],[34,194638,194640],{"className":194639,"style":37774},[31805],[34,194641,194642,194645],{"style":37777},[34,194643],{"className":194644,"style":31814},[31813],[34,194646,194648],{"className":194647},[31818,31819,31820,31821],[34,194649,36732],{"className":194650},[31685,31821],[34,194652],{"className":194653,"style":31846},[31845],[34,194655,202],{"className":194656},[31850],[34,194658],{"className":194659,"style":31846},[31845],[34,194661,194663,194666,194669,194672,194675],{"className":194662},[31676],[34,194664],{"className":194665,"style":35942},[31680],[34,194667,640],{"className":194668},[31685],[34,194670],{"className":194671,"style":32604},[31845],[34,194673,32226],{"className":194674},[32024],[34,194676],{"className":194677,"style":32604},[31845],[34,194679,194681,194684,194687,194690,194693],{"className":194680},[31676],[34,194682],{"className":194683,"style":32617},[31680],[34,194685,194583],{"className":194686},[31685],[34,194688],{"className":194689,"style":31846},[31845],[34,194691,202],{"className":194692},[31850],[34,194694],{"className":194695,"style":31846},[31845],[34,194697,194699,194702],{"className":194698},[31676],[34,194700],{"className":194701,"style":32617},[31680],[34,194703,194588],{"className":194704},[31685],[2627,194706,194707],{},"86.58%",[108,194709,194710,194713,194714],{},[2627,194711,194712],{},"Increase:"," 86.58% - 59.04% = ",[2627,194715,194716],{},"27.54% increase in CPU utilization.",[19,194718,194720],{"id":194719},"q6-concurrent-execution-trajectory","Q6. Concurrent Execution Trajectory",[15,194722,194723],{},[71807,194724,194725],{},"(Note: This requires logical modeling of a timeline which is an external exercise not explicitly contained in the provided source material).",[253,194727,194729],{"id":194728},"_1-working-situation-timeline-of-the-two-programs","(1) Working Situation (Timeline) of the Two Programs",[15,194731,194732],{},"Assuming a uniprocessor system with non-preemptive CPU execution (unless blocked by I\u002FO), the trajectory unfolds as follows:",[105,194734,194735,194741,194747,194756,194762,194768],{},[108,194736,194737,194740],{},[2627,194738,194739],{},"0 ms - 50 ms:"," CPU executes Program A (Calculate: 50ms).",[108,194742,194743,194746],{},[2627,194744,194745],{},"50 ms - 100 ms:"," Program A starts Printing (100ms duration, ends at 150ms). CPU is handed to Program B. CPU executes Program B (Calculate: 50ms).",[108,194748,194749,194752,194753],{},[2627,194750,194751],{},"100 ms - 150 ms:"," Program B finishes calculation and starts Input (80ms duration, ends at 180ms). ",[71807,194754,194755],{},"Both programs are now performing I\u002FO. The CPU is idle.",[108,194757,194758,194761],{},[2627,194759,194760],{},"150 ms - 200 ms:"," Program A finishes Printing and reclaims the CPU. CPU executes Program A (Calculate: 50ms). (Note: At 180ms, Program B finishes Input and becomes ready, but must wait for the CPU).",[108,194763,194764,194767],{},[2627,194765,194766],{},"200 ms - 300 ms:"," Program A finishes calculating and starts its second Print (100ms duration, ends at 300ms). The CPU is handed back to Program B. CPU executes Program B (Calculate: 100ms).",[108,194769,194770,194773],{},[2627,194771,194772],{},"300 ms:"," Both Program A and Program B finish their final operations simultaneously and terminate.",[253,194775,194777],{"id":194776},"_2-cpu-idle-waiting-time","(2) CPU Idle Waiting Time",[15,194779,194780,194783,194784,194787],{},[2627,194781,194782],{},"Yes",", the CPU does have idle waiting time. The CPU is completely idle from ",[2627,194785,194786],{},"100 ms to 150 ms"," (a duration of 50 ms) because Program A is busy using the printer and Program B is busy using the input machine.",{"title":29,"searchDepth":43,"depth":43,"links":194789},[194790,194791,194792,194793,194794,194795],{"id":193853,"depth":43,"text":193854},{"id":193908,"depth":43,"text":193909},{"id":193978,"depth":43,"text":193979},{"id":193998,"depth":43,"text":193999},{"id":194039,"depth":43,"text":194040},{"id":194719,"depth":43,"text":194720,"children":194796},[194797,194798],{"id":194728,"depth":49,"text":194729},{"id":194776,"depth":49,"text":194777},{"order":37},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap2-hw",{"title":193844,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap2-hw","MlWLpeaO6ndFpTkB8S_PG-l_Do24_hei_g6SmLa9hf8",{"id":194805,"title":194806,"body":194807,"description":29,"extension":85,"meta":195520,"navigation":52,"path":195521,"seo":195522,"stem":195523,"__hash__":195524},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap3-hw.md","Chap. 3 HW",{"type":8,"value":194808,"toc":195500},[194809,194813,194817,194820,194853,194857,194860,194892,194896,194906,194909,194913,194916,194961,194965,194971,194974,194981,194985,194992,194999,195003,195007,195285,195289,195364,195368,195372,195375,195389,195393,195396,195413,195416,195420,195423,195427,195457,195461],[11,194810,194812],{"id":194811},"os-chap3-hw","OS Chap.3 HW",[19,194814,194816],{"id":194815},"q1-what-common-events-lead-to-the-creation-of-a-process","Q1. What common events lead to the creation of a process?",[15,194818,194819],{},"There are four common events that lead to the creation of a new process by the operating system (OS):",[105,194821,194822,194828,194834,194840],{},[108,194823,194824,194827],{},[2627,194825,194826],{},"New batch job:"," The OS is provided with a batch job control stream, typically on a disk or tape. When the OS is prepared to handle new work, it reads the next sequence of job control commands and creates a process for it.",[108,194829,194830,194833],{},[2627,194831,194832],{},"Interactive log-on:"," A user attempts to log on to the system from a terminal. The OS accepts authorized users up to the system's saturation point and creates a new process for the user session.",[108,194835,194836,194839],{},[2627,194837,194838],{},"Created by OS to provide a service:"," The OS can autonomously create a process to perform a specific function on behalf of a user program. For example, it might create a background process to control printing so that the user's main application does not have to wait for the printer to finish.",[108,194841,194842,194845,194846,194849,194850,2547],{},[2627,194843,194844],{},"Spawned by existing process:"," A currently running user program can explicitly dictate the creation of new processes to exploit parallelism or improve modularity. When this happens, the original process is called the ",[71807,194847,194848],{},"parent process",", and the newly created one is the ",[71807,194851,194852],{},"child process",[19,194854,194856],{"id":194855},"q2-for-the-five-state-processing-model-briefly-define-each-state","Q2. For the five-state processing model, briefly define each state.",[15,194858,194859],{},"The five-state process model categorizes the lifecycle and execution states of a process as follows:",[105,194861,194862,194868,194874,194880,194886],{},[108,194863,194864,194867],{},[2627,194865,194866],{},"Running:"," The process that is currently being executed by the processor. In a uniprocessor system, at most one process can be in this state at any given time.",[108,194869,194870,194873],{},[2627,194871,194872],{},"Ready:"," A process that is prepared and available to execute as soon as the OS dispatcher gives it the opportunity.",[108,194875,194876,194879],{},[2627,194877,194878],{},"Blocked (Waiting):"," A process that cannot continue execution until some specific external event occurs, such as the completion of an I\u002FO operation or the availability of a requested resource.",[108,194881,194882,194885],{},[2627,194883,194884],{},"New:"," A process that has just been created by the OS. Its Process Control Block (PCB) has been initialized, but it has not yet been loaded into main memory or admitted to the pool of executable processes.",[108,194887,194888,194891],{},[2627,194889,194890],{},"Exit:"," A process that has been permanently released from the pool of executable processes by the OS. This occurs either because it halted naturally upon completion or because it was aborted due to an error or parent request.",[19,194893,194895],{"id":194894},"q3-what-does-it-mean-to-preempt-a-process","Q3. What does it mean to preempt a process?",[15,194897,194898,194901,194902,194905],{},[2627,194899,194900],{},"Process preemption occurs when an executing process is forcefully interrupted by the processor (usually driven by the OS scheduler) so that another process can be executed",". Instead of waiting for the running process to voluntarily yield control or block on an I\u002FO request, the OS moves the running process directly to the ",[2627,194903,194904],{},"Ready"," state.",[15,194907,194908],{},"Preemption is typically triggered by a clock interrupt when a process exhausts its maximum allowable time slice (quantum) for uninterrupted execution. It can also occur if the OS assigns different priority levels; if a higher-priority process transitions out of the Blocked state into the Ready state, the OS may immediately preempt the currently running lower-priority process to dispatch the higher-priority one.",[19,194910,194912],{"id":194911},"q4-what-are-the-steps-performed-by-an-os-to-create-a-new-process","Q4. What are the steps performed by an OS to create a new process?",[15,194914,194915],{},"Once the OS decides to create a new process, it proceeds with the following detailed steps:",[2230,194917,194918,194924,194930,194943,194955],{},[108,194919,194920,194923],{},[2627,194921,194922],{},"Assign a unique process identifier:"," The OS assigns a unique numeric identifier to the new process and adds a new entry for it into the primary process table.",[108,194925,194926,194929],{},[2627,194927,194928],{},"Allocate space for the process:"," The OS allocates the necessary memory for all elements of the process image. This includes the private user address space for the program code and data, the user stack for procedure calls, and the space for the process control block.",[108,194931,194932,194935,194936,194939,194940,194942],{},[2627,194933,194934],{},"Initialize the process control block (PCB):"," The OS populates the PCB with the process ID, the parent process ID, and processor state information (setting registers to zero, setting the program counter to the entry point, and defining stack boundaries). It also initializes process control information, such as setting the initial process state to ",[71807,194937,194938],{},"New"," or ",[71807,194941,194904],{}," and assigning a default priority level.",[108,194944,194945,194948,194949,194939,194951,194954],{},[2627,194946,194947],{},"Set the appropriate linkages:"," The new process is placed into the appropriate scheduling queues (e.g., adding it as a linked list node into the ",[71807,194950,194904],{},[71807,194952,194953],{},"Ready\u002FSuspend"," queue).",[108,194956,194957,194960],{},[2627,194958,194959],{},"Create or expand other data structures:"," The OS establishes any additional data structures needed, such as creating an accounting file to track the process's resource utilization for billing or performance assessment.",[19,194962,194964],{"id":194963},"q5-list-reasons-why-a-mode-switch-between-threads-may-be-cheaper-than-a-mode-switch-between-processes","Q5. List reasons why a mode switch between threads may be cheaper than a mode switch between processes.",[15,194966,194967,194968,2547],{},"Switching between threads is generally much faster and cheaper than switching between processes because ",[2627,194969,194970],{},"less state information must be saved and restored",[15,194972,194973],{},"All threads within a single process share the same virtual address space, files, and execution privileges. Therefore, a thread switch avoids the heavy overhead of updating memory management data structures or swapping out address spaces, which is strictly required during a full process switch.",[15,194975,194976,194977,194980],{},"Additionally, if the system uses User-Level Threads (ULTs), ",[2627,194978,194979],{},"thread switching does not require kernel-mode privileges",". All thread management data structures reside entirely within the user address space of that single process. Because the OS kernel is not involved, the system completely avoids the expensive overhead of performing two mode switches (switching from user mode to kernel mode to process the change, and then back to user mode).",[19,194982,194984],{"id":194983},"q6-why-are-all-threads-blocked-when-a-ult-executes-a-system-call","Q6. Why are all threads blocked when a ULT executes a system call?",[15,194986,194987,194988,194991],{},"This limitation occurs because ",[2627,194989,194990],{},"the thread structure of a process using ULTs is completely invisible to the operating system",". All of the thread management—such as creating, scheduling, and destroying threads—is done by an application-level threads library.",[15,194993,194994,194995,194998],{},"From the perspective of the OS kernel, it only sees and schedules the single overarching process. Consequently, if one User-Level Thread executes a blocking system call (such as waiting for I\u002FO), the OS kernel assumes the ",[71807,194996,194997],{},"entire process"," is waiting for that event. The kernel places the entire process into the Blocked state. Because the process is no longer running, the user-level threads library cannot execute, meaning all other threads within that process are simultaneously blocked from running, even if they are otherwise ready to execute.",[19,195000,195002],{"id":195001},"q7-consider-a-computer-with-n-processes-and-m-processors","Q7. Consider a computer with N processes and M processors.",[253,195004,195006],{"id":195005},"a-maximum-number-of-processes-in-each-state","a. Maximum number of processes in each state",[105,195008,195009,195187,195277],{},[108,195010,195011,195013,195014,195016,195017,195069,195070,195121,195122,195124,195125,195186],{},[2627,195012,194866],{}," At most ",[2627,195015,98360],{}," processes can be in the Running state at any given time (assuming ",[34,195018,195020,195038],{"className":195019},[31642],[34,195021,195023],{"className":195022},[31646],[31648,195024,195025],{"xmlns":31650},[31652,195026,195027,195035],{},[31655,195028,195029,195031,195033],{},[31658,195030,33347],{},[31662,195032,43175],{},[31658,195034,98360],{},[31665,195036,195037],{"encoding":31667},"N \\ge M",[34,195039,195041,195060],{"className":195040,"ariaHidden":758},[31672],[34,195042,195044,195048,195051,195054,195057],{"className":195043},[31676],[34,195045],{"className":195046,"style":195047},[31680],"height:0.8193em;vertical-align:-0.136em;",[34,195049,33347],{"className":195050,"style":33389},[31685,31686],[34,195052],{"className":195053,"style":31846},[31845],[34,195055,43175],{"className":195056},[31850],[34,195058],{"className":195059,"style":31846},[31845],[34,195061,195063,195066],{"className":195062},[31676],[34,195064],{"className":195065,"style":33218},[31680],[34,195067,98360],{"className":195068,"style":33389},[31685,31686],") because there are exactly M processors available to execute instructions. If there are fewer processes than processors (",[34,195071,195073,195091],{"className":195072},[31642],[34,195074,195076],{"className":195075},[31646],[31648,195077,195078],{"xmlns":31650},[31652,195079,195080,195088],{},[31655,195081,195082,195084,195086],{},[31658,195083,33347],{},[31662,195085,932],{},[31658,195087,98360],{},[31665,195089,195090],{"encoding":31667},"N \u003C M",[34,195092,195094,195112],{"className":195093,"ariaHidden":758},[31672],[34,195095,195097,195100,195103,195106,195109],{"className":195096},[31676],[34,195098],{"className":195099,"style":104901},[31680],[34,195101,33347],{"className":195102,"style":33389},[31685,31686],[34,195104],{"className":195105,"style":31846},[31845],[34,195107,932],{"className":195108},[31850],[34,195110],{"className":195111,"style":31846},[31845],[34,195113,195115,195118],{"className":195114},[31676],[34,195116],{"className":195117,"style":33218},[31680],[34,195119,98360],{"className":195120,"style":33389},[31685,31686],"), then at most ",[2627,195123,33347],{}," processes can be running. In general terms, the maximum is ",[2627,195126,195127],{},[34,195128,195130,195156],{"className":195129},[31642],[34,195131,195133],{"className":195132},[31646],[31648,195134,195135],{"xmlns":31650},[31652,195136,195137,195153],{},[31655,195138,195139,195141,195143,195145,195147,195149,195151],{},[31658,195140,56084],{},[31662,195142,32682],{},[31662,195144,456],{"stretchy":702},[31658,195146,33347],{},[31662,195148,31960],{"separator":758},[31658,195150,98360],{},[31662,195152,2557],{"stretchy":702},[31665,195154,195155],{"encoding":31667},"\\min(N, M)",[34,195157,195159],{"className":195158,"ariaHidden":758},[31672],[34,195160,195162,195165,195168,195171,195174,195177,195180,195183],{"className":195161},[31676],[34,195163],{"className":195164,"style":31774},[31680],[34,195166,56084],{"className":195167},[32767],[34,195169,456],{"className":195170},[31782],[34,195172,33347],{"className":195173,"style":33389},[31685,31686],[34,195175,31960],{"className":195176},[32119],[34,195178],{"className":195179,"style":32123},[31845],[34,195181,98360],{"className":195182,"style":33389},[31685,31686],[34,195184,2557],{"className":195185},[31690]," processes.",[108,195188,195189,195013,195191,195193,195194,2547],{},[2627,195190,194872],{},[2627,195192,33347],{}," processes can be in the Ready state. This could happen in a scenario where all N processes have just been admitted or transitioned to Ready, and the OS dispatcher has not yet assigned any of them to the M processors. The maximum number of strictly waiting Ready processes would be ",[2627,195195,195196],{},[34,195197,195199,195229],{"className":195198},[31642],[34,195200,195202],{"className":195201},[31646],[31648,195203,195204],{"xmlns":31650},[31652,195205,195206,195226],{},[31655,195207,195208,195210,195212,195214,195216,195218,195220,195222,195224],{},[31658,195209,33347],{},[31662,195211,32226],{},[31658,195213,56084],{},[31662,195215,32682],{},[31662,195217,456],{"stretchy":702},[31658,195219,33347],{},[31662,195221,31960],{"separator":758},[31658,195223,98360],{},[31662,195225,2557],{"stretchy":702},[31665,195227,195228],{"encoding":31667},"N - \\min(N, M)",[34,195230,195232,195250],{"className":195231,"ariaHidden":758},[31672],[34,195233,195235,195238,195241,195244,195247],{"className":195234},[31676],[34,195236],{"className":195237,"style":34944},[31680],[34,195239,33347],{"className":195240,"style":33389},[31685,31686],[34,195242],{"className":195243,"style":32604},[31845],[34,195245,32226],{"className":195246},[32024],[34,195248],{"className":195249,"style":32604},[31845],[34,195251,195253,195256,195259,195262,195265,195268,195271,195274],{"className":195252},[31676],[34,195254],{"className":195255,"style":31774},[31680],[34,195257,56084],{"className":195258},[32767],[34,195260,456],{"className":195261},[31782],[34,195263,33347],{"className":195264,"style":33389},[31685,31686],[34,195266,31960],{"className":195267},[32119],[34,195269],{"className":195270,"style":32123},[31845],[34,195272,98360],{"className":195273,"style":33389},[31685,31686],[34,195275,2557],{"className":195276},[31690],[108,195278,195279,195013,195282,195284],{},[2627,195280,195281],{},"Blocked:",[2627,195283,33347],{}," processes can be in the Blocked state simultaneously. This happens if every single process in the system has requested an external event (such as an I\u002FO operation) and is currently waiting for it to complete.",[253,195286,195288],{"id":195287},"b-minimum-number-of-processes-in-each-state","b. Minimum number of processes in each state",[105,195290,195291,195298,195357],{},[108,195292,195293,47649,195295,195297],{},[2627,195294,194866],{},[2627,195296,459],{}," processes. The processors could be completely idle if all N processes are in the Blocked state waiting for I\u002FO operations to complete.",[108,195299,195300,47649,195302,195304,195305,195356],{},[2627,195301,194872],{},[2627,195303,459],{}," processes. This occurs if all N processes are currently running (if ",[34,195306,195308,195326],{"className":195307},[31642],[34,195309,195311],{"className":195310},[31646],[31648,195312,195313],{"xmlns":31650},[31652,195314,195315,195323],{},[31655,195316,195317,195319,195321],{},[31658,195318,33347],{},[31662,195320,45972],{},[31658,195322,98360],{},[31665,195324,195325],{"encoding":31667},"N \\le M",[34,195327,195329,195347],{"className":195328,"ariaHidden":758},[31672],[34,195330,195332,195335,195338,195341,195344],{"className":195331},[31676],[34,195333],{"className":195334,"style":195047},[31680],[34,195336,33347],{"className":195337,"style":33389},[31685,31686],[34,195339],{"className":195340,"style":31846},[31845],[34,195342,45972],{"className":195343},[31850],[34,195345],{"className":195346,"style":31846},[31845],[34,195348,195350,195353],{"className":195349},[31676],[34,195351],{"className":195352,"style":33218},[31680],[34,195354,98360],{"className":195355,"style":33389},[31685,31686],"), or if all non-running processes are in the Blocked state.",[108,195358,195359,47649,195361,195363],{},[2627,195360,195281],{},[2627,195362,459],{}," processes. This occurs if all N processes are either actively running on the CPUs or sitting in the Ready queue waiting for processor time.",[19,195365,195367],{"id":195366},"q8-the-queue-model-suggests-that-a-process-can-only-be-in-one-event-queue-at-a-time","Q8. The queue model suggests that a process can only be in one Event queue at a time.",[253,195369,195371],{"id":195370},"a-can-a-process-wait-on-more-than-one-event-simultaneously","a. Can a process wait on more than one event simultaneously?",[15,195373,195374],{},"Yes, it is very possible and common to want an application to wait on multiple events concurrently.",[15,195376,195377,195380,195381,195384,195385,195388],{},[2627,195378,195379],{},"Example:"," An application might be processing data received from one process and then storing the processed results onto a disk. The process may reach a state where it has exhausted its current input and finished its last write. At this point, it needs to wait for ",[2627,195382,195383],{},"both"," additional input data from the other process ",[2627,195386,195387],{},"and"," for the disk to become available to accept a new write.",[253,195390,195392],{"id":195391},"b-how-to-modify-the-queuing-structure","b. How to modify the queuing structure?",[15,195394,195395],{},"There are several ways to modify the queuing structure to handle this:",[105,195397,195398,195407],{},[108,195399,195400,195403,195404,195406],{},[2627,195401,195402],{},"Either\u002FOr Queues:"," You could implement a special type of \"either\u002For\" queue where a process is unblocked if ",[71807,195405,28719],{}," of its designated events occur.",[108,195408,195409,195412],{},[2627,195410,195411],{},"Multiple Queue Assignment:"," The process's control block pointer could be placed into two (or more) separate event queues simultaneously.",[15,195414,195415],{},"In either modified design, the operating system's complexity increases. The OS would have to handle the delicate details of alerting the process to the occurrence of both events (potentially one after the other) and ensuring it is safely removed from all other event queues once it is unblocked.",[19,195417,195419],{"id":195418},"q9-tape-drive-allocation-policy","Q9. Tape Drive Allocation Policy",[15,195421,195422],{},"A multiprocessor with eight processors has 20 attached tape drives. There is a large number of jobs submitted to the system that each require a maximum of four tape drives to complete execution. Assume that each job starts running with only three tape drives for a long period before requiring the fourth tape drive for a short period toward the end of its operation.",[253,195424,195426],{"id":195425},"a-conservative-policy-require-4-tape-drives-upfront","a. Conservative policy (require 4 tape drives upfront)",[105,195428,195429,195439,195448],{},[108,195430,195431,195434,195435,195438],{},[2627,195432,195433],{},"Maximum jobs in progress:"," Because the conservative policy requires allocating all 4 drives up front, the system can support at most ",[2627,195436,195437],{},"5 jobs"," concurrently (20 total drives \u002F 4 drives per job = 5 jobs).",[108,195440,195441,195444,195445,195447],{},[2627,195442,195443],{},"Maximum idle drives:"," Because each job uses only 3 drives for the vast majority of its execution time, 1 drive per job is held but unused. Therefore, the maximum number of idle drives is ",[2627,195446,18136],{}," (5 jobs × 1 idle drive).",[108,195449,195450,195453,195454,195456],{},[2627,195451,195452],{},"Minimum idle drives:"," The minimum number of idle drives is ",[2627,195455,459],{},". This happens in the rare best-case scenario where all 5 active jobs reach the end of their execution at the exact same time and simultaneously utilize their 4th allocated drive.",[253,195458,195460],{"id":195459},"b-alternative-policy-allocate-on-demand","b. Alternative policy (allocate on demand)",[105,195462,195463,195473,195486],{},[108,195464,195465,195468,195469,195472],{},[2627,195466,195467],{},"Alternative Policy:"," To improve utilization, the OS should initially allocate exactly the ",[2627,195470,195471],{},"three tape drives"," each process needs for its long initial phase. The fourth tape drive is kept in a general pool and is only allocated on demand when a job reaches its final short period.",[108,195474,195475,195477,195478,195481,195482,195485],{},[2627,195476,195433],{}," Under this policy, at most ",[2627,195479,195480],{},"6 jobs"," can be active simultaneously (",[31,195483,195484],{},"floor(20\u002F3) = 6","). Giving 3 drives to 6 jobs consumes 18 drives, leaving 2 drives safely in reserve to satisfy the eventual demand for a 4th drive, fully preventing deadlock.",[108,195487,195488,195491,195492,195495,195496,195499],{},[2627,195489,195490],{},"Bounds on idling tape drives:"," The bounds are a ",[2627,195493,195494],{},"maximum of 2"," idle drives and a ",[2627,195497,195498],{},"minimum of 0"," idle drives. The maximum occurs when all 6 jobs are in their long initial phase (using 18 drives, leaving 2 idle). The minimum occurs if 2 of the 6 jobs enter their final phase simultaneously and claim the 2 reserve drives.",{"title":29,"searchDepth":43,"depth":43,"links":195501},[195502,195503,195504,195505,195506,195507,195508,195512,195516],{"id":194815,"depth":43,"text":194816},{"id":194855,"depth":43,"text":194856},{"id":194894,"depth":43,"text":194895},{"id":194911,"depth":43,"text":194912},{"id":194963,"depth":43,"text":194964},{"id":194983,"depth":43,"text":194984},{"id":195001,"depth":43,"text":195002,"children":195509},[195510,195511],{"id":195005,"depth":49,"text":195006},{"id":195287,"depth":49,"text":195288},{"id":195366,"depth":43,"text":195367,"children":195513},[195514,195515],{"id":195370,"depth":49,"text":195371},{"id":195391,"depth":49,"text":195392},{"id":195418,"depth":43,"text":195419,"children":195517},[195518,195519],{"id":195425,"depth":49,"text":195426},{"id":195459,"depth":49,"text":195460},{"order":43},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap3-hw",{"title":194806,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap3-hw","mUaWsMTlRf5dcQu23Mb4Wuiprp992qMEdgTNOS3c2BM",{"id":195526,"title":195527,"body":195528,"description":29,"extension":85,"meta":195720,"navigation":52,"path":195721,"seo":195722,"stem":195723,"__hash__":195724},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap4-hw.md","Chap. 4 HW",{"type":8,"value":195529,"toc":195712},[195530,195534,195538,195546,195548,195552,195566,195568,195572,195579,195581,195585,195602,195604,195608,195615,195617,195621,195626,195633,195638],[11,195531,195533],{"id":195532},"os-chap4-hw","OS Chap.4 HW",[19,195535,195537],{"id":195536},"q1-list-reasons-why-a-mode-switch-between-threads-may-be-cheaper-than-a-mode-switch-between-processes","Q1. List reasons why a mode switch between threads may be cheaper than a mode switch between processes.**",[105,195539,195540],{},[108,195541,195542,195545],{},[2627,195543,195544],{},"Answer:"," Less state information is involved. Thread switching does not require kernel-mode privileges because all of the thread management data structures are within the user address space of a single process. Therefore, the process does not switch to the kernel mode to do thread management, which saves the significant overhead of two mode switches (user to kernel; kernel back to user).",[32188,195547],{},[19,195549,195551],{"id":195550},"q2-what-are-the-two-separate-and-potentially-independent-characteristics-embodied-in-the-concept-of-process","Q2. What are the two separate and potentially independent characteristics embodied in the concept of process?**",[105,195553,195554],{},[108,195555,195556,195558,195559,195562,195563,2547],{},[2627,195557,195544],{}," The concept of a process embodies two separate characteristics: ",[2627,195560,195561],{},"resource ownership"," and ",[2627,195564,195565],{},"scheduling\u002Fexecution",[32188,195567],{},[19,195569,195571],{"id":195570},"q3-what-resources-are-typically-shared-by-all-of-the-threads-of-a-process","Q3. What resources are typically shared by all of the threads of a process?**",[105,195573,195574],{},[108,195575,195576,195578],{},[2627,195577,195544],{}," All of the threads of a process share the state and resources of that process. Examples of typically shared resources include the virtual address space, file resources (such as open files), and execution privileges.",[32188,195580],{},[19,195582,195584],{"id":195583},"q4-list-three-advantages-of-ults-over-klts","Q4. List three advantages of ULTs over KLTs.**",[105,195586,195587],{},[108,195588,195589,195591],{},[2627,195590,195544],{},[2230,195592,195593,195596,195599],{},[108,195594,195595],{},"Thread switching does not require kernel-mode privileges because all of the thread management data structures are within the user address space of a single process.",[108,195597,195598],{},"Scheduling can be application specific. The scheduling algorithm can be tailored to the application without disturbing the underlying OS scheduler.",[108,195600,195601],{},"ULTs (User-Level Threads) can run on any operating system. No changes are required to the underlying kernel to support ULTs because the threads library is simply a set of application-level utilities.",[32188,195603],{},[19,195605,195607],{"id":195606},"q5-it-was-pointed-out-that-two-advantages-of-using-multiple-threads-within-a-process-are-that-1-less-work-is-involved-in-creating-a-new-thread-within-an-existing-process-than-in-creating-a-new-process-and-2-communication-among-threads-within-the-same-process-is-simplified-is-it-also-the-case-that-a-mode-switch-between-two-threads-within-the-same-process-involves-less-work-than-a-mode-switch-between-two-threads-in-different-processes","Q5. It was pointed out that two advantages of using multiple threads within a process are that (1) less work is involved in creating a new thread within an existing process than in creating a new process, and (2) communication among threads within the same process is simplified. Is it also the case that a mode switch between two threads within the same process involves less work than a mode switch between two threads in different processes?**",[105,195609,195610],{},[108,195611,195612,195614],{},[2627,195613,195544],{}," Yes. Switching between two threads in different processes is essentially a process switch. It involves less work to switch between threads in the same process because more state information must be saved and restored to switch from one process to another.",[32188,195616],{},[19,195618,195620],{"id":195619},"q6-a-multiprocessor-with-eight-processors-has-20-attached-tape-drives-there-is-a-large-number-of-jobs-submitted-to-the-system-that-each-require-a-maximum-of-four-tape-drives-to-complete-execution-assume-that-each-job-starts-running-with-only-three-tape-drives-for-a-long-period-before-requiring-the-fourth-tape-drive-for-a-short-period-toward-the-end-of-its-operation-also-assume-an-endless-supply-of-such-jobs","Q6. A multiprocessor with eight processors has 20 attached tape drives. There is a large number of jobs submitted to the system that each require a maximum of four tape drives to complete execution. Assume that each job starts running with only three tape drives for a long period before requiring the fourth tape drive for a short period toward the end of its operation. Also assume an endless supply of such jobs.**",[15,195622,195623],{},[2627,195624,195625],{},"a. Assume the scheduler in the OS will not start a job unless there are four tape drives available. When a job is started, four drives are assigned immediately and are not released until the job finishes. What is the maximum number of jobs that can be in progress at once? What are the maximum and minimum number of tape drives that may be left idle as a result of this policy?",[105,195627,195628],{},[108,195629,195630,195632],{},[2627,195631,195544],{}," If a conservative policy is used, at most 20\u002F4 = 5 processes can be active simultaneously. Because one of the drives allocated to each process can be idle most of the time, at most 5 drives will be idle at a time (Maximum idle = 5). In the best case, none of the drives will be idle (Minimum idle = 0).",[15,195634,195635],{},[2627,195636,195637],{},"b. Suggest an alternative policy to improve tape drive utilization and at the same time avoid system deadlock. What is the maximum number of jobs that can be in progress at once? What are the bounds on the number of idling tape drives?",[105,195639,195640],{},[108,195641,195642,195644,195645,195711],{},[2627,195643,195544],{}," To improve drive utilization, an alternative policy is that each process can be initially allocated with exactly three tape drives, and the fourth one will be allocated on demand. In this policy, at most ",[34,195646,195648,195674],{"className":195647},[31642],[34,195649,195651],{"className":195650},[31646],[31648,195652,195653],{"xmlns":31650},[31652,195654,195655,195671],{},[31655,195656,195657,195659,195661,195663,195665,195667,195669],{},[31662,195658,173327],{"stretchy":702},[31749,195660,3249],{},[31658,195662,38080],{"mathvariant":38079},[31749,195664,786],{},[31662,195666,173336],{"stretchy":702},[31662,195668,202],{},[31749,195670,36656],{},[31665,195672,195673],{"encoding":31667},"\\lfloor 20\u002F3 \\rfloor = 6",[34,195675,195677,195702],{"className":195676,"ariaHidden":758},[31672],[34,195678,195680,195683,195686,195690,195693,195696,195699],{"className":195679},[31676],[34,195681],{"className":195682,"style":31774},[31680],[34,195684,173327],{"className":195685},[31782],[34,195687,195689],{"className":195688},[31685],"20\u002F3",[34,195691,173336],{"className":195692},[31690],[34,195694],{"className":195695,"style":31846},[31845],[34,195697,202],{"className":195698},[31850],[34,195700],{"className":195701,"style":31846},[31845],[34,195703,195705,195708],{"className":195704},[31676],[34,195706],{"className":195707,"style":32617},[31680],[34,195709,36656],{"className":195710},[31685]," processes can be active simultaneously. The bounds on idling tape drives are a minimum of 0 and a maximum of 2.",{"title":29,"searchDepth":43,"depth":43,"links":195713},[195714,195715,195716,195717,195718,195719],{"id":195536,"depth":43,"text":195537},{"id":195550,"depth":43,"text":195551},{"id":195570,"depth":43,"text":195571},{"id":195583,"depth":43,"text":195584},{"id":195606,"depth":43,"text":195607},{"id":195619,"depth":43,"text":195620},{"order":49},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap4-hw",{"title":195527,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap4-hw","pPPiB28baRtYpG029zB04bjFLoqjWR2TSmMY5kp65WI",{"id":195726,"title":195727,"body":195728,"description":29,"extension":85,"meta":199900,"navigation":52,"path":199901,"seo":199902,"stem":199903,"__hash__":199904},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap5-hw.md","Chap. 5 HW",{"type":8,"value":195729,"toc":199885},[195730,195734,195738,195744,195749,195763,195768,195852,195854,195858,195863,195867,195881,195886,195979,195981,195985,196048,196052,196090,196094,196282,196284,196288,196293,196297,196322,196326,196523,196525,196529,196575,196579,196588,196592,196618,196769,196773,196781,196785,196821,197002,197012,197014,197018,197026,197040,197045,197065,197069,197186,197188,197192,197201,197205,197237,197241,197408,197410,197414,197430,197434,197448,197452,197539,197541,197545,197565,197569,197589,197593,197730,197732,197736,197741,197749,197753,197784,197788,197918,197926,197928,197932,197939,197984,197988,198018,198022,198180,198193,198195,198199,198210,198488,198492,198529,198533,198740,198742,198746,198750,198864,198867,198921,198923,198927,198932,199689,199694,199789,199794,199803,199883],[11,195731,195733],{"id":195732},"os-chap5-hw","OS Chap.5 HW",[253,195735,195737],{"id":195736},"_1-阅览室座位与登记问题-reading-room","1. 阅览室座位与登记问题 (Reading Room)",[15,195739,195740,195743],{},[2627,195741,195742],{},"问题分析："," 这是一个带有资源数量限制的互斥问题。座位是有限资源（100个），而“登记表”是临界资源（不能同时有多人一起写），需要互斥访问。",[15,195745,195746],{},[2627,195747,195748],{},"信号量定义与初始化：",[105,195750,195751,195757],{},[108,195752,195753,195756],{},[31,195754,195755],{},"seats = 100;"," \u002F\u002F 资源信号量，表示阅览室空座位的数量。",[108,195758,195759,195762],{},[31,195760,195761],{},"mutex = 1;"," \u002F\u002F 互斥信号量，用于对登记表进行互斥访问。",[15,195764,195765],{},[2627,195766,195767],{},"读者进程 (Reader Process) 的同步结构：",[24,195769,195771],{"className":65734,"code":195770,"language":31965,"meta":29,"style":29},"Reader() {\n while (true) {\n \u002F\u002F 准备进入阅览室\n semWait(seats); \u002F\u002F 申请一个座位，如果没有空位则阻塞等待\n semWait(mutex); \u002F\u002F 申请访问登记表（互斥）\n 登记座位号和姓名; \u002F\u002F 临界区：登记\n semSignal(mutex); \u002F\u002F 释放登记表\n\n 阅读(Reading); \u002F\u002F 在阅览室读书\n\n \u002F\u002F 准备离开阅览室\n semWait(mutex); \u002F\u002F 申请访问登记表（互斥）\n 注销登记记录; \u002F\u002F 临界区：注销\n semSignal(mutex); \u002F\u002F 释放登记表\n semSignal(seats); \u002F\u002F 释放一个座位，唤醒可能在门外等待的读者\n }\n}\n",[31,195772,195773,195778,195783,195788,195793,195798,195803,195808,195812,195817,195821,195826,195830,195835,195839,195844,195848],{"__ignoreMap":29},[34,195774,195775],{"class":36,"line":37},[34,195776,195777],{},"Reader() {\n",[34,195779,195780],{"class":36,"line":43},[34,195781,195782],{}," while (true) {\n",[34,195784,195785],{"class":36,"line":49},[34,195786,195787],{}," \u002F\u002F 准备进入阅览室\n",[34,195789,195790],{"class":36,"line":56},[34,195791,195792],{}," semWait(seats); \u002F\u002F 申请一个座位，如果没有空位则阻塞等待\n",[34,195794,195795],{"class":36,"line":62},[34,195796,195797],{}," semWait(mutex); \u002F\u002F 申请访问登记表（互斥）\n",[34,195799,195800],{"class":36,"line":68},[34,195801,195802],{}," 登记座位号和姓名; \u002F\u002F 临界区：登记\n",[34,195804,195805],{"class":36,"line":74},[34,195806,195807],{}," semSignal(mutex); \u002F\u002F 释放登记表\n",[34,195809,195810],{"class":36,"line":1971},[34,195811,53],{"emptyLinePlaceholder":52},[34,195813,195814],{"class":36,"line":1976},[34,195815,195816],{}," 阅读(Reading); \u002F\u002F 在阅览室读书\n",[34,195818,195819],{"class":36,"line":1985},[34,195820,53],{"emptyLinePlaceholder":52},[34,195822,195823],{"class":36,"line":1990},[34,195824,195825],{}," \u002F\u002F 准备离开阅览室\n",[34,195827,195828],{"class":36,"line":1999},[34,195829,195797],{},[34,195831,195832],{"class":36,"line":2022},[34,195833,195834],{}," 注销登记记录; \u002F\u002F 临界区：注销\n",[34,195836,195837],{"class":36,"line":2036},[34,195838,195807],{},[34,195840,195841],{"class":36,"line":2603},[34,195842,195843],{}," semSignal(seats); \u002F\u002F 释放一个座位，唤醒可能在门外等待的读者\n",[34,195845,195846],{"class":36,"line":5537},[34,195847,5540],{},[34,195849,195850],{"class":36,"line":5543},[34,195851,77],{},[32188,195853],{},[253,195855,195857],{"id":195856},"_2-司机与售票员同步问题-bus-driver-and-conductor","2. 司机与售票员同步问题 (Bus Driver and Conductor)",[15,195859,195860,195862],{},[2627,195861,195742],{}," 这是一个典型的进程同步问题。两者的动作有严格的先后顺序：司机必须等售票员关好门后才能启动车辆；售票员必须等司机停稳车后才能开门。\n初始状态是：车停着，门是开的。下一步应该是售票员关门。",[15,195864,195865],{},[2627,195866,195748],{},[105,195868,195869,195875],{},[108,195870,195871,195874],{},[31,195872,195873],{},"start_bus = 0;"," \u002F\u002F 同步信号量，表示是否可以发车（0表示需等待售票员关门发送信号）。",[108,195876,195877,195880],{},[31,195878,195879],{},"open_door = 0;"," \u002F\u002F 同步信号量，表示是否可以开门（0表示需等待司机停车发送信号）。",[15,195882,195883],{},[2627,195884,195885],{},"同步关系实现：",[24,195887,195889],{"className":65734,"code":195888,"language":31965,"meta":29,"style":29},"Driver() { \u002F\u002F 司机进程\n while (true) {\n semWait(start_bus); \u002F\u002F 等待售票员关门的信号\n 启动车辆(Starting);\n 正常行车(Driving);\n 到站停车(Stopping);\n semSignal(open_door);\u002F\u002F 停车完毕，向售票员发送可以开门的信号\n }\n}\n\nConductor() { \u002F\u002F 售票员进程\n while (true) {\n 关车门(Closing door);\n semSignal(start_bus);\u002F\u002F 关门完毕，向司机发送可以发车的信号\n 售票(Selling tickets);\n semWait(open_door); \u002F\u002F 等待司机停车的信号\n 开车门(Opening door);\n }\n}\n",[31,195890,195891,195896,195900,195905,195910,195915,195920,195925,195929,195933,195937,195942,195946,195951,195956,195961,195966,195971,195975],{"__ignoreMap":29},[34,195892,195893],{"class":36,"line":37},[34,195894,195895],{},"Driver() { \u002F\u002F 司机进程\n",[34,195897,195898],{"class":36,"line":43},[34,195899,195782],{},[34,195901,195902],{"class":36,"line":49},[34,195903,195904],{}," semWait(start_bus); \u002F\u002F 等待售票员关门的信号\n",[34,195906,195907],{"class":36,"line":56},[34,195908,195909],{}," 启动车辆(Starting);\n",[34,195911,195912],{"class":36,"line":62},[34,195913,195914],{}," 正常行车(Driving);\n",[34,195916,195917],{"class":36,"line":68},[34,195918,195919],{}," 到站停车(Stopping);\n",[34,195921,195922],{"class":36,"line":74},[34,195923,195924],{}," semSignal(open_door);\u002F\u002F 停车完毕，向售票员发送可以开门的信号\n",[34,195926,195927],{"class":36,"line":1971},[34,195928,5540],{},[34,195930,195931],{"class":36,"line":1976},[34,195932,77],{},[34,195934,195935],{"class":36,"line":1985},[34,195936,53],{"emptyLinePlaceholder":52},[34,195938,195939],{"class":36,"line":1990},[34,195940,195941],{},"Conductor() { \u002F\u002F 售票员进程\n",[34,195943,195944],{"class":36,"line":1999},[34,195945,195782],{},[34,195947,195948],{"class":36,"line":2022},[34,195949,195950],{}," 关车门(Closing door);\n",[34,195952,195953],{"class":36,"line":2036},[34,195954,195955],{}," semSignal(start_bus);\u002F\u002F 关门完毕，向司机发送可以发车的信号\n",[34,195957,195958],{"class":36,"line":2603},[34,195959,195960],{}," 售票(Selling tickets);\n",[34,195962,195963],{"class":36,"line":5537},[34,195964,195965],{}," semWait(open_door); \u002F\u002F 等待司机停车的信号\n",[34,195967,195968],{"class":36,"line":5543},[34,195969,195970],{}," 开车门(Opening door);\n",[34,195972,195973],{"class":36,"line":5548},[34,195974,5540],{},[34,195976,195977],{"class":36,"line":5564},[34,195978,77],{},[32188,195980],{},[253,195982,195984],{"id":195983},"_3-多级生产者-消费者问题-processes-a-b-and-c","3. 多级生产者-消费者问题 (Processes A, B, and C)",[15,195986,195987,195989,195990,196018,196019,196047],{},[2627,195988,195742],{}," 这是经典生产者\u002F消费者问题 的串联变种。A和B共享大小为 ",[34,195991,195993,196006],{"className":195992},[31642],[34,195994,195996],{"className":195995},[31646],[31648,195997,195998],{"xmlns":31650},[31652,195999,196000,196004],{},[31655,196001,196002],{},[31658,196003,31660],{},[31665,196005,31660],{"encoding":31667},[34,196007,196009],{"className":196008,"ariaHidden":758},[31672],[34,196010,196012,196015],{"className":196011},[31676],[34,196013],{"className":196014,"style":32527},[31680],[34,196016,31660],{"className":196017},[31685,31686]," 的缓冲池，B和C共享大小为 ",[34,196020,196022,196035],{"className":196021},[31642],[34,196023,196025],{"className":196024},[31646],[31648,196026,196027],{"xmlns":31650},[31652,196028,196029,196033],{},[31655,196030,196031],{},[31658,196032,31759],{},[31665,196034,31759],{"encoding":31667},[34,196036,196038],{"className":196037,"ariaHidden":758},[31672],[34,196039,196041,196044],{"className":196040},[31676],[34,196042],{"className":196043,"style":32527},[31680],[34,196045,31759],{"className":196046},[31685,31686]," 的缓冲池。进程 B 既是 A 的消费者，又是 C 的生产者。",[15,196049,196050],{},[2627,196051,195748],{},[105,196053,196054,196060,196066,196072,196078,196084],{},[108,196055,196056,196059],{},[31,196057,196058],{},"empty1 = n;"," \u002F\u002F A和B之间的空闲缓冲区数量",[108,196061,196062,196065],{},[31,196063,196064],{},"full1 = 0;"," \u002F\u002F A和B之间含有数据的缓冲区数量",[108,196067,196068,196071],{},[31,196069,196070],{},"mutex1 = 1;"," \u002F\u002F A和B之间访问缓冲池的互斥信号量",[108,196073,196074,196077],{},[31,196075,196076],{},"empty2 = m;"," \u002F\u002F B和C之间的空闲缓冲区数量",[108,196079,196080,196083],{},[31,196081,196082],{},"full2 = 0;"," \u002F\u002F B和C之间含有数据的缓冲区数量",[108,196085,196086,196089],{},[31,196087,196088],{},"mutex2 = 1;"," \u002F\u002F B和C之间访问缓冲池的互斥信号量",[15,196091,196092],{},[2627,196093,195885],{},[24,196095,196097],{"className":65734,"code":196096,"language":31965,"meta":29,"style":29},"Process_A() { \u002F\u002F 生产者 A\n while (true) {\n 生产数据;\n semWait(empty1); \u002F\u002F 申请第一组的一个空缓冲区\n semWait(mutex1); \u002F\u002F 互斥访问第一组缓冲池\n 将数据放入缓冲池1;\n semSignal(mutex1); \u002F\u002F 释放第一组缓冲池\n semSignal(full1); \u002F\u002F 第一组满缓冲区加1，唤醒 B\n }\n}\n\nProcess_B() { \u002F\u002F 消费者 B & 生产者 B\n while (true) {\n semWait(full1); \u002F\u002F 等待第一组缓冲池中有数据\n semWait(mutex1); \u002F\u002F 互斥访问第一组缓冲池\n 从缓冲池1取出数据;\n semSignal(mutex1); \u002F\u002F 释放第一组缓冲池\n semSignal(empty1); \u002F\u002F 第一组空缓冲区加1，唤醒 A\n \n 加工处理该数据;\n\n semWait(empty2); \u002F\u002F 申请第二组的一个空缓冲区\n semWait(mutex2); \u002F\u002F 互斥访问第二组缓冲池\n 将处理后的数据放入缓冲池2;\n semSignal(mutex2); \u002F\u002F 释放第二组缓冲池\n semSignal(full2); \u002F\u002F 第二组满缓冲区加1，唤醒 C\n }\n}\n\nProcess_C() { \u002F\u002F 消费者 C\n while (true) {\n semWait(full2); \u002F\u002F 等待第二组缓冲池中有数据\n semWait(mutex2); \u002F\u002F 互斥访问第二组缓冲池\n 从缓冲池2取出数据;\n semSignal(mutex2); \u002F\u002F 释放第二组缓冲池\n semSignal(empty2); \u002F\u002F 第二组空缓冲区加1，唤醒 B\n \n 消费数据;\n }\n}\n",[31,196098,196099,196104,196108,196113,196118,196123,196128,196133,196138,196142,196146,196150,196155,196159,196164,196168,196173,196177,196182,196187,196192,196196,196201,196206,196211,196216,196221,196225,196229,196233,196238,196242,196247,196251,196256,196260,196265,196269,196274,196278],{"__ignoreMap":29},[34,196100,196101],{"class":36,"line":37},[34,196102,196103],{},"Process_A() { \u002F\u002F 生产者 A\n",[34,196105,196106],{"class":36,"line":43},[34,196107,195782],{},[34,196109,196110],{"class":36,"line":49},[34,196111,196112],{}," 生产数据;\n",[34,196114,196115],{"class":36,"line":56},[34,196116,196117],{}," semWait(empty1); \u002F\u002F 申请第一组的一个空缓冲区\n",[34,196119,196120],{"class":36,"line":62},[34,196121,196122],{}," semWait(mutex1); \u002F\u002F 互斥访问第一组缓冲池\n",[34,196124,196125],{"class":36,"line":68},[34,196126,196127],{}," 将数据放入缓冲池1;\n",[34,196129,196130],{"class":36,"line":74},[34,196131,196132],{}," semSignal(mutex1); \u002F\u002F 释放第一组缓冲池\n",[34,196134,196135],{"class":36,"line":1971},[34,196136,196137],{}," semSignal(full1); \u002F\u002F 第一组满缓冲区加1，唤醒 B\n",[34,196139,196140],{"class":36,"line":1976},[34,196141,5540],{},[34,196143,196144],{"class":36,"line":1985},[34,196145,77],{},[34,196147,196148],{"class":36,"line":1990},[34,196149,53],{"emptyLinePlaceholder":52},[34,196151,196152],{"class":36,"line":1999},[34,196153,196154],{},"Process_B() { \u002F\u002F 消费者 B & 生产者 B\n",[34,196156,196157],{"class":36,"line":2022},[34,196158,195782],{},[34,196160,196161],{"class":36,"line":2036},[34,196162,196163],{}," semWait(full1); \u002F\u002F 等待第一组缓冲池中有数据\n",[34,196165,196166],{"class":36,"line":2603},[34,196167,196122],{},[34,196169,196170],{"class":36,"line":5537},[34,196171,196172],{}," 从缓冲池1取出数据;\n",[34,196174,196175],{"class":36,"line":5543},[34,196176,196132],{},[34,196178,196179],{"class":36,"line":5548},[34,196180,196181],{}," semSignal(empty1); \u002F\u002F 第一组空缓冲区加1，唤醒 A\n",[34,196183,196184],{"class":36,"line":5564},[34,196185,196186],{}," \n",[34,196188,196189],{"class":36,"line":5576},[34,196190,196191],{}," 加工处理该数据;\n",[34,196193,196194],{"class":36,"line":5606},[34,196195,53],{"emptyLinePlaceholder":52},[34,196197,196198],{"class":36,"line":5616},[34,196199,196200],{}," semWait(empty2); \u002F\u002F 申请第二组的一个空缓冲区\n",[34,196202,196203],{"class":36,"line":5628},[34,196204,196205],{}," semWait(mutex2); \u002F\u002F 互斥访问第二组缓冲池\n",[34,196207,196208],{"class":36,"line":5634},[34,196209,196210],{}," 将处理后的数据放入缓冲池2;\n",[34,196212,196213],{"class":36,"line":6420},[34,196214,196215],{}," semSignal(mutex2); \u002F\u002F 释放第二组缓冲池\n",[34,196217,196218],{"class":36,"line":11294},[34,196219,196220],{}," semSignal(full2); \u002F\u002F 第二组满缓冲区加1，唤醒 C\n",[34,196222,196223],{"class":36,"line":11321},[34,196224,5540],{},[34,196226,196227],{"class":36,"line":11326},[34,196228,77],{},[34,196230,196231],{"class":36,"line":11333},[34,196232,53],{"emptyLinePlaceholder":52},[34,196234,196235],{"class":36,"line":11344},[34,196236,196237],{},"Process_C() { \u002F\u002F 消费者 C\n",[34,196239,196240],{"class":36,"line":11349},[34,196241,195782],{},[34,196243,196244],{"class":36,"line":11354},[34,196245,196246],{}," semWait(full2); \u002F\u002F 等待第二组缓冲池中有数据\n",[34,196248,196249],{"class":36,"line":11363},[34,196250,196205],{},[34,196252,196253],{"class":36,"line":11368},[34,196254,196255],{}," 从缓冲池2取出数据;\n",[34,196257,196258],{"class":36,"line":11377},[34,196259,196215],{},[34,196261,196262],{"class":36,"line":11402},[34,196263,196264],{}," semSignal(empty2); \u002F\u002F 第二组空缓冲区加1，唤醒 B\n",[34,196266,196267],{"class":36,"line":11407},[34,196268,196186],{},[34,196270,196271],{"class":36,"line":11421},[34,196272,196273],{}," 消费数据;\n",[34,196275,196276],{"class":36,"line":11426},[34,196277,5540],{},[34,196279,196280],{"class":36,"line":11447},[34,196281,77],{},[32188,196283],{},[253,196285,196287],{"id":196286},"_4-苹果与橘子问题-father-mother-sons-daughters","4. 苹果与橘子问题 (Father, Mother, Sons, Daughters)",[15,196289,196290,196292],{},[2627,196291,195742],{}," 这是一个多生产者、多消费者且带有产品种类区分的问题。盘子容量为 2（表示最多放2个水果）。每次只能放入或取出一个水果，需要互斥。",[15,196294,196295],{},[2627,196296,195748],{},[105,196298,196299,196305,196311,196317],{},[108,196300,196301,196304],{},[31,196302,196303],{},"plate = 2;"," \u002F\u002F 同步信号量，盘子里的空闲空间，初始可放2个水果",[108,196306,196307,196310],{},[31,196308,196309],{},"apple = 0;"," \u002F\u002F 同步信号量，盘子里苹果的数量",[108,196312,196313,196316],{},[31,196314,196315],{},"orange = 0;"," \u002F\u002F 同步信号量，盘子里橘子的数量",[108,196318,196319,196321],{},[31,196320,195761],{}," \u002F\u002F 互斥信号量，保证每次只有一个动作（放\u002F取）在盘子上进行",[15,196323,196324],{},[2627,196325,195885],{},[24,196327,196329],{"className":65734,"code":196328,"language":31965,"meta":29,"style":29},"Father() {\n while (true) {\n 准备一个苹果;\n semWait(plate); \u002F\u002F 等待盘子有空位\n semWait(mutex); \u002F\u002F 互斥访问盘子\n 把苹果放入盘子;\n semSignal(mutex); \u002F\u002F 释放盘子\n semSignal(apple); \u002F\u002F 苹果数量加1，唤醒女儿\n }\n}\n\nMother() {\n while (true) {\n 准备一个橘子;\n semWait(plate); \u002F\u002F 等待盘子有空位\n semWait(mutex); \u002F\u002F 互斥访问盘子\n 把橘子放入盘子;\n semSignal(mutex); \u002F\u002F 释放盘子\n semSignal(orange); \u002F\u002F 橘子数量加1，唤醒儿子\n }\n}\n\nDaughter_1_and_2() { \u002F\u002F 两个女儿的逻辑相同\n while (true) {\n semWait(apple); \u002F\u002F 等待盘子里有苹果\n semWait(mutex); \u002F\u002F 互斥访问盘子\n 从盘子中取走一个苹果;\n semSignal(mutex); \u002F\u002F 释放盘子\n semSignal(plate); \u002F\u002F 盘子空位加1，唤醒父母\n 吃苹果;\n }\n}\n\nSon_1_and_2() { \u002F\u002F 两个儿子的逻辑相同\n while (true) {\n semWait(orange); \u002F\u002F 等待盘子里有橘子\n semWait(mutex); \u002F\u002F 互斥访问盘子\n 从盘子中取走一个橘子;\n semSignal(mutex); \u002F\u002F 释放盘子\n semSignal(plate); \u002F\u002F 盘子空位加1，唤醒父母\n 吃橘子;\n }\n}\n",[31,196330,196331,196336,196340,196345,196350,196355,196360,196365,196370,196374,196378,196382,196387,196391,196396,196400,196404,196409,196413,196418,196422,196426,196430,196435,196439,196444,196448,196453,196457,196462,196467,196471,196475,196479,196484,196488,196493,196497,196502,196506,196510,196515,196519],{"__ignoreMap":29},[34,196332,196333],{"class":36,"line":37},[34,196334,196335],{},"Father() {\n",[34,196337,196338],{"class":36,"line":43},[34,196339,195782],{},[34,196341,196342],{"class":36,"line":49},[34,196343,196344],{}," 准备一个苹果;\n",[34,196346,196347],{"class":36,"line":56},[34,196348,196349],{}," semWait(plate); \u002F\u002F 等待盘子有空位\n",[34,196351,196352],{"class":36,"line":62},[34,196353,196354],{}," semWait(mutex); \u002F\u002F 互斥访问盘子\n",[34,196356,196357],{"class":36,"line":68},[34,196358,196359],{}," 把苹果放入盘子;\n",[34,196361,196362],{"class":36,"line":74},[34,196363,196364],{}," semSignal(mutex); \u002F\u002F 释放盘子\n",[34,196366,196367],{"class":36,"line":1971},[34,196368,196369],{}," semSignal(apple); \u002F\u002F 苹果数量加1，唤醒女儿\n",[34,196371,196372],{"class":36,"line":1976},[34,196373,5540],{},[34,196375,196376],{"class":36,"line":1985},[34,196377,77],{},[34,196379,196380],{"class":36,"line":1990},[34,196381,53],{"emptyLinePlaceholder":52},[34,196383,196384],{"class":36,"line":1999},[34,196385,196386],{},"Mother() {\n",[34,196388,196389],{"class":36,"line":2022},[34,196390,195782],{},[34,196392,196393],{"class":36,"line":2036},[34,196394,196395],{}," 准备一个橘子;\n",[34,196397,196398],{"class":36,"line":2603},[34,196399,196349],{},[34,196401,196402],{"class":36,"line":5537},[34,196403,196354],{},[34,196405,196406],{"class":36,"line":5543},[34,196407,196408],{}," 把橘子放入盘子;\n",[34,196410,196411],{"class":36,"line":5548},[34,196412,196364],{},[34,196414,196415],{"class":36,"line":5564},[34,196416,196417],{}," semSignal(orange); \u002F\u002F 橘子数量加1，唤醒儿子\n",[34,196419,196420],{"class":36,"line":5576},[34,196421,5540],{},[34,196423,196424],{"class":36,"line":5606},[34,196425,77],{},[34,196427,196428],{"class":36,"line":5616},[34,196429,53],{"emptyLinePlaceholder":52},[34,196431,196432],{"class":36,"line":5628},[34,196433,196434],{},"Daughter_1_and_2() { \u002F\u002F 两个女儿的逻辑相同\n",[34,196436,196437],{"class":36,"line":5634},[34,196438,195782],{},[34,196440,196441],{"class":36,"line":6420},[34,196442,196443],{}," semWait(apple); \u002F\u002F 等待盘子里有苹果\n",[34,196445,196446],{"class":36,"line":11294},[34,196447,196354],{},[34,196449,196450],{"class":36,"line":11321},[34,196451,196452],{}," 从盘子中取走一个苹果;\n",[34,196454,196455],{"class":36,"line":11326},[34,196456,196364],{},[34,196458,196459],{"class":36,"line":11333},[34,196460,196461],{}," semSignal(plate); \u002F\u002F 盘子空位加1，唤醒父母\n",[34,196463,196464],{"class":36,"line":11344},[34,196465,196466],{}," 吃苹果;\n",[34,196468,196469],{"class":36,"line":11349},[34,196470,5540],{},[34,196472,196473],{"class":36,"line":11354},[34,196474,77],{},[34,196476,196477],{"class":36,"line":11363},[34,196478,53],{"emptyLinePlaceholder":52},[34,196480,196481],{"class":36,"line":11368},[34,196482,196483],{},"Son_1_and_2() { \u002F\u002F 两个儿子的逻辑相同\n",[34,196485,196486],{"class":36,"line":11377},[34,196487,195782],{},[34,196489,196490],{"class":36,"line":11402},[34,196491,196492],{}," semWait(orange); \u002F\u002F 等待盘子里有橘子\n",[34,196494,196495],{"class":36,"line":11407},[34,196496,196354],{},[34,196498,196499],{"class":36,"line":11421},[34,196500,196501],{}," 从盘子中取走一个橘子;\n",[34,196503,196504],{"class":36,"line":11426},[34,196505,196364],{},[34,196507,196508],{"class":36,"line":11447},[34,196509,196461],{},[34,196511,196512],{"class":36,"line":11452},[34,196513,196514],{}," 吃橘子;\n",[34,196516,196517],{"class":36,"line":11463},[34,196518,5540],{},[34,196520,196521],{"class":36,"line":11485},[34,196522,77],{},[32188,196524],{},[253,196526,196528],{"id":196527},"_5-进程-p-q-r-同步-concurrent-processes-p-q-r","5. 进程 P, Q, R 同步 (Concurrent processes P, Q, R)",[15,196530,196531,196532,196574],{},"这道题与第3题结构非常相似，分为缓冲池大小为 1 和大小为 ",[34,196533,196535,196553],{"className":196534},[31642],[34,196536,196538],{"className":196537},[31646],[31648,196539,196540],{"xmlns":31650},[31652,196541,196542,196550],{},[31655,196543,196544,196546,196548],{},[31658,196545,31759],{},[31662,196547,31960],{"separator":758},[31658,196549,31660],{},[31665,196551,196552],{"encoding":31667},"m, n",[34,196554,196556],{"className":196555,"ariaHidden":758},[31672],[34,196557,196559,196562,196565,196568,196571],{"className":196558},[31676],[34,196560],{"className":196561,"style":38774},[31680],[34,196563,31759],{"className":196564},[31685,31686],[34,196566,31960],{"className":196567},[32119],[34,196569],{"className":196570,"style":32123},[31845],[34,196572,31660],{"className":196573},[31685,31686]," 两种情况。",[32490,196576,196578],{"id":196577},"_1-共享一个单缓冲-share-one-buffer-each","(1) 共享一个单缓冲 (Share ONE buffer each)",[15,196580,196581,196583,196584,196587],{},[2627,196582,195742],{}," 因为 P 和 Q 之间只有一个缓冲区，Q 和 R 之间也只有一个缓冲区。容量为 1 时，空和满信号量本身就能保证互斥，因此通常可以省略 ",[31,196585,196586],{},"mutex"," 信号量（加上也不算错）。",[15,196589,196590],{},[2627,196591,195748],{},[105,196593,196594,196600,196606,196612],{},[108,196595,196596,196599],{},[31,196597,196598],{},"emptyPQ = 1;"," \u002F\u002F P与Q之间的空缓冲区数",[108,196601,196602,196605],{},[31,196603,196604],{},"fullPQ = 0;"," \u002F\u002F P与Q之间的满缓冲区数",[108,196607,196608,196611],{},[31,196609,196610],{},"emptyQR = 1;"," \u002F\u002F Q与R之间的空缓冲区数",[108,196613,196614,196617],{},[31,196615,196616],{},"fullQR = 0;"," \u002F\u002F Q与R之间的满缓冲区数",[24,196619,196621],{"className":65734,"code":196620,"language":31965,"meta":29,"style":29},"Process_P() {\n while (true) {\n 读取输入设备信息;\n semWait(emptyPQ); \u002F\u002F 等待 PQ 间缓冲区为空\n 将信息放入 PQ 缓冲区;\n semSignal(fullPQ); \u002F\u002F 唤醒 Q\n }\n}\n\nProcess_Q() {\n while (true) {\n semWait(fullPQ); \u002F\u002F 等待 PQ 间缓冲区有信息\n 从 PQ 缓冲区提取信息;\n semSignal(emptyPQ);\u002F\u002F 唤醒 P 继续输入\n \n 加工处理信息;\n\n semWait(emptyQR); \u002F\u002F 等待 QR 间缓冲区为空\n 将处理后信息放入 QR 缓冲区;\n semSignal(fullQR); \u002F\u002F 唤醒 R\n }\n}\n\nProcess_R() {\n while (true) {\n semWait(fullQR); \u002F\u002F 等待 QR 间缓冲区有信息\n 从 QR 缓冲区取信息;\n semSignal(emptyQR);\u002F\u002F 唤醒 Q 继续存放\n \n 打印输出;\n }\n}\n",[31,196622,196623,196628,196632,196637,196642,196647,196652,196656,196660,196664,196669,196673,196678,196683,196688,196692,196697,196701,196706,196711,196716,196720,196724,196728,196733,196737,196742,196747,196752,196756,196761,196765],{"__ignoreMap":29},[34,196624,196625],{"class":36,"line":37},[34,196626,196627],{},"Process_P() {\n",[34,196629,196630],{"class":36,"line":43},[34,196631,195782],{},[34,196633,196634],{"class":36,"line":49},[34,196635,196636],{}," 读取输入设备信息;\n",[34,196638,196639],{"class":36,"line":56},[34,196640,196641],{}," semWait(emptyPQ); \u002F\u002F 等待 PQ 间缓冲区为空\n",[34,196643,196644],{"class":36,"line":62},[34,196645,196646],{}," 将信息放入 PQ 缓冲区;\n",[34,196648,196649],{"class":36,"line":68},[34,196650,196651],{}," semSignal(fullPQ); \u002F\u002F 唤醒 Q\n",[34,196653,196654],{"class":36,"line":74},[34,196655,5540],{},[34,196657,196658],{"class":36,"line":1971},[34,196659,77],{},[34,196661,196662],{"class":36,"line":1976},[34,196663,53],{"emptyLinePlaceholder":52},[34,196665,196666],{"class":36,"line":1985},[34,196667,196668],{},"Process_Q() {\n",[34,196670,196671],{"class":36,"line":1990},[34,196672,195782],{},[34,196674,196675],{"class":36,"line":1999},[34,196676,196677],{}," semWait(fullPQ); \u002F\u002F 等待 PQ 间缓冲区有信息\n",[34,196679,196680],{"class":36,"line":2022},[34,196681,196682],{}," 从 PQ 缓冲区提取信息;\n",[34,196684,196685],{"class":36,"line":2036},[34,196686,196687],{}," semSignal(emptyPQ);\u002F\u002F 唤醒 P 继续输入\n",[34,196689,196690],{"class":36,"line":2603},[34,196691,196186],{},[34,196693,196694],{"class":36,"line":5537},[34,196695,196696],{}," 加工处理信息;\n",[34,196698,196699],{"class":36,"line":5543},[34,196700,53],{"emptyLinePlaceholder":52},[34,196702,196703],{"class":36,"line":5548},[34,196704,196705],{}," semWait(emptyQR); \u002F\u002F 等待 QR 间缓冲区为空\n",[34,196707,196708],{"class":36,"line":5564},[34,196709,196710],{}," 将处理后信息放入 QR 缓冲区;\n",[34,196712,196713],{"class":36,"line":5576},[34,196714,196715],{}," semSignal(fullQR); \u002F\u002F 唤醒 R\n",[34,196717,196718],{"class":36,"line":5606},[34,196719,5540],{},[34,196721,196722],{"class":36,"line":5616},[34,196723,77],{},[34,196725,196726],{"class":36,"line":5628},[34,196727,53],{"emptyLinePlaceholder":52},[34,196729,196730],{"class":36,"line":5634},[34,196731,196732],{},"Process_R() {\n",[34,196734,196735],{"class":36,"line":6420},[34,196736,195782],{},[34,196738,196739],{"class":36,"line":11294},[34,196740,196741],{}," semWait(fullQR); \u002F\u002F 等待 QR 间缓冲区有信息\n",[34,196743,196744],{"class":36,"line":11321},[34,196745,196746],{}," 从 QR 缓冲区取信息;\n",[34,196748,196749],{"class":36,"line":11326},[34,196750,196751],{}," semSignal(emptyQR);\u002F\u002F 唤醒 Q 继续存放\n",[34,196753,196754],{"class":36,"line":11333},[34,196755,196186],{},[34,196757,196758],{"class":36,"line":11344},[34,196759,196760],{}," 打印输出;\n",[34,196762,196763],{"class":36,"line":11349},[34,196764,5540],{},[34,196766,196767],{"class":36,"line":11354},[34,196768,77],{},[32490,196770,196772],{"id":196771},"_2-共享多个缓冲区的缓冲池-share-a-buffer-pool-of-size-m-n","(2) 共享多个缓冲区的缓冲池 (Share a buffer pool of size m, n)",[15,196774,196775,196777,196778,196780],{},[2627,196776,195742],{}," 此时缓冲池容量大于 1，必须引入 ",[31,196779,196586],{}," 互斥信号量来保证同一时间只有一个进程在操作同一个缓冲池的指针。",[15,196782,196783],{},[2627,196784,195748],{},[105,196786,196787,196793,196798,196804,196810,196815],{},[108,196788,196789,196792],{},[31,196790,196791],{},"emptyPQ = m;"," \u002F\u002F P与Q之间的空缓冲区数",[108,196794,196795,196797],{},[31,196796,196604],{}," \u002F\u002F P与Q之间的满缓冲区数",[108,196799,196800,196803],{},[31,196801,196802],{},"mutexPQ = 1;"," \u002F\u002F 互斥访问 P与Q 的缓冲池",[108,196805,196806,196809],{},[31,196807,196808],{},"emptyQR = n;"," \u002F\u002F Q与R之间的空缓冲区数",[108,196811,196812,196814],{},[31,196813,196616],{}," \u002F\u002F Q与R之间的满缓冲区数",[108,196816,196817,196820],{},[31,196818,196819],{},"mutexQR = 1;"," \u002F\u002F 互斥访问 Q与R 的缓冲池",[24,196822,196824],{"className":65734,"code":196823,"language":31965,"meta":29,"style":29},"Process_P() {\n while (true) {\n 读取输入设备信息;\n semWait(emptyPQ);\n semWait(mutexPQ); \u002F\u002F 互斥锁定\n 将信息放入 PQ 缓冲池;\n semSignal(mutexPQ); \u002F\u002F 解锁\n semSignal(fullPQ);\n }\n}\n\nProcess_Q() {\n while (true) {\n semWait(fullPQ);\n semWait(mutexPQ); \u002F\u002F 互斥锁定\n 从 PQ 缓冲池取出信息;\n semSignal(mutexPQ); \u002F\u002F 解锁\n semSignal(emptyPQ);\n \n 加工处理信息;\n\n semWait(emptyQR);\n semWait(mutexQR); \u002F\u002F 互斥锁定\n 将处理后信息放入 QR 缓冲池;\n semSignal(mutexQR); \u002F\u002F 解锁\n semSignal(fullQR);\n }\n}\n\nProcess_R() {\n while (true) {\n semWait(fullQR);\n semWait(mutexQR); \u002F\u002F 互斥锁定\n 从 QR 缓冲池提取信息;\n semSignal(mutexQR); \u002F\u002F 解锁\n semSignal(emptyQR);\n \n 打印输出;\n }\n}\n",[31,196825,196826,196830,196834,196838,196843,196848,196853,196858,196863,196867,196871,196875,196879,196883,196888,196892,196897,196901,196906,196910,196914,196918,196923,196928,196933,196938,196943,196947,196951,196955,196959,196963,196968,196972,196977,196981,196986,196990,196994,196998],{"__ignoreMap":29},[34,196827,196828],{"class":36,"line":37},[34,196829,196627],{},[34,196831,196832],{"class":36,"line":43},[34,196833,195782],{},[34,196835,196836],{"class":36,"line":49},[34,196837,196636],{},[34,196839,196840],{"class":36,"line":56},[34,196841,196842],{}," semWait(emptyPQ);\n",[34,196844,196845],{"class":36,"line":62},[34,196846,196847],{}," semWait(mutexPQ); \u002F\u002F 互斥锁定\n",[34,196849,196850],{"class":36,"line":68},[34,196851,196852],{}," 将信息放入 PQ 缓冲池;\n",[34,196854,196855],{"class":36,"line":74},[34,196856,196857],{}," semSignal(mutexPQ); \u002F\u002F 解锁\n",[34,196859,196860],{"class":36,"line":1971},[34,196861,196862],{}," semSignal(fullPQ);\n",[34,196864,196865],{"class":36,"line":1976},[34,196866,5540],{},[34,196868,196869],{"class":36,"line":1985},[34,196870,77],{},[34,196872,196873],{"class":36,"line":1990},[34,196874,53],{"emptyLinePlaceholder":52},[34,196876,196877],{"class":36,"line":1999},[34,196878,196668],{},[34,196880,196881],{"class":36,"line":2022},[34,196882,195782],{},[34,196884,196885],{"class":36,"line":2036},[34,196886,196887],{}," semWait(fullPQ);\n",[34,196889,196890],{"class":36,"line":2603},[34,196891,196847],{},[34,196893,196894],{"class":36,"line":5537},[34,196895,196896],{}," 从 PQ 缓冲池取出信息;\n",[34,196898,196899],{"class":36,"line":5543},[34,196900,196857],{},[34,196902,196903],{"class":36,"line":5548},[34,196904,196905],{}," semSignal(emptyPQ);\n",[34,196907,196908],{"class":36,"line":5564},[34,196909,196186],{},[34,196911,196912],{"class":36,"line":5576},[34,196913,196696],{},[34,196915,196916],{"class":36,"line":5606},[34,196917,53],{"emptyLinePlaceholder":52},[34,196919,196920],{"class":36,"line":5616},[34,196921,196922],{}," semWait(emptyQR);\n",[34,196924,196925],{"class":36,"line":5628},[34,196926,196927],{}," semWait(mutexQR); \u002F\u002F 互斥锁定\n",[34,196929,196930],{"class":36,"line":5634},[34,196931,196932],{}," 将处理后信息放入 QR 缓冲池;\n",[34,196934,196935],{"class":36,"line":6420},[34,196936,196937],{}," semSignal(mutexQR); \u002F\u002F 解锁\n",[34,196939,196940],{"class":36,"line":11294},[34,196941,196942],{}," semSignal(fullQR);\n",[34,196944,196945],{"class":36,"line":11321},[34,196946,5540],{},[34,196948,196949],{"class":36,"line":11326},[34,196950,77],{},[34,196952,196953],{"class":36,"line":11333},[34,196954,53],{"emptyLinePlaceholder":52},[34,196956,196957],{"class":36,"line":11344},[34,196958,196732],{},[34,196960,196961],{"class":36,"line":11349},[34,196962,195782],{},[34,196964,196965],{"class":36,"line":11354},[34,196966,196967],{}," semWait(fullQR);\n",[34,196969,196970],{"class":36,"line":11363},[34,196971,196927],{},[34,196973,196974],{"class":36,"line":11368},[34,196975,196976],{}," 从 QR 缓冲池提取信息;\n",[34,196978,196979],{"class":36,"line":11377},[34,196980,196937],{},[34,196982,196983],{"class":36,"line":11402},[34,196984,196985],{}," semSignal(emptyQR);\n",[34,196987,196988],{"class":36,"line":11407},[34,196989,196186],{},[34,196991,196992],{"class":36,"line":11421},[34,196993,196760],{},[34,196995,196996],{"class":36,"line":11426},[34,196997,5540],{},[34,196999,197000],{"class":36,"line":11447},[34,197001,77],{},[15,197003,197004,197005,258,197008,197011],{},"以下是为您整理的第五章作业（第6题至第10题）的完整中文解答。这些解答基于操作系统中经典的并发控制原理，并严格使用 ",[31,197006,197007],{},"semWait",[31,197009,197010],{},"semSignal"," 原语来描述。",[32188,197013],{},[253,197015,197017],{"id":197016},"_6-工厂仓库与运输车问题-factory-warehouse-transport-vehicle","6. 工厂仓库与运输车问题 (Factory Warehouse & Transport Vehicle)",[15,197019,197020,197022,197023,347],{},[2627,197021,195742],{}," 这是一个带有额外互斥限制的",[2627,197024,197025],{},"生产者-消费者问题",[2230,197027,197028,197031,197034],{},[108,197029,197030],{},"仓库最多容纳10件设备（有界缓冲）。",[108,197032,197033],{},"生产部门（生产者）入库，销售部门（消费者）出库。",[108,197035,197036,197039],{},[2627,197037,197038],{},"关键限制："," 无论入库还是出库，都必须使用同一辆运输车，且每次只能运一件。因此，“运输车”是一个需要两方互斥使用的共享资源。",[15,197041,197042],{},[2627,197043,197044],{},"信号量与变量定义：",[105,197046,197047,197053,197059],{},[108,197048,197049,197052],{},[31,197050,197051],{},"empty = 10;"," \u002F\u002F 同步信号量，表示仓库还可以存放多少件设备",[108,197054,197055,197058],{},[31,197056,197057],{},"full = 0;"," \u002F\u002F 同步信号量，表示仓库中当前有多少件设备",[108,197060,197061,197064],{},[31,197062,197063],{},"vehicle = 1;"," \u002F\u002F 互斥信号量，表示当前是否有空闲的运输车（1辆）",[15,197066,197067],{},[2627,197068,195885],{},[24,197070,197072],{"className":65734,"code":197071,"language":31965,"meta":29,"style":29},"Producer_Department() { \u002F\u002F 生产部门进程\n while (true) {\n 生产出一件新设备;\n semWait(empty); \u002F\u002F 申请仓库里的一个空位（若仓库满则等待）\n semWait(vehicle); \u002F\u002F 申请唯一的运输车（互斥）\n \n 使用运输车将设备运入仓库;\n \n semSignal(vehicle); \u002F\u002F 释放运输车\n semSignal(full); \u002F\u002F 仓库里的设备总数加1，唤醒可能等待的销售部门\n }\n}\n\nSales_Department() { \u002F\u002F 销售部门进程\n while (true) {\n semWait(full); \u002F\u002F 申请提取一件设备（若仓库空则等待）\n semWait(vehicle); \u002F\u002F 申请唯一的运输车（互斥）\n \n 使用运输车将设备运出仓库;\n \n semSignal(vehicle); \u002F\u002F 释放运输车\n semSignal(empty); \u002F\u002F 仓库里的空位加1，唤醒可能等待的生产部门\n 销售给客户;\n }\n}\n",[31,197073,197074,197079,197083,197088,197093,197098,197102,197107,197111,197116,197121,197125,197129,197133,197138,197142,197147,197151,197155,197160,197164,197168,197173,197178,197182],{"__ignoreMap":29},[34,197075,197076],{"class":36,"line":37},[34,197077,197078],{},"Producer_Department() { \u002F\u002F 生产部门进程\n",[34,197080,197081],{"class":36,"line":43},[34,197082,195782],{},[34,197084,197085],{"class":36,"line":49},[34,197086,197087],{}," 生产出一件新设备;\n",[34,197089,197090],{"class":36,"line":56},[34,197091,197092],{}," semWait(empty); \u002F\u002F 申请仓库里的一个空位（若仓库满则等待）\n",[34,197094,197095],{"class":36,"line":62},[34,197096,197097],{}," semWait(vehicle); \u002F\u002F 申请唯一的运输车（互斥）\n",[34,197099,197100],{"class":36,"line":68},[34,197101,196186],{},[34,197103,197104],{"class":36,"line":74},[34,197105,197106],{}," 使用运输车将设备运入仓库;\n",[34,197108,197109],{"class":36,"line":1971},[34,197110,196186],{},[34,197112,197113],{"class":36,"line":1976},[34,197114,197115],{}," semSignal(vehicle); \u002F\u002F 释放运输车\n",[34,197117,197118],{"class":36,"line":1985},[34,197119,197120],{}," semSignal(full); \u002F\u002F 仓库里的设备总数加1，唤醒可能等待的销售部门\n",[34,197122,197123],{"class":36,"line":1990},[34,197124,5540],{},[34,197126,197127],{"class":36,"line":1999},[34,197128,77],{},[34,197130,197131],{"class":36,"line":2022},[34,197132,53],{"emptyLinePlaceholder":52},[34,197134,197135],{"class":36,"line":2036},[34,197136,197137],{},"Sales_Department() { \u002F\u002F 销售部门进程\n",[34,197139,197140],{"class":36,"line":2603},[34,197141,195782],{},[34,197143,197144],{"class":36,"line":5537},[34,197145,197146],{}," semWait(full); \u002F\u002F 申请提取一件设备（若仓库空则等待）\n",[34,197148,197149],{"class":36,"line":5543},[34,197150,197097],{},[34,197152,197153],{"class":36,"line":5548},[34,197154,196186],{},[34,197156,197157],{"class":36,"line":5564},[34,197158,197159],{}," 使用运输车将设备运出仓库;\n",[34,197161,197162],{"class":36,"line":5576},[34,197163,196186],{},[34,197165,197166],{"class":36,"line":5606},[34,197167,197115],{},[34,197169,197170],{"class":36,"line":5616},[34,197171,197172],{}," semSignal(empty); \u002F\u002F 仓库里的空位加1，唤醒可能等待的生产部门\n",[34,197174,197175],{"class":36,"line":5628},[34,197176,197177],{}," 销售给客户;\n",[34,197179,197180],{"class":36,"line":5634},[34,197181,5540],{},[34,197183,197184],{"class":36,"line":6420},[34,197185,77],{},[32188,197187],{},[253,197189,197191],{"id":197190},"_7-单行桥梁交通拥堵问题-bridge-traffic-management","7. 单行桥梁交通拥堵问题 (Bridge Traffic Management)",[15,197193,197194,197196,197197,197200],{},[2627,197195,195742],{}," 这是一个变种的",[2627,197198,197199],{},"读者-写者问题（同类不互斥，异类互斥）","。\n桥上不允许两车相交（即相反方向的车不能同时在桥上），但允许同向的多辆车依次通过。我们可以将双向的车流分为“南向北（S2N）”和“北向南（N2S）”。两个方向相当于两组不同的“读者”，任何一组在桥上时，另一组必须等待。",[15,197202,197203],{},[2627,197204,197044],{},[105,197206,197207,197213,197219,197225,197231],{},[108,197208,197209,197212],{},[31,197210,197211],{},"bridge = 1;"," \u002F\u002F 互斥信号量，争夺桥梁的使用权",[108,197214,197215,197218],{},[31,197216,197217],{},"count_S2N = 0;"," \u002F\u002F 计数器，记录当前桥上有多少辆南向北的车",[108,197220,197221,197224],{},[31,197222,197223],{},"mutex_S2N = 1;"," \u002F\u002F 互斥信号量，保护 count_S2N 的加减操作",[108,197226,197227,197230],{},[31,197228,197229],{},"count_N2S = 0;"," \u002F\u002F 计数器，记录当前桥上有多少辆北向南的车",[108,197232,197233,197236],{},[31,197234,197235],{},"mutex_N2S = 1;"," \u002F\u002F 互斥信号量，保护 count_N2S 的加减操作",[15,197238,197239],{},[2627,197240,195885],{},[24,197242,197244],{"className":65734,"code":197243,"language":31965,"meta":29,"style":29},"Car_South_to_North() { \u002F\u002F 南向北行驶的车辆\n semWait(mutex_S2N);\n if (count_S2N == 0) {\n semWait(bridge); \u002F\u002F 如果是该方向的第一辆车，负责封锁桥梁，禁止对向车上桥\n }\n count_S2N++; \u002F\u002F 该方向桥上车辆数加1\n semSignal(mutex_S2N);\n\n \u002F\u002F 此时同向车可继续上桥\n 过桥 (Crossing the bridge);\n\n semWait(mutex_S2N);\n count_S2N--; \u002F\u002F 车辆下桥，计数减1\n if (count_S2N == 0) {\n semSignal(bridge); \u002F\u002F 如果是该方向最后一辆下桥的车，释放桥梁使用权\n }\n semSignal(mutex_S2N);\n}\n\nCar_North_to_South() { \u002F\u002F 北向南行驶的车辆\n semWait(mutex_N2S);\n if (count_N2S == 0) {\n semWait(bridge); \u002F\u002F 第一辆车负责封锁桥梁\n }\n count_N2S++;\n semSignal(mutex_N2S);\n\n 过桥 (Crossing the bridge);\n\n semWait(mutex_N2S);\n count_N2S--;\n if (count_N2S == 0) {\n semSignal(bridge); \u002F\u002F 最后一辆车释放桥梁\n }\n semSignal(mutex_N2S);\n}\n",[31,197245,197246,197251,197256,197261,197266,197270,197275,197280,197284,197289,197294,197298,197302,197307,197311,197316,197320,197324,197328,197332,197337,197342,197347,197352,197356,197361,197366,197370,197374,197378,197382,197387,197391,197396,197400,197404],{"__ignoreMap":29},[34,197247,197248],{"class":36,"line":37},[34,197249,197250],{},"Car_South_to_North() { \u002F\u002F 南向北行驶的车辆\n",[34,197252,197253],{"class":36,"line":43},[34,197254,197255],{}," semWait(mutex_S2N);\n",[34,197257,197258],{"class":36,"line":49},[34,197259,197260],{}," if (count_S2N == 0) {\n",[34,197262,197263],{"class":36,"line":56},[34,197264,197265],{}," semWait(bridge); \u002F\u002F 如果是该方向的第一辆车，负责封锁桥梁，禁止对向车上桥\n",[34,197267,197268],{"class":36,"line":62},[34,197269,5540],{},[34,197271,197272],{"class":36,"line":68},[34,197273,197274],{}," count_S2N++; \u002F\u002F 该方向桥上车辆数加1\n",[34,197276,197277],{"class":36,"line":74},[34,197278,197279],{}," semSignal(mutex_S2N);\n",[34,197281,197282],{"class":36,"line":1971},[34,197283,53],{"emptyLinePlaceholder":52},[34,197285,197286],{"class":36,"line":1976},[34,197287,197288],{}," \u002F\u002F 此时同向车可继续上桥\n",[34,197290,197291],{"class":36,"line":1985},[34,197292,197293],{}," 过桥 (Crossing the bridge);\n",[34,197295,197296],{"class":36,"line":1990},[34,197297,53],{"emptyLinePlaceholder":52},[34,197299,197300],{"class":36,"line":1999},[34,197301,197255],{},[34,197303,197304],{"class":36,"line":2022},[34,197305,197306],{}," count_S2N--; \u002F\u002F 车辆下桥，计数减1\n",[34,197308,197309],{"class":36,"line":2036},[34,197310,197260],{},[34,197312,197313],{"class":36,"line":2603},[34,197314,197315],{}," semSignal(bridge); \u002F\u002F 如果是该方向最后一辆下桥的车，释放桥梁使用权\n",[34,197317,197318],{"class":36,"line":5537},[34,197319,5540],{},[34,197321,197322],{"class":36,"line":5543},[34,197323,197279],{},[34,197325,197326],{"class":36,"line":5548},[34,197327,77],{},[34,197329,197330],{"class":36,"line":5564},[34,197331,53],{"emptyLinePlaceholder":52},[34,197333,197334],{"class":36,"line":5576},[34,197335,197336],{},"Car_North_to_South() { \u002F\u002F 北向南行驶的车辆\n",[34,197338,197339],{"class":36,"line":5606},[34,197340,197341],{}," semWait(mutex_N2S);\n",[34,197343,197344],{"class":36,"line":5616},[34,197345,197346],{}," if (count_N2S == 0) {\n",[34,197348,197349],{"class":36,"line":5628},[34,197350,197351],{}," semWait(bridge); \u002F\u002F 第一辆车负责封锁桥梁\n",[34,197353,197354],{"class":36,"line":5634},[34,197355,5540],{},[34,197357,197358],{"class":36,"line":6420},[34,197359,197360],{}," count_N2S++;\n",[34,197362,197363],{"class":36,"line":11294},[34,197364,197365],{}," semSignal(mutex_N2S);\n",[34,197367,197368],{"class":36,"line":11321},[34,197369,53],{"emptyLinePlaceholder":52},[34,197371,197372],{"class":36,"line":11326},[34,197373,197293],{},[34,197375,197376],{"class":36,"line":11333},[34,197377,53],{"emptyLinePlaceholder":52},[34,197379,197380],{"class":36,"line":11344},[34,197381,197341],{},[34,197383,197384],{"class":36,"line":11349},[34,197385,197386],{}," count_N2S--;\n",[34,197388,197389],{"class":36,"line":11354},[34,197390,197346],{},[34,197392,197393],{"class":36,"line":11363},[34,197394,197395],{}," semSignal(bridge); \u002F\u002F 最后一辆车释放桥梁\n",[34,197397,197398],{"class":36,"line":11368},[34,197399,5540],{},[34,197401,197402],{"class":36,"line":11377},[34,197403,197365],{},[34,197405,197406],{"class":36,"line":11402},[34,197407,77],{},[32188,197409],{},[253,197411,197413],{"id":197412},"_8-哲学家就餐问题-the-dining-philosophers-problem","8. 哲学家就餐问题 (The Dining Philosophers Problem)",[15,197415,197416,197418,197419,39984,197422,197425,197426,197429],{},[2627,197417,195742],{}," 5名哲学家，5根筷子。如果5个人同时饿了并同时拿起左手的筷子，所有人都在等右手的筷子，就会发生",[2627,197420,197421],{},"死锁（Deadlock）",[2627,197423,197424],{},"解决方案："," 操作系统中最经典的破除死锁方案之一是",[2627,197427,197428],{},"限制就餐人数","。通过引入一个“门卫”或“座位”限制，最多只允许 4 位哲学家同时去拿筷子。这样至少有1个人能拿到2根筷子顺利吃完，从而打破死锁与饥饿。",[15,197431,197432],{},[2627,197433,197044],{},[105,197435,197436,197442],{},[108,197437,197438,197441],{},[31,197439,197440],{},"fork = {1, 1, 1, 1, 1};"," \u002F\u002F 数组信号量，代表5根筷子，初始均为1",[108,197443,197444,197447],{},[31,197445,197446],{},"room = 4;"," \u002F\u002F 同步信号量，限制最多只允许4名哲学家同时竞争筷子",[15,197449,197450],{},[2627,197451,195885],{},[24,197453,197455],{"className":65734,"code":197454,"language":31965,"meta":29,"style":29},"Philosopher(int i) { \u002F\u002F i 的取值范围是 0 到 4\n while (true) {\n 思考 (Thinking);\n \n \u002F\u002F 准备就餐\n semWait(room); \u002F\u002F 申请进入竞争队列（最多允许4人）\n \n semWait(fork[i]); \u002F\u002F 拿起左边的筷子\n semWait(fork[(i+1) % 5]);\u002F\u002F 拿起右边的筷子\n \n 进餐 (Eating); \u002F\u002F 拿到两根筷子，开始进餐\n \n semSignal(fork[(i+1) % 5]);\u002F\u002F 放下右边的筷子\n semSignal(fork[i]); \u002F\u002F 放下左边的筷子\n \n semSignal(room); \u002F\u002F 离开竞争队列，让出名额给其他人\n }\n}\n",[31,197456,197457,197462,197466,197471,197475,197480,197485,197489,197494,197499,197503,197508,197512,197517,197522,197526,197531,197535],{"__ignoreMap":29},[34,197458,197459],{"class":36,"line":37},[34,197460,197461],{},"Philosopher(int i) { \u002F\u002F i 的取值范围是 0 到 4\n",[34,197463,197464],{"class":36,"line":43},[34,197465,195782],{},[34,197467,197468],{"class":36,"line":49},[34,197469,197470],{}," 思考 (Thinking);\n",[34,197472,197473],{"class":36,"line":56},[34,197474,196186],{},[34,197476,197477],{"class":36,"line":62},[34,197478,197479],{}," \u002F\u002F 准备就餐\n",[34,197481,197482],{"class":36,"line":68},[34,197483,197484],{}," semWait(room); \u002F\u002F 申请进入竞争队列（最多允许4人）\n",[34,197486,197487],{"class":36,"line":74},[34,197488,196186],{},[34,197490,197491],{"class":36,"line":1971},[34,197492,197493],{}," semWait(fork[i]); \u002F\u002F 拿起左边的筷子\n",[34,197495,197496],{"class":36,"line":1976},[34,197497,197498],{}," semWait(fork[(i+1) % 5]);\u002F\u002F 拿起右边的筷子\n",[34,197500,197501],{"class":36,"line":1985},[34,197502,196186],{},[34,197504,197505],{"class":36,"line":1990},[34,197506,197507],{}," 进餐 (Eating); \u002F\u002F 拿到两根筷子，开始进餐\n",[34,197509,197510],{"class":36,"line":1999},[34,197511,196186],{},[34,197513,197514],{"class":36,"line":2022},[34,197515,197516],{}," semSignal(fork[(i+1) % 5]);\u002F\u002F 放下右边的筷子\n",[34,197518,197519],{"class":36,"line":2036},[34,197520,197521],{}," semSignal(fork[i]); \u002F\u002F 放下左边的筷子\n",[34,197523,197524],{"class":36,"line":2603},[34,197525,196186],{},[34,197527,197528],{"class":36,"line":5537},[34,197529,197530],{}," semSignal(room); \u002F\u002F 离开竞争队列，让出名额给其他人\n",[34,197532,197533],{"class":36,"line":5543},[34,197534,5540],{},[34,197536,197537],{"class":36,"line":5548},[34,197538,77],{},[32188,197540],{},[253,197542,197544],{"id":197543},"_9-读者-写者问题-readers-writers-problem","9. 读者-写者问题 (Readers-Writers Problem)",[15,197546,197547,197549,197550,197553,197554,197557,197558,197553,197561,197564],{},[2627,197548,195742],{}," 允许多个读者同时读；只允许一个写者写；写者写的时候禁止任何读者读。此处要求实现“读者优先（Readers Have Priority）”或标准的读者写者模型。需要修改题目中提供的 ",[31,197551,197552],{},"Reader"," 类的 ",[31,197555,197556],{},"Read"," 方法和 ",[31,197559,197560],{},"Writer",[31,197562,197563],{},"Write"," 方法。",[15,197566,197567],{},[2627,197568,197044],{},[105,197570,197571,197577,197583],{},[108,197572,197573,197576],{},[31,197574,197575],{},"readcount = 0;"," \u002F\u002F 全局计数器，记录当前的读者数量",[108,197578,197579,197582],{},[31,197580,197581],{},"x = 1;"," \u002F\u002F 互斥信号量，用于对 readcount 的互斥访问",[108,197584,197585,197588],{},[31,197586,197587],{},"wsem = 1;"," \u002F\u002F 互斥信号量，用于实现“读\u002F写”和“写\u002F写”互斥",[15,197590,197591],{},[2627,197592,195885],{},[24,197594,197596],{"className":65734,"code":197595,"language":31965,"meta":29,"style":29},"class Reader {\n void Read() {\n semWait(x); \u002F\u002F 互斥访问 readcount\n readcount++;\n if (readcount == 1) {\n semWait(wsem); \u002F\u002F 如果是第一个读者，负责阻挡写者\n }\n semSignal(x); \u002F\u002F 释放 readcount 锁\n\n 执行读操作 (Reading data...); \n\n semWait(x); \u002F\u002F 互斥访问 readcount\n readcount--;\n if (readcount == 0) {\n semSignal(wsem); \u002F\u002F 如果是最后一个读者，唤醒\u002F允许写者进入\n }\n semSignal(x); \u002F\u002F 释放 readcount 锁\n }\n}\n\nclass Writer {\n void Write() {\n semWait(wsem); \u002F\u002F 申请对数据区的独占写权限\n\n 执行写操作 (Writing data...);\n\n semSignal(wsem); \u002F\u002F 释放数据区权限\n }\n}\n",[31,197597,197598,197603,197608,197613,197618,197623,197628,197632,197637,197641,197646,197650,197654,197659,197664,197669,197673,197677,197681,197685,197689,197694,197699,197704,197708,197713,197717,197722,197726],{"__ignoreMap":29},[34,197599,197600],{"class":36,"line":37},[34,197601,197602],{},"class Reader {\n",[34,197604,197605],{"class":36,"line":43},[34,197606,197607],{}," void Read() {\n",[34,197609,197610],{"class":36,"line":49},[34,197611,197612],{}," semWait(x); \u002F\u002F 互斥访问 readcount\n",[34,197614,197615],{"class":36,"line":56},[34,197616,197617],{}," readcount++;\n",[34,197619,197620],{"class":36,"line":62},[34,197621,197622],{}," if (readcount == 1) {\n",[34,197624,197625],{"class":36,"line":68},[34,197626,197627],{}," semWait(wsem); \u002F\u002F 如果是第一个读者，负责阻挡写者\n",[34,197629,197630],{"class":36,"line":74},[34,197631,26684],{},[34,197633,197634],{"class":36,"line":1971},[34,197635,197636],{}," semSignal(x); \u002F\u002F 释放 readcount 锁\n",[34,197638,197639],{"class":36,"line":1976},[34,197640,53],{"emptyLinePlaceholder":52},[34,197642,197643],{"class":36,"line":1985},[34,197644,197645],{}," 执行读操作 (Reading data...); \n",[34,197647,197648],{"class":36,"line":1990},[34,197649,53],{"emptyLinePlaceholder":52},[34,197651,197652],{"class":36,"line":1999},[34,197653,197612],{},[34,197655,197656],{"class":36,"line":2022},[34,197657,197658],{}," readcount--;\n",[34,197660,197661],{"class":36,"line":2036},[34,197662,197663],{}," if (readcount == 0) {\n",[34,197665,197666],{"class":36,"line":2603},[34,197667,197668],{}," semSignal(wsem); \u002F\u002F 如果是最后一个读者，唤醒\u002F允许写者进入\n",[34,197670,197671],{"class":36,"line":5537},[34,197672,26684],{},[34,197674,197675],{"class":36,"line":5543},[34,197676,197636],{},[34,197678,197679],{"class":36,"line":5548},[34,197680,5540],{},[34,197682,197683],{"class":36,"line":5564},[34,197684,77],{},[34,197686,197687],{"class":36,"line":5576},[34,197688,53],{"emptyLinePlaceholder":52},[34,197690,197691],{"class":36,"line":5606},[34,197692,197693],{},"class Writer {\n",[34,197695,197696],{"class":36,"line":5616},[34,197697,197698],{}," void Write() {\n",[34,197700,197701],{"class":36,"line":5628},[34,197702,197703],{}," semWait(wsem); \u002F\u002F 申请对数据区的独占写权限\n",[34,197705,197706],{"class":36,"line":5634},[34,197707,53],{"emptyLinePlaceholder":52},[34,197709,197710],{"class":36,"line":6420},[34,197711,197712],{}," 执行写操作 (Writing data...);\n",[34,197714,197715],{"class":36,"line":11294},[34,197716,53],{"emptyLinePlaceholder":52},[34,197718,197719],{"class":36,"line":11321},[34,197720,197721],{}," semSignal(wsem); \u002F\u002F 释放数据区权限\n",[34,197723,197724],{"class":36,"line":11326},[34,197725,5540],{},[34,197727,197728],{"class":36,"line":11333},[34,197729,77],{},[32188,197731],{},[253,197733,197735],{"id":197734},"_10-理发师问题-barber-problem","10. 理发师问题 (Barber Problem)",[15,197737,197738,197740],{},[2627,197739,195742],{}," 这是一个典型的进程同步问题。",[2230,197742,197743,197746],{},[108,197744,197745],{},"理发师没顾客时睡觉（等顾客唤醒）。",[108,197747,197748],{},"顾客来时，如果理发师在忙且有空椅子，就坐下等；如果没有空椅子，就直接离开。",[15,197750,197751],{},[2627,197752,197044],{},[105,197754,197755,197761,197767,197772,197778],{},[108,197756,197757,197760],{},[31,197758,197759],{},"customers = 0;"," \u002F\u002F 同步信号量，记录正在等待理发的顾客数量（用于唤醒理发师）",[108,197762,197763,197766],{},[31,197764,197765],{},"barber = 0;"," \u002F\u002F 同步信号量，表示理发师是否准备好理发（0表示忙\u002F睡，1表示准备好）",[108,197768,197769,197771],{},[31,197770,195761],{}," \u002F\u002F 互斥信号量，保护共享变量 waiting",[108,197773,197774,197777],{},[31,197775,197776],{},"waiting = 0;"," \u002F\u002F 整数变量，记录当前坐在椅子上等待的顾客数量",[108,197779,197780,197783],{},[31,197781,197782],{},"N;"," \u002F\u002F 常量，等候室的椅子总数",[15,197785,197786],{},[2627,197787,195885],{},[24,197789,197791],{"className":65734,"code":197790,"language":31965,"meta":29,"style":29},"Barber() {\n while (true) {\n semWait(customers); \u002F\u002F 理发师等待顾客。若为0则睡觉；若大于0则服务顾客\n \n semWait(mutex); \u002F\u002F 互斥访问 waiting 变量\n waiting--; \u002F\u002F 从等候室叫起一位顾客，等待人数减1\n semSignal(barber); \u002F\u002F 理发师准备就绪，释放理发信号\n semSignal(mutex); \u002F\u002F 释放互斥锁\n \n 理发 (Cutting hair);\n }\n}\n\nCustomer() {\n semWait(mutex); \u002F\u002F 顾客进门，互斥检查椅子数量\n if (waiting \u003C N) { \u002F\u002F 如果有空椅子\n waiting++; \u002F\u002F 坐下等待，等待人数加1\n semSignal(customers); \u002F\u002F 发出顾客到来的信号（如果理发师在睡，就会唤醒他）\n semSignal(mutex); \u002F\u002F 释放互斥锁\n \n semWait(barber); \u002F\u002F 等待理发师叫自己（等待 barber 信号量）\n 被理发 (Getting haircut);\n } else {\n semSignal(mutex); \u002F\u002F 没有空椅子，直接释放互斥锁\n 离开理发店 (Leave the shop); \n }\n}\n",[31,197792,197793,197798,197802,197807,197811,197816,197821,197826,197831,197835,197840,197844,197848,197852,197857,197862,197867,197872,197877,197881,197885,197890,197895,197900,197905,197910,197914],{"__ignoreMap":29},[34,197794,197795],{"class":36,"line":37},[34,197796,197797],{},"Barber() {\n",[34,197799,197800],{"class":36,"line":43},[34,197801,195782],{},[34,197803,197804],{"class":36,"line":49},[34,197805,197806],{}," semWait(customers); \u002F\u002F 理发师等待顾客。若为0则睡觉；若大于0则服务顾客\n",[34,197808,197809],{"class":36,"line":56},[34,197810,196186],{},[34,197812,197813],{"class":36,"line":62},[34,197814,197815],{}," semWait(mutex); \u002F\u002F 互斥访问 waiting 变量\n",[34,197817,197818],{"class":36,"line":68},[34,197819,197820],{}," waiting--; \u002F\u002F 从等候室叫起一位顾客，等待人数减1\n",[34,197822,197823],{"class":36,"line":74},[34,197824,197825],{}," semSignal(barber); \u002F\u002F 理发师准备就绪，释放理发信号\n",[34,197827,197828],{"class":36,"line":1971},[34,197829,197830],{}," semSignal(mutex); \u002F\u002F 释放互斥锁\n",[34,197832,197833],{"class":36,"line":1976},[34,197834,196186],{},[34,197836,197837],{"class":36,"line":1985},[34,197838,197839],{}," 理发 (Cutting hair);\n",[34,197841,197842],{"class":36,"line":1990},[34,197843,5540],{},[34,197845,197846],{"class":36,"line":1999},[34,197847,77],{},[34,197849,197850],{"class":36,"line":2022},[34,197851,53],{"emptyLinePlaceholder":52},[34,197853,197854],{"class":36,"line":2036},[34,197855,197856],{},"Customer() {\n",[34,197858,197859],{"class":36,"line":2603},[34,197860,197861],{}," semWait(mutex); \u002F\u002F 顾客进门，互斥检查椅子数量\n",[34,197863,197864],{"class":36,"line":5537},[34,197865,197866],{}," if (waiting \u003C N) { \u002F\u002F 如果有空椅子\n",[34,197868,197869],{"class":36,"line":5543},[34,197870,197871],{}," waiting++; \u002F\u002F 坐下等待，等待人数加1\n",[34,197873,197874],{"class":36,"line":5548},[34,197875,197876],{}," semSignal(customers); \u002F\u002F 发出顾客到来的信号（如果理发师在睡，就会唤醒他）\n",[34,197878,197879],{"class":36,"line":5564},[34,197880,197830],{},[34,197882,197883],{"class":36,"line":5576},[34,197884,196186],{},[34,197886,197887],{"class":36,"line":5606},[34,197888,197889],{}," semWait(barber); \u002F\u002F 等待理发师叫自己（等待 barber 信号量）\n",[34,197891,197892],{"class":36,"line":5616},[34,197893,197894],{}," 被理发 (Getting haircut);\n",[34,197896,197897],{"class":36,"line":5628},[34,197898,197899],{}," } else {\n",[34,197901,197902],{"class":36,"line":5634},[34,197903,197904],{}," semSignal(mutex); \u002F\u002F 没有空椅子，直接释放互斥锁\n",[34,197906,197907],{"class":36,"line":6420},[34,197908,197909],{}," 离开理发店 (Leave the shop); \n",[34,197911,197912],{"class":36,"line":11294},[34,197913,5540],{},[34,197915,197916],{"class":36,"line":11321},[34,197917,77],{},[15,197919,197920,197921,258,197923,197925],{},"这里为您提供第五章作业剩余题目（第11、12、13题）的完整中文解答。解答继续采用标准的操作系统并发与同步原理，使用 ",[31,197922,197007],{},[31,197924,197010],{}," 原语。",[32188,197927],{},[253,197929,197931],{"id":197930},"_11-少林寺打水问题-shaolin-temple-issue","11. 少林寺打水问题 (Shaolin Temple Issue)",[15,197933,197934,197936,197937,347],{},[2627,197935,195742],{},"\n这是一个带有多个约束条件的复杂",[2627,197938,197025],{},[105,197940,197941,197947,197953,197959,197978],{},[108,197942,197943,197946],{},[2627,197944,197945],{},"生产者："," 小和尚（打水）。",[108,197948,197949,197952],{},[2627,197950,197951],{},"消费者："," 老和尚（饮水）。",[108,197954,197955,197958],{},[2627,197956,197957],{},"共享资源（缓冲）："," 水缸（容量为 10 桶水）。",[108,197960,197961,197964],{},[2627,197962,197963],{},"互斥资源：",[2230,197965,197966,197972],{},[108,197967,197968,197971],{},[2627,197969,197970],{},"水井："," 每次只能容纳一个水桶（打水动作需互斥）。",[108,197973,197974,197977],{},[2627,197975,197976],{},"水缸："," 每次只能倒入或取出一桶水（入水\u002F取水动作需互斥）。",[108,197979,197980,197983],{},[2627,197981,197982],{},"公共限制资源："," 水桶（共 3 个）。不管是小和尚去井里打水，还是老和尚从水缸里取水，都需要先拿一个水桶。",[15,197985,197986],{},[2627,197987,195748],{},[105,197989,197990,197995,198000,198006,198012],{},[108,197991,197992,197994],{},[31,197993,197051],{}," \u002F\u002F 同步信号量：水缸中的剩余空间",[108,197996,197997,197999],{},[31,197998,197057],{}," \u002F\u002F 同步信号量：水缸中当前的水量",[108,198001,198002,198005],{},[31,198003,198004],{},"bucket = 3;"," \u002F\u002F 资源信号量：当前可用的水桶数量",[108,198007,198008,198011],{},[31,198009,198010],{},"mutex_well = 1;"," \u002F\u002F 互斥信号量：对水井的互斥访问",[108,198013,198014,198017],{},[31,198015,198016],{},"mutex_tank = 1;"," \u002F\u002F 互斥信号量：对水缸的互斥访问",[15,198019,198020],{},[2627,198021,195885],{},[24,198023,198025],{"className":65734,"code":198024,"language":31965,"meta":29,"style":29},"Young_Monk() { \u002F\u002F 小和尚进程\n while (true) {\n semWait(empty); \u002F\u002F 申请水缸里的空位（若水缸已满则等待）\n semWait(bucket); \u002F\u002F 申请一个水桶（若无水桶则等待）\n \n semWait(mutex_well); \u002F\u002F 互斥访问水井\n 从水井中打一桶水;\n semSignal(mutex_well); \u002F\u002F 离开水井\n \n semWait(mutex_tank); \u002F\u002F 互斥访问水缸\n 将水倒入水缸中;\n semSignal(mutex_tank); \u002F\u002F 离开水缸\n \n semSignal(bucket); \u002F\u002F 归还水桶\n semSignal(full); \u002F\u002F 水缸里的水量加1，唤醒老和尚\n }\n}\n\nOld_Monk() { \u002F\u002F 老和尚进程\n while (true) {\n semWait(full); \u002F\u002F 申请水缸里的水（若没水则等待）\n semWait(bucket); \u002F\u002F 申请一个水桶（若无水桶则等待）\n \n semWait(mutex_tank); \u002F\u002F 互斥访问水缸\n 从水缸中取出一桶水;\n semSignal(mutex_tank); \u002F\u002F 离开水缸\n \n semSignal(empty); \u002F\u002F 水缸空位加1，唤醒小和尚（可以立刻释放空位）\n \n 喝水 (Drinking water);\n \n semSignal(bucket); \u002F\u002F 喝完后，归还水桶\n }\n}\n",[31,198026,198027,198032,198036,198041,198046,198050,198055,198060,198065,198069,198074,198079,198084,198088,198093,198098,198102,198106,198110,198115,198119,198124,198128,198132,198136,198141,198145,198149,198154,198158,198163,198167,198172,198176],{"__ignoreMap":29},[34,198028,198029],{"class":36,"line":37},[34,198030,198031],{},"Young_Monk() { \u002F\u002F 小和尚进程\n",[34,198033,198034],{"class":36,"line":43},[34,198035,195782],{},[34,198037,198038],{"class":36,"line":49},[34,198039,198040],{}," semWait(empty); \u002F\u002F 申请水缸里的空位（若水缸已满则等待）\n",[34,198042,198043],{"class":36,"line":56},[34,198044,198045],{}," semWait(bucket); \u002F\u002F 申请一个水桶（若无水桶则等待）\n",[34,198047,198048],{"class":36,"line":62},[34,198049,196186],{},[34,198051,198052],{"class":36,"line":68},[34,198053,198054],{}," semWait(mutex_well); \u002F\u002F 互斥访问水井\n",[34,198056,198057],{"class":36,"line":74},[34,198058,198059],{}," 从水井中打一桶水;\n",[34,198061,198062],{"class":36,"line":1971},[34,198063,198064],{}," semSignal(mutex_well); \u002F\u002F 离开水井\n",[34,198066,198067],{"class":36,"line":1976},[34,198068,196186],{},[34,198070,198071],{"class":36,"line":1985},[34,198072,198073],{}," semWait(mutex_tank); \u002F\u002F 互斥访问水缸\n",[34,198075,198076],{"class":36,"line":1990},[34,198077,198078],{}," 将水倒入水缸中;\n",[34,198080,198081],{"class":36,"line":1999},[34,198082,198083],{}," semSignal(mutex_tank); \u002F\u002F 离开水缸\n",[34,198085,198086],{"class":36,"line":2022},[34,198087,196186],{},[34,198089,198090],{"class":36,"line":2036},[34,198091,198092],{}," semSignal(bucket); \u002F\u002F 归还水桶\n",[34,198094,198095],{"class":36,"line":2603},[34,198096,198097],{}," semSignal(full); \u002F\u002F 水缸里的水量加1，唤醒老和尚\n",[34,198099,198100],{"class":36,"line":5537},[34,198101,5540],{},[34,198103,198104],{"class":36,"line":5543},[34,198105,77],{},[34,198107,198108],{"class":36,"line":5548},[34,198109,53],{"emptyLinePlaceholder":52},[34,198111,198112],{"class":36,"line":5564},[34,198113,198114],{},"Old_Monk() { \u002F\u002F 老和尚进程\n",[34,198116,198117],{"class":36,"line":5576},[34,198118,195782],{},[34,198120,198121],{"class":36,"line":5606},[34,198122,198123],{}," semWait(full); \u002F\u002F 申请水缸里的水（若没水则等待）\n",[34,198125,198126],{"class":36,"line":5616},[34,198127,198045],{},[34,198129,198130],{"class":36,"line":5628},[34,198131,196186],{},[34,198133,198134],{"class":36,"line":5634},[34,198135,198073],{},[34,198137,198138],{"class":36,"line":6420},[34,198139,198140],{}," 从水缸中取出一桶水;\n",[34,198142,198143],{"class":36,"line":11294},[34,198144,198083],{},[34,198146,198147],{"class":36,"line":11321},[34,198148,196186],{},[34,198150,198151],{"class":36,"line":11326},[34,198152,198153],{}," semSignal(empty); \u002F\u002F 水缸空位加1，唤醒小和尚（可以立刻释放空位）\n",[34,198155,198156],{"class":36,"line":11333},[34,198157,196186],{},[34,198159,198160],{"class":36,"line":11344},[34,198161,198162],{}," 喝水 (Drinking water);\n",[34,198164,198165],{"class":36,"line":11349},[34,198166,196186],{},[34,198168,198169],{"class":36,"line":11354},[34,198170,198171],{}," semSignal(bucket); \u002F\u002F 喝完后，归还水桶\n",[34,198173,198174],{"class":36,"line":11363},[34,198175,5540],{},[34,198177,198178],{"class":36,"line":11368},[34,198179,77],{},[15,198181,198182],{},[71807,198183,198184,198185,198188,198189,198192],{},"(注意：为防止死锁，小和尚应先申请 ",[31,198186,198187],{},"empty"," 再申请 ",[31,198190,198191],{},"bucket","，否则如果水缸已满，小和尚拿光了所有水桶并等待空位，而老和尚没有水桶无法取水，就会死锁。)",[32188,198194],{},[253,198196,198198],{"id":198197},"_12-自行车生产线问题-bicycle-production-line","12. 自行车生产线问题 (Bicycle Production Line)",[15,198200,198201,198203,198204,198206,198207,347],{},[2627,198202,195742],{},"\n这也是一个",[2627,198205,197025],{},"，难点在于",[2627,198208,198209],{},"防范死锁 (Deadlock Prevention)",[105,198211,198212,198297,198300,198371],{},[108,198213,198214,198217,198218,108987,198246,31901],{},[2627,198215,198216],{},"容量限制："," 箱子总容量为 ",[34,198219,198221,198234],{"className":198220},[31642],[34,198222,198224],{"className":198223},[31646],[31648,198225,198226],{"xmlns":31650},[31652,198227,198228,198232],{},[31655,198229,198230],{},[31658,198231,33347],{},[31665,198233,33347],{"encoding":31667},[34,198235,198237],{"className":198236,"ariaHidden":758},[31672],[34,198238,198240,198243],{"className":198239},[31676],[34,198241],{"className":198242,"style":33218},[31680],[34,198244,33347],{"className":198245,"style":33389},[31685,31686],[34,198247,198249,198267],{"className":198248},[31642],[34,198250,198252],{"className":198251},[31646],[31648,198253,198254],{"xmlns":31650},[31652,198255,198256,198264],{},[31655,198257,198258,198260,198262],{},[31658,198259,33347],{},[31662,198261,43175],{},[31749,198263,786],{},[31665,198265,198266],{"encoding":31667},"N \\ge 3",[34,198268,198270,198288],{"className":198269,"ariaHidden":758},[31672],[34,198271,198273,198276,198279,198282,198285],{"className":198272},[31676],[34,198274],{"className":198275,"style":195047},[31680],[34,198277,33347],{"className":198278,"style":33389},[31685,31686],[34,198280],{"className":198281,"style":31846},[31845],[34,198283,43175],{"className":198284},[31850],[34,198286],{"className":198287,"style":31846},[31845],[34,198289,198291,198294],{"className":198290},[31676],[34,198292],{"className":198293,"style":32617},[31680],[34,198295,786],{"className":198296},[31685],[108,198298,198299],{},"工人 1 生产车架（每次放 1 个）；工人 2 生产车轮（每次放 1 个）；工人 3 组装（每次取 1 个车架、2 个车轮）。",[108,198301,198302,198305],{},[2627,198303,198304],{},"死锁隐患：",[105,198306,198307,198339],{},[108,198308,198309,198310,198338],{},"如果箱子全被“车架”占满（放了 ",[34,198311,198313,198326],{"className":198312},[31642],[34,198314,198316],{"className":198315},[31646],[31648,198317,198318],{"xmlns":31650},[31652,198319,198320,198324],{},[31655,198321,198322],{},[31658,198323,33347],{},[31665,198325,33347],{"encoding":31667},[34,198327,198329],{"className":198328,"ariaHidden":758},[31672],[34,198330,198332,198335],{"className":198331},[31676],[34,198333],{"className":198334,"style":33218},[31680],[34,198336,33347],{"className":198337,"style":33389},[31685,31686]," 个车架），工人 2 放不进车轮，工人 3 取不到车轮，死锁。",[108,198340,198341,198342,198370],{},"如果箱子全被“车轮”占满（放了 ",[34,198343,198345,198358],{"className":198344},[31642],[34,198346,198348],{"className":198347},[31646],[31648,198349,198350],{"xmlns":31650},[31652,198351,198352,198356],{},[31655,198353,198354],{},[31658,198355,33347],{},[31665,198357,33347],{"encoding":31667},[34,198359,198361],{"className":198360,"ariaHidden":758},[31672],[34,198362,198364,198367],{"className":198363},[31676],[34,198365],{"className":198366,"style":33218},[31680],[34,198368,33347],{"className":198369,"style":33389},[31685,31686]," 个车轮），工人 1 放不进车架，工人 3 取不到车架，死锁。",[108,198372,198373,198376,198377,198432,198433,198487],{},[2627,198374,198375],{},"死锁解决方案：","\n限制箱子中",[2627,198378,198379,198380,198431],{},"最多只能放 ",[34,198381,198383,198401],{"className":198382},[31642],[34,198384,198386],{"className":198385},[31646],[31648,198387,198388],{"xmlns":31650},[31652,198389,198390,198398],{},[31655,198391,198392,198394,198396],{},[31658,198393,33347],{},[31662,198395,32226],{},[31749,198397,768],{},[31665,198399,198400],{"encoding":31667},"N-2",[34,198402,198404,198422],{"className":198403,"ariaHidden":758},[31672],[34,198405,198407,198410,198413,198416,198419],{"className":198406},[31676],[34,198408],{"className":198409,"style":34944},[31680],[34,198411,33347],{"className":198412,"style":33389},[31685,31686],[34,198414],{"className":198415,"style":32604},[31845],[34,198417,32226],{"className":198418},[32024],[34,198420],{"className":198421,"style":32604},[31845],[34,198423,198425,198428],{"className":198424},[31676],[34,198426],{"className":198427,"style":32617},[31680],[34,198429,768],{"className":198430},[31685]," 个车架","（保证至少有空间放2个车轮），以及",[2627,198434,198379,198435,198486],{},[34,198436,198438,198456],{"className":198437},[31642],[34,198439,198441],{"className":198440},[31646],[31648,198442,198443],{"xmlns":31650},[31652,198444,198445,198453],{},[31655,198446,198447,198449,198451],{},[31658,198448,33347],{},[31662,198450,32226],{},[31749,198452,640],{},[31665,198454,198455],{"encoding":31667},"N-1",[34,198457,198459,198477],{"className":198458,"ariaHidden":758},[31672],[34,198460,198462,198465,198468,198471,198474],{"className":198461},[31676],[34,198463],{"className":198464,"style":34944},[31680],[34,198466,33347],{"className":198467,"style":33389},[31685,31686],[34,198469],{"className":198470,"style":32604},[31845],[34,198472,32226],{"className":198473},[32024],[34,198475],{"className":198476,"style":32604},[31845],[34,198478,198480,198483],{"className":198479},[31676],[34,198481],{"className":198482,"style":32617},[31680],[34,198484,640],{"className":198485},[31685]," 个车轮","（保证至少有空间放1个车架）。",[15,198489,198490],{},[2627,198491,195748],{},[105,198493,198494,198499,198505,198511,198517,198523],{},[108,198495,198496,198498],{},[31,198497,195761],{}," \u002F\u002F 互斥信号量：互斥访问箱子",[108,198500,198501,198504],{},[31,198502,198503],{},"empty = N;"," \u002F\u002F 同步信号量：箱子中的总空位数",[108,198506,198507,198510],{},[31,198508,198509],{},"frames = 0;"," \u002F\u002F 同步信号量：箱子中的车架数量",[108,198512,198513,198516],{},[31,198514,198515],{},"wheels = 0;"," \u002F\u002F 同步信号量：箱子中的车轮数量",[108,198518,198519,198522],{},[31,198520,198521],{},"empty_frame = N - 2;"," \u002F\u002F 同步信号量：允许放入车架的空位限制",[108,198524,198525,198528],{},[31,198526,198527],{},"empty_wheel = N - 1;"," \u002F\u002F 同步信号量：允许放入车轮的空位限制",[15,198530,198531],{},[2627,198532,195885],{},[24,198534,198536],{"className":65734,"code":198535,"language":31965,"meta":29,"style":29},"Worker_1() { \u002F\u002F 生产车架\n while(1) {\n Processing a frame;\n semWait(empty_frame); \u002F\u002F 检查是否达到车架上限\n semWait(empty); \u002F\u002F 申请一个物理空位\n semWait(mutex); \u002F\u002F 互斥访问箱子\n Put the frame into the box;\n semSignal(mutex); \u002F\u002F 离开箱子\n semSignal(frames); \u002F\u002F 车架数量加1\n }\n}\n\nWorker_2() { \u002F\u002F 生产车轮\n while(1) {\n Processing a wheel;\n semWait(empty_wheel); \u002F\u002F 检查是否达到车轮上限\n semWait(empty); \u002F\u002F 申请一个物理空位\n semWait(mutex); \u002F\u002F 互斥访问箱子\n Put the wheel into the box;\n semSignal(mutex); \u002F\u002F 离开箱子\n semSignal(wheels); \u002F\u002F 车轮数量加1\n }\n}\n\nWorker_3() { \u002F\u002F 组装自行车\n while(1) {\n semWait(frames); \u002F\u002F 申请1个车架\n semWait(wheels); \u002F\u002F 申请第1个车轮\n semWait(wheels); \u002F\u002F 申请第2个车轮\n semWait(mutex); \u002F\u002F 互斥访问箱子\n Take a frame from the box;\n Take two wheels from the box;\n semSignal(mutex); \u002F\u002F 离开箱子\n \n semSignal(empty); \u002F\u002F 释放3个物理空位\n semSignal(empty);\n semSignal(empty);\n semSignal(empty_frame); \u002F\u002F 释放1个车架配额\n semSignal(empty_wheel); \u002F\u002F 释放2个车轮配额\n semSignal(empty_wheel);\n \n Assemble into one vehicle;\n }\n}\n",[31,198537,198538,198543,198548,198553,198558,198563,198568,198573,198578,198583,198587,198591,198595,198600,198604,198609,198614,198618,198622,198627,198631,198636,198640,198644,198648,198653,198657,198662,198667,198672,198676,198681,198686,198690,198694,198699,198704,198708,198713,198718,198723,198727,198732,198736],{"__ignoreMap":29},[34,198539,198540],{"class":36,"line":37},[34,198541,198542],{},"Worker_1() { \u002F\u002F 生产车架\n",[34,198544,198545],{"class":36,"line":43},[34,198546,198547],{}," while(1) {\n",[34,198549,198550],{"class":36,"line":49},[34,198551,198552],{}," Processing a frame;\n",[34,198554,198555],{"class":36,"line":56},[34,198556,198557],{}," semWait(empty_frame); \u002F\u002F 检查是否达到车架上限\n",[34,198559,198560],{"class":36,"line":62},[34,198561,198562],{}," semWait(empty); \u002F\u002F 申请一个物理空位\n",[34,198564,198565],{"class":36,"line":68},[34,198566,198567],{}," semWait(mutex); \u002F\u002F 互斥访问箱子\n",[34,198569,198570],{"class":36,"line":74},[34,198571,198572],{}," Put the frame into the box;\n",[34,198574,198575],{"class":36,"line":1971},[34,198576,198577],{}," semSignal(mutex); \u002F\u002F 离开箱子\n",[34,198579,198580],{"class":36,"line":1976},[34,198581,198582],{}," semSignal(frames); \u002F\u002F 车架数量加1\n",[34,198584,198585],{"class":36,"line":1985},[34,198586,5540],{},[34,198588,198589],{"class":36,"line":1990},[34,198590,77],{},[34,198592,198593],{"class":36,"line":1999},[34,198594,53],{"emptyLinePlaceholder":52},[34,198596,198597],{"class":36,"line":2022},[34,198598,198599],{},"Worker_2() { \u002F\u002F 生产车轮\n",[34,198601,198602],{"class":36,"line":2036},[34,198603,198547],{},[34,198605,198606],{"class":36,"line":2603},[34,198607,198608],{}," Processing a wheel;\n",[34,198610,198611],{"class":36,"line":5537},[34,198612,198613],{}," semWait(empty_wheel); \u002F\u002F 检查是否达到车轮上限\n",[34,198615,198616],{"class":36,"line":5543},[34,198617,198562],{},[34,198619,198620],{"class":36,"line":5548},[34,198621,198567],{},[34,198623,198624],{"class":36,"line":5564},[34,198625,198626],{}," Put the wheel into the box;\n",[34,198628,198629],{"class":36,"line":5576},[34,198630,198577],{},[34,198632,198633],{"class":36,"line":5606},[34,198634,198635],{}," semSignal(wheels); \u002F\u002F 车轮数量加1\n",[34,198637,198638],{"class":36,"line":5616},[34,198639,5540],{},[34,198641,198642],{"class":36,"line":5628},[34,198643,77],{},[34,198645,198646],{"class":36,"line":5634},[34,198647,53],{"emptyLinePlaceholder":52},[34,198649,198650],{"class":36,"line":6420},[34,198651,198652],{},"Worker_3() { \u002F\u002F 组装自行车\n",[34,198654,198655],{"class":36,"line":11294},[34,198656,198547],{},[34,198658,198659],{"class":36,"line":11321},[34,198660,198661],{}," semWait(frames); \u002F\u002F 申请1个车架\n",[34,198663,198664],{"class":36,"line":11326},[34,198665,198666],{}," semWait(wheels); \u002F\u002F 申请第1个车轮\n",[34,198668,198669],{"class":36,"line":11333},[34,198670,198671],{}," semWait(wheels); \u002F\u002F 申请第2个车轮\n",[34,198673,198674],{"class":36,"line":11344},[34,198675,198567],{},[34,198677,198678],{"class":36,"line":11349},[34,198679,198680],{}," Take a frame from the box;\n",[34,198682,198683],{"class":36,"line":11354},[34,198684,198685],{}," Take two wheels from the box;\n",[34,198687,198688],{"class":36,"line":11363},[34,198689,198577],{},[34,198691,198692],{"class":36,"line":11368},[34,198693,196186],{},[34,198695,198696],{"class":36,"line":11377},[34,198697,198698],{}," semSignal(empty); \u002F\u002F 释放3个物理空位\n",[34,198700,198701],{"class":36,"line":11402},[34,198702,198703],{}," semSignal(empty);\n",[34,198705,198706],{"class":36,"line":11407},[34,198707,198703],{},[34,198709,198710],{"class":36,"line":11421},[34,198711,198712],{}," semSignal(empty_frame); \u002F\u002F 释放1个车架配额\n",[34,198714,198715],{"class":36,"line":11426},[34,198716,198717],{}," semSignal(empty_wheel); \u002F\u002F 释放2个车轮配额\n",[34,198719,198720],{"class":36,"line":11447},[34,198721,198722],{}," semSignal(empty_wheel);\n",[34,198724,198725],{"class":36,"line":11452},[34,198726,196186],{},[34,198728,198729],{"class":36,"line":11463},[34,198730,198731],{}," Assemble into one vehicle;\n",[34,198733,198734],{"class":36,"line":11485},[34,198735,5540],{},[34,198737,198738],{"class":36,"line":11495},[34,198739,77],{},[32188,198741],{},[253,198743,198745],{"id":198744},"_13-并发执行路径与时间相关的错误-concurrent-execution-timing-errors","13. 并发执行路径与时间相关的错误 (Concurrent Execution & Timing Errors)",[32490,198747,198749],{"id":198748},"_1-进程-p-和-q-的并发执行路径","(1) 进程 P 和 Q 的并发执行路径",[15,198751,198752,198755,198756,198863],{},[2627,198753,198754],{},"解答：","\n进程 P 包含指令 A、B、C（必须保证顺序：A -> B -> C）。\n进程 Q 包含指令 D、E（必须保证顺序：D -> E）。\n两进程并发执行时，指令在宏观上会发生交替（Interleaving），总指令数为 5 条。符合两种顺序的排列组合数为 ",[34,198757,198759,198783],{"className":198758},[31642],[34,198760,198762],{"className":198761},[31646],[31648,198763,198764],{"xmlns":31650},[31652,198765,198766,198780],{},[31655,198767,198768,198776,198778],{},[38846,198769,198770,198772,198774],{},[31658,198771,47384],{},[31749,198773,18136],{},[31749,198775,768],{},[31662,198777,202],{},[31749,198779,3244],{},[31665,198781,198782],{"encoding":31667},"C_5^2 = 10",[34,198784,198786,198854],{"className":198785,"ariaHidden":758},[31672],[34,198787,198789,198793,198845,198848,198851],{"className":198788},[31676],[34,198790],{"className":198791,"style":198792},[31680],"height:1.0622em;vertical-align:-0.2481em;",[34,198794,198796,198799],{"className":198795},[31685],[34,198797,47384],{"className":198798,"style":38706},[31685,31686],[34,198800,198802],{"className":198801},[31792],[34,198803,198805,198837],{"className":198804},[31796,31797],[34,198806,198808,198834],{"className":198807},[31801],[34,198809,198811,198823],{"className":198810,"style":37774},[31805],[34,198812,198814,198817],{"style":198813},"top:-2.4519em;margin-left:-0.0715em;margin-right:0.05em;",[34,198815],{"className":198816,"style":31814},[31813],[34,198818,198820],{"className":198819},[31818,31819,31820,31821],[34,198821,18136],{"className":198822},[31685,31821],[34,198824,198825,198828],{"style":37777},[34,198826],{"className":198827,"style":31814},[31813],[34,198829,198831],{"className":198830},[31818,31819,31820,31821],[34,198832,768],{"className":198833},[31685,31821],[34,198835,31829],{"className":198836},[31828],[34,198838,198840],{"className":198839},[31801],[34,198841,198843],{"className":198842,"style":39021},[31805],[34,198844],{},[34,198846],{"className":198847,"style":31846},[31845],[34,198849,202],{"className":198850},[31850],[34,198852],{"className":198853,"style":31846},[31845],[34,198855,198857,198860],{"className":198856},[31676],[34,198858],{"className":198859,"style":32617},[31680],[34,198861,3244],{"className":198862},[31685]," 种。",[15,198865,198866],{},"可能的并发执行路径（10种）如下：",[2230,198868,198869,198875,198880,198885,198890,198895,198900,198905,198910,198915],{},[108,198870,198871,198874],{},[2627,198872,198873],{},"A B C D E"," (P 执行完，Q 执行)",[108,198876,198877],{},[2627,198878,198879],{},"A B D C E",[108,198881,198882],{},[2627,198883,198884],{},"A B D E C",[108,198886,198887],{},[2627,198888,198889],{},"A D B C E",[108,198891,198892],{},[2627,198893,198894],{},"A D B E C",[108,198896,198897],{},[2627,198898,198899],{},"A D E B C",[108,198901,198902],{},[2627,198903,198904],{},"D A B C E",[108,198906,198907],{},[2627,198908,198909],{},"D A B E C",[108,198911,198912],{},[2627,198913,198914],{},"D A E B C",[108,198916,198917,198920],{},[2627,198918,198919],{},"D E A B C"," (Q 执行完，P 执行)",[32188,198922],{},[32490,198924,198926],{"id":198925},"_2-变量的并发执行与时间相关的错误","(2) 变量的并发执行与时间相关的错误",[15,198928,198929],{},[2627,198930,198931],{},"背景分析：",[105,198933,198934,198988,199379,199441],{},[108,198935,198936,198937,347],{},"初始值 ",[34,198938,198940,198958],{"className":198939},[31642],[34,198941,198943],{"className":198942},[31646],[31648,198944,198945],{"xmlns":31650},[31652,198946,198947,198955],{},[31655,198948,198949,198951,198953],{},[31658,198950,33759],{},[31662,198952,202],{},[31749,198954,18136],{},[31665,198956,198957],{"encoding":31667},"k = 5",[34,198959,198961,198979],{"className":198960,"ariaHidden":758},[31672],[34,198962,198964,198967,198970,198973,198976],{"className":198963},[31676],[34,198965],{"className":198966,"style":31681},[31680],[34,198968,33759],{"className":198969,"style":33774},[31685,31686],[34,198971],{"className":198972,"style":31846},[31845],[34,198974,202],{"className":198975},[31850],[34,198977],{"className":198978,"style":31846},[31845],[34,198980,198982,198985],{"className":198981},[31676],[34,198983],{"className":198984,"style":32617},[31680],[34,198986,18136],{"className":198987},[31685],[108,198989,198990,198991],{},"P1 独立执行两个循环后：\n",[105,198992,198993,199186],{},[108,198994,198995,198996,36268,199091],{},"第 1 次循环：",[34,198997,198999,199025],{"className":198998},[31642],[34,199000,199002],{"className":199001},[31646],[31648,199003,199004],{"xmlns":31650},[31652,199005,199006,199022],{},[31655,199007,199008,199010,199012,199014,199016,199018,199020],{},[31658,199009,33759],{},[31662,199011,202],{},[31749,199013,18136],{},[31662,199015,42071],{},[31749,199017,768],{},[31662,199019,202],{},[31749,199021,3244],{},[31665,199023,199024],{"encoding":31667},"k = 5 \\times 2 = 10",[34,199026,199028,199046,199064,199082],{"className":199027,"ariaHidden":758},[31672],[34,199029,199031,199034,199037,199040,199043],{"className":199030},[31676],[34,199032],{"className":199033,"style":31681},[31680],[34,199035,33759],{"className":199036,"style":33774},[31685,31686],[34,199038],{"className":199039,"style":31846},[31845],[34,199041,202],{"className":199042},[31850],[34,199044],{"className":199045,"style":31846},[31845],[34,199047,199049,199052,199055,199058,199061],{"className":199048},[31676],[34,199050],{"className":199051,"style":35942},[31680],[34,199053,18136],{"className":199054},[31685],[34,199056],{"className":199057,"style":32604},[31845],[34,199059,42071],{"className":199060},[32024],[34,199062],{"className":199063,"style":32604},[31845],[34,199065,199067,199070,199073,199076,199079],{"className":199066},[31676],[34,199068],{"className":199069,"style":32617},[31680],[34,199071,768],{"className":199072},[31685],[34,199074],{"className":199075,"style":31846},[31845],[34,199077,202],{"className":199078},[31850],[34,199080],{"className":199081,"style":31846},[31845],[34,199083,199085,199088],{"className":199084},[31676],[34,199086],{"className":199087,"style":32617},[31680],[34,199089,3244],{"className":199090},[31685],[34,199092,199094,199120],{"className":199093},[31642],[34,199095,199097],{"className":199096},[31646],[31648,199098,199099],{"xmlns":31650},[31652,199100,199101,199117],{},[31655,199102,199103,199105,199107,199109,199111,199113,199115],{},[31658,199104,33759],{},[31662,199106,202],{},[31749,199108,3244],{},[31662,199110,223],{},[31749,199112,640],{},[31662,199114,202],{},[31749,199116,39620],{},[31665,199118,199119],{"encoding":31667},"k = 10 + 1 = 11",[34,199121,199123,199141,199159,199177],{"className":199122,"ariaHidden":758},[31672],[34,199124,199126,199129,199132,199135,199138],{"className":199125},[31676],[34,199127],{"className":199128,"style":31681},[31680],[34,199130,33759],{"className":199131,"style":33774},[31685,31686],[34,199133],{"className":199134,"style":31846},[31845],[34,199136,202],{"className":199137},[31850],[34,199139],{"className":199140,"style":31846},[31845],[34,199142,199144,199147,199150,199153,199156],{"className":199143},[31676],[34,199145],{"className":199146,"style":35942},[31680],[34,199148,3244],{"className":199149},[31685],[34,199151],{"className":199152,"style":32604},[31845],[34,199154,223],{"className":199155},[32024],[34,199157],{"className":199158,"style":32604},[31845],[34,199160,199162,199165,199168,199171,199174],{"className":199161},[31676],[34,199163],{"className":199164,"style":32617},[31680],[34,199166,640],{"className":199167},[31685],[34,199169],{"className":199170,"style":31846},[31845],[34,199172,202],{"className":199173},[31850],[34,199175],{"className":199176,"style":31846},[31845],[34,199178,199180,199183],{"className":199179},[31676],[34,199181],{"className":199182,"style":32617},[31680],[34,199184,39620],{"className":199185},[31685],[108,199187,199188,199189,36268,199284],{},"第 2 次循环：",[34,199190,199192,199218],{"className":199191},[31642],[34,199193,199195],{"className":199194},[31646],[31648,199196,199197],{"xmlns":31650},[31652,199198,199199,199215],{},[31655,199200,199201,199203,199205,199207,199209,199211,199213],{},[31658,199202,33759],{},[31662,199204,202],{},[31749,199206,39620],{},[31662,199208,42071],{},[31749,199210,768],{},[31662,199212,202],{},[31749,199214,94716],{},[31665,199216,199217],{"encoding":31667},"k = 11 \\times 2 = 22",[34,199219,199221,199239,199257,199275],{"className":199220,"ariaHidden":758},[31672],[34,199222,199224,199227,199230,199233,199236],{"className":199223},[31676],[34,199225],{"className":199226,"style":31681},[31680],[34,199228,33759],{"className":199229,"style":33774},[31685,31686],[34,199231],{"className":199232,"style":31846},[31845],[34,199234,202],{"className":199235},[31850],[34,199237],{"className":199238,"style":31846},[31845],[34,199240,199242,199245,199248,199251,199254],{"className":199241},[31676],[34,199243],{"className":199244,"style":35942},[31680],[34,199246,39620],{"className":199247},[31685],[34,199249],{"className":199250,"style":32604},[31845],[34,199252,42071],{"className":199253},[32024],[34,199255],{"className":199256,"style":32604},[31845],[34,199258,199260,199263,199266,199269,199272],{"className":199259},[31676],[34,199261],{"className":199262,"style":32617},[31680],[34,199264,768],{"className":199265},[31685],[34,199267],{"className":199268,"style":31846},[31845],[34,199270,202],{"className":199271},[31850],[34,199273],{"className":199274,"style":31846},[31845],[34,199276,199278,199281],{"className":199277},[31676],[34,199279],{"className":199280,"style":32617},[31680],[34,199282,94716],{"className":199283},[31685],[34,199285,199287,199313],{"className":199286},[31642],[34,199288,199290],{"className":199289},[31646],[31648,199291,199292],{"xmlns":31650},[31652,199293,199294,199310],{},[31655,199295,199296,199298,199300,199302,199304,199306,199308],{},[31658,199297,33759],{},[31662,199299,202],{},[31749,199301,94716],{},[31662,199303,223],{},[31749,199305,640],{},[31662,199307,202],{},[31749,199309,94703],{},[31665,199311,199312],{"encoding":31667},"k = 22 + 1 = 23",[34,199314,199316,199334,199352,199370],{"className":199315,"ariaHidden":758},[31672],[34,199317,199319,199322,199325,199328,199331],{"className":199318},[31676],[34,199320],{"className":199321,"style":31681},[31680],[34,199323,33759],{"className":199324,"style":33774},[31685,31686],[34,199326],{"className":199327,"style":31846},[31845],[34,199329,202],{"className":199330},[31850],[34,199332],{"className":199333,"style":31846},[31845],[34,199335,199337,199340,199343,199346,199349],{"className":199336},[31676],[34,199338],{"className":199339,"style":35942},[31680],[34,199341,94716],{"className":199342},[31685],[34,199344],{"className":199345,"style":32604},[31845],[34,199347,223],{"className":199348},[32024],[34,199350],{"className":199351,"style":32604},[31845],[34,199353,199355,199358,199361,199364,199367],{"className":199354},[31676],[34,199356],{"className":199357,"style":32617},[31680],[34,199359,640],{"className":199360},[31685],[34,199362],{"className":199363,"style":31846},[31845],[34,199365,202],{"className":199366},[31850],[34,199368],{"className":199369,"style":31846},[31845],[34,199371,199373,199376],{"className":199372},[31676],[34,199374],{"className":199375,"style":32617},[31680],[34,199377,94703],{"className":199378},[31685],[108,199380,199381,199382,199410,199411,347],{},"前两轮执行完毕后，当前 ",[34,199383,199385,199398],{"className":199384},[31642],[34,199386,199388],{"className":199387},[31646],[31648,199389,199390],{"xmlns":31650},[31652,199391,199392,199396],{},[31655,199393,199394],{},[31658,199395,33759],{},[31665,199397,33759],{"encoding":31667},[34,199399,199401],{"className":199400,"ariaHidden":758},[31672],[34,199402,199404,199407],{"className":199403},[31676],[34,199405],{"className":199406,"style":31681},[31680],[34,199408,33759],{"className":199409,"style":33774},[31685,31686]," 的值为 ",[2627,199412,199413],{},[34,199414,199416,199429],{"className":199415},[31642],[34,199417,199419],{"className":199418},[31646],[31648,199420,199421],{"xmlns":31650},[31652,199422,199423,199427],{},[31655,199424,199425],{},[31749,199426,94703],{},[31665,199428,94703],{"encoding":31667},[34,199430,199432],{"className":199431,"ariaHidden":758},[31672],[34,199433,199435,199438],{"className":199434},[31676],[34,199436],{"className":199437,"style":32617},[31680],[34,199439,94703],{"className":199440},[31685],[108,199442,199443,199444],{},"接下来 P1 和 P2 并发执行一次循环，存在 4 条主要语句交替执行：\n",[105,199445,199446,199533,199618,199626],{},[108,199447,199448,8471,199451],{},[31,199449,199450],{},"P1_a",[34,199452,199454,199481],{"className":199453},[31642],[34,199455,199457],{"className":199456},[31646],[31648,199458,199459],{"xmlns":31650},[31652,199460,199461,199478],{},[31655,199462,199463,199465,199467,199469,199471,199473,199475],{},[31658,199464,33759],{},[31662,199466,5598],{},[31662,199468,202],{},[31658,199470,33759],{},[31662,199472,42071],{},[31749,199474,768],{},[31662,199476,199477],{"separator":758},";",[31665,199479,199480],{"encoding":31667},"k := k \\times 2;",[34,199482,199484,199503,199521],{"className":199483,"ariaHidden":758},[31672],[34,199485,199487,199490,199493,199496,199500],{"className":199486},[31676],[34,199488],{"className":199489,"style":31681},[31680],[34,199491,33759],{"className":199492,"style":33774},[31685,31686],[34,199494],{"className":199495,"style":31846},[31845],[34,199497,199499],{"className":199498},[31850],":=",[34,199501],{"className":199502,"style":31846},[31845],[34,199504,199506,199509,199512,199515,199518],{"className":199505},[31676],[34,199507],{"className":199508,"style":34299},[31680],[34,199510,33759],{"className":199511,"style":33774},[31685,31686],[34,199513],{"className":199514,"style":32604},[31845],[34,199516,42071],{"className":199517},[32024],[34,199519],{"className":199520,"style":32604},[31845],[34,199522,199524,199527,199530],{"className":199523},[31676],[34,199525],{"className":199526,"style":40292},[31680],[34,199528,768],{"className":199529},[31685],[34,199531,199477],{"className":199532},[32119],[108,199534,199535,8471,199538],{},[31,199536,199537],{},"P1_b",[34,199539,199541,199567],{"className":199540},[31642],[34,199542,199544],{"className":199543},[31646],[31648,199545,199546],{"xmlns":31650},[31652,199547,199548,199564],{},[31655,199549,199550,199552,199554,199556,199558,199560,199562],{},[31658,199551,33759],{},[31662,199553,5598],{},[31662,199555,202],{},[31658,199557,33759],{},[31662,199559,223],{},[31749,199561,640],{},[31662,199563,199477],{"separator":758},[31665,199565,199566],{"encoding":31667},"k := k + 1;",[34,199568,199570,199588,199606],{"className":199569,"ariaHidden":758},[31672],[34,199571,199573,199576,199579,199582,199585],{"className":199572},[31676],[34,199574],{"className":199575,"style":31681},[31680],[34,199577,33759],{"className":199578,"style":33774},[31685,31686],[34,199580],{"className":199581,"style":31846},[31845],[34,199583,199499],{"className":199584},[31850],[34,199586],{"className":199587,"style":31846},[31845],[34,199589,199591,199594,199597,199600,199603],{"className":199590},[31676],[34,199592],{"className":199593,"style":34299},[31680],[34,199595,33759],{"className":199596,"style":33774},[31685,31686],[34,199598],{"className":199599,"style":32604},[31845],[34,199601,223],{"className":199602},[32024],[34,199604],{"className":199605,"style":32604},[31845],[34,199607,199609,199612,199615],{"className":199608},[31676],[34,199610],{"className":199611,"style":40292},[31680],[34,199613,640],{"className":199614},[31685],[34,199616,199477],{"className":199617},[32119],[108,199619,199620,8471,199623],{},[31,199621,199622],{},"P2_a",[31,199624,199625],{},"print k;",[108,199627,199628,8471,199631],{},[31,199629,199630],{},"P2_b",[34,199632,199634,199656],{"className":199633},[31642],[34,199635,199637],{"className":199636},[31646],[31648,199638,199639],{"xmlns":31650},[31652,199640,199641,199653],{},[31655,199642,199643,199645,199647,199649,199651],{},[31658,199644,33759],{},[31662,199646,5598],{},[31662,199648,202],{},[31749,199650,459],{},[31662,199652,199477],{"separator":758},[31665,199654,199655],{"encoding":31667},"k := 0;",[34,199657,199659,199677],{"className":199658,"ariaHidden":758},[31672],[34,199660,199662,199665,199668,199671,199674],{"className":199661},[31676],[34,199663],{"className":199664,"style":31681},[31680],[34,199666,33759],{"className":199667,"style":33774},[31685,31686],[34,199669],{"className":199670,"style":31846},[31845],[34,199672,199499],{"className":199673},[31850],[34,199675],{"className":199676,"style":31846},[31845],[34,199678,199680,199683,199686],{"className":199679},[31676],[34,199681],{"className":199682,"style":40292},[31680],[34,199684,459],{"className":199685},[31685],[34,199687,199477],{"className":199688},[32119],[15,199690,199691],{},[2627,199692,199693],{},"可能的 print 输出值（如果在高级语言语句级发生交替）：",[2230,199695,199696,199743,199766],{},[108,199697,199698,46324,199701,199703,199704,199732,199733],{},[2627,199699,199700],{},"输出 23",[31,199702,199622],{}," 最先执行（在 P1 还没修改 ",[34,199705,199707,199720],{"className":199706},[31642],[34,199708,199710],{"className":199709},[31646],[31648,199711,199712],{"xmlns":31650},[31652,199713,199714,199718],{},[31655,199715,199716],{},[31658,199717,33759],{},[31665,199719,33759],{"encoding":31667},[34,199721,199723],{"className":199722,"ariaHidden":758},[31672],[34,199724,199726,199729],{"className":199725},[31676],[34,199727],{"className":199728,"style":31681},[31680],[34,199730,33759],{"className":199731,"style":33774},[31685,31686]," 时）。\n",[105,199734,199735],{},[108,199736,199737,47649,199740,199742],{},[71807,199738,199739],{},"执行流：",[31,199741,199622],{}," (print 23) -> 然后无论后面顺序如何。",[108,199744,199745,46324,199748,199750,199751,199753,199754],{},[2627,199746,199747],{},"输出 46",[31,199749,199450],{}," 发生，接着交替到 ",[31,199752,199622],{}," 执行。\n",[105,199755,199756],{},[108,199757,199758,47649,199760,199762,199763,199765],{},[71807,199759,199739],{},[31,199761,199450],{}," (k = 23*2=46) -> ",[31,199764,199622],{}," (print 46) -> ...",[108,199767,199768,199771,199772,199753,199774],{},[2627,199769,199770],{},"输出 47","：如果 P1 完整执行完一遍，再交替到 ",[31,199773,199622],{},[105,199775,199776],{},[108,199777,199778,47649,199780,199782,199783,199785,199786,199788],{},[71807,199779,199739],{},[31,199781,199450],{}," (k=46) -> ",[31,199784,199537],{}," (k=47) -> ",[31,199787,199622],{}," (print 47) -> ...",[15,199790,199791],{},[71807,199792,199793],{},"(注：如果在底层寄存器\u002F汇编指令层面发生中断交替，还可能产生其他异常值，但在通常的考察中，上面三个是基于语句级的合法交替输出。)",[15,199795,199796,199799,199800,347],{},[2627,199797,199798],{},"指出与时间相关的错误 (Errors related to time):","\n这种现象在操作系统中被称为 ",[2627,199801,199802],{},"竞态条件 (Race Condition)",[2230,199804,199805,199873],{},[108,199806,199807,199810,199811,199839,199840,199868,199869,199872],{},[2627,199808,199809],{},"问题本质："," 两个并发进程 P1 和 P2 共享了全局变量 ",[34,199812,199814,199827],{"className":199813},[31642],[34,199815,199817],{"className":199816},[31646],[31648,199818,199819],{"xmlns":31650},[31652,199820,199821,199825],{},[31655,199822,199823],{},[31658,199824,33759],{},[31665,199826,33759],{"encoding":31667},[34,199828,199830],{"className":199829,"ariaHidden":758},[31672],[34,199831,199833,199836],{"className":199832},[31676],[34,199834],{"className":199835,"style":31681},[31680],[34,199837,33759],{"className":199838,"style":33774},[31685,31686],"，但在访问和修改 ",[34,199841,199843,199856],{"className":199842},[31642],[34,199844,199846],{"className":199845},[31646],[31648,199847,199848],{"xmlns":31650},[31652,199849,199850,199854],{},[31655,199851,199852],{},[31658,199853,33759],{},[31665,199855,33759],{"encoding":31667},[34,199857,199859],{"className":199858,"ariaHidden":758},[31672],[34,199860,199862,199865],{"className":199861},[31676],[34,199863],{"className":199864,"style":31681},[31680],[34,199866,33759],{"className":199867,"style":33774},[31685,31686]," 的过程中，",[2627,199870,199871],{},"没有实施互斥 (Mutual Exclusion)","（如未加锁或使用信号量）。",[108,199874,199875,199878,199879,199882],{},[2627,199876,199877],{},"表现："," 程序的最终结果（print 输出）",[2627,199880,199881],{},"高度依赖于 CPU 调度的相对速度和执行时间顺序","。由于进程调度的不可预测性（Nondeterministic），同一个程序在不同时间运行可能得到完全不同的结果，引发了数据不一致性。这违背了程序运行应有的确定性（Determinism）与可再现性（Reproducibility）。",[79,199884,81],{},{"title":29,"searchDepth":43,"depth":43,"links":199886},[199887,199888,199889,199890,199891,199892,199893,199894,199895,199896,199897,199898,199899],{"id":195736,"depth":49,"text":195737},{"id":195856,"depth":49,"text":195857},{"id":195983,"depth":49,"text":195984},{"id":196286,"depth":49,"text":196287},{"id":196527,"depth":49,"text":196528},{"id":197016,"depth":49,"text":197017},{"id":197190,"depth":49,"text":197191},{"id":197412,"depth":49,"text":197413},{"id":197543,"depth":49,"text":197544},{"id":197734,"depth":49,"text":197735},{"id":197930,"depth":49,"text":197931},{"id":198197,"depth":49,"text":198198},{"id":198744,"depth":49,"text":198745},{"order":62},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap5-hw",{"title":195727,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap5-hw","Qy2vRb4pSHxRY7mQAjU7ldklfwy-viNEQjvEt5_LcH8",{"id":199906,"title":199907,"body":199908,"description":29,"extension":85,"meta":200413,"navigation":52,"path":200414,"seo":200415,"stem":200416,"__hash__":200417},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap6-hw.md","Chap. 6 HW",{"type":8,"value":199909,"toc":200404},[199910,199914,199918,199921,199925,199930,199960,199965,200059,200063,200096,200098,200102,200109,200150,200153,200165,200167,200171,200321,200323,200327,200335,200398],[11,199911,199913],{"id":199912},"os-chap6-hw","OS Chap.6 HW",[19,199915,199917],{"id":199916},"problem-1","Problem 1",[15,199919,199920],{},"Consider the following snapshot of a system. There are no outstanding unsatisfied requests for resources.",[253,199922,199924],{"id":199923},"resource-data","Resource Data",[15,199926,199927],{},[2627,199928,199929],{},"Available Resources",[13837,199931,199932,199948],{},[13840,199933,199934],{},[13843,199935,199936,199939,199942,199945],{},[13846,199937,199938],{"align":82355},"r1",[13846,199940,199941],{"align":82355},"r2",[13846,199943,199944],{"align":82355},"r3",[13846,199946,199947],{"align":82355},"r4",[13853,199949,199950],{},[13843,199951,199952,199954,199956,199958],{},[13858,199953,768],{"align":82355},[13858,199955,640],{"align":82355},[13858,199957,459],{"align":82355},[13858,199959,459],{"align":82355},[15,199961,199962],{},[2627,199963,199964],{},"Process Allocation Table",[13837,199966,199967,199983],{},[13840,199968,199969],{},[13843,199970,199971,199974,199977,199980],{},[13846,199972,199973],{"align":82355},"Process",[13846,199975,199976],{"align":82355},"Current Allocation (r1, r2, r3, r4)",[13846,199978,199979],{"align":82355},"Maximum Demand (r1, r2, r3, r4)",[13846,199981,199982],{"align":82355},"Still Needs (r1, r2, r3, r4)",[13853,199984,199985,199999,200014,200029,200044],{},[13843,199986,199987,199992,199995,199997],{},[13858,199988,199989],{"align":82355},[2627,199990,199991],{},"p1",[13858,199993,199994],{"align":82355},"0, 0, 1, 2",[13858,199996,199994],{"align":82355},[13858,199998],{"align":82355},[13843,200000,200001,200006,200009,200012],{},[13858,200002,200003],{"align":82355},[2627,200004,200005],{},"p2",[13858,200007,200008],{"align":82355},"2, 0, 0, 0",[13858,200010,200011],{"align":82355},"2, 7, 5, 0",[13858,200013],{"align":82355},[13843,200015,200016,200021,200024,200027],{},[13858,200017,200018],{"align":82355},[2627,200019,200020],{},"p3",[13858,200022,200023],{"align":82355},"0, 0, 3, 4",[13858,200025,200026],{"align":82355},"6, 6, 5, 6",[13858,200028],{"align":82355},[13843,200030,200031,200036,200039,200042],{},[13858,200032,200033],{"align":82355},[2627,200034,200035],{},"p4",[13858,200037,200038],{"align":82355},"2, 3, 5, 4",[13858,200040,200041],{"align":82355},"4, 3, 5, 6",[13858,200043],{"align":82355},[13843,200045,200046,200051,200054,200057],{},[13858,200047,200048],{"align":82355},[2627,200049,200050],{},"p5",[13858,200052,200053],{"align":82355},"0, 3, 3, 2",[13858,200055,200056],{"align":82355},"0, 6, 5, 2",[13858,200058],{"align":82355},[253,200060,200062],{"id":200061},"questions","Questions",[105,200064,200065,200070,200075,200081,200087],{},[108,200066,200067,200069],{},[2627,200068,181367],{}," Compute what each process still might request and display in the columns labeled “still needs.”",[108,200071,200072,200074],{},[2627,200073,181558],{}," Is this system currently in a safe or unsafe state? Why?",[108,200076,200077,200080],{},[2627,200078,200079],{},"c."," Is this system currently deadlocked? Why or why not?",[108,200082,200083,200086],{},[2627,200084,200085],{},"d."," Which processes, if any, are or may become deadlocked?",[108,200088,200089,200092,200093,200095],{},[2627,200090,200091],{},"e."," If a request from ",[2627,200094,200020],{}," arrives for (0, 1, 0, 0), can that request be safely granted immediately? In what state (deadlocked, safe, unsafe) would immediately granting that whole request leave the system? Which processes, if any, are or may become deadlocked if this whole request is granted immediately?",[32188,200097],{},[19,200099,200101],{"id":200100},"problem-2","Problem 2",[15,200103,200104,200105,200108],{},"Consider a system with a total of ",[2627,200106,200107],{},"150 units of memory",", allocated to three processes as shown:",[13837,200110,200111,200123],{},[13840,200112,200113],{},[13843,200114,200115,200117,200120],{},[13846,200116,199973],{"align":82355},[13846,200118,200119],{"align":82355},"Max Need",[13846,200121,200122],{"align":82355},"Hold",[13853,200124,200125,200134,200142],{},[13843,200126,200127,200129,200131],{},[13858,200128,640],{"align":82355},[13858,200130,98497],{"align":82355},[13858,200132,200133],{"align":82355},"45",[13843,200135,200136,200138,200140],{},[13858,200137,768],{"align":82355},[13858,200139,167664],{"align":82355},[13858,200141,5731],{"align":82355},[13843,200143,200144,200146,200148],{},[13858,200145,786],{"align":82355},[13858,200147,167664],{"align":82355},[13858,200149,95396],{"align":82355},[15,200151,200152],{},"Apply the banker’s algorithm to determine whether it would be safe to grant each of the following requests. If yes, indicate a sequence of terminations that could be guaranteed possible. If no, show the reduction of the resulting allocation table.",[105,200154,200155,200160],{},[108,200156,200157,200159],{},[2627,200158,181367],{}," A fourth process arrives, with a maximum memory need of 60 and an initial need of 25 units.",[108,200161,200162,200164],{},[2627,200163,181558],{}," A fourth process arrives, with a maximum memory need of 60 and an initial need of 35 units.",[32188,200166],{},[19,200168,200170],{"id":200169},"problem-3","Problem 3",[105,200172,200173,200178],{},[108,200174,200175,200177],{},[2627,200176,181367],{}," Three processes share four resource units that can be reserved and released only one at a time. Each process needs a maximum of two units. Show that a deadlock cannot occur.",[108,200179,200180,47649,200182,200210,200211,200239,200240,200268,200269,200320],{},[2627,200181,181558],{},[34,200183,200185,200198],{"className":200184},[31642],[34,200186,200188],{"className":200187},[31646],[31648,200189,200190],{"xmlns":31650},[31652,200191,200192,200196],{},[31655,200193,200194],{},[31658,200195,33347],{},[31665,200197,33347],{"encoding":31667},[34,200199,200201],{"className":200200,"ariaHidden":758},[31672],[34,200202,200204,200207],{"className":200203},[31676],[34,200205],{"className":200206,"style":33218},[31680],[34,200208,33347],{"className":200209,"style":33389},[31685,31686]," processes share ",[34,200212,200214,200227],{"className":200213},[31642],[34,200215,200217],{"className":200216},[31646],[31648,200218,200219],{"xmlns":31650},[31652,200220,200221,200225],{},[31655,200222,200223],{},[31658,200224,98360],{},[31665,200226,98360],{"encoding":31667},[34,200228,200230],{"className":200229,"ariaHidden":758},[31672],[34,200231,200233,200236],{"className":200232},[31676],[34,200234],{"className":200235,"style":33218},[31680],[34,200237,98360],{"className":200238,"style":33389},[31685,31686]," resource units that can be reserved and released only one at a time. The maximum need of each process does not exceed ",[34,200241,200243,200256],{"className":200242},[31642],[34,200244,200246],{"className":200245},[31646],[31648,200247,200248],{"xmlns":31650},[31652,200249,200250,200254],{},[31655,200251,200252],{},[31658,200253,98360],{},[31665,200255,98360],{"encoding":31667},[34,200257,200259],{"className":200258,"ariaHidden":758},[31672],[34,200260,200262,200265],{"className":200261},[31676],[34,200263],{"className":200264,"style":33218},[31680],[34,200266,98360],{"className":200267,"style":33389},[31685,31686],", and the sum of all maximum needs is less than ",[34,200270,200272,200290],{"className":200271},[31642],[34,200273,200275],{"className":200274},[31646],[31648,200276,200277],{"xmlns":31650},[31652,200278,200279,200287],{},[31655,200280,200281,200283,200285],{},[31658,200282,98360],{},[31662,200284,223],{},[31658,200286,33347],{},[31665,200288,200289],{"encoding":31667},"M + N",[34,200291,200293,200311],{"className":200292,"ariaHidden":758},[31672],[34,200294,200296,200299,200302,200305,200308],{"className":200295},[31676],[34,200297],{"className":200298,"style":34944},[31680],[34,200300,98360],{"className":200301,"style":33389},[31685,31686],[34,200303],{"className":200304,"style":32604},[31845],[34,200306,223],{"className":200307},[32024],[34,200309],{"className":200310,"style":32604},[31845],[34,200312,200314,200317],{"className":200313},[31676],[34,200315],{"className":200316,"style":33218},[31680],[34,200318,33347],{"className":200319,"style":33389},[31685,31686],". Show that a deadlock cannot occur.",[32188,200322],{},[19,200324,200326],{"id":200325},"problem-4","Problem 4",[15,200328,200329,200330,195562,200332,200334],{},"The two variables ",[31,200331,111],{},[31,200333,1697],{}," have initial values of 1 and 2, respectively. The following code is for a Linux system:",[13837,200336,200337,200347],{},[13840,200338,200339],{},[13843,200340,200341,200344],{},[13846,200342,200343],{"align":82355},"Thread 1",[13846,200345,200346],{"align":82355},"Thread 2",[13853,200348,200349,200359,200368,200380,200389],{},[13843,200350,200351,200356],{},[13858,200352,200353],{"align":82355},[31,200354,200355],{},"a = 3;",[13858,200357,200358],{"align":82355},"—",[13843,200360,200361,200366],{},[13858,200362,200363],{"align":82355},[31,200364,200365],{},"mb();",[13858,200367,200358],{"align":82355},[13843,200369,200370,200375],{},[13858,200371,200372],{"align":82355},[31,200373,200374],{},"b = 4;",[13858,200376,200377],{"align":82355},[31,200378,200379],{},"c = b;",[13843,200381,200382,200384],{},[13858,200383,200358],{"align":82355},[13858,200385,200386],{"align":82355},[31,200387,200388],{},"rmb();",[13843,200390,200391,200393],{},[13858,200392,200358],{"align":82355},[13858,200394,200395],{"align":82355},[31,200396,200397],{},"d = a;",[15,200399,200400,200403],{},[2627,200401,200402],{},"Question:"," What possible errors are avoided by the use of the memory barriers?",{"title":29,"searchDepth":43,"depth":43,"links":200405},[200406,200410,200411,200412],{"id":199916,"depth":43,"text":199917,"children":200407},[200408,200409],{"id":199923,"depth":49,"text":199924},{"id":200061,"depth":49,"text":200062},{"id":200100,"depth":43,"text":200101},{"id":200169,"depth":43,"text":200170},{"id":200325,"depth":43,"text":200326},{"order":68},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap6-hw",{"title":199907,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap6-hw","cmI4q6XeVyvwdwvc0EuQAwyFRzsUvAlrtXePeoUf7n0",{"id":200419,"title":200420,"body":200421,"description":29,"extension":85,"meta":208633,"navigation":52,"path":208634,"seo":208635,"stem":208636,"__hash__":208637},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap7-hw.md","Chap. 7 HW",{"type":8,"value":200422,"toc":208619},[200423,200427,200431,200436,200441,200446,200451,200457,200491,200496,200501,200543,200548,200551,200672,200674,200678,200687,200692,200698,200754,200782,200786,200792,200851,200862,200872,200934,200944,200950,200996,201006,201015,201068,201078,201080,201084,201089,201094,201157,201174,201178,201181,201321,201327,201408,201414,201420,201424,201618,201809,201813,201833,201837,201842,202113,202118,202254,202259,202559,202564,202701,202706,202802,202804,202808,202813,202858,202861,202866,202904,202907,202973,203255,203259,203262,203288,203291,203296,203630,203635,203964,203969,204168,204173,204502,204507,204833,204837,204843,204848,204888,204890,204894,204899,204903,204959,204961,204965,204970,204974,205119,205121,205125,205130,205134,205689,205691,205695,205700,205704,206084,206086,206090,206095,206099,207497,207499,207503,207511,207515,207530,207535,207656,207661,208230,208235],[11,200424,200426],{"id":200425},"os-chap7-hw","OS Chap.7 HW",[19,200428,200430],{"id":200429},"题目-1动态分区分配算法最坏适应","题目 1：动态分区分配算法（最坏适应）",[15,200432,200433],{},[2627,200434,200435],{},"【英文原题】",[2230,200437,200438],{},[108,200439,200440],{},"Another placement algorithm for dynamic partitioning is referred to as worst-fit. In this case, the largest free block of memory is used for bringing in a process. Discuss the pros and cons of this method compared to first-, next-, and best-fit. What is the average length of the search for worst-fit?",[15,200442,200443],{},[2627,200444,200445],{},"【中文翻译】",[2230,200447,200448],{},[108,200449,200450],{},"动态分区的另一种放置算法被称为最坏适应（worst-fit）。在这种情况下，使用最大的空闲内存块来装入进程。讨论这种方法与首次适应（first-fit）、下次适应（next-fit）和最佳适应（best-fit）相比的优缺点。最坏适应的平均搜索长度是多少？",[15,200452,200453,200456],{},[2627,200454,200455],{},"【知识点讲解】","\n在动态分区内存管理中，当一个进程需要装入内存时，操作系统必须在空闲内存块列表中选择一个足够大的块。不同的选择策略会导致不同的内存碎片分布：",[105,200458,200459,200465,200471,200481],{},[108,200460,200461,200464],{},[2627,200462,200463],{},"首次适应 (First-fit)","：从头开始找，找到第一个能放下的就分配。",[108,200466,200467,200470],{},[2627,200468,200469],{},"下次适应 (Next-fit)","：从上次分配的位置继续往后找，找到第一个能放下的。",[108,200472,200473,200476,200477,200480],{},[2627,200474,200475],{},"最佳适应 (Best-fit)","：遍历所有空闲块，找到能放下且",[2627,200478,200479],{},"最小","的块（即切分后剩下碎片最小的块）。",[108,200482,200483,200486,200487,200490],{},[2627,200484,200485],{},"最坏适应 (Worst-fit)","：遍历所有空闲块，找到能放下的",[2627,200488,200489],{},"最大","的块。",[15,200492,200493],{},[2627,200494,200495],{},"【详尽解析与答案】",[15,200497,200498],{},[2627,200499,200500],{},"1. 优缺点比较分析 (Pros and Cons)",[105,200502,200503,200526],{},[108,200504,200505,200508],{},[2627,200506,200507],{},"与“最佳适应 (Best-fit)”相比：",[105,200509,200510,200520],{},[108,200511,200512,200515,200516,200519],{},[2627,200513,200514],{},"优点 (Pros)","：最佳适应算法总是寻找刚好合适的小块，这会导致切分后留下大量极小、完全无法被利用的“外部碎片”。而",[2627,200517,200518],{},"最坏适应","总是切割最大的块，因此切分后剩余的空闲块通常依然很大，仍有很大机会可以容纳其他后续进程。",[108,200521,200522,200525],{},[2627,200523,200524],{},"缺点 (Cons)","：最坏适应会迅速消耗掉系统中最大的连续空闲内存块。当未来有体积非常巨大的进程请求内存时，系统可能会因为缺乏足够大的单块连续内存而导致分配失败（即使系统总空闲内存足够）。",[108,200527,200528,200531],{},[2627,200529,200530],{},"与“首次适应 (First-fit)”和“下次适应 (Next-fit)”相比：",[105,200532,200533,200538],{},[108,200534,200535,200537],{},[2627,200536,200514],{},"：首次适应容易导致内存低地址端聚集大量小碎片；下次适应容易导致最大空闲块所在的内存末端也被碎片化。最坏适应一定程度上使得内存的分配更加均匀，避免了特定区域的碎片密集。",[108,200539,200540,200542],{},[2627,200541,200524],{},"：在查找速度上处于劣势。首次适应和下次适应通常不需要遍历整个空闲链表（找到合适的就停下），而最坏适应（如果链表未按大小排序）必须遍历完整个空闲链表才能确认哪一个块是最大的，增加了分配的计算开销。",[15,200544,200545],{},[2627,200546,200547],{},"2. 平均搜索长度 (Average length of the search)",[15,200549,200550],{},"平均搜索长度取决于操作系统内部如何维护空闲内存块的链表：",[105,200552,200553,200619],{},[108,200554,200555,200558,200559,200587,200588,347],{},[2627,200556,200557],{},"情况 A（未排序）","：如果空闲块链表是按物理地址顺序排列的，那么为了找到“最大”的块，算法必须盲目地遍历整个链表中的每一个节点。假设当前有 ",[34,200560,200562,200575],{"className":200561},[31642],[34,200563,200565],{"className":200564},[31646],[31648,200566,200567],{"xmlns":31650},[31652,200568,200569,200573],{},[31655,200570,200571],{},[31658,200572,31660],{},[31665,200574,31660],{"encoding":31667},[34,200576,200578],{"className":200577,"ariaHidden":758},[31672],[34,200579,200581,200584],{"className":200580},[31676],[34,200582],{"className":200583,"style":32527},[31680],[34,200585,31660],{"className":200586},[31685,31686]," 个空闲块，那么",[2627,200589,200590,200591],{},"平均搜索长度就是 ",[34,200592,200594,200607],{"className":200593},[31642],[34,200595,200597],{"className":200596},[31646],[31648,200598,200599],{"xmlns":31650},[31652,200600,200601,200605],{},[31655,200602,200603],{},[31658,200604,31660],{},[31665,200606,31660],{"encoding":31667},[34,200608,200610],{"className":200609,"ariaHidden":758},[31672],[34,200611,200613,200616],{"className":200612},[31676],[34,200614],{"className":200615,"style":32527},[31680],[34,200617,31660],{"className":200618},[31685,31686],[108,200620,200621,200624,200625,33775,200628,200671],{},[2627,200622,200623],{},"情况 B（按大小降序排序）","：如果操作系统在后台维护了一个按空间大小从大到小排列的空闲链表，那么最大的块永远在链表的头部。此时，找到最大块的",[2627,200626,200627],{},"搜索长度就是 1",[34,200629,200631,200650],{"className":200630},[31642],[34,200632,200634],{"className":200633},[31646],[31648,200635,200636],{"xmlns":31650},[31652,200637,200638,200648],{},[31655,200639,200640,200642,200644,200646],{},[31658,200641,37890],{},[31662,200643,456],{"stretchy":702},[31749,200645,640],{},[31662,200647,2557],{"stretchy":702},[31665,200649,87861],{"encoding":31667},[34,200651,200653],{"className":200652,"ariaHidden":758},[31672],[34,200654,200656,200659,200662,200665,200668],{"className":200655},[31676],[34,200657],{"className":200658,"style":31774},[31680],[34,200660,37890],{"className":200661,"style":34452},[31685,31686],[34,200663,456],{"className":200664},[31782],[34,200666,640],{"className":200667},[31685],[34,200669,2557],{"className":200670},[31690]," 的时间复杂度）。但代价是每次分配或释放内存时，都需要额外的时间来重新对链表进行排序。",[32188,200673],{},[19,200675,200677],{"id":200676},"题目-2动态分区分配实战推演","题目 2：动态分区分配实战推演",[15,200679,200680,200682,200683,200686],{},[2627,200681,200435],{},"\n2. A dynamic partitioning scheme is being used, and the following is the memory configuration at a given point in time:\n",[71807,200684,200685],{},"(Image provided showing alternating allocated and free blocks. Shaded = allocated, White = free)","\nThe shaded areas are allocated blocks; the white areas are free blocks. The next three memory requests are for 40M, 20M, and 10M. Indicate the starting address for each of the three blocks using the following placement algorithms:\na. First-fit\nb. Best-fit\nc. Next-fit. Assume the most recently added block is at the beginning of memory.\nd. Worst-fit",[15,200688,200689,200691],{},[2627,200690,200445],{},"\n2. 当前正在使用动态分区方案，以下是特定时间点的内存配置：\n（图像显示分配和空闲块交替：20M 分配，20M 空闲，40M 分配，60M 空闲，20M 分配，10M 空闲，60M 分配，40M 空闲，20M 分配，30M 空闲，40M 分配，40M 空闲。阴影区域是已分配块；白色区域是空闲块。）\n接下来的三个内存请求分别为 40M、20M 和 10M。请使用以下放置算法指出这三个块的起始地址：\na. 首次适应 (First-fit)\nb. 最佳适应 (Best-fit)\nc. 下次适应 (Next-fit)。假设最近添加的块位于内存的起始位置。\nd. 最坏适应 (Worst-fit)",[15,200693,200694,200697],{},[2627,200695,200696],{},"【知识点讲解与基础信息提取】","\n要准确计算起始地址，我们首先需要将图像转化为精确的数据模型。假设内存从地址 0 开始，我们将各个块的起始地址推导如下：",[105,200699,200700,200703,200709,200712,200718,200721,200727,200730,200736,200739,200745,200748],{},[108,200701,200702],{},"已分配 20M (地址 [0, 20) )",[108,200704,200705,200708],{},[2627,200706,200707],{},"空闲块 1：20M，起始地址 20"," (范围 [20, 40) )",[108,200710,200711],{},"已分配 40M (地址 [40, 80) )",[108,200713,200714,200717],{},[2627,200715,200716],{},"空闲块 2：60M，起始地址 80"," (范围 [80, 140) )",[108,200719,200720],{},"已分配 20M (地址 [140, 160) )",[108,200722,200723,200726],{},[2627,200724,200725],{},"空闲块 3：10M，起始地址 160"," (范围 [160, 170) )",[108,200728,200729],{},"已分配 60M (地址 [170, 230) )",[108,200731,200732,200735],{},[2627,200733,200734],{},"空闲块 4：40M，起始地址 230"," (范围 [230, 270) )",[108,200737,200738],{},"已分配 20M (地址 [270, 290) )",[108,200740,200741,200744],{},[2627,200742,200743],{},"空闲块 5：30M，起始地址 290"," (范围 [290, 320) )",[108,200746,200747],{},"已分配 40M (地址 [320, 360) )",[108,200749,200750,200753],{},[2627,200751,200752],{},"空闲块 6：40M，起始地址 360"," (范围 [360, 400) )",[15,200755,200756,200759,200760,616,200763,616,200766,616,200769,616,200772,616,200775,200778,200779,347],{},[2627,200757,200758],{},"初始空闲块列表","为：",[31,200761,200762],{},"[20M@20]",[31,200764,200765],{},"[60M@80]",[31,200767,200768],{},"[10M@160]",[31,200770,200771],{},"[40M@230]",[31,200773,200774],{},"[30M@290]",[31,200776,200777],{},"[40M@360]","。\n请求序列：",[2627,200780,200781],{},"40M, 20M, 10M",[15,200783,200784],{},[2627,200785,200495],{},[15,200787,200788,200791],{},[2627,200789,200790],{},"a. 首次适应 (First-fit)","：每次都从内存起始地址（0）开始扫描。",[2230,200793,200794,200815,200833],{},[108,200795,200796,200799,200800,200802,200803,200805,200806],{},[2627,200797,200798],{},"请求 40M","：扫描 ",[31,200801,200762],{}," (太小) -> 找到 ",[31,200804,200765],{}," (满足)。\n",[105,200807,200808],{},[108,200809,200810,200811,200814],{},"分配 40M，",[2627,200812,200813],{},"起始地址为 80","。该空闲块剩余 20M，起始地址变为 120。",[108,200816,200817,200820,200821,200823,200824],{},[2627,200818,200819],{},"请求 20M","：从头开始扫描。找到第一个空闲块 ",[31,200822,200762],{}," (恰好满足)。\n",[105,200825,200826],{},[108,200827,200828,200829,200832],{},"分配 20M，",[2627,200830,200831],{},"起始地址为 20","。该空闲块消耗殆尽。",[108,200834,200835,200838,200839,200842,200843],{},[2627,200836,200837],{},"请求 10M","：从头开始扫描。由于原第一块已用完，遇到现在的第二块 ",[31,200840,200841],{},"[20M@120]"," (原60M剩下的部分，满足)。\n",[105,200844,200845],{},[108,200846,200847,200848,347],{},"分配 10M，",[2627,200849,200850],{},"起始地址为 120",[105,200852,200853],{},[108,200854,200855,200858,200859,347],{},[2627,200856,200857],{},"答案 a","：起始地址分别为 ",[2627,200860,200861],{},"80, 20, 120",[15,200863,200864,200867,200868,200871],{},[2627,200865,200866],{},"b. 最佳适应 (Best-fit)","：每次扫描整个列表，找到满足要求且",[2627,200869,200870],{},"尺寸最小","的空闲块。如果大小相同，通常选最前面的。",[2230,200873,200874,200888,200912],{},[108,200875,200876,200878,200879,39984,200881],{},[2627,200877,200798],{},"：能满足的块有 60M, 40M, 40M。最佳选择是恰好为 40M 的块。第一个 40M 是 ",[31,200880,200771],{},[105,200882,200883],{},[108,200884,200810,200885,200832],{},[2627,200886,200887],{},"起始地址为 230",[108,200889,200890,200892,200893,616,200895,616,200897,616,200899,616,200901,200903,200904,39984,200906],{},[2627,200891,200819],{},"：当前空闲列表为 ",[31,200894,200762],{},[31,200896,200765],{},[31,200898,200768],{},[31,200900,200774],{},[31,200902,200777],{},"。能满足且最小的是 20M 块，即 ",[31,200905,200762],{},[105,200907,200908],{},[108,200909,200828,200910,347],{},[2627,200911,200831],{},[108,200913,200914,200892,200916,616,200918,616,200920,616,200922,200924,200925,39984,200927],{},[2627,200915,200837],{},[31,200917,200765],{},[31,200919,200768],{},[31,200921,200774],{},[31,200923,200777],{},"。能满足且最小的是 10M 块，即 ",[31,200926,200768],{},[105,200928,200929],{},[108,200930,200847,200931,347],{},[2627,200932,200933],{},"起始地址为 160",[105,200935,200936],{},[108,200937,200938,200858,200941,347],{},[2627,200939,200940],{},"答案 b",[2627,200942,200943],{},"230, 20, 160",[15,200945,200946,200949],{},[2627,200947,200948],{},"c. 下次适应 (Next-fit)","：每次从上一次分配结束的位置继续扫描。题目提示初始扫描从内存起始位置开始。",[2230,200951,200952,200969,200983],{},[108,200953,200954,200956,200957,200805,200959],{},[2627,200955,200798],{},"：从起始位置扫描。找到 ",[31,200958,200765],{},[105,200960,200961],{},[108,200962,200810,200963,200965,200966,200968],{},[2627,200964,200813],{},"。剩余空闲块为 ",[31,200967,200841],{},"。下次扫描的指针停留在地址 120 处。",[108,200970,200971,200973,200974,200823,200976],{},[2627,200972,200819],{},"：从地址 120 处继续扫描。发现紧接着的空闲块就是刚才剩下的 ",[31,200975,200841],{},[105,200977,200978],{},[108,200979,200828,200980,200982],{},[2627,200981,200850],{},"。该空闲块用完。下次扫描的指针移动到地址 140 处。",[108,200984,200985,200987,200988,200823,200990],{},[2627,200986,200837],{},"：从地址 140 处继续扫描。找到下一个空闲块 ",[31,200989,200768],{},[105,200991,200992],{},[108,200993,200847,200994,347],{},[2627,200995,200933],{},[105,200997,200998],{},[108,200999,201000,200858,201003,347],{},[2627,201001,201002],{},"答案 c",[2627,201004,201005],{},"80, 120, 160",[15,201007,201008,200867,201011,201014],{},[2627,201009,201010],{},"d. 最坏适应 (Worst-fit)",[2627,201012,201013],{},"尺寸最大","的空闲块。",[2230,201016,201017,201036,201055],{},[108,201018,201019,201021,201022,39984,201024],{},[2627,201020,200798],{},"：当前最大的空闲块是 ",[31,201023,200765],{},[105,201025,201026,201033],{},[108,201027,200810,201028,201030,201031,347],{},[2627,201029,200813],{},"。剩余空闲块变为 ",[31,201032,200841],{},[108,201034,201035],{},"更新后的空闲块大小为：20M, 20M, 10M, 40M, 30M, 40M。",[108,201037,201038,201040,201041,39984,201043],{},[2627,201039,200819],{},"：当前最大的空闲块是 40M（有两个，通常取第一个），即 ",[31,201042,200771],{},[105,201044,201045,201052],{},[108,201046,200828,201047,201030,201049,347],{},[2627,201048,200887],{},[31,201050,201051],{},"[20M@250]",[108,201053,201054],{},"更新后的空闲块大小为：20M, 20M, 10M, 20M, 30M, 40M。",[108,201056,201057,201021,201059,39984,201061],{},[2627,201058,200837],{},[31,201060,200777],{},[105,201062,201063],{},[108,201064,200847,201065,347],{},[2627,201066,201067],{},"起始地址为 360",[105,201069,201070],{},[108,201071,201072,200858,201075,347],{},[2627,201073,201074],{},"答案 d",[2627,201076,201077],{},"80, 230, 360",[32188,201079],{},[19,201081,201083],{"id":201082},"题目-3伙伴系统分配机制-buddy-system","题目 3：伙伴系统分配机制 (Buddy System)",[15,201085,201086,201088],{},[2627,201087,200435],{},"\n3. A 1-Mbyte block of memory is allocated using the buddy system.\na. Show the results of the following sequence in a figure similar to Figure 7.6: Request 70; Request 35; Request 80; Return A; Request 60; Return B; Return D; Return C.\nb. Show the binary tree representation following Return B.",[15,201090,201091,201093],{},[2627,201092,200445],{},"\n3. 使用伙伴系统分配一个 1-Mbyte 的内存块。\na. 用类似于图 7.6 的图形展示以下序列的结果：请求 70（设为 A）；请求 35（设为 B）；请求 80（设为 C）；返回 A；请求 60（设为 D）；返回 B；返回 D；返回 C。（单位：Kbyte）\nb. 展示在“返回 B”操作之后的二叉树表示。",[15,201095,201096,201098,201099,347],{},[2627,201097,200455],{},"\n伙伴系统是一种折中的内存分配方式。它的核心规则是：内存块的大小必须是 ",[34,201100,201102,201119],{"className":201101},[31642],[34,201103,201105],{"className":201104},[31646],[31648,201106,201107],{"xmlns":31650},[31652,201108,201109,201117],{},[31655,201110,201111],{},[37624,201112,201113,201115],{},[31749,201114,768],{},[31658,201116,33759],{},[31665,201118,63208],{"encoding":31667},[34,201120,201122],{"className":201121,"ariaHidden":758},[31672],[34,201123,201125,201128],{"className":201124},[31676],[34,201126],{"className":201127,"style":38486},[31680],[34,201129,201131,201134],{"className":201130},[31685],[34,201132,768],{"className":201133},[31685],[34,201135,201137],{"className":201136},[31792],[34,201138,201140],{"className":201139},[31796],[34,201141,201143],{"className":201142},[31801],[34,201144,201146],{"className":201145,"style":38486},[31805],[34,201147,201148,201151],{"style":37777},[34,201149],{"className":201150,"style":31814},[31813],[34,201152,201154],{"className":201153},[31818,31819,31820,31821],[34,201155,33759],{"className":201156,"style":33774},[31685,31686,31821],[105,201158,201159,201162,201165,201168],{},[108,201160,201161],{},"如果请求的内存大小大于块大小的一半，则分配整个块。",[108,201163,201164],{},"如果请求的内存大小小于或等于块大小的一半，则将该块对半平分为两个“伙伴 (Buddy)”。",[108,201166,201167],{},"当两个属于同一个父节点的空闲“伙伴”都被释放时，它们会立刻合并成一个更大的两倍大小的块。",[108,201169,201170,201173],{},[2627,201171,201172],{},"单位转换","：1 MByte = 1024 KBytes。我们将使用 KB 进行追踪。",[15,201175,201176],{},[2627,201177,200495],{},[15,201179,201180],{},"请求规格分析：",[105,201182,201183,201247,201311,201316],{},[108,201184,201185,201186,61605,201244,347],{},"请求 A (70K)：需要向上取整到最接近的 ",[34,201187,201189,201206],{"className":201188},[31642],[34,201190,201192],{"className":201191},[31646],[31648,201193,201194],{"xmlns":31650},[31652,201195,201196,201204],{},[31655,201197,201198],{},[37624,201199,201200,201202],{},[31749,201201,768],{},[31658,201203,33759],{},[31665,201205,63208],{"encoding":31667},[34,201207,201209],{"className":201208,"ariaHidden":758},[31672],[34,201210,201212,201215],{"className":201211},[31676],[34,201213],{"className":201214,"style":38486},[31680],[34,201216,201218,201221],{"className":201217},[31685],[34,201219,768],{"className":201220},[31685],[34,201222,201224],{"className":201223},[31792],[34,201225,201227],{"className":201226},[31796],[34,201228,201230],{"className":201229},[31801],[34,201231,201233],{"className":201232,"style":38486},[31805],[34,201234,201235,201238],{"style":37777},[34,201236],{"className":201237,"style":31814},[31813],[34,201239,201241],{"className":201240},[31818,31819,31820,31821],[34,201242,33759],{"className":201243,"style":33774},[31685,31686,31821],[2627,201245,201246],{},"128K",[108,201248,201249,201250,61605,201308,347],{},"请求 B (35K)：需要向上取整到最接近的 ",[34,201251,201253,201270],{"className":201252},[31642],[34,201254,201256],{"className":201255},[31646],[31648,201257,201258],{"xmlns":31650},[31652,201259,201260,201268],{},[31655,201261,201262],{},[37624,201263,201264,201266],{},[31749,201265,768],{},[31658,201267,33759],{},[31665,201269,63208],{"encoding":31667},[34,201271,201273],{"className":201272,"ariaHidden":758},[31672],[34,201274,201276,201279],{"className":201275},[31676],[34,201277],{"className":201278,"style":38486},[31680],[34,201280,201282,201285],{"className":201281},[31685],[34,201283,768],{"className":201284},[31685],[34,201286,201288],{"className":201287},[31792],[34,201289,201291],{"className":201290},[31796],[34,201292,201294],{"className":201293},[31801],[34,201295,201297],{"className":201296,"style":38486},[31805],[34,201298,201299,201302],{"style":37777},[34,201300],{"className":201301,"style":31814},[31813],[34,201303,201305],{"className":201304},[31818,31819,31820,31821],[34,201306,33759],{"className":201307,"style":33774},[31685,31686,31821],[2627,201309,201310],{},"64K",[108,201312,201313,201314,347],{},"请求 C (80K)：需要 ",[2627,201315,201246],{},[108,201317,201318,201319,347],{},"请求 D (60K)：需要 ",[2627,201320,201310],{},[15,201322,201323,201326],{},[2627,201324,201325],{},"a. 序列状态跟踪","\n我们将使用文本结构直观展示每次操作后内存块的分布（从左至右，总和为 1024K）。",[2230,201328,201329,201337,201346,201355,201364,201373,201382,201391,201400],{},[108,201330,201331,33183,201334],{},[2627,201332,201333],{},"初始状态",[31,201335,201336],{},"[ 空闲: 1024K ]",[108,201338,201339,201342,201343],{},[2627,201340,201341],{},"请求 A (70K，需 128K)","：\n系统将 1024 分裂为 512, 512；左 512 分裂为 256, 256；左 256 分裂为 128, 128。将 A 分配给第一个 128K。\n",[31,201344,201345],{},"[ A占: 128K ] | [ 空闲: 128K ] | [ 空闲: 256K ] | [ 空闲: 512K ]",[108,201347,201348,201351,201352],{},[2627,201349,201350],{},"请求 B (35K，需 64K)","：\n将左侧第一个空闲的 128K 分裂为 64, 64。B 分配给第一个 64K。\n",[31,201353,201354],{},"[ A占: 128K ] | [ B占: 64K ] | [ 空闲: 64K ] | [ 空闲: 256K ] | [ 空闲: 512K ]",[108,201356,201357,201360,201361],{},[2627,201358,201359],{},"请求 C (80K，需 128K)","：\n没有现成的 128K 空闲块，所以将 256K 块分裂为 128, 128。C 分配给第一个 128K。\n",[31,201362,201363],{},"[ A占: 128K ] | [ B占: 64K ] | [ 空闲: 64K ] | [ C占: 128K ] | [ 空闲: 128K ] | [ 空闲: 512K ]",[108,201365,201366,201369,201370],{},[2627,201367,201368],{},"返回 A","：\nA 释放，变为 128K 空闲。由于其右侧紧邻的是被分割的两个 64K（其中一个被 B 占用），无法合并。\n",[31,201371,201372],{},"[ 空闲: 128K ] | [ B占: 64K ] | [ 空闲: 64K ] | [ C占: 128K ] | [ 空闲: 128K ] | [ 空闲: 512K ]",[108,201374,201375,201378,201379],{},[2627,201376,201377],{},"请求 D (60K，需 64K)","：\n恰好有一个 64K 的空闲块（B 的右侧），将其分配给 D。\n",[31,201380,201381],{},"[ 空闲: 128K ] | [ B占: 64K ] | [ D占: 64K ] | [ C占: 128K ] | [ 空闲: 128K ] | [ 空闲: 512K ]",[108,201383,201384,201387,201388],{},[2627,201385,201386],{},"返回 B","：\nB 释放，变为 64K 空闲。此时其伙伴（D 占据的 64K）并未空闲，因此不能合并。\n",[31,201389,201390],{},"[ 空闲: 128K ] | [ 空闲: 64K ] | [ D占: 64K ] | [ C占: 128K ] | [ 空闲: 128K ] | [ 空闲: 512K ]",[108,201392,201393,201396,201397],{},[2627,201394,201395],{},"返回 D","：\nD 释放，变为 64K 空闲。此时它的伙伴（原 B 的 64K）也是空闲的，两者合并为 128K。\n合并后的 128K 的伙伴是最左侧的空闲 128K，两者再次合并为 256K。\n",[31,201398,201399],{},"[ 空闲: 256K ] | [ C占: 128K ] | [ 空闲: 128K ] | [ 空闲: 512K ]",[108,201401,201402,201405,201406],{},[2627,201403,201404],{},"返回 C","：\nC 释放，变为 128K 空闲。它与右侧的空闲 128K 合并为 256K。\n该 256K 与最左侧的空闲 256K 合并为 512K。\n该 512K 与最右侧的空闲 512K 合并，恢复至整块。\n",[31,201407,201336],{},[15,201409,201410,201413],{},[2627,201411,201412],{},"b. “返回 B” 之后的二叉树表示","\n在步骤 7（返回 B）执行完毕后，内存的切分状态在二叉树结构中表现如下：\n（从根节点 1024K 向下分支）",[24,201415,201418],{"className":201416,"code":201417,"language":2638,"meta":29},[2636]," [1024K]\n \u002F \\\n [512K] [512K: 空闲]\n \u002F \\\n [256K] [256K]\n \u002F \\ \u002F \\\n[128K:空闲] \\ [C占:128K] [128K:空闲]\n \u002F \\\n [64K:空闲(原B)] [D占:64K]\n\n",[31,201419,201417],{"__ignoreMap":29},[19,201421,201423],{"id":201422},"题目-4分页系统地址计算","题目 4：分页系统地址计算",[15,201425,201426,201428,201429,201491,201492,201554,201555,201617],{},[2627,201427,200435],{},"\n4. Consider a simple paging system with the following parameters: ",[34,201430,201432,201450],{"className":201431},[31642],[34,201433,201435],{"className":201434},[31646],[31648,201436,201437],{"xmlns":31650},[31652,201438,201439,201447],{},[31655,201440,201441],{},[37624,201442,201443,201445],{},[31749,201444,768],{},[31749,201446,173486],{},[31665,201448,201449],{"encoding":31667},"2^{32}",[34,201451,201453],{"className":201452,"ariaHidden":758},[31672],[34,201454,201456,201459],{"className":201455},[31676],[34,201457],{"className":201458,"style":37774},[31680],[34,201460,201462,201465],{"className":201461},[31685],[34,201463,768],{"className":201464},[31685],[34,201466,201468],{"className":201467},[31792],[34,201469,201471],{"className":201470},[31796],[34,201472,201474],{"className":201473},[31801],[34,201475,201477],{"className":201476,"style":37774},[31805],[34,201478,201479,201482],{"style":37777},[34,201480],{"className":201481,"style":31814},[31813],[34,201483,201485],{"className":201484},[31818,31819,31820,31821],[34,201486,201488],{"className":201487},[31685,31821],[34,201489,173486],{"className":201490},[31685,31821]," bytes of physical memory; page size of ",[34,201493,201495,201513],{"className":201494},[31642],[34,201496,201498],{"className":201497},[31646],[31648,201499,201500],{"xmlns":31650},[31652,201501,201502,201510],{},[31655,201503,201504],{},[37624,201505,201506,201508],{},[31749,201507,768],{},[31749,201509,3244],{},[31665,201511,201512],{"encoding":31667},"2^{10}",[34,201514,201516],{"className":201515,"ariaHidden":758},[31672],[34,201517,201519,201522],{"className":201518},[31676],[34,201520],{"className":201521,"style":37774},[31680],[34,201523,201525,201528],{"className":201524},[31685],[34,201526,768],{"className":201527},[31685],[34,201529,201531],{"className":201530},[31792],[34,201532,201534],{"className":201533},[31796],[34,201535,201537],{"className":201536},[31801],[34,201538,201540],{"className":201539,"style":37774},[31805],[34,201541,201542,201545],{"style":37777},[34,201543],{"className":201544,"style":31814},[31813],[34,201546,201548],{"className":201547},[31818,31819,31820,31821],[34,201549,201551],{"className":201550},[31685,31821],[34,201552,3244],{"className":201553},[31685,31821]," bytes; ",[34,201556,201558,201576],{"className":201557},[31642],[34,201559,201561],{"className":201560},[31646],[31648,201562,201563],{"xmlns":31650},[31652,201564,201565,201573],{},[31655,201566,201567],{},[37624,201568,201569,201571],{},[31749,201570,768],{},[31749,201572,177869],{},[31665,201574,201575],{"encoding":31667},"2^{16}",[34,201577,201579],{"className":201578,"ariaHidden":758},[31672],[34,201580,201582,201585],{"className":201581},[31676],[34,201583],{"className":201584,"style":37774},[31680],[34,201586,201588,201591],{"className":201587},[31685],[34,201589,768],{"className":201590},[31685],[34,201592,201594],{"className":201593},[31792],[34,201595,201597],{"className":201596},[31796],[34,201598,201600],{"className":201599},[31801],[34,201601,201603],{"className":201602,"style":37774},[31805],[34,201604,201605,201608],{"style":37777},[34,201606],{"className":201607,"style":31814},[31813],[34,201609,201611],{"className":201610},[31818,31819,31820,31821],[34,201612,201614],{"className":201613},[31685,31821],[34,201615,177869],{"className":201616},[31685,31821]," pages of logical address space.\na. How many bits are in a logical address?\nb. How many bytes in a frame?\nc. How many bits in the physical address specify the frame?\nd. How many entries in the page table?\ne. How many bits in each page table entry? Assume each page table entry contains a valid\u002Finvalid bit.",[15,201619,201620,201622,201623,201684,201685,201746,201747,201808],{},[2627,201621,200445],{},"\n4. 考虑一个具有以下参数的简单分页系统：物理内存为 ",[34,201624,201626,201643],{"className":201625},[31642],[34,201627,201629],{"className":201628},[31646],[31648,201630,201631],{"xmlns":31650},[31652,201632,201633,201641],{},[31655,201634,201635],{},[37624,201636,201637,201639],{},[31749,201638,768],{},[31749,201640,173486],{},[31665,201642,201449],{"encoding":31667},[34,201644,201646],{"className":201645,"ariaHidden":758},[31672],[34,201647,201649,201652],{"className":201648},[31676],[34,201650],{"className":201651,"style":37774},[31680],[34,201653,201655,201658],{"className":201654},[31685],[34,201656,768],{"className":201657},[31685],[34,201659,201661],{"className":201660},[31792],[34,201662,201664],{"className":201663},[31796],[34,201665,201667],{"className":201666},[31801],[34,201668,201670],{"className":201669,"style":37774},[31805],[34,201671,201672,201675],{"style":37777},[34,201673],{"className":201674,"style":31814},[31813],[34,201676,201678],{"className":201677},[31818,31819,31820,31821],[34,201679,201681],{"className":201680},[31685,31821],[34,201682,173486],{"className":201683},[31685,31821]," 字节；页大小为 ",[34,201686,201688,201705],{"className":201687},[31642],[34,201689,201691],{"className":201690},[31646],[31648,201692,201693],{"xmlns":31650},[31652,201694,201695,201703],{},[31655,201696,201697],{},[37624,201698,201699,201701],{},[31749,201700,768],{},[31749,201702,3244],{},[31665,201704,201512],{"encoding":31667},[34,201706,201708],{"className":201707,"ariaHidden":758},[31672],[34,201709,201711,201714],{"className":201710},[31676],[34,201712],{"className":201713,"style":37774},[31680],[34,201715,201717,201720],{"className":201716},[31685],[34,201718,768],{"className":201719},[31685],[34,201721,201723],{"className":201722},[31792],[34,201724,201726],{"className":201725},[31796],[34,201727,201729],{"className":201728},[31801],[34,201730,201732],{"className":201731,"style":37774},[31805],[34,201733,201734,201737],{"style":37777},[34,201735],{"className":201736,"style":31814},[31813],[34,201738,201740],{"className":201739},[31818,31819,31820,31821],[34,201741,201743],{"className":201742},[31685,31821],[34,201744,3244],{"className":201745},[31685,31821]," 字节；逻辑地址空间有 ",[34,201748,201750,201767],{"className":201749},[31642],[34,201751,201753],{"className":201752},[31646],[31648,201754,201755],{"xmlns":31650},[31652,201756,201757,201765],{},[31655,201758,201759],{},[37624,201760,201761,201763],{},[31749,201762,768],{},[31749,201764,177869],{},[31665,201766,201575],{"encoding":31667},[34,201768,201770],{"className":201769,"ariaHidden":758},[31672],[34,201771,201773,201776],{"className":201772},[31676],[34,201774],{"className":201775,"style":37774},[31680],[34,201777,201779,201782],{"className":201778},[31685],[34,201780,768],{"className":201781},[31685],[34,201783,201785],{"className":201784},[31792],[34,201786,201788],{"className":201787},[31796],[34,201789,201791],{"className":201790},[31801],[34,201792,201794],{"className":201793,"style":37774},[31805],[34,201795,201796,201799],{"style":37777},[34,201797],{"className":201798,"style":31814},[31813],[34,201800,201802],{"className":201801},[31818,31819,31820,31821],[34,201803,201805],{"className":201804},[31685,31821],[34,201806,177869],{"className":201807},[31685,31821]," 页。\na. 逻辑地址中有多少位（bits）？\nb. 一个页框（frame）中有多少字节？\nc. 物理地址中有多少位用于指定页框？\nd. 页表中有多少个条目（entries）？\ne. 每个页表项中有多少位？假设每个页表项包含一个有效\u002F无效位（valid\u002Finvalid bit）。",[15,201810,201811],{},[2627,201812,200455],{},[105,201814,201815,201821,201827],{},[108,201816,201817,201820],{},[2627,201818,201819],{},"页与页框等大","：物理内存被划分为物理块（Frame，页框），逻辑内存（进程虚拟空间）被划分为逻辑页（Page）。两者的容量大小永远是严格相等的，这样才能实现无缝映射。",[108,201822,201823,201826],{},[2627,201824,201825],{},"逻辑地址结构","：逻辑地址由两部分组成：【页号位数】+【页内偏移量位数】。逻辑地址的总长度决定了进程的最大寻址空间。",[108,201828,201829,201832],{},[2627,201830,201831],{},"物理地址结构","：物理地址由两部分组成：【页框号位数】+【页内偏移量位数】。物理地址的总长度决定了系统能支持的最大物理内存。",[15,201834,201835],{},[2627,201836,200495],{},[15,201838,201839],{},[2627,201840,201841],{},"a. 逻辑地址中有多少位？",[105,201843,201844,202106],{},[108,201845,201846,201848,201849,201910,201911,201939,201940,202001,202002,202030,202031,202105],{},[2627,201847,33190],{},"：逻辑空间由两部分计算得出：页面总数和每页大小。\n已知页面总数为 ",[34,201850,201852,201869],{"className":201851},[31642],[34,201853,201855],{"className":201854},[31646],[31648,201856,201857],{"xmlns":31650},[31652,201858,201859,201867],{},[31655,201860,201861],{},[37624,201862,201863,201865],{},[31749,201864,768],{},[31749,201866,177869],{},[31665,201868,201575],{"encoding":31667},[34,201870,201872],{"className":201871,"ariaHidden":758},[31672],[34,201873,201875,201878],{"className":201874},[31676],[34,201876],{"className":201877,"style":37774},[31680],[34,201879,201881,201884],{"className":201880},[31685],[34,201882,768],{"className":201883},[31685],[34,201885,201887],{"className":201886},[31792],[34,201888,201890],{"className":201889},[31796],[34,201891,201893],{"className":201892},[31801],[34,201894,201896],{"className":201895,"style":37774},[31805],[34,201897,201898,201901],{"style":37777},[34,201899],{"className":201900,"style":31814},[31813],[34,201902,201904],{"className":201903},[31818,31819,31820,31821],[34,201905,201907],{"className":201906},[31685,31821],[34,201908,177869],{"className":201909},[31685,31821]," 页 ",[34,201912,201914,201927],{"className":201913},[31642],[34,201915,201917],{"className":201916},[31646],[31648,201918,201919],{"xmlns":31650},[31652,201920,201921,201925],{},[31655,201922,201923],{},[31662,201924,33239],{},[31665,201926,119689],{"encoding":31667},[34,201928,201930],{"className":201929,"ariaHidden":758},[31672],[34,201931,201933,201936],{"className":201932},[31676],[34,201934],{"className":201935,"style":73089},[31680],[34,201937,33239],{"className":201938},[31850]," 需要 16 位来表示逻辑页号。\n已知页大小为 ",[34,201941,201943,201960],{"className":201942},[31642],[34,201944,201946],{"className":201945},[31646],[31648,201947,201948],{"xmlns":31650},[31652,201949,201950,201958],{},[31655,201951,201952],{},[37624,201953,201954,201956],{},[31749,201955,768],{},[31749,201957,3244],{},[31665,201959,201512],{"encoding":31667},[34,201961,201963],{"className":201962,"ariaHidden":758},[31672],[34,201964,201966,201969],{"className":201965},[31676],[34,201967],{"className":201968,"style":37774},[31680],[34,201970,201972,201975],{"className":201971},[31685],[34,201973,768],{"className":201974},[31685],[34,201976,201978],{"className":201977},[31792],[34,201979,201981],{"className":201980},[31796],[34,201982,201984],{"className":201983},[31801],[34,201985,201987],{"className":201986,"style":37774},[31805],[34,201988,201989,201992],{"style":37777},[34,201990],{"className":201991,"style":31814},[31813],[34,201993,201995],{"className":201994},[31818,31819,31820,31821],[34,201996,201998],{"className":201997},[31685,31821],[34,201999,3244],{"className":202000},[31685,31821]," 字节 ",[34,202003,202005,202018],{"className":202004},[31642],[34,202006,202008],{"className":202007},[31646],[31648,202009,202010],{"xmlns":31650},[31652,202011,202012,202016],{},[31655,202013,202014],{},[31662,202015,33239],{},[31665,202017,119689],{"encoding":31667},[34,202019,202021],{"className":202020,"ariaHidden":758},[31672],[34,202022,202024,202027],{"className":202023},[31676],[34,202025],{"className":202026,"style":73089},[31680],[34,202028,33239],{"className":202029},[31850]," 需要 10 位来表示页内偏移量。\n逻辑地址总位数 = 页号位数 + 偏移量位数 = ",[34,202032,202034,202057],{"className":202033},[31642],[34,202035,202037],{"className":202036},[31646],[31648,202038,202039],{"xmlns":31650},[31652,202040,202041,202054],{},[31655,202042,202043,202045,202047,202049,202051],{},[31749,202044,177869],{},[31662,202046,223],{},[31749,202048,3244],{},[31662,202050,202],{},[31749,202052,202053],{},"26",[31665,202055,202056],{"encoding":31667},"16 + 10 = 26",[34,202058,202060,202078,202096],{"className":202059,"ariaHidden":758},[31672],[34,202061,202063,202066,202069,202072,202075],{"className":202062},[31676],[34,202064],{"className":202065,"style":35942},[31680],[34,202067,177869],{"className":202068},[31685],[34,202070],{"className":202071,"style":32604},[31845],[34,202073,223],{"className":202074},[32024],[34,202076],{"className":202077,"style":32604},[31845],[34,202079,202081,202084,202087,202090,202093],{"className":202080},[31676],[34,202082],{"className":202083,"style":32617},[31680],[34,202085,3244],{"className":202086},[31685],[34,202088],{"className":202089,"style":31846},[31845],[34,202091,202],{"className":202092},[31850],[34,202094],{"className":202095,"style":31846},[31845],[34,202097,202099,202102],{"className":202098},[31676],[34,202100],{"className":202101,"style":32617},[31680],[34,202103,202053],{"className":202104},[31685]," 位。",[108,202107,202108,2664,202110,347],{},[2627,202109,200857],{},[2627,202111,202112],{},"26 bits",[15,202114,202115],{},[2627,202116,202117],{},"b. 一个页框中有多少字节？",[105,202119,202120,202186],{},[108,202121,202122,202124,202125,182446],{},[2627,202123,33190],{},"：分页系统的核心规则是：页框（物理内存划分的块）的大小必须严格等于页（逻辑内存划分的块）的大小。\n题目给出页大小为 ",[34,202126,202128,202145],{"className":202127},[31642],[34,202129,202131],{"className":202130},[31646],[31648,202132,202133],{"xmlns":31650},[31652,202134,202135,202143],{},[31655,202136,202137],{},[37624,202138,202139,202141],{},[31749,202140,768],{},[31749,202142,3244],{},[31665,202144,201512],{"encoding":31667},[34,202146,202148],{"className":202147,"ariaHidden":758},[31672],[34,202149,202151,202154],{"className":202150},[31676],[34,202152],{"className":202153,"style":37774},[31680],[34,202155,202157,202160],{"className":202156},[31685],[34,202158,768],{"className":202159},[31685],[34,202161,202163],{"className":202162},[31792],[34,202164,202166],{"className":202165},[31796],[34,202167,202169],{"className":202168},[31801],[34,202170,202172],{"className":202171,"style":37774},[31805],[34,202173,202174,202177],{"style":37777},[34,202175],{"className":202176,"style":31814},[31813],[34,202178,202180],{"className":202179},[31818,31819,31820,31821],[34,202181,202183],{"className":202182},[31685,31821],[34,202184,3244],{"className":202185},[31685,31821],[108,202187,202188,2664,202190,347],{},[2627,202189,200940],{},[2627,202191,202192,202253],{},[34,202193,202195,202212],{"className":202194},[31642],[34,202196,202198],{"className":202197},[31646],[31648,202199,202200],{"xmlns":31650},[31652,202201,202202,202210],{},[31655,202203,202204],{},[37624,202205,202206,202208],{},[31749,202207,768],{},[31749,202209,3244],{},[31665,202211,201512],{"encoding":31667},[34,202213,202215],{"className":202214,"ariaHidden":758},[31672],[34,202216,202218,202221],{"className":202217},[31676],[34,202219],{"className":202220,"style":37774},[31680],[34,202222,202224,202227],{"className":202223},[31685],[34,202225,768],{"className":202226},[31685],[34,202228,202230],{"className":202229},[31792],[34,202231,202233],{"className":202232},[31796],[34,202234,202236],{"className":202235},[31801],[34,202237,202239],{"className":202238,"style":37774},[31805],[34,202240,202241,202244],{"style":37777},[34,202242],{"className":202243,"style":31814},[31813],[34,202245,202247],{"className":202246},[31818,31819,31820,31821],[34,202248,202250],{"className":202249},[31685,31821],[34,202251,3244],{"className":202252},[31685,31821]," 字节 (或 1024 字节 \u002F 1 KB)",[15,202255,202256],{},[2627,202257,202258],{},"c. 物理地址中有多少位用于指定页框？",[105,202260,202261,202552],{},[108,202262,202263,202265,202266,202327,202328,202488,202489,202551],{},[2627,202264,33190],{},"：物理内存总大小为 ",[34,202267,202269,202286],{"className":202268},[31642],[34,202270,202272],{"className":202271},[31646],[31648,202273,202274],{"xmlns":31650},[31652,202275,202276,202284],{},[31655,202277,202278],{},[37624,202279,202280,202282],{},[31749,202281,768],{},[31749,202283,173486],{},[31665,202285,201449],{"encoding":31667},[34,202287,202289],{"className":202288,"ariaHidden":758},[31672],[34,202290,202292,202295],{"className":202291},[31676],[34,202293],{"className":202294,"style":37774},[31680],[34,202296,202298,202301],{"className":202297},[31685],[34,202299,768],{"className":202300},[31685],[34,202302,202304],{"className":202303},[31792],[34,202305,202307],{"className":202306},[31796],[34,202308,202310],{"className":202309},[31801],[34,202311,202313],{"className":202312,"style":37774},[31805],[34,202314,202315,202318],{"style":37777},[34,202316],{"className":202317,"style":31814},[31813],[34,202319,202321],{"className":202320},[31818,31819,31820,31821],[34,202322,202324],{"className":202323},[31685,31821],[34,202325,173486],{"className":202326},[31685,31821]," 字节。\n物理内存中的页框总数 = 物理内存总大小 \u002F 页框大小 = ",[34,202329,202331,202365],{"className":202330},[31642],[34,202332,202334],{"className":202333},[31646],[31648,202335,202336],{"xmlns":31650},[31652,202337,202338,202362],{},[31655,202339,202340,202346,202348,202354,202356],{},[37624,202341,202342,202344],{},[31749,202343,768],{},[31749,202345,173486],{},[31658,202347,38080],{"mathvariant":38079},[37624,202349,202350,202352],{},[31749,202351,768],{},[31749,202353,3244],{},[31662,202355,202],{},[37624,202357,202358,202360],{},[31749,202359,768],{},[31749,202361,94716],{},[31665,202363,202364],{"encoding":31667},"2^{32} \u002F 2^{10} = 2^{22}",[34,202366,202368,202450],{"className":202367,"ariaHidden":758},[31672],[34,202369,202371,202374,202406,202409,202441,202444,202447],{"className":202370},[31676],[34,202372],{"className":202373,"style":37755},[31680],[34,202375,202377,202380],{"className":202376},[31685],[34,202378,768],{"className":202379},[31685],[34,202381,202383],{"className":202382},[31792],[34,202384,202386],{"className":202385},[31796],[34,202387,202389],{"className":202388},[31801],[34,202390,202392],{"className":202391,"style":37774},[31805],[34,202393,202394,202397],{"style":37777},[34,202395],{"className":202396,"style":31814},[31813],[34,202398,202400],{"className":202399},[31818,31819,31820,31821],[34,202401,202403],{"className":202402},[31685,31821],[34,202404,173486],{"className":202405},[31685,31821],[34,202407,38080],{"className":202408},[31685],[34,202410,202412,202415],{"className":202411},[31685],[34,202413,768],{"className":202414},[31685],[34,202416,202418],{"className":202417},[31792],[34,202419,202421],{"className":202420},[31796],[34,202422,202424],{"className":202423},[31801],[34,202425,202427],{"className":202426,"style":37774},[31805],[34,202428,202429,202432],{"style":37777},[34,202430],{"className":202431,"style":31814},[31813],[34,202433,202435],{"className":202434},[31818,31819,31820,31821],[34,202436,202438],{"className":202437},[31685,31821],[34,202439,3244],{"className":202440},[31685,31821],[34,202442],{"className":202443,"style":31846},[31845],[34,202445,202],{"className":202446},[31850],[34,202448],{"className":202449,"style":31846},[31845],[34,202451,202453,202456],{"className":202452},[31676],[34,202454],{"className":202455,"style":37774},[31680],[34,202457,202459,202462],{"className":202458},[31685],[34,202460,768],{"className":202461},[31685],[34,202463,202465],{"className":202464},[31792],[34,202466,202468],{"className":202467},[31796],[34,202469,202471],{"className":202470},[31801],[34,202472,202474],{"className":202473,"style":37774},[31805],[34,202475,202476,202479],{"style":37777},[34,202477],{"className":202478,"style":31814},[31813],[34,202480,202482],{"className":202481},[31818,31819,31820,31821],[34,202483,202485],{"className":202484},[31685,31821],[34,202486,94716],{"className":202487},[31685,31821]," 个页框。\n为了对这 ",[34,202490,202492,202510],{"className":202491},[31642],[34,202493,202495],{"className":202494},[31646],[31648,202496,202497],{"xmlns":31650},[31652,202498,202499,202507],{},[31655,202500,202501],{},[37624,202502,202503,202505],{},[31749,202504,768],{},[31749,202506,94716],{},[31665,202508,202509],{"encoding":31667},"2^{22}",[34,202511,202513],{"className":202512,"ariaHidden":758},[31672],[34,202514,202516,202519],{"className":202515},[31676],[34,202517],{"className":202518,"style":37774},[31680],[34,202520,202522,202525],{"className":202521},[31685],[34,202523,768],{"className":202524},[31685],[34,202526,202528],{"className":202527},[31792],[34,202529,202531],{"className":202530},[31796],[34,202532,202534],{"className":202533},[31801],[34,202535,202537],{"className":202536,"style":37774},[31805],[34,202538,202539,202542],{"style":37777},[34,202540],{"className":202541,"style":31814},[31813],[34,202543,202545],{"className":202544},[31818,31819,31820,31821],[34,202546,202548],{"className":202547},[31685,31821],[34,202549,94716],{"className":202550},[31685,31821]," 个单独的物理页框进行编号和寻址，我们需要 22 个二进制位。",[108,202553,202554,2664,202556,347],{},[2627,202555,201002],{},[2627,202557,202558],{},"22 bits",[15,202560,202561],{},[2627,202562,202563],{},"d. 页表中有多少个条目？",[105,202565,202566,202633],{},[108,202567,202568,202570,202571,202632],{},[2627,202569,33190],{},"：页表的作用是记录逻辑页到物理页框的映射关系。进程有多少个逻辑页，它的页表就必须提供多少个映射条目。\n题目给出逻辑地址空间有 ",[34,202572,202574,202591],{"className":202573},[31642],[34,202575,202577],{"className":202576},[31646],[31648,202578,202579],{"xmlns":31650},[31652,202580,202581,202589],{},[31655,202582,202583],{},[37624,202584,202585,202587],{},[31749,202586,768],{},[31749,202588,177869],{},[31665,202590,201575],{"encoding":31667},[34,202592,202594],{"className":202593,"ariaHidden":758},[31672],[34,202595,202597,202600],{"className":202596},[31676],[34,202598],{"className":202599,"style":37774},[31680],[34,202601,202603,202606],{"className":202602},[31685],[34,202604,768],{"className":202605},[31685],[34,202607,202609],{"className":202608},[31792],[34,202610,202612],{"className":202611},[31796],[34,202613,202615],{"className":202614},[31801],[34,202616,202618],{"className":202617,"style":37774},[31805],[34,202619,202620,202623],{"style":37777},[34,202621],{"className":202622,"style":31814},[31813],[34,202624,202626],{"className":202625},[31818,31819,31820,31821],[34,202627,202629],{"className":202628},[31685,31821],[34,202630,177869],{"className":202631},[31685,31821]," 页。",[108,202634,202635,2664,202637,347],{},[2627,202636,201074],{},[2627,202638,202639,202700],{},[34,202640,202642,202659],{"className":202641},[31642],[34,202643,202645],{"className":202644},[31646],[31648,202646,202647],{"xmlns":31650},[31652,202648,202649,202657],{},[31655,202650,202651],{},[37624,202652,202653,202655],{},[31749,202654,768],{},[31749,202656,177869],{},[31665,202658,201575],{"encoding":31667},[34,202660,202662],{"className":202661,"ariaHidden":758},[31672],[34,202663,202665,202668],{"className":202664},[31676],[34,202666],{"className":202667,"style":37774},[31680],[34,202669,202671,202674],{"className":202670},[31685],[34,202672,768],{"className":202673},[31685],[34,202675,202677],{"className":202676},[31792],[34,202678,202680],{"className":202679},[31796],[34,202681,202683],{"className":202682},[31801],[34,202684,202686],{"className":202685,"style":37774},[31805],[34,202687,202688,202691],{"style":37777},[34,202689],{"className":202690,"style":31814},[31813],[34,202692,202694],{"className":202693},[31818,31819,31820,31821],[34,202695,202697],{"className":202696},[31685,31821],[34,202698,177869],{"className":202699},[31685,31821]," 个条目 (即 65,536 个条目)",[15,202702,202703],{},[2627,202704,202705],{},"e. 每个页表项中有多少位？",[105,202707,202708,202794],{},[108,202709,202710,202712,202713,202716,202717,202790,202791],{},[2627,202711,33190],{},"：页表项 (PTE, Page Table Entry) 的最基本功能是存储",[2627,202714,202715],{},"映射到的物理页框号","。\n由 c 小题的计算结果可知，物理页框号需要 22 位来存储。\n题目补充要求包含 1 位的有效\u002F无效位（Valid bit，用于判断页面当前是否已加载到真实的物理内存中）。\n每个页表项总位数 = 页框号位数 + 有效位 = ",[34,202718,202720,202742],{"className":202719},[31642],[34,202721,202723],{"className":202722},[31646],[31648,202724,202725],{"xmlns":31650},[31652,202726,202727,202739],{},[31655,202728,202729,202731,202733,202735,202737],{},[31749,202730,94716],{},[31662,202732,223],{},[31749,202734,640],{},[31662,202736,202],{},[31749,202738,94703],{},[31665,202740,202741],{"encoding":31667},"22 + 1 = 23",[34,202743,202745,202763,202781],{"className":202744,"ariaHidden":758},[31672],[34,202746,202748,202751,202754,202757,202760],{"className":202747},[31676],[34,202749],{"className":202750,"style":35942},[31680],[34,202752,94716],{"className":202753},[31685],[34,202755],{"className":202756,"style":32604},[31845],[34,202758,223],{"className":202759},[32024],[34,202761],{"className":202762,"style":32604},[31845],[34,202764,202766,202769,202772,202775,202778],{"className":202765},[31676],[34,202767],{"className":202768,"style":32617},[31680],[34,202770,640],{"className":202771},[31685],[34,202773],{"className":202774,"style":31846},[31845],[34,202776,202],{"className":202777},[31850],[34,202779],{"className":202780,"style":31846},[31845],[34,202782,202784,202787],{"className":202783},[31676],[34,202785],{"className":202786,"style":32617},[31680],[34,202788,94703],{"className":202789},[31685]," 位。\n",[71807,202792,202793],{},"(注：在真实的工业硬件实现中，位数通常会被向上对齐到字节的整数倍，例如 24 位即 3 字节，或者直接分配 32 位以保留用于保护权限、脏位等其他控制位。但从本题的纯理论计算角度，最严谨的答案就是各必须位数的精确总和。)",[108,202795,202796,2664,202799,347],{},[2627,202797,202798],{},"答案 e",[2627,202800,202801],{},"23 bits",[32188,202803],{},[19,202805,202807],{"id":202806},"题目-5分段系统地址转换","题目 5：分段系统地址转换",[15,202809,202810,202812],{},[2627,202811,200435],{},"\n5. Consider a simple segmentation system that has the following segment table:",[13837,202814,202815,202825],{},[13840,202816,202817],{},[13843,202818,202819,202822],{},[13846,202820,202821],{"align":82355},"Starting Address",[13846,202823,202824],{"align":82355},"Length (bytes)",[13853,202826,202827,202834,202842,202850],{},[13843,202828,202829,202832],{},[13858,202830,202831],{"align":82355},"660",[13858,202833,184039],{"align":82355},[13843,202835,202836,202839],{},[13858,202837,202838],{"align":82355},"1752",[13858,202840,202841],{"align":82355},"422",[13843,202843,202844,202847],{},[13858,202845,202846],{"align":82355},"222",[13858,202848,202849],{"align":82355},"198",[13843,202851,202852,202855],{},[13858,202853,202854],{"align":82355},"996",[13858,202856,202857],{"align":82355},"604",[15,202859,202860],{},"For each of the following logical addresses, determine the physical address or indicate if a segment fault occurs:\na. 0, 198 b. 2, 156 c. 1, 530 d. 3, 444 e. 0, 222",[15,202862,202863,202865],{},[2627,202864,200445],{},"\n5. 考虑一个具有以下段表的简单分段系统：",[13837,202867,202868,202878],{},[13840,202869,202870],{},[13843,202871,202872,202875],{},[13846,202873,202874],{"align":82355},"起始物理地址 (Base)",[13846,202876,202877],{"align":82355},"长度\u002F限长 (Limit)",[13853,202879,202880,202886,202892,202898],{},[13843,202881,202882,202884],{},[13858,202883,202831],{"align":82355},[13858,202885,184039],{"align":82355},[13843,202887,202888,202890],{},[13858,202889,202838],{"align":82355},[13858,202891,202841],{"align":82355},[13843,202893,202894,202896],{},[13858,202895,202846],{"align":82355},[13858,202897,202849],{"align":82355},[13843,202899,202900,202902],{},[13858,202901,202854],{"align":82355},[13858,202903,202857],{"align":82355},[15,202905,202906],{},"对于以下每个逻辑地址（以“段号, 偏移量”格式给出），确定其对应的物理地址，或指出是否会发生段错误 (segment fault)：\na. 0, 198 b. 2, 156 c. 1, 530 d. 3, 444 e. 0, 222",[15,202908,202909,202911,202912,202972],{},[2627,202910,200455],{},"\n在分段系统中，逻辑地址通常表示为二元组：",[2627,202913,202914,202915,202943,202944,2557],{},"(段号 ",[34,202916,202918,202931],{"className":202917},[31642],[34,202919,202921],{"className":202920},[31646],[31648,202922,202923],{"xmlns":31650},[31652,202924,202925,202929],{},[31655,202926,202927],{},[31658,202928,31747],{},[31665,202930,31747],{"encoding":31667},[34,202932,202934],{"className":202933,"ariaHidden":758},[31672],[34,202935,202937,202940],{"className":202936},[31676],[34,202938],{"className":202939,"style":32527},[31680],[34,202941,31747],{"className":202942},[31685,31686],", 段内偏移量 ",[34,202945,202947,202960],{"className":202946},[31642],[34,202948,202950],{"className":202949},[31646],[31648,202951,202952],{"xmlns":31650},[31652,202953,202954,202958],{},[31655,202955,202956],{},[31658,202957,46900],{},[31665,202959,46900],{"encoding":31667},[34,202961,202963],{"className":202962,"ariaHidden":758},[31672],[34,202964,202966,202969],{"className":202965},[31676],[34,202967],{"className":202968,"style":31681},[31680],[34,202970,46900],{"className":202971},[31685,31686],"。\n将逻辑地址转换为物理地址的标准硬件步骤如下：",[2230,202974,202975,203020,203114],{},[108,202976,202977,202980,202981,203009,203010,203013,203014,39984,203017],{},[2627,202978,202979],{},"查表","：根据逻辑地址中的段号 ",[34,202982,202984,202997],{"className":202983},[31642],[34,202985,202987],{"className":202986},[31646],[31648,202988,202989],{"xmlns":31650},[31652,202990,202991,202995],{},[31655,202992,202993],{},[31658,202994,31747],{},[31665,202996,31747],{"encoding":31667},[34,202998,203000],{"className":202999,"ariaHidden":758},[31672],[34,203001,203003,203006],{"className":203002},[31676],[34,203004],{"className":203005,"style":32527},[31680],[34,203007,31747],{"className":203008},[31685,31686],"，在段表中找到对应行，提取出该段在物理内存中的",[2627,203011,203012],{},"起始基址 (Base)"," 和该段的最大允许",[2627,203015,203016],{},"长度 (Limit)",[71807,203018,203019],{},"(注：段表通常从 0 开始索引。题目表格从上到下的四行分别隐式对应段号 0, 1, 2, 3)",[108,203021,203022,203025,203026,203054,203055,203110,203111,347],{},[2627,203023,203024],{},"越界检查","：将给定的段内偏移量 ",[34,203027,203029,203042],{"className":203028},[31642],[34,203030,203032],{"className":203031},[31646],[31648,203033,203034],{"xmlns":31650},[31652,203035,203036,203040],{},[31655,203037,203038],{},[31658,203039,46900],{},[31665,203041,46900],{"encoding":31667},[34,203043,203045],{"className":203044,"ariaHidden":758},[31672],[34,203046,203048,203051],{"className":203047},[31676],[34,203049],{"className":203050,"style":31681},[31680],[34,203052,46900],{"className":203053},[31685,31686]," 与该段的长度 Limit 进行比较。\n如果 ",[34,203056,203058,203077],{"className":203057},[31642],[34,203059,203061],{"className":203060},[31646],[31648,203062,203063],{"xmlns":31650},[31652,203064,203065,203074],{},[31655,203066,203067,203069,203071],{},[31658,203068,46900],{},[31662,203070,43175],{},[39361,203072,203073],{},"Limit",[31665,203075,203076],{"encoding":31667},"d \\ge \\text{Limit}",[34,203078,203080,203098],{"className":203079,"ariaHidden":758},[31672],[34,203081,203083,203086,203089,203092,203095],{"className":203082},[31676],[34,203084],{"className":203085,"style":47709},[31680],[34,203087,46900],{"className":203088},[31685,31686],[34,203090],{"className":203091,"style":31846},[31845],[34,203093,43175],{"className":203094},[31850],[34,203096],{"className":203097,"style":31846},[31845],[34,203099,203101,203104],{"className":203100},[31676],[34,203102],{"className":203103,"style":33218},[31680],[34,203105,203107],{"className":203106},[31685,2638],[34,203108,203073],{"className":203109},[31685],"，说明程序正试图访问超出本段范围的内存区域，硬件将立刻拦截此次访问并触发",[2627,203112,203113],{},"段错误 (Segment Fault)",[108,203115,203116,46324,203119,203173,203174],{},[2627,203117,203118],{},"计算物理地址",[34,203120,203122,203140],{"className":203121},[31642],[34,203123,203125],{"className":203124},[31646],[31648,203126,203127],{"xmlns":31650},[31652,203128,203129,203137],{},[31655,203130,203131,203133,203135],{},[31658,203132,46900],{},[31662,203134,932],{},[39361,203136,203073],{},[31665,203138,203139],{"encoding":31667},"d \u003C \\text{Limit}",[34,203141,203143,203161],{"className":203142,"ariaHidden":758},[31672],[34,203144,203146,203149,203152,203155,203158],{"className":203145},[31676],[34,203147],{"className":203148,"style":46991},[31680],[34,203150,46900],{"className":203151},[31685,31686],[34,203153],{"className":203154,"style":31846},[31845],[34,203156,932],{"className":203157},[31850],[34,203159],{"className":203160,"style":31846},[31845],[34,203162,203164,203167],{"className":203163},[31676],[34,203165],{"className":203166,"style":33218},[31680],[34,203168,203170],{"className":203169},[31685,2638],[34,203171,203073],{"className":203172},[31685],"，则该内存访问合法。\n",[34,203175,203177,203201],{"className":203176},[31642],[34,203178,203180],{"className":203179},[31646],[31648,203181,203182],{"xmlns":31650},[31652,203183,203184,203198],{},[31655,203185,203186,203189,203191,203194,203196],{},[39361,203187,203188],{},"Physical Address",[31662,203190,202],{},[39361,203192,203193],{},"Base",[31662,203195,223],{},[31658,203197,46900],{},[31665,203199,203200],{"encoding":31667},"\\text{Physical Address} = \\text{Base} + d",[34,203202,203204,203225,203246],{"className":203203,"ariaHidden":758},[31672],[34,203205,203207,203210,203216,203219,203222],{"className":203206},[31676],[34,203208],{"className":203209,"style":43217},[31680],[34,203211,203213],{"className":203212},[31685,2638],[34,203214,203188],{"className":203215},[31685],[34,203217],{"className":203218,"style":31846},[31845],[34,203220,202],{"className":203221},[31850],[34,203223],{"className":203224,"style":31846},[31845],[34,203226,203228,203231,203237,203240,203243],{"className":203227},[31676],[34,203229],{"className":203230,"style":34944},[31680],[34,203232,203234],{"className":203233},[31685,2638],[34,203235,203193],{"className":203236},[31685],[34,203238],{"className":203239,"style":32604},[31845],[34,203241,223],{"className":203242},[32024],[34,203244],{"className":203245,"style":32604},[31845],[34,203247,203249,203252],{"className":203248},[31676],[34,203250],{"className":203251,"style":31681},[31680],[34,203253,46900],{"className":203254},[31685,31686],[15,203256,203257],{},[2627,203258,200495],{},[15,203260,203261],{},"首先，将表格参数明确列出：",[105,203263,203264,203270,203276,203282],{},[108,203265,203266,203269],{},[2627,203267,203268],{},"段 0","：Base = 660，Limit = 248",[108,203271,203272,203275],{},[2627,203273,203274],{},"段 1","：Base = 1752，Limit = 422",[108,203277,203278,203281],{},[2627,203279,203280],{},"段 2","：Base = 222，Limit = 198",[108,203283,203284,203287],{},[2627,203285,203286],{},"段 3","：Base = 996，Limit = 604",[15,203289,203290],{},"逐题推演：",[15,203292,203293],{},[2627,203294,203295],{},"a. 逻辑地址 (0, 198)",[105,203297,203298,203381,203435,203490,203623],{},[108,203299,203300,203301,47649,203352,203380],{},"提取参数：段号 ",[34,203302,203304,203322],{"className":203303},[31642],[34,203305,203307],{"className":203306},[31646],[31648,203308,203309],{"xmlns":31650},[31652,203310,203311,203319],{},[31655,203312,203313,203315,203317],{},[31658,203314,31747],{},[31662,203316,202],{},[31749,203318,459],{},[31665,203320,203321],{"encoding":31667},"s = 0",[34,203323,203325,203343],{"className":203324,"ariaHidden":758},[31672],[34,203326,203328,203331,203334,203337,203340],{"className":203327},[31676],[34,203329],{"className":203330,"style":32527},[31680],[34,203332,31747],{"className":203333},[31685,31686],[34,203335],{"className":203336,"style":31846},[31845],[34,203338,202],{"className":203339},[31850],[34,203341],{"className":203342,"style":31846},[31845],[34,203344,203346,203349],{"className":203345},[31676],[34,203347],{"className":203348,"style":32617},[31680],[34,203350,459],{"className":203351},[31685],[34,203353,203355,203368],{"className":203354},[31642],[34,203356,203358],{"className":203357},[31646],[31648,203359,203360],{"xmlns":31650},[31652,203361,203362,203366],{},[31655,203363,203364],{},[31662,203365,33239],{},[31665,203367,119689],{"encoding":31667},[34,203369,203371],{"className":203370,"ariaHidden":758},[31672],[34,203372,203374,203377],{"className":203373},[31676],[34,203375],{"className":203376,"style":73089},[31680],[34,203378,33239],{"className":203379},[31850]," Base = 660, Limit = 248。",[108,203382,203383,203384,347],{},"偏移量：",[34,203385,203387,203405],{"className":203386},[31642],[34,203388,203390],{"className":203389},[31646],[31648,203391,203392],{"xmlns":31650},[31652,203393,203394,203402],{},[31655,203395,203396,203398,203400],{},[31658,203397,46900],{},[31662,203399,202],{},[31749,203401,202849],{},[31665,203403,203404],{"encoding":31667},"d = 198",[34,203406,203408,203426],{"className":203407,"ariaHidden":758},[31672],[34,203409,203411,203414,203417,203420,203423],{"className":203410},[31676],[34,203412],{"className":203413,"style":31681},[31680],[34,203415,46900],{"className":203416},[31685,31686],[34,203418],{"className":203419,"style":31846},[31845],[34,203421,202],{"className":203422},[31850],[34,203424],{"className":203425,"style":31846},[31845],[34,203427,203429,203432],{"className":203428},[31676],[34,203430],{"className":203431,"style":32617},[31680],[34,203433,202849],{"className":203434},[31685],[108,203436,203437,203438,203489],{},"越界检查：",[34,203439,203441,203459],{"className":203440},[31642],[34,203442,203444],{"className":203443},[31646],[31648,203445,203446],{"xmlns":31650},[31652,203447,203448,203456],{},[31655,203449,203450,203452,203454],{},[31749,203451,202849],{},[31662,203453,932],{},[31749,203455,184039],{},[31665,203457,203458],{"encoding":31667},"198 \u003C 248",[34,203460,203462,203480],{"className":203461,"ariaHidden":758},[31672],[34,203463,203465,203468,203471,203474,203477],{"className":203464},[31676],[34,203466],{"className":203467,"style":61814},[31680],[34,203469,202849],{"className":203470},[31685],[34,203472],{"className":203473,"style":31846},[31845],[34,203475,932],{"className":203476},[31850],[34,203478],{"className":203479,"style":31846},[31845],[34,203481,203483,203486],{"className":203482},[31676],[34,203484],{"className":203485,"style":32617},[31680],[34,203487,184039],{"className":203488},[31685],"，属于合法访问。",[108,203491,203492,203493,347],{},"物理地址 = ",[34,203494,203496,203533],{"className":203495},[31642],[34,203497,203499],{"className":203498},[31646],[31648,203500,203501],{"xmlns":31650},[31652,203502,203503,203530],{},[31655,203504,203505,203507,203509,203511,203513,203515,203517,203519,203521,203523,203525,203527],{},[31658,203506,54224],{},[31658,203508,111],{},[31658,203510,31747],{},[31658,203512,33244],{},[31662,203514,223],{},[31658,203516,46900],{},[31662,203518,202],{},[31749,203520,202831],{},[31662,203522,223],{},[31749,203524,202849],{},[31662,203526,202],{},[31749,203528,203529],{},"858",[31665,203531,203532],{"encoding":31667},"Base + d = 660 + 198 = 858",[34,203534,203536,203560,203578,203596,203614],{"className":203535,"ariaHidden":758},[31672],[34,203537,203539,203542,203545,203548,203551,203554,203557],{"className":203538},[31676],[34,203540],{"className":203541,"style":34944},[31680],[34,203543,54224],{"className":203544,"style":54252},[31685,31686],[34,203546,111],{"className":203547},[31685,31686],[34,203549,113295],{"className":203550},[31685,31686],[34,203552],{"className":203553,"style":32604},[31845],[34,203555,223],{"className":203556},[32024],[34,203558],{"className":203559,"style":32604},[31845],[34,203561,203563,203566,203569,203572,203575],{"className":203562},[31676],[34,203564],{"className":203565,"style":31681},[31680],[34,203567,46900],{"className":203568},[31685,31686],[34,203570],{"className":203571,"style":31846},[31845],[34,203573,202],{"className":203574},[31850],[34,203576],{"className":203577,"style":31846},[31845],[34,203579,203581,203584,203587,203590,203593],{"className":203580},[31676],[34,203582],{"className":203583,"style":35942},[31680],[34,203585,202831],{"className":203586},[31685],[34,203588],{"className":203589,"style":32604},[31845],[34,203591,223],{"className":203592},[32024],[34,203594],{"className":203595,"style":32604},[31845],[34,203597,203599,203602,203605,203608,203611],{"className":203598},[31676],[34,203600],{"className":203601,"style":32617},[31680],[34,203603,202849],{"className":203604},[31685],[34,203606],{"className":203607,"style":31846},[31845],[34,203609,202],{"className":203610},[31850],[34,203612],{"className":203613,"style":31846},[31845],[34,203615,203617,203620],{"className":203616},[31676],[34,203618],{"className":203619,"style":32617},[31680],[34,203621,203529],{"className":203622},[31685],[108,203624,203625,203627,203628,347],{},[2627,203626,200857],{},"：物理地址为 ",[2627,203629,203529],{},[15,203631,203632],{},[2627,203633,203634],{},"b. 逻辑地址 (2, 156)",[105,203636,203637,203719,203773,203826,203958],{},[108,203638,203300,203639,47649,203690,203718],{},[34,203640,203642,203660],{"className":203641},[31642],[34,203643,203645],{"className":203644},[31646],[31648,203646,203647],{"xmlns":31650},[31652,203648,203649,203657],{},[31655,203650,203651,203653,203655],{},[31658,203652,31747],{},[31662,203654,202],{},[31749,203656,768],{},[31665,203658,203659],{"encoding":31667},"s = 2",[34,203661,203663,203681],{"className":203662,"ariaHidden":758},[31672],[34,203664,203666,203669,203672,203675,203678],{"className":203665},[31676],[34,203667],{"className":203668,"style":32527},[31680],[34,203670,31747],{"className":203671},[31685,31686],[34,203673],{"className":203674,"style":31846},[31845],[34,203676,202],{"className":203677},[31850],[34,203679],{"className":203680,"style":31846},[31845],[34,203682,203684,203687],{"className":203683},[31676],[34,203685],{"className":203686,"style":32617},[31680],[34,203688,768],{"className":203689},[31685],[34,203691,203693,203706],{"className":203692},[31642],[34,203694,203696],{"className":203695},[31646],[31648,203697,203698],{"xmlns":31650},[31652,203699,203700,203704],{},[31655,203701,203702],{},[31662,203703,33239],{},[31665,203705,119689],{"encoding":31667},[34,203707,203709],{"className":203708,"ariaHidden":758},[31672],[34,203710,203712,203715],{"className":203711},[31676],[34,203713],{"className":203714,"style":73089},[31680],[34,203716,33239],{"className":203717},[31850]," Base = 222, Limit = 198。",[108,203720,203383,203721,347],{},[34,203722,203724,203743],{"className":203723},[31642],[34,203725,203727],{"className":203726},[31646],[31648,203728,203729],{"xmlns":31650},[31652,203730,203731,203740],{},[31655,203732,203733,203735,203737],{},[31658,203734,46900],{},[31662,203736,202],{},[31749,203738,203739],{},"156",[31665,203741,203742],{"encoding":31667},"d = 156",[34,203744,203746,203764],{"className":203745,"ariaHidden":758},[31672],[34,203747,203749,203752,203755,203758,203761],{"className":203748},[31676],[34,203750],{"className":203751,"style":31681},[31680],[34,203753,46900],{"className":203754},[31685,31686],[34,203756],{"className":203757,"style":31846},[31845],[34,203759,202],{"className":203760},[31850],[34,203762],{"className":203763,"style":31846},[31845],[34,203765,203767,203770],{"className":203766},[31676],[34,203768],{"className":203769,"style":32617},[31680],[34,203771,203739],{"className":203772},[31685],[108,203774,203437,203775,203489],{},[34,203776,203778,203796],{"className":203777},[31642],[34,203779,203781],{"className":203780},[31646],[31648,203782,203783],{"xmlns":31650},[31652,203784,203785,203793],{},[31655,203786,203787,203789,203791],{},[31749,203788,203739],{},[31662,203790,932],{},[31749,203792,202849],{},[31665,203794,203795],{"encoding":31667},"156 \u003C 198",[34,203797,203799,203817],{"className":203798,"ariaHidden":758},[31672],[34,203800,203802,203805,203808,203811,203814],{"className":203801},[31676],[34,203803],{"className":203804,"style":61814},[31680],[34,203806,203739],{"className":203807},[31685],[34,203809],{"className":203810,"style":31846},[31845],[34,203812,932],{"className":203813},[31850],[34,203815],{"className":203816,"style":31846},[31845],[34,203818,203820,203823],{"className":203819},[31676],[34,203821],{"className":203822,"style":32617},[31680],[34,203824,202849],{"className":203825},[31685],[108,203827,203492,203828,347],{},[34,203829,203831,203868],{"className":203830},[31642],[34,203832,203834],{"className":203833},[31646],[31648,203835,203836],{"xmlns":31650},[31652,203837,203838,203865],{},[31655,203839,203840,203842,203844,203846,203848,203850,203852,203854,203856,203858,203860,203862],{},[31658,203841,54224],{},[31658,203843,111],{},[31658,203845,31747],{},[31658,203847,33244],{},[31662,203849,223],{},[31658,203851,46900],{},[31662,203853,202],{},[31749,203855,202846],{},[31662,203857,223],{},[31749,203859,203739],{},[31662,203861,202],{},[31749,203863,203864],{},"378",[31665,203866,203867],{"encoding":31667},"Base + d = 222 + 156 = 378",[34,203869,203871,203895,203913,203931,203949],{"className":203870,"ariaHidden":758},[31672],[34,203872,203874,203877,203880,203883,203886,203889,203892],{"className":203873},[31676],[34,203875],{"className":203876,"style":34944},[31680],[34,203878,54224],{"className":203879,"style":54252},[31685,31686],[34,203881,111],{"className":203882},[31685,31686],[34,203884,113295],{"className":203885},[31685,31686],[34,203887],{"className":203888,"style":32604},[31845],[34,203890,223],{"className":203891},[32024],[34,203893],{"className":203894,"style":32604},[31845],[34,203896,203898,203901,203904,203907,203910],{"className":203897},[31676],[34,203899],{"className":203900,"style":31681},[31680],[34,203902,46900],{"className":203903},[31685,31686],[34,203905],{"className":203906,"style":31846},[31845],[34,203908,202],{"className":203909},[31850],[34,203911],{"className":203912,"style":31846},[31845],[34,203914,203916,203919,203922,203925,203928],{"className":203915},[31676],[34,203917],{"className":203918,"style":35942},[31680],[34,203920,202846],{"className":203921},[31685],[34,203923],{"className":203924,"style":32604},[31845],[34,203926,223],{"className":203927},[32024],[34,203929],{"className":203930,"style":32604},[31845],[34,203932,203934,203937,203940,203943,203946],{"className":203933},[31676],[34,203935],{"className":203936,"style":32617},[31680],[34,203938,203739],{"className":203939},[31685],[34,203941],{"className":203942,"style":31846},[31845],[34,203944,202],{"className":203945},[31850],[34,203947],{"className":203948,"style":31846},[31845],[34,203950,203952,203955],{"className":203951},[31676],[34,203953],{"className":203954,"style":32617},[31680],[34,203956,203864],{"className":203957},[31685],[108,203959,203960,203627,203962,347],{},[2627,203961,200940],{},[2627,203963,203864],{},[15,203965,203966],{},[2627,203967,203968],{},"c. 逻辑地址 (1, 530)",[105,203970,203971,204053,204107,204161],{},[108,203972,203300,203973,47649,204024,204052],{},[34,203974,203976,203994],{"className":203975},[31642],[34,203977,203979],{"className":203978},[31646],[31648,203980,203981],{"xmlns":31650},[31652,203982,203983,203991],{},[31655,203984,203985,203987,203989],{},[31658,203986,31747],{},[31662,203988,202],{},[31749,203990,640],{},[31665,203992,203993],{"encoding":31667},"s = 1",[34,203995,203997,204015],{"className":203996,"ariaHidden":758},[31672],[34,203998,204000,204003,204006,204009,204012],{"className":203999},[31676],[34,204001],{"className":204002,"style":32527},[31680],[34,204004,31747],{"className":204005},[31685,31686],[34,204007],{"className":204008,"style":31846},[31845],[34,204010,202],{"className":204011},[31850],[34,204013],{"className":204014,"style":31846},[31845],[34,204016,204018,204021],{"className":204017},[31676],[34,204019],{"className":204020,"style":32617},[31680],[34,204022,640],{"className":204023},[31685],[34,204025,204027,204040],{"className":204026},[31642],[34,204028,204030],{"className":204029},[31646],[31648,204031,204032],{"xmlns":31650},[31652,204033,204034,204038],{},[31655,204035,204036],{},[31662,204037,33239],{},[31665,204039,119689],{"encoding":31667},[34,204041,204043],{"className":204042,"ariaHidden":758},[31672],[34,204044,204046,204049],{"className":204045},[31676],[34,204047],{"className":204048,"style":73089},[31680],[34,204050,33239],{"className":204051},[31850]," Base = 1752, Limit = 422。",[108,204054,203383,204055,347],{},[34,204056,204058,204077],{"className":204057},[31642],[34,204059,204061],{"className":204060},[31646],[31648,204062,204063],{"xmlns":31650},[31652,204064,204065,204074],{},[31655,204066,204067,204069,204071],{},[31658,204068,46900],{},[31662,204070,202],{},[31749,204072,204073],{},"530",[31665,204075,204076],{"encoding":31667},"d = 530",[34,204078,204080,204098],{"className":204079,"ariaHidden":758},[31672],[34,204081,204083,204086,204089,204092,204095],{"className":204082},[31676],[34,204084],{"className":204085,"style":31681},[31680],[34,204087,46900],{"className":204088},[31685,31686],[34,204090],{"className":204091,"style":31846},[31845],[34,204093,202],{"className":204094},[31850],[34,204096],{"className":204097,"style":31846},[31845],[34,204099,204101,204104],{"className":204100},[31676],[34,204102],{"className":204103,"style":32617},[31680],[34,204105,204073],{"className":204106},[31685],[108,204108,203437,204109,204160],{},[34,204110,204112,204130],{"className":204111},[31642],[34,204113,204115],{"className":204114},[31646],[31648,204116,204117],{"xmlns":31650},[31652,204118,204119,204127],{},[31655,204120,204121,204123,204125],{},[31749,204122,204073],{},[31662,204124,43175],{},[31749,204126,202841],{},[31665,204128,204129],{"encoding":31667},"530 \\ge 422",[34,204131,204133,204151],{"className":204132,"ariaHidden":758},[31672],[34,204134,204136,204139,204142,204145,204148],{"className":204135},[31676],[34,204137],{"className":204138,"style":55382},[31680],[34,204140,204073],{"className":204141},[31685],[34,204143],{"className":204144,"style":31846},[31845],[34,204146,43175],{"className":204147},[31850],[34,204149],{"className":204150,"style":31846},[31845],[34,204152,204154,204157],{"className":204153},[31676],[34,204155],{"className":204156,"style":32617},[31680],[34,204158,202841],{"className":204159},[31685],"。请求的偏移量超出了段 1 的合法长度边界，这是典型的内存越界违规。",[108,204162,204163,2664,204165,347],{},[2627,204164,201002],{},[2627,204166,204167],{},"发生段错误 (Segment fault)",[15,204169,204170],{},[2627,204171,204172],{},"d. 逻辑地址 (3, 444)",[105,204174,204175,204257,204311,204364,204496],{},[108,204176,203300,204177,47649,204228,204256],{},[34,204178,204180,204198],{"className":204179},[31642],[34,204181,204183],{"className":204182},[31646],[31648,204184,204185],{"xmlns":31650},[31652,204186,204187,204195],{},[31655,204188,204189,204191,204193],{},[31658,204190,31747],{},[31662,204192,202],{},[31749,204194,786],{},[31665,204196,204197],{"encoding":31667},"s = 3",[34,204199,204201,204219],{"className":204200,"ariaHidden":758},[31672],[34,204202,204204,204207,204210,204213,204216],{"className":204203},[31676],[34,204205],{"className":204206,"style":32527},[31680],[34,204208,31747],{"className":204209},[31685,31686],[34,204211],{"className":204212,"style":31846},[31845],[34,204214,202],{"className":204215},[31850],[34,204217],{"className":204218,"style":31846},[31845],[34,204220,204222,204225],{"className":204221},[31676],[34,204223],{"className":204224,"style":32617},[31680],[34,204226,786],{"className":204227},[31685],[34,204229,204231,204244],{"className":204230},[31642],[34,204232,204234],{"className":204233},[31646],[31648,204235,204236],{"xmlns":31650},[31652,204237,204238,204242],{},[31655,204239,204240],{},[31662,204241,33239],{},[31665,204243,119689],{"encoding":31667},[34,204245,204247],{"className":204246,"ariaHidden":758},[31672],[34,204248,204250,204253],{"className":204249},[31676],[34,204251],{"className":204252,"style":73089},[31680],[34,204254,33239],{"className":204255},[31850]," Base = 996, Limit = 604。",[108,204258,203383,204259,347],{},[34,204260,204262,204281],{"className":204261},[31642],[34,204263,204265],{"className":204264},[31646],[31648,204266,204267],{"xmlns":31650},[31652,204268,204269,204278],{},[31655,204270,204271,204273,204275],{},[31658,204272,46900],{},[31662,204274,202],{},[31749,204276,204277],{},"444",[31665,204279,204280],{"encoding":31667},"d = 444",[34,204282,204284,204302],{"className":204283,"ariaHidden":758},[31672],[34,204285,204287,204290,204293,204296,204299],{"className":204286},[31676],[34,204288],{"className":204289,"style":31681},[31680],[34,204291,46900],{"className":204292},[31685,31686],[34,204294],{"className":204295,"style":31846},[31845],[34,204297,202],{"className":204298},[31850],[34,204300],{"className":204301,"style":31846},[31845],[34,204303,204305,204308],{"className":204304},[31676],[34,204306],{"className":204307,"style":32617},[31680],[34,204309,204277],{"className":204310},[31685],[108,204312,203437,204313,203489],{},[34,204314,204316,204334],{"className":204315},[31642],[34,204317,204319],{"className":204318},[31646],[31648,204320,204321],{"xmlns":31650},[31652,204322,204323,204331],{},[31655,204324,204325,204327,204329],{},[31749,204326,204277],{},[31662,204328,932],{},[31749,204330,202857],{},[31665,204332,204333],{"encoding":31667},"444 \u003C 604",[34,204335,204337,204355],{"className":204336,"ariaHidden":758},[31672],[34,204338,204340,204343,204346,204349,204352],{"className":204339},[31676],[34,204341],{"className":204342,"style":61814},[31680],[34,204344,204277],{"className":204345},[31685],[34,204347],{"className":204348,"style":31846},[31845],[34,204350,932],{"className":204351},[31850],[34,204353],{"className":204354,"style":31846},[31845],[34,204356,204358,204361],{"className":204357},[31676],[34,204359],{"className":204360,"style":32617},[31680],[34,204362,202857],{"className":204363},[31685],[108,204365,203492,204366,347],{},[34,204367,204369,204406],{"className":204368},[31642],[34,204370,204372],{"className":204371},[31646],[31648,204373,204374],{"xmlns":31650},[31652,204375,204376,204403],{},[31655,204377,204378,204380,204382,204384,204386,204388,204390,204392,204394,204396,204398,204400],{},[31658,204379,54224],{},[31658,204381,111],{},[31658,204383,31747],{},[31658,204385,33244],{},[31662,204387,223],{},[31658,204389,46900],{},[31662,204391,202],{},[31749,204393,202854],{},[31662,204395,223],{},[31749,204397,204277],{},[31662,204399,202],{},[31749,204401,204402],{},"1440",[31665,204404,204405],{"encoding":31667},"Base + d = 996 + 444 = 1440",[34,204407,204409,204433,204451,204469,204487],{"className":204408,"ariaHidden":758},[31672],[34,204410,204412,204415,204418,204421,204424,204427,204430],{"className":204411},[31676],[34,204413],{"className":204414,"style":34944},[31680],[34,204416,54224],{"className":204417,"style":54252},[31685,31686],[34,204419,111],{"className":204420},[31685,31686],[34,204422,113295],{"className":204423},[31685,31686],[34,204425],{"className":204426,"style":32604},[31845],[34,204428,223],{"className":204429},[32024],[34,204431],{"className":204432,"style":32604},[31845],[34,204434,204436,204439,204442,204445,204448],{"className":204435},[31676],[34,204437],{"className":204438,"style":31681},[31680],[34,204440,46900],{"className":204441},[31685,31686],[34,204443],{"className":204444,"style":31846},[31845],[34,204446,202],{"className":204447},[31850],[34,204449],{"className":204450,"style":31846},[31845],[34,204452,204454,204457,204460,204463,204466],{"className":204453},[31676],[34,204455],{"className":204456,"style":35942},[31680],[34,204458,202854],{"className":204459},[31685],[34,204461],{"className":204462,"style":32604},[31845],[34,204464,223],{"className":204465},[32024],[34,204467],{"className":204468,"style":32604},[31845],[34,204470,204472,204475,204478,204481,204484],{"className":204471},[31676],[34,204473],{"className":204474,"style":32617},[31680],[34,204476,204277],{"className":204477},[31685],[34,204479],{"className":204480,"style":31846},[31845],[34,204482,202],{"className":204483},[31850],[34,204485],{"className":204486,"style":31846},[31845],[34,204488,204490,204493],{"className":204489},[31676],[34,204491],{"className":204492,"style":32617},[31680],[34,204494,204402],{"className":204495},[31685],[108,204497,204498,203627,204500,347],{},[2627,204499,201074],{},[2627,204501,204402],{},[15,204503,204504],{},[2627,204505,204506],{},"e. 逻辑地址 (0, 222)",[105,204508,204509,204589,204642,204695,204827],{},[108,204510,203300,204511,47649,204561,203380],{},[34,204512,204514,204531],{"className":204513},[31642],[34,204515,204517],{"className":204516},[31646],[31648,204518,204519],{"xmlns":31650},[31652,204520,204521,204529],{},[31655,204522,204523,204525,204527],{},[31658,204524,31747],{},[31662,204526,202],{},[31749,204528,459],{},[31665,204530,203321],{"encoding":31667},[34,204532,204534,204552],{"className":204533,"ariaHidden":758},[31672],[34,204535,204537,204540,204543,204546,204549],{"className":204536},[31676],[34,204538],{"className":204539,"style":32527},[31680],[34,204541,31747],{"className":204542},[31685,31686],[34,204544],{"className":204545,"style":31846},[31845],[34,204547,202],{"className":204548},[31850],[34,204550],{"className":204551,"style":31846},[31845],[34,204553,204555,204558],{"className":204554},[31676],[34,204556],{"className":204557,"style":32617},[31680],[34,204559,459],{"className":204560},[31685],[34,204562,204564,204577],{"className":204563},[31642],[34,204565,204567],{"className":204566},[31646],[31648,204568,204569],{"xmlns":31650},[31652,204570,204571,204575],{},[31655,204572,204573],{},[31662,204574,33239],{},[31665,204576,119689],{"encoding":31667},[34,204578,204580],{"className":204579,"ariaHidden":758},[31672],[34,204581,204583,204586],{"className":204582},[31676],[34,204584],{"className":204585,"style":73089},[31680],[34,204587,33239],{"className":204588},[31850],[108,204590,203383,204591,347],{},[34,204592,204594,204612],{"className":204593},[31642],[34,204595,204597],{"className":204596},[31646],[31648,204598,204599],{"xmlns":31650},[31652,204600,204601,204609],{},[31655,204602,204603,204605,204607],{},[31658,204604,46900],{},[31662,204606,202],{},[31749,204608,202846],{},[31665,204610,204611],{"encoding":31667},"d = 222",[34,204613,204615,204633],{"className":204614,"ariaHidden":758},[31672],[34,204616,204618,204621,204624,204627,204630],{"className":204617},[31676],[34,204619],{"className":204620,"style":31681},[31680],[34,204622,46900],{"className":204623},[31685,31686],[34,204625],{"className":204626,"style":31846},[31845],[34,204628,202],{"className":204629},[31850],[34,204631],{"className":204632,"style":31846},[31845],[34,204634,204636,204639],{"className":204635},[31676],[34,204637],{"className":204638,"style":32617},[31680],[34,204640,202846],{"className":204641},[31685],[108,204643,203437,204644,203489],{},[34,204645,204647,204665],{"className":204646},[31642],[34,204648,204650],{"className":204649},[31646],[31648,204651,204652],{"xmlns":31650},[31652,204653,204654,204662],{},[31655,204655,204656,204658,204660],{},[31749,204657,202846],{},[31662,204659,932],{},[31749,204661,184039],{},[31665,204663,204664],{"encoding":31667},"222 \u003C 248",[34,204666,204668,204686],{"className":204667,"ariaHidden":758},[31672],[34,204669,204671,204674,204677,204680,204683],{"className":204670},[31676],[34,204672],{"className":204673,"style":61814},[31680],[34,204675,202846],{"className":204676},[31685],[34,204678],{"className":204679,"style":31846},[31845],[34,204681,932],{"className":204682},[31850],[34,204684],{"className":204685,"style":31846},[31845],[34,204687,204689,204692],{"className":204688},[31676],[34,204690],{"className":204691,"style":32617},[31680],[34,204693,184039],{"className":204694},[31685],[108,204696,203492,204697,347],{},[34,204698,204700,204737],{"className":204699},[31642],[34,204701,204703],{"className":204702},[31646],[31648,204704,204705],{"xmlns":31650},[31652,204706,204707,204734],{},[31655,204708,204709,204711,204713,204715,204717,204719,204721,204723,204725,204727,204729,204731],{},[31658,204710,54224],{},[31658,204712,111],{},[31658,204714,31747],{},[31658,204716,33244],{},[31662,204718,223],{},[31658,204720,46900],{},[31662,204722,202],{},[31749,204724,202831],{},[31662,204726,223],{},[31749,204728,202846],{},[31662,204730,202],{},[31749,204732,204733],{},"882",[31665,204735,204736],{"encoding":31667},"Base + d = 660 + 222 = 882",[34,204738,204740,204764,204782,204800,204818],{"className":204739,"ariaHidden":758},[31672],[34,204741,204743,204746,204749,204752,204755,204758,204761],{"className":204742},[31676],[34,204744],{"className":204745,"style":34944},[31680],[34,204747,54224],{"className":204748,"style":54252},[31685,31686],[34,204750,111],{"className":204751},[31685,31686],[34,204753,113295],{"className":204754},[31685,31686],[34,204756],{"className":204757,"style":32604},[31845],[34,204759,223],{"className":204760},[32024],[34,204762],{"className":204763,"style":32604},[31845],[34,204765,204767,204770,204773,204776,204779],{"className":204766},[31676],[34,204768],{"className":204769,"style":31681},[31680],[34,204771,46900],{"className":204772},[31685,31686],[34,204774],{"className":204775,"style":31846},[31845],[34,204777,202],{"className":204778},[31850],[34,204780],{"className":204781,"style":31846},[31845],[34,204783,204785,204788,204791,204794,204797],{"className":204784},[31676],[34,204786],{"className":204787,"style":35942},[31680],[34,204789,202831],{"className":204790},[31685],[34,204792],{"className":204793,"style":32604},[31845],[34,204795,223],{"className":204796},[32024],[34,204798],{"className":204799,"style":32604},[31845],[34,204801,204803,204806,204809,204812,204815],{"className":204802},[31676],[34,204804],{"className":204805,"style":32617},[31680],[34,204807,202846],{"className":204808},[31685],[34,204810],{"className":204811,"style":31846},[31845],[34,204813,202],{"className":204814},[31850],[34,204816],{"className":204817,"style":31846},[31845],[34,204819,204821,204824],{"className":204820},[31676],[34,204822],{"className":204823,"style":32617},[31680],[34,204825,204733],{"className":204826},[31685],[108,204828,204829,203627,204831,347],{},[2627,204830,202798],{},[2627,204832,204733],{},[19,204834,204836],{"id":204835},"_6-什么是快表它在地址转换中起什么作用","6. 什么是快表？它在地址转换中起什么作用？",[15,204838,204839,204842],{},[2627,204840,204841],{},"【题目】","\n什么是快表？它在地址转换中起什么作用？",[15,204844,204845],{},[2627,204846,204847],{},"【解答与知识点解析】",[105,204849,204850,204860],{},[108,204851,204852,204855,204856,204859],{},[2627,204853,204854],{},"什么是快表 (TLB, Translation Lookaside Buffer)","：\n快表，又称转换后备缓冲器，是一种位于CPU内部的、容量小但访问速度极快的高速缓存（通常由相联存储器 Associative Memory 实现）。它专门用于存放当前进程中最常被访问的",[2627,204857,204858],{},"页表项","（即逻辑页号到物理页框号的映射记录）。",[108,204861,204862,204865,204866,204869,204870],{},[2627,204863,204864],{},"在地址转换中的作用","：\n在没有快表的分页系统中，每次内存访问都需要进行“两次访存”：第一次访问内存中的页表来获取物理地址，第二次才是真正访问该物理地址存取数据。\n快表的作用是",[2627,204867,204868],{},"加速地址转换，减少访存次数","。当CPU给出逻辑地址时，硬件会首先在快表中并行查找对应的页号：\n",[2230,204871,204872,204882],{},[108,204873,204874,204877,204878,204881],{},[2627,204875,204876],{},"若命中 (TLB Hit)","：直接从快表中取出物理页框号，与页内偏移量拼接成物理地址，只需",[2627,204879,204880],{},"一次访存","即可取得数据。",[108,204883,204884,204887],{},[2627,204885,204886],{},"若未命中 (TLB Miss)","：再去主存中查找页表，取出页表项进行地址转换，并同时将该页表项自动更新到快表中，以便下次加速访问。",[32188,204889],{},[19,204891,204893],{"id":204892},"_7-简述存储管理中移动技术的优缺点","7. 简述存储管理中移动技术的优缺点。",[15,204895,204896,204898],{},[2627,204897,204841],{},"\n简述存储管理中移动技术的优缺点。",[15,204900,204901],{},[2627,204902,204847],{},[105,204904,204905,204915,204934],{},[108,204906,204907,204910,204911,204914],{},[2627,204908,204909],{},"概念背景","：移动技术，通常也称为",[2627,204912,204913],{},"内存紧凑 (Compaction)"," 或拼接技术。在动态分区分配中，随着进程的装入和撤出，内存中会产生许多不连续的、极小而无法使用的空闲块，即“外部碎片”。移动技术就是通过在内存中移动进程的位置，把所有的进程靠紧在一起，从而将零散的空闲块拼接成一个大的连续空闲区。",[108,204916,204917,33183,204920],{},[2627,204918,204919],{},"优点",[2230,204921,204922,204928],{},[108,204923,204924,204927],{},[2627,204925,204926],{},"彻底消除外部碎片","：能够将零散的小内存块聚集成一个大内存块，大大提高了主存的利用率。",[108,204929,204930,204933],{},[2627,204931,204932],{},"满足大进程的需求","：当系统剩余内存总和足够，但缺乏连续空间时，通过移动技术可以腾出足够大的连续空间来装入大进程。",[108,204935,204936,33183,204939],{},[2627,204937,204938],{},"缺点",[2230,204940,204941,204947,204953],{},[108,204942,204943,204946],{},[2627,204944,204945],{},"极大的时间与CPU开销","：移动进程意味着要进行大量的数据拷贝，这会大量消耗CPU时间，影响系统性能。",[108,204948,204949,204952],{},[2627,204950,204951],{},"需要动态重定位硬件支持","：进程在内存中的物理位置改变了，因此必须有硬件（如基址寄存器\u002F重定位寄存器）支持动态地址映射。",[108,204954,204955,204958],{},[2627,204956,204957],{},"进程必须暂停","：在移动某个进程期间，该进程必须处于挂起\u002F等待状态，不能进行执行和 I\u002FO 操作。",[32188,204960],{},[19,204962,204964],{"id":204963},"_8-页式和段式内存管理有什么区别怎样才能实现共享和保护","8. 页式和段式内存管理有什么区别？怎样才能实现共享和保护？",[15,204966,204967,204969],{},[2627,204968,204841],{},"\n页式和段式内存管理有什么区别？怎样才能实现共享和保护？",[15,204971,204972],{},[2627,204973,204847],{},[105,204975,204976,205069],{},[108,204977,204978,2664,204981],{},[2627,204979,204980],{},"页式与段式的主要区别",[2230,204982,204983,205015,205034,205055],{},[108,204984,204985,33183,204988],{},[2627,204986,204987],{},"划分的依据与目的",[105,204989,204990,205003],{},[108,204991,204992,204995,204996,204999,205000,31618],{},[2627,204993,204994],{},"页 (Page)"," 是信息的",[2627,204997,204998],{},"物理单位","。分页是为了实现离散分配，解决内存碎片问题，提高内存利用率。它是系统底层管理的需要，对用户是",[2627,205001,205002],{},"透明",[108,205004,205005,204995,205008,205011,205012,31618],{},[2627,205006,205007],{},"段 (Segment)",[2627,205009,205010],{},"逻辑单位","（如主程序、子程序、数据区等）。分段是为了更好地满足用户编程和共享、保护的需要。它是用户",[2627,205013,205014],{},"可见",[108,205016,205017,33183,205020],{},[2627,205018,205019],{},"大小是否固定",[105,205021,205022,205028],{},[108,205023,205024,205027],{},[2627,205025,205026],{},"页的大小是固定的","，由操作系统\u002F硬件决定（如 4KB）。",[108,205029,205030,205033],{},[2627,205031,205032],{},"段的大小是可变的","，取决于用户编写的程序逻辑。",[108,205035,205036,33183,205039],{},[2627,205037,205038],{},"地址空间的维度",[105,205040,205041,205048],{},[108,205042,205043,205044,205047],{},"页式系统的逻辑地址空间是",[2627,205045,205046],{},"一维","的。程序员只需给出一个线性地址，硬件自动将其拆分为页号和偏移量。",[108,205049,205050,205051,205054],{},"段式系统的逻辑地址空间是",[2627,205052,205053],{},"二维","的。程序员（或编译器）必须显式提供“段号”和“段内偏移量”。",[108,205056,205057,205060,205061,205064,205065,205068],{},[2627,205058,205059],{},"碎片问题","：分页会产生",[2627,205062,205063],{},"内部碎片","（最后一页装不满），没有外部碎片；分段会产生",[2627,205066,205067],{},"外部碎片","（段之间留有空隙），没有内部碎片。",[108,205070,205071,2664,205074],{},[2627,205072,205073],{},"如何实现共享和保护",[105,205075,205076,205099],{},[108,205077,205078,33183,205081],{},[2627,205079,205080],{},"共享",[105,205082,205083,205089],{},[108,205084,205085,205088],{},[71807,205086,205087],{},"页式","：让不同进程的页表项指向同一个物理页框号。",[108,205090,205091,205094,205095,205098],{},[71807,205092,205093],{},"段式","：让不同进程的段表项指向同一个物理段的基址。",[2627,205096,205097],{},"分段更容易实现共享","，因为段是逻辑完整的单元（比如一段纯代码），共享一个代码段非常自然。",[108,205100,205101,33183,205104],{},[2627,205102,205103],{},"保护",[105,205105,205106],{},[108,205107,205108,205109,58269,205111,205114,205115,205118],{},"无论页式还是段式，都可以通过在",[2627,205110,204858],{},[2627,205112,205113],{},"段表项","中设置",[2627,205116,205117],{},"保护控制位","（如只读、可读写、可执行等标志位）来实现。当进行地址转换时，硬件会自动检查这些位，若操作非法（如试图写入只读代码段或越界访问），则触发越权中断。",[32188,205120],{},[19,205122,205124],{"id":205123},"_9-虚地址与物理地址的页表计算","9. 虚地址与物理地址的页表计算",[15,205126,205127,205129],{},[2627,205128,204841],{},"\n一台机器有48位虚地址和32位物理地址，若页长为8KB，问页表共有多少个页表项?如果设计一个反置页表，则有多少个页表项?",[15,205131,205132],{},[2627,205133,204847],{},[105,205135,205136,205334,205481],{},[108,205137,205138,33183,205141],{},[2627,205139,205140],{},"基础参数解析",[105,205142,205143,205175,205206],{},[108,205144,205145,205146,202105],{},"虚拟地址 (VA) = ",[34,205147,205149,205163],{"className":205148},[31642],[34,205150,205152],{"className":205151},[31646],[31648,205153,205154],{"xmlns":31650},[31652,205155,205156,205161],{},[31655,205157,205158],{},[31749,205159,205160],{},"48",[31665,205162,205160],{"encoding":31667},[34,205164,205166],{"className":205165,"ariaHidden":758},[31672],[34,205167,205169,205172],{"className":205168},[31676],[34,205170],{"className":205171,"style":32617},[31680],[34,205173,205160],{"className":205174},[31685],[108,205176,205177,205178,202105],{},"物理地址 (PA) = ",[34,205179,205181,205194],{"className":205180},[31642],[34,205182,205184],{"className":205183},[31646],[31648,205185,205186],{"xmlns":31650},[31652,205187,205188,205192],{},[31655,205189,205190],{},[31749,205191,173486],{},[31665,205193,173486],{"encoding":31667},[34,205195,205197],{"className":205196,"ariaHidden":758},[31672],[34,205198,205200,205203],{"className":205199},[31676],[34,205201],{"className":205202,"style":32617},[31680],[34,205204,173486],{"className":205205},[31685],[108,205207,205208,205209,202001,205303,47649,205331,347],{},"页长 (Page Size) = ",[34,205210,205212,205238],{"className":205211},[31642],[34,205213,205215],{"className":205214},[31646],[31648,205216,205217],{"xmlns":31650},[31652,205218,205219,205235],{},[31655,205220,205221,205223,205226,205228],{},[31749,205222,60525],{},[39361,205224,205225],{},"KB",[31662,205227,202],{},[37624,205229,205230,205232],{},[31749,205231,768],{},[31749,205233,205234],{},"13",[31665,205236,205237],{"encoding":31667},"8\\text{KB} = 2^{13}",[34,205239,205241,205265],{"className":205240,"ariaHidden":758},[31672],[34,205242,205244,205247,205250,205256,205259,205262],{"className":205243},[31676],[34,205245],{"className":205246,"style":33218},[31680],[34,205248,60525],{"className":205249},[31685],[34,205251,205253],{"className":205252},[31685,2638],[34,205254,205225],{"className":205255},[31685],[34,205257],{"className":205258,"style":31846},[31845],[34,205260,202],{"className":205261},[31850],[34,205263],{"className":205264,"style":31846},[31845],[34,205266,205268,205271],{"className":205267},[31676],[34,205269],{"className":205270,"style":37774},[31680],[34,205272,205274,205277],{"className":205273},[31685],[34,205275,768],{"className":205276},[31685],[34,205278,205280],{"className":205279},[31792],[34,205281,205283],{"className":205282},[31796],[34,205284,205286],{"className":205285},[31801],[34,205287,205289],{"className":205288,"style":37774},[31805],[34,205290,205291,205294],{"style":37777},[34,205292],{"className":205293,"style":31814},[31813],[34,205295,205297],{"className":205296},[31818,31819,31820,31821],[34,205298,205300],{"className":205299},[31685,31821],[34,205301,205234],{"className":205302},[31685,31821],[34,205304,205306,205319],{"className":205305},[31642],[34,205307,205309],{"className":205308},[31646],[31648,205310,205311],{"xmlns":31650},[31652,205312,205313,205317],{},[31655,205314,205315],{},[31662,205316,33239],{},[31665,205318,119689],{"encoding":31667},[34,205320,205322],{"className":205321,"ariaHidden":758},[31672],[34,205323,205325,205328],{"className":205324},[31676],[34,205326],{"className":205327,"style":73089},[31680],[34,205329,33239],{"className":205330},[31850],[2627,205332,205333],{},"页内偏移量占用 13 位",[108,205335,205336,205339,205340,205343,205344,205417,205418,205480],{},[2627,205337,205338],{},"计算常规页表的页表项数量","：\n常规页表的大小取决于",[2627,205341,205342],{},"虚拟页号 (VPN)"," 的位数。\n虚拟页号位数 = 虚拟地址位数 - 页内偏移位数 = ",[34,205345,205347,205369],{"className":205346},[31642],[34,205348,205350],{"className":205349},[31646],[31648,205351,205352],{"xmlns":31650},[31652,205353,205354,205366],{},[31655,205355,205356,205358,205360,205362,205364],{},[31749,205357,205160],{},[31662,205359,32226],{},[31749,205361,205234],{},[31662,205363,202],{},[31749,205365,5752],{},[31665,205367,205368],{"encoding":31667},"48 - 13 = 35",[34,205370,205372,205390,205408],{"className":205371,"ariaHidden":758},[31672],[34,205373,205375,205378,205381,205384,205387],{"className":205374},[31676],[34,205376],{"className":205377,"style":35942},[31680],[34,205379,205160],{"className":205380},[31685],[34,205382],{"className":205383,"style":32604},[31845],[34,205385,32226],{"className":205386},[32024],[34,205388],{"className":205389,"style":32604},[31845],[34,205391,205393,205396,205399,205402,205405],{"className":205392},[31676],[34,205394],{"className":205395,"style":32617},[31680],[34,205397,205234],{"className":205398},[31685],[34,205400],{"className":205401,"style":31846},[31845],[34,205403,202],{"className":205404},[31850],[34,205406],{"className":205407,"style":31846},[31845],[34,205409,205411,205414],{"className":205410},[31676],[34,205412],{"className":205413,"style":32617},[31680],[34,205415,5752],{"className":205416},[31685]," 位。\n因此，常规页表共有 ",[34,205419,205421,205439],{"className":205420},[31642],[34,205422,205424],{"className":205423},[31646],[31648,205425,205426],{"xmlns":31650},[31652,205427,205428,205436],{},[31655,205429,205430],{},[37624,205431,205432,205434],{},[31749,205433,768],{},[31749,205435,5752],{},[31665,205437,205438],{"encoding":31667},"2^{35}",[34,205440,205442],{"className":205441,"ariaHidden":758},[31672],[34,205443,205445,205448],{"className":205444},[31676],[34,205446],{"className":205447,"style":37774},[31680],[34,205449,205451,205454],{"className":205450},[31685],[34,205452,768],{"className":205453},[31685],[34,205455,205457],{"className":205456},[31792],[34,205458,205460],{"className":205459},[31796],[34,205461,205463],{"className":205462},[31801],[34,205464,205466],{"className":205465,"style":37774},[31805],[34,205467,205468,205471],{"style":37777},[34,205469],{"className":205470,"style":31814},[31813],[34,205472,205474],{"className":205473},[31818,31819,31820,31821],[34,205475,205477],{"className":205476},[31685,31821],[34,205478,5752],{"className":205479},[31685,31821]," 个页表项。",[108,205482,205483,205486,205487,205490,205491,205564,205565,205627,205628,205480],{},[2627,205484,205485],{},"计算反置页表 (Inverted Page Table) 的页表项数量","：\n反置页表的核心思想是：",[2627,205488,205489],{},"不为每个虚拟页建表，而是为物理内存的每个页框 (Frame) 建表","。\n物理页框号 (PFN) 位数 = 物理地址位数 - 页内偏移位数 = ",[34,205492,205494,205516],{"className":205493},[31642],[34,205495,205497],{"className":205496},[31646],[31648,205498,205499],{"xmlns":31650},[31652,205500,205501,205513],{},[31655,205502,205503,205505,205507,205509,205511],{},[31749,205504,173486],{},[31662,205506,32226],{},[31749,205508,205234],{},[31662,205510,202],{},[31749,205512,162590],{},[31665,205514,205515],{"encoding":31667},"32 - 13 = 19",[34,205517,205519,205537,205555],{"className":205518,"ariaHidden":758},[31672],[34,205520,205522,205525,205528,205531,205534],{"className":205521},[31676],[34,205523],{"className":205524,"style":35942},[31680],[34,205526,173486],{"className":205527},[31685],[34,205529],{"className":205530,"style":32604},[31845],[34,205532,32226],{"className":205533},[32024],[34,205535],{"className":205536,"style":32604},[31845],[34,205538,205540,205543,205546,205549,205552],{"className":205539},[31676],[34,205541],{"className":205542,"style":32617},[31680],[34,205544,205234],{"className":205545},[31685],[34,205547],{"className":205548,"style":31846},[31845],[34,205550,202],{"className":205551},[31850],[34,205553],{"className":205554,"style":31846},[31845],[34,205556,205558,205561],{"className":205557},[31676],[34,205559],{"className":205560,"style":32617},[31680],[34,205562,162590],{"className":205563},[31685]," 位。\n因此，系统中实际存在的物理页框总数为 ",[34,205566,205568,205586],{"className":205567},[31642],[34,205569,205571],{"className":205570},[31646],[31648,205572,205573],{"xmlns":31650},[31652,205574,205575,205583],{},[31655,205576,205577],{},[37624,205578,205579,205581],{},[31749,205580,768],{},[31749,205582,162590],{},[31665,205584,205585],{"encoding":31667},"2^{19}",[34,205587,205589],{"className":205588,"ariaHidden":758},[31672],[34,205590,205592,205595],{"className":205591},[31676],[34,205593],{"className":205594,"style":37774},[31680],[34,205596,205598,205601],{"className":205597},[31685],[34,205599,768],{"className":205600},[31685],[34,205602,205604],{"className":205603},[31792],[34,205605,205607],{"className":205606},[31796],[34,205608,205610],{"className":205609},[31801],[34,205611,205613],{"className":205612,"style":37774},[31805],[34,205614,205615,205618],{"style":37777},[34,205616],{"className":205617,"style":31814},[31813],[34,205619,205621],{"className":205620},[31818,31819,31820,31821],[34,205622,205624],{"className":205623},[31685,31821],[34,205625,162590],{"className":205626},[31685,31821]," 个，反置页表共有 ",[34,205629,205631,205648],{"className":205630},[31642],[34,205632,205634],{"className":205633},[31646],[31648,205635,205636],{"xmlns":31650},[31652,205637,205638,205646],{},[31655,205639,205640],{},[37624,205641,205642,205644],{},[31749,205643,768],{},[31749,205645,162590],{},[31665,205647,205585],{"encoding":31667},[34,205649,205651],{"className":205650,"ariaHidden":758},[31672],[34,205652,205654,205657],{"className":205653},[31676],[34,205655],{"className":205656,"style":37774},[31680],[34,205658,205660,205663],{"className":205659},[31685],[34,205661,768],{"className":205662},[31685],[34,205664,205666],{"className":205665},[31792],[34,205667,205669],{"className":205668},[31796],[34,205670,205672],{"className":205671},[31801],[34,205673,205675],{"className":205674,"style":37774},[31805],[34,205676,205677,205680],{"style":37777},[34,205678],{"className":205679,"style":31814},[31813],[34,205681,205683],{"className":205682},[31818,31819,31820,31821],[34,205684,205686],{"className":205685},[31685,31821],[34,205687,162590],{"className":205688},[31685,31821],[32188,205690],{},[19,205692,205694],{"id":205693},"_10-二级页表的逻辑地址计算","10. 二级页表的逻辑地址计算",[15,205696,205697,205699],{},[2627,205698,204841],{},"\n一个32位地址的计算机系统使用二级页表，虚地址被分为9位顶级页表，11位二级页表和偏移。试问：页面长度是多少？虚地址空间共有多少个页面？",[15,205701,205702],{},[2627,205703,204847],{},[105,205705,205706,205716,205938],{},[108,205707,205708,205711,205712,205715],{},[2627,205709,205710],{},"逻辑地址结构解析","：\n在 32 位虚地址系统中，逻辑地址被划分为三部分：",[31,205713,205714],{},"[ 顶级页表号 | 二级页表号 | 页内偏移量 ]","。\n已知：总长 = 32 位，顶级页表 = 9 位，二级页表 = 11 位。",[108,205717,205718,205721,205722,205871,205872,205934,205935,347],{},[2627,205719,205720],{},"计算页面长度","：\n页内偏移量的位数 = 总位数 - (顶级页表位数 + 二级页表位数)\n偏移量位数 = ",[34,205723,205725,205763],{"className":205724},[31642],[34,205726,205728],{"className":205727},[31646],[31648,205729,205730],{"xmlns":31650},[31652,205731,205732,205760],{},[31655,205733,205734,205736,205738,205740,205742,205744,205746,205748,205750,205752,205754,205756,205758],{},[31749,205735,173486],{},[31662,205737,32226],{},[31662,205739,456],{"stretchy":702},[31749,205741,36732],{},[31662,205743,223],{},[31749,205745,39620],{},[31662,205747,2557],{"stretchy":702},[31662,205749,202],{},[31749,205751,173486],{},[31662,205753,32226],{},[31749,205755,3249],{},[31662,205757,202],{},[31749,205759,71869],{},[31665,205761,205762],{"encoding":31667},"32 - (9 + 11) = 32 - 20 = 12",[34,205764,205766,205784,205805,205826,205844,205862],{"className":205765,"ariaHidden":758},[31672],[34,205767,205769,205772,205775,205778,205781],{"className":205768},[31676],[34,205770],{"className":205771,"style":35942},[31680],[34,205773,173486],{"className":205774},[31685],[34,205776],{"className":205777,"style":32604},[31845],[34,205779,32226],{"className":205780},[32024],[34,205782],{"className":205783,"style":32604},[31845],[34,205785,205787,205790,205793,205796,205799,205802],{"className":205786},[31676],[34,205788],{"className":205789,"style":31774},[31680],[34,205791,456],{"className":205792},[31782],[34,205794,36732],{"className":205795},[31685],[34,205797],{"className":205798,"style":32604},[31845],[34,205800,223],{"className":205801},[32024],[34,205803],{"className":205804,"style":32604},[31845],[34,205806,205808,205811,205814,205817,205820,205823],{"className":205807},[31676],[34,205809],{"className":205810,"style":31774},[31680],[34,205812,39620],{"className":205813},[31685],[34,205815,2557],{"className":205816},[31690],[34,205818],{"className":205819,"style":31846},[31845],[34,205821,202],{"className":205822},[31850],[34,205824],{"className":205825,"style":31846},[31845],[34,205827,205829,205832,205835,205838,205841],{"className":205828},[31676],[34,205830],{"className":205831,"style":35942},[31680],[34,205833,173486],{"className":205834},[31685],[34,205836],{"className":205837,"style":32604},[31845],[34,205839,32226],{"className":205840},[32024],[34,205842],{"className":205843,"style":32604},[31845],[34,205845,205847,205850,205853,205856,205859],{"className":205846},[31676],[34,205848],{"className":205849,"style":32617},[31680],[34,205851,3249],{"className":205852},[31685],[34,205854],{"className":205855,"style":31846},[31845],[34,205857,202],{"className":205858},[31850],[34,205860],{"className":205861,"style":31846},[31845],[34,205863,205865,205868],{"className":205864},[31676],[34,205866],{"className":205867,"style":32617},[31680],[34,205869,71869],{"className":205870},[31685]," 位。\n页面长度取决于页内偏移量，因此页面长度 = ",[34,205873,205875,205893],{"className":205874},[31642],[34,205876,205878],{"className":205877},[31646],[31648,205879,205880],{"xmlns":31650},[31652,205881,205882,205890],{},[31655,205883,205884],{},[37624,205885,205886,205888],{},[31749,205887,768],{},[31749,205889,71869],{},[31665,205891,205892],{"encoding":31667},"2^{12}",[34,205894,205896],{"className":205895,"ariaHidden":758},[31672],[34,205897,205899,205902],{"className":205898},[31676],[34,205900],{"className":205901,"style":37774},[31680],[34,205903,205905,205908],{"className":205904},[31685],[34,205906,768],{"className":205907},[31685],[34,205909,205911],{"className":205910},[31792],[34,205912,205914],{"className":205913},[31796],[34,205915,205917],{"className":205916},[31801],[34,205918,205920],{"className":205919,"style":37774},[31805],[34,205921,205922,205925],{"style":37777},[34,205923],{"className":205924,"style":31814},[31813],[34,205926,205928],{"className":205927},[31818,31819,31820,31821],[34,205929,205931],{"className":205930},[31685,31821],[34,205932,71869],{"className":205933},[31685,31821]," 字节 = ",[2627,205936,205937],{},"4 KB",[108,205939,205940,205943,205944,206017,206018,206080,206081,347],{},[2627,205941,205942],{},"计算虚地址空间共有多少个页面","：\n决定页面总数的是表示“页号”的所有位数的总和。\n总页号位数 = 顶级页表位数 + 二级页表位数 = ",[34,205945,205947,205969],{"className":205946},[31642],[34,205948,205950],{"className":205949},[31646],[31648,205951,205952],{"xmlns":31650},[31652,205953,205954,205966],{},[31655,205955,205956,205958,205960,205962,205964],{},[31749,205957,36732],{},[31662,205959,223],{},[31749,205961,39620],{},[31662,205963,202],{},[31749,205965,3249],{},[31665,205967,205968],{"encoding":31667},"9 + 11 = 20",[34,205970,205972,205990,206008],{"className":205971,"ariaHidden":758},[31672],[34,205973,205975,205978,205981,205984,205987],{"className":205974},[31676],[34,205976],{"className":205977,"style":35942},[31680],[34,205979,36732],{"className":205980},[31685],[34,205982],{"className":205983,"style":32604},[31845],[34,205985,223],{"className":205986},[32024],[34,205988],{"className":205989,"style":32604},[31845],[34,205991,205993,205996,205999,206002,206005],{"className":205992},[31676],[34,205994],{"className":205995,"style":32617},[31680],[34,205997,39620],{"className":205998},[31685],[34,206000],{"className":206001,"style":31846},[31845],[34,206003,202],{"className":206004},[31850],[34,206006],{"className":206007,"style":31846},[31845],[34,206009,206011,206014],{"className":206010},[31676],[34,206012],{"className":206013,"style":32617},[31680],[34,206015,3249],{"className":206016},[31685]," 位。\n虚地址空间的总页面数 = ",[34,206019,206021,206039],{"className":206020},[31642],[34,206022,206024],{"className":206023},[31646],[31648,206025,206026],{"xmlns":31650},[31652,206027,206028,206036],{},[31655,206029,206030],{},[37624,206031,206032,206034],{},[31749,206033,768],{},[31749,206035,3249],{},[31665,206037,206038],{"encoding":31667},"2^{20}",[34,206040,206042],{"className":206041,"ariaHidden":758},[31672],[34,206043,206045,206048],{"className":206044},[31676],[34,206046],{"className":206047,"style":37774},[31680],[34,206049,206051,206054],{"className":206050},[31685],[34,206052,768],{"className":206053},[31685],[34,206055,206057],{"className":206056},[31792],[34,206058,206060],{"className":206059},[31796],[34,206061,206063],{"className":206062},[31801],[34,206064,206066],{"className":206065,"style":37774},[31805],[34,206067,206068,206071],{"style":37777},[34,206069],{"className":206070,"style":31814},[31813],[34,206072,206074],{"className":206073},[31818,31819,31820,31821],[34,206075,206077],{"className":206076},[31685,31821],[34,206078,3249],{"className":206079},[31685,31821]," 个 = ",[2627,206082,206083],{},"1M (1,048,576) 个页面",[32188,206085],{},[19,206087,206089],{"id":206088},"_11-快表命中率计算","11. 快表命中率计算",[15,206091,206092,206094],{},[2627,206093,204841],{},"\n一台计算机的内存空间为1024个页面，页表放在内存中，从页表中读一个字的开销是500ns。为了减少开销，使用了有32个字的快表，查找速度为100ns。要把平均开销降到200ns需要的快表命中率是多少?",[15,206096,206097],{},[2627,206098,204847],{},[105,206100,206101,206525,207490],{},[108,206102,206103,206106,206107,347,206110],{},[2627,206104,206105],{},"参数梳理","：\n这里所说的“开销”特指",[2627,206108,206109],{},"地址转换过程的耗时",[105,206111,206112,206238,206358,206494],{},[108,206113,206114,206199,206200,347],{},[34,206115,206117,206141],{"className":206116},[31642],[34,206118,206120],{"className":206119},[31646],[31648,206121,206122],{"xmlns":31650},[31652,206123,206124,206138],{},[31655,206125,206126],{},[31743,206127,206128,206130],{},[31658,206129,33249],{},[31655,206131,206132,206134,206136],{},[31658,206133,37400],{},[31658,206135,33206],{},[31658,206137,54224],{},[31665,206139,206140],{"encoding":31667},"t_{TLB}",[34,206142,206144],{"className":206143,"ariaHidden":758},[31672],[34,206145,206147,206150],{"className":206146},[31676],[34,206148],{"className":206149,"style":83872},[31680],[34,206151,206153,206156],{"className":206152},[31685],[34,206154,33249],{"className":206155},[31685,31686],[34,206157,206159],{"className":206158},[31792],[34,206160,206162,206191],{"className":206161},[31796,31797],[34,206163,206165,206188],{"className":206164},[31801],[34,206166,206168],{"className":206167,"style":116558},[31805],[34,206169,206170,206173],{"style":31809},[34,206171],{"className":206172,"style":31814},[31813],[34,206174,206176],{"className":206175},[31818,31819,31820,31821],[34,206177,206179,206182,206185],{"className":206178},[31685,31821],[34,206180,37400],{"className":206181,"style":37422},[31685,31686,31821],[34,206183,33206],{"className":206184},[31685,31686,31821],[34,206186,54224],{"className":206187,"style":54252},[31685,31686,31821],[34,206189,31829],{"className":206190},[31828],[34,206192,206194],{"className":206193},[31801],[34,206195,206197],{"className":206196,"style":31836},[31805],[34,206198],{}," = 访问快表的时间 = ",[34,206201,206203,206220],{"className":206202},[31642],[34,206204,206206],{"className":206205},[31646],[31648,206207,206208],{"xmlns":31650},[31652,206209,206210,206217],{},[31655,206211,206212,206214],{},[31749,206213,162173],{},[39361,206215,206216],{},"ns",[31665,206218,206219],{"encoding":31667},"100\\text{ns}",[34,206221,206223],{"className":206222,"ariaHidden":758},[31672],[34,206224,206226,206229,206232],{"className":206225},[31676],[34,206227],{"className":206228,"style":32617},[31680],[34,206230,162173],{"className":206231},[31685],[34,206233,206235],{"className":206234},[31685,2638],[34,206236,206216],{"className":206237},[31685],[108,206239,206240,206320,206321,347],{},[34,206241,206243,206265],{"className":206242},[31642],[34,206244,206246],{"className":206245},[31646],[31648,206247,206248],{"xmlns":31650},[31652,206249,206250,206262],{},[31655,206251,206252],{},[31743,206253,206254,206256],{},[31658,206255,33249],{},[31655,206257,206258,206260],{},[31658,206259,105852],{},[31658,206261,37400],{},[31665,206263,206264],{"encoding":31667},"t_{PT}",[34,206266,206268],{"className":206267,"ariaHidden":758},[31672],[34,206269,206271,206274],{"className":206270},[31676],[34,206272],{"className":206273,"style":83872},[31680],[34,206275,206277,206280],{"className":206276},[31685],[34,206278,33249],{"className":206279},[31685,31686],[34,206281,206283],{"className":206282},[31792],[34,206284,206286,206312],{"className":206285},[31796,31797],[34,206287,206289,206309],{"className":206288},[31801],[34,206290,206292],{"className":206291,"style":116558},[31805],[34,206293,206294,206297],{"style":31809},[34,206295],{"className":206296,"style":31814},[31813],[34,206298,206300],{"className":206299},[31818,31819,31820,31821],[34,206301,206303,206306],{"className":206302},[31685,31821],[34,206304,105852],{"className":206305,"style":37422},[31685,31686,31821],[34,206307,37400],{"className":206308,"style":37422},[31685,31686,31821],[34,206310,31829],{"className":206311},[31828],[34,206313,206315],{"className":206314},[31801],[34,206316,206318],{"className":206317,"style":31836},[31805],[34,206319],{}," = 访问内存中页表的时间 = ",[34,206322,206324,206340],{"className":206323},[31642],[34,206325,206327],{"className":206326},[31646],[31648,206328,206329],{"xmlns":31650},[31652,206330,206331,206337],{},[31655,206332,206333,206335],{},[31749,206334,25024],{},[39361,206336,206216],{},[31665,206338,206339],{"encoding":31667},"500\\text{ns}",[34,206341,206343],{"className":206342,"ariaHidden":758},[31672],[34,206344,206346,206349,206352],{"className":206345},[31676],[34,206347],{"className":206348,"style":32617},[31680],[34,206350,25024],{"className":206351},[31685],[34,206353,206355],{"className":206354},[31685,2638],[34,206356,206216],{"className":206357},[31685],[108,206359,206360,206361,206456,206457,347],{},"目标平均地址转换开销 ",[34,206362,206364,206392],{"className":206363},[31642],[34,206365,206367],{"className":206366},[31646],[31648,206368,206369],{"xmlns":31650},[31652,206370,206371,206389],{},[31655,206372,206373],{},[31743,206374,206375,206377],{},[31658,206376,96058],{},[31655,206378,206379,206381,206383,206385,206387],{},[31658,206380,33249],{},[31658,206382,34395],{},[31658,206384,111],{},[31658,206386,31660],{},[31658,206388,31747],{},[31665,206390,206391],{"encoding":31667},"E_{trans}",[34,206393,206395],{"className":206394,"ariaHidden":758},[31672],[34,206396,206398,206401],{"className":206397},[31676],[34,206399],{"className":206400,"style":70915},[31680],[34,206402,206404,206407],{"className":206403},[31685],[34,206405,96058],{"className":206406,"style":106455},[31685,31686],[34,206408,206410],{"className":206409},[31792],[34,206411,206413,206448],{"className":206412},[31796,31797],[34,206414,206416,206445],{"className":206415},[31801],[34,206417,206420],{"className":206418,"style":206419},[31805],"height:0.2806em;",[34,206421,206423,206426],{"style":206422},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[34,206424],{"className":206425,"style":31814},[31813],[34,206427,206429],{"className":206428},[31818,31819,31820,31821],[34,206430,206432,206435,206438,206442],{"className":206431},[31685,31821],[34,206433,33249],{"className":206434},[31685,31686,31821],[34,206436,34395],{"className":206437,"style":34452},[31685,31686,31821],[34,206439,206441],{"className":206440},[31685,31686,31821],"an",[34,206443,31747],{"className":206444},[31685,31686,31821],[34,206446,31829],{"className":206447},[31828],[34,206449,206451],{"className":206450},[31801],[34,206452,206454],{"className":206453,"style":31836},[31805],[34,206455],{}," = ",[34,206458,206460,206476],{"className":206459},[31642],[34,206461,206463],{"className":206462},[31646],[31648,206464,206465],{"xmlns":31650},[31652,206466,206467,206473],{},[31655,206468,206469,206471],{},[31749,206470,168915],{},[39361,206472,206216],{},[31665,206474,206475],{"encoding":31667},"200\\text{ns}",[34,206477,206479],{"className":206478,"ariaHidden":758},[31672],[34,206480,206482,206485,206488],{"className":206481},[31676],[34,206483],{"className":206484,"style":32617},[31680],[34,206486,168915],{"className":206487},[31685],[34,206489,206491],{"className":206490},[31685,2638],[34,206492,206216],{"className":206493},[31685],[108,206495,206496,206497,347],{},"设所需的快表命中率为 ",[34,206498,206500,206513],{"className":206499},[31642],[34,206501,206503],{"className":206502},[31646],[31648,206504,206505],{"xmlns":31650},[31652,206506,206507,206511],{},[31655,206508,206509],{},[31658,206510,118113],{},[31665,206512,118113],{"encoding":31667},[34,206514,206516],{"className":206515,"ariaHidden":758},[31672],[34,206517,206519,206522],{"className":206518},[31676],[34,206520],{"className":206521,"style":33218},[31680],[34,206523,118113],{"className":206524,"style":118235},[31685,31686],[108,206526,206527,206530,206531,206815,206817,206818,207084,207189,207267,207323],{},[2627,206528,206529],{},"公式构建与计算","：\n通常在没有明确说明“并行查找”的情况下，地址转换是顺序进行的。",[2230,206532,206533,206603],{},[108,206534,206535,206566,206567,347],{},[2627,206536,206537,206538,2557],{},"若命中快表 (概率 ",[34,206539,206541,206554],{"className":206540},[31642],[34,206542,206544],{"className":206543},[31646],[31648,206545,206546],{"xmlns":31650},[31652,206547,206548,206552],{},[31655,206549,206550],{},[31658,206551,118113],{},[31665,206553,118113],{"encoding":31667},[34,206555,206557],{"className":206556,"ariaHidden":758},[31672],[34,206558,206560,206563],{"className":206559},[31676],[34,206561],{"className":206562,"style":33218},[31680],[34,206564,118113],{"className":206565,"style":118235},[31685,31686],"：只需查快表，时间开销为 ",[34,206568,206570,206585],{"className":206569},[31642],[34,206571,206573],{"className":206572},[31646],[31648,206574,206575],{"xmlns":31650},[31652,206576,206577,206583],{},[31655,206578,206579,206581],{},[31749,206580,162173],{},[39361,206582,206216],{},[31665,206584,206219],{"encoding":31667},[34,206586,206588],{"className":206587,"ariaHidden":758},[31672],[34,206589,206591,206594,206597],{"className":206590},[31676],[34,206592],{"className":206593,"style":32617},[31680],[34,206595,162173],{"className":206596},[31685],[34,206598,206600],{"className":206599},[31685,2638],[34,206601,206216],{"className":206602},[31685],[108,206604,206605,206659,206660,206696,206697,206733,206734,347],{},[2627,206606,206607,206608,2557],{},"若未命中快表 (概率 ",[34,206609,206611,206629],{"className":206610},[31642],[34,206612,206614],{"className":206613},[31646],[31648,206615,206616],{"xmlns":31650},[31652,206617,206618,206626],{},[31655,206619,206620,206622,206624],{},[31749,206621,640],{},[31662,206623,32226],{},[31658,206625,118113],{},[31665,206627,206628],{"encoding":31667},"1 - H",[34,206630,206632,206650],{"className":206631,"ariaHidden":758},[31672],[34,206633,206635,206638,206641,206644,206647],{"className":206634},[31676],[34,206636],{"className":206637,"style":35942},[31680],[34,206639,640],{"className":206640},[31685],[34,206642],{"className":206643,"style":32604},[31845],[34,206645,32226],{"className":206646},[32024],[34,206648],{"className":206649,"style":32604},[31845],[34,206651,206653,206656],{"className":206652},[31676],[34,206654],{"className":206655,"style":33218},[31680],[34,206657,118113],{"className":206658,"style":118235},[31685,31686],"：先查快表失败耗费 ",[34,206661,206663,206678],{"className":206662},[31642],[34,206664,206666],{"className":206665},[31646],[31648,206667,206668],{"xmlns":31650},[31652,206669,206670,206676],{},[31655,206671,206672,206674],{},[31749,206673,162173],{},[39361,206675,206216],{},[31665,206677,206219],{"encoding":31667},[34,206679,206681],{"className":206680,"ariaHidden":758},[31672],[34,206682,206684,206687,206690],{"className":206683},[31676],[34,206685],{"className":206686,"style":32617},[31680],[34,206688,162173],{"className":206689},[31685],[34,206691,206693],{"className":206692},[31685,2638],[34,206694,206216],{"className":206695},[31685],"，然后再去主存查页表耗费 ",[34,206698,206700,206715],{"className":206699},[31642],[34,206701,206703],{"className":206702},[31646],[31648,206704,206705],{"xmlns":31650},[31652,206706,206707,206713],{},[31655,206708,206709,206711],{},[31749,206710,25024],{},[39361,206712,206216],{},[31665,206714,206339],{"encoding":31667},[34,206716,206718],{"className":206717,"ariaHidden":758},[31672],[34,206719,206721,206724,206727],{"className":206720},[31676],[34,206722],{"className":206723,"style":32617},[31680],[34,206725,25024],{"className":206726},[31685],[34,206728,206730],{"className":206729},[31685,2638],[34,206731,206216],{"className":206732},[31685],"，总开销为 ",[34,206735,206737,206761],{"className":206736},[31642],[34,206738,206740],{"className":206739},[31646],[31648,206741,206742],{"xmlns":31650},[31652,206743,206744,206758],{},[31655,206745,206746,206748,206750,206752,206754,206756],{},[31749,206747,162173],{},[31662,206749,223],{},[31749,206751,25024],{},[31662,206753,202],{},[31749,206755,75661],{},[39361,206757,206216],{},[31665,206759,206760],{"encoding":31667},"100 + 500 = 600\\text{ns}",[34,206762,206764,206782,206800],{"className":206763,"ariaHidden":758},[31672],[34,206765,206767,206770,206773,206776,206779],{"className":206766},[31676],[34,206768],{"className":206769,"style":35942},[31680],[34,206771,162173],{"className":206772},[31685],[34,206774],{"className":206775,"style":32604},[31845],[34,206777,223],{"className":206778},[32024],[34,206780],{"className":206781,"style":32604},[31845],[34,206783,206785,206788,206791,206794,206797],{"className":206784},[31676],[34,206786],{"className":206787,"style":32617},[31680],[34,206789,25024],{"className":206790},[31685],[34,206792],{"className":206793,"style":31846},[31845],[34,206795,202],{"className":206796},[31850],[34,206798],{"className":206799,"style":31846},[31845],[34,206801,206803,206806,206809],{"className":206802},[31676],[34,206804],{"className":206805,"style":32617},[31680],[34,206807,75661],{"className":206808},[31685],[34,206810,206812],{"className":206811},[31685,2638],[34,206813,206216],{"className":206814},[31685],[58531,206816],{},"平均开销公式如下：\n",[34,206819,206821,206885],{"className":206820},[31642],[34,206822,206824],{"className":206823},[31646],[31648,206825,206826],{"xmlns":31650},[31652,206827,206828,206882],{},[31655,206829,206830,206846,206848,206850,206852,206854,206856,206858,206860,206862,206864,206866,206868,206870,206872,206874,206876,206878,206880],{},[31743,206831,206832,206834],{},[31658,206833,96058],{},[31655,206835,206836,206838,206840,206842,206844],{},[31658,206837,33249],{},[31658,206839,34395],{},[31658,206841,111],{},[31658,206843,31660],{},[31658,206845,31747],{},[31662,206847,202],{},[31658,206849,118113],{},[31662,206851,42071],{},[31749,206853,162173],{},[31662,206855,223],{},[31662,206857,456],{"stretchy":702},[31749,206859,640],{},[31662,206861,32226],{},[31658,206863,118113],{},[31662,206865,2557],{"stretchy":702},[31662,206867,42071],{},[31662,206869,456],{"stretchy":702},[31749,206871,162173],{},[31662,206873,223],{},[31749,206875,25024],{},[31662,206877,2557],{"stretchy":702},[31662,206879,202],{},[31749,206881,168915],{},[31665,206883,206884],{"encoding":31667},"E_{trans} = H \\times 100 + (1 - H) \\times (100 + 500) = 200",[34,206886,206888,206955,206973,206991,207012,207033,207054,207075],{"className":206887,"ariaHidden":758},[31672],[34,206889,206891,206894,206946,206949,206952],{"className":206890},[31676],[34,206892],{"className":206893,"style":70915},[31680],[34,206895,206897,206900],{"className":206896},[31685],[34,206898,96058],{"className":206899,"style":106455},[31685,31686],[34,206901,206903],{"className":206902},[31792],[34,206904,206906,206938],{"className":206905},[31796,31797],[34,206907,206909,206935],{"className":206908},[31801],[34,206910,206912],{"className":206911,"style":206419},[31805],[34,206913,206914,206917],{"style":206422},[34,206915],{"className":206916,"style":31814},[31813],[34,206918,206920],{"className":206919},[31818,31819,31820,31821],[34,206921,206923,206926,206929,206932],{"className":206922},[31685,31821],[34,206924,33249],{"className":206925},[31685,31686,31821],[34,206927,34395],{"className":206928,"style":34452},[31685,31686,31821],[34,206930,206441],{"className":206931},[31685,31686,31821],[34,206933,31747],{"className":206934},[31685,31686,31821],[34,206936,31829],{"className":206937},[31828],[34,206939,206941],{"className":206940},[31801],[34,206942,206944],{"className":206943,"style":31836},[31805],[34,206945],{},[34,206947],{"className":206948,"style":31846},[31845],[34,206950,202],{"className":206951},[31850],[34,206953],{"className":206954,"style":31846},[31845],[34,206956,206958,206961,206964,206967,206970],{"className":206957},[31676],[34,206959],{"className":206960,"style":34944},[31680],[34,206962,118113],{"className":206963,"style":118235},[31685,31686],[34,206965],{"className":206966,"style":32604},[31845],[34,206968,42071],{"className":206969},[32024],[34,206971],{"className":206972,"style":32604},[31845],[34,206974,206976,206979,206982,206985,206988],{"className":206975},[31676],[34,206977],{"className":206978,"style":35942},[31680],[34,206980,162173],{"className":206981},[31685],[34,206983],{"className":206984,"style":32604},[31845],[34,206986,223],{"className":206987},[32024],[34,206989],{"className":206990,"style":32604},[31845],[34,206992,206994,206997,207000,207003,207006,207009],{"className":206993},[31676],[34,206995],{"className":206996,"style":31774},[31680],[34,206998,456],{"className":206999},[31782],[34,207001,640],{"className":207002},[31685],[34,207004],{"className":207005,"style":32604},[31845],[34,207007,32226],{"className":207008},[32024],[34,207010],{"className":207011,"style":32604},[31845],[34,207013,207015,207018,207021,207024,207027,207030],{"className":207014},[31676],[34,207016],{"className":207017,"style":31774},[31680],[34,207019,118113],{"className":207020,"style":118235},[31685,31686],[34,207022,2557],{"className":207023},[31690],[34,207025],{"className":207026,"style":32604},[31845],[34,207028,42071],{"className":207029},[32024],[34,207031],{"className":207032,"style":32604},[31845],[34,207034,207036,207039,207042,207045,207048,207051],{"className":207035},[31676],[34,207037],{"className":207038,"style":31774},[31680],[34,207040,456],{"className":207041},[31782],[34,207043,162173],{"className":207044},[31685],[34,207046],{"className":207047,"style":32604},[31845],[34,207049,223],{"className":207050},[32024],[34,207052],{"className":207053,"style":32604},[31845],[34,207055,207057,207060,207063,207066,207069,207072],{"className":207056},[31676],[34,207058],{"className":207059,"style":31774},[31680],[34,207061,25024],{"className":207062},[31685],[34,207064,2557],{"className":207065},[31690],[34,207067],{"className":207068,"style":31846},[31845],[34,207070,202],{"className":207071},[31850],[34,207073],{"className":207074,"style":31846},[31845],[34,207076,207078,207081],{"className":207077},[31676],[34,207079],{"className":207080,"style":32617},[31680],[34,207082,168915],{"className":207083},[31685],[34,207085,207087,207117],{"className":207086},[31642],[34,207088,207090],{"className":207089},[31646],[31648,207091,207092],{"xmlns":31650},[31652,207093,207094,207114],{},[31655,207095,207096,207098,207100,207102,207104,207106,207108,207110,207112],{},[31749,207097,162173],{},[31658,207099,118113],{},[31662,207101,223],{},[31749,207103,75661],{},[31662,207105,32226],{},[31749,207107,75661],{},[31658,207109,118113],{},[31662,207111,202],{},[31749,207113,168915],{},[31665,207115,207116],{"encoding":31667},"100H + 600 - 600H = 200",[34,207118,207120,207141,207159,207180],{"className":207119,"ariaHidden":758},[31672],[34,207121,207123,207126,207129,207132,207135,207138],{"className":207122},[31676],[34,207124],{"className":207125,"style":34944},[31680],[34,207127,162173],{"className":207128},[31685],[34,207130,118113],{"className":207131,"style":118235},[31685,31686],[34,207133],{"className":207134,"style":32604},[31845],[34,207136,223],{"className":207137},[32024],[34,207139],{"className":207140,"style":32604},[31845],[34,207142,207144,207147,207150,207153,207156],{"className":207143},[31676],[34,207145],{"className":207146,"style":35942},[31680],[34,207148,75661],{"className":207149},[31685],[34,207151],{"className":207152,"style":32604},[31845],[34,207154,32226],{"className":207155},[32024],[34,207157],{"className":207158,"style":32604},[31845],[34,207160,207162,207165,207168,207171,207174,207177],{"className":207161},[31676],[34,207163],{"className":207164,"style":33218},[31680],[34,207166,75661],{"className":207167},[31685],[34,207169,118113],{"className":207170,"style":118235},[31685,31686],[34,207172],{"className":207173,"style":31846},[31845],[34,207175,202],{"className":207176},[31850],[34,207178],{"className":207179,"style":31846},[31845],[34,207181,207183,207186],{"className":207182},[31676],[34,207184],{"className":207185,"style":32617},[31680],[34,207187,168915],{"className":207188},[31685],[34,207190,207192,207216],{"className":207191},[31642],[34,207193,207195],{"className":207194},[31646],[31648,207196,207197],{"xmlns":31650},[31652,207198,207199,207213],{},[31655,207200,207201,207203,207205,207207,207209,207211],{},[31749,207202,75661],{},[31662,207204,32226],{},[31749,207206,25024],{},[31658,207208,118113],{},[31662,207210,202],{},[31749,207212,168915],{},[31665,207214,207215],{"encoding":31667},"600 - 500H = 200",[34,207217,207219,207237,207258],{"className":207218,"ariaHidden":758},[31672],[34,207220,207222,207225,207228,207231,207234],{"className":207221},[31676],[34,207223],{"className":207224,"style":35942},[31680],[34,207226,75661],{"className":207227},[31685],[34,207229],{"className":207230,"style":32604},[31845],[34,207232,32226],{"className":207233},[32024],[34,207235],{"className":207236,"style":32604},[31845],[34,207238,207240,207243,207246,207249,207252,207255],{"className":207239},[31676],[34,207241],{"className":207242,"style":33218},[31680],[34,207244,25024],{"className":207245},[31685],[34,207247,118113],{"className":207248,"style":118235},[31685,31686],[34,207250],{"className":207251,"style":31846},[31845],[34,207253,202],{"className":207254},[31850],[34,207256],{"className":207257,"style":31846},[31845],[34,207259,207261,207264],{"className":207260},[31676],[34,207262],{"className":207263,"style":32617},[31680],[34,207265,168915],{"className":207266},[31685],[34,207268,207270,207290],{"className":207269},[31642],[34,207271,207273],{"className":207272},[31646],[31648,207274,207275],{"xmlns":31650},[31652,207276,207277,207287],{},[31655,207278,207279,207281,207283,207285],{},[31749,207280,25024],{},[31658,207282,118113],{},[31662,207284,202],{},[31749,207286,185297],{},[31665,207288,207289],{"encoding":31667},"500H = 400",[34,207291,207293,207314],{"className":207292,"ariaHidden":758},[31672],[34,207294,207296,207299,207302,207305,207308,207311],{"className":207295},[31676],[34,207297],{"className":207298,"style":33218},[31680],[34,207300,25024],{"className":207301},[31685],[34,207303,118113],{"className":207304,"style":118235},[31685,31686],[34,207306],{"className":207307,"style":31846},[31845],[34,207309,202],{"className":207310},[31850],[34,207312],{"className":207313,"style":31846},[31845],[34,207315,207317,207320],{"className":207316},[31676],[34,207318],{"className":207319,"style":32617},[31680],[34,207321,185297],{"className":207322},[31685],[34,207324,207326,207358],{"className":207325},[31642],[34,207327,207329],{"className":207328},[31646],[31648,207330,207331],{"xmlns":31650},[31652,207332,207333,207355],{},[31655,207334,207335,207337,207339,207345,207347,207349,207351,207353],{},[31658,207336,118113],{},[31662,207338,202],{},[41453,207340,207341,207343],{},[31749,207342,185297],{},[31749,207344,25024],{},[31662,207346,202],{},[31749,207348,194366],{},[31662,207350,202],{},[31749,207352,27539],{},[31658,207354,119426],{"mathvariant":38079},[31665,207356,207357],{"encoding":31667},"H = \\frac{400}{500} = 0.8 = 80\\%",[34,207359,207361,207379,207462,207480],{"className":207360,"ariaHidden":758},[31672],[34,207362,207364,207367,207370,207373,207376],{"className":207363},[31676],[34,207365],{"className":207366,"style":33218},[31680],[34,207368,118113],{"className":207369,"style":118235},[31685,31686],[34,207371],{"className":207372,"style":31846},[31845],[34,207374,202],{"className":207375},[31850],[34,207377],{"className":207378,"style":31846},[31845],[34,207380,207382,207385,207453,207456,207459],{"className":207381},[31676],[34,207383],{"className":207384,"style":46727},[31680],[34,207386,207388,207391,207450],{"className":207387},[31685],[34,207389],{"className":207390},[31782,41488],[34,207392,207394],{"className":207393},[41453],[34,207395,207397,207442],{"className":207396},[31796,31797],[34,207398,207400,207439],{"className":207399},[31801],[34,207401,207403,207417,207425],{"className":207402,"style":46746},[31805],[34,207404,207405,207408],{"style":41504},[34,207406],{"className":207407,"style":41508},[31813],[34,207409,207411],{"className":207410},[31818,31819,31820,31821],[34,207412,207414],{"className":207413},[31685,31821],[34,207415,25024],{"className":207416},[31685,31821],[34,207418,207419,207422],{"style":41520},[34,207420],{"className":207421,"style":41508},[31813],[34,207423],{"className":207424,"style":41528},[41527],[34,207426,207427,207430],{"style":46771},[34,207428],{"className":207429,"style":41508},[31813],[34,207431,207433],{"className":207432},[31818,31819,31820,31821],[34,207434,207436],{"className":207435},[31685,31821],[34,207437,185297],{"className":207438},[31685,31821],[34,207440,31829],{"className":207441},[31828],[34,207443,207445],{"className":207444},[31801],[34,207446,207448],{"className":207447,"style":41619},[31805],[34,207449],{},[34,207451],{"className":207452},[31690,41488],[34,207454],{"className":207455,"style":31846},[31845],[34,207457,202],{"className":207458},[31850],[34,207460],{"className":207461,"style":31846},[31845],[34,207463,207465,207468,207471,207474,207477],{"className":207464},[31676],[34,207466],{"className":207467,"style":32617},[31680],[34,207469,194366],{"className":207470},[31685],[34,207472],{"className":207473,"style":31846},[31845],[34,207475,202],{"className":207476},[31850],[34,207478],{"className":207479,"style":31846},[31845],[34,207481,207483,207486],{"className":207482},[31676],[34,207484],{"className":207485,"style":168117},[31680],[34,207487,207489],{"className":207488},[31685],"80%",[108,207491,207492,207494,207495,347],{},[2627,207493,118575],{},"：需要的快表命中率是 ",[2627,207496,207489],{},[32188,207498],{},[19,207500,207502],{"id":207501},"_12-位示图-bitmap-分配主存空间计算","12. 位示图 (Bitmap) 分配主存空间计算",[15,207504,207505,207507,207508,207510],{},[2627,207506,204841],{},"\n现有一台16位字长的专用机，采用页式存储管理。主存储器共有4096块(块号为0～4095)，现用位示图分配主存空间。试问：\n(1)该位示图占用几个字?",[58531,207509],{},"\n(2)主存块号3999对应位示图的字号和位号(均从0开始)各是多少?\n(3)位示图字号199，位号9对应主存的块号是多少?",[15,207512,207513],{},[2627,207514,204847],{},[105,207516,207517],{},[108,207518,207519,207521,207522],{},[2627,207520,204909],{},"：位示图 (Bitmap) 使用二进制的一位 (1 bit) 来表示主存中一个物理块的分配状态（如 0 表示空闲，1 表示已分配）。\n",[105,207523,207524,207527],{},[108,207525,207526],{},"字长：16 位（即 1 个字 = 16 bit）。",[108,207528,207529],{},"块数：4096 块，因此总共需要 4096 个二进制位。",[15,207531,207532],{},[2627,207533,207534],{},"（1）该位示图占用几个字？",[105,207536,207537,207649],{},[108,207538,207539,207542,207543,207573,207574,207648],{},[2627,207540,207541],{},"计算","：位示图所需的总字数 = 总物理块数 ",[34,207544,207546,207561],{"className":207545},[31642],[34,207547,207549],{"className":207548},[31646],[31648,207550,207551],{"xmlns":31650},[31652,207552,207553,207558],{},[31655,207554,207555],{},[31662,207556,207557],{},"÷",[31665,207559,207560],{"encoding":31667},"\\div",[34,207562,207564],{"className":207563,"ariaHidden":758},[31672],[34,207565,207567,207570],{"className":207566},[31676],[34,207568],{"className":207569,"style":32597},[31680],[34,207571,207557],{"className":207572},[31685]," 机器字长。\n总字数 = ",[34,207575,207577,207600],{"className":207576},[31642],[34,207578,207580],{"className":207579},[31646],[31648,207581,207582],{"xmlns":31650},[31652,207583,207584,207597],{},[31655,207585,207586,207589,207591,207593,207595],{},[31749,207587,207588],{},"4096",[31662,207590,207557],{},[31749,207592,177869],{},[31662,207594,202],{},[31749,207596,95418],{},[31665,207598,207599],{"encoding":31667},"4096 \\div 16 = 256",[34,207601,207603,207621,207639],{"className":207602,"ariaHidden":758},[31672],[34,207604,207606,207609,207612,207615,207618],{"className":207605},[31676],[34,207607],{"className":207608,"style":35942},[31680],[34,207610,207588],{"className":207611},[31685],[34,207613],{"className":207614,"style":32604},[31845],[34,207616,207557],{"className":207617},[32024],[34,207619],{"className":207620,"style":32604},[31845],[34,207622,207624,207627,207630,207633,207636],{"className":207623},[31676],[34,207625],{"className":207626,"style":32617},[31680],[34,207628,177869],{"className":207629},[31685],[34,207631],{"className":207632,"style":31846},[31845],[34,207634,202],{"className":207635},[31850],[34,207637],{"className":207638,"style":31846},[31845],[34,207640,207642,207645],{"className":207641},[31676],[34,207643],{"className":207644,"style":32617},[31680],[34,207646,95418],{"className":207647},[31685]," 个字。",[108,207650,207651,207653,207654,207648],{},[2627,207652,118575],{},"：该位示图占用 ",[2627,207655,95418],{},[15,207657,207658],{},[2627,207659,207660],{},"（2）主存块号3999对应位示图的字号和位号(均从0开始)各是多少？",[105,207662,207663,207940,208220],{},[108,207664,207665,207667,207668,207696,207697,207725,207726,207754,207755,207838,207839],{},[2627,207666,207541],{},"：因为字号和位号均从 0 开始，块号 ",[34,207669,207671,207684],{"className":207670},[31642],[34,207672,207674],{"className":207673},[31646],[31648,207675,207676],{"xmlns":31650},[31652,207677,207678,207682],{},[31655,207679,207680],{},[31658,207681,1697],{},[31665,207683,1697],{"encoding":31667},[34,207685,207687],{"className":207686,"ariaHidden":758},[31672],[34,207688,207690,207693],{"className":207689},[31676],[34,207691],{"className":207692,"style":31681},[31680],[34,207694,1697],{"className":207695},[31685,31686]," 与字号 ",[34,207698,207700,207713],{"className":207699},[31642],[34,207701,207703],{"className":207702},[31646],[31648,207704,207705],{"xmlns":31650},[31652,207706,207707,207711],{},[31655,207708,207709],{},[31658,207710,31955],{},[31665,207712,31955],{"encoding":31667},[34,207714,207716],{"className":207715,"ariaHidden":758},[31672],[34,207717,207719,207722],{"className":207718},[31676],[34,207720],{"className":207721,"style":33104},[31680],[34,207723,31955],{"className":207724},[31685,31686],"、位号 ",[34,207727,207729,207742],{"className":207728},[31642],[34,207730,207732],{"className":207731},[31646],[31648,207733,207734],{"xmlns":31650},[31652,207735,207736,207740],{},[31655,207737,207738],{},[31658,207739,31968],{},[31665,207741,31968],{"encoding":31667},[34,207743,207745],{"className":207744,"ariaHidden":758},[31672],[34,207746,207748,207751],{"className":207747},[31676],[34,207749],{"className":207750,"style":33123},[31680],[34,207752,31968],{"className":207753,"style":32153},[31685,31686]," 的关系为：\n字号 ",[34,207756,207758,207784],{"className":207757},[31642],[34,207759,207761],{"className":207760},[31646],[31648,207762,207763],{"xmlns":31650},[31652,207764,207765,207781],{},[31655,207766,207767,207769,207771,207773,207775,207777,207779],{},[31658,207768,31955],{},[31662,207770,202],{},[31662,207772,173327],{"stretchy":702},[31658,207774,1697],{},[31662,207776,207557],{},[31749,207778,177869],{},[31662,207780,173336],{"stretchy":702},[31665,207782,207783],{"encoding":31667},"i = \\lfloor b \\div 16 \\rfloor",[34,207785,207787,207805,207826],{"className":207786,"ariaHidden":758},[31672],[34,207788,207790,207793,207796,207799,207802],{"className":207789},[31676],[34,207791],{"className":207792,"style":33104},[31680],[34,207794,31955],{"className":207795},[31685,31686],[34,207797],{"className":207798,"style":31846},[31845],[34,207800,202],{"className":207801},[31850],[34,207803],{"className":207804,"style":31846},[31845],[34,207806,207808,207811,207814,207817,207820,207823],{"className":207807},[31676],[34,207809],{"className":207810,"style":31774},[31680],[34,207812,173327],{"className":207813},[31782],[34,207815,1697],{"className":207816},[31685,31686],[34,207818],{"className":207819,"style":32604},[31845],[34,207821,207557],{"className":207822},[32024],[34,207824],{"className":207825,"style":32604},[31845],[34,207827,207829,207832,207835],{"className":207828},[31676],[34,207830],{"className":207831,"style":31774},[31680],[34,207833,177869],{"className":207834},[31685],[34,207836,173336],{"className":207837},[31690],"\n位号 ",[34,207840,207842,207878],{"className":207841},[31642],[34,207843,207845],{"className":207844},[31646],[31648,207846,207847],{"xmlns":31650},[31652,207848,207849,207875],{},[31655,207850,207851,207853,207855,207857,207860,207871,207873],{},[31658,207852,31968],{},[31662,207854,202],{},[31658,207856,1697],{},[39361,207858,207859],{}," ",[31662,207861,207863],{"lspace":207862,"rspace":207862},"0.22em",[31655,207864,207865,207867,207869],{},[31658,207866,31759],{"mathvariant":38079},[31658,207868,43383],{"mathvariant":38079},[31658,207870,46900],{"mathvariant":38079},[39361,207872,207859],{},[31749,207874,177869],{},[31665,207876,207877],{"encoding":31667},"j = b \\bmod 16",[34,207879,207881,207899,207931],{"className":207880,"ariaHidden":758},[31672],[34,207882,207884,207887,207890,207893,207896],{"className":207883},[31676],[34,207885],{"className":207886,"style":33123},[31680],[34,207888,31968],{"className":207889,"style":32153},[31685,31686],[34,207891],{"className":207892,"style":31846},[31845],[34,207894,202],{"className":207895},[31850],[34,207897],{"className":207898,"style":31846},[31845],[34,207900,207902,207905,207908,207911,207914,207925,207928],{"className":207901},[31676],[34,207903],{"className":207904,"style":31681},[31680],[34,207906,1697],{"className":207907},[31685,31686],[34,207909],{"className":207910,"style":112322},[31845],[34,207912],{"className":207913,"style":32604},[31845],[34,207915,207917],{"className":207916},[32024],[34,207918,207920],{"className":207919},[31685],[34,207921,207924],{"className":207922},[31685,207923],"mathrm","mod",[34,207926],{"className":207927,"style":112322},[31845],[34,207929],{"className":207930,"style":32604},[31845],[34,207932,207934,207937],{"className":207933},[31676],[34,207935],{"className":207936,"style":32617},[31680],[34,207938,177869],{"className":207939},[31685],[108,207941,207942,207943,207995,207996,207838,208101],{},"代入块号 ",[34,207944,207946,207965],{"className":207945},[31642],[34,207947,207949],{"className":207948},[31646],[31648,207950,207951],{"xmlns":31650},[31652,207952,207953,207962],{},[31655,207954,207955,207957,207959],{},[31658,207956,1697],{},[31662,207958,202],{},[31749,207960,207961],{},"3999",[31665,207963,207964],{"encoding":31667},"b = 3999",[34,207966,207968,207986],{"className":207967,"ariaHidden":758},[31672],[34,207969,207971,207974,207977,207980,207983],{"className":207970},[31676],[34,207972],{"className":207973,"style":31681},[31680],[34,207975,1697],{"className":207976},[31685,31686],[34,207978],{"className":207979,"style":31846},[31845],[34,207981,202],{"className":207982},[31850],[34,207984],{"className":207985,"style":31846},[31845],[34,207987,207989,207992],{"className":207988},[31676],[34,207990],{"className":207991,"style":32617},[31680],[34,207993,207961],{"className":207994},[31685],"：\n字号 ",[34,207997,207999,208029],{"className":207998},[31642],[34,208000,208002],{"className":208001},[31646],[31648,208003,208004],{"xmlns":31650},[31652,208005,208006,208026],{},[31655,208007,208008,208010,208012,208014,208016,208018,208020,208022,208024],{},[31658,208009,31955],{},[31662,208011,202],{},[31662,208013,173327],{"stretchy":702},[31749,208015,207961],{},[31662,208017,207557],{},[31749,208019,177869],{},[31662,208021,173336],{"stretchy":702},[31662,208023,202],{},[31749,208025,184205],{},[31665,208027,208028],{"encoding":31667},"i = \\lfloor 3999 \\div 16 \\rfloor = 249",[34,208030,208032,208050,208071,208092],{"className":208031,"ariaHidden":758},[31672],[34,208033,208035,208038,208041,208044,208047],{"className":208034},[31676],[34,208036],{"className":208037,"style":33104},[31680],[34,208039,31955],{"className":208040},[31685,31686],[34,208042],{"className":208043,"style":31846},[31845],[34,208045,202],{"className":208046},[31850],[34,208048],{"className":208049,"style":31846},[31845],[34,208051,208053,208056,208059,208062,208065,208068],{"className":208052},[31676],[34,208054],{"className":208055,"style":31774},[31680],[34,208057,173327],{"className":208058},[31782],[34,208060,207961],{"className":208061},[31685],[34,208063],{"className":208064,"style":32604},[31845],[34,208066,207557],{"className":208067},[32024],[34,208069],{"className":208070,"style":32604},[31845],[34,208072,208074,208077,208080,208083,208086,208089],{"className":208073},[31676],[34,208075],{"className":208076,"style":31774},[31680],[34,208078,177869],{"className":208079},[31685],[34,208081,173336],{"className":208082},[31690],[34,208084],{"className":208085,"style":31846},[31845],[34,208087,202],{"className":208088},[31850],[34,208090],{"className":208091,"style":31846},[31845],[34,208093,208095,208098],{"className":208094},[31676],[34,208096],{"className":208097,"style":32617},[31680],[34,208099,184205],{"className":208100},[31685],[34,208102,208104,208142],{"className":208103},[31642],[34,208105,208107],{"className":208106},[31646],[31648,208108,208109],{"xmlns":31650},[31652,208110,208111,208139],{},[31655,208112,208113,208115,208117,208119,208121,208131,208133,208135,208137],{},[31658,208114,31968],{},[31662,208116,202],{},[31749,208118,207961],{},[39361,208120,207859],{},[31662,208122,208123],{"lspace":207862,"rspace":207862},[31655,208124,208125,208127,208129],{},[31658,208126,31759],{"mathvariant":38079},[31658,208128,43383],{"mathvariant":38079},[31658,208130,46900],{"mathvariant":38079},[39361,208132,207859],{},[31749,208134,177869],{},[31662,208136,202],{},[31749,208138,95396],{},[31665,208140,208141],{"encoding":31667},"j = 3999 \\bmod 16 = 15",[34,208143,208145,208163,208193,208211],{"className":208144,"ariaHidden":758},[31672],[34,208146,208148,208151,208154,208157,208160],{"className":208147},[31676],[34,208149],{"className":208150,"style":33123},[31680],[34,208152,31968],{"className":208153,"style":32153},[31685,31686],[34,208155],{"className":208156,"style":31846},[31845],[34,208158,202],{"className":208159},[31850],[34,208161],{"className":208162,"style":31846},[31845],[34,208164,208166,208169,208172,208175,208178,208187,208190],{"className":208165},[31676],[34,208167],{"className":208168,"style":31681},[31680],[34,208170,207961],{"className":208171},[31685],[34,208173],{"className":208174,"style":112322},[31845],[34,208176],{"className":208177,"style":32604},[31845],[34,208179,208181],{"className":208180},[32024],[34,208182,208184],{"className":208183},[31685],[34,208185,207924],{"className":208186},[31685,207923],[34,208188],{"className":208189,"style":112322},[31845],[34,208191],{"className":208192,"style":32604},[31845],[34,208194,208196,208199,208202,208205,208208],{"className":208195},[31676],[34,208197],{"className":208198,"style":32617},[31680],[34,208200,177869],{"className":208201},[31685],[34,208203],{"className":208204,"style":31846},[31845],[34,208206,202],{"className":208207},[31850],[34,208209],{"className":208210,"style":31846},[31845],[34,208212,208214,208217],{"className":208213},[31676],[34,208215],{"className":208216,"style":32617},[31680],[34,208218,95396],{"className":208219},[31685],[108,208221,208222,208224,208225,208227,208228,347],{},[2627,208223,118575],{},"：对应的字号是 ",[2627,208226,184205],{},"，位号是 ",[2627,208229,95396],{},[15,208231,208232],{},[2627,208233,208234],{},"（3）位示图字号199，位号9对应主存的块号是多少？",[105,208236,208237,208341,208612],{},[108,208238,208239,208241,208242],{},[2627,208240,207541],{},"：这是第 (2) 问的逆运算。\n块号 ",[34,208243,208245,208272],{"className":208244},[31642],[34,208246,208248],{"className":208247},[31646],[31648,208249,208250],{"xmlns":31650},[31652,208251,208252,208269],{},[31655,208253,208254,208256,208258,208260,208262,208265,208267],{},[31658,208255,1697],{},[31662,208257,202],{},[31658,208259,31955],{},[31662,208261,42071],{},[39361,208263,208264],{},"机器字长",[31662,208266,223],{},[31658,208268,31968],{},[31665,208270,208271],{"encoding":31667},"b = i \\times \\text{机器字长} + j",[34,208273,208275,208293,208311,208332],{"className":208274,"ariaHidden":758},[31672],[34,208276,208278,208281,208284,208287,208290],{"className":208277},[31676],[34,208279],{"className":208280,"style":31681},[31680],[34,208282,1697],{"className":208283},[31685,31686],[34,208285],{"className":208286,"style":31846},[31845],[34,208288,202],{"className":208289},[31850],[34,208291],{"className":208292,"style":31846},[31845],[34,208294,208296,208299,208302,208305,208308],{"className":208295},[31676],[34,208297],{"className":208298,"style":37322},[31680],[34,208300,31955],{"className":208301},[31685,31686],[34,208303],{"className":208304,"style":32604},[31845],[34,208306,42071],{"className":208307},[32024],[34,208309],{"className":208310,"style":32604},[31845],[34,208312,208314,208317,208323,208326,208329],{"className":208313},[31676],[34,208315],{"className":208316,"style":34944},[31680],[34,208318,208320],{"className":208319},[31685,2638],[34,208321,208264],{"className":208322},[31685,105462],[34,208324],{"className":208325,"style":32604},[31845],[34,208327,223],{"className":208328},[32024],[34,208330],{"className":208331,"style":32604},[31845],[34,208333,208335,208338],{"className":208334},[31676],[34,208336],{"className":208337,"style":33123},[31680],[34,208339,31968],{"className":208340,"style":32153},[31685,31686],[108,208342,208343,208344,208396,208397,208448,208449],{},"代入字号 ",[34,208345,208347,208366],{"className":208346},[31642],[34,208348,208350],{"className":208349},[31646],[31648,208351,208352],{"xmlns":31650},[31652,208353,208354,208363],{},[31655,208355,208356,208358,208360],{},[31658,208357,31955],{},[31662,208359,202],{},[31749,208361,208362],{},"199",[31665,208364,208365],{"encoding":31667},"i = 199",[34,208367,208369,208387],{"className":208368,"ariaHidden":758},[31672],[34,208370,208372,208375,208378,208381,208384],{"className":208371},[31676],[34,208373],{"className":208374,"style":33104},[31680],[34,208376,31955],{"className":208377},[31685,31686],[34,208379],{"className":208380,"style":31846},[31845],[34,208382,202],{"className":208383},[31850],[34,208385],{"className":208386,"style":31846},[31845],[34,208388,208390,208393],{"className":208389},[31676],[34,208391],{"className":208392,"style":32617},[31680],[34,208394,208362],{"className":208395},[31685],"，位号 ",[34,208398,208400,208418],{"className":208399},[31642],[34,208401,208403],{"className":208402},[31646],[31648,208404,208405],{"xmlns":31650},[31652,208406,208407,208415],{},[31655,208408,208409,208411,208413],{},[31658,208410,31968],{},[31662,208412,202],{},[31749,208414,36732],{},[31665,208416,208417],{"encoding":31667},"j = 9",[34,208419,208421,208439],{"className":208420,"ariaHidden":758},[31672],[34,208422,208424,208427,208430,208433,208436],{"className":208423},[31676],[34,208425],{"className":208426,"style":33123},[31680],[34,208428,31968],{"className":208429,"style":32153},[31685,31686],[34,208431],{"className":208432,"style":31846},[31845],[34,208434,202],{"className":208435},[31850],[34,208437],{"className":208438,"style":31846},[31845],[34,208440,208442,208445],{"className":208441},[31676],[34,208443],{"className":208444,"style":32617},[31680],[34,208446,36732],{"className":208447},[31685],"：\n块号 ",[34,208450,208452,208492],{"className":208451},[31642],[34,208453,208455],{"className":208454},[31646],[31648,208456,208457],{"xmlns":31650},[31652,208458,208459,208489],{},[31655,208460,208461,208463,208465,208467,208469,208471,208473,208475,208477,208480,208482,208484,208486],{},[31658,208462,1697],{},[31662,208464,202],{},[31749,208466,208362],{},[31662,208468,42071],{},[31749,208470,177869],{},[31662,208472,223],{},[31749,208474,36732],{},[31662,208476,202],{},[31749,208478,208479],{},"3184",[31662,208481,223],{},[31749,208483,36732],{},[31662,208485,202],{},[31749,208487,208488],{},"3193",[31665,208490,208491],{"encoding":31667},"b = 199 \\times 16 + 9 = 3184 + 9 = 3193",[34,208493,208495,208513,208531,208549,208567,208585,208603],{"className":208494,"ariaHidden":758},[31672],[34,208496,208498,208501,208504,208507,208510],{"className":208497},[31676],[34,208499],{"className":208500,"style":31681},[31680],[34,208502,1697],{"className":208503},[31685,31686],[34,208505],{"className":208506,"style":31846},[31845],[34,208508,202],{"className":208509},[31850],[34,208511],{"className":208512,"style":31846},[31845],[34,208514,208516,208519,208522,208525,208528],{"className":208515},[31676],[34,208517],{"className":208518,"style":35942},[31680],[34,208520,208362],{"className":208521},[31685],[34,208523],{"className":208524,"style":32604},[31845],[34,208526,42071],{"className":208527},[32024],[34,208529],{"className":208530,"style":32604},[31845],[34,208532,208534,208537,208540,208543,208546],{"className":208533},[31676],[34,208535],{"className":208536,"style":35942},[31680],[34,208538,177869],{"className":208539},[31685],[34,208541],{"className":208542,"style":32604},[31845],[34,208544,223],{"className":208545},[32024],[34,208547],{"className":208548,"style":32604},[31845],[34,208550,208552,208555,208558,208561,208564],{"className":208551},[31676],[34,208553],{"className":208554,"style":32617},[31680],[34,208556,36732],{"className":208557},[31685],[34,208559],{"className":208560,"style":31846},[31845],[34,208562,202],{"className":208563},[31850],[34,208565],{"className":208566,"style":31846},[31845],[34,208568,208570,208573,208576,208579,208582],{"className":208569},[31676],[34,208571],{"className":208572,"style":35942},[31680],[34,208574,208479],{"className":208575},[31685],[34,208577],{"className":208578,"style":32604},[31845],[34,208580,223],{"className":208581},[32024],[34,208583],{"className":208584,"style":32604},[31845],[34,208586,208588,208591,208594,208597,208600],{"className":208587},[31676],[34,208589],{"className":208590,"style":32617},[31680],[34,208592,36732],{"className":208593},[31685],[34,208595],{"className":208596,"style":31846},[31845],[34,208598,202],{"className":208599},[31850],[34,208601],{"className":208602,"style":31846},[31845],[34,208604,208606,208609],{"className":208605},[31676],[34,208607],{"className":208608,"style":32617},[31680],[34,208610,208488],{"className":208611},[31685],[108,208613,208614,208616,208617,347],{},[2627,208615,118575],{},"：对应主存的块号是 ",[2627,208618,208488],{},{"title":29,"searchDepth":43,"depth":43,"links":208620},[208621,208622,208623,208624,208625,208626,208627,208628,208629,208630,208631,208632],{"id":200429,"depth":43,"text":200430},{"id":200676,"depth":43,"text":200677},{"id":201082,"depth":43,"text":201083},{"id":201422,"depth":43,"text":201423},{"id":202806,"depth":43,"text":202807},{"id":204835,"depth":43,"text":204836},{"id":204892,"depth":43,"text":204893},{"id":204963,"depth":43,"text":204964},{"id":205123,"depth":43,"text":205124},{"id":205693,"depth":43,"text":205694},{"id":206088,"depth":43,"text":206089},{"id":207501,"depth":43,"text":207502},{"order":74},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap7-hw",{"title":200420,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap7-hw","g-rDpzQxhhJNNNOll_8zS4R14ao47f2XZvVqov8cuzQ",{"id":208639,"title":208640,"body":208641,"description":29,"extension":85,"meta":214268,"navigation":52,"path":214269,"seo":214270,"stem":214271,"__hash__":214272},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap8-hw.md","Chap. 8 HW",{"type":8,"value":208642,"toc":214259},[208643,208647,208729,208732,208734,208737,208741,208746,208750,208755,208763,208799,208805,209652,209654,209657,209662,209667,209674,210146,210150,210970,210972,210975,210980,210985,210989,211508,211656,211658,211661,211666,211671,211682,212186,212188,212191,212196,212201,212205,212774,212861,212863,212866,212875,212884,212889,213028,213191,213193,213196,213201,213206,213210,213469,213478,213482,213897],[11,208644,208646],{"id":208645},"os-chap8-hw","OS Chap.8 HW",[105,208648,208649,208655,208661],{},[108,208650,208651,208654],{},[2627,208652,208653],{},"虚拟地址 (Virtual Address, VA)","：CPU生成的地址，由“虚拟页号 (VPN)”和“页内偏移量 (Offset)”两部分组成。",[108,208656,208657,208660],{},[2627,208658,208659],{},"物理地址 (Physical Address, PA)","：内存中的实际地址，由“物理页框号 (PFN)”和“页内偏移量 (Offset)”组成。偏移量在虚拟地址和物理地址中是完全相同的。",[108,208662,208663,208666,208667,208728],{},[2627,208664,208665],{},"页面大小 (Page Size)","：决定了偏移量需要占用多少个二进制位。例如，1024字节（",[34,208668,208670,208687],{"className":208669},[31642],[34,208671,208673],{"className":208672},[31646],[31648,208674,208675],{"xmlns":31650},[31652,208676,208677,208685],{},[31655,208678,208679],{},[37624,208680,208681,208683],{},[31749,208682,768],{},[31749,208684,3244],{},[31665,208686,201512],{"encoding":31667},[34,208688,208690],{"className":208689,"ariaHidden":758},[31672],[34,208691,208693,208696],{"className":208692},[31676],[34,208694],{"className":208695,"style":37774},[31680],[34,208697,208699,208702],{"className":208698},[31685],[34,208700,768],{"className":208701},[31685],[34,208703,208705],{"className":208704},[31792],[34,208706,208708],{"className":208707},[31796],[34,208709,208711],{"className":208710},[31801],[34,208712,208714],{"className":208713,"style":37774},[31805],[34,208715,208716,208719],{"style":37777},[34,208717],{"className":208718,"style":31814},[31813],[34,208720,208722],{"className":208721},[31818,31819,31820,31821],[34,208723,208725],{"className":208724},[31685,31821],[34,208726,3244],{"className":208727},[31685,31821],"）的页面，偏移量就是10位。",[15,208730,208731],{},"以下是所有题目的详细解答。",[32188,208733],{},[253,208735,208736],{"id":208736},"第一题解析",[15,208738,208739],{},[2627,208740,200435],{},[2230,208742,208743],{},[108,208744,208745],{},"Suppose the page table for the process currently executing on the processor looks like the following. All numbers are decimal, everything is numbered starting from zero, and all addresses are memory byte addresses. The page size is 1024 bytes. The following table: (Virtual page number, Valid bit, Reference bit, Modify bit, Page frame number) -> (0, 1, 1, 0, 4), (1, 1, 1, 1, 7), (2, 0, 0, 0, -), (3, 1, 0, 0, 2), (4, 0, 0, 0, -), (5, 1, 0, 1, 0).\na. Describe exactly how, in general, a virtual address generated by the CPU is translated into a physical main memory address.\nb. What physical address, if any, would each of the following virtual addresses correspond to? (Do not try to handle any page faults, if any.) (i) 1052 (ii) 2221 (iii) 5499",[15,208747,208748],{},[2627,208749,200445],{},[2230,208751,208752],{},[108,208753,208754],{},"假设当前正在处理器上执行的进程的页表如下所示。所有数字均为十进制，所有编号均从零开始，所有地址均为内存字节地址。页面大小为 1024 字节。下表： (虚拟页号, 有效位, 访问位, 修改位, 页框号) -> (0, 1, 1, 0, 4), (1, 1, 1, 1, 7), (2, 0, 0, 0, 空), (3, 1, 0, 0, 2), (4, 0, 0, 0, 空), (5, 1, 0, 1, 0)。\na. 准确描述一般情况下，CPU生成的虚拟地址是如何转换为物理主存地址的。\nb. 以下每个虚拟地址（如果有的话）将对应哪个物理地址？（尽量不要处理任何缺页异常，如果有的话。） (i) 1052 (ii) 2221 (iii) 5499",[15,208756,208757,208760],{},[2627,208758,208759],{},"【深入解析】",[2627,208761,208762],{},"a. 地址转换过程：",[2230,208764,208765,208779,208784,208790],{},[108,208766,208767,208770,208771,208774,208775,208778],{},[2627,208768,208769],{},"拆分地址","：当CPU生成一个虚拟地址时，系统首先根据页面大小将该虚拟地址拆分为两部分：虚拟页号（VPN）和页内偏移量（Offset）。拆分公式为：",[31,208772,208773],{},"VPN = 虚拟地址 \u002F 页面大小","（取整数部分）；",[31,208776,208777],{},"Offset = 虚拟地址 % 页面大小","（取余数部分）。",[108,208780,208781,208783],{},[2627,208782,202979],{},"：硬件使用VPN作为索引去查找页表。",[108,208785,208786,208789],{},[2627,208787,208788],{},"检查有效位","：检查该表项的“有效位（Valid bit）”。如果有效位为0，说明该页面不在内存中，触发“缺页异常（Page Fault）”。",[108,208791,208792,208795,208796,347],{},[2627,208793,208794],{},"生成物理地址","：如果有效位为1，则从页表中提取对应的“物理页框号（PFN）”。最终的物理地址计算公式为：",[31,208797,208798],{},"物理地址 = (PFN * 页面大小) + Offset",[15,208800,208801,208804],{},[2627,208802,208803],{},"b. 具体地址计算：","\n已知页面大小为 1024 字节。",[105,208806,208807,208812,208889,209002,209005,209136,209141,209217,209324,209327,209332,209337,209413,209520,209523],{},[108,208808,208809],{},[2627,208810,208811],{},"(i) 虚拟地址 1052",[108,208813,208814,208815],{},"VPN = ",[34,208816,208818,208841],{"className":208817},[31642],[34,208819,208821],{"className":208820},[31646],[31648,208822,208823],{"xmlns":31650},[31652,208824,208825,208838],{},[31655,208826,208827,208830,208832,208834,208836],{},[31749,208828,208829],{},"1052",[31662,208831,207557],{},[31749,208833,177393],{},[31662,208835,202],{},[31749,208837,640],{},[31665,208839,208840],{"encoding":31667},"1052 \\div 1024 = 1",[34,208842,208844,208862,208880],{"className":208843,"ariaHidden":758},[31672],[34,208845,208847,208850,208853,208856,208859],{"className":208846},[31676],[34,208848],{"className":208849,"style":35942},[31680],[34,208851,208829],{"className":208852},[31685],[34,208854],{"className":208855,"style":32604},[31845],[34,208857,207557],{"className":208858},[32024],[34,208860],{"className":208861,"style":32604},[31845],[34,208863,208865,208868,208871,208874,208877],{"className":208864},[31676],[34,208866],{"className":208867,"style":32617},[31680],[34,208869,177393],{"className":208870},[31685],[34,208872],{"className":208873,"style":31846},[31845],[34,208875,202],{"className":208876},[31850],[34,208878],{"className":208879,"style":31846},[31845],[34,208881,208883,208886],{"className":208882},[31676],[34,208884],{"className":208885,"style":32617},[31680],[34,208887,640],{"className":208888},[31685],[108,208890,208891,208892],{},"Offset = ",[34,208893,208895,208936],{"className":208894},[31642],[34,208896,208898],{"className":208897},[31646],[31648,208899,208900],{"xmlns":31650},[31652,208901,208902,208933],{},[31655,208903,208904,208906,208908,208911,208913,208921,208924,208926,208928,208930],{},[31749,208905,208829],{},[31845,208907],{},[31845,208909],{"width":208910},"0.4444em",[31662,208912,456],{"stretchy":702},[31655,208914,208915,208917,208919],{},[31658,208916,31759],{"mathvariant":38079},[31658,208918,43383],{"mathvariant":38079},[31658,208920,46900],{"mathvariant":38079},[31845,208922],{"width":208923},"0.3333em",[31749,208925,177393],{},[31662,208927,2557],{"stretchy":702},[31662,208929,202],{},[31749,208931,208932],{},"28",[31665,208934,208935],{"encoding":31667},"1052 \\pmod{1024} = 28",[34,208937,208939,208956,208993],{"className":208938,"ariaHidden":758},[31672],[34,208940,208942,208945,208948,208952],{"className":208941},[31676],[34,208943],{"className":208944,"style":32617},[31680],[34,208946,208829],{"className":208947},[31685],[34,208949],{"className":208950},[31845,208951],"allowbreak",[34,208953],{"className":208954,"style":208955},[31845],"margin-right:0.4444em;",[34,208957,208959,208962,208965,208974,208978,208981,208984,208987,208990],{"className":208958},[31676],[34,208960],{"className":208961,"style":31774},[31680],[34,208963,456],{"className":208964},[31782],[34,208966,208968],{"className":208967},[31685],[34,208969,208971],{"className":208970},[31685],[34,208972,207924],{"className":208973},[31685,207923],[34,208975],{"className":208976,"style":208977},[31845],"margin-right:0.3333em;",[34,208979,177393],{"className":208980},[31685],[34,208982,2557],{"className":208983},[31690],[34,208985],{"className":208986,"style":31846},[31845],[34,208988,202],{"className":208989},[31850],[34,208991],{"className":208992,"style":31846},[31845],[34,208994,208996,208999],{"className":208995},[31676],[34,208997],{"className":208998,"style":32617},[31680],[34,209000,208932],{"className":209001},[31685],[108,209003,209004],{},"查表：虚拟页号1对应的有效位是1，页框号是7。",[108,209006,203492,209007,47649,209133,347],{},[34,209008,209010,209043],{"className":209009},[31642],[34,209011,209013],{"className":209012},[31646],[31648,209014,209015],{"xmlns":31650},[31652,209016,209017,209040],{},[31655,209018,209019,209021,209023,209025,209027,209029,209031,209034,209036,209038],{},[31749,209020,36723],{},[31662,209022,42071],{},[31749,209024,177393],{},[31662,209026,223],{},[31749,209028,208932],{},[31662,209030,202],{},[31749,209032,209033],{},"7168",[31662,209035,223],{},[31749,209037,208932],{},[31662,209039,202],{},[31665,209041,209042],{"encoding":31667},"7 \\times 1024 + 28 = 7168 + 28 =",[34,209044,209046,209064,209082,209100,209118],{"className":209045,"ariaHidden":758},[31672],[34,209047,209049,209052,209055,209058,209061],{"className":209048},[31676],[34,209050],{"className":209051,"style":35942},[31680],[34,209053,36723],{"className":209054},[31685],[34,209056],{"className":209057,"style":32604},[31845],[34,209059,42071],{"className":209060},[32024],[34,209062],{"className":209063,"style":32604},[31845],[34,209065,209067,209070,209073,209076,209079],{"className":209066},[31676],[34,209068],{"className":209069,"style":35942},[31680],[34,209071,177393],{"className":209072},[31685],[34,209074],{"className":209075,"style":32604},[31845],[34,209077,223],{"className":209078},[32024],[34,209080],{"className":209081,"style":32604},[31845],[34,209083,209085,209088,209091,209094,209097],{"className":209084},[31676],[34,209086],{"className":209087,"style":32617},[31680],[34,209089,208932],{"className":209090},[31685],[34,209092],{"className":209093,"style":31846},[31845],[34,209095,202],{"className":209096},[31850],[34,209098],{"className":209099,"style":31846},[31845],[34,209101,209103,209106,209109,209112,209115],{"className":209102},[31676],[34,209104],{"className":209105,"style":35942},[31680],[34,209107,209033],{"className":209108},[31685],[34,209110],{"className":209111,"style":32604},[31845],[34,209113,223],{"className":209114},[32024],[34,209116],{"className":209117,"style":32604},[31845],[34,209119,209121,209124,209127,209130],{"className":209120},[31676],[34,209122],{"className":209123,"style":32617},[31680],[34,209125,208932],{"className":209126},[31685],[34,209128],{"className":209129,"style":31846},[31845],[34,209131,202],{"className":209132},[31850],[2627,209134,209135],{},"7196",[108,209137,209138],{},[2627,209139,209140],{},"(ii) 虚拟地址 2221",[108,209142,208814,209143],{},[34,209144,209146,209169],{"className":209145},[31642],[34,209147,209149],{"className":209148},[31646],[31648,209150,209151],{"xmlns":31650},[31652,209152,209153,209166],{},[31655,209154,209155,209158,209160,209162,209164],{},[31749,209156,209157],{},"2221",[31662,209159,207557],{},[31749,209161,177393],{},[31662,209163,202],{},[31749,209165,768],{},[31665,209167,209168],{"encoding":31667},"2221 \\div 1024 = 2",[34,209170,209172,209190,209208],{"className":209171,"ariaHidden":758},[31672],[34,209173,209175,209178,209181,209184,209187],{"className":209174},[31676],[34,209176],{"className":209177,"style":35942},[31680],[34,209179,209157],{"className":209180},[31685],[34,209182],{"className":209183,"style":32604},[31845],[34,209185,207557],{"className":209186},[32024],[34,209188],{"className":209189,"style":32604},[31845],[34,209191,209193,209196,209199,209202,209205],{"className":209192},[31676],[34,209194],{"className":209195,"style":32617},[31680],[34,209197,177393],{"className":209198},[31685],[34,209200],{"className":209201,"style":31846},[31845],[34,209203,202],{"className":209204},[31850],[34,209206],{"className":209207,"style":31846},[31845],[34,209209,209211,209214],{"className":209210},[31676],[34,209212],{"className":209213,"style":32617},[31680],[34,209215,768],{"className":209216},[31685],[108,209218,208891,209219],{},[34,209220,209222,209261],{"className":209221},[31642],[34,209223,209225],{"className":209224},[31646],[31648,209226,209227],{"xmlns":31650},[31652,209228,209229,209258],{},[31655,209230,209231,209233,209235,209237,209239,209247,209249,209251,209253,209255],{},[31749,209232,209157],{},[31845,209234],{},[31845,209236],{"width":208910},[31662,209238,456],{"stretchy":702},[31655,209240,209241,209243,209245],{},[31658,209242,31759],{"mathvariant":38079},[31658,209244,43383],{"mathvariant":38079},[31658,209246,46900],{"mathvariant":38079},[31845,209248],{"width":208923},[31749,209250,177393],{},[31662,209252,2557],{"stretchy":702},[31662,209254,202],{},[31749,209256,209257],{},"173",[31665,209259,209260],{"encoding":31667},"2221 \\pmod{1024} = 173",[34,209262,209264,209279,209315],{"className":209263,"ariaHidden":758},[31672],[34,209265,209267,209270,209273,209276],{"className":209266},[31676],[34,209268],{"className":209269,"style":32617},[31680],[34,209271,209157],{"className":209272},[31685],[34,209274],{"className":209275},[31845,208951],[34,209277],{"className":209278,"style":208955},[31845],[34,209280,209282,209285,209288,209297,209300,209303,209306,209309,209312],{"className":209281},[31676],[34,209283],{"className":209284,"style":31774},[31680],[34,209286,456],{"className":209287},[31782],[34,209289,209291],{"className":209290},[31685],[34,209292,209294],{"className":209293},[31685],[34,209295,207924],{"className":209296},[31685,207923],[34,209298],{"className":209299,"style":208977},[31845],[34,209301,177393],{"className":209302},[31685],[34,209304,2557],{"className":209305},[31690],[34,209307],{"className":209308,"style":31846},[31845],[34,209310,202],{"className":209311},[31850],[34,209313],{"className":209314,"style":31846},[31845],[34,209316,209318,209321],{"className":209317},[31676],[34,209319],{"className":209320,"style":32617},[31680],[34,209322,209257],{"className":209323},[31685],[108,209325,209326],{},"查表：虚拟页号2对应的有效位是0。",[108,209328,95791,209329,347],{},[2627,209330,209331],{},"发生缺页异常（Page Fault），无对应的物理地址",[108,209333,209334],{},[2627,209335,209336],{},"(iii) 虚拟地址 5499",[108,209338,208814,209339],{},[34,209340,209342,209365],{"className":209341},[31642],[34,209343,209345],{"className":209344},[31646],[31648,209346,209347],{"xmlns":31650},[31652,209348,209349,209362],{},[31655,209350,209351,209354,209356,209358,209360],{},[31749,209352,209353],{},"5499",[31662,209355,207557],{},[31749,209357,177393],{},[31662,209359,202],{},[31749,209361,18136],{},[31665,209363,209364],{"encoding":31667},"5499 \\div 1024 = 5",[34,209366,209368,209386,209404],{"className":209367,"ariaHidden":758},[31672],[34,209369,209371,209374,209377,209380,209383],{"className":209370},[31676],[34,209372],{"className":209373,"style":35942},[31680],[34,209375,209353],{"className":209376},[31685],[34,209378],{"className":209379,"style":32604},[31845],[34,209381,207557],{"className":209382},[32024],[34,209384],{"className":209385,"style":32604},[31845],[34,209387,209389,209392,209395,209398,209401],{"className":209388},[31676],[34,209390],{"className":209391,"style":32617},[31680],[34,209393,177393],{"className":209394},[31685],[34,209396],{"className":209397,"style":31846},[31845],[34,209399,202],{"className":209400},[31850],[34,209402],{"className":209403,"style":31846},[31845],[34,209405,209407,209410],{"className":209406},[31676],[34,209408],{"className":209409,"style":32617},[31680],[34,209411,18136],{"className":209412},[31685],[108,209414,208891,209415],{},[34,209416,209418,209457],{"className":209417},[31642],[34,209419,209421],{"className":209420},[31646],[31648,209422,209423],{"xmlns":31650},[31652,209424,209425,209454],{},[31655,209426,209427,209429,209431,209433,209435,209443,209445,209447,209449,209451],{},[31749,209428,209353],{},[31845,209430],{},[31845,209432],{"width":208910},[31662,209434,456],{"stretchy":702},[31655,209436,209437,209439,209441],{},[31658,209438,31759],{"mathvariant":38079},[31658,209440,43383],{"mathvariant":38079},[31658,209442,46900],{"mathvariant":38079},[31845,209444],{"width":208923},[31749,209446,177393],{},[31662,209448,2557],{"stretchy":702},[31662,209450,202],{},[31749,209452,209453],{},"379",[31665,209455,209456],{"encoding":31667},"5499 \\pmod{1024} = 379",[34,209458,209460,209475,209511],{"className":209459,"ariaHidden":758},[31672],[34,209461,209463,209466,209469,209472],{"className":209462},[31676],[34,209464],{"className":209465,"style":32617},[31680],[34,209467,209353],{"className":209468},[31685],[34,209470],{"className":209471},[31845,208951],[34,209473],{"className":209474,"style":208955},[31845],[34,209476,209478,209481,209484,209493,209496,209499,209502,209505,209508],{"className":209477},[31676],[34,209479],{"className":209480,"style":31774},[31680],[34,209482,456],{"className":209483},[31782],[34,209485,209487],{"className":209486},[31685],[34,209488,209490],{"className":209489},[31685],[34,209491,207924],{"className":209492},[31685,207923],[34,209494],{"className":209495,"style":208977},[31845],[34,209497,177393],{"className":209498},[31685],[34,209500,2557],{"className":209501},[31690],[34,209503],{"className":209504,"style":31846},[31845],[34,209506,202],{"className":209507},[31850],[34,209509],{"className":209510,"style":31846},[31845],[34,209512,209514,209517],{"className":209513},[31676],[34,209515],{"className":209516,"style":32617},[31680],[34,209518,209453],{"className":209519},[31685],[108,209521,209522],{},"查表：虚拟页号5对应的有效位是1，页框号是0。",[108,209524,203492,209525,47649,209650,347],{},[34,209526,209528,209560],{"className":209527},[31642],[34,209529,209531],{"className":209530},[31646],[31648,209532,209533],{"xmlns":31650},[31652,209534,209535,209557],{},[31655,209536,209537,209539,209541,209543,209545,209547,209549,209551,209553,209555],{},[31749,209538,459],{},[31662,209540,42071],{},[31749,209542,177393],{},[31662,209544,223],{},[31749,209546,209453],{},[31662,209548,202],{},[31749,209550,459],{},[31662,209552,223],{},[31749,209554,209453],{},[31662,209556,202],{},[31665,209558,209559],{"encoding":31667},"0 \\times 1024 + 379 = 0 + 379 =",[34,209561,209563,209581,209599,209617,209635],{"className":209562,"ariaHidden":758},[31672],[34,209564,209566,209569,209572,209575,209578],{"className":209565},[31676],[34,209567],{"className":209568,"style":35942},[31680],[34,209570,459],{"className":209571},[31685],[34,209573],{"className":209574,"style":32604},[31845],[34,209576,42071],{"className":209577},[32024],[34,209579],{"className":209580,"style":32604},[31845],[34,209582,209584,209587,209590,209593,209596],{"className":209583},[31676],[34,209585],{"className":209586,"style":35942},[31680],[34,209588,177393],{"className":209589},[31685],[34,209591],{"className":209592,"style":32604},[31845],[34,209594,223],{"className":209595},[32024],[34,209597],{"className":209598,"style":32604},[31845],[34,209600,209602,209605,209608,209611,209614],{"className":209601},[31676],[34,209603],{"className":209604,"style":32617},[31680],[34,209606,209453],{"className":209607},[31685],[34,209609],{"className":209610,"style":31846},[31845],[34,209612,202],{"className":209613},[31850],[34,209615],{"className":209616,"style":31846},[31845],[34,209618,209620,209623,209626,209629,209632],{"className":209619},[31676],[34,209621],{"className":209622,"style":35942},[31680],[34,209624,459],{"className":209625},[31685],[34,209627],{"className":209628,"style":32604},[31845],[34,209630,223],{"className":209631},[32024],[34,209633],{"className":209634,"style":32604},[31845],[34,209636,209638,209641,209644,209647],{"className":209637},[31676],[34,209639],{"className":209640,"style":32617},[31680],[34,209642,209453],{"className":209643},[31685],[34,209645],{"className":209646,"style":31846},[31845],[34,209648,202],{"className":209649},[31850],[2627,209651,209453],{},[32188,209653],{},[253,209655,209656],{"id":209656},"第二题解析",[15,209658,209659,209661],{},[2627,209660,200435],{},"\n2. Consider a paged virtual memory system with 32-bit virtual addresses and 1K-byte pages. Each page table entry requires 32 bits. It is desired to limit the page table size to one page.\na. How many levels of page tables are required?\nb. What is the size of the page table at each level? Hint: One page table size is smaller.\nc. The smaller page size could be used at the top level or the bottom level of the page table hierarchy. Which strategy consumes the least number of pages?",[15,209663,209664,209666],{},[2627,209665,200445],{},"\n2. 考虑一个具有32位虚拟地址和1KB页面的分页虚拟内存系统。每个页表项需要32位。希望将页表大小限制为一个页面。\na. 需要多少级页表？\nb. 每一级的页表大小是多少？提示：其中一个页表的大小较小。\nc. 较小的页表大小可以用于页表层次结构的顶层或底层。哪种策略消耗的页面数量最少？",[15,209668,209669,209671],{},[2627,209670,208759],{},[2627,209672,209673],{},"前置计算：",[2230,209675,209676,209744,209809,209872,210040],{},[108,209677,209678,209679,209740,209741,347],{},"页面大小 = 1KB = ",[34,209680,209682,209699],{"className":209681},[31642],[34,209683,209685],{"className":209684},[31646],[31648,209686,209687],{"xmlns":31650},[31652,209688,209689,209697],{},[31655,209690,209691],{},[37624,209692,209693,209695],{},[31749,209694,768],{},[31749,209696,3244],{},[31665,209698,201512],{"encoding":31667},[34,209700,209702],{"className":209701,"ariaHidden":758},[31672],[34,209703,209705,209708],{"className":209704},[31676],[34,209706],{"className":209707,"style":37774},[31680],[34,209709,209711,209714],{"className":209710},[31685],[34,209712,768],{"className":209713},[31685],[34,209715,209717],{"className":209716},[31792],[34,209718,209720],{"className":209719},[31796],[34,209721,209723],{"className":209722},[31801],[34,209724,209726],{"className":209725,"style":37774},[31805],[34,209727,209728,209731],{"style":37777},[34,209729],{"className":209730,"style":31814},[31813],[34,209732,209734],{"className":209733},[31818,31819,31820,31821],[34,209735,209737],{"className":209736},[31685,31821],[34,209738,3244],{"className":209739},[31685,31821]," 字节，因此页内偏移量占用 ",[2627,209742,209743],{},"10位",[108,209745,209746,209747,47649,209806,347],{},"虚拟地址总长32位，所以虚拟页号（VPN）占用 ",[34,209748,209750,209770],{"className":209749},[31642],[34,209751,209753],{"className":209752},[31646],[31648,209754,209755],{"xmlns":31650},[31652,209756,209757,209767],{},[31655,209758,209759,209761,209763,209765],{},[31749,209760,173486],{},[31662,209762,32226],{},[31749,209764,3244],{},[31662,209766,202],{},[31665,209768,209769],{"encoding":31667},"32 - 10 =",[34,209771,209773,209791],{"className":209772,"ariaHidden":758},[31672],[34,209774,209776,209779,209782,209785,209788],{"className":209775},[31676],[34,209777],{"className":209778,"style":35942},[31680],[34,209780,173486],{"className":209781},[31685],[34,209783],{"className":209784,"style":32604},[31845],[34,209786,32226],{"className":209787},[32024],[34,209789],{"className":209790,"style":32604},[31845],[34,209792,209794,209797,209800,209803],{"className":209793},[31676],[34,209795],{"className":209796,"style":32617},[31680],[34,209798,3244],{"className":209799},[31685],[34,209801],{"className":209802,"style":31846},[31845],[34,209804,202],{"className":209805},[31850],[2627,209807,209808],{},"22位",[108,209810,209811,209812,209871],{},"每个页表项大小 = 32位 = 4字节（",[34,209813,209815,209833],{"className":209814},[31642],[34,209816,209818],{"className":209817},[31646],[31648,209819,209820],{"xmlns":31650},[31652,209821,209822,209830],{},[31655,209823,209824],{},[37624,209825,209826,209828],{},[31749,209827,768],{},[31749,209829,768],{},[31665,209831,209832],{"encoding":31667},"2^2",[34,209834,209836],{"className":209835,"ariaHidden":758},[31672],[34,209837,209839,209842],{"className":209838},[31676],[34,209840],{"className":209841,"style":37774},[31680],[34,209843,209845,209848],{"className":209844},[31685],[34,209846,768],{"className":209847},[31685],[34,209849,209851],{"className":209850},[31792],[34,209852,209854],{"className":209853},[31796],[34,209855,209857],{"className":209856},[31801],[34,209858,209860],{"className":209859,"style":37774},[31805],[34,209861,209862,209865],{"style":37777},[34,209863],{"className":209864,"style":31814},[31813],[34,209866,209868],{"className":209867},[31818,31819,31820,31821],[34,209869,768],{"className":209870},[31685,31821]," 字节）。",[108,209873,209874,209875,47649,210037,347],{},"一个页面最多可以容纳的表项数 = 页面大小 \u002F 表项大小 = ",[34,209876,209878,209914],{"className":209877},[31642],[34,209879,209881],{"className":209880},[31646],[31648,209882,209883],{"xmlns":31650},[31652,209884,209885,209911],{},[31655,209886,209887,209893,209895,209901,209903,209909],{},[37624,209888,209889,209891],{},[31749,209890,768],{},[31749,209892,3244],{},[31658,209894,38080],{"mathvariant":38079},[37624,209896,209897,209899],{},[31749,209898,768],{},[31749,209900,768],{},[31662,209902,202],{},[37624,209904,209905,209907],{},[31749,209906,768],{},[31749,209908,60525],{},[31662,209910,202],{},[31665,209912,209913],{"encoding":31667},"2^{10} \u002F 2^2 = 2^8 =",[34,209915,209917,209996],{"className":209916,"ariaHidden":758},[31672],[34,209918,209920,209923,209955,209958,209987,209990,209993],{"className":209919},[31676],[34,209921],{"className":209922,"style":37755},[31680],[34,209924,209926,209929],{"className":209925},[31685],[34,209927,768],{"className":209928},[31685],[34,209930,209932],{"className":209931},[31792],[34,209933,209935],{"className":209934},[31796],[34,209936,209938],{"className":209937},[31801],[34,209939,209941],{"className":209940,"style":37774},[31805],[34,209942,209943,209946],{"style":37777},[34,209944],{"className":209945,"style":31814},[31813],[34,209947,209949],{"className":209948},[31818,31819,31820,31821],[34,209950,209952],{"className":209951},[31685,31821],[34,209953,3244],{"className":209954},[31685,31821],[34,209956,38080],{"className":209957},[31685],[34,209959,209961,209964],{"className":209960},[31685],[34,209962,768],{"className":209963},[31685],[34,209965,209967],{"className":209966},[31792],[34,209968,209970],{"className":209969},[31796],[34,209971,209973],{"className":209972},[31801],[34,209974,209976],{"className":209975,"style":37774},[31805],[34,209977,209978,209981],{"style":37777},[34,209979],{"className":209980,"style":31814},[31813],[34,209982,209984],{"className":209983},[31818,31819,31820,31821],[34,209985,768],{"className":209986},[31685,31821],[34,209988],{"className":209989,"style":31846},[31845],[34,209991,202],{"className":209992},[31850],[34,209994],{"className":209995,"style":31846},[31845],[34,209997,209999,210002,210031,210034],{"className":209998},[31676],[34,210000],{"className":210001,"style":37774},[31680],[34,210003,210005,210008],{"className":210004},[31685],[34,210006,768],{"className":210007},[31685],[34,210009,210011],{"className":210010},[31792],[34,210012,210014],{"className":210013},[31796],[34,210015,210017],{"className":210016},[31801],[34,210018,210020],{"className":210019,"style":37774},[31805],[34,210021,210022,210025],{"style":37777},[34,210023],{"className":210024,"style":31814},[31813],[34,210026,210028],{"className":210027},[31818,31819,31820,31821],[34,210029,60525],{"className":210030},[31685,31821],[34,210032],{"className":210033,"style":31846},[31845],[34,210035,202],{"className":210036},[31850],[2627,210038,210039],{},"256 个表项",[108,210041,210042,210043,47649,210142,210145],{},"因为一个页表最多有256个表项，所以每一级页表最多只能映射 ",[34,210044,210046,210076],{"className":210045},[31642],[34,210047,210049],{"className":210048},[31646],[31648,210050,210051],{"xmlns":31650},[31652,210052,210053,210073],{},[31655,210054,210055,210065,210067,210069,210071],{},[31743,210056,210057,210063],{},[31655,210058,210059,210061],{},[31658,210060,885],{},[31662,210062,32682],{},[31749,210064,768],{},[31662,210066,456],{"stretchy":702},[31749,210068,95418],{},[31662,210070,2557],{"stretchy":702},[31662,210072,202],{},[31665,210074,210075],{"encoding":31667},"\\log_2(256) =",[34,210077,210079],{"className":210078,"ariaHidden":758},[31672],[34,210080,210082,210085,210127,210130,210133,210136,210139],{"className":210081},[31676],[34,210083],{"className":210084,"style":31774},[31680],[34,210086,210088,210093],{"className":210087},[32767],[34,210089,38616,210091],{"className":210090},[32767],[34,210092,31942],{"style":38619},[34,210094,210096],{"className":210095},[31792],[34,210097,210099,210119],{"className":210098},[31796,31797],[34,210100,210102,210116],{"className":210101},[31801],[34,210103,210105],{"className":210104,"style":38632},[31805],[34,210106,210107,210110],{"style":38635},[34,210108],{"className":210109,"style":31814},[31813],[34,210111,210113],{"className":210112},[31818,31819,31820,31821],[34,210114,768],{"className":210115},[31685,31821],[34,210117,31829],{"className":210118},[31828],[34,210120,210122],{"className":210121},[31801],[34,210123,210125],{"className":210124,"style":38654},[31805],[34,210126],{},[34,210128,456],{"className":210129},[31782],[34,210131,95418],{"className":210132},[31685],[34,210134,2557],{"className":210135},[31690],[34,210137],{"className":210138,"style":31846},[31845],[34,210140,202],{"className":210141},[31850],[2627,210143,210144],{},"8位"," 的虚拟地址。",[15,210147,210148],{},[2627,210149,198754],{},[105,210151,210152,210332,210612,210624,210797],{},[108,210153,210154,210157,210158,210231,210232,210327,210328,210331],{},[2627,210155,210156],{},"a. 需要的页表级数","\n我们需要用每次最多8位的块来覆盖总共22位的VPN。\n",[34,210159,210161,210183],{"className":210160},[31642],[34,210162,210164],{"className":210163},[31646],[31648,210165,210166],{"xmlns":31650},[31652,210167,210168,210180],{},[31655,210169,210170,210172,210174,210176,210178],{},[31749,210171,94716],{},[31662,210173,207557],{},[31749,210175,60525],{},[31662,210177,202],{},[31749,210179,768],{},[31665,210181,210182],{"encoding":31667},"22 \\div 8 = 2",[34,210184,210186,210204,210222],{"className":210185,"ariaHidden":758},[31672],[34,210187,210189,210192,210195,210198,210201],{"className":210188},[31676],[34,210190],{"className":210191,"style":35942},[31680],[34,210193,94716],{"className":210194},[31685],[34,210196],{"className":210197,"style":32604},[31845],[34,210199,207557],{"className":210200},[32024],[34,210202],{"className":210203,"style":32604},[31845],[34,210205,210207,210210,210213,210216,210219],{"className":210206},[31676],[34,210208],{"className":210209,"style":32617},[31680],[34,210211,60525],{"className":210212},[31685],[34,210214],{"className":210215,"style":31846},[31845],[34,210217,202],{"className":210218},[31850],[34,210220],{"className":210221,"style":31846},[31845],[34,210223,210225,210228],{"className":210224},[31676],[34,210226],{"className":210227,"style":32617},[31680],[34,210229,768],{"className":210230},[31685]," 余 6。这意味着我们需要 ",[34,210233,210235,210261],{"className":210234},[31642],[34,210236,210238],{"className":210237},[31646],[31648,210239,210240],{"xmlns":31650},[31652,210241,210242,210258],{},[31655,210243,210244,210246,210248,210250,210252,210254,210256],{},[31749,210245,60525],{},[31662,210247,223],{},[31749,210249,60525],{},[31662,210251,223],{},[31749,210253,36656],{},[31662,210255,202],{},[31749,210257,94716],{},[31665,210259,210260],{"encoding":31667},"8 + 8 + 6 = 22",[34,210262,210264,210282,210300,210318],{"className":210263,"ariaHidden":758},[31672],[34,210265,210267,210270,210273,210276,210279],{"className":210266},[31676],[34,210268],{"className":210269,"style":35942},[31680],[34,210271,60525],{"className":210272},[31685],[34,210274],{"className":210275,"style":32604},[31845],[34,210277,223],{"className":210278},[32024],[34,210280],{"className":210281,"style":32604},[31845],[34,210283,210285,210288,210291,210294,210297],{"className":210284},[31676],[34,210286],{"className":210287,"style":35942},[31680],[34,210289,60525],{"className":210290},[31685],[34,210292],{"className":210293,"style":32604},[31845],[34,210295,223],{"className":210296},[32024],[34,210298],{"className":210299,"style":32604},[31845],[34,210301,210303,210306,210309,210312,210315],{"className":210302},[31676],[34,210304],{"className":210305,"style":32617},[31680],[34,210307,36656],{"className":210308},[31685],[34,210310],{"className":210311,"style":31846},[31845],[34,210313,202],{"className":210314},[31850],[34,210316],{"className":210317,"style":31846},[31845],[34,210319,210321,210324],{"className":210320},[31676],[34,210322],{"className":210323,"style":32617},[31680],[34,210325,94716],{"className":210326},[31685],"。因此，需要 ",[2627,210329,210330],{},"3级"," 页表。",[108,210333,210334,210337,210338,210419,210420,210471,210472,210475,210476,210419,210557,205934,210608,210611],{},[2627,210335,210336],{},"b. 每一级的页表大小","\n其中两级页表负责映射8位，需要 ",[34,210339,210341,210363],{"className":210340},[31642],[34,210342,210344],{"className":210343},[31646],[31648,210345,210346],{"xmlns":31650},[31652,210347,210348,210360],{},[31655,210349,210350,210356,210358],{},[37624,210351,210352,210354],{},[31749,210353,768],{},[31749,210355,60525],{},[31662,210357,202],{},[31749,210359,95418],{},[31665,210361,210362],{"encoding":31667},"2^8 = 256",[34,210364,210366,210410],{"className":210365,"ariaHidden":758},[31672],[34,210367,210369,210372,210401,210404,210407],{"className":210368},[31676],[34,210370],{"className":210371,"style":37774},[31680],[34,210373,210375,210378],{"className":210374},[31685],[34,210376,768],{"className":210377},[31685],[34,210379,210381],{"className":210380},[31792],[34,210382,210384],{"className":210383},[31796],[34,210385,210387],{"className":210386},[31801],[34,210388,210390],{"className":210389,"style":37774},[31805],[34,210391,210392,210395],{"style":37777},[34,210393],{"className":210394,"style":31814},[31813],[34,210396,210398],{"className":210397},[31818,31819,31820,31821],[34,210399,60525],{"className":210400},[31685,31821],[34,210402],{"className":210403,"style":31846},[31845],[34,210405,202],{"className":210406},[31850],[34,210408],{"className":210409,"style":31846},[31845],[34,210411,210413,210416],{"className":210412},[31676],[34,210414],{"className":210415,"style":32617},[31680],[34,210417,95418],{"className":210418},[31685]," 个表项，大小为 ",[34,210421,210423,210441],{"className":210422},[31642],[34,210424,210426],{"className":210425},[31646],[31648,210427,210428],{"xmlns":31650},[31652,210429,210430,210438],{},[31655,210431,210432,210434,210436],{},[31749,210433,95418],{},[31662,210435,42071],{},[31749,210437,15932],{},[31665,210439,210440],{"encoding":31667},"256 \\times 4",[34,210442,210444,210462],{"className":210443,"ariaHidden":758},[31672],[34,210445,210447,210450,210453,210456,210459],{"className":210446},[31676],[34,210448],{"className":210449,"style":35942},[31680],[34,210451,95418],{"className":210452},[31685],[34,210454],{"className":210455,"style":32604},[31845],[34,210457,42071],{"className":210458},[32024],[34,210460],{"className":210461,"style":32604},[31845],[34,210463,210465,210468],{"className":210464},[31676],[34,210466],{"className":210467,"style":32617},[31680],[34,210469,15932],{"className":210470},[31685]," 字节 = 1024 字节（",[2627,210473,210474],{},"1个页面，即1KB","）。\n有一级较小的页表负责映射剩余的6位，需要 ",[34,210477,210479,210501],{"className":210478},[31642],[34,210480,210482],{"className":210481},[31646],[31648,210483,210484],{"xmlns":31650},[31652,210485,210486,210498],{},[31655,210487,210488,210494,210496],{},[37624,210489,210490,210492],{},[31749,210491,768],{},[31749,210493,36656],{},[31662,210495,202],{},[31749,210497,162212],{},[31665,210499,210500],{"encoding":31667},"2^6 = 64",[34,210502,210504,210548],{"className":210503,"ariaHidden":758},[31672],[34,210505,210507,210510,210539,210542,210545],{"className":210506},[31676],[34,210508],{"className":210509,"style":37774},[31680],[34,210511,210513,210516],{"className":210512},[31685],[34,210514,768],{"className":210515},[31685],[34,210517,210519],{"className":210518},[31792],[34,210520,210522],{"className":210521},[31796],[34,210523,210525],{"className":210524},[31801],[34,210526,210528],{"className":210527,"style":37774},[31805],[34,210529,210530,210533],{"style":37777},[34,210531],{"className":210532,"style":31814},[31813],[34,210534,210536],{"className":210535},[31818,31819,31820,31821],[34,210537,36656],{"className":210538},[31685,31821],[34,210540],{"className":210541,"style":31846},[31845],[34,210543,202],{"className":210544},[31850],[34,210546],{"className":210547,"style":31846},[31845],[34,210549,210551,210554],{"className":210550},[31676],[34,210552],{"className":210553,"style":32617},[31680],[34,210555,162212],{"className":210556},[31685],[34,210558,210560,210578],{"className":210559},[31642],[34,210561,210563],{"className":210562},[31646],[31648,210564,210565],{"xmlns":31650},[31652,210566,210567,210575],{},[31655,210568,210569,210571,210573],{},[31749,210570,162212],{},[31662,210572,42071],{},[31749,210574,15932],{},[31665,210576,210577],{"encoding":31667},"64 \\times 4",[34,210579,210581,210599],{"className":210580,"ariaHidden":758},[31672],[34,210582,210584,210587,210590,210593,210596],{"className":210583},[31676],[34,210585],{"className":210586,"style":35942},[31680],[34,210588,162212],{"className":210589},[31685],[34,210591],{"className":210592,"style":32604},[31845],[34,210594,42071],{"className":210595},[32024],[34,210597],{"className":210598,"style":32604},[31845],[34,210600,210602,210605],{"className":210601},[31676],[34,210603],{"className":210604,"style":32617},[31680],[34,210606,15932],{"className":210607},[31685],[2627,210609,210610],{},"256字节","（占用四分之一个页面）。",[108,210613,210614,210617,39984,210620,210623],{},[2627,210615,210616],{},"c. 较小页表的放置策略",[2627,210618,210619],{},"顶层策略消耗的页面数量最少",[71807,210621,210622],{},"原因分析","：树状数据结构在扩展时，高层节点越少，底层产生的分支总数就越少。",[108,210625,210626,210627,210700,210701,347],{},"如果放在顶层（顶级6位，第二级8位，第三级8位）：顶级有1个页表（大小为256字节），它最多指向64个二级页表；这64个二级页表最多指向 ",[34,210628,210630,210652],{"className":210629},[31642],[34,210631,210633],{"className":210632},[31646],[31648,210634,210635],{"xmlns":31650},[31652,210636,210637,210649],{},[31655,210638,210639,210641,210643,210645,210647],{},[31749,210640,162212],{},[31662,210642,42071],{},[31749,210644,95418],{},[31662,210646,202],{},[31749,210648,182485],{},[31665,210650,210651],{"encoding":31667},"64 \\times 256 = 16384",[34,210653,210655,210673,210691],{"className":210654,"ariaHidden":758},[31672],[34,210656,210658,210661,210664,210667,210670],{"className":210657},[31676],[34,210659],{"className":210660,"style":35942},[31680],[34,210662,162212],{"className":210663},[31685],[34,210665],{"className":210666,"style":32604},[31845],[34,210668,42071],{"className":210669},[32024],[34,210671],{"className":210672,"style":32604},[31845],[34,210674,210676,210679,210682,210685,210688],{"className":210675},[31676],[34,210677],{"className":210678,"style":32617},[31680],[34,210680,95418],{"className":210681},[31685],[34,210683],{"className":210684,"style":31846},[31845],[34,210686,202],{"className":210687},[31850],[34,210689],{"className":210690,"style":31846},[31845],[34,210692,210694,210697],{"className":210693},[31676],[34,210695],{"className":210696,"style":32617},[31680],[34,210698,182485],{"className":210699},[31685]," 个三级页表。极端填满情况下，总页表数为 ",[34,210702,210704,210731],{"className":210703},[31642],[34,210705,210707],{"className":210706},[31646],[31648,210708,210709],{"xmlns":31650},[31652,210710,210711,210728],{},[31655,210712,210713,210715,210717,210719,210721,210723,210725],{},[31749,210714,640],{},[31662,210716,223],{},[31749,210718,162212],{},[31662,210720,223],{},[31749,210722,182485],{},[31662,210724,202],{},[31749,210726,210727],{},"16449",[31665,210729,210730],{"encoding":31667},"1 + 64 + 16384 = 16449",[34,210732,210734,210752,210770,210788],{"className":210733,"ariaHidden":758},[31672],[34,210735,210737,210740,210743,210746,210749],{"className":210736},[31676],[34,210738],{"className":210739,"style":35942},[31680],[34,210741,640],{"className":210742},[31685],[34,210744],{"className":210745,"style":32604},[31845],[34,210747,223],{"className":210748},[32024],[34,210750],{"className":210751,"style":32604},[31845],[34,210753,210755,210758,210761,210764,210767],{"className":210754},[31676],[34,210756],{"className":210757,"style":35942},[31680],[34,210759,162212],{"className":210760},[31685],[34,210762],{"className":210763,"style":32604},[31845],[34,210765,223],{"className":210766},[32024],[34,210768],{"className":210769,"style":32604},[31845],[34,210771,210773,210776,210779,210782,210785],{"className":210772},[31676],[34,210774],{"className":210775,"style":32617},[31680],[34,210777,182485],{"className":210778},[31685],[34,210780],{"className":210781,"style":31846},[31845],[34,210783,202],{"className":210784},[31850],[34,210786],{"className":210787,"style":31846},[31845],[34,210789,210791,210794],{"className":210790},[31676],[34,210792],{"className":210793,"style":32617},[31680],[34,210795,210727],{"className":210796},[31685],[108,210798,210799,210800,210872,210873,210969],{},"如果放在底层（顶级8位，第二级8位，第三级6位）：顶级有1个页表，它最多指向256个二级页表；这256个二级页表最多指向 ",[34,210801,210803,210824],{"className":210802},[31642],[34,210804,210806],{"className":210805},[31646],[31648,210807,210808],{"xmlns":31650},[31652,210809,210810,210822],{},[31655,210811,210812,210814,210816,210818,210820],{},[31749,210813,95418],{},[31662,210815,42071],{},[31749,210817,95418],{},[31662,210819,202],{},[31749,210821,188623],{},[31665,210823,188626],{"encoding":31667},[34,210825,210827,210845,210863],{"className":210826,"ariaHidden":758},[31672],[34,210828,210830,210833,210836,210839,210842],{"className":210829},[31676],[34,210831],{"className":210832,"style":35942},[31680],[34,210834,95418],{"className":210835},[31685],[34,210837],{"className":210838,"style":32604},[31845],[34,210840,42071],{"className":210841},[32024],[34,210843],{"className":210844,"style":32604},[31845],[34,210846,210848,210851,210854,210857,210860],{"className":210847},[31676],[34,210849],{"className":210850,"style":32617},[31680],[34,210852,95418],{"className":210853},[31685],[34,210855],{"className":210856,"style":31846},[31845],[34,210858,202],{"className":210859},[31850],[34,210861],{"className":210862,"style":31846},[31845],[34,210864,210866,210869],{"className":210865},[31676],[34,210867],{"className":210868,"style":32617},[31680],[34,210870,188623],{"className":210871},[31685]," 个三级页表。总页表数为 ",[34,210874,210876,210903],{"className":210875},[31642],[34,210877,210879],{"className":210878},[31646],[31648,210880,210881],{"xmlns":31650},[31652,210882,210883,210900],{},[31655,210884,210885,210887,210889,210891,210893,210895,210897],{},[31749,210886,640],{},[31662,210888,223],{},[31749,210890,95418],{},[31662,210892,223],{},[31749,210894,188623],{},[31662,210896,202],{},[31749,210898,210899],{},"65793",[31665,210901,210902],{"encoding":31667},"1 + 256 + 65536 = 65793",[34,210904,210906,210924,210942,210960],{"className":210905,"ariaHidden":758},[31672],[34,210907,210909,210912,210915,210918,210921],{"className":210908},[31676],[34,210910],{"className":210911,"style":35942},[31680],[34,210913,640],{"className":210914},[31685],[34,210916],{"className":210917,"style":32604},[31845],[34,210919,223],{"className":210920},[32024],[34,210922],{"className":210923,"style":32604},[31845],[34,210925,210927,210930,210933,210936,210939],{"className":210926},[31676],[34,210928],{"className":210929,"style":35942},[31680],[34,210931,95418],{"className":210932},[31685],[34,210934],{"className":210935,"style":32604},[31845],[34,210937,223],{"className":210938},[32024],[34,210940],{"className":210941,"style":32604},[31845],[34,210943,210945,210948,210951,210954,210957],{"className":210944},[31676],[34,210946],{"className":210947,"style":32617},[31680],[34,210949,188623],{"className":210950},[31685],[34,210952],{"className":210953,"style":31846},[31845],[34,210955,202],{"className":210956},[31850],[34,210958],{"className":210959,"style":31846},[31845],[34,210961,210963,210966],{"className":210962},[31676],[34,210964],{"className":210965,"style":32617},[31680],[34,210967,210899],{"className":210968},[31685],"。\n因此，将较小的页表放在顶层可以极大地减少所需的内存开销。",[32188,210971],{},[253,210973,210974],{"id":210974},"第三题解析",[15,210976,210977,210979],{},[2627,210978,200435],{},"\n3. Assume that a task is divided into four equal-sized segments and that the system builds an eight-entry page descriptor table for each segment. Thus, the system has a combination of segmentation and paging. Assume also that the page size is 2 Kbytes.\na. What is the maximum size of each segment?\nb. What is the maximum logical address space for the task?\nc. Assume that an element in physical location 00021ABC is accessed by this task. What is the format of the logical address that the task generates for it? What is the maximum physical address space for the system?",[15,210981,210982,210984],{},[2627,210983,200445],{},"\n3. 假设一个任务被划分为四个大小相等的段，并且系统为每个段构建了一个包含8个表项的页描述符表。因此，系统采用了分段和分页相结合的方式。假设页面大小为 2KB。\na. 每个段的最大大小是多少？\nb. 该任务的最大逻辑地址空间是多少？\nc. 假设该任务访问了物理位置 00021ABC 中的一个元素。该任务为其生成的逻辑地址格式是什么？系统的最大物理地址空间是多少？",[15,210986,210987],{},[2627,210988,208759],{},[105,210990,210991,211030,211068,211213,211321,211429,211501],{},[108,210992,210993,210996,210997,211026,211027,347],{},[2627,210994,210995],{},"a. 每个段的最大大小","\n每个段的页表有8个表项，这意味着每个段最多包含8个页面。每个页面大小为 2KB。\n所以每个段的最大大小 = 8 个页面 ",[34,210998,211000,211014],{"className":210999},[31642],[34,211001,211003],{"className":211002},[31646],[31648,211004,211005],{"xmlns":31650},[31652,211006,211007,211011],{},[31655,211008,211009],{},[31662,211010,42071],{},[31665,211012,211013],{"encoding":31667},"\\times",[34,211015,211017],{"className":211016,"ariaHidden":758},[31672],[34,211018,211020,211023],{"className":211019},[31676],[34,211021],{"className":211022,"style":32597},[31680],[34,211024,42071],{"className":211025},[31685]," 2 KB\u002F页面 = ",[2627,211028,211029],{},"16 KB",[108,211031,211032,211035,211036,211064,211065,347],{},[2627,211033,211034],{},"b. 最大逻辑地址空间","\n任务被分为4个段。\n最大逻辑地址空间 = 4 个段 ",[34,211037,211039,211052],{"className":211038},[31642],[34,211040,211042],{"className":211041},[31646],[31648,211043,211044],{"xmlns":31650},[31652,211045,211046,211050],{},[31655,211047,211048],{},[31662,211049,42071],{},[31665,211051,211013],{"encoding":31667},[34,211053,211055],{"className":211054,"ariaHidden":758},[31672],[34,211056,211058,211061],{"className":211057},[31676],[34,211059],{"className":211060,"style":32597},[31680],[34,211062,42071],{"className":211063},[31685]," 16 KB\u002F段 = ",[2627,211066,211067],{},"64 KB",[108,211069,211070,211073,211076,211077,211212],{},[2627,211071,211072],{},"c. 逻辑地址格式与物理地址空间",[2627,211074,211075],{},"逻辑地址格式："," 逻辑地址空间为64KB，总共需要 ",[34,211078,211080,211116],{"className":211079},[31642],[34,211081,211083],{"className":211082},[31646],[31648,211084,211085],{"xmlns":31650},[31652,211086,211087,211113],{},[31655,211088,211089,211099,211101,211103,211105,211107,211109,211111],{},[31743,211090,211091,211097],{},[31655,211092,211093,211095],{},[31658,211094,885],{},[31662,211096,32682],{},[31749,211098,768],{},[31662,211100,456],{"stretchy":702},[31749,211102,162212],{},[31662,211104,42071],{},[31749,211106,177393],{},[31662,211108,2557],{"stretchy":702},[31662,211110,202],{},[31749,211112,177869],{},[31665,211114,211115],{"encoding":31667},"\\log_2(64 \\times 1024) = 16",[34,211117,211119,211182,211203],{"className":211118,"ariaHidden":758},[31672],[34,211120,211122,211125,211167,211170,211173,211176,211179],{"className":211121},[31676],[34,211123],{"className":211124,"style":31774},[31680],[34,211126,211128,211133],{"className":211127},[32767],[34,211129,38616,211131],{"className":211130},[32767],[34,211132,31942],{"style":38619},[34,211134,211136],{"className":211135},[31792],[34,211137,211139,211159],{"className":211138},[31796,31797],[34,211140,211142,211156],{"className":211141},[31801],[34,211143,211145],{"className":211144,"style":38632},[31805],[34,211146,211147,211150],{"style":38635},[34,211148],{"className":211149,"style":31814},[31813],[34,211151,211153],{"className":211152},[31818,31819,31820,31821],[34,211154,768],{"className":211155},[31685,31821],[34,211157,31829],{"className":211158},[31828],[34,211160,211162],{"className":211161},[31801],[34,211163,211165],{"className":211164,"style":38654},[31805],[34,211166],{},[34,211168,456],{"className":211169},[31782],[34,211171,162212],{"className":211172},[31685],[34,211174],{"className":211175,"style":32604},[31845],[34,211177,42071],{"className":211178},[32024],[34,211180],{"className":211181,"style":32604},[31845],[34,211183,211185,211188,211191,211194,211197,211200],{"className":211184},[31676],[34,211186],{"className":211187,"style":31774},[31680],[34,211189,177393],{"className":211190},[31685],[34,211192,2557],{"className":211193},[31690],[34,211195],{"className":211196,"style":31846},[31845],[34,211198,202],{"className":211199},[31850],[34,211201],{"className":211202,"style":31846},[31845],[34,211204,211206,211209],{"className":211205},[31676],[34,211207],{"className":211208,"style":32617},[31680],[34,211210,177869],{"className":211211},[31685]," 位。具体分配如下：",[108,211214,211215,211218,211219,47649,211318,347],{},[2627,211216,211217],{},"段号 (Segment Number)","：共有4个段，需要 ",[34,211220,211222,211252],{"className":211221},[31642],[34,211223,211225],{"className":211224},[31646],[31648,211226,211227],{"xmlns":31650},[31652,211228,211229,211249],{},[31655,211230,211231,211241,211243,211245,211247],{},[31743,211232,211233,211239],{},[31655,211234,211235,211237],{},[31658,211236,885],{},[31662,211238,32682],{},[31749,211240,768],{},[31662,211242,456],{"stretchy":702},[31749,211244,15932],{},[31662,211246,2557],{"stretchy":702},[31662,211248,202],{},[31665,211250,211251],{"encoding":31667},"\\log_2(4) =",[34,211253,211255],{"className":211254,"ariaHidden":758},[31672],[34,211256,211258,211261,211303,211306,211309,211312,211315],{"className":211257},[31676],[34,211259],{"className":211260,"style":31774},[31680],[34,211262,211264,211269],{"className":211263},[32767],[34,211265,38616,211267],{"className":211266},[32767],[34,211268,31942],{"style":38619},[34,211270,211272],{"className":211271},[31792],[34,211273,211275,211295],{"className":211274},[31796,31797],[34,211276,211278,211292],{"className":211277},[31801],[34,211279,211281],{"className":211280,"style":38632},[31805],[34,211282,211283,211286],{"style":38635},[34,211284],{"className":211285,"style":31814},[31813],[34,211287,211289],{"className":211288},[31818,31819,31820,31821],[34,211290,768],{"className":211291},[31685,31821],[34,211293,31829],{"className":211294},[31828],[34,211296,211298],{"className":211297},[31801],[34,211299,211301],{"className":211300,"style":38654},[31805],[34,211302],{},[34,211304,456],{"className":211305},[31782],[34,211307,15932],{"className":211308},[31685],[34,211310,2557],{"className":211311},[31690],[34,211313],{"className":211314,"style":31846},[31845],[34,211316,202],{"className":211317},[31850],[2627,211319,211320],{},"2位",[108,211322,211323,211326,211327,47649,211426,347],{},[2627,211324,211325],{},"页号 (Page Number)","：每个段有8个页面，需要 ",[34,211328,211330,211360],{"className":211329},[31642],[34,211331,211333],{"className":211332},[31646],[31648,211334,211335],{"xmlns":31650},[31652,211336,211337,211357],{},[31655,211338,211339,211349,211351,211353,211355],{},[31743,211340,211341,211347],{},[31655,211342,211343,211345],{},[31658,211344,885],{},[31662,211346,32682],{},[31749,211348,768],{},[31662,211350,456],{"stretchy":702},[31749,211352,60525],{},[31662,211354,2557],{"stretchy":702},[31662,211356,202],{},[31665,211358,211359],{"encoding":31667},"\\log_2(8) =",[34,211361,211363],{"className":211362,"ariaHidden":758},[31672],[34,211364,211366,211369,211411,211414,211417,211420,211423],{"className":211365},[31676],[34,211367],{"className":211368,"style":31774},[31680],[34,211370,211372,211377],{"className":211371},[32767],[34,211373,38616,211375],{"className":211374},[32767],[34,211376,31942],{"style":38619},[34,211378,211380],{"className":211379},[31792],[34,211381,211383,211403],{"className":211382},[31796,31797],[34,211384,211386,211400],{"className":211385},[31801],[34,211387,211389],{"className":211388,"style":38632},[31805],[34,211390,211391,211394],{"style":38635},[34,211392],{"className":211393,"style":31814},[31813],[34,211395,211397],{"className":211396},[31818,31819,31820,31821],[34,211398,768],{"className":211399},[31685,31821],[34,211401,31829],{"className":211402},[31828],[34,211404,211406],{"className":211405},[31801],[34,211407,211409],{"className":211408,"style":38654},[31805],[34,211410],{},[34,211412,456],{"className":211413},[31782],[34,211415,60525],{"className":211416},[31685],[34,211418,2557],{"className":211419},[31690],[34,211421],{"className":211422,"style":31846},[31845],[34,211424,202],{"className":211425},[31850],[2627,211427,211428],{},"3位",[108,211430,211431,211434,211435,211497,211498,347],{},[2627,211432,211433],{},"页内偏移量 (Offset)","：页面大小为2KB（",[34,211436,211438,211456],{"className":211437},[31642],[34,211439,211441],{"className":211440},[31646],[31648,211442,211443],{"xmlns":31650},[31652,211444,211445,211453],{},[31655,211446,211447],{},[37624,211448,211449,211451],{},[31749,211450,768],{},[31749,211452,39620],{},[31665,211454,211455],{"encoding":31667},"2^{11}",[34,211457,211459],{"className":211458,"ariaHidden":758},[31672],[34,211460,211462,211465],{"className":211461},[31676],[34,211463],{"className":211464,"style":37774},[31680],[34,211466,211468,211471],{"className":211467},[31685],[34,211469,768],{"className":211470},[31685],[34,211472,211474],{"className":211473},[31792],[34,211475,211477],{"className":211476},[31796],[34,211478,211480],{"className":211479},[31801],[34,211481,211483],{"className":211482,"style":37774},[31805],[34,211484,211485,211488],{"style":37777},[34,211486],{"className":211487,"style":31814},[31813],[34,211489,211491],{"className":211490},[31818,31819,31820,31821],[34,211492,211494],{"className":211493},[31685,31821],[34,211495,39620],{"className":211496},[31685,31821]," 字节），需要 ",[2627,211499,211500],{},"11位",[108,211502,211503,211504,211507],{},"格式为：",[2627,211505,211506],{},"2位段号 + 3位页号 + 11位偏移量","（共16位）。",[15,211509,211510,211513,211514,211517,211518,211591,211592,205934,211653,347],{},[2627,211511,211512],{},"最大物理地址空间："," 物理地址给出的示例是十六进制数 ",[31,211515,211516],{},"00021ABC","。 这个数有8位十六进制字符，每个十六进制字符占用4个二进制位，因此该物理地址长度为 ",[34,211519,211521,211543],{"className":211520},[31642],[34,211522,211524],{"className":211523},[31646],[31648,211525,211526],{"xmlns":31650},[31652,211527,211528,211540],{},[31655,211529,211530,211532,211534,211536,211538],{},[31749,211531,60525],{},[31662,211533,42071],{},[31749,211535,15932],{},[31662,211537,202],{},[31749,211539,173486],{},[31665,211541,211542],{"encoding":31667},"8 \\times 4 = 32",[34,211544,211546,211564,211582],{"className":211545,"ariaHidden":758},[31672],[34,211547,211549,211552,211555,211558,211561],{"className":211548},[31676],[34,211550],{"className":211551,"style":35942},[31680],[34,211553,60525],{"className":211554},[31685],[34,211556],{"className":211557,"style":32604},[31845],[34,211559,42071],{"className":211560},[32024],[34,211562],{"className":211563,"style":32604},[31845],[34,211565,211567,211570,211573,211576,211579],{"className":211566},[31676],[34,211568],{"className":211569,"style":32617},[31680],[34,211571,15932],{"className":211572},[31685],[34,211574],{"className":211575,"style":31846},[31845],[34,211577,202],{"className":211578},[31850],[34,211580],{"className":211581,"style":31846},[31845],[34,211583,211585,211588],{"className":211584},[31676],[34,211586],{"className":211587,"style":32617},[31680],[34,211589,173486],{"className":211590},[31685]," 位。\n最大物理地址空间 = ",[34,211593,211595,211612],{"className":211594},[31642],[34,211596,211598],{"className":211597},[31646],[31648,211599,211600],{"xmlns":31650},[31652,211601,211602,211610],{},[31655,211603,211604],{},[37624,211605,211606,211608],{},[31749,211607,768],{},[31749,211609,173486],{},[31665,211611,201449],{"encoding":31667},[34,211613,211615],{"className":211614,"ariaHidden":758},[31672],[34,211616,211618,211621],{"className":211617},[31676],[34,211619],{"className":211620,"style":37774},[31680],[34,211622,211624,211627],{"className":211623},[31685],[34,211625,768],{"className":211626},[31685],[34,211628,211630],{"className":211629},[31792],[34,211631,211633],{"className":211632},[31796],[34,211634,211636],{"className":211635},[31801],[34,211637,211639],{"className":211638,"style":37774},[31805],[34,211640,211641,211644],{"style":37777},[34,211642],{"className":211643,"style":31814},[31813],[34,211645,211647],{"className":211646},[31818,31819,31820,31821],[34,211648,211650],{"className":211649},[31685,31821],[34,211651,173486],{"className":211652},[31685,31821],[2627,211654,211655],{},"4 GB",[32188,211657],{},[253,211659,211660],{"id":211660},"第四题解析",[15,211662,211663,211665],{},[2627,211664,200435],{},"\n4. Four page frames in main memory. The following page trace occurs: 1, 0, 2, 2, 1, 7, 6, 7, 0, 1, 2, 0, 3, 0, 4, 5, 1, 5, 2, 4, 5, 6, 7, 6, 7, 2, 4, 2, 7, 3, 3, 2, 3\na. Show the successive pages residing in the four frames using the LRU replacement policy. Compute the hit ratio in main memory. Assume that the frames are initially empty.\nb. Repeat part (a) for the FIFO replacement policy.\nc. Compare the two hit ratios and comment on the effectiveness of using FIFO to approximate LRU with respect to this particular trace.",[15,211667,211668,211670],{},[2627,211669,200445],{},"\n4. 主存中有四个页框。发生以下页面访问轨迹：1, 0, 2, 2, 1, 7, 6, 7, 0, 1, 2, 0, 3, 0, 4, 5, 1, 5, 2, 4, 5, 6, 7, 6, 7, 2, 4, 2, 7, 3, 3, 2, 3\na. 使用LRU（最近最少使用）替换策略显示连续驻留在四个页框中的页面。计算主存中的命中率。假设页框最初为空。\nb. 对于FIFO（先进先出）替换策略，重复部分(a)。\nc. 比较两种命中率，并评论针对此特定轨迹使用FIFO来近似LRU的有效性。",[15,211672,211673,211675,211676,211678,211679,211681],{},[2627,211674,208759],{},"\n页面访问总次数为 ",[2627,211677,185190],{}," 次。内存有 ",[2627,211680,15932],{}," 个页框。",[105,211683,211684,211689,211694,211700,211706,211714,211722,211728,211735,211743,211749,211756,211762,211770,211777,211784,211791,211797,211803,211811,211816,211824,211831,211836,211841,211849,211855,211861,211867,211874,211881,211886,211893,211900,211907,211946,211951,211955,211959,211963,211969,211975,211980,211986,211992,211998,212004,212010,212016,212022,212028,212034,212039,212044,212050,212056,212062,212068,212074,212080,212086,212092,212098,212103,212109,212115,212120,212126,212131,212137,212172],{},[108,211685,211686],{},[2627,211687,211688],{},"a. LRU (Least Recently Used) 策略：替换最长时间未被访问的页面",[108,211690,211691,211692],{},"1: 未命中 ",[34,211693,640],{},[108,211695,211696,211697],{},"0: 未命中 ",[34,211698,211699],{},"1, 0",[108,211701,211702,211703],{},"2: 未命中 ",[34,211704,211705],{},"1, 0, 2",[108,211707,211708,211709,47649,211712],{},"2: ",[2627,211710,211711],{},"命中",[34,211713,211705],{},[108,211715,211716,211717,47649,211719],{},"1: ",[2627,211718,211711],{},[34,211720,211721],{},"0, 2, 1",[108,211723,211724,211725],{},"7: 未命中 ",[34,211726,211727],{},"0, 2, 1, 7",[108,211729,211730,211731,211734],{},"6: 未命中 ",[34,211732,211733],{},"2, 1, 7, 6"," (替换最久未用的0)",[108,211736,211737,211738,47649,211740],{},"7: ",[2627,211739,211711],{},[34,211741,211742],{},"2, 1, 6, 7",[108,211744,211696,211745,211748],{},[34,211746,211747],{},"1, 6, 7, 0"," (替换最久未用的2)",[108,211750,211716,211751,47649,211753],{},[2627,211752,211711],{},[34,211754,211755],{},"6, 7, 0, 1",[108,211757,211702,211758,211761],{},[34,211759,211760],{},"7, 0, 1, 2"," (替换最久未用的6)",[108,211763,211764,211765,47649,211767],{},"0: ",[2627,211766,211711],{},[34,211768,211769],{},"7, 1, 2, 0",[108,211771,211772,211773,211776],{},"3: 未命中 ",[34,211774,211775],{},"1, 2, 0, 3"," (替换最久未用的7)",[108,211778,211764,211779,47649,211781],{},[2627,211780,211711],{},[34,211782,211783],{},"1, 2, 3, 0",[108,211785,211786,211787,211790],{},"4: 未命中 ",[34,211788,211789],{},"2, 3, 0, 4"," (替换最久未用的1)",[108,211792,211793,211794,211748],{},"5: 未命中 ",[34,211795,211796],{},"3, 0, 4, 5",[108,211798,211691,211799,211802],{},[34,211800,211801],{},"0, 4, 5, 1"," (替换最久未用的3)",[108,211804,211805,211806,47649,211808],{},"5: ",[2627,211807,211711],{},[34,211809,211810],{},"0, 4, 1, 5",[108,211812,211702,211813,211734],{},[34,211814,211815],{},"4, 1, 5, 2",[108,211817,211818,211819,47649,211821],{},"4: ",[2627,211820,211711],{},[34,211822,211823],{},"1, 5, 2, 4",[108,211825,211805,211826,47649,211828],{},[2627,211827,211711],{},[34,211829,211830],{},"1, 2, 4, 5",[108,211832,211730,211833,211790],{},[34,211834,211835],{},"2, 4, 5, 6",[108,211837,211724,211838,211748],{},[34,211839,211840],{},"4, 5, 6, 7",[108,211842,211843,211844,47649,211846],{},"6: ",[2627,211845,211711],{},[34,211847,211848],{},"4, 5, 7, 6",[108,211850,211737,211851,47649,211853],{},[2627,211852,211711],{},[34,211854,211840],{},[108,211856,211702,211857,211860],{},[34,211858,211859],{},"5, 6, 7, 2"," (替换最久未用的4)",[108,211862,211786,211863,211866],{},[34,211864,211865],{},"6, 7, 2, 4"," (替换最久未用的5)",[108,211868,211708,211869,47649,211871],{},[2627,211870,211711],{},[34,211872,211873],{},"6, 7, 4, 2",[108,211875,211737,211876,47649,211878],{},[2627,211877,211711],{},[34,211879,211880],{},"6, 4, 2, 7",[108,211882,211772,211883,211761],{},[34,211884,211885],{},"4, 2, 7, 3",[108,211887,211888,211889,47649,211891],{},"3: ",[2627,211890,211711],{},[34,211892,211885],{},[108,211894,211708,211895,47649,211897],{},[2627,211896,211711],{},[34,211898,211899],{},"4, 7, 3, 2",[108,211901,211888,211902,47649,211904],{},[2627,211903,211711],{},[34,211905,211906],{},"4, 7, 2, 3",[108,211908,211909,211912,211913,47649,211943,347],{},[2627,211910,211911],{},"LRU结果","：总命中次数 = 16次。命中率 = 16 \u002F 33 ",[34,211914,211916,211930],{"className":211915},[31642],[34,211917,211919],{"className":211918},[31646],[31648,211920,211921],{"xmlns":31650},[31652,211922,211923,211927],{},[31655,211924,211925],{},[31662,211926,168560],{},[31665,211928,211929],{"encoding":31667},"\\approx",[34,211931,211933],{"className":211932,"ariaHidden":758},[31672],[34,211934,211936,211940],{"className":211935},[31676],[34,211937],{"className":211938,"style":211939},[31680],"height:0.4831em;",[34,211941,168560],{"className":211942},[31850],[2627,211944,211945],{},"48.48%",[108,211947,211948],{},[2627,211949,211950],{},"b. FIFO (First-In-First-Out) 策略：替换最早进入内存的页面",[108,211952,211691,211953],{},[34,211954,640],{},[108,211956,211696,211957],{},[34,211958,211699],{},[108,211960,211702,211961],{},[34,211962,211705],{},[108,211964,211708,211965,47649,211967],{},[2627,211966,211711],{},[34,211968,211705],{},[108,211970,211716,211971,47649,211973],{},[2627,211972,211711],{},[34,211974,211705],{},[108,211976,211724,211977],{},[34,211978,211979],{},"1, 0, 2, 7",[108,211981,211730,211982,211985],{},[34,211983,211984],{},"0, 2, 7, 6"," (替换最早进入的1)",[108,211987,211737,211988,47649,211990],{},[2627,211989,211711],{},[34,211991,211984],{},[108,211993,211764,211994,47649,211996],{},[2627,211995,211711],{},[34,211997,211984],{},[108,211999,211691,212000,212003],{},[34,212001,212002],{},"2, 7, 6, 1"," (替换最早进入的0)",[108,212005,211708,212006,47649,212008],{},[2627,212007,211711],{},[34,212009,212002],{},[108,212011,211696,212012,212015],{},[34,212013,212014],{},"7, 6, 1, 0"," (替换最早进入的2)",[108,212017,211772,212018,212021],{},[34,212019,212020],{},"6, 1, 0, 3"," (替换最早进入的7)",[108,212023,211764,212024,47649,212026],{},[2627,212025,211711],{},[34,212027,212020],{},[108,212029,211786,212030,212033],{},[34,212031,212032],{},"1, 0, 3, 4"," (替换最早进入的6)",[108,212035,211793,212036,211985],{},[34,212037,212038],{},"0, 3, 4, 5",[108,212040,211691,212041,212003],{},[34,212042,212043],{},"3, 4, 5, 1",[108,212045,211805,212046,47649,212048],{},[2627,212047,211711],{},[34,212049,212043],{},[108,212051,211702,212052,212055],{},[34,212053,212054],{},"4, 5, 1, 2"," (替换最早进入的3)",[108,212057,211818,212058,47649,212060],{},[2627,212059,211711],{},[34,212061,212054],{},[108,212063,211805,212064,47649,212066],{},[2627,212065,211711],{},[34,212067,212054],{},[108,212069,211730,212070,212073],{},[34,212071,212072],{},"5, 1, 2, 6"," (替换最早进入的4)",[108,212075,211724,212076,212079],{},[34,212077,212078],{},"1, 2, 6, 7"," (替换最早进入的5)",[108,212081,211843,212082,47649,212084],{},[2627,212083,211711],{},[34,212085,212078],{},[108,212087,211737,212088,47649,212090],{},[2627,212089,211711],{},[34,212091,212078],{},[108,212093,211708,212094,47649,212096],{},[2627,212095,211711],{},[34,212097,212078],{},[108,212099,211786,212100,211985],{},[34,212101,212102],{},"2, 6, 7, 4",[108,212104,211708,212105,47649,212107],{},[2627,212106,211711],{},[34,212108,212102],{},[108,212110,211737,212111,47649,212113],{},[2627,212112,211711],{},[34,212114,212102],{},[108,212116,211772,212117,212015],{},[34,212118,212119],{},"6, 7, 4, 3",[108,212121,211888,212122,47649,212124],{},[2627,212123,211711],{},[34,212125,212119],{},[108,212127,211702,212128,212033],{},[34,212129,212130],{},"7, 4, 3, 2",[108,212132,211888,212133,47649,212135],{},[2627,212134,211711],{},[34,212136,212130],{},[108,212138,212139,211912,212142,47649,212170,347],{},[2627,212140,212141],{},"FIFO结果",[34,212143,212145,212158],{"className":212144},[31642],[34,212146,212148],{"className":212147},[31646],[31648,212149,212150],{"xmlns":31650},[31652,212151,212152,212156],{},[31655,212153,212154],{},[31662,212155,168560],{},[31665,212157,211929],{"encoding":31667},[34,212159,212161],{"className":212160,"ariaHidden":758},[31672],[34,212162,212164,212167],{"className":212163},[31676],[34,212165],{"className":212166,"style":211939},[31680],[34,212168,168560],{"className":212169},[31850],[2627,212171,211945],{},[108,212173,212174,212177,212178,212181,212182,212185],{},[2627,212175,212176],{},"c. 命中率对比与评论","\n通过严格演算可以看出，针对这个特定的访问轨迹，FIFO和LRU的命中率是",[2627,212179,212180],{},"完全相同","的（均为16\u002F33）。\n",[2627,212183,212184],{},"评论","：这表明在某些特定的内存访问模式下（例如工作集变化剧烈，或者存在淘汰旧页面的循环访问模式），最老的页面恰好也是最近最少使用的页面，此时FIFO可以作为LRU的极佳近似。FIFO的优势在于其实现极其简单（只需要维护一个队列指针），不需要像LRU那样在每次访问时都去更新时间戳或链表，从而大幅降低了系统开销。然而需要指出的是，FIFO存在Belady异常（分配更多内存页框时缺页率反而上升的现象），而LRU不存在。",[32188,212187],{},[253,212189,212190],{"id":212190},"第五题解析",[15,212192,212193,212195],{},[2627,212194,200435],{},"\n5. Assuming a page size of 4 Kbytes and that a page table entry takes 4 bytes, how many levels of page tables would be required to map a 64-bit address space, if the top level page table fits into a single page?",[15,212197,212198,212200],{},[2627,212199,200445],{},"\n5. 假设页面大小为 4KB，且每个页表项占用 4 字节，如果顶级页表正好适合放在一个页面中，那么映射一个 64 位地址空间需要多少级页表？",[15,212202,212203],{},[2627,212204,208759],{},[2230,212206,212207,212210,212277,212343,212404,212586,212690,212693],{},[108,212208,212209],{},"虚拟地址总长度 = 64位。",[108,212211,212212,212213,209740,212274,347],{},"页面大小 = 4KB = ",[34,212214,212216,212233],{"className":212215},[31642],[34,212217,212219],{"className":212218},[31646],[31648,212220,212221],{"xmlns":31650},[31652,212222,212223,212231],{},[31655,212224,212225],{},[37624,212226,212227,212229],{},[31749,212228,768],{},[31749,212230,71869],{},[31665,212232,205892],{"encoding":31667},[34,212234,212236],{"className":212235,"ariaHidden":758},[31672],[34,212237,212239,212242],{"className":212238},[31676],[34,212240],{"className":212241,"style":37774},[31680],[34,212243,212245,212248],{"className":212244},[31685],[34,212246,768],{"className":212247},[31685],[34,212249,212251],{"className":212250},[31792],[34,212252,212254],{"className":212253},[31796],[34,212255,212257],{"className":212256},[31801],[34,212258,212260],{"className":212259,"style":37774},[31805],[34,212261,212262,212265],{"style":37777},[34,212263],{"className":212264,"style":31814},[31813],[34,212266,212268],{"className":212267},[31818,31819,31820,31821],[34,212269,212271],{"className":212270},[31685,31821],[34,212272,71869],{"className":212273},[31685,31821],[2627,212275,212276],{},"12位",[108,212278,212279,212280,47649,212339,212342],{},"虚拟页号 (VPN) 长度 = ",[34,212281,212283,212303],{"className":212282},[31642],[34,212284,212286],{"className":212285},[31646],[31648,212287,212288],{"xmlns":31650},[31652,212289,212290,212300],{},[31655,212291,212292,212294,212296,212298],{},[31749,212293,162212],{},[31662,212295,32226],{},[31749,212297,71869],{},[31662,212299,202],{},[31665,212301,212302],{"encoding":31667},"64 - 12 =",[34,212304,212306,212324],{"className":212305,"ariaHidden":758},[31672],[34,212307,212309,212312,212315,212318,212321],{"className":212308},[31676],[34,212310],{"className":212311,"style":35942},[31680],[34,212313,162212],{"className":212314},[31685],[34,212316],{"className":212317,"style":32604},[31845],[34,212319,32226],{"className":212320},[32024],[34,212322],{"className":212323,"style":32604},[31845],[34,212325,212327,212330,212333,212336],{"className":212326},[31676],[34,212328],{"className":212329,"style":32617},[31680],[34,212331,71869],{"className":212332},[31685],[34,212334],{"className":212335,"style":31846},[31845],[34,212337,202],{"className":212338},[31850],[2627,212340,212341],{},"52位","。我们需要用多级页表来映射这52位。",[108,212344,212345,212346,209871],{},"页表项大小 = 4字节（",[34,212347,212349,212366],{"className":212348},[31642],[34,212350,212352],{"className":212351},[31646],[31648,212353,212354],{"xmlns":31650},[31652,212355,212356,212364],{},[31655,212357,212358],{},[37624,212359,212360,212362],{},[31749,212361,768],{},[31749,212363,768],{},[31665,212365,209832],{"encoding":31667},[34,212367,212369],{"className":212368,"ariaHidden":758},[31672],[34,212370,212372,212375],{"className":212371},[31676],[34,212373],{"className":212374,"style":37774},[31680],[34,212376,212378,212381],{"className":212377},[31685],[34,212379,768],{"className":212380},[31685],[34,212382,212384],{"className":212383},[31792],[34,212385,212387],{"className":212386},[31796],[34,212388,212390],{"className":212389},[31801],[34,212391,212393],{"className":212392,"style":37774},[31805],[34,212394,212395,212398],{"style":37777},[34,212396],{"className":212397,"style":31814},[31813],[34,212399,212401],{"className":212400},[31818,31819,31820,31821],[34,212402,768],{"className":212403},[31685,31821],[108,212405,209874,212406,212585],{},[34,212407,212409,212447],{"className":212408},[31642],[34,212410,212412],{"className":212411},[31646],[31648,212413,212414],{"xmlns":31650},[31652,212415,212416,212444],{},[31655,212417,212418,212424,212426,212432,212434,212440,212442],{},[37624,212419,212420,212422],{},[31749,212421,768],{},[31749,212423,71869],{},[31658,212425,38080],{"mathvariant":38079},[37624,212427,212428,212430],{},[31749,212429,768],{},[31749,212431,768],{},[31662,212433,202],{},[37624,212435,212436,212438],{},[31749,212437,768],{},[31749,212439,3244],{},[31662,212441,202],{},[31749,212443,177393],{},[31665,212445,212446],{"encoding":31667},"2^{12} \u002F 2^2 = 2^{10} = 1024",[34,212448,212450,212529,212576],{"className":212449,"ariaHidden":758},[31672],[34,212451,212453,212456,212488,212491,212520,212523,212526],{"className":212452},[31676],[34,212454],{"className":212455,"style":37755},[31680],[34,212457,212459,212462],{"className":212458},[31685],[34,212460,768],{"className":212461},[31685],[34,212463,212465],{"className":212464},[31792],[34,212466,212468],{"className":212467},[31796],[34,212469,212471],{"className":212470},[31801],[34,212472,212474],{"className":212473,"style":37774},[31805],[34,212475,212476,212479],{"style":37777},[34,212477],{"className":212478,"style":31814},[31813],[34,212480,212482],{"className":212481},[31818,31819,31820,31821],[34,212483,212485],{"className":212484},[31685,31821],[34,212486,71869],{"className":212487},[31685,31821],[34,212489,38080],{"className":212490},[31685],[34,212492,212494,212497],{"className":212493},[31685],[34,212495,768],{"className":212496},[31685],[34,212498,212500],{"className":212499},[31792],[34,212501,212503],{"className":212502},[31796],[34,212504,212506],{"className":212505},[31801],[34,212507,212509],{"className":212508,"style":37774},[31805],[34,212510,212511,212514],{"style":37777},[34,212512],{"className":212513,"style":31814},[31813],[34,212515,212517],{"className":212516},[31818,31819,31820,31821],[34,212518,768],{"className":212519},[31685,31821],[34,212521],{"className":212522,"style":31846},[31845],[34,212524,202],{"className":212525},[31850],[34,212527],{"className":212528,"style":31846},[31845],[34,212530,212532,212535,212567,212570,212573],{"className":212531},[31676],[34,212533],{"className":212534,"style":37774},[31680],[34,212536,212538,212541],{"className":212537},[31685],[34,212539,768],{"className":212540},[31685],[34,212542,212544],{"className":212543},[31792],[34,212545,212547],{"className":212546},[31796],[34,212548,212550],{"className":212549},[31801],[34,212551,212553],{"className":212552,"style":37774},[31805],[34,212554,212555,212558],{"style":37777},[34,212556],{"className":212557,"style":31814},[31813],[34,212559,212561],{"className":212560},[31818,31819,31820,31821],[34,212562,212564],{"className":212563},[31685,31821],[34,212565,3244],{"className":212566},[31685,31821],[34,212568],{"className":212569,"style":31846},[31845],[34,212571,202],{"className":212572},[31850],[34,212574],{"className":212575,"style":31846},[31845],[34,212577,212579,212582],{"className":212578},[31676],[34,212580],{"className":212581,"style":32617},[31680],[34,212583,177393],{"className":212584},[31685]," 个表项。",[108,212587,212588,212589,47649,212688,210145],{},"因此，每一个占满一整页的页表，可以映射 ",[34,212590,212592,212622],{"className":212591},[31642],[34,212593,212595],{"className":212594},[31646],[31648,212596,212597],{"xmlns":31650},[31652,212598,212599,212619],{},[31655,212600,212601,212611,212613,212615,212617],{},[31743,212602,212603,212609],{},[31655,212604,212605,212607],{},[31658,212606,885],{},[31662,212608,32682],{},[31749,212610,768],{},[31662,212612,456],{"stretchy":702},[31749,212614,177393],{},[31662,212616,2557],{"stretchy":702},[31662,212618,202],{},[31665,212620,212621],{"encoding":31667},"\\log_2(1024) =",[34,212623,212625],{"className":212624,"ariaHidden":758},[31672],[34,212626,212628,212631,212673,212676,212679,212682,212685],{"className":212627},[31676],[34,212629],{"className":212630,"style":31774},[31680],[34,212632,212634,212639],{"className":212633},[32767],[34,212635,38616,212637],{"className":212636},[32767],[34,212638,31942],{"style":38619},[34,212640,212642],{"className":212641},[31792],[34,212643,212645,212665],{"className":212644},[31796,31797],[34,212646,212648,212662],{"className":212647},[31801],[34,212649,212651],{"className":212650,"style":38632},[31805],[34,212652,212653,212656],{"style":38635},[34,212654],{"className":212655,"style":31814},[31813],[34,212657,212659],{"className":212658},[31818,31819,31820,31821],[34,212660,768],{"className":212661},[31685,31821],[34,212663,31829],{"className":212664},[31828],[34,212666,212668],{"className":212667},[31801],[34,212669,212671],{"className":212670,"style":38654},[31805],[34,212672],{},[34,212674,456],{"className":212675},[31782],[34,212677,177393],{"className":212678},[31685],[34,212680,2557],{"className":212681},[31690],[34,212683],{"className":212684,"style":31846},[31845],[34,212686,202],{"className":212687},[31850],[2627,212689,209743],{},[108,212691,212692],{},"题目规定“顶级页表适合放在一个页面中”，为了使层级最少，除了顶级页表外，每一级都应该最大化利用（即每级映射10位）。",[108,212694,212695,212696,212770,212771,210331],{},"计算所需级数：",[34,212697,212699,212722],{"className":212698},[31642],[34,212700,212702],{"className":212701},[31646],[31648,212703,212704],{"xmlns":31650},[31652,212705,212706,212719],{},[31655,212707,212708,212710,212712,212714,212716],{},[31749,212709,111450],{},[31662,212711,207557],{},[31749,212713,3244],{},[31662,212715,202],{},[31749,212717,212718],{},"5.2",[31665,212720,212721],{"encoding":31667},"52 \\div 10 = 5.2",[34,212723,212725,212743,212761],{"className":212724,"ariaHidden":758},[31672],[34,212726,212728,212731,212734,212737,212740],{"className":212727},[31676],[34,212729],{"className":212730,"style":35942},[31680],[34,212732,111450],{"className":212733},[31685],[34,212735],{"className":212736,"style":32604},[31845],[34,212738,207557],{"className":212739},[32024],[34,212741],{"className":212742,"style":32604},[31845],[34,212744,212746,212749,212752,212755,212758],{"className":212745},[31676],[34,212747],{"className":212748,"style":32617},[31680],[34,212750,3244],{"className":212751},[31685],[34,212753],{"className":212754,"style":31846},[31845],[34,212756,202],{"className":212757},[31850],[34,212759],{"className":212760,"style":31846},[31845],[34,212762,212764,212767],{"className":212763},[31676],[34,212765],{"className":212766,"style":32617},[31680],[34,212768,212718],{"className":212769},[31685],"。这意味着5级页表只能映射50位，不足以覆盖52位。因此我们必须向上取整，需要 ",[2627,212772,212773],{},"6级",[105,212775,212776],{},[108,212777,212778,212779,212860],{},"具体划分：前五级每级映射10位，最顶级页表映射剩余的2位（顶级页表只有 ",[34,212780,212782,212804],{"className":212781},[31642],[34,212783,212785],{"className":212784},[31646],[31648,212786,212787],{"xmlns":31650},[31652,212788,212789,212801],{},[31655,212790,212791,212797,212799],{},[37624,212792,212793,212795],{},[31749,212794,768],{},[31749,212796,768],{},[31662,212798,202],{},[31749,212800,15932],{},[31665,212802,212803],{"encoding":31667},"2^2=4",[34,212805,212807,212851],{"className":212806,"ariaHidden":758},[31672],[34,212808,212810,212813,212842,212845,212848],{"className":212809},[31676],[34,212811],{"className":212812,"style":37774},[31680],[34,212814,212816,212819],{"className":212815},[31685],[34,212817,768],{"className":212818},[31685],[34,212820,212822],{"className":212821},[31792],[34,212823,212825],{"className":212824},[31796],[34,212826,212828],{"className":212827},[31801],[34,212829,212831],{"className":212830,"style":37774},[31805],[34,212832,212833,212836],{"style":37777},[34,212834],{"className":212835,"style":31814},[31813],[34,212837,212839],{"className":212838},[31818,31819,31820,31821],[34,212840,768],{"className":212841},[31685,31821],[34,212843],{"className":212844,"style":31846},[31845],[34,212846,202],{"className":212847},[31850],[34,212849],{"className":212850,"style":31846},[31845],[34,212852,212854,212857],{"className":212853},[31676],[34,212855],{"className":212856,"style":32617},[31680],[34,212858,15932],{"className":212859},[31685]," 个表项，完全符合“正好适合放在一个页面中”的条件）。",[32188,212862],{},[253,212864,212865],{"id":212865},"第六题解析",[15,212867,212868,212870,212871,212874],{},[2627,212869,200435],{},"\n6. A process has four page frames allocated to it. (All the following numbers are decimal, and everything is numbered starting from zero). The time of the last loading of a page into each page frame, the time of last access to the page in each page frame, the virtual page number in each page frame, and the referenced (R) and modified (M) bits for each page frame are as shown (the times are in clock ticks from the process start at time 0 to the event not the number of ticks since the event to the present). ",[34,212872,212873],{},"Table data: Frame 0(VPN 2, Load 60, Ref 161, R 0, M 1), Frame 1(VPN 1, Load 130, Ref 160, R 1, M 0), Frame 2(VPN 0, Load 26, Ref 162, R 1, M 0), Frame 3(VPN 3, Load 20, Ref 163, R 1, M 1)","\nA page fault to virtual page 4 has occurred at time 164.Which page frame will have its contents replaced for each of the following memory management policies? Explain why in each case.\na. FIFO\nb. LRU\nc. Clock\nd. Optimal (Use the following reference string.)\ne. Given the aforementioned state of memory just before the page fault, consider the following virtual page reference string: 4, 0, 0, 0, 2, 4, 2, 1, 0, 3, 2 How many page faults would occur if the working set policy with LRU were used with a window size of 4 instead of a fixed allocation? Show clearly when each page fault would occur.",[15,212876,212877,212879,212880,212883],{},[2627,212878,200445],{},"\n6. 分配给某个进程四个页框。（以下所有数字均为十进制，一切从零开始编号）。每个页框中页面的最后加载时间、最后访问时间、虚拟页号，以及每个页框的访问(R)位和修改(M)位如下所示（时间是从进程在时间0开始到事件发生的时钟滴答数，而不是从事件发生到现在的滴答数）。",[34,212881,212882],{},"表格数据: 框0(页2, 载入60, 访问161, R=0, M=1), 框1(页1, 载入130, 访问160, R=1, M=0), 框2(页0, 载入26, 访问162, R=1, M=0), 框3(页3, 载入20, 访问163, R=1, M=1)"," 在时间164，发生了对虚拟页面4的缺页异常。对于以下每种内存管理策略，哪个页框的内容将被替换？请分别解释原因。\na. FIFO（先进先出） b. LRU（最近最少使用） c. Clock（时钟算法） d. Optimal（最佳算法） e. 假设在缺页之前的上述内存状态下，考虑引用串：4, 0, 0, 0, 2, 4, 2, 1, 0, 3, 2。如果使用窗口大小为4的LRU工作集策略而不是固定分配，会发生多少次缺页异常？清楚地说明每次缺页发生的时间。",[15,212885,212886,212888],{},[2627,212887,208759],{},"\n我们先整理一下当前在内存中的页面及其属性：",[105,212890,212891,212894,212897,212900,212903,212913,212922,212931,212944,212947,212950,212953,212958],{},[108,212892,212893],{},"框0: 页2 (载入: 60, 访问: 161, R=0, M=1)",[108,212895,212896],{},"框1: 页1 (载入: 130, 访问: 160, R=1, M=0)",[108,212898,212899],{},"框2: 页0 (载入: 26, 访问: 162, R=1, M=0)",[108,212901,212902],{},"框3: 页3 (载入: 20, 访问: 163, R=1, M=1)",[108,212904,212905,212908,212909,212912],{},[2627,212906,212907],{},"a. FIFO 策略","：替换最早被装入内存的页面。比较四个页框的“载入时间(Time loaded)”，框3的时间最小（为20）。解释：所以 ",[2627,212910,212911],{},"页框3"," 的内容将被替换。",[108,212914,212915,212918,212919,212912],{},[2627,212916,212917],{},"b. LRU 策略","：替换最长时间未被访问的页面。比较四个页框的“访问时间(Time referenced)”，框1的时间最小（为160，距离当前时间164最远）。解释：所以 ",[2627,212920,212921],{},"页框1",[108,212923,212924,212927,212928,212912],{},[2627,212925,212926],{},"c. Clock (时钟) 策略","：时钟算法扫描页框集合，寻找引用位(R bit)为0的页面进行替换。当前R位的情况是：框0(R=0)，框1(R=1)，框2(R=1)，框3(R=1)。无论时钟指针从哪里开始扫描，只要遇到R=1就将其清零并向后移动，遇到R=0则直接替换。因为框0是唯一一个当前R=0的页面。解释：所以 ",[2627,212929,212930],{},"页框0",[108,212932,212933,212936,212937,212940,212941,347],{},[2627,212934,212935],{},"d. Optimal (最佳) 策略","：替换在未来最晚才会被访问到的页面。未来的引用序列是：",[31,212938,212939],{},"4, 0, 0, 0, 2, 4, 2, 1, 0, 3, 2","。当前内存里的页面是 ",[31,212942,212943],{},"2, 1, 0, 3",[108,212945,212946],{},"页0在未来第2个被访问。",[108,212948,212949],{},"页2在未来第5个被访问。",[108,212951,212952],{},"页1在未来第8个被访问。",[108,212954,212955,212956,212912],{},"页3在未来第10个被访问。\n解释：因为页3在未来最晚被访问，所以 ",[2627,212957,212911],{},[108,212959,212960,212963,212964,213015,213016,213019,213020,213023,213024,213027],{},[2627,212961,212962],{},"e. 窗口大小为4的工作集策略","：\n工作集策略中，窗口大小 ",[34,212965,212967,212985],{"className":212966},[31642],[34,212968,212970],{"className":212969},[31646],[31648,212971,212972],{"xmlns":31650},[31652,212973,212974,212982],{},[31655,212975,212976,212978,212980],{},[31658,212977,86019],{"mathvariant":38079},[31662,212979,202],{},[31749,212981,15932],{},[31665,212983,212984],{"encoding":31667},"\\Delta = 4",[34,212986,212988,213006],{"className":212987,"ariaHidden":758},[31672],[34,212989,212991,212994,212997,213000,213003],{"className":212990},[31676],[34,212992],{"className":212993,"style":33218},[31680],[34,212995,86019],{"className":212996},[31685],[34,212998],{"className":212999,"style":31846},[31845],[34,213001,202],{"className":213002},[31850],[34,213004],{"className":213005,"style":31846},[31845],[34,213007,213009,213012],{"className":213008},[31676],[34,213010],{"className":213011,"style":32617},[31680],[34,213013,15932],{"className":213014},[31685]," 指的是包含最近 ",[2627,213017,213018],{},"4次内存访问"," 的窗口。\n在时间164之前，最近的4次访问时间分别是163(页3)、162(页0)、161(页2)、160(页1)。所以初始窗口内容是 ",[31,213021,213022],{},"[1, 2, 0, 3]","。工作集（去重后的页面集合）为 ",[31,213025,213026],{},"{1, 2, 0, 3}","。\n接下来我们逐一跟踪未来的访问，窗口不断向前滑动，如果被访问的页面不在当前的工作集中，就会触发一次缺页异常：",[2230,213029,213030,213049,213062,213073,213085,213101,213116,213127,213142,213157,213172],{},[108,213031,213032,213033,213035,213036,213039,213040,347,213043,213045,213046,347],{},"访问 ",[31,213034,15932],{},": 窗口变为 ",[31,213037,213038],{},"[2, 0, 3, 4]","，工作集 ",[31,213041,213042],{},"{2, 0, 3, 4}",[31,213044,15932],{}," 不在原集合中，",[2627,213047,213048],{},"发生第1次缺页异常",[108,213050,213032,213051,213035,213053,213039,213056,347,213059,213061],{},[31,213052,459],{},[31,213054,213055],{},"[0, 3, 4, 0]",[31,213057,213058],{},"{0, 3, 4}",[31,213060,459],{}," 在集合中，命中。",[108,213063,213032,213064,213035,213066,213039,213069,347,213071,213061],{},[31,213065,459],{},[31,213067,213068],{},"[3, 4, 0, 0]",[31,213070,213058],{},[31,213072,459],{},[108,213074,213032,213075,213035,213077,213039,213080,347,213083,213061],{},[31,213076,459],{},[31,213078,213079],{},"[4, 0, 0, 0]",[31,213081,213082],{},"{0, 4}",[31,213084,459],{},[108,213086,213032,213087,213035,213089,213039,213092,347,213095,213097,213098,347],{},[31,213088,768],{},[31,213090,213091],{},"[0, 0, 0, 2]",[31,213093,213094],{},"{0, 2}",[31,213096,768],{}," 不在集合中，",[2627,213099,213100],{},"发生第2次缺页异常",[108,213102,213032,213103,213035,213105,213039,213108,347,213111,213097,213113,347],{},[31,213104,15932],{},[31,213106,213107],{},"[0, 0, 2, 4]",[31,213109,213110],{},"{0, 2, 4}",[31,213112,15932],{},[2627,213114,213115],{},"发生第3次缺页异常",[108,213117,213032,213118,213035,213120,213039,213123,347,213125,213061],{},[31,213119,768],{},[31,213121,213122],{},"[0, 2, 4, 2]",[31,213124,213110],{},[31,213126,768],{},[108,213128,213032,213129,213035,213131,213039,213134,347,213137,213097,213139,347],{},[31,213130,640],{},[31,213132,213133],{},"[2, 4, 2, 1]",[31,213135,213136],{},"{1, 2, 4}",[31,213138,640],{},[2627,213140,213141],{},"发生第4次缺页异常",[108,213143,213032,213144,213035,213146,213039,213149,347,213152,213097,213154,347],{},[31,213145,459],{},[31,213147,213148],{},"[4, 2, 1, 0]",[31,213150,213151],{},"{0, 1, 2, 4}",[31,213153,459],{},[2627,213155,213156],{},"发生第5次缺页异常",[108,213158,213032,213159,213035,213161,213039,213164,347,213167,213097,213169,347],{},[31,213160,786],{},[31,213162,213163],{},"[2, 1, 0, 3]",[31,213165,213166],{},"{0, 1, 2, 3}",[31,213168,786],{},[2627,213170,213171],{},"发生第6次缺页异常",[108,213173,213032,213174,213035,213176,213039,213179,347,213181,213183,213184,213186,213187,213190],{},[31,213175,768],{},[31,213177,213178],{},"[1, 0, 3, 2]",[31,213180,213166],{},[31,213182,768],{}," 在集合中，命中。\n",[2627,213185,114043],{},"：总共会发生 ",[2627,213188,213189],{},"6次"," 缺页异常。分别发生在访问首个页面4、页面2、第二个页面4、页面1、页面0和页面3时。",[32188,213192],{},[253,213194,213195],{"id":213195},"第七题解析",[15,213197,213198,213200],{},[2627,213199,200435],{},"\n7. Consider a paged logical address space (composed of 32 pages of 2 Kbytes each) mapped into a 1-Mbyte physical memory space.\na. What is the format of the processor's logical address?\nb. What is the length and width of the page table (disregarding the \"access rights\"bits)?\nc. What is the effect on the page table if the physical memory space is reduced by half?",[15,213202,213203,213205],{},[2627,213204,200445],{},"\n7. 考虑一个分页逻辑地址空间（由32个页面组成，每个页面2KB），将其映射到1MB的物理内存空间。\na. 处理器逻辑地址的格式是什么？\nb. 页表的长度和宽度是多少（不考虑“访问权限”位）？\nc. 如果物理内存空间减半，对页表有什么影响？",[15,213207,213208],{},[2627,213209,208759],{},[105,213211,213212,213462,213467],{},[108,213213,213214,213217,213218,47649,213317,213320,213321,213382,213383,213385,213386,202790,213459],{},[2627,213215,213216],{},"a. 逻辑地址格式","\n逻辑地址空间共有32个页面。 虚拟页号(VPN)需要 ",[34,213219,213221,213251],{"className":213220},[31642],[34,213222,213224],{"className":213223},[31646],[31648,213225,213226],{"xmlns":31650},[31652,213227,213228,213248],{},[31655,213229,213230,213240,213242,213244,213246],{},[31743,213231,213232,213238],{},[31655,213233,213234,213236],{},[31658,213235,885],{},[31662,213237,32682],{},[31749,213239,768],{},[31662,213241,456],{"stretchy":702},[31749,213243,173486],{},[31662,213245,2557],{"stretchy":702},[31662,213247,202],{},[31665,213249,213250],{"encoding":31667},"\\log_2(32) =",[34,213252,213254],{"className":213253,"ariaHidden":758},[31672],[34,213255,213257,213260,213302,213305,213308,213311,213314],{"className":213256},[31676],[34,213258],{"className":213259,"style":31774},[31680],[34,213261,213263,213268],{"className":213262},[32767],[34,213264,38616,213266],{"className":213265},[32767],[34,213267,31942],{"style":38619},[34,213269,213271],{"className":213270},[31792],[34,213272,213274,213294],{"className":213273},[31796,31797],[34,213275,213277,213291],{"className":213276},[31801],[34,213278,213280],{"className":213279,"style":38632},[31805],[34,213281,213282,213285],{"style":38635},[34,213283],{"className":213284,"style":31814},[31813],[34,213286,213288],{"className":213287},[31818,31819,31820,31821],[34,213289,768],{"className":213290},[31685,31821],[34,213292,31829],{"className":213293},[31828],[34,213295,213297],{"className":213296},[31801],[34,213298,213300],{"className":213299,"style":38654},[31805],[34,213301],{},[34,213303,456],{"className":213304},[31782],[34,213306,173486],{"className":213307},[31685],[34,213309,2557],{"className":213310},[31690],[34,213312],{"className":213313,"style":31846},[31845],[34,213315,202],{"className":213316},[31850],[2627,213318,213319],{},"5位","。\n每个页面大小为2KB（",[34,213322,213324,213341],{"className":213323},[31642],[34,213325,213327],{"className":213326},[31646],[31648,213328,213329],{"xmlns":31650},[31652,213330,213331,213339],{},[31655,213332,213333],{},[37624,213334,213335,213337],{},[31749,213336,768],{},[31749,213338,39620],{},[31665,213340,211455],{"encoding":31667},[34,213342,213344],{"className":213343,"ariaHidden":758},[31672],[34,213345,213347,213350],{"className":213346},[31676],[34,213348],{"className":213349,"style":37774},[31680],[34,213351,213353,213356],{"className":213352},[31685],[34,213354,768],{"className":213355},[31685],[34,213357,213359],{"className":213358},[31792],[34,213360,213362],{"className":213361},[31796],[34,213363,213365],{"className":213364},[31801],[34,213366,213368],{"className":213367,"style":37774},[31805],[34,213369,213370,213373],{"style":37777},[34,213371],{"className":213372,"style":31814},[31813],[34,213374,213376],{"className":213375},[31818,31819,31820,31821],[34,213377,213379],{"className":213378},[31685,31821],[34,213380,39620],{"className":213381},[31685,31821],"字节），所以页内偏移量需要 ",[2627,213384,211500],{},"。\n总逻辑地址长度 = ",[34,213387,213389,213411],{"className":213388},[31642],[34,213390,213392],{"className":213391},[31646],[31648,213393,213394],{"xmlns":31650},[31652,213395,213396,213408],{},[31655,213397,213398,213400,213402,213404,213406],{},[31749,213399,18136],{},[31662,213401,223],{},[31749,213403,39620],{},[31662,213405,202],{},[31749,213407,177869],{},[31665,213409,213410],{"encoding":31667},"5 + 11 = 16",[34,213412,213414,213432,213450],{"className":213413,"ariaHidden":758},[31672],[34,213415,213417,213420,213423,213426,213429],{"className":213416},[31676],[34,213418],{"className":213419,"style":35942},[31680],[34,213421,18136],{"className":213422},[31685],[34,213424],{"className":213425,"style":32604},[31845],[34,213427,223],{"className":213428},[32024],[34,213430],{"className":213431,"style":32604},[31845],[34,213433,213435,213438,213441,213444,213447],{"className":213434},[31676],[34,213436],{"className":213437,"style":32617},[31680],[34,213439,39620],{"className":213440},[31685],[34,213442],{"className":213443,"style":31846},[31845],[34,213445,202],{"className":213446},[31850],[34,213448],{"className":213449,"style":31846},[31845],[34,213451,213453,213456],{"className":213452},[31676],[34,213454],{"className":213455,"style":32617},[31680],[34,213457,177869],{"className":213458},[31685],[2627,213460,213461],{},"格式为：5位页号 + 11位页内偏移量。",[108,213463,213464],{},[2627,213465,213466],{},"b. 页表的长度和宽度",[108,213468],{},[15,213470,213471,213474,213475,347],{},[2627,213472,213473],{},"长度","：页表的长度指的是页表中有多少个表项。因为逻辑地址空间有32个页面，所以页表长度为 ",[2627,213476,213477],{},"32个表项",[105,213479,213480],{},[108,213481],{},[15,213483,213484,213487,213488,213549,213550,213611,213612,213791,213792,47649,213891,213894,213895,347],{},[2627,213485,213486],{},"宽度","：页表的宽度指的是每个表项需要多少个二进制位来存储物理页框号。物理内存空间为 1MB = ",[34,213489,213491,213508],{"className":213490},[31642],[34,213492,213494],{"className":213493},[31646],[31648,213495,213496],{"xmlns":31650},[31652,213497,213498,213506],{},[31655,213499,213500],{},[37624,213501,213502,213504],{},[31749,213503,768],{},[31749,213505,3249],{},[31665,213507,206038],{"encoding":31667},[34,213509,213511],{"className":213510,"ariaHidden":758},[31672],[34,213512,213514,213517],{"className":213513},[31676],[34,213515],{"className":213516,"style":37774},[31680],[34,213518,213520,213523],{"className":213519},[31685],[34,213521,768],{"className":213522},[31685],[34,213524,213526],{"className":213525},[31792],[34,213527,213529],{"className":213528},[31796],[34,213530,213532],{"className":213531},[31801],[34,213533,213535],{"className":213534,"style":37774},[31805],[34,213536,213537,213540],{"style":37777},[34,213538],{"className":213539,"style":31814},[31813],[34,213541,213543],{"className":213542},[31818,31819,31820,31821],[34,213544,213546],{"className":213545},[31685,31821],[34,213547,3249],{"className":213548},[31685,31821]," 字节。页面大小为 ",[34,213551,213553,213570],{"className":213552},[31642],[34,213554,213556],{"className":213555},[31646],[31648,213557,213558],{"xmlns":31650},[31652,213559,213560,213568],{},[31655,213561,213562],{},[37624,213563,213564,213566],{},[31749,213565,768],{},[31749,213567,39620],{},[31665,213569,211455],{"encoding":31667},[34,213571,213573],{"className":213572,"ariaHidden":758},[31672],[34,213574,213576,213579],{"className":213575},[31676],[34,213577],{"className":213578,"style":37774},[31680],[34,213580,213582,213585],{"className":213581},[31685],[34,213583,768],{"className":213584},[31685],[34,213586,213588],{"className":213587},[31792],[34,213589,213591],{"className":213590},[31796],[34,213592,213594],{"className":213593},[31801],[34,213595,213597],{"className":213596,"style":37774},[31805],[34,213598,213599,213602],{"style":37777},[34,213600],{"className":213601,"style":31814},[31813],[34,213603,213605],{"className":213604},[31818,31819,31820,31821],[34,213606,213608],{"className":213607},[31685,31821],[34,213609,39620],{"className":213610},[31685,31821]," 字节。物理内存中的总页框数 = ",[34,213613,213615,213653],{"className":213614},[31642],[34,213616,213618],{"className":213617},[31646],[31648,213619,213620],{"xmlns":31650},[31652,213621,213622,213650],{},[31655,213623,213624,213630,213632,213638,213640,213646,213648],{},[37624,213625,213626,213628],{},[31749,213627,768],{},[31749,213629,3249],{},[31658,213631,38080],{"mathvariant":38079},[37624,213633,213634,213636],{},[31749,213635,768],{},[31749,213637,39620],{},[31662,213639,202],{},[37624,213641,213642,213644],{},[31749,213643,768],{},[31749,213645,36732],{},[31662,213647,202],{},[31749,213649,167308],{},[31665,213651,213652],{"encoding":31667},"2^{20} \u002F 2^{11} = 2^9 = 512",[34,213654,213656,213738,213782],{"className":213655,"ariaHidden":758},[31672],[34,213657,213659,213662,213694,213697,213729,213732,213735],{"className":213658},[31676],[34,213660],{"className":213661,"style":37755},[31680],[34,213663,213665,213668],{"className":213664},[31685],[34,213666,768],{"className":213667},[31685],[34,213669,213671],{"className":213670},[31792],[34,213672,213674],{"className":213673},[31796],[34,213675,213677],{"className":213676},[31801],[34,213678,213680],{"className":213679,"style":37774},[31805],[34,213681,213682,213685],{"style":37777},[34,213683],{"className":213684,"style":31814},[31813],[34,213686,213688],{"className":213687},[31818,31819,31820,31821],[34,213689,213691],{"className":213690},[31685,31821],[34,213692,3249],{"className":213693},[31685,31821],[34,213695,38080],{"className":213696},[31685],[34,213698,213700,213703],{"className":213699},[31685],[34,213701,768],{"className":213702},[31685],[34,213704,213706],{"className":213705},[31792],[34,213707,213709],{"className":213708},[31796],[34,213710,213712],{"className":213711},[31801],[34,213713,213715],{"className":213714,"style":37774},[31805],[34,213716,213717,213720],{"style":37777},[34,213718],{"className":213719,"style":31814},[31813],[34,213721,213723],{"className":213722},[31818,31819,31820,31821],[34,213724,213726],{"className":213725},[31685,31821],[34,213727,39620],{"className":213728},[31685,31821],[34,213730],{"className":213731,"style":31846},[31845],[34,213733,202],{"className":213734},[31850],[34,213736],{"className":213737,"style":31846},[31845],[34,213739,213741,213744,213773,213776,213779],{"className":213740},[31676],[34,213742],{"className":213743,"style":37774},[31680],[34,213745,213747,213750],{"className":213746},[31685],[34,213748,768],{"className":213749},[31685],[34,213751,213753],{"className":213752},[31792],[34,213754,213756],{"className":213755},[31796],[34,213757,213759],{"className":213758},[31801],[34,213760,213762],{"className":213761,"style":37774},[31805],[34,213763,213764,213767],{"style":37777},[34,213765],{"className":213766,"style":31814},[31813],[34,213768,213770],{"className":213769},[31818,31819,31820,31821],[34,213771,36732],{"className":213772},[31685,31821],[34,213774],{"className":213775,"style":31846},[31845],[34,213777,202],{"className":213778},[31850],[34,213780],{"className":213781,"style":31846},[31845],[34,213783,213785,213788],{"className":213784},[31676],[34,213786],{"className":213787,"style":32617},[31680],[34,213789,167308],{"className":213790},[31685]," 个页框。因此，物理页框号需要 ",[34,213793,213795,213825],{"className":213794},[31642],[34,213796,213798],{"className":213797},[31646],[31648,213799,213800],{"xmlns":31650},[31652,213801,213802,213822],{},[31655,213803,213804,213814,213816,213818,213820],{},[31743,213805,213806,213812],{},[31655,213807,213808,213810],{},[31658,213809,885],{},[31662,213811,32682],{},[31749,213813,768],{},[31662,213815,456],{"stretchy":702},[31749,213817,167308],{},[31662,213819,2557],{"stretchy":702},[31662,213821,202],{},[31665,213823,213824],{"encoding":31667},"\\log_2(512) =",[34,213826,213828],{"className":213827,"ariaHidden":758},[31672],[34,213829,213831,213834,213876,213879,213882,213885,213888],{"className":213830},[31676],[34,213832],{"className":213833,"style":31774},[31680],[34,213835,213837,213842],{"className":213836},[32767],[34,213838,38616,213840],{"className":213839},[32767],[34,213841,31942],{"style":38619},[34,213843,213845],{"className":213844},[31792],[34,213846,213848,213868],{"className":213847},[31796,31797],[34,213849,213851,213865],{"className":213850},[31801],[34,213852,213854],{"className":213853,"style":38632},[31805],[34,213855,213856,213859],{"style":38635},[34,213857],{"className":213858,"style":31814},[31813],[34,213860,213862],{"className":213861},[31818,31819,31820,31821],[34,213863,768],{"className":213864},[31685,31821],[34,213866,31829],{"className":213867},[31828],[34,213869,213871],{"className":213870},[31801],[34,213872,213874],{"className":213873,"style":38654},[31805],[34,213875],{},[34,213877,456],{"className":213878},[31782],[34,213880,167308],{"className":213881},[31685],[34,213883,2557],{"className":213884},[31690],[34,213886],{"className":213887,"style":31846},[31845],[34,213889,202],{"className":213890},[31850],[2627,213892,213893],{},"9位","。所以宽度为 ",[2627,213896,213893],{},[105,213898,213899],{},[108,213900,213901,213904,213905,213966,213967,214146,214147,47649,214245,39984,214247,214250,214251,214254,214255,214258],{},[2627,213902,213903],{},"c. 物理内存减半的影响","\n如果物理内存减半，变为 512KB（",[34,213906,213908,213925],{"className":213907},[31642],[34,213909,213911],{"className":213910},[31646],[31648,213912,213913],{"xmlns":31650},[31652,213914,213915,213923],{},[31655,213916,213917],{},[37624,213918,213919,213921],{},[31749,213920,768],{},[31749,213922,162590],{},[31665,213924,205585],{"encoding":31667},[34,213926,213928],{"className":213927,"ariaHidden":758},[31672],[34,213929,213931,213934],{"className":213930},[31676],[34,213932],{"className":213933,"style":37774},[31680],[34,213935,213937,213940],{"className":213936},[31685],[34,213938,768],{"className":213939},[31685],[34,213941,213943],{"className":213942},[31792],[34,213944,213946],{"className":213945},[31796],[34,213947,213949],{"className":213948},[31801],[34,213950,213952],{"className":213951,"style":37774},[31805],[34,213953,213954,213957],{"style":37777},[34,213955],{"className":213956,"style":31814},[31813],[34,213958,213960],{"className":213959},[31818,31819,31820,31821],[34,213961,213963],{"className":213962},[31685,31821],[34,213964,162590],{"className":213965},[31685,31821]," 字节）。\n新的总页框数 = ",[34,213968,213970,214008],{"className":213969},[31642],[34,213971,213973],{"className":213972},[31646],[31648,213974,213975],{"xmlns":31650},[31652,213976,213977,214005],{},[31655,213978,213979,213985,213987,213993,213995,214001,214003],{},[37624,213980,213981,213983],{},[31749,213982,768],{},[31749,213984,162590],{},[31658,213986,38080],{"mathvariant":38079},[37624,213988,213989,213991],{},[31749,213990,768],{},[31749,213992,39620],{},[31662,213994,202],{},[37624,213996,213997,213999],{},[31749,213998,768],{},[31749,214000,60525],{},[31662,214002,202],{},[31749,214004,95418],{},[31665,214006,214007],{"encoding":31667},"2^{19} \u002F 2^{11} = 2^8 = 256",[34,214009,214011,214093,214137],{"className":214010,"ariaHidden":758},[31672],[34,214012,214014,214017,214049,214052,214084,214087,214090],{"className":214013},[31676],[34,214015],{"className":214016,"style":37755},[31680],[34,214018,214020,214023],{"className":214019},[31685],[34,214021,768],{"className":214022},[31685],[34,214024,214026],{"className":214025},[31792],[34,214027,214029],{"className":214028},[31796],[34,214030,214032],{"className":214031},[31801],[34,214033,214035],{"className":214034,"style":37774},[31805],[34,214036,214037,214040],{"style":37777},[34,214038],{"className":214039,"style":31814},[31813],[34,214041,214043],{"className":214042},[31818,31819,31820,31821],[34,214044,214046],{"className":214045},[31685,31821],[34,214047,162590],{"className":214048},[31685,31821],[34,214050,38080],{"className":214051},[31685],[34,214053,214055,214058],{"className":214054},[31685],[34,214056,768],{"className":214057},[31685],[34,214059,214061],{"className":214060},[31792],[34,214062,214064],{"className":214063},[31796],[34,214065,214067],{"className":214066},[31801],[34,214068,214070],{"className":214069,"style":37774},[31805],[34,214071,214072,214075],{"style":37777},[34,214073],{"className":214074,"style":31814},[31813],[34,214076,214078],{"className":214077},[31818,31819,31820,31821],[34,214079,214081],{"className":214080},[31685,31821],[34,214082,39620],{"className":214083},[31685,31821],[34,214085],{"className":214086,"style":31846},[31845],[34,214088,202],{"className":214089},[31850],[34,214091],{"className":214092,"style":31846},[31845],[34,214094,214096,214099,214128,214131,214134],{"className":214095},[31676],[34,214097],{"className":214098,"style":37774},[31680],[34,214100,214102,214105],{"className":214101},[31685],[34,214103,768],{"className":214104},[31685],[34,214106,214108],{"className":214107},[31792],[34,214109,214111],{"className":214110},[31796],[34,214112,214114],{"className":214113},[31801],[34,214115,214117],{"className":214116,"style":37774},[31805],[34,214118,214119,214122],{"style":37777},[34,214120],{"className":214121,"style":31814},[31813],[34,214123,214125],{"className":214124},[31818,31819,31820,31821],[34,214126,60525],{"className":214127},[31685,31821],[34,214129],{"className":214130,"style":31846},[31845],[34,214132,202],{"className":214133},[31850],[34,214135],{"className":214136,"style":31846},[31845],[34,214138,214140,214143],{"className":214139},[31676],[34,214141],{"className":214142,"style":32617},[31680],[34,214144,95418],{"className":214145},[31685]," 个页框。\n此时表示物理页框号所需的位数变为 ",[34,214148,214150,214179],{"className":214149},[31642],[34,214151,214153],{"className":214152},[31646],[31648,214154,214155],{"xmlns":31650},[31652,214156,214157,214177],{},[31655,214158,214159,214169,214171,214173,214175],{},[31743,214160,214161,214167],{},[31655,214162,214163,214165],{},[31658,214164,885],{},[31662,214166,32682],{},[31749,214168,768],{},[31662,214170,456],{"stretchy":702},[31749,214172,95418],{},[31662,214174,2557],{"stretchy":702},[31662,214176,202],{},[31665,214178,210075],{"encoding":31667},[34,214180,214182],{"className":214181,"ariaHidden":758},[31672],[34,214183,214185,214188,214230,214233,214236,214239,214242],{"className":214184},[31676],[34,214186],{"className":214187,"style":31774},[31680],[34,214189,214191,214196],{"className":214190},[32767],[34,214192,38616,214194],{"className":214193},[32767],[34,214195,31942],{"style":38619},[34,214197,214199],{"className":214198},[31792],[34,214200,214202,214222],{"className":214201},[31796,31797],[34,214203,214205,214219],{"className":214204},[31801],[34,214206,214208],{"className":214207,"style":38632},[31805],[34,214209,214210,214213],{"style":38635},[34,214211],{"className":214212,"style":31814},[31813],[34,214214,214216],{"className":214215},[31818,31819,31820,31821],[34,214217,768],{"className":214218},[31685,31821],[34,214220,31829],{"className":214221},[31828],[34,214223,214225],{"className":214224},[31801],[34,214226,214228],{"className":214227,"style":38654},[31805],[34,214229],{},[34,214231,456],{"className":214232},[31782],[34,214234,95418],{"className":214235},[31685],[34,214237,2557],{"className":214238},[31690],[34,214240],{"className":214241,"style":31846},[31845],[34,214243,202],{"className":214244},[31850],[2627,214246,210144],{},[2627,214248,214249],{},"影响","：页表的长度（表项数量）完全取决于逻辑地址空间，因此",[2627,214252,214253],{},"长度保持32个表项不变","；但是由于物理页框数量减少，表项的",[2627,214256,214257],{},"宽度将减少1位","（从9位减少到8位）。",{"title":29,"searchDepth":43,"depth":43,"links":214260},[214261,214262,214263,214264,214265,214266,214267],{"id":208736,"depth":49,"text":208736},{"id":209656,"depth":49,"text":209656},{"id":210974,"depth":49,"text":210974},{"id":211660,"depth":49,"text":211660},{"id":212190,"depth":49,"text":212190},{"id":212865,"depth":49,"text":212865},{"id":213195,"depth":49,"text":213195},{"order":1971},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap8-hw",{"title":208640,"description":29},"source\u002Fcs\u002Fos\u002FHW\u002Fchap8-hw","qTpaR8cOugiuZHGc-5o_sunckBEFS9fQ-kTFKvIJVmc",{"id":214274,"title":214275,"body":214276,"description":214286,"extension":85,"meta":233072,"navigation":52,"path":233073,"seo":233074,"stem":233075,"__hash__":233076},"notes\u002Fsource\u002Fcs\u002Fos\u002FHW\u002Fchap9-hw.md","Chap. 9 HW",{"type":8,"value":214277,"toc":233031},[214278,214282,214287,214293,214298,214347,214357,214363,214367,214697,214701,214708,214713,215623,215628,215665,215671,216921,217018,217334,217338,217349,217351,217357,217362,217619,217907,218223,218287,218882,218885,219114,219118,219152,219404,219438,219801,219804,219965,219967,219973,220048,220197,220232,220557,220560,220906,220910,220915,223074,223077,223277,223280,223283,223293,223295,223299,223917,223921,223958,223962,224293,224701,225285,225366,225950,225952,225956,225961,225964,225967,225972,226498,226503,227026,227031,227553,227558,227978,227980,227984,227992,227995,228000,228302,228307,228604,228609,228908,228920,229103,229105,229109,229114,229197,229205,229208,229211,229216,229489,229494,230198,230206,230208,230212,230215,230344,230357,230360,230475,230478,230623,231073,231075,231079,231084,231150,231153,231240,231243,231424,231426,231430,231554,231557,231806,231927,232954],[11,214279,214281],{"id":214280},"os-chap9-hw","OS Chap.9 HW",[15,214283,214284],{},[2627,214285,214286],{},"本页面内容可能存在错误",[19,214288,214290],{"id":214289},"题目-1",[2627,214291,214292],{},"题目 1",[15,214294,214295],{},[2627,214296,214297],{},"Consider the following set of processes:",[13837,214299,214300,214313],{},[13840,214301,214302],{},[13843,214303,214304,214307,214310],{},[13846,214305,214306],{"align":82355},"Process Name",[13846,214308,214309],{"align":82355},"Arrival Time",[13846,214311,214312],{"align":82355},"Processing Time",[13853,214314,214315,214323,214331,214339],{},[13843,214316,214317,214319,214321],{},[13858,214318,54151],{"align":82355},[13858,214320,459],{"align":82355},[13858,214322,640],{"align":82355},[13843,214324,214325,214327,214329],{},[13858,214326,54224],{"align":82355},[13858,214328,640],{"align":82355},[13858,214330,36732],{"align":82355},[13843,214332,214333,214335,214337],{},[13858,214334,47384],{"align":82355},[13858,214336,768],{"align":82355},[13858,214338,640],{"align":82355},[13843,214340,214341,214343,214345],{},[13858,214342,91994],{"align":82355},[13858,214344,786],{"align":82355},[13858,214346,36732],{"align":82355},[15,214348,214349,214350,195562,214353,214356],{},"Perform the same analysis as depicted in ",[2627,214351,214352],{},"Table 9.5",[2627,214354,214355],{},"Figure 9.5"," for this set.",[15,214358,214359,214362],{},[2627,214360,214361],{},"题目内容：","\n对四个进程（A: 到达0\u002F服务1；B: 到达1\u002F服务9；C: 到达2\u002F服务1；D: 到达3\u002F服务9）进行调度算法分析。",[253,214364,214366],{"id":214365},"概念检查-concept-check","概念检查 (Concept Check)",[105,214368,214369,214570],{},[108,214370,214371,214446,214447,214569],{},[2627,214372,214373,214374,214445],{},"周转时间 (Turnaround Time, ",[34,214375,214377,214395],{"className":214376},[31642],[34,214378,214380],{"className":214379},[31646],[31648,214381,214382],{"xmlns":31650},[31652,214383,214384,214392],{},[31655,214385,214386],{},[31743,214387,214388,214390],{},[31658,214389,37400],{},[31658,214391,34395],{},[31665,214393,214394],{"encoding":31667},"T_r",[34,214396,214398],{"className":214397,"ariaHidden":758},[31672],[34,214399,214401,214404],{"className":214400},[31676],[34,214402],{"className":214403,"style":70915},[31680],[34,214405,214407,214410],{"className":214406},[31685],[34,214408,37400],{"className":214409,"style":37422},[31685,31686],[34,214411,214413],{"className":214412},[31792],[34,214414,214416,214437],{"className":214415},[31796,31797],[34,214417,214419,214434],{"className":214418},[31801],[34,214420,214422],{"className":214421,"style":32362},[31805],[34,214423,214425,214428],{"style":214424},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[34,214426],{"className":214427,"style":31814},[31813],[34,214429,214431],{"className":214430},[31818,31819,31820,31821],[34,214432,34395],{"className":214433,"style":34452},[31685,31686,31821],[34,214435,31829],{"className":214436},[31828],[34,214438,214440],{"className":214439},[31801],[34,214441,214443],{"className":214442,"style":31836},[31805],[34,214444],{},"):"," 进程从提交到完成的总时间 (",[34,214448,214450,214478],{"className":214449},[31642],[34,214451,214453],{"className":214452},[31646],[31648,214454,214455],{"xmlns":31650},[31652,214456,214457,214475],{},[31655,214458,214459,214465,214467,214470,214472],{},[31743,214460,214461,214463],{},[31658,214462,37400],{},[31658,214464,34395],{},[31662,214466,202],{},[39361,214468,214469],{},"完成时间",[31662,214471,32226],{},[39361,214473,214474],{},"到达时间",[31665,214476,214477],{"encoding":31667},"T_r = \\text{完成时间} - \\text{到达时间}",[34,214479,214481,214536,214557],{"className":214480,"ariaHidden":758},[31672],[34,214482,214484,214487,214527,214530,214533],{"className":214483},[31676],[34,214485],{"className":214486,"style":70915},[31680],[34,214488,214490,214493],{"className":214489},[31685],[34,214491,37400],{"className":214492,"style":37422},[31685,31686],[34,214494,214496],{"className":214495},[31792],[34,214497,214499,214519],{"className":214498},[31796,31797],[34,214500,214502,214516],{"className":214501},[31801],[34,214503,214505],{"className":214504,"style":32362},[31805],[34,214506,214507,214510],{"style":214424},[34,214508],{"className":214509,"style":31814},[31813],[34,214511,214513],{"className":214512},[31818,31819,31820,31821],[34,214514,34395],{"className":214515,"style":34452},[31685,31686,31821],[34,214517,31829],{"className":214518},[31828],[34,214520,214522],{"className":214521},[31801],[34,214523,214525],{"className":214524,"style":31836},[31805],[34,214526],{},[34,214528],{"className":214529,"style":31846},[31845],[34,214531,202],{"className":214532},[31850],[34,214534],{"className":214535,"style":31846},[31845],[34,214537,214539,214542,214548,214551,214554],{"className":214538},[31676],[34,214540],{"className":214541,"style":34944},[31680],[34,214543,214545],{"className":214544},[31685,2638],[34,214546,214469],{"className":214547},[31685,105462],[34,214549],{"className":214550,"style":32604},[31845],[34,214552,32226],{"className":214553},[32024],[34,214555],{"className":214556,"style":32604},[31845],[34,214558,214560,214563],{"className":214559},[31676],[34,214561],{"className":214562,"style":33218},[31680],[34,214564,214566],{"className":214565},[31685,2638],[34,214567,214474],{"className":214568},[31685,105462],")。",[108,214571,214572,47649,214575,214696],{},[2627,214573,214574],{},"归一化周转时间 (Normalized Turnaround Time):",[34,214576,214578,214604],{"className":214577},[31642],[34,214579,214581],{"className":214580},[31646],[31648,214582,214583],{"xmlns":31650},[31652,214584,214585,214601],{},[31655,214586,214587,214593,214595],{},[31743,214588,214589,214591],{},[31658,214590,37400],{},[31658,214592,34395],{},[31658,214594,38080],{"mathvariant":38079},[31743,214596,214597,214599],{},[31658,214598,37400],{},[31658,214600,31747],{},[31665,214602,214603],{"encoding":31667},"T_r \u002F T_s",[34,214605,214607],{"className":214606,"ariaHidden":758},[31672],[34,214608,214610,214613,214653,214656],{"className":214609},[31676],[34,214611],{"className":214612,"style":31774},[31680],[34,214614,214616,214619],{"className":214615},[31685],[34,214617,37400],{"className":214618,"style":37422},[31685,31686],[34,214620,214622],{"className":214621},[31792],[34,214623,214625,214645],{"className":214624},[31796,31797],[34,214626,214628,214642],{"className":214627},[31801],[34,214629,214631],{"className":214630,"style":32362},[31805],[34,214632,214633,214636],{"style":214424},[34,214634],{"className":214635,"style":31814},[31813],[34,214637,214639],{"className":214638},[31818,31819,31820,31821],[34,214640,34395],{"className":214641,"style":34452},[31685,31686,31821],[34,214643,31829],{"className":214644},[31828],[34,214646,214648],{"className":214647},[31801],[34,214649,214651],{"className":214650,"style":31836},[31805],[34,214652],{},[34,214654,38080],{"className":214655},[31685],[34,214657,214659,214662],{"className":214658},[31685],[34,214660,37400],{"className":214661,"style":37422},[31685,31686],[34,214663,214665],{"className":214664},[31792],[34,214666,214668,214688],{"className":214667},[31796,31797],[34,214669,214671,214685],{"className":214670},[31801],[34,214672,214674],{"className":214673,"style":32362},[31805],[34,214675,214676,214679],{"style":214424},[34,214677],{"className":214678,"style":31814},[31813],[34,214680,214682],{"className":214681},[31818,31819,31820,31821],[34,214683,31747],{"className":214684},[31685,31686,31821],[34,214686,31829],{"className":214687},[31828],[34,214689,214691],{"className":214690},[31801],[34,214692,214694],{"className":214693,"style":31836},[31805],[34,214695],{},"，即周转时间与实际服务时间的比值。该值越接近1，说明进程遭受的延迟越小。",[253,214698,214700],{"id":214699},"原理解析与步骤推演-the-why","原理解析与步骤推演 (The 'Why')",[15,214702,214703,214704,214707],{},"这道题的数据设计非常巧妙，它完美暴露了",[2627,214705,214706],{},"非抢占式算法的致命缺陷","。我们来推演四种经典算法的执行过程：",[15,214709,214710],{},[2627,214711,214712],{},"(1) FCFS (先来先服务) - 非抢占式",[105,214714,214715,214721,214727,214733,214739],{},[108,214716,214717,214720],{},[2627,214718,214719],{},"t=0:"," A到达并执行，t=1完成。",[108,214722,214723,214726],{},[2627,214724,214725],{},"t=1:"," B到达并执行，t=10完成。",[108,214728,214729,214732],{},[2627,214730,214731],{},"t=10:"," C和D均已到达，C先到，C执行，t=11完成。",[108,214734,214735,214738],{},[2627,214736,214737],{},"t=11:"," D执行，t=20完成。",[108,214740,214741,214743,214744,214963,214964,215183,215184,215403,215404,214569],{},[2627,214742,58554],{}," A(",[34,214745,214747,214789],{"className":214746},[31642],[34,214748,214750],{"className":214749},[31646],[31648,214751,214752],{"xmlns":31650},[31652,214753,214754,214786],{},[31655,214755,214756,214762,214764,214766,214768,214774,214776,214782,214784],{},[31743,214757,214758,214760],{},[31658,214759,37400],{},[31658,214761,34395],{},[31662,214763,202],{},[31749,214765,640],{},[31662,214767,31960],{"separator":758},[31743,214769,214770,214772],{},[31658,214771,37400],{},[31658,214773,34395],{},[31658,214775,38080],{"mathvariant":38079},[31743,214777,214778,214780],{},[31658,214779,37400],{},[31658,214781,31747],{},[31662,214783,202],{},[31749,214785,640],{},[31665,214787,214788],{"encoding":31667},"T_r=1, T_r\u002FT_s=1",[34,214790,214792,214847,214954],{"className":214791,"ariaHidden":758},[31672],[34,214793,214795,214798,214838,214841,214844],{"className":214794},[31676],[34,214796],{"className":214797,"style":70915},[31680],[34,214799,214801,214804],{"className":214800},[31685],[34,214802,37400],{"className":214803,"style":37422},[31685,31686],[34,214805,214807],{"className":214806},[31792],[34,214808,214810,214830],{"className":214809},[31796,31797],[34,214811,214813,214827],{"className":214812},[31801],[34,214814,214816],{"className":214815,"style":32362},[31805],[34,214817,214818,214821],{"style":214424},[34,214819],{"className":214820,"style":31814},[31813],[34,214822,214824],{"className":214823},[31818,31819,31820,31821],[34,214825,34395],{"className":214826,"style":34452},[31685,31686,31821],[34,214828,31829],{"className":214829},[31828],[34,214831,214833],{"className":214832},[31801],[34,214834,214836],{"className":214835,"style":31836},[31805],[34,214837],{},[34,214839],{"className":214840,"style":31846},[31845],[34,214842,202],{"className":214843},[31850],[34,214845],{"className":214846,"style":31846},[31845],[34,214848,214850,214853,214856,214859,214862,214902,214905,214945,214948,214951],{"className":214849},[31676],[34,214851],{"className":214852,"style":31774},[31680],[34,214854,640],{"className":214855},[31685],[34,214857,31960],{"className":214858},[32119],[34,214860],{"className":214861,"style":32123},[31845],[34,214863,214865,214868],{"className":214864},[31685],[34,214866,37400],{"className":214867,"style":37422},[31685,31686],[34,214869,214871],{"className":214870},[31792],[34,214872,214874,214894],{"className":214873},[31796,31797],[34,214875,214877,214891],{"className":214876},[31801],[34,214878,214880],{"className":214879,"style":32362},[31805],[34,214881,214882,214885],{"style":214424},[34,214883],{"className":214884,"style":31814},[31813],[34,214886,214888],{"className":214887},[31818,31819,31820,31821],[34,214889,34395],{"className":214890,"style":34452},[31685,31686,31821],[34,214892,31829],{"className":214893},[31828],[34,214895,214897],{"className":214896},[31801],[34,214898,214900],{"className":214899,"style":31836},[31805],[34,214901],{},[34,214903,38080],{"className":214904},[31685],[34,214906,214908,214911],{"className":214907},[31685],[34,214909,37400],{"className":214910,"style":37422},[31685,31686],[34,214912,214914],{"className":214913},[31792],[34,214915,214917,214937],{"className":214916},[31796,31797],[34,214918,214920,214934],{"className":214919},[31801],[34,214921,214923],{"className":214922,"style":32362},[31805],[34,214924,214925,214928],{"style":214424},[34,214926],{"className":214927,"style":31814},[31813],[34,214929,214931],{"className":214930},[31818,31819,31820,31821],[34,214932,31747],{"className":214933},[31685,31686,31821],[34,214935,31829],{"className":214936},[31828],[34,214938,214940],{"className":214939},[31801],[34,214941,214943],{"className":214942,"style":31836},[31805],[34,214944],{},[34,214946],{"className":214947,"style":31846},[31845],[34,214949,202],{"className":214950},[31850],[34,214952],{"className":214953,"style":31846},[31845],[34,214955,214957,214960],{"className":214956},[31676],[34,214958],{"className":214959,"style":32617},[31680],[34,214961,640],{"className":214962},[31685],")；B(",[34,214965,214967,215009],{"className":214966},[31642],[34,214968,214970],{"className":214969},[31646],[31648,214971,214972],{"xmlns":31650},[31652,214973,214974,215006],{},[31655,214975,214976,214982,214984,214986,214988,214994,214996,215002,215004],{},[31743,214977,214978,214980],{},[31658,214979,37400],{},[31658,214981,34395],{},[31662,214983,202],{},[31749,214985,36732],{},[31662,214987,31960],{"separator":758},[31743,214989,214990,214992],{},[31658,214991,37400],{},[31658,214993,34395],{},[31658,214995,38080],{"mathvariant":38079},[31743,214997,214998,215000],{},[31658,214999,37400],{},[31658,215001,31747],{},[31662,215003,202],{},[31749,215005,640],{},[31665,215007,215008],{"encoding":31667},"T_r=9, T_r\u002FT_s=1",[34,215010,215012,215067,215174],{"className":215011,"ariaHidden":758},[31672],[34,215013,215015,215018,215058,215061,215064],{"className":215014},[31676],[34,215016],{"className":215017,"style":70915},[31680],[34,215019,215021,215024],{"className":215020},[31685],[34,215022,37400],{"className":215023,"style":37422},[31685,31686],[34,215025,215027],{"className":215026},[31792],[34,215028,215030,215050],{"className":215029},[31796,31797],[34,215031,215033,215047],{"className":215032},[31801],[34,215034,215036],{"className":215035,"style":32362},[31805],[34,215037,215038,215041],{"style":214424},[34,215039],{"className":215040,"style":31814},[31813],[34,215042,215044],{"className":215043},[31818,31819,31820,31821],[34,215045,34395],{"className":215046,"style":34452},[31685,31686,31821],[34,215048,31829],{"className":215049},[31828],[34,215051,215053],{"className":215052},[31801],[34,215054,215056],{"className":215055,"style":31836},[31805],[34,215057],{},[34,215059],{"className":215060,"style":31846},[31845],[34,215062,202],{"className":215063},[31850],[34,215065],{"className":215066,"style":31846},[31845],[34,215068,215070,215073,215076,215079,215082,215122,215125,215165,215168,215171],{"className":215069},[31676],[34,215071],{"className":215072,"style":31774},[31680],[34,215074,36732],{"className":215075},[31685],[34,215077,31960],{"className":215078},[32119],[34,215080],{"className":215081,"style":32123},[31845],[34,215083,215085,215088],{"className":215084},[31685],[34,215086,37400],{"className":215087,"style":37422},[31685,31686],[34,215089,215091],{"className":215090},[31792],[34,215092,215094,215114],{"className":215093},[31796,31797],[34,215095,215097,215111],{"className":215096},[31801],[34,215098,215100],{"className":215099,"style":32362},[31805],[34,215101,215102,215105],{"style":214424},[34,215103],{"className":215104,"style":31814},[31813],[34,215106,215108],{"className":215107},[31818,31819,31820,31821],[34,215109,34395],{"className":215110,"style":34452},[31685,31686,31821],[34,215112,31829],{"className":215113},[31828],[34,215115,215117],{"className":215116},[31801],[34,215118,215120],{"className":215119,"style":31836},[31805],[34,215121],{},[34,215123,38080],{"className":215124},[31685],[34,215126,215128,215131],{"className":215127},[31685],[34,215129,37400],{"className":215130,"style":37422},[31685,31686],[34,215132,215134],{"className":215133},[31792],[34,215135,215137,215157],{"className":215136},[31796,31797],[34,215138,215140,215154],{"className":215139},[31801],[34,215141,215143],{"className":215142,"style":32362},[31805],[34,215144,215145,215148],{"style":214424},[34,215146],{"className":215147,"style":31814},[31813],[34,215149,215151],{"className":215150},[31818,31819,31820,31821],[34,215152,31747],{"className":215153},[31685,31686,31821],[34,215155,31829],{"className":215156},[31828],[34,215158,215160],{"className":215159},[31801],[34,215161,215163],{"className":215162,"style":31836},[31805],[34,215164],{},[34,215166],{"className":215167,"style":31846},[31845],[34,215169,202],{"className":215170},[31850],[34,215172],{"className":215173,"style":31846},[31845],[34,215175,215177,215180],{"className":215176},[31676],[34,215178],{"className":215179,"style":32617},[31680],[34,215181,640],{"className":215182},[31685],")；C(",[34,215185,215187,215229],{"className":215186},[31642],[34,215188,215190],{"className":215189},[31646],[31648,215191,215192],{"xmlns":31650},[31652,215193,215194,215226],{},[31655,215195,215196,215202,215204,215206,215208,215214,215216,215222,215224],{},[31743,215197,215198,215200],{},[31658,215199,37400],{},[31658,215201,34395],{},[31662,215203,202],{},[31749,215205,36732],{},[31662,215207,31960],{"separator":758},[31743,215209,215210,215212],{},[31658,215211,37400],{},[31658,215213,34395],{},[31658,215215,38080],{"mathvariant":38079},[31743,215217,215218,215220],{},[31658,215219,37400],{},[31658,215221,31747],{},[31662,215223,202],{},[31749,215225,36732],{},[31665,215227,215228],{"encoding":31667},"T_r=9, T_r\u002FT_s=9",[34,215230,215232,215287,215394],{"className":215231,"ariaHidden":758},[31672],[34,215233,215235,215238,215278,215281,215284],{"className":215234},[31676],[34,215236],{"className":215237,"style":70915},[31680],[34,215239,215241,215244],{"className":215240},[31685],[34,215242,37400],{"className":215243,"style":37422},[31685,31686],[34,215245,215247],{"className":215246},[31792],[34,215248,215250,215270],{"className":215249},[31796,31797],[34,215251,215253,215267],{"className":215252},[31801],[34,215254,215256],{"className":215255,"style":32362},[31805],[34,215257,215258,215261],{"style":214424},[34,215259],{"className":215260,"style":31814},[31813],[34,215262,215264],{"className":215263},[31818,31819,31820,31821],[34,215265,34395],{"className":215266,"style":34452},[31685,31686,31821],[34,215268,31829],{"className":215269},[31828],[34,215271,215273],{"className":215272},[31801],[34,215274,215276],{"className":215275,"style":31836},[31805],[34,215277],{},[34,215279],{"className":215280,"style":31846},[31845],[34,215282,202],{"className":215283},[31850],[34,215285],{"className":215286,"style":31846},[31845],[34,215288,215290,215293,215296,215299,215302,215342,215345,215385,215388,215391],{"className":215289},[31676],[34,215291],{"className":215292,"style":31774},[31680],[34,215294,36732],{"className":215295},[31685],[34,215297,31960],{"className":215298},[32119],[34,215300],{"className":215301,"style":32123},[31845],[34,215303,215305,215308],{"className":215304},[31685],[34,215306,37400],{"className":215307,"style":37422},[31685,31686],[34,215309,215311],{"className":215310},[31792],[34,215312,215314,215334],{"className":215313},[31796,31797],[34,215315,215317,215331],{"className":215316},[31801],[34,215318,215320],{"className":215319,"style":32362},[31805],[34,215321,215322,215325],{"style":214424},[34,215323],{"className":215324,"style":31814},[31813],[34,215326,215328],{"className":215327},[31818,31819,31820,31821],[34,215329,34395],{"className":215330,"style":34452},[31685,31686,31821],[34,215332,31829],{"className":215333},[31828],[34,215335,215337],{"className":215336},[31801],[34,215338,215340],{"className":215339,"style":31836},[31805],[34,215341],{},[34,215343,38080],{"className":215344},[31685],[34,215346,215348,215351],{"className":215347},[31685],[34,215349,37400],{"className":215350,"style":37422},[31685,31686],[34,215352,215354],{"className":215353},[31792],[34,215355,215357,215377],{"className":215356},[31796,31797],[34,215358,215360,215374],{"className":215359},[31801],[34,215361,215363],{"className":215362,"style":32362},[31805],[34,215364,215365,215368],{"style":214424},[34,215366],{"className":215367,"style":31814},[31813],[34,215369,215371],{"className":215370},[31818,31819,31820,31821],[34,215372,31747],{"className":215373},[31685,31686,31821],[34,215375,31829],{"className":215376},[31828],[34,215378,215380],{"className":215379},[31801],[34,215381,215383],{"className":215382,"style":31836},[31805],[34,215384],{},[34,215386],{"className":215387,"style":31846},[31845],[34,215389,202],{"className":215390},[31850],[34,215392],{"className":215393,"style":31846},[31845],[34,215395,215397,215400],{"className":215396},[31676],[34,215398],{"className":215399,"style":32617},[31680],[34,215401,36732],{"className":215402},[31685],")；D(",[34,215405,215407,215449],{"className":215406},[31642],[34,215408,215410],{"className":215409},[31646],[31648,215411,215412],{"xmlns":31650},[31652,215413,215414,215446],{},[31655,215415,215416,215422,215424,215426,215428,215434,215436,215442,215444],{},[31743,215417,215418,215420],{},[31658,215419,37400],{},[31658,215421,34395],{},[31662,215423,202],{},[31749,215425,164836],{},[31662,215427,31960],{"separator":758},[31743,215429,215430,215432],{},[31658,215431,37400],{},[31658,215433,34395],{},[31658,215435,38080],{"mathvariant":38079},[31743,215437,215438,215440],{},[31658,215439,37400],{},[31658,215441,31747],{},[31662,215443,202],{},[31749,215445,61939],{},[31665,215447,215448],{"encoding":31667},"T_r=17, T_r\u002FT_s=1.89",[34,215450,215452,215507,215614],{"className":215451,"ariaHidden":758},[31672],[34,215453,215455,215458,215498,215501,215504],{"className":215454},[31676],[34,215456],{"className":215457,"style":70915},[31680],[34,215459,215461,215464],{"className":215460},[31685],[34,215462,37400],{"className":215463,"style":37422},[31685,31686],[34,215465,215467],{"className":215466},[31792],[34,215468,215470,215490],{"className":215469},[31796,31797],[34,215471,215473,215487],{"className":215472},[31801],[34,215474,215476],{"className":215475,"style":32362},[31805],[34,215477,215478,215481],{"style":214424},[34,215479],{"className":215480,"style":31814},[31813],[34,215482,215484],{"className":215483},[31818,31819,31820,31821],[34,215485,34395],{"className":215486,"style":34452},[31685,31686,31821],[34,215488,31829],{"className":215489},[31828],[34,215491,215493],{"className":215492},[31801],[34,215494,215496],{"className":215495,"style":31836},[31805],[34,215497],{},[34,215499],{"className":215500,"style":31846},[31845],[34,215502,202],{"className":215503},[31850],[34,215505],{"className":215506,"style":31846},[31845],[34,215508,215510,215513,215516,215519,215522,215562,215565,215605,215608,215611],{"className":215509},[31676],[34,215511],{"className":215512,"style":31774},[31680],[34,215514,164836],{"className":215515},[31685],[34,215517,31960],{"className":215518},[32119],[34,215520],{"className":215521,"style":32123},[31845],[34,215523,215525,215528],{"className":215524},[31685],[34,215526,37400],{"className":215527,"style":37422},[31685,31686],[34,215529,215531],{"className":215530},[31792],[34,215532,215534,215554],{"className":215533},[31796,31797],[34,215535,215537,215551],{"className":215536},[31801],[34,215538,215540],{"className":215539,"style":32362},[31805],[34,215541,215542,215545],{"style":214424},[34,215543],{"className":215544,"style":31814},[31813],[34,215546,215548],{"className":215547},[31818,31819,31820,31821],[34,215549,34395],{"className":215550,"style":34452},[31685,31686,31821],[34,215552,31829],{"className":215553},[31828],[34,215555,215557],{"className":215556},[31801],[34,215558,215560],{"className":215559,"style":31836},[31805],[34,215561],{},[34,215563,38080],{"className":215564},[31685],[34,215566,215568,215571],{"className":215567},[31685],[34,215569,37400],{"className":215570,"style":37422},[31685,31686],[34,215572,215574],{"className":215573},[31792],[34,215575,215577,215597],{"className":215576},[31796,31797],[34,215578,215580,215594],{"className":215579},[31801],[34,215581,215583],{"className":215582,"style":32362},[31805],[34,215584,215585,215588],{"style":214424},[34,215586],{"className":215587,"style":31814},[31813],[34,215589,215591],{"className":215590},[31818,31819,31820,31821],[34,215592,31747],{"className":215593},[31685,31686,31821],[34,215595,31829],{"className":215596},[31828],[34,215598,215600],{"className":215599},[31801],[34,215601,215603],{"className":215602,"style":31836},[31805],[34,215604],{},[34,215606],{"className":215607,"style":31846},[31845],[34,215609,202],{"className":215610},[31850],[34,215612],{"className":215613,"style":31846},[31845],[34,215615,215617,215620],{"className":215616},[31676],[34,215618],{"className":215619,"style":32617},[31680],[34,215621,61939],{"className":215622},[31685],[15,215624,215625],{},[2627,215626,215627],{},"(2) SPN (最短进程优先) - 非抢占式",[105,215629,215630,215635,215647,215652,215656],{},[108,215631,215632,215634],{},[2627,215633,214719],{}," A(服务1)执行，t=1完成。",[108,215636,215637,47649,215639,215642,215643,215646],{},[2627,215638,214725],{},[2627,215640,215641],{},"此时系统中只有B(服务9)到达","。因为是",[2627,215644,215645],{},"非抢占式","，B一旦上CPU就不会下来。B执行至t=10完成。",[108,215648,215649,215651],{},[2627,215650,214731],{}," C(服务1)和D(服务9)均在等待。C更短，C执行，t=11完成。",[108,215653,215654,214738],{},[2627,215655,214737],{},[108,215657,215658,215660,215661,215664],{},[2627,215659,58554],{}," 调度顺序和时间与FCFS",[2627,215662,215663],{},"完全一模一样","！短进程C虽然只需要1秒，但被迫等待了整整8秒。",[15,215666,215667,215670],{},[2627,215668,215669],{},"(3) SRT (最短剩余时间) - 抢占式","\n为了解决SPN的缺陷，引入抢占机制。",[105,215672,215673,215678,215683,215693,215699,215704],{},[108,215674,215675,215677],{},[2627,215676,214719],{}," A执行，t=1完成。",[108,215679,215680,215682],{},[2627,215681,214725],{}," B到达，B开始执行。",[108,215684,215685,215688,215689,215692],{},[2627,215686,215687],{},"t=2:"," C到达(需1s)。此时B还剩8s。C的剩余时间更短，",[2627,215690,215691],{},"C抢占CPU","！C执行至t=3完成。",[108,215694,215695,215698],{},[2627,215696,215697],{},"t=3:"," D到达(需9s)。此时B剩8s，D需9s。B更短，B恢复执行，至t=11完成。",[108,215700,215701,215703],{},[2627,215702,214737],{}," D执行至t=20完成。",[108,215705,215706,215708],{},[2627,215707,58554],{},[105,215709,215710,215992,216302,216612],{},[108,215711,215712,215713,616,215849],{},"A: ",[34,215714,215716,215746],{"className":215715},[31642],[34,215717,215719],{"className":215718},[31646],[31648,215720,215721],{"xmlns":31650},[31652,215722,215723,215743],{},[31655,215724,215725,215731,215733,215735,215737,215739,215741],{},[31743,215726,215727,215729],{},[31658,215728,37400],{},[31658,215730,34395],{},[31662,215732,202],{},[31749,215734,640],{},[31662,215736,32226],{},[31749,215738,459],{},[31662,215740,202],{},[31749,215742,640],{},[31665,215744,215745],{"encoding":31667},"T_r = 1-0=1",[34,215747,215749,215804,215822,215840],{"className":215748,"ariaHidden":758},[31672],[34,215750,215752,215755,215795,215798,215801],{"className":215751},[31676],[34,215753],{"className":215754,"style":70915},[31680],[34,215756,215758,215761],{"className":215757},[31685],[34,215759,37400],{"className":215760,"style":37422},[31685,31686],[34,215762,215764],{"className":215763},[31792],[34,215765,215767,215787],{"className":215766},[31796,31797],[34,215768,215770,215784],{"className":215769},[31801],[34,215771,215773],{"className":215772,"style":32362},[31805],[34,215774,215775,215778],{"style":214424},[34,215776],{"className":215777,"style":31814},[31813],[34,215779,215781],{"className":215780},[31818,31819,31820,31821],[34,215782,34395],{"className":215783,"style":34452},[31685,31686,31821],[34,215785,31829],{"className":215786},[31828],[34,215788,215790],{"className":215789},[31801],[34,215791,215793],{"className":215792,"style":31836},[31805],[34,215794],{},[34,215796],{"className":215797,"style":31846},[31845],[34,215799,202],{"className":215800},[31850],[34,215802],{"className":215803,"style":31846},[31845],[34,215805,215807,215810,215813,215816,215819],{"className":215806},[31676],[34,215808],{"className":215809,"style":35942},[31680],[34,215811,640],{"className":215812},[31685],[34,215814],{"className":215815,"style":32604},[31845],[34,215817,32226],{"className":215818},[32024],[34,215820],{"className":215821,"style":32604},[31845],[34,215823,215825,215828,215831,215834,215837],{"className":215824},[31676],[34,215826],{"className":215827,"style":32617},[31680],[34,215829,459],{"className":215830},[31685],[34,215832],{"className":215833,"style":31846},[31845],[34,215835,202],{"className":215836},[31850],[34,215838],{"className":215839,"style":31846},[31845],[34,215841,215843,215846],{"className":215842},[31676],[34,215844],{"className":215845,"style":32617},[31680],[34,215847,640],{"className":215848},[31685],[34,215850,215852,215882],{"className":215851},[31642],[34,215853,215855],{"className":215854},[31646],[31648,215856,215857],{"xmlns":31650},[31652,215858,215859,215879],{},[31655,215860,215861,215867,215869,215875,215877],{},[31743,215862,215863,215865],{},[31658,215864,37400],{},[31658,215866,34395],{},[31658,215868,38080],{"mathvariant":38079},[31743,215870,215871,215873],{},[31658,215872,37400],{},[31658,215874,31747],{},[31662,215876,202],{},[31749,215878,640],{},[31665,215880,215881],{"encoding":31667},"T_r\u002FT_s = 1",[34,215883,215885,215983],{"className":215884,"ariaHidden":758},[31672],[34,215886,215888,215891,215931,215934,215974,215977,215980],{"className":215887},[31676],[34,215889],{"className":215890,"style":31774},[31680],[34,215892,215894,215897],{"className":215893},[31685],[34,215895,37400],{"className":215896,"style":37422},[31685,31686],[34,215898,215900],{"className":215899},[31792],[34,215901,215903,215923],{"className":215902},[31796,31797],[34,215904,215906,215920],{"className":215905},[31801],[34,215907,215909],{"className":215908,"style":32362},[31805],[34,215910,215911,215914],{"style":214424},[34,215912],{"className":215913,"style":31814},[31813],[34,215915,215917],{"className":215916},[31818,31819,31820,31821],[34,215918,34395],{"className":215919,"style":34452},[31685,31686,31821],[34,215921,31829],{"className":215922},[31828],[34,215924,215926],{"className":215925},[31801],[34,215927,215929],{"className":215928,"style":31836},[31805],[34,215930],{},[34,215932,38080],{"className":215933},[31685],[34,215935,215937,215940],{"className":215936},[31685],[34,215938,37400],{"className":215939,"style":37422},[31685,31686],[34,215941,215943],{"className":215942},[31792],[34,215944,215946,215966],{"className":215945},[31796,31797],[34,215947,215949,215963],{"className":215948},[31801],[34,215950,215952],{"className":215951,"style":32362},[31805],[34,215953,215954,215957],{"style":214424},[34,215955],{"className":215956,"style":31814},[31813],[34,215958,215960],{"className":215959},[31818,31819,31820,31821],[34,215961,31747],{"className":215962},[31685,31686,31821],[34,215964,31829],{"className":215965},[31828],[34,215967,215969],{"className":215968},[31801],[34,215970,215972],{"className":215971,"style":31836},[31805],[34,215973],{},[34,215975],{"className":215976,"style":31846},[31845],[34,215978,202],{"className":215979},[31850],[34,215981],{"className":215982,"style":31846},[31845],[34,215984,215986,215989],{"className":215985},[31676],[34,215987],{"className":215988,"style":32617},[31680],[34,215990,640],{"className":215991},[31685],[108,215993,215994,215995,616,216131],{},"B: ",[34,215996,215998,216028],{"className":215997},[31642],[34,215999,216001],{"className":216000},[31646],[31648,216002,216003],{"xmlns":31650},[31652,216004,216005,216025],{},[31655,216006,216007,216013,216015,216017,216019,216021,216023],{},[31743,216008,216009,216011],{},[31658,216010,37400],{},[31658,216012,34395],{},[31662,216014,202],{},[31749,216016,39620],{},[31662,216018,32226],{},[31749,216020,640],{},[31662,216022,202],{},[31749,216024,3244],{},[31665,216026,216027],{"encoding":31667},"T_r = 11-1=10",[34,216029,216031,216086,216104,216122],{"className":216030,"ariaHidden":758},[31672],[34,216032,216034,216037,216077,216080,216083],{"className":216033},[31676],[34,216035],{"className":216036,"style":70915},[31680],[34,216038,216040,216043],{"className":216039},[31685],[34,216041,37400],{"className":216042,"style":37422},[31685,31686],[34,216044,216046],{"className":216045},[31792],[34,216047,216049,216069],{"className":216048},[31796,31797],[34,216050,216052,216066],{"className":216051},[31801],[34,216053,216055],{"className":216054,"style":32362},[31805],[34,216056,216057,216060],{"style":214424},[34,216058],{"className":216059,"style":31814},[31813],[34,216061,216063],{"className":216062},[31818,31819,31820,31821],[34,216064,34395],{"className":216065,"style":34452},[31685,31686,31821],[34,216067,31829],{"className":216068},[31828],[34,216070,216072],{"className":216071},[31801],[34,216073,216075],{"className":216074,"style":31836},[31805],[34,216076],{},[34,216078],{"className":216079,"style":31846},[31845],[34,216081,202],{"className":216082},[31850],[34,216084],{"className":216085,"style":31846},[31845],[34,216087,216089,216092,216095,216098,216101],{"className":216088},[31676],[34,216090],{"className":216091,"style":35942},[31680],[34,216093,39620],{"className":216094},[31685],[34,216096],{"className":216097,"style":32604},[31845],[34,216099,32226],{"className":216100},[32024],[34,216102],{"className":216103,"style":32604},[31845],[34,216105,216107,216110,216113,216116,216119],{"className":216106},[31676],[34,216108],{"className":216109,"style":32617},[31680],[34,216111,640],{"className":216112},[31685],[34,216114],{"className":216115,"style":31846},[31845],[34,216117,202],{"className":216118},[31850],[34,216120],{"className":216121,"style":31846},[31845],[34,216123,216125,216128],{"className":216124},[31676],[34,216126],{"className":216127,"style":32617},[31680],[34,216129,3244],{"className":216130},[31685],[34,216132,216134,216173],{"className":216133},[31642],[34,216135,216137],{"className":216136},[31646],[31648,216138,216139],{"xmlns":31650},[31652,216140,216141,216170],{},[31655,216142,216143,216149,216151,216157,216159,216161,216163,216165,216167],{},[31743,216144,216145,216147],{},[31658,216146,37400],{},[31658,216148,34395],{},[31658,216150,38080],{"mathvariant":38079},[31743,216152,216153,216155],{},[31658,216154,37400],{},[31658,216156,31747],{},[31662,216158,202],{},[31749,216160,3244],{},[31658,216162,38080],{"mathvariant":38079},[31749,216164,36732],{},[31662,216166,168560],{},[31749,216168,216169],{},"1.11",[31665,216171,216172],{"encoding":31667},"T_r\u002FT_s = 10\u002F9 \\approx 1.11",[34,216174,216176,216274,216293],{"className":216175,"ariaHidden":758},[31672],[34,216177,216179,216182,216222,216225,216265,216268,216271],{"className":216178},[31676],[34,216180],{"className":216181,"style":31774},[31680],[34,216183,216185,216188],{"className":216184},[31685],[34,216186,37400],{"className":216187,"style":37422},[31685,31686],[34,216189,216191],{"className":216190},[31792],[34,216192,216194,216214],{"className":216193},[31796,31797],[34,216195,216197,216211],{"className":216196},[31801],[34,216198,216200],{"className":216199,"style":32362},[31805],[34,216201,216202,216205],{"style":214424},[34,216203],{"className":216204,"style":31814},[31813],[34,216206,216208],{"className":216207},[31818,31819,31820,31821],[34,216209,34395],{"className":216210,"style":34452},[31685,31686,31821],[34,216212,31829],{"className":216213},[31828],[34,216215,216217],{"className":216216},[31801],[34,216218,216220],{"className":216219,"style":31836},[31805],[34,216221],{},[34,216223,38080],{"className":216224},[31685],[34,216226,216228,216231],{"className":216227},[31685],[34,216229,37400],{"className":216230,"style":37422},[31685,31686],[34,216232,216234],{"className":216233},[31792],[34,216235,216237,216257],{"className":216236},[31796,31797],[34,216238,216240,216254],{"className":216239},[31801],[34,216241,216243],{"className":216242,"style":32362},[31805],[34,216244,216245,216248],{"style":214424},[34,216246],{"className":216247,"style":31814},[31813],[34,216249,216251],{"className":216250},[31818,31819,31820,31821],[34,216252,31747],{"className":216253},[31685,31686,31821],[34,216255,31829],{"className":216256},[31828],[34,216258,216260],{"className":216259},[31801],[34,216261,216263],{"className":216262,"style":31836},[31805],[34,216264],{},[34,216266],{"className":216267,"style":31846},[31845],[34,216269,202],{"className":216270},[31850],[34,216272],{"className":216273,"style":31846},[31845],[34,216275,216277,216280,216284,216287,216290],{"className":216276},[31676],[34,216278],{"className":216279,"style":31774},[31680],[34,216281,216283],{"className":216282},[31685],"10\u002F9",[34,216285],{"className":216286,"style":31846},[31845],[34,216288,168560],{"className":216289},[31850],[34,216291],{"className":216292,"style":31846},[31845],[34,216294,216296,216299],{"className":216295},[31676],[34,216297],{"className":216298,"style":32617},[31680],[34,216300,216169],{"className":216301},[31685],[108,216303,216304,216305,616,216441,216611],{},"C: ",[34,216306,216308,216338],{"className":216307},[31642],[34,216309,216311],{"className":216310},[31646],[31648,216312,216313],{"xmlns":31650},[31652,216314,216315,216335],{},[31655,216316,216317,216323,216325,216327,216329,216331,216333],{},[31743,216318,216319,216321],{},[31658,216320,37400],{},[31658,216322,34395],{},[31662,216324,202],{},[31749,216326,786],{},[31662,216328,32226],{},[31749,216330,768],{},[31662,216332,202],{},[31749,216334,640],{},[31665,216336,216337],{"encoding":31667},"T_r = 3-2=1",[34,216339,216341,216396,216414,216432],{"className":216340,"ariaHidden":758},[31672],[34,216342,216344,216347,216387,216390,216393],{"className":216343},[31676],[34,216345],{"className":216346,"style":70915},[31680],[34,216348,216350,216353],{"className":216349},[31685],[34,216351,37400],{"className":216352,"style":37422},[31685,31686],[34,216354,216356],{"className":216355},[31792],[34,216357,216359,216379],{"className":216358},[31796,31797],[34,216360,216362,216376],{"className":216361},[31801],[34,216363,216365],{"className":216364,"style":32362},[31805],[34,216366,216367,216370],{"style":214424},[34,216368],{"className":216369,"style":31814},[31813],[34,216371,216373],{"className":216372},[31818,31819,31820,31821],[34,216374,34395],{"className":216375,"style":34452},[31685,31686,31821],[34,216377,31829],{"className":216378},[31828],[34,216380,216382],{"className":216381},[31801],[34,216383,216385],{"className":216384,"style":31836},[31805],[34,216386],{},[34,216388],{"className":216389,"style":31846},[31845],[34,216391,202],{"className":216392},[31850],[34,216394],{"className":216395,"style":31846},[31845],[34,216397,216399,216402,216405,216408,216411],{"className":216398},[31676],[34,216400],{"className":216401,"style":35942},[31680],[34,216403,786],{"className":216404},[31685],[34,216406],{"className":216407,"style":32604},[31845],[34,216409,32226],{"className":216410},[32024],[34,216412],{"className":216413,"style":32604},[31845],[34,216415,216417,216420,216423,216426,216429],{"className":216416},[31676],[34,216418],{"className":216419,"style":32617},[31680],[34,216421,768],{"className":216422},[31685],[34,216424],{"className":216425,"style":31846},[31845],[34,216427,202],{"className":216428},[31850],[34,216430],{"className":216431,"style":31846},[31845],[34,216433,216435,216438],{"className":216434},[31676],[34,216436],{"className":216437,"style":32617},[31680],[34,216439,640],{"className":216440},[31685],[34,216442,216444,216482],{"className":216443},[31642],[34,216445,216447],{"className":216446},[31646],[31648,216448,216449],{"xmlns":31650},[31652,216450,216451,216479],{},[31655,216452,216453,216459,216461,216467,216469,216471,216473,216475,216477],{},[31743,216454,216455,216457],{},[31658,216456,37400],{},[31658,216458,34395],{},[31658,216460,38080],{"mathvariant":38079},[31743,216462,216463,216465],{},[31658,216464,37400],{},[31658,216466,31747],{},[31662,216468,202],{},[31749,216470,640],{},[31658,216472,38080],{"mathvariant":38079},[31749,216474,640],{},[31662,216476,202],{},[31749,216478,640],{},[31665,216480,216481],{"encoding":31667},"T_r\u002FT_s = 1\u002F1 = 1",[34,216483,216485,216583,216602],{"className":216484,"ariaHidden":758},[31672],[34,216486,216488,216491,216531,216534,216574,216577,216580],{"className":216487},[31676],[34,216489],{"className":216490,"style":31774},[31680],[34,216492,216494,216497],{"className":216493},[31685],[34,216495,37400],{"className":216496,"style":37422},[31685,31686],[34,216498,216500],{"className":216499},[31792],[34,216501,216503,216523],{"className":216502},[31796,31797],[34,216504,216506,216520],{"className":216505},[31801],[34,216507,216509],{"className":216508,"style":32362},[31805],[34,216510,216511,216514],{"style":214424},[34,216512],{"className":216513,"style":31814},[31813],[34,216515,216517],{"className":216516},[31818,31819,31820,31821],[34,216518,34395],{"className":216519,"style":34452},[31685,31686,31821],[34,216521,31829],{"className":216522},[31828],[34,216524,216526],{"className":216525},[31801],[34,216527,216529],{"className":216528,"style":31836},[31805],[34,216530],{},[34,216532,38080],{"className":216533},[31685],[34,216535,216537,216540],{"className":216536},[31685],[34,216538,37400],{"className":216539,"style":37422},[31685,31686],[34,216541,216543],{"className":216542},[31792],[34,216544,216546,216566],{"className":216545},[31796,31797],[34,216547,216549,216563],{"className":216548},[31801],[34,216550,216552],{"className":216551,"style":32362},[31805],[34,216553,216554,216557],{"style":214424},[34,216555],{"className":216556,"style":31814},[31813],[34,216558,216560],{"className":216559},[31818,31819,31820,31821],[34,216561,31747],{"className":216562},[31685,31686,31821],[34,216564,31829],{"className":216565},[31828],[34,216567,216569],{"className":216568},[31801],[34,216570,216572],{"className":216571,"style":31836},[31805],[34,216573],{},[34,216575],{"className":216576,"style":31846},[31845],[34,216578,202],{"className":216579},[31850],[34,216581],{"className":216582,"style":31846},[31845],[34,216584,216586,216589,216593,216596,216599],{"className":216585},[31676],[34,216587],{"className":216588,"style":31774},[31680],[34,216590,216592],{"className":216591},[31685],"1\u002F1",[34,216594],{"className":216595,"style":31846},[31845],[34,216597,202],{"className":216598},[31850],[34,216600],{"className":216601,"style":31846},[31845],[34,216603,216605,216608],{"className":216604},[31676],[34,216606],{"className":216607,"style":32617},[31680],[34,216609,640],{"className":216610},[31685]," (完美解决饥饿！)",[108,216613,216614,216615,616,216751],{},"D: ",[34,216616,216618,216648],{"className":216617},[31642],[34,216619,216621],{"className":216620},[31646],[31648,216622,216623],{"xmlns":31650},[31652,216624,216625,216645],{},[31655,216626,216627,216633,216635,216637,216639,216641,216643],{},[31743,216628,216629,216631],{},[31658,216630,37400],{},[31658,216632,34395],{},[31662,216634,202],{},[31749,216636,3249],{},[31662,216638,32226],{},[31749,216640,786],{},[31662,216642,202],{},[31749,216644,164836],{},[31665,216646,216647],{"encoding":31667},"T_r = 20-3=17",[34,216649,216651,216706,216724,216742],{"className":216650,"ariaHidden":758},[31672],[34,216652,216654,216657,216697,216700,216703],{"className":216653},[31676],[34,216655],{"className":216656,"style":70915},[31680],[34,216658,216660,216663],{"className":216659},[31685],[34,216661,37400],{"className":216662,"style":37422},[31685,31686],[34,216664,216666],{"className":216665},[31792],[34,216667,216669,216689],{"className":216668},[31796,31797],[34,216670,216672,216686],{"className":216671},[31801],[34,216673,216675],{"className":216674,"style":32362},[31805],[34,216676,216677,216680],{"style":214424},[34,216678],{"className":216679,"style":31814},[31813],[34,216681,216683],{"className":216682},[31818,31819,31820,31821],[34,216684,34395],{"className":216685,"style":34452},[31685,31686,31821],[34,216687,31829],{"className":216688},[31828],[34,216690,216692],{"className":216691},[31801],[34,216693,216695],{"className":216694,"style":31836},[31805],[34,216696],{},[34,216698],{"className":216699,"style":31846},[31845],[34,216701,202],{"className":216702},[31850],[34,216704],{"className":216705,"style":31846},[31845],[34,216707,216709,216712,216715,216718,216721],{"className":216708},[31676],[34,216710],{"className":216711,"style":35942},[31680],[34,216713,3249],{"className":216714},[31685],[34,216716],{"className":216717,"style":32604},[31845],[34,216719,32226],{"className":216720},[32024],[34,216722],{"className":216723,"style":32604},[31845],[34,216725,216727,216730,216733,216736,216739],{"className":216726},[31676],[34,216728],{"className":216729,"style":32617},[31680],[34,216731,786],{"className":216732},[31685],[34,216734],{"className":216735,"style":31846},[31845],[34,216737,202],{"className":216738},[31850],[34,216740],{"className":216741,"style":31846},[31845],[34,216743,216745,216748],{"className":216744},[31676],[34,216746],{"className":216747,"style":32617},[31680],[34,216749,164836],{"className":216750},[31685],[34,216752,216754,216792],{"className":216753},[31642],[34,216755,216757],{"className":216756},[31646],[31648,216758,216759],{"xmlns":31650},[31652,216760,216761,216789],{},[31655,216762,216763,216769,216771,216777,216779,216781,216783,216785,216787],{},[31743,216764,216765,216767],{},[31658,216766,37400],{},[31658,216768,34395],{},[31658,216770,38080],{"mathvariant":38079},[31743,216772,216773,216775],{},[31658,216774,37400],{},[31658,216776,31747],{},[31662,216778,202],{},[31749,216780,164836],{},[31658,216782,38080],{"mathvariant":38079},[31749,216784,36732],{},[31662,216786,168560],{},[31749,216788,61939],{},[31665,216790,216791],{"encoding":31667},"T_r\u002FT_s = 17\u002F9 \\approx 1.89",[34,216793,216795,216893,216912],{"className":216794,"ariaHidden":758},[31672],[34,216796,216798,216801,216841,216844,216884,216887,216890],{"className":216797},[31676],[34,216799],{"className":216800,"style":31774},[31680],[34,216802,216804,216807],{"className":216803},[31685],[34,216805,37400],{"className":216806,"style":37422},[31685,31686],[34,216808,216810],{"className":216809},[31792],[34,216811,216813,216833],{"className":216812},[31796,31797],[34,216814,216816,216830],{"className":216815},[31801],[34,216817,216819],{"className":216818,"style":32362},[31805],[34,216820,216821,216824],{"style":214424},[34,216822],{"className":216823,"style":31814},[31813],[34,216825,216827],{"className":216826},[31818,31819,31820,31821],[34,216828,34395],{"className":216829,"style":34452},[31685,31686,31821],[34,216831,31829],{"className":216832},[31828],[34,216834,216836],{"className":216835},[31801],[34,216837,216839],{"className":216838,"style":31836},[31805],[34,216840],{},[34,216842,38080],{"className":216843},[31685],[34,216845,216847,216850],{"className":216846},[31685],[34,216848,37400],{"className":216849,"style":37422},[31685,31686],[34,216851,216853],{"className":216852},[31792],[34,216854,216856,216876],{"className":216855},[31796,31797],[34,216857,216859,216873],{"className":216858},[31801],[34,216860,216862],{"className":216861,"style":32362},[31805],[34,216863,216864,216867],{"style":214424},[34,216865],{"className":216866,"style":31814},[31813],[34,216868,216870],{"className":216869},[31818,31819,31820,31821],[34,216871,31747],{"className":216872},[31685,31686,31821],[34,216874,31829],{"className":216875},[31828],[34,216877,216879],{"className":216878},[31801],[34,216880,216882],{"className":216881,"style":31836},[31805],[34,216883],{},[34,216885],{"className":216886,"style":31846},[31845],[34,216888,202],{"className":216889},[31850],[34,216891],{"className":216892,"style":31846},[31845],[34,216894,216896,216899,216903,216906,216909],{"className":216895},[31676],[34,216897],{"className":216898,"style":31774},[31680],[34,216900,216902],{"className":216901},[31685],"17\u002F9",[34,216904],{"className":216905,"style":31846},[31845],[34,216907,168560],{"className":216908},[31850],[34,216910],{"className":216911,"style":31846},[31845],[34,216913,216915,216918],{"className":216914},[31676],[34,216916],{"className":216917,"style":32617},[31680],[34,216919,61939],{"className":216920},[31685],[15,216922,216923,216926,216927,347],{},[2627,216924,216925],{},"(4) HRRN (最高响应比优先) - 非抢占式","\n响应比 ",[34,216928,216930,216958],{"className":216929},[31642],[34,216931,216933],{"className":216932},[31646],[31648,216934,216935],{"xmlns":31650},[31652,216936,216937,216955],{},[31655,216938,216939,216941,216943,216945,216947,216950,216952],{},[31658,216940,34386],{},[31662,216942,202],{},[31749,216944,640],{},[31662,216946,223],{},[39361,216948,216949],{},"等待时间",[31658,216951,38080],{"mathvariant":38079},[39361,216953,216954],{},"服务时间",[31665,216956,216957],{"encoding":31667},"R = 1 + \\text{等待时间}\u002F\\text{服务时间}",[34,216959,216961,216979,216997],{"className":216960,"ariaHidden":758},[31672],[34,216962,216964,216967,216970,216973,216976],{"className":216963},[31676],[34,216965],{"className":216966,"style":33218},[31680],[34,216968,34386],{"className":216969,"style":34427},[31685,31686],[34,216971],{"className":216972,"style":31846},[31845],[34,216974,202],{"className":216975},[31850],[34,216977],{"className":216978,"style":31846},[31845],[34,216980,216982,216985,216988,216991,216994],{"className":216981},[31676],[34,216983],{"className":216984,"style":35942},[31680],[34,216986,640],{"className":216987},[31685],[34,216989],{"className":216990,"style":32604},[31845],[34,216992,223],{"className":216993},[32024],[34,216995],{"className":216996,"style":32604},[31845],[34,216998,217000,217003,217009,217012],{"className":216999},[31676],[34,217001],{"className":217002,"style":31774},[31680],[34,217004,217006],{"className":217005},[31685,2638],[34,217007,216949],{"className":217008},[31685,105462],[34,217010,38080],{"className":217011},[31685],[34,217013,217015],{"className":217014},[31685,2638],[34,217016,216954],{"className":217017},[31685,105462],[105,217019,217020,217025,217030,217326,217329],{},[108,217021,217022,217024],{},[2627,217023,214719],{}," A执行至1。",[108,217026,217027,217029],{},[2627,217028,214725],{}," B执行至10。",[108,217031,217032,217034,217035],{},[2627,217033,214731],{}," 重新计算响应比：\n",[105,217036,217037,217181],{},[108,217038,217039,217040,347],{},"C等待了8s，服务1s， ",[34,217041,217043,217077],{"className":217042},[31642],[34,217044,217046],{"className":217045},[31646],[31648,217047,217048],{"xmlns":31650},[31652,217049,217050,217074],{},[31655,217051,217052,217058,217060,217062,217064,217066,217068,217070,217072],{},[31743,217053,217054,217056],{},[31658,217055,34386],{},[31658,217057,47384],{},[31662,217059,202],{},[31749,217061,640],{},[31662,217063,223],{},[31749,217065,60525],{},[31658,217067,38080],{"mathvariant":38079},[31749,217069,640],{},[31662,217071,202],{},[31749,217073,36732],{},[31665,217075,217076],{"encoding":31667},"R_C = 1 + 8\u002F1 = 9",[34,217078,217080,217135,217153,217172],{"className":217079,"ariaHidden":758},[31672],[34,217081,217083,217086,217126,217129,217132],{"className":217082},[31676],[34,217084],{"className":217085,"style":70915},[31680],[34,217087,217089,217092],{"className":217088},[31685],[34,217090,34386],{"className":217091,"style":34427},[31685,31686],[34,217093,217095],{"className":217094},[31792],[34,217096,217098,217118],{"className":217097},[31796,31797],[34,217099,217101,217115],{"className":217100},[31801],[34,217102,217104],{"className":217103,"style":116558},[31805],[34,217105,217106,217109],{"style":104694},[34,217107],{"className":217108,"style":31814},[31813],[34,217110,217112],{"className":217111},[31818,31819,31820,31821],[34,217113,47384],{"className":217114,"style":38706},[31685,31686,31821],[34,217116,31829],{"className":217117},[31828],[34,217119,217121],{"className":217120},[31801],[34,217122,217124],{"className":217123,"style":31836},[31805],[34,217125],{},[34,217127],{"className":217128,"style":31846},[31845],[34,217130,202],{"className":217131},[31850],[34,217133],{"className":217134,"style":31846},[31845],[34,217136,217138,217141,217144,217147,217150],{"className":217137},[31676],[34,217139],{"className":217140,"style":35942},[31680],[34,217142,640],{"className":217143},[31685],[34,217145],{"className":217146,"style":32604},[31845],[34,217148,223],{"className":217149},[32024],[34,217151],{"className":217152,"style":32604},[31845],[34,217154,217156,217159,217163,217166,217169],{"className":217155},[31676],[34,217157],{"className":217158,"style":31774},[31680],[34,217160,217162],{"className":217161},[31685],"8\u002F1",[34,217164],{"className":217165,"style":31846},[31845],[34,217167,202],{"className":217168},[31850],[34,217170],{"className":217171,"style":31846},[31845],[34,217173,217175,217178],{"className":217174},[31676],[34,217176],{"className":217177,"style":32617},[31680],[34,217179,36732],{"className":217180},[31685],[108,217182,217183,217184,347],{},"D等待了7s，服务9s， ",[34,217185,217187,217222],{"className":217186},[31642],[34,217188,217190],{"className":217189},[31646],[31648,217191,217192],{"xmlns":31650},[31652,217193,217194,217219],{},[31655,217195,217196,217202,217204,217206,217208,217210,217212,217214,217216],{},[31743,217197,217198,217200],{},[31658,217199,34386],{},[31658,217201,91994],{},[31662,217203,202],{},[31749,217205,640],{},[31662,217207,223],{},[31749,217209,36723],{},[31658,217211,38080],{"mathvariant":38079},[31749,217213,36732],{},[31662,217215,168560],{},[31749,217217,217218],{},"1.77",[31665,217220,217221],{"encoding":31667},"R_D = 1 + 7\u002F9 \\approx 1.77",[34,217223,217225,217280,217298,217317],{"className":217224,"ariaHidden":758},[31672],[34,217226,217228,217231,217271,217274,217277],{"className":217227},[31676],[34,217229],{"className":217230,"style":70915},[31680],[34,217232,217234,217237],{"className":217233},[31685],[34,217235,34386],{"className":217236,"style":34427},[31685,31686],[34,217238,217240],{"className":217239},[31792],[34,217241,217243,217263],{"className":217242},[31796,31797],[34,217244,217246,217260],{"className":217245},[31801],[34,217247,217249],{"className":217248,"style":116558},[31805],[34,217250,217251,217254],{"style":104694},[34,217252],{"className":217253,"style":31814},[31813],[34,217255,217257],{"className":217256},[31818,31819,31820,31821],[34,217258,91994],{"className":217259,"style":34452},[31685,31686,31821],[34,217261,31829],{"className":217262},[31828],[34,217264,217266],{"className":217265},[31801],[34,217267,217269],{"className":217268,"style":31836},[31805],[34,217270],{},[34,217272],{"className":217273,"style":31846},[31845],[34,217275,202],{"className":217276},[31850],[34,217278],{"className":217279,"style":31846},[31845],[34,217281,217283,217286,217289,217292,217295],{"className":217282},[31676],[34,217284],{"className":217285,"style":35942},[31680],[34,217287,640],{"className":217288},[31685],[34,217290],{"className":217291,"style":32604},[31845],[34,217293,223],{"className":217294},[32024],[34,217296],{"className":217297,"style":32604},[31845],[34,217299,217301,217304,217308,217311,217314],{"className":217300},[31676],[34,217302],{"className":217303,"style":31774},[31680],[34,217305,217307],{"className":217306},[31685],"7\u002F9",[34,217309],{"className":217310,"style":31846},[31845],[34,217312,168560],{"className":217313},[31850],[34,217315],{"className":217316,"style":31846},[31845],[34,217318,217320,217323],{"className":217319},[31676],[34,217321],{"className":217322,"style":32617},[31680],[34,217324,217218],{"className":217325},[31685],[108,217327,217328],{},"C响应比极高，C执行至11，随后D执行至20。",[108,217330,217331,217333],{},[2627,217332,58554],{}," 顺序依然是 A -> B -> C -> D。与 FCFS 和 SPN 完全一致。",[253,217335,217337],{"id":217336},"常见易错点-common-pitfalls","常见易错点 (Common Pitfalls)",[15,217339,217340,217341,217344,217345,217348],{},"学生在做 ",[2627,217342,217343],{},"SPN (最短进程优先)"," 时，经常会看到 C 只有 1s，就理所当然地认为 C 应该插队在 B 前面。",[2627,217346,217347],{},"绝对不行！"," SPN 是非抢占式的，当 t=1 时，C 还没出生，系统根本不知道未来会有短进程，只能让 B 上台，而 B 一旦上台就必须执行到底。",[32188,217350],{},[19,217352,217354],{"id":217353},"题目-2",[2627,217355,217356],{},"题目 2",[15,217358,217359],{},[2627,217360,217361],{},"Consider the following pair of equations as an alternative to Equation (9.3):",[15,217363,217364],{},[34,217365,217367,217420],{"className":217366},[31642],[34,217368,217370],{"className":217369},[31646],[31648,217371,217372],{"xmlns":31650},[31652,217373,217374,217417],{},[31655,217375,217376,217388,217390,217393,217399,217401,217403,217405,217407,217409,217411],{},[31743,217377,217378,217380],{},[31658,217379,94675],{},[31655,217381,217382,217384,217386],{},[31658,217383,31660],{},[31662,217385,223],{},[31749,217387,640],{},[31662,217389,202],{},[31658,217391,217392],{},"α",[31743,217394,217395,217397],{},[31658,217396,37400],{},[31658,217398,31660],{},[31662,217400,223],{},[31662,217402,456],{"stretchy":702},[31749,217404,640],{},[31662,217406,32226],{},[31658,217408,217392],{},[31662,217410,2557],{"stretchy":702},[31743,217412,217413,217415],{},[31658,217414,94675],{},[31658,217416,31660],{},[31665,217418,217419],{"encoding":31667},"S_{n+1} = \\alpha T_n + (1 - \\alpha) S_n",[34,217421,217423,217487,217546,217567],{"className":217422,"ariaHidden":758},[31672],[34,217424,217426,217429,217478,217481,217484],{"className":217425},[31676],[34,217427],{"className":217428,"style":70992},[31680],[34,217430,217432,217435],{"className":217431},[31685],[34,217433,94675],{"className":217434,"style":106455},[31685,31686],[34,217436,217438],{"className":217437},[31792],[34,217439,217441,217470],{"className":217440},[31796,31797],[34,217442,217444,217467],{"className":217443},[31801],[34,217445,217447],{"className":217446,"style":31806},[31805],[34,217448,217449,217452],{"style":206422},[34,217450],{"className":217451,"style":31814},[31813],[34,217453,217455],{"className":217454},[31818,31819,31820,31821],[34,217456,217458,217461,217464],{"className":217457},[31685,31821],[34,217459,31660],{"className":217460},[31685,31686,31821],[34,217462,223],{"className":217463},[32024,31821],[34,217465,640],{"className":217466},[31685,31821],[34,217468,31829],{"className":217469},[31828],[34,217471,217473],{"className":217472},[31801],[34,217474,217476],{"className":217475,"style":32037},[31805],[34,217477],{},[34,217479],{"className":217480,"style":31846},[31845],[34,217482,202],{"className":217483},[31850],[34,217485],{"className":217486,"style":31846},[31845],[34,217488,217490,217493,217497,217537,217540,217543],{"className":217489},[31676],[34,217491],{"className":217492,"style":70915},[31680],[34,217494,217392],{"className":217495,"style":217496},[31685,31686],"margin-right:0.0037em;",[34,217498,217500,217503],{"className":217499},[31685],[34,217501,37400],{"className":217502,"style":37422},[31685,31686],[34,217504,217506],{"className":217505},[31792],[34,217507,217509,217529],{"className":217508},[31796,31797],[34,217510,217512,217526],{"className":217511},[31801],[34,217513,217515],{"className":217514,"style":32362},[31805],[34,217516,217517,217520],{"style":214424},[34,217518],{"className":217519,"style":31814},[31813],[34,217521,217523],{"className":217522},[31818,31819,31820,31821],[34,217524,31660],{"className":217525},[31685,31686,31821],[34,217527,31829],{"className":217528},[31828],[34,217530,217532],{"className":217531},[31801],[34,217533,217535],{"className":217534,"style":31836},[31805],[34,217536],{},[34,217538],{"className":217539,"style":32604},[31845],[34,217541,223],{"className":217542},[32024],[34,217544],{"className":217545,"style":32604},[31845],[34,217547,217549,217552,217555,217558,217561,217564],{"className":217548},[31676],[34,217550],{"className":217551,"style":31774},[31680],[34,217553,456],{"className":217554},[31782],[34,217556,640],{"className":217557},[31685],[34,217559],{"className":217560,"style":32604},[31845],[34,217562,32226],{"className":217563},[32024],[34,217565],{"className":217566,"style":32604},[31845],[34,217568,217570,217573,217576,217579],{"className":217569},[31676],[34,217571],{"className":217572,"style":31774},[31680],[34,217574,217392],{"className":217575,"style":217496},[31685,31686],[34,217577,2557],{"className":217578},[31690],[34,217580,217582,217585],{"className":217581},[31685],[34,217583,94675],{"className":217584,"style":106455},[31685,31686],[34,217586,217588],{"className":217587},[31792],[34,217589,217591,217611],{"className":217590},[31796,31797],[34,217592,217594,217608],{"className":217593},[31801],[34,217595,217597],{"className":217596,"style":32362},[31805],[34,217598,217599,217602],{"style":206422},[34,217600],{"className":217601,"style":31814},[31813],[34,217603,217605],{"className":217604},[31818,31819,31820,31821],[34,217606,31660],{"className":217607},[31685,31686,31821],[34,217609,31829],{"className":217610},[31828],[34,217612,217614],{"className":217613},[31801],[34,217615,217617],{"className":217616,"style":31836},[31805],[34,217618],{},[15,217620,217621],{},[34,217622,217624,217713],{"className":217623},[31642],[34,217625,217627],{"className":217626},[31646],[31648,217628,217629],{"xmlns":31650},[31652,217630,217631,217710],{},[31655,217632,217633,217645,217647,217649,217651,217653,217655,217657,217659,217661,217663,217665,217667,217669,217671,217673,217675,217677,217679,217681,217683,217685,217687,217689,217692,217704,217706,217708],{},[31743,217634,217635,217637],{},[31658,217636,67467],{},[31655,217638,217639,217641,217643],{},[31658,217640,31660],{},[31662,217642,223],{},[31749,217644,640],{},[31662,217646,202],{},[31658,217648,56084],{},[31662,217650,32682],{},[31662,217652,15972],{"stretchy":702},[31658,217654,33350],{},[31658,217656,1697],{},[31658,217658,43383],{},[31658,217660,94628],{},[31658,217662,31660],{},[31658,217664,46900],{},[31662,217666,31960],{"separator":758},[31658,217668,32679],{},[31662,217670,32682],{},[31662,217672,15972],{"stretchy":702},[31658,217674,33206],{},[31658,217676,1697],{},[31658,217678,43383],{},[31658,217680,94628],{},[31658,217682,31660],{},[31658,217684,46900],{},[31662,217686,31960],{"separator":758},[31662,217688,456],{"stretchy":702},[31658,217690,217691],{},"β",[31743,217693,217694,217696],{},[31658,217695,94675],{},[31655,217697,217698,217700,217702],{},[31658,217699,31660],{},[31662,217701,223],{},[31749,217703,640],{},[31662,217705,2557],{"stretchy":702},[31662,217707,32712],{"stretchy":702},[31662,217709,32712],{"stretchy":702},[31665,217711,217712],{"encoding":31667},"X_{n+1} = \\min[Ubound, \\max[Lbound, (\\beta S_{n+1})]]",[34,217714,217716,217781],{"className":217715,"ariaHidden":758},[31672],[34,217717,217719,217722,217772,217775,217778],{"className":217718},[31676],[34,217720],{"className":217721,"style":70992},[31680],[34,217723,217725,217728],{"className":217724},[31685],[34,217726,67467],{"className":217727,"style":67482},[31685,31686],[34,217729,217731],{"className":217730},[31792],[34,217732,217734,217764],{"className":217733},[31796,31797],[34,217735,217737,217761],{"className":217736},[31801],[34,217738,217740],{"className":217739,"style":31806},[31805],[34,217741,217743,217746],{"style":217742},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[34,217744],{"className":217745,"style":31814},[31813],[34,217747,217749],{"className":217748},[31818,31819,31820,31821],[34,217750,217752,217755,217758],{"className":217751},[31685,31821],[34,217753,31660],{"className":217754},[31685,31686,31821],[34,217756,223],{"className":217757},[32024,31821],[34,217759,640],{"className":217760},[31685,31821],[34,217762,31829],{"className":217763},[31828],[34,217765,217767],{"className":217766},[31801],[34,217768,217770],{"className":217769,"style":32037},[31805],[34,217771],{},[34,217773],{"className":217774,"style":31846},[31845],[34,217776,202],{"className":217777},[31850],[34,217779],{"className":217780,"style":31846},[31845],[34,217782,217784,217787,217790,217793,217796,217799,217802,217805,217808,217811,217814,217817,217820,217823,217826,217829,217832,217835,217838,217841,217844,217847,217850,217854,217903],{"className":217783},[31676],[34,217785],{"className":217786,"style":31774},[31680],[34,217788,56084],{"className":217789},[32767],[34,217791,15972],{"className":217792},[31782],[34,217794,33350],{"className":217795,"style":33389},[31685,31686],[34,217797,1697],{"className":217798},[31685,31686],[34,217800,43383],{"className":217801},[31685,31686],[34,217803,94628],{"className":217804},[31685,31686],[34,217806,31660],{"className":217807},[31685,31686],[34,217809,46900],{"className":217810},[31685,31686],[34,217812,31960],{"className":217813},[32119],[34,217815],{"className":217816,"style":32123},[31845],[34,217818,32679],{"className":217819},[32767],[34,217821,15972],{"className":217822},[31782],[34,217824,33206],{"className":217825},[31685,31686],[34,217827,1697],{"className":217828},[31685,31686],[34,217830,43383],{"className":217831},[31685,31686],[34,217833,94628],{"className":217834},[31685,31686],[34,217836,31660],{"className":217837},[31685,31686],[34,217839,46900],{"className":217840},[31685,31686],[34,217842,31960],{"className":217843},[32119],[34,217845],{"className":217846,"style":32123},[31845],[34,217848,456],{"className":217849},[31782],[34,217851,217691],{"className":217852,"style":217853},[31685,31686],"margin-right:0.0528em;",[34,217855,217857,217860],{"className":217856},[31685],[34,217858,94675],{"className":217859,"style":106455},[31685,31686],[34,217861,217863],{"className":217862},[31792],[34,217864,217866,217895],{"className":217865},[31796,31797],[34,217867,217869,217892],{"className":217868},[31801],[34,217870,217872],{"className":217871,"style":31806},[31805],[34,217873,217874,217877],{"style":206422},[34,217875],{"className":217876,"style":31814},[31813],[34,217878,217880],{"className":217879},[31818,31819,31820,31821],[34,217881,217883,217886,217889],{"className":217882},[31685,31821],[34,217884,31660],{"className":217885},[31685,31686,31821],[34,217887,223],{"className":217888},[32024,31821],[34,217890,640],{"className":217891},[31685,31821],[34,217893,31829],{"className":217894},[31828],[34,217896,217898],{"className":217897},[31801],[34,217899,217901],{"className":217900,"style":32037},[31805],[34,217902],{},[34,217904,217906],{"className":217905},[31690],")]]",[15,217908,217909,217910,195562,217964,218018,218019,218047,218048,218133,218134,218137,218138,2547],{},"Where ",[34,217911,217913,217937],{"className":217912},[31642],[34,217914,217916],{"className":217915},[31646],[31648,217917,217918],{"xmlns":31650},[31652,217919,217920,217934],{},[31655,217921,217922,217924,217926,217928,217930,217932],{},[31658,217923,33350],{},[31658,217925,1697],{},[31658,217927,43383],{},[31658,217929,94628],{},[31658,217931,31660],{},[31658,217933,46900],{},[31665,217935,217936],{"encoding":31667},"Ubound",[34,217938,217940],{"className":217939,"ariaHidden":758},[31672],[34,217941,217943,217946,217949,217952,217955,217958,217961],{"className":217942},[31676],[34,217944],{"className":217945,"style":31681},[31680],[34,217947,33350],{"className":217948,"style":33389},[31685,31686],[34,217950,1697],{"className":217951},[31685,31686],[34,217953,43383],{"className":217954},[31685,31686],[34,217956,94628],{"className":217957},[31685,31686],[34,217959,31660],{"className":217960},[31685,31686],[34,217962,46900],{"className":217963},[31685,31686],[34,217965,217967,217991],{"className":217966},[31642],[34,217968,217970],{"className":217969},[31646],[31648,217971,217972],{"xmlns":31650},[31652,217973,217974,217988],{},[31655,217975,217976,217978,217980,217982,217984,217986],{},[31658,217977,33206],{},[31658,217979,1697],{},[31658,217981,43383],{},[31658,217983,94628],{},[31658,217985,31660],{},[31658,217987,46900],{},[31665,217989,217990],{"encoding":31667},"Lbound",[34,217992,217994],{"className":217993,"ariaHidden":758},[31672],[34,217995,217997,218000,218003,218006,218009,218012,218015],{"className":217996},[31676],[34,217998],{"className":217999,"style":31681},[31680],[34,218001,33206],{"className":218002},[31685,31686],[34,218004,1697],{"className":218005},[31685,31686],[34,218007,43383],{"className":218008},[31685,31686],[34,218010,94628],{"className":218011},[31685,31686],[34,218013,31660],{"className":218014},[31685,31686],[34,218016,46900],{"className":218017},[31685,31686]," are prechosen upper and lower bounds on the estimated value of ",[34,218020,218022,218035],{"className":218021},[31642],[34,218023,218025],{"className":218024},[31646],[31648,218026,218027],{"xmlns":31650},[31652,218028,218029,218033],{},[31655,218030,218031],{},[31658,218032,37400],{},[31665,218034,37400],{"encoding":31667},[34,218036,218038],{"className":218037,"ariaHidden":758},[31672],[34,218039,218041,218044],{"className":218040},[31676],[34,218042],{"className":218043,"style":33218},[31680],[34,218045,37400],{"className":218046,"style":37422},[31685,31686],". The value of ",[34,218049,218051,218075],{"className":218050},[31642],[34,218052,218054],{"className":218053},[31646],[31648,218055,218056],{"xmlns":31650},[31652,218057,218058,218072],{},[31655,218059,218060],{},[31743,218061,218062,218064],{},[31658,218063,67467],{},[31655,218065,218066,218068,218070],{},[31658,218067,31660],{},[31662,218069,223],{},[31749,218071,640],{},[31665,218073,218074],{"encoding":31667},"X_{n+1}",[34,218076,218078],{"className":218077,"ariaHidden":758},[31672],[34,218079,218081,218084],{"className":218080},[31676],[34,218082],{"className":218083,"style":70992},[31680],[34,218085,218087,218090],{"className":218086},[31685],[34,218088,67467],{"className":218089,"style":67482},[31685,31686],[34,218091,218093],{"className":218092},[31792],[34,218094,218096,218125],{"className":218095},[31796,31797],[34,218097,218099,218122],{"className":218098},[31801],[34,218100,218102],{"className":218101,"style":31806},[31805],[34,218103,218104,218107],{"style":217742},[34,218105],{"className":218106,"style":31814},[31813],[34,218108,218110],{"className":218109},[31818,31819,31820,31821],[34,218111,218113,218116,218119],{"className":218112},[31685,31821],[34,218114,31660],{"className":218115},[31685,31686,31821],[34,218117,223],{"className":218118},[32024,31821],[34,218120,640],{"className":218121},[31685,31821],[34,218123,31829],{"className":218124},[31828],[34,218126,218128],{"className":218127},[31801],[34,218129,218131],{"className":218130,"style":32037},[31805],[34,218132],{}," is used in the ",[2627,218135,218136],{},"shortest-process-next"," algorithm, instead of the value of ",[34,218139,218141,218165],{"className":218140},[31642],[34,218142,218144],{"className":218143},[31646],[31648,218145,218146],{"xmlns":31650},[31652,218147,218148,218162],{},[31655,218149,218150],{},[31743,218151,218152,218154],{},[31658,218153,94675],{},[31655,218155,218156,218158,218160],{},[31658,218157,31660],{},[31662,218159,223],{},[31749,218161,640],{},[31665,218163,218164],{"encoding":31667},"S_{n+1}",[34,218166,218168],{"className":218167,"ariaHidden":758},[31672],[34,218169,218171,218174],{"className":218170},[31676],[34,218172],{"className":218173,"style":70992},[31680],[34,218175,218177,218180],{"className":218176},[31685],[34,218178,94675],{"className":218179,"style":106455},[31685,31686],[34,218181,218183],{"className":218182},[31792],[34,218184,218186,218215],{"className":218185},[31796,31797],[34,218187,218189,218212],{"className":218188},[31801],[34,218190,218192],{"className":218191,"style":31806},[31805],[34,218193,218194,218197],{"style":206422},[34,218195],{"className":218196,"style":31814},[31813],[34,218198,218200],{"className":218199},[31818,31819,31820,31821],[34,218201,218203,218206,218209],{"className":218202},[31685,31821],[34,218204,31660],{"className":218205},[31685,31686,31821],[34,218207,223],{"className":218208},[32024,31821],[34,218210,640],{"className":218211},[31685,31821],[34,218213,31829],{"className":218214},[31828],[34,218216,218218],{"className":218217},[31801],[34,218219,218221],{"className":218220,"style":32037},[31805],[34,218222],{},[15,218224,218225],{},[2627,218226,218227,218228,195562,218257,218286],{},"What functions do ",[34,218229,218231,218245],{"className":218230},[31642],[34,218232,218234],{"className":218233},[31646],[31648,218235,218236],{"xmlns":31650},[31652,218237,218238,218242],{},[31655,218239,218240],{},[31658,218241,217392],{},[31665,218243,218244],{"encoding":31667},"\\alpha",[34,218246,218248],{"className":218247,"ariaHidden":758},[31672],[34,218249,218251,218254],{"className":218250},[31676],[34,218252],{"className":218253,"style":32527},[31680],[34,218255,217392],{"className":218256,"style":217496},[31685,31686],[34,218258,218260,218274],{"className":218259},[31642],[34,218261,218263],{"className":218262},[31646],[31648,218264,218265],{"xmlns":31650},[31652,218266,218267,218271],{},[31655,218268,218269],{},[31658,218270,217691],{},[31665,218272,218273],{"encoding":31667},"\\beta",[34,218275,218277],{"className":218276,"ariaHidden":758},[31672],[34,218278,218280,218283],{"className":218279},[31676],[34,218281],{"className":218282,"style":43217},[31680],[34,218284,217691],{"className":218285,"style":217853},[31685,31686]," perform, and what is the effect of higher and lower values on each?",[15,218288,218289,218291,218292,258,218544,93436,218825,258,218853,218881],{},[2627,218290,214361],{},"\n分析公式 ",[34,218293,218295,218346],{"className":218294},[31642],[34,218296,218298],{"className":218297},[31646],[31648,218299,218300],{"xmlns":31650},[31652,218301,218302,218344],{},[31655,218303,218304,218316,218318,218320,218326,218328,218330,218332,218334,218336,218338],{},[31743,218305,218306,218308],{},[31658,218307,94675],{},[31655,218309,218310,218312,218314],{},[31658,218311,31660],{},[31662,218313,223],{},[31749,218315,640],{},[31662,218317,202],{},[31658,218319,217392],{},[31743,218321,218322,218324],{},[31658,218323,37400],{},[31658,218325,31660],{},[31662,218327,223],{},[31662,218329,456],{"stretchy":702},[31749,218331,640],{},[31662,218333,32226],{},[31658,218335,217392],{},[31662,218337,2557],{"stretchy":702},[31743,218339,218340,218342],{},[31658,218341,94675],{},[31658,218343,31660],{},[31665,218345,217419],{"encoding":31667},[34,218347,218349,218413,218471,218492],{"className":218348,"ariaHidden":758},[31672],[34,218350,218352,218355,218404,218407,218410],{"className":218351},[31676],[34,218353],{"className":218354,"style":70992},[31680],[34,218356,218358,218361],{"className":218357},[31685],[34,218359,94675],{"className":218360,"style":106455},[31685,31686],[34,218362,218364],{"className":218363},[31792],[34,218365,218367,218396],{"className":218366},[31796,31797],[34,218368,218370,218393],{"className":218369},[31801],[34,218371,218373],{"className":218372,"style":31806},[31805],[34,218374,218375,218378],{"style":206422},[34,218376],{"className":218377,"style":31814},[31813],[34,218379,218381],{"className":218380},[31818,31819,31820,31821],[34,218382,218384,218387,218390],{"className":218383},[31685,31821],[34,218385,31660],{"className":218386},[31685,31686,31821],[34,218388,223],{"className":218389},[32024,31821],[34,218391,640],{"className":218392},[31685,31821],[34,218394,31829],{"className":218395},[31828],[34,218397,218399],{"className":218398},[31801],[34,218400,218402],{"className":218401,"style":32037},[31805],[34,218403],{},[34,218405],{"className":218406,"style":31846},[31845],[34,218408,202],{"className":218409},[31850],[34,218411],{"className":218412,"style":31846},[31845],[34,218414,218416,218419,218422,218462,218465,218468],{"className":218415},[31676],[34,218417],{"className":218418,"style":70915},[31680],[34,218420,217392],{"className":218421,"style":217496},[31685,31686],[34,218423,218425,218428],{"className":218424},[31685],[34,218426,37400],{"className":218427,"style":37422},[31685,31686],[34,218429,218431],{"className":218430},[31792],[34,218432,218434,218454],{"className":218433},[31796,31797],[34,218435,218437,218451],{"className":218436},[31801],[34,218438,218440],{"className":218439,"style":32362},[31805],[34,218441,218442,218445],{"style":214424},[34,218443],{"className":218444,"style":31814},[31813],[34,218446,218448],{"className":218447},[31818,31819,31820,31821],[34,218449,31660],{"className":218450},[31685,31686,31821],[34,218452,31829],{"className":218453},[31828],[34,218455,218457],{"className":218456},[31801],[34,218458,218460],{"className":218459,"style":31836},[31805],[34,218461],{},[34,218463],{"className":218464,"style":32604},[31845],[34,218466,223],{"className":218467},[32024],[34,218469],{"className":218470,"style":32604},[31845],[34,218472,218474,218477,218480,218483,218486,218489],{"className":218473},[31676],[34,218475],{"className":218476,"style":31774},[31680],[34,218478,456],{"className":218479},[31782],[34,218481,640],{"className":218482},[31685],[34,218484],{"className":218485,"style":32604},[31845],[34,218487,32226],{"className":218488},[32024],[34,218490],{"className":218491,"style":32604},[31845],[34,218493,218495,218498,218501,218504],{"className":218494},[31676],[34,218496],{"className":218497,"style":31774},[31680],[34,218499,217392],{"className":218500,"style":217496},[31685,31686],[34,218502,2557],{"className":218503},[31690],[34,218505,218507,218510],{"className":218506},[31685],[34,218508,94675],{"className":218509,"style":106455},[31685,31686],[34,218511,218513],{"className":218512},[31792],[34,218514,218516,218536],{"className":218515},[31796,31797],[34,218517,218519,218533],{"className":218518},[31801],[34,218520,218522],{"className":218521,"style":32362},[31805],[34,218523,218524,218527],{"style":206422},[34,218525],{"className":218526,"style":31814},[31813],[34,218528,218530],{"className":218529},[31818,31819,31820,31821],[34,218531,31660],{"className":218532},[31685,31686,31821],[34,218534,31829],{"className":218535},[31828],[34,218537,218539],{"className":218538},[31801],[34,218540,218542],{"className":218541,"style":31836},[31805],[34,218543],{},[34,218545,218547,218634],{"className":218546},[31642],[34,218548,218550],{"className":218549},[31646],[31648,218551,218552],{"xmlns":31650},[31652,218553,218554,218632],{},[31655,218555,218556,218568,218570,218572,218574,218576,218578,218580,218582,218584,218586,218588,218590,218592,218594,218596,218598,218600,218602,218604,218606,218608,218610,218612,218614,218626,218628,218630],{},[31743,218557,218558,218560],{},[31658,218559,67467],{},[31655,218561,218562,218564,218566],{},[31658,218563,31660],{},[31662,218565,223],{},[31749,218567,640],{},[31662,218569,202],{},[31658,218571,56084],{},[31662,218573,32682],{},[31662,218575,15972],{"stretchy":702},[31658,218577,33350],{},[31658,218579,1697],{},[31658,218581,43383],{},[31658,218583,94628],{},[31658,218585,31660],{},[31658,218587,46900],{},[31662,218589,31960],{"separator":758},[31658,218591,32679],{},[31662,218593,32682],{},[31662,218595,15972],{"stretchy":702},[31658,218597,33206],{},[31658,218599,1697],{},[31658,218601,43383],{},[31658,218603,94628],{},[31658,218605,31660],{},[31658,218607,46900],{},[31662,218609,31960],{"separator":758},[31662,218611,456],{"stretchy":702},[31658,218613,217691],{},[31743,218615,218616,218618],{},[31658,218617,94675],{},[31655,218619,218620,218622,218624],{},[31658,218621,31660],{},[31662,218623,223],{},[31749,218625,640],{},[31662,218627,2557],{"stretchy":702},[31662,218629,32712],{"stretchy":702},[31662,218631,32712],{"stretchy":702},[31665,218633,217712],{"encoding":31667},[34,218635,218637,218701],{"className":218636,"ariaHidden":758},[31672],[34,218638,218640,218643,218692,218695,218698],{"className":218639},[31676],[34,218641],{"className":218642,"style":70992},[31680],[34,218644,218646,218649],{"className":218645},[31685],[34,218647,67467],{"className":218648,"style":67482},[31685,31686],[34,218650,218652],{"className":218651},[31792],[34,218653,218655,218684],{"className":218654},[31796,31797],[34,218656,218658,218681],{"className":218657},[31801],[34,218659,218661],{"className":218660,"style":31806},[31805],[34,218662,218663,218666],{"style":217742},[34,218664],{"className":218665,"style":31814},[31813],[34,218667,218669],{"className":218668},[31818,31819,31820,31821],[34,218670,218672,218675,218678],{"className":218671},[31685,31821],[34,218673,31660],{"className":218674},[31685,31686,31821],[34,218676,223],{"className":218677},[32024,31821],[34,218679,640],{"className":218680},[31685,31821],[34,218682,31829],{"className":218683},[31828],[34,218685,218687],{"className":218686},[31801],[34,218688,218690],{"className":218689,"style":32037},[31805],[34,218691],{},[34,218693],{"className":218694,"style":31846},[31845],[34,218696,202],{"className":218697},[31850],[34,218699],{"className":218700,"style":31846},[31845],[34,218702,218704,218707,218710,218713,218716,218719,218722,218725,218728,218731,218734,218737,218740,218743,218746,218749,218752,218755,218758,218761,218764,218767,218770,218773,218822],{"className":218703},[31676],[34,218705],{"className":218706,"style":31774},[31680],[34,218708,56084],{"className":218709},[32767],[34,218711,15972],{"className":218712},[31782],[34,218714,33350],{"className":218715,"style":33389},[31685,31686],[34,218717,1697],{"className":218718},[31685,31686],[34,218720,43383],{"className":218721},[31685,31686],[34,218723,94628],{"className":218724},[31685,31686],[34,218726,31660],{"className":218727},[31685,31686],[34,218729,46900],{"className":218730},[31685,31686],[34,218732,31960],{"className":218733},[32119],[34,218735],{"className":218736,"style":32123},[31845],[34,218738,32679],{"className":218739},[32767],[34,218741,15972],{"className":218742},[31782],[34,218744,33206],{"className":218745},[31685,31686],[34,218747,1697],{"className":218748},[31685,31686],[34,218750,43383],{"className":218751},[31685,31686],[34,218753,94628],{"className":218754},[31685,31686],[34,218756,31660],{"className":218757},[31685,31686],[34,218759,46900],{"className":218760},[31685,31686],[34,218762,31960],{"className":218763},[32119],[34,218765],{"className":218766,"style":32123},[31845],[34,218768,456],{"className":218769},[31782],[34,218771,217691],{"className":218772,"style":217853},[31685,31686],[34,218774,218776,218779],{"className":218775},[31685],[34,218777,94675],{"className":218778,"style":106455},[31685,31686],[34,218780,218782],{"className":218781},[31792],[34,218783,218785,218814],{"className":218784},[31796,31797],[34,218786,218788,218811],{"className":218787},[31801],[34,218789,218791],{"className":218790,"style":31806},[31805],[34,218792,218793,218796],{"style":206422},[34,218794],{"className":218795,"style":31814},[31813],[34,218797,218799],{"className":218798},[31818,31819,31820,31821],[34,218800,218802,218805,218808],{"className":218801},[31685,31821],[34,218803,31660],{"className":218804},[31685,31686,31821],[34,218806,223],{"className":218807},[32024,31821],[34,218809,640],{"className":218810},[31685,31821],[34,218812,31829],{"className":218813},[31828],[34,218815,218817],{"className":218816},[31801],[34,218818,218820],{"className":218819,"style":32037},[31805],[34,218821],{},[34,218823,217906],{"className":218824},[31690],[34,218826,218828,218841],{"className":218827},[31642],[34,218829,218831],{"className":218830},[31646],[31648,218832,218833],{"xmlns":31650},[31652,218834,218835,218839],{},[31655,218836,218837],{},[31658,218838,217392],{},[31665,218840,218244],{"enc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的作用及取值影响。",[253,218883,214366],{"id":218884},"概念检查-concept-check-1",[15,218886,218887,218888,218958,218959,219029,219030,31901],{},"在 SPN 或 SRT 等算法中，操作系统无法预知用户进程到底要跑多久。因此，必须利用过去的执行历史（",[34,218889,218891,218909],{"className":218890},[31642],[34,218892,218894],{"className":218893},[31646],[31648,218895,218896],{"xmlns":31650},[31652,218897,218898,218906],{},[31655,218899,218900],{},[31743,218901,218902,218904],{},[31658,218903,37400],{},[31658,218905,31660],{},[31665,218907,218908],{"encoding":31667},"T_n",[34,218910,218912],{"className":218911,"ariaHidden":758},[31672],[34,218913,218915,218918],{"className":218914},[31676],[34,218916],{"className":218917,"style":70915},[31680],[34,218919,218921,218924],{"className":218920},[31685],[34,218922,37400],{"className":218923,"style":37422},[31685,31686],[34,218925,218927],{"className":218926},[31792],[34,218928,218930,218950],{"className":218929},[31796,31797],[34,218931,218933,218947],{"className":218932},[31801],[34,218934,218936],{"className":218935,"style":32362},[31805],[34,218937,218938,218941],{"style":214424},[34,218939],{"className":218940,"style":31814},[31813],[34,218942,218944],{"className":218943},[31818,31819,31820,31821],[34,218945,31660],{"className":218946},[31685,31686,31821],[34,218948,31829],{"className":218949},[31828],[34,218951,218953],{"className":218952},[31801],[34,218954,218956],{"className":218955,"style":31836},[31805],[34,218957],{}," 为第 n 次实际执行时间，",[34,218960,218962,218980],{"className":218961},[31642],[34,218963,218965],{"className":218964},[31646],[31648,218966,218967],{"xmlns":31650},[31652,218968,218969,218977],{},[31655,218970,218971],{},[31743,218972,218973,218975],{},[31658,218974,94675],{},[31658,218976,31660],{},[31665,218978,218979],{"encoding":31667},"S_n",[34,218981,218983],{"className":218982,"ariaHidden":758},[31672],[34,218984,218986,218989],{"className":218985},[31676],[34,218987],{"className":218988,"style":70915},[31680],[34,218990,218992,218995],{"className":218991},[31685],[34,218993,94675],{"className":218994,"style":106455},[31685,31686],[34,218996,218998],{"className":218997},[31792],[34,218999,219001,219021],{"className":219000},[31796,31797],[34,219002,219004,219018],{"className":219003},[31801],[34,219005,219007],{"className":219006,"style":32362},[31805],[34,219008,219009,219012],{"style":206422},[34,219010],{"className":219011,"style":31814},[31813],[34,219013,219015],{"className":219014},[31818,31819,31820,31821],[34,219016,31660],{"className":219017},[31685,31686,31821],[34,219019,31829],{"className":219020},[31828],[34,219022,219024],{"className":219023},[31801],[34,219025,219027],{"className":219026,"style":31836},[31805],[34,219028],{}," 为第 n 次的预测时间）来估算下一次的执行时间（",[34,219031,219033,219056],{"className":219032},[31642],[34,219034,219036],{"className":219035},[31646],[31648,219037,219038],{"xmlns":31650},[31652,219039,219040,219054],{},[31655,219041,219042],{},[31743,219043,219044,219046],{},[31658,219045,94675],{},[31655,219047,219048,219050,219052],{},[31658,219049,31660],{},[31662,219051,223],{},[31749,219053,640],{},[31665,219055,218164],{"encoding":31667},[34,219057,219059],{"className":219058,"ariaHidden":758},[31672],[34,219060,219062,219065],{"className":219061},[31676],[34,219063],{"className":219064,"style":70992},[31680],[34,219066,219068,219071],{"className":219067},[31685],[34,219069,94675],{"className":219070,"style":106455},[31685,31686],[34,219072,219074],{"className":219073},[31792],[34,219075,219077,219106],{"className":219076},[31796,31797],[34,219078,219080,219103],{"className":219079},[31801],[34,219081,219083],{"className":219082,"style":31806},[31805],[34,219084,219085,219088],{"style":206422},[34,219086],{"className":219087,"style":31814},[31813],[34,219089,219091],{"className":219090},[31818,31819,31820,31821],[34,219092,219094,219097,219100],{"className":219093},[31685,31821],[34,219095,31660],{"className":219096},[31685,31686,31821],[34,219098,223],{"className":219099},[32024,31821],[34,219101,640],{"className":219102},[31685,31821],[34,219104,31829],{"className":219105},[31828],[34,219107,219109],{"className":219108},[31801],[34,219110,219112],{"className":219111,"style":32037},[31805],[34,219113],{},[253,219115,219117],{"id":219116},"原理解析-the-why","原理解析 (The 'Why')",[15,219119,219120],{},[2627,219121,219122,219123,219151],{},"1. 参数 ",[34,219124,219126,219139],{"className":219125},[31642],[34,219127,219129],{"className":219128},[31646],[31648,219130,219131],{"xmlns":31650},[31652,219132,219133,219137],{},[31655,219134,219135],{},[31658,219136,217392],{},[31665,219138,218244],{"encoding":31667},[34,219140,219142],{"className":219141,"ariaHidden":758},[31672],[34,219143,219145,219148],{"className":219144},[31676],[34,219146],{"className":219147,"style":32527},[31680],[34,219149,217392],{"className":219150,"style":217496},[31685,31686]," 的作用 (指数平滑因子)",[105,219153,219154,219327],{},[108,219155,219156,47649,219158,219186,219187,219256,219257,219326],{},[2627,219157,30868],{},[34,219159,219161,219174],{"className":219160},[31642],[34,219162,219164],{"className":219163},[31646],[31648,219165,219166],{"xmlns":31650},[31652,219167,219168,219172],{},[31655,219169,219170],{},[31658,219171,217392],{},[31665,219173,218244],{"encoding":31667},[34,219175,219177],{"className":219176,"ariaHidden":758},[31672],[34,219178,219180,219183],{"className":219179},[31676],[34,219181],{"className":219182,"style":32527},[31680],[34,219184,217392],{"className":219185,"style":217496},[31685,31686]," 决定了操作系统在预测时，是更相信“最近一次的实际表现 (",[34,219188,219190,219207],{"className":219189},[31642],[34,219191,219193],{"className":219192},[31646],[31648,219194,219195],{"xmlns":31650},[31652,219196,219197,219205],{},[31655,219198,219199],{},[31743,219200,219201,219203],{},[31658,219202,37400],{},[31658,219204,31660],{},[31665,219206,218908],{"encoding":31667},[34,219208,219210],{"className":219209,"ariaHidden":758},[31672],[34,219211,219213,219216],{"className":219212},[31676],[34,219214],{"className":219215,"style":70915},[31680],[34,219217,219219,219222],{"className":219218},[31685],[34,219220,37400],{"className":219221,"style":37422},[31685,31686],[34,219223,219225],{"className":219224},[31792],[34,219226,219228,219248],{"className":219227},[31796,31797],[34,219229,219231,219245],{"className":219230},[31801],[34,219232,219234],{"className":219233,"style":32362},[31805],[34,219235,219236,219239],{"style":214424},[34,219237],{"className":219238,"style":31814},[31813],[34,219240,219242],{"className":219241},[31818,31819,31820,31821],[34,219243,31660],{"className":219244},[31685,31686,31821],[34,219246,31829],{"className":219247},[31828],[34,219249,219251],{"className":219250},[31801],[34,219252,219254],{"className":219253,"style":31836},[31805],[34,219255],{},")”，还是更相信“长期的历史平均 (",[34,219258,219260,219277],{"className":219259},[31642],[34,219261,219263],{"className":219262},[31646],[31648,219264,219265],{"xmlns":31650},[31652,219266,219267,219275],{},[31655,219268,219269],{},[31743,219270,219271,219273],{},[31658,219272,94675],{},[31658,219274,31660],{},[31665,219276,218979],{"encoding":31667},[34,219278,219280],{"className":219279,"ariaHidden":758},[31672],[34,219281,219283,219286],{"className":219282},[31676],[34,219284],{"className":219285,"style":70915},[31680],[34,219287,219289,219292],{"className":219288},[31685],[34,219290,94675],{"className":219291,"style":106455},[31685,31686],[34,219293,219295],{"className":219294},[31792],[34,219296,219298,219318],{"className":219297},[31796,31797],[34,219299,219301,219315],{"className":219300},[31801],[34,219302,219304],{"className":219303,"style":32362},[31805],[34,219305,219306,219309],{"style":206422},[34,219307],{"className":219308,"style":31814},[31813],[34,219310,219312],{"className":219311},[31818,31819,31820,31821],[34,219313,31660],{"className":219314},[31685,31686,31821],[34,219316,31829],{"className":219317},[31828],[34,219319,219321],{"className":219320},[31801],[34,219322,219324],{"className":219323,"style":31836},[31805],[34,219325],{},")”。",[108,219328,219329,219332],{},[2627,219330,219331],{},"高低取值的影响：",[105,219333,219334,219369],{},[108,219335,219336,219368],{},[2627,219337,219338,219339,219367],{},"较高的 ",[34,219340,219342,219355],{"className":219341},[31642],[34,219343,219345],{"className":219344},[31646],[31648,219346,219347],{"xmlns":31650},[31652,219348,219349,219353],{},[31655,219350,219351],{},[31658,219352,217392],{},[31665,219354,218244],{"encoding":31667},[34,219356,219358],{"className":219357,"ariaHidden":758},[31672],[34,219359,219361,219364],{"className":219360},[31676],[34,219362],{"className":219363,"style":32527},[31680],[34,219365,217392],{"className":219366,"style":217496},[31685,31686]," (接近 1)："," 系统对进程最近的变化反应极快（比如进程突然从 CPU 密集型变成了 I\u002FO 密集型）。但缺点是，如果只是一次偶然的波动，预测值会产生剧烈的波动（抖动）。",[108,219370,219371,219403],{},[2627,219372,219373,219374,219402],{},"较低的 ",[34,219375,219377,219390],{"className":219376},[31642],[34,219378,219380],{"className":219379},[31646],[31648,219381,219382],{"xmlns":31650},[31652,219383,219384,219388],{},[31655,219385,219386],{},[31658,219387,217392],{},[31665,219389,218244],{"encoding":31667},[34,219391,219393],{"className":219392,"ariaHidden":758},[31672],[34,219394,219396,219399],{"className":219395},[31676],[34,219397],{"className":219398,"style":32527},[31680],[34,219400,217392],{"className":219401,"style":217496},[31685,31686]," (接近 0)："," 系统更看重长期历史，预测曲线非常平滑。缺点是对突发变化的适应速度很慢。",[15,219405,219406],{},[2627,219407,219408,219409,219437],{},"2. 参数 ",[34,219410,219412,219425],{"className":219411},[31642],[34,219413,219415],{"className":219414},[31646],[31648,219416,219417],{"xmlns":31650},[31652,219418,219419,219423],{},[31655,219420,219421],{},[31658,219422,217691],{},[31665,219424,218273],{"encoding":31667},[34,219426,219428],{"className":219427,"ariaHidden":758},[31672],[34,219429,219431,219434],{"className":219430},[31676],[34,219432],{"className":219433,"style":43217},[31680],[34,219435,217691],{"className":219436,"style":217853},[31685,31686]," 的作用 (延迟方差\u002F安全冗余因子)",[105,219439,219440,219642],{},[108,219441,219442,47649,219444,219528,219529,347,219613,219641],{},[2627,219443,30868],{},[34,219445,219447,219470],{"className":219446},[31642],[34,219448,219450],{"className":219449},[31646],[31648,219451,219452],{"xmlns":31650},[31652,219453,219454,219468],{},[31655,219455,219456],{},[31743,219457,219458,219460],{},[31658,219459,94675],{},[31655,219461,219462,219464,219466],{},[31658,219463,31660],{},[31662,219465,223],{},[31749,219467,640],{},[31665,219469,218164],{"encoding":31667},[34,219471,219473],{"className":219472,"ariaHidden":758},[31672],[34,219474,219476,219479],{"className":219475},[31676],[34,219477],{"className":219478,"style":70992},[31680],[34,219480,219482,219485],{"className":219481},[31685],[34,219483,94675],{"className":219484,"style":106455},[31685,31686],[34,219486,219488],{"className":219487},[31792],[34,219489,219491,219520],{"className":219490},[31796,31797],[34,219492,219494,219517],{"className":219493},[31801],[34,219495,219497],{"className":219496,"style":31806},[31805],[34,219498,219499,219502],{"style":206422},[34,219500],{"className":219501,"style":31814},[31813],[34,219503,219505],{"className":219504},[31818,31819,31820,31821],[34,219506,219508,219511,219514],{"className":219507},[31685,31821],[34,219509,31660],{"className":219510},[31685,31686,31821],[34,219512,223],{"className":219513},[32024,31821],[34,219515,640],{"className":219516},[31685,31821],[34,219518,31829],{"className":219519},[31828],[34,219521,219523],{"className":219522},[31801],[34,219524,219526],{"className":219525,"style":32037},[31805],[34,219527],{}," 算出来仅仅是一个“平均预测值”。这意味着，在实际运行中，大概有 50% 的概率进程的真实时间会超过 ",[34,219530,219532,219555],{"className":219531},[31642],[34,219533,219535],{"className":219534},[31646],[31648,219536,219537],{"xmlns":31650},[31652,219538,219539,219553],{},[31655,219540,219541],{},[31743,219542,219543,219545],{},[31658,219544,94675],{},[31655,219546,219547,219549,219551],{},[31658,219548,31660],{},[31662,219550,223],{},[31749,219552,640],{},[31665,219554,218164],{"encoding":31667},[34,219556,219558],{"className":219557,"ariaHidden":758},[31672],[34,219559,219561,219564],{"className":219560},[31676],[34,219562],{"className":219563,"style":70992},[31680],[34,219565,219567,219570],{"className":219566},[31685],[34,219568,94675],{"className":219569,"style":106455},[31685,31686],[34,219571,219573],{"className":219572},[31792],[34,219574,219576,219605],{"className":219575},[31796,31797],[34,219577,219579,219602],{"className":219578},[31801],[34,219580,219582],{"className":219581,"style":31806},[31805],[34,219583,219584,219587],{"style":206422},[34,219585],{"className":219586,"style":31814},[31813],[34,219588,219590],{"className":219589},[31818,31819,31820,31821],[34,219591,219593,219596,219599],{"className":219592},[31685,31821],[34,219594,31660],{"className":219595},[31685,31686,31821],[34,219597,223],{"className":219598},[32024,31821],[34,219600,640],{"className":219601},[31685,31821],[34,219603,31829],{"className":219604},[31828],[34,219606,219608],{"className":219607},[31801],[34,219609,219611],{"className":219610,"style":32037},[31805],[34,219612],{},[34,219614,219616,219629],{"className":219615},[31642],[34,219617,219619],{"className":219618},[31646],[31648,219620,219621],{"xmlns":31650},[31652,219622,219623,219627],{},[31655,219624,219625],{},[31658,219626,217691],{},[31665,219628,218273],{"encoding":31667},[34,219630,219632],{"className":219631,"ariaHidden":758},[31672],[34,219633,219635,219638],{"className":219634},[31676],[34,219636],{"className":219637,"style":43217},[31680],[34,219639,217691],{"className":219640,"style":217853},[31685,31686]," 的作用是作为一个乘数因子（通常在 1.3 到 2.0 之间），给预测值加上一层“安全冗余”。",[108,219643,219644,219646],{},[2627,219645,219331],{},[105,219647,219648,219767],{},[108,219649,219650,219681,219682,219766],{},[2627,219651,219338,219652,219680],{},[34,219653,219655,219668],{"className":219654},[31642],[34,219656,219658],{"className":219657},[31646],[31648,219659,219660],{"xmlns":31650},[31652,219661,219662,219666],{},[31655,219663,219664],{},[31658,219665,217691],{},[31665,219667,218273],{"encoding":31667},[34,219669,219671],{"className":219670,"ariaHidden":758},[31672],[34,219672,219674,219677],{"className":219673},[31676],[34,219675],{"className":219676,"style":43217},[31680],[34,219678,217691],{"className":219679,"style":217853},[31685,31686]," (如 2.0)："," 预测值 ",[34,219683,219685,219708],{"className":219684},[31642],[34,219686,219688],{"className":219687},[31646],[31648,219689,219690],{"xmlns":31650},[31652,219691,219692,219706],{},[31655,219693,219694],{},[31743,219695,219696,219698],{},[31658,219697,67467],{},[31655,219699,219700,219702,219704],{},[31658,219701,31660],{},[31662,219703,223],{},[31749,219705,640],{},[31665,219707,218074],{"encoding":31667},[34,219709,219711],{"className":219710,"ariaHidden":758},[31672],[34,219712,219714,219717],{"className":219713},[31676],[34,219715],{"className":219716,"style":70992},[31680],[34,219718,219720,219723],{"className":219719},[31685],[34,219721,67467],{"className":219722,"style":67482},[31685,31686],[34,219724,219726],{"className":219725},[31792],[34,219727,219729,219758],{"className":219728},[31796,31797],[34,219730,219732,219755],{"className":219731},[31801],[34,219733,219735],{"className":219734,"style":31806},[31805],[34,219736,219737,219740],{"style":217742},[34,219738],{"className":219739,"style":31814},[31813],[34,219741,219743],{"className":219742},[31818,31819,31820,31821],[34,219744,219746,219749,219752],{"className":219745},[31685,31821],[34,219747,31660],{"className":219748},[31685,31686,31821],[34,219750,223],{"className":219751},[32024,31821],[34,219753,640],{"className":219754},[31685,31821],[34,219756,31829],{"className":219757},[31828],[34,219759,219761],{"className":219760},[31801],[34,219762,219764],{"className":219763,"style":32037},[31805],[34,219765],{}," 会被放大。这减少了“进程未执行完就被系统误判”的情况，但也削弱了系统挑选“最短进程”的敏锐度。",[108,219768,219769,219800],{},[2627,219770,219373,219771,219799],{},[34,219772,219774,219787],{"className":219773},[31642],[34,219775,219777],{"className":219776},[31646],[31648,219778,219779],{"xmlns":31650},[31652,219780,219781,219785],{},[31655,219782,219783],{},[31658,219784,217691],{},[31665,219786,218273],{"encoding":31667},[34,219788,219790],{"className":219789,"ariaHidden":758},[31672],[34,219791,219793,219796],{"className":219792},[31676],[34,219794],{"className":219795,"style":43217},[31680],[34,219797,217691],{"className":219798,"style":217853},[31685,31686]," (如 1.0 或更低)："," 预测值紧贴平均值，系统能更快速地适应实际观察时间，但会导致预测误差带来的波动更大。",[253,219802,217337],{"id":219803},"常见易错点-common-pitfalls-1",[15,219805,30270,219806,219834,219835,347,219936,219964],{},[34,219807,219809,219822],{"className":219808},[31642],[34,219810,219812],{"className":219811},[31646],[31648,219813,219814],{"xmlns":31650},[31652,219815,219816,219820],{},[31655,219817,219818],{},[31658,219819,217392],{},[31665,219821,218244],{"encoding":31667},[34,219823,219825],{"className":219824,"ariaHidden":758},[31672],[34,219826,219828,219831],{"className":219827},[31676],[34,219829],{"className":219830,"style":32527},[31680],[34,219832,217392],{"className":219833,"style":217496},[31685,31686]," 的作用记反了。记住口诀：",[2627,219836,219837,219865,219866,219935],{},[34,219838,219840,219853],{"className":219839},[31642],[34,219841,219843],{"className":219842},[31646],[31648,219844,219845],{"xmlns":31650},[31652,219846,219847,219851],{},[31655,219848,219849],{},[31658,219850,217392],{},[31665,219852,218244],{"encoding":31667},[34,219854,219856],{"className":219855,"ariaHidden":758},[31672],[34,219857,219859,219862],{"className":219858},[31676],[34,219860],{"className":219861,"style":32527},[31680],[34,219863,217392],{"className":219864,"style":217496},[31685,31686]," 永远与最新的实际值 ",[34,219867,219869,219886],{"className":219868},[31642],[34,219870,219872],{"className":219871},[31646],[31648,219873,219874],{"xmlns":31650},[31652,219875,219876,219884],{},[31655,219877,219878],{},[31743,219879,219880,219882],{},[31658,219881,37400],{},[31658,219883,31660],{},[31665,219885,218908],{"encoding":31667},[34,219887,219889],{"className":219888,"ariaHidden":758},[31672],[34,219890,219892,219895],{"className":219891},[31676],[34,219893],{"className":219894,"style":70915},[31680],[34,219896,219898,219901],{"className":219897},[31685],[34,219899,37400],{"className":219900,"style":37422},[31685,31686],[34,219902,219904],{"className":219903},[31792],[34,219905,219907,219927],{"className":219906},[31796,31797],[34,219908,219910,219924],{"className":219909},[31801],[34,219911,219913],{"className":219912,"style":32362},[31805],[34,219914,219915,219918],{"style":214424},[34,219916],{"className":219917,"style":31814},[31813],[34,219919,219921],{"className":219920},[31818,31819,31820,31821],[34,219922,31660],{"className":219923},[31685,31686,31821],[34,219925,31829],{"className":219926},[31828],[34,219928,219930],{"className":219929},[31801],[34,219931,219933],{"className":219932,"style":31836},[31805],[34,219934],{}," 绑定",[34,219937,219939,219952],{"className":219938},[31642],[34,219940,219942],{"className":219941},[31646],[31648,219943,219944],{"xmlns":31650},[31652,219945,219946,219950],{},[31655,219947,219948],{},[31658,219949,217392],{},[31665,219951,218244],{"encoding":31667},[34,219953,219955],{"className":219954,"ariaHidden":758},[31672],[34,219956,219958,219961],{"className":219957},[31676],[34,219959],{"className":219960,"style":32527},[31680],[34,219962,217392],{"className":219963,"style":217496},[31685,31686]," 越大，系统越“健忘”，越容易被眼前的数据牵着鼻子走。",[32188,219966],{},[19,219968,219970],{"id":219969},"题目-3",[2627,219971,219972],{},"题目 3",[15,219974,219975],{},[2627,219976,219977,219978,220012,220013,220047],{},"Define residence time ",[34,219979,219981,219997],{"className":219980},[31642],[34,219982,219984],{"className":219983},[31646],[31648,219985,219986],{"xmlns":31650},[31652,219987,219988,219994],{},[31655,219989,219990,219992],{},[31658,219991,37400],{},[31658,219993,34395],{},[31665,219995,219996],{"encoding":31667},"Tr",[34,219998,220000],{"className":219999,"ariaHidden":758},[31672],[34,220001,220003,220006,220009],{"className":220002},[31676],[34,220004],{"className":220005,"style":33218},[31680],[34,220007,37400],{"className":220008,"style":37422},[31685,31686],[34,220010,34395],{"className":220011,"style":34452},[31685,31686]," as the average total time a process spends waiting and being served. Show that for FIFO, with mean service time ",[34,220014,220016,220032],{"className":220015},[31642],[34,220017,220019],{"className":220018},[31646],[31648,220020,220021],{"xmlns":31650},[31652,220022,220023,220029],{},[31655,220024,220025,220027],{},[31658,220026,37400],{},[31658,220028,31747],{},[31665,220030,220031],{"encoding":31667},"Ts",[34,220033,220035],{"className":220034,"ariaHidden":758},[31672],[34,220036,220038,220041,220044],{"className":220037},[31676],[34,220039],{"className":220040,"style":33218},[31680],[34,220042,37400],{"className":220043,"style":37422},[31685,31686],[34,220045,31747],{"className":220046},[31685,31686],", we have:",[15,220049,220050],{},[34,220051,220053,220088],{"className":220052},[31642],[34,220054,220056],{"className":220055},[31646],[31648,220057,220058],{"xmlns":31650},[31652,220059,220060,220085],{},[31655,220061,220062,220064,220066,220068],{},[31658,220063,37400],{},[31658,220065,34395],{},[31662,220067,202],{},[41453,220069,220070,220076],{},[31655,220071,220072,220074],{},[31658,220073,37400],{},[31658,220075,31747],{},[31655,220077,220078,220080,220082],{},[31749,220079,640],{},[31662,220081,32226],{},[31658,220083,220084],{},"ρ",[31665,220086,220087],{"encoding":31667},"Tr = \\frac{Ts}{1 - \\rho}",[34,220089,220091,220112],{"className":220090,"ariaHidden":758},[31672],[34,220092,220094,220097,220100,220103,220106,220109],{"className":220093},[31676],[34,220095],{"className":220096,"style":33218},[31680],[34,220098,37400],{"className":220099,"style":37422},[31685,31686],[34,220101,34395],{"className":220102,"style":34452},[31685,31686],[34,220104],{"className":220105,"style":31846},[31845],[34,220107,202],{"className":220108},[31850],[34,220110],{"className":220111,"style":31846},[31845],[34,220113,220115,220119],{"className":220114},[31676],[34,220116],{"className":220117,"style":220118},[31680],"height:1.3534em;vertical-align:-0.4811em;",[34,220120,220122,220125,220194],{"className":220121},[31685],[34,220123],{"className":220124},[31782,41488],[34,220126,220128],{"className":220127},[41453],[34,220129,220131,220185],{"className":220130},[31796,31797],[34,220132,220134,220182],{"className":220133},[31801],[34,220135,220137,220157,220165],{"className":220136,"style":168045},[31805],[34,220138,220139,220142],{"style":41504},[34,220140],{"className":220141,"style":41508},[31813],[34,220143,220145],{"className":220144},[31818,31819,31820,31821],[34,220146,220148,220151,220154],{"className":220147},[31685,31821],[34,220149,640],{"className":220150},[31685,31821],[34,220152,32226],{"className":220153},[32024,31821],[34,220155,220084],{"className":220156},[31685,31686,31821],[34,220158,220159,220162],{"style":41520},[34,220160],{"className":220161,"style":41508},[31813],[34,220163],{"className":220164,"style":41528},[41527],[34,220166,220167,220170],{"style":46771},[34,220168],{"className":220169,"style":41508},[31813],[34,220171,220173],{"className":220172},[31818,31819,31820,31821],[34,220174,220176,220179],{"className":220175},[31685,31821],[34,220177,37400],{"className":220178,"style":37422},[31685,31686,31821],[34,220180,31747],{"className":220181},[31685,31686,31821],[34,220183,31829],{"className":220184},[31828],[34,220186,220188],{"className":220187},[31801],[34,220189,220192],{"className":220190,"style":220191},[31805],"height:0.4811em;",[34,220193],{},[34,220195],{"className":220196},[31690,41488],[15,220198,220199],{},[2627,220200,220201,220202,220231],{},"where ",[34,220203,220205,220219],{"className":220204},[31642],[34,220206,220208],{"className":220207},[31646],[31648,220209,220210],{"xmlns":31650},[31652,220211,220212,220216],{},[31655,220213,220214],{},[31658,220215,220084],{},[31665,220217,220218],{"encoding":31667},"\\rho",[34,220220,220222],{"className":220221,"ariaHidden":758},[31672],[34,220223,220225,220228],{"className":220224},[31676],[34,220226],{"className":220227,"style":38774},[31680],[34,220229,220084],{"className":220230},[31685,31686]," is utilization.",[15,220233,220234,220236,220237,220307,220308,220336,220337,347],{},[2627,220235,214361],{},"\n证明在 FIFO 调度策略中，若平均服务时间为 ",[34,220238,220240,220258],{"className":220239},[31642],[34,220241,220243],{"className":220242},[31646],[31648,220244,220245],{"xmlns":31650},[31652,220246,220247,220255],{},[31655,220248,220249],{},[31743,220250,220251,220253],{},[31658,220252,37400],{},[31658,220254,31747],{},[31665,220256,220257],{"encoding":31667},"T_s",[34,220259,220261],{"className":220260,"ariaHidden":758},[31672],[34,220262,220264,220267],{"className":220263},[31676],[34,220265],{"className":220266,"style":70915},[31680],[34,220268,220270,220273],{"className":220269},[31685],[34,220271,37400],{"className":220272,"style":37422},[31685,31686],[34,220274,220276],{"className":220275},[31792],[34,220277,220279,220299],{"className":220278},[31796,31797],[34,220280,220282,220296],{"className":220281},[31801],[34,220283,220285],{"className":220284,"style":32362},[31805],[34,220286,220287,220290],{"style":214424},[34,220288],{"className":220289,"style":31814},[31813],[34,220291,220293],{"className":220292},[31818,31819,31820,31821],[34,220294,31747],{"className":220295},[31685,31686,31821],[34,220297,31829],{"className":220298},[31828],[34,220300,220302],{"className":220301},[31801],[34,220303,220305],{"className":220304,"style":31836},[31805],[34,220306],{},"，利用率为 ",[34,220309,220311,220324],{"className":220310},[31642],[34,220312,220314],{"className":220313},[31646],[31648,220315,220316],{"xmlns":31650},[31652,220317,220318,220322],{},[31655,220319,220320],{},[31658,220321,220084],{},[31665,220323,220218],{"encoding":31667},[34,220325,220327],{"className":220326,"ariaHidden":758},[31672],[34,220328,220330,220333],{"className":220329},[31676],[34,220331],{"className":220332,"style":38774},[31680],[34,220334,220084],{"className":220335},[31685,31686],"，则平均周转时间 ",[34,220338,220340,220376],{"className":220339},[31642],[34,220341,220343],{"className":220342},[31646],[31648,220344,220345],{"xmlns":31650},[31652,220346,220347,220373],{},[31655,220348,220349,220355,220357],{},[31743,220350,220351,220353],{},[31658,220352,37400],{},[31658,220354,34395],{},[31662,220356,202],{},[41453,220358,220359,220365],{},[31743,220360,220361,220363],{},[31658,220362,37400],{},[31658,220364,31747],{},[31655,220366,220367,220369,220371],{},[31749,220368,640],{},[31662,220370,32226],{},[31658,220372,220084],{},[31665,220374,220375],{"encoding":31667},"T_r = \\frac{T_s}{1 - \\rho}",[34,220377,220379,220434],{"className":220378,"ariaHidden":758},[31672],[34,220380,220382,220385,220425,220428,220431],{"className":220381},[31676],[34,220383],{"className":220384,"style":70915},[31680],[34,220386,220388,220391],{"className":220387},[31685],[34,220389,37400],{"className":220390,"style":37422},[31685,31686],[34,220392,220394],{"className":220393},[31792],[34,220395,220397,220417],{"className":220396},[31796,31797],[34,220398,220400,220414],{"className":220399},[31801],[34,220401,220403],{"className":220402,"style":32362},[31805],[34,220404,220405,220408],{"style":214424},[34,220406],{"className":220407,"style":31814},[31813],[34,220409,220411],{"className":220410},[31818,31819,31820,31821],[34,220412,34395],{"className":220413,"style":34452},[31685,31686,31821],[34,220415,31829],{"className":220416},[31828],[34,220418,220420],{"className":220419},[31801],[34,220421,220423],{"className":220422,"style":31836},[31805],[34,220424],{},[34,220426],{"className":220427,"style":31846},[31845],[34,220429,202],{"className":220430},[31850],[34,220432],{"className":220433,"style":31846},[31845],[34,220435,220437,220441],{"className":220436},[31676],[34,220438],{"className":220439,"style":220440},[31680],"height:1.3695em;vertical-align:-0.4811em;",[34,220442,220444,220447,220554],{"className":220443},[31685],[34,220445],{"className":220446},[31782,41488],[34,220448,220450],{"className":220449},[41453],[34,220451,220453,220546],{"className":220452},[31796,31797],[34,220454,220456,220543],{"className":220455},[31801],[34,220457,220460,220480,220488],{"className":220458,"style":220459},[31805],"height:0.8884em;",[34,220461,220462,220465],{"style":41504},[34,220463],{"className":220464,"style":41508},[31813],[34,220466,220468],{"className":220467},[31818,31819,31820,31821],[34,220469,220471,220474,220477],{"className":220470},[31685,31821],[34,220472,640],{"className":220473},[31685,31821],[34,220475,32226],{"className":220476},[32024,31821],[34,220478,220084],{"className":220479},[31685,31686,31821],[34,220481,220482,220485],{"style":41520},[34,220483],{"className":220484,"style":41508},[31813],[34,220486],{"className":220487,"style":41528},[41527],[34,220489,220491,220494],{"style":220490},"top:-3.4101em;",[34,220492],{"className":220493,"style":41508},[31813],[34,220495,220497],{"className":220496},[31818,31819,31820,31821],[34,220498,220500],{"className":220499},[31685,31821],[34,220501,220503,220506],{"className":220502},[31685,31821],[34,220504,37400],{"className":220505,"style":37422},[31685,31686,31821],[34,220507,220509],{"className":220508},[31792],[34,220510,220512,220534],{"className":220511},[31796,31797],[34,220513,220515,220531],{"className":220514},[31801],[34,220516,220519],{"className":220517,"style":220518},[31805],"height:0.1645em;",[34,220520,220522,220525],{"style":220521},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[34,220523],{"className":220524,"style":43406},[31813],[34,220526,220528],{"className":220527},[31818,43410,43411,31821],[34,220529,31747],{"className":220530},[31685,31686,31821],[34,220532,31829],{"className":220533},[31828],[34,220535,220537],{"className":220536},[31801],[34,220538,220541],{"className":220539,"style":220540},[31805],"height:0.143em;",[34,220542],{},[34,220544,31829],{"className":220545},[31828],[34,220547,220549],{"className":220548},[31801],[34,220550,220552],{"className":220551,"style":220191},[31805],[34,220553],{},[34,220555],{"className":220556},[31690,41488],[253,220558,214366],{"id":220559},"概念检查-concept-check-2",[105,220561,220562,220568],{},[108,220563,220564,220567],{},[2627,220565,220566],{},"排队论基础："," 这是一个典型的 M\u002FM\u002F1 排队系统模型。",[108,220569,220570,220573],{},[2627,220571,220572],{},"变量定义：",[105,220574,220575,220745,220776],{},[108,220576,220577,220646,220647,220675,220676,31901],{},[34,220578,220580,220597],{"className":220579},[31642],[34,220581,220583],{"className":220582},[31646],[31648,220584,220585],{"xmlns":31650},[31652,220586,220587,220595],{},[31655,220588,220589],{},[31743,220590,220591,220593],{},[31658,220592,37400],{},[31658,220594,34395],{},[31665,220596,214394],{"encoding":31667},[34,220598,220600],{"className":220599,"ariaHidden":758},[31672],[34,220601,220603,220606],{"className":220602},[31676],[34,220604],{"className":220605,"style":70915},[31680],[34,220607,220609,220612],{"className":220608},[31685],[34,220610,37400],{"className":220611,"style":37422},[31685,31686],[34,220613,220615],{"className":220614},[31792],[34,220616,220618,220638],{"className":220617},[31796,31797],[34,220619,220621,220635],{"className":220620},[31801],[34,220622,220624],{"className":220623,"style":32362},[31805],[34,220625,220626,220629],{"style":214424},[34,220627],{"className":220628,"style":31814},[31813],[34,220630,220632],{"className":220631},[31818,31819,31820,31821],[34,220633,34395],{"className":220634,"style":34452},[31685,31686,31821],[34,220636,31829],{"className":220637},[31828],[34,220639,220641],{"className":220640},[31801],[34,220642,220644],{"className":220643,"style":31836},[31805],[34,220645],{}," (Residence Time \u002F Turnaround Time)：进程在系统中花费的总时间（包含排队等待时间 ",[34,220648,220650,220663],{"className":220649},[31642],[34,220651,220653],{"className":220652},[31646],[31648,220654,220655],{"xmlns":31650},[31652,220656,220657,220661],{},[31655,220658,220659],{},[31658,220660,56852],{},[31665,220662,56852],{"encoding":31667},[34,220664,220666],{"className":220665,"ariaHidden":758},[31672],[34,220667,220669,220672],{"className":220668},[31676],[34,220670],{"className":220671,"style":33218},[31680],[34,220673,56852],{"className":220674,"style":37422},[31685,31686]," 和被服务的实际执行时间 ",[34,220677,220679,220696],{"className":220678},[31642],[34,220680,220682],{"className":220681},[31646],[31648,220683,220684],{"xmlns":31650},[31652,220685,220686,220694],{},[31655,220687,220688],{},[31743,220689,220690,220692],{},[31658,220691,37400],{},[31658,220693,31747],{},[31665,220695,220257],{"encoding":31667},[34,220697,220699],{"className":220698,"ariaHidden":758},[31672],[34,220700,220702,220705],{"className":220701},[31676],[34,220703],{"className":220704,"style":70915},[31680],[34,220706,220708,220711],{"className":220707},[31685],[34,220709,37400],{"className":220710,"style":37422},[31685,31686],[34,220712,220714],{"className":220713},[31792],[34,220715,220717,220737],{"className":220716},[31796,31797],[34,220718,220720,220734],{"className":220719},[31801],[34,220721,220723],{"className":220722,"style":32362},[31805],[34,220724,220725,220728],{"style":214424},[34,220726],{"className":220727,"style":31814},[31813],[34,220729,220731],{"className":220730},[31818,31819,31820,31821],[34,220732,31747],{"className":220733},[31685,31686,31821],[34,220735,31829],{"className":220736},[31828],[34,220738,220740],{"className":220739},[31801],[34,220741,220743],{"className":220742,"style":31836},[31805],[34,220744],{},[108,220746,220747,220775],{},[34,220748,220750,220763],{"className":220749},[31642],[34,220751,220753],{"className":220752},[31646],[31648,220754,220755],{"xmlns":31650},[31652,220756,220757,220761],{},[31655,220758,220759],{},[31658,220760,220084],{},[31665,220762,220218],{"encoding":31667},[34,220764,220766],{"className":220765,"ariaHidden":758},[31672],[34,220767,220769,220772],{"className":220768},[31676],[34,220770],{"className":220771,"style":38774},[31680],[34,220773,220084],{"className":220774},[31685,31686]," (Utilization)：系统的处理器利用率，表示 CPU 有多大概率在忙碌。",[108,220777,220778,220808,220809,347],{},[34,220779,220781,220796],{"className":220780},[31642],[34,220782,220784],{"className":220783},[31646],[31648,220785,220786],{"xmlns":31650},[31652,220787,220788,220793],{},[31655,220789,220790],{},[31658,220791,220792],{},"λ",[31665,220794,220795],{"encoding":31667},"\\lambda",[34,220797,220799],{"className":220798,"ariaHidden":758},[31672],[34,220800,220802,220805],{"className":220801},[31676],[34,220803],{"className":220804,"style":31681},[31680],[34,220806,220792],{"className":220807},[31685,31686]," (Arrival Rate)：进程到达的速率。根据定义，",[34,220810,220812,220836],{"className":220811},[31642],[34,220813,220815],{"className":220814},[31646],[31648,220816,220817],{"xmlns":31650},[31652,220818,220819,220833],{},[31655,220820,220821,220823,220825,220827],{},[31658,220822,220084],{},[31662,220824,202],{},[31658,220826,220792],{},[31743,220828,220829,220831],{},[31658,220830,37400],{},[31658,220832,31747],{},[31665,220834,220835],{"encoding":31667},"\\rho = \\lambda T_s",[34,220837,220839,220857],{"className":220838,"ariaHidden":758},[31672],[34,220840,220842,220845,220848,220851,220854],{"className":220841},[31676],[34,220843],{"className":220844,"style":38774},[31680],[34,220846,220084],{"className":220847},[31685,31686],[34,220849],{"className":220850,"style":31846},[31845],[34,220852,202],{"className":220853},[31850],[34,220855],{"className":220856,"style":31846},[31845],[34,220858,220860,220863,220866],{"className":220859},[31676],[34,220861],{"className":220862,"style":33814},[31680],[34,220864,220792],{"className":220865},[31685,31686],[34,220867,220869,220872],{"className":220868},[31685],[34,220870,37400],{"className":220871,"style":37422},[31685,31686],[34,220873,220875],{"className":220874},[31792],[34,220876,220878,220898],{"className":220877},[31796,31797],[34,220879,220881,220895],{"className":220880},[31801],[34,220882,220884],{"className":220883,"style":32362},[31805],[34,220885,220886,220889],{"style":214424},[34,220887],{"className":220888,"style":31814},[31813],[34,220890,220892],{"className":220891},[31818,31819,31820,31821],[34,220893,31747],{"className":220894},[31685,31686,31821],[34,220896,31829],{"className":220897},[31828],[34,220899,220901],{"className":220900},[31801],[34,220902,220904],{"className":220903,"style":31836},[31805],[34,220905],{},[253,220907,220909],{"id":220908},"原理解析与推导步骤-the-why","原理解析与推导步骤 (The 'Why')",[15,220911,220912],{},[2627,220913,220914],{},"证明过程：",[2230,220916,220917,221246,221522,221770,222251],{},[108,220918,220919,220922,220923,220992,220993,221021,221022,39984,221091],{},[2627,220920,220921],{},"分解总时间：","\n一个进程进入系统后的总周转时间 ",[34,220924,220926,220943],{"className":220925},[31642],[34,220927,220929],{"className":220928},[31646],[31648,220930,220931],{"xmlns":31650},[31652,220932,220933,220941],{},[31655,220934,220935],{},[31743,220936,220937,220939],{},[31658,220938,37400],{},[31658,220940,34395],{},[31665,220942,214394],{"encoding":31667},[34,220944,220946],{"className":220945,"ariaHidden":758},[31672],[34,220947,220949,220952],{"className":220948},[31676],[34,220950],{"className":220951,"style":70915},[31680],[34,220953,220955,220958],{"className":220954},[31685],[34,220956,37400],{"className":220957,"style":37422},[31685,31686],[34,220959,220961],{"className":220960},[31792],[34,220962,220964,220984],{"className":220963},[31796,31797],[34,220965,220967,220981],{"className":220966},[31801],[34,220968,220970],{"className":220969,"style":32362},[31805],[34,220971,220972,220975],{"style":214424},[34,220973],{"className":220974,"style":31814},[31813],[34,220976,220978],{"className":220977},[31818,31819,31820,31821],[34,220979,34395],{"className":220980,"style":34452},[31685,31686,31821],[34,220982,31829],{"className":220983},[31828],[34,220985,220987],{"className":220986},[31801],[34,220988,220990],{"className":220989,"style":31836},[31805],[34,220991],{},"，等于它在就绪队列中排队的等待时间 ",[34,220994,220996,221009],{"className":220995},[31642],[34,220997,220999],{"className":220998},[31646],[31648,221000,221001],{"xmlns":31650},[31652,221002,221003,221007],{},[31655,221004,221005],{},[31658,221006,56852],{},[31665,221008,56852],{"encoding":31667},[34,221010,221012],{"className":221011,"ariaHidden":758},[31672],[34,221013,221015,221018],{"className":221014},[31676],[34,221016],{"className":221017,"style":33218},[31680],[34,221019,56852],{"className":221020,"style":37422},[31685,31686],"，加上它自己实际占用 CPU 的服务时间 ",[34,221023,221025,221042],{"className":221024},[31642],[34,221026,221028],{"className":221027},[31646],[31648,221029,221030],{"xmlns":31650},[31652,221031,221032,221040],{},[31655,221033,221034],{},[31743,221035,221036,221038],{},[31658,221037,37400],{},[31658,221039,31747],{},[31665,221041,220257],{"encoding":31667},[34,221043,221045],{"className":221044,"ariaHidden":758},[31672],[34,221046,221048,221051],{"className":221047},[31676],[34,221049],{"className":221050,"style":70915},[31680],[34,221052,221054,221057],{"className":221053},[31685],[34,221055,37400],{"className":221056,"style":37422},[31685,31686],[34,221058,221060],{"className":221059},[31792],[34,221061,221063,221083],{"className":221062},[31796,31797],[34,221064,221066,221080],{"className":221065},[31801],[34,221067,221069],{"className":221068,"style":32362},[31805],[34,221070,221071,221074],{"style":214424},[34,221072],{"className":221073,"style":31814},[31813],[34,221075,221077],{"className":221076},[31818,31819,31820,31821],[34,221078,31747],{"className":221079},[31685,31686,31821],[34,221081,31829],{"className":221082},[31828],[34,221084,221086],{"className":221085},[31801],[34,221087,221089],{"className":221088,"style":31836},[31805],[34,221090],{},[34,221092,221094,221124],{"className":221093},[31642],[34,221095,221097],{"className":221096},[31646],[31648,221098,221099],{"xmlns":31650},[31652,221100,221101,221121],{},[31655,221102,221103,221109,221111,221113,221115],{},[31743,221104,221105,221107],{},[31658,221106,37400],{},[31658,221108,34395],{},[31662,221110,202],{},[31658,221112,56852],{},[31662,221114,223],{},[31743,221116,221117,221119],{},[31658,221118,37400],{},[31658,221120,31747],{},[31665,221122,221123],{"encoding":31667},"T_r = W + T_s",[34,221125,221127,221182,221200],{"className":221126,"ariaHidden":758},[31672],[34,221128,221130,221133,221173,221176,221179],{"className":221129},[31676],[34,221131],{"className":221132,"style":70915},[31680],[34,221134,221136,221139],{"className":221135},[31685],[34,221137,37400],{"className":221138,"style":37422},[31685,31686],[34,221140,221142],{"className":221141},[31792],[34,221143,221145,221165],{"className":221144},[31796,31797],[34,221146,221148,221162],{"className":221147},[31801],[34,221149,221151],{"className":221150,"style":32362},[31805],[34,221152,221153,221156],{"style":214424},[34,221154],{"className":221155,"style":31814},[31813],[34,221157,221159],{"className":221158},[31818,31819,31820,31821],[34,221160,34395],{"className":221161,"style":34452},[31685,31686,31821],[34,221163,31829],{"className":221164},[31828],[34,221166,221168],{"className":221167},[31801],[34,221169,221171],{"className":221170,"style":31836},[31805],[34,221172],{},[34,221174],{"className":221175,"style":31846},[31845],[34,221177,202],{"className":221178},[31850],[34,221180],{"className":221181,"style":31846},[31845],[34,221183,221185,221188,221191,221194,221197],{"className":221184},[31676],[34,221186],{"className":221187,"style":34944},[31680],[34,221189,56852],{"className":221190,"style":37422},[31685,31686],[34,221192],{"className":221193,"style":32604},[31845],[34,221195,223],{"className":221196},[32024],[34,221198],{"className":221199,"style":32604},[31845],[34,221201,221203,221206],{"className":221202},[31676],[34,221204],{"className":221205,"style":70915},[31680],[34,221207,221209,221212],{"className":221208},[31685],[34,221210,37400],{"className":221211,"style":37422},[31685,31686],[34,221213,221215],{"className":221214},[31792],[34,221216,221218,221238],{"className":221217},[31796,31797],[34,221219,221221,221235],{"className":221220},[31801],[34,221222,221224],{"className":221223,"style":32362},[31805],[34,221225,221226,221229],{"style":214424},[34,221227],{"className":221228,"style":31814},[31813],[34,221230,221232],{"className":221231},[31818,31819,31820,31821],[34,221233,31747],{"className":221234},[31685,31686,31821],[34,221236,31829],{"className":221237},[31828],[34,221239,221241],{"className":221240},[31801],[34,221242,221244],{"className":221243,"style":31836},[31805],[34,221245],{},[108,221247,221248,221279,221280,221308,221309,221378,221379,221407,221408],{},[2627,221249,221250,221251,2664],{},"计算等待时间 ",[34,221252,221254,221267],{"className":221253},[31642],[34,221255,221257],{"className":221256},[31646],[31648,221258,221259],{"xmlns":31650},[31652,221260,221261,221265],{},[31655,221262,221263],{},[31658,221264,56852],{},[31665,221266,56852],{"encoding":31667},[34,221268,221270],{"className":221269,"ariaHidden":758},[31672],[34,221271,221273,221276],{"className":221272},[31676],[34,221274],{"className":221275,"style":33218},[31680],[34,221277,56852],{"className":221278,"style":37422},[31685,31686],"\n当一个新进程到达时，它需要等待排在它前面的所有进程都被处理完。根据排队论的 PASTA 定理（泊松到达见时间平均），新到达的进程看到的系统内平均进程数（包括正在排队和正在执行的）就是整个系统的平均进程数，设为 ",[34,221281,221283,221296],{"className":221282},[31642],[34,221284,221286],{"className":221285},[31646],[31648,221287,221288],{"xmlns":31650},[31652,221289,221290,221294],{},[31655,221291,221292],{},[31658,221293,38759],{},[31665,221295,38759],{"encoding":31667},[34,221297,221299],{"className":221298,"ariaHidden":758},[31672],[34,221300,221302,221305],{"className":221301},[31676],[34,221303],{"className":221304,"style":38774},[31680],[34,221306,38759],{"className":221307,"style":32062},[31685,31686],"。\n因为是 FIFO 且处于平衡状态，每个进程的平均处理时间都是 ",[34,221310,221312,221329],{"className":221311},[31642],[34,221313,221315],{"className":221314},[31646],[31648,221316,221317],{"xmlns":31650},[31652,221318,221319,221327],{},[31655,221320,221321],{},[31743,221322,221323,221325],{},[31658,221324,37400],{},[31658,221326,31747],{},[31665,221328,220257],{"encoding":31667},[34,221330,221332],{"className":221331,"ariaHidden":758},[31672],[34,221333,221335,221338],{"className":221334},[31676],[34,221336],{"className":221337,"style":70915},[31680],[34,221339,221341,221344],{"className":221340},[31685],[34,221342,37400],{"className":221343,"style":37422},[31685,31686],[34,221345,221347],{"className":221346},[31792],[34,221348,221350,221370],{"className":221349},[31796,31797],[34,221351,221353,221367],{"className":221352},[31801],[34,221354,221356],{"className":221355,"style":32362},[31805],[34,221357,221358,221361],{"style":214424},[34,221359],{"className":221360,"style":31814},[31813],[34,221362,221364],{"className":221363},[31818,31819,31820,31821],[34,221365,31747],{"className":221366},[31685,31686,31821],[34,221368,31829],{"className":221369},[31828],[34,221371,221373],{"className":221372},[31801],[34,221374,221376],{"className":221375,"style":31836},[31805],[34,221377],{},"。所以，清空前面 ",[34,221380,221382,221395],{"className":221381},[31642],[34,221383,221385],{"className":221384},[31646],[31648,221386,221387],{"xmlns":31650},[31652,221388,221389,221393],{},[31655,221390,221391],{},[31658,221392,38759],{},[31665,221394,38759],{"encoding":31667},[34,221396,221398],{"className":221397,"ariaHidden":758},[31672],[34,221399,221401,221404],{"className":221400},[31676],[34,221402],{"className":221403,"style":38774},[31680],[34,221405,38759],{"className":221406,"style":32062},[31685,31686]," 个进程所需的总时间就是：\n",[34,221409,221411,221437],{"className":221410},[31642],[34,221412,221414],{"className":221413},[31646],[31648,221415,221416],{"xmlns":31650},[31652,221417,221418,221434],{},[31655,221419,221420,221422,221424,221426,221428],{},[31658,221421,56852],{},[31662,221423,202],{},[31658,221425,38759],{},[31662,221427,42071],{},[31743,221429,221430,221432],{},[31658,221431,37400],{},[31658,221433,31747],{},[31665,221435,221436],{"encoding":31667},"W = q \\times T_s",[34,221438,221440,221458,221476],{"className":221439,"ariaHidden":758},[31672],[34,221441,221443,221446,221449,221452,221455],{"className":221442},[31676],[34,221444],{"className":221445,"style":33218},[31680],[34,221447,56852],{"className":221448,"style":37422},[31685,31686],[34,221450],{"className":221451,"style":31846},[31845],[34,221453,202],{"className":221454},[31850],[34,221456],{"className":221457,"style":31846},[31845],[34,221459,221461,221464,221467,221470,221473],{"className":221460},[31676],[34,221462],{"className":221463,"style":55949},[31680],[34,221465,38759],{"className":221466,"style":32062},[31685,31686],[34,221468],{"className":221469,"style":32604},[31845],[34,221471,42071],{"className":221472},[32024],[34,221474],{"className":221475,"style":32604},[31845],[34,221477,221479,221482],{"className":221478},[31676],[34,221480],{"className":221481,"style":70915},[31680],[34,221483,221485,221488],{"className":221484},[31685],[34,221486,37400],{"className":221487,"style":37422},[31685,31686],[34,221489,221491],{"className":221490},[31792],[34,221492,221494,221514],{"className":221493},[31796,31797],[34,221495,221497,221511],{"className":221496},[31801],[34,221498,221500],{"className":221499,"style":32362},[31805],[34,221501,221502,221505],{"style":214424},[34,221503],{"className":221504,"style":31814},[31813],[34,221506,221508],{"className":221507},[31818,31819,31820,31821],[34,221509,31747],{"className":221510},[31685,31686,31821],[34,221512,31829],{"className":221513},[31828],[34,221515,221517],{"className":221516},[31801],[34,221518,221520],{"className":221519,"style":31836},[31805],[34,221521],{},[108,221523,221524,221527,221528,221556,221557,221585,221586,221655,221656],{},[2627,221525,221526],{},"引入利特尔法则 (Little's Law)：","\n利特尔法则告诉我们，系统内的平均进程数 ",[34,221529,221531,221544],{"className":221530},[31642],[34,221532,221534],{"className":221533},[31646],[31648,221535,221536],{"xmlns":31650},[31652,221537,221538,221542],{},[31655,221539,221540],{},[31658,221541,38759],{},[31665,221543,38759],{"encoding":31667},[34,221545,221547],{"className":221546,"ariaHidden":758},[31672],[34,221548,221550,221553],{"className":221549},[31676],[34,221551],{"className":221552,"style":38774},[31680],[34,221554,38759],{"className":221555,"style":32062},[31685,31686]," 等于到达率 ",[34,221558,221560,221573],{"className":221559},[31642],[34,221561,221563],{"className":221562},[31646],[31648,221564,221565],{"xmlns":31650},[31652,221566,221567,221571],{},[31655,221568,221569],{},[31658,221570,220792],{},[31665,221572,220795],{"encoding":31667},[34,221574,221576],{"className":221575,"ariaHidden":758},[31672],[34,221577,221579,221582],{"className":221578},[31676],[34,221580],{"className":221581,"style":31681},[31680],[34,221583,220792],{"className":221584},[31685,31686]," 乘以进程在系统内的平均总逗留时间 ",[34,221587,221589,221606],{"className":221588},[31642],[34,221590,221592],{"className":221591},[31646],[31648,221593,221594],{"xmlns":31650},[31652,221595,221596,221604],{},[31655,221597,221598],{},[31743,221599,221600,221602],{},[31658,221601,37400],{},[31658,221603,34395],{},[31665,221605,214394],{"encoding":31667},[34,221607,221609],{"className":221608,"ariaHidden":758},[31672],[34,221610,221612,221615],{"className":221611},[31676],[34,221613],{"className":221614,"style":70915},[31680],[34,221616,221618,221621],{"className":221617},[31685],[34,221619,37400],{"className":221620,"style":37422},[31685,31686],[34,221622,221624],{"className":221623},[31792],[34,221625,221627,221647],{"className":221626},[31796,31797],[34,221628,221630,221644],{"className":221629},[31801],[34,221631,221633],{"className":221632,"style":32362},[31805],[34,221634,221635,221638],{"style":214424},[34,221636],{"className":221637,"style":31814},[31813],[34,221639,221641],{"className":221640},[31818,31819,31820,31821],[34,221642,34395],{"className":221643,"style":34452},[31685,31686,31821],[34,221645,31829],{"className":221646},[31828],[34,221648,221650],{"className":221649},[31801],[34,221651,221653],{"className":221652,"style":31836},[31805],[34,221654],{},"。即：\n",[34,221657,221659,221685],{"className":221658},[31642],[34,221660,221662],{"className":221661},[31646],[31648,221663,221664],{"xmlns":31650},[31652,221665,221666,221682],{},[31655,221667,221668,221670,221672,221674,221676],{},[31658,221669,38759],{},[31662,221671,202],{},[31658,221673,220792],{},[31662,221675,42071],{},[31743,221677,221678,221680],{},[31658,221679,37400],{},[31658,221681,34395],{},[31665,221683,221684],{"encoding":31667},"q = \\lambda \\times T_r",[34,221686,221688,221706,221724],{"className":221687,"ariaHidden":758},[31672],[34,221689,221691,221694,221697,221700,221703],{"className":221690},[31676],[34,221692],{"className":221693,"style":38774},[31680],[34,221695,38759],{"className":221696,"style":32062},[31685,31686],[34,221698],{"className":221699,"style":31846},[31845],[34,221701,202],{"className":221702},[31850],[34,221704],{"className":221705,"style":31846},[31845],[34,221707,221709,221712,221715,221718,221721],{"className":221708},[31676],[34,221710],{"className":221711,"style":34299},[31680],[34,221713,220792],{"className":221714},[31685,31686],[34,221716],{"className":221717,"style":32604},[31845],[34,221719,42071],{"className":221720},[32024],[34,221722],{"className":221723,"style":32604},[31845],[34,221725,221727,221730],{"className":221726},[31676],[34,221728],{"className":221729,"style":70915},[31680],[34,221731,221733,221736],{"className":221732},[31685],[34,221734,37400],{"className":221735,"style":37422},[31685,31686],[34,221737,221739],{"className":221738},[31792],[34,221740,221742,221762],{"className":221741},[31796,31797],[34,221743,221745,221759],{"className":221744},[31801],[34,221746,221748],{"className":221747,"style":32362},[31805],[34,221749,221750,221753],{"style":214424},[34,221751],{"className":221752,"style":31814},[31813],[34,221754,221756],{"className":221755},[31818,31819,31820,31821],[34,221757,34395],{"className":221758,"style":34452},[31685,31686,31821],[34,221760,31829],{"className":221761},[31828],[34,221763,221765],{"className":221764},[31801],[34,221766,221768],{"className":221767,"style":31836},[31805],[34,221769],{},[108,221771,221772,221775,221776,221804,221805,221833,221834,222021,222022,222136,222137],{},[2627,221773,221774],{},"合并替换：","\n将 ",[34,221777,221779,221792],{"className":221778},[31642],[34,221780,221782],{"className":221781},[31646],[31648,221783,221784],{"xmlns":31650},[31652,221785,221786,221790],{},[31655,221787,221788],{},[31658,221789,38759],{},[31665,221791,38759],{"encoding":31667},[34,221793,221795],{"className":221794,"ariaHidden":758},[31672],[34,221796,221798,221801],{"className":221797},[31676],[34,221799],{"className":221800,"style":38774},[31680],[34,221802,38759],{"className":221803,"style":32062},[31685,31686]," 的表达式代入 ",[34,221806,221808,221821],{"className":221807},[31642],[34,221809,221811],{"className":221810},[31646],[31648,221812,221813],{"xmlns":31650},[31652,221814,221815,221819],{},[31655,221816,221817],{},[31658,221818,56852],{},[31665,221820,56852],{"encoding":31667},[34,221822,221824],{"className":221823,"ariaHidden":758},[31672],[34,221825,221827,221830],{"className":221826},[31676],[34,221828],{"className":221829,"style":33218},[31680],[34,221831,56852],{"className":221832,"style":37422},[31685,31686]," 的公式中：\n",[34,221835,221837,221875],{"className":221836},[31642],[34,221838,221840],{"className":221839},[31646],[31648,221841,221842],{"xmlns":31650},[31652,221843,221844,221872],{},[31655,221845,221846,221848,221850,221852,221854,221856,221862,221864,221866],{},[31658,221847,56852],{},[31662,221849,202],{},[31662,221851,456],{"stretchy":702},[31658,221853,220792],{},[31662,221855,42071],{},[31743,221857,221858,221860],{},[31658,221859,37400],{},[31658,221861,34395],{},[31662,221863,2557],{"stretchy":702},[31662,221865,42071],{},[31743,221867,221868,221870],{},[31658,221869,37400],{},[31658,221871,31747],{},[31665,221873,221874],{"encoding":31667},"W = (\\lambda \\times T_r) \\times T_s",[34,221876,221878,221896,221917,221975],{"className":221877,"ariaHidden":758},[31672],[34,221879,221881,221884,221887,221890,221893],{"className":221880},[31676],[34,221882],{"className":221883,"style":33218},[31680],[34,221885,56852],{"className":221886,"style":37422},[31685,31686],[34,221888],{"className":221889,"style":31846},[31845],[34,221891,202],{"className":221892},[31850],[34,221894],{"className":221895,"style":31846},[31845],[34,221897,221899,221902,221905,221908,221911,221914],{"className":221898},[31676],[34,221900],{"className":221901,"style":31774},[31680],[34,221903,456],{"className":221904},[31782],[34,221906,220792],{"className":221907},[31685,31686],[34,221909],{"className":221910,"style":32604},[31845],[34,221912,42071],{"className":221913},[32024],[34,221915],{"className":221916,"style":32604},[31845],[34,221918,221920,221923,221963,221966,221969,221972],{"className":221919},[31676],[34,221921],{"className":221922,"style":31774},[31680],[34,221924,221926,221929],{"className":221925},[31685],[34,221927,37400],{"className":221928,"style":37422},[31685,31686],[34,221930,221932],{"className":221931},[31792],[34,221933,221935,221955],{"className":221934},[31796,31797],[34,221936,221938,221952],{"className":221937},[31801],[34,221939,221941],{"className":221940,"style":32362},[31805],[34,221942,221943,221946],{"style":214424},[34,221944],{"className":221945,"style":31814},[31813],[34,221947,221949],{"className":221948},[31818,31819,31820,31821],[34,221950,34395],{"className":221951,"style":34452},[31685,31686,31821],[34,221953,31829],{"className":221954},[31828],[34,221956,221958],{"className":221957},[31801],[34,221959,221961],{"className":221960,"style":31836},[31805],[34,221962],{},[34,221964,2557],{"className":221965},[31690],[34,221967],{"className":221968,"style":32604},[31845],[34,221970,42071],{"className":221971},[32024],[34,221973],{"className":221974,"style":32604},[31845],[34,221976,221978,221981],{"className":221977},[31676],[34,221979],{"className":221980,"style":70915},[31680],[34,221982,221984,221987],{"className":221983},[31685],[34,221985,37400],{"className":221986,"style":37422},[31685,31686],[34,221988,221990],{"className":221989},[31792],[34,221991,221993,222013],{"className":221992},[31796,31797],[34,221994,221996,222010],{"className":221995},[31801],[34,221997,221999],{"className":221998,"style":32362},[31805],[34,222000,222001,222004],{"style":214424},[34,222002],{"className":222003,"style":31814},[31813],[34,222005,222007],{"className":222006},[31818,31819,31820,31821],[34,222008,31747],{"className":222009},[31685,31686,31821],[34,222011,31829],{"className":222012},[31828],[34,222014,222016],{"className":222015},[31801],[34,222017,222019],{"className":222018,"style":31836},[31805],[34,222020],{},"\n我们知道利用率 ",[34,222023,222025,222051],{"className":222024},[31642],[34,222026,222028],{"className":222027},[31646],[31648,222029,222030],{"xmlns":31650},[31652,222031,222032,222048],{},[31655,222033,222034,222036,222038,222040,222042],{},[31658,222035,220084],{},[31662,222037,202],{},[31658,222039,220792],{},[31662,222041,42071],{},[31743,222043,222044,222046],{},[31658,222045,37400],{},[31658,222047,31747],{},[31665,222049,222050],{"encoding":31667},"\\rho = \\lambda \\times T_s",[34,222052,222054,222072,222090],{"className":222053,"ariaHidden":758},[31672],[34,222055,222057,222060,222063,222066,222069],{"className":222056},[31676],[34,222058],{"className":222059,"style":38774},[31680],[34,222061,220084],{"className":222062},[31685,31686],[34,222064],{"className":222065,"style":31846},[31845],[34,222067,202],{"className":222068},[31850],[34,222070],{"className":222071,"style":31846},[31845],[34,222073,222075,222078,222081,222084,222087],{"className":222074},[31676],[34,222076],{"className":222077,"style":34299},[31680],[34,222079,220792],{"className":222080},[31685,31686],[34,222082],{"className":222083,"style":32604},[31845],[34,222085,42071],{"className":222086},[32024],[34,222088],{"className":222089,"style":32604},[31845],[34,222091,222093,222096],{"className":222092},[31676],[34,222094],{"className":222095,"style":70915},[31680],[34,222097,222099,222102],{"className":222098},[31685],[34,222100,37400],{"className":222101,"style":37422},[31685,31686],[34,222103,222105],{"className":222104},[31792],[34,222106,222108,222128],{"className":222107},[31796,31797],[34,222109,222111,222125],{"className":222110},[31801],[34,222112,222114],{"className":222113,"style":32362},[31805],[34,222115,222116,222119],{"style":214424},[34,222117],{"className":222118,"style":31814},[31813],[34,222120,222122],{"className":222121},[31818,31819,31820,31821],[34,222123,31747],{"className":222124},[31685,31686,31821],[34,222126,31829],{"className":222127},[31828],[34,222129,222131],{"className":222130},[31801],[34,222132,222134],{"className":222133,"style":31836},[31805],[34,222135],{},"。整理可得：\n",[34,222138,222140,222166],{"className":222139},[31642],[34,222141,222143],{"className":222142},[31646],[31648,222144,222145],{"xmlns":31650},[31652,222146,222147,222163],{},[31655,222148,222149,222151,222153,222155,222157],{},[31658,222150,56852],{},[31662,222152,202],{},[31658,222154,220084],{},[31662,222156,42071],{},[31743,222158,222159,222161],{},[31658,222160,37400],{},[31658,222162,34395],{},[31665,222164,222165],{"encoding":31667},"W = \\rho \\times T_r",[34,222167,222169,222187,222205],{"className":222168,"ariaHidden":758},[31672],[34,222170,222172,222175,222178,222181,222184],{"className":222171},[31676],[34,222173],{"className":222174,"style":33218},[31680],[34,222176,56852],{"className":222177,"style":37422},[31685,31686],[34,222179],{"className":222180,"style":31846},[31845],[34,222182,202],{"className":222183},[31850],[34,222185],{"className":222186,"style":31846},[31845],[34,222188,222190,222193,222196,222199,222202],{"className":222189},[31676],[34,222191],{"className":222192,"style":55949},[31680],[34,222194,220084],{"className":222195},[31685,31686],[34,222197],{"className":222198,"style":32604},[31845],[34,222200,42071],{"className":222201},[32024],[34,222203],{"className":222204,"style":32604},[31845],[34,222206,222208,222211],{"className":222207},[31676],[34,222209],{"className":222210,"style":70915},[31680],[34,222212,222214,222217],{"className":222213},[31685],[34,222215,37400],{"className":222216,"style":37422},[31685,31686],[34,222218,222220],{"className":222219},[31792],[34,222221,222223,222243],{"className":222222},[31796,31797],[34,222224,222226,222240],{"className":222225},[31801],[34,222227,222229],{"className":222228,"style":32362},[31805],[34,222230,222231,222234],{"style":214424},[34,222232],{"className":222233,"style":31814},[31813],[34,222235,222237],{"className":222236},[31818,31819,31820,31821],[34,222238,34395],{"className":222239,"style":34452},[31685,31686,31821],[34,222241,31829],{"className":222242},[31828],[34,222244,222246],{"className":222245},[31801],[34,222247,222249],{"className":222248,"style":31836},[31805],[34,222250],{},[108,222252,222253,222256,222257,222285,222286,222487,222688,222858,223071],{},[2627,222254,222255],{},"得出最终公式：","\n把 ",[34,222258,222260,222273],{"className":222259},[31642],[34,222261,222263],{"className":222262},[31646],[31648,222264,222265],{"xmlns":31650},[31652,222266,222267,222271],{},[31655,222268,222269],{},[31658,222270,56852],{},[31665,222272,56852],{"encoding":31667},[34,222274,222276],{"className":222275,"ariaHidden":758},[31672],[34,222277,222279,222282],{"className":222278},[31676],[34,222280],{"className":222281,"style":33218},[31680],[34,222283,56852],{"className":222284,"style":37422},[31685,31686]," 的结果带回步骤1的公式中：\n",[34,222287,222289,222325],{"className":222288},[31642],[34,222290,222292],{"className":222291},[31646],[31648,222293,222294],{"xmlns":31650},[31652,222295,222296,222322],{},[31655,222297,222298,222304,222306,222308,222314,222316],{},[31743,222299,222300,222302],{},[31658,222301,37400],{},[31658,222303,34395],{},[31662,222305,202],{},[31658,222307,220084],{},[31743,222309,222310,222312],{},[31658,222311,37400],{},[31658,222313,34395],{},[31662,222315,223],{},[31743,222317,222318,222320],{},[31658,222319,37400],{},[31658,222321,31747],{},[31665,222323,222324],{"encoding":31667},"T_r = \\rho T_r + T_s",[34,222326,222328,222383,222441],{"className":222327,"ariaHidden":758},[31672],[34,222329,222331,222334,222374,222377,222380],{"className":222330},[31676],[34,222332],{"className":222333,"style":70915},[31680],[34,222335,222337,222340],{"className":222336},[31685],[34,222338,37400],{"className":222339,"style":37422},[31685,31686],[34,222341,222343],{"className":222342},[31792],[34,222344,222346,222366],{"className":222345},[31796,31797],[34,222347,222349,222363],{"className":222348},[31801],[34,222350,222352],{"className":222351,"style":32362},[31805],[34,222353,222354,222357],{"style":214424},[34,222355],{"className":222356,"style":31814},[31813],[34,222358,222360],{"className":222359},[31818,31819,31820,31821],[34,222361,34395],{"className":222362,"style":34452},[31685,31686,31821],[34,222364,31829],{"className":222365},[31828],[34,222367,222369],{"className":222368},[31801],[34,222370,222372],{"className":222371,"style":31836},[31805],[34,222373],{},[34,222375],{"className":222376,"style":31846},[31845],[34,222378,202],{"className":222379},[31850],[34,222381],{"className":222382,"style":31846},[31845],[34,222384,222386,222389,222392,222432,222435,222438],{"className":222385},[31676],[34,222387],{"className":222388,"style":70739},[31680],[34,222390,220084],{"className":222391},[31685,31686],[34,222393,222395,222398],{"className":222394},[31685],[34,222396,37400],{"className":222397,"style":37422},[31685,31686],[34,222399,222401],{"className":222400},[31792],[34,222402,222404,222424],{"className":222403},[31796,31797],[34,222405,222407,222421],{"className":222406},[31801],[34,222408,222410],{"className":222409,"style":32362},[31805],[34,222411,222412,222415],{"style":214424},[34,222413],{"className":222414,"style":31814},[31813],[34,222416,222418],{"className":222417},[31818,31819,31820,31821],[34,222419,34395],{"className":222420,"style":34452},[31685,31686,31821],[34,222422,31829],{"className":222423},[31828],[34,222425,222427],{"className":222426},[31801],[34,222428,222430],{"className":222429,"style":31836},[31805],[34,222431],{},[34,222433],{"className":222434,"style":32604},[31845],[34,222436,223],{"className":222437},[32024],[34,222439],{"className":222440,"style":32604},[31845],[34,222442,222444,222447],{"className":222443},[31676],[34,222445],{"className":222446,"style":70915},[31680],[34,222448,222450,222453],{"className":222449},[31685],[34,222451,37400],{"className":222452,"style":37422},[31685,31686],[34,222454,222456],{"className":222455},[31792],[34,222457,222459,222479],{"className":222458},[31796,31797],[34,222460,222462,222476],{"className":222461},[31801],[34,222463,222465],{"className":222464,"style":32362},[31805],[34,222466,222467,222470],{"style":214424},[34,222468],{"className":222469,"style":31814},[31813],[34,222471,222473],{"className":222472},[31818,31819,31820,31821],[34,222474,31747],{"className":222475},[31685,31686,31821],[34,222477,31829],{"className":222478},[31828],[34,222480,222482],{"className":222481},[31801],[34,222483,222485],{"className":222484,"style":31836},[31805],[34,222486],{},[34,222488,222490,222526],{"className":222489},[31642],[34,222491,222493],{"className":222492},[31646],[31648,222494,222495],{"xmlns":31650},[31652,222496,222497,222523],{},[31655,222498,222499,222505,222507,222509,222515,222517],{},[31743,222500,222501,222503],{},[31658,222502,37400],{},[31658,222504,34395],{},[31662,222506,32226],{},[31658,222508,220084],{},[31743,222510,222511,222513],{},[31658,222512,37400],{},[31658,222514,34395],{},[31662,222516,202],{},[31743,222518,222519,222521],{},[31658,222520,37400],{},[31658,222522,31747],{},[31665,222524,222525],{"encoding":31667},"T_r - \\rho T_r = T_s",[34,222527,222529,222584,222642],{"className":222528,"ariaHidden":758},[31672],[34,222530,222532,222535,222575,222578,222581],{"className":222531},[31676],[34,222533],{"className":222534,"style":70915},[31680],[34,222536,222538,222541],{"className":222537},[31685],[34,222539,37400],{"className":222540,"style":37422},[31685,31686],[34,222542,222544],{"className":222543},[31792],[34,222545,222547,222567],{"className":222546},[31796,31797],[34,222548,222550,222564],{"className":222549},[31801],[34,222551,222553],{"className":222552,"style":32362},[31805],[34,222554,222555,222558],{"style":214424},[34,222556],{"className":222557,"style":31814},[31813],[34,222559,222561],{"className":222560},[31818,31819,31820,31821],[34,222562,34395],{"className":222563,"style":34452},[31685,31686,31821],[34,222565,31829],{"className":222566},[31828],[34,222568,222570],{"className":222569},[31801],[34,222571,222573],{"className":222572,"style":31836},[31805],[34,222574],{},[34,222576],{"className":222577,"style":32604},[31845],[34,222579,32226],{"className":222580},[32024],[34,222582],{"className":222583,"style":32604},[31845],[34,222585,222587,222590,222593,222633,222636,222639],{"className":222586},[31676],[34,222588],{"className":222589,"style":70739},[31680],[34,222591,220084],{"className":222592},[31685,31686],[34,222594,222596,222599],{"className":222595},[31685],[34,222597,37400],{"className":222598,"style":37422},[31685,31686],[34,222600,222602],{"className":222601},[31792],[34,222603,222605,222625],{"className":222604},[31796,31797],[34,222606,222608,222622],{"className":222607},[31801],[34,222609,222611],{"className":222610,"style":32362},[31805],[34,222612,222613,222616],{"style":214424},[34,222614],{"className":222615,"style":31814},[31813],[34,222617,222619],{"className":222618},[31818,31819,31820,31821],[34,222620,34395],{"className":222621,"style":34452},[31685,31686,31821],[34,222623,31829],{"className":222624},[31828],[34,222626,222628],{"className":222627},[31801],[34,222629,222631],{"className":222630,"style":31836},[31805],[34,222632],{},[34,222634],{"className":222635,"style":31846},[31845],[34,222637,202],{"className":222638},[31850],[34,222640],{"className":222641,"style":31846},[31845],[34,222643,222645,222648],{"className":222644},[31676],[34,222646],{"className":222647,"style":70915},[31680],[34,222649,222651,222654],{"className":222650},[31685],[34,222652,37400],{"className":222653,"style":37422},[31685,31686],[34,222655,222657],{"className":222656},[31792],[34,222658,222660,222680],{"className":222659},[31796,31797],[34,222661,222663,222677],{"className":222662},[31801],[34,222664,222666],{"className":222665,"style":32362},[31805],[34,222667,222668,222671],{"style":214424},[34,222669],{"className":222670,"style":31814},[31813],[34,222672,222674],{"className":222673},[31818,31819,31820,31821],[34,222675,31747],{"className":222676},[31685,31686,31821],[34,222678,31829],{"className":222679},[31828],[34,222681,222683],{"className":222682},[31801],[34,222684,222686],{"className":222685,"style":31836},[31805],[34,222687],{},[34,222689,222691,222727],{"className":222690},[31642],[34,222692,222694],{"className":222693},[31646],[31648,222695,222696],{"xmlns":31650},[31652,222697,222698,222724],{},[31655,222699,222700,222706,222708,222710,222712,222714,222716,222718],{},[31743,222701,222702,222704],{},[31658,222703,37400],{},[31658,222705,34395],{},[31662,222707,456],{"stretchy":702},[31749,222709,640],{},[31662,222711,32226],{},[31658,222713,220084],{},[31662,222715,2557],{"stretchy":702},[31662,222717,202],{},[31743,222719,222720,222722],{},[31658,222721,37400],{},[31658,222723,31747],{},[31665,222725,222726],{"encoding":31667},"T_r (1 - \\rho) = T_s",[34,222728,222730,222791,222812],{"className":222729,"ariaHidden":758},[31672],[34,222731,222733,222736,222776,222779,222782,222785,222788],{"className":222732},[31676],[34,222734],{"className":222735,"style":31774},[31680],[34,222737,222739,222742],{"className":222738},[31685],[34,222740,37400],{"className":222741,"style":37422},[31685,31686],[34,222743,222745],{"className":222744},[31792],[34,222746,222748,222768],{"className":222747},[31796,31797],[34,222749,222751,222765],{"className":222750},[31801],[34,222752,222754],{"className":222753,"style":32362},[31805],[34,222755,222756,222759],{"style":214424},[34,222757],{"className":222758,"style":31814},[31813],[34,222760,222762],{"className":222761},[31818,31819,31820,31821],[34,222763,34395],{"className":222764,"style":34452},[31685,31686,31821],[34,222766,31829],{"className":222767},[31828],[34,222769,222771],{"className":222770},[31801],[34,222772,222774],{"className":222773,"style":31836},[31805],[34,222775],{},[34,222777,456],{"className":222778},[31782],[34,222780,640],{"className":222781},[31685],[34,222783],{"className":222784,"style":32604},[31845],[34,222786,32226],{"className":222787},[32024],[34,222789],{"className":222790,"style":32604},[31845],[34,222792,222794,222797,222800,222803,222806,222809],{"className":222793},[31676],[34,222795],{"className":222796,"style":31774},[31680],[34,222798,220084],{"className":222799},[31685,31686],[34,222801,2557],{"className":222802},[31690],[34,222804],{"className":222805,"style":31846},[31845],[34,222807,202],{"className":222808},[31850],[34,222810],{"className":222811,"style":31846},[31845],[34,222813,222815,222818],{"className":222814},[31676],[34,222816],{"className":222817,"style":70915},[31680],[34,222819,222821,222824],{"className":222820},[31685],[34,222822,37400],{"className":222823,"style":37422},[31685,31686],[34,222825,222827],{"className":222826},[31792],[34,222828,222830,222850],{"className":222829},[31796,31797],[34,222831,222833,222847],{"className":222832},[31801],[34,222834,222836],{"className":222835,"style":32362},[31805],[34,222837,222838,222841],{"style":214424},[34,222839],{"className":222840,"style":31814},[31813],[34,222842,222844],{"className":222843},[31818,31819,31820,31821],[34,222845,31747],{"className":222846},[31685,31686,31821],[34,222848,31829],{"className":222849},[31828],[34,222851,222853],{"className":222852},[31801],[34,222854,222856],{"className":222855,"style":31836},[31805],[34,222857],{},[34,222859,222861,222896],{"className":222860},[31642],[34,222862,222864],{"className":222863},[31646],[31648,222865,222866],{"xmlns":31650},[31652,222867,222868,222894],{},[31655,222869,222870,222876,222878],{},[31743,222871,222872,222874],{},[31658,222873,37400],{},[31658,222875,34395],{},[31662,222877,202],{},[41453,222879,222880,222886],{},[31743,222881,222882,222884],{},[31658,222883,37400],{},[31658,222885,31747],{},[31655,222887,222888,222890,222892],{},[31749,222889,640],{},[31662,222891,32226],{},[31658,222893,220084],{},[31665,222895,220375],{"encoding":31667},[34,222897,222899,222954],{"className":222898,"ariaHidden":758},[31672],[34,222900,222902,222905,222945,222948,222951],{"className":222901},[31676],[34,222903],{"className":222904,"style":70915},[31680],[34,222906,222908,222911],{"className":222907},[31685],[34,222909,37400],{"className":222910,"style":37422},[31685,31686],[34,222912,222914],{"className":222913},[31792],[34,222915,222917,222937],{"className":222916},[31796,31797],[34,222918,222920,222934],{"className":222919},[31801],[34,222921,222923],{"className":222922,"style":32362},[31805],[34,222924,222925,222928],{"style":214424},[34,222926],{"className":222927,"style":31814},[31813],[34,222929,222931],{"className":222930},[31818,31819,31820,31821],[34,222932,34395],{"className":222933,"style":34452},[31685,31686,31821],[34,222935,31829],{"className":222936},[31828],[34,222938,222940],{"className":222939},[31801],[34,222941,222943],{"className":222942,"style":31836},[31805],[34,222944],{},[34,222946],{"className":222947,"style":31846},[31845],[34,222949,202],{"className":222950},[31850],[34,222952],{"className":222953,"style":31846},[31845],[34,222955,222957,222960],{"className":222956},[31676],[34,222958],{"className":222959,"style":220440},[31680],[34,222961,222963,222966,223068],{"className":222962},[31685],[34,222964],{"className":222965},[31782,41488],[34,222967,222969],{"className":222968},[41453],[34,222970,222972,223060],{"className":222971},[31796,31797],[34,222973,222975,223057],{"className":222974},[31801],[34,222976,222978,222998,223006],{"className":222977,"style":220459},[31805],[34,222979,222980,222983],{"style":41504},[34,222981],{"className":222982,"style":41508},[31813],[34,222984,222986],{"className":222985},[31818,31819,31820,31821],[34,222987,222989,222992,222995],{"className":222988},[31685,31821],[34,222990,640],{"className":222991},[31685,31821],[34,222993,32226],{"className":222994},[32024,31821],[34,222996,220084],{"className":2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Little's Law (",[34,223081,223083,223107],{"className":223082},[31642],[34,223084,223086],{"className":223085},[31646],[31648,223087,223088],{"xmlns":31650},[31652,223089,223090,223104],{},[31655,223091,223092,223094,223096,223098],{},[31658,223093,38759],{},[31662,223095,202],{},[31658,223097,220792],{},[31743,223099,223100,223102],{},[31658,223101,37400],{},[31658,223103,34395],{},[31665,223105,223106],{"encoding":31667},"q = \\lambda T_r",[34,223108,223110,223128],{"className":223109,"ariaHidden":758},[31672],[34,223111,223113,223116,223119,223122,223125],{"className":223112},[31676],[34,223114],{"className":223115,"style":38774},[31680],[34,223117,38759],{"className":223118,"style":32062},[31685,31686],[34,223120],{"className":223121,"style":31846},[31845],[34,223123,202],{"className":223124},[31850],[34,223126],{"className":223127,"style":31846},[31845],[34,223129,223131,223134,223137],{"className":223130},[31676],[34,223132],{"className":223133,"style":33814},[31680],[34,223135,220792],{"className":223136},[31685,31686],[34,223138,223140,223143],{"className":223139},[31685],[34,223141,37400],{"className":223142,"style":37422},[31685,31686],[34,223144,223146],{"className":223145},[31792],[34,223147,223149,223169],{"className":223148},[31796,31797],[34,223150,223152,223166],{"className":223151},[31801],[34,223153,223155],{"className":223154,"style":32362},[31805],[34,223156,223157,223160],{"style":214424},[34,223158],{"className":223159,"style":31814},[31813],[34,223161,223163],{"className":223162},[31818,31819,31820,31821],[34,223164,34395],{"className":223165,"style":34452},[31685,31686,31821],[34,223167,31829],{"className":223168},[31828],[34,223170,223172],{"className":223171},[31801],[34,223173,223175],{"className":223174,"style":31836},[31805],[34,223176],{},") 将未知的等待队列长度 ",[34,223179,223181,223194],{"className":223180},[31642],[34,223182,223184],{"className":223183},[31646],[31648,223185,223186],{"xmlns":31650},[31652,223187,223188,223192],{},[31655,223189,223190],{},[31658,223191,38759],{},[31665,223193,38759],{"encoding":31667},[34,223195,223197],{"className":223196,"ariaHidden":758},[31672],[34,223198,223200,223203],{"className":223199},[31676],[34,223201],{"className":223202,"style":38774},[31680],[34,223204,38759],{"className":223205,"style":32062},[31685,31686]," 转换成了包含 ",[34,223208,223210,223227],{"className":223209},[31642],[34,223211,223213],{"className":223212},[31646],[31648,223214,223215],{"xmlns":31650},[31652,223216,223217,223225],{},[31655,223218,223219],{},[31743,223220,223221,223223],{},[31658,223222,37400],{},[31658,223224,34395],{},[31665,223226,214394],{"encoding":31667},[34,223228,223230],{"className":223229,"ariaHidden":758},[31672],[34,223231,223233,223236],{"className":223232},[31676],[34,223234],{"className":223235,"style":70915},[31680],[34,223237,223239,223242],{"className":223238},[31685],[34,223240,37400],{"className":223241,"style":37422},[31685,31686],[34,223243,223245],{"className":223244},[31792],[34,223246,223248,223268],{"className":223247},[31796,31797],[34,223249,223251,223265],{"className":223250},[31801],[34,223252,223254],{"className":223253,"style":32362},[31805],[34,223255,223256,223259],{"style":214424},[34,223257],{"className":223258,"style":31814},[31813],[34,223260,223262],{"className":223261},[31818,31819,31820,31821],[34,223263,34395],{"className":223264,"style":34452},[31685,31686,31821],[34,223266,31829],{"className":223267},[31828],[34,223269,223271],{"className":223270},[31801],[34,223272,223274],{"className":223273,"style":31836},[31805],[34,223275],{}," 的表达式。很多学生在推导时，试图用微积分或复杂的概率密度函数去硬解，反而会陷入死胡同。",[15,223278,223279],{},"你好！我是你的专属操作系统专家导师。你带来的这批题目（4~10题）非常精彩，它们不仅是操作系统中“进程调度”与“多道程序系统”的精髓，更是各类考试中最容易“挖坑”的计算大题。",[15,223281,223282],{},"尤其是**“多道限制下的双层调度（第8、9题）”**，它要求你同时扮演“长程调度器（决定谁进内存）”和“短程调度器（决定谁上CPU）”两个角色。",[15,223284,223285,223286,223288,223289,223292],{},"请放心，我会继续运用**“概念核对 (Concept Check)”",[2627,223287,146],{},"“底层的为什么 (The 'Why')”",[2627,223290,223291],{},"以及","“易错点 (Common Pitfalls)”**的模式，为你逐一拆解这些硬核题目。",[32188,223294],{},[19,223296,223298],{"id":223297},"_4证明短作业优先-sjf-算法能获得最小平均周转时间","4、证明短作业优先 (SJF) 算法能获得最小平均周转时间",[15,223300,223301,223303,223304,223501,223502,223697,223698,223916],{},[2627,223302,214361],{},"\n若后备作业队列中等待运行的同时有N个作业 ",[34,223305,223307,223346],{"className":223306},[31642],[34,223308,223310],{"className":223309},[31646],[31648,223311,223312],{"xmlns":31650},[31652,223313,223314,223343],{},[31655,223315,223316,223323,223325,223331,223333,223335,223337],{},[31743,223317,223318,223321],{},[31658,223319,223320],{},"J",[31749,223322,640],{},[31662,223324,31960],{"separator":758},[31743,223326,223327,223329],{},[31658,223328,223320],{},[31749,223330,768],{},[31662,223332,31960],{"separator":758},[31662,223334,33934],{},[31662,223336,31960],{"separator":758},[31743,223338,223339,223341],{},[31658,223340,223320],{},[31658,223342,31660],{},[31665,223344,223345],{"encoding":31667},"J_1, J_2, \\dots, J_n",[34,223347,223349],{"className":223348,"ariaHidden":758},[31672],[34,223350,223352,223355,223397,223400,223403,223443,223446,223449,223452,223455,223458,223461],{"className":223351},[31676],[34,223353],{"className":223354,"style":70739},[31680],[34,223356,223358,223362],{"className":223357},[31685],[34,223359,223320],{"className":223360,"style":223361},[31685,31686],"margin-right:0.0962em;",[34,223363,223365],{"className":223364},[31792],[34,223366,223368,223389],{"className":223367},[31796,31797],[34,223369,223371,223386],{"className":223370},[31801],[34,223372,223374],{"className":223373,"style":31806},[31805],[34,223375,223377,223380],{"style":223376},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[34,223378],{"className":223379,"style":31814},[31813],[34,223381,223383],{"className":223382},[31818,31819,31820,31821],[34,223384,640],{"className":223385},[31685,31821],[34,223387,31829],{"className":223388},[31828],[34,223390,223392],{"className":223391},[31801],[34,223393,223395],{"className":223394,"style":31836},[31805],[34,223396],{},[34,223398,31960],{"className":223399},[32119],[34,223401],{"className":223402,"style":32123},[31845],[34,223404,223406,223409],{"className":223405},[31685],[34,223407,223320],{"className":223408,"style":223361},[31685,31686],[34,223410,223412],{"className":223411},[31792],[34,223413,223415,223435],{"className":223414},[31796,31797],[34,223416,223418,223432],{"className":223417},[31801],[34,223419,223421],{"className":223420,"style":31806},[31805],[34,223422,223423,223426],{"style":223376},[34,223424],{"className":223425,"style":31814},[31813],[34,223427,223429],{"className":223428},[31818,31819,31820,31821],[34,223430,768],{"className":223431},[31685,31821],[34,223433,31829],{"className":223434},[31828],[34,223436,223438],{"className":223437},[31801],[34,223439,223441],{"className":223440,"style":31836},[31805],[34,223442],{},[34,223444,31960],{"className":223445},[32119],[34,223447],{"className":223448,"style":32123},[31845],[34,223450,33934],{"className":223451},[34006],[34,223453],{"className":223454,"style":32123},[31845],[34,223456,31960],{"className":223457},[32119],[34,223459],{"className":223460,"style":32123},[31845],[34,223462,223464,223467],{"className":223463},[31685],[34,223465,223320],{"className":223466,"style":223361},[31685,31686],[34,223468,223470],{"className":223469},[31792],[34,223471,223473,223493],{"className":223472},[31796,31797],[34,223474,223476,223490],{"className":223475},[31801],[34,223477,223479],{"className":223478,"style":32362},[31805],[34,223480,223481,223484],{"style":223376},[34,223482],{"className":223483,"style":31814},[31813],[34,223485,223487],{"className":223486},[31818,31819,31820,31821],[34,223488,31660],{"className":223489},[31685,31686,31821],[34,223491,31829],{"className":223492},[31828],[34,223494,223496],{"className":223495},[31801],[34,223497,223499],{"className":223498,"style":31836},[31805],[34,223500],{},"，已知它们各自的运行时间为 ",[34,223503,223505,223543],{"className":223504},[31642],[34,223506,223508],{"className":223507},[31646],[31648,223509,223510],{"xmlns":31650},[31652,223511,223512,223540],{},[31655,223513,223514,223520,223522,223528,223530,223532,223534],{},[31743,223515,223516,223518],{},[31658,223517,33249],{},[31749,223519,640],{},[31662,223521,31960],{"separator":758},[31743,223523,223524,223526],{},[31658,223525,33249],{},[31749,223527,768],{},[31662,223529,31960],{"separator":758},[31662,223531,33934],{},[31662,223533,31960],{"separator":758},[31743,223535,223536,223538],{},[31658,223537,33249],{},[31658,223539,31660],{},[31665,223541,223542],{"encoding":31667},"t_1, t_2, \\dots, t_n",[34,223544,223546],{"className":223545,"ariaHidden":758},[31672],[34,223547,223549,223553,223593,223596,223599,223639,223642,223645,223648,223651,223654,223657],{"className":223548},[31676],[34,223550],{"className":223551,"style":223552},[31680],"height:0.8095em;vertical-align:-0.1944em;",[34,223554,223556,223559],{"className":223555},[31685],[34,223557,33249],{"className":223558},[31685,31686],[34,223560,223562],{"className":223561},[31792],[34,223563,223565,223585],{"className":223564},[31796,31797],[34,223566,223568,223582],{"className":223567},[31801],[34,223569,223571],{"className":223570,"style":31806},[31805],[34,223572,223573,223576],{"style":31809},[34,223574],{"className":223575,"style":31814},[31813],[34,223577,223579],{"className":223578},[31818,31819,31820,31821],[34,223580,640],{"className":223581},[31685,31821],[34,223583,31829],{"className":223584},[31828],[34,223586,223588],{"className":223587},[31801],[34,223589,223591],{"className":223590,"style":31836},[31805],[34,223592],{},[34,223594,31960],{"className":223595},[32119],[34,223597],{"className":223598,"style":32123},[31845],[34,223600,223602,223605],{"className":223601},[31685],[34,223603,33249],{"className":223604},[31685,31686],[34,223606,223608],{"className":223607},[31792],[34,223609,223611,223631],{"className":223610},[31796,31797],[34,223612,223614,223628],{"className":223613},[31801],[34,223615,223617],{"className":223616,"style":31806},[31805],[34,223618,223619,223622],{"style":31809},[34,223620],{"className":223621,"style":31814},[31813],[34,223623,223625],{"className":223624},[31818,31819,31820,31821],[34,223626,768],{"className":223627},[31685,31821],[34,223629,31829],{"className":223630},[31828],[34,223632,223634],{"className":223633},[31801],[34,223635,223637],{"className":223636,"style":31836},[31805],[34,223638],{},[34,223640,31960],{"className":223641},[32119],[34,223643],{"className":223644,"style":32123},[31845],[34,223646,33934],{"className":223647},[34006],[34,223649],{"className":223650,"style":32123},[31845],[34,223652,31960],{"className":223653},[32119],[34,223655],{"className":223656,"style":32123},[31845],[34,223658,223660,223663],{"className":223659},[31685],[34,223661,33249],{"className":223662},[31685,31686],[34,223664,223666],{"className":223665},[31792],[34,223667,223669,223689],{"className":223668},[31796,31797],[34,223670,223672,223686],{"className":223671},[31801],[34,223673,223675],{"className":223674,"style":32362},[31805],[34,223676,223677,223680],{"style":31809},[34,223678],{"className":223679,"style":31814},[31813],[34,223681,223683],{"className":223682},[31818,31819,31820,31821],[34,223684,31660],{"className":223685},[31685,31686,31821],[34,223687,31829],{"className":223688},[31828],[34,223690,223692],{"className":223691},[31801],[34,223693,223695],{"className":223694,"style":31836},[31805],[34,223696],{},"，且满足 ",[34,223699,223701,223739],{"className":223700},[31642],[34,223702,223704],{"className":223703},[31646],[31648,223705,223706],{"xmlns":31650},[31652,223707,223708,223736],{},[31655,223709,223710,223716,223718,223724,223726,223728,223730],{},[31743,223711,223712,223714],{},[31658,223713,33249],{},[31749,223715,640],{},[31662,223717,932],{},[31743,223719,223720,223722],{},[31658,223721,33249],{},[31749,223723,768],{},[31662,223725,932],{},[31662,223727,38874],{},[31662,223729,932],{},[31743,223731,223732,223734],{},[31658,223733,33249],{},[31658,223735,31660],{},[31665,223737,223738],{"encoding":31667},"t_1 \u003C t_2 \u003C \\dots \u003C t_n",[34,223740,223742,223797,223852,223870],{"className":223741,"ariaHidden":758},[31672],[34,223743,223745,223748,223788,223791,223794],{"className":223744},[31676],[34,223746],{"className":223747,"style":83872},[31680],[34,223749,223751,223754],{"className":223750},[31685],[34,223752,33249],{"className":223753},[31685,31686],[34,223755,223757],{"className":223756},[31792],[34,223758,223760,223780],{"className":223759},[31796,31797],[34,223761,223763,223777],{"className":223762},[31801],[34,223764,223766],{"className":223765,"style":31806},[31805],[34,223767,223768,223771],{"style":31809},[34,223769],{"className":223770,"style":31814},[31813],[34,223772,223774],{"className":223773},[31818,31819,31820,31821],[34,223775,640],{"className":223776},[31685,31821],[34,223778,31829],{"className":223779},[31828],[34,223781,223783],{"className":223782},[31801],[34,223784,223786],{"className":223785,"style":31836},[31805],[34,223787],{},[34,223789],{"className":223790,"style":31846},[31845],[34,223792,932],{"className":223793},[31850],[34,223795],{"className":223796,"style":31846},[31845],[34,223798,223800,223803,223843,223846,223849],{"className":223799},[31676],[34,223801],{"className":223802,"style":83872},[31680],[34,223804,223806,223809],{"className":223805},[31685],[34,223807,33249],{"className":223808},[31685,31686],[34,223810,223812],{"className":223811},[31792],[34,223813,223815,223835],{"className":223814},[31796,31797],[34,223816,223818,223832],{"className":223817},[31801],[34,223819,223821],{"className":223820,"style":31806},[31805],[34,223822,223823,223826],{"style":31809},[34,223824],{"className":223825,"style":31814},[31813],[34,223827,223829],{"className":223828},[31818,31819,31820,31821],[34,223830,768],{"className":223831},[31685,31821],[34,223833,31829],{"className":223834},[31828],[34,223836,223838],{"className":223837},[31801],[34,223839,223841],{"className":223840,"style":31836},[31805],[34,223842],{},[34,223844],{"className":223845,"style":31846},[31845],[34,223847,932],{"className":223848},[31850],[34,223850],{"className":223851,"style":31846},[31845],[34,223853,223855,223858,223861,223864,223867],{"className":223854},[31676],[34,223856],{"className":223857,"style":34812},[31680],[34,223859,38874],{"className":223860},[34006],[34,223862],{"className":223863,"style":31846},[31845],[34,223865,932],{"className":223866},[31850],[34,223868],{"className":223869,"style":31846},[31845],[34,223871,223873,223876],{"className":223872},[31676],[34,223874],{"className":223875,"style":83872},[31680],[34,223877,223879,223882],{"className":223878},[31685],[34,223880,33249],{"className":223881},[31685,31686],[34,223883,223885],{"className":223884},[31792],[34,223886,223888,223908],{"className":223887},[31796,31797],[34,223889,223891,223905],{"className":223890},[31801],[34,223892,223894],{"className":223893,"style":32362},[31805],[34,223895,223896,223899],{"style":31809},[34,223897],{"className":223898,"style":31814},[31813],[34,223900,223902],{"className":223901},[31818,31819,31820,31821],[34,223903,31660],{"className":223904},[31685,31686,31821],[34,223906,31829],{"className":223907},[31828],[34,223909,223911],{"className":223910},[31801],[34,223912,223914],{"className":223913,"style":31836},[31805],[34,223915],{},"。试证明采用短作业优先算法调度能获得最小平均周转时间。",[253,223918,223920],{"id":223919},"概念核对-concept-check","概念核对 (Concept Check)",[105,223922,223923],{},[108,223924,223925,223928,223929,223957],{},[2627,223926,223927],{},"周转时间 (Turnaround Time)："," 作业从提交到完成的总时间。由于这 ",[34,223930,223932,223945],{"className":223931},[31642],[34,223933,223935],{"className":223934},[31646],[31648,223936,223937],{"xmlns":31650},[31652,223938,223939,223943],{},[31655,223940,223941],{},[31658,223942,33347],{},[31665,223944,33347],{"encoding":31667},[34,223946,223948],{"className":223947,"ariaHidden":758},[31672],[34,223949,223951,223954],{"className":223950},[31676],[34,223952],{"className":223953,"style":33218},[31680],[34,223955,33347],{"className":223956,"style":33389},[31685,31686]," 个作业是“同时”在队列中等待的，所以某个作业的周转时间 = 它自己的等待时间 + 它的运行时间。",[253,223959,223961],{"id":223960},"原理解析与证明-the-why","原理解析与证明 (The 'Why')",[15,223963,223964,223965,223993,223994,224064,224065,224135,224136,224164,224165,224235,224236,224264,224265,31901],{},"假设这 ",[34,223966,223968,223981],{"className":223967},[31642],[34,223969,223971],{"className":223970},[31646],[31648,223972,223973],{"xmlns":31650},[31652,223974,223975,223979],{},[31655,223976,223977],{},[31658,223978,33347],{},[31665,223980,33347],{"encoding":31667},[34,223982,223984],{"className":223983,"ariaHidden":758},[31672],[34,223985,223987,223990],{"className":223986},[31676],[34,223988],{"className":223989,"style":33218},[31680],[34,223991,33347],{"className":223992,"style":33389},[31685,31686]," 个作业按照某种任意的顺序执行，我们把它们实际执行的顺序记为：第1个执行的运行时间为 ",[34,223995,223997,224015],{"className":223996},[31642],[34,223998,224000],{"className":223999},[31646],[31648,224001,224002],{"xmlns":31650},[31652,224003,224004,224012],{},[31655,224005,224006],{},[31743,224007,224008,224010],{},[31658,224009,32936],{},[31749,224011,640],{},[31665,224013,224014],{"encoding":31667},"x_1",[34,224016,224018],{"className":224017,"ariaHidden":758},[31672],[34,224019,224021,224024],{"className":224020},[31676],[34,224022],{"className":224023,"style":31860},[31680],[34,224025,224027,224030],{"className":224026},[31685],[34,224028,32936],{"className":224029},[31685,31686],[34,224031,224033],{"className":224032},[31792],[34,224034,224036,224056],{"className":224035},[31796,31797],[34,224037,224039,224053],{"className":224038},[31801],[34,224040,224042],{"className":224041,"style":31806},[31805],[34,224043,224044,224047],{"style":31809},[34,224045],{"className":224046,"style":31814},[31813],[34,224048,224050],{"className":224049},[31818,31819,31820,31821],[34,224051,640],{"className":224052},[31685,31821],[34,224054,31829],{"className":224055},[31828],[34,224057,224059],{"className":224058},[31801],[34,224060,224062],{"className":224061,"style":31836},[31805],[34,224063],{},"，第2个为 ",[34,224066,224068,224086],{"className":224067},[31642],[34,224069,224071],{"className":224070},[31646],[31648,224072,224073],{"xmlns":31650},[31652,224074,224075,224083],{},[31655,224076,224077],{},[31743,224078,224079,224081],{},[31658,224080,32936],{},[31749,224082,768],{},[31665,224084,224085],{"encoding":31667},"x_2",[34,224087,224089],{"className":224088,"ariaHidden":758},[31672],[34,224090,224092,224095],{"className":224091},[31676],[34,224093],{"className":224094,"style":31860},[31680],[34,224096,224098,224101],{"className":224097},[31685],[34,224099,32936],{"className":224100},[31685,31686],[34,224102,224104],{"className":224103},[31792],[34,224105,224107,224127],{"className":224106},[31796,31797],[34,224108,224110,224124],{"className":224109},[31801],[34,224111,224113],{"className":224112,"style":31806},[31805],[34,224114,224115,224118],{"style":31809},[34,224116],{"className":224117,"style":31814},[31813],[34,224119,224121],{"className":224120},[31818,31819,31820,31821],[34,224122,768],{"className":224123},[31685,31821],[34,224125,31829],{"className":224126},[31828],[34,224128,224130],{"className":224129},[31801],[34,224131,224133],{"className":224132,"style":31836},[31805],[34,224134],{},"，...，第 ",[34,224137,224139,224152],{"className":224138},[31642],[34,224140,224142],{"className":224141},[31646],[31648,224143,224144],{"xmlns":31650},[31652,224145,224146,224150],{},[31655,224147,224148],{},[31658,224149,31660],{},[31665,224151,31660],{"encoding":31667},[34,224153,224155],{"className":224154,"ariaHidden":758},[31672],[34,224156,224158,224161],{"className":224157},[31676],[34,224159],{"className":224160,"style":32527},[31680],[34,224162,31660],{"className":224163},[31685,31686]," 个为 ",[34,224166,224168,224186],{"className":224167},[31642],[34,224169,224171],{"className":224170},[31646],[31648,224172,224173],{"xmlns":31650},[31652,224174,224175,224183],{},[31655,224176,224177],{},[31743,224178,224179,224181],{},[31658,224180,32936],{},[31658,224182,31660],{},[31665,224184,224185],{"encoding":31667},"x_n",[34,224187,224189],{"className":224188,"ariaHidden":758},[31672],[34,224190,224192,224195],{"className":224191},[31676],[34,224193],{"className":224194,"style":31860},[31680],[34,224196,224198,224201],{"className":224197},[31685],[34,224199,32936],{"className":224200},[31685,31686],[34,224202,224204],{"className":224203},[31792],[34,224205,224207,224227],{"className":224206},[31796,31797],[34,224208,224210,224224],{"className":224209},[31801],[34,224211,224213],{"className":224212,"style":32362},[31805],[34,224214,224215,224218],{"style":31809},[34,224216],{"className":224217,"style":31814},[31813],[34,224219,224221],{"className":224220},[31818,31819,31820,31821],[34,224222,31660],{"className":224223},[31685,31686,31821],[34,224225,31829],{"className":224226},[31828],[34,224228,224230],{"className":224229},[31801],[34,224231,224233],{"className":224232,"style":31836},[31805],[34,224234],{},"（注意这里的 ",[34,224237,224239,224252],{"className":224238},[31642],[34,224240,224242],{"className":224241},[31646],[31648,224243,224244],{"xmlns":31650},[31652,224245,224246,224250],{},[31655,224247,224248],{},[31658,224249,32936],{},[31665,224251,32936],{"encoding":31667},[34,224253,224255],{"className":224254,"ariaHidden":758},[31672],[34,224256,224258,224261],{"className":224257},[31676],[34,224259],{"className":224260,"style":32527},[31680],[34,224262,32936],{"className":224263},[31685,31686]," 是打乱后的 ",[34,224266,224268,224281],{"className":224267},[31642],[34,224269,224271],{"className":224270},[31646],[31648,224272,224273],{"xmlns":31650},[31652,224274,224275,224279],{},[31655,224276,224277],{},[31658,224278,33249],{},[31665,224280,33249],{"encoding":31667},[34,224282,224284],{"className":224283,"ariaHidden":758},[31672],[34,224285,224287,224290],{"className":224286},[31676],[34,224288],{"className":224289,"style":33280},[31680],[34,224291,33249],{"className":224292},[31685,31686],[2230,224294,224295,224448,224571],{},[108,224296,224297,224298,224348,224349,224418,224419,224447],{},"第1个作业执行时，后面 ",[34,224299,224301,224318],{"className":224300},[31642],[34,224302,224304],{"className":224303},[31646],[31648,224305,224306],{"xmlns":31650},[31652,224307,224308,224316],{},[31655,224309,224310,224312,224314],{},[31658,224311,31660],{},[31662,224313,32226],{},[31749,224315,640],{},[31665,224317,32587],{"encoding":31667},[34,224319,224321,224339],{"className":224320,"ariaHidden":758},[31672],[34,224322,224324,224327,224330,224333,224336],{"className":224323},[31676],[34,224325],{"className":224326,"style":32597},[31680],[34,224328,31660],{"className":224329},[31685,31686],[34,224331],{"className":224332,"style":32604},[31845],[34,224334,32226],{"className":224335},[32024],[34,224337],{"className":224338,"style":32604},[31845],[34,224340,224342,224345],{"className":224341},[31676],[34,224343],{"className":224344,"style":32617},[31680],[34,224346,640],{"className":224347},[31685]," 个作业都在陪着它等。因此，",[34,224350,224352,224369],{"className":224351},[31642],[34,224353,224355],{"className":224354},[31646],[31648,224356,224357],{"xmlns":31650},[31652,224358,224359,224367],{},[31655,224360,224361],{},[31743,224362,224363,224365],{},[31658,224364,32936],{},[31749,224366,640],{},[31665,224368,224014],{"encoding":31667},[34,224370,224372],{"className":224371,"ariaHidden":758},[31672],[34,224373,224375,224378],{"className":224374},[31676],[34,224376],{"className":224377,"style":31860},[31680],[34,224379,224381,224384],{"className":224380},[31685],[34,224382,32936],{"className":224383},[31685,31686],[34,224385,224387],{"className":224386},[31792],[34,224388,224390,224410],{"className":224389},[31796,31797],[34,224391,224393,224407],{"className":224392},[31801],[34,224394,224396],{"className":224395,"style":31806},[31805],[34,224397,224398,224401],{"style":31809},[34,224399],{"className":224400,"style":31814},[31813],[34,224402,224404],{"className":224403},[31818,31819,31820,31821],[34,224405,640],{"className":224406},[31685,31821],[34,224408,31829],{"className":224409},[31828],[34,224411,224413],{"className":224412},[31801],[34,224414,224416],{"className":224415,"style":31836},[31805],[34,224417],{}," 这段执行时间，被累计计入了 ",[34,224420,224422,224435],{"className":224421},[31642],[34,224423,224425],{"className":224424},[31646],[31648,224426,224427],{"xmlns":31650},[31652,224428,224429,224433],{},[31655,224430,224431],{},[31658,224432,31660],{},[31665,224434,31660],{"encoding":31667},[34,224436,224438],{"className":224437,"ariaHidden":758},[31672],[34,224439,224441,224444],{"className":224440},[31676],[34,224442],{"className":224443,"style":32527},[31680],[34,224445,31660],{"className":224446},[31685,31686]," 个作业的周转时间中。",[108,224449,224450,224451,224520,224521,224447],{},"第2个作业执行时，",[34,224452,224454,224471],{"className":224453},[31642],[34,224455,224457],{"className":224456},[31646],[31648,224458,224459],{"xmlns":31650},[31652,224460,224461,224469],{},[31655,224462,224463],{},[31743,224464,224465,224467],{},[31658,224466,32936],{},[31749,224468,768],{},[31665,224470,224085],{"encoding":31667},[34,224472,224474],{"className":224473,"ariaHidden":758},[31672],[34,224475,224477,224480],{"className":224476},[31676],[34,224478],{"className":224479,"style":31860},[31680],[34,224481,224483,224486],{"className":224482},[31685],[34,224484,32936],{"className":224485},[31685,31686],[34,224487,224489],{"className":224488},[31792],[34,224490,224492,224512],{"className":224491},[31796,31797],[34,224493,224495,224509],{"className":224494},[31801],[34,224496,224498],{"className":224497,"style":31806},[31805],[34,224499,224500,224503],{"style":31809},[34,224501],{"className":224502,"style":31814},[31813],[34,224504,224506],{"className":224505},[31818,31819,31820,31821],[34,224507,768],{"className":224508},[31685,31821],[34,224510,31829],{"className":224511},[31828],[34,224513,224515],{"className":224514},[31801],[34,224516,224518],{"className":224517,"style":31836},[31805],[34,224519],{}," 被累计计入了 ",[34,224522,224524,224541],{"className":224523},[31642],[34,224525,224527],{"className":224526},[31646],[31648,224528,224529],{"xmlns":31650},[31652,224530,224531,224539],{},[31655,224532,224533,224535,224537],{},[31658,224534,31660],{},[31662,224536,32226],{},[31749,224538,640],{},[31665,224540,32587],{"encoding":31667},[34,224542,224544,224562],{"className":224543,"ariaHidden":758},[31672],[34,224545,224547,224550,224553,224556,224559],{"className":224546},[31676],[34,224548],{"className":224549,"style":32597},[31680],[34,224551,31660],{"className":224552},[31685,31686],[34,224554],{"className":224555,"style":32604},[31845],[34,224557,32226],{"className":224558},[32024],[34,224560],{"className":224561,"style":32604},[31845],[34,224563,224565,224568],{"className":224564},[31676],[34,224566],{"className":224567,"style":32617},[31680],[34,224569,640],{"className":224570},[31685],[108,224572,224573,224574,224602,224603,224672,224673,89490],{},"依此类推，第 ",[34,224575,224577,224590],{"className":224576},[31642],[34,224578,224580],{"className":224579},[31646],[31648,224581,224582],{"xmlns":31650},[31652,224583,224584,224588],{},[31655,224585,224586],{},[31658,224587,31660],{},[31665,224589,31660],{"encoding":31667},[34,224591,224593],{"className":224592,"ariaHidden":758},[31672],[34,224594,224596,224599],{"className":224595},[31676],[34,224597],{"className":224598,"style":32527},[31680],[34,224600,31660],{"className":224601},[31685,31686]," 个作业执行时，只有它自己在跑， ",[34,224604,224606,224623],{"className":224605},[31642],[34,224607,224609],{"className":224608},[31646],[31648,224610,224611],{"xmlns":31650},[31652,224612,224613,224621],{},[31655,224614,224615],{},[31743,224616,224617,224619],{},[31658,224618,32936],{},[31658,224620,31660],{},[31665,224622,224185],{"encoding":31667},[34,224624,224626],{"className":224625,"ariaHidden":758},[31672],[34,224627,224629,224632],{"className":224628},[31676],[34,224630],{"className":224631,"style":31860},[31680],[34,224633,224635,224638],{"className":224634},[31685],[34,224636,32936],{"className":224637},[31685,31686],[34,224639,224641],{"className":224640},[31792],[34,224642,224644,224664],{"className":224643},[31796,31797],[34,224645,224647,224661],{"className":224646},[31801],[34,224648,224650],{"className":224649,"style":32362},[31805],[34,224651,224652,224655],{"style":31809},[34,224653],{"className":224654,"style":31814},[31813],[34,224656,224658],{"className":224657},[31818,31819,31820,31821],[34,224659,31660],{"className":224660},[31685,31686,31821],[34,224662,31829],{"className":224663},[31828],[34,224665,224667],{"className":224666},[31801],[34,224668,224670],{"className":224669,"style":31836},[31805],[34,224671],{}," 被累计计入 ",[34,224674,224676,224689],{"className":224675},[31642],[34,224677,224679],{"className":224678},[31646],[31648,224680,224681],{"xmlns":31650},[31652,224682,224683,224687],{},[31655,224684,224685],{},[31749,224686,640],{},[31665,224688,640],{"encoding":31667},[34,224690,224692],{"className":224691,"ariaHidden":758},[31672],[34,224693,224695,224698],{"className":224694},[31676],[34,224696],{"className":224697,"style":32617},[31680],[34,224699,640],{"className":224700},[31685],[15,224702,224703,224704,224743,224744,33183,224747,225213,225214,347],{},"总周转时间 ",[34,224705,224707,224725],{"className":224706},[31642],[34,224708,224710],{"className":224709},[31646],[31648,224711,224712],{"xmlns":31650},[31652,224713,224714,224722],{},[31655,224715,224716,224718,224720],{},[31658,224717,94675],{},[31658,224719,94628],{},[31658,224721,31759],{},[31665,224723,224724],{"encoding":31667},"Sum",[34,224726,224728],{"className":224727,"ariaHidden":758},[31672],[34,224729,224731,224734,224737,224740],{"className":224730},[31676],[34,224732],{"className":224733,"style":33218},[31680],[34,224735,94675],{"className":224736,"style":106455},[31685,31686],[34,224738,94628],{"className":224739},[31685,31686],[34,224741,31759],{"className":224742},[31685,31686]," 的公式可以写为 ",[34,224745,224746],{},"cite: 160",[34,224748,224750,224836],{"className":224749},[31642],[34,224751,224753],{"className":224752},[31646],[31648,224754,224755],{"xmlns":31650},[31652,224756,224757,224833],{},[31655,224758,224759,224761,224763,224765,224767,224769,224771,224777,224779,224781,224783,224785,224787,224789,224791,224797,224799,224801,224803,224805,224807,224809,224811,224817,224819,224821,224823,224825,224827],{},[31658,224760,94675],{},[31658,224762,94628],{},[31658,224764,31759],{},[31662,224766,202],{},[31658,224768,31660],{},[31662,224770,71191],{},[31743,224772,224773,224775],{},[31658,224774,32936],{},[31749,224776,640],{},[31662,224778,223],{},[31662,224780,456],{"stretchy":702},[31658,224782,31660],{},[31662,224784,32226],{},[31749,224786,640],{},[31662,224788,2557],{"stretchy":702},[31662,224790,71191],{},[31743,224792,224793,224795],{},[31658,224794,32936],{},[31749,224796,768],{},[31662,224798,223],{},[31662,224800,456],{"stretchy":702},[31658,224802,31660],{},[31662,224804,32226],{},[31749,224806,768],{},[31662,224808,2557],{"stretchy":702},[31662,224810,71191],{},[31743,224812,224813,224815],{},[31658,224814,32936],{},[31749,224816,786],{},[31662,224818,223],{},[31662,224820,38874],{},[31662,224822,223],{},[31749,224824,640],{},[31662,224826,71191],{},[31743,224828,224829,224831],{},[31658,224830,32936],{},[31658,224832,31660],{},[31665,224834,224835],{"encoding":31667},"Sum = n \\cdot x_1 + (n-1) \\cdot x_2 + (n-2) \\cdot x_3 + \\dots + 1 \\cdot x_n",[34,224837,224839,224863,224882,224937,224958,224979,225034,225055,225076,225131,225149,225167],{"className":224838,"ariaHidden":758},[31672],[34,224840,224842,224845,224848,224851,224854,224857,224860],{"className":224841},[31676],[34,224843],{"className":224844,"style":33218},[31680],[34,224846,94675],{"className":224847,"style":106455},[31685,31686],[34,224849,94628],{"className":224850},[31685,31686],[34,224852,31759],{"className":224853},[31685,31686],[34,224855],{"className":224856,"style":31846},[31845],[34,224858,202],{"className":224859},[31850],[34,224861],{"className":224862,"style":31846},[31845],[34,224864,224866,224870,224873,224876,224879],{"className":224865},[31676],[34,224867],{"className":224868,"style":224869},[31680],"height:0.4445em;",[34,224871,31660],{"className":224872},[31685,31686],[34,224874],{"className":224875,"style":32604},[31845],[34,224877,71191],{"className":224878},[32024],[34,224880],{"className":224881,"style":32604},[31845],[34,224883,224885,224888,224928,224931,224934],{"className":224884},[31676],[34,224886],{"className":224887,"style":40522},[31680],[34,224889,224891,224894],{"className":224890},[31685],[34,224892,32936],{"className":224893},[31685,31686],[34,224895,224897],{"className":224896},[31792],[34,224898,224900,224920],{"className":224899},[31796,31797],[34,224901,224903,224917],{"className":224902},[31801],[34,224904,224906],{"className":224905,"style":31806},[31805],[34,224907,224908,224911],{"style":31809},[34,224909],{"className":224910,"style":31814},[31813],[34,224912,224914],{"className":224913},[31818,31819,31820,31821],[34,224915,640],{"className":224916},[31685,31821],[34,224918,31829],{"className":224919},[31828],[34,224921,224923],{"className":224922},[31801],[34,224924,224926],{"className":224925,"style":31836},[31805],[34,224927],{},[34,224929],{"className":224930,"style":32604},[31845],[34,224932,223],{"className":224933},[32024],[34,224935],{"className":224936,"style":32604},[31845],[34,224938,224940,224943,224946,224949,224952,224955],{"className":224939},[31676],[34,224941],{"className":224942,"style":31774},[31680],[34,224944,456],{"className":224945},[31782],[34,224947,31660],{"className":224948},[31685,31686],[34,224950],{"className":224951,"style":32604},[31845],[34,224953,32226],{"className":224954},[32024],[34,224956],{"className":224957,"style":32604},[31845],[34,224959,224961,224964,224967,224970,224973,224976],{"className":224960},[31676],[34,224962],{"className":224963,"style":31774},[31680],[34,224965,640],{"className":224966},[31685],[34,224968,2557],{"className":224969},[31690],[34,224971],{"className":224972,"style":32604},[31845],[34,224974,71191],{"className":224975},[32024],[34,224977],{"className":224978,"style":32604},[31845],[34,224980,224982,224985,225025,225028,225031],{"className":224981},[31676],[34,224983],{"className":224984,"style":40522},[31680],[34,224986,224988,224991],{"className":224987},[31685],[34,224989,32936],{"className":224990},[31685,31686],[34,224992,224994],{"className":224993},[31792],[34,224995,224997,225017],{"className":224996},[31796,31797],[34,224998,225000,225014],{"className":224999},[31801],[34,225001,225003],{"className":225002,"style":31806},[31805],[34,225004,225005,225008],{"style":31809},[34,225006],{"className":225007,"style":31814},[31813],[34,225009,225011],{"className":225010},[31818,31819,31820,31821],[34,225012,768],{"className":225013},[31685,31821],[34,225015,31829],{"className":225016},[31828],[34,225018,225020],{"className":225019},[31801],[34,225021,225023],{"className":225022,"style":31836},[31805],[34,225024],{},[34,225026],{"className":225027,"style":32604},[31845],[34,225029,223],{"className":225030},[32024],[34,225032],{"className":225033,"style":32604},[31845],[34,225035,225037,225040,225043,225046,225049,225052],{"className":225036},[31676],[34,225038],{"className":225039,"style":31774},[31680],[34,225041,456],{"className":225042},[31782],[34,225044,31660],{"className":225045},[31685,31686],[34,225047],{"className":225048,"style":32604},[31845],[34,225050,32226],{"className":225051},[32024],[34,225053],{"className":225054,"style":32604},[31845],[34,225056,225058,225061,225064,225067,225070,225073],{"className":225057},[31676],[34,225059],{"className":225060,"style":31774},[31680],[34,225062,768],{"className":225063},[31685],[34,225065,2557],{"className":225066},[31690],[34,225068],{"className":225069,"style":32604},[31845],[34,225071,71191],{"className":225072},[32024],[34,225074],{"className":225075,"style":32604},[31845],[34,225077,225079,225082,225122,225125,225128],{"className":225078},[31676],[34,225080],{"className":225081,"style":40522},[31680],[34,225083,225085,225088],{"className":225084},[31685],[34,225086,32936],{"className":225087},[31685,31686],[34,225089,225091],{"className":225090},[31792],[34,225092,225094,225114],{"className":225093},[31796,31797],[34,225095,225097,225111],{"className":225096},[31801],[34,225098,225100],{"className":225099,"style":31806},[31805],[34,225101,225102,225105],{"style":31809},[34,225103],{"className":225104,"style":31814},[31813],[34,225106,225108],{"className":225107},[31818,31819,31820,31821],[34,225109,786],{"className":225110},[31685,31821],[34,225112,31829],{"className":225113},[31828],[34,225115,225117],{"className":225116},[31801],[34,225118,225120],{"className":225119,"style":31836},[31805],[34,225121],{},[34,225123],{"className":225124,"style":32604},[31845],[34,225126,223],{"className":225127},[32024],[34,225129],{"className":225130,"style":32604},[31845],[34,225132,225134,225137,225140,225143,225146],{"className":225133},[31676],[34,225135],{"className":225136,"style":32597},[31680],[34,225138,38874],{"className":225139},[34006],[34,225141],{"className":225142,"style":32604},[31845],[34,225144,223],{"className":225145},[32024],[34,225147],{"className":225148,"style":32604},[31845],[34,225150,225152,225155,225158,225161,225164],{"className":225151},[31676],[34,225153],{"className":225154,"style":32617},[31680],[34,225156,640],{"className":225157},[31685],[34,225159],{"className":225160,"style":32604},[31845],[34,225162,71191],{"className":225163},[32024],[34,225165],{"className":225166,"style":32604},[31845],[34,225168,225170,225173],{"className":225169},[31676],[34,225171],{"className":225172,"style":31860},[31680],[34,225174,225176,225179],{"className":225175},[31685],[34,225177,32936],{"className":225178},[31685,31686],[34,225180,225182],{"className":225181},[31792],[34,225183,225185,225205],{"className":225184},[31796,31797],[34,225186,225188,225202],{"className":225187},[31801],[34,225189,225191],{"className":225190,"style":32362},[31805],[34,225192,225193,225196],{"style":31809},[34,225194],{"className":225195,"style":31814},[31813],[34,225197,225199],{"className":225198},[31818,31819,31820,31821],[34,225200,31660],{"className":225201},[31685,31686,31821],[34,225203,31829],{"className":225204},[31828],[34,225206,225208],{"className":225207},[31801],[34,225209,225211],{"className":225210,"style":31836},[31805],[34,225212],{},"\n平均周转时间 ",[34,225215,225217,225243],{"className":225216},[31642],[34,225218,225220],{"className":225219},[31646],[31648,225221,225222],{"xmlns":31650},[31652,225223,225224,225240],{},[31655,225225,225226,225228,225230,225232,225234,225236,225238],{},[31658,225227,37400],{},[31662,225229,202],{},[31658,225231,94675],{},[31658,225233,94628],{},[31658,225235,31759],{},[31658,225237,38080],{"mathvariant":38079},[31658,225239,31660],{},[31665,225241,225242],{"encoding":31667},"T = Sum \u002F n",[34,225244,225246,225264],{"className":225245,"ariaHidden":758},[31672],[34,225247,225249,225252,225255,225258,225261],{"className":225248},[31676],[34,225250],{"className":225251,"style":33218},[31680],[34,225253,37400],{"className":225254,"style":37422},[31685,31686],[34,225256],{"className":225257,"style":31846},[31845],[34,225259,202],{"className":225260},[31850],[34,225262],{"className":225263,"style":31846},[31845],[34,225265,225267,225270,225273,225276,225279,225282],{"className":225266},[31676],[34,225268],{"className":225269,"style":31774},[31680],[34,225271,94675],{"className":225272,"style":106455},[31685,31686],[34,225274,94628],{"className":225275},[31685,31686],[34,225277,31759],{"className":225278},[31685,31686],[34,225280,38080],{"className":225281},[31685],[34,225283,31660],{"className":225284},[31685,31686],[15,225286,225287,225290,225291,225329,225330,225358,225359,225362,225363,347],{},[2627,225288,225289],{},"证明：","\n为了让 ",[34,225292,225294,225311],{"className":225293},[31642],[34,225295,225297],{"className":225296},[31646],[31648,225298,225299],{"xmlns":31650},[31652,225300,225301,225309],{},[31655,225302,225303,225305,225307],{},[31658,225304,94675],{},[31658,225306,94628],{},[31658,225308,31759],{},[31665,225310,224724],{"encoding":31667},[34,225312,225314],{"className":225313,"ariaHidden":758},[31672],[34,225315,225317,225320,225323,225326],{"className":225316},[31676],[34,225318],{"className":225319,"style":33218},[31680],[34,225321,94675],{"className":225322,"style":106455},[31685,31686],[34,225324,94628],{"className":225325},[31685,31686],[34,225327,31759],{"className":225328},[31685,31686],"（也就是平均周转时间 ",[34,225331,225333,225346],{"className":225332},[31642],[34,225334,225336],{"className":225335},[31646],[31648,225337,225338],{"xmlns":31650},[31652,225339,225340,225344],{},[31655,225341,225342],{},[31658,225343,37400],{},[31665,225345,37400],{"encoding":31667},[34,225347,225349],{"className":225348,"ariaHidden":758},[31672],[34,225350,225352,225355],{"className":225351},[31676],[34,225353],{"className":225354,"style":33218},[31680],[34,225356,37400],{"className":225357,"style":37422},[31685,31686],"）尽可能地小，根据数学上的",[2627,225360,225361],{},"排序不等式原理","（或贪心思想）：我们必须把",[2627,225364,225365],{},"最小的值分配给最大的权重（乘数）",[105,225367,225368,225533,225720],{},[108,225369,225370,225371,225399,225400,347],{},"最大的乘数是 ",[34,225372,225374,225387],{"className":225373},[31642],[34,225375,225377],{"className":225376},[31646],[31648,225378,225379],{"xmlns":31650},[31652,225380,225381,225385],{},[31655,225382,225383],{},[31658,225384,31660],{},[31665,225386,31660],{"encoding":31667},[34,225388,225390],{"className":225389,"ariaHidden":758},[31672],[34,225391,225393,225396],{"className":225392},[31676],[34,225394],{"className":225395,"style":32527},[31680],[34,225397,31660],{"className":225398},[31685,31686],"，所以应该把最短的时间分配给它，即 ",[34,225401,225403,225429],{"className":225402},[31642],[34,225404,225406],{"className":225405},[31646],[31648,225407,225408],{"xmlns":31650},[31652,225409,225410,225426],{},[31655,225411,225412,225418,225420],{},[31743,225413,225414,225416],{},[31658,225415,32936],{},[31749,225417,640],{},[31662,225419,202],{},[31743,225421,225422,225424],{},[31658,225423,33249],{},[31749,225425,640],{},[31665,225427,225428],{"encoding":31667},"x_1 = t_1",[34,225430,225432,225487],{"className":225431,"ariaHidden":758},[31672],[34,225433,225435,225438,225478,225481,225484],{"className":225434},[31676],[34,225436],{"className":225437,"style":31860},[31680],[34,225439,225441,225444],{"className":225440},[31685],[34,225442,32936],{"className":225443},[31685,31686],[34,225445,225447],{"className":225446},[31792],[34,225448,225450,225470],{"className":225449},[31796,31797],[34,225451,225453,225467],{"className":225452},[31801],[34,225454,225456],{"className":225455,"style":31806},[31805],[34,225457,225458,225461],{"style":31809},[34,225459],{"className":225460,"style":31814},[31813],[34,225462,225464],{"className":225463},[31818,31819,31820,31821],[34,225465,640],{"className":225466},[31685,31821],[34,225468,31829],{"className":225469},[31828],[34,225471,225473],{"className":225472},[31801],[34,225474,225476],{"className":225475,"style":31836},[31805],[34,225477],{},[34,225479],{"className":225480,"style":31846},[31845],[34,225482,202],{"className":225483},[31850],[34,225485],{"className":225486,"style":31846},[31845],[34,225488,225490,225493],{"className":225489},[31676],[34,225491],{"className":225492,"style":83872},[31680],[34,225494,225496,225499],{"className":225495},[31685],[34,225497,33249],{"className":225498},[31685,31686],[34,225500,225502],{"className":225501},[31792],[34,225503,225505,225525],{"className":225504},[31796,31797],[34,225506,225508,225522],{"className":225507},[31801],[34,225509,225511],{"className":225510,"style":31806},[31805],[34,225512,225513,225516],{"style":31809},[34,225514],{"className":225515,"style":31814},[31813],[34,225517,225519],{"className":225518},[31818,31819,31820,31821],[34,225520,640],{"className":225521},[31685,31821],[34,225523,31829],{"className":225524},[31828],[34,225526,225528],{"className":225527},[31801],[34,225529,225531],{"className":225530,"style":31836},[31805],[34,225532],{},[108,225534,225535,225536,225586,225587,347],{},"第二大的乘数是 ",[34,225537,225539,225556],{"className":225538},[31642],[34,225540,225542],{"className":225541},[31646],[31648,225543,225544],{"xmlns":31650},[31652,225545,225546,225554],{},[31655,225547,225548,225550,225552],{},[31658,225549,31660],{},[31662,225551,32226],{},[31749,225553,640],{},[31665,225555,32587],{"encoding":31667},[34,225557,225559,225577],{"className":225558,"ariaHidden":758},[31672],[34,225560,225562,225565,225568,225571,225574],{"className":225561},[31676],[34,225563],{"className":225564,"style":32597},[31680],[34,225566,31660],{"className":225567},[31685,31686],[34,225569],{"className":225570,"style":32604},[31845],[34,225572,32226],{"className":225573},[32024],[34,225575],{"className":225576,"style":32604},[31845],[34,225578,225580,225583],{"className":225579},[31676],[34,225581],{"className":225582,"style":32617},[31680],[34,225584,640],{"className":225585},[31685],"，所以应该把第二短的时间分配给它，即 ",[34,225588,225590,225616],{"className":225589},[31642],[34,225591,225593],{"className":225592},[31646],[31648,225594,225595],{"xmlns":31650},[31652,225596,225597,225613],{},[31655,225598,225599,225605,225607],{},[31743,225600,225601,225603],{},[31658,225602,32936],{},[31749,225604,768],{},[31662,225606,202],{},[31743,225608,225609,225611],{},[31658,225610,33249],{},[31749,225612,768],{},[31665,225614,225615],{"encoding":31667},"x_2 = t_2",[34,225617,225619,225674],{"className":225618,"ariaHidden":758},[31672],[34,225620,225622,225625,225665,225668,225671],{"className":225621},[31676],[34,225623],{"className":225624,"style":31860},[31680],[34,225626,225628,225631],{"className":225627},[31685],[34,225629,32936],{"className":225630},[31685,31686],[34,225632,225634],{"className":225633},[31792],[34,225635,225637,225657],{"className":225636},[31796,31797],[34,225638,225640,225654],{"className":225639},[31801],[34,225641,225643],{"className":225642,"style":31806},[31805],[34,225644,225645,225648],{"style":31809},[34,225646],{"className":225647,"style":31814},[31813],[34,225649,225651],{"className":225650},[31818,31819,31820,31821],[34,225652,768],{"className":225653},[31685,31821],[34,225655,31829],{"className":225656},[31828],[34,225658,225660],{"className":225659},[31801],[34,225661,225663],{"className":225662,"style":31836},[31805],[34,225664],{},[34,225666],{"className":225667,"style":31846},[31845],[34,225669,202],{"className":225670},[31850],[34,225672],{"className":225673,"style":31846},[31845],[34,225675,225677,225680],{"className":225676},[31676],[34,225678],{"className":225679,"style":83872},[31680],[34,225681,225683,225686],{"className":225682},[31685],[34,225684,33249],{"className":225685},[31685,31686],[34,225687,225689],{"className":225688},[31792],[34,225690,225692,225712],{"className":225691},[31796,31797],[34,225693,225695,225709],{"className":225694},[31801],[34,225696,225698],{"className":225697,"style":31806},[31805],[34,225699,225700,225703],{"style":31809},[34,225701],{"className":225702,"style":31814},[31813],[34,225704,225706],{"className":225705},[31818,31819,31820,31821],[34,225707,768],{"className":225708},[31685,31821],[34,225710,31829],{"className":225711},[31828],[34,225713,225715],{"className":225714},[31801],[34,225716,225718],{"className":225717,"style":31836},[31805],[34,225719],{},[108,225721,225722,225723,225942,225943,225946,225947],{},"以此类推，最终的执行顺序必须严格遵守 ",[34,225724,225726,225764],{"className":225725},[31642],[34,225727,225729],{"className":225728},[31646],[31648,225730,225731],{"xmlns":31650},[31652,225732,225733,225761],{},[31655,225734,225735,225741,225743,225749,225751,225753,225755],{},[31743,225736,225737,225739],{},[31658,225738,32936],{},[31749,225740,640],{},[31662,225742,932],{},[31743,225744,225745,225747],{},[31658,225746,32936],{},[31749,225748,768],{},[31662,225750,932],{},[31662,225752,38874],{},[31662,225754,932],{},[31743,225756,225757,225759],{},[31658,225758,32936],{},[31658,225760,31660],{},[31665,225762,225763],{"encoding":31667},"x_1 \u003C x_2 \u003C \\dots \u003C x_n",[34,225765,225767,225823,225878,225896],{"className":225766,"ariaHidden":758},[31672],[34,225768,225770,225774,225814,225817,225820],{"className":225769},[31676],[34,225771],{"className":225772,"style":225773},[31680],"height:0.6891em;vertical-align:-0.15em;",[34,225775,225777,225780],{"className":225776},[31685],[34,225778,32936],{"className":225779},[31685,31686],[34,225781,225783],{"className":225782},[31792],[34,225784,225786,225806],{"className":225785},[31796,31797],[34,225787,225789,225803],{"className":225788},[31801],[34,225790,225792],{"className":225791,"style":31806},[31805],[34,225793,225794,225797],{"style":31809},[34,225795],{"className":225796,"style":31814},[31813],[34,225798,225800],{"className":225799},[31818,31819,31820,31821],[34,225801,640],{"className":225802},[31685,31821],[34,225804,31829],{"className":225805},[31828],[34,225807,225809],{"className":225808},[31801],[34,225810,225812],{"className":225811,"style":31836},[31805],[34,225813],{},[34,225815],{"className":225816,"style":31846},[31845],[34,225818,932],{"className":225819},[31850],[34,225821],{"className":225822,"style":31846},[31845],[34,225824,225826,225829,225869,225872,225875],{"className":225825},[31676],[34,225827],{"className":225828,"style":225773},[31680],[34,225830,225832,225835],{"className":225831},[31685],[34,225833,32936],{"className":225834},[31685,31686],[34,225836,225838],{"className":225837},[31792],[34,225839,225841,225861],{"className":225840},[31796,31797],[34,225842,225844,225858],{"className":225843},[31801],[34,225845,225847],{"className":225846,"style":31806},[31805],[34,225848,225849,225852],{"style":31809},[34,225850],{"className":225851,"style":31814},[31813],[34,225853,225855],{"className":225854},[31818,31819,31820,31821],[34,225856,768],{"className":225857},[31685,31821],[34,225859,31829],{"className":225860},[31828],[34,225862,225864],{"className":225863},[31801],[34,225865,225867],{"className":225866,"style":31836},[31805],[34,225868],{},[34,225870],{"className":225871,"style":31846},[31845],[34,225873,932],{"className":225874},[31850],[34,225876],{"className":225877,"style":31846},[31845],[34,225879,225881,225884,225887,225890,225893],{"className":225880},[31676],[34,225882],{"className":225883,"style":34812},[31680],[34,225885,38874],{"className":225886},[34006],[34,225888],{"className":225889,"style":31846},[31845],[34,225891,932],{"className":225892},[31850],[34,225894],{"className":225895,"style":31846},[31845],[34,225897,225899,225902],{"className":225898},[31676],[34,225900],{"className":225901,"style":31860},[31680],[34,225903,225905,225908],{"className":225904},[31685],[34,225906,32936],{"className":225907},[31685,31686],[34,225909,225911],{"className":225910},[31792],[34,225912,225914,225934],{"className":225913},[31796,31797],[34,225915,225917,225931],{"className":225916},[31801],[34,225918,225920],{"className":225919,"style":32362},[31805],[34,225921,225922,225925],{"style":31809},[34,225923],{"className":225924,"style":31814},[31813],[34,225926,225928],{"className":225927},[31818,31819,31820,31821],[34,225929,31660],{"className":225930},[31685,31686,31821],[34,225932,31829],{"className":225933},[31828],[34,225935,225937],{"className":225936},[31801],[34,225938,225940],{"className":225939,"style":31836},[31805],[34,225941],{},"。\n这恰恰就是",[2627,225944,225945],{},"短作业优先 (SJF)"," 的调度逻辑。",[2627,225948,225949],{},"证毕。",[32188,225951],{},[19,225953,225955],{"id":225954},"_5单处理器经典调度算法模拟","5、单处理器经典调度算法模拟",[15,225957,225958,225960],{},[2627,225959,214361],{},"\n5个作业在时刻0按次序1、2、3、4、5进入系统。运行时间分别为10, 1, 2, 1, 5；优先权分别为3, 1, 3, 4, 2（数值小优先权高）。求 FCFS、RR、SJF、非抢占优先权调度算法的执行次序、平均周转时间和平均带权周转时间。",[253,225962,225963],{"id":225963},"详细讲解与答案",[15,225965,225966],{},"由于作业同时在时刻0到达，周转时间 = 完成时间，带权周转时间 = 周转时间 \u002F 执行时间。",[15,225968,225969],{},[2627,225970,225971],{},"(1) 先来先服务 (FCFS)",[105,225973,225974,226096,226102,226108,226303],{},[108,225975,225976,225979,225980,226008,226009,226037,226038,226066,226067,226095],{},[2627,225977,225978],{},"执行次序："," 1 ",[34,225981,225983,225996],{"className":225982},[31642],[34,225984,225986],{"className":225985},[31646],[31648,225987,225988],{"xmlns":31650},[31652,225989,225990,225994],{},[31655,225991,225992],{},[31662,225993,33239],{},[31665,225995,119689],{"encoding":31667},[34,225997,225999],{"className":225998,"ariaHidden":758},[31672],[34,226000,226002,226005],{"className":226001},[31676],[34,226003],{"className":226004,"style":73089},[31680],[34,226006,33239],{"className":226007},[31850]," 2 ",[34,226010,226012,226025],{"className":226011},[31642],[34,226013,226015],{"className":226014},[31646],[31648,226016,226017],{"xmlns":31650},[31652,226018,226019,226023],{},[31655,226020,226021],{},[31662,226022,33239],{},[31665,226024,119689],{"encoding":31667},[34,226026,226028],{"className":226027,"ariaHidden":758},[31672],[34,226029,226031,226034],{"className":226030},[31676],[34,226032],{"className":226033,"style":73089},[31680],[34,226035,33239],{"className":226036},[31850]," 3 ",[34,226039,226041,226054],{"className":226040},[31642],[34,226042,226044],{"className":226043},[31646],[31648,226045,226046],{"xmlns":31650},[31652,226047,226048,226052],{},[31655,226049,226050],{},[31662,226051,33239],{},[31665,226053,119689],{"encoding":31667},[34,226055,226057],{"className":226056,"ariaHidden":758},[31672],[34,226058,226060,226063],{"className":226059},[31676],[34,226061],{"className":226062,"style":73089},[31680],[34,226064,33239],{"className":226065},[31850]," 4 ",[34,226068,226070,226083],{"className":226069},[31642],[34,226071,226073],{"className":226072},[31646],[31648,226074,226075],{"xmlns":31650},[31652,226076,226077,226081],{},[31655,226078,226079],{},[31662,226080,33239],{},[31665,226082,119689],{"encoding":31667},[34,226084,226086],{"className":226085,"ariaHidden":758},[31672],[34,226087,226089,226092],{"className":226088},[31676],[34,226090],{"className":226091,"style":73089},[31680],[34,226093,33239],{"className":226094},[31850]," 5",[108,226097,226098,226101],{},[2627,226099,226100],{},"完成时间 (周转时间)："," 1:10, 2:11, 3:13, 4:14, 5:19",[108,226103,226104,226107],{},[2627,226105,226106],{},"带权周转："," 1:(10\u002F10=1), 2:(11\u002F1=11), 3:(13\u002F2=6.5), 4:(14\u002F1=14), 5:(19\u002F5=3.8)",[108,226109,226110,47649,226113],{},[2627,226111,226112],{},"平均周转时间：",[34,226114,226116,226168],{"className":226115},[31642],[34,226117,226119],{"className":226118},[31646],[31648,226120,226121],{"xmlns":31650},[31652,226122,226123,226165],{},[31655,226124,226125,226127,226129,226131,226133,226135,226137,226139,226141,226143,226145,226147,226149,226151,226153,226155,226157,226159,226161],{},[31662,226126,456],{"stretchy":702},[31749,226128,3244],{},[31662,226130,223],{},[31749,226132,39620],{},[31662,226134,223],{},[31749,226136,205234],{},[31662,226138,223],{},[31749,226140,164756],{},[31662,226142,223],{},[31749,226144,162590],{},[31662,226146,2557],{"stretchy":702},[31658,226148,38080],{"mathvariant":38079},[31749,226150,18136],{},[31662,226152,202],{},[31749,226154,99870],{},[31658,226156,38080],{"mathvariant":38079},[31749,226158,18136],{},[31662,226160,202],{},[31749,226162,226164],{"mathvariant":226163},"bold","13.4",[31665,226166,226167],{"encoding":31667},"(10+11+13+14+19) \u002F 5 = 67 \u002F 5 = \\mathbf{13.4}",[34,226169,226171,226192,226210,226228,226246,226271,226290],{"className":226170,"ariaHidden":758},[31672],[34,226172,226174,226177,226180,226183,226186,226189],{"className":226173},[31676],[34,226175],{"className":226176,"style":31774},[31680],[34,226178,456],{"className":226179},[31782],[34,226181,3244],{"className":226182},[31685],[34,226184],{"className":226185,"style":32604},[31845],[34,226187,223],{"className":226188},[32024],[34,226190],{"className":226191,"style":32604},[31845],[34,226193,226195,226198,226201,226204,226207],{"className":226194},[31676],[34,226196],{"className":226197,"style":35942},[31680],[34,226199,39620],{"className":226200},[31685],[34,226202],{"className":226203,"style":32604},[31845],[34,226205,223],{"className":226206},[32024],[34,226208],{"className":226209,"style":32604},[31845],[34,226211,226213,226216,226219,226222,226225],{"className":226212},[31676],[34,226214],{"className":226215,"style":35942},[31680],[34,226217,205234],{"className":226218},[31685],[34,226220],{"className":226221,"style":32604},[31845],[34,226223,223],{"className":226224},[32024],[34,226226],{"className":226227,"style":32604},[31845],[34,226229,226231,226234,226237,226240,226243],{"className":226230},[31676],[34,226232],{"className":226233,"style":35942},[31680],[34,226235,164756],{"className":226236},[31685],[34,226238],{"className":226239,"style":32604},[31845],[34,226241,223],{"className":226242},[32024],[34,226244],{"className":226245,"style":32604},[31845],[34,226247,226249,226252,226255,226258,226262,226265,226268],{"className":226248},[31676],[34,226250],{"className":226251,"style":31774},[31680],[34,226253,162590],{"className":226254},[31685],[34,226256,2557],{"className":226257},[31690],[34,226259,226261],{"className":226260},[31685],"\u002F5",[34,226263],{"className":226264,"style":31846},[31845],[34,226266,202],{"className":226267},[31850],[34,226269],{"className":226270,"style":31846},[31845],[34,226272,226274,226277,226281,226284,226287],{"className":226273},[31676],[34,226275],{"className":226276,"style":31774},[31680],[34,226278,226280],{"className":226279},[31685],"67\u002F5",[34,226282],{"className":226283,"style":31846},[31845],[34,226285,202],{"className":226286},[31850],[34,226288],{"className":226289,"style":31846},[31845],[34,226291,226293,226296],{"className":226292},[31676],[34,226294],{"className":226295,"style":32617},[31680],[34,226297,226299],{"className":226298},[31685],[34,226300,226164],{"className":226301},[31685,226302],"mathbf",[108,226304,226305,47649,226308],{},[2627,226306,226307],{},"平均带权周转：",[34,226309,226311,226365],{"className":226310},[31642],[34,226312,226314],{"className":226313},[31646],[31648,226315,226316],{"xmlns":31650},[31652,226317,226318,226362],{},[31655,226319,226320,226322,226324,226326,226328,226330,226333,226335,226337,226339,226342,226344,226346,226348,226350,226353,226355,226357,226359],{},[31662,226321,456],{"stretchy":702},[31749,226323,640],{},[31662,226325,223],{},[31749,226327,39620],{},[31662,226329,223],{},[31749,226331,226332],{},"6.5",[31662,226334,223],{},[31749,226336,164756],{},[31662,226338,223],{},[31749,226340,226341],{},"3.8",[31662,226343,2557],{"stretchy":702},[31658,226345,38080],{"mathvariant":38079},[31749,226347,18136],{},[31662,226349,202],{},[31749,226351,226352],{},"36.3",[31658,226354,38080],{"mathvariant":38079},[31749,226356,18136],{},[31662,226358,202],{},[31749,226360,226361],{"mathvariant":226163},"7.26",[31665,226363,226364],{"encoding":31667},"(1+11+6.5+14+3.8) \u002F 5 = 36.3 \u002F 5 = \\mathbf{7.26}",[34,226366,226368,226389,226407,226425,226443,226467,226486],{"className":226367,"ariaHidden":758},[31672],[34,226369,226371,226374,226377,226380,226383,226386],{"className":226370},[31676],[34,226372],{"className":226373,"style":31774},[31680],[34,226375,456],{"className":226376},[31782],[34,226378,640],{"className":226379},[31685],[34,226381],{"className":226382,"style":32604},[31845],[34,226384,223],{"className":226385},[32024],[34,226387],{"className":226388,"style":32604},[31845],[34,226390,226392,226395,226398,226401,226404],{"className":226391},[31676],[34,226393],{"className":226394,"style":35942},[31680],[34,226396,39620],{"className":226397},[31685],[34,226399],{"className":226400,"style":32604},[31845],[34,226402,223],{"className":226403},[32024],[34,226405],{"className":226406,"style":32604},[31845],[34,226408,226410,226413,226416,226419,226422],{"className":226409},[31676],[34,226411],{"className":226412,"style":35942},[31680],[34,226414,226332],{"className":226415},[31685],[34,226417],{"className":226418,"style":32604},[31845],[34,226420,223],{"className":226421},[32024],[34,226423],{"className":226424,"style":32604},[31845],[34,226426,226428,226431,226434,226437,226440],{"className":226427},[31676],[34,226429],{"className":226430,"style":35942},[31680],[34,226432,164756],{"className":226433},[31685],[34,226435],{"className":226436,"style":32604},[31845],[34,226438,223],{"className":226439},[32024],[34,226441],{"className":226442,"style":32604},[31845],[34,226444,226446,226449,226452,226455,226458,226461,226464],{"className":226445},[31676],[34,226447],{"className":226448,"style":31774},[31680],[34,226450,226341],{"className":226451},[31685],[34,226453,2557],{"className":226454},[31690],[34,226456,226261],{"className":226457},[31685],[34,226459],{"className":226460,"style":31846},[31845],[34,226462,202],{"className":226463},[31850],[34,226465],{"className":226466,"style":31846},[31845],[34,226468,226470,226473,226477,226480,226483],{"className":226469},[31676],[34,226471],{"className":226472,"style":31774},[31680],[34,226474,226476],{"className":226475},[31685],"36.3\u002F5",[34,226478],{"className":226479,"style":31846},[31845],[34,226481,202],{"className":226482},[31850],[34,226484],{"className":226485,"style":31846},[31845],[34,226487,226489,226492],{"className":226488},[31676],[34,226490],{"className":226491,"style":32617},[31680],[34,226493,226495],{"className":226494},[31685],[34,226496,226361],{"className":226497},[31685,226302],[15,226499,226500],{},[2627,226501,226502],{},"(2) 短作业优先 (SJF)",[105,226504,226505,226511,226632,226637,226642,226833],{},[108,226506,226507,226510],{},[2627,226508,226509],{},"排序原则："," 按照执行时间从短到长，时间相同则按到达次序(FCFS)。",[108,226512,226513,226515,226516,226544,226545,226573,226574,226602,226603,226631],{},[2627,226514,225978],{}," 2(1) ",[34,226517,226519,226532],{"className":226518},[31642],[34,226520,226522],{"className":226521},[31646],[31648,226523,226524],{"xmlns":31650},[31652,226525,226526,226530],{},[31655,226527,226528],{},[31662,226529,33239],{},[31665,226531,119689],{"encoding":31667},[34,226533,226535],{"className":226534,"ariaHidden":758},[31672],[34,226536,226538,226541],{"className":226537},[31676],[34,226539],{"className":226540,"style":73089},[31680],[34,226542,33239],{"className":226543},[31850]," 4(1) ",[34,226546,226548,226561],{"className":226547},[31642],[34,226549,226551],{"className":226550},[31646],[31648,226552,226553],{"xmlns":31650},[31652,226554,226555,226559],{},[31655,226556,226557],{},[31662,226558,33239],{},[31665,226560,119689],{"encoding":31667},[34,226562,226564],{"className":226563,"ariaHidden":758},[31672],[34,226565,226567,226570],{"className":226566},[31676],[34,226568],{"className":226569,"style":73089},[31680],[34,226571,33239],{"className":226572},[31850]," 3(2) ",[34,226575,226577,226590],{"className":226576},[31642],[34,226578,226580],{"className":226579},[31646],[31648,226581,226582],{"xmlns":31650},[31652,226583,226584,226588],{},[31655,226585,226586],{},[31662,226587,33239],{},[31665,226589,119689],{"encoding":31667},[34,226591,226593],{"className":226592,"ariaHidden":758},[31672],[34,226594,226596,226599],{"className":226595},[31676],[34,226597],{"className":226598,"style":73089},[31680],[34,226600,33239],{"className":226601},[31850]," 5(5) ",[34,226604,226606,226619],{"className":226605},[31642],[34,226607,226609],{"className":226608},[31646],[31648,226610,226611],{"xmlns":31650},[31652,226612,226613,226617],{},[31655,226614,226615],{},[31662,226616,33239],{},[31665,226618,119689],{"encoding":31667},[34,226620,226622],{"className":226621,"ariaHidden":758},[31672],[34,226623,226625,226628],{"className":226624},[31676],[34,226626],{"className":226627,"style":73089},[31680],[34,226629,33239],{"className":226630},[31850]," 1(10)",[108,226633,226634,226636],{},[2627,226635,226100],{}," 2:1, 4:2, 3:4, 5:9, 1:19",[108,226638,226639,226641],{},[2627,226640,226106],{}," 2:(1\u002F1=1), 4:(2\u002F1=2), 3:(4\u002F2=2), 5:(9\u002F5=1.8), 1:(19\u002F10=1.9)",[108,226643,226644,47649,226646],{},[2627,226645,226112],{},[34,226647,226649,226700],{"className":226648},[31642],[34,226650,226652],{"className":226651},[31646],[31648,226653,226654],{"xmlns":31650},[31652,226655,226656,226697],{},[31655,226657,226658,226660,226662,226664,226666,226668,226670,226672,226674,226676,226678,226680,226682,226684,226686,226688,226690,226692,226694],{},[31662,226659,456],{"stretchy":702},[31749,226661,640],{},[31662,226663,223],{},[31749,226665,768],{},[31662,226667,223],{},[31749,226669,15932],{},[31662,226671,223],{},[31749,226673,36732],{},[31662,226675,223],{},[31749,226677,162590],{},[31662,226679,2557],{"stretchy":702},[31658,226681,38080],{"mathvariant":38079},[31749,226683,18136],{},[31662,226685,202],{},[31749,226687,5752],{},[31658,226689,38080],{"mathvariant":38079},[31749,226691,18136],{},[31662,226693,202],{},[31749,226695,226696],{"mathvariant":226163},"7.0",[31665,226698,226699],{"encoding":31667},"(1+2+4+9+19) \u002F 5 = 35 \u002F 5 = \\mathbf{7.0}",[34,226701,226703,226724,226742,226760,226778,226802,226821],{"className":226702,"ariaHidden":758},[31672],[34,226704,226706,226709,226712,226715,226718,226721],{"className":226705},[31676],[34,226707],{"className":226708,"style":31774},[31680],[34,226710,456],{"className":226711},[31782],[34,226713,640],{"className":226714},[31685],[34,226716],{"className":226717,"style":32604},[31845],[34,226719,223],{"className":226720},[32024],[34,226722],{"className":226723,"style":32604},[31845],[34,226725,226727,226730,226733,226736,226739],{"className":226726},[31676],[34,226728],{"className":226729,"style":35942},[31680],[34,226731,768],{"className":226732},[31685],[34,226734],{"className":226735,"style":32604},[31845],[34,226737,223],{"className":226738},[32024],[34,226740],{"className":226741,"style":32604},[31845],[34,226743,226745,226748,226751,226754,226757],{"className":226744},[31676],[34,226746],{"className":226747,"style":35942},[31680],[34,226749,15932],{"className":226750},[31685],[34,226752],{"className":226753,"style":32604},[31845],[34,226755,223],{"className":226756},[32024],[34,226758],{"className":226759,"style":32604},[31845],[34,226761,226763,226766,226769,226772,226775],{"className":226762},[31676],[34,226764],{"className":226765,"style":35942},[31680],[34,226767,36732],{"className":226768},[31685],[34,226770],{"className":226771,"style":32604},[31845],[34,226773,223],{"className":226774},[32024],[34,226776],{"className":226777,"style":32604},[31845],[34,226779,226781,226784,226787,226790,226793,226796,226799],{"className":226780},[31676],[34,226782],{"className":226783,"style":31774},[31680],[34,226785,162590],{"className":226786},[31685],[34,226788,2557],{"className":226789},[31690],[34,226791,226261],{"className":226792},[31685],[34,226794],{"className":226795,"style":31846},[31845],[34,226797,202],{"className":226798},[31850],[34,226800],{"className":226801,"style":31846},[31845],[34,226803,226805,226808,226812,226815,226818],{"className":226804},[31676],[34,226806],{"className":226807,"style":31774},[31680],[34,226809,226811],{"className":226810},[31685],"35\u002F5",[34,226813],{"className":226814,"style":31846},[31845],[34,226816,202],{"className":226817},[31850],[34,226819],{"className":226820,"style":31846},[31845],[34,226822,226824,226827],{"className":226823},[31676],[34,226825],{"className":226826,"style":32617},[31680],[34,226828,226830],{"className":226829},[31685],[34,226831,226696],{"className":226832},[31685,226302],[108,226834,226835,47649,226837],{},[2627,226836,226307],{},[34,226838,226840,226893],{"className":226839},[31642],[34,226841,226843],{"className":226842},[31646],[31648,226844,226845],{"xmlns":31650},[31652,226846,226847,226890],{},[31655,226848,226849,226851,226853,226855,226857,226859,226861,226863,226865,226867,226870,226872,226874,226876,226878,226881,226883,226885,226887],{},[31662,226850,456],{"stretchy":702},[31749,226852,640],{},[31662,226854,223],{},[31749,226856,768],{},[31662,226858,223],{},[31749,226860,768],{},[31662,226862,223],{},[31749,226864,171494],{},[31662,226866,223],{},[31749,226868,226869],{},"1.9",[31662,226871,2557],{"stretchy":702},[31658,226873,38080],{"mathvariant":38079},[31749,226875,18136],{},[31662,226877,202],{},[31749,226879,226880],{},"8.7",[31658,226882,38080],{"mathvariant":38079},[31749,226884,18136],{},[31662,226886,202],{},[31749,226888,226889],{"mathvariant":226163},"1.74",[31665,226891,226892],{"encoding":31667},"(1+2+2+1.8+1.9) \u002F 5 = 8.7 \u002F 5 = \\mathbf{1.74}",[34,226894,226896,226917,226935,226953,226971,226995,227014],{"className":226895,"ariaHidden":758},[31672],[34,226897,226899,226902,226905,226908,226911,226914],{"className":226898},[31676],[34,226900],{"className":226901,"style":31774},[31680],[34,226903,456],{"className":226904},[31782],[34,226906,640],{"className":226907},[31685],[34,226909],{"className":226910,"style":32604},[31845],[34,226912,223],{"className":226913},[32024],[34,226915],{"className":226916,"style":32604},[31845],[34,226918,226920,226923,226926,226929,226932],{"className":226919},[31676],[34,226921],{"className":226922,"style":35942},[31680],[34,226924,768],{"className":226925},[31685],[34,226927],{"className":226928,"style":32604},[31845],[34,226930,223],{"className":226931},[32024],[34,226933],{"className":226934,"style":32604},[31845],[34,226936,226938,226941,226944,226947,226950],{"className":226937},[31676],[34,226939],{"className":226940,"style":35942},[31680],[34,226942,768],{"className":226943},[31685],[34,226945],{"className":226946,"style":32604},[31845],[34,226948,223],{"className":226949},[32024],[34,226951],{"className":226952,"style":32604},[31845],[34,226954,226956,226959,226962,226965,226968],{"className":226955},[31676],[34,226957],{"className":226958,"style":35942},[31680],[34,226960,171494],{"className":226961},[31685],[34,226963],{"className":226964,"style":32604},[31845],[34,226966,223],{"className":226967},[32024],[34,226969],{"className":226970,"style":32604},[31845],[34,226972,226974,226977,226980,226983,226986,226989,226992],{"className":226973},[31676],[34,226975],{"className":226976,"style":31774},[31680],[34,226978,226869],{"className":226979},[31685],[34,226981,2557],{"className":226982},[31690],[34,226984,226261],{"className":226985},[31685],[34,226987],{"className":226988,"style":31846},[31845],[34,226990,202],{"className":226991},[31850],[34,226993],{"className":226994,"style":31846},[31845],[34,226996,226998,227001,227005,227008,227011],{"className":226997},[31676],[34,226999],{"className":227000,"style":31774},[31680],[34,227002,227004],{"className":227003},[31685],"8.7\u002F5",[34,227006],{"className":227007,"style":31846},[31845],[34,227009,202],{"className":227010},[31850],[34,227012],{"className":227013,"style":31846},[31845],[34,227015,227017,227020],{"className":227016},[31676],[34,227018],{"className":227019,"style":32617},[31680],[34,227021,227023],{"className":227022},[31685],[34,227024,226889],{"className":227025},[31685,226302],[15,227027,227028],{},[2627,227029,227030],{},"(3) 非抢占优先权调度",[105,227032,227033,227038,227159,227164,227169,227360],{},[108,227034,227035,227037],{},[2627,227036,226509],{}," 优先权数值越小优先级越高。若优先级相同（如作业1和3），按到达次序先调度1。",[108,227039,227040,227042,227043,227071,227072,227100,227101,227129,227130,227158],{},[2627,227041,225978],{}," 2(优1) ",[34,227044,227046,227059],{"className":227045},[31642],[34,227047,227049],{"className":227048},[31646],[31648,227050,227051],{"xmlns":31650},[31652,227052,227053,227057],{},[31655,227054,227055],{},[31662,227056,33239],{},[31665,227058,119689],{"encoding":31667},[34,227060,227062],{"className":227061,"ariaHidden":758},[31672],[34,227063,227065,227068],{"className":227064},[31676],[34,227066],{"className":227067,"style":73089},[31680],[34,227069,33239],{"className":227070},[31850]," 5(优2) ",[34,227073,227075,227088],{"className":227074},[31642],[34,227076,227078],{"className":227077},[31646],[31648,227079,227080],{"xmlns":31650},[31652,227081,227082,227086],{},[31655,227083,227084],{},[31662,227085,33239],{},[31665,227087,119689],{"encoding":31667},[34,227089,227091],{"className":227090,"ariaHidden":758},[31672],[34,227092,227094,227097],{"className":227093},[31676],[34,227095],{"className":227096,"style":73089},[31680],[34,227098,33239],{"className":227099},[31850]," 1(优3) ",[34,227102,227104,227117],{"className":227103},[31642],[34,227105,227107],{"className":227106},[31646],[31648,227108,227109],{"xmlns":31650},[31652,227110,227111,227115],{},[31655,227112,227113],{},[31662,227114,33239],{},[31665,227116,119689],{"encoding":31667},[34,227118,227120],{"className":227119,"ariaHidden":758},[31672],[34,227121,227123,227126],{"className":227122},[31676],[34,227124],{"className":227125,"style":73089},[31680],[34,227127,33239],{"className":227128},[31850]," 3(优3) ",[34,227131,227133,227146],{"className":227132},[31642],[34,227134,227136],{"className":227135},[31646],[31648,227137,227138],{"xmlns":31650},[31652,227139,227140,227144],{},[31655,227141,227142],{},[31662,227143,33239],{},[31665,227145,119689],{"encoding":31667},[34,227147,227149],{"className":227148,"ariaHidden":758},[31672],[34,227150,227152,227155],{"className":227151},[31676],[34,227153],{"className":227154,"style":73089},[31680],[34,227156,33239],{"className":227157},[31850]," 4(优4)",[108,227160,227161,227163],{},[2627,227162,226100],{}," 2:1, 5:6, 1:16, 3:18, 4:19",[108,227165,227166,227168],{},[2627,227167,226106],{}," 2:(1\u002F1=1), 5:(6\u002F5=1.2), 1:(16\u002F10=1.6), 3:(18\u002F2=9), 4:(19\u002F1=19)",[108,227170,227171,47649,227173],{},[2627,227172,226112],{},[34,227174,227176,227227],{"className":227175},[31642],[34,227177,227179],{"className":227178},[31646],[31648,227180,227181],{"xmlns":31650},[31652,227182,227183,227224],{},[31655,227184,227185,227187,227189,227191,227193,227195,227197,227199,227201,227203,227205,227207,227209,227211,227213,227215,227217,227219,227221],{},[31662,227186,456],{"stretchy":702},[31749,227188,640],{},[31662,227190,223],{},[31749,227192,36656],{},[31662,227194,223],{},[31749,227196,177869],{},[31662,227198,223],{},[31749,227200,95611],{},[31662,227202,223],{},[31749,227204,162590],{},[31662,227206,2557],{"stretchy":702},[31658,227208,38080],{"mathvariant":38079},[31749,227210,18136],{},[31662,227212,202],{},[31749,227214,167664],{},[31658,227216,38080],{"mathvariant":38079},[31749,227218,18136],{},[31662,227220,202],{},[31749,227222,227223],{"mathvariant":226163},"12.0",[31665,227225,227226],{"encoding":31667},"(1+6+16+18+19) \u002F 5 = 60 \u002F 5 = \\mathbf{12.0}",[34,227228,227230,227251,227269,227287,227305,227329,227348],{"className":227229,"ariaHidden":758},[31672],[34,227231,227233,227236,227239,227242,227245,227248],{"className":227232},[31676],[34,227234],{"className":227235,"style":31774},[31680],[34,227237,456],{"className":227238},[31782],[34,227240,640],{"className":227241},[31685],[34,227243],{"className":227244,"style":32604},[31845],[34,227246,223],{"className":227247},[32024],[34,227249],{"className":227250,"style":32604},[31845],[34,227252,227254,227257,227260,227263,227266],{"className":227253},[31676],[34,227255],{"className":227256,"style":35942},[31680],[34,227258,36656],{"className":227259},[31685],[34,227261],{"className":227262,"style":32604},[31845],[34,227264,223],{"className":227265},[32024],[34,227267],{"className":227268,"style":32604},[31845],[34,227270,227272,227275,227278,227281,227284],{"className":227271},[31676],[34,227273],{"className":227274,"style":35942},[31680],[34,227276,177869],{"className":227277},[31685],[34,227279],{"className":227280,"style":32604},[31845],[34,227282,223],{"className":227283},[32024],[34,227285],{"className":227286,"style":32604},[31845],[34,227288,227290,227293,227296,227299,227302],{"className":227289},[31676],[34,227291],{"className":227292,"style":35942},[31680],[34,227294,95611],{"className":227295},[31685],[34,227297],{"className":227298,"style":32604},[31845],[34,227300,223],{"className":227301},[32024],[34,227303],{"className":227304,"style":32604},[31845],[34,227306,227308,227311,227314,227317,227320,227323,227326],{"className":227307},[31676],[34,227309],{"className":227310,"style":31774},[31680],[34,227312,162590],{"className":227313},[31685],[34,227315,2557],{"className":227316},[31690],[34,227318,226261],{"className":227319},[31685],[34,227321],{"className":227322,"style":31846},[31845],[34,227324,202],{"className":227325},[31850],[34,227327],{"className":227328,"style":31846},[31845],[34,227330,227332,227335,227339,227342,227345],{"className":227331},[31676],[34,227333],{"className":227334,"style":31774},[31680],[34,227336,227338],{"className":227337},[31685],"60\u002F5",[34,227340],{"className":227341,"style":31846},[31845],[34,227343,202],{"className":227344},[31850],[34,227346],{"className":227347,"style":31846},[31845],[34,227349,227351,227354],{"className":227350},[31676],[34,227352],{"className":227353,"style":32617},[31680],[34,227355,227357],{"className":227356},[31685],[34,227358,227223],{"className":227359},[31685,226302],[108,227361,227362,47649,227364],{},[2627,227363,226307],{},[34,227365,227367,227420],{"className":227366},[31642],[34,227368,227370],{"className":227369},[31646],[31648,227371,227372],{"xmlns":31650},[31652,227373,227374,227417],{},[31655,227375,227376,227378,227380,227382,227384,227386,227389,227391,227393,227395,227397,227399,227401,227403,227405,227408,227410,227412,227414],{},[31662,227377,456],{"stretchy":702},[31749,227379,640],{},[31662,227381,223],{},[31749,227383,169377],{},[31662,227385,223],{},[31749,227387,227388],{},"1.6",[31662,227390,223],{},[31749,227392,36732],{},[31662,227394,223],{},[31749,227396,162590],{},[31662,227398,2557],{"stretchy":702},[31658,227400,38080],{"mathvariant":38079},[31749,227402,18136],{},[31662,227404,202],{},[31749,227406,227407],{},"31.8",[31658,227409,38080],{"mathvariant":38079},[31749,227411,18136],{},[31662,227413,202],{},[31749,227415,227416],{"mathvariant":226163},"6.36",[31665,227418,227419],{"encoding":31667},"(1+1.2+1.6+9+19) \u002F 5 = 31.8 \u002F 5 = \\mathbf{6.36}",[34,227421,227423,227444,227462,227480,227498,227522,227541],{"className":227422,"ariaHidden":758},[31672],[34,227424,227426,227429,227432,227435,227438,227441],{"className":227425},[31676],[34,227427],{"className":227428,"style":31774},[31680],[34,227430,456],{"className":227431},[31782],[34,227433,640],{"className":227434},[31685],[34,227436],{"className":227437,"style":32604},[31845],[34,227439,223],{"className":227440},[32024],[34,227442],{"className":227443,"style":32604},[31845],[34,227445,227447,227450,227453,227456,227459],{"className":227446},[31676],[34,227448],{"className":227449,"style":35942},[31680],[34,227451,169377],{"className":227452},[31685],[34,227454],{"className":227455,"style":32604},[31845],[34,227457,223],{"className":227458},[32024],[34,227460],{"className":227461,"style":32604},[31845],[34,227463,227465,227468,227471,227474,227477],{"className":227464},[31676],[34,227466],{"className":227467,"style":35942},[31680],[34,227469,227388],{"className":227470},[31685],[34,227472],{"className":227473,"style":32604},[31845],[34,227475,223],{"className":227476},[32024],[34,227478],{"className":227479,"style":32604},[31845],[34,227481,227483,227486,227489,227492,227495],{"className":227482},[31676],[34,227484],{"className":227485,"style":35942},[31680],[34,227487,36732],{"className":227488},[31685],[34,227490],{"className":227491,"style":32604},[31845],[34,227493,223],{"className":227494},[32024],[34,227496],{"className":227497,"style":32604},[31845],[34,227499,227501,227504,227507,227510,227513,227516,227519],{"className":227500},[31676],[34,227502],{"className":227503,"style":31774},[31680],[34,227505,162590],{"className":227506},[31685],[34,227508,2557],{"className":227509},[31690],[34,227511,226261],{"className":227512},[31685],[34,227514],{"className":227515,"style":31846},[31845],[34,227517,202],{"className":227518},[31850],[34,227520],{"className":227521,"style":31846},[31845],[34,227523,227525,227528,227532,227535,227538],{"className":227524},[31676],[34,227526],{"className":227527,"style":31774},[31680],[34,227529,227531],{"className":227530},[31685],"31.8\u002F5",[34,227533],{"className":227534,"style":31846},[31845],[34,227536,202],{"className":227537},[31850],[34,227539],{"className":227540,"style":31846},[31845],[34,227542,227544,227547],{"className":227543},[31676],[34,227545],{"className":227546,"style":32617},[31680],[34,227548,227550],{"className":227549},[31685],[34,227551,227416],{"className":227552},[31685,226302],[15,227554,227555],{},[2627,227556,227557],{},"(4) 时间片轮转 (RR，假设时间片 q=1)",[105,227559,227560,227582,227587,227592,227784],{},[108,227561,227562,227565],{},[2627,227563,227564],{},"执行过程模拟 (队列轮转)：",[105,227566,227567,227570,227573,227576,227579],{},[108,227568,227569],{},"t=1: 1(剩9)，2完成。",[108,227571,227572],{},"t=2: 3(剩1)。",[108,227574,227575],{},"t=3: 4完成。",[108,227577,227578],{},"t=4: 5(剩4)。",[108,227580,227581],{},"后续由于2和4已完成，只剩下1、3、5轮转。t=7时3完成。t=14时5完成。最后剩下1跑完剩下的5个时间片（t=19完成）。",[108,227583,227584,227586],{},[2627,227585,226100],{}," 2:2, 4:4, 3:7, 5:14, 1:19",[108,227588,227589,227591],{},[2627,227590,226106],{}," 2(2), 4(4), 3(3.5), 5(2.8), 1(1.9)",[108,227593,227594,47649,227596],{},[2627,227595,226112],{},[34,227597,227599,227651],{"className":227598},[31642],[34,227600,227602],{"className":227601},[31646],[31648,227603,227604],{"xmlns":31650},[31652,227605,227606,227648],{},[31655,227607,227608,227610,227612,227614,227616,227618,227620,227622,227624,227626,227628,227630,227632,227634,227636,227639,227641,227643,227645],{},[31662,227609,456],{"stretchy":702},[31749,227611,768],{},[31662,227613,223],{},[31749,227615,15932],{},[31662,227617,223],{},[31749,227619,36723],{},[31662,227621,223],{},[31749,227623,164756],{},[31662,227625,223],{},[31749,227627,162590],{},[31662,227629,2557],{"stretchy":702},[31658,227631,38080],{"mathvariant":38079},[31749,227633,18136],{},[31662,227635,202],{},[31749,227637,227638],{},"46",[31658,227640,38080],{"mathvariant":38079},[31749,227642,18136],{},[31662,227644,202],{},[31749,227646,227647],{"mathvariant":226163},"9.2",[31665,227649,227650],{"encoding":31667},"(2+4+7+14+19) \u002F 5 = 46 \u002F 5 = \\mathbf{9.2}",[34,227652,227654,227675,227693,227711,227729,227753,227772],{"className":227653,"ariaHidden":758},[31672],[34,227655,227657,227660,227663,227666,227669,227672],{"className":227656},[31676],[34,227658],{"className":227659,"style":31774},[31680],[34,227661,456],{"className":227662},[31782],[34,227664,768],{"className":227665},[31685],[34,227667],{"className":227668,"style":32604},[31845],[34,227670,223],{"className":227671},[32024],[34,227673],{"className":227674,"style":32604},[31845],[34,227676,227678,227681,227684,227687,227690],{"className":227677},[31676],[34,227679],{"className":227680,"style":35942},[31680],[34,227682,15932],{"className":227683},[31685],[34,227685],{"className":227686,"style":32604},[31845],[34,227688,223],{"className":227689},[32024],[34,227691],{"className":227692,"style":32604},[31845],[34,227694,227696,227699,227702,227705,227708],{"className":227695},[31676],[34,227697],{"className":227698,"style":35942},[31680],[34,227700,36723],{"className":227701},[31685],[34,227703],{"className":227704,"style":32604},[31845],[34,227706,223],{"className":227707},[32024],[34,227709],{"className":227710,"style":32604},[31845],[34,227712,227714,227717,227720,227723,227726],{"className":227713},[31676],[34,227715],{"className":227716,"style":35942},[31680],[34,227718,164756],{"className":227719},[31685],[34,227721],{"className":227722,"style":32604},[31845],[34,227724,223],{"className":227725},[32024],[34,227727],{"className":227728,"style":32604},[31845],[34,227730,227732,227735,227738,227741,227744,227747,227750],{"className":227731},[31676],[34,227733],{"className":227734,"style":31774},[31680],[34,227736,162590],{"className":227737},[31685],[34,227739,2557],{"className":227740},[31690],[34,227742,226261],{"className":227743},[31685],[34,227745],{"className":227746,"style":31846},[31845],[34,227748,202],{"className":227749},[31850],[34,227751],{"className":227752,"style":31846},[31845],[34,227754,227756,227759,227763,227766,227769],{"className":227755},[31676],[34,227757],{"className":227758,"style":31774},[31680],[34,227760,227762],{"className":227761},[31685],"46\u002F5",[34,227764],{"className":227765,"style":31846},[31845],[34,227767,202],{"className":227768},[31850],[34,227770],{"className":227771,"style":31846},[31845],[34,227773,227775,227778],{"className":227774},[31676],[34,227776],{"className":227777,"style":32617},[31680],[34,227779,227781],{"className":227780},[31685],[34,227782,227647],{"className":227783},[31685,226302],[108,227785,227786,47649,227788],{},[2627,227787,226307],{},[34,227789,227791,227845],{"className":227790},[31642],[34,227792,227794],{"className":227793},[31646],[31648,227795,227796],{"xmlns":31650},[31652,227797,227798,227842],{},[31655,227799,227800,227802,227804,227806,227808,227810,227813,227815,227818,227820,227822,227824,227826,227828,227830,227833,227835,227837,227839],{},[31662,227801,456],{"stretchy":702},[31749,227803,768],{},[31662,227805,223],{},[31749,227807,15932],{},[31662,227809,223],{},[31749,227811,227812],{},"3.5",[31662,227814,223],{},[31749,227816,227817],{},"2.8",[31662,227819,223],{},[31749,227821,226869],{},[31662,227823,2557],{"stretchy":702},[31658,227825,38080],{"mathvariant":38079},[31749,227827,18136],{},[31662,227829,202],{},[31749,227831,227832],{},"14.2",[31658,227834,38080],{"mathvariant":38079},[31749,227836,18136],{},[31662,227838,202],{},[31749,227840,227841],{"mathvariant":226163},"2.84",[31665,227843,227844],{"encoding":31667},"(2+4+3.5+2.8+1.9) \u002F 5 = 14.2 \u002F 5 = \\mathbf{2.84}",[34,227846,227848,227869,227887,227905,227923,227947,227966],{"className":227847,"ariaHidden":758},[31672],[34,227849,227851,227854,227857,227860,227863,227866],{"className":227850},[31676],[34,227852],{"className":227853,"style":31774},[31680],[34,227855,456],{"className":227856},[31782],[34,227858,768],{"className":227859},[31685],[34,227861],{"className":227862,"style":32604},[31845],[34,227864,223],{"className":227865},[32024],[34,227867],{"className":227868,"style":32604},[31845],[34,227870,227872,227875,227878,227881,227884],{"className":227871},[31676],[34,227873],{"className":227874,"style":35942},[31680],[34,227876,15932],{"className":227877},[31685],[34,227879],{"className":227880,"style":32604},[31845],[34,227882,223],{"className":227883},[32024],[34,227885],{"className":227886,"style":32604},[31845],[34,227888,227890,227893,227896,227899,227902],{"className":227889},[31676],[34,227891],{"className":227892,"style":35942},[31680],[34,227894,227812],{"className":227895},[31685],[34,227897],{"className":227898,"style":32604},[31845],[34,227900,223],{"className":227901},[32024],[34,227903],{"className":227904,"style":32604},[31845],[34,227906,227908,227911,227914,227917,227920],{"className":227907},[31676],[34,227909],{"className":227910,"style":35942},[31680],[34,227912,227817],{"className":227913},[31685],[34,227915],{"className":227916,"style":32604},[31845],[34,227918,223],{"className":227919},[32024],[34,227921],{"className":227922,"style":32604},[31845],[34,227924,227926,227929,227932,227935,227938,227941,227944],{"className":227925},[31676],[34,227927],{"className":227928,"style":31774},[31680],[34,227930,226869],{"className":227931},[31685],[34,227933,2557],{"className":227934},[31690],[34,227936,226261],{"className":227937},[31685],[34,227939],{"className":227940,"style":31846},[31845],[34,227942,202],{"className":227943},[31850],[34,227945],{"className":227946,"style":31846},[31845],[34,227948,227950,227953,227957,227960,227963],{"className":227949},[31676],[34,227951],{"className":227952,"style":31774},[31680],[34,227954,227956],{"className":227955},[31685],"14.2\u002F5",[34,227958],{"className":227959,"style":31846},[31845],[34,227961,202],{"className":227962},[31850],[34,227964],{"className":227965,"style":31846},[31845],[34,227967,227969,227972],{"className":227968},[31676],[34,227970],{"className":227971,"style":32617},[31680],[34,227973,227975],{"className":227974},[31685],[34,227976,227841],{"className":227977},[31685,226302],[32188,227979],{},[19,227981,227983],{"id":227982},"_6同时到达的5个作业调度-包含-rr-q2","6、同时到达的5个作业调度 (包含 RR q=2)",[15,227985,227986,227988,227989,347],{},[2627,227987,214361],{},"\nA(2), B(4), C(6), D(8), E(10) 同时到达。优先级 1~5（5为最高）。求下列情况的",[2627,227990,227991],{},"平均周转时间",[253,227993,225963],{"id":227994},"详细讲解与答案-1",[15,227996,227997],{},[2627,227998,227999],{},"(4) FCFS (按指定次序 C, D, B, E, A)",[105,228001,228002,228121,228124],{},[108,228003,228004,228005,228033,228034,228062,228063,228091,228092,228120],{},"顺序：C(6) ",[34,228006,228008,228021],{"className":228007},[31642],[34,228009,228011],{"className":228010},[31646],[31648,228012,228013],{"xmlns":31650},[31652,228014,228015,228019],{},[31655,228016,228017],{},[31662,228018,33239],{},[31665,228020,119689],{"encoding":31667},[34,228022,228024],{"className":228023,"ariaHidden":758},[31672],[34,228025,228027,228030],{"className":228026},[31676],[34,228028],{"className":228029,"style":73089},[31680],[34,228031,33239],{"className":228032},[31850]," D(8) ",[34,228035,228037,228050],{"className":228036},[31642],[34,228038,228040],{"className":228039},[31646],[31648,228041,228042],{"xmlns":31650},[31652,228043,228044,228048],{},[31655,228045,228046],{},[31662,228047,33239],{},[31665,228049,119689],{"encoding":31667},[34,228051,228053],{"className":228052,"ariaHidden":758},[31672],[34,228054,228056,228059],{"className":228055},[31676],[34,228057],{"className":228058,"style":73089},[31680],[34,228060,33239],{"className":228061},[31850]," B(4) ",[34,228064,228066,228079],{"className":228065},[31642],[34,228067,228069],{"className":228068},[31646],[31648,228070,228071],{"xmlns":31650},[31652,228072,228073,228077],{},[31655,228074,228075],{},[31662,228076,33239],{},[31665,228078,119689],{"encoding":31667},[34,228080,228082],{"className":228081,"ariaHidden":758},[31672],[34,228083,228085,228088],{"className":228084},[31676],[34,228086],{"className":228087,"style":73089},[31680],[34,228089,33239],{"className":228090},[31850]," E(10) ",[34,228093,228095,228108],{"className":228094},[31642],[34,228096,228098],{"className":228097},[31646],[31648,228099,228100],{"xmlns":31650},[31652,228101,228102,228106],{},[31655,228103,228104],{},[31662,228105,33239],{},[31665,228107,119689],{"encoding":31667},[34,228109,228111],{"className":228110,"ariaHidden":758},[31672],[34,228112,228114,228117],{"className":228113},[31676],[34,228115],{"className":228116,"style":73089},[31680],[34,228118,33239],{"className":228119},[31850]," A(2)",[108,228122,228123],{},"完成时间：C:6, D:14, B:18, E:28, A:30",[108,228125,228126,228127],{},"平均周转时间： ",[34,228128,228130,228178],{"className":228129},[31642],[34,228131,228133],{"className":228132},[31646],[31648,228134,228135],{"xmlns":31650},[31652,228136,228137,228175],{},[31655,228138,228139,228141,228143,228145,228147,228149,228151,228153,228155,228157,228159,228161,228163,228165,228167],{},[31662,228140,456],{"stretchy":702},[31749,228142,36656],{},[31662,228144,223],{},[31749,228146,164756],{},[31662,228148,223],{},[31749,228150,95611],{},[31662,228152,223],{},[31749,228154,208932],{},[31662,228156,223],{},[31749,228158,9314],{},[31662,228160,2557],{"stretchy":702},[31658,228162,38080],{"mathvariant":38079},[31749,228164,18136],{},[31662,228166,202],{},[31655,228168,228169,228172],{},[31749,228170,228171],{"mathvariant":226163},"19.2",[39361,228173,228174],{}," 分钟",[31665,228176,228177],{"encoding":31667},"(6+14+18+28+30) \u002F 5 = \\mathbf{19.2 \\text{ 分钟}}",[34,228179,228181,228202,228220,228238,228256,228280],{"className":228180,"ariaHidden":758},[31672],[34,228182,228184,228187,228190,228193,228196,228199],{"className":228183},[31676],[34,228185],{"className":228186,"style":31774},[31680],[34,228188,456],{"className":228189},[31782],[34,228191,36656],{"className":228192},[31685],[34,228194],{"className":228195,"style":32604},[31845],[34,228197,223],{"className":228198},[32024],[34,228200],{"className":228201,"style":32604},[31845],[34,228203,228205,228208,228211,228214,228217],{"className":228204},[31676],[34,228206],{"className":228207,"style":35942},[31680],[34,228209,164756],{"className":228210},[31685],[34,228212],{"className":228213,"style":32604},[31845],[34,228215,223],{"className":228216},[32024],[34,228218],{"className":228219,"style":32604},[31845],[34,228221,228223,228226,228229,228232,228235],{"className":228222},[31676],[34,228224],{"className":228225,"style":35942},[31680],[34,228227,95611],{"className":228228},[31685],[34,228230],{"className":228231,"style":32604},[31845],[34,228233,223],{"className":228234},[32024],[34,228236],{"className":228237,"style":32604},[31845],[34,228239,228241,228244,228247,228250,228253],{"className":228240},[31676],[34,228242],{"className":228243,"style":35942},[31680],[34,228245,208932],{"className":228246},[31685],[34,228248],{"className":228249,"style":32604},[31845],[34,228251,223],{"className":228252},[32024],[34,228254],{"className":228255,"style":32604},[31845],[34,228257,228259,228262,228265,228268,228271,228274,228277],{"className":228258},[31676],[34,228260],{"className":228261,"style":31774},[31680],[34,228263,9314],{"className":228264},[31685],[34,228266,2557],{"className":228267},[31690],[34,228269,226261],{"className":228270},[31685],[34,228272],{"className":228273,"style":31846},[31845],[34,228275,202],{"className":228276},[31850],[34,228278],{"className":228279,"style":31846},[31845],[34,228281,228283,228286],{"className":228282},[31676],[34,228284],{"className":228285,"style":33218},[31680],[34,228287,228289,228292],{"className":228288},[31685],[34,228290,228171],{"className":228291},[31685,226302],[34,228293,228295,228298],{"className":228294},[31685,2638],[34,228296,130298],{"className":228297},[31685],[34,228299,228301],{"className":228300},[31685,105462],"分钟",[15,228303,228304],{},[2627,228305,228306],{},"(3) 短作业优先 (SJF)",[105,228308,228309,228426,228429],{},[108,228310,228311,228312,228062,228340,228368,228369,228033,228397,228425],{},"顺序：A(2) ",[34,228313,228315,228328],{"className":228314},[31642],[34,228316,228318],{"className":228317},[31646],[31648,228319,228320],{"xmlns":31650},[31652,228321,228322,228326],{},[31655,228323,228324],{},[31662,228325,33239],{},[31665,228327,119689],{"encoding":31667},[34,228329,228331],{"className":228330,"ariaHidden":758},[31672],[34,228332,228334,228337],{"className":228333},[31676],[34,228335],{"className":228336,"style":73089},[31680],[34,228338,33239],{"className":228339},[31850],[34,228341,228343,228356],{"className":228342},[31642],[34,228344,228346],{"className":228345},[31646],[31648,228347,228348],{"xmlns":31650},[31652,228349,228350,228354],{},[31655,228351,228352],{},[31662,228353,33239],{},[31665,228355,119689],{"encoding":31667},[34,228357,228359],{"className":228358,"ariaHidden":758},[31672],[34,228360,228362,228365],{"className":228361},[31676],[34,228363],{"className":228364,"style":73089},[31680],[34,228366,33239],{"className":228367},[31850]," C(6) ",[34,228370,228372,228385],{"className":228371},[31642],[34,228373,228375],{"className":228374},[31646],[31648,228376,228377],{"xmlns":31650},[31652,228378,228379,228383],{},[31655,228380,228381],{},[31662,228382,33239],{},[31665,228384,119689],{"encoding":31667},[34,228386,228388],{"className":228387,"ariaHidden":758},[31672],[34,228389,228391,228394],{"className":228390},[31676],[34,228392],{"className":228393,"style":73089},[31680],[34,228395,33239],{"className":228396},[31850],[34,228398,228400,228413],{"className":228399},[31642],[34,228401,228403],{"className":228402},[31646],[31648,228404,228405],{"xmlns":31650},[31652,228406,228407,228411],{},[31655,228408,228409],{},[31662,228410,33239],{},[31665,228412,119689],{"encoding":31667},[34,228414,228416],{"className":228415,"ariaHidden":758},[31672],[34,228417,228419,228422],{"className":228418},[31676],[34,228420],{"className":228421,"style":73089},[31680],[34,228423,33239],{"className":228424},[31850]," E(10)",[108,228427,228428],{},"完成时间：A:2, B:6, C:12, D:20, E:30",[108,228430,228126,228431],{},[34,228432,228434,228481],{"className":228433},[31642],[34,228435,228437],{"className":228436},[31646],[31648,228438,228439],{"xmlns":31650},[31652,228440,228441,228478],{},[31655,228442,228443,228445,228447,228449,228451,228453,228455,228457,228459,228461,228463,228465,228467,228469,228471],{},[31662,228444,456],{"stretchy":702},[31749,228446,768],{},[31662,228448,223],{},[31749,228450,36656],{},[31662,228452,223],{},[31749,228454,71869],{},[31662,228456,223],{},[31749,228458,3249],{},[31662,228460,223],{},[31749,228462,9314],{},[31662,228464,2557],{"stretchy":702},[31658,228466,38080],{"mathvariant":38079},[31749,228468,18136],{},[31662,228470,202],{},[31655,228472,228473,228476],{},[31749,228474,228475],{"mathvariant":226163},"14.0",[39361,228477,228174],{},[31665,228479,228480],{"encoding":31667},"(2+6+12+20+30) \u002F 5 = \\mathbf{14.0 \\text{ 分钟}}",[34,228482,228484,228505,228523,228541,228559,228583],{"className":228483,"ariaHidden":758},[31672],[34,228485,228487,228490,228493,228496,228499,228502],{"className":228486},[31676],[34,228488],{"className":228489,"style":31774},[31680],[34,228491,456],{"className":228492},[31782],[34,228494,768],{"className":228495},[31685],[34,228497],{"className":228498,"style":32604},[31845],[34,228500,223],{"className":228501},[32024],[34,228503],{"className":228504,"style":32604},[31845],[34,228506,228508,228511,228514,228517,228520],{"className":228507},[31676],[34,228509],{"className":228510,"style":35942},[31680],[34,228512,36656],{"className":228513},[31685],[34,228515],{"className":228516,"style":32604},[31845],[34,228518,223],{"className":228519},[32024],[34,228521],{"className":228522,"style":32604},[31845],[34,228524,228526,228529,228532,228535,228538],{"className":228525},[31676],[34,228527],{"className":228528,"style":35942},[31680],[34,228530,71869],{"className":228531},[31685],[34,228533],{"className":228534,"style":32604},[31845],[34,228536,223],{"className":228537},[32024],[34,228539],{"className":228540,"style":32604},[31845],[34,228542,228544,228547,228550,228553,228556],{"className":228543},[31676],[34,228545],{"className":228546,"style":35942},[31680],[34,228548,3249],{"className":228549},[31685],[34,228551],{"className":228552,"style":32604},[31845],[34,228554,223],{"className":228555},[32024],[34,228557],{"className":228558,"style":32604},[31845],[34,228560,228562,228565,228568,228571,228574,228577,228580],{"className":228561},[31676],[34,228563],{"className":228564,"style":31774},[31680],[34,228566,9314],{"className":228567},[31685],[34,228569,2557],{"className":228570},[31690],[34,228572,226261],{"className":228573},[31685],[34,228575],{"className":228576,"style":31846},[31845],[34,228578,202],{"className":228579},[31850],[34,228581],{"className":228582,"style":31846},[31845],[34,228584,228586,228589],{"className":228585},[31676],[34,228587],{"className":228588,"style":33218},[31680],[34,228590,228592,228595],{"className":228591},[31685],[34,228593,228475],{"className":228594},[31685,226302],[34,228596,228598,228601],{"className":228597},[31685,2638],[34,228599,130298],{"className":228600},[31685],[34,228602,228301],{"className":228603},[31685,105462],[15,228605,228606],{},[2627,228607,228608],{},"(2) 优先数算法 (5为最高级)",[105,228610,228611,228730,228733],{},[108,228612,228613,228614,228642,228643,228671,228672,228700,228701,228729],{},"顺序：E(5级,10) ",[34,228615,228617,228630],{"className":228616},[31642],[34,228618,228620],{"className":228619},[31646],[31648,228621,228622],{"xmlns":31650},[31652,228623,228624,228628],{},[31655,228625,228626],{},[31662,228627,33239],{},[31665,228629,119689],{"encoding":31667},[34,228631,228633],{"className":228632,"ariaHidden":758},[31672],[34,228634,228636,228639],{"className":228635},[31676],[34,228637],{"className":228638,"style":73089},[31680],[34,228640,33239],{"className":228641},[31850]," D(4级,8) ",[34,228644,228646,228659],{"className":228645},[31642],[34,228647,228649],{"className":228648},[31646],[31648,228650,228651],{"xmlns":31650},[31652,228652,228653,228657],{},[31655,228654,228655],{},[31662,228656,33239],{},[31665,228658,119689],{"encoding":31667},[34,228660,228662],{"className":228661,"ariaHidden":758},[31672],[34,228663,228665,228668],{"className":228664},[31676],[34,228666],{"className":228667,"style":73089},[31680],[34,228669,33239],{"className":228670},[31850]," C(3级,6) ",[34,228673,228675,228688],{"className":228674},[31642],[34,228676,228678],{"className":228677},[31646],[31648,228679,228680],{"xmlns":31650},[31652,228681,228682,228686],{},[31655,228683,228684],{},[31662,228685,33239],{},[31665,228687,119689],{"encoding":31667},[34,228689,228691],{"className":228690,"ariaHidden":758},[31672],[34,228692,228694,228697],{"className":228693},[31676],[34,228695],{"className":228696,"style":73089},[31680],[34,228698,33239],{"className":228699},[31850]," B(2级,4) ",[34,228702,228704,228717],{"className":228703},[31642],[34,228705,228707],{"className":228706},[31646],[31648,228708,228709],{"xmlns":31650},[31652,228710,228711,228715],{},[31655,228712,228713],{},[31662,228714,33239],{},[31665,228716,119689],{"encoding":31667},[34,228718,228720],{"className":228719,"ariaHidden":758},[31672],[34,228721,228723,228726],{"className":228722},[31676],[34,228724],{"className":228725,"style":73089},[31680],[34,228727,33239],{"className":228728},[31850]," A(1级,2)",[108,228731,228732],{},"完成时间：E:10, D:18, C:24, B:28, A:30",[108,228734,228126,228735],{},[34,228736,228738,228785],{"className":228737},[31642],[34,228739,228741],{"className":228740},[31646],[31648,228742,228743],{"xmlns":31650},[31652,228744,228745,228782],{},[31655,228746,228747,228749,228751,228753,228755,228757,228759,228761,228763,228765,228767,228769,228771,228773,228775],{},[31662,228748,456],{"stretchy":702},[31749,228750,3244],{},[31662,228752,223],{},[31749,228754,95611],{},[31662,228756,223],{},[31749,228758,96087],{},[31662,228760,223],{},[31749,228762,208932],{},[31662,228764,223],{},[31749,228766,9314],{},[31662,228768,2557],{"stretchy":702},[31658,228770,38080],{"mathvariant":38079},[31749,228772,18136],{},[31662,228774,202],{},[31655,228776,228777,228780],{},[31749,228778,228779],{"mathvariant":226163},"22.0",[39361,228781,228174],{},[31665,228783,228784],{"encoding":31667},"(10+18+24+28+30) \u002F 5 = \\mathbf{22.0 \\text{ 分钟}}",[34,228786,228788,228809,228827,228845,228863,228887],{"className":228787,"ariaHidden":758},[31672],[34,228789,228791,228794,228797,228800,228803,228806],{"className":228790},[31676],[34,228792],{"className":228793,"style":31774},[31680],[34,228795,456],{"className":228796},[31782],[34,228798,3244],{"className":228799},[31685],[34,228801],{"className":228802,"style":32604},[31845],[34,228804,223],{"className":228805},[32024],[34,228807],{"className":228808,"style":32604},[31845],[34,228810,228812,228815,228818,228821,228824],{"className":228811},[31676],[34,228813],{"className":228814,"style":35942},[31680],[34,228816,95611],{"className":228817},[31685],[34,228819],{"className":228820,"style":32604},[31845],[34,228822,223],{"className":228823},[32024],[34,228825],{"className":228826,"style":32604},[31845],[34,228828,228830,228833,228836,228839,228842],{"className":228829},[31676],[34,228831],{"className":228832,"style":35942},[31680],[34,228834,96087],{"className":228835},[31685],[34,228837],{"className":228838,"style":32604},[31845],[34,228840,223],{"className":228841},[32024],[34,228843],{"className":228844,"style":32604},[31845],[34,228846,228848,228851,228854,228857,228860],{"className":228847},[31676],[34,228849],{"className":228850,"style":35942},[31680],[34,228852,208932],{"className":228853},[31685],[34,228855],{"className":228856,"style":32604},[31845],[34,228858,223],{"className":228859},[32024],[34,228861],{"className":228862,"style":32604},[31845],[34,228864,228866,228869,228872,228875,228878,228881,228884],{"className":228865},[31676],[34,228867],{"className":228868,"style":31774},[31680],[34,228870,9314],{"className":228871},[31685],[34,228873,2557],{"className":228874},[31690],[34,228876,226261],{"className":228877},[31685],[34,228879],{"className":228880,"style":31846},[31845],[34,228882,202],{"className":228883},[31850],[34,228885],{"className":228886,"style":31846},[31845],[34,228888,228890,228893],{"className":228889},[31676],[34,228891],{"className":228892,"style":33218},[31680],[34,228894,228896,228899],{"className":228895},[31685],[34,228897,228779],{"className":228898},[31685,226302],[34,228900,228902,228905],{"className":228901},[31685,2638],[34,228903,130298],{"className":228904},[31685],[34,228906,228301],{"className":228907},[31685,105462],[15,228909,228910,228913,228916,228917],{},[2627,228911,228912],{},"(1) 时间片轮转算法 (RR, q=2) 获得两分钟时间片",[71807,228914,228915],{},"假设按 A,B,C,D,E 的初始顺序入队：","\n第一轮：A完成(2)；B剩2；C剩4；D剩6；E剩8。\n第二轮：B完成(10)；C剩2；D剩4；E剩6。\n第三轮：C完成(16)；D剩2；E剩4。\n第四轮：D完成(20)；E剩2。\n第五轮：E完成(22)。",[71807,228918,228919],{},"(注：总时间应该为30，前面算错：A2+B2+C2+D2+E2=10；第二轮 B2(12完成)+C2+D2+E2=18；第三轮 C2(20完成)+D2+E2=24；第四轮 D2(26完成)+E2=28；第五轮 E2(30完成)。)",[105,228921,228922,228928],{},[108,228923,228924,228927],{},[2627,228925,228926],{},"正确完成时间："," A:2, B:12, C:20, D:26, E:30",[108,228929,228126,228930],{},[34,228931,228933,228980],{"className":228932},[31642],[34,228934,228936],{"className":228935},[31646],[31648,228937,228938],{"xmlns":31650},[31652,228939,228940,228977],{},[31655,228941,228942,228944,228946,228948,228950,228952,228954,228956,228958,228960,228962,228964,228966,228968,228970],{},[31662,228943,456],{"stretchy":702},[31749,228945,768],{},[31662,228947,223],{},[31749,228949,71869],{},[31662,228951,223],{},[31749,228953,3249],{},[31662,228955,223],{},[31749,228957,202053],{},[31662,228959,223],{},[31749,228961,9314],{},[31662,228963,2557],{"stretchy":702},[31658,228965,38080],{"mathvariant":38079},[31749,228967,18136],{},[31662,228969,202],{},[31655,228971,228972,228975],{},[31749,228973,228974],{"mathvariant":226163},"18.0",[39361,228976,228174],{},[31665,228978,228979],{"encoding":31667},"(2+12+20+26+30) \u002F 5 = \\mathbf{18.0 \\text{ 分钟}}",[34,228981,228983,229004,229022,229040,229058,229082],{"className":228982,"ariaHidden":758},[31672],[34,228984,228986,228989,228992,228995,228998,229001],{"className":228985},[31676],[34,228987],{"className":228988,"style":31774},[31680],[34,228990,456],{"className":228991},[31782],[34,228993,768],{"className":228994},[31685],[34,228996],{"className":228997,"style":32604},[31845],[34,228999,223],{"className":229000},[32024],[34,229002],{"className":229003,"style":32604},[31845],[34,229005,229007,229010,229013,229016,229019],{"className":229006},[31676],[34,229008],{"className":229009,"style":35942},[31680],[34,229011,71869],{"className":229012},[31685],[34,229014],{"className":229015,"style":32604},[31845],[34,229017,223],{"className":229018},[32024],[34,229020],{"className":229021,"style":32604},[31845],[34,229023,229025,229028,229031,229034,229037],{"className":229024},[31676],[34,229026],{"className":229027,"style":35942},[31680],[34,229029,3249],{"className":229030},[31685],[34,229032],{"className":229033,"style":32604},[31845],[34,229035,223],{"className":229036},[32024],[34,229038],{"className":229039,"style":32604},[31845],[34,229041,229043,229046,229049,229052,229055],{"className":229042},[31676],[34,229044],{"className":229045,"style":35942},[31680],[34,229047,202053],{"className":229048},[31685],[34,229050],{"className":229051,"style":32604},[31845],[34,229053,223],{"className":229054},[32024],[34,229056],{"className":229057,"style":32604},[31845],[34,229059,229061,229064,229067,229070,229073,229076,229079],{"className":229060},[31676],[34,229062],{"className":229063,"style":31774},[31680],[34,229065,9314],{"className":229066},[31685],[34,229068,2557],{"className":229069},[31690],[34,229071,226261],{"className":229072},[31685],[34,229074],{"className":229075,"style":31846},[31845],[34,229077,202],{"className":229078},[31850],[34,229080],{"className":229081,"style":31846},[31845],[34,229083,229085,229088],{"className":229084},[31676],[34,229086],{"className":229087,"style":33218},[31680],[34,229089,229091,229094],{"className":229090},[31685],[34,229092,228974],{"className":229093},[31685,226302],[34,229095,229097,229100],{"className":229096},[31685,2638],[34,229098,130298],{"className":229099},[31685],[34,229101,228301],{"className":229102},[31685,105462],[32188,229104],{},[19,229106,229108],{"id":229107},"_7fcfs-与-hrrn-性能对比","7、FCFS 与 HRRN 性能对比",[15,229110,229111,229113],{},[2627,229112,214361],{},"\n单道批处理，3个作业。比较 FCFS 和 HRRN（最高响应比）性能。完成表格",[13837,229115,229116,229141],{},[13840,229117,229118],{},[13843,229119,229120,229123,229126,229129,229132,229135,229138],{},[13846,229121,229122],{"align":82355},"作業",[13846,229124,229125],{"align":82355},"提交時間",[13846,229127,229128],{"align":82355},"運行時間",[13846,229130,229131],{"align":82355},"開始時間",[13846,229133,229134],{"align":82355},"完成時間",[13846,229136,229137],{"align":82355},"周轉時間",[13846,229139,229140],{"align":82355},"帶權周轉時間",[13853,229142,229143,229161,229179],{},[13843,229144,229145,229147,229150,229153,229155,229157,229159],{},[13858,229146,640],{"align":82355},[13858,229148,229149],{"align":82355},"10:00",[13858,229151,229152],{"align":82355},"2:00",[13858,229154],{"align":82355},[13858,229156],{"align":82355},[13858,229158],{"align":82355},[13858,229160],{"align":82355},[13843,229162,229163,229165,229168,229171,229173,229175,229177],{},[13858,229164,768],{"align":82355},[13858,229166,229167],{"align":82355},"10:10",[13858,229169,229170],{"align":82355},"1:00",[13858,229172],{"align":82355},[13858,229174],{"align":82355},[13858,229176],{"align":82355},[13858,229178],{"align":82355},[13843,229180,229181,229183,229186,229189,229191,229193,229195],{},[13858,229182,786],{"align":82355},[13858,229184,229185],{"align":82355},"10:25",[13858,229187,229188],{"align":82355},"0:25",[13858,229190],{"align":82355},[13858,229192],{"align":82355},[13858,229194],{"align":82355},[13858,229196],{"align":82355},[15,229198,229199,229202],{},[2627,229200,229201],{},"平均作業周轉時間 T =",[2627,229203,229204],{},"平均作業帶權周轉時間 W =",[253,229206,225963],{"id":229207},"详细讲解与答案-2",[15,229209,229210],{},"为了方便计算，将运行时间统一转换为分钟：1(120m), 2(60m), 3(15m)。",[15,229212,229213],{},[2627,229214,229215],{},"(1) FCFS 算法",[105,229217,229218,229224,229230,229236],{},[108,229219,229220,229223],{},[2627,229221,229222],{},"J1:"," 10:00~12:00。周转=120m。带权=1.0。",[108,229225,229226,229229],{},[2627,229227,229228],{},"J2:"," 12:00~13:00。周转=180m (10:10到达)。带权=180\u002F60=3.0。",[108,229231,229232,229235],{},[2627,229233,229234],{},"J3:"," 13:00~13:15。周转=170m (10:25到达)。带权=170\u002F15≈11.33。",[108,229237,229238,229240,229241,229371,229372,347],{},[2627,229239,29799],{}," 平均周转 = ",[34,229242,229244,229284],{"className":229243},[31642],[34,229245,229247],{"className":229246},[31646],[31648,229248,229249],{"xmlns":31650},[31652,229250,229251,229281],{},[31655,229252,229253,229255,229257,229259,229261,229263,229266,229268,229270,229272,229274],{},[31662,229254,456],{"stretchy":702},[31749,229256,162217],{},[31662,229258,223],{},[31749,229260,74303],{},[31662,229262,223],{},[31749,229264,229265],{},"170",[31662,229267,2557],{"stretchy":702},[31658,229269,38080],{"mathvariant":38079},[31749,229271,786],{},[31662,229273,202],{},[31655,229275,229276,229279],{},[31749,229277,229278],{"mathvariant":226163},"156.67",[39361,229280,228174],{},[31665,229282,229283],{"encoding":31667},"(120+180+170)\u002F3 = \\mathbf{156.67 \\text{ 分钟}}",[34,229285,229287,229308,229326,229350],{"className":229286,"ariaHidden":758},[31672],[34,229288,229290,229293,229296,229299,229302,229305],{"className":229289},[31676],[34,229291],{"className":229292,"style":31774},[31680],[34,229294,456],{"className":229295},[31782],[34,229297,162217],{"className":229298},[31685],[34,229300],{"className":229301,"style":32604},[31845],[34,229303,223],{"className":229304},[32024],[34,229306],{"className":229307,"style":32604},[31845],[34,229309,229311,229314,229317,229320,229323],{"className":229310},[31676],[34,229312],{"className":229313,"style":35942},[31680],[34,229315,74303],{"className":229316},[31685],[34,229318],{"className":229319,"style":32604},[31845],[34,229321,223],{"className":229322},[32024],[34,229324],{"className":229325,"style":32604},[31845],[34,229327,229329,229332,229335,229338,229341,229344,229347],{"className":229328},[31676],[34,229330],{"className":229331,"style":31774},[31680],[34,229333,229265],{"className":229334},[31685],[34,229336,2557],{"className":229337},[31690],[34,229339,42614],{"className":229340},[31685],[34,229342],{"className":229343,"style":31846},[31845],[34,229345,202],{"className":229346},[31850],[34,229348],{"className":229349,"style":31846},[31845],[34,229351,229353,229356],{"className":229352},[31676],[34,229354],{"className":229355,"style":33218},[31680],[34,229357,229359,229362],{"className":229358},[31685],[34,229360,229278],{"className":229361},[31685,226302],[34,229363,229365,229368],{"className":229364},[31685,2638],[34,229366,130298],{"className":229367},[31685],[34,229369,228301],{"className":229370},[31685,105462],"；平均带权 = ",[34,229373,229375,229411],{"className":229374},[31642],[34,229376,229378],{"className":229377},[31646],[31648,229379,229380],{"xmlns":31650},[31652,229381,229382,229408],{},[31655,229383,229384,229386,229388,229390,229392,229394,229397,229399,229401,229403,229405],{},[31662,229385,456],{"stretchy":702},[31749,229387,640],{},[31662,229389,223],{},[31749,229391,786],{},[31662,229393,223],{},[31749,229395,229396],{},"11.33",[31662,229398,2557],{"stretchy":702},[31658,229400,38080],{"mathvariant":38079},[31749,229402,786],{},[31662,229404,202],{},[31749,229406,229407],{"mathvariant":226163},"5.11",[31665,229409,229410],{"encoding":31667},"(1+3+11.33)\u002F3 = \\mathbf{5.11}",[34,229412,229414,229435,229453,229477],{"className":229413,"ariaHidden":758},[31672],[34,229415,229417,229420,229423,229426,229429,229432],{"className":229416},[31676],[34,229418],{"className":229419,"style":31774},[31680],[34,229421,456],{"className":229422},[31782],[34,229424,640],{"className":229425},[31685],[34,229427],{"className":229428,"style":32604},[31845],[34,229430,223],{"className":229431},[32024],[34,229433],{"className":229434,"style":32604},[31845],[34,229436,229438,229441,229444,229447,229450],{"className":229437},[31676],[34,229439],{"className":229440,"style":35942},[31680],[34,229442,786],{"className":229443},[31685],[34,229445],{"className":229446,"style":32604},[31845],[34,229448,223],{"className":229449},[32024],[34,229451],{"className":229452,"style":32604},[31845],[34,229454,229456,229459,229462,229465,229468,229471,229474],{"className":229455},[31676],[34,229457],{"className":229458,"style":31774},[31680],[34,229460,229396],{"className":229461},[31685],[34,229463,2557],{"className":229464},[31690],[34,229466,42614],{"className":229467},[31685],[34,229469],{"className":229470,"style":31846},[31845],[34,229472,202],{"className":229473},[31850],[34,229475],{"className":229476,"style":31846},[31845],[34,229478,229480,229483],{"className":229479},[31676],[34,229481],{"className":229482,"style":32617},[31680],[34,229484,229486],{"className":229485},[31685],[34,229487,229407],{"className":229488},[31685,226302],[15,229490,229491],{},[2627,229492,229493],{},"(2) HRRN (最高响应比优先) 算法",[105,229495,229496,229502,229895,229901],{},[108,229497,229498,229501],{},[2627,229499,229500],{},"10:00:"," J1执行，至 12:00 结束。",[108,229503,229504,229597],{},[2627,229505,229506,229507,2664],{},"12:00 重新计算响应比 ",[34,229508,229510,229537],{"className":229509},[31642],[34,229511,229513],{"className":229512},[31646],[31648,229514,229515],{"xmlns":31650},[31652,229516,229517,229534],{},[31655,229518,229519,229521,229523,229525,229527,229529,229531],{},[31658,229520,34386],{},[31662,229522,202],{},[31749,229524,640],{},[31662,229526,223],{},[39361,229528,216949],{},[31658,229530,38080],{"mathvariant":38079},[39361,229532,229533],{},"运行时间",[31665,229535,229536],{"encoding":31667},"R = 1 + \\text{等待时间}\u002F\\text{运行时间}",[34,229538,229540,229558,229576],{"className":229539,"ariaHidden":758},[31672],[34,229541,229543,229546,229549,229552,229555],{"className":229542},[31676],[34,229544],{"className":229545,"style":33218},[31680],[34,229547,34386],{"className":229548,"style":34427},[31685,31686],[34,229550],{"className":229551,"style":31846},[31845],[34,229553,202],{"className":229554},[31850],[34,229556],{"className":229557,"style":31846},[31845],[34,229559,229561,229564,229567,229570,229573],{"className":229560},[31676],[34,229562],{"className":229563,"style":35942},[31680],[34,229565,640],{"className":229566},[31685],[34,229568],{"className":229569,"style":32604},[31845],[34,229571,223],{"className":229572},[32024],[34,229574],{"className":229575,"style":32604},[31845],[34,229577,229579,229582,229588,229591],{"className":229578},[31676],[34,229580],{"className":229581,"style":31774},[31680],[34,229583,229585],{"className":229584},[31685,2638],[34,229586,216949],{"className":229587},[31685,105462],[34,229589,38080],{"className":229590},[31685],[34,229592,229594],{"className":229593},[31685,2638],[34,229595,229533],{"className":229596},[31685,105462],[105,229598,229599,229744,229889],{},[108,229600,229601,229602,347],{},"J2 已等 110m (12:00 - 10:10)。",[34,229603,229605,229640],{"className":229604},[31642],[34,229606,229608],{"className":229607},[31646],[31648,229609,229610],{"xmlns":31650},[31652,229611,229612,229637],{},[31655,229613,229614,229620,229622,229624,229626,229628,229630,229632,229634],{},[31743,229615,229616,229618],{},[31658,229617,34386],{},[31749,229619,768],{},[31662,229621,202],{},[31749,229623,640],{},[31662,229625,223],{},[31749,229627,162188],{},[31658,229629,38080],{"mathvariant":38079},[31749,229631,167664],{},[31662,229633,202],{},[31749,229635,229636],{},"2.83",[31665,229638,229639],{"encoding":31667},"R_2 = 1 + 110\u002F60 = 2.83",[34,229641,229643,229698,229716,229735],{"className":229642,"ariaHidden":758},[31672],[34,229644,229646,229649,229689,229692,229695],{"className":229645},[31676],[34,229647],{"className":229648,"style":70915},[31680],[34,229650,229652,229655],{"className":229651},[31685],[34,229653,34386],{"className":229654,"style":34427},[31685,31686],[34,229656,229658],{"className":229657},[31792],[34,229659,229661,229681],{"className":229660},[31796,31797],[34,229662,229664,229678],{"className":229663},[31801],[34,229665,229667],{"className":229666,"style":31806},[31805],[34,229668,229669,229672],{"style":104694},[34,229670],{"className":229671,"style":31814},[31813],[34,229673,229675],{"className":229674},[31818,31819,31820,31821],[34,229676,768],{"className":229677},[31685,31821],[34,229679,31829],{"className":229680},[31828],[34,229682,229684],{"className":229683},[31801],[34,229685,229687],{"className":229686,"style":31836},[31805],[34,229688],{},[34,229690],{"className":229691,"style":31846},[31845],[34,229693,202],{"className":229694},[31850],[34,229696],{"className":229697,"style":31846},[31845],[34,229699,229701,229704,229707,229710,229713],{"className":229700},[31676],[34,229702],{"className":229703,"style":35942},[31680],[34,229705,640],{"className":229706},[31685],[34,229708],{"className":229709,"style":32604},[31845],[34,229711,223],{"className":229712},[32024],[34,229714],{"className":229715,"style":32604},[31845],[34,229717,229719,229722,229726,229729,229732],{"className":229718},[31676],[34,229720],{"className":229721,"style":31774},[31680],[34,229723,229725],{"className":229724},[31685],"110\u002F60",[34,229727],{"className":229728,"style":31846},[31845],[34,229730,202],{"className":229731},[31850],[34,229733],{"className":229734,"style":31846},[31845],[34,229736,229738,229741],{"className":229737},[31676],[34,229739],{"className":229740,"style":32617},[31680],[34,229742,229636],{"className":229743},[31685],[108,229745,229746,229747,347],{},"J3 已等 95m (12:00 - 10:25)。",[34,229748,229750,229785],{"className":229749},[31642],[34,229751,229753],{"className":229752},[31646],[31648,229754,229755],{"xmlns":31650},[31652,229756,229757,229782],{},[31655,229758,229759,229765,229767,229769,229771,229773,229775,229777,229779],{},[31743,229760,229761,229763],{},[31658,229762,34386],{},[31749,229764,786],{},[31662,229766,202],{},[31749,229768,640],{},[31662,229770,223],{},[31749,229772,96817],{},[31658,229774,38080],{"mathvariant":38079},[31749,229776,95396],{},[31662,229778,202],{},[31749,229780,229781],{},"7.33",[31665,229783,229784],{"encoding":31667},"R_3 = 1 + 95\u002F15 = 7.33",[34,229786,229788,229843,229861,229880],{"className":229787,"ariaHidden":758},[31672],[34,229789,229791,229794,229834,229837,229840],{"className":229790},[31676],[34,229792],{"className":229793,"style":70915},[31680],[34,229795,229797,229800],{"className":229796},[31685],[34,229798,34386],{"className":229799,"style":34427},[31685,31686],[34,229801,229803],{"className":229802},[31792],[34,229804,229806,229826],{"className":229805},[31796,31797],[34,229807,229809,229823],{"className":229808},[31801],[34,229810,229812],{"className":229811,"style":31806},[31805],[34,229813,229814,229817],{"style":104694},[34,229815],{"className":229816,"style":31814},[31813],[34,229818,229820],{"className":229819},[31818,31819,31820,31821],[34,229821,786],{"className":229822},[31685,31821],[34,229824,31829],{"className":229825},[31828],[34,229827,229829],{"className":229828},[31801],[34,229830,229832],{"className":229831,"style":31836},[31805],[34,229833],{},[34,229835],{"className":229836,"style":31846},[31845],[34,229838,202],{"className":229839},[31850],[34,229841],{"className":229842,"style":31846},[31845],[34,229844,229846,229849,229852,229855,229858],{"className":229845},[31676],[34,229847],{"className":229848,"style":35942},[31680],[34,229850,640],{"className":229851},[31685],[34,229853],{"className":229854,"style":32604},[31845],[34,229856,223],{"className":229857},[32024],[34,229859],{"className":229860,"style":32604},[31845],[34,229862,229864,229867,229871,229874,229877],{"className":229863},[31676],[34,229865],{"className":229866,"style":31774},[31680],[34,229868,229870],{"className":229869},[31685],"95\u002F15",[34,229872],{"className":229873,"style":31846},[31845],[34,229875,202],{"className":229876},[31850],[34,229878],{"className":229879,"style":31846},[31845],[34,229881,229883,229886],{"className":229882},[31676],[34,229884],{"className":229885,"style":32617},[31680],[34,229887,229781],{"className":229888},[31685],[108,229890,229891,229894],{},[2627,229892,229893],{},"J3 响应比最高，抢得CPU","。J3 运行 15m，至 12:15 结束。",[108,229896,229897,229900],{},[2627,229898,229899],{},"12:15:"," 仅剩 J2，J2 运行 60m，至 13:15 结束。",[108,229902,229903,229906],{},[2627,229904,229905],{},"性能指标：",[105,229907,229908,229911,229914,229917,230078],{},[108,229909,229910],{},"J1: 周转=120m, 带权=1.0。",[108,229912,229913],{},"J3: 周转=110m (10:25~12:15), 带权=110\u002F15=7.33。",[108,229915,229916],{},"J2: 周转=185m (10:10~13:15), 带权=185\u002F60=3.08。",[108,229918,229919,229920],{},"平均周转 = ",[34,229921,229923,229972],{"className":229922},[31642],[34,229924,229926],{"className":229925},[31646],[31648,229927,229928],{"xmlns":31650},[31652,229929,229930,229969],{},[31655,229931,229932,229934,229936,229938,229940,229942,229945,229947,229949,229951,229953,229956,229958,229960,229962],{},[31662,229933,456],{"stretchy":702},[31749,229935,162217],{},[31662,229937,223],{},[31749,229939,162188],{},[31662,229941,223],{},[31749,229943,229944],{},"185",[31662,229946,2557],{"stretchy":702},[31658,229948,38080],{"mathvariant":38079},[31749,229950,786],{},[31662,229952,202],{},[31749,229954,229955],{},"415",[31658,229957,38080],{"mathvariant":38079},[31749,229959,786],{},[31662,229961,202],{},[31655,229963,229964,229967],{},[31749,229965,229966],{"mathvariant":226163},"138.33",[39361,229968,228174],{},[31665,229970,229971],{"encoding":31667},"(120+110+185)\u002F3 = 415\u002F3 = \\mathbf{138.33 \\text{ 分钟}}",[34,229973,229975,229996,230014,230038,230057],{"className":229974,"ariaHidden":758},[31672],[34,229976,229978,229981,229984,229987,229990,229993],{"className":229977},[31676],[34,229979],{"className":229980,"style":31774},[31680],[34,229982,456],{"className":229983},[31782],[34,229985,162217],{"className":229986},[31685],[34,229988],{"className":229989,"style":32604},[31845],[34,229991,223],{"className":229992},[32024],[34,229994],{"className":229995,"style":32604},[31845],[34,229997,229999,230002,230005,230008,230011],{"className":229998},[31676],[34,230000],{"className":230001,"style":35942},[31680],[34,230003,162188],{"className":230004},[31685],[34,230006],{"className":230007,"style":32604},[31845],[34,230009,223],{"className":230010},[32024],[34,230012],{"className":230013,"style":32604},[31845],[34,230015,230017,230020,230023,230026,230029,230032,230035],{"className":230016},[31676],[34,230018],{"className":230019,"style":31774},[31680],[34,230021,229944],{"className":230022},[31685],[34,230024,2557],{"className":230025},[31690],[34,230027,42614],{"className":230028},[31685],[34,230030],{"className":230031,"style":31846},[31845],[34,230033,202],{"className":230034},[31850],[34,230036],{"className":230037,"style":31846},[31845],[34,230039,230041,230044,230048,230051,230054],{"className":230040},[31676],[34,230042],{"className":230043,"style":31774},[31680],[34,230045,230047],{"className":230046},[31685],"415\u002F3",[34,230049],{"className":230050,"style":31846},[31845],[34,230052,202],{"className":230053},[31850],[34,230055],{"className":230056,"style":31846},[31845],[34,230058,230060,230063],{"className":230059},[31676],[34,230061],{"className":230062,"style":33218},[31680],[34,230064,230066,230069],{"className":230065},[31685],[34,230067,229966],{"className":230068},[31685,226302],[34,230070,230072,230075],{"className":230071},[31685,2638],[34,230073,130298],{"className":230074},[31685],[34,230076,228301],{"className":230077},[31685,105462],[108,230079,230080,230081],{},"平均带权 = ",[34,230082,230084,230120],{"className":230083},[31642],[34,230085,230087],{"className":230086},[31646],[31648,230088,230089],{"xmlns":31650},[31652,230090,230091,230117],{},[31655,230092,230093,230095,230097,230099,230101,230103,230106,230108,230110,230112,230114],{},[31662,230094,456],{"stretchy":702},[31749,230096,640],{},[31662,230098,223],{},[31749,230100,229781],{},[31662,230102,223],{},[31749,230104,230105],{},"3.08",[31662,230107,2557],{"stretchy":702},[31658,230109,38080],{"mathvariant":38079},[31749,230111,786],{},[31662,230113,202],{},[31749,230115,230116],{"mathvariant":226163},"3.80",[31665,230118,230119],{"encoding":31667},"(1+7.33+3.08)\u002F3 = \\mathbf{3.80}",[34,230121,230123,230144,230162,230186],{"className":230122,"ariaHidden":758},[31672],[34,230124,230126,230129,230132,230135,230138,230141],{"className":230125},[31676],[34,230127],{"className":230128,"style":31774},[31680],[34,230130,456],{"className":230131},[31782],[34,230133,640],{"className":230134},[31685],[34,230136],{"className":230137,"style":32604},[31845],[34,230139,223],{"className":230140},[32024],[34,230142],{"className":230143,"style":32604},[31845],[34,230145,230147,230150,230153,230156,230159],{"className":230146},[31676],[34,230148],{"className":230149,"style":35942},[31680],[34,230151,229781],{"className":230152},[31685],[34,230154],{"className":230155,"style":32604},[31845],[34,230157,223],{"className":230158},[32024],[34,230160],{"className":230161,"style":32604},[31845],[34,230163,230165,230168,230171,230174,230177,230180,230183],{"className":230164},[31676],[34,230166],{"className":230167,"style":31774},[31680],[34,230169,230105],{"className":230170},[31685],[34,230172,2557],{"className":230173},[31690],[34,230175,42614],{"className":230176},[31685],[34,230178],{"className":230179,"style":31846},[31845],[34,230181,202],{"className":230182},[31850],[34,230184],{"className":230185,"style":31846},[31845],[34,230187,230189,230192],{"className":230188},[31676],[34,230190],{"className":230191,"style":32617},[31680],[34,230193,230195],{"className":230194},[31685],[34,230196,230116],{"className":230197},[31685,226302],[15,230199,230200,47649,230202,230205],{},[2627,230201,62532],{},[2627,230203,230204],{},"HRRN 性能更好","，因为它成功让极其短小的作业 J3 免受了长时间的饥饿等待，大幅拉低了平均值。",[32188,230207],{},[19,230209,230211],{"id":230210},"_8四道系统的双层调度-内存容量限制-srt-抢占","8、四道系统的双层调度 (内存容量限制 + SRT 抢占)",[15,230213,230214],{},"若有一个四道作业系统，如果在一段时间内先后有6个作业，它们提交和运行时间由下表给出(时间单位为分钟)。系统采用短作业优先(SJF)调度算法，作业被调度进入系统后中途不会退出，但是作业会被更短的作业抢占。问题：\n(1) 画图给出6个作业的执行时间序列；\n(2) 完成各个作业的开始时间、完成时间、周转时间和带权周转时间。\n(3) 计算平均作业周转时间和平均作业带权周转时间。",[13837,230216,230217,230240],{},[13840,230218,230219],{},[13843,230220,230221,230224,230227,230229,230232,230234,230237],{},[13846,230222,230223],{"align":82355},"作业序号",[13846,230225,230226],{"align":82355},"提交时间",[13846,230228,229533],{"align":82355},[13846,230230,230231],{"align":82355},"开始时间",[13846,230233,214469],{"align":82355},[13846,230235,230236],{"align":82355},"周转时间",[13846,230238,230239],{"align":82355},"带权周转时间",[13853,230241,230242,230259,230276,230293,230310,230327],{},[13843,230243,230244,230246,230249,230251,230253,230255,230257],{},[13858,230245,640],{"align":82355},[13858,230247,230248],{"align":82355},"8:00",[13858,230250,167664],{"align":82355},[13858,230252],{"align":82355},[13858,230254],{"align":82355},[13858,230256],{"align":82355},[13858,230258],{"align":82355},[13843,230260,230261,230263,230266,230268,230270,230272,230274],{},[13858,230262,768],{"align":82355},[13858,230264,230265],{"align":82355},"8:20",[13858,230267,5752],{"align":82355},[13858,230269],{"align":82355},[13858,230271],{"align":82355},[13858,230273],{"align":82355},[13858,230275],{"align":82355},[13843,230277,230278,230280,230283,230285,230287,230289,230291],{},[13858,230279,786],{"align":82355},[13858,230281,230282],{"align":82355},"8:25",[13858,230284,3249],{"align":82355},[13858,230286],{"align":82355},[13858,230288],{"align":82355},[13858,230290],{"align":82355},[13858,230292],{"align":82355},[13843,230294,230295,230297,230300,230302,230304,230306,230308],{},[13858,230296,15932],{"align":82355},[13858,230298,230299],{"align":82355},"8:30",[13858,230301,95556],{"align":82355},[13858,230303],{"align":82355},[13858,230305],{"align":82355},[13858,230307],{"align":82355},[13858,230309],{"align":82355},[13843,230311,230312,230314,230317,230319,230321,230323,230325],{},[13858,230313,18136],{"align":82355},[13858,230315,230316],{"align":82355},"8:35",[13858,230318,18136],{"align":82355},[13858,230320],{"align":82355},[13858,230322],{"align":82355},[13858,230324],{"align":82355},[13858,230326],{"align":82355},[13843,230328,230329,230331,230334,230336,230338,230340,230342],{},[13858,230330,36656],{"align":82355},[13858,230332,230333],{"align":82355},"8:40",[13858,230335,3244],{"align":82355},[13858,230337],{"align":82355},[13858,230339],{"align":82355},[13858,230341],{"align":82355},[13858,230343],{"align":82355},[15,230345,230346,230349,230350,230353,230354,347],{},[2627,230347,230348],{},"注意：易错点 (Common Pitfalls)：","\n这是本卷最难的题目！题目明确说明这是**“四道作业系统”",[2627,230351,230352],{},"，这意味着","内存中最多只能装入4个作业**。当第5个作业到达时，即便它再短，只要内存没空位，它就只能在“外存后备队列”死等！\n同时，“中途不退出但会被抢占”意味着进程调度是",[2627,230355,230356],{},"最短剩余时间优先 (SRT)",[253,230358,230359],{"id":230359},"详细讲解推演",[105,230361,230362,230367,230376,230385,230394,230403,230420,230437,230451,230460,230469],{},[108,230363,230364,230366],{},[2627,230365,230248],{}," J1(60) 到达。内存(1\u002F4)。J1 运行。",[108,230368,230369,230371,230372,230375],{},[2627,230370,230265],{}," J2(35) 到达。内存(2\u002F4)。J1还剩40，J2比J1短，",[2627,230373,230374],{},"J2 抢占 J1","。J2 运行。",[108,230377,230378,230380,230381,230384],{},[2627,230379,230282],{}," J3(20) 到达。内存(3\u002F4)。J2还剩30，J3比J2短，",[2627,230382,230383],{},"J3 抢占 J2","。J3 运行。",[108,230386,230387,230389,230390,230393],{},[2627,230388,230299],{}," J4(25) 到达。内存(4\u002F4)。",[2627,230391,230392],{},"此时内存已满！"," J3还剩15，继续运行。",[108,230395,230396,230399,230400,347],{},[2627,230397,230398],{},"8:35 & 8:40"," J5(5) 和 J6(10) 到达。",[2627,230401,230402],{},"因为内存已满，它们被关在门外排队",[108,230404,230405,230408,230409,39984,230412],{},[2627,230406,230407],{},"8:45"," J3 执行完毕！内存腾出1个位置。作业调度器使用 SJF 检查门外排队的 J5(5) 和 J6(10)。J5 更短，",[2627,230410,230411],{},"J5 被调入内存",[105,230413,230414],{},[108,230415,230416,230417,347],{},"此时内存有 J1(剩40), J2(剩30), J4(25), J5(5)。",[2627,230418,230419],{},"J5 最短，J5 抢得CPU开始运行",[108,230421,230422,230425,230426,39984,230429],{},[2627,230423,230424],{},"8:50"," J5 执行完毕！内存腾出。",[2627,230427,230428],{},"J6 被调入内存",[105,230430,230431],{},[108,230432,230433,230434,347],{},"内存有 J1(40), J2(30), J4(25), J6(10)。",[2627,230435,230436],{},"J6 最短，J6 运行",[108,230438,230439,230442,230443],{},[2627,230440,230441],{},"9:00"," J6 执行完毕！外存没作业了。\n",[105,230444,230445],{},[108,230446,230447,230448,347],{},"内存有 J1(40), J2(30), J4(25)。",[2627,230449,230450],{},"J4 运行",[108,230452,230453,230456,230457,347],{},[2627,230454,230455],{},"9:25"," J4 完毕。",[2627,230458,230459],{},"J2 恢复运行",[108,230461,230462,230465,230466,347],{},[2627,230463,230464],{},"9:55"," J2 完毕。",[2627,230467,230468],{},"J1 恢复运行",[108,230470,230471,230474],{},[2627,230472,230473],{},"10:35"," J1 完毕。全剧终。",[253,230476,230477],{"id":230477},"答案表格",[13837,230479,230480,230499],{},[13840,230481,230482],{},[13843,230483,230484,230487,230489,230491,230493,230495,230497],{},[13846,230485,230486],{"align":82355},"作业",[13846,230488,230226],{"align":82355},[13846,230490,229533],{"align":82355},[13846,230492,230231],{"align":82355},[13846,230494,214469],{"align":82355},[13846,230496,230236],{"align":82355},[13846,230498,230239],{"align":82355},[13853,230500,230501,230519,230536,230553,230570,230587,230604],{},[13843,230502,230503,230505,230507,230509,230511,230513,230516],{},[13858,230504,640],{"align":82355},[13858,230506,230248],{"align":82355},[13858,230508,167664],{"align":82355},[13858,230510,230248],{"align":82355},[13858,230512,230473],{"align":82355},[13858,230514,230515],{"align":82355},"155",[13858,230517,230518],{"align":82355},"155\u002F60 ≈ 2.58",[13843,230520,230521,230523,230525,230527,230529,230531,230533],{},[13858,230522,768],{"align":82355},[13858,230524,230265],{"align":82355},[13858,230526,5752],{"align":82355},[13858,230528,230265],{"align":82355},[13858,230530,230464],{"align":82355},[13858,230532,96817],{"align":82355},[13858,230534,230535],{"align":82355},"95\u002F35 ≈ 2.71",[13843,230537,230538,230540,230542,230544,230546,230548,230550],{},[13858,230539,786],{"align":82355},[13858,230541,230282],{"align":82355},[13858,230543,3249],{"align":82355},[13858,230545,230282],{"align":82355},[13858,230547,230407],{"align":82355},[13858,230549,3249],{"align":82355},[13858,230551,230552],{"align":82355},"20\u002F20 = 1.00",[13843,230554,230555,230557,230559,230561,230563,230565,230567],{},[13858,230556,15932],{"align":82355},[13858,230558,230299],{"align":82355},[13858,230560,95556],{"align":82355},[13858,230562,230441],{"align":82355},[13858,230564,230455],{"align":82355},[13858,230566,96327],{"align":82355},[13858,230568,230569],{"align":82355},"55\u002F25 = 2.20",[13843,230571,230572,230574,230576,230578,230580,230582,230584],{},[13858,230573,18136],{"align":82355},[13858,230575,230316],{"align":82355},[13858,230577,18136],{"align":82355},[13858,230579,230407],{"align":82355},[13858,230581,230424],{"align":82355},[13858,230583,95396],{"align":82355},[13858,230585,230586],{"align":82355},"15\u002F5 = 3.00",[13843,230588,230589,230591,230593,230595,230597,230599,230601],{},[13858,230590,36656],{"align":82355},[13858,230592,230333],{"align":82355},[13858,230594,3244],{"align":82355},[13858,230596,230424],{"align":82355},[13858,230598,230441],{"align":82355},[13858,230600,3249],{"align":82355},[13858,230602,230603],{"align":82355},"20\u002F10 = 2.00",[13843,230605,230606,230611,230613,230615,230617,230619,230621],{},[13858,230607,230608],{"align":82355},[71807,230609,230610],{},"(注意：J4虽然8:30进内存，但9:00才真正“开始”分配到CPU)",[13858,230612],{"align":82355},[13858,230614],{"align":82355},[13858,230616],{"align":82355},[13858,230618],{"align":82355},[13858,230620],{"align":82355},[13858,230622],{"align":82355},[105,230624,230625,230852],{},[108,230626,230627,47649,230629,347],{},[2627,230628,226112],{},[34,230630,230632,230691],{"className":230631},[31642],[34,230633,230635],{"className":230634},[31646],[31648,230636,230637],{"xmlns":31650},[31652,230638,230639,230688],{},[31655,230640,230641,230643,230645,230647,230649,230651,230653,230655,230657,230659,230661,230663,230665,230667,230669,230671,230673,230675,230677,230679,230681],{},[31662,230642,456],{"stretchy":702},[31749,230644,230515],{},[31662,230646,223],{},[31749,230648,96817],{},[31662,230650,223],{},[31749,230652,3249],{},[31662,230654,223],{},[31749,230656,96327],{},[31662,230658,223],{},[31749,230660,95396],{},[31662,230662,223],{},[31749,230664,3249],{},[31662,230666,2557],{"stretchy":702},[31658,230668,38080],{"mathvariant":38079},[31749,230670,36656],{},[31662,230672,202],{},[31749,230674,73878],{},[31658,230676,38080],{"mathvariant":38079},[31749,230678,36656],{},[31662,230680,202],{},[31655,230682,230683,230686],{},[31749,230684,230685],{"mathvariant":226163},"60.0",[39361,230687,228174],{},[31665,230689,230690],{"encoding":31667},"(155+95+20+55+15+20) \u002F 6 = 360 \u002F 6 = \\mathbf{60.0 \\text{ 分钟}}",[34,230692,230694,230715,230733,230751,230769,230787,230812,230831],{"className":230693,"ariaHidden":758},[31672],[34,230695,230697,230700,230703,230706,230709,230712],{"className":230696},[31676],[34,230698],{"className":230699,"style":31774},[31680],[34,230701,456],{"className":230702},[31782],[34,230704,230515],{"className":230705},[31685],[34,230707],{"className":230708,"style":32604},[31845],[34,230710,223],{"className":230711},[32024],[34,230713],{"className":230714,"style":32604},[31845],[34,230716,230718,230721,230724,230727,230730],{"className":230717},[31676],[34,230719],{"className":230720,"style":35942},[31680],[34,230722,96817],{"className":230723},[31685],[34,230725],{"className":230726,"style":32604},[31845],[34,230728,223],{"className":230729},[32024],[34,230731],{"className":230732,"style":32604},[31845],[34,230734,230736,230739,230742,230745,230748],{"className":230735},[31676],[34,230737],{"className":230738,"style":35942},[31680],[34,230740,3249],{"className":230741},[31685],[34,230743],{"className":230744,"style":32604},[31845],[34,230746,223],{"className":230747},[32024],[34,230749],{"className":230750,"style":32604},[31845],[34,230752,230754,230757,230760,230763,230766],{"className":230753},[31676],[34,230755],{"className":230756,"style":35942},[31680],[34,230758,96327],{"className":230759},[31685],[34,230761],{"className":230762,"style":32604},[31845],[34,230764,223],{"className":230765},[32024],[34,230767],{"className":230768,"style":32604},[31845],[34,230770,230772,230775,230778,230781,230784],{"className":230771},[31676],[34,230773],{"className":230774,"style":35942},[31680],[34,230776,95396],{"className":230777},[31685],[34,230779],{"className":230780,"style":32604},[31845],[34,230782,223],{"className":230783},[32024],[34,230785],{"className":230786,"style":32604},[31845],[34,230788,230790,230793,230796,230799,230803,230806,230809],{"className":230789},[31676],[34,230791],{"className":230792,"style":31774},[31680],[34,230794,3249],{"className":230795},[31685],[34,230797,2557],{"className":230798},[31690],[34,230800,230802],{"className":230801},[31685],"\u002F6",[34,230804],{"className":230805,"style":31846},[31845],[34,230807,202],{"className":230808},[31850],[34,230810],{"className":230811,"style":31846},[31845],[34,230813,230815,230818,230822,230825,230828],{"className":230814},[31676],[34,230816],{"className":230817,"style":31774},[31680],[34,230819,230821],{"className":230820},[31685],"360\u002F6",[34,230823],{"className":230824,"style":31846},[31845],[34,230826,202],{"className":230827},[31850],[34,230829],{"className":230830,"style":31846},[31845],[34,230832,230834,230837],{"className":230833},[31676],[34,230835],{"className":230836,"style":33218},[31680],[34,230838,230840,230843],{"className":230839},[31685],[34,230841,230685],{"className":230842},[31685,226302],[34,230844,230846,230849],{"className":230845},[31685,2638],[34,230847,130298],{"className":230848},[31685],[34,230850,228301],{"className":230851},[31685,105462],[108,230853,230854,47649,230857,347],{},[2627,230855,230856],{},"平均带权周转时间：",[34,230858,230860,230922],{"className":230859},[31642],[34,230861,230863],{"className":230862},[31646],[31648,230864,230865],{"xmlns":31650},[31652,230866,230867,230919],{},[31655,230868,230869,230871,230874,230876,230879,230881,230884,230886,230889,230891,230894,230896,230899,230901,230903,230905,230907,230910,230912,230914,230916],{},[31662,230870,456],{"stretchy":702},[31749,230872,230873],{},"2.58",[31662,230875,223],{},[31749,230877,230878],{},"2.71",[31662,230880,223],{},[31749,230882,230883],{},"1.00",[31662,230885,223],{},[31749,230887,230888],{},"2.20",[31662,230890,223],{},[31749,230892,230893],{},"3.00",[31662,230895,223],{},[31749,230897,230898],{},"2.00",[31662,230900,2557],{"stretchy":702},[31658,230902,38080],{"mathvariant":38079},[31749,230904,36656],{},[31662,230906,202],{},[31749,230908,230909],{},"13.49",[31658,230911,38080],{"mathvariant":38079},[31749,230913,36656],{},[31662,230915,168560],{},[31749,230917,230918],{"mathvariant":226163},"2.25",[31665,230920,230921],{"encoding":31667},"(2.58+2.71+1.00+2.20+3.00+2.00) \u002F 6 = 13.49 \u002F 6 \\approx \\mathbf{2.25}",[34,230923,230925,230946,230964,230982,231000,231018,231042,231061],{"className":230924,"ariaHidden":758},[31672],[34,230926,230928,230931,230934,230937,230940,230943],{"className":230927},[31676],[34,230929],{"className":230930,"style":31774},[31680],[34,230932,456],{"className":230933},[31782],[34,230935,230873],{"className":230936},[31685],[34,230938],{"className":230939,"style":32604},[31845],[34,230941,223],{"className":230942},[32024],[34,230944],{"className":230945,"style":32604},[31845],[34,230947,230949,230952,230955,230958,230961],{"className":230948},[31676],[34,230950],{"className":230951,"style":35942},[31680],[34,230953,230878],{"className":230954},[31685],[34,230956],{"className":230957,"style":32604},[31845],[34,230959,223],{"className":230960},[32024],[34,230962],{"className":230963,"style":32604},[31845],[34,230965,230967,230970,230973,230976,230979],{"className":230966},[31676],[34,230968],{"className":230969,"style":35942},[31680],[34,230971,230883],{"className":230972},[31685],[34,230974],{"className":230975,"style":32604},[31845],[34,230977,223],{"className":230978},[32024],[34,230980],{"className":230981,"style":32604},[31845],[34,230983,230985,230988,230991,230994,230997],{"className":230984},[31676],[34,230986],{"className":230987,"style":35942},[31680],[34,230989,230888],{"className":230990},[31685],[34,230992],{"className":230993,"style":32604},[31845],[34,230995,223],{"className":230996},[32024],[34,230998],{"className":230999,"style":32604},[31845],[34,231001,231003,231006,231009,231012,231015],{"className":231002},[31676],[34,231004],{"className":231005,"style":35942},[31680],[34,231007,230893],{"className":231008},[31685],[34,231010],{"className":231011,"style":32604},[31845],[34,231013,223],{"className":231014},[32024],[34,231016],{"className":231017,"style":32604},[31845],[34,231019,231021,231024,231027,231030,231033,231036,231039],{"className":231020},[31676],[34,231022],{"className":231023,"style":31774},[31680],[34,231025,230898],{"className":231026},[31685],[34,231028,2557],{"className":231029},[31690],[34,231031,230802],{"className":231032},[31685],[34,231034],{"className":231035,"style":31846},[31845],[34,231037,202],{"className":231038},[31850],[34,231040],{"className":231041,"style":31846},[31845],[34,231043,231045,231048,231052,231055,231058],{"className":231044},[31676],[34,231046],{"className":231047,"style":31774},[31680],[34,231049,231051],{"className":231050},[31685],"13.49\u002F6",[34,231053],{"className":231054,"style":31846},[31845],[34,231056,168560],{"className":231057},[31850],[34,231059],{"className":231060,"style":31846},[31845],[34,231062,231064,231067],{"className":231063},[31676],[34,231065],{"className":231066,"style":32617},[31680],[34,231068,231070],{"className":231069},[31685],[34,231071,230918],{"className":231072},[31685,226302],[32188,231074],{},[19,231076,231078],{"id":231077},"_9两道系统双层调度-内存容量-2-抢占式优先权","9、两道系统双层调度 (内存容量 2 + 抢占式优先权)",[15,231080,231081,231083],{},[2627,231082,214361],{},"\n两道作业系统。作业调度: SJF；进程调度: 抢占式优先数（越小越高）。A(10:00, 40m, 优5), B(10:20, 30m, 优3), C(10:30, 50m, 优4), D(10:50, 20m, 优6)。",[13837,231085,231086,231101],{},[13840,231087,231088],{},[13843,231089,231090,231093,231095,231098],{},[13846,231091,231092],{"align":82355},"作业名",[13846,231094,230226],{"align":82355},[13846,231096,231097],{"align":82355},"估计运行时间",[13846,231099,231100],{"align":82355},"优先数",[13853,231102,231103,231114,231126,231138],{},[13843,231104,231105,231107,231109,231112],{},[13858,231106,54151],{"align":82355},[13858,231108,229149],{"align":82355},[13858,231110,231111],{"align":82355},"40 分",[13858,231113,18136],{"align":82355},[13843,231115,231116,231118,231121,231124],{},[13858,231117,54224],{"align":82355},[13858,231119,231120],{"align":82355},"10:20",[13858,231122,231123],{"align":82355},"30 分",[13858,231125,786],{"align":82355},[13843,231127,231128,231130,231133,231136],{},[13858,231129,47384],{"align":82355},[13858,231131,231132],{"align":82355},"10:30",[13858,231134,231135],{"align":82355},"50 分",[13858,231137,15932],{"align":82355},[13843,231139,231140,231142,231145,231148],{},[13858,231141,91994],{"align":82355},[13858,231143,231144],{"align":82355},"10:50",[13858,231146,231147],{"align":82355},"20 分",[13858,231149,36656],{"align":82355},[253,231151,230359],{"id":231152},"详细讲解推演-1",[105,231154,231155,231160,231170,231176,231202,231225,231234],{},[108,231156,231157,231159],{},[2627,231158,229500],{}," A 到达。进内存 (1\u002F2)。A(优5) 运行。",[108,231161,231162,231165,231166,231169],{},[2627,231163,231164],{},"10:20:"," B 到达。进内存 (2\u002F2，已满)。A 还剩 20m。B 的优先级(3)高于 A(5)，",[2627,231167,231168],{},"B 抢占 A","。B 运行。",[108,231171,231172,231175],{},[2627,231173,231174],{},"10:30 & 10:50:"," C 和 D 到达。因为内存已满，在外存队列等待。",[108,231177,231178,231181,231182],{},[2627,231179,231180],{},"10:50:"," B 运行 30m 结束！内存腾出 1 个空位。\n",[105,231183,231184,231193],{},[108,231185,231186,231189,231190,347],{},[2627,231187,231188],{},"作业调度 (SJF)："," 外存队列有 C(50m) 和 D(20m)。D 更短，",[2627,231191,231192],{},"D 被选中调入内存",[108,231194,231195,231198,231199,347],{},[2627,231196,231197],{},"进程调度 (抢占优先权)："," 当前内存有 A(剩20m, 优5) 和 D(20m, 优6)。A 优先级比 D 高 (5\u003C6)，所以 ",[2627,231200,231201],{},"A 恢复运行",[108,231203,231204,231207,231208],{},[2627,231205,231206],{},"11:10:"," A 执行结束！内存腾出 1 个空位。\n",[105,231209,231210,231216],{},[108,231211,231212,231215],{},[2627,231213,231214],{},"作业调度："," 将 C(50m) 调入内存。",[108,231217,231218,231221,231222,347],{},[2627,231219,231220],{},"进程调度："," 内存有 D(20m, 优6) 和 C(50m, 优4)。C 优先级更高 (4\u003C6)，",[2627,231223,231224],{},"C 抢占执行",[108,231226,231227,231230,231231,347],{},[2627,231228,231229],{},"12:00:"," C 执行结束！",[2627,231232,231233],{},"D 开始执行",[108,231235,231236,231239],{},[2627,231237,231238],{},"12:20:"," D 执行结束！全剧终。",[253,231241,231242],{"id":231242},"答案整理",[105,231244,231245,231251,231257,231263,231269],{},[108,231246,231247,231250],{},[2627,231248,231249],{},"A:"," 10:00 进内存，11:10 结束。周转 = 70 分钟。",[108,231252,231253,231256],{},[2627,231254,231255],{},"B:"," 10:20 进内存，10:50 结束。周转 = 30 分钟。",[108,231258,231259,231262],{},[2627,231260,231261],{},"C:"," 11:10 进内存，12:00 结束。周转 = 90 分钟 (10:30~12:00)。",[108,231264,231265,231268],{},[2627,231266,231267],{},"D:"," 10:50 进内存，12:20 结束。周转 = 90 分钟 (10:50~12:20)。",[108,231270,231271,47649,231273,347],{},[2627,231272,226112],{},[34,231274,231276,231319],{"className":231275},[31642],[34,231277,231279],{"className":231278},[31646],[31648,231280,231281],{"xmlns":31650},[31652,231282,231283,231316],{},[31655,231284,231285,231287,231289,231291,231293,231295,231297,231299,231301,231303,231305,231307,231309],{},[31662,231286,456],{"stretchy":702},[31749,231288,98497],{},[31662,231290,223],{},[31749,231292,9314],{},[31662,231294,223],{},[31749,231296,854],{},[31662,231298,223],{},[31749,231300,854],{},[31662,231302,2557],{"stretchy":702},[31658,231304,38080],{"mathvariant":38079},[31749,231306,15932],{},[31662,231308,202],{},[31655,231310,231311,231314],{},[31749,231312,231313],{"mathvariant":226163},"70.0",[39361,231315,228174],{},[31665,231317,231318],{"encoding":31667},"(70+30+90+90) \u002F 4 = \\mathbf{70.0 \\text{ 分钟}}",[34,231320,231322,231343,231361,231379,231403],{"className":231321,"ariaHidden":758},[31672],[34,231323,231325,231328,231331,231334,231337,231340],{"className":231324},[31676],[34,231326],{"className":231327,"style":31774},[31680],[34,231329,456],{"className":231330},[31782],[34,231332,98497],{"className":231333},[31685],[34,231335],{"className":231336,"style":32604},[31845],[34,231338,223],{"className":231339},[32024],[34,231341],{"className":231342,"style":32604},[31845],[34,231344,231346,231349,231352,231355,231358],{"className":231345},[31676],[34,231347],{"className":231348,"style":35942},[31680],[34,231350,9314],{"className":231351},[31685],[34,231353],{"className":231354,"style":32604},[31845],[34,231356,223],{"className":231357},[32024],[34,231359],{"className":231360,"style":32604},[31845],[34,231362,231364,231367,231370,231373,231376],{"className":231363},[31676],[34,231365],{"className":231366,"style":35942},[31680],[34,231368,854],{"className":231369},[31685],[34,231371],{"className":231372,"style":32604},[31845],[34,231374,223],{"className":231375},[32024],[34,231377],{"className":231378,"style":32604},[31845],[34,231380,231382,231385,231388,231391,231394,231397,231400],{"className":231381},[31676],[34,231383],{"className":231384,"style":31774},[31680],[34,231386,854],{"className":231387},[31685],[34,231389,2557],{"className":231390},[31690],[34,231392,42280],{"className":231393},[31685],[34,231395],{"className":231396,"style":31846},[31845],[34,231398,202],{"className":231399},[31850],[34,231401],{"className":231402,"style":31846},[31845],[34,231404,231406,231409],{"className":231405},[31676],[34,231407],{"className":231408,"style":33218},[31680],[34,231410,231412,231415],{"className":231411},[31685],[34,231413,231313],{"className":231414},[31685,226302],[34,231416,231418,231421],{"className":231417},[31685,2638],[34,231419,130298],{"className":231420},[31685],[34,231422,228301],{"className":231423},[31685,105462],[32188,231425],{},[19,231427,231429],{"id":231428},"_10实时系统调度允许的最大执行时间","10、实时系统调度允许的最大执行时间",[15,231431,231432,231434,231435,231463,231464,231492,231493,231524,231525,231553],{},[2627,231433,214361],{},"\n4个周期性事件，周期 ",[34,231436,231438,231451],{"className":231437},[31642],[34,231439,231441],{"className":231440},[31646],[31648,231442,231443],{"xmlns":31650},[31652,231444,231445,231449],{},[31655,231446,231447],{},[31658,231448,37400],{},[31665,231450,37400],{"encoding":31667},[34,231452,231454],{"className":231453,"ariaHidden":758},[31672],[34,231455,231457,231460],{"className":231456},[31676],[34,231458],{"className":231459,"style":33218},[31680],[34,231461,37400],{"className":231462,"style":37422},[31685,31686]," 为 50, 100, 300, 250ms；处理时间 ",[34,231465,231467,231480],{"className":231466},[31642],[34,231468,231470],{"className":231469},[31646],[31648,231471,231472],{"xmlns":31650},[31652,231473,231474,231478],{},[31655,231475,231476],{},[31658,231477,47384],{},[31665,231479,47384],{"encoding":31667},[34,231481,231483],{"className":231482,"ariaHidden":758},[31672],[34,231484,231486,231489],{"className":231485},[31676],[34,231487],{"className":231488,"style":33218},[31680],[34,231490,47384],{"className":231491,"style":38706},[31685,31686]," 为 35, 20, 10, ",[34,231494,231496,231511],{"className":231495},[31642],[34,231497,231499],{"className":231498},[31646],[31648,231500,231501],{"xmlns":31650},[31652,231502,231503,231508],{},[31655,231504,231505],{},[31658,231506,231507],{},"δ",[31665,231509,231510],{"encoding":31667},"\\delta",[34,231512,231514],{"className":231513,"ariaHidden":758},[31672],[34,231515,231517,231520],{"className":231516},[31676],[34,231518],{"className":231519,"style":31681},[31680],[34,231521,231507],{"className":231522,"style":231523},[31685,31686],"margin-right:0.0379em;"," ms。系统可调度的最大 ",[34,231526,231528,231541],{"className":231527},[31642],[34,231529,231531],{"className":231530},[31646],[31648,231532,231533],{"xmlns":31650},[31652,231534,231535,231539],{},[31655,231536,231537],{},[31658,231538,231507],{},[31665,231540,231510],{"encoding":31667},[34,231542,231544],{"className":231543,"ariaHidden":758},[31672],[34,231545,231547,231550],{"className":231546},[31676],[34,231548],{"className":231549,"style":31681},[31680],[34,231551,231507],{"className":231552,"style":231523},[31685,31686]," 是多少？",[253,231555,219117],{"id":231556},"原理解析-the-why-1",[15,231558,231559,231560,231591,231592,347],{},"实时系统的核心指标是 ",[2627,231561,231562,231563,2557],{},"CPU 利用率 (",[34,231564,231566,231579],{"className":231565},[31642],[34,231567,231569],{"className":231568},[31646],[31648,231570,231571],{"xmlns":31650},[31652,231572,231573,231577],{},[31655,231574,231575],{},[31658,231576,33350],{},[31665,231578,33350],{"encoding":31667},[34,231580,231582],{"className":231581,"ariaHidden":758},[31672],[34,231583,231585,231588],{"className":231584},[31676],[34,231586],{"className":231587,"style":33218},[31680],[34,231589,33350],{"className":231590,"style":33389},[31685,31686],"。利用率定义为所有任务处理时间与其周期的比值之和：",[34,231593,231595,231627],{"className":231594},[31642],[34,231596,231598],{"className":231597},[31646],[31648,231599,231600],{"xmlns":31650},[31652,231601,231602,231624],{},[31655,231603,231604,231606,231608,231610],{},[31658,231605,33350],{},[31662,231607,202],{},[31662,231609,131138],{},[41453,231611,231612,231618],{},[31743,231613,231614,231616],{},[31658,231615,47384],{},[31658,231617,31955],{},[31743,231619,231620,231622],{},[31658,231621,37400],{},[31658,231623,31955],{},[31665,231625,231626],{"encoding":31667},"U = \\sum \\frac{C_i}{T_i}",[34,231628,231630,231648],{"className":231629,"ariaHidden":758},[31672],[34,231631,231633,231636,231639,231642,231645],{"className":231632},[31676],[34,231634],{"className":231635,"style":33218},[31680],[34,231637,33350],{"className":231638,"style":33389},[31685,31686],[34,231640],{"className":231641,"style":31846},[31845],[34,231643,202],{"className":231644},[31850],[34,231646],{"className":231647,"style":31846},[31845],[34,231649,231651,231655,231658,231661],{"className":231650},[31676],[34,231652],{"className":231653,"style":231654},[31680],"height:1.3335em;vertical-align:-0.4451em;",[34,231656,131138],{"className":231657,"style":156561},[32767,131249,156560],[34,231659],{"className":231660,"style":32123},[31845],[34,231662,231664,231667,231803],{"className":231663},[31685],[34,231665],{"className":231666},[31782,41488],[34,231668,231670],{"className":231669},[41453],[34,231671,231673,231794],{"className":231672},[31796,31797],[34,231674,231676,231791],{"className":231675},[31801],[34,231677,231679,231731,231739],{"className":231678,"style":220459},[31805],[34,231680,231681,231684],{"style":41504},[34,231682],{"className":231683,"style":41508},[31813],[34,231685,231687],{"className":231686},[31818,31819,31820,31821],[34,231688,231690],{"className":231689},[31685,31821],[34,231691,231693,231696],{"className":231692},[31685,31821],[34,231694,37400],{"className":231695,"style":37422},[31685,31686,31821],[34,231697,231699],{"className":231698},[31792],[34,231700,231702,231723],{"className":231701},[31796,31797],[34,231703,231705,231720],{"className":231704},[31801],[34,231706,231709],{"className":231707,"style":231708},[31805],"height:0.3281em;",[34,231710,231711,231714],{"style":220521},[34,231712],{"className":231713,"style":43406},[31813],[34,231715,231717],{"className":231716},[31818,43410,43411,31821],[34,231718,31955],{"className":231719},[31685,31686,31821],[34,231721,31829],{"className":231722},[31828],[34,231724,231726],{"className":231725},[31801],[34,231727,231729],{"className":231728,"style":220540},[31805],[34,231730],{},[34,231732,231733,231736],{"style":41520},[34,231734],{"className":231735,"style":41508},[31813],[34,231737],{"className":231738,"style":41528},[41527],[34,231740,231741,231744],{"style":220490},[34,231742],{"className":231743,"style":41508},[31813],[34,231745,231747],{"className":231746},[31818,31819,31820,31821],[34,231748,231750],{"className":231749},[31685,31821],[34,231751,231753,231756],{"className":231752},[31685,31821],[34,231754,47384],{"className":231755,"style":38706},[31685,31686,31821],[34,231757,231759],{"className":231758},[31792],[34,231760,231762,231783],{"className":231761},[31796,31797],[34,231763,231765,231780],{"className":231764},[31801],[34,231766,231768],{"className":231767,"style":231708},[31805],[34,231769,231771,231774],{"style":231770},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[34,231772],{"className":231773,"style":43406},[31813],[34,231775,231777],{"className":231776},[31818,43410,43411,31821],[34,231778,31955],{"className":231779},[31685,31686,31821],[34,231781,31829],{"className":231782},[31828],[34,231784,231786],{"className":231785},[31801],[34,231787,231789],{"className":231788,"style":220540},[31805],[34,231790],{},[34,231792,31829],{"className":231793},[31828],[34,231795,231797],{"className":231796},[31801],[34,231798,231801],{"className":231799,"style":231800},[31805],"height:0.4451em;",[34,231802],{},[34,231804],{"className":231805},[31690,41488],[105,231807,231808,231815],{},[108,231809,231810,231811,231814],{},"如果采用固定的",[2627,231812,231813],{},"速率单调调度 (RMS)","，能保证调度的理论安全上限大约是 0.69（针对无限个任务），本题若严格按RMS算将不满足。",[108,231816,231817,231818,231821,231822,231923,231924,347],{},"通常在这种不指定具体算法求绝对理论上限的考题中，我们使用",[2627,231819,231820],{},"动态调度算法（如最早截止时间优先 EDF）的理论极限","，即只要总CPU利用率不超过 100%（",[34,231823,231825,231849],{"className":231824},[31642],[34,231826,231828],{"className":231827},[31646],[31648,231829,231830],{"xmlns":31650},[31652,231831,231832,231846],{},[31655,231833,231834,231836,231842,231844],{},[31662,231835,131138],{},[31743,231837,231838,231840],{},[31658,231839,33350],{},[31658,231841,31955],{},[31662,231843,45972],{},[31749,231845,640],{},[31665,231847,231848],{"encoding":31667},"\\sum U_i \\le 1",[34,231850,231852,231914],{"className":231851,"ariaHidden":758},[31672],[34,231853,231855,231858,231861,231864,231905,231908,231911],{"className":231854},[31676],[34,231856],{"className":231857,"style":31774},[31680],[34,231859,131138],{"className":231860,"style":156561},[32767,131249,156560],[34,231862],{"className":231863,"style":32123},[31845],[34,231865,231867,231870],{"className":231866},[31685],[34,231868,33350],{"className":231869,"style":33389},[31685,31686],[34,231871,231873],{"className":231872},[31792],[34,231874,231876,231897],{"className":231875},[31796,31797],[34,231877,231879,231894],{"className":231878},[31801],[34,231880,231882],{"className":231881,"style":32090},[31805],[34,231883,231885,231888],{"style":231884},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[34,231886],{"className":231887,"style":31814},[31813],[34,231889,231891],{"className":231890},[31818,31819,31820,31821],[34,231892,31955],{"className":231893},[31685,31686,31821],[34,231895,31829],{"className":231896},[31828],[34,231898,231900],{"className":231899},[31801],[34,231901,231903],{"className":231902,"style":31836},[31805],[34,231904],{},[34,231906],{"className":231907,"style":31846},[31845],[34,231909,45972],{"className":231910},[31850],[34,231912],{"className":231913,"style":31846},[31845],[34,231915,231917,231920],{"className":231916},[31676],[34,231918],{"className":231919,"style":32617},[31680],[34,231921,640],{"className":231922},[31685],"），系统在理论上就是“可调度的” ",[34,231925,231926],{},"cite: 120",[15,231928,231929,231931,231932,232392,232393,33183,232436,232580,232745,232746,232779,232780],{},[2627,231930,182565],{},"\n前 3 个任务的利用率为：\n",[34,231933,231935,232007],{"className":231934},[31642],[34,231936,231938],{"className":231937},[31646],[31648,231939,231940],{"xmlns":31650},[31652,231941,231942,232004],{},[31655,231943,231944,231960,231962,231968,231970,231976,231978,231984,231986,231989,231991,231994,231996,231999,232001],{},[31743,231945,231946,231948],{},[31658,231947,33350],{},[31655,231949,231950,231952,231954,231956,231958],{},[31749,231951,640],{},[31662,231953,31960],{"separator":758},[31749,231955,768],{},[31662,231957,31960],{"separator":758},[31749,231959,786],{},[31662,231961,202],{},[41453,231963,231964,231966],{},[31749,231965,5752],{},[31749,231967,5774],{},[31662,231969,223],{},[41453,231971,231972,231974],{},[31749,231973,3249],{},[31749,231975,162173],{},[31662,231977,223],{},[41453,231979,231980,231982],{},[31749,231981,3244],{},[31749,231983,25357],{},[31662,231985,202],{},[31749,231987,231988],{},"0.7",[31662,231990,223],{},[31749,231992,231993],{},"0.2",[31662,231995,223],{},[31749,231997,231998],{},"0.0333",[31662,232000,168560],{},[31749,232002,232003],{},"0.9333",[31665,232005,232006],{"encoding":31667},"U_{1,2,3} = \\frac{35}{50} + \\frac{20}{100} + \\frac{10}{300} = 0.7 + 0.2 + 0.0333 \\approx 0.9333",[34,232008,232010,232080,232163,232246,232329,232347,232365,232383],{"className":232009,"ariaHidden":758},[31672],[34,232011,232013,232016,232071,232074,232077],{"className":232012},[31676],[34,232014],{"className":232015,"style":55484},[31680],[34,232017,232019,232022],{"className":232018},[31685],[34,232020,33350],{"className":232021,"style":33389},[31685,31686],[34,232023,232025],{"className":232024},[31792],[34,232026,232028,232063],{"className":232027},[31796,31797],[34,232029,232031,232060],{"className":232030},[31801],[34,232032,232034],{"className":232033,"style":31806},[31805],[34,232035,232036,232039],{"style":231884},[34,232037],{"className":232038,"style":31814},[31813],[34,232040,232042],{"className":232041},[31818,31819,31820,31821],[34,232043,232045,232048,232051,232054,232057],{"className":232044},[31685,31821],[34,232046,640],{"className":232047},[31685,31821],[34,232049,31960],{"className":232050},[32119,31821],[34,232052,768],{"className":232053},[31685,31821],[34,232055,31960],{"className":232056},[32119,31821],[34,232058,786],{"className":232059},[31685,31821],[34,232061,31829],{"className":232062},[31828],[34,232064,232066],{"className":232065},[31801],[34,232067,232069],{"className":232068,"style":32163},[31805],[34,232070],{},[34,232072],{"className":232073,"style":31846},[31845],[34,232075,202],{"className":232076},[31850],[34,232078],{"className":232079,"style":31846},[31845],[34,232081,232083,232086,232154,232157,232160],{"className":232082},[31676],[34,232084],{"className":232085,"style":46727},[31680],[34,232087,232089,232092,232151],{"className":232088},[31685],[34,232090],{"className":232091},[31782,41488],[34,232093,232095],{"className":232094},[41453],[34,232096,232098,232143],{"className":232097},[31796,31797],[34,232099,232101,232140],{"className":232100},[31801],[34,232102,232104,232118,232126],{"className":232103,"style":46746},[31805],[34,232105,232106,232109],{"style":41504},[34,232107],{"className":232108,"style":41508},[31813],[34,232110,232112],{"className":232111},[31818,31819,31820,31821],[34,232113,232115],{"className":232114},[31685,31821],[34,232116,5774],{"className":232117},[31685,31821],[34,232119,232120,232123],{"style":41520},[34,232121],{"className":232122,"style":41508},[31813],[34,232124],{"className":232125,"style":41528},[41527],[34,232127,232128,232131],{"style":46771},[34,232129],{"className":232130,"style":41508},[31813],[34,232132,232134],{"className":232133},[31818,31819,31820,31821],[34,232135,232137],{"className":232136},[31685,31821],[34,232138,5752],{"className":232139},[31685,31821],[34,232141,31829],{"className":232142},[31828],[34,232144,232146],{"className":232145},[31801],[34,232147,232149],{"className":232148,"style":41619},[31805],[34,232150],{},[34,232152],{"className":232153},[31690,41488],[34,232155],{"className":232156,"style":32604},[31845],[34,232158,223],{"className":232159},[32024],[34,232161],{"className":232162,"style":32604},[31845],[34,232164,232166,232169,232237,232240,232243],{"className":232165},[31676],[34,232167],{"className":232168,"style":46727},[31680],[34,232170,232172,232175,232234],{"className":232171},[31685],[34,232173],{"className":232174},[31782,41488],[34,232176,232178],{"className":232177},[41453],[34,232179,232181,232226],{"className":232180},[31796,31797],[34,232182,232184,232223],{"className":232183},[31801],[34,232185,232187,232201,232209],{"className":232186,"style":46746},[31805],[34,232188,232189,232192],{"style":41504},[34,232190],{"className":232191,"style":41508},[31813],[34,232193,232195],{"className":232194},[31818,31819,31820,31821],[34,232196,232198],{"className":232197},[31685,31821],[34,232199,162173],{"className":232200},[31685,31821],[34,232202,232203,232206],{"style":41520},[34,232204],{"className":232205,"style":41508},[31813],[34,232207],{"className":232208,"style":41528},[41527],[34,232210,232211,232214],{"style":46771},[34,232212],{"className":232213,"style":41508},[31813],[34,232215,232217],{"className":232216},[31818,31819,31820,31821],[34,232218,232220],{"className":232219},[31685,31821],[34,232221,3249],{"className":232222},[31685,31821],[34,232224,31829],{"className":232225},[31828],[34,232227,232229],{"className":232228},[31801],[34,232230,232232],{"className":232231,"style":41619},[31805],[34,232233],{},[34,232235],{"className":232236},[31690,41488],[34,232238],{"className":232239,"style":32604},[31845],[34,232241,223],{"className":232242},[32024],[34,232244],{"className":232245,"style":32604},[31845],[34,232247,232249,232252,232320,232323,232326],{"className":232248},[31676],[34,232250],{"className":232251,"style":46727},[31680],[34,232253,232255,232258,232317],{"className":232254},[31685],[34,232256],{"className":232257},[31782,41488],[34,232259,232261],{"className":232260},[41453],[34,232262,232264,232309],{"className":232263},[31796,31797],[34,232265,232267,232306],{"className":232266},[31801],[34,232268,232270,232284,232292],{"className":232269,"style":46746},[31805],[34,232271,232272,232275],{"style":41504},[34,232273],{"className":232274,"style":41508},[31813],[34,232276,232278],{"className":232277},[31818,31819,31820,31821],[34,232279,232281],{"className":232280},[31685,31821],[34,232282,25357],{"className":232283},[31685,31821],[34,232285,232286,232289],{"style":41520},[34,232287],{"className":232288,"style":41508},[31813],[34,232290],{"className":232291,"style":41528},[41527],[34,232293,232294,232297],{"style":46771},[34,232295],{"className":232296,"style":41508},[31813],[34,232298,232300],{"className":232299},[31818,31819,31820,31821],[34,232301,232303],{"className":232302},[31685,31821],[34,232304,3244],{"className":232305},[31685,31821],[34,232307,31829],{"className":232308},[31828],[34,232310,232312],{"className":232311},[31801],[34,232313,232315],{"className":232314,"style":41619},[31805],[34,232316],{},[34,232318],{"className":232319},[31690,41488],[34,232321],{"className":232322,"style":31846},[31845],[34,232324,202],{"className":232325},[31850],[34,232327],{"className":232328,"style":31846},[31845],[34,232330,232332,232335,232338,232341,232344],{"className":232331},[31676],[34,232333],{"className":232334,"style":35942},[31680],[34,232336,231988],{"className":232337},[31685],[34,232339],{"className":232340,"style":32604},[31845],[34,232342,223],{"className":232343},[32024],[34,232345],{"className":232346,"style":32604},[31845],[34,232348,232350,232353,232356,232359,232362],{"className":232349},[31676],[34,232351],{"className":232352,"style":35942},[31680],[34,232354,231993],{"className":232355},[31685],[34,232357],{"className":232358,"style":32604},[31845],[34,232360,223],{"className":232361},[32024],[34,232363],{"className":232364,"style":32604},[31845],[34,232366,232368,232371,232374,232377,232380],{"className":232367},[31676],[34,232369],{"className":232370,"style":32617},[31680],[34,232372,231998],{"className":232373},[31685],[34,232375],{"className":232376,"style":31846},[31845],[34,232378,168560],{"className":232379},[31850],[34,232381],{"className":232382,"style":31846},[31845],[34,232384,232386,232389],{"className":232385},[31676],[34,232387],{"className":232388,"style":32617},[31680],[34,232390,232003],{"className":232391},[31685],"\n要让系统可调度，总利用率必须 ",[34,232394,232396,232412],{"className":232395},[31642],[34,232397,232399],{"className":232398},[31646],[31648,232400,232401],{"xmlns":31650},[31652,232402,232403,232409],{},[31655,232404,232405,232407],{},[31662,232406,45972],{},[31749,232408,640],{},[31665,232410,232411],{"encoding":31667},"\\le 1",[34,232413,232415,232427],{"className":232414,"ariaHidden":758},[31672],[34,232416,232418,232421,232424],{"className":232417},[31676],[34,232419],{"className":232420,"style":43198},[31680],[34,232422,45972],{"className":232423},[31850],[34,232425],{"className":232426,"style":31846},[31845],[34,232428,232430,232433],{"className":232429},[31676],[34,232431],{"className":232432,"style":32617},[31680],[34,232434,640],{"className":232435},[31685],[34,232437,232439,232465],{"className":232438},[31642],[34,232440,232442],{"className":232441},[31646],[31648,232443,232444],{"xmlns":31650},[31652,232445,232446,232462],{},[31655,232447,232448,232450,232452,232458,232460],{},[31749,232449,232003],{},[31662,232451,223],{},[41453,232453,232454,232456],{},[31658,232455,231507],{},[31749,232457,97890],{},[31662,232459,45972],{},[31749,232461,640],{},[31665,232463,232464],{"encoding":31667},"0.9333 + \\frac{\\delta}{250} \\le 1",[34,232466,232468,232486,232571],{"className":232467,"ariaHidden":758},[31672],[34,232469,232471,232474,232477,232480,232483],{"className":232470},[31676],[34,232472],{"className":232473,"style":35942},[31680],[34,232475,232003],{"className":232476},[31685],[34,232478],{"className":232479,"style":32604},[31845],[34,232481,223],{"className":232482},[32024],[34,232484],{"className":232485,"style":32604},[31845],[34,232487,232489,232493,232562,232565,232568],{"className":232488},[31676],[34,232490],{"className":232491,"style":232492},[31680],"height:1.2251em;vertical-align:-0.345em;",[34,232494,232496,232499,232559],{"className":232495},[31685],[34,232497],{"className":232498},[31782,41488],[34,232500,232502],{"className":232501},[41453],[34,232503,232505,232551],{"className":232504},[31796,31797],[34,232506,232508,232548],{"className":232507},[31801],[34,232509,232512,232526,232534],{"className":232510,"style":232511},[31805],"height:0.8801em;",[34,232513,232514,232517],{"style":41504},[34,232515],{"className":232516,"style":41508},[31813],[34,232518,232520],{"className":232519},[31818,31819,31820,31821],[34,232521,232523],{"className":232522},[31685,31821],[34,232524,97890],{"className":232525},[31685,31821],[34,232527,232528,232531],{"style":41520},[34,232529],{"className":232530,"style":41508},[31813],[34,232532],{"className":232533,"style":41528},[41527],[34,232535,232536,232539],{"style":46771},[34,232537],{"className":232538,"style":41508},[31813],[34,232540,232542],{"className":232541},[31818,31819,31820,31821],[34,232543,232545],{"className":232544},[31685,31821],[34,232546,231507],{"className":232547,"style":231523},[31685,31686,31821],[34,232549,31829],{"className":232550},[31828],[34,232552,232554],{"className":232553},[31801],[34,232555,232557],{"className":232556,"style":41619},[31805],[34,232558],{},[34,232560],{"className":232561},[31690,41488],[34,232563],{"className":232564,"style":31846},[31845],[34,232566,45972],{"className":232567},[31850],[34,232569],{"className":232570,"style":31846},[31845],[34,232572,232574,232577],{"className":232573},[31676],[34,232575],{"className":232576,"style":32617},[31680],[34,232578,640],{"className":232579},[31685],[34,232581,232583,232614],{"className":232582},[31642],[34,232584,232586],{"className":232585},[31646],[31648,232587,232588],{"xmlns":31650},[31652,232589,232590,232611],{},[31655,232591,232592,232598,232600,232602,232604,232606,232608],{},[41453,232593,232594,232596],{},[31658,232595,231507],{},[31749,232597,97890],{},[31662,232599,45972],{},[31749,232601,640],{},[31662,232603,32226],{},[31749,232605,232003],{},[31662,232607,202],{},[31749,232609,232610],{},"0.0667",[31665,232612,232613],{"encoding":31667},"\\frac{\\delta}{250} \\le 1 - 0.9333 = 0.0667",[34,232615,232617,232700,232718,232736],{"className":232616,"ariaHidden":758},[31672],[34,232618,232620,232623,232691,232694,232697],{"className":232619},[31676],[34,232621],{"className":232622,"style":232492},[31680],[34,232624,232626,232629,232688],{"className":232625},[31685],[34,232627],{"className":232628},[31782,41488],[34,232630,232632],{"className":232631},[41453],[34,232633,232635,232680],{"className":232634},[31796,31797],[34,232636,232638,232677],{"className":232637},[31801],[34,232639,232641,232655,232663],{"className":232640,"style":232511},[31805],[34,232642,232643,232646],{"style":41504},[34,232644],{"className":232645,"style":41508},[31813],[34,232647,232649],{"className":232648},[31818,31819,31820,31821],[34,232650,232652],{"className":232651},[31685,31821],[34,232653,97890],{"className":232654},[31685,31821],[34,232656,232657,232660],{"style":41520},[34,232658],{"className":232659,"style":41508},[31813],[34,232661],{"className":232662,"style":41528},[41527],[34,232664,232665,232668],{"style":46771},[34,232666],{"className":232667,"style":41508},[31813],[34,232669,232671],{"className":232670},[31818,31819,31820,31821],[34,232672,232674],{"className":232673},[31685,31821],[34,232675,231507],{"className":232676,"style":231523},[31685,31686,31821],[34,232678,31829],{"className":232679},[31828],[34,232681,232683],{"className":232682},[31801],[34,232684,232686],{"className":232685,"style":41619},[31805],[34,232687],{},[34,232689],{"className":232690},[31690,41488],[34,232692],{"className":232693,"style":31846},[31845],[34,232695,45972],{"className":232696},[31850],[34,232698],{"className":232699,"style":31846},[31845],[34,232701,232703,232706,232709,232712,232715],{"className":232702},[31676],[34,232704],{"className":232705,"style":35942},[31680],[34,232707,640],{"className":232708},[31685],[34,232710],{"className":232711,"style":32604},[31845],[34,232713,32226],{"className":232714},[32024],[34,232716],{"className":232717,"style":32604},[31845],[34,232719,232721,232724,232727,232730,232733],{"className":232720},[31676],[34,232722],{"className":232723,"style":32617},[31680],[34,232725,232003],{"className":232726},[31685],[34,232728],{"className":232729,"style":31846},[31845],[34,232731,202],{"className":232732},[31850],[34,232734],{"className":232735,"style":31846},[31845],[34,232737,232739,232742],{"className":232738},[31676],[34,232740],{"className":232741,"style":32617},[31680],[34,232743,232610],{"className":232744},[31685]," （即 ",[34,232747,232749,232767],{"className":232748},[31642],[34,232750,232752],{"className":232751},[31646],[31648,232753,232754],{"xmlns":31650},[31652,232755,232756,232764],{},[31655,232757,232758,232760,232762],{},[31749,232759,640],{},[31658,232761,38080],{"mathvariant":38079},[31749,232763,95396],{},[31665,232765,232766],{"encoding":31667},"1\u002F15",[34,232768,232770],{"className":232769,"ariaHidden":758},[31672],[34,232771,232773,232776],{"className":232772},[31676],[34,232774],{"className":232775,"style":31774},[31680],[34,232777,232766],{"className":232778},[31685],"）\n",[34,232781,232783,232817],{"className":232782},[31642],[34,232784,232786],{"className":232785},[31646],[31648,232787,232788],{"xmlns":31650},[31652,232789,232790,232814],{},[31655,232791,232792,232794,232796,232798,232800,232806,232808,232811],{},[31658,232793,231507],{},[31662,232795,45972],{},[31749,232797,97890],{},[31662,232799,42071],{},[41453,232801,232802,232804],{},[31749,232803,640],{},[31749,232805,95396],{},[31662,232807,168560],{},[31749,232809,232810],{},"16.67",[39361,232812,232813],{}," ms",[31665,232815,232816],{"encoding":31667},"\\delta \\le 250 \\times \\frac{1}{15} \\approx 16.67 \\text{ ms}",[34,232818,232820,232838,232856,232939],{"className":232819,"ariaHidden":758},[31672],[34,232821,232823,232826,232829,232832,232835],{"className":232822},[31676],[34,232824],{"className":232825,"style":47709},[31680],[34,232827,231507],{"className":232828,"style":231523},[31685,31686],[34,232830],{"className":232831,"style":31846},[31845],[34,232833,45972],{"className":232834},[31850],[34,232836],{"className":232837,"style":31846},[31845],[34,232839,232841,232844,232847,232850,232853],{"className":232840},[31676],[34,232842],{"className":232843,"style":35942},[31680],[34,232845,97890],{"className":232846},[31685],[34,232848],{"className":232849,"style":32604},[31845],[34,232851,42071],{"className":232852},[32024],[34,232854],{"className":232855,"style":32604},[31845],[34,232857,232859,232862,232930,232933,232936],{"className":232858},[31676],[34,232860],{"className":232861,"style":46727},[31680],[34,232863,232865,232868,232927],{"className":232864},[31685],[34,232866],{"className":232867},[31782,41488],[34,232869,232871],{"className":232870},[41453],[34,232872,232874,232919],{"className":232873},[31796,31797],[34,232875,232877,232916],{"className":232876},[31801],[34,232878,232880,232894,232902],{"className":232879,"style":46746},[31805],[34,232881,232882,232885],{"style":41504},[34,232883],{"className":232884,"style":41508},[31813],[34,232886,232888],{"className":232887},[31818,31819,31820,31821],[34,232889,232891],{"className":232890},[31685,31821],[34,232892,95396],{"className":232893},[31685,31821],[34,232895,232896,232899],{"style":41520},[34,232897],{"className":232898,"style":41508},[31813],[34,232900],{"className":232901,"style":41528},[41527],[34,232903,232904,232907],{"style":46771},[34,232905],{"className":232906,"style":41508},[31813],[34,232908,232910],{"className":232909},[31818,31819,31820,31821],[34,232911,232913],{"className":232912},[31685,31821],[34,232914,640],{"className":232915},[31685,31821],[34,232917,31829],{"className":232918},[31828],[34,232920,232922],{"className":232921},[31801],[34,232923,232925],{"className":232924,"style":41619},[31805],[34,232926],{},[34,232928],{"className":232929},[31690,41488],[34,232931],{"className":232932,"style":31846},[31845],[34,232934,168560],{"className":232935},[31850],[34,232937],{"className":232938,"style":31846},[31845],[34,232940,232942,232945,232948],{"className":232941},[31676],[34,232943],{"className":232944,"style":32617},[31680],[34,232946,232810],{"className":232947},[31685],[34,232949,232951],{"className":232950},[31685,2638],[34,232952,232813],{"className":232953},[31685],[15,232955,232956,232958,232959,232987,232988,233027,233028,31901],{},[2627,232957,58173],{}," 该系统可调度允许的 ",[34,232960,232962,232975],{"className":232961},[31642],[34,232963,232965],{"className":232964},[31646],[31648,232966,232967],{"xmlns":31650},[31652,232968,232969,232973],{},[31655,232970,232971],{},[31658,232972,231507],{},[31665,232974,231510],{"encoding":31667},[34,232976,232978],{"className":232977,"ariaHidden":758},[31672],[34,232979,232981,232984],{"className":232980},[31676],[34,232982],{"className":232983,"style":31681},[31680],[34,232985,231507],{"className":232986,"style":231523},[31685,31686]," 最大值为 ",[2627,232989,232990],{},[34,232991,232993,233009],{"className":232992},[31642],[34,232994,232996],{"className":232995},[31646],[31648,232997,232998],{"xmlns":31650},[31652,232999,233000,233006],{},[31655,233001,233002,233004],{},[31749,233003,232810],{},[39361,233005,232813],{},[31665,233007,233008],{"encoding":31667},"16.67 \\text{ ms}",[34,233010,233012],{"className":233011,"ariaHidden":758},[31672],[34,233013,233015,233018,233021],{"className":233014},[31676],[34,233016],{"className":233017,"style":32617},[31680],[34,233019,232810],{"className":233020},[31685],[34,233022,233024],{"className":233023},[31685,2638],[34,233025,232813],{"className":233026},[31685]," （若要求整数则为 ",[2627,233029,233030],{},"16 ms",{"title":29,"searchDepth":43,"depth":43,"links":233032},[233033,233038,233043,233048,233052,233055,233058,233061,233065,233069],{"id":214289,"depth":43,"text":214292,"children":233034},[233035,233036,233037],{"id":214365,"depth":49,"text":214366},{"id":214699,"depth":49,"text":214700},{"id":217336,"depth":49,"text":217337},{"id":217353,"depth":43,"text":217356,"children":233039},[233040,233041,233042],{"id":218884,"depth":49,"text":214366},{"id":219116,"depth":49,"text":219117},{"id":219803,"depth":49,"text":217337},{"id":219969,"depth":43,"text":219972,"children":233044},[233045,233046,233047],{"id":220559,"depth":49,"text":214366},{"id":220908,"depth":49,"text":220909},{"id":223076,"depth":49,"text":217337},{"id":223297,"depth":43,"text":223298,"children":233049},[233050,233051],{"id":223919,"depth":49,"text":223920},{"id":223960,"depth":49,"text":223961},{"id":225954,"depth":43,"text":225955,"children":233053},[233054],{"id":225963,"depth":49,"text":225963},{"id":227982,"depth":43,"text":227983,"children":233056},[233057],{"id":227994,"depth":49,"text":225963},{"id":229107,"depth":43,"text":229108,"children":233059},[233060],{"id":229207,"depth":49,"text":225963},{"id":230210,"depth":43,"text":230211,"children":233062},[233063,233064],{"id":230359,"depth":49,"text":230359},{"id":230477,"depth":49,"text":230477},{"id":231077,"depth":43,"text":231078,"children":233066},[233067,233068],{"id":231152,"depth":49,"text":230359},{"id":231242,"depth":49,"text":231242},{"id":231428,"depth":43,"text":231429,"children":233070},[233071],{"id":231556,"depth":49,"text":219117},{"order":1976},"\u002Fsource\u002Fcs\u002Fos\u002Fhw\u002Fchap9-hw",{"title":214275,"description":214286},"source\u002Fcs\u002Fos\u002FHW\u002Fchap9-hw","2CX025Cxdy0RV1DkSJEJV6Jmr8OAaSUZ3QXo9e_aBZ4",{"id":233078,"title":233079,"body":233080,"description":233101,"extension":85,"meta":233102,"navigation":52,"path":233103,"seo":233104,"stem":233105,"__hash__":233106},"notes\u002Fsource\u002Fcs\u002Fos\u002Findex.md","Operating Systems",{"type":8,"value":233081,"toc":233099},[233082,233096],[16016,233083,31533,233085,31533,233088],{"className":233084},[31532],[11,233086,233079],{"id":233087},"operating-systems",[15,233089,233091,233092,233095],{"className":233090},[31540],"课程教材：",[71807,233093,233094],{},"Operating Systems: Internals and Design Principles"," — William Stallings, 9th Edition (2018)",[31543,233097],{"path":233098,"unwrap-single-group":29},"\u002Fcs\u002Fos\u002F",{"title":29,"searchDepth":43,"depth":43,"links":233100},[],"操作系统学习笔记、作业、复习题目，教材 William Stallings Operating Systems Internals and Design Principles 第9版，涵盖进程、线程、内存管理、文件系统等核心章节。",{},"\u002Fsource\u002Fcs\u002Fos",{"title":233079,"description":233101},"source\u002Fcs\u002Fos\u002Findex","xzfhbm0INSN_kjqpHC--Gbe2ACc2x9Qg0Nbc9wDTjSQ",{"id":233108,"title":233109,"body":233110,"description":29,"extension":85,"meta":237704,"navigation":52,"path":237705,"seo":237706,"stem":237707,"__hash__":237708},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap11.md","Chap. 11 Note",{"type":8,"value":233111,"toc":237625},[233112,233116,233120,233123,233128,233131,233181,233183,233187,233191,233194,233242,233246,233249,233311,233315,233318,233328,233332,233378,233381,233385,233421,233423,233427,233431,233434,233494,233498,233504,233507,233532,233535,233539,233542,233545,233580,233583,233587,233593,233596,233599,233619,233623,233630,233634,233641,233644,233664,233668,233671,233724,233730,233732,233736,233740,233749,233778,233785,233789,233792,233795,233832,233835,233855,233859,233865,233868,233870,233895,233902,233906,233909,233957,233960,233975,233978,233982,233985,234044,234047,234075,234078,234080,234084,234088,234094,234097,234117,234120,234124,234160,234164,234167,234187,234191,234194,234196,234216,234219,234272,234275,234358,234361,234363,234383,234387,234416,234419,234423,234429,234431,234451,234454,234515,234518,234533,234537,234540,234543,234563,234567,234570,234572,234576,234580,234583,234587,234590,234634,234637,234857,234860,235191,235193,235228,235232,235235,235250,235254,235257,235430,235434,235437,235440,235465,235469,235472,235474,235499,235502,235506,235509,235512,235540,235543,235546,235550,235553,235556,235571,235575,235581,235584,235587,235617,235620,235645,235649,235652,235699,235701,235716,235719,235744,235746,235750,235753,235778,235785,235805,235808,235848,235850,235854,235858,235865,235868,235897,235901,235904,235906,235931,235934,235938,235941,235943,235968,235970,235985,235989,235992,235994,236014,236017,236021,236024,236026,236051,236054,236057,236061,236064,236066,236091,236095,236098,236100,236130,236133,236137,236140,236142,236167,236171,236309,236311,236315,236319,236325,236350,236354,236413,236417,236420,236439,236442,236446,236449,236451,236466,236470,236473,236475,236495,236499,236502,236506,236509,236512,236514,236518,236522,236525,236534,236537,236541,236569,236572,236587,236591,236594,236598,236601,236605,236608,236610,236614,236617,236621,236624,236627,236660,236664,236667,236670,236690,236694,236697,236701,236704,236724,236728,236731,236735,236738,236741,236756,236758,236762,236766,236769,236773,236776,236801,236805,236833,236836,236839,236869,236873,236876,236880,236909,236911,236915,236919,236922,236925,236940,236944,236947,236977,236980,237005,237009,237034,237037,237039,237042,237323,237325,237328,237332,237459,237463,237530,237534,237612,237614,237617,237622],[11,233113,233115],{"id":233114},"chap-11-io-management-and-disk-scheduling","Chap 11. I\u002FO Management and Disk Scheduling",[19,233117,233119],{"id":233118},"_0-本章总线索","0. 本章总线索",[15,233121,233122],{},"这一章的核心不是“会背设备名”，而是理解：",[15,233124,233125],{},[2627,233126,233127],{},"外设很慢、种类很多 → 操作系统要统一管理 I\u002FO → 用中断\u002FDMA\u002F通道减少 CPU 等待 → 用缓冲和缓存提高效率 → 用磁盘调度和 RAID 改善磁盘性能。",[15,233129,233130],{},"可以按这条线记：",[24,233132,233134],{"className":30504,"code":233133,"language":85,"meta":29,"style":29},"I\u002FO 设备差异\n→ I\u002FO 控制方式：轮询 \u002F 中断 \u002F DMA \u002F 通道\n→ OS 如何抽象和管理 I\u002FO\n→ 缓冲技术：单缓冲 \u002F 双缓冲 \u002F 循环缓冲\n→ 磁盘性能：寻道 + 旋转延迟 + 传输时间\n→ 磁盘调度：FIFO \u002F SSTF \u002F SCAN \u002F C-SCAN \u002F N-step-SCAN \u002F FSCAN\n→ RAID：用多个磁盘提高性能和可靠性\n→ Disk Cache：用内存缓存磁盘块\n→ UNIX \u002F Linux \u002F Windows 的 I\u002FO 实现\n",[31,233135,233136,233141,233146,233151,233156,233161,233166,233171,233176],{"__ignoreMap":29},[34,233137,233138],{"class":36,"line":37},[34,233139,233140],{"class":190},"I\u002FO 设备差异\n",[34,233142,233143],{"class":36,"line":43},[34,233144,233145],{"class":190},"→ I\u002FO 控制方式：轮询 \u002F 中断 \u002F DMA \u002F 通道\n",[34,233147,233148],{"class":36,"line":49},[34,233149,233150],{"class":190},"→ OS 如何抽象和管理 I\u002FO\n",[34,233152,233153],{"class":36,"line":56},[34,233154,233155],{"class":190},"→ 缓冲技术：单缓冲 \u002F 双缓冲 \u002F 循环缓冲\n",[34,233157,233158],{"class":36,"line":62},[34,233159,233160],{"class":190},"→ 磁盘性能：寻道 + 旋转延迟 + 传输时间\n",[34,233162,233163],{"class":36,"line":68},[34,233164,233165],{"class":190},"→ 磁盘调度：FIFO \u002F SSTF \u002F SCAN \u002F C-SCAN \u002F N-step-SCAN \u002F FSCAN\n",[34,233167,233168],{"class":36,"line":74},[34,233169,233170],{"class":190},"→ RAID：用多个磁盘提高性能和可靠性\n",[34,233172,233173],{"class":36,"line":1971},[34,233174,233175],{"class":190},"→ Disk Cache：用内存缓存磁盘块\n",[34,233177,233178],{"class":36,"line":1976},[34,233179,233180],{"class":190},"→ UNIX \u002F Linux \u002F Windows 的 I\u002FO 实现\n",[32188,233182],{},[11,233184,233186],{"id":233185},"_111-io-devices","11.1 I\u002FO Devices",[19,233188,233190],{"id":233189},"_1-io-设备的三大类","1. I\u002FO 设备的三大类",[15,233192,233193],{},"教材把外部 I\u002FO 设备分为三类，课件也采用同样分类：",[13837,233195,233196,233207],{},[13840,233197,233198],{},[13843,233199,233200,233202,233204],{},[13846,233201,114061],{},[13846,233203,126615],{},[13846,233205,233206],{},"例子",[13853,233208,233209,233220,233231],{},[13843,233210,233211,233214,233217],{},[13858,233212,233213],{},"Human readable",[13858,233215,233216],{},"面向用户，人能直接读写",[13858,233218,233219],{},"显示器、键盘、鼠标、打印机、终端",[13843,233221,233222,233225,233228],{},[13858,233223,233224],{},"Machine readable",[13858,233226,233227],{},"面向机器或电子设备",[13858,233229,233230],{},"磁盘、USB、传感器、控制器、执行器",[13843,233232,233233,233236,233239],{},[13858,233234,233235],{},"Communication",[13858,233237,233238],{},"面向远程通信",[13858,233240,233241],{},"调制解调器、数字线路驱动器、网络通信设备",[19,233243,233245],{"id":233244},"_2-io-设备差异","2. I\u002FO 设备差异",[15,233247,233248],{},"不同 I\u002FO 设备差异很大，所以 OS 很难用一种完全统一的方式管理所有设备。重点记 6 个差异：",[13837,233250,233251,233261],{},[13840,233252,233253],{},[13843,233254,233255,233258],{},[13846,233256,233257],{},"差异",[13846,233259,233260],{},"解释",[13853,233262,233263,233271,233279,233287,233295,233303],{},[13843,233264,233265,233268],{},[13858,233266,233267],{},"Data rate 数据率",[13858,233269,233270],{},"键盘、磁盘、网络、内存速度差距巨大",[13843,233272,233273,233276],{},[13858,233274,233275],{},"Application 应用场景",[13858,233277,233278],{},"同一个磁盘用于文件系统、虚拟内存交换区时，管理策略不同",[13843,233280,233281,233284],{},[13858,233282,233283],{},"Complexity of control 控制复杂度",[13858,233285,233286],{},"打印机简单，磁盘控制复杂",[13843,233288,233289,233292],{},[13858,233290,233291],{},"Unit of transfer 传输单位",[13858,233293,233294],{},"字节流或块",[13843,233296,233297,233300],{},[13858,233298,233299],{},"Data representation 数据表示",[13858,233301,233302],{},"字符编码、校验位、数据格式可能不同",[13843,233304,233305,233308],{},[13858,233306,233307],{},"Error conditions 错误条件",[13858,233309,233310],{},"不同设备错误类型、报告方式、恢复方式不同",[19,233312,233314],{"id":233313},"_3-课件补充io-系统与-io-操作","3. 课件补充：I\u002FO 系统与 I\u002FO 操作",[15,233316,233317],{},"课件给出的中文定义也要背：",[15,233319,233320,233323,233324,233327],{},[2627,233321,233322],{},"I\u002FO 系统","：I\u002FO 设备及其接口线路、控制部件、通道和管理软件的总称。\n",[2627,233325,233326],{},"I\u002FO 操作","：计算机内存和外围设备介质之间的信息传送操作。",[19,233329,233331],{"id":233330},"_4-字符设备与块设备","4. 字符设备与块设备",[13837,233333,233334,233348],{},[13840,233335,233336],{},[13843,233337,233338,233340,233343,233346],{},[13846,233339,114061],{},[13846,233341,233342],{},"传输单位",[13846,233344,233345],{},"特点",[13846,233347,233206],{},[13853,233349,233350,233364],{},[13843,233351,233352,233355,233358,233361],{},[13858,233353,233354],{},"字符设备 Character device",[13858,233356,233357],{},"字节 \u002F 字符",[13858,233359,233360],{},"按字符流传输，无固定块结构",[13858,233362,233363],{},"键盘、鼠标、终端、打印机",[13843,233365,233366,233369,233372,233375],{},[13858,233367,233368],{},"块设备 Block device",[13858,233370,233371],{},"数据块",[13858,233373,233374],{},"以固定大小块传输，可按块号访问",[13858,233376,233377],{},"磁盘、USB、SSD",[15,233379,233380],{},"课件还强调：输入型、输出型外围设备一般是字符设备；存储型外围设备一般是块设备。",[19,233382,233384],{"id":233383},"_5-顺序存取与直接存取","5. 顺序存取与直接存取",[13837,233386,233387,233397],{},[13840,233388,233389],{},[13843,233390,233391,233393,233395],{},[13846,233392,114061],{},[13846,233394,233345],{},[13846,233396,233206],{},[13853,233398,233399,233410],{},[13843,233400,233401,233404,233407],{},[13858,233402,233403],{},"顺序存取 Sequential access",[13858,233405,233406],{},"依赖物理顺序，访问后面的数据必须经过前面的数据",[13858,233408,233409],{},"磁带",[13843,233411,233412,233415,233418],{},[13858,233413,233414],{},"直接存取 Direct access",[13858,233416,233417],{},"可以直接定位到某个物理块",[13858,233419,233420],{},"磁盘",[32188,233422],{},[11,233424,233426],{"id":233425},"_112-organization-of-the-io-function","11.2 Organization of the I\u002FO Function",[19,233428,233430],{"id":233429},"_1-三种基本-io-技术","1. 三种基本 I\u002FO 技术",[15,233432,233433],{},"教材和课件都强调三种 I\u002FO 执行方式：Programmed I\u002FO、Interrupt-driven I\u002FO、DMA。",[13837,233435,233436,233450],{},[13840,233437,233438],{},[13843,233439,233440,233443,233446,233448],{},[13846,233441,233442],{},"技术",[13846,233444,233445],{},"CPU 做什么",[13846,233447,204919],{},[13846,233449,204938],{},[13853,233451,233452,233466,233480],{},[13843,233453,233454,233457,233460,233463],{},[13858,233455,233456],{},"Programmed I\u002FO 程序控制 I\u002FO \u002F 轮询",[13858,233458,233459],{},"CPU 发出 I\u002FO 命令后不断检查设备状态",[13858,233461,233462],{},"简单",[13858,233464,233465],{},"CPU 忙等，浪费处理器时间",[13843,233467,233468,233471,233474,233477],{},[13858,233469,233470],{},"Interrupt-driven I\u002FO 中断驱动 I\u002FO",[13858,233472,233473],{},"CPU 发出命令后继续执行，设备完成后中断 CPU",[13858,233475,233476],{},"CPU 不必一直等",[13858,233478,233479],{},"每次中断有开销",[13843,233481,233482,233485,233488,233491],{},[13858,233483,233484],{},"DMA 直接存储器访问",[13858,233486,233487],{},"DMA 模块负责内存与 I\u002FO 模块之间的数据块传输",[13858,233489,233490],{},"CPU 只参与开始和结束",[13858,233492,233493],{},"需要 DMA 控制器，占用总线周期",[19,233495,233497],{"id":233496},"_2-轮询方式-programmed-io-polling","2. 轮询方式 Programmed I\u002FO \u002F Polling",[15,233499,233500,233501,2664],{},"课件中文叫",[2627,233502,233503],{},"轮询方式",[15,233505,233506],{},"CPU 反复查询设备控制器的状态位，判断设备是否就绪。\n基本过程：",[24,233508,233510],{"className":30504,"code":233509,"language":85,"meta":29,"style":29},"查询设备状态\n→ 如果未就绪，继续查询\n→ 如果就绪，执行读\u002F写\n→ 再查询下一个设备\n",[31,233511,233512,233517,233522,233527],{"__ignoreMap":29},[34,233513,233514],{"class":36,"line":37},[34,233515,233516],{"class":190},"查询设备状态\n",[34,233518,233519],{"class":36,"line":43},[34,233520,233521],{"class":190},"→ 如果未就绪，继续查询\n",[34,233523,233524],{"class":36,"line":49},[34,233525,233526],{"class":190},"→ 如果就绪，执行读\u002F写\n",[34,233528,233529],{"class":36,"line":56},[34,233530,233531],{"class":190},"→ 再查询下一个设备\n",[15,233533,233534],{},"缺点很明显：CPU 会被查询过程拖住，设备慢时效率很低。",[19,233536,233538],{"id":233537},"_3-中断方式-interrupt-driven-io","3. 中断方式 Interrupt-driven I\u002FO",[15,233540,233541],{},"中断方式下，CPU 启动 I\u002FO 后可以去执行别的程序。设备完成后，通过中断请求线通知 CPU。课件强调：设备控制器执行 I\u002FO 操作，完成后发出 I\u002FO 中断信号，CPU 转去执行中断处理程序，再恢复原进程。",[15,233543,233544],{},"重点理解：",[24,233546,233548],{"className":30504,"code":233547,"language":85,"meta":29,"style":29},"CPU 启动 I\u002FO\n→ 设备自己工作\n→ CPU 执行其他任务\n→ 设备完成，发中断\n→ CPU 执行中断处理程序\n→ 恢复原程序\n",[31,233549,233550,233555,233560,233565,233570,233575],{"__ignoreMap":29},[34,233551,233552],{"class":36,"line":37},[34,233553,233554],{"class":190},"CPU 启动 I\u002FO\n",[34,233556,233557],{"class":36,"line":43},[34,233558,233559],{"class":190},"→ 设备自己工作\n",[34,233561,233562],{"class":36,"line":49},[34,233563,233564],{"class":190},"→ CPU 执行其他任务\n",[34,233566,233567],{"class":36,"line":56},[34,233568,233569],{"class":190},"→ 设备完成，发中断\n",[34,233571,233572],{"class":36,"line":62},[34,233573,233574],{"class":190},"→ CPU 执行中断处理程序\n",[34,233576,233577],{"class":36,"line":68},[34,233578,233579],{"class":190},"→ 恢复原程序\n",[15,233581,233582],{},"优点：减少 CPU 空等。\n缺点：如果每传一个字节就中断一次，中断开销会很大。",[19,233584,233586],{"id":233585},"_4-dma-方式","4. DMA 方式",[15,233588,233589,233590],{},"DMA 的核心是：",[2627,233591,233592],{},"设备和内存之间直接传输数据块，不经过 CPU 逐字节搬运。",[15,233594,233595],{},"CPU 给 DMA 模块的信息包括：读\u002F写类型、I\u002FO 设备地址、内存起始地址、传输字数。之后 DMA 负责搬运整个数据块，完成后再中断 CPU。课件还列出 DMA 需要的硬件：内存地址寄存器、字计数器、数据缓冲寄存器或缓冲区、设备地址寄存器、中断机制和控制逻辑。",[15,233597,233598],{},"DMA 的关键点：",[24,233600,233602],{"className":30504,"code":233601,"language":85,"meta":29,"style":29},"CPU：只负责发起请求和处理完成中断\nDMA：负责实际数据传输\n设备 ↔ DMA ↔ 内存\n",[31,233603,233604,233609,233614],{"__ignoreMap":29},[34,233605,233606],{"class":36,"line":37},[34,233607,233608],{"class":190},"CPU：只负责发起请求和处理完成中断\n",[34,233610,233611],{"class":36,"line":43},[34,233612,233613],{"class":190},"DMA：负责实际数据传输\n",[34,233615,233616],{"class":36,"line":49},[34,233617,233618],{"class":190},"设备 ↔ DMA ↔ 内存\n",[19,233620,233622],{"id":233621},"_5-周期窃用-cycle-stealing","5. 周期窃用 Cycle Stealing",[15,233624,233625,233626,233629],{},"课件提到：当 DMA 和 CPU 同时要访问内存总线时，CPU 会把总线让给 DMA，这叫",[2627,233627,233628],{},"周期窃用","。这样 DMA 能完成设备和内存之间的数据交换，但会占用一些总线周期。",[19,233631,233633],{"id":233632},"_6-通道方式-io-channel","6. 通道方式 I\u002FO Channel",[15,233635,233636,233637,233640],{},"通道方式比 DMA 更进一步。DMA 只是搬数据，而",[2627,233638,233639],{},"通道像一个专门的 I\u002FO 小处理器","，能执行 I\u002FO 程序。CPU 启动通道后，通道控制设备完成 I\u002FO，CPU 可以继续执行其他任务，直到通道发出完成中断。",[15,233642,233643],{},"通道方式的意义：",[24,233645,233647],{"className":30504,"code":233646,"language":85,"meta":29,"style":29},"CPU 和外设并行程度更高\nI\u002FO 控制更独立\n适合管理大量、复杂、多类型设备\n",[31,233648,233649,233654,233659],{"__ignoreMap":29},[34,233650,233651],{"class":36,"line":37},[34,233652,233653],{"class":190},"CPU 和外设并行程度更高\n",[34,233655,233656],{"class":36,"line":43},[34,233657,233658],{"class":190},"I\u002FO 控制更独立\n",[34,233660,233661],{"class":36,"line":49},[34,233662,233663],{"class":190},"适合管理大量、复杂、多类型设备\n",[19,233665,233667],{"id":233666},"_7-io-功能的发展阶段","7. I\u002FO 功能的发展阶段",[15,233669,233670],{},"教材列出 I\u002FO 功能从简单到复杂的 6 个阶段：",[24,233672,233674],{"className":30504,"code":233673,"language":85,"meta":29,"style":29},"1. CPU 直接控制外设\n2. 加入控制器 \u002F I\u002FO 模块，使用程序控制 I\u002FO\n3. 加入中断，CPU 不必忙等\n4. 使用 DMA，I\u002FO 模块可直接访问内存\n5. I\u002FO 模块发展成 I\u002FO Channel，可执行专门 I\u002FO 指令\n6. I\u002FO 模块有自己的局部内存，成为 I\u002FO Processor\n",[31,233675,233676,233684,233692,233700,233708,233716],{"__ignoreMap":29},[34,233677,233678,233681],{"class":36,"line":37},[34,233679,233680],{"class":1326},"1.",[34,233682,233683],{"class":190}," CPU 直接控制外设\n",[34,233685,233686,233689],{"class":36,"line":43},[34,233687,233688],{"class":1326},"2.",[34,233690,233691],{"class":190}," 加入控制器 \u002F I\u002FO 模块，使用程序控制 I\u002FO\n",[34,233693,233694,233697],{"class":36,"line":49},[34,233695,233696],{"class":1326},"3.",[34,233698,233699],{"class":190}," 加入中断，CPU 不必忙等\n",[34,233701,233702,233705],{"class":36,"line":56},[34,233703,233704],{"class":1326},"4.",[34,233706,233707],{"class":190}," 使用 DMA，I\u002FO 模块可直接访问内存\n",[34,233709,233710,233713],{"class":36,"line":62},[34,233711,233712],{"class":1326},"5.",[34,233714,233715],{"class":190}," I\u002FO 模块发展成 I\u002FO Channel，可执行专门 I\u002FO 指令\n",[34,233717,233718,233721],{"class":36,"line":68},[34,233719,233720],{"class":1326},"6.",[34,233722,233723],{"class":190}," I\u002FO 模块有自己的局部内存，成为 I\u002FO Processor\n",[15,233725,233726,233727],{},"一句话总结：",[2627,233728,233729],{},"越往后，CPU 参与越少，I\u002FO 独立性越强，系统并行性越高。",[32188,233731],{},[11,233733,233735],{"id":233734},"_113-operating-system-design-issues","11.3 Operating System Design Issues",[19,233737,233739],{"id":233738},"_1-两个设计目标efficiency-与-generality","1. 两个设计目标：Efficiency 与 Generality",[15,233741,233742,233743,119123,233746,347],{},"教材和课件都强调 I\u002FO 系统设计的两个目标：",[2627,233744,233745],{},"效率",[2627,233747,233748],{},"通用性",[13837,233750,233751,233760],{},[13840,233752,233753],{},[13843,233754,233755,233758],{},[13846,233756,233757],{},"目标",[13846,233759,126615],{},[13853,233761,233762,233770],{},[13843,233763,233764,233767],{},[13858,233765,233766],{},"Efficiency 效率",[13858,233768,233769],{},"I\u002FO 很容易成为瓶颈，尤其磁盘 I\u002FO，所以要减少等待、提高并行",[13843,233771,233772,233775],{},[13858,233773,233774],{},"Generality 通用性",[13858,233776,233777],{},"希望用统一方式管理各种设备，例如 read、write、open、close",[15,233779,233780,233781,233784],{},"难点在于：设备差异太大，所以不能完全统一，只能通过",[2627,233782,233783],{},"层次化、模块化设计","隐藏底层差异。",[19,233786,233788],{"id":233787},"_2-io-层次结构","2. I\u002FO 层次结构",[15,233790,233791],{},"教材提出 I\u002FO 功能应该分层：低层处理硬件细节，高层提供通用接口。这样硬件变化不会影响大部分 I\u002FO 软件。",[15,233793,233794],{},"对简单本地外设，通常有三层：",[13837,233796,233797,233806],{},[13840,233798,233799],{},[13843,233800,233801,233804],{},[13846,233802,233803],{},"层次",[13846,233805,153527],{},[13853,233807,233808,233816,233824],{},[13843,233809,233810,233813],{},[13858,233811,233812],{},"Logical I\u002FO 逻辑 I\u002FO",[13858,233814,233815],{},"把设备看作逻辑资源，提供 open、close、read、write 等接口",[13843,233817,233818,233821],{},[13858,233819,233820],{},"Device I\u002FO 设备 I\u002FO",[13858,233822,233823],{},"把逻辑请求转换为 I\u002FO 指令、通道命令、控制器命令",[13843,233825,233826,233829],{},[13858,233827,233828],{},"Scheduling and Control 调度与控制",[13858,233830,233831],{},"排队、调度、处理中断、收集状态，直接和硬件交互",[15,233833,233834],{},"对通信设备，逻辑 I\u002FO 层可能被通信协议结构替代，例如 TCP\u002FIP。\n对文件系统支持的二级存储，还要加：",[24,233836,233838],{"className":30504,"code":233837,"language":85,"meta":29,"style":29},"Directory management 目录管理\nFile system 文件系统\nPhysical organization 物理组织\n",[31,233839,233840,233845,233850],{"__ignoreMap":29},[34,233841,233842],{"class":36,"line":37},[34,233843,233844],{"class":190},"Directory management 目录管理\n",[34,233846,233847],{"class":36,"line":43},[34,233848,233849],{"class":190},"File system 文件系统\n",[34,233851,233852],{"class":36,"line":49},[34,233853,233854],{"class":190},"Physical organization 物理组织\n",[19,233856,233858],{"id":233857},"_3-设备独立性-device-independence","3. 设备独立性 Device Independence",[15,233860,233861,233862,347],{},"课件补充了一个考试常考点：",[2627,233863,233864],{},"设备独立性",[15,233866,233867],{},"设备独立性指：用户程序不直接指定物理设备，而是使用逻辑设备名；系统再把逻辑设备映射到具体物理设备。这样程序不依赖某台具体设备。",[15,233869,14543],{},[24,233871,233873],{"className":30504,"code":233872,"language":85,"meta":29,"style":29},"用户程序与具体物理设备解耦\n系统增加、更换设备时，程序不必修改\n设备故障时可以切换到其他物理设备\n提高设备分配灵活性\n",[31,233874,233875,233880,233885,233890],{"__ignoreMap":29},[34,233876,233877],{"class":36,"line":37},[34,233878,233879],{"class":190},"用户程序与具体物理设备解耦\n",[34,233881,233882],{"class":36,"line":43},[34,233883,233884],{"class":190},"系统增加、更换设备时，程序不必修改\n",[34,233886,233887],{"class":36,"line":49},[34,233888,233889],{"class":190},"设备故障时可以切换到其他物理设备\n",[34,233891,233892],{"class":36,"line":56},[34,233893,233894],{"class":190},"提高设备分配灵活性\n",[15,233896,233897,233898,233901],{},"逻辑设备名到物理设备名的映射通过 ",[2627,233899,233900],{},"LUT，Logical Unit Table，逻辑设备表"," 实现。单用户系统可以全系统一张 LUT；多用户系统可以每个用户一张 LUT，允许不同用户使用相同逻辑设备名。",[19,233903,233905],{"id":233904},"_4-设备分配","4. 设备分配",[15,233907,233908],{},"课件按设备特性把设备分为三类：",[13837,233910,233911,233922],{},[13840,233912,233913],{},[13843,233914,233915,233917,233919],{},[13846,233916,114061],{},[13846,233918,126615],{},[13846,233920,233921],{},"管理方式",[13853,233923,233924,233935,233946],{},[13843,233925,233926,233929,233932],{},[13858,233927,233928],{},"独占设备",[13858,233930,233931],{},"一段时间只能给一个进程用",[13858,233933,233934],{},"独占分配",[13843,233936,233937,233940,233943],{},[13858,233938,233939],{},"共享设备",[13858,233941,233942],{},"可被多个进程共享",[13858,233944,233945],{},"共享分配",[13843,233947,233948,233951,233954],{},[13858,233949,233950],{},"虚拟设备",[13858,233952,233953],{},"通过软件把独占设备改造成可共享效果",[13858,233955,233956],{},"虚拟分配，例如 SPOOLING",[15,233958,233959],{},"常用分配算法：",[24,233961,233963],{"className":30504,"code":233962,"language":85,"meta":29,"style":29},"先请求先服务 FCFS\n优先级高者先服务\n",[31,233964,233965,233970],{"__ignoreMap":29},[34,233966,233967],{"class":36,"line":37},[34,233968,233969],{"class":190},"先请求先服务 FCFS\n",[34,233971,233972],{"class":36,"line":43},[34,233973,233974],{"class":190},"优先级高者先服务\n",[15,233976,233977],{},"还要注意：多进程申请设备时可能发生死锁，所以需要安全性检查。",[19,233979,233981],{"id":233980},"_5-设备分配的数据结构","5. 设备分配的数据结构",[15,233983,233984],{},"课件列出几个关键表：",[13837,233986,233987,233998],{},[13840,233988,233989],{},[13843,233990,233991,233993,233996],{},[13846,233992,118590],{},[13846,233994,233995],{},"全称",[13846,233997,153527],{},[13853,233999,234000,234011,234022,234033],{},[13843,234001,234002,234005,234008],{},[13858,234003,234004],{},"DCT",[13858,234006,234007],{},"Device Control Table",[13858,234009,234010],{},"每个设备一张，记录设备类型、地址、状态、等待队列、控制器指针等",[13843,234012,234013,234016,234019],{},[13858,234014,234015],{},"SDT",[13858,234017,234018],{},"System Device Table",[13858,234020,234021],{},"全系统一张，记录所有物理设备",[13843,234023,234024,234027,234030],{},[13858,234025,234026],{},"COCT",[13858,234028,234029],{},"Controller Control Table",[13858,234031,234032],{},"每个控制器一张，记录控制器状态和连接情况",[13843,234034,234035,234038,234041],{},[13858,234036,234037],{},"CHCT",[13858,234039,234040],{},"Channel Control Table",[13858,234042,234043],{},"每个通道一张，只在通道方式系统中存在",[15,234045,234046],{},"一个进程要真正执行 I\u002FO，通常需要获得：",[24,234048,234050],{"className":30504,"code":234049,"language":85,"meta":29,"style":29},"设备 Device\n\n- 控制器 Controller\n- 通道 Channel\n",[31,234051,234052,234057,234061,234068],{"__ignoreMap":29},[34,234053,234054],{"class":36,"line":37},[34,234055,234056],{"class":190},"设备 Device\n",[34,234058,234059],{"class":36,"line":43},[34,234060,53],{"emptyLinePlaceholder":52},[34,234062,234063,234065],{"class":36,"line":49},[34,234064,2511],{"class":1326},[34,234066,234067],{"class":190}," 控制器 Controller\n",[34,234069,234070,234072],{"class":36,"line":56},[34,234071,2511],{"class":1326},[34,234073,234074],{"class":190}," 通道 Channel\n",[15,234076,234077],{},"三者都具备后，才有进行 I\u002FO 的物理条件。",[32188,234079],{},[11,234081,234083],{"id":234082},"_114-io-buffering","11.4 I\u002FO Buffering",[19,234085,234087],{"id":234086},"_1-缓冲的目的","1. 缓冲的目的",[15,234089,234090,234091],{},"缓冲技术的本质：",[2627,234092,234093],{},"在内存中开辟缓冲区，让 CPU、内存、设备之间速度不匹配的问题变小。",[15,234095,234096],{},"课件给出的目的很重要：",[24,234098,234100],{"className":30504,"code":234099,"language":85,"meta":29,"style":29},"改善 CPU 与外设速度不匹配\n协调逻辑记录大小和物理记录大小不一致\n提高 CPU 与 I\u002FO 设备的并行性\n",[31,234101,234102,234107,234112],{"__ignoreMap":29},[34,234103,234104],{"class":36,"line":37},[34,234105,234106],{"class":190},"改善 CPU 与外设速度不匹配\n",[34,234108,234109],{"class":36,"line":43},[34,234110,234111],{"class":190},"协调逻辑记录大小和物理记录大小不一致\n",[34,234113,234114],{"class":36,"line":49},[34,234115,234116],{"class":190},"提高 CPU 与 I\u002FO 设备的并行性\n",[15,234118,234119],{},"教材也强调，缓冲可以提前执行输入传输，或延后执行输出传输，从而改善系统性能。",[19,234121,234123],{"id":234122},"_2-block-oriented-与-stream-oriented","2. Block-oriented 与 Stream-oriented",[13837,234125,234126,234136],{},[13840,234127,234128],{},[13843,234129,234130,234132,234134],{},[13846,234131,114061],{},[13846,234133,233345],{},[13846,234135,233206],{},[13853,234137,234138,234149],{},[13843,234139,234140,234143,234146],{},[13858,234141,234142],{},"Block-oriented device",[13858,234144,234145],{},"固定大小块，一次传一块，可按块号访问",[13858,234147,234148],{},"磁盘、USB",[13843,234150,234151,234154,234157],{},[13858,234152,234153],{},"Stream-oriented device",[13858,234155,234156],{},"字节流，没有固定块结构",[13858,234158,234159],{},"终端、打印机、鼠标、通信端口",[19,234161,234163],{"id":234162},"_3-无缓冲-no-buffer","3. 无缓冲 No Buffer",[15,234165,234166],{},"无缓冲时，OS 直接访问设备。\n问题：",[24,234168,234170],{"className":30504,"code":234169,"language":85,"meta":29,"style":29},"进程容易被 I\u002FO 阻塞\n设备慢，CPU 等待时间长\n进程 I\u002FO 期间可能不能被换出\n",[31,234171,234172,234177,234182],{"__ignoreMap":29},[34,234173,234174],{"class":36,"line":37},[34,234175,234176],{"class":190},"进程容易被 I\u002FO 阻塞\n",[34,234178,234179],{"class":36,"line":43},[34,234180,234181],{"class":190},"设备慢，CPU 等待时间长\n",[34,234183,234184],{"class":36,"line":49},[34,234185,234186],{"class":190},"进程 I\u002FO 期间可能不能被换出\n",[19,234188,234190],{"id":234189},"_4-单缓冲-single-buffer","4. 单缓冲 Single Buffer",[15,234192,234193],{},"单缓冲：OS 为一次 I\u002FO 请求分配一个系统缓冲区。块设备输入时，数据先进入系统缓冲区，再复制到用户空间；同时可以预读下一块。",[15,234195,131588],{},[24,234197,234199],{"className":30504,"code":234198,"language":85,"meta":29,"style":29},"T = 输入一块数据所需时间\nC = 两次输入请求之间的计算时间\nM = 系统缓冲区到用户内存的数据移动时间\n",[31,234200,234201,234206,234211],{"__ignoreMap":29},[34,234202,234203],{"class":36,"line":37},[34,234204,234205],{"class":190},"T = 输入一块数据所需时间\n",[34,234207,234208],{"class":36,"line":43},[34,234209,234210],{"class":190},"C = 两次输入请求之间的计算时间\n",[34,234212,234213],{"class":36,"line":49},[34,234214,234215],{"class":190},"M = 系统缓冲区到用户内存的数据移动时间\n",[15,234217,234218],{},"无缓冲每块大约：",[15,234220,234221],{},[34,234222,234224,234242],{"className":234223},[31642],[34,234225,234227],{"className":234226},[31646],[31648,234228,234229],{"xmlns":31650},[31652,234230,234231,234239],{},[31655,234232,234233,234235,234237],{},[31658,234234,37400],{},[31662,234236,223],{},[31658,234238,47384],{},[31665,234240,234241],{"encoding":31667},"T + C",[34,234243,234245,234263],{"className":234244,"ariaHidden":758},[31672],[34,234246,234248,234251,234254,234257,234260],{"className":234247},[31676],[34,234249],{"className":234250,"style":34944},[31680],[34,234252,37400],{"className":234253,"style":37422},[31685,31686],[34,234255],{"className":234256,"style":32604},[31845],[34,234258,223],{"className":234259},[32024],[34,234261],{"className":234262,"style":32604},[31845],[34,234264,234266,234269],{"className":234265},[31676],[34,234267],{"className":234268,"style":33218},[31680],[34,234270,47384],{"className":234271,"style":38706},[31685,31686],[15,234273,234274],{},"单缓冲每块大约：",[15,234276,234277],{},[34,234278,234280,234310],{"className":234279},[31642],[34,234281,234283],{"className":234282},[31646],[31648,234284,234285],{"xmlns":31650},[31652,234286,234287,234307],{},[31655,234288,234289,234291,234293,234295,234297,234299,234301,234303,234305],{},[31658,234290,32679],{},[31662,234292,32682],{},[31662,234294,456],{"stretchy":702},[31658,234296,47384],{},[31662,234298,31960],{"separator":758},[31658,234300,37400],{},[31662,234302,2557],{"stretchy":702},[31662,234304,223],{},[31658,234306,98360],{},[31665,234308,234309],{"encoding":31667},"\\max(C, T) + M",[34,234311,234313,234349],{"className":234312,"ariaHidden":758},[31672],[34,234314,234316,234319,234322,234325,234328,234331,234334,234337,234340,234343,234346],{"className":234315},[31676],[34,234317],{"className":234318,"style":31774},[31680],[34,234320,32679],{"className":234321},[32767],[34,234323,456],{"className":234324},[31782],[34,234326,47384],{"className":234327,"style":38706},[31685,31686],[34,234329,31960],{"className":234330},[32119],[34,234332],{"className":234333,"style":32123},[31845],[34,234335,37400],{"className":234336,"style":37422},[31685,31686],[34,234338,2557],{"className":234339},[31690],[34,234341],{"className":234342,"style":32604},[31845],[34,234344,223],{"className":234345},[32024],[34,234347],{"className":234348,"style":32604},[31845],[34,234350,234352,234355],{"className":234351},[31676],[34,234353],{"className":234354,"style":33218},[31680],[34,234356,98360],{"className":234357,"style":33389},[31685,31686],[15,234359,234360],{},"所以单缓冲通常比无缓冲快。",[15,234362,177826],{},[24,234364,234366],{"className":30504,"code":234365,"language":85,"meta":29,"style":29},"OS 缓冲区管理更复杂\n换入换出逻辑更复杂\n如果 I\u002FO 设备和交换区在同一磁盘上，调度更麻烦\n",[31,234367,234368,234373,234378],{"__ignoreMap":29},[34,234369,234370],{"class":36,"line":37},[34,234371,234372],{"class":190},"OS 缓冲区管理更复杂\n",[34,234374,234375],{"class":36,"line":43},[34,234376,234377],{"class":190},"换入换出逻辑更复杂\n",[34,234379,234380],{"class":36,"line":49},[34,234381,234382],{"class":190},"如果 I\u002FO 设备和交换区在同一磁盘上，调度更麻烦\n",[19,234384,234386],{"id":234385},"_5-面向流设备的单缓冲","5. 面向流设备的单缓冲",[13837,234388,234389,234398],{},[13840,234390,234391],{},[13843,234392,234393,234396],{},[13846,234394,234395],{},"方式",[13846,234397,13848],{},[13853,234399,234400,234408],{},[13843,234401,234402,234405],{},[13858,234403,234404],{},"Line-at-a-time",[13858,234406,234407],{},"终端一行一行输入，回车表示一行结束",[13843,234409,234410,234413],{},[13858,234411,234412],{},"Byte-at-a-time",[13858,234414,234415],{},"每个按键都有意义，例如交互式终端、传感器、控制器",[15,234417,234418],{},"Byte-at-a-time 本质上很像生产者—消费者模型。",[19,234420,234422],{"id":234421},"_6-双缓冲-double-buffer","6. 双缓冲 Double Buffer",[15,234424,234425,234426,347],{},"双缓冲：使用两个系统缓冲区。进程处理一个缓冲区时，OS 或设备可以填充\u002F清空另一个缓冲区。也叫 ",[2627,234427,234428],{},"buffer swapping",[15,234430,139555],{},[24,234432,234434],{"className":30504,"code":234433,"language":85,"meta":29,"style":29},"缓冲区 1：交给进程处理\n缓冲区 2：同时由设备填充\n处理完后交换角色\n",[31,234435,234436,234441,234446],{"__ignoreMap":29},[34,234437,234438],{"class":36,"line":37},[34,234439,234440],{"class":190},"缓冲区 1：交给进程处理\n",[34,234442,234443],{"class":36,"line":43},[34,234444,234445],{"class":190},"缓冲区 2：同时由设备填充\n",[34,234447,234448],{"class":36,"line":49},[34,234449,234450],{"class":190},"处理完后交换角色\n",[15,234452,234453],{},"块设备双缓冲时间大致接近：",[15,234455,234456],{},[34,234457,234459,234485],{"className":234458},[31642],[34,234460,234462],{"className":234461},[31646],[31648,234463,234464],{"xmlns":31650},[31652,234465,234466,234482],{},[31655,234467,234468,234470,234472,234474,234476,234478,234480],{},[31658,234469,32679],{},[31662,234471,32682],{},[31662,234473,456],{"stretchy":702},[31658,234475,47384],{},[31662,234477,31960],{"separator":758},[31658,234479,37400],{},[31662,234481,2557],{"stretchy":702},[31665,234483,234484],{"encoding":31667},"\\max(C, T)",[34,234486,234488],{"className":234487,"ariaHidden":758},[31672],[34,234489,234491,234494,234497,234500,234503,234506,234509,234512],{"className":234490},[31676],[34,234492],{"className":234493,"style":31774},[31680],[34,234495,32679],{"className":234496},[32767],[34,234498,456],{"className":234499},[31782],[34,234501,47384],{"className":234502,"style":38706},[31685,31686],[34,234504,31960],{"className":234505},[32119],[34,234507],{"className":234508,"style":32123},[31845],[34,234510,37400],{"className":234511,"style":37422},[31685,31686],[34,234513,2557],{"className":234514},[31690],[15,234516,234517],{},"课件还补充了两种情况：",[24,234519,234521],{"className":30504,"code":234520,"language":85,"meta":29,"style":29},"若 C \u003C T，且 M \u003C\u003C T，则一块数据处理时间约为 T，可保证设备连续工作。\n若 C > T，且 M \u003C\u003C T，则一块数据处理时间约为 C + M，可保证进程不必等待 I\u002FO。\n",[31,234522,234523,234528],{"__ignoreMap":29},[34,234524,234525],{"class":36,"line":37},[34,234526,234527],{"class":190},"若 C \u003C T，且 M \u003C\u003C T，则一块数据处理时间约为 T，可保证设备连续工作。\n",[34,234529,234530],{"class":36,"line":43},[34,234531,234532],{"class":190},"若 C > T，且 M \u003C\u003C T，则一块数据处理时间约为 C + M，可保证进程不必等待 I\u002FO。\n",[19,234534,234536],{"id":234535},"_7-循环缓冲-circular-buffer","7. 循环缓冲 Circular Buffer",[15,234538,234539],{},"如果两个缓冲区还不够，就使用多个缓冲区组成循环队列。每个缓冲区是循环缓冲中的一个单元。它本质上就是有限缓冲区的生产者—消费者模型。",[15,234541,234542],{},"适用场景：",[24,234544,234546],{"className":30504,"code":234545,"language":85,"meta":29,"style":29},"进程有突发 I\u002FO\n设备和进程速度波动较大\n需要平滑 I\u002FO 峰值\n",[31,234547,234548,234553,234558],{"__ignoreMap":29},[34,234549,234550],{"class":36,"line":37},[34,234551,234552],{"class":190},"进程有突发 I\u002FO\n",[34,234554,234555],{"class":36,"line":43},[34,234556,234557],{"class":190},"设备和进程速度波动较大\n",[34,234559,234560],{"class":36,"line":49},[34,234561,234562],{"class":190},"需要平滑 I\u002FO 峰值\n",[19,234564,234566],{"id":234565},"_8-缓冲的局限","8. 缓冲的局限",[15,234568,234569],{},"缓冲只能平滑短期峰值。\n如果进程长期平均产生数据的速度大于设备处理速度，那么再多缓冲区也会被填满，进程最终还是要等待。",[32188,234571],{},[11,234573,234575],{"id":234574},"_115-disk-scheduling","11.5 Disk Scheduling",[19,234577,234579],{"id":234578},"_1-为什么磁盘-io-特别重要","1. 为什么磁盘 I\u002FO 特别重要？",[15,234581,234582],{},"磁盘速度远慢于 CPU 和内存，所以磁盘 I\u002FO 很容易成为系统瓶颈。教材强调，磁盘性能对整体系统性能影响极大，因此本章花大量篇幅讨论磁盘调度、RAID、磁盘缓存。",[19,234584,234586],{"id":234585},"_2-磁盘访问时间组成","2. 磁盘访问时间组成",[15,234588,234589],{},"一次磁盘读写主要包括：",[13837,234591,234592,234600],{},[13840,234593,234594],{},[13843,234595,234596,234598],{},[13846,234597,100174],{},[13846,234599,126615],{},[13853,234601,234602,234610,234618,234626],{},[13843,234603,234604,234607],{},[13858,234605,234606],{},"Seek time 寻道时间",[13858,234608,234609],{},"磁头移动到目标磁道的时间",[13843,234611,234612,234615],{},[13858,234613,234614],{},"Rotational delay \u002F latency 旋转延迟",[13858,234616,234617],{},"目标扇区旋转到磁头下方的时间",[13843,234619,234620,234623],{},[13858,234621,234622],{},"Transfer time 传输时间",[13858,234624,234625],{},"数据真正经过磁头读写的时间",[13843,234627,234628,234631],{},[13858,234629,234630],{},"Queueing delay 排队延迟",[13858,234632,234633],{},"请求等待设备、通道、控制器空闲的时间",[15,234635,234636],{},"访问时间：",[15,234638,234639],{},[34,234640,234642,234730],{"className":234641},[31642],[34,234643,234645],{"className":234644},[31646],[31648,234646,234647],{"xmlns":31650},[31652,234648,234649,234727],{},[31655,234650,234651,234653,234655,234657,234659,234661,234663,234665,234667,234669,234671,234673,234675,234677,234679,234681,234683,234685,234687,234689,234691,234693,234695,234697,234699,234701,234703,234705,234707,234709,234711,234713,234715,234717,234719,234721,234723,234725],{},[31658,234652,54151],{},[31658,234654,31965],{},[31658,234656,31965],{},[31658,234658,33244],{},[31658,234660,31747],{},[31658,234662,31747],{},[39361,234664,130298],{},[31658,234666,37400],{},[31658,234668,31955],{},[31658,234670,31759],{},[31658,234672,33244],{},[31662,234674,202],{},[31658,234676,94675],{},[31658,234678,33244],{},[31658,234680,33244],{},[31658,234682,33759],{},[39361,234684,130298],{},[31658,234686,37400],{},[31658,234688,31955],{},[31658,234690,31759],{},[31658,234692,33244],{},[31662,234694,223],{},[31658,234696,34386],{},[31658,234698,43383],{},[31658,234700,33249],{},[31658,234702,111],{},[31658,234704,33249],{},[31658,234706,31955],{},[31658,234708,43383],{},[31658,234710,31660],{},[31658,234712,111],{},[31658,234714,43379],{},[39361,234716,130298],{},[31658,234718,91994],{},[31658,234720,33244],{},[31658,234722,43379],{},[31658,234724,111],{},[31658,234726,67764],{},[31665,234728,234729],{"encoding":31667},"Access\\ Time = Seek\\ Time + Rotational\\ Delay",[34,234731,234733,234768,234805],{"className":234732,"ariaHidden":758},[31672],[34,234734,234736,234739,234742,234746,234749,234752,234756,234759,234762,234765],{"className":234735},[31676],[34,234737],{"className":234738,"style":33218},[31680],[34,234740,54151],{"className":234741},[31685,31686],[34,234743,234745],{"className":234744},[31685,31686],"ccess",[34,234747,130298],{"className":234748},[31845],[34,234750,37400],{"className":234751,"style":37422},[31685,31686],[34,234753,234755],{"className":234754},[31685,31686],"im",[34,234757,33244],{"className":234758},[31685,31686],[34,234760],{"className":234761,"style":31846},[31845],[34,234763,202],{"className":234764},[31850],[34,234766],{"className":234767,"style":31846},[31845],[34,234769,234771,234774,234777,234781,234784,234787,234790,234793,234796,234799,234802],{"className":234770},[31676],[34,234772],{"className":234773,"style":34299},[31680],[34,234775,94675],{"className":234776,"style":106455},[31685,31686],[34,234778,234780],{"className":234779},[31685,31686],"ee",[34,234782,33759],{"className":234783,"style":33774},[31685,31686],[34,234785,130298],{"className":234786},[31845],[34,234788,37400],{"className":234789,"style":37422},[31685,31686],[34,234791,234755],{"className":234792},[31685,31686],[34,234794,33244],{"className":234795},[31685,31686],[34,234797],{"className":234798,"style":32604},[31845],[34,234800,223],{"className":234801},[32024],[34,234803],{"className":234804,"style":32604},[31845],[34,234806,234808,234811,234814,234817,234820,234823,234826,234829,234832,234836,234839,234842,234845,234848,234851,234854],{"className":234807},[31676],[34,234809],{"className":234810,"style":43217},[31680],[34,234812,34386],{"className":234813,"style":34427},[31685,31686],[34,234815,43383],{"className":234816},[31685,31686],[34,234818,33249],{"className":234819},[31685,31686],[34,234821,111],{"className":234822},[31685,31686],[34,234824,33249],{"className":234825},[31685,31686],[34,234827,31955],{"className":234828},[31685,31686],[34,234830,43383],{"className":234831},[31685,31686],[34,234833,234835],{"className":234834},[31685,31686],"na",[34,234837,43379],{"className":234838,"style":47113},[31685,31686],[34,234840,130298],{"className":234841},[31845],[34,234843,91994],{"className":234844,"style":34452},[31685,31686],[34,234846,33244],{"className":234847},[31685,31686],[34,234849,43379],{"className":234850,"style":47113},[31685,31686],[34,234852,111],{"className":234853},[31685,31686],[34,234855,67764],{"className":234856,"style":32062},[31685,31686],[15,234858,234859],{},"总平均访问时间常写成：",[15,234861,234862],{},[34,234863,234865,234915],{"className":234864},[31642],[34,234866,234868],{"className":234867},[31646],[31648,234869,234870],{"xmlns":31650},[31652,234871,234872,234912],{},[31655,234873,234874,234880,234882,234888,234890,234900,234902],{},[31743,234875,234876,234878],{},[31658,234877,37400],{},[31658,234879,111],{},[31662,234881,202],{},[31743,234883,234884,234886],{},[31658,234885,37400],{},[31658,234887,31747],{},[31662,234889,223],{},[41453,234891,234892,234894],{},[31749,234893,640],{},[31655,234895,234896,234898],{},[31749,234897,768],{},[31658,234899,34395],{},[31662,234901,223],{},[41453,234903,234904,234906],{},[31658,234905,1697],{},[31655,234907,234908,234910],{},[31658,234909,34395],{},[31658,234911,33347],{},[31665,234913,234914],{"encoding":31667},"T_a = T_s + \\frac{1}{2r} + \\frac{b}{rN}",[34,234916,234918,234973,235028,235114],{"className":234917,"ariaHidden":758},[31672],[34,234919,234921,234924,234964,234967,234970],{"className":234920},[31676],[34,234922],{"className":234923,"style":70915},[31680],[34,234925,234927,234930],{"className":234926},[31685],[34,234928,37400],{"className":234929,"style":37422},[31685,31686],[34,234931,234933],{"className":234932},[31792],[34,234934,234936,234956],{"className":234935},[31796,31797],[34,234937,234939,234953],{"className":234938},[31801],[34,234940,234942],{"className":234941,"style":32362},[31805],[34,234943,234944,234947],{"style":214424},[34,234945],{"className":234946,"style":31814},[31813],[34,234948,234950],{"className":234949},[31818,31819,31820,31821],[34,234951,111],{"className":234952},[31685,31686,31821],[34,234954,31829],{"className":234955},[31828],[34,234957,234959],{"className":234958},[31801],[34,234960,234962],{"className":234961,"style":31836},[31805],[34,234963],{},[34,234965],{"className":234966,"style":31846},[31845],[34,234968,202],{"className":234969},[31850],[34,234971],{"className":234972,"style":31846},[31845],[34,234974,234976,234979,235019,235022,235025],{"className":234975},[31676],[34,234977],{"className":234978,"style":70915},[31680],[34,234980,234982,234985],{"className":234981},[31685],[34,234983,37400],{"className":234984,"style":37422},[31685,31686],[34,234986,234988],{"className":234987},[31792],[34,234989,234991,235011],{"className":234990},[31796,31797],[34,234992,234994,235008],{"className":234993},[31801],[34,234995,234997],{"className":234996,"style":32362},[31805],[34,234998,234999,235002],{"style":214424},[34,235000],{"className":235001,"style":31814},[31813],[34,235003,235005],{"className":235004},[31818,31819,31820,31821],[34,235006,31747],{"className":235007},[31685,31686,31821],[34,235009,31829],{"className":235010},[31828],[34,235012,235014],{"className":235013},[31801],[34,235015,235017],{"className":235016,"style":31836},[31805],[34,235018],{},[34,235020],{"className":235021,"style":32604},[31845],[34,235023,223],{"className":235024},[32024],[34,235026],{"className":235027,"style":32604},[31845],[34,235029,235031,235034,235105,235108,235111],{"className":235030},[31676],[34,235032],{"className":235033,"style":46727},[31680],[34,235035,235037,235040,235102],{"className":235036},[31685],[34,235038],{"className":235039},[31782,41488],[34,235041,235043],{"className":235042},[41453],[34,235044,235046,235094],{"className":235045},[31796,31797],[34,235047,235049,235091],{"className":235048},[31801],[34,235050,235052,235069,235077],{"className":235051,"style":46746},[31805],[34,235053,235054,235057],{"style":41504},[34,235055],{"className":235056,"style":41508},[31813],[34,235058,235060],{"className":235059},[31818,31819,31820,31821],[34,235061,235063,235066],{"className":235062},[31685,31821],[34,235064,768],{"className":235065},[31685,31821],[34,235067,34395],{"className":235068,"style":34452},[31685,31686,31821],[34,235070,235071,235074],{"style":41520},[34,235072],{"className":235073,"style":41508},[31813],[34,235075],{"className":235076,"style":41528},[41527],[34,235078,235079,235082],{"style":46771},[34,235080],{"className":235081,"style":41508},[31813],[34,235083,235085],{"className":235084},[31818,31819,31820,31821],[34,235086,235088],{"className":235087},[31685,31821],[34,235089,640],{"className":235090},[31685,31821],[34,235092,31829],{"className":235093},[31828],[34,235095,235097],{"className":235096},[31801],[34,235098,235100],{"className":235099,"style":41619},[31805],[34,235101],{},[34,235103],{"className":235104},[31690,41488],[34,235106],{"className":235107,"style":32604},[31845],[34,235109,223],{"className":235110},[32024],[34,235112],{"className":235113,"style":32604},[31845],[34,235115,235117,235120],{"className":235116},[31676],[34,235118],{"className":235119,"style":232492},[31680],[34,235121,235123,235126,235188],{"className":235122},[31685],[34,235124],{"className":235125},[31782,41488],[34,235127,235129],{"className":235128},[41453],[34,235130,235132,235180],{"className":235131},[31796,31797],[34,235133,235135,235177],{"className":235134},[31801],[34,235136,235138,235155,235163],{"className":235137,"style":232511},[31805],[34,235139,235140,235143],{"style":41504},[34,235141],{"className":235142,"style":41508},[31813],[34,235144,235146],{"className":235145},[31818,31819,31820,31821],[34,235147,235149,235152],{"className":235148},[31685,31821],[34,235150,34395],{"className":235151,"style":34452},[31685,31686,31821],[34,235153,33347],{"className":235154,"style":33389},[31685,31686,31821],[34,235156,235157,235160],{"style":41520},[34,235158],{"className":235159,"style":41508},[31813],[34,235161],{"className":235162,"style":41528},[41527],[34,235164,235165,235168],{"style":46771},[34,235166],{"className":235167,"style":41508},[31813],[34,235169,235171],{"className":235170},[31818,31819,31820,31821],[34,235172,235174],{"className":235173},[31685,31821],[34,235175,1697],{"className":235176},[31685,31686,31821],[34,235178,31829],{"className":235179},[31828],[34,235181,235183],{"className":235182},[31801],[34,235184,235186],{"className":235185,"style":41619},[31805],[34,235187],{},[34,235189],{"className":235190},[31690,41488],[15,235192,104944],{},[24,235194,235196],{"className":30504,"code":235195,"language":85,"meta":29,"style":29},"Ts = 平均寻道时间\nr = 每秒转数\nb = 要传输的字节数\nN = 每磁道字节数\nb \u002F rN = 传输时间\n1 \u002F 2r = 平均旋转延迟\n",[31,235197,235198,235203,235208,235213,235218,235223],{"__ignoreMap":29},[34,235199,235200],{"class":36,"line":37},[34,235201,235202],{"class":190},"Ts = 平均寻道时间\n",[34,235204,235205],{"class":36,"line":43},[34,235206,235207],{"class":190},"r = 每秒转数\n",[34,235209,235210],{"class":36,"line":49},[34,235211,235212],{"class":190},"b = 要传输的字节数\n",[34,235214,235215],{"class":36,"line":56},[34,235216,235217],{"class":190},"N = 每磁道字节数\n",[34,235219,235220],{"class":36,"line":62},[34,235221,235222],{"class":190},"b \u002F rN = 传输时间\n",[34,235224,235225],{"class":36,"line":68},[34,235226,235227],{"class":190},"1 \u002F 2r = 平均旋转延迟\n",[19,235229,235231],{"id":235230},"_3-顺序访问-vs-随机访问","3. 顺序访问 vs 随机访问",[15,235233,235234],{},"教材给过一个很典型的对比：读同样大小的文件，若文件连续存放，寻道和旋转开销少；若随机分布，每个扇区都可能重新寻道和等待旋转，时间会差很多。核心结论是：",[24,235236,235238],{"className":30504,"code":235237,"language":85,"meta":29,"style":29},"磁盘性能不只取决于数据量，\n更取决于访问顺序和物理位置。\n",[31,235239,235240,235245],{"__ignoreMap":29},[34,235241,235242],{"class":36,"line":37},[34,235243,235244],{"class":190},"磁盘性能不只取决于数据量，\n",[34,235246,235247],{"class":36,"line":43},[34,235248,235249],{"class":190},"更取决于访问顺序和物理位置。\n",[19,235251,235253],{"id":235252},"_4-磁盘调度算法总表","4. 磁盘调度算法总表",[15,235255,235256],{},"教材表 11.3 总结了常见磁盘调度策略：Random、FIFO、PRI、LIFO、SSTF、SCAN、C-SCAN、N-step-SCAN、FSCAN。",[13837,235258,235259,235275],{},[13840,235260,235261],{},[13843,235262,235263,235265,235268,235271,235273],{},[13846,235264,153524],{},[13846,235266,235267],{},"中文名",[13846,235269,235270],{},"核心规则",[13846,235272,204919],{},[13846,235274,204938],{},[13853,235276,235277,235294,235311,235328,235345,235362,235379,235396,235413],{},[13843,235278,235279,235282,235285,235288,235291],{},[13858,235280,235281],{},"Random",[13858,235283,235284],{},"随机",[13858,235286,235287],{},"随机选请求",[13858,235289,235290],{},"可作基准",[13858,235292,235293],{},"性能差",[13843,235295,235296,235299,235302,235305,235308],{},[13858,235297,235298],{},"FIFO",[13858,235300,235301],{},"先来先服务",[13858,235303,235304],{},"按请求到达顺序服务",[13858,235306,235307],{},"公平，简单",[13858,235309,235310],{},"可能寻道距离大",[13843,235312,235313,235316,235319,235322,235325],{},[13858,235314,235315],{},"PRI",[13858,235317,235318],{},"优先级",[13858,235320,235321],{},"按进程优先级服务",[13858,235323,235324],{},"满足系统优先级策略",[13858,235326,235327],{},"不一定优化磁盘，可能饥饿",[13843,235329,235330,235333,235336,235339,235342],{},[13858,235331,235332],{},"LIFO",[13858,235334,235335],{},"后来先服务",[13858,235337,235338],{},"最新请求先服务",[13858,235340,235341],{},"利用局部性",[13858,235343,235344],{},"老请求可能饥饿",[13843,235346,235347,235350,235353,235356,235359],{},[13858,235348,235349],{},"SSTF",[13858,235351,235352],{},"最短服务时间优先",[13858,235354,235355],{},"选离当前磁头最近的请求",[13858,235357,235358],{},"平均寻道较短",[13858,235360,235361],{},"远处请求可能饥饿",[13843,235363,235364,235367,235370,235373,235376],{},[13858,235365,235366],{},"SCAN",[13858,235368,235369],{},"扫描 \u002F 电梯算法",[13858,235371,235372],{},"沿一个方向服务，到端点或无请求后反向",[13858,235374,235375],{},"避免严重饥饿，分布较好",[13858,235377,235378],{},"对刚扫过区域不友好",[13843,235380,235381,235384,235387,235390,235393],{},[13858,235382,235383],{},"C-SCAN",[13858,235385,235386],{},"循环扫描",[13858,235388,235389],{},"只沿一个方向服务，到末端后快速回起点",[13858,235391,235392],{},"等待时间更均匀",[13858,235394,235395],{},"回程不服务",[13843,235397,235398,235401,235404,235407,235410],{},[13858,235399,235400],{},"N-step-SCAN",[13858,235402,235403],{},"N 步扫描",[13858,235405,235406],{},"将请求队列分成长度 N 的子队列，逐个 SCAN",[13858,235408,235409],{},"防止磁臂黏着",[13858,235411,235412],{},"N 选择影响性能",[13843,235414,235415,235418,235421,235424,235427],{},[13858,235416,235417],{},"FSCAN",[13858,235419,235420],{},"双队列扫描",[13858,235422,235423],{},"扫描时新请求放入另一个队列",[13858,235425,235426],{},"对负载变化更稳定",[13858,235428,235429],{},"实现略复杂",[19,235431,235433],{"id":235432},"_5-fifo","5. FIFO",[15,235435,235436],{},"FIFO 按请求到达顺序服务。\n优点是公平；缺点是请求如果分布很散，磁头会来回跳，寻道开销大。教材例子中 FIFO 平均寻道长度较大。",[15,235438,235439],{},"计算方法：",[24,235441,235443],{"className":30504,"code":235442,"language":85,"meta":29,"style":29},"从初始磁道开始\n按请求序列逐个访问\n每一步移动距离 = |当前磁道 - 下一个请求磁道|\n总移动距离 = 所有移动距离求和\n",[31,235444,235445,235450,235455,235460],{"__ignoreMap":29},[34,235446,235447],{"class":36,"line":37},[34,235448,235449],{"class":190},"从初始磁道开始\n",[34,235451,235452],{"class":36,"line":43},[34,235453,235454],{"class":190},"按请求序列逐个访问\n",[34,235456,235457],{"class":36,"line":49},[34,235458,235459],{"class":190},"每一步移动距离 = |当前磁道 - 下一个请求磁道|\n",[34,235461,235462],{"class":36,"line":56},[34,235463,235464],{"class":190},"总移动距离 = 所有移动距离求和\n",[19,235466,235468],{"id":235467},"_6-sstf","6. SSTF",[15,235470,235471],{},"SSTF 每次选择离当前磁头位置最近的请求。",[15,235473,235439],{},[24,235475,235477],{"className":30504,"code":235476,"language":85,"meta":29,"style":29},"当前位置出发\n在未完成请求中找距离最近的\n移动过去并删除该请求\n重复直到请求为空\n",[31,235478,235479,235484,235489,235494],{"__ignoreMap":29},[34,235480,235481],{"class":36,"line":37},[34,235482,235483],{"class":190},"当前位置出发\n",[34,235485,235486],{"class":36,"line":43},[34,235487,235488],{"class":190},"在未完成请求中找距离最近的\n",[34,235490,235491],{"class":36,"line":49},[34,235492,235493],{"class":190},"移动过去并删除该请求\n",[34,235495,235496],{"class":36,"line":56},[34,235497,235498],{"class":190},"重复直到请求为空\n",[15,235500,235501],{},"优点：通常比 FIFO 寻道距离短。\n缺点：如果一直有近处请求，远处请求可能长期得不到服务，也就是饥饿。",[19,235503,235505],{"id":235504},"_7-scan-电梯算法","7. SCAN \u002F 电梯算法",[15,235507,235508],{},"SCAN 像电梯一样：磁头沿一个方向移动，沿途服务所有请求，到达端点或该方向无请求后，再反向服务。",[15,235510,235511],{},"计算时注意题目给的方向：",[24,235513,235515],{"className":30504,"code":235514,"language":85,"meta":29,"style":29},"当前磁头位置\n\n- 当前移动方向\n- 请求序列\n",[31,235516,235517,235522,235526,235533],{"__ignoreMap":29},[34,235518,235519],{"class":36,"line":37},[34,235520,235521],{"class":190},"当前磁头位置\n",[34,235523,235524],{"class":36,"line":43},[34,235525,53],{"emptyLinePlaceholder":52},[34,235527,235528,235530],{"class":36,"line":49},[34,235529,2511],{"class":1326},[34,235531,235532],{"class":190}," 当前移动方向\n",[34,235534,235535,235537],{"class":36,"line":56},[34,235536,2511],{"class":1326},[34,235538,235539],{"class":190}," 请求序列\n",[15,235541,235542],{},"如果题目说“刚服务过 120，当前在 100”，说明磁头正在向小号磁道移动。",[15,235544,235545],{},"课件还提醒：有些中文题里的“电梯调度算法”更接近 LOOK，即没有请求时不一定走到磁盘最边界，而是到该方向最后一个请求后反向。",[19,235547,235549],{"id":235548},"_8-c-scan","8. C-SCAN",[15,235551,235552],{},"C-SCAN 只在一个方向上服务请求。到达末端后，磁头直接回到另一端，回程不服务。这样可以减少新请求的最大等待时间，使服务更均匀。",[15,235554,235555],{},"理解：",[24,235557,235559],{"className":30504,"code":235558,"language":85,"meta":29,"style":29},"SCAN：来回都服务\nC-SCAN：只单向服务，回程不服务\n",[31,235560,235561,235566],{"__ignoreMap":29},[34,235562,235563],{"class":36,"line":37},[34,235564,235565],{"class":190},"SCAN：来回都服务\n",[34,235567,235568],{"class":36,"line":43},[34,235569,235570],{"class":190},"C-SCAN：只单向服务，回程不服务\n",[19,235572,235574],{"id":235573},"_9-n-step-scan-与-fscan","9. N-step-SCAN 与 FSCAN",[15,235576,235577,235578,347],{},"SSTF、SCAN、C-SCAN 都可能出现一种现象：某些进程不断请求同一附近磁道，磁臂长期停留在一小片区域，叫 ",[2627,235579,235580],{},"arm stickiness，磁臂黏着",[15,235582,235583],{},"为了解决它，可以把队列分段：",[253,235585,235400],{"id":235586},"n-step-scan",[24,235588,235590],{"className":30504,"code":235589,"language":85,"meta":29,"style":29},"把磁盘请求队列分成长度 N 的子队列\n每次对一个子队列使用 SCAN\n处理当前子队列时，新请求加入其他队列\nN 很大时接近 SCAN\nN = 1 时接近 FIFO\n",[31,235591,235592,235597,235602,235607,235612],{"__ignoreMap":29},[34,235593,235594],{"class":36,"line":37},[34,235595,235596],{"class":190},"把磁盘请求队列分成长度 N 的子队列\n",[34,235598,235599],{"class":36,"line":43},[34,235600,235601],{"class":190},"每次对一个子队列使用 SCAN\n",[34,235603,235604],{"class":36,"line":49},[34,235605,235606],{"class":190},"处理当前子队列时，新请求加入其他队列\n",[34,235608,235609],{"class":36,"line":56},[34,235610,235611],{"class":190},"N 很大时接近 SCAN\n",[34,235613,235614],{"class":36,"line":62},[34,235615,235616],{"class":190},"N = 1 时接近 FIFO\n",[253,235618,235417],{"id":235619},"fscan",[24,235621,235623],{"className":30504,"code":235622,"language":85,"meta":29,"style":29},"使用两个队列\n扫描开始时，所有旧请求在一个队列\n扫描过程中，新请求进入另一个队列\n旧请求全部处理完后，再处理新队列\n",[31,235624,235625,235630,235635,235640],{"__ignoreMap":29},[34,235626,235627],{"class":36,"line":37},[34,235628,235629],{"class":190},"使用两个队列\n",[34,235631,235632],{"class":36,"line":43},[34,235633,235634],{"class":190},"扫描开始时，所有旧请求在一个队列\n",[34,235636,235637],{"class":36,"line":49},[34,235638,235639],{"class":190},"扫描过程中，新请求进入另一个队列\n",[34,235641,235642],{"class":36,"line":56},[34,235643,235644],{"class":190},"旧请求全部处理完后，再处理新队列\n",[19,235646,235648],{"id":235647},"_10-磁盘调度题做法模板","10. 磁盘调度题做法模板",[15,235650,235651],{},"遇到磁盘调度计算题，按这个流程：",[24,235653,235655],{"className":30504,"code":235654,"language":85,"meta":29,"style":29},"1. 写出初始磁头位置\n2. 判断初始方向\n3. 写出请求序列\n4. 根据算法排序访问顺序\n5. 每一步计算移动距离\n6. 求总移动磁道数\n",[31,235656,235657,235664,235671,235678,235685,235692],{"__ignoreMap":29},[34,235658,235659,235661],{"class":36,"line":37},[34,235660,233680],{"class":1326},[34,235662,235663],{"class":190}," 写出初始磁头位置\n",[34,235665,235666,235668],{"class":36,"line":43},[34,235667,233688],{"class":1326},[34,235669,235670],{"class":190}," 判断初始方向\n",[34,235672,235673,235675],{"class":36,"line":49},[34,235674,233696],{"class":1326},[34,235676,235677],{"class":190}," 写出请求序列\n",[34,235679,235680,235682],{"class":36,"line":56},[34,235681,233704],{"class":1326},[34,235683,235684],{"class":190}," 根据算法排序访问顺序\n",[34,235686,235687,235689],{"class":36,"line":62},[34,235688,233712],{"class":1326},[34,235690,235691],{"class":190}," 每一步计算移动距离\n",[34,235693,235694,235696],{"class":36,"line":68},[34,235695,233720],{"class":1326},[34,235697,235698],{"class":190}," 求总移动磁道数\n",[15,235700,2632],{},[24,235702,235704],{"className":30504,"code":235703,"language":85,"meta":29,"style":29},"移动距离 = |当前磁道号 - 下一磁道号|\n总移动距离 = 每一步移动距离之和\n",[31,235705,235706,235711],{"__ignoreMap":29},[34,235707,235708],{"class":36,"line":37},[34,235709,235710],{"class":190},"移动距离 = |当前磁道号 - 下一磁道号|\n",[34,235712,235713],{"class":36,"line":43},[34,235714,235715],{"class":190},"总移动距离 = 每一步移动距离之和\n",[15,235717,235718],{},"考试最容易错的是：",[24,235720,235722],{"className":30504,"code":235721,"language":85,"meta":29,"style":29},"SCAN 是否走到 0 \u002F 最大磁道？\nLOOK 是否只走到该方向最后一个请求？\nC-SCAN 回程是否计入移动距离？\n题目有没有说“刚服务过某磁道”，这会影响初始方向。\n",[31,235723,235724,235729,235734,235739],{"__ignoreMap":29},[34,235725,235726],{"class":36,"line":37},[34,235727,235728],{"class":190},"SCAN 是否走到 0 \u002F 最大磁道？\n",[34,235730,235731],{"class":36,"line":43},[34,235732,235733],{"class":190},"LOOK 是否只走到该方向最后一个请求？\n",[34,235735,235736],{"class":36,"line":49},[34,235737,235738],{"class":190},"C-SCAN 回程是否计入移动距离？\n",[34,235740,235741],{"class":36,"line":56},[34,235742,235743],{"class":190},"题目有没有说“刚服务过某磁道”，这会影响初始方向。\n",[32188,235745],{},[11,235747,235749],{"id":235748},"课件补充旋转优化与记录分布","课件补充：旋转优化与记录分布",[15,235751,235752],{},"课件有一组“循环排序 \u002F 优化分布”的例题，核心思想是：",[24,235754,235756],{"className":30504,"code":235755,"language":85,"meta":29,"style":29},"磁盘不是只考虑磁道移动，还要考虑旋转延迟。\n如果记录顺序存放，但处理一个记录需要时间，\n处理完后下一个扇区可能已经转过去了，\n就需要等几乎一整圈。\n",[31,235757,235758,235763,235768,235773],{"__ignoreMap":29},[34,235759,235760],{"class":36,"line":37},[34,235761,235762],{"class":190},"磁盘不是只考虑磁道移动，还要考虑旋转延迟。\n",[34,235764,235765],{"class":36,"line":43},[34,235766,235767],{"class":190},"如果记录顺序存放，但处理一个记录需要时间，\n",[34,235769,235770],{"class":36,"line":49},[34,235771,235772],{"class":190},"处理完后下一个扇区可能已经转过去了，\n",[34,235774,235775],{"class":36,"line":56},[34,235776,235777],{"class":190},"就需要等几乎一整圈。\n",[15,235779,235780,235781,235784],{},"所以可以通过",[2627,235782,235783],{},"交错存放记录","优化读取时间：",[24,235786,235788],{"className":30504,"code":235787,"language":85,"meta":29,"style":29},"逻辑顺序：A B C D E ...\n物理顺序：不一定连续摆放\n目标：让 CPU 处理完 A 后，B 正好快转到磁头下\n",[31,235789,235790,235795,235800],{"__ignoreMap":29},[34,235791,235792],{"class":36,"line":37},[34,235793,235794],{"class":190},"逻辑顺序：A B C D E ...\n",[34,235796,235797],{"class":36,"line":43},[34,235798,235799],{"class":190},"物理顺序：不一定连续摆放\n",[34,235801,235802],{"class":36,"line":49},[34,235803,235804],{"class":190},"目标：让 CPU 处理完 A 后，B 正好快转到磁头下\n",[15,235806,235807],{},"这类题步骤：",[24,235809,235811],{"className":30504,"code":235810,"language":85,"meta":29,"style":29},"1. 算每转一圈时间\n2. 算每个扇区经过磁头的时间\n3. 算处理一个记录需要跨过几个扇区\n4. 按这个间隔重新排列逻辑记录\n5. 计算总读取时间\n",[31,235812,235813,235820,235827,235834,235841],{"__ignoreMap":29},[34,235814,235815,235817],{"class":36,"line":37},[34,235816,233680],{"class":1326},[34,235818,235819],{"class":190}," 算每转一圈时间\n",[34,235821,235822,235824],{"class":36,"line":43},[34,235823,233688],{"class":1326},[34,235825,235826],{"class":190}," 算每个扇区经过磁头的时间\n",[34,235828,235829,235831],{"class":36,"line":49},[34,235830,233696],{"class":1326},[34,235832,235833],{"class":190}," 算处理一个记录需要跨过几个扇区\n",[34,235835,235836,235838],{"class":36,"line":56},[34,235837,233704],{"class":1326},[34,235839,235840],{"class":190}," 按这个间隔重新排列逻辑记录\n",[34,235842,235843,235845],{"class":36,"line":62},[34,235844,233712],{"class":1326},[34,235846,235847],{"class":190}," 计算总读取时间\n",[32188,235849],{},[11,235851,235853],{"id":235852},"_116-raid","11.6 RAID",[19,235855,235857],{"id":235856},"_1-raid-的思想","1. RAID 的思想",[15,235859,235860,235861,235864],{},"RAID 的核心是：",[2627,235862,235863],{},"用多个磁盘并行工作，提高性能，并通过冗余提高可靠性。"," 教材指出，RAID 0、1、5、6 是最常用的几个级别。",[15,235866,235867],{},"两个关键词：",[13837,235869,235870,235879],{},[13840,235871,235872],{},[13843,235873,235874,235877],{},[13846,235875,235876],{},"词",[13846,235878,126615],{},[13853,235880,235881,235889],{},[13843,235882,235883,235886],{},[13858,235884,235885],{},"Strip",[13858,235887,235888],{},"条带单元，逻辑磁盘上的一小块连续数据",[13843,235890,235891,235894],{},[13858,235892,235893],{},"Stripe",[13858,235895,235896],{},"条带组，一组 strip 分布到多个磁盘上",[19,235898,235900],{"id":235899},"_2-raid-0","2. RAID 0",[15,235902,235903],{},"RAID 0：条带化，无冗余。它不是真正意义上的“冗余阵列”，因为没有容错。",[15,235905,4160],{},[24,235907,235909],{"className":30504,"code":235908,"language":85,"meta":29,"style":29},"数据分散到多个磁盘\n多个磁盘可并行读写\n性能高，容量利用率高\n没有冗余，一个盘坏可能导致数据丢失\n",[31,235910,235911,235916,235921,235926],{"__ignoreMap":29},[34,235912,235913],{"class":36,"line":37},[34,235914,235915],{"class":190},"数据分散到多个磁盘\n",[34,235917,235918],{"class":36,"line":43},[34,235919,235920],{"class":190},"多个磁盘可并行读写\n",[34,235922,235923],{"class":36,"line":49},[34,235924,235925],{"class":190},"性能高，容量利用率高\n",[34,235927,235928],{"class":36,"line":56},[34,235929,235930],{"class":190},"没有冗余，一个盘坏可能导致数据丢失\n",[15,235932,235933],{},"适合追求性能、不重视可靠性的场景。",[19,235935,235937],{"id":235936},"_3-raid-1","3. RAID 1",[15,235939,235940],{},"RAID 1：镜像。每份数据写到两个磁盘。",[15,235942,14543],{},[24,235944,235946],{"className":30504,"code":235945,"language":85,"meta":29,"style":29},"读请求可以从两个镜像盘中选择更快的一个\n写请求需要两个盘都写\n恢复简单，一个盘坏了另一个还能用\n可靠性高\n",[31,235947,235948,235953,235958,235963],{"__ignoreMap":29},[34,235949,235950],{"class":36,"line":37},[34,235951,235952],{"class":190},"读请求可以从两个镜像盘中选择更快的一个\n",[34,235954,235955],{"class":36,"line":43},[34,235956,235957],{"class":190},"写请求需要两个盘都写\n",[34,235959,235960],{"class":36,"line":49},[34,235961,235962],{"class":190},"恢复简单，一个盘坏了另一个还能用\n",[34,235964,235965],{"class":36,"line":56},[34,235966,235967],{"class":190},"可靠性高\n",[15,235969,177826],{},[24,235971,235973],{"className":30504,"code":235972,"language":85,"meta":29,"style":29},"成本高，需要 2 倍磁盘空间\n容量利用率约 50%\n",[31,235974,235975,235980],{"__ignoreMap":29},[34,235976,235977],{"class":36,"line":37},[34,235978,235979],{"class":190},"成本高，需要 2 倍磁盘空间\n",[34,235981,235982],{"class":36,"line":43},[34,235983,235984],{"class":190},"容量利用率约 50%\n",[19,235986,235988],{"id":235987},"_4-raid-2","4. RAID 2",[15,235990,235991],{},"RAID 2：使用 Hamming Code 进行纠错，按位或字节级条带化。",[15,235993,4160],{},[24,235995,235997],{"className":30504,"code":235996,"language":85,"meta":29,"style":29},"所有磁盘并行参与每次 I\u002FO\n使用多个校验盘存放纠错码\n能纠正单比特错误、检测双比特错误\n",[31,235998,235999,236004,236009],{"__ignoreMap":29},[34,236000,236001],{"class":36,"line":37},[34,236002,236003],{"class":190},"所有磁盘并行参与每次 I\u002FO\n",[34,236005,236006],{"class":36,"line":43},[34,236007,236008],{"class":190},"使用多个校验盘存放纠错码\n",[34,236010,236011],{"class":36,"line":49},[34,236012,236013],{"class":190},"能纠正单比特错误、检测双比特错误\n",[15,236015,236016],{},"现实中很少使用，因为现代磁盘自身可靠性较高，RAID 2 成本偏高，有点“过度设计”。",[19,236018,236020],{"id":236019},"_5-raid-3","5. RAID 3",[15,236022,236023],{},"RAID 3：小条带 + 单独奇偶校验盘。",[15,236025,4160],{},[24,236027,236029],{"className":30504,"code":236028,"language":85,"meta":29,"style":29},"数据以很小单位分散到多个数据盘\n一个专门的校验盘存储 parity\n适合大块连续数据传输\n小 I\u002FO 并发能力差，因为每次 I\u002FO 都牵涉多个磁盘\n",[31,236030,236031,236036,236041,236046],{"__ignoreMap":29},[34,236032,236033],{"class":36,"line":37},[34,236034,236035],{"class":190},"数据以很小单位分散到多个数据盘\n",[34,236037,236038],{"class":36,"line":43},[34,236039,236040],{"class":190},"一个专门的校验盘存储 parity\n",[34,236042,236043],{"class":36,"line":49},[34,236044,236045],{"class":190},"适合大块连续数据传输\n",[34,236047,236048],{"class":36,"line":56},[34,236049,236050],{"class":190},"小 I\u002FO 并发能力差，因为每次 I\u002FO 都牵涉多个磁盘\n",[15,236052,236053],{},"奇偶校验恢复思想：",[15,236055,236056],{},"如果有数据盘损坏，可以通过剩余数据盘和校验盘 XOR 恢复数据。",[19,236058,236060],{"id":236059},"_6-raid-4","6. RAID 4",[15,236062,236063],{},"RAID 4：块级条带 + 专用校验盘。",[15,236065,4160],{},[24,236067,236069],{"className":30504,"code":236068,"language":85,"meta":29,"style":29},"数据按块分布\n读操作可独立访问不同磁盘\n写操作需要更新校验盘\n校验盘容易成为写瓶颈\n",[31,236070,236071,236076,236081,236086],{"__ignoreMap":29},[34,236072,236073],{"class":36,"line":37},[34,236074,236075],{"class":190},"数据按块分布\n",[34,236077,236078],{"class":36,"line":43},[34,236079,236080],{"class":190},"读操作可独立访问不同磁盘\n",[34,236082,236083],{"class":36,"line":49},[34,236084,236085],{"class":190},"写操作需要更新校验盘\n",[34,236087,236088],{"class":36,"line":56},[34,236089,236090],{"class":190},"校验盘容易成为写瓶颈\n",[19,236092,236094],{"id":236093},"_7-raid-5","7. RAID 5",[15,236096,236097],{},"RAID 5：块级条带 + 分布式奇偶校验。",[15,236099,4160],{},[24,236101,236103],{"className":30504,"code":236102,"language":85,"meta":29,"style":29},"没有固定校验盘\n校验信息分布到所有磁盘\n读性能接近 RAID 0\n写操作有 parity 更新开销\n比 RAID 4 更避免单一校验盘瓶颈\n",[31,236104,236105,236110,236115,236120,236125],{"__ignoreMap":29},[34,236106,236107],{"class":36,"line":37},[34,236108,236109],{"class":190},"没有固定校验盘\n",[34,236111,236112],{"class":36,"line":43},[34,236113,236114],{"class":190},"校验信息分布到所有磁盘\n",[34,236116,236117],{"class":36,"line":49},[34,236118,236119],{"class":190},"读性能接近 RAID 0\n",[34,236121,236122],{"class":36,"line":56},[34,236123,236124],{"class":190},"写操作有 parity 更新开销\n",[34,236126,236127],{"class":36,"line":62},[34,236128,236129],{"class":190},"比 RAID 4 更避免单一校验盘瓶颈\n",[15,236131,236132],{},"这是实际系统中很常见的 RAID 级别之一。",[19,236134,236136],{"id":236135},"_8-raid-6","8. RAID 6",[15,236138,236139],{},"RAID 6：块级条带 + 双重分布式奇偶校验。",[15,236141,4160],{},[24,236143,236145],{"className":30504,"code":236144,"language":85,"meta":29,"style":29},"能容忍两个磁盘同时故障\n可靠性最高\n写开销比 RAID 5 更大\n至少需要 N + 2 个磁盘\n",[31,236146,236147,236152,236157,236162],{"__ignoreMap":29},[34,236148,236149],{"class":36,"line":37},[34,236150,236151],{"class":190},"能容忍两个磁盘同时故障\n",[34,236153,236154],{"class":36,"line":43},[34,236155,236156],{"class":190},"可靠性最高\n",[34,236158,236159],{"class":36,"line":49},[34,236160,236161],{"class":190},"写开销比 RAID 5 更大\n",[34,236163,236164],{"class":36,"line":56},[34,236165,236166],{"class":190},"至少需要 N + 2 个磁盘\n",[19,236168,236170],{"id":236169},"_9-raid-级别速记表","9. RAID 级别速记表",[13837,236172,236173,236192],{},[13840,236174,236175],{},[13843,236176,236177,236180,236183,236186,236189],{},[13846,236178,236179],{},"RAID",[13846,236181,236182],{},"核心",[13846,236184,236185],{},"冗余",[13846,236187,236188],{},"性能",[13846,236190,236191],{},"可靠性",[13853,236193,236194,236210,236227,236244,236261,236276,236292],{},[13843,236195,236196,236199,236202,236204,236207],{},[13858,236197,236198],{},"RAID 0",[13858,236200,236201],{},"条带化",[13858,236203,109153],{},[13858,236205,236206],{},"很高",[13858,236208,236209],{},"低",[13843,236211,236212,236215,236218,236221,236224],{},[13858,236213,236214],{},"RAID 1",[13858,236216,236217],{},"镜像",[13858,236219,236220],{},"完整复制",[13858,236222,236223],{},"读好，写一般",[13858,236225,236226],{},"高",[13843,236228,236229,236232,236235,236238,236241],{},[13858,236230,236231],{},"RAID 2",[13858,236233,236234],{},"Hamming Code",[13858,236236,236237],{},"多校验盘",[13858,236239,236240],{},"并行访问",[13858,236242,236243],{},"高但少用",[13843,236245,236246,236249,236252,236255,236258],{},[13858,236247,236248],{},"RAID 3",[13858,236250,236251],{},"小条带 + 专用 parity",[13858,236253,236254],{},"1 个校验盘",[13858,236256,236257],{},"大块传输好",[13858,236259,236260],{},"可容错 1 盘",[13843,236262,236263,236266,236269,236271,236274],{},[13858,236264,236265],{},"RAID 4",[13858,236267,236268],{},"块条带 + 专用 parity",[13858,236270,236254],{},[13858,236272,236273],{},"读好，写瓶颈",[13858,236275,236260],{},[13843,236277,236278,236281,236284,236287,236290],{},[13858,236279,236280],{},"RAID 5",[13858,236282,236283],{},"块条带 + 分布 parity",[13858,236285,236286],{},"分布式校验",[13858,236288,236289],{},"综合较好",[13858,236291,236260],{},[13843,236293,236294,236297,236300,236303,236306],{},[13858,236295,236296],{},"RAID 6",[13858,236298,236299],{},"双 parity",[13858,236301,236302],{},"双重校验",[13858,236304,236305],{},"写较慢",[13858,236307,236308],{},"可容错 2 盘",[32188,236310],{},[11,236312,236314],{"id":236313},"_117-disk-cache","11.7 Disk Cache",[19,236316,236318],{"id":236317},"_1-disk-cache-是什么","1. Disk Cache 是什么？",[15,236320,236321,236322,347],{},"磁盘缓存是主存中的一块缓冲区，用来保存磁盘扇区的副本。请求某个扇区时，先查 cache：命中则直接从内存取，未命中再读磁盘。它利用的是",[2627,236323,236324],{},"局部性原理",[24,236326,236328],{"className":30504,"code":236327,"language":85,"meta":29,"style":29},"请求磁盘块\n→ 查 Disk Cache\n→ 命中：从内存返回\n→ 未命中：从磁盘读入 cache，再返回\n",[31,236329,236330,236335,236340,236345],{"__ignoreMap":29},[34,236331,236332],{"class":36,"line":37},[34,236333,236334],{"class":190},"请求磁盘块\n",[34,236336,236337],{"class":36,"line":43},[34,236338,236339],{"class":190},"→ 查 Disk Cache\n",[34,236341,236342],{"class":36,"line":49},[34,236343,236344],{"class":190},"→ 命中：从内存返回\n",[34,236346,236347],{"class":36,"line":56},[34,236348,236349],{"class":190},"→ 未命中：从磁盘读入 cache，再返回\n",[19,236351,236353],{"id":236352},"_2-disk-cache-与-buffering-的区别","2. Disk Cache 与 Buffering 的区别",[13837,236355,236356,236369],{},[13840,236357,236358],{},[13843,236359,236360,236363,236366],{},[13846,236361,236362],{},"项目",[13846,236364,236365],{},"Buffering",[13846,236367,236368],{},"Disk Cache",[13853,236370,236371,236382,236392,236402],{},[13843,236372,236373,236376,236379],{},[13858,236374,236375],{},"目的",[13858,236377,236378],{},"平滑速度差，协调传输",[13858,236380,236381],{},"减少重复磁盘访问",[13843,236383,236384,236386,236389],{},[13858,236385,90875],{},[13858,236387,236388],{},"当前 I\u002FO 数据",[13858,236390,236391],{},"可能未来会再次访问的磁盘块",[13843,236393,236394,236396,236399],{},[13858,236395,37443],{},[13858,236397,236398],{},"CPU 与设备并行",[13858,236400,236401],{},"时间局部性、空间局部性",[13843,236403,236404,236407,236410],{},[13858,236405,236406],{},"典型策略",[13858,236408,236409],{},"单缓冲、双缓冲、循环缓冲",[13858,236411,236412],{},"LRU、LFU、Frequency-based",[19,236414,236416],{"id":236415},"_3-cache-数据交付方式","3. Cache 数据交付方式",[15,236418,236419],{},"当 I\u002FO 请求在 disk cache 中命中时，数据可以：",[24,236421,236423],{"className":30504,"code":236422,"language":85,"meta":29,"style":29},"1. 从 cache 复制到用户进程空间\n2. 通过共享内存传递 cache 中位置的指针\n",[31,236424,236425,236432],{"__ignoreMap":29},[34,236426,236427,236429],{"class":36,"line":37},[34,236428,233680],{"class":1326},[34,236430,236431],{"class":190}," 从 cache 复制到用户进程空间\n",[34,236433,236434,236436],{"class":36,"line":43},[34,236435,233688],{"class":1326},[34,236437,236438],{"class":190}," 通过共享内存传递 cache 中位置的指针\n",[15,236440,236441],{},"第二种方式避免内存到内存复制，效率更高。",[19,236443,236445],{"id":236444},"_4-lru","4. LRU",[15,236447,236448],{},"LRU，Least Recently Used，最近最少使用。\n规则：替换最长时间没有被访问的块。教材和课件都说它是常用的替换策略。",[15,236450,235555],{},[24,236452,236454],{"className":30504,"code":236453,"language":85,"meta":29,"style":29},"最近用过 → 更可能马上再用\n很久没用 → 更适合淘汰\n",[31,236455,236456,236461],{"__ignoreMap":29},[34,236457,236458],{"class":36,"line":37},[34,236459,236460],{"class":190},"最近用过 → 更可能马上再用\n",[34,236462,236463],{"class":36,"line":43},[34,236464,236465],{"class":190},"很久没用 → 更适合淘汰\n",[19,236467,236469],{"id":236468},"_5-lfu","5. LFU",[15,236471,236472],{},"LFU，Least Frequently Used，最不经常使用。\n规则：替换访问次数最少的块。每个块有计数器，每访问一次计数加 1。",[15,236474,11832],{},[24,236476,236478],{"className":30504,"code":236477,"language":85,"meta":29,"style":29},"某些块曾经短时间高频访问，计数很高；\n但之后可能不再使用；\nLFU 仍可能保留它们，导致替换效果差。\n",[31,236479,236480,236485,236490],{"__ignoreMap":29},[34,236481,236482],{"class":36,"line":37},[34,236483,236484],{"class":190},"某些块曾经短时间高频访问，计数很高；\n",[34,236486,236487],{"class":36,"line":43},[34,236488,236489],{"class":190},"但之后可能不再使用；\n",[34,236491,236492],{"class":36,"line":49},[34,236493,236494],{"class":190},"LFU 仍可能保留它们，导致替换效果差。\n",[19,236496,236498],{"id":236497},"_6-frequency-based-replacement","6. Frequency-based Replacement",[15,236500,236501],{},"为解决 LFU 的问题，教材介绍了基于频率的改进策略：把 cache 栈分成 new、middle、old 等区域，只有 old 区块才适合替换，这样能避免刚进入 cache 的块因为计数低而马上被淘汰。",[19,236503,236505],{"id":236504},"_7-写回问题","7. 写回问题",[15,236507,236508],{},"如果缓存块只是被读过，替换时不必写回磁盘。\n如果缓存块被修改过，它就是 dirty block，替换前必须写回磁盘。",[15,236510,236511],{},"为了减少寻道，系统通常会把多个 dirty block 聚集起来，按较优顺序写回。",[32188,236513],{},[11,236515,236517],{"id":236516},"_118-unix-svr4-io","11.8 UNIX SVR4 I\u002FO",[19,236519,236521],{"id":236520},"_1-unix-的统一接口思想","1. UNIX 的统一接口思想",[15,236523,236524],{},"UNIX 把每个 I\u002FO 设备都关联为一个特殊文件。进程通过读写特殊文件来读写设备。这体现了 UNIX 的经典思想：",[24,236526,236528],{"className":30504,"code":236527,"language":85,"meta":29,"style":29},"Everything is a file.\n",[31,236529,236530],{"__ignoreMap":29},[34,236531,236532],{"class":36,"line":37},[34,236533,236527],{"class":190},[15,236535,236536],{},"这样用户进程看到的是统一接口，而底层设备差异由内核隐藏。",[19,236538,236540],{"id":236539},"_2-unix-的两类-io","2. UNIX 的两类 I\u002FO",[13837,236542,236543,236551],{},[13840,236544,236545],{},[13843,236546,236547,236549],{},[13846,236548,114061],{},[13846,236550,126615],{},[13853,236552,236553,236561],{},[13843,236554,236555,236558],{},[13858,236556,236557],{},"Buffered I\u002FO",[13858,236559,236560],{},"经过系统缓冲区",[13843,236562,236563,236566],{},[13858,236564,236565],{},"Unbuffered I\u002FO",[13858,236567,236568],{},"通常通过 DMA，直接在设备和进程 I\u002FO 区之间传输",[15,236570,236571],{},"Buffered I\u002FO 主要使用：",[24,236573,236575],{"className":30504,"code":236574,"language":85,"meta":29,"style":29},"Buffer cache\nCharacter queue\n",[31,236576,236577,236582],{"__ignoreMap":29},[34,236578,236579],{"class":36,"line":37},[34,236580,236581],{"class":190},"Buffer cache\n",[34,236583,236584],{"class":36,"line":43},[34,236585,236586],{"class":190},"Character queue\n",[19,236588,236590],{"id":236589},"_3-buffer-cache","3. Buffer Cache",[15,236592,236593],{},"UNIX buffer cache 本质上是磁盘缓存。它用来缓存磁盘块。管理时维护三类列表：free list、device list、driver I\u002FO queue；查找时使用按 device 和 block 组织的哈希链表，替换策略使用 LRU。",[19,236595,236597],{"id":236596},"_4-character-queue","4. Character Queue",[15,236599,236600],{},"字符设备如终端、打印机适合 character queue。字符队列体现生产者—消费者模型：设备写入、进程读取，或进程写入、设备读取。字符被读走后就“消耗掉”，不像 buffer cache 可以多次读取。",[19,236602,236604],{"id":236603},"_5-unbuffered-io","5. Unbuffered I\u002FO",[15,236606,236607],{},"Unbuffered I\u002FO 是设备和进程空间之间通过 DMA 直接传输，速度快，但进程必须锁定在内存中，不能被换出，会降低系统整体调度和内存管理灵活性。",[32188,236609],{},[11,236611,236613],{"id":236612},"_119-linux-io","11.9 Linux I\u002FO",[15,236615,236616],{},"Linux I\u002FO 与 UNIX 类似，也区分块设备和字符设备。教材重点讲 Linux 磁盘调度和 page cache。",[19,236618,236620],{"id":236619},"_1-linux-elevator-scheduler","1. Linux Elevator Scheduler",[15,236622,236623],{},"Linux 2.4 默认磁盘调度器是 Linux Elevator，类似 LOOK 算法。它维护一个请求队列，并按块号排序，同时进行请求合并。",[15,236625,236626],{},"插入新请求时大致考虑：",[24,236628,236630],{"className":30504,"code":236629,"language":85,"meta":29,"style":29},"1. 能否和已有相邻请求合并\n2. 是否有旧请求需要避免饥饿\n3. 能否按块号有序插入\n4. 否则放到队尾\n",[31,236631,236632,236639,236646,236653],{"__ignoreMap":29},[34,236633,236634,236636],{"class":36,"line":37},[34,236635,233680],{"class":1326},[34,236637,236638],{"class":190}," 能否和已有相邻请求合并\n",[34,236640,236641,236643],{"class":36,"line":43},[34,236642,233688],{"class":1326},[34,236644,236645],{"class":190}," 是否有旧请求需要避免饥饿\n",[34,236647,236648,236650],{"class":36,"line":49},[34,236649,233696],{"class":1326},[34,236651,236652],{"class":190}," 能否按块号有序插入\n",[34,236654,236655,236657],{"class":36,"line":56},[34,236656,233704],{"class":1326},[34,236658,236659],{"class":190}," 否则放到队尾\n",[19,236661,236663],{"id":236662},"_2-deadline-scheduler","2. Deadline Scheduler",[15,236665,236666],{},"Deadline 调度器是为了解决 Elevator 的饥饿问题和读写不公平问题。它使用 sorted queue 和 FIFO queue，读请求和写请求分别有过期时间；读请求默认过期时间比写请求短，因为读通常会阻塞进程。",[15,236668,236669],{},"重点：",[24,236671,236673],{"className":30504,"code":236672,"language":85,"meta":29,"style":29},"读请求更影响进程响应时间\n写请求通常可以异步延后\nDeadline 通过过期时间保证请求不会被无限推迟\n",[31,236674,236675,236680,236685],{"__ignoreMap":29},[34,236676,236677],{"class":36,"line":37},[34,236678,236679],{"class":190},"读请求更影响进程响应时间\n",[34,236681,236682],{"class":36,"line":43},[34,236683,236684],{"class":190},"写请求通常可以异步延后\n",[34,236686,236687],{"class":36,"line":49},[34,236688,236689],{"class":190},"Deadline 通过过期时间保证请求不会被无限推迟\n",[19,236691,236693],{"id":236692},"_3-anticipatory-scheduler","3. Anticipatory Scheduler",[15,236695,236696],{},"预期调度器的思想：读请求完成后，稍等很短时间，看同一进程是否马上发出附近位置的新读请求。因为局部性原理，新请求很可能就在附近。教材提到 Linux 2.6 中它叠加在 Deadline Scheduler 上，读请求后最多等待约 6ms。",[19,236698,236700],{"id":236699},"_4-noop-scheduler","4. NOOP Scheduler",[15,236702,236703],{},"NOOP 是最简单的调度器：",[24,236705,236707],{"className":30504,"code":236706,"language":85,"meta":29,"style":29},"FIFO 队列\n只做请求合并\n适合本身已有调度能力的设备或非传统磁盘设备\n",[31,236708,236709,236714,236719],{"__ignoreMap":29},[34,236710,236711],{"class":36,"line":37},[34,236712,236713],{"class":190},"FIFO 队列\n",[34,236715,236716],{"class":36,"line":43},[34,236717,236718],{"class":190},"只做请求合并\n",[34,236720,236721],{"class":36,"line":49},[34,236722,236723],{"class":190},"适合本身已有调度能力的设备或非传统磁盘设备\n",[19,236725,236727],{"id":236726},"_5-cfq-scheduler","5. CFQ Scheduler",[15,236729,236730],{},"CFQ，Completely Fair Queuing，完全公平队列。\n它为每个进程维护 I\u002FO 队列，并给每个队列时间片，以 round-robin 的方式分配磁盘 I\u002FO 带宽，目标是公平。",[19,236732,236734],{"id":236733},"_6-linux-page-cache","6. Linux Page Cache",[15,236736,236737],{},"Linux 2.4 以后使用统一 page cache，所有磁盘与主存之间的数据流都经过它。好处：dirty pages 可以排序后批量写回；由于时间局部性，缓存页可能再次被访问，从而减少磁盘 I\u002FO。",[15,236739,236740],{},"dirty pages 写回场景：",[24,236742,236744],{"className":30504,"code":236743,"language":85,"meta":29,"style":29},"空闲内存低于阈值\n脏页年龄超过阈值\n",[31,236745,236746,236751],{"__ignoreMap":29},[34,236747,236748],{"class":36,"line":37},[34,236749,236750],{"class":190},"空闲内存低于阈值\n",[34,236752,236753],{"class":36,"line":43},[34,236754,236755],{"class":190},"脏页年龄超过阈值\n",[32188,236757],{},[11,236759,236761],{"id":236760},"_1110-windows-io","11.10 Windows I\u002FO",[19,236763,236765],{"id":236764},"_1-windows-io-manager","1. Windows I\u002FO Manager",[15,236767,236768],{},"Windows 的 I\u002FO Manager 负责操作系统的所有 I\u002FO，并给各种驱动提供统一接口。教材列出的 Chapter 11 Windows I\u002FO 重点包括 Basic I\u002FO Facilities、Asynchronous and Synchronous I\u002FO、Software RAID、Volume Shadow Copies、Volume Encryption。",[19,236770,236772],{"id":236771},"_2-windows-io-的核心组件","2. Windows I\u002FO 的核心组件",[15,236774,236775],{},"教材中 Windows I\u002FO Manager 主要配合四类组件：",[24,236777,236779],{"className":30504,"code":236778,"language":85,"meta":29,"style":29},"Cache Manager：文件缓存，lazy writer 批量写回\nFile System Drivers：文件系统驱动\nNetwork Drivers：网络驱动和远程文件系统支持\nHardware Device Drivers：硬件设备驱动，经 HAL 访问硬件寄存器\n",[31,236780,236781,236786,236791,236796],{"__ignoreMap":29},[34,236782,236783],{"class":36,"line":37},[34,236784,236785],{"class":190},"Cache Manager：文件缓存，lazy writer 批量写回\n",[34,236787,236788],{"class":36,"line":43},[34,236789,236790],{"class":190},"File System Drivers：文件系统驱动\n",[34,236792,236793],{"class":36,"line":49},[34,236794,236795],{"class":190},"Network Drivers：网络驱动和远程文件系统支持\n",[34,236797,236798],{"class":36,"line":56},[34,236799,236800],{"class":190},"Hardware Device Drivers：硬件设备驱动，经 HAL 访问硬件寄存器\n",[19,236802,236804],{"id":236803},"_3-同步-io-与异步-io","3. 同步 I\u002FO 与异步 I\u002FO",[13837,236806,236807,236815],{},[13840,236808,236809],{},[13843,236810,236811,236813],{},[13846,236812,114061],{},[13846,236814,126615],{},[13853,236816,236817,236825],{},[13843,236818,236819,236822],{},[13858,236820,236821],{},"Synchronous I\u002FO",[13858,236823,236824],{},"发出 I\u002FO 后线程阻塞，直到完成",[13843,236826,236827,236830],{},[13858,236828,236829],{},"Asynchronous I\u002FO",[13858,236831,236832],{},"发出 I\u002FO 后线程继续执行，完成后通过某种机制通知",[15,236834,236835],{},"异步 I\u002FO 更适合提高应用性能，但需要完成通知机制。",[15,236837,236838],{},"Windows 常见完成通知方式：",[24,236840,236842],{"className":30504,"code":236841,"language":85,"meta":29,"style":29},"File object signaling\nEvent object signaling\nAsynchronous Procedure Call, APC\nI\u002FO Completion Port\nPolling\n",[31,236843,236844,236849,236854,236859,236864],{"__ignoreMap":29},[34,236845,236846],{"class":36,"line":37},[34,236847,236848],{"class":190},"File object signaling\n",[34,236850,236851],{"class":36,"line":43},[34,236852,236853],{"class":190},"Event object signaling\n",[34,236855,236856],{"class":36,"line":49},[34,236857,236858],{"class":190},"Asynchronous Procedure Call, APC\n",[34,236860,236861],{"class":36,"line":56},[34,236862,236863],{"class":190},"I\u002FO Completion Port\n",[34,236865,236866],{"class":36,"line":62},[34,236867,236868],{"class":190},"Polling\n",[19,236870,236872],{"id":236871},"_4-windows-software-raid","4. Windows Software RAID",[15,236874,236875],{},"Windows 支持硬件 RAID 和软件 RAID。软件 RAID 由 OS 中的容错磁盘驱动实现，Windows Server 可实现 RAID 1 和 RAID 5。",[19,236877,236879],{"id":236878},"_5-volume-shadow-copies-与-volume-encryption","5. Volume Shadow Copies 与 Volume Encryption",[13837,236881,236882,236891],{},[13840,236883,236884],{},[13843,236885,236886,236889],{},[13846,236887,236888],{},"功能",[13846,236890,153527],{},[13853,236892,236893,236901],{},[13843,236894,236895,236898],{},[13858,236896,236897],{},"Volume Shadow Copies",[13858,236899,236900],{},"创建卷的一致性快照，用于备份和恢复旧版本文件",[13843,236902,236903,236906],{},[13858,236904,236905],{},"Volume Encryption",[13858,236907,236908],{},"使用 BitLocker 对整个卷加密，比只加密单个文件更完整",[32188,236910],{},[11,236912,236914],{"id":236913},"课件补充spooling-与虚拟设备","课件补充：SPOOLING 与虚拟设备",[19,236916,236918],{"id":236917},"_1-什么是-spooling","1. 什么是 SPOOLING？",[15,236920,236921],{},"SPOOLING 又叫假脱机技术。它的思想是：用磁盘上的缓冲区模拟外围设备，把独占设备虚拟成可共享设备。课件中把用于缓冲的存储区域称为“井”。",[15,236923,236924],{},"典型例子：",[24,236926,236928],{"className":30504,"code":236927,"language":85,"meta":29,"style":29},"打印机 SPOOLING\n网络通信 SPOOLING\n",[31,236929,236930,236935],{"__ignoreMap":29},[34,236931,236932],{"class":36,"line":37},[34,236933,236934],{"class":190},"打印机 SPOOLING\n",[34,236936,236937],{"class":36,"line":43},[34,236938,236939],{"class":190},"网络通信 SPOOLING\n",[19,236941,236943],{"id":236942},"_2-spooling-的组成","2. SPOOLING 的组成",[15,236945,236946],{},"课件列出：",[24,236948,236950],{"className":30504,"code":236949,"language":85,"meta":29,"style":29},"输入井\n输出井\n预输入程序\n缓输出程序\n井管理程序\n",[31,236951,236952,236957,236962,236967,236972],{"__ignoreMap":29},[34,236953,236954],{"class":36,"line":37},[34,236955,236956],{"class":190},"输入井\n",[34,236958,236959],{"class":36,"line":43},[34,236960,236961],{"class":190},"输出井\n",[34,236963,236964],{"class":36,"line":49},[34,236965,236966],{"class":190},"预输入程序\n",[34,236968,236969],{"class":36,"line":56},[34,236970,236971],{"class":190},"缓输出程序\n",[34,236973,236974],{"class":36,"line":62},[34,236975,236976],{"class":190},"井管理程序\n",[15,236978,236979],{},"工作理解：",[24,236981,236983],{"className":30504,"code":236982,"language":85,"meta":29,"style":29},"用户作业输入 → 先进入输入井\n作业运行 → 从输入井取数据\n作业输出 → 先写入输出井\n输出设备空闲 → 缓输出程序慢慢输出\n",[31,236984,236985,236990,236995,237000],{"__ignoreMap":29},[34,236986,236987],{"class":36,"line":37},[34,236988,236989],{"class":190},"用户作业输入 → 先进入输入井\n",[34,236991,236992],{"class":36,"line":43},[34,236993,236994],{"class":190},"作业运行 → 从输入井取数据\n",[34,236996,236997],{"class":36,"line":49},[34,236998,236999],{"class":190},"作业输出 → 先写入输出井\n",[34,237001,237002],{"class":36,"line":56},[34,237003,237004],{"class":190},"输出设备空闲 → 缓输出程序慢慢输出\n",[19,237006,237008],{"id":237007},"_3-spooling-的意义","3. SPOOLING 的意义",[24,237010,237012],{"className":30504,"code":237011,"language":85,"meta":29,"style":29},"提高独占设备利用率\n减少人工干预\n实现虚拟设备\n让多个作业看起来可以同时使用独占设备\n",[31,237013,237014,237019,237024,237029],{"__ignoreMap":29},[34,237015,237016],{"class":36,"line":37},[34,237017,237018],{"class":190},"提高独占设备利用率\n",[34,237020,237021],{"class":36,"line":43},[34,237022,237023],{"class":190},"减少人工干预\n",[34,237025,237026],{"class":36,"line":49},[34,237027,237028],{"class":190},"实现虚拟设备\n",[34,237030,237031],{"class":36,"line":56},[34,237032,237033],{"class":190},"让多个作业看起来可以同时使用独占设备\n",[15,237035,237036],{},"最经典例子：多个进程都“打印”，其实只是把输出放入输出井，真正的打印机按队列慢慢打印。",[32188,237038],{},[11,237040,237041],{"id":237041},"重点概念速记",[24,237043,237045],{"className":30504,"code":237044,"language":85,"meta":29,"style":29},"I\u002FO 系统：\nI\u002FO 设备 + 接口线路 + 控制部件 + 通道 + 管理软件\n\n字符设备：\n按字节\u002F字符流传输，如键盘、终端、打印机\n\n块设备：\n按块传输，可随机访问，如磁盘、USB\n\nProgrammed I\u002FO：\nCPU 发命令后忙等\n\nInterrupt-driven I\u002FO：\n设备完成后中断 CPU\n\nDMA：\n设备与内存直接传输数据块，CPU 只管开始和结束\n\nI\u002FO Channel：\n能执行 I\u002FO 程序的专用 I\u002FO 处理结构\n\nBuffering：\n平滑 CPU 和设备速度差\n\nDisk Cache：\n用内存缓存磁盘块，减少磁盘访问\n\nSeek Time：\n磁头移动到目标磁道时间\n\nRotational Delay：\n目标扇区转到磁头下方的时间\n\nTransfer Time：\n真正读写数据的时间\n\nFIFO：\n按到达顺序服务\n\nSSTF：\n每次选最近磁道\n\nSCAN：\n电梯算法，单方向扫，到端点或无请求后反向\n\nC-SCAN：\n循环扫描，只单向服务，回程不服务\n\nRAID 0：\n条带化，无冗余，高性能低可靠\n\nRAID 1：\n镜像，高可靠，成本高\n\nRAID 5：\n分布式 parity，常用\n\nRAID 6：\n双 parity，可坏两个盘\n",[31,237046,237047,237052,237057,237061,237066,237071,237075,237080,237085,237089,237094,237099,237103,237108,237113,237117,237122,237127,237131,237136,237141,237145,237150,237155,237159,237164,237169,237173,237178,237183,237187,237192,237197,237201,237206,237211,237215,237220,237225,237229,237234,237239,237243,237248,237253,237257,237262,237267,237271,237276,237281,237285,237290,237295,237299,237304,237309,237313,237318],{"__ignoreMap":29},[34,237048,237049],{"class":36,"line":37},[34,237050,237051],{"class":190},"I\u002FO 系统：\n",[34,237053,237054],{"class":36,"line":43},[34,237055,237056],{"class":190},"I\u002FO 设备 + 接口线路 + 控制部件 + 通道 + 管理软件\n",[34,237058,237059],{"class":36,"line":49},[34,237060,53],{"emptyLinePlaceholder":52},[34,237062,237063],{"class":36,"line":56},[34,237064,237065],{"class":190},"字符设备：\n",[34,237067,237068],{"class":36,"line":62},[34,237069,237070],{"class":190},"按字节\u002F字符流传输，如键盘、终端、打印机\n",[34,237072,237073],{"class":36,"line":68},[34,237074,53],{"emptyLinePlaceholder":52},[34,237076,237077],{"class":36,"line":74},[34,237078,237079],{"class":190},"块设备：\n",[34,237081,237082],{"class":36,"line":1971},[34,237083,237084],{"class":190},"按块传输，可随机访问，如磁盘、USB\n",[34,237086,237087],{"class":36,"line":1976},[34,237088,53],{"emptyLinePlaceholder":52},[34,237090,237091],{"class":36,"line":1985},[34,237092,237093],{"class":190},"Programmed I\u002FO：\n",[34,237095,237096],{"class":36,"line":1990},[34,237097,237098],{"class":190},"CPU 发命令后忙等\n",[34,237100,237101],{"class":36,"line":1999},[34,237102,53],{"emptyLinePlaceholder":52},[34,237104,237105],{"class":36,"line":2022},[34,237106,237107],{"class":190},"Interrupt-driven I\u002FO：\n",[34,237109,237110],{"class":36,"line":2036},[34,237111,237112],{"class":190},"设备完成后中断 CPU\n",[34,237114,237115],{"class":36,"line":2603},[34,237116,53],{"emptyLinePlaceholder":52},[34,237118,237119],{"class":36,"line":5537},[34,237120,237121],{"class":190},"DMA：\n",[34,237123,237124],{"class":36,"line":5543},[34,237125,237126],{"class":190},"设备与内存直接传输数据块，CPU 只管开始和结束\n",[34,237128,237129],{"class":36,"line":5548},[34,237130,53],{"emptyLinePlaceholder":52},[34,237132,237133],{"class":36,"line":5564},[34,237134,237135],{"class":190},"I\u002FO Channel：\n",[34,237137,237138],{"class":36,"line":5576},[34,237139,237140],{"class":190},"能执行 I\u002FO 程序的专用 I\u002FO 处理结构\n",[34,237142,237143],{"class":36,"line":5606},[34,237144,53],{"emptyLinePlaceholder":52},[34,237146,237147],{"class":36,"line":5616},[34,237148,237149],{"class":190},"Buffering：\n",[34,237151,237152],{"class":36,"line":5628},[34,237153,237154],{"class":190},"平滑 CPU 和设备速度差\n",[34,237156,237157],{"class":36,"line":5634},[34,237158,53],{"emptyLinePlaceholder":52},[34,237160,237161],{"class":36,"line":6420},[34,237162,237163],{"class":190},"Disk Cache：\n",[34,237165,237166],{"class":36,"line":11294},[34,237167,237168],{"class":190},"用内存缓存磁盘块，减少磁盘访问\n",[34,237170,237171],{"class":36,"line":11321},[34,237172,53],{"emptyLinePlaceholder":52},[34,237174,237175],{"class":36,"line":11326},[34,237176,237177],{"class":190},"Seek Time：\n",[34,237179,237180],{"class":36,"line":11333},[34,237181,237182],{"class":190},"磁头移动到目标磁道时间\n",[34,237184,237185],{"class":36,"line":11344},[34,237186,53],{"emptyLinePlaceholder":52},[34,237188,237189],{"class":36,"line":11349},[34,237190,237191],{"class":190},"Rotational Delay：\n",[34,237193,237194],{"class":36,"line":11354},[34,237195,237196],{"class":190},"目标扇区转到磁头下方的时间\n",[34,237198,237199],{"class":36,"line":11363},[34,237200,53],{"emptyLinePlaceholder":52},[34,237202,237203],{"class":36,"line":11368},[34,237204,237205],{"class":190},"Transfer Time：\n",[34,237207,237208],{"class":36,"line":11377},[34,237209,237210],{"class":190},"真正读写数据的时间\n",[34,237212,237213],{"class":36,"line":11402},[34,237214,53],{"emptyLinePlaceholder":52},[34,237216,237217],{"class":36,"line":11407},[34,237218,237219],{"class":190},"FIFO：\n",[34,237221,237222],{"class":36,"line":11421},[34,237223,237224],{"class":190},"按到达顺序服务\n",[34,237226,237227],{"class":36,"line":11426},[34,237228,53],{"emptyLinePlaceholder":52},[34,237230,237231],{"class":36,"line":11447},[34,237232,237233],{"class":190},"SSTF：\n",[34,237235,237236],{"class":36,"line":11452},[34,237237,237238],{"class":190},"每次选最近磁道\n",[34,237240,237241],{"class":36,"line":11463},[34,237242,53],{"emptyLinePlaceholder":52},[34,237244,237245],{"class":36,"line":11485},[34,237246,237247],{"class":190},"SCAN：\n",[34,237249,237250],{"class":36,"line":11495},[34,237251,237252],{"class":190},"电梯算法，单方向扫，到端点或无请求后反向\n",[34,237254,237255],{"class":36,"line":11528},[34,237256,53],{"emptyLinePlaceholder":52},[34,237258,237259],{"class":36,"line":11534},[34,237260,237261],{"class":190},"C-SCAN：\n",[34,237263,237264],{"class":36,"line":11544},[34,237265,237266],{"class":190},"循环扫描，只单向服务，回程不服务\n",[34,237268,237269],{"class":36,"line":11553},[34,237270,53],{"emptyLinePlaceholder":52},[34,237272,237273],{"class":36,"line":11562},[34,237274,237275],{"class":190},"RAID 0：\n",[34,237277,237278],{"class":36,"line":16859},[34,237279,237280],{"class":190},"条带化，无冗余，高性能低可靠\n",[34,237282,237283],{"class":36,"line":16885},[34,237284,53],{"emptyLinePlaceholder":52},[34,237286,237287],{"class":36,"line":16890},[34,237288,237289],{"class":190},"RAID 1：\n",[34,237291,237292],{"class":36,"line":16911},[34,237293,237294],{"class":190},"镜像，高可靠，成本高\n",[34,237296,237297],{"class":36,"line":16932},[34,237298,53],{"emptyLinePlaceholder":52},[34,237300,237301],{"class":36,"line":16955},[34,237302,237303],{"class":190},"RAID 5：\n",[34,237305,237306],{"class":36,"line":16975},[34,237307,237308],{"class":190},"分布式 parity，常用\n",[34,237310,237311],{"class":36,"line":16996},[34,237312,53],{"emptyLinePlaceholder":52},[34,237314,237315],{"class":36,"line":17005},[34,237316,237317],{"class":190},"RAID 6：\n",[34,237319,237320],{"class":36,"line":17010},[34,237321,237322],{"class":190},"双 parity，可坏两个盘\n",[32188,237324],{},[11,237326,237327],{"id":237327},"考试高频点",[19,237329,237331],{"id":237330},"_1-简答题高频","1. 简答题高频",[24,237333,237335],{"className":30504,"code":237334,"language":85,"meta":29,"style":29},"1. I\u002FO 设备分为哪三类？\n2. I\u002FO 设备有哪些差异？\n3. Programmed I\u002FO、Interrupt-driven I\u002FO、DMA 区别？\n4. DMA 的工作过程和优点？\n5. 通道方式和 DMA 的区别？\n6. I\u002FO 设计目标为什么是效率和通用性？\n7. Logical I\u002FO、Device I\u002FO、Scheduling and Control 分别做什么？\n8. 单缓冲、双缓冲、循环缓冲区别？\n9. 磁盘访问时间由哪些部分组成？\n10. FIFO、SSTF、SCAN、C-SCAN 区别？\n11. N-step-SCAN 和 FSCAN 解决什么问题？\n12. RAID 0、1、5、6 区别？\n13. Disk Cache 和 Buffering 区别？\n14. UNIX 为什么把设备看作特殊文件？\n15. SPOOLING 如何把独占设备虚拟成共享设备？\n16. 什么是设备独立性？\n",[31,237336,237337,237344,237351,237358,237365,237372,237379,237387,237395,237403,237411,237419,237427,237435,237443,237451],{"__ignoreMap":29},[34,237338,237339,237341],{"class":36,"line":37},[34,237340,233680],{"class":1326},[34,237342,237343],{"class":190}," I\u002FO 设备分为哪三类？\n",[34,237345,237346,237348],{"class":36,"line":43},[34,237347,233688],{"class":1326},[34,237349,237350],{"class":190}," I\u002FO 设备有哪些差异？\n",[34,237352,237353,237355],{"class":36,"line":49},[34,237354,233696],{"class":1326},[34,237356,237357],{"class":190}," Programmed I\u002FO、Interrupt-driven I\u002FO、DMA 区别？\n",[34,237359,237360,237362],{"class":36,"line":56},[34,237361,233704],{"class":1326},[34,237363,237364],{"class":190}," DMA 的工作过程和优点？\n",[34,237366,237367,237369],{"class":36,"line":62},[34,237368,233712],{"class":1326},[34,237370,237371],{"class":190}," 通道方式和 DMA 的区别？\n",[34,237373,237374,237376],{"class":36,"line":68},[34,237375,233720],{"class":1326},[34,237377,237378],{"class":190}," I\u002FO 设计目标为什么是效率和通用性？\n",[34,237380,237381,237384],{"class":36,"line":74},[34,237382,237383],{"class":1326},"7.",[34,237385,237386],{"class":190}," Logical I\u002FO、Device I\u002FO、Scheduling and Control 分别做什么？\n",[34,237388,237389,237392],{"class":36,"line":1971},[34,237390,237391],{"class":1326},"8.",[34,237393,237394],{"class":190}," 单缓冲、双缓冲、循环缓冲区别？\n",[34,237396,237397,237400],{"class":36,"line":1976},[34,237398,237399],{"class":1326},"9.",[34,237401,237402],{"class":190}," 磁盘访问时间由哪些部分组成？\n",[34,237404,237405,237408],{"class":36,"line":1985},[34,237406,237407],{"class":1326},"10.",[34,237409,237410],{"class":190}," FIFO、SSTF、SCAN、C-SCAN 区别？\n",[34,237412,237413,237416],{"class":36,"line":1990},[34,237414,237415],{"class":1326},"11.",[34,237417,237418],{"class":190}," N-step-SCAN 和 FSCAN 解决什么问题？\n",[34,237420,237421,237424],{"class":36,"line":1999},[34,237422,237423],{"class":1326},"12.",[34,237425,237426],{"class":190}," RAID 0、1、5、6 区别？\n",[34,237428,237429,237432],{"class":36,"line":2022},[34,237430,237431],{"class":1326},"13.",[34,237433,237434],{"class":190}," Disk Cache 和 Buffering 区别？\n",[34,237436,237437,237440],{"class":36,"line":2036},[34,237438,237439],{"class":1326},"14.",[34,237441,237442],{"class":190}," UNIX 为什么把设备看作特殊文件？\n",[34,237444,237445,237448],{"class":36,"line":2603},[34,237446,237447],{"class":1326},"15.",[34,237449,237450],{"class":190}," SPOOLING 如何把独占设备虚拟成共享设备？\n",[34,237452,237453,237456],{"class":36,"line":5537},[34,237454,237455],{"class":1326},"16.",[34,237457,237458],{"class":190}," 什么是设备独立性？\n",[19,237460,237462],{"id":237461},"_2-计算题高频","2. 计算题高频",[24,237464,237466],{"className":30504,"code":237465,"language":85,"meta":29,"style":29},"磁盘调度：\n给初始磁头位置、方向、请求序列，求 FIFO \u002F SSTF \u002F SCAN \u002F C-SCAN 总移动磁道数。\n\n旋转优化：\n给转速、扇区数、处理时间，求顺序存放读取时间和优化分布读取时间。\n\n缓冲时间：\n无缓冲 T + C\n单缓冲 max(C, T) + M\n双缓冲根据 C 和 T 判断接近 T 或 C + M\n\n磁盘访问时间：\nTa = Ts + 1\u002F(2r) + b\u002F(rN)\n",[31,237467,237468,237473,237478,237482,237487,237492,237496,237501,237506,237511,237516,237520,237525],{"__ignoreMap":29},[34,237469,237470],{"class":36,"line":37},[34,237471,237472],{"class":190},"磁盘调度：\n",[34,237474,237475],{"class":36,"line":43},[34,237476,237477],{"class":190},"给初始磁头位置、方向、请求序列，求 FIFO \u002F SSTF \u002F SCAN \u002F C-SCAN 总移动磁道数。\n",[34,237479,237480],{"class":36,"line":49},[34,237481,53],{"emptyLinePlaceholder":52},[34,237483,237484],{"class":36,"line":56},[34,237485,237486],{"class":190},"旋转优化：\n",[34,237488,237489],{"class":36,"line":62},[34,237490,237491],{"class":190},"给转速、扇区数、处理时间，求顺序存放读取时间和优化分布读取时间。\n",[34,237493,237494],{"class":36,"line":68},[34,237495,53],{"emptyLinePlaceholder":52},[34,237497,237498],{"class":36,"line":74},[34,237499,237500],{"class":190},"缓冲时间：\n",[34,237502,237503],{"class":36,"line":1971},[34,237504,237505],{"class":190},"无缓冲 T + C\n",[34,237507,237508],{"class":36,"line":1976},[34,237509,237510],{"class":190},"单缓冲 max(C, T) + M\n",[34,237512,237513],{"class":36,"line":1985},[34,237514,237515],{"class":190},"双缓冲根据 C 和 T 判断接近 T 或 C + M\n",[34,237517,237518],{"class":36,"line":1990},[34,237519,53],{"emptyLinePlaceholder":52},[34,237521,237522],{"class":36,"line":1999},[34,237523,237524],{"class":190},"磁盘访问时间：\n",[34,237526,237527],{"class":36,"line":2022},[34,237528,237529],{"class":190},"Ta = Ts + 1\u002F(2r) + b\u002F(rN)\n",[19,237531,237533],{"id":237532},"_3-最容易混淆的点","3. 最容易混淆的点",[13837,237535,237536,237546],{},[13840,237537,237538],{},[13843,237539,237540,237543],{},[13846,237541,237542],{},"易混点",[13846,237544,237545],{},"正确区分",[13853,237547,237548,237556,237564,237572,237580,237588,237596,237604],{},[13843,237549,237550,237553],{},[13858,237551,237552],{},"Buffer vs Cache",[13858,237554,237555],{},"Buffer 为了解决速度差；Cache 为了减少重复访问",[13843,237557,237558,237561],{},[13858,237559,237560],{},"Interrupt vs DMA",[13858,237562,237563],{},"中断仍可能频繁让 CPU 介入；DMA 传一整块后才中断",[13843,237565,237566,237569],{},[13858,237567,237568],{},"DMA vs Channel",[13858,237570,237571],{},"DMA 负责数据块搬运；Channel 可执行 I\u002FO 程序",[13843,237573,237574,237577],{},[13858,237575,237576],{},"SCAN vs LOOK",[13858,237578,237579],{},"SCAN 通常到端点；LOOK 到该方向最后一个请求",[13843,237581,237582,237585],{},[13858,237583,237584],{},"SCAN vs C-SCAN",[13858,237586,237587],{},"SCAN 双向服务；C-SCAN 单向服务，回程不服务",[13843,237589,237590,237593],{},[13858,237591,237592],{},"RAID 0 vs RAID 1",[13858,237594,237595],{},"RAID 0 高性能无冗余；RAID 1 镜像高可靠",[13843,237597,237598,237601],{},[13858,237599,237600],{},"RAID 4 vs RAID 5",[13858,237602,237603],{},"RAID 4 专用 parity 盘；RAID 5 parity 分布到各盘",[13843,237605,237606,237609],{},[13858,237607,237608],{},"字符设备 vs 块设备",[13858,237610,237611],{},"字符设备按流；块设备按块，可随机访问",[32188,237613],{},[11,237615,237616],{"id":237616},"一句话总复习",[15,237618,237619],{},[2627,237620,237621],{},"Chapter 11 的核心是：操作系统通过抽象、分层、缓冲、中断、DMA、通道、调度、RAID 和缓存，把复杂且慢速的 I\u002FO 设备管理成统一、高效、可靠的系统资源。",[79,237623,237624],{},"html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html .default .shiki span {color: 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.s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}",{"title":29,"searchDepth":43,"depth":43,"links":237626},[237627,237628,237629,237630,237631,237632,237633,237634,237635,237636,237637,237638,237639,237640,237641,237642,237643,237644,237645,237646,237647,237648,237649,237650,237651,237652,237653,237654,237655,237656,237657,237658,237659,237660,237661,237665,237666,237667,237668,237669,237670,237671,237672,237673,237674,237675,237676,237677,237678,237679,237680,237681,237682,237683,237684,237685,237686,237687,237688,237689,237690,237691,237692,237693,237694,237695,237696,237697,237698,237699,237700,237701,237702,237703],{"id":233118,"depth":43,"text":233119},{"id":233189,"depth":43,"text":233190},{"id":233244,"depth":43,"text":233245},{"id":233313,"depth":43,"text":233314},{"id":233330,"depth":43,"text":233331},{"id":233383,"depth":43,"text":233384},{"id":233429,"depth":43,"text":233430},{"id":233496,"depth":43,"text":233497},{"id":233537,"depth":43,"text":233538},{"id":233585,"depth":43,"text":233586},{"id":233621,"depth":43,"text":233622},{"id":233632,"depth":43,"text":233633},{"id":233666,"depth":43,"text":233667},{"id":233738,"depth":43,"text":233739},{"id":233787,"depth":43,"text":233788},{"id":233857,"depth":43,"text":233858},{"id":233904,"depth":43,"text":233905},{"id":233980,"depth":43,"text":233981},{"id":234086,"depth":43,"text":234087},{"id":234122,"depth":43,"text":234123},{"id":234162,"depth":43,"text":234163},{"id":234189,"depth":43,"text":234190},{"id":234385,"depth":43,"text":234386},{"id":234421,"depth":43,"text":234422},{"id":234535,"depth":43,"text":234536},{"id":234565,"depth":43,"text":234566},{"id":234578,"depth":43,"text":234579},{"id":234585,"depth":43,"text":234586},{"id":235230,"depth":43,"text":235231},{"id":235252,"depth":43,"text":235253},{"id":235432,"depth":43,"text":235433},{"id":235467,"depth":43,"text":235468},{"id":235504,"depth":43,"text":235505},{"id":235548,"depth":43,"text":235549},{"id":235573,"depth":43,"text":235574,"children":237662},[237663,237664],{"id":235586,"depth":49,"text":235400},{"id":235619,"depth":49,"text":235417},{"id":235647,"depth":43,"text":235648},{"id":235856,"depth":43,"text":235857},{"id":235899,"depth":43,"text":235900},{"id":235936,"depth":43,"text":235937},{"id":235987,"depth":43,"text":235988},{"id":236019,"depth":43,"text":236020},{"id":236059,"depth":43,"text":236060},{"id":236093,"depth":43,"text":236094},{"id":236135,"depth":43,"text":236136},{"id":236169,"depth":43,"text":236170},{"id":236317,"depth":43,"text":236318},{"id":236352,"depth":43,"text":236353},{"id":236415,"depth":43,"text":236416},{"id":236444,"depth":43,"text":236445},{"id":236468,"depth":43,"text":236469},{"id":236497,"depth":43,"text":236498},{"id":236504,"depth":43,"text":236505},{"id":236520,"depth":43,"text":236521},{"id":236539,"depth":43,"text":236540},{"id":236589,"depth":43,"text":236590},{"id":236596,"depth":43,"text":236597},{"id":236603,"depth":43,"text":236604},{"id":236619,"depth":43,"text":236620},{"id":236662,"depth":43,"text":236663},{"id":236692,"depth":43,"text":236693},{"id":236699,"depth":43,"text":236700},{"id":236726,"depth":43,"text":236727},{"id":236733,"depth":43,"text":236734},{"id":236764,"depth":43,"text":236765},{"id":236771,"depth":43,"text":236772},{"id":236803,"depth":43,"text":236804},{"id":236871,"depth":43,"text":236872},{"id":236878,"depth":43,"text":236879},{"id":236917,"depth":43,"text":236918},{"id":236942,"depth":43,"text":236943},{"id":237007,"depth":43,"text":237008},{"id":237330,"depth":43,"text":237331},{"id":237461,"depth":43,"text":237462},{"id":237532,"depth":43,"text":237533},{"order":1990},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap11",{"title":233109,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap11","VNKpSZ44gjsjb-EF9W4R8JumdHWgu7kHFwRJSnwTOhk",{"id":237710,"title":237711,"body":237712,"description":29,"extension":85,"meta":240786,"navigation":52,"path":240787,"seo":240788,"stem":240789,"__hash__":240790},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap12.md","Chap. 12 Note",{"type":8,"value":237713,"toc":240679},[237714,237718,237722,237725,237736,237739,237750,237753,237773,237776,237779,237804,237807,237821,237824,237835,237838,237864,237868,237876,237880,237883,237886,237909,237912,237932,237935,237938,237995,237998,238021,238025,238028,238036,238039,238056,238059,238067,238070,238087,238091,238093,238110,238113,238121,238123,238137,238139,238147,238151,238153,238179,238181,238192,238194,238205,238207,238218,238221,238235,238239,238241,238252,238255,238266,238269,238280,238282,238293,238296,238307,238309,238320,238324,238326,238337,238340,238358,238360,238374,238376,238387,238390,238398,238402,238404,238418,238420,238431,238433,238441,238443,238457,238461,238465,238479,238483,238509,238513,238516,238536,238540,238560,238564,238592,238596,238610,238614,238617,238628,238631,238634,238645,238648,238662,238665,238676,238679,238705,238708,238725,238728,238731,238745,238748,238765,238768,238785,238788,238796,238800,238811,238815,238829,238833,238836,238844,238847,238850,238916,238919,238927,238930,238941,238944,238961,238965,238968,238979,238982,238985,238996,238999,239010,239013,239017,239034,239038,239052,239054,239062,239064,239075,239079,239089,239091,239099,239101,239109,239112,239123,239127,239130,239138,239141,239152,239156,239167,239169,239177,239179,239187,239191,239202,239206,239214,239217,239220,239234,239237,239251,239254,239280,239283,239287,239289,239300,239302,239313,239315,239326,239330,239332,239345,239347,239358,239360,239374,239378,239380,239391,239393,239407,239409,239420,239424,239427,239431,239454,239457,239462,239466,239477,239479,239487,239489,239500,239504,239518,239522,239547,239551,239565,239569,239572,239580,239583,239600,239603,239614,239618,239622,239684,239688,239702,239705,239740,239744,239760,239763,239800,239802,239813,239817,239828,239832,239835,239869,239873,239877,239891,239895,239915,239919,239953,239957,239960,239989,239992,240009,240013,240027,240031,240048,240052,240089,240093,240119,240123,240127,240135,240139,240206,240210,240242,240246,240257,240261,240264,240298,240302,240322,240326,240329,240395,240399,240402,240436,240439,240453,240457,240461,240469,240473,240484,240488,240521,240525,240539,240543,240554,240558,240575,240579,240582,240602,240605],[11,237715,237717],{"id":237716},"chapter-12-file-management","Chapter 12 File Management",[19,237719,237721],{"id":237720},"_121-overview文件与文件管理系统","12.1 Overview：文件与文件管理系统",[253,237723,237724],{"id":237724},"文件系统的核心作用",[105,237726,237727,237730,237733],{},[108,237728,237729],{},"文件系统是操作系统中用户最直接接触的部分之一。",[108,237731,237732],{},"文件用于长期保存输入、输出和程序运行结果。",[108,237734,237735],{},"文件具有独立于进程的生命周期：进程结束后文件仍可存在。",[253,237737,237738],{"id":237738},"文件的三个重要性质",[105,237740,237741,237744,237747],{},[108,237742,237743],{},"长期存在：文件存储在磁盘等二级存储器上，不随用户注销或进程结束而消失。",[108,237745,237746],{},"可共享：文件有名字和访问权限，可被多个进程或用户受控共享。",[108,237748,237749],{},"有结构：文件内部可以有记录、字段等结构；文件之间也可通过目录形成层次结构。",[253,237751,237752],{"id":237752},"文件系统常见操作",[105,237754,237755,237758,237761,237764,237767,237770],{},[108,237756,237757],{},"Create：创建文件，并放入文件结构中。",[108,237759,237760],{},"Delete：删除文件。",[108,237762,237763],{},"Open：进程打开文件，之后才能读写。",[108,237765,237766],{},"Close：关闭文件，进程不再访问它。",[108,237768,237769],{},"Read：读取文件的全部或部分内容。",[108,237771,237772],{},"Write：写入或修改文件内容，可扩展文件大小。",[253,237774,237775],{"id":237775},"文件属性",[15,237777,237778],{},"文件系统通常维护以下属性：",[105,237780,237781,237784,237787,237790,237793,237795,237798,237801],{},[108,237782,237783],{},"文件所有者",[108,237785,237786],{},"创建时间",[108,237788,237789],{},"最近修改时间",[108,237791,237792],{},"最近访问时间",[108,237794,182589],{},[108,237796,237797],{},"文件类型",[108,237799,237800],{},"访问权限",[108,237802,237803],{},"存储位置",[253,237805,237806],{"id":237806},"数据层次结构",[105,237808,237809,237812,237815,237818],{},[108,237810,237811],{},"Field 字段：最基本的数据元素，只包含一个值，如姓名、日期、传感器读数。",[108,237813,237814],{},"Record 记录：一组相关字段，可被应用程序作为一个单位处理。",[108,237816,237817],{},"File 文件：一组相似记录的集合，用户和程序按名字引用。",[108,237819,237820],{},"Database 数据库：相关数据的集合，强调数据之间关系明确，并可被多个应用共享。",[253,237822,237823],{"id":237823},"文件与数据库的区别",[105,237825,237826,237829,237832],{},[108,237827,237828],{},"文件通常是一组相似记录的集合。",[108,237830,237831],{},"数据库不仅包含数据，还显式维护数据之间的关系。",[108,237833,237834],{},"数据库通常由独立的 DBMS 管理，但底层仍依赖操作系统文件管理功能。",[253,237836,237837],{"id":237837},"文件可支持的典型记录级操作",[105,237839,237840,237843,237846,237849,237852,237855,237858,237861],{},[108,237841,237842],{},"顺序检索所有记录。",[108,237844,237845],{},"检索单条记录。",[108,237847,237848],{},"检索下一条记录。",[108,237850,237851],{},"检索上一条记录。",[108,237853,237854],{},"插入新记录。",[108,237856,237857],{},"删除记录。",[108,237859,237860],{},"更新记录。",[108,237862,237863],{},"按条件检索多条记录。",[253,237865,237867],{"id":237866},"unix-类系统的特殊点","UNIX 类系统的特殊点",[105,237869,237870,237873],{},[108,237871,237872],{},"UNIX 内核把普通文件看作字节流。",[108,237874,237875],{},"字段、记录等逻辑结构由应用程序自己解释，不由操作系统强制规定。",[253,237877,237879],{"id":237878},"文件管理系统-file-management-system","文件管理系统 File Management System",[15,237881,237882],{},"文件管理系统是一组系统软件，为用户和应用程序提供文件服务。",[15,237884,237885],{},"主要目标：",[105,237887,237888,237891,237894,237897,237900,237903,237906],{},[108,237889,237890],{},"满足用户的数据管理需求。",[108,237892,237893],{},"尽可能保证文件数据有效。",[108,237895,237896],{},"优化系统吞吐量和用户响应时间。",[108,237898,237899],{},"支持多种存储设备。",[108,237901,237902],{},"减少或避免数据丢失。",[108,237904,237905],{},"为用户进程提供标准 I\u002FO 接口。",[108,237907,237908],{},"在多用户系统中支持多用户文件访问。",[253,237910,237911],{"id":237911},"文件管理系统的基本用户需求",[105,237913,237914,237917,237920,237923,237926,237929],{},[108,237915,237916],{},"用户能创建、删除、读、写、修改文件。",[108,237918,237919],{},"用户可受控访问其他用户文件。",[108,237921,237922],{},"用户可控制别人对自己文件的访问类型。",[108,237924,237925],{},"用户可在文件之间移动数据。",[108,237927,237928],{},"用户可备份和恢复文件。",[108,237930,237931],{},"用户可通过文件名而非数字编号访问文件。",[253,237933,237934],{"id":237934},"文件系统软件层次结构",[15,237936,237937],{},"从低到高：",[105,237939,237940,237951,237962,237973,237984],{},[108,237941,237942,237943],{},"Device Drivers 设备驱动：\n",[105,237944,237945,237948],{},[108,237946,237947],{},"直接与设备、控制器或通道通信。",[108,237949,237950],{},"启动 I\u002FO 操作并处理中断或完成事件。",[108,237952,237953,237954],{},"Basic File System \u002F Physical I\u002FO 基本文件系统：\n",[105,237955,237956,237959],{},[108,237957,237958],{},"处理磁盘或磁带上的数据块。",[108,237960,237961],{},"关心块的位置和内存缓冲，不理解文件内容。",[108,237963,237964,237965],{},"Basic I\u002FO Supervisor 基本 I\u002FO 管理器：\n",[105,237966,237967,237970],{},[108,237968,237969],{},"负责文件 I\u002FO 的启动和结束。",[108,237971,237972],{},"维护设备 I\u002FO、调度、文件状态、缓冲区和空间分配。",[108,237974,237975,237976],{},"Logical I\u002FO 逻辑 I\u002FO：\n",[105,237977,237978,237981],{},[108,237979,237980],{},"面向记录而不是物理块。",[108,237982,237983],{},"维护文件的基本数据。",[108,237985,237986,237987],{},"Access Method 访问方法：\n",[105,237988,237989,237992],{},[108,237990,237991],{},"最接近用户和应用。",[108,237993,237994],{},"为不同文件结构提供统一接口，如顺序访问、索引访问、散列访问。",[253,237996,237997],{"id":237997},"文件管理主要功能",[105,237999,238000,238003,238006,238009,238012,238015,238018],{},[108,238001,238002],{},"目录管理：通过目录定位文件并维护属性。",[108,238004,238005],{},"访问控制：限制哪些用户可以怎样访问文件。",[108,238007,238008],{},"记录处理：把用户看到的记录映射到文件操作。",[108,238010,238011],{},"块管理：把逻辑记录组织为物理块。",[108,238013,238014],{},"二级存储管理：分配和回收磁盘块。",[108,238016,238017],{},"I\u002FO 调度：提升磁盘访问性能。",[108,238019,238020],{},"空闲空间管理：记录哪些磁盘块尚未使用。",[19,238022,238024],{"id":238023},"_122-file-organization-and-access文件组织与访问","12.2 File Organization and Access：文件组织与访问",[253,238026,238027],{"id":238027},"文件组织的含义",[105,238029,238030,238033],{},[108,238031,238032],{},"文件组织指记录的逻辑结构，取决于记录的访问方式。",[108,238034,238035],{},"物理组织还受到记录阻塞和文件分配策略影响。",[253,238037,238038],{"id":238038},"选择文件组织的重要标准",[105,238040,238041,238044,238047,238050,238053],{},[108,238042,238043],{},"访问时间短。",[108,238045,238046],{},"易于更新。",[108,238048,238049],{},"节省存储空间。",[108,238051,238052],{},"维护简单。",[108,238054,238055],{},"可靠性高。",[15,238057,238058],{},"这些标准可能冲突：",[105,238060,238061,238064],{},[108,238062,238063],{},"索引能加快访问，但会增加冗余和空间开销。",[108,238065,238066],{},"顺序结构简单省空间，但随机访问慢。",[253,238068,238069],{"id":238069},"五种基本文件组织",[105,238071,238072,238075,238078,238081,238084],{},[108,238073,238074],{},"Pile 堆文件。",[108,238076,238077],{},"Sequential File 顺序文件。",[108,238079,238080],{},"Indexed Sequential File 索引顺序文件。",[108,238082,238083],{},"Indexed File 索引文件。",[108,238085,238086],{},"Direct \u002F Hashed File 直接文件或散列文件。",[253,238088,238090],{"id":238089},"pile-堆文件","Pile 堆文件",[15,238092,4160],{},[105,238094,238095,238098,238101,238104,238107],{},[108,238096,238097],{},"最简单的文件组织。",[108,238099,238100],{},"数据按到达顺序收集。",[108,238102,238103],{},"每条记录可以字段不同、顺序不同。",[108,238105,238106],{},"字段通常需要自描述，包括字段名和值。",[108,238108,238109],{},"字段长度可通过分隔符、长度子字段或默认规则表示。",[15,238111,238112],{},"访问方式：",[105,238114,238115,238118],{},[108,238116,238117],{},"没有结构，只能穷尽搜索。",[108,238119,238120],{},"查找某字段或某值时，可能必须扫描整个文件。",[15,238122,14543],{},[105,238124,238125,238128,238131,238134],{},[108,238126,238127],{},"易于插入和更新。",[108,238129,238130],{},"适合收集原始数据。",[108,238132,238133],{},"适合数据大小和结构变化很大的场景。",[108,238135,238136],{},"空间利用较好。",[15,238138,177826],{},[105,238140,238141,238144],{},[108,238142,238143],{},"查找效率差。",[108,238145,238146],{},"不适合大多数需要快速检索的应用。",[253,238148,238150],{"id":238149},"sequential-file-顺序文件","Sequential File 顺序文件",[15,238152,4160],{},[105,238154,238155,238158,238161,238164,238167,238170,238173,238176],{},[108,238156,238157],{},"最常见的文件结构。",[108,238159,238160],{},"记录格式固定。",[108,238162,238163],{},"所有记录长度相同。",[108,238165,238166],{},"字段数量、字段顺序和字段长度固定。",[108,238168,238169],{},"只需存储字段值，字段名和字段长度作为文件结构属性保存。",[108,238171,238172],{},"通常有一个 key field 关键字段。",[108,238174,238175],{},"关键字段唯一标识记录。",[108,238177,238178],{},"记录按关键字段顺序存储。",[15,238180,234542],{},[105,238182,238183,238186,238189],{},[108,238184,238185],{},"批处理应用。",[108,238187,238188],{},"每次处理全部或大部分记录，如工资、账单。",[108,238190,238191],{},"唯一容易同时存储在磁带和磁盘上的组织方式。",[15,238193,14543],{},[105,238195,238196,238199,238202],{},[108,238197,238198],{},"结构简单。",[108,238200,238201],{},"顺序处理效率高。",[108,238203,238204],{},"适合全文件扫描。",[15,238206,177826],{},[105,238208,238209,238212,238215],{},[108,238210,238211],{},"单条记录查询慢，需要顺序搜索。",[108,238213,238214],{},"插入新记录困难。",[108,238216,238217],{},"常用 log file \u002F transaction file 暂存新记录，再周期性批量合并。",[15,238219,238220],{},"链表式顺序文件：",[105,238222,238223,238226,238229,238232],{},[108,238224,238225],{},"每个物理块保存一个或多个记录。",[108,238227,238228],{},"块中包含指向下一块的指针。",[108,238230,238231],{},"插入时只需修改指针。",[108,238233,238234],{},"代价是额外指针开销和处理开销。",[253,238236,238238],{"id":238237},"indexed-sequential-file-索引顺序文件","Indexed Sequential File 索引顺序文件",[15,238240,148881],{},[105,238242,238243,238246,238249],{},[108,238244,238245],{},"保留顺序文件按关键字排序的特点。",[108,238247,238248],{},"增加索引以支持随机访问。",[108,238250,238251],{},"增加溢出文件 overflow file 处理插入。",[15,238253,238254],{},"结构：",[105,238256,238257,238260,238263],{},[108,238258,238259],{},"主文件按关键字段顺序排列。",[108,238261,238262],{},"索引文件包含关键字段和指向主文件的指针。",[108,238264,238265],{},"溢出文件保存新插入但尚未合并进主文件的记录。",[15,238267,238268],{},"查找过程：",[105,238270,238271,238274,238277],{},[108,238272,238273],{},"先查索引，找到小于等于目标键的最大索引项。",[108,238275,238276],{},"根据指针进入主文件附近位置。",[108,238278,238279],{},"再顺序搜索少量记录。",[15,238281,14543],{},[105,238283,238284,238287,238290],{},[108,238285,238286],{},"比普通顺序文件大幅减少平均查找时间。",[108,238288,238289],{},"仍保留顺序处理能力。",[108,238291,238292],{},"可用多级索引进一步提升效率。",[15,238294,238295],{},"插入处理：",[105,238297,238298,238301,238304],{},[108,238299,238300],{},"新记录先放入溢出文件。",[108,238302,238303],{},"前驱记录维护指向溢出记录的指针。",[108,238305,238306],{},"定期把主文件和溢出文件批量合并。",[15,238308,177826],{},[105,238310,238311,238314,238317],{},[108,238312,238313],{},"溢出文件过大时性能下降。",[108,238315,238316],{},"需要定期重组。",[108,238318,238319],{},"主要仍围绕一个关键字段组织。",[253,238321,238323],{"id":238322},"indexed-file-索引文件","Indexed File 索引文件",[15,238325,148881],{},[105,238327,238328,238331,238334],{},[108,238329,238330],{},"放弃单一关键字段和顺序性。",[108,238332,238333],{},"记录通过多个索引访问。",[108,238335,238336],{},"每个可能被搜索的字段都可有索引。",[15,238338,238339],{},"索引类型：",[105,238341,238342,238350],{},[108,238343,238344,238345],{},"Exhaustive Index 穷尽索引：\n",[105,238346,238347],{},[108,238348,238349],{},"每条记录都有一个索引项。",[108,238351,238352,238353],{},"Partial Index 部分索引：\n",[105,238354,238355],{},[108,238356,238357],{},"只为包含某字段的记录建立索引。",[15,238359,14543],{},[105,238361,238362,238365,238368,238371],{},[108,238363,238364],{},"可按多个属性高效查询。",[108,238366,238367],{},"记录位置不受限制。",[108,238369,238370],{},"支持变长记录。",[108,238372,238373],{},"适合查询频繁、实时性强的应用。",[15,238375,177826],{},[105,238377,238378,238381,238384],{},[108,238379,238380],{},"新增记录时必须更新相关索引。",[108,238382,238383],{},"索引维护成本高。",[108,238385,238386],{},"全文件顺序处理不如顺序文件自然。",[15,238388,238389],{},"典型应用：",[105,238391,238392,238395],{},[108,238393,238394],{},"航空订票系统。",[108,238396,238397],{},"库存控制系统。",[253,238399,238401],{"id":238400},"direct-hashed-file-直接文件或散列文件","Direct \u002F Hashed File 直接文件或散列文件",[15,238403,4160],{},[105,238405,238406,238409,238412,238415],{},[108,238407,238408],{},"利用磁盘可直接访问已知地址块的能力。",[108,238410,238411],{},"每条记录仍需要关键字段。",[108,238413,238414],{},"不强调记录顺序。",[108,238416,238417],{},"对关键值应用散列函数，得到记录存储位置。",[15,238419,14543],{},[105,238421,238422,238425,238428],{},[108,238423,238424],{},"单条记录访问非常快。",[108,238426,238427],{},"适合固定长度记录。",[108,238429,238430],{},"适合每次只访问一条记录的场景。",[15,238432,177826],{},[105,238434,238435,238438],{},[108,238436,238437],{},"不适合顺序处理。",[108,238439,238440],{},"需要处理散列冲突，通常配合溢出区。",[15,238442,238389],{},[105,238444,238445,238448,238451,238454],{},[108,238446,238447],{},"目录。",[108,238449,238450],{},"价格表。",[108,238452,238453],{},"日程表。",[108,238455,238456],{},"名单。",[19,238458,238460],{"id":238459},"_123-b-treesb-树","12.3 B-Trees：B 树",[253,238462,238464],{"id":238463},"为什么需要-b-树","为什么需要 B 树",[105,238466,238467,238470,238473,238476],{},[108,238468,238469],{},"大文件或数据库中，单级顺序索引仍然太慢。",[108,238471,238472],{},"多级索引可提高访问效率。",[108,238474,238475],{},"普通树若不平衡，会出现长短不一的搜索路径。",[108,238477,238478],{},"B 树是一种平衡的多路搜索树，适合作为大型索引结构。",[253,238480,238482],{"id":238481},"b-树特点","B 树特点",[105,238484,238485,238488,238491,238494,238497,238500,238503,238506],{},[108,238486,238487],{},"是树结构，没有闭环。",[108,238489,238490],{},"由节点和叶子组成。",[108,238492,238493],{},"每个节点包含若干关键字和指向子节点的指针。",[108,238495,238496],{},"节点内关键字按非降序排列。",[108,238498,238499],{},"每个节点最多有相同数量的关键字。",[108,238501,238502],{},"所有叶子在同一层。",[108,238504,238505],{},"树保持平衡，搜索路径长度稳定。",[108,238507,238508],{},"分支因子大，因此树通常很矮。",[253,238510,238512],{"id":238511},"b-树的度-d","B 树的度 d",[15,238514,238515],{},"若 B 树最小度为 d：",[105,238517,238518,238521,238524,238527,238530,238533],{},[108,238519,238520],{},"每个节点最多有 2d - 1 个关键字。",[108,238522,238523],{},"每个节点最多有 2d 个子指针。",[108,238525,238526],{},"除根节点外，每个内部节点至少有 d - 1 个关键字。",[108,238528,238529],{},"除根节点外，每个内部节点至少有 d 个子指针。",[108,238531,238532],{},"根节点至少有 1 个关键字和 2 个子节点。",[108,238534,238535],{},"有 k 个指针的非叶节点包含 k - 1 个关键字。",[253,238537,238539],{"id":238538},"b-树查找","B 树查找",[105,238541,238542,238545,238548,238551,238554,238557],{},[108,238543,238544],{},"从根节点开始。",[108,238546,238547],{},"若关键字在当前节点中，查找成功。",[108,238549,238550],{},"若目标小于当前节点最小关键字，走最左指针。",[108,238552,238553],{},"若目标大于当前节点最大关键字，走最右指针。",[108,238555,238556],{},"若目标位于两个相邻关键字之间，走中间指针。",[108,238558,238559],{},"重复直到找到或到达叶子。",[253,238561,238563],{"id":238562},"b-树插入","B 树插入",[105,238565,238566,238569,238572,238586,238589],{},[108,238567,238568],{},"先按查找过程找到应插入的最低层节点。",[108,238570,238571],{},"若该节点未满，直接按顺序插入。",[108,238573,238574,238575],{},"若节点已满：\n",[105,238576,238577,238580,238583],{},[108,238578,238579],{},"围绕中位关键字分裂节点。",[108,238581,238582],{},"中位关键字提升到父节点。",[108,238584,238585],{},"左右两边形成两个节点。",[108,238587,238588],{},"若父节点也满，继续向上分裂。",[108,238590,238591],{},"若根节点分裂，则产生新根，树高加 1。",[253,238593,238595],{"id":238594},"b-树优点","B 树优点",[105,238597,238598,238601,238604,238607],{},[108,238599,238600],{},"查找、插入、删除效率高。",[108,238602,238603],{},"树高低，磁盘访问次数少。",[108,238605,238606],{},"所有叶子同层，性能稳定。",[108,238608,238609],{},"广泛用于数据库和操作系统文件系统索引。",[19,238611,238613],{"id":238612},"_124-file-directories文件目录","12.4 File Directories：文件目录",[253,238615,238616],{"id":238616},"目录的作用",[105,238618,238619,238622,238625],{},[108,238620,238621],{},"目录保存文件信息，包括属性、位置、所有者等。",[108,238623,238624],{},"用户通过文件名引用文件，目录负责把文件名映射到实际文件。",[108,238626,238627],{},"目录本身也是一种文件，但通常只能通过系统例程间接访问。",[253,238629,238630],{"id":238630},"文件目录中的信息",[15,238632,238633],{},"Basic Information 基本信息：",[105,238635,238636,238639,238642],{},[108,238637,238638],{},"文件名。",[108,238640,238641],{},"文件类型。",[108,238643,238644],{},"文件组织方式。",[15,238646,238647],{},"Address Information 地址信息：",[105,238649,238650,238653,238656,238659],{},[108,238651,238652],{},"所在卷 volume。",[108,238654,238655],{},"起始地址。",[108,238657,238658],{},"已用大小。",[108,238660,238661],{},"已分配大小。",[15,238663,238664],{},"Access Control Information 访问控制信息：",[105,238666,238667,238670,238673],{},[108,238668,238669],{},"文件所有者。",[108,238671,238672],{},"授权用户信息。",[108,238674,238675],{},"允许的操作，如读、写、执行、网络传输。",[15,238677,238678],{},"Usage Information 使用信息：",[105,238680,238681,238684,238687,238690,238693,238696,238699,238702],{},[108,238682,238683],{},"创建日期。",[108,238685,238686],{},"创建者身份。",[108,238688,238689],{},"最近读取日期。",[108,238691,238692],{},"最近读取者。",[108,238694,238695],{},"最近修改日期。",[108,238697,238698],{},"最近修改者。",[108,238700,238701],{},"最近备份日期。",[108,238703,238704],{},"当前使用状态，如是否打开、是否锁定、是否已在内存修改但尚未写回磁盘。",[253,238706,238707],{"id":238707},"目录操作",[105,238709,238710,238713,238716,238719,238722],{},[108,238711,238712],{},"Search：搜索目录以定位文件。",[108,238714,238715],{},"Create file：创建文件时添加目录项。",[108,238717,238718],{},"Delete file：删除文件时移除目录项。",[108,238720,238721],{},"List directory：列出目录内容和属性。",[108,238723,238724],{},"Update directory：文件属性变化时更新目录项。",[253,238726,238727],{"id":238727},"目录结构",[32490,238729,238730],{"id":238730},"简单列表目录",[105,238732,238733,238736,238739,238742],{},[108,238734,238735],{},"每个文件一个目录项。",[108,238737,238738],{},"文件名作为关键字。",[108,238740,238741],{},"实现简单。",[108,238743,238744],{},"缺点是搜索慢、命名冲突严重、不适合多用户和大量文件。",[32490,238746,238747],{"id":238747},"二级目录",[105,238749,238750,238753,238756,238759,238762],{},[108,238751,238752],{},"有一个主目录。",[108,238754,238755],{},"每个用户有自己的用户目录。",[108,238757,238758],{},"文件名只需在单个用户目录内唯一。",[108,238760,238761],{},"更容易做访问控制。",[108,238763,238764],{},"缺点是仍不能很好支持用户按项目、类型等组织文件。",[32490,238766,238767],{"id":238767},"树形目录",[105,238769,238770,238773,238776,238779,238782],{},[108,238771,238772],{},"几乎所有现代系统采用。",[108,238774,238775],{},"主目录下有用户目录。",[108,238777,238778],{},"用户目录下可有子目录和文件。",[108,238780,238781],{},"任意目录都可包含文件或子目录。",[108,238783,238784],{},"支持层次化组织、权限隔离和更自然的命名。",[253,238786,238787],{"id":238787},"目录内部组织",[105,238789,238790,238793],{},[108,238791,238792],{},"小目录可用顺序文件。",[108,238794,238795],{},"大目录若条目很多，用散列结构可减少搜索时间。",[253,238797,238799],{"id":238798},"pathname-路径名","Pathname 路径名",[105,238801,238802,238805,238808],{},[108,238803,238804],{},"从根目录到目标文件的一系列目录名加文件名组成 pathname。",[108,238806,238807],{},"同名文件可以存在于不同目录中，只要路径名唯一。",[108,238809,238810],{},"路径名解决全系统唯一命名问题。",[253,238812,238814],{"id":238813},"working-directory-工作目录","Working Directory 工作目录",[105,238816,238817,238820,238823,238826],{},[108,238818,238819],{},"进程或交互用户当前所在目录。",[108,238821,238822],{},"文件可相对于工作目录引用。",[108,238824,238825],{},"登录或进程创建时，默认工作目录通常是用户主目录。",[108,238827,238828],{},"用户可通过切换目录改变工作目录。",[19,238830,238832],{"id":238831},"_125-file-sharing文件共享","12.5 File Sharing：文件共享",[253,238834,238835],{"id":238835},"文件共享中的两个核心问题",[105,238837,238838,238841],{},[108,238839,238840],{},"Access Rights 访问权限。",[108,238842,238843],{},"Simultaneous Access 同时访问控制。",[253,238845,238846],{"id":238846},"访问权限层次",[15,238848,238849],{},"常见权限从弱到强：",[105,238851,238852,238860,238868,238876,238884,238892,238900,238908],{},[108,238853,238854,238855],{},"None：\n",[105,238856,238857],{},[108,238858,238859],{},"用户甚至不知道文件存在。",[108,238861,238862,238863],{},"Knowledge：\n",[105,238864,238865],{},[108,238866,238867],{},"用户知道文件存在和所有者。",[108,238869,238870,238871],{},"Execution：\n",[105,238872,238873],{},[108,238874,238875],{},"用户可加载并执行程序，但不能复制。",[108,238877,238878,238879],{},"Reading：\n",[105,238880,238881],{},[108,238882,238883],{},"用户可读取文件，包括复制和执行。",[108,238885,238886,238887],{},"Appending：\n",[105,238888,238889],{},[108,238890,238891],{},"用户可在文件末尾追加数据，但不能修改或删除已有内容。",[108,238893,238894,238895],{},"Updating：\n",[105,238896,238897],{},[108,238898,238899],{},"用户可修改、删除、添加文件内容。",[108,238901,238902,238903],{},"Changing Protection：\n",[105,238904,238905],{},[108,238906,238907],{},"用户可改变其他用户的访问权限。",[108,238909,238910,238911],{},"Deletion：\n",[105,238912,238913],{},[108,238914,238915],{},"用户可删除文件。",[15,238917,238918],{},"权限通常构成层次：",[105,238920,238921,238924],{},[108,238922,238923],{},"较强权限隐含较弱权限。",[108,238925,238926],{},"例如 Updating 通常隐含 Knowledge、Execution、Reading、Appending。",[253,238928,238929],{"id":238929},"用户分类",[105,238931,238932,238935,238938],{},[108,238933,238934],{},"Specific user：指定用户。",[108,238936,238937],{},"User groups：用户组。",[108,238939,238940],{},"All：所有用户，即公共文件。",[253,238942,238943],{"id":238943},"同时访问",[105,238945,238946,238949,238952,238955,238958],{},[108,238947,238948],{},"多个用户拥有 append 或 update 权限时，系统必须控制并发。",[108,238950,238951],{},"简单方法：更新时锁定整个文件。",[108,238953,238954],{},"更细粒度方法：只锁定被更新的记录。",[108,238956,238957],{},"本质上类似 readers\u002Fwriters problem。",[108,238959,238960],{},"必须处理互斥和死锁问题。",[19,238962,238964],{"id":238963},"_126-record-blocking记录阻塞","12.6 Record Blocking：记录阻塞",[253,238966,238967],{"id":238967},"记录与块",[105,238969,238970,238973,238976],{},[108,238971,238972],{},"记录是结构化文件的逻辑访问单位。",[108,238974,238975],{},"块是二级存储器 I\u002FO 的基本单位。",[108,238977,238978],{},"文件 I\u002FO 必须把记录组织成块。",[253,238980,238981],{"id":238981},"块大小的权衡",[15,238983,238984],{},"较大块的优点：",[105,238986,238987,238990,238993],{},[108,238988,238989],{},"一次 I\u002FO 可传输更多记录。",[108,238991,238992],{},"顺序处理时减少 I\u002FO 次数。",[108,238994,238995],{},"可能提升整体传输效率。",[15,238997,238998],{},"较大块的缺点：",[105,239000,239001,239004,239007],{},[108,239002,239003],{},"随机访问时可能读入大量无用记录。",[108,239005,239006],{},"需要更大的 I\u002FO 缓冲区。",[108,239008,239009],{},"缓冲管理更复杂。",[253,239011,239012],{"id":239012},"三种记录阻塞方式",[32490,239014,239016],{"id":239015},"fixed-blocking-固定阻塞","Fixed Blocking 固定阻塞",[105,239018,239019,239022,239025,239028,239031],{},[108,239020,239021],{},"使用定长记录。",[108,239023,239024],{},"一个块中存放整数个记录。",[108,239026,239027],{},"块末尾可能有未用空间。",[108,239029,239030],{},"未用空间称为内部碎片。",[108,239032,239033],{},"常用于定长记录的顺序文件。",[32490,239035,239037],{"id":239036},"variable-length-spanned-blocking-变长跨块阻塞","Variable-Length Spanned Blocking 变长跨块阻塞",[105,239039,239040,239043,239046,239049],{},[108,239041,239042],{},"使用变长记录。",[108,239044,239045],{},"记录紧密打包进块中，几乎不浪费空间。",[108,239047,239048],{},"一个记录可以跨越两个块。",[108,239050,239051],{},"需要指针指向后续块。",[15,239053,14543],{},[105,239055,239056,239059],{},[108,239057,239058],{},"存储利用率高。",[108,239060,239061],{},"不限制记录大小。",[15,239063,177826],{},[105,239065,239066,239069,239072],{},[108,239067,239068],{},"实现复杂。",[108,239070,239071],{},"跨块记录读取可能需要两次 I\u002FO。",[108,239073,239074],{},"更新困难。",[32490,239076,239078],{"id":239077},"variable-length-unspanned-blocking-变长非跨块阻塞","Variable-Length Unspanned Blocking 变长非跨块阻塞",[105,239080,239081,239083,239086],{},[108,239082,239042],{},[108,239084,239085],{},"记录不能跨块。",[108,239087,239088],{},"若剩余空间不足以放下下一条记录，则浪费该空间。",[15,239090,14543],{},[105,239092,239093,239096],{},[108,239094,239095],{},"比跨块方式简单。",[108,239097,239098],{},"单条记录不会分散到多个块。",[15,239100,177826],{},[105,239102,239103,239106],{},[108,239104,239105],{},"会产生空间浪费。",[108,239107,239108],{},"记录大小不能超过块大小。",[253,239110,239111],{"id":239111},"与虚拟内存的关系",[105,239113,239114,239117,239120],{},[108,239115,239116],{},"虚拟内存系统中，页通常是基本传输单位。",[108,239118,239119],{},"页较小，不适合直接作为大文件 I\u002FO 的块。",[108,239121,239122],{},"一些系统会把多个页组合为一个较大的文件 I\u002FO 块。",[19,239124,239126],{"id":239125},"_127-secondary-storage-management二级存储管理","12.7 Secondary Storage Management：二级存储管理",[253,239128,239129],{"id":239129},"两个基本任务",[105,239131,239132,239135],{},[108,239133,239134],{},"为文件分配磁盘空间。",[108,239136,239137],{},"跟踪哪些磁盘空间仍可分配。",[253,239139,239140],{"id":239140},"文件分配需要考虑的问题",[105,239142,239143,239146,239149],{},[108,239144,239145],{},"创建文件时是否一次分配最大空间？",[108,239147,239148],{},"分配单位是一个块、多个连续块，还是整个文件？",[108,239150,239151],{},"用什么数据结构跟踪文件已分配的部分？",[253,239153,239155],{"id":239154},"preallocation-预分配","Preallocation 预分配",[105,239157,239158,239161,239164],{},[108,239159,239160],{},"创建文件时声明最大大小。",[108,239162,239163],{},"一次分配所需空间。",[108,239165,239166],{},"适合能预估文件大小的场景，如编译输出、文件传输。",[15,239168,14543],{},[105,239170,239171,239174],{},[108,239172,239173],{},"运行时分配开销小。",[108,239175,239176],{},"可提高连续性。",[15,239178,177826],{},[105,239180,239181,239184],{},[108,239182,239183],{},"很多应用无法准确预测最大文件大小。",[108,239185,239186],{},"用户容易高估大小，造成空间浪费。",[253,239188,239190],{"id":239189},"dynamic-allocation-动态分配","Dynamic Allocation 动态分配",[105,239192,239193,239196,239199],{},[108,239194,239195],{},"文件增长时按需分配空间。",[108,239197,239198],{},"更灵活，减少预留浪费。",[108,239200,239201],{},"但可能导致文件分散，影响性能。",[253,239203,239205],{"id":239204},"portion-分配片段","Portion 分配片段",[105,239207,239208,239211],{},[108,239209,239210],{},"portion 是连续的一组已分配块。",[108,239212,239213],{},"可以小到一个块，大到整个文件。",[253,239215,239216],{"id":239216},"分配片段大小的权衡",[15,239218,239219],{},"大连续片段：",[105,239221,239222,239225,239228,239231],{},[108,239223,239224],{},"顺序访问性能好。",[108,239226,239227],{},"分配表小。",[108,239229,239230],{},"空间重用困难。",[108,239232,239233],{},"可能产生外部碎片。",[15,239235,239236],{},"小固定块：",[105,239238,239239,239242,239245,239248],{},[108,239240,239241],{},"灵活。",[108,239243,239244],{},"易于重用。",[108,239246,239247],{},"分配表可能很大。",[108,239249,239250],{},"连续性差，可能影响顺序访问。",[253,239252,239253],{"id":239253},"空闲空间分配策略",[105,239255,239256,239264,239272],{},[108,239257,239258,239259],{},"First Fit：\n",[105,239260,239261],{},[108,239262,239263],{},"选择第一个足够大的空闲连续块组。",[108,239265,239266,239267],{},"Best Fit：\n",[105,239268,239269],{},[108,239270,239271],{},"选择满足要求的最小空闲块组。",[108,239273,239274,239275],{},"Nearest Fit：\n",[105,239276,239277],{},[108,239278,239279],{},"选择离该文件上次分配位置最近的空闲块组，提高局部性。",[253,239281,239282],{"id":239282},"三种文件分配方法",[32490,239284,239286],{"id":239285},"contiguous-allocation-连续分配","Contiguous Allocation 连续分配",[15,239288,4160],{},[105,239290,239291,239294,239297],{},[108,239292,239293],{},"文件创建时分配一整段连续块。",[108,239295,239296],{},"属于预分配。",[108,239298,239299],{},"文件分配表只需记录起始块和长度。",[15,239301,14543],{},[105,239303,239304,239307,239310],{},[108,239305,239306],{},"顺序访问性能最好。",[108,239308,239309],{},"随机访问也简单。",[108,239311,239312],{},"第 i 个块的位置为 b + i - 1。",[15,239314,177826],{},[105,239316,239317,239320,239323],{},[108,239318,239319],{},"必须预先知道文件大小。",[108,239321,239322],{},"会产生外部碎片。",[108,239324,239325],{},"可能需要磁盘紧缩 compaction。",[32490,239327,239329],{"id":239328},"chained-allocation-链式分配","Chained Allocation 链式分配",[15,239331,4160],{},[105,239333,239334,239337,239340,239342],{},[108,239335,239336],{},"通常按单个块分配。",[108,239338,239339],{},"每个块包含指向下一块的指针。",[108,239341,239299],{},[108,239343,239344],{},"可预分配，但更常见的是按需分配。",[15,239346,14543],{},[105,239348,239349,239352,239355],{},[108,239350,239351],{},"不需要连续空间。",[108,239353,239354],{},"没有外部碎片。",[108,239356,239357],{},"增长方便。",[15,239359,177826],{},[105,239361,239362,239365,239368,239371],{},[108,239363,239364],{},"随机访问差，必须沿链查找。",[108,239366,239367],{},"指针占用空间。",[108,239369,239370],{},"局部性差，顺序读取可能频繁移动磁头。",[108,239372,239373],{},"可通过周期性文件整理改善局部性。",[32490,239375,239377],{"id":239376},"indexed-allocation-索引分配","Indexed Allocation 索引分配",[15,239379,4160],{},[105,239381,239382,239385,239388],{},[108,239383,239384],{},"每个文件有一个索引。",[108,239386,239387],{},"索引中每项指向文件的一个块或一个片段。",[108,239389,239390],{},"文件分配表指向该文件的索引块。",[15,239392,14543],{},[105,239394,239395,239398,239401,239404],{},[108,239396,239397],{},"支持顺序访问和直接访问。",[108,239399,239400],{},"避免连续分配的外部碎片问题。",[108,239402,239403],{},"比链式分配随机访问更好。",[108,239405,239406],{},"是最流行的分配方式之一。",[15,239408,177826],{},[105,239410,239411,239414,239417],{},[108,239412,239413],{},"需要索引块。",[108,239415,239416],{},"小文件也有索引开销。",[108,239418,239419],{},"索引过大时需要多级索引。",[253,239421,239423],{"id":239422},"free-space-management-空闲空间管理","Free Space Management 空闲空间管理",[15,239425,239426],{},"为了分配文件块，系统必须知道哪些块空闲，因此需要 Disk Allocation Table。",[32490,239428,239430],{"id":239429},"bit-table-位表","Bit Table 位表",[105,239432,239433,239436,239439,239442,239445,239448,239451],{},[108,239434,239435],{},"每个磁盘块对应 1 位。",[108,239437,239438],{},"0 表示空闲，1 表示已使用。",[108,239440,239441],{},"空间开销最小。",[108,239443,239444],{},"易于查找连续空闲块。",[108,239446,239447],{},"若磁盘很大，位表也可能很大。",[108,239449,239450],{},"通常希望位表驻留内存，否则搜索成本高。",[108,239452,239453],{},"可用辅助摘要表记录每个范围的空闲块数量和最大连续空闲块长度。",[15,239455,239456],{},"位表大小公式：",[105,239458,239459],{},[108,239460,239461],{},"所需字节数 = disk size in bytes \u002F (8 × file system block size)",[32490,239463,239465],{"id":239464},"chained-free-portions-空闲片段链","Chained Free Portions 空闲片段链",[105,239467,239468,239471,239474],{},[108,239469,239470],{},"每个空闲片段保存指针和长度。",[108,239472,239473],{},"只需保存链表头指针和首片段长度。",[108,239475,239476],{},"空间开销小。",[15,239478,14543],{},[105,239480,239481,239484],{},[108,239482,239483],{},"适合多种分配方法。",[108,239485,239486],{},"不需要完整磁盘分配表。",[15,239488,177826],{},[105,239490,239491,239494,239497],{},[108,239492,239493],{},"磁盘用久后会碎片化。",[108,239495,239496],{},"分配一个块前可能要先读该空闲块以得到下一个指针。",[108,239498,239499],{},"大量分配和删除碎片文件时很慢。",[32490,239501,239503],{"id":239502},"indexing-索引法","Indexing 索引法",[105,239505,239506,239509,239512,239515],{},[108,239507,239508],{},"把空闲空间当成一个文件。",[108,239510,239511],{},"用索引表记录每个空闲片段。",[108,239513,239514],{},"通常按变长片段记录更高效。",[108,239516,239517],{},"能有效支持各种文件分配方法。",[32490,239519,239521],{"id":239520},"free-block-list-空闲块列表","Free Block List 空闲块列表",[105,239523,239524,239527,239530,239533,239544],{},[108,239525,239526],{},"每个块编号。",[108,239528,239529],{},"在磁盘保留区域维护所有空闲块编号列表。",[108,239531,239532],{},"比位表大很多，通常存磁盘上。",[108,239534,239535,239536],{},"可把列表的一小部分放入内存：\n",[105,239537,239538,239541],{},[108,239539,239540],{},"栈方式：分配 pop，释放 push。",[108,239542,239543],{},"FIFO 队列方式：从队首分配，释放到队尾。",[108,239545,239546],{},"后台线程可对内存中的空闲块号排序，以便连续分配。",[253,239548,239550],{"id":239549},"volume-卷","Volume 卷",[105,239552,239553,239556,239559,239562],{},[108,239554,239555],{},"卷是操作系统或应用可用于数据存储的一组可寻址扇区。",[108,239557,239558],{},"最简单情况：一个磁盘就是一个卷。",[108,239560,239561],{},"一个磁盘可分多个分区，每个分区是一个卷。",[108,239563,239564],{},"多个磁盘或多个分区也可组合成一个卷。",[253,239566,239568],{"id":239567},"reliability-可靠性","Reliability 可靠性",[15,239570,239571],{},"潜在问题：",[105,239573,239574,239577],{},[108,239575,239576],{},"系统为了效率把磁盘分配表和文件分配表缓存在内存中。",[108,239578,239579],{},"若内存表已更新但磁盘表未更新时系统崩溃，可能导致空间重复分配或文件损坏。",[15,239581,239582],{},"安全分配步骤：",[105,239584,239585,239588,239591,239594,239597],{},[108,239586,239587],{},"锁定磁盘上的磁盘分配表。",[108,239589,239590],{},"搜索空闲空间。",[108,239592,239593],{},"分配空间并更新磁盘分配表到磁盘。",[108,239595,239596],{},"更新文件分配表到磁盘。",[108,239598,239599],{},"解锁磁盘分配表。",[15,239601,239602],{},"性能问题：",[105,239604,239605,239608,239611],{},[108,239606,239607],{},"每次小分配都同步写磁盘会很慢。",[108,239609,239610],{},"可用批量分配减少开销。",[108,239612,239613],{},"崩溃后需清理已标记使用但未真正归属文件的空间。",[19,239615,239617],{"id":239616},"_128-unix-file-managementunix-文件管理","12.8 UNIX File Management：UNIX 文件管理",[253,239619,239621],{"id":239620},"unix-文件类型","UNIX 文件类型",[105,239623,239624,239635,239649,239657,239668,239676],{},[108,239625,239626,239627],{},"Regular \u002F Ordinary 普通文件：\n",[105,239628,239629,239632],{},[108,239630,239631],{},"包含任意数据。",[108,239633,239634],{},"文件系统不规定内部结构，只看作字节流。",[108,239636,239637,239638],{},"Directory 目录文件：\n",[105,239639,239640,239643,239646],{},[108,239641,239642],{},"包含文件名和 inode 指针。",[108,239644,239645],{},"目录按层次结构组织。",[108,239647,239648],{},"目录本质是特殊保护的普通文件。",[108,239650,239651,239652],{},"Special 特殊文件：\n",[105,239653,239654],{},[108,239655,239656],{},"用文件名映射物理设备。",[108,239658,239659,239660],{},"Named Pipes 命名管道：\n",[105,239661,239662,239665],{},[108,239663,239664],{},"用于进程间通信。",[108,239666,239667],{},"先进先出传输数据。",[108,239669,239670,239671],{},"Links 硬链接：\n",[105,239672,239673],{},[108,239674,239675],{},"已有文件的另一个名字。",[108,239677,239678,239679],{},"Symbolic Links 符号链接：\n",[105,239680,239681],{},[108,239682,239683],{},"一个数据文件，内容是被链接文件的名字。",[253,239685,239687],{"id":239686},"inode","Inode",[105,239689,239690,239693,239696,239699],{},[108,239691,239692],{},"inode 是 UNIX 管理文件的核心控制结构。",[108,239694,239695],{},"每个文件由一个 inode 控制。",[108,239697,239698],{},"多个文件名可以关联到同一个 inode。",[108,239700,239701],{},"文件名不在 inode 中，目录项负责把文件名映射到 inode。",[15,239703,239704],{},"inode 中通常包含：",[105,239706,239707,239710,239713,239716,239719,239722,239725,239728,239731,239734,239737],{},[108,239708,239709],{},"文件类型和访问模式。",[108,239711,239712],{},"所有者和用户组。",[108,239714,239715],{},"创建、读取、写入、inode 更新时间。",[108,239717,239718],{},"文件大小。",[108,239720,239721],{},"数据块指针。",[108,239723,239724],{},"文件使用的物理块数量。",[108,239726,239727],{},"指向该文件的目录项数量。",[108,239729,239730],{},"文件标志。",[108,239732,239733],{},"generation number，用于检测对已删除文件的旧引用。",[108,239735,239736],{},"块大小。",[108,239738,239739],{},"扩展属性。",[253,239741,239743],{"id":239742},"unix-文件分配","UNIX 文件分配",[105,239745,239746,239749,239751,239754,239757],{},[108,239747,239748],{},"按块动态分配。",[108,239750,178140],{},[108,239752,239753],{},"使用索引分配。",[108,239755,239756],{},"inode 中保存部分索引。",[108,239758,239759],{},"inode 包含直接指针和间接指针。",[15,239761,239762],{},"典型结构：",[105,239764,239765,239776,239784,239792],{},[108,239766,239767,239768],{},"直接指针：\n",[105,239769,239770,239773],{},[108,239771,239772],{},"直接指向文件数据块。",[108,239774,239775],{},"小文件访问快。",[108,239777,239778,239779],{},"单级间接指针：\n",[105,239780,239781],{},[108,239782,239783],{},"指向一个保存数据块地址的块。",[108,239785,239786,239787],{},"二级间接指针：\n",[105,239788,239789],{},[108,239790,239791],{},"指向保存单级间接块地址的块。",[108,239793,239794,239795],{},"三级间接指针：\n",[105,239796,239797],{},[108,239798,239799],{},"指向保存二级间接块地址的块。",[15,239801,14543],{},[105,239803,239804,239807,239810],{},[108,239805,239806],{},"inode 固定且较小，可长期留在内存。",[108,239808,239809],{},"小文件访问几乎不需要间接寻址。",[108,239811,239812],{},"最大文件大小可非常大。",[253,239814,239816],{"id":239815},"unix-目录","UNIX 目录",[105,239818,239819,239822,239825],{},[108,239820,239821],{},"目录是包含文件名和 inode 编号的文件。",[108,239823,239824],{},"目录项 dentry 包含文件名和 i-number。",[108,239826,239827],{},"i-number 用作 inode 表索引。",[253,239829,239831],{"id":239830},"unix-卷结构","UNIX 卷结构",[15,239833,239834],{},"一个 UNIX 文件系统通常包含：",[105,239836,239837,239845,239853,239861],{},[108,239838,239839,239840],{},"Boot block：\n",[105,239841,239842],{},[108,239843,239844],{},"启动操作系统所需代码。",[108,239846,239847,239848],{},"Superblock：\n",[105,239849,239850],{},[108,239851,239852],{},"文件系统属性，如分区大小、inode 表大小。",[108,239854,239855,239856],{},"Inode table：\n",[105,239857,239858],{},[108,239859,239860],{},"所有文件 inode 的集合。",[108,239862,239863,239864],{},"Data blocks：\n",[105,239865,239866],{},[108,239867,239868],{},"存放文件数据和子目录。",[19,239870,239872],{"id":239871},"_129-linux-virtual-file-systemlinux-虚拟文件系统","12.9 Linux Virtual File System：Linux 虚拟文件系统",[253,239874,239876],{"id":239875},"vfs-的目的","VFS 的目的",[105,239878,239879,239882,239885,239888],{},[108,239880,239881],{},"Linux 支持多种真实文件系统。",[108,239883,239884],{},"VFS 为用户进程提供统一文件系统接口。",[108,239886,239887],{},"各真实文件系统通过映射模块接入 VFS。",[108,239889,239890],{},"用户调用如 read、write、open 时，先进入 VFS，再被转换为具体文件系统操作。",[253,239892,239894],{"id":239893},"vfs-工作过程","VFS 工作过程",[105,239896,239897,239900,239903,239906,239909,239912],{},[108,239898,239899],{},"用户进程发出文件系统调用。",[108,239901,239902],{},"VFS 处理文件系统无关部分。",[108,239904,239905],{},"VFS 调用具体文件系统的映射函数。",[108,239907,239908],{},"具体文件系统把请求转换为设备操作。",[108,239910,239911],{},"设备驱动执行底层 I\u002FO。",[108,239913,239914],{},"结果沿相反路径返回用户。",[253,239916,239918],{"id":239917},"vfs-的四类核心对象","VFS 的四类核心对象",[105,239920,239921,239929,239937,239945],{},[108,239922,239923,239924],{},"Superblock object：\n",[105,239925,239926],{},[108,239927,239928],{},"表示一个已挂载文件系统。",[108,239930,239931,239932],{},"Inode object：\n",[105,239933,239934],{},[108,239935,239936],{},"表示一个具体文件。",[108,239938,239939,239940],{},"Dentry object：\n",[105,239941,239942],{},[108,239943,239944],{},"表示路径中的一个目录项。",[108,239946,239947,239948],{},"File object：\n",[105,239949,239950],{},[108,239951,239952],{},"表示一个进程打开的文件。",[253,239954,239956],{"id":239955},"superblock-object","Superblock Object",[15,239958,239959],{},"保存一个文件系统的信息：",[105,239961,239962,239965,239968,239971,239974,239977,239980,239983,239986],{},[108,239963,239964],{},"挂载设备。",[108,239966,239967],{},"文件系统基本块大小。",[108,239969,239970],{},"dirty 标志。",[108,239972,239973],{},"文件系统类型。",[108,239975,239976],{},"只读等标志。",[108,239978,239979],{},"根目录指针。",[108,239981,239982],{},"打开文件列表。",[108,239984,239985],{},"访问控制信号量。",[108,239987,239988],{},"superblock 操作函数列表。",[15,239990,239991],{},"常见操作：",[105,239993,239994,239997,240000,240003,240006],{},[108,239995,239996],{},"分配 inode。",[108,239998,239999],{},"写 inode 到磁盘。",[108,240001,240002],{},"卸载时释放 superblock。",[108,240004,240005],{},"获取文件系统统计信息。",[108,240007,240008],{},"重新挂载时更新挂载选项。",[253,240010,240012],{"id":240011},"inode-object","Inode Object",[105,240014,240015,240018,240021,240024],{},[108,240016,240017],{},"每个文件对应一个 inode object。",[108,240019,240020],{},"保存文件名和实际数据内容之外的文件信息。",[108,240022,240023],{},"包含所有者、组、权限、访问时间、大小、链接数等。",[108,240025,240026],{},"包含 inode operations，如创建文件、查找文件名、创建目录。",[253,240028,240030],{"id":240029},"dentry-object","Dentry Object",[105,240032,240033,240036,240039,240042,240045],{},[108,240034,240035],{},"表示路径中的一个组成部分。",[108,240037,240038],{},"可以是目录名或文件名。",[108,240040,240041],{},"加速路径查找。",[108,240043,240044],{},"用于 dentry cache。",[108,240046,240047],{},"包含指向 inode、superblock、父 dentry、子 dentry 的指针。",[253,240049,240051],{"id":240050},"file-object","File Object",[105,240053,240054,240057,240060,240063],{},[108,240055,240056],{},"表示进程已经打开的文件。",[108,240058,240059],{},"open 系统调用时创建。",[108,240061,240062],{},"close 系统调用时销毁。",[108,240064,240065,240066],{},"包含：\n",[105,240067,240068,240071,240074,240077,240080,240083,240086],{},[108,240069,240070],{},"对应 dentry。",[108,240072,240073],{},"所属文件系统。",[108,240075,240076],{},"使用计数。",[108,240078,240079],{},"用户 ID。",[108,240081,240082],{},"用户组 ID。",[108,240084,240085],{},"当前文件指针。",[108,240087,240088],{},"read、write、open、release、lock 等操作函数。",[253,240090,240092],{"id":240091},"vfs-缓存","VFS 缓存",[105,240094,240095,240103,240111],{},[108,240096,240097,240098],{},"Inode cache：\n",[105,240099,240100],{},[108,240101,240102],{},"缓存最近访问过的 inode。",[108,240104,240105,240106],{},"Directory cache：\n",[105,240107,240108],{},[108,240109,240110],{},"缓存目录名到 inode 编号的映射。",[108,240112,240113,240114],{},"Buffer cache：\n",[105,240115,240116],{},[108,240117,240118],{},"缓存物理块数据，减少重复磁盘读取。",[19,240120,240122],{"id":240121},"_1210-windows-file-systemwindows-ntfs","12.10 Windows File System：Windows NTFS",[253,240124,240126],{"id":240125},"ntfs-目标","NTFS 目标",[105,240128,240129,240132],{},[108,240130,240131],{},"面向高端工作站和服务器。",[108,240133,240134],{},"支持大型文件、可靠恢复、安全访问和网络应用。",[253,240136,240138],{"id":240137},"ntfs-关键特性","NTFS 关键特性",[105,240140,240141,240155,240166,240174,240182,240190,240198],{},[108,240142,240143,240144],{},"Recoverability 可恢复性：\n",[105,240145,240146,240149,240152],{},[108,240147,240148],{},"崩溃或磁盘故障后恢复卷到一致状态。",[108,240150,240151],{},"文件系统修改被视为事务。",[108,240153,240154],{},"事务要么完成，要么回滚。",[108,240156,240157,240158],{},"Security 安全性：\n",[105,240159,240160,240163],{},[108,240161,240162],{},"使用 Windows 对象模型。",[108,240164,240165],{},"文件对象带有安全描述符。",[108,240167,240168,240169],{},"Large disks and large files：\n",[105,240170,240171],{},[108,240172,240173],{},"高效支持大磁盘和大文件。",[108,240175,240176,240177],{},"Multiple data streams：\n",[105,240178,240179],{},[108,240180,240181],{},"一个文件可有多个数据流。",[108,240183,240184,240185],{},"Journaling 日志：\n",[105,240186,240187],{},[108,240188,240189],{},"记录卷上文件变化。",[108,240191,240192,240193],{},"Compression and encryption：\n",[105,240194,240195],{},[108,240196,240197],{},"支持透明压缩和加密。",[108,240199,240200,240201],{},"Hard links and symbolic links：\n",[105,240202,240203],{},[108,240204,240205],{},"支持硬链接、符号链接和挂载点。",[253,240207,240209],{"id":240208},"ntfs-存储概念","NTFS 存储概念",[105,240211,240212,240220,240231],{},[108,240213,240214,240215],{},"Sector 扇区：\n",[105,240216,240217],{},[108,240218,240219],{},"磁盘最小物理存储单位，通常 512 字节。",[108,240221,240222,240223],{},"Cluster 簇：\n",[105,240224,240225,240228],{},[108,240226,240227],{},"一个或多个连续扇区。",[108,240229,240230],{},"NTFS 的基本分配单位。",[108,240232,240233,240234],{},"Volume 卷：\n",[105,240235,240236,240239],{},[108,240237,240238],{},"一个逻辑分区，由一个或多个簇组成。",[108,240240,240241],{},"可位于单磁盘或跨多个磁盘。",[253,240243,240245],{"id":240244},"cluster-的意义","Cluster 的意义",[105,240247,240248,240251,240254],{},[108,240249,240250],{},"NTFS 不直接按扇区分配，而按簇分配。",[108,240252,240253],{},"簇大小越大，管理项越少，适合大磁盘。",[108,240255,240256],{},"文件的簇不必连续，因此可产生碎片。",[253,240258,240260],{"id":240259},"ntfs-卷布局","NTFS 卷布局",[15,240262,240263],{},"NTFS 卷主要包括：",[105,240265,240266,240274,240282,240290],{},[108,240267,240268,240269],{},"Partition boot sector：\n",[105,240270,240271],{},[108,240272,240273],{},"卷布局、文件系统结构、启动信息和代码。",[108,240275,240276,240277],{},"Master File Table，MFT：\n",[105,240278,240279],{},[108,240280,240281],{},"保存该卷所有文件和文件夹的信息。",[108,240283,240284,240285],{},"System files：\n",[105,240286,240287],{},[108,240288,240289],{},"包括 MFT 镜像、日志文件、簇位图、属性定义表等。",[108,240291,240292,240293],{},"File area：\n",[105,240294,240295],{},[108,240296,240297],{},"存放普通文件数据。",[253,240299,240301],{"id":240300},"mft-主文件表","MFT 主文件表",[105,240303,240304,240307,240310,240313,240316,240319],{},[108,240305,240306],{},"NTFS 的核心。",[108,240308,240309],{},"MFT 是由 1024 字节记录组成的表。",[108,240311,240312],{},"每行描述一个文件或文件夹。",[108,240314,240315],{},"MFT 自身也被当作文件。",[108,240317,240318],{},"小文件可完全存放在 MFT 记录中。",[108,240320,240321],{},"大文件内容溢出到其他簇，MFT 记录保存指针。",[253,240323,240325],{"id":240324},"ntfs-文件属性","NTFS 文件属性",[15,240327,240328],{},"文件和目录由属性集合定义，包括：",[105,240330,240331,240339,240347,240355,240363,240371,240379,240387],{},[108,240332,240333,240334],{},"Standard information：\n",[105,240335,240336],{},[108,240337,240338],{},"访问属性、时间戳、链接计数等。",[108,240340,240341,240342],{},"Attribute list：\n",[105,240343,240344],{},[108,240345,240346],{},"属性列表，适用于属性无法放入单个 MFT 记录的情况。",[108,240348,240349,240350],{},"File name：\n",[105,240351,240352],{},[108,240353,240354],{},"文件或目录名。",[108,240356,240357,240358],{},"Security descriptor：\n",[105,240359,240360],{},[108,240361,240362],{},"所有者和访问控制信息。",[108,240364,240365,240366],{},"Data：\n",[105,240367,240368],{},[108,240369,240370],{},"文件内容，可有默认未命名数据流和命名数据流。",[108,240372,240373,240374],{},"Index root \u002F Index allocation：\n",[105,240375,240376],{},[108,240377,240378],{},"用于实现文件夹。",[108,240380,240381,240382],{},"Volume information：\n",[105,240383,240384],{},[108,240385,240386],{},"卷版本、卷名等。",[108,240388,240389,240390],{},"Bitmap：\n",[105,240391,240392],{},[108,240393,240394],{},"表示 MFT 或文件夹中记录使用情况。",[253,240396,240398],{"id":240397},"ntfs-可恢复性","NTFS 可恢复性",[15,240400,240401],{},"关键组件：",[105,240403,240404,240412,240420,240428],{},[108,240405,240406,240407],{},"I\u002FO Manager：\n",[105,240408,240409],{},[108,240410,240411],{},"包含 NTFS 驱动，处理 open、close、read、write。",[108,240413,240414,240415],{},"Log File Service：\n",[105,240416,240417],{},[108,240418,240419],{},"记录文件系统元数据修改日志。",[108,240421,240422,240423],{},"Cache Manager：\n",[105,240424,240425],{},[108,240426,240427],{},"缓存文件读写，提高性能。",[108,240429,240430,240431],{},"Virtual Memory Manager：\n",[105,240432,240433],{},[108,240434,240435],{},"将缓存文件映射到虚拟内存。",[15,240437,240438],{},"重要考点：",[105,240440,240441,240444,240447,240450],{},[108,240442,240443],{},"NTFS 恢复的是文件系统元数据，不保证用户文件内容完全恢复。",[108,240445,240446],{},"NTFS 使用日志。",[108,240448,240449],{},"修改文件系统的重要子操作要先写日志，再写磁盘结构。",[108,240451,240452],{},"崩溃后可根据日志 redo 或 undo 未完成事务。",[19,240454,240456],{"id":240455},"_1211-android-file-managementandroid-文件管理","12.11 Android File Management：Android 文件管理",[253,240458,240460],{"id":240459},"android-文件系统","Android 文件系统",[105,240462,240463,240466],{},[108,240464,240465],{},"Android 使用 Linux 的文件管理能力。",[108,240467,240468],{},"目录结构类似 Linux，但有 Android 特有目录。",[253,240470,240472],{"id":240471},"system-目录","system 目录",[105,240474,240475,240478,240481],{},[108,240476,240477],{},"包含操作系统核心部分。",[108,240479,240480],{},"包括系统二进制文件、系统库、配置文件和基础应用。",[108,240482,240483],{},"系统镜像通常只读。",[253,240485,240487],{"id":240486},"data-目录","data 目录",[105,240489,240490,240493,240496,240499,240516],{},[108,240491,240492],{},"应用保存私有数据的主要位置。",[108,240494,240495],{},"包含用户数据，如联系人、短信、设置、安装的应用。",[108,240497,240498],{},"恢复出厂设置会清除此分区。",[108,240500,240501,240502],{},"安装应用时：\n",[105,240503,240504,240507,240513],{},[108,240505,240506],{},".apk 放入 \u002Fdata\u002Fapp。",[108,240508,240509,240510,347],{},"应用私有库放入 ",[31,240511,240512],{},"\u002Fdata\u002Fdata\u002F\u003Capplication name>",[108,240514,240515],{},"建立应用相关文件和数据库。",[108,240517,240518,240520],{},[31,240519,240512],{}," 是应用沙箱，其他应用不可访问。",[253,240522,240524],{"id":240523},"cache-目录","cache 目录",[105,240526,240527,240530,240533,240536],{},[108,240528,240529],{},"用于操作系统临时存储。",[108,240531,240532],{},"保存频繁访问的数据和应用组件。",[108,240534,240535],{},"清除 cache 不影响用户个人数据。",[108,240537,240538],{},"缓存会随着使用自动重建。",[253,240540,240542],{"id":240541},"mntsdcard-目录","mnt\u002Fsdcard 目录",[105,240544,240545,240548,240551],{},[108,240546,240547],{},"表示 SD 卡或用户可读写存储空间。",[108,240549,240550],{},"用于音频、视频和用户数据。",[108,240552,240553],{},"在有内置和外置 SD 卡的设备上，\u002Fsdcard 通常指内置 SD 卡。",[253,240555,240557],{"id":240556},"sqlite","SQLite",[105,240559,240560,240563,240566,240569,240572],{},[108,240561,240562],{},"Android 广泛使用 SQLite。",[108,240564,240565],{},"SQLite 是基于 SQL 的轻量级关系数据库引擎。",[108,240567,240568],{},"适合嵌入式和内存受限系统。",[108,240570,240571],{},"SQLite 不是独立数据库服务器进程。",[108,240573,240574],{},"SQLite 库直接链接进应用程序，成为应用的一部分。",[19,240576,240578],{"id":240577},"_1212-summary本章总结","12.12 Summary：本章总结",[253,240580,240581],{"id":240581},"总体脉络",[105,240583,240584,240587,240590,240593,240596,240599],{},[108,240585,240586],{},"文件管理系统为用户和应用提供文件访问、目录维护和访问控制。",[108,240588,240589],{},"文件由记录组成，记录访问方式决定文件逻辑组织。",[108,240591,240592],{},"文件目录提供文件命名、定位、层次组织和访问控制基础。",[108,240594,240595],{},"记录与磁盘块大小通常不一致，因此需要记录阻塞策略。",[108,240597,240598],{},"磁盘空间管理包括文件分配和空闲空间管理。",[108,240600,240601],{},"现代文件系统还必须考虑并发访问、可靠性、安全性和缓存性能。",[253,240603,240604],{"id":240604},"易考判断点",[105,240606,240607,240610,240613,240616,240619,240622,240625,240628,240631,240634,240637,240640,240643,240646,240649,240652,240655,240658,240661,240664,240667,240670,240673,240676],{},[108,240608,240609],{},"文件是相似记录的集合，不是字段的集合。",[108,240611,240612],{},"字段是最基本数据元素，记录是相关字段集合。",[108,240614,240615],{},"数据库强调数据关系显式，并服务于多个应用。",[108,240617,240618],{},"堆文件按到达顺序存储，不是排序存储。",[108,240620,240621],{},"顺序文件的关键字段必须唯一。",[108,240623,240624],{},"顺序文件最适合全文件批处理。",[108,240626,240627],{},"索引顺序文件增加索引和溢出文件。",[108,240629,240630],{},"一般索引文件可有多个索引，并支持变长记录。",[108,240632,240633],{},"直接文件使用对关键值的散列。",[108,240635,240636],{},"目录的访问控制信息包含所有者、访问用户和允许动作。",[108,240638,240639],{},"working directory 是当前目录。",[108,240641,240642],{},"文件共享需要处理互斥和死锁。",[108,240644,240645],{},"固定阻塞可能产生内部碎片。",[108,240647,240648],{},"连续分配必须预分配，易产生外部碎片。",[108,240650,240651],{},"链式分配无外部碎片，但随机访问差。",[108,240653,240654],{},"索引分配支持顺序和直接访问。",[108,240656,240657],{},"空闲空间管理需要磁盘分配表。",[108,240659,240660],{},"bit table 中每个块对应一位。",[108,240662,240663],{},"UNIX 普通文件是字节流。",[108,240665,240666],{},"UNIX 使用 inode 保存文件控制信息。",[108,240668,240669],{},"Linux VFS 提供统一文件接口。",[108,240671,240672],{},"NTFS 的最小物理单位是 sector，但基本分配单位是 cluster。",[108,240674,240675],{},"NTFS 核心结构是 MFT。",[108,240677,240678],{},"NTFS 日志主要恢复元数据，不保证用户数据内容。",{"title":29,"searchDepth":43,"depth":43,"links":240680},[240681,240695,240705,240713,240722,240728,240734,240747,240754,240764,240774,240782],{"id":237720,"depth":43,"text":237721,"children":240682},[240683,240684,240685,240686,240687,240688,240689,240690,240691,240692,240693,240694],{"id":237724,"depth":49,"text":237724},{"id":237738,"depth":49,"text":237738},{"id":237752,"depth":49,"text":237752},{"id":237775,"depth":49,"text":237775},{"id":237806,"depth":49,"text":237806},{"id":237823,"depth":49,"text":237823},{"id":237837,"depth":49,"text":237837},{"id":237866,"depth":49,"text":237867},{"id":237878,"depth":49,"text":237879},{"id":237911,"depth":49,"text":237911},{"id":237934,"depth":49,"text":237934},{"id":237997,"depth":49,"text":237997},{"id":238023,"depth":43,"text":238024,"children":240696},[240697,240698,240699,240700,240701,240702,240703,240704],{"id":238027,"depth":49,"text":238027},{"id":238038,"depth":49,"text":238038},{"id":238069,"depth":49,"text":238069},{"id":238089,"depth":49,"text":238090},{"id":238149,"depth":49,"text":238150},{"id":238237,"depth":49,"text":238238},{"id":238322,"depth":49,"text":238323},{"id":238400,"depth":49,"text":238401},{"id":238459,"depth":43,"text":238460,"children":240706},[240707,240708,240709,240710,240711,240712],{"id":238463,"depth":49,"text":238464},{"id":238481,"depth":49,"text":238482},{"id":238511,"depth":49,"text":238512},{"id":238538,"depth":49,"text":238539},{"id":238562,"depth":49,"text":238563},{"id":238594,"depth":49,"text":238595},{"id":238612,"depth":43,"text":238613,"children":240714},[240715,240716,240717,240718,240719,240720,240721],{"id":238616,"depth":49,"text":238616},{"id":238630,"depth":49,"text":238630},{"id":238707,"depth":49,"text":238707},{"id":238727,"depth":49,"text":238727},{"id":238787,"depth":49,"text":238787},{"id":238798,"depth":49,"text":238799},{"id":238813,"depth":49,"text":238814},{"id":238831,"depth":43,"text":238832,"children":240723},[240724,240725,240726,240727],{"id":238835,"depth":49,"text":238835},{"id":238846,"depth":49,"text":238846},{"id":238929,"depth":49,"text":238929},{"id":238943,"depth":49,"text":238943},{"id":238963,"depth":43,"text":238964,"children":240729},[240730,240731,240732,240733],{"id":238967,"depth":49,"text":238967},{"id":238981,"depth":49,"text":238981},{"id":239012,"depth":49,"text":239012},{"id":239111,"depth":49,"text":239111},{"id":239125,"depth":43,"text":239126,"children":240735},[240736,240737,240738,240739,240740,240741,240742,240743,240744,240745,240746],{"id":239129,"depth":49,"text":239129},{"id":239140,"depth":49,"text":239140},{"id":239154,"depth":49,"text":239155},{"id":239189,"depth":49,"text":239190},{"id":239204,"depth":49,"text":239205},{"id":239216,"depth":49,"text":239216},{"id":239253,"depth":49,"text":239253},{"id":239282,"depth":49,"text":239282},{"id":239422,"depth":49,"text":239423},{"id":239549,"depth":49,"text":239550},{"id":239567,"depth":49,"text":239568},{"id":239616,"depth":43,"text":239617,"children":240748},[240749,240750,240751,240752,240753],{"id":239620,"depth":49,"text":239621},{"id":239686,"depth":49,"text":239687},{"id":239742,"depth":49,"text":239743},{"id":239815,"depth":49,"text":239816},{"id":239830,"depth":49,"text":239831},{"id":239871,"depth":43,"text":239872,"children":240755},[240756,240757,240758,240759,240760,240761,240762,240763],{"id":239875,"depth":49,"text":239876},{"id":239893,"depth":49,"text":239894},{"id":239917,"depth":49,"text":239918},{"id":239955,"depth":49,"text":239956},{"id":240011,"depth":49,"text":240012},{"id":240029,"depth":49,"text":240030},{"id":240050,"depth":49,"text":240051},{"id":240091,"depth":49,"text":240092},{"id":240121,"depth":43,"text":240122,"children":240765},[240766,240767,240768,240769,240770,240771,240772,240773],{"id":240125,"depth":49,"text":240126},{"id":240137,"depth":49,"text":240138},{"id":240208,"depth":49,"text":240209},{"id":240244,"depth":49,"text":240245},{"id":240259,"depth":49,"text":240260},{"id":240300,"depth":49,"text":240301},{"id":240324,"depth":49,"text":240325},{"id":240397,"depth":49,"text":240398},{"id":240455,"depth":43,"text":240456,"children":240775},[240776,240777,240778,240779,240780,240781],{"id":240459,"depth":49,"text":240460},{"id":240471,"depth":49,"text":240472},{"id":240486,"depth":49,"text":240487},{"id":240523,"depth":49,"text":240524},{"id":240541,"depth":49,"text":240542},{"id":240556,"depth":49,"text":240557},{"id":240577,"depth":43,"text":240578,"children":240783},[240784,240785],{"id":240581,"depth":49,"text":240581},{"id":240604,"depth":49,"text":240604},{"order":1999},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap12",{"title":237711,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap12","aMmAAhyW35le_3xgHe2vKUFVo16g9Y-E3DlgvUZ7ScI",{"id":240792,"title":240793,"body":240794,"description":29,"extension":85,"meta":241323,"navigation":52,"path":241324,"seo":241325,"stem":241326,"__hash__":241327},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap2.md","Chap. 2 Note",{"type":8,"value":240795,"toc":241298},[240796,240800,240804,240810,240830,240834,240837,240888,240892,240895,240910,240914,240917,240919,240923,240927,240930,240948,240952,240959,240990,240994,240997,241014,241018,241021,241036,241038,241042,241046,241053,241075,241079,241082,241114,241130,241134,241137,241163,241167,241170,241189,241191,241195,241231,241233,241237,241243,241245,241249,241253,241256,241273,241277,241280],[11,240797,240799],{"id":240798},"chapter-2-operating-system-overview","Chapter 2: Operating System Overview",[19,240801,240803],{"id":240802},"_1-operating-system-objectives-and-functions","1. Operating System Objectives and Functions",[15,240805,240806,240809],{},[2627,240807,240808],{},"Def",": An Operating System is a program that controls the execution of application programs and acts as an interface between applications and the computer hardware. It has three main objectives:",[105,240811,240812,240818,240824],{},[108,240813,240814,240817],{},[2627,240815,240816],{},"Convenience (方便):"," Makes a computer more convenient to use for the end user,.",[108,240819,240820,240823],{},[2627,240821,240822],{},"Efficiency (高效):"," Allows the computer system resources to be used in an efficient manner,.",[108,240825,240826,240829],{},[2627,240827,240828],{},"Ability to evolve (可演进能力):"," Permits the effective development, testing, and introduction of new system functions without interfering with service,.",[253,240831,240833],{"id":240832},"_11-the-os-as-a-usercomputer-interface","1.1 The OS as a User\u002FComputer Interface",[15,240835,240836],{},"The OS masks the details of the hardware from the programmer and provides a convenient interface. It typically provides services in the following areas:",[105,240838,240839,240845,240851,240857,240863,240869,240875,240881],{},[108,240840,240841,240844],{},[2627,240842,240843],{},"Program development:"," Provides tools like editors and debuggers (often via utilities).",[108,240846,240847,240850],{},[2627,240848,240849],{},"Program execution:"," Handles scheduling duties, loads instructions\u002Fdata into main memory, and initializes files and I\u002FO devices.",[108,240852,240853,240856],{},[2627,240854,240855],{},"Access to I\u002FO devices:"," Hides specific hardware details and provides a uniform interface for reading and writing.",[108,240858,240859,240862],{},[2627,240860,240861],{},"Controlled access to files:"," Understands storage media and data structures, and provides protection mechanisms in multi-user environments.",[108,240864,240865,240868],{},[2627,240866,240867],{},"System access:"," Resolves resource conflicts and protects resources\u002Fdata from unauthorized access.",[108,240870,240871,240874],{},[2627,240872,240873],{},"Error detection and response:"," Responds to internal\u002Fexternal hardware errors or software errors with minimal impact on running applications-.",[108,240876,240877,240880],{},[2627,240878,240879],{},"Accounting (记账):"," Collects usage statistics to tune system performance and bill users.",[108,240882,240883,240884,240887],{},"It provides an ",[2627,240885,240886],{},"Application Programming Interface (API, 应用程序编程接口)",", allowing software to be ported easily across systems-.",[253,240889,240891],{"id":240890},"_12-the-os-as-resource-manager","1.2 The OS as Resource Manager",[15,240893,240894],{},"The OS is responsible for controlling the use of a computer's resources (I\u002FO, main\u002Fsecondary memory, processor execution time).",[105,240896,240897,240900,240903],{},[108,240898,240899],{},"Unlike typical control mechanisms, the OS functions as an ordinary computer program executed by the processor.",[108,240901,240902],{},"It frequently relinquishes control to let the processor perform \"useful\" work and depends on the processor to regain control.",[108,240904,240905,240906,240909],{},"The ",[2627,240907,240908],{},"Kernel (内核)",", or nucleus, contains the most frequently used OS functions and typically remains permanently in main memory,-.",[253,240911,240913],{"id":240912},"_13-ease-of-evolution","1.3 Ease of Evolution",[15,240915,240916],{},"An OS will evolve over time due to hardware upgrades, new types of hardware, new services, and software fixes-. This requires the OS to be modular in construction with clearly defined interfaces.",[32188,240918],{},[19,240920,240922],{"id":240921},"_2-the-evolution-of-operating-systems","2. The Evolution of Operating Systems",[253,240924,240926],{"id":240925},"_21-serial-processing-串行处理","2.1 Serial Processing (串行处理)",[15,240928,240929],{},"In early computers (late 1940s to mid-1950s), there was no OS; programmers interacted directly with hardware.",[105,240931,240932,240938],{},[108,240933,240934,240937],{},[2627,240935,240936],{},"Scheduling:"," Users used a hardcopy sign-up sheet, which often resulted in wasted computer processing time-.",[108,240939,240940,240943,240944,240947],{},[2627,240941,240942],{},"Setup time:"," A single ",[2627,240945,240946],{},"job (作业)"," involved mounting tapes and setting up card decks manually, causing a considerable waste of time.",[253,240949,240951],{"id":240950},"_22-simple-batch-systems-简单批处理系统","2.2 Simple Batch Systems (简单批处理系统)",[15,240953,240954,240955,240958],{},"To maximize processor utilization, early batch OSs used a ",[2627,240956,240957],{},"Monitor (监控程序)","-.",[105,240960,240961,240964,240967,240973,240979],{},[108,240962,240963],{},"Users submitted jobs on cards\u002Ftape to an operator, who batched them together on an input device for the monitor.",[108,240965,240966],{},"The monitor automatically loaded the next program after a job was completed, avoiding idle time-.",[108,240968,240969,240972],{},[2627,240970,240971],{},"Job Control Language (JCL, 作业控制语言)"," was used to provide instructions to the monitor.",[108,240974,240975,240978],{},[2627,240976,240977],{},"Hardware Features Required:"," Memory protection (内存保护), Timer (定时器), Privileged instructions (特权指令), and Interrupts (中断).",[108,240980,240981,240982,240985,240986,240989],{},"Execution modes were split into ",[2627,240983,240984],{},"User mode (用户态)"," (with restricted memory and instruction access) and ",[2627,240987,240988],{},"Kernel mode \u002F System mode (内核态\u002F系统态)"," (for the monitor to execute privileged instructions),.",[253,240991,240993],{"id":240992},"_23-multiprogrammed-batch-systems-多道批处理系统","2.3 Multiprogrammed Batch Systems (多道批处理系统)",[15,240995,240996],{},"In a simple batch system, the processor is often idle waiting for slow I\u002FO devices.",[105,240998,240999,241005,241011],{},[108,241000,241001,241004],{},[2627,241002,241003],{},"Multiprogramming \u002F Multitasking (多道程序设计\u002F多任务处理):"," Expands memory to hold multiple programs; when one job waits for I\u002FO, the processor switches to another job,.",[108,241006,241007,241008,2547],{},"This relies on hardware supporting I\u002FO interrupts and ",[2627,241009,241010],{},"Direct Memory Access (DMA, 直接内存访问)",[108,241012,241013],{},"It maximizes processor utilization but requires advanced memory management and scheduling algorithms,.",[253,241015,241017],{"id":241016},"_24-time-sharing-systems-分时系统","2.4 Time-Sharing Systems (分时系统)",[15,241019,241020],{},"Used to handle multiple interactive jobs by interleaving execution in short bursts or time slices.",[105,241022,241023,241026,241033],{},[108,241024,241025],{},"Multiple users simultaneously access the system through terminals.",[108,241027,241028,241029,241032],{},"While multiprogramming maximizes processor use, time-sharing aims to minimize ",[2627,241030,241031],{},"Response time (响应时间)",",.",[108,241034,241035],{},"Examples include the Compatible Time-Sharing System (CTSS).",[32188,241037],{},[19,241039,241041],{"id":241040},"_3-major-achievements","3. Major Achievements",[253,241043,241045],{"id":241044},"_31-the-process-进程","3.1 The Process (进程)",[15,241047,241048,241049,241052],{},"A ",[2627,241050,241051],{},"Process (进程)"," is defined as a program in execution, an instance of a running program, or a unit of activity characterized by a single sequential thread of execution, a current state, and an associated set of system resources,.",[105,241054,241055,241066,241069],{},[108,241056,241057,241058,241061,241062,241065],{},"It consists of three components: an executable program, associated data, and the ",[2627,241059,241060],{},"Execution context (执行上下文)"," (or ",[2627,241063,241064],{},"Process state (进程状态)",")-,-.",[108,241067,241068],{},"The execution context includes contents of processor registers and information the OS needs to manage the process, allowing the OS to suspend and resume the process correctly,.",[108,241070,241071,241074],{},[2627,241072,241073],{},"Thread (线程):"," A single process can be broken up into multiple, concurrent threads that execute cooperatively.",[253,241076,241078],{"id":241077},"_32-memory-management-内存管理","3.2 Memory Management (内存管理)",[15,241080,241081],{},"The OS has five principal storage management responsibilities-,:",[2230,241083,241084,241090,241096,241102,241108],{},[108,241085,241086,241089],{},[2627,241087,241088],{},"Process isolation (进程隔离):"," Preventing independent processes from interfering with each other's memory,.",[108,241091,241092,241095],{},[2627,241093,241094],{},"Automatic allocation and management:"," Dynamically allocating memory across the hierarchy transparently to the programmer,.",[108,241097,241098,241101],{},[2627,241099,241100],{},"Support of modular programming:"," Allowing dynamic creation and alteration of program modules,.",[108,241103,241104,241107],{},[2627,241105,241106],{},"Protection and access control:"," Permitting shared memory while protecting the integrity of programs\u002FOS,.",[108,241109,241110,241113],{},[2627,241111,241112],{},"Long-term storage:"," Storing information persistently (e.g., via file systems),.",[105,241115,241116],{},[108,241117,241118,241121,241122,241125,241126,241129],{},[2627,241119,241120],{},"Virtual memory (虚拟内存):"," Allows programs to address memory logically without worrying about physical limitations. It maps a ",[2627,241123,241124],{},"Virtual address (虚拟地址)"," to a ",[2627,241127,241128],{},"Real address \u002F Physical address (实地址\u002F物理地址)"," dynamically-,.",[253,241131,241133],{"id":241132},"_33-information-protection-and-security-信息保护与安全","3.3 Information Protection and Security (信息保护与安全)",[15,241135,241136],{},"Security policies generally target four categories:",[2230,241138,241139,241145,241151,241157],{},[108,241140,241141,241144],{},[2627,241142,241143],{},"Availability (可用性):"," Protecting against interruption.",[108,241146,241147,241150],{},[2627,241148,241149],{},"Confidentiality (机密性):"," Assuring unauthorized users cannot read data.",[108,241152,241153,241156],{},[2627,241154,241155],{},"Data integrity (数据完整性):"," Protecting data from unauthorized modification.",[108,241158,241159,241162],{},[2627,241160,241161],{},"Authenticity (真实性):"," Verifying user identity and message validity.",[253,241164,241166],{"id":241165},"_34-scheduling-and-resource-management-调度和资源管理","3.4 Scheduling and Resource Management (调度和资源管理)",[15,241168,241169],{},"The OS must schedule process execution and allocate resources based on three factors-:",[105,241171,241172,241178,241184],{},[108,241173,241174,241177],{},[2627,241175,241176],{},"Fairness (公平性):"," Giving processes equal and fair access to resources.",[108,241179,241180,241183],{},[2627,241181,241182],{},"Differential responsiveness (差异响应性):"," Discriminating among different classes of jobs to meet different service requirements dynamically.",[108,241185,241186,241188],{},[2627,241187,240822],{}," Maximizing throughput, minimizing response time, and accommodating the maximum number of users.",[32188,241190],{},[19,241192,241194],{"id":241193},"_4-developments-leading-to-modern-operating-systems","4. Developments Leading to Modern Operating Systems",[105,241196,241197,241207,241213,241219,241225],{},[108,241198,241199,241202,241203,241206],{},[2627,241200,241201],{},"Microkernel architecture (微内核架构):"," Assigns only a few essential functions (e.g., address space management, Interprocess Communication (IPC), basic scheduling) to the kernel,. Other services run in user mode, unlike a ",[2627,241204,241205],{},"Monolithic kernel (单体内核)"," where all elements share the same address space,.",[108,241208,241209,241212],{},[2627,241210,241211],{},"Multithreading (多线程):"," A technique where a process is divided into threads that run concurrently,. A thread is a dispatchable unit of work; a process is a collection of threads and associated resources.",[108,241214,241215,241218],{},[2627,241216,241217],{},"Symmetric Multiprocessing (SMP, 对称多处理):"," The OS schedules threads across multiple processors, offering advantages in Performance, Availability, Incremental growth, and Scaling-.",[108,241220,241221,241224],{},[2627,241222,241223],{},"Distributed Operating Systems (分布式操作系统):"," Provides the illusion of a single main and secondary memory space across a cluster of separate computers.",[108,241226,241227,241230],{},[2627,241228,241229],{},"Object-oriented design (面向对象设计):"," Allows programmers to customize the OS without disrupting system integrity.",[32188,241232],{},[19,241234,241236],{"id":241235},"_5-fault-tolerance-容错","5. Fault Tolerance (容错)",[15,241238,241239,241242],{},[2627,241240,241241],{},"Fault tolerance (容错)"," is the ability of a system to continue normal operations despite the presence of hardware or software faults. This involves redundancy to increase reliability, balancing the costs against the critical nature of the resources.",[32188,241244],{},[19,241246,241248],{"id":241247},"_6-os-design-considerations-for-multiprocessor-and-multicore","6. OS Design Considerations for Multiprocessor and Multicore",[253,241250,241252],{"id":241251},"_61-smp-os-considerations","6.1 SMP OS Considerations",[15,241254,241255],{},"An SMP OS must manage processor and computer resources to provide the illusion of a uniprocessor system. Key design issues include-,:",[105,241257,241258,241261,241264,241267,241270],{},[108,241259,241260],{},"Simultaneous concurrent processes or threads,.",[108,241262,241263],{},"Scheduling,.",[108,241265,241266],{},"Synchronization,.",[108,241268,241269],{},"Memory management,.",[108,241271,241272],{},"Reliability and fault tolerance,.",[253,241274,241276],{"id":241275},"_62-multicore-os-considerations","6.2 Multicore OS Considerations",[15,241278,241279],{},"With \"many-core\" systems, the challenge is extracting parallelism from workloads-.",[105,241281,241282,241288,241295],{},[108,241283,241284,241287],{},[2627,241285,241286],{},"Parallelism within Applications:"," Developers divide applications into parallel tasks.",[108,241289,241290,241291,241294],{},"OS approaches like Apple's ",[2627,241292,241293],{},"Grand Central Dispatch (GCD)"," map tasks onto a pool of available threads.",[108,241296,241297],{},"Future architectures might drop the distinction between user and kernel mode, treating the OS more like a hypervisor while applications self-manage resources-.",{"title":29,"searchDepth":43,"depth":43,"links":241299},[241300,241305,241311,241317,241318,241319],{"id":240802,"depth":43,"text":240803,"children":241301},[241302,241303,241304],{"id":240832,"depth":49,"text":240833},{"id":240890,"depth":49,"text":240891},{"id":240912,"depth":49,"text":240913},{"id":240921,"depth":43,"text":240922,"children":241306},[241307,241308,241309,241310],{"id":240925,"depth":49,"text":240926},{"id":240950,"depth":49,"text":240951},{"id":240992,"depth":49,"text":240993},{"id":241016,"depth":49,"text":241017},{"id":241040,"depth":43,"text":241041,"children":241312},[241313,241314,241315,241316],{"id":241044,"depth":49,"text":241045},{"id":241077,"depth":49,"text":241078},{"id":241132,"depth":49,"text":241133},{"id":241165,"depth":49,"text":241166},{"id":241193,"depth":43,"text":241194},{"id":241235,"depth":43,"text":241236},{"id":241247,"depth":43,"text":241248,"children":241320},[241321,241322],{"id":241251,"depth":49,"text":241252},{"id":241275,"depth":49,"text":241276},{"order":37},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap2",{"title":240793,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap2","mMs1rIXfNOsq5oLjymjP19jWu_QGHYyT9YZlr7fM8kI",{"id":241329,"title":241330,"body":241331,"description":29,"extension":85,"meta":241799,"navigation":52,"path":241800,"seo":241801,"stem":241802,"__hash__":241803},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap3.md","Chap. 3 Note",{"type":8,"value":241332,"toc":241775},[241333,241337,241341,241344,241349,241353,241364,241406,241410,241417,241419,241423,241426,241440,241444,241447,241461,241465,241468,241499,241503,241514,241528,241534,241536,241540,241544,241550,241567,241571,241574,241576,241580,241596,241600,241607,241633,241637,241640,241664,241666,241670,241674,241677,241691,241695,241698,241715,241719,241722,241742,241752],[11,241334,241336],{"id":241335},"chapter-3-process-description-and-control","Chapter 3: Process Description and Control",[19,241338,241340],{"id":241339},"_1-the-concept-of-process-进程的概念","1. The Concept of Process (进程的概念)",[15,241342,241343],{},"The design of a multiprogramming operating system is built around the concept of the process. The OS must interleave the execution of multiple processes to maximize processor utilization, allocate resources avoiding deadlock, and support interprocess communication and user creation of processes.",[15,241345,241048,241346,241348],{},[2627,241347,241051],{}," can be defined as a program in execution, an instance of a running program, the entity assigned to a processor, or a unit of activity characterized by a single sequential thread of execution, a current state, and an associated set of system resources.",[253,241350,241352],{"id":241351},"_11-process-elements-进程的组成","1.1 Process Elements (进程的组成)",[15,241354,241355,241356,241359,241360,241363],{},"While a program is executing, the executing entity is called a process, which consists of two essential elements: ",[2627,241357,241358],{},"Program code (程序代码)"," and a ",[2627,241361,241362],{},"Set of data (数据集)"," associated with that code. It is uniquely characterized by elements including:",[105,241365,241366,241371,241376,241381,241386,241391,241396,241401],{},[108,241367,241368],{},[2627,241369,241370],{},"Identifier (标识符)",[108,241372,241373],{},[2627,241374,241375],{},"State (状态)",[108,241377,241378],{},[2627,241379,241380],{},"Priority (优先级)",[108,241382,241383],{},[2627,241384,241385],{},"Program counter (程序计数器)",[108,241387,241388],{},[2627,241389,241390],{},"Memory pointers (内存指针)",[108,241392,241393],{},[2627,241394,241395],{},"Context data (上下文数据)",[108,241397,241398],{},[2627,241399,241400],{},"I\u002FO status information (I\u002FO状态信息)",[108,241402,241403],{},[2627,241404,241405],{},"Accounting information (记账信息)",[253,241407,241409],{"id":241408},"_12-process-control-block-pcb-进程控制块","1.2 Process Control Block (PCB, 进程控制块)",[15,241411,241412,241413,241416],{},"The OS creates and manages a data structure called the ",[2627,241414,241415],{},"Process Control Block (PCB)",", which contains all the process elements mentioned above. It is the most important data structure in the OS because it allows the OS to interrupt a running process and later resume it exactly as if the interruption had not occurred.",[32188,241418],{},[19,241420,241422],{"id":241421},"_2-process-states-进程的状态","2. Process States (进程的状态)",[15,241424,241425],{},"To understand process states, we first define two terms:",[105,241427,241428,241434],{},[108,241429,241430,241433],{},[2627,241431,241432],{},"Trace (轨迹):"," The behavior of an individual process characterized by the sequence of instructions that execute for that process.",[108,241435,241436,241439],{},[2627,241437,241438],{},"Dispatcher (调度器):"," A small OS program that switches the processor from one process to another.",[253,241441,241443],{"id":241442},"_21-two-state-process-model-两态进程模型","2.1 Two-State Process Model (两态进程模型)",[15,241445,241446],{},"The simplest model defines two states:",[105,241448,241449,241455],{},[108,241450,241451,241454],{},[2627,241452,241453],{},"Running (运行态):"," The process is currently being executed.",[108,241456,241457,241460],{},[2627,241458,241459],{},"Not Running (非运行态):"," The process is waiting for an opportunity to execute.\nProcesses in the \"Not Running\" state are typically kept in a queue waiting their turn to be dispatched.",[253,241462,241464],{"id":241463},"_22-five-state-process-model-五态进程模型","2.2 Five-State Process Model (五态进程模型)",[15,241466,241467],{},"To handle processes that are waiting for I\u002FO or events, the \"Not Running\" state is split, leading to the five-state model:",[2230,241469,241470,241475,241481,241487,241493],{},[108,241471,241472,241474],{},[2627,241473,241453],{}," The process is currently executing.",[108,241476,241477,241480],{},[2627,241478,241479],{},"Ready (就绪态):"," The process is prepared to execute when given the opportunity.",[108,241482,241483,241486],{},[2627,241484,241485],{},"Blocked \u002F Waiting (阻塞态\u002F等待态):"," The process cannot execute until some event occurs (e.g., I\u002FO completion).",[108,241488,241489,241492],{},[2627,241490,241491],{},"New (新建态):"," A process just created but not yet admitted to the pool of executable processes by the OS.",[108,241494,241495,241498],{},[2627,241496,241497],{},"Exit (退出态):"," A process released from the executable pool because it halted or aborted.",[253,241500,241502],{"id":241501},"_23-suspended-processes-挂起进程","2.3 Suspended Processes (挂起进程)",[15,241504,241505,241506,241509,241510,241513],{},"If all processes in main memory are blocked, the processor becomes idle. To solve this, the OS uses ",[2627,241507,241508],{},"Swapping (交换)"," to move part or all of a process from main memory to disk, creating a ",[2627,241511,241512],{},"Suspend queue (挂起队列)",". This introduces two new states:",[105,241515,241516,241522],{},[108,241517,241518,241521],{},[2627,241519,241520],{},"Blocked\u002FSuspend (阻塞\u002F挂起):"," The process is in secondary memory and awaiting an event.",[108,241523,241524,241527],{},[2627,241525,241526],{},"Ready\u002FSuspend (就绪\u002F挂起):"," The process is in secondary memory but is available for execution as soon as it is loaded into main memory.",[15,241529,241530,241533],{},[2627,241531,241532],{},"Characteristics of a Suspended Process:"," It is not immediately available for execution, may or may not be waiting on an event, was placed in this state by an agent (itself, parent, or OS), and cannot be removed until explicitly ordered. Reasons for suspension include swapping, interactive user requests, timing, or parent process requests.",[32188,241535],{},[19,241537,241539],{"id":241538},"_3-process-creation-and-termination-进程的创建与终止","3. Process Creation and Termination (进程的创建与终止)",[253,241541,241543],{"id":241542},"_31-process-creation-进程创建","3.1 Process Creation (进程创建)",[15,241545,241546,241547,2547],{},"Reasons for creating a process include: new batch jobs, interactive logons, created by the OS to provide a service, or ",[2627,241548,241549],{},"Process Spawning (进程派生)",[105,241551,241552,241558],{},[108,241553,241554,241557],{},[2627,241555,241556],{},"Process Spawning:"," When the OS creates a process at the explicit request of another process.",[108,241559,241560,241561,194849,241564,2547],{},"The original process is the ",[2627,241562,241563],{},"Parent process (父进程)",[2627,241565,241566],{},"Child process (子进程)",[253,241568,241570],{"id":241569},"_32-process-termination-进程终止","3.2 Process Termination (进程终止)",[15,241572,241573],{},"A process terminates when it reaches a normal completion, time limit exceeded, memory unavailable, bounds violation, protection error, arithmetic error, I\u002FO failure, invalid instruction, or parent termination.",[32188,241575],{},[19,241577,241579],{"id":241578},"_4-process-description-and-control-structures-进程描述与控制结构","4. Process Description and Control Structures (进程描述与控制结构)",[15,241581,241582,241583,616,241586,616,241589,241592,241593,2547],{},"The OS manages system resources by maintaining four types of control tables: ",[2627,241584,241585],{},"Memory tables",[2627,241587,241588],{},"I\u002FO tables",[2627,241590,241591],{},"File tables",", and ",[2627,241594,241595],{},"Process tables",[253,241597,241599],{"id":241598},"_41-process-image-进程映像","4.1 Process Image (进程映像)",[15,241601,241602,241603,241606],{},"The collection of program, data, stack, and attributes is referred to as the ",[2627,241604,241605],{},"Process Image (进程映像)",". It typically includes:",[105,241608,241609,241615,241621,241627],{},[108,241610,241611,241614],{},[2627,241612,241613],{},"User Data:"," Modifiable part of user space.",[108,241616,241617,241620],{},[2627,241618,241619],{},"User Program:"," Program to be executed.",[108,241622,241623,241626],{},[2627,241624,241625],{},"System Stack:"," Used for procedure calls and parameter passing.",[108,241628,241629,241632],{},[2627,241630,241631],{},"Process Control Block (PCB):"," Information needed by the OS to control the process.",[253,241634,241636],{"id":241635},"_42-elements-of-the-process-control-block","4.2 Elements of the Process Control Block",[15,241638,241639],{},"The PCB information falls into three categories:",[2230,241641,241642,241648,241658],{},[108,241643,241644,241647],{},[2627,241645,241646],{},"Process Identification (进程标识):"," Numeric identifiers for the process, its parent, and the user.",[108,241649,241650,241653,241654,241657],{},[2627,241651,241652],{},"Processor State Information (处理器状态信息):"," User-visible registers, control and status registers (e.g., the ",[2627,241655,241656],{},"Program Status Word \u002F PSW (程序状态字)","), and stack pointers.",[108,241659,241660,241663],{},[2627,241661,241662],{},"Process Control Information (进程控制信息):"," Scheduling and state information, data structuring (pointers to queues), IPC flags, process privileges, memory management pointers, and resource ownership.",[32188,241665],{},[19,241667,241669],{"id":241668},"_5-process-control-进程控制","5. Process Control (进程控制)",[253,241671,241673],{"id":241672},"_51-modes-of-execution-执行模式","5.1 Modes of Execution (执行模式)",[15,241675,241676],{},"To protect OS tables and resources, processors support at least two execution modes:",[105,241678,241679,241685],{},[108,241680,241681,241684],{},[2627,241682,241683],{},"User Mode (用户态):"," A less-privileged mode where user programs typically execute.",[108,241686,241687,241690],{},[2627,241688,241689],{},"System Mode \u002F Kernel Mode \u002F Control Mode (系统态\u002F内核态\u002F控制态):"," A more-privileged mode where the OS kernel executes, granting complete control of the processor and memory.",[253,241692,241694],{"id":241693},"_52-steps-of-process-creation-进程创建的步骤","5.2 Steps of Process Creation (进程创建的步骤)",[15,241696,241697],{},"When the OS creates a new process, it typically:",[2230,241699,241700,241703,241706,241709,241712],{},[108,241701,241702],{},"Assigns a unique process identifier.",[108,241704,241705],{},"Allocates memory space for the process image.",[108,241707,241708],{},"Initializes the Process Control Block (PCB).",[108,241710,241711],{},"Sets the appropriate linkages (e.g., inserts it into the Ready queue).",[108,241713,241714],{},"Creates or expands other data structures (e.g., accounting files).",[253,241716,241718],{"id":241717},"_53-process-switching-进程切换","5.3 Process Switching (进程切换)",[15,241720,241721],{},"A process switch occurs when the OS gains control from the currently running process. The mechanisms triggering this are:",[105,241723,241724,241730,241736],{},[108,241725,241726,241729],{},[2627,241727,241728],{},"Interrupt (中断):"," External to the current instruction (e.g., Clock interrupt\u002Ftime slice expiration, I\u002FO interrupt, Memory fault).",[108,241731,241732,241735],{},[2627,241733,241734],{},"Trap (陷阱):"," An error or exception generated within the current process.",[108,241737,241738,241741],{},[2627,241739,241740],{},"Supervisor Call (系统调用):"," Explicit request to an OS function by the user program.",[15,241743,241744,241747,241748,241751],{},[2627,241745,241746],{},"Mode Switch vs. Process Switch (模式切换与进程切换):","\nA mode switch (user to kernel) does not necessarily change the state of the running process and has low overhead. A ",[2627,241749,241750],{},"Full Process Switch"," is much heavier and requires:",[2230,241753,241754,241757,241760,241763,241766,241769,241772],{},[108,241755,241756],{},"Saving the context of the processor.",[108,241758,241759],{},"Updating the PCB of the currently running process.",[108,241761,241762],{},"Moving the PCB to the appropriate queue (Ready, Blocked, etc.).",[108,241764,241765],{},"Selecting another process for execution.",[108,241767,241768],{},"Updating the PCB of the selected process.",[108,241770,241771],{},"Updating memory management data structures.",[108,241773,241774],{},"Restoring the context of the newly selected process.",{"title":29,"searchDepth":43,"depth":43,"links":241776},[241777,241781,241786,241790,241794],{"id":241339,"depth":43,"text":241340,"children":241778},[241779,241780],{"id":241351,"depth":49,"text":241352},{"id":241408,"depth":49,"text":241409},{"id":241421,"depth":43,"text":241422,"children":241782},[241783,241784,241785],{"id":241442,"depth":49,"text":241443},{"id":241463,"depth":49,"text":241464},{"id":241501,"depth":49,"text":241502},{"id":241538,"depth":43,"text":241539,"children":241787},[241788,241789],{"id":241542,"depth":49,"text":241543},{"id":241569,"depth":49,"text":241570},{"id":241578,"depth":43,"text":241579,"children":241791},[241792,241793],{"id":241598,"depth":49,"text":241599},{"id":241635,"depth":49,"text":241636},{"id":241668,"depth":43,"text":241669,"children":241795},[241796,241797,241798],{"id":241672,"depth":49,"text":241673},{"id":241693,"depth":49,"text":241694},{"id":241717,"depth":49,"text":241718},{"order":43},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap3",{"title":241330,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap3","8YNvkuAvOHOkGPQW2Fp3h1R0R2SyWKUua-N6qDRiJgU",{"id":241805,"title":241806,"body":241807,"description":29,"extension":85,"meta":242342,"navigation":52,"path":242343,"seo":242344,"stem":242345,"__hash__":242346},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap4.md","Chap. 4 Note",{"type":8,"value":241808,"toc":242317},[241809,241813,241817,241820,241834,241849,241853,241859,241873,241878,241892,241898,241902,241915,241941,241943,241947,241956,241960,241967,242004,242008,242011,242023,242027,242054,242056,242060,242064,242085,242089,242092,242112,242114,242118,242122,242160,242164,242167,242205,242207,242211,242214,242240,242244,242247,242255,242257,242261,242265],[11,241810,241812],{"id":241811},"chapter-4-threads-smp-and-microkernels-review-notes","Chapter 4: Threads, SMP, and Microkernels - Review Notes",[19,241814,241816],{"id":241815},"_1-processes-and-threads-进程与线程","1. Processes and Threads (进程与线程)",[15,241818,241819],{},"In traditional operating systems, the concept of a process embodies two independent characteristics:",[105,241821,241822,241828],{},[108,241823,241824,241827],{},[2627,241825,241826],{},"Resource ownership (资源所有权):"," A process includes a virtual address space to hold the process image and is allocated control or ownership of resources such as main memory, I\u002FO channels, I\u002FO devices, and files.",[108,241829,241830,241833],{},[2627,241831,241832],{},"Scheduling\u002Fexecution (调度\u002F执行):"," The execution of a process follows an execution path (trace) through one or more programs, and it is the entity that is scheduled and dispatched by the OS.",[15,241835,241836,241837,194939,241840,241843,241844,194939,241846,2547],{},"To distinguish between these two characteristics, the unit of dispatching is usually referred to as a ",[2627,241838,241839],{},"Thread (线程)",[2627,241841,241842],{},"Lightweight process (轻量级进程)",", while the unit of resource ownership is referred to as a ",[2627,241845,241051],{},[2627,241847,241848],{},"Task (任务)",[253,241850,241852],{"id":241851},"_11-multithreading-多线程","1.1 Multithreading (多线程)",[15,241854,241855,241858],{},[2627,241856,241857],{},"Multithreading"," refers to the ability of an OS to support multiple, concurrent paths of execution within a single process.",[105,241860,241861,241867],{},[108,241862,241863,241866],{},[2627,241864,241865],{},"In a multithreaded environment:"," A process is defined as the unit of resource allocation and a unit of protection. All of the threads of a process share the state and resources of that process.",[108,241868,241869,241872],{},[2627,241870,241871],{},"Thread Elements:"," Within a process, each thread has an execution state (Running, Ready, etc.), a saved thread context, an execution stack, per-thread static storage for local variables, and access to the shared memory and resources of its process.",[15,241874,241875],{},[2627,241876,241877],{},"Benefits of Threads (线程的优势):",[2230,241879,241880,241883,241886,241889],{},[108,241881,241882],{},"It takes far less time to create a new thread in an existing process than to create a brand-new process.",[108,241884,241885],{},"It takes less time to terminate a thread than a process.",[108,241887,241888],{},"It takes less time to switch between two threads within the same process than to switch between processes.",[108,241890,241891],{},"Threads enhance efficiency in communication between different executing programs because threads within the same process share memory and files, allowing them to communicate without invoking the kernel.",[15,241893,241894,241897],{},[2627,241895,241896],{},"Uses of Threads:","\nThreads are useful for foreground and background work, asynchronous processing, speeding up execution via parallel computation, and organizing a modular program structure.",[253,241899,241901],{"id":241900},"_12-thread-functionality-线程功能","1.2 Thread Functionality (线程功能)",[15,241903,241904,241905,616,241908,241592,241911,241914],{},"The key execution states for a thread are ",[2627,241906,241907],{},"Running (运行)",[2627,241909,241910],{},"Ready (就绪)",[2627,241912,241913],{},"Blocked (阻塞)",".\nThere are four basic thread operations associated with a change in thread state:",[105,241916,241917,241923,241929,241935],{},[108,241918,241919,241922],{},[2627,241920,241921],{},"Spawn (派生):"," When a new process is spawned, a thread is also spawned. A thread can subsequently spawn another thread within the same process.",[108,241924,241925,241928],{},[2627,241926,241927],{},"Block (阻塞):"," A thread blocks when it needs to wait for an event, saving its user registers, program counter, and stack pointers.",[108,241930,241931,241934],{},[2627,241932,241933],{},"Unblock (解除阻塞):"," When the event occurs, the thread is moved to the Ready queue.",[108,241936,241937,241940],{},[2627,241938,241939],{},"Finish (结束):"," When a thread completes, its register context and stacks are deallocated.",[32188,241942],{},[19,241944,241946],{"id":241945},"_2-types-of-threads-线程的类型","2. Types of Threads (线程的类型)",[15,241948,241949,241950,195562,241953,2547],{},"There are two broad categories of thread implementation: ",[2627,241951,241952],{},"User-Level Threads (ULT, 用户级线程)",[2627,241954,241955],{},"Kernel-Level Threads (KLT, 内核级线程)",[253,241957,241959],{"id":241958},"_21-user-level-threads-ult","2.1 User-Level Threads (ULT)",[15,241961,241962,241963,241966],{},"In a pure ULT facility, all of the work of thread management is done by the application (using a ",[2627,241964,241965],{},"threads library (线程库)","), and the kernel is not aware of the existence of threads.",[105,241968,241969,241985,241998],{},[108,241970,241971,241974],{},[2627,241972,241973],{},"Advantages:",[2230,241975,241976,241979,241982],{},[108,241977,241978],{},"Thread switching does not require kernel-mode privileges, saving the overhead of two mode switches.",[108,241980,241981],{},"Scheduling can be application-specific and tailored to the application's needs.",[108,241983,241984],{},"ULTs can run on any OS because no changes are required to the underlying kernel.",[108,241986,241987,241990],{},[2627,241988,241989],{},"Disadvantages:",[2230,241991,241992,241995],{},[108,241993,241994],{},"When a ULT executes a blocking system call, all of the threads within the entire process are blocked.",[108,241996,241997],{},"A multithreaded application cannot take advantage of multiprocessing since the kernel assigns one process to only one processor at a time.",[108,241999,242000,242003],{},[2627,242001,242002],{},"Jacketing (封装\u002F套套):"," A technique to overcome blocking threads by converting a blocking system call into a nonblocking application-level I\u002FO jacket routine.",[253,242005,242007],{"id":242006},"_22-kernel-level-threads-klt","2.2 Kernel-Level Threads (KLT)",[15,242009,242010],{},"In a pure KLT facility, all thread management work is done by the kernel, with applications using an API to access the kernel thread facility.",[105,242012,242013,242018],{},[108,242014,242015,242017],{},[2627,242016,241973],{}," The kernel can simultaneously schedule multiple threads from the same process on multiple processors, and blocking one thread does not block the entire process.",[108,242019,242020,242022],{},[2627,242021,241989],{}," The transfer of control from one thread to another within the same process requires a mode switch to the kernel, incurring much greater latency overhead than ULTs.",[253,242024,242026],{"id":242025},"_23-combined-approaches-and-other-arrangements","2.3 Combined Approaches and Other Arrangements",[105,242028,242029,242035],{},[108,242030,242031,242034],{},[2627,242032,242033],{},"Combined Approaches (组合方法):"," Thread creation is done entirely in user space, but multiple ULTs from a single application are mapped onto a smaller or equal number of KLTs (e.g., Solaris).",[108,242036,242037,242040],{},[2627,242038,242039],{},"Other Relationships:",[105,242041,242042,242048],{},[108,242043,242044,242047],{},[2627,242045,242046],{},"M:N (Many-to-Many):"," Explored in systems like TRIX where a thread can move from one domain (address space) to another.",[108,242049,242050,242053],{},[2627,242051,242052],{},"1:M (One-to-Many):"," A thread can migrate from one process environment to another, such as in distributed operating systems like Clouds (Ra kernel) and Emerald.",[32188,242055],{},[19,242057,242059],{"id":242058},"_3-multicore-and-multithreading-多核与多线程","3. Multicore and Multithreading (多核与多线程)",[253,242061,242063],{"id":242062},"_31-performance-on-multicore","3.1 Performance on Multicore",[105,242065,242066,242079],{},[108,242067,242068,242071,242072,242075,242076,242078],{},[2627,242069,242070],{},"Amdahl's Law (阿姆达尔定律):"," Defines the potential speedup of a program on multiple processors as ",[31,242073,242074],{},"Speedup = 1 \u002F ((1 - f) + f\u002FN)",", where ",[31,242077,31739],{}," is the infinitely parallelizable fraction. Even a small amount of serial code (e.g., 10%) severely limits the performance gain on multicore architectures.",[108,242080,242081,242084],{},[2627,242082,242083],{},"Beneficial Applications:"," Database management systems, multithreaded native applications, multiprocess applications, Java applications (running on the multithreaded JVM), and multi-instance applications scale well on multicore systems.",[253,242086,242088],{"id":242087},"_32-threading-granularity-线程粒度","3.2 Threading Granularity (线程粒度)",[15,242090,242091],{},"Using Valve's game software as an example, there are three options for threading granularity:",[2230,242093,242094,242100,242106],{},[108,242095,242096,242099],{},[2627,242097,242098],{},"Coarse threading (粗粒度线程):"," Individual modules (e.g., rendering, AI, physics) are assigned to individual processors.",[108,242101,242102,242105],{},[2627,242103,242104],{},"Fine-grained threading (细粒度线程):"," Many similar tasks (e.g., a loop) are spread across multiple processors.",[108,242107,242108,242111],{},[2627,242109,242110],{},"Hybrid threading (混合线程):"," A selective use of fine-grained threading for some systems, and single-threaded for other systems. This approach scales the best.",[32188,242113],{},[19,242115,242117],{"id":242116},"_4-windows-process-and-thread-management-windows进程与线程管理","4. Windows Process and Thread Management (Windows进程与线程管理)",[253,242119,242121],{"id":242120},"_41-process-and-thread-objects","4.1 Process and Thread Objects",[105,242123,242124,242130,242136,242142,242148,242154],{},[108,242125,242126,242129],{},[2627,242127,242128],{},"Windows Process:"," Represents an application and contains a virtual address space, executable code, open handles, a security access token, and at least one thread.",[108,242131,242132,242135],{},[2627,242133,242134],{},"Windows Thread:"," The dispatchable entity within a process that can be scheduled for execution. It maintains exception handlers, a scheduling priority, and a saved context.",[108,242137,242138,242141],{},[2627,242139,242140],{},"Job Object (作业对象):"," Allows groups of processes to be managed as a unit (e.g., enforcing limits or terminating them together).",[108,242143,242144,242147],{},[2627,242145,242146],{},"Thread Pool (线程池):"," A collection of worker threads used to efficiently execute asynchronous callbacks.",[108,242149,242150,242153],{},[2627,242151,242152],{},"Fiber (纤程):"," A unit of execution that must be manually scheduled by the application, running in the context of the thread that scheduled it.",[108,242155,242156,242159],{},[2627,242157,242158],{},"User-Mode Scheduling (UMS, 用户模式调度):"," A lightweight mechanism that allows applications to switch between UMS threads in user mode without involving the system scheduler.",[253,242161,242163],{"id":242162},"_42-thread-states-线程状态","4.2 Thread States (线程状态)",[15,242165,242166],{},"An existing Windows thread is in one of six states:",[105,242168,242169,242175,242181,242187,242193,242199],{},[108,242170,242171,242174],{},[2627,242172,242173],{},"Ready (就绪):"," May be scheduled for execution.",[108,242176,242177,242180],{},[2627,242178,242179],{},"Standby (备用):"," Selected to run next on a particular processor.",[108,242182,242183,242186],{},[2627,242184,242185],{},"Running (运行):"," Currently executing.",[108,242188,242189,242192],{},[2627,242190,242191],{},"Waiting (等待):"," Blocked on an event or voluntarily waiting for synchronization.",[108,242194,242195,242198],{},[2627,242196,242197],{},"Transition (过渡):"," Ready to run, but resources (like paged-out stack) are not currently available.",[108,242200,242201,242204],{},[2627,242202,242203],{},"Terminated (终止):"," Thread has completed or been killed.",[32188,242206],{},[19,242208,242210],{"id":242209},"_5-solaris-thread-and-smp-management-solaris线程与smp管理","5. Solaris Thread and SMP Management (Solaris线程与SMP管理)",[15,242212,242213],{},"Solaris employs a multilevel thread architecture utilizing four concepts:",[2230,242215,242216,242222,242228,242234],{},[108,242217,242218,242221],{},[2627,242219,242220],{},"Process (进程):"," The normal UNIX process with an address space.",[108,242223,242224,242227],{},[2627,242225,242226],{},"User-Level Threads (ULT, 用户级线程):"," Invisible to the OS, implemented via a threads library in the user address space.",[108,242229,242230,242233],{},[2627,242231,242232],{},"Lightweight Processes (LWP, 轻量级进程):"," Acts as a mapping between ULTs and kernel threads.",[108,242235,242236,242239],{},[2627,242237,242238],{},"Kernel Threads (内核线程):"," The fundamental entities scheduled and dispatched to system processors.",[253,242241,242243],{"id":242242},"_51-interrupts-as-threads-作为线程的中断","5.1 Interrupts as Threads (作为线程的中断)",[15,242245,242246],{},"Solaris converts hardware interrupts to kernel threads to reduce overhead and simplify synchronization.",[105,242248,242249,242252],{},[108,242250,242251],{},"Interrupt threads have their own identifier, priority, context, and stack.",[108,242253,242254],{},"They use standard mutual exclusion primitives for synchronization and are assigned higher priorities than all other kernel threads.",[32188,242256],{},[19,242258,242260],{"id":242259},"_6-linux-process-and-thread-management-linux进程与线程管理","6. Linux Process and Thread Management (Linux进程与线程管理)",[253,242262,242264],{"id":242263},"_61-linux-tasks-and-threads","6.1 Linux Tasks and Threads",[105,242266,242267,242292,242298],{},[108,242268,242269,242272,242273,242276,242277,616,242280,616,242283,616,242286,241592,242289,2547],{},[2627,242270,242271],{},"Task (任务):"," A process or task in Linux is represented by a ",[31,242274,242275],{},"task_struct"," data structure. The execution states include ",[2627,242278,242279],{},"Running (执行\u002F就绪)",[2627,242281,242282],{},"Interruptible (可中断阻塞)",[2627,242284,242285],{},"Uninterruptible (不可中断阻塞)",[2627,242287,242288],{},"Stopped (停止)",[2627,242290,242291],{},"Zombie (僵尸)",[108,242293,242294,242297],{},[2627,242295,242296],{},"Threads in Linux:"," Linux does not recognize a strict distinction between threads and processes. User-level threads are mapped into kernel-level processes that share the same group ID.",[108,242299,242300,242306,242307,242309,242310,242313,242314,242316],{},[2627,242301,240905,242302,242305],{},[31,242303,242304],{},"clone()"," System Call:"," A new process is created using the ",[31,242308,242304],{}," command, which allows the cloned process to share resources (like files and virtual memory) with the parent. The traditional ",[31,242311,242312],{},"fork()"," is just ",[31,242315,242304],{}," with all sharing flags cleared.",{"title":29,"searchDepth":43,"depth":43,"links":242318},[242319,242323,242328,242332,242336,242339],{"id":241815,"depth":43,"text":241816,"children":242320},[242321,242322],{"id":241851,"depth":49,"text":241852},{"id":241900,"depth":49,"text":241901},{"id":241945,"depth":43,"text":241946,"children":242324},[242325,242326,242327],{"id":241958,"depth":49,"text":241959},{"id":242006,"depth":49,"text":242007},{"id":242025,"depth":49,"text":242026},{"id":242058,"depth":43,"text":242059,"children":242329},[242330,242331],{"id":242062,"depth":49,"text":242063},{"id":242087,"depth":49,"text":242088},{"id":242116,"depth":43,"text":242117,"children":242333},[242334,242335],{"id":242120,"depth":49,"text":242121},{"id":242162,"depth":49,"text":242163},{"id":242209,"depth":43,"text":242210,"children":242337},[242338],{"id":242242,"depth":49,"text":242243},{"id":242259,"depth":43,"text":242260,"children":242340},[242341],{"id":242263,"depth":49,"text":242264},{"order":56},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap4",{"title":241806,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap4","ztcm11doFA1H6S3V5-BuqAXB6z1_7t9v1EMM-1UEed8",{"id":242348,"title":242349,"body":242350,"description":29,"extension":85,"meta":242822,"navigation":52,"path":242823,"seo":242824,"stem":242825,"__hash__":242826},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap5.md","Chap. 5 Note",{"type":8,"value":242351,"toc":242803},[242352,242356,242360,242363,242367,242411,242415,242418,242438,242440,242444,242447,242479,242481,242485,242488,242502,242504,242508,242512,242515,242519,242522,242557,242559,242563,242571,242601,242603,242607,242610,242663,242665,242669,242679,242742,242744,242748,242752,242768,242772],[11,242353,242355],{"id":242354},"chapter-5-concurrency-mutual-exclusion-and-synchronization-review-notes","Chapter 5: Concurrency: Mutual Exclusion and Synchronization - Review Notes",[19,242357,242359],{"id":242358},"_1-principles-of-concurrency-并发的原理","1. Principles of Concurrency (并发的原理)",[15,242361,242362],{},"Concurrency is fundamental to operating system design and arises in three different contexts: multiple applications (multiprogramming), structured applications, and operating system structure.",[253,242364,242366],{"id":242365},"_11-key-terminology-核心术语","1.1 Key Terminology (核心术语)",[105,242368,242369,242375,242381,242387,242393,242399,242405],{},[108,242370,242371,242374],{},[2627,242372,242373],{},"Atomic operation (原子操作):"," A function or action implemented as a sequence of one or more instructions that appears to be indivisible; no other process can see an intermediate state or interrupt the operation.",[108,242376,242377,242380],{},[2627,242378,242379],{},"Critical section (临界区):"," A section of code within a process that requires access to shared resources and must not be executed while another process is in a corresponding section of code.",[108,242382,242383,242386],{},[2627,242384,242385],{},"Deadlock (死锁):"," A situation in which two or more processes are unable to proceed because each is waiting for one of the others to do something.",[108,242388,242389,242392],{},[2627,242390,242391],{},"Livelock (活锁):"," A situation in which two or more processes continuously change their states in response to changes in the other process(es) without doing any useful work.",[108,242394,242395,242398],{},[2627,242396,242397],{},"Mutual exclusion (互斥):"," The requirement that when one process is in a critical section that accesses shared resources, no other process may be in a critical section that accesses any of those shared resources.",[108,242400,242401,242404],{},[2627,242402,242403],{},"Race condition (竞态条件):"," A situation in which multiple threads or processes read and write a shared data item, and the final result depends on the relative timing of their execution.",[108,242406,242407,242410],{},[2627,242408,242409],{},"Starvation (饥饿):"," A situation in which a runnable process is overlooked indefinitely by the scheduler; although it is able to proceed, it is never chosen.",[253,242412,242414],{"id":242413},"_12-process-interaction-进程交互","1.2 Process Interaction (进程交互)",[15,242416,242417],{},"Processes interact in three main ways based on their degree of awareness of each other:",[2230,242419,242420,242426,242432],{},[108,242421,242422,242425],{},[2627,242423,242424],{},"Processes unaware of each other (Competition \u002F 竞争):"," These are independent processes that are not intended to work together. The OS must regulate access to shared resources, which can lead to control problems like mutual exclusion, deadlock (with renewable resources), and starvation.",[108,242427,242428,242431],{},[2627,242429,242430],{},"Processes indirectly aware of each other (Cooperation by sharing \u002F 通过共享合作):"," These processes share access to some object, such as an I\u002FO buffer, and exhibit cooperation in sharing. The control problems include mutual exclusion, deadlock, starvation, and the added requirement of data coherence.",[108,242433,242434,242437],{},[2627,242435,242436],{},"Processes directly aware of each other (Cooperation by communication \u002F 通过通信合作):"," These processes communicate by process ID and are designed to work jointly on some activity. The control problems here are deadlock (with consumable resources) and starvation, but mutual exclusion is not required since nothing is explicitly shared.",[32188,242439],{},[19,242441,242443],{"id":242442},"_2-requirements-for-mutual-exclusion-互斥的要求","2. Requirements for Mutual Exclusion (互斥的要求)",[15,242445,242446],{},"Any facility or capability that provides support for mutual exclusion must meet six requirements:",[2230,242448,242449,242455,242458,242464,242467,242473],{},[108,242450,242451,242454],{},[2627,242452,242453],{},"Mutual exclusion must be enforced:"," Only one process at a time is allowed into its critical section.",[108,242456,242457],{},"A process that halts in its noncritical section must do so without interfering with other processes.",[108,242459,242460,242461,26192],{},"It must not be possible for a process requiring access to a critical section to be delayed indefinitely (i.e., ",[2627,242462,242463],{},"no deadlock or starvation",[108,242465,242466],{},"When no process is in a critical section, any process that requests entry must be permitted to enter without delay.",[108,242468,242469,242472],{},[2627,242470,242471],{},"No assumptions"," are made about relative process speeds or the number of processors.",[108,242474,242475,242476,2547],{},"A process remains inside its critical section for a ",[2627,242477,242478],{},"finite time only",[32188,242480],{},[19,242482,242484],{"id":242483},"_3-mutual-exclusion-software-approaches-互斥的软件方法","3. Mutual Exclusion: Software Approaches (互斥的软件方法)",[15,242486,242487],{},"Software approaches rely entirely on the processes coordinating with each other without explicit support from the OS or programming language.",[105,242489,242490,242496],{},[108,242491,242492,242495],{},[2627,242493,242494],{},"Dekker’s Algorithm:"," A progressively developed solution for two processes that avoids the pitfalls of strict alternation, deadlock, and livelock.",[108,242497,242498,242501],{},[2627,242499,242500],{},"Peterson’s Algorithm:"," A simpler, elegant solution for two processes that uses a flag array to indicate intent to enter the critical section and a turn variable to resolve simultaneity conflicts.",[32188,242503],{},[19,242505,242507],{"id":242506},"_4-mutual-exclusion-hardware-support-互斥的硬件支持","4. Mutual Exclusion: Hardware Support (互斥的硬件支持)",[253,242509,242511],{"id":242510},"_41-interrupt-disabling-禁用中断","4.1 Interrupt Disabling (禁用中断)",[15,242513,242514],{},"In a uniprocessor system, mutual exclusion can be guaranteed by preventing a process from being interrupted. A process can disable interrupts before entering its critical section and enable them after exiting. However, this degrades execution efficiency and does not work in a multiprocessor architecture where multiple processes can execute simultaneously.",[253,242516,242518],{"id":242517},"_42-special-machine-instructions-专用机器指令","4.2 Special Machine Instructions (专用机器指令)",[15,242520,242521],{},"Processors offer machine instructions that carry out two actions atomically (in a single instruction fetch cycle).",[105,242523,242524,242530,242536],{},[108,242525,242526,242529],{},[2627,242527,242528],{},"Compare and Swap Instruction (比较并交换指令):"," Compares a memory value with a test value, and if they are the same, a new value is written to the memory location.",[108,242531,242532,242535],{},[2627,242533,242534],{},"Exchange Instruction (交换指令):"," Exchanges the contents of a register with that of a memory location.",[108,242537,242538,242541],{},[2627,242539,242540],{},"Properties of Machine-Instruction Approach (优缺点):",[105,242542,242543,242548],{},[108,242544,242545,242547],{},[71807,242546,241973],{}," Applicable to any number of processes on either a single processor or multiple processors sharing memory, simple to verify, and supports multiple critical sections.",[108,242549,242550,242552,242553,242556],{},[71807,242551,241989],{}," Employs ",[2627,242554,242555],{},"Busy waiting \u002F Spin waiting (忙等待\u002F自旋等待)"," which consumes processor time, starvation is possible since process selection is arbitrary, and deadlock is possible if a higher-priority process interrupts a lower-priority one holding the lock.",[32188,242558],{},[19,242560,242562],{"id":242561},"_5-semaphores-信号量","5. Semaphores (信号量)",[15,242564,242565,242566,242568,242569,26192],{},"A semaphore is an integer value used for signaling among processes, operated on by only three atomic operations: initialize, decrement (",[31,242567,197007],{},"), and increment (",[31,242570,197010],{},[105,242572,242573,242578,242583,242589,242595],{},[108,242574,240905,242575,242577],{},[31,242576,197007],{}," operation decrements the semaphore value; if the value becomes negative, the executing process is blocked.",[108,242579,240905,242580,242582],{},[31,242581,197010],{}," operation increments the semaphore value; if the resulting value is less than or equal to zero, a blocked process is unblocked.",[108,242584,242585,242588],{},[2627,242586,242587],{},"Binary semaphore (二元信号量):"," A semaphore that takes on only the values 0 and 1.",[108,242590,242591,242594],{},[2627,242592,242593],{},"Mutex (互斥锁):"," Similar to a binary semaphore, but the key difference is that the process that locks the mutex must be the one to unlock it.",[108,242596,242597,242600],{},[2627,242598,242599],{},"Strong vs. Weak Semaphores (强信号量与弱信号量):"," A strong semaphore uses a first-in-first-out (FIFO) policy to unblock processes, guaranteeing freedom from starvation. A weak semaphore does not specify the order in which processes are removed from the queue.",[32188,242602],{},[19,242604,242606],{"id":242605},"_6-monitors-管程","6. Monitors (管程)",[15,242608,242609],{},"A monitor is a programming language construct that encapsulates variables, access procedures, and initialization code within an abstract data type. It provides equivalent functionality to semaphores but is easier to control.",[105,242611,242612,242615,242618,242632],{},[108,242613,242614],{},"Local data variables are accessible only by the monitor's procedures.",[108,242616,242617],{},"Only one process may be executing in the monitor at a time, inherently enforcing mutual exclusion.",[108,242619,242620,242623,242624,242627,242628,242631],{},[2627,242621,242622],{},"Condition variables (条件变量):"," Used to synchronize processes within the monitor; operated on by ",[31,242625,242626],{},"cwait"," (suspends the calling process) and ",[31,242629,242630],{},"csignal"," (resumes a blocked process).",[108,242633,242634,242637],{},[2627,242635,242636],{},"Hoare Monitor vs. Mesa Monitor (两种管程模型):",[105,242638,242639,242645],{},[108,242640,242641,242644],{},[71807,242642,242643],{},"Hoare Monitor:"," Requires that when a process issues a signal, the waiting process immediately runs, meaning the signaling process must exit or block.",[108,242646,242647,242650,242651,242654,242655,242658,242659,242662],{},[71807,242648,242649],{},"Mesa Monitor (Lampson\u002FRedell):"," Uses a ",[31,242652,242653],{},"cnotify"," primitive instead of a signal, allowing the signaling process to continue executing. The waiting process is placed on a ready queue and must recheck the condition using a ",[31,242656,242657],{},"while"," loop when it resumes. It is less prone to error and supports a ",[31,242660,242661],{},"cbroadcast"," primitive to wake all waiting processes.",[32188,242664],{},[19,242666,242668],{"id":242667},"_7-message-passing-消息传递","7. Message Passing (消息传递)",[15,242670,242671,242672,195562,242675,242678],{},"Message passing provides a means for processes to exchange information and perform synchronization, typically utilizing ",[31,242673,242674],{},"send",[31,242676,242677],{},"receive"," primitives.",[105,242680,242681,242707,242736],{},[108,242682,242683,242686,242687],{},[2627,242684,242685],{},"Synchronization (同步):"," Both sender and receiver can be blocking or nonblocking.\n",[105,242688,242689,242695,242701],{},[108,242690,242691,242694],{},[71807,242692,242693],{},"Blocking send, blocking receive:"," A rendezvous allowing tight synchronization.",[108,242696,242697,242700],{},[71807,242698,242699],{},"Nonblocking send, blocking receive:"," The most useful combination, often used for server processes providing services.",[108,242702,242703,242706],{},[71807,242704,242705],{},"Nonblocking send, nonblocking receive:"," Neither party is required to wait.",[108,242708,242709,242712],{},[2627,242710,242711],{},"Addressing (寻址):",[105,242713,242714,242723],{},[108,242715,242716,242719,242720,242722],{},[71807,242717,242718],{},"Direct addressing (直接寻址):"," The ",[31,242721,242674],{}," primitive includes a specific identifier of the destination process.",[108,242724,242725,242728,242729,194939,242732,242735],{},[71807,242726,242727],{},"Indirect addressing (间接寻址):"," Messages are sent to a shared data structure consisting of queues, generally referred to as ",[2627,242730,242731],{},"mailboxes (信箱)",[2627,242733,242734],{},"ports (端口)",". This allows for one-to-one, many-to-one, one-to-many, or many-to-many relationships.",[108,242737,242738,242741],{},[2627,242739,242740],{},"Message Format (消息格式):"," Generally divided into a header (containing source, destination, length, type, and control info) and a body (the actual contents).",[32188,242743],{},[19,242745,242747],{"id":242746},"_8-classic-concurrency-problems-经典并发问题","8. Classic Concurrency Problems (经典并发问题)",[253,242749,242751],{"id":242750},"_81-the-producerconsumer-problem-生产者消费者问题","8.1 The Producer\u002FConsumer Problem (生产者\u002F消费者问题)",[105,242753,242754,242759,242765],{},[108,242755,242756,242758],{},[2627,242757,193861],{}," One or more producers generate data and place it in a buffer, while a single consumer takes items out of the buffer one at a time.",[108,242760,242761,242764],{},[2627,242762,242763],{},"Constraints:"," The system must prevent overlap of buffer operations, ensure the producer does not add to a full buffer, and ensure the consumer does not remove data from an empty buffer.",[108,242766,242767],{},"Can be solved using semaphores, monitors, or message passing.",[253,242769,242771],{"id":242770},"_82-the-readerswriters-problem-读者写者问题","8.2 The Readers\u002FWriters Problem (读者\u002F写者问题)",[105,242773,242774,242779,242784],{},[108,242775,242776,242778],{},[2627,242777,193861],{}," A data area is shared among multiple processes. Readers only read the data area, and writers only write to it.",[108,242780,242781,242783],{},[2627,242782,242763],{}," Any number of readers may simultaneously read the file; only one writer at a time may write to the file; if a writer is writing, no reader may read it.",[108,242785,242786,242789],{},[2627,242787,242788],{},"Priority Strategies:",[105,242790,242791,242797],{},[108,242792,242793,242796],{},[71807,242794,242795],{},"Readers Have Priority:"," Writers are subject to starvation as long as at least one reader is continuously reading.",[108,242798,242799,242802],{},[71807,242800,242801],{},"Writers Have Priority:"," Guarantees no new readers are allowed access once a writer has declared a desire to write.",{"title":29,"searchDepth":43,"depth":43,"links":242804},[242805,242809,242810,242811,242815,242816,242817,242818],{"id":242358,"depth":43,"text":242359,"children":242806},[242807,242808],{"id":242365,"depth":49,"text":242366},{"id":242413,"depth":49,"text":242414},{"id":242442,"depth":43,"text":242443},{"id":242483,"depth":43,"text":242484},{"id":242506,"depth":43,"text":242507,"children":242812},[242813,242814],{"id":242510,"depth":49,"text":242511},{"id":242517,"depth":49,"text":242518},{"id":242561,"depth":43,"text":242562},{"id":242605,"depth":43,"text":242606},{"id":242667,"depth":43,"text":242668},{"id":242746,"depth":43,"text":242747,"children":242819},[242820,242821],{"id":242750,"depth":49,"text":242751},{"id":242770,"depth":49,"text":242771},{"order":62},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap5",{"title":242349,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap5","GP9MEBbP3KVgvkzQ1TULVnh_PBAiqrNRNiIp9CXwdm8",{"id":242828,"title":242829,"body":242830,"description":29,"extension":85,"meta":243378,"navigation":52,"path":243379,"seo":243380,"stem":243381,"__hash__":243382},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap6.md","Chap. 6 Note",{"type":8,"value":242831,"toc":243353},[242832,242836,242840,242846,242850,242864,242868,242874,242899,242903,242906,242934,242936,242940,242943,242975,242977,242981,242988,242992,242995,242999,243005,243035,243037,243041,243044,243048,243055,243078,243082,243085,243111,243113,243117,243120,243146,243148,243152,243155,243169,243171,243175,243179,243182,243218,243222,243252,243256,243281,243285,243333,243337],[11,242833,242835],{"id":242834},"chapter-6-concurrency-deadlock-and-starvation-review-notes","Chapter 6: Concurrency: Deadlock and Starvation - Review Notes",[19,242837,242839],{"id":242838},"_1-principles-of-deadlock-死锁的原理","1. Principles of Deadlock (死锁的原理)",[15,242841,242842,242845],{},[2627,242843,242844],{},"Deadlock (死锁)"," is defined as the permanent blocking of a set of processes that either compete for system resources or communicate with each other. It occurs when each process in the set is blocked awaiting an event (typically the freeing of a requested resource) that can only be triggered by another blocked process in the same set.",[253,242847,242849],{"id":242848},"_11-types-of-resources-资源类型","1.1 Types of Resources (资源类型)",[105,242851,242852,242858],{},[108,242853,242854,242857],{},[2627,242855,242856],{},"Reusable Resources (可重用资源):"," A resource that can be safely used by only one process at a time and is not depleted by that use. Examples include processors, I\u002FO channels, main and secondary memory, files, databases, and semaphores.",[108,242859,242860,242863],{},[2627,242861,242862],{},"Consumable Resources (可消耗资源):"," A resource that can be created (produced) and destroyed (consumed). There is typically no limit on the number of consumable resources of a particular type. Examples include interrupts, signals, messages, and information in I\u002FO buffers.",[253,242865,242867],{"id":242866},"_12-resource-allocation-graphs-资源分配图","1.2 Resource Allocation Graphs (资源分配图)",[15,242869,241048,242870,242873],{},[2627,242871,242872],{},"Resource Allocation Graph (RAG)"," is a directed graph that depicts the state of the system regarding resources and processes.",[105,242875,242876,242882,242896],{},[108,242877,242878,242881],{},[2627,242879,242880],{},"Nodes:"," Processes (represented as circles) and Resources (represented as squares with dots inside for each instance).",[108,242883,242884,242887,242888,242891,242892,242895],{},[2627,242885,242886],{},"Edges:"," A directed edge from a process to a resource indicates a ",[2627,242889,242890],{},"Request (请求)","; an edge from a resource to a process indicates it is ",[2627,242893,242894],{},"Held by (被持有\u002F分配给)"," that process.",[108,242897,242898],{},"A circular chain of processes and resources in the graph typically illustrates a deadlock situation.",[253,242900,242902],{"id":242901},"_13-the-conditions-for-deadlock-死锁的条件","1.3 The Conditions for Deadlock (死锁的条件)",[15,242904,242905],{},"For a deadlock to exist, four conditions must be met:",[2230,242907,242908,242913,242919,242925],{},[108,242909,242910,242912],{},[2627,242911,242397],{}," Only one process may use a resource at a time.",[108,242914,242915,242918],{},[2627,242916,242917],{},"Hold and wait (占有并等待):"," A process may hold allocated resources while awaiting assignment of other resources.",[108,242920,242921,242924],{},[2627,242922,242923],{},"No preemption (非抢占):"," No resource can be forcibly removed from a process holding it.",[108,242926,242927,242930,242931],{},[2627,242928,242929],{},"Circular wait (循环等待):"," A closed chain of processes exists, such that each process holds at least one resource needed by the next process in the chain.\n",[71807,242932,242933],{},"(Note: Conditions 1-3 are necessary, but Condition 4 makes them sufficient for a deadlock to actually occur.)",[32188,242935],{},[19,242937,242939],{"id":242938},"_2-deadlock-prevention-死锁预防","2. Deadlock Prevention (死锁预防)",[15,242941,242942],{},"Deadlock prevention involves designing the system to exclude the possibility of deadlock by breaking at least one of the four necessary conditions.",[105,242944,242945,242951,242960,242966],{},[108,242946,242947,242950],{},[2627,242948,242949],{},"Preventing Mutual Exclusion:"," Generally cannot be disallowed. If a resource requires exclusive access, the OS must support it.",[108,242952,242953,242956,242957,242959],{},[2627,242954,242955],{},"Preventing Hold and Wait:"," Require that a process request all of its required resources at one time and block it until all can be granted simultaneously. ",[71807,242958,241989],{}," Highly inefficient, resources may remain unused for long periods, and modular applications may not know all required resources in advance.",[108,242961,242962,242965],{},[2627,242963,242964],{},"Preventing No Preemption:"," If a process holding resources is denied a further request, it must release its original resources and request them again later. Alternatively, the OS may preempt a lower-priority process.",[108,242967,242968,242971,242972,242974],{},[2627,242969,242970],{},"Preventing Circular Wait:"," Define a linear ordering of resource types. A process can only request resources that follow its currently held resources in the ordering. ",[71807,242973,241989],{}," Can be inefficient and deny resource access unnecessarily.",[32188,242976],{},[19,242978,242980],{"id":242979},"_3-deadlock-avoidance-死锁避免","3. Deadlock Avoidance (死锁避免)",[15,242982,242983,242984,242987],{},"Unlike prevention, ",[2627,242985,242986],{},"Deadlock Avoidance"," allows the first three necessary conditions but makes dynamic, judicious choices to assure that the deadlock point is never reached. It requires knowledge of future process resource requests.",[253,242989,242991],{"id":242990},"_31-process-initiation-denial-进程启动拒绝","3.1 Process Initiation Denial (进程启动拒绝)",[15,242993,242994],{},"Do not start a process if its total demands, added to the maximum claims of all current processes, might exceed the total resources in the system. It is conservative and assumes all processes will make their maximum claims simultaneously.",[253,242996,242998],{"id":242997},"_32-resource-allocation-denial-bankers-algorithm-资源分配拒绝-银行家算法","3.2 Resource Allocation Denial \u002F Banker's Algorithm (资源分配拒绝 \u002F 银行家算法)",[15,243000,243001,243002,2547],{},"Proposed by Dijkstra, this algorithm dynamically checks if granting a request will leave the system in a ",[2627,243003,243004],{},"Safe state (安全状态)",[105,243006,243007,243013,243023,243029],{},[108,243008,243009,243012],{},[2627,243010,243011],{},"Safe state:"," A state where there is at least one sequence of resource allocations to processes that allows all processes to run to completion without deadlock.",[108,243014,243015,243018,243019,243022],{},[2627,243016,243017],{},"Unsafe state (不安全状态):"," A state that is not safe, which has the ",[71807,243020,243021],{},"potential"," for deadlock.",[108,243024,243025,243028],{},[2627,243026,243027],{},"Mechanism:"," When a process requests resources, the OS temporarily assumes the request is granted, updates the system state, and checks if the new state is safe. If safe, it grants the request; if unsafe, the process is blocked.",[108,243030,243031,243034],{},[71807,243032,243033],{},"Restrictions:"," Maximum requirements must be stated in advance, processes must be independent, and the number of resources must be fixed.",[32188,243036],{},[19,243038,243040],{"id":243039},"_4-deadlock-detection-and-recovery-死锁检测与恢复","4. Deadlock Detection and Recovery (死锁检测与恢复)",[15,243042,243043],{},"Deadlock detection strategies do not limit resource access. Instead, the OS periodically executes an algorithm to check for the presence of the circular wait condition.",[253,243045,243047],{"id":243046},"_41-deadlock-detection-algorithm-死锁检测算法","4.1 Deadlock Detection Algorithm (死锁检测算法)",[15,243049,243050,243051,243054],{},"The algorithm marks processes that are ",[71807,243052,243053],{},"not"," part of a deadlocked set.",[2230,243056,243057,243060,243066,243072],{},[108,243058,243059],{},"Mark processes with no allocated resources.",[108,243061,243062,243063,243065],{},"Initialize a temporary vector ",[31,243064,56852],{}," with the currently available resources.",[108,243067,243068,243069,243071],{},"Find an unmarked process whose requested resources are less than or equal to ",[31,243070,56852],{},". If none exists, terminate.",[108,243073,243074,243075,243077],{},"If found, mark the process, add its currently allocated resources to ",[31,243076,56852],{},", and return to step 3.\nIf any unmarked processes remain when the algorithm terminates, they are deadlocked.",[253,243079,243081],{"id":243080},"_42-recovery-strategies-恢复策略","4.2 Recovery Strategies (恢复策略)",[15,243083,243084],{},"Once deadlock is detected, the OS can recover by:",[2230,243086,243087,243093,243099,243105],{},[108,243088,243089,243092],{},[2627,243090,243091],{},"Abort all deadlocked processes (终止所有死锁进程):"," The most common solution.",[108,243094,243095,243098],{},[2627,243096,243097],{},"Back up to a checkpoint and restart:"," Rollback all processes, risking recurring deadlock.",[108,243100,243101,243104],{},[2627,243102,243103],{},"Successively abort deadlocked processes:"," Terminate one by one based on minimum cost until the deadlock is resolved.",[108,243106,243107,243110],{},[2627,243108,243109],{},"Successively preempt resources:"," Preempt resources one by one and rollback the affected processes until resolved.",[32188,243112],{},[19,243114,243116],{"id":243115},"_5-an-integrated-deadlock-strategy-综合死锁策略","5. An Integrated Deadlock Strategy (综合死锁策略)",[15,243118,243119],{},"Instead of relying on a single strategy, systems can group resources into classes and apply the most appropriate technique for each:",[105,243121,243122,243128,243134,243140],{},[108,243123,243124,243127],{},[2627,243125,243126],{},"Swappable space:"," Deadlock prevention (hold-and-wait) or avoidance.",[108,243129,243130,243133],{},[2627,243131,243132],{},"Process resources (devices, files):"," Deadlock avoidance or prevention by resource ordering.",[108,243135,243136,243139],{},[2627,243137,243138],{},"Main memory:"," Deadlock prevention by preemption (swapping).",[108,243141,243142,243145],{},[2627,243143,243144],{},"Internal resources:"," Prevention by resource ordering.",[32188,243147],{},[19,243149,243151],{"id":243150},"_6-the-dining-philosophers-problem-哲学家就餐问题","6. The Dining Philosophers Problem (哲学家就餐问题)",[15,243153,243154],{},"A classic synchronization problem illustrating deadlock and starvation issues. Five philosophers sit around a table with a bowl of spaghetti and five forks. A philosopher needs two forks to eat.",[105,243156,243157,243163],{},[108,243158,243159,243162],{},[2627,243160,243161],{},"Solution using Semaphores (使用信号量):"," If all pick up their left fork simultaneously, deadlock occurs. A fix is to limit the room to four philosophers at a time, ensuring at least one can get two forks.",[108,243164,243165,243168],{},[2627,243166,243167],{},"Solution using a Monitor (使用管程):"," Uses condition variables for the forks. A philosopher can only pick up forks if both are available, and because only one process can enter the monitor at a time, deadlock is prevented.",[32188,243170],{},[19,243172,243174],{"id":243173},"_7-os-concurrency-mechanisms-操作系统的并发机制","7. OS Concurrency Mechanisms (操作系统的并发机制)",[253,243176,243178],{"id":243177},"_71-unix","7.1 UNIX",[15,243180,243181],{},"UNIX provides several mechanisms for Interprocess Communication (IPC) and synchronization:",[105,243183,243184,243190,243196,243202,243212],{},[108,243185,243186,243189],{},[2627,243187,243188],{},"Pipes (管道):"," A circular buffer allowing two processes to communicate on the producer-consumer model.",[108,243191,243192,243195],{},[2627,243193,243194],{},"Messages (消息):"," Data exchange via message queues.",[108,243197,243198,243201],{},[2627,243199,243200],{},"Shared Memory (共享内存):"," Direct memory sharing between processes.",[108,243203,243204,243207,243208,243211],{},[2627,243205,243206],{},"Semaphores (信号量):"," Usually created in sets. An atomic ",[31,243209,243210],{},"semop"," system call can perform operations on multiple semaphores simultaneously.",[108,243213,243214,243217],{},[2627,243215,243216],{},"Signals (信号):"," Software mechanisms informing a process of asynchronous events.",[253,243219,243221],{"id":243220},"_72-linux-kernel","7.2 Linux Kernel",[105,243223,243224,243230,243240,243246],{},[108,243225,243226,243229],{},[2627,243227,243228],{},"Atomic Operations (原子操作):"," Simple synchronization executing without interruption (e.g., atomic integer or bitmap operations).",[108,243231,243232,243235,243236,243239],{},[2627,243233,243234],{},"Spinlocks (自旋锁):"," A lock where a waiting thread continuously polls (spins) in a busy-waiting loop. It includes plain spinlocks and ",[2627,243237,243238],{},"Reader-Writer Spinlocks (读写自旋锁)"," (which favor readers).",[108,243241,243242,243245],{},[2627,243243,243244],{},"Semaphores:"," Includes traditional and reader-writer semaphores.",[108,243247,243248,243251],{},[2627,243249,243250],{},"Barriers (屏障):"," Enforce the order in which memory access instructions (loads and stores) are executed.",[253,243253,243255],{"id":243254},"_73-solaris","7.3 Solaris",[105,243257,243258,243264,243269,243275],{},[108,243259,243260,243263],{},[2627,243261,243262],{},"Mutual Exclusion Locks (互斥锁\u002FMutex):"," Defaults to spinlocks, but can use interrupt-based blocking queues.",[108,243265,243266,243268],{},[2627,243267,243244],{}," Traditional counting semaphores.",[108,243270,243271,243274],{},[2627,243272,243273],{},"Reader\u002FWriter Locks:"," Allows multiple concurrent readers or a single writer.",[108,243276,243277,243280],{},[2627,243278,243279],{},"Condition Variables:"," Used with a mutex lock to wait until a particular condition becomes true.",[253,243282,243284],{"id":243283},"_74-windows","7.4 Windows",[105,243286,243287,243296,243302,243312,243318,243323],{},[108,243288,243289,243292,243293,26192],{},[2627,243290,243291],{},"Wait Functions (等待函数):"," Allows a thread to block its execution on one or multiple objects (e.g., ",[31,243294,243295],{},"WaitForSingleObject",[108,243297,243298,243301],{},[2627,243299,243300],{},"Dispatcher Objects (调度器对象):"," Includes Mutexes, Semaphores, Notification\u002FSynchronization Events, Waitable Timers, Processes, and Threads.",[108,243303,243304,243307,243308,243311],{},[2627,243305,243306],{},"Critical Sections (临界区):"," Similar to mutexes but can only be used by threads within a ",[71807,243309,243310],{},"single process",". It uses spinlocks first, then falls back to dispatcher objects if blocked.",[108,243313,243314,243317],{},[2627,243315,243316],{},"Slim Reader-Writer (SRW) Locks:"," Highly optimized locks for reader-writer scenarios.",[108,243319,243320,243322],{},[2627,243321,243279],{}," Used with Critical Sections or SRW Locks.",[108,243324,243325,243328,243329,243332],{},[2627,243326,243327],{},"Lock-free Synchronization (无锁同步):"," Utilizes hardware ",[31,243330,243331],{},"Interlocked"," operations to perform atomic read-modify-write actions, ensuring threads are never blocked while holding a lock.",[253,243334,243336],{"id":243335},"_75-android","7.5 Android",[105,243338,243339],{},[108,243340,243341,243344,243345,243348,243349,243352],{},[2627,243342,243343],{},"Interprocess Communication (IPC):"," Android extensively uses the ",[2627,243346,243347],{},"Binder"," framework for RPC (Remote Procedure Call). A client proxy marshals transaction data into a parcel, sends it to the Binder via ",[31,243350,243351],{},"ioctl",", and the Binder wakes up the server stub to process it.",{"title":29,"searchDepth":43,"depth":43,"links":243354},[243355,243360,243361,243365,243369,243370,243371],{"id":242838,"depth":43,"text":242839,"children":243356},[243357,243358,243359],{"id":242848,"depth":49,"text":242849},{"id":242866,"depth":49,"text":242867},{"id":242901,"depth":49,"text":242902},{"id":242938,"depth":43,"text":242939},{"id":242979,"depth":43,"text":242980,"children":243362},[243363,243364],{"id":242990,"depth":49,"text":242991},{"id":242997,"depth":49,"text":242998},{"id":243039,"depth":43,"text":243040,"children":243366},[243367,243368],{"id":243046,"depth":49,"text":243047},{"id":243080,"depth":49,"text":243081},{"id":243115,"depth":43,"text":243116},{"id":243150,"depth":43,"text":243151},{"id":243173,"depth":43,"text":243174,"children":243372},[243373,243374,243375,243376,243377],{"id":243177,"depth":49,"text":243178},{"id":243220,"depth":49,"text":243221},{"id":243254,"depth":49,"text":243255},{"id":243283,"depth":49,"text":243284},{"id":243335,"depth":49,"text":243336},{"order":68},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap6",{"title":242829,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap6","X8h234zV59cvGCYXyTWrzySPluv1H4ZNYVXWCPsmgq0",{"id":243384,"title":243385,"body":243386,"description":29,"extension":85,"meta":243698,"navigation":52,"path":243699,"seo":243700,"stem":243701,"__hash__":243702},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap7.md","Chap. 7 Note",{"type":8,"value":243387,"toc":243684},[243388,243392,243396,243399,243425,243429,243432,243452,243456,243459,243463,243466,243480,243484,243487,243507,243511,243514,243564,243568,243575,243579,243592,243596,243599,243641,243645,243648],[11,243389,243391],{"id":243390},"chap7-内存管理-memory-management","Chap7. 内存管理 (Memory Management)",[19,243393,243395],{"id":243394},"_1-内存管理的核心基础","1. 内存管理的核心基础",[15,243397,243398],{},"在多道程序设计系统（Multiprogramming system）中，主内存（Main memory）必须被合理地划分和管理。",[105,243400,243401,243407,243413,243419],{},[108,243402,243403,243406],{},[2627,243404,243405],{},"内存管理 (Memory Management) 的定义","：由操作系统自动完成的、在操作系统与用户进程之间划分和分配内存空间的任务被称为内存管理。",[108,243408,243409,243412],{},[2627,243410,243411],{},"内存管理满足的系统需求","：内存管理机制需要满足多种系统层面的需求，主要包括保护（Protection）、重定位（Relocation）以及物理组织（Physical organization）等。",[108,243414,243415,243418],{},[2627,243416,243417],{},"重定位 (Relocation)","：这是内存管理中的一项重要任务，指的是为了支持交换（Swapping）操作，将进程映像（Process image）在内存的不同区域之间进行移动的过程。",[108,243420,243421,243424],{},[2627,243422,243423],{},"覆盖技术 (Overlaying)","：这是一种程序和数据的组织实践，通过这种方式，可以将程序的不同模块分配到内存的同一个区域中，从而节省空间。",[19,243426,243428],{"id":243427},"_2-内存地址的分类","2. 内存地址的分类",[15,243430,243431],{},"理解内存地址的区别是学习内存管理的关键前提。",[105,243433,243434,243440,243446],{},[108,243435,243436,243439],{},[2627,243437,243438],{},"逻辑地址 (Logical Address)","：指对内存位置的引用，这种引用是独立于当前数据在内存中的实际物理分配的。物理内存地址绝不能与逻辑地址混淆。",[108,243441,243442,243445],{},[2627,243443,243444],{},"绝对地址 (Absolute Address)","：指代主内存中真实存在的、实际的物理存储位置。",[108,243447,243448,243451],{},[2627,243449,243450],{},"相对地址 (Relative Address)","：使用相对地址的程序通常是通过动态运行时加载（Dynamic run-time loading）机制被加载到内存中的。",[19,243453,243455],{"id":243454},"_3-内存分区技术-partitioning-techniques","3. 内存分区技术 (Partitioning Techniques)",[15,243457,243458],{},"早期的内存管理主要通过对内存进行分区来实现。了解它们的缺陷（尤其是“碎片”问题）有助于理解现代内存管理技术的演进。",[253,243460,243462],{"id":243461},"_31-碎片化问题-fragmentation","3.1 碎片化问题 (Fragmentation)",[15,243464,243465],{},"在讲解具体技术前，必须先理清两种“内存浪费”的现象：",[105,243467,243468,243474],{},[108,243469,243470,243473],{},[2627,243471,243472],{},"内部碎片 (Internal Fragmentation)","：当加载到分区中的数据块小于该分区的大小时，分区内部会产生未被使用的浪费空间，这种现象称为内部碎片。",[108,243475,243476,243479],{},[2627,243477,243478],{},"外部碎片 (External Fragmentation)","：在内存分配过程中，现有分区之外产生的、由于太小而无法被利用的空闲内存块，这种现象称为外部碎片。",[253,243481,243483],{"id":243482},"_32-固定分区-fixed-partitioning","3.2 固定分区 (Fixed Partitioning)",[15,243485,243486],{},"这是一种很大程度上已经被淘汰的早期技术。",[105,243488,243489,243498],{},[108,243490,243491,243494,243495,243497],{},[2627,243492,243493],{},"主要问题","：固定分区面临着三大问题：只允许固定数量的进程运行、内存使用效率低下，以及严重的",[2627,243496,205063],{},"问题。固定分区的主要问题并非外部碎片。",[108,243499,243500,243503,243504,101018],{},[2627,243501,243502],{},"优化方案","：为了减轻固定分区带来的内部碎片问题，系统可以采用",[2627,243505,243506],{},"大小不等的分区 (Unequal size partitions)",[253,243508,243510],{"id":243509},"_33-动态分区-dynamic-partitioning","3.3 动态分区 (Dynamic Partitioning)",[15,243512,243513],{},"动态分区根据进程的实际需要为其分配确切大小的内存。",[105,243515,243516,243524,243534],{},[108,243517,243518,243520,243521,243523],{},[2627,243519,243493],{},"：动态分区技术会导致",[2627,243522,205067],{},"的产生。",[108,243525,243526,243529,243530,243533],{},[2627,243527,243528],{},"解决方案","：使用",[2627,243531,243532],{},"压缩技术 (Compaction)","。压缩是指将所有进程移动到连续的内存块中，从而将所有空闲的内存聚合汇总成一个单一的大空闲块。",[108,243535,243536,243539,243540],{},[2627,243537,243538],{},"放置算法 (Placement Algorithms)","：当有新进程需要载入时，系统需要决定将其放入哪个空闲块中。\n",[105,243541,243542,243552,243558],{},[108,243543,243544,243547,243548,243551],{},[2627,243545,243546],{},"最佳适应算法 (Best-fit)","：选择大小最接近请求大小的空闲块。注意：最佳适应算法通常",[2627,243549,243550],{},"不是","可用算法中性能最好的，因为它会产生大量微小的外部碎片。",[108,243553,243554,243557],{},[2627,243555,243556],{},"首次适应算法 (First-fit)","：该算法并非从上一次放置的位置开始扫描，而是通常从头开始扫描内存，寻找第一个足够大的空闲块。",[108,243559,243560,243563],{},[2627,243561,243562],{},"下次适应算法 (Next-fit)","：从上一次放置发生的位置开始扫描内存，并选择下一个足够满足请求的可用空闲块。",[253,243565,243567],{"id":243566},"_34-伙伴系统-buddy-system","3.4 伙伴系统 (Buddy System)",[105,243569,243570],{},[108,243571,243572,243574],{},[2627,243573,31577],{},"：伙伴系统是一种结合了固定分区和动态分区（可变分区）两者特点的方案，它是克服这两种方案各自缺点的合理折中选择。",[19,243576,243578],{"id":243577},"_4-现代内存管理技术分页与分段-paging-segmentation","4. 现代内存管理技术：分页与分段 (Paging & Segmentation)",[15,243580,243581,243582,243585,243586,258,243589,347],{},"在几乎所有现代的多道程序设计系统中，内存都是通过一种极其复杂的、被称为",[2627,243583,243584],{},"虚拟内存 (Virtual Memory)"," 的技术来进行管理的。虚拟内存的概念建立在两种基本技术的基础之上（或二者兼有）：",[2627,243587,243588],{},"分段 (Segmentation)",[2627,243590,243591],{},"分页 (Paging)",[253,243593,243595],{"id":243594},"_41-分页系统-paging","4.1 分页系统 (Paging)",[15,243597,243598],{},"分页系统将进程和内存都划分为大小固定且相等的块。",[105,243600,243601,243615,243628],{},[108,243602,243603,243606,243607,243610,243611,243614],{},[2627,243604,243605],{},"核心概念","：进程的块被称为",[2627,243608,243609],{},"页 (Pages)","，而可用内存的块被称为",[2627,243612,243613],{},"帧 (Frames)","。在分配时，进程的页会被分配到主存中可用的帧里。",[108,243616,243617,243620,243621,243623,243624,243627],{},[2627,243618,243619],{},"页表 (Page Table)","：系统会为每个进程维护一个页表。页表的作用是记录进程中每一个",[2627,243622,204994],{}," 所对应的物理",[2627,243625,243626],{},"帧 (Frame)"," 的位置。",[108,243629,243630,243633,243634,243636,243637,243640],{},[2627,243631,243632],{},"碎片情况","：采用分页策略的系统会遭受轻微的",[2627,243635,205063],{},"（通常发生在进程的最后一页），但",[2627,243638,243639],{},"完全没有外部碎片","。系统浪费的空间主要是由内部碎片引起的。",[253,243642,243644],{"id":243643},"_42-分段系统-segmentation","4.2 分段系统 (Segmentation)",[15,243646,243647],{},"分段系统根据程序的逻辑结构（如函数、对象等）将其划分为不同的部分。",[105,243649,243650,243663,243672],{},[108,243651,243652,243654,243655,243658,243659,243662],{},[2627,243653,243605],{},"：在分段系统中，程序及其相关数据被划分为若干个",[2627,243656,243657],{},"段 (Segments)","。关键点在于，这些段的长度",[2627,243660,243661],{},"不需要","相等（即不必大小一致）。",[108,243664,243665,243668,243669,347],{},[2627,243666,243667],{},"段表 (Segment Table)","：分段系统利用段表来工作，段表提供了每个对应段在主内存中的",[2627,243670,243671],{},"起始地址 (Starting address)",[108,243673,243674,243676,243677,243679,243680,243683],{},[2627,243675,243632],{},"：与分页系统相反，采用分段策略的系统会遭受轻微的",[2627,243678,205067],{},"，但",[2627,243681,243682],{},"完全没有内部碎片","。在分段系统中，被浪费的内存空间主要是由于外部碎片导致的。",{"title":29,"searchDepth":43,"depth":43,"links":243685},[243686,243687,243688,243694],{"id":243394,"depth":43,"text":243395},{"id":243427,"depth":43,"text":243428},{"id":243454,"depth":43,"text":243455,"children":243689},[243690,243691,243692,243693],{"id":243461,"depth":49,"text":243462},{"id":243482,"depth":49,"text":243483},{"id":243509,"depth":49,"text":243510},{"id":243566,"depth":49,"text":243567},{"id":243577,"depth":43,"text":243578,"children":243695},[243696,243697],{"id":243594,"depth":49,"text":243595},{"id":243643,"depth":49,"text":243644},{"order":74},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap7",{"title":243385,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap7","xqSIws7O7fOO_SWWI7GQjqlzqv1bSxRqzOqFtiODuSo",{"id":243704,"title":243705,"body":243706,"description":29,"extension":85,"meta":244065,"navigation":52,"path":244066,"seo":244067,"stem":244068,"__hash__":244069},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap8.md","Chap. 8 Note",{"type":8,"value":243707,"toc":244048},[243708,243712,243714,243718,243721,243725,243764,243768,243798,243800,243804,243807,243810,243813,243878,243881,243895,243897,243934,243936,243940,243943,243947,243950,243976,243980,243994,243998,244032,244034,244038],[11,243709,243711],{"id":243710},"chap8-虚拟内存-virtual-memory","Chap.8 虚拟内存 (Virtual Memory)",[32188,243713],{},[19,243715,243717],{"id":243716},"_81-硬件和控制结构-hardware-and-control-structures","8.1 硬件和控制结构 (Hardware and Control Structures)",[15,243719,243720],{},"虚拟内存的实现是硬件（如MMU）和操作系统软件共同配合的奇迹。",[253,243722,243724],{"id":243723},"核心概念核对-concept-check","核心概念核对 (Concept Check)",[105,243726,243727,243733,243739],{},[108,243728,243729,243732],{},[2627,243730,243731],{},"虚拟内存 (Virtual Memory):"," 一种存储分配方案，允许将辅助存储器（如磁盘）作为主存来寻址。虚拟大小受限于计算机的地址方案和磁盘空间，而非实际的物理内存。",[108,243734,243735,243738],{},[2627,243736,243737],{},"局部性原理 (Principle of Locality):"," 算法利用局部性原理，预测哪些驻留页面在不久的将来最不可能被访问，从而将它们换出。局部性分为空间局部性和时间局部性。",[108,243740,243741,243744],{},[2627,243742,243743],{},"页表项控制位 (Page Table Entry Bits):",[105,243745,243746,243752,243758],{},[108,243747,243748,243751],{},[2627,243749,243750],{},"页框号 (Frame number):"," 标识主存中页面的顺序编号。",[108,243753,243754,243757],{},[2627,243755,243756],{},"存在位 (Present\u002FValid bit):"," 指示该页当前是否在主存中。",[108,243759,243760,243763],{},[2627,243761,243762],{},"修改位 (Modify\u002FDirty bit):"," 指示该页自装入主存后是否被修改过。如果没有修改过，置换时就不需要将其写回磁盘。",[253,243765,243767],{"id":243766},"底层逻辑剖析-the-why","底层逻辑剖析 (The 'Why')",[105,243769,243770,243776,243782,243792],{},[108,243771,243772,243775],{},[2627,243773,243774],{},"为什么要用转换后备缓冲区 (TLB)?","\n每次虚拟地址转换都需要访问内存中的页表，导致访存时间翻倍。TLB 是一个专用的高速缓存，保存了最近使用的页表项。其目的是在绝大多数情况下，避免去磁盘或主存中检索页表项，极大提升地址转换速度。",[108,243777,243778,243781],{},[2627,243779,243780],{},"为什么要设计多级页表 (Multilevel Page Tables)?","\n对于巨大的虚拟地址空间（例如 32 位或 64 位），单级页表会占用极其庞大的连续物理内存。多级页表允许页表本身也被分页并存放在虚拟内存中，只将顶层目录（Root page table）常驻主存。",[108,243783,243784,243787,243788,243791],{},[2627,243785,243786],{},"反置页表 (Inverted Page Table) 的意义是什么?","\n传统页表大小与虚拟地址空间成正比，而反置页表的大小",[2627,243789,243790],{},"与物理内存成正比","。它利用哈希表映射，每个物理页框只对应一个页表项。页表项通常包含：页号、进程标识符、控制位和用于解决哈希冲突的链指针。",[108,243793,243794,243797],{},[2627,243795,243796],{},"为什么要结合分页和分段 (Combined Paging and Segmentation)?","\n分页对程序员透明，消除了外部碎片；而分段对程序员可见，支持不断增长的数据结构、模块化、共享和保护。段页式结合方案中，进程拥有一个段表，而每个段拥有自己的页表。",[32188,243799],{},[19,243801,243803],{"id":243802},"_82-操作系统软件-operating-system-software","8.2 操作系统软件 (Operating System Software)",[15,243805,243806],{},"操作系统需要制定一系列策略来管理虚拟内存，以最小化缺页中断（Page Fault）带来的巨大开销。",[253,243808,243724],{"id":243809},"核心概念核对-concept-check-1",[15,243811,243812],{},"操作系统的虚拟内存策略主要包含以下六个方面：",[2230,243814,243815,243835,243841,243847,243853,243872],{},[108,243816,243817,243820,243821],{},[2627,243818,243819],{},"读取策略 (Fetch Policy):"," 决定何时将页面调入主存。\n",[105,243822,243823,243829],{},[108,243824,243825,243828],{},[2627,243826,243827],{},"请求分页 (Demand Paging):"," 只有当访问到该页某个位置而发生缺页时，才将其调入主存。",[108,243830,243831,243834],{},[2627,243832,243833],{},"预分页 (Prepaging):"," 除了引发缺页的页面，还会将预期可能用到的其他页面一并调入。",[108,243836,243837,243840],{},[2627,243838,243839],{},"放置策略 (Placement Policy):"," 决定页面放在哪个物理框（主要在NUMA架构中重要）。",[108,243842,243843,243846],{},[2627,243844,243845],{},"置换策略 (Replacement Policy):"," 决定淘汰哪个页面。常见算法包括 OPT, LRU, FIFO, Clock。",[108,243848,243849,243852],{},[2627,243850,243851],{},"驻留集管理 (Resident Set Management):"," 决定分配给进程多少个页框，以及置换范围。",[108,243854,243855,243858],{},[2627,243856,243857],{},"清除策略 (Cleaning Policy):",[105,243859,243860,243866],{},[108,243861,243862,243865],{},[2627,243863,243864],{},"请求清除 (Demand Cleaning):"," 只有当页面被选中替换时才写回磁盘。",[108,243867,243868,243871],{},[2627,243869,243870],{},"预清除 (Precleaning):"," 在页面框被需要之前，将修改过的页面成批写回磁盘。",[108,243873,243874,243877],{},[2627,243875,243876],{},"负载控制 (Load Control):"," 决定系统多道程序度，防止抖动。",[253,243879,243767],{"id":243880},"底层逻辑剖析-the-why-1",[105,243882,243883,243889],{},[108,243884,243885,243888],{},[2627,243886,243887],{},"为什么 Clock 算法比 LRU 更实用？","\n真正的 LRU (最近最久未使用) 算法需要为每次内存访问记录时间戳或维护链表，硬件开销极大。Clock (时钟) 策略通过“使用位 (Use bit)”来近似 LRU：当发生置换时，指针扫描环形缓冲区，跳过使用位为 1 的页（将其置 0），替换掉遇到第一个使用位为 0 的页。",[108,243890,243891,243894],{},[2627,243892,243893],{},"局部置换与全局置换 (Local vs. Global Replacement):","\n固定分配策略要求分配给进程的页框数固定，因此发生缺页时，只能替换该进程自己的页面（局部置换）。",[253,243896,217337],{"id":217336},[105,243898,243899,243909,243928],{},[108,243900,243901,243904,243905,243908],{},[2627,243902,243903],{},"Belady 异常 (Belady's Anomaly):"," 学生常认为分配给进程的物理页框越多，缺页率一定越低。但在 ",[2627,243906,243907],{},"FIFO 置换算法"," 中，增加页框数有时反而会导致更多的缺页传输，这就是著名的 Belady 异常。",[108,243910,243911,243914],{},[2627,243912,243913],{},"驻留集 (Resident Set) vs 工作集 (Working Set):",[105,243915,243916,243922],{},[108,243917,243918,243921],{},[71807,243919,243920],{},"驻留集"," 指的是一个进程当前实际存在于主存中的页面集合。",[108,243923,243924,243927],{},[71807,243925,243926],{},"工作集"," 指的是一个进程在最近一段时间内被引用过的页面集合。系统应当试图让驻留集包含工作集，以降低缺页率。",[108,243929,243930,243933],{},[2627,243931,243932],{},"抖动 (Thrashing):"," 这不是硬件故障！抖动是指在虚拟内存机制中，处理器将大部分时间耗费在换入换出页面上，而不是执行指令上的系统过载现象。通常因为多道程序度过高、进程分配的页框不足以覆盖其局部性引发。",[32188,243935],{},[19,243937,243939],{"id":243938},"_83-86-具体操作系统的内存管理","8.3 - 8.6 具体操作系统的内存管理",[15,243941,243942],{},"不同的操作系统在实现虚拟内存时有各自的特色结构。",[253,243944,243946],{"id":243945},"_1-unix-和-solaris-内存管理","1. UNIX 和 Solaris 内存管理",[15,243948,243949],{},"UNIX SVR4 采用了四个关键数据结构来管理分页：",[105,243951,243952,243958,243964,243970],{},[108,243953,243954,243957],{},[2627,243955,243956],{},"页表项 (Page Table Entry):"," 包含页框号、年龄、修改位、访问位、存在位等。",[108,243959,243960,243963],{},[2627,243961,243962],{},"磁盘块描述符 (Disk Block Descriptor):"," 记录页面在交换设备（Swap device）上的逻辑设备号和块号。",[108,243965,243966,243969],{},[2627,243967,243968],{},"页框数据表项 (Page Frame Data Table Entry):"," 描述物理页框的状态和引用计数。",[108,243971,243972,243975],{},[2627,243973,243974],{},"交换使用表项 (Swap-Use Table Entry):"," 记录交换设备页面的使用计数。",[253,243977,243979],{"id":243978},"_2-linux-内存管理","2. Linux 内存管理",[105,243981,243982,243988],{},[108,243983,243984,243987],{},[2627,243985,243986],{},"四级页表结构:"," 为了保持平台独立性并支持大地址空间，Linux 将虚拟地址分为四个字段：页目录 (Page Directory) 索引、页中间目录 (Page Middle Directory) 索引、页表 (Page Table) 索引，以及偏移量。",[108,243989,243990,243993],{},[2627,243991,243992],{},"页大小:"," 基础单位通常是 4KB，但也支持大页（Hugepages，如 2MB），后者可以显著减少 TLB 访问次数从而提高性能。",[253,243995,243997],{"id":243996},"_3-windows-内存管理","3. Windows 内存管理",[105,243999,244000],{},[108,244001,244002,244005,244006],{},[2627,244003,244004],{},"32位默认虚拟地址映射:"," Windows 为正常的 32 位用户进程提供了 4GB 的虚拟地址空间，默认划分为四个区域：\n",[2230,244007,244008,244014,244020,244026],{},[108,244009,244010,244013],{},[31,244011,244012],{},"0x00000000 - 0x0000FFFF"," (64KB): NULL 指针分配区，不可访问，用于捕获空指针错误。",[108,244015,244016,244019],{},[31,244017,244018],{},"0x00010000 - 0x7FFEFFFF"," (~2GB): 用户可用地址空间。",[108,244021,244022,244025],{},[31,244023,244024],{},"0x7FFF0000 - 0x7FFFFFFF"," (64KB): 保护页（Guard page），不可访问，用于检查指针越界。",[108,244027,244028,244031],{},[31,244029,244030],{},"0x80000000 - 0xFFFFFFFF"," (2GB): 操作系统内核专属区域，用户态不可访问。",[32188,244033],{},[19,244035,244037],{"id":244036},"总结-summary","总结 (Summary)",[15,244039,244040,244041,244043,244044,244047],{},"虚拟内存不仅让程序员摆脱了主存容量的物理限制，更通过分页、分段硬件与复杂的页面调度算法（置换策略、驻留集管理等），在**性能（时间和缺页率）",[2627,244042,150586],{},"空间（内存利用率）**之间实现了极其微妙的平衡。理解本章的核心在于牢记 ",[2627,244045,244046],{},"“局部性原理 (Locality)”"," 以及如何用最少的开销（如 TLB、Clock 算法）去逼近完美的预测。",{"title":29,"searchDepth":43,"depth":43,"links":244049},[244050,244054,244059,244064],{"id":243716,"depth":43,"text":243717,"children":244051},[244052,244053],{"id":243723,"depth":49,"text":243724},{"id":243766,"depth":49,"text":243767},{"id":243802,"depth":43,"text":243803,"children":244055},[244056,244057,244058],{"id":243809,"depth":49,"text":243724},{"id":243880,"depth":49,"text":243767},{"id":217336,"depth":49,"text":217337},{"id":243938,"depth":43,"text":243939,"children":244060},[244061,244062,244063],{"id":243945,"depth":49,"text":243946},{"id":243978,"depth":49,"text":243979},{"id":243996,"depth":49,"text":243997},{"id":244036,"depth":43,"text":244037},{"order":1971},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap8",{"title":243705,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap8","IaYgg7Ojwxeu_ckNyYAaeOawmi_eBUvIMIfRYbtlIyM",{"id":244071,"title":244072,"body":244073,"description":29,"extension":85,"meta":245987,"navigation":52,"path":245988,"seo":245989,"stem":245990,"__hash__":245991},"notes\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap9.md","Chap. 9 Note",{"type":8,"value":244074,"toc":245982},[244075,244079,244081,244085,244088,244092,244095,244118,244122,244290,244292,244296,244300,244317,244321,244343,244347,244682,244686,244698,244702,245006,245010,245031,245035,245452,245454,245458],[11,244076,244078],{"id":244077},"chap-9-单处理器调度uniprocessor-scheduling","Chap 9. 单处理器调度（Uniprocessor Scheduling）",[32188,244080],{},[253,244082,244084],{"id":244083},"一-concept","一、 Concept",[15,244086,244087],{},"在学习具体的调度算法之前，先理清调度的“三大层次”和“核心考核指标”。",[32490,244089,244091],{"id":244090},"_1-调度的三个层次","1. 调度的三个层次",[15,244093,244094],{},"操作系统中不仅有CPU调度，实际上包含了四个维度的调度（其中 I\u002FO 调度在第11章讨论），本章重点关注与进程状态转换密切相关的三种处理器调度：",[105,244096,244097,244106,244112],{},[108,244098,244099,244102,244103,31901],{},[2627,244100,244101],{},"长程调度 (Long-term Scheduling)："," 决定是否将一个新进程加入到待执行的进程池中（控制系统的",[2627,244104,244105],{},"多道程序调度",[108,244107,244108,244111],{},[2627,244109,244110],{},"中程调度 (Medium-term Scheduling)："," 属于对换（Swapping）功能的一部分，决定是否将进程部分或全部调入主存（处理挂起状态 Suspend）。",[108,244113,244114,244117],{},[2627,244115,244116],{},"短程调度 (Short-term Scheduling \u002F Dispatcher)："," 发生频率最高，决定当前主存中哪个就绪进程将获得 CPU 的执行权。",[32490,244119,244121],{"id":244120},"_2-短程调度的核心性能指标","2. 短程调度的核心性能指标",[105,244123,244124,244222,244232],{},[108,244125,244126,244129,244130,244221],{},[2627,244127,244128],{},"周转时间 (Turnaround Time, TAT)："," 进程从提交到完成的总时间。",[2627,244131,244132],{},[34,244133,244135,244161],{"className":244134},[31642],[34,244136,244138],{"className":244137},[31646],[31648,244139,244140],{"xmlns":31650},[31652,244141,244142,244158],{},[31655,244143,244144,244146,244148,244150,244152,244154,244156],{},[31658,244145,37400],{},[31658,244147,54151],{},[31658,244149,37400],{},[31662,244151,202],{},[39361,244153,216949],{},[31662,244155,223],{},[39361,244157,216954],{},[31665,244159,244160],{"encoding":31667},"TAT = \\text{等待时间} + \\text{服务时间}",[34,244162,244164,244188,244209],{"className":244163,"ariaHidden":758},[31672],[34,244165,244167,244170,244173,244176,244179,244182,244185],{"className":244166},[31676],[34,244168],{"className":244169,"style":33218},[31680],[34,244171,37400],{"className":244172,"style":37422},[31685,31686],[34,244174,54151],{"className":244175},[31685,31686],[34,244177,37400],{"className":244178,"style":37422},[31685,31686],[34,244180],{"className":244181,"style":31846},[31845],[34,244183,202],{"className":244184},[31850],[34,244186],{"className":244187,"style":31846},[31845],[34,244189,244191,244194,244200,244203,244206],{"className":244190},[31676],[34,244192],{"className":244193,"style":34944},[31680],[34,244195,244197],{"className":244196},[31685,2638],[34,244198,216949],{"className":244199},[31685,105462],[34,244201],{"className":244202,"style":32604},[31845],[34,244204,223],{"className":244205},[32024],[34,244207],{"className":244208,"style":32604},[31845],[34,244210,244212,244215],{"className":244211},[31676],[34,244213],{"className":244214,"style":33218},[31680],[34,244216,244218],{"className":244217},[31685,2638],[34,244219,216954],{"className":244220},[31685,105462],"。这是批处理系统最看重的指标。",[108,244223,244224,244227,244228,244231],{},[2627,244225,244226],{},"响应时间 (Response Time)："," 进程从提交请求到系统",[2627,244229,244230],{},"首次","产生响应的时间。这是交互式系统最看重的指标。",[108,244233,244234,47649,244237,244289],{},[2627,244235,244236],{},"归一化周转时间 (Normalized Turnaround Time)：",[34,244238,244240,244262],{"className":244239},[31642],[34,244241,244243],{"className":244242},[31646],[31648,244244,244245],{"xmlns":31650},[31652,244246,244247,244259],{},[31655,244248,244249,244251,244253,244255,244257],{},[31658,244250,37400],{},[31658,244252,54151],{},[31658,244254,37400],{},[31658,244256,38080],{"mathvariant":38079},[39361,244258,216954],{},[31665,244260,244261],{"encoding":31667},"TAT \u002F \\text{服务时间}",[34,244263,244265],{"className":244264,"ariaHidden":758},[31672],[34,244266,244268,244271,244274,244277,244280,244283],{"className":244267},[31676],[34,244269],{"className":244270,"style":31774},[31680],[34,244272,37400],{"className":244273,"style":37422},[31685,31686],[34,244275,54151],{"className":244276},[31685,31686],[34,244278,37400],{"className":244279,"style":37422},[31685,31686],[34,244281,38080],{"className":244282},[31685],[34,244284,244286],{"className":244285},[31685,2638],[34,244287,216954],{"className":244288},[31685,105462],"。该值越接近1越好，表明进程没有被无谓地等待。",[32188,244291],{},[253,244293,244295],{"id":244294},"二-调度算法的演进","二、 调度算法的演进",[32490,244297,244299],{"id":244298},"_1-先来先服务-fcfs-first-come-first-served","1. 先来先服务 (FCFS - First-Come-First-Served)",[105,244301,244302,244308],{},[108,244303,244304,244307],{},[2627,244305,244306],{},"机制："," 非抢占式。谁先进入就绪队列，谁先执行，直到完成或阻塞。",[108,244309,244310,244312,244313,244316],{},[2627,244311,177826],{}," 虽然简单公平，但如果一个长进程排在前面，后面的短进程会被严重拖累（即所谓的“护航效应”）。此外，它极度",[2627,244314,244315],{},"偏袒 CPU密集型进程","，导致 I\u002FO密集型进程经常吃亏。",[32490,244318,244320],{"id":244319},"_2-时间片轮转-round-robin-rr","2. 时间片轮转 (Round Robin, RR)",[105,244322,244323,244328,244337],{},[108,244324,244325,244327],{},[2627,244326,244306],{}," 抢占式。每个进程分配一个时间片（Time Quantum \u002F Slice），时间片用完后触发时钟中断，强行剥夺 CPU，进程回到队列尾部排队。",[108,244329,244330,244332,244333,244336],{},[2627,244331,177826],{}," FCFS 响应时间太差了！RR 的引入是为了在",[2627,244334,244335],{},"分时系统","中提供良好的交互响应时间。",[108,244338,244339,244342],{},[2627,244340,244341],{},"设计权衡："," 时间片太长，就退化成了 FCFS；时间片太短，CPU的大部分时间都浪费在进程上下文切换上了。",[32490,244344,244346],{"id":244345},"_3-最短进程优先-spn-shortest-process-next","3. 最短进程优先 (SPN - Shortest Process Next)",[105,244348,244349,244354,244363],{},[108,244350,244351,244353],{},[2627,244352,244306],{}," 非抢占式。每次挑选预计处理时间最短的进程优先执行。",[108,244355,244356,244358,244359,244362],{},[2627,244357,4160],{}," 能够提供所有非抢占式算法中",[2627,244360,244361],{},"最优的平均等待时间","和平均周转时间。",[108,244364,244365,244368,244369,67359,244624,244652,244653,244681],{},[2627,244366,244367],{},"预测技术（指数平滑法 Exponential Averaging）："," 我们怎么提前知道进程要跑多久？操作系统通过历史数据来预测未来。公式为：",[2627,244370,244371],{},[34,244372,244374,244426],{"className":244373},[31642],[34,244375,244377],{"className":244376},[31646],[31648,244378,244379],{"xmlns":31650},[31652,244380,244381,244423],{},[31655,244382,244383,244395,244397,244399,244405,244407,244409,244411,244413,244415,244417],{},[31743,244384,244385,244387],{},[31658,244386,94675],{},[31655,244388,244389,244391,244393],{},[31658,244390,31660],{},[31662,244392,223],{},[31749,244394,640],{},[31662,244396,202],{},[31658,244398,217392],{},[31743,244400,244401,244403],{},[31658,244402,37400],{},[31658,244404,31660],{},[31662,244406,223],{},[31662,244408,456],{"stretchy":702},[31749,244410,640],{},[31662,244412,32226],{},[31658,244414,217392],{},[31662,244416,2557],{"stretchy":702},[31743,244418,244419,244421],{},[31658,244420,94675],{},[31658,244422,31660],{},[31665,244424,244425],{"encoding":31667},"S_{n+1} = \\alpha T_n + (1-\\alpha)S_n",[34,244427,244429,244493,244551,244572],{"className":244428,"ariaHidden":758},[31672],[34,244430,244432,244435,244484,244487,244490],{"className":244431},[31676],[34,244433],{"className":244434,"style":70992},[31680],[34,244436,244438,244441],{"className":244437},[31685],[34,244439,94675],{"className":244440,"style":106455},[31685,31686],[34,244442,244444],{"className":244443},[31792],[34,244445,244447,244476],{"className":244446},[31796,31797],[34,244448,244450,244473],{"className":244449},[31801],[34,244451,244453],{"className":244452,"style":31806},[31805],[34,244454,244455,244458],{"style":206422},[34,244456],{"className":244457,"style":31814},[31813],[34,244459,244461],{"className":244460},[31818,31819,31820,31821],[34,244462,244464,244467,244470],{"className":244463},[31685,31821],[34,244465,31660],{"className":244466},[31685,31686,31821],[34,244468,223],{"className":244469},[32024,31821],[34,244471,640],{"className":244472},[31685,31821],[34,244474,31829],{"className":244475},[31828],[34,244477,244479],{"className":244478},[31801],[34,244480,244482],{"className":244481,"style":32037},[31805],[34,244483],{},[34,244485],{"className":244486,"style":31846},[31845],[34,244488,202],{"className":244489},[31850],[34,244491],{"className":244492,"style":31846},[31845],[34,244494,244496,244499,244502,244542,244545,244548],{"className":244495},[31676],[34,244497],{"className":244498,"style":70915},[31680],[34,244500,217392],{"className":244501,"style":217496},[31685,31686],[34,244503,244505,244508],{"className":244504},[31685],[34,244506,37400],{"className":244507,"style":37422},[31685,31686],[34,244509,244511],{"className":244510},[31792],[34,244512,244514,244534],{"className":244513},[31796,31797],[34,244515,244517,244531],{"className":244516},[31801],[34,244518,244520],{"className":244519,"style":32362},[31805],[34,244521,244522,244525],{"style":214424},[34,244523],{"className":244524,"style":31814},[31813],[34,244526,244528],{"className":244527},[31818,31819,31820,31821],[34,244529,31660],{"className":244530},[31685,31686,31821],[34,244532,31829],{"className":244533},[31828],[34,244535,244537],{"className":244536},[31801],[34,244538,244540],{"className":244539,"style":31836},[31805],[34,244541],{},[34,244543],{"className":244544,"style":32604},[31845],[34,244546,223],{"className":244547},[32024],[34,244549],{"className":244550,"style":32604},[31845],[34,244552,244554,244557,244560,244563,244566,244569],{"className":244553},[31676],[34,244555],{"className":244556,"style":31774},[31680],[34,244558,456],{"className":244559},[31782],[34,244561,640],{"className":244562},[31685],[34,244564],{"className":244565,"style":32604},[31845],[34,244567,32226],{"className":244568},[32024],[34,244570],{"className":244571,"style":32604},[31845],[34,244573,244575,244578,244581,244584],{"className":244574},[31676],[34,244576],{"className":244577,"style":31774},[31680],[34,244579,217392],{"className":244580,"style":217496},[31685,31686],[34,244582,2557],{"className":244583},[31690],[34,244585,244587,244590],{"className":244586},[31685],[34,244588,94675],{"className":244589,"style":106455},[31685,31686],[34,244591,244593],{"className":244592},[31792],[34,244594,244596,244616],{"className":244595},[31796,31797],[34,244597,244599,244613],{"className":244598},[31801],[34,244600,244602],{"className":244601,"style":32362},[31805],[34,244603,244604,244607],{"style":206422},[34,244605],{"className":244606,"style":31814},[31813],[34,244608,244610],{"className":244609},[31818,31819,31820,31821],[34,244611,31660],{"className":244612},[31685,31686,31821],[34,244614,31829],{"className":244615},[31828],[34,244617,244619],{"className":244618},[31801],[34,244620,244622],{"className":244621,"style":31836},[31805],[34,244623],{},[34,244625,244627,244640],{"className":244626},[31642],[34,244628,244630],{"className":244629},[31646],[31648,244631,244632],{"xmlns":31650},[31652,244633,244634,244638],{},[31655,244635,244636],{},[31658,244637,217392],{},[31665,244639,218244],{"encoding":31667},[34,244641,244643],{"className":244642,"ariaHidden":758},[31672],[34,244644,244646,244649],{"className":244645},[31676],[34,244647],{"className":244648,"style":32527},[31680],[34,244650,217392],{"className":244651,"style":217496},[31685,31686]," 控制着权重的分配：",[34,244654,244656,244669],{"className":244655},[31642],[34,244657,244659],{"className":244658},[31646],[31648,244660,244661],{"xmlns":31650},[31652,244662,244663,244667],{},[31655,244664,244665],{},[31658,244666,217392],{},[31665,244668,218244],{"encoding":31667},[34,244670,244672],{"className":244671,"ariaHidden":758},[31672],[34,244673,244675,244678],{"className":244674},[31676],[34,244676],{"className":244677,"style":32527},[31680],[34,244679,217392],{"className":244680,"style":217496},[31685,31686]," 越大（如0.8），系统对近期的突发变化反应越快。",[32490,244683,244685],{"id":244684},"_4-最短剩余时间优先-srt-shortest-remaining-time","4. 最短剩余时间优先 (SRT - Shortest Remaining Time)",[105,244687,244688,244693],{},[108,244689,244690,244692],{},[2627,244691,244306],{}," 抢占式。SPN的升级版。如果新到达的进程比当前运行进程的剩余时间还要短，立即抢占 CPU。",[108,244694,244695,244697],{},[2627,244696,177826],{}," SPN 一旦开始执行长进程就不会停，新来的短进程依然要等。SRT 通过引入抢占进一步优化了短进程的响应速度，但带来了更高的系统开销。",[32490,244699,244701],{"id":244700},"_5-最高响应比优先-hrrn-highest-response-ratio-next","5. 最高响应比优先 (HRRN - Highest Response Ratio Next)",[105,244703,244704,244996],{},[108,244705,244706,244708,244709,244737,244738,347],{},[2627,244707,244306],{}," 非抢占式。每次调度时，计算所有就绪进程的响应比 ",[34,244710,244712,244725],{"className":244711},[31642],[34,244713,244715],{"className":244714},[31646],[31648,244716,244717],{"xmlns":31650},[31652,244718,244719,244723],{},[31655,244720,244721],{},[31658,244722,34386],{},[31665,244724,34386],{"encoding":31667},[34,244726,244728],{"className":244727,"ariaHidden":758},[31672],[34,244729,244731,244734],{"className":244730},[31676],[34,244732],{"className":244733,"style":33218},[31680],[34,244735,34386],{"className":244736,"style":34427},[31685,31686],"，选最大的执行。公式：",[2627,244739,244740],{},[34,244741,244743,244783],{"className":244742},[31642],[34,244744,244746],{"className":244745},[31646],[31648,244747,244748],{"xmlns":31650},[31652,244749,244750,244780],{},[31655,244751,244752,244754,244756,244768,244770,244772,244774],{},[31658,244753,34386],{},[31662,244755,202],{},[41453,244757,244758,244766],{},[31655,244759,244760,244762,244764],{},[39361,244761,216949],{},[31662,244763,223],{},[39361,244765,216954],{},[39361,244767,216954],{},[31662,244769,202],{},[31749,244771,640],{},[31662,244773,223],{},[41453,244775,244776,244778],{},[31658,244777,47077],{},[31658,244779,31747],{},[31665,244781,244782],{"encoding":31667},"R = \\frac{\\text{等待时间} + \\text{服务时间}}{\\text{服务时间}} = 1 + \\frac{w}{s}",[34,244784,244786,244804,244902,244920],{"className":244785,"ariaHidden":758},[31672],[34,244787,244789,244792,244795,244798,244801],{"className":244788},[31676],[34,244790],{"className":244791,"style":33218},[31680],[34,244793,34386],{"className":244794,"style":34427},[31685,31686],[34,244796],{"className":244797,"style":31846},[31845],[34,244799,202],{"className":244800},[31850],[34,244802],{"className":244803,"style":31846},[31845],[34,244805,244807,244810,244893,244896,244899],{"className":244806},[31676],[34,244808],{"className":244809,"style":168026},[31680],[34,244811,244813,244816,244890],{"className":244812},[31685],[34,244814],{"className":244815},[31782,41488],[34,244817,244819],{"className":244818},[41453],[34,244820,244822,244882],{"className":244821},[31796,31797],[34,244823,244825,244879],{"className":244824},[31801],[34,244826,244828,244845,244853],{"className":244827,"style":168045},[31805],[34,244829,244830,244833],{"style":41504},[34,244831],{"className":244832,"style":41508},[31813],[34,244834,244836],{"className":244835},[31818,31819,31820,31821],[34,244837,244839],{"className":244838},[31685,31821],[34,244840,244842],{"className":244841},[31685,2638,31821],[34,244843,216954],{"className":244844},[31685,105462,31821],[34,244846,244847,244850],{"style":41520},[34,244848],{"className":244849,"style":41508},[31813],[34,244851],{"className":244852,"style":41528},[41527],[34,244854,244855,244858],{"style":46771},[34,244856],{"className":244857,"style":41508},[31813],[34,244859,244861],{"className":244860},[31818,31819,31820,31821],[34,244862,244864,244870,244873],{"className":244863},[31685,31821],[34,244865,244867],{"className":244866},[31685,2638,31821],[34,244868,216949],{"className":244869},[31685,105462,31821],[34,244871,223],{"className":244872},[32024,31821],[34,244874,244876],{"className":244875},[31685,2638,31821],[34,244877,216954],{"className":244878},[31685,105462,31821],[34,244880,31829],{"className":244881},[31828],[34,244883,244885],{"className":244884},[31801],[34,244886,244888],{"className":244887,"style":41619},[31805],[34,244889],{},[34,244891],{"className":244892},[31690,41488],[34,244894],{"className":244895,"style":31846},[31845],[34,244897,202],{"className":244898},[31850],[34,244900],{"className":244901,"style":31846},[31845],[34,244903,244905,244908,244911,244914,244917],{"className":244904},[31676],[34,244906],{"className":244907,"style":35942},[31680],[34,244909,640],{"className":244910},[31685],[34,244912],{"className":244913,"style":32604},[31845],[34,244915,223],{"className":244916},[32024],[34,244918],{"className":244919,"style":32604},[31845],[34,244921,244923,244927],{"className":244922},[31676],[34,244924],{"className":244925,"style":244926},[31680],"height:1.0404em;vertical-align:-0.345em;",[34,244928,244930,244933,244993],{"className":244929},[31685],[34,244931],{"className":244932},[31782,41488],[34,244934,244936],{"className":244935},[41453],[34,244937,244939,244985],{"className":244938},[31796,31797],[34,244940,244942,244982],{"className":244941},[31801],[34,244943,244946,244960,244968],{"className":244944,"style":244945},[31805],"height:0.6954em;",[34,244947,244948,244951],{"style":41504},[34,244949],{"className":244950,"style":41508},[31813],[34,244952,244954],{"className":244953},[31818,31819,31820,31821],[34,244955,244957],{"className":244956},[31685,31821],[34,244958,31747],{"className":244959},[31685,31686,31821],[34,244961,244962,244965],{"style":41520},[34,244963],{"className":244964,"style":41508},[31813],[34,244966],{"className":244967,"style":41528},[41527],[34,244969,244970,244973],{"style":46771},[34,244971],{"className":244972,"style":41508},[31813],[34,244974,244976],{"className":244975},[31818,31819,31820,31821],[34,244977,244979],{"className":244978},[31685,31821],[34,244980,47077],{"className":244981,"style":47120},[31685,31686,31821],[34,244983,31829],{"className":244984},[31828],[34,244986,244988],{"className":244987},[31801],[34,244989,244991],{"className":244990,"style":41619},[31805],[34,244992],{},[34,244994],{"className":244995},[31690,41488],[108,244997,244998,245001,245002,245005],{},[2627,244999,245000],{},"The 'Why' (解决饥饿)："," SPN 和 SRT 有一个致命缺陷——",[2627,245003,245004],{},"长进程会无限期饥饿 (Starvation)","。HRRN 巧妙地引入了“年龄（等待时间 w）”因素。随着等待时间的增加，即使是长进程，其响应比也会越来越大，最终一定会获得 CPU。",[32490,245007,245009],{"id":245008},"_6-多级反馈队列-multilevel-feedback-queue","6. 多级反馈队列 (Multilevel Feedback Queue)",[105,245011,245012,245017],{},[108,245013,245014,245016],{},[2627,245015,244306],{}," 抢占式。设立多个优先级队列。新进程进入最高优先级队列（时间片最短）；如果在时间片内没跑完，就被“降级”惩罚到下一级队列。",[108,245018,245019,245022,245023,245026,245027,245030],{},[2627,245020,245021],{},"The 'Why'："," 之前那些优秀的算法（SPN, SRT, HRRN）都有一个共同前提——",[2627,245024,245025],{},"必须预先知道或估算进程的运行时间","。但在实际中很难精准估算。Feedback 算法",[2627,245028,245029],{},"不需要知道进程多长","。它通过“惩罚耗时进程，奖励快速I\u002FO进程”的动态调整机制，完美兼顾了短进程响应、I\u002FO 密集型效率和不用预知长度的难题，是现代操作系统的基石。",[32490,245032,245034],{"id":245033},"_7-传统-unix-调度-traditional-unix-scheduling","7. 传统 UNIX 调度 (Traditional UNIX Scheduling)",[105,245036,245037,245042],{},[108,245038,245039,245041],{},[2627,245040,244306],{}," 采用带有多级反馈和循环轮转（Round Robin）的抢占式调度。",[108,245043,245044,47649,245047,245451],{},[2627,245045,245046],{},"优先数计算：",[34,245048,245050,245126],{"className":245049},[31642],[34,245051,245053],{"className":245052},[31646],[31648,245054,245055],{"xmlns":31650},[31652,245056,245057,245123],{},[31655,245058,245059,245065,245067,245069,245071,245073,245075,245077,245079,245085,245087,245109,245111,245113,245115,245117],{},[31743,245060,245061,245063],{},[31658,245062,105852],{},[31658,245064,31968],{},[31662,245066,456],{"stretchy":702},[31658,245068,31955],{},[31662,245070,2557],{"stretchy":702},[31662,245072,202],{},[31658,245074,54224],{},[31658,245076,111],{},[31658,245078,31747],{},[31743,245080,245081,245083],{},[31658,245082,33244],{},[31658,245084,31968],{},[31662,245086,223],{},[41453,245088,245089,245107],{},[31655,245090,245091,245093,245095,245101,245103,245105],{},[31658,245092,47384],{},[31658,245094,105852],{},[31743,245096,245097,245099],{},[31658,245098,33350],{},[31658,245100,31968],{},[31662,245102,456],{"stretchy":702},[31658,245104,31955],{},[31662,245106,2557],{"stretchy":702},[31749,245108,768],{},[31662,245110,223],{},[31658,245112,31660],{},[31658,245114,31955],{},[31658,245116,31965],{},[31743,245118,245119,245121],{},[31658,245120,33244],{},[31658,245122,31968],{},[31665,245124,245125],{"encoding":31667},"P_j(i) = Base_j + \\frac{CPU_j(i)}{2} + nice_j",[34,245127,245129,245193,245257,245397],{"className":245128,"ariaHidden":758},[31672],[34,245130,245132,245135,245175,245178,245181,245184,245187,245190],{"className":245131},[31676],[34,245133],{"className":245134,"style":32058},[31680],[34,245136,245138,245141],{"className":245137},[31685],[34,245139,105852],{"className":245140,"style":37422},[31685,31686],[34,245142,245144],{"className":245143},[31792],[34,245145,245147,245167],{"className":245146},[31796,31797],[34,245148,245150,245164],{"className":245149},[31801],[34,245151,245153],{"className":245152,"style":32090},[31805],[34,245154,245155,245158],{"style":214424},[34,245156],{"className":245157,"style":31814},[31813],[34,245159,245161],{"className":245160},[31818,31819,31820,31821],[34,245162,31968],{"className":245163,"style":32153},[31685,31686,31821],[34,245165,31829],{"className":245166},[31828],[34,245168,245170],{"className":245169},[31801],[34,245171,245173],{"className":245172,"style":32163},[31805],[34,245174],{},[34,245176,456],{"className":245177},[31782],[34,245179,31955],{"className":245180},[31685,31686],[34,245182,2557],{"className":245183},[31690],[34,245185],{"className":245186,"style":31846},[31845],[34,245188,202],{"className":245189},[31850],[34,245191],{"className":245192,"style":31846},[31845],[34,245194,245196,245199,245202,245205,245208,245248,245251,245254],{"className":245195},[31676],[34,245197],{"className":245198,"style":55484},[31680],[34,245200,54224],{"className":245201,"style":54252},[31685,31686],[34,245203,111],{"className":245204},[31685,31686],[34,245206,31747],{"className":245207},[31685,31686],[34,245209,245211,245214],{"className":245210},[31685],[34,245212,33244],{"className":245213},[31685,31686],[34,245215,245217],{"className":245216},[31792],[34,245218,245220,245240],{"className":245219},[31796,31797],[34,245221,245223,245237],{"className":245222},[31801],[34,245224,245226],{"className":245225,"style":32090},[31805],[34,245227,245228,245231],{"style":31809},[34,245229],{"className":245230,"style":31814},[31813],[34,245232,245234],{"className":245233},[31818,31819,31820,31821],[34,245235,31968],{"className":245236,"style":32153},[31685,31686,31821],[34,245238,31829],{"className":245239},[31828],[34,245241,245243],{"className":245242},[31801],[34,245244,245246],{"className":245245,"style":32163},[31805],[34,245247],{},[34,245249],{"className":245250,"style":32604},[31845],[34,245252,223],{"className":245253},[32024],[34,245255],{"className":245256,"style":32604},[31845],[34,245258,245260,245264,245388,245391,245394],{"className":245259},[31676],[34,245261],{"className":245262,"style":245263},[31680],"height:1.3773em;vertical-align:-0.345em;",[34,245265,245267,245270,245385],{"className":245266},[31685],[34,245268],{"className":245269},[31782,41488],[34,245271,245273],{"className":245272},[41453],[34,245274,245276,245377],{"className":245275},[31796,31797],[34,245277,245279,245374],{"className":245278},[31801],[34,245280,245283,245297,245305],{"className":245281,"style":245282},[31805],"height:1.0323em;",[34,245284,245285,245288],{"style":41504},[34,245286],{"className":245287,"style":41508},[31813],[34,245289,245291],{"className":245290},[31818,31819,31820,31821],[34,245292,245294],{"className":245293},[31685,31821],[34,245295,768],{"className":245296},[31685,31821],[34,245298,245299,245302],{"style":41520},[34,245300],{"className":245301,"style":41508},[31813],[34,245303],{"className":245304,"style":41528},[41527],[34,245306,245308,245311],{"style":245307},"top:-3.5073em;",[34,245309],{"className":245310,"style":41508},[31813],[34,245312,245314],{"className":245313},[31818,31819,31820,31821],[34,245315,245317,245320,245323,245365,245368,245371],{"className":245316},[31685,31821],[34,245318,47384],{"className":245319,"style":38706},[31685,31686,31821],[34,245321,105852],{"className":245322,"style":37422},[31685,31686,31821],[34,245324,245326,245329],{"className":245325},[31685,31821],[34,245327,33350],{"className":245328,"style":33389},[31685,31686,31821],[34,245330,245332],{"className":245331},[31792],[34,245333,245335,245356],{"className":245334},[31796,31797],[34,245336,245338,245353],{"className":245337},[31801],[34,245339,245341],{"className":245340,"style":231708},[31805],[34,245342,245344,245347],{"style":245343},"top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;",[34,245345],{"className":245346,"style":43406},[31813],[34,245348,245350],{"className":245349},[31818,43410,43411,31821],[34,245351,31968],{"className":245352,"style":32153},[31685,31686,31821],[34,245354,31829],{"className":245355},[31828],[34,245357,245359],{"className":245358},[31801],[34,245360,245363],{"className":245361,"style":245362},[31805],"height:0.2819em;",[34,245364],{},[34,245366,456],{"className":245367},[31782,31821],[34,245369,31955],{"className":245370},[31685,31686,31821],[34,245372,2557],{"className":245373},[31690,31821],[34,245375,31829],{"className":245376},[31828],[34,245378,245380],{"className":245379},[31801],[34,245381,245383],{"className":245382,"style":41619},[31805],[34,245384],{},[34,245386],{"className":245387},[31690,41488],[34,245389],{"className":245390,"style":32604},[31845],[34,245392,223],{"className":245393},[32024],[34,245395],{"className":245396,"style":32604},[31845],[34,245398,245400,245404,245408,245411],{"className":245399},[31676],[34,245401],{"className":245402,"style":245403},[31680],"height:0.9456em;vertical-align:-0.2861em;",[34,245405,245407],{"className":245406},[31685,31686],"ni",[34,245409,31965],{"className":245410},[31685,31686],[34,245412,245414,245417],{"className":245413},[31685],[34,245415,33244],{"className":245416},[31685,31686],[34,245418,245420],{"className":245419},[31792],[34,245421,245423,245443],{"className":245422},[31796,31797],[34,245424,245426,245440],{"className":245425},[31801],[34,245427,245429],{"className":245428,"style":32090},[31805],[34,245430,245431,245434],{"style":31809},[34,245432],{"className":245433,"style":31814},[31813],[34,245435,245437],{"className":245436},[31818,31819,31820,31821],[34,245438,31968],{"className":245439,"style":32153},[31685,31686,31821],[34,245441,31829],{"className":245442},[31828],[34,245444,245446],{"className":245445},[31801],[34,245447,245449],{"className":245448,"style":32163},[31805],[34,245450],{},"。系统根据进程的历史 CPU 使用情况动态调整优先级，确保 I\u002FO 密集型和交互式进程的高优先级。",[32188,245453],{},[253,245455,245457],{"id":245456},"三-common-pitfalls","三、 Common Pitfalls",[2230,245459,245460,245478,245501,245525],{},[108,245461,245462,245465,245466,245469,245470,245473,245474,245477],{},[2627,245463,245464],{},"混淆“优先数大小”与“优先级高低”：","\n在 ",[2627,245467,245468],{},"UNIX\u002FLinux 系统","和大量的学术计算题中，",[2627,245471,245472],{},"数值越小，优先级越高","（即 Priority 0 比 Priority 10 优先）；但在 ",[2627,245475,245476],{},"Windows 系统","中，数值越大代表优先级越高。做题时务必先看清题目设定！",[108,245479,245480,245483],{},[2627,245481,245482],{},"搞混 周转时间 (Turnaround) 与 响应时间 (Response)：",[105,245484,245485,245491],{},[108,245486,245487,245490],{},[71807,245488,245489],{},"陷阱："," 计算响应时间时，算成了进程执行结束的时间。",[108,245492,245493,245496,245497,245500],{},[71807,245494,245495],{},"正解："," 周转时间 = 整个任务全部做完的时间 - 到达时间。响应时间 = 任务",[2627,245498,245499],{},"第一次获得 CPU 开始执行","的时间 - 到达时间。",[108,245502,245503,245506],{},[2627,245504,245505],{},"忽略了“非抢占式 (Nonpreemptive)”的锁死效应：",[105,245507,245508,245513],{},[108,245509,245510,245512],{},[71807,245511,245489],{}," 在做 SPN 或 HRRN 推演时，一旦有新的短进程到达，就立刻让当前长进程下台。",[108,245514,245515,245517,245518,245520,245521,245524],{},[71807,245516,245495],{}," SPN, HRRN, FCFS 是",[2627,245519,215645],{},"的！这意味着只要一个进程上了 CPU，除非它自己执行完毕或主动阻塞，否则",[2627,245522,245523],{},"天王老子来了也不能把它赶下来","。抢占只发生在时间片到期（RR, Feedback）或特定事件（SRT）发生时。",[108,245526,245527,245559],{},[2627,245528,245529,245530,245558],{},"死板地计算指数平滑法 (",[34,245531,245533,245546],{"className":245532},[31642],[34,245534,245536],{"className":245535},[31646],[31648,245537,245538],{"xmlns":31650},[31652,245539,245540,245544],{},[31655,245541,245542],{},[31658,245543,217392],{},[31665,245545,218244],{"encoding":31667},[34,245547,245549],{"className":245548,"ariaHidden":758},[31672],[34,245550,245552,245555],{"className":245551},[31676],[34,245553],{"className":245554,"style":32527},[31680],[34,245556,217392],{"className":245557,"style":217496},[31685,31686],")：",[105,245560,245561,245595],{},[108,245562,245563,245565,245566,245594],{},[71807,245564,245489],{}," 记不住公式中 ",[34,245567,245569,245582],{"className":245568},[31642],[34,245570,245572],{"className":245571},[31646],[31648,245573,245574],{"xmlns":31650},[31652,245575,245576,245580],{},[31655,245577,245578],{},[31658,245579,217392],{},[31665,245581,218244],{"encoding":31667},[34,245583,245585],{"className":245584,"ariaHidden":758},[31672],[34,245586,245588,245591],{"className":245587},[31676],[34,245589],{"className":245590,"style":32527},[31680],[34,245592,217392],{"className":245593,"style":217496},[31685,31686]," 到底乘的是过去真实的观测值还是预测值。",[108,245596,245597,47649,245599,347,245851,347,245953,245981],{},[71807,245598,245495],{},[34,245600,245602,245653],{"className":245601},[31642],[34,245603,245605],{"className":245604},[31646],[31648,245606,245607],{"xmlns":31650},[31652,245608,245609,245651],{},[31655,245610,245611,245623,245625,245627,245633,245635,245637,245639,245641,245643,245645],{},[31743,245612,245613,245615],{},[31658,245614,94675],{},[31655,245616,245617,245619,245621],{},[31658,245618,31660],{},[31662,245620,223],{},[31749,245622,640],{},[31662,245624,202],{},[31658,245626,217392],{},[31743,245628,245629,245631],{},[31658,245630,37400],{},[31658,245632,31660],{},[31662,245634,223],{},[31662,245636,456],{"stretchy":702},[31749,245638,640],{},[31662,245640,32226],{},[31658,245642,217392],{},[31662,245644,2557],{"stretchy":702},[31743,245646,245647,245649],{},[31658,245648,94675],{},[31658,245650,31660],{},[31665,245652,244425],{"encoding":31667},[34,245654,245656,245720,245778,245799],{"className":245655,"ariaHidden":758},[31672],[34,245657,245659,245662,245711,245714,245717],{"className":245658},[31676],[34,245660],{"className":245661,"style":70992},[31680],[34,245663,245665,245668],{"className":245664},[31685],[34,245666,94675],{"className":245667,"style":106455},[31685,31686],[34,245669,245671],{"className":245670},[31792],[34,245672,245674,245703],{"className":245673},[31796,31797],[34,245675,245677,245700],{"className":245676},[31801],[34,245678,245680],{"className":245679,"style":31806},[31805],[34,245681,245682,245685],{"style":206422},[34,245683],{"className":245684,"style":31814},[31813],[34,245686,245688],{"className":245687},[31818,31819,31820,31821],[34,245689,245691,245694,245697],{"className":245690},[31685,31821],[34,245692,31660],{"className":245693},[31685,31686,31821],[34,245695,223],{"className":245696},[32024,31821],[34,245698,640],{"className":245699},[31685,31821],[34,245701,31829],{"className":245702},[31828],[34,245704,245706],{"className":245705},[31801],[34,245707,245709],{"className":245708,"style":32037},[31805],[34,245710],{},[34,245712],{"className":245713,"style":31846},[31845],[34,245715,202],{"className":245716},[31850],[34,245718],{"className":245719,"style":31846},[31845],[34,245721,245723,245726,245729,245769,245772,245775],{"className":245722},[31676],[34,245724],{"className":245725,"style":70915},[31680],[34,245727,217392],{"className":245728,"style":217496},[31685,31686],[34,245730,245732,245735],{"className":245731},[31685],[34,245733,37400],{"className":245734,"style":37422},[31685,31686],[34,245736,245738],{"className":245737},[31792],[34,245739,245741,245761],{"className":245740},[31796,31797],[34,245742,245744,245758],{"className":245743},[31801],[34,245745,245747],{"className":245746,"style":32362},[31805],[34,245748,245749,245752],{"style":214424},[34,245750],{"className":245751,"style":31814},[31813],[34,245753,245755],{"className":245754},[31818,31819,31820,31821],[34,245756,31660],{"className":245757},[31685,31686,31821],[34,245759,31829],{"className":245760},[31828],[34,245762,245764],{"className":245763},[31801],[34,245765,245767],{"className":245766,"style":31836},[31805],[34,245768],{},[34,245770],{"className":245771,"style":32604},[31845],[34,245773,223],{"className":245774},[32024],[34,245776],{"className":245777,"style":32604},[31845],[34,245779,245781,245784,245787,245790,245793,245796],{"className":245780},[31676],[34,245782],{"className":245783,"style":31774},[31680],[34,245785,456],{"className":245786},[31782],[34,245788,640],{"className":245789},[31685],[34,245791],{"className":245792,"style":32604},[31845],[34,245794,32226],{"className":245795},[32024],[34,245797],{"className":245798,"style":32604},[31845],[34,245800,245802,245805,245808,245811],{"className":245801},[31676],[34,245803],{"className":245804,"style":31774},[31680],[34,245806,217392],{"className":245807,"style":217496},[31685,31686],[34,245809,2557],{"className":245810},[31690],[34,245812,245814,245817],{"className":245813},[31685],[34,245815,94675],{"className":245816,"style":106455},[31685,31686],[34,245818,245820],{"className":245819},[31792],[34,245821,245823,245843],{"className":245822},[31796,31797],[34,245824,245826,245840],{"className":245825},[31801],[34,245827,245829],{"className":245828,"style":32362},[31805],[34,245830,245831,245834],{"style":206422},[34,245832],{"className":245833,"style":31814},[31813],[34,245835,245837],{"className":245836},[31818,31819,31820,31821],[34,245838,31660],{"className":245839},[31685,31686,31821],[34,245841,31829],{"className":245842},[31828],[34,245844,245846],{"className":245845},[31801],[34,245847,245849],{"className":245848,"style":31836},[31805],[34,245850],{},[2627,245852,245853,245854,245882,245883,245952],{},"记住口诀：“",[34,245855,245857,245870],{"className":245856},[31642],[34,245858,245860],{"className":245859},[31646],[31648,245861,245862],{"xmlns":31650},[31652,245863,245864,245868],{},[31655,245865,245866],{},[31658,245867,217392],{},[31665,245869,218244],{"encoding":31667},[34,245871,245873],{"className":245872,"ariaHidden":758},[31672],[34,245874,245876,245879],{"className":245875},[31676],[34,245877],{"className":245878,"style":32527},[31680],[34,245880,217392],{"className":245881,"style":217496},[31685,31686]," 永远跟最新的真实观测值 ",[34,245884,245886,245903],{"className":245885},[31642],[34,245887,245889],{"className":245888},[31646],[31648,245890,245891],{"xmlns":31650},[31652,245892,245893,245901],{},[31655,245894,245895],{},[31743,245896,245897,245899],{},[31658,245898,37400],{},[31658,245900,31660],{},[31665,245902,218908],{"encoding":31667},[34,245904,245906],{"className":245905,"ariaHidden":758},[31672],[34,245907,245909,245912],{"className":245908},[31676],[34,245910],{"className":245911,"style":70915},[31680],[34,245913,245915,245918],{"className":245914},[31685],[34,245916,37400],{"className":245917,"style":37422},[31685,31686],[34,245919,245921],{"className":245920},[31792],[34,245922,245924,245944],{"className":245923},[31796,31797],[34,245925,245927,245941],{"className":245926},[31801],[34,245928,245930],{"className":245929,"style":32362},[31805],[34,245931,245932,245935],{"style":214424},[34,245933],{"className":245934,"style":31814},[31813],[34,245936,245938],{"className":245937},[31818,31819,31820,31821],[34,245939,31660],{"className":245940},[31685,31686,31821],[34,245942,31829],{"className":245943},[31828],[34,245945,245947],{"className":245946},[31801],[34,245948,245950],{"className":245949,"style":31836},[31805],[34,245951],{}," 在一起”",[34,245954,245956,245969],{"className":245955},[31642],[34,245957,245959],{"className":245958},[31646],[31648,245960,245961],{"xmlns":31650},[31652,245962,245963,245967],{},[31655,245964,245965],{},[31658,245966,217392],{},[31665,245968,218244],{"encoding":31667},[34,245970,245972],{"className":245971,"ariaHidden":758},[31672],[34,245973,245975,245978],{"className":245974},[31676],[34,245976],{"className":245977,"style":32527},[31680],[34,245979,217392],{"className":245980,"style":217496},[31685,31686]," 越大，表示操作系统越“健忘”，越相信最近刚刚发生的事。",{"title":29,"searchDepth":43,"depth":43,"links":245983},[245984,245985,245986],{"id":244083,"depth":49,"text":244084},{"id":244294,"depth":49,"text":244295},{"id":245456,"depth":49,"text":245457},{"order":1976},"\u002Fsource\u002Fcs\u002Fos\u002Fnote\u002Fchap9",{"title":244072,"description":29},"source\u002Fcs\u002Fos\u002Fnote\u002Fchap9","fCdIIlwprGFcrBU8TSjEytn41khvMRNpJJJwqjjW-lM",{"id":245993,"title":245994,"body":245995,"description":29,"extension":85,"meta":246486,"navigation":52,"path":246487,"seo":246488,"stem":246489,"__hash__":246490},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002FDef.md","名词解释",{"type":8,"value":245996,"toc":246464},[245997,245999,246003,246023,246027,246044,246048,246065,246069,246086,246090,246107,246111,246128,246132,246149,246153,246170,246174,246191,246195,246212,246216,246233,246237,246254,246258,246275,246279,246296,246300,246317,246321,246338,246342,246359,246363,246380,246384,246401,246405,246425,246428,246431,246434,246437,246440,246443,246446,246449,246452,246455,246458,246461],[11,245998,245994],{"id":245994},[19,246000,246002],{"id":246001},"_1文件系统-file-system","1．文件系统 (File System)",[105,246004,246005,246011,246017],{},[108,246006,246007,246010],{},[2627,246008,246009],{},"概念核对 (Concept Check):"," 文件管理系统是一组系统软件，主要负责为用户和应用程序提供管理、组织和使用文件的服务。",[108,246012,246013,246016],{},[2627,246014,246015],{},"底层的“为什么” (The 'Why'):"," 裸磁盘只能以物理的“扇区”或“块”为单位存储数据，这对于人类和上层应用程序来说极其难以管理。操作系统引入文件系统这一抽象概念，屏蔽了底层的物理复杂性，使得数据能够以“按名存取”的逻辑方式进行读写、组织和长久保存。",[108,246018,246019,246022],{},[2627,246020,246021],{},"常见易错点 (Common Pitfalls):"," 学生常将“文件系统”误认为是“磁盘上所有文件的集合”。实际上，文件系统不仅仅包括文件本身，更重要的是包含了管理这些文件的核心软件代码模块以及维护磁盘空间的数据结构（如目录结构、文件分配表等）。",[19,246024,246026],{"id":246025},"_2抖动-thrashing","2．抖动 (Thrashing)",[105,246028,246029,246034,246039],{},[108,246030,246031,246033],{},[2627,246032,246009],{}," 抖动是虚拟内存机制中的一种系统过载现象，指处理器把绝大部分时间都花在将页面在主存和辅存之间频繁换入换出上，而不是真正在执行程序指令。",[108,246035,246036,246038],{},[2627,246037,246015],{}," 当分配给某个进程的物理页面过少，或者页面置换算法没有很好地符合程序的“局部性原理”时，刚才被换出到磁盘的页面立刻又被需要，导致系统陷入无休止的缺页中断中，性能急剧下降。",[108,246040,246041,246043],{},[2627,246042,246021],{}," 容易将其与硬件故障混淆。抖动纯粹是由于内存资源极度紧张且页面调度策略失败引发的软件逻辑现象。",[19,246045,246047],{"id":246046},"_3设备独立性-device-independence","3．设备独立性 (Device Independence)",[105,246049,246050,246055,246060],{},[108,246051,246052,246054],{},[2627,246053,246009],{}," 也称设备无关性，指逻辑 I\u002FO 模块将底层硬件设备视为一种抽象的逻辑资源，允许用户进程使用统一的逻辑设备标识符和简单的命令（如 open、read）来操作设备，而无需关心控制该具体设备的物理细节。",[108,246056,246057,246059],{},[2627,246058,246015],{}," 为了保证操作系统的通用性和应用程序的兼容性。有了设备独立性，无论底层连接的是什么品牌或型号的磁盘、打印机，应用代码都不需要修改，操作系统通过内部的逻辑设备表进行动态映射即可。",[108,246061,246062,246064],{},[2627,246063,246021],{}," 误以为设备独立性意味着设备不需要操作系统管理。它实际上是指“上层用户程序独立于具体的物理设备特征”。",[19,246066,246068],{"id":246067},"_4原语-primitive-atomic-operation","4．原语 (Primitive \u002F Atomic operation)",[105,246070,246071,246076,246081],{},[108,246072,246073,246075],{},[2627,246074,246009],{}," 原语是指由一个或多个指令组成的函数或动作，在执行过程中表现为“不可分割（indivisible）”的原子操作。这意味着没有其他进程可以中断其执行或看到它的中间状态。",[108,246077,246078,246080],{},[2627,246079,246015],{}," 在并发编程环境中，操作系统需要构建绝对可靠的底层同步机制（如信号量的 Wait\u002FSignal 操作）。如果这些基础指令在执行一半时被调度器切走，会导致严重的数据竞争，因此必须利用硬件级手段保证其原子性。",[108,246082,246083,246085],{},[2627,246084,246021],{}," 常常忽略其最核心的“不可被中断”的特性，将其与普通的系统调用或 C 语言函数混为一谈。",[19,246087,246089],{"id":246088},"_5置换策略-replacement-policy","5．置换策略 (Replacement Policy)",[105,246091,246092,246097,246102],{},[108,246093,246094,246096],{},[2627,246095,246009],{}," 在请求分页式虚拟内存中，当内存已被占满而又发生缺页时，用于决定从内存中挑选出哪一个特定页面将其淘汰（换出）的算法机制。",[108,246098,246099,246101],{},[2627,246100,246015],{}," 物理内存空间非常宝贵。一个好的置换策略（例如 LRU 或 Clock 算法）会试图预测未来，把“未来最不可能被用到”的页面换出，从而尽最大努力降低缺页中断率。",[108,246103,246104,246106],{},[2627,246105,246021],{}," 学生易将“置换策略（换掉谁）”与“驻留集管理（分配给进程多少个内存块）”这两个概念混淆，它们是虚拟内存管理的两个不同层面。",[19,246108,246110],{"id":246109},"_6物理地址-physical-address","6．物理地址 (Physical Address)",[105,246112,246113,246118,246123],{},[108,246114,246115,246117],{},[2627,246116,246009],{}," 物理地址，也称为绝对地址，是主存储器（内存）中真实的、硬件级别的物理存储位置编号。",[108,246119,246120,246122],{},[2627,246121,246015],{}," CPU 和内存条进行最终的电信号数据传输时，必须认准硬件上的真实物理引脚位置，因此一切高级语言中的逻辑地址，在执行前或执行时最终都必须转换成物理地址。",[108,246124,246125,246127],{},[2627,246126,246021],{}," 在虚拟内存语境中，容易将处理器内寄存器指向的“逻辑地址”误当成物理地址。用户程序永远看不到真实的物理地址。",[19,246129,246131],{"id":246130},"_7dma-direct-memory-access直接内存访问","7．DMA (Direct Memory Access，直接内存访问)",[105,246133,246134,246139,246144],{},[108,246135,246136,246138],{},[2627,246137,246009],{}," 一种高级的 I\u002FO 控制技术，由 DMA 模块负责控制主存与 I\u002FO 模块之间的数据交换。处理器只需向 DMA 发送传输数据块的请求，直到整个数据块全部传输完毕后，DMA 才会中断处理器。",[108,246140,246141,246143],{},[2627,246142,246015],{}," 在程序直接控制或纯中断控制 I\u002FO 中，每个字节或字的传输都需要 CPU 亲自搬运。引入 DMA 是为了把 CPU 从这种繁杂的数据搬运中彻底解放出来，极大地提高系统并行处理能力。",[108,246145,246146,246148],{},[2627,246147,246021],{}," 误认为使用 DMA 时 CPU 完全不参与。实际上 CPU 在 I\u002FO 的“启动阶段”和“结束阶段（响应中断）”仍需介入，只是中途的循环搬运工作交给了 DMA 硬件。",[19,246150,246152],{"id":246151},"_8地址重定位-address-relocation","8．地址重定位 (Address Relocation)",[105,246154,246155,246160,246165],{},[108,246156,246157,246159],{},[2627,246158,246009],{}," 将程序代码中使用的逻辑地址（独立于数据在内存中当前分配位置的地址引用）转换映射为主存中绝对物理地址的过程。",[108,246161,246162,246164],{},[2627,246163,246015],{}," 程序员和编译器在编写生成代码时，通常假设程序是从 0 地址开始存放的。但在多道程序系统中，程序会被加载到内存的随机空闲区域，甚至在执行过程中被移动（Swapping）。因此，必须对地址进行相对偏移的调整，这就是重定位的意义。",[108,246166,246167,246169],{},[2627,246168,246021],{}," 忘记区分“静态重定位”（程序装入内存时一次性修改所有地址）和“动态重定位”（运行时由硬件 MMU 动态加基址转换）。现代操作系统基本都依赖后者。",[19,246171,246173],{"id":246172},"_9文件-file","9．文件 (File)",[105,246175,246176,246181,246186],{},[108,246177,246178,246180],{},[2627,246179,246009],{}," 文件是相似记录的集合或相关数据的有序序列，被用户和应用程序视为一个具有唯一名称的单一逻辑实体。",[108,246182,246183,246185],{},[2627,246184,246015],{}," 提供一种结构化、易于人类识别和程序引用的方式来持久保存数据。它将底层复杂的柱面、磁道、扇区等物理概念抽象为对用户友好的信息集合。",[108,246187,246188,246190],{},[2627,246189,246021],{}," 忽略文件的本质是操作系统提供给用户的一层“逻辑抽象”，而并非硬件天然存在的形态。",[19,246192,246194],{"id":246193},"_10操作系统-operating-system","10．操作系统 (Operating System)",[105,246196,246197,246202,246207],{},[108,246198,246199,246201],{},[2627,246200,246009],{}," 操作系统是一套最基本的系统软件，它利用一个或多个处理器的硬件资源为系统用户提供一组服务，并代表用户去调度管理辅助存储器、I\u002FO 设备等所有计算机资源。",[108,246203,246204,246206],{},[2627,246205,246015],{}," 第一，硬件接口太复杂，操作系统为应用程序提供了一个更便捷、一致的抽象接口（虚拟机假象）；第二，资源是有限的，必须有一个权威的“管家”来公平、高效地在并发的多个程序之间分配资源。",[108,246208,246209,246211],{},[2627,246210,246021],{}," 认为操作系统和普通软件一样运行。操作系统具有极高的特权级别，其核心代码运行在“内核态”，拥有对所有硬件的直接控制权。",[19,246213,246215],{"id":246214},"_11临界资源-critical-resource","11．临界资源 (Critical Resource)",[105,246217,246218,246223,246228],{},[108,246219,246220,246222],{},[2627,246221,246009],{}," 在计算机系统中，一次仅允许一个进程安全使用的共享资源。",[108,246224,246225,246227],{},[2627,246226,246015],{}," 如果允许多个进程同时读取和修改此类资源（如共享内存变量、打印机等），会导致指令交错执行，产生数据不一致或逻辑混乱的严重错误（竞态条件）。",[108,246229,246230,246232],{},[2627,246231,246021],{}," 极易将“临界资源（具体的硬件设备或共享变量）”和“临界区（访问这些资源的那段程序代码）”混淆,。",[19,246234,246236],{"id":246235},"_12死锁-deadlock","12．死锁 (Deadlock)",[105,246238,246239,246244,246249],{},[108,246240,246241,246243],{},[2627,246242,246009],{}," 一组进程因为都在相互等待对方所占有的系统资源或通信事件，而导致它们全部永久阻塞的现象,。",[108,246245,246246,246248],{},[2627,246247,246015],{}," 在多道程序并发环境中，由于进程推进顺序不可预测，当多个进程对不可抢占的资源提出互斥且请求和保持的申请，并形成了一个闭合的循环等待链时，系统就彻底僵死了。",[108,246250,246251,246253],{},[2627,246252,246021],{}," 容易和“饥饿（Starvation）”混淆。饥饿是某一个进程由于优先级低迟迟得不到资源，但系统整体还在正常运转；死锁则是涉及的进程群体“全部互相卡死”。",[19,246255,246257],{"id":246256},"_13信号量-semaphore","13．信号量 (Semaphore)",[105,246259,246260,246265,246270],{},[108,246261,246262,246264],{},[2627,246263,246009],{}," 信号量是一种用于进程间发送信号和实现同步机制的整数变量。对它只能执行三种操作：初始化、原子递减（Wait\u002FP操作，可能导致进程阻塞）和原子递增（Signal\u002FV操作，可能唤醒进程）。",[108,246266,246267,246269],{},[2627,246268,246015],{}," 它是解决高并发下互斥和同步问题的最经典工具。通过底层的原子计数，可以精确地表示目前可用资源的个数（正数），或者正在排队等待该资源的进程数（负数的绝对值），从而协调复杂进程间的推进顺序。",[108,246271,246272,246274],{},[2627,246273,246021],{}," 在书写代码时，最容易将 Wait 操作和 Signal 操作的顺序颠倒，导致无法实现互斥，甚至引发死锁。",[19,246276,246278],{"id":246277},"_14进程-process","14．进程 (Process)",[105,246280,246281,246286,246291],{},[108,246282,246283,246285],{},[2627,246284,246009],{}," 进程是程序在计算机上的一次动态执行过程。它不仅包含二进制的程序代码，还包含与代码相关的数据集，以及由操作系统维护的进程控制块（PCB）等执行上下文,。",[108,246287,246288,246290],{},[2627,246289,246015],{}," 在分时与多道程序系统中，程序是走走停停并发执行的。静态的“程序”概念已经无法反映这种动态的推进过程。引入进程作为资源分配和系统独立调度的基本单位，才能让操作系统精确记录当前任务的状态、断点和资源占有情况,。",[108,246292,246293,246295],{},[2627,246294,246021],{}," 将进程等同于程序。程序存在硬盘上，是静态的代码实体；进程存在内存中，是动态运行的生命实体。一个程序可以被多次启动，产生多个不同的进程。",[19,246297,246299],{"id":246298},"_15块设备-block-oriented-device","15．块设备 (Block-oriented Device)",[105,246301,246302,246307,246312],{},[108,246303,246304,246306],{},[2627,246305,246009],{}," 块设备是将信息存储在固定大小的“块（Blocks）”中，并且所有的数据传输也都是一次一个块来进行的 I\u002FO 设备，磁盘和磁带是其典型代表。",[108,246308,246309,246311],{},[2627,246310,246015],{}," 操作系统在管理大规模数据（特别是文件系统和虚拟内存的页交换）时，按“块”进行处理能提供随机寻址的能力，并且大幅度提升数据批量传输的效率。",[108,246313,246314,246316],{},[2627,246315,246021],{}," 常常与“流设备\u002F字符设备（如打印机、终端，按数据流依次传输）”的特性搞混。",[19,246318,246320],{"id":246319},"_16用户接口-user-interface","16．用户接口 (User Interface)",[105,246322,246323,246328,246333],{},[108,246324,246325,246327],{},[2627,246326,246009],{}," 操作系统向用户提供的一层操作边界（通常被称为 Shell 或外壳），它将用户与操作系统的底层复杂细节分离开来，使得操作系统纯粹体现为一个服务集合。",[108,246329,246330,246332],{},[2627,246331,246015],{}," 普通人类用户无法直接记忆和输入复杂的机器指令或系统调用函数。用户接口（包括命令行解释器和图形用户界面 GUI）充当了翻译官，让用户只需下达直观的指令即可控制计算机运行。",[108,246334,246335,246337],{},[2627,246336,246021],{}," 认为这仅仅是指 GUI 界面。实际上传统的命令行终端（CLI）乃至早期的批处理作业控制语言也是用户接口的重要形式。",[19,246339,246341],{"id":246340},"_17并发-concurrency","17．并发 (Concurrency)",[105,246343,246344,246349,246354],{},[108,246345,246346,246348],{},[2627,246347,246009],{}," 指在单个处理器或多处理器的计算机系统中，多个进程在宏观时间段内都处于活动和执行状态，产生一种多任务同时推进的现象。",[108,246350,246351,246353],{},[2627,246352,246015],{}," CPU 的运算速度远远大于 I\u002FO 操作的速度。为了不让 CPU 闲置等待，操作系统通过快速的时间片轮转切换进程，使得多个任务“交替”使用资源，从而极大提升系统吞吐量和资源利用率。",[108,246355,246356,246358],{},[2627,246357,246021],{}," 极易将并发与并行（Parallelism）混为一谈。并发是指在单核上通过交替执行“假装”同时发生；并行则是依托多核硬件“真正”在同一物理时刻执行多条指令。",[19,246360,246362],{"id":246361},"_18快表-tlb-translation-lookaside-buffer","18．快表 (TLB \u002F Translation Lookaside Buffer)",[105,246364,246365,246370,246375],{},[108,246366,246367,246369],{},[2627,246368,246009],{}," TLB 是一种位于处理器内、专门用于虚拟内存分页管理的极高速缓存。它专门保存那些最近刚刚被使用过的页表项。",[108,246371,246372,246374],{},[2627,246373,246015],{}," 虚拟地址到物理地址的转换必须查页表，而页表存放在相对较慢的主存中。如果没有 TLB，每次读写数据都要访问两次内存（一次查表，一次拿数据）。利用程序的局部性原理，把最常用的查表结果存入 TLB 硬件缓存，可以瞬间完成地址转换，解决系统性能瓶颈。",[108,246376,246377,246379],{},[2627,246378,246021],{}," 误以为 TLB 缓存的是普通的应用“数据”。它缓存的仅仅是“逻辑页号到物理框号的地址映射记录（页表项）”。",[19,246381,246383],{"id":246382},"_19文件目录-file-directory","19．文件目录 (File Directory)",[105,246385,246386,246391,246396],{},[108,246387,246388,246390],{},[2627,246389,246009],{}," 文件目录是一种系统数据结构，它使得大量文件能够基于所有者、应用程序等标准被组织和清晰地分离，里面记录了文件的名称及其物理位置信息的映射。",[108,246392,246393,246395],{},[2627,246394,246015],{}," 在现代大容量存储器中，存在成千上万的文件。如果没有目录系统，不仅查找极慢，还会面临严重的文件名冲突（同名覆盖）。目录有效提升了检索速度、安全性，并保证了系统完整性。",[108,246397,246398,246400],{},[2627,246399,246021],{}," 很多初学者不认为目录本身也是“文件”。在大多数系统（如 UNIX）的设计哲学中，目录本质上就是一种包含文件名和 inode 映射指针列表的特殊结构文件。",[19,246402,246404],{"id":246403},"_20树形结构目录-tree-structured-directory","20．树形结构目录 (Tree-structured Directory)",[105,246406,246407,246412,246417,246422],{},[108,246408,246409,246411],{},[2627,246410,246009],{}," 这是一种多级目录管理体系。在这种结构中，所有目录和文件形成了一棵倒置的树，到达特定文件的显式路径序列被称为该文件的“路径名（Pathname）”。",[108,246413,246414,246416],{},[2627,246415,246015],{}," 相比于死板的单级或两级目录，树形结构赋予了用户极大的组织灵活性。用户可以在任意层级自由创建子文件夹对项目进行归类，完美隔离了不同分支下的同名文件，同时也极大地优化了搜索效率和权限管理。",[108,246418,246419,246421],{},[2627,246420,246021],{}," 在树形结构中，唯一标识一个文件的标识符不再是短短的“文件名”，而是其在树中的“绝对路径名”。学生常在编程和概念题中忽略这一点。",[108,246423,246424],{},"逻辑地址\n程序中使用的地址，又称虚拟地址，是进程看到的地址，需要经过地址转换才能变成物理地址。",[15,246426,246427],{},"外部碎片\n内存中存在许多不连续的小空闲区，总空闲空间可能足够，但因为不连续，无法分配给需要连续空间的进程。",[15,246429,246430],{},"i-node\nUNIX\u002FLinux 文件系统中的索引节点，用来保存文件的元数据和数据块地址信息，如文件类型、权限、大小、时间、链接计数、直接\u002F间接地址项等。",[15,246432,246433],{},"活锁\n进程没有被阻塞，也在不断改变状态，但由于彼此反复让步或重试，始终无法向前推进完成任务。",[15,246435,246436],{},"临界区\n进程中访问共享资源或共享数据的那段代码。为了避免竞争条件，同一时刻通常只允许一个进程进入相关临界区。",[15,246438,246439],{},"中断\n由硬件或软件事件引起的处理器控制转移机制，使 CPU 暂停当前执行，转去执行相应的中断处理程序。",[15,246441,246442],{},"字符设备\n以字符流为单位进行输入输出的设备，通常不支持随机访问，如键盘、鼠标、串口、打印机等。",[15,246444,246445],{},"用户接口\n操作系统提供给用户与计算机系统交互的方式，包括命令行接口、图形用户接口和系统调用接口等。",[15,246447,246448],{},"并行\n多个任务在同一时刻真正同时执行，通常需要多个处理器或多个核心支持。",[15,246450,246451],{},"饥饿\n某个进程长期得不到所需资源或 CPU 服务，虽然系统整体在运行，但该进程一直无法继续执行。",[15,246453,246454],{},"文件管理系统\n操作系统中负责文件存储、组织、访问、保护和目录管理的部分，为用户提供按名存取文件的功能。",[15,246456,246457],{},"FAT\nFile Allocation Table，文件分配表。它记录磁盘中各文件数据块之间的链接关系，用于管理文件的磁盘空间分配。",[15,246459,246460],{},"索引文件\n通过索引表来组织和访问记录的文件。索引项通常保存关键字和对应记录地址，可以提高查找速度。",[15,246462,246463],{},"相对地址\n相对于某个基准地址或程序起始位置的地址，也称偏移地址。实际访问时需与基址相加形成物理地址或有效地址。",{"title":29,"searchDepth":43,"depth":43,"links":246465},[246466,246467,246468,246469,246470,246471,246472,246473,246474,246475,246476,246477,246478,246479,246480,246481,246482,246483,246484,246485],{"id":246001,"depth":43,"text":246002},{"id":246025,"depth":43,"text":246026},{"id":246046,"depth":43,"text":246047},{"id":246067,"depth":43,"text":246068},{"id":246088,"depth":43,"text":246089},{"id":246109,"depth":43,"text":246110},{"id":246130,"depth":43,"text":246131},{"id":246151,"depth":43,"text":246152},{"id":246172,"depth":43,"text":246173},{"id":246193,"depth":43,"text":246194},{"id":246214,"depth":43,"text":246215},{"id":246235,"depth":43,"text":246236},{"id":246256,"depth":43,"text":246257},{"id":246277,"depth":43,"text":246278},{"id":246298,"depth":43,"text":246299},{"id":246319,"depth":43,"text":246320},{"id":246340,"depth":43,"text":246341},{"id":246361,"depth":43,"text":246362},{"id":246382,"depth":43,"text":246383},{"id":246403,"depth":43,"text":246404},{"order":43},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fdef",{"title":245994,"description":29},"source\u002Fcs\u002Fos\u002Freview\u002FDef","uxN9UY7hW-zfQUG8o0dGUnGW3UmC_ZPQoUReaGbaUNI",{"id":246492,"title":246493,"body":246494,"description":29,"extension":85,"meta":248244,"navigation":52,"path":248245,"seo":248246,"stem":248247,"__hash__":248248},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002FTorF.md","TorF",{"type":8,"value":246495,"toc":248181},[246496,246498,246502,246505,246509,246514,246534,246538,246543,246560,246564,246568,246585,246589,246593,246610,246614,246618,246635,246639,246643,246660,246664,246668,246689,246693,246697,246714,246718,246722,246746,246750,246754,246775,246779,246783,246800,246804,246808,246825,246829,246833,246854,246858,246862,246879,246883,246887,246907,246911,246915,246932,246936,246940,246969,246973,246977,247002,247006,247010,247031,247035,247039,247063,247067,247071,247088,247092,247096,247117,247121,247125,247146,247150,247154,247175,247179,247183,247208,247212,247216,247233,247237,247241,247266,247270,247274,247291,247295,247299,247324,247328,247332,247349,247353,247357,247379,247383,247387,247412,247416,247420,247441,247445,247449,247474,247478,247482,247503,247507,247511,247528,247532,247536,247557,247561,247565,247590,247594,247598,247622,247626,247630,247647,247651,247655,247676,247680,247684,247709,247713,247717,247734,247738,247745,247752,247756,247760,247777,247781,247788,247795,247799,247806,247813,247817,247821,247838,247842,247846,247863,247867,247871,247888,247892,247896,247924,247928,247932,247963,247967,247971,247992,247996,248000,248021,248025,248029,248046,248050,248054,248088,248092,248096,248113,248117,248124,248131,248135,248139,248156,248160,248164],[32188,246497],{},[19,246499,246501],{"id":246500},"title-判断题order-5","title: 判断题\norder: 5",[11,246503,246504],{"id":246504},"判断题",[19,246506,246508],{"id":246507},"_1文件系统中分配存储空间的基本单位不是记录","1．文件系统中分配存储空间的基本单位不是记录。",[15,246510,246511,246513],{},[2627,246512,58173],{}," 正确 (True)",[105,246515,246516,246522,246528],{},[108,246517,246518,246521],{},[2627,246519,246520],{},"概念核对："," 文件系统的物理分配与逻辑组织。",[108,246523,246524,246527],{},[2627,246525,246526],{},"原理解析："," 从用户的逻辑视角来看，文件可以是由“记录”组成的。但是，当文件系统将数据真正存入磁盘时，它是以“物理块（Block）”为基本分配和传输单位的。磁盘是典型的面向块的设备（Block-oriented device），数据传输总是以块为单位进行，而不是以逻辑记录为单位。",[108,246529,246530,246533],{},[2627,246531,246532],{},"常见易错点："," 很多学生会混淆“逻辑视图”和“物理视图”。记录是讲给程序员听的逻辑单位，而物理块是讲给磁盘和操作系统听的物理分配单位。",[19,246535,246537],{"id":246536},"_2具有多道功能的操作系统一定是多用户操作系统","2．具有多道功能的操作系统一定是多用户操作系统。",[15,246539,246540,246542],{},[2627,246541,58173],{}," 错误 (False)",[105,246544,246545,246550,246555],{},[108,246546,246547,246549],{},[2627,246548,246520],{}," 多道程序设计 (Multiprogramming) 与 多用户 (Multi-user) 的区别。",[108,246551,246552,246554],{},[2627,246553,246526],{}," 多道程序设计指的是在内存中同时驻留多个进程，让它们交替使用CPU以提高资源利用率。这仅仅是一种“运行机制”。一个系统完全可以是单用户的但具备多道功能（例如，一个用户在个人电脑上一边听音乐，一边下载文件，一边写文档）。参考资料也明确指出，多道程序设计是现代操作系统的中心主题，但并未强制绑定必须有多个用户终端。",[108,246556,246557,246559],{},[2627,246558,246532],{}," 把“多任务（多个进程同时跑）”和“多用户（多个账号同时登录）”画等号。",[19,246561,246563],{"id":246562},"_3虚拟存储器是由操作系统提供的一个假想的特大存储器它并不是实际的内存其大小可比内存空间大得多","3．虚拟存储器是由操作系统提供的一个假想的特大存储器，它并不是实际的内存，其大小可比内存空间大得多。",[15,246565,246566,246513],{},[2627,246567,58173],{},[105,246569,246570,246575,246580],{},[108,246571,246572,246574],{},[2627,246573,246520],{}," 虚拟内存 (Virtual Memory) 的定义。",[108,246576,246577,246579],{},[2627,246578,246526],{}," 虚拟内存是一种存储分配方案，它允许辅存（如磁盘）被当作主存来寻址。程序员感觉到自己拥有一个极其庞大的内存空间（其大小受限于计算机的地址位数和磁盘容量，而不是实际物理内存的大小）。操作系统和硬件通过动态地址转换和缺页中断机制，在这个假想的存储器和真实的物理内存之间玩了“移花接木”的魔术。",[108,246581,246582,246584],{},[2627,246583,246532],{}," 误以为插在主板上的内存条才是虚拟内存。虚拟内存本质上是“磁盘空间+地址映射机制”营造的幻觉。",[19,246586,246588],{"id":246587},"_4批处理系统的主要优点是系统的吞吐量大资源利用率高系统的开销较小","4．批处理系统的主要优点是系统的吞吐量大、资源利用率高、系统的开销较小。",[15,246590,246591,246513],{},[2627,246592,58173],{},[105,246594,246595,246600,246605],{},[108,246596,246597,246599],{},[2627,246598,246520],{}," 批处理系统 (Batch Systems) 的设计目标。",[108,246601,246602,246604],{},[2627,246603,246526],{}," 简单批处理系统和多道批处理系统的核心目标，就是为了最大限度地提高处理器的利用率和系统吞吐量。通过让作业无缝衔接运行，消除了人类交互带来的缓慢等待时间。因为不需要频繁处理交互式中断，其调度和进程切换开销相对时间分时系统来说确实较小。",[108,246606,246607,246609],{},[2627,246608,246532],{}," 将批处理系统与分时系统（强调响应时间和交互性）的优缺点弄反。",[19,246611,246613],{"id":246612},"_5文件系统中源程序是有结构的记录式文件","5．文件系统中源程序是有结构的记录式文件。",[15,246615,246616,246542],{},[2627,246617,58173],{},[105,246619,246620,246625,246630],{},[108,246621,246622,246624],{},[2627,246623,246520],{}," 文件的逻辑结构：流式文件 vs. 记录式文件。",[108,246626,246627,246629],{},[2627,246628,246526],{}," 源程序（如C语言、Python代码）本质上是一连串无格式的字符或字节流（Stream of bytes），系统并不关心其内部的逻辑界限。因此，源程序属于“无结构文件（流式文件）”。而有结构的“记录式文件”通常是由一组结构相似的记录（包含特定字段）组成，常见于数据库文件。",[108,246631,246632,246634],{},[2627,246633,246532],{}," 以为代码有“缩进、行号”就是系统眼中的“有结构”。在OS眼里，只要不是固定字段组成的记录，统统视为无结构的字节流。",[19,246636,246638],{"id":246637},"_6在多道程序环境下普通用户也能设计用内存物理地址直接访问内存的程序","6．在多道程序环境下，普通用户也能设计用内存物理地址直接访问内存的程序。",[15,246640,246641,246542],{},[2627,246642,58173],{},[105,246644,246645,246650,246655],{},[108,246646,246647,246649],{},[2627,246648,246520],{}," 逻辑地址与物理地址、内存保护。",[108,246651,246652,246654],{},[2627,246653,246526],{}," 在多道程序系统中，程序会被动态加载甚至换入换出，程序员无法提前知道程序会被放在哪个物理地址。因此，用户程序只能使用“逻辑地址（虚拟地址）”。硬件的内存管理单元（MMU）会在运行时将逻辑地址翻译为物理地址。如果允许用户直接操作物理地址，就彻底破坏了操作系统的“内存保护（Protection）”机制，恶意程序可以轻易篡改操作系统或其他用户的内存数据。",[108,246656,246657,246659],{},[2627,246658,246532],{}," 以为C语言中的指针拿到的就是真实的物理引脚地址。其实指针里存的依然是逻辑地址。",[19,246661,246663],{"id":246662},"_7顺序文件适合建立在顺序存储设备上而不适合建立在磁盘上","7．顺序文件适合建立在顺序存储设备上，而不适合建立在磁盘上。",[15,246665,246666,246542],{},[2627,246667,58173],{},[105,246669,246670,246675,246684],{},[108,246671,246672,246674],{},[2627,246673,246520],{}," 文件物理结构与存储介质特性的匹配。",[108,246676,246677,246679,246680,246683],{},[2627,246678,246526],{}," （注：此解析结合了给定的参考资料以及操作系统的通用外部补充知识进行讲解）顺序存储设备（如磁带）确实只能建立顺序文件；但是，磁盘作为一种支持随机存取的直接存储设备，它",[2627,246681,246682],{},"同样可以","用来建立顺序文件，并且存取速度远快于磁带。只是说磁盘还能支持更高级的索引文件而已。",[108,246685,246686,246688],{},[2627,246687,246532],{}," 认为“高级设备不能兼容低级文件组织”。实际上磁盘能够完美兼容顺序读写。",[19,246690,246692],{"id":246691},"_8spooling系统实现设备管理的虚拟技术即将独占设备改造为共享设备它由专门负责io的常驻内存进程以及输入输出井组成","8．SPOOLing系统实现设备管理的虚拟技术，即：将独占设备改造为共享设备。它由专门负责I\u002FO的常驻内存进程以及输入、输出井组成。",[15,246694,246695,246513],{},[2627,246696,58173],{},[105,246698,246699,246704,246709],{},[108,246700,246701,246703],{},[2627,246702,246520],{}," 假脱机技术 (SPOOLing)。",[108,246705,246706,246708],{},[2627,246707,246526],{}," SPOOLing的核心功能正是利用高速大容量的磁盘，在上面开辟出输入井和输出井。当用户进程想使用独占设备（比如打印机）时，它实际上只是把数据发送到了磁盘的输出井中。后台的常驻守护进程会负责排队并将这些数据真正发送到物理设备。这样，所有进程都不会因为等待独占设备而阻塞，独占设备在逻辑上被改造为了虚拟的共享设备。",[108,246710,246711,246713],{},[2627,246712,246532],{}," 误以为输入\u002F输出井是开辟在内存（RAM）中的。它们必须在磁盘上，以便提供巨大缓冲。",[19,246715,246717],{"id":246716},"_9系统调用是操作系统与外界程序之间的接口它属于核心程序在层次结构设计中它最靠近硬件","9．系统调用是操作系统与外界程序之间的接口，它属于核心程序。在层次结构设计中，它最靠近硬件。",[15,246719,246720,246542],{},[2627,246721,58173],{},[105,246723,246724,246729,246741],{},[108,246725,246726,246728],{},[2627,246727,246520],{}," 系统调用 (System Call) 与 OS 层次结构。",[108,246730,246731,246733,246734,246737,246738,246740],{},[2627,246732,246526],{}," 系统调用确实是应用程序请求OS服务的接口。但是，在层次结构设计中，系统调用接口位于操作系统的",[2627,246735,246736],{},"最外层","（紧贴着用户程序），而",[2627,246739,243550],{},"最靠近硬件！最靠近硬件的应该是硬件控制层（如硬件抽象层HAL、设备驱动程序、中断处理程序等）。",[108,246742,246743,246745],{},[2627,246744,246532],{}," 觉得系统调用权限大，就一定放在底层。实际上它是高层的API，用于屏蔽底层的复杂性。",[19,246747,246749],{"id":246748},"_10若系统中存在一个循环等待的进程集合则必定会死锁","10．若系统中存在一个循环等待的进程集合，则必定会死锁。",[15,246751,246752,246542],{},[2627,246753,58173],{},[105,246755,246756,246761,246770],{},[108,246757,246758,246760],{},[2627,246759,246520],{}," 死锁的四个必要条件 (Necessary conditions for Deadlock)。",[108,246762,246763,246765,246766,246769],{},[2627,246764,246526],{}," 产生死锁有四个条件：互斥、请求和保持、不剥夺、循环等待。这四个条件是死锁的",[2627,246767,246768],{},"必要条件","。（注：结合通用OS知识补充：只有当每种资源都只有一个实例时，循环等待才是死锁的充分条件。如果某种资源有多个实例，即使出现了循环等待环路，环路外的某个进程依然可能释放资源从而打破僵局，此时并不会死锁）。",[108,246771,246772,246774],{},[2627,246773,246532],{}," 将必要条件当成了充分条件。",[19,246776,246778],{"id":246777},"_11并发性是指若干事件在同一时刻发生","11．并发性是指若干事件在同一时刻发生。",[15,246780,246781,246542],{},[2627,246782,58173],{},[105,246784,246785,246790,246795],{},[108,246786,246787,246789],{},[2627,246788,246520],{}," 并发 (Concurrency) vs. 并行 (Parallelism)。",[108,246791,246792,246794],{},[2627,246793,246526],{}," 在计算机科学中，“并发”是指在宏观的同一个时间段（时间间隔）内有多个程序交替推进，它们是在同一实体上通过时分复用（走走停停）实现的。“在同一时刻（微观上的同时）发生”的物理概念叫做“并行（Parallelism）”，这通常需要多处理器（Multiprocessor）或多核支持。",[108,246796,246797,246799],{},[2627,246798,246532],{}," 日常生活中这两个词通用，但在OS学科中它们界限森严。",[19,246801,246803],{"id":246802},"_12虚存容量的扩大是以牺牲cpu工作时间以及内外存交换时间为代价的","12．虚存容量的扩大是以牺牲CPU工作时间以及内、外存交换时间为代价的。",[15,246805,246806,246513],{},[2627,246807,58173],{},[105,246809,246810,246815,246820],{},[108,246811,246812,246814],{},[2627,246813,246520],{}," 虚拟内存的性能开销。",[108,246816,246817,246819],{},[2627,246818,246526],{}," 虚拟内存通过将程序切分为页或段并在主存和磁盘之间动态换入换出，从而扩大了逻辑上的内存容量。然而，这个魔术的代价是：一旦发生缺页（Page fault），系统必须挂起进程，执行磁盘I\u002FO（内外存交换），并且操作系统必须花费CPU时间来执行复杂的页置换算法、更新页表和处理中断。如果管理不当，甚至会导致抖动（Thrashing）使得CPU利用率暴跌。",[108,246821,246822,246824],{},[2627,246823,246532],{}," 认为虚拟内存是“免费的午餐”。一切虚拟化都是以时间换空间。",[19,246826,246828],{"id":246827},"_13用户为每个自己的进程创建pcb并控制进程的执行过程","13．用户为每个自己的进程创建PCB，并控制进程的执行过程。",[15,246830,246831,246542],{},[2627,246832,58173],{},[105,246834,246835,246840,246849],{},[108,246836,246837,246839],{},[2627,246838,246520],{}," 进程控制块 (Process Control Block, PCB) 的管理权限。",[108,246841,246842,246844,246845,246848],{},[2627,246843,246526],{}," PCB包含了处理器寄存器、优先级、内存指针等极度核心的信息。它由",[2627,246846,246847],{},"操作系统","负责创建和专门管理，以防止用户程序互相干扰。用户程序无权直接操作PCB，也不能直接控制进程的底层调度与执行过程，这些都属于操作系统的特权功能（在内核态执行）。",[108,246850,246851,246853],{},[2627,246852,246532],{}," 以为“自己的进程”就可以自己控制。在OS的“家长”模式下，用户只是提交任务，一切管理权归OS。",[19,246855,246857],{"id":246856},"_14树型目录结构能够解决文件重名问题","14．树型目录结构能够解决文件重名问题。",[15,246859,246860,246513],{},[2627,246861,58173],{},[105,246863,246864,246869,246874],{},[108,246865,246866,246868],{},[2627,246867,246520],{}," 目录结构的隔离性。",[108,246870,246871,246873],{},[2627,246872,246526],{}," 单级目录所有文件挤在一起，非常容易重名。而树型目录允许不同的用户或项目创建自己的子目录（分支），不同子目录下的文件即使名字相同，其完整的绝对路径（Pathname）也是独一无二的。这种层次结构不仅便于分离管理，也彻底避免了文件名冲突（Name clashes）的问题。",[108,246875,246876,246878],{},[2627,246877,246532],{}," 对文件系统的基础认识不足，可能会误以为系统强行要求所有磁盘文件不能重名。",[19,246880,246882],{"id":246881},"_15原语是一种不可分割的操作","15．原语是一种不可分割的操作。",[15,246884,246885,246513],{},[2627,246886,58173],{},[105,246888,246889,246894,246899],{},[108,246890,246891,246893],{},[2627,246892,246520],{}," 原语 (Atomic Operation) 的定义。",[108,246895,246896,246898],{},[2627,246897,246526],{}," 原语（原子操作）是指由一个或多个指令组成的函数或动作，在执行过程中表现为“不可分割（indivisible）”。即它要么全做，要么全不做，且在执行过程中绝对不会被硬件中断或其它进程打断，这保证了并发环境下的隔离性。",[108,246900,246901,246903,246904,347],{},[2627,246902,246532],{}," 误以为原语只是某种基础函数。它的灵魂在于它的",[2627,246905,246906],{},"不可中断性",[19,246908,246910],{"id":246909},"_16通道一旦被启动就能独立于cpu运行这样可使cpu和通道并行操作","16．通道一旦被启动就能独立于CPU运行，这样可使CPU和通道并行操作。",[15,246912,246913,246513],{},[2627,246914,58173],{},[105,246916,246917,246922,246927],{},[108,246918,246919,246921],{},[2627,246920,246520],{}," I\u002FO 通道 (I\u002FO Channel) 的工作原理。",[108,246923,246924,246926],{},[2627,246925,246526],{}," I\u002FO通道（或I\u002FO处理器）本质上是一台专门负责I\u002FO的“小电脑”。CPU只需要向通道发出一组I\u002FO程序的起始地址，然后就可以去执行其他进程了。通道会自动从内存中取出指令并控制外设执行，直到整个复杂的I\u002FO任务全部完成后，才向CPU发出一次中断。这极大释放了CPU，实现了真正的CPU与I\u002FO并行。",[108,246928,246929,246931],{},[2627,246930,246532],{}," 低估了通道的自主性，认为仍需CPU循环介入。",[19,246933,246935],{"id":246934},"_17页式的地址是一维的段式的地址是二维的","17．页式的地址是一维的，段式的地址是二维的",[15,246937,246938,246513],{},[2627,246939,58173],{},[105,246941,246942,246947,246964],{},[108,246943,246944,246946],{},[2627,246945,246520],{}," 分页与分段的逻辑地址结构差异。",[108,246948,246949,246951,246952,246955,246956,246959,246960,246963],{},[2627,246950,246526],{},"\n对于分页：分页是由",[2627,246953,246954],{},"硬件","自动完成的，对程序员透明。程序员写代码时使用的是一个单一的、线性的连续地址（一维）。硬件MMU会自动把这个一维地址截断成“页号”和“页内偏移”。\n对于分段：分段是",[2627,246957,246958],{},"程序员可见","的。程序员（或编译器）必须显式地指出模块（段）和偏移量，即地址形式为 ",[31,246961,246962],{},"(段名\u002F段号, 段内偏移)","。因为它保留了程序的逻辑结构，因此逻辑上被称为二维地址。",[108,246965,246966,246968],{},[2627,246967,246532],{}," 因为两者的物理翻译机制都有“号+偏移”，就误以为两者对程序员来说都是二维的。",[19,246970,246972],{"id":246971},"_18位示图方法可用于磁盘的调度管理","18．位示图方法可用于磁盘的调度管理。",[15,246974,246975,246542],{},[2627,246976,58173],{},[105,246978,246979,246984,246997],{},[108,246980,246981,246983],{},[2627,246982,246520],{}," 位示图 (Bit map) 的用途。",[108,246985,246986,246988,246989,246992,246993,246996],{},[2627,246987,246526],{}," （注：此解析结合了给定的参考资料以及操作系统的通用外部补充知识进行讲解）位示图是一种利用二进制位（0或1）来表示资源使用情况的数据结构，主要用于",[2627,246990,246991],{},"磁盘空闲存储空间的分配与管理","（记录哪个块被用了，哪个块是空的）。而磁盘的",[2627,246994,246995],{},"调度管理","（Disk Scheduling）指的是决定磁头移动顺序以优化寻道时间的算法（如FCFS、SSTF、SCAN等）。两者概念风马牛不相及。",[108,246998,246999,247001],{},[2627,247000,246532],{}," 看到“磁盘”和“管理”就觉得是对的。",[19,247003,247005],{"id":247004},"_19虚拟设备是指把一个物理设备变换成多个对应的逻辑设备它通过逻辑设备表来实现的","19．虚拟设备是指把一个物理设备变换成多个对应的逻辑设备，它通过逻辑设备表来实现的。",[15,247007,247008,246542],{},[2627,247009,58173],{},[105,247011,247012,247017,247026],{},[108,247013,247014,247016],{},[2627,247015,246520],{}," 虚拟设备 (Virtual Device) 与 设备独立性 (Device Independence)。",[108,247018,247019,247021,247022,247025],{},[2627,247020,246526],{}," 命题的前半句描述的是虚拟设备（即通过SPOOLing技术，将一个物理打印机变为了多个逻辑打印机，让每个进程感觉自己独占设备）。但是，这",[2627,247023,247024],{},"并非","通过“逻辑设备表（LUT）”实现的。SPOOLing依靠的是大容量的辅助存储器（磁盘井）和多道程序设计来截获缓冲；而“逻辑设备表”主要是用来实现“设备独立性”（让代码中的逻辑设备名映射到当前系统插着的物理设备驱动上）。两者机制被张冠李戴了。",[108,247027,247028,247030],{},[2627,247029,246532],{}," 将操作系统中的不同映射机制弄混。",[19,247032,247034],{"id":247033},"_20页式管理易于实现不同进程间的信息共享","20．页式管理易于实现不同进程间的信息共享。",[15,247036,247037,246542],{},[2627,247038,58173],{},[105,247040,247041,247046,247058],{},[108,247042,247043,247045],{},[2627,247044,246520],{}," 分段与分页在共享和保护上的优劣。",[108,247047,247048,247050,247051,247054,247055,347],{},[2627,247049,246526],{}," 页是物理单位，大小固定，一页里面可能前半截是代码，后半截是私有数据。如果把这页共享出去，就会把不该共享的数据也暴露了。相反，",[2627,247052,247053],{},"分段管理","（Segmentation）按照程序的逻辑结构（如某个完整的函数、某个数组）进行切分。因为段具有完整的逻辑意义，所以",[2627,247056,247057],{},"段式管理才易于实现进程间的信息共享与保护",[108,247059,247060,247062],{},[2627,247061,246532],{}," 以为分页是最现代的技术，就在任何指标上都优于分段。",[19,247064,247066],{"id":247065},"_21在虚拟存储方式下程序员编制程序时不必考虑主存的容量但系统的吞吐量在很大程度上依赖于主存储器的容量","21．在虚拟存储方式下，程序员编制程序时不必考虑主存的容量，但系统的吞吐量在很大程度上依赖于主存储器的容量；",[15,247068,247069,246513],{},[2627,247070,58173],{},[105,247072,247073,247078,247083],{},[108,247074,247075,247077],{},[2627,247076,246520],{}," 虚拟内存的编程透明性及其物理瓶颈。",[108,247079,247080,247082],{},[2627,247081,246526],{}," 虚拟内存给程序员提供了一个比实际物理内存大得多的地址空间（哪怕只有2GB内存，也能跑4GB的程序）。因此编程时不必考虑物理内存限制。但如果在多道程序下物理内存太小，进程被分配的页框不足，就会引发极高的缺页中断率，导致CPU大量时间浪费在等磁盘I\u002FO上（即系统“抖动” Thrashing），吞吐量会急剧下降。",[108,247084,247085,247087],{},[2627,247086,246532],{}," 认为虚拟内存完全摆脱了对物理硬件的依赖。",[19,247089,247091],{"id":247090},"_22可重定位分区管理可以对作业分配不连续的内存单元","22．可重定位分区管理可以对作业分配不连续的内存单元；",[15,247093,247094,246542],{},[2627,247095,58173],{},[105,247097,247098,247103,247112],{},[108,247099,247100,247102],{},[2627,247101,246520],{}," 分区管理 (Partitioning) 与 连续分配。",[108,247104,247105,247107,247108,247111],{},[2627,247106,246526],{}," “可重定位分区管理（通过紧凑\u002FCompaction技术解决碎片）”依然属于",[2627,247109,247110],{},"连续分配方式","。它要求一个作业在内存中必须占有一整块连续的区域。紧凑技术只是把作业搬来搬去，但在任意时刻，作业在内存里依然是紧挨着的。只有“分页”或“分段”技术才能实现真正的非连续分配。",[108,247113,247114,247116],{},[2627,247115,246532],{}," 把动态重定位中的“可移动性”与“非连续分配”混淆。",[19,247118,247120],{"id":247119},"_23采用动态重定位技术的系统目标程序可以不经任何改动而装入物理内存","23．采用动态重定位技术的系统，目标程序可以不经任何改动，而装入物理内存；",[15,247122,247123,246513],{},[2627,247124,58173],{},[105,247126,247127,247132,247141],{},[108,247128,247129,247131],{},[2627,247130,246520],{}," 动态重定位 (Dynamic Relocation)。",[108,247133,247134,247136,247137,247140],{},[2627,247135,246526],{}," 采用动态重定位时，程序装入内存的过程不需要修改二进制代码中的逻辑地址。一切地址翻译工作都被推迟到了",[2627,247138,247139],{},"指令真正执行的那一瞬间","。此时，处理器硬件（MMU）会自动将指令中的相对地址加上基址寄存器中的值，实时生成绝对物理地址。正因如此，目标程序自身不经任何改动即可执行，且可以随时在内存中搬家。",[108,247142,247143,247145],{},[2627,247144,246532],{}," 分不清静态重定位（装入时必须修改程序内部地址）和动态重定位。",[19,247147,247149],{"id":247148},"_24页式存储管理中一个作业可以占用不连续的内存空间而段式存储管理一个作业则是占用连续的内存空间","24．页式存储管理中，一个作业可以占用不连续的内存空间，而段式存储管理，一个作业则是占用连续的内存空间。",[15,247151,247152,246542],{},[2627,247153,58173],{},[105,247155,247156,247161,247170],{},[108,247157,247158,247160],{},[2627,247159,246520],{}," 分段技术的非连续性。",[108,247162,247163,247165,247166,247169],{},[2627,247164,246526],{}," 在段式存储管理中，一个段内部是连续的，但是",[2627,247167,247168],{},"同一个作业的不同段是无需连续存放的","（它们可以散落在内存的各个不同分区里）。无论是页式还是段式管理，它们相比于老式的分区管理，最大的突破就在于允许一个进程被打散并存放在非连续的内存空间中。",[108,247171,247172,247174],{},[2627,247173,246532],{}," 误以为只有分页能打散，分段就必须整块连续。",[19,247176,247178],{"id":247177},"_25线程是最小的拥有资源的单位","25．线程是最小的拥有资源的单位。",[15,247180,247181,246542],{},[2627,247182,58173],{},[105,247184,247185,247190,247203],{},[108,247186,247187,247189],{},[2627,247188,246520],{}," 进程与线程的职能划分。",[108,247191,247192,247194,247195,247198,247199,247202],{},[2627,247193,246526],{}," 传统的操作系统设计中，",[2627,247196,247197],{},"进程","（Process）是资源分配和保护的最小单位（它拥有自己独立的虚拟地址空间、文件句柄等）；而",[2627,247200,247201],{},"线程","（Thread）是CPU调度和执行的最小单位。同一个进程内的所有线程都共享该进程拥有的资源，线程自身基本不拥有独立的系统资源（除了一点必不可少的栈和寄存器上下文）。",[108,247204,247205,247207],{},[2627,247206,246532],{}," 概念颠倒。记住：进程管“家产（资源）”，线程管“干活（执行）”。",[19,247209,247211],{"id":247210},"_26文件系统最基本的功能是实现按名存取","26．文件系统最基本的功能是实现按名存取。",[15,247213,247214,246513],{},[2627,247215,58173],{},[105,247217,247218,247223,247228],{},[108,247219,247220,247222],{},[2627,247221,246520],{}," 文件系统抽象。",[108,247224,247225,247227],{},[2627,247226,246526],{}," 文件是对磁盘数据的逻辑抽象，用户只需要给文件起个“名字”即可。文件系统的核心机制就是维护“文件名”到“物理磁盘块位置”的映射目录。因此，将复杂的物理块寻址隐藏起来，向用户提供“按名存取”的服务，是文件系统最根本的功能和存在意义。",[108,247229,247230,247232],{},[2627,247231,246532],{}," 纠结于读写权限或缓存优化，这些都是附加的高级功能。",[19,247234,247236],{"id":247235},"_27存取控制表是每个用户一张表明该用户对不同文件的存取权限","27．存取控制表是每个用户一张，表明该用户对不同文件的存取权限。",[15,247238,247239,246542],{},[2627,247240,58173],{},[105,247242,247243,247248,247261],{},[108,247244,247245,247247],{},[2627,247246,246520],{}," 访问控制列表 (Access Control List, ACL)。",[108,247249,247250,247252,247253,247256,247257,247260],{},[2627,247251,246526],{}," （注：此解析结合了给定的参考资料以及操作系统的通用外部补充知识进行讲解）存取控制表（ACL）通常是",[2627,247254,247255],{},"每个文件（或客体）一张","，表中列出了系统里哪些用户对“该特定文件”拥有哪些权限。与之相对的，",[2627,247258,247259],{},"按用户","来维护的叫做“能力表”（Capability List），它才是给每个用户发一张清单，上面列出了该用户能访问哪些文件。",[108,247262,247263,247265],{},[2627,247264,246532],{}," 弄混了访问矩阵的行和列。ACL是“挂在门上的名单”，能力表是“揣在身上的钥匙串”。",[19,247267,247269],{"id":247268},"_28spooling技术可以解决进程使用设备死锁问题","28．SPOOLing技术可以解决进程使用设备死锁问题。",[15,247271,247272,246513],{},[2627,247273,58173],{},[105,247275,247276,247281,247286],{},[108,247277,247278,247280],{},[2627,247279,246520],{}," 死锁预防机制与SPOOLing。",[108,247282,247283,247285],{},[2627,247284,246526],{}," 发生死锁的一个重要原因是进程申请了独占性设备，如果不放手又去申请别的资源，就容易满足“请求和保持”条件。SPOOLing技术将打印机这种独占设备改造成了虚拟的共享设备。进程不再直接去独占物理设备，而是迅速将数据写入到磁盘缓存井中然后立刻返回。这就彻底消除了进程由于争夺此类独占设备而长期阻塞的现象，从而从根本上预防了该环节的死锁。",[108,247287,247288,247290],{},[2627,247289,246532],{}," 认为SPOOLing只是用来提高打印速度的。在系统层面上，它也是死锁防范的利器。",[19,247292,247294],{"id":247293},"_29对于一个具有三级索引表的文件存取一个记录需要访问三次磁盘","29．对于一个具有三级索引表的文件，存取一个记录需要访问三次磁盘。",[15,247296,247297,246542],{},[2627,247298,58173],{},[105,247300,247301,247306,247319],{},[108,247302,247303,247305],{},[2627,247304,246520],{}," 多级索引（如UNIX inode）的访存次数。",[108,247307,247308,247310,247311,247314,247315,247318],{},[2627,247309,246526],{}," （注：此解析结合了给定的参考资料以及操作系统的通用外部补充知识进行讲解）如果文件采用三级（间接）索引，要拿到最终的数据块，一共需要访问",[2627,247312,247313],{},"四次","磁盘：第一次读取一级索引表所在块，第二次读取二级索引表所在块，第三次读取三级索引表所在块，",[2627,247316,247317],{},"第四次才是真正去读取最终指向的那个数据记录块本身","。（前提是假设这些索引块均没有缓存在内存中）。",[108,247320,247321,247323],{},[2627,247322,246532],{}," 忘记了“索引只是指路牌”，找完路之后最后还要踏出那一步去拿“数据本身”。",[19,247325,247327],{"id":247326},"_30在io控制的多种方式中传输速率高对主机影响少的方式最好","30．在I\u002FO控制的多种方式中，传输速率高，对主机影响少的方式最好。",[15,247329,247330,246513],{},[2627,247331,58173],{},[105,247333,247334,247339,247344],{},[108,247335,247336,247338],{},[2627,247337,246520],{}," I\u002FO 控制方式的演进目标。",[108,247340,247341,247343],{},[2627,247342,246526],{}," 从早期的程序直接控制（CPU死循环轮询），到中断控制（解除了死等，但CPU还要搬运数据），再到DMA（直接内存访问，只需在开始和结束时打扰CPU），再到独立性极高的通道\u002FI\u002FO处理器技术。整个I\u002FO发展史的核心目标就是两个：提高外设与内存之间的数据传输速率，以及尽可能地减少CPU（主机）的介入和开销。",[108,247345,247346,247348],{},[2627,247347,246532],{}," 对于设计原则的把控不够宏观，系统设计的终极追求永远是解放CPU和提高效率。",[19,247350,247352],{"id":247351},"_31进程可以删除自己的pcb表","31．进程可以删除自己的PCB表。",[15,247354,247355,246542],{},[2627,247356,58173],{},[105,247358,247359,247364,247373],{},[108,247360,247361,247363],{},[2627,247362,246520],{}," 进程控制块（PCB，Process Control Block）的作用与管理权限。",[108,247365,247366,247368,247369,247372],{},[2627,247367,246526],{}," PCB 包含了处理器寄存器、进程优先级、内存指针等极其核心的系统信息，它不仅是操作系统（OS）管理进程的唯一凭证，而且",[2627,247370,247371],{},"完全由操作系统负责创建和维护","。用户进程只能在“用户态（User mode）”下运行，而读取和修改 PCB 属于特权操作，必须在“内核态（Kernel mode）”下由 OS 来执行。如果允许进程自己删除自己的 PCB，将会直接导致操作系统崩溃或系统状态彻底失控。",[108,247374,247375,247378],{},[2627,247376,247377],{},"易错点："," 学生常误以为“既然是‘我的’进程，我就有权限掌控它的所有数据”。在操作系统的“家长式”管理下，用户进程连自己的“身份证（PCB）”都摸不到，更别提删除了。",[19,247380,247382],{"id":247381},"_32可重定位分区法能够支持虚拟存储器的技术","32．可重定位分区法能够支持虚拟存储器的技术。",[15,247384,247385,246542],{},[2627,247386,58173],{},[105,247388,247389,247394,247407],{},[108,247390,247391,247393],{},[2627,247392,246520],{}," 虚拟内存（Virtual Memory）的实现前提。",[108,247395,247396,247398,247399,247402,247403,247406],{},[2627,247397,246526],{}," 虚拟内存的核心突破在于：允许进程被切分成许多小块（页或段），并且",[2627,247400,247401],{},"不需要连续存放，甚至不需要全部装入内存","即可运行。而可重定位分区法（即带有内存紧凑\u002FCompaction技术的动态分区）虽然允许进程在内存中移动，但它本质上依然属于",[2627,247404,247405],{},"连续分配","方案，要求一个进程必须完整且连续地霸占一整块内存区域。因此，它无法支持按需调页的虚拟存储技术。",[108,247408,247409,247411],{},[2627,247410,247377],{}," 极易将“动态重定位（能在内存里搬家）”与“虚拟内存（能把部分身子放在硬盘上）”这两个概念混淆。",[19,247413,247415],{"id":247414},"_33单级目录结构能够解决文件重名问题","33．单级目录结构能够解决文件重名问题。",[15,247417,247418,246542],{},[2627,247419,58173],{},[105,247421,247422,247427,247436],{},[108,247423,247424,247426],{},[2627,247425,246520],{}," 目录结构对命名空间的隔离作用。",[108,247428,247429,247431,247432,247435],{},[2627,247430,246526],{}," 在单级目录（Single-level directory）中，整个系统只有一个存放所有文件的“大平层”列表。这意味着系统中所有的文件都必须拥有独一无二的名字，一旦两个用户恰好取了相同的名字，就会发生冲突或覆盖。只有引入",[2627,247433,247434],{},"树形结构目录（Tree-structured Directory）","，允许不同用户或项目创建自己的子文件夹，才能通过不同的“绝对路径”来完美隔离并解决同名冲突（前文背景）。",[108,247437,247438,247440],{},[2627,247439,247377],{}," 高估了单级目录的管理能力。只要都在同一个列表里，就不可能允许重名。",[19,247442,247444],{"id":247443},"_34分页式存储管理中页的大小是可以不相等的","34．分页式存储管理中，页的大小是可以不相等的。",[15,247446,247447,246542],{},[2627,247448,58173],{},[105,247450,247451,247456,247469],{},[108,247452,247453,247455],{},[2627,247454,246520],{}," 分页（Paging）与 分段（Segmentation）的基本定义。",[108,247457,247458,247460,247461,247468],{},[2627,247459,246526],{}," 分页的本质就是将内存和进程生硬地切割成",[2627,247462,247463,247464,247467],{},"固定且相等大小（fixed-size, equal-size）",[2627,247465,247466],{},"的块（物理上叫页框 frame，逻辑上叫页 page）。如果大小不相等，就失去了分页用来消除外部碎片、简化硬件地址映射的初衷。大小可以不相等的是","分段（Segmentation）","，因为段是根据程序的逻辑意义切分的。",[108,247470,247471,247473],{},[2627,247472,247377],{}," 概念张冠李戴。记住：页是硬件强制的“均分蛋糕”，段是程序员按逻辑切的“大小不一的披萨”。",[19,247475,247477],{"id":247476},"_35执行原语时不会响应任何中断","35．执行原语时不会响应任何中断。",[15,247479,247480,246513],{},[2627,247481,58173],{},[105,247483,247484,247489,247498],{},[108,247485,247486,247488],{},[2627,247487,246520],{}," 原语（Primitive \u002F Atomic Operation）的原子性。",[108,247490,247491,247493,247494,247497],{},[2627,247492,246526],{}," 原语是由若干条机器指令组成的一个过程，它被设计为",[2627,247495,247496],{},"不可分割的原子操作（indivisible）","。这意味着在原语执行的整个过程中，处理器会通过硬件手段（例如关中断）来屏蔽所有外部干扰，绝对不会响应任何中断。如果原语被打断，可能会导致系统内核的数据结构（如信号量队列）处于不一致的中间状态。",[108,247499,247500,247502],{},[2627,247501,247377],{}," 以为原语只是一种普通的 C 语言函数。它的核心灵魂就在于“要么全做，要么全不做，且绝不被打断”。",[19,247504,247506],{"id":247505},"_36段页式管理实现了段式页式两种存储方式的优势互补","36．段页式管理实现了段式、页式两种存储方式的优势互补。",[15,247508,247509,246513],{},[2627,247510,58173],{},[105,247512,247513,247518,247523],{},[108,247514,247515,247517],{},[2627,247516,246520],{}," 段页式虚拟内存（Combined Paging and Segmentation）。",[108,247519,247520,247522],{},[2627,247521,246526],{}," 分页的优势在于对程序员透明且彻底消灭了外部碎片，使得内存利用率高；分段的优势在于可见于程序员，方便实现数据共享、保护以及动态增长。操作系统设计者通过将用户地址空间“先分段，段内再分页”，完美地将这两种方案的优势结合在了一起。",[108,247524,247525,247527],{},[2627,247526,247377],{}," 这是操作系统设计中“折中与融合”的经典案例。",[19,247529,247531],{"id":247530},"_37对临界资源应采取互斥访问方式来实现共享","37．对临界资源应采取互斥访问方式来实现共享。",[15,247533,247534,246513],{},[2627,247535,58173],{},[105,247537,247538,247543,247552],{},[108,247539,247540,247542],{},[2627,247541,246520],{}," 临界资源（Critical Resource）与互斥（Mutual Exclusion）。",[108,247544,247545,247547,247548,247551],{},[2627,247546,246526],{}," 临界资源是指那些一次只允许一个进程安全使用的资源（如打印机、共享全局变量）。如果大家同时去乱抢乱写，数据就会崩溃。因此，我们必须在代码的临界区（Critical section）施加",[2627,247549,247550],{},"互斥访问机制","——即“排队轮流使用”，以此在保证数据一致性的前提下，实现宏观上的系统资源共享。",[108,247553,247554,247556],{},[2627,247555,247377],{}," 有些同学会觉得“互斥”和“共享”是反义词。其实在 OS 中，“互斥”恰恰是为了实现安全的“分时共享”。",[19,247558,247560],{"id":247559},"_38文件系统中分配存储空间的基本单位是记录","38．文件系统中分配存储空间的基本单位是记录。",[15,247562,247563,246542],{},[2627,247564,58173],{},[105,247566,247567,247572,247585],{},[108,247568,247569,247571],{},[2627,247570,246520],{}," 逻辑单位 vs. 物理分配单位。",[108,247573,247574,247576,247577,247580,247581,247584],{},[2627,247575,246526],{}," “记录（Record）”是文件在",[2627,247578,247579],{},"逻辑","上呈现给用户的有结构单位。但是，从底层文件系统对磁盘存储空间的分配和传输来看，所有的外围存储设备（特别是磁盘）都是面向块的设备（Block-oriented devices）。因此，操作系统与磁盘交换数据、分配空间的",[2627,247582,247583],{},"基本物理单位是块（Block）或物理块","（前文背景）。",[108,247586,247587,247589],{},[2627,247588,247377],{}," 将人类视角的逻辑结构强加给底层的物理硬件。",[19,247591,247593],{"id":247592},"_39若系统中存在一个循环等待的进程集合则必定会死锁","39．若系统中存在一个循环等待的进程集合，则必定会死锁。",[15,247595,247596,246542],{},[2627,247597,58173],{},[105,247599,247600,247605,247617],{},[108,247601,247602,247604],{},[2627,247603,246520],{}," 死锁的四个必要条件（Necessary conditions for Deadlock）。",[108,247606,247607,247609,247610,247612,247613,247616],{},[2627,247608,246526],{}," 循环等待（Circular wait）仅仅是产生死锁的四个",[2627,247611,246768],{},"之一。只有当系统中的每类资源都",[2627,247614,247615],{},"只有 1 个实例","时，循环等待才是必定死锁的充分条件。如果某些资源拥有多个实例，即便进程之间构成了循环等待的闭环图，环路外的其他进程依然可能释放资源，从而打破僵局，这种情况下就不会发生死锁 (图 6.5d 即为有环但无死锁的例子)。",[108,247618,247619,247621],{},[2627,247620,247377],{}," 逻辑学中的“必要条件”不等于“充分条件”。有环不一定死锁，除非资源都是单实例的。",[19,247623,247625],{"id":247624},"_40具有多道功能的操作系统一定是多用户操作系统","40．具有多道功能的操作系统一定是多用户操作系统。",[15,247627,247628,246542],{},[2627,247629,58173],{},[105,247631,247632,247637,247642],{},[108,247633,247634,247636],{},[2627,247635,246520],{}," 多道程序设计（Multiprogramming）的概念。",[108,247638,247639,247641],{},[2627,247640,246526],{}," 多道功能指的是内存中能够同时存放多个进程，让处理器交替执行以提高效率。这与是否有多个用户终端（多用户）没有必然联系。你手头的个人电脑（Windows\u002FmacOS）只有一个用户（你），但它绝对是一个支持多道程序并发的系统。",[108,247643,247644,247646],{},[2627,247645,247377],{}," 混淆了“任务数量（多任务）”和“终端\u002F登录账号数量（多用户）”。",[19,247648,247650],{"id":247649},"_41用磁带作为文件存储介质时文件只能组织成顺序文件","41．用磁带作为文件存储介质时，文件只能组织成顺序文件。",[15,247652,247653,246513],{},[2627,247654,58173],{},[105,247656,247657,247662,247671],{},[108,247658,247659,247661],{},[2627,247660,246520],{}," 存储介质物理特性对文件结构的影响。",[108,247663,247664,247666,247667,247670],{},[2627,247665,246526],{}," 磁带是一种典型的",[2627,247668,247669],{},"顺序存取设备","。受限于物理机械结构，磁带只能通过卷轴向前或向后连续卷动来读取数据，根本无法实现瞬间跳跃式的随机访问。因此，建立在磁带上的文件，其逻辑和物理结构都只能是严格的顺序文件。",[108,247672,247673,247675],{},[2627,247674,247377],{}," 忽略了底层硬件的机械限制。",[19,247677,247679],{"id":247678},"_42交换技术的单位是段","42．交换技术的单位是段。",[15,247681,247682,246542],{},[2627,247683,58173],{},[105,247685,247686,247691,247704],{},[108,247687,247688,247690],{},[2627,247689,246520],{}," 交换（Swapping）的传统定义。",[108,247692,247693,247695,247696,247703],{},[2627,247694,246526],{}," 传统的交换技术（Swapping）是指将",[2627,247697,247698,247699,247702],{},"整个进程（part or all of a process）",[2627,247700,247701],{},"从主存移出到磁盘的挂起队列中，或者把整个进程从磁盘移回主存。虽然在虚拟内存的细化管理下出现了请求调页\u002F调段，但宏观意义上的“交换技术”的基本单位是","整个进程","，而不是单独的段。",[108,247705,247706,247708],{},[2627,247707,247377],{}," 将内存管理策略（分段）与中级调度的换入换出技术（交换进程）混淆。",[19,247710,247712],{"id":247711},"_43spooling将独占设备改造为共享设备","43．SPOOLing将独占设备改造为共享设备。",[15,247714,247715,246513],{},[2627,247716,58173],{},[105,247718,247719,247724,247729],{},[108,247720,247721,247723],{},[2627,247722,246520],{}," 假脱机技术（SPOOLing）的核心目的。",[108,247725,247726,247728],{},[2627,247727,246526],{}," SPOOLing 通过在高速磁盘上开辟缓冲井，并安排专门的后台进程来接管所有的 I\u002FO 请求。这样，多个进程想要使用打印机（独占设备）时，就不必死死阻塞等待，而是直接把数据输出到磁盘井中即可继续运行。在宏观逻辑上，原本只能一人独占的设备被成功改造成了允许多人同时发请求的“虚拟共享设备”。",[108,247730,247731,247733],{},[2627,247732,247377],{}," 这是一个十分经典的正确结论，必须牢记。",[19,247735,247737],{"id":247736},"_44具有多道功能的操作系统一定是多用户操作系统","44．具有多道功能的操作系统一定是多用户操作系统。",[15,247739,247740,247743,246542],{},[71807,247741,247742],{},"(注：此题与第40题完全重复)",[2627,247744,58173],{},[105,247746,247747],{},[108,247748,247749],{},[2627,247750,247751],{},"解析参考第40题。",[19,247753,247755],{"id":247754},"_45页式请求是以段为单位进行内存与外存的信息交换","45．页式请求是以段为单位进行内存与外存的信息交换。",[15,247757,247758,246542],{},[2627,247759,58173],{},[105,247761,247762,247767,247772],{},[108,247763,247764,247766],{},[2627,247765,246520],{}," 请求分页虚拟内存（Demand Paging）。",[108,247768,247769,247771],{},[2627,247770,246526],{}," 顾名思义，“请求页式”管理（Demand Paging）的机制是：当程序运行缺页时，操作系统会向磁盘发出请求，将缺失的数据以**页（Page）**为单位调入内存。它与“段”没有任何关系。",[108,247773,247774,247776],{},[2627,247775,247377],{}," 题目本身前后矛盾，页式机制的交换单位必然是页框\u002F页。",[19,247778,247780],{"id":247779},"_46段页式管理实现了段式页式两种存储方式的优势互补","46．段页式管理实现了段式、页式两种存储方式的优势互补。",[15,247782,247783,247786,246513],{},[71807,247784,247785],{},"(注：此题与第36题完全重复)",[2627,247787,58173],{},[105,247789,247790],{},[108,247791,247792],{},[2627,247793,247794],{},"解析参考第36题。",[19,247796,247798],{"id":247797},"_47对临界资源应采取互斥访问方式来实现共享","47．对临界资源应采取互斥访问方式来实现共享。",[15,247800,247801,247804,246513],{},[71807,247802,247803],{},"(注：此题与第37题完全重复)",[2627,247805,58173],{},[105,247807,247808],{},[108,247809,247810],{},[2627,247811,247812],{},"解析参考第37题。",[19,247814,247816],{"id":247815},"_48文件系统中文件的内容只能是源代码","48．文件系统中文件的内容只能是源代码。",[15,247818,247819,246542],{},[2627,247820,58173],{},[105,247822,247823,247828,247833],{},[108,247824,247825,247827],{},[2627,247826,246520],{}," 文件（File）的定义。",[108,247829,247830,247832],{},[2627,247831,246526],{}," 文件是对持久化数据的一种通用逻辑抽象。在文件系统中，文件的内容包罗万象，既可以是源代码，也可以是编译后的可执行机器指令、图片、视频、音频流、甚至是目录本身（在 UNIX 中目录也是文件）。",[108,247834,247835,247837],{},[2627,247836,247377],{}," 对文件的理解过于狭隘。",[19,247839,247841],{"id":247840},"_49外存对换空间保存的是虚拟内存管理系统调出的程序","49．外存对换空间保存的是虚拟内存管理系统调出的程序。",[15,247843,247844,246513],{},[2627,247845,58173],{},[105,247847,247848,247853,247858],{},[108,247849,247850,247852],{},[2627,247851,246520],{}," 对换区（Swap Space）的用途。",[108,247854,247855,247857],{},[2627,247856,246526],{}," 磁盘通常被划分为文件区和对换区。对换空间（Swappable space）专门被操作系统用来存放那些被中级调度器挂起（Swapped out）或被虚拟内存管理器置换出内存的进程页面和程序段。它充当了主存的直接逻辑扩充。",[108,247859,247860,247862],{},[2627,247861,247377],{}," 极少部分同学会误以为对换空间存的是操作系统的备份，实际上它是用户进程在硬盘上的“临时休息室”。",[19,247864,247866],{"id":247865},"_50计算机系统为每一台设备确定一个用以标识它的编号被称为设备的绝对号","50．计算机系统为每一台设备确定一个用以标识它的编号，被称为设备的绝对号。",[15,247868,247869,246513],{},[2627,247870,58173],{},[105,247872,247873,247878,247883],{},[108,247874,247875,247877],{},[2627,247876,246520],{}," 设备的绝对号与逻辑设备名。",[108,247879,247880,247882],{},[2627,247881,246526],{}," 在计算机底层硬件中，为了让 I\u002FO 控制器能够精准地寻址到挂载的每一台物理外设，系统会为每台具体的设备分配一个硬性的、唯一的物理标识编号，这在操作系统理论中统称为“设备绝对号”。而在编程时我们使用的是“逻辑设备名”，OS负责将逻辑名映射为这个绝对号。",[108,247884,247885,247887],{},[2627,247886,247377],{}," 这是设备独立性原理的基础知识点，绝对号代表物理实体。",[19,247889,247891],{"id":247890},"_51linux文件分为用户文件目录文件和特殊文件","51．Linux文件分为用户文件、目录文件和特殊文件。",[15,247893,247894,246513],{},[2627,247895,58173],{},[105,247897,247898,247903,247919],{},[108,247899,247900,247902],{},[2627,247901,246520],{}," UNIX\u002FLinux 文件分类。",[108,247904,247905,247907,247908],{},[2627,247906,246526],{}," 这是经典 UNIX\u002FLinux 文件系统的“三分法”。\n",[2230,247909,247910,247913,247916],{},[108,247911,247912],{},"用户文件（通常也译作“普通文件” Regular file）：包含普通数据、代码、文本等。",[108,247914,247915],{},"目录文件（Directory file）：包含文件名到 inode 指针的映射表。",[108,247917,247918],{},"特殊文件（Special file）：代表系统中的硬件设备（包括字符设备文件和块设备文件），这也是 Linux “一切皆文件”哲学的体现。",[108,247920,247921,247923],{},[2627,247922,247377],{}," 很多从 Windows 过来的用户不知道 Linux 把鼠标、键盘、硬盘统统抽象成了“特殊文件”。",[19,247925,247927],{"id":247926},"_52unixlinux系统中的文件名不区分大小写","52．UNIX\u002FLinux系统中的文件名不区分大小写。",[15,247929,247930,246542],{},[2627,247931,58173],{},[105,247933,247934,247939,247958],{},[108,247935,247936,247938],{},[2627,247937,246520],{}," Linux 命名约定。",[108,247940,247941,247943,247944,247947,247948,146,247951,258,247954,247957],{},[2627,247942,246526],{}," 与 Windows (DOS\u002FNTFS 默认习惯) 完全不同，",[2627,247945,247946],{},"UNIX\u002FLinux 系统是严格区分大小写（Case-sensitive）的","。在 Linux 中，",[31,247949,247950],{},"File.txt",[31,247952,247953],{},"file.txt",[31,247955,247956],{},"FILE.TXT"," 被视为三个完全独立的不同文件。",[108,247959,247960,247962],{},[2627,247961,247377],{}," 被 Windows 操作系统的长年使用习惯所误导。",[19,247964,247966],{"id":247965},"_53文件系统要负责文件存储空间管理但不能完成从文件名到物理地址的转换","53．文件系统要负责文件存储空间管理，但不能完成从文件名到物理地址的转换。",[15,247968,247969,246542],{},[2627,247970,58173],{},[105,247972,247973,247978,247987],{},[108,247974,247975,247977],{},[2627,247976,246520],{}," 文件系统的核心功能（目录管理）。",[108,247979,247980,247982,247983,247986],{},[2627,247981,246526],{}," 恰恰相反！实现“按名存取”正是文件系统存在的最根本原因。文件系统通过维护",[2627,247984,247985],{},"目录表（File Directories）","，将用户提供的逻辑文件名，翻译转换为对应文件在磁盘上的物理地址（或对应的 FCB\u002Finode，进而在其中找到物理块号）（前文背景）。",[108,247988,247989,247991],{},[2627,247990,247377],{}," 目录管理是文件系统的中枢神经，没有名字到物理地址的转换，文件系统就毫无意义。",[19,247993,247995],{"id":247994},"_54在文件系统的支持下用户需要知道文件存放的物理地址","54．在文件系统的支持下，用户需要知道文件存放的物理地址。",[15,247997,247998,246542],{},[2627,247999,58173],{},[105,248001,248002,248007,248016],{},[108,248003,248004,248006],{},[2627,248005,246520],{}," 操作系统提供的高层抽象（Abstraction）。",[108,248008,248009,248011,248012,248015],{},[2627,248010,246526],{}," 操作系统通过文件系统向上层掩盖了磁盘的物理结构（磁道、扇区、柱面等恶心细节）。程序员和用户被这种抽象“保护”了起来，他们",[2627,248013,248014],{},"完全不需要","知道文件的物理地址，只需要记住一个具有人类逻辑的“文件名”即可。",[108,248017,248018,248020],{},[2627,248019,247377],{}," 操作系统存在的最大价值之一就是隐藏硬件细节。",[19,248022,248024],{"id":248023},"_55顺序结构是一种逻辑记录顺序和物理块的顺序相一致的文件结构","55．顺序结构是一种逻辑记录顺序和物理块的顺序相一致的文件结构。",[15,248026,248027,246513],{},[2627,248028,58173],{},[105,248030,248031,248036,248041],{},[108,248032,248033,248035],{},[2627,248034,246520],{}," 连续分配（顺序文件结构）的特征。",[108,248037,248038,248040],{},[2627,248039,246526],{}," 如果一个文件采用连续分配的物理结构（即所谓的顺序物理结构），文件中的第一条逻辑记录存放在分配到的第一个物理块中，第二条记录紧挨着放在随后的物理空间中。此时，逻辑记录的排列顺序与它们在磁盘物理块上的排列顺序是完全一一对应的。",[108,248042,248043,248045],{},[2627,248044,247377],{}," 在链接分配和索引分配中，逻辑相邻的记录在物理磁盘上往往是散乱分布的。只有顺序分配能做到高度一致。",[19,248047,248049],{"id":248048},"_56可顺序存取的文件不一定能随机存取但可随机存取的文件都可以顺序存取","56．可顺序存取的文件不一定能随机存取；但可随机存取的文件都可以顺序存取。",[15,248051,248052,246513],{},[2627,248053,58173],{},[105,248055,248056,248061,248083],{},[108,248057,248058,248060],{},[2627,248059,246520],{}," 文件的存取方式（Sequential Access vs. Random\u002FDirect Access）。",[108,248062,248063,248065],{},[2627,248064,246526],{},[105,248066,248067,248077],{},[108,248068,248069,248072,248073,248076],{},[2627,248070,248071],{},"前半句："," 建立在磁带这种严格顺序介质上的文件，只能顺序存取，",[2627,248074,248075],{},"不能","高效随机存取。",[108,248078,248079,248082],{},[2627,248080,248081],{},"后半句："," 如果一个文件支持随机存取（比如磁盘上的索引文件，你可以直接跳到第100条记录），那么你当然可以选择“随机地”先访问第1条，再访问第2条、第3条……通过这种笨方法，完全可以实现逻辑上的“顺序存取”。",[108,248084,248085,248087],{},[2627,248086,247377],{}," 逻辑推理题，高阶功能总是向下兼容低阶行为。",[19,248089,248091],{"id":248090},"_57一般的文件系统都是基于磁盘设备的而磁带设备可以作为转储设备使用以提高系统的可靠性","57．一般的文件系统都是基于磁盘设备的，而磁带设备可以作为转储设备使用，以提高系统的可靠性。",[15,248093,248094,246513],{},[2627,248095,58173],{},[105,248097,248098,248103,248108],{},[108,248099,248100,248102],{},[2627,248101,246520],{}," 二级\u002F辅助存储设备的职能划分。",[108,248104,248105,248107],{},[2627,248106,246526],{}," 磁盘设备支持高效的随机访问，因此现代的活跃文件系统全都建立在磁盘（或闪存SSD）上。而磁带容量巨大、成本极低，虽然读取速度慢且只能顺序访问，但它非常适合做定期的冷备份和灾难恢复转储（Dump），从而极大提高系统数据的持久可靠性。",[108,248109,248110,248112],{},[2627,248111,247377],{}," 不了解企业级服务器和大型机对磁带（Tape）的大量使用现状。",[19,248114,248116],{"id":248115},"_58在文件系统的支持下用户需要知道文件存放的物理地址","58．在文件系统的支持下，用户需要知道文件存放的物理地址。",[15,248118,248119,248122,246542],{},[71807,248120,248121],{},"(注：此题与第54题完全重复)",[2627,248123,58173],{},[105,248125,248126],{},[108,248127,248128],{},[2627,248129,248130],{},"解析参考第54题。",[19,248132,248134],{"id":248133},"_59随机访问文件也能顺序访问但一般效率较差","59．随机访问文件也能顺序访问，但一般效率较差。",[15,248136,248137,246513],{},[2627,248138,58173],{},[105,248140,248141,248146,248151],{},[108,248142,248143,248145],{},[2627,248144,246520],{}," 特殊文件结构（如哈希文件\u002F直接文件）的遍历成本。",[108,248147,248148,248150],{},[2627,248149,246526],{}," 诸如哈希文件（Hashed File）这样的结构，专门为了极致的“随机访问”而生——它是通过哈希函数将逻辑记录随机打散在磁盘的各个物理块中。如果你想按某种逻辑顺序（如ID从小到大）把它们遍历（顺序访问）一遍，磁头就必须在磁盘上剧烈地来回跳跃、疯狂寻道，因此其顺序访问的效率往往极其低下。",[108,248152,248153,248155],{},[2627,248154,247377],{}," 学生容易忽略物理存储的连续性对磁盘寻道时间的巨大影响。",[19,248157,248159],{"id":248158},"_60在索引文件中建立索引表会占用额外的存储空间和访问时间","60．在索引文件中，建立索引表会占用额外的存储空间和访问时间。",[15,248161,248162,246513],{},[2627,248163,58173],{},[105,248165,248166,248171,248176],{},[108,248167,248168,248170],{},[2627,248169,246520],{}," 索引分配的性能与空间权衡（Trade-offs）。",[108,248172,248173,248175],{},[2627,248174,246526],{}," 天下没有免费的午餐。为了实现高效的定长或变长记录的随机查找，系统必须为索引文件构建专门的“索引表”。这个索引表本身是一个数据结构，首先，它需要存储在磁盘的额外物理块上（占用额外空间）；其次，在读取最终数据前，系统必须先发起一次磁盘 I\u002FO 去读取这个索引块查表，这就导致了额外的访存\u002F访盘时间开销（类比多级页表的额外开销）。",[108,248177,248178,248180],{},[2627,248179,247377],{}," 对数据结构的运行开销缺乏实感，误以为引入机制都是百利而无一害的。",{"title":29,"searchDepth":43,"depth":43,"links":248182},[248183,248184,248185,248186,248187,248188,248189,248190,248191,248192,248193,248194,248195,248196,248197,248198,248199,248200,248201,248202,248203,248204,248205,248206,248207,248208,248209,248210,248211,248212,248213,248214,248215,248216,248217,248218,248219,248220,248221,248222,248223,248224,248225,248226,248227,248228,248229,248230,248231,248232,248233,248234,248235,248236,248237,248238,248239,248240,248241,248242,248243],{"id":246500,"depth":43,"text":246501},{"id":246507,"depth":43,"text":246508},{"id":246536,"depth":43,"text":246537},{"id":246562,"depth":43,"text":246563},{"id":246587,"depth":43,"text":246588},{"id":246612,"depth":43,"text":246613},{"id":246637,"depth":43,"text":246638},{"id":246662,"depth":43,"text":246663},{"id":246691,"depth":43,"text":246692},{"id":246716,"depth":43,"text":246717},{"id":246748,"depth":43,"text":246749},{"id":246777,"depth":43,"text":246778},{"id":246802,"depth":43,"text":246803},{"id":246827,"depth":43,"text":246828},{"id":246856,"depth":43,"text":246857},{"id":246881,"depth":43,"text":246882},{"id":246909,"depth":43,"text":246910},{"id":246934,"depth":43,"text":246935},{"id":246971,"depth":43,"text":246972},{"id":247004,"depth":43,"text":247005},{"id":247033,"depth":43,"text":247034},{"id":247065,"depth":43,"text":247066},{"id":247090,"depth":43,"text":247091},{"id":247119,"depth":43,"text":247120},{"id":247148,"depth":43,"text":247149},{"id":247177,"depth":43,"text":247178},{"id":247210,"depth":43,"text":247211},{"id":247235,"depth":43,"text":247236},{"id":247268,"depth":43,"text":247269},{"id":247293,"depth":43,"text":247294},{"id":247326,"depth":43,"text":247327},{"id":247351,"depth":43,"text":247352},{"id":247381,"depth":43,"text":247382},{"id":247414,"depth":43,"text":247415},{"id":247443,"depth":43,"text":247444},{"id":247476,"depth":43,"text":247477},{"id":247505,"depth":43,"text":247506},{"id":247530,"depth":43,"text":247531},{"id":247559,"depth":43,"text":247560},{"id":247592,"depth":43,"text":247593},{"id":247624,"depth":43,"text":247625},{"id":247649,"depth":43,"text":247650},{"id":247678,"depth":43,"text":247679},{"id":247711,"depth":43,"text":247712},{"id":247736,"depth":43,"text":247737},{"id":247754,"depth":43,"text":247755},{"id":247779,"depth":43,"text":247780},{"id":247797,"depth":43,"text":247798},{"id":247815,"depth":43,"text":247816},{"id":247840,"depth":43,"text":247841},{"id":247865,"depth":43,"text":247866},{"id":247890,"depth":43,"text":247891},{"id":247926,"depth":43,"text":247927},{"id":247965,"depth":43,"text":247966},{"id":247994,"depth":43,"text":247995},{"id":248023,"depth":43,"text":248024},{"id":248048,"depth":43,"text":248049},{"id":248090,"depth":43,"text":248091},{"id":248115,"depth":43,"text":248116},{"id":248133,"depth":43,"text":248134},{"id":248158,"depth":43,"text":248159},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Ftorf",{"description":29},"source\u002Fcs\u002Fos\u002Freview\u002FTorF","AJZZzcKm8-QNwpl6jPxaqgArNQ5o-EGj2KglUIl5VlY",{"id":248250,"title":248251,"body":248252,"description":29,"extension":85,"meta":257992,"navigation":52,"path":257993,"seo":257994,"stem":257995,"__hash__":257996},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Ffill-in-the-blanks.md","填空题",{"type":8,"value":248253,"toc":257920},[248254,248256,248260,249295,249299,249310,249327,249331,249345,249349,249360,249405,249409,249420,249443,249447,249458,249481,249485,249495,249616,249620,249631,249650,249654,249672,249676,249687,249704,249708,249718,249738,249742,249753,249773,249777,249795,249799,249810,249827,249831,249842,249862,249866,249877,249897,249901,249912,249935,249939,249957,249961,249979,249983,249994,250008,250028,250032,250045,250049,250147,250151,250310,250336,250340,250425,250461,250465,251034,251038,251049,252697,252701,252978,252982,252996,253000,253370,254431,254435,254444,254519,254523,254534,254988,254992,255006,255010,255136,255140,255151,255168,255172,255183,255197,255201,255212,255226,255230,255244,255248,255266,255270,255434,255438,255452,255456,255470,255474,255488,255492,255506,255510,255524,255528,255550,255554,255568,255572,255583,257306,257310,257347,257351,257373,257377,257395,257399,257410,257431,257435,257449,257453,257481,257485,257510,257514,257535,257539,257550,257571,257575,257590,257594,257611,257615,257636,257640,257658,257662,257679,257683,257713,257717,257738,257742,257758,257762,257781,257785,257796,257826,257830,257845,257849,257860,257876,257880,257901,257905],[11,248255,248251],{"id":248251},[19,248257,248259],{"id":248258},"_1实时系统有4个周期性事件周期分别为50100200和150ms其处理分别需要252020和δms则该系统可调度允许的δ最大值为-ms","1．实时系统有4个周期性事件，周期分别为50、100、200和150ms，其处理分别需要25、20、20和δms，则该系统可调度允许的δ最大值为( )ms。",[15,248261,248262,248264,248265,248268,248269,248297,248298,248300,248301,70884,248459,248529,248530,248600,248601,248652,248653,248793,248794,248919,248999,249057,249058,39984,249109,249112,249113,249244,249245,31901],{},[2627,248263,58173],{}," 30\n",[2627,248266,248267],{},"概念核對："," 實時系統的可調度性 (Schedulability)。對於多個週期性任務，要保證它們都能在截止時間前完成，其總 CPU 使用率 (Utilization, ",[34,248270,248272,248285],{"className":248271},[31642],[34,248273,248275],{"className":248274},[31646],[31648,248276,248277],{"xmlns":31650},[31652,248278,248279,248283],{},[31655,248280,248281],{},[31658,248282,33350],{},[31665,248284,33350],{"encoding":31667},[34,248286,248288],{"className":248287,"ariaHidden":758},[31672],[34,248289,248291,248294],{"className":248290},[31676],[34,248292],{"className":248293,"style":33218},[31680],[34,248295,33350],{"className":248296,"style":33389},[31685,31686],") 不能超過系統的最大處理能力。\n",[2627,248299,246526],{}," 根據系統資源利用率的公式，總使用率 ",[34,248302,248304,248340],{"className":248303},[31642],[34,248305,248307],{"className":248306},[31646],[31648,248308,248309],{"xmlns":31650},[31652,248310,248311,248337],{},[31655,248312,248313,248315,248317,248319,248321,248327,248329,248335],{},[31658,248314,33350],{},[31662,248316,202],{},[31662,248318,131138],{},[31662,248320,456],{"stretchy":702},[31743,248322,248323,248325],{},[31658,248324,47384],{},[31658,248326,31955],{},[31658,248328,38080],{"mathvariant":38079},[31743,248330,248331,248333],{},[31658,248332,37400],{},[31658,248334,31955],{},[31662,248336,2557],{"stretchy":702},[31665,248338,248339],{"encoding":31667},"U = \\sum (C_i \u002F T_i)",[34,248341,248343,248361],{"className":248342,"ariaHidden":758},[31672],[34,248344,248346,248349,248352,248355,248358],{"className":248345},[31676],[34,248347],{"className":248348,"style":33218},[31680],[34,248350,33350],{"className":248351,"style":33389},[31685,31686],[34,248353],{"className":248354,"style":31846},[31845],[34,248356,202],{"className":248357},[31850],[34,248359],{"className":248360,"style":31846},[31845],[34,248362,248364,248367,248370,248373,248413,248416,248456],{"className":248363},[31676],[34,248365],{"className":248366,"style":31774},[31680],[34,248368,131138],{"className":248369,"style":156561},[32767,131249,156560],[34,248371,456],{"className":248372},[31782],[34,248374,248376,248379],{"className":248375},[31685],[34,248377,47384],{"className":248378,"style":38706},[31685,31686],[34,248380,248382],{"className":248381},[31792],[34,248383,248385,248405],{"className":248384},[31796,31797],[34,248386,248388,248402],{"className":248387},[31801],[34,248389,248391],{"className":248390,"style":32090},[31805],[34,248392,248393,248396],{"style":129511},[34,248394],{"className":248395,"style":31814},[31813],[34,248397,248399],{"className":248398},[31818,31819,31820,31821],[34,248400,31955],{"className":248401},[31685,31686,31821],[34,248403,31829],{"className":248404},[31828],[34,248406,248408],{"className":248407},[31801],[34,248409,248411],{"className":248410,"style":31836},[31805],[34,248412],{},[34,248414,38080],{"className":248415},[31685],[34,248417,248419,248422],{"className":248418},[31685],[34,248420,37400],{"className":248421,"style":37422},[31685,31686],[34,248423,248425],{"className":248424},[31792],[34,248426,248428,248448],{"className":248427},[31796,31797],[34,248429,248431,248445],{"className":248430},[31801],[34,248432,248434],{"className":248433,"style":32090},[31805],[34,248435,248436,248439],{"style":214424},[34,248437],{"className":248438,"style":31814},[31813],[34,248440,248442],{"className":248441},[31818,31819,31820,31821],[34,248443,31955],{"className":248444},[31685,31686,31821],[34,248446,31829],{"className":248447},[31828],[34,248449,248451],{"className":248450},[31801],[34,248452,248454],{"className":248453,"style":31836},[31805],[34,248455],{},[34,248457,2557],{"className":248458},[31690],[34,248460,248462,248480],{"className":248461},[31642],[34,248463,248465],{"className":248464},[31646],[31648,248466,248467],{"xmlns":31650},[31652,248468,248469,248477],{},[31655,248470,248471],{},[31743,248472,248473,248475],{},[31658,248474,47384],{},[31658,248476,31955],{},[31665,248478,248479],{"encoding":31667},"C_i",[34,248481,248483],{"className":248482,"ariaHidden":758},[31672],[34,248484,248486,248489],{"className":248485},[31676],[34,248487],{"className":248488,"style":70915},[31680],[34,248490,248492,248495],{"className":248491},[31685],[34,248493,47384],{"className":248494,"style":38706},[31685,31686],[34,248496,248498],{"className":248497},[31792],[34,248499,248501,248521],{"className":248500},[31796,31797],[34,248502,248504,248518],{"className":248503},[31801],[34,248505,248507],{"className":248506,"style":32090},[31805],[34,248508,248509,248512],{"style":129511},[34,248510],{"className":248511,"style":31814},[31813],[34,248513,248515],{"className":248514},[31818,31819,31820,31821],[34,248516,31955],{"className":248517},[31685,31686,31821],[34,248519,31829],{"className":248520},[31828],[34,248522,248524],{"className":248523},[31801],[34,248525,248527],{"className":248526,"style":31836},[31805],[34,248528],{}," 是執行時間，",[34,248531,248533,248551],{"className":248532},[31642],[34,248534,248536],{"className":248535},[31646],[31648,248537,248538],{"xmlns":31650},[31652,248539,248540,248548],{},[31655,248541,248542],{},[31743,248543,248544,248546],{},[31658,248545,37400],{},[31658,248547,31955],{},[31665,248549,248550],{"encoding":31667},"T_i",[34,248552,248554],{"className":248553,"ariaHidden":758},[31672],[34,248555,248557,248560],{"className":248556},[31676],[34,248558],{"className":248559,"style":70915},[31680],[34,248561,248563,248566],{"className":248562},[31685],[34,248564,37400],{"className":248565,"style":37422},[31685,31686],[34,248567,248569],{"className":248568},[31792],[34,248570,248572,248592],{"className":248571},[31796,31797],[34,248573,248575,248589],{"className":248574},[31801],[34,248576,248578],{"className":248577,"style":32090},[31805],[34,248579,248580,248583],{"style":214424},[34,248581],{"className":248582,"style":31814},[31813],[34,248584,248586],{"className":248585},[31818,31819,31820,31821],[34,248587,31955],{"className":248588},[31685,31686,31821],[34,248590,31829],{"className":248591},[31828],[34,248593,248595],{"className":248594},[31801],[34,248596,248598],{"className":248597,"style":31836},[31805],[34,248599],{}," 是週期。在理想的動態優先級排程（如最早截止時間優先 EDF）下，系統可調度的理論上限為 ",[34,248602,248604,248622],{"className":248603},[31642],[34,248605,248607],{"className":248606},[31646],[31648,248608,248609],{"xmlns":31650},[31652,248610,248611,248619],{},[31655,248612,248613,248615,248617],{},[31658,248614,33350],{},[31662,248616,45972],{},[31749,248618,640],{},[31665,248620,248621],{"encoding":31667},"U \\le 1",[34,248623,248625,248643],{"className":248624,"ariaHidden":758},[31672],[34,248626,248628,248631,248634,248637,248640],{"className":248627},[31676],[34,248629],{"className":248630,"style":195047},[31680],[34,248632,33350],{"className":248633,"style":33389},[31685,31686],[34,248635],{"className":248636,"style":31846},[31845],[34,248638,45972],{"className":248639},[31850],[34,248641],{"className":248642,"style":31846},[31845],[34,248644,248646,248649],{"className":248645},[31676],[34,248647],{"className":248648,"style":32617},[31680],[34,248650,640],{"className":248651},[31685],"。\n代入數值：",[34,248654,248656,248702],{"className":248655},[31642],[34,248657,248659],{"className":248658},[31646],[31648,248660,248661],{"xmlns":31650},[31652,248662,248663,248699],{},[31655,248664,248665,248667,248669,248671,248673,248675,248677,248679,248681,248683,248685,248687,248689,248691,248693,248695,248697],{},[31749,248666,95556],{},[31658,248668,38080],{"mathvariant":38079},[31749,248670,5774],{},[31662,248672,223],{},[31749,248674,3249],{},[31658,248676,38080],{"mathvariant":38079},[31749,248678,162173],{},[31662,248680,223],{},[31749,248682,3249],{},[31658,248684,38080],{"mathvariant":38079},[31749,248686,168915],{},[31662,248688,223],{},[31658,248690,231507],{},[31658,248692,38080],{"mathvariant":38079},[31749,248694,73451],{},[31662,248696,45972],{},[31749,248698,640],{},[31665,248700,248701],{"encoding":31667},"25\u002F50 + 20\u002F100 + 20\u002F200 + \\delta\u002F150 \\le 1",[34,248703,248705,248724,248743,248762,248784],{"className":248704,"ariaHidden":758},[31672],[34,248706,248708,248711,248715,248718,248721],{"className":248707},[31676],[34,248709],{"className":248710,"style":31774},[31680],[34,248712,248714],{"className":248713},[31685],"25\u002F50",[34,248716],{"className":248717,"style":32604},[31845],[34,248719,223],{"className":248720},[32024],[34,248722],{"className":248723,"style":32604},[31845],[34,248725,248727,248730,248734,248737,248740],{"className":248726},[31676],[34,248728],{"className":248729,"style":31774},[31680],[34,248731,248733],{"className":248732},[31685],"20\u002F100",[34,248735],{"className":248736,"style":32604},[31845],[34,248738,223],{"className":248739},[32024],[34,248741],{"className":248742,"style":32604},[31845],[34,248744,248746,248749,248753,248756,248759],{"className":248745},[31676],[34,248747],{"className":248748,"style":31774},[31680],[34,248750,248752],{"className":248751},[31685],"20\u002F200",[34,248754],{"className":248755,"style":32604},[31845],[34,248757,223],{"className":248758},[32024],[34,248760],{"className":248761,"style":32604},[31845],[34,248763,248765,248768,248771,248775,248778,248781],{"className":248764},[31676],[34,248766],{"className":248767,"style":31774},[31680],[34,248769,231507],{"className":248770,"style":231523},[31685,31686],[34,248772,248774],{"className":248773},[31685],"\u002F150",[34,248776],{"className":248777,"style":31846},[31845],[34,248779,45972],{"className":248780},[31850],[34,248782],{"className":248783,"style":31846},[31845],[34,248785,248787,248790],{"className":248786},[31676],[34,248788],{"className":248789,"style":32617},[31680],[34,248791,640],{"className":248792},[31685],"\n即：",[34,248795,248797,248832],{"className":248796},[31642],[34,248798,248800],{"className":248799},[31646],[31648,248801,248802],{"xmlns":31650},[31652,248803,248804,248829],{},[31655,248805,248806,248808,248810,248812,248814,248817,248819,248821,248823,248825,248827],{},[31749,248807,65507],{},[31662,248809,223],{},[31749,248811,231993],{},[31662,248813,223],{},[31749,248815,248816],{},"0.1",[31662,248818,223],{},[31658,248820,231507],{},[31658,248822,38080],{"mathvariant":38079},[31749,248824,73451],{},[31662,248826,45972],{},[31749,248828,640],{},[31665,248830,248831],{"encoding":31667},"0.5 + 0.2 + 0.1 + \\delta\u002F150 \\le 1",[34,248833,248835,248853,248871,248889,248910],{"className":248834,"ariaHidden":758},[31672],[34,248836,248838,248841,248844,248847,248850],{"className":248837},[31676],[34,248839],{"className":248840,"style":35942},[31680],[34,248842,65507],{"className":248843},[31685],[34,248845],{"className":248846,"style":32604},[31845],[34,248848,223],{"className":248849},[32024],[34,248851],{"className":248852,"style":32604},[31845],[34,248854,248856,248859,248862,248865,248868],{"className":248855},[31676],[34,248857],{"className":248858,"style":35942},[31680],[34,248860,231993],{"className":248861},[31685],[34,248863],{"className":248864,"style":32604},[31845],[34,248866,223],{"className":248867},[32024],[34,248869],{"className":248870,"style":32604},[31845],[34,248872,248874,248877,248880,248883,248886],{"className":248873},[31676],[34,248875],{"className":248876,"style":35942},[31680],[34,248878,248816],{"className":248879},[31685],[34,248881],{"className":248882,"style":32604},[31845],[34,248884,223],{"className":248885},[32024],[34,248887],{"className":248888,"style":32604},[31845],[34,248890,248892,248895,248898,248901,248904,248907],{"className":248891},[31676],[34,248893],{"className":248894,"style":31774},[31680],[34,248896,231507],{"className":248897,"style":231523},[31685,31686],[34,248899,248774],{"className":248900},[31685],[34,248902],{"className":248903,"style":31846},[31845],[34,248905,45972],{"className":248906},[31850],[34,248908],{"className":248909,"style":31846},[31845],[34,248911,248913,248916],{"className":248912},[31676],[34,248914],{"className":248915,"style":32617},[31680],[34,248917,640],{"className":248918},[31685],[34,248920,248922,248948],{"className":248921},[31642],[34,248923,248925],{"className":248924},[31646],[31648,248926,248927],{"xmlns":31650},[31652,248928,248929,248945],{},[31655,248930,248931,248933,248935,248937,248939,248941,248943],{},[31749,248932,194366],{},[31662,248934,223],{},[31658,248936,231507],{},[31658,248938,38080],{"mathvariant":38079},[31749,248940,73451],{},[31662,248942,45972],{},[31749,248944,640],{},[31665,248946,248947],{"encoding":31667},"0.8 + \\delta\u002F150 \\le 1",[34,248949,248951,248969,248990],{"className":248950,"ariaHidden":758},[31672],[34,248952,248954,248957,248960,248963,248966],{"className":248953},[31676],[34,248955],{"className":248956,"style":35942},[31680],[34,248958,194366],{"className":248959},[31685],[34,248961],{"className":248962,"style":32604},[31845],[34,248964,223],{"className":248965},[32024],[34,248967],{"className":248968,"style":32604},[31845],[34,248970,248972,248975,248978,248981,248984,248987],{"className":248971},[31676],[34,248973],{"className":248974,"style":31774},[31680],[34,248976,231507],{"className":248977,"style":231523},[31685,31686],[34,248979,248774],{"className":248980},[31685],[34,248982],{"className":248983,"style":31846},[31845],[34,248985,45972],{"className":248986},[31850],[34,248988],{"className":248989,"style":31846},[31845],[34,248991,248993,248996],{"className":248992},[31676],[34,248994],{"className":248995,"style":32617},[31680],[34,248997,640],{"className":248998},[31685],[34,249000,249002,249024],{"className":249001},[31642],[34,249003,249005],{"className":249004},[31646],[31648,249006,249007],{"xmlns":31650},[31652,249008,249009,249021],{},[31655,249010,249011,249013,249015,249017,249019],{},[31658,249012,231507],{},[31658,249014,38080],{"mathvariant":38079},[31749,249016,73451],{},[31662,249018,45972],{},[31749,249020,231993],{},[31665,249022,249023],{"encoding":31667},"\\delta\u002F150 \\le 0.2",[34,249025,249027,249048],{"className":249026,"ariaHidden":758},[31672],[34,249028,249030,249033,249036,249039,249042,249045],{"className":249029},[31676],[34,249031],{"className":249032,"style":31774},[31680],[34,249034,231507],{"className":249035,"style":231523},[31685,31686],[34,249037,248774],{"className":249038},[31685],[34,249040],{"className":249041,"style":31846},[31845],[34,249043,45972],{"className":249044},[31850],[34,249046],{"className":249047,"style":31846},[31845],[34,249049,249051,249054],{"className":249050},[31676],[34,249052],{"className":249053,"style":32617},[31680],[34,249055,231993],{"className":249056},[31685],"，解得 ",[34,249059,249061,249079],{"className":249060},[31642],[34,249062,249064],{"className":249063},[31646],[31648,249065,249066],{"xmlns":31650},[31652,249067,249068,249076],{},[31655,249069,249070,249072,249074],{},[31658,249071,231507],{},[31662,249073,45972],{},[31749,249075,9314],{},[31665,249077,249078],{"encoding":31667},"\\delta \\le 30",[34,249080,249082,249100],{"className":249081,"ariaHidden":758},[31672],[34,249083,249085,249088,249091,249094,249097],{"className":249084},[31676],[34,249086],{"className":249087,"style":47709},[31680],[34,249089,231507],{"className":249090,"style":231523},[31685,31686],[34,249092],{"className":249093,"style":31846},[31845],[34,249095,45972],{"className":249096},[31850],[34,249098],{"className":249099,"style":31846},[31845],[34,249101,249103,249106],{"className":249102},[31676],[34,249104],{"className":249105,"style":32617},[31680],[34,249107,9314],{"className":249108},[31685],[2627,249110,249111],{},"易錯點："," 學生常會將此題與速率單調排程 (RMS) 的充分條件 ",[34,249114,249116,249154],{"className":249115},[31642],[34,249117,249119],{"className":249118},[31646],[31648,249120,249121],{"xmlns":31650},[31652,249122,249123,249151],{},[31655,249124,249125,249127,249129,249131,249133,249145,249147,249149],{},[31658,249126,33350],{},[31662,249128,45972],{},[31658,249130,31660],{},[31662,249132,456],{"stretchy":702},[37624,249134,249135,249137],{},[31749,249136,768],{},[31655,249138,249139,249141,249143],{},[31749,249140,640],{},[31658,249142,38080],{"mathvariant":38079},[31658,249144,31660],{},[31662,249146,32226],{},[31749,249148,640],{},[31662,249150,2557],{"stretchy":702},[31665,249152,249153],{"encoding":31667},"U \\le n(2^{1\u002Fn}-1)",[34,249155,249157,249175,249232],{"className":249156,"ariaHidden":758},[31672],[34,249158,249160,249163,249166,249169,249172],{"className":249159},[31676],[34,249161],{"className":249162,"style":195047},[31680],[34,249164,33350],{"className":249165,"style":33389},[31685,31686],[34,249167],{"className":249168,"style":31846},[31845],[34,249170,45972],{"className":249171},[31850],[34,249173],{"className":249174,"style":31846},[31845],[34,249176,249178,249181,249184,249187,249223,249226,249229],{"className":249177},[31676],[34,249179],{"className":249180,"style":61012},[31680],[34,249182,31660],{"className":249183},[31685,31686],[34,249185,456],{"className":249186},[31782],[34,249188,249190,249193],{"className":249189},[31685],[34,249191,768],{"className":249192},[31685],[34,249194,249196],{"className":249195},[31792],[34,249197,249199],{"className":249198},[31796],[34,249200,249202],{"className":249201},[31801],[34,249203,249205],{"className":249204,"style":61037},[31805],[34,249206,249207,249210],{"style":37777},[34,249208],{"className":249209,"style":31814},[31813],[34,249211,249213],{"className":249212},[31818,31819,31820,31821],[34,249214,249216,249220],{"className":249215},[31685,31821],[34,249217,249219],{"className":249218},[31685,31821],"1\u002F",[34,249221,31660],{"className":249222},[31685,31686,31821],[34,249224],{"className":249225,"style":32604},[31845],[34,249227,32226],{"className":249228},[32024],[34,249230],{"className":249231,"style":32604},[31845],[34,249233,249235,249238,249241],{"className":249234},[31676],[34,249236],{"className":249237,"style":31774},[31680],[34,249239,640],{"className":249240},[31685],[34,249242,2557],{"className":249243},[31690]," 搞混。除非題目明確指定使用「靜態優先級\u002FRMS 排程」，否則求「系統可調度允許的最大值」時，預設採用理論最佳情況（即 ",[34,249246,249248,249265],{"className":249247},[31642],[34,249249,249251],{"className":249250},[31646],[31648,249252,249253],{"xmlns":31650},[31652,249254,249255,249263],{},[31655,249256,249257,249259,249261],{},[31658,249258,33350],{},[31662,249260,45972],{},[31749,249262,640],{},[31665,249264,248621],{"encoding":31667},[34,249266,249268,249286],{"className":249267,"ariaHidden":758},[31672],[34,249269,249271,249274,249277,249280,249283],{"className":249270},[31676],[34,249272],{"className":249273,"style":195047},[31680],[34,249275,33350],{"className":249276,"style":33389},[31685,31686],[34,249278],{"className":249279,"style":31846},[31845],[34,249281,45972],{"className":249282},[31850],[34,249284],{"className":249285,"style":31846},[31845],[34,249287,249289,249292],{"className":249288},[31676],[34,249290],{"className":249291,"style":32617},[31680],[34,249293,640],{"className":249294},[31685],[19,249296,249298],{"id":249297},"_2进程调度的方式通常有-和-两种方式","2．进程调度的方式通常有（ ）和（ ）两种方式。",[15,249300,249301,249303,249304,249306,249307,249309],{},[2627,249302,58173],{}," 非抢占式 (Nonpreemptive) 、 抢占式 (Preemptive)\n",[2627,249305,248267],{}," 行程排程的決策模式 (Decision Mode)。\n",[2627,249308,246526],{}," 作業系統在決定何時切換行程時分為兩大類：",[2230,249311,249312,249318],{},[108,249313,249314,249317],{},[2627,249315,249316],{},"非搶占式 (Nonpreemptive)："," 行程一旦獲得 CPU，就會一直執行直到終止，或者因為等待 I\u002FO 而主動阻塞。",[108,249319,249320,249323,249324,249326],{},[2627,249321,249322],{},"搶占式 (Preemptive)："," 作業系統可以在新行程到達、中斷發生或時間片耗盡時，強行中斷當前正在執行的行程，將其移回就緒佇列。\n",[2627,249325,249111],{}," 不要與「排程演算法」（如 FCFS, RR 等）混淆，這裡問的是「方式\u002F機制」。",[19,249328,249330],{"id":249329},"_3每个索引文件都必须有一张-表其中的地址登记项用来指出文件在外存上的位置信息","3．每个索引文件都必须有一张（ ）表，其中的地址登记项用来指出文件在外存上的位置信息。",[15,249332,249333,249335,249336,249338,249339,249341,249342,249344],{},[2627,249334,58173],{}," 索引 (Index)\n",[2627,249337,248267],{}," 索引檔案結構 (Indexed file structure)。\n",[2627,249340,246526],{}," 為了能隨機且快速地存取檔案中特定長度或變長的記錄，檔案系統會為該檔案建立一張「索引表」。索引表中的每一個項目都指向對應記錄在磁碟（外存）上的實際物理位址，這突破了順序檔案必須連續讀取的限制。\n",[2627,249343,249111],{}," 記得區分「索引表 (Index table)」與「檔案配置表 (FAT)」。FAT 記錄的是整個磁碟區塊的配置，而索引表是針對「單一索引檔案」內部記錄的定位。",[19,249346,249348],{"id":249347},"_4在一请求分页系统中假如一个作业的页面走向为432143543215当分配给该作业的物理块数为4时开始时没有装入页面采用lru页面淘汰算法将产生-次缺页中断","4．在一请求分页系统中，假如一个作业的页面走向为：4、3、2、1、4、3、5、4、3、2、1、5，当分配给该作业的物理块数为4时（开始时没有装入页面），采用LRU页面淘汰算法将产生( )次缺页中断。",[15,249350,249351,249353,249354,249356,249357,249359],{},[2627,249352,58173],{}," 8\n",[2627,249355,248267],{}," 最近最少使用演算法 (Least Recently Used, LRU)。\n",[2627,249358,246526],{}," LRU 演算法會在需要淘汰分頁時，選擇「最長時間未被存取」的分頁進行替換。物理區塊數為 4：",[105,249361,249362,249368,249374,249380,249385,249391,249397],{},[108,249363,249364,249367],{},[2627,249365,249366],{},"4, 3, 2, 1:"," 皆不在記憶體，產生 4 次缺頁。目前記憶體為 ``。",[108,249369,249370,249373],{},[2627,249371,249372],{},"4, 3:"," 皆命中，不缺頁。最近使用順序更新為 `` (3最常使用)。",[108,249375,249376,249379],{},[2627,249377,249378],{},"5:"," 缺頁 (第5次)。淘汰最久未用的 2。記憶體變為 ``。",[108,249381,249382,249384],{},[2627,249383,249372],{}," 命中。使用順序更新為 ``。",[108,249386,249387,249390],{},[2627,249388,249389],{},"2:"," 缺頁 (第6次)。淘汰最久未用的 1。記憶體變為 ``。",[108,249392,249393,249396],{},[2627,249394,249395],{},"1:"," 缺頁 (第7次)。淘汰 5。記憶體變為 ``。",[108,249398,249399,249401,249402,249404],{},[2627,249400,249378],{}," 缺頁 (第8次)。淘汰 4。記憶體變為 ``。\n總缺页次数 = 初始4次 + 替換4次 = 8次。\n",[2627,249403,249111],{}," 模擬 LRU 時，千萬要記得在「命中 (Hit)」時，必須將該分頁的「新鮮度」更新為最新，否則會把它誤當成舊分頁淘汰。",[19,249406,249408],{"id":249407},"_5信号量被广泛用于三个目的是-和描述前趋关系","5．信号量被广泛用于三个目的是( )、( )和描述前趋关系。",[15,249410,249411,249413,249414,249416,249417,249419],{},[2627,249412,58173],{}," 互斥 (Mutual exclusion) 、 同步 (Synchronization)\n",[2627,249415,248267],{}," 號誌 (Semaphores) 的應用場景。\n",[2627,249418,246526],{}," 號誌是 Dijkstra 提出的一種非常靈活的整數變數，主要用來解決並發行程間的三大問題：",[2230,249421,249422,249428,249434],{},[108,249423,249424,249427],{},[2627,249425,249426],{},"互斥："," 確保同時只有一個行程能進入臨界區（保護共用資源）。",[108,249429,249430,249433],{},[2627,249431,249432],{},"同步："," 協調多個行程的執行順序（例如生產者\u002F消費者模型，必須先產生資料才能消費）。",[108,249435,249436,249439,249440,249442],{},[2627,249437,249438],{},"前趨關係："," 是同步的一種延伸，用來規定哪段程式碼必須在另一段之前執行。\n",[2627,249441,249111],{}," 互斥與同步經常被學生混為一談。互斥是「排他性（你進我就不能進）」，同步是「協作性（我等你做完才做）」。",[19,249444,249446],{"id":249445},"_6程序并发执行时的特征是-和独立性","6．程序并发执行时的特征是( )、( )、( )和独立性。",[15,249448,249449,249451,249452,249454,249455,249457],{},[2627,249450,58173],{}," 间断性 (Asynchrony\u002FPause-and-go) 、 失去封闭性 (Loss of closedness) 、 不可再现性 (Non-reproducibility)\n",[2627,249453,248267],{}," 行程並發執行的基礎特徵。\n",[2627,249456,246526],{}," 當多個程式在記憶體中交錯執行（並發）時，它們會競爭資源。這導致了：",[2230,249459,249460,249466,249472],{},[108,249461,249462,249465],{},[2627,249463,249464],{},"間斷性："," 程式走走停停，執行速度不可預測。",[108,249467,249468,249471],{},[2627,249469,249470],{},"失去封閉性："," 程式共享全域資源，其狀態可能會受到其他並發程式的干擾。",[108,249473,249474,249477,249478,249480],{},[2627,249475,249476],{},"不可再現性："," 因為執行順序（Race condition）不可預測，相同的輸入可能因為干擾而產生不同的結果。\n",[2627,249479,249111],{}," 這些特徵正是作業系統為什麼必須引入「行程(Process)」和「同步機制」的根本原因。",[19,249482,249484],{"id":249483},"_7如果信号量的当前值为3表示可用的资源数目为3如果信号量的当前值为-3则表示","7．如果信号量的当前值为3，表示可用的资源数目为3，如果信号量的当前值为-3，则表示( )。",[15,249486,249487,249489,249490,249492,249493],{},[2627,249488,58173],{}," 有3个进程正在等待该资源 (There are 3 processes waiting to be unblocked)\n",[2627,249491,248267],{}," 計數型號誌 (Counting Semaphore) 數值的物理意義。\n",[2627,249494,246526],{},[105,249496,249497,249555],{},[108,249498,249499,249500,249551,249552,347],{},"當 ",[34,249501,249503,249521],{"className":249502},[31642],[34,249504,249506],{"className":249505},[31646],[31648,249507,249508],{"xmlns":31650},[31652,249509,249510,249518],{},[31655,249511,249512,249514,249516],{},[31658,249513,94675],{},[31662,249515,29622],{},[31749,249517,459],{},[31665,249519,249520],{"encoding":31667},"S > 0",[34,249522,249524,249542],{"className":249523,"ariaHidden":758},[31672],[34,249525,249527,249530,249533,249536,249539],{"className":249526},[31676],[34,249528],{"className":249529,"style":104901},[31680],[34,249531,94675],{"className":249532,"style":106455},[31685,31686],[34,249534],{"className":249535,"style":31846},[31845],[34,249537,29622],{"className":249538},[31850],[34,249540],{"className":249541,"style":31846},[31845],[34,249543,249545,249548],{"className":249544},[31676],[34,249546],{"className":249547,"style":32617},[31680],[34,249549,459],{"className":249550},[31685]," 時，數值代表目前系統中",[2627,249553,249554],{},"可用資源的數量",[108,249556,249499,249557,249608,249609,249612,249613,249615],{},[34,249558,249560,249578],{"className":249559},[31642],[34,249561,249563],{"className":249562},[31646],[31648,249564,249565],{"xmlns":31650},[31652,249566,249567,249575],{},[31655,249568,249569,249571,249573],{},[31658,249570,94675],{},[31662,249572,45972],{},[31749,249574,459],{},[31665,249576,249577],{"encoding":31667},"S \\le 0",[34,249579,249581,249599],{"className":249580,"ariaHidden":758},[31672],[34,249582,249584,249587,249590,249593,249596],{"className":249583},[31676],[34,249585],{"className":249586,"style":195047},[31680],[34,249588,94675],{"className":249589,"style":106455},[31685,31686],[34,249591],{"className":249592,"style":31846},[31845],[34,249594,45972],{"className":249595},[31850],[34,249597],{"className":249598,"style":31846},[31845],[34,249600,249602,249605],{"className":249601},[31676],[34,249603],{"className":249604,"style":32617},[31680],[34,249606,459],{"className":249607},[31685]," 時，該數值的絕對值代表因為請求此資源而",[2627,249610,249611],{},"被阻塞（在等待佇列中）的行程數量","。因此，-3 表示有 3 個行程正在排隊等待。\n",[2627,249614,249111],{}," 負號代表「欠債」，欠了3個資源，就意味著有3個行程在排隊。",[19,249617,249619],{"id":249618},"_8io控制的方式有程序直接控制方式中断控制方式-和通道方式","8．I\u002FO控制的方式有程序直接控制方式、中断控制方式、( )和通道方式。",[15,249621,249622,249624,249625,249627,249628,249630],{},[2627,249623,58173],{}," DMA方式 (直接存储器访问 \u002F Direct Memory Access)\n",[2627,249626,248267],{}," I\u002FO 設備資料傳輸的演進。\n",[2627,249629,246526],{}," 為了不斷減少 CPU 在 I\u002FO 操作上的介入，I\u002FO 控制方式演進了四個階段：",[2230,249632,249633,249636,249639,249644],{},[108,249634,249635],{},"程式直接控制（輪詢，CPU 佔用率最高）。",[108,249637,249638],{},"中斷控制（CPU 啟動後去忙別的，好了再回來）。",[108,249640,249641,347],{},[2627,249642,249643],{},"DMA（由硬體控制器接管，讓 I\u002FO 設備與記憶體直接交換一整塊資料，CPU 只在開頭和結尾介入）",[108,249645,249646,249647,249649],{},"通道控制（有了專門的 I\u002FO 處理器，自主性最高）。\n",[2627,249648,249111],{}," 留意 DMA 是 \"Direct Memory Access\" 的縮寫。",[19,249651,249653],{"id":249652},"_9-在首次适应算法中要求空闲分区按地址递增顺序链接成空闲分区链在最佳适应算法中是按空闲分区-形成空闲分区链","9. 在首次适应算法中，要求空闲分区按地址递增顺序链接成空闲分区链；在最佳适应算法中是按空闲分区( )形成空闲分区链。",[15,249655,249656,249658,249659,249661,249662,249664,249665,249668,249669,249671],{},[2627,249657,58173],{}," 大小递增顺序 (Increasing size)\n",[2627,249660,248267],{}," 動態分區配置演算法 (Dynamic partitioning algorithms)。\n",[2627,249663,246526],{}," 最佳適應 (Best-fit) 的目標是找到「最接近需求大小」的空閒區塊。為了讓搜尋速度最快，系統會預先將所有空閒區塊按照",[2627,249666,249667],{},"容量（大小）由小到大","進行排序。這樣只要從頭開始找，找到的第一個能放得下的區塊，就一定是最佳區塊。\n",[2627,249670,249111],{}," 最佳適應算法雖然名字叫「最佳」，但它反而最容易產生大量極小且無法利用的記憶體碎片 (Fragmentation)。",[19,249673,249675],{"id":249674},"_10-文件的物理结构有顺序文件链接文件和-三种","10. 文件的物理结构有顺序文件、链接文件和( )三种。",[15,249677,249678,249680,249681,249683,249684,249686],{},[2627,249679,58173],{}," 索引文件 (Indexed file)\n",[2627,249682,248267],{}," 檔案的物理結構 (Physical File Organization)。\n",[2627,249685,246526],{}," 檔案資料在磁碟上有幾種典型的實體存放方式：",[2230,249688,249689,249692,249695],{},[108,249690,249691],{},"連續\u002F順序配置（要求連續的磁碟區塊）。",[108,249693,249694],{},"連結配置（利用指標將散落的區塊串起來，如 FAT 表）。",[108,249696,249697,249700,249701,249703],{},[2627,249698,249699],{},"索引配置","（用一個專門的索引區塊來登記檔案所有資料區塊的磁碟位址，支援高效的隨機存取）。\n",[2627,249702,249111],{}," 請將「邏輯結構」（流式、記錄式）與「物理結構」（順序、連結、索引）明確區分開來。",[19,249705,249707],{"id":249706},"_11-现代操作系统的特征是并发-虚拟和异步性","11. 现代操作系统的特征是并发、( )、虚拟和异步性。",[15,249709,249710,249712,249713,249715,249716],{},[2627,249711,58173],{}," 共享 (Sharing)\n",[2627,249714,248267],{}," 作業系統的四大基本特徵。\n",[2627,249717,246526],{},[105,249719,249720,249723,249729,249732],{},[108,249721,249722],{},"並發：多個行程在宏觀上同時執行。",[108,249724,249725,249728],{},[2627,249726,249727],{},"共享："," 系統中的資源供記憶體中的多個並發行程共同使用。",[108,249730,249731],{},"虛擬：將實體實體（如 CPU、記憶體）映射為多個邏輯實體。",[108,249733,249734,249735,249737],{},"非同步：行程執行的走走停停。\n",[2627,249736,249111],{}," 並發和共享是作業系統最基礎的兩個特徵，兩者互為存在條件。",[19,249739,249741],{"id":249740},"_12产生死锁的四个必要条件是互斥条件和请求和保持-和环路条件","12.产生死锁的四个必要条件是互斥条件和请求和保持，( )和环路条件。",[15,249743,249744,249746,249747,249749,249750,249752],{},[2627,249745,58173],{}," 不可剥夺条件 (No preemption \u002F 非抢占条件)\n",[2627,249748,248267],{}," 死結的四個必要條件 (Necessary conditions for Deadlock)。\n",[2627,249751,246526],{}," 如果系統發生死結，必然同時滿足四個條件：",[2230,249754,249755,249758,249761,249767],{},[108,249756,249757],{},"互斥：資源一次只能給一個行程用。",[108,249759,249760],{},"請求和保持 (Hold and wait)：拿著部分資源不放，同時請求新資源。",[108,249762,249763,249766],{},[2627,249764,249765],{},"不可剝奪 (No preemption)："," 行程持有的資源在它用完主動釋放前，系統不能強行搶走。",[108,249768,249769,249770,249772],{},"循環等待 (Circular wait)：存在一個等待環路。\n",[2627,249771,249111],{}," 「不可剝奪」的意思是，OS 不具備強制奪取該類資源的權力（例如印表機正在列印時不能被搶走）。",[19,249774,249776],{"id":249775},"_13操作系统的五大功能是-存储管理设备管理文件系统和用户接口","13.操作系统的五大功能是( )、存储管理、设备管理、文件系统和用户接口。",[15,249778,249779,249781,249782,249784,249785,249787,249788,249791,249792,249794],{},[2627,249780,58173],{}," 处理机管理 (Processor management \u002F 进程管理)\n",[2627,249783,248267],{}," OS 的核心功能模組。\n",[2627,249786,246526],{}," 作業系統的核心任務是管理硬體資源。這包括對 CPU 的分配與排程，也就是",[2627,249789,249790],{},"處理機管理","（或稱行程\u002F執行緒管理）。\n",[2627,249793,249111],{}," 在多道程式系統中，處理機管理實質上就是對「行程(Process)」的管理。",[19,249796,249798],{"id":249797},"_14按逻辑结构可把文件分为-和-两类","14．按逻辑结构可把文件分为（ ）和（ ）两类。",[15,249800,249801,249803,249804,249806,249807,249809],{},[2627,249802,58173],{}," 流式文件 (Stream file \u002F 无结构文件) 、 记录式文件 (Record file \u002F 有结构文件)\n",[2627,249805,248267],{}," 檔案的邏輯結構。\n",[2627,249808,246526],{}," 從使用者的角度來看，檔案有兩種：",[2230,249811,249812,249818],{},[108,249813,249814,249817],{},[2627,249815,249816],{},"無結構\u002F流式檔案："," 將檔案視為一串連續的字元或位元組流（如 UNIX 系統的文字檔）。",[108,249819,249820,249823,249824,249826],{},[2627,249821,249822],{},"有結構\u002F記錄式檔案："," 檔案由一組相似的記錄 (Records) 組合而成，每條記錄由多個欄位組成（如資料庫表格）。\n",[2627,249825,249111],{}," 邏輯結構是使用者眼中的檔案，而實體結構是磁碟眼中的檔案，不要混淆。",[19,249828,249830],{"id":249829},"_15unix系统中提供了-异步写和-三种写方式","15．UNIX系统中提供了（ ）、异步写和（ ）三种写方式。",[15,249832,249833,249835,249836,249838,249839,249841],{},[2627,249834,58173],{}," 同步写 (Synchronous write) 、 延迟写 (Delayed write)\n",[2627,249837,248267],{}," UNIX 的磁碟 I\u002FO 寫入策略。\n",[2627,249840,246526],{}," UNIX 檔案系統為了兼顧效能與資料安全性，提供了不同的寫入策略：",[2230,249843,249844,249847,249853],{},[108,249845,249846],{},"異步寫：寫入緩衝區後直接返回，由系統稍後自動寫入磁碟。",[108,249848,249849,249852],{},[2627,249850,249851],{},"同步寫："," 程式必須等待資料確確實實地寫入物理磁碟後，操作才算完成，這能保證最高的安全性。",[108,249854,249855,249858,249859,249861],{},[2627,249856,249857],{},"延遲寫："," 寫入緩衝區後只做個標記，直到緩衝區需要被替換時才真正寫入磁碟（極大地減少了磁碟 I\u002FO 次數）。\n",[2627,249860,249111],{}," 延遲寫 (Lazy write) 是現代作業系統提升磁碟效能最核心的緩衝機制。",[19,249863,249865],{"id":249864},"_16请求分页式虚拟存储系统必须至少具有三种硬件支持即-和地址变换机构","16．请求分页式虚拟存储系统必须至少具有三种硬件支持，即（ ）、（ ）和地址变换机构。",[15,249867,249868,249870,249871,249873,249874,249876],{},[2627,249869,58173],{}," 页表机制 (Page table mechanism) 、 缺页中断机构 (Page fault interrupt mechanism)\n",[2627,249872,248267],{}," 虛擬記憶體硬體支援。\n",[2627,249875,246526],{}," 要實現「請求分頁（程式不必全裝入，用到了再調入）」，硬體必須支援：",[2230,249878,249879,249885,249891],{},[108,249880,249881,249884],{},[2627,249882,249883],{},"頁表機制："," 頁表不僅要記錄塊號，還必須增加「存在位元」、「修改位元」等來追蹤頁面狀態。",[108,249886,249887,249890],{},[2627,249888,249889],{},"缺頁中斷機構："," 當地址變換發現頁面不在記憶體時，能觸發中斷，通知 OS 去磁碟抓取。",[108,249892,249893,249894,249896],{},"位址變換機構：用來在執行時將邏輯位址快速轉換為實體位址。\n",[2627,249895,249111],{}," 缺頁中斷是一種特殊的「內部異常 (Trap)」，它是在指令執行「期間」發生的，且處理完後會重新執行該指令。",[19,249898,249900],{"id":249899},"_17解决死锁的基本方法有-检测死锁和解除死锁","17．解决死锁的基本方法有（ ）、（ ）、检测死锁和解除死锁。",[15,249902,249903,249905,249906,249908,249909,249911],{},[2627,249904,58173],{}," 预防死锁 (Deadlock prevention) 、 避免死锁 (Deadlock avoidance)\n",[2627,249907,248267],{}," 死結處理策略。\n",[2627,249910,246526],{}," 作業系統應對死結的方法按「嚴格程度」遞減分為四種：",[2230,249913,249914,249920,249926,249929],{},[108,249915,249916,249919],{},[2627,249917,249918],{},"預防死鎖："," 從機制上破壞死結的四個必要條件之一（例如要求一次性申請所有資源）。",[108,249921,249922,249925],{},[2627,249923,249924],{},"避免死锁："," 允許資源動態申請，但在分配前用演算法（如銀行家演算法）檢查是否會進入不安全狀態。",[108,249927,249928],{},"檢測：允許死結發生，定期檢查。",[108,249930,249931,249932,249934],{},"解除：發現死結後，強行剝奪資源或殺死行程。\n",[2627,249933,249111],{}," 預防 (Prevention) 是一種靜態約束；避免 (Avoidance) 是一種動態檢查，兩者概念不同。",[19,249936,249938],{"id":249937},"_18如果把一本词典的内容作为一个文件存放每个单词和对它的解释组成一个记录为了便于该词典的使用者迅速查到所需的单词这个文件的存储结构采用-文件结构比较合适","18．如果把一本词典的内容作为一个文件存放，每个单词和对它的解释组成一个记录。为了便于该词典的使用者迅速查到所需的单词，这个文件的存储结构采用（ ）文件结构比较合适。",[15,249940,249941,249943,249944,249946,249947,249949,249950,249953,249954,249956],{},[2627,249942,58173],{}," 索引顺序 (Indexed sequential) (或 直接\u002F散列 Hashed 文件)\n",[2627,249945,248267],{}," 檔案邏輯組織。\n",[2627,249948,246526],{}," 詞典需要按照字母順序排列（順序性），同時又需要支援對單一單詞的快速「隨機查找」。",[2627,249951,249952],{},"索引順序檔案"," 為順序排列的記錄建立一個多級索引表，極大地加快了搜尋速度，是最符合這種情境的結構。\n",[2627,249955,249111],{}," 單純的順序檔案尋找極慢；單純的索引檔案又浪費空間。索引順序檔案結合了兩者的優點。",[19,249958,249960],{"id":249959},"_19通过操作系统对外围设备的管理可以实现外围设备和计算机系统的-之间的并行操作","19．通过操作系统对外围设备的管理，可以实现外围设备和计算机系统的（ ）之间的并行操作。",[15,249962,249963,249965,249966,249968,249969,249971,249972,249975,249976,249978],{},[2627,249964,58173],{}," 处理机 (CPU \u002F 处理器)\n",[2627,249967,248267],{}," I\u002FO 管理的目標。\n",[2627,249970,246526],{}," 多道程式設計和中斷、DMA 機制的核心目的，就是要讓慢速的外部設備與高速的 ",[2627,249973,249974],{},"CPU (處理機)"," 能夠同時工作（並行操作），避免 CPU 陷入無謂的等待 (busy-waiting)。\n",[2627,249977,249111],{}," 這裡強調的是 CPU 運算與 I\u002FO 傳輸的重疊 (Overlapping)。",[19,249980,249982],{"id":249981},"_20如果某文件系统以成组方式存放记录每个磁盘块最多可以存放8个记录用于记录成组和分解的主存缓冲区的大小与磁盘块大小相同若0-7号记录存放在第0个磁盘块815号记录存放在第1个磁盘块那么为了依次读出第23242517号记录需要进行-次读盘操作","20．如果某文件系统以成组方式存放记录，每个磁盘块最多可以存放8个记录，用于记录成组和分解的主存缓冲区的大小与磁盘块大小相同。若0-7号记录存放在第0个磁盘块，8—15号记录存放在第1个磁盘块……，那么为了依次读出第23、24、25、17号记录，需要进行（ ）次读盘操作。",[15,249984,249985,249987,249988,249990,249991,249993],{},[2627,249986,58173],{}," 3\n",[2627,249989,248267],{}," 記錄的成組與分解 (Blocking & Unblocking)。\n",[2627,249992,246526],{},"\n每個磁碟塊 (Block) 包含 8 個記錄：",[105,249995,249996,249999,250002,250005],{},[108,249997,249998],{},"Block 0: 記錄 0~7",[108,250000,250001],{},"Block 1: 記錄 8~15",[108,250003,250004],{},"Block 2: 記錄 16~23",[108,250006,250007],{},"Block 3: 記錄 24~31\n讀取順序與磁碟 I\u002FO：",[2230,250009,250010,250013,250016,250022],{},[108,250011,250012],{},"讀記錄 23：位於 Block 2。系統將 Block 2 讀入緩衝區（第 1 次讀盤）。",[108,250014,250015],{},"讀記錄 24：位於 Block 3。系統將 Block 3 讀入緩衝區，覆蓋舊資料（第 2 次讀盤）。",[108,250017,250018,250019,347],{},"讀記錄 25：位於 Block 3。此時 Block 3 已在緩衝區中，",[2627,250020,250021],{},"直接命中，不需讀盤",[108,250023,250024,250025,250027],{},"讀記錄 17：位於 Block 2。此時緩衝區是 Block 3，所以必須重新將 Block 2 讀入緩衝區（第 3 次讀盤）。\n因此，共需 3 次實體讀盤操作。\n",[2627,250026,249111],{}," 緩衝區只能存放一個磁碟塊的資料，當請求跨越磁碟塊又折返時，會產生重複讀取的開銷（這也是為何作業系統會引入多重緩衝或 LRU 緩衝池的原因）。",[19,250029,250031],{"id":250030},"_21若信号量s的初值定义为10则在s上调用了12次p操作和10次v操作后s的值应该为","21．若信号量S的初值定义为10，则在S上调用了12次P操作和10次V操作后S的值应该为（ ）。",[15,250033,250034,249353,250036,250038,250039,250041,250042,250044],{},[2627,250035,58173],{},[2627,250037,248267],{}," 號誌 P\u002FV 操作的數值計算。\n",[2627,250040,246526],{},"\nP 操作 (Wait) 會消耗資源：S = S - 1。\nV 操作 (Signal) 會釋放資源：S = S + 1。\nS 最終值 = 初始值 - P操作次數 + V操作次數 = 10 - 12 + 10 = 8。\n",[2627,250043,249111],{}," 由於 S > 0，這意味著還有 8 個資源可以使用，沒有行程被阻塞。",[19,250046,250048],{"id":250047},"_22如果系统中有n个进程则在就绪队列中进程的个数最多为","22．如果系统中有n个进程，则在就绪队列中进程的个数最多为( )。",[15,250050,250051,250053,250054,250056,250057,250059,250060,250062,250063,250114,250115,250117,250118,250146],{},[2627,250052,58173],{}," n-1\n",[2627,250055,248267],{}," 行程狀態轉換與資源限制。\n",[2627,250058,246526],{}," 假設這是一個單處理機（單核心 CPU）系統。同一時刻只能有 ",[2627,250061,640],{}," 個行程處於「執行 (Running)」狀態。所以剩下的行程最多只能有 ",[34,250064,250066,250084],{"className":250065},[31642],[34,250067,250069],{"className":250068},[31646],[31648,250070,250071],{"xmlns":31650},[31652,250072,250073,250081],{},[31655,250074,250075,250077,250079],{},[31658,250076,31660],{},[31662,250078,32226],{},[31749,250080,640],{},[31665,250082,250083],{"encoding":31667},"n - 1",[34,250085,250087,250105],{"className":250086,"ariaHidden":758},[31672],[34,250088,250090,250093,250096,250099,250102],{"className":250089},[31676],[34,250091],{"className":250092,"style":32597},[31680],[34,250094,31660],{"className":250095},[31685,31686],[34,250097],{"className":250098,"style":32604},[31845],[34,250100,32226],{"className":250101},[32024],[34,250103],{"className":250104,"style":32604},[31845],[34,250106,250108,250111],{"className":250107},[31676],[34,250109],{"className":250110,"style":32617},[31680],[34,250112,640],{"className":250113},[31685]," 個處於「就緒 (Ready)」狀態排隊等待 CPU。\n",[2627,250116,249111],{}," 不要回答 ",[34,250119,250121,250134],{"className":250120},[31642],[34,250122,250124],{"className":250123},[31646],[31648,250125,250126],{"xmlns":31650},[31652,250127,250128,250132],{},[31655,250129,250130],{},[31658,250131,31660],{},[31665,250133,31660],{"encoding":31667},[34,250135,250137],{"className":250136,"ariaHidden":758},[31672],[34,250138,250140,250143],{"className":250139},[31676],[34,250141],{"className":250142,"style":32527},[31680],[34,250144,31660],{"className":250145},[31685,31686],"，因為必然要有 1 個行程正在執行，才會有排隊的現象產生。",[19,250148,250150],{"id":250149},"_23计算机有缓存内存辅存实现虚拟存储器如果数据在缓存中访问它需要20ns如果在内存但不在缓存需要60ns将其装入缓存然后才能访问如果不在内存而在辅存需要12μs将其读入内存用60ns再读入缓存然后才能访问假设缓存命中率为09内存命中率为06数据平均访问时间为-ns","23．计算机有缓存、内存、辅存实现虚拟存储器。如果数据在缓存中，访问它需要20ns；如果在内存但不在缓存，需要60ns将其装入缓存，然后才能访问；如果不在内存而在辅存，需要12μs将其读入内存，用60ns再读入缓存，然后才能访问。假设缓存命中率为0.9，内存命中率为0.6。数据平均访问时间为（ ）ns。",[15,250152,250153,250155,250156,250158,250159,250161,250162,250234,250235,250309],{},[2627,250154,58173],{}," 506\n",[2627,250157,248267],{}," 階層式記憶體的有效存取時間 (Effective Access Time, EAT)。\n",[2627,250160,246526],{},"\n請注意時間單位的統一：",[34,250163,250165,250192],{"className":250164},[31642],[34,250166,250168],{"className":250167},[31646],[31648,250169,250170],{"xmlns":31650},[31652,250171,250172,250189],{},[31655,250173,250174,250176,250178,250180,250182,250185,250187],{},[31749,250175,71869],{},[31658,250177,135107],{},[31658,250179,31747],{},[31662,250181,202],{},[31749,250183,250184],{},"12000",[31658,250186,31660],{},[31658,250188,31747],{},[31665,250190,250191],{"encoding":31667},"12 \\mu s = 12000 ns",[34,250193,250195,250219],{"className":250194,"ariaHidden":758},[31672],[34,250196,250198,250201,250204,250207,250210,250213,250216],{"className":250197},[31676],[34,250199],{"className":250200,"style":40292},[31680],[34,250202,71869],{"className":250203},[31685],[34,250205,135107],{"className":250206},[31685,31686],[34,250208,31747],{"className":250209},[31685,31686],[34,250211],{"className":250212,"style":31846},[31845],[34,250214,202],{"className":250215},[31850],[34,250217],{"className":250218,"style":31846},[31845],[34,250220,250222,250225,250228,250231],{"className":250221},[31676],[34,250223],{"className":250224,"style":32617},[31680],[34,250226,250184],{"className":250227},[31685],[34,250229,31660],{"className":250230},[31685,31686],[34,250232,31747],{"className":250233},[31685,31686],"。\n平均存取時間 = ",[34,250236,250238,250264],{"className":250237},[31642],[34,250239,250241],{"className":250240},[31646],[31648,250242,250243],{"xmlns":31650},[31652,250244,250245,250261],{},[31655,250246,250247,250249,250251,250254,250256,250259],{},[31662,250248,131138],{},[31662,250250,456],{"stretchy":702},[39361,250252,250253],{},"各階層命中率",[31662,250255,42071],{},[39361,250257,250258],{},"該階層的存取耗時",[31662,250260,2557],{"stretchy":702},[31665,250262,250263],{"encoding":31667},"\\sum (\\text{各階層命中率} \\times \\text{該階層的存取耗時})",[34,250265,250267,250294],{"className":250266,"ariaHidden":758},[31672],[34,250268,250270,250273,250276,250279,250285,250288,250291],{"className":250269},[31676],[34,250271],{"className":250272,"style":31774},[31680],[34,250274,131138],{"className":250275,"style":156561},[32767,131249,156560],[34,250277,456],{"className":250278},[31782],[34,250280,250282],{"className":250281},[31685,2638],[34,250283,250253],{"className":250284},[31685,105462],[34,250286],{"className":250287,"style":32604},[31845],[34,250289,42071],{"className":250290},[32024],[34,250292],{"className":250293,"style":32604},[31845],[34,250295,250297,250300,250306],{"className":250296},[31676],[34,250298],{"className":250299,"style":31774},[31680],[34,250301,250303],{"className":250302},[31685,2638],[34,250304,250258],{"className":250305},[31685,105462],[34,250307,2557],{"className":250308},[31690],"。\n這是一個階層讀取：如果錯過上一層，就必須付出把資料搬上來的懲罰時間。",[2230,250311,250312,250318,250324],{},[108,250313,250314,250317],{},[2627,250315,250316],{},"快取 (Cache) 命中："," 命中率 = 0.9。耗時 = 20ns。",[108,250319,250320,250323],{},[2627,250321,250322],{},"快取未命中，但記憶體 (Memory) 命中："," 機率 = (1 - 0.9) × 0.6 = 0.06。\n耗時 = 載入快取(60) + 存取(20) = 80ns。",[108,250325,250326,250329,250330,250333,250335],{},[2627,250327,250328],{},"記憶體也未命中（需找輔存）："," 機率 = (1 - 0.9) × (1 - 0.6) = 0.1 × 0.4 = 0.04。\n耗時 = 從輔存讀入記憶體(12000) + 從記憶體載入快取(60) + 存取(20) = 12080ns。\n",[2627,250331,250332],{},"EAT = (0.9 × 20) + (0.06 × 80) + (0.04 × 12080) = 18 + 4.8 + 483.2 = 506 ns。",[2627,250334,249111],{}," 資料從下層搬到上層後，最終還是要從最上層（快取）進行存取。所以在未命中的耗時計算中，千萬別漏加了上一層的載入時間和最終的存取時間。",[19,250337,250339],{"id":250338},"_24设文件索引节点中有7个地址项其中4个为直接地址索引2个是一级间接地址索引1个是二级间接地址索引地址项大小为4字节若磁盘索引块和磁盘数据块大小均为256字节则可表示的单个文件的最大长度是","24．设文件索引节点中有7个地址项，其中4个为直接地址索引，2个是一级间接地址索引，1个是二级间接地址索引，地址项大小为4字节，若磁盘索引块和磁盘数据块大小均为256字节，则可表示的单个文件的最大长度是（ ）。",[15,250341,250342,250344,250345,250347,250348,250350,250351,250424],{},[2627,250343,58173],{}," 1057 KB (或 1082368 字节)\n",[2627,250346,248267],{}," UNIX 的多級索引節點 (inode) 容量計算。\n",[2627,250349,246526],{},"\n每個區塊可以容納的指標數量 = 磁碟塊大小 \u002F 指標大小 = 256 B \u002F 4 B = 64 個指標。\n這意味著一個一級間接區塊可以指向 64 個資料區塊；一個二級間接區塊可以指向 ",[34,250352,250354,250376],{"className":250353},[31642],[34,250355,250357],{"className":250356},[31646],[31648,250358,250359],{"xmlns":31650},[31652,250360,250361,250373],{},[31655,250362,250363,250365,250367,250369,250371],{},[31749,250364,162212],{},[31662,250366,42071],{},[31749,250368,162212],{},[31662,250370,202],{},[31749,250372,207588],{},[31665,250374,250375],{"encoding":31667},"64 \\times 64 = 4096",[34,250377,250379,250397,250415],{"className":250378,"ariaHidden":758},[31672],[34,250380,250382,250385,250388,250391,250394],{"className":250381},[31676],[34,250383],{"className":250384,"style":35942},[31680],[34,250386,162212],{"className":250387},[31685],[34,250389],{"className":250390,"style":32604},[31845],[34,250392,42071],{"className":250393},[32024],[34,250395],{"className":250396,"style":32604},[31845],[34,250398,250400,250403,250406,250409,250412],{"className":250399},[31676],[34,250401],{"className":250402,"style":32617},[31680],[34,250404,162212],{"className":250405},[31685],[34,250407],{"className":250408,"style":31846},[31845],[34,250410,202],{"className":250411},[31850],[34,250413],{"className":250414,"style":31846},[31845],[34,250416,250418,250421],{"className":250417},[31676],[34,250419],{"className":250420,"style":32617},[31680],[34,250422,207588],{"className":250423},[31685]," 個資料區塊。",[2230,250426,250427,250436,250445],{},[108,250428,250429,250432,250433,347],{},[2627,250430,250431],{},"4 個直接索引："," 指向 4 個資料區塊。容量 = 4 × 256 B = 1024 B = ",[2627,250434,250435],{},"1 KB",[108,250437,250438,250441,250442,347],{},[2627,250439,250440],{},"2 個一級間接索引："," 指向 2 × 64 = 128 個資料區塊。容量 = 128 × 256 B = 32768 B = ",[2627,250443,250444],{},"32 KB",[108,250446,250447,250450,250451,250454,250455,39984,250458,250460],{},[2627,250448,250449],{},"1 個二級間接索引："," 指向 1 × 64 × 64 = 4096 個資料區塊。容量 = 4096 × 256 B = 1048576 B = ",[2627,250452,250453],{},"1024 KB"," (也就是 1 MB)。\n總最大長度 = 1 KB + 32 KB + 1024 KB = ",[2627,250456,250457],{},"1057 KB",[2627,250459,249111],{}," 記得先算出「一個區塊能裝幾個指標」，再依據層級計算次方。這類題目要特別注意單位是 Byte、KB 還是 MB。",[19,250462,250464],{"id":250463},"_25实时系统有4个周期性事件周期分别为50100200和200ms其处理分别需要302020和δms则该系统可调度允许的δ最大值为","25．实时系统有4个周期性事件，周期分别为50、100、200和200ms，其处理分别需要30、20、20和δms，则该系统可调度允许的δ最大值为（ ）。",[15,250466,250467,250469,250470,250472,250473,250475,250476,250656,250657,250685,250686,250714,250715,250841,250922,249057,250980,39984,251031,251033],{},[2627,250468,58173],{}," 20\n",[2627,250471,248267],{}," 實時系統的可調度性（與第 1 題類似）。\n",[2627,250474,246526],{}," 最大允許 CPU 總使用率 ",[34,250477,250479,250519],{"className":250478},[31642],[34,250480,250482],{"className":250481},[31646],[31648,250483,250484],{"xmlns":31650},[31652,250485,250486,250516],{},[31655,250487,250488,250490,250492,250494,250496,250502,250504,250510,250512,250514],{},[31658,250489,33350],{},[31662,250491,202],{},[31662,250493,131138],{},[31662,250495,456],{"stretchy":702},[31743,250497,250498,250500],{},[31658,250499,47384],{},[31658,250501,31955],{},[31658,250503,38080],{"mathvariant":38079},[31743,250505,250506,250508],{},[31658,250507,37400],{},[31658,250509,31955],{},[31662,250511,2557],{"stretchy":702},[31662,250513,45972],{},[31749,250515,640],{},[31665,250517,250518],{"encoding":31667},"U = \\sum (C_i \u002F T_i) \\le 1",[34,250520,250522,250540,250647],{"className":250521,"ariaHidden":758},[31672],[34,250523,250525,250528,250531,250534,250537],{"className":250524},[31676],[34,250526],{"className":250527,"style":33218},[31680],[34,250529,33350],{"className":250530,"style":33389},[31685,31686],[34,250532],{"className":250533,"style":31846},[31845],[34,250535,202],{"className":250536},[31850],[34,250538],{"className":250539,"style":31846},[31845],[34,250541,250543,250546,250549,250552,250592,250595,250635,250638,250641,250644],{"className":250542},[31676],[34,250544],{"className":250545,"style":31774},[31680],[34,250547,131138],{"className":250548,"style":156561},[32767,131249,156560],[34,250550,456],{"className":250551},[31782],[34,250553,250555,250558],{"className":250554},[31685],[34,250556,47384],{"className":250557,"style":38706},[31685,31686],[34,250559,250561],{"className":250560},[31792],[34,250562,250564,250584],{"className":250563},[31796,31797],[34,250565,250567,250581],{"className":250566},[31801],[34,250568,250570],{"className":250569,"style":32090},[31805],[34,250571,250572,250575],{"style":129511},[34,250573],{"className":250574,"style":31814},[31813],[34,250576,250578],{"className":250577},[31818,31819,31820,31821],[34,250579,31955],{"className":250580},[31685,31686,31821],[34,250582,31829],{"className":250583},[31828],[34,250585,250587],{"className":250586},[31801],[34,250588,250590],{"className":250589,"style":31836},[31805],[34,250591],{},[34,250593,38080],{"className":250594},[31685],[34,250596,250598,250601],{"className":250597},[31685],[34,250599,37400],{"className":250600,"style":37422},[31685,31686],[34,250602,250604],{"className":250603},[31792],[34,250605,250607,250627],{"className":250606},[31796,31797],[34,250608,250610,250624],{"className":250609},[31801],[34,250611,250613],{"className":250612,"style":32090},[31805],[34,250614,250615,250618],{"style":214424},[34,250616],{"className":250617,"style":31814},[31813],[34,250619,250621],{"className":250620},[31818,31819,31820,31821],[34,250622,31955],{"className":250623},[31685,31686,31821],[34,250625,31829],{"className":250626},[31828],[34,250628,250630],{"className":250629},[31801],[34,250631,250633],{"className":250632,"style":31836},[31805],[34,250634],{},[34,250636,2557],{"className":250637},[31690],[34,250639],{"className":250640,"style":31846},[31845],[34,250642,45972],{"className":250643},[31850],[34,250645],{"className":250646,"style":31846},[31845],[34,250648,250650,250653],{"className":250649},[31676],[34,250651],{"className":250652,"style":32617},[31680],[34,250654,640],{"className":250655},[31685],"。\n計算：30\u002F50 + 20\u002F100 + 20\u002F200 + ",[34,250658,250660,250673],{"className":250659},[31642],[34,250661,250663],{"className":250662},[31646],[31648,250664,250665],{"xmlns":31650},[31652,250666,250667,250671],{},[31655,250668,250669],{},[31658,250670,231507],{},[31665,250672,231510],{"encoding":31667},[34,250674,250676],{"className":250675,"ariaHidden":758},[31672],[34,250677,250679,250682],{"className":250678},[31676],[34,250680],{"className":250681,"style":31681},[31680],[34,250683,231507],{"className":250684,"style":231523},[31685,31686],"\u002F200 ",[34,250687,250689,250702],{"className":250688},[31642],[34,250690,250692],{"className":250691},[31646],[31648,250693,250694],{"xmlns":31650},[31652,250695,250696,250700],{},[31655,250697,250698],{},[31662,250699,45972],{},[31665,250701,45975],{"encoding":31667},[34,250703,250705],{"className":250704,"ariaHidden":758},[31672],[34,250706,250708,250711],{"className":250707},[31676],[34,250709],{"className":250710,"style":43198},[31680],[34,250712,45972],{"className":250713},[31850]," 1\n即：",[34,250716,250718,250753],{"className":250717},[31642],[34,250719,250721],{"className":250720},[31646],[31648,250722,250723],{"xmlns":31650},[31652,250724,250725,250750],{},[31655,250726,250727,250730,250732,250734,250736,250738,250740,250742,250744,250746,250748],{},[31749,250728,250729],{},"0.6",[31662,250731,223],{},[31749,250733,231993],{},[31662,250735,223],{},[31749,250737,248816],{},[31662,250739,223],{},[31658,250741,231507],{},[31658,250743,38080],{"mathvariant":38079},[31749,250745,168915],{},[31662,250747,45972],{},[31749,250749,640],{},[31665,250751,250752],{"encoding":31667},"0.6 + 0.2 + 0.1 + \\delta\u002F200 \\le 1",[34,250754,250756,250774,250792,250810,250832],{"className":250755,"ariaHidden":758},[31672],[34,250757,250759,250762,250765,250768,250771],{"className":250758},[31676],[34,250760],{"className":250761,"style":35942},[31680],[34,250763,250729],{"className":250764},[31685],[34,250766],{"className":250767,"style":32604},[31845],[34,250769,223],{"className":250770},[32024],[34,250772],{"className":250773,"style":32604},[31845],[34,250775,250777,250780,250783,250786,250789],{"className":250776},[31676],[34,250778],{"className":250779,"style":35942},[31680],[34,250781,231993],{"className":250782},[31685],[34,250784],{"className":250785,"style":32604},[31845],[34,250787,223],{"className":250788},[32024],[34,250790],{"className":250791,"style":32604},[31845],[34,250793,250795,250798,250801,250804,250807],{"className":250794},[31676],[34,250796],{"className":250797,"style":35942},[31680],[34,250799,248816],{"className":250800},[31685],[34,250802],{"className":250803,"style":32604},[31845],[34,250805,223],{"className":250806},[32024],[34,250808],{"className":250809,"style":32604},[31845],[34,250811,250813,250816,250819,250823,250826,250829],{"className":250812},[31676],[34,250814],{"className":250815,"style":31774},[31680],[34,250817,231507],{"className":250818,"style":231523},[31685,31686],[34,250820,250822],{"className":250821},[31685],"\u002F200",[34,250824],{"className":250825,"style":31846},[31845],[34,250827,45972],{"className":250828},[31850],[34,250830],{"className":250831,"style":31846},[31845],[34,250833,250835,250838],{"className":250834},[31676],[34,250836],{"className":250837,"style":32617},[31680],[34,250839,640],{"className":250840},[31685],[34,250842,250844,250871],{"className":250843},[31642],[34,250845,250847],{"className":250846},[31646],[31648,250848,250849],{"xmlns":31650},[31652,250850,250851,250868],{},[31655,250852,250853,250856,250858,250860,250862,250864,250866],{},[31749,250854,250855],{},"0.9",[31662,250857,223],{},[31658,250859,231507],{},[31658,250861,38080],{"mathvariant":38079},[31749,250863,168915],{},[31662,250865,45972],{},[31749,250867,640],{},[31665,250869,250870],{"encoding":31667},"0.9 + \\delta\u002F200 \\le 1",[34,250872,250874,250892,250913],{"className":250873,"ariaHidden":758},[31672],[34,250875,250877,250880,250883,250886,250889],{"className":250876},[31676],[34,250878],{"className":250879,"style":35942},[31680],[34,250881,250855],{"className":250882},[31685],[34,250884],{"className":250885,"style":32604},[31845],[34,250887,223],{"className":250888},[32024],[34,250890],{"className":250891,"style":32604},[31845],[34,250893,250895,250898,250901,250904,250907,250910],{"className":250894},[31676],[34,250896],{"className":250897,"style":31774},[31680],[34,250899,231507],{"className":250900,"style":231523},[31685,31686],[34,250902,250822],{"className":250903},[31685],[34,250905],{"className":250906,"style":31846},[31845],[34,250908,45972],{"className":250909},[31850],[34,250911],{"className":250912,"style":31846},[31845],[34,250914,250916,250919],{"className":250915},[31676],[34,250917],{"className":250918,"style":32617},[31680],[34,250920,640],{"className":250921},[31685],[34,250923,250925,250947],{"className":250924},[31642],[34,250926,250928],{"className":250927},[31646],[31648,250929,250930],{"xmlns":31650},[31652,250931,250932,250944],{},[31655,250933,250934,250936,250938,250940,250942],{},[31658,250935,231507],{},[31658,250937,38080],{"mathvariant":38079},[31749,250939,168915],{},[31662,250941,45972],{},[31749,250943,248816],{},[31665,250945,250946],{"encoding":31667},"\\delta\u002F200 \\le 0.1",[34,250948,250950,250971],{"className":250949,"ariaHidden":758},[31672],[34,250951,250953,250956,250959,250962,250965,250968],{"className":250952},[31676],[34,250954],{"className":250955,"style":31774},[31680],[34,250957,231507],{"className":250958,"style":231523},[31685,31686],[34,250960,250822],{"className":250961},[31685],[34,250963],{"className":250964,"style":31846},[31845],[34,250966,45972],{"className":250967},[31850],[34,250969],{"className":250970,"style":31846},[31845],[34,250972,250974,250977],{"className":250973},[31676],[34,250975],{"className":250976,"style":32617},[31680],[34,250978,248816],{"className":250979},[31685],[34,250981,250983,251001],{"className":250982},[31642],[34,250984,250986],{"className":250985},[31646],[31648,250987,250988],{"xmlns":31650},[31652,250989,250990,250998],{},[31655,250991,250992,250994,250996],{},[31658,250993,231507],{},[31662,250995,45972],{},[31749,250997,3249],{},[31665,250999,251000],{"encoding":31667},"\\delta \\le 20",[34,251002,251004,251022],{"className":251003,"ariaHidden":758},[31672],[34,251005,251007,251010,251013,251016,251019],{"className":251006},[31676],[34,251008],{"className":251009,"style":47709},[31680],[34,251011,231507],{"className":251012,"style":231523},[31685,31686],[34,251014],{"className":251015,"style":31846},[31845],[34,251017,45972],{"className":251018},[31850],[34,251020],{"className":251021,"style":31846},[31845],[34,251023,251025,251028],{"className":251024},[31676],[34,251026],{"className":251027,"style":32617},[31680],[34,251029,3249],{"className":251030},[31685],[2627,251032,249111],{}," 只要列對利用率方程求和即可，非常簡單，但要小心分數加法。",[19,251035,251037],{"id":251036},"_26系统提供24位虚存空间主存为218b分页式虚拟存储管理页面尺寸为1kb用户程序虚拟地址11123456八进制页面分得块号为200八进制物理地址","26．系统提供24位虚存空间，主存为218B，分页式虚拟存储管理，页面尺寸为1KB。用户程序虚拟地址11123456(八进制)，页面分得块号为200(八进制)，物理地址（ ）。",[15,251039,251040,251042,251043,251045,251046,251048],{},[2627,251041,58173],{}," 201456 (八进制)\n",[2627,251044,248267],{}," 虛擬位址到實體位址的轉換。\n",[2627,251047,246526],{},"\n這種八進位轉換題，利用八進位運算會比轉回二進位更安全。",[2230,251050,251051,251955],{},[108,251052,251053,251056,251057,251118,251119,251195,251196,1840,251270,251417,251418,39984,251500,251672,251673,251881,251882,347],{},[2627,251054,251055],{},"找出偏移量 (Offset)："," 頁面大小為 ",[34,251058,251060,251082],{"className":251059},[31642],[34,251061,251063],{"className":251062},[31646],[31648,251064,251065],{"xmlns":31650},[31652,251066,251067,251079],{},[31655,251068,251069,251071,251073,251075,251077],{},[31749,251070,640],{},[31658,251072,38691],{},[31658,251074,54224],{},[31662,251076,202],{},[31749,251078,177393],{},[31665,251080,251081],{"encoding":31667},"1KB = 1024",[34,251083,251085,251109],{"className":251084,"ariaHidden":758},[31672],[34,251086,251088,251091,251094,251097,251100,251103,251106],{"className":251087},[31676],[34,251089],{"className":251090,"style":33218},[31680],[34,251092,640],{"className":251093},[31685],[34,251095,38691],{"className":251096,"style":38706},[31685,31686],[34,251098,54224],{"className":251099,"style":54252},[31685,31686],[34,251101],{"className":251102,"style":31846},[31845],[34,251104,202],{"className":251105},[31850],[34,251107],{"className":251108,"style":31846},[31845],[34,251110,251112,251115],{"className":251111},[31676],[34,251113],{"className":251114,"style":32617},[31680],[34,251116,177393],{"className":251117},[31685]," Bytes。\n在八進位中，",[34,251120,251122,251140],{"className":251121},[31642],[34,251123,251125],{"className":251124},[31646],[31648,251126,251127],{"xmlns":31650},[31652,251128,251129,251137],{},[31655,251130,251131],{},[31743,251132,251133,251135],{},[31749,251134,177393],{},[31749,251136,3244],{},[31665,251138,251139],{"encoding":31667},"1024_{10}",[34,251141,251143],{"className":251142,"ariaHidden":758},[31672],[34,251144,251146,251149,251152],{"className":251145},[31676],[34,251147],{"className":251148,"style":38357},[31680],[34,251150,162508],{"className":251151},[31685],[34,251153,251155,251158],{"className":251154},[31685],[34,251156,15932],{"className":251157},[31685],[34,251159,251161],{"className":251160},[31792],[34,251162,251164,251187],{"className":251163},[31796,31797],[34,251165,251167,251184],{"className":251166},[31801],[34,251168,251170],{"className":251169,"style":31806},[31805],[34,251171,251172,251175],{"style":31809},[34,251173],{"className":251174,"style":31814},[31813],[34,251176,251178],{"className":251177},[31818,31819,31820,31821],[34,251179,251181],{"className":251180},[31685,31821],[34,251182,3244],{"className":251183},[31685,31821],[34,251185,31829],{"className":251186},[31828],[34,251188,251190],{"className":251189},[31801],[34,251191,251193],{"className":251192,"style":31836},[31805],[34,251194],{}," 剛好是 ",[34,251197,251199,251218],{"className":251198},[31642],[34,251200,251202],{"className":251201},[31646],[31648,251203,251204],{"xmlns":31650},[31652,251205,251206,251215],{},[31655,251207,251208],{},[31743,251209,251210,251213],{},[31749,251211,251212],{},"2000",[31749,251214,60525],{},[31665,251216,251217],{"encoding":31667},"2000_8",[34,251219,251221],{"className":251220,"ariaHidden":758},[31672],[34,251222,251224,251227,251230],{"className":251223},[31676],[34,251225],{"className":251226,"style":38357},[31680],[34,251228,168915],{"className":251229},[31685],[34,251231,251233,251236],{"className":251232},[31685],[34,251234,459],{"className":251235},[31685],[34,251237,251239],{"className":251238},[31792],[34,251240,251242,251262],{"className":251241},[31796,31797],[34,251243,251245,251259],{"className":251244},[31801],[34,251246,251248],{"className":251247,"style":31806},[31805],[34,251249,251250,251253],{"style":31809},[34,251251],{"className":251252,"style":31814},[31813],[34,251254,251256],{"className":251255},[31818,31819,31820,31821],[34,251257,60525],{"className":251258},[31685,31821],[34,251260,31829],{"className":251261},[31828],[34,251263,251265],{"className":251264},[31801],[34,251266,251268],{"className":251267,"style":31836},[31805],[34,251269],{},[34,251271,251273,251307],{"className":251272},[31642],[34,251274,251276],{"className":251275},[31646],[31648,251277,251278],{"xmlns":31650},[31652,251279,251280,251304],{},[31655,251281,251282,251284,251286,251292,251294,251296,251298,251300,251302],{},[31749,251283,768],{},[31662,251285,42071],{},[37624,251287,251288,251290],{},[31749,251289,60525],{},[31749,251291,786],{},[31662,251293,202],{},[31749,251295,768],{},[31662,251297,42071],{},[31749,251299,167308],{},[31662,251301,202],{},[31749,251303,177393],{},[31665,251305,251306],{"encoding":31667},"2 \\times 8^3 = 2 \\times 512 = 1024",[34,251308,251310,251328,251372,251390,251408],{"className":251309,"ariaHidden":758},[31672],[34,251311,251313,251316,251319,251322,251325],{"className":251312},[31676],[34,251314],{"className":251315,"style":35942},[31680],[34,251317,768],{"className":251318},[31685],[34,251320],{"className":251321,"style":32604},[31845],[34,251323,42071],{"className":251324},[32024],[34,251326],{"className":251327,"style":32604},[31845],[34,251329,251331,251334,251363,251366,251369],{"className":251330},[31676],[34,251332],{"className":251333,"style":37774},[31680],[34,251335,251337,251340],{"className":251336},[31685],[34,251338,60525],{"className":251339},[31685],[34,251341,251343],{"className":251342},[31792],[34,251344,251346],{"className":251345},[31796],[34,251347,251349],{"className":251348},[31801],[34,251350,251352],{"className":251351,"style":37774},[31805],[34,251353,251354,251357],{"style":37777},[34,251355],{"className":251356,"style":31814},[31813],[34,251358,251360],{"className":251359},[31818,31819,31820,31821],[34,251361,786],{"className":251362},[31685,31821],[34,251364],{"className":251365,"style":31846},[31845],[34,251367,202],{"className":251368},[31850],[34,251370],{"className":251371,"style":31846},[31845],[34,251373,251375,251378,251381,251384,251387],{"className":251374},[31676],[34,251376],{"className":251377,"style":35942},[31680],[34,251379,768],{"className":251380},[31685],[34,251382],{"className":251383,"style":32604},[31845],[34,251385,42071],{"className":251386},[32024],[34,251388],{"className":251389,"style":32604},[31845],[34,251391,251393,251396,251399,251402,251405],{"className":251392},[31676],[34,251394],{"className":251395,"style":32617},[31680],[34,251397,167308],{"className":251398},[31685],[34,251400],{"className":251401,"style":31846},[31845],[34,251403,202],{"className":251404},[31850],[34,251406],{"className":251407,"style":31846},[31845],[34,251409,251411,251414],{"className":251410},[31676],[34,251412],{"className":251413,"style":32617},[31680],[34,251415,177393],{"className":251416},[31685],")。\n虛擬位址的「頁內偏移量」等於 虛擬位址 ",[34,251419,251421,251454],{"className":251420},[31642],[34,251422,251424],{"className":251423},[31646],[31648,251425,251426],{"xmlns":31650},[31652,251427,251428,251451],{},[31655,251429,251430,251432,251434,251436,251444,251446,251449],{},[31845,251431],{},[31845,251433],{"width":208910},[31662,251435,456],{"stretchy":702},[31655,251437,251438,251440,251442],{},[31658,251439,31759],{"mathvariant":38079},[31658,251441,43383],{"mathvariant":38079},[31658,251443,46900],{"mathvariant":38079},[31845,251445],{"width":208923},[39361,251447,251448],{},"頁面大小",[31662,251450,2557],{"stretchy":702},[31665,251452,251453],{"encoding":31667},"\\pmod{\\text{頁面大小}}",[34,251455,251457,251470],{"className":251456,"ariaHidden":758},[31672],[34,251458,251460,251464,251467],{"className":251459},[31676],[34,251461],{"className":251462,"style":251463},[31680],"height:0em;",[34,251465],{"className":251466},[31845,208951],[34,251468],{"className":251469,"style":208955},[31845],[34,251471,251473,251476,251479,251488,251491,251497],{"className":251472},[31676],[34,251474],{"className":251475,"style":31774},[31680],[34,251477,456],{"className":251478},[31782],[34,251480,251482],{"className":251481},[31685],[34,251483,251485],{"className":251484},[31685],[34,251486,207924],{"className":251487},[31685,207923],[34,251489],{"className":251490,"style":208977},[31845],[34,251492,251494],{"className":251493},[31685,2638],[34,251495,251448],{"className":251496},[31685,105462],[34,251498,2557],{"className":251499},[31690],[34,251501,251503,251546],{"className":251502},[31642],[34,251504,251506],{"className":251505},[31646],[31648,251507,251508],{"xmlns":31650},[31652,251509,251510,251543],{},[31655,251511,251512,251519,251521,251523,251525,251533,251535,251541],{},[31743,251513,251514,251517],{},[31749,251515,251516],{},"11123456",[31749,251518,60525],{},[31845,251520],{},[31845,251522],{"width":208910},[31662,251524,456],{"stretchy":702},[31655,251526,251527,251529,251531],{},[31658,251528,31759],{"mathvariant":38079},[31658,251530,43383],{"mathvariant":38079},[31658,251532,46900],{"mathvariant":38079},[31845,251534],{"width":208923},[31743,251536,251537,251539],{},[31749,251538,251212],{},[31749,251540,60525],{},[31662,251542,2557],{"stretchy":702},[31665,251544,251545],{"encoding":31667},"11123456_8 \\pmod{2000_8}",[34,251547,251549,251605],{"className":251548,"ariaHidden":758},[31672],[34,251550,251552,251555,251559,251599,251602],{"className":251551},[31676],[34,251553],{"className":251554,"style":38357},[31680],[34,251556,251558],{"className":251557},[31685],"1112345",[34,251560,251562,251565],{"className":251561},[31685],[34,251563,36656],{"className":251564},[31685],[34,251566,251568],{"className":251567},[31792],[34,251569,251571,251591],{"className":251570},[31796,31797],[34,251572,251574,251588],{"className":251573},[31801],[34,251575,251577],{"className":251576,"style":31806},[31805],[34,251578,251579,251582],{"style":31809},[34,251580],{"className":251581,"style":31814},[31813],[34,251583,251585],{"className":251584},[31818,31819,31820,31821],[34,251586,60525],{"className":251587},[31685,31821],[34,251589,31829],{"className":251590},[31828],[34,251592,251594],{"className":251593},[31801],[34,251595,251597],{"className":251596,"style":31836},[31805],[34,251598],{},[34,251600],{"className":251601},[31845,208951],[34,251603],{"className":251604,"style":208955},[31845],[34,251606,251608,251611,251614,251623,251626,251629,251669],{"className":251607},[31676],[34,251609],{"className":251610,"style":31774},[31680],[34,251612,456],{"className":251613},[31782],[34,251615,251617],{"className":251616},[31685],[34,251618,251620],{"className":251619},[31685],[34,251621,207924],{"className":251622},[31685,207923],[34,251624],{"className":251625,"style":208977},[31845],[34,251627,168915],{"className":251628},[31685],[34,251630,251632,251635],{"className":251631},[31685],[34,251633,459],{"className":251634},[31685],[34,251636,251638],{"className":251637},[31792],[34,251639,251641,251661],{"className":251640},[31796,31797],[34,251642,251644,251658],{"className":251643},[31801],[34,251645,251647],{"className":251646,"style":31806},[31805],[34,251648,251649,251652],{"style":31809},[34,251650],{"className":251651,"style":31814},[31813],[34,251653,251655],{"className":251654},[31818,31819,31820,31821],[34,251656,60525],{"className":251657},[31685,31821],[34,251659,31829],{"className":251660},[31828],[34,251662,251664],{"className":251663},[31801],[34,251665,251667],{"className":251666,"style":31836},[31805],[34,251668],{},[34,251670,2557],{"className":251671},[31690],"，只要取低位的數字，注意 ",[34,251674,251676,251712],{"className":251675},[31642],[34,251677,251679],{"className":251678},[31646],[31648,251680,251681],{"xmlns":31650},[31652,251682,251683,251709],{},[31655,251684,251685,251692,251694,251700,251702],{},[31743,251686,251687,251690],{},[31749,251688,251689],{},"3456",[31749,251691,60525],{},[31662,251693,202],{},[31743,251695,251696,251698],{},[31749,251697,251212],{},[31749,251699,60525],{},[31662,251701,223],{},[31743,251703,251704,251707],{},[31749,251705,251706],{},"1456",[31749,251708,60525],{},[31665,251710,251711],{"encoding":31667},"3456_8 = 2000_8 + 1456_8",[34,251713,251715,251773,251831],{"className":251714,"ariaHidden":758},[31672],[34,251716,251718,251721,251724,251764,251767,251770],{"className":251717},[31676],[34,251719],{"className":251720,"style":38357},[31680],[34,251722,95431],{"className":251723},[31685],[34,251725,251727,251730],{"className":251726},[31685],[34,251728,36656],{"className":251729},[31685],[34,251731,251733],{"className":251732},[31792],[34,251734,251736,251756],{"className":251735},[31796,31797],[34,251737,251739,251753],{"className":251738},[31801],[34,251740,251742],{"className":251741,"style":31806},[31805],[34,251743,251744,251747],{"style":31809},[34,251745],{"className":251746,"style":31814},[31813],[34,251748,251750],{"className":251749},[31818,31819,31820,31821],[34,251751,60525],{"className":251752},[31685,31821],[34,251754,31829],{"className":251755},[31828],[34,251757,251759],{"className":251758},[31801],[34,251760,251762],{"className":251761,"style":31836},[31805],[34,251763],{},[34,251765],{"className":251766,"style":31846},[31845],[34,251768,202],{"className":251769},[31850],[34,251771],{"className":251772,"style":31846},[31845],[34,251774,251776,251779,251782,251822,251825,251828],{"className":251775},[31676],[34,251777],{"className":251778,"style":38357},[31680],[34,251780,168915],{"className":251781},[31685],[34,251783,251785,251788],{"className":251784},[31685],[34,251786,459],{"className":251787},[31685],[34,251789,251791],{"className":251790},[31792],[34,251792,251794,251814],{"className":251793},[31796,31797],[34,251795,251797,251811],{"className":251796},[31801],[34,251798,251800],{"className":251799,"style":31806},[31805],[34,251801,251802,251805],{"style":31809},[34,251803],{"className":251804,"style":31814},[31813],[34,251806,251808],{"className":251807},[31818,31819,31820,31821],[34,251809,60525],{"className":251810},[31685,31821],[34,251812,31829],{"className":251813},[31828],[34,251815,251817],{"className":251816},[31801],[34,251818,251820],{"className":251819,"style":31836},[31805],[34,251821],{},[34,251823],{"className":251824,"style":32604},[31845],[34,251826,223],{"className":251827},[32024],[34,251829],{"className":251830,"style":32604},[31845],[34,251832,251834,251837,251841],{"className":251833},[31676],[34,251835],{"className":251836,"style":38357},[31680],[34,251838,251840],{"className":251839},[31685],"145",[34,251842,251844,251847],{"className":251843},[31685],[34,251845,36656],{"className":251846},[31685],[34,251848,251850],{"className":251849},[31792],[34,251851,251853,251873],{"className":251852},[31796,31797],[34,251854,251856,251870],{"className":251855},[31801],[34,251857,251859],{"className":251858,"style":31806},[31805],[34,251860,251861,251864],{"style":31809},[34,251862],{"className":251863,"style":31814},[31813],[34,251865,251867],{"className":251866},[31818,31819,31820,31821],[34,251868,60525],{"className":251869},[31685,31821],[34,251871,31829],{"className":251872},[31828],[34,251874,251876],{"className":251875},[31801],[34,251877,251879],{"className":251878,"style":31836},[31805],[34,251880],{},"。\n所以，偏移量 (Offset) = ",[34,251883,251885,251903],{"className":251884},[31642],[34,251886,251888],{"className":251887},[31646],[31648,251889,251890],{"xmlns":31650},[31652,251891,251892,251900],{},[31655,251893,251894],{},[31743,251895,251896,251898],{},[31749,251897,251706],{},[31749,251899,60525],{},[31665,251901,251902],{"encoding":31667},"1456_8",[34,251904,251906],{"className":251905,"ariaHidden":758},[31672],[34,251907,251909,251912,251915],{"className":251908},[31676],[34,251910],{"className":251911,"style":38357},[31680],[34,251913,251840],{"className":251914},[31685],[34,251916,251918,251921],{"className":251917},[31685],[34,251919,36656],{"className":251920},[31685],[34,251922,251924],{"className":251923},[31792],[34,251925,251927,251947],{"className":251926},[31796,31797],[34,251928,251930,251944],{"className":251929},[31801],[34,251931,251933],{"className":251932,"style":31806},[31805],[34,251934,251935,251938],{"style":31809},[34,251936],{"className":251937,"style":31814},[31813],[34,251939,251941],{"className":251940},[31818,31819,31820,31821],[34,251942,60525],{"className":251943},[31685,31821],[34,251945,31829],{"className":251946},[31828],[34,251948,251950],{"className":251949},[31801],[34,251951,251953],{"className":251952,"style":31836},[31805],[34,251954],{},[108,251956,251957,251960,251961,251989,251990,252063,252064,252271,252272,39984,252479,252481,252482,252521,252522,347,252595,252623,252624,252696],{},[2627,251958,251959],{},"計算實體位址 (Physical Address)：","\n實體位址 = (物理塊號 ",[34,251962,251964,251977],{"className":251963},[31642],[34,251965,251967],{"className":251966},[31646],[31648,251968,251969],{"xmlns":31650},[31652,251970,251971,251975],{},[31655,251972,251973],{},[31662,251974,42071],{},[31665,251976,211013],{"encoding":31667},[34,251978,251980],{"className":251979,"ariaHidden":758},[31672],[34,251981,251983,251986],{"className":251982},[31676],[34,251984],{"className":251985,"style":32597},[31680],[34,251987,42071],{"className":251988},[31685]," 頁面大小) + 偏移量。\n塊號為 ",[34,251991,251993,252011],{"className":251992},[31642],[34,251994,251996],{"className":251995},[31646],[31648,251997,251998],{"xmlns":31650},[31652,251999,252000,252008],{},[31655,252001,252002],{},[31743,252003,252004,252006],{},[31749,252005,168915],{},[31749,252007,60525],{},[31665,252009,252010],{"encoding":31667},"200_8",[34,252012,252014],{"className":252013,"ariaHidden":758},[31672],[34,252015,252017,252020,252023],{"className":252016},[31676],[34,252018],{"className":252019,"style":38357},[31680],[34,252021,3249],{"className":252022},[31685],[34,252024,252026,252029],{"className":252025},[31685],[34,252027,459],{"className":252028},[31685],[34,252030,252032],{"className":252031},[31792],[34,252033,252035,252055],{"className":252034},[31796,31797],[34,252036,252038,252052],{"className":252037},[31801],[34,252039,252041],{"className":252040,"style":31806},[31805],[34,252042,252043,252046],{"style":31809},[34,252044],{"className":252045,"style":31814},[31813],[34,252047,252049],{"className":252048},[31818,31819,31820,31821],[34,252050,60525],{"className":252051},[31685,31821],[34,252053,31829],{"className":252054},[31828],[34,252056,252058],{"className":252057},[31801],[34,252059,252061],{"className":252060,"style":31836},[31805],[34,252062],{},"。\n塊號的基礎位址 = ",[34,252065,252067,252102],{"className":252066},[31642],[34,252068,252070],{"className":252069},[31646],[31648,252071,252072],{"xmlns":31650},[31652,252073,252074,252099],{},[31655,252075,252076,252082,252084,252090,252092],{},[31743,252077,252078,252080],{},[31749,252079,168915],{},[31749,252081,60525],{},[31662,252083,42071],{},[31743,252085,252086,252088],{},[31749,252087,251212],{},[31749,252089,60525],{},[31662,252091,202],{},[31743,252093,252094,252097],{},[31749,252095,252096],{},"400000",[31749,252098,60525],{},[31665,252100,252101],{"encoding":31667},"200_8 \\times 2000_8 = 400000_8",[34,252103,252105,252163,252221],{"className":252104,"ariaHidden":758},[31672],[34,252106,252108,252111,252114,252154,252157,252160],{"className":252107},[31676],[34,252109],{"className":252110,"style":38357},[31680],[34,252112,3249],{"className":252113},[31685],[34,252115,252117,252120],{"className":252116},[31685],[34,252118,459],{"className":252119},[31685],[34,252121,252123],{"className":252122},[31792],[34,252124,252126,252146],{"className":252125},[31796,31797],[34,252127,252129,252143],{"className":252128},[31801],[34,252130,252132],{"className":252131,"style":31806},[31805],[34,252133,252134,252137],{"style":31809},[34,252135],{"className":252136,"style":31814},[31813],[34,252138,252140],{"className":252139},[31818,31819,31820,31821],[34,252141,60525],{"className":252142},[31685,31821],[34,252144,31829],{"className":252145},[31828],[34,252147,252149],{"className":252148},[31801],[34,252150,252152],{"className":252151,"style":31836},[31805],[34,252153],{},[34,252155],{"className":252156,"style":32604},[31845],[34,252158,42071],{"className":252159},[32024],[34,252161],{"className":252162,"style":32604},[31845],[34,252164,252166,252169,252172,252212,252215,252218],{"className":252165},[31676],[34,252167],{"className":252168,"style":38357},[31680],[34,252170,168915],{"className":252171},[31685],[34,252173,252175,252178],{"className":252174},[31685],[34,252176,459],{"className":252177},[31685],[34,252179,252181],{"className":252180},[31792],[34,252182,252184,252204],{"className":252183},[31796,31797],[34,252185,252187,252201],{"className":252186},[31801],[34,252188,252190],{"className":252189,"style":31806},[31805],[34,252191,252192,252195],{"style":31809},[34,252193],{"className":252194,"style":31814},[31813],[34,252196,252198],{"className":252197},[31818,31819,31820,31821],[34,252199,60525],{"className":252200},[31685,31821],[34,252202,31829],{"className":252203},[31828],[34,252205,252207],{"className":252206},[31801],[34,252208,252210],{"className":252209,"style":31836},[31805],[34,252211],{},[34,252213],{"className":252214,"style":31846},[31845],[34,252216,202],{"className":252217},[31850],[34,252219],{"className":252220,"style":31846},[31845],[34,252222,252224,252227,252231],{"className":252223},[31676],[34,252225],{"className":252226,"style":38357},[31680],[34,252228,252230],{"className":252229},[31685],"40000",[34,252232,252234,252237],{"className":252233},[31685],[34,252235,459],{"className":252236},[31685],[34,252238,252240],{"className":252239},[31792],[34,252241,252243,252263],{"className":252242},[31796,31797],[34,252244,252246,252260],{"className":252245},[31801],[34,252247,252249],{"className":252248,"style":31806},[31805],[34,252250,252251,252254],{"style":31809},[34,252252],{"className":252253,"style":31814},[31813],[34,252255,252257],{"className":252256},[31818,31819,31820,31821],[34,252258,60525],{"className":252259},[31685,31821],[34,252261,31829],{"className":252262},[31828],[34,252264,252266],{"className":252265},[31801],[34,252267,252269],{"className":252268,"style":31836},[31805],[34,252270],{},"。\n實體位址 = ",[34,252273,252275,252310],{"className":252274},[31642],[34,252276,252278],{"className":252277},[31646],[31648,252279,252280],{"xmlns":31650},[31652,252281,252282,252307],{},[31655,252283,252284,252290,252292,252298,252300],{},[31743,252285,252286,252288],{},[31749,252287,252096],{},[31749,252289,60525],{},[31662,252291,223],{},[31743,252293,252294,252296],{},[31749,252295,251706],{},[31749,252297,60525],{},[31662,252299,202],{},[31743,252301,252302,252305],{},[31749,252303,252304],{},"401456",[31749,252306,60525],{},[31665,252308,252309],{"encoding":31667},"400000_8 + 1456_8 = 401456_8",[34,252311,252313,252371,252429],{"className":252312,"ariaHidden":758},[31672],[34,252314,252316,252319,252322,252362,252365,252368],{"className":252315},[31676],[34,252317],{"className":252318,"style":38357},[31680],[34,252320,252230],{"className":252321},[31685],[34,252323,252325,252328],{"className":252324},[31685],[34,252326,459],{"className":252327},[31685],[34,252329,252331],{"className":252330},[31792],[34,252332,252334,252354],{"className":252333},[31796,31797],[34,252335,252337,252351],{"className":252336},[31801],[34,252338,252340],{"className":252339,"style":31806},[31805],[34,252341,252342,252345],{"style":31809},[34,252343],{"className":252344,"style":31814},[31813],[34,252346,252348],{"className":252347},[31818,31819,31820,31821],[34,252349,60525],{"className":252350},[31685,31821],[34,252352,31829],{"className":252353},[31828],[34,252355,252357],{"className":252356},[31801],[34,252358,252360],{"className":252359,"style":31836},[31805],[34,252361],{},[34,252363],{"className":252364,"style":32604},[31845],[34,252366,223],{"className":252367},[32024],[34,252369],{"className":252370,"style":32604},[31845],[34,252372,252374,252377,252380,252420,252423,252426],{"className":252373},[31676],[34,252375],{"className":252376,"style":38357},[31680],[34,252378,251840],{"className":252379},[31685],[34,252381,252383,252386],{"className":252382},[31685],[34,252384,36656],{"className":252385},[31685],[34,252387,252389],{"className":252388},[31792],[34,252390,252392,252412],{"className":252391},[31796,31797],[34,252393,252395,252409],{"className":252394},[31801],[34,252396,252398],{"className":252397,"style":31806},[31805],[34,252399,252400,252403],{"style":31809},[34,252401],{"className":252402,"style":31814},[31813],[34,252404,252406],{"className":252405},[31818,31819,31820,31821],[34,252407,60525],{"className":252408},[31685,31821],[34,252410,31829],{"className":252411},[31828],[34,252413,252415],{"className":252414},[31801],[34,252416,252418],{"className":252417,"style":31836},[31805],[34,252419],{},[34,252421],{"className":252422,"style":31846},[31845],[34,252424,202],{"className":252425},[31850],[34,252427],{"className":252428,"style":31846},[31845],[34,252430,252432,252435,252439],{"className":252431},[31676],[34,252433],{"className":252434,"style":38357},[31680],[34,252436,252438],{"className":252437},[31685],"40145",[34,252440,252442,252445],{"className":252441},[31685],[34,252443,36656],{"className":252444},[31685],[34,252446,252448],{"className":252447},[31792],[34,252449,252451,252471],{"className":252450},[31796,31797],[34,252452,252454,252468],{"className":252453},[31801],[34,252455,252457],{"className":252456,"style":31806},[31805],[34,252458,252459,252462],{"style":31809},[34,252460],{"className":252461,"style":31814},[31813],[34,252463,252465],{"className":252464},[31818,31819,31820,31821],[34,252466,60525],{"className":252467},[31685,31821],[34,252469,31829],{"className":252470},[31828],[34,252472,252474],{"className":252473},[31801],[34,252475,252477],{"className":252476,"style":31836},[31805],[34,252478],{},[2627,252480,249111],{}," 切忌將八進位的 ",[34,252483,252485,252503],{"className":252484},[31642],[34,252486,252488],{"className":252487},[31646],[31648,252489,252490],{"xmlns":31650},[31652,252491,252492,252500],{},[31655,252493,252494,252496,252498],{},[31749,252495,640],{},[31658,252497,38691],{},[31658,252499,54224],{},[31665,252501,252502],{"encoding":31667},"1KB",[34,252504,252506],{"className":252505,"ariaHidden":758},[31672],[34,252507,252509,252512,252515,252518],{"className":252508},[31676],[34,252510],{"className":252511,"style":33218},[31680],[34,252513,640],{"className":252514},[31685],[34,252516,38691],{"className":252517,"style":38706},[31685,31686],[34,252519,54224],{"className":252520,"style":54252},[31685,31686]," 直覺當成 ",[34,252523,252525,252543],{"className":252524},[31642],[34,252526,252528],{"className":252527},[31646],[31648,252529,252530],{"xmlns":31650},[31652,252531,252532,252540],{},[31655,252533,252534],{},[31743,252535,252536,252538],{},[31749,252537,5249],{},[31749,252539,60525],{},[31665,252541,252542],{"encoding":31667},"1000_8",[34,252544,252546],{"className":252545,"ariaHidden":758},[31672],[34,252547,252549,252552,252555],{"className":252548},[31676],[34,252550],{"className":252551,"style":38357},[31680],[34,252553,162173],{"className":252554},[31685],[34,252556,252558,252561],{"className":252557},[31685],[34,252559,459],{"className":252560},[31685],[34,252562,252564],{"className":252563},[31792],[34,252565,252567,252587],{"className":252566},[31796,31797],[34,252568,252570,252584],{"className":252569},[31801],[34,252571,252573],{"className":252572,"style":31806},[31805],[34,252574,252575,252578],{"style":31809},[34,252576],{"className":252577,"style":31814},[31813],[34,252579,252581],{"className":252580},[31818,31819,31820,31821],[34,252582,60525],{"className":252583},[31685,31821],[34,252585,31829],{"className":252586},[31828],[34,252588,252590],{"className":252589},[31801],[34,252591,252593],{"className":252592,"style":31836},[31805],[34,252594],{},[34,252596,252598,252611],{"className":252597},[31642],[34,252599,252601],{"className":252600},[31646],[31648,252602,252603],{"xmlns":31650},[31652,252604,252605,252609],{},[31655,252606,252607],{},[31749,252608,177393],{},[31665,252610,177393],{"encoding":31667},[34,252612,252614],{"className":252613,"ariaHidden":758},[31672],[34,252615,252617,252620],{"className":252616},[31676],[34,252618],{"className":252619,"style":32617},[31680],[34,252621,177393],{"className":252622},[31685]," 轉換為八進位是 ",[34,252625,252627,252644],{"className":252626},[31642],[34,252628,252630],{"className":252629},[31646],[31648,252631,252632],{"xmlns":31650},[31652,252633,252634,252642],{},[31655,252635,252636],{},[31743,252637,252638,252640],{},[31749,252639,251212],{},[31749,252641,60525],{},[31665,252643,251217],{"encoding":31667},[34,252645,252647],{"className":252646,"ariaHidden":758},[31672],[34,252648,252650,252653,252656],{"className":252649},[31676],[34,252651],{"className":252652,"style":38357},[31680],[34,252654,168915],{"className":252655},[31685],[34,252657,252659,252662],{"className":252658},[31685],[34,252660,459],{"className":252661},[31685],[34,252663,252665],{"className":252664},[31792],[34,252666,252668,252688],{"className":252667},[31796,31797],[34,252669,252671,252685],{"className":252670},[31801],[34,252672,252674],{"className":252673,"style":31806},[31805],[34,252675,252676,252679],{"style":31809},[34,252677],{"className":252678,"style":31814},[31813],[34,252680,252682],{"className":252681},[31818,31819,31820,31821],[34,252683,60525],{"className":252684},[31685,31821],[34,252686,31829],{"className":252687},[31828],[34,252689,252691],{"className":252690},[31801],[34,252692,252694],{"className":252693,"style":31836},[31805],[34,252695],{},"。這一步是解題的關鍵。",[19,252698,252700],{"id":252699},"_27计算机系统中屏幕显示分辨率为640480若要存储一屏256彩色的图像需要-kb存储空间","27．计算机系统中，屏幕显示分辨率为640×480，若要存储一屏256彩色的图像，需要( )KB存储空间。",[15,252702,252703,252705,252706,252708,252709,252711,252712,252792,252793,252796,252797,252910,252911,39984,252975,252977],{},[2627,252704,58173],{}," 300\n",[2627,252707,248267],{}," 影像記憶體容量計算。\n",[2627,252710,246526],{},"\n256 彩色表示每個像素需要 8 bits（即 ",[34,252713,252715,252736],{"className":252714},[31642],[34,252716,252718],{"className":252717},[31646],[31648,252719,252720],{"xmlns":31650},[31652,252721,252722,252734],{},[31655,252723,252724,252730,252732],{},[37624,252725,252726,252728],{},[31749,252727,768],{},[31749,252729,60525],{},[31662,252731,202],{},[31749,252733,95418],{},[31665,252735,210362],{"encoding":31667},[34,252737,252739,252783],{"className":252738,"ariaHidden":758},[31672],[34,252740,252742,252745,252774,252777,252780],{"className":252741},[31676],[34,252743],{"className":252744,"style":37774},[31680],[34,252746,252748,252751],{"className":252747},[31685],[34,252749,768],{"className":252750},[31685],[34,252752,252754],{"className":252753},[31792],[34,252755,252757],{"className":252756},[31796],[34,252758,252760],{"className":252759},[31801],[34,252761,252763],{"className":252762,"style":37774},[31805],[34,252764,252765,252768],{"style":37777},[34,252766],{"className":252767,"style":31814},[31813],[34,252769,252771],{"className":252770},[31818,31819,31820,31821],[34,252772,60525],{"className":252773},[31685,31821],[34,252775],{"className":252776,"style":31846},[31845],[34,252778,202],{"className":252779},[31850],[34,252781],{"className":252782,"style":31846},[31845],[34,252784,252786,252789],{"className":252785},[31676],[34,252787],{"className":252788,"style":32617},[31680],[34,252790,95418],{"className":252791},[31685],"），也就是 ",[2627,252794,252795],{},"1 Byte"," 的存儲空間。\n總位元組數 = ",[34,252798,252800,252832],{"className":252799},[31642],[34,252801,252803],{"className":252802},[31646],[31648,252804,252805],{"xmlns":31650},[31652,252806,252807,252829],{},[31655,252808,252809,252811,252813,252815,252817,252819,252822,252824,252826],{},[31749,252810,176999],{},[31662,252812,42071],{},[31749,252814,173407],{},[31662,252816,42071],{},[31749,252818,640],{},[39361,252820,252821],{}," Byte",[31662,252823,202],{},[31749,252825,177064],{},[39361,252827,252828],{}," Bytes",[31665,252830,252831],{"encoding":31667},"640 \\times 480 \\times 1 \\text{ Byte} = 307200 \\text{ Bytes}",[34,252833,252835,252853,252871,252895],{"className":252834,"ariaHidden":758},[31672],[34,252836,252838,252841,252844,252847,252850],{"className":252837},[31676],[34,252839],{"className":252840,"style":35942},[31680],[34,252842,176999],{"className":252843},[31685],[34,252845],{"className":252846,"style":32604},[31845],[34,252848,42071],{"className":252849},[32024],[34,252851],{"className":252852,"style":32604},[31845],[34,252854,252856,252859,252862,252865,252868],{"className":252855},[31676],[34,252857],{"className":252858,"style":35942},[31680],[34,252860,173407],{"className":252861},[31685],[34,252863],{"className":252864,"style":32604},[31845],[34,252866,42071],{"className":252867},[32024],[34,252869],{"className":252870,"style":32604},[31845],[34,252872,252874,252877,252880,252886,252889,252892],{"className":252873},[31676],[34,252875],{"className":252876,"style":70739},[31680],[34,252878,640],{"className":252879},[31685],[34,252881,252883],{"className":252882},[31685,2638],[34,252884,252821],{"className":252885},[31685],[34,252887],{"className":252888,"style":31846},[31845],[34,252890,202],{"className":252891},[31850],[34,252893],{"className":252894,"style":31846},[31845],[34,252896,252898,252901,252904],{"className":252897},[31676],[34,252899],{"className":252900,"style":70739},[31680],[34,252902,177064],{"className":252903},[31685],[34,252905,252907],{"className":252906},[31685,2638],[34,252908,252828],{"className":252909},[31685],"。\n換算為 KB = ",[34,252912,252914,252939],{"className":252913},[31642],[34,252915,252917],{"className":252916},[31646],[31648,252918,252919],{"xmlns":31650},[31652,252920,252921,252936],{},[31655,252922,252923,252925,252927,252929,252931,252933],{},[31749,252924,177064],{},[31658,252926,38080],{"mathvariant":38079},[31749,252928,177393],{},[31662,252930,202],{},[31749,252932,25357],{},[39361,252934,252935],{}," KB",[31665,252937,252938],{"encoding":31667},"307200 \u002F 1024 = 300 \\text{ KB}",[34,252940,252942,252960],{"className":252941,"ariaHidden":758},[31672],[34,252943,252945,252948,252951,252954,252957],{"className":252944},[31676],[34,252946],{"className":252947,"style":31774},[31680],[34,252949,177483],{"className":252950},[31685],[34,252952],{"className":252953,"style":31846},[31845],[34,252955,202],{"className":252956},[31850],[34,252958],{"className":252959,"style":31846},[31845],[34,252961,252963,252966,252969],{"className":252962},[31676],[34,252964],{"className":252965,"style":33218},[31680],[34,252967,25357],{"className":252968},[31685],[34,252970,252972],{"className":252971},[31685,2638],[34,252973,252935],{"className":252974},[31685],[2627,252976,249111],{}," 要除以 1024，而不是 1000。",[19,252979,252981],{"id":252980},"_28信号量s初值10则在s上调用16-次p操作和15-次v操作后s的值应该为","28．信号量S初值10，则在S上调用16 次P操作和15 次V操作后，S的值应该为( )。",[15,252983,252984,252986,252987,252989,252990,252992,252993,252995],{},[2627,252985,58173],{}," 9\n",[2627,252988,248267],{}," 號誌值的累加（與第 21 題類似）。\n",[2627,252991,246526],{}," S = 初始值 - P操作次數 + V操作次數 = 10 - 16 + 15 = 9。\n",[2627,252994,249111],{}," 這是一道送分題，細心加減即可。",[19,252997,252999],{"id":252998},"_29系统提供24-位虚存空间主存为218b分页式虚拟存储管理页面尺寸为2kb用户程序虚拟地址11124457八进制页面分得块号为100八进制物理地址","29．系统提供24 位虚存空间，主存为2^18B，分页式虚拟存储管理，页面尺寸为2KB。用户程序虚拟地址11124457(八进制)，页面分得块号为100(八进制)，物理地址( )。",[15,253001,253002,253004,253005,253007,253008,253010,253011,251118,253073,251195,253149,1840,253223,214569],{},[2627,253003,58173],{}," 400457 (八进制)\n",[2627,253006,248267],{}," 虛擬位址到實體位址的轉換（同第 26 題）。\n",[2627,253009,246526],{},"\n頁面大小為 ",[34,253012,253014,253037],{"className":253013},[31642],[34,253015,253017],{"className":253016},[31646],[31648,253018,253019],{"xmlns":31650},[31652,253020,253021,253034],{},[31655,253022,253023,253025,253027,253029,253031],{},[31749,253024,768],{},[31658,253026,38691],{},[31658,253028,54224],{},[31662,253030,202],{},[31749,253032,253033],{},"2048",[31665,253035,253036],{"encoding":31667},"2KB = 2048",[34,253038,253040,253064],{"className":253039,"ariaHidden":758},[31672],[34,253041,253043,253046,253049,253052,253055,253058,253061],{"className":253042},[31676],[34,253044],{"className":253045,"style":33218},[31680],[34,253047,768],{"className":253048},[31685],[34,253050,38691],{"className":253051,"style":38706},[31685,31686],[34,253053,54224],{"className":253054,"style":54252},[31685,31686],[34,253056],{"className":253057,"style":31846},[31845],[34,253059,202],{"className":253060},[31850],[34,253062],{"className":253063,"style":31846},[31845],[34,253065,253067,253070],{"className":253066},[31676],[34,253068],{"className":253069,"style":32617},[31680],[34,253071,253033],{"className":253072},[31685],[34,253074,253076,253094],{"className":253075},[31642],[34,253077,253079],{"className":253078},[31646],[31648,253080,253081],{"xmlns":31650},[31652,253082,253083,253091],{},[31655,253084,253085],{},[31743,253086,253087,253089],{},[31749,253088,253033],{},[31749,253090,3244],{},[31665,253092,253093],{"encoding":31667},"2048_{10}",[34,253095,253097],{"className":253096,"ariaHidden":758},[31672],[34,253098,253100,253103,253106],{"className":253099},[31676],[34,253101],{"className":253102,"style":38357},[31680],[34,253104,95404],{"className":253105},[31685],[34,253107,253109,253112],{"className":253108},[31685],[34,253110,60525],{"className":253111},[31685],[34,253113,253115],{"className":253114},[31792],[34,253116,253118,253141],{"className":253117},[31796,31797],[34,253119,253121,253138],{"className":253120},[31801],[34,253122,253124],{"className":253123,"style":31806},[31805],[34,253125,253126,253129],{"style":31809},[34,253127],{"className":253128,"style":31814},[31813],[34,253130,253132],{"className":253131},[31818,31819,31820,31821],[34,253133,253135],{"className":253134},[31685,31821],[34,253136,3244],{"className":253137},[31685,31821],[34,253139,31829],{"className":253140},[31828],[34,253142,253144],{"className":253143},[31801],[34,253145,253147],{"className":253146,"style":31836},[31805],[34,253148],{},[34,253150,253152,253171],{"className":253151},[31642],[34,253153,253155],{"className":253154},[31646],[31648,253156,253157],{"xmlns":31650},[31652,253158,253159,253168],{},[31655,253160,253161],{},[31743,253162,253163,253166],{},[31749,253164,253165],{},"4000",[31749,253167,60525],{},[31665,253169,253170],{"encoding":31667},"4000_8",[34,253172,253174],{"className":253173,"ariaHidden":758},[31672],[34,253175,253177,253180,253183],{"className":253176},[31676],[34,253178],{"className":253179,"style":38357},[31680],[34,253181,185297],{"className":253182},[31685],[34,253184,253186,253189],{"className":253185},[31685],[34,253187,459],{"className":253188},[31685],[34,253190,253192],{"className":253191},[31792],[34,253193,253195,253215],{"className":253194},[31796,31797],[34,253196,253198,253212],{"className":253197},[31801],[34,253199,253201],{"className":253200,"style":31806},[31805],[34,253202,253203,253206],{"style":31809},[34,253204],{"className":253205,"style":31814},[31813],[34,253207,253209],{"className":253208},[31818,31819,31820,31821],[34,253210,60525],{"className":253211},[31685,31821],[34,253213,31829],{"className":253214},[31828],[34,253216,253218],{"className":253217},[31801],[34,253219,253221],{"className":253220,"style":31836},[31805],[34,253222],{},[34,253224,253226,253260],{"className":253225},[31642],[34,253227,253229],{"className":253228},[31646],[31648,253230,253231],{"xmlns":31650},[31652,253232,253233,253257],{},[31655,253234,253235,253237,253239,253245,253247,253249,253251,253253,253255],{},[31749,253236,15932],{},[31662,253238,42071],{},[37624,253240,253241,253243],{},[31749,253242,60525],{},[31749,253244,786],{},[31662,253246,202],{},[31749,253248,15932],{},[31662,253250,42071],{},[31749,253252,167308],{},[31662,253254,202],{},[31749,253256,253033],{},[31665,253258,253259],{"encoding":31667},"4 \\times 8^3 = 4 \\times 512 = 2048",[34,253261,253263,253281,253325,253343,253361],{"className":253262,"ariaHidden":758},[31672],[34,253264,253266,253269,253272,253275,253278],{"className":253265},[31676],[34,253267],{"className":253268,"style":35942},[31680],[34,253270,15932],{"className":253271},[31685],[34,253273],{"className":253274,"style":32604},[31845],[34,253276,42071],{"className":253277},[32024],[34,253279],{"className":253280,"style":32604},[31845],[34,253282,253284,253287,253316,253319,253322],{"className":253283},[31676],[34,253285],{"className":253286,"style":37774},[31680],[34,253288,253290,253293],{"className":253289},[31685],[34,253291,60525],{"className":253292},[31685],[34,253294,253296],{"className":253295},[31792],[34,253297,253299],{"className":253298},[31796],[34,253300,253302],{"className":253301},[31801],[34,253303,253305],{"className":253304,"style":37774},[31805],[34,253306,253307,253310],{"style":37777},[34,253308],{"className":253309,"style":31814},[31813],[34,253311,253313],{"className":253312},[31818,31819,31820,31821],[34,253314,786],{"className":253315},[31685,31821],[34,253317],{"className":253318,"style":31846},[31845],[34,253320,202],{"className":253321},[31850],[34,253323],{"className":253324,"style":31846},[31845],[34,253326,253328,253331,253334,253337,253340],{"className":253327},[31676],[34,253329],{"className":253330,"style":35942},[31680],[34,253332,15932],{"className":253333},[31685],[34,253335],{"className":253336,"style":32604},[31845],[34,253338,42071],{"className":253339},[32024],[34,253341],{"className":253342,"style":32604},[31845],[34,253344,253346,253349,253352,253355,253358],{"className":253345},[31676],[34,253347],{"className":253348,"style":32617},[31680],[34,253350,167308],{"className":253351},[31685],[34,253353],{"className":253354,"style":31846},[31845],[34,253356,202],{"className":253357},[31850],[34,253359],{"className":253360,"style":31846},[31845],[34,253362,253364,253367],{"className":253363},[31676],[34,253365],{"className":253366,"style":32617},[31680],[34,253368,253033],{"className":253369},[31685],[2230,253371,253372,253907],{},[108,253373,253374,47649,253377,253549,253550,253625,253626,253833,253834,347],{},[2627,253375,253376],{},"偏移量 (Offset)：",[34,253378,253380,253423],{"className":253379},[31642],[34,253381,253383],{"className":253382},[31646],[31648,253384,253385],{"xmlns":31650},[31652,253386,253387,253420],{},[31655,253388,253389,253396,253398,253400,253402,253410,253412,253418],{},[31743,253390,253391,253394],{},[31749,253392,253393],{},"11124457",[31749,253395,60525],{},[31845,253397],{},[31845,253399],{"width":208910},[31662,253401,456],{"stretchy":702},[31655,253403,253404,253406,253408],{},[31658,253405,31759],{"mathvariant":38079},[31658,253407,43383],{"mathvariant":38079},[31658,253409,46900],{"mathvariant":38079},[31845,253411],{"width":208923},[31743,253413,253414,253416],{},[31749,253415,253165],{},[31749,253417,60525],{},[31662,253419,2557],{"stretchy":702},[31665,253421,253422],{"encoding":31667},"11124457_8 \\pmod{4000_8}",[34,253424,253426,253482],{"className":253425,"ariaHidden":758},[31672],[34,253427,253429,253432,253436,253476,253479],{"className":253428},[31676],[34,253430],{"className":253431,"style":38357},[31680],[34,253433,253435],{"className":253434},[31685],"1112445",[34,253437,253439,253442],{"className":253438},[31685],[34,253440,36723],{"className":253441},[31685],[34,253443,253445],{"className":253444},[31792],[34,253446,253448,253468],{"className":253447},[31796,31797],[34,253449,253451,253465],{"className":253450},[31801],[34,253452,253454],{"className":253453,"style":31806},[31805],[34,253455,253456,253459],{"style":31809},[34,253457],{"className":253458,"style":31814},[31813],[34,253460,253462],{"className":253461},[31818,31819,31820,31821],[34,253463,60525],{"className":253464},[31685,31821],[34,253466,31829],{"className":253467},[31828],[34,253469,253471],{"className":253470},[31801],[34,253472,253474],{"className":253473,"style":31836},[31805],[34,253475],{},[34,253477],{"className":253478},[31845,208951],[34,253480],{"className":253481,"style":208955},[31845],[34,253483,253485,253488,253491,253500,253503,253506,253546],{"className":253484},[31676],[34,253486],{"className":253487,"style":31774},[31680],[34,253489,456],{"className":253490},[31782],[34,253492,253494],{"className":253493},[31685],[34,253495,253497],{"className":253496},[31685],[34,253498,207924],{"className":253499},[31685,207923],[34,253501],{"className":253502,"style":208977},[31845],[34,253504,185297],{"className":253505},[31685],[34,253507,253509,253512],{"className":253508},[31685],[34,253510,459],{"className":253511},[31685],[34,253513,253515],{"className":253514},[31792],[34,253516,253518,253538],{"className":253517},[31796,31797],[34,253519,253521,253535],{"className":253520},[31801],[34,253522,253524],{"className":253523,"style":31806},[31805],[34,253525,253526,253529],{"style":31809},[34,253527],{"className":253528,"style":31814},[31813],[34,253530,253532],{"className":253531},[31818,31819,31820,31821],[34,253533,60525],{"className":253534},[31685,31821],[34,253536,31829],{"className":253537},[31828],[34,253539,253541],{"className":253540},[31801],[34,253542,253544],{"className":253543,"style":31836},[31805],[34,253545],{},[34,253547,2557],{"className":253548},[31690],"。\n只看末四碼 ",[34,253551,253553,253572],{"className":253552},[31642],[34,253554,253556],{"className":253555},[31646],[31648,253557,253558],{"xmlns":31650},[31652,253559,253560,253569],{},[31655,253561,253562],{},[31743,253563,253564,253567],{},[31749,253565,253566],{},"4457",[31749,253568,60525],{},[31665,253570,253571],{"encoding":31667},"4457_8",[34,253573,253575],{"className":253574,"ariaHidden":758},[31672],[34,253576,253578,253581,253585],{"className":253577},[31676],[34,253579],{"className":253580,"style":38357},[31680],[34,253582,253584],{"className":253583},[31685],"445",[34,253586,253588,253591],{"className":253587},[31685],[34,253589,36723],{"className":253590},[31685],[34,253592,253594],{"className":253593},[31792],[34,253595,253597,253617],{"className":253596},[31796,31797],[34,253598,253600,253614],{"className":253599},[31801],[34,253601,253603],{"className":253602,"style":31806},[31805],[34,253604,253605,253608],{"style":31809},[34,253606],{"className":253607,"style":31814},[31813],[34,253609,253611],{"className":253610},[31818,31819,31820,31821],[34,253612,60525],{"className":253613},[31685,31821],[34,253615,31829],{"className":253616},[31828],[34,253618,253620],{"className":253619},[31801],[34,253621,253623],{"className":253622,"style":31836},[31805],[34,253624],{},"。因為 ",[34,253627,253629,253664],{"className":253628},[31642],[34,253630,253632],{"className":253631},[31646],[31648,253633,253634],{"xmlns":31650},[31652,253635,253636,253661],{},[31655,253637,253638,253644,253646,253652,253654],{},[31743,253639,253640,253642],{},[31749,253641,253566],{},[31749,253643,60525],{},[31662,253645,202],{},[31743,253647,253648,253650],{},[31749,253649,253165],{},[31749,253651,60525],{},[31662,253653,223],{},[31743,253655,253656,253659],{},[31749,253657,253658],{},"0457",[31749,253660,60525],{},[31665,253662,253663],{"encoding":31667},"4457_8 = 4000_8 + 0457_8",[34,253665,253667,253725,253783],{"className":253666,"ariaHidden":758},[31672],[34,253668,253670,253673,253676,253716,253719,253722],{"className":253669},[31676],[34,253671],{"className":253672,"style":38357},[31680],[34,253674,253584],{"className":253675},[31685],[34,253677,253679,253682],{"className":253678},[31685],[34,253680,36723],{"className":253681},[31685],[34,253683,253685],{"className":253684},[31792],[34,253686,253688,253708],{"className":253687},[31796,31797],[34,253689,253691,253705],{"className":253690},[31801],[34,253692,253694],{"className":253693,"style":31806},[31805],[34,253695,253696,253699],{"style":31809},[34,253697],{"className":253698,"style":31814},[31813],[34,253700,253702],{"className":253701},[31818,31819,31820,31821],[34,253703,60525],{"className":253704},[31685,31821],[34,253706,31829],{"className":253707},[31828],[34,253709,253711],{"className":253710},[31801],[34,253712,253714],{"className":253713,"style":31836},[31805],[34,253715],{},[34,253717],{"className":253718,"style":31846},[31845],[34,253720,202],{"className":253721},[31850],[34,253723],{"className":253724,"style":31846},[31845],[34,253726,253728,253731,253734,253774,253777,253780],{"className":253727},[31676],[34,253729],{"className":253730,"style":38357},[31680],[34,253732,185297],{"className":253733},[31685],[34,253735,253737,253740],{"className":253736},[31685],[34,253738,459],{"className":253739},[31685],[34,253741,253743],{"className":253742},[31792],[34,253744,253746,253766],{"className":253745},[31796,31797],[34,253747,253749,253763],{"className":253748},[31801],[34,253750,253752],{"className":253751,"style":31806},[31805],[34,253753,253754,253757],{"style":31809},[34,253755],{"className":253756,"style":31814},[31813],[34,253758,253760],{"className":253759},[31818,31819,31820,31821],[34,253761,60525],{"className":253762},[31685,31821],[34,253764,31829],{"className":253765},[31828],[34,253767,253769],{"className":253768},[31801],[34,253770,253772],{"className":253771,"style":31836},[31805],[34,253773],{},[34,253775],{"className":253776,"style":32604},[31845],[34,253778,223],{"className":253779},[32024],[34,253781],{"className":253782,"style":32604},[31845],[34,253784,253786,253789,253793],{"className":253785},[31676],[34,253787],{"className":253788,"style":38357},[31680],[34,253790,253792],{"className":253791},[31685],"045",[34,253794,253796,253799],{"className":253795},[31685],[34,253797,36723],{"className":253798},[31685],[34,253800,253802],{"className":253801},[31792],[34,253803,253805,253825],{"className":253804},[31796,31797],[34,253806,253808,253822],{"className":253807},[31801],[34,253809,253811],{"className":253810,"style":31806},[31805],[34,253812,253813,253816],{"style":31809},[34,253814],{"className":253815,"style":31814},[31813],[34,253817,253819],{"className":253818},[31818,31819,31820,31821],[34,253820,60525],{"className":253821},[31685,31821],[34,253823,31829],{"className":253824},[31828],[34,253826,253828],{"className":253827},[31801],[34,253829,253831],{"className":253830,"style":31836},[31805],[34,253832],{},"。\n所以偏移量 = ",[34,253835,253837,253855],{"className":253836},[31642],[34,253838,253840],{"className":253839},[31646],[31648,253841,253842],{"xmlns":31650},[31652,253843,253844,253852],{},[31655,253845,253846],{},[31743,253847,253848,253850],{},[31749,253849,253658],{},[31749,253851,60525],{},[31665,253853,253854],{"encoding":31667},"0457_8",[34,253856,253858],{"className":253857,"ariaHidden":758},[31672],[34,253859,253861,253864,253867],{"className":253860},[31676],[34,253862],{"className":253863,"style":38357},[31680],[34,253865,253792],{"className":253866},[31685],[34,253868,253870,253873],{"className":253869},[31685],[34,253871,36723],{"className":253872},[31685],[34,253874,253876],{"className":253875},[31792],[34,253877,253879,253899],{"className":253878},[31796,31797],[34,253880,253882,253896],{"className":253881},[31801],[34,253883,253885],{"className":253884,"style":31806},[31805],[34,253886,253887,253890],{"style":31809},[34,253888],{"className":253889,"style":31814},[31813],[34,253891,253893],{"className":253892},[31818,31819,31820,31821],[34,253894,60525],{"className":253895},[31685,31821],[34,253897,31829],{"className":253898},[31828],[34,253900,253902],{"className":253901},[31801],[34,253903,253905],{"className":253904,"style":31836},[31805],[34,253906],{},[108,253908,253909,253912,253913,253941,253942,254015,254016,252271,254221,39984,254428,254430],{},[2627,253910,253911],{},"實體位址："," (物理塊號 ",[34,253914,253916,253929],{"className":253915},[31642],[34,253917,253919],{"className":253918},[31646],[31648,253920,253921],{"xmlns":31650},[31652,253922,253923,253927],{},[31655,253924,253925],{},[31662,253926,42071],{},[31665,253928,211013],{"encoding":31667},[34,253930,253932],{"className":253931,"ariaHidden":758},[31672],[34,253933,253935,253938],{"className":253934},[31676],[34,253936],{"className":253937,"style":32597},[31680],[34,253939,42071],{"className":253940},[31685]," 頁面大小) + 偏移量。\n塊號 = ",[34,253943,253945,253963],{"className":253944},[31642],[34,253946,253948],{"className":253947},[31646],[31648,253949,253950],{"xmlns":31650},[31652,253951,253952,253960],{},[31655,253953,253954],{},[31743,253955,253956,253958],{},[31749,253957,162173],{},[31749,253959,60525],{},[31665,253961,253962],{"encoding":31667},"100_8",[34,253964,253966],{"className":253965,"ariaHidden":758},[31672],[34,253967,253969,253972,253975],{"className":253968},[31676],[34,253970],{"className":253971,"style":38357},[31680],[34,253973,3244],{"className":253974},[31685],[34,253976,253978,253981],{"className":253977},[31685],[34,253979,459],{"className":253980},[31685],[34,253982,253984],{"className":253983},[31792],[34,253985,253987,254007],{"className":253986},[31796,31797],[34,253988,253990,254004],{"className":253989},[31801],[34,253991,253993],{"className":253992,"style":31806},[31805],[34,253994,253995,253998],{"style":31809},[34,253996],{"className":253997,"style":31814},[31813],[34,253999,254001],{"className":254000},[31818,31819,31820,31821],[34,254002,60525],{"className":254003},[31685,31821],[34,254005,31829],{"className":254006},[31828],[34,254008,254010],{"className":254009},[31801],[34,254011,254013],{"className":254012,"style":31836},[31805],[34,254014],{},"。\n塊號基礎位址 = ",[34,254017,254019,254053],{"className":254018},[31642],[34,254020,254022],{"className":254021},[31646],[31648,254023,254024],{"xmlns":31650},[31652,254025,254026,254050],{},[31655,254027,254028,254034,254036,254042,254044],{},[31743,254029,254030,254032],{},[31749,254031,162173],{},[31749,254033,60525],{},[31662,254035,42071],{},[31743,254037,254038,254040],{},[31749,254039,253165],{},[31749,254041,60525],{},[31662,254043,202],{},[31743,254045,254046,254048],{},[31749,254047,252096],{},[31749,254049,60525],{},[31665,254051,254052],{"encoding":31667},"100_8 \\times 4000_8 = 400000_8",[34,254054,254056,254114,254172],{"className":254055,"ariaHidden":758},[31672],[34,254057,254059,254062,254065,254105,254108,254111],{"className":254058},[31676],[34,254060],{"className":254061,"style":38357},[31680],[34,254063,3244],{"className":254064},[31685],[34,254066,254068,254071],{"className":254067},[31685],[34,254069,459],{"className":254070},[31685],[34,254072,254074],{"className":254073},[31792],[34,254075,254077,254097],{"className":254076},[31796,31797],[34,254078,254080,254094],{"className":254079},[31801],[34,254081,254083],{"className":254082,"style":31806},[31805],[34,254084,254085,254088],{"style":31809},[34,254086],{"className":254087,"style":31814},[31813],[34,254089,254091],{"className":254090},[31818,31819,31820,31821],[34,254092,60525],{"className":254093},[31685,31821],[34,254095,31829],{"className":254096},[31828],[34,254098,254100],{"className":254099},[31801],[34,254101,254103],{"className":254102,"style":31836},[31805],[34,254104],{},[34,254106],{"className":254107,"style":32604},[31845],[34,254109,42071],{"className":254110},[32024],[34,254112],{"className":254113,"style":32604},[31845],[34,254115,254117,254120,254123,254163,254166,254169],{"className":254116},[31676],[34,254118],{"className":254119,"style":38357},[31680],[34,254121,185297],{"className":254122},[31685],[34,254124,254126,254129],{"className":254125},[31685],[34,254127,459],{"className":254128},[31685],[34,254130,254132],{"className":254131},[31792],[34,254133,254135,254155],{"className":254134},[31796,31797],[34,254136,254138,254152],{"className":254137},[31801],[34,254139,254141],{"className":254140,"style":31806},[31805],[34,254142,254143,254146],{"style":31809},[34,254144],{"className":254145,"style":31814},[31813],[34,254147,254149],{"className":254148},[31818,31819,31820,31821],[34,254150,60525],{"className":254151},[31685,31821],[34,254153,31829],{"className":254154},[31828],[34,254156,254158],{"className":254157},[31801],[34,254159,254161],{"className":254160,"style":31836},[31805],[34,254162],{},[34,254164],{"className":254165,"style":31846},[31845],[34,254167,202],{"className":254168},[31850],[34,254170],{"className":254171,"style":31846},[31845],[34,254173,254175,254178,254181],{"className":254174},[31676],[34,254176],{"className":254177,"style":38357},[31680],[34,254179,252230],{"className":254180},[31685],[34,254182,254184,254187],{"className":254183},[31685],[34,254185,459],{"className":254186},[31685],[34,254188,254190],{"className":254189},[31792],[34,254191,254193,254213],{"className":254192},[31796,31797],[34,254194,254196,254210],{"className":254195},[31801],[34,254197,254199],{"className":254198,"style":31806},[31805],[34,254200,254201,254204],{"style":31809},[34,254202],{"className":254203,"style":31814},[31813],[34,254205,254207],{"className":254206},[31818,31819,31820,31821],[34,254208,60525],{"className":254209},[31685,31821],[34,254211,31829],{"className":254212},[31828],[34,254214,254216],{"className":254215},[31801],[34,254217,254219],{"className":254218,"style":31836},[31805],[34,254220],{},[34,254222,254224,254259],{"className":254223},[31642],[34,254225,254227],{"className":254226},[31646],[31648,254228,254229],{"xmlns":31650},[31652,254230,254231,254256],{},[31655,254232,254233,254239,254241,254247,254249],{},[31743,254234,254235,254237],{},[31749,254236,252096],{},[31749,254238,60525],{},[31662,254240,223],{},[31743,254242,254243,254245],{},[31749,254244,253658],{},[31749,254246,60525],{},[31662,254248,202],{},[31743,254250,254251,254254],{},[31749,254252,254253],{},"400457",[31749,254255,60525],{},[31665,254257,254258],{"encoding":31667},"400000_8 + 0457_8 = 400457_8",[34,254260,254262,254320,254378],{"className":254261,"ariaHidden":758},[31672],[34,254263,254265,254268,254271,254311,254314,254317],{"className":254264},[31676],[34,254266],{"className":254267,"style":38357},[31680],[34,254269,252230],{"className":254270},[31685],[34,254272,254274,254277],{"className":254273},[31685],[34,254275,459],{"className":254276},[31685],[34,254278,254280],{"className":254279},[31792],[34,254281,254283,254303],{"className":254282},[31796,31797],[34,254284,254286,254300],{"className":254285},[31801],[34,254287,254289],{"className":254288,"style":31806},[31805],[34,254290,254291,254294],{"style":31809},[34,254292],{"className":254293,"style":31814},[31813],[34,254295,254297],{"className":254296},[31818,31819,31820,31821],[34,254298,60525],{"className":254299},[31685,31821],[34,254301,31829],{"className":254302},[31828],[34,254304,254306],{"className":254305},[31801],[34,254307,254309],{"className":254308,"style":31836},[31805],[34,254310],{},[34,254312],{"className":254313,"style":32604},[31845],[34,254315,223],{"className":254316},[32024],[34,254318],{"className":254319,"style":32604},[31845],[34,254321,254323,254326,254329,254369,254372,254375],{"className":254322},[31676],[34,254324],{"className":254325,"style":38357},[31680],[34,254327,253792],{"className":254328},[31685],[34,254330,254332,254335],{"className":254331},[31685],[34,254333,36723],{"className":254334},[31685],[34,254336,254338],{"className":254337},[31792],[34,254339,254341,254361],{"className":254340},[31796,31797],[34,254342,254344,254358],{"className":254343},[31801],[34,254345,254347],{"className":254346,"style":31806},[31805],[34,254348,254349,254352],{"style":31809},[34,254350],{"className":254351,"style":31814},[31813],[34,254353,254355],{"className":254354},[31818,31819,31820,31821],[34,254356,60525],{"className":254357},[31685,31821],[34,254359,31829],{"className":254360},[31828],[34,254362,254364],{"className":254363},[31801],[34,254365,254367],{"className":254366,"style":31836},[31805],[34,254368],{},[34,254370],{"className":254371,"style":31846},[31845],[34,254373,202],{"className":254374},[31850],[34,254376],{"className":254377,"style":31846},[31845],[34,254379,254381,254384,254388],{"className":254380},[31676],[34,254382],{"className":254383,"style":38357},[31680],[34,254385,254387],{"className":254386},[31685],"40045",[34,254389,254391,254394],{"className":254390},[31685],[34,254392,36723],{"className":254393},[31685],[34,254395,254397],{"className":254396},[31792],[34,254398,254400,254420],{"className":254399},[31796,31797],[34,254401,254403,254417],{"className":254402},[31801],[34,254404,254406],{"className":254405,"style":31806},[31805],[34,254407,254408,254411],{"style":31809},[34,254409],{"className":254410,"style":31814},[31813],[34,254412,254414],{"className":254413},[31818,31819,31820,31821],[34,254415,60525],{"className":254416},[31685,31821],[34,254418,31829],{"className":254419},[31828],[34,254421,254423],{"className":254422},[31801],[34,254424,254426],{"className":254425,"style":31836},[31805],[34,254427],{},[2627,254429,249111],{}," 靈活運用八進位乘法和求餘數，比起轉換成二進位再去截斷位元，不僅速度快幾倍，而且極難出錯。",[19,254432,254434],{"id":254433},"_30设分区存储管理系统有45kb作业a分配15kb作业b分配20kb系统释放作业a有作业c申请8kb和作业d申请6kb按照最佳分配算法则最大碎片是-kb","30．设分区存储管理系统有45KB，作业A分配15KB，作业B分配20KB。系统释放作业A,有作业C申请8KB和作业D申请6KB，按照最佳分配算法，则最大碎片是( )KB。",[15,254436,254437,252986,254439,254441,254442],{},[2627,254438,58173],{},[2627,254440,248267],{}," 最佳適應 (Best-Fit) 分配演算法的推演。\n",[2627,254443,246526],{},[2230,254445,254446,254452,254467,254485,254501],{},[108,254447,254448,254451],{},[2627,254449,254450],{},"初始狀態："," 45KB 空閒。",[108,254453,254454,254457,254458,47649,254461,47649,254464,347],{},[2627,254455,254456],{},"A=15, B=20："," 記憶體佈局變為 ",[34,254459,254460],{},"A:15",[34,254462,254463],{},"B:20",[34,254465,254466],{},"空閒:10",[108,254468,254469,254472,254473,47649,254478,47649,254480,254484],{},[2627,254470,254471],{},"釋放 A："," 記憶體變為 ",[2627,254474,254475],{},[34,254476,254477],{},"空閒:15",[34,254479,254463],{},[2627,254481,254482],{},[34,254483,254466],{},"。因為中間隔著 B，兩個空閒區無法合併。",[108,254486,254487,254490,254491,254494,254495,616,254498,347],{},[2627,254488,254489],{},"C 申請 8KB (最佳適應)："," 在 15KB 和 10KB 之間，選擇最接近 8KB 的 ",[2627,254492,254493],{},"10KB"," 區塊。分配後，該區塊剩餘 2KB。目前空閒：",[34,254496,254497],{},"15KB",[34,254499,254500],{},"2KB",[108,254502,254503,254506,254507,254509,254510,616,254513,254515,254516,254518],{},[2627,254504,254505],{},"D 申請 6KB (最佳適應)："," 只能分配到 ",[2627,254508,254497],{}," 區塊。分配後剩餘 9KB。目前空閒：",[34,254511,254512],{},"9KB",[34,254514,254500],{},"。\n因此，系統中剩下的最大碎片是 9KB。\n",[2627,254517,249111],{}," 分區釋放時，必須注意它與相鄰分區是否能合併。這裡由於作業 B 還在，A 釋放的空間無法與最後尾部的空閒空間合併。",[19,254520,254522],{"id":254521},"_31每执行一次v操作信号量的数值s加1若-则该进程继续执行否则从对应的-队列中移出一个进程并将-状态赋予该进程","31．每执行一次V操作，信号量的数值S加1。若( )，则该进程继续执行；否则，从对应的( )队列中移出一个进程并将( )状态赋予该进程。",[15,254524,254525,254527,254528,254530,254531,254533],{},[2627,254526,58173],{}," S > 0 、 阻塞 (或等待) 、 就绪 (Ready)\n",[2627,254529,248267],{}," V 操作 (Signal) 的底層邏輯。\n",[2627,254532,246526],{},"\nV 操作的核心是「釋放資源」。",[105,254535,254536,254612,254722],{},[108,254537,254538,254539,347],{},"執行 ",[34,254540,254542,254564],{"className":254541},[31642],[34,254543,254545],{"className":254544},[31646],[31648,254546,254547],{"xmlns":31650},[31652,254548,254549,254561],{},[31655,254550,254551,254553,254555,254557,254559],{},[31658,254552,94675],{},[31662,254554,202],{},[31658,254556,94675],{},[31662,254558,223],{},[31749,254560,640],{},[31665,254562,254563],{"encoding":31667},"S = S + 1",[34,254565,254567,254585,254603],{"className":254566,"ariaHidden":758},[31672],[34,254568,254570,254573,254576,254579,254582],{"className":254569},[31676],[34,254571],{"className":254572,"style":33218},[31680],[34,254574,94675],{"className":254575,"style":106455},[31685,31686],[34,254577],{"className":254578,"style":31846},[31845],[34,254580,202],{"className":254581},[31850],[34,254583],{"className":254584,"style":31846},[31845],[34,254586,254588,254591,254594,254597,254600],{"className":254587},[31676],[34,254589],{"className":254590,"style":34944},[31680],[34,254592,94675],{"className":254593,"style":106455},[31685,31686],[34,254595],{"className":254596,"style":32604},[31845],[34,254598,223],{"className":254599},[32024],[34,254601],{"className":254602,"style":32604},[31845],[34,254604,254606,254609],{"className":254605},[31676],[34,254607],{"className":254608,"style":32617},[31680],[34,254610,640],{"className":254611},[31685],[108,254613,254614,254615,254665,254666,254717,254718,254721],{},"如果加 1 後 ",[34,254616,254618,254635],{"className":254617},[31642],[34,254619,254621],{"className":254620},[31646],[31648,254622,254623],{"xmlns":31650},[31652,254624,254625,254633],{},[31655,254626,254627,254629,254631],{},[31658,254628,94675],{},[31662,254630,29622],{},[31749,254632,459],{},[31665,254634,249520],{"encoding":31667},[34,254636,254638,254656],{"className":254637,"ariaHidden":758},[31672],[34,254639,254641,254644,254647,254650,254653],{"className":254640},[31676],[34,254642],{"className":254643,"style":104901},[31680],[34,254645,94675],{"className":254646,"style":106455},[31685,31686],[34,254648],{"className":254649,"style":31846},[31845],[34,254651,29622],{"className":254652},[31850],[34,254654],{"className":254655,"style":31846},[31845],[34,254657,254659,254662],{"className":254658},[31676],[34,254660],{"className":254661,"style":32617},[31680],[34,254663,459],{"className":254664},[31685],"，說明原本 ",[34,254667,254669,254687],{"className":254668},[31642],[34,254670,254672],{"className":254671},[31646],[31648,254673,254674],{"xmlns":31650},[31652,254675,254676,254684],{},[31655,254677,254678,254680,254682],{},[31658,254679,94675],{},[31662,254681,43175],{},[31749,254683,459],{},[31665,254685,254686],{"encoding":31667},"S \\ge 0",[34,254688,254690,254708],{"className":254689,"ariaHidden":758},[31672],[34,254691,254693,254696,254699,254702,254705],{"className":254692},[31676],[34,254694],{"className":254695,"style":195047},[31680],[34,254697,94675],{"className":254698,"style":106455},[31685,31686],[34,254700],{"className":254701,"style":31846},[31845],[34,254703,43175],{"className":254704},[31850],[34,254706],{"className":254707,"style":31846},[31845],[34,254709,254711,254714],{"className":254710},[31676],[34,254712],{"className":254713,"style":32617},[31680],[34,254715,459],{"className":254716},[31685],"，表示",[2627,254719,254720],{},"根本沒有行程在排隊等待這個資源","，所以當前行程直接繼續執行即可。",[108,254723,254724,254725,254775,254776,254778,254779,254829,254830,254665,254881,254937,254938,347],{},"否則（也就是加 1 後 ",[34,254726,254728,254745],{"className":254727},[31642],[34,254729,254731],{"className":254730},[31646],[31648,254732,254733],{"xmlns":31650},[31652,254734,254735,254743],{},[31655,254736,254737,254739,254741],{},[31658,254738,94675],{},[31662,254740,45972],{},[31749,254742,459],{},[31665,254744,249577],{"encoding":31667},[34,254746,254748,254766],{"className":254747,"ariaHidden":758},[31672],[34,254749,254751,254754,254757,254760,254763],{"className":254750},[31676],[34,254752],{"className":254753,"style":195047},[31680],[34,254755,94675],{"className":254756,"style":106455},[31685,31686],[34,254758],{"className":254759,"style":31846},[31845],[34,254761,45972],{"className":254762},[31850],[34,254764],{"className":254765,"style":31846},[31845],[34,254767,254769,254772],{"className":254768},[31676],[34,254770],{"className":254771,"style":32617},[31680],[34,254773,459],{"className":254774},[31685],"），說明原本 S 是負數，有行程正在阻塞佇列中。這時要喚醒佇列中的一個行程，將其狀態從「阻塞」改為「就緒」，讓它有機會被 CPU 執行。\n",[2627,254777,249111],{}," 若寫成 ",[34,254780,254782,254799],{"className":254781},[31642],[34,254783,254785],{"className":254784},[31646],[31648,254786,254787],{"xmlns":31650},[31652,254788,254789,254797],{},[31655,254790,254791,254793,254795],{},[31658,254792,94675],{},[31662,254794,43175],{},[31749,254796,459],{},[31665,254798,254686],{"encoding":31667},[34,254800,254802,254820],{"className":254801,"ariaHidden":758},[31672],[34,254803,254805,254808,254811,254814,254817],{"className":254804},[31676],[34,254806],{"className":254807,"style":195047},[31680],[34,254809,94675],{"className":254810,"style":106455},[31685,31686],[34,254812],{"className":254813,"style":31846},[31845],[34,254815,43175],{"className":254816},[31850],[34,254818],{"className":254819,"style":31846},[31845],[34,254821,254823,254826],{"className":254822},[31676],[34,254824],{"className":254825,"style":32617},[31680],[34,254827,459],{"className":254828},[31685]," 會不夠精確。因為如果加 1 後 ",[34,254831,254833,254851],{"className":254832},[31642],[34,254834,254836],{"className":254835},[31646],[31648,254837,254838],{"xmlns":31650},[31652,254839,254840,254848],{},[31655,254841,254842,254844,254846],{},[31658,254843,94675],{},[31662,254845,202],{},[31749,254847,459],{},[31665,254849,254850],{"encoding":31667},"S = 0",[34,254852,254854,254872],{"className":254853,"ariaHidden":758},[31672],[34,254855,254857,254860,254863,254866,254869],{"className":254856},[31676],[34,254858],{"className":254859,"style":33218},[31680],[34,254861,94675],{"className":254862,"style":106455},[31685,31686],[34,254864],{"className":254865,"style":31846},[31845],[34,254867,202],{"className":254868},[31850],[34,254870],{"className":254871,"style":31846},[31845],[34,254873,254875,254878],{"className":254874},[31676],[34,254876],{"className":254877,"style":32617},[31680],[34,254879,459],{"className":254880},[31685],[34,254882,254884,254904],{"className":254883},[31642],[34,254885,254887],{"className":254886},[31646],[31648,254888,254889],{"xmlns":31650},[31652,254890,254891,254901],{},[31655,254892,254893,254895,254897,254899],{},[31658,254894,94675],{},[31662,254896,202],{},[31662,254898,32226],{},[31749,254900,640],{},[31665,254902,254903],{"encoding":31667},"S = -1",[34,254905,254907,254925],{"className":254906,"ariaHidden":758},[31672],[34,254908,254910,254913,254916,254919,254922],{"className":254909},[31676],[34,254911],{"className":254912,"style":33218},[31680],[34,254914,94675],{"className":254915,"style":106455},[31685,31686],[34,254917],{"className":254918,"style":31846},[31845],[34,254920,202],{"className":254921},[31850],[34,254923],{"className":254924,"style":31846},[31845],[34,254926,254928,254931,254934],{"className":254927},[31676],[34,254929],{"className":254930,"style":35942},[31680],[34,254932,32226],{"className":254933},[31685],[34,254935,640],{"className":254936},[31685],"，表示有 1 個行程在等，此時「必須」喚醒它，所以喚醒的條件包含了 ",[34,254939,254941,254958],{"className":254940},[31642],[34,254942,254944],{"className":254943},[31646],[31648,254945,254946],{"xmlns":31650},[31652,254947,254948,254956],{},[31655,254949,254950,254952,254954],{},[31658,254951,94675],{},[31662,254953,45972],{},[31749,254955,459],{},[31665,254957,249577],{"encoding":31667},[34,254959,254961,254979],{"className":254960,"ariaHidden":758},[31672],[34,254962,254964,254967,254970,254973,254976],{"className":254963},[31676],[34,254965],{"className":254966,"style":195047},[31680],[34,254968,94675],{"className":254969,"style":106455},[31685,31686],[34,254971],{"className":254972,"style":31846},[31845],[34,254974,45972],{"className":254975},[31850],[34,254977],{"className":254978,"style":31846},[31845],[34,254980,254982,254985],{"className":254981},[31676],[34,254983],{"className":254984,"style":32617},[31680],[34,254986,459],{"className":254987},[31685],[19,254989,254991],{"id":254990},"_32利用信号量实现进程的-应为临界区设置一个信号量mutex其初值为1表示该资源尚未使用临界区应置于-和-原语之间","32．利用信号量实现进程的( )，应为临界区设置一个信号量mutex，其初值为1，表示该资源尚未使用，临界区应置于( )和( )原语之间。",[15,254993,254994,254996,254997,254999,255000,255002,255003,255005],{},[2627,254995,58173],{}," 互斥 (Mutual exclusion) 、 P操作 (Wait \u002F semWait) 、 V操作 (Signal \u002F semSignal)\n",[2627,254998,248267],{}," 號誌實現互斥的標準範式。\n",[2627,255001,246526],{}," 為了保證每次只有一個行程能進入臨界區（即互斥），我們會將號誌初始化為 1（代表有 1 把鎖）。行程進入臨界區前必須執行 P 操作（拿鎖），離開時執行 V 操作（還鎖）。\n",[2627,255004,249111],{}," 順序絕不能反。P 操作相當於「申請」，V 操作相當於「釋放」。如果先 V 再 P，就徹底失去保護作用了。",[19,255007,255009],{"id":255008},"_33计算机系统中屏幕显示分辨率为1024x768若要存储一屏256彩色的图像需要-字节存储空间","33．计算机系统中，屏幕显示分辨率为1024x768，若要存储一屏256彩色的图像，需要（ ）字节存储空间。",[15,255011,255012,255014,255015,255017,255018,255020,255021,39984,255133,255135],{},[2627,255013,58173],{}," 786432\n",[2627,255016,248267],{}," 影像記憶體容量計算（同第 27 題）。\n",[2627,255019,246526],{},"\n256 色 = 8 bits = 1 Byte。\n總位元組數 = ",[34,255022,255024,255055],{"className":255023},[31642],[34,255025,255027],{"className":255026},[31646],[31648,255028,255029],{"xmlns":31650},[31652,255030,255031,255052],{},[31655,255032,255033,255035,255037,255039,255041,255043,255045,255047,255050],{},[31749,255034,177393],{},[31662,255036,42071],{},[31749,255038,184127],{},[31662,255040,42071],{},[31749,255042,640],{},[39361,255044,252821],{},[31662,255046,202],{},[31749,255048,255049],{},"786432",[39361,255051,252828],{},[31665,255053,255054],{"encoding":31667},"1024 \\times 768 \\times 1 \\text{ Byte} = 786432 \\text{ Bytes}",[34,255056,255058,255076,255094,255118],{"className":255057,"ariaHidden":758},[31672],[34,255059,255061,255064,255067,255070,255073],{"className":255060},[31676],[34,255062],{"className":255063,"style":35942},[31680],[34,255065,177393],{"className":255066},[31685],[34,255068],{"className":255069,"style":32604},[31845],[34,255071,42071],{"className":255072},[32024],[34,255074],{"className":255075,"style":32604},[31845],[34,255077,255079,255082,255085,255088,255091],{"className":255078},[31676],[34,255080],{"className":255081,"style":35942},[31680],[34,255083,184127],{"className":255084},[31685],[34,255086],{"className":255087,"style":32604},[31845],[34,255089,42071],{"className":255090},[32024],[34,255092],{"className":255093,"style":32604},[31845],[34,255095,255097,255100,255103,255109,255112,255115],{"className":255096},[31676],[34,255098],{"className":255099,"style":70739},[31680],[34,255101,640],{"className":255102},[31685],[34,255104,255106],{"className":255105},[31685,2638],[34,255107,252821],{"className":255108},[31685],[34,255110],{"className":255111,"style":31846},[31845],[34,255113,202],{"className":255114},[31850],[34,255116],{"className":255117,"style":31846},[31845],[34,255119,255121,255124,255127],{"className":255120},[31676],[34,255122],{"className":255123,"style":70739},[31680],[34,255125,255049],{"className":255126},[31685],[34,255128,255130],{"className":255129},[31685,2638],[34,255131,252828],{"className":255132},[31685],[2627,255134,249111],{}," 題目問的是「字節 (Bytes)」，不需要再除以 1024 轉換成 KB。仔細看清題目要求的單位。",[19,255137,255139],{"id":255138},"_34在一个单处理机系统中若有4个用户进程且假定当前时刻有一个进程处于执行状态则处于就绪状态的进程最多有-个最少有-个","34．在一个单处理机系统中，若有4个用户进程且假定当前时刻有一个进程处于执行状态，则处于就绪状态的进程最多有（ ）个，最少有（ ）个。",[15,255141,255142,255144,255145,255147,255148,255150],{},[2627,255143,58173],{}," 3 、 0\n",[2627,255146,248267],{}," 單處理機的行程狀態數量分佈（同第 22 題）。\n",[2627,255149,246526],{},"\n總共 4 個行程，1 個在執行。剩下 3 個行程。",[105,255152,255153,255159],{},[108,255154,255155,255158],{},[2627,255156,255157],{},"最多："," 如果這 3 個行程都不需要等待 I\u002FO，它們都在排隊搶 CPU，那就全在就緒狀態，最多為 3。",[108,255160,255161,255164,255165,255167],{},[2627,255162,255163],{},"最少："," 如果這 3 個行程剛好都發起了 I\u002FO 請求，那它們全都會陷入「阻塞\u002F等待」狀態。這時就緒佇列裡空無一人，最少為 0。\n",[2627,255166,249111],{}," 極端情況的考量是作業系統排程分析的基本功。",[19,255169,255171],{"id":255170},"_35按使用情况文件可分为-和档案文件","35．按使用情况，文件可分为（ ）、（ ）和档案文件。",[15,255173,255174,255176,255177,255179,255180,255182],{},[2627,255175,58173],{}," 临时文件 (Temporary file) 、 永久文件 (Permanent file)\n",[2627,255178,248267],{}," 檔案的分類。\n",[2627,255181,246526],{}," 從檔案的生命週期和使用目的來分，通常會分為：",[2230,255184,255185,255188,255191],{},[108,255186,255187],{},"臨時檔案（如程式編譯過程產生的中間檔，用完即刪）。",[108,255189,255190],{},"永久檔案（如使用者的文件、系統執行檔，長期保留）。",[108,255192,255193,255194,255196],{},"檔案\u002F歸檔檔案 (Archive file)（如歷史備份、日誌）。\n",[2627,255195,249111],{}," 這屬於較傳統作業系統課程的分類法，記住生命週期的三個階段即可。",[19,255198,255200],{"id":255199},"_36面对一般用户通过-方式控制操作系统面对编程人员通过-控制","36．面对一般用户，通过（ ）方式控制操作系统；面对编程人员，通过（ ）控制。",[15,255202,255203,255205,255206,255208,255209,255211],{},[2627,255204,58173],{}," 操作命令 (Command line \u002F 命令行 \u002F 终端界面) 、 系统调用 (System call)\n",[2627,255207,248267],{}," 作業系統的使用者介面 (User Interface)。\n",[2627,255210,246526],{}," 作業系統對外提供兩種類型的介面：",[2230,255213,255214,255217],{},[108,255215,255216],{},"給「人類」用的：終端機輸入指令（Command line）或是圖形化介面（GUI）。",[108,255218,255219,255220,39984,255223,255225],{},"給「程式」用的：程式設計師在寫程式時，如果需要調用 OS 的特權服務（如讀寫磁碟），必須使用 OS 提供的應用程式介面，即",[2627,255221,255222],{},"系統呼叫 (System calls)",[2627,255224,249111],{}," 系統呼叫是程式從「使用者態」切換到「核心態」的唯一合法橋樑。",[19,255227,255229],{"id":255228},"_37在动态分区算法中-倾向与优先利用内存中的低地址部分的空闲分区从而保留了高地址部分的大空闲分区","37．在动态分区算法中，（ ）倾向与优先利用内存中的低地址部分的空闲分区，从而保留了高地址部分的大空闲分区。",[15,255231,255232,255234,255235,255237,255238,255240,255241,255243],{},[2627,255233,58173],{}," 首次适应算法 (First-fit algorithm)\n",[2627,255236,248267],{}," 首次適應演算法的特徵。\n",[2627,255239,246526],{}," 首次適應要求空閒區塊按「實體位址遞增」的順序連結。每次分配時都從鏈表頭（低位址）開始找，只要夠大就切。這導致低位址不斷被切碎，但卻能有效保護高位址的連續大區塊不被動到。\n",[2627,255242,249111],{}," 與最佳適應（產生最多碎塊）、最差適應（專切最大區塊）的特徵作好比較。",[19,255245,255247],{"id":255246},"_38作业执行期间当访问到指令或数据时才进行地址变换的方式为","38．作业执行期间，当访问到指令或数据时才进行地址变换的方式为（ ）。",[15,255249,255250,255252,255253,255255,255256,255258,255259,255262,255263,255265],{},[2627,255251,58173],{}," 动态重定位 (Dynamic relocation \u002F 动态地址映射)\n",[2627,255254,248267],{}," 記憶體位址轉換的時機。\n",[2627,255257,246526],{}," 位址轉換可以分為靜態（裝入時一次性全改好）和動態。",[2627,255260,255261],{},"動態重定位","依賴硬體 MMU（記憶體管理單元），程式在記憶體中始終維持邏輯位址不變，只有在 CPU 真正抓取指令執行的「那一瞬間」，才加上基址暫存器算出實體位址。這允許程式在記憶體中浮動和進行虛擬記憶體管理。\n",[2627,255264,249111],{}," 這是現代作業系統分頁與分段機制的絕對基礎。",[19,255267,255269],{"id":255268},"_39在有m个进程的系统中出现死锁时死锁进程的个数k应该满足的条件是","39．在有m个进程的系统中出现死锁时，死锁进程的个数k应该满足的条件是（ ）。",[15,255271,255272,47649,255274,255347,255349,255350,255352,255353,255356,255357,255359,255360,255433],{},[2627,255273,58173],{},[34,255275,255277,255299],{"className":255276},[31642],[34,255278,255280],{"className":255279},[31646],[31648,255281,255282],{"xmlns":31650},[31652,255283,255284,255296],{},[31655,255285,255286,255288,255290,255292,255294],{},[31749,255287,768],{},[31662,255289,45972],{},[31658,255291,33759],{},[31662,255293,45972],{},[31658,255295,31759],{},[31665,255297,255298],{"encoding":31667},"2 \\le k \\le m",[34,255300,255302,255320,255338],{"className":255301,"ariaHidden":758},[31672],[34,255303,255305,255308,255311,255314,255317],{"className":255304},[31676],[34,255306],{"className":255307,"style":55382},[31680],[34,255309,768],{"className":255310},[31685],[34,255312],{"className":255313,"style":31846},[31845],[34,255315,45972],{"className":255316},[31850],[34,255318],{"className":255319,"style":31846},[31845],[34,255321,255323,255326,255329,255332,255335],{"className":255322},[31676],[34,255324],{"className":255325,"style":47709},[31680],[34,255327,33759],{"className":255328,"style":33774},[31685,31686],[34,255330],{"className":255331,"style":31846},[31845],[34,255333,45972],{"className":255334},[31850],[34,255336],{"className":255337,"style":31846},[31845],[34,255339,255341,255344],{"className":255340},[31676],[34,255342],{"className":255343,"style":32527},[31680],[34,255345,31759],{"className":255346},[31685,31686],[2627,255348,248267],{}," 死結的定義。\n",[2627,255351,246526],{}," 死結是指一組行程互相等待對方釋放資源，形成了一個「循環等待」的殭局。一個行程自己等自己那叫無限迴圈或普通阻塞，要構成「環路」，",[2627,255354,255355],{},"至少需要 2 個行程","參與。因此死結行程數 k 的範圍是從 2 到所有系統行程數 m。\n",[2627,255358,249111],{}," 不能寫 ",[34,255361,255363,255385],{"className":255362},[31642],[34,255364,255366],{"className":255365},[31646],[31648,255367,255368],{"xmlns":31650},[31652,255369,255370,255382],{},[31655,255371,255372,255374,255376,255378,255380],{},[31749,255373,640],{},[31662,255375,45972],{},[31658,255377,33759],{},[31662,255379,45972],{},[31658,255381,31759],{},[31665,255383,255384],{"encoding":31667},"1 \\le k \\le m",[34,255386,255388,255406,255424],{"className":255387,"ariaHidden":758},[31672],[34,255389,255391,255394,255397,255400,255403],{"className":255390},[31676],[34,255392],{"className":255393,"style":55382},[31680],[34,255395,640],{"className":255396},[31685],[34,255398],{"className":255399,"style":31846},[31845],[34,255401,45972],{"className":255402},[31850],[34,255404],{"className":255405,"style":31846},[31845],[34,255407,255409,255412,255415,255418,255421],{"className":255408},[31676],[34,255410],{"className":255411,"style":47709},[31680],[34,255413,33759],{"className":255414,"style":33774},[31685,31686],[34,255416],{"className":255417,"style":31846},[31845],[34,255419,45972],{"className":255420},[31850],[34,255422],{"className":255423,"style":31846},[31845],[34,255425,255427,255430],{"className":255426},[31676],[34,255428],{"className":255429,"style":32527},[31680],[34,255431,31759],{"className":255432},[31685,31686],"，死結無法由單一行程構成。",[19,255435,255437],{"id":255436},"_40-是指由于一个作业装入到与其地址空间不一致的存储空间所引起的对有关地址部分的调整过程","40．（ ）是指由于一个作业装入到与其地址空间不一致的存储空间所引起的对有关地址部分的调整过程。",[15,255439,255440,255442,255443,255445,255446,255448,255449,255451],{},[2627,255441,58173],{}," 重定位 (Relocation)\n",[2627,255444,248267],{}," 重定位的定義。\n",[2627,255447,246526],{}," 程式編譯出來後，它的邏輯位址通常是從 0 開始的。但主記憶體的 0 號位址通常是留給作業系統核心的。將程式裝入到記憶體中的某個實際位置時，必須將程式內部所有的位址參照進行相應的偏移調整，這個調整的過程就叫做「重定位 (Relocation)」。\n",[2627,255450,249111],{}," 記得結合第 38 題，重定位又分為「靜態重定位」與「動態重定位」。",[19,255453,255455],{"id":255454},"_41单个分区存储管理仅适用于-和-系统","41．单个分区存储管理仅适用于（ ）和（ ）系统。",[15,255457,255458,255460,255461,255463,255464,255466,255467,255469],{},[2627,255459,58173],{}," 单用户 (Single-user) 和 单道程序 (Uniprogramming)\n",[2627,255462,246520],{}," 单一连续分区（Single Partitioning）是最早、最简单的内存管理方式。整个内存被分为两个部分：操作系统区和用户区。\n",[2627,255465,246526],{}," 在这种机制下，主存中永远只能同时驻留一个用户程序。正如参考资料指出，在单道程序系统（uniprogramming system）中，主存被划分为两部分，一部分给操作系统内核，另一部分给当前正在执行的唯一程序。因此，它完全不支持多道程序并发，仅适用于早期的单用户、单任务操作系统（如MS-DOS）。\n",[2627,255468,247377],{}," 不要填成“批处理”，因为多道批处理系统（Multiprogrammed Batch Systems）需要内存中同时存在多个作业，这必须依赖于多分区（Multi-partition）管理。",[19,255471,255473],{"id":255472},"_42每个索引文件都必须有一张-表其中每个登记项用来指出一个逻辑记录的","42．每个索引文件都必须有一张（ ）表，其中每个登记项用来指出一个逻辑记录的（ ）。",[15,255475,255476,255478,255479,255481,255482,255484,255485,255487],{},[2627,255477,58173],{}," 索引 (Index) \u002F 物理地址 (Physical address)\n",[2627,255480,246520],{}," 索引文件（Indexed file）是一种为了支持快速随机访问而设计的文件逻辑与物理结构。\n",[2627,255483,246526],{}," 当我们需要快速定位文件中的某一条特定记录时（尤其是在记录长度可变时），顺序查找效率极低。参考资料表明，索引文件通过建立一个附加的“索引（index）”，使得记录只能通过其索引被访问。这个索引表中的每一行（登记项）都包含了一个指针，直接指向该逻辑记录在磁盘上的“物理地址（或逻辑块号\u002F相对位置）”。\n",[2627,255486,247377],{}," 学生容易把“物理地址”写成“逻辑地址”。索引表的根本目的就是完成从“逻辑记录关键字”到实际存储的“物理地址”的映射。",[19,255489,255491],{"id":255490},"_43操作系统是运行在计算机-系统上的最基本的系统软件","43．操作系统是运行在计算机（ ）系统上的最基本的系统软件。",[15,255493,255494,255496,255497,255499,255500,255502,255503,255505],{},[2627,255495,58173],{}," 硬件 (Hardware)\n",[2627,255498,246520],{}," 操作系统的定义与层次结构。\n",[2627,255501,246526],{}," 计算机平台由一系列硬件资源组成。操作系统是直接覆盖在“裸机（硬件）”之上的第一层软件。它屏蔽了底层硬件的复杂细节，为上层的应用程序和用户提供了一个一致的、易于使用的抽象接口。\n",[2627,255504,247377],{}," 注意题目问的是运行在什么“系统”上，底层即是“硬件系统（Hardware System）”。",[19,255507,255509],{"id":255508},"_44实现spool系统时必须在磁盘上辟出称为-和-的专门区域以存放作业信息和作业执行结果","44．实现SPOOL系统时必须在磁盘上辟出称为（ ）和（ ）的专门区域，以存放作业信息和作业执行结果。",[15,255511,255512,255514,255515,255517,255518,255520,255521,255523],{},[2627,255513,58173],{}," 输入井 (Input well \u002F buffer) 和 输出井 (Output well \u002F buffer)\n",[2627,255516,246520],{}," SPOOLing (Simultaneous Peripheral Operations On-Line) 即假脱机技术。\n",[2627,255519,246526],{}," SPOOLing 的核心思想是利用高速的磁盘空间来模拟慢速的独占设备（如打印机）。参考资料中描述了一个SPOOLing系统的模型，它由输入进程、用户进程和输出进程组成，这些进程通过“缓冲（buffers）”进行连接，并且这些块被缓存在磁盘上。在标准的操作系统术语中，磁盘上开辟出来的这两块巨大的缓冲区域分别被称为“输入井（用于存放尚未处理的作业信息）”和“输出井（用于存放作业产生的输出结果）”。\n",[2627,255522,247377],{}," 不要与内存中的“输入\u002F输出缓冲区（I\u002FO buffers）”混淆。SPOOLing 的“井”是建立在**磁盘（Secondary Storage）**上的大容量区域。",[19,255525,255527],{"id":255526},"_45用户调用-和-文件操作来申请对文件的使用权","45．用户调用（ ）和（ ）文件操作来申请对文件的使用权。",[15,255529,255530,255532,255533,255535,255536,255538,255539,255542,255543,255546,255547,255549],{},[2627,255531,58173],{}," 打开 (Open) 和 关闭 (Close)\n",[2627,255534,246520],{}," 逻辑I\u002FO与文件操作API。\n",[2627,255537,246526],{}," 操作系统通过逻辑I\u002FO模块，允许用户进程以设备标识符和简单的命令（诸如 open, close, read, write）来处理设备和文件。其中，",[31,255540,255541],{},"open"," 操作（如参考资料列出的UNIX系统调用）的作用是建立进程与文件之间的连接，实际上就是“申请使用权”并加载文件的属性到内存；而 ",[31,255544,255545],{},"close"," 则是释放使用权，断开连接。\n",[2627,255548,247377],{}," read和write是进行数据交换的操作，而申请\u002F释放使用权的动作必须由 open 和 close 来完成。",[19,255551,255553],{"id":255552},"_46主存储器与外围设备之间的信息传送操作称为","46．主存储器与外围设备之间的信息传送操作称为（ ）。",[15,255555,255556,255558,255559,255561,255562,255564,255565,255567],{},[2627,255557,58173],{}," I\u002FO操作 (Input\u002FOutput operation 或 输入输出操作)\n",[2627,255560,246520],{}," I\u002FO 模块的职责。\n",[2627,255563,246526],{}," 计算机系统四大主要结构元素之一就是 I\u002FO 模块（I\u002FO Modules），它的核心作用就是负责在计算机（主要是主存或处理器）与它的外部环境（外围设备）之间“移动数据（Move data）”。这种信息传送的过程统称为I\u002FO操作。\n",[2627,255566,247377],{}," 此处不应填DMA。DMA只是执行I\u002FO操作的其中一种“方式”，而题目问的是这种传送操作的“统称”。",[19,255569,255571],{"id":255570},"_47若干个等待访问磁盘者依次要访问的磁道为2044404801276移动臂当前位于40号柱面则先来先服务算法的平均寻道长度为-最短寻道时间优先算法的平均寻道长度为-扫描算法当前磁头移动的方向为磁道递增的平均寻道长度为","47．若干个等待访问磁盘者依次要访问的磁道为20，44，40，4，80，12，76，移动臂当前位于40号柱面，则先来先服务算法的平均寻道长度为（ ）； 最短寻道时间优先算法的平均寻道长度为（ ）；扫描算法（当前磁头移动的方向为磁道递增）的平均寻道长度为（ ）。",[15,255573,255574,255576,255577,255579,255580,255582],{},[2627,255575,58173],{}," 41.7 (或 292\u002F7) ； 17.1 (或 120\u002F7) ； 16.6 (或 116\u002F7)\n",[2627,255578,246520],{}," 磁盘调度算法 (Disk Scheduling Policies) 的性能评估。\n",[2627,255581,246526],{},"\n总共有 7 个请求需要满足。我们分别计算总移动距离，然后除以 7：",[2230,255584,255585,256407,256856],{},[108,255586,255587,255590,255591,255619,255620,255648,255649,255677,255678,226066,255706,255734,255735,255763,255764,255792,255793,256150,256349,256350,347],{},[2627,255588,255589],{},"先来先服务 (FCFS - First-Come-First-Served):"," 严格按照队列顺序移动。\n路线：40(起点) ",[34,255592,255594,255607],{"className":255593},[31642],[34,255595,255597],{"className":255596},[31646],[31648,255598,255599],{"xmlns":31650},[31652,255600,255601,255605],{},[31655,255602,255603],{},[31662,255604,33239],{},[31665,255606,119689],{"encoding":31667},[34,255608,255610],{"className":255609,"ariaHidden":758},[31672],[34,255611,255613,255616],{"className":255612},[31676],[34,255614],{"className":255615,"style":73089},[31680],[34,255617,33239],{"className":255618},[31850]," 20 ",[34,255621,255623,255636],{"className":255622},[31642],[34,255624,255626],{"className":255625},[31646],[31648,255627,255628],{"xmlns":31650},[31652,255629,255630,255634],{},[31655,255631,255632],{},[31662,255633,33239],{},[31665,255635,119689],{"encoding":31667},[34,255637,255639],{"className":255638,"ariaHidden":758},[31672],[34,255640,255642,255645],{"className":255641},[31676],[34,255643],{"className":255644,"style":73089},[31680],[34,255646,33239],{"className":255647},[31850]," 44 ",[34,255650,255652,255665],{"className":255651},[31642],[34,255653,255655],{"className":255654},[31646],[31648,255656,255657],{"xmlns":31650},[31652,255658,255659,255663],{},[31655,255660,255661],{},[31662,255662,33239],{},[31665,255664,119689],{"encoding":31667},[34,255666,255668],{"className":255667,"ariaHidden":758},[31672],[34,255669,255671,255674],{"className":255670},[31676],[34,255672],{"className":255673,"style":73089},[31680],[34,255675,33239],{"className":255676},[31850]," 40 ",[34,255679,255681,255694],{"className":255680},[31642],[34,255682,255684],{"className":255683},[31646],[31648,255685,255686],{"xmlns":31650},[31652,255687,255688,255692],{},[31655,255689,255690],{},[31662,255691,33239],{},[31665,255693,119689],{"encoding":31667},[34,255695,255697],{"className":255696,"ariaHidden":758},[31672],[34,255698,255700,255703],{"className":255699},[31676],[34,255701],{"className":255702,"style":73089},[31680],[34,255704,33239],{"className":255705},[31850],[34,255707,255709,255722],{"className":255708},[31642],[34,255710,255712],{"className":255711},[31646],[31648,255713,255714],{"xmlns":31650},[31652,255715,255716,255720],{},[31655,255717,255718],{},[31662,255719,33239],{},[31665,255721,119689],{"encoding":31667},[34,255723,255725],{"className":255724,"ariaHidden":758},[31672],[34,255726,255728,255731],{"className":255727},[31676],[34,255729],{"className":255730,"style":73089},[31680],[34,255732,33239],{"className":255733},[31850]," 80 ",[34,255736,255738,255751],{"className":255737},[31642],[34,255739,255741],{"className":255740},[31646],[31648,255742,255743],{"xmlns":31650},[31652,255744,255745,255749],{},[31655,255746,255747],{},[31662,255748,33239],{},[31665,255750,119689],{"encoding":31667},[34,255752,255754],{"className":255753,"ariaHidden":758},[31672],[34,255755,255757,255760],{"className":255756},[31676],[34,255758],{"className":255759,"style":73089},[31680],[34,255761,33239],{"className":255762},[31850]," 12 ",[34,255765,255767,255780],{"className":255766},[31642],[34,255768,255770],{"className":255769},[31646],[31648,255771,255772],{"xmlns":31650},[31652,255773,255774,255778],{},[31655,255775,255776],{},[31662,255777,33239],{},[31665,255779,119689],{"encoding":31667},[34,255781,255783],{"className":255782,"ariaHidden":758},[31672],[34,255784,255786,255789],{"className":255785},[31676],[34,255787],{"className":255788,"style":73089},[31680],[34,255790,33239],{"className":255791},[31850]," 76。\n距离：",[34,255794,255796,255891],{"className":255795},[31642],[34,255797,255799],{"className":255798},[31646],[31648,255800,255801],{"xmlns":31650},[31652,255802,255803,255888],{},[31655,255804,255805,255807,255809,255811,255813,255815,255817,255819,255821,255823,255826,255828,255830,255832,255834,255836,255838,255840,255842,255844,255846,255848,255850,255852,255854,255856,255858,255860,255862,255864,255866,255868,255870,255872,255874,255876,255878,255880,255882,255884,255886],{},[31658,255806,162086],{"mathvariant":38079},[31749,255808,5731],{},[31662,255810,32226],{},[31749,255812,3249],{},[31658,255814,162086],{"mathvariant":38079},[31662,255816,223],{},[31658,255818,162086],{"mathvariant":38079},[31749,255820,3249],{},[31662,255822,32226],{},[31749,255824,255825],{},"44",[31658,255827,162086],{"mathvariant":38079},[31662,255829,223],{},[31658,255831,162086],{"mathvariant":38079},[31749,255833,255825],{},[31662,255835,32226],{},[31749,255837,5731],{},[31658,255839,162086],{"mathvariant":38079},[31662,255841,223],{},[31658,255843,162086],{"mathvariant":38079},[31749,255845,5731],{},[31662,255847,32226],{},[31749,255849,15932],{},[31658,255851,162086],{"mathvariant":38079},[31662,255853,223],{},[31658,255855,162086],{"mathvariant":38079},[31749,255857,15932],{},[31662,255859,32226],{},[31749,255861,27539],{},[31658,255863,162086],{"mathvariant":38079},[31662,255865,223],{},[31658,255867,162086],{"mathvariant":38079},[31749,255869,27539],{},[31662,255871,32226],{},[31749,255873,71869],{},[31658,255875,162086],{"mathvariant":38079},[31662,255877,223],{},[31658,255879,162086],{"mathvariant":38079},[31749,255881,71869],{},[31662,255883,32226],{},[31749,255885,95437],{},[31658,255887,162086],{"mathvariant":38079},[31665,255889,255890],{"encoding":31667},"|40-20| + |20-44| + |44-40| + |40-4| + |4-80| + |80-12| + |12-76|",[34,255892,255894,255913,255932,255951,255970,255989,256008,256026,256045,256064,256083,256102,256121,256140],{"className":255893,"ariaHidden":758},[31672],[34,255895,255897,255900,255904,255907,255910],{"className":255896},[31676],[34,255898],{"className":255899,"style":31774},[31680],[34,255901,255903],{"className":255902},[31685],"∣40",[34,255905],{"className":255906,"style":32604},[31845],[34,255908,32226],{"className":255909},[32024],[34,255911],{"className":255912,"style":32604},[31845],[34,255914,255916,255919,255923,255926,255929],{"className":255915},[31676],[34,255917],{"className":255918,"style":31774},[31680],[34,255920,255922],{"className":255921},[31685],"20∣",[34,255924],{"className":255925,"style":32604},[31845],[34,255927,223],{"className":255928},[32024],[34,255930],{"className":255931,"style":32604},[31845],[34,255933,255935,255938,255942,255945,255948],{"className":255934},[31676],[34,255936],{"className":255937,"style":31774},[31680],[34,255939,255941],{"className":255940},[31685],"∣20",[34,255943],{"className":255944,"style":32604},[31845],[34,255946,32226],{"className":255947},[32024],[34,255949],{"className":255950,"style":32604},[31845],[34,255952,255954,255957,255961,255964,255967],{"className":255953},[31676],[34,255955],{"className":255956,"style":31774},[31680],[34,255958,255960],{"className":255959},[31685],"44∣",[34,255962],{"className":255963,"style":32604},[31845],[34,255965,223],{"className":255966},[32024],[34,255968],{"className":255969,"style":32604},[31845],[34,255971,255973,255976,255980,255983,255986],{"className":255972},[31676],[34,255974],{"className":255975,"style":31774},[31680],[34,255977,255979],{"className":255978},[31685],"∣44",[34,255981],{"className":255982,"style":32604},[31845],[34,255984,32226],{"className":255985},[32024],[34,255987],{"className":255988,"style":32604},[31845],[34,255990,255992,255995,255999,256002,256005],{"className":255991},[31676],[34,255993],{"className":255994,"style":31774},[31680],[34,255996,255998],{"className":255997},[31685],"40∣",[34,256000],{"className":256001,"style":32604},[31845],[34,256003,223],{"className":256004},[32024],[34,256006],{"className":256007,"style":32604},[31845],[34,256009,256011,256014,256017,256020,256023],{"className":256010},[31676],[34,256012],{"className":256013,"style":31774},[31680],[34,256015,255903],{"className":256016},[31685],[34,256018],{"className":256019,"style":32604},[31845],[34,256021,32226],{"className":256022},[32024],[34,256024],{"className":256025,"style":32604},[31845],[34,256027,256029,256032,256036,256039,256042],{"className":256028},[31676],[34,256030],{"className":256031,"style":31774},[31680],[34,256033,256035],{"className":256034},[31685],"4∣",[34,256037],{"className":256038,"style":32604},[31845],[34,256040,223],{"className":256041},[32024],[34,256043],{"className":256044,"style":32604},[31845],[34,256046,256048,256051,256055,256058,256061],{"className":256047},[31676],[34,256049],{"className":256050,"style":31774},[31680],[34,256052,256054],{"className":256053},[31685],"∣4",[34,256056],{"className":256057,"style":32604},[31845],[34,256059,32226],{"className":256060},[32024],[34,256062],{"className":256063,"style":32604},[31845],[34,256065,256067,256070,256074,256077,256080],{"className":256066},[31676],[34,256068],{"className":256069,"style":31774},[31680],[34,256071,256073],{"className":256072},[31685],"80∣",[34,256075],{"className":256076,"style":32604},[31845],[34,256078,223],{"className":256079},[32024],[34,256081],{"className":256082,"style":32604},[31845],[34,256084,256086,256089,256093,256096,256099],{"className":256085},[31676],[34,256087],{"className":256088,"style":31774},[31680],[34,256090,256092],{"className":256091},[31685],"∣80",[34,256094],{"className":256095,"style":32604},[31845],[34,256097,32226],{"className":256098},[32024],[34,256100],{"className":256101,"style":32604},[31845],[34,256103,256105,256108,256112,256115,256118],{"className":256104},[31676],[34,256106],{"className":256107,"style":31774},[31680],[34,256109,256111],{"className":256110},[31685],"12∣",[34,256113],{"className":256114,"style":32604},[31845],[34,256116,223],{"className":256117},[32024],[34,256119],{"className":256120,"style":32604},[31845],[34,256122,256124,256127,256131,256134,256137],{"className":256123},[31676],[34,256125],{"className":256126,"style":31774},[31680],[34,256128,256130],{"className":256129},[31685],"∣12",[34,256132],{"className":256133,"style":32604},[31845],[34,256135,32226],{"className":256136},[32024],[34,256138],{"className":256139,"style":32604},[31845],[34,256141,256143,256146],{"className":256142},[31676],[34,256144],{"className":256145,"style":31774},[31680],[34,256147,256149],{"className":256148},[31685],"76∣",[34,256151,256153,256199],{"className":256152},[31642],[34,256154,256156],{"className":256155},[31646],[31648,256157,256158],{"xmlns":31650},[31652,256159,256160,256196],{},[31655,256161,256162,256164,256166,256168,256170,256172,256174,256176,256178,256180,256182,256184,256187,256189,256191,256193],{},[31662,256163,202],{},[31749,256165,3249],{},[31662,256167,223],{},[31749,256169,96087],{},[31662,256171,223],{},[31749,256173,15932],{},[31662,256175,223],{},[31749,256177,165563],{},[31662,256179,223],{},[31749,256181,95437],{},[31662,256183,223],{},[31749,256185,256186],{},"68",[31662,256188,223],{},[31749,256190,162212],{},[31662,256192,202],{},[31749,256194,256195],{},"292",[31665,256197,256198],{"encoding":31667},"= 20 + 24 + 4 + 36 + 76 + 68 + 64 = 292",[34,256200,256202,256214,256232,256250,256268,256286,256304,256322,256340],{"className":256201,"ariaHidden":758},[31672],[34,256203,256205,256208,256211],{"className":256204},[31676],[34,256206],{"className":256207,"style":73089},[31680],[34,256209,202],{"className":256210},[31850],[34,256212],{"className":256213,"style":31846},[31845],[34,256215,256217,256220,256223,256226,256229],{"className":256216},[31676],[34,256218],{"className":256219,"style":35942},[31680],[34,256221,3249],{"className":256222},[31685],[34,256224],{"className":256225,"style":32604},[31845],[34,256227,223],{"className":256228},[32024],[34,256230],{"className":256231,"style":32604},[31845],[34,256233,256235,256238,256241,256244,256247],{"className":256234},[31676],[34,256236],{"className":256237,"style":35942},[31680],[34,256239,96087],{"className":256240},[31685],[34,256242],{"className":256243,"style":32604},[31845],[34,256245,223],{"className":256246},[32024],[34,256248],{"className":256249,"style":32604},[31845],[34,256251,256253,256256,256259,256262,256265],{"className":256252},[31676],[34,256254],{"className":256255,"style":35942},[31680],[34,256257,15932],{"className":256258},[31685],[34,256260],{"className":256261,"style":32604},[31845],[34,256263,223],{"className":256264},[32024],[34,256266],{"className":256267,"style":32604},[31845],[34,256269,256271,256274,256277,256280,256283],{"className":256270},[31676],[34,256272],{"className":256273,"style":35942},[31680],[34,256275,165563],{"className":256276},[31685],[34,256278],{"className":256279,"style":32604},[31845],[34,256281,223],{"className":256282},[32024],[34,256284],{"className":256285,"style":32604},[31845],[34,256287,256289,256292,256295,256298,256301],{"className":256288},[31676],[34,256290],{"className":256291,"style":35942},[31680],[34,256293,95437],{"className":256294},[31685],[34,256296],{"className":256297,"style":32604},[31845],[34,256299,223],{"className":256300},[32024],[34,256302],{"className":256303,"style":32604},[31845],[34,256305,256307,256310,256313,256316,256319],{"className":256306},[31676],[34,256308],{"className":256309,"style":35942},[31680],[34,256311,256186],{"className":256312},[31685],[34,256314],{"className":256315,"style":32604},[31845],[34,256317,223],{"className":256318},[32024],[34,256320],{"className":256321,"style":32604},[31845],[34,256323,256325,256328,256331,256334,256337],{"className":256324},[31676],[34,256326],{"className":256327,"style":32617},[31680],[34,256329,162212],{"className":256330},[31685],[34,256332],{"className":256333,"style":31846},[31845],[34,256335,202],{"className":256336},[31850],[34,256338],{"className":256339,"style":31846},[31845],[34,256341,256343,256346],{"className":256342},[31676],[34,256344],{"className":256345,"style":32617},[31680],[34,256347,256195],{"className":256348},[31685],"。\n平均长度：",[34,256351,256353,256376],{"className":256352},[31642],[34,256354,256356],{"className":256355},[31646],[31648,256357,256358],{"xmlns":31650},[31652,256359,256360,256373],{},[31655,256361,256362,256364,256366,256368,256370],{},[31749,256363,256195],{},[31658,256365,38080],{"mathvariant":38079},[31749,256367,36723],{},[31662,256369,168560],{},[31749,256371,256372],{},"41.7",[31665,256374,256375],{"encoding":31667},"292 \u002F 7 \\approx 41.7",[34,256377,256379,256398],{"className":256378,"ariaHidden":758},[31672],[34,256380,256382,256385,256389,256392,256395],{"className":256381},[31676],[34,256383],{"className":256384,"style":31774},[31680],[34,256386,256388],{"className":256387},[31685],"292\u002F7",[34,256390],{"className":256391,"style":31846},[31845],[34,256393,168560],{"className":256394},[31850],[34,256396],{"className":256397,"style":31846},[31845],[34,256399,256401,256404],{"className":256400},[31676],[34,256402],{"className":256403,"style":32617},[31680],[34,256405,256372],{"className":256406},[31685],[108,256408,256409,256412,256413,256441,256442,256470,256471,256499,256500,256528,256529,256557,256558,256586,256587,256615,256616,256349,256799,347],{},[2627,256410,256411],{},"最短寻道时间优先 (SSTF - Shortest Seek Time First):"," 每次选择离当前磁头最近的磁道。\n初始磁头在 40。队列中有 40 本身，所以首先满足 40（移动 0）。\n路线：40 ",[34,256414,256416,256429],{"className":256415},[31642],[34,256417,256419],{"className":256418},[31646],[31648,256420,256421],{"xmlns":31650},[31652,256422,256423,256427],{},[31655,256424,256425],{},[31662,256426,33239],{},[31665,256428,119689],{"encoding":31667},[34,256430,256432],{"className":256431,"ariaHidden":758},[31672],[34,256433,256435,256438],{"className":256434},[31676],[34,256436],{"className":256437,"style":73089},[31680],[34,256439,33239],{"className":256440},[31850]," 40(距离0) ",[34,256443,256445,256458],{"className":256444},[31642],[34,256446,256448],{"className":256447},[31646],[31648,256449,256450],{"xmlns":31650},[31652,256451,256452,256456],{},[31655,256453,256454],{},[31662,256455,33239],{},[31665,256457,119689],{"encoding":31667},[34,256459,256461],{"className":256460,"ariaHidden":758},[31672],[34,256462,256464,256467],{"className":256463},[31676],[34,256465],{"className":256466,"style":73089},[31680],[34,256468,33239],{"className":256469},[31850]," 44(距离4) ",[34,256472,256474,256487],{"className":256473},[31642],[34,256475,256477],{"className":256476},[31646],[31648,256478,256479],{"xmlns":31650},[31652,256480,256481,256485],{},[31655,256482,256483],{},[31662,256484,33239],{},[31665,256486,119689],{"encoding":31667},[34,256488,256490],{"className":256489,"ariaHidden":758},[31672],[34,256491,256493,256496],{"className":256492},[31676],[34,256494],{"className":256495,"style":73089},[31680],[34,256497,33239],{"className":256498},[31850]," 20(距离24) ",[34,256501,256503,256516],{"className":256502},[31642],[34,256504,256506],{"className":256505},[31646],[31648,256507,256508],{"xmlns":31650},[31652,256509,256510,256514],{},[31655,256511,256512],{},[31662,256513,33239],{},[31665,256515,119689],{"encoding":31667},[34,256517,256519],{"className":256518,"ariaHidden":758},[31672],[34,256520,256522,256525],{"className":256521},[31676],[34,256523],{"className":256524,"style":73089},[31680],[34,256526,33239],{"className":256527},[31850]," 12(距离8) ",[34,256530,256532,256545],{"className":256531},[31642],[34,256533,256535],{"className":256534},[31646],[31648,256536,256537],{"xmlns":31650},[31652,256538,256539,256543],{},[31655,256540,256541],{},[31662,256542,33239],{},[31665,256544,119689],{"encoding":31667},[34,256546,256548],{"className":256547,"ariaHidden":758},[31672],[34,256549,256551,256554],{"className":256550},[31676],[34,256552],{"className":256553,"style":73089},[31680],[34,256555,33239],{"className":256556},[31850]," 4(距离8) ",[34,256559,256561,256574],{"className":256560},[31642],[34,256562,256564],{"className":256563},[31646],[31648,256565,256566],{"xmlns":31650},[31652,256567,256568,256572],{},[31655,256569,256570],{},[31662,256571,33239],{},[31665,256573,119689],{"encoding":31667},[34,256575,256577],{"className":256576,"ariaHidden":758},[31672],[34,256578,256580,256583],{"className":256579},[31676],[34,256581],{"className":256582,"style":73089},[31680],[34,256584,33239],{"className":256585},[31850]," 76(距离72) ",[34,256588,256590,256603],{"className":256589},[31642],[34,256591,256593],{"className":256592},[31646],[31648,256594,256595],{"xmlns":31650},[31652,256596,256597,256601],{},[31655,256598,256599],{},[31662,256600,33239],{},[31665,256602,119689],{"encoding":31667},[34,256604,256606],{"className":256605,"ariaHidden":758},[31672],[34,256607,256609,256612],{"className":256608},[31676],[34,256610],{"className":256611,"style":73089},[31680],[34,256613,33239],{"className":256614},[31850]," 80(距离4)。\n总距离：",[34,256617,256619,256661],{"className":256618},[31642],[34,256620,256622],{"className":256621},[31646],[31648,256623,256624],{"xmlns":31650},[31652,256625,256626,256658],{},[31655,256627,256628,256630,256632,256634,256636,256638,256640,256642,256644,256646,256648,256650,256652,256654,256656],{},[31749,256629,459],{},[31662,256631,223],{},[31749,256633,15932],{},[31662,256635,223],{},[31749,256637,96087],{},[31662,256639,223],{},[31749,256641,60525],{},[31662,256643,223],{},[31749,256645,60525],{},[31662,256647,223],{},[31749,256649,96800],{},[31662,256651,223],{},[31749,256653,15932],{},[31662,256655,202],{},[31749,256657,162217],{},[31665,256659,256660],{"encoding":31667},"0 + 4 + 24 + 8 + 8 + 72 + 4 = 120",[34,256662,256664,256682,256700,256718,256736,256754,256772,256790],{"className":256663,"ariaHidden":758},[31672],[34,256665,256667,256670,256673,256676,256679],{"className":256666},[31676],[34,256668],{"className":256669,"style":35942},[31680],[34,256671,459],{"className":256672},[31685],[34,256674],{"className":256675,"style":32604},[31845],[34,256677,223],{"className":256678},[32024],[34,256680],{"className":256681,"style":32604},[31845],[34,256683,256685,256688,256691,256694,256697],{"className":256684},[31676],[34,256686],{"className":256687,"style":35942},[31680],[34,256689,15932],{"className":256690},[31685],[34,256692],{"className":256693,"style":32604},[31845],[34,256695,223],{"className":256696},[32024],[34,256698],{"className":256699,"style":32604},[31845],[34,256701,256703,256706,256709,256712,256715],{"className":256702},[31676],[34,256704],{"className":256705,"style":35942},[31680],[34,256707,96087],{"className":256708},[31685],[34,256710],{"className":256711,"style":32604},[31845],[34,256713,223],{"className":256714},[32024],[34,256716],{"className":256717,"style":32604},[31845],[34,256719,256721,256724,256727,256730,256733],{"className":256720},[31676],[34,256722],{"className":256723,"style":35942},[31680],[34,256725,60525],{"className":256726},[31685],[34,256728],{"className":256729,"style":32604},[31845],[34,256731,223],{"className":256732},[32024],[34,256734],{"className":256735,"style":32604},[31845],[34,256737,256739,256742,256745,256748,256751],{"className":256738},[31676],[34,256740],{"className":256741,"style":35942},[31680],[34,256743,60525],{"className":256744},[31685],[34,256746],{"className":256747,"style":32604},[31845],[34,256749,223],{"className":256750},[32024],[34,256752],{"className":256753,"style":32604},[31845],[34,256755,256757,256760,256763,256766,256769],{"className":256756},[31676],[34,256758],{"className":256759,"style":35942},[31680],[34,256761,96800],{"className":256762},[31685],[34,256764],{"className":256765,"style":32604},[31845],[34,256767,223],{"className":256768},[32024],[34,256770],{"className":256771,"style":32604},[31845],[34,256773,256775,256778,256781,256784,256787],{"className":256774},[31676],[34,256776],{"className":256777,"style":32617},[31680],[34,256779,15932],{"className":256780},[31685],[34,256782],{"className":256783,"style":31846},[31845],[34,256785,202],{"className":256786},[31850],[34,256788],{"className":256789,"style":31846},[31845],[34,256791,256793,256796],{"className":256792},[31676],[34,256794],{"className":256795,"style":32617},[31680],[34,256797,162217],{"className":256798},[31685],[34,256800,256802,256825],{"className":256801},[31642],[34,256803,256805],{"className":256804},[31646],[31648,256806,256807],{"xmlns":31650},[31652,256808,256809,256822],{},[31655,256810,256811,256813,256815,256817,256819],{},[31749,256812,162217],{},[31658,256814,38080],{"mathvariant":38079},[31749,256816,36723],{},[31662,256818,168560],{},[31749,256820,256821],{},"17.1",[31665,256823,256824],{"encoding":31667},"120 \u002F 7 \\approx 17.1",[34,256826,256828,256847],{"className":256827,"ariaHidden":758},[31672],[34,256829,256831,256834,256838,256841,256844],{"className":256830},[31676],[34,256832],{"className":256833,"style":31774},[31680],[34,256835,256837],{"className":256836},[31685],"120\u002F7",[34,256839],{"className":256840,"style":31846},[31845],[34,256842,168560],{"className":256843},[31850],[34,256845],{"className":256846,"style":31846},[31845],[34,256848,256850,256853],{"className":256849},[31676],[34,256851],{"className":256852,"style":32617},[31680],[34,256854,256821],{"className":256855},[31685],[108,256857,256858,256861,256862,256441,256890,256470,256918,256946,256947,256975,256976,257004,257005,256528,257033,257061,257062,256349,257246,39984,257303,257305],{},[2627,256859,256860],{},"扫描算法 (SCAN \u002F Elevator Algorithm):"," 磁头单向移动直到遇到最高需求，再反向。\n初始磁头在 40，方向递增。\n路线：40 ",[34,256863,256865,256878],{"className":256864},[31642],[34,256866,256868],{"className":256867},[31646],[31648,256869,256870],{"xmlns":31650},[31652,256871,256872,256876],{},[31655,256873,256874],{},[31662,256875,33239],{},[31665,256877,119689],{"encoding":31667},[34,256879,256881],{"className":256880,"ariaHidden":758},[31672],[34,256882,256884,256887],{"className":256883},[31676],[34,256885],{"className":256886,"style":73089},[31680],[34,256888,33239],{"className":256889},[31850],[34,256891,256893,256906],{"className":256892},[31642],[34,256894,256896],{"className":256895},[31646],[31648,256897,256898],{"xmlns":31650},[31652,256899,256900,256904],{},[31655,256901,256902],{},[31662,256903,33239],{},[31665,256905,119689],{"encoding":31667},[34,256907,256909],{"className":256908,"ariaHidden":758},[31672],[34,256910,256912,256915],{"className":256911},[31676],[34,256913],{"className":256914,"style":73089},[31680],[34,256916,33239],{"className":256917},[31850],[34,256919,256921,256934],{"className":256920},[31642],[34,256922,256924],{"className":256923},[31646],[31648,256925,256926],{"xmlns":31650},[31652,256927,256928,256932],{},[31655,256929,256930],{},[31662,256931,33239],{},[31665,256933,119689],{"encoding":31667},[34,256935,256937],{"className":256936,"ariaHidden":758},[31672],[34,256938,256940,256943],{"className":256939},[31676],[34,256941],{"className":256942,"style":73089},[31680],[34,256944,33239],{"className":256945},[31850]," 76(距离32) ",[34,256948,256950,256963],{"className":256949},[31642],[34,256951,256953],{"className":256952},[31646],[31648,256954,256955],{"xmlns":31650},[31652,256956,256957,256961],{},[31655,256958,256959],{},[31662,256960,33239],{},[31665,256962,119689],{"encoding":31667},[34,256964,256966],{"className":256965,"ariaHidden":758},[31672],[34,256967,256969,256972],{"className":256968},[31676],[34,256970],{"className":256971,"style":73089},[31680],[34,256973,33239],{"className":256974},[31850]," 80(距离4)。到达80后，递增方向再无请求，反向递减。\n",[34,256977,256979,256992],{"className":256978},[31642],[34,256980,256982],{"className":256981},[31646],[31648,256983,256984],{"xmlns":31650},[31652,256985,256986,256990],{},[31655,256987,256988],{},[31662,256989,33239],{},[31665,256991,119689],{"encoding":31667},[34,256993,256995],{"className":256994,"ariaHidden":758},[31672],[34,256996,256998,257001],{"className":256997},[31676],[34,256999],{"className":257000,"style":73089},[31680],[34,257002,33239],{"className":257003},[31850]," 20(距离60) ",[34,257006,257008,257021],{"className":257007},[31642],[34,257009,257011],{"className":257010},[31646],[31648,257012,257013],{"xmlns":31650},[31652,257014,257015,257019],{},[31655,257016,257017],{},[31662,257018,33239],{},[31665,257020,119689],{"encoding":31667},[34,257022,257024],{"className":257023,"ariaHidden":758},[31672],[34,257025,257027,257030],{"className":257026},[31676],[34,257028],{"className":257029,"style":73089},[31680],[34,257031,33239],{"className":257032},[31850],[34,257034,257036,257049],{"className":257035},[31642],[34,257037,257039],{"className":257038},[31646],[31648,257040,257041],{"xmlns":31650},[31652,257042,257043,257047],{},[31655,257044,257045],{},[31662,257046,33239],{},[31665,257048,119689],{"encoding":31667},[34,257050,257052],{"className":257051,"ariaHidden":758},[31672],[34,257053,257055,257058],{"className":257054},[31676],[34,257056],{"className":257057,"style":73089},[31680],[34,257059,33239],{"className":257060},[31850]," 4(距离8)。\n总距离：",[34,257063,257065,257108],{"className":257064},[31642],[34,257066,257068],{"className":257067},[31646],[31648,257069,257070],{"xmlns":31650},[31652,257071,257072,257105],{},[31655,257073,257074,257076,257078,257080,257082,257084,257086,257088,257090,257092,257094,257096,257098,257100,257102],{},[31749,257075,459],{},[31662,257077,223],{},[31749,257079,15932],{},[31662,257081,223],{},[31749,257083,173486],{},[31662,257085,223],{},[31749,257087,15932],{},[31662,257089,223],{},[31749,257091,167664],{},[31662,257093,223],{},[31749,257095,60525],{},[31662,257097,223],{},[31749,257099,60525],{},[31662,257101,202],{},[31749,257103,257104],{},"116",[31665,257106,257107],{"encoding":31667},"0 + 4 + 32 + 4 + 60 + 8 + 8 = 116",[34,257109,257111,257129,257147,257165,257183,257201,257219,257237],{"className":257110,"ariaHidden":758},[31672],[34,257112,257114,257117,257120,257123,257126],{"className":257113},[31676],[34,257115],{"className":257116,"style":35942},[31680],[34,257118,459],{"className":257119},[31685],[34,257121],{"className":257122,"style":32604},[31845],[34,257124,223],{"className":257125},[32024],[34,257127],{"className":257128,"style":32604},[31845],[34,257130,257132,257135,257138,257141,257144],{"className":257131},[31676],[34,257133],{"className":257134,"style":35942},[31680],[34,257136,15932],{"className":257137},[31685],[34,257139],{"className":257140,"style":32604},[31845],[34,257142,223],{"className":257143},[32024],[34,257145],{"className":257146,"style":32604},[31845],[34,257148,257150,257153,257156,257159,257162],{"className":257149},[31676],[34,257151],{"className":257152,"style":35942},[31680],[34,257154,173486],{"className":257155},[31685],[34,257157],{"className":257158,"style":32604},[31845],[34,257160,223],{"className":257161},[32024],[34,257163],{"className":257164,"style":32604},[31845],[34,257166,257168,257171,257174,257177,257180],{"className":257167},[31676],[34,257169],{"className":257170,"style":35942},[31680],[34,257172,15932],{"className":257173},[31685],[34,257175],{"className":257176,"style":32604},[31845],[34,257178,223],{"className":257179},[32024],[34,257181],{"className":257182,"style":32604},[31845],[34,257184,257186,257189,257192,257195,257198],{"className":257185},[31676],[34,257187],{"className":257188,"style":35942},[31680],[34,257190,167664],{"className":257191},[31685],[34,257193],{"className":257194,"style":32604},[31845],[34,257196,223],{"className":257197},[32024],[34,257199],{"className":257200,"style":32604},[31845],[34,257202,257204,257207,257210,257213,257216],{"className":257203},[31676],[34,257205],{"className":257206,"style":35942},[31680],[34,257208,60525],{"className":257209},[31685],[34,257211],{"className":257212,"style":32604},[31845],[34,257214,223],{"className":257215},[32024],[34,257217],{"className":257218,"style":32604},[31845],[34,257220,257222,257225,257228,257231,257234],{"className":257221},[31676],[34,257223],{"className":257224,"style":32617},[31680],[34,257226,60525],{"className":257227},[31685],[34,257229],{"className":257230,"style":31846},[31845],[34,257232,202],{"className":257233},[31850],[34,257235],{"className":257236,"style":31846},[31845],[34,257238,257240,257243],{"className":257239},[31676],[34,257241],{"className":257242,"style":32617},[31680],[34,257244,257104],{"className":257245},[31685],[34,257247,257249,257272],{"className":257248},[31642],[34,257250,257252],{"className":257251},[31646],[31648,257253,257254],{"xmlns":31650},[31652,257255,257256,257269],{},[31655,257257,257258,257260,257262,257264,257266],{},[31749,257259,257104],{},[31658,257261,38080],{"mathvariant":38079},[31749,257263,36723],{},[31662,257265,168560],{},[31749,257267,257268],{},"16.6",[31665,257270,257271],{"encoding":31667},"116 \u002F 7 \\approx 16.6",[34,257273,257275,257294],{"className":257274,"ariaHidden":758},[31672],[34,257276,257278,257281,257285,257288,257291],{"className":257277},[31676],[34,257279],{"className":257280,"style":31774},[31680],[34,257282,257284],{"className":257283},[31685],"116\u002F7",[34,257286],{"className":257287,"style":31846},[31845],[34,257289,168560],{"className":257290},[31850],[34,257292],{"className":257293,"style":31846},[31845],[34,257295,257297,257300],{"className":257296},[31676],[34,257298],{"className":257299,"style":32617},[31680],[34,257301,257268],{"className":257302},[31685],[2627,257304,247377],{}," 在 SSTF 和 SCAN 计算时，学生经常忘记“40”既是起点也是请求队列中的一个元素，处理40到40的距离应计入完成了一次请求（距离为0）。除数必须是请求队列的总数 7。",[19,257307,257309],{"id":257308},"_48在生产者消费者问题中消费者进程的两个wait原语的正确顺序为-和","48．在生产者—消费者问题中，消费者进程的两个wait原语的正确顺序为（ ）和（ ）。",[15,257311,257312,257314,257315,257317,257318,257320,257321,257324,257325,257327,257328,68853,257331,257334,257335,257337,257338,39984,257341,257343,257344,347],{},[2627,257313,58173],{}," wait(full) \u002F P(full) 和 wait(mutex) \u002F P(mutex)\n",[2627,257316,246520],{}," 生产者\u002F消费者模型与信号量同步（Semaphore Synchronization）。\n",[2627,257319,246526],{}," 在并发编程中，消费者需要从缓冲区取走数据。它必须先确保缓冲区中有数据可取，即检查资源信号量 ",[31,257322,257323],{},"full","；然后再锁住缓冲区，即申请互斥信号量 ",[31,257326,196586],{},"。\n如果顺序颠倒（先 ",[31,257329,257330],{},"wait(mutex)",[31,257332,257333],{},"wait(full)","），一旦消费者抢到了锁（mutex）但发现缓冲区为空（full=0），它就会抱着锁进入睡眠。而此时生产者想要生产数据，却因为拿不到互斥锁 ",[31,257336,196586],{}," 而被永远阻塞，这就造成了",[2627,257339,257340],{},"死锁 (Deadlock)",[2627,257342,247377],{}," P操作（wait）的顺序生死攸关：",[2627,257345,257346],{},"先同步（检查资源），后互斥（加锁）",[19,257348,257350],{"id":257349},"_49在请求页式管理中当硬件变换机构发现所需的页不在内存时产生-中断处理程序作相应的处理","49．在请求页式管理中，当硬件变换机构发现所需的页不在内存时，产生（ ），中断处理程序作相应的处理。",[15,257352,257353,257355,257356,257358,257359,257361,257362,257365,257366,257368,257369,257372],{},[2627,257354,58173],{}," 缺页中断 (Page fault)\n",[2627,257357,246520],{}," 虚拟内存中的请求调页（Demand paging）。\n",[2627,257360,246526],{}," 虚拟内存允许程序执行时不必全部加载进内存。当硬件的地址变换机构在页表中查找某个页，但发现其“有效位 (Valid bit)”指示该页不在主存中时，它会触发一种特殊的异常或中断——即",[2627,257363,257364],{},"缺页中断 (Page Fault)","。随后，操作系统的缺页中断处理程序会接管CPU，去磁盘将缺失的页面调入主存。\n",[2627,257367,247377],{}," 缺页中断属于内部异常（Trap \u002F 陷入），它是在指令执行中途发生的，并且在处理完成后，系统会",[2627,257370,257371],{},"重新执行","那条引发缺页的指令。",[19,257374,257376],{"id":257375},"_50置换算法是在内存中没有-时被调用的它的目的是选出一个被淘汰的页面","50．置换算法是在内存中没有（ ）时被调用的，它的目的是选出一个被淘汰的页面。",[15,257378,257379,257381,257382,257384,257385,257387,257388,257391,257392,257394],{},[2627,257380,58173],{}," 空闲物理块 (Free frames) 或 空闲页面\n",[2627,257383,246520],{}," 页面置换策略 (Page Replacement Policy)。\n",[2627,257386,246526],{}," 当发生缺页中断且系统需要将新页调入主存时，操作系统首先会去检查“空闲物理块链表 (free-frame list)”。如果内存中仍有充足的空闲帧，就可以直接装入。只有当内存已经塞满，",[2627,257389,257390],{},"没有空闲物理块","时，才必须调用页面置换算法（如LRU、FIFO等），“牺牲”掉一个已在内存中的旧页面以为新页面腾出空间。\n",[2627,257393,247377],{}," 置换不是每次缺页都会发生，它只在“内存已满（无 free frames）”的极端情况下才被激活。",[19,257396,257398],{"id":257397},"_51在段页式存储管理系统中面向-的地址空间是段式划分面向物理实现的地址空间是页式划分","51．在段页式存储管理系统中，面向（ ）的地址空间是段式划分，面向物理实现的地址空间是页式划分。",[15,257400,257401,257403,257404,257406,257407,257409],{},[2627,257402,58173],{}," 用户 (User) 或 程序员 (Programmer)\n",[2627,257405,246520],{}," 段页式结合的虚拟内存 (Combined Paging and Segmentation)。\n",[2627,257408,246526],{}," 分段和分页的设计初衷不同：",[105,257411,257412,257418],{},[108,257413,257414,257417],{},[2627,257415,257416],{},"分页","对用户是透明的，它是为了消除外部碎片，由硬件\u002FOS负责切分，面向的是物理内存的管理效率。",[108,257419,257420,257423,257424,257427,257428,257430],{},[2627,257421,257422],{},"分段","则是对程序员可见的（visible to the programmer）。它将程序按逻辑意义（如代码段、数据段、堆栈段）切分，面向",[2627,257425,257426],{},"用户\u002F程序员","，便于实现模块化、共享与保护。在段页式系统中，逻辑地址依然呈现出段式结构给用户，但在底层被操作系统进一步切分为页以存入物理内存。\n",[2627,257429,247377],{}," 分页是为了“方便系统”，分段是为了“方便用户”。牢记这一设计哲学。",[19,257432,257434],{"id":257433},"_52文件的存储器是分成大小相等的-并以它为单位交换信息","52．文件的存储器是分成大小相等的（ ），并以它为单位交换信息。",[15,257436,257437,257439,257440,257442,257443,257445,257446,257448],{},[2627,257438,58173],{}," 物理块 (Blocks \u002F Disk blocks)\n",[2627,257441,246520],{}," 块设备 (Block-oriented devices) 的信息交互单元。\n",[2627,257444,246526],{}," 磁盘和磁带是典型的块设备。参考资料明确指出：“面向块的设备将信息存储在通常为固定大小的块（blocks）中，并且数据传输也是一次一个块地进行”。文件系统的存储以及主存与磁盘之间的I\u002FO，均是以物理块（也称磁盘块）为最小单位进行的。\n",[2627,257447,247377],{}," 这里的单位不是“字节(Byte)”或“扇区(Sector)”。在文件系统和操作系统的宏观管理中，标准交换单位是逻辑化的“物理块 (Block)”。",[19,257450,257452],{"id":257451},"_53为文件-file-增加执行权限的-unix-命令为","53．为文件 file 增加执行权限的 UNIX 命令为（ ）。",[15,257454,257455,47649,257457,257460,257461,6111,257464,257466,257467,257469,257470,257473,257474,257477,257478,257480],{},[2627,257456,58173],{},[31,257458,257459],{},"chmod +x file"," (或 ",[31,257462,257463],{},"chmod a+x file",[2627,257465,246520],{}," UNIX 权限管理。\n",[2627,257468,246526],{}," 在UNIX系统中，每个文件都有一组权限掩码字段（Permission modes fields）用于控制创建和访问。改变权限的命令是 ",[31,257471,257472],{},"chmod"," (Change Mode)。",[31,257475,257476],{},"+x"," 代表增加（+）执行（eXecute）权限。\n",[2627,257479,247377],{}," 不熟悉 Linux\u002FUNIX 基本命令的学生可能只会写纯理论知识。注意操作系统的实操题。",[19,257482,257484],{"id":257483},"_54显示目录-mydir-中文件的详细信息的-unix-命令为","54．显示目录 mydir 中文件的详细信息的 UNIX 命令为（ ）。",[15,257486,257487,47649,257489,257492,257494,257495,47649,257497,257500,257501,257504,257505,47649,257507,257509],{},[2627,257488,58173],{},[31,257490,257491],{},"ls -l mydir",[2627,257493,246520],{}," UNIX 目录查看命令。\n",[2627,257496,246526],{},[31,257498,257499],{},"ls"," (List) 命令用于列出目录内容。带上 ",[31,257502,257503],{},"-l"," (long format) 选项后，除了文件名，还会详细列出文件的读写权限、硬链接数、所有者、文件大小以及最后修改时间等长格式信息。\n",[2627,257506,247377],{},[31,257508,257503],{}," 是小写的 L，不是数字 1。",[19,257511,257513],{"id":257512},"_55-操作系统提供给编程人员的唯一接口是","55. 操作系统提供给编程人员的唯一接口是（ ）。",[15,257515,257516,257518,257519,257521,257522,257524,257525,257528,257529,39984,257532,257534],{},[2627,257517,58173],{}," 系统调用 (System calls)\n",[2627,257520,246520],{}," 操作系统的双重模式与接口。\n",[2627,257523,246526],{}," 一般用户可以通过命令行或图形界面使用计算机；但对于",[2627,257526,257527],{},"应用程序的编程人员 (Programmers)","，若他们的程序需要在用户模式（User mode）下请求操作系统内核代表其执行特权操作（如读写文件、申请内存），他们能够跨越用户态到内核态的唯一合法桥梁就是发起",[2627,257530,257531],{},"系统调用 (System calls)",[2627,257533,247377],{}," 系统调用本质上是由一组用C\u002F汇编语言写成的 API 组成的。",[19,257536,257538],{"id":257537},"_56-按文件的逻辑存储结构分文件分为有结构文件又称为-和无结构文件又称","56. 按文件的逻辑存储结构分，文件分为有结构文件，又称为（ ）和无结构文件，又称（ ）。",[15,257540,257541,257543,257544,257546,257547,257549],{},[2627,257542,58173],{}," 记录式文件 (Record file) 和 流式文件 (Stream file)\n",[2627,257545,246520],{}," 文件的逻辑结构组织。\n",[2627,257548,246526],{},"\n从操作系统的逻辑视角来看：",[105,257551,257552,257560],{},[108,257553,257554,257555,58269,257558,347],{},"如果文件由一条条具有固定格式的相似数据记录（Record）组成（例如数据库表），我们称之为",[2627,257556,257557],{},"有结构文件",[2627,257559,178212],{},[108,257561,257562,257563,58269,257566,39984,257568,257570],{},"如果操作系统对文件内部内容一无所知，仅仅把它看作是一连串无格式、无间隔的字节流（Stream of bytes）——就像典型的UNIX处理方式那样，我们称之为",[2627,257564,257565],{},"无结构文件",[2627,257567,175354],{},[2627,257569,247377],{}," 这是操作系统的经典二分法，不要和“顺序文件、索引文件”等具体的组织方式混淆。",[19,257572,257574],{"id":257573},"_57-unix的文件系统空闲空间的管理是采用","57. UNIX的文件系统空闲空间的管理是采用（ ）。",[15,257576,257577,257579,257580,257582,257583,257585,257586,257589],{},[2627,257578,58173],{}," 成组链接法 (Grouped Link Method)\n",[2627,257581,246520],{}," 磁盘空闲空间管理策略。\n",[2627,257584,246526],{}," 为了记录哪些磁盘块是空闲的，普通的操作系统可能采用空闲表法（Free block list）或位示图法（Bit tables）。然而在经典的 UNIX 系统 V (SVR4) 文件系统中，为了兼顾大容量磁盘和内存开销，采用的是一种巧妙的结合体：",[2627,257587,257588],{},"成组链接法","。它将空闲块分组，每组第一块记录下一组的所有空闲块号，大幅减少了文件操作时的磁盘 I\u002FO 寻道时间。",[19,257591,257593],{"id":257592},"_58-逻辑设备表lut的主要功能是实现","58. 逻辑设备表（LUT）的主要功能是实现（ ）。",[15,257595,257596,257598,257599,257601,257602,257604,257605,257607,257608,257610],{},[2627,257597,58173],{}," 设备独立性 (Device independence) 或 逻辑设备到物理设备的映射\n",[2627,257600,246520],{}," 逻辑I\u002FO模块的职责设计。\n",[2627,257603,246526],{}," 参考资料指出：“逻辑 I\u002FO 模块将设备视为逻辑资源，并允许用户进程通过设备标识符与之交互，而不用关心底层具体物理控制细节”。操作系统通过引入逻辑设备表（LUT），在程序中使用“逻辑设备名”，而在运行时通过LUT将其动态映射到真正的“物理设备名”上。这实现了",[2627,257606,233864],{},"，使得用户程序可以不加修改地适配不同的外设。\n",[2627,257609,247377],{}," 深刻理解“独立性”一词，即软件与具体硬件设备的解耦。",[19,257612,257614],{"id":257613},"_59-文件系统中用于文件的描述和控制并与文件一一对应的是","59. 文件系统中，用于文件的描述和控制并与文件一一对应的是（ ）。",[15,257616,257617,257619,257620,257622,257623,257625,257626,257629,257630,39984,257633,257635],{},[2627,257618,58173],{}," 文件控制块 (FCB - File Control Block) 或 索引节点 (inode)\n",[2627,257621,246520],{}," 文件的控制结构。\n",[2627,257624,246526],{}," 正如进程需要“进程控制块 (PCB)”来被操作系统管理一样，每一个文件也必须有一个专属的数据结构来存储它的属性信息（如大小、权限、磁盘位置指针等）。在一般OS理论中称为",[2627,257627,257628],{},"文件控制块 (FCB)","。而在 UNIX\u002FLinux 系统中，这个包含了操作系统所需关键控制信息的结构被称为 ",[2627,257631,257632],{},"索引节点 (inode)",[2627,257634,247377],{}," FCB与文件是绝对一一对应的。它是文件存在的唯一标志。",[19,257637,257639],{"id":257638},"_60-所谓-是一块能控制一台或多台外围设备与cpu并行工作的硬件","60. 所谓（ ），是一块能控制一台或多台外围设备与CPU并行工作的硬件。",[15,257641,257642,257644,257645,257647,257648,257650,257651,257654,257655,257657],{},[2627,257643,58173],{}," 通道 (I\u002FO Channel) 或 I\u002FO 处理器 (I\u002FO Processor)\n",[2627,257646,246520],{}," I\u002FO 模块的演进。\n",[2627,257649,246526],{}," 为了彻底将 CPU 从繁重的 I\u002FO 控制中解放出来，I\u002FO 模块进化成了一个拥有自己定制专用指令集的独立处理器。系统配置中这种独立的I\u002FO处理器，通常被称为",[2627,257652,257653],{},"通道 (Channel)","。它能执行通道程序，独立控制一台或多台设备，从而实现设备与主CPU的高度并行工作。\n",[2627,257656,247377],{}," DMA控制器也能并行工作，但通道比DMA更高级：DMA只能实现一次块传输，而通道可以通过执行一套“通道程序”完成复杂的、多次跨设备的传输。",[19,257659,257661],{"id":257660},"_61多道程序系统中有关程序的执行可能受到另一个程序的","61．多道程序系统中，有关程序的执行可能受到另一个程序的（ ）。",[15,257663,257664,257666,257667,257669,257670,257672,257673,39984,257676,257678],{},[2627,257665,58173],{}," 制约 (Constraint) 或 干扰 (Interference)\n",[2627,257668,246520],{}," 并发执行时的制约关系。\n",[2627,257671,246526],{}," 当多个进程在内存中共享系统资源时（多道程序设计），如果没有有效的同步和互斥机制，一个程序的执行就必然会受到其他程序执行的相互影响。参考资料提到，操作系统必须保护每个进程的数据和物理资源免受其他进程的“意外干扰（unintended interference）”。这种由于竞争资源或需要协作而产生的互相等待或互相阻塞现象，统称为程序的",[2627,257674,257675],{},"制约",[2627,257677,247377],{}," 制约分为两种：直接制约（因合作同步而等待）和间接制约（因竞争互斥资源而等待）。",[19,257680,257682],{"id":257681},"_62进程有三个特性它们是动态性并发性和","62．进程有三个特性，它们是动态性、并发性和（ ）。",[15,257684,257685,257687,257688,257690,257691,257693,257694,257697,257698,257701,257702,257705,257706,257709,257710,257712],{},[2627,257686,58173],{}," 独立性 (Independence) 或 异步性 (Asynchrony)\n",[2627,257689,246520],{}," 进程 (Process) 的基本属性。\n",[2627,257692,246526],{}," 虽然“进程”的确切特征总结因教材而异，但在经典的计算机科学考点中，进程的主要特性通常概括为：",[2627,257695,257696],{},"动态性","（进程是程序的一次执行过程，有生命周期）、",[2627,257699,257700],{},"并发性","（多个进程能在内存中同时推进）、",[2627,257703,257704],{},"独立性","（进程是系统进行资源分配和调度的一个独立单位）、以及",[2627,257707,257708],{},"异步性","（进程按照各自独立的、不可预知的速度向前推进）。\n",[2627,257711,247377],{}," “异步性”正是操作系统必须引入信号量、管程等同步机制的根本原因，因为你无法预判哪个进程跑得更快。",[19,257714,257716],{"id":257715},"_63计算机系统中处理器是按程序状态字寄存器中的指示控制","63．计算机系统中，处理器是按程序状态字寄存器中的指示控制（ ）。",[15,257718,257719,257721,257722,257724,257725,257727,257728,257734,257735,257737],{},[2627,257720,58173],{}," 执行模式 (Execution mode \u002F 工作状态) 或 程序的执行\n",[2627,257723,246520],{}," 程序状态字 (Program Status Word, PSW) 寄存器的功能。\n",[2627,257726,246526],{}," PSW 寄存器包含了当前运行进程的状态信息。除了保存数学运算的条件码（如溢出、进位），它最重要的任务是",[2627,257729,257730,257731,257733],{},"控制处理器的当前执行模式（Kernel mode 还是 User mode）",[2627,257732,223291],{},"中断是否被允许（Interrupt enable\u002Fdisable）","。处理器根据PSW中的这一位来决定当前是否有权限执行特权指令。\n",[2627,257736,247377],{}," PSW不仅仅是一个“结果存储器”，它是一个控制CPU行为最高权限的“开关面板”。",[19,257739,257741],{"id":257740},"_64把逻辑地址转换成绝对地址的工作称为","64．把逻辑地址转换成绝对地址的工作称为（ ）。",[15,257743,257744,257746,257747,257749,257750,257752,257753,39984,257755,257757],{},[2627,257745,58173],{}," 重定位 (Relocation) 或 地址映射 (Address Mapping)\n",[2627,257748,246520],{}," 内存地址转换 (Address Translation)。\n",[2627,257751,246526],{}," 程序员编写程序和编译器生成代码时，使用的通常是从0开始的相对\u002F逻辑地址。然而，当程序被装入到物理内存中的某个随机区域时，所有的地址都必须加上基址进行调整。操作系统自动划分内存并在程序运行中调整相关地址的过程，就叫做",[2627,257754,243417],{},[2627,257756,247377],{}," 重定位分为静态重定位（装入内存时一次性修改）和动态重定位（执行每条指令时由硬件MMU实时修改）。",[19,257759,257761],{"id":257760},"_65-如果把一本词典的内容作为一个文件存放每个单词和对它的解释组成一个记录为了便于该词典的使用者迅速查到所需的单词这个文件的存储结构采用-文件结构比较合适","65. 如果把一本词典的内容作为一个文件存放，每个单词和对它的解释组成一个记录。为了便于该词典的使用者迅速查到所需的单词，这个文件的存储结构采用（ ）文件结构比较合适。",[15,257763,257764,257766,257767,257769,257770,257772,257773,257780],{},[2627,257765,58173],{}," 索引顺序 (Indexed sequential) (或者 哈希\u002F直接文件 Hashed file)\n",[2627,257768,246520],{}," 文件的逻辑组织方式。\n",[2627,257771,246526],{}," 词典具有两个非常显著的特点：第一，它的记录是按字典序（如A到Z的字母顺序）排列的；第二，用户极其需要“随机、快速的单条记录查找”。\n仅仅是“顺序文件”，查找时间会随着文件变大而极其漫长。参考资料建议，如果在顺序文件的基础上，保留记录按关键字段排序的特性，同时增加一个",[2627,257774,257775,257776,257779],{},"索引表（Index）",[2627,257777,257778],{},"来支持随机快速访问，这种结构即为","索引顺序文件 (Indexed sequential file)","。它完美契合了词典的存取需求。",[19,257782,257784],{"id":257783},"_66-操作系统为用户提供了三种接口它们是-接口-接口和图形接口","66. 操作系统为用户提供了三种接口，它们是( )接口，( )接口和图形接口。",[15,257786,257787,257789,257790,257792,257793,257795],{},[2627,257788,58173],{}," 命令(行) (Command \u002F Command line) 接口，系统调用 (System call) 或 程序 (Program) 接口\n",[2627,257791,246520],{}," 操作系统提供给外界的服务接口类型。\n",[2627,257794,246526],{}," 操作系统需要服务两种受众：",[2230,257797,257798,257813],{},[108,257799,257800,257801,257804,257805,257808,257809,257812],{},"面对",[2627,257802,257803],{},"普通用户","，系统提供了方便操作的 ",[2627,257806,257807],{},"命令接口"," (如 Bash, CMD) 和 ",[2627,257810,257811],{},"图形用户接口"," (GUI，如 Windows 桌面)。",[108,257814,257800,257815,257818,257819,257822,257823,257825],{},[2627,257816,257817],{},"程序员\u002F应用程序","，系统提供了一套由核心函数构成的 ",[2627,257820,257821],{},"程序接口 \u002F 系统调用 (System calls)"," 供其在代码中请求操作系统服务。\n",[2627,257824,247377],{}," “程序接口”实际上就是指API（应用编程接口）或者系统调用。",[19,257827,257829],{"id":257828},"_67-计算机系统能及时处理过程控制数据并做出响应的操作系统称为","67. 计算机系统能及时处理过程控制数据并做出响应的操作系统称为( )。",[15,257831,257832,257834,257835,257837,257838,257840,257841,257844],{},[2627,257833,58173],{}," 实时操作系统 (Real-Time Operating System \u002F RTOS)\n",[2627,257836,246520],{}," 操作系统分类。\n",[2627,257839,246526],{}," 参考资料在探讨调度时提到了一种特殊的系统：如果任务未能满足其“截止时间（deadline）”，可能会对系统造成不期望的破坏甚至致命错误，这类任务属于“硬实时任务（Hard real-time task）”。能够在这个严苛的时间约束内及时接收、处理外部控制数据并做出确定性响应的OS，就是",[2627,257842,257843],{},"实时操作系统","。常见于工业自动控制、航空航天等领域。",[19,257846,257848],{"id":257847},"_68-和-是操作系统的两个基本特征两者之间互为条件","68. ( )和( )是操作系统的两个基本特征，两者之间互为条件。",[15,257850,257851,257853,257854,257856,257857,257859],{},[2627,257852,58173],{}," 并发 (Concurrency) 和 共享 (Sharing)\n",[2627,257855,246520],{}," 操作系统的基石特性。\n",[2627,257858,246526],{}," 现代操作系统最核心的两个特征：",[105,257861,257862,257868],{},[108,257863,257864,257867],{},[2627,257865,257866],{},"并发："," 多个程序在宏观时间段内同时处于运行状态。",[108,257869,257870,257872,257873,257875],{},[2627,257871,249727],{}," 系统中的软硬件资源可供内存中多个并发执行的程序共同使用。\n两者互为依存条件：如果系统不能并发，就谈不上让多个程序去共享资源；而如果系统不能提供有效的资源共享机制，进程之间必然冲突崩溃，并发也就无法实现。\n",[2627,257874,247377],{}," 并发（Concurrency）和并行（Parallelism）不同。并发是在单个CPU上交替执行，而并行是在多个CPU上同时执行。",[19,257877,257879],{"id":257878},"_69-在一段时间内只允许一个进程访问的资源称为","69. 在一段时间内只允许一个进程访问的资源，称为( )。",[15,257881,257882,257884,257885,257887,257888,257890,257891,39984,257894,257896,257897,257900],{},[2627,257883,58173],{}," 临界资源 (Critical resource)\n",[2627,257886,246520],{}," 互斥与共享资源。\n",[2627,257889,246526],{}," 系统中有许多资源（如打印机、共享内存变量等），如果允许多个进程同时去写或修改，就会导致数据错乱。参考资料中在讨论互斥（Mutual exclusion）时强调：“一次仅允许一个进程使用的资源（Only one process may use a resource at a time）”。这种必须排他性、互斥地被访问的共享资源，在操作系统中被定义为",[2627,257892,257893],{},"临界资源",[2627,257895,247377],{}," 访问临界资源的那段“程序代码”，则被称为",[2627,257898,257899],{},"临界区 (Critical Section)","。务必分清“资源”和“代码”的概念区别。",[19,257902,257904],{"id":257903},"_70-为了使多个进程能有效地同时处理输入和输出最好使用","70. 为了使多个进程能有效地同时处理输入和输出，最好使用( )。",[15,257906,257907,257909,257910,257912,257913,257915,257916,257919],{},[2627,257908,58173],{}," SPOOLing技术 (或 缓冲池技术 Buffering)\n",[2627,257911,246520],{}," 改善 I\u002FO 效率并实现设备共享的高级技术。\n",[2627,257914,246526],{}," 当多个进程同时想要向同一台独占设备（如打印机）输出时，如果让它们排队抢夺该设备的物理控制权，会导致CPU被极大地浪费。通过 ",[2627,257917,257918],{},"SPOOLing（假脱机）"," 技术（参考资料中描述的带有输入和输出缓冲区的 spooling 系统），操作系统在高速磁盘上划出区域作为缓冲，截获所有进程的输出请求并暂存。这样，对用户进程而言，就像它们“同时”完成了输出，随后由后台的输出守护进程慢慢处理。SPOOLing 将独占设备成功地改造成了可共享的虚拟设备。",{"title":29,"searchDepth":43,"depth":43,"links":257921},[257922,257923,257924,257925,257926,257927,257928,257929,257930,257931,257932,257933,257934,257935,257936,257937,257938,257939,257940,257941,257942,257943,257944,257945,257946,257947,257948,257949,257950,257951,257952,257953,257954,257955,257956,257957,257958,257959,257960,257961,257962,257963,257964,257965,257966,257967,257968,257969,257970,257971,257972,257973,257974,257975,257976,257977,257978,257979,257980,257981,257982,257983,257984,257985,257986,257987,257988,257989,257990,257991],{"id":248258,"depth":43,"text":248259},{"id":249297,"depth":43,"text":249298},{"id":249329,"depth":43,"text":249330},{"id":249347,"depth":43,"text":249348},{"id":249407,"depth":43,"text":249408},{"id":249445,"depth":43,"text":249446},{"id":249483,"depth":43,"text":249484},{"id":249618,"depth":43,"text":249619},{"id":249652,"depth":43,"text":249653},{"id":249674,"depth":43,"text":249675},{"id":249706,"depth":43,"text":249707},{"id":249740,"depth":43,"text":249741},{"id":249775,"depth":43,"text":249776},{"id":249797,"depth":43,"text":249798},{"id":249829,"depth":43,"text":249830},{"id":249864,"depth":43,"text":249865},{"id":249899,"depth":43,"text":249900},{"id":249937,"depth":43,"text":249938},{"id":249959,"depth":43,"text":249960},{"id":249981,"depth":43,"text":249982},{"id":250030,"depth":43,"text":250031},{"id":250047,"depth":43,"text":250048},{"id":250149,"depth":43,"text":250150},{"id":250338,"depth":43,"text":250339},{"id":250463,"depth":43,"text":250464},{"id":251036,"depth":43,"text":251037},{"id":252699,"depth":43,"text":252700},{"id":252980,"depth":43,"text":252981},{"id":252998,"depth":43,"text":252999},{"id":254433,"depth":43,"text":254434},{"id":254521,"depth":43,"text":254522},{"id":254990,"depth":43,"text":254991},{"id":255008,"depth":43,"text":255009},{"id":255138,"depth":43,"text":255139},{"id":255170,"depth":43,"text":255171},{"id":255199,"depth":43,"text":255200},{"id":255228,"depth":43,"text":255229},{"id":255246,"depth":43,"text":255247},{"id":255268,"depth":43,"text":255269},{"id":255436,"depth":43,"text":255437},{"id":255454,"depth":43,"text":255455},{"id":255472,"depth":43,"text":255473},{"id":255490,"depth":43,"text":255491},{"id":255508,"depth":43,"text":255509},{"id":255526,"depth":43,"text":255527},{"id":255552,"depth":43,"text":255553},{"id":255570,"depth":43,"text":255571},{"id":257308,"depth":43,"text":257309},{"id":257349,"depth":43,"text":257350},{"id":257375,"depth":43,"text":257376},{"id":257397,"depth":43,"text":257398},{"id":257433,"depth":43,"text":257434},{"id":257451,"depth":43,"text":257452},{"id":257483,"depth":43,"text":257484},{"id":257512,"depth":43,"text":257513},{"id":257537,"depth":43,"text":257538},{"id":257573,"depth":43,"text":257574},{"id":257592,"depth":43,"text":257593},{"id":257613,"depth":43,"text":257614},{"id":257638,"depth":43,"text":257639},{"id":257660,"depth":43,"text":257661},{"id":257681,"depth":43,"text":257682},{"id":257715,"depth":43,"text":257716},{"id":257740,"depth":43,"text":257741},{"id":257760,"depth":43,"text":257761},{"id":257783,"depth":43,"text":257784},{"id":257828,"depth":43,"text":257829},{"id":257847,"depth":43,"text":257848},{"id":257878,"depth":43,"text":257879},{"id":257903,"depth":43,"text":257904},{"order":49},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Ffill-in-the-blanks",{"title":248251,"description":29},"source\u002Fcs\u002Fos\u002Freview\u002Ffill-in-the-blanks","IMbg6j_0mByJgqmYKwgVETE6BH7JmH9Q0U8QU9OenWQ",{"id":257998,"title":257999,"body":258000,"description":29,"extension":85,"meta":262973,"navigation":52,"path":262974,"seo":262975,"stem":262976,"__hash__":262977},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fmcq.md","选择题",{"type":8,"value":258001,"toc":262901},[258002,258004,258008,258013,258018,258086,258090,258095,258100,258105,258109,258114,258119,258124,258128,258133,258138,258143,258147,258155,258159,258164,258168,258173,258177,258182,258202,258206,258216,258220,258225,258229,258243,258247,258252,258256,258261,258265,258374,258378,258383,258387,258392,258396,258401,258405,258410,258426,258430,258435,258439,258444,258448,258457,258461,258466,259241,259245,259250,259254,259530,259534,259539,259543,259548,259552,259557,259561,259566,259570,259575,259579,259584,259588,259595,259599,259604,259608,259616,259620,259625,259654,259658,259672,259676,259681,259685,259690,259694,259699,259703,259708,259712,259721,259725,259730,259734,259739,259743,259993,259997,260138,260142,260147,260151,260156,260419,260423,260428,260432,260781,260785,260790,260794,260799,260837,260841,260846,260850,260859,260863,260868,260872,260877,260898,260902,260909,260913,260925,260929,260934,260938,260943,260947,260952,260956,260961,260965,260970,260974,260979,260983,260988,260992,260997,261001,261006,261010,261015,261019,261033,261037,261042,261046,261054,261058,261063,261067,261081,261085,261090,261108,261112,261117,261121,261126,261130,261135,261139,261144,261148,261156,261160,261165,261169,261174,261178,261183,261187,261192,261196,261201,261205,261213,261217,261222,261226,261231,261235,261240,261244,261249,261253,261258,261262,261267,261271,261276,261280,261285,261289,261294,261298,261303,261307,261312,261316,261321,261325,261330,261334,261339,261343,261348,261352,261357,261361,261368,261372,261380,261384,261482,261486,261491,261495,261500,261504,261512,261516,261521,261525,261533,261537,261542,261546,261551,261555,261560,261564,261569,261573,261578,261582,261587,261591,261596,261600,261604,261608,261613,261617,261625,261629,261637,261660,261664,261669,261673,261685,261689,261694,261698,261711,261715,261720,261724,261736,261740,261745,261749,261765,261769,261783,261787,261804,261808,261822,261826,261831,261971,261975,261980,261984,262824,262828,262833,262837,262850,262854,262868,262872,262881],[11,258003,257999],{"id":257999},[19,258005,258007],{"id":258006},"_1若处理器有32位地址则它的虚拟地址空间为-字节","1．若处理器有32位地址，则它的虚拟地址空间为（ ）字节。",[15,258009,258010],{},[2627,258011,258012],{},"A 2GB B 4GB C 100KB D 640KB",[15,258014,258015,258017],{},[2627,258016,58173],{}," B",[15,258019,258020,258023,258024,258085],{},[2627,258021,258022],{},"详细解析：","\n在计算机系统中，虚拟地址空间的大小由地址总线的位数决定。根据参考资料，“在32位平台上，每个Windows用户进程拥有独立的32位地址空间，允许每个进程使用4GB的虚拟内存（",[34,258025,258027,258044],{"className":258026},[31642],[34,258028,258030],{"className":258029},[31646],[31648,258031,258032],{"xmlns":31650},[31652,258033,258034,258042],{},[31655,258035,258036],{},[37624,258037,258038,258040],{},[31749,258039,768],{},[31749,258041,173486],{},[31665,258043,201449],{"encoding":31667},[34,258045,258047],{"className":258046,"ariaHidden":758},[31672],[34,258048,258050,258053],{"className":258049},[31676],[34,258051],{"className":258052,"style":37774},[31680],[34,258054,258056,258059],{"className":258055},[31685],[34,258057,768],{"className":258058},[31685],[34,258060,258062],{"className":258061},[31792],[34,258063,258065],{"className":258064},[31796],[34,258066,258068],{"className":258067},[31801],[34,258069,258071],{"className":258070,"style":37774},[31805],[34,258072,258073,258076],{"style":37777},[34,258074],{"className":258075,"style":31814},[31813],[34,258077,258079],{"className":258078},[31818,31819,31820,31821],[34,258080,258082],{"className":258081},[31685,31821],[34,258083,173486],{"className":258084},[31685,31821]," 字节 = 4GB）”。因此，32位地址处理器可以寻址的最大虚拟地址空间为4GB。",[19,258087,258089],{"id":258088},"_2支持程序浮动的地址转换机制是","2．支持程序浮动的地址转换机制是( )",[15,258091,258092],{},[2627,258093,258094],{},"A 动态重定位 B 段式地址转换 C 页式地址转换 D 静态重定位",[15,258096,258097,258099],{},[2627,258098,58173],{}," A",[15,258101,258102,258104],{},[2627,258103,258022],{},"\n“程序浮动”指的是程序在执行过程中可以在内存中移动或被换出到磁盘后换入到新的物理位置。参考资料指出：“为了让程序能在运行过程中被换出并在随后加载到内存的其它位置，不能在初始加载时就把内存引用绑定为绝对地址... 只有在指令真正执行时，才由特殊的处理器硬件动态地进行物理地址的计算”。这种在运行时将相对逻辑地址翻译成物理主存地址的机制，即为动态重定位（动态运行时加载\u002FDynamic run-time loading），它为程序在内存中浮动提供了完全的灵活性。",[19,258106,258108],{"id":258107},"_3unix中的文件系统采用","3．UNIX中的文件系统采用（ ）。",[15,258110,258111],{},[2627,258112,258113],{},"A 网状文件 B 记录式文件 C 索引文件 D 流式文件",[15,258115,258116,258118],{},[2627,258117,58173],{}," D",[15,258120,258121,258123],{},[2627,258122,258022],{},"\n参考资料指出：“UNIX内核将所有的文件视为字节流（streams of bytes），文件内部的任何逻辑结构完全由具体的应用程序来决定”。因此，UNIX文件系统采用的是无结构字节流组成的流式文件（Stream file）。",[19,258125,258127],{"id":258126},"_4段页式管理每取一数据要访问-次内存","4．段页式管理每取一数据，要访问（ ）次内存。",[15,258129,258130],{},[2627,258131,258132],{},"A 1 B 2 C 3 D 4",[15,258134,258135,258137],{},[2627,258136,58173],{}," C",[15,258139,258140,258142],{},[2627,258141,258022],{},"\n在结合了分段和分页的存储管理系统（段页式管理）中，逻辑地址被划分为段号、页号和页内偏移量。根据参考资料中段页式地址转换的流程图及说明，系统首先要访问驻留在内存中的“段表”（第一次访存）以获取该段对应的页表基址；然后利用页号访问所在的“页表”（第二次访存）获取对应的物理页框号；最后，将物理页框号加上偏移量结合成真实物理地址，去访问内存中所需的数据（第三次访存）。所以共需要访问3次内存。",[19,258144,258146],{"id":258145},"_5文件系统的主要目的是","5．文件系统的主要目的是（ ）。",[15,258148,258149,258152],{},[2627,258150,258151],{},"A 实现对文件的按名存取 B 实现虚拟存贮器",[2627,258153,258154],{},"C 提高外围设备的输入输出速度 D 用于存贮系统文档",[15,258156,258157,258099],{},[2627,258158,58173],{},[15,258160,258161,258163],{},[2627,258162,258022],{},"\n文件系统的核心目的在于方便用户和程序对数据进行组织与访问。参考资料指出：“文件系统的主要职能是实现长期存储，将信息存储在被称为‘文件’的命名对象中。文件对程序员而言是个极其方便的概念”，并且“文件被用户和应用程序视为单个实体，可以通过名称被引用（referenced by name）”。这就意味着其主要目的是提供“按名存取”的机制。",[19,258165,258167],{"id":258166},"_6-某基于动态分区存储管理的计算机其主存容量为55mb初始为空采用最佳适配算法分配和释放的顺序为分配15mb分配30mb释放15mb分配8mb分配6mb此时主存中最大空闲分区的大小是","6. 某基于动态分区存储管理的计算机，其主存容量为55mb（初始为空），采用最佳适配算法，分配和释放的顺序为：分配15mb，分配30mb，释放15mb，分配8mb，分配6mb，此时主存中最大空闲分区的大小是( )",[15,258169,258170],{},[2627,258171,258172],{},"A 7mb B 9mb C 10mb D 15mb",[15,258174,258175,258017],{},[2627,258176,58173],{},[15,258178,258179,258181],{},[2627,258180,258022],{},"\n我们逐步推演动态分区和最佳适配（Best-fit）算法的分配过程：",[2230,258183,258184,258187,258190,258193,258196,258199],{},[108,258185,258186],{},"初始状态：空闲 55MB。",[108,258188,258189],{},"分配15MB：主存分为 占用15MB，剩余空闲 40MB。",[108,258191,258192],{},"分配30MB：主存分为 占用15MB，占用30MB，剩余空闲 10MB。",[108,258194,258195],{},"释放15MB：此时主存有两个空闲区：15MB 和 10MB（中间隔着被占用的30MB，无法合并）。",[108,258197,258198],{},"分配8MB：最佳适配算法要求选择与请求大小最接近的空闲块。因此在15MB和10MB中，选择10MB的空闲区。分配后，该区域剩余 2MB 的空闲区。此时空闲区为：15MB 和 2MB。",[108,258200,258201],{},"分配6MB：同样使用最佳适配算法，在15MB和2MB中只能选择15MB。分配后，该区域剩余 9MB。\n最终此时系统中的空闲区大小分别为 9MB 和 2MB，其中最大空闲分区的大小为 9MB。",[19,258203,258205],{"id":258204},"_7设计批处理多道系统时首先要考虑的是","7．设计批处理多道系统时，首先要考虑的是( )。",[15,258207,258208,258211,258213],{},[2627,258209,258210],{},"A 灵活性和可适应性 B 系统效率和吞吐量",[58531,258212],{},[2627,258214,258215],{},"C 交互性和响应时间 D 实时性和可靠性",[15,258217,258218,258017],{},[2627,258219,58173],{},[15,258221,258222,258224],{},[2627,258223,258022],{},"\n操作系统的调度和设计目标因系统类型而异。参考资料在对比操作系统的调度策略时指出：“分时多道系统（time sharing, multiprogramming system）的主要目的是优化响应时间，而批处理多道系统（multiprogrammed batch system）的目的是优化系统效率和吞吐量”。",[19,258226,258228],{"id":258227},"_8进程调度的对象和任务分别是","8．进程调度的对象和任务分别是( )。",[15,258230,258231,258234,258237,258240],{},[2627,258232,258233],{},"A 作业，从就绪队列中按一定的调度策略选择一个进程占用CPU",[2627,258235,258236],{},"B 进程，从后备作业队列中按调度策略选择一个作业占用CPU",[2627,258238,258239],{},"C 进程，从就绪队列中按一定的调度策略选择一个进程占用CPU",[2627,258241,258242],{},"D 作业，从后备作业队列中调度策略选择一个作业占用CPU",[15,258244,258245,258137],{},[2627,258246,58173],{},[15,258248,258249,258251],{},[2627,258250,258022],{},"\n操作系统的调度分为多个层级。参考资料指出，“短程调度器（short-term scheduler，即进程调度器）是最频繁执行的调度”。“调度的任务就是随着时间推移将处理器分配给进程”，“进程调度策略会从就绪队列（Ready queue）中选择一个给定的进程来执行，并让其占用处理器”。因此，进程调度的对象是“进程”，任务是“从就绪队列中选择一个进程占用CPU”。",[19,258253,258255],{"id":258254},"_9一种既有利于短小作业又兼顾到长作业的作业调度算法是","9．一种既有利于短小作业又兼顾到长作业的作业调度算法是( )。",[15,258257,258258],{},[2627,258259,258260],{},"A 先来先服务 B 轮转 C 最高响应比优先 D 均衡调度",[15,258262,258263,258137],{},[2627,258264,58173],{},[15,258266,258267,258269,258270,258373],{},[2627,258268,258022],{},"\n最高响应比优先（HRRN）算法每次调度前都会计算响应比：",[34,258271,258273,258304],{"className":258272},[31642],[34,258274,258276],{"className":258275},[31646],[31648,258277,258278],{"xmlns":31650},[31652,258279,258280,258301],{},[31655,258281,258282,258285,258287,258289,258291,258293,258295,258297,258299],{},[39361,258283,258284],{},"响应比",[31662,258286,202],{},[31749,258288,640],{},[31662,258290,223],{},[31662,258292,456],{"stretchy":702},[39361,258294,216949],{},[31658,258296,38080],{"mathvariant":38079},[39361,258298,216954],{},[31662,258300,2557],{"stretchy":702},[31665,258302,258303],{"encoding":31667},"\\text{响应比} = 1 + (\\text{等待时间} \u002F \\text{服务时间})",[34,258305,258307,258328,258346],{"className":258306,"ariaHidden":758},[31672],[34,258308,258310,258313,258319,258322,258325],{"className":258309},[31676],[34,258311],{"className":258312,"style":33218},[31680],[34,258314,258316],{"className":258315},[31685,2638],[34,258317,258284],{"className":258318},[31685,105462],[34,258320],{"className":258321,"style":31846},[31845],[34,258323,202],{"className":258324},[31850],[34,258326],{"className":258327,"style":31846},[31845],[34,258329,258331,258334,258337,258340,258343],{"className":258330},[31676],[34,258332],{"className":258333,"style":35942},[31680],[34,258335,640],{"className":258336},[31685],[34,258338],{"className":258339,"style":32604},[31845],[34,258341,223],{"className":258342},[32024],[34,258344],{"className":258345,"style":32604},[31845],[34,258347,258349,258352,258355,258361,258364,258370],{"className":258348},[31676],[34,258350],{"className":258351,"style":31774},[31680],[34,258353,456],{"className":258354},[31782],[34,258356,258358],{"className":258357},[31685,2638],[34,258359,216949],{"className":258360},[31685,105462],[34,258362,38080],{"className":258363},[31685],[34,258365,258367],{"className":258366},[31685,2638],[34,258368,216954],{"className":258369},[31685,105462],[34,258371,2557],{"className":258372},[31690],"。因为包含了服务时间在分母，所以对短小作业非常有利；而因为包含了等待时间在分子，长作业随着等待时间的延长，响应比也会逐渐升高，从而避免了长作业发生“饥饿”现象（兼顾了长作业）。其他选项中，先来先服务对短作业不利；轮转调度主要用于分时系统。",[19,258375,258377],{"id":258376},"_10两个进程合作完成任务在并发执行中一个进程要等待其合作伙伴发来消息或者建立某个条件后再向前执行这种制约性合作关系称为进程的","10．两个进程合作完成任务。在并发执行中，一个进程要等待其合作伙伴发来消息，或者建立某个条件后再向前执行，这种制约性合作关系称为进程的（ ）。",[15,258379,258380],{},[2627,258381,258382],{},"A 互斥 B 同步 C 调度 D 伙伴",[15,258384,258385,258017],{},[2627,258386,58173],{},[15,258388,258389,258391],{},[2627,258390,258022],{},"\n参考资料提到：“当一个进程因为等待某个事件（如子进程终止或输入\u002F输出操作等某种条件的建立）而无法继续前进时，它会阻塞并把控制权交还给调度器”。为了协调这类并发事件的合作和先后顺序，需要由进程同步（Process synchronization）与进程间通信来支持。由于协作进程需等待条件满足，这种明确的制约关系即为进程“同步”。",[19,258393,258395],{"id":258394},"_11当每类资源只有一个个体时下列说法中不正确的是","11．当每类资源只有一个个体时，下列说法中不正确的是（ ）。",[15,258397,258398],{},[2627,258399,258400],{},"A 有环必死锁 B 死锁必有环 C 有环不一定死锁 D 被锁者一定全在环中",[15,258402,258403,258137],{},[2627,258404,58173],{},[15,258406,258407,258409],{},[2627,258408,258022],{},"\n在资源分配图中，如果图中包含环路：",[2230,258411,258412,258423],{},[108,258413,258414,258415,258418,258419,258422],{},"若每类资源只有",[2627,258416,258417],{},"一个","个体，环路是死锁存在的",[2627,258420,258421],{},"充分必要条件","。也就是说，“有环必死锁”，“死锁必有环”。",[108,258424,258425],{},"若某类资源有多个个体，环路只是死锁存在的必要条件，此时有环不一定死锁。\n题目中明确前提是“每类资源只有一个个体”，所以有环则必定发生死锁，因此“有环不一定死锁”的说法是错误的。",[19,258427,258429],{"id":258428},"_12在现代操作系统中引入了-从而使并发和共享成为可能","12．在现代操作系统中引入了（ ），从而使并发和共享成为可能。",[15,258431,258432],{},[2627,258433,258434],{},"A 单道程序 B 磁盘 C 对象 D 多道程序",[15,258436,258437,258118],{},[2627,258438,58173],{},[15,258440,258441,258443],{},[2627,258442,258022],{},"\n参考资料指出：“现代操作系统的中心主题，建立在内存中多个程序间进行切换的概念之上，这种机制被称为多道程序设计（multiprogramming）”。正是因为多道程序设计的引入，多个应用程序能够同时处于活动状态（并发），并高效共享处理器、内存及其他硬件资源。",[19,258445,258447],{"id":258446},"_13设有3个作业它们同时到达运行时间分别为t1t2和t3且t1t2t3若它们在单处理机系统中按单道运行采用短作业优先调度算法则平均周转时间为","13．设有3个作业，它们同时到达，运行时间分别为T1、T2和T3,且T1≤T2≤T3，若它们在单处理机系统中按单道运行，采用短作业优先调度算法，则平均周转时间为（ ）",[15,258449,258450],{},[2627,258451,258452,258453,258456],{},"A T1+T2+T3 B (T1+T2+T3)\u002F3 C T1+T2\u002F3+2",[71807,258454,258455],{},"T3\u002F3 D T3\u002F3+2","T2\u002F3+T1",[15,258458,258459,258118],{},[2627,258460,58173],{},[15,258462,258463,258465],{},[2627,258464,258022],{},"\n按照短作业优先（SJF）调度算法，由于同时到达，系统会按运行时间从小到大执行，即顺序为：作业1、作业2、作业3。",[105,258467,258468,258504,258569,258660],{},[108,258469,258470,258471],{},"作业1的周转时间为：",[34,258472,258474,258489],{"className":258473},[31642],[34,258475,258477],{"className":258476},[31646],[31648,258478,258479],{"xmlns":31650},[31652,258480,258481,258487],{},[31655,258482,258483,258485],{},[31658,258484,37400],{},[31749,258486,640],{},[31665,258488,100187],{"encoding":31667},[34,258490,258492],{"className":258491,"ariaHidden":758},[31672],[34,258493,258495,258498,258501],{"className":258494},[31676],[34,258496],{"className":258497,"style":33218},[31680],[34,258499,37400],{"className":258500,"style":37422},[31685,31686],[34,258502,640],{"className":258503},[31685],[108,258505,258506,258507,258568],{},"作业2的周转时间为：",[34,258508,258510,258532],{"className":258509},[31642],[34,258511,258513],{"className":258512},[31646],[31648,258514,258515],{"xmlns":31650},[31652,258516,258517,258529],{},[31655,258518,258519,258521,258523,258525,258527],{},[31658,258520,37400],{},[31749,258522,640],{},[31662,258524,223],{},[31658,258526,37400],{},[31749,258528,768],{},[31665,258530,258531],{"encoding":31667},"T1 + T2",[34,258533,258535,258556],{"className":258534,"ariaHidden":758},[31672],[34,258536,258538,258541,258544,258547,258550,258553],{"className":258537},[31676],[34,258539],{"className":258540,"style":34944},[31680],[34,258542,37400],{"className":258543,"style":37422},[31685,31686],[34,258545,640],{"className":258546},[31685],[34,258548],{"className":258549,"style":32604},[31845],[34,258551,223],{"className":258552},[32024],[34,258554],{"className":258555,"style":32604},[31845],[34,258557,258559,258562,258565],{"className":258558},[31676],[34,258560],{"className":258561,"style":33218},[31680],[34,258563,37400],{"className":258564,"style":37422},[31685,31686],[34,258566,768],{"className":258567},[31685]," （作业1结束后开始）",[108,258570,258571,258572],{},"作业3的周转时间为：",[34,258573,258575,258603],{"className":258574},[31642],[34,258576,258578],{"className":258577},[31646],[31648,258579,258580],{"xmlns":31650},[31652,258581,258582,258600],{},[31655,258583,258584,258586,258588,258590,258592,258594,258596,258598],{},[31658,258585,37400],{},[31749,258587,640],{},[31662,258589,223],{},[31658,258591,37400],{},[31749,258593,768],{},[31662,258595,223],{},[31658,258597,37400],{},[31749,258599,786],{},[31665,258601,258602],{"encoding":31667},"T1 + T2 + T3",[34,258604,258606,258627,258648],{"className":258605,"ariaHidden":758},[31672],[34,258607,258609,258612,258615,258618,258621,258624],{"className":258608},[31676],[34,258610],{"className":258611,"style":34944},[31680],[34,258613,37400],{"className":258614,"style":37422},[31685,31686],[34,258616,640],{"className":258617},[31685],[34,258619],{"className":258620,"style":32604},[31845],[34,258622,223],{"className":258623},[32024],[34,258625],{"className":258626,"style":32604},[31845],[34,258628,258630,258633,258636,258639,258642,258645],{"className":258629},[31676],[34,258631],{"className":258632,"style":34944},[31680],[34,258634,37400],{"className":258635,"style":37422},[31685,31686],[34,258637,768],{"className":258638},[31685],[34,258640],{"className":258641,"style":32604},[31845],[34,258643,223],{"className":258644},[32024],[34,258646],{"className":258647,"style":32604},[31845],[34,258649,258651,258654,258657],{"className":258650},[31676],[34,258652],{"className":258653,"style":33218},[31680],[34,258655,37400],{"className":258656,"style":37422},[31685,31686],[34,258658,786],{"className":258659},[31685],[108,258661,258662,258663,259240],{},"平均周转时间 = ",[34,258664,258666,258788],{"className":258665},[31642],[34,258667,258669],{"className":258668},[31646],[31648,258670,258671],{"xmlns":31650},[31652,258672,258673,258785],{},[31655,258674,258675,258723,258725,258755,258757,258759,258761,258763,258769,258771,258773,258775,258781,258783],{},[41453,258676,258677,258721],{},[31655,258678,258679,258681,258683,258685,258687,258689,258691,258693,258695,258697,258699,258701,258703,258705,258707,258709,258711,258713,258715,258717,258719],{},[31658,258680,37400],{},[31749,258682,640],{},[31662,258684,223],{},[31662,258686,456],{"stretchy":702},[31658,258688,37400],{},[31749,258690,640],{},[31662,258692,223],{},[31658,258694,37400],{},[31749,258696,768],{},[31662,258698,2557],{"stretchy":702},[31662,258700,223],{},[31662,258702,456],{"stretchy":702},[31658,258704,37400],{},[31749,258706,640],{},[31662,258708,223],{},[31658,258710,37400],{},[31749,258712,768],{},[31662,258714,223],{},[31658,258716,37400],{},[31749,258718,786],{},[31662,258720,2557],{"stretchy":702},[31749,258722,786],{},[31662,258724,202],{},[41453,258726,258727,258753],{},[31655,258728,258729,258731,258733,258735,258737,258739,258741,258743,258745,258747,258749,258751],{},[31749,258730,786],{},[31662,258732,71191],{},[31658,258734,37400],{},[31749,258736,640],{},[31662,258738,223],{},[31749,258740,768],{},[31662,258742,71191],{},[31658,258744,37400],{},[31749,258746,768],{},[31662,258748,223],{},[31658,258750,37400],{},[31749,258752,786],{},[31749,258754,786],{},[31662,258756,202],{},[31658,258758,37400],{},[31749,258760,640],{},[31662,258762,223],{},[41453,258764,258765,258767],{},[31749,258766,768],{},[31749,258768,786],{},[31658,258770,37400],{},[31749,258772,768],{},[31662,258774,223],{},[41453,258776,258777,258779],{},[31749,258778,640],{},[31749,258780,786],{},[31658,258782,37400],{},[31749,258784,786],{},[31665,258786,258787],{"encoding":31667},"\\frac{T1 + (T1 + T2) + (T1 + T2 + T3)}{3} = \\frac{3 \\cdot T1 + 2 \\cdot T2 + T3}{3} = T1 + \\frac{2}{3}T2 + \\frac{1}{3}T3",[34,258789,258791,258934,259050,259071,259160],{"className":258790,"ariaHidden":758},[31672],[34,258792,258794,258797,258925,258928,258931],{"className":258793},[31676],[34,258795],{"className":258796,"style":70485},[31680],[34,258798,258800,258803,258922],{"className":258799},[31685],[34,258801],{"className":258802},[31782,41488],[34,258804,258806],{"className":258805},[41453],[34,258807,258809,258914],{"className":258808},[31796,31797],[34,258810,258812,258911],{"className":258811},[31801],[34,258813,258815,258829,258837],{"className":258814,"style":70504},[31805],[34,258816,258817,258820],{"style":41504},[34,258818],{"className":258819,"style":41508},[31813],[34,258821,258823],{"className":258822},[31818,31819,31820,31821],[34,258824,258826],{"className":258825},[31685,31821],[34,258827,786],{"className":258828},[31685,31821],[34,258830,258831,258834],{"style":41520},[34,258832],{"className":258833,"style":41508},[31813],[34,258835],{"className":258836,"style":41528},[41527],[34,258838,258839,258842],{"style":70529},[34,258840],{"className":258841,"style":41508},[31813],[34,258843,258845],{"className":258844},[31818,31819,31820,31821],[34,258846,258848,258851,258854,258857,258860,258863,258866,258869,258872,258875,258878,258881,258884,258887,258890,258893,258896,258899,258902,258905,258908],{"className":258847},[31685,31821],[34,258849,37400],{"className":258850,"style":37422},[31685,31686,31821],[34,258852,640],{"className":258853},[31685,31821],[34,258855,223],{"className":258856},[32024,31821],[34,258858,456],{"className":258859},[31782,31821],[34,258861,37400],{"className":258862,"style":37422},[31685,31686,31821],[34,258864,640],{"className":258865},[31685,31821],[34,258867,223],{"className":258868},[32024,31821],[34,258870,37400],{"className":258871,"style":37422},[31685,31686,31821],[34,258873,768],{"className":258874},[31685,31821],[34,258876,2557],{"className":258877},[31690,31821],[34,258879,223],{"className":258880},[32024,31821],[34,258882,456],{"className":258883},[31782,31821],[34,258885,37400],{"className":258886,"style":37422},[31685,31686,31821],[34,258888,640],{"className":258889},[31685,31821],[34,258891,223],{"className":258892},[32024,31821],[34,258894,37400],{"className":258895,"style":37422},[31685,31686,31821],[34,258897,768],{"className":258898},[31685,31821],[34,258900,223],{"className":258901},[32024,31821],[34,258903,37400],{"className":258904,"style":37422},[31685,31686,31821],[34,258906,786],{"className":258907},[31685,31821],[34,258909,2557],{"className":258910},[31690,31821],[34,258912,31829],{"className":258913},[31828],[34,258915,258917],{"className":258916},[31801],[34,258918,258920],{"className":258919,"style":41619},[31805],[34,258921],{},[34,258923],{"className":258924},[31690,41488],[34,258926],{"className":258927,"style":31846},[31845],[34,258929,202],{"className":258930},[31850],[34,258932],{"className":258933,"style":31846},[31845],[34,258935,258937,258940,259041,259044,259047],{"className":258936},[31676],[34,258938],{"className":258939,"style":168026},[31680],[34,258941,258943,258946,259038],{"className":258942},[31685],[34,258944],{"className":258945},[31782,41488],[34,258947,258949],{"className":258948},[41453],[34,258950,258952,259030],{"className":258951},[31796,31797],[34,258953,258955,259027],{"className":258954},[31801],[34,258956,258958,258972,258980],{"className":258957,"style":168045},[31805],[34,258959,258960,258963],{"style":41504},[34,258961],{"className":258962,"style":41508},[31813],[34,258964,258966],{"className":258965},[31818,31819,31820,31821],[34,258967,258969],{"className":258968},[31685,31821],[34,258970,786],{"className":258971},[31685,31821],[34,258973,258974,258977],{"style":41520},[34,258975],{"className":258976,"style":41508},[31813],[34,258978],{"className":258979,"style":41528},[41527],[34,258981,258982,258985],{"style":46771},[34,258983],{"className":258984,"style":41508},[31813],[34,258986,258988],{"className":258987},[31818,31819,31820,31821],[34,258989,258991,258994,258997,259000,259003,259006,259009,259012,259015,259018,259021,259024],{"className":258990},[31685,31821],[34,258992,786],{"className":258993},[31685,31821],[34,258995,71191],{"className":258996},[32024,31821],[34,258998,37400],{"className":258999,"style":37422},[31685,31686,31821],[34,259001,640],{"className":259002},[31685,31821],[34,259004,223],{"className":259005},[32024,31821],[34,259007,768],{"className":259008},[31685,31821],[34,259010,71191],{"className":259011},[32024,31821],[34,259013,37400],{"className":259014,"style":37422},[31685,31686,31821],[34,259016,768],{"className":259017},[31685,31821],[34,259019,223],{"className":259020},[32024,31821],[34,259022,37400],{"className":259023,"style":37422},[31685,31686,31821],[34,259025,786],{"className":259026},[31685,31821],[34,259028,31829],{"className":259029},[31828],[34,259031,259033],{"className":259032},[31801],[34,259034,259036],{"className":259035,"style":41619},[31805],[34,259037],{},[34,259039],{"className":259040},[31690,41488],[34,259042],{"className":259043,"style":31846},[31845],[34,259045,202],{"className":259046},[31850],[34,259048],{"className":259049,"style":31846},[31845],[34,259051,259053,259056,259059,259062,259065,259068],{"className":259052},[31676],[34,259054],{"className":259055,"style":34944},[31680],[34,259057,37400],{"className":259058,"style":37422},[31685,31686],[34,259060,640],{"className":259061},[31685],[34,259063],{"className":259064,"style":32604},[31845],[34,259066,223],{"className":259067},[32024],[34,259069],{"className":259070,"style":32604},[31845],[34,259072,259074,259077,259145,259148,259151,259154,259157],{"className":259073},[31676],[34,259075],{"className":259076,"style":46727},[31680],[34,259078,259080,259083,259142],{"className":259079},[31685],[34,259081],{"className":259082},[31782,41488],[34,259084,259086],{"className":259085},[41453],[34,259087,259089,259134],{"className":259088},[31796,31797],[34,259090,259092,259131],{"className":259091},[31801],[34,259093,259095,259109,259117],{"className":259094,"style":46746},[31805],[34,259096,259097,259100],{"style":41504},[34,259098],{"className":259099,"style":41508},[31813],[34,259101,259103],{"className":259102},[31818,31819,31820,31821],[34,259104,259106],{"className":259105},[31685,31821],[34,259107,786],{"className":259108},[31685,31821],[34,259110,259111,259114],{"style":41520},[34,259112],{"className":259113,"style":41508},[31813],[34,259115],{"className":259116,"style":41528},[41527],[34,259118,259119,259122],{"style":46771},[34,259120],{"className":259121,"style":41508},[31813],[34,259123,259125],{"className":259124},[31818,31819,31820,31821],[34,259126,259128],{"className":259127},[31685,31821],[34,259129,768],{"className":259130},[31685,31821],[34,259132,31829],{"className":259133},[31828],[34,259135,259137],{"className":259136},[31801],[34,259138,259140],{"className":259139,"style":41619},[31805],[34,259141],{},[34,259143],{"className":259144},[31690,41488],[34,259146,37400],{"className":259147,"style":37422},[31685,31686],[34,259149,768],{"className":259150},[31685],[34,259152],{"className":259153,"style":32604},[31845],[34,259155,223],{"className":259156},[32024],[34,259158],{"className":259159,"style":32604},[31845],[34,259161,259163,259166,259234,259237],{"className":259162},[31676],[34,259164],{"className":259165,"style":46727},[31680],[34,259167,259169,259172,259231],{"className":259168},[31685],[34,259170],{"className":259171},[31782,41488],[34,259173,259175],{"className":259174},[41453],[34,259176,259178,259223],{"className":259177},[31796,31797],[34,259179,259181,259220],{"className":259180},[31801],[34,259182,259184,259198,259206],{"className":259183,"style":46746},[31805],[34,259185,259186,259189],{"style":41504},[34,259187],{"className":259188,"style":41508},[31813],[34,259190,259192],{"className":259191},[31818,31819,31820,31821],[34,259193,259195],{"className":259194},[31685,31821],[34,259196,786],{"className":259197},[31685,31821],[34,259199,259200,259203],{"style":41520},[34,259201],{"className":259202,"style":41508},[31813],[34,259204],{"className":259205,"style":41528},[41527],[34,259207,259208,259211],{"style":46771},[34,259209],{"className":259210,"style":41508},[31813],[34,259212,259214],{"className":259213},[31818,31819,31820,31821],[34,259215,259217],{"className":259216},[31685,31821],[34,259218,640],{"className":259219},[31685,31821],[34,259221,31829],{"className":259222},[31828],[34,259224,259226],{"className":259225},[31801],[34,259227,259229],{"className":259228,"style":41619},[31805],[34,259230],{},[34,259232],{"className":259233},[31690,41488],[34,259235,37400],{"className":259236,"style":37422},[31685,31686],[34,259238,786],{"className":259239},[31685],"\n该计算结果正是选项 D 所列公式的重新排列。",[19,259242,259244],{"id":259243},"_14若系统中有五台绘图仪有多个进程均需要使用两台规定每个进程一次仅允许申请一台则至多允许-个进程参于竞争而不会发生死锁","14．若系统中有五台绘图仪，有多个进程均需要使用两台，规定每个进程一次仅允许申请一台，则至多允许（ ）个进程参于竞争，而不会发生死锁。",[15,259246,259247],{},[2627,259248,259249],{},"A 5 B 2 C 3 D 4",[15,259251,259252,258118],{},[2627,259253,58173],{},[15,259255,259256,259258,259259,259287,259288,259316,259317,259345,259346,259419,259420,259448,259449,259500,259501,259529],{},[2627,259257,258022],{},"\n假设系统中有 ",[34,259260,259262,259275],{"className":259261},[31642],[34,259263,259265],{"className":259264},[31646],[31648,259266,259267],{"xmlns":31650},[31652,259268,259269,259273],{},[31655,259270,259271],{},[31658,259272,38691],{},[31665,259274,38691],{"encoding":31667},[34,259276,259278],{"className":259277,"ariaHidden":758},[31672],[34,259279,259281,259284],{"className":259280},[31676],[34,259282],{"className":259283,"style":33218},[31680],[34,259285,38691],{"className":259286,"style":38706},[31685,31686]," 个进程参与竞争，采用极端的最坏情况分析（预防死锁）：每个进程都已经申请并获得了 ",[34,259289,259291,259304],{"className":259290},[31642],[34,259292,259294],{"className":259293},[31646],[31648,259295,259296],{"xmlns":31650},[31652,259297,259298,259302],{},[31655,259299,259300],{},[31749,259301,640],{},[31665,259303,640],{"encoding":31667},[34,259305,259307],{"className":259306,"ariaHidden":758},[31672],[34,259308,259310,259313],{"className":259309},[31676],[34,259311],{"className":259312,"style":32617},[31680],[34,259314,640],{"className":259315},[31685]," 台绘图仪，且全都在请求各自的第 ",[34,259318,259320,259333],{"className":259319},[31642],[34,259321,259323],{"className":259322},[31646],[31648,259324,259325],{"xmlns":31650},[31652,259326,259327,259331],{},[31655,259328,259329],{},[31749,259330,768],{},[31665,259332,768],{"encoding":31667},[34,259334,259336],{"className":259335,"ariaHidden":758},[31672],[34,259337,259339,259342],{"className":259338},[31676],[34,259340],{"className":259341,"style":32617},[31680],[34,259343,768],{"className":259344},[31685]," 台绘图仪。\n如果此时被占用的绘图仪总数为 ",[34,259347,259349,259371],{"className":259348},[31642],[34,259350,259352],{"className":259351},[31646],[31648,259353,259354],{"xmlns":31650},[31652,259355,259356,259368],{},[31655,259357,259358,259360,259362,259364,259366],{},[31749,259359,640],{},[31662,259361,42071],{},[31658,259363,38691],{},[31662,259365,202],{},[31658,259367,38691],{},[31665,259369,259370],{"encoding":31667},"1 \\times K = K",[34,259372,259374,259392,259410],{"className":259373,"ariaHidden":758},[31672],[34,259375,259377,259380,259383,259386,259389],{"className":259376},[31676],[34,259378],{"className":259379,"style":35942},[31680],[34,259381,640],{"className":259382},[31685],[34,259384],{"className":259385,"style":32604},[31845],[34,259387,42071],{"className":259388},[32024],[34,259390],{"className":259391,"style":32604},[31845],[34,259393,259395,259398,259401,259404,259407],{"className":259394},[31676],[34,259396],{"className":259397,"style":33218},[31680],[34,259399,38691],{"className":259400,"style":38706},[31685,31686],[34,259402],{"className":259403,"style":31846},[31845],[34,259405,202],{"className":259406},[31850],[34,259408],{"className":259409,"style":31846},[31845],[34,259411,259413,259416],{"className":259412},[31676],[34,259414],{"className":259415,"style":33218},[31680],[34,259417,38691],{"className":259418,"style":38706},[31685,31686]," 台。为了打破可能发生的死锁（避免循环等待），至少要剩下 1 台绘图仪能分配给其中一个进程，使其完成任务后释放所占资源给其它进程。\n因此，总资源数 5 需要大于等于极端情况下已分配出去的 ",[34,259421,259423,259436],{"className":259422},[31642],[34,259424,259426],{"className":259425},[31646],[31648,259427,259428],{"xmlns":31650},[31652,259429,259430,259434],{},[31655,259431,259432],{},[31658,259433,38691],{},[31665,259435,38691],{"encoding":31667},[34,259437,259439],{"className":259438,"ariaHidden":758},[31672],[34,259440,259442,259445],{"className":259441},[31676],[34,259443],{"className":259444,"style":33218},[31680],[34,259446,38691],{"className":259447,"style":38706},[31685,31686]," 台绘图仪，即 ",[34,259450,259452,259470],{"className":259451},[31642],[34,259453,259455],{"className":259454},[31646],[31648,259456,259457],{"xmlns":31650},[31652,259458,259459,259467],{},[31655,259460,259461,259463,259465],{},[31749,259462,18136],{},[31662,259464,29622],{},[31658,259466,38691],{},[31665,259468,259469],{"encoding":31667},"5 > K",[34,259471,259473,259491],{"className":259472,"ariaHidden":758},[31672],[34,259474,259476,259479,259482,259485,259488],{"className":259475},[31676],[34,259477],{"className":259478,"style":61814},[31680],[34,259480,18136],{"className":259481},[31685],[34,259483],{"className":259484,"style":31846},[31845],[34,259486,29622],{"className":259487},[31850],[34,259489],{"className":259490,"style":31846},[31845],[34,259492,259494,259497],{"className":259493},[31676],[34,259495],{"className":259496,"style":33218},[31680],[34,259498,38691],{"className":259499,"style":38706},[31685,31686],"，于是 ",[34,259502,259504,259517],{"className":259503},[31642],[34,259505,259507],{"className":259506},[31646],[31648,259508,259509],{"xmlns":31650},[31652,259510,259511,259515],{},[31655,259512,259513],{},[31658,259514,38691],{},[31665,259516,38691],{"encoding":31667},[34,259518,259520],{"className":259519,"ariaHidden":758},[31672],[34,259521,259523,259526],{"className":259522},[31676],[34,259524],{"className":259525,"style":33218},[31680],[34,259527,38691],{"className":259528,"style":38706},[31685,31686]," 的最大值为 4。如果有5个进程就会陷入全部分配1台、谁也无法拿到第二台的死锁状态。",[19,259531,259533],{"id":259532},"_15cpu输出数据速度远远高于打印机的打印速度为解决矛盾可采用","15．CPU输出数据速度远远高于打印机的打印速度，为解决矛盾，可采用（ ）",[15,259535,259536],{},[2627,259537,259538],{},"A 并行技术 B 缓冲技术 C 虚拟存储器技术 D 覆盖技术",[15,259540,259541,258017],{},[2627,259542,58173],{},[15,259544,259545,259547],{},[2627,259546,258022],{},"\n参考资料在I\u002FO管理一节明确指出：“使用多重缓冲试图缓解吸收快速I\u002FO突发问题的情景，通常被称为缓冲技术（buffering）”。“为了改善I\u002FO设备的利用率和解决处理器管理I\u002FO传输效率低下的问题，可以利用缓冲技术（Buffering techniques）”。因此引入缓冲技术可平滑CPU和慢速外设（打印机）之间的速度不匹配。",[19,259549,259551],{"id":259550},"_16为了允许不同用户的文件具有相同的文件名通常在文件系统中采用","16.为了允许不同用户的文件具有相同的文件名，通常在文件系统中采用（ ）",[15,259553,259554],{},[2627,259555,259556],{},"A 重名翻译 B 多级目录 C 约定 D 文件名",[15,259558,259559,258017],{},[2627,259560,58173],{},[15,259562,259563,259565],{},[2627,259564,258022],{},"\n参考资料指出，多级目录或树形结构的目录使得不同所有者的文件可以被有效组织并隔离。“这不仅提高了安全性、完整性，也避免了文件名冲突（name clashes）的问题”。每个用户可以拥有自己的子目录，从而在不同目录中可以存在同名的文件。",[19,259567,259569],{"id":259568},"_17在可变分区存储管理中最优适应分配算法要求对空闲区表项按-排列","17．在可变分区存储管理中，最优适应分配算法要求对空闲区表项按( )排列。",[15,259571,259572],{},[2627,259573,259574],{},"A 地址从大到小 B 地址从小到大 C 尺寸从小到大 D 尺寸从大到小",[15,259576,259577,258137],{},[2627,259578,58173],{},[15,259580,259581,259583],{},[2627,259582,258022],{},"\n“最佳适应（Best-fit）布局算法指的是选择那些在大小上最接近请求内存量（closest in size to the request）的空闲内存块进行分配”。为了加快查找最合适空闲块的速度并快速停止遍历，实际实现中往往将空闲区表项按照内存块尺寸（大小）从最小到最大进行排列。",[19,259585,259587],{"id":259586},"_18支持程序浮动的地址转换机制是","18．支持程序浮动的地址转换机制是( )",[15,259589,259590,259593],{},[2627,259591,259592],{},"(此题与第2题重复)",[2627,259594,258094],{},[15,259596,259597,258099],{},[2627,259598,58173],{},[15,259600,259601,259603],{},[2627,259602,258022],{},"\n同第2题。使用“动态运行时加载”和动态地址转换使得程序在内存中浮动和动态重定位变为可能。",[19,259605,259607],{"id":259606},"_19在可变式分区分配方案中某一作业完成后系统收回其主存空间并与相邻空闲区合并为此需修改空闲区表造成空闲区数减1的情况是","19．在可变式分区分配方案中，某一作业完成后，系统收回其主存空间，并与相邻空闲区合并，为此需修改空闲区表，造成空闲区数减1的情况是（ ）",[15,259609,259610,259613],{},[2627,259611,259612],{},"A 无上邻空闲区，也无下邻空闲区 B 有上邻空闲区，但无下邻空闲区",[2627,259614,259615],{},"C 有下邻空闲区，但无上邻空闲区 D 有上邻空闲区，也有下邻空闲区",[15,259617,259618,258118],{},[2627,259619,58173],{},[15,259621,259622,259624],{},[2627,259623,258022],{},"\n作业释放内存时产生的合并有4种情况：",[2230,259626,259627,259636,259645],{},[108,259628,259629,259632,259633,347],{},[2627,259630,259631],{},"无上邻且无下邻："," 回收区将作为新的独立空闲区加入表中，空闲区总数 ",[2627,259634,259635],{},"+1",[108,259637,259638,259641,259642,347],{},[2627,259639,259640],{},"仅有上邻 或 仅有下邻："," 回收区仅与上一个（或下一个）空闲区合并，使原空闲区块变大，空闲区总数 ",[2627,259643,259644],{},"不变",[108,259646,259647,259650,259651,347],{},[2627,259648,259649],{},"有上邻空闲区，也有下邻空闲区："," 回收区充当了桥梁，将它上面的空闲区与它下面的空闲区连成了一片。原先表中的两个空闲区合并成了一个大空闲区，所以空闲区总数 ",[2627,259652,259653],{},"-1",[19,259655,259657],{"id":259656},"_20在下面关于虚拟存储器的叙述中正确的是","20．在下面关于虚拟存储器的叙述中，正确的是（ ）",[15,259659,259660,259663,259666,259669],{},[2627,259661,259662],{},"A 要求程序运行前必须全部装入内存且在运行过程中一直驻留在内存",[2627,259664,259665],{},"B 要求程序运行前不必全部装入内存且在运行过程中不必一直驻留在内存",[2627,259667,259668],{},"C 要求程序运行前不必全部装入内存但是在运行过程中必须一直驻留在内存",[2627,259670,259671],{},"D 要求程序运行前必须全部装入内存但在运行过程中不必一直驻留在内存",[15,259673,259674,258017],{},[2627,259675,58173],{},[15,259677,259678,259680],{},[2627,259679,258022],{},"\n根据参考资料中虚拟内存的论述：“由于动态运行时的地址转换...没必要在程序执行时将其所有的页或段全部驻留在内存中。进程可以被打散成各个部分，且在执行时无需全部在内存中”。并且“如果不再被使用或为了腾出空间，内存中的页面会被操作系统显式或隐式地调出”。因此，虚拟内存既不要求运行前全部装入，也不要求运行过程中一直全部驻留。",[19,259682,259684],{"id":259683},"_21文件系统中用-管理文件","21．文件系统中用（ ）管理文件。",[15,259686,259687],{},[2627,259688,259689],{},"A 堆栈结构 B 指针 C 页表 D 目录",[15,259691,259692,258118],{},[2627,259693,58173],{},[15,259695,259696,259698],{},[2627,259697,258022],{},"\n参考资料明确表明文件系统的组织：“文件目录的地址信息元素会维护文件所有者的信息以及其它控制权限信息”，“目录允许文件被很好地组织并清晰地按照不同准则分离...”。操作系统依靠文件目录（Directory）来实现文件系统对大量文件的组织、存取和管理。",[19,259700,259702],{"id":259701},"_22在多进程的并发系统中肯定不会因竞争-而产生死锁","22．在多进程的并发系统中，肯定不会因竞争( )而产生死锁。",[15,259704,259705],{},[2627,259706,259707],{},"A 打印机 B 磁带机 C CPU D 磁盘",[15,259709,259710,258137],{},[2627,259711,58173],{},[15,259713,259714,259716,259717,259720],{},[2627,259715,258022],{},"\n死锁常常涉及",[2627,259718,259719],{},"不可剥夺","的临界排他资源（例如打印机、磁带机等）。CPU属于典型的“可剥夺性（Preemptible）资源”，即操作系统可以通过时间片中断（调度程序）随时从正在运行的进程手中收回CPU使用权并分配给其他进程。因此，多进程永远不会因为竞争CPU而永久阻塞陷入死锁。",[19,259722,259724],{"id":259723},"_23程序员利用系统调用打开io设备时通常使用的设备标识","23．程序员利用系统调用打开I\u002FO设备时，通常使用的设备标识( )",[15,259726,259727],{},[2627,259728,259729],{},"A 从设备号 B 物理设备名 C 主设备号 D 逻辑设备名",[15,259731,259732,258118],{},[2627,259733,58173],{},[15,259735,259736,259738],{},[2627,259737,258022],{},"\n根据I\u002FO管理的逻辑结构：“逻辑I\u002FO模块（Logical I\u002FO）把设备视为逻辑资源...允许用户进程仅仅通过‘设备标识符（即逻辑设备名）’和简单的命令（如open、close、read、write）来进行I\u002FO访问，而无需关注具体的物理细节”。这样实现了设备独立性。",[19,259740,259742],{"id":259741},"_24分段存储管理系统中地址长度为32位其中段号占8位则段长最大","24．分段存储管理系统中，地址长度为32位，其中段号占8位，则段长最大( )",[15,259744,259745],{},[2627,259746,241048,259747,259806,259807,259868,259869,259931,259932],{},[34,259748,259750,259768],{"className":259749},[31642],[34,259751,259753],{"className":259752},[31646],[31648,259754,259755],{"xmlns":31650},[31652,259756,259757,259765],{},[31655,259758,259759],{},[37624,259760,259761,259763],{},[31749,259762,768],{},[31749,259764,60525],{},[31665,259766,259767],{"encoding":31667},"2^8",[34,259769,259771],{"className":259770,"ariaHidden":758},[31672],[34,259772,259774,259777],{"className":259773},[31676],[34,259775],{"className":259776,"style":37774},[31680],[34,259778,259780,259783],{"className":259779},[31685],[34,259781,768],{"className":259782},[31685],[34,259784,259786],{"className":259785},[31792],[34,259787,259789],{"className":259788},[31796],[34,259790,259792],{"className":259791},[31801],[34,259793,259795],{"className":259794,"style":37774},[31805],[34,259796,259797,259800],{"style":37777},[34,259798],{"className":259799,"style":31814},[31813],[34,259801,259803],{"className":259802},[31818,31819,31820,31821],[34,259804,60525],{"className":259805},[31685,31821]," B ",[34,259808,259810,259827],{"className":259809},[31642],[34,259811,259813],{"className":259812},[31646],[31648,259814,259815],{"xmlns":31650},[31652,259816,259817,259825],{},[31655,259818,259819],{},[37624,259820,259821,259823],{},[31749,259822,768],{},[31749,259824,177869],{},[31665,259826,201575],{"encoding":31667},[34,259828,259830],{"className":259829,"ariaHidden":758},[31672],[34,259831,259833,259836],{"className":259832},[31676],[34,259834],{"className":259835,"style":37774},[31680],[34,259837,259839,259842],{"className":259838},[31685],[34,259840,768],{"className":259841},[31685],[34,259843,259845],{"className":259844},[31792],[34,259846,259848],{"className":259847},[31796],[34,259849,259851],{"className":259850},[31801],[34,259852,259854],{"className":259853,"style":37774},[31805],[34,259855,259856,259859],{"style":37777},[34,259857],{"className":259858,"style":31814},[31813],[34,259860,259862],{"className":259861},[31818,31819,31820,31821],[34,259863,259865],{"className":259864},[31685,31821],[34,259866,177869],{"className":259867},[31685,31821]," C ",[34,259870,259872,259890],{"className":259871},[31642],[34,259873,259875],{"className":259874},[31646],[31648,259876,259877],{"xmlns":31650},[31652,259878,259879,259887],{},[31655,259880,259881],{},[37624,259882,259883,259885],{},[31749,259884,768],{},[31749,259886,96087],{},[31665,259888,259889],{"encoding":31667},"2^{24}",[34,259891,259893],{"className":259892,"ariaHidden":758},[31672],[34,259894,259896,259899],{"className":259895},[31676],[34,259897],{"className":259898,"style":37774},[31680],[34,259900,259902,259905],{"className":259901},[31685],[34,259903,768],{"className":259904},[31685],[34,259906,259908],{"className":259907},[31792],[34,259909,259911],{"className":259910},[31796],[34,259912,259914],{"className":259913},[31801],[34,259915,259917],{"className":259916,"style":37774},[31805],[34,259918,259919,259922],{"style":37777},[34,259920],{"className":259921,"style":31814},[31813],[34,259923,259925],{"className":259924},[31818,31819,31820,31821],[34,259926,259928],{"className":259927},[31685,31821],[34,259929,96087],{"className":259930},[31685,31821]," D ",[34,259933,259935,259952],{"className":259934},[31642],[34,259936,259938],{"className":259937},[31646],[31648,259939,259940],{"xmlns":31650},[31652,259941,259942,259950],{},[31655,259943,259944],{},[37624,259945,259946,259948],{},[31749,259947,768],{},[31749,259949,173486],{},[31665,259951,201449],{"encoding":31667},[34,259953,259955],{"className":259954,"ariaHidden":758},[31672],[34,259956,259958,259961],{"className":259957},[31676],[34,259959],{"className":259960,"style":37774},[31680],[34,259962,259964,259967],{"className":259963},[31685],[34,259965,768],{"className":259966},[31685],[34,259968,259970],{"className":259969},[31792],[34,259971,259973],{"className":259972},[31796],[34,259974,259976],{"className":259975},[31801],[34,259977,259979],{"className":259978,"style":37774},[31805],[34,259980,259981,259984],{"style":37777},[34,259982],{"className":259983,"style":31814},[31813],[34,259985,259987],{"className":259986},[31818,31819,31820,31821],[34,259988,259990],{"className":259989},[31685,31821],[34,259991,173486],{"className":259992},[31685,31821],[15,259994,259995,258137],{},[2627,259996,58173],{},[15,259998,259999,260001,260002,260075,260076,260137],{},[2627,260000,258022],{},"\n在分段存储系统中，逻辑地址由“段号”和“段内偏移量”两部分组成。已知地址总长为32位，段号占据其中的8位，那么留给段内偏移量的位数为：",[34,260003,260005,260027],{"className":260004},[31642],[34,260006,260008],{"className":260007},[31646],[31648,260009,260010],{"xmlns":31650},[31652,260011,260012,260024],{},[31655,260013,260014,260016,260018,260020,260022],{},[31749,260015,173486],{},[31662,260017,32226],{},[31749,260019,60525],{},[31662,260021,202],{},[31749,260023,96087],{},[31665,260025,260026],{"encoding":31667},"32 - 8 = 24",[34,260028,260030,260048,260066],{"className":260029,"ariaHidden":758},[31672],[34,260031,260033,260036,260039,260042,260045],{"className":260032},[31676],[34,260034],{"className":260035,"style":35942},[31680],[34,260037,173486],{"className":260038},[31685],[34,260040],{"className":260041,"style":32604},[31845],[34,260043,32226],{"className":260044},[32024],[34,260046],{"className":260047,"style":32604},[31845],[34,260049,260051,260054,260057,260060,260063],{"className":260050},[31676],[34,260052],{"className":260053,"style":32617},[31680],[34,260055,60525],{"className":260056},[31685],[34,260058],{"className":260059,"style":31846},[31845],[34,260061,202],{"className":260062},[31850],[34,260064],{"className":260065,"style":31846},[31845],[34,260067,260069,260072],{"className":260068},[31676],[34,260070],{"className":260071,"style":32617},[31680],[34,260073,96087],{"className":260074},[31685],"位。段内偏移量决定了段能够寻址的最大长度范围，因此每个段的段长最大为 ",[34,260077,260079,260096],{"className":260078},[31642],[34,260080,260082],{"className":260081},[31646],[31648,260083,260084],{"xmlns":31650},[31652,260085,260086,260094],{},[31655,260087,260088],{},[37624,260089,260090,260092],{},[31749,260091,768],{},[31749,260093,96087],{},[31665,260095,259889],{"encoding":31667},[34,260097,260099],{"className":260098,"ariaHidden":758},[31672],[34,260100,260102,260105],{"className":260101},[31676],[34,260103],{"className":260104,"style":37774},[31680],[34,260106,260108,260111],{"className":260107},[31685],[34,260109,768],{"className":260110},[31685],[34,260112,260114],{"className":260113},[31792],[34,260115,260117],{"className":260116},[31796],[34,260118,260120],{"className":260119},[31801],[34,260121,260123],{"className":260122,"style":37774},[31805],[34,260124,260125,260128],{"style":37777},[34,260126],{"className":260127,"style":31814},[31813],[34,260129,260131],{"className":260130},[31818,31819,31820,31821],[34,260132,260134],{"className":260133},[31685,31821],[34,260135,96087],{"className":260136},[31685,31821]," 个存储单元。",[19,260139,260141],{"id":260140},"_25设与某资源相关联的信号量初值为3当前值为1若m表示该资源的可用个数n表示等待资源的进程数则mn分别是","25．设与某资源相关联的信号量初值为3，当前值为1，若M表示该资源的可用个数，N表示等待资源的进程数，则M,N分别是( )",[15,260143,260144],{},[2627,260145,260146],{},"A 1，0 B 0，1 C 1，2 D 2，0",[15,260148,260149,258099],{},[2627,260150,58173],{},[15,260152,260153,260155],{},[2627,260154,258022],{},"\n信号量（Semaphore）的值代表资源的当前可用状况：",[105,260157,260158,260208],{},[108,260159,260160,260161,260204,260205,347],{},"当信号量值 ",[34,260162,260164,260180],{"className":260163},[31642],[34,260165,260167],{"className":260166},[31646],[31648,260168,260169],{"xmlns":31650},[31652,260170,260171,260177],{},[31655,260172,260173,260175],{},[31662,260174,29622],{},[31749,260176,459],{},[31665,260178,260179],{"encoding":31667},"> 0",[34,260181,260183,260195],{"className":260182,"ariaHidden":758},[31672],[34,260184,260186,260189,260192],{"className":260185},[31676],[34,260187],{"className":260188,"style":34812},[31680],[34,260190,29622],{"className":260191},[31850],[34,260193],{"className":260194,"style":31846},[31845],[34,260196,260198,260201],{"className":260197},[31676],[34,260199],{"className":260200,"style":32617},[31680],[34,260202,459],{"className":260203},[31685]," 时，该数值代表系统中",[2627,260206,260207],{},"目前可用的资源个数",[108,260209,260160,260210,260253,260254,260257,260258,260286,260287,260315,260316,260367,260368,214569],{},[34,260211,260213,260229],{"className":260212},[31642],[34,260214,260216],{"className":260215},[31646],[31648,260217,260218],{"xmlns":31650},[31652,260219,260220,260226],{},[31655,260221,260222,260224],{},[31662,260223,45972],{},[31749,260225,459],{},[31665,260227,260228],{"encoding":31667},"\\le 0",[34,260230,260232,260244],{"className":260231,"ariaHidden":758},[31672],[34,260233,260235,260238,260241],{"className":260234},[31676],[34,260236],{"className":260237,"style":43198},[31680],[34,260239,45972],{"className":260240},[31850],[34,260242],{"className":260243,"style":31846},[31845],[34,260245,260247,260250],{"className":260246},[31676],[34,260248],{"className":260249,"style":32617},[31680],[34,260251,459],{"className":260252},[31685]," 时，它的绝对值代表",[2627,260255,260256],{},"正在排队等待该资源的阻塞进程数","。\n当前信号量的值为正数 ",[34,260259,260261,260274],{"className":260260},[31642],[34,260262,260264],{"className":260263},[31646],[31648,260265,260266],{"xmlns":31650},[31652,260267,260268,260272],{},[31655,260269,260270],{},[31749,260271,640],{},[31665,260273,640],{"encoding":31667},[34,260275,260277],{"className":260276,"ariaHidden":758},[31672],[34,260278,260280,260283],{"className":260279},[31676],[34,260281],{"className":260282,"style":32617},[31680],[34,260284,640],{"className":260285},[31685],"，说明该资源尚有 ",[34,260288,260290,260303],{"className":260289},[31642],[34,260291,260293],{"className":260292},[31646],[31648,260294,260295],{"xmlns":31650},[31652,260296,260297,260301],{},[31655,260298,260299],{},[31749,260300,640],{},[31665,260302,640],{"encoding":31667},[34,260304,260306],{"className":260305,"ariaHidden":758},[31672],[34,260307,260309,260312],{"className":260308},[31676],[34,260310],{"className":260311,"style":32617},[31680],[34,260313,640],{"className":260314},[31685]," 个可用 (",[34,260317,260319,260337],{"className":260318},[31642],[34,260320,260322],{"className":260321},[31646],[31648,260323,260324],{"xmlns":31650},[31652,260325,260326,260334],{},[31655,260327,260328,260330,260332],{},[31658,260329,98360],{},[31662,260331,202],{},[31749,260333,640],{},[31665,260335,260336],{"encoding":31667},"M=1",[34,260338,260340,260358],{"className":260339,"ariaHidden":758},[31672],[34,260341,260343,260346,260349,260352,260355],{"className":260342},[31676],[34,260344],{"className":260345,"style":33218},[31680],[34,260347,98360],{"className":260348,"style":33389},[31685,31686],[34,260350],{"className":260351,"style":31846},[31845],[34,260353,202],{"className":260354},[31850],[34,260356],{"className":260357,"style":31846},[31845],[34,260359,260361,260364],{"className":260360},[31676],[34,260362],{"className":260363,"style":32617},[31680],[34,260365,640],{"className":260366},[31685],")，目前没有任何进程因为请求该资源而被阻塞等待 (",[34,260369,260371,260389],{"className":260370},[31642],[34,260372,260374],{"className":260373},[31646],[31648,260375,260376],{"xmlns":31650},[31652,260377,260378,260386],{},[31655,260379,260380,260382,260384],{},[31658,260381,33347],{},[31662,260383,202],{},[31749,260385,459],{},[31665,260387,260388],{"encoding":31667},"N=0",[34,260390,260392,260410],{"className":260391,"ariaHidden":758},[31672],[34,260393,260395,260398,260401,260404,260407],{"className":260394},[31676],[34,260396],{"className":260397,"style":33218},[31680],[34,260399,33347],{"className":260400,"style":33389},[31685,31686],[34,260402],{"className":260403,"style":31846},[31845],[34,260405,202],{"className":260406},[31850],[34,260408],{"className":260409,"style":31846},[31845],[34,260411,260413,260416],{"className":260412},[31676],[34,260414],{"className":260415,"style":32617},[31680],[34,260417,459],{"className":260418},[31685],[19,260420,260422],{"id":260421},"_26某计算机系统中有8台打印机有k个进程竞争使用每个进程最多需要3台打印机该系统可能会发生死锁的k的最小值","26．某计算机系统中有8台打印机，有K个进程竞争使用，每个进程最多需要3台打印机。该系统可能会发生死锁的K的最小值( )",[15,260424,260425],{},[2627,260426,260427],{},"A 2 B 3 C 4 D 5",[15,260429,260430,258137],{},[2627,260431,58173],{},[15,260433,260434,260436,260437,260465,260466,260517,260518,260546,260547,260603,260604,93231,260655,260706,260707,260780],{},[2627,260435,258022],{},"\n发生死锁的最坏情况是：每个进程都只差最后一台打印机就能运行（即每个进程都占有了2台）。\n如果此时系统中所有的8台打印机都已被分配完毕，且没有剩余（或不够），所有进程都将相互等待。\n对于 ",[34,260438,260440,260453],{"className":260439},[31642],[34,260441,260443],{"className":260442},[31646],[31648,260444,260445],{"xmlns":31650},[31652,260446,260447,260451],{},[31655,260448,260449],{},[31658,260450,38691],{},[31665,260452,38691],{"encoding":31667},[34,260454,260456],{"className":260455,"ariaHidden":758},[31672],[34,260457,260459,260462],{"className":260458},[31676],[34,260460],{"className":260461,"style":33218},[31680],[34,260463,38691],{"className":260464,"style":38706},[31685,31686]," 个进程，每人都占用2台，总共需要 ",[34,260467,260469,260487],{"className":260468},[31642],[34,260470,260472],{"className":260471},[31646],[31648,260473,260474],{"xmlns":31650},[31652,260475,260476,260484],{},[31655,260477,260478,260480,260482],{},[31749,260479,768],{},[31662,260481,42071],{},[31658,260483,38691],{},[31665,260485,260486],{"encoding":31667},"2 \\times K",[34,260488,260490,260508],{"className":260489,"ariaHidden":758},[31672],[34,260491,260493,260496,260499,260502,260505],{"className":260492},[31676],[34,260494],{"className":260495,"style":35942},[31680],[34,260497,768],{"className":260498},[31685],[34,260500],{"className":260501,"style":32604},[31845],[34,260503,42071],{"className":260504},[32024],[34,260506],{"className":260507,"style":32604},[31845],[34,260509,260511,260514],{"className":260510},[31676],[34,260512],{"className":260513,"style":33218},[31680],[34,260515,38691],{"className":260516,"style":38706},[31685,31686]," 台。要让这 ",[34,260519,260521,260534],{"className":260520},[31642],[34,260522,260524],{"className":260523},[31646],[31648,260525,260526],{"xmlns":31650},[31652,260527,260528,260532],{},[31655,260529,260530],{},[31658,260531,38691],{},[31665,260533,38691],{"encoding":31667},[34,260535,260537],{"className":260536,"ariaHidden":758},[31672],[34,260538,260540,260543],{"className":260539},[31676],[34,260541],{"className":260542,"style":33218},[31680],[34,260544,38691],{"className":260545,"style":38706},[31685,31686]," 个进程可能陷入死锁，必须满足：它们占有的资源数大于等于系统的总资源数，即 ",[34,260548,260550,260570],{"className":260549},[31642],[34,260551,260553],{"className":260552},[31646],[31648,260554,260555],{"xmlns":31650},[31652,260556,260557,260567],{},[31655,260558,260559,260561,260563,260565],{},[31749,260560,768],{},[31658,260562,38691],{},[31662,260564,43175],{},[31749,260566,60525],{},[31665,260568,260569],{"encoding":31667},"2K \\ge 8",[34,260571,260573,260594],{"className":260572,"ariaHidden":758},[31672],[34,260574,260576,260579,260582,260585,260588,260591],{"className":260575},[31676],[34,260577],{"className":260578,"style":195047},[31680],[34,260580,768],{"className":260581},[31685],[34,260583,38691],{"className":260584,"style":38706},[31685,31686],[34,260586],{"className":260587,"style":31846},[31845],[34,260589,43175],{"className":260590},[31850],[34,260592],{"className":260593,"style":31846},[31845],[34,260595,260597,260600],{"className":260596},[31676],[34,260598],{"className":260599,"style":32617},[31680],[34,260601,60525],{"className":260602},[31685],"。\n由此解出 ",[34,260605,260607,260625],{"className":260606},[31642],[34,260608,260610],{"className":260609},[31646],[31648,260611,260612],{"xmlns":31650},[31652,260613,260614,260622],{},[31655,260615,260616,260618,260620],{},[31658,260617,38691],{},[31662,260619,43175],{},[31749,260621,15932],{},[31665,260623,260624],{"encoding":31667},"K \\ge 4",[34,260626,260628,260646],{"className":260627,"ariaHidden":758},[31672],[34,260629,260631,260634,260637,260640,260643],{"className":260630},[31676],[34,260632],{"className":260633,"style":195047},[31680],[34,260635,38691],{"className":260636,"style":38706},[31685,31686],[34,260638],{"className":260639,"style":31846},[31845],[34,260641,43175],{"className":260642},[31850],[34,260644],{"className":260645,"style":31846},[31845],[34,260647,260649,260652],{"className":260648},[31676],[34,260650],{"className":260651,"style":32617},[31680],[34,260653,15932],{"className":260654},[31685],[34,260656,260658,260676],{"className":260657},[31642],[34,260659,260661],{"className":260660},[31646],[31648,260662,260663],{"xmlns":31650},[31652,260664,260665,260673],{},[31655,260666,260667,260669,260671],{},[31658,260668,38691],{},[31662,260670,202],{},[31749,260672,786],{},[31665,260674,260675],{"encoding":31667},"K=3",[34,260677,260679,260697],{"className":260678,"ariaHidden":758},[31672],[34,260680,260682,260685,260688,260691,260694],{"className":260681},[31676],[34,260683],{"className":260684,"style":33218},[31680],[34,260686,38691],{"className":260687,"style":38706},[31685,31686],[34,260689],{"className":260690,"style":31846},[31845],[34,260692,202],{"className":260693},[31850],[34,260695],{"className":260696,"style":31846},[31845],[34,260698,260700,260703],{"className":260699},[31676],[34,260701],{"className":260702,"style":32617},[31680],[34,260704,786],{"className":260705},[31685],"，它们最坏总共占有 ",[34,260708,260710,260732],{"className":260709},[31642],[34,260711,260713],{"className":260712},[31646],[31648,260714,260715],{"xmlns":31650},[31652,260716,260717,260729],{},[31655,260718,260719,260721,260723,260725,260727],{},[31749,260720,786],{},[31662,260722,42071],{},[31749,260724,768],{},[31662,260726,202],{},[31749,260728,36656],{},[31665,260730,260731],{"encoding":31667},"3 \\times 2 = 6",[34,260733,260735,260753,260771],{"className":260734,"ariaHidden":758},[31672],[34,260736,260738,260741,260744,260747,260750],{"className":260737},[31676],[34,260739],{"className":260740,"style":35942},[31680],[34,260742,786],{"className":260743},[31685],[34,260745],{"className":260746,"style":32604},[31845],[34,260748,42071],{"className":260749},[32024],[34,260751],{"className":260752,"style":32604},[31845],[34,260754,260756,260759,260762,260765,260768],{"className":260755},[31676],[34,260757],{"className":260758,"style":32617},[31680],[34,260760,768],{"className":260761},[31685],[34,260763],{"className":260764,"style":31846},[31845],[34,260766,202],{"className":260767},[31850],[34,260769],{"className":260770,"style":31846},[31845],[34,260772,260774,260777],{"className":260773},[31676],[34,260775],{"className":260776,"style":32617},[31680],[34,260778,36656],{"className":260779},[31685],"台，系统还剩2台可以完全满足其中一个进程让其运行结束，就不会发生死锁。所以最小的可能死锁的K为4。",[19,260782,260784],{"id":260783},"_27设文件f1当前引用计数值为1先建立f1的符号链接文件f2再建立f1的硬链接文件f3然后删除f1此时f2和f3的引用计数值分别是","27．设文件F1当前引用计数值为1，先建立F1的符号链接文件F2，再建立F1的硬链接文件F3，然后删除F1。此时，F2和F3的引用计数值分别是( )",[15,260786,260787],{},[2627,260788,260789],{},"A 0，1 B 1，2 C 1，1 D 2，1",[15,260791,260792,258137],{},[2627,260793,58173],{},[15,260795,260796,260798],{},[2627,260797,258022],{},"\n在文件系统（如UNIX）中，引用计数（Reference count）通常指的是指向 inode（索引节点）的硬链接数：",[2230,260800,260801,260806,260818,260826],{},[108,260802,260803,260805],{},[31,260804,175510],{}," 初始计数值为 1。",[108,260807,260808,260809,260811,260812,260814,260815,260817],{},"建立 ",[31,260810,175510],{}," 的符号链接（软链接）",[31,260813,175651],{},"。这不会改变 ",[31,260816,175510],{}," inode的硬链接计数；软链接本身是一个全新文件并有自己的独立inode，其计数初始为 1。",[108,260819,260808,260820,260822,260823,260825],{},[31,260821,175510],{}," 的硬链接 ",[31,260824,175903],{},"。由于硬链接直接指向目标文件的 inode，因此原 inode（即 F1\u002FF3 共用的数据节点）的引用计数加1，变为 2。",[108,260827,101509,260828,260830,260831,260833,260834,260836],{},[31,260829,175510],{}," 目录项。该操作移除了名字为 F1 的硬链接，原 inode 引用计数减 1，变为 1（因为 F3 仍指向它）。\n因此，",[31,260832,175651],{}," 本身的引用计数依然是它自己目录项的 1，而 ",[31,260835,175903],{}," 所在的 inode 引用计数也是 1。两者均为 1。",[19,260838,260840],{"id":260839},"_28当进程因时间片用完而让出处理机时该进程应转变为-状态","28．当进程因时间片用完而让出处理机时，该进程应转变为( )状态。",[15,260842,260843],{},[2627,260844,260845],{},"A 等待 B 就绪 C 运行 D 完成",[15,260847,260848,258017],{},[2627,260849,58173],{},[15,260851,260852,260854,260855,260858],{},[2627,260853,258022],{},"\n参考资料指出：“在轮转调度（Round-robin）策略中，每个队列里的进程都被分配一个时间片（a certain amount of time）来执行，除非它发生阻塞，否则时间片用完时它会被",[2627,260856,260857],{},"返回到调度队列中","（returned to the queue）”。“由于不是因为等待某些输入\u002F输出事件而被阻塞，它会将控制权交回给调度程序，状态转为可被再次调度的就绪状态”。因此，进程因时间片用完被迫挂起时，会转入就绪（Ready）状态，等待下一次分配CPU。",[19,260860,260862],{"id":260861},"_29文件的保密是指防止文件被","29．文件的保密是指防止文件被( )。",[15,260864,260865],{},[2627,260866,260867],{},"A 篡改 B 破坏 C 窃取 D 删除",[15,260869,260870,258137],{},[2627,260871,58173],{},[15,260873,260874,260876],{},[2627,260875,258022],{},"\n文件安全涵盖了多个方面：",[105,260878,260879,260889,260895],{},[108,260880,260881,260882,260885,260886,347],{},"防止未经授权的读取或复制，通常称为",[2627,260883,260884],{},"保密性（Confidentiality）","，针对的是被",[2627,260887,260888],{},"窃取",[108,260890,260891,260892,260894],{},"防止未经授权的修改，被称为",[2627,260893,119279],{},"，针对的是被篡改或破坏。",[108,260896,260897],{},"防止数据丢失，涉及的是可用性和可靠性，针对的是被删除或破坏。\n因此，“文件的保密”明确对应的是防止未经授权的浏览、访问或被窃取。",[19,260899,260901],{"id":260900},"_30为了允许不同用户的文件具有相同的文件名通常在文件系统中采用","30.为了允许不同用户的文件具有相同的文件名，通常在文件系统中采用（ ）。",[15,260903,260904,260907],{},[2627,260905,260906],{},"(此题与第16题完全一致)",[2627,260908,259556],{},[15,260910,260911,258017],{},[2627,260912,58173],{},[15,260914,260915,260917,260918,43298,260921,260924],{},[2627,260916,258022],{},"\n参见第16题解析。通过构建树形的层次化多级目录结构（如",[31,260919,260920],{},"\u002Fhome\u002FuserA\u002Ffile.txt",[31,260922,260923],{},"\u002Fhome\u002FuserB\u002Ffile.txt","），可以很好地分离具有同样名称的文件，从根本上解决文件名冲突的问题。",[19,260926,260928],{"id":260927},"_31-用户程序读取文件第100个逻辑块时使用操作系统提供-接口","31. 用户程序读取文件第100个逻辑块时，使用操作系统提供（ ）接口。",[15,260930,260931],{},[2627,260932,260933],{},"A 系统调用 B 图形用户接口 C 原语 D 键盘命令",[15,260935,260936,258099],{},[2627,260937,58173],{},[15,260939,260940,260942],{},[2627,260941,258022],{},"\n根据参考资料指出：“操作系统向程序员屏蔽了硬件的具体细节，并提供了一个方便的接口供程序员使用……应用程序可以通过系统调用（System calls）接口，使用简单的命令（如read、write、open等）来访问系统提供的服务”。因此，当用户程序需要读取文件的逻辑块时，必须通过操作系统提供的“系统调用（API）”接口来向内核发出读取请求。",[19,260944,260946],{"id":260945},"_32数据文件存放在到存储介质上时采用的逻辑组织形式是与-有关的","32．数据文件存放在到存储介质上时，采用的逻辑组织形式是与( )有关的。",[15,260948,260949],{},[2627,260950,260951],{},"A 文件逻辑结构 B 存储介质特性 C 主存储器管理方式 D 分配外设方式",[15,260953,260954,258017],{},[2627,260955,58173],{},[15,260957,260958,260960],{},[2627,260959,258022],{},"\n在操作系统中，文件的组织形式分为逻辑结构和物理结构。当数据文件具体“存放到存储介质上”时，讨论的实际上是其物理组织形式（如顺序、链接、索引等）。参考资料在介绍文件组织时特别提到：“顺序文件组织形式（Sequential file）是唯一一种既容易存储在磁带上，又容易存储在磁盘上的形式”。这说明，文件在存储介质上采用何种组织结构，受到具体“存储介质特性”（如磁带仅支持顺序存取，而磁盘支持随机存取）的强烈限制和影响。",[19,260962,260964],{"id":260963},"_33实时操作系统必须在-内处理完来自外部的事件","33．实时操作系统必须在 ( ) 内处理完来自外部的事件。",[15,260966,260967],{},[2627,260968,260969],{},"A. 响应时间\t\tB. 周转时间\t\tC. 规定时间\t\tD. 调度时间",[15,260971,260972,258137],{},[2627,260973,58173],{},[15,260975,260976,260978],{},[2627,260977,258022],{},"\n参考资料对实时操作系统的定义是：“实时系统是指其正确性不仅取决于计算的逻辑结果，还取决于产生结果的时间。因此，实时任务通常都有一个相关的截止时间（Deadline）”。对于硬实时任务，必须在“规定的截止时间（规定时间）”内满足要求并完成事件处理；否则将导致不可接受的破坏或致命的系统错误。",[19,260980,260982],{"id":260981},"_34用户程序向系统提出使用外设的请求方式是","34．用户程序向系统提出使用外设的请求方式是()。",[15,260984,260985],{},[2627,260986,260987],{},"A. 作业申请\t\tB. 原语\t\tC. 系统调用\t\tD. I\u002FO指令",[15,260989,260990,258137],{},[2627,260991,58173],{},[15,260993,260994,260996],{},[2627,260995,258022],{},"\n参考资料指出，I\u002FO指令属于特权指令（Privileged instructions），“特权指令只能由操作系统中的监控程序（内核）执行……如果用户程序希望执行I\u002FO操作，它必须请求监控程序代为执行”。用户程序提出这种请求的标准方式就是通过“系统调用（System call）”。",[19,260998,261000],{"id":260999},"_35-是一种只能进行p操作和v操作的特殊变量","35．( ) 是一种只能进行P操作和V操作的特殊变量。",[15,261002,261003],{},[2627,261004,261005],{},"A. 同步\t\tB. 互斥\t\tC. 信号量\tD. 管程",[15,261007,261008,258137],{},[2627,261009,58173],{},[15,261011,261012,261014],{},[2627,261013,258022],{},"\n参考资料明确定义：“信号量（Semaphore）是一个用于进程间发送信号的整数值。对信号量只能执行三种操作（且均为原子操作）：初始化、递减和递增”。在Dijkstra最初的论文以及大量的文献中，“字母P被用于等待（递减）操作，而字母V被用于发信号（递增）操作”。因此，只能进行P操作和V操作的特殊变量是信号量。",[19,261016,261018],{"id":261017},"_36以下关于死锁的必要条件的叙述中错误的是","36．以下关于死锁的必要条件的叙述中错误的是 ( ) 。",[15,261020,261021,261024,261027,261030],{},[2627,261022,261023],{},"A. 只要具备了死锁的必要条件，就一定发生死锁现象",[2627,261025,261026],{},"B. 解决死锁问题可以从死锁的必要条件出发",[2627,261028,261029],{},"C. 一旦出现死锁现象，处于死锁状态的进程一定同时具备死锁的必要条件",[2627,261031,261032],{},"D. 死锁的四个必要条件之间不是完全独立的，但也不是等价的",[15,261034,261035,258099],{},[2627,261036,58173],{},[15,261038,261039,261041],{},[2627,261040,258022],{},"\n死锁产生的四个条件（互斥、占有且等待、非剥夺、循环等待）是死锁发生的“必要条件”。参考资料解释道：“前三个条件是死锁存在的必要条件，但不是充分条件……只有当第四个条件（循环等待）也发生时，这四个条件放在一起才构成了死锁存在的充分必要条件”。更准确地说，即便四个必要条件可能达成，也必须是进程资源请求的具体推进顺序进入了“致命区域（Fatal region）”才会真实触发死锁。因此，“只要具备了死锁的必要条件，就一定发生死锁”的说法是错误的。",[19,261043,261045],{"id":261044},"_37在-中不可能产生系统抖动现象","37．在 ( ) 中，不可能产生系统抖动现象。",[15,261047,261048,261051],{},[2627,261049,261050],{},"A. 请求页式存储管理\t\t\tB. 段式存储管理",[2627,261052,261053],{},"C. 固定式分区存储管理\t\tD. 段页式存储管理",[15,261055,261056,258137],{},[2627,261057,58173],{},[15,261059,261060,261062],{},[2627,261061,258022],{},"\n系统抖动（Thrashing）是指在虚拟内存系统中，系统花费了大量的时间在内外存之间频繁地进行页面或段的换入换出，而实际执行指令的时间却很少的现象。参考资料指出，抖动问题是由按需调入碎片的“虚拟内存机制”引发的。而“固定式分区存储管理（Fixed Partitioning）”在装入程序时会直接把整个程序装入到一个固定的内存分区中，不涉及运行时程序部分页面的动态置换，所以根本不可能产生页面置换引发的系统抖动现象。",[19,261064,261066],{"id":261065},"_38下面是关于重定位的有关描述其中错误的是","38．下面是关于重定位的有关描述，其中错误的是 ( ) 。",[15,261068,261069,261072,261075,261078],{},[2627,261070,261071],{},"A. 绝对地址是主存空间的地址编号",[2627,261073,261074],{},"B. 用户程序中使用的从0地址开始的地址编号是逻辑地址",[2627,261076,261077],{},"C. 动态重定位中装入主存的作业仍保持原来的逻辑地址",[2627,261079,261080],{},"D. 静态重定位中装人主存的作业仍保持原来的逻辑地址",[15,261082,261083,258118],{},[2627,261084,58173],{},[15,261086,261087,261089],{},[2627,261088,258022],{},"\n这道题考查程序的地址重定位技术。",[105,261091,261092,261102],{},[108,261093,261094,261097,261098,261101],{},[2627,261095,261096],{},"静态重定位："," 是在程序装入内存时，由装入程序一次性把指令和数据的逻辑地址全部修改为内存中的绝对（物理）地址。因此，装入主存后的作业",[2627,261099,261100],{},"不再保持","原来的逻辑地址。所以选项D错误。（注：此特性属于操作系统的通用外部基础理论）。",[108,261103,261104,261107],{},[2627,261105,261106],{},"动态重定位："," 参考资料中提到，“系统提供了虚拟地址（Logical Address）到真实物理地址（Real Address）的动态映射（Dynamic mapping）”。在动态重定位下，程序装入内存时依旧保持原来的逻辑地址，地址转换是在每条指令真正执行时，由硬件（MMU）临时完成的。",[19,261109,261111],{"id":261110},"_39通过硬件和软件的功能扩充把原来独占的设备改造成若干用户共享的设备这种设备称为","39．通过硬件和软件的功能扩充，把原来独占的设备改造成若干用户共享的设备，这种设备称为 ( ) 。",[15,261113,261114],{},[2627,261115,261116],{},"A. 存储设备\tB. 系统设备\tC. 虚拟设备\tD. 用户设备",[15,261118,261119,258137],{},[2627,261120,58173],{},[15,261122,261123,261125],{},[2627,261124,258022],{},"\n（注：此题涉及的SPOOLing概念属于操作系统通用外部知识）。操作系统通过SPOOLing（假脱机）技术，例如在打印机等独占设备前设置磁盘缓冲区，使得多个进程可以同时向磁盘发送打印请求。操作系统对这些请求进行排队和调度，从而在逻辑上把一台独占的物理设备改造成了多台可以同时供多个用户使用的设备。在操作系统术语中，这种被改造后的共享设备称为“虚拟设备（Virtual Device）”。",[19,261127,261129],{"id":261128},"_40对磁盘而言输入输出操作的信息传送单位为","40．对磁盘而言，输入输出操作的信息传送单位为 ( ) 。",[15,261131,261132],{},[2627,261133,261134],{},"A. 字符\tB. 字\tC. 块\t\tD. 文件",[15,261136,261137,258137],{},[2627,261138,58173],{},[15,261140,261141,261143],{},[2627,261142,258022],{},"\n参考资料在对I\u002FO设备分类的讲解中指出：“在数据传输单位（Unit of transfer）方面：数据可以作为字节或字符流进行传输（例如终端I\u002FO），也可以在更大的‘块（Blocks）’中进行传输（例如磁盘I\u002FO）”。此外也明确指出，磁盘和磁带是典型的“面向块的设备（Block-oriented devices）”，信息的存取与传输均是以“块”为单位进行的。",[19,261145,261147],{"id":261146},"_41进程所请求的一次打印输出结束后将使进程状态从","41．进程所请求的一次打印输出结束后，将使进程状态从（ ）",[15,261149,261150,261153],{},[2627,261151,261152],{},"A、运行态变为就绪态 B、运行态变为等待态",[2627,261154,261155],{},"C、就绪态变为运行态 D、等待态变为就绪态",[15,261157,261158,258118],{},[2627,261159,58173],{},[15,261161,261162,261164],{},[2627,261163,258022],{},"\n当进程请求打印机（I\u002FO操作）时，它会被阻塞，进入“阻塞\u002F等待（Blocked）”状态。参考资料指出：“当I\u002FO中断发生时，操作系统会判断对应的I\u002FO操作是否完成，如果已完成，操作系统会将所有等待该事件的阻塞进程移动到就绪态（Ready state）”。因此，I\u002FO操作结束会使进程从等待态转变为就绪态。",[19,261166,261168],{"id":261167},"_42-不是基本的操作系统","42．（ ）不是基本的操作系统。",[15,261170,261171],{},[2627,261172,261173],{},"A、批处理操作系统 B、分时操作系统 C、实时操作系统 D、网络操作系统",[15,261175,261176,258118],{},[2627,261177,58173],{},[15,261179,261180,261182],{},[2627,261181,258022],{},"\n在操作系统的发展历程中，最具代表性、也是最基本的三大操作系统类型是：批处理操作系统（Batch Systems）、分时操作系统（Time-sharing Systems）和实时操作系统（Real-time Systems）。网络操作系统是在这三大基本操作系统的基础上，随着计算机网络技术的发展而衍生出来的现代操作系统形式，不属于传统意义上的“基本操作系统”。",[19,261184,261186],{"id":261185},"_43-不是分时系统的基本特征","43．（ ）不是分时系统的基本特征：",[15,261188,261189],{},[2627,261190,261191],{},"A、同时性 B、独立性 C、实时性 D、交互性",[15,261193,261194,258137],{},[2627,261195,58173],{},[15,261197,261198,261200],{},[2627,261199,258022],{},"\n分时操作系统（Time-sharing system）的核心目标是优化响应时间，它具有多路性（同时性）、独立性、及时性和交互性（Interactive）四大特征。而“实时性”要求系统在严苛的“规定时间（Deadline）”内对外部事件做出响应，这是“实时操作系统（Real-time OS）”的典型特征，并非分时系统的特征。",[19,261202,261204],{"id":261203},"_44采用动态重定位方式装入的作业在执行中允许-将其移动","44．采用动态重定位方式装入的作业，在执行中允许（ ）将其移动。",[15,261206,261207,261210],{},[2627,261208,261209],{},"A、用户有条件地 \t B、用户无条件地",[2627,261211,261212],{},"C、操作系统有条件地 D、操作系统无条件地",[15,261214,261215,258137],{},[2627,261216,58173],{},[15,261218,261219,261221],{},[2627,261220,258022],{},"\n在使用动态重定位（以及动态分区）时，操作系统可以通过一种称为“内存紧凑（Compaction）”的技术来移动内存中的进程，以消除外部碎片。这种移动是对用户透明的（不由用户控制）。但操作系统进行移动也需要满足一定的“条件”，例如参考资料提到的“如果进程正在被锁定在主存中（temporarily locked into main memory）用于某些I\u002FO交互”，则系统暂时无法将其移动或换出。因此是“操作系统有条件地”允许将其移动。",[19,261223,261225],{"id":261224},"_45分页式存储管理中地址转换工作是由-完成的","45．分页式存储管理中，地址转换工作是由（ ）完成的。",[15,261227,261228],{},[2627,261229,261230],{},"A、硬件 B、地址转换程序 C、用户程序 D、装入程序",[15,261232,261233,258099],{},[2627,261234,58173],{},[15,261236,261237,261239],{},[2627,261238,258022],{},"\n在虚拟内存与分页式存储管理中，由于每次内存访问都需要进行虚拟地址到物理地址的转换（Logical-to-Physical translation），为了保证执行效率，这种地址映射“必须是由特殊的硬件机制（Special hardware）来实现的”。具体硬件部件通常被称为内存管理单元（MMU）。",[19,261241,261243],{"id":261242},"_46如果允许不同用户的文件可以具有相同的文件名通常采用-来保证按名存取的安全","46．如果允许不同用户的文件可以具有相同的文件名，通常采用（ ）来保证按名存取的安全。",[15,261245,261246],{},[2627,261247,261248],{},"A、重名翻译机构 B、建立索引表 C、建立指针 \t D、多级目录结构",[15,261250,261251,258118],{},[2627,261252,58173],{},[15,261254,261255,261257],{},[2627,261256,258022],{},"\n（此题与上一批的Q16及Q30重复）。参考资料中指出，通过构建多级目录结构（如树形目录），不仅可以依据所有者、应用程序等清晰地分离文件，“这也避免了名称冲突（name clashes）的问题”。因此多级目录结构允许不同用户的目录下具有同名文件而不会引发混淆。",[19,261259,261261],{"id":261260},"_47对记录式文件操作系统为用户存取文件信息的最小单位是","47．对记录式文件，操作系统为用户存取文件信息的最小单位是（ ）。",[15,261263,261264],{},[2627,261265,261266],{},"A、字符 \t B、数据项 C、记录 D、文件",[15,261268,261269,258137],{},[2627,261270,58173],{},[15,261272,261273,261275],{},[2627,261274,258022],{},"\n参考资料在文件管理的定义中说明：“记录（Record）是一组相关字段的集合，能够被应用程序视为一个单一的单元（treated as a unit by some application program）。而文件是相似记录的集合”。因此，对于逻辑上具有结构的“记录式文件”，操作系统为应用层提供的存取、处理最小逻辑单元即是“记录”。",[19,261277,261279],{"id":261278},"_48为了提高设备分配的灵活性用户申请设备时应指定-号","48．为了提高设备分配的灵活性，用户申请设备时应指定（ ）号。",[15,261281,261282],{},[2627,261283,261284],{},"A、设备类相对 \t B、设备类绝对 C、相对 \t D、绝对",[15,261286,261287,258099],{},[2627,261288,58173],{},[15,261290,261291,261293],{},[2627,261292,258022],{},"\n（注：此题涉及的是操作系统中的设备独立性原理）。为了保证设备独立性（Device Independence）与设备分配的灵活性，用户在编程申请外设时，不应该指定某台具体的物理设备（绝对号），而应当指定需要哪一种类型的设备（如“打印机类”）以及在该类设备中的“相对号”（如要求第一台打印机）。由操作系统负责把该“设备类相对号”映射到系统中当前可用的物理设备的绝对号上。",[19,261295,261297],{"id":261296},"_49一作业进入内存后则所属该作业的进程初始时处于-状态","49．一作业进入内存后，则所属该作业的进程初始时处于（ ）状态。",[15,261299,261300],{},[2627,261301,261302],{},"A、运行 \t B、等待 C、就绪 \t\tD、收容",[15,261304,261305,258137],{},[2627,261306,58173],{},[15,261308,261309,261311],{},[2627,261310,258022],{},"\n参考资料对进程状态的描述为：“新建态（New）是指刚刚被创建但尚未被操作系统接纳进入可执行进程池的进程。就绪态（Ready）是指已经准备好，只要得到机会就可以立即执行的进程”。当一个作业被完全接纳并装入内存中（即脱离了新建态\u002F后备状态）时，它的初始活动状态是“就绪（Ready）”状态。",[19,261313,261315],{"id":261314},"_50共享变量是指-访问的变量","50．共享变量是指（ ）访问的变量。",[15,261317,261318],{},[2627,261319,261320],{},"A、只能被系统进程 B、只能被多个进程互斥 C、只能被用户进程 D、可被多个进程",[15,261322,261323,258017],{},[2627,261324,58173],{},[15,261326,261327,261329],{},[2627,261328,258022],{},"\n参考资料在讨论并发控制时指出：“我们必须保护共享的全局变量（shared global variables）……我们必须施加一种纪律，即同一时刻只允许一个进程进入该过程去访问共享变量”。也就是说，对于会被多个并发进程使用的共享变量（也称为临界资源），它们必须受到控制，即“只能被多个进程互斥地访问”，以避免竞态条件和数据不一致。",[19,261331,261333],{"id":261332},"_51-批处理系统的主要缺点是","51. 批处理系统的主要缺点是（ ）。",[15,261335,261336],{},[2627,261337,261338],{},"A.CPU的利用率不高 B.失去了交互性 C.不具备并行性 D.以上都不是",[15,261340,261341,258017],{},[2627,261342,58173],{},[15,261344,261345,261347],{},[2627,261346,258022],{},"\n参考资料指出，为了最大化处理器利用率，开发出了简单的批处理操作系统（Simple Batch Systems）。在此类系统中，“用户不再对处理器拥有直接访问权限”，而是由操作员将作业分批次送入系统顺序执行。因为作业一旦提交便脱离了用户控制并自动连续处理，使得用户无法在程序运行过程中与之交互，这是它最大的缺点（失去交互性）。",[19,261349,261351],{"id":261350},"_52-树型目录结构的第一级称为目录树的","52. 树型目录结构的第一级称为目录树的（ ）。",[15,261353,261354],{},[2627,261355,261356],{},"A.分支节点 B.根节点 C.叶节点 D.终节点",[15,261358,261359,258017],{},[2627,261360,58173],{},[15,261362,261363,261365,261366,347],{},[2627,261364,258022],{},"\n（注：这属于操作系统文件系统基础概念，且与下文的第60题重复）。在树型目录结构中，所有目录和文件都挂载在一个唯一的最高层级上。这第一级的起始目录在数据结构上被称为树的“根节点（Root directory \u002F Root node）”，例如在UNIX\u002FLinux系统中就是根目录 ",[31,261367,38080],{},[19,261369,261371],{"id":261370},"_53-虚拟内存的容量只受-的限制","53. 虚拟内存的容量只受（ ）的限制。",[15,261373,261374,261377],{},[2627,261375,261376],{},"A.物理内存的大小 B.磁盘空间的大小",[2627,261378,261379],{},"C.数据存放的实际地址 D.计算机地址位数",[15,261381,261382,258118],{},[2627,261383,58173],{},[15,261385,261386,261388,261389,261481],{},[2627,261387,258022],{},"\n虚拟内存（Virtual memory）允许程序从逻辑的角度寻址内存，而不需要关心物理内存的实际大小。理论上，程序能够生成的最大虚拟地址空间，严格取决于处理器体系结构所提供的逻辑地址总线的宽度（即“计算机地址位数”）。例如32位处理器寻址能力上限为 ",[34,261390,261392,261418],{"className":261391},[31642],[34,261393,261395],{"className":261394},[31646],[31648,261396,261397],{"xmlns":31650},[31652,261398,261399,261415],{},[31655,261400,261401,261407,261409,261411,261413],{},[37624,261402,261403,261405],{},[31749,261404,768],{},[31749,261406,173486],{},[31662,261408,202],{},[31749,261410,15932],{},[31658,261412,97709],{},[31658,261414,54224],{},[31665,261416,261417],{"encoding":31667},"2^{32} = 4GB",[34,261419,261421,261468],{"className":261420,"ariaHidden":758},[31672],[34,261422,261424,261427,261459,261462,261465],{"className":261423},[31676],[34,261425],{"className":261426,"style":37774},[31680],[34,261428,261430,261433],{"className":261429},[31685],[34,261431,768],{"className":261432},[31685],[34,261434,261436],{"className":261435},[31792],[34,261437,261439],{"className":261438},[31796],[34,261440,261442],{"className":261441},[31801],[34,261443,261445],{"className":261444,"style":37774},[31805],[34,261446,261447,261450],{"style":37777},[34,261448],{"className":261449,"style":31814},[31813],[34,261451,261453],{"className":261452},[31818,31819,31820,31821],[34,261454,261456],{"className":261455},[31685,31821],[34,261457,173486],{"className":261458},[31685,31821],[34,261460],{"className":261461,"style":31846},[31845],[34,261463,202],{"className":261464},[31850],[34,261466],{"className":261467,"style":31846},[31845],[34,261469,261471,261474,261477],{"className":261470},[31676],[34,261472],{"className":261473,"style":33218},[31680],[34,261475,15932],{"className":261476},[31685],[34,261478,261480],{"className":261479,"style":54252},[31685,31686],"GB","。虽然它也受外部磁盘容量的实际制约，但在定义上，其最大逻辑容量“仅受限于计算机架构的地址位数”。",[19,261483,261485],{"id":261484},"_54通道是一种","54.通道是一种（ ）。",[15,261487,261488],{},[2627,261489,261490],{},"A.I\u002FO端口 B.数据通道 C.I\u002FO专用处理机 D.软件工具",[15,261492,261493,258137],{},[2627,261494,58173],{},[15,261496,261497,261499],{},[2627,261498,258022],{},"\n参考资料追溯了I\u002FO功能的演进过程，指出随着技术的发展，“I\u002FO模块被增强成为一个独立的处理器（a separate processor），拥有针对I\u002FO定制的专门指令集”。“这种能够独立获取并执行I\u002FO指令、具有自己指令集并控制外设的I\u002FO模块通常被称作I\u002FO通道（I\u002FO channel）或I\u002FO处理器（I\u002FO processor）”。",[19,261501,261503],{"id":261502},"_55-缓冲技术用于","55. 缓冲技术用于（ ）。",[15,261505,261506,261509],{},[2627,261507,261508],{},"A 提高主机和设备交换信息的速度 B 提供主、辅存接口",[2627,261510,261511],{},"C 提高设备利用率 D 扩充相对地址空间",[15,261513,261514,258137],{},[2627,261515,58173],{},[15,261517,261518,261520],{},[2627,261519,258022],{},"\n参考资料在I\u002FO缓存（I\u002FO Buffering）部分指出，缓冲（如磁盘缓存）能显著提升系统性能：“通过使用缓冲池，系统可以合并大量的磁盘读写操作（使得CPU不用长时间阻塞），使得慢速设备保持持续运转”。更准确地说，缓冲技术并没有真正改变硬件本身的传输速度（A不准确），它的根本作用是缓解CPU和I\u002FO设备间的速度不匹配，实现两者的并行工作，从而“提高设备和CPU的利用率”。在标准OS理论中，此题通常最优选为C。",[19,261522,261524],{"id":261523},"_56-采用spooling技术的目的是","56. 采用SPOOLing技术的目的是（ ）。",[15,261526,261527,261530],{},[2627,261528,261529],{},"A.提高独占设备的利用率 B.提高主机效率",[2627,261531,261532],{},"C.减轻用户编程负担 D.提高程序的运行速度",[15,261534,261535,258099],{},[2627,261536,58173],{},[15,261538,261539,261541],{},[2627,261540,258022],{},"\n（注：此部分内容为操作系统经典外围知识）。SPOOLing（Simultaneous Peripheral Operations On-Line，假脱机技术）的核心目的就是将原本只能由一个进程占用的“独占设备”（例如打印机）改造成可以通过输入输出井进行逻辑排队的“共享设备（虚拟设备）”。这避免了进程因长时间等待独占设备而阻塞，极大地提高了“独占设备的利用率”。",[19,261543,261545],{"id":261544},"_57在unix-系统中对空闲磁盘空间管理的方法是","57．在UNIX 系统中对空闲磁盘空间管理的方法是（ ）。",[15,261547,261548],{},[2627,261549,261550],{},"A 位示图 B 空闲空间链 C 成组链接法 D 空闲表",[15,261552,261553,258137],{},[2627,261554,58173],{},[15,261556,261557,261559],{},[2627,261558,258022],{},"\n（根据操作系统通用理论补充）：在经典的文件系统实现中，UNIX System V系统采用的空闲磁盘块管理方法是“成组链接法（Grouped Link Method）”。这是一种将“空闲空间链”和“空闲表”两种机制综合起来的方法，它把磁盘上每100个空闲块分为一组，极大减少了存取文件时的频繁磁盘寻道操作，提升了分配和回收速度。",[19,261561,261563],{"id":261562},"_58实现虚拟存储器最关键的技术是","58．实现虚拟存储器最关键的技术是（ ）。",[15,261565,261566],{},[2627,261567,261568],{},"A 内存分配 B 置换算法 C 请求调页(段) D 对换空间管理",[15,261570,261571,258137],{},[2627,261572,58173],{},[15,261574,261575,261577],{},[2627,261576,258022],{},"\n虚拟存储器的核心思想是不必在程序运行前将其全部加载入内存。参考资料表明：“如果处理器遇到一个不再主存中的虚拟地址引用（内存缺页），操作系统必须从副存储器（磁盘）将包含该引用的页面或段‘调入（bring in）’主存”。这种在运行过程中，一旦缺页就向系统发出请求从而将数据拉入内存的机制即为“请求调页（Demand Paging）或请求调段技术”，它是实现虚拟内存最核心和基础的机制。置换算法等则是其优化手段。",[19,261579,261581],{"id":261580},"_59-如果文件系统中有两个文件重名不应采用","59. 如果文件系统中有两个文件重名，不应采用（ ）。",[15,261583,261584],{},[2627,261585,261586],{},"A.一级目录结构 B.树型目录结构 C.二级目录结构 D. A和C",[15,261588,261589,258099],{},[2627,261590,58173],{},[15,261592,261593,261595],{},[2627,261594,258022],{},"\n这道题与目录的文件命名隔离能力有关。\n在一级目录结构（Single-level directory）中，整个文件系统只维护一张包含系统中所有文件的总表。这意味着系统中所有的文件都必须拥有绝对独特的名字，完全不允许产生任何形式的重名现象（不论是否是同一用户）。而二级目录或树型结构（Tree-structured directory）通过为不同用户分配不同的文件夹或允许建立子文件夹，隔离了命名空间，“有效地避免了名称冲突问题”。因此，若要允许重名，绝对不能采用一级目录结构。",[19,261597,261599],{"id":261598},"_60-树型目录结构的第一级称为目录树的","60. 树型目录结构的第一级称为目录树的（ ）。",[15,261601,261602],{},[2627,261603,261356],{},[15,261605,261606,258017],{},[2627,261607,58173],{},[15,261609,261610,261612],{},[2627,261611,258022],{},"\n（此题与第52题完全一致）。根据参考资料，对文件系统树的遍历从文件系统的最高层级也就是起点开始，这个统领所有子目录与文件的第一级节点被称为“根节点（Root directory）”。",[19,261614,261616],{"id":261615},"_61在配置多道批处理操作系统的计算机系统中","61．在配置多道批处理操作系统的计算机系统中(　　　)",[15,261618,261619,261622],{},[2627,261620,261621],{},"A．用户可联机、调试自己的程序\tB．允许用户直接干预作业的执行",[2627,261623,261624],{},"C．能对外部事件实时响应\tD．允许多个作业同时使用不同的外围设备",[15,261626,261627,258118],{},[2627,261628,58173],{},[15,261630,261631,261633,261634,347],{},[2627,261632,258022],{},"\n多道批处理系统（Multiprogrammed Batch Systems）的核心设计目标是",[2627,261635,261636],{},"提高系统资源的利用率和系统吞吐量",[105,261638,261639,261650,261653],{},[108,261640,261641,261642,261645,261646,261649],{},"在该系统中，用户将作业提交给操作员或系统后，作业便脱离了用户的直接控制，由系统自动调度执行。因此，用户",[2627,261643,261644],{},"无法联机调试","自己的程序（排除A），也",[2627,261647,261648],{},"不允许直接干预","作业的执行（排除B）。",[108,261651,261652],{},"对外部事件的实时响应是“实时操作系统（Real-time OS）”的特征，批处理系统无法做到实时响应（排除C）。",[108,261654,261655,261656,261659],{},"引入“多道（Multiprogramming）”技术的目的，正是为了让多个作业同时驻留在内存中，并交替使用CPU和",[2627,261657,261658],{},"并发使用不同的外围I\u002FO设备","，从而避免CPU等待慢速I\u002FO设备而处于空闲状态。因此D是正确答案。",[19,261661,261663],{"id":261662},"_62unix操作系统是一个","62．UNIX操作系统是一个(　　　)",[15,261665,261666],{},[2627,261667,261668],{},"A．交互式分时操作系统 B．多道批处理操作系统 C．实时操作系统 D．分布式操作系统",[15,261670,261671,258099],{},[2627,261672,58173],{},[15,261674,261675,261677,261678,261681,261682,347],{},[2627,261676,258022],{},"\n根据操作系统的分类，UNIX 操作系统最初被设计并广泛应用于提供多用户、多任务的交互式环境。参考资料也明确指出：“传统的 UNIX 调度器主要针对",[2627,261679,261680],{},"分时交互式环境","（time-sharing interactive environment）设计，旨在为交互式用户提供良好的响应时间”，且“UNIX是一个多用户操作系统（multiuser operating system）”。因此，UNIX 的经典定义是一个典型的",[2627,261683,261684],{},"交互式分时操作系统",[19,261686,261688],{"id":261687},"_63若操作系统管理的某用户程序当前正占有中央处理器该用户程序欲读磁盘上的文件信息那么用户程序中相应的指令应该是","63．若操作系统管理的某用户程序当前正占有中央处理器，该用户程序欲读磁盘上的文件信息，那么用户程序中相应的指令应该是(　　　)",[15,261690,261691],{},[2627,261692,261693],{},"A．启动I\u002FO指令 B．等待I\u002FO指令 C．转移指令 D．访管指令",[15,261695,261696,258118],{},[2627,261697,58173],{},[15,261699,261700,261702,261703,261706,261707,261710],{},[2627,261701,258022],{},"\n在现代操作系统中，由于安全和资源保护的需要，",[2627,261704,261705],{},"直接操作硬件设备的指令（如启动I\u002FO指令）属于特权指令","，只能由操作系统在内核态（管态）下执行，严禁用户程序直接调用。\n当运行在用户态（目态）的用户程序需要读取磁盘文件等系统服务时，必须通过执行一种特殊的指令——",[2627,261708,261709],{},"访管指令","（Supervisor Call，或称为陷入指令\u002FTrap、系统调用指令\u002FSystem Call）来主动触发中断，将处理器的状态从用户态切换到内核态，请求操作系统代为执行I\u002FO操作。",[19,261712,261714],{"id":261713},"_64当一次系统调用功能完成后中央处理器的工作状态应","64．当一次系统调用功能完成后，中央处理器的工作状态应(　　　)",[15,261716,261717],{},[2627,261718,261719],{},"A．保持管态 B．保持目态 C．从管态转换成目态 D．从目态转换成管态",[15,261721,261722,258137],{},[2627,261723,58173],{},[15,261725,261726,261728,261729,261732,261733,347],{},[2627,261727,258022],{},"\n处理器的状态通常分为用户态（目态\u002FUser Mode）和内核态（管态\u002FSystem Mode 或 Kernel Mode）。\n当用户程序发起系统调用时，CPU 会从目态切换到管态，以便操作系统内核执行特权指令完成相应的服务。参考资料指出：“当用户调用操作系统服务时……模式被设置为内核态（管态）；而",[2627,261730,261731],{},"从服务返回到用户进程时，模式被设置为用户态（目态）","”。因此，当系统调用功能完成，要把控制权交还给用户程序时，CPU 的工作状态必须",[2627,261734,261735],{},"从管态转换成目态",[19,261737,261739],{"id":261738},"_65分布式操作系统的特点是","65．分布式操作系统的特点是(　　　)",[15,261741,261742],{},[2627,261743,261744],{},"A．资源共享 B．资源地理位置分散 C．资源位置透明 D．多个用户的程序并行运行",[15,261746,261747,258137],{},[2627,261748,58173],{},[15,261750,261751,261753,261754,261757,261758,261761,261762,347],{},[2627,261752,258022],{},"\n虽然分布式系统确实具有资源共享、地理位置分散以及多程序并发等特征（计算机网络也具备这些特征），但",[2627,261755,261756],{},"分布式操作系统（Distributed Operating System）最核心和独有的设计特点是“透明性（Transparency）”","。\n参考资料指出：“分布式操作系统为用户提供了一个",[2627,261759,261760],{},"单一主存空间和单一辅助存储空间的假象（illusion of a single system）","，以及其他统一的访问设施”。这意味着用户和应用程序在访问网络上的资源时，无需关心资源具体位于哪一台物理机器上，就像访问本地资源一样。这就是",[2627,261763,261764],{},"资源位置透明",[19,261766,261768],{"id":261767},"_66引入进程的原因是","66．引入进程的原因是(　　　)",[15,261770,261771,261774,261777,261780],{},[2627,261772,261773],{},"A．提高资源的利用率和控制程序的执行",[2627,261775,261776],{},"B．提高资源的利用率和正确描述程序的执行情况",[2627,261778,261779],{},"C．提高程序的执行速度和控制程序的执行",[2627,261781,261782],{},"D．提高程序的执行速度和正确描述程序的执行情况",[15,261784,261785,258017],{},[2627,261786,58173],{},[15,261788,261789,261791,261792,261795,261796,261799,261800,261803],{},[2627,261790,258022],{},"\n在引入多道程序设计后，多个程序在内存中并发执行，这极大地",[2627,261793,261794],{},"提高了系统资源的利用率","（如CPU和I\u002FO设备的重叠使用）。\n然而，并发执行带来了程序执行速度不可预测、资源竞争以及相互制约等复杂问题。静态的“程序”概念已经无法反映这种动态的、走走停停的执行上下文。参考资料指出：“为了解决这些问题，我们需要一种系统化的方法来监控和控制这些程序……",[2627,261797,261798],{},"进程的概念提供了这一基础","”。引入“进程”就是为了",[2627,261801,261802],{},"正确地描述和记录","程序在并发环境下的动态执行情况（如保存寄存器上下文、程序计数器、状态等）。因此，引入进程的原因是提高资源利用率并正确描述程序的执行情况。",[19,261805,261807],{"id":261806},"_67进程有三种基本状态可能的状态转换是","67．进程有三种基本状态，可能的状态转换是(　　　)",[15,261809,261810,261813,261816,261819],{},[2627,261811,261812],{},"A．就绪态到运行态、等待态到就绪态、运行态到等待态",[2627,261814,261815],{},"B．就绪态到运行态、就绪态到等待态、等待态到运行态",[2627,261817,261818],{},"C．就绪态到运行态、等待态到就绪态、等待态到运行态",[2627,261820,261821],{},"D．运行态到就绪态、就绪态到等待态、等待态到运行态",[15,261823,261824,258099],{},[2627,261825,58173],{},[15,261827,261828,261830],{},[2627,261829,258022],{},"\n进程的三种基本状态是：运行态（Running）、就绪态（Ready）和等待态（Blocked \u002F 等待 \u002F 阻塞）。根据参考资料中的进程状态转换模型，合法的状态转换包括：",[2230,261832,261833,261868,261903,261937],{},[108,261834,261835,261867],{},[2627,261836,261837,261838,261866],{},"就绪态 ",[34,261839,261841,261854],{"className":261840},[31642],[34,261842,261844],{"className":261843},[31646],[31648,261845,261846],{"xmlns":31650},[31652,261847,261848,261852],{},[31655,261849,261850],{},[31662,261851,33239],{},[31665,261853,119689],{"encoding":31667},[34,261855,261857],{"className":261856,"ariaHidden":758},[31672],[34,261858,261860,261863],{"className":261859},[31676],[34,261861],{"className":261862,"style":73089},[31680],[34,261864,33239],{"className":261865},[31850]," 运行态","：调度程序（Dispatcher）选择一个就绪进程分配CPU。",[108,261869,261870,261902],{},[2627,261871,261872,261873,261901],{},"运行态 ",[34,261874,261876,261889],{"className":261875},[31642],[34,261877,261879],{"className":261878},[31646],[31648,261880,261881],{"xmlns":31650},[31652,261882,261883,261887],{},[31655,261884,261885],{},[31662,261886,33239],{},[31665,261888,119689],{"encoding":31667},[34,261890,261892],{"className":261891,"ariaHidden":758},[31672],[34,261893,261895,261898],{"className":261894},[31676],[34,261896],{"className":261897,"style":73089},[31680],[34,261899,33239],{"className":261900},[31850]," 就绪态","：进程的时间片用完，或被高优先级进程抢占。",[108,261904,261905,261936],{},[2627,261906,261872,261907,261935],{},[34,261908,261910,261923],{"className":261909},[31642],[34,261911,261913],{"className":261912},[31646],[31648,261914,261915],{"xmlns":31650},[31652,261916,261917,261921],{},[31655,261918,261919],{},[31662,261920,33239],{},[31665,261922,119689],{"encoding":31667},[34,261924,261926],{"className":261925,"ariaHidden":758},[31672],[34,261927,261929,261932],{"className":261928},[31676],[34,261930],{"className":261931,"style":73089},[31680],[34,261933,33239],{"className":261934},[31850]," 等待态","：进程因请求I\u002FO操作或等待某个事件而主动阻塞自己。",[108,261938,261939,261970],{},[2627,261940,261941,261942,261901],{},"等待态 ",[34,261943,261945,261958],{"className":261944},[31642],[34,261946,261948],{"className":261947},[31646],[31648,261949,261950],{"xmlns":31650},[31652,261951,261952,261956],{},[31655,261953,261954],{},[31662,261955,33239],{},[31665,261957,119689],{"encoding":31667},[34,261959,261961],{"className":261960,"ariaHidden":758},[31672],[34,261962,261964,261967],{"className":261963},[31676],[34,261965],{"className":261966,"style":73089},[31680],[34,261968,33239],{"className":261969},[31850],"：进程等待的事件发生（如I\u002FO完成），被唤醒进入就绪队列。\n选项A完全符合上述合法的转换关系。选项B、C、D中包含了非法的转换（如“就绪态到等待态”是不可能的，因为只有正在运行的进程才能主动请求阻塞；“等待态到运行态”也是不可能的，必须先转入就绪态等待调度）。",[19,261972,261974],{"id":261973},"_68系统有某类资源5个供3个进程共享为保证系统的安全应限定每个进程申请的资源数不超过","68．系统有某类资源5个，供3个进程共享，为保证系统的安全，应限定每个进程申请的资源数不超过(　　　)",[15,261976,261977],{},[2627,261978,261979],{},"A．1个 B．2个 C．3个 D．4个",[15,261981,261982,258017],{},[2627,261983,58173],{},[15,261985,261986,261988,261989,262017,262018,262069,262070,262098,262099,262150,262151,262179,262180,262230,262231,262336,262337,262340,262341,262468,262469,262596,262791,262792,262820,262821,262823],{},[2627,261987,258022],{},"\n这道题考查的是死锁的预防与避免。我们可以使用极端的最坏情况（每个进程都拿到了它所需要的资源数减1，并且都在请求最后1个资源）来分析。\n设系统有 ",[34,261990,261992,262005],{"className":261991},[31642],[34,261993,261995],{"className":261994},[31646],[31648,261996,261997],{"xmlns":31650},[31652,261998,261999,262003],{},[31655,262000,262001],{},[31658,262002,98360],{},[31665,262004,98360],{"encoding":31667},[34,262006,262008],{"className":262007,"ariaHidden":758},[31672],[34,262009,262011,262014],{"className":262010},[31676],[34,262012],{"className":262013,"style":33218},[31680],[34,262015,98360],{"className":262016,"style":33389},[31685,31686]," 个资源（",[34,262019,262021,262039],{"className":262020},[31642],[34,262022,262024],{"className":262023},[31646],[31648,262025,262026],{"xmlns":31650},[31652,262027,262028,262036],{},[31655,262029,262030,262032,262034],{},[31658,262031,98360],{},[31662,262033,202],{},[31749,262035,18136],{},[31665,262037,262038],{"encoding":31667},"M=5",[34,262040,262042,262060],{"className":262041,"ariaHidden":758},[31672],[34,262043,262045,262048,262051,262054,262057],{"className":262044},[31676],[34,262046],{"className":262047,"style":33218},[31680],[34,262049,98360],{"className":262050,"style":33389},[31685,31686],[34,262052],{"className":262053,"style":31846},[31845],[34,262055,202],{"className":262056},[31850],[34,262058],{"className":262059,"style":31846},[31845],[34,262061,262063,262066],{"className":262062},[31676],[34,262064],{"className":262065,"style":32617},[31680],[34,262067,18136],{"className":262068},[31685],"），有 ",[34,262071,262073,262086],{"className":262072},[31642],[34,262074,262076],{"className":262075},[31646],[31648,262077,262078],{"xmlns":31650},[31652,262079,262080,262084],{},[31655,262081,262082],{},[31658,262083,105852],{},[31665,262085,105852],{"encoding":31667},[34,262087,262089],{"className":262088,"ariaHidden":758},[31672],[34,262090,262092,262095],{"className":262091},[31676],[34,262093],{"className":262094,"style":33218},[31680],[34,262096,105852],{"className":262097,"style":37422},[31685,31686]," 个进程（",[34,262100,262102,262120],{"className":262101},[31642],[34,262103,262105],{"className":262104},[31646],[31648,262106,262107],{"xmlns":31650},[31652,262108,262109,262117],{},[31655,262110,262111,262113,262115],{},[31658,262112,105852],{},[31662,262114,202],{},[31749,262116,786],{},[31665,262118,262119],{"encoding":31667},"P=3",[34,262121,262123,262141],{"className":262122,"ariaHidden":758},[31672],[34,262124,262126,262129,262132,262135,262138],{"className":262125},[31676],[34,262127],{"className":262128,"style":33218},[31680],[34,262130,105852],{"className":262131,"style":37422},[31685,31686],[34,262133],{"className":262134,"style":31846},[31845],[34,262136,202],{"className":262137},[31850],[34,262139],{"className":262140,"style":31846},[31845],[34,262142,262144,262147],{"className":262143},[31676],[34,262145],{"className":262146,"style":32617},[31680],[34,262148,786],{"className":262149},[31685],"），限定每个进程最多申请 ",[34,262152,262154,262167],{"className":262153},[31642],[34,262155,262157],{"className":262156},[31646],[31648,262158,262159],{"xmlns":31650},[31652,262160,262161,262165],{},[31655,262162,262163],{},[31658,262164,33347],{},[31665,262166,33347],{"encoding":31667},[34,262168,262170],{"className":262169,"ariaHidden":758},[31672],[34,262171,262173,262176],{"className":262172},[31676],[34,262174],{"className":262175,"style":33218},[31680],[34,262177,33347],{"className":262178,"style":33389},[31685,31686]," 个资源。\n发生死锁的极端情况是：3个进程每个都获取了 ",[34,262181,262183,262200],{"className":262182},[31642],[34,262184,262186],{"className":262185},[31646],[31648,262187,262188],{"xmlns":31650},[31652,262189,262190,262198],{},[31655,262191,262192,262194,262196],{},[31658,262193,33347],{},[31662,262195,32226],{},[31749,262197,640],{},[31665,262199,198455],{"encoding":31667},[34,262201,262203,262221],{"className":262202,"ariaHidden":758},[31672],[34,262204,262206,262209,262212,262215,262218],{"className":262205},[31676],[34,262207],{"className":262208,"style":34944},[31680],[34,262210,33347],{"className":262211,"style":33389},[31685,31686],[34,262213],{"className":262214,"style":32604},[31845],[34,262216,32226],{"className":262217},[32024],[34,262219],{"className":262220,"style":32604},[31845],[34,262222,262224,262227],{"className":262223},[31676],[34,262225],{"className":262226,"style":32617},[31680],[34,262228,640],{"className":262229},[31685]," 个资源，且系统再也没有剩余资源可以分配。即已分配资源总数 ",[34,262232,262234,262264],{"className":262233},[31642],[34,262235,262237],{"className":262236},[31646],[31648,262238,262239],{"xmlns":31650},[31652,262240,262241,262261],{},[31655,262242,262243,262245,262247,262249,262251,262253,262255,262257,262259],{},[31749,262244,786],{},[31662,262246,42071],{},[31662,262248,456],{"stretchy":702},[31658,262250,33347],{},[31662,262252,32226],{},[31749,262254,640],{},[31662,262256,2557],{"stretchy":702},[31662,262258,43175],{},[31749,262260,18136],{},[31665,262262,262263],{"encoding":31667},"3 \\times (N-1) \\ge 5",[34,262265,262267,262285,262306,262327],{"className":262266,"ariaHidden":758},[31672],[34,262268,262270,262273,262276,262279,262282],{"className":262269},[31676],[34,262271],{"className":262272,"style":35942},[31680],[34,262274,786],{"className":262275},[31685],[34,262277],{"className":262278,"style":32604},[31845],[34,262280,42071],{"className":262281},[32024],[34,262283],{"className":262284,"style":32604},[31845],[34,262286,262288,262291,262294,262297,262300,262303],{"className":262287},[31676],[34,262289],{"className":262290,"style":31774},[31680],[34,262292,456],{"className":262293},[31782],[34,262295,33347],{"className":262296,"style":33389},[31685,31686],[34,262298],{"className":262299,"style":32604},[31845],[34,262301,32226],{"className":262302},[32024],[34,262304],{"className":262305,"style":32604},[31845],[34,262307,262309,262312,262315,262318,262321,262324],{"className":262308},[31676],[34,262310],{"className":262311,"style":31774},[31680],[34,262313,640],{"className":262314},[31685],[34,262316,2557],{"className":262317},[31690],[34,262319],{"className":262320,"style":31846},[31845],[34,262322,43175],{"className":262323},[31850],[34,262325],{"className":262326,"style":31846},[31845],[34,262328,262330,262333],{"className":262329},[31676],[34,262331],{"className":262332,"style":32617},[31680],[34,262334,18136],{"className":262335},[31685],"。\n要保证系统绝对安全（不会发生死锁），必须让系统在最坏分配情况下至少还剩下 ",[2627,262338,262339],{},"1个"," 资源，以保证至少有一个进程能拿到所有资源并执行完毕释放资源。\n因此，安全条件为：",[34,262342,262344,262378],{"className":262343},[31642],[34,262345,262347],{"className":262346},[31646],[31648,262348,262349],{"xmlns":31650},[31652,262350,262351,262375],{},[31655,262352,262353,262355,262357,262359,262361,262363,262365,262367,262369,262371,262373],{},[31658,262354,105852],{},[31662,262356,42071],{},[31662,262358,456],{"stretchy":702},[31658,262360,33347],{},[31662,262362,32226],{},[31749,262364,640],{},[31662,262366,2557],{"stretchy":702},[31662,262368,223],{},[31749,262370,640],{},[31662,262372,45972],{},[31658,262374,98360],{},[31665,262376,262377],{"encoding":31667},"P \\times (N - 1) + 1 \\le M",[34,262379,262381,262399,262420,262441,262459],{"className":262380,"ariaHidden":758},[31672],[34,262382,262384,262387,262390,262393,262396],{"className":262383},[31676],[34,262385],{"className":262386,"style":34944},[31680],[34,262388,105852],{"className":262389,"style":37422},[31685,31686],[34,262391],{"className":262392,"style":32604},[31845],[34,262394,42071],{"className":262395},[32024],[34,262397],{"className":262398,"style":32604},[31845],[34,262400,262402,262405,262408,262411,262414,262417],{"className":262401},[31676],[34,262403],{"className":262404,"style":31774},[31680],[34,262406,456],{"className":262407},[31782],[34,262409,33347],{"className":262410,"style":33389},[31685,31686],[34,262412],{"className":262413,"style":32604},[31845],[34,262415,32226],{"className":262416},[32024],[34,262418],{"className":262419,"style":32604},[31845],[34,262421,262423,262426,262429,262432,262435,262438],{"className":262422},[31676],[34,262424],{"className":262425,"style":31774},[31680],[34,262427,640],{"className":262428},[31685],[34,262430,2557],{"className":262431},[31690],[34,262433],{"className":262434,"style":32604},[31845],[34,262436,223],{"className":262437},[32024],[34,262439],{"className":262440,"style":32604},[31845],[34,262442,262444,262447,262450,262453,262456],{"className":262443},[31676],[34,262445],{"className":262446,"style":55382},[31680],[34,262448,640],{"className":262449},[31685],[34,262451],{"className":262452,"style":31846},[31845],[34,262454,45972],{"className":262455},[31850],[34,262457],{"className":262458,"style":31846},[31845],[34,262460,262462,262465],{"className":262461},[31676],[34,262463],{"className":262464,"style":33218},[31680],[34,262466,98360],{"className":262467,"style":33389},[31685,31686],"。\n代入数据：",[34,262470,262472,262506],{"className":262471},[31642],[34,262473,262475],{"className":262474},[31646],[31648,262476,262477],{"xmlns":31650},[31652,262478,262479,262503],{},[31655,262480,262481,262483,262485,262487,262489,262491,262493,262495,262497,262499,262501],{},[31749,262482,786],{},[31662,262484,42071],{},[31662,262486,456],{"stretchy":702},[31658,262488,33347],{},[31662,262490,32226],{},[31749,262492,640],{},[31662,262494,2557],{"stretchy":702},[31662,262496,223],{},[31749,262498,640],{},[31662,262500,45972],{},[31749,262502,18136],{},[31665,262504,262505],{"encoding":31667},"3 \\times (N - 1) + 1 \\le 5",[34,262507,262509,262527,262548,262569,262587],{"className":262508,"ariaHidden":758},[31672],[34,262510,262512,262515,262518,262521,262524],{"className":262511},[31676],[34,262513],{"className":262514,"style":35942},[31680],[34,262516,786],{"className":262517},[31685],[34,262519],{"className":262520,"style":32604},[31845],[34,262522,42071],{"className":262523},[32024],[34,262525],{"className":262526,"style":32604},[31845],[34,262528,262530,262533,262536,262539,262542,262545],{"className":262529},[31676],[34,262531],{"className":262532,"style":31774},[31680],[34,262534,456],{"className":262535},[31782],[34,262537,33347],{"className":262538,"style":33389},[31685,31686],[34,262540],{"className":262541,"style":32604},[31845],[34,262543,32226],{"className":262544},[32024],[34,262546],{"className":262547,"style":32604},[31845],[34,262549,262551,262554,262557,262560,262563,262566],{"className":262550},[31676],[34,262552],{"className":262553,"style":31774},[31680],[34,262555,640],{"className":262556},[31685],[34,262558,2557],{"className":262559},[31690],[34,262561],{"className":262562,"style":32604},[31845],[34,262564,223],{"className":262565},[32024],[34,262567],{"className":262568,"style":32604},[31845],[34,262570,262572,262575,262578,262581,262584],{"className":262571},[31676],[34,262573],{"className":262574,"style":55382},[31680],[34,262576,640],{"className":262577},[31685],[34,262579],{"className":262580,"style":31846},[31845],[34,262582,45972],{"className":262583},[31850],[34,262585],{"className":262586,"style":31846},[31845],[34,262588,262590,262593],{"className":262589},[31676],[34,262591],{"className":262592,"style":32617},[31680],[34,262594,18136],{"className":262595},[31685],[34,262597,262599,262652],{"className":262598},[31642],[34,262600,262602],{"className":262601},[31646],[31648,262603,262604],{"xmlns":31650},[31652,262605,262606,262649],{},[31655,262607,262608,262610,262612,262614,262616,262618,262620,262623,262626,262628,262630,262632,262634,262636,262638,262640,262642,262644,262646],{},[31749,262609,786],{},[31658,262611,33347],{},[31662,262613,32226],{},[31749,262615,786],{},[31662,262617,45972],{},[31749,262619,15932],{},[39361,262621,262622],{},"  ",[31662,262624,262625],{},"⟹",[39361,262627,262622],{},[31749,262629,786],{},[31658,262631,33347],{},[31662,262633,45972],{},[31749,262635,36723],{},[39361,262637,262622],{},[31662,262639,262625],{},[39361,262641,262622],{},[31658,262643,33347],{},[31662,262645,45972],{},[31749,262647,262648],{},"2.33",[31665,262650,262651],{"encoding":31667},"3N - 3 \\le 4 \\implies 3N \\le 7 \\implies N \\le 2.33",[34,262653,262655,262676,262694,262719,262740,262764,262782],{"className":262654,"ariaHidden":758},[31672],[34,262656,262658,262661,262664,262667,262670,262673],{"className":262657},[31676],[34,262659],{"className":262660,"style":34944},[31680],[34,262662,786],{"className":262663},[31685],[34,262665,33347],{"className":262666,"style":33389},[31685,31686],[34,262668],{"className":262669,"style":32604},[31845],[34,262671,32226],{"className":262672},[32024],[34,262674],{"className":262675,"style":32604},[31845],[34,262677,262679,262682,262685,262688,262691],{"className":262678},[31676],[34,262680],{"className":262681,"style":55382},[31680],[34,262683,786],{"className":262684},[31685],[34,262686],{"className":262687,"style":31846},[31845],[34,262689,45972],{"className":262690},[31850],[34,262692],{"className":262693,"style":31846},[31845],[34,262695,262697,262701,262704,262707,262710,262713,262716],{"className":262696},[31676],[34,262698],{"className":262699,"style":262700},[31680],"height:0.6684em;vertical-align:-0.024em;",[34,262702,15932],{"className":262703},[31685],[34,262705],{"className":262706,"style":31846},[31845],[34,262708],{"className":262709,"style":31846},[31845],[34,262711,262625],{"className":262712},[31850],[34,262714],{"className":262715,"style":31846},[31845],[34,262717],{"className":262718,"style":31846},[31845],[34,262720,262722,262725,262728,262731,262734,262737],{"className":262721},[31676],[34,262723],{"className":262724,"style":195047},[31680],[34,262726,786],{"className":262727},[31685],[34,262729,33347],{"className":262730,"style":33389},[31685,31686],[34,262732],{"className":262733,"style":31846},[31845],[34,262735,45972],{"className":262736},[31850],[34,262738],{"className":262739,"style":31846},[31845],[34,262741,262743,262746,262749,262752,262755,262758,262761],{"className":262742},[31676],[34,262744],{"className":262745,"style":262700},[31680],[34,262747,36723],{"className":262748},[31685],[34,262750],{"className":262751,"style":31846},[31845],[34,262753],{"className":262754,"style":31846},[31845],[34,262756,262625],{"className":262757},[31850],[34,262759],{"className":262760,"style":31846},[31845],[34,262762],{"className":262763,"style":31846},[31845],[34,262765,262767,262770,262773,262776,262779],{"className":262766},[31676],[34,262768],{"className":262769,"style":195047},[31680],[34,262771,33347],{"className":262772,"style":33389},[31685,31686],[34,262774],{"className":262775,"style":31846},[31845],[34,262777,45972],{"className":262778},[31850],[34,262780],{"className":262781,"style":31846},[31845],[34,262783,262785,262788],{"className":262784},[31676],[34,262786],{"className":262787,"style":32617},[31680],[34,262789,262648],{"className":262790},[31685],"。\n由于申请的资源数必须是整数，所以 ",[34,262793,262795,262808],{"className":262794},[31642],[34,262796,262798],{"className":262797},[31646],[31648,262799,262800],{"xmlns":31650},[31652,262801,262802,262806],{},[31655,262803,262804],{},[31658,262805,33347],{},[31665,262807,33347],{"encoding":31667},[34,262809,262811],{"className":262810,"ariaHidden":758},[31672],[34,262812,262814,262817],{"className":262813},[31676],[34,262815],{"className":262816,"style":33218},[31680],[34,262818,33347],{"className":262819,"style":33389},[31685,31686]," 的最大值为 ",[2627,262822,768],{},"。\n（验证：如果限定每人最多申请2个，最坏情况3个进程各占1个，共耗用3个资源，系统还剩2个，任意进程均可再申请1个满足需求，不会死锁；若限定申请3个，最坏情况其中两个进程各占2个，一个占1个，共耗用5个资源，此时它们都还需资源，从而死锁。）",[19,262825,262827],{"id":262826},"_69-在指令系统中只能由操作系统使用的指令称为","69. 在指令系统中只能由操作系统使用的指令称为(　　　)。",[15,262829,262830],{},[2627,262831,262832],{},"A系统指令 B设备指令 C非特权指令 D特权指令",[15,262834,262835,258118],{},[2627,262836,58173],{},[15,262838,262839,262841,262842,262845,262846,262849],{},[2627,262840,258022],{},"\n计算机指令系统将指令分为两类以支持双重模式（目态和管态）操作以保护系统安全。\n参考资料明确指出：“",[2627,262843,262844],{},"特权指令（Privileged instructions）","：某些机器级指令被指定为特权指令，",[2627,262847,262848],{},"只能由监控程序（操作系统\u002Fmonitor）执行","。如果处理器在执行用户程序时遇到此类指令，就会发生错误并将控制权转移给操作系统”。这些指令通常涉及I\u002FO操作、内存管理、寄存器状态修改等核心功能。",[19,262851,262853],{"id":262852},"_70-操作系统的基本类型主要有","70. 操作系统的基本类型主要有(　　　)。",[15,262855,262856,262859,262862,262865],{},[2627,262857,262858],{},"A批处理系统、分时系统和多任务系统",[2627,262860,262861],{},"B单用户系统、多用户系统和批处理系统",[2627,262863,262864],{},"C批处理系统、分时系统和实时系统",[2627,262866,262867],{},"D实时系统、分时系统和多用户系统",[15,262869,262870,258137],{},[2627,262871,58173],{},[15,262873,262874,262876,262877,262880],{},[2627,262875,258022],{},"\n从操作系统的发展历史和分类来看，最具代表性且被广泛认同的",[2627,262878,262879],{},"三大基本操作系统类型","是：",[2230,262882,262883,262889,262895],{},[108,262884,262885,262888],{},[2627,262886,262887],{},"批处理系统（Batch Systems）","：如多道批处理系统，旨在最大化吞吐量和资源利用率。",[108,262890,262891,262894],{},[2627,262892,262893],{},"分时系统（Time-Sharing Systems）","：允许多个交互式用户同时通过终端共享处理器时间，旨在最小化响应时间。",[108,262896,262897,262900],{},[2627,262898,262899],{},"实时系统（Real-Time Systems）","：需要在严苛的截止时间内对外部事件做出确定性响应，如事务处理系统和工业控制系统。\n其他的如多任务系统、多用户系统、分布式系统等，均是在这三大基本类型基础上衍生或复合而来的概念。",{"title":29,"searchDepth":43,"depth":43,"links":262902},[262903,262904,262905,262906,262907,262908,262909,262910,262911,262912,262913,262914,262915,262916,262917,262918,262919,262920,262921,262922,262923,262924,262925,262926,262927,262928,262929,262930,262931,262932,262933,262934,262935,262936,262937,262938,262939,262940,262941,262942,262943,262944,262945,262946,262947,262948,262949,262950,262951,262952,262953,262954,262955,262956,262957,262958,262959,262960,262961,262962,262963,262964,262965,262966,262967,262968,262969,262970,262971,262972],{"id":258006,"depth":43,"text":258007},{"id":258088,"depth":43,"text":258089},{"id":258107,"depth":43,"text":258108},{"id":258126,"depth":43,"text":258127},{"id":258145,"depth":43,"text":258146},{"id":258166,"depth":43,"text":258167},{"id":258204,"depth":43,"text":258205},{"id":258227,"depth":43,"text":258228},{"id":258254,"depth":43,"text":258255},{"id":258376,"depth":43,"text":258377},{"id":258394,"depth":43,"text":258395},{"id":258428,"depth":43,"text":258429},{"id":258446,"depth":43,"text":258447},{"id":259243,"depth":43,"text":259244},{"id":259532,"depth":43,"text":259533},{"id":259550,"depth":43,"text":259551},{"id":259568,"depth":43,"text":259569},{"id":259586,"depth":43,"text":259587},{"id":259606,"depth":43,"text":259607},{"id":259656,"depth":43,"text":259657},{"id":259683,"depth":43,"text":259684},{"id":259701,"depth":43,"text":259702},{"id":259723,"depth":43,"text":259724},{"id":259741,"depth":43,"text":259742},{"id":260140,"depth":43,"text":260141},{"id":260421,"depth":43,"text":260422},{"id":260783,"depth":43,"text":260784},{"id":260839,"depth":43,"text":260840},{"id":260861,"depth":43,"text":260862},{"id":260900,"depth":43,"text":260901},{"id":260927,"depth":43,"text":260928},{"id":260945,"depth":43,"text":260946},{"id":260963,"depth":43,"text":260964},{"id":260981,"depth":43,"text":260982},{"id":260999,"depth":43,"text":261000},{"id":261017,"depth":43,"text":261018},{"id":261044,"depth":43,"text":261045},{"id":261065,"depth":43,"text":261066},{"id":261110,"depth":43,"text":261111},{"id":261128,"depth":43,"text":261129},{"id":261146,"depth":43,"text":261147},{"id":261167,"depth":43,"text":261168},{"id":261185,"depth":43,"text":261186},{"id":261203,"depth":43,"text":261204},{"id":261224,"depth":43,"text":261225},{"id":261242,"depth":43,"text":261243},{"id":261260,"depth":43,"text":261261},{"id":261278,"depth":43,"text":261279},{"id":261296,"depth":43,"text":261297},{"id":261314,"depth":43,"text":261315},{"id":261332,"depth":43,"text":261333},{"id":261350,"depth":43,"text":261351},{"id":261370,"depth":43,"text":261371},{"id":261484,"depth":43,"text":261485},{"id":261502,"depth":43,"text":261503},{"id":261523,"depth":43,"text":261524},{"id":261544,"depth":43,"text":261545},{"id":261562,"depth":43,"text":261563},{"id":261580,"depth":43,"text":261581},{"id":261598,"depth":43,"text":261599},{"id":261615,"depth":43,"text":261616},{"id":261662,"depth":43,"text":261663},{"id":261687,"depth":43,"text":261688},{"id":261713,"depth":43,"text":261714},{"id":261738,"depth":43,"text":261739},{"id":261767,"depth":43,"text":261768},{"id":261806,"depth":43,"text":261807},{"id":261973,"depth":43,"text":261974},{"id":262826,"depth":43,"text":262827},{"id":262852,"depth":43,"text":262853},{"order":56},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fmcq",{"title":257999,"description":29},"source\u002Fcs\u002Fos\u002Freview\u002Fmcq","_YZnLcDPj-2Aui1PuSpgEaFbTnXdYC34Buk5WeWZOMk",{"id":262979,"title":262980,"body":262981,"description":29,"extension":85,"meta":264267,"navigation":52,"path":264268,"seo":264269,"stem":264270,"__hash__":264271},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fmidterm-exercise.md","Midterm Exercise",{"type":8,"value":262982,"toc":264255},[262983,262985,262989,262993,262997,263079,263081,263085,263120,263123,263152,263154,263158,263163,263180,263182,263187,263203,263205,263210,263226,263228,263233,263249,263251,263256,263272,263274,263279,263295,263297,263302,263318,263320,263325,263341,263343,263345,263350,263358,263360,263365,263372,263374,263379,263399,263401,263406,263425,263427,263432,263458,263460,263465,263479,263481,263485,263489,263492,263591,263597,263603,263609,263624,263632,263646,263648,263652,263655,263700,263703,263706,263709,263712,263714,263718,263801,263831,263836,263866,263869,264066,264071,264077,264082,264253],[32188,262984],{},[19,262986,262988],{"id":262987},"title-ch1ch6-期中复习习题order-1","title: CH1–CH6 期中复习习题\norder: 1",[11,262990,262992],{"id":262991},"ch1ch6-期中复习习题","CH1–CH6 期中复习习题",[19,262994,262996],{"id":262995},"basic-concepts","Basic Concepts",[2230,262998,262999,263007,263015,263023,263031,263039,263047,263055,263063,263071],{},[108,263000,263001,263004,263006],{},[2627,263002,263003],{},"Operating System（操作系统）",[58531,263005],{},"\n首先是一个程序（系统程序），其功能是控制应用程序的执行；一个接口，连接应用程序和计算机硬件。",[108,263008,263009,263012,263014],{},[2627,263010,263011],{},"Process（进程）",[58531,263013],{},"\n一个正在运行的程序；一个运行程序的实例；分配且能够在处理器上执行的实体；由一组执行的指令、一个当前状态和一组相关的系统资源表征的活动单元。",[108,263016,263017,263020,263022],{},[2627,263018,263019],{},"Thread（线程）",[58531,263021],{},"\n操作系统中能够运算调度的最小单位。",[108,263024,263025,263028,263030],{},[2627,263026,263027],{},"Concurrency（并发）",[58531,263029],{},"\n指在一个时间段内，操作系统中存在多个程序能够在同一个处理机上运行，但在任一个时刻点上只有一个程序在处理机上运行。",[108,263032,263033,263036,263038],{},[2627,263034,263035],{},"Critical Section（临界区）",[58531,263037],{},"\n指一个访问临界资源的程序片段，临界资源指不可同时被多个进程或线程访问的共享资源。",[108,263040,263041,263044,263046],{},[2627,263042,263043],{},"Mutual Exclusion（互斥）",[58531,263045],{},"\n进程之间因相互竞争共享资源所产生的一种制约关系，当一个进程进入临界区使用共享资源时，另一个进程必须等待。",[108,263048,263049,263052,263054],{},[2627,263050,263051],{},"Synchronization（同步）",[58531,263053],{},"\n进程之间需要在执行的某个位置上协调工作次序、传递信息所产生的一种制约关系。",[108,263056,263057,263060,263062],{},[2627,263058,263059],{},"Semaphore（信号量）",[58531,263061],{},"\n一个整数值，用于进程间信号的传递，作用于阻塞进程或解除阻塞。",[108,263064,263065,263068,263070],{},[2627,263066,263067],{},"Deadlock（死锁）",[58531,263069],{},"\n一组相互竞争系统资源或进行通信的进程间的\"永久\"阻塞。",[108,263072,263073,263076,263078],{},[2627,263074,263075],{},"Starvation（饥饿）",[58531,263077],{},"\n进程资源竞争而导致长时间得不到处理器时间的一种状态。",[32188,263080],{},[19,263082,263084],{"id":263083},"fill-in-the-blanks","Fill in the Blanks",[2230,263086,263087,263096,263099,263102,263105,263108,263111,263114,263117],{},[108,263088,263089,263090,263095],{},"The main objectives of the OS are: (",[71807,263091,263092],{},[2627,263093,263094],{},") (",") and (___).",[108,263097,263098],{},"(___) was introduced in modern operating systems, making concurrency and sharing possible.",[108,263100,263101],{},"When a processor time slice of a process is exhausted, the process should change to the (___) state.",[108,263103,263104],{},"When creating a new process, the first step is to assign a unique (___) to the new process.",[108,263106,263107],{},"The first step in a full process switch is to save the (___) of the process.",[108,263109,263110],{},"In the instruction system, (___) can only be used by the Operating System.",[108,263112,263113],{},"(___) is a special integer variable to be used for signaling among processes.",[108,263115,263116],{},"(___) is a special semaphore that takes on only the values 0 and 1.",[108,263118,263119],{},"The operating system's (___) refers to its inherent flexibility in permitting functional modifications to the system without interfering with service.",[15,263121,263122],{},"::: details Reference Answers",[2230,263124,263125,263128,263131,263134,263137,263140,263143,263146,263149],{},[108,263126,263127],{},"Convenience（方便）、Efficiency（有效）、Ability to evolve（扩展能力）",[108,263129,263130],{},"Multiprogramming（多道程序设计）",[108,263132,263133],{},"Ready（就绪）",[108,263135,263136],{},"Identifier（标识符）",[108,263138,263139],{},"Context（上下文 \u002F 处理机状态字）",[108,263141,263142],{},"Privileged instructions（特权指令）",[108,263144,263145],{},"Semaphore \u002F Counting semaphore \u002F General semaphore（信号量）",[108,263147,263148],{},"Binary semaphore（二值信号量）",[108,263150,263151],{},"Ability to evolve（可扩展性）\n:::",[32188,263153],{},[19,263155,263157],{"id":263156},"single-choice","Single Choice",[15,263159,263160,263162],{},[2627,263161,233680],{}," A process is in (___) state when it is in the main memory and awaiting an event.",[105,263164,263165,263168,263171,263177],{},[108,263166,263167],{},"A) Ready",[108,263169,263170],{},"B) Blocked\u002FSuspend",[108,263172,263173,263176],{},[2627,263174,263175],{},"C) Blocked"," (正确)",[108,263178,263179],{},"D) Ready\u002FSuspend",[32188,263181],{},[15,263183,263184,263186],{},[2627,263185,233688],{}," Requested resources are granted to processes whenever possible with (___).",[105,263188,263189,263194,263197,263200],{},[108,263190,263191,263176],{},[2627,263192,263193],{},"A) Deadlock detection",[108,263195,263196],{},"B) Deadlock avoidance",[108,263198,263199],{},"C) Mutual exclusion",[108,263201,263202],{},"D) Preemption",[32188,263204],{},[15,263206,263207,263209],{},[2627,263208,233696],{}," In a three state Process Model, (___) is not a valid state transition.",[105,263211,263212,263217,263220,263223],{},[108,263213,263214,263176],{},[2627,263215,263216],{},"A) Block -> Running",[108,263218,263219],{},"B) Running → Block",[108,263221,263222],{},"C) Ready → Running",[108,263224,263225],{},"D) Running → Ready",[32188,263227],{},[15,263229,263230,263232],{},[2627,263231,233704],{}," The permanent blocking of a set of processes that either compete for system resources or communicate with each other is called: (___).",[105,263234,263235,263238,263243,263246],{},[108,263236,263237],{},"A) Starvation",[108,263239,263240,263176],{},[2627,263241,263242],{},"B) Deadlock",[108,263244,263245],{},"C) Prioritization",[108,263247,263248],{},"D) Mutual exclusion",[32188,263250],{},[15,263252,263253,263255],{},[2627,263254,233712],{}," Suppose the initial value of a semaphore associated with a resource is 4 and the current value is −2. If V is the number of available resources and W is the number of processes waiting for the resources, then V and W are (___) respectively.",[105,263257,263258,263263,263266,263269],{},[108,263259,263260,263176],{},[2627,263261,263262],{},"A) 0, 2",[108,263264,263265],{},"B) 4, 0",[108,263267,263268],{},"C) 2, 0",[108,263270,263271],{},"D) −2, 0",[32188,263273],{},[15,263275,263276,263278],{},[2627,263277,233720],{}," The end of a printout requested by a process will cause the process state to change from (___).",[105,263280,263281,263284,263287,263290],{},[108,263282,263283],{},"A) running → ready",[108,263285,263286],{},"B) running → blocking",[108,263288,263289],{},"C) ready → running",[108,263291,263292,263176],{},[2627,263293,263294],{},"D) blocking -> ready",[32188,263296],{},[15,263298,263299,263301],{},[2627,263300,237383],{}," If there are five plotters in the system and all processes need to use two of them. Suppose each process is allowed to request only one at a time, then at most (___) processes are allowed to compete without deadlock.",[105,263303,263304,263307,263310,263313],{},[108,263305,263306],{},"A) 5",[108,263308,263309],{},"B) 2",[108,263311,263312],{},"C) 3",[108,263314,263315,263176],{},[2627,263316,263317],{},"D) 4",[32188,263319],{},[15,263321,263322,263324],{},[2627,263323,237391],{}," In a concurrent system with multiple processes, there is certainly no deadlock due to contention over (___).",[105,263326,263327,263330,263333,263336],{},[108,263328,263329],{},"A) Printer",[108,263331,263332],{},"B) Tape drive",[108,263334,263335],{},"C) CPU",[108,263337,263338,263176],{},[2627,263339,263340],{},"D) Disk",[32188,263342],{},[19,263344,200062],{"id":200061},[15,263346,263347],{},[2627,263348,263349],{},"1. What are the differences between process and thread?",[104833,263351,263352],{},[15,263353,263354,263357],{},[2627,263355,263356],{},"答："," 进程是资源分配的基本单位；线程是处理器调度的基本单位。",[32188,263359],{},[15,263361,263362],{},[2627,263363,263364],{},"2. Explain the relationship between program and process.",[104833,263366,263367],{},[15,263368,263369,263371],{},[2627,263370,263356],{}," 程序是静态的，未装入内存中；而进程是动态的，是正在被执行的程序。一个程序可以对应多个进程，但一个进程只能对应一个程序。",[32188,263373],{},[15,263375,263376],{},[2627,263377,263378],{},"3. What are the differences between reusable resources and consumable resources?",[104833,263380,263381,263385],{},[15,263382,263383],{},[2627,263384,263356],{},[105,263386,263387,263393],{},[108,263388,263389,263392],{},[2627,263390,263391],{},"可重用资源（Reusable Resource）","：一次仅供一个进程安全使用且不因使用而耗尽的资源。进程得到资源单元并使用后，会释放这些单元供其他进程再次使用。例如：处理器、I\u002FO 通道、内存和外存、设备，以及文件、数据库和信号量等数据结构。",[108,263394,263395,263398],{},[2627,263396,263397],{},"可消耗资源（Consumable Resource）","：可被创建（生产）和销毁（消耗）的资源。消费进程得到一个资源时，该资源就不再存在。例如：中断、信号、消息和 I\u002FO 缓冲区中的信息。",[32188,263400],{},[15,263402,263403],{},[2627,263404,263405],{},"4. What are deadlock and livelock? Please briefly explain their difference.",[104833,263407,263408,263412],{},[15,263409,263410],{},[2627,263411,263356],{},[105,263413,263414,263419],{},[108,263415,263416,263418],{},[2627,263417,197421],{},"：两个或两个以上的线程在执行过程中，由于竞争资源或彼此通信而造成的一种永久阻塞现象，若无外力作用，它们都将无法推进下去。",[108,263420,263421,263424],{},[2627,263422,263423],{},"活锁（Livelock）","：两个或两个以上的进程为响应其他进程中的变化而持续改变自己的状态，但不做有用工作的情况。进程没有被阻塞，但由于某些条件没有满足，导致一直重复\"尝试—失败—尝试—失败\"的过程。处于活锁的实体不断改变状态，活锁有可能自行解开。",[32188,263426],{},[15,263428,263429],{},[2627,263430,263431],{},"5. List the requirements for mutual exclusion.",[104833,263433,263434,263438],{},[15,263435,263436],{},[2627,263437,263356],{},[2230,263439,263440,263443,263446,263449,263452,263455],{},[108,263441,263442],{},"必须强制实施互斥：一次只允许一个进程进入临界区",[108,263444,263445],{},"一个在临界区停止的进程不能干涉其他进程",[108,263447,263448],{},"不能产生死锁或饥饿",[108,263450,263451],{},"在没有进程在临界区时，任何需要进入临界区的进程必须能够立即进入（公平性）",[108,263453,263454],{},"对进程的执行速度和处理器数量没有任何限制和要求",[108,263456,263457],{},"进程驻留在临界区的时间必须是有限的",[32188,263459],{},[15,263461,263462],{},[2627,263463,263464],{},"6. List reasons why a mode switch between threads may be cheaper than a mode switch between processes.",[104833,263466,263467,263471],{},[15,263468,263469],{},[2627,263470,263356],{},[105,263472,263473,263476],{},[108,263474,263475],{},"线程共享内存和文件，线程切换不需要内核参与；",[108,263477,263478],{},"线程上下文切换所需开销比进程上下文切换开销小。",[32188,263480],{},[19,263482,263484],{"id":263483},"problem-solving","Problem Solving",[253,263486,263488],{"id":263487},"problem-1-银行家算法-bankers-algorithm","Problem 1 — 银行家算法 Banker's Algorithm",[15,263490,263491],{},"Suppose current system states are shown as in the following table, at this time Available = (1,2,1).",[13837,263493,31533,263494,31533,263523],{},[13840,263495,263496,263497,263496,263508,31533],{},"\n ",[13843,263498,263499,263500,263499,263502,263499,263505,263496],{},"\n ",[13846,263501,199973],{"rowSpan":43},[13846,263503,263504],{"colSpan":49},"Claim",[13846,263506,263507],{"colSpan":49},"Allocation",[13843,263509,263499,263510,263512,263514,263499,263517,263519,263521,263496],{},[13846,263511,93353],{},[13846,263513,93359],{},[13846,263515,263516],{},"R3",[13846,263518,93353],{},[13846,263520,93359],{},[13846,263522,263516],{},[13853,263524,263496,263525,263496,263541,263496,263557,263496,263574,31533],{},[13843,263526,263527,263529,263531,263533,263535,263537,263539],{},[13858,263528,97876],{},[13858,263530,786],{},[13858,263532,768],{},[13858,263534,768],{},[13858,263536,640],{},[13858,263538,459],{},[13858,263540,459],{},[13843,263542,263543,263545,263547,263549,263551,263553,263555],{},[13858,263544,97925],{},[13858,263546,36656],{},[13858,263548,786],{},[13858,263550,640],{},[13858,263552,18136],{},[13858,263554,640],{},[13858,263556,640],{},[13843,263558,263559,263562,263564,263566,263568,263570,263572],{},[13858,263560,263561],{},"P3",[13858,263563,786],{},[13858,263565,15932],{},[13858,263567,640],{},[13858,263569,768],{},[13858,263571,640],{},[13858,263573,640],{},[13843,263575,263576,263579,263581,263583,263585,263587,263589],{},[13858,263577,263578],{},"P4",[13858,263580,15932],{},[13858,263582,768],{},[13858,263584,768],{},[13858,263586,459],{},[13858,263588,768],{},[13858,263590,459],{},[15,263592,263593,263596],{},[2627,263594,263595],{},"(1)"," When P2 issues request₂(1,1,0), can the system allocate the resource? If yes, give a safety sequence.",[15,263598,263599,263602],{},[2627,263600,263601],{},"(2)"," After P2 issues request₂(1,1,0), P1 then issues request₁(0,1,0). Can the system allocate?",[15,263604,263605,263608],{},[2627,263606,263607],{},"(3)"," After P2 issues request₂(1,1,0), P3 then issues request₃(0,0,1). Can the system allocate?",[15,263610,263611,263612,263615,263617,263618,263620,263623],{},"::: details Reference Answers\n",[2627,263613,263614],{},"(1) 可以分配。",[58531,263616],{},"\nP2 分配后仍可执行完毕，释放资源，不存在死锁情况。",[58531,263619],{},[2627,263621,263622],{},"安全序列：P2 → P1 → P3 → P4","（P2 在最前即正确）",[15,263625,263626,263629,263631],{},[2627,263627,263628],{},"(2) 不可以分配。",[58531,263630],{},"\nP2 和 P1 均分配后，剩余可用资源无法满足任何一个进程继续运行，会产生死锁。",[15,263633,263634,263637,263639,263640,263642,263645],{},[2627,263635,263636],{},"(3) 可以分配。",[58531,263638],{},"\nP2 仍能获得足够资源运行完毕，运行结束后释放资源，不会产生死锁。",[58531,263641],{},[2627,263643,263644],{},"安全序列：P2 → P3 → P1 → P4","\n:::",[32188,263647],{},[253,263649,263651],{"id":263650},"problem-2-信号量同步","Problem 2 — 信号量同步",[15,263653,263654],{},"两个进程并发执行，优先级相同，信号量 s1、s2 初值均为 0：",[24,263656,263658],{"className":65734,"code":263657,"language":31965,"meta":29,"style":29},"P1() { P2() {\n y = 1; x = 1;\n y = y + 1; x = x + 3;\n semSignal(s1); semWait(s1);\n z = y + 3; x = x + y;\n semWait(s2); semSignal(s2);\n y = z + y; z = y + x;\n} }\n",[31,263659,263660,263665,263670,263675,263680,263685,263690,263695],{"__ignoreMap":29},[34,263661,263662],{"class":36,"line":37},[34,263663,263664],{},"P1() { P2() {\n",[34,263666,263667],{"class":36,"line":43},[34,263668,263669],{}," y = 1; x = 1;\n",[34,263671,263672],{"class":36,"line":49},[34,263673,263674],{}," y = y + 1; x = x + 3;\n",[34,263676,263677],{"class":36,"line":56},[34,263678,263679],{}," semSignal(s1); semWait(s1);\n",[34,263681,263682],{"class":36,"line":62},[34,263683,263684],{}," z = y + 3; x = x + y;\n",[34,263686,263687],{"class":36,"line":68},[34,263688,263689],{}," semWait(s2); semSignal(s2);\n",[34,263691,263692],{"class":36,"line":74},[34,263693,263694],{}," y = z + y; z = y + x;\n",[34,263696,263697],{"class":36,"line":1971},[34,263698,263699],{},"} }\n",[15,263701,263702],{},"请给出并发执行后 x、y、z 的值。",[15,263704,263705],{},"::: details Reference Answers\ny=2;x=4",[15,263707,263708],{},"Z=5;x=6",[15,263710,263711],{},"z=8;y=10;x=6 或者 y=7;z=13;x=6\n:::",[32188,263713],{},[253,263715,263717],{"id":263716},"problem-3-生产者消费者问题","Problem 3 — 生产者消费者问题",[15,263719,263720,263721,83913,263749,263800],{},"三个工人 A、B、C 共享一个大小为 ",[34,263722,263724,263737],{"className":263723},[31642],[34,263725,263727],{"className":263726},[31646],[31648,263728,263729],{"xmlns":31650},[31652,263730,263731,263735],{},[31655,263732,263733],{},[31658,263734,98360],{},[31665,263736,98360],{"encoding":31667},[34,263738,263740],{"className":263739,"ariaHidden":758},[31672],[34,263741,263743,263746],{"className":263742},[31676],[34,263744],{"className":263745,"style":33218},[31680],[34,263747,98360],{"className":263748,"style":33389},[31685,31686],[34,263750,263752,263770],{"className":263751},[31642],[34,263753,263755],{"className":263754},[31646],[31648,263756,263757],{"xmlns":31650},[31652,263758,263759,263767],{},[31655,263760,263761,263763,263765],{},[31658,263762,98360],{},[31662,263764,43175],{},[31749,263766,786],{},[31665,263768,263769],{"encoding":31667},"M \\geq 3",[34,263771,263773,263791],{"className":263772,"ariaHidden":758},[31672],[34,263774,263776,263779,263782,263785,263788],{"className":263775},[31676],[34,263777],{"className":263778,"style":195047},[31680],[34,263780,98360],{"className":263781,"style":33389},[31685,31686],[34,263783],{"className":263784,"style":31846},[31845],[34,263786,43175],{"className":263787},[31850],[34,263789],{"className":263790,"style":31846},[31845],[34,263792,263794,263797],{"className":263793},[31676],[34,263795],{"className":263796,"style":32617},[31680],[34,263798,786],{"className":263799},[31685],"）的箱子：",[105,263802,263803,263813,263821],{},[108,263804,263805,263808,263809,263812],{},[2627,263806,263807],{},"Worker A","：持续生产 ",[2627,263810,263811],{},"车架（frame）"," 并放入箱子",[108,263814,263815,263808,263818,263812],{},[2627,263816,263817],{},"Worker B",[2627,263819,263820],{},"车轮（wheel）",[108,263822,263823,263826,263827,263830],{},[2627,263824,263825],{},"Worker C","：每次取 ",[2627,263828,263829],{},"1 个车架 + 2 个车轮","，组装成自行车",[15,263832,263833],{},[2627,263834,263835],{},"原始伪代码（未同步）：",[24,263837,263839],{"className":65734,"code":263838,"language":31965,"meta":29,"style":29},"Worker A { while(true) { produce a frame; put frame into box; } }\nWorker B { while(true) { produce a wheel; put wheel into box; } }\nWorker C { while(true) { take a frame from box;\n take two wheels from box;\n assemble a bike; } }\n",[31,263840,263841,263846,263851,263856,263861],{"__ignoreMap":29},[34,263842,263843],{"class":36,"line":37},[34,263844,263845],{},"Worker A { while(true) { produce a frame; put frame into box; } }\n",[34,263847,263848],{"class":36,"line":43},[34,263849,263850],{},"Worker B { while(true) { produce a wheel; put wheel into box; } }\n",[34,263852,263853],{"class":36,"line":49},[34,263854,263855],{},"Worker C { while(true) { take a frame from box;\n",[34,263857,263858],{"class":36,"line":56},[34,263859,263860],{}," take two wheels from box;\n",[34,263862,263863],{"class":36,"line":62},[34,263864,263865],{}," assemble a bike; } }\n",[15,263867,263868],{},"Please use semWait and semSignal operation to achieve the cooperation of three workers respectively, which requires that the solution does not contain deadlock.",[15,263870,263871,263874,263876,263877,263905,263906,263957,263958,263960,263961,347],{},[2627,263872,263873],{},"死锁分析：",[58531,263875],{},"\n若箱中 ",[34,263878,263880,263893],{"className":263879},[31642],[34,263881,263883],{"className":263882},[31646],[31648,263884,263885],{"xmlns":31650},[31652,263886,263887,263891],{},[31655,263888,263889],{},[31658,263890,98360],{},[31665,263892,98360],{"encoding":31667},[34,263894,263896],{"className":263895,"ariaHidden":758},[31672],[34,263897,263899,263902],{"className":263898},[31676],[34,263900],{"className":263901,"style":33218},[31680],[34,263903,98360],{"className":263904,"style":33389},[31685,31686]," 个槽全部为车轮，或 ",[34,263907,263909,263927],{"className":263908},[31642],[34,263910,263912],{"className":263911},[31646],[31648,263913,263914],{"xmlns":31650},[31652,263915,263916,263924],{},[31655,263917,263918,263920,263922],{},[31658,263919,98360],{},[31662,263921,32226],{},[31749,263923,640],{},[31665,263925,263926],{"encoding":31667},"M-1",[34,263928,263930,263948],{"className":263929,"ariaHidden":758},[31672],[34,263931,263933,263936,263939,263942,263945],{"className":263932},[31676],[34,263934],{"className":263935,"style":34944},[31680],[34,263937,98360],{"className":263938,"style":33389},[31685,31686],[34,263940],{"className":263941,"style":32604},[31845],[34,263943,32226],{"className":263944},[32024],[34,263946],{"className":263947,"style":32604},[31845],[34,263949,263951,263954],{"className":263950},[31676],[34,263952],{"className":263953,"style":32617},[31680],[34,263955,640],{"className":263956},[31685]," 个槽全为车架（剩余空间不足放第 2 个车轮），Worker C 均无法组装。",[58531,263959],{},"\n因此：",[2627,263962,263963,263964,264015,264016],{},"车架上限 = ",[34,263965,263967,263985],{"className":263966},[31642],[34,263968,263970],{"className":263969},[31646],[31648,263971,263972],{"xmlns":31650},[31652,263973,263974,263982],{},[31655,263975,263976,263978,263980],{},[31658,263977,98360],{},[31662,263979,32226],{},[31749,263981,768],{},[31665,263983,263984],{"encoding":31667},"M-2",[34,263986,263988,264006],{"className":263987,"ariaHidden":758},[31672],[34,263989,263991,263994,263997,264000,264003],{"className":263990},[31676],[34,263992],{"className":263993,"style":34944},[31680],[34,263995,98360],{"className":263996,"style":33389},[31685,31686],[34,263998],{"className":263999,"style":32604},[31845],[34,264001,32226],{"className":264002},[32024],[34,264004],{"className":264005,"style":32604},[31845],[34,264007,264009,264012],{"className":264008},[31676],[34,264010],{"className":264011,"style":32617},[31680],[34,264013,768],{"className":264014},[31685],"，车轮上限 = ",[34,264017,264019,264036],{"className":264018},[31642],[34,264020,264022],{"className":264021},[31646],[31648,264023,264024],{"xmlns":31650},[31652,264025,264026,264034],{},[31655,264027,264028,264030,264032],{},[31658,264029,98360],{},[31662,264031,32226],{},[31749,264033,640],{},[31665,264035,263926],{"encoding":31667},[34,264037,264039,264057],{"className":264038,"ariaHidden":758},[31672],[34,264040,264042,264045,264048,264051,264054],{"className":264041},[31676],[34,264043],{"className":264044,"style":34944},[31680],[34,264046,98360],{"className":264047,"style":33389},[31685,31686],[34,264049],{"className":264050,"style":32604},[31845],[34,264052,32226],{"className":264053},[32024],[34,264055],{"className":264056,"style":32604},[31845],[34,264058,264060,264063],{"className":264059},[31676],[34,264061],{"className":264062,"style":32617},[31680],[34,264064,640],{"className":264065},[31685],[15,264067,264068],{},[2627,264069,264070],{},"信号量定义：",[24,264072,264075],{"className":264073,"code":264074,"language":2638},[2636],"semaphore empty = M \u002F\u002F 空槽数量\nsemaphore frame = 0 \u002F\u002F 箱中车架数\nsemaphore wheel = 0 \u002F\u002F 箱中车轮数\nsemaphore maxframe = M - 2 \u002F\u002F 车架数上限\nsemaphore maxwheel = M - 1 \u002F\u002F 车轮数上限\n",[31,264076,264074],{"__ignoreMap":29},[15,264078,264079],{},[2627,264080,264081],{},"完整同步伪代码：",[24,264083,264085],{"className":65734,"code":264084,"language":31965,"meta":29,"style":29},"Worker A {\n while (true) {\n produce a frame;\n semWait(maxframe); \u002F\u002F 车架未超上限\n semWait(empty); \u002F\u002F 有空槽\n put frame into box;\n semSignal(frame); \u002F\u002F 车架数 +1\n }\n}\n\nWorker B {\n while (true) {\n produce a wheel;\n semWait(maxwheel); \u002F\u002F 车轮未超上限\n semWait(empty); \u002F\u002F 有空槽\n put wheel into box;\n semSignal(wheel); \u002F\u002F 车轮数 +1\n }\n}\n\nWorker C {\n while (true) {\n semWait(frame); \u002F\u002F 等待至少 1 个车架\n take a frame from box;\n semWait(wheel); \u002F\u002F 等待至少 2 个车轮\n semWait(wheel);\n take two wheels from box;\n assemble a bike;\n semSignal(maxframe); \u002F\u002F 车架槽释放\n semSignal(maxwheel); \u002F\u002F 车轮槽释放 ×2\n semSignal(maxwheel);\n semSignal(empty); \u002F\u002F 空槽 +3\n semSignal(empty);\n semSignal(empty);\n }\n}\n",[31,264086,264087,264092,264096,264101,264106,264111,264116,264121,264125,264129,264133,264138,264142,264147,264152,264156,264161,264166,264170,264174,264178,264183,264187,264192,264197,264202,264207,264212,264217,264222,264227,264232,264237,264241,264245,264249],{"__ignoreMap":29},[34,264088,264089],{"class":36,"line":37},[34,264090,264091],{},"Worker A {\n",[34,264093,264094],{"class":36,"line":43},[34,264095,195782],{},[34,264097,264098],{"class":36,"line":49},[34,264099,264100],{}," produce a frame;\n",[34,264102,264103],{"class":36,"line":56},[34,264104,264105],{}," semWait(maxframe); \u002F\u002F 车架未超上限\n",[34,264107,264108],{"class":36,"line":62},[34,264109,264110],{}," semWait(empty); \u002F\u002F 有空槽\n",[34,264112,264113],{"class":36,"line":68},[34,264114,264115],{}," put frame into box;\n",[34,264117,264118],{"class":36,"line":74},[34,264119,264120],{}," semSignal(frame); \u002F\u002F 车架数 +1\n",[34,264122,264123],{"class":36,"line":1971},[34,264124,5540],{},[34,264126,264127],{"class":36,"line":1976},[34,264128,77],{},[34,264130,264131],{"class":36,"line":1985},[34,264132,53],{"emptyLinePlaceholder":52},[34,264134,264135],{"class":36,"line":1990},[34,264136,264137],{},"Worker B {\n",[34,264139,264140],{"class":36,"line":1999},[34,264141,195782],{},[34,264143,264144],{"class":36,"line":2022},[34,264145,264146],{}," produce a wheel;\n",[34,264148,264149],{"class":36,"line":2036},[34,264150,264151],{}," semWait(maxwheel); \u002F\u002F 车轮未超上限\n",[34,264153,264154],{"class":36,"line":2603},[34,264155,264110],{},[34,264157,264158],{"class":36,"line":5537},[34,264159,264160],{}," put wheel into box;\n",[34,264162,264163],{"class":36,"line":5543},[34,264164,264165],{}," semSignal(wheel); \u002F\u002F 车轮数 +1\n",[34,264167,264168],{"class":36,"line":5548},[34,264169,5540],{},[34,264171,264172],{"class":36,"line":5564},[34,264173,77],{},[34,264175,264176],{"class":36,"line":5576},[34,264177,53],{"emptyLinePlaceholder":52},[34,264179,264180],{"class":36,"line":5606},[34,264181,264182],{},"Worker C {\n",[34,264184,264185],{"class":36,"line":5616},[34,264186,195782],{},[34,264188,264189],{"class":36,"line":5628},[34,264190,264191],{}," semWait(frame); \u002F\u002F 等待至少 1 个车架\n",[34,264193,264194],{"class":36,"line":5634},[34,264195,264196],{}," take a frame from box;\n",[34,264198,264199],{"class":36,"line":6420},[34,264200,264201],{}," semWait(wheel); \u002F\u002F 等待至少 2 个车轮\n",[34,264203,264204],{"class":36,"line":11294},[34,264205,264206],{}," semWait(wheel);\n",[34,264208,264209],{"class":36,"line":11321},[34,264210,264211],{}," take two wheels from box;\n",[34,264213,264214],{"class":36,"line":11326},[34,264215,264216],{}," assemble a bike;\n",[34,264218,264219],{"class":36,"line":11333},[34,264220,264221],{}," semSignal(maxframe); \u002F\u002F 车架槽释放\n",[34,264223,264224],{"class":36,"line":11344},[34,264225,264226],{}," semSignal(maxwheel); \u002F\u002F 车轮槽释放 ×2\n",[34,264228,264229],{"class":36,"line":11349},[34,264230,264231],{}," semSignal(maxwheel);\n",[34,264233,264234],{"class":36,"line":11354},[34,264235,264236],{}," semSignal(empty); \u002F\u002F 空槽 +3\n",[34,264238,264239],{"class":36,"line":11363},[34,264240,198703],{},[34,264242,264243],{"class":36,"line":11368},[34,264244,198703],{},[34,264246,264247],{"class":36,"line":11377},[34,264248,5540],{},[34,264250,264251],{"class":36,"line":11402},[34,264252,77],{},[79,264254,81],{},{"title":29,"searchDepth":43,"depth":43,"links":264256},[264257,264258,264259,264260,264261,264262],{"id":262987,"depth":43,"text":262988},{"id":262995,"depth":43,"text":262996},{"id":263083,"depth":43,"text":263084},{"id":263156,"depth":43,"text":263157},{"id":200061,"depth":43,"text":200062},{"id":263483,"depth":43,"text":263484,"children":264263},[264264,264265,264266],{"id":263487,"depth":49,"text":263488},{"id":263650,"depth":49,"text":263651},{"id":263716,"depth":49,"text":263717},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fmidterm-exercise",{"description":29},"source\u002Fcs\u002Fos\u002Freview\u002Fmidterm-exercise","WkikqUCl_MIqRPCWnGEhvRkaI3iWr7bBk88w5S2980c",{"id":264273,"title":264274,"body":264275,"description":29,"extension":85,"meta":283798,"navigation":52,"path":283799,"seo":283800,"stem":283801,"__hash__":283802},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fprob.md","Prob",{"type":8,"value":264276,"toc":283697},[264277,264279,264283,264286,264290,264293,264301,264304,264307,264311,264314,264380,264383,264394,264396,264398,264402,264405,264407,264409,264413,264416,264430,264432,264434,264438,264441,264443,264445,264449,264452,264455,264457,264459,264463,264466,264587,264589,264597,264599,264601,264605,264608,264664,264666,264677,264679,264681,264685,264688,264696,264698,264700,264704,264707,264709,264711,264715,264718,264720,264722,264726,264729,264737,264739,264750,264752,264754,264758,264761,264763,264765,264769,264772,264780,264782,264784,265006,265008,265344,265723,265795,266817,266826,266830,266833,266841,266843,266845,266848,266893,267091,267205,267208,267213,267421,267425,267579,267583,267586,267588,267590,267610,267614,267617,267622,267791,267869,267874,267877,267985,267990,268150,268226,268230,268233,268235,268238,268241,269257,269260,269265,269602,269607,270266,270270,270273,270275,270278,271225,271229,271232,271364,271366,271372,271374,271377,271391,271394,271399,271554,271559,271978,271987,271991,271994,271996,271999,272327,272331,272334,272436,272438,272446,272448,272451,272644,272647,272652,272894,272899,273018,273252,273257,273260,273264,273267,273269,273271,273275,273278,273286,273288,273290,273294,273297,273302,273304,273306,273310,273313,273318,273326,273328,273330,273334,273337,273343,273346,273352,273355,273357,273359,273363,273366,273368,273370,273374,273377,273379,273381,273385,273392,273394,273396,273400,273403,273405,273407,273411,273414,273421,273423,273425,273429,273432,273511,273514,273522,273524,273526,273530,273533,273544,273546,273548,273567,273570,273575,273595,273600,273618,273623,273639,273644,273648,273651,273653,273656,273659,273802,273807,273811,273814,273931,273933,273936,275308,275314,275519,275524,275528,275531,275627,275638,275640,275643,275648,275739,275744,275838,275851,275855,275858,275869,275871,275874,275879,275958,275963,276153,276158,276377,276382,276386,276389,276391,276394,276396,276434,276612,276623,276627,276630,276632,276635,276688,276693,276697,276700,276702,276705,276708,276734,276901,276961,276965,276968,276979,276981,276983,276987,276990,276992,276994,276998,277001,277009,277011,277013,277017,277020,277028,277030,277032,277036,277039,277079,277082,277090,277092,277094,277098,277101,277196,277199,277207,277209,277211,277215,277217,277219,277223,277226,277234,277236,277238,277242,277245,277262,277264,277266,277270,277273,277275,277278,277311,277314,277395,277399,277952,277956,278037,279066,279070,279150,279778,279780,279783,279787,279790,279867,279875,279877,279880,280136,280139,280143,281680,281684,281838,283603,283605,283695],[32188,264278],{},[19,264280,264282],{"id":264281},"title-应用题order-7","title: 应用题\norder: 7",[11,264284,264285],{"id":264285},"应用题",[19,264287,264289],{"id":264288},"_1-磁道调度scan-与-sstf-算法","1. 磁道调度：SCAN 与 SSTF 算法",[15,264291,264292],{},"假定当前磁头位于 100 号磁道，刚服务过 80 号磁道。进程对磁道的请求序列依次为 55，65，39，28，90，155，145，38，170。当采用扫描算法和最短寻道时间优先算法时，总的移动的磁道数分别是多少？（请给出寻道次序和每步移动磁道数）",[105,264294,264295,264298],{},[108,264296,264297],{},"(1) 采用扫描算法（寻道次序、每步移动磁道数和总的移动磁道数）。",[108,264299,264300],{},"(2) 最短寻道时间优先算法（寻道次序、每步移动磁道数和总的移动磁道数）。",[15,264302,264303],{},"::: details 解析",[15,264305,264306],{},":::",[19,264308,264310],{"id":264309},"_2-银行家算法四类资源五进程","2. 银行家算法：四类资源、五进程",[15,264312,264313],{},"假设 UNIX 系统有 A、B、C、D 四类资源可供五个进程 P1、P2、P3、P4、P5 共享。系统对这四类资源的拥有量为: (3,14,12,12)。进程对资源的需求和分配情况如下：",[13837,264315,264316,264328],{},[13840,264317,264318],{},[13843,264319,264320,264322,264325],{},[13846,264321,247197],{"align":82355},[13846,264323,264324],{"align":82355},"已占有资源 (A, B, C, D)",[13846,264326,264327],{"align":82355},"最大需求数 (A, B, C, D)",[13853,264329,264330,264339,264349,264359,264369],{},[13843,264331,264332,264334,264337],{},[13858,264333,97876],{"align":82355},[13858,264335,264336],{"align":82355},"(0, 0, 1, 2)",[13858,264338,264336],{"align":82355},[13843,264340,264341,264343,264346],{},[13858,264342,97925],{"align":82355},[13858,264344,264345],{"align":82355},"(1, 0, 0, 0)",[13858,264347,264348],{"align":82355},"(1, 7, 5, 0)",[13843,264350,264351,264353,264356],{},[13858,264352,263561],{"align":82355},[13858,264354,264355],{"align":82355},"(1, 3, 5, 4)",[13858,264357,264358],{"align":82355},"(2, 3, 5, 6)",[13843,264360,264361,264363,264366],{},[13858,264362,263578],{"align":82355},[13858,264364,264365],{"align":82355},"(0, 6, 3, 2)",[13858,264367,264368],{"align":82355},"(0, 6, 5, 2)",[13843,264370,264371,264374,264377],{},[13858,264372,264373],{"align":82355},"P5",[13858,264375,264376],{"align":82355},"(0, 0, 1, 4)",[13858,264378,264379],{"align":82355},"(0, 6, 5, 6)",[15,264381,264382],{},"按银行家算法回答下列问题：",[105,264384,264385,264388,264391],{},[108,264386,264387],{},"(1) 系统是否处于安全状态？为什么？",[108,264389,264390],{},"(2) 如果 P2 提出请求 (0,4,2,0)，系统能否满足请求？请说明原因。",[108,264392,264393],{},"(3) 如果在 P2 提出资源需求 (0,4,2,0) 后，P3 提出资源需求 (0,0,0,1)，系统能否满足其请求？请说明原因。",[15,264395,264303],{},[15,264397,264306],{},[19,264399,264401],{"id":264400},"_3-打印机共享设备的判断","3. 打印机：共享设备的判断",[15,264403,264404],{},"办公室只有一台打印机通过局域网为办公室人员提供打印服务。该打印机是共享设备的观点是否正确？为什么？",[15,264406,264303],{},[15,264408,264306],{},[19,264410,264412],{"id":264411},"_4-请求分页逻辑地址转换与-lru","4. 请求分页：逻辑地址转换与 LRU",[15,264414,264415],{},"在请求分页系统中，有一个长度为 6 页的进程，页大小为 1KB。假如为它分配 3 个物理块，逻辑地址为：2049, 3098, 2078, 1066, 5018, 5163, 2088, 6010, 5069, 5287, 4000, 3068。试求出进程访问的页面次序，并用 LRU 页面置换算法计算出程序访问过程中所发生的缺页次数和缺页中断率。（说明：初始页面为空，计算为缺页。）",[105,264417,264418,264421,264424,264427],{},[108,264419,264420],{},"(1) 进程访问的页面次序。",[108,264422,264423],{},"(2) LRU 页面置换算法计算出程序访问过程。",[108,264425,264426],{},"(3) 访问过程中所发生的缺页次数。",[108,264428,264429],{},"(4) 缺页中断率。",[15,264431,264303],{},[15,264433,264306],{},[19,264435,264437],{"id":264436},"_5-页表项数正置页表与反置页表","5. 页表项数：正置页表与反置页表",[15,264439,264440],{},"一台机器有 48 位虚地址和 32 位物理地址，若页长为 8KB，问页表共有多少个页表项? 如果设计一个反置页表，则有多少个页表项?",[15,264442,264303],{},[15,264444,264306],{},[19,264446,264448],{"id":264447},"_6-页面置换fifooptlru缺页次数","6. 页面置换（FIFO、OPT、LRU）缺页次数",[15,264450,264451],{},"在一个请求分页虚拟存储管理系统中，一个程序运行的页面走向是：\n1、2、3、4、2、1、5、6、2、1、2、3、7、6、3、2、1、2、3、6。",[15,264453,264454],{},"分别用 FIFO、OPT 和 LRU 算法，对分配给程序 3 个页框，求出缺页中断次数和缺页中断率。",[15,264456,264303],{},[15,264458,264306],{},[19,264460,264462],{"id":264461},"_7-短作业优先抢占式调度时间表","7. 短作业优先（抢占式）调度时间表",[15,264464,264465],{},"若有一个四道作业系统，如果在一段时间内先后有 6 个作业，它们提交和运行时间由下表给出。作业采用短作业优先的调度算法，进程采用以剩余时间最短优先的抢占式调度算法。（说明：计算结果保留一位小数。）",[13837,264467,264468,264488],{},[13840,264469,264470],{},[13843,264471,264472,264474,264476,264479,264481,264484,264486],{},[13846,264473,230223],{"align":82355},[13846,264475,230226],{"align":82355},[13846,264477,264478],{"align":82355},"运行时间（分钟）",[13846,264480,230231],{"align":82355},[13846,264482,264483],{"align":82355},"结束时间",[13846,264485,230236],{"align":82355},[13846,264487,230239],{"align":82355},[13853,264489,264490,264506,264523,264539,264555,264571],{},[13843,264491,264492,264494,264496,264498,264500,264502,264504],{},[13858,264493,640],{"align":82355},[13858,264495,230248],{"align":82355},[13858,264497,5774],{"align":82355},[13858,264499],{"align":82355},[13858,264501],{"align":82355},[13858,264503],{"align":82355},[13858,264505],{"align":82355},[13843,264507,264508,264510,264513,264515,264517,264519,264521],{},[13858,264509,768],{"align":82355},[13858,264511,264512],{"align":82355},"8:15",[13858,264514,9314],{"align":82355},[13858,264516],{"align":82355},[13858,264518],{"align":82355},[13858,264520],{"align":82355},[13858,264522],{"align":82355},[13843,264524,264525,264527,264529,264531,264533,264535,264537],{},[13858,264526,786],{"align":82355},[13858,264528,230265],{"align":82355},[13858,264530,3249],{"align":82355},[13858,264532],{"align":82355},[13858,264534],{"align":82355},[13858,264536],{"align":82355},[13858,264538],{"align":82355},[13843,264540,264541,264543,264545,264547,264549,264551,264553],{},[13858,264542,15932],{"align":82355},[13858,264544,230282],{"align":82355},[13858,264546,3249],{"align":82355},[13858,264548],{"align":82355},[13858,264550],{"align":82355},[13858,264552],{"align":82355},[13858,264554],{"align":82355},[13843,264556,264557,264559,264561,264563,264565,264567,264569],{},[13858,264558,18136],{"align":82355},[13858,264560,230299],{"align":82355},[13858,264562,3244],{"align":82355},[13858,264564],{"align":82355},[13858,264566],{"align":82355},[13858,264568],{"align":82355},[13858,264570],{"align":82355},[13843,264572,264573,264575,264577,264579,264581,264583,264585],{},[13858,264574,36656],{"align":82355},[13858,264576,230316],{"align":82355},[13858,264578,18136],{"align":82355},[13858,264580],{"align":82355},[13858,264582],{"align":82355},[13858,264584],{"align":82355},[13858,264586],{"align":82355},[15,264588,11832],{},[105,264590,264591,264594],{},[108,264592,264593],{},"(1) 完成上表，给出各个作业的开始时间、完成时间、周转时间和带权周转时间；",[108,264595,264596],{},"(2) 计算平均作业周转时间和平均作业带权周转时间。",[15,264598,264303],{},[15,264600,264306],{},[19,264602,264604],{"id":264603},"_8-银行家算法三类资源r1r2r3","8. 银行家算法：三类资源（R1\u002FR2\u002FR3）",[15,264606,264607],{},"设当前的系统状态如下表所示，系统此时 Available == (1,1,2)。",[13837,264609,264610,264622],{},[13840,264611,264612],{},[13843,264613,264614,264616,264619],{},[13846,264615,247197],{"align":82355},[13846,264617,264618],{"align":82355},"Claim (R1, R2, R3)",[13846,264620,264621],{"align":82355},"Allocation (R1, R2, R3)",[13853,264623,264624,264634,264644,264654],{},[13843,264625,264626,264628,264631],{},[13858,264627,97876],{"align":82355},[13858,264629,264630],{"align":82355},"(3, 2, 2)",[13858,264632,264633],{"align":82355},"(1, 0, 0)",[13843,264635,264636,264638,264641],{},[13858,264637,97925],{"align":82355},[13858,264639,264640],{"align":82355},"(6, 1, 3)",[13858,264642,264643],{"align":82355},"(5, 1, 1)",[13843,264645,264646,264648,264651],{},[13858,264647,263561],{"align":82355},[13858,264649,264650],{"align":82355},"(3, 1, 4)",[13858,264652,264653],{"align":82355},"(2, 1, 1)",[13843,264655,264656,264658,264661],{},[13858,264657,263578],{"align":82355},[13858,264659,264660],{"align":82355},"(4, 2, 2)",[13858,264662,264663],{"align":82355},"(0, 0, 2)",[15,264665,11832],{},[105,264667,264668,264671,264674],{},[108,264669,264670],{},"(1) 当 P2 发出资源请求向量 request2(1,0,1)，此时系统能否把资源分配给它？为什么？如果能分配给它，则请给出一个安全序列；",[108,264672,264673],{},"(2) 若在 P2 发出资源请求向量 request2(1,0,1) 后，若 P1 发出资源请求向量 request1(1,0,1)，此时系统能否把资源分配给它？为什么？",[108,264675,264676],{},"(3) 若在 P2 发出资源请求向量 request2(1,0,1) 后，若 P3 发出资源请求向量 request3(0,0,1)，此时系统能否把资源分配给它？为什么？",[15,264678,264303],{},[15,264680,264306],{},[19,264682,264684],{"id":264683},"_9-磁道调度fcfsscansstf","9. 磁道调度（FCFS、SCAN、SSTF）",[15,264686,264687],{},"假定当前磁头位于 100 号磁道，刚服务过 105 号磁道。进程对磁道的请求序列依次为 55，34，58，39，18，90，160，45，145，38，170，180。当采用先来先服务算法、扫描算法和最短寻道时间优先算法时，总的移动的磁道数分别是多少？（请给出寻道次序、每步移动磁道数和总的移动磁道数）",[105,264689,264690,264693],{},[108,264691,264692],{},"(1) 扫描算法",[108,264694,264695],{},"(2) 最短寻道时间优先算法",[15,264697,264303],{},[15,264699,264306],{},[19,264701,264703],{"id":264702},"_10-二级页表虚地址结构","10. 二级页表虚地址结构",[15,264705,264706],{},"一个 32 位地址的计算机系统使用二级页表，虚地址被分为 9 位顶级页表，11 位二级页表和偏移。试问：页面长度是多少？虚地址空间共有多少个页面？",[15,264708,264303],{},[15,264710,264306],{},[19,264712,264714],{"id":264713},"_11-虚地址转实地址十六进制","11. 虚地址转实地址（十六进制）",[15,264716,264717],{},"假设某虚存的用户空间为 1024KB，页面大小为 4KB，内存空间为 512KB。已知用户的虚页 10、11、12、13 页分得内存页框号为 62、78、25、36，求出虚地址 0BEBC（16 进制）的实地址（16 进制）是多少?",[15,264719,264303],{},[15,264721,264306],{},[19,264723,264725],{"id":264724},"_12-两道程序并发运行时间图","12. 两道程序并发运行时间图",[15,264727,264728],{},"一个计算机系统，有一台输入机和一台打印机，现有两道程序设计投入运行，且程序 A 先开始做，程序 B 后开始运行。",[105,264730,264731,264734],{},[108,264732,264733],{},"程序 A 的运行轨迹为：计算 50ms、打印 40ms、计算 80ms、打印 80ms，结束。",[108,264735,264736],{},"程序 B 的运行轨迹为：计算 50ms、输入 100ms、计算 40ms、打印 40ms，结束。",[15,264738,11832],{},[105,264740,264741,264744,264747],{},[108,264742,264743],{},"(1) 画出相应的运行图；",[108,264745,264746],{},"(2) 两道程序运行时，CPU 有无空闲等待？若有，在哪段时间等待？为什么？",[108,264748,264749],{},"(3) 程序 A、B 有无等待的情况？若有，指出发生等待的时刻。",[15,264751,264303],{},[15,264753,264306],{},[19,264755,264757],{"id":264756},"_13-请求分页-lru页大小-100","13. 请求分页 LRU（页大小 100）",[15,264759,264760],{},"假设 UNIX 系统采用请求式分页存储，有一个长度为 6 页的进程，页大小为 100。假如为它分配 3 个物理块，逻辑地址为：021, 256, 301, 508, 279, 098, 334, 302, 578, 412。试求出进程访问的页面次序，并用 LRU 页面置换算法计算出程序访问过程，以及访问过程中所发生的缺页次数和缺页中断率。（说明：初始页面为空，计算为缺页。）",[15,264762,264303],{},[15,264764,264306],{},[19,264766,264768],{"id":264767},"_14-磁盘交错存储6000rmin9扇区","14. 磁盘交错存储：6000r\u002Fmin、9扇区",[15,264770,264771],{},"假定磁盘转速为 6000r\u002Fmin（转\u002F分），磁盘格式化时每个盘面被分为 9 个扇区，现有一个文件共有 A，B，C，D，E，F，G，H，I 九个逻辑记录要存放在同一磁道上供处理程序使用，假设每个记录的大小与扇区的大小相同，处理程序每次从磁盘读出一个记录后要花 2.5ms 处理时间。若忽略其他辅助时间，请回答下列问题：",[105,264773,264774,264777],{},[108,264775,264776],{},"（1）现在假设已经顺序存放好这 9 个记录，记录逆时针排列，磁盘顺时针旋转。那么读出该文件需要多少时间？",[108,264778,264779],{},"（2）为了使读出文件需要的时间最短，请重新调整各个记录的存放位置，画出各个记录的存放位置，计算该文件的读出时间。",[15,264781,264303],{},[253,264783,214366],{"id":214365},[105,264785,264786,264893,265000],{},[108,264787,264788,264791,264792,347],{},[2627,264789,264790],{},"磁盘周期计算："," 6000 r\u002Fmin = 100 转\u002F秒。因此，磁盘旋转一圈（1转）的时间 ",[34,264793,264795,264829],{"className":264794},[31642],[34,264796,264798],{"className":264797},[31646],[31648,264799,264800],{"xmlns":31650},[31652,264801,264802,264826],{},[31655,264803,264804,264806,264808,264810,264812,264814,264816,264818,264820,264822,264824],{},[31658,264805,37400],{},[31662,264807,202],{},[31749,264809,5249],{},[31658,264811,31759],{},[31658,264813,31747],{},[31658,264815,38080],{"mathvariant":38079},[31749,264817,162173],{},[31662,264819,202],{},[31749,264821,3244],{},[31658,264823,31759],{},[31658,264825,31747],{},[31665,264827,264828],{"encoding":31667},"T = 1000ms \u002F 100 = 10ms",[34,264830,264832,264850,264878],{"className":264831,"ariaHidden":758},[31672],[34,264833,264835,264838,264841,264844,264847],{"className":264834},[31676],[34,264836],{"className":264837,"style":33218},[31680],[34,264839,37400],{"className":264840,"style":37422},[31685,31686],[34,264842],{"className":264843,"style":31846},[31845],[34,264845,202],{"className":264846},[31850],[34,264848],{"className":264849,"style":31846},[31845],[34,264851,264853,264856,264859,264862,264865,264869,264872,264875],{"className":264852},[31676],[34,264854],{"className":264855,"style":31774},[31680],[34,264857,5249],{"className":264858},[31685],[34,264860,31759],{"className":264861},[31685,31686],[34,264863,31747],{"className":264864},[31685,31686],[34,264866,264868],{"className":264867},[31685],"\u002F100",[34,264870],{"className":264871,"style":31846},[31845],[34,264873,202],{"className":264874},[31850],[34,264876],{"className":264877,"style":31846},[31845],[34,264879,264881,264884,264887,264890],{"className":264880},[31676],[34,264882],{"className":264883,"style":32617},[31680],[34,264885,3244],{"className":264886},[31685],[34,264888,31759],{"className":264889},[31685,31686],[34,264891,31747],{"className":264892},[31685,31686],[108,264894,264895,264898,264899,347],{},[2627,264896,264897],{},"扇区读取时间："," 1圈有9个扇区，读取1个扇区（记录）的时间 ",[34,264900,264902,264936],{"className":264901},[31642],[34,264903,264905],{"className":264904},[31646],[31648,264906,264907],{"xmlns":31650},[31652,264908,264909,264933],{},[31655,264910,264911,264913,264915,264917,264919,264921,264923,264925,264927,264929,264931],{},[31658,264912,33249],{},[31662,264914,202],{},[31749,264916,3244],{},[31658,264918,31759],{},[31658,264920,31747],{},[31658,264922,38080],{"mathvariant":38079},[31749,264924,36732],{},[31662,264926,168560],{},[31749,264928,216169],{},[31658,264930,31759],{},[31658,264932,31747],{},[31665,264934,264935],{"encoding":31667},"t = 10ms \u002F 9 \\approx 1.11ms",[34,264937,264939,264957,264985],{"className":264938,"ariaHidden":758},[31672],[34,264940,264942,264945,264948,264951,264954],{"className":264941},[31676],[34,264943],{"className":264944,"style":33280},[31680],[34,264946,33249],{"className":264947},[31685,31686],[34,264949],{"className":264950,"style":31846},[31845],[34,264952,202],{"className":264953},[31850],[34,264955],{"className":264956,"style":31846},[31845],[34,264958,264960,264963,264966,264969,264972,264976,264979,264982],{"className":264959},[31676],[34,264961],{"className":264962,"style":31774},[31680],[34,264964,3244],{"className":264965},[31685],[34,264967,31759],{"className":264968},[31685,31686],[34,264970,31747],{"className":264971},[31685,31686],[34,264973,264975],{"className":264974},[31685],"\u002F9",[34,264977],{"className":264978,"style":31846},[31845],[34,264980,168560],{"className":264981},[31850],[34,264983],{"className":264984,"style":31846},[31845],[34,264986,264988,264991,264994,264997],{"className":264987},[31676],[34,264989],{"className":264990,"style":32617},[31680],[34,264992,216169],{"className":264993},[31685],[34,264995,31759],{"className":264996},[31685,31686],[34,264998,31747],{"className":264999},[31685,31686],[108,265001,265002,265005],{},[2627,265003,265004],{},"处理时间："," 给定为 2.5ms。",[253,265007,219117],{"id":219116},[15,265009,265010,265013,265014,265122,265123,265161,265162,265200,265201,265339,265340,265343],{},[2627,265011,265012],{},"(1) 顺序存放时的困境：","\n当读完记录A（耗时 ",[34,265015,265017,265039],{"className":265016},[31642],[34,265018,265020],{"className":265019},[31646],[31648,265021,265022],{"xmlns":31650},[31652,265023,265024,265036],{},[31655,265025,265026,265032,265034],{},[41453,265027,265028,265030],{},[31749,265029,3244],{},[31749,265031,36732],{},[31658,265033,31759],{},[31658,265035,31747],{},[31665,265037,265038],{"encoding":31667},"\\frac{10}{9} ms",[34,265040,265042],{"className":265041,"ariaHidden":758},[31672],[34,265043,265045,265048,265116,265119],{"className":265044},[31676],[34,265046],{"className":265047,"style":46727},[31680],[34,265049,265051,265054,265113],{"className":265050},[31685],[34,265052],{"className":265053},[31782,41488],[34,265055,265057],{"className":265056},[41453],[34,265058,265060,265105],{"className":265059},[31796,31797],[34,265061,265063,265102],{"className":265062},[31801],[34,265064,265066,265080,265088],{"className":265065,"style":46746},[31805],[34,265067,265068,265071],{"style":41504},[34,265069],{"className":265070,"style":41508},[31813],[34,265072,265074],{"className":265073},[31818,31819,31820,31821],[34,265075,265077],{"className":265076},[31685,31821],[34,265078,36732],{"className":265079},[31685,31821],[34,265081,265082,265085],{"style":41520},[34,265083],{"className":265084,"style":41508},[31813],[34,265086],{"className":265087,"style":41528},[41527],[34,265089,265090,265093],{"style":46771},[34,265091],{"className":265092,"style":41508},[31813],[34,265094,265096],{"className":265095},[31818,31819,31820,31821],[34,265097,265099],{"className":265098},[31685,31821],[34,265100,3244],{"className":265101},[31685,31821],[34,265103,31829],{"className":265104},[31828],[34,265106,265108],{"className":265107},[31801],[34,265109,265111],{"className":265110,"style":41619},[31805],[34,265112],{},[34,265114],{"className":265115},[31690,41488],[34,265117,31759],{"className":265118},[31685,31686],[34,265120,31747],{"className":265121},[31685,31686],"）后，磁头来到了扇区1的末尾（即扇区2的开头）。此时处理程序开始处理A，耗时 ",[34,265124,265126,265143],{"className":265125},[31642],[34,265127,265129],{"className":265128},[31646],[31648,265130,265131],{"xmlns":31650},[31652,265132,265133,265141],{},[31655,265134,265135,265137,265139],{},[31749,265136,167259],{},[31658,265138,31759],{},[31658,265140,31747],{},[31665,265142,170166],{"encoding":31667},[34,265144,265146],{"className":265145,"ariaHidden":758},[31672],[34,265147,265149,265152,265155,265158],{"className":265148},[31676],[34,265150],{"className":265151,"style":32617},[31680],[34,265153,167259],{"className":265154},[31685],[34,265156,31759],{"className":265157},[31685,31686],[34,265159,31747],{"className":265160},[31685,31686],"。\n在这 ",[34,265163,265165,265182],{"className":265164},[31642],[34,265166,265168],{"className":265167},[31646],[31648,265169,265170],{"xmlns":31650},[31652,265171,265172,265180],{},[31655,265173,265174,265176,265178],{},[31749,265175,167259],{},[31658,265177,31759],{},[31658,265179,31747],{},[31665,265181,170166],{"encoding":31667},[34,265183,265185],{"className":265184,"ariaHidden":758},[31672],[34,265186,265188,265191,265194,265197],{"className":265187},[31676],[34,265189],{"className":265190,"style":32617},[31680],[34,265192,167259],{"className":265193},[31685],[34,265195,31759],{"className":265196},[31685,31686],[34,265198,31747],{"className":265199},[31685,31686]," 内，磁盘没有停下，它继续旋转了 ",[34,265202,265204,265234],{"className":265203},[31642],[34,265205,265207],{"className":265206},[31646],[31648,265208,265209],{"xmlns":31650},[31652,265210,265211,265231],{},[31655,265212,265213,265215,265217,265219,265225,265227,265229],{},[31749,265214,167259],{},[31658,265216,38080],{"mathvariant":38079},[31662,265218,456],{"stretchy":702},[41453,265220,265221,265223],{},[31749,265222,3244],{},[31749,265224,36732],{},[31662,265226,2557],{"stretchy":702},[31662,265228,202],{},[31749,265230,230918],{},[31665,265232,265233],{"encoding":31667},"2.5 \u002F (\\frac{10}{9}) = 2.25",[34,265235,265237,265330],{"className":265236,"ariaHidden":758},[31672],[34,265238,265240,265243,265247,265250,265318,265321,265324,265327],{"className":265239},[31676],[34,265241],{"className":265242,"style":46727},[31680],[34,265244,265246],{"className":265245},[31685],"2.5\u002F",[34,265248,456],{"className":265249},[31782],[34,265251,265253,265256,265315],{"className":265252},[31685],[34,265254],{"className":265255},[31782,41488],[34,265257,265259],{"className":265258},[41453],[34,265260,265262,265307],{"className":265261},[31796,31797],[34,265263,265265,265304],{"className":265264},[31801],[34,265266,265268,265282,265290],{"className":265267,"style":46746},[31805],[34,265269,265270,265273],{"style":41504},[34,265271],{"className":265272,"style":41508},[31813],[34,265274,265276],{"className":265275},[31818,31819,31820,31821],[34,265277,265279],{"className":265278},[31685,31821],[34,265280,36732],{"className":265281},[31685,31821],[34,265283,265284,265287],{"style":41520},[34,265285],{"className":265286,"style":41508},[31813],[34,265288],{"className":265289,"style":41528},[41527],[34,265291,265292,265295],{"style":46771},[34,265293],{"className":265294,"style":41508},[31813],[34,265296,265298],{"className":265297},[31818,31819,31820,31821],[34,265299,265301],{"className":265300},[31685,31821],[34,265302,3244],{"className":265303},[31685,31821],[34,265305,31829],{"className":265306},[31828],[34,265308,265310],{"className":265309},[31801],[34,265311,265313],{"className":265312,"style":41619},[31805],[34,265314],{},[34,265316],{"className":265317},[31690,41488],[34,265319,2557],{"className":265320},[31690],[34,265322],{"className":265323,"style":31846},[31845],[34,265325,202],{"className":265326},[31850],[34,265328],{"className":265329,"style":31846},[31845],[34,265331,265333,265336],{"className":265332},[31676],[34,265334],{"className":265335,"style":32617},[31680],[34,265337,230918],{"className":265338},[31685]," 个扇区。\n也就是说，当A处理完时，磁头已经越过了记录B（紧挨着A），来到了扇区4的内部！为了读取B，磁头必须",[2627,265341,265342],{},"傻傻地等待磁盘再转一整圈","，直到B再次转到磁头下方。",[105,265345,265346,265355,265451,265634],{},[108,265347,265348,265351,265352,347],{},[2627,265349,265350],{},"读取并处理1个记录的总周期"," = 读出时间 + 处理时间 + 延迟等待下一个记录的时间 = ",[2627,265353,265354],{},"刚好是磁盘旋转一圈的时间 (10ms)",[108,265356,265357,265358,347],{},"读前8个记录并找到下一个：",[34,265359,265361,265391],{"className":265360},[31642],[34,265362,265364],{"className":265363},[31646],[31648,265365,265366],{"xmlns":31650},[31652,265367,265368,265388],{},[31655,265369,265370,265372,265374,265376,265378,265380,265382,265384,265386],{},[31749,265371,60525],{},[31662,265373,42071],{},[31749,265375,3244],{},[31658,265377,31759],{},[31658,265379,31747],{},[31662,265381,202],{},[31749,265383,27539],{},[31658,265385,31759],{},[31658,265387,31747],{},[31665,265389,265390],{"encoding":31667},"8 \\times 10ms = 80ms",[34,265392,265394,265412,265436],{"className":265393,"ariaHidden":758},[31672],[34,265395,265397,265400,265403,265406,265409],{"className":265396},[31676],[34,265398],{"className":265399,"style":35942},[31680],[34,265401,60525],{"className":265402},[31685],[34,265404],{"className":265405,"style":32604},[31845],[34,265407,42071],{"className":265408},[32024],[34,265410],{"className":265411,"style":32604},[31845],[34,265413,265415,265418,265421,265424,265427,265430,265433],{"className":265414},[31676],[34,265416],{"className":265417,"style":32617},[31680],[34,265419,3244],{"className":265420},[31685],[34,265422,31759],{"className":265423},[31685,31686],[34,265425,31747],{"className":265426},[31685,31686],[34,265428],{"className":265429,"style":31846},[31845],[34,265431,202],{"className":265432},[31850],[34,265434],{"className":265435,"style":31846},[31845],[34,265437,265439,265442,265445,265448],{"className":265438},[31676],[34,265440],{"className":265441,"style":32617},[31680],[34,265443,27539],{"className":265444},[31685],[34,265446,31759],{"className":265447},[31685,31686],[34,265449,31747],{"className":265450},[31685,31686],[108,265452,265453,265454,265561,265562,347],{},"读出最后一个记录I读取 ",[34,265455,265457,265478],{"className":265456},[31642],[34,265458,265460],{"className":265459},[31646],[31648,265461,265462],{"xmlns":31650},[31652,265463,265464,265476],{},[31655,265465,265466,265472,265474],{},[41453,265467,265468,265470],{},[31749,265469,3244],{},[31749,265471,36732],{},[31658,265473,31759],{},[31658,265475,31747],{},[31665,265477,265038],{"encoding":31667},[34,265479,265481],{"className":265480,"ariaHidden":758},[31672],[34,265482,265484,265487,265555,265558],{"className":265483},[31676],[34,265485],{"className":265486,"style":46727},[31680],[34,265488,265490,265493,265552],{"className":265489},[31685],[34,265491],{"className":265492},[31782,41488],[34,265494,265496],{"className":265495},[41453],[34,265497,265499,265544],{"className":265498},[31796,31797],[34,265500,265502,265541],{"className":265501},[31801],[34,265503,265505,265519,265527],{"className":265504,"style":46746},[31805],[34,265506,265507,265510],{"style":41504},[34,265508],{"className":265509,"style":41508},[31813],[34,265511,265513],{"className":265512},[31818,31819,31820,31821],[34,265514,265516],{"className":265515},[31685,31821],[34,265517,36732],{"className":265518},[31685,31821],[34,265520,265521,265524],{"style":41520},[34,265522],{"className":265523,"style":41508},[31813],[34,265525],{"className":265526,"style":41528},[41527],[34,265528,265529,265532],{"style":46771},[34,265530],{"className":265531,"style":41508},[31813],[34,265533,265535],{"className":265534},[31818,31819,31820,31821],[34,265536,265538],{"className":265537},[31685,31821],[34,265539,3244],{"className":265540},[31685,31821],[34,265542,31829],{"className":265543},[31828],[34,265545,265547],{"className":265546},[31801],[34,265548,265550],{"className":265549,"style":41619},[31805],[34,265551],{},[34,265553],{"className":265554},[31690,41488],[34,265556,31759],{"className":265557},[31685,31686],[34,265559,31747],{"className":265560},[31685,31686]," + 处理 ",[34,265563,265565,265592],{"className":265564},[31642],[34,265566,265568],{"className":265567},[31646],[31648,265569,265570],{"xmlns":31650},[31652,265571,265572,265589],{},[31655,265573,265574,265576,265578,265580,265582,265585,265587],{},[31749,265575,167259],{},[31658,265577,31759],{},[31658,265579,31747],{},[31662,265581,202],{},[31749,265583,265584],{},"3.61",[31658,265586,31759],{},[31658,265588,31747],{},[31665,265590,265591],{"encoding":31667},"2.5ms = 3.61ms",[34,265593,265595,265619],{"className":265594,"ariaHidden":758},[31672],[34,265596,265598,265601,265604,265607,265610,265613,265616],{"className":265597},[31676],[34,265599],{"className":265600,"style":32617},[31680],[34,265602,167259],{"className":265603},[31685],[34,265605,31759],{"className":265606},[31685,31686],[34,265608,31747],{"className":265609},[31685,31686],[34,265611],{"className":265612,"style":31846},[31845],[34,265614,202],{"className":265615},[31850],[34,265617],{"className":265618,"style":31846},[31845],[34,265620,265622,265625,265628,265631],{"className":265621},[31676],[34,265623],{"className":265624,"style":32617},[31680],[34,265626,265584],{"className":265627},[31685],[34,265629,31759],{"className":265630},[31685,31686],[34,265632,31747],{"className":265633},[31685,31686],[108,265635,265636,206456,265639,347],{},[2627,265637,265638],{},"总时间",[34,265640,265642,265669],{"className":265641},[31642],[34,265643,265645],{"className":265644},[31646],[31648,265646,265647],{"xmlns":31650},[31652,265648,265649,265666],{},[31655,265650,265651,265653,265655,265657,265659,265662,265664],{},[31749,265652,27539],{},[31662,265654,223],{},[31749,265656,265584],{},[31662,265658,202],{},[31749,265660,265661],{},"83.61",[31658,265663,31759],{},[31658,265665,31747],{},[31665,265667,265668],{"encoding":31667},"80 + 3.61 = 83.61ms",[34,265670,265672,265690,265708],{"className":265671,"ariaHidden":758},[31672],[34,265673,265675,265678,265681,265684,265687],{"className":265674},[31676],[34,265676],{"className":265677,"style":35942},[31680],[34,265679,27539],{"className":265680},[31685],[34,265682],{"className":265683,"style":32604},[31845],[34,265685,223],{"className":265686},[32024],[34,265688],{"className":265689,"style":32604},[31845],[34,265691,265693,265696,265699,265702,265705],{"className":265692},[31676],[34,265694],{"className":265695,"style":32617},[31680],[34,265697,265584],{"className":265698},[31685],[34,265700],{"className":265701,"style":31846},[31845],[34,265703,202],{"className":265704},[31850],[34,265706],{"className":265707,"style":31846},[31845],[34,265709,265711,265714,265717,265720],{"className":265710},[31676],[34,265712],{"className":265713,"style":32617},[31680],[34,265715,265661],{"className":265716},[31685],[34,265718,31759],{"className":265719},[31685,31686],[34,265721,31747],{"className":265722},[31685,31686],[15,265724,265725,265728,265729,265757,265758,265761,265762,265790,265791,265794],{},[2627,265726,265727],{},"(2) 最优化分布的设计：","\n既然处理A需要 ",[34,265730,265732,265745],{"className":265731},[31642],[34,265733,265735],{"className":265734},[31646],[31648,265736,265737],{"xmlns":31650},[31652,265738,265739,265743],{},[31655,265740,265741],{},[31749,265742,230918],{},[31665,265744,230918],{"encoding":31667},[34,265746,265748],{"className":265747,"ariaHidden":758},[31672],[34,265749,265751,265754],{"className":265750},[31676],[34,265752],{"className":265753,"style":32617},[31680],[34,265755,230918],{"className":265756},[31685]," 个扇区的时间，那我们何不",[2627,265759,265760],{},"跳过这3个扇区","，直接把B放在磁头处理完A后刚刚好到达的地方呢？\n读完A，处理A用掉 ",[34,265763,265765,265778],{"className":265764},[31642],[34,265766,265768],{"className":265767},[31646],[31648,265769,265770],{"xmlns":31650},[31652,265771,265772,265776],{},[31655,265773,265774],{},[31749,265775,230918],{},[31665,265777,230918],{"encoding":31667},[34,265779,265781],{"className":265780,"ariaHidden":758},[31672],[34,265782,265784,265787],{"className":265783},[31676],[34,265785],{"className":265786,"style":32617},[31680],[34,265788,230918],{"className":265789},[31685]," 个扇区的时间，磁头滑过了 3 个扇区的起始位置。因此，逻辑上相邻的记录在物理位置上应该",[2627,265792,265793],{},"间隔 3 个扇区","（即放在第4个位置）。",[105,265796,265797,265808],{},[108,265798,265799,47649,265802,265805],{},[2627,265800,265801],{},"存放位置（按扇区0~8顺序列出）：",[31,265803,265804],{},"A, H, F, D, B, I, G, E, C",[71807,265806,265807],{},"(推理：A在0，B在4，C在8，D在3，E在7，F在2，G在6，H在1，I在5)",[108,265809,265810,265813,265814,266045,266046,266268,266269,39984,266491,206456,266493,347],{},[2627,265811,265812],{},"计算时间：","\n每个记录读取+处理+对齐耗时恰好为 4 个扇区的时间（",[34,265815,265817,265855],{"className":265816},[31642],[34,265818,265820],{"className":265819},[31646],[31648,265821,265822],{"xmlns":31650},[31652,265823,265824,265852],{},[31655,265825,265826,265828,265830,265836,265838,265840,265842,265848,265850],{},[31749,265827,15932],{},[31662,265829,42071],{},[41453,265831,265832,265834],{},[31749,265833,3244],{},[31749,265835,36732],{},[31658,265837,31759],{},[31658,265839,31747],{},[31662,265841,202],{},[41453,265843,265844,265846],{},[31749,265845,5731],{},[31749,265847,36732],{},[31658,265849,31759],{},[31658,265851,31747],{},[31665,265853,265854],{"encoding":31667},"4 \\times \\frac{10}{9}ms = \\frac{40}{9}ms",[34,265856,265858,265876,265965],{"className":265857,"ariaHidden":758},[31672],[34,265859,265861,265864,265867,265870,265873],{"className":265860},[31676],[34,265862],{"className":265863,"style":35942},[31680],[34,265865,15932],{"className":265866},[31685],[34,265868],{"className":265869,"style":32604},[31845],[34,265871,42071],{"className":265872},[32024],[34,265874],{"className":265875,"style":32604},[31845],[34,265877,265879,265882,265950,265953,265956,265959,265962],{"className":265878},[31676],[34,265880],{"className":265881,"style":46727},[31680],[34,265883,265885,265888,265947],{"className":265884},[31685],[34,265886],{"className":265887},[31782,41488],[34,265889,265891],{"className":265890},[41453],[34,265892,265894,265939],{"className":265893},[31796,31797],[34,265895,265897,265936],{"className":265896},[31801],[34,265898,265900,265914,265922],{"className":265899,"style":46746},[31805],[34,265901,265902,265905],{"style":41504},[34,265903],{"className":265904,"style":41508},[31813],[34,265906,265908],{"className":265907},[31818,31819,31820,31821],[34,265909,265911],{"className":265910},[31685,31821],[34,265912,36732],{"className":265913},[31685,31821],[34,265915,265916,265919],{"style":41520},[34,265917],{"className":265918,"style":41508},[31813],[34,265920],{"className":265921,"style":41528},[41527],[34,265923,265924,265927],{"style":46771},[34,265925],{"className":265926,"style":41508},[31813],[34,265928,265930],{"className":265929},[31818,31819,31820,31821],[34,265931,265933],{"className":265932},[31685,31821],[34,265934,3244],{"className":265935},[31685,31821],[34,265937,31829],{"className":265938},[31828],[34,265940,265942],{"className":265941},[31801],[34,265943,265945],{"className":265944,"style":41619},[31805],[34,265946],{},[34,265948],{"className":265949},[31690,41488],[34,265951,31759],{"className":265952},[31685,31686],[34,265954,31747],{"className":265955},[31685,31686],[34,265957],{"className":265958,"style":31846},[31845],[34,265960,202],{"className":265961},[31850],[34,265963],{"className":265964,"style":31846},[31845],[34,265966,265968,265971,266039,266042],{"className":265967},[31676],[34,265969],{"className":265970,"style":46727},[31680],[34,265972,265974,265977,266036],{"className":265973},[31685],[34,265975],{"className":265976},[31782,41488],[34,265978,265980],{"className":265979},[41453],[34,265981,265983,266028],{"className":265982},[31796,31797],[34,265984,265986,266025],{"className":265985},[31801],[34,265987,265989,266003,266011],{"className":265988,"style":46746},[31805],[34,265990,265991,265994],{"style":41504},[34,265992],{"className":265993,"style":41508},[31813],[34,265995,265997],{"className":265996},[31818,31819,31820,31821],[34,265998,266000],{"className":265999},[31685,31821],[34,266001,36732],{"className":266002},[31685,31821],[34,266004,266005,266008],{"style":41520},[34,266006],{"className":266007,"style":41508},[31813],[34,266009],{"className":266010,"style":41528},[41527],[34,266012,266013,266016],{"style":46771},[34,266014],{"className":266015,"style":41508},[31813],[34,266017,266019],{"className":266018},[31818,31819,31820,31821],[34,266020,266022],{"className":266021},[31685,31821],[34,266023,5731],{"className":266024},[31685,31821],[34,266026,31829],{"className":266027},[31828],[34,266029,266031],{"className":266030},[31801],[34,266032,266034],{"className":266033,"style":41619},[31805],[34,266035],{},[34,266037],{"className":266038},[31690,41488],[34,266040,31759],{"className":266041},[31685,31686],[34,266043,31747],{"className":266044},[31685,31686],"）。\n前 8 个记录耗时：",[34,266047,266049,266084],{"className":266048},[31642],[34,266050,266052],{"className":266051},[31646],[31648,266053,266054],{"xmlns":31650},[31652,266055,266056,266081],{},[31655,266057,266058,266060,266062,266068,266070,266077,266079],{},[31749,266059,60525],{},[31662,266061,42071],{},[41453,266063,266064,266066],{},[31749,266065,5731],{},[31749,266067,36732],{},[31662,266069,202],{},[41453,266071,266072,266075],{},[31749,266073,266074],{},"320",[31749,266076,36732],{},[31658,266078,31759],{},[31658,266080,31747],{},[31665,266082,266083],{"encoding":31667},"8 \\times \\frac{40}{9} = \\frac{320}{9} ms",[34,266085,266087,266105,266188],{"className":266086,"ariaHidden":758},[31672],[34,266088,266090,266093,266096,266099,266102],{"className":266089},[31676],[34,266091],{"className":266092,"style":35942},[31680],[34,266094,60525],{"className":266095},[31685],[34,266097],{"className":266098,"style":32604},[31845],[34,266100,42071],{"className":266101},[32024],[34,266103],{"className":266104,"style":32604},[31845],[34,266106,266108,266111,266179,266182,266185],{"className":266107},[31676],[34,266109],{"className":266110,"style":46727},[31680],[34,266112,266114,266117,266176],{"className":266113},[31685],[34,266115],{"className":266116},[31782,41488],[34,266118,266120],{"className":266119},[41453],[34,266121,266123,266168],{"className":266122},[31796,31797],[34,266124,266126,266165],{"className":266125},[31801],[34,266127,266129,266143,266151],{"className":266128,"style":46746},[31805],[34,266130,266131,266134],{"style":41504},[34,266132],{"className":266133,"style":41508},[31813],[34,266135,266137],{"className":266136},[31818,31819,31820,31821],[34,266138,266140],{"className":266139},[31685,31821],[34,266141,36732],{"className":266142},[31685,31821],[34,266144,266145,266148],{"style":41520},[34,266146],{"className":266147,"style":41508},[31813],[34,266149],{"className":266150,"style":41528},[41527],[34,266152,266153,266156],{"style":46771},[34,266154],{"className":266155,"style":41508},[31813],[34,266157,266159],{"className":266158},[31818,31819,31820,31821],[34,266160,266162],{"className":266161},[31685,31821],[34,266163,5731],{"className":266164},[31685,31821],[34,266166,31829],{"className":266167},[31828],[34,266169,266171],{"className":266170},[31801],[34,266172,266174],{"className":266173,"style":41619},[31805],[34,266175],{},[34,266177],{"className":266178},[31690,41488],[34,266180],{"className":266181,"style":31846},[31845],[34,266183,202],{"className":266184},[31850],[34,266186],{"className":266187,"style":31846},[31845],[34,266189,266191,266194,266262,266265],{"className":266190},[31676],[34,266192],{"className":266193,"style":46727},[31680],[34,266195,266197,266200,266259],{"className":266196},[31685],[34,266198],{"className":266199},[31782,41488],[34,266201,266203],{"className":266202},[41453],[34,266204,266206,266251],{"className":266205},[31796,31797],[34,266207,266209,266248],{"className":266208},[31801],[34,266210,266212,266226,266234],{"className":266211,"style":46746},[31805],[34,266213,266214,266217],{"style":41504},[34,266215],{"className":266216,"style":41508},[31813],[34,266218,266220],{"className":266219},[31818,31819,31820,31821],[34,266221,266223],{"className":266222},[31685,31821],[34,266224,36732],{"className":266225},[31685,31821],[34,266227,266228,266231],{"style":41520},[34,266229],{"className":266230,"style":41508},[31813],[34,266232],{"className":266233,"style":41528},[41527],[34,266235,266236,266239],{"style":46771},[34,266237],{"className":266238,"style":41508},[31813],[34,266240,266242],{"className":266241},[31818,31819,31820,31821],[34,266243,266245],{"className":266244},[31685,31821],[34,266246,266074],{"className":266247},[31685,31821],[34,266249,31829],{"className":266250},[31828],[34,266252,266254],{"className":266253},[31801],[34,266255,266257],{"className":266256,"style":41619},[31805],[34,266258],{},[34,266260],{"className":266261},[31690,41488],[34,266263,31759],{"className":266264},[31685,31686],[34,266266,31747],{"className":266267},[31685,31686],"。\n第 9 个记录读出并处理：",[34,266270,266272,266307],{"className":266271},[31642],[34,266273,266275],{"className":266274},[31646],[31648,266276,266277],{"xmlns":31650},[31652,266278,266279,266304],{},[31655,266280,266281,266287,266289,266291,266293,266300,266302],{},[41453,266282,266283,266285],{},[31749,266284,3244],{},[31749,266286,36732],{},[31662,266288,223],{},[31749,266290,167259],{},[31662,266292,202],{},[41453,266294,266295,266298],{},[31749,266296,266297],{},"32.5",[31749,266299,36732],{},[31658,266301,31759],{},[31658,266303,31747],{},[31665,266305,266306],{"encoding":31667},"\\frac{10}{9} + 2.5 = \\frac{32.5}{9} ms",[34,266308,266310,266393,266411],{"className":266309,"ariaHidden":758},[31672],[34,266311,266313,266316,266384,266387,266390],{"className":266312},[31676],[34,266314],{"className":266315,"style":46727},[31680],[34,266317,266319,266322,266381],{"className":266318},[31685],[34,266320],{"className":266321},[31782,41488],[34,266323,266325],{"className":266324},[41453],[34,266326,266328,266373],{"className":266327},[31796,31797],[34,266329,266331,266370],{"className":266330},[31801],[34,266332,266334,266348,266356],{"className":266333,"style":46746},[31805],[34,266335,266336,266339],{"style":41504},[34,266337],{"className":266338,"style":41508},[31813],[34,266340,266342],{"className":266341},[31818,31819,31820,31821],[34,266343,266345],{"className":266344},[31685,31821],[34,266346,36732],{"className":266347},[31685,31821],[34,266349,266350,266353],{"style":41520},[34,266351],{"className":266352,"style":41508},[31813],[34,266354],{"className":266355,"style":41528},[41527],[34,266357,266358,266361],{"style":46771},[34,266359],{"className":266360,"style":41508},[31813],[34,266362,266364],{"className":266363},[31818,31819,31820,31821],[34,266365,266367],{"className":266366},[31685,31821],[34,266368,3244],{"className":266369},[31685,31821],[34,266371,31829],{"className":266372},[31828],[34,266374,266376],{"className":266375},[31801],[34,266377,266379],{"className":266378,"style":41619},[31805],[34,266380],{},[34,266382],{"className":266383},[31690,41488],[34,266385],{"className":266386,"style":32604},[31845],[34,266388,223],{"className":266389},[32024],[34,266391],{"className":266392,"style":32604},[31845],[34,266394,266396,266399,266402,266405,266408],{"className":266395},[31676],[34,266397],{"className":266398,"style":32617},[31680],[34,266400,167259],{"className":266401},[31685],[34,266403],{"className":266404,"style":31846},[31845],[34,266406,202],{"className":266407},[31850],[34,266409],{"className":266410,"style":31846},[31845],[34,266412,266414,266417,266485,266488],{"className":266413},[31676],[34,266415],{"className":266416,"style":46727},[31680],[34,266418,266420,266423,266482],{"className":266419},[31685],[34,266421],{"className":266422},[31782,41488],[34,266424,266426],{"className":266425},[41453],[34,266427,266429,266474],{"className":266428},[31796,31797],[34,266430,266432,266471],{"className":266431},[31801],[34,266433,266435,266449,266457],{"className":266434,"style":46746},[31805],[34,266436,266437,266440],{"style":41504},[34,266438],{"className":266439,"style":41508},[31813],[34,266441,266443],{"className":266442},[31818,31819,31820,31821],[34,266444,266446],{"className":266445},[31685,31821],[34,266447,36732],{"className":266448},[31685,31821],[34,266450,266451,266454],{"style":41520},[34,266452],{"className":266453,"style":41508},[31813],[34,266455],{"className":266456,"style":41528},[41527],[34,266458,266459,266462],{"style":46771},[34,266460],{"className":266461,"style":41508},[31813],[34,266463,266465],{"className":266464},[31818,31819,31820,31821],[34,266466,266468],{"className":266467},[31685,31821],[34,266469,266297],{"className":266470},[31685,31821],[34,266472,31829],{"className":266473},[31828],[34,266475,266477],{"className":266476},[31801],[34,266478,266480],{"className":266479,"style":41619},[31805],[34,266481],{},[34,266483],{"className":266484},[31690,41488],[34,266486,31759],{"className":266487},[31685,31686],[34,266489,31747],{"className":266490},[31685,31686],[2627,266492,265638],{},[34,266494,266496,266546],{"className":266495},[31642],[34,266497,266499],{"className":266498},[31646],[31648,266500,266501],{"xmlns":31650},[31652,266502,266503,266543],{},[31655,266504,266505,266511,266513,266519,266521,266528,266530,266532,266534],{},[41453,266506,266507,266509],{},[31749,266508,266074],{},[31749,266510,36732],{},[31662,266512,223],{},[41453,266514,266515,266517],{},[31749,266516,266297],{},[31749,266518,36732],{},[31662,266520,202],{},[41453,266522,266523,266526],{},[31749,266524,266525],{},"352.5",[31749,266527,36732],{},[31658,266529,31759],{},[31658,266531,31747],{},[31662,266533,168560],{},[31655,266535,266536,266539,266541],{},[31749,266537,266538],{"mathvariant":226163},"39.17",[31658,266540,31759],{"mathvariant":226163},[31658,266542,31747],{"mathvariant":226163},[31665,266544,266545],{"encoding":31667},"\\frac{320}{9} + \\frac{32.5}{9} = \\frac{352.5}{9} ms \\approx \\mathbf{39.17 ms}",[34,266547,266549,266632,266715,266804],{"className":266548,"ariaHidden":758},[31672],[34,266550,266552,266555,266623,266626,266629],{"className":266551},[31676],[34,266553],{"className":266554,"style":46727},[31680],[34,266556,266558,266561,266620],{"className":266557},[31685],[34,266559],{"className":266560},[31782,41488],[34,266562,266564],{"className":266563},[41453],[34,266565,266567,266612],{"className":266566},[31796,31797],[34,266568,266570,266609],{"className":266569},[31801],[34,266571,266573,266587,266595],{"className":266572,"style":46746},[31805],[34,266574,266575,266578],{"style":41504},[34,266576],{"className":266577,"style":41508},[31813],[34,266579,266581],{"className":266580},[31818,31819,31820,31821],[34,266582,266584],{"className":266583},[31685,31821],[34,266585,36732],{"className":266586},[31685,31821],[34,266588,266589,266592],{"style":41520},[34,266590],{"className":266591,"style":41508},[31813],[34,266593],{"className":266594,"style":41528},[41527],[34,266596,266597,266600],{"style":46771},[34,266598],{"className":266599,"style":41508},[31813],[34,266601,266603],{"className":266602},[31818,31819,31820,31821],[34,266604,266606],{"className":266605},[31685,31821],[34,266607,266074],{"className":266608},[31685,31821],[34,266610,31829],{"className":266611},[31828],[34,266613,266615],{"className":266614},[31801],[34,266616,266618],{"className":266617,"style":41619},[31805],[34,266619],{},[34,266621],{"className":266622},[31690,41488],[34,266624],{"className":266625,"style":32604},[31845],[34,266627,223],{"className":266628},[32024],[34,266630],{"className":266631,"style":32604},[31845],[34,266633,266635,266638,266706,266709,266712],{"className":266634},[31676],[34,266636],{"className":266637,"style":46727},[31680],[34,266639,266641,266644,266703],{"className":266640},[31685],[34,266642],{"className":266643},[31782,41488],[34,266645,266647],{"className":266646},[41453],[34,266648,266650,266695],{"className":266649},[31796,31797],[34,266651,266653,266692],{"className":266652},[31801],[34,266654,266656,266670,266678],{"className":266655,"style":46746},[31805],[34,266657,266658,266661],{"style":41504},[34,266659],{"className":266660,"style":41508},[31813],[34,266662,266664],{"className":266663},[31818,31819,31820,31821],[34,266665,266667],{"className":266666},[31685,31821],[34,266668,36732],{"className":266669},[31685,31821],[34,266671,266672,266675],{"style":41520},[34,266673],{"className":266674,"style":41508},[31813],[34,266676],{"className":266677,"style":41528},[41527],[34,266679,266680,266683],{"style":46771},[34,266681],{"className":266682,"style":41508},[31813],[34,266684,266686],{"className":266685},[31818,31819,31820,31821],[34,266687,266689],{"className":266688},[31685,31821],[34,266690,266297],{"className":266691},[31685,31821],[34,266693,31829],{"className":266694},[31828],[34,266696,266698],{"className":266697},[31801],[34,266699,266701],{"className":266700,"style":41619},[31805],[34,266702],{},[34,266704],{"className":266705},[31690,41488],[34,266707],{"className":266708,"style":31846},[31845],[34,266710,202],{"className":266711},[31850],[34,266713],{"className":266714,"style":31846},[31845],[34,266716,266718,266721,266789,266792,266795,266798,266801],{"className":266717},[31676],[34,266719],{"className":266720,"style":46727},[31680],[34,266722,266724,266727,266786],{"className":266723},[31685],[34,266725],{"className":266726},[31782,41488],[34,266728,266730],{"className":266729},[41453],[34,266731,266733,266778],{"className":266732},[31796,31797],[34,266734,266736,266775],{"className":266735},[31801],[34,266737,266739,266753,266761],{"className":266738,"style":46746},[31805],[34,266740,266741,266744],{"style":41504},[34,266742],{"className":266743,"style":41508},[31813],[34,266745,266747],{"className":266746},[31818,31819,31820,31821],[34,266748,266750],{"className":266749},[31685,31821],[34,266751,36732],{"className":266752},[31685,31821],[34,266754,266755,266758],{"style":41520},[34,266756],{"className":266757,"style":41508},[31813],[34,266759],{"className":266760,"style":41528},[41527],[34,266762,266763,266766],{"style":46771},[34,266764],{"className":266765,"style":41508},[31813],[34,266767,266769],{"className":266768},[31818,31819,31820,31821],[34,266770,266772],{"className":266771},[31685,31821],[34,266773,266525],{"className":266774},[31685,31821],[34,266776,31829],{"className":266777},[31828],[34,266779,266781],{"className":266780},[31801],[34,266782,266784],{"className":266783,"style":41619},[31805],[34,266785],{},[34,266787],{"className":266788},[31690,41488],[34,266790,31759],{"className":266791},[31685,31686],[34,266793,31747],{"className":266794},[31685,31686],[34,266796],{"className":266797,"style":31846},[31845],[34,266799,168560],{"className":266800},[31850],[34,266802],{"className":266803,"style":31846},[31845],[34,266805,266807,266810],{"className":266806},[31676],[34,266808],{"className":266809,"style":32617},[31680],[34,266811,266813],{"className":266812},[31685],[34,266814,266816],{"className":266815},[31685,226302],"39.17ms",[15,266818,266819,266821,266822,266825],{},[2627,266820,247377],{}," 学生常常忘记磁盘在CPU处理数据时是",[2627,266823,266824],{},"不会停下等你的","。必须把\"错过的扇区\"换算成下一圈的延迟时间。\n:::",[19,266827,266829],{"id":266828},"_15-磁盘交错存储20msr10扇区","15. 磁盘交错存储：20ms\u002Fr、10扇区",[15,266831,266832],{},"假定磁盘转速为 20ms\u002Fr，每个磁道被划分为 10 个扇区。现有 10 条记录存放在同一磁道上(一条记录正好与一个扇区的大小相等)，处理程序从磁盘读出一条记录需要 4ms，现要求按从 1 到 10 的顺序处理这 10 条记录。若磁头处于首条记录的起点位置，则：",[105,266834,266835,266838],{},[108,266836,266837],{},"(1) 按逆时针方向依次存放这 10 条记录(磁盘顺时针方向旋转)，处理程序读取这 10 条记录需要多长时间？",[108,266839,266840],{},"(2) 按最优化分布重新安排这 10 条记录，写出记录的逆时针存放顺序，并计算处理这 10 条记录需要的时间。",[15,266842,264303],{},[253,266844,219117],{"id":231556},[15,266846,266847],{},"这也是典型的**交错因子（Interleaving）**问题。",[105,266849,266850,266888],{},[108,266851,266852,266855,266856,47649,266885,347],{},[2627,266853,266854],{},"转速与扇区："," 20ms\u002F圈，10个扇区\u002F圈 ",[34,266857,266859,266873],{"className":266858},[31642],[34,266860,266862],{"className":266861},[31646],[31648,266863,266864],{"xmlns":31650},[31652,266865,266866,266870],{},[31655,266867,266868],{},[31662,266869,140094],{},[31665,266871,266872],{"encoding":31667},"\\Rightarrow",[34,266874,266876],{"className":266875,"ariaHidden":758},[31672],[34,266877,266879,266882],{"className":266878},[31676],[34,266880],{"className":266881,"style":73089},[31680],[34,266883,140094],{"className":266884},[31850],[2627,266886,266887],{},"读取 1 个扇区需要 2ms",[108,266889,266890,266892],{},[2627,266891,265004],{}," 给定为 4ms（注意，这4ms等于转过 2 个扇区的时间）。",[15,266894,266895,266898,266899,266927,266928,266967,266968,267051,267052,347],{},[2627,266896,266897],{},"(1) 顺序存放：","\n读R1(2ms) ",[34,266900,266902,266915],{"className":266901},[31642],[34,266903,266905],{"className":266904},[31646],[31648,266906,266907],{"xmlns":31650},[31652,266908,266909,266913],{},[31655,266910,266911],{},[31662,266912,33239],{},[31665,266914,119689],{"encoding":31667},[34,266916,266918],{"className":266917,"ariaHidden":758},[31672],[34,266919,266921,266924],{"className":266920},[31676],[34,266922],{"className":266923,"style":73089},[31680],[34,266925,33239],{"className":266926},[31850]," 处理R1(4ms)。此时磁盘转过了2个扇区，磁头来到了扇区3的起始位置。\n但我们需要读R2（在扇区1）。错过了！只能等磁盘再转一圈转回来。\n所以，处理一条记录的周期 = ",[34,266929,266931,266949],{"className":266930},[31642],[34,266932,266934],{"className":266933},[31646],[31648,266935,266936],{"xmlns":31650},[31652,266937,266938,266946],{},[31655,266939,266940,266942,266944],{},[31749,266941,3249],{},[31658,266943,31759],{},[31658,266945,31747],{},[31665,266947,266948],{"encoding":31667},"20ms",[34,266950,266952],{"className":266951,"ariaHidden":758},[31672],[34,266953,266955,266958,266961,266964],{"className":266954},[31676],[34,266956],{"className":266957,"style":32617},[31680],[34,266959,3249],{"className":266960},[31685],[34,266962,31759],{"className":266963},[31685,31686],[34,266965,31747],{"className":266966},[31685,31686]," (刚好一圈)。前9条需 ",[34,266969,266971,266997],{"className":266970},[31642],[34,266972,266974],{"className":266973},[31646],[31648,266975,266976],{"xmlns":31650},[31652,266977,266978,266994],{},[31655,266979,266980,266982,266984,266986,266988,266990,266992],{},[31749,266981,36732],{},[31662,266983,42071],{},[31749,266985,3249],{},[31662,266987,202],{},[31749,266989,74303],{},[31658,266991,31759],{},[31658,266993,31747],{},[31665,266995,266996],{"encoding":31667},"9 \\times 20 = 180ms",[34,266998,267000,267018,267036],{"className":266999,"ariaHidden":758},[31672],[34,267001,267003,267006,267009,267012,267015],{"className":267002},[31676],[34,267004],{"className":267005,"style":35942},[31680],[34,267007,36732],{"className":267008},[31685],[34,267010],{"className":267011,"style":32604},[31845],[34,267013,42071],{"className":267014},[32024],[34,267016],{"className":267017,"style":32604},[31845],[34,267019,267021,267024,267027,267030,267033],{"className":267020},[31676],[34,267022],{"className":267023,"style":32617},[31680],[34,267025,3249],{"className":267026},[31685],[34,267028],{"className":267029,"style":31846},[31845],[34,267031,202],{"className":267032},[31850],[34,267034],{"className":267035,"style":31846},[31845],[34,267037,267039,267042,267045,267048],{"className":267038},[31676],[34,267040],{"className":267041,"style":32617},[31680],[34,267043,74303],{"className":267044},[31685],[34,267046,31759],{"className":267047},[31685,31686],[34,267049,31747],{"className":267050},[31685,31686],"。最后一条R10只需读取(2ms)和处理(4ms)，共6ms。总计 ",[34,267053,267055,267073],{"className":267054},[31642],[34,267056,267058],{"className":267057},[31646],[31648,267059,267060],{"xmlns":31650},[31652,267061,267062,267070],{},[31655,267063,267064,267066,267068],{},[31749,267065,162193],{},[31658,267067,31759],{},[31658,267069,31747],{},[31665,267071,267072],{"encoding":31667},"186ms",[34,267074,267076],{"className":267075,"ariaHidden":758},[31672],[34,267077,267079,267082,267085,267088],{"className":267078},[31676],[34,267080],{"className":267081,"style":32617},[31680],[34,267083,162193],{"className":267084},[31685],[34,267086,31759],{"className":267087},[31685,31686],[34,267089,31747],{"className":267090},[31685,31686],[15,267092,267093,267096,267097,267200,267201,267204],{},[2627,267094,267095],{},"(2) 最优化分布：","\n读完并处理完R1，一共耗时 ",[34,267098,267100,267134],{"className":267099},[31642],[34,267101,267103],{"className":267102},[31646],[31648,267104,267105],{"xmlns":31650},[31652,267106,267107,267131],{},[31655,267108,267109,267111,267113,267115,267117,267119,267121,267123,267125,267127,267129],{},[31749,267110,768],{},[31658,267112,31759],{},[31658,267114,31747],{},[31662,267116,223],{},[31749,267118,15932],{},[31658,267120,31759],{},[31658,267122,31747],{},[31662,267124,202],{},[31749,267126,36656],{},[31658,267128,31759],{},[31658,267130,31747],{},[31665,267132,267133],{"encoding":31667},"2ms + 4ms = 6ms",[34,267135,267137,267161,267185],{"className":267136,"ariaHidden":758},[31672],[34,267138,267140,267143,267146,267149,267152,267155,267158],{"className":267139},[31676],[34,267141],{"className":267142,"style":35942},[31680],[34,267144,768],{"className":267145},[31685],[34,267147,31759],{"className":267148},[31685,31686],[34,267150,31747],{"className":267151},[31685,31686],[34,267153],{"className":267154,"style":32604},[31845],[34,267156,223],{"className":267157},[32024],[34,267159],{"className":267160,"style":32604},[31845],[34,267162,267164,267167,267170,267173,267176,267179,267182],{"className":267163},[31676],[34,267165],{"className":267166,"style":32617},[31680],[34,267168,15932],{"className":267169},[31685],[34,267171,31759],{"className":267172},[31685,31686],[34,267174,31747],{"className":267175},[31685,31686],[34,267177],{"className":267178,"style":31846},[31845],[34,267180,202],{"className":267181},[31850],[34,267183],{"className":267184,"style":31846},[31845],[34,267186,267188,267191,267194,267197],{"className":267187},[31676],[34,267189],{"className":267190,"style":32617},[31680],[34,267192,36656],{"className":267193},[31685],[34,267195,31759],{"className":267196},[31685,31686],[34,267198,31747],{"className":267199},[31685,31686],"。此时磁头刚好完美对准了",[2627,267202,267203],{},"第3个扇区","（偏移量为3，即扇区号3，如果起始为0）的起点！\n因此，不需要任何延迟，我们直接把 R2 放在扇区3即可。物理间隔为 3。",[253,267206,267207],{"id":267207},"答案与详细解答",[15,267209,267210],{},[2627,267211,267212],{},"(1) 顺序存放的时间：",[105,267214,267215,267311,267416],{},[108,267216,267217,267218,347],{},"前9条记录，每条触发一次完整的20ms旋转周期：",[34,267219,267221,267251],{"className":267220},[31642],[34,267222,267224],{"className":267223},[31646],[31648,267225,267226],{"xmlns":31650},[31652,267227,267228,267248],{},[31655,267229,267230,267232,267234,267236,267238,267240,267242,267244,267246],{},[31749,267231,36732],{},[31662,267233,42071],{},[31749,267235,3249],{},[31658,267237,31759],{},[31658,267239,31747],{},[31662,267241,202],{},[31749,267243,74303],{},[31658,267245,31759],{},[31658,267247,31747],{},[31665,267249,267250],{"encoding":31667},"9 \\times 20ms = 180ms",[34,267252,267254,267272,267296],{"className":267253,"ariaHidden":758},[31672],[34,267255,267257,267260,267263,267266,267269],{"className":267256},[31676],[34,267258],{"className":267259,"style":35942},[31680],[34,267261,36732],{"className":267262},[31685],[34,267264],{"className":267265,"style":32604},[31845],[34,267267,42071],{"className":267268},[32024],[34,267270],{"className":267271,"style":32604},[31845],[34,267273,267275,267278,267281,267284,267287,267290,267293],{"className":267274},[31676],[34,267276],{"className":267277,"style":32617},[31680],[34,267279,3249],{"className":267280},[31685],[34,267282,31759],{"className":267283},[31685,31686],[34,267285,31747],{"className":267286},[31685,31686],[34,267288],{"className":267289,"style":31846},[31845],[34,267291,202],{"className":267292},[31850],[34,267294],{"className":267295,"style":31846},[31845],[34,267297,267299,267302,267305,267308],{"className":267298},[31676],[34,267300],{"className":267301,"style":32617},[31680],[34,267303,74303],{"className":267304},[31685],[34,267306,31759],{"className":267307},[31685,31686],[34,267309,31747],{"className":267310},[31685,31686],[108,267312,267313,267314,347],{},"第10条记录读取并处理：",[34,267315,267317,267350],{"className":267316},[31642],[34,267318,267320],{"className":267319},[31646],[31648,267321,267322],{"xmlns":31650},[31652,267323,267324,267348],{},[31655,267325,267326,267328,267330,267332,267334,267336,267338,267340,267342,267344,267346],{},[31749,267327,768],{},[31658,267329,31759],{},[31658,267331,31747],{},[31662,267333,223],{},[31749,267335,15932],{},[31658,267337,31759],{},[31658,267339,31747],{},[31662,267341,202],{},[31749,267343,36656],{},[31658,267345,31759],{},[31658,267347,31747],{},[31665,267349,267133],{"encoding":31667},[34,267351,267353,267377,267401],{"className":267352,"ariaHidden":758},[31672],[34,267354,267356,267359,267362,267365,267368,267371,267374],{"className":267355},[31676],[34,267357],{"className":267358,"style":35942},[31680],[34,267360,768],{"className":267361},[31685],[34,267363,31759],{"className":267364},[31685,31686],[34,267366,31747],{"className":267367},[31685,31686],[34,267369],{"className":267370,"style":32604},[31845],[34,267372,223],{"className":267373},[32024],[34,267375],{"className":267376,"style":32604},[31845],[34,267378,267380,267383,267386,267389,267392,267395,267398],{"className":267379},[31676],[34,267381],{"className":267382,"style":32617},[31680],[34,267384,15932],{"className":267385},[31685],[34,267387,31759],{"className":267388},[31685,31686],[34,267390,31747],{"className":267391},[31685,31686],[34,267393],{"className":267394,"style":31846},[31845],[34,267396,202],{"className":267397},[31850],[34,267399],{"className":267400,"style":31846},[31845],[34,267402,267404,267407,267410,267413],{"className":267403},[31676],[34,267405],{"className":267406,"style":32617},[31680],[34,267408,36656],{"className":267409},[31685],[34,267411,31759],{"className":267412},[31685,31686],[34,267414,31747],{"className":267415},[31685,31686],[108,267417,267418,347],{},[2627,267419,267420],{},"总时间 = 186 ms",[15,267422,267423],{},[2627,267424,267095],{},[105,267426,267427,267438],{},[108,267428,267429,47649,267432,39984,267435],{},[2627,267430,267431],{},"存放顺序（按物理扇区 0~9 列出）：",[31,267433,267434],{},"1, 8, 5, 2, 9, 6, 3, 10, 7, 4",[71807,267436,267437],{},"(推理：R1在0，R2在3，R3在6，R4在9，R5在2，R6在5，R7在8，R8在1，R9在4，R10在7)",[108,267439,267440,267442,267443,267578],{},[2627,267441,265004],{},"\n因为布局完美无缝衔接，没有任何旋转延迟。\n10条记录的 (读取 + 处理) = ",[34,267444,267446,267490],{"className":267445},[31642],[34,267447,267449],{"className":267448},[31646],[31648,267450,267451],{"xmlns":31650},[31652,267452,267453,267487],{},[31655,267454,267455,267457,267459,267461,267463,267465,267467,267469,267471,267473,267475,267477,267479],{},[31749,267456,3244],{},[31662,267458,42071],{},[31662,267460,456],{"stretchy":702},[31749,267462,768],{},[31658,267464,31759],{},[31658,267466,31747],{},[31662,267468,223],{},[31749,267470,15932],{},[31658,267472,31759],{},[31658,267474,31747],{},[31662,267476,2557],{"stretchy":702},[31662,267478,202],{},[31655,267480,267481,267483,267485],{},[31749,267482,167664],{"mathvariant":226163},[31658,267484,31759],{"mathvariant":226163},[31658,267486,31747],{"mathvariant":226163},[31665,267488,267489],{"encoding":31667},"10 \\times (2ms + 4ms) = \\mathbf{60 ms}",[34,267491,267493,267511,267538,267565],{"className":267492,"ariaHidden":758},[31672],[34,267494,267496,267499,267502,267505,267508],{"className":267495},[31676],[34,267497],{"className":267498,"style":35942},[31680],[34,267500,3244],{"className":267501},[31685],[34,267503],{"className":267504,"style":32604},[31845],[34,267506,42071],{"className":267507},[32024],[34,267509],{"className":267510,"style":32604},[31845],[34,267512,267514,267517,267520,267523,267526,267529,267532,267535],{"className":267513},[31676],[34,267515],{"className":267516,"style":31774},[31680],[34,267518,456],{"className":267519},[31782],[34,267521,768],{"className":267522},[31685],[34,267524,31759],{"className":267525},[31685,31686],[34,267527,31747],{"className":267528},[31685,31686],[34,267530],{"className":267531,"style":32604},[31845],[34,267533,223],{"className":267534},[32024],[34,267536],{"className":267537,"style":32604},[31845],[34,267539,267541,267544,267547,267550,267553,267556,267559,267562],{"className":267540},[31676],[34,267542],{"className":267543,"style":31774},[31680],[34,267545,15932],{"className":267546},[31685],[34,267548,31759],{"className":267549},[31685,31686],[34,267551,31747],{"className":267552},[31685,31686],[34,267554,2557],{"className":267555},[31690],[34,267557],{"className":267558,"style":31846},[31845],[34,267560,202],{"className":267561},[31850],[34,267563],{"className":267564,"style":31846},[31845],[34,267566,267568,267571],{"className":267567},[31676],[34,267569],{"className":267570,"style":32617},[31680],[34,267572,267574],{"className":267573},[31685],[34,267575,267577],{"className":267576},[31685,226302],"60ms","。\n:::",[19,267580,267582],{"id":267581},"_16-页面置换fifooptlru缺页中断率","16. 页面置换（FIFO\u002FOPT\u002FLRU）缺页中断率",[15,267584,267585],{},"一个页式存储管理系统使用 FIFO、OPT 和 LRU 页面替换算法，如果作业的页面走向为：2、3、2、1、5、2、4、5、3、2、5、2。分配给作业的物理块数分别为 3。假设初始页面为空，给出页面置换过程，并计算访问过程中发生的缺页中断次数和缺页中断率。",[15,267587,264303],{},[253,267589,214366],{"id":218884},[105,267591,267592,267598,267604],{},[108,267593,267594,267597],{},[2627,267595,267596],{},"FIFO (先进先出):"," 淘汰驻留内存时间最长的页。",[108,267599,267600,267603],{},[2627,267601,267602],{},"OPT (最佳置换):"," 淘汰未来最长时间内不再被访问的页（神仙算法，往后看）。",[108,267605,267606,267609],{},[2627,267607,267608],{},"LRU (最近最久未使用):"," 淘汰历史中最长时间没有被访问过的页（往前看）。",[253,267611,267613],{"id":267612},"答案与详细解答-过程模拟表","答案与详细解答 (过程模拟表)",[15,267615,267616],{},"序列： 2, 3, 2, 1, 5, 2, 4, 5, 3, 2, 5, 2 （总长 12）",[15,267618,267619],{},[2627,267620,267621],{},"(1) FIFO 算法",[13837,267623,267624,267655],{},[13840,267625,267626],{},[13843,267627,267628,267631,267633,267635,267637,267639,267641,267643,267645,267647,267649,267651,267653],{},[13846,267629,267630],{"align":82355},"页面走向",[13846,267632,768],{"align":82355},[13846,267634,786],{"align":82355},[13846,267636,768],{"align":82355},[13846,267638,640],{"align":82355},[13846,267640,18136],{"align":82355},[13846,267642,768],{"align":82355},[13846,267644,15932],{"align":82355},[13846,267646,18136],{"align":82355},[13846,267648,786],{"align":82355},[13846,267650,768],{"align":82355},[13846,267652,18136],{"align":82355},[13846,267654,768],{"align":82355},[13853,267656,267657,267692,267727,267762],{},[13843,267658,267659,267662,267666,267668,267670,267672,267676,267678,267680,267682,267686,267688,267690],{},[13858,267660,267661],{"align":82355},"物理块1",[13858,267663,267664],{"align":82355},[2627,267665,768],{},[13858,267667,768],{"align":82355},[13858,267669,768],{"align":82355},[13858,267671,768],{"align":82355},[13858,267673,267674],{"align":82355},[2627,267675,18136],{},[13858,267677,18136],{"align":82355},[13858,267679,18136],{"align":82355},[13858,267681,18136],{"align":82355},[13858,267683,267684],{"align":82355},[2627,267685,786],{},[13858,267687,786],{"align":82355},[13858,267689,786],{"align":82355},[13858,267691,786],{"align":82355},[13843,267693,267694,267697,267699,267703,267705,267707,267709,267713,267715,267717,267719,267723,267725],{},[13858,267695,267696],{"align":82355},"物理块2",[13858,267698],{"align":82355},[13858,267700,267701],{"align":82355},[2627,267702,786],{},[13858,267704,786],{"align":82355},[13858,267706,786],{"align":82355},[13858,267708,786],{"align":82355},[13858,267710,267711],{"align":82355},[2627,267712,768],{},[13858,267714,768],{"align":82355},[13858,267716,768],{"align":82355},[13858,267718,768],{"align":82355},[13858,267720,267721],{"align":82355},[2627,267722,18136],{},[13858,267724,18136],{"align":82355},[13858,267726,18136],{"align":82355},[13843,267728,267729,267732,267734,267736,267738,267742,267744,267746,267750,267752,267754,267756,267760],{},[13858,267730,267731],{"align":82355},"物理块3",[13858,267733],{"align":82355},[13858,267735],{"align":82355},[13858,267737],{"align":82355},[13858,267739,267740],{"align":82355},[2627,267741,640],{},[13858,267743,640],{"align":82355},[13858,267745,640],{"align":82355},[13858,267747,267748],{"align":82355},[2627,267749,15932],{},[13858,267751,15932],{"align":82355},[13858,267753,15932],{"align":82355},[13858,267755,15932],{"align":82355},[13858,267757,267758],{"align":82355},[2627,267759,768],{},[13858,267761,768],{"align":82355},[13843,267763,267764,267767,267769,267771,267773,267775,267777,267779,267781,267783,267785,267787,267789],{},[13858,267765,267766],{"align":82355},"是否缺页",[13858,267768,51834],{"align":82355},[13858,267770,51834],{"align":82355},[13858,267772],{"align":82355},[13858,267774,51834],{"align":82355},[13858,267776,51834],{"align":82355},[13858,267778,51834],{"align":82355},[13858,267780,51834],{"align":82355},[13858,267782],{"align":82355},[13858,267784,51834],{"align":82355},[13858,267786,51834],{"align":82355},[13858,267788],{"align":82355},[13858,267790,51834],{"align":82355},[105,267792,267793,267799],{},[108,267794,267795,267798],{},[2627,267796,267797],{},"缺页次数："," 9次",[108,267800,267801,47649,267804],{},[2627,267802,267803],{},"缺页中断率：",[34,267805,267807,267834],{"className":267806},[31642],[34,267808,267810],{"className":267809},[31646],[31648,267811,267812],{"xmlns":31650},[31652,267813,267814,267831],{},[31655,267815,267816,267818,267820,267822,267824],{},[31749,267817,36732],{},[31658,267819,38080],{"mathvariant":38079},[31749,267821,71869],{},[31662,267823,202],{},[31655,267825,267826,267829],{},[31749,267827,267828],{"mathvariant":226163},"75",[31658,267830,119426],{"mathvariant":226163},[31665,267832,267833],{"encoding":31667},"9 \u002F 12 = \\mathbf{75\\%}",[34,267835,267837,267856],{"className":267836,"ariaHidden":758},[31672],[34,267838,267840,267843,267847,267850,267853],{"className":267839},[31676],[34,267841],{"className":267842,"style":31774},[31680],[34,267844,267846],{"className":267845},[31685],"9\u002F12",[34,267848],{"className":267849,"style":31846},[31845],[34,267851,202],{"className":267852},[31850],[34,267854],{"className":267855,"style":31846},[31845],[34,267857,267859,267862],{"className":267858},[31676],[34,267860],{"className":267861,"style":168117},[31680],[34,267863,267865],{"className":267864},[31685],[34,267866,267868],{"className":267867},[31685,226302],"75%",[15,267870,267871],{},[2627,267872,267873],{},"(2) OPT 算法 (向右看，淘汰最晚出现的)",[15,267875,267876],{},"修正后的OPT过程：",[105,267878,267879,267882,267889,267892,267899,267902,267909,267912,267917],{},[108,267880,267881],{},"进1: 内存有2,3,1。",[108,267883,267884,267885,267888],{},"进5: 往后看未来是 ",[31,267886,267887],{},"2,4,5,3...","，1再也不出现，淘汰1。块变为2,3,5。",[108,267890,267891],{},"进2: 命中。",[108,267893,267894,267895,267898],{},"进4: 往后看 ",[31,267896,267897],{},"5,3,2,5,2","，3和2和5都会用，3最晚出现，淘汰3。块变为2,5,4。",[108,267900,267901],{},"进5: 命中。",[108,267903,267904,267905,267908],{},"进3: 往后看 ",[31,267906,267907],{},"2,5,2","，4再也不用，淘汰4。块变为2,5,3。",[108,267910,267911],{},"后续全部命中。",[108,267913,267914,267916],{},[2627,267915,267797],{}," 6次",[108,267918,267919,47649,267921],{},[2627,267920,267803],{},[34,267922,267924,267950],{"className":267923},[31642],[34,267925,267927],{"className":267926},[31646],[31648,267928,267929],{"xmlns":31650},[31652,267930,267931,267947],{},[31655,267932,267933,267935,267937,267939,267941],{},[31749,267934,36656],{},[31658,267936,38080],{"mathvariant":38079},[31749,267938,71869],{},[31662,267940,202],{},[31655,267942,267943,267945],{},[31749,267944,5774],{"mathvariant":226163},[31658,267946,119426],{"mathvariant":226163},[31665,267948,267949],{"encoding":31667},"6 \u002F 12 = \\mathbf{50\\%}",[34,267951,267953,267972],{"className":267952,"ariaHidden":758},[31672],[34,267954,267956,267959,267963,267966,267969],{"className":267955},[31676],[34,267957],{"className":267958,"style":31774},[31680],[34,267960,267962],{"className":267961},[31685],"6\u002F12",[34,267964],{"className":267965,"style":31846},[31845],[34,267967,202],{"className":267968},[31850],[34,267970],{"className":267971,"style":31846},[31845],[34,267973,267975,267978],{"className":267974},[31676],[34,267976],{"className":267977,"style":168117},[31680],[34,267979,267981],{"className":267980},[31685],[34,267982,267984],{"className":267983},[31685,226302],"50%",[15,267986,267987],{},[2627,267988,267989],{},"(3) LRU 算法 (向左看，淘汰最久未用的)",[13837,267991,267992,268022],{},[13840,267993,267994],{},[13843,267995,267996,267998,268000,268002,268004,268006,268008,268010,268012,268014,268016,268018,268020],{},[13846,267997,267630],{"align":82355},[13846,267999,768],{"align":82355},[13846,268001,786],{"align":82355},[13846,268003,768],{"align":82355},[13846,268005,640],{"align":82355},[13846,268007,18136],{"align":82355},[13846,268009,768],{"align":82355},[13846,268011,15932],{"align":82355},[13846,268013,18136],{"align":82355},[13846,268015,786],{"align":82355},[13846,268017,768],{"align":82355},[13846,268019,18136],{"align":82355},[13846,268021,768],{"align":82355},[13853,268023,268024,268056,268088,268122],{},[13843,268025,268026,268028,268032,268034,268036,268038,268040,268042,268044,268046,268050,268052,268054],{},[13858,268027,267661],{"align":82355},[13858,268029,268030],{"align":82355},[2627,268031,768],{},[13858,268033,768],{"align":82355},[13858,268035,768],{"align":82355},[13858,268037,768],{"align":82355},[13858,268039,768],{"align":82355},[13858,268041,768],{"align":82355},[13858,268043,768],{"align":82355},[13858,268045,768],{"align":82355},[13858,268047,268048],{"align":82355},[2627,268049,786],{},[13858,268051,786],{"align":82355},[13858,268053,786],{"align":82355},[13858,268055,786],{"align":82355},[13843,268057,268058,268060,268062,268066,268068,268070,268074,268076,268078,268080,268082,268084,268086],{},[13858,268059,267696],{"align":82355},[13858,268061],{"align":82355},[13858,268063,268064],{"align":82355},[2627,268065,786],{},[13858,268067,786],{"align":82355},[13858,268069,786],{"align":82355},[13858,268071,268072],{"align":82355},[2627,268073,18136],{},[13858,268075,18136],{"align":82355},[13858,268077,18136],{"align":82355},[13858,268079,18136],{"align":82355},[13858,268081,18136],{"align":82355},[13858,268083,18136],{"align":82355},[13858,268085,18136],{"align":82355},[13858,268087,18136],{"align":82355},[13843,268089,268090,268092,268094,268096,268098,268102,268104,268106,268110,268112,268114,268118,268120],{},[13858,268091,267731],{"align":82355},[13858,268093],{"align":82355},[13858,268095],{"align":82355},[13858,268097],{"align":82355},[13858,268099,268100],{"align":82355},[2627,268101,640],{},[13858,268103,640],{"align":82355},[13858,268105,640],{"align":82355},[13858,268107,268108],{"align":82355},[2627,268109,15932],{},[13858,268111,15932],{"align":82355},[13858,268113,15932],{"align":82355},[13858,268115,268116],{"align":82355},[2627,268117,768],{},[13858,268119,768],{"align":82355},[13858,268121,768],{"align":82355},[13843,268123,268124,268126,268128,268130,268132,268134,268136,268138,268140,268142,268144,268146,268148],{},[13858,268125,267766],{"align":82355},[13858,268127,51834],{"align":82355},[13858,268129,51834],{"align":82355},[13858,268131],{"align":82355},[13858,268133,51834],{"align":82355},[13858,268135,51834],{"align":82355},[13858,268137],{"align":82355},[13858,268139,51834],{"align":82355},[13858,268141],{"align":82355},[13858,268143,51834],{"align":82355},[13858,268145,51834],{"align":82355},[13858,268147],{"align":82355},[13858,268149],{"align":82355},[105,268151,268152,268157],{},[108,268153,268154,268156],{},[2627,268155,267797],{}," 7次",[108,268158,268159,47649,268161,263645],{},[2627,268160,267803],{},[34,268162,268164,268191],{"className":268163},[31642],[34,268165,268167],{"className":268166},[31646],[31648,268168,268169],{"xmlns":31650},[31652,268170,268171,268188],{},[31655,268172,268173,268175,268177,268179,268181],{},[31749,268174,36723],{},[31658,268176,38080],{"mathvariant":38079},[31749,268178,71869],{},[31662,268180,168560],{},[31655,268182,268183,268186],{},[31749,268184,268185],{"mathvariant":226163},"58.3",[31658,268187,119426],{"mathvariant":226163},[31665,268189,268190],{"encoding":31667},"7 \u002F 12 \\approx \\mathbf{58.3\\%}",[34,268192,268194,268213],{"className":268193,"ariaHidden":758},[31672],[34,268195,268197,268200,268204,268207,268210],{"className":268196},[31676],[34,268198],{"className":268199,"style":31774},[31680],[34,268201,268203],{"className":268202},[31685],"7\u002F12",[34,268205],{"className":268206,"style":31846},[31845],[34,268208,168560],{"className":268209},[31850],[34,268211],{"className":268212,"style":31846},[31845],[34,268214,268216,268219],{"className":268215},[31676],[34,268217],{"className":268218,"style":168117},[31680],[34,268220,268222],{"className":268221},[31685],[34,268223,268225],{"className":268224},[31685,226302],"58.3%",[19,268227,268229],{"id":268228},"_17-多级存储器有效访问时间eat","17. 多级存储器有效访问时间（EAT）",[15,268231,268232],{},"某计算机有缓存、内存、辅存来实现虚拟存储器。如果数据在缓存中，访问它需要 Ans；如果在内存但不在缓存，需要 Bns 将其装入缓存，然后才能访问；如果不在内存而在辅存，需要 Cns 将其读入内存，然后，用 Bns 再读入缓存，然后才能访问。假设缓存命中率为 (n-1)\u002Fn，内存命中率为 (m-1)\u002Fm。给出数据在缓存、内存、以及辅存中比率，并计算数据平均访问时间。",[15,268234,264303],{},[253,268236,219117],{"id":268237},"原理解析-the-why-2",[15,268239,268240],{},"这是经典的多级存储器有效访问时间（EAT）计算。",[105,268242,268243,268475,268811],{},[108,268244,268245,47649,268248,268446,268447,347],{},[2627,268246,268247],{},"命中缓存的概率：",[34,268249,268251,268293],{"className":268250},[31642],[34,268252,268254],{"className":268253},[31646],[31648,268255,268256],{"xmlns":31650},[31652,268257,268258,268290],{},[31655,268259,268260,268276,268278],{},[31743,268261,268262,268264],{},[31658,268263,105852],{},[31655,268265,268266,268268,268270,268272,268274],{},[31658,268267,31965],{},[31658,268269,111],{},[31658,268271,31965],{},[31658,268273,47292],{},[31658,268275,33244],{},[31662,268277,202],{},[41453,268279,268280,268288],{},[31655,268281,268282,268284,268286],{},[31658,268283,31660],{},[31662,268285,32226],{},[31749,268287,640],{},[31658,268289,31660],{},[31665,268291,268292],{"encoding":31667},"P_{cache} = \\frac{n-1}{n}",[34,268294,268296,268366],{"className":268295,"ariaHidden":758},[31672],[34,268297,268299,268302,268357,268360,268363],{"className":268298},[31676],[34,268300],{"className":268301,"style":70915},[31680],[34,268303,268305,268308],{"className":268304},[31685],[34,268306,105852],{"className":268307,"style":37422},[31685,31686],[34,268309,268311],{"className":268310},[31792],[34,268312,268314,268349],{"className":268313},[31796,31797],[34,268315,268317,268346],{"className":268316},[31801],[34,268318,268320],{"className":268319,"style":33833},[31805],[34,268321,268322,268325],{"style":214424},[34,268323],{"className":268324,"style":31814},[31813],[34,268326,268328],{"className":268327},[31818,31819,31820,31821],[34,268329,268331,268334,268337,268340,268343],{"className":268330},[31685,31821],[34,268332,31965],{"className":268333},[31685,31686,31821],[34,268335,111],{"className":268336},[31685,31686,31821],[34,268338,31965],{"className":268339},[31685,31686,31821],[34,268341,47292],{"className":268342},[31685,31686,31821],[34,268344,33244],{"className":268345},[31685,31686,31821],[34,268347,31829],{"className":268348},[31828],[34,268350,268352],{"className":268351},[31801],[34,268353,268355],{"className":268354,"style":31836},[31805],[34,268356],{},[34,268358],{"className":268359,"style":31846},[31845],[34,268361,202],{"className":268362},[31850],[34,268364],{"className":268365,"style":31846},[31845],[34,268367,268369,268372],{"className":268368},[31676],[34,268370],{"className":268371,"style":46727},[31680],[34,268373,268375,268378,268443],{"className":268374},[31685],[34,268376],{"className":268377},[31782,41488],[34,268379,268381],{"className":268380},[41453],[34,268382,268384,268435],{"className":268383},[31796,31797],[34,268385,268387,268432],{"className":268386},[31801],[34,268388,268390,268404,268412],{"className":268389,"style":46746},[31805],[34,268391,268392,268395],{"style":41504},[34,268393],{"className":268394,"style":41508},[31813],[34,268396,268398],{"className":268397},[31818,31819,31820,31821],[34,268399,268401],{"className":268400},[31685,31821],[34,268402,31660],{"className":268403},[31685,31686,31821],[34,268405,268406,268409],{"style":41520},[34,268407],{"className":268408,"style":41508},[31813],[34,268410],{"className":268411,"style":41528},[41527],[34,268413,268414,268417],{"style":46771},[34,268415],{"className":268416,"style":41508},[31813],[34,268418,268420],{"className":268419},[31818,31819,31820,31821],[34,268421,268423,268426,268429],{"className":268422},[31685,31821],[34,268424,31660],{"className":268425},[31685,31686,31821],[34,268427,32226],{"className":268428},[32024,31821],[34,268430,640],{"className":268431},[31685,31821],[34,268433,31829],{"className":268434},[31828],[34,268436,268438],{"className":268437},[31801],[34,268439,268441],{"className":268440,"style":41619},[31805],[34,268442],{},[34,268444],{"className":268445},[31690,41488],"。耗时：只需 ",[34,268448,268450,268463],{"className":268449},[31642],[34,268451,268453],{"className":268452},[31646],[31648,268454,268455],{"xmlns":31650},[31652,268456,268457,268461],{},[31655,268458,268459],{},[31658,268460,54151],{},[31665,268462,54151],{"encoding":31667},[34,268464,268466],{"className":268465,"ariaHidden":758},[31672],[34,268467,268469,268472],{"className":268468},[31676],[34,268470],{"className":268471,"style":33218},[31680],[34,268473,54151],{"className":268474},[31685,31686],[108,268476,268477,47649,268480,268759,268760,347],{},[2627,268478,268479],{},"缓存未命中，但命中内存的概率：",[34,268481,268483,268529],{"className":268482},[31642],[34,268484,268486],{"className":268485},[31646],[31648,268487,268488],{"xmlns":31650},[31652,268489,268490,268526],{},[31655,268491,268492,268504,268506,268512,268514],{},[31743,268493,268494,268496],{},[31658,268495,105852],{},[31655,268497,268498,268500,268502],{},[31658,268499,31759],{},[31658,268501,33244],{},[31658,268503,31759],{},[31662,268505,202],{},[41453,268507,268508,268510],{},[31749,268509,640],{},[31658,268511,31660],{},[31662,268513,42071],{},[41453,268515,268516,268524],{},[31655,268517,268518,268520,268522],{},[31658,268519,31759],{},[31662,268521,32226],{},[31749,268523,640],{},[31658,268525,31759],{},[31665,268527,268528],{"encoding":31667},"P_{mem} = \\frac{1}{n} \\times \\frac{m-1}{m}",[34,268530,268532,268596,268679],{"className":268531,"ariaHidden":758},[31672],[34,268533,268535,268538,268587,268590,268593],{"className":268534},[31676],[34,268536],{"className":268537,"style":70915},[31680],[34,268539,268541,268544],{"className":268540},[31685],[34,268542,105852],{"className":268543,"style":37422},[31685,31686],[34,268545,268547],{"className":268546},[31792],[34,268548,268550,268579],{"className":268549},[31796,31797],[34,268551,268553,268576],{"className":268552},[31801],[34,268554,268556],{"className":268555,"style":32362},[31805],[34,268557,268558,268561],{"style":214424},[34,268559],{"className":268560,"style":31814},[31813],[34,268562,268564],{"className":268563},[31818,31819,31820,31821],[34,268565,268567,268570,268573],{"className":268566},[31685,31821],[34,268568,31759],{"className":268569},[31685,31686,31821],[34,268571,33244],{"className":268572},[31685,31686,31821],[34,268574,31759],{"className":268575},[31685,31686,31821],[34,268577,31829],{"className":268578},[31828],[34,268580,268582],{"className":268581},[31801],[34,268583,268585],{"className":268584,"style":31836},[31805],[34,268586],{},[34,268588],{"className":268589,"style":31846},[31845],[34,268591,202],{"className":268592},[31850],[34,268594],{"className":268595,"style":31846},[31845],[34,268597,268599,268602,268670,268673,268676],{"className":268598},[31676],[34,268600],{"className":268601,"style":46727},[31680],[34,268603,268605,268608,268667],{"className":268604},[31685],[34,268606],{"className":268607},[31782,41488],[34,268609,268611],{"className":268610},[41453],[34,268612,268614,268659],{"className":268613},[31796,31797],[34,268615,268617,268656],{"className":268616},[31801],[34,268618,268620,268634,268642],{"className":268619,"style":46746},[31805],[34,268621,268622,268625],{"style":41504},[34,268623],{"className":268624,"style":41508},[31813],[34,268626,268628],{"className":268627},[31818,31819,31820,31821],[34,268629,268631],{"className":268630},[31685,31821],[34,268632,31660],{"className":268633},[31685,31686,31821],[34,268635,268636,268639],{"style":41520},[34,268637],{"className":268638,"style":41508},[31813],[34,268640],{"className":268641,"style":41528},[41527],[34,268643,268644,268647],{"style":46771},[34,268645],{"className":268646,"style":41508},[31813],[34,268648,268650],{"className":268649},[31818,31819,31820,31821],[34,268651,268653],{"className":268652},[31685,31821],[34,268654,640],{"className":268655},[31685,31821],[34,268657,31829],{"className":268658},[31828],[34,268660,268662],{"className":268661},[31801],[34,268663,268665],{"className":268664,"style":41619},[31805],[34,268666],{},[34,268668],{"className":268669},[31690,41488],[34,268671],{"className":268672,"style":32604},[31845],[34,268674,42071],{"className":268675},[32024],[34,268677],{"className":268678,"style":32604},[31845],[34,268680,268682,268685],{"className":268681},[31676],[34,268683],{"className":268684,"style":46727},[31680],[34,268686,268688,268691,268756],{"className":268687},[31685],[34,268689],{"className":268690},[31782,41488],[34,268692,268694],{"className":268693},[41453],[34,268695,268697,268748],{"className":268696},[31796,31797],[34,268698,268700,268745],{"className":268699},[31801],[34,268701,268703,268717,268725],{"className":268702,"style":46746},[31805],[34,268704,268705,268708],{"style":41504},[34,268706],{"className":268707,"style":41508},[31813],[34,268709,268711],{"className":268710},[31818,31819,31820,31821],[34,268712,268714],{"className":268713},[31685,31821],[34,268715,31759],{"className":268716},[31685,31686,31821],[34,268718,268719,268722],{"style":41520},[34,268720],{"className":268721,"style":41508},[31813],[34,268723],{"className":268724,"style":41528},[41527],[34,268726,268727,268730],{"style":46771},[34,268728],{"className":268729,"style":41508},[31813],[34,268731,268733],{"className":268732},[31818,31819,31820,31821],[34,268734,268736,268739,268742],{"className":268735},[31685,31821],[34,268737,31759],{"className":268738},[31685,31686,31821],[34,268740,32226],{"className":268741},[32024,31821],[34,268743,640],{"className":268744},[31685,31821],[34,268746,31829],{"className":268747},[31828],[34,268749,268751],{"className":268750},[31801],[34,268752,268754],{"className":268753,"style":41619},[31805],[34,268755],{},[34,268757],{"className":268758},[31690,41488],"。耗时：",[34,268761,268763,268781],{"className":268762},[31642],[34,268764,268766],{"className":268765},[31646],[31648,268767,268768],{"xmlns":31650},[31652,268769,268770,268778],{},[31655,268771,268772,268774,268776],{},[31658,268773,54151],{},[31662,268775,223],{},[31658,268777,54224],{},[31665,268779,268780],{"encoding":31667},"A+B",[34,268782,268784,268802],{"className":268783,"ariaHidden":758},[31672],[34,268785,268787,268790,268793,268796,268799],{"className":268786},[31676],[34,268788],{"className":268789,"style":34944},[31680],[34,268791,54151],{"className":268792},[31685,31686],[34,268794],{"className":268795,"style":32604},[31845],[34,268797,223],{"className":268798},[32024],[34,268800],{"className":268801,"style":32604},[31845],[34,268803,268805,268808],{"className":268804},[31676],[34,268806],{"className":268807,"style":33218},[31680],[34,268809,54224],{"className":268810,"style":54252},[31685,31686],[108,268812,268813,47649,268816,268759,269184,347],{},[2627,268814,268815],{},"全未命中，在辅存中的概率：",[34,268817,268819,268873],{"className":268818},[31642],[34,268820,268822],{"className":268821},[31646],[31648,268823,268824],{"xmlns":31650},[31652,268825,268826,268870],{},[31655,268827,268828,268842,268844,268850,268852,268858,268860],{},[31743,268829,268830,268832],{},[31658,268831,105852],{},[31655,268833,268834,268836,268838,268840],{},[31658,268835,46900],{},[31658,268837,31955],{},[31658,268839,31747],{},[31658,268841,33759],{},[31662,268843,202],{},[41453,268845,268846,268848],{},[31749,268847,640],{},[31658,268849,31660],{},[31662,268851,42071],{},[41453,268853,268854,268856],{},[31749,268855,640],{},[31658,268857,31759],{},[31662,268859,202],{},[41453,268861,268862,268864],{},[31749,268863,640],{},[31655,268865,268866,268868],{},[31658,268867,31660],{},[31658,268869,31759],{},[31665,268871,268872],{"encoding":31667},"P_{disk} = \\frac{1}{n} \\times \\frac{1}{m} = \\frac{1}{nm}",[34,268874,268876,268943,269026,269109],{"className":268875,"ariaHidden":758},[31672],[34,268877,268879,268882,268934,268937,268940],{"className":268878},[31676],[34,268880],{"className":268881,"style":70915},[31680],[34,268883,268885,268888],{"className":268884},[31685],[34,268886,105852],{"className":268887,"style":37422},[31685,31686],[34,268889,268891],{"className":268890},[31792],[34,268892,268894,268926],{"className":268893},[31796,31797],[34,268895,268897,268923],{"className":268896},[31801],[34,268898,268900],{"className":268899,"style":33833},[31805],[34,268901,268902,268905],{"style":214424},[34,268903],{"className":268904,"style":31814},[31813],[34,268906,268908],{"className":268907},[31818,31819,31820,31821],[34,268909,268911,268914,268917,268920],{"className":268910},[31685,31821],[34,268912,46900],{"className":268913},[31685,31686,31821],[34,268915,31955],{"className":268916},[31685,31686,31821],[34,268918,31747],{"className":268919},[31685,31686,31821],[34,268921,33759],{"className":268922,"style":33774},[31685,31686,31821],[34,268924,31829],{"className":268925},[31828],[34,268927,268929],{"className":268928},[31801],[34,268930,268932],{"className":268931,"style":31836},[31805],[34,268933],{},[34,268935],{"className":268936,"style":31846},[31845],[34,268938,202],{"className":268939},[31850],[34,268941],{"className":268942,"style":31846},[31845],[34,268944,268946,268949,269017,269020,269023],{"className":268945},[31676],[34,268947],{"className":268948,"style":46727},[31680],[34,268950,268952,268955,269014],{"className":268951},[31685],[34,268953],{"className":268954},[31782,41488],[34,268956,268958],{"className":268957},[41453],[34,268959,268961,269006],{"className":268960},[31796,31797],[34,268962,268964,269003],{"className":268963},[31801],[34,268965,268967,268981,268989],{"className":268966,"style":46746},[31805],[34,268968,268969,268972],{"style":41504},[34,268970],{"className":268971,"style":41508},[31813],[34,268973,268975],{"className":268974},[31818,31819,31820,31821],[34,268976,268978],{"className":268977},[31685,31821],[34,268979,31660],{"className":268980},[31685,31686,31821],[34,268982,268983,268986],{"style":41520},[34,268984],{"className":268985,"style":41508},[31813],[34,268987],{"className":268988,"style":41528},[41527],[34,268990,268991,268994],{"style":46771},[34,268992],{"className":268993,"style":41508},[31813],[34,268995,268997],{"className":268996},[31818,31819,31820,31821],[34,268998,269000],{"className":268999},[31685,31821],[34,269001,640],{"className":269002},[31685,31821],[34,269004,31829],{"className":269005},[31828],[34,269007,269009],{"className":269008},[31801],[34,269010,269012],{"className":269011,"style":41619},[31805],[34,269013],{},[34,269015],{"className":269016},[31690,41488],[34,269018],{"className":269019,"style":32604},[31845],[34,269021,42071],{"className":269022},[32024],[34,269024],{"className":269025,"style":32604},[31845],[34,269027,269029,269032,269100,269103,269106],{"className":269028},[31676],[34,269030],{"className":269031,"style":46727},[31680],[34,269033,269035,269038,269097],{"className":269034},[31685],[34,269036],{"className":269037},[31782,41488],[34,269039,269041],{"className":269040},[41453],[34,269042,269044,269089],{"className":269043},[31796,31797],[34,269045,269047,269086],{"className":269046},[31801],[34,269048,269050,269064,269072],{"className":269049,"style":46746},[31805],[34,269051,269052,269055],{"style":41504},[34,269053],{"className":269054,"style":41508},[31813],[34,269056,269058],{"className":269057},[31818,31819,31820,31821],[34,269059,269061],{"className":269060},[31685,31821],[34,269062,31759],{"className":269063},[31685,31686,31821],[34,269065,269066,269069],{"style":41520},[34,269067],{"className":269068,"style":41508},[31813],[34,269070],{"className":269071,"style":41528},[41527],[34,269073,269074,269077],{"style":46771},[34,269075],{"className":269076,"style":41508},[31813],[34,269078,269080],{"className":269079},[31818,31819,31820,31821],[34,269081,269083],{"className":269082},[31685,31821],[34,269084,640],{"className":269085},[31685,31821],[34,269087,31829],{"className":269088},[31828],[34,269090,269092],{"className":269091},[31801],[34,269093,269095],{"className":269094,"style":41619},[31805],[34,269096],{},[34,269098],{"className":269099},[31690,41488],[34,269101],{"className":269102,"style":31846},[31845],[34,269104,202],{"className":269105},[31850],[34,269107],{"className":269108,"style":31846},[31845],[34,269110,269112,269115],{"className":269111},[31676],[34,269113],{"className":269114,"style":46727},[31680],[34,269116,269118,269121,269181],{"className":269117},[31685],[34,269119],{"className":269120},[31782,41488],[34,269122,269124],{"className":269123},[41453],[34,269125,269127,269173],{"className":269126},[31796,31797],[34,269128,269130,269170],{"className":269129},[31801],[34,269131,269133,269148,269156],{"className":269132,"style":46746},[31805],[34,269134,269135,269138],{"style":41504},[34,269136],{"className":269137,"style":41508},[31813],[34,269139,269141],{"className":269140},[31818,31819,31820,31821],[34,269142,269144],{"className":269143},[31685,31821],[34,269145,269147],{"className":269146},[31685,31686,31821],"nm",[34,269149,269150,269153],{"style":41520},[34,269151],{"className":269152,"style":41508},[31813],[34,269154],{"className":269155,"style":41528},[41527],[34,269157,269158,269161],{"style":46771},[34,269159],{"className":269160,"style":41508},[31813],[34,269162,269164],{"className":269163},[31818,31819,31820,31821],[34,269165,269167],{"className":269166},[31685,31821],[34,269168,640],{"className":269169},[31685,31821],[34,269171,31829],{"className":269172},[31828],[34,269174,269176],{"className":269175},[31801],[34,269177,269179],{"className":269178,"style":41619},[31805],[34,269180],{},[34,269182],{"className":269183},[31690,41488],[34,269185,269187,269209],{"className":269186},[31642],[34,269188,269190],{"className":269189},[31646],[31648,269191,269192],{"xmlns":31650},[31652,269193,269194,269206],{},[31655,269195,269196,269198,269200,269202,269204],{},[31658,269197,54151],{},[31662,269199,223],{},[31658,269201,54224],{},[31662,269203,223],{},[31658,269205,47384],{},[31665,269207,269208],{"encoding":31667},"A+B+C",[34,269210,269212,269230,269248],{"className":269211,"ariaHidden":758},[31672],[34,269213,269215,269218,269221,269224,269227],{"className":269214},[31676],[34,269216],{"className":269217,"style":34944},[31680],[34,269219,54151],{"className":269220},[31685,31686],[34,269222],{"className":269223,"style":32604},[31845],[34,269225,223],{"className":269226},[32024],[34,269228],{"className":269229,"style":32604},[31845],[34,269231,269233,269236,269239,269242,269245],{"className":269232},[31676],[34,269234],{"className":269235,"style":34944},[31680],[34,269237,54224],{"className":269238,"style":54252},[31685,31686],[34,269240],{"className":269241,"style":32604},[31845],[34,269243,223],{"className":269244},[32024],[34,269246],{"className":269247,"style":32604},[31845],[34,269249,269251,269254],{"className":269250},[31676],[34,269252],{"className":269253,"style":33218},[31680],[34,269255,47384],{"className":269256,"style":38706},[31685,31686],[253,269258,267207],{"id":269259},"答案与详细解答-1",[15,269261,269262],{},[2627,269263,269264],{},"(1) 数据所在层级的比率：",[105,269266,269267,269380,269497],{},[108,269268,269269,269270],{},"数据在缓存中的比率： ",[34,269271,269273,269297],{"className":269272},[31642],[34,269274,269276],{"className":269275},[31646],[31648,269277,269278],{"xmlns":31650},[31652,269279,269280,269294],{},[31655,269281,269282],{},[41453,269283,269284,269292],{},[31655,269285,269286,269288,269290],{},[31658,269287,31660],{"mathvariant":226163},[31662,269289,32226],{},[31749,269291,640],{"mathvariant":226163},[31658,269293,31660],{"mathvariant":226163},[31665,269295,269296],{"encoding":31667},"\\mathbf{\\frac{n-1}{n}}",[34,269298,269300],{"className":269299,"ariaHidden":758},[31672],[34,269301,269303,269306],{"className":269302},[31676],[34,269304],{"className":269305,"style":46727},[31680],[34,269307,269309,269312,269377],{"className":269308},[31685],[34,269310],{"className":269311},[31782,41488],[34,269313,269315],{"className":269314},[41453],[34,269316,269318,269369],{"className":269317},[31796,31797],[34,269319,269321,269366],{"className":269320},[31801],[34,269322,269324,269338,269346],{"className":269323,"style":46746},[31805],[34,269325,269326,269329],{"style":41504},[34,269327],{"className":269328,"style":41508},[31813],[34,269330,269332],{"className":269331},[31818,31819,31820,31821],[34,269333,269335],{"className":269334},[31685,31821],[34,269336,31660],{"className":269337},[31685,226302,31821],[34,269339,269340,269343],{"style":41520},[34,269341],{"className":269342,"style":41508},[31813],[34,269344],{"className":269345,"style":41528},[41527],[34,269347,269348,269351],{"style":46771},[34,269349],{"className":269350,"style":41508},[31813],[34,269352,269354],{"className":269353},[31818,31819,31820,31821],[34,269355,269357,269360,269363],{"className":269356},[31685,31821],[34,269358,31660],{"className":269359},[31685,226302,31821],[34,269361,32226],{"className":269362},[32024,31821],[34,269364,640],{"className":269365},[31685,226302,31821],[34,269367,31829],{"className":269368},[31828],[34,269370,269372],{"className":269371},[31801],[34,269373,269375],{"className":269374,"style":41619},[31805],[34,269376],{},[34,269378],{"className":269379},[31690,41488],[108,269381,269382,269383],{},"数据在内存（但不在缓存）中的比率： ",[34,269384,269386,269414],{"className":269385},[31642],[34,269387,269389],{"className":269388},[31646],[31648,269390,269391],{"xmlns":31650},[31652,269392,269393,269411],{},[31655,269394,269395],{},[41453,269396,269397,269405],{},[31655,269398,269399,269401,269403],{},[31658,269400,31759],{"mathvariant":226163},[31662,269402,32226],{},[31749,269404,640],{"mathvariant":226163},[31655,269406,269407,269409],{},[31658,269408,31660],{"mathvariant":226163},[31658,269410,31759],{"mathvariant":226163},[31665,269412,269413],{"encoding":31667},"\\mathbf{\\frac{m-1}{nm}}",[34,269415,269417],{"className":269416,"ariaHidden":758},[31672],[34,269418,269420,269423],{"className":269419},[31676],[34,269421],{"className":269422,"style":46727},[31680],[34,269424,269426,269429,269494],{"className":269425},[31685],[34,269427],{"className":269428},[31782,41488],[34,269430,269432],{"className":269431},[41453],[34,269433,269435,269486],{"className":269434},[31796,31797],[34,269436,269438,269483],{"className":269437},[31801],[34,269439,269441,269455,269463],{"className":269440,"style":46746},[31805],[34,269442,269443,269446],{"style":41504},[34,269444],{"className":269445,"style":41508},[31813],[34,269447,269449],{"className":269448},[31818,31819,31820,31821],[34,269450,269452],{"className":269451},[31685,31821],[34,269453,269147],{"className":269454},[31685,226302,31821],[34,269456,269457,269460],{"style":41520},[34,269458],{"className":269459,"style":41508},[31813],[34,269461],{"className":269462,"style":41528},[41527],[34,269464,269465,269468],{"style":46771},[34,269466],{"className":269467,"style":41508},[31813],[34,269469,269471],{"className":269470},[31818,31819,31820,31821],[34,269472,269474,269477,269480],{"className":269473},[31685,31821],[34,269475,31759],{"className":269476},[31685,226302,31821],[34,269478,32226],{"className":269479},[32024,31821],[34,269481,640],{"className":269482},[31685,226302,31821],[34,269484,31829],{"className":269485},[31828],[34,269487,269489],{"className":269488},[31801],[34,269490,269492],{"className":269491,"style":41619},[31805],[34,269493],{},[34,269495],{"className":269496},[31690,41488],[108,269498,269499,269500],{},"数据在辅存（不在内存）中的比率： ",[34,269501,269503,269525],{"className":269502},[31642],[34,269504,269506],{"className":269505},[31646],[31648,269507,269508],{"xmlns":31650},[31652,269509,269510,269522],{},[31655,269511,269512],{},[41453,269513,269514,269516],{},[31749,269515,640],{"mathvariant":226163},[31655,269517,269518,269520],{},[31658,269519,31660],{"mathvariant":226163},[31658,269521,31759],{"mathvariant":226163},[31665,269523,269524],{"encoding":31667},"\\mathbf{\\frac{1}{nm}}",[34,269526,269528],{"className":269527,"ariaHidden":758},[31672],[34,269529,269531,269534],{"className":269530},[31676],[34,269532],{"className":269533,"style":46727},[31680],[34,269535,269537,269540,269599],{"className":269536},[31685],[34,269538],{"className":269539},[31782,41488],[34,269541,269543],{"className":269542},[41453],[34,269544,269546,269591],{"className":269545},[31796,31797],[34,269547,269549,269588],{"className":269548},[31801],[34,269550,269552,269566,269574],{"className":269551,"style":46746},[31805],[34,269553,269554,269557],{"style":41504},[34,269555],{"className":269556,"style":41508},[31813],[34,269558,269560],{"className":269559},[31818,31819,31820,31821],[34,269561,269563],{"className":269562},[31685,31821],[34,269564,269147],{"className":269565},[31685,226302,31821],[34,269567,269568,269571],{"style":41520},[34,269569],{"className":269570,"style":41508},[31813],[34,269572],{"className":269573,"style":41528},[41527],[34,269575,269576,269579],{"style":46771},[34,269577],{"className":269578,"style":41508},[31813],[34,269580,269582],{"className":269581},[31818,31819,31820,31821],[34,269583,269585],{"className":269584},[31685,31821],[34,269586,640],{"className":269587},[31685,226302,31821],[34,269589,31829],{"className":269590},[31828],[34,269592,269594],{"className":269593},[31801],[34,269595,269597],{"className":269596,"style":41619},[31805],[34,269598],{},[34,269600],{"className":269601},[31690,41488],[15,269603,269604],{},[2627,269605,269606],{},"(2) 平均访问时间 (EAT) 计算：",[15,269608,269609,263645],{},[34,269610,269612,269729],{"className":269611},[31642],[34,269613,269615],{"className":269614},[31646],[31648,269616,269617],{"xmlns":31650},[31652,269618,269619,269726],{},[31655,269620,269621,269623,269625,269627,269629,269641,269643,269645,269661,269663,269665,269667,269669,269671,269673,269683,269685,269687,269689,269691,269693,269695,269697,269699,269723],{},[31658,269622,96058],{},[31658,269624,54151],{},[31658,269626,37400],{},[31662,269628,202],{},[41453,269630,269631,269639],{},[31655,269632,269633,269635,269637],{},[31658,269634,31660],{},[31662,269636,32226],{},[31749,269638,640],{},[31658,269640,31660],{},[31658,269642,54151],{},[31662,269644,223],{},[41453,269646,269647,269655],{},[31655,269648,269649,269651,269653],{},[31658,269650,31759],{},[31662,269652,32226],{},[31749,269654,640],{},[31655,269656,269657,269659],{},[31658,269658,31660],{},[31658,269660,31759],{},[31662,269662,456],{"stretchy":702},[31658,269664,54151],{},[31662,269666,223],{},[31658,269668,54224],{},[31662,269670,2557],{"stretchy":702},[31662,269672,223],{},[41453,269674,269675,269677],{},[31749,269676,640],{},[31655,269678,269679,269681],{},[31658,269680,31660],{},[31658,269682,31759],{},[31662,269684,456],{"stretchy":702},[31658,269686,54151],{},[31662,269688,223],{},[31658,269690,54224],{},[31662,269692,223],{},[31658,269694,47384],{},[31662,269696,2557],{"stretchy":702},[31662,269698,202],{},[31655,269700,269701,269703,269705,269711,269713],{},[31658,269702,54151],{"mathvariant":226163},[31662,269704,223],{},[41453,269706,269707,269709],{},[31658,269708,54224],{"mathvariant":226163},[31658,269710,31660],{"mathvariant":226163},[31662,269712,223],{},[41453,269714,269715,269717],{},[31658,269716,47384],{"mathvariant":226163},[31655,269718,269719,269721],{},[31658,269720,31660],{"mathvariant":226163},[31658,269722,31759],{"mathvariant":226163},[39361,269724,269725],{}," (ns)",[31665,269727,269728],{"encoding":31667},"EAT = \\frac{n-1}{n}A + \\frac{m-1}{nm}(A+B) + \\frac{1}{nm}(A+B+C) = \\mathbf{A + \\frac{B}{n} + \\frac{C}{nm}} \\text{ (ns)}",[34,269730,269732,269756,269848,269943,269964,270053,270071,270092],{"className":269731,"ariaHidden":758},[31672],[34,269733,269735,269738,269741,269744,269747,269750,269753],{"className":269734},[31676],[34,269736],{"className":269737,"style":33218},[31680],[34,269739,96058],{"className":269740,"style":106455},[31685,31686],[34,269742,54151],{"className":269743},[31685,31686],[34,269745,37400],{"className":269746,"style":37422},[31685,31686],[34,269748],{"className":269749,"style":31846},[31845],[34,269751,202],{"className":269752},[31850],[34,269754],{"className":269755,"style":31846},[31845],[34,269757,269759,269762,269836,269839,269842,269845],{"className":269758},[31676],[34,269760],{"className":269761,"style":46727},[31680],[34,269763,269765,269768,269833],{"className":269764},[31685],[34,269766],{"className":269767},[31782,41488],[34,269769,269771],{"className":269770},[41453],[34,269772,269774,269825],{"className":269773},[31796,31797],[34,269775,269777,269822],{"className":269776},[31801],[34,269778,269780,269794,269802],{"className":269779,"style":46746},[31805],[34,269781,269782,269785],{"style":41504},[34,269783],{"className":269784,"style":41508},[31813],[34,269786,269788],{"className":269787},[31818,31819,31820,31821],[34,269789,269791],{"className":269790},[31685,31821],[34,269792,31660],{"className":269793},[31685,31686,31821],[34,269795,269796,269799],{"style":41520},[34,269797],{"className":269798,"style":41508},[31813],[34,269800],{"className":269801,"style":41528},[41527],[34,269803,269804,269807],{"style":46771},[34,269805],{"className":269806,"style":41508},[31813],[34,269808,269810],{"className":269809},[31818,31819,31820,31821],[34,269811,269813,269816,269819],{"className":269812},[31685,31821],[34,269814,31660],{"className":269815},[31685,31686,31821],[34,269817,32226],{"className":269818},[32024,31821],[34,269820,640],{"className":269821},[31685,31821],[34,269823,31829],{"className":269824},[31828],[34,269826,269828],{"className":269827},[31801],[34,269829,269831],{"className":269830,"style":41619},[31805],[34,269832],{},[34,269834],{"className":269835},[31690,41488],[34,269837,54151],{"className":269838},[31685,31686],[34,269840],{"className":269841,"style":32604},[31845],[34,269843,223],{"className":269844},[32024],[34,269846],{"className":269847,"style":32604},[31845],[34,269849,269851,269854,269928,269931,269934,269937,269940],{"className":269850},[31676],[34,269852],{"className":269853,"style":46727},[31680],[34,269855,269857,269860,269925],{"className":269856},[31685],[34,269858],{"className":269859},[31782,41488],[34,269861,269863],{"className":269862},[41453],[34,269864,269866,269917],{"className":269865},[31796,31797],[34,269867,269869,269914],{"className":269868},[31801],[34,269870,269872,269886,269894],{"className":269871,"style":46746},[31805],[34,269873,269874,269877],{"style":41504},[34,269875],{"className":269876,"style":41508},[31813],[34,269878,269880],{"className":269879},[31818,31819,31820,31821],[34,269881,269883],{"className":269882},[31685,31821],[34,269884,269147],{"className":269885},[31685,31686,31821],[34,269887,269888,269891],{"style":41520},[34,269889],{"className":269890,"style":41508},[31813],[34,269892],{"className":269893,"style":41528},[41527],[34,269895,269896,269899],{"style":46771},[34,269897],{"className":269898,"style":41508},[31813],[34,269900,269902],{"className":269901},[31818,31819,31820,31821],[34,269903,269905,269908,269911],{"className":269904},[31685,31821],[34,269906,31759],{"className":269907},[31685,31686,31821],[34,269909,32226],{"className":269910},[32024,31821],[34,269912,640],{"className":269913},[31685,31821],[34,269915,31829],{"className":269916},[31828],[34,269918,269920],{"className":269919},[31801],[34,269921,269923],{"className":269922,"style":41619},[31805],[34,269924],{},[34,269926],{"className":269927},[31690,41488],[34,269929,456],{"className":269930},[31782],[34,269932,54151],{"className":269933},[31685,31686],[34,269935],{"className":269936,"style":32604},[31845],[34,269938,223],{"className":269939},[32024],[34,269941],{"className":269942,"style":32604},[31845],[34,269944,269946,269949,269952,269955,269958,269961],{"className":269945},[31676],[34,269947],{"className":269948,"style":31774},[31680],[34,269950,54224],{"className":269951,"style":54252},[31685,31686],[34,269953,2557],{"className":269954},[31690],[34,269956],{"className":269957,"style":32604},[31845],[34,269959,223],{"className":269960},[32024],[34,269962],{"className":269963,"style":32604},[31845],[34,269965,269967,269970,270038,270041,270044,270047,270050],{"className":269966},[31676],[34,269968],{"className":269969,"style":46727},[31680],[34,269971,269973,269976,270035],{"className":269972},[31685],[34,269974],{"className":269975},[31782,41488],[34,269977,269979],{"className":269978},[41453],[34,269980,269982,270027],{"className":269981},[31796,31797],[34,269983,269985,270024],{"className":269984},[31801],[34,269986,269988,270002,270010],{"className":269987,"style":46746},[31805],[34,269989,269990,269993],{"style":41504},[34,269991],{"className":269992,"style":41508},[31813],[34,269994,269996],{"className":269995},[31818,31819,31820,31821],[34,269997,269999],{"className":269998},[31685,31821],[34,270000,269147],{"className":270001},[31685,31686,31821],[34,270003,270004,270007],{"style":41520},[34,270005],{"className":270006,"style":41508},[31813],[34,270008],{"className":270009,"style":41528},[41527],[34,270011,270012,270015],{"style":46771},[34,270013],{"className":270014,"style":41508},[31813],[34,270016,270018],{"className":270017},[31818,31819,31820,31821],[34,270019,270021],{"className":270020},[31685,31821],[34,270022,640],{"className":270023},[31685,31821],[34,270025,31829],{"className":270026},[31828],[34,270028,270030],{"className":270029},[31801],[34,270031,270033],{"className":270032,"style":41619},[31805],[34,270034],{},[34,270036],{"className":270037},[31690,41488],[34,270039,456],{"className":270040},[31782],[34,270042,54151],{"className":270043},[31685,31686],[34,270045],{"className":270046,"style":32604},[31845],[34,270048,223],{"className":270049},[32024],[34,270051],{"className":270052,"style":32604},[31845],[34,270054,270056,270059,270062,270065,270068],{"className":270055},[31676],[34,270057],{"className":270058,"style":34944},[31680],[34,270060,54224],{"className":270061,"style":54252},[31685,31686],[34,270063],{"className":270064,"style":32604},[31845],[34,270066,223],{"className":270067},[32024],[34,270069],{"className":270070,"style":32604},[31845],[34,270072,270074,270077,270080,270083,270086,270089],{"className":270073},[31676],[34,270075],{"className":270076,"style":31774},[31680],[34,270078,47384],{"className":270079,"style":38706},[31685,31686],[34,270081,2557],{"className":270082},[31690],[34,270084],{"className":270085,"style":31846},[31845],[34,270087,202],{"className":270088},[31850],[34,270090],{"className":270091,"style":31846},[31845],[34,270093,270095,270099,270260],{"className":270094},[31676],[34,270096],{"className":270097,"style":270098},[31680],"height:1.2193em;vertical-align:-0.345em;",[34,270100,270102,270105,270108,270111,270114,270183,270186,270189,270192],{"className":270101},[31685],[34,270103,54151],{"className":270104},[31685,226302],[34,270106],{"className":270107,"style":32604},[31845],[34,270109,223],{"className":270110},[32024],[34,270112],{"className":270113,"style":32604},[31845],[34,270115,270117,270120,270180],{"className":270116},[31685],[34,270118],{"className":270119},[31782,41488],[34,270121,270123],{"className":270122},[41453],[34,270124,270126,270172],{"className":270125},[31796,31797],[34,270127,270129,270169],{"className":270128},[31801],[34,270130,270133,270147,270155],{"className":270131,"style":270132},[31805],"height:0.8743em;",[34,270134,270135,270138],{"style":41504},[34,270136],{"className":270137,"style":41508},[31813],[34,270139,270141],{"className":270140},[31818,31819,31820,31821],[34,270142,270144],{"className":270143},[31685,31821],[34,270145,31660],{"className":270146},[31685,226302,31821],[34,270148,270149,270152],{"style":41520},[34,270150],{"className":270151,"style":41508},[31813],[34,270153],{"className":270154,"style":41528},[41527],[34,270156,270157,270160],{"style":46771},[34,270158],{"className":270159,"style":41508},[31813],[34,270161,270163],{"className":270162},[31818,31819,31820,31821],[34,270164,270166],{"className":270165},[31685,31821],[34,270167,54224],{"className":270168},[31685,226302,31821],[34,270170,31829],{"className":270171},[31828],[34,270173,270175],{"className":270174},[31801],[34,270176,270178],{"className":270177,"style":41619},[31805],[34,270179],{},[34,270181],{"className":270182},[31690,41488],[34,270184],{"className":270185,"style":32604},[31845],[34,270187,223],{"className":270188},[32024],[34,270190],{"className":270191,"style":32604},[31845],[34,270193,270195,270198,270257],{"className":270194},[31685],[34,270196],{"className":270197},[31782,41488],[34,270199,270201],{"className":270200},[41453],[34,270202,270204,270249],{"className":270203},[31796,31797],[34,270205,270207,270246],{"className":270206},[31801],[34,270208,270210,270224,270232],{"className":270209,"style":270132},[31805],[34,270211,270212,270215],{"style":41504},[34,270213],{"className":270214,"style":41508},[31813],[34,270216,270218],{"className":270217},[31818,31819,31820,31821],[34,270219,270221],{"className":270220},[31685,31821],[34,270222,269147],{"className":270223},[31685,226302,31821],[34,270225,270226,270229],{"style":41520},[34,270227],{"className":270228,"style":41508},[31813],[34,270230],{"className":270231,"style":41528},[41527],[34,270233,270234,270237],{"style":46771},[34,270235],{"className":270236,"style":41508},[31813],[34,270238,270240],{"className":270239},[31818,31819,31820,31821],[34,270241,270243],{"className":270242},[31685,31821],[34,270244,47384],{"className":270245},[31685,226302,31821],[34,270247,31829],{"className":270248},[31828],[34,270250,270252],{"className":270251},[31801],[34,270253,270255],{"className":270254,"style":41619},[31805],[34,270256],{},[34,270258],{"className":270259},[31690,41488],[34,270261,270263],{"className":270262},[31685,2638],[34,270264,269725],{"className":270265},[31685],[19,270267,270269],{"id":270268},"_18-磁盘组柱面数总容量与平均等待时间","18. 磁盘组柱面数、总容量与平均等待时间",[15,270271,270272],{},"某磁盘组有 6 片盘片，每片有两个记录面，存储区域内径为 22cm，外径为 33cm，道存储密度为 40 道\u002Fcm，内层位存储密度为 400b\u002Fcm，转速为 3000r\u002Fmin（转\u002F分），问共有多少柱面？盘组总存储量为多少？平均等待时间为多少？",[15,270274,264303],{},[253,270276,267207],{"id":270277},"答案与详细解答-2",[105,270279,270280,270542,270991],{},[108,270281,270282,270284,270285,270426,270427,347],{},[2627,270283,187367],{},"\n有效半径范围 = ",[34,270286,270288,270319],{"className":270287},[31642],[34,270289,270291],{"className":270290},[31646],[31648,270292,270293],{"xmlns":31650},[31652,270294,270295,270316],{},[31655,270296,270297,270309,270311,270313],{},[41453,270298,270299,270307],{},[31655,270300,270301,270303,270305],{},[31749,270302,185190],{},[31662,270304,32226],{},[31749,270306,94716],{},[31749,270308,768],{},[31662,270310,202],{},[31749,270312,185571],{},[39361,270314,270315],{}," cm",[31665,270317,270318],{"encoding":31667},"\\frac{33 - 22}{2} = 5.5 \\text{ cm}",[34,270320,270322,270411],{"className":270321,"ariaHidden":758},[31672],[34,270323,270325,270328,270402,270405,270408],{"className":270324},[31676],[34,270326],{"className":270327,"style":46727},[31680],[34,270329,270331,270334,270399],{"className":270330},[31685],[34,270332],{"className":270333},[31782,41488],[34,270335,270337],{"className":270336},[41453],[34,270338,270340,270391],{"className":270339},[31796,31797],[34,270341,270343,270388],{"className":270342},[31801],[34,270344,270346,270360,270368],{"className":270345,"style":46746},[31805],[34,270347,270348,270351],{"style":41504},[34,270349],{"className":270350,"style":41508},[31813],[34,270352,270354],{"className":270353},[31818,31819,31820,31821],[34,270355,270357],{"className":270356},[31685,31821],[34,270358,768],{"className":270359},[31685,31821],[34,270361,270362,270365],{"style":41520},[34,270363],{"className":270364,"style":41508},[31813],[34,270366],{"className":270367,"style":41528},[41527],[34,270369,270370,270373],{"style":46771},[34,270371],{"className":270372,"style":41508},[31813],[34,270374,270376],{"className":270375},[31818,31819,31820,31821],[34,270377,270379,270382,270385],{"className":270378},[31685,31821],[34,270380,185190],{"className":270381},[31685,31821],[34,270383,32226],{"className":270384},[32024,31821],[34,270386,94716],{"className":270387},[31685,31821],[34,270389,31829],{"className":270390},[31828],[34,270392,270394],{"className":270393},[31801],[34,270395,270397],{"className":270396,"style":41619},[31805],[34,270398],{},[34,270400],{"className":270401},[31690,41488],[34,270403],{"className":270404,"style":31846},[31845],[34,270406,202],{"className":270407},[31850],[34,270409],{"className":270410,"style":31846},[31845],[34,270412,270414,270417,270420],{"className":270413},[31676],[34,270415],{"className":270416,"style":32617},[31680],[34,270418,185571],{"className":270419},[31685],[34,270421,270423],{"className":270422},[31685,2638],[34,270424,270315],{"className":270425},[31685],"\n柱面数 = ",[34,270428,270430,270462],{"className":270429},[31642],[34,270431,270433],{"className":270432},[31646],[31648,270434,270435],{"xmlns":31650},[31652,270436,270437,270459],{},[31655,270438,270439,270441,270443,270445,270447,270450,270452],{},[31749,270440,185571],{},[39361,270442,270315],{},[31662,270444,42071],{},[31749,270446,5731],{},[39361,270448,270449],{}," 道\u002Fcm",[31662,270451,202],{},[31655,270453,270454,270456],{},[31749,270455,185719],{"mathvariant":226163},[39361,270457,270458],{}," 柱面",[31665,270460,270461],{"encoding":31667},"5.5 \\text{ cm} \\times 40 \\text{ 道\u002Fcm} = \\mathbf{220 \\text{ 柱面}}",[34,270463,270465,270489,270520],{"className":270464,"ariaHidden":758},[31672],[34,270466,270468,270471,270474,270480,270483,270486],{"className":270467},[31676],[34,270469],{"className":270470,"style":35942},[31680],[34,270472,185571],{"className":270473},[31685],[34,270475,270477],{"className":270476},[31685,2638],[34,270478,270315],{"className":270479},[31685],[34,270481],{"className":270482,"style":32604},[31845],[34,270484,42071],{"className":270485},[32024],[34,270487],{"className":270488,"style":32604},[31845],[34,270490,270492,270495,270498,270511,270514,270517],{"className":270491},[31676],[34,270493],{"className":270494,"style":31774},[31680],[34,270496,5731],{"className":270497},[31685],[34,270499,270501,270504,270507],{"className":270500},[31685,2638],[34,270502,130298],{"className":270503},[31685],[34,270505,185268],{"className":270506},[31685,105462],[34,270508,270510],{"className":270509},[31685],"\u002Fcm",[34,270512],{"className":270513,"style":31846},[31845],[34,270515,202],{"className":270516},[31850],[34,270518],{"className":270519,"style":31846},[31845],[34,270521,270523,270526],{"className":270522},[31676],[34,270524],{"className":270525,"style":33218},[31680],[34,270527,270529,270532],{"className":270528},[31685],[34,270530,185719],{"className":270531},[31685,226302],[34,270533,270535,270538],{"className":270534},[31685,2638],[34,270536,130298],{"className":270537},[31685],[34,270539,270541],{"className":270540},[31685,105462],"柱面",[108,270543,270544,270546,270547,270606,270607,270716,270717,270745,270746,270774,270775,47649,270803,270837,270838,270907,270908,4374,270950,31901],{},[2627,270545,193043],{},"\n最内层磁道周长 = ",[34,270548,270550,270570],{"className":270549},[31642],[34,270551,270553],{"className":270552},[31646],[31648,270554,270555],{"xmlns":31650},[31652,270556,270557,270567],{},[31655,270558,270559,270561,270563,270565],{},[31658,270560,58442],{},[31662,270562,42071],{},[31749,270564,94716],{},[39361,270566,270315],{},[31665,270568,270569],{"encoding":31667},"\\pi \\times 22 \\text{ cm}",[34,270571,270573,270591],{"className":270572,"ariaHidden":758},[31672],[34,270574,270576,270579,270582,270585,270588],{"className":270575},[31676],[34,270577],{"className":270578,"style":32597},[31680],[34,270580,58442],{"className":270581,"style":32062},[31685,31686],[34,270583],{"className":270584,"style":32604},[31845],[34,270586,42071],{"className":270587},[32024],[34,270589],{"className":270590,"style":32604},[31845],[34,270592,270594,270597,270600],{"className":270593},[31676],[34,270595],{"className":270596,"style":32617},[31680],[34,270598,94716],{"className":270599},[31685],[34,270601,270603],{"className":270602},[31685,2638],[34,270604,270315],{"className":270605},[31685],"\n每条磁道容量 = ",[34,270608,270610,270644],{"className":270609},[31642],[34,270611,270613],{"className":270612},[31646],[31648,270614,270615],{"xmlns":31650},[31652,270616,270617,270641],{},[31655,270618,270619,270621,270623,270625,270627,270629,270632,270634,270636,270638],{},[31749,270620,94716],{},[31658,270622,58442],{},[39361,270624,270315],{},[31662,270626,42071],{},[31749,270628,185297],{},[39361,270630,270631],{}," b\u002Fcm",[31662,270633,202],{},[31749,270635,186049],{},[31658,270637,58442],{},[39361,270639,270640],{}," bits",[31665,270642,270643],{"encoding":31667},"22\\pi \\text{ cm} \\times 400 \\text{ b\u002Fcm} = 8800\\pi \\text{ bits}",[34,270645,270647,270674,270698],{"className":270646,"ariaHidden":758},[31672],[34,270648,270650,270653,270656,270659,270665,270668,270671],{"className":270649},[31676],[34,270651],{"className":270652,"style":35942},[31680],[34,270654,94716],{"className":270655},[31685],[34,270657,58442],{"className":270658,"style":32062},[31685,31686],[34,270660,270662],{"className":270661},[31685,2638],[34,270663,270315],{"className":270664},[31685],[34,270666],{"className":270667,"style":32604},[31845],[34,270669,42071],{"className":270670},[32024],[34,270672],{"className":270673,"style":32604},[31845],[34,270675,270677,270680,270683,270689,270692,270695],{"className":270676},[31676],[34,270678],{"className":270679,"style":31774},[31680],[34,270681,185297],{"className":270682},[31685],[34,270684,270686],{"className":270685},[31685,2638],[34,270687,270631],{"className":270688},[31685],[34,270690],{"className":270691,"style":31846},[31845],[34,270693,202],{"className":270694},[31850],[34,270696],{"className":270697,"style":31846},[31845],[34,270699,270701,270704,270707,270710],{"className":270700},[31676],[34,270702],{"className":270703,"style":31681},[31680],[34,270705,186049],{"className":270706},[31685],[34,270708,58442],{"className":270709,"style":32062},[31685,31686],[34,270711,270713],{"className":270712},[31685,2638],[34,270714,270640],{"className":270715},[31685],"。\n总磁头\u002F盘面数 = 6片 ",[34,270718,270720,270733],{"className":270719},[31642],[34,270721,270723],{"className":270722},[31646],[31648,270724,270725],{"xmlns":31650},[31652,270726,270727,270731],{},[31655,270728,270729],{},[31662,270730,42071],{},[31665,270732,211013],{"encoding":31667},[34,270734,270736],{"className":270735,"ariaHidden":758},[31672],[34,270737,270739,270742],{"className":270738},[31676],[34,270740],{"className":270741,"style":32597},[31680],[34,270743,42071],{"className":270744},[31685]," 2面 = 12面。\n总容量 = 12面 ",[34,270747,270749,270762],{"className":270748},[31642],[34,270750,270752],{"className":270751},[31646],[31648,270753,270754],{"xmlns":31650},[31652,270755,270756,270760],{},[31655,270757,270758],{},[31662,270759,42071],{},[31665,270761,211013],{"encoding":31667},[34,270763,270765],{"className":270764,"ariaHidden":758},[31672],[34,270766,270768,270771],{"className":270767},[31676],[34,270769],{"className":270770,"style":32597},[31680],[34,270772,42071],{"className":270773},[31685]," 220道 ",[34,270776,270778,270791],{"className":270777},[31642],[34,270779,270781],{"className":270780},[31646],[31648,270782,270783],{"xmlns":31650},[31652,270784,270785,270789],{},[31655,270786,270787],{},[31662,270788,42071],{},[31665,270790,211013],{"encoding":31667},[34,270792,270794],{"className":270793,"ariaHidden":758},[31672],[34,270795,270797,270800],{"className":270796},[31676],[34,270798],{"className":270799,"style":32597},[31680],[34,270801,42071],{"className":270802},[31685],[34,270804,270806,270822],{"className":270805},[31642],[34,270807,270809],{"className":270808},[31646],[31648,270810,270811],{"xmlns":31650},[31652,270812,270813,270819],{},[31655,270814,270815,270817],{},[31749,270816,186049],{},[31658,270818,58442],{},[31665,270820,270821],{"encoding":31667},"8800\\pi",[34,270823,270825],{"className":270824,"ariaHidden":758},[31672],[34,270826,270828,270831,270834],{"className":270827},[31676],[34,270829],{"className":270830,"style":32617},[31680],[34,270832,186049],{"className":270833},[31685],[34,270835,58442],{"className":270836,"style":32062},[31685,31686]," bits = ",[34,270839,270841,270868],{"className":270840},[31642],[34,270842,270844],{"className":270843},[31646],[31648,270845,270846],{"xmlns":31650},[31652,270847,270848,270865],{},[31655,270849,270850,270852,270854,270857,270859,270861,270863],{},[31749,270851,94703],{},[31662,270853,31960],{"separator":758},[31749,270855,270856],{},"232",[31662,270858,31960],{"separator":758},[31749,270860,182364],{},[31658,270862,58442],{},[39361,270864,270640],{},[31665,270866,270867],{"encoding":31667},"23,232,000 \\pi \\text{ bits}",[34,270869,270871],{"className":270870,"ariaHidden":758},[31672],[34,270872,270874,270877,270880,270883,270886,270889,270892,270895,270898,270901],{"className":270873},[31676],[34,270875],{"className":270876,"style":43217},[31680],[34,270878,94703],{"className":270879},[31685],[34,270881,31960],{"className":270882},[32119],[34,270884],{"className":270885,"style":32123},[31845],[34,270887,270856],{"className":270888},[31685],[34,270890,31960],{"className":270891},[32119],[34,270893],{"className":270894,"style":32123},[31845],[34,270896,182364],{"className":270897},[31685],[34,270899,58442],{"className":270900,"style":32062},[31685,31686],[34,270902,270904],{"className":270903},[31685,2638],[34,270905,270640],{"className":270906},[31685]," （约 ",[34,270909,270911,270929],{"className":270910},[31642],[34,270912,270914],{"className":270913},[31646],[31648,270915,270916],{"xmlns":31650},[31652,270917,270918,270926],{},[31655,270919,270920,270923],{},[31749,270921,270922],{"mathvariant":226163},"72.98",[39361,270924,270925],{}," Mbits",[31665,270927,270928],{"encoding":31667},"\\mathbf{72.98 \\text{ Mbits}}",[34,270930,270932],{"className":270931,"ariaHidden":758},[31672],[34,270933,270935,270938],{"className":270934},[31676],[34,270936],{"className":270937,"style":31681},[31680],[34,270939,270941,270944],{"className":270940},[31685],[34,270942,270922],{"className":270943},[31685,226302],[34,270945,270947],{"className":270946},[31685,2638],[34,270948,270925],{"className":270949},[31685],[34,270951,270953,270970],{"className":270952},[31642],[34,270954,270956],{"className":270955},[31646],[31648,270957,270958],{"xmlns":31650},[31652,270959,270960,270967],{},[31655,270961,270962,270964],{},[31749,270963,186541],{"mathvariant":226163},[39361,270965,270966],{}," MB",[31665,270968,270969],{"encoding":31667},"\\mathbf{9.12 \\text{ MB}}",[34,270971,270973],{"className":270972,"ariaHidden":758},[31672],[34,270974,270976,270979],{"className":270975},[31676],[34,270977],{"className":270978,"style":33218},[31680],[34,270980,270982,270985],{"className":270981},[31685],[34,270983,186541],{"className":270984},[31685,226302],[34,270986,270988],{"className":270987},[31685,2638],[34,270989,270966],{"className":270990},[31685],[108,270992,270993,270995,270996,271062,271063,267578],{},[2627,270994,187599],{},"\n转速 = 3000 r\u002Fmin = 50 r\u002Fs。旋转一圈需 ",[34,270997,270999,271025],{"className":270998},[31642],[34,271000,271002],{"className":271001},[31646],[31648,271003,271004],{"xmlns":31650},[31652,271005,271006,271022],{},[31655,271007,271008,271010,271012,271014,271016,271018,271020],{},[31749,271009,5249],{},[31658,271011,38080],{"mathvariant":38079},[31749,271013,5774],{},[31662,271015,202],{},[31749,271017,3249],{},[31658,271019,31759],{},[31658,271021,31747],{},[31665,271023,271024],{"encoding":31667},"1000 \u002F 50 = 20 ms",[34,271026,271028,271047],{"className":271027,"ariaHidden":758},[31672],[34,271029,271031,271034,271038,271041,271044],{"className":271030},[31676],[34,271032],{"className":271033,"style":31774},[31680],[34,271035,271037],{"className":271036},[31685],"1000\u002F50",[34,271039],{"className":271040,"style":31846},[31845],[34,271042,202],{"className":271043},[31850],[34,271045],{"className":271046,"style":31846},[31845],[34,271048,271050,271053,271056,271059],{"className":271049},[31676],[34,271051],{"className":271052,"style":32617},[31680],[34,271054,3249],{"className":271055},[31685],[34,271057,31759],{"className":271058},[31685,31686],[34,271060,31747],{"className":271061},[31685,31686],"。\n平均等待时间 = ",[34,271064,271066,271102],{"className":271065},[31642],[34,271067,271069],{"className":271068},[31646],[31648,271070,271071],{"xmlns":31650},[31652,271072,271073,271099],{},[31655,271074,271075,271081,271083,271085,271087,271089,271091],{},[41453,271076,271077,271079],{},[31749,271078,640],{},[31749,271080,768],{},[31662,271082,42071],{},[31749,271084,3249],{},[31658,271086,31759],{},[31658,271088,31747],{},[31662,271090,202],{},[31655,271092,271093,271095,271097],{},[31749,271094,3244],{"mathvariant":226163},[31658,271096,31759],{"mathvariant":226163},[31658,271098,31747],{"mathvariant":226163},[31665,271100,271101],{"encoding":31667},"\\frac{1}{2} \\times 20ms = \\mathbf{10 ms}",[34,271103,271105,271188,271212],{"className":271104,"ariaHidden":758},[31672],[34,271106,271108,271111,271179,271182,271185],{"className":271107},[31676],[34,271109],{"className":271110,"style":46727},[31680],[34,271112,271114,271117,271176],{"className":271113},[31685],[34,271115],{"className":271116},[31782,41488],[34,271118,271120],{"className":271119},[41453],[34,271121,271123,271168],{"className":271122},[31796,31797],[34,271124,271126,271165],{"className":271125},[31801],[34,271127,271129,271143,271151],{"className":271128,"style":46746},[31805],[34,271130,271131,271134],{"style":41504},[34,271132],{"className":271133,"style":41508},[31813],[34,271135,271137],{"className":271136},[31818,31819,31820,31821],[34,271138,271140],{"className":271139},[31685,31821],[34,271141,768],{"className":271142},[31685,31821],[34,271144,271145,271148],{"style":41520},[34,271146],{"className":271147,"style":41508},[31813],[34,271149],{"className":271150,"style":41528},[41527],[34,271152,271153,271156],{"style":46771},[34,271154],{"className":271155,"style":41508},[31813],[34,271157,271159],{"className":271158},[31818,31819,31820,31821],[34,271160,271162],{"className":271161},[31685,31821],[34,271163,640],{"className":271164},[31685,31821],[34,271166,31829],{"className":271167},[31828],[34,271169,271171],{"className":271170},[31801],[34,271172,271174],{"className":271173,"style":41619},[31805],[34,271175],{},[34,271177],{"className":271178},[31690,41488],[34,271180],{"className":271181,"style":32604},[31845],[34,271183,42071],{"className":271184},[32024],[34,271186],{"className":271187,"style":32604},[31845],[34,271189,271191,271194,271197,271200,271203,271206,271209],{"className":271190},[31676],[34,271192],{"className":271193,"style":32617},[31680],[34,271195,3249],{"className":271196},[31685],[34,271198,31759],{"className":271199},[31685,31686],[34,271201,31747],{"className":271202},[31685,31686],[34,271204],{"className":271205,"style":31846},[31845],[34,271207,202],{"className":271208},[31850],[34,271210],{"className":271211,"style":31846},[31845],[34,271213,271215,271218],{"className":271214},[31676],[34,271216],{"className":271217,"style":32617},[31680],[34,271219,271221],{"className":271220},[31685],[34,271222,271224],{"className":271223},[31685,226302],"10ms",[19,271226,271228],{"id":271227},"_19-四道作业系统sjf-抢占式优先数调度","19. 四道作业系统：SJF + 抢占式优先数调度",[15,271230,271231],{},"若有一个四道作业系统，如果在一段时间内先后有 6 个作业，它们提交和运行时间由下表给出。作业采用短作业优先的调度算法，进程采用以优先数的抢占式调度算法，作业优先数就是进程优先数，优先数越小，优先级越高。",[13837,271233,271234,271254],{},[13840,271235,271236],{},[13843,271237,271238,271240,271242,271244,271246,271248,271250,271252],{},[13846,271239,230223],{"align":82355},[13846,271241,230226],{"align":82355},[13846,271243,264478],{"align":82355},[13846,271245,231100],{"align":82355},[13846,271247,230231],{"align":82355},[13846,271249,264483],{"align":82355},[13846,271251,230236],{"align":82355},[13846,271253,230239],{"align":82355},[13853,271255,271256,271274,271292,271310,271328,271346],{},[13843,271257,271258,271260,271262,271264,271266,271268,271270,271272],{},[13858,271259,640],{"align":82355},[13858,271261,230248],{"align":82355},[13858,271263,5731],{"align":82355},[13858,271265,18136],{"align":82355},[13858,271267],{"align":82355},[13858,271269],{"align":82355},[13858,271271],{"align":82355},[13858,271273],{"align":82355},[13843,271275,271276,271278,271280,271282,271284,271286,271288,271290],{},[13858,271277,768],{"align":82355},[13858,271279,264512],{"align":82355},[13858,271281,9314],{"align":82355},[13858,271283,15932],{"align":82355},[13858,271285],{"align":82355},[13858,271287],{"align":82355},[13858,271289],{"align":82355},[13858,271291],{"align":82355},[13843,271293,271294,271296,271298,271300,271302,271304,271306,271308],{},[13858,271295,786],{"align":82355},[13858,271297,230265],{"align":82355},[13858,271299,3249],{"align":82355},[13858,271301,768],{"align":82355},[13858,271303],{"align":82355},[13858,271305],{"align":82355},[13858,271307],{"align":82355},[13858,271309],{"align":82355},[13843,271311,271312,271314,271316,271318,271320,271322,271324,271326],{},[13858,271313,15932],{"align":82355},[13858,271315,230282],{"align":82355},[13858,271317,3249],{"align":82355},[13858,271319,786],{"align":82355},[13858,271321],{"align":82355},[13858,271323],{"align":82355},[13858,271325],{"align":82355},[13858,271327],{"align":82355},[13843,271329,271330,271332,271334,271336,271338,271340,271342,271344],{},[13858,271331,18136],{"align":82355},[13858,271333,230299],{"align":82355},[13858,271335,3244],{"align":82355},[13858,271337,768],{"align":82355},[13858,271339],{"align":82355},[13858,271341],{"align":82355},[13858,271343],{"align":82355},[13858,271345],{"align":82355},[13843,271347,271348,271350,271352,271354,271356,271358,271360,271362],{},[13858,271349,36656],{"align":82355},[13858,271351,230316],{"align":82355},[13858,271353,18136],{"align":82355},[13858,271355,768],{"align":82355},[13858,271357],{"align":82355},[13858,271359],{"align":82355},[13858,271361],{"align":82355},[13858,271363],{"align":82355},[15,271365,11832],{},[105,271367,271368,271370],{},[108,271369,264593],{},[108,271371,264596],{},[15,271373,264303],{},[253,271375,219117],{"id":271376},"原理解析-the-why-3",[105,271378,271379,271385],{},[108,271380,271381,271384],{},[2627,271382,271383],{},"\"四道\"作业系统："," 意味着内存中最多只能同时存在 4 个进程。如果有第 5 个作业到达，哪怕它是极短的作业，也必须在\"外存后备队列\"中排队，直到内存有人腾出位置！",[108,271386,271387,271390],{},[2627,271388,271389],{},"双层调度："," 高级调度（进内存）看 SJF；低级调度（抢 CPU）看 优先数。",[253,271392,267207],{"id":271393},"答案与详细解答-3",[15,271395,271396],{},[2627,271397,271398],{},"(1) 时间表格计算",[13837,271400,271401,271422],{},[13840,271402,271403],{},[13843,271404,271405,271407,271409,271411,271414,271416,271418,271420],{},[13846,271406,230486],{"align":82355},[13846,271408,230226],{"align":82355},[13846,271410,229533],{"align":82355},[13846,271412,271413],{"align":82355},"优先",[13846,271415,230231],{"align":82355},[13846,271417,214469],{"align":82355},[13846,271419,230236],{"align":82355},[13846,271421,230239],{"align":82355},[13853,271423,271424,271446,271468,271489,271512,271533],{},[13843,271425,271426,271430,271432,271434,271436,271438,271441,271443],{},[13858,271427,271428],{"align":82355},[2627,271429,640],{},[13858,271431,230248],{"align":82355},[13858,271433,5731],{"align":82355},[13858,271435,18136],{"align":82355},[13858,271437,230248],{"align":82355},[13858,271439,271440],{"align":82355},"10:05",[13858,271442,95630],{"align":82355},[13858,271444,271445],{"align":82355},"125\u002F40 = 3.125",[13843,271447,271448,271452,271454,271456,271458,271460,271463,271465],{},[13858,271449,271450],{"align":82355},[2627,271451,768],{},[13858,271453,264512],{"align":82355},[13858,271455,9314],{"align":82355},[13858,271457,15932],{"align":82355},[13858,271459,264512],{"align":82355},[13858,271461,271462],{"align":82355},"9:40",[13858,271464,94815],{"align":82355},[13858,271466,271467],{"align":82355},"85\u002F30 ≈ 2.833",[13843,271469,271470,271474,271476,271478,271480,271482,271484,271486],{},[13858,271471,271472],{"align":82355},[2627,271473,786],{},[13858,271475,230265],{"align":82355},[13858,271477,3249],{"align":82355},[13858,271479,768],{"align":82355},[13858,271481,230265],{"align":82355},[13858,271483,230333],{"align":82355},[13858,271485,3249],{"align":82355},[13858,271487,271488],{"align":82355},"20\u002F20 = 1.0",[13843,271490,271491,271495,271497,271499,271501,271504,271507,271509],{},[13858,271492,271493],{"align":82355},[2627,271494,15932],{},[13858,271496,230282],{"align":82355},[13858,271498,3249],{"align":82355},[13858,271500,786],{"align":82355},[13858,271502,271503],{"align":82355},"8:55",[13858,271505,271506],{"align":82355},"9:15",[13858,271508,5774],{"align":82355},[13858,271510,271511],{"align":82355},"50\u002F20 = 2.5",[13843,271513,271514,271518,271520,271522,271524,271526,271528,271530],{},[13858,271515,271516],{"align":82355},[2627,271517,18136],{},[13858,271519,230299],{"align":82355},[13858,271521,3244],{"align":82355},[13858,271523,768],{"align":82355},[13858,271525,230407],{"align":82355},[13858,271527,271503],{"align":82355},[13858,271529,95556],{"align":82355},[13858,271531,271532],{"align":82355},"25\u002F10 = 2.5",[13843,271534,271535,271539,271541,271543,271545,271547,271549,271551],{},[13858,271536,271537],{"align":82355},[2627,271538,36656],{},[13858,271540,230316],{"align":82355},[13858,271542,18136],{"align":82355},[13858,271544,768],{"align":82355},[13858,271546,230333],{"align":82355},[13858,271548,230407],{"align":82355},[13858,271550,3244],{"align":82355},[13858,271552,271553],{"align":82355},"10\u002F5 = 2.0",[15,271555,271556],{},[2627,271557,271558],{},"(2) 平均计算",[105,271560,271561,271788],{},[108,271562,271563,47649,271565,347],{},[2627,271564,226112],{},[34,271566,271568,271628],{"className":271567},[31642],[34,271569,271571],{"className":271570},[31646],[31648,271572,271573],{"xmlns":31650},[31652,271574,271575,271625],{},[31655,271576,271577,271579,271581,271583,271585,271587,271589,271591,271593,271595,271597,271599,271601,271603,271605,271607,271609,271612,271614,271616,271618],{},[31662,271578,456],{"stretchy":702},[31749,271580,95630],{},[31662,271582,223],{},[31749,271584,94815],{},[31662,271586,223],{},[31749,271588,3249],{},[31662,271590,223],{},[31749,271592,5774],{},[31662,271594,223],{},[31749,271596,95556],{},[31662,271598,223],{},[31749,271600,3244],{},[31662,271602,2557],{"stretchy":702},[31658,271604,38080],{"mathvariant":38079},[31749,271606,36656],{},[31662,271608,202],{},[31749,271610,271611],{},"315",[31658,271613,38080],{"mathvariant":38079},[31749,271615,36656],{},[31662,271617,202],{},[31655,271619,271620,271623],{},[31749,271621,271622],{"mathvariant":226163},"52.5",[39361,271624,228174],{},[31665,271626,271627],{"encoding":31667},"(125 + 85 + 20 + 50 + 25 + 10) \u002F 6 = 315 \u002F 6 = \\mathbf{52.5 \\text{ 分钟}}",[34,271629,271631,271652,271670,271688,271706,271724,271748,271767],{"className":271630,"ariaHidden":758},[31672],[34,271632,271634,271637,271640,271643,271646,271649],{"className":271633},[31676],[34,271635],{"className":271636,"style":31774},[31680],[34,271638,456],{"className":271639},[31782],[34,271641,95630],{"className":271642},[31685],[34,271644],{"className":271645,"style":32604},[31845],[34,271647,223],{"className":271648},[32024],[34,271650],{"className":271651,"style":32604},[31845],[34,271653,271655,271658,271661,271664,271667],{"className":271654},[31676],[34,271656],{"className":271657,"style":35942},[31680],[34,271659,94815],{"className":271660},[31685],[34,271662],{"className":271663,"style":32604},[31845],[34,271665,223],{"className":271666},[32024],[34,271668],{"className":271669,"style":32604},[31845],[34,271671,271673,271676,271679,271682,271685],{"className":271672},[31676],[34,271674],{"className":271675,"style":35942},[31680],[34,271677,3249],{"className":271678},[31685],[34,271680],{"className":271681,"style":32604},[31845],[34,271683,223],{"className":271684},[32024],[34,271686],{"className":271687,"style":32604},[31845],[34,271689,271691,271694,271697,271700,271703],{"className":271690},[31676],[34,271692],{"className":271693,"style":35942},[31680],[34,271695,5774],{"className":271696},[31685],[34,271698],{"className":271699,"style":32604},[31845],[34,271701,223],{"className":271702},[32024],[34,271704],{"className":271705,"style":32604},[31845],[34,271707,271709,271712,271715,271718,271721],{"className":271708},[31676],[34,271710],{"className":271711,"style":35942},[31680],[34,271713,95556],{"className":271714},[31685],[34,271716],{"className":271717,"style":32604},[31845],[34,271719,223],{"className":271720},[32024],[34,271722],{"className":271723,"style":32604},[31845],[34,271725,271727,271730,271733,271736,271739,271742,271745],{"className":271726},[31676],[34,271728],{"className":271729,"style":31774},[31680],[34,271731,3244],{"className":271732},[31685],[34,271734,2557],{"className":271735},[31690],[34,271737,230802],{"className":271738},[31685],[34,271740],{"className":271741,"style":31846},[31845],[34,271743,202],{"className":271744},[31850],[34,271746],{"className":271747,"style":31846},[31845],[34,271749,271751,271754,271758,271761,271764],{"className":271750},[31676],[34,271752],{"className":271753,"style":31774},[31680],[34,271755,271757],{"className":271756},[31685],"315\u002F6",[34,271759],{"className":271760,"style":31846},[31845],[34,271762,202],{"className":271763},[31850],[34,271765],{"className":271766,"style":31846},[31845],[34,271768,271770,271773],{"className":271769},[31676],[34,271771],{"className":271772,"style":33218},[31680],[34,271774,271776,271779],{"className":271775},[31685],[34,271777,271622],{"className":271778},[31685,226302],[34,271780,271782,271785],{"className":271781},[31685,2638],[34,271783,130298],{"className":271784},[31685],[34,271786,228301],{"className":271787},[31685,105462],[108,271789,271790,47649,271792,347],{},[2627,271791,230856],{},[34,271793,271795,271846],{"className":271794},[31642],[34,271796,271798],{"className":271797},[31646],[31648,271799,271800],{"xmlns":31650},[31652,271801,271802,271843],{},[31655,271803,271804,271806,271809,271811,271814,271816,271819,271821,271823,271825,271827,271829,271832,271834,271836,271838,271840],{},[31662,271805,456],{"stretchy":702},[31749,271807,271808],{},"3.125",[31662,271810,223],{},[31749,271812,271813],{},"2.833",[31662,271815,223],{},[31749,271817,271818],{},"1.0",[31662,271820,223],{},[31749,271822,167259],{},[31662,271824,223],{},[31749,271826,167259],{},[31662,271828,223],{},[31749,271830,271831],{},"2.0",[31662,271833,2557],{"stretchy":702},[31658,271835,38080],{"mathvariant":38079},[31749,271837,36656],{},[31662,271839,168560],{},[31749,271841,271842],{"mathvariant":226163},"2.326",[31665,271844,271845],{"encoding":31667},"(3.125 + 2.833 + 1.0 + 2.5 + 2.5 + 2.0) \u002F 6 \\approx \\mathbf{2.326}",[34,271847,271849,271870,271888,271906,271924,271942,271966],{"className":271848,"ariaHidden":758},[31672],[34,271850,271852,271855,271858,271861,271864,271867],{"className":271851},[31676],[34,271853],{"className":271854,"style":31774},[31680],[34,271856,456],{"className":271857},[31782],[34,271859,271808],{"className":271860},[31685],[34,271862],{"className":271863,"style":32604},[31845],[34,271865,223],{"className":271866},[32024],[34,271868],{"className":271869,"style":32604},[31845],[34,271871,271873,271876,271879,271882,271885],{"className":271872},[31676],[34,271874],{"className":271875,"style":35942},[31680],[34,271877,271813],{"className":271878},[31685],[34,271880],{"className":271881,"style":32604},[31845],[34,271883,223],{"className":271884},[32024],[34,271886],{"className":271887,"style":32604},[31845],[34,271889,271891,271894,271897,271900,271903],{"className":271890},[31676],[34,271892],{"className":271893,"style":35942},[31680],[34,271895,271818],{"className":271896},[31685],[34,271898],{"className":271899,"style":32604},[31845],[34,271901,223],{"className":271902},[32024],[34,271904],{"className":271905,"style":32604},[31845],[34,271907,271909,271912,271915,271918,271921],{"className":271908},[31676],[34,271910],{"className":271911,"style":35942},[31680],[34,271913,167259],{"className":271914},[31685],[34,271916],{"className":271917,"style":32604},[31845],[34,271919,223],{"className":271920},[32024],[34,271922],{"className":271923,"style":32604},[31845],[34,271925,271927,271930,271933,271936,271939],{"className":271926},[31676],[34,271928],{"className":271929,"style":35942},[31680],[34,271931,167259],{"className":271932},[31685],[34,271934],{"className":271935,"style":32604},[31845],[34,271937,223],{"className":271938},[32024],[34,271940],{"className":271941,"style":32604},[31845],[34,271943,271945,271948,271951,271954,271957,271960,271963],{"className":271944},[31676],[34,271946],{"className":271947,"style":31774},[31680],[34,271949,271831],{"className":271950},[31685],[34,271952,2557],{"className":271953},[31690],[34,271955,230802],{"className":271956},[31685],[34,271958],{"className":271959,"style":31846},[31845],[34,271961,168560],{"className":271962},[31850],[34,271964],{"className":271965,"style":31846},[31845],[34,271967,271969,271972],{"className":271968},[31676],[34,271970],{"className":271971,"style":32617},[31680],[34,271973,271975],{"className":271974},[31685],[34,271976,271842],{"className":271977},[31685,226302],[15,271979,271980,271982,271983,271986],{},[2627,271981,247377],{}," 完全忽视\"四道\"的限制，把5和6早早放进内存引发错误的抢占。记住，",[2627,271984,271985],{},"多道度满了之后，大门焊死","，新作业只能在外边排队等。\n:::",[19,271988,271990],{"id":271989},"_20-unix-多级索引节点inode最大文件长度","20. UNIX 多级索引节点（Inode）最大文件长度",[15,271992,271993],{},"设文件索引节点中有 7 个地址项，其中 4 个为直接地址索引，2 个是一级间接地址索引，1 个是二级间接地址索引，地址项大小为 4B，若磁盘索引块和磁盘数据块大小均为 1KB，求可表示的单个文件的最大长度。",[15,271995,264303],{},[253,271997,267207],{"id":271998},"答案与详细解答-4",[105,272000,272001,272080,272086,272121,272185,272285],{},[108,272002,272003,47649,272006,272079],{},[2627,272004,272005],{},"每个索引块能容纳的指针数：",[34,272007,272009,272037],{"className":272008},[31642],[34,272010,272012],{"className":272011},[31646],[31648,272013,272014],{"xmlns":31650},[31652,272015,272016,272034],{},[31655,272017,272018,272020,272022,272024,272026,272028,272030,272032],{},[31749,272019,640],{},[31658,272021,38691],{},[31658,272023,54224],{},[31658,272025,38080],{"mathvariant":38079},[31749,272027,15932],{},[31658,272029,54224],{},[31662,272031,202],{},[31749,272033,95418],{},[31665,272035,272036],{"encoding":31667},"1KB \u002F 4B = 256",[34,272038,272040,272070],{"className":272039,"ariaHidden":758},[31672],[34,272041,272043,272046,272049,272052,272055,272058,272061,272064,272067],{"className":272042},[31676],[34,272044],{"className":272045,"style":31774},[31680],[34,272047,640],{"className":272048},[31685],[34,272050,38691],{"className":272051,"style":38706},[31685,31686],[34,272053,54224],{"className":272054,"style":54252},[31685,31686],[34,272056,42280],{"className":272057},[31685],[34,272059,54224],{"className":272060,"style":54252},[31685,31686],[34,272062],{"className":272063,"style":31846},[31845],[34,272065,202],{"className":272066},[31850],[34,272068],{"className":272069,"style":31846},[31845],[34,272071,272073,272076],{"className":272072},[31676],[34,272074],{"className":272075,"style":32617},[31680],[34,272077,95418],{"className":272078},[31685]," 个指针。",[108,272081,272082,272085],{},[2627,272083,272084],{},"直接寻址："," 4 个指针，指向 4 个物理块。",[108,272087,272088,272091,272092,272120],{},[2627,272089,272090],{},"一级间接："," 2 个指针 ",[34,272093,272095,272108],{"className":272094},[31642],[34,272096,272098],{"className":272097},[31646],[31648,272099,272100],{"xmlns":31650},[31652,272101,272102,272106],{},[31655,272103,272104],{},[31662,272105,42071],{},[31665,272107,211013],{"encoding":31667},[34,272109,272111],{"className":272110,"ariaHidden":758},[31672],[34,272112,272114,272117],{"className":272113},[31676],[34,272115],{"className":272116,"style":32597},[31680],[34,272118,42071],{"className":272119},[31685]," 256 = 512 个物理块。",[108,272122,272123,272126,272127,272155,272156,272184],{},[2627,272124,272125],{},"二级间接："," 1 个指针 ",[34,272128,272130,272143],{"className":272129},[31642],[34,272131,272133],{"className":272132},[31646],[31648,272134,272135],{"xmlns":31650},[31652,272136,272137,272141],{},[31655,272138,272139],{},[31662,272140,42071],{},[31665,272142,211013],{"encoding":31667},[34,272144,272146],{"className":272145,"ariaHidden":758},[31672],[34,272147,272149,272152],{"className":272148},[31676],[34,272150],{"className":272151,"style":32597},[31680],[34,272153,42071],{"className":272154},[31685]," 256 ",[34,272157,272159,272172],{"className":272158},[31642],[34,272160,272162],{"className":272161},[31646],[31648,272163,272164],{"xmlns":31650},[31652,272165,272166,272170],{},[31655,272167,272168],{},[31662,272169,42071],{},[31665,272171,211013],{"encoding":31667},[34,272173,272175],{"className":272174,"ariaHidden":758},[31672],[34,272176,272178,272181],{"className":272177},[31676],[34,272179],{"className":272180,"style":32597},[31680],[34,272182,42071],{"className":272183},[31685]," 256 = 65536 个物理块。",[108,272186,272187,47649,272190,272284],{},[2627,272188,272189],{},"文件最大块数：",[34,272191,272193,272218],{"className":272192},[31642],[34,272194,272196],{"className":272195},[31646],[31648,272197,272198],{"xmlns":31650},[31652,272199,272200,272216],{},[31655,272201,272202,272204,272206,272208,272210,272212,272214],{},[31749,272203,15932],{},[31662,272205,223],{},[31749,272207,167308],{},[31662,272209,223],{},[31749,272211,188623],{},[31662,272213,202],{},[31749,272215,188930],{},[31665,272217,188933],{"encoding":31667},[34,272219,272221,272239,272257,272275],{"className":272220,"ariaHidden":758},[31672],[34,272222,272224,272227,272230,272233,272236],{"className":272223},[31676],[34,272225],{"className":272226,"style":35942},[31680],[34,272228,15932],{"className":272229},[31685],[34,272231],{"className":272232,"style":32604},[31845],[34,272234,223],{"className":272235},[32024],[34,272237],{"className":272238,"style":32604},[31845],[34,272240,272242,272245,272248,272251,272254],{"className":272241},[31676],[34,272243],{"className":272244,"style":35942},[31680],[34,272246,167308],{"className":272247},[31685],[34,272249],{"className":272250,"style":32604},[31845],[34,272252,223],{"className":272253},[32024],[34,272255],{"className":272256,"style":32604},[31845],[34,272258,272260,272263,272266,272269,272272],{"className":272259},[31676],[34,272261],{"className":272262,"style":32617},[31680],[34,272264,188623],{"className":272265},[31685],[34,272267],{"className":272268,"style":31846},[31845],[34,272270,202],{"className":272271},[31850],[34,272273],{"className":272274,"style":31846},[31845],[34,272276,272278,272281],{"className":272277},[31676],[34,272279],{"className":272280,"style":32617},[31680],[34,272282,188930],{"className":272283},[31685]," 块。",[108,272286,272287,272290,272291,272319,272320,33775,272323,272326],{},[2627,272288,272289],{},"文件最大长度："," 66052 块 ",[34,272292,272294,272307],{"className":272293},[31642],[34,272295,272297],{"className":272296},[31646],[31648,272298,272299],{"xmlns":31650},[31652,272300,272301,272305],{},[31655,272302,272303],{},[31662,272304,42071],{},[31665,272306,211013],{"encoding":31667},[34,272308,272310],{"className":272309,"ariaHidden":758},[31672],[34,272311,272313,272316],{"className":272312},[31676],[34,272314],{"className":272315,"style":32597},[31680],[34,272317,42071],{"className":272318},[31685]," 1KB = ",[2627,272321,272322],{},"66052 KB",[2627,272324,272325],{},"64MB + 516KB","）。\n:::",[19,272328,272330],{"id":272329},"_21-单道调度fcfs-sjf-hrrn-对比","21. 单道调度：FCFS \u002F SJF \u002F HRRN 对比",[15,272332,272333],{},"如果在一段时间内先后有 5 个作业，它们提交和运行时间由下表给出。",[13837,272335,272336,272354],{},[13840,272337,272338],{},[13843,272339,272340,272342,272344,272346,272348,272350,272352],{},[13846,272341,230223],{"align":82355},[13846,272343,230226],{"align":82355},[13846,272345,264478],{"align":82355},[13846,272347,230231],{"align":82355},[13846,272349,264483],{"align":82355},[13846,272351,230236],{"align":82355},[13846,272353,230239],{"align":82355},[13853,272355,272356,272372,272388,272404,272420],{},[13843,272357,272358,272360,272362,272364,272366,272368,272370],{},[13858,272359,640],{"align":82355},[13858,272361,230248],{"align":82355},[13858,272363,5774],{"align":82355},[13858,272365],{"align":82355},[13858,272367],{"align":82355},[13858,272369],{"align":82355},[13858,272371],{"align":82355},[13843,272373,272374,272376,272378,272380,272382,272384,272386],{},[13858,272375,768],{"align":82355},[13858,272377,264512],{"align":82355},[13858,272379,9314],{"align":82355},[13858,272381],{"align":82355},[13858,272383],{"align":82355},[13858,272385],{"align":82355},[13858,272387],{"align":82355},[13843,272389,272390,272392,272394,272396,272398,272400,272402],{},[13858,272391,786],{"align":82355},[13858,272393,230265],{"align":82355},[13858,272395,3249],{"align":82355},[13858,272397],{"align":82355},[13858,272399],{"align":82355},[13858,272401],{"align":82355},[13858,272403],{"align":82355},[13843,272405,272406,272408,272410,272412,272414,272416,272418],{},[13858,272407,15932],{"align":82355},[13858,272409,230282],{"align":82355},[13858,272411,3249],{"align":82355},[13858,272413],{"align":82355},[13858,272415],{"align":82355},[13858,272417],{"align":82355},[13858,272419],{"align":82355},[13843,272421,272422,272424,272426,272428,272430,272432,272434],{},[13858,272423,18136],{"align":82355},[13858,272425,230299],{"align":82355},[13858,272427,3244],{"align":82355},[13858,272429],{"align":82355},[13858,272431],{"align":82355},[13858,272433],{"align":82355},[13858,272435],{"align":82355},[15,272437,11832],{},[105,272439,272440,272443],{},[108,272441,272442],{},"(1) 分别使用先来先服务、最短进程优先和最高响应比算法完成上表，给出各个作业的开始时间、完成时间、周转时间和带权周转时间；",[108,272444,272445],{},"(2) 计算各算法的平均作业周转时间和平均作业带权周转时间。",[15,272447,264303],{},[253,272449,219117],{"id":272450},"原理解析-the-why-4",[15,272452,272453,272454,272643],{},"响应比 ",[34,272455,272457,272487],{"className":272456},[31642],[34,272458,272460],{"className":272459},[31646],[31648,272461,272462],{"xmlns":31650},[31652,272463,272464,272484],{},[31655,272465,272466,272472,272474,272476,272478],{},[31743,272467,272468,272470],{},[31658,272469,34386],{},[31658,272471,34386],{},[31662,272473,202],{},[31749,272475,640],{},[31662,272477,223],{},[41453,272479,272480,272482],{},[39361,272481,216949],{},[39361,272483,216954],{},[31665,272485,272486],{"encoding":31667},"R_R = 1 + \\frac{\\text{等待时间}}{\\text{服务时间}}",[34,272488,272490,272545,272563],{"className":272489,"ariaHidden":758},[31672],[34,272491,272493,272496,272536,272539,272542],{"className":272492},[31676],[34,272494],{"className":272495,"style":70915},[31680],[34,272497,272499,272502],{"className":272498},[31685],[34,272500,34386],{"className":272501,"style":34427},[31685,31686],[34,272503,272505],{"className":272504},[31792],[34,272506,272508,272528],{"className":272507},[31796,31797],[34,272509,272511,272525],{"className":272510},[31801],[34,272512,272514],{"className":272513,"style":116558},[31805],[34,272515,272516,272519],{"style":104694},[34,272517],{"className":272518,"style":31814},[31813],[34,272520,272522],{"className":272521},[31818,31819,31820,31821],[34,272523,34386],{"className":272524,"style":34427},[31685,31686,31821],[34,272526,31829],{"className":272527},[31828],[34,272529,272531],{"className":272530},[31801],[34,272532,272534],{"className":272533,"style":31836},[31805],[34,272535],{},[34,272537],{"className":272538,"style":31846},[31845],[34,272540,202],{"className":272541},[31850],[34,272543],{"className":272544,"style":31846},[31845],[34,272546,272548,272551,272554,272557,272560],{"className":272547},[31676],[34,272549],{"className":272550,"style":35942},[31680],[34,272552,640],{"className":272553},[31685],[34,272555],{"className":272556,"style":32604},[31845],[34,272558,223],{"className":272559},[32024],[34,272561],{"className":272562,"style":32604},[31845],[34,272564,272566,272569],{"className":272565},[31676],[34,272567],{"className":272568,"style":168026},[31680],[34,272570,272572,272575,272640],{"className":272571},[31685],[34,272573],{"className":272574},[31782,41488],[34,272576,272578],{"className":272577},[41453],[34,272579,272581,272632],{"className":272580},[31796,31797],[34,272582,272584,272629],{"className":272583},[31801],[34,272585,272587,272604,272612],{"className":272586,"style":168045},[31805],[34,272588,272589,272592],{"style":41504},[34,272590],{"className":272591,"style":41508},[31813],[34,272593,272595],{"className":272594},[31818,31819,31820,31821],[34,272596,272598],{"className":272597},[31685,31821],[34,272599,272601],{"className":272600},[31685,2638,31821],[34,272602,216954],{"className":272603},[31685,105462,31821],[34,272605,272606,272609],{"style":41520},[34,272607],{"className":272608,"style":41508},[31813],[34,272610],{"className":272611,"style":41528},[41527],[34,272613,272614,272617],{"style":46771},[34,272615],{"className":272616,"style":41508},[31813],[34,272618,272620],{"className":272619},[31818,31819,31820,31821],[34,272621,272623],{"className":272622},[31685,31821],[34,272624,272626],{"className":272625},[31685,2638,31821],[34,272627,216949],{"className":272628},[31685,105462,31821],[34,272630,31829],{"className":272631},[31828],[34,272633,272635],{"className":272634},[31801],[34,272636,272638],{"className":272637,"style":41619},[31805],[34,272639],{},[34,272641],{"className":272642},[31690,41488],"。HRRN在每次有作业完成时，动态计算队列中所有人的响应比，挑最大的执行。",[253,272645,267207],{"id":272646},"答案与详细解答-5",[15,272648,272649],{},[2627,272650,272651],{},"(1) FCFS (先来先服务)",[13837,272653,272654,272675],{},[13840,272655,272656],{},[13843,272657,272658,272660,272663,272666,272668,272670,272672],{},[13846,272659,230486],{"align":82355},[13846,272661,272662],{"align":82355},"提交",[13846,272664,272665],{"align":82355},"运行",[13846,272667,230231],{"align":82355},[13846,272669,214469],{"align":82355},[13846,272671,230236],{"align":82355},[13846,272673,272674],{"align":82355},"带权周转",[13853,272676,272677,272695,272716,272735,272754,272773],{},[13843,272678,272679,272683,272685,272687,272689,272691,272693],{},[13858,272680,272681],{"align":82355},[2627,272682,640],{},[13858,272684,230248],{"align":82355},[13858,272686,5774],{"align":82355},[13858,272688,230248],{"align":82355},[13858,272690,230424],{"align":82355},[13858,272692,5774],{"align":82355},[13858,272694,271818],{"align":82355},[13843,272696,272697,272701,272703,272705,272707,272710,272713],{},[13858,272698,272699],{"align":82355},[2627,272700,768],{},[13858,272702,264512],{"align":82355},[13858,272704,9314],{"align":82355},[13858,272706,230424],{"align":82355},[13858,272708,272709],{"align":82355},"9:20",[13858,272711,272712],{"align":82355},"65",[13858,272714,272715],{"align":82355},"2.167",[13843,272717,272718,272722,272724,272726,272728,272730,272732],{},[13858,272719,272720],{"align":82355},[2627,272721,786],{},[13858,272723,230265],{"align":82355},[13858,272725,3249],{"align":82355},[13858,272727,272709],{"align":82355},[13858,272729,271462],{"align":82355},[13858,272731,27539],{"align":82355},[13858,272733,272734],{"align":82355},"4.0",[13843,272736,272737,272741,272743,272745,272747,272749,272751],{},[13858,272738,272739],{"align":82355},[2627,272740,15932],{},[13858,272742,230282],{"align":82355},[13858,272744,3249],{"align":82355},[13858,272746,271462],{"align":82355},[13858,272748,229149],{"align":82355},[13858,272750,96817],{"align":82355},[13858,272752,272753],{"align":82355},"4.75",[13843,272755,272756,272760,272762,272764,272766,272768,272770],{},[13858,272757,272758],{"align":82355},[2627,272759,18136],{},[13858,272761,230299],{"align":82355},[13858,272763,3244],{"align":82355},[13858,272765,229149],{"align":82355},[13858,272767,229167],{"align":82355},[13858,272769,162173],{"align":82355},[13858,272771,272772],{"align":82355},"10.0",[13843,272774,272775,272882,272884,272886,272888,272890,272892],{},[13858,272776,272777,47649,272780,36268,272846,47649,272849],{"align":82355},[71807,272778,272779],{},"平均周转:",[34,272781,272783,272811],{"className":272782},[31642],[34,272784,272786],{"className":272785},[31646],[31648,272787,272788],{"xmlns":31650},[31652,272789,272790,272808],{},[31655,272791,272792,272795,272797,272799,272801],{},[31749,272793,272794],{},"390",[31658,272796,38080],{"mathvariant":38079},[31749,272798,18136],{},[31662,272800,202],{},[31655,272802,272803,272806],{},[31749,272804,272805],{"mathvariant":226163},"78",[31658,272807,31759],{"mathvariant":226163},[31665,272809,272810],{"encoding":31667},"390 \u002F 5 = \\mathbf{78 m}",[34,272812,272814,272833],{"className":272813,"ariaHidden":758},[31672],[34,272815,272817,272820,272824,272827,272830],{"className":272816},[31676],[34,272818],{"className":272819,"style":31774},[31680],[34,272821,272823],{"className":272822},[31685],"390\u002F5",[34,272825],{"className":272826,"style":31846},[31845],[34,272828,202],{"className":272829},[31850],[34,272831],{"className":272832,"style":31846},[31845],[34,272834,272836,272839],{"className":272835},[31676],[34,272837],{"className":272838,"style":32617},[31680],[34,272840,272842],{"className":272841},[31685],[34,272843,272845],{"className":272844},[31685,226302],"78m",[71807,272847,272848],{},"平均带权:",[34,272850,272852,272867],{"className":272851},[31642],[34,272853,272855],{"className":272854},[31646],[31648,272856,272857],{"xmlns":31650},[31652,272858,272859,272864],{},[31655,272860,272861],{},[31749,272862,272863],{"mathvariant":226163},"4.383",[31665,272865,272866],{"encoding":31667},"\\mathbf{4.383}",[34,272868,272870],{"className":272869,"ariaHidden":758},[31672],[34,272871,272873,272876],{"className":272872},[31676],[34,272874],{"className":272875,"style":32617},[31680],[34,272877,272879],{"className":272878},[31685],[34,272880,272863],{"className":272881},[31685,226302],[13858,272883],{"align":82355},[13858,272885],{"align":82355},[13858,272887],{"align":82355},[13858,272889],{"align":82355},[13858,272891],{"align":82355},[13858,272893],{"align":82355},[15,272895,272896],{},[2627,272897,272898],{},"(2) SJF (最短作业优先 - 非抢占)",[15,272900,272901,272902,272930,272931,272959,272960,272988,272989,273017],{},"J1(8:00~8:50)。8:50时队列中全员到齐，按运行时间从短到长选：J5(10) > J3(20) = J4(20) > J2(30)。J3比J4早到，故J3先。执行顺序：J1 ",[34,272903,272905,272918],{"className":272904},[31642],[34,272906,272908],{"className":272907},[31646],[31648,272909,272910],{"xmlns":31650},[31652,272911,272912,272916],{},[31655,272913,272914],{},[31662,272915,33239],{},[31665,272917,119689],{"encoding":31667},[34,272919,272921],{"className":272920,"ariaHidden":758},[31672],[34,272922,272924,272927],{"className":272923},[31676],[34,272925],{"className":272926,"style":73089},[31680],[34,272928,33239],{"className":272929},[31850]," J5 ",[34,272932,272934,272947],{"className":272933},[31642],[34,272935,272937],{"className":272936},[31646],[31648,272938,272939],{"xmlns":31650},[31652,272940,272941,272945],{},[31655,272942,272943],{},[31662,272944,33239],{},[31665,272946,119689],{"encoding":31667},[34,272948,272950],{"className":272949,"ariaHidden":758},[31672],[34,272951,272953,272956],{"className":272952},[31676],[34,272954],{"className":272955,"style":73089},[31680],[34,272957,33239],{"className":272958},[31850]," J3 ",[34,272961,272963,272976],{"className":272962},[31642],[34,272964,272966],{"className":272965},[31646],[31648,272967,272968],{"xmlns":31650},[31652,272969,272970,272974],{},[31655,272971,272972],{},[31662,272973,33239],{},[31665,272975,119689],{"encoding":31667},[34,272977,272979],{"className":272978,"ariaHidden":758},[31672],[34,272980,272982,272985],{"className":272981},[31676],[34,272983],{"className":272984,"style":73089},[31680],[34,272986,33239],{"className":272987},[31850]," J4 ",[34,272990,272992,273005],{"className":272991},[31642],[34,272993,272995],{"className":272994},[31646],[31648,272996,272997],{"xmlns":31650},[31652,272998,272999,273003],{},[31655,273000,273001],{},[31662,273002,33239],{},[31665,273004,119689],{"encoding":31667},[34,273006,273008],{"className":273007,"ariaHidden":758},[31672],[34,273009,273011,273014],{"className":273010},[31676],[34,273012],{"className":273013,"style":73089},[31680],[34,273015,33239],{"className":273016},[31850]," J2。",[13837,273019,273020,273038],{},[13840,273021,273022],{},[13843,273023,273024,273026,273028,273030,273032,273034,273036],{},[13846,273025,230486],{"align":82355},[13846,273027,272662],{"align":82355},[13846,273029,272665],{"align":82355},[13846,273031,230231],{"align":82355},[13846,273033,214469],{"align":82355},[13846,273035,230236],{"align":82355},[13846,273037,272674],{"align":82355},[13853,273039,273040,273058,273077,273095,273114,273134],{},[13843,273041,273042,273046,273048,273050,273052,273054,273056],{},[13858,273043,273044],{"align":82355},[2627,273045,640],{},[13858,273047,230248],{"align":82355},[13858,273049,5774],{"align":82355},[13858,273051,230248],{"align":82355},[13858,273053,230424],{"align":82355},[13858,273055,5774],{"align":82355},[13858,273057,271818],{"align":82355},[13843,273059,273060,273064,273066,273068,273070,273072,273074],{},[13858,273061,273062],{"align":82355},[2627,273063,18136],{},[13858,273065,230299],{"align":82355},[13858,273067,3244],{"align":82355},[13858,273069,230424],{"align":82355},[13858,273071,230441],{"align":82355},[13858,273073,9314],{"align":82355},[13858,273075,273076],{"align":82355},"3.0",[13843,273078,273079,273083,273085,273087,273089,273091,273093],{},[13858,273080,273081],{"align":82355},[2627,273082,786],{},[13858,273084,230265],{"align":82355},[13858,273086,3249],{"align":82355},[13858,273088,230441],{"align":82355},[13858,273090,272709],{"align":82355},[13858,273092,167664],{"align":82355},[13858,273094,273076],{"align":82355},[13843,273096,273097,273101,273103,273105,273107,273109,273111],{},[13858,273098,273099],{"align":82355},[2627,273100,15932],{},[13858,273102,230282],{"align":82355},[13858,273104,3249],{"align":82355},[13858,273106,272709],{"align":82355},[13858,273108,271462],{"align":82355},[13858,273110,267828],{"align":82355},[13858,273112,273113],{"align":82355},"3.75",[13843,273115,273116,273120,273122,273124,273126,273128,273131],{},[13858,273117,273118],{"align":82355},[2627,273119,768],{},[13858,273121,264512],{"align":82355},[13858,273123,9314],{"align":82355},[13858,273125,271462],{"align":82355},[13858,273127,229167],{"align":82355},[13858,273129,273130],{"align":82355},"115",[13858,273132,273133],{"align":82355},"3.833",[13843,273135,273136,273240,273242,273244,273246,273248,273250],{},[13858,273137,273138,47649,273140,36268,273205,47649,273207],{"align":82355},[71807,273139,272779],{},[34,273141,273143,273170],{"className":273142},[31642],[34,273144,273146],{"className":273145},[31646],[31648,273147,273148],{"xmlns":31650},[31652,273149,273150,273167],{},[31655,273151,273152,273154,273156,273158,273160],{},[31749,273153,75745],{},[31658,273155,38080],{"mathvariant":38079},[31749,273157,18136],{},[31662,273159,202],{},[31655,273161,273162,273165],{},[31749,273163,273164],{"mathvariant":226163},"66",[31658,273166,31759],{"mathvariant":226163},[31665,273168,273169],{"encoding":31667},"330 \u002F 5 = \\mathbf{66 m}",[34,273171,273173,273192],{"className":273172,"ariaHidden":758},[31672],[34,273174,273176,273179,273183,273186,273189],{"className":273175},[31676],[34,273177],{"className":273178,"style":31774},[31680],[34,273180,273182],{"className":273181},[31685],"330\u002F5",[34,273184],{"className":273185,"style":31846},[31845],[34,273187,202],{"className":273188},[31850],[34,273190],{"className":273191,"style":31846},[31845],[34,273193,273195,273198],{"className":273194},[31676],[34,273196],{"className":273197,"style":32617},[31680],[34,273199,273201],{"className":273200},[31685],[34,273202,273204],{"className":273203},[31685,226302],"66m",[71807,273206,272848],{},[34,273208,273210,273225],{"className":273209},[31642],[34,273211,273213],{"className":273212},[31646],[31648,273214,273215],{"xmlns":31650},[31652,273216,273217,273222],{},[31655,273218,273219],{},[31749,273220,273221],{"mathvariant":226163},"2.917",[31665,273223,273224],{"encoding":31667},"\\mathbf{2.917}",[34,273226,273228],{"className":273227,"ariaHidden":758},[31672],[34,273229,273231,273234],{"className":273230},[31676],[34,273232],{"className":273233,"style":32617},[31680],[34,273235,273237],{"className":273236},[31685],[34,273238,273221],{"className":273239},[31685,226302],[13858,273241],{"align":82355},[13858,273243],{"align":82355},[13858,273245],{"align":82355},[13858,273247],{"align":82355},[13858,273249],{"align":82355},[13858,273251],{"align":82355},[15,273253,273254],{},[2627,273255,273256],{},"(3) HRRN (最高响应比优先)",[15,273258,273259],{},"对于这组数据，HRRN 的调度顺序与 SJF 完全一致（依然是 1-5-3-4-2），时间表和平均时间同 SJF 算法。\n:::",[19,273261,273263],{"id":273262},"_22-请求分页4页框页面走向-3456356375","22. 请求分页：4页框，页面走向 3,4,5,6,3,5,6,3,7,5",[15,273265,273266],{},"在请求分页管理系统中, 一个程序的页面走向为: 3, 4, 5, 6, 3, 5, 6, 3, 7, 5，设分配给该程序的存储块为 4。所有内存开始时都是空的，凡第一次用到的页面都产生一次缺页中断。",[15,273268,264303],{},[15,273270,264306],{},[19,273272,273274],{"id":273273},"_23-磁盘顺序文件逻辑记录定位","23. 磁盘顺序文件：逻辑记录定位",[15,273276,273277],{},"假设有一个磁盘组共有 100 个柱面，每个柱面上有 8 个磁道，每个盘面被分成 8 个扇区。现有一个含有 6400 逻辑记录的文件，逻辑记录的大小与扇区一致，该文件以顺序结构的形式被存储到磁盘上。柱面、磁道、扇区的编号从\"0\"开始，逻辑记录的编号也从\"0\"开始。文件信息从 0 柱面、0 磁道、0 扇区开始存放，试问：",[105,273279,273280,273283],{},[108,273281,273282],{},"(1) 该文件的 3680 个逻辑记录应该存放在什么位置？",[108,273284,273285],{},"(2) 78 柱面的 6 磁道的 6 扇区中存放了该文件的第几号逻辑记录？",[15,273287,264303],{},[15,273289,264306],{},[19,273291,273293],{"id":273292},"_24-生产者-消费者getproput-同步互斥","24. 生产者-消费者：GET\u002FPRO\u002FPUT 同步互斥",[15,273295,273296],{},"如图所示，系统中有三个进程 GET、PRO 和 PUT，共用两个缓冲区 BUF1 和 BUF2。假设 BUF1 中最多可放 11 个信息，现已放入了两个信息；BUF2 最多可放 5 个信息。GET 进程负责不断地将输入信息送入 BUF1 中，PRO 进程负责从 BUF1 中取出信息进行处理，并将处理结果送到 BUF2 中，PUT 进程负责从 BUF2 中读取结果并输出。试写出正确实现 GET、PRO、PUT 的同步与互斥的算法（要求：（1）用类 C 语言描述，条理清楚，注释恰当；（2）信号量原语统一使用 wait 和 signal。）",[104833,273298,273299],{},[15,273300,273301],{},"图 进程合作：GET → BUF1 → PRO → BUF2 → PUT",[15,273303,264303],{},[15,273305,264306],{},[19,273307,273309],{"id":273308},"_25-死锁判断r1r2-设备申请顺序","25. 死锁判断：R1\u002FR2 设备申请顺序",[15,273311,273312],{},"某系统有 R1 设备 3 台，R2 设备 4 台，它们被 P1、P2、P3 和 P4 进程共享，且已知这 4 个进程均按以下顺序使用设备：",[15,273314,273315],{},[31,273316,273317],{},"→申请R1→申请R2→申请R1→释放R1→释放R2→释放R1",[105,273319,273320,273323],{},[108,273321,273322],{},"(1) 系统运行中可能产生死锁吗？为什么？",[108,273324,273325],{},"(2) 若可能的话，请举出一种情况，并画出表示该死锁状态的进程—资源图。",[15,273327,264303],{},[15,273329,264306],{},[19,273331,273333],{"id":273332},"_26-并发进程执行路径与与时间相关的错误","26. 并发进程执行路径与与时间相关的错误",[15,273335,273336],{},"(1) 两个并发进程并发执行，其中，A、B、C、D、E 是原语，试给出可能的并发执行路径。",[24,273338,273341],{"className":273339,"code":273340,"language":2638},[2636],"Process P Process Q\nbegin begin\n A; D;\n B; E;\n C; end;\nend;\n",[31,273342,273340],{"__ignoreMap":29},[15,273344,273345],{},"(2) 两个并发进程 P1 和 P2 并发执行，它们的程序分别如下：",[24,273347,273350],{"className":273348,"code":273349,"language":2638},[2636]," P1 P2\n repeat repeat\n k:=k×2; print k;\n k:=k+1; k:=0;\n until false; until false;\n",[31,273351,273349],{"__ignoreMap":29},[15,273353,273354],{},"若令 k 的初值为 5，让 P1 先执行两个循环，然后，P1 和 P2 又并发执行了一个循环，写出可能的打印值，指出与时间有关的错误。",[15,273356,264303],{},[15,273358,264306],{},[19,273360,273362],{"id":273361},"_27-快表命中率计算","27. 快表命中率计算",[15,273364,273365],{},"一台计算机的内存空间为 1024 个页面，页表放在内存中，从页表中读一个字的开销是 500ns。为了减少开销，使用了有 32 个字的快表，查找速度为 100ns。要把平均开销降到 200ns 需要的快表命中率是多少?",[15,273367,264303],{},[15,273369,264306],{},[19,273371,273373],{"id":273372},"_28-磁带分块存储与利用率","28. 磁带分块存储与利用率",[15,273375,273376],{},"假定磁带记录密度为每英寸 800 字符，每一逻辑记录为 200 字符，块间隔为 0.6 英寸。现有 3200 个逻辑记录需要存储，如果不考虑存储记录，则不成组处理和以 8 个逻辑记录为一组的成组处理时磁带的利用率各是多少？两种情况下，3200 个逻辑记录需要占用多少磁带空间？",[15,273378,264303],{},[15,273380,264306],{},[19,273382,273384],{"id":273383},"_29-unix-文件权限判断","29. UNIX 文件权限判断",[15,273386,273387,273388,273391],{},"一个 UNIX 文件 F 的存取权限为：",[31,273389,273390],{},"rwxr-x---","，该文件的文件主 uid＝12，gid=1，另一个用户的 uid=6，gid=1，是否允许该用户执行文件 F？",[15,273393,264303],{},[15,273395,264306],{},[19,273397,273399],{"id":273398},"_30-unixlinux-文件访问偏移间接次数","30. UNIX\u002FLinux 文件访问偏移：间接次数",[15,273401,273402],{},"一个 UNIX\u002FLinux 文件，如果一个盘块的大小为 1KB，每个盘块占 4 个字节，那么，若进程欲访问偏移为 263168 字节处的数据，需经过几次间接？",[15,273404,264303],{},[15,273406,264306],{},[19,273408,273410],{"id":273409},"_31-磁盘交错存储20msr8扇区","31. 磁盘交错存储：20ms\u002Fr、8扇区",[15,273412,273413],{},"假设有 8 个记录 A、B，C、D、E、F、G、H 存放在磁盘里，每个磁道有 8 个扇区，正好可以存放 8 个记录。假设磁盘旋转速度为 20ms\u002F转，处理程序每读出一个记录后，用 2ms 的时间进行处理，请问：",[105,273415,273416,273418],{},[108,273417,170045],{},[108,273419,273420],{},"b. 如何采取优化方法，使处理这些记录所花费的总时间最短？求出该最短时间。",[15,273422,264303],{},[15,273424,264306],{},[19,273426,273428],{"id":273427},"_32-可变分区管理sjf-与-hrrn-调度","32. 可变分区管理：SJF 与 HRRN 调度",[15,273430,273431],{},"系统采用不能移动的可变分区管理方案，现有可供用户使用的主存空间为 100K，设有四个作业 J1, J2, J3, J4 它们的到达时间和计算时间如下表:",[13837,273433,273434,273450],{},[13840,273435,273436],{},[13843,273437,273438,273440,273442,273445,273448],{},[13846,273439,230486],{"align":82355},[13846,273441,214474],{"align":82355},[13846,273443,273444],{"align":82355},"计算时间",[13846,273446,273447],{"align":82355},"需要主存容量",[13846,273449,230236],{"align":82355},[13853,273451,273452,273467,273482,273496],{},[13843,273453,273454,273457,273459,273462,273465],{},[13858,273455,273456],{"align":82355},"J1",[13858,273458,230248],{"align":82355},[13858,273460,273461],{"align":82355},"40分钟",[13858,273463,273464],{"align":82355},"30K",[13858,273466],{"align":82355},[13843,273468,273469,273472,273474,273477,273480],{},[13858,273470,273471],{"align":82355},"J2",[13858,273473,230265],{"align":82355},[13858,273475,273476],{"align":82355},"35分钟",[13858,273478,273479],{"align":82355},"70K",[13858,273481],{"align":82355},[13843,273483,273484,273487,273489,273492,273494],{},[13858,273485,273486],{"align":82355},"J3",[13858,273488,230299],{"align":82355},[13858,273490,273491],{"align":82355},"20分钟",[13858,273493,273464],{"align":82355},[13858,273495],{"align":82355},[13843,273497,273498,273501,273503,273506,273509],{},[13858,273499,273500],{"align":82355},"J4",[13858,273502,230333],{"align":82355},[13858,273504,273505],{"align":82355},"10分钟",[13858,273507,273508],{"align":82355},"20K",[13858,273510],{"align":82355},[15,273512,273513],{},"若作业在处理机上按单道方式运行，请分别写出：",[105,273515,273516,273519],{},[108,273517,273518],{},"（1）最短者优先算法选中作业的执行顺序，并计算周转时间和平均周转时间。",[108,273520,273521],{},"（2）响应比高者优先算法选中作业的执行顺序，并计算周转时间和平均周转时间。",[15,273523,264303],{},[15,273525,264306],{},[19,273527,273529],{"id":273528},"_33-分页式虚存fifo-与-lru-缺页中断对比","33. 分页式虚存：FIFO 与 LRU 缺页中断对比",[15,273531,273532],{},"使用\"分页式\"虚拟存储管理技术，假设一个进程 P 的页面访问顺序如下：0 1 2 3 0 1 4 0 1 2 3 4。该进程创建时没有加载任何页面，即该进程启动时其所有指令和数据都不在内存中。",[105,273534,273535,273538,273541],{},[108,273536,273537],{},"(1) 设分配给该进程的物理页帧为 3 个，使用 FIFO 页面置换算法时，请问会发生多少次缺页中断？使用硬件实现的 LRU 算法，会发生多少次缺页中断？",[108,273539,273540],{},"(2) 对于以上两种页面置换算法，如希望减少缺页中断的次数，是否可以通过增加物理页帧来解决？为什么？",[108,273542,273543],{},"(3) 在分页系统中将 I\u002FO 设备的数据缓冲区映射到内存空间后，其对应的页面是否能够被替换？为什么？",[15,273545,264303],{},[253,273547,223920],{"id":223919},[105,273549,273550,273556,273562],{},[108,273551,273552,273555],{},[2627,273553,273554],{},"FIFO (先进先出)","：淘汰在内存中驻留时间最久的页面。",[108,273557,273558,273561],{},[2627,273559,273560],{},"LRU (最近最少使用)","：淘汰最长时间没有被访问过的页面。",[108,273563,273564,273566],{},[2627,273565,257364],{},"：当 CPU 访问的页面不在实体内存中时触发。",[253,273568,267207],{"id":273569},"答案与详细解答-6",[15,273571,273572],{},[2627,273573,273574],{},"(1) 缺页中断次数模拟：",[105,273576,273577,273586],{},[108,273578,273579,273582,273583],{},[2627,273580,273581],{},"FIFO 算法 (3个页框)：","\n0(缺) → 1(缺) → 2(缺) → 3(缺，淘汰0) → 0(缺，淘汰1) → 1(缺，淘汰2) → 4(缺，淘汰3) → 0(命中) → 1(命中) → 2(缺，淘汰0) → 3(缺，淘汰1) → 4(命中)\n",[2627,273584,273585],{},"总计：发生 9 次缺页中断。",[108,273587,273588,273591,273592],{},[2627,273589,273590],{},"LRU 算法 (3个页框)：","\n0(缺) → 1(缺) → 2(缺) → 3(缺，淘汰0) → 0(缺，淘汰1) → 1(缺，淘汰2) → 4(缺，淘汰3) → 0(命中，更新) → 1(命中，更新) → 2(缺，淘汰4) → 3(缺，淘汰0) → 4(缺，淘汰1)\n",[2627,273593,273594],{},"总计：发生 10 次缺页中断。",[15,273596,273597],{},[2627,273598,273599],{},"(2) 增加物理页框是否能减少缺页？",[105,273601,273602,273608],{},[108,273603,273604,273607],{},[2627,273605,273606],{},"LRU：可以。"," 增加页框一定能减少或维持缺页次数（LRU 符合堆栈属性 Stack Property）。",[108,273609,273610,273613,273614,273617],{},[2627,273611,273612],{},"FIFO：不一定。"," FIFO 算法可能会遭遇 ",[2627,273615,273616],{},"Belady 异常 (Belady's Anomaly)","。对于某些特定的访问序列，增加分配的物理页框数量反而会导致缺页次数上升。",[15,273619,273620],{},[2627,273621,273622],{},"(3) 映射到内存的 I\u002FO 缓冲页面能否被替换？",[105,273624,273625,273630],{},[108,273626,273627],{},[2627,273628,273629],{},"不能。",[108,273631,273632,273634,273635,273638],{},[2627,273633,7819],{}," 如果正在进行 I\u002FO 操作（如 DMA 传输）的页面被置换到磁盘，而另一个进程的数据刚好载入这个物理页框，那么硬件 I\u002FO 设备会将数据直接覆盖掉无辜进程的内存，造成严重的数据损坏。因此，涉及 I\u002FO 的页面必须被",[2627,273636,273637],{},"锁定 (Pinned\u002FLocked)"," 在主存中，禁止被替换。",[15,273640,273641,273643],{},[2627,273642,246532],{}," 学生常会直觉认为\"更好的算法\"在任何情况下都会比\"简单的算法\"表现好。这题恰好反直觉，LRU（10次）的缺页次数反而高于 FIFO（9次）。\n:::",[19,273645,273647],{"id":273646},"_34-连接文件逻辑字节访问对应物理块","34. 连接文件：逻辑字节访问对应物理块",[15,273649,273650],{},"设某文件为连接文件，由 5 个逻辑记录组成，每个逻辑记录的大小与磁盘块大小相等，均为 512 字节，并依次存放在 50、121、75、80、63 号磁盘块上。若要存取文件的第 1569 逻辑字节处的信息，问要访问哪一个磁盘块？",[15,273652,264303],{},[253,273654,267207],{"id":273655},"答案与详细解答-7",[15,273657,273658],{},"已知每个逻辑记录和磁盘块的大小为 512 Bytes。",[2230,273660,273661,273776,273796],{},[108,273662,273663,273664,273767,273768,273771,273772,273775],{},"计算目标字节位于第几个逻辑记录：\n",[34,273665,273667,273697],{"className":273666},[31642],[34,273668,273670],{"className":273669},[31646],[31648,273671,273672],{"xmlns":31650},[31652,273673,273674,273694],{},[31655,273675,273676,273679,273681,273683,273685,273687,273689,273692],{},[31749,273677,273678],{},"1569",[31662,273680,207557],{},[31749,273682,167308],{},[31662,273684,202],{},[31749,273686,786],{},[31662,273688,33934],{},[39361,273690,273691],{},"余数 ",[31749,273693,185190],{},[31665,273695,273696],{"encoding":31667},"1569 \\div 512 = 3 \\ldots \\text{余数 } 33",[34,273698,273700,273718,273736],{"className":273699,"ariaHidden":758},[31672],[34,273701,273703,273706,273709,273712,273715],{"className":273702},[31676],[34,273704],{"className":273705,"style":35942},[31680],[34,273707,273678],{"className":273708},[31685],[34,273710],{"className":273711,"style":32604},[31845],[34,273713,207557],{"className":273714},[32024],[34,273716],{"className":273717,"style":32604},[31845],[34,273719,273721,273724,273727,273730,273733],{"className":273720},[31676],[34,273722],{"className":273723,"style":32617},[31680],[34,273725,167308],{"className":273726},[31685],[34,273728],{"className":273729,"style":31846},[31845],[34,273731,202],{"className":273732},[31850],[34,273734],{"className":273735,"style":31846},[31845],[34,273737,273739,273742,273745,273748,273751,273754,273764],{"className":273738},[31676],[34,273740],{"className":273741,"style":33218},[31680],[34,273743,786],{"className":273744},[31685],[34,273746],{"className":273747,"style":32123},[31845],[34,273749,33934],{"className":273750},[34006],[34,273752],{"className":273753,"style":32123},[31845],[34,273755,273757,273761],{"className":273756},[31685,2638],[34,273758,273760],{"className":273759},[31685,105462],"余数",[34,273762,130298],{"className":273763},[31685],[34,273765,185190],{"className":273766},[31685],"。\n这表示第 1569 个字节位在",[2627,273769,273770],{},"第 3 个完整的逻辑块之后","（也就是进入了",[2627,273773,273774],{},"第 4 个逻辑记录","，索引为 3），偏移量为 33。",[108,273777,273778,273779],{},"对应物理磁盘块顺序（从0开始索引）：\n",[105,273780,273781,273784,273787,273790],{},[108,273782,273783],{},"索引 0：第 50 号块",[108,273785,273786],{},"索引 1：第 121 号块",[108,273788,273789],{},"索引 2：第 75 号块",[108,273791,273792,273793],{},"索引 3：",[2627,273794,273795],{},"第 80 号块",[108,273797,273798,273799,347],{},"因此，需要访问的是",[2627,273800,273801],{},"第 80 号磁盘块",[15,273803,273804,273806],{},[2627,273805,246532],{}," 商为 3 代表前面已经有 3 个完整的块（索引0, 1, 2），数据落在下一个块（索引 3），也就是清单中的第 4 个块。\n:::",[19,273808,273810],{"id":273809},"_35-hrrn-调度五个作业响应比与周转时间","35. HRRN 调度：五个作业响应比与周转时间",[15,273812,273813],{},"设有某系统可供用户使用的主存空间为 100K，有五个作业 J1, J2, J3, J4, J5 进入输入井的时间、计算时间和内存要求如下表所示。若作业在处理机上按单道方式运行，且作业按响应比高者优先调度算法，进程按先来先服务算法。试写出作业的执行顺序，计算响应比、作业的周转时间和平均周转时间。",[13837,273815,273816,273835],{},[13840,273817,273818],{},[13843,273819,273820,273822,273825,273827,273829,273831,273833],{},[13846,273821,230486],{"align":82355},[13846,273823,273824],{"align":82355},"进入输入井时间",[13846,273826,273444],{"align":82355},[13846,273828,273447],{"align":82355},[13846,273830,230231],{"align":82355},[13846,273832,264483],{"align":82355},[13846,273834,230236],{"align":82355},[13853,273836,273837,273856,273875,273893,273911],{},[13843,273838,273839,273841,273844,273847,273850,273852,273854],{},[13858,273840,273456],{"align":82355},[13858,273842,273843],{"align":82355},"10:06",[13858,273845,273846],{"align":82355},"42分钟",[13858,273848,273849],{"align":82355},"18K",[13858,273851],{"align":82355},[13858,273853],{"align":82355},[13858,273855],{"align":82355},[13843,273857,273858,273860,273863,273866,273869,273871,273873],{},[13858,273859,273471],{"align":82355},[13858,273861,273862],{"align":82355},"10:19",[13858,273864,273865],{"align":82355},"30分钟",[13858,273867,273868],{"align":82355},"65K",[13858,273870],{"align":82355},[13858,273872],{"align":82355},[13858,273874],{"align":82355},[13843,273876,273877,273879,273881,273884,273887,273889,273891],{},[13858,273878,273486],{"align":82355},[13858,273880,231132],{"align":82355},[13858,273882,273883],{"align":82355},"24分钟",[13858,273885,273886],{"align":82355},"57K",[13858,273888],{"align":82355},[13858,273890],{"align":82355},[13858,273892],{"align":82355},[13843,273894,273895,273897,273900,273902,273905,273907,273909],{},[13858,273896,273500],{"align":82355},[13858,273898,273899],{"align":82355},"10:36",[13858,273901,273883],{"align":82355},[13858,273903,273904],{"align":82355},"15K",[13858,273906],{"align":82355},[13858,273908],{"align":82355},[13858,273910],{"align":82355},[13843,273912,273913,273916,273919,273922,273925,273927,273929],{},[13858,273914,273915],{"align":82355},"J5",[13858,273917,273918],{"align":82355},"10:42",[13858,273920,273921],{"align":82355},"12分钟",[13858,273923,273924],{"align":82355},"25K",[13858,273926],{"align":82355},[13858,273928],{"align":82355},[13858,273930],{"align":82355},[15,273932,264303],{},[253,273934,267207],{"id":273935},"答案与详细解答-8",[105,273937,273938,273947,274543,274992,275297],{},[108,273939,273940,273942,273943,273946],{},[2627,273941,273843],{},"，只有 J1 到达。J1 开始执行。J1 执行 42 分钟，结束时间为 ",[2627,273944,273945],{},"10:48","。J1 周转时间 = 42 分。",[108,273948,273949,273951,273952],{},[2627,273950,273945],{},"，此时 J2, J3, J4, J5 皆已到达，计算响应比：\n",[105,273953,273954,274100,274245,274390,274533],{},[108,273955,273956,273957],{},"J2 等待 = 29分。",[34,273958,273960,273996],{"className":273959},[31642],[34,273961,273963],{"className":273962},[31646],[31648,273964,273965],{"xmlns":31650},[31652,273966,273967,273993],{},[31655,273968,273969,273975,273977,273979,273981,273984,273986,273988,273990],{},[31743,273970,273971,273973],{},[31658,273972,34386],{},[31749,273974,768],{},[31662,273976,202],{},[31749,273978,640],{},[31662,273980,223],{},[31749,273982,273983],{},"29",[31658,273985,38080],{"mathvariant":38079},[31749,273987,9314],{},[31662,273989,202],{},[31749,273991,273992],{},"1.967",[31665,273994,273995],{"encoding":31667},"R_2 = 1 + 29\u002F30 = 1.967",[34,273997,273999,274054,274072,274091],{"className":273998,"ariaHidden":758},[31672],[34,274000,274002,274005,274045,274048,274051],{"className":274001},[31676],[34,274003],{"className":274004,"style":70915},[31680],[34,274006,274008,274011],{"className":274007},[31685],[34,274009,34386],{"className":274010,"style":34427},[31685,31686],[34,274012,274014],{"className":274013},[31792],[34,274015,274017,274037],{"className":274016},[31796,31797],[34,274018,274020,274034],{"className":274019},[31801],[34,274021,274023],{"className":274022,"style":31806},[31805],[34,274024,274025,274028],{"style":104694},[34,274026],{"className":274027,"style":31814},[31813],[34,274029,274031],{"className":274030},[31818,31819,31820,31821],[34,274032,768],{"className":274033},[31685,31821],[34,274035,31829],{"className":274036},[31828],[34,274038,274040],{"className":274039},[31801],[34,274041,274043],{"className":274042,"style":31836},[31805],[34,274044],{},[34,274046],{"className":274047,"style":31846},[31845],[34,274049,202],{"className":274050},[31850],[34,274052],{"className":274053,"style":31846},[31845],[34,274055,274057,274060,274063,274066,274069],{"className":274056},[31676],[34,274058],{"className":274059,"style":35942},[31680],[34,274061,640],{"className":274062},[31685],[34,274064],{"className":274065,"style":32604},[31845],[34,274067,223],{"className":274068},[32024],[34,274070],{"className":274071,"style":32604},[31845],[34,274073,274075,274078,274082,274085,274088],{"className":274074},[31676],[34,274076],{"className":274077,"style":31774},[31680],[34,274079,274081],{"className":274080},[31685],"29\u002F30",[34,274083],{"className":274084,"style":31846},[31845],[34,274086,202],{"className":274087},[31850],[34,274089],{"className":274090,"style":31846},[31845],[34,274092,274094,274097],{"className":274093},[31676],[34,274095],{"className":274096,"style":32617},[31680],[34,274098,273992],{"className":274099},[31685],[108,274101,274102,274103],{},"J3 等待 = 18分。",[34,274104,274106,274141],{"className":274105},[31642],[34,274107,274109],{"className":274108},[31646],[31648,274110,274111],{"xmlns":31650},[31652,274112,274113,274138],{},[31655,274114,274115,274121,274123,274125,274127,274129,274131,274133,274135],{},[31743,274116,274117,274119],{},[31658,274118,34386],{},[31749,274120,786],{},[31662,274122,202],{},[31749,274124,640],{},[31662,274126,223],{},[31749,274128,95611],{},[31658,274130,38080],{"mathvariant":38079},[31749,274132,96087],{},[31662,274134,202],{},[31749,274136,274137],{},"1.75",[31665,274139,274140],{"encoding":31667},"R_3 = 1 + 18\u002F24 = 1.75",[34,274142,274144,274199,274217,274236],{"className":274143,"ariaHidden":758},[31672],[34,274145,274147,274150,274190,274193,274196],{"className":274146},[31676],[34,274148],{"className":274149,"style":70915},[31680],[34,274151,274153,274156],{"className":274152},[31685],[34,274154,34386],{"className":274155,"style":34427},[31685,31686],[34,274157,274159],{"className":274158},[31792],[34,274160,274162,274182],{"className":274161},[31796,31797],[34,274163,274165,274179],{"className":274164},[31801],[34,274166,274168],{"className":274167,"style":31806},[31805],[34,274169,274170,274173],{"style":104694},[34,274171],{"className":274172,"style":31814},[31813],[34,274174,274176],{"className":274175},[31818,31819,31820,31821],[34,274177,786],{"className":274178},[31685,31821],[34,274180,31829],{"className":274181},[31828],[34,274183,274185],{"className":274184},[31801],[34,274186,274188],{"className":274187,"style":31836},[31805],[34,274189],{},[34,274191],{"className":274192,"style":31846},[31845],[34,274194,202],{"className":274195},[31850],[34,274197],{"className":274198,"style":31846},[31845],[34,274200,274202,274205,274208,274211,274214],{"className":274201},[31676],[34,274203],{"className":274204,"style":35942},[31680],[34,274206,640],{"className":274207},[31685],[34,274209],{"className":274210,"style":32604},[31845],[34,274212,223],{"className":274213},[32024],[34,274215],{"className":274216,"style":32604},[31845],[34,274218,274220,274223,274227,274230,274233],{"className":274219},[31676],[34,274221],{"className":274222,"style":31774},[31680],[34,274224,274226],{"className":274225},[31685],"18\u002F24",[34,274228],{"className":274229,"style":31846},[31845],[34,274231,202],{"className":274232},[31850],[34,274234],{"className":274235,"style":31846},[31845],[34,274237,274239,274242],{"className":274238},[31676],[34,274240],{"className":274241,"style":32617},[31680],[34,274243,274137],{"className":274244},[31685],[108,274246,274247,274248],{},"J4 等待 = 12分。",[34,274249,274251,274286],{"className":274250},[31642],[34,274252,274254],{"className":274253},[31646],[31648,274255,274256],{"xmlns":31650},[31652,274257,274258,274283],{},[31655,274259,274260,274266,274268,274270,274272,274274,274276,274278,274280],{},[31743,274261,274262,274264],{},[31658,274263,34386],{},[31749,274265,15932],{},[31662,274267,202],{},[31749,274269,640],{},[31662,274271,223],{},[31749,274273,71869],{},[31658,274275,38080],{"mathvariant":38079},[31749,274277,96087],{},[31662,274279,202],{},[31749,274281,274282],{},"1.5",[31665,274284,274285],{"encoding":31667},"R_4 = 1 + 12\u002F24 = 1.5",[34,274287,274289,274344,274362,274381],{"className":274288,"ariaHidden":758},[31672],[34,274290,274292,274295,274335,274338,274341],{"className":274291},[31676],[34,274293],{"className":274294,"style":70915},[31680],[34,274296,274298,274301],{"className":274297},[31685],[34,274299,34386],{"className":274300,"style":34427},[31685,31686],[34,274302,274304],{"className":274303},[31792],[34,274305,274307,274327],{"className":274306},[31796,31797],[34,274308,274310,274324],{"className":274309},[31801],[34,274311,274313],{"className":274312,"style":31806},[31805],[34,274314,274315,274318],{"style":104694},[34,274316],{"className":274317,"style":31814},[31813],[34,274319,274321],{"className":274320},[31818,31819,31820,31821],[34,274322,15932],{"className":274323},[31685,31821],[34,274325,31829],{"className":274326},[31828],[34,274328,274330],{"className":274329},[31801],[34,274331,274333],{"className":274332,"style":31836},[31805],[34,274334],{},[34,274336],{"className":274337,"style":31846},[31845],[34,274339,202],{"className":274340},[31850],[34,274342],{"className":274343,"style":31846},[31845],[34,274345,274347,274350,274353,274356,274359],{"className":274346},[31676],[34,274348],{"className":274349,"style":35942},[31680],[34,274351,640],{"className":274352},[31685],[34,274354],{"className":274355,"style":32604},[31845],[34,274357,223],{"className":274358},[32024],[34,274360],{"className":274361,"style":32604},[31845],[34,274363,274365,274368,274372,274375,274378],{"className":274364},[31676],[34,274366],{"className":274367,"style":31774},[31680],[34,274369,274371],{"className":274370},[31685],"12\u002F24",[34,274373],{"className":274374,"style":31846},[31845],[34,274376,202],{"className":274377},[31850],[34,274379],{"className":274380,"style":31846},[31845],[34,274382,274384,274387],{"className":274383},[31676],[34,274385],{"className":274386,"style":32617},[31680],[34,274388,274282],{"className":274389},[31685],[108,274391,274392,274393],{},"J5 等待 = 6分。",[34,274394,274396,274430],{"className":274395},[31642],[34,274397,274399],{"className":274398},[31646],[31648,274400,274401],{"xmlns":31650},[31652,274402,274403,274427],{},[31655,274404,274405,274411,274413,274415,274417,274419,274421,274423,274425],{},[31743,274406,274407,274409],{},[31658,274408,34386],{},[31749,274410,18136],{},[31662,274412,202],{},[31749,274414,640],{},[31662,274416,223],{},[31749,274418,36656],{},[31658,274420,38080],{"mathvariant":38079},[31749,274422,71869],{},[31662,274424,202],{},[31749,274426,274282],{},[31665,274428,274429],{"encoding":31667},"R_5 = 1 + 6\u002F12 = 1.5",[34,274431,274433,274488,274506,274524],{"className":274432,"ariaHidden":758},[31672],[34,274434,274436,274439,274479,274482,274485],{"className":274435},[31676],[34,274437],{"className":274438,"style":70915},[31680],[34,274440,274442,274445],{"className":274441},[31685],[34,274443,34386],{"className":274444,"style":34427},[31685,31686],[34,274446,274448],{"className":274447},[31792],[34,274449,274451,274471],{"className":274450},[31796,31797],[34,274452,274454,274468],{"className":274453},[31801],[34,274455,274457],{"className":274456,"style":31806},[31805],[34,274458,274459,274462],{"style":104694},[34,274460],{"className":274461,"style":31814},[31813],[34,274463,274465],{"className":274464},[31818,31819,31820,31821],[34,274466,18136],{"className":274467},[31685,31821],[34,274469,31829],{"className":274470},[31828],[34,274472,274474],{"className":274473},[31801],[34,274475,274477],{"className":274476,"style":31836},[31805],[34,274478],{},[34,274480],{"className":274481,"style":31846},[31845],[34,274483,202],{"className":274484},[31850],[34,274486],{"className":274487,"style":31846},[31845],[34,274489,274491,274494,274497,274500,274503],{"className":274490},[31676],[34,274492],{"className":274493,"style":35942},[31680],[34,274495,640],{"className":274496},[31685],[34,274498],{"className":274499,"style":32604},[31845],[34,274501,223],{"className":274502},[32024],[34,274504],{"className":274505,"style":32604},[31845],[34,274507,274509,274512,274515,274518,274521],{"className":274508},[31676],[34,274510],{"className":274511,"style":31774},[31680],[34,274513,267962],{"className":274514},[31685],[34,274516],{"className":274517,"style":31846},[31845],[34,274519,202],{"className":274520},[31850],[34,274522],{"className":274523,"style":31846},[31845],[34,274525,274527,274530],{"className":274526},[31676],[34,274528],{"className":274529,"style":32617},[31680],[34,274531,274282],{"className":274532},[31685],[108,274534,274535,274536,274538,274539,274542],{},"最高者为 ",[2627,274537,273471],{},"。J2 执行 30 分钟，结束时间为 ",[2627,274540,274541],{},"11:18","。J2 周转时间 = 59分。",[108,274544,274545,274547,274548],{},[2627,274546,274541],{},"，J3, J4, J5 继续竞争：\n",[105,274549,274550,274694,274839,274983],{},[108,274551,274552,274553],{},"J3 等待 = 48分。",[34,274554,274556,274590],{"className":274555},[31642],[34,274557,274559],{"className":274558},[31646],[31648,274560,274561],{"xmlns":31650},[31652,274562,274563,274587],{},[31655,274564,274565,274571,274573,274575,274577,274579,274581,274583,274585],{},[31743,274566,274567,274569],{},[31658,274568,34386],{},[31749,274570,786],{},[31662,274572,202],{},[31749,274574,640],{},[31662,274576,223],{},[31749,274578,205160],{},[31658,274580,38080],{"mathvariant":38079},[31749,274582,96087],{},[31662,274584,202],{},[31749,274586,273076],{},[31665,274588,274589],{"encoding":31667},"R_3 = 1 + 48\u002F24 = 3.0",[34,274591,274593,274648,274666,274685],{"className":274592,"ariaHidden":758},[31672],[34,274594,274596,274599,274639,274642,274645],{"className":274595},[31676],[34,274597],{"className":274598,"style":70915},[31680],[34,274600,274602,274605],{"className":274601},[31685],[34,274603,34386],{"className":274604,"style":34427},[31685,31686],[34,274606,274608],{"className":274607},[31792],[34,274609,274611,274631],{"className":274610},[31796,31797],[34,274612,274614,274628],{"className":274613},[31801],[34,274615,274617],{"className":274616,"style":31806},[31805],[34,274618,274619,274622],{"style":104694},[34,274620],{"className":274621,"style":31814},[31813],[34,274623,274625],{"className":274624},[31818,31819,31820,31821],[34,274626,786],{"className":274627},[31685,31821],[34,274629,31829],{"className":274630},[31828],[34,274632,274634],{"className":274633},[31801],[34,274635,274637],{"className":274636,"style":31836},[31805],[34,274638],{},[34,274640],{"className":274641,"style":31846},[31845],[34,274643,202],{"className":274644},[31850],[34,274646],{"className":274647,"style":31846},[31845],[34,274649,274651,274654,274657,274660,274663],{"className":274650},[31676],[34,274652],{"className":274653,"style":35942},[31680],[34,274655,640],{"className":274656},[31685],[34,274658],{"className":274659,"style":32604},[31845],[34,274661,223],{"className":274662},[32024],[34,274664],{"className":274665,"style":32604},[31845],[34,274667,274669,274672,274676,274679,274682],{"className":274668},[31676],[34,274670],{"className":274671,"style":31774},[31680],[34,274673,274675],{"className":274674},[31685],"48\u002F24",[34,274677],{"className":274678,"style":31846},[31845],[34,274680,202],{"className":274681},[31850],[34,274683],{"className":274684,"style":31846},[31845],[34,274686,274688,274691],{"className":274687},[31676],[34,274689],{"className":274690,"style":32617},[31680],[34,274692,273076],{"className":274693},[31685],[108,274695,274696,274697],{},"J4 等待 = 42分。",[34,274698,274700,274735],{"className":274699},[31642],[34,274701,274703],{"className":274702},[31646],[31648,274704,274705],{"xmlns":31650},[31652,274706,274707,274732],{},[31655,274708,274709,274715,274717,274719,274721,274723,274725,274727,274729],{},[31743,274710,274711,274713],{},[31658,274712,34386],{},[31749,274714,15932],{},[31662,274716,202],{},[31749,274718,640],{},[31662,274720,223],{},[31749,274722,95547],{},[31658,274724,38080],{"mathvariant":38079},[31749,274726,96087],{},[31662,274728,202],{},[31749,274730,274731],{},"2.75",[31665,274733,274734],{"encoding":31667},"R_4 = 1 + 42\u002F24 = 2.75",[34,274736,274738,274793,274811,274830],{"className":274737,"ariaHidden":758},[31672],[34,274739,274741,274744,274784,274787,274790],{"className":274740},[31676],[34,274742],{"className":274743,"style":70915},[31680],[34,274745,274747,274750],{"className":274746},[31685],[34,274748,34386],{"className":274749,"style":34427},[31685,31686],[34,274751,274753],{"className":274752},[31792],[34,274754,274756,274776],{"className":274755},[31796,31797],[34,274757,274759,274773],{"className":274758},[31801],[34,274760,274762],{"className":274761,"style":31806},[31805],[34,274763,274764,274767],{"style":104694},[34,274765],{"className":274766,"style":31814},[31813],[34,274768,274770],{"className":274769},[31818,31819,31820,31821],[34,274771,15932],{"className":274772},[31685,31821],[34,274774,31829],{"className":274775},[31828],[34,274777,274779],{"className":274778},[31801],[34,274780,274782],{"className":274781,"style":31836},[31805],[34,274783],{},[34,274785],{"className":274786,"style":31846},[31845],[34,274788,202],{"className":274789},[31850],[34,274791],{"className":274792,"style":31846},[31845],[34,274794,274796,274799,274802,274805,274808],{"className":274795},[31676],[34,274797],{"className":274798,"style":35942},[31680],[34,274800,640],{"className":274801},[31685],[34,274803],{"className":274804,"style":32604},[31845],[34,274806,223],{"className":274807},[32024],[34,274809],{"className":274810,"style":32604},[31845],[34,274812,274814,274817,274821,274824,274827],{"className":274813},[31676],[34,274815],{"className":274816,"style":31774},[31680],[34,274818,274820],{"className":274819},[31685],"42\u002F24",[34,274822],{"className":274823,"style":31846},[31845],[34,274825,202],{"className":274826},[31850],[34,274828],{"className":274829,"style":31846},[31845],[34,274831,274833,274836],{"className":274832},[31676],[34,274834],{"className":274835,"style":32617},[31680],[34,274837,274731],{"className":274838},[31685],[108,274840,274841,274842],{},"J5 等待 = 36分。",[34,274843,274845,274879],{"className":274844},[31642],[34,274846,274848],{"className":274847},[31646],[31648,274849,274850],{"xmlns":31650},[31652,274851,274852,274876],{},[31655,274853,274854,274860,274862,274864,274866,274868,274870,274872,274874],{},[31743,274855,274856,274858],{},[31658,274857,34386],{},[31749,274859,18136],{},[31662,274861,202],{},[31749,274863,640],{},[31662,274865,223],{},[31749,274867,165563],{},[31658,274869,38080],{"mathvariant":38079},[31749,274871,71869],{},[31662,274873,202],{},[31749,274875,272734],{},[31665,274877,274878],{"encoding":31667},"R_5 = 1 + 36\u002F12 = 4.0",[34,274880,274882,274937,274955,274974],{"className":274881,"ariaHidden":758},[31672],[34,274883,274885,274888,274928,274931,274934],{"className":274884},[31676],[34,274886],{"className":274887,"style":70915},[31680],[34,274889,274891,274894],{"className":274890},[31685],[34,274892,34386],{"className":274893,"style":34427},[31685,31686],[34,274895,274897],{"className":274896},[31792],[34,274898,274900,274920],{"className":274899},[31796,31797],[34,274901,274903,274917],{"className":274902},[31801],[34,274904,274906],{"className":274905,"style":31806},[31805],[34,274907,274908,274911],{"style":104694},[34,274909],{"className":274910,"style":31814},[31813],[34,274912,274914],{"className":274913},[31818,31819,31820,31821],[34,274915,18136],{"className":274916},[31685,31821],[34,274918,31829],{"className":274919},[31828],[34,274921,274923],{"className":274922},[31801],[34,274924,274926],{"className":274925,"style":31836},[31805],[34,274927],{},[34,274929],{"className":274930,"style":31846},[31845],[34,274932,202],{"className":274933},[31850],[34,274935],{"className":274936,"style":31846},[31845],[34,274938,274940,274943,274946,274949,274952],{"className":274939},[31676],[34,274941],{"className":274942,"style":35942},[31680],[34,274944,640],{"className":274945},[31685],[34,274947],{"className":274948,"style":32604},[31845],[34,274950,223],{"className":274951},[32024],[34,274953],{"className":274954,"style":32604},[31845],[34,274956,274958,274961,274965,274968,274971],{"className":274957},[31676],[34,274959],{"className":274960,"style":31774},[31680],[34,274962,274964],{"className":274963},[31685],"36\u002F12",[34,274966],{"className":274967,"style":31846},[31845],[34,274969,202],{"className":274970},[31850],[34,274972],{"className":274973,"style":31846},[31845],[34,274975,274977,274980],{"className":274976},[31676],[34,274978],{"className":274979,"style":32617},[31680],[34,274981,272734],{"className":274982},[31685],[108,274984,274535,274985,274987,274988,274991],{},[2627,274986,273915],{},"。J5 执行 12 分钟，结束时间为 ",[2627,274989,274990],{},"11:30","。J5 周转时间 = 48分。",[108,274993,274994,274996,274997],{},[2627,274995,274990],{},"，J3, J4 继续竞争：\n",[105,274998,274999,275143,275288],{},[108,275000,275001,275002],{},"J3 等待 = 60分。",[34,275003,275005,275039],{"className":275004},[31642],[34,275006,275008],{"className":275007},[31646],[31648,275009,275010],{"xmlns":31650},[31652,275011,275012,275036],{},[31655,275013,275014,275020,275022,275024,275026,275028,275030,275032,275034],{},[31743,275015,275016,275018],{},[31658,275017,34386],{},[31749,275019,786],{},[31662,275021,202],{},[31749,275023,640],{},[31662,275025,223],{},[31749,275027,167664],{},[31658,275029,38080],{"mathvariant":38079},[31749,275031,96087],{},[31662,275033,202],{},[31749,275035,227812],{},[31665,275037,275038],{"encoding":31667},"R_3 = 1 + 60\u002F24 = 3.5",[34,275040,275042,275097,275115,275134],{"className":275041,"ariaHidden":758},[31672],[34,275043,275045,275048,275088,275091,275094],{"className":275044},[31676],[34,275046],{"className":275047,"style":70915},[31680],[34,275049,275051,275054],{"className":275050},[31685],[34,275052,34386],{"className":275053,"style":34427},[31685,31686],[34,275055,275057],{"className":275056},[31792],[34,275058,275060,275080],{"className":275059},[31796,31797],[34,275061,275063,275077],{"className":275062},[31801],[34,275064,275066],{"className":275065,"style":31806},[31805],[34,275067,275068,275071],{"style":104694},[34,275069],{"className":275070,"style":31814},[31813],[34,275072,275074],{"className":275073},[31818,31819,31820,31821],[34,275075,786],{"className":275076},[31685,31821],[34,275078,31829],{"className":275079},[31828],[34,275081,275083],{"className":275082},[31801],[34,275084,275086],{"className":275085,"style":31836},[31805],[34,275087],{},[34,275089],{"className":275090,"style":31846},[31845],[34,275092,202],{"className":275093},[31850],[34,275095],{"className":275096,"style":31846},[31845],[34,275098,275100,275103,275106,275109,275112],{"className":275099},[31676],[34,275101],{"className":275102,"style":35942},[31680],[34,275104,640],{"className":275105},[31685],[34,275107],{"className":275108,"style":32604},[31845],[34,275110,223],{"className":275111},[32024],[34,275113],{"className":275114,"style":32604},[31845],[34,275116,275118,275121,275125,275128,275131],{"className":275117},[31676],[34,275119],{"className":275120,"style":31774},[31680],[34,275122,275124],{"className":275123},[31685],"60\u002F24",[34,275126],{"className":275127,"style":31846},[31845],[34,275129,202],{"className":275130},[31850],[34,275132],{"className":275133,"style":31846},[31845],[34,275135,275137,275140],{"className":275136},[31676],[34,275138],{"className":275139,"style":32617},[31680],[34,275141,227812],{"className":275142},[31685],[108,275144,275145,275146],{},"J4 等待 = 54分。",[34,275147,275149,275184],{"className":275148},[31642],[34,275150,275152],{"className":275151},[31646],[31648,275153,275154],{"xmlns":31650},[31652,275155,275156,275181],{},[31655,275157,275158,275164,275166,275168,275170,275172,275174,275176,275178],{},[31743,275159,275160,275162],{},[31658,275161,34386],{},[31749,275163,15932],{},[31662,275165,202],{},[31749,275167,640],{},[31662,275169,223],{},[31749,275171,162999],{},[31658,275173,38080],{"mathvariant":38079},[31749,275175,96087],{},[31662,275177,202],{},[31749,275179,275180],{},"3.25",[31665,275182,275183],{"encoding":31667},"R_4 = 1 + 54\u002F24 = 3.25",[34,275185,275187,275242,275260,275279],{"className":275186,"ariaHidden":758},[31672],[34,275188,275190,275193,275233,275236,275239],{"className":275189},[31676],[34,275191],{"className":275192,"style":70915},[31680],[34,275194,275196,275199],{"className":275195},[31685],[34,275197,34386],{"className":275198,"style":34427},[31685,31686],[34,275200,275202],{"className":275201},[31792],[34,275203,275205,275225],{"className":275204},[31796,31797],[34,275206,275208,275222],{"className":275207},[31801],[34,275209,275211],{"className":275210,"style":31806},[31805],[34,275212,275213,275216],{"style":104694},[34,275214],{"className":275215,"style":31814},[31813],[34,275217,275219],{"className":275218},[31818,31819,31820,31821],[34,275220,15932],{"className":275221},[31685,31821],[34,275223,31829],{"className":275224},[31828],[34,275226,275228],{"className":275227},[31801],[34,275229,275231],{"className":275230,"style":31836},[31805],[34,275232],{},[34,275234],{"className":275235,"style":31846},[31845],[34,275237,202],{"className":275238},[31850],[34,275240],{"className":275241,"style":31846},[31845],[34,275243,275245,275248,275251,275254,275257],{"className":275244},[31676],[34,275246],{"className":275247,"style":35942},[31680],[34,275249,640],{"className":275250},[31685],[34,275252],{"className":275253,"style":32604},[31845],[34,275255,223],{"className":275256},[32024],[34,275258],{"className":275259,"style":32604},[31845],[34,275261,275263,275266,275270,275273,275276],{"className":275262},[31676],[34,275264],{"className":275265,"style":31774},[31680],[34,275267,275269],{"className":275268},[31685],"54\u002F24",[34,275271],{"className":275272,"style":31846},[31845],[34,275274,202],{"className":275275},[31850],[34,275277],{"className":275278,"style":31846},[31845],[34,275280,275282,275285],{"className":275281},[31676],[34,275283],{"className":275284,"style":32617},[31680],[34,275286,275180],{"className":275287},[31685],[108,275289,274535,275290,275292,275293,275296],{},[2627,275291,273486],{},"。J3 执行 24 分钟，结束时间为 ",[2627,275294,275295],{},"11:54","。J3 周转时间 = 84分。",[108,275298,275299,46528,275301,275303,275304,275307],{},[2627,275300,275295],{},[2627,275302,273500],{}," 开始执行。J4 执行 24 分钟，结束时间为 ",[2627,275305,275306],{},"12:18","。J4 周转时间 = 102分。",[15,275309,275310,275313],{},[2627,275311,275312],{},"执行顺序","：J1 → J2 → J5 → J3 → J4",[15,275315,275316,2664,275318,347],{},[2627,275317,227991],{},[34,275319,275321,275377],{"className":275320},[31642],[34,275322,275324],{"className":275323},[31646],[31648,275325,275326],{"xmlns":31650},[31652,275327,275328,275374],{},[31655,275329,275330,275332,275334,275336,275339,275341,275343,275345,275347,275349,275351,275353,275355,275357,275359,275362,275364,275366,275368],{},[31662,275331,456],{"stretchy":702},[31749,275333,95547],{},[31662,275335,223],{},[31749,275337,275338],{},"59",[31662,275340,223],{},[31749,275342,94841],{},[31662,275344,223],{},[31749,275346,162508],{},[31662,275348,223],{},[31749,275350,205160],{},[31662,275352,2557],{"stretchy":702},[31658,275354,38080],{"mathvariant":38079},[31749,275356,18136],{},[31662,275358,202],{},[31749,275360,275361],{},"335",[31658,275363,38080],{"mathvariant":38079},[31749,275365,18136],{},[31662,275367,202],{},[31655,275369,275370,275372],{},[31749,275371,99870],{"mathvariant":226163},[39361,275373,228174],{},[31665,275375,275376],{"encoding":31667},"(42 + 59 + 84 + 102 + 48) \u002F 5 = 335 \u002F 5 = \\mathbf{67 \\text{ 分钟}}",[34,275378,275380,275401,275419,275437,275455,275479,275498],{"className":275379,"ariaHidden":758},[31672],[34,275381,275383,275386,275389,275392,275395,275398],{"className":275382},[31676],[34,275384],{"className":275385,"style":31774},[31680],[34,275387,456],{"className":275388},[31782],[34,275390,95547],{"className":275391},[31685],[34,275393],{"className":275394,"style":32604},[31845],[34,275396,223],{"className":275397},[32024],[34,275399],{"className":275400,"style":32604},[31845],[34,275402,275404,275407,275410,275413,275416],{"className":275403},[31676],[34,275405],{"className":275406,"style":35942},[31680],[34,275408,275338],{"className":275409},[31685],[34,275411],{"className":275412,"style":32604},[31845],[34,275414,223],{"className":275415},[32024],[34,275417],{"className":275418,"style":32604},[31845],[34,275420,275422,275425,275428,275431,275434],{"className":275421},[31676],[34,275423],{"className":275424,"style":35942},[31680],[34,275426,94841],{"className":275427},[31685],[34,275429],{"className":275430,"style":32604},[31845],[34,275432,223],{"className":275433},[32024],[34,275435],{"className":275436,"style":32604},[31845],[34,275438,275440,275443,275446,275449,275452],{"className":275439},[31676],[34,275441],{"className":275442,"style":35942},[31680],[34,275444,162508],{"className":275445},[31685],[34,275447],{"className":275448,"style":32604},[31845],[34,275450,223],{"className":275451},[32024],[34,275453],{"className":275454,"style":32604},[31845],[34,275456,275458,275461,275464,275467,275470,275473,275476],{"className":275457},[31676],[34,275459],{"className":275460,"style":31774},[31680],[34,275462,205160],{"className":275463},[31685],[34,275465,2557],{"className":275466},[31690],[34,275468,226261],{"className":275469},[31685],[34,275471],{"className":275472,"style":31846},[31845],[34,275474,202],{"className":275475},[31850],[34,275477],{"className":275478,"style":31846},[31845],[34,275480,275482,275485,275489,275492,275495],{"className":275481},[31676],[34,275483],{"className":275484,"style":31774},[31680],[34,275486,275488],{"className":275487},[31685],"335\u002F5",[34,275490],{"className":275491,"style":31846},[31845],[34,275493,202],{"className":275494},[31850],[34,275496],{"className":275497,"style":31846},[31845],[34,275499,275501,275504],{"className":275500},[31676],[34,275502],{"className":275503,"style":33218},[31680],[34,275505,275507,275510],{"className":275506},[31685],[34,275508,99870],{"className":275509},[31685,226302],[34,275511,275513,275516],{"className":275512},[31685,2638],[34,275514,130298],{"className":275515},[31685],[34,275517,228301],{"className":275518},[31685,105462],[15,275520,275521,275523],{},[2627,275522,246532],{}," 内存 100K 是一个干扰信息！题目明确\"按单道方式运行\"，不需要考虑内存分配并发情况。\n:::",[19,275525,275527],{"id":275526},"_36-银行家算法三类资源abc五进程","36. 银行家算法：三类资源（A\u002FB\u002FC）、五进程",[15,275529,275530],{},"设系统中资源类集合为 {A，B，C}，资源类 A 中含有 10 个资源实例，B 中含有 5 个，C 中含有 7 个。在 T0 时刻状态如下：",[13837,275532,275533,275551],{},[13840,275534,275535],{},[13843,275536,275537,275539,275542,275545,275548],{},[13846,275538,247197],{"align":82355},[13846,275540,275541],{"align":82355},"Max (A,B,C)",[13846,275543,275544],{"align":82355},"Allocation (A,B,C)",[13846,275546,275547],{"align":82355},"Need (A,B,C)",[13846,275549,275550],{"align":82355},"Available (A,B,C)",[13853,275552,275553,275570,275584,275599,275613],{},[13843,275554,275555,275558,275561,275564,275567],{},[13858,275556,275557],{"align":82355},"P0",[13858,275559,275560],{"align":82355},"(7, 5, 3)",[13858,275562,275563],{"align":82355},"(0, 1, 0)",[13858,275565,275566],{"align":82355},"(7, 4, 3)",[13858,275568,275569],{"align":82355},"(3, 3, 2)",[13843,275571,275572,275574,275576,275579,275582],{},[13858,275573,97876],{"align":82355},[13858,275575,264630],{"align":82355},[13858,275577,275578],{"align":82355},"(2, 0, 0)",[13858,275580,275581],{"align":82355},"(1, 2, 2)",[13858,275583],{"align":82355},[13843,275585,275586,275588,275591,275594,275597],{},[13858,275587,97925],{"align":82355},[13858,275589,275590],{"align":82355},"(9, 0, 2)",[13858,275592,275593],{"align":82355},"(3, 0, 2)",[13858,275595,275596],{"align":82355},"(6, 0, 0)",[13858,275598],{"align":82355},[13843,275600,275601,275603,275606,275608,275611],{},[13858,275602,263561],{"align":82355},[13858,275604,275605],{"align":82355},"(2, 2, 2)",[13858,275607,264653],{"align":82355},[13858,275609,275610],{"align":82355},"(0, 1, 1)",[13858,275612],{"align":82355},[13843,275614,275615,275617,275620,275622,275625],{},[13858,275616,263578],{"align":82355},[13858,275618,275619],{"align":82355},"(4, 3, 3)",[13858,275621,264663],{"align":82355},[13858,275623,275624],{"align":82355},"(4, 3, 1)",[13858,275626],{"align":82355},[105,275628,275629,275635],{},[108,275630,275631,275632,275634],{},"(1) 假如现在进程 P1 发出新的资源申请 Request",[34,275633,640],{},"=(1,0,2)，系统是否可以实施资源分配，为什么？",[108,275636,275637],{},"(2) 在上面新状态下，对于进程 P0 所发出的资源请求（0, 2, 0）系统是否能实施资源分配？要求写出计算步骤及过程。",[15,275639,264303],{},[253,275641,267207],{"id":275642},"答案与详细解答-9",[15,275644,275645],{},[2627,275646,275647],{},"(1) P1 申请 (1, 0, 2)",[2230,275649,275650,275682,275714,275730],{},[108,275651,275652,275653,275681],{},"检查合法性：Request (1,0,2) ",[34,275654,275656,275669],{"className":275655},[31642],[34,275657,275659],{"className":275658},[31646],[31648,275660,275661],{"xmlns":31650},[31652,275662,275663,275667],{},[31655,275664,275665],{},[31662,275666,45972],{},[31665,275668,45975],{"encoding":31667},[34,275670,275672],{"className":275671,"ariaHidden":758},[31672],[34,275673,275675,275678],{"className":275674},[31676],[34,275676],{"className":275677,"style":43198},[31680],[34,275679,45972],{"className":275680},[31850]," P1 Need (1,2,2) 成立。",[108,275683,275684,275685,275713],{},"检查可用性：Request (1,0,2) ",[34,275686,275688,275701],{"className":275687},[31642],[34,275689,275691],{"className":275690},[31646],[31648,275692,275693],{"xmlns":31650},[31652,275694,275695,275699],{},[31655,275696,275697],{},[31662,275698,45972],{},[31665,275700,45975],{"encoding":31667},[34,275702,275704],{"className":275703,"ariaHidden":758},[31672],[34,275705,275707,275710],{"className":275706},[31676],[34,275708],{"className":275709,"style":43198},[31680],[34,275711,45972],{"className":275712},[31850]," Available (3,3,2) 成立。",[108,275715,275716,275719],{},[2627,275717,275718],{},"试探分配后的新状态：",[105,275720,275721,275727],{},[108,275722,275723,275724],{},"Available = (3,3,2) - (1,0,2) = ",[2627,275725,275726],{},"(2, 3, 0)",[108,275728,275729],{},"P1 Allocation = (3, 0, 2)，P1 Need = (0, 2, 0)",[108,275731,275732,275735,275736],{},[2627,275733,275734],{},"安全性检查","（Available=(2,3,0)）：\nP1 Need=(0,2,0) 满足，P1 执行完释放，Available=(5,3,2)。\n后续可满足 P3 → P4 → P0 → P2。\n",[2627,275737,275738],{},"结论：存在安全序列（P1→P3→P4→P0→P2），可以实施资源分配。",[15,275740,275741],{},[2627,275742,275743],{},"(2) 随后 P0 请求 (0, 2, 0)",[2230,275745,275746,275751,275812,275827],{},[108,275747,275748,275749,347],{},"在上述新状态下，Available = ",[2627,275750,275726],{},[108,275752,275753,275754,275782,275783,275811],{},"检查：Request ",[34,275755,275757,275770],{"className":275756},[31642],[34,275758,275760],{"className":275759},[31646],[31648,275761,275762],{"xmlns":31650},[31652,275763,275764,275768],{},[31655,275765,275766],{},[31662,275767,45972],{},[31665,275769,45975],{"encoding":31667},[34,275771,275773],{"className":275772,"ariaHidden":758},[31672],[34,275774,275776,275779],{"className":275775},[31676],[34,275777],{"className":275778,"style":43198},[31680],[34,275780,45972],{"className":275781},[31850]," P0 Need 成立；Request ",[34,275784,275786,275799],{"className":275785},[31642],[34,275787,275789],{"className":275788},[31646],[31648,275790,275791],{"xmlns":31650},[31652,275792,275793,275797],{},[31655,275794,275795],{},[31662,275796,45972],{},[31665,275798,45975],{"encoding":31667},[34,275800,275802],{"className":275801,"ariaHidden":758},[31672],[34,275803,275805,275808],{"className":275804},[31676],[34,275806],{"className":275807,"style":43198},[31680],[34,275809,45972],{"className":275810},[31850]," Available 成立。",[108,275813,275814,275816],{},[2627,275815,275718],{},[105,275817,275818,275824],{},[108,275819,275820,275821],{},"Available = (2,3,0) - (0,2,0) = ",[2627,275822,275823],{},"(2, 1, 0)",[108,275825,275826],{},"P0 Allocation = (0,3,0)，P0 Need = (7,2,3)",[108,275828,275829,275831,275832,39984,275835],{},[2627,275830,275734],{},"（Available=(2,1,0)）：\n检查所有进程当前的 Need：没有任何一个进程的 Need 可以被 (2,1,0) 满足，无法找到安全序列，陷入",[2627,275833,275834],{},"不安全状态",[2627,275836,275837],{},"结论：拒绝分配，P0 必须等待。",[15,275839,275840,275842,275843,275846,275847,275850],{},[2627,275841,246532],{}," 许多人在第二步忘了把 Available 基础值更新为第一步算完的 ",[31,275844,275845],{},"(2,3,0)","，而是继续拿最初的 ",[31,275848,275849],{},"(3,3,2)"," 去算。\n:::",[19,275852,275854],{"id":275853},"_37-位示图字号和位号计算","37. 位示图：字号和位号计算",[15,275856,275857],{},"现有一台 16 位字长的专用机，采用页式存储管理。主存储器共有 4096 块 (块号为 0～4095)，现用位示图分配主存空间。试问：",[105,275859,275860,275863,275866],{},[108,275861,275862],{},"(1) 该位示图占用几个字?",[108,275864,275865],{},"(2) 主存块号 3999 对应位示图的字号和位号 (均从 0 开始) 各是多少?",[108,275867,275868],{},"(3) 位示图字号 199，位号 9 对应主存的块号是多少?",[15,275870,264303],{},[253,275872,267207],{"id":275873},"答案与详细解答-10",[15,275875,275876],{},[2627,275877,275878],{},"(1) 位示图占用几个字？",[15,275880,275881,275882,207648],{},"共有 4096 个块，需要 4096 个 Bit。机器字长为 16 位，每个 Word 有 16 个 Bit。\n佔用字数 = ",[34,275883,275885,275907],{"className":275884},[31642],[34,275886,275888],{"className":275887},[31646],[31648,275889,275890],{"xmlns":31650},[31652,275891,275892,275904],{},[31655,275893,275894,275896,275898,275900,275902],{},[31749,275895,207588],{},[31662,275897,207557],{},[31749,275899,177869],{},[31662,275901,202],{},[31749,275903,95418],{"mathvariant":226163},[31665,275905,275906],{"encoding":31667},"4096 \\div 16 = \\mathbf{256}",[34,275908,275910,275928,275946],{"className":275909,"ariaHidden":758},[31672],[34,275911,275913,275916,275919,275922,275925],{"className":275912},[31676],[34,275914],{"className":275915,"style":35942},[31680],[34,275917,207588],{"className":275918},[31685],[34,275920],{"className":275921,"style":32604},[31845],[34,275923,207557],{"className":275924},[32024],[34,275926],{"className":275927,"style":32604},[31845],[34,275929,275931,275934,275937,275940,275943],{"className":275930},[31676],[34,275932],{"className":275933,"style":32617},[31680],[34,275935,177869],{"className":275936},[31685],[34,275938],{"className":275939,"style":31846},[31845],[34,275941,202],{"className":275942},[31850],[34,275944],{"className":275945,"style":31846},[31845],[34,275947,275949,275952],{"className":275948},[31676],[34,275950],{"className":275951,"style":32617},[31680],[34,275953,275955],{"className":275954},[31685],[34,275956,95418],{"className":275957},[31685,226302],[15,275959,275960],{},[2627,275961,275962],{},"(2) 块号 3999 对应的字号和位号？",[15,275964,275965,275966,276052,276053,347],{},"字号 = ",[34,275967,275969,275995],{"className":275968},[31642],[34,275970,275972],{"className":275971},[31646],[31648,275973,275974],{"xmlns":31650},[31652,275975,275976,275992],{},[31655,275977,275978,275980,275982,275984,275986,275988,275990],{},[31662,275979,173327],{"stretchy":702},[31749,275981,207961],{},[31662,275983,207557],{},[31749,275985,177869],{},[31662,275987,173336],{"stretchy":702},[31662,275989,202],{},[31749,275991,184205],{"mathvariant":226163},[31665,275993,275994],{"encoding":31667},"\\lfloor 3999 \\div 16 \\rfloor = \\mathbf{249}",[34,275996,275998,276019,276040],{"className":275997,"ariaHidden":758},[31672],[34,275999,276001,276004,276007,276010,276013,276016],{"className":276000},[31676],[34,276002],{"className":276003,"style":31774},[31680],[34,276005,173327],{"className":276006},[31782],[34,276008,207961],{"className":276009},[31685],[34,276011],{"className":276012,"style":32604},[31845],[34,276014,207557],{"className":276015},[32024],[34,276017],{"className":276018,"style":32604},[31845],[34,276020,276022,276025,276028,276031,276034,276037],{"className":276021},[31676],[34,276023],{"className":276024,"style":31774},[31680],[34,276026,177869],{"className":276027},[31685],[34,276029,173336],{"className":276030},[31690],[34,276032],{"className":276033,"style":31846},[31845],[34,276035,202],{"className":276036},[31850],[34,276038],{"className":276039,"style":31846},[31845],[34,276041,276043,276046],{"className":276042},[31676],[34,276044],{"className":276045,"style":32617},[31680],[34,276047,276049],{"className":276048},[31685],[34,276050,184205],{"className":276051},[31685,226302],"。\n位号 = ",[34,276054,276056,276090],{"className":276055},[31642],[34,276057,276059],{"className":276058},[31646],[31648,276060,276061],{"xmlns":31650},[31652,276062,276063,276087],{},[31655,276064,276065,276067,276069,276079,276081,276083,276085],{},[31749,276066,207961],{},[39361,276068,207859],{},[31662,276070,276071],{"lspace":207862,"rspace":207862},[31655,276072,276073,276075,276077],{},[31658,276074,31759],{"mathvariant":38079},[31658,276076,43383],{"mathvariant":38079},[31658,276078,46900],{"mathvariant":38079},[39361,276080,207859],{},[31749,276082,177869],{},[31662,276084,202],{},[31749,276086,95396],{"mathvariant":226163},[31665,276088,276089],{"encoding":31667},"3999 \\bmod 16 = \\mathbf{15}",[34,276091,276093,276123,276141],{"className":276092,"ariaHidden":758},[31672],[34,276094,276096,276099,276102,276105,276108,276117,276120],{"className":276095},[31676],[34,276097],{"className":276098,"style":31681},[31680],[34,276100,207961],{"className":276101},[31685],[34,276103],{"className":276104,"style":112322},[31845],[34,276106],{"className":276107,"style":32604},[31845],[34,276109,276111],{"className":276110},[32024],[34,276112,276114],{"className":276113},[31685],[34,276115,207924],{"className":276116},[31685,207923],[34,276118],{"className":276119,"style":112322},[31845],[34,276121],{"className":276122,"style":32604},[31845],[34,276124,276126,276129,276132,276135,276138],{"className":276125},[31676],[34,276127],{"className":276128,"style":32617},[31680],[34,276130,177869],{"className":276131},[31685],[34,276133],{"className":276134,"style":31846},[31845],[34,276136,202],{"className":276137},[31850],[34,276139],{"className":276140,"style":31846},[31845],[34,276142,276144,276147],{"className":276143},[31676],[34,276145],{"className":276146,"style":32617},[31680],[34,276148,276150],{"className":276149},[31685],[34,276151,95396],{"className":276152},[31685,226302],[15,276154,276155],{},[2627,276156,276157],{},"(3) 字号 199，位号 9 对应的主存块号？",[15,276159,276160,276161,347],{},"块号 = ",[34,276162,276164,276212],{"className":276163},[31642],[34,276165,276167],{"className":276166},[31646],[31648,276168,276169],{"xmlns":31650},[31652,276170,276171,276209],{},[31655,276172,276173,276176,276178,276180,276182,276185,276187,276189,276191,276193,276195,276197,276199,276201,276203,276205,276207],{},[39361,276174,276175],{},"字号",[31662,276177,42071],{},[31749,276179,177869],{},[31662,276181,223],{},[39361,276183,276184],{},"位号",[31662,276186,202],{},[31749,276188,208362],{},[31662,276190,42071],{},[31749,276192,177869],{},[31662,276194,223],{},[31749,276196,36732],{},[31662,276198,202],{},[31749,276200,208479],{},[31662,276202,223],{},[31749,276204,36732],{},[31662,276206,202],{},[31749,276208,208488],{"mathvariant":226163},[31665,276210,276211],{"encoding":31667},"\\text{字号} \\times 16 + \\text{位号} = 199 \\times 16 + 9 = 3184 + 9 = \\mathbf{3193}",[34,276213,276215,276236,276254,276275,276293,276311,276329,276347,276365],{"className":276214,"ariaHidden":758},[31672],[34,276216,276218,276221,276227,276230,276233],{"className":276217},[31676],[34,276219],{"className":276220,"style":34944},[31680],[34,276222,276224],{"className":276223},[31685,2638],[34,276225,276175],{"className":276226},[31685,105462],[34,276228],{"className":276229,"style":32604},[31845],[34,276231,42071],{"className":276232},[32024],[34,276234],{"className":276235,"style":32604},[31845],[34,276237,276239,276242,276245,276248,276251],{"className":276238},[31676],[34,276240],{"className":276241,"style":35942},[31680],[34,276243,177869],{"className":276244},[31685],[34,276246],{"className":276247,"style":32604},[31845],[34,276249,223],{"className":276250},[32024],[34,276252],{"className":276253,"style":32604},[31845],[34,276255,276257,276260,276266,276269,276272],{"className":276256},[31676],[34,276258],{"className":276259,"style":33218},[31680],[34,276261,276263],{"className":276262},[31685,2638],[34,276264,276184],{"className":276265},[31685,105462],[34,276267],{"className":276268,"style":31846},[31845],[34,276270,202],{"className":276271},[31850],[34,276273],{"className":276274,"style":31846},[31845],[34,276276,276278,276281,276284,276287,276290],{"className":276277},[31676],[34,276279],{"className":276280,"style":35942},[31680],[34,276282,208362],{"className":276283},[31685],[34,276285],{"className":276286,"style":32604},[31845],[34,276288,42071],{"className":276289},[32024],[34,276291],{"className":276292,"style":32604},[31845],[34,276294,276296,276299,276302,276305,276308],{"className":276295},[31676],[34,276297],{"className":276298,"style":35942},[31680],[34,276300,177869],{"className":276301},[31685],[34,276303],{"className":276304,"style":32604},[31845],[34,276306,223],{"className":276307},[32024],[34,276309],{"className":276310,"style":32604},[31845],[34,276312,276314,276317,276320,276323,276326],{"className":276313},[31676],[34,276315],{"className":276316,"style":32617},[31680],[34,276318,36732],{"className":276319},[31685],[34,276321],{"className":276322,"style":31846},[31845],[34,276324,202],{"className":276325},[31850],[34,276327],{"className":276328,"style":31846},[31845],[34,276330,276332,276335,276338,276341,276344],{"className":276331},[31676],[34,276333],{"className":276334,"style":35942},[31680],[34,276336,208479],{"className":276337},[31685],[34,276339],{"className":276340,"style":32604},[31845],[34,276342,223],{"className":276343},[32024],[34,276345],{"className":276346,"style":32604},[31845],[34,276348,276350,276353,276356,276359,276362],{"className":276349},[31676],[34,276351],{"className":276352,"style":32617},[31680],[34,276354,36732],{"className":276355},[31685],[34,276357],{"className":276358,"style":31846},[31845],[34,276360,202],{"className":276361},[31850],[34,276363],{"className":276364,"style":31846},[31845],[34,276366,276368,276371],{"className":276367},[31676],[34,276369],{"className":276370,"style":32617},[31680],[34,276372,276374],{"className":276373},[31685],[34,276375,208488],{"className":276376},[31685,226302],[15,276378,276379,276381],{},[2627,276380,246532],{}," 注意题目强调\"均从 0 开始\"。如果从 1 开始，计算公式需要进行平移。\n:::",[19,276383,276385],{"id":276384},"_38-信号量同步数据采集-计算-输出三进程","38. 信号量同步：数据采集-计算-输出三进程",[15,276387,276388],{},"某数据处理系统由数据采集、数据计算和数据输出三个进程组成，采集进程把采集到的数据送入由 M 个缓冲块组成的输入缓冲区（每次向一个缓冲块送数据），计算进程从输入缓冲区取数据计算（每次取一个缓冲块的数据），并将计算结果送入到由 N 个缓冲块组成的输出缓冲区（每次向一个缓冲块送数据），输出进程每次从输出缓冲区取一个结果输出。编写利用信号量机制实现的三者之间同步算法，要求写出信号量的含义和初值。",[15,276390,264303],{},[253,276392,267207],{"id":276393},"答案与详细解答-11",[15,276395,264070],{},[105,276397,276398,276404,276410,276416,276422,276428],{},[108,276399,276400,276403],{},[31,276401,276402],{},"emptyM = M","：输入缓冲区的空位数。",[108,276405,276406,276409],{},[31,276407,276408],{},"fullM = 0","：输入缓冲区中已有数据的数量。",[108,276411,276412,276415],{},[31,276413,276414],{},"mutexM = 1","：对输入缓冲池进行互斥访问。",[108,276417,276418,276421],{},[31,276419,276420],{},"emptyN = N","：输出缓冲区的空位数。",[108,276423,276424,276427],{},[31,276425,276426],{},"fullN = 0","：输出缓冲区中已有结果的数量。",[108,276429,276430,276433],{},[31,276431,276432],{},"mutexN = 1","：对输出缓冲池进行互斥访问。",[24,276435,276437],{"className":65734,"code":276436,"language":31965,"meta":29,"style":29},"\u002F\u002F 采集进程 (Producer 1)\nwhile(true) {\n 采集数据;\n wait(emptyM); \u002F\u002F 申请输入缓冲区空位 (同步)\n wait(mutexM); \u002F\u002F 锁定输入缓冲区 (互斥)\n 将数据放入输入缓冲区;\n signal(mutexM); \u002F\u002F 解锁输入缓冲区\n signal(fullM); \u002F\u002F 通知输入缓冲区已有数据\n}\n\n\u002F\u002F 计算进程 (Consumer 1 & Producer 2)\nwhile(true) {\n wait(fullM); \u002F\u002F 等待输入缓冲区有数据\n wait(mutexM); \u002F\u002F 锁定输入缓冲区\n 从输入缓冲区取出数据;\n signal(mutexM);\n signal(emptyM); \u002F\u002F 释放输入缓冲区空位\n\n 计算数据;\n\n wait(emptyN); \u002F\u002F 申请输出缓冲区空位\n wait(mutexN); \u002F\u002F 锁定输出缓冲区\n 将结果放入输出缓冲区;\n signal(mutexN);\n signal(fullN); \u002F\u002F 通知输出缓冲区已有结果\n}\n\n\u002F\u002F 输出进程 (Consumer 2)\nwhile(true) {\n wait(fullN); \u002F\u002F 等待输出缓冲区有结果\n wait(mutexN); \u002F\u002F 锁定输出缓冲区\n 从输出缓冲区取出结果;\n signal(mutexN);\n signal(emptyN); \u002F\u002F 释放输出缓冲区空位\n \n 输出结果;\n}\n",[31,276438,276439,276444,276449,276454,276459,276464,276469,276474,276479,276483,276487,276492,276496,276501,276506,276511,276516,276521,276525,276530,276534,276539,276544,276549,276554,276559,276563,276567,276572,276576,276581,276585,276590,276594,276599,276603,276608],{"__ignoreMap":29},[34,276440,276441],{"class":36,"line":37},[34,276442,276443],{},"\u002F\u002F 采集进程 (Producer 1)\n",[34,276445,276446],{"class":36,"line":43},[34,276447,276448],{},"while(true) {\n",[34,276450,276451],{"class":36,"line":49},[34,276452,276453],{}," 采集数据;\n",[34,276455,276456],{"class":36,"line":56},[34,276457,276458],{}," wait(emptyM); \u002F\u002F 申请输入缓冲区空位 (同步)\n",[34,276460,276461],{"class":36,"line":62},[34,276462,276463],{}," wait(mutexM); \u002F\u002F 锁定输入缓冲区 (互斥)\n",[34,276465,276466],{"class":36,"line":68},[34,276467,276468],{}," 将数据放入输入缓冲区;\n",[34,276470,276471],{"class":36,"line":74},[34,276472,276473],{}," signal(mutexM); \u002F\u002F 解锁输入缓冲区\n",[34,276475,276476],{"class":36,"line":1971},[34,276477,276478],{}," signal(fullM); \u002F\u002F 通知输入缓冲区已有数据\n",[34,276480,276481],{"class":36,"line":1976},[34,276482,77],{},[34,276484,276485],{"class":36,"line":1985},[34,276486,53],{"emptyLinePlaceholder":52},[34,276488,276489],{"class":36,"line":1990},[34,276490,276491],{},"\u002F\u002F 计算进程 (Consumer 1 & Producer 2)\n",[34,276493,276494],{"class":36,"line":1999},[34,276495,276448],{},[34,276497,276498],{"class":36,"line":2022},[34,276499,276500],{}," wait(fullM); \u002F\u002F 等待输入缓冲区有数据\n",[34,276502,276503],{"class":36,"line":2036},[34,276504,276505],{}," wait(mutexM); \u002F\u002F 锁定输入缓冲区\n",[34,276507,276508],{"class":36,"line":2603},[34,276509,276510],{}," 从输入缓冲区取出数据;\n",[34,276512,276513],{"class":36,"line":5537},[34,276514,276515],{}," signal(mutexM);\n",[34,276517,276518],{"class":36,"line":5543},[34,276519,276520],{}," signal(emptyM); \u002F\u002F 释放输入缓冲区空位\n",[34,276522,276523],{"class":36,"line":5548},[34,276524,53],{"emptyLinePlaceholder":52},[34,276526,276527],{"class":36,"line":5564},[34,276528,276529],{}," 计算数据;\n",[34,276531,276532],{"class":36,"line":5576},[34,276533,53],{"emptyLinePlaceholder":52},[34,276535,276536],{"class":36,"line":5606},[34,276537,276538],{}," wait(emptyN); \u002F\u002F 申请输出缓冲区空位\n",[34,276540,276541],{"class":36,"line":5616},[34,276542,276543],{}," wait(mutexN); \u002F\u002F 锁定输出缓冲区\n",[34,276545,276546],{"class":36,"line":5628},[34,276547,276548],{}," 将结果放入输出缓冲区;\n",[34,276550,276551],{"class":36,"line":5634},[34,276552,276553],{}," signal(mutexN);\n",[34,276555,276556],{"class":36,"line":6420},[34,276557,276558],{}," signal(fullN); \u002F\u002F 通知输出缓冲区已有结果\n",[34,276560,276561],{"class":36,"line":11294},[34,276562,77],{},[34,276564,276565],{"class":36,"line":11321},[34,276566,53],{"emptyLinePlaceholder":52},[34,276568,276569],{"class":36,"line":11326},[34,276570,276571],{},"\u002F\u002F 输出进程 (Consumer 2)\n",[34,276573,276574],{"class":36,"line":11333},[34,276575,276448],{},[34,276577,276578],{"class":36,"line":11344},[34,276579,276580],{}," wait(fullN); \u002F\u002F 等待输出缓冲区有结果\n",[34,276582,276583],{"class":36,"line":11349},[34,276584,276543],{},[34,276586,276587],{"class":36,"line":11354},[34,276588,276589],{}," 从输出缓冲区取出结果;\n",[34,276591,276592],{"class":36,"line":11363},[34,276593,276553],{},[34,276595,276596],{"class":36,"line":11368},[34,276597,276598],{}," signal(emptyN); \u002F\u002F 释放输出缓冲区空位\n",[34,276600,276601],{"class":36,"line":11377},[34,276602,90595],{},[34,276604,276605],{"class":36,"line":11402},[34,276606,276607],{}," 输出结果;\n",[34,276609,276610],{"class":36,"line":11407},[34,276611,77],{},[15,276613,276614,47649,276616,258,276619,276622],{},[2627,276615,246532],{},[31,276617,276618],{},"wait(互斥信号量)",[31,276620,276621],{},"wait(同步信号量)"," 的顺序不能颠倒！必须先同步（申请资源 empty\u002Ffull），再互斥（锁定 mutex），否则会导致死锁。\n:::",[19,276624,276626],{"id":276625},"_39-文件系统设计60g-磁盘商用-io","39. 文件系统设计：60G 磁盘、商用 I\u002FO",[15,276628,276629],{},"假定你负责设计一个基于 32 位计算机的文件系统，如果存储磁盘的容量是 60G，磁盘扇区大小为 1M，文件的最大容量为 2GB，文件名仅支持 8.3 格式。该文件系统主要满足商用 I\u002FO 操作，因此空间变化比较频繁，请设计一种合理的文件系统磁盘空间管理方式。包括目录、文件的逻辑结构与物理实现。",[15,276631,264303],{},[253,276633,267207],{"id":276634},"答案与详细解答-12",[105,276636,276637,276651,276664,276674],{},[108,276638,276639,276642,276643,276646,276647,276650],{},[2627,276640,276641],{},"物理实现 (Physical Structure)：","\n因为空间变化频繁，动态扩容要求高，应采用",[2627,276644,276645],{},"索引分配 (Indexed Allocation)","。\n磁盘块大小为 1MB，最大文件 2GB，代表一个文件最多占用 2048 个块。一个索引块完全可以装得下这 2048 个指针，因此采用",[2627,276648,276649],{},"单级索引","结构即可。",[108,276652,276653,276656,276657,276660,276661,347],{},[2627,276654,276655],{},"逻辑结构 (Logical Structure)：","\n因为主要满足商用 I\u002FO，这类场景通常涉及数据库和结构化查询，应该采用",[2627,276658,276659],{},"有结构文件（记录式文件）","，为了加速查询，设计为",[2627,276662,276663],{},"索引顺序文件",[108,276665,276666,276669,276670,276673],{},[2627,276667,276668],{},"磁盘空间管理 (Free Space Management)：","\n总块数 = 60GB \u002F 1MB = 60,000 块。\n可以使用",[2627,276671,276672],{},"位示图 (Bit map)"," 来管理空闲块。只有 60,000 块，位示图仅需约 7.5 KB 的空间，可以常驻内存，提供极高的分配效率。",[108,276675,276676,276679,276680,276683,276684,276687],{},[2627,276677,276678],{},"目录结构 (Directory Structure)：","\n采用",[2627,276681,276682],{},"多级树状目录结构 (Tree-structured Directory)","。\n由于只支持 8.3 格式（文件名 8 字符 + 扩展名 3 字符，共 11 bytes），目录项可以设计为包含 ",[31,276685,276686],{},"11 Bytes文件名 + 4 Bytes索引节点\u002F首块指针"," 等简洁结构。",[15,276689,276690,276692],{},[2627,276691,246532],{}," 不要盲目写多级索引。这里块大小有 1MB 那么大，一个块装几万个指针绰绰有余，完全不需要 UNIX 那种多层间接索引。\n:::",[19,276694,276696],{"id":276695},"_40-信号量同步readmoveprint-三进程单容量缓冲","40. 信号量同步：read\u002Fmove\u002Fprint 三进程单容量缓冲",[15,276698,276699],{},"假定系统有三个并发进程 read, move 和 print 共享缓冲器 B1 和 B2。进程 read 负责从输入设备上读信息，每读出一个记录后把它存放到缓冲器 B1 中。进程 move 从缓冲器 B1 中取出一记录，加工后存入缓冲器 B2。进程 print 将 B2 中的记录取出打印输出。缓冲器 B1 和 B2 每次只能存放一个记录。要求三个进程协调完成任务，使打印出来的与读入的记录的个数，次序完全一样。请用 wait 和 signal 原语写出它们的并发程序。",[15,276701,264303],{},[253,276703,267207],{"id":276704},"答案与详细解答-13",[15,276706,276707],{},"由于 B1 和 B2 的容量都严格为 1，为每个缓冲区定义一对同步信号量：",[105,276709,276710,276716,276722,276728],{},[108,276711,276712,276715],{},[31,276713,276714],{},"empty1 = 1","：B1 的空位数。",[108,276717,276718,276721],{},[31,276719,276720],{},"full1 = 0","：B1 的数据数。",[108,276723,276724,276727],{},[31,276725,276726],{},"empty2 = 1","：B2 的空位数。",[108,276729,276730,276733],{},[31,276731,276732],{},"full2 = 0","：B2 的数据数。",[24,276735,276737],{"className":65734,"code":276736,"language":31965,"meta":29,"style":29},"\u002F\u002F 读入进程\nvoid read_process() {\n while(true) {\n 读入一个记录 info;\n wait(empty1); \u002F\u002F 确保 B1 为空\n 将 info 存入 B1;\n signal(full1); \u002F\u002F 通知 move 进程 B1 已有数据\n }\n}\n\n\u002F\u002F 搬运\u002F加工进程\nvoid move_process() {\n while(true) {\n wait(full1); \u002F\u002F 确保 B1 有数据\n 从 B1 取出记录 info;\n signal(empty1); \u002F\u002F 释放 B1\n\n 加工 info;\n\n wait(empty2); \u002F\u002F 确保 B2 为空\n 将 info 存入 B2;\n signal(full2); \u002F\u002F 通知 print 进程 B2 已有数据\n }\n}\n\n\u002F\u002F 打印进程\nvoid print_process() {\n while(true) {\n wait(full2); \u002F\u002F 确保 B2 有数据\n 从 B2 取出记录 info;\n signal(empty2); \u002F\u002F 释放 B2\n\n 打印 info;\n }\n}\n",[31,276738,276739,276744,276749,276754,276759,276764,276769,276774,276778,276782,276786,276791,276796,276800,276805,276810,276815,276819,276824,276828,276833,276838,276843,276847,276851,276855,276860,276865,276869,276874,276879,276884,276888,276893,276897],{"__ignoreMap":29},[34,276740,276741],{"class":36,"line":37},[34,276742,276743],{},"\u002F\u002F 读入进程\n",[34,276745,276746],{"class":36,"line":43},[34,276747,276748],{},"void read_process() {\n",[34,276750,276751],{"class":36,"line":49},[34,276752,276753],{}," while(true) {\n",[34,276755,276756],{"class":36,"line":56},[34,276757,276758],{}," 读入一个记录 info;\n",[34,276760,276761],{"class":36,"line":62},[34,276762,276763],{}," wait(empty1); \u002F\u002F 确保 B1 为空\n",[34,276765,276766],{"class":36,"line":68},[34,276767,276768],{}," 将 info 存入 B1;\n",[34,276770,276771],{"class":36,"line":74},[34,276772,276773],{}," signal(full1); \u002F\u002F 通知 move 进程 B1 已有数据\n",[34,276775,276776],{"class":36,"line":1971},[34,276777,5540],{},[34,276779,276780],{"class":36,"line":1976},[34,276781,77],{},[34,276783,276784],{"class":36,"line":1985},[34,276785,53],{"emptyLinePlaceholder":52},[34,276787,276788],{"class":36,"line":1990},[34,276789,276790],{},"\u002F\u002F 搬运\u002F加工进程\n",[34,276792,276793],{"class":36,"line":1999},[34,276794,276795],{},"void move_process() {\n",[34,276797,276798],{"class":36,"line":2022},[34,276799,276753],{},[34,276801,276802],{"class":36,"line":2036},[34,276803,276804],{}," wait(full1); \u002F\u002F 确保 B1 有数据\n",[34,276806,276807],{"class":36,"line":2603},[34,276808,276809],{}," 从 B1 取出记录 info;\n",[34,276811,276812],{"class":36,"line":5537},[34,276813,276814],{}," signal(empty1); \u002F\u002F 释放 B1\n",[34,276816,276817],{"class":36,"line":5543},[34,276818,53],{"emptyLinePlaceholder":52},[34,276820,276821],{"class":36,"line":5548},[34,276822,276823],{}," 加工 info;\n",[34,276825,276826],{"class":36,"line":5564},[34,276827,53],{"emptyLinePlaceholder":52},[34,276829,276830],{"class":36,"line":5576},[34,276831,276832],{}," wait(empty2); \u002F\u002F 确保 B2 为空\n",[34,276834,276835],{"class":36,"line":5606},[34,276836,276837],{}," 将 info 存入 B2;\n",[34,276839,276840],{"class":36,"line":5616},[34,276841,276842],{}," signal(full2); \u002F\u002F 通知 print 进程 B2 已有数据\n",[34,276844,276845],{"class":36,"line":5628},[34,276846,5540],{},[34,276848,276849],{"class":36,"line":5634},[34,276850,77],{},[34,276852,276853],{"class":36,"line":6420},[34,276854,53],{"emptyLinePlaceholder":52},[34,276856,276857],{"class":36,"line":11294},[34,276858,276859],{},"\u002F\u002F 打印进程\n",[34,276861,276862],{"class":36,"line":11321},[34,276863,276864],{},"void print_process() {\n",[34,276866,276867],{"class":36,"line":11326},[34,276868,276753],{},[34,276870,276871],{"class":36,"line":11333},[34,276872,276873],{}," wait(full2); \u002F\u002F 确保 B2 有数据\n",[34,276875,276876],{"class":36,"line":11344},[34,276877,276878],{}," 从 B2 取出记录 info;\n",[34,276880,276881],{"class":36,"line":11349},[34,276882,276883],{}," signal(empty2); \u002F\u002F 释放 B2\n",[34,276885,276886],{"class":36,"line":11354},[34,276887,53],{"emptyLinePlaceholder":52},[34,276889,276890],{"class":36,"line":11363},[34,276891,276892],{}," 打印 info;\n",[34,276894,276895],{"class":36,"line":11368},[34,276896,5540],{},[34,276898,276899],{"class":36,"line":11377},[34,276900,77],{},[15,276902,276903,276905,276906,276908,276909,276952,276953,276955,276956,258,276958,276960],{},[2627,276904,246532],{}," 画蛇添足加上 ",[31,276907,196586],{},"。当缓冲区大小 ",[34,276910,276912,276928],{"className":276911},[31642],[34,276913,276915],{"className":276914},[31646],[31648,276916,276917],{"xmlns":31650},[31652,276918,276919,276925],{},[31655,276920,276921,276923],{},[31662,276922,29622],{},[31749,276924,640],{},[31665,276926,276927],{"encoding":31667},">1",[34,276929,276931,276943],{"className":276930,"ariaHidden":758},[31672],[34,276932,276934,276937,276940],{"className":276933},[31676],[34,276935],{"className":276936,"style":34812},[31680],[34,276938,29622],{"className":276939},[31850],[34,276941],{"className":276942,"style":31846},[31845],[34,276944,276946,276949],{"className":276945},[31676],[34,276947],{"className":276948,"style":32617},[31680],[34,276950,640],{"className":276951},[31685]," 或存在多个竞争者时才必须加上 ",[31,276954,196586],{},"。对于严格的一对一且容量为 1 的通道，",[31,276957,198187],{},[31,276959,257323],{}," 的 0\u002F1 切换本身就是完美的互斥锁。\n:::",[19,276962,276964],{"id":276963},"_41-磁道调度fcfs-sstf-scan递增方向","41. 磁道调度：FCFS \u002F SSTF \u002F SCAN（递增方向）",[15,276966,276967],{},"若干个等待访问磁盘者依次要访问的磁道为 20，44，40，4，80，12，76，假设每移动一个磁道需要 3 毫秒时间，移动臂当前位于 40 号柱面，请按下列算法分别写出访问序列并计算为完成上述各次访问总共花费的寻道时间。",[105,276969,276970,276973,276976],{},[108,276971,276972],{},"（1）先来先服务算法；",[108,276974,276975],{},"（2）最短寻道时间优先算法；",[108,276977,276978],{},"（3）扫描算法（当前磁头移动的方向为磁道递增）。",[15,276980,264303],{},[15,276982,264306],{},[19,276984,276986],{"id":276985},"_42-文件目录检索目录项分解的平均磁盘访问次数","42. 文件目录检索：目录项分解的平均磁盘访问次数",[15,276988,276989],{},"在实现文件系统时，一般为加快文件目录的检索速度，可利用\"文件控制块部分装入\"的方法。假设目录文件（即文件控制块）存放在磁盘上，磁盘的每个盘块为 512B，每个目录项占 128B，其中文件名占 11B。为提高检索速度，通常将目录项分解成两部分，第一部分（包括文件名和文件内部号）占 16B，第二部分（包括文件内部号和文件其他描述信息）占 122B。假设某一目录共有 254 个目录项（文件控制块），试分别给出前、后二种方法查找该目录文件某一目录项的平均访问磁盘次数。",[15,276991,264303],{},[15,276993,264306],{},[19,276995,276997],{"id":276996},"_43-磁鼓优化存储20个区1ms读取2ms处理","43. 磁鼓优化存储：20个区，1ms读取，2ms处理",[15,276999,277000],{},"旋转型设备上信息的优化分布能减少为若干个 I\u002FO 服务的总时间。设磁鼓上分为 20 个区，每区存放一个记录，磁鼓旋转一周需 20 毫秒，读出每个记录平均需用 1 毫秒，读出后经 2 毫秒处理，再继续处理下一个记录。在不知当前磁鼓位置的情况下：",[105,277002,277003,277006],{},[108,277004,277005],{},"（1）顺序存放记录 1、……，记录 20 时，试计算读出并处理 20 个记录的总时间；",[108,277007,277008],{},"（2）给出优先分布 20 个记录的一种方案，使得所花的总处理时间减少，且计算出这个方案所花的总时间。",[15,277010,264303],{},[15,277012,264306],{},[19,277014,277016],{"id":277015},"_44-分时操作系统设计重要进程与-io-瓶颈","44. 分时操作系统设计：重要进程与 I\u002FO 瓶颈",[15,277018,277019],{},"假设你需要为某企业设计一个分时操作系统，请回答以下问题：",[105,277021,277022,277025],{},[108,277023,277024],{},"(1) 你认为哪些系统进程最重要？",[108,277026,277027],{},"(2) 如果该操作系统主要负责处理 IO-Bounded 进程，你认为系统性能的瓶颈可能是什么？如何解决？",[15,277029,264303],{},[15,277031,264306],{},[19,277033,277035],{"id":277034},"_45-银行家算法单类资源s类-150个","45. 银行家算法：单类资源（S类 150个）",[15,277037,277038],{},"系统中具有 S 类资源 150 个，在 T0 时刻按下表所示分配给 3 个进程：",[13837,277040,277041,277053],{},[13840,277042,277043],{},[13843,277044,277045,277047,277050],{},[13846,277046,247197],{"align":82355},[13846,277048,277049],{"align":82355},"Maximum demand",[13846,277051,277052],{"align":82355},"Current allocation",[13853,277054,277055,277063,277071],{},[13843,277056,277057,277059,277061],{},[13858,277058,97876],{"align":82355},[13858,277060,98497],{"align":82355},[13858,277062,95556],{"align":82355},[13843,277064,277065,277067,277069],{},[13858,277066,97925],{"align":82355},[13858,277068,167664],{"align":82355},[13858,277070,5731],{"align":82355},[13843,277072,277073,277075,277077],{},[13858,277074,263561],{"align":82355},[13858,277076,167664],{"align":82355},[13858,277078,200133],{"align":82355},[15,277080,277081],{},"对下列请求应用银行家算法逐步分别分析判定是否安全，如果是安全的，请给出一个可能的进程安全执行序列；如果不是安全的，请说明原因。",[105,277083,277084,277087],{},[108,277085,277086],{},"(1) 第 4 个进程 P4 到达，对资源 S 的最大需求为 60 个，当前请求分配 25 个；",[108,277088,277089],{},"(2) 第 4 个进程 P4 到达，对资源 S 的最大需求 50 个，当前请求分配 35 个。",[15,277091,264303],{},[15,277093,264306],{},[19,277095,277097],{"id":277096},"_46-请求分页快表地址转换过程与虚地址计算","46. 请求分页：快表地址转换过程与虚地址计算",[15,277099,277100],{},"设正在处理器上执行的一个进程的页表如下表所示，表中的虚页号和物理块号是十进制数，起始页号（块号）均为 0。所有的地址均是存储器字节地址。页的大小为 1024 字节。",[13837,277102,277103,277122],{},[13840,277104,277105],{},[13843,277106,277107,277110,277113,277116,277119],{},[13846,277108,277109],{"align":82355},"虚页号",[13846,277111,277112],{"align":82355},"状态位",[13846,277114,277115],{"align":82355},"访问位",[13846,277117,277118],{"align":82355},"修改位",[13846,277120,277121],{"align":82355},"物理块号",[13853,277123,277124,277136,277148,277160,277172,277184],{},[13843,277125,277126,277128,277130,277132,277134],{},[13858,277127,459],{"align":82355},[13858,277129,640],{"align":82355},[13858,277131,640],{"align":82355},[13858,277133,459],{"align":82355},[13858,277135,15932],{"align":82355},[13843,277137,277138,277140,277142,277144,277146],{},[13858,277139,640],{"align":82355},[13858,277141,640],{"align":82355},[13858,277143,640],{"align":82355},[13858,277145,640],{"align":82355},[13858,277147,36723],{"align":82355},[13843,277149,277150,277152,277154,277156,277158],{},[13858,277151,768],{"align":82355},[13858,277153,459],{"align":82355},[13858,277155,459],{"align":82355},[13858,277157,459],{"align":82355},[13858,277159,2511],{"align":82355},[13843,277161,277162,277164,277166,277168,277170],{},[13858,277163,786],{"align":82355},[13858,277165,640],{"align":82355},[13858,277167,459],{"align":82355},[13858,277169,459],{"align":82355},[13858,277171,768],{"align":82355},[13843,277173,277174,277176,277178,277180,277182],{},[13858,277175,15932],{"align":82355},[13858,277177,459],{"align":82355},[13858,277179,459],{"align":82355},[13858,277181,459],{"align":82355},[13858,277183,2511],{"align":82355},[13843,277185,277186,277188,277190,277192,277194],{},[13858,277187,18136],{"align":82355},[13858,277189,640],{"align":82355},[13858,277191,459],{"align":82355},[13858,277193,640],{"align":82355},[13858,277195,459],{"align":82355},[15,277197,277198],{},"注：当某页被访问时，其访问位置 1。",[105,277200,277201,277204],{},[108,277202,277203],{},"(1) 详述在设有快表的请求分页存储管理系统中，一个虚地址转换成物理内存地址的过程。",[108,277205,277206],{},"(2) 下列十进制虚地址对应于什么物理地址：5579，2232",[15,277208,264303],{},[15,277210,264306],{},[19,277212,277214],{"id":277213},"_47题目缺失","47.（题目缺失）",[15,277216,264303],{},[15,277218,264306],{},[19,277220,277222],{"id":277221},"_48-sstf-与-scan-磁道调度200个柱面","48. SSTF 与 SCAN 磁道调度（200个柱面）",[15,277224,277225],{},"假定一个磁盘有 200 个柱面（编号 0-199），磁盘请求队列由对如下柱面的请求序列构成：50, 60, 30, 18, 90, 165, 150, 38, 12, 190. 已知磁头当前位于 95 号柱面，移动方向为向内。",[105,277227,277228,277231],{},[108,277229,277230],{},"(1) 如果系统采用 SSTF 算法进行调度，那么系统处理完上述所有的磁盘请求所需的寻道距离是多少？",[108,277232,277233],{},"(2) 如果系统采用 SCAN 算法进行调度，那么系统处理完上述所有的磁盘请求所需的寻道距离是多少？",[15,277235,264303],{},[15,277237,264306],{},[19,277239,277241],{"id":277240},"_49-请求分页地址位数与物理块数计算","49. 请求分页：地址位数与物理块数计算",[15,277243,277244],{},"一个采用请求式存储管理的计算机系统，其主存（实存）容量为 256M 字节，虚存容量（给用户的最大地址空间）为 4G 字节，页面大小为 4K 字节，试问：",[105,277246,277247,277250,277253,277256,277259],{},[108,277248,277249],{},"(1) 主存物理地址应设为多少位？",[108,277251,277252],{},"(2) 主存中有多少物理块？",[108,277254,277255],{},"(3) 虚拟地址应该设多少位？",[108,277257,277258],{},"(4) 虚拟地址空间最多可以有多少页？",[108,277260,277261],{},"(5) 页内最大和最小偏移量是多少？",[15,277263,264303],{},[15,277265,264306],{},[19,277267,277269],{"id":277268},"_50-页面置换算法计算fifo-lru-opt","50. 页面置换算法计算（FIFO \u002F LRU \u002F OPT）",[15,277271,277272],{},"设某作业占有 7 个页面，如果在主存中只允许装入 4 个工作页面（即工作集为 4），作业运行时，实际访问页面的顺序是：1, 2, 3, 6, 4, 7, 3, 2, 1, 4, 7, 5, 6, 5, 2, 1。试用 FIFO、LRU 和 OPT 页面置换算法，列出各自的页面淘汰顺序和页面置换次数。",[15,277274,264303],{},[253,277276,223920],{"id":277277},"概念核对-concept-check-1",[105,277279,277280,277286,277295],{},[108,277281,277282,277285],{},[2627,277283,277284],{},"工作集大小："," 题目规定只允许装入 4 个工作页面，意味着内存中最多只有 4 个物理块（Frame）供该作业使用。",[108,277287,277288,277291,277292,347],{},[2627,277289,277290],{},"缺页中断 (Page Fault) 与 页面置换 (Page Replacement) 的区别："," 当内存块未满时，调入新页面只会发生\"缺页中断\"，但不会发生\"页面置换\"（因为有空闲位置）。只有当 4 个块都装满后，再调入新页面才会发生\"置换\"。题目明确要求计算的是",[2627,277293,277294],{},"置换次数",[108,277296,277297,277300],{},[2627,277298,277299],{},"三大算法：",[105,277301,277302,277305,277308],{},[108,277303,277304],{},"FIFO (先进先出)：淘汰在内存中驻留时间最长的页面。",[108,277306,277307],{},"LRU (最近最久未使用)：淘汰在最近一段时间内最久没有被访问过的页面（向前看）。",[108,277309,277310],{},"OPT (最佳置换)：淘汰在未来最长时间内不再被访问的页面（向后看，理想算法）。",[253,277312,277313],{"id":277313},"原理解析与计算过程",[15,277315,277316,277317,347],{},"给定的访问序列长度为16：1, 2, 3, 6, 4, 7, 3, 2, 1, 4, 7, 5, 6, 5, 2, 1。\n初始 4 个空闲块，装入 1, 2, 3, 6 时发生 4 次缺页，0 次置换。此时内存为 ",[34,277318,277320,277350],{"className":277319},[31642],[34,277321,277323],{"className":277322},[31646],[31648,277324,277325],{"xmlns":31650},[31652,277326,277327,277347],{},[31655,277328,277329,277331,277333,277335,277337,277339,277341,277343,277345],{},[31662,277330,126529],{"stretchy":702},[31749,277332,640],{},[31662,277334,31960],{"separator":758},[31749,277336,768],{},[31662,277338,31960],{"separator":758},[31749,277340,786],{},[31662,277342,31960],{"separator":758},[31749,277344,36656],{},[31662,277346,126544],{"stretchy":702},[31665,277348,277349],{"encoding":31667},"\\langle1, 2, 3, 6\\rangle",[34,277351,277353],{"className":277352,"ariaHidden":758},[31672],[34,277354,277356,277359,277362,277365,277368,277371,277374,277377,277380,277383,277386,277389,277392],{"className":277355},[31676],[34,277357],{"className":277358,"style":31774},[31680],[34,277360,126529],{"className":277361},[31782],[34,277363,640],{"className":277364},[31685],[34,277366,31960],{"className":277367},[32119],[34,277369],{"className":277370,"style":32123},[31845],[34,277372,768],{"className":277373},[31685],[34,277375,31960],{"className":277376},[32119],[34,277378],{"className":277379,"style":32123},[31845],[34,277381,786],{"className":277382},[31685],[34,277384,31960],{"className":277385},[32119],[34,277387],{"className":277388,"style":32123},[31845],[34,277390,36656],{"className":277391},[31685],[34,277393,126544],{"className":277394},[31690],[32490,277396,277398],{"id":277397},"_1-fifo-先进先出","1. FIFO (先进先出)",[105,277400,277401,277488,277574,277579,277665,277751,277757,277843,277929,277934],{},[108,277402,277403,277405,277406,277409,277410,347],{},[2627,277404,15932],{},"：内存满，最先进入的是 1。",[2627,277407,277408],{},"淘汰 1","。内存变为 ",[34,277411,277413,277443],{"className":277412},[31642],[34,277414,277416],{"className":277415},[31646],[31648,277417,277418],{"xmlns":31650},[31652,277419,277420,277440],{},[31655,277421,277422,277424,277426,277428,277430,277432,277434,277436,277438],{},[31662,277423,126529],{"stretchy":702},[31749,277425,768],{},[31662,277427,31960],{"separator":758},[31749,277429,786],{},[31662,277431,31960],{"separator":758},[31749,277433,36656],{},[31662,277435,31960],{"separator":758},[31749,277437,15932],{},[31662,277439,126544],{"stretchy":702},[31665,277441,277442],{"encoding":31667},"\\langle2, 3, 6, 4\\rangle",[34,277444,277446],{"className":277445,"ariaHidden":758},[31672],[34,277447,277449,277452,277455,277458,277461,277464,277467,277470,277473,277476,277479,277482,277485],{"className":277448},[31676],[34,277450],{"className":277451,"style":31774},[31680],[34,277453,126529],{"className":277454},[31782],[34,277456,768],{"className":277457},[31685],[34,277459,31960],{"className":277460},[32119],[34,277462],{"className":277463,"style":32123},[31845],[34,277465,786],{"className":277466},[31685],[34,277468,31960],{"className":277469},[32119],[34,277471],{"className":277472,"style":32123},[31845],[34,277474,36656],{"className":277475},[31685],[34,277477,31960],{"className":277478},[32119],[34,277480],{"className":277481,"style":32123},[31845],[34,277483,15932],{"className":277484},[31685],[34,277486,126544],{"className":277487},[31690],[108,277489,277490,277492,277493,277409,277496,347],{},[2627,277491,36723],{},"：最早进入的是 2。",[2627,277494,277495],{},"淘汰 2",[34,277497,277499,277529],{"className":277498},[31642],[34,277500,277502],{"className":277501},[31646],[31648,277503,277504],{"xmlns":31650},[31652,277505,277506,277526],{},[31655,277507,277508,277510,277512,277514,277516,277518,277520,277522,277524],{},[31662,277509,126529],{"stretchy":702},[31749,277511,786],{},[31662,277513,31960],{"separator":758},[31749,277515,36656],{},[31662,277517,31960],{"separator":758},[31749,277519,15932],{},[31662,277521,31960],{"separator":758},[31749,277523,36723],{},[31662,277525,126544],{"stretchy":702},[31665,277527,277528],{"encoding":31667},"\\langle3, 6, 4, 7\\rangle",[34,277530,277532],{"className":277531,"ariaHidden":758},[31672],[34,277533,277535,277538,277541,277544,277547,277550,277553,277556,277559,277562,277565,277568,277571],{"className":277534},[31676],[34,277536],{"className":277537,"style":31774},[31680],[34,277539,126529],{"className":277540},[31782],[34,277542,786],{"className":277543},[31685],[34,277545,31960],{"className":277546},[32119],[34,277548],{"className":277549,"style":32123},[31845],[34,277551,36656],{"className":277552},[31685],[34,277554,31960],{"className":277555},[32119],[34,277557],{"className":277558,"style":32123},[31845],[34,277560,15932],{"className":277561},[31685],[34,277563,31960],{"className":277564},[32119],[34,277566],{"className":277567,"style":32123},[31845],[34,277569,36723],{"className":277570},[31685],[34,277572,126544],{"className":277573},[31690],[108,277575,277576,277578],{},[2627,277577,786],{},"：命中。",[108,277580,277581,277583,277584,277409,277587,347],{},[2627,277582,768],{},"：最早进入的是 3。",[2627,277585,277586],{},"淘汰 3",[34,277588,277590,277620],{"className":277589},[31642],[34,277591,277593],{"className":277592},[31646],[31648,277594,277595],{"xmlns":31650},[31652,277596,277597,277617],{},[31655,277598,277599,277601,277603,277605,277607,277609,277611,277613,277615],{},[31662,277600,126529],{"stretchy":702},[31749,277602,36656],{},[31662,277604,31960],{"separator":758},[31749,277606,15932],{},[31662,277608,31960],{"separator":758},[31749,277610,36723],{},[31662,277612,31960],{"separator":758},[31749,277614,768],{},[31662,277616,126544],{"stretchy":702},[31665,277618,277619],{"encoding":31667},"\\langle6, 4, 7, 2\\rangle",[34,277621,277623],{"className":277622,"ariaHidden":758},[31672],[34,277624,277626,277629,277632,277635,277638,277641,277644,277647,277650,277653,277656,277659,277662],{"className":277625},[31676],[34,277627],{"className":277628,"style":31774},[31680],[34,277630,126529],{"className":277631},[31782],[34,277633,36656],{"className":277634},[31685],[34,277636,31960],{"className":277637},[32119],[34,277639],{"className":277640,"style":32123},[31845],[34,277642,15932],{"className":277643},[31685],[34,277645,31960],{"className":277646},[32119],[34,277648],{"className":277649,"style":32123},[31845],[34,277651,36723],{"className":277652},[31685],[34,277654,31960],{"className":277655},[32119],[34,277657],{"className":277658,"style":32123},[31845],[34,277660,768],{"className":277661},[31685],[34,277663,126544],{"className":277664},[31690],[108,277666,277667,277669,277670,277409,277673,347],{},[2627,277668,640],{},"：最早进入的是 6。",[2627,277671,277672],{},"淘汰 6",[34,277674,277676,277706],{"className":277675},[31642],[34,277677,277679],{"className":277678},[31646],[31648,277680,277681],{"xmlns":31650},[31652,277682,277683,277703],{},[31655,277684,277685,277687,277689,277691,277693,277695,277697,277699,277701],{},[31662,277686,126529],{"stretchy":702},[31749,277688,15932],{},[31662,277690,31960],{"separator":758},[31749,277692,36723],{},[31662,277694,31960],{"separator":758},[31749,277696,768],{},[31662,277698,31960],{"separator":758},[31749,277700,640],{},[31662,277702,126544],{"stretchy":702},[31665,277704,277705],{"encoding":31667},"\\langle4, 7, 2, 1\\rangle",[34,277707,277709],{"className":277708,"ariaHidden":758},[31672],[34,277710,277712,277715,277718,277721,277724,277727,277730,277733,277736,277739,277742,277745,277748],{"className":277711},[31676],[34,277713],{"className":277714,"style":31774},[31680],[34,277716,126529],{"className":277717},[31782],[34,277719,15932],{"className":277720},[31685],[34,277722,31960],{"className":277723},[32119],[34,277725],{"className":277726,"style":32123},[31845],[34,277728,36723],{"className":277729},[31685],[34,277731,31960],{"className":277732},[32119],[34,277734],{"className":277735,"style":32123},[31845],[34,277737,768],{"className":277738},[31685],[34,277740,31960],{"className":277741},[32119],[34,277743],{"className":277744,"style":32123},[31845],[34,277746,640],{"className":277747},[31685],[34,277749,126544],{"className":277750},[31690],[108,277752,277753,277756],{},[2627,277754,277755],{},"4, 7","：连续命中。",[108,277758,277759,277761,277762,277409,277765,347],{},[2627,277760,18136],{},"：最早进入的是 4。",[2627,277763,277764],{},"淘汰 4",[34,277766,277768,277798],{"className":277767},[31642],[34,277769,277771],{"className":277770},[31646],[31648,277772,277773],{"xmlns":31650},[31652,277774,277775,277795],{},[31655,277776,277777,277779,277781,277783,277785,277787,277789,277791,277793],{},[31662,277778,126529],{"stretchy":702},[31749,277780,36723],{},[31662,277782,31960],{"separator":758},[31749,277784,768],{},[31662,277786,31960],{"separator":758},[31749,277788,640],{},[31662,277790,31960],{"separator":758},[31749,277792,18136],{},[31662,277794,126544],{"stretchy":702},[31665,277796,277797],{"encoding":31667},"\\langle7, 2, 1, 5\\rangle",[34,277799,277801],{"className":277800,"ariaHidden":758},[31672],[34,277802,277804,277807,277810,277813,277816,277819,277822,277825,277828,277831,277834,277837,277840],{"className":277803},[31676],[34,277805],{"className":277806,"style":31774},[31680],[34,277808,126529],{"className":277809},[31782],[34,277811,36723],{"className":277812},[31685],[34,277814,31960],{"className":277815},[32119],[34,277817],{"className":277818,"style":32123},[31845],[34,277820,768],{"className":277821},[31685],[34,277823,31960],{"className":277824},[32119],[34,277826],{"className":277827,"style":32123},[31845],[34,277829,640],{"className":277830},[31685],[34,277832,31960],{"className":277833},[32119],[34,277835],{"className":277836,"style":32123},[31845],[34,277838,18136],{"className":277839},[31685],[34,277841,126544],{"className":277842},[31690],[108,277844,277845,277847,277848,277409,277851,347],{},[2627,277846,36656],{},"：最早进入的是 7。",[2627,277849,277850],{},"淘汰 7",[34,277852,277854,277884],{"className":277853},[31642],[34,277855,277857],{"className":277856},[31646],[31648,277858,277859],{"xmlns":31650},[31652,277860,277861,277881],{},[31655,277862,277863,277865,277867,277869,277871,277873,277875,277877,277879],{},[31662,277864,126529],{"stretchy":702},[31749,277866,768],{},[31662,277868,31960],{"separator":758},[31749,277870,640],{},[31662,277872,31960],{"separator":758},[31749,277874,18136],{},[31662,277876,31960],{"separator":758},[31749,277878,36656],{},[31662,277880,126544],{"stretchy":702},[31665,277882,277883],{"encoding":31667},"\\langle2, 1, 5, 6\\rangle",[34,277885,277887],{"className":277886,"ariaHidden":758},[31672],[34,277888,277890,277893,277896,277899,277902,277905,277908,277911,277914,277917,277920,277923,277926],{"className":277889},[31676],[34,277891],{"className":277892,"style":31774},[31680],[34,277894,126529],{"className":277895},[31782],[34,277897,768],{"className":277898},[31685],[34,277900,31960],{"className":277901},[32119],[34,277903],{"className":277904,"style":32123},[31845],[34,277906,640],{"className":277907},[31685],[34,277909,31960],{"className":277910},[32119],[34,277912],{"className":277913,"style":32123},[31845],[34,277915,18136],{"className":277916},[31685],[34,277918,31960],{"className":277919},[32119],[34,277921],{"className":277922,"style":32123},[31845],[34,277924,36656],{"className":277925},[31685],[34,277927,126544],{"className":277928},[31690],[108,277930,277931,277756],{},[2627,277932,277933],{},"5, 2, 1",[108,277935,277936,277938],{},[2627,277937,95791],{},[105,277939,277940,277946],{},[108,277941,277942,277945],{},[2627,277943,277944],{},"淘汰顺序："," 1, 2, 3, 6, 4, 7",[108,277947,277948,277951],{},[2627,277949,277950],{},"置换次数："," 6 次（总缺页10次）",[32490,277953,277955],{"id":277954},"_2-lru-最近最久未使用","2. LRU (最近最久未使用)",[15,277957,277958,277959,278036],{},"初始装入后为 ",[34,277960,277962,277991],{"className":277961},[31642],[34,277963,277965],{"className":277964},[31646],[31648,277966,277967],{"xmlns":31650},[31652,277968,277969,277989],{},[31655,277970,277971,277973,277975,277977,277979,277981,277983,277985,277987],{},[31662,277972,126529],{"stretchy":702},[31749,277974,640],{},[31662,277976,31960],{"separator":758},[31749,277978,768],{},[31662,277980,31960],{"separator":758},[31749,277982,786],{},[31662,277984,31960],{"separator":758},[31749,277986,36656],{},[31662,277988,126544],{"stretchy":702},[31665,277990,277349],{"encoding":31667},[34,277992,277994],{"className":277993,"ariaHidden":758},[31672],[34,277995,277997,278000,278003,278006,278009,278012,278015,278018,278021,278024,278027,278030,278033],{"className":277996},[31676],[34,277998],{"className":277999,"style":31774},[31680],[34,278001,126529],{"className":278002},[31782],[34,278004,640],{"className":278005},[31685],[34,278007,31960],{"className":278008},[32119],[34,278010],{"className":278011,"style":32123},[31845],[34,278013,768],{"className":278014},[31685],[34,278016,31960],{"className":278017},[32119],[34,278019],{"className":278020,"style":32123},[31845],[34,278022,786],{"className":278023},[31685],[34,278025,31960],{"className":278026},[32119],[34,278028],{"className":278029,"style":32123},[31845],[34,278031,36656],{"className":278032},[31685],[34,278034,126544],{"className":278035},[31690],"（右侧为最新使用）。",[105,278038,278039,278124,278208,278291,278376,278461,278546,278631,278715,278799,278882,278966,279050],{},[108,278040,278041,278043,278044,278046,278047,347],{},[2627,278042,15932],{},"：最久未用是 1。",[2627,278045,277408],{},"。变为 ",[34,278048,278050,278079],{"className":278049},[31642],[34,278051,278053],{"className":278052},[31646],[31648,278054,278055],{"xmlns":31650},[31652,278056,278057,278077],{},[31655,278058,278059,278061,278063,278065,278067,278069,278071,278073,278075],{},[31662,278060,126529],{"stretchy":702},[31749,278062,768],{},[31662,278064,31960],{"separator":758},[31749,278066,786],{},[31662,278068,31960],{"separator":758},[31749,278070,36656],{},[31662,278072,31960],{"separator":758},[31749,278074,15932],{},[31662,278076,126544],{"stretchy":702},[31665,278078,277442],{"encoding":31667},[34,278080,278082],{"className":278081,"ariaHidden":758},[31672],[34,278083,278085,278088,278091,278094,278097,278100,278103,278106,278109,278112,278115,278118,278121],{"className":278084},[31676],[34,278086],{"className":278087,"style":31774},[31680],[34,278089,126529],{"className":278090},[31782],[34,278092,768],{"className":278093},[31685],[34,278095,31960],{"className":278096},[32119],[34,278098],{"className":278099,"style":32123},[31845],[34,278101,786],{"className":278102},[31685],[34,278104,31960],{"className":278105},[32119],[34,278107],{"className":278108,"style":32123},[31845],[34,278110,36656],{"className":278111},[31685],[34,278113,31960],{"className":278114},[32119],[34,278116],{"className":278117,"style":32123},[31845],[34,278119,15932],{"className":278120},[31685],[34,278122,126544],{"className":278123},[31690],[108,278125,278126,278128,278129,278046,278131,347],{},[2627,278127,36723],{},"：最久未用是 2。",[2627,278130,277495],{},[34,278132,278134,278163],{"className":278133},[31642],[34,278135,278137],{"className":278136},[31646],[31648,278138,278139],{"xmlns":31650},[31652,278140,278141,278161],{},[31655,278142,278143,278145,278147,278149,278151,278153,278155,278157,278159],{},[31662,278144,126529],{"stretchy":702},[31749,278146,786],{},[31662,278148,31960],{"separator":758},[31749,278150,36656],{},[31662,278152,31960],{"separator":758},[31749,278154,15932],{},[31662,278156,31960],{"separator":758},[31749,278158,36723],{},[31662,278160,126544],{"stretchy":702},[31665,278162,277528],{"encoding":31667},[34,278164,278166],{"className":278165,"ariaHidden":758},[31672],[34,278167,278169,278172,278175,278178,278181,278184,278187,278190,278193,278196,278199,278202,278205],{"className":278168},[31676],[34,278170],{"className":278171,"style":31774},[31680],[34,278173,126529],{"className":278174},[31782],[34,278176,786],{"className":278177},[31685],[34,278179,31960],{"className":278180},[32119],[34,278182],{"className":278183,"style":32123},[31845],[34,278185,36656],{"className":278186},[31685],[34,278188,31960],{"className":278189},[32119],[34,278191],{"className":278192,"style":32123},[31845],[34,278194,15932],{"className":278195},[31685],[34,278197,31960],{"className":278198},[32119],[34,278200],{"className":278201,"style":32123},[31845],[34,278203,36723],{"className":278204},[31685],[34,278206,126544],{"className":278207},[31690],[108,278209,278210,278212,278213,347],{},[2627,278211,786],{},"：命中。更新顺序变为 ",[34,278214,278216,278246],{"className":278215},[31642],[34,278217,278219],{"className":278218},[31646],[31648,278220,278221],{"xmlns":31650},[31652,278222,278223,278243],{},[31655,278224,278225,278227,278229,278231,278233,278235,278237,278239,278241],{},[31662,278226,126529],{"stretchy":702},[31749,278228,36656],{},[31662,278230,31960],{"separator":758},[31749,278232,15932],{},[31662,278234,31960],{"separator":758},[31749,278236,36723],{},[31662,278238,31960],{"separator":758},[31749,278240,786],{},[31662,278242,126544],{"stretchy":702},[31665,278244,278245],{"encoding":31667},"\\langle6, 4, 7, 3\\rangle",[34,278247,278249],{"className":278248,"ariaHidden":758},[31672],[34,278250,278252,278255,278258,278261,278264,278267,278270,278273,278276,278279,278282,278285,278288],{"className":278251},[31676],[34,278253],{"className":278254,"style":31774},[31680],[34,278256,126529],{"className":278257},[31782],[34,278259,36656],{"className":278260},[31685],[34,278262,31960],{"className":278263},[32119],[34,278265],{"className":278266,"style":32123},[31845],[34,278268,15932],{"className":278269},[31685],[34,278271,31960],{"className":278272},[32119],[34,278274],{"className":278275,"style":32123},[31845],[34,278277,36723],{"className":278278},[31685],[34,278280,31960],{"className":278281},[32119],[34,278283],{"className":278284,"style":32123},[31845],[34,278286,786],{"className":278287},[31685],[34,278289,126544],{"className":278290},[31690],[108,278292,278293,278295,278296,278046,278298,347],{},[2627,278294,768],{},"：最久未用是 6。",[2627,278297,277672],{},[34,278299,278301,278331],{"className":278300},[31642],[34,278302,278304],{"className":278303},[31646],[31648,278305,278306],{"xmlns":31650},[31652,278307,278308,278328],{},[31655,278309,278310,278312,278314,278316,278318,278320,278322,278324,278326],{},[31662,278311,126529],{"stretchy":702},[31749,278313,15932],{},[31662,278315,31960],{"separator":758},[31749,278317,36723],{},[31662,278319,31960],{"separator":758},[31749,278321,786],{},[31662,278323,31960],{"separator":758},[31749,278325,768],{},[31662,278327,126544],{"stretchy":702},[31665,278329,278330],{"encoding":31667},"\\langle4, 7, 3, 2\\rangle",[34,278332,278334],{"className":278333,"ariaHidden":758},[31672],[34,278335,278337,278340,278343,278346,278349,278352,278355,278358,278361,278364,278367,278370,278373],{"className":278336},[31676],[34,278338],{"className":278339,"style":31774},[31680],[34,278341,126529],{"className":278342},[31782],[34,278344,15932],{"className":278345},[31685],[34,278347,31960],{"className":278348},[32119],[34,278350],{"className":278351,"style":32123},[31845],[34,278353,36723],{"className":278354},[31685],[34,278356,31960],{"className":278357},[32119],[34,278359],{"className":278360,"style":32123},[31845],[34,278362,786],{"className":278363},[31685],[34,278365,31960],{"className":278366},[32119],[34,278368],{"className":278369,"style":32123},[31845],[34,278371,768],{"className":278372},[31685],[34,278374,126544],{"className":278375},[31690],[108,278377,278378,278380,278381,278046,278383,347],{},[2627,278379,640],{},"：最久未用是 4。",[2627,278382,277764],{},[34,278384,278386,278416],{"className":278385},[31642],[34,278387,278389],{"className":278388},[31646],[31648,278390,278391],{"xmlns":31650},[31652,278392,278393,278413],{},[31655,278394,278395,278397,278399,278401,278403,278405,278407,278409,278411],{},[31662,278396,126529],{"stretchy":702},[31749,278398,36723],{},[31662,278400,31960],{"separator":758},[31749,278402,786],{},[31662,278404,31960],{"separator":758},[31749,278406,768],{},[31662,278408,31960],{"separator":758},[31749,278410,640],{},[31662,278412,126544],{"stretchy":702},[31665,278414,278415],{"encoding":31667},"\\langle7, 3, 2, 1\\rangle",[34,278417,278419],{"className":278418,"ariaHidden":758},[31672],[34,278420,278422,278425,278428,278431,278434,278437,278440,278443,278446,278449,278452,278455,278458],{"className":278421},[31676],[34,278423],{"className":278424,"style":31774},[31680],[34,278426,126529],{"className":278427},[31782],[34,278429,36723],{"className":278430},[31685],[34,278432,31960],{"className":278433},[32119],[34,278435],{"className":278436,"style":32123},[31845],[34,278438,786],{"className":278439},[31685],[34,278441,31960],{"className":278442},[32119],[34,278444],{"className":278445,"style":32123},[31845],[34,278447,768],{"className":278448},[31685],[34,278450,31960],{"className":278451},[32119],[34,278453],{"className":278454,"style":32123},[31845],[34,278456,640],{"className":278457},[31685],[34,278459,126544],{"className":278460},[31690],[108,278462,278463,278465,278466,278046,278468,347],{},[2627,278464,15932],{},"：最久未用是 7。",[2627,278467,277850],{},[34,278469,278471,278501],{"className":278470},[31642],[34,278472,278474],{"className":278473},[31646],[31648,278475,278476],{"xmlns":31650},[31652,278477,278478,278498],{},[31655,278479,278480,278482,278484,278486,278488,278490,278492,278494,278496],{},[31662,278481,126529],{"stretchy":702},[31749,278483,786],{},[31662,278485,31960],{"separator":758},[31749,278487,768],{},[31662,278489,31960],{"separator":758},[31749,278491,640],{},[31662,278493,31960],{"separator":758},[31749,278495,15932],{},[31662,278497,126544],{"stretchy":702},[31665,278499,278500],{"encoding":31667},"\\langle3, 2, 1, 4\\rangle",[34,278502,278504],{"className":278503,"ariaHidden":758},[31672],[34,278505,278507,278510,278513,278516,278519,278522,278525,278528,278531,278534,278537,278540,278543],{"className":278506},[31676],[34,278508],{"className":278509,"style":31774},[31680],[34,278511,126529],{"className":278512},[31782],[34,278514,786],{"className":278515},[31685],[34,278517,31960],{"className":278518},[32119],[34,278520],{"className":278521,"style":32123},[31845],[34,278523,768],{"className":278524},[31685],[34,278526,31960],{"className":278527},[32119],[34,278529],{"className":278530,"style":32123},[31845],[34,278532,640],{"className":278533},[31685],[34,278535,31960],{"className":278536},[32119],[34,278538],{"className":278539,"style":32123},[31845],[34,278541,15932],{"className":278542},[31685],[34,278544,126544],{"className":278545},[31690],[108,278547,278548,278550,278551,278046,278553,347],{},[2627,278549,36723],{},"：最久未用是 3。",[2627,278552,277586],{},[34,278554,278556,278586],{"className":278555},[31642],[34,278557,278559],{"className":278558},[31646],[31648,278560,278561],{"xmlns":31650},[31652,278562,278563,278583],{},[31655,278564,278565,278567,278569,278571,278573,278575,278577,278579,278581],{},[31662,278566,126529],{"stretchy":702},[31749,278568,768],{},[31662,278570,31960],{"separator":758},[31749,278572,640],{},[31662,278574,31960],{"separator":758},[31749,278576,15932],{},[31662,278578,31960],{"separator":758},[31749,278580,36723],{},[31662,278582,126544],{"stretchy":702},[31665,278584,278585],{"encoding":31667},"\\langle2, 1, 4, 7\\rangle",[34,278587,278589],{"className":278588,"ariaHidden":758},[31672],[34,278590,278592,278595,278598,278601,278604,278607,278610,278613,278616,278619,278622,278625,278628],{"className":278591},[31676],[34,278593],{"className":278594,"style":31774},[31680],[34,278596,126529],{"className":278597},[31782],[34,278599,768],{"className":278600},[31685],[34,278602,31960],{"className":278603},[32119],[34,278605],{"className":278606,"style":32123},[31845],[34,278608,640],{"className":278609},[31685],[34,278611,31960],{"className":278612},[32119],[34,278614],{"className":278615,"style":32123},[31845],[34,278617,15932],{"className":278618},[31685],[34,278620,31960],{"className":278621},[32119],[34,278623],{"className":278624,"style":32123},[31845],[34,278626,36723],{"className":278627},[31685],[34,278629,126544],{"className":278630},[31690],[108,278632,278633,278128,278635,278046,278637,347],{},[2627,278634,18136],{},[2627,278636,277495],{},[34,278638,278640,278670],{"className":278639},[31642],[34,278641,278643],{"className":278642},[31646],[31648,278644,278645],{"xmlns":31650},[31652,278646,278647,278667],{},[31655,278648,278649,278651,278653,278655,278657,278659,278661,278663,278665],{},[31662,278650,126529],{"stretchy":702},[31749,278652,640],{},[31662,278654,31960],{"separator":758},[31749,278656,15932],{},[31662,278658,31960],{"separator":758},[31749,278660,36723],{},[31662,278662,31960],{"separator":758},[31749,278664,18136],{},[31662,278666,126544],{"stretchy":702},[31665,278668,278669],{"encoding":31667},"\\langle1, 4, 7, 5\\rangle",[34,278671,278673],{"className":278672,"ariaHidden":758},[31672],[34,278674,278676,278679,278682,278685,278688,278691,278694,278697,278700,278703,278706,278709,278712],{"className":278675},[31676],[34,278677],{"className":278678,"style":31774},[31680],[34,278680,126529],{"className":278681},[31782],[34,278683,640],{"className":278684},[31685],[34,278686,31960],{"className":278687},[32119],[34,278689],{"className":278690,"style":32123},[31845],[34,278692,15932],{"className":278693},[31685],[34,278695,31960],{"className":278696},[32119],[34,278698],{"className":278699,"style":32123},[31845],[34,278701,36723],{"className":278702},[31685],[34,278704,31960],{"className":278705},[32119],[34,278707],{"className":278708,"style":32123},[31845],[34,278710,18136],{"className":278711},[31685],[34,278713,126544],{"className":278714},[31690],[108,278716,278717,278043,278719,278046,278721,347],{},[2627,278718,36656],{},[2627,278720,277408],{},[34,278722,278724,278754],{"className":278723},[31642],[34,278725,278727],{"className":278726},[31646],[31648,278728,278729],{"xmlns":31650},[31652,278730,278731,278751],{},[31655,278732,278733,278735,278737,278739,278741,278743,278745,278747,278749],{},[31662,278734,126529],{"stretchy":702},[31749,278736,15932],{},[31662,278738,31960],{"separator":758},[31749,278740,36723],{},[31662,278742,31960],{"separator":758},[31749,278744,18136],{},[31662,278746,31960],{"separator":758},[31749,278748,36656],{},[31662,278750,126544],{"stretchy":702},[31665,278752,278753],{"encoding":31667},"\\langle4, 7, 5, 6\\rangle",[34,278755,278757],{"className":278756,"ariaHidden":758},[31672],[34,278758,278760,278763,278766,278769,278772,278775,278778,278781,278784,278787,278790,278793,278796],{"className":278759},[31676],[34,278761],{"className":278762,"style":31774},[31680],[34,278764,126529],{"className":278765},[31782],[34,278767,15932],{"className":278768},[31685],[34,278770,31960],{"className":278771},[32119],[34,278773],{"className":278774,"style":32123},[31845],[34,278776,36723],{"className":278777},[31685],[34,278779,31960],{"className":278780},[32119],[34,278782],{"className":278783,"style":32123},[31845],[34,278785,18136],{"className":278786},[31685],[34,278788,31960],{"className":278789},[32119],[34,278791],{"className":278792,"style":32123},[31845],[34,278794,36656],{"className":278795},[31685],[34,278797,126544],{"className":278798},[31690],[108,278800,278801,278803,278804,347],{},[2627,278802,18136],{},"：命中。变为 ",[34,278805,278807,278837],{"className":278806},[31642],[34,278808,278810],{"className":278809},[31646],[31648,278811,278812],{"xmlns":31650},[31652,278813,278814,278834],{},[31655,278815,278816,278818,278820,278822,278824,278826,278828,278830,278832],{},[31662,278817,126529],{"stretchy":702},[31749,278819,15932],{},[31662,278821,31960],{"separator":758},[31749,278823,36723],{},[31662,278825,31960],{"separator":758},[31749,278827,36656],{},[31662,278829,31960],{"separator":758},[31749,278831,18136],{},[31662,278833,126544],{"stretchy":702},[31665,278835,278836],{"encoding":31667},"\\langle4, 7, 6, 5\\rangle",[34,278838,278840],{"className":278839,"ariaHidden":758},[31672],[34,278841,278843,278846,278849,278852,278855,278858,278861,278864,278867,278870,278873,278876,278879],{"className":278842},[31676],[34,278844],{"className":278845,"style":31774},[31680],[34,278847,126529],{"className":278848},[31782],[34,278850,15932],{"className":278851},[31685],[34,278853,31960],{"className":278854},[32119],[34,278856],{"className":278857,"style":32123},[31845],[34,278859,36723],{"className":278860},[31685],[34,278862,31960],{"className":278863},[32119],[34,278865],{"className":278866,"style":32123},[31845],[34,278868,36656],{"className":278869},[31685],[34,278871,31960],{"className":278872},[32119],[34,278874],{"className":278875,"style":32123},[31845],[34,278877,18136],{"className":278878},[31685],[34,278880,126544],{"className":278881},[31690],[108,278883,278884,278380,278886,278046,278888,347],{},[2627,278885,768],{},[2627,278887,277764],{},[34,278889,278891,278921],{"className":278890},[31642],[34,278892,278894],{"className":278893},[31646],[31648,278895,278896],{"xmlns":31650},[31652,278897,278898,278918],{},[31655,278899,278900,278902,278904,278906,278908,278910,278912,278914,278916],{},[31662,278901,126529],{"stretchy":702},[31749,278903,36723],{},[31662,278905,31960],{"separator":758},[31749,278907,36656],{},[31662,278909,31960],{"separator":758},[31749,278911,18136],{},[31662,278913,31960],{"separator":758},[31749,278915,768],{},[31662,278917,126544],{"stretchy":702},[31665,278919,278920],{"encoding":31667},"\\langle7, 6, 5, 2\\rangle",[34,278922,278924],{"className":278923,"ariaHidden":758},[31672],[34,278925,278927,278930,278933,278936,278939,278942,278945,278948,278951,278954,278957,278960,278963],{"className":278926},[31676],[34,278928],{"className":278929,"style":31774},[31680],[34,278931,126529],{"className":278932},[31782],[34,278934,36723],{"className":278935},[31685],[34,278937,31960],{"className":278938},[32119],[34,278940],{"className":278941,"style":32123},[31845],[34,278943,36656],{"className":278944},[31685],[34,278946,31960],{"className":278947},[32119],[34,278949],{"className":278950,"style":32123},[31845],[34,278952,18136],{"className":278953},[31685],[34,278955,31960],{"className":278956},[32119],[34,278958],{"className":278959,"style":32123},[31845],[34,278961,768],{"className":278962},[31685],[34,278964,126544],{"className":278965},[31690],[108,278967,278968,278465,278970,278046,278972,347],{},[2627,278969,640],{},[2627,278971,277850],{},[34,278973,278975,279005],{"className":278974},[31642],[34,278976,278978],{"className":278977},[31646],[31648,278979,278980],{"xmlns":31650},[31652,278981,278982,279002],{},[31655,278983,278984,278986,278988,278990,278992,278994,278996,278998,279000],{},[31662,278985,126529],{"stretchy":702},[31749,278987,36656],{},[31662,278989,31960],{"separator":758},[31749,278991,18136],{},[31662,278993,31960],{"separator":758},[31749,278995,768],{},[31662,278997,31960],{"separator":758},[31749,278999,640],{},[31662,279001,126544],{"stretchy":702},[31665,279003,279004],{"encoding":31667},"\\langle6, 5, 2, 1\\rangle",[34,279006,279008],{"className":279007,"ariaHidden":758},[31672],[34,279009,279011,279014,279017,279020,279023,279026,279029,279032,279035,279038,279041,279044,279047],{"className":279010},[31676],[34,279012],{"className":279013,"style":31774},[31680],[34,279015,126529],{"className":279016},[31782],[34,279018,36656],{"className":279019},[31685],[34,279021,31960],{"className":279022},[32119],[34,279024],{"className":279025,"style":32123},[31845],[34,279027,18136],{"className":279028},[31685],[34,279030,31960],{"className":279031},[32119],[34,279033],{"className":279034,"style":32123},[31845],[34,279036,768],{"className":279037},[31685],[34,279039,31960],{"className":279040},[32119],[34,279042],{"className":279043,"style":32123},[31845],[34,279045,640],{"className":279046},[31685],[34,279048,126544],{"className":279049},[31690],[108,279051,279052,279054],{},[2627,279053,95791],{},[105,279055,279056,279061],{},[108,279057,279058,279060],{},[2627,279059,277944],{}," 1, 2, 6, 4, 7, 3, 2, 1, 4, 7",[108,279062,279063,279065],{},[2627,279064,277950],{}," 10 次（总缺页14次）",[32490,279067,279069],{"id":279068},"_3-opt-最佳置换","3. OPT (最佳置换)",[15,279071,279072,279073,347],{},"初始装入 ",[34,279074,279076,279105],{"className":279075},[31642],[34,279077,279079],{"className":279078},[31646],[31648,279080,279081],{"xmlns":31650},[31652,279082,279083,279103],{},[31655,279084,279085,279087,279089,279091,279093,279095,279097,279099,279101],{},[31662,279086,126529],{"stretchy":702},[31749,279088,640],{},[31662,279090,31960],{"separator":758},[31749,279092,768],{},[31662,279094,31960],{"separator":758},[31749,279096,786],{},[31662,279098,31960],{"separator":758},[31749,279100,36656],{},[31662,279102,126544],{"stretchy":702},[31665,279104,277349],{"encoding":31667},[34,279106,279108],{"className":279107,"ariaHidden":758},[31672],[34,279109,279111,279114,279117,279120,279123,279126,279129,279132,279135,279138,279141,279144,279147],{"className":279110},[31676],[34,279112],{"className":279113,"style":31774},[31680],[34,279115,126529],{"className":279116},[31782],[34,279118,640],{"className":279119},[31685],[34,279121,31960],{"className":279122},[32119],[34,279124],{"className":279125,"style":32123},[31845],[34,279127,768],{"className":279128},[31685],[34,279130,31960],{"className":279131},[32119],[34,279133],{"className":279134,"style":32123},[31845],[34,279136,786],{"className":279137},[31685],[34,279139,31960],{"className":279140},[32119],[34,279142],{"className":279143,"style":32123},[31845],[34,279145,36656],{"className":279146},[31685],[34,279148,126544],{"className":279149},[31690],[105,279151,279152,279241,279330,279335,279501,279505,279670,279758,279762],{},[108,279153,279154,279156,279157,279160,279161,278046,279163,347],{},[2627,279155,15932],{},"：向后看后续序列 ",[31,279158,279159],{},"(7,3,2,1,4,7,5,6,5,2,1)","。1, 2, 3 很快被用到，6 在倒数第4个才用到。",[2627,279162,277672],{},[34,279164,279166,279196],{"className":279165},[31642],[34,279167,279169],{"className":279168},[31646],[31648,279170,279171],{"xmlns":31650},[31652,279172,279173,279193],{},[31655,279174,279175,279177,279179,279181,279183,279185,279187,279189,279191],{},[31662,279176,126529],{"stretchy":702},[31749,279178,640],{},[31662,279180,31960],{"separator":758},[31749,279182,768],{},[31662,279184,31960],{"separator":758},[31749,279186,786],{},[31662,279188,31960],{"separator":758},[31749,279190,15932],{},[31662,279192,126544],{"stretchy":702},[31665,279194,279195],{"encoding":31667},"\\langle1, 2, 3, 4\\rangle",[34,279197,279199],{"className":279198,"ariaHidden":758},[31672],[34,279200,279202,279205,279208,279211,279214,279217,279220,279223,279226,279229,279232,279235,279238],{"className":279201},[31676],[34,279203],{"className":279204,"style":31774},[31680],[34,279206,126529],{"className":279207},[31782],[34,279209,640],{"className":279210},[31685],[34,279212,31960],{"className":279213},[32119],[34,279215],{"className":279216,"style":32123},[31845],[34,279218,768],{"className":279219},[31685],[34,279221,31960],{"className":279222},[32119],[34,279224],{"className":279225,"style":32123},[31845],[34,279227,786],{"className":279228},[31685],[34,279230,31960],{"className":279231},[32119],[34,279233],{"className":279234,"style":32123},[31845],[34,279236,15932],{"className":279237},[31685],[34,279239,126544],{"className":279240},[31690],[108,279242,279243,279245,279246,279249,279250,278046,279252,347],{},[2627,279244,36723],{},"：向后看 ",[31,279247,279248],{},"(3,2,1,4,7,...)","。1, 2, 3 都比 4 先用到。",[2627,279251,277764],{},[34,279253,279255,279285],{"className":279254},[31642],[34,279256,279258],{"className":279257},[31646],[31648,279259,279260],{"xmlns":31650},[31652,279261,279262,279282],{},[31655,279263,279264,279266,279268,279270,279272,279274,279276,279278,279280],{},[31662,279265,126529],{"stretchy":702},[31749,279267,640],{},[31662,279269,31960],{"separator":758},[31749,279271,768],{},[31662,279273,31960],{"separator":758},[31749,279275,786],{},[31662,279277,31960],{"separator":758},[31749,279279,36723],{},[31662,279281,126544],{"stretchy":702},[31665,279283,279284],{"encoding":31667},"\\langle1, 2, 3, 7\\rangle",[34,279286,279288],{"className":279287,"ariaHidden":758},[31672],[34,279289,279291,279294,279297,279300,279303,279306,279309,279312,279315,279318,279321,279324,279327],{"className":279290},[31676],[34,279292],{"className":279293,"style":31774},[31680],[34,279295,126529],{"className":279296},[31782],[34,279298,640],{"className":279299},[31685],[34,279301,31960],{"className":279302},[32119],[34,279304],{"className":279305,"style":32123},[31845],[34,279307,768],{"className":279308},[31685],[34,279310,31960],{"className":279311},[32119],[34,279313],{"className":279314,"style":32123},[31845],[34,279316,786],{"className":279317},[31685],[34,279319,31960],{"className":279320},[32119],[34,279322],{"className":279323,"style":32123},[31845],[34,279325,36723],{"className":279326},[31685],[34,279328,126544],{"className":279329},[31690],[108,279331,279332,277756],{},[2627,279333,279334],{},"3, 2, 1",[108,279336,279337,279245,279339,279342,279343,279420,279421,278046,279423,347],{},[2627,279338,15932],{},[31,279340,279341],{},"(7,5,6,5,2,1)","。当前内存 ",[34,279344,279346,279375],{"className":279345},[31642],[34,279347,279349],{"className":279348},[31646],[31648,279350,279351],{"xmlns":31650},[31652,279352,279353,279373],{},[31655,279354,279355,279357,279359,279361,279363,279365,279367,279369,279371],{},[31662,279356,126529],{"stretchy":702},[31749,279358,640],{},[31662,279360,31960],{"separator":758},[31749,279362,768],{},[31662,279364,31960],{"separator":758},[31749,279366,786],{},[31662,279368,31960],{"separator":758},[31749,279370,36723],{},[31662,279372,126544],{"stretchy":702},[31665,279374,279284],{"encoding":31667},[34,279376,279378],{"className":279377,"ariaHidden":758},[31672],[34,279379,279381,279384,279387,279390,279393,279396,279399,279402,279405,279408,279411,279414,279417],{"className":279380},[31676],[34,279382],{"className":279383,"style":31774},[31680],[34,279385,126529],{"className":279386},[31782],[34,279388,640],{"className":279389},[31685],[34,279391,31960],{"className":279392},[32119],[34,279394],{"className":279395,"style":32123},[31845],[34,279397,768],{"className":279398},[31685],[34,279400,31960],{"className":279401},[32119],[34,279403],{"className":279404,"style":32123},[31845],[34,279406,786],{"className":279407},[31685],[34,279409,31960],{"className":279410},[32119],[34,279412],{"className":279413,"style":32123},[31845],[34,279415,36723],{"className":279416},[31685],[34,279418,126544],{"className":279419},[31690]," 中，3 以后再也用不到了。",[2627,279422,277586],{},[34,279424,279426,279456],{"className":279425},[31642],[34,279427,279429],{"className":279428},[31646],[31648,279430,279431],{"xmlns":31650},[31652,279432,279433,279453],{},[31655,279434,279435,279437,279439,279441,279443,279445,279447,279449,279451],{},[31662,279436,126529],{"stretchy":702},[31749,279438,640],{},[31662,279440,31960],{"separator":758},[31749,279442,768],{},[31662,279444,31960],{"separator":758},[31749,279446,36723],{},[31662,279448,31960],{"separator":758},[31749,279450,15932],{},[31662,279452,126544],{"stretchy":702},[31665,279454,279455],{"encoding":31667},"\\langle1, 2, 7, 4\\rangle",[34,279457,279459],{"className":279458,"ariaHidden":758},[31672],[34,279460,279462,279465,279468,279471,279474,279477,279480,279483,279486,279489,279492,279495,279498],{"className":279461},[31676],[34,279463],{"className":279464,"style":31774},[31680],[34,279466,126529],{"className":279467},[31782],[34,279469,640],{"className":279470},[31685],[34,279472,31960],{"className":279473},[32119],[34,279475],{"className":279476,"style":32123},[31845],[34,279478,768],{"className":279479},[31685],[34,279481,31960],{"className":279482},[32119],[34,279484],{"className":279485,"style":32123},[31845],[34,279487,36723],{"className":279488},[31685],[34,279490,31960],{"className":279491},[32119],[34,279493],{"className":279494,"style":32123},[31845],[34,279496,15932],{"className":279497},[31685],[34,279499,126544],{"className":279500},[31690],[108,279502,279503,277578],{},[2627,279504,36723],{},[108,279506,279507,279245,279509,279342,279512,279589,279590,278046,279592,347],{},[2627,279508,18136],{},[31,279510,279511],{},"(6,5,2,1)",[34,279513,279515,279544],{"className":279514},[31642],[34,279516,279518],{"className":279517},[31646],[31648,279519,279520],{"xmlns":31650},[31652,279521,279522,279542],{},[31655,279523,279524,279526,279528,279530,279532,279534,279536,279538,279540],{},[31662,279525,126529],{"stretchy":702},[31749,279527,640],{},[31662,279529,31960],{"separator":758},[31749,279531,768],{},[31662,279533,31960],{"separator":758},[31749,279535,36723],{},[31662,279537,31960],{"separator":758},[31749,279539,15932],{},[31662,279541,126544],{"stretchy":702},[31665,279543,279455],{"encoding":31667},[34,279545,279547],{"className":279546,"ariaHidden":758},[31672],[34,279548,279550,279553,279556,279559,279562,279565,279568,279571,279574,279577,279580,279583,279586],{"className":279549},[31676],[34,279551],{"className":279552,"style":31774},[31680],[34,279554,126529],{"className":279555},[31782],[34,279557,640],{"className":279558},[31685],[34,279560,31960],{"className":279561},[32119],[34,279563],{"className":279564,"style":32123},[31845],[34,279566,768],{"className":279567},[31685],[34,279569,31960],{"className":279570},[32119],[34,279572],{"className":279573,"style":32123},[31845],[34,279575,36723],{"className":279576},[31685],[34,279578,31960],{"className":279579},[32119],[34,279581],{"className":279582,"style":32123},[31845],[34,279584,15932],{"className":279585},[31685],[34,279587,126544],{"className":279588},[31690]," 中，4 再也用不到了，淘汰 4。",[2627,279591,277764],{},[34,279593,279595,279625],{"className":279594},[31642],[34,279596,279598],{"className":279597},[31646],[31648,279599,279600],{"xmlns":31650},[31652,279601,279602,279622],{},[31655,279603,279604,279606,279608,279610,279612,279614,279616,279618,279620],{},[31662,279605,126529],{"stretchy":702},[31749,279607,640],{},[31662,279609,31960],{"separator":758},[31749,279611,768],{},[31662,279613,31960],{"separator":758},[31749,279615,36723],{},[31662,279617,31960],{"separator":758},[31749,279619,18136],{},[31662,279621,126544],{"stretchy":702},[31665,279623,279624],{"encoding":31667},"\\langle1, 2, 7, 5\\rangle",[34,279626,279628],{"className":279627,"ariaHidden":758},[31672],[34,279629,279631,279634,279637,279640,279643,279646,279649,279652,279655,279658,279661,279664,279667],{"className":279630},[31676],[34,279632],{"className":279633,"style":31774},[31680],[34,279635,126529],{"className":279636},[31782],[34,279638,640],{"className":279639},[31685],[34,279641,31960],{"className":279642},[32119],[34,279644],{"className":279645,"style":32123},[31845],[34,279647,768],{"className":279648},[31685],[34,279650,31960],{"className":279651},[32119],[34,279653],{"className":279654,"style":32123},[31845],[34,279656,36723],{"className":279657},[31685],[34,279659,31960],{"className":279660},[32119],[34,279662],{"className":279663,"style":32123},[31845],[34,279665,18136],{"className":279666},[31685],[34,279668,126544],{"className":279669},[31690],[108,279671,279672,279245,279674,279677,279678,278046,279680,347],{},[2627,279673,36656],{},[31,279675,279676],{},"(5,2,1)","。7 再也用不到了。",[2627,279679,277850],{},[34,279681,279683,279713],{"className":279682},[31642],[34,279684,279686],{"className":279685},[31646],[31648,279687,279688],{"xmlns":31650},[31652,279689,279690,279710],{},[31655,279691,279692,279694,279696,279698,279700,279702,279704,279706,279708],{},[31662,279693,126529],{"stretchy":702},[31749,279695,640],{},[31662,279697,31960],{"separator":758},[31749,279699,768],{},[31662,279701,31960],{"separator":758},[31749,279703,18136],{},[31662,279705,31960],{"separator":758},[31749,279707,36656],{},[31662,279709,126544],{"stretchy":702},[31665,279711,279712],{"encoding":31667},"\\langle1, 2, 5, 6\\rangle",[34,279714,279716],{"className":279715,"ariaHidden":758},[31672],[34,279717,279719,279722,279725,279728,279731,279734,279737,279740,279743,279746,279749,279752,279755],{"className":279718},[31676],[34,279720],{"className":279721,"style":31774},[31680],[34,279723,126529],{"className":279724},[31782],[34,279726,640],{"className":279727},[31685],[34,279729,31960],{"className":279730},[32119],[34,279732],{"className":279733,"style":32123},[31845],[34,279735,768],{"className":279736},[31685],[34,279738,31960],{"className":279739},[32119],[34,279741],{"className":279742,"style":32123},[31845],[34,279744,18136],{"className":279745},[31685],[34,279747,31960],{"className":279748},[32119],[34,279750],{"className":279751,"style":32123},[31845],[34,279753,36656],{"className":279754},[31685],[34,279756,126544],{"className":279757},[31690],[108,279759,279760,277756],{},[2627,279761,277933],{},[108,279763,279764,279766],{},[2627,279765,95791],{},[105,279767,279768,279773],{},[108,279769,279770,279772],{},[2627,279771,277944],{}," 6, 4, 3, 4, 7",[108,279774,279775,279777],{},[2627,279776,277950],{}," 5 次（总缺页9次）",[253,279779,217337],{"id":217336},[15,279781,279782],{},"在 LRU 算法中，一旦某个页面\"命中（Hit）\"，一定要记得将其\"新鲜度\"提升到最高。很多同学在发生命中时忘记更新该页面的时间戳，导致后续淘汰了错误的页面。\n:::",[19,279784,279786],{"id":279785},"_51-旋转型磁盘-io-优化分布计算","51. 旋转型磁盘 I\u002FO 优化分布计算",[15,279788,279789],{},"旋转型磁盘上的信息优化分布能减少若干 I\u002FO 服务的总时间。假如有 13 个记录存放在磁盘的某一磁道上，每个磁道划分成 13 块，每块存放一个记录，如图所示。磁盘旋转速度为 30ms 转 1 周，处理程序每读一个记录后花 5ms 进行处理。",[13837,279791,279792,279825],{},[13840,279793,279794],{},[13843,279795,279796,279799,279801,279803,279805,279807,279809,279811,279813,279815,279817,279819,279821,279823],{},[13846,279797,279798],{"align":82355},"块号",[13846,279800,640],{"align":82355},[13846,279802,768],{"align":82355},[13846,279804,786],{"align":82355},[13846,279806,15932],{"align":82355},[13846,279808,18136],{"align":82355},[13846,279810,36656],{"align":82355},[13846,279812,36723],{"align":82355},[13846,279814,60525],{"align":82355},[13846,279816,36732],{"align":82355},[13846,279818,3244],{"align":82355},[13846,279820,39620],{"align":82355},[13846,279822,71869],{"align":82355},[13846,279824,205234],{"align":82355},[13853,279826,279827],{},[13843,279828,279829,279831,279833,279835,279837,279840,279843,279846,279849,279852,279855,279858,279861,279864],{},[13858,279830,171689],{"align":82355},[13858,279832,93353],{"align":82355},[13858,279834,93359],{"align":82355},[13858,279836,263516],{"align":82355},[13858,279838,279839],{"align":82355},"R4",[13858,279841,279842],{"align":82355},"R5",[13858,279844,279845],{"align":82355},"R6",[13858,279847,279848],{"align":82355},"R7",[13858,279850,279851],{"align":82355},"R8",[13858,279853,279854],{"align":82355},"R9",[13858,279856,279857],{"align":82355},"R10",[13858,279859,279860],{"align":82355},"R11",[13858,279862,279863],{"align":82355},"R12",[13858,279865,279866],{"align":82355},"R13",[105,279868,279869,279872],{},[108,279870,279871],{},"（1）处理完 13 个记录的总时间是多少？",[108,279873,279874],{},"（2）为缩短处理时间应如何排列这些记录？计算重新排列后的总时间。",[15,279876,264303],{},[253,279878,223920],{"id":279879},"概念核对-concept-check-2",[105,279881,279882,280036,280129],{},[108,279883,279884,279977,279978,280035],{},[2627,279885,279886,279887,245558],{},"单块读取时间 (",[34,279888,279890,279916],{"className":279889},[31642],[34,279891,279893],{"className":279892},[31646],[31648,279894,279895],{"xmlns":31650},[31652,279896,279897,279913],{},[31655,279898,279899],{},[31743,279900,279901,279903],{},[31658,279902,33249],{},[31655,279904,279905,279907,279909,279911],{},[31658,279906,34395],{},[31658,279908,33244],{},[31658,279910,111],{},[31658,279912,46900],{},[31665,279914,279915],{"encoding":31667},"t_{read}",[34,279917,279919],{"className":279918,"ariaHidden":758},[31672],[34,279920,279922,279925],{"className":279921},[31676],[34,279923],{"className":279924,"style":83872},[31680],[34,279926,279928,279931],{"className":279927},[31685],[34,279929,33249],{"className":279930},[31685,31686],[34,279932,279934],{"className":279933},[31792],[34,279935,279937,279969],{"className":279936},[31796,31797],[34,279938,279940,279966],{"className":279939},[31801],[34,279941,279943],{"className":279942,"style":33833},[31805],[34,279944,279945,279948],{"style":31809},[34,279946],{"className":279947,"style":31814},[31813],[34,279949,279951],{"className":279950},[31818,31819,31820,31821],[34,279952,279954,279957,279960,279963],{"className":279953},[31685,31821],[34,279955,34395],{"className":279956,"style":34452},[31685,31686,31821],[34,279958,33244],{"className":279959},[31685,31686,31821],[34,279961,111],{"className":279962},[31685,31686,31821],[34,279964,46900],{"className":279965},[31685,31686,31821],[34,279967,31829],{"className":279968},[31828],[34,279970,279972],{"className":279971},[31801],[34,279973,279975],{"className":279974,"style":31836},[31805],[34,279976],{}," 磁盘转 1 周 30ms，共 13 块，读取 1 块的时间为 ",[34,279979,279981,280004],{"className":279980},[31642],[34,279982,279984],{"className":279983},[31646],[31648,279985,279986],{"xmlns":31650},[31652,279987,279988,280001],{},[31655,279989,279990,279992,279994,279996,279998],{},[31749,279991,9314],{},[31658,279993,38080],{"mathvariant":38079},[31749,279995,205234],{},[31662,279997,168560],{},[31749,279999,280000],{},"2.31",[31665,280002,280003],{"encoding":31667},"30 \u002F 13 \\approx 2.31",[34,280005,280007,280026],{"className":280006,"ariaHidden":758},[31672],[34,280008,280010,280013,280017,280020,280023],{"className":280009},[31676],[34,280011],{"className":280012,"style":31774},[31680],[34,280014,280016],{"className":280015},[31685],"30\u002F13",[34,280018],{"className":280019,"style":31846},[31845],[34,280021,168560],{"className":280022},[31850],[34,280024],{"className":280025,"style":31846},[31845],[34,280027,280029,280032],{"className":280028},[31676],[34,280030],{"className":280031,"style":32617},[31680],[34,280033,280000],{"className":280034},[31685]," ms。",[108,280037,280038,280128],{},[2627,280039,280040,280041,245558],{},"处理时间 (",[34,280042,280044,280070],{"className":280043},[31642],[34,280045,280047],{"className":280046},[31646],[31648,280048,280049],{"xmlns":31650},[31652,280050,280051,280067],{},[31655,280052,280053],{},[31743,280054,280055,280057],{},[31658,280056,33249],{},[31655,280058,280059,280061,280063,280065],{},[31658,280060,15],{},[31658,280062,34395],{},[31658,280064,43383],{},[31658,280066,31965],{},[31665,280068,280069],{"encoding":31667},"t_{proc}",[34,280071,280073],{"className":280072,"ariaHidden":758},[31672],[34,280074,280076,280079],{"className":280075},[31676],[34,280077],{"className":280078,"style":85752},[31680],[34,280080,280082,280085],{"className":280081},[31685],[34,280083,33249],{"className":280084},[31685,31686],[34,280086,280088],{"className":280087},[31792],[34,280089,280091,280120],{"className":280090},[31796,31797],[34,280092,280094,280117],{"className":280093},[31801],[34,280095,280097],{"className":280096,"style":32362},[31805],[34,280098,280099,280102],{"style":31809},[34,280100],{"className":280101,"style":31814},[31813],[34,280103,280105],{"className":280104},[31818,31819,31820,31821],[34,280106,280108,280111,280114],{"className":280107},[31685,31821],[34,280109,15],{"className":280110},[31685,31686,31821],[34,280112,34395],{"className":280113,"style":34452},[31685,31686,31821],[34,280115,176862],{"className":280116},[31685,31686,31821],[34,280118,31829],{"className":280119},[31828],[34,280121,280123],{"className":280122},[31801],[34,280124,280126],{"className":280125,"style":32163},[31805],[34,280127],{}," 每读完一个记录后，CPU 需花费 5ms 处理。",[108,280130,280131,280132,280135],{},"磁盘在 CPU 处理记录时",[2627,280133,280134],{},"不会停止旋转","。如果在处理完毕后，磁头已经错过了下一个逻辑记录的开头，就必须等待磁盘再转一圈才能读取。",[253,280137,277313],{"id":280138},"原理解析与计算过程-1",[32490,280140,280142],{"id":280141},"_1处理完-13-个记录的总时间未优化","（1）处理完 13 个记录的总时间（未优化）",[2230,280144,280145,280638,280945],{},[108,280146,280147,280150,280151,280220,280221,280253,280254,280323,280324,280352,280353,280381,280382,280497,280498,280567,280568,280637],{},[2627,280148,280149],{},"分析单次循环：","\n读取 ",[34,280152,280154,280171],{"className":280153},[31642],[34,280155,280157],{"className":280156},[31646],[31648,280158,280159],{"xmlns":31650},[31652,280160,280161,280169],{},[31655,280162,280163],{},[31743,280164,280165,280167],{},[31658,280166,34386],{},[31749,280168,640],{},[31665,280170,107510],{"encoding":31667},[34,280172,280174],{"className":280173,"ariaHidden":758},[31672],[34,280175,280177,280180],{"className":280176},[31676],[34,280178],{"className":280179,"style":70915},[31680],[34,280181,280183,280186],{"className":280182},[31685],[34,280184,34386],{"className":280185,"style":34427},[31685,31686],[34,280187,280189],{"className":280188},[31792],[34,280190,280192,280212],{"className":280191},[31796,31797],[34,280193,280195,280209],{"className":280194},[31801],[34,280196,280198],{"className":280197,"style":31806},[31805],[34,280199,280200,280203],{"style":104694},[34,280201],{"className":280202,"style":31814},[31813],[34,280204,280206],{"className":280205},[31818,31819,31820,31821],[34,280207,640],{"className":280208},[31685,31821],[34,280210,31829],{"className":280211},[31828],[34,280213,280215],{"className":280214},[31801],[34,280216,280218],{"className":280217,"style":31836},[31805],[34,280219],{}," 需要 ",[34,280222,280224,280241],{"className":280223},[31642],[34,280225,280227],{"className":280226},[31646],[31648,280228,280229],{"xmlns":31650},[31652,280230,280231,280239],{},[31655,280232,280233,280235,280237],{},[31749,280234,9314],{},[31658,280236,38080],{"mathvariant":38079},[31749,280238,205234],{},[31665,280240,280016],{"encoding":31667},[34,280242,280244],{"className":280243,"ariaHidden":758},[31672],[34,280245,280247,280250],{"className":280246},[31676],[34,280248],{"className":280249,"style":31774},[31680],[34,280251,280016],{"className":280252},[31685]," ms。读取结束后，磁头位于第 2 块的开头。此时 CPU 开始处理 ",[34,280255,280257,280274],{"className":280256},[31642],[34,280258,280260],{"className":280259},[31646],[31648,280261,280262],{"xmlns":31650},[31652,280263,280264,280272],{},[31655,280265,280266],{},[31743,280267,280268,280270],{},[31658,280269,34386],{},[31749,280271,640],{},[31665,280273,107510],{"encoding":31667},[34,280275,280277],{"className":280276,"ariaHidden":758},[31672],[34,280278,280280,280283],{"className":280279},[31676],[34,280281],{"className":280282,"style":70915},[31680],[34,280284,280286,280289],{"className":280285},[31685],[34,280287,34386],{"className":280288,"style":34427},[31685,31686],[34,280290,280292],{"className":280291},[31792],[34,280293,280295,280315],{"className":280294},[31796,31797],[34,280296,280298,280312],{"className":280297},[31801],[34,280299,280301],{"className":280300,"style":31806},[31805],[34,280302,280303,280306],{"style":104694},[34,280304],{"className":280305,"style":31814},[31813],[34,280307,280309],{"className":280308},[31818,31819,31820,31821],[34,280310,640],{"className":280311},[31685,31821],[34,280313,31829],{"className":280314},[31828],[34,280316,280318],{"className":280317},[31801],[34,280319,280321],{"className":280320,"style":31836},[31805],[34,280322],{},"，耗时 ",[34,280325,280327,280340],{"className":280326},[31642],[34,280328,280330],{"className":280329},[31646],[31648,280331,280332],{"xmlns":31650},[31652,280333,280334,280338],{},[31655,280335,280336],{},[31749,280337,18136],{},[31665,280339,18136],{"encoding":31667},[34,280341,280343],{"className":280342,"ariaHidden":758},[31672],[34,280344,280346,280349],{"className":280345},[31676],[34,280347],{"className":280348,"style":32617},[31680],[34,280350,18136],{"className":280351},[31685]," ms。\n在这 ",[34,280354,280356,280369],{"className":280355},[31642],[34,280357,280359],{"className":280358},[31646],[31648,280360,280361],{"xmlns":31650},[31652,280362,280363,280367],{},[31655,280364,280365],{},[31749,280366,18136],{},[31665,280368,18136],{"encoding":31667},[34,280370,280372],{"className":280371,"ariaHidden":758},[31672],[34,280373,280375,280378],{"className":280374},[31676],[34,280376],{"className":280377,"style":32617},[31680],[34,280379,18136],{"className":280380},[31685]," ms 内，磁盘转过了 ",[34,280383,280385,280424],{"className":280384},[31642],[34,280386,280388],{"className":280387},[31646],[31648,280389,280390],{"xmlns":31650},[31652,280391,280392,280421],{},[31655,280393,280394,280396,280398,280400,280402,280404,280406,280408,280410,280412,280414,280416,280418],{},[31749,280395,18136],{},[31662,280397,207557],{},[31662,280399,456],{"stretchy":702},[31749,280401,9314],{},[31658,280403,38080],{"mathvariant":38079},[31749,280405,205234],{},[31662,280407,2557],{"stretchy":702},[31662,280409,202],{},[31749,280411,272712],{},[31658,280413,38080],{"mathvariant":38079},[31749,280415,9314],{},[31662,280417,202],{},[31749,280419,280420],{},"2.16",[31665,280422,280423],{"encoding":31667},"5 \\div (30\u002F13) = 65\u002F30 = 2.16",[34,280425,280427,280445,280469,280488],{"className":280426,"ariaHidden":758},[31672],[34,280428,280430,280433,280436,280439,280442],{"className":280429},[31676],[34,280431],{"className":280432,"style":35942},[31680],[34,280434,18136],{"className":280435},[31685],[34,280437],{"className":280438,"style":32604},[31845],[34,280440,207557],{"className":280441},[32024],[34,280443],{"className":280444,"style":32604},[31845],[34,280446,280448,280451,280454,280457,280460,280463,280466],{"className":280447},[31676],[34,280449],{"className":280450,"style":31774},[31680],[34,280452,456],{"className":280453},[31782],[34,280455,280016],{"className":280456},[31685],[34,280458,2557],{"className":280459},[31690],[34,280461],{"className":280462,"style":31846},[31845],[34,280464,202],{"className":280465},[31850],[34,280467],{"className":280468,"style":31846},[31845],[34,280470,280472,280475,280479,280482,280485],{"className":280471},[31676],[34,280473],{"className":280474,"style":31774},[31680],[34,280476,280478],{"className":280477},[31685],"65\u002F30",[34,280480],{"className":280481,"style":31846},[31845],[34,280483,202],{"className":280484},[31850],[34,280486],{"className":280487,"style":31846},[31845],[34,280489,280491,280494],{"className":280490},[31676],[34,280492],{"className":280493,"style":32617},[31680],[34,280495,280420],{"className":280496},[31685]," 块。\n也就是说，当 CPU 处理完 ",[34,280499,280501,280518],{"className":280500},[31642],[34,280502,280504],{"className":280503},[31646],[31648,280505,280506],{"xmlns":31650},[31652,280507,280508,280516],{},[31655,280509,280510],{},[31743,280511,280512,280514],{},[31658,280513,34386],{},[31749,280515,640],{},[31665,280517,107510],{"encoding":31667},[34,280519,280521],{"className":280520,"ariaHidden":758},[31672],[34,280522,280524,280527],{"className":280523},[31676],[34,280525],{"className":280526,"style":70915},[31680],[34,280528,280530,280533],{"className":280529},[31685],[34,280531,34386],{"className":280532,"style":34427},[31685,31686],[34,280534,280536],{"className":280535},[31792],[34,280537,280539,280559],{"className":280538},[31796,31797],[34,280540,280542,280556],{"className":280541},[31801],[34,280543,280545],{"className":280544,"style":31806},[31805],[34,280546,280547,280550],{"style":104694},[34,280548],{"className":280549,"style":31814},[31813],[34,280551,280553],{"className":280552},[31818,31819,31820,31821],[34,280554,640],{"className":280555},[31685,31821],[34,280557,31829],{"className":280558},[31828],[34,280560,280562],{"className":280561},[31801],[34,280563,280565],{"className":280564,"style":31836},[31805],[34,280566],{}," 时，磁头已经越过了第 2 块、第 3 块，正处于第 4 块的上方。\n此时 CPU 想要读取连续存放的 ",[34,280569,280571,280588],{"className":280570},[31642],[34,280572,280574],{"className":280573},[31646],[31648,280575,280576],{"xmlns":31650},[31652,280577,280578,280586],{},[31655,280579,280580],{},[31743,280581,280582,280584],{},[31658,280583,34386],{},[31749,280585,768],{},[31665,280587,107707],{"encoding":31667},[34,280589,280591],{"className":280590,"ariaHidden":758},[31672],[34,280592,280594,280597],{"className":280593},[31676],[34,280595],{"className":280596,"style":70915},[31680],[34,280598,280600,280603],{"className":280599},[31685],[34,280601,34386],{"className":280602,"style":34427},[31685,31686],[34,280604,280606],{"className":280605},[31792],[34,280607,280609,280629],{"className":280608},[31796,31797],[34,280610,280612,280626],{"className":280611},[31801],[34,280613,280615],{"className":280614,"style":31806},[31805],[34,280616,280617,280620],{"style":104694},[34,280618],{"className":280619,"style":31814},[31813],[34,280621,280623],{"className":280622},[31818,31819,31820,31821],[34,280624,768],{"className":280625},[31685,31821],[34,280627,31829],{"className":280628},[31828],[34,280630,280632],{"className":280631},[31801],[34,280633,280635],{"className":280634,"style":31836},[31805],[34,280636],{},"（位于第 2 块），必须等待磁盘继续旋转，直到第 2 块的开头再次来到磁头下方。",[108,280639,280640,280731,280732,280801,280802,280871,280872,280035],{},[2627,280641,280642,280643,245558],{},"计算等待时间 (",[34,280644,280646,280672],{"className":280645},[31642],[34,280647,280649],{"className":280648},[31646],[31648,280650,280651],{"xmlns":31650},[31652,280652,280653,280669],{},[31655,280654,280655],{},[31743,280656,280657,280659],{},[31658,280658,33249],{},[31655,280660,280661,280663,280665,280667],{},[31658,280662,47077],{},[31658,280664,111],{},[31658,280666,31955],{},[31658,280668,33249],{},[31665,280670,280671],{"encoding":31667},"t_{wait}",[34,280673,280675],{"className":280674,"ariaHidden":758},[31672],[34,280676,280678,280681],{"className":280677},[31676],[34,280679],{"className":280680,"style":83872},[31680],[34,280682,280684,280687],{"className":280683},[31685],[34,280685,33249],{"className":280686},[31685,31686],[34,280688,280690],{"className":280689},[31792],[34,280691,280693,280723],{"className":280692},[31796,31797],[34,280694,280696,280720],{"className":280695},[31801],[34,280697,280699],{"className":280698,"style":32090},[31805],[34,280700,280701,280704],{"style":31809},[34,280702],{"className":280703,"style":31814},[31813],[34,280705,280707],{"className":280706},[31818,31819,31820,31821],[34,280708,280710,280713,280717],{"className":280709},[31685,31821],[34,280711,47077],{"className":280712,"style":47120},[31685,31686,31821],[34,280714,280716],{"className":280715},[31685,31686,31821],"ai",[34,280718,33249],{"className":280719},[31685,31686,31821],[34,280721,31829],{"className":280722},[31828],[34,280724,280726],{"className":280725},[31801],[34,280727,280729],{"className":280728,"style":31836},[31805],[34,280730],{},"\n从读取 ",[34,280733,280735,280752],{"className":280734},[31642],[34,280736,280738],{"className":280737},[31646],[31648,280739,280740],{"xmlns":31650},[31652,280741,280742,280750],{},[31655,280743,280744],{},[31743,280745,280746,280748],{},[31658,280747,34386],{},[31749,280749,640],{},[31665,280751,107510],{"encoding":31667},[34,280753,280755],{"className":280754,"ariaHidden":758},[31672],[34,280756,280758,280761],{"className":280757},[31676],[34,280759],{"className":280760,"style":70915},[31680],[34,280762,280764,280767],{"className":280763},[31685],[34,280765,34386],{"className":280766,"style":34427},[31685,31686],[34,280768,280770],{"className":280769},[31792],[34,280771,280773,280793],{"className":280772},[31796,31797],[34,280774,280776,280790],{"className":280775},[31801],[34,280777,280779],{"className":280778,"style":31806},[31805],[34,280780,280781,280784],{"style":104694},[34,280782],{"className":280783,"style":31814},[31813],[34,280785,280787],{"className":280786},[31818,31819,31820,31821],[34,280788,640],{"className":280789},[31685,31821],[34,280791,31829],{"className":280792},[31828],[34,280794,280796],{"className":280795},[31801],[34,280797,280799],{"className":280798,"style":31836},[31805],[34,280800],{}," 结束 到 再次遇到 ",[34,280803,280805,280822],{"className":280804},[31642],[34,280806,280808],{"className":280807},[31646],[31648,280809,280810],{"xmlns":31650},[31652,280811,280812,280820],{},[31655,280813,280814],{},[31743,280815,280816,280818],{},[31658,280817,34386],{},[31749,280819,768],{},[31665,280821,107707],{"encoding":31667},[34,280823,280825],{"className":280824,"ariaHidden":758},[31672],[34,280826,280828,280831],{"className":280827},[31676],[34,280829],{"className":280830,"style":70915},[31680],[34,280832,280834,280837],{"className":280833},[31685],[34,280835,34386],{"className":280836,"style":34427},[31685,31686],[34,280838,280840],{"className":280839},[31792],[34,280841,280843,280863],{"className":280842},[31796,31797],[34,280844,280846,280860],{"className":280845},[31801],[34,280847,280849],{"className":280848,"style":31806},[31805],[34,280850,280851,280854],{"style":104694},[34,280852],{"className":280853,"style":31814},[31813],[34,280855,280857],{"className":280856},[31818,31819,31820,31821],[34,280858,768],{"className":280859},[31685,31821],[34,280861,31829],{"className":280862},[31828],[34,280864,280866],{"className":280865},[31801],[34,280867,280869],{"className":280868,"style":31836},[31805],[34,280870],{}," 开头，磁盘正好需要转动一整圈（30ms）。\n因为 CPU 处理耗去了 5ms，所以额外需要等待的时间为 ",[34,280873,280875,280897],{"className":280874},[31642],[34,280876,280878],{"className":280877},[31646],[31648,280879,280880],{"xmlns":31650},[31652,280881,280882,280894],{},[31655,280883,280884,280886,280888,280890,280892],{},[31749,280885,9314],{},[31662,280887,32226],{},[31749,280889,18136],{},[31662,280891,202],{},[31749,280893,95556],{},[31665,280895,280896],{"encoding":31667},"30 - 5 = 25",[34,280898,280900,280918,280936],{"className":280899,"ariaHidden":758},[31672],[34,280901,280903,280906,280909,280912,280915],{"className":280902},[31676],[34,280904],{"className":280905,"style":35942},[31680],[34,280907,9314],{"className":280908},[31685],[34,280910],{"className":280911,"style":32604},[31845],[34,280913,32226],{"className":280914},[32024],[34,280916],{"className":280917,"style":32604},[31845],[34,280919,280921,280924,280927,280930,280933],{"className":280920},[31676],[34,280922],{"className":280923,"style":32617},[31680],[34,280925,18136],{"className":280926},[31685],[34,280928],{"className":280929,"style":31846},[31845],[34,280931,202],{"className":280932},[31850],[34,280934],{"className":280935,"style":31846},[31845],[34,280937,280939,280942],{"className":280938},[31676],[34,280940],{"className":280941,"style":32617},[31680],[34,280943,95556],{"className":280944},[31685],[108,280946,280947,280950,280951,281120,281121,281194,281195,281341,281342,206456,281344,280035],{},[2627,280948,280949],{},"计算总时间：","\n对于前 12 个记录，每个记录的周期 = 读取 + 处理 + 等待 = ",[34,280952,280954,280989],{"className":280953},[31642],[34,280955,280957],{"className":280956},[31646],[31648,280958,280959],{"xmlns":31650},[31652,280960,280961,280986],{},[31655,280962,280963,280965,280967,280969,280971,280973,280975,280977,280979],{},[31749,280964,9314],{},[31658,280966,38080],{"mathvariant":38079},[31749,280968,205234],{},[31662,280970,223],{},[31749,280972,18136],{},[31662,280974,223],{},[31749,280976,95556],{},[31662,280978,202],{},[41453,280980,280981,280984],{},[31749,280982,280983],{},"420",[31749,280985,205234],{},[31665,280987,280988],{"encoding":31667},"30\u002F13 + 5 + 25 = \\frac{420}{13}",[34,280990,280992,281010,281028,281046],{"className":280991,"ariaHidden":758},[31672],[34,280993,280995,280998,281001,281004,281007],{"className":280994},[31676],[34,280996],{"className":280997,"style":31774},[31680],[34,280999,280016],{"className":281000},[31685],[34,281002],{"className":281003,"style":32604},[31845],[34,281005,223],{"className":281006},[32024],[34,281008],{"className":281009,"style":32604},[31845],[34,281011,281013,281016,281019,281022,281025],{"className":281012},[31676],[34,281014],{"className":281015,"style":35942},[31680],[34,281017,18136],{"className":281018},[31685],[34,281020],{"className":281021,"style":32604},[31845],[34,281023,223],{"className":281024},[32024],[34,281026],{"className":281027,"style":32604},[31845],[34,281029,281031,281034,281037,281040,281043],{"className":281030},[31676],[34,281032],{"className":281033,"style":32617},[31680],[34,281035,95556],{"className":281036},[31685],[34,281038],{"className":281039,"style":31846},[31845],[34,281041,202],{"className":281042},[31850],[34,281044],{"className":281045,"style":31846},[31845],[34,281047,281049,281052],{"className":281048},[31676],[34,281050],{"className":281051,"style":46727},[31680],[34,281053,281055,281058,281117],{"className":281054},[31685],[34,281056],{"className":281057},[31782,41488],[34,281059,281061],{"className":281060},[41453],[34,281062,281064,281109],{"className":281063},[31796,31797],[34,281065,281067,281106],{"className":281066},[31801],[34,281068,281070,281084,281092],{"className":281069,"style":46746},[31805],[34,281071,281072,281075],{"style":41504},[34,281073],{"className":281074,"style":41508},[31813],[34,281076,281078],{"className":281077},[31818,31819,31820,31821],[34,281079,281081],{"className":281080},[31685,31821],[34,281082,205234],{"className":281083},[31685,31821],[34,281085,281086,281089],{"style":41520},[34,281087],{"className":281088,"style":41508},[31813],[34,281090],{"className":281091,"style":41528},[41527],[34,281093,281094,281097],{"style":46771},[34,281095],{"className":281096,"style":41508},[31813],[34,281098,281100],{"className":281099},[31818,31819,31820,31821],[34,281101,281103],{"className":281102},[31685,31821],[34,281104,280983],{"className":281105},[31685,31821],[34,281107,31829],{"className":281108},[31828],[34,281110,281112],{"className":281111},[31801],[34,281113,281115],{"className":281114,"style":41619},[31805],[34,281116],{},[34,281118],{"className":281119},[31690,41488]," ms。\n对于最后 1 个记录（",[34,281122,281124,281142],{"className":281123},[31642],[34,281125,281127],{"className":281126},[31646],[31648,281128,281129],{"xmlns":31650},[31652,281130,281131,281139],{},[31655,281132,281133],{},[31743,281134,281135,281137],{},[31658,281136,34386],{},[31749,281138,205234],{},[31665,281140,281141],{"encoding":31667},"R_{13}",[34,281143,281145],{"className":281144,"ariaHidden":758},[31672],[34,281146,281148,281151],{"className":281147},[31676],[34,281149],{"className":281150,"style":70915},[31680],[34,281152,281154,281157],{"className":281153},[31685],[34,281155,34386],{"className":281156,"style":34427},[31685,31686],[34,281158,281160],{"className":281159},[31792],[34,281161,281163,281186],{"className":281162},[31796,31797],[34,281164,281166,281183],{"className":281165},[31801],[34,281167,281169],{"className":281168,"style":31806},[31805],[34,281170,281171,281174],{"style":104694},[34,281172],{"className":281173,"style":31814},[31813],[34,281175,281177],{"className":281176},[31818,31819,31820,31821],[34,281178,281180],{"className":281179},[31685,31821],[34,281181,205234],{"className":281182},[31685,31821],[34,281184,31829],{"className":281185},[31828],[34,281187,281189],{"className":281188},[31801],[34,281190,281192],{"className":281191,"style":31836},[31805],[34,281193],{},"），读完并处理完即结束，不需要再等待。耗时 = ",[34,281196,281198,281228],{"className":281197},[31642],[34,281199,281201],{"className":281200},[31646],[31648,281202,281203],{"xmlns":31650},[31652,281204,281205,281225],{},[31655,281206,281207,281209,281211,281213,281215,281217,281219],{},[31749,281208,9314],{},[31658,281210,38080],{"mathvariant":38079},[31749,281212,205234],{},[31662,281214,223],{},[31749,281216,18136],{},[31662,281218,202],{},[41453,281220,281221,281223],{},[31749,281222,96817],{},[31749,281224,205234],{},[31665,281226,281227],{"encoding":31667},"30\u002F13 + 5 = \\frac{95}{13}",[34,281229,281231,281249,281267],{"className":281230,"ariaHidden":758},[31672],[34,281232,281234,281237,281240,281243,281246],{"className":281233},[31676],[34,281235],{"className":281236,"style":31774},[31680],[34,281238,280016],{"className":281239},[31685],[34,281241],{"className":281242,"style":32604},[31845],[34,281244,223],{"className":281245},[32024],[34,281247],{"className":281248,"style":32604},[31845],[34,281250,281252,281255,281258,281261,281264],{"className":281251},[31676],[34,281253],{"className":281254,"style":32617},[31680],[34,281256,18136],{"className":281257},[31685],[34,281259],{"className":281260,"style":31846},[31845],[34,281262,202],{"className":281263},[31850],[34,281265],{"className":281266,"style":31846},[31845],[34,281268,281270,281273],{"className":281269},[31676],[34,281271],{"className":281272,"style":46727},[31680],[34,281274,281276,281279,281338],{"className":281275},[31685],[34,281277],{"className":281278},[31782,41488],[34,281280,281282],{"className":281281},[41453],[34,281283,281285,281330],{"className":281284},[31796,31797],[34,281286,281288,281327],{"className":281287},[31801],[34,281289,281291,281305,281313],{"className":281290,"style":46746},[31805],[34,281292,281293,281296],{"style":41504},[34,281294],{"className":281295,"style":41508},[31813],[34,281297,281299],{"className":281298},[31818,31819,31820,31821],[34,281300,281302],{"className":281301},[31685,31821],[34,281303,205234],{"className":281304},[31685,31821],[34,281306,281307,281310],{"style":41520},[34,281308],{"className":281309,"style":41508},[31813],[34,281311],{"className":281312,"style":41528},[41527],[34,281314,281315,281318],{"style":46771},[34,281316],{"className":281317,"style":41508},[31813],[34,281319,281321],{"className":281320},[31818,31819,31820,31821],[34,281322,281324],{"className":281323},[31685,31821],[34,281325,96817],{"className":281326},[31685,31821],[34,281328,31829],{"className":281329},[31828],[34,281331,281333],{"className":281332},[31801],[34,281334,281336],{"className":281335,"style":41619},[31805],[34,281337],{},[34,281339],{"className":281340},[31690,41488]," ms。\n",[2627,281343,265638],{},[34,281345,281347,281395],{"className":281346},[31642],[34,281348,281350],{"className":281349},[31646],[31648,281351,281352],{"xmlns":31650},[31652,281353,281354,281392],{},[31655,281355,281356,281358,281360,281362,281368,281370,281372,281378,281380,281387,281389],{},[31749,281357,71869],{},[31662,281359,42071],{},[31662,281361,456],{"stretchy":702},[41453,281363,281364,281366],{},[31749,281365,280983],{},[31749,281367,205234],{},[31662,281369,2557],{"stretchy":702},[31662,281371,223],{},[41453,281373,281374,281376],{},[31749,281375,96817],{},[31749,281377,205234],{},[31662,281379,202],{},[41453,281381,281382,281385],{},[31749,281383,281384],{},"5135",[31749,281386,205234],{},[31662,281388,168560],{},[31749,281390,281391],{},"395",[31665,281393,281394],{"encoding":31667},"12 \\times (\\frac{420}{13}) + \\frac{95}{13} = \\frac{5135}{13} \\approx 395",[34,281396,281398,281416,281505,281588,281671],{"className":281397,"ariaHidden":758},[31672],[34,281399,281401,281404,281407,281410,281413],{"className":281400},[31676],[34,281402],{"className":281403,"style":35942},[31680],[34,281405,71869],{"className":281406},[31685],[34,281408],{"className":281409,"style":32604},[31845],[34,281411,42071],{"className":281412},[32024],[34,281414],{"className":281415,"style":32604},[31845],[34,281417,281419,281422,281425,281493,281496,281499,281502],{"className":281418},[31676],[34,281420],{"className":281421,"style":46727},[31680],[34,281423,456],{"className":281424},[31782],[34,281426,281428,281431,281490],{"className":281427},[31685],[34,281429],{"className":281430},[31782,41488],[34,281432,281434],{"className":281433},[41453],[34,281435,281437,281482],{"className":281436},[31796,31797],[34,281438,281440,281479],{"className":281439},[31801],[34,281441,281443,281457,281465],{"className":281442,"style":46746},[31805],[34,281444,281445,281448],{"style":41504},[34,281446],{"className":281447,"style":41508},[31813],[34,281449,281451],{"className":281450},[31818,31819,31820,31821],[34,281452,281454],{"className":281453},[31685,31821],[34,281455,205234],{"className":281456},[31685,31821],[34,281458,281459,281462],{"style":41520},[34,281460],{"className":281461,"style":41508},[31813],[34,281463],{"className":281464,"style":41528},[41527],[34,281466,281467,281470],{"style":46771},[34,281468],{"className":281469,"style":41508},[31813],[34,281471,281473],{"className":281472},[31818,31819,31820,31821],[34,281474,281476],{"className":281475},[31685,31821],[34,281477,280983],{"className":281478},[31685,31821],[34,281480,31829],{"className":281481},[31828],[34,281483,281485],{"className":281484},[31801],[34,281486,281488],{"className":281487,"style":41619},[31805],[34,281489],{},[34,281491],{"className":281492},[31690,41488],[34,281494,2557],{"className":281495},[31690],[34,281497],{"className":281498,"style":32604},[31845],[34,281500,223],{"className":281501},[32024],[34,281503],{"className":281504,"style":32604},[31845],[34,281506,281508,281511,281579,281582,281585],{"className":281507},[31676],[34,281509],{"className":281510,"style":46727},[31680],[34,281512,281514,281517,281576],{"className":281513},[31685],[34,281515],{"className":281516},[31782,41488],[34,281518,281520],{"className":281519},[41453],[34,281521,281523,281568],{"className":281522},[31796,31797],[34,281524,281526,281565],{"className":281525},[31801],[34,281527,281529,281543,281551],{"className":281528,"style":46746},[31805],[34,281530,281531,281534],{"style":41504},[34,281532],{"className":281533,"style":41508},[31813],[34,281535,281537],{"className":281536},[31818,31819,31820,31821],[34,281538,281540],{"className":281539},[31685,31821],[34,281541,205234],{"className":281542},[31685,31821],[34,281544,281545,281548],{"style":41520},[34,281546],{"className":281547,"style":41508},[31813],[34,281549],{"className":281550,"style":41528},[41527],[34,281552,281553,281556],{"style":46771},[34,281554],{"className":281555,"style":41508},[31813],[34,281557,281559],{"className":281558},[31818,31819,31820,31821],[34,281560,281562],{"className":281561},[31685,31821],[34,281563,96817],{"className":281564},[31685,31821],[34,281566,31829],{"className":281567},[31828],[34,281569,281571],{"className":281570},[31801],[34,281572,281574],{"className":281573,"style":41619},[31805],[34,281575],{},[34,281577],{"className":281578},[31690,41488],[34,281580],{"className":281581,"style":31846},[31845],[34,281583,202],{"className":281584},[31850],[34,281586],{"className":281587,"style":31846},[31845],[34,281589,281591,281594,281662,281665,281668],{"className":281590},[31676],[34,281592],{"className":281593,"style":46727},[31680],[34,281595,281597,281600,281659],{"className":281596},[31685],[34,281598],{"className":281599},[31782,41488],[34,281601,281603],{"className":281602},[41453],[34,281604,281606,281651],{"className":281605},[31796,31797],[34,281607,281609,281648],{"className":281608},[31801],[34,281610,281612,281626,281634],{"className":281611,"style":46746},[31805],[34,281613,281614,281617],{"style":41504},[34,281615],{"className":281616,"style":41508},[31813],[34,281618,281620],{"className":281619},[31818,31819,31820,31821],[34,281621,281623],{"className":281622},[31685,31821],[34,281624,205234],{"className":281625},[31685,31821],[34,281627,281628,281631],{"style":41520},[34,281629],{"className":281630,"style":41508},[31813],[34,281632],{"className":281633,"style":41528},[41527],[34,281635,281636,281639],{"style":46771},[34,281637],{"className":281638,"style":41508},[31813],[34,281640,281642],{"className":281641},[31818,31819,31820,31821],[34,281643,281645],{"className":281644},[31685,31821],[34,281646,281384],{"className":281647},[31685,31821],[34,281649,31829],{"className":281650},[31828],[34,281652,281654],{"className":281653},[31801],[34,281655,281657],{"className":281656,"style":41619},[31805],[34,281658],{},[34,281660],{"className":281661},[31690,41488],[34,281663],{"className":281664,"style":31846},[31845],[34,281666,168560],{"className":281667},[31850],[34,281669],{"className":281670,"style":31846},[31845],[34,281672,281674,281677],{"className":281673},[31676],[34,281675],{"className":281676,"style":32617},[31680],[34,281678,281391],{"className":281679},[31685],[32490,281681,281683],{"id":281682},"_2重新排列记录以缩短处理时间","（2）重新排列记录以缩短处理时间",[15,281685,281686,281689,281690,281759,281760,281829,281830,281833,281834,281837],{},[2627,281687,281688],{},"优化逻辑："," 为了消除高达 25ms 的无谓等待时间，我们应该把 ",[34,281691,281693,281710],{"className":281692},[31642],[34,281694,281696],{"className":281695},[31646],[31648,281697,281698],{"xmlns":31650},[31652,281699,281700,281708],{},[31655,281701,281702],{},[31743,281703,281704,281706],{},[31658,281705,34386],{},[31749,281707,768],{},[31665,281709,107707],{"encoding":31667},[34,281711,281713],{"className":281712,"ariaHidden":758},[31672],[34,281714,281716,281719],{"className":281715},[31676],[34,281717],{"className":281718,"style":70915},[31680],[34,281720,281722,281725],{"className":281721},[31685],[34,281723,34386],{"className":281724,"style":34427},[31685,31686],[34,281726,281728],{"className":281727},[31792],[34,281729,281731,281751],{"className":281730},[31796,31797],[34,281732,281734,281748],{"className":281733},[31801],[34,281735,281737],{"className":281736,"style":31806},[31805],[34,281738,281739,281742],{"style":104694},[34,281740],{"className":281741,"style":31814},[31813],[34,281743,281745],{"className":281744},[31818,31819,31820,31821],[34,281746,768],{"className":281747},[31685,31821],[34,281749,31829],{"className":281750},[31828],[34,281752,281754],{"className":281753},[31801],[34,281755,281757],{"className":281756,"style":31836},[31805],[34,281758],{}," 放在 CPU 刚好处理完 ",[34,281761,281763,281780],{"className":281762},[31642],[34,281764,281766],{"className":281765},[31646],[31648,281767,281768],{"xmlns":31650},[31652,281769,281770,281778],{},[31655,281771,281772],{},[31743,281773,281774,281776],{},[31658,281775,34386],{},[31749,281777,640],{},[31665,281779,107510],{"encoding":31667},[34,281781,281783],{"className":281782,"ariaHidden":758},[31672],[34,281784,281786,281789],{"className":281785},[31676],[34,281787],{"className":281788,"style":70915},[31680],[34,281790,281792,281795],{"className":281791},[31685],[34,281793,34386],{"className":281794,"style":34427},[31685,31686],[34,281796,281798],{"className":281797},[31792],[34,281799,281801,281821],{"className":281800},[31796,31797],[34,281802,281804,281818],{"className":281803},[31801],[34,281805,281807],{"className":281806,"style":31806},[31805],[34,281808,281809,281812],{"style":104694},[34,281810],{"className":281811,"style":31814},[31813],[34,281813,281815],{"className":281814},[31818,31819,31820,31821],[34,281816,640],{"className":281817},[31685,31821],[34,281819,31829],{"className":281820},[31828],[34,281822,281824],{"className":281823},[31801],[34,281825,281827],{"className":281826,"style":31836},[31805],[34,281828],{}," 时磁头所到达的那个物理块上。\n根据刚才的分析，CPU 处理耗时 5ms，相当于转过了 2.16 块。\n从第 1 块读完（处于第 2 块开头）算起，转过 2.16 块后，磁头位于第 4 块内部。因此，",[2627,281831,281832],{},"下一个能从头开始完整读取的物理块是第 5 块","。\n这就意味着，物理块的",[2627,281835,281836],{},"间隔应为 4","（即 1, 1+4=5, 5+4=9...）。",[2230,281839,281840,282575],{},[108,281841,281842,281845],{},[2627,281843,281844],{},"最优排列方式（按物理块1~13列出记录）：",[34,281846,281848,281962],{"className":281847},[31642],[34,281849,281851],{"className":281850},[31646],[31648,281852,281853],{"xmlns":31650},[31652,281854,281855,281959],{},[31655,281856,281857,281863,281865,281871,281873,281879,281881,281887,281889,281895,281897,281903,281905,281911,281913,281919,281921,281927,281929,281935,281937,281943,281945,281951,281953],{},[31743,281858,281859,281861],{},[31658,281860,34386],{},[31749,281862,640],{},[31662,281864,31960],{"separator":758},[31743,281866,281867,281869],{},[31658,281868,34386],{},[31749,281870,39620],{},[31662,281872,31960],{"separator":758},[31743,281874,281875,281877],{},[31658,281876,34386],{},[31749,281878,60525],{},[31662,281880,31960],{"separator":758},[31743,281882,281883,281885],{},[31658,281884,34386],{},[31749,281886,18136],{},[31662,281888,31960],{"separator":758},[31743,281890,281891,281893],{},[31658,281892,34386],{},[31749,281894,768],{},[31662,281896,31960],{"separator":758},[31743,281898,281899,281901],{},[31658,281900,34386],{},[31749,281902,71869],{},[31662,281904,31960],{"separator":758},[31743,281906,281907,281909],{},[31658,281908,34386],{},[31749,281910,36732],{},[31662,281912,31960],{"separator":758},[31743,281914,281915,281917],{},[31658,281916,34386],{},[31749,281918,36656],{},[31662,281920,31960],{"separator":758},[31743,281922,281923,281925],{},[31658,281924,34386],{},[31749,281926,786],{},[31662,281928,31960],{"separator":758},[31743,281930,281931,281933],{},[31658,281932,34386],{},[31749,281934,205234],{},[31662,281936,31960],{"separator":758},[31743,281938,281939,281941],{},[31658,281940,34386],{},[31749,281942,3244],{},[31662,281944,31960],{"separator":758},[31743,281946,281947,281949],{},[31658,281948,34386],{},[31749,281950,36723],{},[31662,281952,31960],{"separator":758},[31743,281954,281955,281957],{},[31658,281956,34386],{},[31749,281958,15932],{},[31665,281960,281961],{"encoding":31667},"R_1, R_{11}, R_8, R_5, R_2, R_{12}, R_9, R_6, R_3, R_{13}, R_{10}, R_7, R_4",[34,281963,281965],{"className":281964,"ariaHidden":758},[31672],[34,281966,281968,281971,282011,282014,282017,282060,282063,282066,282106,282109,282112,282152,282155,282158,282198,282201,282204,282247,282250,282253,282293,282296,282299,282339,282342,282345,282385,282388,282391,282434,282437,282440,282483,282486,282489,282529,282532,282535],{"className":281967},[31676],[34,281969],{"className":281970,"style":70739},[31680],[34,281972,281974,281977],{"className":281973},[31685],[34,281975,34386],{"className":281976,"style":34427},[31685,31686],[34,281978,281980],{"className":281979},[31792],[34,281981,281983,282003],{"className":281982},[31796,31797],[34,281984,281986,282000],{"className":281985},[31801],[34,281987,281989],{"className":281988,"style":31806},[31805],[34,281990,281991,281994],{"style":104694},[34,281992],{"className":281993,"style":31814},[31813],[34,281995,281997],{"className":281996},[31818,31819,31820,31821],[34,281998,640],{"className":281999},[31685,31821],[34,282001,31829],{"className":282002},[31828],[34,282004,282006],{"className":282005},[31801],[34,282007,282009],{"className":282008,"style":31836},[31805],[34,282010],{},[34,282012,31960],{"className":282013},[32119],[34,282015],{"className":282016,"style":32123},[31845],[34,282018,282020,282023],{"className":282019},[31685],[34,282021,34386],{"className":282022,"style":34427},[31685,31686],[34,282024,282026],{"className":282025},[31792],[34,282027,282029,282052],{"className":282028},[31796,31797],[34,282030,282032,282049],{"className":282031},[31801],[34,282033,282035],{"className":282034,"style":31806},[31805],[34,282036,282037,282040],{"style":104694},[34,282038],{"className":282039,"style":31814},[31813],[34,282041,282043],{"className":282042},[31818,31819,31820,31821],[34,282044,282046],{"className":282045},[31685,31821],[34,282047,39620],{"className":282048},[31685,31821],[34,282050,31829],{"className":282051},[31828],[34,282053,282055],{"className":282054},[31801],[34,282056,282058],{"className":282057,"style":31836},[31805],[34,282059],{},[34,282061,31960],{"className":282062},[32119],[34,282064],{"className":282065,"style":32123},[31845],[34,282067,282069,282072],{"className":282068},[31685],[34,282070,34386],{"className":282071,"style":34427},[31685,31686],[34,282073,282075],{"className":282074},[31792],[34,282076,282078,282098],{"className":282077},[31796,31797],[34,282079,282081,282095],{"className":282080},[31801],[34,282082,282084],{"className":282083,"style":31806},[31805],[34,282085,282086,282089],{"style":104694},[34,282087],{"className":282088,"style":31814},[31813],[34,282090,282092],{"className":282091},[31818,31819,31820,31821],[34,282093,60525],{"className":282094},[31685,31821],[34,282096,31829],{"className":282097},[31828],[34,282099,282101],{"className":282100},[31801],[34,282102,282104],{"className":282103,"style":31836},[31805],[34,282105],{},[34,282107,31960],{"className":282108},[32119],[34,282110],{"className":282111,"style":32123},[31845],[34,282113,282115,282118],{"className":282114},[31685],[34,282116,34386],{"className":282117,"style":34427},[31685,31686],[34,282119,282121],{"className":282120},[31792],[34,282122,282124,282144],{"className":282123},[31796,31797],[34,282125,282127,282141],{"className":282126},[31801],[34,282128,282130],{"className":282129,"style":31806},[31805],[34,282131,282132,282135],{"style":104694},[34,282133],{"className":282134,"style":31814},[31813],[34,282136,282138],{"className":282137},[31818,31819,31820,31821],[34,282139,18136],{"className":282140},[31685,31821],[34,282142,31829],{"className":282143},[31828],[34,282145,282147],{"className":282146},[31801],[34,282148,282150],{"className":282149,"style":31836},[31805],[34,282151],{},[34,282153,31960],{"className":282154},[32119],[34,282156],{"className":282157,"style":32123},[31845],[34,282159,282161,282164],{"className":282160},[31685],[34,282162,34386],{"className":282163,"style":34427},[31685,31686],[34,282165,282167],{"className":282166},[31792],[34,282168,282170,282190],{"className":282169},[31796,31797],[34,282171,282173,282187],{"className":282172},[31801],[34,282174,282176],{"className":282175,"style":31806},[31805],[34,282177,282178,282181],{"style":104694},[34,282179],{"className":282180,"style":31814},[31813],[34,282182,282184],{"className":282183},[31818,31819,31820,31821],[34,282185,768],{"className":282186},[31685,31821],[34,282188,31829],{"className":282189},[31828],[34,282191,282193],{"className":282192},[31801],[34,282194,282196],{"className":282195,"style":31836},[31805],[34,282197],{},[34,282199,31960],{"className":282200},[32119],[34,282202],{"className":282203,"style":32123},[31845],[34,282205,282207,282210],{"className":282206},[31685],[34,282208,34386],{"className":282209,"style":34427},[31685,31686],[34,282211,282213],{"className":282212},[31792],[34,282214,282216,282239],{"className":282215},[31796,31797],[34,282217,282219,282236],{"className":282218},[31801],[34,282220,282222],{"className":282221,"style":31806},[31805],[34,282223,282224,282227],{"style":104694},[34,282225],{"className":282226,"style":31814},[31813],[34,282228,282230],{"className":282229},[31818,31819,31820,31821],[34,282231,282233],{"className":282232},[31685,31821],[34,282234,71869],{"className":282235},[31685,31821],[34,282237,31829],{"className":282238},[31828],[34,282240,282242],{"className":282241},[31801],[34,282243,282245],{"className":282244,"style":31836},[31805],[34,282246],{},[34,282248,31960],{"className":282249},[32119],[34,282251],{"className":282252,"style":32123},[31845],[34,282254,282256,282259],{"className":282255},[31685],[34,282257,34386],{"className":282258,"style":34427},[31685,31686],[34,282260,282262],{"className":282261},[31792],[34,282263,282265,282285],{"className":282264},[31796,31797],[34,282266,282268,282282],{"className":282267},[31801],[34,282269,282271],{"className":282270,"style":31806},[31805],[34,282272,282273,282276],{"style":104694},[34,282274],{"className":282275,"style":31814},[31813],[34,282277,282279],{"className":282278},[31818,31819,31820,31821],[34,282280,36732],{"className":282281},[31685,31821],[34,282283,31829],{"className":282284},[31828],[34,282286,282288],{"className":282287},[31801],[34,282289,282291],{"className":282290,"style":31836},[31805],[34,282292],{},[34,282294,31960],{"className":282295},[32119],[34,282297],{"className":282298,"style":32123},[31845],[34,282300,282302,282305],{"className":282301},[31685],[34,282303,34386],{"className":282304,"style":34427},[31685,31686],[34,282306,282308],{"className":282307},[31792],[34,282309,282311,282331],{"className":282310},[31796,31797],[34,282312,282314,282328],{"className":282313},[31801],[34,282315,282317],{"className":282316,"style":31806},[31805],[34,282318,282319,282322],{"style":104694},[34,282320],{"className":282321,"style":31814},[31813],[34,282323,282325],{"className":282324},[31818,31819,31820,31821],[34,282326,36656],{"className":282327},[31685,31821],[34,282329,31829],{"className":282330},[31828],[34,282332,282334],{"className":282333},[31801],[34,282335,282337],{"className":282336,"style":31836},[31805],[34,282338],{},[34,282340,31960],{"className":282341},[32119],[34,282343],{"className":282344,"style":32123},[31845],[34,282346,282348,282351],{"className":282347},[31685],[34,282349,34386],{"className":282350,"style":34427},[31685,31686],[34,282352,282354],{"className":282353},[31792],[34,282355,282357,282377],{"className":282356},[31796,31797],[34,282358,282360,282374],{"className":282359},[31801],[34,282361,282363],{"className":282362,"style":31806},[31805],[34,282364,282365,282368],{"style":104694},[34,282366],{"className":282367,"style":31814},[31813],[34,282369,282371],{"className":282370},[31818,31819,31820,31821],[34,282372,786],{"className":282373},[31685,31821],[34,282375,31829],{"className":282376},[31828],[34,282378,282380],{"className":282379},[31801],[34,282381,282383],{"className":282382,"style":31836},[31805],[34,282384],{},[34,282386,31960],{"className":282387},[32119],[34,282389],{"className":282390,"style":32123},[31845],[34,282392,282394,282397],{"className":282393},[31685],[34,282395,34386],{"className":282396,"style":34427},[31685,31686],[34,282398,282400],{"className":282399},[31792],[34,282401,282403,282426],{"className":282402},[31796,31797],[34,282404,282406,282423],{"className":282405},[31801],[34,282407,282409],{"className":282408,"style":31806},[31805],[34,282410,282411,282414],{"style":104694},[34,282412],{"className":282413,"style":31814},[31813],[34,282415,282417],{"className":282416},[31818,31819,31820,31821],[34,282418,282420],{"className":282419},[31685,31821],[34,282421,205234],{"className":282422},[31685,31821],[34,282424,31829],{"className":282425},[31828],[34,282427,282429],{"className":282428},[31801],[34,282430,282432],{"className":282431,"style":31836},[31805],[34,282433],{},[34,282435,31960],{"className":282436},[32119],[34,282438],{"className":282439,"style":32123},[31845],[34,282441,282443,282446],{"className":282442},[31685],[34,282444,34386],{"className":282445,"style":34427},[31685,31686],[34,282447,282449],{"className":282448},[31792],[34,282450,282452,282475],{"className":282451},[31796,31797],[34,282453,282455,282472],{"className":282454},[31801],[34,282456,282458],{"className":282457,"style":31806},[31805],[34,282459,282460,282463],{"style":104694},[34,282461],{"className":282462,"style":31814},[31813],[34,282464,282466],{"className":282465},[31818,31819,31820,31821],[34,282467,282469],{"className":282468},[31685,31821],[34,282470,3244],{"className":282471},[31685,31821],[34,282473,31829],{"className":282474},[31828],[34,282476,282478],{"className":282477},[31801],[34,282479,282481],{"className":282480,"style":31836},[31805],[34,282482],{},[34,282484,31960],{"className":282485},[32119],[34,282487],{"className":282488,"style":32123},[31845],[34,282490,282492,282495],{"className":282491},[31685],[34,282493,34386],{"className":282494,"style":34427},[31685,31686],[34,282496,282498],{"className":282497},[31792],[34,282499,282501,282521],{"className":282500},[31796,31797],[34,282502,282504,282518],{"className":282503},[31801],[34,282505,282507],{"className":282506,"style":31806},[31805],[34,282508,282509,282512],{"style":104694},[34,282510],{"className":282511,"style":31814},[31813],[34,282513,282515],{"className":282514},[31818,31819,31820,31821],[34,282516,36723],{"className":282517},[31685,31821],[34,282519,31829],{"className":282520},[31828],[34,282522,282524],{"className":282523},[31801],[34,282525,282527],{"className":282526,"style":31836},[31805],[34,282528],{},[34,282530,31960],{"className":282531},[32119],[34,282533],{"className":282534,"style":32123},[31845],[34,282536,282538,282541],{"className":282537},[31685],[34,282539,34386],{"className":282540,"style":34427},[31685,31686],[34,282542,282544],{"className":282543},[31792],[34,282545,282547,282567],{"className":282546},[31796,31797],[34,282548,282550,282564],{"className":282549},[31801],[34,282551,282553],{"className":282552,"style":31806},[31805],[34,282554,282555,282558],{"style":104694},[34,282556],{"className":282557,"style":31814},[31813],[34,282559,282561],{"className":282560},[31818,31819,31820,31821],[34,282562,15932],{"className":282563},[31685,31821],[34,282565,31829],{"className":282566},[31828],[34,282568,282570],{"className":282569},[31801],[34,282571,282573],{"className":282572,"style":31836},[31805],[34,282574],{},[108,282576,282577,282580,282581,282650,282651,282720,282721,282877,282878,282976,282977,283049,283050,281341,283261,206456,283264,280035],{},[2627,282578,282579],{},"优化后的总时间计算：","\n从 ",[34,282582,282584,282601],{"className":282583},[31642],[34,282585,282587],{"className":282586},[31646],[31648,282588,282589],{"xmlns":31650},[31652,282590,282591,282599],{},[31655,282592,282593],{},[31743,282594,282595,282597],{},[31658,282596,34386],{},[31749,282598,640],{},[31665,282600,107510],{"encoding":31667},[34,282602,282604],{"className":282603,"ariaHidden":758},[31672],[34,282605,282607,282610],{"className":282606},[31676],[34,282608],{"className":282609,"style":70915},[31680],[34,282611,282613,282616],{"className":282612},[31685],[34,282614,34386],{"className":282615,"style":34427},[31685,31686],[34,282617,282619],{"className":282618},[31792],[34,282620,282622,282642],{"className":282621},[31796,31797],[34,282623,282625,282639],{"className":282624},[31801],[34,282626,282628],{"className":282627,"style":31806},[31805],[34,282629,282630,282633],{"style":104694},[34,282631],{"className":282632,"style":31814},[31813],[34,282634,282636],{"className":282635},[31818,31819,31820,31821],[34,282637,640],{"className":282638},[31685,31821],[34,282640,31829],{"className":282641},[31828],[34,282643,282645],{"className":282644},[31801],[34,282646,282648],{"className":282647,"style":31836},[31805],[34,282649],{}," 的开头 到 ",[34,282652,282654,282671],{"className":282653},[31642],[34,282655,282657],{"className":282656},[31646],[31648,282658,282659],{"xmlns":31650},[31652,282660,282661,282669],{},[31655,282662,282663],{},[31743,282664,282665,282667],{},[31658,282666,34386],{},[31749,282668,768],{},[31665,282670,107707],{"encoding":31667},[34,282672,282674],{"className":282673,"ariaHidden":758},[31672],[34,282675,282677,282680],{"className":282676},[31676],[34,282678],{"className":282679,"style":70915},[31680],[34,282681,282683,282686],{"className":282682},[31685],[34,282684,34386],{"className":282685,"style":34427},[31685,31686],[34,282687,282689],{"className":282688},[31792],[34,282690,282692,282712],{"className":282691},[31796,31797],[34,282693,282695,282709],{"className":282694},[31801],[34,282696,282698],{"className":282697,"style":31806},[31805],[34,282699,282700,282703],{"style":104694},[34,282701],{"className":282702,"style":31814},[31813],[34,282704,282706],{"className":282705},[31818,31819,31820,31821],[34,282707,768],{"className":282708},[31685,31821],[34,282710,31829],{"className":282711},[31828],[34,282713,282715],{"className":282714},[31801],[34,282716,282718],{"className":282717,"style":31836},[31805],[34,282719],{}," 的开头（间隔 4 块），磁盘转动的时间是固定的：",[34,282722,282724,282758],{"className":282723},[31642],[34,282725,282727],{"className":282726},[31646],[31648,282728,282729],{"xmlns":31650},[31652,282730,282731,282755],{},[31655,282732,282733,282735,282737,282739,282741,282743,282745,282747,282749],{},[31749,282734,15932],{},[31662,282736,42071],{},[31662,282738,456],{"stretchy":702},[31749,282740,9314],{},[31658,282742,38080],{"mathvariant":38079},[31749,282744,205234],{},[31662,282746,2557],{"stretchy":702},[31662,282748,202],{},[41453,282750,282751,282753],{},[31749,282752,162217],{},[31749,282754,205234],{},[31665,282756,282757],{"encoding":31667},"4 \\times (30\u002F13) = \\frac{120}{13}",[34,282759,282761,282779,282803],{"className":282760,"ariaHidden":758},[31672],[34,282762,282764,282767,282770,282773,282776],{"className":282763},[31676],[34,282765],{"className":282766,"style":35942},[31680],[34,282768,15932],{"className":282769},[31685],[34,282771],{"className":282772,"style":32604},[31845],[34,282774,42071],{"className":282775},[32024],[34,282777],{"className":282778,"style":32604},[31845],[34,282780,282782,282785,282788,282791,282794,282797,282800],{"className":282781},[31676],[34,282783],{"className":282784,"style":31774},[31680],[34,282786,456],{"className":282787},[31782],[34,282789,280016],{"className":282790},[31685],[34,282792,2557],{"className":282793},[31690],[34,282795],{"className":282796,"style":31846},[31845],[34,282798,202],{"className":282799},[31850],[34,282801],{"className":282802,"style":31846},[31845],[34,282804,282806,282809],{"className":282805},[31676],[34,282807],{"className":282808,"style":46727},[31680],[34,282810,282812,282815,282874],{"className":282811},[31685],[34,282813],{"className":282814},[31782,41488],[34,282816,282818],{"className":282817},[41453],[34,282819,282821,282866],{"className":282820},[31796,31797],[34,282822,282824,282863],{"className":282823},[31801],[34,282825,282827,282841,282849],{"className":282826,"style":46746},[31805],[34,282828,282829,282832],{"style":41504},[34,282830],{"className":282831,"style":41508},[31813],[34,282833,282835],{"className":282834},[31818,31819,31820,31821],[34,282836,282838],{"className":282837},[31685,31821],[34,282839,205234],{"className":282840},[31685,31821],[34,282842,282843,282846],{"style":41520},[34,282844],{"className":282845,"style":41508},[31813],[34,282847],{"className":282848,"style":41528},[41527],[34,282850,282851,282854],{"style":46771},[34,282852],{"className":282853,"style":41508},[31813],[34,282855,282857],{"className":282856},[31818,31819,31820,31821],[34,282858,282860],{"className":282859},[31685,31821],[34,282861,162217],{"className":282862},[31685,31821],[34,282864,31829],{"className":282865},[31828],[34,282867,282869],{"className":282868},[31801],[34,282870,282872],{"className":282871,"style":41619},[31805],[34,282873],{},[34,282875],{"className":282876},[31690,41488]," ms。\n前 12 个记录，每次读取间隔精确为 ",[34,282879,282881,282899],{"className":282880},[31642],[34,282882,282884],{"className":282883},[31646],[31648,282885,282886],{"xmlns":31650},[31652,282887,282888,282896],{},[31655,282889,282890],{},[41453,282891,282892,282894],{},[31749,282893,162217],{},[31749,282895,205234],{},[31665,282897,282898],{"encoding":31667},"\\frac{120}{13}",[34,282900,282902],{"className":282901,"ariaHidden":758},[31672],[34,282903,282905,282908],{"className":282904},[31676],[34,282906],{"className":282907,"style":46727},[31680],[34,282909,282911,282914,282973],{"className":282910},[31685],[34,282912],{"className":282913},[31782,41488],[34,282915,282917],{"className":282916},[41453],[34,282918,282920,282965],{"className":282919},[31796,31797],[34,282921,282923,282962],{"className":282922},[31801],[34,282924,282926,282940,282948],{"className":282925,"style":46746},[31805],[34,282927,282928,282931],{"style":41504},[34,282929],{"className":282930,"style":41508},[31813],[34,282932,282934],{"className":282933},[31818,31819,31820,31821],[34,282935,282937],{"className":282936},[31685,31821],[34,282938,205234],{"className":282939},[31685,31821],[34,282941,282942,282945],{"style":41520},[34,282943],{"className":282944,"style":41508},[31813],[34,282946],{"className":282947,"style":41528},[41527],[34,282949,282950,282953],{"style":46771},[34,282951],{"className":282952,"style":41508},[31813],[34,282954,282956],{"className":282955},[31818,31819,31820,31821],[34,282957,282959],{"className":282958},[31685,31821],[34,282960,162217],{"className":282961},[31685,31821],[34,282963,31829],{"className":282964},[31828],[34,282966,282968],{"className":282967},[31801],[34,282969,282971],{"className":282970,"style":41619},[31805],[34,282972],{},[34,282974],{"className":282975},[31690,41488]," ms。\n最后一个记录 ",[34,282978,282980,282997],{"className":282979},[31642],[34,282981,282983],{"className":282982},[31646],[31648,282984,282985],{"xmlns":31650},[31652,282986,282987,282995],{},[31655,282988,282989],{},[31743,282990,282991,282993],{},[31658,282992,34386],{},[31749,282994,205234],{},[31665,282996,281141],{"encoding":31667},[34,282998,283000],{"className":282999,"ariaHidden":758},[31672],[34,283001,283003,283006],{"className":283002},[31676],[34,283004],{"className":283005,"style":70915},[31680],[34,283007,283009,283012],{"className":283008},[31685],[34,283010,34386],{"className":283011,"style":34427},[31685,31686],[34,283013,283015],{"className":283014},[31792],[34,283016,283018,283041],{"className":283017},[31796,31797],[34,283019,283021,283038],{"className":283020},[31801],[34,283022,283024],{"className":283023,"style":31806},[31805],[34,283025,283026,283029],{"style":104694},[34,283027],{"className":283028,"style":31814},[31813],[34,283030,283032],{"className":283031},[31818,31819,31820,31821],[34,283033,283035],{"className":283034},[31685,31821],[34,283036,205234],{"className":283037},[31685,31821],[34,283039,31829],{"className":283040},[31828],[34,283042,283044],{"className":283043},[31801],[34,283045,283047],{"className":283046,"style":31836},[31805],[34,283048],{}," 只需读取并处理，耗时 ",[34,283051,283053,283083],{"className":283052},[31642],[34,283054,283056],{"className":283055},[31646],[31648,283057,283058],{"xmlns":31650},[31652,283059,283060,283080],{},[31655,283061,283062,283068,283070,283072,283074],{},[41453,283063,283064,283066],{},[31749,283065,9314],{},[31749,283067,205234],{},[31662,283069,223],{},[31749,283071,18136],{},[31662,283073,202],{},[41453,283075,283076,283078],{},[31749,283077,96817],{},[31749,283079,205234],{},[31665,283081,283082],{"encoding":31667},"\\frac{30}{13} + 5 = \\frac{95}{13}",[34,283084,283086,283169,283187],{"className":283085,"ariaHidden":758},[31672],[34,283087,283089,283092,283160,283163,283166],{"className":283088},[31676],[34,283090],{"className":283091,"style":46727},[31680],[34,283093,283095,283098,283157],{"className":283094},[31685],[34,283096],{"className":283097},[31782,41488],[34,283099,283101],{"className":283100},[41453],[34,283102,283104,283149],{"className":283103},[31796,31797],[34,283105,283107,283146],{"className":283106},[31801],[34,283108,283110,283124,283132],{"className":283109,"style":46746},[31805],[34,283111,283112,283115],{"style":41504},[34,283113],{"className":283114,"style":41508},[31813],[34,283116,283118],{"className":283117},[31818,31819,31820,31821],[34,283119,283121],{"className":283120},[31685,31821],[34,283122,205234],{"className":283123},[31685,31821],[34,283125,283126,283129],{"style":41520},[34,283127],{"className":283128,"style":41508},[31813],[34,283130],{"className":283131,"style":41528},[41527],[34,283133,283134,283137],{"style":46771},[34,283135],{"className":283136,"style":41508},[31813],[34,283138,283140],{"className":283139},[31818,31819,31820,31821],[34,283141,283143],{"className":283142},[31685,31821],[34,283144,9314],{"className":283145},[31685,31821],[34,283147,31829],{"className":283148},[31828],[34,283150,283152],{"className":283151},[31801],[34,283153,283155],{"className":283154,"style":41619},[31805],[34,283156],{},[34,283158],{"className":283159},[31690,41488],[34,283161],{"className":283162,"style":32604},[31845],[34,283164,223],{"className":283165},[32024],[34,283167],{"className":283168,"style":32604},[31845],[34,283170,283172,283175,283178,283181,283184],{"className":283171},[31676],[34,283173],{"className":283174,"style":32617},[31680],[34,283176,18136],{"className":283177},[31685],[34,283179],{"className":283180,"style":31846},[31845],[34,283182,202],{"className":283183},[31850],[34,283185],{"className":283186,"style":31846},[31845],[34,283188,283190,283193],{"className":283189},[31676],[34,283191],{"className":283192,"style":46727},[31680],[34,283194,283196,283199,283258],{"className":283195},[31685],[34,283197],{"className":283198},[31782,41488],[34,283200,283202],{"className":283201},[41453],[34,283203,283205,283250],{"className":283204},[31796,31797],[34,283206,283208,283247],{"className":283207},[31801],[34,283209,283211,283225,283233],{"className":283210,"style":46746},[31805],[34,283212,283213,283216],{"style":41504},[34,283214],{"className":283215,"style":41508},[31813],[34,283217,283219],{"className":283218},[31818,31819,31820,31821],[34,283220,283222],{"className":283221},[31685,31821],[34,283223,205234],{"className":283224},[31685,31821],[34,283226,283227,283230],{"style":41520},[34,283228],{"className":283229,"style":41508},[31813],[34,283231],{"className":283232,"style":41528},[41527],[34,283234,283235,283238],{"style":46771},[34,283236],{"className":283237,"style":41508},[31813],[34,283239,283241],{"className":283240},[31818,31819,31820,31821],[34,283242,283244],{"className":283243},[31685,31821],[34,283245,96817],{"className":283246},[31685,31821],[34,283248,31829],{"className":283249},[31828],[34,283251,283253],{"className":283252},[31801],[34,283254,283256],{"className":283255,"style":41619},[31805],[34,283257],{},[34,283259],{"className":283260},[31690,41488],[2627,283262,283263],{},"优化后的总时间",[34,283265,283267,283315],{"className":283266},[31642],[34,283268,283270],{"className":283269},[31646],[31648,283271,283272],{"xmlns":31650},[31652,283273,283274,283312],{},[31655,283275,283276,283278,283280,283282,283288,283290,283292,283298,283300,283307,283309],{},[31749,283277,71869],{},[31662,283279,42071],{},[31662,283281,456],{"stretchy":702},[41453,283283,283284,283286],{},[31749,283285,162217],{},[31749,283287,205234],{},[31662,283289,2557],{"stretchy":702},[31662,283291,223],{},[41453,283293,283294,283296],{},[31749,283295,96817],{},[31749,283297,205234],{},[31662,283299,202],{},[41453,283301,283302,283305],{},[31749,283303,283304],{},"1535",[31749,283306,205234],{},[31662,283308,168560],{},[31749,283310,283311],{"mathvariant":226163},"118.08",[31665,283313,283314],{"encoding":31667},"12 \\times (\\frac{120}{13}) + \\frac{95}{13} = \\frac{1535}{13} \\approx \\mathbf{118.08}",[34,283316,283318,283336,283425,283508,283591],{"className":283317,"ariaHidden":758},[31672],[34,283319,283321,283324,283327,283330,283333],{"className":283320},[31676],[34,283322],{"className":283323,"style":35942},[31680],[34,283325,71869],{"className":283326},[31685],[34,283328],{"className":283329,"style":32604},[31845],[34,283331,42071],{"className":283332},[32024],[34,283334],{"className":283335,"style":32604},[31845],[34,283337,283339,283342,283345,283413,283416,283419,283422],{"className":283338},[31676],[34,283340],{"className":283341,"style":46727},[31680],[34,283343,456],{"className":283344},[31782],[34,283346,283348,283351,283410],{"className":283347},[31685],[34,283349],{"className":283350},[31782,41488],[34,283352,283354],{"className":283353},[41453],[34,283355,283357,283402],{"className":283356},[31796,31797],[34,283358,283360,283399],{"className":283359},[31801],[34,283361,283363,283377,283385],{"className":283362,"style":46746},[31805],[34,283364,283365,283368],{"style":41504},[34,283366],{"className":283367,"style":41508},[31813],[34,283369,283371],{"className":283370},[31818,31819,31820,31821],[34,283372,283374],{"className":283373},[31685,31821],[34,283375,205234],{"className":283376},[31685,31821],[34,283378,283379,283382],{"style":41520},[34,283380],{"className":283381,"style":41508},[31813],[34,283383],{"className":283384,"style":41528},[41527],[34,283386,283387,283390],{"style":46771},[34,283388],{"className":283389,"style":41508},[31813],[34,283391,283393],{"className":283392},[31818,31819,31820,31821],[34,283394,283396],{"className":283395},[31685,31821],[34,283397,162217],{"className":283398},[31685,31821],[34,283400,31829],{"className":283401},[31828],[34,283403,283405],{"className":283404},[31801],[34,283406,283408],{"className":283407,"style":41619},[31805],[34,283409],{},[34,283411],{"className":283412},[31690,41488],[34,283414,2557],{"className":283415},[31690],[34,283417],{"className":283418,"style":32604},[31845],[34,283420,223],{"className":283421},[32024],[34,283423],{"className":283424,"style":32604},[31845],[34,283426,283428,283431,283499,283502,283505],{"className":283427},[31676],[34,283429],{"className":283430,"style":46727},[31680],[34,283432,283434,283437,283496],{"className":283433},[31685],[34,283435],{"className":283436},[31782,41488],[34,283438,283440],{"className":283439},[41453],[34,283441,283443,283488],{"className":283442},[31796,31797],[34,283444,283446,283485],{"className":283445},[31801],[34,283447,283449,283463,283471],{"className":283448,"style":46746},[31805],[34,283450,283451,283454],{"style":41504},[34,283452],{"className":283453,"style":41508},[31813],[34,283455,283457],{"className":283456},[31818,31819,31820,31821],[34,283458,283460],{"className":283459},[31685,31821],[34,283461,205234],{"className":283462},[31685,31821],[34,283464,283465,283468],{"style":41520},[34,283466],{"className":283467,"style":41508},[31813],[34,283469],{"className":283470,"style":41528},[41527],[34,283472,283473,283476],{"style":46771},[34,283474],{"className":283475,"style":41508},[31813],[34,283477,283479],{"className":283478},[31818,31819,31820,31821],[34,283480,283482],{"className":283481},[31685,31821],[34,283483,96817],{"className":283484},[31685,31821],[34,283486,31829],{"className":283487},[31828],[34,283489,283491],{"className":283490},[31801],[34,283492,283494],{"className":283493,"style":41619},[31805],[34,283495],{},[34,283497],{"className":283498},[31690,41488],[34,283500],{"className":283501,"style":31846},[31845],[34,283503,202],{"className":283504},[31850],[34,283506],{"className":283507,"style":31846},[31845],[34,283509,283511,283514,283582,283585,283588],{"className":283510},[31676],[34,283512],{"className":283513,"style":46727},[31680],[34,283515,283517,283520,283579],{"className":283516},[31685],[34,283518],{"className":283519},[31782,41488],[34,283521,283523],{"className":283522},[41453],[34,283524,283526,283571],{"className":283525},[31796,31797],[34,283527,283529,283568],{"className":283528},[31801],[34,283530,283532,283546,283554],{"className":283531,"style":46746},[31805],[34,283533,283534,283537],{"style":41504},[34,283535],{"className":283536,"style":41508},[31813],[34,283538,283540],{"className":283539},[31818,31819,31820,31821],[34,283541,283543],{"className":283542},[31685,31821],[34,283544,205234],{"className":283545},[31685,31821],[34,283547,283548,283551],{"style":41520},[34,283549],{"className":283550,"style":41508},[31813],[34,283552],{"className":283553,"style":41528},[41527],[34,283555,283556,283559],{"style":46771},[34,283557],{"className":283558,"style":41508},[31813],[34,283560,283562],{"className":283561},[31818,31819,31820,31821],[34,283563,283565],{"className":283564},[31685,31821],[34,283566,283304],{"className":283567},[31685,31821],[34,283569,31829],{"className":283570},[31828],[34,283572,283574],{"className":283573},[31801],[34,283575,283577],{"className":283576,"style":41619},[31805],[34,283578],{},[34,283580],{"className":283581},[31690,41488],[34,283583],{"className":283584,"style":31846},[31845],[34,283586,168560],{"className":283587},[31850],[34,283589],{"className":283590,"style":31846},[31845],[34,283592,283594,283597],{"className":283593},[31676],[34,283595],{"className":283596,"style":32617},[31680],[34,283598,283600],{"className":283599},[31685],[34,283601,283311],{"className":283602},[31685,226302],[253,283604,217337],{"id":219803},[15,283606,283607,283608,283641,283642,283645,283646,283694],{},"在计算优化后的总时间时，学生容易错误地直接把\"等待时间归零\"来计算单步时间。实际上，由于处理时间 (5ms) 并没有刚好踩准物理块的边界，磁头到达第 5 块之前依然有非常短暂的等待对齐时间（本题为 ",[34,283609,283611,283629],{"className":283610},[31642],[34,283612,283614],{"className":283613},[31646],[31648,283615,283616],{"xmlns":31650},[31652,283617,283618,283626],{},[31655,283619,283620,283622,283624],{},[31749,283621,95556],{},[31658,283623,38080],{"mathvariant":38079},[31749,283625,205234],{},[31665,283627,283628],{"encoding":31667},"25\u002F13",[34,283630,283632],{"className":283631,"ariaHidden":758},[31672],[34,283633,283635,283638],{"className":283634},[31676],[34,283636],{"className":283637,"style":31774},[31680],[34,283639,283628],{"className":283640},[31685]," ms）。最稳妥的计算方式是直接看两个记录",[2627,283643,283644],{},"开头之间的物理距离","（4 块的旋转时间 ",[34,283647,283649,283669],{"className":283648},[31642],[34,283650,283652],{"className":283651},[31646],[31648,283653,283654],{"xmlns":31650},[31652,283655,283656,283666],{},[31655,283657,283658,283660,283662,283664],{},[31662,283659,202],{},[31749,283661,162217],{},[31658,283663,38080],{"mathvariant":38079},[31749,283665,205234],{},[31665,283667,283668],{"encoding":31667},"= 120\u002F13",[34,283670,283672,283684],{"className":283671,"ariaHidden":758},[31672],[34,283673,283675,283678,283681],{"className":283674},[31676],[34,283676],{"className":283677,"style":73089},[31680],[34,283679,202],{"className":283680},[31850],[34,283682],{"className":283683,"style":31846},[31845],[34,283685,283687,283690],{"className":283686},[31676],[34,283688],{"className":283689,"style":31774},[31680],[34,283691,283693],{"className":283692},[31685],"120\u002F13"," ms）。\n:::",[79,283696,81],{},{"title":29,"searchDepth":43,"depth":43,"links":283698},[283699,283700,283701,283702,283703,283704,283705,283706,283707,283708,283709,283710,283711,283712,283713,283717,283721,283725,283729,283732,283736,283739,283743,283744,283745,283746,283747,283748,283749,283750,283751,283752,283753,283754,283758,283761,283764,283767,283770,283773,283776,283779,283780,283781,283782,283783,283784,283785,283786,283787,283788,283793],{"id":264281,"depth":43,"text":264282},{"id":264288,"depth":43,"text":264289},{"id":264309,"depth":43,"text":264310},{"id":264400,"depth":43,"text":264401},{"id":264411,"depth":43,"text":264412},{"id":264436,"depth":43,"text":264437},{"id":264447,"depth":43,"text":264448},{"id":264461,"depth":43,"text":264462},{"id":264603,"depth":43,"text":264604},{"id":264683,"depth":43,"text":264684},{"id":264702,"depth":43,"text":264703},{"id":264713,"depth":43,"text":264714},{"id":264724,"depth":43,"text":264725},{"id":264756,"depth":43,"text":264757},{"id":264767,"depth":43,"text":264768,"children":283714},[283715,283716],{"id":214365,"depth":49,"text":214366},{"id":219116,"depth":49,"text":219117},{"id":266828,"depth":43,"text":266829,"children":283718},[283719,283720],{"id":231556,"depth":49,"text":219117},{"id":267207,"depth":49,"text":267207},{"id":267581,"depth":43,"text":267582,"children":283722},[283723,283724],{"id":218884,"depth":49,"text":214366},{"id":267612,"depth":49,"text":267613},{"id":268228,"depth":43,"text":268229,"children":283726},[283727,283728],{"id":268237,"depth":49,"text":219117},{"id":269259,"depth":49,"text":267207},{"id":270268,"depth":43,"text":270269,"children":283730},[283731],{"id":270277,"depth":49,"text":267207},{"id":271227,"depth":43,"text":271228,"children":283733},[283734,283735],{"id":271376,"depth":49,"text":219117},{"id":271393,"depth":49,"text":267207},{"id":271989,"depth":43,"text":271990,"children":283737},[283738],{"id":271998,"depth":49,"text":267207},{"id":272329,"depth":43,"text":272330,"children":283740},[283741,283742],{"id":272450,"depth":49,"text":219117},{"id":272646,"depth":49,"text":267207},{"id":273262,"depth":43,"text":273263},{"id":273273,"depth":43,"text":273274},{"id":273292,"depth":43,"text":273293},{"id":273308,"depth":43,"text":273309},{"id":273332,"depth":43,"text":273333},{"id":273361,"depth":43,"text":273362},{"id":273372,"depth":43,"text":273373},{"id":273383,"depth":43,"text":273384},{"id":273398,"depth":43,"text":273399},{"id":273409,"depth":43,"text":273410},{"id":273427,"depth":43,"text":273428},{"id":273528,"depth":43,"text":273529,"children":283755},[283756,283757],{"id":223919,"depth":49,"text":223920},{"id":273569,"depth":49,"text":267207},{"id":273646,"depth":43,"text":273647,"children":283759},[283760],{"id":273655,"depth":49,"text":267207},{"id":273809,"depth":43,"text":273810,"children":283762},[283763],{"id":273935,"depth":49,"text":267207},{"id":275526,"depth":43,"text":275527,"children":283765},[283766],{"id":275642,"depth":49,"text":267207},{"id":275853,"depth":43,"text":275854,"children":283768},[283769],{"id":275873,"depth":49,"text":267207},{"id":276384,"depth":43,"text":276385,"children":283771},[283772],{"id":276393,"depth":49,"text":267207},{"id":276625,"depth":43,"text":276626,"children":283774},[283775],{"id":276634,"depth":49,"text":267207},{"id":276695,"depth":43,"text":276696,"children":283777},[283778],{"id":276704,"depth":49,"text":267207},{"id":276963,"depth":43,"text":276964},{"id":276985,"depth":43,"text":276986},{"id":276996,"depth":43,"text":276997},{"id":277015,"depth":43,"text":277016},{"id":277034,"depth":43,"text":277035},{"id":277096,"depth":43,"text":277097},{"id":277213,"depth":43,"text":277214},{"id":277221,"depth":43,"text":277222},{"id":277240,"depth":43,"text":277241},{"id":277268,"depth":43,"text":277269,"children":283789},[283790,283791,283792],{"id":277277,"depth":49,"text":223920},{"id":277313,"depth":49,"text":277313},{"id":217336,"depth":49,"text":217337},{"id":279785,"depth":43,"text":279786,"children":283794},[283795,283796,283797],{"id":279879,"depth":49,"text":223920},{"id":280138,"depth":49,"text":277313},{"id":219803,"depth":49,"text":217337},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fprob",{"description":29},"source\u002Fcs\u002Fos\u002Freview\u002Fprob","2-2JoObW6kw71e0s83S0-P_SPvXLwfwwwvIDnoPLRzs",{"id":283804,"title":283805,"body":283806,"description":29,"extension":85,"meta":285722,"navigation":52,"path":285723,"seo":285724,"stem":285725,"__hash__":285726},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fshortans.md","简答题",{"type":8,"value":283807,"toc":285685},[283808,283810,283814,283819,283823,283855,283857,283861,283866,283870,283897,283899,283903,283908,283912,283956,283958,283962,283967,283971,284011,284013,284017,284022,284026,284073,284075,284079,284084,284088,284144,284146,284150,284155,284159,284202,284204,284208,284213,284217,284269,284271,284275,284286,284290,284324,284326,284330,284341,284345,284395,284397,284401,284412,284416,284474,284476,284480,284485,284505,284509,284545,284547,284551,284556,284560,284589,284591,284595,284600,284626,284630,284670,284672,284676,284691,284695,284719,284721,284725,284732,284784,284788,284809,284811,284815,284820,284824,284872,284874,284878,284883,284887,284911,284913,284917,284929,284933,284983,284985,284989,284994,284998,285045,285047,285050,285053,285055,285059,285065,285117,285119,285123,285128,285145,285147,285151,285159,285212,285214,285218,285226,285243,285245,285249,285254,285275,285277,285281,285285,285314,285316,285320,285331,285349,285351,285355,285359,285391,285393,285397,285405,285431,285438,285440,285444,285448,285472,285474,285478,285483,285509,285511,285515,285520,285537,285539,285543,285547,285571,285573,285577,285584,285603,285608,285619,285621,285625,285633],[11,283809,283805],{"id":283805},[19,283811,283813],{"id":283812},"_1为什么分页技术不能实现共享而分段技术可以","1．为什么分页技术不能实现共享，而分段技术可以？",[15,283815,283816,283818],{},[2627,283817,58173],{},"\n分页技术对程序员透明且按物理固定大小切分，极易截断逻辑上完整的代码或数据，导致难以安全隔离和共享；而分段技术按程序的逻辑结构（如函数、数组）划分，具有完整的逻辑意义，因此非常适合实现共享。",[15,283820,283821],{},[2627,283822,258022],{},[105,283824,283825,283831,283845],{},[108,283826,283827,283830],{},[2627,283828,283829],{},"Concept Check (概念核对)："," 分页（Paging）是将内存和程序划分为固定大小的“页”，以消除外部碎片；分段（Segmentation）则是根据程序的逻辑结构（如代码段、数据段、堆栈段）将程序划分为大小不一的“段”。",[108,283832,283833,283836,283837,283840,283841,283844],{},[2627,283834,283835],{},"The 'Why' (原理解析)：","\n根据操作系统的内存管理理论，",[2627,283838,283839],{},"分页是面向物理系统、对程序员透明的","。一个页面可能前半部分属于函数A的代码，后半部分属于函数B的数据。如果另一个进程想要共享函数A，操作系统如果把这整个页共享出去，就会被迫把不该共享的函数B的数据也暴露给对方，这在安全和逻辑上是不可行的。\n相反，",[2627,283842,283843],{},"分段是面向程序员可见的逻辑组织","。一个段可能是一个完整的数学库函数，或者一个共享数组。因为段拥有完整的逻辑意义和独立的保护属性，操作系统只需在两个进程的段表中，将某个段表项指向同一个物理内存基址，就能干净利落地实现代码或数据的共享。",[108,283846,283847,283850,283851,283854],{},[2627,283848,283849],{},"Common Pitfalls (易错点)："," 很多学生误以为分页“绝对不能”物理共享。实际上，分页底层是可以物理共享页面的（例如写时复制机制），但这题探讨的是",[2627,283852,283853],{},"逻辑实体的共享能力","。分段在原生设计上就是为了支持逻辑共享和保护而生的。",[32188,283856],{},[19,283858,283860],{"id":283859},"_2引入缓冲的目的","2．引入缓冲的目的。",[15,283862,283863,283865],{},[2627,283864,58173],{},"\n引入缓冲的主要目的是为了缓和CPU与I\u002FO设备之间速度极度不匹配的矛盾，从而提高CPU和I\u002FO设备的并行工作程度（利用率）。",[15,283867,283868],{},[2627,283869,258022],{},[105,283871,283872,283877,283885],{},[108,283873,283874,283876],{},[2627,283875,283829],{}," 缓冲（Buffering）是在内存中开辟的一块临时存储区，用于在数据到达最终目的地之前暂存数据。",[108,283878,283879,283881,283882,347],{},[2627,283880,283835],{},"\n计算机系统中，CPU的运算速度通常比外部I\u002FO设备（如磁盘、打印机）快好几个数量级。如果没有缓冲，CPU在发起I\u002FO请求后，要么必须“停机”苦苦等待慢速设备完成传输，要么就是设备跟不上CPU吐出数据的速度。\n引入缓冲后（例如双重缓冲 Double Buffering），设备可以将数据先慢吞吞地填入缓冲区，填满后CPU可以瞬间将其取走处理；或者CPU将数据瞬间抛入缓冲区后继续做其他事情，设备在后台慢慢读取。这极大地",[2627,283883,283884],{},"掩盖了速度差异，减少了CPU的中断次数，并实现了CPU计算与I\u002FO传输的高度重叠（并行）",[108,283886,283887,283889,283890,283893,283894,347],{},[2627,283888,283849],{}," 容易将“缓冲(Buffering)”和“缓存(Caching)”混淆。缓冲是为了",[2627,283891,283892],{},"平滑速度差异、吸收突发数据流","；缓存是为了",[2627,283895,283896],{},"保留热点数据以加速未来的重复访问",[32188,283898],{},[19,283900,283902],{"id":283901},"_3什么是文件的物理结构有哪几种组织方式","3．什么是文件的物理结构？有哪几种组织方式？",[15,283904,283905,283907],{},[2627,283906,58173],{},"\n文件的物理结构（也称物理组织形式）是指文件数据在物理存储介质（如磁盘）上的实际排列和分配方式。主要有三种组织方式：连续分配（顺序结构）、链接分配（链接结构）和索引分配（索引结构）。",[15,283909,283910],{},[2627,283911,258022],{},[105,283913,283914,283919,283943],{},[108,283915,283916,283918],{},[2627,283917,283829],{}," 物理结构决定了操作系统的文件系统如何将逻辑上的文件字节流映射到磁盘上大小固定的“物理块”中。",[108,283920,283921,283923],{},[2627,283922,283835],{},[2230,283924,283925,283931,283937],{},[108,283926,283927,283930],{},[2627,283928,283929],{},"连续分配 (Contiguous Allocation)："," 文件在磁盘上占据一整块连续的物理块。优点是读取极快（磁头只需移动一次），缺点是容易产生严重的外部碎片，且文件难以动态扩容。",[108,283932,283933,283936],{},[2627,283934,283935],{},"链接分配 (Chained\u002FLinked Allocation)："," 文件按块存储，每一块包含一个指向下一块的指针。优点是彻底消除了外部碎片，便于扩容；缺点是只能顺序访问，不支持高效的随机查找，且指针占用一定空间。",[108,283938,283939,283942],{},[2627,283940,283941],{},"索引分配 (Indexed Allocation)："," 系统为每个文件分配一个特殊的“索引块”，里面存放了该文件所有数据块的物理地址指针。这种方式既支持高效的随机访问，又允许文件动态增长，是现代操作系统（如UNIX的inode）采用的主流结构。",[108,283944,283945,283947,283948,283951,283952,283955],{},[2627,283946,283849],{}," 极易将“物理结构”与“逻辑结构”搞混。物理结构是",[2627,283949,283950],{},"磁盘视角的分配策略","（连续、链接、索引）；逻辑结构是",[2627,283953,283954],{},"用户视角的格式","（流式、记录式）。",[32188,283957],{},[19,283959,283961],{"id":283960},"_4分析操作系统特征","4．分析操作系统特征。",[15,283963,283964,283966],{},[2627,283965,58173],{},"\n现代操作系统具有四个基本特征：并发（Concurrency）、共享（Sharing）、虚拟（Virtualization）和异步性（Asynchrony）。其中并发和共享是最基础的两个特征，且互为存在条件。",[15,283968,283969],{},[2627,283970,258022],{},[105,283972,283973,283978,284006],{},[108,283974,283975,283977],{},[2627,283976,283829],{}," 操作系统特征描绘了多道程序系统在宏观运行时的根本表现。",[108,283979,283980,283982],{},[2627,283981,283835],{},[2230,283983,283984,283989,283994,284000],{},[108,283985,283986,283988],{},[2627,283987,257866],{}," 指两个或多个事件在同一时间间隔内发生。操作系统通过CPU调度，让多个程序在内存中“走走停停”交替执行，宏观上给用户“同时运行”的假象。",[108,283990,283991,283993],{},[2627,283992,249727],{}," 系统中的软硬件资源可供内存中多个并发执行的进程共同使用。分为“互斥共享”（如打印机）和“同时访问”（如可重入代码）。并发和共享互为条件，无并发则无共享的必要，无共享机制则并发会崩溃。",[108,283995,283996,283999],{},[2627,283997,283998],{},"虚拟："," 将物理实体（如单核CPU、有限内存）通过时分复用或空分复用技术，映射为多个逻辑上的对应物（如虚拟处理器、虚拟内存）。",[108,284001,284002,284005],{},[2627,284003,284004],{},"异步性："," 由于资源竞争和进程同步，并发进程的执行速度和暂停时机是不可预知的，具有走走停停的异步特征。",[108,284007,284008,284010],{},[2627,284009,283849],{}," 在理解时，常把“并发（Concurrency）”和“并行（Parallelism）”等同。并发是单核上的“时分复用”幻觉，并行是多核上的“物理同时”执行。",[32188,284012],{},[19,284014,284016],{"id":284015},"_5试述临界区访问遵循的原则","5．试述临界区访问遵循的原则。",[15,284018,284019,284021],{},[2627,284020,58173],{},"\n临界区访问必须遵循四大原则：空闲让进、忙则等待（互斥）、有限等待（防止死锁和饥饿）、让权等待。",[15,284023,284024],{},[2627,284025,258022],{},[105,284027,284028,284033,284064],{},[108,284029,284030,284032],{},[2627,284031,283829],{}," 临界区（Critical Section）是指进程代码中访问临界资源（排他性共享资源）的那段代码。",[108,284034,284035,284037,284038],{},[2627,284036,283835],{}," 为了保证多进程环境下的数据一致性与系统活跃度，必须满足：\n",[2230,284039,284040,284046,284052,284058],{},[108,284041,284042,284045],{},[2627,284043,284044],{},"空闲让进："," 如果临界区没有任何进程在执行，应允许一个请求进入的进程立即进入。",[108,284047,284048,284051],{},[2627,284049,284050],{},"忙则等待 (Mutual Exclusion)："," 如果已经有进程在临界区内，其他试图进入的进程必须阻塞等待，保证互斥。",[108,284053,284054,284057],{},[2627,284055,284056],{},"有限等待："," 任何请求进入临界区的进程，必须能在有限的时间内获得许可，绝对不能出现无限期的延迟（即防止死锁和饥饿现象）。",[108,284059,284060,284063],{},[2627,284061,284062],{},"让权等待："," （硬件实现有时无法保证，但在OS高级原语中要求）如果进程因为“忙则等待”无法进入临界区，它应当立即释放CPU（放弃处理器调度权），而不是在CPU上进行毫无意义的“忙等待（Busy Waiting）”。",[108,284065,284066,284068,284069,284072],{},[2627,284067,283849],{}," 学生常把“临界资源”和“临界区”混淆。临界区是",[2627,284070,284071],{},"代码段","，不是硬件或变量本身。访问原则是用来约束这些代码段执行逻辑的。",[32188,284074],{},[19,284076,284078],{"id":284077},"_6什么是文件的逻辑结构有哪几种组织方式","6．什么是文件的逻辑结构？有哪几种组织方式？",[15,284080,284081,284083],{},[2627,284082,58173],{},"\n文件的逻辑结构是指从用户和应用程序的角度所观察到的文件的内部组织形式。主要分为两大类：无结构文件（流式文件）和有结构文件（记录式文件），其中有结构文件又包含顺序文件、索引顺序文件、索引文件和直接（哈希）文件。",[15,284085,284086],{},[2627,284087,258022],{},[105,284089,284090,284095,284139],{},[108,284091,284092,284094],{},[2627,284093,283829],{}," 逻辑结构完全不关心文件在磁盘上如何存放，它只关心数据在业务逻辑上是如何定界和关联的。",[108,284096,284097,284099],{},[2627,284098,283835],{},[2230,284100,284101,284107],{},[108,284102,284103,284106],{},[2627,284104,284105],{},"无结构文件（流式文件）："," 最基本的结构，操作系统将其视为一连串无格式的字符或字节流（Stream of bytes）。比如UNIX中的TXT文本文件、可执行文件。",[108,284108,284109,284112,284113],{},[2627,284110,284111],{},"有结构文件（记录式文件）："," 文件由一条条相关联的数据记录（Record）组成（类似于数据库的一张表）。\n",[105,284114,284115,284121,284127,284133],{},[108,284116,284117,284120],{},[71807,284118,284119],{},"顺序文件 (Sequential File)："," 记录按某种顺序（如按到达时间或关键字）线性排列。",[108,284122,284123,284126],{},[71807,284124,284125],{},"索引顺序文件 (Indexed Sequential File)："," 顺序排列的同时，加入了一个轻量级索引表，以加速定位。",[108,284128,284129,284132],{},[71807,284130,284131],{},"索引文件 (Indexed File)："," 为文件中的记录建立完整的索引表，通过索引指针进行随机访问。",[108,284134,284135,284138],{},[71807,284136,284137],{},"直接或哈希文件 (Direct\u002FHashed File)："," 通过哈希函数直接由关键字计算出记录的逻辑地址，查询速度极快。",[108,284140,284141,284143],{},[2627,284142,283849],{}," 再次强调，此题与第3题是考点双子星，回答时切忌将逻辑结构的“索引文件”和物理结构的“索引分配”混为一谈。",[32188,284145],{},[19,284147,284149],{"id":284148},"_7试述设备管理的基本功能","7．试述设备管理的基本功能。",[15,284151,284152,284154],{},[2627,284153,58173],{},"\n设备管理的基本功能是隐藏物理设备的复杂细节，为用户提供一个统一、简单、独立于设备的接口，并高效地分配和管理系统中的I\u002FO设备。",[15,284156,284157],{},[2627,284158,258022],{},[105,284160,284161,284166,284197],{},[108,284162,284163,284165],{},[2627,284164,283829],{}," 设备管理（I\u002FO管理）是OS内核的一个核心子系统，负责介于CPU\u002F内存和外设之间的数据传输。",[108,284167,284168,284170,284171],{},[2627,284169,283835],{},"\n根据操作系统的I\u002FO管理目标（效率和通用性），其核心功能包括：\n",[2230,284172,284173,284179,284185,284191],{},[108,284174,284175,284178],{},[2627,284176,284177],{},"设备分配与回收："," 根据设备类型（独占、共享或虚拟）和调度算法，公平且安全地将设备分配给请求的进程。",[108,284180,284181,284184],{},[2627,284182,284183],{},"缓冲区管理 (Buffering)："," 管理内存中的I\u002FO缓冲区，平滑CPU与外设的速度差，缓解总线瓶颈。",[108,284186,284187,284190],{},[2627,284188,284189],{},"设备独立性 (Device Independence)："," 通过逻辑设备表（LUT），使得用户程序可以使用“逻辑设备名”请求I\u002FO，操作系统将其映射为实际的物理设备驱动。",[108,284192,284193,284196],{},[2627,284194,284195],{},"中断处理："," 捕获并处理由设备发出的I\u002FO完成或错误中断，唤醒相关的阻塞进程。",[108,284198,284199,284201],{},[2627,284200,283849],{}," 容易忽略“设备独立性”。实际上，屏蔽硬件差异是OS设备管理最具价值的功能之一。",[32188,284203],{},[19,284205,284207],{"id":284206},"_8分区存储管理技术有哪些分配策略","8．分区存储管理技术有哪些分配策略？",[15,284209,284210,284212],{},[2627,284211,58173],{},"\n动态分区存储管理主要有四种放置（分配）策略算法：首次适应算法（First-fit）、下次适应算法（Next-fit）、最佳适应算法（Best-fit）和最坏适应算法（Worst-fit）。",[15,284214,284215],{},[2627,284216,258022],{},[105,284218,284219,284224,284260],{},[108,284220,284221,284223],{},[2627,284222,283829],{}," 分区存储管理是指系统在进程调入时，从连续的空闲内存区域中切出一块合适大小的内存分配给它。",[108,284225,284226,284228],{},[2627,284227,283835],{},[2230,284229,284230,284236,284242,284251],{},[108,284231,284232,284235],{},[2627,284233,284234],{},"首次适应 (First-fit)："," 按地址递增扫描空闲链表，找到第一个足够大的空闲区即分配。速度快，保留了高地址大块，但低地址易产生碎片。",[108,284237,284238,284241],{},[2627,284239,284240],{},"下次适应 (Next-fit)："," 从上次分配的位置继续向后扫描。分配均匀，但易破坏尾部的大空闲区。",[108,284243,284244,284247,284248,284250],{},[2627,284245,284246],{},"最佳适应 (Best-fit)："," 扫描全表，找到满足要求且",[2627,284249,200479],{},"的空闲区。看似最节省，但每次切分后剩下的“细小碎片（外部碎片）”极多，极其难以再利用。",[108,284252,284253,284256,284257,284259],{},[2627,284254,284255],{},"最坏适应 (Worst-fit)："," 扫描全表，选择",[2627,284258,200489],{},"的空闲区进行分配。切分后剩下的空间依然很大，碎片不易产生，但缺乏对大作业的预留能力。",[108,284261,284262,284264,284265,284268],{},[2627,284263,283849],{}," 名字具有迷惑性。",[2627,284266,284267],{},"最佳","适应（Best-fit）在实际运行中产生的外部碎片问题往往是“最糟”的；它反而不如简单的首次适应算法效率高。",[32188,284270],{},[19,284272,284274],{"id":284273},"_9什么是文件和文件系统","9．什么是文件和文件系统？",[15,284276,284277,284279,284281,284282,284285],{},[2627,284278,58173],{},[2627,284280,90872],{},"是操作系统提供的一种逻辑抽象，是由用户命名的、具有一定逻辑结构的相似记录或数据的集合；",[2627,284283,284284],{},"文件系统","则是操作系统中负责管理、组织和存取文件的所有软件模块、数据结构及外存空间的统称。",[15,284287,284288],{},[2627,284289,258022],{},[105,284291,284292,284297,284319],{},[108,284293,284294,284296],{},[2627,284295,283829],{}," 这是操作系统为外存（磁盘、磁带）提供的一层高级抽象模型。",[108,284298,284299,284301,284302,284304,284305,284308,284309,284311,284312,284315,284316,347],{},[2627,284300,283835],{},"\n裸磁盘是一大堆无序的磁道和扇区。\n",[2627,284303,90872],{},"的出现，使得应用程序和人类用户可以通过直观的“名称”和逻辑边界来持久化保存数据，而不需要记忆数据存放在磁盘的哪一号柱面上。\n",[2627,284306,284307],{},"文件系统 (File System)"," 不仅包含了存储在磁盘上的文件本身，更重要的是包含了维护“名称到物理地址映射”的",[2627,284310,238727],{},"，管理磁盘空闲块的",[2627,284313,284314],{},"分配表","，以及提供打开、读写保护的",[2627,284317,284318],{},"软件驱动代码",[108,284320,284321,284323],{},[2627,284322,283849],{}," 认为文件系统就是磁盘上的数据。文件系统实际上是“软件+数据结构+数据实体”的三位一体。",[32188,284325],{},[19,284327,284329],{"id":284328},"_10为什么要采用虚拟存储器管理实现虚拟存储器的条件是什么","10．为什么要采用虚拟存储器管理，实现虚拟存储器的条件是什么？",[15,284331,284332,284334,284336,284337,284340],{},[2627,284333,58173],{},[2627,284335,7819],{}," 为了突破物理内存大小的限制，允许执行比物理内存更大的程序；同时能够让更多的进程并发常驻内存，极大提高CPU的多道程序度与系统吞吐量。\n",[2627,284338,284339],{},"实现条件："," 硬件必须提供分页\u002F分段机构、地址转换机制（MMU）和缺页中断机构；软件必须提供页\u002F段置换算法和外存交换空间（Swap space）。",[15,284342,284343],{},[2627,284344,258022],{},[105,284346,284347,284352,284387],{},[108,284348,284349,284351],{},[2627,284350,283829],{}," 虚拟存储器（Virtual Memory）允许程序仅将当前所需的极少部分页面装入内存即可运行，遇到不在内存的页面时再动态调入。",[108,284353,284354,284356,284359,284360,284363,284364,284367],{},[2627,284355,283835],{},[71807,284357,284358],{},"为什么要用？"," 在没有虚拟内存时，程序必须“全量”装入内存才能运行。一旦程序比内存大，或者想跑多个大程序，系统就会崩溃。采用虚拟内存后，由于程序具有",[2627,284361,284362],{},"局部性原理 (Locality)","，OS只需装入当前正在用的几个页面，这就在逻辑上给用户提供了一个极其庞大的“虚拟空间”。\n",[71807,284365,284366],{},"需要什么条件？",[2230,284368,284369,284375,284380],{},[108,284370,284371,284372,347],{},"底层支持非连续分配的",[2627,284373,284374],{},"分页或分段硬件",[108,284376,284377,284378,347],{},"当CPU访问到不在内存的页面时，硬件必须能触发",[2627,284379,257364],{},[108,284381,284382,284383,284386],{},"操作系统必须包含负责换入换出的",[2627,284384,284385],{},"软件管理机制","，以决定何时调入、淘汰谁（置换策略）。",[108,284388,284389,284391,284392,347],{},[2627,284390,283849],{}," 虚拟内存实际上会因为页面换入换出带来巨大的I\u002FO开销，它并没有提升单体程序的“执行速度”，而是提升了",[2627,284393,284394],{},"资源利用率和系统并发能力",[32188,284396],{},[19,284398,284400],{"id":284399},"_11内存利用率不高主要表现在哪些方面可通过哪些途径来提高内存利用率","11．内存利用率不高主要表现在哪些方面？可通过哪些途径来提高内存利用率？",[15,284402,284403,284405,284407,284408,284411],{},[2627,284404,58173],{},[2627,284406,199877],{}," 内存利用率不高主要表现在产生大量的“内部碎片（分配给进程但未被使用的空间）”和“外部碎片（散落在内存中太小而无法分配的空闲块）”。\n",[2627,284409,284410],{},"途径："," 可以采用内存紧凑（Compaction）技术、采用允许非连续分配的分页或分段技术、以及引入虚拟内存技术来大幅提高内存利用率。",[15,284413,284414],{},[2627,284415,258022],{},[105,284417,284418,284423,284469],{},[108,284419,284420,284422],{},[2627,284421,283829],{}," 碎片（Fragmentation）是吃掉内存容量的最大元凶。",[108,284424,284425,284427],{},[2627,284426,283835],{},[105,284428,284429,284435,284441],{},[108,284430,284431,284434],{},[2627,284432,284433],{},"内部碎片："," 固定分区或分页系统中，分配给作业的块比作业实际需要的大，多出来的那部分无法给别人用。",[108,284436,284437,284440],{},[2627,284438,284439],{},"外部碎片："," 动态分区系统中，作业频繁进出，留下许多不连续的小洞，加起来虽然很大，但无法分配给一个大作业。",[108,284442,284443,284446],{},[2627,284444,284445],{},"提升途径：",[2230,284447,284448,284455,284462],{},[108,284449,284450,284451,284454],{},"对于动态分区，通过",[2627,284452,284453],{},"紧凑 (Compaction)"," 将作业向一端移动，把小洞拼成大洞。",[108,284456,284457,284458,284461],{},"彻底放弃连续分配，改用",[2627,284459,284460],{},"分页系统","。将程序打散存放到物理帧中，从根本上消灭外部碎片。",[108,284463,284464,284465,284468],{},"引入",[2627,284466,284467],{},"虚拟内存技术","。没用到的代码连内存都不进，直接呆在磁盘上，实现内存的极致利用。",[108,284470,284471,284473],{},[2627,284472,283849],{}," “紧凑”技术虽然能拼合碎片，但因为要大量移动数据，消耗极高的CPU时间，在现代OS中已被硬件支持的分页技术彻底取代。",[32188,284475],{},[19,284477,284479],{"id":284478},"_12什么是高级调度中级调度和低级调度","12．什么是高级调度、中级调度和低级调度？",[15,284481,284482,284484],{},[2627,284483,58173],{},"\n这是操作系统针对不同层级资源进行分配的三级调度模型：",[105,284486,284487,284493,284499],{},[108,284488,284489,284492],{},[2627,284490,284491],{},"高级调度（长程调度）："," 决定将哪些外存后备队列中的作业调入内存，并为它们创建进程。",[108,284494,284495,284498],{},[2627,284496,284497],{},"中级调度（中程调度）："," 决定将哪些内存中暂时不能运行的进程挂起（换出）到磁盘，或将磁盘上具备运行条件的就绪挂起进程换入内存，以调节系统并发度和内存压力。",[108,284500,284501,284504],{},[2627,284502,284503],{},"低级调度（短程\u002F进程调度）："," 决定将CPU实际分配给内存就绪队列中的哪一个进程。",[15,284506,284507],{},[2627,284508,258022],{},[105,284510,284511,284516,284540],{},[108,284512,284513,284515],{},[2627,284514,283829],{}," 调度（Scheduling）本质上是决定“谁排队、谁获得资源”的策略器。",[108,284517,284518,284520],{},[2627,284519,283835],{},[2230,284521,284522,284528,284534],{},[108,284523,284524,284527],{},[2627,284525,284526],{},"高级 (Long-term)："," 发生频率最低。它的任务是控制系统的“多道程序度”，即内存里到底装多少个作业合适。装少了CPU吃不饱，装多了内存要爆。",[108,284529,284530,284533],{},[2627,284531,284532],{},"中级 (Medium-term)："," 相当于内存的“缓冲阀”。当内存吃紧时，它负责把一些处于阻塞态的进程“剥夺”内存并放到硬盘缓存区（挂起态\u002FSuspend）；等内存宽松了再接回来。",[108,284535,284536,284539],{},[2627,284537,284538],{},"低级 (Short-term)："," 发生频率极高（毫秒级）。它直接操作CPU，从就绪队列挑选一个进程上CPU执行。",[108,284541,284542,284544],{},[2627,284543,283849],{}," 容易混淆中级调度和高级调度。高级调度管的是作业的“生与死”（创建进程），中级调度管的是进程在内存的“进与出”（Swap in\u002Fout）。",[32188,284546],{},[19,284548,284550],{"id":284549},"_13什么是设备独立性它是如何实现的","13．什么是设备独立性，它是如何实现的？",[15,284552,284553,284555],{},[2627,284554,58173],{},"\n设备独立性（Device Independence）是指应用程序应独立于具体使用的物理设备。即用户程序可以使用统一的“逻辑设备名”来请求I\u002FO操作，而无需关心底层实际连接的物理设备型号或控制细节。它是通过操作系统中的逻辑设备表（LUT）和分层的设备驱动程序来实现的。",[15,284557,284558],{},[2627,284559,258022],{},[105,284561,284562,284567,284584],{},[108,284563,284564,284566],{},[2627,284565,283829],{}," 设备独立性使得你的代码不需要因为换了一台新牌子的打印机就要重新编译。",[108,284568,284569,284571,284572,284575,284576,284579,284580,284583],{},[2627,284570,283835],{},"\n在没有设备独立性的时代，程序需要针对具体的硬件接口编写控制逻辑。引入该机制后：\n用户在程序中只声明要使用“打印机1（逻辑设备名）”并调用标准的",[31,284573,284574],{},"write()","命令。\n操作系统收到系统调用后，查找",[2627,284577,284578],{},"逻辑设备表 (LUT, Logical Unit Table)","。LUT中记录了“打印机1”当前映射到的“USB接口惠普打印机（物理设备名）”以及对应",[2627,284581,284582],{},"设备驱动程序","的入口地址。\n系统再调用该特定驱动程序将标准命令转化为具体的硬件电信号。",[108,284585,284586,284588],{},[2627,284587,283849],{}," 设备独立性并不意味着不需要驱动程序。相反，它正是依靠底层“设备驱动层”对上屏蔽了差异，才为应用层提供了统一的独立接口。",[32188,284590],{},[19,284592,284594],{"id":284593},"_14进程的基本状态有哪些这些状态之间是如何转换的","14．进程的基本状态有哪些？这些状态之间是如何转换的？",[15,284596,284597,284599],{},[2627,284598,58173],{},"\n进程的基本状态有五种：新建态（New）、就绪态（Ready）、运行态（Running）、阻塞\u002F等待态（Blocked\u002FWaiting）和退出态（Exit）。\n主要转换关系为：",[105,284601,284602,284608,284614,284620],{},[108,284603,284604,284607],{},[2627,284605,284606],{},"就绪 -> 运行："," 进程被调度程序选中分配了CPU。",[108,284609,284610,284613],{},[2627,284611,284612],{},"运行 -> 就绪："," 进程时间片用完，或被高优先级抢占。",[108,284615,284616,284619],{},[2627,284617,284618],{},"运行 -> 阻塞："," 进程请求I\u002FO或等待某事件发生。",[108,284621,284622,284625],{},[2627,284623,284624],{},"阻塞 -> 就绪："," 进程等待的I\u002FO完成或事件已发生。",[15,284627,284628],{},[2627,284629,258022],{},[105,284631,284632,284637,284665],{},[108,284633,284634,284636],{},[2627,284635,283829],{}," 状态模型描述了进程从出生到死亡的生命周期。",[108,284638,284639,284641,284642,284645,284646,284649,284650,284653,284654,284657,284658,284660,284661,284664],{},[2627,284640,283835],{},"\n操作系统之所以要维护这些状态，是因为CPU只有一个（或少数几个），而进程有很多。\n当程序刚被调入时处于",[2627,284643,284644],{},"新建态","。准备好资源后进入",[2627,284647,284648],{},"就绪态","排队。当低级调度器分配CPU时，进程转入",[2627,284651,284652],{},"运行态","。\n如果进程在运行中需要读取磁盘文件，因为磁盘极慢，进程必须交出CPU，转入",[2627,284655,284656],{},"阻塞态","去睡觉。此时OS会从就绪队列挑另一个进程去运行。等磁盘读完发出中断，OS再把睡着的进程叫醒，放回",[2627,284659,284648],{},"（注意，绝不能直接回到运行态，必须重新排队）。完成所有任务后转入",[2627,284662,284663],{},"退出态","清理资源。",[108,284666,284667,284669],{},[2627,284668,283849],{}," 绝不存在“阻塞态 -> 运行态”的直接转换。进程被唤醒后只能先回到就绪队列排队等待下一次调度。",[32188,284671],{},[19,284673,284675],{"id":284674},"_15spooling的含义是什么试述spooling系统的特点功能以及控制过程","15．SPOOLing的含义是什么？试述SPOOLing系统的特点、功能以及控制过程。",[15,284677,284678,284680,284682,284683,284686,284687,284690],{},[2627,284679,58173],{},[2627,284681,13689],{}," SPOOLing（Simultaneous Peripheral Operations On-Line）即假脱机技术，它利用多道程序设计和高速磁盘缓存，使CPU计算与慢速外设的I\u002FO操作同时在线进行。\n",[2627,284684,284685],{},"特点与功能："," 提高了I\u002FO速度，将独占设备（如打印机）改造成了多个进程可以同时请求的“虚拟共享设备”。\n",[2627,284688,284689],{},"控制过程："," 当用户进程请求打印时，SPOOLing系统并不会直接将其接入物理打印机，而是在磁盘上的“输出井”中为其分配一块空间暂存打印数据，并在队列中添加一个请求表项。后台的“打印守护进程”会依照队列顺序，将磁盘井中的数据慢慢输出到物理打印机。",[15,284692,284693],{},[2627,284694,258022],{},[105,284696,284697,284702,284714],{},[108,284698,284699,284701],{},[2627,284700,283829],{}," SPOOLing是用软件在磁盘上模拟硬件排队机制的经典设计。",[108,284703,284704,284706,284707,119123,284710,284713],{},[2627,284705,283835],{},"\n如果没有SPOOLing，进程A请求独占打印机时，进程B如果也想打印，就会被死死阻塞，甚至发生死锁。\nSPOOLing利用了磁盘极大的容量，划分了",[2627,284708,284709],{},"输入井",[2627,284711,284712],{},"输出井","。进程A和进程B可以“瞬间”把数据丢入磁盘上的输出井，系统告诉它们“打印完成了”，于是A和B继续高速执行计算任务（这就极大地提升了系统效率）。实际上，数据被操作系统悄悄排队了，由一个不知疲倦的后台进程逐个发往物理设备完成真实输出。",[108,284715,284716,284718],{},[2627,284717,283849],{}," SPOOLing不是在主存（RAM）中做缓冲，它的“输入井\u002F输出井”必须开辟在大容量的**辅助存储器（磁盘）**上。",[32188,284720],{},[19,284722,284724],{"id":284723},"_16简述进程的几种状态和引起状态转换的典型原因以及相关的操作原语","16．简述进程的几种状态和引起状态转换的典型原因，以及相关的操作原语。",[15,284726,284727,284729],{},[2627,284728,58173],{},[71807,284730,284731],{},"(注：状态与转换原因同第14题，此处重点补充操作原语)",[2230,284733,284734,284742,284750,284758,284766,284775],{},[108,284735,284736,284738,284739,347],{},[2627,284737,284606],{}," 原因是被调度器选中。原语：",[31,284740,284741],{},"调度 (Dispatch)",[108,284743,284744,284746,284747,347],{},[2627,284745,284612],{}," 原因是时间片耗尽或被抢占。原语：",[31,284748,284749],{},"时间片到期 (Timeout\u002FYield)",[108,284751,284752,284754,284755,347],{},[2627,284753,284618],{}," 原因是进程需要等待I\u002FO或某信号量。原语：",[31,284756,284757],{},"阻塞 (Block \u002F wait \u002F P操作)",[108,284759,284760,284762,284763,347],{},[2627,284761,284624],{}," 原因是等待的I\u002FO操作完成，中断发生。原语：",[31,284764,284765],{},"唤醒 (Wakeup \u002F signal \u002F V操作)",[108,284767,284768,284771,284772,347],{},[2627,284769,284770],{},"新建 -> 就绪："," 系统接纳新作业。原语：",[31,284773,284774],{},"创建 (Create)",[108,284776,284777,284780,284781,347],{},[2627,284778,284779],{},"运行 -> 退出："," 进程执行结束或异常。原语：",[31,284782,284783],{},"终止 (Destroy\u002FExit)",[15,284785,284786],{},[2627,284787,258022],{},[105,284789,284790,284795,284804],{},[108,284791,284792,284794],{},[2627,284793,283829],{}," 原语（Primitive）是操作系统内核中不可中断的一段执行代码，用于安全地改变进程控制块（PCB）的状态和链表归属。",[108,284796,284797,284799,284800,284803],{},[2627,284798,283835],{},"\n操作系统的进程转换不是瞬间完成的。比如",[31,284801,284802],{},"Block","原语：系统需要先把进程保护现场（保存寄存器），将其PCB中的状态位改为Blocked，将其PCB插入到相应的等待队列，最后调用调度器选一个新进程运行。因为涉及修改关键数据结构，这个过程必须用原子性的“原语”来保证不被中断打断。",[108,284805,284806,284808],{},[2627,284807,283849],{}," 引起状态转换的是“事件（如中断）”，而执行转换的具体代码实现则是“原语”。",[32188,284810],{},[19,284812,284814],{"id":284813},"_17设备管理中的数据传送控制方式有哪几种分别简述如何实现的","17．设备管理中的数据传送控制方式有哪几种？分别简述如何实现的。",[15,284816,284817,284819],{},[2627,284818,58173],{},"\n主要有三种：程序直接控制方式（Programmed I\u002FO）、中断驱动控制方式（Interrupt-driven I\u002FO）、直接存储器访问方式（DMA）。",[15,284821,284822],{},[2627,284823,258022],{},[105,284825,284826,284831,284867],{},[108,284827,284828,284830],{},[2627,284829,283829],{}," 这体现了计算机架构中为了解放CPU而进行的I\u002FO演进。",[108,284832,284833,284835],{},[2627,284834,283835],{},[2230,284836,284837,284846,284855],{},[108,284838,284839,284842,284843,347],{},[2627,284840,284841],{},"程序直接控制方式 (Programmed I\u002FO)："," CPU发出I\u002FO指令后，陷入“轮询（Busy Waiting）”状态，不断循环读取状态寄存器，直到外设准备好数据，CPU再将其读入。",[2627,284844,284845],{},"极其浪费CPU",[108,284847,284848,284851,284852,347],{},[2627,284849,284850],{},"中断驱动方式 (Interrupt-driven I\u002FO)："," CPU发出I\u002FO命令后直接去执行其他进程。当外设准备好数据时，向CPU发一个硬件中断信号，CPU再暂停当前工作，把这一个字节\u002F字的数据读入内存。",[2627,284853,284854],{},"解放了等待时间，但数据仍需CPU以字为单位搬运",[108,284856,284857,284860,284861,347,284864,347],{},[2627,284858,284859],{},"DMA (直接存储器访问)："," 硬件中增加了一个DMA控制器。CPU告诉DMA“把磁盘X扇区的N个字节搬到内存Y地址”，然后去干别的。DMA控制器接管总线，直接将一块数据在磁盘和内存间搬运，",[2627,284862,284863],{},"全部搬完后才向CPU发一次中断",[2627,284865,284866],{},"彻底将CPU从繁重的数据搬运中解放出来",[108,284868,284869,284871],{},[2627,284870,283849],{}," 认为DMA完全不需要CPU。实际上，DMA的启动指令和完成后的收尾工作依然需要CPU介入响应中断，它只是省去了中间循环搬运的开销。",[32188,284873],{},[19,284875,284877],{"id":284876},"_18什么是快表它在地址转换中起什么作用","18．什么是快表？它在地址转换中起什么作用？",[15,284879,284880,284882],{},[2627,284881,58173],{},"\n快表（TLB, Translation Lookaside Buffer）是集成在CPU内部（MMU中）的一块极高速的相联存储器缓存，专门用于存储最近被频繁访问的“页表项”（即虚拟页号到物理页框号的映射记录）。\n它的作用是大幅加速虚拟地址到物理地址的转换过程，避免每次寻址都去访问主存中的慢速页表。",[15,284884,284885],{},[2627,284886,258022],{},[105,284888,284889,284894,284902],{},[108,284890,284891,284893],{},[2627,284892,283829],{}," TLB是虚拟内存系统性能不崩盘的核心救命稻草。",[108,284895,284896,284898,284899,284901],{},[2627,284897,283835],{},"\n在虚拟分页系统中，页表是存放在主存（RAM）中的。如果没有快表，CPU执行一条指令需要访问两次主存：第一次去主存查页表算物理地址，第二次再去主存取真实数据。这会让机器速度直接降低一半。\n由于程序具有",[2627,284900,236324],{},"，CPU在短时间内往往会频繁访问同一个页。TLB作为硬件缓存，记住了最近的页表映射。CPU进行地址转换时，硬件会同时在TLB中并行比对虚拟页号：如果“命中（TLB hit）”，则直接拿到物理地址去读数据（只需1次内存访问）；如果“未命中（TLB miss）”，再去主存查页表，并将结果更新进TLB。",[108,284903,284904,284906,284907,284910],{},[2627,284905,283849],{}," 快表（TLB）缓存的是",[2627,284908,284909],{},"页表项（地址映射表）","，而不是程序的实际数据或代码！缓存实际数据的是通常意义上的CPU L1\u002FL2 Cache。",[32188,284912],{},[19,284914,284916],{"id":284915},"_19什么是死锁产生死锁的四个必要条件是什么","19．什么是死锁？产生死锁的四个必要条件是什么？",[15,284918,284919,284921,284924,284925,284928],{},[2627,284920,58173],{},[2627,284922,284923],{},"死锁","是指两个或两个以上的进程在执行过程中，由于相互竞争系统资源或通信而导致的一种永久阻塞的僵局；若无外力作用，它们都将永远无法向前推进。\n",[2627,284926,284927],{},"四个必要条件："," 互斥条件、请求和保持条件、不剥夺（非抢占）条件、循环等待条件。",[15,284930,284931],{},[2627,284932,258022],{},[105,284934,284935,284940,284971],{},[108,284936,284937,284939],{},[2627,284938,283829],{}," 死锁是并发编程中最具毁灭性的软件灾难。",[108,284941,284942,284944,284945],{},[2627,284943,283835],{},"\n产生死锁必须同时满足以下四条：\n",[2230,284946,284947,284953,284959,284965],{},[108,284948,284949,284952],{},[2627,284950,284951],{},"互斥 (Mutual exclusion)："," 至少有一个资源必须是排他性使用的。如果是可同时读取的共享资源，就不会引发阻塞。",[108,284954,284955,284958],{},[2627,284956,284957],{},"请求和保持 (Hold and wait)："," 进程手里已经死死攥着至少一个资源，同时又去申请别人占用的新资源，并且在等待期间绝不放下自己手里的东西。",[108,284960,284961,284964],{},[2627,284962,284963],{},"不剥夺 (No preemption)："," 进程所获得的资源在未使用完毕之前，操作系统不能强行把它抢走，只能由进程自愿释放。",[108,284966,284967,284970],{},[2627,284968,284969],{},"循环等待 (Circular wait)："," 在系统中必然存在一条由两个或以上进程组成的闭合等待链。例如，P1等P2的资源，P2等P3的资源，P3又转回头等P1的资源。",[108,284972,284973,284975,284976,284978,284979,284982],{},[2627,284974,283849],{}," 这四个条件是死锁的",[2627,284977,246768],{},"（死锁发生了，这四条一定成立），但不是",[2627,284980,284981],{},"充分条件","（即便这四条成立，只要还有额外的资源可供打破循环，死锁也不一定真的发生）。",[32188,284984],{},[19,284986,284988],{"id":284987},"_20在哲学家算法中是否能防止或解除死锁为什么","20．在哲学家算法中，是否能防止或解除死锁？为什么？",[15,284990,284991,284993],{},[2627,284992,58173],{},"\n能。可以通过破坏死锁的四个必要条件之一来预防死锁，或者利用算法机制避免死锁。这是因为死锁的产生必须严格依赖四个条件的共同成立，只要用程序逻辑打破“循环等待”或“请求和保持”，死锁就不会发生。",[15,284995,284996],{},[2627,284997,258022],{},[105,284999,285000,285005,285037],{},[108,285001,285002,285004],{},[2627,285003,283829],{}," 哲学家就餐问题是Dijkstra提出的经典同步问题。5个哲学家围坐，每两人中间有1根筷子（共5根）。需要同时拿到左右两根才能吃饭。",[108,285006,285007,285009,285010,285012,285013],{},[2627,285008,283835],{},"\n如果所有哲学家同时拿起左边的筷子，此时5根筷子全被占用。接着他们同时去拿右边的筷子，就会陷入无限期互相等待，满足了“循环等待”条件，发生",[2627,285011,284923],{},"。\n防止死锁的常用解决策略包括：\n",[2230,285014,285015,285021,285031],{},[108,285016,285017,285020],{},[2627,285018,285019],{},"破坏循环等待条件："," 规定奇数号哲学家先拿左筷子再拿右筷子，偶数号哲学家先拿右筷子再拿左筷子（不对称策略）。这样总有两个人竞争同一根筷子，必有一人能拿到两根，从而打破僵局。",[108,285022,285023,285026,285027,285030],{},[2627,285024,285025],{},"破坏请求和保持条件："," 要求哲学家必须",[2627,285028,285029],{},"同时","抓起两根筷子（使用AND型信号量或用互斥锁保护抓筷子的过程），如果不能同时拿到，就一根也不拿。",[108,285032,285033,285036],{},[2627,285034,285035],{},"资源预留："," 限制最多只允许4名哲学家同时入座。这样至少有1名哲学家能拿到两根筷子。",[108,285038,285039,285041,285042,285044],{},[2627,285040,283849],{}," 死锁虽然可以解决，但如果在解决时设计不当（例如大家都发现没筷子，同时放下左手，过一秒又同时拿起左手），会导致**活锁（Livelock）",[2627,285043,58269],{},"饥饿（Starvation）**现象，即进程虽然状态在变，但依然永远吃不到饭。这是更深层次的并发问题。",[32188,285046],{},[15,285048,285049],{},"你好！我是你的专属操作系统辅导老师。接下来，我将继续运用“概念核对 (Concept Check)”、“原理解析 (The 'Why')”以及“易错点 (Common Pitfalls)”的结构，为你深入浅出地详细讲解第21至35题。",[15,285051,285052],{},"请注意：对于部分在给定参考资料中提及较简略的经典概念（如分时系统的四大特征总结、软硬链接对比），我会在解析中辅以操作系统的标准理论（External Knowledge）来确保你能全面掌握。",[32188,285054],{},[19,285056,285058],{"id":285057},"_21-在单处理机环境下进程间有哪几种通信方式是如何实现的","21. 在单处理机环境下，进程间有哪几种通信方式，是如何实现的？",[15,285060,285061,285064],{},[2627,285062,285063],{},"答案与详细解析：","\n在单处理机环境下，UNIX等操作系统通常提供以下几种进程间通信（IPC）机制：管道（Pipes）、消息传递（Messages）、共享内存（Shared memory）、信号量（Semaphores）和信号（Signals）。",[105,285066,285067,285073,285111],{},[108,285068,285069,285072],{},[2627,285070,285071],{},"概念核对 (Concept Check)："," 进程间通信（IPC）是指操作系统提供的机制，使得独立的进程能够交换数据或同步动作。",[108,285074,285075,285078],{},[2627,285076,285077],{},"原理解析 (The 'Why')：",[105,285079,285080,285086,285099,285105],{},[108,285081,285082,285085],{},[2627,285083,285084],{},"管道 (Pipes)："," 是一个环形缓冲区，允许两个进程以“生产者-消费者”模型进行通信（先进先出）。一个进程写，另一个进程读。",[108,285087,285088,285091,285092,258,285095,285098],{},[2627,285089,285090],{},"消息传递 (Messages)："," 进程通过系统调用（如UNIX中的 ",[31,285093,285094],{},"msgsnd",[31,285096,285097],{},"msgrcv","）发送和接收带有类型的字节块。通常结合“信箱（Mailbox）”或消息队列来实现。",[108,285100,285101,285104],{},[2627,285102,285103],{},"共享内存 (Shared memory)："," 最快的通信方式。操作系统将同一块物理内存映射到多个进程的虚拟地址空间中，进程像读写自身内存一样读写这块区域。",[108,285106,285107,285110],{},[2627,285108,285109],{},"信号量 (Semaphores) \u002F 信号 (Signals)："," 不主要用于传输海量数据，而是用于进程间的动作触发和状态同步。",[108,285112,285113,285116],{},[2627,285114,285115],{},"易错点 (Common Pitfalls)："," 容易把“同步机制”（如信号量）和“数据传输机制”（如共享内存）混为一谈。共享内存虽然快，但它本身不提供互斥约束，必须结合信号量来防止数据读写冲突。、",[32188,285118],{},[19,285120,285122],{"id":285121},"_22-在段页式虚拟存储系统中不同进程之间是如何实现程序共享的","22. 在段页式虚拟存储系统中，不同进程之间是如何实现程序共享的？",[15,285124,285125,285127],{},[2627,285126,285063],{},"\n通过让多个进程的段表（Segment tables）中的某个表项指向同一个共享的物理内存区域（或同一个页表）来实现。",[105,285129,285130,285135,285140],{},[108,285131,285132,285134],{},[2627,285133,285071],{}," 分段与分页结合的系统中，分段面向用户逻辑，分页面向物理内存管理。共享通常在“段”这一层级进行。",[108,285136,285137,285139],{},[2627,285138,285077],{},"\n为什么用段来共享而不是页？因为“分页”是按照固定大小生硬切分的，一个页里可能同时包含不可共享的私人数据和可共享的代码；而“分段”是按照程序的逻辑结构（如某个数学函数库、某个全局数组）划分的，具有完整的逻辑意义。因此，当两个进程需要共享某个函数库时，操作系统只需将这两个进程各自的段表中代表该函数库的表项，指向主存中同一块物理基址即可。",[108,285141,285142,285144],{},[2627,285143,285115],{}," 误以为单纯的页式管理也很容易实现共享。实际上，由于页会切断逻辑边界，单纯的页式系统很难安全、干净地实现共享。",[32188,285146],{},[19,285148,285150],{"id":285149},"_23-在生产者消费者问题中能否将生产者进程的waitempty和waitmutex语句互换为什么","23. 在生产者—消费者问题中，能否将生产者进程的wait(empty)和wait(mutex)语句互换，为什么？",[15,285152,285153,285155,285158],{},[2627,285154,285063],{},[2627,285156,285157],{},"绝对不能互换","，因为这会导致死锁（Deadlock）。",[105,285160,285161,285178,285203],{},[108,285162,285163,47649,285165,285168,285169,285171,285172,285174,285175,285177],{},[2627,285164,285071],{},[31,285166,285167],{},"wait","操作（也称",[31,285170,105852],{},"操作）用于申请资源。",[31,285173,198187],{},"用于同步（检查缓冲区是否有空位），",[31,285176,196586],{},"用于互斥（锁定缓冲区）。",[108,285179,285180,285182,285183,285185,285186,285189,285190,285193,285194,285196,285197,285199,285200,285202],{},[2627,285181,285077],{},"\n如果将生产者的操作互换，即先执行 ",[31,285184,257330],{}," 再执行 ",[31,285187,285188],{},"wait(empty)","：假设此时缓冲区已满（",[31,285191,285192],{},"empty = 0","），生产者进程首先成功拿到了互斥锁（",[31,285195,196586],{},"），然后它去申请空位（",[31,285198,285188],{},"），因为没有空位，生产者抱着互斥锁进入阻塞睡眠状态。\n此时，消费者进程想要取走数据以腾出空位，但它在执行 ",[31,285201,257330],{}," 时，发现锁被生产者死死抱着不放，于是消费者也被阻塞。双方互相等待，形成死锁。",[108,285204,285205,285207,285208,285211],{},[2627,285206,285115],{}," 记住一个黄金法则：",[2627,285209,285210],{},"P操作（wait）的顺序生死攸关，必须“先同步，后互斥”","。而 V操作（signal）的顺序通常不会引起死锁。",[32188,285213],{},[19,285215,285217],{"id":285216},"_24-目前操作系统采用的目录结构是什么它具有什么优点","24. 目前操作系统采用的目录结构是什么？它具有什么优点？",[15,285219,285220,285222,285223,347],{},[2627,285221,285063],{},"\n目前主流操作系统采用的是",[2627,285224,285225],{},"树形目录结构（Tree-structured directory）",[105,285227,285228,285233,285238],{},[108,285229,285230,285232],{},[2627,285231,285071],{}," 树形结构是一种多级目录体系，所有目录和文件构成一棵倒置的树，通过“路径名（Pathname）”来唯一定位文件。",[108,285234,285235,285237],{},[2627,285236,285077],{},"\n早期的单级目录把所有文件混在一起，极易重名。树形结构允许用户自由创建子文件夹，这种设计极大地提高了灵活性。它的主要优点是：能够基于所有者或应用项目对文件进行清晰的分类和隔离，从而提高了安全性、数据完整性（便于备份），并且彻底避免了不同分支下的文件名冲突（Name clashes）问题。",[108,285239,285240,285242],{},[2627,285241,285115],{}," 很多学生忽略了树形结构中“唯一标识符不再是文件名，而是绝对路径名”这一关键事实。",[32188,285244],{},[19,285246,285248],{"id":285247},"_25-文件目录有何作用","25. 文件目录有何作用？",[15,285250,285251,285253],{},[2627,285252,285063],{},"\n文件目录的主要作用是维护“逻辑文件名”到“物理存储地址”的映射，并记录文件的各种控制属性。",[105,285255,285256,285261,285270],{},[108,285257,285258,285260],{},[2627,285259,285071],{}," 目录是文件系统用于组织和检索文件的一种特殊数据结构。",[108,285262,285263,285265,285266,285269],{},[2627,285264,285077],{},"\n人类只记得住“文件名”（如 ",[31,285267,285268],{},"report.docx","），但磁盘只认得“物理块号”。文件目录充当了翻译官，它提供了关于文件是否存在、在辅助存储器（磁盘）上的具体物理位置、当前状态、访问权限（读写执行）、创建者等关键信息。",[108,285271,285272,285274],{},[2627,285273,285115],{}," 误以为目录只是一层“可视化文件夹”。在操作系统底层，目录本身也是一种文件，其内容就是一张“文件名与文件控制块（FCB\u002Finode）对应关系”的映射表。",[32188,285276],{},[19,285278,285280],{"id":285279},"_26-什么是文件的逻辑结构和物理结构","26. 什么是文件的逻辑结构和物理结构？",[15,285282,285283],{},[2627,285284,285063],{},[105,285286,285287,285293,285299,285304,285309],{},[108,285288,285289,285292],{},[2627,285290,285291],{},"逻辑结构："," 是从用户和应用程序视角看到的文件组织形式（如无结构的流式文件，或有结构的记录式文件）。",[108,285294,285295,285298],{},[2627,285296,285297],{},"物理结构："," 是从操作系统和磁盘视角看到的文件数据在物理存储介质上的实际排列和分配方式（如连续分配、链接分配、索引分配）。",[108,285300,285301,285303],{},[2627,285302,285071],{}," 逻辑结构关乎“怎么用”，物理结构关乎“怎么存”。",[108,285305,285306,285308],{},[2627,285307,285077],{},"\n用户写代码时，只关心数据是像水流一样连续读取（流式\u002FStream file），还是像数据库那样按关键字查找一条条记录（记录式\u002FRecord file）。\n但操作系统必须面对现实：磁盘是分块的。OS必须决定是把文件连续存放在一起（读取快，但有外部碎片），还是用链表串起来（无碎片，但不支持随机访问），或者是建立索引块（支持随机访问，是现代OS的首选）。",[108,285310,285311,285313],{},[2627,285312,285115],{}," 极其容易混淆这两个概念。比如“索引顺序文件”属于逻辑结构；而“索引分配”属于物理结构。",[32188,285315],{},[19,285317,285319],{"id":285318},"_27-在操作系统中为什么要引入进程概念它与程序的差别和关系是怎样的","27. 在操作系统中为什么要引入进程概念？它与程序的差别和关系是怎样的？",[15,285321,285322,285324,285327,285328],{},[2627,285323,285063],{},[2627,285325,285326],{},"引入原因："," 在多道程序设计中，多程序并发执行会导致资源竞争、执行速度不可预测。为了系统化地监控、控制并协调这些并发实体的推进，操作系统引入了“进程”作为资源分配和独立调度的基本单位。\n",[2627,285329,285330],{},"差别与关系：",[105,285332,285333,285339,285344],{},[108,285334,285335,285338],{},[2627,285336,285337],{},"差别："," 程序是静态的二进制代码（存放在磁盘上）；进程是动态的、处于执行状态的实体（存在于内存中）。",[108,285340,285341,285343],{},[2627,285342,120100],{}," 进程是程序的一次执行过程。一个进程包含了可执行的程序代码、相关的数据集，以及用于记录执行上下文的进程控制块（PCB）。一个程序可以被多次启动，从而产生多个不同的进程。",[108,285345,285346,285348],{},[2627,285347,285115],{}," 把进程和程序等同。记住口诀：“程序是菜谱，进程是正在炒菜的过程”。",[32188,285350],{},[19,285352,285354],{"id":285353},"_28-什么是进程的互斥与同步","28. 什么是进程的互斥与同步？",[15,285356,285357],{},[2627,285358,285063],{},[105,285360,285361,285367,285373,285386],{},[108,285362,285363,285366],{},[2627,285364,285365],{},"互斥（Mutual Exclusion）："," 确保在任何时刻，只允许一个进程访问某个临界资源（即排他性资源的代码段）。",[108,285368,285369,285372],{},[2627,285370,285371],{},"同步（Synchronization）："," 协调多个并发进程的执行顺序，确保它们按照特定的逻辑先后顺序推进（例如，必须先生产数据，才能消费数据）。",[108,285374,285375,285377,285378,285381,285382,285385],{},[2627,285376,285077],{},"\n因为并发进程的执行速度是“异步”的（走走停停，不可预知）。如果不加",[2627,285379,285380],{},"互斥","，两个进程同时修改一个全局变量就会导致数据损坏（Race condition）；如果不加",[2627,285383,285384],{},"同步","，依赖别人计算结果的进程可能会在数据还没准备好时就提前执行，导致逻辑错误。",[108,285387,285388,285390],{},[2627,285389,285115],{}," 互斥是“排他”（你进我就不能进，防止冲突）；同步是“协作”（我等你做完我再做，保证顺序）。",[32188,285392],{},[19,285394,285396],{"id":285395},"_29-简述分时系统的四个基本特征","29. 简述分时系统的四个基本特征？",[15,285398,285399,285401,285404],{},[2627,285400,285063],{},[71807,285402,285403],{},"(注：结合参考资料与标准OS理论补充)","\n分时系统（Time-sharing system）的核心目标是最大化响应时间，让多个用户能够直接与计算机交互。其基本特征通常总结为四点：",[2230,285406,285407,285413,285419,285425],{},[108,285408,285409,285412],{},[2627,285410,285411],{},"多路性 \u002F 同时性："," 多个用户通过各自的终端同时连接并使用同一台主机。",[108,285414,285415,285418],{},[2627,285416,285417],{},"独立性："," 每个用户感觉自己独占了整个计算机，彼此之间互不干扰。",[108,285420,285421,285424],{},[2627,285422,285423],{},"及时性 \u002F 响应快："," 系统通过时间片轮转（Round-robin\u002FTime-slicing）调度进程，保证在用户能接受的短暂时间内给出响应。",[108,285426,285427,285430],{},[2627,285428,285429],{},"交互性："," 用户可以通过终端直接向系统发出指令并立刻获得结果，这是它区别于批处理系统的最大特征。",[105,285432,285433],{},[108,285434,285435,285437],{},[2627,285436,285115],{}," 不要把分时系统和“实时系统（Real-time system）”混淆。分时系统追求的是“人类感觉够快即可”，实时系统追求的是“必须在严格的毫秒级截止时间内完成”。",[32188,285439],{},[19,285441,285443],{"id":285442},"_30-文件的逻辑结构分为几种形式文件的磁盘布局分为几种形式","30. 文件的逻辑结构分为几种形式？文件的磁盘布局分为几种形式？",[15,285445,285446],{},[2627,285447,285063],{},[105,285449,285450,285456,285462,285467],{},[108,285451,285452,285455],{},[2627,285453,285454],{},"逻辑结构形式："," 根据参考资料，主要分为两大类：无结构文件（流式文件）和 有结构文件（记录式文件）。其中记录式文件进一步细分为：堆型文件（Pile）、顺序文件（Sequential）、索引顺序文件（Indexed sequential）、索引文件（Indexed）和直接\u002F哈希文件（Direct\u002FHashed）。",[108,285457,285458,285461],{},[2627,285459,285460],{},"磁盘布局（物理结构\u002F分配策略）形式："," 连续分配（Contiguous allocation）、链式分配（Chained allocation）和索引分配（Indexed allocation）。",[108,285463,285464,285466],{},[2627,285465,285077],{},"\n见第26题。这一题是对第26题的具体分类列举。",[108,285468,285469,285471],{},[2627,285470,285115],{}," 在答题时如果被问到“逻辑结构”，很多学生会漏掉最简单的“流式文件（Stream file）”，这是UNIX\u002FLinux最喜欢用的形式（把一切看作字节流）。",[32188,285473],{},[19,285475,285477],{"id":285476},"_31-什么是计算机操作系统","31. 什么是计算机操作系统？",[15,285479,285480,285482],{},[2627,285481,285063],{},"\n操作系统（Operating System, OS）是计算机系统中最基本的系统软件。它利用并控制一个或多个处理器的硬件资源，负责控制应用程序的执行，并充当应用程序\u002F用户与计算机硬件之间的接口。",[105,285484,285485,285504],{},[108,285486,285487,285489,285490],{},[2627,285488,285077],{},"\n操作系统的两大核心身份：\n",[2230,285491,285492,285498],{},[108,285493,285494,285497],{},[2627,285495,285496],{},"资源管理者 (Resource Manager)："," 公平、高效地管理CPU、内存、I\u002FO设备，防止多个程序打架。",[108,285499,285500,285503],{},[2627,285501,285502],{},"扩展机器\u002F虚拟机 (Virtual Machine\u002FInterface)："," 硬件接口太难用了，OS为程序员提供了一套方便、丰富、安全且一致的接口（如系统调用），屏蔽了硬件的丑陋细节。",[108,285505,285506,285508],{},[2627,285507,285115],{}," 认为OS只是提供用户界面的软件。UI（外壳）只是OS的表层，OS的真正核心是底层的内核（Kernel）。",[32188,285510],{},[19,285512,285514],{"id":285513},"_32-简述存储管理中移动技术compaction的优缺点","32. 简述存储管理中移动技术（Compaction）的优缺点。",[15,285516,285517,285519],{},[2627,285518,285063],{},"\n移动技术（或称内存紧凑\u002F压缩技术，Compaction）用于动态分区内存管理中。",[105,285521,285522,285527,285532],{},[108,285523,285524,285526],{},[2627,285525,14543],{}," 能够彻底解决“外部碎片（External fragmentation）”问题。它通过将内存中所有的进程向一端移动，将原本散落在各处的微小空闲孔洞聚集成一个巨大的空闲内存块，从而可以容纳新的大作业。",[108,285528,285529,285531],{},[2627,285530,177826],{}," 极其浪费处理器时间。因为需要在内存中搬运大量的数据字节，这使得CPU无法用于执行有用的计算任务，导致系统效率极度低下。",[108,285533,285534,285536],{},[2627,285535,285115],{}," 现代操作系统基本上已经抛弃了这种耗时的移动技术，转而使用硬件支持的“分页（Paging）”技术，从根本上消灭了外部碎片。",[32188,285538],{},[19,285540,285542],{"id":285541},"_33-简述并发与并行之间的关系","33. 简述并发与并行之间的关系。",[15,285544,285545],{},[2627,285546,285063],{},[105,285548,285549,285555,285561,285566],{},[108,285550,285551,285554],{},[2627,285552,285553],{},"并发 (Concurrency)："," 指多个进程在宏观时间段内看起来是同时推进的。在单核处理器上，这是通过进程的交替执行（时间片轮转）来实现的。",[108,285556,285557,285560],{},[2627,285558,285559],{},"并行 (Parallelism)："," 指多个进程在微观的同一物理时刻，真正地同时执行。这必须依赖于多核或多处理器（Multiprocessor）硬件架构。",[108,285562,285563,285565],{},[2627,285564,285077],{},"\n可以把“并发”想象成一个厨师同时处理三个锅（交替翻炒，同一瞬间只在一个锅前）；而“并行”是三个厨师分别站在三个锅前同时炒菜。并行是并发的一个物理真子集。",[108,285567,285568,285570],{},[2627,285569,285115],{}," 在日常用语中这两个词常混用，但在计算机科学中，并行代表硬件上的绝对同时，而并发是软件调度营造的“宏观同时”假象。",[32188,285572],{},[19,285574,285576],{"id":285575},"_34-页式和段式内存管理有什么区别怎样才能实现共享和保护","34. 页式和段式内存管理有什么区别？怎样才能实现共享和保护？",[15,285578,285579,285581],{},[2627,285580,285063],{},[2627,285582,285583],{},"主要区别：",[2230,285585,285586,285597],{},[108,285587,285588,285591,285592,285594,285595,347],{},[2627,285589,285590],{},"目的与可见性："," 分页是为了消除外部碎片、提高内存利用率，由系统硬件自动完成，对程序员",[2627,285593,205002],{},"；分段是为了适应程序的逻辑结构（如模块化），方便数据和代码共享，由程序员划分，对程序员",[2627,285596,205014],{},[108,285598,285599,285602],{},[2627,285600,285601],{},"大小："," 页面大小是固定的（由硬件决定）；段的大小是可变的（由程序逻辑决定）。",[15,285604,285605],{},[2627,285606,285607],{},"实现共享和保护：",[105,285609,285610,285616],{},[108,285611,285612,285613,347],{},"由于分段拥有完整的逻辑意义（例如整个段都是某个只读函数），因此",[2627,285614,285615],{},"共享和保护通常在“段”这一级别来实现最为理想",[108,285617,285618],{},"实现方法是：操作系统通过在不同进程的**段表（Segment Tables）**中，设置表项指向主存中相同的物理基址（或同一套页表），从而让它们共享同一块逻辑代码或数据。同时，在段表中设置权限位（如只读、读写、执行）来实现安全保护。",[32188,285620],{},[19,285622,285624],{"id":285623},"_35-比较基于索引节点和基于符号链的文件共享方法","35. 比较基于索引节点和基于符号链的文件共享方法。",[15,285626,285627,285629,285632],{},[2627,285628,285063],{},[71807,285630,285631],{},"(注：本题深入考察文件共享机制，辅以标准操作系统的软硬链接理论知识进行比对说明)","\n在类 UNIX 系统中，文件共享主要通过“硬链接（基于索引节点）”和“软链接\u002F符号链接（基于符号链）”实现：",[105,285634,285635,285658,285680],{},[108,285636,285637,285640],{},[2627,285638,285639],{},"基于索引节点（Hard Link \u002F 硬链接）：",[105,285641,285642,285648,285653],{},[108,285643,285644,285647],{},[2627,285645,285646],{},"原理："," 多个不同目录下的目录项（文件名）直接指向同一个物理文件的索引节点（inode）。inode内部维护一个引用计数（Reference count）。",[108,285649,285650,285652],{},[2627,285651,14543],{}," 访问速度快（直接通过inode找物理块）；即使删除了原文件名，只要还有其他硬链接指向该inode，文件数据就不会丢失。",[108,285654,285655,285657],{},[2627,285656,177826],{}," 不能跨越不同的文件系统创建硬链接；为防止形成循环环路，通常不允许对目录创建硬链接。",[108,285659,285660,285663],{},[2627,285661,285662],{},"基于符号链（Symbolic Link \u002F 软链接）：",[105,285664,285665,285670,285675],{},[108,285666,285667,285669],{},[2627,285668,285646],{}," 创建一个特殊类型的新文件（软链接文件），该文件的内容是目标文件的“绝对路径或相对路径”。",[108,285671,285672,285674],{},[2627,285673,14543],{}," 极其灵活，可以跨越不同的文件系统共享文件；甚至可以对目录建立链接。",[108,285676,285677,285679],{},[2627,285678,177826],{}," 访问速度较慢，因为操作系统每次都要去解析并遍历该路径字符串；如果原文件被删除，符号链接就会变成指向虚无的“悬空指针（Dangling pointer）”。",[108,285681,285682,285684],{},[2627,285683,285115],{}," 考试时常考的一点是：删除源文件后，哪种方式会出错？答案是符号链会出错（找不到文件），而基于索引节点的方式依然能通过剩余的硬链接正常访问文件。",{"title":29,"searchDepth":43,"depth":43,"links":285686},[285687,285688,285689,285690,285691,285692,285693,285694,285695,285696,285697,285698,285699,285700,285701,285702,285703,285704,285705,285706,285707,285708,285709,285710,285711,285712,285713,285714,285715,285716,285717,285718,285719,285720,285721],{"id":283812,"depth":43,"text":283813},{"id":283859,"depth":43,"text":283860},{"id":283901,"depth":43,"text":283902},{"id":283960,"depth":43,"text":283961},{"id":284015,"depth":43,"text":284016},{"id":284077,"depth":43,"text":284078},{"id":284148,"depth":43,"text":284149},{"id":284206,"depth":43,"text":284207},{"id":284273,"depth":43,"text":284274},{"id":284328,"depth":43,"text":284329},{"id":284399,"depth":43,"text":284400},{"id":284478,"depth":43,"text":284479},{"id":284549,"depth":43,"text":284550},{"id":284593,"depth":43,"text":284594},{"id":284674,"depth":43,"text":284675},{"id":284723,"depth":43,"text":284724},{"id":284813,"depth":43,"text":284814},{"id":284876,"depth":43,"text":284877},{"id":284915,"depth":43,"text":284916},{"id":284987,"depth":43,"text":284988},{"id":285057,"depth":43,"text":285058},{"id":285121,"depth":43,"text":285122},{"id":285149,"depth":43,"text":285150},{"id":285216,"depth":43,"text":285217},{"id":285247,"depth":43,"text":285248},{"id":285279,"depth":43,"text":285280},{"id":285318,"depth":43,"text":285319},{"id":285353,"depth":43,"text":285354},{"id":285395,"depth":43,"text":285396},{"id":285442,"depth":43,"text":285443},{"id":285476,"depth":43,"text":285477},{"id":285513,"depth":43,"text":285514},{"id":285541,"depth":43,"text":285542},{"id":285575,"depth":43,"text":285576},{"id":285623,"depth":43,"text":285624},{"order":68},"\u002Fsource\u002Fcs\u002Fos\u002Freview\u002Fshortans",{"title":283805,"description":29},"source\u002Fcs\u002Fos\u002Freview\u002Fshortans","T5hyf_3xJduRRr1ywp1bLidKAPSAz89ebm7lbBaT16U",{"id":285728,"title":285729,"body":285730,"description":29,"extension":85,"meta":286637,"navigation":52,"path":286638,"seo":286639,"stem":286640,"__hash__":286641},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F01-基本概念1-10.md","review202505 答案讲解 01：基本概念 1-10",{"type":8,"value":285731,"toc":286625},[285732,285735,285739,285744,285749,285754,285757,285760,285771,285774,285777,285829,285832,285846,285849,285853,285857,285862,285866,285869,285872,285886,285889,285893,285897,285902,285906,285909,286001,286004,286018,286021,286025,286029,286034,286038,286041,286055,286058,286061,286123,286126,286130,286134,286139,286143,286146,286160,286163,286231,286233,286247,286250,286254,286258,286263,286267,286270,286281,286284,286290,286293,286299,286302,286306,286310,286315,286319,286322,286325,286345,286348,286352,286356,286361,286365,286368,286371,286385,286387,286393,286396,286399,286410,286414,286418,286423,286427,286430,286433,286447,286449,286476,286479,286483,286487,286492,286496,286499,286519,286522,286539,286542,286611,286614,286622],[11,285733,285729],{"id":285734},"review202505-答案讲解-01基本概念-1-10",[19,285736,285738],{"id":285737},"_1-操作系统定义目标功能和特征","1. 操作系统定义、目标、功能和特征",[15,285740,285741],{},[2627,285742,285743],{},"题意：",[104833,285745,285746],{},[15,285747,285748],{},"1、阐述操作系统定义、目标、功能、和特征。",[15,285750,285751],{},[2627,285752,285753],{},"答案讲解：",[15,285755,285756],{},"操作系统是配置在计算机硬件上的第一层系统软件，是用户与计算机硬件之间的接口，也是计算机系统资源的管理者和控制者。",[15,285758,285759],{},"按 Stallings 教材第 2 章的口径，操作系统的主要目标是三项：",[2230,285761,285762,285765,285768],{},[108,285763,285764],{},"方便性：让用户更方便地使用计算机。",[108,285766,285767],{},"有效性：提高 CPU、内存、外设等资源利用率，提高系统吞吐量。",[108,285769,285770],{},"可演化性：系统结构应便于开发、测试和引入新功能，并尽量不影响已有服务。",[15,285772,285773],{},"有些中文教材还会把“开放性”列为目标之一。若按本课程使用的 Stallings 资料，应优先写“方便性、有效性、可演化性”。",[15,285775,285776],{},"操作系统的主要功能：",[13837,285778,285779,285787],{},[13840,285780,285781],{},[13843,285782,285783,285785],{},[13846,285784,236888],{},[13846,285786,111426],{},[13853,285788,285789,285797,285805,285813,285821],{},[13843,285790,285791,285794],{},[13858,285792,285793],{},"处理机管理",[13858,285795,285796],{},"进程创建、撤销、调度、同步、通信",[13843,285798,285799,285802],{},[13858,285800,285801],{},"存储器管理",[13858,285803,285804],{},"内存分配、地址转换、内存保护、虚拟存储",[13843,285806,285807,285810],{},[13858,285808,285809],{},"设备管理",[13858,285811,285812],{},"设备分配、驱动程序、缓冲、I\u002FO 控制",[13843,285814,285815,285818],{},[13858,285816,285817],{},"文件管理",[13858,285819,285820],{},"文件创建、删除、读写、目录、权限、磁盘空间管理",[13843,285822,285823,285826],{},[13858,285824,285825],{},"用户接口",[13858,285827,285828],{},"命令接口、图形界面、系统调用",[15,285830,285831],{},"操作系统的基本特征：",[2230,285833,285834,285837,285840,285843],{},[108,285835,285836],{},"并发：多个程序在同一时间段内推进。",[108,285838,285839],{},"共享：系统资源可被多个用户或多个进程共同使用。",[108,285841,285842],{},"虚拟：把物理资源抽象成逻辑资源，例如虚拟内存、虚拟设备。",[108,285844,285845],{},"异步：进程推进速度不可预知，运行顺序受调度和事件影响。",[15,285847,285848],{},"一句话记忆：操作系统向上提供方便接口，向下管理硬件资源，中间负责让多个程序安全、高效地并发运行。",[19,285850,285852],{"id":285851},"_2-并发与并行","2. 并发与并行",[15,285854,285855],{},[2627,285856,285743],{},[104833,285858,285859],{},[15,285860,285861],{},"2、简述并发与并行之间的关系。",[15,285863,285864],{},[2627,285865,285753],{},[15,285867,285868],{},"并发是指多个任务在同一时间段内都在推进；并行是指多个任务在同一时刻真正同时执行。",[15,285870,285871],{},"二者关系：",[2230,285873,285874,285877,285880,285883],{},[108,285875,285876],{},"并发强调“时间段”，并行强调“时间点”。",[108,285878,285879],{},"单核 CPU 可以通过时间片轮转实现并发，但不能真正并行执行多个 CPU 指令流。",[108,285881,285882],{},"多核或多处理机系统可以实现真正并行。",[108,285884,285885],{},"并行一定属于并发的表现形式，但并发不一定是并行。",[15,285887,285888],{},"例子：单核 CPU 上 A、B 两个进程轮流运行，用户感觉它们同时运行，这是并发；双核 CPU 上 A、B 分别占用一个核心同时执行，这是并行。",[19,285890,285892],{"id":285891},"_3-进程和线程区别线程的优越性","3. 进程和线程区别，线程的优越性",[15,285894,285895],{},[2627,285896,285743],{},[104833,285898,285899],{},[15,285900,285901],{},"3、进程和线程区别？线程相比如进程有何优越性？",[15,285903,285904],{},[2627,285905,285753],{},[15,285907,285908],{},"进程是程序的一次执行过程，是系统进行资源分配和保护的基本单位。线程是进程中的一条执行流，是 CPU 调度和执行的基本单位。",[13837,285910,285911,285922],{},[13840,285912,285913],{},[13843,285914,285915,285918,285920],{},[13846,285916,285917],{},"比较项",[13846,285919,247197],{},[13846,285921,247201],{},[13853,285923,285924,285935,285946,285957,285968,285979,285990],{},[13843,285925,285926,285929,285932],{},[13858,285927,285928],{},"基本性质",[13858,285930,285931],{},"资源分配单位",[13858,285933,285934],{},"CPU 调度和执行单位",[13843,285936,285937,285940,285943],{},[13858,285938,285939],{},"地址空间",[13858,285941,285942],{},"通常拥有独立地址空间",[13858,285944,285945],{},"同一进程内线程共享地址空间",[13843,285947,285948,285951,285954],{},[13858,285949,285950],{},"资源拥有",[13858,285952,285953],{},"拥有文件、内存等资源",[13858,285955,285956],{},"通常不单独拥有系统资源，只拥有少量运行现场",[13843,285958,285959,285962,285965],{},[13858,285960,285961],{},"创建撤销开销",[13858,285963,285964],{},"较大",[13858,285966,285967],{},"较小",[13843,285969,285970,285973,285976],{},[13858,285971,285972],{},"上下文切换",[13858,285974,285975],{},"较慢",[13858,285977,285978],{},"较快",[13843,285980,285981,285984,285987],{},[13858,285982,285983],{},"通信方式",[13858,285985,285986],{},"需要管道、消息队列、共享内存等 IPC 机制",[13858,285988,285989],{},"同进程线程可直接访问共享变量",[13843,285991,285992,285995,285998],{},[13858,285993,285994],{},"稳定性",[13858,285996,285997],{},"隔离性较强",[13858,285999,286000],{},"一个线程出错可能影响整个进程",[15,286002,286003],{},"线程相对进程的优越性：",[2230,286005,286006,286009,286012,286015],{},[108,286007,286008],{},"创建、撤销、切换开销小。",[108,286010,286011],{},"同一进程内线程通信方便。",[108,286013,286014],{},"可以提高程序响应速度。",[108,286016,286017],{},"适合多核并行，提高吞吐量。",[15,286019,286020],{},"记忆方式：进程像“资源容器”，线程像“容器里真正执行的工人”。",[19,286022,286024],{"id":286023},"_4-进程的组成元素","4. 进程的组成元素",[15,286026,286027],{},[2627,286028,285743],{},[104833,286030,286031],{},[15,286032,286033],{},"4、进程的组成元素；",[15,286035,286036],{},[2627,286037,285753],{},[15,286039,286040],{},"按 Stallings 第 3 章，进程的程序、数据、栈和属性合起来称为进程映像。典型进程映像包含：",[2230,286042,286043,286046,286049,286052],{},[108,286044,286045],{},"用户程序：将要执行的程序代码。",[108,286047,286048],{},"用户数据：用户空间中可修改的数据，如程序数据、用户栈、可修改程序区等。",[108,286050,286051],{},"系统栈：用于保存过程调用、系统调用的参数和调用地址等。",[108,286053,286054],{},"进程控制块 PCB：操作系统控制该进程所需的数据。",[15,286056,286057],{},"从操作系统实现角度，也常把 CPU 现场、打开文件、内存空间、I\u002FO 状态等作为进程的重要组成信息。",[15,286059,286060],{},"PCB 中常见内容：",[13837,286062,286063,286073],{},[13840,286064,286065],{},[13843,286066,286067,286070],{},[13846,286068,286069],{},"类别",[13846,286071,286072],{},"典型内容",[13853,286074,286075,286083,286091,286099,286107,286115],{},[13843,286076,286077,286080],{},[13858,286078,286079],{},"标识信息",[13858,286081,286082],{},"PID、父进程 ID、用户 ID",[13843,286084,286085,286088],{},[13858,286086,286087],{},"状态信息",[13858,286089,286090],{},"新建、就绪、运行、阻塞、终止",[13843,286092,286093,286096],{},[13858,286094,286095],{},"调度信息",[13858,286097,286098],{},"优先级、时间片、队列指针",[13843,286100,286101,286104],{},[13858,286102,286103],{},"CPU 现场",[13858,286105,286106],{},"程序计数器、通用寄存器、状态寄存器",[13843,286108,286109,286112],{},[13858,286110,286111],{},"内存管理信息",[13858,286113,286114],{},"页表、段表、基址、限长",[13843,286116,286117,286120],{},[13858,286118,286119],{},"资源信息",[13858,286121,286122],{},"打开文件、设备、信号量、I\u002FO 状态",[15,286124,286125],{},"PCB 是进程存在的标志。一个程序只有被操作系统创建 PCB、分配资源并纳入调度后，才成为进程。",[19,286127,286129],{"id":286128},"_5-为什么引入进程概念进程与程序的差别和关系","5. 为什么引入进程概念，进程与程序的差别和关系",[15,286131,286132],{},[2627,286133,285743],{},[104833,286135,286136],{},[15,286137,286138],{},"5、在操作系统中为什么要引入进程概念？它与程序的差别和关系是怎样的？",[15,286140,286141],{},[2627,286142,285753],{},[15,286144,286145],{},"引入进程的原因：",[2230,286147,286148,286151,286154,286157],{},[108,286149,286150],{},"描述程序的并发执行。",[108,286152,286153],{},"作为资源分配和保护的基本单位。",[108,286155,286156],{},"便于操作系统进行调度、同步、通信和控制。",[108,286158,286159],{},"反映多道程序环境中程序之间的制约关系。",[15,286161,286162],{},"程序和进程的区别：",[13837,286164,286165,286176],{},[13840,286166,286167],{},[13843,286168,286169,286171,286174],{},[13846,286170,285917],{},[13846,286172,286173],{},"程序",[13846,286175,247197],{},[13853,286177,286178,286189,286199,286210,286220],{},[13843,286179,286180,286183,286186],{},[13858,286181,286182],{},"性质",[13858,286184,286185],{},"静态",[13858,286187,286188],{},"动态",[13843,286190,286191,286193,286196],{},[13858,286192,178344],{},[13858,286194,286195],{},"通常存放在外存文件中",[13858,286197,286198],{},"运行时存在于内存和系统数据结构中",[13843,286200,286201,286204,286207],{},[13858,286202,286203],{},"生命周期",[13858,286205,286206],{},"可长期保存",[13858,286208,286209],{},"有创建、运行、阻塞、终止等过程",[13843,286211,286212,286214,286217],{},[13858,286213,257700],{},[13858,286215,286216],{},"本身不并发",[13858,286218,286219],{},"可与其他进程并发",[13843,286221,286222,286225,286228],{},[13858,286223,286224],{},"组成",[13858,286226,286227],{},"指令和数据",[13858,286229,286230],{},"程序、数据、PCB、资源和执行现场",[15,286232,285871],{},[2230,286234,286235,286238,286241,286244],{},[108,286236,286237],{},"程序是进程运行的基础。",[108,286239,286240],{},"进程是程序的一次执行过程。",[108,286242,286243],{},"同一个程序可以对应多个进程。",[108,286245,286246],{},"一个进程执行过程中也可能调用多个程序模块。",[15,286248,286249],{},"例子：磁盘上的浏览器可执行文件是程序；打开浏览器后，系统中运行的浏览器实例是进程。",[19,286251,286253],{"id":286252},"_6-信号量的操作","6. 信号量的操作",[15,286255,286256],{},[2627,286257,285743],{},[104833,286259,286260],{},[15,286261,286262],{},"6、信号量的操作有哪些？",[15,286264,286265],{},[2627,286266,285753],{},[15,286268,286269],{},"信号量是用于进程同步和互斥的整型变量，只能通过原子操作访问。主要操作有：",[2230,286271,286272,286275,286278],{},[108,286273,286274],{},"初始化：设置信号量初值。",[108,286276,286277],{},"P 操作，也称 wait、down、申请操作。",[108,286279,286280],{},"V 操作，也称 signal、up、释放操作。",[15,286282,286283],{},"P 操作：",[24,286285,286288],{"className":286286,"code":286287,"language":2638,"meta":29},[2636],"P(S):\n S = S - 1\n if S \u003C 0:\n 阻塞当前进程，并把它放入等待队列\n",[31,286289,286287],{"__ignoreMap":29},[15,286291,286292],{},"V 操作：",[24,286294,286297],{"className":286295,"code":286296,"language":2638,"meta":29},[2636],"V(S):\n S = S + 1\n if S \u003C= 0:\n 唤醒一个等待该信号量的进程\n",[31,286298,286296],{"__ignoreMap":29},[15,286300,286301],{},"若信号量用于互斥，初值一般为 1；若信号量用于资源计数，初值一般为资源数量。例如 3 台同类设备对应的资源信号量初值可设为 3。",[19,286303,286305],{"id":286304},"_7-临界区访问原则","7. 临界区访问原则",[15,286307,286308],{},[2627,286309,285743],{},[104833,286311,286312],{},[15,286313,286314],{},"7、试述临界区访问遵循的原则。",[15,286316,286317],{},[2627,286318,285753],{},[15,286320,286321],{},"临界区是访问临界资源的代码段。临界资源是一次只允许一个进程访问的资源。",[15,286323,286324],{},"按 Stallings\u002FSolutions-OS8e 和课程习题课件，互斥或临界区访问应满足六条要求：",[2230,286326,286327,286330,286333,286336,286339,286342],{},[108,286328,286329],{},"必须强制实施互斥：对同一资源或共享对象的临界区，同一时刻只允许一个进程进入。",[108,286331,286332],{},"停在非临界区的进程不能干涉其他进程。",[108,286334,286335],{},"需要进入临界区的进程不能被无限期推迟，即不能出现死锁或饥饿。",[108,286337,286338],{},"当没有进程在临界区内时，任何请求进入临界区的进程应能立即进入。",[108,286340,286341],{},"不能对进程相对执行速度或处理器数量作任何假设。",[108,286343,286344],{},"进程在临界区内停留的时间必须有限。",[15,286346,286347],{},"中文教材常把它概括为“空闲让进、忙则等待、有限等待、让权等待”。如果考试问“requirements for mutual exclusion”，优先写上面六条；如果问“四原则”，再写四原则。",[19,286349,286351],{"id":286350},"_8-死锁及四个必要条件","8. 死锁及四个必要条件",[15,286353,286354],{},[2627,286355,285743],{},[104833,286357,286358],{},[15,286359,286360],{},"8、什么是死锁？产生死锁的四个必要条件是什么？",[15,286362,286363],{},[2627,286364,285753],{},[15,286366,286367],{},"按 Stallings 教材，死锁是指一组进程中的每个进程都被阻塞，等待一个只能由该组中另一个被阻塞进程触发的事件；若无外力作用，这些进程将永久阻塞。",[15,286369,286370],{},"产生死锁的四个必要条件：",[2230,286372,286373,286376,286379,286382],{},[108,286374,286375],{},"互斥条件：一次只有一个进程可以使用某个资源。",[108,286377,286378],{},"占有并等待条件：进程已持有资源，同时还在等待分配其他资源。",[108,286380,286381],{},"不可剥夺条件：资源不能从持有它的进程中强行剥夺。",[108,286383,286384],{},"循环等待条件：存在一个闭合进程链，链中每个进程至少持有下一个进程所需的一个资源。",[15,286386,19553],{},[24,286388,286391],{"className":286389,"code":286390,"language":2638,"meta":29},[2636],"P1 持有 R1，等待 R2\nP2 持有 R2，等待 R1\n",[31,286392,286390],{"__ignoreMap":29},[15,286394,286395],{},"此时 P1、P2 相互等待，可能形成死锁。",[15,286397,286398],{},"处理死锁的思路：",[2230,286400,286401,286404,286407],{},[108,286402,286403],{},"预防死锁：破坏四个必要条件之一。",[108,286405,286406],{},"避免死锁：分配资源前判断是否会进入不安全状态，例如银行家算法。",[108,286408,286409],{},"检测并解除死锁：允许死锁发生，再检测等待环并撤销进程或剥夺资源。",[19,286411,286413],{"id":286412},"_9-为什么分页技术不便共享而分段技术可以","9. 为什么分页技术不便共享，而分段技术可以",[15,286415,286416],{},[2627,286417,285743],{},[104833,286419,286420],{},[15,286421,286422],{},"9、为什么分页技术不能实现共享，而分段技术可以？",[15,286424,286425],{},[2627,286426,285753],{},[15,286428,286429],{},"更严谨地说，分页技术也可以通过让多个进程的页表项指向同一物理页框来实现共享。但在教材常见语境中，分页“不便于按逻辑意义共享”，分段“天然适合共享”。",[15,286431,286432],{},"原因如下：",[2230,286434,286435,286438,286441,286444],{},[108,286436,286437],{},"页是固定大小的物理划分单位，不反映程序的逻辑结构。",[108,286439,286440],{},"一个函数、过程、数组或共享库可能跨多个页面，按页共享时管理和保护不方便。",[108,286442,286443],{},"段是按程序逻辑单位划分的，例如代码段、数据段、栈段、共享库段。",[108,286445,286446],{},"多个进程可让段表项指向同一个共享段，并设置读、写、执行权限。",[15,286448,159962],{},[13837,286450,286451,286460],{},[13840,286452,286453],{},[13843,286454,286455,286457],{},[13846,286456,233921],{},[13846,286458,286459],{},"共享特点",[13853,286461,286462,286469],{},[13843,286463,286464,286466],{},[13858,286465,257416],{},[13858,286467,286468],{},"以固定大小页面为单位，共享粒度机械，不便表达逻辑对象",[13843,286470,286471,286473],{},[13858,286472,257422],{},[13858,286474,286475],{},"以逻辑模块为单位，天然适合共享和保护",[15,286477,286478],{},"答题时可以写：分页主要面向内存离散分配，分段主要面向程序逻辑结构，因此分段更便于共享和保护。",[19,286480,286482],{"id":286481},"_10-页式和段式内存管理区别共享与保护","10. 页式和段式内存管理区别，共享与保护",[15,286484,286485],{},[2627,286486,285743],{},[104833,286488,286489],{},[15,286490,286491],{},"10、页式和段式内存管理有什么区别？怎样才能实现共享和保护？",[15,286493,286494],{},[2627,286495,285753],{},[15,286497,286498],{},"页式内存管理：",[2230,286500,286501,286504,286507,286510,286513,286516],{},[108,286502,286503],{},"逻辑地址空间划分为固定大小的页。",[108,286505,286506],{},"主存划分为同样大小的页框。",[108,286508,286509],{},"逻辑地址通常为“页号 + 页内偏移”。",[108,286511,286512],{},"通过页表把页号映射为页框号。",[108,286514,286515],{},"优点是无外部碎片，便于离散分配。",[108,286517,286518],{},"缺点是可能产生内部碎片，不体现程序逻辑结构。",[15,286520,286521],{},"段式内存管理：",[2230,286523,286524,286527,286530,286533,286536],{},[108,286525,286526],{},"地址空间按逻辑单位划分为长度可变的段。",[108,286528,286529],{},"逻辑地址通常为“段号 + 段内偏移”。",[108,286531,286532],{},"段表项包含段基址、段长和访问权限。",[108,286534,286535],{},"优点是便于共享、保护和动态链接。",[108,286537,286538],{},"缺点是可能产生外部碎片，分配和回收较复杂。",[15,286540,286541],{},"对比表：",[13837,286543,286544,286554],{},[13840,286545,286546],{},[13843,286547,286548,286550,286552],{},[13846,286549,285917],{},[13846,286551,205087],{},[13846,286553,205093],{},[13853,286555,286556,286567,286578,286589,286600],{},[13843,286557,286558,286561,286564],{},[13858,286559,286560],{},"划分依据",[13858,286562,286563],{},"固定大小",[13858,286565,286566],{},"程序逻辑单位",[13843,286568,286569,286572,286575],{},[13858,286570,286571],{},"大小",[13858,286573,286574],{},"页大小固定",[13858,286576,286577],{},"段长可变",[13843,286579,286580,286583,286586],{},[13858,286581,286582],{},"地址结构",[13858,286584,286585],{},"页号 + 页内偏移",[13858,286587,286588],{},"段号 + 段内偏移",[13843,286590,286591,286594,286597],{},[13858,286592,286593],{},"碎片",[13858,286595,286596],{},"有内部碎片，无外部碎片",[13858,286598,286599],{},"有外部碎片",[13843,286601,286602,286605,286608],{},[13858,286603,286604],{},"共享保护",[13858,286606,286607],{},"可实现但不直观",[13858,286609,286610],{},"更自然",[15,286612,286613],{},"共享和保护的实现：",[2230,286615,286616,286619],{},[108,286617,286618],{},"页式系统中，让多个进程的页表项指向同一页框，并在页表项设置读、写、执行权限位。",[108,286620,286621],{},"段式系统中，让多个进程的段表项指向同一共享段，并通过段长检查和权限位实现越界保护、读写保护和执行保护。",[15,286623,286624],{},"现代系统常结合两者思想，采用段页式管理或以分页为主、配合页表权限实现共享和保护。",{"title":29,"searchDepth":43,"depth":43,"links":286626},[286627,286628,286629,286630,286631,286632,286633,286634,286635,286636],{"id":285737,"depth":43,"text":285738},{"id":285851,"depth":43,"text":285852},{"id":285891,"depth":43,"text":285892},{"id":286023,"depth":43,"text":286024},{"id":286128,"depth":43,"text":286129},{"id":286252,"depth":43,"text":286253},{"id":286304,"depth":43,"text":286305},{"id":286350,"depth":43,"text":286351},{"id":286412,"depth":43,"text":286413},{"id":286481,"depth":43,"text":286482},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F01-1-10",{"title":285729,"description":29},"source\u002Fcs\u002Fos\u002Freview202505\u002F01-基本概念1-10","Rm9OZqYwqCv5AyE9HwOyqsE6BQqy-VsfLyaZruZoOP8",{"id":286643,"title":286644,"body":286645,"description":29,"extension":85,"meta":287551,"navigation":52,"path":287552,"seo":287553,"stem":287554,"__hash__":287555},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F02-基本概念11-20.md","review202505 答案讲解 02：基本概念 11-20",{"type":8,"value":286646,"toc":287539},[286647,286650,286654,286658,286663,286667,286670,286687,286690,286734,286737,286741,286745,286750,286754,286757,286760,286777,286780,286826,286829,286833,286837,286842,286846,286849,286852,286869,286872,286875,286924,286927,286931,286935,286940,286944,286947,286950,286953,286973,286976,286982,286986,286990,286995,286999,287002,287005,287025,287028,287031,287042,287046,287050,287055,287059,287062,287120,287123,287137,287141,287145,287150,287154,287157,287160,287242,287245,287253,287256,287260,287264,287269,287273,287276,287279,287282,287285,287291,287294,287298,287302,287307,287311,287314,287317,287331,287334,287340,287343,287346,287354,287358,287362,287367,287371,287374,287391,287394,287456,287459,287528,287531],[11,286648,286644],{"id":286649},"review202505-答案讲解-02基本概念-11-20",[19,286651,286653],{"id":286652},"_11-引入缓冲的目的和种类","11. 引入缓冲的目的和种类",[15,286655,286656],{},[2627,286657,285743],{},[104833,286659,286660],{},[15,286661,286662],{},"11、引入缓冲的目的，种类。",[15,286664,286665],{},[2627,286666,285753],{},[15,286668,286669],{},"按 CH 11 课件，缓冲的基本思想是：在请求发生之前预先执行输入传输，或在输出请求发生之后延后一段时间执行实际输出。引入缓冲的主要目的：",[2230,286671,286672,286675,286678,286681,286684],{},[108,286673,286674],{},"缓和 CPU 与 I\u002FO 设备速度不匹配。",[108,286676,286677],{},"减少 CPU 被中断的次数。",[108,286679,286680],{},"提高 CPU 与设备、设备与设备之间的并行程度。",[108,286682,286683],{},"平滑 I\u002FO 需求峰值，解决数据到达速度不稳定的问题。",[108,286685,286686],{},"支持记录到块、块到记录等数据格式转换。",[15,286688,286689],{},"Stallings CH 11 明确讲到的常见缓冲种类：",[13837,286691,286692,286700],{},[13840,286693,286694],{},[13843,286695,286696,286698],{},[13846,286697,114061],{},[13846,286699,111426],{},[13853,286701,286702,286710,286718,286726],{},[13843,286703,286704,286707],{},[13858,286705,286706],{},"单缓冲",[13858,286708,286709],{},"设置一个缓冲区，结构简单，但并行程度有限",[13843,286711,286712,286715],{},[13858,286713,286714],{},"双缓冲",[13858,286716,286717],{},"设置两个缓冲区，一个输入或输出，一个处理，提高并行度",[13843,286719,286720,286723],{},[13858,286721,286722],{},"循环缓冲",[13858,286724,286725],{},"多个缓冲区首尾相接，适合连续数据流",[13843,286727,286728,286731],{},[13858,286729,286730],{},"多缓冲或缓冲池",[13858,286732,286733],{},"多个缓冲区统一管理，适合更复杂或更高峰值的 I\u002FO 需求",[15,286735,286736],{},"核心理解：缓冲的作用是“解耦”。输入设备可以先把数据放入缓冲区，CPU 不必一直等待；输出时 CPU 把数据放入缓冲区后即可继续执行，设备慢慢输出。",[19,286738,286740],{"id":286739},"_12-文件的物理结构及组织方式","12. 文件的物理结构及组织方式",[15,286742,286743],{},[2627,286744,285743],{},[104833,286746,286747],{},[15,286748,286749],{},"12、什么是文件的物理结构？有哪几种组织方式？",[15,286751,286752],{},[2627,286753,285753],{},[15,286755,286756],{},"文件的物理结构是文件在外存上的存放组织方式，也称文件的存储结构或文件分配方式。它回答的问题是：文件的数据块在磁盘上怎样存放。",[15,286758,286759],{},"常见组织方式：",[2230,286761,286762,286765,286768,286771,286774],{},[108,286763,286764],{},"连续分配：文件占用一组连续磁盘块。",[108,286766,286767],{},"链接分配：文件块可分散存放，每块保存下一块指针。",[108,286769,286770],{},"索引分配：为文件建立索引块，索引块中保存文件各数据块地址。",[108,286772,286773],{},"混合索引或多级索引：例如 UNIX inode 中的直接索引、一级间接、二级间接、三级间接。",[108,286775,286776],{},"扩展区分配：用一段连续块作为一个 extent，一个文件由一个或多个 extent 组成。",[15,286778,286779],{},"对比：",[13837,286781,286782,286792],{},[13840,286783,286784],{},[13843,286785,286786,286788,286790],{},[13846,286787,234395],{},[13846,286789,204919],{},[13846,286791,204938],{},[13853,286793,286794,286804,286815],{},[13843,286795,286796,286798,286801],{},[13858,286797,247405],{},[13858,286799,286800],{},"顺序访问快，随机访问也方便",[13858,286802,286803],{},"文件增长困难，容易产生外部碎片",[13843,286805,286806,286809,286812],{},[13858,286807,286808],{},"链接分配",[13858,286810,286811],{},"无外部碎片，文件容易增长",[13858,286813,286814],{},"随机访问差，指针有额外开销",[13843,286816,286817,286820,286823],{},[13858,286818,286819],{},"索引分配",[13858,286821,286822],{},"随机访问方便，文件可离散存放",[13858,286824,286825],{},"索引块占空间，小文件可能浪费",[15,286827,286828],{},"考试中若问“哪几种组织方式”，一般答“连续、链接、索引”三类即可，再补充多级索引更完整。",[19,286830,286832],{"id":286831},"_13-文件的逻辑结构及组织方式","13. 文件的逻辑结构及组织方式",[15,286834,286835],{},[2627,286836,285743],{},[104833,286838,286839],{},[15,286840,286841],{},"13、什么是文件的逻辑结构？有哪几种组织方式？",[15,286843,286844],{},[2627,286845,285753],{},[15,286847,286848],{},"文件的逻辑结构是从用户或应用程序角度看到的文件组织形式，即文件内部记录或字节之间的逻辑关系。它回答的问题是：应用认为文件由什么组成、怎样查找和访问。",[15,286850,286851],{},"按 Stallings 第 12 章，常见文件组织方式包括：",[2230,286853,286854,286857,286860,286863,286866],{},[108,286855,286856],{},"堆文件或堆式文件：数据按到达顺序收集，每条记录是一组数据。",[108,286858,286859],{},"顺序文件：使用固定记录格式，记录通常按关键字段顺序组织。",[108,286861,286862],{},"索引顺序文件：保持顺序文件按关键字段组织的特点，同时增加索引和溢出文件以支持较快查找。",[108,286864,286865],{},"索引文件：使用索引结构支持按关键字快速访问，可支持变长记录和多个索引。",[108,286867,286868],{},"直接文件或散列文件：根据关键字通过散列函数直接定位记录。",[15,286870,286871],{},"中文操作系统教材还常把文件逻辑结构概括为“无结构字节流文件”和“记录式文件”。若题目问“逻辑结构”，可以先写这两大类；若按 Stallings 的文件组织章节，应补充上面的堆、顺序、索引顺序、索引和直接文件。",[15,286873,286874],{},"逻辑结构和物理结构区别：",[13837,286876,286877,286889],{},[13840,286878,286879],{},[13843,286880,286881,286883,286886],{},[13846,286882,285917],{},[13846,286884,286885],{},"逻辑结构",[13846,286887,286888],{},"物理结构",[13853,286890,286891,286902,286913],{},[13843,286892,286893,286896,286899],{},[13858,286894,286895],{},"观察角度",[13858,286897,286898],{},"用户或程序员角度",[13858,286900,286901],{},"操作系统和磁盘存储角度",[13843,286903,286904,286907,286910],{},[13858,286905,286906],{},"关注问题",[13858,286908,286909],{},"文件内部信息如何组织",[13858,286911,286912],{},"文件块在外存上如何存放",[13843,286914,286915,286918,286921],{},[13858,286916,286917],{},"典型形式",[13858,286919,286920],{},"字节流、记录、堆文件、顺序文件、索引顺序文件、索引文件、直接文件",[13858,286922,286923],{},"连续、链接、索引分配",[15,286925,286926],{},"同一个逻辑文件可以采用不同物理结构存储。例如一个记录文件在磁盘上可以用连续分配，也可以用索引分配。",[19,286928,286930],{"id":286929},"_14-文件和文件系统","14. 文件和文件系统",[15,286932,286933],{},[2627,286934,285743],{},[104833,286936,286937],{},[15,286938,286939],{},"14、什么是文件和文件系统？",[15,286941,286942],{},[2627,286943,285753],{},[15,286945,286946],{},"按 Stallings 第 12 章，文件是相似记录的集合，被用户和应用程序当作一个单一实体处理，并且可以通过名字引用。更一般地说，文件是具有文件名的一组相关信息的集合，是操作系统管理外存数据的基本单位。",[15,286948,286949],{},"文件系统或文件管理系统是向用户和应用程序提供文件使用服务的一组系统软件和数据结构。它向用户提供按文件名访问数据的能力，并负责底层磁盘空间管理。",[15,286951,286952],{},"文件系统的主要功能：",[2230,286954,286955,286958,286961,286964,286967,286970],{},[108,286956,286957],{},"文件创建、删除、打开、关闭、读写。",[108,286959,286960],{},"目录管理和路径名解析。",[108,286962,286963],{},"磁盘空间分配与回收。",[108,286965,286966],{},"文件共享和访问控制。",[108,286968,286969],{},"文件系统一致性维护和故障恢复。",[108,286971,286972],{},"文件属性管理，例如大小、权限、时间戳、所有者。",[15,286974,286975],{},"可这样理解：",[24,286977,286980],{"className":286978,"code":286979,"language":2638,"meta":29},[2636],"文件：用户看到的数据单位\n文件系统：管理这些文件、目录和磁盘空间的操作系统模块\n",[31,286981,286979],{"__ignoreMap":29},[19,286983,286985],{"id":286984},"_15-设备独立性及实现","15. 设备独立性及实现",[15,286987,286988],{},[2627,286989,285743],{},[104833,286991,286992],{},[15,286993,286994],{},"15、什么是设备独立性，它是如何实现的？",[15,286996,286997],{},[2627,286998,285753],{},[15,287000,287001],{},"设备独立性是指用户程序使用设备时，不必关心具体物理设备的型号、位置、厂商和驱动细节，只需通过统一的逻辑设备名或统一接口进行 I\u002FO。Stallings 在 I\u002FO 设计目标中把这一思想称为 generality，即希望以统一方式处理各种 I\u002FO 设备。",[15,287003,287004],{},"实现方法：",[2230,287006,287007,287010,287013,287016,287019,287022],{},[108,287008,287009],{},"逻辑设备名与物理设备名分离。",[108,287011,287012],{},"使用设备驱动程序屏蔽硬件差异。",[108,287014,287015],{},"操作系统提供统一 I\u002FO 接口和系统调用。",[108,287017,287018],{},"通过设备表维护逻辑设备到物理设备的映射。",[108,287020,287021],{},"通过逻辑 I\u002FO 层把设备看作逻辑资源，使用户进程只面对设备标识符和 open、close、read、write 等简单命令。",[108,287023,287024],{},"通过缓冲、缓存、SPOOLing 等技术改善设备使用方式。",[15,287026,287027],{},"例子：应用程序只需向“打印机”输出，不需要知道打印机型号、连接端口和控制命令。更换打印机时，只要驱动和逻辑映射正确，应用程序通常不用修改。",[15,287029,287030],{},"设备独立性的意义：",[2230,287032,287033,287036,287039],{},[108,287034,287035],{},"提高程序可移植性。",[108,287037,287038],{},"简化用户程序设计。",[108,287040,287041],{},"便于设备分配、替换和管理。",[19,287043,287045],{"id":287044},"_16-抖动地址重定位碎片逻辑地址和物理地址","16. 抖动、地址重定位、碎片、逻辑地址和物理地址",[15,287047,287048],{},[2627,287049,285743],{},[104833,287051,287052],{},[15,287053,287054],{},"16、抖动、地址重定位、内部碎片、外部碎片、逻辑地址、物理地址",[15,287056,287057],{},[2627,287058,285753],{},[15,287060,287061],{},"这些是存储管理中的基础概念。",[13837,287063,287064,287072],{},[13840,287065,287066],{},[13843,287067,287068,287070],{},[13846,287069,31577],{},[13846,287071,126615],{},[13853,287073,287074,287082,287090,287097,287104,287112],{},[13843,287075,287076,287079],{},[13858,287077,287078],{},"抖动",[13858,287080,287081],{},"系统频繁发生页面调入调出，大量时间用于缺页和磁盘 I\u002FO，真正执行用户程序的时间很少",[13843,287083,287084,287087],{},[13858,287085,287086],{},"地址重定位",[13858,287088,287089],{},"把程序中的逻辑地址转换为实际物理地址的过程",[13843,287091,287092,287094],{},[13858,287093,205063],{},[13858,287095,287096],{},"已分配给进程但没有被实际使用的空间",[13843,287098,287099,287101],{},[13858,287100,205067],{},[13858,287102,287103],{},"空闲空间总量足够，但被分散成许多小块，无法满足较大连续空间请求",[13843,287105,287106,287109],{},[13858,287107,287108],{},"逻辑地址",[13858,287110,287111],{},"程序运行时产生的地址，也称虚拟地址或相对地址",[13843,287113,287114,287117],{},[13858,287115,287116],{},"物理地址",[13858,287118,287119],{},"主存中的真实地址，由地址转换机构转换得到",[15,287121,287122],{},"补充理解：",[2230,287124,287125,287128,287131,287134],{},[108,287126,287127],{},"分页主要消除外部碎片，但可能产生内部碎片。",[108,287129,287130],{},"分段符合逻辑结构，但可能产生外部碎片。",[108,287132,287133],{},"虚拟存储系统中，逻辑地址到物理地址的转换通常由 MMU 硬件和操作系统共同完成。",[108,287135,287136],{},"抖动常由内存不足、进程过多或页面置换不合理导致。",[19,287138,287140],{"id":287139},"_17-inode-文件共享与符号链接文件共享","17. inode 文件共享与符号链接文件共享",[15,287142,287143],{},[2627,287144,285743],{},[104833,287146,287147],{},[15,287148,287149],{},"17、比较基于索引节点和基于符号链的文件共享方法",[15,287151,287152],{},[2627,287153,285753],{},[15,287155,287156],{},"基于索引节点的共享通常指硬链接。多个目录项指向同一个 inode，文件数据和 inode 只有一份。",[15,287158,287159],{},"基于符号链的共享指创建一个特殊文件，该文件内容保存被链接文件的路径名，也就是符号链接或软链接。",[13837,287161,287162,287174],{},[13840,287163,287164],{},[13843,287165,287166,287168,287171],{},[13846,287167,285917],{},[13846,287169,287170],{},"基于 inode 的硬链接",[13846,287172,287173],{},"符号链接",[13853,287175,287176,287187,287198,287209,287220,287231],{},[13843,287177,287178,287181,287184],{},[13858,287179,287180],{},"本质",[13858,287182,287183],{},"多个目录项指向同一个 inode",[13858,287185,287186],{},"一个独立文件，内容是目标路径",[13843,287188,287189,287192,287195],{},[13858,287190,287191],{},"是否增加原文件链接计数",[13858,287193,287194],{},"增加",[13858,287196,287197],{},"不增加",[13843,287199,287200,287203,287206],{},[13858,287201,287202],{},"是否可跨文件系统",[13858,287204,287205],{},"通常不可以",[13858,287207,287208],{},"可以",[13843,287210,287211,287214,287217],{},[13858,287212,287213],{},"是否可链接目录",[13858,287215,287216],{},"通常限制较多",[13858,287218,287219],{},"可以链接目录",[13843,287221,287222,287225,287228],{},[13858,287223,287224],{},"删除原文件名后的影响",[13858,287226,287227],{},"只要还有硬链接，文件数据仍存在",[13858,287229,287230],{},"符号链接可能变成悬空链接",[13843,287232,287233,287236,287239],{},[13858,287234,287235],{},"访问开销",[13858,287237,287238],{},"较小，直接定位 inode",[13858,287240,287241],{},"需要解析路径，开销较大",[15,287243,287244],{},"记忆方式：",[2230,287246,287247,287250],{},[108,287248,287249],{},"硬链接像“给同一个文件实体起多个名字”。",[108,287251,287252],{},"符号链接像“快捷方式”，指向目标路径。",[15,287254,287255],{},"如果删除原文件名，硬链接仍能访问原数据；符号链接可能因为路径目标不存在而失效。",[19,287257,287259],{"id":287258},"_18-局域网打印机是否是共享设备","18. 局域网打印机是否是共享设备",[15,287261,287262],{},[2627,287263,285743],{},[104833,287265,287266],{},[15,287267,287268],{},"18、办公室只有一台打印机通过局域网为办公室人员提供打印服务。该打印机是共享设备的观点是否正确？为什么？",[15,287270,287271],{},[2627,287272,285753],{},[15,287274,287275],{},"这个说法需要区分“物理设备性质”和“用户使用效果”。",[15,287277,287278],{},"从物理设备分类看，打印机不是严格意义上的共享设备，而是独占设备。因为打印机在某一时刻只能打印一个作业，不能让多个作业真正同时使用。",[15,287280,287281],{},"从用户使用效果看，通过局域网打印服务和 SPOOLing 技术，它可以表现为共享设备。多个用户可以同时提交打印任务，系统先把任务放入打印队列，再由打印机逐个输出。",[15,287283,287284],{},"所以更准确的答案是：",[24,287286,287289],{"className":287287,"code":287288,"language":2638,"meta":29},[2636],"打印机物理上是独占设备，逻辑上可通过 SPOOLing 技术被多个用户共享。\n",[31,287290,287288],{"__ignoreMap":29},[15,287292,287293],{},"答题时不要只写“正确”或“不正确”，应说明原因。若题目强调“打印机是共享设备”这个观点，直接说它是共享设备并不严谨。",[19,287295,287297],{"id":287296},"_19-快表及其在地址转换中的作用","19. 快表及其在地址转换中的作用",[15,287299,287300],{},[2627,287301,285743],{},[104833,287303,287304],{},[15,287305,287306],{},"19、什么是快表？它在地址转换中起什么作用？",[15,287308,287309],{},[2627,287310,285753],{},[15,287312,287313],{},"快表是页表项的高速缓存，英文常称 TLB，即 Translation Lookaside Buffer。Solutions-OS8e 将其解释为保存最近使用过的页表项的 cache。",[15,287315,287316],{},"作用：",[2230,287318,287319,287322,287325,287328],{},[108,287320,287321],{},"缓存最近使用过的页表项。",[108,287323,287324],{},"地址转换时先查快表，若命中则直接得到页框号。",[108,287326,287327],{},"若快表未命中，再访问内存中的页表，并把新页表项装入快表。",[108,287329,287330],{},"减少访问页表带来的内存访问次数，提高地址转换速度。",[15,287332,287333],{},"普通分页系统中，访问一个数据通常需要：",[24,287335,287338],{"className":287336,"code":287337,"language":2638,"meta":29},[2636],"访问页表一次 + 访问数据一次\n",[31,287339,287337],{"__ignoreMap":29},[15,287341,287342],{},"如果快表命中，则页表查询由高速硬件完成，实际只需访问数据所在内存位置，速度明显提高。",[15,287344,287345],{},"快表能有效工作的原因是程序访问具有局部性：",[2230,287347,287348,287351],{},[108,287349,287350],{},"时间局部性：最近访问过的页可能很快再次访问。",[108,287352,287353],{},"空间局部性：相邻地址可能落在同一页或相邻页中。",[19,287355,287357],{"id":287356},"_20-进程状态状态转换原因和相关原语","20. 进程状态、状态转换原因和相关原语",[15,287359,287360],{},[2627,287361,285743],{},[104833,287363,287364],{},[15,287365,287366],{},"20、简述进程的几种状态和引起状态转换的典型原因，以及相关的操作原语。",[15,287368,287369],{},[2627,287370,285753],{},[15,287372,287373],{},"常见五状态模型：",[2230,287375,287376,287379,287382,287385,287388],{},[108,287377,287378],{},"新建态：进程正在被创建。",[108,287380,287381],{},"就绪态：进程已具备运行条件，等待 CPU。",[108,287383,287384],{},"运行态：进程正在 CPU 上执行。",[108,287386,287387],{},"阻塞态：进程等待某事件，暂时不能运行。",[108,287389,287390],{},"终止态：进程运行结束或被撤销。",[15,287392,287393],{},"典型状态转换：",[13837,287395,287396,287406],{},[13840,287397,287398],{},[13843,287399,287400,287403],{},[13846,287401,287402],{},"转换",[13846,287404,287405],{},"典型原因",[13853,287407,287408,287416,287424,287432,287440,287448],{},[13843,287409,287410,287413],{},[13858,287411,287412],{},"新建态 -> 就绪态",[13858,287414,287415],{},"创建完成，被接纳进入就绪队列",[13843,287417,287418,287421],{},[13858,287419,287420],{},"就绪态 -> 运行态",[13858,287422,287423],{},"调度程序分配 CPU",[13843,287425,287426,287429],{},[13858,287427,287428],{},"运行态 -> 就绪态",[13858,287430,287431],{},"时间片用完、更高优先级进程到来、主动让出 CPU",[13843,287433,287434,287437],{},[13858,287435,287436],{},"运行态 -> 阻塞态",[13858,287438,287439],{},"等待 I\u002FO、等待信号量、等待事件",[13843,287441,287442,287445],{},[13858,287443,287444],{},"阻塞态 -> 就绪态",[13858,287446,287447],{},"I\u002FO 完成、资源可用、事件发生",[13843,287449,287450,287453],{},[13858,287451,287452],{},"运行态 -> 终止态",[13858,287454,287455],{},"正常结束或异常终止",[15,287457,287458],{},"相关操作原语：",[13837,287460,287461,287470],{},[13840,287462,287463],{},[13843,287464,287465,287468],{},[13846,287466,287467],{},"原语",[13846,287469,153527],{},[13853,287471,287472,287480,287488,287496,287504,287512,287520],{},[13843,287473,287474,287477],{},[13858,287475,287476],{},"创建原语",[13858,287478,287479],{},"创建进程和 PCB",[13843,287481,287482,287485],{},[13858,287483,287484],{},"撤销原语",[13858,287486,287487],{},"终止进程并回收资源",[13843,287489,287490,287493],{},[13858,287491,287492],{},"阻塞原语",[13858,287494,287495],{},"使运行进程进入阻塞态",[13843,287497,287498,287501],{},[13858,287499,287500],{},"唤醒原语",[13858,287502,287503],{},"把等待事件完成的进程转为就绪态",[13843,287505,287506,287509],{},[13858,287507,287508],{},"挂起原语",[13858,287510,287511],{},"把进程移出活动队列或换出内存",[13843,287513,287514,287517],{},[13858,287515,287516],{},"激活原语",[13858,287518,287519],{},"恢复挂起进程，使其重新参与调度",[13843,287521,287522,287525],{},[13858,287523,287524],{},"调度或切换原语",[13858,287526,287527],{},"保存当前进程现场，恢复下一个进程现场",[15,287529,287530],{},"关键区别：",[2230,287532,287533,287536],{},[108,287534,287535],{},"就绪态：没有 CPU，但只要获得 CPU 就能运行。",[108,287537,287538],{},"阻塞态：即使给它 CPU 也不能运行，因为它还在等待某事件。",{"title":29,"searchDepth":43,"depth":43,"links":287540},[287541,287542,287543,287544,287545,287546,287547,287548,287549,287550],{"id":286652,"depth":43,"text":286653},{"id":286739,"depth":43,"text":286740},{"id":286831,"depth":43,"text":286832},{"id":286929,"depth":43,"text":286930},{"id":286984,"depth":43,"text":286985},{"id":287044,"depth":43,"text":287045},{"id":287139,"depth":43,"text":287140},{"id":287258,"depth":43,"text":287259},{"id":287296,"depth":43,"text":287297},{"id":287356,"depth":43,"text":287357},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F02-11-20",{"title":286644,"description":29},"source\u002Fcs\u002Fos\u002Freview202505\u002F02-基本概念11-20","3EhohPpyaBXdJscJIC0i26iFCRpgKkygHhRv_MAhZhg",{"id":287557,"title":287558,"body":287559,"description":29,"extension":85,"meta":289999,"navigation":52,"path":290000,"seo":290001,"stem":290002,"__hash__":290003},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F03-计算题1-10.md","review202505 答案讲解 03：计算题 1-10",{"type":8,"value":287560,"toc":289971},[287561,287564,287568,287572,287577,287581,287584,287685,287688,287694,287697,287703,287706,287712,287714,287731,287735,287739,287744,287748,287751,287754,287762,287768,287779,287787,287793,287799,287805,287808,287814,287820,287826,287835,287838,287842,287846,287851,287855,287857,287863,287866,287872,287875,287881,287884,287890,287892,287898,287909,287913,287917,287931,287935,287938,287944,287947,287953,287957,287960,287966,287969,287975,287978,287984,287987,287993,288005,288009,288012,288018,288021,288027,288035,288048,288052,288055,288061,288068,288071,288077,288080,288086,288092,288095,288098,288106,288113,288117,288121,288132,288136,288139,288145,288148,288154,288158,288164,288167,288173,288176,288179,288185,288188,288194,288201,288204,288207,288213,288219,288222,288225,288231,288234,288299,288308,288318,288322,288326,288336,288455,288459,288462,288465,288562,288565,288690,288692,288698,288700,288706,288718,288722,288726,288731,288865,288878,288882,288885,288891,288894,288900,288903,288909,288915,288997,289001,289004,289010,289013,289085,289093,289101,289103,289109,289112,289118,289121,289127,289130,289196,289203,289210,289213,289219,289222,289228,289231,289237,289242,289246,289250,289255,289259,289268,289274,289277,289283,289286,289289,289295,289298,289304,289307,289310,289316,289319,289325,289328,289334,289352,289356,289360,289365,289369,289372,289378,289381,289405,289408,289414,289416,289422,289434,289444,289448,289452,289460,289464,289467,289473,289476,289480,289483,289679,289686,289689,289695,289698,289704,289708,289711,289899,289902,289908,289911,289917,289922],[11,287562,287558],{"id":287563},"review202505-答案讲解-03计算题-1-10",[19,287565,287567],{"id":287566},"_1-两级生产者消费者同步","1. 两级生产者消费者同步",[15,287569,287570],{},[2627,287571,285743],{},[104833,287573,287574],{},[15,287575,287576],{},"1、设有三个进程A、B、C，其中A与B构成一对生产者与消费者（A为生产者，B为消费者），共享一个由n个缓冲块组成的缓冲池；B与C也构成一对生产者与消费者（此时B为生产者，C为消费者），共享另一个由m个缓冲块组成的缓冲池。用P、V操作描述它们之间的同步关系。",[15,287578,287579],{},[2627,287580,285753],{},[15,287582,287583],{},"设置 6 个信号量：",[13837,287585,287586,287599],{},[13840,287587,287588],{},[13843,287589,287590,287593,287597],{},[13846,287591,287592],{},"信号量",[13846,287594,287596],{"align":287595},"right","初值",[13846,287598,126615],{},[13853,287600,287601,287615,287629,287643,287657,287671],{},[13843,287602,287603,287608,287612],{},[13858,287604,287605],{},[31,287606,287607],{},"empty1",[13858,287609,287610],{"align":287595},[31,287611,31660],{},[13858,287613,287614],{},"缓冲池 1 中空缓冲块数",[13843,287616,287617,287622,287626],{},[13858,287618,287619],{},[31,287620,287621],{},"full1",[13858,287623,287624],{"align":287595},[31,287625,459],{},[13858,287627,287628],{},"缓冲池 1 中满缓冲块数",[13843,287630,287631,287636,287640],{},[13858,287632,287633],{},[31,287634,287635],{},"mutex1",[13858,287637,287638],{"align":287595},[31,287639,640],{},[13858,287641,287642],{},"缓冲池 1 的互斥访问",[13843,287644,287645,287650,287654],{},[13858,287646,287647],{},[31,287648,287649],{},"empty2",[13858,287651,287652],{"align":287595},[31,287653,31759],{},[13858,287655,287656],{},"缓冲池 2 中空缓冲块数",[13843,287658,287659,287664,287668],{},[13858,287660,287661],{},[31,287662,287663],{},"full2",[13858,287665,287666],{"align":287595},[31,287667,459],{},[13858,287669,287670],{},"缓冲池 2 中满缓冲块数",[13843,287672,287673,287678,287682],{},[13858,287674,287675],{},[31,287676,287677],{},"mutex2",[13858,287679,287680],{"align":287595},[31,287681,640],{},[13858,287683,287684],{},"缓冲池 2 的互斥访问",[15,287686,287687],{},"A 进程：",[24,287689,287692],{"className":287690,"code":287691,"language":2638,"meta":29},[2636],"while true:\n 生产一个产品 x\n P(empty1)\n P(mutex1)\n 将 x 放入缓冲池 1\n V(mutex1)\n V(full1)\n",[31,287693,287691],{"__ignoreMap":29},[15,287695,287696],{},"B 进程：",[24,287698,287701],{"className":287699,"code":287700,"language":2638,"meta":29},[2636],"while true:\n P(full1)\n P(mutex1)\n 从缓冲池 1 取出产品 x\n V(mutex1)\n V(empty1)\n\n 对 x 进行处理，得到 y\n\n P(empty2)\n P(mutex2)\n 将 y 放入缓冲池 2\n V(mutex2)\n V(full2)\n",[31,287702,287700],{"__ignoreMap":29},[15,287704,287705],{},"C 进程：",[24,287707,287710],{"className":287708,"code":287709,"language":2638,"meta":29},[2636],"while true:\n P(full2)\n P(mutex2)\n 从缓冲池 2 取出产品 y\n V(mutex2)\n V(empty2)\n 消费 y\n",[31,287711,287709],{"__ignoreMap":29},[15,287713,22058],{},[2230,287715,287716,287721,287726],{},[108,287717,287718,287720],{},[31,287719,198187],{}," 用于防止生产者向满缓冲池继续放入数据。",[108,287722,287723,287725],{},[31,287724,257323],{}," 用于防止消费者从空缓冲池取数据。",[108,287727,287728,287730],{},[31,287729,196586],{}," 用于保证同一个缓冲池一次只被一个进程访问。",[19,287732,287734],{"id":287733},"_2-证明-sjf-平均周转时间最小","2. 证明 SJF 平均周转时间最小",[15,287736,287737],{},[2627,287738,285743],{},[104833,287740,287741],{},[15,287742,287743],{},"2、若后备作业队列中等待运行的同时有4个作业J1、J2, …, J4，已知它们各自的运行为t1,t2,…, t4，且满足t1\u003Ct2\u003C…\u003Ct4。试证明采用短作业优先算法调度能获得最小平均周转时间",[15,287745,287746],{},[2627,287747,285753],{},[15,287749,287750],{},"由于 4 个作业同时等待运行，所以周转时间等于完成时间。平均周转时间最小等价于总周转时间最小。",[15,287752,287753],{},"用相邻交换法证明：",[15,287755,287756,287757,146,287759,287761],{},"假设某个调度序列中有两个相邻作业 ",[31,287758,31955],{},[31,287760,31968],{},"，并且：",[24,287763,287766],{"className":287764,"code":287765,"language":2638,"meta":29},[2636],"ti > tj\n",[31,287767,287765],{"__ignoreMap":29},[15,287769,287770,287771,68853,287773,287775,287776,287778],{},"但调度顺序是先 ",[31,287772,31955],{},[31,287774,31968],{},"。设这两个作业开始前已经运行了 ",[31,287777,37400],{}," 时间。",[15,287780,287781,287782,68853,287784,287786],{},"先 ",[31,287783,31955],{},[31,287785,31968],{}," 时，两者完成时间之和为：",[24,287788,287791],{"className":287789,"code":287790,"language":2638,"meta":29},[2636],"(T + ti) + (T + ti + tj)\n= 2T + 2ti + tj\n",[31,287792,287790],{"__ignoreMap":29},[15,287794,287781,287795,68853,287797,287786],{},[31,287796,31968],{},[31,287798,31955],{},[24,287800,287803],{"className":287801,"code":287802,"language":2638,"meta":29},[2636],"(T + tj) + (T + tj + ti)\n= 2T + 2tj + ti\n",[31,287804,287802],{"__ignoreMap":29},[15,287806,287807],{},"两种顺序相减：",[24,287809,287812],{"className":287810,"code":287811,"language":2638,"meta":29},[2636],"(2T + 2ti + tj) - (2T + 2tj + ti)\n= ti - tj\n",[31,287813,287811],{"__ignoreMap":29},[15,287815,93310,287816,287819],{},[31,287817,287818],{},"ti > tj","，所以：",[24,287821,287824],{"className":287822,"code":287823,"language":2638,"meta":29},[2636],"ti - tj > 0\n",[31,287825,287823],{"__ignoreMap":29},[15,287827,287828,287829,287831,287832,287834],{},"说明把短作业 ",[31,287830,31968],{}," 放到长作业 ",[31,287833,31955],{}," 前面，会使总完成时间减小。不断交换所有“长作业在短作业前面”的逆序对，最后得到按运行时间从短到长排列的顺序，也就是短作业优先 SJF。",[15,287836,287837],{},"因此 SJF 能获得最小平均周转时间。",[19,287839,287841],{"id":287840},"_3-inode-地址项计算最大文件长度","3. inode 地址项计算最大文件长度",[15,287843,287844],{},[2627,287845,285743],{},[104833,287847,287848],{},[15,287849,287850],{},"3、设文件索引节点中有7个地址项，其中4个为直接地址索引，2个是一级间接地址索引，1个是二级间接地址索引，地址项大小为4B，若磁盘索引块和磁盘数据块大小均为1KB，求可表示的单个文件的最大长度。",[15,287852,287853],{},[2627,287854,285753],{},[15,287856,173129],{},[24,287858,287861],{"className":287859,"code":287860,"language":2638,"meta":29},[2636],"索引块大小 = 1KB = 1024B\n地址项大小 = 4B\n",[31,287862,287860],{"__ignoreMap":29},[15,287864,287865],{},"每个索引块可存放地址项数：",[24,287867,287870],{"className":287868,"code":287869,"language":2638,"meta":29},[2636],"1024 \u002F 4 = 256\n",[31,287871,287869],{"__ignoreMap":29},[15,287873,287874],{},"可寻址的数据块数：",[24,287876,287879],{"className":287877,"code":287878,"language":2638,"meta":29},[2636],"直接地址：4 块\n一级间接：2 × 256 = 512 块\n二级间接：1 × 256 × 256 = 65536 块\n总块数：4 + 512 + 65536 = 66052 块\n",[31,287880,287878],{"__ignoreMap":29},[15,287882,287883],{},"每个数据块大小为 1KB，所以最大文件长度为：",[24,287885,287888],{"className":287886,"code":287887,"language":2638,"meta":29},[2636],"66052 × 1KB = 66052KB\n",[31,287889,287887],{"__ignoreMap":29},[15,287891,186458],{},[24,287893,287896],{"className":287894,"code":287895,"language":2638,"meta":29},[2636],"66052 × 1024 = 67,637,248B\n",[31,287897,287895],{"__ignoreMap":29},[15,287899,287900,287902,287903,61605,287906,347],{},[2627,287901,58173],{}," 单个文件最大长度为 ",[31,287904,287905],{},"66052KB",[31,287907,287908],{},"67,637,248B",[19,287910,287912],{"id":287911},"_4-unix-inode-最大文件最大分区和磁盘访问次数","4. UNIX inode 最大文件、最大分区和磁盘访问次数",[15,287914,287915],{},[2627,287916,285743],{},[104833,287918,287919,287922,287925,287928],{},[15,287920,287921],{},"4、考虑由索引节点表示的UNIX文件的组织。在每个节点中，假定有12个直接块指针，以及一个一级、二级和三级间接指针。此外，假定系统磁盘块的大小为8KB。如果盘块指针用32位表示，其中8位用于标识物理磁盘号，24位用于标识磁盘块号，那么：",[15,287923,287924],{},"(1)该系统支持的最大文件大小是多少",[15,287926,287927],{},"(2)该系统支持的最大文件系统分区是多少",[15,287929,287930],{},"(3)假定主存中除了文件索引节点外没有别的信息，访问在位置12423956的字节时，需要访问磁盘多少次",[15,287932,287933],{},[2627,287934,285753],{},[15,287936,287937],{},"每个块指针大小：",[24,287939,287942],{"className":287940,"code":287941,"language":2638,"meta":29},[2636],"32 位 = 4B\n",[31,287943,287941],{"__ignoreMap":29},[15,287945,287946],{},"每个索引块可容纳指针数：",[24,287948,287951],{"className":287949,"code":287950,"language":2638,"meta":29},[2636],"8KB \u002F 4B = 8192 \u002F 4 = 2048\n",[31,287952,287950],{"__ignoreMap":29},[253,287954,287956],{"id":287955},"_1-最大文件大小","(1) 最大文件大小",[15,287958,287959],{},"可寻址数据块数：",[24,287961,287964],{"className":287962,"code":287963,"language":2638,"meta":29},[2636],"直接块：12\n一级间接：2048\n二级间接：2048^2\n三级间接：2048^3\n",[31,287965,287963],{"__ignoreMap":29},[15,287967,287968],{},"总数据块数：",[24,287970,287973],{"className":287971,"code":287972,"language":2638,"meta":29},[2636],"12 + 2048 + 2048^2 + 2048^3\n= 8,594,130,956 块\n",[31,287974,287972],{"__ignoreMap":29},[15,287976,287977],{},"按 Solutions-OS8e 12.13 的表达，可以把最大文件大小分项写为：",[24,287979,287982],{"className":287980,"code":287981,"language":2638,"meta":29},[2636],"直接块：12 × 8KB = 96KB\n一级间接：2048 × 8KB = 16MB\n二级间接：2048^2 × 8KB = 32GB\n三级间接：2048^3 × 8KB = 64TB\n",[31,287983,287981],{"__ignoreMap":29},[15,287985,287986],{},"精确总量为：",[24,287988,287991],{"className":287989,"code":287990,"language":2638,"meta":29},[2636],"8,594,130,956 × 8KB = 70,403,120,791,552B ≈ 64.031TiB\n",[31,287992,287990],{"__ignoreMap":29},[15,287994,287995,287997,287998,288001,288002,347],{},[2627,287996,58173],{}," 最大文件大小约为 ",[31,287999,288000],{},"64TB + 32GB + 16MB + 96KB","，精确为 ",[31,288003,288004],{},"70,403,120,791,552B",[253,288006,288008],{"id":288007},"_2-最大文件系统分区","(2) 最大文件系统分区",[15,288010,288011],{},"块号字段为 24 位，所以一个分区内最多可编号：",[24,288013,288016],{"className":288014,"code":288015,"language":2638,"meta":29},[2636],"2^24 个磁盘块\n",[31,288017,288015],{"__ignoreMap":29},[15,288019,288020],{},"每块 8KB：",[24,288022,288025],{"className":288023,"code":288024,"language":2638,"meta":29},[2636],"2^24 × 8KB = 2^24 × 2^13B = 2^37B = 128GiB\n",[31,288026,288024],{"__ignoreMap":29},[15,288028,288029,288031,288032,347],{},[2627,288030,58173],{}," 最大文件系统分区为 ",[31,288033,288034],{},"128GiB",[15,288036,288037,288038,288041,288042,288045,288046,347],{},"补充：若把 8 位物理磁盘号也计入整个系统寻址能力，则总磁盘寻址空间可达 ",[31,288039,288040],{},"2^8 × 128GiB = 32TiB","。但 Solutions-OS8e 12.13 对“partition”的答案为 ",[31,288043,288044],{},"128GB"," 量级，因此本题按分区答 ",[31,288047,288034],{},[253,288049,288051],{"id":288050},"_3-访问位置-12423956-的字节需要访问磁盘多少次","(3) 访问位置 12423956 的字节需要访问磁盘多少次",[15,288053,288054],{},"逻辑块号：",[24,288056,288059],{"className":288057,"code":288058,"language":2638,"meta":29},[2636],"12423956 \u002F 8192 = 1516 余 4884\n",[31,288060,288058],{"__ignoreMap":29},[15,288062,288063,288064,288067],{},"所以该字节在逻辑块 ",[31,288065,288066],{},"1516"," 内。",[15,288069,288070],{},"直接块范围：",[24,288072,288075],{"className":288073,"code":288074,"language":2638,"meta":29},[2636],"0 - 11\n",[31,288076,288074],{"__ignoreMap":29},[15,288078,288079],{},"一级间接范围：",[24,288081,288084],{"className":288082,"code":288083,"language":2638,"meta":29},[2636],"12 - 2059\n",[31,288085,288083],{"__ignoreMap":29},[15,288087,288088,288089,288091],{},"逻辑块 ",[31,288090,288066],{}," 属于一级间接范围。",[15,288093,288094],{},"题目说主存中除了 inode 外没有别的信息，因此 inode 已在内存中，但一级间接索引块和目标数据块都不在内存中。",[15,288096,288097],{},"需要访问磁盘：",[2230,288099,288100,288103],{},[108,288101,288102],{},"读一级间接索引块。",[108,288104,288105],{},"读目标数据块。",[15,288107,288108,288110,288111,89490],{},[2627,288109,58173],{}," 需要访问磁盘 ",[31,288112,768],{},[19,288114,288116],{"id":288115},"_5-磁盘记录顺序处理与优化排列","5. 磁盘记录顺序处理与优化排列",[15,288118,288119],{},[2627,288120,285743],{},[104833,288122,288123,288126,288129],{},[15,288124,288125],{},"5、假设有8个记录A、B，C、D、E、F、G、H存放在磁盘里，每个磁道有8个扇区，正好可以存放8个记录。假设磁盘旋转速度为20ms\u002F转，处理程序每读出一个记录后，用2ms的时间进行处理，请问：",[15,288127,288128],{},"a．当记录A、B、C、D、E、F、G、H按顺序放在磁道上时，顺序处理这8个记录花费的总时间是多少？假设启动时的位置正好在A扇区的起点。",[15,288130,288131],{},"b．如何采取优化方法，使处理这些记录所花费的总时间最短？求出该最短时间。",[15,288133,288134],{},[2627,288135,285753],{},[15,288137,288138],{},"每磁道 8 个扇区，旋转一周 20ms，所以每个扇区经过磁头的时间：",[24,288140,288143],{"className":288141,"code":288142,"language":2638,"meta":29},[2636],"20ms \u002F 8 = 2.5ms\n",[31,288144,288142],{"__ignoreMap":29},[15,288146,288147],{},"每读出一个记录后处理 2ms，因此读完一个记录再处理完成所需时间：",[24,288149,288152],{"className":288150,"code":288151,"language":2638,"meta":29},[2636],"2.5ms + 2ms = 4.5ms\n",[31,288153,288151],{"__ignoreMap":29},[253,288155,288157],{"id":288156},"a-顺序存放","(a) 顺序存放",[15,288159,288160,288161,288163],{},"记录按 ",[31,288162,170444],{}," 顺序存放。启动时正好在 A 扇区起点。",[15,288165,288166],{},"读 A：",[24,288168,288171],{"className":288169,"code":288170,"language":2638,"meta":29},[2636],"读 A：0 - 2.5ms\n处理 A：2.5 - 4.5ms\n",[31,288172,288170],{"__ignoreMap":29},[15,288174,288175],{},"处理完 A 时，B 扇区已经转过磁头，只能等下一圈 B 再来。之后每读一个记录都会发生类似情况。",[15,288177,288178],{},"相邻两个逻辑记录开始读取的间隔为：",[24,288180,288183],{"className":288181,"code":288182,"language":2638,"meta":29},[2636],"2.5ms 读当前记录 + 2ms 处理 + 等待下一圈对应扇区\n= 22.5ms\n",[31,288184,288182],{"__ignoreMap":29},[15,288186,288187],{},"总时间：",[24,288189,288192],{"className":288190,"code":288191,"language":2638,"meta":29},[2636],"A 的读和处理：4.5ms\n后续 7 个记录：7 × 22.5ms = 157.5ms\n总时间：4.5 + 157.5 = 162ms\n",[31,288193,288191],{"__ignoreMap":29},[15,288195,288196,288198,288199,347],{},[2627,288197,58173],{}," 顺序存放时处理 8 个记录共需 ",[31,288200,171351],{},[253,288202,288203],{"id":171374},"(b) 优化方法",[15,288205,288206],{},"这类题对应教材解答中“记录交错存放”的思想：让相邻逻辑记录在磁道上的间隔匹配处理时间，避免每次处理完都错过下一个扇区。",[15,288208,288209,288210,288212],{},"因为读一个记录并处理完成需要 ",[31,288211,170408],{},"，约等于 2 个扇区时间：",[24,288214,288217],{"className":288215,"code":288216,"language":2638,"meta":29},[2636],"2 × 2.5ms = 5ms\n",[31,288218,288216],{"__ignoreMap":29},[15,288220,288221],{},"所以逻辑上的下一个记录应放在物理位置约 2 个扇区之后，使处理完成后尽快遇到下一个要读的记录。",[15,288223,288224],{},"一种最优排列：",[24,288226,288229],{"className":288227,"code":288228,"language":2638,"meta":29},[2636],"扇区位置：1 2 3 4 5 6 7 8\n存放记录：A E B F C G D H\n",[31,288230,288228],{"__ignoreMap":29},[15,288232,288233],{},"读取开始时间：",[13837,288235,288236,288245],{},[13840,288237,288238],{},[13843,288239,288240,288242],{},[13846,288241,171689],{},[13846,288243,288244],{"align":287595},"开始读时间",[13853,288246,288247,288253,288259,288265,288271,288278,288285,288292],{},[13843,288248,288249,288251],{},[13858,288250,54151],{},[13858,288252,171746],{"align":287595},[13843,288254,288255,288257],{},[13858,288256,54224],{},[13858,288258,171932],{"align":287595},[13843,288260,288261,288263],{},[13858,288262,47384],{},[13858,288264,271224],{"align":287595},[13843,288266,288267,288269],{},[13858,288268,91994],{},[13858,288270,172066],{"align":287595},[13843,288272,288273,288275],{},[13858,288274,96058],{},[13858,288276,288277],{"align":287595},"22.5ms",[13843,288279,288280,288282],{},[13858,288281,51834],{},[13858,288283,288284],{"align":287595},"27.5ms",[13843,288286,288287,288289],{},[13858,288288,97709],{},[13858,288290,288291],{"align":287595},"32.5ms",[13843,288293,288294,288296],{},[13858,288295,118113],{},[13858,288297,288298],{"align":287595},"37.5ms",[15,288300,288301,288302,288305,288306,347],{},"H 读取结束为 ",[31,288303,288304],{},"40ms","，处理 H 再用 ",[31,288307,170210],{},[15,288309,288310,288312,288313,288315,288316,347],{},[2627,288311,58173],{}," 最短总时间为 ",[31,288314,172994],{},"，一种优化排列为 ",[31,288317,171654],{},[19,288319,288321],{"id":288320},"_6-四道作业系统str-与-sjf","6. 四道作业系统：STR 与 SJF",[15,288323,288324],{},[2627,288325,285743],{},[104833,288327,288328,288331,288334],{},[15,288329,288330],{},"6、四道作业系统，如果在一段时间内先后有6个作业，它们提交和运行时间由下表给出(时间单位为分钟)。系统采用进程剩余时间优先(STR)调度算法，作业采用最短作业优先算法。问题：",[15,288332,288333],{},"(1) 完成各个作业的开始时间、完成时间、周转时间和带权周转时间。",[15,288335,264596],{},[13837,288337,288338,288356],{},[13840,288339,288340],{},[13843,288341,288342,288344,288346,288348,288350,288352,288354],{},[13846,288343,230223],{"align":287595},[13846,288345,230226],{"align":287595},[13846,288347,229533],{"align":287595},[13846,288349,230231],{"align":287595},[13846,288351,214469],{"align":287595},[13846,288353,230236],{"align":287595},[13846,288355,230239],{"align":287595},[13853,288357,288358,288374,288391,288407,288423,288439],{},[13843,288359,288360,288362,288364,288366,288368,288370,288372],{},[13858,288361,640],{"align":287595},[13858,288363,230248],{"align":287595},[13858,288365,5774],{"align":287595},[13858,288367],{"align":287595},[13858,288369],{"align":287595},[13858,288371],{"align":287595},[13858,288373],{"align":287595},[13843,288375,288376,288378,288381,288383,288385,288387,288389],{},[13858,288377,768],{"align":287595},[13858,288379,288380],{"align":287595},"8:10",[13858,288382,5752],{"align":287595},[13858,288384],{"align":287595},[13858,288386],{"align":287595},[13858,288388],{"align":287595},[13858,288390],{"align":287595},[13843,288392,288393,288395,288397,288399,288401,288403,288405],{},[13858,288394,786],{"align":287595},[13858,288396,264512],{"align":287595},[13858,288398,95556],{"align":287595},[13858,288400],{"align":287595},[13858,288402],{"align":287595},[13858,288404],{"align":287595},[13858,288406],{"align":287595},[13843,288408,288409,288411,288413,288415,288417,288419,288421],{},[13858,288410,15932],{"align":287595},[13858,288412,230265],{"align":287595},[13858,288414,95556],{"align":287595},[13858,288416],{"align":287595},[13858,288418],{"align":287595},[13858,288420],{"align":287595},[13858,288422],{"align":287595},[13843,288424,288425,288427,288429,288431,288433,288435,288437],{},[13858,288426,18136],{"align":287595},[13858,288428,230316],{"align":287595},[13858,288430,95396],{"align":287595},[13858,288432],{"align":287595},[13858,288434],{"align":287595},[13858,288436],{"align":287595},[13858,288438],{"align":287595},[13843,288440,288441,288443,288445,288447,288449,288451,288453],{},[13858,288442,36656],{"align":287595},[13858,288444,230333],{"align":287595},[13858,288446,3244],{"align":287595},[13858,288448],{"align":287595},[13858,288450],{"align":287595},[13858,288452],{"align":287595},[13858,288454],{"align":287595},[15,288456,288457],{},[2627,288458,285753],{},[15,288460,288461],{},"四道作业系统表示内存中最多同时容纳 4 个作业。进程调度采用剩余时间优先 STR，会发生抢占；作业调度采用短作业优先，选择后备队列中运行时间最短的作业进入内存。",[15,288463,288464],{},"执行过程：",[13837,288466,288467,288479],{},[13840,288468,288469],{},[13843,288470,288471,288474,288477],{},[13846,288472,288473],{},"时间段",[13846,288475,288476],{"align":287595},"运行作业",[13846,288478,111426],{},[13853,288480,288481,288491,288501,288511,288522,288532,288542,288552],{},[13843,288482,288483,288486,288488],{},[13858,288484,288485],{},"8:00-8:10",[13858,288487,273456],{"align":287595},[13858,288489,288490],{},"只有 J1 到达",[13843,288492,288493,288496,288498],{},[13858,288494,288495],{},"8:10-8:15",[13858,288497,273471],{"align":287595},[13858,288499,288500],{},"J2 运行时间 35，小于 J1 剩余 40",[13843,288502,288503,288506,288508],{},[13858,288504,288505],{},"8:15-8:40",[13858,288507,273486],{"align":287595},[13858,288509,288510],{},"J3 运行时间 25，小于 J2 剩余 30",[13843,288512,288513,288516,288519],{},[13858,288514,288515],{},"8:40-8:50",[13858,288517,288518],{"align":287595},"J6",[13858,288520,288521],{},"J3 完成，后备队列中 J5 与 J6，SJF 选 J6",[13843,288523,288524,288527,288529],{},[13858,288525,288526],{},"8:50-9:05",[13858,288528,273915],{"align":287595},[13858,288530,288531],{},"J6 完成后 J5 进入并运行",[13843,288533,288534,288537,288539],{},[13858,288535,288536],{},"9:05-9:30",[13858,288538,273500],{"align":287595},[13858,288540,288541],{},"剩余时间最短",[13843,288543,288544,288547,288549],{},[13858,288545,288546],{},"9:30-10:00",[13858,288548,273471],{"align":287595},[13858,288550,288551],{},"J2 剩余 30",[13843,288553,288554,288557,288559],{},[13858,288555,288556],{},"10:00-10:40",[13858,288558,273456],{"align":287595},[13858,288560,288561],{},"J1 剩余 40",[15,288563,288564],{},"结果表：",[13837,288566,288567,288585],{},[13840,288568,288569],{},[13843,288570,288571,288573,288575,288577,288579,288581,288583],{},[13846,288572,230486],{"align":287595},[13846,288574,230226],{"align":287595},[13846,288576,229533],{"align":287595},[13846,288578,230231],{"align":287595},[13846,288580,214469],{"align":287595},[13846,288582,230236],{"align":287595},[13846,288584,230239],{"align":287595},[13853,288586,288587,288606,288623,288639,288658,288674],{},[13843,288588,288589,288591,288593,288595,288597,288600,288603],{},[13858,288590,640],{"align":287595},[13858,288592,230248],{"align":287595},[13858,288594,5774],{"align":287595},[13858,288596,230248],{"align":287595},[13858,288598,288599],{"align":287595},"10:40",[13858,288601,288602],{"align":287595},"160",[13858,288604,288605],{"align":287595},"3.20",[13843,288607,288608,288610,288612,288614,288616,288618,288620],{},[13858,288609,768],{"align":287595},[13858,288611,288380],{"align":287595},[13858,288613,5752],{"align":287595},[13858,288615,288380],{"align":287595},[13858,288617,229149],{"align":287595},[13858,288619,162188],{"align":287595},[13858,288621,288622],{"align":287595},"3.14",[13843,288624,288625,288627,288629,288631,288633,288635,288637],{},[13858,288626,786],{"align":287595},[13858,288628,264512],{"align":287595},[13858,288630,95556],{"align":287595},[13858,288632,264512],{"align":287595},[13858,288634,230333],{"align":287595},[13858,288636,95556],{"align":287595},[13858,288638,230883],{"align":287595},[13843,288640,288641,288643,288645,288647,288650,288653,288655],{},[13858,288642,15932],{"align":287595},[13858,288644,230265],{"align":287595},[13858,288646,95556],{"align":287595},[13858,288648,288649],{"align":287595},"9:05",[13858,288651,288652],{"align":287595},"9:30",[13858,288654,98497],{"align":287595},[13858,288656,288657],{"align":287595},"2.80",[13843,288659,288660,288662,288664,288666,288668,288670,288672],{},[13858,288661,18136],{"align":287595},[13858,288663,230316],{"align":287595},[13858,288665,95396],{"align":287595},[13858,288667,230424],{"align":287595},[13858,288669,288649],{"align":287595},[13858,288671,9314],{"align":287595},[13858,288673,230898],{"align":287595},[13843,288675,288676,288678,288680,288682,288684,288686,288688],{},[13858,288677,36656],{"align":287595},[13858,288679,230333],{"align":287595},[13858,288681,3244],{"align":287595},[13858,288683,230333],{"align":287595},[13858,288685,230424],{"align":287595},[13858,288687,3244],{"align":287595},[13858,288689,230883],{"align":287595},[15,288691,226112],{},[24,288693,288696],{"className":288694,"code":288695,"language":2638,"meta":29},[2636],"(160 + 110 + 25 + 70 + 30 + 10) \u002F 6 = 67.5 分钟\n",[31,288697,288695],{"__ignoreMap":29},[15,288699,230856],{},[24,288701,288704],{"className":288702,"code":288703,"language":2638,"meta":29},[2636],"(3.20 + 3.14 + 1.00 + 2.80 + 2.00 + 1.00) \u002F 6 ≈ 2.19\n",[31,288705,288703],{"__ignoreMap":29},[15,288707,288708,288710,288711,288714,288715,347],{},[2627,288709,58173],{}," 平均周转时间为 ",[31,288712,288713],{},"67.5"," 分钟，平均带权周转时间约为 ",[31,288716,288717],{},"2.19",[19,288719,288721],{"id":288720},"_7-银行家算法","7. 银行家算法",[15,288723,288724],{},[2627,288725,285743],{},[104833,288727,288728],{},[15,288729,288730],{},"7、假设系统有A、B、C、D四类资源可供五个进程P1、P2、P3、P4、P5共享。系统对这四类资源的拥有量为:(3,14,12,12)。进程对资源的需求和分配情况如下：",[13837,288732,288733,288763],{},[13840,288734,288735],{},[13843,288736,288737,288739,288742,288745,288748,288751,288754,288757,288760],{},[13846,288738,247197],{},[13846,288740,288741],{"align":287595},"已占 A",[13846,288743,288744],{"align":287595},"已占 B",[13846,288746,288747],{"align":287595},"已占 C",[13846,288749,288750],{"align":287595},"已占 D",[13846,288752,288753],{"align":287595},"最大 A",[13846,288755,288756],{"align":287595},"最大 B",[13846,288758,288759],{"align":287595},"最大 C",[13846,288761,288762],{"align":287595},"最大 D",[13853,288764,288765,288785,288805,288825,288845],{},[13843,288766,288767,288769,288771,288773,288775,288777,288779,288781,288783],{},[13858,288768,97876],{},[13858,288770,459],{"align":287595},[13858,288772,459],{"align":287595},[13858,288774,640],{"align":287595},[13858,288776,768],{"align":287595},[13858,288778,459],{"align":287595},[13858,288780,459],{"align":287595},[13858,288782,640],{"align":287595},[13858,288784,768],{"align":287595},[13843,288786,288787,288789,288791,288793,288795,288797,288799,288801,288803],{},[13858,288788,97925],{},[13858,288790,640],{"align":287595},[13858,288792,459],{"align":287595},[13858,288794,459],{"align":287595},[13858,288796,459],{"align":287595},[13858,288798,640],{"align":287595},[13858,288800,36723],{"align":287595},[13858,288802,18136],{"align":287595},[13858,288804,459],{"align":287595},[13843,288806,288807,288809,288811,288813,288815,288817,288819,288821,288823],{},[13858,288808,263561],{},[13858,288810,640],{"align":287595},[13858,288812,786],{"align":287595},[13858,288814,18136],{"align":287595},[13858,288816,15932],{"align":287595},[13858,288818,768],{"align":287595},[13858,288820,786],{"align":287595},[13858,288822,18136],{"align":287595},[13858,288824,36656],{"align":287595},[13843,288826,288827,288829,288831,288833,288835,288837,288839,288841,288843],{},[13858,288828,263578],{},[13858,288830,459],{"align":287595},[13858,288832,36656],{"align":287595},[13858,288834,786],{"align":287595},[13858,288836,768],{"align":287595},[13858,288838,459],{"align":287595},[13858,288840,36656],{"align":287595},[13858,288842,18136],{"align":287595},[13858,288844,768],{"align":287595},[13843,288846,288847,288849,288851,288853,288855,288857,288859,288861,288863],{},[13858,288848,264373],{},[13858,288850,459],{"align":287595},[13858,288852,459],{"align":287595},[13858,288854,640],{"align":287595},[13858,288856,15932],{"align":287595},[13858,288858,459],{"align":287595},[13858,288860,36656],{"align":287595},[13858,288862,18136],{"align":287595},[13858,288864,36656],{"align":287595},[104833,288866,288867,288869,288872,288875],{},[15,288868,264382],{},[15,288870,288871],{},"(1)系统是否处于安全状态？为什么？",[15,288873,288874],{},"(2)如果P2提出请求(0,4,2,0)，系统能否满足请求？请说明原因。",[15,288876,288877],{},"(3)如果在P2提出资源需求(0,4,2,0)后，P3提出资源需求(0,0,0,1)，系统能否满足其请求？请说明原因。",[15,288879,288880],{},[2627,288881,285753],{},[15,288883,288884],{},"总资源：",[24,288886,288889],{"className":288887,"code":288888,"language":2638,"meta":29},[2636],"Total = (3,14,12,12)\n",[31,288890,288888],{"__ignoreMap":29},[15,288892,288893],{},"已分配资源总和：",[24,288895,288898],{"className":288896,"code":288897,"language":2638,"meta":29},[2636],"Allocation 总和 = (2,9,10,12)\n",[31,288899,288897],{"__ignoreMap":29},[15,288901,288902],{},"可用资源：",[24,288904,288907],{"className":288905,"code":288906,"language":2638,"meta":29},[2636],"Available = (3,14,12,12) - (2,9,10,12)\n = (1,5,2,0)\n",[31,288908,288906],{"__ignoreMap":29},[15,288910,288911,288912,2664],{},"需求矩阵 ",[31,288913,288914],{},"Need = Max - Allocation",[13837,288916,288917,288935],{},[13840,288918,288919],{},[13843,288920,288921,288923,288926,288929,288932],{},[13846,288922,247197],{},[13846,288924,288925],{"align":287595},"Need A",[13846,288927,288928],{"align":287595},"Need B",[13846,288930,288931],{"align":287595},"Need C",[13846,288933,288934],{"align":287595},"Need D",[13853,288936,288937,288949,288961,288973,288985],{},[13843,288938,288939,288941,288943,288945,288947],{},[13858,288940,97876],{},[13858,288942,459],{"align":287595},[13858,288944,459],{"align":287595},[13858,288946,459],{"align":287595},[13858,288948,459],{"align":287595},[13843,288950,288951,288953,288955,288957,288959],{},[13858,288952,97925],{},[13858,288954,459],{"align":287595},[13858,288956,36723],{"align":287595},[13858,288958,18136],{"align":287595},[13858,288960,459],{"align":287595},[13843,288962,288963,288965,288967,288969,288971],{},[13858,288964,263561],{},[13858,288966,640],{"align":287595},[13858,288968,459],{"align":287595},[13858,288970,459],{"align":287595},[13858,288972,768],{"align":287595},[13843,288974,288975,288977,288979,288981,288983],{},[13858,288976,263578],{},[13858,288978,459],{"align":287595},[13858,288980,459],{"align":287595},[13858,288982,768],{"align":287595},[13858,288984,459],{"align":287595},[13843,288986,288987,288989,288991,288993,288995],{},[13858,288988,264373],{},[13858,288990,459],{"align":287595},[13858,288992,36656],{"align":287595},[13858,288994,15932],{"align":287595},[13858,288996,768],{"align":287595},[253,288998,289000],{"id":288999},"_1-是否安全","(1) 是否安全",[15,289002,289003],{},"初始：",[24,289005,289008],{"className":289006,"code":289007,"language":2638,"meta":29},[2636],"Work = Available = (1,5,2,0)\n",[31,289009,289007],{"__ignoreMap":29},[15,289011,289012],{},"可取安全序列：",[13837,289014,289015,289028],{},[13840,289016,289017],{},[13843,289018,289019,289022,289025],{},[13846,289020,289021],{"align":287595},"步骤",[13846,289023,289024],{},"可完成进程",[13846,289026,289027],{},"完成后 Work",[13853,289029,289030,289041,289052,289063,289074],{},[13843,289031,289032,289034,289036],{},[13858,289033,640],{"align":287595},[13858,289035,97876],{},[13858,289037,289038],{},[31,289039,289040],{},"(1,5,3,2)",[13843,289042,289043,289045,289047],{},[13858,289044,768],{"align":287595},[13858,289046,263561],{},[13858,289048,289049],{},[31,289050,289051],{},"(2,8,8,6)",[13843,289053,289054,289056,289058],{},[13858,289055,786],{"align":287595},[13858,289057,97925],{},[13858,289059,289060],{},[31,289061,289062],{},"(3,8,8,6)",[13843,289064,289065,289067,289069],{},[13858,289066,15932],{"align":287595},[13858,289068,263578],{},[13858,289070,289071],{},[31,289072,289073],{},"(3,14,11,8)",[13843,289075,289076,289078,289080],{},[13858,289077,18136],{"align":287595},[13858,289079,264373],{},[13858,289081,289082],{},[31,289083,289084],{},"(3,14,12,12)",[15,289086,289087,289089,289090,347],{},[2627,289088,58173],{}," 系统处于安全状态，安全序列可为 ",[31,289091,289092],{},"P1 -> P3 -> P2 -> P4 -> P5",[253,289094,289096,289097,289100],{"id":289095},"_2-p2-请求-0420-是否可满足","(2) P2 请求 ",[31,289098,289099],{},"(0,4,2,0)"," 是否可满足",[15,289102,27083],{},[24,289104,289107],{"className":289105,"code":289106,"language":2638,"meta":29},[2636],"Request2 = (0,4,2,0)\nNeed2 = (0,7,5,0)\nAvailable = (1,5,2,0)\n",[31,289108,289106],{"__ignoreMap":29},[15,289110,289111],{},"满足：",[24,289113,289116],{"className":289114,"code":289115,"language":2638,"meta":29},[2636],"Request2 \u003C= Need2\nRequest2 \u003C= Available\n",[31,289117,289115],{"__ignoreMap":29},[15,289119,289120],{},"试分配后：",[24,289122,289125],{"className":289123,"code":289124,"language":2638,"meta":29},[2636],"Available = (1,5,2,0) - (0,4,2,0) = (1,1,0,0)\nP2 Allocation = (1,4,2,0)\nP2 Need = (0,3,3,0)\n",[31,289126,289124],{"__ignoreMap":29},[15,289128,289129],{},"安全性检查：",[13837,289131,289132,289142],{},[13840,289133,289134],{},[13843,289135,289136,289138,289140],{},[13846,289137,289021],{"align":287595},[13846,289139,289024],{},[13846,289141,289027],{},[13853,289143,289144,289155,289166,289176,289186],{},[13843,289145,289146,289148,289150],{},[13858,289147,640],{"align":287595},[13858,289149,97876],{},[13858,289151,289152],{},[31,289153,289154],{},"(1,1,1,2)",[13843,289156,289157,289159,289161],{},[13858,289158,768],{"align":287595},[13858,289160,263561],{},[13858,289162,289163],{},[31,289164,289165],{},"(2,4,6,6)",[13843,289167,289168,289170,289172],{},[13858,289169,786],{"align":287595},[13858,289171,97925],{},[13858,289173,289174],{},[31,289175,289062],{},[13843,289177,289178,289180,289182],{},[13858,289179,15932],{"align":287595},[13858,289181,263578],{},[13858,289183,289184],{},[31,289185,289073],{},[13843,289187,289188,289190,289192],{},[13858,289189,18136],{"align":287595},[13858,289191,264373],{},[13858,289193,289194],{},[31,289195,289084],{},[15,289197,289198,289200,289201,347],{},[2627,289199,58173],{}," 可以满足。试分配后仍安全，安全序列可为 ",[31,289202,289092],{},[253,289204,289206,289207,289100],{"id":289205},"_3-p2-请求后p3-请求-0001-是否可满足","(3) P2 请求后，P3 请求 ",[31,289208,289209],{},"(0,0,0,1)",[15,289211,289212],{},"在 P2 请求被满足后：",[24,289214,289217],{"className":289215,"code":289216,"language":2638,"meta":29},[2636],"Available = (1,1,0,0)\n",[31,289218,289216],{"__ignoreMap":29},[15,289220,289221],{},"P3 请求：",[24,289223,289226],{"className":289224,"code":289225,"language":2638,"meta":29},[2636],"Request3 = (0,0,0,1)\n",[31,289227,289225],{"__ignoreMap":29},[15,289229,289230],{},"虽然该请求不超过 P3 的剩余需求，但：",[24,289232,289235],{"className":289233,"code":289234,"language":2638,"meta":29},[2636],"Request3 的 D = 1 > Available 的 D = 0\n",[31,289236,289234],{"__ignoreMap":29},[15,289238,289239,289241],{},[2627,289240,58173],{}," 不能立即满足，因为 D 类资源当前可用数为 0。",[19,289243,289245],{"id":289244},"_8-增加内存后的-cpu-利用率","8. 增加内存后的 CPU 利用率",[15,289247,289248],{},[2627,289249,285743],{},[104833,289251,289252],{},[15,289253,289254],{},"8、有一台计算机，具有1MB内存，操作系统占用200KB，每个用户占用200KB。如果用户进程等待I\u002FO的时间为80%，若增加1MB内存，则CPU的利用率提高多少？",[15,289256,289257],{},[2627,289258,285753],{},[15,289260,289261,289262,289264,289265,289267],{},"多道程序环境下，若每个进程等待 I\u002FO 的概率为 ",[31,289263,15],{},"，内存中有 ",[31,289266,31660],{}," 个用户进程，则 CPU 利用率近似为：",[24,289269,289272],{"className":289270,"code":289271,"language":2638,"meta":29},[2636],"U = 1 - p^n\n",[31,289273,289271],{"__ignoreMap":29},[15,289275,289276],{},"这里：",[24,289278,289281],{"className":289279,"code":289280,"language":2638,"meta":29},[2636],"p = 0.8\n",[31,289282,289280],{"__ignoreMap":29},[15,289284,289285],{},"初始内存 1MB，即 1024KB。操作系统占用 200KB，每个用户进程占 200KB。",[15,289287,289288],{},"初始可容纳用户进程数：",[24,289290,289293],{"className":289291,"code":289292,"language":2638,"meta":29},[2636],"floor((1024 - 200) \u002F 200) = 4\n",[31,289294,289292],{"__ignoreMap":29},[15,289296,289297],{},"初始 CPU 利用率：",[24,289299,289302],{"className":289300,"code":289301,"language":2638,"meta":29},[2636],"U1 = 1 - 0.8^4\n = 1 - 0.4096\n = 0.5904\n = 59.04%\n",[31,289303,289301],{"__ignoreMap":29},[15,289305,289306],{},"增加 1MB 后，总内存为 2MB，即 2048KB。",[15,289308,289309],{},"新可容纳用户进程数：",[24,289311,289314],{"className":289312,"code":289313,"language":2638,"meta":29},[2636],"floor((2048 - 200) \u002F 200) = 9\n",[31,289315,289313],{"__ignoreMap":29},[15,289317,289318],{},"新的 CPU 利用率：",[24,289320,289323],{"className":289321,"code":289322,"language":2638,"meta":29},[2636],"U2 = 1 - 0.8^9\n = 1 - 0.134217728\n = 0.865782272\n ≈ 86.58%\n",[31,289324,289322],{"__ignoreMap":29},[15,289326,289327],{},"提高：",[24,289329,289332],{"className":289330,"code":289331,"language":2638,"meta":29},[2636],"U2 - U1 = 0.865782272 - 0.5904\n = 0.275382272\n ≈ 27.54 个百分点\n",[31,289333,289331],{"__ignoreMap":29},[15,289335,289336,289338,289339,289341,289342,289344,289345,289348,289349,347],{},[2627,289337,58173],{}," CPU 利用率从 ",[31,289340,194452],{}," 提高到约 ",[31,289343,194707],{},"，提高约 ",[31,289346,289347],{},"27.54"," 个百分点。若按相对提高比例，则约为 ",[31,289350,289351],{},"46.65%",[19,289353,289355],{"id":289354},"_9-cache主存辅存平均访问时间","9. cache、主存、辅存平均访问时间",[15,289357,289358],{},[2627,289359,285743],{},[104833,289361,289362],{},[15,289363,289364],{},"9、某计算机有cache、主存、辅存来实现虚拟存储器。如果数据在cache中，访问它需要20ns;如果在主存但不在cache，需要60ns将其装入缓存，然后才能访问;如果不在主存而在辅存，需要12s将其读入主存，然后，用60ns再读入cache，然后才能访问。假设cache合中率为0.9，主存命中率为0.6，则数据平均访问时间是多少(ns)?",[15,289366,289367],{},[2627,289368,285753],{},[15,289370,289371],{},"单位换算：",[24,289373,289376],{"className":289374,"code":289375,"language":2638,"meta":29},[2636],"12s = 12,000,000,000ns\n",[31,289377,289375],{"__ignoreMap":29},[15,289379,289380],{},"按题意理解：",[2230,289382,289383,289392,289397,289402],{},[108,289384,289385,289386,289388,289389,347],{},"cache 命中概率为 ",[31,289387,250855],{},"，访问时间 ",[31,289390,289391],{},"20ns",[108,289393,289394,289395,347],{},"cache 未命中概率为 ",[31,289396,248816],{},[108,289398,289399,289400,347],{},"在 cache 未命中情况下，主存命中概率为 ",[31,289401,250729],{},[108,289403,289404],{},"在主存未命中情况下，需要从辅存读入主存，再读入 cache，再访问。",[15,289406,289407],{},"平均访问时间：",[24,289409,289412],{"className":289410,"code":289411,"language":2638,"meta":29},[2636],"E = 0.9 × 20\n + 0.1 × 0.6 × (60 + 20)\n + 0.1 × 0.4 × (12,000,000,000 + 60 + 20)\n",[31,289413,289411],{"__ignoreMap":29},[15,289415,174094],{},[24,289417,289420],{"className":289418,"code":289419,"language":2638,"meta":29},[2636],"E = 18 + 4.8 + 480,000,003.2\n = 480,000,026ns\n",[31,289421,289419],{"__ignoreMap":29},[15,289423,289424,289426,289427,289430,289431,347],{},[2627,289425,58173],{}," 数据平均访问时间约为 ",[31,289428,289429],{},"480,000,026ns","，即约 ",[31,289432,289433],{},"0.480000026s",[15,289435,289436,289437,289439,289440,289443],{},"说明：若教材把“装入 cache 后才能访问”的最后一次 ",[31,289438,289391],{}," 省略，则结果为 ",[31,289441,289442],{},"480,000,024ns","，只差 2ns，不影响结论。",[19,289445,289447],{"id":289446},"_10-页面置换opt-和-lru","10. 页面置换：OPT 和 LRU",[15,289449,289450],{},[2627,289451,285743],{},[104833,289453,289454,289457],{},[15,289455,289456],{},"10、在一个请求分页虚拟存储管理系统中，一个作业共有5页，执行时其访问页面次序为: 1、4、3、1、2、5、1、4、2、1、4、5。",[15,289458,289459],{},"若分配给该作业三个页框，分别采用OPT和 LRU页面替换算法，求出各自的缺页中断次数、淘汰的页面、和缺页中断率。",[15,289461,289462],{},[2627,289463,285753],{},[15,289465,289466],{},"访问序列：",[24,289468,289471],{"className":289469,"code":289470,"language":2638,"meta":29},[2636],"1, 4, 3, 1, 2, 5, 1, 4, 2, 1, 4, 5\n",[31,289472,289470],{"__ignoreMap":29},[15,289474,289475],{},"页框数为 3，访问总次数为 12。",[253,289477,289479],{"id":289478},"opt-算法","OPT 算法",[15,289481,289482],{},"OPT 淘汰未来最长时间不再访问的页面。",[13837,289484,289485,289502],{},[13840,289486,289487],{},[13843,289488,289489,289491,289494,289496,289499],{},[13846,289490,289021],{"align":287595},[13846,289492,289493],{"align":287595},"访问页",[13846,289495,267766],{},[13846,289497,289498],{"align":287595},"淘汰页",[13846,289500,289501],{},"页框状态",[13853,289503,289504,289519,289534,289549,289564,289579,289594,289608,289622,289636,289650,289664],{},[13843,289505,289506,289508,289510,289513,289515],{},[13858,289507,640],{"align":287595},[13858,289509,640],{"align":287595},[13858,289511,289512],{},"是",[13858,289514,2511],{"align":287595},[13858,289516,289517],{},[31,289518,640],{},[13843,289520,289521,289523,289525,289527,289529],{},[13858,289522,768],{"align":287595},[13858,289524,15932],{"align":287595},[13858,289526,289512],{},[13858,289528,2511],{"align":287595},[13858,289530,289531],{},[31,289532,289533],{},"1,4",[13843,289535,289536,289538,289540,289542,289544],{},[13858,289537,786],{"align":287595},[13858,289539,786],{"align":287595},[13858,289541,289512],{},[13858,289543,2511],{"align":287595},[13858,289545,289546],{},[31,289547,289548],{},"1,4,3",[13843,289550,289551,289553,289555,289558,289560],{},[13858,289552,15932],{"align":287595},[13858,289554,640],{"align":287595},[13858,289556,289557],{},"否",[13858,289559,2511],{"align":287595},[13858,289561,289562],{},[31,289563,289548],{},[13843,289565,289566,289568,289570,289572,289574],{},[13858,289567,18136],{"align":287595},[13858,289569,768],{"align":287595},[13858,289571,289512],{},[13858,289573,786],{"align":287595},[13858,289575,289576],{},[31,289577,289578],{},"1,4,2",[13843,289580,289581,289583,289585,289587,289589],{},[13858,289582,36656],{"align":287595},[13858,289584,18136],{"align":287595},[13858,289586,289512],{},[13858,289588,768],{"align":287595},[13858,289590,289591],{},[31,289592,289593],{},"1,4,5",[13843,289595,289596,289598,289600,289602,289604],{},[13858,289597,36723],{"align":287595},[13858,289599,640],{"align":287595},[13858,289601,289557],{},[13858,289603,2511],{"align":287595},[13858,289605,289606],{},[31,289607,289593],{},[13843,289609,289610,289612,289614,289616,289618],{},[13858,289611,60525],{"align":287595},[13858,289613,15932],{"align":287595},[13858,289615,289557],{},[13858,289617,2511],{"align":287595},[13858,289619,289620],{},[31,289621,289593],{},[13843,289623,289624,289626,289628,289630,289632],{},[13858,289625,36732],{"align":287595},[13858,289627,768],{"align":287595},[13858,289629,289512],{},[13858,289631,18136],{"align":287595},[13858,289633,289634],{},[31,289635,289578],{},[13843,289637,289638,289640,289642,289644,289646],{},[13858,289639,3244],{"align":287595},[13858,289641,640],{"align":287595},[13858,289643,289557],{},[13858,289645,2511],{"align":287595},[13858,289647,289648],{},[31,289649,289578],{},[13843,289651,289652,289654,289656,289658,289660],{},[13858,289653,39620],{"align":287595},[13858,289655,15932],{"align":287595},[13858,289657,289557],{},[13858,289659,2511],{"align":287595},[13858,289661,289662],{},[31,289663,289578],{},[13843,289665,289666,289668,289670,289672,289674],{},[13858,289667,71869],{"align":287595},[13858,289669,18136],{"align":287595},[13858,289671,289512],{},[13858,289673,640],{"align":287595},[13858,289675,289676],{},[31,289677,289678],{},"5,4,2",[15,289680,289681,289682,289685],{},"最后一步中，",[31,289683,289684],{},"1、4、2"," 后面都不再访问，淘汰任意一个都不影响缺页次数。",[15,289687,289688],{},"OPT 缺页次数：",[24,289690,289693],{"className":289691,"code":289692,"language":2638,"meta":29},[2636],"7 次\n",[31,289694,289692],{"__ignoreMap":29},[15,289696,289697],{},"OPT 缺页率：",[24,289699,289702],{"className":289700,"code":289701,"language":2638,"meta":29},[2636],"7 \u002F 12 ≈ 58.33%\n",[31,289703,289701],{"__ignoreMap":29},[253,289705,289707],{"id":289706},"lru-算法","LRU 算法",[15,289709,289710],{},"LRU 淘汰最长时间未被访问的页面。",[13837,289712,289713,289727],{},[13840,289714,289715],{},[13843,289716,289717,289719,289721,289723,289725],{},[13846,289718,289021],{"align":287595},[13846,289720,289493],{"align":287595},[13846,289722,267766],{},[13846,289724,289498],{"align":287595},[13846,289726,289501],{},[13853,289728,289729,289743,289757,289771,289785,289800,289815,289829,289843,289857,289871,289885],{},[13843,289730,289731,289733,289735,289737,289739],{},[13858,289732,640],{"align":287595},[13858,289734,640],{"align":287595},[13858,289736,289512],{},[13858,289738,2511],{"align":287595},[13858,289740,289741],{},[31,289742,640],{},[13843,289744,289745,289747,289749,289751,289753],{},[13858,289746,768],{"align":287595},[13858,289748,15932],{"align":287595},[13858,289750,289512],{},[13858,289752,2511],{"align":287595},[13858,289754,289755],{},[31,289756,289533],{},[13843,289758,289759,289761,289763,289765,289767],{},[13858,289760,786],{"align":287595},[13858,289762,786],{"align":287595},[13858,289764,289512],{},[13858,289766,2511],{"align":287595},[13858,289768,289769],{},[31,289770,289548],{},[13843,289772,289773,289775,289777,289779,289781],{},[13858,289774,15932],{"align":287595},[13858,289776,640],{"align":287595},[13858,289778,289557],{},[13858,289780,2511],{"align":287595},[13858,289782,289783],{},[31,289784,289548],{},[13843,289786,289787,289789,289791,289793,289795],{},[13858,289788,18136],{"align":287595},[13858,289790,768],{"align":287595},[13858,289792,289512],{},[13858,289794,15932],{"align":287595},[13858,289796,289797],{},[31,289798,289799],{},"1,2,3",[13843,289801,289802,289804,289806,289808,289810],{},[13858,289803,36656],{"align":287595},[13858,289805,18136],{"align":287595},[13858,289807,289512],{},[13858,289809,786],{"align":287595},[13858,289811,289812],{},[31,289813,289814],{},"1,2,5",[13843,289816,289817,289819,289821,289823,289825],{},[13858,289818,36723],{"align":287595},[13858,289820,640],{"align":287595},[13858,289822,289557],{},[13858,289824,2511],{"align":287595},[13858,289826,289827],{},[31,289828,289814],{},[13843,289830,289831,289833,289835,289837,289839],{},[13858,289832,60525],{"align":287595},[13858,289834,15932],{"align":287595},[13858,289836,289512],{},[13858,289838,768],{"align":287595},[13858,289840,289841],{},[31,289842,289593],{},[13843,289844,289845,289847,289849,289851,289853],{},[13858,289846,36732],{"align":287595},[13858,289848,768],{"align":287595},[13858,289850,289512],{},[13858,289852,18136],{"align":287595},[13858,289854,289855],{},[31,289856,289578],{},[13843,289858,289859,289861,289863,289865,289867],{},[13858,289860,3244],{"align":287595},[13858,289862,640],{"align":287595},[13858,289864,289557],{},[13858,289866,2511],{"align":287595},[13858,289868,289869],{},[31,289870,289578],{},[13843,289872,289873,289875,289877,289879,289881],{},[13858,289874,39620],{"align":287595},[13858,289876,15932],{"align":287595},[13858,289878,289557],{},[13858,289880,2511],{"align":287595},[13858,289882,289883],{},[31,289884,289578],{},[13843,289886,289887,289889,289891,289893,289895],{},[13858,289888,71869],{"align":287595},[13858,289890,18136],{"align":287595},[13858,289892,289512],{},[13858,289894,768],{"align":287595},[13858,289896,289897],{},[31,289898,289593],{},[15,289900,289901],{},"LRU 缺页次数：",[24,289903,289906],{"className":289904,"code":289905,"language":2638,"meta":29},[2636],"8 次\n",[31,289907,289905],{"__ignoreMap":29},[15,289909,289910],{},"LRU 缺页率：",[24,289912,289915],{"className":289913,"code":289914,"language":2638,"meta":29},[2636],"8 \u002F 12 ≈ 66.67%\n",[31,289916,289914],{"__ignoreMap":29},[15,289918,289919],{},[2627,289920,289921],{},"答案汇总：",[13837,289923,289924,289939],{},[13840,289925,289926],{},[13843,289927,289928,289930,289933,289936],{},[13846,289929,153524],{},[13846,289931,289932],{"align":287595},"缺页次数",[13846,289934,289935],{},"淘汰页面序列",[13846,289937,289938],{"align":287595},"缺页率",[13853,289940,289941,289956],{},[13843,289942,289943,289946,289948,289953],{},[13858,289944,289945],{},"OPT",[13858,289947,36723],{"align":287595},[13858,289949,289950],{},[31,289951,289952],{},"3, 2, 5, 1",[13858,289954,289955],{"align":287595},"58.33%",[13843,289957,289958,289961,289963,289968],{},[13858,289959,289960],{},"LRU",[13858,289962,60525],{"align":287595},[13858,289964,289965],{},[31,289966,289967],{},"4, 3, 2, 5, 2",[13858,289969,289970],{"align":287595},"66.67%",{"title":29,"searchDepth":43,"depth":43,"links":289972},[289973,289974,289975,289976,289981,289985,289986,289993,289994,289995],{"id":287566,"depth":43,"text":287567},{"id":287733,"depth":43,"text":287734},{"id":287840,"depth":43,"text":287841},{"id":287911,"depth":43,"text":287912,"children":289977},[289978,289979,289980],{"id":287955,"depth":49,"text":287956},{"id":288007,"depth":49,"text":288008},{"id":288050,"depth":49,"text":288051},{"id":288115,"depth":43,"text":288116,"children":289982},[289983,289984],{"id":288156,"depth":49,"text":288157},{"id":171374,"depth":49,"text":288203},{"id":288320,"depth":43,"text":288321},{"id":288720,"depth":43,"text":288721,"children":289987},[289988,289989,289991],{"id":288999,"depth":49,"text":289000},{"id":289095,"depth":49,"text":289990},"(2) P2 请求 (0,4,2,0) 是否可满足",{"id":289205,"depth":49,"text":289992},"(3) P2 请求后，P3 请求 (0,0,0,1) 是否可满足",{"id":289244,"depth":43,"text":289245},{"id":289354,"depth":43,"text":289355},{"id":289446,"depth":43,"text":289447,"children":289996},[289997,289998],{"id":289478,"depth":49,"text":289479},{"id":289706,"depth":49,"text":289707},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F03-1-10",{"title":287558,"description":29},"source\u002Fcs\u002Fos\u002Freview202505\u002F03-计算题1-10","KeSs7nNg6Rdm3KfWhbiSOqYpHD5i2uwOlQ43vMDpz-c",{"id":290005,"title":290006,"body":290007,"description":29,"extension":85,"meta":291924,"navigation":52,"path":291925,"seo":291926,"stem":291927,"__hash__":291928},"notes\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F04-计算题11-20.md","review202505 答案讲解 04：计算题 11-20",{"type":8,"value":290008,"toc":291899},[290009,290012,290016,290020,290029,290033,290036,290039,290045,290049,290052,290058,290061,290067,290070,290076,290079,290085,290087,290093,290102,290105,290111,290115,290118,290120,290126,290128,290134,290136,290142,290150,290154,290158,290166,290170,290173,290179,290182,290185,290191,290194,290200,290203,290205,290396,290404,290406,290412,290415,290421,290423,290429,290431,290621,290623,290628,290631,290637,290639,290644,290648,290690,290694,290698,290703,290707,290710,290729,290732,290738,290742,290748,290755,290759,290763,290768,290772,290775,290781,290784,290787,290793,290796,290802,290805,290811,290821,290825,290829,290834,290838,290841,290847,290850,290856,290859,290865,290868,290874,290877,290883,290886,290892,290903,290907,290911,290916,290920,290923,290937,290939,290947,290961,290965,290969,290974,290978,290980,290985,290988,290994,290997,291003,291006,291012,291015,291021,291024,291030,291038,291042,291046,291051,291055,291058,291061,291067,291069,291075,291077,291083,291086,291095,291101,291104,291112,291116,291120,291125,291230,291237,291241,291244,291250,291254,291257,291263,291368,291370,291376,291378,291384,291388,291391,291393,291399,291502,291504,291510,291512,291518,291522,291525,291531,291534,291591,291594,291599,291602,291704,291706,291712,291714,291720,291724,291728,291741,291745,291748,291754,291757,291763,291766,291772,291775,291778,291781,291798,291801,291805,291808,291814,291817,291823,291826,291829,291892,291897],[11,290010,290006],{"id":290011},"review202505-答案讲解-04计算题-11-20",[19,290013,290015],{"id":290014},"_11-磁盘调度scan-与-sstf","11. 磁盘调度：SCAN 与 SSTF",[15,290017,290018],{},[2627,290019,285743],{},[104833,290021,290022,290025,290027],{},[15,290023,290024],{},"11、假定当前磁头位于100号磁道，刚服务过80号磁道。进程对磁道的请求序列依次为55，65，39，28，90，155，145，38，170。当采用扫描算法和最短寻道时间优先算法时，总的移动的磁道数分别是多少？",[15,290026,264297],{},[15,290028,264300],{},[15,290030,290031],{},[2627,290032,285753],{},[15,290034,290035],{},"当前磁头位于 100，刚服务过 80，说明磁头当前移动方向是向磁道号增大的方向。",[15,290037,290038],{},"请求序列：",[24,290040,290043],{"className":290041,"code":290042,"language":2638,"meta":29},[2636],"55, 65, 39, 28, 90, 155, 145, 38, 170\n",[31,290044,290042],{"__ignoreMap":29},[253,290046,290048],{"id":290047},"_1-scan-算法","(1) SCAN 算法",[15,290050,290051],{},"先沿增大方向服务：",[24,290053,290056],{"className":290054,"code":290055,"language":2638,"meta":29},[2636],"145, 155, 170\n",[31,290057,290055],{"__ignoreMap":29},[15,290059,290060],{},"再反向服务：",[24,290062,290065],{"className":290063,"code":290064,"language":2638,"meta":29},[2636],"90, 65, 55, 39, 38, 28\n",[31,290066,290064],{"__ignoreMap":29},[15,290068,290069],{},"寻道次序：",[24,290071,290074],{"className":290072,"code":290073,"language":2638,"meta":29},[2636],"100 -> 145 -> 155 -> 170 -> 90 -> 65 -> 55 -> 39 -> 38 -> 28\n",[31,290075,290073],{"__ignoreMap":29},[15,290077,290078],{},"每步移动磁道数：",[24,290080,290083],{"className":290081,"code":290082,"language":2638,"meta":29},[2636],"45, 10, 15, 80, 25, 10, 16, 1, 10\n",[31,290084,290082],{"__ignoreMap":29},[15,290086,163137],{},[24,290088,290091],{"className":290089,"code":290090,"language":2638,"meta":29},[2636],"45 + 10 + 15 + 80 + 25 + 10 + 16 + 1 + 10 = 212\n",[31,290092,290090],{"__ignoreMap":29},[15,290094,290095,290097,290098,290101],{},[2627,290096,58173],{}," SCAN 总移动 ",[31,290099,290100],{},"212"," 个磁道。",[15,290103,290104],{},"说明：Solutions-OS8e 11.6 对 SCAN 的定义是：磁臂沿一个方向移动，服务途中所有请求，直到到达该方向最后一个磁道或该方向已无请求，然后反向。因此题目没有给磁盘最大磁道号时，按“服务到当前方向最后一个请求 170 后折返”最贴合教材定义。若老师额外要求必须先移动到磁盘端点，例如 199 号磁道，再折返，则总移动数会变为：",[24,290106,290109],{"className":290107,"code":290108,"language":2638,"meta":29},[2636],"100 -> 199 -> 28，总移动 = 99 + 171 = 270\n",[31,290110,290108],{"__ignoreMap":29},[253,290112,290114],{"id":290113},"_2-sstf-最短寻道时间优先","(2) SSTF 最短寻道时间优先",[15,290116,290117],{},"每次选择离当前磁头最近的请求。",[15,290119,290069],{},[24,290121,290124],{"className":290122,"code":290123,"language":2638,"meta":29},[2636],"100 -> 90 -> 65 -> 55 -> 39 -> 38 -> 28 -> 145 -> 155 -> 170\n",[31,290125,290123],{"__ignoreMap":29},[15,290127,290078],{},[24,290129,290132],{"className":290130,"code":290131,"language":2638,"meta":29},[2636],"10, 25, 10, 16, 1, 10, 117, 10, 15\n",[31,290133,290131],{"__ignoreMap":29},[15,290135,163137],{},[24,290137,290140],{"className":290138,"code":290139,"language":2638,"meta":29},[2636],"10 + 25 + 10 + 16 + 1 + 10 + 117 + 10 + 15 = 214\n",[31,290141,290139],{"__ignoreMap":29},[15,290143,290144,290146,290147,290101],{},[2627,290145,58173],{}," SSTF 总移动 ",[31,290148,290149],{},"214",[19,290151,290153],{"id":290152},"_12-请求分页地址序列lru-与-opt","12. 请求分页：地址序列、LRU 与 OPT",[15,290155,290156],{},[2627,290157,285743],{},[104833,290159,290160,290163],{},[15,290161,290162],{},"12、在请求分页系统中，有一个长度为6页的进程，页大小为4KB。假如为它分配3个物理块，逻辑地址为：2F49, 3E98, 2A78, 1C66, 5D18, 5F63, 2C88, 6A10, 5A69, 5D87, 4F00, 3F68。地址为16进制。",[15,290164,290165],{},"问题：试求出进程访问的页面次序，采用LRU和OPT页面置换算法计算出程序访问过程中所发生的缺页次数、淘汰的页面、和缺页中断率。（说明：初始页面为空，计算为缺页。）",[15,290167,290168],{},[2627,290169,285753],{},[15,290171,290172],{},"页大小为 4KB：",[24,290174,290177],{"className":290175,"code":290176,"language":2638,"meta":29},[2636],"4KB = 4096B = 0x1000\n",[31,290178,290176],{"__ignoreMap":29},[15,290180,290181],{},"因此每个十六进制地址的高位十六进制数就是页号，低 3 位十六进制数是页内偏移。",[15,290183,290184],{},"地址序列：",[24,290186,290189],{"className":290187,"code":290188,"language":2638,"meta":29},[2636],"2F49, 3E98, 2A78, 1C66, 5D18, 5F63, 2C88, 6A10, 5A69, 5D87, 4F00, 3F68\n",[31,290190,290188],{"__ignoreMap":29},[15,290192,290193],{},"页面访问序列：",[24,290195,290198],{"className":290196,"code":290197,"language":2638,"meta":29},[2636],"2, 3, 2, 1, 5, 5, 2, 6, 5, 5, 4, 3\n",[31,290199,290197],{"__ignoreMap":29},[15,290201,290202],{},"说明：题面说进程长度为 6 页，但出现页号 6。若页号严格从 0 开始，则页号 6 越界；若按教材题目意图继续做页面置换，则把它作为一次访问页 6 处理。",[253,290204,289479],{"id":289478},[13837,290206,290207,290221],{},[13840,290208,290209],{},[13843,290210,290211,290213,290215,290217,290219],{},[13846,290212,289021],{"align":287595},[13846,290214,289493],{"align":287595},[13846,290216,267766],{},[13846,290218,289498],{"align":287595},[13846,290220,289501],{},[13853,290222,290223,290237,290252,290266,290281,290296,290310,290324,290339,290353,290367,290382],{},[13843,290224,290225,290227,290229,290231,290233],{},[13858,290226,640],{"align":287595},[13858,290228,768],{"align":287595},[13858,290230,289512],{},[13858,290232,2511],{"align":287595},[13858,290234,290235],{},[31,290236,768],{},[13843,290238,290239,290241,290243,290245,290247],{},[13858,290240,768],{"align":287595},[13858,290242,786],{"align":287595},[13858,290244,289512],{},[13858,290246,2511],{"align":287595},[13858,290248,290249],{},[31,290250,290251],{},"2,3",[13843,290253,290254,290256,290258,290260,290262],{},[13858,290255,786],{"align":287595},[13858,290257,768],{"align":287595},[13858,290259,289557],{},[13858,290261,2511],{"align":287595},[13858,290263,290264],{},[31,290265,290251],{},[13843,290267,290268,290270,290272,290274,290276],{},[13858,290269,15932],{"align":287595},[13858,290271,640],{"align":287595},[13858,290273,289512],{},[13858,290275,2511],{"align":287595},[13858,290277,290278],{},[31,290279,290280],{},"2,3,1",[13843,290282,290283,290285,290287,290289,290291],{},[13858,290284,18136],{"align":287595},[13858,290286,18136],{"align":287595},[13858,290288,289512],{},[13858,290290,640],{"align":287595},[13858,290292,290293],{},[31,290294,290295],{},"2,3,5",[13843,290297,290298,290300,290302,290304,290306],{},[13858,290299,36656],{"align":287595},[13858,290301,18136],{"align":287595},[13858,290303,289557],{},[13858,290305,2511],{"align":287595},[13858,290307,290308],{},[31,290309,290295],{},[13843,290311,290312,290314,290316,290318,290320],{},[13858,290313,36723],{"align":287595},[13858,290315,768],{"align":287595},[13858,290317,289557],{},[13858,290319,2511],{"align":287595},[13858,290321,290322],{},[31,290323,290295],{},[13843,290325,290326,290328,290330,290332,290334],{},[13858,290327,60525],{"align":287595},[13858,290329,36656],{"align":287595},[13858,290331,289512],{},[13858,290333,768],{"align":287595},[13858,290335,290336],{},[31,290337,290338],{},"6,3,5",[13843,290340,290341,290343,290345,290347,290349],{},[13858,290342,36732],{"align":287595},[13858,290344,18136],{"align":287595},[13858,290346,289557],{},[13858,290348,2511],{"align":287595},[13858,290350,290351],{},[31,290352,290338],{},[13843,290354,290355,290357,290359,290361,290363],{},[13858,290356,3244],{"align":287595},[13858,290358,18136],{"align":287595},[13858,290360,289557],{},[13858,290362,2511],{"align":287595},[13858,290364,290365],{},[31,290366,290338],{},[13843,290368,290369,290371,290373,290375,290377],{},[13858,290370,39620],{"align":287595},[13858,290372,15932],{"align":287595},[13858,290374,289512],{},[13858,290376,36656],{"align":287595},[13858,290378,290379],{},[31,290380,290381],{},"4,3,5",[13843,290383,290384,290386,290388,290390,290392],{},[13858,290385,71869],{"align":287595},[13858,290387,786],{"align":287595},[13858,290389,289557],{},[13858,290391,2511],{"align":287595},[13858,290393,290394],{},[31,290395,290381],{},[15,290397,290398,290399,258,290401,290403],{},"第 11 步中，",[31,290400,36656],{},[31,290402,18136],{}," 后面都不再访问，淘汰任意一个都可以，不影响缺页次数。",[15,290405,289688],{},[24,290407,290410],{"className":290408,"code":290409,"language":2638,"meta":29},[2636],"6 次\n",[31,290411,290409],{"__ignoreMap":29},[15,290413,290414],{},"OPT 淘汰页面序列：",[24,290416,290419],{"className":290417,"code":290418,"language":2638,"meta":29},[2636],"1, 2, 6\n",[31,290420,290418],{"__ignoreMap":29},[15,290422,289697],{},[24,290424,290427],{"className":290425,"code":290426,"language":2638,"meta":29},[2636],"6 \u002F 12 = 50%\n",[31,290428,290426],{"__ignoreMap":29},[253,290430,289707],{"id":289706},[13837,290432,290433,290447],{},[13840,290434,290435],{},[13843,290436,290437,290439,290441,290443,290445],{},[13846,290438,289021],{"align":287595},[13846,290440,289493],{"align":287595},[13846,290442,267766],{},[13846,290444,289498],{"align":287595},[13846,290446,289501],{},[13853,290448,290449,290463,290477,290491,290505,290520,290534,290548,290563,290577,290591,290606],{},[13843,290450,290451,290453,290455,290457,290459],{},[13858,290452,640],{"align":287595},[13858,290454,768],{"align":287595},[13858,290456,289512],{},[13858,290458,2511],{"align":287595},[13858,290460,290461],{},[31,290462,768],{},[13843,290464,290465,290467,290469,290471,290473],{},[13858,290466,768],{"align":287595},[13858,290468,786],{"align":287595},[13858,290470,289512],{},[13858,290472,2511],{"align":287595},[13858,290474,290475],{},[31,290476,290251],{},[13843,290478,290479,290481,290483,290485,290487],{},[13858,290480,786],{"align":287595},[13858,290482,768],{"align":287595},[13858,290484,289557],{},[13858,290486,2511],{"align":287595},[13858,290488,290489],{},[31,290490,290251],{},[13843,290492,290493,290495,290497,290499,290501],{},[13858,290494,15932],{"align":287595},[13858,290496,640],{"align":287595},[13858,290498,289512],{},[13858,290500,2511],{"align":287595},[13858,290502,290503],{},[31,290504,290280],{},[13843,290506,290507,290509,290511,290513,290515],{},[13858,290508,18136],{"align":287595},[13858,290510,18136],{"align":287595},[13858,290512,289512],{},[13858,290514,786],{"align":287595},[13858,290516,290517],{},[31,290518,290519],{},"2,5,1",[13843,290521,290522,290524,290526,290528,290530],{},[13858,290523,36656],{"align":287595},[13858,290525,18136],{"align":287595},[13858,290527,289557],{},[13858,290529,2511],{"align":287595},[13858,290531,290532],{},[31,290533,290519],{},[13843,290535,290536,290538,290540,290542,290544],{},[13858,290537,36723],{"align":287595},[13858,290539,768],{"align":287595},[13858,290541,289557],{},[13858,290543,2511],{"align":287595},[13858,290545,290546],{},[31,290547,290519],{},[13843,290549,290550,290552,290554,290556,290558],{},[13858,290551,60525],{"align":287595},[13858,290553,36656],{"align":287595},[13858,290555,289512],{},[13858,290557,640],{"align":287595},[13858,290559,290560],{},[31,290561,290562],{},"2,5,6",[13843,290564,290565,290567,290569,290571,290573],{},[13858,290566,36732],{"align":287595},[13858,290568,18136],{"align":287595},[13858,290570,289557],{},[13858,290572,2511],{"align":287595},[13858,290574,290575],{},[31,290576,290562],{},[13843,290578,290579,290581,290583,290585,290587],{},[13858,290580,3244],{"align":287595},[13858,290582,18136],{"align":287595},[13858,290584,289557],{},[13858,290586,2511],{"align":287595},[13858,290588,290589],{},[31,290590,290562],{},[13843,290592,290593,290595,290597,290599,290601],{},[13858,290594,39620],{"align":287595},[13858,290596,15932],{"align":287595},[13858,290598,289512],{},[13858,290600,768],{"align":287595},[13858,290602,290603],{},[31,290604,290605],{},"4,5,6",[13843,290607,290608,290610,290612,290614,290616],{},[13858,290609,71869],{"align":287595},[13858,290611,786],{"align":287595},[13858,290613,289512],{},[13858,290615,36656],{"align":287595},[13858,290617,290618],{},[31,290619,290620],{},"4,5,3",[15,290622,289901],{},[24,290624,290626],{"className":290625,"code":289692,"language":2638,"meta":29},[2636],[31,290627,289692],{"__ignoreMap":29},[15,290629,290630],{},"LRU 淘汰页面序列：",[24,290632,290635],{"className":290633,"code":290634,"language":2638,"meta":29},[2636],"3, 1, 2, 6\n",[31,290636,290634],{"__ignoreMap":29},[15,290638,289910],{},[24,290640,290642],{"className":290641,"code":289701,"language":2638,"meta":29},[2636],[31,290643,289701],{"__ignoreMap":29},[15,290645,290646],{},[2627,290647,289921],{},[13837,290649,290650,290662],{},[13840,290651,290652],{},[13843,290653,290654,290656,290658,290660],{},[13846,290655,153524],{},[13846,290657,289932],{"align":287595},[13846,290659,289935],{},[13846,290661,289938],{"align":287595},[13853,290663,290664,290677],{},[13843,290665,290666,290668,290670,290675],{},[13858,290667,289945],{},[13858,290669,36656],{"align":287595},[13858,290671,290672],{},[31,290673,290674],{},"1, 2, 6",[13858,290676,267984],{"align":287595},[13843,290678,290679,290681,290683,290688],{},[13858,290680,289960],{},[13858,290682,36723],{"align":287595},[13858,290684,290685],{},[31,290686,290687],{},"3, 1, 2, 6",[13858,290689,289955],{"align":287595},[19,290691,290693],{"id":290692},"_13-信号量当前值与资源等待进程数","13. 信号量当前值与资源、等待进程数",[15,290695,290696],{},[2627,290697,285743],{},[104833,290699,290700],{},[15,290701,290702],{},"13、设与某资源相关联的信号量初值为3，当前值为1，若M表示该资源的可用个数，N表示等待资源的进程数，则M,N分别是多少？",[15,290704,290705],{},[2627,290706,285753],{},[15,290708,290709],{},"信号量含义：",[2230,290711,290712,290720],{},[108,290713,35328,290714,4504,290717,290719],{},[31,290715,290716],{},"S >= 0",[31,290718,94675],{}," 表示当前可用资源数，等待进程数为 0。",[108,290721,35328,290722,290725,290726,347],{},[31,290723,290724],{},"S \u003C 0"," 时，资源可用数为 0，等待进程数为 ",[31,290727,290728],{},"-S",[15,290730,290731],{},"当前信号量值：",[24,290733,290736],{"className":290734,"code":290735,"language":2638,"meta":29},[2636],"S = 1\n",[31,290737,290735],{"__ignoreMap":29},[15,290739,93310,290740,287819],{},[31,290741,290716],{},[24,290743,290746],{"className":290744,"code":290745,"language":2638,"meta":29},[2636],"M = 1\nN = 0\n",[31,290747,290745],{"__ignoreMap":29},[15,290749,290750,47649,290752,347],{},[2627,290751,58173],{},[31,290753,290754],{},"M = 1，N = 0",[19,290756,290758],{"id":290757},"_14-图像存储空间","14. 图像存储空间",[15,290760,290761],{},[2627,290762,285743],{},[104833,290764,290765],{},[15,290766,290767],{},"14、计算机系统中，屏幕显示分辨率为1024x768，若要存储一屏256彩色的图像，需要多少字节存储空间？",[15,290769,290770],{},[2627,290771,285753],{},[15,290773,290774],{},"256 色需要的位数：",[24,290776,290779],{"className":290777,"code":290778,"language":2638,"meta":29},[2636],"log2(256) = 8 bit\n",[31,290780,290778],{"__ignoreMap":29},[15,290782,290783],{},"8 bit 等于 1 byte，所以每个像素需要 1 字节。",[15,290785,290786],{},"像素总数：",[24,290788,290791],{"className":290789,"code":290790,"language":2638,"meta":29},[2636],"1024 × 768 = 786,432\n",[31,290792,290790],{"__ignoreMap":29},[15,290794,290795],{},"存储空间：",[24,290797,290800],{"className":290798,"code":290799,"language":2638,"meta":29},[2636],"786,432 × 1B = 786,432B\n",[31,290801,290799],{"__ignoreMap":29},[15,290803,290804],{},"换算为 KB：",[24,290806,290809],{"className":290807,"code":290808,"language":2638,"meta":29},[2636],"786,432B \u002F 1024 = 768KB\n",[31,290810,290808],{"__ignoreMap":29},[15,290812,290813,280220,290815,61605,290818,347],{},[2627,290814,58173],{},[31,290816,290817],{},"786,432B",[31,290819,290820],{},"768KB",[19,290822,290824],{"id":290823},"_15-八进制虚拟地址转换为物理地址","15. 八进制虚拟地址转换为物理地址",[15,290826,290827],{},[2627,290828,285743],{},[104833,290830,290831],{},[15,290832,290833],{},"15、系统提供24位虚存空间，主存为2^18B，分页式虚拟存储管理，页面尺寸为2KB。用户程序虚拟地址11124457(八进制)，页面分得块号为100(八进制)，求物理地址？",[15,290835,290836],{},[2627,290837,285753],{},[15,290839,290840],{},"页面大小：",[24,290842,290845],{"className":290843,"code":290844,"language":2638,"meta":29},[2636],"2KB = 2048B = 2^11 = 4000(8)\n",[31,290846,290844],{"__ignoreMap":29},[15,290848,290849],{},"虚拟地址为：",[24,290851,290854],{"className":290852,"code":290853,"language":2638,"meta":29},[2636],"11124457(8)\n",[31,290855,290853],{"__ignoreMap":29},[15,290857,290858],{},"按页大小拆分虚拟地址：",[24,290860,290863],{"className":290861,"code":290862,"language":2638,"meta":29},[2636],"11124457(8) \u002F 4000(8)\n= 页号 2225(8)，页内偏移 0457(8)\n",[31,290864,290862],{"__ignoreMap":29},[15,290866,290867],{},"题目给出该页分得物理块号：",[24,290869,290872],{"className":290870,"code":290871,"language":2638,"meta":29},[2636],"100(8)\n",[31,290873,290871],{"__ignoreMap":29},[15,290875,290876],{},"物理地址：",[24,290878,290881],{"className":290879,"code":290880,"language":2638,"meta":29},[2636],"物理地址 = 物理块号 × 页面大小 + 页内偏移\n = 100(8) × 4000(8) + 0457(8)\n = 400457(8)\n",[31,290882,290880],{"__ignoreMap":29},[15,290884,290885],{},"换成十六进制：",[24,290887,290890],{"className":290888,"code":290889,"language":2638,"meta":29},[2636],"400457(8) = 0x2012F\n",[31,290891,290889],{"__ignoreMap":29},[15,290893,290894,290896,290897,61605,290900,347],{},[2627,290895,58173],{}," 物理地址为 ",[31,290898,290899],{},"400457(8)",[31,290901,290902],{},"0x2012F",[19,290904,290906],{"id":290905},"_16-符号链接硬链接和引用计数","16. 符号链接、硬链接和引用计数",[15,290908,290909],{},[2627,290910,285743],{},[104833,290912,290913],{},[15,290914,290915],{},"16、设文件F1当前引用计数值为1，先建立F1的符号链接文件F2，再建立F1的硬链接文件F3，然后删除F1。此时，F2和F3的引用计数值分别是多少？",[15,290917,290918],{},[2627,290919,285753],{},[15,290921,290922],{},"分析过程：",[2230,290924,290925,290928,290931,290934],{},[108,290926,290927],{},"初始时，F1 的引用计数为 1。",[108,290929,290930],{},"建立 F1 的符号链接 F2。符号链接是一个独立文件，内容保存目标路径，不增加 F1 的 inode 引用计数。",[108,290932,290933],{},"建立 F1 的硬链接 F3。硬链接与 F1 指向同一个 inode，使该 inode 引用计数从 1 变为 2。",[108,290935,290936],{},"删除 F1。删除的只是 F1 这个目录项，该 inode 引用计数从 2 减为 1。",[15,290938,175712],{},[2230,290940,290941,290944],{},[108,290942,290943],{},"F2 是符号链接文件，它自身的引用计数仍为 1；但由于路径名 F1 被删除，F2 可能成为悬空符号链接。",[108,290945,290946],{},"F3 指向原 F1 的 inode，该 inode 当前引用计数为 1。",[15,290948,290949,47649,290951,290953,290954,46528,290956,290958,290959,347],{},[2627,290950,58173],{},[31,290952,175651],{}," 的引用计数为 ",[31,290955,640],{},[31,290957,175903],{}," 指向的文件 inode 引用计数为 ",[31,290960,640],{},[19,290962,290964],{"id":290963},"_17-虚地址-0bebc-转换为实地址","17. 虚地址 0BEBC 转换为实地址",[15,290966,290967],{},[2627,290968,285743],{},[104833,290970,290971],{},[15,290972,290973],{},"17、假设某虚存的用户空间为1024KB，页面大小为4KB，内存空间为512KB。已知用户的虚页10、11、12、13页分得内存页框号为62、78、25、36，求出虚地址0BEBC(16进制)的实地址(16进制)是多少?",[15,290975,290976],{},[2627,290977,285753],{},[15,290979,290840],{},[24,290981,290983],{"className":290982,"code":290176,"language":2638,"meta":29},[2636],[31,290984,290176],{"__ignoreMap":29},[15,290986,290987],{},"虚地址：",[24,290989,290992],{"className":290990,"code":290991,"language":2638,"meta":29},[2636],"0BEBC(16)\n",[31,290993,290991],{"__ignoreMap":29},[15,290995,290996],{},"拆分为页号和页内偏移：",[24,290998,291001],{"className":290999,"code":291000,"language":2638,"meta":29},[2636],"页号 = 0xB = 11\n页内偏移 = 0xEBC\n",[31,291002,291000],{"__ignoreMap":29},[15,291004,291005],{},"题目给出：",[24,291007,291010],{"className":291008,"code":291009,"language":2638,"meta":29},[2636],"虚页 11 -> 页框 78\n",[31,291011,291009],{"__ignoreMap":29},[15,291013,291014],{},"页框号 78 转为十六进制：",[24,291016,291019],{"className":291017,"code":291018,"language":2638,"meta":29},[2636],"78(10) = 0x4E\n",[31,291020,291018],{"__ignoreMap":29},[15,291022,291023],{},"实地址：",[24,291025,291028],{"className":291026,"code":291027,"language":2638,"meta":29},[2636],"0x4E000 + 0xEBC = 0x4EEBC\n",[31,291029,291027],{"__ignoreMap":29},[15,291031,291032,291034,291035,347],{},[2627,291033,58173],{}," 实地址为 ",[31,291036,291037],{},"4EEBC(16)",[19,291039,291041],{"id":291040},"_18-unixlinux-文件偏移对应几次间接","18. UNIX\u002FLinux 文件偏移对应几次间接",[15,291043,291044],{},[2627,291045,285743],{},[104833,291047,291048],{},[15,291049,291050],{},"18、一个UNIX\u002FLinux文件，如果一个盘块的大小为1KB，每个盘块占4个字节，那么，若进程欲访问偏移为263168字节处的数据，需经过几次间接？",[15,291052,291053],{},[2627,291054,285753],{},[15,291056,291057],{},"题目中的“每个盘块占 4 个字节”通常应理解为“每个盘块指针或盘块地址占 4 个字节”。",[15,291059,291060],{},"每个间接索引块可存放的指针数：",[24,291062,291065],{"className":291063,"code":291064,"language":2638,"meta":29},[2636],"1KB \u002F 4B = 1024 \u002F 4 = 256\n",[31,291066,291064],{"__ignoreMap":29},[15,291068,190893],{},[24,291070,291073],{"className":291071,"code":291072,"language":2638,"meta":29},[2636],"263168B\n",[31,291074,291072],{"__ignoreMap":29},[15,291076,190931],{},[24,291078,291081],{"className":291079,"code":291080,"language":2638,"meta":29},[2636],"263168 \u002F 1024 = 257\n",[31,291082,291080],{"__ignoreMap":29},[15,291084,291085],{},"即访问第 257 号逻辑块，按从 0 开始编号。",[15,291087,291088,291089,291091,291092,291094],{},"Solutions-OS8e 12.12 的 UNIX inode 例题采用 ",[31,291090,3244],{}," 个直接块；本复习卷第 4 题采用 ",[31,291093,71869],{}," 个直接块。无论取 10 还是 12，本题逻辑块 257 都落在一级间接范围内。若按 Solutions-OS8e 12.12 的 10 个直接块：",[24,291096,291099],{"className":291097,"code":291098,"language":2638,"meta":29},[2636],"直接块：0 - 9\n一级间接：10 - 265\n",[31,291100,291098],{"__ignoreMap":29},[15,291102,291103],{},"逻辑块 257 落在一级间接范围内。",[15,291105,291106,291108,291109,291111],{},[2627,291107,58173],{}," 需经过 ",[31,291110,640],{}," 次间接，即一级间接。",[19,291113,291115],{"id":291114},"_19-5-个作业的-fcfssjfhrrn-调度","19. 5 个作业的 FCFS、SJF、HRRN 调度",[15,291117,291118],{},[2627,291119,285743],{},[104833,291121,291122],{},[15,291123,291124],{},"19、如果在一段时间内先后有5个作业，它们提交和运行时间由下表给出。",[13837,291126,291127,291148],{},[13840,291128,291129],{},[13843,291130,291131,291134,291136,291138,291141,291144,291146],{},[13846,291132,291133],{"align":287595},"序号",[13846,291135,272662],{"align":287595},[13846,291137,229533],{"align":287595},[13846,291139,291140],{"align":287595},"开始",[13846,291142,291143],{"align":287595},"结束",[13846,291145,230236],{"align":287595},[13846,291147,230239],{"align":287595},[13853,291149,291150,291166,291182,291198,291214],{},[13843,291151,291152,291154,291156,291158,291160,291162,291164],{},[13858,291153,640],{"align":287595},[13858,291155,230248],{"align":287595},[13858,291157,5774],{"align":287595},[13858,291159],{"align":287595},[13858,291161],{"align":287595},[13858,291163],{"align":287595},[13858,291165],{"align":287595},[13843,291167,291168,291170,291172,291174,291176,291178,291180],{},[13858,291169,768],{"align":287595},[13858,291171,264512],{"align":287595},[13858,291173,9314],{"align":287595},[13858,291175],{"align":287595},[13858,291177],{"align":287595},[13858,291179],{"align":287595},[13858,291181],{"align":287595},[13843,291183,291184,291186,291188,291190,291192,291194,291196],{},[13858,291185,786],{"align":287595},[13858,291187,230265],{"align":287595},[13858,291189,3249],{"align":287595},[13858,291191],{"align":287595},[13858,291193],{"align":287595},[13858,291195],{"align":287595},[13858,291197],{"align":287595},[13843,291199,291200,291202,291204,291206,291208,291210,291212],{},[13858,291201,15932],{"align":287595},[13858,291203,230282],{"align":287595},[13858,291205,3249],{"align":287595},[13858,291207],{"align":287595},[13858,291209],{"align":287595},[13858,291211],{"align":287595},[13858,291213],{"align":287595},[13843,291215,291216,291218,291220,291222,291224,291226,291228],{},[13858,291217,18136],{"align":287595},[13858,291219,230299],{"align":287595},[13858,291221,3244],{"align":287595},[13858,291223],{"align":287595},[13858,291225],{"align":287595},[13858,291227],{"align":287595},[13858,291229],{"align":287595},[104833,291231,291232,291235],{},[15,291233,291234],{},"问题：(1) 分别使用先来先服务、最短进程优先和最高响应比算法完成上表，给出各个作业的开始时间、完成时间、周转时间和带权周转时间；",[15,291236,272445],{},[15,291238,291239],{},[2627,291240,285753],{},[15,291242,291243],{},"计算公式：",[24,291245,291248],{"className":291246,"code":291247,"language":2638,"meta":29},[2636],"周转时间 = 完成时间 - 提交时间\n带权周转时间 = 周转时间 \u002F 运行时间\n",[31,291249,291247],{"__ignoreMap":29},[253,291251,291253],{"id":291252},"_1-fcfs-先来先服务","(1) FCFS 先来先服务",[15,291255,291256],{},"调度顺序：",[24,291258,291261],{"className":291259,"code":291260,"language":2638,"meta":29},[2636],"J1 -> J2 -> J3 -> J4 -> J5\n",[31,291262,291260],{"__ignoreMap":29},[13837,291264,291265,291283],{},[13840,291266,291267],{},[13843,291268,291269,291271,291273,291275,291277,291279,291281],{},[13846,291270,230486],{"align":287595},[13846,291272,272662],{"align":287595},[13846,291274,229533],{"align":287595},[13846,291276,291140],{"align":287595},[13846,291278,291143],{"align":287595},[13846,291280,230236],{"align":287595},[13846,291282,230239],{"align":287595},[13853,291284,291285,291301,291318,291335,291351],{},[13843,291286,291287,291289,291291,291293,291295,291297,291299],{},[13858,291288,640],{"align":287595},[13858,291290,230248],{"align":287595},[13858,291292,5774],{"align":287595},[13858,291294,230248],{"align":287595},[13858,291296,230424],{"align":287595},[13858,291298,5774],{"align":287595},[13858,291300,230883],{"align":287595},[13843,291302,291303,291305,291307,291309,291311,291313,291315],{},[13858,291304,768],{"align":287595},[13858,291306,264512],{"align":287595},[13858,291308,9314],{"align":287595},[13858,291310,230424],{"align":287595},[13858,291312,272709],{"align":287595},[13858,291314,272712],{"align":287595},[13858,291316,291317],{"align":287595},"2.17",[13843,291319,291320,291322,291324,291326,291328,291330,291332],{},[13858,291321,786],{"align":287595},[13858,291323,230265],{"align":287595},[13858,291325,3249],{"align":287595},[13858,291327,272709],{"align":287595},[13858,291329,271462],{"align":287595},[13858,291331,27539],{"align":287595},[13858,291333,291334],{"align":287595},"4.00",[13843,291336,291337,291339,291341,291343,291345,291347,291349],{},[13858,291338,15932],{"align":287595},[13858,291340,230282],{"align":287595},[13858,291342,3249],{"align":287595},[13858,291344,271462],{"align":287595},[13858,291346,229149],{"align":287595},[13858,291348,96817],{"align":287595},[13858,291350,272753],{"align":287595},[13843,291352,291353,291355,291357,291359,291361,291363,291365],{},[13858,291354,18136],{"align":287595},[13858,291356,230299],{"align":287595},[13858,291358,3244],{"align":287595},[13858,291360,229149],{"align":287595},[13858,291362,229167],{"align":287595},[13858,291364,162173],{"align":287595},[13858,291366,291367],{"align":287595},"10.00",[15,291369,226112],{},[24,291371,291374],{"className":291372,"code":291373,"language":2638,"meta":29},[2636],"(50 + 65 + 80 + 95 + 100) \u002F 5 = 78\n",[31,291375,291373],{"__ignoreMap":29},[15,291377,230856],{},[24,291379,291382],{"className":291380,"code":291381,"language":2638,"meta":29},[2636],"(1.00 + 2.17 + 4.00 + 4.75 + 10.00) \u002F 5 ≈ 4.38\n",[31,291383,291381],{"__ignoreMap":29},[253,291385,291387],{"id":291386},"_2-sjf-最短进程优先","(2) SJF 最短进程优先",[15,291389,291390],{},"J1 到达时只有 J1，因此先运行 J1。J1 完成后所有作业均已到达，按运行时间从短到长选择。",[15,291392,291256],{},[24,291394,291397],{"className":291395,"code":291396,"language":2638,"meta":29},[2636],"J1 -> J5 -> J3 -> J4 -> J2\n",[31,291398,291396],{"__ignoreMap":29},[13837,291400,291401,291419],{},[13840,291402,291403],{},[13843,291404,291405,291407,291409,291411,291413,291415,291417],{},[13846,291406,230486],{"align":287595},[13846,291408,272662],{"align":287595},[13846,291410,229533],{"align":287595},[13846,291412,291140],{"align":287595},[13846,291414,291143],{"align":287595},[13846,291416,230236],{"align":287595},[13846,291418,230239],{"align":287595},[13853,291420,291421,291437,291454,291470,291486],{},[13843,291422,291423,291425,291427,291429,291431,291433,291435],{},[13858,291424,640],{"align":287595},[13858,291426,230248],{"align":287595},[13858,291428,5774],{"align":287595},[13858,291430,230248],{"align":287595},[13858,291432,230424],{"align":287595},[13858,291434,5774],{"align":287595},[13858,291436,230883],{"align":287595},[13843,291438,291439,291441,291443,291445,291447,291449,291451],{},[13858,291440,768],{"align":287595},[13858,291442,264512],{"align":287595},[13858,291444,9314],{"align":287595},[13858,291446,271462],{"align":287595},[13858,291448,229167],{"align":287595},[13858,291450,273130],{"align":287595},[13858,291452,291453],{"align":287595},"3.83",[13843,291455,291456,291458,291460,291462,291464,291466,291468],{},[13858,291457,786],{"align":287595},[13858,291459,230265],{"align":287595},[13858,291461,3249],{"align":287595},[13858,291463,230441],{"align":287595},[13858,291465,272709],{"align":287595},[13858,291467,167664],{"align":287595},[13858,291469,230893],{"align":287595},[13843,291471,291472,291474,291476,291478,291480,291482,291484],{},[13858,291473,15932],{"align":287595},[13858,291475,230282],{"align":287595},[13858,291477,3249],{"align":287595},[13858,291479,272709],{"align":287595},[13858,291481,271462],{"align":287595},[13858,291483,267828],{"align":287595},[13858,291485,273113],{"align":287595},[13843,291487,291488,291490,291492,291494,291496,291498,291500],{},[13858,291489,18136],{"align":287595},[13858,291491,230299],{"align":287595},[13858,291493,3244],{"align":287595},[13858,291495,230424],{"align":287595},[13858,291497,230441],{"align":287595},[13858,291499,9314],{"align":287595},[13858,291501,230893],{"align":287595},[15,291503,226112],{},[24,291505,291508],{"className":291506,"code":291507,"language":2638,"meta":29},[2636],"(50 + 115 + 60 + 75 + 30) \u002F 5 = 66\n",[31,291509,291507],{"__ignoreMap":29},[15,291511,230856],{},[24,291513,291516],{"className":291514,"code":291515,"language":2638,"meta":29},[2636],"(1.00 + 3.83 + 3.00 + 3.75 + 3.00) \u002F 5 ≈ 2.92\n",[31,291517,291515],{"__ignoreMap":29},[253,291519,291521],{"id":291520},"_3-hrrn-最高响应比优先","(3) HRRN 最高响应比优先",[15,291523,291524],{},"响应比公式：",[24,291526,291529],{"className":291527,"code":291528,"language":2638,"meta":29},[2636],"响应比 = (等待时间 + 服务时间) \u002F 服务时间\n = 1 + 等待时间 \u002F 服务时间\n",[31,291530,291528],{"__ignoreMap":29},[15,291532,291533],{},"J1 到达时只有 J1，因此先运行 J1。8:50 时各作业响应比：",[13837,291535,291536,291548],{},[13840,291537,291538],{},[13843,291539,291540,291542,291544,291546],{},[13846,291541,230486],{"align":287595},[13846,291543,216949],{"align":287595},[13846,291545,216954],{"align":287595},[13846,291547,258284],{"align":287595},[13853,291549,291550,291560,291571,291581],{},[13843,291551,291552,291554,291556,291558],{},[13858,291553,273471],{"align":287595},[13858,291555,5752],{"align":287595},[13858,291557,9314],{"align":287595},[13858,291559,291317],{"align":287595},[13843,291561,291562,291564,291566,291568],{},[13858,291563,273486],{"align":287595},[13858,291565,9314],{"align":287595},[13858,291567,3249],{"align":287595},[13858,291569,291570],{"align":287595},"2.50",[13843,291572,291573,291575,291577,291579],{},[13858,291574,273500],{"align":287595},[13858,291576,95556],{"align":287595},[13858,291578,3249],{"align":287595},[13858,291580,230918],{"align":287595},[13843,291582,291583,291585,291587,291589],{},[13858,291584,273915],{"align":287595},[13858,291586,3249],{"align":287595},[13858,291588,3244],{"align":287595},[13858,291590,230893],{"align":287595},[15,291592,291593],{},"选择 J5。继续计算可得调度顺序：",[24,291595,291597],{"className":291596,"code":291396,"language":2638,"meta":29},[2636],[31,291598,291396],{"__ignoreMap":29},[15,291600,291601],{},"因此本题 HRRN 与 SJF 结果相同：",[13837,291603,291604,291622],{},[13840,291605,291606],{},[13843,291607,291608,291610,291612,291614,291616,291618,291620],{},[13846,291609,230486],{"align":287595},[13846,291611,272662],{"align":287595},[13846,291613,229533],{"align":287595},[13846,291615,291140],{"align":287595},[13846,291617,291143],{"align":287595},[13846,291619,230236],{"align":287595},[13846,291621,230239],{"align":287595},[13853,291623,291624,291640,291656,291672,291688],{},[13843,291625,291626,291628,291630,291632,291634,291636,291638],{},[13858,291627,640],{"align":287595},[13858,291629,230248],{"align":287595},[13858,291631,5774],{"align":287595},[13858,291633,230248],{"align":287595},[13858,291635,230424],{"align":287595},[13858,291637,5774],{"align":287595},[13858,291639,230883],{"align":287595},[13843,291641,291642,291644,291646,291648,291650,291652,291654],{},[13858,291643,768],{"align":287595},[13858,291645,264512],{"align":287595},[13858,291647,9314],{"align":287595},[13858,291649,271462],{"align":287595},[13858,291651,229167],{"align":287595},[13858,291653,273130],{"align":287595},[13858,291655,291453],{"align":287595},[13843,291657,291658,291660,291662,291664,291666,291668,291670],{},[13858,291659,786],{"align":287595},[13858,291661,230265],{"align":287595},[13858,291663,3249],{"align":287595},[13858,291665,230441],{"align":287595},[13858,291667,272709],{"align":287595},[13858,291669,167664],{"align":287595},[13858,291671,230893],{"align":287595},[13843,291673,291674,291676,291678,291680,291682,291684,291686],{},[13858,291675,15932],{"align":287595},[13858,291677,230282],{"align":287595},[13858,291679,3249],{"align":287595},[13858,291681,272709],{"align":287595},[13858,291683,271462],{"align":287595},[13858,291685,267828],{"align":287595},[13858,291687,273113],{"align":287595},[13843,291689,291690,291692,291694,291696,291698,291700,291702],{},[13858,291691,18136],{"align":287595},[13858,291693,230299],{"align":287595},[13858,291695,3244],{"align":287595},[13858,291697,230424],{"align":287595},[13858,291699,230441],{"align":287595},[13858,291701,9314],{"align":287595},[13858,291703,230893],{"align":287595},[15,291705,226112],{},[24,291707,291710],{"className":291708,"code":291709,"language":2638,"meta":29},[2636],"66\n",[31,291711,291709],{"__ignoreMap":29},[15,291713,230856],{},[24,291715,291718],{"className":291716,"code":291717,"language":2638,"meta":29},[2636],"约 2.92\n",[31,291719,291717],{"__ignoreMap":29},[19,291721,291723],{"id":291722},"_20-固定资源申请顺序与死锁","20. 固定资源申请顺序与死锁",[15,291725,291726],{},[2627,291727,285743],{},[104833,291729,291730,291733,291735,291738],{},[15,291731,291732],{},"20、某系统有R1设备3台，R2设备4台，它们被P1、P2、P3和P4进程共享，且已知这4个进程均按以下顺序使用设备：",[15,291734,273317],{},[15,291736,291737],{},"系统运行中可能产生死锁吗？为什么？",[15,291739,291740],{},"若可能的话，请举出一种情况，并画出表示该死锁状态的进程—资源图。",[15,291742,291743],{},[2627,291744,285753],{},[15,291746,291747],{},"资源数量：",[24,291749,291752],{"className":291750,"code":291751,"language":2638,"meta":29},[2636],"R1 有 3 台\nR2 有 4 台\n",[31,291753,291751],{"__ignoreMap":29},[15,291755,291756],{},"每个进程的资源使用顺序：",[24,291758,291761],{"className":291759,"code":291760,"language":2638,"meta":29},[2636],"申请 R1 -> 申请 R2 -> 申请 R1 -> 释放 R1 -> 释放 R2 -> 释放 R1\n",[31,291762,291760],{"__ignoreMap":29},[15,291764,291765],{},"每个进程最多需要：",[24,291767,291770],{"className":291768,"code":291769,"language":2638,"meta":29},[2636],"R1: 2 台\nR2: 1 台\n",[31,291771,291769],{"__ignoreMap":29},[253,291773,291774],{"id":291774},"是否可能死锁",[15,291776,291777],{},"可能产生死锁。",[15,291779,291780],{},"构造一种状态：",[2230,291782,291783,291786,291789,291792,291795],{},[108,291784,291785],{},"P1、P2、P3 各自已经申请到第 1 台 R1。",[108,291787,291788],{},"P1、P2、P3 又各自申请到 1 台 R2。",[108,291790,291791],{},"此时 3 台 R1 全部被 P1、P2、P3 占有。",[108,291793,291794],{},"P1、P2、P3 都继续申请第 2 台 R1。",[108,291796,291797],{},"系统中已无空闲 R1，而这些进程只有申请到第 2 台 R1 后才会执行释放操作。",[15,291799,291800],{},"于是 P1、P2、P3 都持有资源并等待 R1，没有任何一个能继续运行到释放资源的位置，因此形成死锁。",[253,291802,291804],{"id":291803},"进程资源图文字表示","进程—资源图文字表示",[15,291806,291807],{},"已分配边：",[24,291809,291812],{"className":291810,"code":291811,"language":2638,"meta":29},[2636],"R1_1 -> P1\nR1_2 -> P2\nR1_3 -> P3\n\nR2_1 -> P1\nR2_2 -> P2\nR2_3 -> P3\n",[31,291813,291811],{"__ignoreMap":29},[15,291815,291816],{},"请求边：",[24,291818,291821],{"className":291819,"code":291820,"language":2638,"meta":29},[2636],"P1 -> R1\nP2 -> R1\nP3 -> R1\n",[31,291822,291820],{"__ignoreMap":29},[15,291824,291825],{},"由于 R1 的 3 个实例都已分配出去，P1、P2、P3 又都请求新的 R1 实例，且它们不释放已占资源，因此死锁成立。",[15,291827,291828],{},"可用 Mermaid 图表示：",[24,291830,291832],{"className":98778,"code":291831,"language":98780,"meta":29,"style":29},"flowchart LR\n R11[\"R1 实例1\"] --> P1[\"P1\"]\n R12[\"R1 实例2\"] --> P2[\"P2\"]\n R13[\"R1 实例3\"] --> P3[\"P3\"]\n\n R21[\"R2 实例1\"] --> P1\n R22[\"R2 实例2\"] --> P2\n R23[\"R2 实例3\"] --> P3\n\n P1 -.请求.-> R1[\"R1 无空闲实例\"]\n P2 -.请求.-> R1\n P3 -.请求.-> R1\n",[31,291833,291834,291839,291844,291849,291854,291858,291863,291868,291873,291877,291882,291887],{"__ignoreMap":29},[34,291835,291836],{"class":36,"line":37},[34,291837,291838],{},"flowchart LR\n",[34,291840,291841],{"class":36,"line":43},[34,291842,291843],{}," R11[\"R1 实例1\"] --> P1[\"P1\"]\n",[34,291845,291846],{"class":36,"line":49},[34,291847,291848],{}," R12[\"R1 实例2\"] --> P2[\"P2\"]\n",[34,291850,291851],{"class":36,"line":56},[34,291852,291853],{}," R13[\"R1 实例3\"] --> P3[\"P3\"]\n",[34,291855,291856],{"class":36,"line":62},[34,291857,53],{"emptyLinePlaceholder":52},[34,291859,291860],{"class":36,"line":68},[34,291861,291862],{}," R21[\"R2 实例1\"] --> P1\n",[34,291864,291865],{"class":36,"line":74},[34,291866,291867],{}," R22[\"R2 实例2\"] --> P2\n",[34,291869,291870],{"class":36,"line":1971},[34,291871,291872],{}," R23[\"R2 实例3\"] --> P3\n",[34,291874,291875],{"class":36,"line":1976},[34,291876,53],{"emptyLinePlaceholder":52},[34,291878,291879],{"class":36,"line":1985},[34,291880,291881],{}," P1 -.请求.-> R1[\"R1 无空闲实例\"]\n",[34,291883,291884],{"class":36,"line":1990},[34,291885,291886],{}," P2 -.请求.-> R1\n",[34,291888,291889],{"class":36,"line":1999},[34,291890,291891],{}," P3 -.请求.-> R1\n",[15,291893,291894,291896],{},[2627,291895,58173],{}," 系统运行中可能产生死锁。典型死锁状态是 P1、P2、P3 各持有 1 台 R1 和 1 台 R2，同时都请求第 2 台 R1，而 R1 已无空闲实例，导致它们均无法继续执行并释放资源。",[79,291898,81],{},{"title":29,"searchDepth":43,"depth":43,"links":291900},[291901,291905,291909,291910,291911,291912,291913,291914,291915,291920],{"id":290014,"depth":43,"text":290015,"children":291902},[291903,291904],{"id":290047,"depth":49,"text":290048},{"id":290113,"depth":49,"text":290114},{"id":290152,"depth":43,"text":290153,"children":291906},[291907,291908],{"id":289478,"depth":49,"text":289479},{"id":289706,"depth":49,"text":289707},{"id":290692,"depth":43,"text":290693},{"id":290757,"depth":43,"text":290758},{"id":290823,"depth":43,"text":290824},{"id":290905,"depth":43,"text":290906},{"id":290963,"depth":43,"text":290964},{"id":291040,"depth":43,"text":291041},{"id":291114,"depth":43,"text":291115,"children":291916},[291917,291918,291919],{"id":291252,"depth":49,"text":291253},{"id":291386,"depth":49,"text":291387},{"id":291520,"depth":49,"text":291521},{"id":291722,"depth":43,"text":291723,"children":291921},[291922,291923],{"id":291774,"depth":49,"text":291774},{"id":291803,"depth":49,"text":291804},{},"\u002Fsource\u002Fcs\u002Fos\u002Freview202505\u002F04-11-20",{"title":290006,"description":29},"source\u002Fcs\u002Fos\u002Freview202505\u002F04-计算题11-20","m2XyId4OM1d19Q5V-HWvlD0mJTlODPj5fsdOw6S_BeU",{"id":291930,"title":291931,"body":291932,"description":29,"extension":85,"meta":293352,"navigation":52,"path":293353,"seo":293354,"stem":293355,"__hash__":293356},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap1.md","Chap. 1",{"type":8,"value":291933,"toc":293347},[291934,291938,291942,291947,291968,291973,291991,291996,292023,292028,292046,292051,292077,292082,292111,292116,292138,292143,292164,292169,292191,292196,292222,292227,292253,292258,292280,292285,292303,292308,292329,292334,292360,292365,292387,292389,292393,292399,292418,292424,292459,292465,292491,292497,292519,292525,292547,292553,292577,292583,292605,292611,292645,292651,292674,292680,292711,292717,292740,292746,292765,292771,292801,292807,292829,292835,292862,292868,292898,292900,292904,292909,292931,292936,292958,292963,292986,292991,293014,293019,293038,293043,293062,293067,293092,293097,293130,293135,293161,293166,293188,293193,293215,293220,293239,293244,293267,293272,293291,293296,293319,293324],[11,291935,291937],{"id":291936},"chapter-1-computer-system-overview","Chapter 1 - Computer System Overview",[19,291939,291941],{"id":291940},"一-true-false-questions-判断题","一、 True \u002F False Questions (判断题)",[15,291943,291944],{},[2627,291945,291946],{},"1. The operating system acts as an interface between the computer hardware and the human user.",[105,291948,291949,291955,291962],{},[108,291950,291951,291954],{},[2627,291952,291953],{},"中文翻译","：操作系统充当计算机硬件和人类用户之间的接口。",[108,291956,291957,2664,291959],{},[2627,291958,118575],{},[2627,291960,291961],{},"True (正确)",[108,291963,291964,291967],{},[2627,291965,291966],{},"知识点讲解","：这是操作系统的核心定义之一。操作系统向用户（以及应用程序员）提供一个方便、抽象的接口，屏蔽了底层硬件的复杂细节（例如，你不需要知道如何控制硬盘磁头就能保存文件）。",[15,291969,291970],{},[2627,291971,291972],{},"2. One of the processor's main functions is to exchange data with memory.",[105,291974,291975,291980,291986],{},[108,291976,291977,291979],{},[2627,291978,291953],{},"：处理器的主要功能之一是与内存交换数据。",[108,291981,291982,2664,291984],{},[2627,291983,118575],{},[2627,291985,291961],{},[108,291987,291988,291990],{},[2627,291989,291966],{},"：处理器的基本操作包括：读取指令、读取数据（操作数）、处理数据、写回结果。在这个过程中，它必须通过系统总线频繁地与主存（Main Memory）进行数据交换。",[15,291992,291993],{},[2627,291994,291995],{},"3. User-visible registers are typically accessible to system programs but are not typically available to application programs.",[105,291997,291998,292003,292010],{},[108,291999,292000,292002],{},[2627,292001,291953],{},"：用户可见寄存器通常可由系统程序访问，但通常不可供应用程序访问。",[108,292004,292005,2664,292007],{},[2627,292006,118575],{},[2627,292008,292009],{},"False (错误)",[108,292011,292012,292014,292015,292018,292019,292022],{},[2627,292013,291966],{},"：名字已经说明了，“",[2627,292016,292017],{},"用户可见寄存器 (User-visible registers)","”正是设计给所有程序员（包括应用程序员）使用的，例如通用数据寄存器和地址寄存器。而",[2627,292020,292021],{},"控制和状态寄存器 (Control\u002FStatus registers)","（如程序计数器 PC、指令寄存器 IR、程序状态字 PSW 的某些位）才是通常只允许系统程序（操作系统内核）在特权模式下修改的。",[15,292024,292025],{},[2627,292026,292027],{},"4. Data registers are general purpose in nature, but may be restricted to specific tasks such as performing floating-point operations.",[105,292029,292030,292035,292041],{},[108,292031,292032,292034],{},[2627,292033,291953],{},"：数据寄存器本质上是通用的，但可能会被限制于特定任务，例如执行浮点运算。",[108,292036,292037,2664,292039],{},[2627,292038,118575],{},[2627,292040,291961],{},[108,292042,292043,292045],{},[2627,292044,291966],{},"：数据寄存器可用于各种算术和逻辑操作。但在很多处理器架构中，为了效率，会将通用整数寄存器和专用的浮点寄存器（Floating-point registers）分开，后者专门用于执行特定的浮点运算任务。",[15,292047,292048],{},[2627,292049,292050],{},"5. The Program Status Word contains status information in the form of condition codes, which are bits typically set by the programmer as a result of program operation.",[105,292052,292053,292058,292064],{},[108,292054,292055,292057],{},[2627,292056,291953],{},"：程序状态字 (PSW) 以条件码的形式包含状态信息，这些位通常是由程序员作为程序操作的结果设置的。",[108,292059,292060,2664,292062],{},[2627,292061,118575],{},[2627,292063,292009],{},[108,292065,292066,292068,292069,292072,292073,292076],{},[2627,292067,291966],{},"：PSW 中的条件码（如进位标志 Carry、溢出标志 Overflow、零标志 Zero、负数标志 Sign）是由",[2627,292070,292071],{},"处理器硬件自动设置的","，以反映最近一次算术或逻辑操作的结果。程序员可以测试（读取）这些位来进行条件跳转，但通常",[2627,292074,292075],{},"不直接手动设置","它们。",[15,292078,292079],{},[2627,292080,292081],{},"6. The processing required for a single instruction on a typical computer system is called the Execute Cycle.",[105,292083,292084,292089,292095],{},[108,292085,292086,292088],{},[2627,292087,291953],{},"：在典型的计算机系统上处理单个指令所需的过程被称为执行周期 (Execute Cycle)。",[108,292090,292091,2664,292093],{},[2627,292092,118575],{},[2627,292094,292009],{},[108,292096,292097,292099,292100,292103,292104,258,292107,292110],{},[2627,292098,291966],{},"：处理单个指令所需的完整过程被称为",[2627,292101,292102],{},"指令周期 (Instruction Cycle)","。一个基本的指令周期至少包含两个步骤：",[2627,292105,292106],{},"获取周期 (Fetch Cycle)",[2627,292108,292109],{},"执行周期 (Execute Cycle)","。Execute Cycle 只是其中的一半。",[15,292112,292113],{},[2627,292114,292115],{},"7. A fetched instruction is normally loaded into the Instruction Register (IR).",[105,292117,292118,292123,292129],{},[108,292119,292120,292122],{},[2627,292121,291953],{},"：获取的指令通常被加载到指令寄存器 (IR) 中。",[108,292124,292125,2664,292127],{},[2627,292126,118575],{},[2627,292128,291961],{},[108,292130,292131,292133,292134,292137],{},[2627,292132,291966],{},"：这是指令执行的标准流程。处理器从内存中读取指令后，会将其存放在 CPU 内部的",[2627,292135,292136],{},"指令寄存器 (Instruction Register, IR)"," 中，随后解码并执行该指令。",[15,292139,292140],{},[2627,292141,292142],{},"8. An interrupt is a mechanism used by system modules to signal the processor that normal processing should be temporarily suspended.",[105,292144,292145,292150,292156],{},[108,292146,292147,292149],{},[2627,292148,291953],{},"：中断是系统模块用来向处理器发出信号，指示应暂时中止正常处理的机制。",[108,292151,292152,2664,292154],{},[2627,292153,118575],{},[2627,292155,291961],{},[108,292157,292158,2664,292160,292163],{},[2627,292159,291966],{},[2627,292161,292162],{},"中断 (Interrupt)"," 允许 I\u002FO 设备、计时器或其他硬件组件打断 CPU 当前的顺序执行流程，使其立刻去处理更高优先级或突发的紧急事件（即执行中断处理程序），处理完毕后再恢复原程序的执行。",[15,292165,292166],{},[2627,292167,292168],{},"9. To accommodate interrupts, an extra fetch cycle is added to the instruction cycle.",[105,292170,292171,292176,292182],{},[108,292172,292173,292175],{},[2627,292174,291953],{},"：为了适应中断，指令周期中增加了一个额外的获取周期 (fetch cycle)。",[108,292177,292178,2664,292180],{},[2627,292179,118575],{},[2627,292181,292009],{},[108,292183,292184,292186,292187,292190],{},[2627,292185,291966],{},"：为了处理中断，指令周期中增加的不是 fetch cycle，而是",[2627,292188,292189],{},"中断周期 (Interrupt Cycle)","。在这个周期中，处理器会检查是否有中断挂起，如果有，则保存当前状态并跳转到中断处理程序。",[15,292192,292193],{},[2627,292194,292195],{},"10. The minimum information that must be saved before the processor transfers control to the interrupt handler routine is the program status word (PSW) and the location of the current instruction.",[105,292197,292198,292203,292209],{},[108,292199,292200,292202],{},[2627,292201,291953],{},"：处理器将控制权转移给中断处理程序之前必须保存的最小信息是程序状态字 (PSW) 和当前指令的位置。",[108,292204,292205,2664,292207],{},[2627,292206,118575],{},[2627,292208,292009],{},[108,292210,292211,292213,292214,292217,292218,292221],{},[2627,292212,291966],{},"：处理器必须保存 PSW 和",[2627,292215,292216],{},"下一条将要执行的指令的位置","（即",[2627,292219,292220],{},"程序计数器 PC"," 的值），而不是“当前指令的位置”。如果保存的是当前指令，中断返回后就会重复执行同一条指令。",[15,292223,292224],{},[2627,292225,292226],{},"11. One approach to dealing with multiple interrupts is to disable all interrupts while an interrupt is being processed.",[105,292228,292229,292234,292240],{},[108,292230,292231,292233],{},[2627,292232,291953],{},"：处理多个中断的一种方法是在处理中断时禁用所有中断。",[108,292235,292236,2664,292238],{},[2627,292237,118575],{},[2627,292239,291961],{},[108,292241,292242,292244,292245,292248,292249,292252],{},[2627,292243,291966],{},"：这是处理多重中断最简单的方法：",[2627,292246,292247],{},"顺序中断处理 (Sequential interrupt processing)","。在处理一个中断时关闭中断响应，其他中断只能排队等待。另一种更高级的方法是",[2627,292250,292251],{},"嵌套中断 (Nested interrupts)","，允许高优先级中断打断低优先级中断。",[15,292254,292255],{},[2627,292256,292257],{},"12. Multiprogramming allows the processor to make use of idle time caused by long-wait interrupt handling.",[105,292259,292260,292265,292271],{},[108,292261,292262,292264],{},[2627,292263,291953],{},"：多道程序设计允许处理器利用因长时间等待中断处理而造成的空闲时间。",[108,292266,292267,2664,292269],{},[2627,292268,118575],{},[2627,292270,291961],{},[108,292272,292273,292275,292276,292279],{},[2627,292274,291966],{},"：这是",[2627,292277,292278],{},"多道程序设计 (Multiprogramming)"," 的核心优势。当一个程序（A）因为发起 I\u002FO 操作而必须长时间等待时，CPU 可以切换去执行内存中的另一个程序（B），从而最大限度地减少了 CPU 的空闲浪费。",[15,292281,292282],{},[2627,292283,292284],{},"13. In a two-level memory hierarchy, the Hit Ratio is defined as the fraction of all memory accesses found in the slower memory.",[105,292286,292287,292292,292298],{},[108,292288,292289,292291],{},[2627,292290,291953],{},"：在两级内存层次结构中，命中率 (Hit Ratio) 被定义为在较慢内存中找到的所有内存访问的比例。",[108,292293,292294,2664,292296],{},[2627,292295,118575],{},[2627,292297,292009],{},[108,292299,292300,292302],{},[2627,292301,291966],{},"：命中率定义为在**较快\u002F较高层内存（例如 Cache）**中找到所需数据的访问次数占总内存访问次数的比例。如果在慢速内存中才找到，那叫缺失 (Miss)。",[15,292304,292305],{},[2627,292306,292307],{},"14. Cache memory exploits the principle of locality by providing a small, fast memory between the processor and main memory.",[105,292309,292310,292315,292321],{},[108,292311,292312,292314],{},[2627,292313,291953],{},"：高速缓存 (Cache memory) 利用局部性原理，在处理器和主存之间提供了一个小型快速的内存。",[108,292316,292317,2664,292319],{},[2627,292318,118575],{},[2627,292320,291961],{},[108,292322,292323,2664,292325,292328],{},[2627,292324,291966],{},[2627,292326,292327],{},"局部性原理 (Principle of Locality)","（分为时间局部性和空间局部性）表明程序在短期内倾向于集中访问一小部分内存区域。Cache 正是利用这一点，将最可能被访问的数据保留在靠近 CPU 的高速区域，极大提升了系统速度。",[15,292330,292331],{},[2627,292332,292333],{},"15. In cache memory design, block size refers to the unit of data exchanged between cache and main memory.",[105,292335,292336,292341,292347],{},[108,292337,292338,292340],{},[2627,292339,291953],{},"：在高速缓存设计中，块大小 (block size) 指的是在缓存和主存之间交换数据的单位。",[108,292342,292343,2664,292345],{},[2627,292344,118575],{},[2627,292346,291961],{},[108,292348,292349,292351,292352,292355,292356,292359],{},[2627,292350,291966],{},"：主存被划分为固定大小的",[2627,292353,292354],{},"块 (Blocks)","，Cache 被划分为同样大小的",[2627,292357,292358],{},"槽\u002F行 (Slots\u002FLines)","。Cache 和主存之间的数据传输不是按字节，而是按整个 Block 为单位进行的。",[15,292361,292362],{},[2627,292363,292364],{},"16. The primary problem with programmed I\u002FO is that the processor must wait for the I\u002FO module to become ready and must repeatedly interrogate the status of the I\u002FO module while waiting.",[105,292366,292367,292372,292378],{},[108,292368,292369,292371],{},[2627,292370,291953],{},"：可编程 I\u002FO (programmed I\u002FO) 的主要问题是，处理器必须等待 I\u002FO 模块准备就绪，并在等待时必须重复询问 I\u002FO 模块的状态。",[108,292373,292374,2664,292376],{},[2627,292375,118575],{},[2627,292377,291961],{},[108,292379,292380,292382,292383,292386],{},[2627,292381,291966],{},"：在",[2627,292384,292385],{},"程序控制 I\u002FO (Programmed I\u002FO)"," 中，CPU 发出 I\u002FO 指令后，必须通过一个循环不断读取状态寄存器来检查 I\u002FO 是否完成（即轮询 Polling）。这会导致 CPU 时间被大量浪费在无意义的等待上（忙等 busy-waiting）。",[32188,292388],{},[19,292390,292392],{"id":292391},"二-multiple-choice-questions-单选题","二、 Multiple Choice Questions (单选题)",[15,292394,292395,292398],{},[2627,292396,292397],{},"1. The general role of an operating system is to:","\na. Act as an interface between various computers\nb. Provide a set of services to system users\nc. Manage files for application programs\nd. None of the above",[105,292400,292401,292406,292413],{},[108,292402,292403,292405],{},[2627,292404,291953],{},"：操作系统的一般作用是：",[108,292407,292408,2664,292410],{},[2627,292409,118575],{},[2627,292411,292412],{},"b. Provide a set of services to system users",[108,292414,292415,292417],{},[2627,292416,291966],{},"：操作系统最宏观的作用是作为资源管理器和用户接口，为系统用户（包括终端用户和应用程序）提供一组系统服务（如文件管理、进程调度等）。虽然 c 也是 OS 的功能之一，但 b 是最全面和准确的概括。",[15,292419,292420,292423],{},[2627,292421,292422],{},"2. The four main structural elements of a computer system are:","\na. Processor, Registers, I\u002FO Modules & Main Memory\nb. Processor, Registers, Main Memory & System Bus\nc. Processor, Main Memory, I\u002FO Modules & System Bus\nd. None of the above",[105,292425,292426,292431,292438],{},[108,292427,292428,292430],{},[2627,292429,291953],{},"：计算机系统的四个主要结构元素是：",[108,292432,292433,2664,292435],{},[2627,292434,118575],{},[2627,292436,292437],{},"c. Processor, Main Memory, I\u002FO Modules & System Bus",[108,292439,292440,292442,292443,292446,292447,292450,292451,292454,292455,292458],{},[2627,292441,291966],{},"：计算机顶层架构的四大基本组件是：执行处理的",[2627,292444,292445],{},"处理器 (Processor\u002FCPU)","、存储指令和数据的",[2627,292448,292449],{},"主存 (Main Memory)","、负责外部通信的 ",[2627,292452,292453],{},"I\u002FO 模块 (I\u002FO Modules)","，以及连接它们的",[2627,292456,292457],{},"系统总线 (System Bus)","。寄存器是包含在 Processor 内部的。",[15,292460,292461,292464],{},[2627,292462,292463],{},"3. The two basic types of processor registers are:","\na. User-visible and Control\u002FStatus registers\nb. Control and Status registers\nc. User-visible and user-invisible registers\nd. None of the above",[105,292466,292467,292472,292479],{},[108,292468,292469,292471],{},[2627,292470,291953],{},"：处理器寄存器的两种基本类型是：",[108,292473,292474,2664,292476],{},[2627,292475,118575],{},[2627,292477,292478],{},"a. User-visible and Control\u002FStatus registers",[108,292480,292481,292483,292484,292486,292487,292490],{},[2627,292482,291966],{},"：按照用途分类，寄存器分为",[2627,292485,292017],{},"（如数据、地址寄存器，供汇编程序员使用）和",[2627,292488,292489],{},"控制与状态寄存器 (Control\u002FStatus registers)","（如 PC、IR、PSW，由 CPU 硬件和内核使用控制系统操作）。",[15,292492,292493,292496],{},[2627,292494,292495],{},"4. Address registers may contain:","\na. Memory addresses of data\nb. Memory addresses of instructions\nc. Partial memory addresses\nd. All of the above",[105,292498,292499,292504,292511],{},[108,292500,292501,292503],{},[2627,292502,291953],{},"：地址寄存器可能包含：",[108,292505,292506,2664,292508],{},[2627,292507,118575],{},[2627,292509,292510],{},"d. All of the above",[108,292512,292513,2664,292515,292518],{},[2627,292514,291966],{},[2627,292516,292517],{},"地址寄存器","用于存放主存地址。它可以存放数据的地址、指令的地址，或者在某些寻址方式（如基址寻址或段式寻址）中存放部分地址（偏移量或基址），因此以上皆有可能。",[15,292520,292521,292524],{},[2627,292522,292523],{},"5. A Control\u002FStatus register that contains the address of the next instruction to be fetched is called the:","\na. Instruction Register (IR)\nb. Program Counter (PC)\nc. Program Status Word (PSW)\nd. All of the above",[105,292526,292527,292532,292539],{},[108,292528,292529,292531],{},[2627,292530,291953],{},"：包含要获取的下一条指令地址的控制\u002F状态寄存器被称为：",[108,292533,292534,2664,292536],{},[2627,292535,118575],{},[2627,292537,292538],{},"b. Program Counter (PC)",[108,292540,292541,2664,292543,292546],{},[2627,292542,291966],{},[2627,292544,292545],{},"程序计数器 (Program Counter, PC)"," 的唯一职责就是保存下一条将要被 CPU 从内存中取出的指令的地址。",[15,292548,292549,292552],{},[2627,292550,292551],{},"6. The two basic steps used by the processor in instruction processing are:","\na. Fetch and Instruction cycles\nb. Instruction and Execute cycles\nc. Fetch and Execute cycles\nd. None of the above",[105,292554,292555,292560,292567],{},[108,292556,292557,292559],{},[2627,292558,291953],{},"：处理器在指令处理中使用的两个基本步骤是：",[108,292561,292562,2664,292564],{},[2627,292563,118575],{},[2627,292565,292566],{},"c. Fetch and Execute cycles",[108,292568,292569,292571,292572,292574,292575,347],{},[2627,292570,291966],{},"：最简单的指令周期由两部分组成：第一步从内存读取指令，即",[2627,292573,292106],{},"；第二步解码并执行该指令，即",[2627,292576,292109],{},[15,292578,292579,292582],{},[2627,292580,292581],{},"7. A fetched instruction is normally loaded into the:","\na. Instruction Register (IR)\nb. Program Counter (PC)\nc. Accumulator (AC)\nd. None of the above",[105,292584,292585,292590,292597],{},[108,292586,292587,292589],{},[2627,292588,291953],{},"：获取的指令通常被加载到：",[108,292591,292592,2664,292594],{},[2627,292593,118575],{},[2627,292595,292596],{},"a. Instruction Register (IR)",[108,292598,292599,292601,292602,292604],{},[2627,292600,291966],{},"：指令被获取后，会存放在 CPU 的",[2627,292603,292136],{}," 中，由控制单元进行解码分析。",[15,292606,292607,292610],{},[2627,292608,292609],{},"8. A common class of interrupts is:","\na. Program\nb. Timer\nc. I\u002FO\nd. All of the above",[105,292612,292613,292618,292624],{},[108,292614,292615,292617],{},[2627,292616,291953],{},"：常见的一类中断是：",[108,292619,292620,2664,292622],{},[2627,292621,118575],{},[2627,292623,292510],{},[108,292625,292626,292628,292629,292632,292633,292636,292637,292640,292641,292644],{},[2627,292627,291966],{},"：中断通常分为四大类：",[2627,292630,292631],{},"程序中断","（如除以零、越界）、",[2627,292634,292635],{},"定时器中断","（由操作系统时钟产生，用于分时调度）、",[2627,292638,292639],{},"I\u002FO 中断","（I\u002FO 设备完成操作或出错时产生）以及",[2627,292642,292643],{},"硬件故障中断","（如断电、内存奇偶校验错）。",[15,292646,292647,292650],{},[2627,292648,292649],{},"9. When an external device becomes ready to be serviced by the processor, the device sends this type of signal to the processor:","\na. Interrupt signal\nb. Halt signal\nc. Handler signal\nd. None of the above",[105,292652,292653,292658,292665],{},[108,292654,292655,292657],{},[2627,292656,291953],{},"：当外部设备准备好由处理器提供服务时，该设备向处理器发送此类型的信号：",[108,292659,292660,2664,292662],{},[2627,292661,118575],{},[2627,292663,292664],{},"a. Interrupt signal",[108,292666,292667,292669,292670,292673],{},[2627,292668,291966],{},"：外部设备（如网卡接收到数据、硬盘读写完成）通过向 CPU 的特定引脚发送",[2627,292671,292672],{},"中断信号 (Interrupt signal)"," 来通知 CPU 需要服务。",[15,292675,292676,292679],{},[2627,292677,292678],{},"10. Information that must be saved prior to the processor transferring control to the interrupt handler routine includes:","\na. Processor Status Word (PSW)\nb. Processor Status Word (PSW) & Location of next instruction\nc. Processor Status Word (PSW) & Contents of processor registers\nd. None of the above",[105,292681,292682,292687,292694],{},[108,292683,292684,292686],{},[2627,292685,291953],{},"：处理器在将控制权转移到中断处理程序之前必须保存的信息包括：",[108,292688,292689,2664,292691],{},[2627,292690,118575],{},[2627,292692,292693],{},"b. Processor Status Word (PSW) & Location of next instruction",[108,292695,292696,292698,292699,292702,292703,292706,292707,292710],{},[2627,292697,291966],{},"：在响应中断的硬件层面上，CPU 必须自动压栈保存",[2627,292700,292701],{},"最小集信息","，以便中断结束后能够准确恢复。这包括当前的执行状态 ",[2627,292704,292705],{},"PSW"," 和被打断程序的断点，即",[2627,292708,292709],{},"下一条指令的地址 (PC)","。其他通用寄存器的内容通常由中断处理程序（软件）在开始执行时手动保存。",[15,292712,292713,292716],{},[2627,292714,292715],{},"11. One accepted method of dealing with multiple interrupts is to:","\na. Define priorities for the interrupts\nb. Disable all interrupts except those of highest priority\nc. Service them in round-robin fashion\nd. None of the above",[105,292718,292719,292724,292731],{},[108,292720,292721,292723],{},[2627,292722,291953],{},"：处理多重中断的一种公认方法是：",[108,292725,292726,2664,292728],{},[2627,292727,118575],{},[2627,292729,292730],{},"a. Define priorities for the interrupts",[108,292732,292733,292735,292736,292739],{},[2627,292734,291966],{},"：为了更高效地处理多个突发事件，现代系统会为不同类型的中断",[2627,292737,292738],{},"定义优先级 (Define priorities)","。高优先级的中断（如硬件时钟、磁盘 I\u002FO）可以打断正在执行的低优先级中断（如键盘输入），实现嵌套中断处理。",[15,292741,292742,292745],{},[2627,292743,292744],{},"12. In a uniprocessor system, multiprogramming increases processor efficiency by:","\na. Increasing processor speed\nb. Taking advantage of time wasted by long wait interrupt handling\nc. Eliminating all idle processor cycles\nd. All of the above",[105,292747,292748,292753,292760],{},[108,292749,292750,292752],{},[2627,292751,291953],{},"：在单处理器系统中，多道程序设计通过以下方式提高处理器效率：",[108,292754,292755,2664,292757],{},[2627,292756,118575],{},[2627,292758,292759],{},"b. Taking advantage of time wasted by long wait interrupt handling",[108,292761,292762,292764],{},[2627,292763,291966],{},"：多道程序设计无法物理上提升 CPU 速度，也无法完全消除所有空闲周期。它的核心原理是利用程序在等待慢速 I\u002FO 操作时产生的空闲时间，让 CPU 去执行其他准备好的程序。",[15,292766,292767,292770],{},[2627,292768,292769],{},"13. As one proceeds down the memory hierarchy (i.e., from inboard memory to offline storage), the following condition(s) apply:","\na. Increasing cost per bit\nb. Decreasing capacity\nc. Increasing access time\nd. All of the above",[105,292772,292773,292778,292785],{},[108,292774,292775,292777],{},[2627,292776,291953],{},"：当沿着内存层次结构向下（即从板载内存到离线存储）时，适用以下情况：",[108,292779,292780,2664,292782],{},[2627,292781,118575],{},[2627,292783,292784],{},"c. Increasing access time",[108,292786,292787,292789,292790,46528,292793,292796,292797,292800],{},[2627,292788,291966],{},"：内存金字塔向下（从寄存器 -> Cache -> 主存 -> 硬盘 -> 磁带）：",[2627,292791,292792],{},"访问时间变长 (Increasing access time)",[2627,292794,292795],{},"容量变大 (Increasing capacity)","，并且 ",[2627,292798,292799],{},"每位成本下降 (Decreasing cost per bit)","。因此只有 c 是正确的。",[15,292802,292803,292806],{},[2627,292804,292805],{},"14. Small, fast memory located between the processor and main memory is called:","\na. WORM memory\nb. Cache memory\nc. CD-RW memory\nd. None of the above",[105,292808,292809,292814,292821],{},[108,292810,292811,292813],{},[2627,292812,291953],{},"：位于处理器和主存之间的小型快速内存被称为：",[108,292815,292816,2664,292818],{},[2627,292817,118575],{},[2627,292819,292820],{},"b. Cache memory",[108,292822,292823,292825,292826,347],{},[2627,292824,291966],{},"：为了弥合高速 CPU 和相对慢速的主存之间的速度差异，引入了基于 SRAM 技术的",[2627,292827,292828],{},"高速缓存 (Cache memory)",[15,292830,292831,292834],{},[2627,292832,292833],{},"15. When a new block of data is written into cache memory, the following determines which cache location the block will occupy:","\na. Block size\nb. Cache size\nc. Write policy\nd. None of the above",[105,292836,292837,292842,292849],{},[108,292838,292839,292841],{},[2627,292840,291953],{},"：当新数据块写入缓存时，以下哪项决定了该块将占据哪个缓存位置：",[108,292843,292844,2664,292846],{},[2627,292845,118575],{},[2627,292847,292848],{},"d. None of the above",[108,292850,292851,292853,292854,292857,292858,292861],{},[2627,292852,291966],{},"：决定内存块存放在 Cache 哪个位置的是 ",[2627,292855,292856],{},"映射函数 (Mapping function)","，例如直接映射、全相联映射或组相联映射。而如果缓存已满，决定替换谁的是 ",[2627,292859,292860],{},"置换算法 (Replacement algorithm)","。选项中并未给出正确答案。",[15,292863,292864,292867],{},[2627,292865,292866],{},"16. Direct Memory Access (DMA) operations require the following information from the processor:","\na. Address of I\u002FO device\nb. Starting memory location to read from or write to\nc. Number of words to be read or written\nd. All of the above",[105,292869,292870,292875,292881],{},[108,292871,292872,292874],{},[2627,292873,291953],{},"：直接内存访问 (DMA) 操作需要处理器提供以下信息：",[108,292876,292877,2664,292879],{},[2627,292878,118575],{},[2627,292880,292510],{},[108,292882,292883,292885,292886,292889,292890,292893,292894,292897],{},[2627,292884,291966],{},"：为了让 DMA 控制器自主完成大块数据的传输，CPU 必须初始化 DMA，告诉它：操作的是哪个 ",[2627,292887,292888],{},"I\u002FO 设备的地址","，主存的",[2627,292891,292892],{},"起始地址","，是读还是写，以及要传输的",[2627,292895,292896],{},"数据字数","。设置完毕后，CPU 就可以离开，DMA 完成传输后会发出中断。",[32188,292899],{},[19,292901,292903],{"id":292902},"三-fill-in-the-blank-questions-填空题","三、 Fill-In-The-Blank Questions (填空题)",[15,292905,292906],{},[2627,292907,292908],{},"1. An operating system exploits the hardware resources of one or more processors to provide a set of services to ________.",[105,292910,292911,292916,292926],{},[108,292912,292913,292915],{},[2627,292914,291953],{},"：操作系统利用一个或多个处理器的硬件资源，为 ________ 提供一组服务。",[108,292917,292918,2664,292920,4374,292923],{},[2627,292919,118575],{},[2627,292921,292922],{},"system users (系统用户)",[2627,292924,292925],{},"applications (应用程序)",[108,292927,292928,292930],{},[2627,292929,291966],{},"：操作系统作为一个管理者和服务提供者，主要服务对象是使用计算机的用户以及运行在计算机上的各种应用程序。",[15,292932,292933],{},[2627,292934,292935],{},"2. Each location in Main Memory contains a ________ that can be interpreted as either an instruction or data.",[105,292937,292938,292943,292953],{},[108,292939,292940,292942],{},[2627,292941,291953],{},"：主存中的每个位置都包含一个 ________，它可以被解释为指令或数据。",[108,292944,292945,2664,292947,4374,292950],{},[2627,292946,118575],{},[2627,292948,292949],{},"bit pattern (位模式\u002F二进制模式)",[2627,292951,292952],{},"word (字)",[108,292954,292955,292957],{},[2627,292956,291966],{},"：冯·诺依曼架构的核心特性就是“存储程序”，指令和数据以相同的二进制形式（位模式）混合存储在内存中。CPU 取决于执行周期的阶段来决定如何解释这些二进制位。",[15,292959,292960],{},[2627,292961,292962],{},"3. Registers that are used by system programs to minimize main memory references by optimizing register use are called ________.",[105,292964,292965,292970,292977],{},[108,292966,292967,292969],{},[2627,292968,291953],{},"：由系统程序用来通过优化寄存器使用来最大限度地减少主存引用的寄存器被称为 ________。",[108,292971,292972,2664,292974],{},[2627,292973,118575],{},[2627,292975,292976],{},"user-visible registers (用户可见寄存器)",[108,292978,292979,292981,292982,292985],{},[2627,292980,291966],{},"：编译器或汇编程序员会尽可能多地将高频使用的变量分配到通用的",[2627,292983,292984],{},"用户可见寄存器","中，因为访问寄存器的速度远快于访问主存。",[15,292987,292988],{},[2627,292989,292990],{},"4. A special type of address register, required by a system that implements user-visible stack addressing, is called a ________.",[105,292992,292993,292998,293005],{},[108,292994,292995,292997],{},[2627,292996,291953],{},"：实现用户可见堆栈寻址的系统所需的一种特殊类型的地址寄存器被称为 ________。",[108,292999,293000,2664,293002],{},[2627,293001,118575],{},[2627,293003,293004],{},"stack pointer (堆栈指针 \u002F SP)",[108,293006,293007,293009,293010,293013],{},[2627,293008,291966],{},"：现代 CPU 硬件直接支持堆栈结构（用于函数调用和局部变量），并提供了一个专门的地址寄存器：",[2627,293011,293012],{},"堆栈指针 (Stack Pointer, SP)","，它总是指向当前栈顶的内存地址。",[15,293015,293016],{},[2627,293017,293018],{},"5. The ________ contains the most recently fetched instruction.",[105,293020,293021,293026,293033],{},[108,293022,293023,293025],{},[2627,293024,291953],{},"：________ 包含最近获取的指令。",[108,293027,293028,2664,293030],{},[2627,293029,118575],{},[2627,293031,293032],{},"Instruction Register \u002F IR (指令寄存器)",[108,293034,293035,293037],{},[2627,293036,291966],{},"：再次巩固前文知识，CPU 从内存中 Fetch 出来的指令，在解码和执行前存放在 IR 中。",[15,293039,293040],{},[2627,293041,293042],{},"6. The processing required for a single instruction is called a(n) ________.",[105,293044,293045,293050,293057],{},[108,293046,293047,293049],{},[2627,293048,291953],{},"：单条指令所需的处理过程称为 ________。",[108,293051,293052,2664,293054],{},[2627,293053,118575],{},[2627,293055,293056],{},"instruction cycle (指令周期)",[108,293058,293059,293061],{},[2627,293060,291966],{},"：一条指令的生命周期，包括 Fetch 和 Execute 等阶段，统称为 Instruction Cycle。",[15,293063,293064],{},[2627,293065,293066],{},"7. A fetched instruction is normally loaded into the ________ cycle.",[105,293068,293069,293077,293087],{},[108,293070,293071,293073,293074],{},[2627,293072,291953],{},"：获取的指令通常被加载到 ________ 周期中。\n",[71807,293075,293076],{},"(注：这道填空题的原题表达有些语义问题，通常应问加载到哪里(IR)。但按照其后跟 \"cycle\" 一词，可以理解为它想问这发生在什么周期。若题目旨在询问放入何处，应填 IR。基于教材上下文，最合理的断句可能是填 Instruction Register，无视其后的冗余单词，或填写 execute 表示随后进入执行。我们给出标准器件答案)",[108,293078,293079,2664,293081,47649,293084],{},[2627,293080,118575],{},[2627,293082,293083],{},"Instruction Register (IR)",[71807,293085,293086],{},"(题干中 cycle 可能是多余的错别字，若必须填阶段，则是 fetch )",[108,293088,293089,293091],{},[2627,293090,291966],{},"：如多选题所述， fetched instruction 是被加载到 IR 中的。",[15,293093,293094],{},[2627,293095,293096],{},"8. An arithmetic overflow condition resulting from some instructional execution will generate a(n) ________ interrupt.",[105,293098,293099,293104,293118],{},[108,293100,293101,293103],{},[2627,293102,291953],{},"：由某些指令执行产生的算术溢出情况将生成一个 ________ 中断。",[108,293105,293106,2664,293108,4374,293111,293114,293115,347],{},[2627,293107,118575],{},[2627,293109,293110],{},"program (程序 \u002F 异常)",[2627,293112,293113],{},"hardware failure (硬件故障)"," \u002F 通常分类为 ",[2627,293116,293117],{},"program",[108,293119,293120,293122,293123,293126,293127,347],{},[2627,293121,291966],{},"：算术溢出、除以零、内存越界访问等都是由正在执行的",[2627,293124,293125],{},"程序指令本身引起","的错误，它们会触发硬件抛出一种同步中断，统称为",[2627,293128,293129],{},"程序中断 (Program Interrupt) 或异常\u002F陷入 (Exception\u002FTrap)",[15,293131,293132],{},[2627,293133,293134],{},"9. To accommodate interrupts, a(n) ________ is added to the basic instruction cycle.",[105,293136,293137,293142,293152],{},[108,293138,293139,293141],{},[2627,293140,291953],{},"：为了适应中断，基础指令周期中增加了一个 ________。",[108,293143,293144,2664,293146,4374,293149],{},[2627,293145,118575],{},[2627,293147,293148],{},"interrupt stage (中断阶段)",[2627,293150,293151],{},"interrupt cycle (中断周期)",[108,293153,293154,293156,293157,293160],{},[2627,293155,291966],{},"：在 Fetch 和 Execute 周期之后，CPU 进入 ",[2627,293158,293159],{},"Interrupt Cycle","，检查中断请求引脚是否有信号。",[15,293162,293163],{},[2627,293164,293165],{},"10. The ________ is part of the information that must be saved prior to the processor transferring control to the interrupt handler routine, and it tells the processor where to return control to the previously interrupted program.",[105,293167,293168,293173,293180],{},[108,293169,293170,293172],{},[2627,293171,291953],{},"：________ 是处理器将控制权转移到中断处理程序之前必须保存的信息的一部分，它告诉处理器应将控制权返回到先前被中断程序的位置。",[108,293174,293175,2664,293177],{},[2627,293176,118575],{},[2627,293178,293179],{},"program counter \u002F PC (程序计数器)",[108,293181,293182,2664,293184,293187],{},[2627,293183,291966],{},[2627,293185,293186],{},"程序计数器 (PC)"," 保存着被打断程序下一条要执行的指令的地址。中断返回指令（如 IRET）会恢复这个值，使程序无缝继续运行。",[15,293189,293190],{},[2627,293191,293192],{},"11. A drawback to the disable interrupt strategy of dealing with multiple interrupts is that it doesn't account for ________.",[105,293194,293195,293200,293210],{},[108,293196,293197,293199],{},[2627,293198,291953],{},"：处理多重中断的禁用中断策略的一个缺点是它没有考虑到 ________。",[108,293201,293202,2664,293204,4374,293207],{},[2627,293203,118575],{},[2627,293205,293206],{},"relative priority (相对优先级)",[2627,293208,293209],{},"time-critical needs (时间关键型需求)",[108,293211,293212,293214],{},[2627,293213,291966],{},"：如果处理一个中断时完全关闭其他所有中断，那么一些对时间极度敏感、优先级极高的紧急中断（如电源故障警告或高速网卡数据）将无法得到及时响应，可能导致数据丢失或系统崩溃。",[15,293216,293217],{},[2627,293218,293219],{},"12. The concept of multiple programs taking turns in execution is known as ________.",[105,293221,293222,293227,293234],{},[108,293223,293224,293226],{},[2627,293225,291953],{},"：多个程序轮流执行的概念被称为 ________。",[108,293228,293229,2664,293231],{},[2627,293230,118575],{},[2627,293232,293233],{},"multiprogramming (多道程序设计)",[108,293235,293236,293238],{},[2627,293237,291966],{},"：内存中驻留多个程序，当一个程序 I\u002FO 阻塞时，CPU 切换给另一个程序执行，这就是多道程序设计的本质。",[15,293240,293241],{},[2627,293242,293243],{},"13. The memory design dilemma (regarding cost vs. capacity vs. access time) is solved by employing a(n) ________.",[105,293245,293246,293251,293258],{},[108,293247,293248,293250],{},[2627,293249,291953],{},"：内存设计的困境（关于成本、容量与访问时间）通过采用 ________ 来解决。",[108,293252,293253,2664,293255],{},[2627,293254,118575],{},[2627,293256,293257],{},"memory hierarchy (内存层次结构)",[108,293259,293260,293262,293263,293266],{},[2627,293261,291966],{},"：人们想要速度快、容量大又便宜的内存，物理上无法做到。解决之道就是构建",[2627,293264,293265],{},"内存金字塔 (Memory Hierarchy)","，结合少量昂贵的高速缓存和大量廉价的低速存储。",[15,293268,293269],{},[2627,293270,293271],{},"14. ________ exploits the principle of locality by providing a small, fast memory between the processor and main memory.",[105,293273,293274,293279,293286],{},[108,293275,293276,293278],{},[2627,293277,291953],{},"：________ 利用局部性原理，在处理器和主存之间提供了一个小型快速的内存。",[108,293280,293281,2664,293283],{},[2627,293282,118575],{},[2627,293284,293285],{},"Cache memory (高速缓存)",[108,293287,293288,293290],{},[2627,293289,291966],{},"：这是 Cache 设计的根本原理。",[15,293292,293293],{},[2627,293294,293295],{},"15. The ________ chooses, within the constraints of the mapping function, which block to replace when a new block is to be loaded and all cache slots are already filled.",[105,293297,293298,293303,293310],{},[108,293299,293300,293302],{},[2627,293301,291953],{},"：当要加载一个新块且所有缓存槽都已填满时，________ 在映射函数的约束下，选择要置换哪个块。",[108,293304,293305,2664,293307],{},[2627,293306,118575],{},[2627,293308,293309],{},"replacement algorithm (置换算法)",[108,293311,293312,293314,293315,293318],{},[2627,293313,291966],{},"：当 Cache 发生冲突（空间满）时，需要决策丢弃谁。常用的",[2627,293316,293317],{},"置换算法","有 LRU（最近最少使用）、FIFO、随机等。",[15,293320,293321],{},[2627,293322,293323],{},"16. Interrupt-driven I\u002FO, although more efficient than simple Programmed I\u002FO, still requires the use of the ________ to transfer data between memory and an I\u002FO module.",[105,293325,293326,293331,293338],{},[108,293327,293328,293330],{},[2627,293329,291953],{},"：中断驱动的 I\u002FO 虽然比简单的可编程 I\u002FO 更有效，但仍然需要使用 ________ 来在内存和 I\u002FO 模块之间传输数据。",[108,293332,293333,2664,293335],{},[2627,293334,118575],{},[2627,293336,293337],{},"processor \u002F CPU (处理器)",[108,293339,293340,293342,293343,293346],{},[2627,293341,291966],{},"：无论是 Programmed I\u002FO 还是 Interrupt-driven I\u002FO，",[2627,293344,293345],{},"数据的实际搬运工作（从 I\u002FO 读到寄存器，再从寄存器写到内存）依然是由 CPU 执行指令完成的","，这会占用 CPU 周期。这也是为什么后来发明了 DMA 技术的原因（DMA 允许 I\u002FO 设备绕过 CPU 直接访问内存）。",{"title":29,"searchDepth":43,"depth":43,"links":293348},[293349,293350,293351],{"id":291940,"depth":43,"text":291941},{"id":292391,"depth":43,"text":292392},{"id":292902,"depth":43,"text":292903},{"order":37},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap1",{"title":291931,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap1","k0dgDbKTlimeOUtcR8wIyrwgndJiLHQ4TbAylzFS9bo",{"id":293358,"title":293359,"body":293360,"description":29,"extension":85,"meta":294851,"navigation":52,"path":294852,"seo":294853,"stem":294854,"__hash__":294855},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap10.md","Chap. 10",{"type":8,"value":293361,"toc":294846},[293362,293366,293370,293375,293381,293387,293391,293412,293417,293422,293428,293432,293444,293449,293454,293460,293464,293476,293481,293486,293492,293496,293512,293517,293522,293528,293532,293552,293557,293562,293568,293572,293584,293589,293594,293600,293604,293616,293621,293626,293632,293636,293648,293653,293658,293664,293668,293684,293689,293694,293700,293704,293716,293721,293726,293732,293736,293752,293757,293762,293768,293772,293791,293796,293801,293807,293811,293827,293832,293837,293843,293847,293871,293876,293881,293887,293891,293911,293913,293917,293922,293927,293934,293938,293950,293955,293960,293967,293971,293983,293988,293993,293999,294003,294015,294020,294025,294032,294036,294048,294053,294058,294064,294068,294088,294093,294098,294105,294109,294121,294126,294131,294138,294142,294154,294159,294164,294171,294175,294187,294192,294197,294204,294208,294220,294225,294230,294237,294241,294261,294266,294271,294278,294282,294301,294306,294311,294317,294321,294333,294338,294343,294350,294354,294370,294375,294380,294387,294391,294411,294416,294421,294428,294432,294444,294446,294450,294455,294460,294467,294472,294477,294482,294489,294494,294499,294504,294511,294516,294521,294526,294533,294538,294543,294548,294555,294560,294565,294570,294577,294582,294587,294592,294599,294604,294609,294614,294621,294626,294631,294636,294643,294648,294653,294658,294665,294670,294675,294680,294687,294692,294697,294702,294709,294714,294719,294724,294731,294802,294807,294812,294819,294824,294829,294834,294841],[11,293363,293365],{"id":293364},"chapter-10-multiprocessor-and-real-time-scheduling","Chapter 10 - Multiprocessor and Real-Time Scheduling",[19,293367,293369],{"id":293368},"true-false-questions-判断题","True \u002F False Questions (判断题)",[15,293371,293372,293374],{},[2627,293373,115273],{},": The term cluster refers to a set of processors that share a common main memory and are under the integrated control of an operating system.",[15,293376,293377,293380],{},[2627,293378,293379],{},"翻译",": 术语“集群（cluster）”指的是一组共享公共主存并在操作系统的统一控制下运行的处理器集合。",[15,293382,293383,8471,293385],{},[2627,293384,118575],{},[2627,293386,292009],{},[15,293388,293389,5598],{},[2627,293390,118582],{},[105,293392,293393,293398],{},[108,293394,293395,293397],{},[2627,293396,156287],{},": 多处理器系统的分类（集群与紧密耦合）。",[108,293399,293400,293403,293404,293407,293408,293411],{},[2627,293401,293402],{},"详解",": 集群（Cluster）或松散耦合多处理器（Loosely coupled multiprocessor）是由一组相对自治的系统组成的，",[2627,293405,293406],{},"每个处理器都有自己独立的主存","和 I\u002FO 通道。共享公共主存并在同一个 OS 控制下的是",[2627,293409,293410],{},"紧密耦合多处理器（Tightly coupled multiprocessor）","，如 SMP（对称多处理器）。",[15,293413,293414,293416],{},[2627,293415,115295],{},": A good way of classifying multiprocessors and placing them in context with other architectures is to consider the synchronization granularity between processes in a system.",[15,293418,293419,293421],{},[2627,293420,293379],{},": 对多处理器进行分类并将其与其他架构联系起来的一个好方法是，考虑系统中进程之间的同步粒度（synchronization granularity）。",[15,293423,293424,8471,293426],{},[2627,293425,118575],{},[2627,293427,291961],{},[15,293429,293430,5598],{},[2627,293431,118582],{},[105,293433,293434,293439],{},[108,293435,293436,293438],{},[2627,293437,156287],{},": 多处理器的同步粒度。",[108,293440,293441,293443],{},[2627,293442,293402],{},": 教科书中指出，根据进程间交互的频率和同步的粒度，可以很好地对并行性进行分类，包括细粒度（Fine-grained）、中粒度（Medium-grained）、粗粒度（Coarse-grained）、非常粗粒度（Very coarse-grained）和独立并行（Independent parallelism）。",[15,293445,293446,293448],{},[2627,293447,115317],{},": In a system in which many processors are available, the primary concern shifts from ensuring maximum processor usage to providing the best average performance to all applications.",[15,293450,293451,293453],{},[2627,293452,293379],{},": 在有许多可用处理器的系统中，主要关注点从确保最大的处理器使用率转向为所有应用程序提供最佳的平均性能。",[15,293455,293456,8471,293458],{},[2627,293457,118575],{},[2627,293459,291961],{},[15,293461,293462,5598],{},[2627,293463,118582],{},[105,293465,293466,293471],{},[108,293467,293468,293470],{},[2627,293469,156287],{},": 多处理器调度的目标。",[108,293472,293473,293475],{},[2627,293474,293402],{},": 在单处理器系统中，由于 CPU 是稀缺资源，调度器的重要目标是保持 CPU 忙碌（高利用率）。而在多处理器系统中，处理器的充裕使得“压榨每一个 CPU 周期”不再是首要任务，核心目标变成了如何高效地分配应用，降低开销，并优化整体应用的平均响应时间和性能。",[15,293477,293478,293480],{},[2627,293479,115339],{},": A general conclusion of studies comparing multiprocessor and uniprocessor systems under various conditions is that the specific process scheduling discipline employed becomes much more important as the number of processors increase.",[15,293482,293483,293485],{},[2627,293484,293379],{},": 在各种条件下比较多处理器和单处理器系统的研究得出的一个普遍结论是：随着处理器数量的增加，所采用的具体进程调度策略变得越来越重要。",[15,293487,293488,8471,293490],{},[2627,293489,118575],{},[2627,293491,292009],{},[15,293493,293494,5598],{},[2627,293495,118582],{},[105,293497,293498,293503],{},[108,293499,293500,293502],{},[2627,293501,156287],{},": 多处理器调度的影响。",[108,293504,293505,293507,293508,293511],{},[2627,293506,293402],{},": 结论恰恰相反。研究表明，随着系统中处理器数量的增加，单处理器中极其重要的那些复杂调度策略（如计算各种动态优先级）的影响力",[2627,293509,293510],{},"会随之下降（decreases）","。在多处理器系统中，使用简单的 FCFS（先来先服务）配合合理的负载均衡，往往比复杂的单核调度算法效果更好。",[15,293513,293514,293516],{},[2627,293515,115391],{},": In the Load Sharing multiprocessor scheduling and assignment approach, processes are assigned to a specific processor.",[15,293518,293519,293521],{},[2627,293520,293379],{},": 在负载共享（Load Sharing）多处理器调度和分配方法中，进程被分配给特定的处理器。",[15,293523,293524,8471,293526],{},[2627,293525,118575],{},[2627,293527,292009],{},[15,293529,293530,5598],{},[2627,293531,118582],{},[105,293533,293534,293539],{},[108,293535,293536,293538],{},[2627,293537,156287],{},": 负载共享（Load Sharing）。",[108,293540,293541,293543,293544,293547,293548,293551],{},[2627,293542,293402],{},": 负载共享是多处理器调度中最常用的机制之一。它的核心是维护一个",[2627,293545,293546],{},"全局的就绪队列","。进程\u002F线程",[2627,293549,293550],{},"并不","绑定到任何特定的处理器上。当任何一个处理器空闲时，它就会去全局队列中抓取下一个可用的线程来执行。",[15,293553,293554,293556],{},[2627,293555,115413],{},": One advantage to the Load Sharing multiprocessor scheduling and assignment approach is that no centralized scheduler is required.",[15,293558,293559,293561],{},[2627,293560,293379],{},": 负载共享多处理器调度和分配方法的一个优点是它不需要集中式的调度器。",[15,293563,293564,8471,293566],{},[2627,293565,118575],{},[2627,293567,291961],{},[15,293569,293570,5598],{},[2627,293571,118582],{},[105,293573,293574,293579],{},[108,293575,293576,293578],{},[2627,293577,156287],{},": 负载共享的优势。",[108,293580,293581,293583],{},[2627,293582,293402],{},": 负载共享机制下，每个处理器自己负责进行上下文切换和从全局队列中挑选线程。这意味着系统不需要一个专职的、唯一的“中央调度器”进程来发号施令，这避免了调度器成为单一瓶颈。",[15,293585,293586,293588],{},[2627,293587,115435],{},": The processor allocation problem on a multiprocessor more closely resembles the memory allocation problem on a uniprocessor than the scheduling problem on a uniprocessor.",[15,293590,293591,293593],{},[2627,293592,293379],{},": 多处理器上的处理器分配问题比单处理器上的调度问题更类似于单处理器上的内存分配问题。",[15,293595,293596,8471,293598],{},[2627,293597,118575],{},[2627,293599,291961],{},[15,293601,293602,5598],{},[2627,293603,118582],{},[105,293605,293606,293611],{},[108,293607,293608,293610],{},[2627,293609,156287],{},": 处理器分配。",[108,293612,293613,293615],{},[2627,293614,293402],{},": 在单处理器中，内存是分成多个块分配给程序的。而在多处理器系统中，多个同构的处理器就像多个可用的硬件块一样，需要将它们“分配（allocate）”给不同的并发应用。这种资源维度的分配逻辑确实与内存分配高度相似。",[15,293617,293618,293620],{},[2627,293619,115457],{},": In Dynamic Scheduling, the operating system leaves all scheduling decisions to the application.",[15,293622,293623,293625],{},[2627,293624,293379],{},": 在动态调度（Dynamic Scheduling）中，操作系统将所有的调度决策留给应用程序。",[15,293627,293628,8471,293630],{},[2627,293629,118575],{},[2627,293631,292009],{},[15,293633,293634,5598],{},[2627,293635,118582],{},[105,293637,293638,293643],{},[108,293639,293640,293642],{},[2627,293641,156287],{},": 动态调度（Dynamic Scheduling）。",[108,293644,293645,293647],{},[2627,293646,293402],{},": 在动态调度中，调度是由操作系统和应用程序**共同参与（joint effort）**的。操作系统负责动态地将一组处理器分配给一个作业，而应用程序（或运行库）自身负责判断哪些线程可以在这些分配到的处理器上执行。OS 并没有“完全撒手不管”。",[15,293649,293650,293652],{},[2627,293651,115479],{},": A soft real-time task is one which will be terminated if the deadline is missed.",[15,293654,293655,293657],{},[2627,293656,293379],{},": 软实时任务（soft real-time task）是指如果错过了截止时间就会被终止的任务。",[15,293659,293660,8471,293662],{},[2627,293661,118575],{},[2627,293663,292009],{},[15,293665,293666,5598],{},[2627,293667,118582],{},[105,293669,293670,293675],{},[108,293671,293672,293674],{},[2627,293673,156287],{},": 软实时与硬实时。",[108,293676,293677,293679,293680,293683],{},[2627,293678,293402],{},": 如果错过了截止时间会导致系统崩溃或致命错误，并且任务必须严格完成的任务是",[2627,293681,293682],{},"硬实时任务（Hard real-time task）","。**软实时任务（Soft real-time task）**即使错过了截止时间，虽然不希望发生，但系统仍能容忍其继续执行并产生价值，不会被强制终止。",[15,293685,293686,293688],{},[2627,293687,115501],{},": An operating system is deterministic to the extent that it performs operations at fixed, predetermined times or within predetermined time intervals.",[15,293690,293691,293693],{},[2627,293692,293379],{},": 操作系统在一定程度上是确定性的（deterministic），即它在固定的、预定的时间或在预定的时间间隔内执行操作。",[15,293695,293696,8471,293698],{},[2627,293697,118575],{},[2627,293699,291961],{},[15,293701,293702,5598],{},[2627,293703,118582],{},[105,293705,293706,293711],{},[108,293707,293708,293710],{},[2627,293709,156287],{},": 实时操作系统的确定性。",[108,293712,293713,293715],{},[2627,293714,293402],{},": 确定性是实时操作系统（RTOS）的重要指标。它指的是系统响应内部或外部事件的时序是可预测的、固定的或有明确的上限时间保证的。",[15,293717,293718,293720],{},[2627,293719,115523],{},": Dynamic best effort scheduling is the approach used by many real-time systems that are currently commercially available.",[15,293722,293723,293725],{},[2627,293724,293379],{},": 动态尽力而为（Dynamic best effort）调度是目前许多商业可用的实时系统所采用的方法。",[15,293727,293728,8471,293730],{},[2627,293729,118575],{},[2627,293731,292009],{},[15,293733,293734,5598],{},[2627,293735,118582],{},[105,293737,293738,293743],{},[108,293739,293740,293742],{},[2627,293741,156287],{},": 实时调度方法。",[108,293744,293745,293747,293748,293751],{},[2627,293746,293402],{},": 目前绝大多数商用的实时操作系统采用的并非是动态尽力而为，而是",[2627,293749,293750],{},"静态优先级驱动抢占式调度（Static priority-driven preemptive approach）","，因为这种方式的开销小且行为完全可预测。",[15,293753,293754,293756],{},[2627,293755,115545],{},": T\u002FF-A particularly useful metric in evaluating real-time operating systems is the frequency of task dispatching and interrupt handling.",[15,293758,293759,293761],{},[2627,293760,293379],{},": 评估实时操作系统的一个特别有用的指标是任务分派和中断处理的频率（frequency）。",[15,293763,293764,8471,293766],{},[2627,293765,118575],{},[2627,293767,292009],{},[15,293769,293770,5598],{},[2627,293771,118582],{},[105,293773,293774,293779],{},[108,293775,293776,293778],{},[2627,293777,156287],{},": 实时操作系统的评估指标。",[108,293780,293781,293783,293784,293790],{},[2627,293782,293402],{},": 评估实时操作系统的关键不仅是事件发生的“频率”，更重要的是",[2627,293785,293786,293787,293789],{},"延迟（Latency）",[2627,293788,58269],{},"响应时间（Responsiveness）","——即从中断发生到操作系统处理该中断并分派相关任务所需的时间（上下文切换时间等）。",[15,293792,293793,293795],{},[2627,293794,115567],{},": A promising method of resolving multitask scheduling conflicts for periodic tasks is called Deadline Scheduling.",[15,293797,293798,293800],{},[2627,293799,293379],{},": 解决周期性任务的多任务调度冲突的一种很有前途的方法叫做截止时间调度（Deadline Scheduling）。",[15,293802,293803,8471,293805],{},[2627,293804,118575],{},[2627,293806,292009],{},[15,293808,293809,5598],{},[2627,293810,118582],{},[105,293812,293813,293818],{},[108,293814,293815,293817],{},[2627,293816,156287],{},": 周期性任务的实时调度。",[108,293819,293820,293822,293823,293826],{},[2627,293821,293402],{},": 针对周期性任务（Periodic tasks），业界最经典、最被广泛认可和应用的方法是",[2627,293824,293825],{},"速率单调调度（Rate Monotonic Scheduling, RMS）","，而不是泛泛而指的 Deadline Scheduling。",[15,293828,293829,293831],{},[2627,293830,115589],{},": Linux builds on the traditional UNIX scheduling described in Chapter 9 by adding two scheduling classes for soft real-time scheduling: SCHED_FIFO and SCHED_RR.",[15,293833,293834,293836],{},[2627,293835,293379],{},": Linux 在第 9 章描述的传统 UNIX 调度的基础上，通过添加两个用于软实时调度的调度类进行构建：SCHED_FIFO 和 SCHED_RR。",[15,293838,293839,8471,293841],{},[2627,293840,118575],{},[2627,293842,291961],{},[15,293844,293845,5598],{},[2627,293846,118582],{},[105,293848,293849,293854],{},[108,293850,293851,293853],{},[2627,293852,156287],{},": Linux 实时调度策略。",[108,293855,293856,293858,293859,293862,293863,293866,293867,293870],{},[2627,293857,293402],{},": POSIX 标准（Linux 遵循）定义了三种调度策略：",[31,293860,293861],{},"SCHED_OTHER"," 用于普通的分时任务，而 ",[31,293864,293865],{},"SCHED_FIFO","（先进先出）和 ",[31,293868,293869],{},"SCHED_RR","（轮转）则是专门为了满足软实时（POSIX realtime）任务调度而添加的。",[15,293872,293873,293875],{},[2627,293874,115611],{},": Priorities in W2K are organized into two bands, or classes: non-real-time and variable.",[15,293877,293878,293880],{},[2627,293879,293379],{},": 在 W2K（Windows 2000\u002FWindows NT内核）中，优先级被组织成两个频段或类：非实时（non-real-time）和可变（variable）。",[15,293882,293883,8471,293885],{},[2627,293884,118575],{},[2627,293886,292009],{},[15,293888,293889,5598],{},[2627,293890,118582],{},[105,293892,293893,293898],{},[108,293894,293895,293897],{},[2627,293896,156287],{},": Windows 优先级类。",[108,293899,293900,293902,293903,293906,293907,293910],{},[2627,293901,293402],{},": Windows 的优先级分为 0 到 31。它们被组织为两个大类：16-31 为",[2627,293904,293905],{},"实时类（Real-time class）","，1-15 为",[2627,293908,293909],{},"可变类（Variable class）","（0是用于空闲线程的特殊级别）。所以应该是 Real-time 和 variable。",[32188,293912],{},[19,293914,293916],{"id":293915},"multiple-choice-questions-单选题","Multiple Choice Questions (单选题)",[15,293918,293919,293921],{},[2627,293920,115273],{},": A multiprocessor system in which there is a master, general-purpose processor that controls and receives services from other processors in the system is referred to as:\na. Tightly coupled multiprocessing\nb. Loosely coupled multiprocessing\nc. Functionally specialized multiprocessing\nd. None of the above",[15,293923,293924,293926],{},[2627,293925,293379],{},": 在一个多处理器系统中，如果存在一个通用的主处理器来控制系统中的其他处理器并从中接收服务，这种系统被称为：\na. 紧密耦合多处理\nb. 松散耦合多处理\nc. 功能专用多处理（Functionally specialized multiprocessing）\nd. 以上都不是",[15,293928,293929,8471,293931],{},[2627,293930,118575],{},[2627,293932,293933],{},"c. Functionally specialized multiprocessing",[15,293935,293936,5598],{},[2627,293937,118582],{},[105,293939,293940,293945],{},[108,293941,293942,293944],{},[2627,293943,156287],{},": 多处理器的硬件架构分类。",[108,293946,293947,293949],{},[2627,293948,293402],{},": 当系统具有一个主 CPU（Master）而辅以其他专门用于特定任务（如 I\u002FO 处理、协处理）的从处理器时，这种不对称的、具有专职分工的系统被分类为“功能专用多处理系统”。",[15,293951,293952,293954],{},[2627,293953,115295],{},": The category of parallelism in multiprocessors in which there is no explicit synchronization among processes is termed:\na. Independent parallelism\nb. Coarse-grained parallelism\nc. Fine-grained parallelism\nd. None of the above",[15,293956,293957,293959],{},[2627,293958,293379],{},": 在多处理器并行的类别中，进程之间没有显式同步的并行类别被称为：\na. 独立并行（Independent parallelism）\nb. 粗粒度并行\nc. 细粒度并行\nd. 以上都不是",[15,293961,293962,8471,293964],{},[2627,293963,118575],{},[2627,293965,293966],{},"a. Independent parallelism",[15,293968,293969,5598],{},[2627,293970,118582],{},[105,293972,293973,293978],{},[108,293974,293975,293977],{},[2627,293976,156287],{},": 同步粒度。",[108,293979,293980,293982],{},[2627,293981,293402],{},": 独立并行（Independent parallelism）是指在系统上同时运行多个毫无关联的进程（如多个不同用户运行各自的程序）。它们之间不需要通信，也不需要显式同步。",[15,293984,293985,293987],{},[2627,293986,115317],{},": Scheduling on a multiprocessor involves which of the following design issues:\na. The assignment of processes to processors\nb. The use of multiprogramming on individual processors\nc. The actual dispatching of a process\nd. All of the above",[15,293989,293990,293992],{},[2627,293991,293379],{},": 多处理器上的调度涉及以下哪些设计问题：\na. 将进程分配给处理器\nb. 在各个处理器上使用多道程序设计\nc. 进程的实际分派（dispatching）\nd. 以上所有",[15,293994,293995,8471,293997],{},[2627,293996,118575],{},[2627,293998,292510],{},[15,294000,294001,5598],{},[2627,294002,118582],{},[105,294004,294005,294010],{},[108,294006,294007,294009],{},[2627,294008,156287],{},": 多处理器调度的复杂性。",[108,294011,294012,294014],{},[2627,294013,293402],{},": 多处理器调度必须解决三大核心问题：将哪个进程分配给哪个 CPU（分配）、CPU 内部是否以及如何采用分时切换（多道程序）、以及分派器具体的行为逻辑。这三个维度缺一不可。",[15,294016,294017,294019],{},[2627,294018,115339],{},": Studies show that, as the number of processors in a system increases, the importance of the selection of the specific process scheduling discipline employed:\na. Increases\nb. Decreases\nc. Stays the same - no impact\nd. All of the above",[15,294021,294022,294024],{},[2627,294023,293379],{},": 研究表明，随着系统中处理器数量的增加，所采用的具体进程调度策略选择的重要性：\na. 增加\nb. 减少（Decreases）\nc. 保持不变 - 没有影响\nd. 以上所有",[15,294026,294027,8471,294029],{},[2627,294028,118575],{},[2627,294030,294031],{},"b. Decreases",[15,294033,294034,5598],{},[2627,294035,118582],{},[105,294037,294038,294043],{},[108,294039,294040,294042],{},[2627,294041,156287],{},": 调度策略在多处理器中的影响。",[108,294044,294045,294047],{},[2627,294046,293402],{},": 如前文所述，在多核\u002F多处理器中，算力不再是绝对瓶颈。复杂的调度算法带来的微小优化会被其带来的开销抵消，因此具体的进程调度规则（如用 SJF 还是 FCFS）的重要性比在单核中要低。",[15,294049,294050,294052],{},[2627,294051,115391],{},": Which of the following statements about the Dedicated Processor Assignment multiprocessor scheduling and assignment approach is true?\na. Each program is assigned one processor\nb. Related threads are scheduled to run on a set of processors at the same time\nc. Each processor is returned to the available pool as its assigned thread completes\nd. None of the above",[15,294054,294055,294057],{},[2627,294056,293379],{},": 关于专用处理器分配（Dedicated Processor Assignment）多处理器调度和分配方法，以下哪项陈述是正确的？\na. 每个程序被分配一个处理器\nb. 相关的线程被安排在同一时间在一组处理器上运行\nc. 每个处理器在其分配的线程完成时被返回到可用池中\nd. 以上都不是",[15,294059,294060,8471,294062],{},[2627,294061,118575],{},[2627,294063,292848],{},[15,294065,294066,5598],{},[2627,294067,118582],{},[105,294069,294070,294075],{},[108,294071,294072,294074],{},[2627,294073,156287],{},": 专用处理器分配。",[108,294076,294077,294079,294080,294083,294084,294087],{},[2627,294078,293402],{},": 在专用处理器分配中，当一个应用程序被调度时，它的",[2627,294081,294082],{},"每一个线程都会被分配一个独占的处理器","（a 错误）。b 描述的是群集调度（Gang Scheduling）的一般特征。c 错误的原因是，在 Dedicated Processor Assignment 中，即使某个线程提前结束，该处理器也会",[2627,294085,294086],{},"保持空闲状态（被应用霸占）直到整个应用程序完成","，而不会立即归还给可用池。因此，所有选项都不准确。",[15,294089,294090,294092],{},[2627,294091,115413],{},": A disadvantage of the Load Sharing multiprocessor scheduling and assignment approach is:\na. No centralized scheduler is required\nb. The central queue can be a bottleneck due to mutual exclusion\nc. The load must be distributed evenly across all processors\nd. All of the above",[15,294094,294095,294097],{},[2627,294096,293379],{},": 负载共享多处理器调度和分配方法的一个缺点是：\na. 不需要集中式的调度器\nb. 由于互斥，中央队列可能成为瓶颈\nc. 负载必须在所有处理器上均匀分布\nd. 以上所有",[15,294099,294100,8471,294102],{},[2627,294101,118575],{},[2627,294103,294104],{},"b. The central queue can be a bottleneck due to mutual exclusion",[15,294106,294107,5598],{},[2627,294108,118582],{},[105,294110,294111,294116],{},[108,294112,294113,294115],{},[2627,294114,156287],{},": 负载共享机制的缺点。",[108,294117,294118,294120],{},[2627,294119,293402],{},": 负载共享所有的空闲 CPU 都去读取一个全局的就绪队列。为了保证数据一致性，必须在读取和修改队列时加上互斥锁（Mutual exclusion）。当处理器数量众多时，这个锁争用会非常激烈，导致中央队列成为系统性能瓶颈。",[15,294122,294123,294125],{},[2627,294124,115435],{},": An extreme form of Gang Scheduling in which a group of processors is dedicated to a particular application for the duration of its execution is called:\na. Load Sharing\nb. Dynamic Scheduling\nc. Dedicated Processor Assignment\nd. None of the above",[15,294127,294128,294130],{},[2627,294129,293379],{},": 一种群集调度（Gang Scheduling）的极端形式，其中一组处理器在特定应用程序执行期间专门为其专用，这被称为：\na. 负载共享\nb. 动态调度\nc. 专用处理器分配（Dedicated Processor Assignment）\nd. 以上都不是",[15,294132,294133,8471,294135],{},[2627,294134,118575],{},[2627,294136,294137],{},"c. Dedicated Processor Assignment",[15,294139,294140,5598],{},[2627,294141,118582],{},[105,294143,294144,294149],{},[108,294145,294146,294148],{},[2627,294147,156287],{},": 多处理器调度的分配策略。",[108,294150,294151,294153],{},[2627,294152,293402],{},": 群集调度是指将进程的所有线程同时调度运行。而它的“极端形式”就是给该应用的每个线程分配一个物理 CPU，并完全霸占这些 CPU 直到应用结束。这就是 Dedicated Processor Assignment。",[15,294155,294156,294158],{},[2627,294157,115457],{},": An approach to multiprocessor scheduling and assignment in which both the operating system and the application are involved in making scheduling decisions is called:\na. Load Sharing\nb. Dynamic Scheduling\nc. Dedicated Processor Assignment\nd. None of the above",[15,294160,294161,294163],{},[2627,294162,293379],{},": 一种操作系统和应用程序都参与制定调度决策的多处理器调度和分配方法被称为：\na. 负载共享\nb. 动态调度（Dynamic Scheduling）\nc. 专用处理器分配\nd. 以上都不是",[15,294165,294166,8471,294168],{},[2627,294167,118575],{},[2627,294169,294170],{},"b. Dynamic Scheduling",[15,294172,294173,5598],{},[2627,294174,118582],{},[105,294176,294177,294182],{},[108,294178,294179,294181],{},[2627,294180,156287],{},": 动态调度。",[108,294183,294184,294186],{},[2627,294185,293402],{},": 动态调度改变了单方面的调度逻辑。OS 负责把处理器资源分配给各个作业，而在作业内部，由应用程序自己的运行时库决定把哪个线程派发到哪个被分配到的处理器上去。",[15,294188,294189,294191],{},[2627,294190,115479],{},": A task that, unless it meets its deadline may cause undesirable damage or a fatal error to the system, is called a(n):\na. Hard real-time task\nb. Soft real-time task\nc. Aperiodic task\nd. None of the above",[15,294193,294194,294196],{},[2627,294195,293379],{},": 除非满足其截止时间，否则可能会对系统造成不良损坏或致命错误的任务被称为：\na. 硬实时任务（Hard real-time task）\nb. 软实时任务\nc. 非周期性任务\nd. 以上都不是",[15,294198,294199,8471,294201],{},[2627,294200,118575],{},[2627,294202,294203],{},"a. Hard real-time task",[15,294205,294206,5598],{},[2627,294207,118582],{},[105,294209,294210,294215],{},[108,294211,294212,294214],{},[2627,294213,156287],{},": 硬实时的定义。",[108,294216,294217,294219],{},[2627,294218,293402],{},": 硬实时（Hard real-time）系统的约束极其严格。例如医疗起搏器控制或导弹拦截系统，一旦错过 Deadline 就等同于系统失效甚至引发灾难。",[15,294221,294222,294224],{},[2627,294223,115501],{},": Which of the following requirements of real-time operating systems is concerned with how long, after acknowledgement, it takes the O\u002FS to service the interrupt?\na. Determinism\nb. Responsiveness\nc. Reliability\nd. None of the above",[15,294226,294227,294229],{},[2627,294228,293379],{},": 实时操作系统的以下哪项要求与操作系统在确认后需要多长时间来处理中断有关？\na. 确定性（Determinism）\nb. 响应性（Responsiveness）\nc. 可靠性（Reliability）\nd. 以上都不是",[15,294231,294232,8471,294234],{},[2627,294233,118575],{},[2627,294235,294236],{},"b. Responsiveness",[15,294238,294239,5598],{},[2627,294240,118582],{},[105,294242,294243,294248],{},[108,294244,294245,294247],{},[2627,294246,156287],{},": 实时系统特性。",[108,294249,294250,294252,294253,294256,294257,294260],{},[2627,294251,293402],{},": 确定性（Determinism）是指中断多久之后能被",[2627,294254,294255],{},"识别并确认","；而响应性（Responsiveness）是指中断确认之后，系统进行上下文切换并",[2627,294258,294259],{},"实际开始执行","对应中断处理服务所需的时间。",[15,294262,294263,294265],{},[2627,294264,115523],{},": The result of the schedulability analysis of the following approach is a schedule that determines, at run time, when a task must begin execution.\na. Static table-driven approach\nb. Static priority-driven preemptive approach\nc. Dynamic planning-based approach\nd. None of the above",[15,294267,294268,294270],{},[2627,294269,293379],{},": 以下哪种方法的可调度性分析结果是产生一个在运行时（run time）决定任务何时必须开始执行的调度？\na. 静态表驱动方法\nb. 静态优先级驱动抢占式方法\nc. 基于动态规划的方法（Dynamic planning-based approach）\nd. 以上都不是",[15,294272,294273,8471,294275],{},[2627,294274,118575],{},[2627,294276,294277],{},"c. Dynamic planning-based approach",[15,294279,294280,5598],{},[2627,294281,118582],{},[105,294283,294284,294289],{},[108,294285,294286,294288],{},[2627,294287,156287],{},": 实时调度方法的分类。",[108,294290,294291,5598,294293],{},[2627,294292,293402],{},[105,294294,294295,294298],{},[108,294296,294297],{},"静态表驱动：在离线编译时就算好了整张时间表。",[108,294299,294300],{},"基于动态规划：允许任务在**运行时（run time）**动态到达。到达时，系统会动态地规划计算，看是否有可行的调度方案来满足它的截止时间。",[15,294302,294303,294305],{},[2627,294304,115545],{},": Useful information about a task that might be used in real-time task scheduling includes:\na. Starting deadline\nb. Resource requirements\nc. Priority\nd. All of the above",[15,294307,294308,294310],{},[2627,294309,293379],{},": 可用于实时任务调度的、关于任务的有用信息包括：\na. 开始截止时间\nb. 资源需求\nc. 优先级\nd. 以上所有",[15,294312,294313,8471,294315],{},[2627,294314,118575],{},[2627,294316,292510],{},[15,294318,294319,5598],{},[2627,294320,118582],{},[105,294322,294323,294328],{},[108,294324,294325,294327],{},[2627,294326,156287],{},": 实时任务属性。",[108,294329,294330,294332],{},[2627,294331,293402],{},": 为了构建可行的实时调度计划，调度器需要知道任务的确切限制：它必须在什么时候开始\u002F结束、需要多少内存和 I\u002FO 资源、以及它的重要级（优先级）。",[15,294334,294335,294337],{},[2627,294336,115567],{},": A promising method of resolving multitask scheduling conflicts for periodic tasks is:\na. Deadline Scheduling (DS)\nb. Rate Monotonic Scheduling (RMS)\nc. Real-Time Scheduling (RTS)\nd. All of the above",[15,294339,294340,294342],{},[2627,294341,293379],{},": 解决周期性任务的多任务调度冲突的一种很有前途的方法是：\na. 截止时间调度 (DS)\nb. 速率单调调度 (Rate Monotonic Scheduling, RMS)\nc. 实时调度 (RTS)\nd. 以上所有",[15,294344,294345,8471,294347],{},[2627,294346,118575],{},[2627,294348,294349],{},"b. Rate Monotonic Scheduling (RMS)",[15,294351,294352,5598],{},[2627,294353,118582],{},[105,294355,294356,294361],{},[108,294357,294358,294360],{},[2627,294359,156287],{},": RMS 算法。",[108,294362,294363,294365,294366,294369],{},[2627,294364,293402],{},": RMS（速率单调调度）是处理",[2627,294367,294368],{},"周期性任务","的静态优先级最优算法。它根据任务周期的长短来分配优先级——周期越短（执行频率越快），优先级越高。",[15,294371,294372,294374],{},[2627,294373,115589],{},": Linux builds on the traditional UNIX scheduling described in Chapter 9 by adding two scheduling classes for soft real-time scheduling:\na. SCHED_FIFO and SCHED_RR\nb. SCHED_RR and SCHED_OTHER\nc. SCHED_FIFO and SCHED_OTHER\nd. All of the above",[15,294376,294377,294379],{},[2627,294378,293379],{},": Linux 在第 9 章描述的传统 UNIX 调度的基础上，通过添加哪两个调度类来实现软实时调度：\na. SCHED_FIFO 和 SCHED_RR\nb. SCHED_RR 和 SCHED_OTHER\nc. SCHED_FIFO 和 SCHED_OTHER\nd. 以上所有",[15,294381,294382,8471,294384],{},[2627,294383,118575],{},[2627,294385,294386],{},"a. SCHED_FIFO and SCHED_RR",[15,294388,294389,5598],{},[2627,294390,118582],{},[105,294392,294393,294398],{},[108,294394,294395,294397],{},[2627,294396,156287],{},": Linux 的 POSIX 调度策略。",[108,294399,294400,8471,294402,294404,294405,294407,294408,294410],{},[2627,294401,293402],{},[31,294403,293861],{}," 是传统的非实时分时调度。",[31,294406,293865],{},"（先进先出直到主动放弃或被更高优先级抢占）和 ",[31,294409,293869],{},"（按时间片轮转的 FIFO）是专门针对实时任务的两种策略。",[15,294412,294413,294415],{},[2627,294414,115611],{},": Priorities in W2K are organized into two bands, or classes:\na. Real-time and static\nb. Real-time and variable\nc. Non-real-time and variable\nd. All of the above",[15,294417,294418,294420],{},[2627,294419,293379],{},": 在 W2K（Windows）中，优先级被组织成两个频段或类：\na. 实时和静态\nb. 实时和可变（Real-time and variable）\nc. 非实时和可变\nd. 以上所有",[15,294422,294423,8471,294425],{},[2627,294424,118575],{},[2627,294426,294427],{},"b. Real-time and variable",[15,294429,294430,5598],{},[2627,294431,118582],{},[105,294433,294434,294439],{},[108,294435,294436,294438],{},[2627,294437,156287],{},": Windows 优先级系统。",[108,294440,294441,294443],{},[2627,294442,293402],{},": Windows 的 32 个优先级级别被分为 Real-time class（16-31，优先级固定，不会因等待而提升）和 Variable class（1-15，动态优先级，会为了防止饥饿而进行优先级提升）。",[32188,294445],{},[19,294447,294449],{"id":294448},"fill-in-the-blank-questions-填空题","Fill-In-The-Blank Questions (填空题)",[15,294451,294452,294454],{},[2627,294453,115273],{},": A collection of relatively autonomous systems, where each processor has its own main memory and I\u002FO channels is called __________.",[15,294456,294457,294459],{},[2627,294458,293379],{},": 一个由相对自治的系统组成的集合，其中每个处理器都有自己的主存和 I\u002FO 通道，被称为 __________。",[15,294461,294462,8471,294464],{},[2627,294463,118575],{},[2627,294465,294466],{},"cluster (或 loosely coupled multiprocessor)",[15,294468,294469,294471],{},[2627,294470,118582],{},": 具有独立主存和通道的系统通过网络连接起来就是集群（Cluster）或松散耦合多处理系统。",[15,294473,294474,294476],{},[2627,294475,115295],{},": A system that implements a high degree of coordination and interaction among the threads of an application, as specified by the programmer, falls into the category of __________ parallelism.",[15,294478,294479,294481],{},[2627,294480,293379],{},": 一个实现了应用程序线程之间高度协调和交互的系统（如程序员所指定的那样），属于 __________ 并行类别。",[15,294483,294484,8471,294486],{},[2627,294485,118575],{},[2627,294487,294488],{},"medium-grained",[15,294490,294491,294493],{},[2627,294492,118582],{},": 教科书对并行粒度的定义中指出，单个应用程序的多个线程之间的高度交互并行属于**中等粒度（Medium-grained）**并行。细粒度通常指指令级别的并行。",[15,294495,294496,294498],{},[2627,294497,115317],{},": The process assignment strategy where a process is permanently assigned to one processor from activation to completion is referred to as __________ assignment.",[15,294500,294501,294503],{},[2627,294502,293379],{},": 从激活到完成，将一个进程永久地分配给一个处理器的进程分配策略被称为 __________ 分配。",[15,294505,294506,8471,294508],{},[2627,294507,118575],{},[2627,294509,294510],{},"Dedicated Processor (或 dedicated)",[15,294512,294513,294515],{},[2627,294514,118582],{},": 将处理器从始至终死死绑定到一个应用的所有线程上，这就是“专用处理器分配（Dedicated Processor Assignment）”。",[15,294517,294518,294520],{},[2627,294519,115339],{},": As the number of processors in a system increases, the impact of the specific scheduling discipline __________.",[15,294522,294523,294525],{},[2627,294524,293379],{},": 随着系统中处理器数量的增加，特定调度策略的影响力会 __________。",[15,294527,294528,8471,294530],{},[2627,294529,118575],{},[2627,294531,294532],{},"decreases",[15,294534,294535,294537],{},[2627,294536,118582],{},": 这是多处理器调度的一个重要结论。有了更多的 CPU，即使是最简单的 FCFS 也能跑得很好，复杂调度算法的边际效益在降低（decreases）。",[15,294539,294540,294542],{},[2627,294541,115391],{},": The multiprocessor scheduling and assignment approach in which a set of related threads is scheduled to run on a set of processors at the same time is referred to as __________.",[15,294544,294545,294547],{},[2627,294546,293379],{},": 一组相关的线程被安排在同一时间在一组处理器上运行的多处理器调度和分配方法被称为 __________。",[15,294549,294550,8471,294552],{},[2627,294551,118575],{},[2627,294553,294554],{},"gang scheduling (或 co-scheduling)",[15,294556,294557,294559],{},[2627,294558,118582],{},": 为了避免并发线程间相互等待对方的消息而在阻塞和唤醒之间浪费大量上下文切换时间，系统将这组相关的线程一起调度（同时执行）。这种做法称为群集调度（Gang Scheduling）。",[15,294561,294562,294564],{},[2627,294563,115413],{},": Using simulation methods, it has been determined that the __________ load balancing policy is superior to both the preemptive and non-preemptive versions of the Smallest Number of Threads First load balancing policy.",[15,294566,294567,294569],{},[2627,294568,293379],{},": 使用仿真方法已经确定，__________ 负载均衡策略优于最少线程数优先（SNTF）负载均衡策略的抢占式和非抢占式版本。",[15,294571,294572,8471,294574],{},[2627,294573,118575],{},[2627,294575,294576],{},"FCFS (First-Come-First-Served)",[15,294578,294579,294581],{},[2627,294580,118582],{},": 在动态多处理器调度研究中，先来先服务（FCFS）因其极低的系统开销和不需复杂的队列搜索，实际上优于那些看似更聪明的策略（如 SNTF）。",[15,294583,294584,294586],{},[2627,294585,115435],{},": The term __________ has been applied to the simultaneous scheduling of the threads that make up a single process.",[15,294588,294589,294591],{},[2627,294590,293379],{},": 术语 __________ 已被应用于对构成单个进程的多个线程进行同时调度的场景。",[15,294593,294594,8471,294596],{},[2627,294595,118575],{},[2627,294597,294598],{},"Gang Scheduling (或 co-scheduling)",[15,294600,294601,294603],{},[2627,294602,118582],{},": 同第 5 题，同时调度同一进程下所有相关线程的技术称为群集调度。",[15,294605,294606,294608],{},[2627,294607,115457],{},": Both Gang Scheduling and Dedicated Processor Assignment attack the scheduling problem by addressing the issue of __________ allocation.",[15,294610,294611,294613],{},[2627,294612,293379],{},": 群集调度和专用处理器分配这两种机制，都是通过解决 __________ 分配问题来应对调度问题的。",[15,294615,294616,8471,294618],{},[2627,294617,118575],{},[2627,294619,294620],{},"processor",[15,294622,294623,294625],{},[2627,294624,118582],{},": 这两种策略的本质思路是将问题转化为空间上的分配，它们更关心如何在空间上将并行的**处理器（Processor）**一次性地分配给应用程序。",[15,294627,294628,294630],{},[2627,294629,115479],{},": A real-time task that has a deadline by which it must finish or start, or that may have a constraint on both start and finish time is called a(n) __________ task.",[15,294632,294633,294635],{},[2627,294634,293379],{},": 一个具有必须完成或必须开始的截止时间，或者可能对开始和完成时间都有约束的实时任务被称为 __________ 任务。",[15,294637,294638,8471,294640],{},[2627,294639,118575],{},[2627,294641,294642],{},"aperiodic",[15,294644,294645,294647],{},[2627,294646,118582],{},": 教科书根据时间到达模式将任务分为周期性和非周期性任务。非周期性任务（Aperiodic task）是突发的事件，它伴随着一个必须执行或结束的确切截止期限。",[15,294649,294650,294652],{},[2627,294651,115501],{},": __________ is a characteristic that refers to the ability of a system to fail in such a way as to preserve as much capability and data as possible.",[15,294654,294655,294657],{},[2627,294656,293379],{},": __________ 是一种特性，指的是系统以能够尽可能多地保留系统能力和数据的方式发生故障的能力。",[15,294659,294660,8471,294662],{},[2627,294661,118575],{},[2627,294663,294664],{},"Fail-soft operation",[15,294666,294667,294669],{},[2627,294668,118582],{},": “故障弱化操作（Fail-soft operation）”是实时系统的一个关键特征。当系统无法承受所有负载或发生部分组件故障时，它会优雅地降级服务（例如优先保证最关键的硬实时任务运行），而不是全面崩溃。",[15,294671,294672,294674],{},[2627,294673,115523],{},": A major drawback to the __________ real-time scheduling approach is that it cannot be known whether a timing constraint will be met until a deadline arrives or the task completes.",[15,294676,294677,294679],{},[2627,294678,293379],{},": __________ 实时调度方法的一个主要缺点是：只有当截止时间到来或任务完成时，才能知道是否满足了时序约束。",[15,294681,294682,8471,294684],{},[2627,294683,118575],{},[2627,294685,294686],{},"dynamic best effort",[15,294688,294689,294691],{},[2627,294690,118582],{},": 在动态尽力而为（dynamic best effort）调度中，系统不进行可行性分析，仅仅尝试尽快运行它认为最重要的任务。因此，你无法提前预知任务会不会超时。",[15,294693,294694,294696],{},[2627,294695,115545],{},": It can be shown, for a given preemption strategy, using either starting or completion deadlines, that a policy of scheduling the task with the __________ deadline minimizes the fraction of tasks that miss their deadlines.",[15,294698,294699,294701],{},[2627,294700,293379],{},": 可以证明，对于给定的抢占策略，无论使用开始截止时间还是完成截止时间，调度具有 __________ 截止时间的任务的策略能够使错过截止时间的任务比例最小化。",[15,294703,294704,8471,294706],{},[2627,294705,118575],{},[2627,294707,294708],{},"earliest",[15,294710,294711,294713],{},[2627,294712,118582],{},": 这是最早截止期优先（Earliest Deadline First, EDF）调度算法的理论基础。总是优先调度最快要到期的任务。",[15,294715,294716,294718],{},[2627,294717,115567],{},": The scheduling scheme that assigns priorities to tasks on the basis of their periods (i.e., amount of time between the arrival of instances of the task) is referred to as __________.",[15,294720,294721,294723],{},[2627,294722,293379],{},": 基于任务周期（即任务实例到达之间的时间量）为任务分配优先级的调度方案被称为 __________。",[15,294725,294726,8471,294728],{},[2627,294727,118575],{},[2627,294729,294730],{},"Rate Monotonic (或 RM, Rate Monotonic Scheduling)",[15,294732,294733,294735,294736,294764,294765,294801],{},[2627,294734,118582],{},": 速率单调（Rate Monotonic）调度是一种静态优先级调度策略，其核心原则就是：周期 ",[34,294737,294739,294752],{"className":294738},[31642],[34,294740,294742],{"className":294741},[31646],[31648,294743,294744],{"xmlns":31650},[31652,294745,294746,294750],{},[31655,294747,294748],{},[31658,294749,37400],{},[31665,294751,37400],{"encoding":31667},[34,294753,294755],{"className":294754,"ariaHidden":758},[31672],[34,294756,294758,294761],{"className":294757},[31676],[34,294759],{"className":294760,"style":33218},[31680],[34,294762,37400],{"className":294763,"style":37422},[31685,31686]," 越短（速率 ",[34,294766,294768,294786],{"className":294767},[31642],[34,294769,294771],{"className":294770},[31646],[31648,294772,294773],{"xmlns":31650},[31652,294774,294775,294783],{},[31655,294776,294777,294779,294781],{},[31749,294778,640],{},[31658,294780,38080],{"mathvariant":38079},[31658,294782,37400],{},[31665,294784,294785],{"encoding":31667},"1\u002FT",[34,294787,294789],{"className":294788,"ariaHidden":758},[31672],[34,294790,294792,294795,294798],{"className":294791},[31676],[34,294793],{"className":294794,"style":31774},[31680],[34,294796,249219],{"className":294797},[31685],[34,294799,37400],{"className":294800,"style":37422},[31685,31686]," 越快），分配的静态优先级越高。",[15,294803,294804,294806],{},[2627,294805,115589],{},": Linux builds on the traditional UNIX scheduling described in Chapter 9 by adding two additional scheduling classes (SCHED_FIFO and SCHED_RR) to accommodate __________ scheduling.",[15,294808,294809,294811],{},[2627,294810,293379],{},": Linux 在第 9 章描述的传统 UNIX 调度的基础上，通过添加两个额外的调度类（SCHED_FIFO 和 SCHED_RR）来适应 __________ 调度。",[15,294813,294814,8471,294816],{},[2627,294815,118575],{},[2627,294817,294818],{},"real-time (或 soft real-time)",[15,294820,294821,294823],{},[2627,294822,118582],{},": 这两个 POSIX 调度策略是专为实时系统应用（在 Linux 中主要体现为软实时任务）设计的。",[15,294825,294826,294828],{},[2627,294827,115611],{},": On a uniprocessor system running W2K, if there is more than one thread at the highest priority level, then the processor is utilized in a __________ fashion amongst the available threads at that highest priority level.",[15,294830,294831,294833],{},[2627,294832,293379],{},": 在运行 W2K 的单处理器系统上，如果在最高优先级上有多个线程，那么处理器将在该最高优先级上的可用线程之间以 __________ 方式被利用。",[15,294835,294836,8471,294838],{},[2627,294837,118575],{},[2627,294839,294840],{},"round-robin",[15,294842,294843,294845],{},[2627,294844,118582],{},": 当多个具有相同优先级的线程处于就绪状态且它是全系统最高优先级时，Windows 分派器会在它们之间通过时间片轮转（Round-Robin）的方式进行调度。",{"title":29,"searchDepth":43,"depth":43,"links":294847},[294848,294849,294850],{"id":293368,"depth":43,"text":293369},{"id":293915,"depth":43,"text":293916},{"id":294448,"depth":43,"text":294449},{"order":1985},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap10",{"title":293359,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap10","Owgyg6JggItKeQSfifbQRLquLBMUI4bGD9X4H72AYIE",{"id":294857,"title":294858,"body":294859,"description":29,"extension":85,"meta":296315,"navigation":52,"path":296316,"seo":296317,"stem":296318,"__hash__":296319},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap11.md","Chap. 11",{"type":8,"value":294860,"toc":296310},[294861,294865,294867,294872,294877,294883,294887,294923,294928,294933,294939,294943,294955,294960,294965,294971,294975,294987,294992,294997,295003,295007,295023,295028,295033,295039,295043,295062,295067,295072,295078,295082,295094,295099,295104,295110,295114,295126,295131,295136,295142,295146,295158,295163,295168,295174,295178,295194,295199,295204,295210,295214,295226,295231,295236,295242,295246,295270,295275,295280,295286,295290,295309,295314,295319,295325,295329,295349,295354,295359,295365,295369,295381,295386,295391,295397,295401,295417,295419,295421,295426,295431,295437,295441,295453,295458,295463,295470,295474,295489,295494,295499,295509,295513,295533,295538,295543,295550,295554,295566,295571,295576,295583,295587,295612,295617,295622,295628,295632,295655,295660,295665,295672,295676,295688,295693,295698,295705,295709,295721,295726,295731,295738,295742,295757,295762,295767,295774,295778,295790,295795,295800,295807,295811,295829,295834,295839,295845,295849,295877,295882,295887,295894,295898,295910,295915,295920,295927,295931,295943,295948,295953,295960,295964,295980,295982,295984,295989,295994,296001,296006,296011,296016,296023,296028,296033,296038,296045,296050,296055,296060,296067,296072,296077,296082,296088,296093,296098,296103,296110,296115,296120,296125,296131,296136,296141,296146,296153,296158,296163,296168,296175,296180,296185,296190,296197,296202,296207,296212,296218,296223,296228,296233,296239,296244,296249,296254,296261,296266,296271,296276,296283,296288,296293,296298,296305],[11,294862,294864],{"id":294863},"chapter-11-io-management-and-disk-scheduling","Chapter 11 - I\u002FO Management and Disk Scheduling",[19,294866,293369],{"id":293368},[15,294868,294869,294871],{},[2627,294870,115273],{},": One general grouping of external I\u002FO devices that communicate with computer systems is the Human Readable category, which includes disk drives and tape drives.",[15,294873,294874,294876],{},[2627,294875,293379],{},": 与计算机系统通信的外部 I\u002FO 设备的一个通用分组是人类可读（Human Readable）类别，其中包括磁盘驱动器和磁带驱动器。",[15,294878,294879,8471,294881],{},[2627,294880,118575],{},[2627,294882,292009],{},[15,294884,294885,5598],{},[2627,294886,118582],{},[105,294888,294889,294894],{},[108,294890,294891,294893],{},[2627,294892,156287],{},": I\u002FO 设备的分类。",[108,294895,294896,294898,294899],{},[2627,294897,293402],{},": 外部 I\u002FO 设备通常被分为三大类：",[2230,294900,294901,294907,294917],{},[108,294902,294903,294906],{},[2627,294904,294905],{},"人类可读（Human Readable）","：适合与人类用户交互的设备，如屏幕、打印机、键盘、鼠标。",[108,294908,294909,294912,294913,294916],{},[2627,294910,294911],{},"机器可读（Machine Readable）","：适合与电子设备通信的设备，如",[2627,294914,294915],{},"磁盘驱动器、磁带驱动器","、传感器等。",[108,294918,294919,294922],{},[2627,294920,294921],{},"通信（Communication）","：适用于与远程设备通信的设备，如调制解调器、网卡。\n因此，磁盘和磁带属于机器可读类别，而非人类可读类别。",[15,294924,294925,294927],{},[2627,294926,115295],{},": In the Direct Memory Access (DMA) I\u002FO technique, a DMA module controls the exchange of data between main memory and the I\u002FO module.",[15,294929,294930,294932],{},[2627,294931,293379],{},": 在直接内存访问（DMA）I\u002FO 技术中，DMA 模块控制主存和 I\u002FO 模块之间的数据交换。",[15,294934,294935,8471,294937],{},[2627,294936,118575],{},[2627,294938,291961],{},[15,294940,294941,5598],{},[2627,294942,118582],{},[105,294944,294945,294950],{},[108,294946,294947,294949],{},[2627,294948,156287],{},": DMA (Direct Memory Access) 工作原理。",[108,294951,294952,294954],{},[2627,294953,293402],{},": DMA 技术的设计初衷就是将处理器从繁重的数据搬运中解放出来。处理器只需向 DMA 模块发送一条包含读\u002F写类型、设备地址、内存起始地址和数据量等信息的命令，随后 DMA 模块将完全接管系统总线，直接在主存和 I\u002FO 设备之间传送数据块。",[15,294956,294957,294959],{},[2627,294958,115317],{},": In the evolution of the I\u002FO function, the general tendency is for the processor to become more involved in I\u002FO operations.",[15,294961,294962,294964],{},[2627,294963,293379],{},": 在 I\u002FO 功能的演变中，总体趋势是处理器越来越多地参与到 I\u002FO 操作中。",[15,294966,294967,8471,294969],{},[2627,294968,118575],{},[2627,294970,292009],{},[15,294972,294973,5598],{},[2627,294974,118582],{},[105,294976,294977,294982],{},[108,294978,294979,294981],{},[2627,294980,156287],{},": I\u002FO 功能的演进。",[108,294983,294984,294986],{},[2627,294985,293402],{},": 趋势恰恰相反。I\u002FO 系统的演进方向是让处理器**越来越少（less involved）**地参与 I\u002FO 操作。演进路径为：程序控制 I\u002FO -> 中断驱动 I\u002FO -> DMA -> I\u002FO 通道 -> I\u002FO 处理器。每一步都在进一步将 I\u002FO 控制权下放给独立的硬件，以提高系统的并行性和 CPU 的利用率。",[15,294988,294989,294991],{},[2627,294990,115339],{},": A separate bus, other than the system bus, is provided for DMA transfers in the \"Single bus, detached DMA\" configuration.",[15,294993,294994,294996],{},[2627,294995,293379],{},": 在“单总线，分离 DMA（Single bus, detached DMA）”配置中，除了系统总线外，还提供了一条独立的总线用于 DMA 传输。",[15,294998,294999,8471,295001],{},[2627,295000,118575],{},[2627,295002,292009],{},[15,295004,295005,5598],{},[2627,295006,118582],{},[105,295008,295009,295014],{},[108,295010,295011,295013],{},[2627,295012,156287],{},": DMA 总线配置。",[108,295015,295016,295018,295019,295022],{},[2627,295017,293402],{},": “单总线，分离 DMA”指的是所有模块（包括处理器、主存、DMA 模块和所有 I\u002FO 模块）都共享",[2627,295020,295021],{},"同一条","系统总线。当 DMA 需要传输数据时，它必须与处理器竞争这条唯一的系统总线。没有提供额外的独立总线。",[15,295024,295025,295027],{},[2627,295026,115391],{},": The two primary objectives that are paramount in designing the I\u002FO facility of a computer system are effectiveness and generality.",[15,295029,295030,295032],{},[2627,295031,293379],{},": 在设计计算机系统的 I\u002FO 设施时，至关重要的两个主要目标是有效性（effectiveness）和通用性（generality）。",[15,295034,295035,8471,295037],{},[2627,295036,118575],{},[2627,295038,292009],{},[15,295040,295041,5598],{},[2627,295042,118582],{},[105,295044,295045,295050],{},[108,295046,295047,295049],{},[2627,295048,156287],{},": I\u002FO 设计目标。",[108,295051,295052,295054,295055,295061],{},[2627,295053,293402],{},": 教科书中明确指出，I\u002FO 设计的两个首要目标是",[2627,295056,295057,295058,295060],{},"效率（Efficiency）",[2627,295059,119123],{},"通用性（Generality）","。I\u002FO 操作往往是计算机系统的性能瓶颈，因此极度强调“效率”；同时为了操作系统能以统一的接口管理五花八门的设备，强调“通用性”。题目使用的词汇是 effectiveness，严格的术语应为 efficiency。",[15,295063,295064,295066],{},[2627,295065,115413],{},": The logical I\u002FO module, as defined in the hierarchical structure that manages I\u002FO on local peripheral devices, deals with the device as a logical resource and is not concerned with the details of actually controlling the device.",[15,295068,295069,295071],{},[2627,295070,293379],{},": 在管理本地外围设备 I\u002FO 的分层结构中定义的逻辑 I\u002FO 模块，将设备作为逻辑资源来处理，并且不关心实际控制该设备的细节。",[15,295073,295074,8471,295076],{},[2627,295075,118575],{},[2627,295077,291961],{},[15,295079,295080,5598],{},[2627,295081,118582],{},[105,295083,295084,295089],{},[108,295085,295086,295088],{},[2627,295087,156287],{},": I\u002FO 软件的层次结构。",[108,295090,295091,295093],{},[2627,295092,293402],{},": “逻辑 I\u002FO (Logical I\u002FO)”层提供了一个简单的、高级的接口来操作设备（将其视为逻辑上的块集合或字符流），并隐藏了特定于物理设备的底层控制细节（如磁头移动、寻道等细节交由 Device I\u002FO 层处理）。",[15,295095,295096,295098],{},[2627,295097,115435],{},": Disks and tapes are examples of stream-oriented I\u002FO devices.",[15,295100,295101,295103],{},[2627,295102,293379],{},": 磁盘和磁带是面向流的（stream-oriented）I\u002FO 设备的例子。",[15,295105,295106,8471,295108],{},[2627,295107,118575],{},[2627,295109,292009],{},[15,295111,295112,5598],{},[2627,295113,118582],{},[105,295115,295116,295121],{},[108,295117,295118,295120],{},[2627,295119,156287],{},": 面向块与面向流的 I\u002FO 设备。",[108,295122,295123,295125],{},[2627,295124,293402],{},": 磁盘和磁带是将信息保存在固定大小的数据块中，并以块为单位进行寻址和传输的，它们是典型的**面向块（Block-oriented）**设备。面向流（Stream-oriented）设备是以不可寻址的字节流形式传输数据的设备，如终端、打印机、通信端口和鼠标。",[15,295127,295128,295130],{},[2627,295129,115457],{},": Double buffering refers to the concept of using two buffers to alternatively fill and empty in order to facilitate the buffering of an I\u002FO request.",[15,295132,295133,295135],{},[2627,295134,293379],{},": 双缓冲（Double buffering）指的是使用两个缓冲区交替填充和清空的概念，以促进 I\u002FO 请求的缓冲。",[15,295137,295138,8471,295140],{},[2627,295139,118575],{},[2627,295141,291961],{},[15,295143,295144,5598],{},[2627,295145,118582],{},[105,295147,295148,295153],{},[108,295149,295150,295152],{},[2627,295151,156287],{},": 双缓冲技术。",[108,295154,295155,295157],{},[2627,295156,293402],{},": 这也称为缓冲交换（Buffer swapping）。当一个进程从一个缓冲区读取数据时，操作系统可以将后续的 I\u002FO 数据同时填入第二个缓冲区。这种交替并行极大地平滑了数据流，避免了进程因等待数据而挂起。",[15,295159,295160,295162],{},[2627,295161,115479],{},": In the operation of a disk drive, Seek Time is the sum of rotational delay and access time.",[15,295164,295165,295167],{},[2627,295166,293379],{},": 在磁盘驱动器的操作中，寻道时间（Seek Time）是旋转延迟（rotational delay）和存取时间（access time）的总和。",[15,295169,295170,8471,295172],{},[2627,295171,118575],{},[2627,295173,292009],{},[15,295175,295176,5598],{},[2627,295177,118582],{},[105,295179,295180,295185],{},[108,295181,295182,295184],{},[2627,295183,156287],{},": 磁盘性能参数。",[108,295186,295187,295189,295190,295193],{},[2627,295188,293402],{},": 物理公式颠倒了。正确的公式是：",[2627,295191,295192],{},"存取时间（Access Time） = 寻道时间（Seek Time） + 旋转延迟（Rotational Delay）","。存取时间是将读写头定位到正确扇区所需等待的总时间。",[15,295195,295196,295198],{},[2627,295197,115501],{},": Random scheduling is useful as a benchmark against which to evaluate other disk scheduling policies because it provides a worst-case scenario.",[15,295200,295201,295203],{},[2627,295202,293379],{},": 随机调度（Random scheduling）作为评估其他磁盘调度策略的基准很有用，因为它提供了一个最坏情况的场景。",[15,295205,295206,8471,295208],{},[2627,295207,118575],{},[2627,295209,291961],{},[15,295211,295212,5598],{},[2627,295213,118582],{},[105,295215,295216,295221],{},[108,295217,295218,295220],{},[2627,295219,156287],{},": 随机调度策略的作用。",[108,295222,295223,295225],{},[2627,295224,293402],{},": 在生产环境中没有人会真正使用完全随机的顺序去读取磁盘磁道。它仅仅在理论分析和基准测试中被使用，用来模拟调度效率的最下限（最差情况）。如果一个新算法的表现不比随机调度强多少，说明该算法极其失败。",[15,295227,295228,295230],{},[2627,295229,115523],{},": In the C-SCAN disk scheduling algorithm, the disk arm is required to move in one direction only until it reaches the last track or there are no more requests to service in that direction, then it reverses direction and the scan proceeds in the opposite direction in the same fashion.",[15,295232,295233,295235],{},[2627,295234,293379],{},": 在 C-SCAN 磁盘调度算法中，磁盘臂需要仅沿一个方向移动，直到到达最后一个磁道或该方向上没有更多要服务的请求，然后它反转方向并以相同的方式沿相反方向进行扫描。",[15,295237,295238,8471,295240],{},[2627,295239,118575],{},[2627,295241,292009],{},[15,295243,295244,5598],{},[2627,295245,118582],{},[105,295247,295248,295253],{},[108,295249,295250,295252],{},[2627,295251,156287],{},": 磁盘调度算法（SCAN 与 C-SCAN）。",[108,295254,295255,295257,295258,295261,295262,295265,295266,295269],{},[2627,295256,293402],{},": 题目描述的算法是 ",[2627,295259,295260],{},"SCAN（电梯算法）","，它在两个方向上都进行处理。而 ",[2627,295263,295264],{},"C-SCAN（循环扫描）"," 的行为是：磁头只在一个方向上响应请求。当磁头移动到磁盘一端时，它会立刻“空载”返回到磁盘的另一端起点，",[2627,295267,295268],{},"在返回途中不处理任何请求","，形成一个逻辑上的单向循环环路，这减少了极端磁道请求的饥饿时间。",[15,295271,295272,295274],{},[2627,295273,115545],{},": The unique contribution of RAID is that it effectively addresses the need for redundancy in data storage.",[15,295276,295277,295279],{},[2627,295278,293379],{},": RAID 的独特贡献在于它有效地满足了数据存储中对冗余（redundancy）的需求。",[15,295281,295282,8471,295284],{},[2627,295283,118575],{},[2627,295285,292009],{},[15,295287,295288,5598],{},[2627,295289,118582],{},[105,295291,295292,295297],{},[108,295293,295294,295296],{},[2627,295295,156287],{},": RAID 的设计目标。",[108,295298,295299,295301,295302,295308],{},[2627,295300,293402],{},": RAID（独立磁盘冗余阵列）解决的绝不仅仅是冗余需求。它有两个同样重要的独特贡献：",[2627,295303,295304,295305,295307],{},"性能提升（Performance）",[2627,295306,119123],{},"冗余（Redundancy）","。通过条带化（Striping）在多个磁盘上并行读写，它极大地提高了 I\u002FO 速度（甚至 RAID 0 根本就没有提供冗余，完全是为了性能）。",[15,295310,295311,295313],{},[2627,295312,115567],{},": Cache memory is smaller and faster than main memory and is interposed between main memory and secondary storage.",[15,295315,295316,295318],{},[2627,295317,293379],{},": 高速缓存（Cache memory）比主存更小、更快，它介于主存和辅助存储之间。",[15,295320,295321,8471,295323],{},[2627,295322,118575],{},[2627,295324,292009],{},[15,295326,295327,5598],{},[2627,295328,118582],{},[105,295330,295331,295336],{},[108,295332,295333,295335],{},[2627,295334,156287],{},": 存储器层次结构。",[108,295337,295338,295340,295341,295344,295345,295348],{},[2627,295339,293402],{},": 介于主存和辅助存储（如硬盘）之间，用于平滑磁盘 I\u002FO 速度差异的缓存被称为",[2627,295342,295343],{},"磁盘缓存（Disk Cache）","。而**高速缓存（Cache memory，如 L1\u002FL2 Cache）**是介于 ",[2627,295346,295347],{},"处理器（CPU）和主存（Main Memory）"," 之间的。",[15,295350,295351,295353],{},[2627,295352,115589],{},": For buffered I\u002FO in a UNIX system, two types of buffers are used: system buffer caches and character queues.",[15,295355,295356,295358],{},[2627,295357,293379],{},": 对于 UNIX 系统中的缓冲 I\u002FO，使用两种类型的缓冲区：系统缓冲缓存（system buffer caches）和字符队列（character queues）。",[15,295360,295361,8471,295363],{},[2627,295362,118575],{},[2627,295364,291961],{},[15,295366,295367,5598],{},[2627,295368,118582],{},[105,295370,295371,295376],{},[108,295372,295373,295375],{},[2627,295374,156287],{},": UNIX 的 I\u002FO 缓冲机制。",[108,295377,295378,295380],{},[2627,295379,293402],{},": UNIX 将 I\u002FO 设备分为块设备（如磁盘和磁带）和字符设备（如终端和通信端口）。对应地，操作系统采用“系统缓冲缓存（System Buffer Cache）”来处理块 I\u002FO，采用“字符队列（Character Queues）”来处理字符流 I\u002FO。",[15,295382,295383,295385],{},[2627,295384,115611],{},": In a W2K system, the synchronous mode of I\u002FO operation is used whenever possible to optimize application performance.",[15,295387,295388,295390],{},[2627,295389,293379],{},": 在 W2K 系统中，只要可能，就会使用同步模式的 I\u002FO 操作来优化应用程序的性能。",[15,295392,295393,8471,295395],{},[2627,295394,118575],{},[2627,295396,292009],{},[15,295398,295399,5598],{},[2627,295400,118582],{},[105,295402,295403,295408],{},[108,295404,295405,295407],{},[2627,295406,156287],{},": Windows 的 I\u002FO 架构。",[108,295409,295410,295412,295413,295416],{},[2627,295411,293402],{},": 为了优化性能，Windows 提供了丰富的",[2627,295414,295415],{},"异步（Asynchronous） I\u002FO"," 操作能力。异步 I\u002FO 允许应用程序发起读写请求后立刻去执行其他计算任务，等 I\u002FO 完成后再通过回调或事件得到通知，这能够极大地提升系统的并发吞吐量。因此系统鼓励使用的是异步而非同步模式。",[32188,295418],{},[19,295420,293916],{"id":293915},[15,295422,295423,295425],{},[2627,295424,115273],{},": An example of the key differences that can exist across (and even in) classes of I\u002FO devices is:\na. Data rate\nb. Data representation\nc. Error conditions\nd. All of the above",[15,295427,295428,295430],{},[2627,295429,293379],{},": 在各类 I\u002FO 设备之间（甚至在类别内部）可能存在的关键差异的例子是：\na. 数据传输率 (Data rate)\nb. 数据表示 (Data representation)\nc. 错误条件 (Error conditions)\nd. 以上所有",[15,295432,295433,8471,295435],{},[2627,295434,118575],{},[2627,295436,292510],{},[15,295438,295439,5598],{},[2627,295440,118582],{},[105,295442,295443,295448],{},[108,295444,295445,295447],{},[2627,295446,156287],{},": I\u002FO 设备的多样性。",[108,295449,295450,295452],{},[2627,295451,293402],{},": 外部设备极具多样性。比如千兆网卡和鼠标的传输率（Data rate）有着天壤之别；不同的设备使用不同的数据编码形式（Data representation）；它们上报的机械或硬件故障（Error conditions）也截然不同。操作系统的 I\u002FO 模块的一项大工程就是将这些差异屏蔽掉。",[15,295454,295455,295457],{},[2627,295456,115295],{},": The I\u002FO technique where the processor busy waits for an I\u002FO operation to complete is called:\na. Programmed I\u002FO\nb. Interrupt-driven I\u002FO\nc. Direct memory access (DMA)\nd. None of the above",[15,295459,295460,295462],{},[2627,295461,293379],{},": 处理器通过“忙等待（busy waits）”来等待 I\u002FO 操作完成的 I\u002FO 技术被称为：\na. 程序控制 I\u002FO (Programmed I\u002FO)\nb. 中断驱动 I\u002FO\nc. 直接内存访问 (DMA)\nd. 以上都不是",[15,295464,295465,8471,295467],{},[2627,295466,118575],{},[2627,295468,295469],{},"a. Programmed I\u002FO",[15,295471,295472,5598],{},[2627,295473,118582],{},[105,295475,295476,295481],{},[108,295477,295478,295480],{},[2627,295479,156287],{},": I\u002FO 控制技术演进。",[108,295482,295483,295485,295486,295488],{},[2627,295484,293402],{},": 在最原始的",[2627,295487,292385],{}," 模式下，处理器向 I\u002FO 模块发送命令后，会进入一个死循环（忙等待），不断读取 I\u002FO 模块的状态寄存器，直到操作完成。这极大地浪费了 CPU 时间。",[15,295490,295491,295493],{},[2627,295492,115317],{},": The system configuration that includes an I\u002FO module which is a separate processor with a specialized instruction set can be referred to using the following terminology:\na. Direct Memory Access (DMA)\nb. I\u002FO Channel\nc. I\u002FO Processor\nd. All of the above",[15,295495,295496,295498],{},[2627,295497,293379],{},": 包含一个作为独立处理器并具有专用指令集的 I\u002FO 模块的系统配置，可以使用以下术语来指代：\na. DMA\nb. I\u002FO 通道 (I\u002FO Channel)\nc. I\u002FO 处理器 (I\u002FO Processor)\nd. 以上所有",[15,295500,295501,8471,295503,47649,295506],{},[2627,295502,118575],{},[2627,295504,295505],{},"b. I\u002FO Channel",[71807,295507,295508],{},"(注：I\u002FO Channel 和 I\u002FO Processor 在教科书演化步骤中是递进的，通常具有专门 I\u002FO 指令集的代名词是 I\u002FO 通道)",[15,295510,295511,5598],{},[2627,295512,118582],{},[105,295514,295515,295520],{},[108,295516,295517,295519],{},[2627,295518,156287],{},": I\u002FO 通道（I\u002FO Channel）。",[108,295521,295522,295524,295525,295528,295529,295532],{},[2627,295523,293402],{},": 这是 I\u002FO 演化的第四步。I\u002FO 模块被增强为一个",[2627,295526,295527],{},"独立的处理器（带有专门为 I\u002FO 定制的指令集）","。中央处理器（CPU）指示该模块在内存中执行一个 I\u002FO 程序，这个独立的模块就被称为 ",[2627,295530,295531],{},"I\u002FO 通道（I\u002FO Channel）","。如果它进一步获得了自己的本地内存，则被称为 I\u002FO 处理器（I\u002FO Processor）。",[15,295534,295535,295537],{},[2627,295536,115339],{},": The bus configuration for DMA that provides no path other than the system bus between the DMA module(s) and I\u002FO devices is:\na. Single bus, detached DMA\nb. Single bus, integrated DMA-I\u002FO\nc. I\u002FO bus\nd. None of the above",[15,295539,295540,295542],{},[2627,295541,293379],{},": DMA 模块和 I\u002FO 设备之间除了系统总线外不提供其他路径的 DMA 总线配置是：\na. 单总线，分离 DMA (Single bus, detached DMA)\nb. 单总线，集成 DMA-I\u002FO\nc. I\u002FO 总线\nd. 以上都不是",[15,295544,295545,8471,295547],{},[2627,295546,118575],{},[2627,295548,295549],{},"a. Single bus, detached DMA",[15,295551,295552,5598],{},[2627,295553,118582],{},[105,295555,295556,295561],{},[108,295557,295558,295560],{},[2627,295559,156287],{},": DMA 的系统架构方案。",[108,295562,295563,295565],{},[2627,295564,293402],{},": 如果 DMA 控制器完全独立于各个 I\u002FO 模块，并且挂载在唯一的系统总线上，任何 I\u002FO 模块与 DMA 模块之间交换数据都要通过系统总线。这就是开销最大的“单总线，分离 DMA”架构。",[15,295567,295568,295570],{},[2627,295569,115391],{},": The primary objective in designing the I\u002FO facility of a computer system that deals with the desire to handle all I\u002FO devices in a uniform manner is referred to as:\na. Efficiency\nb. Generality\nc. Directory management\nd. None of the above",[15,295572,295573,295575],{},[2627,295574,293379],{},": 在设计计算机系统 I\u002FO 设施时，处理希望以统一方式处理所有 I\u002FO 设备的主要目标被称为：\na. 效率 (Efficiency)\nb. 通用性 (Generality)\nc. 目录管理\nd. 以上都不是",[15,295577,295578,8471,295580],{},[2627,295579,118575],{},[2627,295581,295582],{},"b. Generality",[15,295584,295585,5598],{},[2627,295586,118582],{},[105,295588,295589,295594],{},[108,295590,295591,295593],{},[2627,295592,156287],{},": 通用性目标。",[108,295595,295596,295598,295599,616,295601,616,295604,616,295607,295609,295610,347],{},[2627,295597,293402],{},": 操作系统希望无论底层是磁盘、网卡还是打印机，用户程序都能用相同的接口（如 ",[31,295600,255541],{},[31,295602,295603],{},"read",[31,295605,295606],{},"write",[31,295608,255545],{},"）进行操作，这种对异构性进行统一抽象的性质就是",[2627,295611,295060],{},[15,295613,295614,295616],{},[2627,295615,115413],{},": In a hierarchical structure for managing I\u002FO on a secondary storage device that supports a file system, the layer that is closest to the hardware is the:\na. Directory management layer\nb. Device I\u002FO layer\nc. Physical organization layer\nd. None of the above",[15,295618,295619,295621],{},[2627,295620,293379],{},": 在管理支持文件系统的辅助存储设备 I\u002FO 的分层结构中，最靠近硬件的层是：\na. 目录管理层\nb. 设备 I\u002FO 层 (Device I\u002FO layer)\nc. 物理组织层\nd. 以上都不是",[15,295623,295624,8471,295626],{},[2627,295625,118575],{},[2627,295627,292848],{},[15,295629,295630,5598],{},[2627,295631,118582],{},[105,295633,295634,295639],{},[108,295635,295636,295638],{},[2627,295637,156287],{},": I\u002FO 层次结构模型。",[108,295640,295641,295643,295644,295647,295648,295651,295652],{},[2627,295642,293402],{},": 对于支持文件系统的辅助存储，教科书列出的逻辑层次由上至下通常为：目录管理 (Directory Management) -> 逻辑文件系统 (Logical File System) -> 物理组织 (Physical Organization) -> 设备 I\u002FO (Device I\u002FO) -> ",[2627,295645,295646],{},"调度和控制 (Scheduling and Control)","。最底层（最靠近硬件）是",[2627,295649,295650],{},"调度和控制层","。由于 a, b, c 中没有该选项（Device I\u002FO 只是倒数第二层），最严谨的答案是“以上都不是”。",[71807,295653,295654],{},"(注：部分教辅资料可能直接选择 Device I\u002FO，但严格遵循结构模型应为缺失底层)",[15,295656,295657,295659],{},[2627,295658,115435],{},": An example of a block-oriented I\u002FO device is:\na. CD-ROM\nb. Printer\nc. Modem\nd. All of the above",[15,295661,295662,295664],{},[2627,295663,293379],{},": 面向块（block-oriented）的 I\u002FO 设备的例子是：\na. CD-ROM\nb. 打印机\nc. 调制解调器\nd. 以上所有",[15,295666,295667,8471,295669],{},[2627,295668,118575],{},[2627,295670,295671],{},"a. CD-ROM",[15,295673,295674,5598],{},[2627,295675,118582],{},[105,295677,295678,295683],{},[108,295679,295680,295682],{},[2627,295681,156287],{},": 块设备。",[108,295684,295685,295687],{},[2627,295686,293402],{},": 块设备是以固定大小的“块（Block）”为单位存储和传输数据的，并且可以通过地址进行随机访问。CD-ROM（光盘）和磁盘都是典型的块设备。打印机和调制解调器是传输字符流的字符设备。",[15,295689,295690,295692],{},[2627,295691,115457],{},": The scenario where multiple buffers are used in an attempt to alleviate the problem of absorbing rapid bursts of I\u002FO is typically referred to as:\na. Single buffering\nb. Double buffering\nc. Circular buffering\nd. None of the above",[15,295694,295695,295697],{},[2627,295696,293379],{},": 使用多个缓冲区试图缓解吸收快速爆发的 I\u002FO 问题的场景通常被称为：\na. 单缓冲\nb. 双缓冲\nc. 循环缓冲 (Circular buffering)\nd. 以上都不是",[15,295699,295700,8471,295702],{},[2627,295701,118575],{},[2627,295703,295704],{},"c. Circular buffering",[15,295706,295707,5598],{},[2627,295708,118582],{},[105,295710,295711,295716],{},[108,295712,295713,295715],{},[2627,295714,156287],{},": 循环缓冲。",[108,295717,295718,295720],{},[2627,295719,293402],{},": 虽然双缓冲能解决大部分问题，但当系统偶尔发生极其急促的 I\u002FO 爆发（rapid bursts）且单个处理速度跟不上时，即使双缓冲也会溢出。此时将两个以上的缓冲区组织成环形队列使用，就是**循环缓冲（Circular buffering）**技术。",[15,295722,295723,295725],{},[2627,295724,115479],{},": The aspect of disk performance that represents the time it takes to position the head at the desired track is known as:\na. Seek time\nb. Rotational delay\nc. Access time\nd. None of the above",[15,295727,295728,295730],{},[2627,295729,293379],{},": 代表将磁头定位到所需磁道所花费的时间的磁盘性能方面被称为：\na. 寻道时间 (Seek time)\nb. 旋转延迟\nc. 存取时间\nd. 以上都不是",[15,295732,295733,8471,295735],{},[2627,295734,118575],{},[2627,295736,295737],{},"a. Seek time",[15,295739,295740,5598],{},[2627,295741,118582],{},[105,295743,295744,295749],{},[108,295745,295746,295748],{},[2627,295747,156287],{},": 寻道时间。",[108,295750,295751,295753,295754,347],{},[2627,295752,293402],{},": 磁盘读写的机械动作中，移动磁盘臂使得读写磁头精确对准指定磁道（Track）或柱面的动作，其耗时就被称为",[2627,295755,295756],{},"寻道时间（Seek time）",[15,295758,295759,295761],{},[2627,295760,115501],{},": The following disk scheduling policy is useful as a benchmark against which to evaluate other disk scheduling policies because it provides a worst-case scenario:\na. FIFO scheduling\nb. Priority scheduling\nc. Random scheduling\nd. None of the above",[15,295763,295764,295766],{},[2627,295765,293379],{},": 以下磁盘调度策略作为评估其他磁盘调度策略的基准很有用，因为它提供了一个最坏情况场景：\na. FIFO 调度\nb. 优先级调度\nc. 随机调度 (Random scheduling)\nd. 以上都不是",[15,295768,295769,8471,295771],{},[2627,295770,118575],{},[2627,295772,295773],{},"c. Random scheduling",[15,295775,295776,5598],{},[2627,295777,118582],{},[105,295779,295780,295785],{},[108,295781,295782,295784],{},[2627,295783,156287],{},": 随机调度算法。",[108,295786,295787,295789],{},[2627,295788,293402],{},": 同判断题第 10 题。完全随机的调度能产生最恶劣的平均寻道长度，是其他所有调度算法效果评估的反面基准（Worst-case benchmark）。",[15,295791,295792,295794],{},[2627,295793,115523],{},": The disk scheduling algorithm that implements two subqueues in a measure to avoid the problem of \"arm stickiness\" is the:\na. C-SCAN policy\nb. FSCAN policy\nc. N-step-SCAN policy\nd. All of the above",[15,295796,295797,295799],{},[2627,295798,293379],{},": 为了避免“磁臂粘滞（arm stickiness）”问题而实现两个子队列的磁盘调度算法是：\na. C-SCAN 策略\nb. FSCAN 策略\nc. N-step-SCAN 策略\nd. 以上所有",[15,295801,295802,8471,295804],{},[2627,295803,118575],{},[2627,295805,295806],{},"b. FSCAN policy",[15,295808,295809,5598],{},[2627,295810,118582],{},[105,295812,295813,295818],{},[108,295814,295815,295817],{},[2627,295816,156287],{},": FSCAN 算法原理。",[108,295819,295820,295822,295823,347,295826,295828],{},[2627,295821,293402],{},": 如果在磁头扫描经过某个磁道时，系统不断有请求访问该磁道，磁头就会被“粘”在原地，这叫",[2627,295824,295825],{},"磁臂粘滞现象（Arm stickiness）",[2627,295827,235417],{}," 算法通过将请求队列严格分成两个独立的子队列来解决这个问题：磁头只处理第一个队列里的请求，而在扫描期间新到达的所有请求都被强行放入第二个空队列中等待下一轮。它使用了正好两个（Two subqueues）队列。",[15,295830,295831,295833],{},[2627,295832,115545],{},": Which of the following RAID levels implement some form of parity calculation to introduce redundancy:\na. RAID Level 2\nb. RAID Level 4\nc. RAID Level 6\nd. All of the above",[15,295835,295836,295838],{},[2627,295837,293379],{},": 以下哪些 RAID 级别实现了某种形式的奇偶校验计算以引入冗余：\na. RAID 级别 2\nb. RAID 级别 4\nc. RAID 级别 6\nd. 以上所有",[15,295840,295841,8471,295843],{},[2627,295842,118575],{},[2627,295844,292510],{},[15,295846,295847,5598],{},[2627,295848,118582],{},[105,295850,295851,295856],{},[108,295852,295853,295855],{},[2627,295854,156287],{},": 各种 RAID 的冗余校验技术。",[108,295857,295858,5598,295860],{},[2627,295859,293402],{},[105,295861,295862,295867,295872],{},[108,295863,295864,295866],{},[2627,295865,236231],{}," 使用海明码（Hamming code），这本质上是多重奇偶校验位的计算。",[108,295868,295869,295871],{},[2627,295870,236265],{}," 使用独立的一个校验盘存放数据的块级奇偶校验（Block-level parity）。",[108,295873,295874,295876],{},[2627,295875,236296],{}," 使用双重分布式校验算法进行复杂的双校验计算以容忍两块磁盘同时损坏。它们都使用了校验。",[15,295878,295879,295881],{},[2627,295880,115567],{},": The disk cache replacement strategy that replaces the block that has experienced the fewest references is called:\na. Least Recently Used (LRU)\nb. Least Referenced (LR)\nc. Least Frequently Used (LFU)\nd. All of the above",[15,295883,295884,295886],{},[2627,295885,293379],{},": 替换经历引用次数最少的块的磁盘缓存替换策略被称为：\na. 最近最少使用 (LRU)\nb. 最少引用 (LR)\nc. 最不经常使用 (Least Frequently Used, LFU)\nd. 以上所有",[15,295888,295889,8471,295891],{},[2627,295890,118575],{},[2627,295892,295893],{},"c. Least Frequently Used (LFU)",[15,295895,295896,5598],{},[2627,295897,118582],{},[105,295899,295900,295905],{},[108,295901,295902,295904],{},[2627,295903,156287],{},": 缓存替换算法。",[108,295906,295907,295909],{},[2627,295908,293402],{},": 基于“最少引用次数（fewest references）”进行淘汰的策略称为最不经常使用算法（LFU）。它为每个块维护一个计数器，替换计数器值最低的缓存块。",[15,295911,295912,295914],{},[2627,295913,115589],{},": In a UNIX system, which of the following types of I\u002FO devices make use of character queues:\na. Disk drive\nb. Tape drive\nc. Communications lines\nd. All of the above",[15,295916,295917,295919],{},[2627,295918,293379],{},": 在 UNIX 系统中，以下哪种类型的 I\u002FO 设备利用字符队列：\na. 磁盘驱动器\nb. 磁带驱动器\nc. 通信线路 (Communications lines)\nd. 以上所有",[15,295921,295922,8471,295924],{},[2627,295923,118575],{},[2627,295925,295926],{},"c. Communications lines",[15,295928,295929,5598],{},[2627,295930,118582],{},[105,295932,295933,295938],{},[108,295934,295935,295937],{},[2627,295936,156287],{},": UNIX 缓冲 I\u002FO 结构。",[108,295939,295940,295942],{},[2627,295941,293402],{},": 在 UNIX 中，磁盘和磁带等面向块的设备使用的是“系统缓冲缓存（System Buffer Caches）”。而通信线路、终端等面向字节流的字符设备，它们不处理定长数据块，因此系统为它们分配的是“字符队列（Character Queues）”。",[15,295944,295945,295947],{},[2627,295946,115611],{},": In a W2K system, the I\u002FO manager module that includes lazy write and lazy commit services to improve overall performance is the:\na. Cache manager\nb. File system drivers\nc. Hardware device drivers\nd. None of the above",[15,295949,295950,295952],{},[2627,295951,293379],{},": 在 W2K（Windows 2000内核系列）系统中，包含“延迟写入（lazy write）”和“延迟提交（lazy commit）”服务以提高整体性能的 I\u002FO 管理器模块是：\na. 缓存管理器 (Cache manager)\nb. 文件系统驱动程序\nc. 硬件设备驱动程序\nd. 以上都不是",[15,295954,295955,8471,295957],{},[2627,295956,118575],{},[2627,295958,295959],{},"a. Cache manager",[15,295961,295962,5598],{},[2627,295963,118582],{},[105,295965,295966,295971],{},[108,295967,295968,295970],{},[2627,295969,156287],{},": Windows I\u002FO 管理器组件。",[108,295972,295973,295975,295976,295979],{},[2627,295974,293402],{},": Windows 的 ",[2627,295977,295978],{},"缓存管理器（Cache Manager）"," 负责整合和管理整个系统的内存到磁盘的缓存架构。它提供的“延迟写（Lazy Write）”机制允许系统在 CPU 负载低时再将被修改的缓存数据刷回磁盘，极大地提高了感知上的写入性能。",[32188,295981],{},[19,295983,294449],{"id":294448},[15,295985,295986,295988],{},[2627,295987,115273],{},": The term __________ refers to the speed with which data moves to and from the individual I\u002FO device.",[15,295990,295991,295993],{},[2627,295992,293379],{},": 术语 __________ 指的是数据流入和流出单个 I\u002FO 设备的速度。",[15,295995,295996,8471,295998],{},[2627,295997,118575],{},[2627,295999,296000],{},"data rate",[15,296002,296003,296005],{},[2627,296004,118582],{},": 数据传输率（Data rate）是衡量外部 I\u002FO 性能最关键的物理指标。",[15,296007,296008,296010],{},[2627,296009,115295],{},": In the __________ I\u002FO technique, the processor issues an I\u002FO request, continues with other work and eventually receives notification that the request was fulfilled.",[15,296012,296013,296015],{},[2627,296014,293379],{},": 在 __________ I\u002FO 技术中，处理器发出 I\u002FO 请求，继续进行其他工作，并最终收到请求已完成的通知。",[15,296017,296018,8471,296020],{},[2627,296019,118575],{},[2627,296021,296022],{},"interrupt-driven",[15,296024,296025,296027],{},[2627,296026,118582],{},": 中断驱动 I\u002FO（Interrupt-driven I\u002FO）解决了程序控制 I\u002FO 的忙等待问题。设备完成工作后，通过硬件电平信号触发一个中断来“通知（notification）”处理器。",[15,296029,296030,296032],{},[2627,296031,115317],{},": The term __________ characterizes a system configuration that includes an I\u002FO module that is a separate processor with a specialized instruction set.",[15,296034,296035,296037],{},[2627,296036,293379],{},": 术语 __________ 描述了一种系统配置，该配置包含一个作为具有专门指令集的独立处理器的 I\u002FO 模块。",[15,296039,296040,8471,296042],{},[2627,296041,118575],{},[2627,296043,296044],{},"I\u002FO channel (或 I\u002FO processor)",[15,296046,296047,296049],{},[2627,296048,118582],{},": 这种带有专属 I\u002FO 指令的高级架构被称为 I\u002FO 通道（I\u002FO Channel）。",[15,296051,296052,296054],{},[2627,296053,115339],{},": The bus configuration for DMA that provides no path other than the system bus between the DMA module(s) and I\u002FO devices is called __________.",[15,296056,296057,296059],{},[2627,296058,293379],{},": DMA 模块和 I\u002FO 设备之间除了系统总线外不提供其他路径的 DMA 总线配置被称为 __________。",[15,296061,296062,8471,296064],{},[2627,296063,118575],{},[2627,296065,296066],{},"single-bus, detached DMA",[15,296068,296069,296071],{},[2627,296070,118582],{},": 即单总线、分离 DMA。这种设计由于必须两度竞争系统总线（设备到 DMA，DMA 到内存），效率受到总线的严重制约。",[15,296073,296074,296076],{},[2627,296075,115391],{},": The term __________ refers to the I\u002FO design objective that focuses on preventing I\u002FO from becoming a bottleneck in the system.",[15,296078,296079,296081],{},[2627,296080,293379],{},": 术语 __________ 指的是致力于防止 I\u002FO 成为系统瓶颈的 I\u002FO 设计目标。",[15,296083,296084,8471,296086],{},[2627,296085,118575],{},[2627,296087,193875],{},[15,296089,296090,296092],{},[2627,296091,118582],{},": 计算机操作中大多数等待时间都是由于缓慢的外部设备造成的，因此操作系统的 I\u002FO 设计在各个层面上都极致追求效率（Efficiency）。",[15,296094,296095,296097],{},[2627,296096,115413],{},": The __________ layer is the hierarchical layer that is closest to the hardware in most I\u002FO management structures.",[15,296099,296100,296102],{},[2627,296101,293379],{},": __________ 层是在大多数 I\u002FO 管理结构中最靠近硬件的分层层。",[15,296104,296105,8471,296107],{},[2627,296106,118575],{},[2627,296108,296109],{},"scheduling and control",[15,296111,296112,296114],{},[2627,296113,118582],{},": 在 I\u002FO 软件的逻辑分层模型的最底部，“调度和控制层（Scheduling and Control layer）”直接和硬件的控制器\u002F寄存器打交道，负责实际的中断处理和命令分派。",[15,296116,296117,296119],{},[2627,296118,115435],{},": A hard drive is an example of a __________-oriented I\u002FO device.",[15,296121,296122,296124],{},[2627,296123,293379],{},": 硬盘驱动器是 __________ 导向的 I\u002FO 设备的例子。",[15,296126,296127,8471,296129],{},[2627,296128,118575],{},[2627,296130,126394],{},[15,296132,296133,296135],{},[2627,296134,118582],{},": 磁盘（硬盘）是面向块（Block-oriented）存取数据的典型设备。",[15,296137,296138,296140],{},[2627,296139,115457],{},": An improvement over single buffering is the technique known as __________, where two system buffers are assigned to the operation and one can be emptied while the other is filled.",[15,296142,296143,296145],{},[2627,296144,293379],{},": 相对于单缓冲的一种改进是被称为 __________ 的技术，其中为操作分配了两个系统缓冲区，一个可以在填充时清空另一个。",[15,296147,296148,8471,296150],{},[2627,296149,118575],{},[2627,296151,296152],{},"double buffering (或 buffer swapping)",[15,296154,296155,296157],{},[2627,296156,118582],{},": 双缓冲（Double buffering）允许 I\u002FO 硬件和处理进程实现一定程度的真正并行操作。",[15,296159,296160,296162],{},[2627,296161,115479],{},": The __________ of data to or from a disk depends on the rotation speed of the disk, the number of bytes on a track and the number of bytes to be transferred.",[15,296164,296165,296167],{},[2627,296166,293379],{},": 进出磁盘的数据的 __________ 决于磁盘的旋转速度、磁道上的字节数以及要传输的字节数。",[15,296169,296170,8471,296172],{},[2627,296171,118575],{},[2627,296173,296174],{},"transfer time",[15,296176,296177,296179],{},[2627,296178,118582],{},": 一旦磁头定位在目标扇区，数据开始通过磁头感应被读出，此期间消耗的时间称为传输时间（Transfer Time）。",[15,296181,296182,296184],{},[2627,296183,115501],{},": __________ is useful as a benchmark against which to evaluate other disk scheduling policies because it provides a worst-case scenario.",[15,296186,296187,296189],{},[2627,296188,293379],{},": __________ 作为评估其他磁盘调度策略的基准很有用，因为它提供了一个最坏情况场景。",[15,296191,296192,8471,296194],{},[2627,296193,118575],{},[2627,296195,296196],{},"Random scheduling",[15,296198,296199,296201],{},[2627,296200,118582],{},": 随机调度（Random scheduling）作为理论上的性能地板基准。",[15,296203,296204,296206],{},[2627,296205,115523],{},": The disk scheduling algorithm that implements exactly 2 subqueues in a measure to avoid the problem of \"arm stickiness\" is the __________ policy.",[15,296208,296209,296211],{},[2627,296210,293379],{},": 为了避免“磁臂粘滞”问题而精确实现 2 个子队列的磁盘调度算法是 __________ 策略。",[15,296213,296214,8471,296216],{},[2627,296215,118575],{},[2627,296217,235417],{},[15,296219,296220,296222],{},[2627,296221,118582],{},": FSCAN 就是为了根治短距离高频请求造成的 SCAN\u002FC-SCAN 的粘滞现象而设计的，严格将请求区分为当前队列（处理中）和下一轮队列（收集新请求）。",[15,296224,296225,296227],{},[2627,296226,115545],{},": RAID Level __________ employs a Hamming code to correct single-bit errors and detect double-bit errors.",[15,296229,296230,296232],{},[2627,296231,293379],{},": RAID 级别 __________ 采用海明码来纠正单比特错误并检测双比特错误。",[15,296234,296235,8471,296237],{},[2627,296236,118575],{},[2627,296238,768],{},[15,296240,296241,296243],{},[2627,296242,118582],{},": RAID 2 在多块数据盘旁边使用多个独立的校验盘来存储海明码（Hamming Code），提供强大的内存级纠错能力（但因商业成本和现代硬盘内部自带ECC的原因极少被使用）。",[15,296245,296246,296248],{},[2627,296247,115567],{},": A __________ is a buffer in main memory designed for the temporary storage of disk sectors.",[15,296250,296251,296253],{},[2627,296252,293379],{},": __________ 是主存中的一个缓冲区，专为磁盘扇区的临时存储而设计。",[15,296255,296256,8471,296258],{},[2627,296257,118575],{},[2627,296259,296260],{},"disk cache",[15,296262,296263,296265],{},[2627,296264,118582],{},": 磁盘缓存（Disk cache）是由操作系统在内存里划分出的一片区域，用来缓存近期被读取的磁盘扇区或预读后续扇区，大幅提升命中时的访问速度。",[15,296267,296268,296270],{},[2627,296269,115589],{},": In a UNIX system, __________ I\u002FO typically involves the DMA facility, with the transfer taking place directly between the I\u002FO module and the process I\u002FO area.",[15,296272,296273,296275],{},[2627,296274,293379],{},": 在 UNIX 系统中，__________ I\u002FO 通常涉及 DMA 设施，传输直接在 I\u002FO 模块和进程 I\u002FO 区域之间进行。",[15,296277,296278,8471,296280],{},[2627,296279,118575],{},[2627,296281,296282],{},"unbuffered",[15,296284,296285,296287],{},[2627,296286,118582],{},": 当程序为了最高性能追求自己接管缓存管理时，可以使用无缓冲 I\u002FO（Unbuffered I\u002FO 或 Raw I\u002FO），此时数据直接通过 DMA 从磁盘“绕过系统缓冲区”直达用户进程的地址空间。",[15,296289,296290,296292],{},[2627,296291,115611],{},": In a W2K system, the term __________ refers to the fact that the system improves performance by recording updates in the cache and later writing the changes to disk when demand on the processor is low.",[15,296294,296295,296297],{},[2627,296296,293379],{},": 在 W2K 系统中，术语 __________ 指的是系统通过在缓存中记录更新，然后在处理器需求低时将更改写入磁盘来提高性能的事实。",[15,296299,296300,8471,296302],{},[2627,296301,118575],{},[2627,296303,296304],{},"lazy write",[15,296306,296307,296309],{},[2627,296308,118582],{},": 延迟写（Lazy write）是现代操作系统缓存管理的神兵利器，它将耗时的慢速写入聚合，并在系统空闲后台周期性地刷入磁盘。",{"title":29,"searchDepth":43,"depth":43,"links":296311},[296312,296313,296314],{"id":293368,"depth":43,"text":293369},{"id":293915,"depth":43,"text":293916},{"id":294448,"depth":43,"text":294449},{"order":1990},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap11",{"title":294858,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap11","z3tNfhJYKrsnghz6KLSyfsl4ziun1zedIXct3VwSAXs",{"id":296321,"title":296322,"body":296323,"description":29,"extension":85,"meta":297755,"navigation":52,"path":297756,"seo":297757,"stem":297758,"__hash__":297759},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap12.md","Chap. 12",{"type":8,"value":296324,"toc":297750},[296325,296328,296330,296335,296340,296346,296350,296366,296371,296376,296382,296386,296398,296403,296408,296414,296418,296434,296439,296444,296450,296454,296466,296471,296476,296482,296486,296498,296503,296508,296514,296518,296538,296543,296548,296554,296558,296574,296579,296584,296590,296594,296606,296611,296616,296622,296626,296643,296648,296653,296659,296663,296675,296680,296685,296691,296695,296711,296716,296721,296727,296731,296743,296748,296753,296759,296763,296778,296783,296788,296794,296798,296810,296815,296820,296826,296830,296846,296848,296850,296855,296860,296867,296871,296886,296891,296896,296903,296907,296919,296924,296929,296936,296940,296955,296960,296965,296972,296976,296992,296997,297002,297009,297013,297033,297038,297043,297049,297053,297065,297070,297075,297082,297086,297098,297103,297108,297115,297119,297138,297143,297148,297155,297159,297171,297176,297181,297188,297192,297207,297212,297217,297224,297228,297240,297245,297250,297256,297260,297272,297277,297282,297289,297293,297309,297314,297319,297326,297330,297356,297361,297366,297373,297377,297396,297398,297400,297405,297410,297417,297422,297427,297432,297439,297444,297449,297454,297461,297466,297471,297476,297483,297488,297493,297498,297505,297510,297515,297520,297527,297536,297541,297546,297553,297558,297563,297568,297575,297580,297585,297590,297597,297606,297611,297616,297623,297632,297637,297642,297649,297654,297659,297664,297671,297676,297681,297686,297693,297698,297703,297708,297715,297720,297725,297730,297737],[11,296326,296327],{"id":237716},"Chapter 12 - File Management",[19,296329,293369],{"id":293368},[15,296331,296332,296334],{},[2627,296333,115273],{},": A file is a collection of fields, which is in turn a collection of related records.",[15,296336,296337,296339],{},[2627,296338,293379],{},": 文件是字段的集合，而字段又是相关记录的集合。",[15,296341,296342,8471,296344],{},[2627,296343,118575],{},[2627,296345,292009],{},[15,296347,296348,5598],{},[2627,296349,118582],{},[105,296351,296352,296357],{},[108,296353,296354,296356],{},[2627,296355,156287],{},": 文件系统的数据层次结构。",[108,296358,296359,296361,296362,296365],{},[2627,296360,293402],{},": 概念的层级关系反了。正确的层级是从小到大：",[2627,296363,296364],{},"字段（Field） -> 记录（Record） -> 文件（File） -> 数据库（Database）","。也就是说，记录（Record）是相关字段的集合，而文件（File）是相关记录的集合。",[15,296367,296368,296370],{},[2627,296369,115295],{},": At the lowest level of the file system architecture, device drivers communicate directly with peripheral devices or their controllers or channels.",[15,296372,296373,296375],{},[2627,296374,293379],{},": 在文件系统架构的最低层，设备驱动程序直接与外围设备或其控制器或通道通信。",[15,296377,296378,8471,296380],{},[2627,296379,118575],{},[2627,296381,291961],{},[15,296383,296384,5598],{},[2627,296385,118582],{},[105,296387,296388,296393],{},[108,296389,296390,296392],{},[2627,296391,156287],{},": 文件系统架构。",[108,296394,296395,296397],{},[2627,296396,293402],{},": 文件系统通常分为多个抽象层。最底层是设备驱动程序（Device Drivers），它们不关心文件的逻辑结构，而是直接向 I\u002FO 硬件（控制器、通道）发送特定的机器指令，实现物理数据的读写。",[15,296399,296400,296402],{},[2627,296401,115317],{},": A pile file refers to the least complicated form of file organization, where data are collected in sorted order and each record consists of one burst of data.",[15,296404,296405,296407],{},[2627,296406,293379],{},": 堆文件（pile file）指的是最不复杂的文件组织形式，其中数据按排序顺序收集，每条记录由一阵数据组成。",[15,296409,296410,8471,296412],{},[2627,296411,118575],{},[2627,296413,292009],{},[15,296415,296416,5598],{},[2627,296417,118582],{},[105,296419,296420,296425],{},[108,296421,296422,296424],{},[2627,296423,156287],{},": 堆文件（Pile File）组织。",[108,296426,296427,296429,296430,296433],{},[2627,296428,293402],{},": 堆文件确实是最不复杂的形式，但它的核心特征是",[2627,296431,296432],{},"数据完全没有特定的结构或排序（not sorted）","。数据只是按照到达的先后顺序被连续追加（堆积）到文件中。题目中称其“按排序顺序收集（collected in sorted order）”是错误的。",[15,296435,296436,296438],{},[2627,296437,115339],{},": The key field in a sequential file acts as an identifier for each record and must therefore be unique to that record.",[15,296440,296441,296443],{},[2627,296442,293379],{},": 顺序文件中的关键字段充当每条记录的标识符，因此对于该记录必须是唯一的。",[15,296445,296446,8471,296448],{},[2627,296447,118575],{},[2627,296449,291961],{},[15,296451,296452,5598],{},[2627,296453,118582],{},[105,296455,296456,296461],{},[108,296457,296458,296460],{},[2627,296459,156287],{},": 顺序文件（Sequential File）。",[108,296462,296463,296465],{},[2627,296464,293402],{},": 在顺序文件中，记录通常按照某个特定的字段（称为关键字段\u002FKey field）的值进行排序和存储。为了准确标识和查找某条特定记录，这个关键字段通常要求必须具有唯一性。",[15,296467,296468,296470],{},[2627,296469,115391],{},": In the indexed sequential file structure, multiple indexes (each referencing a defined portion of the file) can be used to increase record access efficiency.",[15,296472,296473,296475],{},[2627,296474,293379],{},": 在索引顺序文件结构中，可以使用多个索引（每个索引引用定义的文件的某部分）来提高记录访问效率。",[15,296477,296478,8471,296480],{},[2627,296479,118575],{},[2627,296481,291961],{},[15,296483,296484,5598],{},[2627,296485,118582],{},[105,296487,296488,296493],{},[108,296489,296490,296492],{},[2627,296491,156287],{},": 索引顺序文件（Indexed Sequential File）。",[108,296494,296495,296497],{},[2627,296496,293402],{},": 为了避免对大型顺序文件进行缓慢的线性搜索，索引顺序文件引入了多级索引（如主索引引用部分索引，部分索引引用具体的记录块），这种类似树状结构的索引大大缩短了搜索时间，极大地提高了访问效率。",[15,296499,296500,296502],{},[2627,296501,115413],{},": In the general indexed file structure, there are no key fields and variable-length records are allowed.",[15,296504,296505,296507],{},[2627,296506,293379],{},": 在通用索引文件（general indexed file）结构中，没有关键字段，并且允许可变长度记录。",[15,296509,296510,8471,296512],{},[2627,296511,118575],{},[2627,296513,292009],{},[15,296515,296516,5598],{},[2627,296517,118582],{},[105,296519,296520,296525],{},[108,296521,296522,296524],{},[2627,296523,156287],{},": 索引文件（Indexed File）。",[108,296526,296527,296529,296530,296533,296534,296537],{},[2627,296528,293402],{},": 通用索引文件放弃了顺序文件的单一物理排序，通常用于只通过索引访问记录的场景。它允许可变长度记录，但它",[2627,296531,296532],{},"绝对有关键字段（key fields）","——事实上，它通常具有",[2627,296535,296536],{},"多个","关键字段，并且会为这些不同的关键字段分别建立不同的索引表。因此说它“没有关键字段”是错的。",[15,296539,296540,296542],{},[2627,296541,115435],{},": The address information element of a file directory maintains data about the owner of the file, the file access information, and a list of actions permitted on the file.",[15,296544,296545,296547],{},[2627,296546,293379],{},": 文件目录的地址信息（address information）元素维护关于文件所有者、文件访问信息以及文件允许执行的操作列表的数据。",[15,296549,296550,8471,296552],{},[2627,296551,118575],{},[2627,296553,292009],{},[15,296555,296556,5598],{},[2627,296557,118582],{},[105,296559,296560,296565],{},[108,296561,296562,296564],{},[2627,296563,156287],{},": 文件目录的信息元素。",[108,296566,296567,296569,296570,296573],{},[2627,296568,293402],{},": 文件目录中包含所有者、访问信息、允许的操作等内容的部分被称为",[2627,296571,296572],{},"访问控制信息（Access Control Information）","。而**地址信息（Address Information）**维护的是文件存放的卷、起始物理地址、文件大小和块分配指针等。",[15,296575,296576,296578],{},[2627,296577,115457],{},": Typically, an interactive user or a process has associated with it a current directory, often referred to as the working directory.",[15,296580,296581,296583],{},[2627,296582,293379],{},": 通常，交互式用户或进程具有与之关联的当前目录，通常称为工作目录（working directory）。",[15,296585,296586,8471,296588],{},[2627,296587,118575],{},[2627,296589,291961],{},[15,296591,296592,5598],{},[2627,296593,118582],{},[105,296595,296596,296601],{},[108,296597,296598,296600],{},[2627,296599,156287],{},": 目录结构。",[108,296602,296603,296605],{},[2627,296604,293402],{},": 为了方便用户和进程通过相对路径访问文件而无需每次都输入长长的绝对路径，操作系统为每个活动进程\u002F用户维护一个当前关联的目录，即工作目录（Working Directory）。",[15,296607,296608,296610],{},[2627,296609,115479],{},": The primary issues involved in the simultaneous access of files for updating or appending are mutual exclusion and starvation.",[15,296612,296613,296615],{},[2627,296614,293379],{},": 涉及同时访问文件进行更新或追加的主要问题是互斥（mutual exclusion）和饥饿（starvation）。",[15,296617,296618,8471,296620],{},[2627,296619,118575],{},[2627,296621,292009],{},[15,296623,296624,5598],{},[2627,296625,118582],{},[105,296627,296628,296633],{},[108,296629,296630,296632],{},[2627,296631,156287],{},": 文件的并发访问控制。",[108,296634,296635,296637,296638,347],{},[2627,296636,293402],{},": 尽管饥饿是一个普遍的并发问题，但在教科书讨论文件共享更新的上下文中，文件系统需要解决的两个首要问题是",[2627,296639,296640,296641,197421],{},"互斥（Mutual Exclusion）",[2627,296642,119123],{},[15,296644,296645,296647],{},[2627,296646,115501],{},": The larger the file block size the more records are passed per I\u002FO transaction, but this requires larger I\u002FO buffers making buffer management more difficult.",[15,296649,296650,296652],{},[2627,296651,293379],{},": 文件块大小越大，每次 I\u002FO 事务传递的记录就越多，但这需要更大的 I\u002FO 缓冲区，使得缓冲区管理更加困难。",[15,296654,296655,8471,296657],{},[2627,296656,118575],{},[2627,296658,291961],{},[15,296660,296661,5598],{},[2627,296662,118582],{},[105,296664,296665,296670],{},[108,296666,296667,296669],{},[2627,296668,156287],{},": 文件阻塞（File Blocking）。",[108,296671,296672,296674],{},[2627,296673,293402],{},": 使用大块（Large Block Size）可以提高磁盘 I\u002FO 的吞吐量，因为一次物理传输能带出多条记录。但代价是操作系统必须在内存中分配同样巨大的 I\u002FO 缓冲区，这增加了系统内存的压力和缓冲池管理的复杂性。",[15,296676,296677,296679],{},[2627,296678,115523],{},": A preallocation policy for secondary storage management requires that the minimum size of a file be declared at the time of the file creation request.",[15,296681,296682,296684],{},[2627,296683,293379],{},": 辅助存储管理的预分配（preallocation）策略要求在文件创建请求时声明文件的最小大小。",[15,296686,296687,8471,296689],{},[2627,296688,118575],{},[2627,296690,292009],{},[15,296692,296693,5598],{},[2627,296694,118582],{},[105,296696,296697,296702],{},[108,296698,296699,296701],{},[2627,296700,156287],{},": 空间分配策略。",[108,296703,296704,296706,296707,296710],{},[2627,296705,293402],{},": 预分配策略（Preallocation）要求用户或应用程序在创建文件时声明该文件将来可能达到的",[2627,296708,296709],{},"最大大小（Maximum size）","，以便操作系统能一次性为该文件分配足够的空间，而不是最小大小。",[15,296712,296713,296715],{},[2627,296714,115545],{},": In the Free Block List strategy of free disk space management, each block is assigned a number sequentially and the list of the numbers of all free blocks is maintained in a reserved portion of the disk.",[15,296717,296718,296720],{},[2627,296719,293379],{},": 在空闲磁盘空间管理的空闲块列表（Free Block List）策略中，每个块被顺序分配一个编号，并且所有空闲块编号的列表被维护在磁盘的保留部分中。",[15,296722,296723,8471,296725],{},[2627,296724,118575],{},[2627,296726,291961],{},[15,296728,296729,5598],{},[2627,296730,118582],{},[105,296732,296733,296738],{},[108,296734,296735,296737],{},[2627,296736,156287],{},": 磁盘空闲空间管理。",[108,296739,296740,296742],{},[2627,296741,293402],{},": 在 Free Block List 策略中，磁盘上每一个空闲的物理块编号（Block number）都被收集起来存放在一张列表中，这张表保存在磁盘的一个固定管理区域中，操作系统按需从中摘取可用块。",[15,296744,296745,296747],{},[2627,296746,115567],{},": Reliability problems can occur when the disk allocation table and file allocation table are stored on disk.",[15,296749,296750,296752],{},[2627,296751,293379],{},": 当磁盘分配表（DAT）和文件分配表（FAT）存储在磁盘上时，可能会出现可靠性问题。",[15,296754,296755,8471,296757],{},[2627,296756,118575],{},[2627,296758,292009],{},[15,296760,296761,5598],{},[2627,296762,118582],{},[105,296764,296765,296770],{},[108,296766,296767,296769],{},[2627,296768,156287],{},": 文件系统的可靠性。",[108,296771,296772,296774,296775,296777],{},[2627,296773,293402],{},": 事实上，将 DAT 和 FAT 存放在",[2627,296776,233420],{},"上是最安全的。可靠性问题恰恰发生在操作系统为了提高性能，将这两张表**缓存在主存（Main Memory）**中进行修改时。如果在这个过程中系统崩溃或突然断电，内存中的数据还没来得及写回磁盘，就会导致磁盘结构彻底损坏（严重的不一致）。",[15,296779,296780,296782],{},[2627,296781,115589],{},": The UNIX kernel views all files as streams of bytes, any internal logical structure is application specific.",[15,296784,296785,296787],{},[2627,296786,293379],{},": UNIX 内核将所有文件视为字节流，任何内部逻辑结构都是特定于应用程序的。",[15,296789,296790,8471,296792],{},[2627,296791,118575],{},[2627,296793,291961],{},[15,296795,296796,5598],{},[2627,296797,118582],{},[105,296799,296800,296805],{},[108,296801,296802,296804],{},[2627,296803,156287],{},": UNIX 文件系统特性。",[108,296806,296807,296809],{},[2627,296808,293402],{},": UNIX 不像早期大型机系统那样在操作系统层面区分“顺序文件”、“索引文件”等结构。对于 UNIX 内核而言，所有的普通文件仅仅是一个没有格式的、连续的“字节流（streams of bytes）”。如何解释这些字节（如当作文本、数据库记录或视频），完全由打开该文件的应用程序自行决定。",[15,296811,296812,296814],{},[2627,296813,115611],{},": The first few sectors of any W2K disk partition using the NTFS file system is occupied by the master file table (MFT).",[15,296816,296817,296819],{},[2627,296818,293379],{},": 任何使用 NTFS 文件系统的 W2K 磁盘分区的前几个扇区被主文件表（MFT）占据。",[15,296821,296822,8471,296824],{},[2627,296823,118575],{},[2627,296825,292009],{},[15,296827,296828,5598],{},[2627,296829,118582],{},[105,296831,296832,296837],{},[108,296833,296834,296836],{},[2627,296835,156287],{},": Windows NTFS 卷结构。",[108,296838,296839,296841,296842,296845],{},[2627,296840,293402],{},": 在 NTFS 卷（分区）的开头，占据前几个扇区的是",[2627,296843,296844],{},"分区引导扇区（Partition Boot Sector）","，它包含启动系统的代码以及有关该卷的基本信息。主文件表（MFT）并不固定在开头，而是可以在分区中的任何位置，其具体位置指针记录在引导扇区中。",[32188,296847],{},[19,296849,293916],{"id":293915},[15,296851,296852,296854],{},[2627,296853,115273],{},": A file is generally defined to be:\na. A basic element of data\nb. A collection of related fields\nc. A collection of similar records\nd. All of the above",[15,296856,296857,296859],{},[2627,296858,293379],{},": 文件通常被定义为：\na. 数据的基本元素\nb. 相关字段的集合\nc. 相似记录的集合\nd. 以上所有",[15,296861,296862,8471,296864],{},[2627,296863,118575],{},[2627,296865,296866],{},"c. A collection of similar records",[15,296868,296869,5598],{},[2627,296870,118582],{},[105,296872,296873,296878],{},[108,296874,296875,296877],{},[2627,296876,156287],{},": 文件系统的数据结构层次。",[108,296879,296880,296882,296883,347],{},[2627,296881,293402],{},": 在经典文件系统中，数据的基本元素是字符\u002F字节；相关字符组成字段（Field）；相关字段的集合构成记录（Record）；而",[2627,296884,296885],{},"文件（File）是相似记录（records）的集合",[15,296887,296888,296890],{},[2627,296889,115295],{},": The level of the file system architecture that enables users and applications to access file records is called the:\na. Basic file system level\nb. Basic I\u002FO supervisor level\nc. Logical I\u002FO level\nd. All of the above",[15,296892,296893,296895],{},[2627,296894,293379],{},": 使得用户和应用程序能够访问文件记录的文件系统架构级别被称为：\na. 基本文件系统级别\nb. 基本 I\u002FO 监督程序级别\nc. 逻辑 I\u002FO 级别\nd. 以上所有",[15,296897,296898,8471,296900],{},[2627,296899,118575],{},[2627,296901,296902],{},"c. Logical I\u002FO level",[15,296904,296905,5598],{},[2627,296906,118582],{},[105,296908,296909,296914],{},[108,296910,296911,296913],{},[2627,296912,156287],{},": 文件系统软件架构。",[108,296915,296916,296918],{},[2627,296917,293402],{},": 逻辑 I\u002FO（Logical I\u002FO）层为用户和应用程序提供了一个高级接口。它负责处理记录的存取请求，并将这些逻辑上的记录访问转换为底层可以理解的具体的块（block）和物理层操作。",[15,296920,296921,296923],{},[2627,296922,115317],{},": Record access in a pile file can be conducted by:\na. Exhaustive search\nb. Key field\nc. Partial index\nd. All of the above",[15,296925,296926,296928],{},[2627,296927,293379],{},": 在堆文件（pile file）中的记录访问可以通过以下方式进行：\na. 穷举搜索 (Exhaustive search)\nb. 关键字段\nc. 部分索引\nd. 以上所有",[15,296930,296931,8471,296933],{},[2627,296932,118575],{},[2627,296934,296935],{},"a. Exhaustive search",[15,296937,296938,5598],{},[2627,296939,118582],{},[105,296941,296942,296947],{},[108,296943,296944,296946],{},[2627,296945,156287],{},": 堆文件查询方式。",[108,296948,296949,296951,296952,347],{},[2627,296950,293402],{},": 堆文件中的数据记录是无序累加的，并且记录的格式和长度都不固定。因为没有索引也没有按关键字段排序，所以要查找某条特定记录，唯一的办法就是从头到尾对文件进行",[2627,296953,296954],{},"穷举搜索（Exhaustive search）",[15,296956,296957,296959],{},[2627,296958,115339],{},": Sequential files are optimal in scenarios involving:\na. Applications that require frequent queries\nb. Applications that require the processing of all records in the file\nc. Applications that require infrequent updates\nd. All of the above",[15,296961,296962,296964],{},[2627,296963,293379],{},": 顺序文件在涉及以下场景时是最佳的：\na. 需要频繁查询的应用程序\nb. 需要处理文件中所有记录的应用程序\nc. 需要不频繁更新的应用程序\nd. 以上所有",[15,296966,296967,8471,296969],{},[2627,296968,118575],{},[2627,296970,296971],{},"b. Applications that require the processing of all records in the file",[15,296973,296974,5598],{},[2627,296975,118582],{},[105,296977,296978,296983],{},[108,296979,296980,296982],{},[2627,296981,156287],{},": 顺序文件（Sequential file）的最佳用例。",[108,296984,296985,296987,296988,296991],{},[2627,296986,293402],{},": 如果你的应用（例如月末的工资单批处理程序）需要依次处理文件中的",[2627,296989,296990],{},"每一条记录（all records）","，那么顺序文件是性能最好、开销最低的设计。对于单条记录的随机\u002F频繁查询，顺序文件效率极低。",[15,296993,296994,296996],{},[2627,296995,115391],{},": Indexed sequential files similar to sequential files, but contain two added features:\na. Hash function and an overflow file\nb. Hash function and file index\nc. File index and overflow file\nd. All of the above",[15,296998,296999,297001],{},[2627,297000,293379],{},": 索引顺序文件类似于顺序文件，但包含两个增加的特征：\na. 哈希函数和溢出文件\nb. 哈希函数和文件索引\nc. 文件索引和溢出文件 (File index and overflow file)\nd. 以上所有",[15,297003,297004,8471,297006],{},[2627,297005,118575],{},[2627,297007,297008],{},"c. File index and overflow file",[15,297010,297011,5598],{},[2627,297012,118582],{},[105,297014,297015,297020],{},[108,297016,297017,297019],{},[2627,297018,156287],{},": 索引顺序文件结构。",[108,297021,297022,297024,297025,297032],{},[2627,297023,293402],{},": 索引顺序文件在顺序文件的基础上，保留了主文件的排序，同时增加了两个关键组件：一个是",[2627,297026,297027,297028,297031],{},"文件索引（File index）",[2627,297029,297030],{},"以支持快速的随机查找；另一个是","溢出文件（Overflow file）","，用于存放因主文件块装满而无法插入的新记录，以后再择机与主文件合并。",[15,297034,297035,297037],{},[2627,297036,115413],{},": Direct or hashed files are often used where:\na. Very rapid access is required\nb. Fixed length records are used\nc. Records are always accessed one at a time\nd. All of the above",[15,297039,297040,297042],{},[2627,297041,293379],{},": 直接或哈希文件（hashed files）常用于以下情况：\na. 需要非常快速的访问\nb. 使用固定长度的记录\nc. 总是每次访问一条记录\nd. 以上所有",[15,297044,297045,8471,297047],{},[2627,297046,118575],{},[2627,297048,292510],{},[15,297050,297051,5598],{},[2627,297052,118582],{},[105,297054,297055,297060],{},[108,297056,297057,297059],{},[2627,297058,156287],{},": 哈希\u002F直接文件（Direct\u002FHashed file）。",[108,297061,297062,297064],{},[2627,297063,293402],{},": 直接\u002F哈希文件通过计算主键的哈希值直接定位数据块。这种方式能够提供极快的单条记录访问速度（a, c），并且这种映射通常要求记录是定长的（b），常用于目录、定价表或名单的快速查询。",[15,297066,297067,297069],{},[2627,297068,115435],{},": The file directory information element that holds information such as the identity of the creator of the file is the:\na. Address information element\nb. Access control information element\nc. Usage information element\nd. All of the above",[15,297071,297072,297074],{},[2627,297073,293379],{},": 包含诸如文件创建者身份等信息的文件目录信息元素是：\na. 地址信息元素\nb. 访问控制信息元素 (Access control information element)\nc. 使用信息元素\nd. 以上所有",[15,297076,297077,8471,297079],{},[2627,297078,118575],{},[2627,297080,297081],{},"b. Access control information element",[15,297083,297084,5598],{},[2627,297085,118582],{},[105,297087,297088,297093],{},[108,297089,297090,297092],{},[2627,297091,156287],{},": 目录条目结构。",[108,297094,297095,297097],{},[2627,297096,293402],{},": 文件的所有者（Owner）或创建者的身份信息，被归类在**访问控制信息（Access control information）**中，操作系统通过它来判定谁有权给该文件分配或更改权限。",[15,297099,297100,297102],{},[2627,297101,115457],{},": In a tree-structured directory, the series of directory names that culminates in a file name is referred to as the:\na. Pathname\nb. Working directory\nc. Symbolic name\nd. None of the above",[15,297104,297105,297107],{},[2627,297106,293379],{},": 在树形结构目录中，以文件名结尾的一系列目录名称被称为：\na. 路径名 (Pathname)\nb. 工作目录\nc. 符号名\nd. 以上都不是",[15,297109,297110,8471,297112],{},[2627,297111,118575],{},[2627,297113,297114],{},"a. Pathname",[15,297116,297117,5598],{},[2627,297118,118582],{},[105,297120,297121,297126],{},[108,297122,297123,297125],{},[2627,297124,156287],{},": 路径名的定义。",[108,297127,297128,297130,297131,297134,297135,347],{},[2627,297129,293402],{},": 从根目录或当前工作目录开始，穿过一层层目录（如 ",[31,297132,297133],{},"\u002Fusr\u002Flocal\u002Fbin\u002Fpython","）最终到达特定文件名的整个字符串链条，就叫做",[2627,297136,297137],{},"路径名（Pathname）",[15,297139,297140,297142],{},[2627,297141,115479],{},": Access rights on a file typically are considered to constitute a hierarchy, with each right implying those that:\na. Supercede it\nb. Precede it\nc. Succeed it\nd. None of the above",[15,297144,297145,297147],{},[2627,297146,293379],{},": 文件的访问权限通常被认为构成一个层次结构，其中每种权利都暗示着那些：\na. 取代它的权限\nb. 在它之前的权限 (Precede it)\nc. 在它之后的权限\nd. 以上都不是",[15,297149,297150,8471,297152],{},[2627,297151,118575],{},[2627,297153,297154],{},"b. Precede it",[15,297156,297157,5598],{},[2627,297158,118582],{},[105,297160,297161,297166],{},[108,297162,297163,297165],{},[2627,297164,156287],{},": 访问权限层级。",[108,297167,297168,297170],{},[2627,297169,293402],{},": 访问权限构成了一个逐步增强的层级，比如：无访问权 -> 读权限 -> 写权限 -> 执行\u002F更新权限。如果你被授予了“写”权限，它自然暗示着你也拥有“在它之前的（precede it）”底层权限——即“读”权限。高权限涵盖低权限。",[15,297172,297173,297175],{},[2627,297174,115501],{},": Fixed file blocking experiences the following potential problem:\na. Gaps due to hardware design\nb. External fragmentation\nc. Internal fragmentation\nd. None of the above",[15,297177,297178,297180],{},[2627,297179,293379],{},": 固定文件分块（Fixed file blocking）可能会遇到以下潜在问题：\na. 由于硬件设计造成的间隙\nb. 外部碎片\nc. 内部碎片 (Internal fragmentation)\nd. 以上都不是",[15,297182,297183,8471,297185],{},[2627,297184,118575],{},[2627,297186,297187],{},"c. Internal fragmentation",[15,297189,297190,5598],{},[2627,297191,118582],{},[105,297193,297194,297199],{},[108,297195,297196,297198],{},[2627,297197,156287],{},": 文件块分配与内部碎片。",[108,297200,297201,297203,297204,347],{},[2627,297202,293402],{},": 在固定阻塞（Fixed blocking）中，若干条记录被打包进一个固定大小的块（Block）中。如果最后几条记录无法填满整个块的剩余空间，该块中剩下的空间就会被白白浪费，这被称为",[2627,297205,297206],{},"内部碎片（Internal fragmentation）",[15,297208,297209,297211],{},[2627,297210,115523],{},": In which of the following file allocation methods is preallocation required:\na. Contiguous\nb. Chained\nc. Indexed\nd. None of the above",[15,297213,297214,297216],{},[2627,297215,293379],{},": 在以下哪种文件分配方法中，需要进行预分配（preallocation）：\na. 连续分配 (Contiguous)\nb. 链接分配 (Chained)\nc. 索引分配 (Indexed)\nd. 以上都不是",[15,297218,297219,8471,297221],{},[2627,297220,118575],{},[2627,297222,297223],{},"a. Contiguous",[15,297225,297226,5598],{},[2627,297227,118582],{},[105,297229,297230,297235],{},[108,297231,297232,297234],{},[2627,297233,156287],{},": 连续分配（Contiguous Allocation）。",[108,297236,297237,297239],{},[2627,297238,293402],{},": 连续分配要求文件在磁盘上占据一段完全连续的块集。为了实现这一点，操作系统在创建文件时必须提前知道（或被告知）这个文件的最大预期大小，以便在磁盘上寻找并**预分配（Preallocation）**一块足够大的连续空闲空间。链接分配和索引分配都是动态分配的。",[15,297241,297242,297244],{},[2627,297243,115545],{},": The technique of free disk space management that employs a pointer and length value of each free portion is the:\na. Free block list\nb. Bit tables\nc. Indexing\nd. None of the above",[15,297246,297247,297249],{},[2627,297248,293379],{},": 采用针对每个空闲部分使用指针和长度值（length value）的空闲磁盘空间管理技术是：\na. 空闲块列表\nb. 位图\u002F位表\nc. 索引\nd. 以上都不是",[15,297251,297252,8471,297254],{},[2627,297253,118575],{},[2627,297255,292848],{},[15,297257,297258,5598],{},[2627,297259,118582],{},[105,297261,297262,297267],{},[108,297263,297264,297266],{},[2627,297265,156287],{},": 磁盘空闲空间管理策略。",[108,297268,297269,297271],{},[2627,297270,293402],{},": 教科书中，使用一个指针指向一段连续空闲块的起始位置，并附带一个长度值（length value，表示这段空闲空间包含多少个块）的方法被称为**链式空闲区（Chained Free Portions）**或者基于区段（Portions）的管理。由于选项中给出的 a, b, c 都不是这个名字（a 是纯粹记录单个块号的列表，不是记录长度的），因此答案是 d。",[15,297273,297274,297276],{},[2627,297275,115567],{},": The data structure that maintains information on available disk space is called the:\na. File Allocation Table (FAT)\nb. Disk Allocation Table\nc. Bit Table\nd. None of the above",[15,297278,297279,297281],{},[2627,297280,293379],{},": 维护可用磁盘空间信息的数据结构被称为：\na. 文件分配表 (FAT)\nb. 磁盘分配表 (Disk Allocation Table)\nc. 位表\nd. 以上都不是",[15,297283,297284,8471,297286],{},[2627,297285,118575],{},[2627,297287,297288],{},"b. Disk Allocation Table",[15,297290,297291,5598],{},[2627,297292,118582],{},[105,297294,297295,297300],{},[108,297296,297297,297299],{},[2627,297298,156287],{},": 磁盘空间跟踪。",[108,297301,297302,297304,297305,297308],{},[2627,297303,293402],{},": FAT (File Allocation Table) 用于追踪已分配给特定文件的那些块；而用于追踪全盘所有空闲（可用）块的总表，在学术上统一被称为",[2627,297306,297307],{},"磁盘分配表（Disk Allocation Table, DAT）","，尽管其底层实现可能是位图或空闲块链。",[15,297310,297311,297313],{},[2627,297312,115589],{},": File allocation in a UNIX system has the following characteristics:\na. Dynamic allocation using non-contiguous blocks with indexing\nb. Dynamic allocation using contiguous blocks without indexing\nc. Preallocation using non-contiguous blocks without indexing\nd. None of the above",[15,297315,297316,297318],{},[2627,297317,293379],{},": UNIX 系统中的文件分配具有以下特征：\na. 动态分配，使用带索引的非连续块\nb. 动态分配，使用不带索引的连续块\nc. 预分配，使用不带索引的非连续块\nd. 以上都不是",[15,297320,297321,8471,297323],{},[2627,297322,118575],{},[2627,297324,297325],{},"a. Dynamic allocation using non-contiguous blocks with indexing",[15,297327,297328,5598],{},[2627,297329,118582],{},[105,297331,297332,297337],{},[108,297333,297334,297336],{},[2627,297335,156287],{},": UNIX 文件分配（inode 机制）。",[108,297338,297339,297341,297342,297344,297345,347],{},[2627,297340,293402],{},": UNIX 文件系统使用著名的 ",[31,297343,239686],{},"（索引节点）结构进行空间分配。它完全是",[2627,297346,297347,297348,297355],{},"动态分配（Dynamic allocation）",[2627,297349,297350,297351,297354],{},"的，数据块可以散布在磁盘各处（",[2627,297352,297353],{},"非连续非连续块\u002Fnon-contiguous blocks","），并通过 inode 中的多级指针树（直接、一次间接、二次间接等）进行","索引（indexing）",[15,297357,297358,297360],{},[2627,297359,115611],{},": In a W2K NTFS file system, the smallest physical storage unit on the disk (almost always 512 bytes) is called a:\na. Cluster\nb. Sector\nc. Volume\nd. None of the above",[15,297362,297363,297365],{},[2627,297364,293379],{},": 在 W2K NTFS 文件系统中，磁盘上最小的物理存储单元（几乎总是 512 字节）被称为：\na. 簇\nb. 扇区 (Sector)\nc. 卷\nd. 以上都不是",[15,297367,297368,8471,297370],{},[2627,297369,118575],{},[2627,297371,297372],{},"b. Sector",[15,297374,297375,5598],{},[2627,297376,118582],{},[105,297378,297379,297384],{},[108,297380,297381,297383],{},[2627,297382,156287],{},": 硬盘的物理与逻辑单位。",[108,297385,297386,297388,297389,297392,297393,297395],{},[2627,297387,293402],{},": 硬盘上硬件级别的、最小的不可分割的物理存储单位是",[2627,297390,297391],{},"扇区（Sector）","，传统大小就是 512 字节。而 NTFS 等文件系统管理空间时使用的最小",[2627,297394,247579],{},"单位是簇（Cluster），它通常是由一个或多个连续的扇区组成的。",[32188,297397],{},[19,297399,294449],{"id":294448},[15,297401,297402,297404],{},[2627,297403,115273],{},": The essential aspects of a __________ are that relationships among data are explicit and that it is designed for use by a number of different applications.",[15,297406,297407,297409],{},[2627,297408,293379],{},": __________ 的本质特征在于，数据之间的关系是显式的，并且它是为了供许多不同的应用程序使用而设计的。",[15,297411,297412,8471,297414],{},[2627,297413,118575],{},[2627,297415,297416],{},"database (或 database system)",[15,297418,297419,297421],{},[2627,297420,118582],{},": 区分文件系统和数据库系统的标志在于：数据库（Database）不仅存储数据本身，还结构化地存储数据之间的复杂关系，并被设计为独立于单个应用程序之外，为整个企业的一系列应用提供统一服务。",[15,297423,297424,297426],{},[2627,297425,115295],{},": The level of the file system architecture that provides a standard interface between applications and the file systems and devices is often termed the __________.",[15,297428,297429,297431],{},[2627,297430,293379],{},": 在应用程序与文件系统及设备之间提供标准接口的文件系统架构层级通常被称为 __________。",[15,297433,297434,8471,297436],{},[2627,297435,118575],{},[2627,297437,297438],{},"logical I\u002FO",[15,297440,297441,297443],{},[2627,297442,118582],{},": 逻辑 I\u002FO（Logical I\u002FO）层抽象出了通用接口，让应用程序只需要关心“记录”，而不需要了解文件是如何存储或设备是如何运作的。",[15,297445,297446,297448],{},[2627,297447,115317],{},": A basic requirement for a file management system states that each user should have __________ access to other user's files.",[15,297450,297451,297453],{},[2627,297452,293379],{},": 文件管理系统的一个基本要求指出，每个用户应具有对其他用户文件的 __________ 访问。",[15,297455,297456,8471,297458],{},[2627,297457,118575],{},[2627,297459,297460],{},"controlled",[15,297462,297463,297465],{},[2627,297464,118582],{},": 操作系统不应该完全禁止共享，也不应该允许随意访问。基本原则是在明确的授权和权限控制下，提供**受控的（controlled）**访问。",[15,297467,297468,297470],{},[2627,297469,115339],{},": The __________ file organization is the only one that is easily stored on tape as well as disk.",[15,297472,297473,297475],{},[2627,297474,293379],{},": __________ 文件组织是唯一一种可以轻松存储在磁带以及磁盘上的文件组织。",[15,297477,297478,8471,297480],{},[2627,297479,118575],{},[2627,297481,297482],{},"sequential",[15,297484,297485,297487],{},[2627,297486,118582],{},": 磁带是一种纯粹的顺序存取物理介质，它不支持磁头的来回随机跳跃寻址。因此，只有顺序追加、顺序读取的**顺序文件（Sequential file）**架构能很好地在磁带上运作。",[15,297489,297490,297492],{},[2627,297491,115391],{},": In the indexed sequential file structure, multiple level of indexing can be used to increase efficiency in __________.",[15,297494,297495,297497],{},[2627,297496,293379],{},": 在索引顺序文件结构中，可以使用多级索引来提高 __________ 的效率。",[15,297499,297500,8471,297502],{},[2627,297501,118575],{},[2627,297503,297504],{},"record access (或 search)",[15,297506,297507,297509],{},[2627,297508,118582],{},": 多级索引极大地缩小了搜索的范围，其主要目的就是为了提高记录访问（Record access）或搜索的效率。",[15,297511,297512,297514],{},[2627,297513,115413],{},": The direct file makes use of __________ on the key value.",[15,297516,297517,297519],{},[2627,297518,293379],{},": 直接文件（direct file）利用基于键值的 __________。",[15,297521,297522,8471,297524],{},[2627,297523,118575],{},[2627,297525,297526],{},"hashing (或 a hash function)",[15,297528,297529,297531,297532,297535],{},[2627,297530,118582],{},": 直接文件没有索引表，它利用",[2627,297533,297534],{},"哈希算法（Hashing）","，将查询键的值通过哈希函数计算，直接映射出该记录在磁盘上的物理块地址。",[15,297537,297538,297540],{},[2627,297539,115435],{},": The file directory information element that holds information such as the permitted actions on the file (e.g., reading, writing, executing, etc.) is the __________ information element.",[15,297542,297543,297545],{},[2627,297544,293379],{},": 包含关于文件允许的操作（例如，读取、写入、执行等）等信息的文件目录信息元素是 __________ 信息元素。",[15,297547,297548,8471,297550],{},[2627,297549,118575],{},[2627,297551,297552],{},"access control",[15,297554,297555,297557],{},[2627,297556,118582],{},": “允许读、写、执行”这就是典型的权限系统，这些数据被保存在目录的访问控制（Access control）信息元素中。",[15,297559,297560,297562],{},[2627,297561,115457],{},": Typically, an interactive user or a process has associated with it a current directory, often referred to as the __________.",[15,297564,297565,297567],{},[2627,297566,293379],{},": 通常，交互式用户或进程具有与之关联的当前目录，通常称为 __________。",[15,297569,297570,8471,297572],{},[2627,297571,118575],{},[2627,297573,297574],{},"working directory",[15,297576,297577,297579],{},[2627,297578,118582],{},": 当前所在的路径状态称为工作目录（Working directory）。",[15,297581,297582,297584],{},[2627,297583,115479],{},": Issues of __________ and deadlock must be addressed in designing the shared access capability for a file.",[15,297586,297587,297589],{},[2627,297588,293379],{},": 在设计文件的共享访问能力时，必须解决 __________ 和死锁的问题。",[15,297591,297592,8471,297594],{},[2627,297593,118575],{},[2627,297595,297596],{},"mutual exclusion",[15,297598,297599,297601,297602,297605],{},[2627,297600,118582],{},": 当多个进程想同时写入同一个文件时，系统必须提供互斥锁（如文件锁、记录锁）来保证修改的原子性，即",[2627,297603,297604],{},"互斥（mutual exclusion）","，随之而来的也是必须防范的死锁问题。",[15,297607,297608,297610],{},[2627,297609,115501],{},": __________ blocking is the common mode of file blocking for sequential files with fixed-length records.",[15,297612,297613,297615],{},[2627,297614,293379],{},": __________ 分块是具有固定长度记录的顺序文件的常见文件分块模式。",[15,297617,297618,8471,297620],{},[2627,297619,118575],{},[2627,297621,297622],{},"Fixed",[15,297624,297625,297627,297628,297631],{},[2627,297626,118582],{},": 既然记录长度是固定的，最自然、最高效的分块方式就是",[2627,297629,297630],{},"固定分块（Fixed blocking）","，即将固定数量的记录紧密打包成一个大小固定的数据块。",[15,297633,297634,297636],{},[2627,297635,115523],{},": The data structure or table that is used to keep track of the portions assigned to a file is referred to as a __________.",[15,297638,297639,297641],{},[2627,297640,293379],{},": 用于跟踪分配给文件的那些部分的数据结构或表被称为 __________。",[15,297643,297644,8471,297646],{},[2627,297645,118575],{},[2627,297647,297648],{},"File Allocation Table (或 FAT)",[15,297650,297651,297653],{},[2627,297652,118582],{},": 记录哪个磁盘块属于哪个文件的跟踪表，经典的名称就是文件分配表（File Allocation Table, FAT）。",[15,297655,297656,297658],{},[2627,297657,115545],{},": In order to keep track of the free portions of the disk, a __________ is needed in addition to the file allocation table.",[15,297660,297661,297663],{},[2627,297662,293379],{},": 为了跟踪磁盘的空闲部分，除了文件分配表之外，还需要一个 __________。",[15,297665,297666,8471,297668],{},[2627,297667,118575],{},[2627,297669,297670],{},"Disk Allocation Table (或 DAT)",[15,297672,297673,297675],{},[2627,297674,118582],{},": FAT 管的是已被文件占用的块；而追踪剩余可用空间的结构被称为磁盘分配表（Disk Allocation Table, DAT）或空闲空间管理表。",[15,297677,297678,297680],{},[2627,297679,115567],{},": Reliability problems can arise when copies of the File Allocation Table and Disk Allocation Table are maintained in __________.",[15,297682,297683,297685],{},[2627,297684,293379],{},": 当文件分配表和磁盘分配表的副本被维护在 __________ 中时，可能会出现可靠性问题。",[15,297687,297688,8471,297690],{},[2627,297689,118575],{},[2627,297691,297692],{},"main memory",[15,297694,297695,297697],{},[2627,297696,118582],{},": 如果这些关键的系统表长时间驻留在易失性的**主存（main memory）**中，一旦断电，内存数据丢失，磁盘上的结构就与实际状态脱节，从而引发致命的可靠性危机。",[15,297699,297700,297702],{},[2627,297701,115589],{},": UNIX employs __________, which is a control structure that contains the key information needed by the operating system for a particular file.",[15,297704,297705,297707],{},[2627,297706,293379],{},": UNIX 采用 __________，这是一种控制结构，包含操作系统针对特定文件所需的关键信息。",[15,297709,297710,8471,297712],{},[2627,297711,118575],{},[2627,297713,297714],{},"inodes (或 an inode \u002F index node)",[15,297716,297717,297719],{},[2627,297718,118582],{},": 在 UNIX\u002FLinux 系统中，与特定文件绑定的控制结构（存放除了文件名以外的所有元数据及物理数据块指针的结构）被称为索引节点（inode）。",[15,297721,297722,297724],{},[2627,297723,115611],{},": In a W2K NTFS file system, the __________ contains information about files and folders on a particular volume as well as information about available unallocated space.",[15,297726,297727,297729],{},[2627,297728,293379],{},": 在 W2K NTFS 文件系统中，__________ 包含有关特定卷上文件和文件夹的信息，以及有关可用未分配空间的信息。",[15,297731,297732,8471,297734],{},[2627,297733,118575],{},[2627,297735,297736],{},"Master File Table (或 MFT)",[15,297738,297739,297741,297742,297745,297746,297749],{},[2627,297740,118582],{},": NTFS 的核心就是",[2627,297743,297744],{},"主文件表（Master File Table, MFT）","。在 NTFS 的哲学中“一切皆文件”，甚至连追踪空闲空间的 bitmap（位图）也是以一个特殊文件（",[31,297747,297748],{},"$Bitmap","）的形式存储，而这个文件的元数据记录同样保存在 MFT 中。",{"title":29,"searchDepth":43,"depth":43,"links":297751},[297752,297753,297754],{"id":293368,"depth":43,"text":293369},{"id":293915,"depth":43,"text":293916},{"id":294448,"depth":43,"text":294449},{"order":1999},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap12",{"title":296322,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap12","9_dsZl0ojcxGpFyv0U8YDAmg0MXrQn4Q5lJXlFwTM3c",{"id":297761,"title":297762,"body":297763,"description":29,"extension":85,"meta":299053,"navigation":52,"path":299054,"seo":299055,"stem":299056,"__hash__":299057},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap2.md","Chap. 2",{"type":8,"value":297764,"toc":299048},[297765,297768,297770,297775,297794,297799,297821,297826,297852,297857,297875,297880,297902,297907,297925,297930,297959,297964,297985,297990,298012,298017,298042,298047,298068,298073,298095,298100,298126,298131,298152,298157,298183,298185,298187,298193,298211,298217,298240,298246,298269,298275,298294,298300,298323,298329,298352,298358,298384,298390,298419,298425,298454,298460,298485,298491,298513,298519,298541,298547,298569,298575,298598,298604,298627,298629,298631,298636,298662,298667,298690,298695,298717,298722,298745,298750,298772,298777,298799,298804,298824,298829,298851,298856,298883,298888,298910,298915,298937,298942,298965,298970,298993,298998,299021,299026],[11,297766,297767],{"id":240798},"Chapter 2 - Operating System Overview",[19,297769,291941],{"id":291940},[15,297771,297772],{},[2627,297773,297774],{},"1. An operating system controls the execution of applications and acts as an interface between applications and the computer hardware.",[105,297776,297777,297782,297788],{},[108,297778,297779,297781],{},[2627,297780,291953],{},"：操作系统控制应用程序的执行，并作为应用程序与计算机硬件之间的接口。",[108,297783,297784,2664,297786],{},[2627,297785,118575],{},[2627,297787,291961],{},[108,297789,297790,297793],{},[2627,297791,297792],{},"详细解析","：这是操作系统的最基本定义。操作系统的两个主要目标是：为应用程序和用户提供一个方便的抽象接口（屏蔽底层硬件的复杂性），以及作为资源管理器来控制硬件和软件资源的分配与执行。",[15,297795,297796],{},[2627,297797,297798],{},"2. The operating system maintains information that can be used for billing purposes on multi-user systems.",[105,297800,297801,297806,297812],{},[108,297802,297803,297805],{},[2627,297804,291953],{},"：操作系统维护的信息可用于多用户系统上的计费目的。",[108,297807,297808,2664,297810],{},[2627,297809,118575],{},[2627,297811,291961],{},[108,297813,297814,297816,297817,297820],{},[2627,297815,297792],{},"：操作系统的服务之一是",[2627,297818,297819],{},"记账 (Accounting)","。在一个共享的多用户或多任务系统中，操作系统会收集各类资源（如 CPU 时间、内存使用量、I\u002FO 操作次数）的使用统计信息。这些信息既可以用于性能监控和优化，也可以用于对用户的资源消耗进行计费。",[15,297822,297823],{},[2627,297824,297825],{},"3. The operating system typically runs in parallel with application programs, on it's own special O\u002FS processor.",[105,297827,297828,297833,297839],{},[108,297829,297830,297832],{},[2627,297831,291953],{},"：操作系统通常与应用程序并行运行在它自己专用的 OS 处理器上。",[108,297834,297835,2664,297837],{},[2627,297836,118575],{},[2627,297838,292009],{},[108,297840,297841,297843,297844,297847,297848,297851],{},[2627,297842,297792],{},"：在传统的计算机体系结构中，操作系统",[2627,297845,297846],{},"并不是","运行在一个完全隔离的、专用的处理器上。相反，操作系统和用户应用程序共享同一个（或同一组）CPU。操作系统通过",[2627,297849,297850],{},"中断 (Interrupts)"," 和上下文切换机制，交替地与应用程序分时使用 CPU。当操作系统放弃 CPU 时，它必须依赖硬件中断才能重新夺回对处理器的控制权。",[15,297853,297854],{},[2627,297855,297856],{},"4. One of the driving forces in operating system evolution is advancement in the underlying hardware technology.",[105,297858,297859,297864,297870],{},[108,297860,297861,297863],{},[2627,297862,291953],{},"：操作系统演进的驱动力之一是底层硬件技术的进步。",[108,297865,297866,2664,297868],{},[2627,297867,118575],{},[2627,297869,291961],{},[108,297871,297872,297874],{},[2627,297873,297792],{},"：操作系统的演进主要由三个因素驱动：底层硬件的升级和新硬件的出现（例如从单核到多核、新型存储设备）、新服务的需求，以及修复系统中的软件缺陷 (Bugs)。因此，硬件的进步绝对是核心驱动力。",[15,297876,297877],{},[2627,297878,297879],{},"5. In the first computers, users interacted directly with the hardware and operating systems did not exist.",[105,297881,297882,297887,297893],{},[108,297883,297884,297886],{},[2627,297885,291953],{},"：在早期的计算机中，用户直接与硬件交互，不存在操作系统。",[108,297888,297889,2664,297891],{},[2627,297890,118575],{},[2627,297892,291961],{},[108,297894,297895,297897,297898,297901],{},[2627,297896,297792],{},"：在第一代计算机（20世纪40年代末到50年代中期的真空管计算机）时期，还没有操作系统的概念。程序员直接在控制台（Console）上通过拨动开关和插入接线板来与纯硬件交互，这被称为",[2627,297899,297900],{},"串行处理 (Serial Processing)"," 时代。",[15,297903,297904],{},[2627,297905,297906],{},"6. In a batch-processing system, the phrase \"control is passed to a job\" means that the processor is now fetching and executing instructions in a user program.",[105,297908,297909,297914,297920],{},[108,297910,297911,297913],{},[2627,297912,291953],{},"：在批处理系统中，“控制权传递给作业”这句话意味着处理器现在正在获取并执行用户程序中的指令。",[108,297915,297916,2664,297918],{},[2627,297917,118575],{},[2627,297919,291961],{},[108,297921,297922,297924],{},[2627,297923,297792],{},"：在早期的简单批处理系统中，存在一个常驻监控程序 (Resident Monitor)。当监控程序完成了一个作业的加载和设置后，它会将程序计数器 (PC) 指向用户程序的起始地址。这就是所谓的“交出控制权”，此时 CPU 开始执行用户的代码，直到程序结束或出错，控制权才交还给监控程序。",[15,297926,297927],{},[2627,297928,297929],{},"7. Uniprogramming typically provides better utilization of system resources than multiprogramming.",[105,297931,297932,297937,297943],{},[108,297933,297934,297936],{},[2627,297935,291953],{},"：单道程序设计通常比多道程序设计提供更好的系统资源利用率。",[108,297938,297939,2664,297941],{},[2627,297940,118575],{},[2627,297942,292009],{},[108,297944,297945,297947,297948,297951,297952,297954,297955,297958],{},[2627,297946,297792],{},"：恰恰相反。",[2627,297949,297950],{},"单道程序设计 (Uniprogramming)"," 意味着内存中只有一个程序，当该程序进行 I\u002FO 操作时，CPU 只能闲置等待，资源利用率极低。",[2627,297953,292278],{}," 允许多个程序同时驻留内存，当一个程序等待 I\u002FO 时，CPU 可以切换去执行另一个程序，从而极大",[2627,297956,297957],{},"提高","了 CPU 和系统资源的利用率。",[15,297960,297961],{},[2627,297962,297963],{},"8. In a time sharing system, a user's program is preempted at regular intervals, but due to relatively slow human reaction time this occurrence is usually transparent to the user.",[105,297965,297966,297971,297977],{},[108,297967,297968,297970],{},[2627,297969,291953],{},"：在分时系统中，用户的程序会定期被抢占，但由于人类的反应时间相对较慢，这种情况对用户来说通常是透明的。",[108,297972,297973,2664,297975],{},[2627,297974,118575],{},[2627,297976,291961],{},[108,297978,297979,2664,297981,297984],{},[2627,297980,297792],{},[2627,297982,297983],{},"分时系统 (Time Sharing System)"," 通过时钟中断将 CPU 时间划分为极短的时间片 (Time slices)。每个用户的程序轮流执行一个时间片。由于时间片非常短（如几十毫秒），切换速度远快于人类的感知和反应速度，因此每个用户都感觉自己独占了整个计算机。",[15,297986,297987],{},[2627,297988,297989],{},"9. A process can be defined as a unit of activity characterized by a single sequential thread of execution, a current state, and an associated set of system resources.",[105,297991,297992,297997,298003],{},[108,297993,297994,297996],{},[2627,297995,291953],{},"：进程可以定义为一个活动单元，其特征是具有单一的顺序执行线程、当前状态以及一组关联的系统资源。",[108,297998,297999,2664,298001],{},[2627,298000,118575],{},[2627,298002,291961],{},[108,298004,298005,298007,298008,298011],{},[2627,298006,297792],{},"：这是经典操作系统中对",[2627,298009,298010],{},"进程 (Process)"," 的标准定义。一个进程包含了可执行代码、程序数据、执行上下文（即状态和寄存器内容）以及它所拥有的系统资源（如内存、文件句柄等）。",[15,298013,298014],{},[2627,298015,298016],{},"10. A virtual memory address typically consists of a page number and an offset within the page.",[105,298018,298019,298024,298030],{},[108,298020,298021,298023],{},[2627,298022,291953],{},"：虚拟内存地址通常由页号和页内偏移量组成。",[108,298025,298026,2664,298028],{},[2627,298027,118575],{},[2627,298029,291961],{},[108,298031,298032,298034,298035,298037,298038,298041],{},[2627,298033,297792],{},"：在分页 (Paging) 虚拟内存管理中，逻辑地址被划分为两部分：",[2627,298036,211325],{},"，用于在页表中查找对应的物理页框号；以及",[2627,298039,298040],{},"偏移量 (Offset)","，表示在这一页内部的相对地址。偏移量在虚拟地址和物理地址中是完全相同的。",[15,298043,298044],{},[2627,298045,298046],{},"11. Implementing priority levels is a common strategy for short-term scheduling, which involves assigning each process in the queue to the processor according to its level of importance.",[105,298048,298049,298054,298060],{},[108,298050,298051,298053],{},[2627,298052,291953],{},"：实现优先级是短期调度的常见策略，这涉及根据队列中每个进程的重要性级别将其分配给处理器。",[108,298055,298056,2664,298058],{},[2627,298057,118575],{},[2627,298059,291961],{},[108,298061,298062,2664,298064,298067],{},[2627,298063,297792],{},[2627,298065,298066],{},"短期调度器 (Short-term scheduler \u002F Dispatcher)"," 决定哪个处于“就绪”状态的进程将获得 CPU。为其分配优先级（Priority levels）是操作系统的核心调度策略之一，确保关键或高优先级的任务能优先执行。",[15,298069,298070],{},[2627,298071,298072],{},"12. Complex operating systems today typically consist of a few thousand lines of instructions.",[105,298074,298075,298080,298086],{},[108,298076,298077,298079],{},[2627,298078,291953],{},"：如今复杂的操作系统通常包含几千行指令。",[108,298081,298082,2664,298084],{},[2627,298083,118575],{},[2627,298085,292009],{},[108,298087,298088,298090,298091,298094],{},[2627,298089,297792],{},"：现代操作系统是极其庞大且复杂的软件工程。例如 Windows 或 Linux 内核的代码量都是在",[2627,298092,298093],{},"数千万行 (Millions of lines)"," 级别，而不是区区几千行。早期的“软件危机”正是源于这种庞大规模带来的维护困难。",[15,298096,298097],{},[2627,298098,298099],{},"13. A monolithic kernel architecture assigns only a few essential functions to the kernel, including address spaces, interprocess communication and basic scheduling.",[105,298101,298102,298107,298113],{},[108,298103,298104,298106],{},[2627,298105,291953],{},"：单体内核 (monolithic kernel) 架构只将少数基本功能分配给内核，包括地址空间、进程间通信和基本调度。",[108,298108,298109,2664,298111],{},[2627,298110,118575],{},[2627,298112,292009],{},[108,298114,298115,298117,298118,298121,298122,298125],{},[2627,298116,297792],{},"：题目描述的是",[2627,298119,298120],{},"微内核 (Microkernel)"," 架构的特点。",[2627,298123,298124],{},"单体内核 (Monolithic Kernel)"," 正好相反，它将几乎所有的操作系统服务（如文件系统、设备驱动、网络协议栈等）全部打包在一个巨大的内核地址空间中运行。",[15,298127,298128],{},[2627,298129,298130],{},"14. The hardware abstraction layer (HAL) maps between generic hardware commands\u002Fresponses and those unique to a specific platform.",[105,298132,298133,298138,298144],{},[108,298134,298135,298137],{},[2627,298136,291953],{},"：硬件抽象层 (HAL) 在通用硬件命令\u002F响应与特定平台独有的命令\u002F响应之间进行映射。",[108,298139,298140,2664,298142],{},[2627,298141,118575],{},[2627,298143,291961],{},[108,298145,298146,2664,298148,298151],{},[2627,298147,297792],{},[2627,298149,298150],{},"HAL (Hardware Abstraction Layer)"," 是一层位于操作系统内核底部的软件。它隔离了硬件的差异，为上层操作系统提供了一个统一的、通用的硬件接口，使得操作系统具有高度的可移植性 (Portability)。",[15,298153,298154],{},[2627,298155,298156],{},"15. Linux is one example of a modern UNIX system that implements a modular architecture.",[105,298158,298159,298164,298170],{},[108,298160,298161,298163],{},[2627,298162,291953],{},"：Linux 是实现了模块化架构的现代 UNIX 系统的例子之一。",[108,298165,298166,2664,298168],{},[2627,298167,118575],{},[2627,298169,291961],{},[108,298171,298172,298174,298175,298178,298179,298182],{},[2627,298173,297792],{},"：虽然 Linux 本质上是一个",[2627,298176,298177],{},"单体内核 (Monolithic kernel)","，但为了解决单体内核过于庞大、不够灵活的问题，Linux 引入了",[2627,298180,298181],{},"可加载内核模块 (Loadable Kernel Modules, LKMs)"," 机制。这赋予了 Linux 模块化 (Modular) 的架构特征，允许在运行时动态加载和卸载功能（如驱动程序）。",[32188,298184],{},[19,298186,292392],{"id":292391},[15,298188,298189,298192],{},[2627,298190,298191],{},"1. A primary objective of an operating system is:","\na. Convenience\nb. Efficiency\nc. Ability to evolve\nd. All of the above",[105,298194,298195,298200,298206],{},[108,298196,298197,298199],{},[2627,298198,291953],{},"：操作系统的主要目标是：\na. 方便性 (Convenience)\nb. 高效性 (Efficiency)\nc. 演进能力 (Ability to evolve)\nd. 以上皆是",[108,298201,298202,2664,298204],{},[2627,298203,118575],{},[2627,298205,292510],{},[108,298207,298208,298210],{},[2627,298209,297792],{},"：这三个正是操作系统设计的三大核心目标。方便性指使得计算机更容易被用户使用；高效性指更好地管理计算机系统资源；演进能力指操作系统结构应该允许在不影响服务的情况下，有效开发、测试和引入新的系统功能。",[15,298212,298213,298216],{},[2627,298214,298215],{},"2. The operating system provides many types of services to end-users, programmers and system designers, including:","\na. Built-in user applications\nb. Error detection and response\nc. Relational database capabilities with the internal file system\nd. All of the above",[105,298218,298219,298224,298231],{},[108,298220,298221,298223],{},[2627,298222,291953],{},"：操作系统向终端用户、程序员和系统设计者提供多种类型的服务，包括：\na. 内置用户应用程序\nb. 错误检测和响应\nc. 带有内部文件系统的关系数据库功能\nd. 以上皆是",[108,298225,298226,2664,298228],{},[2627,298227,118575],{},[2627,298229,298230],{},"b. Error detection and response",[108,298232,298233,298235,298236,298239],{},[2627,298234,297792],{},"：操作系统的核心服务包括程序开发、程序执行、访问 I\u002FO 设备、文件访问控制、系统访问、",[2627,298237,298238],{},"错误检测和响应 (Error detection and response)"," 以及记账。它通常不包含也不提供内置的用户级应用（如文字处理）或关系数据库服务（这些属于应用层或中间件层）。",[15,298241,298242,298245],{},[2627,298243,298244],{},"3. The operating system is unusual in it's role as a control mechanism, in that:","\na. It runs on a special processor, completely separated from the rest of the system\nb. It frequently relinquishes control of the system processor and must depend on the processor to regain control of the system\nc. It never relinquishes control of the system processor\nd. None of the above",[105,298247,298248,298253,298260],{},[108,298249,298250,298252],{},[2627,298251,291953],{},"：操作系统作为一种控制机制，其不寻常之处在于：",[108,298254,298255,2664,298257],{},[2627,298256,118575],{},[2627,298258,298259],{},"b. It frequently relinquishes control of the system processor and must depend on the processor to regain control of the system",[108,298261,298262,298264,298265,298268],{},[2627,298263,297792],{},"：操作系统本身也是一套程序代码。为了让用户程序能够运行，操作系统必须将 CPU 的",[2627,298266,298267],{},"控制权交出 (relinquish control)","。一旦交出，操作系统本身就不在运行了，它必须依靠底层硬件的定时器中断或系统调用等机制，才能强行打断用户程序，重新夺回对 CPU 的控制。",[15,298270,298271,298274],{},[2627,298272,298273],{},"4. Operating systems must evolve over time because:","\na. Hardware must be replaced when it fails\nb. Users will only purchase software that has a current copyright date\nc. New hardware is designed and implemented in the computer system\nd. All of the above",[105,298276,298277,298282,298289],{},[108,298278,298279,298281],{},[2627,298280,291953],{},"：操作系统必须随着时间演进，因为：",[108,298283,298284,2664,298286],{},[2627,298285,118575],{},[2627,298287,298288],{},"c. New hardware is designed and implemented in the computer system",[108,298290,298291,298293],{},[2627,298292,297792],{},"：操作系统的演进主要由于硬件的升级和新型硬件的出现（例如从机械硬盘到 SSD，从单核到多核架构）。选项 a 描述的是硬件故障替换，不需要 OS 演进；选项 b 是无稽之谈。因此 c 是正确的。",[15,298295,298296,298299],{},[2627,298297,298298],{},"5. A major problem with early serial processing systems was:","\na. Setup time\nb. Lack of input devices\nc. Inability to get hardcopy output\nd. All of the above",[105,298301,298302,298307,298314],{},[108,298303,298304,298306],{},[2627,298305,291953],{},"：早期串行处理系统的一个主要问题是：",[108,298308,298309,2664,298311],{},[2627,298310,118575],{},[2627,298312,298313],{},"a. Setup time",[108,298315,298316,298318,298319,298322],{},[2627,298317,297792],{},"：早期的计算机没有操作系统，程序员必须手动挂载磁带、设置开关。",[2627,298320,298321],{},"建立时间 (Setup time)"," （即准备运行一个程序所花费的物理人工时间）往往比程序实际执行的时间还要长，导致极大的浪费。这是推动早期批处理系统诞生的核心痛点。",[15,298324,298325,298328],{},[2627,298326,298327],{},"6. An example of a hardware feature that is desirable in a batch-processing system is:","\na. Privileged instructions\nb. A completely accessible memory area\nc. Large clock cycles\nd. None of the above",[105,298330,298331,298336,298343],{},[108,298332,298333,298335],{},[2627,298334,291953],{},"：在批处理系统中，期望拥有的硬件特性示例是：",[108,298337,298338,2664,298340],{},[2627,298339,118575],{},[2627,298341,298342],{},"a. Privileged instructions",[108,298344,298345,298347,298348,298351],{},[2627,298346,297792],{},"：为了让常驻监控程序 (Monitor) 能够控制系统并防止用户程序恶意干扰，硬件必须支持",[2627,298349,298350],{},"特权指令 (Privileged instructions)","。只有在监控程序运行（内核态）时才能执行这些指令（如控制 I\u002FO），用户程序试图执行时将引发异常。这保证了批处理监控程序的权威性。",[15,298353,298354,298357],{},[2627,298355,298356],{},"7. A computer hardware feature that is vital to the effective operation of a multiprogramming operating system is:","\na. Very large memory\nb. Multiple processors\nc. I\u002FO interrupts and DMA\nd. All of the above",[105,298359,298360,298365,298372],{},[108,298361,298362,298364],{},[2627,298363,291953],{},"：对于多道程序操作系统有效运行至关重要的计算机硬件特征是：",[108,298366,298367,2664,298369],{},[2627,298368,118575],{},[2627,298370,298371],{},"c. I\u002FO interrupts and DMA",[108,298373,298374,298376,298377,258,298380,298383],{},[2627,298375,297792],{},"：多道程序设计的核心思想是：当一个进程进行 I\u002FO 时，CPU 可以切换去执行另一个进程。要实现这一点，必须依赖 ",[2627,298378,298379],{},"I\u002FO 中断 (I\u002FO interrupts)",[2627,298381,298382],{},"DMA (直接内存访问)"," 技术。有了 DMA，CPU 可以将 I\u002FO 工作委托给控制器，自己去执行其他程序；当 I\u002FO 完成时，控制器通过中断通知 CPU。",[15,298385,298386,298389],{},[2627,298387,298388],{},"8. The principle objective of a time sharing, multiprogramming system is to:","\na. Maximize response time\nb. Maximize processor use\nc. Provide exclusive access to hardware\nd. None of the above",[105,298391,298392,298397,298403],{},[108,298393,298394,298396],{},[2627,298395,291953],{},"：分时、多道程序系统的主要目标是：",[108,298398,298399,2664,298401],{},[2627,298400,118575],{},[2627,298402,292848],{},[108,298404,298405,2664,298407,298410,298411,298414,298415,298418],{},[2627,298406,297792],{},[2627,298408,298409],{},"分时系统 (Time sharing system)"," 的主要目标是",[2627,298412,298413],{},"最小化响应时间 (Minimize response time)","，让每个交互式用户都感觉系统响应迅速。选项 a 荒谬地说要“最大化响应时间”。选项 b（最大化处理器使用率）是",[2627,298416,298417],{},"批处理 (Batch multiprogramming)"," 系统的主要目标。因此选项均不正确。",[15,298420,298421,298424],{},[2627,298422,298423],{},"9. Which of the following major line of computer system development created problems in timing and synchronization that contributed to the development of the concept of the process?","\na. Multiprogramming batch operation systems\nb. Time sharing systems\nc. Real time transaction systems\nd. All of the above",[105,298426,298427,298432,298438],{},[108,298428,298429,298431],{},[2627,298430,291953],{},"：以下哪条主要的计算机系统发展路线产生了时序和同步问题，从而促成了“进程”概念的发展？",[108,298433,298434,2664,298436],{},[2627,298435,118575],{},[2627,298437,292510],{},[108,298439,298440,2664,298442,146,298445,223291,298447,298450,298451,298453],{},[2627,298441,297792],{},[2627,298443,298444],{},"多道批处理系统",[2627,298446,244335],{},[2627,298448,298449],{},"实时事务系统","，由于在内存中并发执行多个程序，并且频繁交替使用 CPU，导致系统状态极度复杂。中断处理、并发资源的争抢导致了严重的同步、死锁和时序错乱问题。这直接促使计算机科学家们抽象出了",[2627,298452,298010],{}," 的概念，以便更系统地管理并发状态。",[15,298455,298456,298459],{},[2627,298457,298458],{},"10. The paging system in a memory management system provides for dynamic mapping between a virtual address used in a program and:","\na. A virtual address in main memory\nb. A real address in main memory\nc. A real address in a program\nd. None of the above",[105,298461,298462,298467,298474],{},[108,298463,298464,298466],{},[2627,298465,291953],{},"：内存管理系统中的分页机制提供了程序中使用的虚拟地址与什么之间的动态映射：",[108,298468,298469,2664,298471],{},[2627,298470,118575],{},[2627,298472,298473],{},"b. A real address in main memory",[108,298475,298476,2664,298478,298481,298482,347],{},[2627,298477,297792],{},[2627,298479,298480],{},"分页机制 (Paging system)"," 的核心功能就是通过硬件中的页表，将程序看到的虚拟地址（逻辑地址）在运行时动态地映射为",[2627,298483,298484],{},"主存中的实地址 \u002F 物理地址 (A real address in main memory)",[15,298486,298487,298490],{},[2627,298488,298489],{},"11. Relative to information protection and security in computer systems, access control typically refers to:","\na. Proving that security mechanisms perform according to specification\nb. The flow of data within the system\nc. Regulating user and process access to various aspects of the system\nd. None of the above",[105,298492,298493,298498,298505],{},[108,298494,298495,298497],{},[2627,298496,291953],{},"：相对于计算机系统中的信息保护和安全性，访问控制通常指：",[108,298499,298500,2664,298502],{},[2627,298501,118575],{},[2627,298503,298504],{},"c. Regulating user and process access to various aspects of the system",[108,298506,298507,2664,298509,298512],{},[2627,298508,297792],{},[2627,298510,298511],{},"访问控制 (Access Control)"," 指的是对用户、程序或进程在计算机系统中访问各类资源（如文件、内存段等）的权限进行监管和限制。选项 a 描述的是保证\u002F证明 (Assurance)，选项 b 描述的是信息流控制。",[15,298514,298515,298518],{},[2627,298516,298517],{},"12. A common problem with full-featured operating systems, due to their size and difficulty of the tasks they address, is:","\na. Chronically late in delivery\nb. Latent bugs that show up in the field\nc. Sub-par performance\nd. All of the above",[105,298520,298521,298526,298532],{},[108,298522,298523,298525],{},[2627,298524,291953],{},"：由于其规模庞大以及所处理任务的难度，全功能操作系统存在的一个共同问题是：",[108,298527,298528,2664,298530],{},[2627,298529,118575],{},[2627,298531,292510],{},[108,298533,298534,298536,298537,298540],{},[2627,298535,297792],{},"：历史上著名的",[2627,298538,298539],{},"软件危机 (Software Crisis)"," 表明，开发庞大的操作系统（如 IBM 的 OS\u002F360 时代）往往伴随着严重的管理问题：开发总是延期 (Chronically late)，上线后暴露出无数潜伏的 Bug (Latent bugs)，并且由于系统过于庞杂而导致性能低下 (Sub-par performance)。",[15,298542,298543,298546],{},[2627,298544,298545],{},"13. A technique in which a process, executing an application, is divided into threads that can run concurrently is called:","\na. Multithreading\nb. Multiprocessing\nc. Symmetric multiprocessing (SMP)\nd. None of the above",[105,298548,298549,298554,298561],{},[108,298550,298551,298553],{},[2627,298552,291953],{},"：将执行应用程序的进程划分为可以并发运行的多个线程的技术被称为：",[108,298555,298556,2664,298558],{},[2627,298557,118575],{},[2627,298559,298560],{},"a. Multithreading",[108,298562,298563,2664,298565,298568],{},[2627,298564,297792],{},[2627,298566,298567],{},"多线程 (Multithreading)"," 是一种技术，它将执行应用程序的一个大进程拆分成多个可以并发独立调度的执行线索（线程）。这极大提高了程序的并发性和响应能力。",[15,298570,298571,298574],{},[2627,298572,298573],{},"14. WIN2K supports several types of user applications, including:","\na. WIN32\nb. Linux\nc. System 10\nd. None of the above",[105,298576,298577,298582,298589],{},[108,298578,298579,298581],{},[2627,298580,291953],{},"：Windows 2000 (WIN2K) 支持几种类型的用户应用程序，包括：",[108,298583,298584,2664,298586],{},[2627,298585,118575],{},[2627,298587,298588],{},"a. WIN32",[108,298590,298591,298593,298594,298597],{},[2627,298592,297792],{},"：Windows 2000 基于 Windows NT 架构，它设计了环境子系统 (Environmental Subsystems) 以支持多种应用程序 API。其中最主要的是原生的 ",[2627,298595,298596],{},"Win32"," 子系统。它还包含 POSIX 和 OS\u002F2 子系统，但并不支持原生运行 Linux 应用。",[15,298599,298600,298603],{},[2627,298601,298602],{},"15. Key to the success of Linux has been it's character as a free software package available under the auspices of the:","\na. World Wide Web Consortium\nb. Free Software Foundation\nc. Berkeley Software Distribution\nd. None of the above",[105,298605,298606,298611,298618],{},[108,298607,298608,298610],{},[2627,298609,291953],{},"：Linux 成功的关键在于其作为免费软件包的特性，该软件包在以下哪个组织的主持下提供：",[108,298612,298613,2664,298615],{},[2627,298614,118575],{},[2627,298616,298617],{},"b. Free Software Foundation",[108,298619,298620,298622,298623,298626],{},[2627,298621,297792],{},"：Linux 能够取得巨大成功并形成完整的操作系统生态，极大程度上是因为它采用了",[2627,298624,298625],{},"自由软件基金会 (Free Software Foundation, FSF)"," 制定的 GNU 通用公共许可证 (GPL)。此外，Linux 内核与 FSF 的 GNU 软件工具（如 GCC 编译器、Shell 等）结合，才构成了完整的 GNU\u002FLinux 操作系统。",[32188,298628],{},[19,298630,292903],{"id":292902},[15,298632,298633],{},[2627,298634,298635],{},"1. The operating system's ________ refers to its inherent flexibility in permitting functional modifications to the system without interruption of services.",[105,298637,298638,298643,298653],{},[108,298639,298640,298642],{},[2627,298641,291953],{},"：操作系统的 ________ 指的是其固有的灵活性，允许在不中断服务的情况下对系统进行功能修改。",[108,298644,298645,2664,298647,4374,298650],{},[2627,298646,118575],{},[2627,298648,298649],{},"ability to evolve (演进能力)",[2627,298651,298652],{},"extensibility (可扩展性)",[108,298654,298655,298657,298658,298661],{},[2627,298656,297792],{},"：前面提到，操作系统的三大目标是方便、高效以及 ",[2627,298659,298660],{},"演进能力 (Ability to evolve)","。它意味着系统架构有足够的弹性，可以在不造成破坏的情况下集成新特性。",[15,298663,298664],{},[2627,298665,298666],{},"2. The operating system masks the details of the ________ from the application programmer.",[105,298668,298669,298674,298681],{},[108,298670,298671,298673],{},[2627,298672,291953],{},"：操作系统向应用程序员屏蔽了 ________ 的细节。",[108,298675,298676,2664,298678],{},[2627,298677,118575],{},[2627,298679,298680],{},"hardware (硬件)",[108,298682,298683,298685,298686,298689],{},[2627,298684,297792],{},"：应用程序员不需要知道硬盘的磁道和扇区如何转动，也不需要知道网卡如何发包，因为 OS 提供了一层抽象，完全屏蔽了底层",[2627,298687,298688],{},"硬件 (hardware)"," 的复杂性。",[15,298691,298692],{},[2627,298693,298694],{},"3. The ________ is the portion of the operating system that remains in main memory during system operation.",[105,298696,298697,298702,298709],{},[108,298698,298699,298701],{},[2627,298700,291953],{},"：________ 是操作系统在系统运行期间驻留在主存中的那部分。",[108,298703,298704,2664,298706],{},[2627,298705,118575],{},[2627,298707,298708],{},"kernel (内核) \u002F nucleus (核心)",[108,298710,298711,298713,298714,347],{},[2627,298712,297792],{},"：操作系统包含大量工具和实用程序，但真正控制系统并始终常驻内存（不可被换出到磁盘）的核心代码块被称为",[2627,298715,298716],{},"内核 (Kernel)",[15,298718,298719],{},[2627,298720,298721],{},"4. An operating system should be ________ in construction, allowing it greater flexibility in the evolutionary process.",[105,298723,298724,298729,298736],{},[108,298725,298726,298728],{},[2627,298727,291953],{},"：操作系统的构建应该是 ________ 的，从而使其在演进过程中具有更大的灵活性。",[108,298730,298731,2664,298733],{},[2627,298732,118575],{},[2627,298734,298735],{},"modular (模块化)",[108,298737,298738,298740,298741,298744],{},[2627,298739,297792],{},"：为了应对不断演进的需求，操作系统必须打破单块结构的僵化，采用 ",[2627,298742,298743],{},"模块化 (modular)"," 构建。这使得可以独立替换或升级某个特定的子系统（如文件系统、网络栈）。",[15,298746,298747],{},[2627,298748,298749],{},"5. The earliest computers employed ________ processing, a name derived by the way the users were forced to access the systems.",[105,298751,298752,298757,298764],{},[108,298753,298754,298756],{},[2627,298755,291953],{},"：最早的计算机采用 ________ 处理，该名称来源于用户被迫访问系统的方式。",[108,298758,298759,2664,298761],{},[2627,298760,118575],{},[2627,298762,298763],{},"serial (串行)",[108,298765,298766,298768,298769,347],{},[2627,298767,297792],{},"：第一代计算机中，用户必须提前预约机房时间，一个人接一个人地排队使用，程序的执行是一个接着一个的串行流，故称 ",[2627,298770,298771],{},"串行处理 (Serial processing)",[15,298773,298774],{},[2627,298775,298776],{},"6. The special type of programming language used to provide instructions to a monitor in a batch-processing scheme is called ________.",[105,298778,298779,298784,298791],{},[108,298780,298781,298783],{},[2627,298782,291953],{},"：在批处理方案中，用于向监控程序提供指令的特殊类型的编程语言称为 ________。",[108,298785,298786,2664,298788],{},[2627,298787,118575],{},[2627,298789,298790],{},"Job Control Language \u002F JCL (作业控制语言)",[108,298792,298793,298795,298796,347],{},[2627,298794,297792],{},"：在打孔卡片的批处理时代，为了告诉监控程序该加载什么编译器、执行哪个用户程序、输出到哪里，操作员会编写一组特定的控制卡片，这些卡片上使用的控制语法被称为 ",[2627,298797,298798],{},"作业控制语言 (JCL)",[15,298800,298801],{},[2627,298802,298803],{},"7. The central theme of modern operating systems, based on the concept of switching among multiple programs in memory, is called ________.",[105,298805,298806,298811,298817],{},[108,298807,298808,298810],{},[2627,298809,291953],{},"：现代操作系统的核心主题是基于在内存中的多个程序之间进行切换的概念，这被称为 ________。",[108,298812,298813,2664,298815],{},[2627,298814,118575],{},[2627,298816,293233],{},[108,298818,298819,2664,298821,298823],{},[2627,298820,297792],{},[2627,298822,292278],{}," 是现代 OS 的基石。它利用硬件的 I\u002FO 阻塞时间差，在内存的多个作业之间穿梭执行，是提升单处理机性能的最伟大飞跃。",[15,298825,298826],{},[2627,298827,298828],{},"8. In a time-sharing, multiprogramming system, users interact with the system through ________.",[105,298830,298831,298836,298843],{},[108,298832,298833,298835],{},[2627,298834,291953],{},"：在分时多道程序系统中，用户通过 ________ 与系统交互。",[108,298837,298838,2664,298840],{},[2627,298839,118575],{},[2627,298841,298842],{},"terminals (终端)",[108,298844,298845,298847,298848,347],{},[2627,298846,297792],{},"：分时系统的初衷就是让多个用户能够同时、交互式地使用主机。用户坐在各自的显示器\u002F键盘设备面前与主机通信，这些设备被称为 ",[2627,298849,298850],{},"终端 (terminals)",[15,298852,298853],{},[2627,298854,298855],{},"9. A process consists of three elements: an executable program, associated data, and a(n) ________ which includes all information needed by the operating system and processor to manage and execute the process.",[105,298857,298858,298863,298870],{},[108,298859,298860,298862],{},[2627,298861,291953],{},"：一个进程由三个元素组成：可执行程序、相关数据，以及一个 ________，其中包含操作系统和处理器管理和执行该进程所需的所有信息。",[108,298864,298865,2664,298867],{},[2627,298866,118575],{},[2627,298868,298869],{},"execution context (执行上下文) \u002F process control block (进程控制块)",[108,298871,298872,298874,298875,298878,298879,298882],{},[2627,298873,297792],{},"：根据经典教材的定义，构成进程的核心三要素是程序代码、数据，以及",[2627,298876,298877],{},"执行上下文 (Execution Context)","。这个上下文在操作系统内部由数据结构 ",[2627,298880,298881],{},"PCB (进程控制块)"," 来表示，包含了寄存器状态、优先级、内存映射等重要元数据。",[15,298884,298885],{},[2627,298886,298887],{},"10. ________ is a facility that allows programs to address memory from a logical point of view, without regard to the physical amount of main memory.",[105,298889,298890,298895,298902],{},[108,298891,298892,298894],{},[2627,298893,291953],{},"：________ 是一种机制，它允许程序从逻辑角度寻址内存，而不必考虑主存的实际物理容量。",[108,298896,298897,2664,298899],{},[2627,298898,118575],{},[2627,298900,298901],{},"Virtual memory (虚拟内存)",[108,298903,298904,2664,298906,298909],{},[2627,298905,297792],{},[2627,298907,298908],{},"虚拟内存 (Virtual memory)"," 通过磁盘充当后备存储，将物理内存扩展成一个庞大且统一的逻辑空间。程序员只需面对这块逻辑空间，无需担心内存不够用或物理地址冲突的问题。",[15,298911,298912],{},[2627,298913,298914],{},"11. The ________ queue in the operating system scheduling system consists of processes that are in main memory.",[105,298916,298917,298922,298929],{},[108,298918,298919,298921],{},[2627,298920,291953],{},"：在操作系统调度系统中，________ 队列包含位于主存中的进程。",[108,298923,298924,2664,298926],{},[2627,298925,118575],{},[2627,298927,298928],{},"ready (就绪)",[108,298930,298931,298933,298934,347],{},[2627,298932,297792],{},"：进程调度模型中，所有已被加载进主存、万事俱备只欠 CPU 资源的进程，会被操作系统组织在一个等待队列中，这个队列被称为 ",[2627,298935,298936],{},"就绪队列 (Ready queue)",[15,298938,298939],{},[2627,298940,298941],{},"12. The interface to an operating system is often referred to as a ________ because it separates the user from O\u002FS details and presents the O\u002FS simply as a collection of services.",[105,298943,298944,298949,298956],{},[108,298945,298946,298948],{},[2627,298947,291953],{},"：操作系统的接口通常被称为 ________，因为它将用户与 OS 细节分离开来，并仅将 OS 呈现为一系列服务。",[108,298950,298951,2664,298953],{},[2627,298952,118575],{},[2627,298954,298955],{},"shell (外壳)",[108,298957,298958,298960,298961,298964],{},[2627,298959,297792],{},"：正如坚果的外壳包裹着核心一样，操作系统的外围接口程序（无论是命令行接口 CLI 还是图形用户接口 GUI）被称为 ",[2627,298962,298963],{},"Shell","。它负责接收用户命令，解释后传递给内部的 Kernel (内核)。",[15,298966,298967],{},[2627,298968,298969],{},"13. A ________ operating system provides the illusion of a single main memory space and a single secondary memory space, plus other unified access facilities.",[105,298971,298972,298977,298984],{},[108,298973,298974,298976],{},[2627,298975,291953],{},"：________ 操作系统提供了单一主存空间和单一辅助存储空间的假象，以及其他统一的访问设施。",[108,298978,298979,2664,298981],{},[2627,298980,118575],{},[2627,298982,298983],{},"distributed (分布式)",[108,298985,298986,298988,298989,298992],{},[2627,298987,297792],{},"：在多台物理独立的计算机通过网络相连时，",[2627,298990,298991],{},"分布式操作系统 (Distributed OS)"," 可以将它们聚合成一个虚拟的单机系统。用户感觉就像在使用一台拥有巨大内存和硬盘的巨型计算机，这是分布式系统最高级的透明性表现。",[15,298994,298995],{},[2627,298996,298997],{},"14. The executive, protected subsystems and applications in a WIN2K system are structured using the ________ computing model, which is a common model for distributed computing.",[105,298999,299000,299005,299012],{},[108,299001,299002,299004],{},[2627,299003,291953],{},"：WIN2K 系统中的执行体、受保护的子系统和应用程序是使用 ________ 计算模型构建的，这是一种分布式计算的常见模型。",[108,299006,299007,2664,299009],{},[2627,299008,118575],{},[2627,299010,299011],{},"client\u002Fserver (客户端\u002F服务器 或 C\u002FS)",[108,299013,299014,299016,299017,299020],{},[2627,299015,297792],{},"：Windows NT \u002F 2000 的微内核风格架构深度应用了 ",[2627,299018,299019],{},"客户端\u002F服务器 (Client\u002FServer)"," 模型。用户应用程序（客户端）通过发送消息或 RPC，向环境子系统（服务器）请求服务（如文件读写、创建进程），即使它们运行在同一台机器上。",[15,299022,299023],{},[2627,299024,299025],{},"15. Most UNIX systems are ________ in that they include virtually all of the O\u002FS functionality in a single large block of code that runs in a single process with a single address space.",[105,299027,299028,299033,299040],{},[108,299029,299030,299032],{},[2627,299031,291953],{},"：大多数 UNIX 系统都是 ________ 的，因为它们几乎将所有的操作系统功能包含在一个庞大的代码块中，该代码块运行在具有单一地址空间的单个进程中。",[108,299034,299035,2664,299037],{},[2627,299036,118575],{},[2627,299038,299039],{},"monolithic (单体 \u002F 宏内核)",[108,299041,299042,299044,299045,299047],{},[2627,299043,297792],{},"：传统的 UNIX 操作系统（如 System V、BSD、Linux 的基础核心）采用的都是 ",[2627,299046,298124],{}," 架构。所有的驱动、文件系统、调度、IPC 等都在一个特权级下的连续地址空间中执行。它的优点是内部调用效率极高，缺点是体积庞大、其中一个模块崩溃可能导致整个内核崩溃。",{"title":29,"searchDepth":43,"depth":43,"links":299049},[299050,299051,299052],{"id":291940,"depth":43,"text":291941},{"id":292391,"depth":43,"text":292392},{"id":292902,"depth":43,"text":292903},{"order":43},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap2",{"title":297762,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap2","KqemutSYq2vpYh3GWJlBK7ApbmwW7-1D3XUbWURbSHg",{"id":299059,"title":299060,"body":299061,"description":29,"extension":85,"meta":300481,"navigation":52,"path":300482,"seo":300483,"stem":300484,"__hash__":300485},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap3.md","Chap. 3",{"type":8,"value":299062,"toc":300476},[299063,299066,299068,299073,299078,299084,299088,299100,299105,299110,299116,299120,299132,299137,299142,299148,299152,299164,299169,299174,299180,299184,299196,299201,299206,299212,299216,299228,299233,299238,299244,299248,299260,299265,299270,299276,299280,299292,299297,299302,299308,299312,299324,299329,299334,299340,299344,299356,299361,299366,299372,299376,299391,299396,299401,299407,299411,299423,299428,299433,299439,299443,299459,299464,299469,299475,299479,299491,299496,299501,299507,299511,299523,299528,299533,299539,299543,299555,299557,299559,299564,299569,299576,299580,299596,299601,299606,299613,299617,299629,299634,299639,299646,299650,299662,299667,299672,299678,299682,299694,299699,299704,299711,299715,299752,299757,299762,299768,299772,299803,299808,299813,299820,299824,299836,299841,299846,299852,299856,299868,299873,299878,299885,299889,299901,299906,299919,299925,299929,299941,299946,299951,299958,299962,299974,299979,299984,299990,299994,300006,300011,300016,300022,300026,300038,300043,300048,300054,300058,300080,300085,300090,300097,300101,300113,300115,300117,300122,300127,300134,300139,300144,300149,300156,300161,300166,300171,300178,300183,300188,300193,300200,300209,300214,300219,300226,300231,300236,300241,300248,300253,300258,300263,300270,300275,300280,300285,300292,300297,300302,300307,300314,300319,300324,300329,300335,300340,300345,300350,300357,300362,300367,300372,300379,300384,300389,300394,300401,300406,300411,300416,300423,300428,300436,300440,300446,300451,300457,300462,300468],[11,299064,299065],{"id":241335},"Chapter 3 - Process Description and Control",[19,299067,293369],{"id":293368},[15,299069,299070,299072],{},[2627,299071,115273],{},": The principal function of the processor is to execute machine instructions residing in main memory.",[15,299074,299075,299077],{},[2627,299076,293379],{},": 处理器的主要功能是执行驻留在主存中的机器指令。",[15,299079,299080,8471,299082],{},[2627,299081,118575],{},[2627,299083,291961],{},[15,299085,299086,5598],{},[2627,299087,118582],{},[105,299089,299090,299095],{},[108,299091,299092,299094],{},[2627,299093,156287],{},": 处理器的基本功能。",[108,299096,299097,299099],{},[2627,299098,293402],{},": 计算机系统的核心目标是执行程序，而程序是由一系列机器指令构成的。处理器（CPU）的基本工作周期就是从主存（Main Memory）中不断地获取（Fetch）指令并执行（Execute）它们。",[15,299101,299102,299104],{},[2627,299103,115295],{},": A process trace is a listing of the sequence of instructions that execute for that process.",[15,299106,299107,299109],{},[2627,299108,293379],{},": 进程轨迹（Process trace）是该进程执行的指令序列的列表。",[15,299111,299112,8471,299114],{},[2627,299113,118575],{},[2627,299115,291961],{},[15,299117,299118,5598],{},[2627,299119,118582],{},[105,299121,299122,299127],{},[108,299123,299124,299126],{},[2627,299125,156287],{},": 进程轨迹（Process Trace）的定义。",[108,299128,299129,299131],{},[2627,299130,293402],{},": 在操作系统中，为了描述和理解单个进程的行为，我们通常使用“轨迹（Trace）”这一概念。它精确地记录了该进程在处理器上运行时的机器指令序列。",[15,299133,299134,299136],{},[2627,299135,115317],{},": The principal responsibility of the operating system is to control the execution of processes.",[15,299138,299139,299141],{},[2627,299140,293379],{},": 操作系统的主要职责是控制进程的执行。",[15,299143,299144,8471,299146],{},[2627,299145,118575],{},[2627,299147,291961],{},[15,299149,299150,5598],{},[2627,299151,118582],{},[105,299153,299154,299159],{},[108,299155,299156,299158],{},[2627,299157,156287],{},": 操作系统的职责。",[108,299160,299161,299163],{},[2627,299162,293402],{},": 操作系统作为资源管理器，其最核心的任务之一就是管理和调度多个进程的执行，确保它们公平、高效且互不干扰地使用处理器和其他系统资源。",[15,299165,299166,299168],{},[2627,299167,115339],{},": When one process spawns another, the spawning process is referred to as the child process and the spawned process is referred to as the parent process.",[15,299170,299171,299173],{},[2627,299172,293379],{},": 当一个进程派生（spawn）另一个进程时，派生进程被称为子进程，被派生的进程被称为父进程。",[15,299175,299176,8471,299178],{},[2627,299177,118575],{},[2627,299179,292009],{},[15,299181,299182,5598],{},[2627,299183,118582],{},[105,299185,299186,299191],{},[108,299187,299188,299190],{},[2627,299189,156287],{},": 进程的创建（Process Spawning）。",[108,299192,299193,299195],{},[2627,299194,293402],{},": 概念颠倒了。发起创建动作（spawning）的原进程是“父进程（Parent process）”，而被创建出来的新进程是“子进程（Child process）”。",[15,299197,299198,299200],{},[2627,299199,115391],{},": Round-Robin processing refers to a method of thread prioritization for scheduling.",[15,299202,299203,299205],{},[2627,299204,293379],{},": 轮转（Round-Robin）处理指的是一种用于调度的线程优先级排序方法。",[15,299207,299208,8471,299210],{},[2627,299209,118575],{},[2627,299211,292009],{},[15,299213,299214,5598],{},[2627,299215,118582],{},[105,299217,299218,299223],{},[108,299219,299220,299222],{},[2627,299221,156287],{},": 轮转调度算法（Round-Robin Scheduling）。",[108,299224,299225,299227],{},[2627,299226,293402],{},": 轮转（Round-Robin）是一种基于时间片（Time Slicing）的公平调度算法，它不依赖于静态或动态的“优先级（prioritization）”。在这种机制下，就绪队列中的每个进程（或线程）被轮流分配一个固定大小的时间片来执行，时间片用完即被剥夺执行权。",[15,299229,299230,299232],{},[2627,299231,115413],{},": The primary difference between the Two-State Process Model and the Five-State Process Model is that the latter splits the Running state into two new states: Ready and Blocked.",[15,299234,299235,299237],{},[2627,299236,293379],{},": 两态进程模型和五态进程模型之间的主要区别在于，后者将“运行（Running）”状态拆分为两个新状态：就绪（Ready）和阻塞（Blocked）。",[15,299239,299240,8471,299242],{},[2627,299241,118575],{},[2627,299243,292009],{},[15,299245,299246,5598],{},[2627,299247,118582],{},[105,299249,299250,299255],{},[108,299251,299252,299254],{},[2627,299253,156287],{},": 进程状态模型。",[108,299256,299257,299259],{},[2627,299258,293402],{},": 描述错误。两态模型包含“运行（Running）”和“非运行（Not Running）”。五态模型实际上是将“非运行（Not Running）”状态进行了细分，拆分成了“就绪（Ready）”和“阻塞\u002F等待（Blocked\u002FWaiting）”，而不是拆分“运行”状态。",[15,299261,299262,299264],{},[2627,299263,115435],{},": One solution to the problem of limited main memory space is swapping, which involves moving all or part of a process from main memory to secondary memory.",[15,299266,299267,299269],{},[2627,299268,293379],{},": 解决主存空间有限问题的一种方法是交换（swapping），它涉及将进程的全部或部分从主存移动到辅助存储器。",[15,299271,299272,8471,299274],{},[2627,299273,118575],{},[2627,299275,291961],{},[15,299277,299278,5598],{},[2627,299279,118582],{},[105,299281,299282,299287],{},[108,299283,299284,299286],{},[2627,299285,156287],{},": 内存交换（Swapping）与挂起状态（Suspend）。",[108,299288,299289,299291],{},[2627,299290,293402],{},": 当主存中所有的进程都处于阻塞状态，且没有足够的空闲内存来加载新的就绪进程时，操作系统会将一部分被阻塞的进程“交换（Swap）”到磁盘（辅助存储器）上，以腾出物理内存，这一过程是解决内存短缺的经典技术。",[15,299293,299294,299296],{},[2627,299295,115457],{},": In order to define the control structures (e.g., tables) that the O\u002FS needs to manage processes and resources, it must have access to configuration data during initialization.",[15,299298,299299,299301],{},[2627,299300,293379],{},": 为了定义操作系统管理进程和资源所需的控制结构（例如，表），它必须在初始化期间能够访问配置数据。",[15,299303,299304,8471,299306],{},[2627,299305,118575],{},[2627,299307,291961],{},[15,299309,299310,5598],{},[2627,299311,118582],{},[105,299313,299314,299319],{},[108,299315,299316,299318],{},[2627,299317,156287],{},": 操作系统控制结构的建立。",[108,299320,299321,299323],{},[2627,299322,293402],{},": 操作系统在启动和初始化（Initialization）阶段，需要读取系统的硬件配置信息（如内存容量、设备类型和数量等），才能准确地在内存中分配和建立起内存表、I\u002FO表、文件表和进程表等控制结构。",[15,299325,299326,299328],{},[2627,299327,115479],{},": The Process Image refers to the binary form of the program code.",[15,299330,299331,299333],{},[2627,299332,293379],{},": 进程映像（Process Image）指的是程序代码的二进制形式。",[15,299335,299336,8471,299338],{},[2627,299337,118575],{},[2627,299339,292009],{},[15,299341,299342,5598],{},[2627,299343,118582],{},[105,299345,299346,299351],{},[108,299347,299348,299350],{},[2627,299349,156287],{},": 进程映像的组成。",[108,299352,299353,299355],{},[2627,299354,293402],{},": 进程映像不仅仅包含程序代码的二进制形式（User Program）。根据教科书定义，它是一个包含了四个要素的完整实体：用户程序（User Program）、用户数据（User Data）、系统栈（System Stack）以及进程控制块（Process Control Block, PCB）。",[15,299357,299358,299360],{},[2627,299359,115501],{},": The portion of the Process Control Block that consists of the contents of the processor registers is called the Process Control Information.",[15,299362,299363,299365],{},[2627,299364,293379],{},": 进程控制块中包含处理器寄存器内容的部分称为进程控制信息（Process Control Information）。",[15,299367,299368,8471,299370],{},[2627,299369,118575],{},[2627,299371,292009],{},[15,299373,299374,5598],{},[2627,299375,118582],{},[105,299377,299378,299383],{},[108,299379,299380,299382],{},[2627,299381,156287],{},": 进程控制块（PCB）的结构。",[108,299384,299385,299387,299388,347],{},[2627,299386,293402],{},": 教科书中将 PCB 分为三大类：进程标识（Process Identification）、处理器状态信息（Processor State Information）和进程控制信息（Process Control Information）。其中，保存处理器寄存器（如程序计数器、PSW等）内容的区域被称为",[2627,299389,299390],{},"处理器状态信息（Processor State Information）",[15,299392,299393,299395],{},[2627,299394,115523],{},": The less-privileged processor execution mode is often referred to as kernel mode.",[15,299397,299398,299400],{},[2627,299399,293379],{},": 权限较低的处理器执行模式通常被称为内核模式（kernel mode）。",[15,299402,299403,8471,299405],{},[2627,299404,118575],{},[2627,299406,292009],{},[15,299408,299409,5598],{},[2627,299410,118582],{},[105,299412,299413,299418],{},[108,299414,299415,299417],{},[2627,299416,156287],{},": 处理器执行模式。",[108,299419,299420,299422],{},[2627,299421,293402],{},": 权限较低的模式是“用户模式（User mode）”。“内核模式（Kernel mode）”（或称系统模式、控制模式）是权限最高的模式，允许执行特权指令和访问受保护的内存区域。",[15,299424,299425,299427],{},[2627,299426,115545],{},": The primary process table contains one entry per process, unless the process spawns a new process, in which case the table contains multiple entries for the parent process.",[15,299429,299430,299432],{},[2627,299431,293379],{},": 主进程表为每个进程包含一个条目，除非该进程派生了一个新进程，在这种情况下，表中将为父进程包含多个条目。",[15,299434,299435,8471,299437],{},[2627,299436,118575],{},[2627,299438,292009],{},[15,299440,299441,5598],{},[2627,299442,118582],{},[105,299444,299445,299450],{},[108,299446,299447,299449],{},[2627,299448,156287],{},": 进程表（Process Table）。",[108,299451,299452,299454,299455,299458],{},[2627,299453,293402],{},": 无论进程是否派生了新进程，",[2627,299456,299457],{},"每个进程在进程表中只占据一个条目（Entry）","。当父进程派生子进程时，操作系统会为子进程在进程表中分配一个全新的、独立的条目，而不是给父进程增加条目。",[15,299460,299461,299463],{},[2627,299462,115567],{},": One kind of system interrupt, the trap, relates to an error or exception condition in the currently running process.",[15,299465,299466,299468],{},[2627,299467,293379],{},": 一种系统中断——陷阱（trap），与当前正在运行的进程中的错误或异常情况有关。",[15,299470,299471,8471,299473],{},[2627,299472,118575],{},[2627,299474,291961],{},[15,299476,299477,5598],{},[2627,299478,118582],{},[105,299480,299481,299486],{},[108,299482,299483,299485],{},[2627,299484,156287],{},": 陷阱（Trap）的定义。",[108,299487,299488,299490],{},[2627,299489,293402],{},": 中断分为异步中断（通常由外部 I\u002FO 硬件引发）和同步中断。同步中断通常被称为“陷阱（Trap）”，它是由正在执行的指令引发的异常情况，如除以零、非法内存访问等。",[15,299492,299493,299495],{},[2627,299494,115589],{},": In the Nonprocess Kernel approach to defining the relationship between the O\u002FS and the User Process, the O\u002FS code is executed as a separate entity that operates in privileged mode.",[15,299497,299498,299500],{},[2627,299499,293379],{},": 在定义操作系统与用户进程之间关系的“无进程内核（Nonprocess Kernel）”方法中，操作系统代码作为在特权模式下运行的独立实体来执行。",[15,299502,299503,8471,299505],{},[2627,299504,118575],{},[2627,299506,291961],{},[15,299508,299509,5598],{},[2627,299510,118582],{},[105,299512,299513,299518],{},[108,299514,299515,299517],{},[2627,299516,156287],{},": 操作系统的执行模型。",[108,299519,299520,299522],{},[2627,299521,293402],{},": 传统系统（如早期系统）采用 Nonprocess Kernel 模型。在这种模型下，操作系统的内核独立于任何用户进程之外运行。当发生中断或系统调用时，当前进程的状态被保存，控制权转移到内核，内核作为一个享有特权的独立代码实体进行处理。",[15,299524,299525,299527],{},[2627,299526,115611],{},": A typical UNIX system employs two Running states, to indicate whether the process is executing in user mode or kernel mode.",[15,299529,299530,299532],{},[2627,299531,293379],{},": 典型的 UNIX 系统使用两种运行状态，以指示进程是在用户模式还是内核模式下执行。",[15,299534,299535,8471,299537],{},[2627,299536,118575],{},[2627,299538,291961],{},[15,299540,299541,5598],{},[2627,299542,118582],{},[105,299544,299545,299550],{},[108,299546,299547,299549],{},[2627,299548,156287],{},": UNIX 进程状态模型。",[108,299551,299552,299554],{},[2627,299553,293402],{},": UNIX SVR4 进程状态模型在“Running”状态上进行了细化，分为“User Running（用户态运行）”和“Kernel Running（内核态运行）”。进程发生系统调用或被中断时，会从 User Running 切换到 Kernel Running。",[32188,299556],{},[19,299558,293916],{"id":293915},[15,299560,299561,299563],{},[2627,299562,115273],{},": The behavior of a processor can be characterized by examining:\na. A single process trace\nb. Multiple process traces\nc. The interleaving of the process traces\nd. All of the above",[15,299565,299566,299568],{},[2627,299567,293379],{},": 处理器的行为可以通过检查以下哪项来表征：\na. 单个进程的轨迹\nb. 多个进程的轨迹\nc. 进程轨迹的交织（interleaving）\nd. 以上所有",[15,299570,299571,8471,299573],{},[2627,299572,118575],{},[2627,299574,299575],{},"c. The interleaving of the process traces",[15,299577,299578,5598],{},[2627,299579,118582],{},[105,299581,299582,299587],{},[108,299583,299584,299586],{},[2627,299585,156287],{},": 处理器的行为。",[108,299588,299589,299591,299592,299595],{},[2627,299590,293402],{},": 单个进程的轨迹只能描述该进程本身的行为。处理器是在多个进程之间不断切换的，因此要描述",[2627,299593,299594],{},"处理器","在宏观上的整体行为，必须观察所有活跃进程的轨迹是如何在时间轴上交织（interleaving）执行的。",[15,299597,299598,299600],{},[2627,299599,115295],{},": The behavior of an individual process can be characterized by examining:\na. A single process trace\nb. Multiple process traces\nc. The interleaving of the process traces\nd. All of the above",[15,299602,299603,299605],{},[2627,299604,293379],{},": 单个进程的行为可以通过检查以下哪项来表征：\na. 单个进程的轨迹\nb. 多个进程的轨迹\nc. 进程轨迹的交织\nd. 以上所有",[15,299607,299608,8471,299610],{},[2627,299609,118575],{},[2627,299611,299612],{},"a. A single process trace",[15,299614,299615,5598],{},[2627,299616,118582],{},[105,299618,299619,299624],{},[108,299620,299621,299623],{},[2627,299622,156287],{},": 进程的行为。",[108,299625,299626,299628],{},[2627,299627,293402],{},": 进程的“轨迹（Trace）”是针对特定进程的指令执行序列，只需研究这一个特定的 Trace 就可以完全了解该进程个体的行为。",[15,299630,299631,299633],{},[2627,299632,115317],{},": The basic Two-State Process Model defines two possible states for a process in relationship to the processor:\na. Running and Executing\nb. Running and Not Running\nc. Executing and Waiting\nd. None of the above",[15,299635,299636,299638],{},[2627,299637,293379],{},": 基本的两态进程模型定义了进程相对于处理器的两种可能状态：\na. 运行（Running）和执行（Executing）\nb. 运行（Running）和非运行（Not Running）\nc. 执行（Executing）和等待（Waiting）\nd. 以上都不是",[15,299640,299641,8471,299643],{},[2627,299642,118575],{},[2627,299644,299645],{},"b. Running and Not Running",[15,299647,299648,5598],{},[2627,299649,118582],{},[105,299651,299652,299657],{},[108,299653,299654,299656],{},[2627,299655,156287],{},": 两态进程模型（Two-State Process Model）。",[108,299658,299659,299661],{},[2627,299660,293402],{},": 最简单的操作系统进程模型只区分进程当前是否占用 CPU，即“Running（正在运行）”和“Not Running（未在运行，位于队列中排队）”。",[15,299663,299664,299666],{},[2627,299665,115339],{},": There are a number of conditions that can lead to process termination, including:\na. Normal completion\nb. Bounds violation\nc. Parent termination\nd. All of the above",[15,299668,299669,299671],{},[2627,299670,293379],{},": 有许多条件可能导致进程终止，包括：\na. 正常完成\nb. 越界违规（内存越界）\nc. 父进程终止\nd. 以上所有",[15,299673,299674,8471,299676],{},[2627,299675,118575],{},[2627,299677,292510],{},[15,299679,299680,5598],{},[2627,299681,118582],{},[105,299683,299684,299689],{},[108,299685,299686,299688],{},[2627,299687,156287],{},": 进程终止的原因。",[108,299690,299691,299693],{},[2627,299692,293402],{},": 进程可能因为多种原因退出，包括：自身执行完毕（Normal completion）、出现致命错误如访问了不属于自己的内存（Bounds violation），或者因为它的父进程被终止，操作系统连带终止了所有子进程（Parent termination）。",[15,299695,299696,299698],{},[2627,299697,115391],{},": In the Five-State Process Model, the following represents a valid state transition:\na. Running -> Blocked\nb. New -> Running\nc. New -> Blocked\nd. All of the above",[15,299700,299701,299703],{},[2627,299702,293379],{},": 在五态进程模型中，以下哪项代表有效的状态转换：\na. 运行 -> 阻塞\nb. 新建 -> 运行\nc. 新建 -> 阻塞\nd. 以上所有",[15,299705,299706,8471,299708],{},[2627,299707,118575],{},[2627,299709,299710],{},"a. Running -> Blocked",[15,299712,299713,5598],{},[2627,299714,118582],{},[105,299716,299717,299722],{},[108,299718,299719,299721],{},[2627,299720,156287],{},": 五态模型中的状态转换。",[108,299723,299724,5598,299726],{},[2627,299725,293402],{},[105,299727,299728,299734,299746],{},[108,299729,299730,299733],{},[31,299731,299732],{},"Running -> Blocked"," 是有效的：正在运行的进程发起 I\u002FO 请求或等待某事件时会发生此转换。",[108,299735,299736,299739,299740,299742,299743,347],{},[31,299737,299738],{},"New -> Running"," 无效：新建进程必须先进入 ",[31,299741,194904],{}," 队列，由调度器选中后才能 ",[31,299744,299745],{},"Running",[108,299747,299748,299751],{},[31,299749,299750],{},"New -> Blocked"," 无效：新建进程还未执行，不可能发起等待事件。",[15,299753,299754,299756],{},[2627,299755,115413],{},": In a Process Model that implements two suspend states, a valid state transition is represented by:\na. Ready\u002FSuspend -> Ready\nb. Running -> Ready\u002FSuspend\nc. Ready -> Ready\u002FSuspend\nd. All of the above",[15,299758,299759,299761],{},[2627,299760,293379],{},": 在实现了两个挂起状态的进程模型中，有效的状态转换由以下哪项表示：\na. 就绪\u002F挂起 -> 就绪\nb. 运行 -> 就绪\u002F挂起\nc. 就绪 -> 就绪\u002F挂起\nd. 以上所有",[15,299763,299764,8471,299766],{},[2627,299765,118575],{},[2627,299767,292510],{},[15,299769,299770,5598],{},[2627,299771,118582],{},[105,299773,299774,299779],{},[108,299775,299776,299778],{},[2627,299777,156287],{},": 七态模型（引入挂起状态）。",[108,299780,299781,5598,299783],{},[2627,299782,293402],{},[105,299784,299785,299791,299797],{},[108,299786,299787,299790],{},[31,299788,299789],{},"Ready\u002FSuspend -> Ready","：当主存有足够空间，操作系统将外存中挂起的就绪进程换入主存。",[108,299792,299793,299796],{},[31,299794,299795],{},"Running -> Ready\u002FSuspend","：一个正在运行的进程可能因为优先级较低或为了腾出内存而被直接抢占并挂起到外存。",[108,299798,299799,299802],{},[31,299800,299801],{},"Ready -> Ready\u002FSuspend","：主存空间紧张时，操作系统会将内存中等待执行的就绪进程挂起到外存。",[15,299804,299805,299807],{},[2627,299806,115435],{},": The scheduling strategy where each process in the queue is given a certain amount of time, in turn, to execute and then returned to the queue, unless blocked is referred to as:\na. Prioritization\nb. Round-Robin\nc. LIFO\nd. All of the above",[15,299809,299810,299812],{},[2627,299811,293379],{},": 队列中的每个进程依次获得一定的时间量来执行，除非被阻塞，否则随后返回队列，这种调度策略被称为：\na. 优先级排序\nb. 轮转（Round-Robin）\nc. 后进先出（LIFO）\nd. 以上所有",[15,299814,299815,8471,299817],{},[2627,299816,118575],{},[2627,299818,299819],{},"b. Round-Robin",[15,299821,299822,5598],{},[2627,299823,118582],{},[105,299825,299826,299831],{},[108,299827,299828,299830],{},[2627,299829,156287],{},": 轮转调度算法。",[108,299832,299833,299835],{},[2627,299834,293402],{},": 这正是轮转算法（Round-Robin）的经典定义。操作系统为每个进程分配一个时间片（Time Slice\u002FQuantum），时间耗尽发生时钟中断，进程被剥夺执行权并送回就绪队列的末尾排队。",[15,299837,299838,299840],{},[2627,299839,115457],{},": A Memory Table is an O\u002FS control structure that is used by the O\u002FS to:\na. Manage I\u002FO devices\nb. Manage processes\nc. Provide information about system files\nd. None of the above",[15,299842,299843,299845],{},[2627,299844,293379],{},": 内存表（Memory Table）是操作系统的一种控制结构，被操作系统用来：\na. 管理 I\u002FO 设备\nb. 管理进程\nc. 提供有关系统文件的信息\nd. 以上都不是",[15,299847,299848,8471,299850],{},[2627,299849,118575],{},[2627,299851,292848],{},[15,299853,299854,5598],{},[2627,299855,118582],{},[105,299857,299858,299863],{},[108,299859,299860,299862],{},[2627,299861,156287],{},": 操作系统的控制表。",[108,299864,299865,299867],{},[2627,299866,293402],{},": 内存表（Memory Table）专门用于跟踪主存（Main Memory）和外存（Secondary Memory）的分配情况、保护属性及虚拟内存映射，而不是用来管理 I\u002FO（I\u002FO表）、管理进程（进程表）或提供文件信息（文件表）。因此前三个选项均不正确。",[15,299869,299870,299872],{},[2627,299871,115479],{},": The Process Image element that contains the collection of attributes needed by the O\u002FS to control a particular process is called the:\na. User Data\nb. System Stack\nc. Process Control Block\nd. None of the above",[15,299874,299875,299877],{},[2627,299876,293379],{},": 进程映像中包含操作系统控制特定进程所需的一组属性的元素被称为：\na. 用户数据\nb. 系统栈\nc. 进程控制块\nd. 以上都不是",[15,299879,299880,8471,299882],{},[2627,299881,118575],{},[2627,299883,299884],{},"c. Process Control Block",[15,299886,299887,5598],{},[2627,299888,118582],{},[105,299890,299891,299896],{},[108,299892,299893,299895],{},[2627,299894,156287],{},": 进程控制块（PCB）的作用。",[108,299897,299898,299900],{},[2627,299899,293402],{},": 操作系统为了管理和调度进程，需要一种数据结构来记录进程的所有关键属性（如状态、PC、寄存器、优先级、内存指针等），这个数据结构就是 PCB。",[15,299902,299903,299905],{},[2627,299904,115501],{},": The Process Image element that contains the modifiable part of the user space is called the:\na. User Program\nb. System Stack\nc. Process Control Block\nd. None of the above",[15,299907,299908,299910,299911,299914,299915,299918],{},[2627,299909,293379],{},": 进程映像中包含用户空间可修改部分的元素被称为：\na. 用户数据（User Program 在此题选项为 a，但题意指向 User Data。Wait! 选项 a 是 User Program。但前面提到的四大元素是 User Data, User Program, System Stack, PCB）\n",[71807,299912,299913],{},"注意原文选项","：a. User Program \u002F b. System Stack \u002F c. Process Control Block \u002F d. None of the above。如果题目选项中没有 \"User Data\"，那么可修改的部分应该是 User Data，答案就是 d。\n",[2627,299916,299917],{},"仔细审题","：\n题目：\"The Process Image element that contains the modifiable part of the user space is called the:\"\na. User Program （不可修改的代码段）\nb. System Stack （属于内核控制，用于系统调用上下文）\nc. Process Control Block （属于操作系统内核管理）\nd. None of the above",[15,299920,299921,8471,299923],{},[2627,299922,118575],{},[2627,299924,292848],{},[15,299926,299927,5598],{},[2627,299928,118582],{},[105,299930,299931,299936],{},[108,299932,299933,299935],{},[2627,299934,156287],{},": 进程映像的组成部分。",[108,299937,299938,299940],{},[2627,299939,293402],{},": 进程映像由四部分组成：User Program（代码，通常是只读的）、User Data（用户数据，是可修改的用户空间部分）、System Stack 和 PCB。因为选项中没有提供“User Data”，故选 d。",[15,299942,299943,299945],{},[2627,299944,115523],{},": The processor execution mode that user programs typically execute in is referred to as:\na. User mode\nb. System mode\nc. Kernel mode\nd. None of the above",[15,299947,299948,299950],{},[2627,299949,293379],{},": 用户程序通常在其中执行的处理器执行模式被称为：\na. 用户模式\nb. 系统模式\nc. 内核模式\nd. 以上都不是",[15,299952,299953,8471,299955],{},[2627,299954,118575],{},[2627,299956,299957],{},"a. User mode",[15,299959,299960,5598],{},[2627,299961,118582],{},[105,299963,299964,299969],{},[108,299965,299966,299968],{},[2627,299967,156287],{},": 处理器特权级。",[108,299970,299971,299973],{},[2627,299972,293402],{},": 为了安全性和保护操作系统不受恶意或错误程序的破坏，普通的用户级应用程序都运行在受限的“用户模式（User mode）”下。",[15,299975,299976,299978],{},[2627,299977,115545],{},": One step in the procedure for creating a new process involves:\na. Initializing the process control block\nb. Allocating space for the process\nc. Assigning a unique identifier\nd. All of the above",[15,299980,299981,299983],{},[2627,299982,293379],{},": 创建新进程过程中的一个步骤涉及：\na. 初始化进程控制块\nb. 为进程分配空间\nc. 分配一个唯一的标识符\nd. 以上所有",[15,299985,299986,8471,299988],{},[2627,299987,118575],{},[2627,299989,292510],{},[15,299991,299992,5598],{},[2627,299993,118582],{},[105,299995,299996,300001],{},[108,299997,299998,300000],{},[2627,299999,156287],{},": 进程的创建步骤。",[108,300002,300003,300005],{},[2627,300004,293402],{},": 当操作系统创建新进程时，必须：为新进程分配一个唯一的进程ID (c)；为进程的代码、数据、栈和PCB分配内存空间 (b)；最后初始化该进程的PCB结构 (a) 等。因此全选。",[15,300007,300008,300010],{},[2627,300009,115567],{},": A process switch may occur when the system encounters an interrupt condition, such as that generated by a:\na. Memory fault\nb. Supervisor call\nc. Trap\nd. All of the above",[15,300012,300013,300015],{},[2627,300014,293379],{},": 当系统遇到中断条件时可能会发生进程切换，例如由以下情况生成的中断：\na. 内存错误（Memory fault）\nb. 访管调用（Supervisor call，即系统调用）\nc. 陷阱（Trap）\nd. 以上所有",[15,300017,300018,8471,300020],{},[2627,300019,118575],{},[2627,300021,292510],{},[15,300023,300024,5598],{},[2627,300025,118582],{},[105,300027,300028,300033],{},[108,300029,300030,300032],{},[2627,300031,156287],{},": 引起进程切换的事件。",[108,300034,300035,300037],{},[2627,300036,293402],{},": 能够中断当前进程执行并将控制权交还给操作系统的机制包括：普通中断（时钟或I\u002FO）、陷阱\u002F异常（Trap，例如除零错误或缺页引发的 Memory fault），以及进程主动发起的访管调用（Supervisor call）。这些事件发生后，操作系统内核可能会根据情况决定进行进程切换。",[15,300039,300040,300042],{},[2627,300041,115589],{},": In the Process Based O\u002FS:\na. Major kernel functions are organized as separate functions\nb. The User Process Image includes a kernel stack\nc. O\u002FS code and data are contained in the shared address space\nd. None of the above",[15,300044,300045,300047],{},[2627,300046,293379],{},": 在基于进程的操作系统（Process Based O\u002FS）中：\na. 主要内核功能被组织为独立的函数\nb. 用户进程映像包括一个内核栈\nc. 操作系统代码和数据包含在共享地址空间中\nd. 以上都不是",[15,300049,300050,8471,300052],{},[2627,300051,118575],{},[2627,300053,292848],{},[15,300055,300056,5598],{},[2627,300057,118582],{},[105,300059,300060,300065],{},[108,300061,300062,300064],{},[2627,300063,156287],{},": 操作系统架构模型（Process-based OS）。",[108,300066,300067,5598,300069],{},[2627,300068,293402],{},[105,300070,300071,300074,300077],{},[108,300072,300073],{},"在教科书中，“基于进程的操作系统（Process-based OS）”是将主要的内核功能组织为“独立的系统进程（separate system processes）”，而不是选项 a 所说的“独立的函数（functions）”。",[108,300075,300076],{},"选项 b 和 c 描述的是另一种称为“Execution Within User Processes（在用户进程内执行）”的模型特征。",[108,300078,300079],{},"由于选项 a 具有迷惑性的“functions”而不是“processes”，严格来说应选 d。",[15,300081,300082,300084],{},[2627,300083,115611],{},": In a typical UNIX system, the element of the process image that contains the processor status information is the:\na. System-level context\nb. Register context\nc. User-level context\nd. All of the above",[15,300086,300087,300089],{},[2627,300088,293379],{},": 在典型的 UNIX 系统中，包含处理器状态信息的进程映像元素是：\na. 系统级上下文\nb. 寄存器上下文\nc. 用户级上下文\nd. 以上所有",[15,300091,300092,8471,300094],{},[2627,300093,118575],{},[2627,300095,300096],{},"b. Register context",[15,300098,300099,5598],{},[2627,300100,118582],{},[105,300102,300103,300108],{},[108,300104,300105,300107],{},[2627,300106,156287],{},": UNIX System V 的进程控制块结构。",[108,300109,300110,300112],{},[2627,300111,293402],{},": 典型的 UNIX 系统将进程映像拆分为不同层次的上下文。包含 PC、程序状态字 PSW 和各通用寄存器值的部分，被称为“寄存器上下文（Register context）”。",[32188,300114],{},[19,300116,294449],{"id":294448},[15,300118,300119,300121],{},[2627,300120,115273],{},": The listing of a sequence of instructions that execute for a particular process is called a __________.",[15,300123,300124,300126],{},[2627,300125,293379],{},": 为特定进程执行的一系列指令序列的列表被称为__________。",[15,300128,300129,8471,300131],{},[2627,300130,118575],{},[2627,300132,300133],{},"trace (或 process trace)",[15,300135,300136,300138],{},[2627,300137,118582],{},": 在操作系统中，我们用轨迹（trace）这个术语来宏观跟踪或刻画一个进程随着时间推移的指令执行序列。",[15,300140,300141,300143],{},[2627,300142,115295],{},": The behavior of a processor can be characterized by examining the interleaving of the process __________ for the processes currently running on the system.",[15,300145,300146,300148],{},[2627,300147,293379],{},": 处理器的行为可以通过检查系统上当前正在运行的进程的进程__________的交织情况来表征。",[15,300150,300151,8471,300153],{},[2627,300152,118575],{},[2627,300154,300155],{},"traces",[15,300157,300158,300160],{},[2627,300159,118582],{},": 同上，处理器的宏观行为是通过多个进程的指令轨迹（traces）是如何被调度和交替（interleaving）执行来刻画的。",[15,300162,300163,300165],{},[2627,300164,115317],{},": The portion of the operating system that selects the next process to run is called the __________.",[15,300167,300168,300170],{},[2627,300169,293379],{},": 操作系统中负责选择下一个要运行的进程的部分被称为__________。",[15,300172,300173,8471,300175],{},[2627,300174,118575],{},[2627,300176,300177],{},"dispatcher (或 scheduler)",[15,300179,300180,300182],{},[2627,300181,118582],{},": 调度器（scheduler）决定谁可以运行，而分派器（dispatcher）是实际执行状态切换并把CPU控制权交给该进程的内核组件。教科书通常将短期调度和实际上下文切换统称为 dispatcher 或 short-term scheduler。",[15,300184,300185,300187],{},[2627,300186,115339],{},": When the O\u002FS creates a process at the explicit request of an existing process, the action is referred to as __________.",[15,300189,300190,300192],{},[2627,300191,293379],{},": 当操作系统在现有进程的显式请求下创建一个进程时，该操作被称为__________。",[15,300194,300195,8471,300197],{},[2627,300196,118575],{},[2627,300198,300199],{},"process spawning",[15,300201,300202,300204,300205,300208],{},[2627,300203,118582],{},": 这是派生（spawning）的标准定义。当一个父进程调用 OS 服务（如 ",[31,300206,300207],{},"fork","）来生成子进程时，我们称之为 process spawning。",[15,300210,300211,300213],{},[2627,300212,115391],{},": A process that cannot execute until some event occurs is said to be in the __________ state.",[15,300215,300216,300218],{},[2627,300217,293379],{},": 一个直到某个事件发生后才能执行的进程，据说处于__________状态。",[15,300220,300221,8471,300223],{},[2627,300222,118575],{},[2627,300224,300225],{},"Blocked (或 Waiting)",[15,300227,300228,300230],{},[2627,300229,118582],{},": 当进程因为等待 I\u002FO 完成或等待某种资源而无法继续执行时，操作系统会将其移入阻塞（Blocked）队列中，不再给它分配 CPU。",[15,300232,300233,300235],{},[2627,300234,115413],{},": In a system that implements two suspend states, a process that has been swapped out of main memory and into secondary memory and that is also awaiting an event is in the __________ state.",[15,300237,300238,300240],{},[2627,300239,293379],{},": 在实现了两个挂起状态的系统中，一个被换出主存并进入辅助存储器，且同时在等待一个事件的进程处于__________状态。",[15,300242,300243,8471,300245],{},[2627,300244,118575],{},[2627,300246,300247],{},"Blocked\u002FSuspend",[15,300249,300250,300252],{},[2627,300251,118582],{},": 在七态模型中，一个原本在内存中且被阻塞（Blocked）的进程，如果为了腾出内存空间被操作系统转移到了磁盘外存中，该进程就进入了阻塞\u002F挂起（Blocked\u002FSuspend）状态。",[15,300254,300255,300257],{},[2627,300256,115435],{},": The scheduling strategy where each process in the queue is given a certain amount of time, in turn, to execute and then returned to the queue, unless blocked is referred to as __________.",[15,300259,300260,300262],{},[2627,300261,293379],{},": 队列中的每个进程依次获得一定的时间量来执行，除非被阻塞，否则随后返回队列，这种调度策略被称为__________。",[15,300264,300265,8471,300267],{},[2627,300266,118575],{},[2627,300268,300269],{},"Round-Robin",[15,300271,300272,300274],{},[2627,300273,118582],{},": 这是轮转算法的经典定义，每个进程获得的固定时间量称为时间片（Time Quantum）。",[15,300276,300277,300279],{},[2627,300278,115457],{},": The O\u002FS control structure that the O\u002FS uses to manage system processes is called the __________.",[15,300281,300282,300284],{},[2627,300283,293379],{},": 操作系统用来管理系统进程的控制结构被称为__________。",[15,300286,300287,8471,300289],{},[2627,300288,118575],{},[2627,300290,300291],{},"Process Table",[15,300293,300294,300296],{},[2627,300295,118582],{},": 操作系统使用进程表（Process Table）来维护系统中所有进程的目录。每个活跃进程在表中有一个条目，包含指向该进程映像（Process Image）的指针或其实际信息。",[15,300298,300299,300301],{},[2627,300300,115479],{},": The User Data, User Program, System Stack and Process Control Block elements collectively make up what is referred to as the __________.",[15,300303,300304,300306],{},[2627,300305,293379],{},": 用户数据、用户程序、系统栈和进程控制块元素共同构成了所谓的__________。",[15,300308,300309,8471,300311],{},[2627,300310,118575],{},[2627,300312,300313],{},"Process Image",[15,300315,300316,300318],{},[2627,300317,118582],{},": 这四部分要素组合在一起构成了进程在物理内存或虚拟内存中存在的一个完整实体，操作系统理论中称之为“进程映像（Process Image）”。",[15,300320,300321,300323],{},[2627,300322,115501],{},": The Process Identification, Processor State Information and the Process Control Information are the general categories that collectively make up what is referred to as the __________.",[15,300325,300326,300328],{},[2627,300327,293379],{},": 进程标识、处理器状态信息和进程控制信息这几个通用类别，共同构成了所谓的__________。",[15,300330,300331,8471,300333],{},[2627,300332,118575],{},[2627,300334,241415],{},[15,300336,300337,300339],{},[2627,300338,118582],{},": 这是 PCB 的标准内部逻辑结构，PCB 是操作系统中最重要的控制数据结构。",[15,300341,300342,300344],{},[2627,300343,115523],{},": The processor typically maintains the current operating mode (i.e., user or kernel) in the __________.",[15,300346,300347,300349],{},[2627,300348,293379],{},": 处理器通常在__________中维护当前的运行模式（即用户模式或内核模式）。",[15,300351,300352,8471,300354],{},[2627,300353,118575],{},[2627,300355,300356],{},"Program Status Word (PSW)",[15,300358,300359,300361],{},[2627,300360,118582],{},": 程序状态字（PSW）是一个硬件寄存器，其中包含条件码、中断使能位以及指示当前 CPU 执行处于核心态还是用户态的特权级别位（Mode bit）。",[15,300363,300364,300366],{},[2627,300365,115545],{},": The first step in creating a new process is to assign a unique __________ to the new process.",[15,300368,300369,300371],{},[2627,300370,293379],{},": 创建新进程的第一步是为新进程分配一个唯一的__________。",[15,300373,300374,8471,300376],{},[2627,300375,118575],{},[2627,300377,300378],{},"process identifier (或 process ID)",[15,300380,300381,300383],{},[2627,300382,118582],{},": 为了能在进程表中添加新记录，操作系统必须首先生成一个当前未被使用的唯一的数字标识符（PID）。",[15,300385,300386,300388],{},[2627,300387,115567],{},": The execution of a user process may be interrupted by a __________, which might be generated by the process requesting an I\u002FO operation.",[15,300390,300391,300393],{},[2627,300392,293379],{},": 用户进程的执行可能会被一个__________打断，这可能是由该进程请求 I\u002FO 操作而生成的。",[15,300395,300396,8471,300398],{},[2627,300397,118575],{},[2627,300399,300400],{},"Supervisor call (或 system call)",[15,300402,300403,300405],{},[2627,300404,118582],{},": 当用户进程需要操作系统服务（如请求 I\u002FO 操作）时，它必须通过触发软中断来发起一个访管调用（Supervisor Call \u002F System Call），这将引起 CPU 特权级切换并陷入操作系统。",[15,300407,300408,300410],{},[2627,300409,115589],{},": In the __________ model for illustrating the relationship between the O\u002FS and User Processes, the O\u002FS has its own region of memory to use and its own system stack for controlling procedure calls and returns.",[15,300412,300413,300415],{},[2627,300414,293379],{},": 在阐明操作系统与用户进程之间关系的__________模型中，操作系统拥有自己使用的内存区域，并拥有自己的系统栈用于控制过程调用和返回。",[15,300417,300418,8471,300420],{},[2627,300419,118575],{},[2627,300421,300422],{},"Nonprocess Kernel",[15,300424,300425,300427],{},[2627,300426,118582],{},": “无进程内核（Nonprocess Kernel）”模型是一种传统的内核设计方式，其中内核在其专用的内存空间中作为一个实体执行，不运行在任何用户进程的上下文中。",[15,300429,300430,300432,300433],{},[2627,300431,115611],{},": In the __________ model for illustrating the relationship between the O\u002FS and User Processes, the O\u002FS has its own region of memory to use and its own system stack for controlling procedure calls and returns.\n",[71807,300434,300435],{},"(注：第 15 题在源文档中与第 14 题完全相同，故答案与解析一致)",[15,300437,300438,300415],{},[2627,300439,293379],{},[15,300441,300442,8471,300444],{},[2627,300443,118575],{},[2627,300445,300422],{},[15,300447,300448,300450],{},[2627,300449,118582],{},": 题目与前一题重复。考查的依然是 Nonprocess Kernel 结构模型。",[15,300452,300453,300456],{},[2627,300454,300455],{},"16. 题目",": Process creation in a typical UNIX system is made by means of a kernel system call named __________.",[15,300458,300459,300461],{},[2627,300460,293379],{},": 在典型的 UNIX 系统中，进程创建是通过名为__________的内核系统调用来实现的。",[15,300463,300464,8471,300466],{},[2627,300465,118575],{},[2627,300467,300207],{},[15,300469,300470,300472,300473,300475],{},[2627,300471,118582],{},": 在 UNIX 系操作系统中，",[31,300474,242312],{}," 是唯一的创建新进程的方式。它会精确克隆调用者（父进程）的内存空间，生成一个近乎完全相同的子进程映像。",{"title":29,"searchDepth":43,"depth":43,"links":300477},[300478,300479,300480],{"id":293368,"depth":43,"text":293369},{"id":293915,"depth":43,"text":293916},{"id":294448,"depth":43,"text":294449},{"order":49},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap3",{"title":299060,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap3","uQLUjQPY79zSLKRj38jFKY-3AeI5OnuiXEjnP6Z7qPQ",{"id":300487,"title":300488,"body":300489,"description":29,"extension":85,"meta":301780,"navigation":52,"path":301781,"seo":301782,"stem":301783,"__hash__":301784},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap4.md","Chap. 4",{"type":8,"value":300490,"toc":301775},[300491,300495,300497,300502,300524,300529,300551,300556,300574,300579,300604,300609,300631,300636,300654,300659,300677,300682,300704,300709,300727,300732,300758,300763,300781,300786,300804,300809,300837,300842,300863,300868,300890,300892,300894,300900,300927,300933,300952,300958,300981,300987,301017,301023,301042,301048,301074,301080,301099,301105,301128,301134,301163,301169,301191,301197,301227,301233,301263,301269,301292,301298,301321,301327,301360,301362,301364,301369,301391,301396,301415,301420,301442,301447,301470,301475,301498,301503,301522,301527,301550,301555,301574,301579,301602,301607,301630,301635,301656,301661,301684,301689,301719,301724,301747,301752],[11,300492,300494],{"id":300493},"chapter-4-threads-smp-and-microkernels","Chapter 4 - Threads, SMP, and Microkernels",[19,300496,291941],{"id":291940},[15,300498,300499],{},[2627,300500,300501],{},"1. The basic unit of dispatching in an operating system is usually referred to as a thread or lightweight process.",[105,300503,300504,300509,300515],{},[108,300505,300506,300508],{},[2627,300507,291953],{},"：操作系统中的基本调度单位通常被称为线程（thread）或轻量级进程（lightweight process）。",[108,300510,300511,2664,300513],{},[2627,300512,118575],{},[2627,300514,291961],{},[108,300516,300517,300519,300520,300523],{},[2627,300518,291966],{},"：在现代操作系统中，**进程（Process）",[2627,300521,300522],{},"是资源分配和保护的基本单位，而","线程（Thread）**或轻量级进程是 CPU 调度和执行的基本单位。一个进程可以包含一个或多个线程。",[15,300525,300526],{},[2627,300527,300528],{},"2. An example of a system that implements a single process with multiple threads is MS-DOS.",[105,300530,300531,300536,300542],{},[108,300532,300533,300535],{},[2627,300534,291953],{},"：实现单进程多线程的系统的一个例子是 MS-DOS。",[108,300537,300538,2664,300540],{},[2627,300539,118575],{},[2627,300541,292009],{},[108,300543,300544,300546,300547,300550],{},[2627,300545,291966],{},"：MS-DOS 是典型的",[2627,300548,300549],{},"单进程单线程","（Single process, single thread）操作系统。**Java 运行时环境（JVM）**才是单进程多线程（Single process, multiple threads）的典型代表。",[15,300552,300553],{},[2627,300554,300555],{},"3. In a multithreaded environment, a process is defined as the unit of resource allocation and a unit of protection.",[105,300557,300558,300563,300569],{},[108,300559,300560,300562],{},[2627,300561,291953],{},"：在多线程环境中，进程被定义为资源分配单位和保护单位。",[108,300564,300565,2664,300567],{},[2627,300566,118575],{},[2627,300568,291961],{},[108,300570,300571,300573],{},[2627,300572,291966],{},"：多线程模型将进程的两个核心属性分开了。进程依然拥有独立的虚拟地址空间、文件句柄和其他系统资源，并负责资源隔离与保护；而线程只拥有执行所必需的最小状态（如程序计数器、栈、寄存器上下文）。",[15,300575,300576],{},[2627,300577,300578],{},"4. The concept of thread synchronization is required in multithreaded systems because threads of a single process share the process's process control block (PCB).",[105,300580,300581,300586,300592],{},[108,300582,300583,300585],{},[2627,300584,291953],{},"：多线程系统需要线程同步的概念，因为单个进程的线程共享该进程的进程控制块 (PCB)。",[108,300587,300588,2664,300590],{},[2627,300589,118575],{},[2627,300591,292009],{},[108,300593,300594,300596,300597,300599,300600,300603],{},[2627,300595,291966],{},"：线程需要同步的根本原因",[2627,300598,243550],{},"共享 PCB，而是因为它们",[2627,300601,300602],{},"共享该进程的内存地址空间（Address Space）和全局变量等资源","。并发访问共享内存中的数据会导致竞争条件，因此需要互斥和同步机制。每个线程实际上有自己的线程控制块（TCB）。",[15,300605,300606],{},[2627,300607,300608],{},"5. In a pure User-Level Thread (ULT) facility, all of the work of thread management is done by the application, but the kernel is aware of the existence of threads.",[105,300610,300611,300616,300622],{},[108,300612,300613,300615],{},[2627,300614,291953],{},"：在纯用户级线程 (ULT) 机制中，所有线程管理工作都由应用程序完成，但内核知道线程的存在。",[108,300617,300618,2664,300620],{},[2627,300619,118575],{},[2627,300621,292009],{},[108,300623,300624,300626,300627,300630],{},[2627,300625,291966],{},"：在纯用户级线程 (ULT) 模型中，线程管理完全由用户空间的线程库（Thread Library）负责，",[2627,300628,300629],{},"内核对 ULT 一无所知（The kernel is unaware of the existence of threads）","。内核依然把该应用程序当成一个单线程的普通进程进行调度。",[15,300632,300633],{},[2627,300634,300635],{},"6. In the field of distributed operating system design, the One-to-Many (Thread-to-Process) relationship is particularly interesting because it involves the concept of thread migration.",[105,300637,300638,300643,300649],{},[108,300639,300640,300642],{},[2627,300641,291953],{},"：在分布式操作系统设计领域，一对多（线程对进程）关系特别有趣，因为它涉及线程迁移（thread migration）的概念。",[108,300644,300645,2664,300647],{},[2627,300646,118575],{},[2627,300648,291961],{},[108,300650,300651,300653],{},[2627,300652,291966],{},"：在“一对多”模型（如 Ra 操作系统）中，一个逻辑线程可以跨越多个进程甚至多台机器的边界。这种机制允许一个线程在不同节点的地址空间中移动执行，这就是线程迁移。",[15,300655,300656],{},[2627,300657,300658],{},"7. One disadvantage to the master\u002Fslave shared-memory multiprocessor architecture is that the failure of the master brings down the whole system.",[105,300660,300661,300666,300672],{},[108,300662,300663,300665],{},[2627,300664,291953],{},"：主\u002F从（master\u002Fslave）共享内存多处理器架构的一个缺点是，主处理器的故障会导致整个系统崩溃。",[108,300667,300668,2664,300670],{},[2627,300669,118575],{},[2627,300671,291961],{},[108,300673,300674,300676],{},[2627,300675,291966],{},"：在主从架构中，主处理器（Master）负责所有的内核功能、调度和 I\u002FO 管理，从处理器（Slaves）仅执行用户程序。如果 Master 宕机，整个系统将失去核心管理能力，从而全面崩溃，存在单点故障（Single Point of Failure）。",[15,300678,300679],{},[2627,300680,300681],{},"8. In a symmetric multiprocessing (SMP) system, each processor has access only to a private main memory area.",[105,300683,300684,300689,300695],{},[108,300685,300686,300688],{},[2627,300687,291953],{},"：在对称多处理 (SMP) 系统中，每个处理器只能访问私有的主存区域。",[108,300690,300691,2664,300693],{},[2627,300692,118575],{},[2627,300694,292009],{},[108,300696,300697,300699,300700,300703],{},[2627,300698,291966],{},"：SMP 架构的核心特征之一就是",[2627,300701,300702],{},"共享主存（Shared Memory）","。所有的处理器不仅有均等的地位，还可以通过同一个系统总线或互连网络访问同一个全局主存空间（以及共享的 I\u002FO 设备）。",[15,300705,300706],{},[2627,300707,300708],{},"9. An SMP O\u002FS manages processor and other resources so that the user may view the system in the same fashion as a multiprogramming uniprocessor system.",[105,300710,300711,300716,300722],{},[108,300712,300713,300715],{},[2627,300714,291953],{},"：SMP 操作系统管理处理器和其他资源，以便用户可以像查看多道程序单处理器系统一样查看该系统。",[108,300717,300718,2664,300720],{},[2627,300719,118575],{},[2627,300721,291961],{},[108,300723,300724,300726],{},[2627,300725,291966],{},"：SMP 操作系统必须对用户提供透明性。底层的多处理器对用户程序是隐藏的，程序员和用户感觉就像在使用一个速度更快的单处理器系统，不需要修改应用程序代码。",[15,300728,300729],{},[2627,300730,300731],{},"10. The primary advantage of the basic microkernel design over layered kernel designs involves increased performance.",[105,300733,300734,300739,300745],{},[108,300735,300736,300738],{},[2627,300737,291953],{},"：与分层内核设计相比，基本微内核设计的主要优势在于提高了性能。",[108,300740,300741,2664,300743],{},[2627,300742,118575],{},[2627,300744,292009],{},[108,300746,300747,300749,300750,300753,300754,300757],{},[2627,300748,291966],{},"：微内核（Microkernel）的",[2627,300751,300752],{},"主要缺点恰恰是性能开销大","。由于大多数系统服务被移到了用户空间，服务请求需要通过内核的进程间通信（IPC，通常是消息传递）来进行，导致频繁的上下文切换和模式切换。微内核的优势在于",[2627,300755,300756],{},"扩展性、灵活性、可移植性和可靠性","，而非性能。",[15,300759,300760],{},[2627,300761,300762],{},"11. The philosophy underlying the microkernel is that only absolutely essential core operating system functions should be in the kernel.",[105,300764,300765,300770,300776],{},[108,300766,300767,300769],{},[2627,300768,291953],{},"：微内核的基础哲学是，内核中只应包含绝对必要的操作系统核心功能。",[108,300771,300772,2664,300774],{},[2627,300773,118575],{},[2627,300775,291961],{},[108,300777,300778,300780],{},[2627,300779,291966],{},"：微内核设计将文件系统、设备驱动、网络协议栈等功能移出内核空间，内核仅保留最基本的功能，如地址空间管理、进程间通信（IPC）和基本的调度。",[15,300782,300783],{},[2627,300784,300785],{},"12. The basic form of communication between processes or threads in a microkernel O\u002FS is messages.",[105,300787,300788,300793,300799],{},[108,300789,300790,300792],{},[2627,300791,291953],{},"：在微内核操作系统中，进程或线程之间基本的通信形式是消息（messages）。",[108,300794,300795,2664,300797],{},[2627,300796,118575],{},[2627,300798,291961],{},[108,300800,300801,300803],{},[2627,300802,291966],{},"：在微内核中，由于各个系统服务作为独立的用户级服务器运行，它们之间以及它们与客户端程序之间的通信必须依赖于底层的消息传递机制（Message Passing IPC）。",[15,300805,300806],{},[2627,300807,300808],{},"13. Linux makes no distinction between a process and a thread.",[105,300810,300811,300816,300822],{},[108,300812,300813,300815],{},[2627,300814,291953],{},"：Linux 对进程和线程不作区分。",[108,300817,300818,2664,300820],{},[2627,300819,118575],{},[2627,300821,291961],{},[108,300823,300824,300826,300827,300830,300831,300833,300834,300836],{},[2627,300825,291966],{},"：在 Linux 内核层面，没有传统意义上独立的“线程”数据结构。它将进程和线程统称为",[2627,300828,300829],{},"任务（Task）","，都使用 ",[31,300832,242275],{}," 结构来表示。线程只是被视作与其他任务共享某些资源（如虚拟内存、打开的文件）的进程（通常通过 ",[31,300835,242304],{}," 系统调用创建）。",[15,300838,300839],{},[2627,300840,300841],{},"14. Windows 2000 is an object-oriented O\u002FS, but only processes (not threads) are implemented as objects in the WIN2K O\u002FS.",[105,300843,300844,300849,300855],{},[108,300845,300846,300848],{},[2627,300847,291953],{},"：Windows 2000 是一个面向对象的操作系统，但在 WIN2K OS 中只有进程（而非线程）被实现为对象。",[108,300850,300851,2664,300853],{},[2627,300852,118575],{},[2627,300854,292009],{},[108,300856,300857,300859,300860,347],{},[2627,300858,291966],{},"：Windows (从 NT\u002F2000 起) 高度依赖对象架构。不仅进程是对象（Process Object），",[2627,300861,300862],{},"线程本身也是被内核管理的对象（Thread Object）",[15,300864,300865],{},[2627,300866,300867],{},"15. In the Solaris O\u002FS, a User-Level Thread (ULT) in the active state is assigned to a Light-Weight Process (LWP) and executes while the underlying kernel thread executes.",[105,300869,300870,300875,300881],{},[108,300871,300872,300874],{},[2627,300873,291953],{},"：在 Solaris 操作系统中，处于活动状态的用户级线程 (ULT) 被分配给轻量级进程 (LWP)，并在底层内核线程执行时执行。",[108,300876,300877,2664,300879],{},[2627,300878,118575],{},[2627,300880,291961],{},[108,300882,300883,300885,300886,300889],{},[2627,300884,291966],{},"：Solaris 采用经典的",[2627,300887,300888],{},"多对多（M:N）模型","。应用程序创建许多 ULT，而操作系统维护一组 LWP（每个 LWP 映射到一个真正的 Kernel Thread）。活跃的 ULT 会被复用绑定（绑定执行）到某个 LWP 上从而获得 CPU 时间。",[32188,300891],{},[19,300893,292392],{"id":292391},[15,300895,300896,300899],{},[2627,300897,300898],{},"1. The concept of a process in an operating system embodies two primary characteristics, one of which is:","\na. Multithreading\nb. Resource ownership\nc. Symmetric multiprocessing\nd. None of the above",[105,300901,300902,300907,300914],{},[108,300903,300904,300906],{},[2627,300905,291953],{},"：操作系统中的进程概念体现了两个主要特征，其中之一是：\na. 多线程\nb. 资源所有权 (Resource ownership)\nc. 对称多处理\nd. 以上都不是",[108,300908,300909,2664,300911],{},[2627,300910,118575],{},[2627,300912,300913],{},"b. Resource ownership",[108,300915,300916,300918,300919,300922,300923,300926],{},[2627,300917,291966],{},"：传统的进程概念包含两个核心独立特征：一是",[2627,300920,300921],{},"资源所有权 (Resource ownership)","，二是",[2627,300924,300925],{},"调度\u002F执行 (Scheduling\u002FExecution)","。多线程技术就是把这两者分离，进程拥有资源，线程负责执行。",[15,300928,300929,300932],{},[2627,300930,300931],{},"2. An example of a system that implements a single process with multiple threads is:","\na. WIN 2000\nb. Solaris\nc. Java\nd. All of the above",[105,300934,300935,300940,300947],{},[108,300936,300937,300939],{},[2627,300938,291953],{},"：实现单进程多线程的系统的一个例子是：\na. Windows 2000 (支持多进程多线程)\nb. Solaris (支持多进程多线程)\nc. Java (运行时环境是单进程多线程)\nd. 以上都是",[108,300941,300942,2664,300944],{},[2627,300943,118575],{},[2627,300945,300946],{},"c. Java",[108,300948,300949,300951],{},[2627,300950,291966],{},"：早期的系统分类中，MS-DOS 是 1:1（单进程单线程）；传统的 UNIX 是 N:1（多进程单线程）；Java 运行环境（JVM 本身运行为一个进程）内部支持多个执行流，体现了 1:N（单进程多线程）的概念；现代 OS (Win\u002FSolaris\u002FLinux) 则是 N:M（多进程多线程）。",[15,300953,300954,300957],{},[2627,300955,300956],{},"3. Which of the following is true regarding the relationship between processes and threads:","\na. It takes far less time to create a new thread in an existing process than to create a new process\nb. It takes less time to terminate a process than a thread\nc. It takes less time to switch between two different processes than to switch between two threads within the same process\nd. All of the above",[105,300959,300960,300965,300972],{},[108,300961,300962,300964],{},[2627,300963,291953],{},"：关于进程和线程之间的关系，以下哪项是正确的：\na. 在现有进程中创建新线程比创建新进程花费的时间要少得多。\nb. 终止进程比终止线程花费的时间少。（错误，线程轻量）\nc. 在两个不同进程之间切换的时间比在同一进程的两个线程之间切换的时间少。（错误，进程上下文切换重于线程）\nd. 以上都是",[108,300966,300967,2664,300969],{},[2627,300968,118575],{},[2627,300970,300971],{},"a. It takes far less time to create a new thread in an existing process than to create a new process",[108,300973,300974,300976,300977,300980],{},[2627,300975,291966],{},"：线程是轻量级的。由于线程共享了地址空间和系统资源，创建线程无需复制页表、分配大块内存等，因此创建、终止、切换线程的开销都远",[2627,300978,300979],{},"低于","进程。",[15,300982,300983,300986],{},[2627,300984,300985],{},"4. The basic thread operation related to the change in thread state that occurs when a thread needs to wait for an event is referred to as the:","\na. Unblock operation\nb. Spawn operation\nc. Block operation\nd. None of the above",[105,300988,300989,300994,301001],{},[108,300990,300991,300993],{},[2627,300992,291953],{},"：当线程需要等待某个事件时发生的状态更改相关的基本线程操作被称为：",[108,300995,300996,2664,300998],{},[2627,300997,118575],{},[2627,300999,301000],{},"c. Block operation",[108,301002,301003,301005,301006,301009,301010,301013,301014,347],{},[2627,301004,291966],{},"：线程与进程类似，有基本的生命周期操作。等待事件（如 I\u002FO 或锁）时，会执行 ",[2627,301007,301008],{},"阻塞 (Block)"," 操作；当事件发生时，会执行 ",[2627,301011,301012],{},"解除阻塞 (Unblock)"," 操作；派生新线程称为 ",[2627,301015,301016],{},"产生 (Spawn)",[15,301018,301019,301022],{},[2627,301020,301021],{},"5. One of the disadvantages of User-Level Threads (ULTs) compared to Kernel-Level Threads (KLTs) is:","\na. Scheduling is application specific\nb. When a ULT executes a system call, all threads in the process are blocked\nc. Thread switching does not require kernel mode privileges\nd. All of the above",[105,301024,301025,301030,301037],{},[108,301026,301027,301029],{},[2627,301028,291953],{},"：与内核级线程 (KLT) 相比，用户级线程 (ULT) 的缺点之一是：\na. 调度是特定于应用程序的（这是优点）\nb. 当一个 ULT 执行系统调用时，进程中的所有线程都会被阻塞（致命缺点）\nc. 线程切换不需要内核模式权限（这是优点）\nd. 以上都是",[108,301031,301032,2664,301034],{},[2627,301033,118575],{},[2627,301035,301036],{},"b. When a ULT executes a system call, all threads in the process are blocked",[108,301038,301039,301041],{},[2627,301040,291966],{},"：ULT 对内核不可见，内核只知道包含这些线程的整个进程。因此，如果进程中的某一个 ULT 发起了阻塞式的系统调用（如读取文件），内核就会阻塞整个进程，导致进程内的其他 ULT 全部停滞，无法利用多处理器的并发优势。",[15,301043,301044,301047],{},[2627,301045,301046],{},"6. In the Linux O\u002FS, multiple threads may be created and executed within a single process. This is an example of the following Thread-to-Process relationship:","\na. 1:1\nb. 1:M\nc. M:N\nd. None of the above",[105,301049,301050,301055,301062],{},[108,301051,301052,301054],{},[2627,301053,291953],{},"：在 Linux 操作系统中，可以在单个进程内创建和执行多个线程。这是以下哪种线程与进程关系的示例：",[108,301056,301057,2664,301059],{},[2627,301058,118575],{},[2627,301060,301061],{},"a. 1:1",[108,301063,301064,301066,301067,301070,301071,301073],{},[2627,301065,291966],{},"：在 Linux 的设计中（通过 NPTL，Native POSIX Thread Library），实现了 ",[2627,301068,301069],{},"1:1（One-to-One）模型","。每一个用户空间的线程都直接对应一个内核空间的 ",[31,301072,242275],{},"（即内核可调度的独立实体）。",[15,301075,301076,301079],{},[2627,301077,301078],{},"7. The computer system category where a single processor executes a single instruction stream to operate on data stored in a single memory is called:","\na. Single Instruction Single Data (SISD) stream\nb. Single Instruction Multiple Data (SIMD) stream\nc. Multiple Instruction Single Data (MISD) stream\nd. None of the above",[105,301081,301082,301087,301094],{},[108,301083,301084,301086],{},[2627,301085,291953],{},"：单处理器执行单条指令流来操作存储在单一内存中的数据的计算机系统类别被称为：",[108,301088,301089,2664,301091],{},[2627,301090,118575],{},[2627,301092,301093],{},"a. Single Instruction Single Data (SISD) stream",[108,301095,301096,301098],{},[2627,301097,291966],{},"：这是弗林分类法（Flynn's Taxonomy）中最传统的单机架构：单指令流、单数据流 (SISD)。常见的桌面单核 CPU 即属于此类。",[15,301100,301101,301104],{},[2627,301102,301103],{},"8. In a SMP system, each processor maintains a local cache and must alert all other processors that a change to cache update has taken place. This is referred to as the:","\na. Interconnection mechanism problem\nb. Synchronization mechanism problem\nc. Cache coherency problem\nd. None of the above",[105,301106,301107,301112,301119],{},[108,301108,301109,301111],{},[2627,301110,291953],{},"：在 SMP 系统中，每个处理器维护一个本地缓存，并且必须向所有其他处理器发出警报，表明缓存更新已发生更改。这被称为：",[108,301113,301114,2664,301116],{},[2627,301115,118575],{},[2627,301117,301118],{},"c. Cache coherency problem",[108,301120,301121,301123,301124,301127],{},[2627,301122,291966],{},"：这是多处理器系统中经典的",[2627,301125,301126],{},"缓存一致性问题（Cache Coherency Problem）","。由于内存数据可能缓存在多个处理器的私有 L1\u002FL2 缓存中，如果一个处理器修改了数据，必须通过总线嗅探（Snooping）或目录协议等机制让其他处理器的缓存失效或更新，以防止读取脏数据。",[15,301129,301130,301133],{},[2627,301131,301132],{},"9. Key issues involved in the design of multiprocessor operating systems include:","\na. Scheduling\nb. Synchronization\nc. Reliability and fault tolerance\nd. All of the above",[105,301135,301136,301141,301147],{},[108,301137,301138,301140],{},[2627,301139,291953],{},"：多处理器操作系统设计中涉及的关键问题包括：",[108,301142,301143,2664,301145],{},[2627,301144,118575],{},[2627,301146,292510],{},[108,301148,301149,301151,301152,301155,301156,301159,301160,214569],{},[2627,301150,291966],{},"：SMP 操作系统要比单核复杂得多。必须处理好进程\u002F线程分配到哪个 CPU (",[2627,301153,301154],{},"Scheduling",")、并发访问内核资源的锁冲突 (",[2627,301157,301158],{},"Synchronization",") 以及某个 CPU 宕机不影响整体系统的容错机制 (",[2627,301161,301162],{},"Reliability and fault tolerance",[15,301164,301165,301168],{},[2627,301166,301167],{},"10. Early operating systems that were designed with little concern about structure are typically referred to as:","\na. Monolithic operating systems\nb. Layered operating systems\nc. Kernel operating systems\nd. All of the above",[105,301170,301171,301176,301183],{},[108,301172,301173,301175],{},[2627,301174,291953],{},"：早期设计时很少考虑结构的操作系统通常被称为：",[108,301177,301178,2664,301180],{},[2627,301179,118575],{},[2627,301181,301182],{},"a. Monolithic operating systems",[108,301184,301185,301187,301188,347],{},[2627,301186,291966],{},"：早期的操作系统功能混杂在一起，全部代码运行在内核态，缺乏清晰的模块化边界，这种庞大且无明确结构的系统被称为",[2627,301189,301190],{},"单体内核（Monolithic kernel \u002F OS）",[15,301192,301193,301196],{},[2627,301194,301195],{},"11. A benefit of the microkernel organization is:","\na. Extensibility\nb. Portability\nc. Flexibility\nd. All of the above",[105,301198,301199,301204,301210],{},[108,301200,301201,301203],{},[2627,301202,291953],{},"：微内核组织的优点是：",[108,301205,301206,2664,301208],{},[2627,301207,118575],{},[2627,301209,292510],{},[108,301211,301212,301214,301215,301218,301219,301222,301223,301226],{},[2627,301213,291966],{},"：微内核的主要优点：",[2627,301216,301217],{},"可扩展性 (Extensibility)"," 方便增加新功能（只需新增用户态服务器）；",[2627,301220,301221],{},"可移植性 (Portability)"," 因为硬件相关代码最小化；",[2627,301224,301225],{},"灵活性 (Flexibility)"," 允许根据需求裁剪系统组件。",[15,301228,301229,301232],{},[2627,301230,301231],{},"12. In low-level microkernel memory management, an example of an operation that can support external paging and virtual memory management is the:","\na. Grant operation\nb. Map operation\nc. Flush operation\nd. All of the above",[105,301234,301235,301240,301246],{},[108,301236,301237,301239],{},[2627,301238,291953],{},"：在底层的微内核内存管理中，能够支持外部调页和虚拟内存管理的内存操作例子是：",[108,301241,301242,2664,301244],{},[2627,301243,118575],{},[2627,301245,292510],{},[108,301247,301248,301250,301251,301254,301255,301258,301259,301262],{},[2627,301249,291966],{},"：以著名的 Mach 微内核为例，它为了让用户空间程序接管复杂的虚拟内存管理，提供了三条基本的底层原语：",[2627,301252,301253],{},"Grant (授权)"," 允许传递页框使用权；",[2627,301256,301257],{},"Map (映射)"," 允许共享页框；",[2627,301260,301261],{},"Flush (刷新)"," 允许回收页框。",[15,301264,301265,301268],{},[2627,301266,301267],{},"13. In a W2K system, the state that a thread enters when it has been unblocked and the resource for which it has been blocked is not yet available is called the:","\na. Transition state\nb. Waiting state\nc. Standby state\nd. None of the above",[105,301270,301271,301276,301283],{},[108,301272,301273,301275],{},[2627,301274,291953],{},"：在 Windows 2000 系统中，当线程被解除阻塞但其所需资源（通常指内存）尚不可用时，它进入的状态被称为：",[108,301277,301278,2664,301280],{},[2627,301279,118575],{},[2627,301281,301282],{},"a. Transition state",[108,301284,301285,301287,301288,301291],{},[2627,301286,291966],{},"：在 Windows 线程状态机中，当线程从阻塞（Wait）中被唤醒准备执行，但它的内核栈被换出到磁盘（Page Out）还没载入物理内存时，它会处于一种短暂的 ",[2627,301289,301290],{},"转换状态 (Transition)","，待内存加载完毕后才转入就绪态 (Ready)。",[15,301293,301294,301297],{},[2627,301295,301296],{},"14. In a Solaris system, a User-Level Thread (ULT) that enters the active state is assigned to a:","\na. Kernel thread\nb. Heavy-Weight Process (HWP)\nc. Light-Weight Process (LWP)\nd. None of the above",[105,301299,301300,301305,301312],{},[108,301301,301302,301304],{},[2627,301303,291953],{},"：在 Solaris 系统中，进入活动状态的用户级线程 (ULT) 被分配给：",[108,301306,301307,2664,301309],{},[2627,301308,118575],{},[2627,301310,301311],{},"c. Light-Weight Process (LWP)",[108,301313,301314,301316,301317,301320],{},[2627,301315,291966],{},"：在 Solaris M:N 模型中，ULT 不直接与底层 Kernel Thread 交互，它们被复用和调度到中间层的 ",[2627,301318,301319],{},"轻量级进程 (LWP)"," 上执行。",[15,301322,301323,301326],{},[2627,301324,301325],{},"15. In a Linux system, when a new process is cloned, the two processes share the same:","\na. Process identifier\nb. Virtual memory\nc. task_struct data structure\nd. All of the above",[105,301328,301329,301334,301341],{},[108,301330,301331,301333],{},[2627,301332,291953],{},"：在 Linux 系统中，当使用 clone 克隆一个新进程时，这两个进程共享相同的：",[108,301335,301336,2664,301338],{},[2627,301337,118575],{},[2627,301339,301340],{},"b. Virtual memory",[108,301342,301343,301345,301346,301348,301349,301352,301353,301356,301357,301359],{},[2627,301344,291966],{},"：Linux 使用 ",[31,301347,242304],{}," 系统调用来创建线程。通过传递特定标志（如 ",[31,301350,301351],{},"CLONE_VM","），新老任务（Task）可以",[2627,301354,301355],{},"共享同一个虚拟内存空间 (Virtual memory)","。但它们依然有各自的 ",[31,301358,242275],{}," 结构和内部唯一的进程标识符 (PID\u002FTID)。",[32188,301361],{},[19,301363,292903],{"id":292902},[15,301365,301366],{},[2627,301367,301368],{},"1. In an operating system, the unit of dispatching is usually referred to as a ________ while the unit of resource ownership is usually referred to as a process or task.",[105,301370,301371,301376,301383],{},[108,301372,301373,301375],{},[2627,301374,291953],{},"：在操作系统中，调度的单位通常被称为 ________，而资源所有权的单位通常被称为进程或任务。",[108,301377,301378,2664,301380],{},[2627,301379,118575],{},[2627,301381,301382],{},"thread (或 lightweight process)",[108,301384,301385,301387,301388,301390],{},[2627,301386,291966],{},"：再次强调了现代 OS 中将调度单位（",[2627,301389,247201],{},"）与资源分配单位（进程）进行解耦的设计。",[15,301392,301393],{},[2627,301394,301395],{},"2. An example of an operating system that supports a single user process and a single thread is ________.",[105,301397,301398,301403,301410],{},[108,301399,301400,301402],{},[2627,301401,291953],{},"：支持单用户进程和单线程的操作系统的一个例子是 ________。",[108,301404,301405,2664,301407],{},[2627,301406,118575],{},[2627,301408,301409],{},"MS-DOS",[108,301411,301412,301414],{},[2627,301413,291966],{},"：MS-DOS 是典型的单任务、单线程操作系统，同一时间只能执行一个程序流。",[15,301416,301417],{},[2627,301418,301419],{},"3. An example of an operating system that supports multiple user processes and multiple threads is ________.",[105,301421,301422,301427,301437],{},[108,301423,301424,301426],{},[2627,301425,291953],{},"：支持多个用户进程和多个线程的操作系统的一个例子是 ________。",[108,301428,301429,2664,301431,47649,301434],{},[2627,301430,118575],{},[2627,301432,301433],{},"Windows 2000 \u002F Solaris \u002F Linux \u002F Mac OS",[71807,301435,301436],{},"(填任何一个现代 OS 即可)",[108,301438,301439,301441],{},[2627,301440,291966],{},"：目前主流的通用操作系统全部采用了多进程且每个进程支持多线程（M:N 架构）的设计模型。",[15,301443,301444],{},[2627,301445,301446],{},"4. It is necessary to ________ the activities of various threads so they do not interfere with each other or corrupt data structures.",[105,301448,301449,301454,301461],{},[108,301450,301451,301453],{},[2627,301452,291953],{},"：必须 ________ 各个线程的活动，以便它们不会相互干扰或破坏数据结构。",[108,301455,301456,2664,301458],{},[2627,301457,118575],{},[2627,301459,301460],{},"synchronize (同步)",[108,301462,301463,301465,301466,301469],{},[2627,301464,291966],{},"：由于线程共享内存，并发访问共享数据会引发竞态条件。必须采用互斥锁、信号量等机制对线程行为进行",[2627,301467,301468],{},"同步 (synchronize)","，保证数据一致性。",[15,301471,301472],{},[2627,301473,301474],{},"5. A process that cannot execute until some event occurs is said to be in the ________ state.",[105,301476,301477,301482,301489],{},[108,301478,301479,301481],{},[2627,301480,291953],{},"：在发生某事件之前无法执行的进程被称为处于 ________ 状态。",[108,301483,301484,2664,301486],{},[2627,301485,118575],{},[2627,301487,301488],{},"blocked (阻塞) \u002F waiting (等待)",[108,301490,301491,301493,301494,301497],{},[2627,301492,291966],{},"：当进程或线程请求 I\u002FO 操作或试图获取一把被占用的锁时，它将被 OS 挂起，交出 CPU 控制权，进入",[2627,301495,301496],{},"阻塞 (blocked)"," 状态，直到等待的事件完成。",[15,301499,301500],{},[2627,301501,301502],{},"6. The Clouds O\u002FS implements the concept of a thread as primarily an entity that can move among address spaces which represents the ________ Thread-to-Process relationship.",[105,301504,301505,301510,301517],{},[108,301506,301507,301509],{},[2627,301508,291953],{},"：Clouds 操作系统主要将线程的概念实现为可以在地址空间之间移动的实体，这代表了 ________ 线程对进程关系。",[108,301511,301512,2664,301514],{},[2627,301513,118575],{},[2627,301515,301516],{},"One-to-Many (1:M \u002F 一对多)",[108,301518,301519,301521],{},[2627,301520,291966],{},"：在分布式\u002F基于对象的操作系统（如 Clouds、Ra）中，一个执行线索（Thread）可以从一个对象的方法调用进入另一个对象，也就是穿梭于不同进程的地址空间中，形成**一（线程）对多（进程）**的模型。",[15,301523,301524],{},[2627,301525,301526],{},"7. In a ________ system, the kernel can execute on any processor, and typically each processor does self-scheduling from the pool of available processes or threads.",[105,301528,301529,301534,301541],{},[108,301530,301531,301533],{},[2627,301532,291953],{},"：在 ________ 系统中，内核可以在任何处理器上执行，并且通常每个处理器都从可用进程或线程池中进行自我调度。",[108,301535,301536,2664,301538],{},[2627,301537,118575],{},[2627,301539,301540],{},"Symmetric Multiprocessing (SMP \u002F 对称多处理)",[108,301542,301543,301545,301546,301549],{},[2627,301544,291966],{},"：区别于主从架构，在 ",[2627,301547,301548],{},"SMP"," 系统中所有 CPU 都是平等的，任何 CPU 都能执行操作系统内核代码，它们共享一个全局的就绪队列，自主进行调度和任务获取。",[15,301551,301552],{},[2627,301553,301554],{},"8. In most modern computer systems, processors generally have at least one level of ________ that is private to the processor.",[105,301556,301557,301562,301569],{},[108,301558,301559,301561],{},[2627,301560,291953],{},"：在大多数现代计算机系统中，处理器通常具有至少一级对处理器私有的 ________。",[108,301563,301564,2664,301566],{},[2627,301565,118575],{},[2627,301567,301568],{},"cache (缓存)",[108,301570,301571,301573],{},[2627,301572,291966],{},"：现代多核处理器中，L1 Cache（甚至 L2 Cache）通常是每个核心（Core）私有的，而 L3 Cache 则是多个核心共享的。这就引出了前文提到的缓存一致性问题。",[15,301575,301576],{},[2627,301577,301578],{},"9. With multiple active processes in an SMP system having potential access to shared address space or shared I\u002FO resources, care must be taken to provide effective ________.",[105,301580,301581,301586,301593],{},[108,301582,301583,301585],{},[2627,301584,291953],{},"：由于 SMP 系统中有多个活动进程潜在地访问共享地址空间或共享 I\u002FO 资源，因此必须小心提供有效的 ________。",[108,301587,301588,2664,301590],{},[2627,301589,118575],{},[2627,301591,301592],{},"synchronization (同步机制)",[108,301594,301595,301597,301598,301601],{},[2627,301596,291966],{},"：在真正的硬件并行（SMP）下，不仅同一进程的线程需要同步，整个 OS 内核级别的自旋锁（Spinlock）、互斥量等",[2627,301599,301600],{},"同步 (synchronization)"," 原语对于保护共享内核数据结构至关重要。",[15,301603,301604],{},[2627,301605,301606],{},"10. In the ________ O\u002FS architecture, functions are organized hierarchically and interaction only takes place between adjacent sections.",[105,301608,301609,301614,301621],{},[108,301610,301611,301613],{},[2627,301612,291953],{},"：在 ________ 操作系统架构中，功能是按层次化组织的，并且交互仅在相邻部分之间发生。",[108,301615,301616,2664,301618],{},[2627,301617,118575],{},[2627,301619,301620],{},"layered (分层)",[108,301622,301623,301625,301626,301629],{},[2627,301624,291966],{},"：这是早期的",[2627,301627,301628],{},"分层式设计 (Layered architecture)","（如 Multics 早期的某些概念）。上层仅依赖下层提供的服务，限制了各模块之间的无序调用，方便调试和维护，但严格的层次跨越会损害性能。",[15,301631,301632],{},[2627,301633,301634],{},"11. One advantage of the microkernel architecture is ________ allowing the addition of new services as well as the provision of multiple services in the same functional area.",[105,301636,301637,301642,301648],{},[108,301638,301639,301641],{},[2627,301640,291953],{},"：微内核架构的一个优点是 ________，允许添加新服务以及在同一功能区域提供多个服务。",[108,301643,301644,2664,301646],{},[2627,301645,118575],{},[2627,301647,298652],{},[108,301649,301650,301652,301653,347],{},[2627,301651,291966],{},"：因为微内核的绝大多数服务是以用户态进程（服务器）运行的，要支持一个新的文件系统或网络协议，只需要添加一个新的用户态服务器即可，不需要修改内核代码，这就是极佳的",[2627,301654,301655],{},"可扩展性 (extensibility)",[15,301657,301658],{},[2627,301659,301660],{},"12. The basic form of communication between processes or threads in a microkernel O\u002FS is ________.",[105,301662,301663,301668,301675],{},[108,301664,301665,301667],{},[2627,301666,291953],{},"：在微内核操作系统中，进程或线程之间基本的通信形式是 ________。",[108,301669,301670,2664,301672],{},[2627,301671,118575],{},[2627,301673,301674],{},"messages (或 message passing \u002F 消息)",[108,301676,301677,301679,301680,301683],{},[2627,301678,291966],{},"：所有系统服务请求（如客户端请求文件服务）在微内核中都被打包成",[2627,301681,301682],{},"消息 (messages)"," 进行发送和接收。",[15,301685,301686],{},[2627,301687,301688],{},"13. In a Linux system, if the process has been terminated but, for some reason, still must have its task structure in the process table is in the ________ state.",[105,301690,301691,301696,301703],{},[108,301692,301693,301695],{},[2627,301694,291953],{},"：在 Linux 系统中，如果进程已终止，但由于某种原因仍必须将其任务结构保存在进程表中，则它处于 ________ 状态。",[108,301697,301698,2664,301700],{},[2627,301699,118575],{},[2627,301701,301702],{},"zombie (僵尸)",[108,301704,301705,301707,301708,301710,301711,301714,301715,301718],{},[2627,301706,291966],{},"：当 Linux 进程终止后，内核会释放它的绝大部分资源（内存、文件），但必须保留一个空的骨架（退出码等在 ",[31,301709,242275],{}," 中），等待它的父进程调用 ",[31,301712,301713],{},"wait()"," 来收集。在这期间，它处于 ",[2627,301716,301717],{},"僵尸 (zombie)"," 状态。",[15,301720,301721],{},[2627,301722,301723],{},"14. In a Solaris system, a User-Level Thread (ULT) in the active state is assigned to a(n) ________ and executes while the underlying kernel thread executes.",[105,301725,301726,301731,301738],{},[108,301727,301728,301730],{},[2627,301729,291953],{},"：在 Solaris 系统中，处于活动状态的用户级线程 (ULT) 被分配给 ________，并在底层内核线程执行时执行。",[108,301732,301733,2664,301735],{},[2627,301734,118575],{},[2627,301736,301737],{},"LWP (Light-Weight Process \u002F 轻量级进程)",[108,301739,301740,301742,301743,301746],{},[2627,301741,291966],{},"：Solaris 线程库调度器将处于就绪态的 ULT 放置在一个活动队列中，然后将它们一对一地绑定到可用的 ",[2627,301744,301745],{},"LWP"," 虚拟处理器上执行。",[15,301748,301749],{},[2627,301750,301751],{},"15. In a Windows 2000 system, a process that has been selected to run next on a particular process moves from the Ready state to the ________ state.",[105,301753,301754,301759,301766],{},[108,301755,301756,301758],{},[2627,301757,291953],{},"：在 Windows 2000 系统中，被选择为要在特定处理器上接下来运行的线程（原题表述有微瑕，此处应指 thread 被调度）从就绪状态移动到 ________ 状态。",[108,301760,301761,2664,301763],{},[2627,301762,118575],{},[2627,301764,301765],{},"Standby (备用 \u002F 挂起待命)",[108,301767,301768,301770,301771,301774],{},[2627,301769,291966],{},"：在 Windows 操作系统的线程状态机中，当一个就绪（Ready）线程被调度器选中，准备在某颗 CPU 上执行，但在进行实际上下文切换的极短时间窗口内，它会处于 ",[2627,301772,301773],{},"备用 (Standby)"," 状态。一旦上下文切换完成，即变为运行态 (Running)。",{"title":29,"searchDepth":43,"depth":43,"links":301776},[301777,301778,301779],{"id":291940,"depth":43,"text":291941},{"id":292391,"depth":43,"text":292392},{"id":292902,"depth":43,"text":292903},{"order":56},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap4",{"title":300488,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap4","YW1nBabUxdDMZIsSO3zdS0zaw9gvlVqbD6xBFWTjJlM",{"id":301786,"title":301787,"body":301788,"description":29,"extension":85,"meta":303056,"navigation":52,"path":303057,"seo":303058,"stem":303059,"__hash__":303060},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap5.md","Chap. 5",{"type":8,"value":301789,"toc":303051},[301790,301794,301796,301801,301806,301812,301816,301828,301833,301838,301844,301848,301860,301865,301870,301876,301880,301892,301897,301902,301908,301912,301924,301929,301934,301940,301944,301960,301965,301970,301976,301980,302021,302026,302031,302037,302041,302053,302058,302063,302069,302073,302093,302098,302103,302109,302113,302125,302130,302135,302141,302145,302164,302169,302174,302180,302184,302196,302201,302206,302212,302216,302228,302233,302238,302244,302248,302264,302269,302274,302280,302284,302296,302301,302306,302312,302316,302332,302334,302336,302341,302346,302352,302357,302362,302367,302374,302379,302384,302389,302396,302408,302413,302418,302425,302430,302435,302440,302446,302451,302456,302461,302468,302473,302478,302483,302490,302495,302500,302505,302512,302517,302522,302527,302534,302539,302544,302549,302556,302560,302575,302580,302585,302592,302597,302602,302607,302614,302619,302624,302629,302635,302643,302648,302653,302660,302665,302673,302678,302683,302690,302699,302701,302703,302708,302713,302720,302725,302730,302735,302741,302746,302751,302756,302763,302772,302777,302782,302789,302794,302799,302804,302810,302815,302820,302825,302831,302836,302841,302846,302853,302858,302863,302868,302875,302880,302885,302890,302897,302906,302911,302916,302923,302935,302940,302945,302952,302960,302965,302970,302977,302982,302987,302992,302999,303008,303013,303018,303025,303030,303035,303040,303046],[11,301791,301793],{"id":301792},"chapter-5-concurrency-mutual-exclusion-and-synchronization","Chapter 5 - Concurrency: Mutual Exclusion and Synchronization",[19,301795,293369],{"id":293368},[15,301797,301798,301800],{},[2627,301799,115273],{},": Distributed processing can be defined as the management of multiple processes executing on multiple, distributed computer systems.",[15,301802,301803,301805],{},[2627,301804,293379],{},": 分布式处理可以被定义为对在多个分布式计算机系统上执行的多个进程的管理。",[15,301807,301808,8471,301810],{},[2627,301809,118575],{},[2627,301811,291961],{},[15,301813,301814,5598],{},[2627,301815,118582],{},[105,301817,301818,301823],{},[108,301819,301820,301822],{},[2627,301821,156287],{},": 并发（Concurrency）的上下文。",[108,301824,301825,301827],{},[2627,301826,293402],{},": 并发出现在三种不同的上下文中：多应用程序（Multiple applications）、结构化应用程序（Structured applications）和操作系统结构（OS structure）。而在硬件层面，它可以运行在单处理器、多处理器以及**分布式处理（Distributed processing）**系统中。分布式处理正是指管理在多个物理上分离的计算机节点上执行的多个进程。",[15,301829,301830,301832],{},[2627,301831,115295],{},": Both process interleaving and process overlapping are examples of concurrent processes and both present the same basic problems.",[15,301834,301835,301837],{},[2627,301836,293379],{},": 进程交替（interleaving）和进程重叠（overlapping）都是并发进程的例子，并且它们都呈现出相同的基本问题。",[15,301839,301840,8471,301842],{},[2627,301841,118575],{},[2627,301843,291961],{},[15,301845,301846,5598],{},[2627,301847,118582],{},[105,301849,301850,301855],{},[108,301851,301852,301854],{},[2627,301853,156287],{},": 并发的特性。",[108,301856,301857,301859],{},[2627,301858,293402],{},": 在单处理器系统中，并发表现为进程的交替执行（Interleaving）；在多处理器系统中，并发表现为进程的重叠执行（Overlapping）。尽管物理执行方式不同，但由于它们都可以以不可预测的相对速度运行并共享资源，因此它们面临着完全相同的基本同步和互斥问题。",[15,301861,301862,301864],{},[2627,301863,115317],{},": Concurrency issues are a concern on multiprocessor systems, but do not impact uniprocessor systems.",[15,301866,301867,301869],{},[2627,301868,293379],{},": 并发问题是多处理器系统中的一个隐患，但不会影响单处理器系统。",[15,301871,301872,8471,301874],{},[2627,301873,118575],{},[2627,301875,292009],{},[15,301877,301878,5598],{},[2627,301879,118582],{},[105,301881,301882,301887],{},[108,301883,301884,301886],{},[2627,301885,156287],{},": 单处理器中的并发。",[108,301888,301889,301891],{},[2627,301890,293402],{},": 即使在单处理器系统中，由于进程的交替执行（Interleaving）被中断机制打断，一个进程在更新共享变量的中间状态时可能会被挂起，另一个进程随之介入，这同样会导致严重的数据不一致和并发问题。",[15,301893,301894,301896],{},[2627,301895,115339],{},": Starvation refers to the situation where competing processes are denied access to a resource due to scheduling problems.",[15,301898,301899,301901],{},[2627,301900,293379],{},": 饥饿（Starvation）指的是竞争进程由于调度问题而被拒绝访问资源的情况。",[15,301903,301904,8471,301906],{},[2627,301905,118575],{},[2627,301907,291961],{},[15,301909,301910,5598],{},[2627,301911,118582],{},[105,301913,301914,301919],{},[108,301915,301916,301918],{},[2627,301917,156287],{},": 饥饿的定义。",[108,301920,301921,301923],{},[2627,301922,293402],{},": 当三个或更多进程竞争资源时，如果某两个或多个进程不断地互相交替获得资源，导致另一个处于等待状态的进程无限期地被调度器“忽略”，从而永远得不到执行机会，这种情况就称为饥饿。",[15,301925,301926,301928],{},[2627,301927,115391],{},": Any facility or capability that is to provide support for mutual exclusion must make certain assumptions about relative process speeds and the number of processors in the system.",[15,301930,301931,301933],{},[2627,301932,293379],{},": 任何提供互斥支持的设施或能力都必须对相对进程速度和系统中处理器的数量做出一定的假设。",[15,301935,301936,8471,301938],{},[2627,301937,118575],{},[2627,301939,292009],{},[15,301941,301942,5598],{},[2627,301943,118582],{},[105,301945,301946,301951],{},[108,301947,301948,301950],{},[2627,301949,156287],{},": 互斥（Mutual Exclusion）的必要条件。",[108,301952,301953,301955,301956,301959],{},[2627,301954,293402],{},": 教科书中明确列出了互斥机制必须满足的六个条件，其中之一就是：",[2627,301957,301958],{},"绝不能对相对的进程执行速度或处理器的数量做任何假设（No assumptions can be made about relative process speeds or number of processors）","。互斥机制必须在任何情况下都坚如磐石。",[15,301961,301962,301964],{},[2627,301963,115413],{},": Peterson's Algorithm for solving mutual exclusion is only valid for two processes and cannot be generalized to the case of n processes.",[15,301966,301967,301969],{},[2627,301968,293379],{},": 用于解决互斥问题的 Peterson 算法仅对两个进程有效，不能推广到 n 个进程的情况。",[15,301971,301972,8471,301974],{},[2627,301973,118575],{},[2627,301975,292009],{},[15,301977,301978,5598],{},[2627,301979,118582],{},[105,301981,301982,301987],{},[108,301983,301984,301986],{},[2627,301985,156287],{},": 软件互斥解决方案（Peterson's Algorithm）。",[108,301988,301989,301991,301992,302020],{},[2627,301990,293402],{},": Peterson 算法最初是为了优雅地解决两个进程的互斥问题而设计的，但它在理论上是可以被扩展并推广到解决 ",[34,301993,301995,302008],{"className":301994},[31642],[34,301996,301998],{"className":301997},[31646],[31648,301999,302000],{"xmlns":31650},[31652,302001,302002,302006],{},[31655,302003,302004],{},[31658,302005,31660],{},[31665,302007,31660],{"encoding":31667},[34,302009,302011],{"className":302010,"ariaHidden":758},[31672],[34,302012,302014,302017],{"className":302013},[31676],[34,302015],{"className":302016,"style":32527},[31680],[34,302018,31660],{"className":302019},[31685,31686]," 个进程的互斥问题的（尽管在实际现代系统中我们通常倾向于使用硬件原语或信号量，因为推广后的软件算法开销较大）。",[15,302022,302023,302025],{},[2627,302024,115435],{},": In a uniprocessor machine, concurrent processes cannot be overlapped; they can only be interleaved.",[15,302027,302028,302030],{},[2627,302029,293379],{},": 在单处理器机器中，并发进程不能重叠；它们只能交替执行。",[15,302032,302033,8471,302035],{},[2627,302034,118575],{},[2627,302036,291961],{},[15,302038,302039,5598],{},[2627,302040,118582],{},[105,302042,302043,302048],{},[108,302044,302045,302047],{},[2627,302046,156287],{},": 单处理器 vs 多处理器并发。",[108,302049,302050,302052],{},[2627,302051,293402],{},": 因为单处理器（Uniprocessor）在任何一个给定的物理时刻只能执行一条指令，所以多个进程只能通过时间片轮转进行上下文切换（交替执行），无法像在多核系统上那样真正地在同一时刻一起执行（重叠执行）。",[15,302054,302055,302057],{},[2627,302056,115457],{},": Weak semaphores guarantee freedom from starvation, but strong semaphores do not.",[15,302059,302060,302062],{},[2627,302061,293379],{},": 弱信号量（Weak semaphores）保证免于饥饿，但强信号量（strong semaphores）不能。",[15,302064,302065,8471,302067],{},[2627,302066,118575],{},[2627,302068,292009],{},[15,302070,302071,5598],{},[2627,302072,118582],{},[105,302074,302075,302080],{},[108,302076,302077,302079],{},[2627,302078,156287],{},": 信号量（Semaphores）的分类。",[108,302081,302082,302084,302085,302092],{},[2627,302083,293402],{},": 完全说反了。",[2627,302086,302087,302088,302091],{},"强信号量（Strong semaphores）",[2627,302089,302090],{},"要求被阻塞的进程队列遵循先进先出（FIFO）原则，因此保证了排队的进程最终一定会得到资源，从而","避免饥饿","。而**弱信号量（Weak semaphores）**对阻塞队列的唤醒顺序没有规定，因此可能会导致某个进程被无限期滞留（发生饥饿）。",[15,302094,302095,302097],{},[2627,302096,115479],{},": A finite circular buffer and an infinite buffer are two ways to implement a data storage area for the classic Producer\u002FConsumer Problem.",[15,302099,302100,302102],{},[2627,302101,293379],{},": 有限循环缓冲区和无限缓冲区是为经典的生产者\u002F消费者问题实现数据存储区的两种方法。",[15,302104,302105,8471,302107],{},[2627,302106,118575],{},[2627,302108,291961],{},[15,302110,302111,5598],{},[2627,302112,118582],{},[105,302114,302115,302120],{},[108,302116,302117,302119],{},[2627,302118,156287],{},": 生产者\u002F消费者问题（Producer\u002FConsumer Problem）。",[108,302121,302122,302124],{},[2627,302123,293402],{},": 教科书中在探讨生产者\u002F消费者问题时，针对共享缓冲区的设计给出了两种经典的理论场景：一种是无限大缓冲区（不需要考虑缓冲区满的问题），另一种是更符合实际情况的有限循环缓冲区（需要同时处理缓冲区空和缓冲区满的同步问题）。",[15,302126,302127,302129],{},[2627,302128,115501],{},": The major difficulty with semaphores is that wait and signal operations may be scattered throughout a program and it is difficult to see the overall effect of these operations on the semaphores they affect.",[15,302131,302132,302134],{},[2627,302133,293379],{},": 信号量的主要困难在于 wait 和 signal 操作可能散布在整个程序中，很难看出这些操作对它们所影响的信号量的整体效果。",[15,302136,302137,8471,302139],{},[2627,302138,118575],{},[2627,302140,291961],{},[15,302142,302143,5598],{},[2627,302144,118582],{},[105,302146,302147,302152],{},[108,302148,302149,302151],{},[2627,302150,156287],{},": 信号量的缺点与管程（Monitors）的引入动机。",[108,302153,302154,302156,302157,302159,302160,302163],{},[2627,302155,293402],{},": 信号量非常强大且灵活，但这也是它的致命弱点。程序员必须小心翼翼地在代码的各个地方配对使用 ",[31,302158,285167],{},"（P操作）和 ",[31,302161,302162],{},"signal","（V操作），一旦漏掉一个或顺序写反，就会导致死锁或互斥失效。这种分散性使得程序极难调试和证明正确性，这也促使了**管程（Monitor）**这种更高级的结构化同步机制的诞生。",[15,302165,302166,302168],{},[2627,302167,115523],{},": Message passing provides both synchronization and communication, which are fundamental requirements for interacting processes.",[15,302170,302171,302173],{},[2627,302172,293379],{},": 消息传递（Message passing）同时提供了同步和通信功能，这是交互进程的基本要求。",[15,302175,302176,8471,302178],{},[2627,302177,118575],{},[2627,302179,291961],{},[15,302181,302182,5598],{},[2627,302183,118582],{},[105,302185,302186,302191],{},[108,302187,302188,302190],{},[2627,302189,156287],{},": 消息传递（Message Passing）。",[108,302192,302193,302195],{},[2627,302194,293402],{},": 消息传递机制非常优雅，它将互斥（通过阻塞接收等同步机制实现）和数据交换（通信）合二为一，使得它在分布式系统、共享内存多处理器和单处理器系统中都非常适用。",[15,302197,302198,302200],{},[2627,302199,115545],{},": In a message passing system, one queuing discipline alternative is to allow the receiver to inspect the message queue and select which message to receive next.",[15,302202,302203,302205],{},[2627,302204,293379],{},": 在消息传递系统中，一种队列规则的替代方案是允许接收者检查消息队列并选择下一步接收哪条消息。",[15,302207,302208,8471,302210],{},[2627,302209,118575],{},[2627,302211,291961],{},[15,302213,302214,5598],{},[2627,302215,118582],{},[105,302217,302218,302223],{},[108,302219,302220,302222],{},[2627,302221,156287],{},": 消息队列的管理。",[108,302224,302225,302227],{},[2627,302226,293402],{},": 虽然最简单的队列规则是 FIFO（先进先出），但许多消息传递系统允许基于优先级或者消息类型进行选择性接收（Selective Receipt）。接收者可以检查邮箱\u002F队列，并根据特定的条件（如发件人ID或消息标签）提取消息。",[15,302229,302230,302232],{},[2627,302231,115567],{},": In the communications mechanism of a message passing system, only the receiver of the communication can be blocking.",[15,302234,302235,302237],{},[2627,302236,293379],{},": 在消息传递系统的通信机制中，只有通信的接收者可以被阻塞（blocking）。",[15,302239,302240,8471,302242],{},[2627,302241,118575],{},[2627,302243,292009],{},[15,302245,302246,5598],{},[2627,302247,118582],{},[105,302249,302250,302255],{},[108,302251,302252,302254],{},[2627,302253,156287],{},": 消息传递的同步类型。",[108,302256,302257,302259,302260,302263],{},[2627,302258,293402],{},": 消息传递分为阻塞（Blocking）和非阻塞（Non-blocking）。不仅仅是接收者可以被阻塞（等待消息到来），",[2627,302261,302262],{},"发送者同样也可以被阻塞","（例如：Blocking Send要求发送者阻塞，直到接收者确实收到了该消息）。",[15,302265,302266,302268],{},[2627,302267,115589],{},": In indirect addressing, as applied to message passing, messages are sent to a temporary shared data structure typically known as a mailbox.",[15,302270,302271,302273],{},[2627,302272,293379],{},": 在应用于消息传递的间接寻址中，消息被发送到一个通常被称为邮箱（mailbox）的临时共享数据结构中。",[15,302275,302276,8471,302278],{},[2627,302277,118575],{},[2627,302279,291961],{},[15,302281,302282,5598],{},[2627,302283,118582],{},[105,302285,302286,302291],{},[108,302287,302288,302290],{},[2627,302289,156287],{},": 消息传递的寻址方式（Indirect Addressing）。",[108,302292,302293,302295],{},[2627,302294,293402],{},": 与直接寻址（明确指定目标进程ID）不同，间接寻址（Indirect Addressing）解耦了发送方和接收方。发送方将消息发往一个称为**邮箱（Mailbox）或端口（Port）**的共享结构，接收方从中读取，这支持了一对多、多对一等复杂的通信模式。",[15,302297,302298,302300],{},[2627,302299,115611],{},": The Producer\u002FConsumer problem is typically considered a special case of the Readers\u002FWrites problem, with only one reader and one writer.",[15,302302,302303,302305],{},[2627,302304,293379],{},": 生产者\u002F消费者问题通常被认为是读者\u002F写者问题的一个特例，即只有一个读者和一个写者。",[15,302307,302308,8471,302310],{},[2627,302309,118575],{},[2627,302311,292009],{},[15,302313,302314,5598],{},[2627,302315,118582],{},[105,302317,302318,302323],{},[108,302319,302320,302322],{},[2627,302321,156287],{},": 读者\u002F写者问题与生产者\u002F消费者问题的区别。",[108,302324,302325,302327,302328,302331],{},[2627,302326,293402],{},": 它们有本质区别。在读者\u002F写者问题中，读者仅仅是“读取”数据，不会改变数据，因此允许多个读者同时访问。而在生产者\u002F消费者问题中，消费者读取数据的同时实际上是将数据从缓冲区中“消耗（删除）”并修改指针，这是一种",[2627,302329,302330],{},"写操作（写入状态）","。因此，不能将消费者简单等同于不改变数据的“读者”。",[32188,302333],{},[19,302335,293916],{"id":293915},[15,302337,302338,302340],{},[2627,302339,115273],{},": Concurrency plays a major part in which of the following specific contexts:\na. Multiple applications\nb. Structured applications\nc. O\u002FS structure\nd. All of the above",[15,302342,302343,302345],{},[2627,302344,293379],{},": 并发在以下哪些特定的上下文中扮演着主要角色：\na. 多应用程序（Multiple applications）\nb. 结构化应用程序（Structured applications）\nc. 操作系统结构（O\u002FS structure）\nd. 以上所有",[15,302347,302348,8471,302350],{},[2627,302349,118575],{},[2627,302351,292510],{},[15,302353,302354,302356],{},[2627,302355,118582],{},": 根据教科书第一节“并发原理”，并发的概念适用于这三种上下文：允许多个应用程序分时运行、将单个应用程序设计为一组并发进程以实现模块化（Structured applications），以及将操作系统本身实现为一组并发的系统进程。",[15,302358,302359,302361],{},[2627,302360,115295],{},": Examples of solutions to the concurrency problem that do not involve busy waiting are the following:\na. Semaphores and monitors\nb. Message passing and caching\nc. Producers and consumers\nd. None of the above",[15,302363,302364,302366],{},[2627,302365,293379],{},": 以下是不涉及忙等待（busy waiting）的并发问题解决方案的例子：\na. 信号量和管程（Semaphores and monitors）\nb. 消息传递和缓存（Message passing and caching）\nc. 生产者和消费者（Producers and consumers）\nd. 以上都不是",[15,302368,302369,8471,302371],{},[2627,302370,118575],{},[2627,302372,302373],{},"a. Semaphores and monitors",[15,302375,302376,302378],{},[2627,302377,118582],{},": 忙等待（如 Peterson 算法或基于自旋锁的硬件指令）会白白浪费 CPU 周期。而信号量（Semaphores）和管程（Monitors）通常基于操作系统的阻塞\u002F唤醒机制实现，如果进程无法进入临界区，它会被挂起放入等待队列，从而避免了忙等待。",[15,302380,302381,302383],{},[2627,302382,115317],{},": A basic echo procedure (that echoes a typed character to the screen) running on a multiprocessor system can produce erroneous output if:\na. Two processes deadlock while in the echo code\nb. Access to the echo procedure is unsynchronized\nc. Access to the echo procedure is synchronized\nd. None of the above",[15,302385,302386,302388],{},[2627,302387,293379],{},": 在多处理器系统上运行的基本回显程序（将键入的字符回显到屏幕）如果在以下情况下会产生错误的输出：\na. 两个进程在回显代码中发生死锁\nb. 对回显程序的访问未同步（unsynchronized）\nc. 对回显程序的访问已同步（synchronized）\nd. 以上都不是",[15,302390,302391,8471,302393],{},[2627,302392,118575],{},[2627,302394,302395],{},"b. Access to the echo procedure is unsynchronized",[15,302397,302398,302400,302401,302404,302405,302407],{},[2627,302399,118582],{},": 教科书中用了一个经典的 ",[31,302402,302403],{},"echo"," 程序的例子。如果两个进程并发且未同步地访问同一个 ",[31,302406,302403],{}," 函数（共享全局变量或输入\u002F输出缓冲区），它们的指令可能会交错执行，导致字符被覆盖或打印混乱的错误输出。",[15,302409,302410,302412],{},[2627,302411,115339],{},": In order to implement mutual exclusion on a critical resource for competing processes, only one program at a time should be allowed:\na. In the critical section of the program\nb. To perform message passing\nc. To Exhibit cooperation\nd. None of the above",[15,302414,302415,302417],{},[2627,302416,293379],{},": 为了对竞争进程的关键资源实施互斥，在同一时间只应允许一个程序：\na. 处于程序的临界区（critical section）中\nb. 执行消息传递\nc. 展现协作\nd. 以上都不是",[15,302419,302420,8471,302422],{},[2627,302421,118575],{},[2627,302423,302424],{},"a. In the critical section of the program",[15,302426,302427,302429],{},[2627,302428,118582],{},": 临界区（Critical Section）指的是访问共享资源的那段代码。互斥的核心原则就是确保任何时刻最多只能有一个进程在执行它自己的那段临界区代码。",[15,302431,302432,302434],{},[2627,302433,115391],{},": The following requirement must be met by any facility or capability that is to provide support for mutual exclusion:\na. Only one process at a time can be allowed into a critical code section\nb. A process remains in its critical code section for a finite time only\nc. No assumptions can be made about relative process speeds\nd. All of the above",[15,302436,302437,302439],{},[2627,302438,293379],{},": 任何提供互斥支持的设施或能力都必须满足以下要求：\na. 一次只允许一个进程进入关键代码区\nb. 进程在其关键代码区中只停留有限的时间\nc. 不能对相对进程速度做任何假设\nd. 以上所有",[15,302441,302442,8471,302444],{},[2627,302443,118575],{},[2627,302445,292510],{},[15,302447,302448,302450],{},[2627,302449,118582],{},": 互斥的六大基本要求中包含了：必须强制互斥（a）、不能产生死锁或饥饿（意味着进程只能在临界区停留有限时间 b）、以及不能对处理器速度做假设（c）。故全选。",[15,302452,302453,302455],{},[2627,302454,115413],{},": Processes that are designed to be able to pass execution control back and forth between themselves are referred to as:\na. Threads\nb. Coroutines\nc. Busy waiting processes\nd. None of the above",[15,302457,302458,302460],{},[2627,302459,293379],{},": 被设计为能够在彼此之间来回传递执行控制权的进程被称为：\na. 线程（Threads）\nb. 协程（Coroutines）\nc. 忙等待进程（Busy waiting processes）\nd. 以上都不是",[15,302462,302463,8471,302465],{},[2627,302464,118575],{},[2627,302466,302467],{},"b. Coroutines",[15,302469,302470,302472],{},[2627,302471,118582],{},": 协程（Coroutine）是一种程序组件，它们通过显式的移交控制权（如 yield 操作）在彼此之间协作切换。教科书在探讨并发机制（特别是消息传递或单线程内的并发逻辑）时提到了这种协作模型。",[15,302474,302475,302477],{},[2627,302476,115435],{},": In a uniprocessor system, mutual exclusion can be guaranteed by:\na. Overlapping processes\nb. Interleaving processes\nc. Disabling interrupts\nd. All of the above",[15,302479,302480,302482],{},[2627,302481,293379],{},": 在单处理器系统中，互斥可以通过以下方式来保证：\na. 重叠进程\nb. 交替进程\nc. 禁用中断（Disabling interrupts）\nd. 以上所有",[15,302484,302485,8471,302487],{},[2627,302486,118575],{},[2627,302488,302489],{},"c. Disabling interrupts",[15,302491,302492,302494],{},[2627,302493,118582],{},": 在单处理器上，并发进程仅仅是通过被时钟中断等机制打断来实现交替执行的。如果在进程进入临界区前关闭系统中断（Disabling interrupts），那么该进程就不会被抢占，从而简单粗暴地保证了互斥（但不适用于多核系统，且代价高昂）。",[15,302496,302497,302499],{},[2627,302498,115457],{},": A semaphore that does not specify the order in which processes are removed from the queue is called a:\na. Weak semaphore\nb. Strong semaphore\nc. Binary semaphore\nd. None of the above",[15,302501,302502,302504],{},[2627,302503,293379],{},": 未指定进程从队列中移除的顺序的信号量被称为：\na. 弱信号量（Weak semaphore）\nb. 强信号量（Strong semaphore）\nc. 二元信号量（Binary semaphore）\nd. 以上都不是",[15,302506,302507,8471,302509],{},[2627,302508,118575],{},[2627,302510,302511],{},"a. Weak semaphore",[15,302513,302514,302516],{},[2627,302515,118582],{},": 如前述判断题所述，不规定唤醒策略（如不保证 FIFO）的信号量称为弱信号量（Weak semaphore），可能导致饥饿。",[15,302518,302519,302521],{},[2627,302520,115479],{},": The finite circular buffer is used to implement which of the following basic queuing strategies:\na. FILO\nb. LIFO\nc. FIFO\nd. None of the above",[15,302523,302524,302526],{},[2627,302525,293379],{},": 有限循环缓冲区用于实现以下哪种基本的队列策略：\na. 先进后出（FILO）\nb. 后进先出（LIFO）\nc. 先进先出（FIFO）\nd. 以上都不是",[15,302528,302529,8471,302531],{},[2627,302530,118575],{},[2627,302532,302533],{},"c. FIFO",[15,302535,302536,302538],{},[2627,302537,118582],{},": 在生产者\u002F消费者模型中，有限循环缓冲区通过头指针（消费者读）和尾指针（生产者写）进行管理。最先被生产进入缓冲区的数据项总是最先被消费者取出，因此它实现的是先进先出（FIFO）队列。",[15,302540,302541,302543],{},[2627,302542,115501],{},": A chief characteristic of a monitor is:\na. A maximum of two processes may be executing in a monitor at a time\nb. Local data variables of the monitor are accessible by any procedure requesting use of the monitor\nc. A process enters the monitor by invoking one of its procedures\nd. All of the above",[15,302545,302546,302548],{},[2627,302547,293379],{},": 管程（monitor）的主要特征是：\na. 最多可以有两个进程同时在管程中执行\nb. 请求使用管程的任何过程都可以访问管程的局部数据变量\nc. 进程通过调用管程的某个过程（procedure）来进入管程\nd. 以上所有",[15,302550,302551,8471,302553],{},[2627,302552,118575],{},[2627,302554,302555],{},"c. A process enters the monitor by invoking one of its procedures",[15,302557,302558,5598],{},[2627,302559,118582],{},[105,302561,302562,302565,302572],{},[108,302563,302564],{},"a 错误：管程的本质特性是任何时刻**最多只允许一个（ONE）**进程在管程内部执行。",[108,302566,302567,302568,302571],{},"b 错误：管程的局部数据变量是",[2627,302569,302570],{},"私有","的，外部无法直接访问，只能通过管程提供的入口过程来操作。",[108,302573,302574],{},"c 正确：进程只有通过调用管程封装好、暴露给外部的特定入口函数（procedures）才能进入管程。",[15,302576,302577,302579],{},[2627,302578,115523],{},": In synchronization involving message passing, the sender of a message can be:\na. Either blocking or non-blocking\nb. Only blocking\nc. Only non-blocking\nd. All of the above",[15,302581,302582,302584],{},[2627,302583,293379],{},": 在涉及消息传递的同步中，消息的发送者可以是：\na. 阻塞的或非阻塞的（Either blocking or non-blocking）\nb. 仅阻塞的\nc. 仅非阻塞的\nd. 以上所有",[15,302586,302587,8471,302589],{},[2627,302588,118575],{},[2627,302590,302591],{},"a. Either blocking or non-blocking",[15,302593,302594,302596],{},[2627,302595,118582],{},": 消息传递的发送原语（Send Primitive）有两种形式：阻塞发送（发送方挂起直到消息被接收方提取）和非阻塞发送（发送方把消息丢给系统后立刻继续执行后续代码）。",[15,302598,302599,302601],{},[2627,302600,115545],{},": In a system employing message passing, when a message is sent to a shared temporary data structure, this general approach is known as:\na. Direct addressing\nb. Indirect addressing\nc. Blocking\nd. None of the above",[15,302603,302604,302606],{},[2627,302605,293379],{},": 在采用消息传递的系统中，当消息被发送到一个共享的临时数据结构时，这种通用方法被称为：\na. 直接寻址\nb. 间接寻址（Indirect addressing）\nc. 阻塞\nd. 以上都不是",[15,302608,302609,8471,302611],{},[2627,302610,118575],{},[2627,302612,302613],{},"b. Indirect addressing",[15,302615,302616,302618],{},[2627,302617,118582],{},": 发送到一个作为中介的临时数据结构（如邮箱 Mailbox），而不是直接指向目标进程的进程标识符，这就是间接寻址（Indirect Addressing）的定义。",[15,302620,302621,302623],{},[2627,302622,115567],{},": In a system employing message passing, the typical message is divided into two primary sections:\na. Header and mailbox\nb. Body and mailbox\nc. Destination ID and Source ID\nd. None of the above",[15,302625,302626,302628],{},[2627,302627,293379],{},": 在采用消息传递的系统中，典型的消息分为两个主要部分：\na. 头部和邮箱\nb. 正文和邮箱\nc. 目的地 ID 和源 ID\nd. 以上都不是",[15,302630,302631,8471,302633],{},[2627,302632,118575],{},[2627,302634,292848],{},[15,302636,302637,302639,302640,302642],{},[2627,302638,118582],{},": 在典型的消息传递模型中，一条消息通常由**首部（Header）",[2627,302641,119123],{},"主体（Body）**两部分组成。Header 包含了控制信息（如目标 ID、源 ID、消息长度、控制类型），Body 包含了实际的有效载荷数据。由于选项中没有提供 \"Header and Body\" 这一组合，故选 d。",[15,302644,302645,302647],{},[2627,302646,115589],{},": The Reader\u002FWriter problem requires that certain conditions be satisfied, such as:\na. Readers may read from the file while writers are writing to it\nb. Multiple writers may write to the file simultaneously\nc. Any number of readers may simultaneously read from the file\nd. None of the above",[15,302649,302650,302652],{},[2627,302651,293379],{},": 读者\u002F写者问题要求满足某些条件，例如：\na. 读者可以在写者写入文件时读取该文件\nb. 多个写者可以同时向文件写入\nc. 任意数量的读者可以同时读取该文件\nd. 以上都不是",[15,302654,302655,8471,302657],{},[2627,302656,118575],{},[2627,302658,302659],{},"c. Any number of readers may simultaneously read from the file",[15,302661,302662,302664],{},[2627,302663,118582],{},": 读者\u002F写者问题的基本约束条件是：",[2230,302666,302667,302670],{},[108,302668,302669],{},"允许多个读者（Any number of readers）同时读取共享数据区（因此 c 正确）。",[108,302671,302672],{},"一次只允许一个写者（排斥其他任何写者和读者，因此 a 和 b 错误）。",[15,302674,302675,302677],{},[2627,302676,115611],{},": A reason why the Producer\u002FConsumer problem cannot be considered a special case of the Reader\u002FWriter problem with a single writer (the producer) and a single reader (the consumer) is:\na. The producer and consumer must be both reader and writer\nb. The consumer must perform writes while the reader performs reads\nc. The Producer\u002FConsumer problem doesn't deal with concurrency issues\nd. None of the above",[15,302679,302680,302682],{},[2627,302681,293379],{},": 为什么不能将生产者\u002F消费者问题视为拥有单个写者（生产者）和单个读者（消费者）的读者\u002F写者问题的特例，其原因是：\na. 生产者和消费者都必须同时是读者和写者\nb. 消费者必须执行写操作，而读者执行读操作\nc. 生产者\u002F消费者问题不涉及并发问题\nd. 以上都不是",[15,302684,302685,8471,302687],{},[2627,302686,118575],{},[2627,302688,302689],{},"a. The producer and consumer must be both reader and writer",[15,302691,302692,302694,302695,302698],{},[2627,302693,118582],{},": 在缓冲区的操作中，消费者看似在“读取（消费）”数据，但实际上它必须调整队列的队头指针（或计数器），这是一次对共享状态的",[2627,302696,302697],{},"写操作","。同理，生产者放入数据也是写操作。由于双方都在修改（Write）共享控制变量，这打破了单纯的“读者不修改任何状态”的前提，故 a 解释了这一本质差别。",[32188,302700],{},[19,302702,294449],{"id":294448},[15,302704,302705,302707],{},[2627,302706,115273],{},": Many approaches to achieving mutual exclusion are software solutions that require the use of a technique called __________.",[15,302709,302710,302712],{},[2627,302711,293379],{},": 实现互斥的许多方法都是软件解决方案，这些方案要求使用一种称为 __________ 的技术。",[15,302714,302715,8471,302717],{},[2627,302716,118575],{},[2627,302718,302719],{},"busy waiting (或 spin waiting)",[15,302721,302722,302724],{},[2627,302723,118582],{},": 早期的纯软件算法（如 Dekker 算法、Peterson 算法）为了探测另一个进程是否已经退出临界区，必须在一个 while 循环中不断测试某个变量。这种不让出 CPU 而是持续消耗执行周期的技术叫做忙等待（Busy waiting）。",[15,302726,302727,302729],{},[2627,302728,115295],{},": The basic requirement for support of concurrent process is the ability to enforce __________.",[15,302731,302732,302734],{},[2627,302733,293379],{},": 支持并发进程的基本要求是能够强制执行 __________。",[15,302736,302737,8471,302739],{},[2627,302738,118575],{},[2627,302740,297596],{},[15,302742,302743,302745],{},[2627,302744,118582],{},": 互斥（Mutual Exclusion）是所有并发控制理论的基石。没有互斥，就无法保护共享状态免受由于交替执行引发的数据竞争破坏。",[15,302747,302748,302750],{},[2627,302749,115317],{},": In order to protect shared variables (and other shared global resources) the system must control the __________.",[15,302752,302753,302755],{},[2627,302754,293379],{},": 为了保护共享变量（以及其他共享全局资源），系统必须控制 __________。",[15,302757,302758,8471,302760],{},[2627,302759,118575],{},[2627,302761,302762],{},"access (或 critical sections)",[15,302764,302765,302767,302768,302771],{},[2627,302766,118582],{},": 系统必须严格控制对这些共享资源的",[2627,302769,302770],{},"访问（access）","，通常通过确保只有处于**临界区（critical sections）**的单个进程才能进行访问来实现。",[15,302773,302774,302776],{},[2627,302775,115339],{},": The situation where Process 1 (P1) holds Resource 1 (R1), while P2 holds R2, and P1 needs R2 to complete and P2 needs R1 to complete is referred to as __________.",[15,302778,302779,302781],{},[2627,302780,293379],{},": 进程 1（P1）持有资源 1（R1），同时 P2 持有 R2，并且 P1 需要 R2 才能完成，而 P2 需要 R1 才能完成的情况被称为 __________。",[15,302783,302784,8471,302786],{},[2627,302785,118575],{},[2627,302787,302788],{},"deadlock",[15,302790,302791,302793],{},[2627,302792,118582],{},": 这是经典的死锁（Deadlock）状态的定义：两个或多个进程互相持有对方需要的资源，形成循环等待，导致系统永远停滞。",[15,302795,302796,302798],{},[2627,302797,115391],{},": When only one process is allowed in its critical code section at a time, then __________ is enforced.",[15,302800,302801,302803],{},[2627,302802,293379],{},": 当一次只允许一个进程进入其关键代码区时，这就强制执行了 __________。",[15,302805,302806,8471,302808],{},[2627,302807,118575],{},[2627,302809,297596],{},[15,302811,302812,302814],{},[2627,302813,118582],{},": 这是互斥（Mutual Exclusion）的标准定义。",[15,302816,302817,302819],{},[2627,302818,115413],{},": The technique in which a process can do nothing until it gets permission to enter its critical section but continues to test the appropriate variable to gain entrance is called __________.",[15,302821,302822,302824],{},[2627,302823,293379],{},": 进程在获得进入临界区的许可之前什么也做不了，但仍继续测试适当的变量以试图进入的技术称为 __________。",[15,302826,302827,8471,302829],{},[2627,302828,118575],{},[2627,302830,302719],{},[15,302832,302833,302835],{},[2627,302834,118582],{},": 与第一题类似，这种在一个死循环中不断询问“我可以进去了吗？”的行为就是忙等待（Busy waiting）。",[15,302837,302838,302840],{},[2627,302839,115435],{},": In multiprocessor configurations, special machine instructions that carry out two actions in a single instruction cycle are said to do so __________.",[15,302842,302843,302845],{},[2627,302844,293379],{},": 在多处理器配置中，在单个指令周期中执行两个操作的特殊机器指令被认为是以 __________ 方式执行的。",[15,302847,302848,8471,302850],{},[2627,302849,118575],{},[2627,302851,302852],{},"atomically",[15,302854,302855,302857],{},[2627,302856,118582],{},": 硬件互斥机制（如 Test-and-Set 或 Exchange 指令）之所以能在多核下工作，是因为它们被硬件设计为**原子地（atomically）**执行。即读取和写入在同一个不可分割的周期内完成，不会被其他处理器打断。",[15,302859,302860,302862],{},[2627,302861,115457],{},": A semaphore whose definition includes the FIFO policy for releasing blocked processes from the queue is called a __________.",[15,302864,302865,302867],{},[2627,302866,293379],{},": 包含从队列中释放阻塞进程的 FIFO 策略的信号量定义被称为 __________。",[15,302869,302870,8471,302872],{},[2627,302871,118575],{},[2627,302873,302874],{},"strong semaphore",[15,302876,302877,302879],{},[2627,302878,118582],{},": 强制使用先进先出（FIFO）队列以防止饥饿的信号量被归类为强信号量（Strong Semaphore）。",[15,302881,302882,302884],{},[2627,302883,115479],{},": The Barbershop Problem uses __________ to implement concurrency.",[15,302886,302887,302889],{},[2627,302888,293379],{},": 理发店问题（Barbershop Problem）使用 __________ 来实现并发控制。",[15,302891,302892,8471,302894],{},[2627,302893,118575],{},[2627,302895,302896],{},"semaphores",[15,302898,302899,302901,302902,302905],{},[2627,302900,118582],{},": 理发店问题是经典的同步问题之一，教科书在附录等章节中通常用",[2627,302903,302904],{},"信号量（semaphores）","（包括控制等待座椅数量、理发师就绪状态等信号量）来演示其解决方案。",[15,302907,302908,302910],{},[2627,302909,115501],{},": A monitor supports __________ by the use of condition variables that are contained within the monitor and accessible only within the monitor.",[15,302912,302913,302915],{},[2627,302914,293379],{},": 管程通过使用包含在管程内且仅在管程内可访问的条件变量（condition variables）来支持 __________。",[15,302917,302918,8471,302920],{},[2627,302919,118575],{},[2627,302921,302922],{},"synchronization",[15,302924,302925,302927,302928,347],{},[2627,302926,118582],{},": 管程本身利用管程锁（互斥锁）提供了互斥功能，而它额外引入",[2627,302929,302930,302931,302934],{},"条件变量（Condition variables，带有 wait 和 signal 操作）",[2627,302932,302933],{},"则是为了支持进程间的","同步（synchronization）",[15,302936,302937,302939],{},[2627,302938,115523],{},": A blocking send, blocking receive message passing scenario is sometimes referred to as a __________.",[15,302941,302942,302944],{},[2627,302943,293379],{},": 阻塞发送、阻塞接收的消息传递场景有时被称为 __________。",[15,302946,302947,8471,302949],{},[2627,302948,118575],{},[2627,302950,302951],{},"rendezvous",[15,302953,302954,302956,302957,347],{},[2627,302955,118582],{},": 当发送者必须等待接收者，且接收者必须等待发送者，两方在消息成功交接的那一刻在时间上实现了精确对齐，这种强同步的情况被称为",[2627,302958,302959],{},"汇合（rendezvous）",[15,302961,302962,302964],{},[2627,302963,115545],{},": The shared data structures that temporarily hold messages in a message passing system employing indirect addressing are generally referred to as __________.",[15,302966,302967,302969],{},[2627,302968,293379],{},": 在采用间接寻址的消息传递系统中，临时保存消息的共享数据结构通常被称为 __________。",[15,302971,302972,8471,302974],{},[2627,302973,118575],{},[2627,302975,302976],{},"mailboxes (或 ports)",[15,302978,302979,302981],{},[2627,302980,118582],{},": 间接通信需要一个中间媒介存放消息，最常见的叫法就是邮箱（mailboxes）或端口（ports）。",[15,302983,302984,302986],{},[2627,302985,115567],{},": In the __________ addressing implementation of message passing, the \"send\" primitive includes a specific identifier of the destination process.",[15,302988,302989,302991],{},[2627,302990,293379],{},": 在消息传递的 __________ 寻址实现中，“发送（send）”原语包含目标进程的特定标识符。",[15,302993,302994,8471,302996],{},[2627,302995,118575],{},[2627,302997,302998],{},"direct",[15,303000,303001,303003,303004,303007],{},[2627,303002,118582],{},": 如果你需要指名道姓地把消息发给目标（如 ",[31,303005,303006],{},"send(P, message)","），这被称为直接寻址（direct addressing）。",[15,303009,303010,303012],{},[2627,303011,115589],{},": The classic concurrency problem that involves readers and writers that can both read from and write to a shared data area is the __________ Problem.",[15,303014,303015,303017],{},[2627,303016,293379],{},": 涉及都可以读取和写入共享数据区的读者和写者的经典并发问题是 __________ 问题。",[15,303019,303020,8471,303022],{},[2627,303021,118575],{},[2627,303023,303024],{},"Readers\u002FWriters",[15,303026,303027,303029],{},[2627,303028,118582],{},": 探讨数据库或文件共享时，将进程分为只读和只写两大类的经典模型称为读者\u002F写者问题（Readers\u002FWriters Problem）。",[15,303031,303032,303034],{},[2627,303033,115611],{},": The classic concurrency problem that involves multiple readers that can read from a shared data area when no single writer is exclusively writing to it is the __________ Problem.",[15,303036,303037,303039],{},[2627,303038,293379],{},": 涉及当没有单一写者专门对其进行写入时、多个读者可以从共享数据区中进行读取的经典并发问题是 __________ 问题。",[15,303041,303042,8471,303044],{},[2627,303043,118575],{},[2627,303045,303024],{},[15,303047,303048,303050],{},[2627,303049,118582],{},": 这个问题和上一个是同一个术语（仅仅是对条件的进一步细化描述）。它仍然是指读者\u002F写者问题（Readers\u002FWriters Problem）。",{"title":29,"searchDepth":43,"depth":43,"links":303052},[303053,303054,303055],{"id":293368,"depth":43,"text":293369},{"id":293915,"depth":43,"text":293916},{"id":294448,"depth":43,"text":294449},{"order":62},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap5",{"title":301787,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap5","HJPtZiDxu_c0z96_qqQwx-IeKe8OFrW0qGvR-ghU-SM",{"id":303062,"title":303063,"body":303064,"description":29,"extension":85,"meta":304087,"navigation":52,"path":304088,"seo":304089,"stem":304090,"__hash__":304091},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap6.md","Chap. 6",{"type":8,"value":303065,"toc":304082},[303066,303070,303074,303079,303110,303115,303137,303142,303163,303168,303194,303199,303220,303225,303246,303251,303272,303277,303303,303308,303329,303334,303355,303360,303381,303386,303407,303412,303433,303438,303459,303464,303490,303492,303496,303502,303519,303525,303541,303547,303564,303570,303587,303593,303610,303616,303632,303638,303654,303660,303676,303682,303698,303704,303720,303726,303742,303748,303764,303770,303786,303792,303808,303814,303830,303832,303836,303841,303853,303858,303870,303875,303887,303892,303904,303909,303921,303926,303943,303948,303960,303965,303982,303987,304004,304009,304021,304026,304043,304048,304060,304065],[11,303067,303069],{"id":303068},"chapter-6-concurrency-deadlock-and-starvation","Chapter 6 – Concurrency: Deadlock and Starvation",[19,303071,303073],{"id":303072},"一-true-false-questions-是非題","一、 True \u002F False Questions (是非題)",[15,303075,303076],{},[2627,303077,303078],{},"1. Deadlock can be defined as the periodic blocking of a set of processes that either compete for system resources or communicate with each other.",[105,303080,303081,303087,303092,303098,303104],{},[108,303082,303083,303086],{},[2627,303084,303085],{},"中文翻譯","：死結可以定義為一組競爭系統資源或相互通訊的進程的週期性阻塞。",[108,303088,303089,303091],{},[2627,303090,118575],{},"：False",[108,303093,303094,303097],{},[2627,303095,303096],{},"Concept Check","：死結 (Deadlock) 的定義。",[108,303099,303100,303103],{},[2627,303101,303102],{},"The 'Why'","：死結是指進程陷入了「永久性 (permanent)」的阻塞。如果沒有作業系統的強制介入，這些進程永遠無法繼續執行，因此絕對不是可以自行恢復的「週期性 (periodic)」現象。",[108,303105,303106,303109],{},[2627,303107,303108],{},"Common Pitfalls","：學生常因前半段定義正確而忽略了「periodic」這個形容詞。",[15,303111,303112],{},[2627,303113,303114],{},"2. All deadlocks involve conflicting needs for resources by two or more processes.",[105,303116,303117,303122,303127,303132],{},[108,303118,303119,303121],{},[2627,303120,303085],{},"：所有死結都涉及兩個或多個進程對資源的衝突需求。",[108,303123,303124,303126],{},[2627,303125,118575],{},"：True",[108,303128,303129,303131],{},[2627,303130,303096],{},"：產生死結的參與者條件。",[108,303133,303134,303136],{},[2627,303135,303102],{},"：死結的本質是互相等待（形成循環），因此在邏輯上至少需要兩個或兩個以上的進程互相持有對方需要的資源才能構成死結。",[15,303138,303139],{},[2627,303140,303141],{},"3. A reusable resource is one that can be safely used by only one process at a time and is not depleted by that use.",[105,303143,303144,303149,303153,303158],{},[108,303145,303146,303148],{},[2627,303147,303085],{},"：可重用資源是指一次只能由一個進程安全使用，且不會因該使用而被耗盡的資源。",[108,303150,303151,303126],{},[2627,303152,118575],{},[108,303154,303155,303157],{},[2627,303156,303096],{},"：可重用資源 (Reusable Resource) 的定義。",[108,303159,303160,303162],{},[2627,303161,303102],{},"：這正是可重用資源的標準定義（例如處理器、記憶體、I\u002FO 設備）。它們在使用後會被歸還給系統，供其他進程再次使用，不會消失。",[15,303164,303165],{},[2627,303166,303167],{},"4. A consumable resource is one that can be safely used by only one process at a time and is not depleted by that use.",[105,303169,303170,303175,303179,303184,303189],{},[108,303171,303172,303174],{},[2627,303173,303085],{},"：可消耗資源是指一次只能由一個進程安全使用，且不會因該使用而被耗盡的資源。",[108,303176,303177,303091],{},[2627,303178,118575],{},[108,303180,303181,303183],{},[2627,303182,303096],{},"：可消耗資源 (Consumable Resource) 的定義。",[108,303185,303186,303188],{},[2627,303187,303102],{},"：題目描述的是可重用資源。可消耗資源（如訊息、中斷、信號）一旦被獲取（接收），就會「被耗盡 \u002F 銷毀 (depleted \u002F destroyed)」，不再存在於系統中。",[108,303190,303191,303193],{},[2627,303192,303108],{},"：搞混 Reusable 和 Consumable 的特性。Consumable 是一次性的。",[15,303195,303196],{},[2627,303197,303198],{},"5. Although deadlock can potentially exist without it, the condition known as Circular Wait is required for deadlock to actually take place.",[105,303200,303201,303206,303210,303215],{},[108,303202,303203,303205],{},[2627,303204,303085],{},"：儘管沒有循環等待（Circular Wait）也可能潛在地存在死結的風險，但要讓死結實際發生，循環等待條件是必須的。",[108,303207,303208,303126],{},[2627,303209,118575],{},[108,303211,303212,303214],{},[2627,303213,303096],{},"：死結的四個必要條件與充分條件。",[108,303216,303217,303219],{},[2627,303218,303102],{},"：前三個條件（互斥、佔有且等待、不可剝奪）是死結的「必要條件」，它們構成了潛在的死結風險（致命區域）；但只有當第四個條件「循環等待」實際發生時，死結才真正形成（充分條件）。",[15,303221,303222],{},[2627,303223,303224],{},"6. The strategy of deadlock prevention is to design a system in such a way that the possibility of deadlock is minimized.",[105,303226,303227,303232,303236,303241],{},[108,303228,303229,303231],{},[2627,303230,303085],{},"：死結預防的策略是設計一個系統，使得死結的可能性降到最低。",[108,303233,303234,303091],{},[2627,303235,118575],{},[108,303237,303238,303240],{},[2627,303239,303096],{},"：死結預防 (Deadlock Prevention) 的嚴格性。",[108,303242,303243,303245],{},[2627,303244,303102],{},"：死結預防的目標不是「最小化 (minimized)」，而是「完全排除 (excluded)」。它透過嚴格破壞四個必要條件之一，確保死結「絕對不可能」發生。",[15,303247,303248],{},[2627,303249,303250],{},"7. The Deadlock Avoidance approach to solving the deadlock problem allows the three necessary conditions for deadlocks to exist.",[105,303252,303253,303258,303262,303267],{},[108,303254,303255,303257],{},[2627,303256,303085],{},"：用於解決死結問題的「死結避免」方法允許死結存在的三個必要條件。",[108,303259,303260,303126],{},[2627,303261,118575],{},[108,303263,303264,303266],{},[2627,303265,303096],{},"：死結避免 (Deadlock Avoidance) 的特徵。",[108,303268,303269,303271],{},[2627,303270,303102],{},"：與死結預防不同，死結避免允許前三個條件（互斥、佔有且等待、不剝奪）存在以提高資源利用率，但它會在每次分配資源前動態檢查，透過演算法確保系統永遠不會進入導致循環等待的不安全狀態。",[15,303273,303274],{},[2627,303275,303276],{},"8. In the Resource Allocation Denial approach to Deadlock Avoidance, a safe state is defined as one in which all potential process sequences do not result in a deadlock.",[105,303278,303279,303284,303288,303293,303298],{},[108,303280,303281,303283],{},[2627,303282,303085],{},"：在死結避免的資源分配拒絕方法（銀行家演算法）中，安全狀態被定義為：所有潛在的進程執行序列都不會導致死結。",[108,303285,303286,303091],{},[2627,303287,118575],{},[108,303289,303290,303292],{},[2627,303291,303096],{},"：安全狀態 (Safe State) 的定義。",[108,303294,303295,303297],{},[2627,303296,303102],{},"：安全狀態的定義是「存在『至少一個 (at least one)』執行序列不會導致死結」，而不是「所有 (all)」。只要系統能找到一條安全出路，它就是安全的。",[108,303299,303300,303302],{},[2627,303301,303108],{},"：學生常誤以為安全狀態代表隨便怎麼執行都安全。實際上只要有一條解法即可。",[15,303304,303305],{},[2627,303306,303307],{},"9. Deadlock Detection strategies do not limit resource access or restrict process actions.",[105,303309,303310,303315,303319,303324],{},[108,303311,303312,303314],{},[2627,303313,303085],{},"：死結偵測策略不限制資源存取或約束進程行為。",[108,303316,303317,303126],{},[2627,303318,118575],{},[108,303320,303321,303323],{},[2627,303322,303096],{},"：死結偵測 (Deadlock Detection) 的哲學。",[108,303325,303326,303328],{},[2627,303327,303102],{},"：死結偵測是一種極度寬鬆的策略（Very liberal）。只要資源夠，系統就立刻分配，不加任何限制，然後只是定期檢查是否發生了死結。",[15,303330,303331],{},[2627,303332,303333],{},"10. One of the most common approaches for recovery from deadlocked processes is to abort all deadlocked processes.",[105,303335,303336,303341,303345,303350],{},[108,303337,303338,303340],{},[2627,303339,303085],{},"：從死結進程中恢復的最常見方法之一是中止所有死結的進程。",[108,303342,303343,303126],{},[2627,303344,118575],{},[108,303346,303347,303349],{},[2627,303348,303096],{},"：死結恢復 (Recovery) 策略。",[108,303351,303352,303354],{},[2627,303353,303102],{},"：雖然聽起來很粗暴，但「中止所有死結進程 (Abort all deadlocked processes)」確實是現實作業系統中最常用、最容易實作的死結恢復方案。",[15,303356,303357],{},[2627,303358,303359],{},"11. Although each strategy that deals with deadlocks has its advantages and disadvantages, the best solution to the problem is to choose one and stick with it.",[105,303361,303362,303367,303371,303376],{},[108,303363,303364,303366],{},[2627,303365,303085],{},"：儘管處理死結的每種策略都有其優缺點，但解決該問題的最佳方法是選擇其中一種並堅持使用。",[108,303368,303369,303091],{},[2627,303370,118575],{},[108,303372,303373,303375],{},[2627,303374,303096],{},"：綜合死結策略 (Integrated Deadlock Strategy)。",[108,303377,303378,303380],{},[2627,303379,303102],{},"：沒有任何單一策略是完美的。最佳的實務做法是「綜合策略 (Integrated Strategy)」，將系統資源分類（如記憶體、I\u002FO設備等），然後針對不同類別的資源特性，混合使用預防、避免和偵測。",[15,303382,303383],{},[2627,303384,303385],{},"12. The Dining Philosopher’s Problem illustrates basic problems in deadlock and starvation.",[105,303387,303388,303393,303397,303402],{},[108,303389,303390,303392],{},[2627,303391,303085],{},"：哲學家就餐問題說明了死結和飢餓的基本問題。",[108,303394,303395,303126],{},[2627,303396,118575],{},[108,303398,303399,303401],{},[2627,303400,303096],{},"：哲學家就餐問題的目的。",[108,303403,303404,303406],{},[2627,303405,303102],{},"：如果所有哲學家同時拿起左手叉子會發生死結；如果不停放下再拿起，則可能導致沒有人能吃到飯的飢餓（活鎖）。它是測試這些併發問題的標準模型。",[15,303408,303409],{},[2627,303410,303411],{},"13. A pipe in UNIX is a circular buffer that allows two processes to communicate on the producer-consumer model.",[105,303413,303414,303419,303423,303428],{},[108,303415,303416,303418],{},[2627,303417,303085],{},"：UNIX 中的管道 (pipe) 是一個循環緩衝區，允許兩個進程以生產者-消費者模型進行通訊。",[108,303420,303421,303126],{},[2627,303422,118575],{},[108,303424,303425,303427],{},[2627,303426,303096],{},"：UNIX 並發機制的 Pipe。",[108,303429,303430,303432],{},[2627,303431,303102],{},"：UNIX 管道的本質就是實作了生產者（寫入端）與消費者（讀取端）模型的先進先出 (FIFO) 循環緩衝區。",[15,303434,303435],{},[2627,303436,303437],{},"14. One thread synchronization primitive supported by Solaris is the Mutual Exclusion (mutex) lock.",[105,303439,303440,303445,303449,303454],{},[108,303441,303442,303444],{},[2627,303443,303085],{},"：Solaris 支援的一種執行緒同步原語是互斥 (mutex) 鎖。",[108,303446,303447,303126],{},[2627,303448,118575],{},[108,303450,303451,303453],{},[2627,303452,303096],{},"：Solaris 的同步機制。",[108,303455,303456,303458],{},[2627,303457,303102],{},"：Solaris 作業系統提供了多種同步原語，其中最基礎的就是互斥鎖 (mutex locks)。",[15,303460,303461],{},[2627,303462,303463],{},"15. Each synchronization object in a W2K system can either be in a signaled or signaling state.",[105,303465,303466,303471,303475,303480,303485],{},[108,303467,303468,303470],{},[2627,303469,303085],{},"：W2K (Windows 2000) 系統中的每個同步物件可以處於「發信號的 (signaled)」或「發信號中 (signaling)」狀態。",[108,303472,303473,303091],{},[2627,303474,118575],{},[108,303476,303477,303479],{},[2627,303478,303096],{},"：Windows 分派器物件 (Dispatcher Objects) 的狀態。",[108,303481,303482,303484],{},[2627,303483,303102],{},"：Windows 同步物件的兩種狀態是「發信號的 (signaled)」與「未發信號的 (nonsignaled)」，並沒有所謂的 signaling state。",[108,303486,303487,303489],{},[2627,303488,303108],{},"：專有名詞陷阱。記住是 Signaled vs Nonsignaled。",[32188,303491],{},[19,303493,303495],{"id":303494},"二-multiple-choice-questions-選擇題","二、 Multiple Choice Questions (選擇題)",[15,303497,303498,303501],{},[2627,303499,303500],{},"1. The permanent blocking of a set of processes that either compete for system resources or communicate with each other is called:","\na. Starvation b. Deadlock c. Prioritization d. All of the above",[105,303503,303504,303509,303514],{},[108,303505,303506,303508],{},[2627,303507,303085],{},"：一組競爭系統資源或相互通訊的進程發生永久阻塞被稱為：a. 飢餓 b. 死結 c. 優先級化 d. 以上皆是",[108,303510,303511,303513],{},[2627,303512,118575],{},"：b",[108,303515,303516,303518],{},[2627,303517,303102],{},"：這是死結 (Deadlock) 的標準定義。",[15,303520,303521,303524],{},[2627,303522,303523],{},"2. All deadlocks involve conflicting needs for resources by:","\na. One or more processes b. Two or more processes c. Three or more processes d. None of the above",[105,303526,303527,303532,303536],{},[108,303528,303529,303531],{},[2627,303530,303085],{},"：所有死結都涉及多少進程對資源的衝突需求：",[108,303533,303534,303513],{},[2627,303535,118575],{},[108,303537,303538,303540],{},[2627,303539,303102],{},"：死結必須形成循環等待，至少需要兩個或兩個以上的進程互相牽制。",[15,303542,303543,303546],{},[2627,303544,303545],{},"3. A resource that can be created and destroyed is called a:","\na. Reusable resource b. Producible resource c. Consumable resource d. All of the above",[105,303548,303549,303554,303559],{},[108,303550,303551,303553],{},[2627,303552,303085],{},"：可以被創建和銷毀的資源被稱為：",[108,303555,303556,303558],{},[2627,303557,118575],{},"：c",[108,303560,303561,303563],{},[2627,303562,303102],{},"：一旦被獲取就會消失的資源，在 OS 術語中稱為「可消耗資源 (Consumable resource)」。",[15,303565,303566,303569],{},[2627,303567,303568],{},"4. An example of a consumable resource is the following:","\na. Messages b. Printers c. Main Memory d. All of the above",[105,303571,303572,303577,303582],{},[108,303573,303574,303576],{},[2627,303575,303085],{},"：以下哪一項是可消耗資源的例子：",[108,303578,303579,303581],{},[2627,303580,118575],{},"：a",[108,303583,303584,303586],{},[2627,303585,303102],{},"：印表機和主記憶體用完要歸還，是可重用的。訊息 (Messages) 接收後就被消耗了。",[15,303588,303589,303592],{},[2627,303590,303591],{},"5. A condition of policy that must be present for a deadlock to be possible is:","\na. Mutual exclusion b. Hold and wait c. No preemption d. All of the above",[105,303594,303595,303600,303605],{},[108,303596,303597,303599],{},[2627,303598,303085],{},"：要讓死結成為可能，必須存在哪種策略條件：",[108,303601,303602,303604],{},[2627,303603,118575],{},"：d",[108,303606,303607,303609],{},[2627,303608,303102],{},"：產生死結的前三個必要條件（政策條件）就是互斥、佔有且等待、不剝奪。",[15,303611,303612,303615],{},[2627,303613,303614],{},"6. A direct method of deadlock prevention is to prevent the occurrence of:","\na. Mutual exclusion b. Hold and wait c. Circular wait d. All of the above",[105,303617,303618,303623,303627],{},[108,303619,303620,303622],{},[2627,303621,303085],{},"：死結預防的「直接」方法是防止發生：",[108,303624,303625,303558],{},[2627,303626,118575],{},[108,303628,303629,303631],{},[2627,303630,303102],{},"：破壞前三個條件被稱為「間接方法 (indirect method)」。破壞第四個條件（循環等待）被稱為「直接方法 (direct method)」。這是一個容易忽略的分類細節。",[15,303633,303634,303637],{},[2627,303635,303636],{},"7. One approach to deadlock avoidance is called:","\na. Process Termination Denial b. Resource Allocation Denial c. Hold and wait d. None of the above",[105,303639,303640,303645,303649],{},[108,303641,303642,303644],{},[2627,303643,303085],{},"：一種死結避免的方法被稱為：",[108,303646,303647,303513],{},[2627,303648,118575],{},[108,303650,303651,303653],{},[2627,303652,303102],{},"：死結避免有兩種主要方法：進程啟動拒絕 (Process Initiation Denial) 與資源分配拒絕 (Resource Allocation Denial，即銀行家演算法)。",[15,303655,303656,303659],{},[2627,303657,303658],{},"8. In the Resource Allocation Denial approach to Deadlock Avoidance, a safe state is defined as one in which:","\na. At least one potential process sequence does not result in a deadlock\nb. All potential process sequences do not result in a deadlock\nc. Several potential process sequences do not result in a deadlock\nd. None of the above",[105,303661,303662,303667,303671],{},[108,303663,303664,303666],{},[2627,303665,303085],{},"：在死結避免的資源分配拒絕方法中，安全狀態定義為：",[108,303668,303669,303581],{},[2627,303670,118575],{},[108,303672,303673,303675],{},[2627,303674,303102],{},"：安全狀態只要能找到「至少一條 (at least one)」讓所有進程順利完成的序列即可。",[15,303677,303678,303681],{},[2627,303679,303680],{},"9. A conservative strategy for dealing with deadlocks that involves limiting access to resources and imposing restrictions on processes is called:","\na. Deadlock Prevention b. Deadlock Avoidance c. Deadlock Detection d. None of the above",[105,303683,303684,303689,303693],{},[108,303685,303686,303688],{},[2627,303687,303085],{},"：一種處理死結的保守策略，涉及限制資源存取和對進程施加限制，被稱為：",[108,303690,303691,303581],{},[2627,303692,118575],{},[108,303694,303695,303697],{},[2627,303696,303102],{},"：預防 (Prevention) 是最保守的策略，透過直接或間接破壞必要條件來限制進程。",[15,303699,303700,303703],{},[2627,303701,303702],{},"10. In deadlocked process recovery, selection criteria for choosing a particular process to abort or rollback includes designating the process with the:","\na. Most estimated time remaining b. Lowest priority c. Least total resources allocated so far d. All of the above",[105,303705,303706,303711,303715],{},[108,303707,303708,303710],{},[2627,303709,303085],{},"：在死結進程恢復中，選擇中止或回滾特定進程的標準包含指定具有下列特徵的進程：",[108,303712,303713,303604],{},[2627,303714,118575],{},[108,303716,303717,303719],{},[2627,303718,303102],{},"：為了將恢復成本降到最低，OS 會選擇：剩餘時間最多的（剛開始執行的）、優先級最低的、或已經分配到資源最少的進程作為犧牲品。",[15,303721,303722,303725],{},[2627,303723,303724],{},"11. One approach to an integrated strategy for dealing with deadlocks involves the implementation of:","\na. Resource classes b. Process rollbacks c. Virtual memory d. None of the above",[105,303727,303728,303733,303737],{},[108,303729,303730,303732],{},[2627,303731,303085],{},"：處理死結的綜合策略之一涉及實作：",[108,303734,303735,303581],{},[2627,303736,118575],{},[108,303738,303739,303741],{},[2627,303740,303102],{},"：綜合策略 (Integrated Strategy) 的核心是將資源劃分為不同的「資源類別 (Resource classes)」，並針對每類資源採用不同的死結處理方法。",[15,303743,303744,303747],{},[2627,303745,303746],{},"12. The Dining Philosopher’s Problem is a standard test case for evaluating approaches to implementing:","\na. Deadlock b. Starvation c. Synchronization d. All of the above",[105,303749,303750,303755],{},[108,303751,303752,303754],{},[2627,303753,303085],{},"：哲學家就餐問題是評估實作什麼方法的標準測試案例：",[108,303756,303757,303759,303760,303763],{},[2627,303758,118575],{},"：c (依據 Stallings 課本上下文，雖然它展示死結與飢餓問題，但它主要是作為評估",[2627,303761,303762],{},"同步 (Synchronization)"," 機制設計是否良好的標準測試案例。)",[15,303765,303766,303769],{},[2627,303767,303768],{},"13. A software mechanism that informs a process of the occurrences of asynchronous events in UNIX are called:","\na. Pipes b. Messages c. Signals d. All of the above",[105,303771,303772,303777,303781],{},[108,303773,303774,303776],{},[2627,303775,303085],{},"：UNIX 中通知進程發生非同步事件的軟體機制被稱為：",[108,303778,303779,303558],{},[2627,303780,118575],{},[108,303782,303783,303785],{},[2627,303784,303102],{},"：信號 (Signals) 是 UNIX 用來處理非同步事件（如按下 Ctrl+C、除以零）的軟體中斷機制。",[15,303787,303788,303791],{},[2627,303789,303790],{},"14. Thread synchronization primitives supported by Solaris include:","\na. Mutual exclusion (mutex) locks b. Semaphores c. Condition variables d. All of the above",[105,303793,303794,303799,303803],{},[108,303795,303796,303798],{},[2627,303797,303085],{},"：Solaris 支援的執行緒同步原語包含：",[108,303800,303801,303604],{},[2627,303802,118575],{},[108,303804,303805,303807],{},[2627,303806,303102],{},"：Solaris 提供全面的同步工具，包含互斥鎖、信號量、讀寫鎖以及條件變數等。",[15,303809,303810,303813],{},[2627,303811,303812],{},"15. The family of synchronization objects implemented by W2K include:","\na. Mutex objects b. Semaphore objects c. Event objects d. All of the above",[105,303815,303816,303821,303825],{},[108,303817,303818,303820],{},[2627,303819,303085],{},"：W2K 實作的同步物件系列包含：",[108,303822,303823,303604],{},[2627,303824,118575],{},[108,303826,303827,303829],{},[2627,303828,303102],{},"：Windows (W2K) 提供了多種分派器物件用於同步，包含 Mutex、Semaphore、Event、Timer 等。",[32188,303831],{},[19,303833,303835],{"id":303834},"三-fill-in-the-blank-questions-填充題","三、 Fill-In-The-Blank Questions (填充題)",[15,303837,303838],{},[2627,303839,303840],{},"1. The permanent blocking of a set of processes that either compete for system resources or communicate with each other is called _________.",[105,303842,303843,303848],{},[108,303844,303845,303847],{},[2627,303846,303085],{},"：一組競爭系統資源或互相通訊的進程發生的永久性阻塞稱為________。",[108,303849,303850,303852],{},[2627,303851,118575],{},"：deadlock (死結)",[15,303854,303855],{},[2627,303856,303857],{},"2. All deadlocks involve conflicting needs for resources by _________ or more processes.",[105,303859,303860,303865],{},[108,303861,303862,303864],{},[2627,303863,303085],{},"：所有死結都涉及________個或多個進程對資源的衝突需求。",[108,303866,303867,303869],{},[2627,303868,118575],{},"：two (兩個)",[15,303871,303872],{},[2627,303873,303874],{},"3. A resource that can be created (produced) and destroyed (consumed) is called a _________ resource.",[105,303876,303877,303882],{},[108,303878,303879,303881],{},[2627,303880,303085],{},"：可以被創建（生產）和銷毀（消耗）的資源被稱為________資源。",[108,303883,303884,303886],{},[2627,303885,118575],{},"：consumable (可消耗)",[15,303888,303889],{},[2627,303890,303891],{},"4. An example of a reusable resource is a _______________.",[105,303893,303894,303899],{},[108,303895,303896,303898],{},[2627,303897,303085],{},"：可重用資源的一個例子是_______________。",[108,303900,303901,303903],{},[2627,303902,118575],{},"：processor \u002F main memory \u002F I\u002FO channel \u002F printer (處理器\u002F主記憶體\u002F磁碟等皆可)",[15,303905,303906],{},[2627,303907,303908],{},"5. The ___________________ policy condition, which says a process may hold allocated resources while awaiting assignment of others, must be present for a deadlock to be possible.",[105,303910,303911,303916],{},[108,303912,303913,303915],{},[2627,303914,303085],{},"：___________________策略條件，指的是進程可以持有已分配的資源同時等待分配其他資源，這必須存在才能使死結成為可能。",[108,303917,303918,303920],{},[2627,303919,118575],{},"：hold and wait (佔有且等待)",[15,303922,303923],{},[2627,303924,303925],{},"6. A(n) _____________ method of deadlock prevention is to prevent the occurrence of one of the three necessary conditions for deadlock.",[105,303927,303928,303933,303938],{},[108,303929,303930,303932],{},[2627,303931,303085],{},"：一種死結預防的_____________方法是防止死結的三個必要條件之一發生。",[108,303934,303935,303937],{},[2627,303936,118575],{},"：indirect (間接)",[108,303939,303940,303942],{},[2627,303941,303102],{},"：預防前三個條件屬於間接方法；預防循環等待則屬於直接方法。",[15,303944,303945],{},[2627,303946,303947],{},"7. In Deadlock Avoidance, the Resource Allocation Denial strategy is also referred to as the _______________.",[105,303949,303950,303955],{},[108,303951,303952,303954],{},[2627,303953,303085],{},"：在死結避免中，資源分配拒絕策略也被稱為_______________。",[108,303956,303957,303959],{},[2627,303958,118575],{},"：banker's algorithm (銀行家演算法)",[15,303961,303962],{},[2627,303963,303964],{},"8. A(n) _________________ state is one in which every potential sequence of allocation of resources to processes results in a deadlock.",[105,303966,303967,303972,303977],{},[108,303968,303969,303971],{},[2627,303970,303085],{},"：一個_________________狀態是指資源分配給進程的每一個潛在序列都會導致死結的狀態。",[108,303973,303974,303976],{},[2627,303975,118575],{},"：unsafe (不安全) 或 fatal (致命)",[108,303978,303979,303981],{},[2627,303980,303102],{},"：如果在該狀態下找不到任何一條可以避免死結的執行序列，該狀態即定義為不安全狀態 (Unsafe State)。",[15,303983,303984],{},[2627,303985,303986],{},"9. In Deadlock Detection, the O\u002FS periodically performs an algorithm that allows it to detect the ________________ condition.",[105,303988,303989,303994,303999],{},[108,303990,303991,303993],{},[2627,303992,303085],{},"：在死結偵測中，作業系統定期執行一種演算法，使其能夠偵測________________條件。",[108,303995,303996,303998],{},[2627,303997,118575],{},"：circular wait (循環等待)",[108,304000,304001,304003],{},[2627,304002,303102],{},"：死結偵測演算法的核心任務，就是去資源分配圖中尋找是否存在構成環路的「循環等待」。",[15,304005,304006],{},[2627,304007,304008],{},"10. Once a deadlock has been detected, some strategy is needed for ___________.",[105,304010,304011,304016],{},[108,304012,304013,304015],{},[2627,304014,303085],{},"：一旦偵測到死結，就需要某種策略來進行___________。",[108,304017,304018,304020],{},[2627,304019,118575],{},"：recovery (恢復)",[15,304022,304023],{},[2627,304024,304025],{},"11. One approach to an integrated strategy for dealing with deadlocks involves the implementation of __________________.",[105,304027,304028,304033,304038],{},[108,304029,304030,304032],{},[2627,304031,303085],{},"：處理死結的綜合策略的一種方法涉及實作__________________。",[108,304034,304035,304037],{},[2627,304036,118575],{},"：resource classes (資源類別)",[108,304039,304040,304042],{},[2627,304041,303102],{},"：將資源依照分配層次分門別類，各個擊破。",[15,304044,304045],{},[2627,304046,304047],{},"12. The Dining Philosopher’s Problem illustrates basic problems in __________________ and __________________.",[105,304049,304050,304055],{},[108,304051,304052,304054],{},[2627,304053,303085],{},"：哲學家就餐問題說明了__________________和__________________的基本問題。",[108,304056,304057,304059],{},[2627,304058,118575],{},"：deadlock (死結), starvation (飢餓)",[15,304061,304062],{},[2627,304063,304064],{},"13. The type of UNIX pipe that can be shared by unrelated processes is called a(n) __________________ pipe.",[105,304066,304067,304072,304077],{},[108,304068,304069,304071],{},[2627,304070,303085],{},"：可以由不相關進程共享的 UNIX 管道類型被稱為__________________管道。",[108,304073,304074,304076],{},[2627,304075,118575],{},"：named (命名) 或 FIFO",[108,304078,304079,304081],{},[2627,304080,303102],{},"：一般的匿名管道只能用於有血緣關係的父子進程；具名管道 (Named Pipe) 在系統中有對應名稱，因此任意進程皆可存取。",{"title":29,"searchDepth":43,"depth":43,"links":304083},[304084,304085,304086],{"id":303072,"depth":43,"text":303073},{"id":303494,"depth":43,"text":303495},{"id":303834,"depth":43,"text":303835},{"order":68},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap6",{"title":303063,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap6","XERXjLgoXFVRyKkhSFJCUHGZURckoh255QWbYpl2ddU",{"id":304093,"title":304094,"body":304095,"description":29,"extension":85,"meta":305351,"navigation":52,"path":305352,"seo":305353,"stem":305354,"__hash__":305355},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap7.md","Chap. 7",{"type":8,"value":304096,"toc":305346},[304097,304101,304103,304108,304126,304131,304156,304161,304183,304188,304206,304211,304237,304242,304260,304265,304287,304292,304314,304319,304349,304354,304372,304377,304406,304411,304431,304436,304462,304467,304491,304496,304522,304524,304526,304532,304554,304560,304578,304584,304603,304609,304628,304634,304659,304665,304684,304690,304711,304717,304736,304741,304760,304766,304787,304792,304818,304824,304854,304860,304881,304887,304908,304914,304936,304938,304940,304945,304964,304969,304991,304996,305019,305024,305047,305052,305074,305079,305098,305103,305125,305130,305152,305157,305179,305184,305210,305215,305237,305242,305264,305269,305291,305296,305319,305324],[11,304098,304100],{"id":304099},"chapter-7-memory-management","Chapter 7 Memory Management",[19,304102,293369],{"id":293368},[15,304104,304105],{},[2627,304106,304107],{},"1. In a multiprogramming system, main memory is divided into multiple sections: one for the operating system (resident monitor, kernel) and one for the set of processes currently being executed.",[105,304109,304110,304115,304121],{},[108,304111,304112,304114],{},[2627,304113,291953],{},"：在多道程序系统中，主内存被分为多个部分：一部分用于操作系统（常驻监控程序、内核），另一部分用于当前正在执行的进程集。",[108,304116,304117,2664,304119],{},[2627,304118,118575],{},[2627,304120,291961],{},[108,304122,304123,304125],{},[2627,304124,297792],{},"：内存管理的基本要求之一是将内存划分为系统区和用户区。操作系统占据固定的一部分内存，剩余的内存用于容纳一个或多个正在执行的用户进程。这正是多道程序设计的基础，使得 CPU 可以交替执行多个进程。",[15,304127,304128],{},[2627,304129,304130],{},"2. The task of subdividing memory between the O\u002FS and processes is performed automatically by the O\u002FS and is called relocation.",[105,304132,304133,304138,304144],{},[108,304134,304135,304137],{},[2627,304136,291953],{},"：操作系统自动执行在 OS 和进程之间细分内存的任务，这被称为重定位 (relocation)。",[108,304139,304140,2664,304142],{},[2627,304141,118575],{},[2627,304143,292009],{},[108,304145,304146,304148,304149,304152,304153,304155],{},[2627,304147,297792],{},"：题目描述的概念混淆了。细分内存、分配内存的任务统称为",[2627,304150,304151],{},"内存管理 (Memory Management)","。而",[2627,304154,243417],{}," 具体指的是当进程在内存中被换出又换入时，由于可能不会被加载回相同的物理地址，操作系统需要将程序中的逻辑地址动态地转换为实际的物理地址的过程。",[15,304157,304158],{},[2627,304159,304160],{},"3. The practice in which a program and data are organized in such a way that various modules can be assigned the same region of memory is called overlaying.",[105,304162,304163,304168,304174],{},[108,304164,304165,304167],{},[2627,304166,291953],{},"：将程序和数据组织成特定方式，使得各个模块可以被分配到同一内存区域的做法称为覆盖 (overlaying)。",[108,304169,304170,2664,304172],{},[2627,304171,118575],{},[2627,304173,291961],{},[108,304175,304176,304178,304179,304182],{},[2627,304177,297792],{},"：当一个程序比可用内存大时，程序员可以将程序分为若干模块。在执行过程中，不需要同时存在于内存中的模块可以轮流加载到同一个内存区域，这就是",[2627,304180,304181],{},"覆盖 (overlaying)","。这是一种早期的技术，主要由程序员手动完成。",[15,304184,304185],{},[2627,304186,304187],{},"4. The concept of virtual memory is based on one or both of two basic techniques: segmentation and paging.",[105,304189,304190,304195,304201],{},[108,304191,304192,304194],{},[2627,304193,291953],{},"：虚拟内存的概念基于分段 (segmentation) 和分页 (paging) 这两种基本技术的一种或两种。",[108,304196,304197,2664,304199],{},[2627,304198,118575],{},[2627,304200,291961],{},[108,304202,304203,304205],{},[2627,304204,297792],{},"：现代操作系统中，虚拟内存的实现几乎完全依赖于分页机制，或者分段机制，或者两者的结合（如段页式内存管理）。这两种技术允许进程的物理地址在内存中不连续，为虚拟内存打下了硬件基础。",[15,304207,304208],{},[2627,304209,304210],{},"5. A major problem with the largely obsolete Fixed Partitioning memory management technique is that of external fragmentation.",[105,304212,304213,304218,304224],{},[108,304214,304215,304217],{},[2627,304216,291953],{},"：基本上已过时的固定分区 (Fixed Partitioning) 内存管理技术的一个主要问题是外部碎片 (external fragmentation)。",[108,304219,304220,2664,304222],{},[2627,304221,118575],{},[2627,304223,292009],{},[108,304225,304226,304228,304229,304232,304233,304236],{},[2627,304227,297792],{},"：固定分区技术的主要问题是",[2627,304230,304231],{},"内部碎片 (Internal fragmentation)","。因为分区大小是固定的，当一个相对较小的进程被加载到一个较大的分区时，分区内部剩余的空间就会被浪费，这种现象称为内部碎片。",[2627,304234,304235],{},"外部碎片 (External fragmentation)"," 主要是动态分区 (Dynamic Partitioning) 的问题。",[15,304238,304239],{},[2627,304240,304241],{},"6. The problem of internal fragmentation can be lessened in a system employing a fixed-partition memory management scheme by using unequal size partitions.",[105,304243,304244,304249,304255],{},[108,304245,304246,304248],{},[2627,304247,291953],{},"：在采用固定分区内存管理方案的系统中，可以通过使用大小不等的分区来减轻内部碎片的问题。",[108,304250,304251,2664,304253],{},[2627,304252,118575],{},[2627,304254,291961],{},[108,304256,304257,304259],{},[2627,304258,297792],{},"：如果所有固定分区大小相同，小进程会浪费大量空间（内部碎片）。通过提供大小不等的固定分区（Unequal-size partitions），操作系统可以尝试将进程分配到能够容纳它且最小的分区中，从而显著减少内部碎片的浪费。",[15,304261,304262],{},[2627,304263,304264],{},"7. In the Dynamic Partitioning technique of memory management, the best-fit placement algorithm is usually the best performer of the available algorithms.",[105,304266,304267,304272,304278],{},[108,304268,304269,304271],{},[2627,304270,291953],{},"：在内存管理的动态分区技术中，最佳适应 (best-fit) 放置算法通常是所有可用算法中性能最好的。",[108,304273,304274,2664,304276],{},[2627,304275,118575],{},[2627,304277,292009],{},[108,304279,304280,304282,304283,304286],{},[2627,304281,297792],{},"：虽然名字叫 \"best-fit\" (最佳适应)，但它通常是",[2627,304284,304285],{},"性能最差","的。因为它总是寻找与请求大小最接近的空闲块，这会导致留下许多非常小且难以利用的内存碎片。此外，它每次都需要扫描整个或大部分空闲块列表，速度较慢。通常，First-fit (首次适应) 或 Next-fit (下次适应) 表现更好。",[15,304288,304289],{},[2627,304290,304291],{},"8. In the Dynamic Partitioning technique of memory management, compaction refers to shifting the processes into a contiguous block, resulting in all the free memory aggregated into in a single block.",[105,304293,304294,304299,304305],{},[108,304295,304296,304298],{},[2627,304297,291953],{},"：在内存管理的动态分区技术中，压缩 (compaction) 指的是移动进程使其成为一个连续的块，从而导致所有空闲内存聚集成一个单一的块。",[108,304300,304301,2664,304303],{},[2627,304302,118575],{},[2627,304304,291961],{},[108,304306,304307,304309,304310,304313],{},[2627,304308,297792],{},"：为了解决动态分区产生的外部碎片问题，操作系统会定期进行",[2627,304311,304312],{},"压缩 (Compaction)","。这涉及移动内存中正在运行的进程，使它们紧凑地排列在一起，从而将所有原本零散的空闲空间合并成一个巨大的、连续的空闲块，以便加载更大的新进程。",[15,304315,304316],{},[2627,304317,304318],{},"9. In the Dynamic Partitioning technique of memory management, the first-fit placement algorithm scans memory from the location of the last placement and chooses the first available block it finds that satisfies the request.",[105,304320,304321,304326,304332],{},[108,304322,304323,304325],{},[2627,304324,291953],{},"：在内存管理的动态分区技术中，首次适应 (first-fit) 放置算法从最后一次放置的位置开始扫描内存，并选择它找到的第一个满足请求的可用块。",[108,304327,304328,2664,304330],{},[2627,304329,118575],{},[2627,304331,292009],{},[108,304333,304334,304336,304337,304340,304341,304344,304345,304348],{},[2627,304335,297792],{},"：题目描述的是 ",[2627,304338,304339],{},"Next-fit (下次适应)"," 算法。",[2627,304342,304343],{},"First-fit (首次适应)"," 算法总是从内存空间的",[2627,304346,304347],{},"起始位置","开始扫描，选择第一个足够大的空闲块。Next-fit 才是从上次分配结束的地方开始扫描。",[15,304350,304351],{},[2627,304352,304353],{},"10. The Buddy System is a reasonable compromise to overcome the disadvantages of both the fixed and variable partition schemes.",[105,304355,304356,304361,304367],{},[108,304357,304358,304360],{},[2627,304359,291953],{},"：伙伴系统 (Buddy System) 是一个合理的折衷方案，用于克服固定分区和可变（动态）分区方案的缺点。",[108,304362,304363,2664,304365],{},[2627,304364,118575],{},[2627,304366,291961],{},[108,304368,304369,304371],{},[2627,304370,297792],{},"：伙伴系统通过不断将大块内存一分为二（互为伙伴）来满足较小的请求，并在内存释放时将相邻的伙伴合并。它避免了动态分区中复杂的碎片整理（压缩），同时比固定分区提供了更多的灵活性和较少的内部碎片。",[15,304373,304374],{},[2627,304375,304376],{},"11. A physical memory address is a reference to a memory location independent of the current assignment of data to memory.",[105,304378,304379,304384,304390],{},[108,304380,304381,304383],{},[2627,304382,291953],{},"：物理内存地址是对内存位置的引用，它独立于当前数据到内存的分配。",[108,304385,304386,2664,304388],{},[2627,304387,118575],{},[2627,304389,292009],{},[108,304391,304392,298117,304394,304397,304398,304401,304402,304405],{},[2627,304393,297792],{},[2627,304395,304396],{},"逻辑地址 (Logical address)"," 或者是相对地址。",[2627,304399,304400],{},"物理地址 (Physical address) \u002F 绝对地址 (Absolute address)"," 恰恰是主内存中",[2627,304403,304404],{},"实际的、绝对的位置","。逻辑地址才独立于数据当前是否以及被分配在内存的具体哪个位置。",[15,304407,304408],{},[2627,304409,304410],{},"12. A memory system employing paging may suffer slightly from internal fragmentation and experiences no external fragmentation.",[105,304412,304413,304418,304424],{},[108,304414,304415,304417],{},[2627,304416,291953],{},"：采用分页 (paging) 的内存系统可能会遭受轻微的内部碎片，但没有外部碎片。",[108,304419,304420,2664,304422],{},[2627,304421,118575],{},[2627,304423,291961],{},[108,304425,304426,304428,304429,347],{},[2627,304427,297792],{},"：在分页机制中，进程被划分为固定大小的页 (pages)，物理内存被划分为同样大小的页框 (frames)。因为分配的最小单位是页，所以不会有任何外部碎片。但是，进程的最后一部分通常不能刚好填满一个完整的页，这会导致该页的剩余部分被浪费，这就是轻微的",[2627,304430,205063],{},[15,304432,304433],{},[2627,304434,304435],{},"13. In a memory system employing paging, the chunks of a process (called frames) can be assigned to available chunks of memory (called pages).",[105,304437,304438,304443,304449],{},[108,304439,304440,304442],{},[2627,304441,291953],{},"：在采用分页的内存系统中，进程的块（称为页框 frames）可以分配给可用的内存块（称为页 pages）。",[108,304444,304445,2664,304447],{},[2627,304446,118575],{},[2627,304448,292009],{},[108,304450,304451,304453,304454,304457,304458,304461],{},[2627,304452,297792],{},"：术语弄反了。进程被划分的块称为",[2627,304455,304456],{},"页 (pages)","，而主内存被划分的物理块称为",[2627,304459,304460],{},"页框 (frames)","。正确的说法是：进程的 pages 被分配到主内存的 available frames 中。",[15,304463,304464],{},[2627,304465,304466],{},"14. A memory system employing segmentation may suffer slightly from external fragmentation and experience no internal fragmentation.",[105,304468,304469,304474,304480],{},[108,304470,304471,304473],{},[2627,304472,291953],{},"：采用分段 (segmentation) 的内存系统可能会遭受轻微的外部碎片，但没有内部碎片。",[108,304475,304476,2664,304478],{},[2627,304477,118575],{},[2627,304479,291961],{},[108,304481,304482,304484,304485,304488,304489,347],{},[2627,304483,297792],{},"：在分段机制中，进程被划分为大小不等的段 (segments)，如代码段、数据段等。分配时为每个段分配正好需要的内存量，因此",[2627,304486,304487],{},"没有内部碎片","。但是，由于段被加载和卸载，内存中会留下大小不一的空隙，类似于动态分区，因此会导致",[2627,304490,205067],{},[15,304492,304493],{},[2627,304494,304495],{},"15. A memory system employing segmentation consists of a number of user program segments that must be of the same length and have a maximum segment length.",[105,304497,304498,304503,304509],{},[108,304499,304500,304502],{},[2627,304501,291953],{},"：采用分段的内存系统由多个用户程序段组成，这些段必须具有相同的长度，并且有一个最大段长度。",[108,304504,304505,2664,304507],{},[2627,304506,118575],{},[2627,304508,292009],{},[108,304510,304511,304513,304514,304517,304518,304521],{},[2627,304512,297792],{},"：分段技术的核心特征之一就是段是",[2627,304515,304516],{},"大小不等 (unequal size)"," 的。它们对应于程序员眼中的逻辑实体（如函数、数组），因此长度不可能相同。虽然通常会受限于硬件有一个最大段长度限制，但它们绝对不要求具备相同长度。要求相同长度的是",[2627,304519,304520],{},"分页 (paging)"," 技术。",[32188,304523],{},[19,304525,293916],{"id":293915},[15,304527,304528,304531],{},[2627,304529,304530],{},"1. The task of subdividing memory between the O\u002FS and processes is performed automatically by the O\u002FS and is called:","\na. Protection\nb. Relocation\nc. Memory Management\nd. All of the above",[105,304533,304534,304539,304546],{},[108,304535,304536,304538],{},[2627,304537,291953],{},"：操作系统自动执行在 OS 和进程之间细分内存的任务，这被称为：\na. 保护 (Protection)\nb. 重定位 (Relocation)\nc. 内存管理 (Memory Management)\nd. 以上皆是",[108,304540,304541,2664,304543],{},[2627,304542,118575],{},[2627,304544,304545],{},"c. Memory Management",[108,304547,304548,304550,304551,304553],{},[2627,304549,297792],{},"：细分和分配内存的整体任务被称为",[2627,304552,304151],{},"。保护和重定位是内存管理的具体需求和机制，但“细分内存”这个任务本身是内存管理的核心定义。",[15,304555,304556,304559],{},[2627,304557,304558],{},"2. The concept of Memory Management satisfies certain system requirements, including:","\na. Protection\nb. Relocation\nc. Physical organization\nd. All of the above",[105,304561,304562,304567,304573],{},[108,304563,304564,304566],{},[2627,304565,291953],{},"：内存管理的概念满足特定的系统要求，包括：\na. 保护 (Protection)\nb. 重定位 (Relocation)\nc. 物理组织 (Physical organization)\nd. 以上皆是",[108,304568,304569,2664,304571],{},[2627,304570,118575],{},[2627,304572,292510],{},[108,304574,304575,304577],{},[2627,304576,297792],{},"：根据操作系统的原理，内存管理需要满足五个基本要求：重定位 (Relocation)、保护 (Protection)、共享 (Sharing)、逻辑组织 (Logical organization) 和 物理组织 (Physical organization)。因此，a、b、c 都包含在内。",[15,304579,304580,304583],{},[2627,304581,304582],{},"3. The practice in which a program and data are organized in such a way that various modules can be assigned the same region of memory is called:","\na. Overlaying\nb. Sharing\nc. Relocation\nd. None of the above",[105,304585,304586,304591,304598],{},[108,304587,304588,304590],{},[2627,304589,291953],{},"：将程序和数据组织成特定方式，使得各个模块可以被分配到同一内存区域的做法称为：\na. 覆盖 (Overlaying)\nb. 共享 (Sharing)\nc. 重定位 (Relocation)\nd. 以上皆非",[108,304592,304593,2664,304595],{},[2627,304594,118575],{},[2627,304596,304597],{},"a. Overlaying",[108,304599,304600,304602],{},[2627,304601,297792],{},"：如前所述，当程序过大时，将不同时执行的模块加载到内存的同一位置的技术称为覆盖。",[15,304604,304605,304608],{},[2627,304606,304607],{},"4. The concept of virtual memory is based on one or both of two basic techniques:","\na. Overlaying and relocation\nb. Segmentation and paging\nc. Segmentation and partitioning\nd. None of the above",[105,304610,304611,304616,304623],{},[108,304612,304613,304615],{},[2627,304614,291953],{},"：虚拟内存的概念基于以下两种基本技术的一种或两种：\na. 覆盖和重定位\nb. 分段和分页 (Segmentation and paging)\nc. 分段和分区\nd. 以上皆非",[108,304617,304618,2664,304620],{},[2627,304619,118575],{},[2627,304621,304622],{},"b. Segmentation and paging",[108,304624,304625,304627],{},[2627,304626,297792],{},"：虚拟内存的实现硬件支持是分页 (paging) 或 分段 (segmentation)，或者将两者结合的段页式机制。",[15,304629,304630,304633],{},[2627,304631,304632],{},"5. A problem with the largely obsolete Fixed Partitioning memory management technique is that of:","\na. Allowing only a fixed number of Processes\nb. Inefficient use of memory\nc. Internal fragmentation\nd. All of the above",[105,304635,304636,304641,304647],{},[108,304637,304638,304640],{},[2627,304639,291953],{},"：基本上已过时的固定分区内存管理技术的问题在于：\na. 只允许固定数量的进程\nb. 内存使用效率低下\nc. 内部碎片 (Internal fragmentation)\nd. 以上皆是",[108,304642,304643,2664,304645],{},[2627,304644,118575],{},[2627,304646,292510],{},[108,304648,304649,304651,304652,304654,304655,304658],{},[2627,304650,297792],{},"：固定分区在系统初始化时就划分好。由于分区数量固定，所以同时能在内存中的进程数被限制（多道程序度受限）。由于进程很难刚好填满分区，导致严重的",[2627,304653,205063],{},"，进而造成",[2627,304656,304657],{},"内存使用效率低下","。因此所有选项都是其问题。",[15,304660,304661,304664],{},[2627,304662,304663],{},"6. The problem of internal fragmentation can be lessened in systems employing a fixed-partition memory management scheme by using:","\na. Random size partitions\nb. Equal size partitions\nc. Unequal size partitions\nd. None of the above",[105,304666,304667,304672,304679],{},[108,304668,304669,304671],{},[2627,304670,291953],{},"：在采用固定分区内存管理方案的系统中，可以通过使用什么来减轻内部碎片的问题：\na. 随机大小分区\nb. 相等大小分区\nc. 大小不等分区 (Unequal size partitions)\nd. 以上皆非",[108,304673,304674,2664,304676],{},[2627,304675,118575],{},[2627,304677,304678],{},"c. Unequal size partitions",[108,304680,304681,304683],{},[2627,304682,297792],{},"：使用大小不等的固定分区，可以将小进程放入小分区，大进程放入大分区，以此来降低内部碎片的产生。",[15,304685,304686,304689],{},[2627,304687,304688],{},"7. In the Dynamic Partitioning technique of memory management, the phenomenon that results in unused blocks of memory outside of existing partitions is called:","\na. Internal fragmentation\nb. External fragmentation\nc. Compaction\nd. None of the above",[105,304691,304692,304697,304704],{},[108,304693,304694,304696],{},[2627,304695,291953],{},"：在内存管理的动态分区技术中，导致现有分区外部出现未使用内存块的现象被称为：\na. 内部碎片\nb. 外部碎片 (External fragmentation)\nc. 压缩\nd. 以上皆非",[108,304698,304699,2664,304701],{},[2627,304700,118575],{},[2627,304702,304703],{},"b. External fragmentation",[108,304705,304706,304708,304709,347],{},[2627,304707,297792],{},"：动态分区中，随着进程不断被载入和移出，内存中会留下许多过小而无法装入新进程的空闲孔洞（未使用的块）。因为这些孔洞在分配给进程的区域“外部”，所以称为",[2627,304710,205067],{},[15,304712,304713,304716],{},[2627,304714,304715],{},"8. In the Dynamic Partitioning technique of memory management, the placement algorithm that chooses the block that is closest in size to the request is called:","\na. Best-fit\nb. First-fit\nc. Next-fit\nd. All of the above",[105,304718,304719,304724,304731],{},[108,304720,304721,304723],{},[2627,304722,291953],{},"：在内存管理的动态分区技术中，选择大小最接近请求的空闲块的放置算法被称为：\na. 最佳适应 (Best-fit)\nb. 首次适应 (First-fit)\nc. 下次适应 (Next-fit)\nd. 以上皆是",[108,304725,304726,2664,304728],{},[2627,304727,118575],{},[2627,304729,304730],{},"a. Best-fit",[108,304732,304733,304735],{},[2627,304734,297792],{},"：这是 Best-fit (最佳适应) 算法的定义：遍历内存，找到满足要求且尺寸最接近的空闲块。",[15,304737,304738,304716],{},[2627,304739,304740],{},"9. In the Dynamic Partitioning technique of memory management, the placement algorithm that scans memory from the location of the last placement and chooses the next available block that large enough to satisfy the request is called:",[105,304742,304743,304748,304755],{},[108,304744,304745,304747],{},[2627,304746,291953],{},"：在内存管理的动态分区技术中，从最后一次放置的位置开始扫描内存，并选择下一个足够大以满足请求的可用块的放置算法被称为：\na. 最佳适应\nb. 首次适应\nc. 下次适应 (Next-fit)\nd. 以上皆是",[108,304749,304750,2664,304752],{},[2627,304751,118575],{},[2627,304753,304754],{},"c. Next-fit",[108,304756,304757,304759],{},[2627,304758,297792],{},"：这是 Next-fit (下次适应) 算法的定义，其目的是为了平均使用整个内存空间，而不是像 First-fit 那样总是集中在内存低端。",[15,304761,304762,304765],{},[2627,304763,304764],{},"10. A reference to a memory location independent of the current assignment of data to memory is called a(n):","\na. Relative address\nb. Logical address\nc. Absolute address\nd. None of the above",[105,304767,304768,304773,304780],{},[108,304769,304770,304772],{},[2627,304771,291953],{},"：独立于当前数据到内存的分配的对内存位置的引用被称为：\na. 相对地址\nb. 逻辑地址 (Logical address)\nc. 绝对地址\nd. 以上皆非",[108,304774,304775,2664,304777],{},[2627,304776,118575],{},[2627,304778,304779],{},"b. Logical address",[108,304781,304782,2664,304784,304786],{},[2627,304783,297792],{},[2627,304785,287108],{},"是对内存位置的一种抽象引用，它和进程数据当前到底实际存放在物理内存的哪里无关。它需要在执行时被转换（映射）为物理地址。",[15,304788,304789,304765],{},[2627,304790,304791],{},"11. An actual location in main memory is called a(n):",[105,304793,304794,304799,304806],{},[108,304795,304796,304798],{},[2627,304797,291953],{},"：主内存中的实际位置被称为：\na. 相对地址\nb. 逻辑地址\nc. 绝对地址 (Absolute address)\nd. 以上皆非",[108,304800,304801,2664,304803],{},[2627,304802,118575],{},[2627,304804,304805],{},"c. Absolute address",[108,304807,304808,304810,304811,304814,304815,347],{},[2627,304809,297792],{},"：实际的物理内存硬件位置被称为",[2627,304812,304813],{},"物理地址 (Physical address)"," 或",[2627,304816,304817],{},"绝对地址 (Absolute address)",[15,304819,304820,304823],{},[2627,304821,304822],{},"12. The page table for each process maintains:","\na. The frame location for each page of the process\nb. The page location for each frame of the process\nc. The physical memory location of the process\nd. None of the above",[105,304825,304826,304831,304838],{},[108,304827,304828,304830],{},[2627,304829,291953],{},"：每个进程的页表 (page table) 维护了：\na. 进程的每个页的页框位置 (The frame location for each page of the process)\nb. 进程的每个页框的页位置\nc. 进程的物理内存位置\nd. 以上皆非",[108,304832,304833,2664,304835],{},[2627,304834,118575],{},[2627,304836,304837],{},"a. The frame location for each page of the process",[108,304839,304840,304842,304843,304846,304847,304850,304851,50492],{},[2627,304841,297792],{},"：在分页系统中，操作系统为每个进程维护一个",[2627,304844,304845],{},"页表 (page table)","。页表的作用是记录该进程的逻辑",[2627,304848,304849],{},"页 (page)"," 被装载到了物理内存的哪一个",[2627,304852,304853],{},"页框 (frame)",[15,304855,304856,304859],{},[2627,304857,304858],{},"13. In a system employing a paging scheme for memory management, wasted space is due to:","\na. External fragmentation\nb. Internal fragmentation\nc. Pages and frames of different specified sizes\nd. None of the above",[105,304861,304862,304867,304874],{},[108,304863,304864,304866],{},[2627,304865,291953],{},"：在采用分页方案进行内存管理的系统中，浪费的空间是由于：\na. 外部碎片\nb. 内部碎片 (Internal fragmentation)\nc. 指定了不同大小的页和页框\nd. 以上皆非",[108,304868,304869,2664,304871],{},[2627,304870,118575],{},[2627,304872,304873],{},"b. Internal fragmentation",[108,304875,304876,304878,304879,347],{},[2627,304877,297792],{},"：分页系统消除了外部碎片，但进程的最后一页通常无法被填满，从而在分配给它的那个 frame 内部造成了空间浪费，即",[2627,304880,205063],{},[15,304882,304883,304886],{},[2627,304884,304885],{},"14. In a system employing a segmentation scheme for memory management, wasted space is due to:","\na. External fragmentation\nb. Internal fragmentation\nc. Segments of different sizes\nd. None of the above",[105,304888,304889,304894,304901],{},[108,304890,304891,304893],{},[2627,304892,291953],{},"：在采用分段方案进行内存管理的系统中，浪费的空间是由于：\na. 外部碎片 (External fragmentation)\nb. 内部碎片\nc. 不同大小的段\nd. 以上皆非",[108,304895,304896,2664,304898],{},[2627,304897,118575],{},[2627,304899,304900],{},"a. External fragmentation",[108,304902,304903,304905,304906,347],{},[2627,304904,297792],{},"：分段系统中，内存是按段的实际大小分配的，所以没有内部碎片。但由于段长短不一，在反复装入和移出后，内存中会留下很多小的空隙，形成",[2627,304907,205067],{},[15,304909,304910,304913],{},[2627,304911,304912],{},"15. In a system employing a segmentation scheme for memory management, a process is divided into:","\na. One segment per thread\nb. A number of segments which must be of equal size\nc. A number of segments which need not be of equal size\nd. None of the above",[105,304915,304916,304921,304928],{},[108,304917,304918,304920],{},[2627,304919,291953],{},"：在采用分段方案进行内存管理的系统中，一个进程被分为：\na. 每个线程一个段\nb. 多个必须大小相等的段\nc. 多个不需要大小相等的段 (A number of segments which need not be of equal size)\nd. 以上皆非",[108,304922,304923,2664,304925],{},[2627,304924,118575],{},[2627,304926,304927],{},"c. A number of segments which need not be of equal size",[108,304929,304930,304932,304933,31618],{},[2627,304931,297792],{},"：分段的核心特性就是将程序划分为反映逻辑结构的段，例如代码段、数据段、堆栈段等，这些段的长度自然是",[2627,304934,304935],{},"大小不等",[32188,304937],{},[19,304939,294449],{"id":294448},[15,304941,304942],{},[2627,304943,304944],{},"1. The task of subdividing memory between the O\u002FS and processes is performed automatically by the O\u002FS and is called ________.",[105,304946,304947,304952,304959],{},[108,304948,304949,304951],{},[2627,304950,291953],{},"：操作系统自动执行在 OS 和进程之间细分内存的任务，这被称为 ________。",[108,304953,304954,2664,304956],{},[2627,304955,118575],{},[2627,304957,304958],{},"Memory Management",[108,304960,304961,304963],{},[2627,304962,297792],{},"：如同前面选择题第 1 题，细分和管理主存的核心任务统称为 Memory Management（内存管理）。",[15,304965,304966],{},[2627,304967,304968],{},"2. The Memory Management task of moving the process image between different areas of memory as required to support swapping is referred to as ________.",[105,304970,304971,304976,304983],{},[108,304972,304973,304975],{},[2627,304974,291953],{},"：为支持交换 (swapping)，根据需要在内存的不同区域之间移动进程映像的内存管理任务被称为 ________。",[108,304977,304978,2664,304980],{},[2627,304979,118575],{},[2627,304981,304982],{},"Relocation",[108,304984,304985,304987,304988,347],{},[2627,304986,297792],{},"：当一个进程被换出到磁盘，之后再被换入时，由于原本的位置可能已被占用，它会被加载到内存的不同区域。操作系统必须能够解决地址转换的问题，这个过程称为 ",[2627,304989,304990],{},"Relocation (重定位)",[15,304992,304993],{},[2627,304994,304995],{},"3. The practice in which a program and data are organized in such a way that various modules can be assigned the same region of memory is called ________.",[105,304997,304998,305003,305010],{},[108,304999,305000,305002],{},[2627,305001,291953],{},"：将程序和数据组织成特定方式，使得各个模块可以被分配到同一内存区域的做法称为 ________。",[108,305004,305005,2664,305007],{},[2627,305006,118575],{},[2627,305008,305009],{},"Overlaying",[108,305011,305012,305014,305015,305018],{},[2627,305013,297792],{},"：这是 ",[2627,305016,305017],{},"Overlaying (覆盖)"," 技术的标准定义。",[15,305020,305021],{},[2627,305022,305023],{},"4. In almost all modern multiprogramming systems, memory is managed using a sophisticated technique known as ________.",[105,305025,305026,305031,305038],{},[108,305027,305028,305030],{},[2627,305029,291953],{},"：在几乎所有现代多道程序系统中，内存都是使用一种称为 ________ 的复杂技术来管理的。",[108,305032,305033,2664,305035],{},[2627,305034,118575],{},[2627,305036,305037],{},"Virtual Memory",[108,305039,305040,305042,305043,305046],{},[2627,305041,297792],{},"：现代操作系统摒弃了简单的分区和单纯的分页分段，而是普遍采用了 ",[2627,305044,305045],{},"Virtual Memory (虚拟内存)"," 技术，它允许进程的执行空间比实际物理内存大得多。",[15,305048,305049],{},[2627,305050,305051],{},"5. The phenomenon, in which there is wasted space internal to a partition due to the fact that the block of data loaded is smaller than the partition, is referred to as ________.",[105,305053,305054,305059,305066],{},[108,305055,305056,305058],{},[2627,305057,291953],{},"：由于加载的数据块小于分区，导致分区内部存在浪费空间的现象被称为 ________。",[108,305060,305061,2664,305063],{},[2627,305062,118575],{},[2627,305064,305065],{},"Internal fragmentation",[108,305067,305068,305070,305071,347],{},[2627,305069,297792],{},"：由于固定分区或分页机制中分配的单元大于进程实际所需，导致单元内部剩余空间被浪费，这明确定义了 ",[2627,305072,305073],{},"Internal fragmentation (内部碎片)",[15,305075,305076],{},[2627,305077,305078],{},"6. The problem of internal fragmentation can be lessened in system employing a fixed-partition memory management scheme by using ________ partitions.",[105,305080,305081,305086,305093],{},[108,305082,305083,305085],{},[2627,305084,291953],{},"：在采用固定分区内存管理方案的系统中，可以通过使用 ________ 分区来减轻内部碎片的问题。",[108,305087,305088,2664,305090],{},[2627,305089,118575],{},[2627,305091,305092],{},"Unequal-size",[108,305094,305095,305097],{},[2627,305096,297792],{},"：通过提供大小不一 (Unequal-size) 的固定分区，系统可以更精确地匹配进程的需求，从而减少内部碎片。",[15,305099,305100],{},[2627,305101,305102],{},"7. In the Dynamic Partitioning technique of memory management, the process of shifting processes so they occupy a single contiguous block in memory is called ________.",[105,305104,305105,305110,305117],{},[108,305106,305107,305109],{},[2627,305108,291953],{},"：在内存管理的动态分区技术中，移动进程使其占据内存中单个连续块的过程称为 ________。",[108,305111,305112,2664,305114],{},[2627,305113,118575],{},[2627,305115,305116],{},"Compaction",[108,305118,305119,305121,305122,347],{},[2627,305120,297792],{},"：为了消除动态分区产生的外部碎片，操作系统执行的内存紧缩、移动进程的操作称为 ",[2627,305123,305124],{},"Compaction (压缩\u002F紧凑)",[15,305126,305127],{},[2627,305128,305129],{},"8. In the Dynamic Partitioning technique of memory management, the placement algorithm that chooses the block that is closest in size to the request is called ________.",[105,305131,305132,305137,305144],{},[108,305133,305134,305136],{},[2627,305135,291953],{},"：在内存管理的动态分区技术中，选择大小最接近请求的块的放置算法被称为 ________。",[108,305138,305139,2664,305141],{},[2627,305140,118575],{},[2627,305142,305143],{},"Best-fit",[108,305145,305146,305148,305149,347],{},[2627,305147,297792],{},"：寻找大小最契合的空闲块的算法名为 ",[2627,305150,305151],{},"Best-fit (最佳适应)",[15,305153,305154],{},[2627,305155,305156],{},"9. In the Dynamic Partitioning technique of memory management, the phenomenon that results in unused blocks of memory outside of existing partitions is called ________.",[105,305158,305159,305164,305171],{},[108,305160,305161,305163],{},[2627,305162,291953],{},"：在内存管理的动态分区技术中，导致现有分区外部出现未使用内存块的现象被称为 ________。",[108,305165,305166,2664,305168],{},[2627,305167,118575],{},[2627,305169,305170],{},"External fragmentation",[108,305172,305173,305175,305176,347],{},[2627,305174,297792],{},"：动态分区由于不断的分配和回收，在进程分区之间会产生大量零散无法使用的空闲块，称为 ",[2627,305177,305178],{},"External fragmentation (外部碎片)",[15,305180,305181],{},[2627,305182,305183],{},"10. Programs that employ ________ addresses in memory are loaded using dynamic run-time loading.",[105,305185,305186,305191,305198],{},[108,305187,305188,305190],{},[2627,305189,291953],{},"：在内存中使用 ________ 地址的程序是使用动态运行时加载 (dynamic run-time loading) 的。",[108,305192,305193,2664,305195],{},[2627,305194,118575],{},[2627,305196,305197],{},"Logical \u002F Relative",[108,305199,305200,305202,305203,4374,305206,305209],{},[2627,305201,297792],{},"：为了支持进程在内存中的移动（重定位）和动态加载，程序代码不能使用绝对物理地址，而必须使用 ",[2627,305204,305205],{},"Logical (逻辑)",[2627,305207,305208],{},"Relative (相对)"," 地址。这些地址在执行指令时才由硬件（MMU）转换为绝对物理地址。",[15,305211,305212],{},[2627,305213,305214],{},"11. A compromise between the fixed and dynamic partitioning schemes for memory management that employs aspects of both is called the ________.",[105,305216,305217,305222,305229],{},[108,305218,305219,305221],{},[2627,305220,291953],{},"：内存管理的固定分区和动态分区方案之间的一种折衷方案，它利用了两者优点的被称为 ________。",[108,305223,305224,2664,305226],{},[2627,305225,118575],{},[2627,305227,305228],{},"Buddy System",[108,305230,305231,2664,305233,305236],{},[2627,305232,297792],{},[2627,305234,305235],{},"Buddy System (伙伴系统)"," 结合了固定分区的快速和动态分区的灵活性，它分配内存块的大小总是 2 的幂。",[15,305238,305239],{},[2627,305240,305241],{},"12. In a system that employs a paging memory management scheme, the ________ shows the frame location for each page of the process.",[105,305243,305244,305249,305256],{},[108,305245,305246,305248],{},[2627,305247,291953],{},"：在采用分页内存管理方案的系统中，________ 显示了进程每个页的页框位置。",[108,305250,305251,2664,305253],{},[2627,305252,118575],{},[2627,305254,305255],{},"Page table",[108,305257,305258,2664,305260,305263],{},[2627,305259,297792],{},[2627,305261,305262],{},"Page table (页表)"," 是操作系统维护的数据结构，用于记录逻辑页 (page) 到物理页框 (frame) 的映射关系。",[15,305265,305266],{},[2627,305267,305268],{},"13. In a system that employs a paging memory management scheme, the chunks of a process can be assigned to available chunks of memory, which are called ________.",[105,305270,305271,305276,305283],{},[108,305272,305273,305275],{},[2627,305274,291953],{},"：在采用分页内存管理方案的系统中，进程的块可以分配给可用的内存块，这些内存块被称为 ________。",[108,305277,305278,2664,305280],{},[2627,305279,118575],{},[2627,305281,305282],{},"Frames",[108,305284,305285,305287,305288,347],{},[2627,305286,297792],{},"：在分页中，主物理内存被划分为固定大小的块，称为 ",[2627,305289,305290],{},"Frames (页框 \u002F 物理块)",[15,305292,305293],{},[2627,305294,305295],{},"14. A system that employs a segmentation memory management scheme makes use of a ________ that provides the starting address of the corresponding segment in main memory.",[105,305297,305298,305303,305310],{},[108,305299,305300,305302],{},[2627,305301,291953],{},"：采用分段内存管理方案的系统利用 ________ 来提供主内存中相应段的起始地址。",[108,305304,305305,2664,305307],{},[2627,305306,118575],{},[2627,305308,305309],{},"Segment table",[108,305311,305312,305314,305315,305318],{},[2627,305313,297792],{},"：与分页系统类似，分段系统使用 ",[2627,305316,305317],{},"Segment table (段表)"," 来记录每个段的基址（起始物理地址）和段长 (Length)。",[15,305320,305321],{},[2627,305322,305323],{},"15. A system that employs a segmentation memory management scheme, the program and its associated data are divided into a number of ________ that need not be of the same length.",[105,305325,305326,305331,305338],{},[108,305327,305328,305330],{},[2627,305329,291953],{},"：采用分段内存管理方案的系统，程序及其相关数据被划分为若干个不需要具有相同长度的 ________。",[108,305332,305333,2664,305335],{},[2627,305334,118575],{},[2627,305336,305337],{},"Segments",[108,305339,305340,305342,305343,347],{},[2627,305341,297792],{},"：分段机制按程序的逻辑结构将其划分为大小不等的 ",[2627,305344,305345],{},"Segments (段)",{"title":29,"searchDepth":43,"depth":43,"links":305347},[305348,305349,305350],{"id":293368,"depth":43,"text":293369},{"id":293915,"depth":43,"text":293916},{"id":294448,"depth":43,"text":294449},{"order":74},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap7",{"title":304094,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap7","YkV6VbUu0r1pIC7G55ohjU4Fjjmp1YjaK95RPU0-ZBg",{"id":305357,"title":305358,"body":305359,"description":29,"extension":85,"meta":306659,"navigation":52,"path":306660,"seo":306661,"stem":306662,"__hash__":306663},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap8.md","Chap. 8",{"type":8,"value":305360,"toc":306654},[305361,305365,305367,305372,305399,305404,305425,305430,305452,305457,305481,305486,305506,305511,305538,305543,305561,305566,305588,305593,305623,305628,305650,305655,305680,305685,305715,305720,305746,305751,305773,305778,305858,305860,305862,305868,305888,305894,305916,305922,305945,305951,305976,305982,306000,306006,306033,306039,306058,306064,306082,306088,306111,306117,306139,306145,306166,306172,306194,306200,306222,306228,306247,306253,306272,306274,306276,306281,306300,306305,306324,306329,306348,306353,306372,306377,306400,306405,306424,306429,306448,306453,306472,306477,306496,306501,306520,306525,306544,306549,306568,306573,306592,306597,306616,306621],[11,305362,305364],{"id":305363},"chapter-8-virtual-memory","Chapter 8 - Virtual Memory",[19,305366,291941],{"id":291940},[15,305368,305369],{},[2627,305370,305371],{},"1. In a system employing a memory management strategy that doesn't require an entire process to be in main memory at one time, the portion of a process that is actually in main memory at any given time is defined to be the resident set of the process.",[105,305373,305374,305379,305385],{},[108,305375,305376,305378],{},[2627,305377,291953],{},"：在一个不要求进程一次性全部驻留在主存中的内存管理系统中，进程在任意给定时间实际存在于主存中的那部分被称为该进程的驻留集 (resident set)。",[108,305380,305381,2664,305383],{},[2627,305382,118575],{},[2627,305384,291961],{},[108,305386,305387,305390,305391,305394,305395,305398],{},[2627,305388,305389],{},"知识点解析","：在虚拟内存系统中，进程被划分为多个块（如页或段）。",[2627,305392,305393],{},"驻留集 (Resident Set)"," 就是指当前真正在物理内存（主存）中的那些页面的集合。与其对应的一个概念是 ",[2627,305396,305397],{},"工作集 (Working Set)","，它指的是进程在最近一段时间内实际访问过的页面集合。",[15,305400,305401],{},[2627,305402,305403],{},"2. The condition known as thrashing occurs when the majority of the processes in main memory require repetitive blocking on a single shared I\u002FO device in the system.",[105,305405,305406,305411,305417],{},[108,305407,305408,305410],{},[2627,305409,291953],{},"：当主存中的大多数进程都需要在系统中单个共享 I\u002FO 设备上重复阻塞时，就会发生称为“抖动 (thrashing)”的情况。",[108,305412,305413,2664,305415],{},[2627,305414,118575],{},[2627,305416,292009],{},[108,305418,305419,2664,305421,305424],{},[2627,305420,305389],{},[2627,305422,305423],{},"抖动 (Thrashing)"," 并不是因为争抢 I\u002FO 设备引起的。抖动是指由于物理内存不足，操作系统频繁地将页面换出到磁盘又换入到内存，导致处理器把大部分时间花在页面置换（缺页中断处理）上，而不是执行实际的指令。这通常是因为主存中的进程数量过多（多道程序度过高），导致每个进程分不到足够的页框来容纳其工作集。",[15,305426,305427],{},[2627,305428,305429],{},"3. The modify (M) bit is a control bit in a page table entry that indicates whether the contents of the corresponding page have been altered since the page was last loaded into memory.",[105,305431,305432,305437,305443],{},[108,305433,305434,305436],{},[2627,305435,291953],{},"：修改位 (M bit) 是页表项中的一个控制位，用于指示对应页面的内容自上次加载到内存后是否被更改过。",[108,305438,305439,2664,305441],{},[2627,305440,118575],{},[2627,305442,291961],{},[108,305444,305445,305447,305448,305451],{},[2627,305446,305389],{},"：修改位通常也被称为",[2627,305449,305450],{},"脏位 (Dirty Bit)","。当页面被选中要被替换出内存时，操作系统会检查这个位。如果该位为 0，说明页面未被修改过，可以直接丢弃（因为磁盘上已经有相同的副本）；如果为 1，说明页面在内存中被写过，必须将其写回磁盘以更新副本。这极大地减少了不必要的磁盘 I\u002FO 写入。",[15,305453,305454],{},[2627,305455,305456],{},"4. A Page Fault occurs when the desired page table entry is not found in the Translation Lookaside Buffer (TLB).",[105,305458,305459,305464,305470],{},[108,305460,305461,305463],{},[2627,305462,291953],{},"：当在转换后备缓冲器 (TLB) 中找不到所需的页表项时，就会发生缺页中断 (Page Fault)。",[108,305465,305466,2664,305468],{},[2627,305467,118575],{},[2627,305469,292009],{},[108,305471,305472,305474,305475,305478,305479,347],{},[2627,305473,305389],{},"：在 TLB 中找不到页表项被称为 ",[2627,305476,305477],{},"TLB Miss（TLB 缺失）","，发生这种情况时，硬件（或操作系统）会去访问内存中的页表。如果在内存的页表中发现该页面的“存在位 (Present Bit)”为 0（即该页面不在物理内存中），这才叫 ",[2627,305480,257364],{},[15,305482,305483],{},[2627,305484,305485],{},"5. One of the advantages to the programmer of virtual memory using segmentation is that it simplifies the handling of growing data structures by allowing the segment to grow or shrink as necessary.",[105,305487,305488,305493,305499],{},[108,305489,305490,305492],{},[2627,305491,291953],{},"：使用分段技术的虚拟内存对程序员的优势之一是，它允许段根据需要增长或缩小，从而简化了对不断增长的数据结构的处理。",[108,305494,305495,2664,305497],{},[2627,305496,118575],{},[2627,305498,291961],{},[108,305500,305501,2664,305503,305505],{},[2627,305502,305389],{},[2627,305504,243588],{}," 是对程序员可见的。每个段代表一个逻辑实体（如数组、堆栈）。如果一个数据结构（比如堆栈）需要增长，操作系统可以分配一个更大的物理区域并更新段表，而不需要干扰进程的其他段，这极大地方便了动态数据结构的管理。",[15,305507,305508],{},[2627,305509,305510],{},"6. In a combined paging\u002Fsegmentation system, a user's address space is broken up into a number of fixed-size pages, which in turn are broken up into a number of segments.",[105,305512,305513,305518,305524],{},[108,305514,305515,305517],{},[2627,305516,291953],{},"：在结合了分页\u002F分段的系统中，用户的地址空间被分成许多固定大小的页，这些页又被分成许多段。",[108,305519,305520,2664,305522],{},[2627,305521,118575],{},[2627,305523,292009],{},[108,305525,305526,305528,305529,305532,305533,305535,305536,347],{},[2627,305527,305389],{},"：顺序完全反了。在",[2627,305530,305531],{},"段页式 (Combined Paging\u002FSegmentation)"," 系统中，用户的地址空间首先根据逻辑结构被划分为多个大小可变的",[2627,305534,243657],{},"，然后为了便于物理内存的分配和避免外部碎片，每个段又被划分为多个固定大小的",[2627,305537,243609],{},[15,305539,305540],{},[2627,305541,305542],{},"7. To achieve sharing in a segmentation system, it is possible for a segment to be referenced in the segment tables of more than one process.",[105,305544,305545,305550,305556],{},[108,305546,305547,305549],{},[2627,305548,291953],{},"：为了在分段系统中实现共享，可以在多个进程的段表中引用同一个段。",[108,305551,305552,2664,305554],{},[2627,305553,118575],{},[2627,305555,291961],{},[108,305557,305558,305560],{},[2627,305559,305389],{},"：分段非常适合共享。如果多个进程需要执行同一段代码（如共享库），只需在每个进程的段表中添加一个条目，将该条目的基地址指向内存中该共享代码段的物理基址即可。",[15,305562,305563],{},[2627,305564,305565],{},"8. Linux is an example of an operating system that does not provide virtual memory.",[105,305567,305568,305573,305579],{},[108,305569,305570,305572],{},[2627,305571,291953],{},"：Linux 是不提供虚拟内存的操作系统的一个例子。",[108,305574,305575,2664,305577],{},[2627,305576,118575],{},[2627,305578,292009],{},[108,305580,305581,305583,305584,305587],{},[2627,305582,305389],{},"：Linux 作为一个现代的成熟操作系统，",[2627,305585,305586],{},"重度依赖虚拟内存管理","。它使用按需分页机制来管理进程和物理内存，绝对不是不提供虚拟内存的系统。",[15,305589,305590],{},[2627,305591,305592],{},"9. The fetch policy determines when a page should be brought into main memory.",[105,305594,305595,305600,305606],{},[108,305596,305597,305599],{},[2627,305598,291953],{},"：获取策略 (fetch policy) 决定了页面应该何时被引入主存。",[108,305601,305602,2664,305604],{},[2627,305603,118575],{},[2627,305605,291961],{},[108,305607,305608,305610,305611,305614,305615,305618,305619,305622],{},[2627,305609,305389],{},"：虚拟内存操作系统的内存管理设计有几个核心策略：",[2627,305612,305613],{},"获取策略 (Fetch Policy)"," 决定何时将页读入内存（如请求调页 Demand Paging，或预调页 Prepaging）；",[2627,305616,305617],{},"放置策略 (Placement Policy)"," 决定将页放在哪里；",[2627,305620,305621],{},"置换策略 (Replacement Policy)"," 决定内存满时换出哪个页。",[15,305624,305625],{},[2627,305626,305627],{},"10. The Least Recently Used (LRU) replacement policy replaces the page in memory that has been referenced most recently.",[105,305629,305630,305635,305641],{},[108,305631,305632,305634],{},[2627,305633,291953],{},"：最近最少使用 (LRU) 置换策略替换内存中最近被引用过的页面。",[108,305636,305637,2664,305639],{},[2627,305638,118575],{},[2627,305640,292009],{},[108,305642,305643,305645,305646,305649],{},[2627,305644,305389],{},"：名字已经说明了一切。LRU (Least Recently Used) 策略替换的是在内存中",[2627,305647,305648],{},"最长时间未被访问过 (least recently)"," 的页面，而不是最近刚访问过 (most recently) 的页面。LRU 基于时间局部性原理，认为很久没用的页面近期也不太会被用。",[15,305651,305652],{},[2627,305653,305654],{},"11. A global replacement policy considers all unlocked pages in main memory as candidates for replacement, regardless of which process owns a particular page.",[105,305656,305657,305662,305668],{},[108,305658,305659,305661],{},[2627,305660,291953],{},"：全局置换策略将主存中所有未锁定的页面视为置换的候选者，而不管该页面属于哪个进程。",[108,305663,305664,2664,305666],{},[2627,305665,118575],{},[2627,305667,291961],{},[108,305669,305670,2664,305672,305675,305676,305679],{},[2627,305671,305389],{},[2627,305673,305674],{},"全局置换 (Global Replacement)"," 允许一个进程引发缺页中断时，从系统中其他进程的物理页框里抢夺内存。相对的是",[2627,305677,305678],{},"局部置换 (Local Replacement)","，进程只能从分配给自己进程的驻留集里挑选页面进行置换。",[15,305681,305682],{},[2627,305683,305684],{},"12. In a precleaning policy, modified pages are written to secondary memory once they have been selected for replacement.",[105,305686,305687,305692,305698],{},[108,305688,305689,305691],{},[2627,305690,291953],{},"：在预清除策略中，一旦修改过的页面被选中进行置换，它们就会被写入辅助存储器（磁盘）。",[108,305693,305694,2664,305696],{},[2627,305695,118575],{},[2627,305697,292009],{},[108,305699,305700,305702,305703,305706,305707,305710,305711,305714],{},[2627,305701,305389],{},"：被选中置换时才写回磁盘的做法叫做 ",[2627,305704,305705],{},"按需清除 (Demand Cleaning)","。而 ",[2627,305708,305709],{},"预清除 (Precleaning)"," 策略是指在这些脏页面被选中置换出内存",[2627,305712,305713],{},"之前","，操作系统就成批地将它们写回磁盘，这样当之后真正需要替换页面时，这些页面的内容和磁盘上是一致的，就可以直接丢弃，加快了置换速度。",[15,305716,305717],{},[2627,305718,305719],{},"13. SVR4 and Solaris systems use two separate schemes for memory management, one for user processes and disk I\u002FO, and another for kernel memory allocation.",[105,305721,305722,305727,305733],{},[108,305723,305724,305726],{},[2627,305725,291953],{},"：SVR4 和 Solaris 系统使用两种独立的内存管理方案，一种用于用户进程和磁盘 I\u002FO，另一种用于内核内存分配。",[108,305728,305729,2664,305731],{},[2627,305730,118575],{},[2627,305732,291961],{},[108,305734,305735,305737,305738,305741,305742,305745],{},[2627,305736,305389],{},"：这两种经典的 UNIX 系统使用",[2627,305739,305740],{},"分页系统 (Paging System)"," 来管理用户进程和磁盘 I\u002FO，而内核往往需要分配许多小型的、经常使用的数据结构（如 inode、进程表等），因此使用了另一套称为 ",[2627,305743,305744],{},"内核内存分配器 (Kernel Memory Allocator, KMA)"," 的机制（通常基于 Slab 分配器）来优化内核对象的分配。",[15,305747,305748],{},[2627,305749,305750],{},"14. Linux makes use of a two-level page table structure, consisting of a page directory and a page table.",[105,305752,305753,305758,305764],{},[108,305754,305755,305757],{},[2627,305756,291953],{},"：Linux 利用两级页表结构，由页目录和页表组成。",[108,305759,305760,2664,305762],{},[2627,305761,118575],{},[2627,305763,292009],{},[108,305765,305766,305768,305769,305772],{},[2627,305767,305389],{},"：为了兼容 64 位架构（如最初的 Alpha 架构），Linux 早期设计了",[2627,305770,305771],{},"三级页表结构","：页全局目录 (PGD)、页中间目录 (PMD) 和页表 (PTE)。在 32 位系统上，Linux 通过将中间目录的长度设为 1，使其逻辑上等同于两级页表。现代 Linux 在 64 位架构上甚至扩展到了四级或五级页表。因此，说它仅由页目录和页表两级组成是错误的。",[15,305774,305775],{},[2627,305776,305777],{},"15. Every W2K user process sees a separate 32-bit address space, allowing 4 GB of memory per process.",[105,305779,305780,305785,305791],{},[108,305781,305782,305784],{},[2627,305783,291953],{},"：每个 Windows 2000 (W2K) 用户进程都看到一个独立的 32 位地址空间，允许每个进程拥有 4 GB 的内存。",[108,305786,305787,2664,305789],{},[2627,305788,118575],{},[2627,305790,291961],{},[108,305792,305793,305795,305796,305857],{},[2627,305794,305389],{},"：在 32 位 Windows 操作系统中，寻址空间为 ",[34,305797,305799,305816],{"className":305798},[31642],[34,305800,305802],{"className":305801},[31646],[31648,305803,305804],{"xmlns":31650},[31652,305805,305806,305814],{},[31655,305807,305808],{},[37624,305809,305810,305812],{},[31749,305811,768],{},[31749,305813,173486],{},[31665,305815,201449],{"encoding":31667},[34,305817,305819],{"className":305818,"ariaHidden":758},[31672],[34,305820,305822,305825],{"className":305821},[31676],[34,305823],{"className":305824,"style":37774},[31680],[34,305826,305828,305831],{"className":305827},[31685],[34,305829,768],{"className":305830},[31685],[34,305832,305834],{"className":305833},[31792],[34,305835,305837],{"className":305836},[31796],[34,305838,305840],{"className":305839},[31801],[34,305841,305843],{"className":305842,"style":37774},[31805],[34,305844,305845,305848],{"style":37777},[34,305846],{"className":305847,"style":31814},[31813],[34,305849,305851],{"className":305850},[31818,31819,31820,31821],[34,305852,305854],{"className":305853},[31685,31821],[34,305855,173486],{"className":305856},[31685,31821]," 字节，即 4 GB。每个进程都有自己独立的 4 GB 虚拟地址空间，通常默认情况下，低 2 GB 分配给用户进程代码和数据，高 2 GB 留给操作系统内核。",[32188,305859],{},[19,305861,292392],{"id":292391},[15,305863,305864,305867],{},[2627,305865,305866],{},"1. The type of memory that allows for very effective multiprogramming and relieves the user of memory size constraints is referred to as:","\na. Real memory\nb. Virtual memory\nc. Main memory\nd. All of the above",[105,305869,305870,305875,305881],{},[108,305871,305872,305874],{},[2627,305873,291953],{},"：允许非常有效的多道程序设计并免除用户内存大小限制的内存类型被称为：",[108,305876,305877,2664,305879],{},[2627,305878,118575],{},[2627,305880,301340],{},[108,305882,305883,2664,305885,305887],{},[2627,305884,305389],{},[2627,305886,243584],{}," 通过将外存（磁盘）和主存结合，给程序员提供了一个比实际物理内存大得多的逻辑内存空间。这使得庞大的程序可以运行，同时能将更多进程放入内存中，从而实现有效的多道程序设计。",[15,305889,305890,305893],{},[2627,305891,305892],{},"2. The situation where the processor spends most of its time swapping process pieces rather than executing instructions is called:","\na. Paging\nb. The Principle of Locality\nc. Thrashing\nd. None of the above",[105,305895,305896,305901,305908],{},[108,305897,305898,305900],{},[2627,305899,291953],{},"：处理器将大部分时间花在交换进程片段而不是执行指令上的情况被称为：",[108,305902,305903,2664,305905],{},[2627,305904,118575],{},[2627,305906,305907],{},"c. Thrashing",[108,305909,305910,305912,305913,305915],{},[2627,305911,305389],{},"：这就是",[2627,305914,305423],{}," 的标准定义。当系统中的内存过度承诺 (overcommitted) 时，进程不断发生缺页中断，导致系统资源几乎全部耗费在磁盘 I\u002FO 的换入换出上，CPU 的实际利用率急剧下降。",[15,305917,305918,305921],{},[2627,305919,305920],{},"3. The situation that occurs when the desired page table entry is not found in the Translation Lookaside Buffer (TLB) is called a:","\na. TLB miss\nb. TLB hit\nc. Page fault\nd. None of the above",[105,305923,305924,305929,305936],{},[108,305925,305926,305928],{},[2627,305927,291953],{},"：当在转换后备缓冲器 (TLB) 中找不到所需的页表项时发生的情况被称为：",[108,305930,305931,2664,305933],{},[2627,305932,118575],{},[2627,305934,305935],{},"a. TLB miss",[108,305937,305938,305940,305941,305944],{},[2627,305939,305389],{},"：TLB 是页表的高速缓存。查找 TLB 时没有找到目标页表项，称为 ",[2627,305942,305943],{},"TLB 缺失 (TLB Miss)","。此时硬件需要去访问主存中完整的页表来获取映射信息，并将该条目加载进 TLB。",[15,305946,305947,305950],{},[2627,305948,305949],{},"4. The real address of a word in memory is translated from the following portions of a virtual address:","\na. Page number and frame number\nb. Page number and offset\nc. Frame number and offset\nd. None of the above",[105,305952,305953,305958,305965],{},[108,305954,305955,305957],{},[2627,305956,291953],{},"：内存中一个字的实地址（物理地址）是从虚拟地址的以下部分转换而来的：",[108,305959,305960,2664,305962],{},[2627,305961,118575],{},[2627,305963,305964],{},"b. Page number and offset",[108,305966,305967,305969,305970,258,305973,305975],{},[2627,305968,305389],{},"：虚拟地址（逻辑地址）由两部分组成：",[2627,305971,305972],{},"页号 (Page number)",[2627,305974,211433],{},"。MMU（内存管理单元）通过查找页表，将“页号”转换为物理内存中的“页框号 (Frame number)”，然后将页框号与原封不动的“页内偏移量”拼接起来，就得到了最终的实际物理地址。",[15,305977,305978,305981],{},[2627,305979,305980],{},"5. Segmentation has a number of advantages to the programmer over a non-segmented address space, including:","\na. Simplifying the handling of growing data structures\nb. Sharing among processes\nc. Protection\nd. All of the above",[105,305983,305984,305989,305995],{},[108,305985,305986,305988],{},[2627,305987,291953],{},"：与非分段地址空间相比，分段对程序员有许多优点，包括：",[108,305990,305991,2664,305993],{},[2627,305992,118575],{},[2627,305994,292510],{},[108,305996,305997,305999],{},[2627,305998,305389],{},"：分段由于符合程序的逻辑视图，因此：(a) 可以让数组或堆栈段独立增长；(b) 可以轻松实现进程间的代码或数据共享（只需共享特定的段）；(c) 可以对不同的段施加不同的保护权限（例如代码段设为只读，数据段设为读写）。",[15,306001,306002,306005],{},[2627,306003,306004],{},"6. In a combined paging\u002Fsegmentation system, a user's address space is broken up into a number of:","\na. Segments or pages, at the discretion of the programmer\nb. Fixed-size pages, which are in turn broken down into variable-sized segments\nc. Variable-sized Segments, which are in turn broken down into fixed-size pages\nd. All of the above",[105,306007,306008,306013,306020],{},[108,306009,306010,306012],{},[2627,306011,291953],{},"：在结合了分页\u002F分段的系统中，用户的地址空间被分为许多：",[108,306014,306015,2664,306017],{},[2627,306016,118575],{},[2627,306018,306019],{},"c. Variable-sized Segments, which are in turn broken down into fixed-size pages",[108,306021,306022,306024,306025,306028,306029,306032],{},[2627,306023,305389],{},"：这就是段页式管理的精髓。",[2627,306026,306027],{},"宏观上","对程序员来说，程序被划分为大小可变的逻辑段 (Segments)；",[2627,306030,306031],{},"微观上","对操作系统来说，为了消除外部碎片，每个段在内部又被切割成了多个固定大小的页 (Pages) 进行物理内存管理。",[15,306034,306035,306038],{},[2627,306036,306037],{},"7. Sharing is achieved in a segmentation system by:","\na. Referencing a segment in the segment tables of more than one process\nb. Each process segment table having a reference to the dispatcher main memory area\nc. Having a common data area that all processes can share\nd. All of the above",[105,306040,306041,306046,306053],{},[108,306042,306043,306045],{},[2627,306044,291953],{},"：在分段系统中，共享是通过什么实现的：",[108,306047,306048,2664,306050],{},[2627,306049,118575],{},[2627,306051,306052],{},"a. Referencing a segment in the segment tables of more than one process",[108,306054,306055,306057],{},[2627,306056,305389],{},"：通过在不同进程的段表 (Segment Table) 中建立一个表项，让这些表项指向同一个物理内存区域（同一个段），即可轻松实现共享。",[15,306059,306060,306063],{},[2627,306061,306062],{},"8. A fundamental choice in the design of the memory-management portion of an O\u002FS is:","\na. Whether or not to use virtual memory techniques\nb. Whether to use paging, segmentation of a combination of the two\nc. The algorithms employed for various aspects of memory management\nd. All of the above",[105,306065,306066,306071,306077],{},[108,306067,306068,306070],{},[2627,306069,291953],{},"：操作系统内存管理部分设计的一个基本选择是：",[108,306072,306073,2664,306075],{},[2627,306074,118575],{},[2627,306076,292510],{},[108,306078,306079,306081],{},[2627,306080,305389],{},"：设计 OS 的内存管理涉及宏观架构到微观算法的选择，包括：要不要虚拟内存（现代通用 OS 肯定要，但一些嵌入式 OS 可能不需要）、采用何种硬件映射结构（分页、分段还是混合）、以及具体的放置、获取、置换等算法的选择。",[15,306083,306084,306087],{},[2627,306085,306086],{},"9. The fetch policy that exploits the characteristics of most secondary memory devices, such as disks, which have seek time and rotational latency is called:","\na. Demand paging\nb. Prepaging\nc. Swapping\nd. None of the above",[105,306089,306090,306095,306102],{},[108,306091,306092,306094],{},[2627,306093,291953],{},"：利用磁盘等辅助存储设备具有寻道时间和旋转延迟这一特性的获取策略被称为：",[108,306096,306097,2664,306099],{},[2627,306098,118575],{},[2627,306100,306101],{},"b. Prepaging",[108,306103,306104,306106,306107,306110],{},[2627,306105,305389],{},"：磁盘 I\u002FO 中最耗时的是寻道时间和旋转延迟，而实际的顺序数据传输非常快。",[2627,306108,306109],{},"预调页 (Prepaging)"," 利用了这一点，当发生缺页中断时，不仅读入发生缺页的那个页面，还将它在磁盘上相邻的几个页面一起顺便读入内存。这极大摊销了 I\u002FO 开销，并且根据空间局部性原理，相邻页面大概率很快会被用到。",[15,306112,306113,306116],{},[2627,306114,306115],{},"10. The replacement policy that is impossible to implement because it would require the O\u002FS to have perfect knowledge of future events is called the:","\na. Optimal policy\nb. Least recently used (LRU) policy\nc. Clock policy\nd. None of the above",[105,306118,306119,306124,306131],{},[108,306120,306121,306123],{},[2627,306122,291953],{},"：因为要求操作系统拥有对未来事件的完美了解而不可能实现的置换策略被称为：",[108,306125,306126,2664,306128],{},[2627,306127,118575],{},[2627,306129,306130],{},"a. Optimal policy",[108,306132,306133,2664,306135,306138],{},[2627,306134,305389],{},[2627,306136,306137],{},"最佳置换策略 (Optimal Policy, OPT)"," 要求置换出那个在“未来最长时间内不会被访问到”的页面。由于 OS 无法预知进程未来的执行轨迹，所以它无法真正在生产系统中实现，仅用来作为评估其他算法性能的理论上限（标杆）。",[15,306140,306141,306144],{},[2627,306142,306143],{},"11. The replacement policy that chooses only among the resident pages of the process that generated the page fault in selecting a page to replace is referred to as a:","\na. Global replacement policy\nb. Local replacement policy\nc. Variable replacement policy\nd. None of the above",[105,306146,306147,306152,306159],{},[108,306148,306149,306151],{},[2627,306150,291953],{},"：在选择要置换的页面时，仅在产生缺页中断的进程自身的驻留页面中进行选择的置换策略被称为：",[108,306153,306154,2664,306156],{},[2627,306155,118575],{},[2627,306157,306158],{},"b. Local replacement policy",[108,306160,306161,305014,306163,306165],{},[2627,306162,305389],{},[2627,306164,305678],{}," 的定义。它的好处是进程之间不会因为抢夺页框而相互影响，缺点是如果系统初始分配给该进程的页框太少，它会频繁发生内部的缺页中断。",[15,306167,306168,306171],{},[2627,306169,306170],{},"12. The concept associated with determining the number of processes that will be resident in main memory is referred to as:","\na. A cleaning policy\nb. The page fault frequency\nc. Load Control\nd. None of the above",[105,306173,306174,306179,306186],{},[108,306175,306176,306178],{},[2627,306177,291953],{},"：决定将有多少个进程驻留在主存中的相关概念被称为：",[108,306180,306181,2664,306183],{},[2627,306182,118575],{},[2627,306184,306185],{},"c. Load Control",[108,306187,306188,2664,306190,306193],{},[2627,306189,305389],{},[2627,306191,306192],{},"负载控制 (Load Control)"," 决定了系统的多道程序度 (Multiprogramming level)。如果驻留进程太少，CPU 经常会因为所有进程都在等待 I\u002FO 而空闲；如果驻留进程太多，每个进程分到的页框太少，又会导致严重的抖动 (Thrashing)。因此必须精确控制。",[15,306195,306196,306199],{},[2627,306197,306198],{},"13. In SVR4 and Solaris systems, the memory management scheme that manages user processes and disk I\u002FO is called the:","\na. Paging system\nb. Virtual memory manager\nc. Kernel memory allocator\nd. None of the above",[105,306201,306202,306207,306214],{},[108,306203,306204,306206],{},[2627,306205,291953],{},"：在 SVR4 和 Solaris 系统中，管理用户进程和磁盘 I\u002FO 的内存管理方案被称为：",[108,306208,306209,2664,306211],{},[2627,306210,118575],{},[2627,306212,306213],{},"a. Paging system",[108,306215,306216,306218,306219,306221],{},[2627,306217,305389],{},"：如前所述，Unix系统如 SVR4 的内存管理主要分两部分：用于用户和磁盘缓冲的 ",[2627,306220,305740],{},"，以及专用于内核小对象分配的内核内存分配器 (KMA)。",[15,306223,306224,306227],{},[2627,306225,306226],{},"14. The multi-level memory management scheme implemented in Linux was designed to minimize large page tables and directories in which of the following line of processors:","\na. 16-bit X86 architecture\nb. 32-bit Pentium\u002FX86 architecture\nc. 64-bit Alpha architecture\nd. None of the above",[105,306229,306230,306235,306242],{},[108,306231,306232,306234],{},[2627,306233,291953],{},"：在 Linux 中实现的多级内存管理方案旨在最大限度地减少大页表和目录，这最初是为了适应下列哪种处理器架构：",[108,306236,306237,2664,306239],{},[2627,306238,118575],{},[2627,306240,306241],{},"c. 64-bit Alpha architecture",[108,306243,306244,306246],{},[2627,306245,305389],{},"：最初的 Linux 只支持 32 位的 x86 处理器，使用了两级页表。当 Linux 移植到 64 位的 DEC Alpha 架构时，巨大的地址空间使得两级页表会占用庞大的连续内存，因此内核开发者引入了三级页表结构来稀疏化并减少页表的内存占用。",[15,306248,306249,306252],{},[2627,306250,306251],{},"15. The Windows 2000 virtual memory manager can use page sizes ranging from:","\na. 4 KB to 64 KB\nb. 64 KB to 4 GB\nc. 4 GB to 4 TB\nd. None of the above",[105,306254,306255,306260,306267],{},[108,306256,306257,306259],{},[2627,306258,291953],{},"：Windows 2000 虚拟内存管理器可以使用的页面大小范围是：",[108,306261,306262,2664,306264],{},[2627,306263,118575],{},[2627,306265,306266],{},"a. 4 KB to 64 KB",[108,306268,306269,306271],{},[2627,306270,305389],{},"：操作系统通常支持不同架构下的多种页面大小。在 Windows 2000 中，根据底层处理器的不同，页面大小的范围是从 4 KB (例如 x86 系统) 到 64 KB (某些其他支持的架构) 不等。同时现代硬件通常也支持利用大页面 (Large Pages)，例如 x86 下的 2MB 或 4MB 页面，但在本题考查的章节上下文细节中，选项 a 反映了不同基础架构的常规页面大小设定范围。",[32188,306273],{},[19,306275,292903],{"id":292902},[15,306277,306278],{},[2627,306279,306280],{},"1. In a system employing a memory management strategy that doesn't require an entire process to be in main memory at one time, the portion of a process that is actually in main memory at any given time is defined to be the ________ of the process.",[105,306282,306283,306288,306295],{},[108,306284,306285,306287],{},[2627,306286,291953],{},"：进程在任意给定时间实际存在于主存中的那部分被称为该进程的 ________。",[108,306289,306290,2664,306292],{},[2627,306291,118575],{},[2627,306293,306294],{},"resident set (驻留集)",[108,306296,306297,306299],{},[2627,306298,305389],{},"：进程的所有页面集合构成它的虚拟地址空间，而真正加载到物理内存里的那个子集就叫做驻留集。",[15,306301,306302],{},[2627,306303,306304],{},"2. The situation where the processor spends most of its time swapping process pieces rather than executing instructions is called ________.",[105,306306,306307,306312,306319],{},[108,306308,306309,306311],{},[2627,306310,291953],{},"：处理器将大部分时间花在交换进程片段而不是执行指令上的情况被称为 ________。",[108,306313,306314,2664,306316],{},[2627,306315,118575],{},[2627,306317,306318],{},"thrashing (抖动)",[108,306320,306321,306323],{},[2627,306322,305389],{},"：这是一种极度恶化的系统状态，多道程序度越过了最优边界后性能断崖式下跌的表现。",[15,306325,306326],{},[2627,306327,306328],{},"3. Most virtual memory schemes make use of a special high-speed cache for page table entries, called a ________.",[105,306330,306331,306336,306343],{},[108,306332,306333,306335],{},[2627,306334,291953],{},"：大多数虚拟内存方案都使用一个特殊的用于页表项的高速缓存，称为 ________。",[108,306337,306338,2664,306340],{},[2627,306339,118575],{},[2627,306341,306342],{},"Translation Lookaside Buffer \u002F TLB (转换后备缓冲器)",[108,306344,306345,306347],{},[2627,306346,305389],{},"：由于每次虚拟地址到物理地址的转换都需要访问内存中的页表，为了加速这一过程，TLB 会缓存最近使用过的页表项，其硬件本质是相联存储器 (Associative Memory)。",[15,306349,306350],{},[2627,306351,306352],{},"4. Each entry in a ________ contains control bits and the corresponding frame number if the page is resident in memory.",[105,306354,306355,306360,306367],{},[108,306356,306357,306359],{},[2627,306358,291953],{},"：如果页面驻留在内存中，________ 中的每个条目都包含控制位和对应的页框号。",[108,306361,306362,2664,306364],{},[2627,306363,118575],{},[2627,306365,306366],{},"page table (页表)",[108,306368,306369,306371],{},[2627,306370,305389],{},"：页表将逻辑页映射到物理页框。其条目 (PTE) 除了包含页框号，通常还包括存在位 (Present Bit)、修改位 (Modify Bit)、访问位 (Accessed Bit) 以及读写权限等控制位。",[15,306373,306374],{},[2627,306375,306376],{},"5. In a segmentation system, each entry in a ________ contains control bits and the starting address and the length of the segment.",[105,306378,306379,306384,306391],{},[108,306380,306381,306383],{},[2627,306382,291953],{},"：在分段系统中，________ 中的每个条目都包含控制位以及该段的起始地址和长度。",[108,306385,306386,2664,306388],{},[2627,306387,118575],{},[2627,306389,306390],{},"segment table (段表)",[108,306392,306393,306395,306396,306399],{},[2627,306394,305389],{},"：因为段是大小可变的，因此不仅要记录它在内存中的起始物理位置（基址），还必须记录它的边界即",[2627,306397,306398],{},"段长 (length \u002F limit)","。如果程序访问的偏移量超出了段长，硬件会立刻触发越界异常（这也是保护机制的核心）。",[15,306401,306402],{},[2627,306403,306404],{},"6. ________ which is transparent to the programmer, eliminates external fragmentation, provides efficient use of main memory, and has pieces of fixed, equal size.",[105,306406,306407,306412,306419],{},[108,306408,306409,306411],{},[2627,306410,291953],{},"：________ 对程序员是透明的，它消除了外部碎片，提供了主存的高效利用，并且具有固定、相等大小的块。",[108,306413,306414,2664,306416],{},[2627,306415,118575],{},[2627,306417,306418],{},"Paging (分页)",[108,306420,306421,306423],{},[2627,306422,305389],{},"：由于页的大小是固定的，操作系统可以直接将任何一页放进任意一个空闲的页框中，无需连续分配，彻底消灭了外部碎片，但它是纯底层的硬件\u002F操作系统机制，程序员通常感知不到它的存在。",[15,306425,306426],{},[2627,306427,306428],{},"7. ________ which is visible to the programmer, has the ability to handle growing data structures, modularity, and support for sharing and protection.",[105,306430,306431,306436,306443],{},[108,306432,306433,306435],{},[2627,306434,291953],{},"：________ 对程序员是可见的，具备处理不断增长的数据结构的能力，支持模块化，并支持共享和保护。",[108,306437,306438,2664,306440],{},[2627,306439,118575],{},[2627,306441,306442],{},"Segmentation (分段)",[108,306444,306445,306447],{},[2627,306446,305389],{},"：分段完全符合程序模块的逻辑划分，是高级语言及程序员眼中的内存视图。",[15,306449,306450],{},[2627,306451,306452],{},"8. An example of an O\u002FS that doesn't provide virtual memory is ________.",[105,306454,306455,306460,306467],{},[108,306456,306457,306459],{},[2627,306458,291953],{},"：不提供虚拟内存的操作系统的一个例子是 ________。",[108,306461,306462,2664,306464],{},[2627,306463,118575],{},[2627,306465,306466],{},"MS-DOS (或早期操作系统 \u002F 某些嵌入式操作系统)",[108,306468,306469,306471],{},[2627,306470,305389],{},"：像 MS-DOS 这类早期的、在资源极其受限环境下运行的单任务操作系统，并没有实现复杂的虚拟内存管理。",[15,306473,306474],{},[2627,306475,306476],{},"9. The fetch policy where a page is brought into main memory only if a reference is made to a location on that page is called ________.",[105,306478,306479,306484,306491],{},[108,306480,306481,306483],{},[2627,306482,291953],{},"：只有在对页面上的位置进行引用时才将该页面引入主存的获取策略被称为 ________。",[108,306485,306486,2664,306488],{},[2627,306487,118575],{},[2627,306489,306490],{},"demand paging (请求调页)",[108,306492,306493,306495],{},[2627,306494,305389],{},"：这是最经典的虚拟内存拉取策略。进程启动时不装入所有的页，而是发生缺页中断时，才“按需”从磁盘上把对应的页调入内存。",[15,306497,306498],{},[2627,306499,306500],{},"10. The replacement policy that treats the page frames allocated to a process as a circular buffer is called ________.",[105,306502,306503,306508,306515],{},[108,306504,306505,306507],{},[2627,306506,291953],{},"：将分配给进程的页框视为循环缓冲区的置换策略被称为 ________。",[108,306509,306510,2664,306512],{},[2627,306511,118575],{},[2627,306513,306514],{},"FIFO (先进先出 \u002F First-In-First-Out) 或 Clock (时钟策略)",[108,306516,306517,306519],{},[2627,306518,305389],{},"：FIFO 策略本质上就是维护一个队列（循环缓冲区），换出在内存中驻留时间最久的页面。Clock 策略也使用循环缓冲区的结构，配合一个指针（像钟表指针）和使用位 (Use Bit) 来扫描和置换。填 FIFO 是最基础直接的答案。",[15,306521,306522],{},[2627,306523,306524],{},"11. A ________ replacement policy chooses only among the resident ages of the process that generated the page fault in selecting a page to replace.",[105,306526,306527,306532,306539],{},[108,306528,306529,306531],{},[2627,306530,291953],{},"：________ 置换策略在选择要置换的页面时，仅在产生缺页中断的进程自身的驻留页面中进行选择。",[108,306533,306534,2664,306536],{},[2627,306535,118575],{},[2627,306537,306538],{},"local (局部)",[108,306540,306541,306543],{},[2627,306542,305389],{},"：局部置换意味着各个进程独立管理各自的页面池，相互不影响。",[15,306545,306546],{},[2627,306547,306548],{},"12. A ________ policy writes modified pages before their page frames are needed so that pages can be written out in batches.",[105,306550,306551,306556,306563],{},[108,306552,306553,306555],{},[2627,306554,291953],{},"：________ 策略在需要页框之前先写入修改过的页面，这样页面就可以分批写出。",[108,306557,306558,2664,306560],{},[2627,306559,118575],{},[2627,306561,306562],{},"precleaning (预清除)",[108,306564,306565,306567],{},[2627,306566,305389],{},"：与之相对的 Demand Cleaning（按需清除）是在置换发生时才写磁盘。Precleaning 可以利用磁盘批处理写入来提高吞吐量。",[15,306569,306570],{},[2627,306571,306572],{},"13. In SVR4 and Solaris systems, the memory management scheme that manages memory allocation for the kernel is called the ________.",[105,306574,306575,306580,306587],{},[108,306576,306577,306579],{},[2627,306578,291953],{},"：在 SVR4 和 Solaris 系统中，为内核管理内存分配的内存管理方案被称为 ________。",[108,306581,306582,2664,306584],{},[2627,306583,118575],{},[2627,306585,306586],{},"kernel memory allocator \u002F KMA (内核内存分配器)",[108,306588,306589,306591],{},[2627,306590,305389],{},"：内核自身不能被换出（通常常驻物理内存），且它需要频繁地创建和销毁微小的结构（远远小于一个 Page 的大小），因此操作系统需要 KMA 这样的特殊设施来高效管理这一层。",[15,306593,306594],{},[2627,306595,306596],{},"14. Linux systems use a ________-level page table structure in its memory management scheme to accommodate large addresses.",[105,306598,306599,306604,306611],{},[108,306600,306601,306603],{},[2627,306602,291953],{},"：Linux 系统在其内存管理方案中使用 ________ 级页表结构来容纳大地址。",[108,306605,306606,2664,306608],{},[2627,306607,118575],{},[2627,306609,306610],{},"three (三) \u002F (或 multi\u002F多)",[108,306612,306613,306615],{},[2627,306614,305389],{},"：根据原版教材上下文，为了适应 64 位寻址，Linux 确立了经典的三级页表结构：PGD、PMD、PTE。（注：最新内核已发展到四级或五级，但作为这本教材的填空考核点，通常标准答案是 three）。",[15,306617,306618],{},[2627,306619,306620],{},"15. In a W2K system, although each user sees a 32-bit address space, allowing 4 GB of memory per process, a portion of this memory is reserved for O\u002FS use, so a user process actually has access to ________ of virtual address space.",[105,306622,306623,306628,306635],{},[108,306624,306625,306627],{},[2627,306626,291953],{},"：在 W2K 系统中，虽然每个用户都能看到一个 32 位的地址空间（允许每个进程有 4GB 的内存），但这部分内存有一部分是保留给 OS 使用的，因此用户进程实际可访问的虚拟地址空间为 ________。",[108,306629,306630,2664,306632],{},[2627,306631,118575],{},[2627,306633,306634],{},"2 GB",[108,306636,306637,306639,306640,53834,306643,306646,306647,53834,306650,306653],{},[2627,306638,305389],{},"：在默认情况下的 32 位 Windows 系统中，4 GB 的总虚拟地址空间被切半：",[31,306641,306642],{},"0x00000000",[31,306644,306645],{},"0x7FFFFFFF"," 的 2 GB 给用户态程序自由使用，",[31,306648,306649],{},"0x80000000",[31,306651,306652],{},"0xFFFFFFFF"," 的 2 GB 强制保留给操作系统内核以及硬件驱动程序。",{"title":29,"searchDepth":43,"depth":43,"links":306655},[306656,306657,306658],{"id":291940,"depth":43,"text":291941},{"id":292391,"depth":43,"text":292392},{"id":292902,"depth":43,"text":292903},{"order":1971},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap8",{"title":305358,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap8","X9ywU8f1CAdwBdAwi-xvJBkIu2K9oXsR34EjJtGX45U",{"id":306665,"title":306666,"body":306667,"description":29,"extension":85,"meta":308464,"navigation":52,"path":308465,"seo":308466,"stem":308467,"__hash__":308468},"notes\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap9.md","Chap. 9",{"type":8,"value":306668,"toc":308459},[306669,306673,306677,306682,306717,306722,306762,306767,306794,306799,306830,306835,306862,306867,306903,306908,306935,306940,306971,306976,307003,307008,307042,307047,307074,307079,307351,307356,307383,307388,307415,307420,307451,307453,307457,307463,307493,307499,307526,307532,307562,307568,307599,307605,307642,307648,307679,307685,307716,307722,307752,307758,307789,307795,307822,307828,307863,307869,308019,308025,308052,308058,308088,308094,308125,308127,308129,308134,308151,308156,308173,308178,308195,308200,308217,308222,308239,308244,308261,308266,308283,308288,308305,308310,308327,308332,308349,308354,308371,308376,308393,308398,308415,308420,308437,308442],[11,306670,306672],{"id":306671},"chapter-9-uniprocessor-scheduling","Chapter 9 Uniprocessor Scheduling",[19,306674,306676],{"id":306675},"true-false-questions-是非题","True \u002F False Questions (是非题)",[15,306678,306679],{},[2627,306680,306681],{},"1. Scheduling affects the performance of a system because it determines which processes will wait and which will progress.",[105,306683,306684,306689,306693],{},[108,306685,306686,306688],{},[2627,306687,291953],{},"：调度会影响系统的性能，因为它决定了哪些进程需要等待，哪些进程可以继续执行。",[108,306690,306691,303126],{},[2627,306692,118575],{},[108,306694,306695,33183,306697],{},[2627,306696,118582],{},[105,306698,306699,306705,306711],{},[108,306700,306701,306704],{},[2627,306702,306703],{},"Concept Check (概念检查)","：调度的核心目的是在多道程序环境中分配处理器资源。",[108,306706,306707,306710],{},[2627,306708,306709],{},"The 'Why' (为什么)","：系统性能（如吞吐量、响应时间）直接受进程队列管理的影响，调度决定了进程状态的转换。",[108,306712,306713,306716],{},[2627,306714,306715],{},"Common Pitfalls (常见陷阱)","：不要误以为调度只是简单的“排队”，它实际上是决定系统整体效率的核心机制。",[15,306718,306719],{},[2627,306720,306721],{},"2. The short-term scheduler may limit the degree of multiprogramming to provide satisfactory service to the current set of processes.",[105,306723,306724,306729,306733],{},[108,306725,306726,306728],{},[2627,306727,291953],{},"：短期调度程序可能会限制多道程序设计的程度，以向当前进程集提供令人满意的服务。",[108,306730,306731,303091],{},[2627,306732,118575],{},[108,306734,306735,33183,306737],{},[2627,306736,118582],{},[105,306738,306739,306744,306757],{},[108,306740,306741,306743],{},[2627,306742,306703],{},"：操作系统的三级调度：长期调度（Long-term）、中期调度（Medium-term）和短期调度（Short-term）。",[108,306745,306746,306748,306749,306752,306753,306756],{},[2627,306747,306709],{},"：控制多道程序设计程度（Degree of Multiprogramming）的是",[2627,306750,306751],{},"长期调度程序","（决定接纳多少新进程）和",[2627,306754,306755],{},"中期调度程序","（决定多少进程驻留主存）。短期调度（分派器）只负责在当前“就绪”进程中挑一个执行。",[108,306758,306759,306761],{},[2627,306760,306715],{},"：极易混淆三种调度的职责。记住：长期管“门槛”，中期管“内外（内存\u002F外存）”，短期管“上CPU”。",[15,306763,306764],{},[2627,306765,306766],{},"3. Medium-term scheduling is part of the system swapping function.",[105,306768,306769,306774,306778],{},[108,306770,306771,306773],{},[2627,306772,291953],{},"：中期调度是系统交换（Swapping）功能的一部分。",[108,306775,306776,303126],{},[2627,306777,118575],{},[108,306779,306780,33183,306782],{},[2627,306781,118582],{},[105,306783,306784,306789],{},[108,306785,306786,306788],{},[2627,306787,306703],{},"：中期调度决定将哪些进程部分或全部换入（Swap-in）或换出（Swap-out）主存。",[108,306790,306791,306793],{},[2627,306792,306709],{},"：通过交换，中期调度能够有效管理主存空间和多道程序设计的程度，当内存不足时将进程挂起至磁盘。",[15,306795,306796],{},[2627,306797,306798],{},"4. The long-term scheduler is invoked whenever an event occurs that may lead to the suspension or preemption of the currently running process.",[105,306800,306801,306806,306810],{},[108,306802,306803,306805],{},[2627,306804,291953],{},"：每当发生可能导致当前运行进程挂起或被抢占的事件时，都会调用长期调度程序。",[108,306807,306808,303091],{},[2627,306809,118575],{},[108,306811,306812,33183,306814],{},[2627,306813,118582],{},[105,306815,306816,306821],{},[108,306817,306818,306820],{},[2627,306819,306703],{},"：调度触发的时机。",[108,306822,306823,306825,306826,306829],{},[2627,306824,306709],{},"：处理中断、挂起或抢占等频繁发生的事件时，调用的是",[2627,306827,306828],{},"短期调度程序 (Short-term scheduler)","。长期调度只有在作业\u002F进程创建时才被偶尔调用。",[15,306831,306832],{},[2627,306833,306834],{},"5. The main objective of short-term scheduling is to allocate processor time in such a way as to optimize one or more aspects of system behavior.",[105,306836,306837,306842,306846],{},[108,306838,306839,306841],{},[2627,306840,291953],{},"：短期调度的主要目标是以优化系统行为的一个或多个方面的方式来分配处理器时间。",[108,306843,306844,303126],{},[2627,306845,118575],{},[108,306847,306848,33183,306850],{},[2627,306849,118582],{},[105,306851,306852,306857],{},[108,306853,306854,306856],{},[2627,306855,306703],{},"：短期调度标准（Scheduling Criteria），如响应时间、周转时间、吞吐量等。",[108,306858,306859,306861],{},[2627,306860,306709],{},"：分派CPU资源时，必须基于某种策略（如FCFS、SPN）来最大化CPU利用率或最小化用户等待时间。",[15,306863,306864],{},[2627,306865,306866],{},"6. One problem with a pure priority scheduling scheme is that lower-priority processes may suffer deadlock.",[105,306868,306869,306874,306878],{},[108,306870,306871,306873],{},[2627,306872,291953],{},"：纯优先级调度方案的一个问题是，较低优先级的进程可能会遭受死锁。",[108,306875,306876,303091],{},[2627,306877,118575],{},[108,306879,306880,33183,306882],{},[2627,306881,118582],{},[105,306883,306884,306889,306898],{},[108,306885,306886,306888],{},[2627,306887,306703],{},"：死锁 (Deadlock) 与饥饿 (Starvation) 的区别。",[108,306890,306891,306893,306894,306897],{},[2627,306892,306709],{},"：低优先级进程因为一直有高优先级进程抢占CPU而无法运行，这叫做",[2627,306895,306896],{},"饥饿 (Starvation)","，而不是死锁。死锁通常指多个进程互相等待对方持有的资源而永远僵持。",[108,306899,306900,306902],{},[2627,306901,306715],{},"：把“一直等不到资源”一律叫死锁。死锁是“互相等”，饥饿是“单方面一直被插队”。",[15,306904,306905],{},[2627,306906,306907],{},"7. The selection function determines which process, among ready processes, is selected next for execution.",[105,306909,306910,306915,306919],{},[108,306911,306912,306914],{},[2627,306913,291953],{},"：选择函数（Selection function）决定在就绪进程中选择哪一个进行下一次执行。",[108,306916,306917,303126],{},[2627,306918,118575],{},[108,306920,306921,33183,306923],{},[2627,306922,118582],{},[105,306924,306925,306930],{},[108,306926,306927,306929],{},[2627,306928,306703],{},"：调度算法的两个核心维度：选择函数 (Selection Function) 和决策模式 (Decision Mode，如抢占\u002F非抢占)。",[108,306931,306932,306934],{},[2627,306933,306709],{},"：选择函数通过某种评估（如等待时间长短、所需服务时间长短）挑选出胜出者。",[15,306936,306937],{},[2627,306938,306939],{},"8. First-come-first-served (FCFS) is a simple scheduling policy that tends to favor I\u002FO-bound processes over processor bound processes.",[105,306941,306942,306947,306951],{},[108,306943,306944,306946],{},[2627,306945,291953],{},"：先来先服务（FCFS）是一种简单的调度策略，它倾向于偏好 I\u002FO 密集型进程，而不是处理器密集型进程。",[108,306948,306949,303091],{},[2627,306950,118575],{},[108,306952,306953,33183,306955],{},[2627,306954,118582],{},[105,306956,306957,306962],{},[108,306958,306959,306961],{},[2627,306960,306703],{},"：FCFS对不同类型进程的公平性。",[108,306963,306964,306966,306967,306970],{},[2627,306965,306709],{},"：FCFS 实际上偏好",[2627,306968,306969],{},"处理器密集型 (Processor-bound)"," 进程。因为非抢占式的特性，一个长CPU密集型进程一旦获取CPU就会一直运行，导致后面短的 I\u002FO 密集型进程长时间等待，极大降低了 I\u002FO 设备的利用率。",[15,306972,306973],{},[2627,306974,306975],{},"9. Round Robin is a scheduling technique is also known as time slicing, because each process is given a slice of time before being preempted.",[105,306977,306978,306983,306987],{},[108,306979,306980,306982],{},[2627,306981,291953],{},"：轮转（Round Robin）是一种调度技术，也称为时间片轮转（time slicing），因为每个进程在被抢占之前都会被分配一个时间片。",[108,306984,306985,303126],{},[2627,306986,118575],{},[108,306988,306989,33183,306991],{},[2627,306990,118582],{},[105,306992,306993,306998],{},[108,306994,306995,306997],{},[2627,306996,306703],{},"：Round Robin (RR) 的基本原理。",[108,306999,307000,307002],{},[2627,307001,306709],{},"：为防止长进程霸占CPU，系统设置一个固定的时间量子 (Time Quantum)，时间一到强制切走，这就是 Time Slicing。",[15,307004,307005],{},[2627,307006,307007],{},"10. The Shortest Process Next (SPN) scheduling policy is often used for time-sharing and transaction processing environments because of the lack of preemption.",[105,307009,307010,307015,307019],{},[108,307011,307012,307014],{},[2627,307013,291953],{},"：由于缺乏抢占性，最短进程优先（SPN）调度策略通常用于分时和事务处理环境。",[108,307016,307017,303091],{},[2627,307018,118575],{},[108,307020,307021,33183,307023],{},[2627,307022,118582],{},[105,307024,307025,307030],{},[108,307026,307027,307029],{},[2627,307028,306703],{},"：不同调度环境的需求。",[108,307031,307032,307034,307035,307038,307039,307041],{},[2627,307033,306709],{},"：分时和交互式环境",[2627,307036,307037],{},"必须","具有抢占性（如Round Robin），以保证用户的响应时间。SPN是",[2627,307040,215645],{},"的，会导致长进程一旦执行就不会被中断，无法满足分时环境对快速响应的要求。",[15,307043,307044],{},[2627,307045,307046],{},"11. The Shortest Remaining Time (SRT) scheduling policy is a preemptive version of the Shortest Process Next (SPN) scheduling policy.",[105,307048,307049,307054,307058],{},[108,307050,307051,307053],{},[2627,307052,291953],{},"：最短剩余时间（SRT）调度策略是最短进程优先（SPN）调度策略的抢占式版本。",[108,307055,307056,303126],{},[2627,307057,118575],{},[108,307059,307060,33183,307062],{},[2627,307061,118582],{},[105,307063,307064,307069],{},[108,307065,307066,307068],{},[2627,307067,306703],{},"：SRT与SPN的联系与区别。",[108,307070,307071,307073],{},[2627,307072,306709],{},"：SPN是在CPU空闲时挑最短的执行且不可剥夺；SRT不仅挑最短的，如果新来的进程比当前进程剩余的时间还短，它会**剥夺（Preempt）**当前进程。",[15,307075,307076],{},[2627,307077,307078],{},"12. In the Highest Response Ratio Next (HRRN) scheduling policy, longer jobs are favored, because they yield a larger ratio from the smaller denominator in the equation.",[105,307080,307081,307086,307090],{},[108,307082,307083,307085],{},[2627,307084,291953],{},"：在最高响应比优先（HRRN）调度策略中，较长的作业更受偏好，因为它们在公式中由较小的分母产生较大的比率。",[108,307087,307088,303091],{},[2627,307089,118575],{},[108,307091,307092,33183,307094],{},[2627,307093,118582],{},[105,307095,307096,307252],{},[108,307097,307098,307100,307101,70884,307194,307222,307223,307251],{},[2627,307099,306703],{},"：响应比公式 ",[34,307102,307104,307134],{"className":307103},[31642],[34,307105,307107],{"className":307106},[31646],[31648,307108,307109],{"xmlns":31650},[31652,307110,307111,307131],{},[31655,307112,307113,307115,307117,307119,307121,307123,307125,307127,307129],{},[31658,307114,34386],{},[31662,307116,202],{},[31662,307118,456],{"stretchy":702},[31658,307120,47077],{},[31662,307122,223],{},[31658,307124,31747],{},[31662,307126,2557],{"stretchy":702},[31658,307128,38080],{"mathvariant":38079},[31658,307130,31747],{},[31665,307132,307133],{"encoding":31667},"R = (w + s) \u002F s",[34,307135,307137,307155,307176],{"className":307136,"ariaHidden":758},[31672],[34,307138,307140,307143,307146,307149,307152],{"className":307139},[31676],[34,307141],{"className":307142,"style":33218},[31680],[34,307144,34386],{"className":307145,"style":34427},[31685,31686],[34,307147],{"className":307148,"style":31846},[31845],[34,307150,202],{"className":307151},[31850],[34,307153],{"className":307154,"style":31846},[31845],[34,307156,307158,307161,307164,307167,307170,307173],{"className":307157},[31676],[34,307159],{"className":307160,"style":31774},[31680],[34,307162,456],{"className":307163},[31782],[34,307165,47077],{"className":307166,"style":47120},[31685,31686],[34,307168],{"className":307169,"style":32604},[31845],[34,307171,223],{"className":307172},[32024],[34,307174],{"className":307175,"style":32604},[31845],[34,307177,307179,307182,307185,307188,307191],{"className":307178},[31676],[34,307180],{"className":307181,"style":31774},[31680],[34,307183,31747],{"className":307184},[31685,31686],[34,307186,2557],{"className":307187},[31690],[34,307189,38080],{"className":307190},[31685],[34,307192,31747],{"className":307193},[31685,31686],[34,307195,307197,307210],{"className":307196},[31642],[34,307198,307200],{"className":307199},[31646],[31648,307201,307202],{"xmlns":31650},[31652,307203,307204,307208],{},[31655,307205,307206],{},[31658,307207,47077],{},[31665,307209,47077],{"encoding":31667},[34,307211,307213],{"className":307212,"ariaHidden":758},[31672],[34,307214,307216,307219],{"className":307215},[31676],[34,307217],{"className":307218,"style":32527},[31680],[34,307220,47077],{"className":307221,"style":47120},[31685,31686]," 是等待时间，",[34,307224,307226,307239],{"className":307225},[31642],[34,307227,307229],{"className":307228},[31646],[31648,307230,307231],{"xmlns":31650},[31652,307232,307233,307237],{},[31655,307234,307235],{},[31658,307236,31747],{},[31665,307238,31747],{"encoding":31667},[34,307240,307242],{"className":307241,"ariaHidden":758},[31672],[34,307243,307245,307248],{"className":307244},[31676],[34,307246],{"className":307247,"style":32527},[31680],[34,307249,31747],{"className":307250},[31685,31686]," 是预期的服务时间（作业长度）。",[108,307253,307254,307256,307257,34781,307285,307313,307314,307317,307318,307321,307322,307350],{},[2627,307255,306709],{},"：公式中分母是 ",[34,307258,307260,307273],{"className":307259},[31642],[34,307261,307263],{"className":307262},[31646],[31648,307264,307265],{"xmlns":31650},[31652,307266,307267,307271],{},[31655,307268,307269],{},[31658,307270,31747],{},[31665,307272,31747],{"encoding":31667},[34,307274,307276],{"className":307275,"ariaHidden":758},[31672],[34,307277,307279,307282],{"className":307278},[31676],[34,307280],{"className":307281,"style":32527},[31680],[34,307283,31747],{"className":307284},[31685,31686],[34,307286,307288,307301],{"className":307287},[31642],[34,307289,307291],{"className":307290},[31646],[31648,307292,307293],{"xmlns":31650},[31652,307294,307295,307299],{},[31655,307296,307297],{},[31658,307298,31747],{},[31665,307300,31747],{"encoding":31667},[34,307302,307304],{"className":307303,"ariaHidden":758},[31672],[34,307305,307307,307310],{"className":307306},[31676],[34,307308],{"className":307309,"style":32527},[31680],[34,307311,31747],{"className":307312},[31685,31686]," 较",[2627,307315,307316],{},"小","（意味着是",[2627,307319,307320],{},"短作业","）时，分数才更大。因此HRRN是偏好短作业的，但随着 ",[34,307323,307325,307338],{"className":307324},[31642],[34,307326,307328],{"className":307327},[31646],[31648,307329,307330],{"xmlns":31650},[31652,307331,307332,307336],{},[31655,307333,307334],{},[31658,307335,47077],{},[31665,307337,47077],{"encoding":31667},[34,307339,307341],{"className":307340,"ariaHidden":758},[31672],[34,307342,307344,307347],{"className":307343},[31676],[34,307345],{"className":307346,"style":32527},[31680],[34,307348,47077],{"className":307349,"style":47120},[31685,31686],"（等待时间）的增加，长作业的响应比也会提高，从而避免了长作业饥饿。",[15,307352,307353],{},[2627,307354,307355],{},"13. A disadvantage of simulation is that results for a given “run” only apply to that particular collection of processes under that particular set of assumptions.",[105,307357,307358,307363,307367],{},[108,307359,307360,307362],{},[2627,307361,291953],{},"：模拟的一个缺点是，给定“运行”的结果仅适用于该特定假设集下的特定进程集合。",[108,307364,307365,303126],{},[2627,307366,118575],{},[108,307368,307369,33183,307371],{},[2627,307370,118582],{},[105,307372,307373,307378],{},[108,307374,307375,307377],{},[2627,307376,306703],{},"：模拟建模（Simulation Modeling）的局限性。",[108,307379,307380,307382],{},[2627,307381,306709],{},"：模拟使用的是特制的随机数和特定的概率分布。因此，结论往往只对本次输入的特征集有效，缺乏像数学分析模型那样普适的解析解。",[15,307384,307385],{},[2627,307386,307387],{},"14. In “fair share” scheduling, each user is assigned a weighting of some sort that defines that user’s share of system resources as a fraction of the total usage of those resources.",[105,307389,307390,307395,307399],{},[108,307391,307392,307394],{},[2627,307393,291953],{},"：在“公平共享（fair share）”调度中，会为每个用户分配某种权重，该权重定义了该用户在系统资源总使用量中所占的份额。",[108,307396,307397,303126],{},[2627,307398,118575],{},[108,307400,307401,33183,307403],{},[2627,307402,118582],{},[105,307404,307405,307410],{},[108,307406,307407,307409],{},[2627,307408,306703],{},"：Fair-share Scheduling 的资源分配理念。",[108,307411,307412,307414],{},[2627,307413,306709],{},"：在多用户系统中，有时不仅需要考虑进程级别的公平，还需要考虑“用户（或进程组）”级别的公平，所以按照用户权重分配 CPU 时间是 Fair-share 的核心。",[15,307416,307417],{},[2627,307418,307419],{},"15. The traditional UNIX scheduler, such as those used in SVR3 and 4.3 BSD UNIX systems, employs single level feedback using round robin.",[105,307421,307422,307427,307431],{},[108,307423,307424,307426],{},[2627,307425,291953],{},"：传统的 UNIX 调度程序（例如在 SVR3 和 4.3 BSD UNIX 系统中使用的调度程序）采用使用轮转的单级反馈。",[108,307428,307429,303091],{},[2627,307430,118575],{},[108,307432,307433,33183,307435],{},[2627,307434,118582],{},[105,307436,307437,307442],{},[108,307438,307439,307441],{},[2627,307440,306703],{},"：传统 UNIX 调度的机制。",[108,307443,307444,307446,307447,307450],{},[2627,307445,306709],{},"：传统 UNIX 使用的是",[2627,307448,307449],{},"多级反馈队列（Multilevel Feedback）","，它将进程划分为不同的固定优先级带（如 Swapper, Block I\u002FO, User processes等），并在带内进行轮转。",[32188,307452],{},[19,307454,307456],{"id":307455},"multiple-choice-questions-选择题","Multiple Choice Questions (选择题)",[15,307458,307459,307462],{},[2627,307460,307461],{},"1. The type of scheduling that involves the decision to add a process to those that are at least partially in main memory and therefore available for execution is referred to as:","\na. Long-term scheduling b. Medium-term scheduling c. I\u002FO scheduling d. None of the above",[105,307464,307465,307470,307474],{},[108,307466,307467,307469],{},[2627,307468,291953],{},"：决定将进程添加到至少部分驻留在主存中，因此可以被执行的进程池中的调度类型被称为：\na. 长期调度 b. 中期调度 c. I\u002FO 调度 d. 以上皆非",[108,307471,307472,303513],{},[2627,307473,118575],{},[108,307475,307476,33183,307478],{},[2627,307477,118582],{},[105,307479,307480,307485],{},[108,307481,307482,307484],{},[2627,307483,306703],{},"：三级调度体系。",[108,307486,307487,307489,307490,347],{},[2627,307488,306709],{},"：将部分或全部映像载入“主存（Main Memory）”，这是内存管理中交换（Swapping-in）功能的一部分，属于",[2627,307491,307492],{},"中期调度 (Medium-term scheduling)",[15,307494,307495,307498],{},[2627,307496,307497],{},"2. The decision as to which job to admit to the system next can be based on which of the following criteria:","\na. Simple FIFO b. Priority c. I\u002FO requirements d. All of the above",[105,307500,307501,307506,307510],{},[108,307502,307503,307505],{},[2627,307504,291953],{},"：关于接下来接纳哪个作业进入系统的决定，可以基于以下哪些标准：\na. 简单先进先出 (FIFO) b. 优先级 c. I\u002FO 需求 d. 以上皆是",[108,307507,307508,303604],{},[2627,307509,118575],{},[108,307511,307512,33183,307514],{},[2627,307513,118582],{},[105,307515,307516,307521],{},[108,307517,307518,307520],{},[2627,307519,306703],{},"：长期调度（作业调度）的准入策略。",[108,307522,307523,307525],{},[2627,307524,306709],{},"：接纳作业时不仅可以按排队顺序，也可以考虑优先级，甚至是I\u002FO和CPU的平衡（混合I\u002FO密集与CPU密集型以最大化利用率），因此所有策略都可以应用。",[15,307527,307528,307531],{},[2627,307529,307530],{},"3. Typically, the swapping-in function for processes is based on the need to manage:","\na. Process priorities b. Virtual memory c. The degree of multiprogramming d. None of the above",[105,307533,307534,307539,307543],{},[108,307535,307536,307538],{},[2627,307537,291953],{},"：通常，进程的换入（swapping-in）功能是基于管理什么的需求：\na. 进程优先级 b. 虚拟内存 c. 多道程序设计的程度 d. 以上皆非",[108,307540,307541,303558],{},[2627,307542,118575],{},[108,307544,307545,33183,307547],{},[2627,307546,118582],{},[105,307548,307549,307554],{},[108,307550,307551,307553],{},[2627,307552,306703],{},"：中期调度的目的。",[108,307555,307556,307558,307559,347],{},[2627,307557,306709],{},"：换出是为了腾出内存，换入（Swapping-in）是为了维持当前在内存中活跃竞跑的进程数量，即",[2627,307560,307561],{},"多道程序设计的程度 (Degree of multiprogramming)",[15,307563,307564,307567],{},[2627,307565,307566],{},"4. In terms of frequency of execution, the short-term scheduler is usually the one that executes:","\na. Most frequently b. Least frequently c. About the same as the other schedulers d. None of the above",[105,307569,307570,307575,307579],{},[108,307571,307572,307574],{},[2627,307573,291953],{},"：就执行频率而言，短期调度程序通常是：\na. 执行最频繁的 b. 执行最不频繁的 c. 与其他调度程序差不多 d. 以上皆非",[108,307576,307577,303581],{},[2627,307578,118575],{},[108,307580,307581,33183,307583],{},[2627,307582,118582],{},[105,307584,307585,307590],{},[108,307586,307587,307589],{},[2627,307588,306703],{},"：各级调度器的工作频率。",[108,307591,307592,307594,307595,307598],{},[2627,307593,306709],{},"：短期调度（分派器）需要在每次时钟中断、I\u002FO中断、系统调用等可能导致CPU易手的时刻被调用，因此是执行",[2627,307596,307597],{},"最频繁 (Most frequently)"," 的调度程序。",[15,307600,307601,307604],{},[2627,307602,307603],{},"5. Response time in an interactive system is an example of:","\na. System-oriented criteria for short-term scheduling policies\nb. User-oriented criteria for short-term scheduling policies\nc. System-oriented criteria for long-term scheduling policies\nd. None of the above",[105,307606,307607,307612,307616],{},[108,307608,307609,307611],{},[2627,307610,291953],{},"：交互式系统中的响应时间是以下的例子：\na. 面向系统的短期调度策略标准\nb. 面向用户的短期调度策略标准\nc. 面向系统的长期调度策略标准\nd. 以上皆非",[108,307613,307614,303513],{},[2627,307615,118575],{},[108,307617,307618,33183,307620],{},[2627,307619,118582],{},[105,307621,307622,307627],{},[108,307623,307624,307626],{},[2627,307625,306703],{},"：调度标准分类 (User-oriented vs. System-oriented)。",[108,307628,307629,2664,307631,258,307634,307637,307638,307641],{},[2627,307630,306709],{},[2627,307632,307633],{},"响应时间 (Response time)",[2627,307635,307636],{},"周转时间 (Turnaround time)"," 是用户最直接能感受到的性能指标，因此属于",[2627,307639,307640],{},"面向用户 (User-oriented)"," 的标准；而吞吐量、CPU利用率则属于面向系统的标准。",[15,307643,307644,307647],{},[2627,307645,307646],{},"6. A typical way to overcome starvation of lower-priority processes in a priority-based scheduling system is to:","\na. Change a process priority randomly\nb. Change a process priority with its age\nc. Round-robin cycling of processes in a priority queue\nd. All of the above",[105,307649,307650,307655,307659],{},[108,307651,307652,307654],{},[2627,307653,291953],{},"：在基于优先级的调度系统中，克服低优先级进程饥饿的典型方法是：\na. 随机改变进程优先级\nb. 随进程年龄（等待时间）改变优先级\nc. 对优先级队列中的进程进行轮转循环\nd. 以上皆是",[108,307656,307657,303513],{},[2627,307658,118575],{},[108,307660,307661,33183,307663],{},[2627,307662,118582],{},[105,307664,307665,307670],{},[108,307666,307667,307669],{},[2627,307668,306703],{},"：老化技术（Aging）。",[108,307671,307672,307674,307675,307678],{},[2627,307673,306709],{},"：解决优先级调度引发的饥饿问题的最佳方式就是",[2627,307676,307677],{},"老化 (Aging)","，即随着进程在就绪队列中等待时间的增长（年龄变大），逐渐提升其优先级，最终它总能获得CPU。",[15,307680,307681,307684],{},[2627,307682,307683],{},"7. Which of the following scheduling policies allow the O\u002FS to interrupt the currently running process and move it to the Ready state?","\na. Preemptive b. Non-Preemptive c. First-come-first-served d. None of the above",[105,307686,307687,307692,307696],{},[108,307688,307689,307691],{},[2627,307690,291953],{},"：以下哪种调度策略允许操作系统中断当前运行的进程并将其移动到就绪状态？\na. 抢占式 (Preemptive) b. 非抢占式 (Non-Preemptive) c. 先来先服务 (FCFS) d. 以上皆非",[108,307693,307694,303581],{},[2627,307695,118575],{},[108,307697,307698,33183,307700],{},[2627,307699,118582],{},[105,307701,307702,307707],{},[108,307703,307704,307706],{},[2627,307705,306703],{},"：决策模式（Decision Mode）。",[108,307708,307709,307711,307712,307715],{},[2627,307710,306709],{},"：定义就是",[2627,307713,307714],{},"抢占式 (Preemptive)","：操作系统有权强行剥夺正在运行进程的CPU使用权，以分配给另一个（往往是优先级更高或时间片已到的）进程。",[15,307717,307718,307721],{},[2627,307719,307720],{},"8. In terms of the queuing model, the total time that a process spends in a system (waiting time plus service time) is called:","\na. Normalized turnaround time (TAT) b. Finish time (FT) c. Turnaround or residence time (TAT) d. None of the above",[105,307723,307724,307729,307733],{},[108,307725,307726,307728],{},[2627,307727,291953],{},"：在排队模型中，一个进程在系统中花费的总时间（等待时间加上服务时间）被称为：\na. 归一化周转时间 b. 完成时间 c. 周转时间或驻留时间 d. 以上皆非",[108,307730,307731,303558],{},[2627,307732,118575],{},[108,307734,307735,33183,307737],{},[2627,307736,118582],{},[105,307738,307739,307744],{},[108,307740,307741,307743],{},[2627,307742,306703],{},"：性能评估的时间指标。",[108,307745,307746,307748,307749,307751],{},[2627,307747,306709],{},"：驻留时间 (Residence time) 或 ",[2627,307750,307636],{}," 指的是一个任务从进入系统到完成并离开系统的总时长，包括了在就绪队列里排队等待的时间以及真正在CPU上运行的服务时间。",[15,307753,307754,307757],{},[2627,307755,307756],{},"9. In the Round Robin scheduling technique, the principle design issue is:","\na. Determining the fair distribution of time quanta to individual processes\nb. Determining the method of cycling through a given set of processes\nc. Determining the length of the time quantum\nd. None of the above",[105,307759,307760,307765,307769],{},[108,307761,307762,307764],{},[2627,307763,291953],{},"：在轮转 (Round Robin) 调度技术中，主要的系统设计问题是：\na. 确定向各个进程公平分配时间片\nb. 确定循环遍历给定进程集的方法\nc. 确定时间片（quantum）的长度\nd. 以上皆非",[108,307766,307767,303558],{},[2627,307768,118575],{},[108,307770,307771,33183,307773],{},[2627,307772,118582],{},[105,307774,307775,307780],{},[108,307776,307777,307779],{},[2627,307778,306703],{},"：Round Robin (RR) 算法的设计。",[108,307781,307782,307784,307785,307788],{},[2627,307783,306709],{},"：RR算法唯一且最重要的参数就是",[2627,307786,307787],{},"时间片的大小 (length of the time quantum)","。如果时间片太大，RR退化为FCFS；如果时间片太小，上下文切换（Context Switch）造成的开销将严重拖垮系统性能。",[15,307790,307791,307794],{},[2627,307792,307793],{},"10. One difficulty with the Shortest Process Next (SPN) scheduling technique is:","\na. The need to know or estimate required processing times for each process\nb. The starvation of longer processes\nc. The lack of preemption\nd. All of the above",[105,307796,307797,307802,307806],{},[108,307798,307799,307801],{},[2627,307800,291953],{},"：最短进程优先（SPN）调度技术的一个困难在于：\na. 需要知道或估计每个进程所需的处理时间\nb. 长进程的饥饿\nc. 缺乏抢占\nd. 以上皆是",[108,307803,307804,303604],{},[2627,307805,118575],{},[108,307807,307808,33183,307810],{},[2627,307809,118582],{},[105,307811,307812,307817],{},[108,307813,307814,307816],{},[2627,307815,306703],{},"：SPN算法的缺点。",[108,307818,307819,307821],{},[2627,307820,306709],{},"：SPN是非抢占的（缺乏抢占），它会导致系统响应不佳；它绝对偏好短进程，因此不断涌入的短任务会导致长任务饿死（长进程饥饿）；最致命的是它在做决策时必须预先知道任务有多长，而在现实中这需要极其复杂的指数平均预测（需要估计时间）。所以三个选项都是它的短板。",[15,307823,307824,307827],{},[2627,307825,307826],{},"11. One difficulty with the Shortest Remaining Time (SRT) scheduling technique is:","\na. The need to know or estimate required processing times for each process\nb. The starvation of shorter processes\nc. The lack of preemption\nd. All of the above",[105,307829,307830,307835,307839],{},[108,307831,307832,307834],{},[2627,307833,291953],{},"：最短剩余时间（SRT）调度技术的一个困难在于：\na. 需要知道或估计每个进程所需的处理时间\nb. 短进程的饥饿\nc. 缺乏抢占\nd. 以上皆是",[108,307836,307837,303581],{},[2627,307838,118575],{},[108,307840,307841,33183,307843],{},[2627,307842,118582],{},[105,307844,307845,307850],{},[108,307846,307847,307849],{},[2627,307848,306703],{},"：SRT算法的特性与局限性。",[108,307851,307852,307854,307855,307858,307859,307862],{},[2627,307853,306709],{},"：首先，SRT ",[2627,307856,307857],{},"具有"," 抢占性（排除c）。其次，SRT偏好短进程，所以它会导致",[2627,307860,307861],{},"长进程饥饿","，而不是短进程饥饿（排除b）。它唯一符合选项的难点和 SPN 一样，就是操作系统必须事先预测或知道进程到底需要执行多长时间。",[15,307864,307865,307868],{},[2627,307866,307867],{},"12. Which of the following scheduling policies require prior knowledge or estimation of process length:","\na. Shortest Remaining Time (SRT) b. Shortest Process Next (SPN) c. Highest Response Ratio Next (HRRN) d. All of the above",[105,307870,307871,307876,307880],{},[108,307872,307873,307875],{},[2627,307874,291953],{},"：以下哪种调度策略需要事先知道或估计进程长度：\na. SRT b. SPN c. HRRN d. 以上皆是",[108,307877,307878,303604],{},[2627,307879,118575],{},[108,307881,307882,33183,307884],{},[2627,307883,118582],{},[105,307885,307886,307891],{},[108,307887,307888,307890],{},[2627,307889,306703],{},"：基于处理时间调度的算法共性。",[108,307892,307893,307895,307896,307924,307925,308018],{},[2627,307894,306709],{},"：SPN挑总时间最短的，SRT挑剩余时间最短的，HRRN需要预估执行时间 ",[34,307897,307899,307912],{"className":307898},[31642],[34,307900,307902],{"className":307901},[31646],[31648,307903,307904],{"xmlns":31650},[31652,307905,307906,307910],{},[31655,307907,307908],{},[31658,307909,31747],{},[31665,307911,31747],{"encoding":31667},[34,307913,307915],{"className":307914,"ariaHidden":758},[31672],[34,307916,307918,307921],{"className":307917},[31676],[34,307919],{"className":307920,"style":32527},[31680],[34,307922,31747],{"className":307923},[31685,31686]," 来计算响应比 ",[34,307926,307928,307958],{"className":307927},[31642],[34,307929,307931],{"className":307930},[31646],[31648,307932,307933],{"xmlns":31650},[31652,307934,307935,307955],{},[31655,307936,307937,307939,307941,307943,307945,307947,307949,307951,307953],{},[31658,307938,34386],{},[31662,307940,202],{},[31662,307942,456],{"stretchy":702},[31658,307944,47077],{},[31662,307946,223],{},[31658,307948,31747],{},[31662,307950,2557],{"stretchy":702},[31658,307952,38080],{"mathvariant":38079},[31658,307954,31747],{},[31665,307956,307957],{"encoding":31667},"R=(w+s)\u002Fs",[34,307959,307961,307979,308000],{"className":307960,"ariaHidden":758},[31672],[34,307962,307964,307967,307970,307973,307976],{"className":307963},[31676],[34,307965],{"className":307966,"style":33218},[31680],[34,307968,34386],{"className":307969,"style":34427},[31685,31686],[34,307971],{"className":307972,"style":31846},[31845],[34,307974,202],{"className":307975},[31850],[34,307977],{"className":307978,"style":31846},[31845],[34,307980,307982,307985,307988,307991,307994,307997],{"className":307981},[31676],[34,307983],{"className":307984,"style":31774},[31680],[34,307986,456],{"className":307987},[31782],[34,307989,47077],{"className":307990,"style":47120},[31685,31686],[34,307992],{"className":307993,"style":32604},[31845],[34,307995,223],{"className":307996},[32024],[34,307998],{"className":307999,"style":32604},[31845],[34,308001,308003,308006,308009,308012,308015],{"className":308002},[31676],[34,308004],{"className":308005,"style":31774},[31680],[34,308007,31747],{"className":308008},[31685,31686],[34,308010,2557],{"className":308011},[31690],[34,308013,38080],{"className":308014},[31685],[34,308016,31747],{"className":308017},[31685,31686],"。这三种策略底层都极度依赖对执行时间的预测。",[15,308020,308021,308024],{},[2627,308022,308023],{},"13. It is impossible to make definitive comparisons of various scheduling policies due to dependence on factors such as:","\na. The probability distribution of service times of the various processes\nb. The efficiency of the scheduling and context switching mechanisms\nc. The nature of the I\u002FO demand and performance of the I\u002FO subsystem\nd. All of the above",[105,308026,308027,308032,308036],{},[108,308028,308029,308031],{},[2627,308030,291953],{},"：由于依赖以下因素，不可能对各种调度策略进行确定的（绝对的）比较：\na. 各种进程服务时间的概率分布\nb. 调度和上下文切换机制的效率\nc. I\u002FO需求的性质和I\u002FO子系统的性能\nd. 以上皆是",[108,308033,308034,303604],{},[2627,308035,118575],{},[108,308037,308038,33183,308040],{},[2627,308039,118582],{},[105,308041,308042,308047],{},[108,308043,308044,308046],{},[2627,308045,306703],{},"：调度算法性能评估的复杂性。",[108,308048,308049,308051],{},[2627,308050,306709],{},"：在真实操作系统中，脱离具体的使用场景（如I\u002FO情况、上下文切换硬件开销、任务到达分布）空谈哪种调度算法最优秀毫无意义，必须结合具体环境。",[15,308053,308054,308057],{},[2627,308055,308056],{},"14. The strategy that schedules processes based on their group affiliation is generally referred to as:","\na. Queuing analysis b. Simulation modeling c. Fair share scheduling d. All of the above",[105,308059,308060,308065,308069],{},[108,308061,308062,308064],{},[2627,308063,291953],{},"：基于进程的组属关系来调度进程的策略通常被称为：\na. 排队分析 b. 模拟建模 c. 公平共享调度 d. 以上皆非",[108,308066,308067,303558],{},[2627,308068,118575],{},[108,308070,308071,33183,308073],{},[2627,308072,118582],{},[105,308074,308075,308080],{},[108,308076,308077,308079],{},[2627,308078,306703],{},"：Fair-share Scheduling 的概念。",[108,308081,308082,308084,308085,347],{},[2627,308083,306709],{},"：当我们将同一用户（或同一部门）的多个进程绑定为一个Group，并基于组分配CPU权重时，这种在用户组层面而非单进程层面体现公平的调度就是",[2627,308086,308087],{},"公平共享调度 (Fair Share Scheduling)",[15,308089,308090,308093],{},[2627,308091,308092],{},"15. The traditional UNIX scheduler divides processes into fixed bands of priority levels, with the highest priority band being the:","\na. Swapper band b. File manipulation band c. User process band d. None of the above",[105,308095,308096,308101,308105],{},[108,308097,308098,308100],{},[2627,308099,291953],{},"：传统 UNIX 调度程序将进程划分为固定优先级级别的带（bands），其中最高优先级带是：\na. 交换程序带 (Swapper band) b. 文件操作带 c. 用户进程带 d. 以上皆非",[108,308102,308103,303581],{},[2627,308104,118575],{},[108,308106,308107,33183,308109],{},[2627,308108,118582],{},[105,308110,308111,308116],{},[108,308112,308113,308115],{},[2627,308114,306703],{},"：传统 UNIX SVR3\u002F4.3 BSD 的多级反馈队列带划分。",[108,308117,308118,308120,308121,308124],{},[2627,308119,306709],{},"：为了系统整体性能，传统UNIX严格划分优先级层次，其中管理系统底层内存流转的 ",[2627,308122,308123],{},"交换程序带 (Swapper band)"," 拥有最高优先级，其次是块设备I\u002FO控制，最后才是普通的用户进程带 (User process band)。",[32188,308126],{},[19,308128,294449],{"id":294448},[15,308130,308131],{},[2627,308132,308133],{},"1. The task of assigning processes to the processor or processors over time, in a way that meets system objectives is called _________.",[105,308135,308136,308141,308146],{},[108,308137,308138,308140],{},[2627,308139,291953],{},"：随着时间的推移，以满足系统目标的方式将进程分配给一个或多个处理器的任务被称为_________。",[108,308142,308143,308145],{},[2627,308144,118575],{},"：processor scheduling (处理器调度)",[108,308147,308148,308150],{},[2627,308149,118582],{},"：这是调度的标准定义。",[15,308152,308153],{},[2627,308154,308155],{},"2. The decision as to when to create a new process is generally driven by the desired degree of _________.",[105,308157,308158,308163,308168],{},[108,308159,308160,308162],{},[2627,308161,291953],{},"：关于何时创建新进程的决策通常由所需的_________程度所驱动。",[108,308164,308165,308167],{},[2627,308166,118575],{},"：multiprogramming (多道程序设计)",[108,308169,308170,308172],{},[2627,308171,118582],{},"：长期调度负责控制接纳的新进程数量，核心目的就是控制内存中并发进程的总量（多道程序设计度）。",[15,308174,308175],{},[2627,308176,308177],{},"3. _________-term scheduling is part of the system swapping function.",[105,308179,308180,308185,308190],{},[108,308181,308182,308184],{},[2627,308183,291953],{},"：_________期调度是系统交换功能的一部分。",[108,308186,308187,308189],{},[2627,308188,118575],{},"：Medium (中)",[108,308191,308192,308194],{},[2627,308193,118582],{},"：中期调度控制进程在内存和外存之间的调入与调出。",[15,308196,308197],{},[2627,308198,308199],{},"4. The _________-term scheduler is invoked whenever an event occurs that may lead to the suspension or preemption of the currently running process.",[105,308201,308202,308207,308212],{},[108,308203,308204,308206],{},[2627,308205,291953],{},"：每当发生可能导致当前运行进程挂起或被抢占的事件时，就会调用_________期调度程序。",[108,308208,308209,308211],{},[2627,308210,118575],{},"：short (短)",[108,308213,308214,308216],{},[2627,308215,118582],{},"：也就是分派器（Dispatcher），它在发生时钟中断、I\u002FO中断等非常频繁的底层事件时触发。",[15,308218,308219],{},[2627,308220,308221],{},"5. Response time and throughput are examples of _________-related criteria for short-term scheduling.",[105,308223,308224,308229,308234],{},[108,308225,308226,308228],{},[2627,308227,291953],{},"：响应时间和吞吐量是短期调度的与_________相关标准的例子。",[108,308230,308231,308233],{},[2627,308232,118575],{},"：performance (性能)",[108,308235,308236,308238],{},[2627,308237,118582],{},"：调度准则主要分为系统\u002F用户导向（System-oriented vs. User-oriented）和与性能相关\u002F无关（Performance-related vs. Non-performance-related）。这两个指标直接反映系统效能。",[15,308240,308241],{},[2627,308242,308243],{},"6. In a system employing priority scheduling, the scheduler always selects the process with the _________ priority level for processing.",[105,308245,308246,308251,308256],{},[108,308247,308248,308250],{},[2627,308249,291953],{},"：在使用优先级调度的系统中，调度程序总是选择具有_________优先级级别的进程进行处理。",[108,308252,308253,308255],{},[2627,308254,118575],{},"：highest (最高)",[108,308257,308258,308260],{},[2627,308259,118582],{},"：显而易见，调度器始终从队列中挑最高优先级执行。",[15,308262,308263],{},[2627,308264,308265],{},"7. The _________, which has two general categories, specifies the instants in time at which the selection function is exercised.",[105,308267,308268,308273,308278],{},[108,308269,308270,308272],{},[2627,308271,291953],{},"：包含两大类的_________，指定了在何时执行选择函数的时刻。",[108,308274,308275,308277],{},[2627,308276,118575],{},"：decision mode (决策模式)",[108,308279,308280,308282],{},[2627,308281,118582],{},"：Decision Mode 决定了操作系统什么时候去剥夺CPU使用权，两大类即为抢占式 (Preemptive) 和非抢占式 (Non-preemptive)。",[15,308284,308285],{},[2627,308286,308287],{},"8. In terms of the queuing model, the total time that a process spends in a system (waiting time plus service time) is called the _________.",[105,308289,308290,308295,308300],{},[108,308291,308292,308294],{},[2627,308293,291953],{},"：在排队模型中，一个进程在系统中花费的总时间（等待时间加上服务时间）被称为_________。",[108,308296,308297,308299],{},[2627,308298,118575],{},"：turnaround time (或 residence time) (周转时间 \u002F 驻留时间)",[108,308301,308302,308304],{},[2627,308303,118582],{},"：这在计算系统各项评价指标时极其重要。",[15,308306,308307],{},[2627,308308,308309],{},"9. The Round Robin scheduling technique is also known as _________, because each process is given a set amount of processor time before being preempted.",[105,308311,308312,308317,308322],{},[108,308313,308314,308316],{},[2627,308315,291953],{},"：轮转 (Round Robin) 调度技术也称为_________，因为每个进程在被抢占之前都被赋予了一定数量的处理器时间。",[108,308318,308319,308321],{},[2627,308320,118575],{},"：time slicing (时间片轮转 \u002F 时间分片)",[108,308323,308324,308326],{},[2627,308325,118582],{},"：通过固定的时间片（Quantum）来轮流交替执行机制。",[15,308328,308329],{},[2627,308330,308331],{},"10. _________ is a scheduling policy in which the process with the shortest expected processing time is selected next, but there is no preemption.",[105,308333,308334,308339,308344],{},[108,308335,308336,308338],{},[2627,308337,291953],{},"：_________是一种调度策略，其中接下来选择预期处理时间最短的进程，但没有抢占。",[108,308340,308341,308343],{},[2627,308342,118575],{},"：Shortest process next 或 SPN (最短进程优先)",[108,308345,308346,308348],{},[2627,308347,118582],{},"：区别于抢占版的SRT。",[15,308350,308351],{},[2627,308352,308353],{},"11. _________ is a scheduling policy in which the process with the shortest expected processing time is selected next, and if a shorter process becomes ready in the system, the currently running process is preempted.",[105,308355,308356,308361,308366],{},[108,308357,308358,308360],{},[2627,308359,291953],{},"：_________是一种调度策略，其中接下来选择预期处理时间最短的进程，并且如果系统中有更短的进程就绪，则当前正在运行的进程将被抢占。",[108,308362,308363,308365],{},[2627,308364,118575],{},"：Shortest remaining time 或 SRT (最短剩余时间)",[108,308367,308368,308370],{},[2627,308369,118582],{},"：SPN的抢占版，极其偏好短任务，并提供良好的响应特性，但容易导致长任务饥饿。",[15,308372,308373],{},[2627,308374,308375],{},"12. A scheduling mechanism that requires no prior knowledge of process length, yet can nevertheless favor shorter jobs, is known as the _________ scheduling mechanism.",[105,308377,308378,308383,308388],{},[108,308379,308380,308382],{},[2627,308381,291953],{},"：一种不需要事先知道进程长度，但仍然可以偏好较短作业的调度机制被称为_________调度机制。",[108,308384,308385,308387],{},[2627,308386,118575],{},"：feedback (反馈 \u002F 多级反馈队列)",[108,308389,308390,308392],{},[2627,308391,118582],{},"：Feedback 将进程在多个不同时间片长度的优先级队列中降级流转。短任务会在最初的小时间片高优先级队列中迅速完成，而长任务会“沉底”，完美做到不知长短也能优待短任务。",[15,308394,308395],{},[2627,308396,308397],{},"13. Some of the difficulties of analytic modeling are overcome by using _________, which allows a wide range of policies to be modeled.",[105,308399,308400,308405,308410],{},[108,308401,308402,308404],{},[2627,308403,291953],{},"：通过使用_________克服了解析建模的一些困难，它允许对广泛的策略进行建模。",[108,308406,308407,308409],{},[2627,308408,118575],{},"：simulation (模拟 \u002F 模拟建模)",[108,308411,308412,308414],{},[2627,308413,118582],{},"：面对排队论和数学推导解决不了的复杂调度场景，通过跑代码做随机模拟（Simulation）是直观的手段。",[15,308416,308417],{},[2627,308418,308419],{},"14. In _________ scheduling, each user is assigned a weighting of some sort that defines that user’s share of system resources as a fraction of the total usage of those resources.",[105,308421,308422,308427,308432],{},[108,308423,308424,308426],{},[2627,308425,291953],{},"：在_________调度中，为每个用户分配某种权重，该权重定义了该用户在系统资源总使用量中所占的份额。",[108,308428,308429,308431],{},[2627,308430,118575],{},"：fair share (公平共享)",[108,308433,308434,308436],{},[2627,308435,118582],{},"：面向用户群组维度的调度方法。",[15,308438,308439],{},[2627,308440,308441],{},"15. The traditional UNIX scheduler, such as those used in SVR3 and 4.3 BSD UNIX systems, divides processes into fixed _________ of priority levels.",[105,308443,308444,308449,308454],{},[108,308445,308446,308448],{},[2627,308447,291953],{},"：传统的 UNIX 调度程序（例如在 SVR3 和 4.3 BSD UNIX 系统中使用的那些）将进程划分为固定优先级级别的_________。",[108,308450,308451,308453],{},[2627,308452,118575],{},"：bands (带)",[108,308455,308456,308458],{},[2627,308457,118582],{},"：如前述，划分为 Swapper band, User process band 等优先级固定带。",{"title":29,"searchDepth":43,"depth":43,"links":308460},[308461,308462,308463],{"id":306675,"depth":43,"text":306676},{"id":307455,"depth":43,"text":307456},{"id":294448,"depth":43,"text":294449},{"order":1976},"\u002Fsource\u002Fcs\u002Fos\u002Ftestbank\u002Fchap9",{"title":306666,"description":29},"source\u002Fcs\u002Fos\u002Ftestbank\u002Fchap9","GIfgzHYOnE3db7wGLKzQRuiO8X5Da-HRlBDtPoBgD1A",{"id":308470,"title":308471,"body":308472,"description":308696,"extension":85,"meta":308697,"navigation":52,"path":308698,"seo":308699,"stem":308700,"__hash__":308701},"notes\u002Fsource\u002Fcs\u002Fse\u002Findex.md","Software Engineering",{"type":8,"value":308473,"toc":308687},[308474,308484],[16016,308475,31533,308477,31533,308480],{"className":308476},[31532],[11,308478,308471],{"id":308479},"software-engineering",[15,308481,308483],{"className":308482},[31540],"软件工程",[31543,308485,308487,308491,308501,308505,308511,308567,308573,308605,308611,308655,308661],{"path":308486,"unwrap-single-group":29},"\u002Fcs\u002Fse\u002F",[19,308488,308490],{"id":308489},"course-book","Course Book",[105,308492,308493],{},[108,308494,308495],{},[111,308496,308500],{"href":308497,"rel":308498},"https:\u002F\u002Fdn790001.ca.archive.org\u002F0\u002Fitems\u002Fbme-vik-konyvek\u002FSoftware%20Engineering%20-%20Ian%20Sommerville.pdf",[308499],"nofollow","Software Engineering — Ian Sommerville",[19,308502,308504],{"id":308503},"book-contents","Book Contents",[253,308506,308508],{"id":308507},"part-1-introduction-to-software-engineering-第一部分軟體工程簡介",[2627,308509,308510],{},"Part 1: Introduction to Software Engineering (第一部分：軟體工程簡介)",[105,308512,308513,308519,308525,308531,308537,308543,308549,308555,308561],{},[108,308514,308515,308518],{},[2627,308516,308517],{},"Chapter 1:"," Introduction (第1章：簡介)",[108,308520,308521,308524],{},[2627,308522,308523],{},"Chapter 2:"," Software processes (第2章：軟體流程)",[108,308526,308527,308530],{},[2627,308528,308529],{},"Chapter 3:"," Agile software development (第3章：敏捷軟體開發)",[108,308532,308533,308536],{},[2627,308534,308535],{},"Chapter 4:"," Requirements engineering (第4章：需求工程)",[108,308538,308539,308542],{},[2627,308540,308541],{},"Chapter 5:"," System modeling (第5章：系統塑模)",[108,308544,308545,308548],{},[2627,308546,308547],{},"Chapter 6:"," Architectural design (第6章：架構設計)",[108,308550,308551,308554],{},[2627,308552,308553],{},"Chapter 7:"," Design and implementation (第7章：設計與實作)",[108,308556,308557,308560],{},[2627,308558,308559],{},"Chapter 8:"," Software testing (第8章：軟體測試)",[108,308562,308563,308566],{},[2627,308564,308565],{},"Chapter 9:"," Software evolution (第9章：軟體演進)",[253,308568,308570],{"id":308569},"part-2-system-dependability-and-security-第二部分系統可靠度與安全性",[2627,308571,308572],{},"Part 2: System Dependability and Security (第二部分：系統可靠度與安全性)",[105,308574,308575,308581,308587,308593,308599],{},[108,308576,308577,308580],{},[2627,308578,308579],{},"Chapter 10:"," Dependable systems (第10章：可靠系統)",[108,308582,308583,308586],{},[2627,308584,308585],{},"Chapter 11:"," Reliability engineering (第11章：可靠度工程)",[108,308588,308589,308592],{},[2627,308590,308591],{},"Chapter 12:"," Safety engineering (第12章：安全性工程)",[108,308594,308595,308598],{},[2627,308596,308597],{},"Chapter 13:"," Security engineering (第13章：資安工程)",[108,308600,308601,308604],{},[2627,308602,308603],{},"Chapter 14:"," Resilience engineering (第14章：韌性工程)",[253,308606,308608],{"id":308607},"part-3-advanced-software-engineering-第三部分進階軟體工程",[2627,308609,308610],{},"Part 3: Advanced Software Engineering (第三部分：進階軟體工程)",[105,308612,308613,308619,308625,308631,308637,308643,308649],{},[108,308614,308615,308618],{},[2627,308616,308617],{},"Chapter 15:"," Software reuse (第15章：軟體重複使用)",[108,308620,308621,308624],{},[2627,308622,308623],{},"Chapter 16:"," Component-based software engineering (第16章：基於組件的軟體工程)",[108,308626,308627,308630],{},[2627,308628,308629],{},"Chapter 17:"," Distributed software engineering (第17章：分散式軟體工程)",[108,308632,308633,308636],{},[2627,308634,308635],{},"Chapter 18:"," Service-oriented software engineering (第18章：服務導向的軟體工程)",[108,308638,308639,308642],{},[2627,308640,308641],{},"Chapter 19:"," Systems engineering (第19章：系統工程)",[108,308644,308645,308648],{},[2627,308646,308647],{},"Chapter 20:"," Systems of systems (第20章：系統之系統)",[108,308650,308651,308654],{},[2627,308652,308653],{},"Chapter 21:"," Real-time software engineering (第21章：即時軟體工程)",[253,308656,308658],{"id":308657},"part-4-software-management-第四部分軟體管理",[2627,308659,308660],{},"Part 4: Software management (第四部分：軟體管理)",[105,308662,308663,308669,308675,308681],{},[108,308664,308665,308668],{},[2627,308666,308667],{},"Chapter 22:"," Project management (第22章：專案管理)",[108,308670,308671,308674],{},[2627,308672,308673],{},"Chapter 23:"," Project planning (第23章：專案規劃)",[108,308676,308677,308680],{},[2627,308678,308679],{},"Chapter 24:"," Quality management (第24章：品質管理)",[108,308682,308683,308686],{},[2627,308684,308685],{},"Chapter 25:"," Configuration management (第25章：組態管理)",{"title":29,"searchDepth":43,"depth":43,"links":308688},[308689,308690],{"id":308489,"depth":43,"text":308490},{"id":308503,"depth":43,"text":308504,"children":308691},[308692,308693,308694,308695],{"id":308507,"depth":49,"text":308510},{"id":308569,"depth":49,"text":308572},{"id":308607,"depth":49,"text":308610},{"id":308657,"depth":49,"text":308660},"软件工程学习笔记，基于 Ian Sommerville 教材，涵盖软件过程模型、需求获取与分析、架构模式、需求追溯矩阵（RTM）等内容。",{},"\u002Fsource\u002Fcs\u002Fse",{"title":308471,"description":308696},"source\u002Fcs\u002Fse\u002Findex","r31CnV5vV5gXbHY63KFapiEEHvdTdxrC6TzAReDK_RU",{"id":308703,"title":308704,"body":308705,"description":29,"extension":85,"meta":308946,"navigation":52,"path":308947,"seo":308948,"stem":308949,"__hash__":308950},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002F2.1.md","軟體開發流程模型 (Software Process Models)",{"type":8,"value":308706,"toc":308938},[308707,308710,308714,308717,308753,308757,308760,308806,308810,308813,308830,308834,308837,308879,308883,308886,308913,308917,308920],[11,308708,308704],{"id":308709},"軟體開發流程模型-software-process-models",[19,308711,308713],{"id":308712},"_1-瀑布模型-the-waterfall-model","1. 瀑布模型 (The Waterfall Model)",[15,308715,308716],{},"瀑布模型是第一個被公佈的軟體開發流程模型，源自於大型軍事系統工程。它是一種**計畫驅動（Plan-driven）**的流程，在開始開發前，必須先規劃與排定所有活動的時程。",[105,308718,308719,308725,308731,308737,308747],{},[108,308720,308721,308724],{},[2627,308722,308723],{},"核心階段","：該模型將開發流程分為獨立的階段，包含：需求分析與定義、系統和軟體設計、實作和單元測試、整合與系統測試，以及運行與維護。",[108,308726,308727,308730],{},[2627,308728,308729],{},"運作方式","：每一個階段的產出通常是經過核准（簽核）的文件。原則上，前一個階段完成後，下一個階段才能開始，如同瀑布般向下流動。",[108,308732,308733,308736],{},[2627,308734,308735],{},"優點","：架構嚴謹，每個階段都有明確的產出與文件，對於硬體開發或管理規範嚴格的專案非常有用。",[108,308738,308739,308742,308743,308746],{},[2627,308740,308741],{},"缺點","：在實際運作中，各個階段往往會重疊且資訊會相互回饋。其最大的缺點是",[2627,308744,308745],{},"缺乏彈性","，過早凍結需求會導致無法適應客戶不斷改變的需求，最終開發出不符期望的系統。",[108,308748,308749,308752],{},[2627,308750,308751],{},"適用場景","：需要與硬體介接的嵌入式系統、對安全性與可靠性要求極高（需經過詳細分析與記錄）的關鍵系統，以及由多家合作夥伴共同開發的大型軟體系統。",[19,308754,308756],{"id":308755},"_2-增量開發模型-incremental-development","2. 增量開發模型 (Incremental Development)",[15,308758,308759],{},"增量開發是目前應用程式和軟體產品中最常見的方法。它交錯進行規格制定、開發和驗證活動。",[105,308761,308762,308767,308772,308796,308801],{},[108,308763,308764,308766],{},[2627,308765,243605],{},"：系統透過一系列的版本（增量）來開發，每一個版本都比前一個版本增加新的功能。早期的增量通常會先實作最重要或最迫切需要的功能。",[108,308768,308769,308771],{},[2627,308770,308729],{},"：可以採用計畫驅動、敏捷開發，或是兩者的混合。它反映了人類解決問題的常見模式：先採取出步解決方案，當發現錯誤時再回溯修改。",[108,308773,308774,33183,308776],{},[2627,308775,308735],{},[105,308777,308778,308784,308790],{},[108,308779,308780,308783],{},[2627,308781,308782],{},"降低變更成本","：修改需求的成本大幅降低，需要重做的分析與文件比瀑布模型少很多。",[108,308785,308786,308789],{},[2627,308787,308788],{},"易於獲取回饋","：客戶可以對軟體展示提出意見，比從設計文件中判斷進度容易得多。",[108,308791,308792,308795],{},[2627,308793,308794],{},"及早交付","：即便功能尚未完全，客戶仍能及早使用並從中獲得價值。",[108,308797,308798,308800],{},[2627,308799,308741],{},"：對於管理層而言「流程不可見」，缺乏定期產出的文件以衡量進度；此外，隨著增量不斷加入，系統結構容易退化，程式碼可能會變得凌亂。",[108,308802,308803,308805],{},[2627,308804,308751],{},"：需求可能會在開發過程中不斷變化的業務系統或軟體產品。",[19,308807,308809],{"id":308808},"_3-v模型-the-v-model","3. V模型 (The V-model)",[15,308811,308812],{},"V模型是瀑布模型與計畫驅動流程中的一個變體，它特別強調**驗證（Verification）與確認（Validation）**的階段。",[105,308814,308815,308820,308825],{},[108,308816,308817,308819],{},[2627,308818,243605],{},"：V模型將軟體的驗證活動與瀑布流程中對應的開發階段連接起來。",[108,308821,308822,308824],{},[2627,308823,308729],{},"：在嚴謹的系統開發中，測試是由一系列測試計畫驅動的。V模型顯示了需求規格、系統規格、系統設計和元件設計等階段，如何分別對應並產生客戶測試計畫、系統整合測試計畫、以及子系統整合測試計畫。將傳統線性模型折疊成 V 字型後，可以清楚看出每一個開發階段都有一個對應的測試與驗證階段。",[108,308826,308827,308829],{},[2627,308828,308751],{},"：需要嚴格品質控管與獨立測試團隊的關鍵系統開發。",[19,308831,308833],{"id":308832},"_4-敏捷開發-agile-methods","4. 敏捷開發 (Agile Methods)",[15,308835,308836],{},"為了克服重量級、計畫驅動方法（如瀑布模型）在應對變更時的龐大負擔，敏捷方法於1990年代後期被提出。",[105,308838,308839,308844,308850,308869,308874],{},[108,308840,308841,308843],{},[2627,308842,243605],{},"：敏捷方法將設計與實作視為軟體流程的核心活動，把需求引出和測試整合到設計與實作中，並以快速產出可用軟體為目標。",[108,308845,308846,308849],{},[2627,308847,308848],{},"敏捷宣言核心價值","：個人與互動重於流程與工具；可用的軟體重於詳盡的文件；客戶合作重於合約協商；回應變化重於遵循計畫。",[108,308851,308852,33183,308855],{},[2627,308853,308854],{},"代表實踐與框架",[105,308856,308857,308863],{},[108,308858,308859,308862],{},[2627,308860,308861],{},"極限程式設計 (Extreme Programming, XP)","：將迭代開發推向極致，包含使用者故事 (User stories)、測試驅動開發 (Test-first development)、結對程式設計 (Pair programming) 以及持續重構 (Refactoring) 等實踐。",[108,308864,308865,308868],{},[2627,308866,308867],{},"Scrum","：專注於敏捷專案管理的框架，包含衝刺 (Sprint)、產品待辦清單 (Product backlog) 和 ScrumMaster 等概念，是目前最廣泛使用的敏捷方法。",[108,308870,308871,308873],{},[2627,308872,308735],{},"：能快速交付功能，適應需求的快速變更，將客戶納入開發團隊以確保系統符合真實需求。",[108,308875,308876,308878],{},[2627,308877,308751],{},"：中小型軟體產品的開發，或是組織內部有客戶強力參與且法規限制較少的客製化系統開發。",[19,308880,308882],{"id":308881},"_5-整合與配置-面向復用的軟體工程-integration-and-configuration-reuse-oriented","5. 整合與配置 \u002F 面向復用的軟體工程 (Integration and Configuration \u002F Reuse-oriented)",[15,308884,308885],{},"這種模型建立在現有軟體元件與系統可以被重複使用的基礎上。",[105,308887,308888,308893,308903,308908],{},[108,308889,308890,308892],{},[2627,308891,243605],{},"：不要從頭開發，而是透過挑選、設定並整合現有的可重複使用元件（如獨立應用系統、套件框架、Web 服務）來打造新系統。",[108,308894,308895,308898,308899,308902],{},[2627,308896,308897],{},"階段","：包含需求規格制定、軟體發現與評估、利用可用元件來",[2627,308900,308901],{},"精煉需求","、應用系統配置、以及元件調整與整合。",[108,308904,308905,308907],{},[2627,308906,308735],{},"：減少了必須開發的軟體數量，因此能顯著降低成本與風險，並帶來更快的交付速度。",[108,308909,308910,308912],{},[2627,308911,308741],{},"：無可避免地必須在需求上做出妥協（為了遷就現成軟體），且因為新版本的現成軟體不受組織控制，組織會失去部分系統演進的控制權。",[19,308914,308916],{"id":308915},"_6-螺旋模型-boehms-spiral-model","6. 螺旋模型 (Boehm's Spiral Model)",[15,308918,308919],{},"由軟體工程先驅 Barry Boehm 所提出，是一種**風險驅動（Risk-driven）**的增量開發模型。",[105,308921,308922,308927,308933],{},[108,308923,308924,308926],{},[2627,308925,243605],{},"：軟體流程不表示為一系列活動，而是表示為一個螺旋。螺旋中的每一個迴圈代表軟體流程中的一個階段（例如最內圈評估可行性，下一圈定義需求，再下一圈系統設計）。",[108,308928,308929,308932],{},[2627,308930,308931],{},"特點","：它結合了「避免變更」與「容忍變更」的概念，並假設所有變更都是專案風險所造成的。因此，該模型納入了明確的「風險管理活動」來降低專案風險。在開發過程中，螺旋模型可以根據細節程度容納不同的開發方法，甚至可以結合原型設計 (Prototyping) 以釐清需求並降低風險。",[108,308934,308935,308937],{},[2627,308936,308751],{},"：高風險、大型、複雜，且在初期難以釐清所有潛在問題的軟體專案。",{"title":29,"searchDepth":43,"depth":43,"links":308939},[308940,308941,308942,308943,308944,308945],{"id":308712,"depth":43,"text":308713},{"id":308755,"depth":43,"text":308756},{"id":308808,"depth":43,"text":308809},{"id":308832,"depth":43,"text":308833},{"id":308881,"depth":43,"text":308882},{"id":308915,"depth":43,"text":308916},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002F2.1",{"title":308704,"description":29},"source\u002Fcs\u002Fse\u002Fnotes\u002F2.1","RNixl6jiYph15n8qHHZshoMHjB0L9gWGVolBPcRGZTM",{"id":308952,"title":308953,"body":308954,"description":29,"extension":85,"meta":309234,"navigation":52,"path":309235,"seo":309236,"stem":309237,"__hash__":309238},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002F4.3.md","需求分析與誘導（Requirements elicitation and analysis）",{"type":8,"value":308955,"toc":309224},[308956,308959,308963,308966,308986,308990,308993,309017,309021,309024,309048,309052,309055,309097,309101,309104,309153,309167,309171,309174,309188,309192,309195,309203,309207,309210,309218],[11,308957,308953],{"id":308958},"需求分析與誘導requirements-elicitation-and-analysis",[253,308960,308962],{"id":308961},"_1-訪談-interviewing","1. 訪談 (Interviewing)",[15,308964,308965],{},"訪談是多數需求工程流程中的核心活動，開發團隊透過向利害關係人提問來獲取現有系統與未來新系統的相關需求。",[105,308967,308968,308974,308980],{},[108,308969,308970,308973],{},[2627,308971,308972],{},"封閉式訪談 (Closed interviews)："," 讓利害關係人回答一組預先定義好的特定問題。",[108,308975,308976,308979],{},[2627,308977,308978],{},"開放式訪談 (Open interviews)："," 沒有預設議程，開發團隊與利害關係人針對各種議題進行探索性討論，以更深入了解他們的需求。",[108,308981,308982,308985],{},[2627,308983,308984],{},"實務應用："," 實務上通常會混合使用這兩種方式。訪談非常適合用來獲取對使用者工作內容的整體理解。然而，因為使用者很難憑空想像系統未來的樣貌，單靠訪談容易遺漏重要資訊，且領域專家經常使用特定術語（jargon），容易造成工程師的誤解，因此必須搭配其他技術使用。",[253,308987,308989],{"id":308988},"_2-民族誌-觀察法-ethnography","2. 民族誌 \u002F 觀察法 (Ethnography)",[15,308991,308992],{},"許多軟體系統失敗的原因在於其需求忽略了社會與組織因素的影響。民族誌是一種觀察技術，分析師會將自己沉浸在系統未來要運作的工作環境中，觀察人們日常的實際工作任務。",[105,308994,308995,309005,309011],{},[108,308996,308997,309000,309001,309004],{},[2627,308998,308999],{},"發掘隱性需求："," 民族誌的價值在於能發掘出反映人們",[2627,309002,309003],{},"實際工作方式","的「隱性」需求，而非僅僅依據組織定義的正式流程。例如，航空管制員可能會因為警報系統過於敏感而將其關閉，轉而依賴其他策略。",[108,309006,309007,309010],{},[2627,309008,309009],{},"發掘協作需求："," 它能敏銳捕捉到團隊合作與「感知他人活動」的需求（例如觀察同事的工作量來調整自己的策略）。",[108,309012,309013,309016],{},[2627,309014,309015],{},"結合雛型設計："," 民族誌經常與系統雛型設計（Prototyping）結合使用，透過觀察結果來引導雛型的開發，同時也能針對觀察中發現的問題在下一次的評估中進行驗證。",[253,309018,309020],{"id":309019},"_3-故事與情境-stories-and-scenarios","3. 故事與情境 (Stories and Scenarios)",[15,309022,309023],{},"人們通常難以直接說出抽象的系統功能需求，但很容易對真實生活中的互動例子產生共鳴。",[105,309025,309026,309032,309038],{},[108,309027,309028,309031],{},[2627,309029,309030],{},"故事 (Stories)："," 以敘事文本的方式撰寫，提供系統使用的高階描述（Big picture）。它的優勢在於任何人都能輕易理解，非常適合作為引導討論的起點，甚至可以放在共享平台上讓廣泛的使用者群體提供回饋。",[108,309033,309034,309037],{},[2627,309035,309036],{},"情境 (Scenarios)："," 情境是從故事發展而來的具體互動範例。它通常是以結構化的方式呈現，內容包含：情境開始時的初始假設與狀態、正常的事件流程、可能發生的錯誤及其處理方式、其他同時進行的活動，以及情境結束時的系統狀態。",[108,309039,309040,309043,309044,309047],{},[2627,309041,309042],{},"敏捷開發的應用："," 在敏捷方法（如極限程式設計 XP）中，廣泛使用的",[2627,309045,309046],{},"使用者故事 (User stories)"," 即是情境的應用。客戶與開發團隊將需求寫在故事卡上，接著分解為具體的開發任務並進行優先排序。",[253,309049,309051],{"id":309050},"_4-标杆法-benchmarking-基准分析法","4. 标杆法 (Benchmarking \u002F 基准分析法)",[15,309053,309054],{},"标杆法是一种通过与行业内现有的、优秀的竞争对手或相似系统进行对比，从而获取和定义需求的方法。",[105,309056,309057,309062,309087],{},[108,309058,309059,309061],{},[2627,309060,243605],{},"：不要闭门造车。如果市场上已经有成熟的同类产品或最佳实践（Best Practices），开发团队应该系统性地分析这些“标杆”，了解它们的功能、性能指标和用户界面。",[108,309063,309064,33183,309067],{},[2627,309065,309066],{},"运作方式",[105,309068,309069,309075,309081],{},[108,309070,309071,309074],{},[2627,309072,309073],{},"识别标杆","：找出市场上最受欢迎或评价最高的竞品（例如，如果您要开发一个电商系统，可以拿淘宝或亚马逊作为标杆）。",[108,309076,309077,309080],{},[2627,309078,309079],{},"评估与对比","：系统性地测试这些标杆系统，记录它们的优点和缺点。",[108,309082,309083,309086],{},[2627,309084,309085],{},"提炼需求","：将标杆系统中的优秀功能转化为新系统的基准需求（Base requirements），并针对标杆系统的不足之处提出改进或创新需求。",[108,309088,309089,309092,309093,309096],{},[2627,309090,309091],{},"适用场景","：适用于开发商业软件产品、SaaS 应用，或当客户的目标是“做一个比 XXX 更好的系统”时。它特别有助于明确",[2627,309094,309095],{},"非功能性需求","（例如：系统响应时间必须比竞品快 20%）。",[253,309098,309100],{"id":309099},"_5-学徒法-apprenticeship","5. 学徒法 (Apprenticeship)",[15,309102,309103],{},"学徒法是情境访谈（Contextual Inquiry）中的一种核心技术。它重新定义了分析师与用户之间的关系。",[105,309105,309106,309111,309129,309147],{},[108,309107,309108,309110],{},[2627,309109,243605],{},"：分析师（需求工程师）扮演“学徒”（Apprentice）的角色，而终端用户扮演“师傅”（Master）的角色。师傅在真实的工作环境中执行日常任务，学徒在旁边观察、学习并适时提问。",[108,309112,309113,33183,309115],{},[2627,309114,309066],{},[105,309116,309117,309123],{},[108,309118,309119,309122],{},[2627,309120,309121],{},"在情境中学习","：分析师不是在会议室里问“您平时怎么工作？”，而是直接坐在用户的办公桌旁，看用户如何操作现有的系统或处理业务。",[108,309124,309125,309128],{},[2627,309126,309127],{},"师徒互动","：当用户执行某个特定动作（例如，在一个表格旁贴上便利贴）时，学徒会立刻提问：“师傅，您刚才为什么要把这个数据抄在纸上？” 师傅会基于当下发生的动作进行解释。",[108,309130,309131,33183,309133],{},[2627,309132,204919],{},[105,309134,309135,309141],{},[108,309136,309137,309140],{},[2627,309138,309139],{},"发掘隐性知识（Tacit Knowledge）","：用户往往对自己烂熟于心的工作流程习以为常，在传统会议访谈中根本想不起来提（正如参考资料中指出，领域专家经常觉得某些知识太基础而不值得一提，这容易导致需求遗漏）。学徒法能有效捕捉这些隐秘细节。",[108,309142,309143,309146],{},[2627,309144,309145],{},"避免脱离实际的假设","：让分析师真正理解用户的痛点和实际工作环境。",[108,309148,309149,309152],{},[2627,309150,309151],{},"与资料中“民族志\u002F观察法”的联系","：学徒法在理念上与参考资料中提到的**民族志\u002F观察法（Ethnography）**非常相似。参考资料指出，民族志要求分析师“沉浸在系统未来要运作的工作环境中”，去发掘那些反映人们实际工作方式的“隐性”需求（例如空管员觉得警报太吵而关掉它）。学徒法可以看作是这种观察法的一种高度互动和结构化的具体实践形式。",[15,309154,309155,309158,309159,309166],{},[2627,309156,309157],{},"学徒法","在精神上高度契合资料中强调的",[2627,309160,309161,309162,309165],{},"沉浸式观察（Ethnography）",[2627,309163,309164],{},"和发掘隐性需求的理念。而","标杆法","则通常作为资料中提到的“软件发现与评估（Software discovery and evaluation）”或竞品分析过程的一个自然延伸。",[253,309168,309170],{"id":309169},"_6-使用案例-use-cases","6. 使用案例 (Use Cases)",[15,309172,309173],{},"使用案例是一種透過圖形模型與結構化文本來描述系統與其使用者（或其他系統）之間互動的方法，是統一塑模語言（UML）的基礎功能。",[105,309175,309176,309182],{},[108,309177,309178,309181],{},[2627,309179,309180],{},"核心元素："," 它能識別出參與互動的參與者（Actors，通常以火柴人表示，可為人類、外部硬體或其他系統）以及互動的任務（以橢圓形表示）。",[108,309183,309184,309187],{},[2627,309185,309186],{},"細節補充："," 每個使用案例都代表一個獨立的外部互動，通常會附上詳細的文字描述。為了補充更詳細的流程，設計師會搭配 UML 的順序圖（Sequence diagrams）或狀態圖（State charts）來展示參與者與系統物件之間的具體訊息傳遞與互動序列。",[253,309189,309191],{"id":309190},"_7-觀點分析-viewpoints","7. 觀點分析 (Viewpoints)",[15,309193,309194],{},"大型系統會有非常多具有不同背景的利害關係人。為了簡化複雜的需求分析，可以將利害關係人的資訊進行分組與結構化。",[105,309196,309197],{},[108,309198,309199,309202],{},[2627,309200,309201],{},"運作方式："," 將具有共同特徵的利害關係人（如終端使用者、管理者）或外部系統約束視為一個「觀點」（Viewpoint），並將相關的需求收集到該觀點中。這有助於明確辨識哪些人可以提供特定資訊，並更有系統地組織需求以進行分析。",[253,309204,309206],{"id":309205},"_8-系統雛型設計-system-prototyping","8. 系統雛型設計 (System Prototyping)",[15,309208,309209],{},"雛型是軟體系統的早期可執行版本，主要用來展示概念、嘗試設計選項，並幫助利害關係人釐清問題。",[105,309211,309212],{},[108,309213,309214,309217],{},[2627,309215,309216],{},"引出與驗證需求："," 透過讓使用者直接與雛型系統互動，他們可以發現系統的優缺點，並提出原本沒想到的新需求；同時，這也能揭露原始需求規格中的錯誤或遺漏。雛型設計是一種有效「預期變更」的策略，能大幅減少系統交付後才提出的修改要求。",[15,309219,309220,309223],{},[2627,309221,309222],{},"需求分析過程中的常見挑戰：","\n在使用上述方法時，分析人員必須意識到需求分析是一個充滿挑戰的過程。利害關係人往往不知道自己真正需要什麼，或者會提出不切實際的要求；他們容易使用特定領域的行話；不同部門的利害關係人可能會提出相互衝突的需求；甚至組織內部的政治因素或權力鬥爭也會干預需求的制定。因此，需求分析通常需要召開跨部門會議來進行優先排序與妥協協商（Requirements prioritization and negotiation），才能產出最終的需求文件。",{"title":29,"searchDepth":43,"depth":43,"links":309225},[309226,309227,309228,309229,309230,309231,309232,309233],{"id":308961,"depth":49,"text":308962},{"id":308988,"depth":49,"text":308989},{"id":309019,"depth":49,"text":309020},{"id":309050,"depth":49,"text":309051},{"id":309099,"depth":49,"text":309100},{"id":309169,"depth":49,"text":309170},{"id":309190,"depth":49,"text":309191},{"id":309205,"depth":49,"text":309206},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002F4.3",{"title":308953,"description":29},"source\u002Fcs\u002Fse\u002Fnotes\u002F4.3","JvANtMCbVWe377OQDqrgg4n85C1oVQlr8hvywtubQ_I",{"id":309240,"title":309241,"body":309242,"description":29,"extension":85,"meta":309524,"navigation":52,"path":309525,"seo":309526,"stem":309527,"__hash__":309528},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002F6.3.md","6.3 架构模式 (Architectural patterns)",{"type":8,"value":309243,"toc":309516},[309244,309247,309251,309261,309264,309266,309270,309322,309324,309328,309366,309368,309372,309406,309408,309412,309458,309460,309464],[11,309245,309241],{"id":309246},"_63-架构模式-architectural-patterns",[253,309248,309250],{"id":309249},"架构模式概述-introduction-to-architectural-patterns","架构模式概述 (Introduction to Architectural Patterns)",[15,309252,309253,309256,309257,309260],{},[2627,309254,309255],{},"架构模式 (Architectural patterns)","（也常被称为",[2627,309258,309259],{},"架构风格 \u002F Architectural styles","）是对在不同软件系统和环境中经过尝试与测试的良好实践（good practice）的风格化、抽象化描述。\n一个架构模式不仅描述了系统的组织结构，还会详细说明在什么情况下适合或不适合使用该模式，并指出其优缺点（strengths and weaknesses）。",[15,309262,309263],{},"第 6.3 节详细介绍了五种最常用的架构模式：",[32188,309265],{},[19,309267,309269],{"id":309268},"_1-模型-视图-控制器模式-model-view-controller-mvc","1. 模型-视图-控制器模式 (Model-View-Controller, MVC)",[105,309271,309272,309298,309304,309310,309316],{},[108,309273,309274,309277,309278],{},[2627,309275,309276],{},"结构与形式 (Description\u002FStructure)","：\n系统被构造为三个相互交互的逻辑组件，将系统数据与表现形式（presentation）及用户交互（interaction）分离。\n",[105,309279,309280,309286,309292],{},[108,309281,309282,309285],{},[2627,309283,309284],{},"模型 (Model)","：管理系统数据及相关的数据操作，封装了应用程序的状态（application state）。",[108,309287,309288,309291],{},[2627,309289,309290],{},"视图 (View)","：定义并管理数据如何呈现（renders model）给用户。",[108,309293,309294,309297],{},[2627,309295,309296],{},"控制器 (Controller)","：管理用户交互（如按键、鼠标点击），并将这些交互事件（user events）映射并传递给视图和模型。",[108,309299,309300,309303],{},[2627,309301,309302],{},"适用场景 (When used)","：\n当存在多种查看和交互数据的方式时；或者当未来对数据展示和交互的需求还不明确时使用。",[108,309305,309306,309309],{},[2627,309307,309308],{},"优点 (Advantages)","：\n允许数据与其表现形式独立变更。支持用不同的方式呈现相同的数据，并且在一个表现形式中做出的更改会同步反映在所有的表现形式中。",[108,309311,309312,309315],{},[2627,309313,309314],{},"缺点 (Disadvantages)","：\n当数据模型和业务交互非常简单时，使用 MVC 可能会产生冗余的代码，反而增加代码的复杂性（code complexity）。",[108,309317,309318,309321],{},[2627,309319,309320],{},"示例 (Example)","：\n许多语言框架支持的基于 Web 的应用系统（Web-based application system）常使用该模式进行交互管理。",[32188,309323],{},[19,309325,309327],{"id":309326},"_2-分层架构-layered-architecture","2. 分层架构 (Layered Architecture)",[105,309329,309330,309339,309344,309349,309354],{},[108,309331,309332,309334,309335,309338],{},[2627,309333,309276],{},"：\n将系统组织成多个",[2627,309336,309337],{},"层 (layers)","，每个层关联特定的功能。一个层只为其直接位于其上方的层提供服务（services）。最底层的层通常代表整个系统可能用到的核心服务（core services），如操作系统或数据库支持。",[108,309340,309341,309343],{},[2627,309342,309302],{},"：\n需要在现有系统之上构建新设施时；开发工作由多个团队分散进行，且每个团队负责一层功能时；或者系统有多级安全需求（multilevel security）时。",[108,309345,309346,309348],{},[2627,309347,309308],{},"：\n支持系统的增量开发。只要维持好接口不变，整个层可以被独立替换。它将机器依赖性（machine dependencies）局部化，使系统更容易移植。可以在每一层提供冗余的设施（如认证 authentication）来提高系统的可靠性。",[108,309350,309351,309353],{},[2627,309352,309314],{},"：\n在实践中，很难在层与层之间提供绝对干净的分离，高层往往需要越过其直接下层，直接与更低层交互。此外，由于服务请求在每一层都需要被解释和处理，多层传递可能会导致性能问题（performance problems）。",[108,309355,309356,309358,309359,309362,309363,31901],{},[2627,309357,309320],{},"：\n用于支持学校全学科学习的数字学习系统（如 ",[2627,309360,309361],{},"iLearn system","），以及精神健康护理系统（",[2627,309364,309365],{},"Mentcare system",[32188,309367],{},[19,309369,309371],{"id":309370},"_3-仓库架构-存储库架构-repository-architecture","3. 仓库架构 \u002F 存储库架构 (Repository Architecture)",[105,309373,309374,309379,309384,309389,309398],{},[108,309375,309376,309378],{},[2627,309377,309276],{},"：\n系统中的所有数据都在一个中央仓库（central repository）或共享数据库（shared database）中管理，所有系统组件都可以访问它。组件之间不直接交互，只能通过仓库进行通信。",[108,309380,309381,309383],{},[2627,309382,309302],{},"：\n当系统产生并需要长期存储大量信息（large volumes of information）时；或者适用于数据驱动系统（data-driven systems），即仓库中包含新数据时会触发组件执行特定动作。",[108,309385,309386,309388],{},[2627,309387,309308],{},"：\n组件之间保持独立（independent），它们不需要知道其他组件的存在。一个组件所做的数据更改会立刻对所有组件可见。因为所有数据都在一个地方，所以数据的管理（如一致性备份 backups）非常方便统一。",[108,309390,309391,309393,309394,309397],{},[2627,309392,309314],{},"：\n中央仓库是一个",[2627,309395,309396],{},"单点故障 (single point of failure)","，仓库出现问题会影响整个系统。通过仓库组织所有通信可能会导致效率低下。将仓库分布到多台计算机上通常很困难。",[108,309399,309400,33183,309402,309405],{},[2627,309401,309320],{},[2627,309403,309404],{},"集成开发环境 (Integrated Development Environment, IDE)","，其中的各种软件工具（如 Java 编辑器、UML 编辑器、代码生成器等）通过一个共享的系统设计信息项目仓库进行协作。还包括指挥控制系统、管理信息系统、CAD系统等。",[32188,309407],{},[19,309409,309411],{"id":309410},"_4-客户端-服务器架构-client-server-architecture","4. 客户端-服务器架构 (Client-Server Architecture)",[105,309413,309414,309426,309431,309440,309449],{},[108,309415,309416,309418,309419,309422,309423,347],{},[2627,309417,309276],{},"：\n展示了分布式系统的运行时组织形式。系统被组织为一组提供服务的",[2627,309420,309421],{},"服务器 (servers)","，以及通过网络访问和使用这些服务的",[2627,309424,309425],{},"客户端 (clients)",[108,309427,309428,309430],{},[2627,309429,309302],{},"：\n当共享数据库中的数据需要从多个不同的位置被访问时；当系统的负载（load）多变且不确定时。",[108,309432,309433,309435,309436,309439],{},[2627,309434,309308],{},"：\n最大的优势是它是一个",[2627,309437,309438],{},"分布式架构 (distributed architecture)","，能有效利用网络化系统中的多个分布式处理器。添加新服务器或透明地升级服务器非常容易，且不会影响系统的其他部分。",[108,309441,309442,309444,309445,309448],{},[2627,309443,309314],{},"：\n每一个服务都是一个单点故障。系统容易受到",[2627,309446,309447],{},"拒绝服务攻击 (denial-of-service attacks)"," 和服务器故障的影响。系统性能依赖于网络，难以预测。如果服务器由不同的组织所有，可能会出现管理问题。",[108,309450,309451,309453,309454,309457],{},[2627,309452,309320],{},"：\n一个在互联网上提供视频、照片和目录服务的",[2627,309455,309456],{},"电影库系统 (film library)","。客户端可以是集成的 Web 浏览器界面，它通过网络访问后端的各种服务器。",[32188,309459],{},[19,309461,309463],{"id":309462},"_5-管道和过滤器架构-pipe-and-filter-architecture","5. 管道和过滤器架构 (Pipe and Filter Architecture)",[105,309465,309466,309479,309484,309489,309498],{},[108,309467,309468,309470,309471,309474,309475,309478],{},[2627,309469,309276],{},"：\n系统的数据处理被组织成多个离散的组件（即",[2627,309472,309473],{},"过滤器 \u002F filter","），每个组件执行一种数据转换。数据像在",[2627,309476,309477],{},"管道 (pipe)"," 中一样从一个组件流向另一个组件。这些转换可以串行（sequentially）执行，也可以并行（in parallel）执行。",[108,309480,309481,309483],{},[2627,309482,309302],{},"：\n常用于数据处理应用（data-processing applications），包括批处理或基于事务的系统，在这些系统中，输入数据经过分阶段处理后生成相关的输出。",[108,309485,309486,309488],{},[2627,309487,309308],{},"：\n易于理解，并支持转换组件的复用（transformation reuse）。这种工作流风格与许多业务流程的结构相匹配。通过添加新的转换可以很直接地使系统演进（evolution）。可以被实现为顺序系统或并发系统。",[108,309490,309491,309493,309494,309497],{},[2627,309492,309314],{},"：\n通信的转换组件之间必须事先商定数据传输的格式（data transfer format）。每个过滤器都必须解析（parse）其输入，并反解析（unparse）其输出为商定格式，这增加了",[2627,309495,309496],{},"系统开销 (system overhead)","。对于使用不兼容数据结构的架构组件，无法进行复用。对于基于事件控制的交互式系统（interactive systems），很难以这种顺序流的方式来实现。",[108,309499,309500,33183,309502],{},[2627,309501,309320],{},[105,309503,309504,309510],{},[108,309505,309506,309509],{},[2627,309507,309508],{},"Unix \u002F Linux 系统","：\"管道和过滤器\"的名字正是来源于早期的 Unix 系统。在 Unix shell 中，可以通过“管道（pipes）”将进程链接起来，把一个进程的文本流（text stream）传递给下一个进程处理。",[108,309511,309512,309515],{},[2627,309513,309514],{},"发票批处理系统 (batch sequential model)","：读取已开具的发票、识别付款、为已付清的发票开具收据，并向未付款客户发出催款单。",{"title":29,"searchDepth":43,"depth":43,"links":309517},[309518,309519,309520,309521,309522,309523],{"id":309249,"depth":49,"text":309250},{"id":309268,"depth":43,"text":309269},{"id":309326,"depth":43,"text":309327},{"id":309370,"depth":43,"text":309371},{"id":309410,"depth":43,"text":309411},{"id":309462,"depth":43,"text":309463},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002F6.3",{"title":309241,"description":29},"source\u002Fcs\u002Fse\u002Fnotes\u002F6.3","fGPQMJjQcByssx_lW9k4z9F_hkiLUTS3hnnpLii9zBY",{"id":309530,"title":309531,"body":309532,"description":29,"extension":85,"meta":309813,"navigation":52,"path":309814,"seo":309815,"stem":309816,"__hash__":309817},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002Frtm.md","RTM（Requirements Traceability Matrix，需求跟踪矩阵）",{"type":8,"value":309533,"toc":309807},[309534,309537,309541,309547,309552,309572,309574,309578,309585,309625,309627,309631,309634,309748,309750,309754,309757,309787],[11,309535,309531],{"id":309536},"rtmrequirements-traceability-matrix需求跟踪矩阵",[19,309538,309540],{"id":309539},"一-什么是需求跟踪-requirements-traceability","一、 什么是需求跟踪 (Requirements Traceability)？",[15,309542,309543,309546],{},[2627,309544,309545],{},"本质："," 它就是一种“记录和追踪”的机制。专门用来记录“需求之间”、“需求与它的来源之间”、以及“需求与系统设计\u002F代码之间”的相互依赖和对应关系。",[15,309548,309549],{},[2627,309550,309551],{},"三大核心价值：",[2230,309553,309554,309560,309566],{},[108,309555,309556,309559],{},[2627,309557,309558],{},"评估变更影响 (Impact Analysis)："," 软件系统在开发过程中，需求变更是家常便饭。当有人提出修改时，跟踪信息能帮团队分析为什么要改，并准确评估这个变动会波及到系统的哪些部分。",[108,309561,309562,309565],{},[2627,309563,309564],{},"掌控连锁反应："," 就像蝴蝶效应一样，单点修改可能会在整个系统中引发连锁反应。追踪机制能让开发和维护人员对这种波及范围一目了然。",[108,309567,309568,309571],{},[2627,309569,309570],{},"协助风险管理："," 掌握了变更的冲击范围，团队就能在系统设计时做到更好的信息隐藏与隔离，从而大幅降低由于需求变更带来的开发风险。",[32188,309573],{},[19,309575,309577],{"id":309576},"二-如何落实与管理需求跟踪","二、 如何落实与管理需求跟踪？",[15,309579,309580,309581,309584],{},"要在项目中落地 RTM，必须在",[2627,309582,309583],{},"需求文档草稿阶段","就开始动手规划，主要包括以下三点：",[105,309586,309587,309593,309599],{},[108,309588,309589,309592],{},[2627,309590,309591],{},"唯一标识符 (Unique ID)："," 必须为每一个需求分配一个独一无二的 ID（就像人的身份证号），这是后续能够进行交叉引用 (Cross-referenced) 的基础。",[108,309594,309595,309598],{},[2627,309596,309597],{},"制定跟踪策略 (Traceability Policies)："," 团队需要立下规矩：哪些需求之间必须建立关联？需求跟设计之间怎么关联？这些记录谁来维护？",[108,309600,309601,309604,309605],{},[2627,309602,309603],{},"引入工具支持："," 真实项目的需求信息量极其庞大，靠人脑根本记不住。因此必须借助工具：\n",[105,309606,309607,309613,309619],{},[108,309608,309609,309612],{},[2627,309610,309611],{},"最常见："," 共享的电子表格 (Spreadsheets，比如 Excel)，通常直接用来绘制 RTM。",[108,309614,309615,309618],{},[2627,309616,309617],{},"更专业："," 数据库系统或专业的软件生命周期管理工具（如 DOORS）。",[108,309620,309621,309624],{},[2627,309622,309623],{},"最前沿："," 一些进阶工具甚至会运用 NLP（自然语言处理）技术，自动帮你发掘需求之间的潜在关联。",[32188,309626],{},[19,309628,309630],{"id":309629},"三-rtm-的五大核心元素矩阵的骨架","三、 RTM 的五大核心元素（矩阵的骨架）",[15,309632,309633],{},"如果你把 RTM 想象成一个巨大的 Excel 表格，那么资料中提到的这五个元素，就是表格里的五个核心“列 (Columns)”。它们通过唯一 ID 一脉相承地串联在一起：",[2230,309635,309636,309658,309679,309700,309724],{},[108,309637,309638,309641],{},[2627,309639,309640],{},"User Case (用例 \u002F 用户需求)",[105,309642,309643,309653],{},[108,309644,309645,309648,309649,309652],{},[2627,309646,309647],{},"角色："," 跟踪的",[2627,309650,309651],{},"源头（起点）","。它代表了高阶的业务目标，描述用户到底期望系统能做什么。",[108,309654,309655,309657],{},[2627,309656,287316],{}," 在 RTM 中，每个 User Case 都有专属 ID，用来向下追踪它到底衍生出了哪些具体的功能。",[108,309659,309660,309663],{},[2627,309661,309662],{},"Function (系统功能需求)",[105,309664,309665,309674],{},[108,309666,309667,309669,309670,309673],{},[2627,309668,309647],{}," 用户期望的",[2627,309671,309672],{},"具体化","。详细规定了系统要怎么输入、怎么输出、出现异常怎么处理。",[108,309675,309676,309678],{},[2627,309677,287316],{}," RTM 会记录“功能 ID”对应哪个“User Case ID”。这能确保开发团队没有凭空捏造无关的功能（防止范围蔓延），也确保了用户的所有期望都落到了实处。",[108,309680,309681,309684],{},[2627,309682,309683],{},"Design (系统\u002F组件设计)",[105,309685,309686,309695],{},[108,309687,309688,309690,309691,309694],{},[2627,309689,309647],{}," 描述该功能在系统底层是",[2627,309692,309693],{},"如何被规划的","（架构设计、数据库表设计、API 接口等）。",[108,309696,309697,309699],{},[2627,309698,287316],{}," 功能 ID 会链接到具体的“设计文档或模块 ID”。这帮助架构师确认，每一个口头上的功能需求，在底层都有坚实的架构来支撑它。",[108,309701,309702,309705],{},[2627,309703,309704],{},"Code (代码 \u002F 实现)",[105,309706,309707,309715],{},[108,309708,309709,309711,309712,347],{},[2627,309710,309647],{}," 设计的",[2627,309713,309714],{},"最终实体化",[108,309716,309717,309719,309720,309723],{},[2627,309718,287316],{}," 记录实现该功能的模块名、类 (Class) 或具体的代码提交版本 (Commit ID)。当需求发生变更时，工程师通过 RTM 能",[2627,309721,309722],{},"瞬间定位","到需要修改哪几行代码，这也是评估连连锁反应最直接的手段。",[108,309725,309726,309729],{},[2627,309727,309728],{},"Test Case (测试用例)",[105,309730,309731,309739],{},[108,309732,309733,309735,309736,347],{},[2627,309734,309647],{}," 最关键的",[2627,309737,309738],{},"验证环节",[108,309740,309741,309743,309744,309747],{},[2627,309742,287316],{}," 每一个需求都应该至少对应一个测试用例。RTM 会明确列出每个功能或设计 ID 对应的 Test Case ID，确保",[2627,309745,309746],{},"测试覆盖率达到 100%","，没有任何功能被漏测（即基于需求的测试）。",[32188,309749],{},[19,309751,309753],{"id":309752},"四-总结rtm-的追踪链是如何运作的","四、 总结：RTM 的追踪链是如何运作的？",[15,309755,309756],{},"当把这五个元素组合在一起，RTM 就形成了一条极具控制力的追踪链，看起来就像这样：",[104833,309758,309759],{},[15,309760,309761,309766,309767,309766,309772,309766,309777,309766,309782],{},[2627,309762,309763],{},[34,309764,309765],{},"用例 ID: UC-01 登录"," -> ",[2627,309768,309769],{},[34,309770,309771],{},"功能 ID: F-011 密码验证",[2627,309773,309774],{},[34,309775,309776],{},"设计 ID: D-Auth-Module",[2627,309778,309779],{},[34,309780,309781],{},"代码 ID: AuthController.java",[2627,309783,309784],{},[34,309785,309786],{},"测试用例 ID: TC-011-A 错误密码测试",[105,309788,309789,309798],{},[108,309790,309791,309794,309795,31901],{},[2627,309792,309793],{},"向前追踪 (从左向右看)："," 确保客户提出的每一个原始需求，最终都被开发出来，并且被测试通过了（",[2627,309796,309797],{},"确保不漏",[108,309799,309800,309803,309804,31901],{},[2627,309801,309802],{},"向后追踪 (从右向左看)："," 确保程序员写的每一行代码、测试员执行的每一个用例，都是为了解决真实存在的用户需求（",[2627,309805,309806],{},"确保不多余",{"title":29,"searchDepth":43,"depth":43,"links":309808},[309809,309810,309811,309812],{"id":309539,"depth":43,"text":309540},{"id":309576,"depth":43,"text":309577},{"id":309629,"depth":43,"text":309630},{"id":309752,"depth":43,"text":309753},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotes\u002Frtm",{"title":309531,"description":29},"source\u002Fcs\u002Fse\u002Fnotes\u002Frtm","hj2cN4uCl1gxjK2FXrZWodEEbp2YWAfoAlT4YFSyvec",{"id":309819,"title":309820,"body":309821,"description":29,"extension":85,"meta":309825,"navigation":52,"path":309826,"seo":309827,"stem":309828,"__hash__":309829},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap1.md","Chap1",{"type":8,"value":309822,"toc":309823},[],{"title":29,"searchDepth":43,"depth":43,"links":309824},[],{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap1",{"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap1","6vkmljMKH9lPj570z2UVoOzFjTNB7o7AECz9rV4hCOs",{"id":309831,"title":309832,"body":309833,"description":29,"extension":85,"meta":309837,"navigation":52,"path":309838,"seo":309839,"stem":309840,"__hash__":309841},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap2.md","Chap2",{"type":8,"value":309834,"toc":309835},[],{"title":29,"searchDepth":43,"depth":43,"links":309836},[],{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap2",{"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap2","061TP0JB7l-PETZXdmqcl1UYQ-EpTPIUeHlrBaOUjJY",{"id":309843,"title":309844,"body":309845,"description":29,"extension":85,"meta":310357,"navigation":52,"path":310358,"seo":310359,"stem":310360,"__hash__":310361},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap3.md","Chapter 3: Agile Software Development",{"type":8,"value":309846,"toc":310335},[309847,309850,309854,309861,309863,309867,309874,309878,309902,309906,309913,309939,309943,309975,309977,309981,309988,309992,309999,310033,310037,310044,310063,310067,310074,310155,310159,310162,310169,310171,310175,310189,310193,310221,310225,310261,310273,310275,310279,310282,310286,310306,310310,310313,310333],[11,309848,309844],{"id":309849},"chapter-3-agile-software-development",[19,309851,309853],{"id":309852},"executive-summary","Executive Summary",[15,309855,309856,309857,309860],{},"In the 1980s and 1990s, software engineering was dominated by \"plan-driven\" approaches characterized by rigorous planning, extensive documentation, and strict sequential phases. While effective for massive, decade-long aerospace or government projects, these heavyweight methods proved too rigid for the fast-paced, rapidly changing landscape of modern business. This chapter explores ",[2627,309858,309859],{},"Agile Software Development",", a paradigm shift designed to drastically reduce process overhead, embrace changing requirements, and deliver business value through rapid, iterative increments. We will dissect the core agile techniques (such as Extreme Programming), explore agile project management (Scrum), and analyze the challenges of scaling agile for large enterprise systems.",[32188,309862],{},[19,309864,309866],{"id":309865},"_31-the-shift-plan-driven-vs-agile-methods","3.1 The Shift: Plan-Driven vs. Agile Methods",[15,309868,309869,309870,309873],{},"To understand ",[71807,309871,309872],{},"why"," agile exists, we must first understand what it replaced.",[253,309875,309877],{"id":309876},"the-plan-driven-traditional-approach","The Plan-Driven (Traditional) Approach",[105,309879,309880,309886,309892],{},[108,309881,309882,309885],{},[2627,309883,309884],{},"The Core Assumption:"," All system requirements can be known, specified, and documented before any design or coding begins.",[108,309887,309888,309891],{},[2627,309889,309890],{},"The Process:"," A linear progression (e.g., Waterfall). You finish Requirements completely, hand off a massive document to Designers, who hand off schematics to Programmers, who hand off code to Testers.",[108,309893,309894,309897,309898,309901],{},[2627,309895,309896],{},"The Problem:"," In reality, ",[2627,309899,309900],{},"requirements always change",". Business needs evolve, competitors release new products, and customers rarely know exactly what they want until they see a working prototype. When changes occur in a plan-driven model, you must go back and update hundreds of pages of documentation, causing severe delays.",[253,309903,309905],{"id":309904},"the-agile-paradigm","The Agile Paradigm",[15,309907,309908,309909,309912],{},"Agile methods reject the idea of complete upfront planning. Instead, they propose a highly adaptive, iterative lifecycle. The core philosophy is captured in the ",[2627,309910,309911],{},"Agile Manifesto",", which values:",[2230,309914,309915,309921,309927,309933],{},[108,309916,309917,309920],{},[2627,309918,309919],{},"Individuals and interactions"," over processes and tools.",[108,309922,309923,309926],{},[2627,309924,309925],{},"Working software"," over comprehensive documentation.",[108,309928,309929,309932],{},[2627,309930,309931],{},"Customer collaboration"," over contract negotiation.",[108,309934,309935,309938],{},[2627,309936,309937],{},"Responding to change"," over following a plan.",[253,309940,309942],{"id":309941},"five-key-agile-principles","Five Key Agile Principles",[2230,309944,309945,309951,309957,309963,309969],{},[108,309946,309947,309950],{},[2627,309948,309949],{},"Customer Involvement:"," The customer is a continuous, active member of the development team, prioritizing features and evaluating increments.",[108,309952,309953,309956],{},[2627,309954,309955],{},"Embrace Change:"," The architecture and process are specifically designed to easily accommodate new requirements at any stage.",[108,309958,309959,309962],{},[2627,309960,309961],{},"Incremental Delivery:"," Software is built and released in small, frequent chunks (e.g., every 2–3 weeks).",[108,309964,309965,309968],{},[2627,309966,309967],{},"Maintain Simplicity:"," Do enough design for the current requirements and absolutely no more.",[108,309970,309971,309974],{},[2627,309972,309973],{},"People, Not Process:"," Exploit the skills of the development team through practices like pair programming, avoiding rigid administrative workflows.",[32188,309976],{},[19,309978,309980],{"id":309979},"_32-agile-development-techniques-extreme-programming","3.2 Agile Development Techniques (Extreme Programming)",[15,309982,309983,309984,309987],{},"While \"Agile\" is the philosophy, ",[2627,309985,309986],{},"Extreme Programming (XP)"," provided the first widely adopted set of concrete technical practices. XP takes recognized good software engineering practices and turns the dial up to \"extreme\".",[253,309989,309991],{"id":309990},"_1-user-stories-replacing-requirements-documents","1. User Stories (Replacing Requirements Documents)",[15,309993,309994,309995,309998],{},"Instead of formal \"The system shall...\" statements, XP uses ",[2627,309996,309997],{},"User Stories",": brief, plain-language scenarios written on physical or digital cards.",[105,310000,310001,310014],{},[108,310002,310003,310006,310007,310010,310011,310013],{},[2627,310004,310005],{},"The \"How\":"," They act as a ",[71807,310008,310009],{},"placeholder for a conversation",". During an iteration planning meeting, developers discuss the story with the customer, break it down into technical ",[2627,310012,1345],{},", and estimate the effort.",[108,310015,310016,310019],{},[2627,310017,310018],{},"Mastery Application:",[105,310020,310021,310027],{},[108,310022,310023,310026],{},[71807,310024,310025],{},"Story:"," \"As a doctor prescribing medication, I want the system to warn me if a prescribed dose is outside safe limits.\"",[108,310028,310029,310032],{},[71807,310030,310031],{},"Tasks derived:"," (1) Build UI alert box. (2) Write SQL query to fetch max\u002Fmin dosage limits from the formulary database. (3) Write validation logic to compare input against limits.",[253,310034,310036],{"id":310035},"_2-refactoring-continuous-code-improvement","2. Refactoring (Continuous Code Improvement)",[15,310038,310039,310040,310043],{},"Traditional engineering says \"design for change\"—anticipate future needs and build generic frameworks. XP says this is wasted effort because anticipated changes rarely happen. Instead, build the simplest thing today, but ",[2627,310041,310042],{},"Refactor"," constantly.",[105,310045,310046,310055],{},[108,310047,310048,310051,310052,2547],{},[2627,310049,310050],{},"The \"Why\":"," As software is incrementally patched, its structure naturally degrades (this is called \"technical debt\"). Refactoring is the process of cleaning up the code (removing duplicates, renaming variables, simplifying loops) ",[71807,310053,310054],{},"without changing its external behavior",[108,310056,310057,310059,310060,26192],{},[2627,310058,310005],{}," If a developer spots clunky code, they fix it immediately, regardless of who originally wrote it (a principle called ",[2627,310061,310062],{},"Collective Ownership",[253,310064,310066],{"id":310065},"_3-test-first-development-tdd","3. Test-First Development (TDD)",[15,310068,310069,310070,310073],{},"Perhaps the most crucial technical legacy of XP. You do not write code and then test it. You write the test ",[71807,310071,310072],{},"before"," the code exists.",[105,310075,310076,310081,310089],{},[108,310077,310078,310080],{},[2627,310079,309890],{}," 1. Write a test for a tiny piece of functionality.\n2. Run it (it fails, because the feature isn't built yet).\n3. Write the exact amount of code needed to make the test pass.\n4. Refactor the code to clean it up.",[108,310082,310083,310085,310086,310088],{},[2627,310084,310050],{}," It forces the developer to explicitly define the interface and behavior ",[71807,310087,310072],{}," getting lost in implementation logic. Furthermore, it results in a comprehensive suite of automated tests. If a developer accidentally breaks existing logic while adding a new feature, the automated test instantly fails, catching the bug before deployment.",[108,310090,310091,310094],{},[2627,310092,310093],{},"Mastery Application (Python TDD Example):",[24,310095,310099],{"className":310096,"code":310097,"language":310098,"meta":29,"style":29},"language-python shiki shiki-themes github-light github-dark","# 1. The Test (Written First)\ndef test_dose_checking():\n assert is_dose_safe(drug=\"Aspirin\", amount=500) == True\n assert is_dose_safe(drug=\"Aspirin\", amount=5000) == False # Overdose\n\n# 2. The Implementation (Written Second)\ndef is_dose_safe(drug, amount):\n max_limits = {\"Aspirin\": 1000, \"Ibuprofen\": 800}\n if amount > max_limits.get(drug, 0):\n return False\n return True\n","python",[31,310100,310101,310106,310111,310116,310121,310125,310130,310135,310140,310145,310150],{"__ignoreMap":29},[34,310102,310103],{"class":36,"line":37},[34,310104,310105],{},"# 1. The Test (Written First)\n",[34,310107,310108],{"class":36,"line":43},[34,310109,310110],{},"def test_dose_checking():\n",[34,310112,310113],{"class":36,"line":49},[34,310114,310115],{}," assert is_dose_safe(drug=\"Aspirin\", amount=500) == True\n",[34,310117,310118],{"class":36,"line":56},[34,310119,310120],{}," assert is_dose_safe(drug=\"Aspirin\", amount=5000) == False # Overdose\n",[34,310122,310123],{"class":36,"line":62},[34,310124,53],{"emptyLinePlaceholder":52},[34,310126,310127],{"class":36,"line":68},[34,310128,310129],{},"# 2. The Implementation (Written Second)\n",[34,310131,310132],{"class":36,"line":74},[34,310133,310134],{},"def is_dose_safe(drug, amount):\n",[34,310136,310137],{"class":36,"line":1971},[34,310138,310139],{}," max_limits = {\"Aspirin\": 1000, \"Ibuprofen\": 800}\n",[34,310141,310142],{"class":36,"line":1976},[34,310143,310144],{}," if amount > max_limits.get(drug, 0):\n",[34,310146,310147],{"class":36,"line":1985},[34,310148,310149],{}," return False\n",[34,310151,310152],{"class":36,"line":1990},[34,310153,310154],{}," return True\n",[253,310156,310158],{"id":310157},"_4-pair-programming","4. Pair Programming",[15,310160,310161],{},"Developers work in pairs at a single computer.",[105,310163,310164],{},[108,310165,310166,310168],{},[2627,310167,310050],{}," It provides real-time, continuous code review. While one developer types (focusing on syntax), the other navigates (focusing on overall strategic logic and spotting typos). It also ensures that at least two people thoroughly understand every piece of the codebase, eliminating \"islands of expertise.\"",[32188,310170],{},[19,310172,310174],{"id":310173},"_33-agile-project-management-scrum","3.3 Agile Project Management (Scrum)",[15,310176,310177,310178,310181,310182,310184,310185,310188],{},"While XP focuses on ",[71807,310179,310180],{},"technical"," execution, ",[2627,310183,308867],{}," provides an overarching ",[71807,310186,310187],{},"management"," framework. Traditional project managers need visibility (schedules, budgets), but agile teams need autonomy. Scrum bridges this gap.",[253,310190,310192],{"id":310191},"core-scrum-terminology-roles","Core Scrum Terminology & Roles",[105,310194,310195,310205,310211],{},[108,310196,310197,310200,310201,310204],{},[2627,310198,310199],{},"Product Owner:"," Represents the business and customers. They own the \"Product Backlog\" and dictate ",[71807,310202,310203],{},"what"," gets built based on business value.",[108,310206,310207,310210],{},[2627,310208,310209],{},"ScrumMaster:"," Replaces the traditional \"Project Manager.\" They do not command the team; they act as a facilitator and shield the team from external distractions.",[108,310212,310213,310216,310217,310220],{},[2627,310214,310215],{},"Development Team:"," A self-organizing group of max 7-9 people. They alone decide ",[71807,310218,310219],{},"how"," to build the software.",[253,310222,310224],{"id":310223},"the-scrum-process-sequential-execution","The Scrum Process (Sequential Execution)",[2230,310226,310227,310233,310239,310249,310255],{},[108,310228,310229,310232],{},[2627,310230,310231],{},"Product Backlog:"," A master list of all desired features (user stories), prioritized by the Product Owner.",[108,310234,310235,310238],{},[2627,310236,310237],{},"Sprint Planning:"," The team decides how much work they can commit to pulling from the top of the Product Backlog into the upcoming \"Sprint\".",[108,310240,310241,310244,310245,310248],{},[2627,310242,310243],{},"The Sprint:"," A fixed time-box (e.g., 2–4 weeks) where the team executes the work to create a potentially shippable increment. ",[2627,310246,310247],{},"Crucial Rule:"," No outside changes can be added to a Sprint once it begins.",[108,310250,310251,310254],{},[2627,310252,310253],{},"Daily Scrum (Stand-up):"," A 15-minute daily meeting where members quickly state: What they did yesterday, what they will do today, and any blockers in their way.",[108,310256,310257,310260],{},[2627,310258,310259],{},"Sprint Review & Retrospective:"," The team demonstrates the working software to stakeholders and then reflects internally on how to improve their processes for the next Sprint.",[105,310262,310263],{},[108,310264,310265,310268,310269,310272],{},[2627,310266,310267],{},"Mastery Application (Velocity):"," How does a Scrum team know what they can achieve in a Sprint? They track ",[2627,310270,310271],{},"Velocity","—a measure of how many \"story points\" they historically complete per iteration. If their average velocity is 40 points, they only pull 40 points worth of work into the next Sprint.",[32188,310274],{},[19,310276,310278],{"id":310277},"_34-scaling-agile-methods","3.4 Scaling Agile Methods",[15,310280,310281],{},"Agile was designed for small, co-located teams (e.g., 8 people in one room). However, modern enterprises build \"Systems of Systems\" requiring hundreds of developers. Scaling agile is a massive industry challenge.",[253,310283,310285],{"id":310284},"the-challenges-of-scale","The Challenges of Scale",[105,310287,310288,310294,310300],{},[108,310289,310290,310293],{},[2627,310291,310292],{},"System Size & Architecture:"," You cannot simply \"emerge\" an architecture for a massive system; some upfront architectural design is mandatory before coding begins.",[108,310295,310296,310299],{},[2627,310297,310298],{},"Geographic Distribution:"," Global teams cannot easily rely on informal, face-to-face communication.",[108,310301,310302,310305],{},[2627,310303,310304],{},"Regulatory Compliance:"," Aviation, medical, and financial systems legally require extensive documentation and safety cases, contradicting the \"working software over documentation\" manifesto principle.",[253,310307,310309],{"id":310308},"solutions-eg-ibms-agility-at-scale-model","Solutions (e.g., IBM's Agility at Scale Model)",[15,310311,310312],{},"To scale, organizations must adopt a hybrid approach, blending agile execution with plan-driven governance:",[2230,310314,310315,310321,310327],{},[108,310316,310317,310320],{},[2627,310318,310319],{},"Upfront Requirements:"," A complete, detailed requirements document isn't needed, but a high-level system outline is mandatory to define contracts and separate subsystems.",[108,310322,310323,310326],{},[2627,310324,310325],{},"Multiple Product Owners:"," A single customer cannot understand a massive enterprise system. A hierarchy of Product Owners is required.",[108,310328,310329,310332],{},[2627,310330,310331],{},"Cross-Team Synchronization:"," Using frameworks like \"Scrum of Scrums\" (where representatives from multiple individual Scrum teams meet to coordinate) and aligning release schedules across all teams.",[79,310334,81],{},{"title":29,"searchDepth":43,"depth":43,"links":310336},[310337,310338,310343,310349,310353],{"id":309852,"depth":43,"text":309853},{"id":309865,"depth":43,"text":309866,"children":310339},[310340,310341,310342],{"id":309876,"depth":49,"text":309877},{"id":309904,"depth":49,"text":309905},{"id":309941,"depth":49,"text":309942},{"id":309979,"depth":43,"text":309980,"children":310344},[310345,310346,310347,310348],{"id":309990,"depth":49,"text":309991},{"id":310035,"depth":49,"text":310036},{"id":310065,"depth":49,"text":310066},{"id":310157,"depth":49,"text":310158},{"id":310173,"depth":43,"text":310174,"children":310350},[310351,310352],{"id":310191,"depth":49,"text":310192},{"id":310223,"depth":49,"text":310224},{"id":310277,"depth":43,"text":310278,"children":310354},[310355,310356],{"id":310284,"depth":49,"text":310285},{"id":310308,"depth":49,"text":310309},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap3",{"title":309844,"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap3","nmMWzQL3Hnuu0JkW3IB5jNbDwJ3wMYZA7_a_qYc9P0w",{"id":310363,"title":310364,"body":310365,"description":310905,"extension":85,"meta":310906,"navigation":52,"path":310907,"seo":310908,"stem":310909,"__hash__":310910},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap4.md","Chapter 4 - Requirements Engineering",{"type":8,"value":310366,"toc":310885},[310367,310370,310376,310395,310397,310401,310404,310410,310414,310445,310449,310471,310473,310477,310480,310484,310491,310525,310529,310535,310571,310575,310585,310600,310602,310606,310613,310617,310620,310640,310645,310681,310685,310691,310696,310724,310728,310737,310776,310781,310801,310803,310807,310814,310818,310841,310845,310851,310873],[11,310368,310364],{"id":310369},"chapter-4-requirements-engineering",[15,310371,310372,310375],{},[2627,310373,310374],{},"Requirements Engineering (RE)"," is the foundational stage of the software development lifecycle. It is the systematic process of discovering, analyzing, documenting, and validating the services that a system should provide and the constraints under which it must operate. Failing to properly engineer requirements leads to systems that do not meet user needs, exceed budgets, and are plagued by constant rework.",[15,310377,310378,310379,310382,310383,310386,310387,310390,310391,310394],{},"This chapter breaks down RE into four deeply interconnected phases: ",[2627,310380,310381],{},"Elicitation"," (finding out what is needed), ",[2627,310384,310385],{},"Specification"," (writing it down precisely), ",[2627,310388,310389],{},"Validation"," (ensuring it is exactly what the customer wants), and ",[2627,310392,310393],{},"Requirements Management"," (controlling the inevitable changes that arise during and after development).",[32188,310396],{},[19,310398,310400],{"id":310399},"_1-the-anatomy-of-a-requirement-start-from-zero","1. The Anatomy of a Requirement: Start from Zero",[15,310402,310403],{},"Before diving into the processes of RE, we must define what a \"requirement\" actually is. In software engineering, the term is highly elastic. A requirement can range from a high-level, abstract statement of a feature (\"The system shall track patient records\") to a mathematically precise definition of an algorithm.",[15,310405,310406,310407,5598],{},"To prevent confusion, software engineers separate requirements into two distinct layers based on the ",[2627,310408,310409],{},"audience",[253,310411,310413],{"id":310412},"_11-user-requirements","1.1 User Requirements",[105,310415,310416,310422,310428,310434],{},[108,310417,310418,310421],{},[2627,310419,310420],{},"What it is:"," High-level, abstract statements of the services the system is expected to provide and its operational constraints.",[108,310423,310424,310427],{},[2627,310425,310426],{},"Target Audience:"," Client managers, system end-users, and business stakeholders.",[108,310429,310430,310433],{},[2627,310431,310432],{},"Format:"," Written in simple, natural language supplemented by intuitive diagrams. They deliberately avoid technical jargon or software architecture details.",[108,310435,310436,310438,310439,310441,310442,310444],{},[2627,310437,310050],{}," A business manager needs to understand ",[71807,310440,310203],{}," the system will do for the company without getting bogged down by ",[71807,310443,310219],{}," the database is structured.",[253,310446,310448],{"id":310447},"_12-system-requirements","1.2 System Requirements",[105,310450,310451,310456,310461,310466],{},[108,310452,310453,310455],{},[2627,310454,310420],{}," A highly detailed and precise description of the software system's functions, services, and constraints. This document (often called the functional specification) serves as the contract between the buyer and the software developers.",[108,310457,310458,310460],{},[2627,310459,310426],{}," System architects, software developers, and client engineers.",[108,310462,310463,310465],{},[2627,310464,310432],{}," Structured natural language, graphical models (like UML diagrams), or formal mathematical specifications.",[108,310467,310468,310470],{},[2627,310469,310050],{}," Developers need unambiguous, exact instructions to write code. A vague user requirement like \"The system should be fast\" is useless to a developer; they need a system requirement like \"The system shall process a database query within 500 milliseconds.\"",[32188,310472],{},[19,310474,310476],{"id":310475},"_2-deep-dive-functional-vs-non-functional-requirements","2. Deep Dive: Functional vs. Non-Functional Requirements",[15,310478,310479],{},"A core analytical framework in RE is classifying requirements into functional and non-functional categories.",[253,310481,310483],{"id":310482},"_21-functional-requirements","2.1 Functional Requirements",[15,310485,310486,310487,310490],{},"These describe ",[2627,310488,310489],{},"what the system should do",". They detail the system's exact reactions to specific inputs and its behavior in specific situations.",[105,310492,310493,310507],{},[108,310494,310495,310498,310499,310502,310503,310506],{},[2627,310496,310497],{},"Traits:"," They must ideally be ",[2627,310500,310501],{},"complete"," (all required services are defined) and ",[2627,310504,310505],{},"consistent"," (no contradictory definitions).",[108,310508,310509,310512,310513,310516,310517,310520,310521,310524],{},[2627,310510,310511],{},"Mastery Application (Edge Case):"," In a mental health care system (like the textbook's ",[71807,310514,310515],{},"Mentcare"," example), a functional requirement might be: ",[71807,310518,310519],{},"\"The system shall allow users to search for patient appointments across all clinics.\""," However, if the specification lacks precision, a developer might implement a search function that requires the user to select a clinic ",[71807,310522,310523],{},"first"," before searching, slowing down the medical staff. Precision in functional requirements prevents developer assumptions.",[253,310526,310528],{"id":310527},"_22-non-functional-requirements-nfrs","2.2 Non-Functional Requirements (NFRs)",[15,310530,310486,310531,310534],{},[2627,310532,310533],{},"how the system should behave",". They are constraints on the services offered by the system, such as timing, security, and performance.",[105,310536,310537,310546],{},[108,310538,310539,310541,310542,310545],{},[2627,310540,310050],{}," NFRs are often ",[71807,310543,310544],{},"more critical"," than functional requirements. If a system lacks a minor functional feature, users can usually find a workaround. But if an NFR fails—for example, if a heart monitor's real-time processing fails, or an aircraft's reliability constraint is breached—the entire system is useless or dangerous.",[108,310547,310548,310551],{},[2627,310549,310550],{},"Categories of NFRs:",[2230,310552,310553,310559,310565],{},[108,310554,310555,310558],{},[2627,310556,310557],{},"Product Requirements:"," Dictate the software's runtime behavior (e.g., execution speed, memory footprint, reliability).",[108,310560,310561,310564],{},[2627,310562,310563],{},"Organizational Requirements:"," Derived from the customer's or developer's policies (e.g., the system must be written in Java, or users must authenticate via smart-card swipe).",[108,310566,310567,310570],{},[2627,310568,310569],{},"External Requirements:"," Derived from external laws or environments (e.g., the system must comply with HIPAA privacy legislation).",[253,310572,310574],{"id":310573},"_23-the-verifiability-problem","2.3 The Verifiability Problem",[15,310576,310577,310578,310581,310582,2547],{},"A common pitfall is writing NFRs as \"goals\" (e.g., ",[71807,310579,310580],{},"\"The system shall be easy to use\"","). Goals are impossible to test. To master NFRs, you must translate goals into ",[2627,310583,310584],{},"verifiable metrics",[105,310586,310587],{},[108,310588,310589,310591,310592,310595,310596,310599],{},[2627,310590,310018],{}," Instead of ",[71807,310593,310594],{},"\"easy to use,\""," translate it to: ",[71807,310597,310598],{},"\"Medical staff shall be able to use all core system functions after two hours of training. Thereafter, the average error rate shall not exceed two per hour.\""," This is now testable via software instrumentation and observation.",[32188,310601],{},[19,310603,310605],{"id":310604},"_3-the-requirements-engineering-process","3. The Requirements Engineering Process",[15,310607,310608,310609,310612],{},"RE is not a strict, linear \"waterfall\" phase. It is an ",[2627,310610,310611],{},"iterative spiral",". You elicit basic requirements, specify them, validate them, and then cycle back to elicit deeper, more complex requirements.",[253,310614,310616],{"id":310615},"_31-requirements-elicitation","3.1 Requirements Elicitation",[15,310618,310619],{},"This is the process of discovering requirements by interacting with stakeholders. It is notoriously difficult because of several underlying assumptions about human psychology and business:",[105,310621,310622,310628,310634],{},[108,310623,310624,310627],{},[2627,310625,310626],{},"Tacit Knowledge:"," Users know their jobs so well they leave out crucial, implicit details when explaining them to an outsider.",[108,310629,310630,310633],{},[2627,310631,310632],{},"Conflicting Goals:"," Different stakeholders want different things (e.g., management wants strict data tracking; nurses want quick data entry).",[108,310635,310636,310639],{},[2627,310637,310638],{},"Political Factors:"," Managers may demand features simply to increase their department's influence.",[15,310641,310642],{},[2627,310643,310644],{},"Elicitation Techniques:",[2230,310646,310647,310653,310675],{},[108,310648,310649,310652],{},[2627,310650,310651],{},"Interviewing (Open and Closed):"," Directly asking stakeholders about their needs. Good for baseline understanding, but poor for uncovering tacit knowledge.",[108,310654,310655,310658,310659],{},[2627,310656,310657],{},"Ethnography (Observation):"," An analyst immerses themselves in the users' working environment.\n",[105,310660,310661,310670],{},[108,310662,310663,310665,310666,310669],{},[71807,310664,310050],{}," Ethnography reveals how people ",[71807,310667,310668],{},"actually"," work, rather than how the formal company manual says they work.",[108,310671,310672,310674],{},[71807,310673,310018],{}," An ethnographic study of air traffic controllers might reveal that they deliberately switch off a conflict-alert system because it triggers false alarms. A pure interview might have yielded a requirement to \"replicate the conflict-alert system,\" but ethnography reveals the requirement should actually be \"design an alert system with a lower false-positive threshold.\"",[108,310676,310677,310680],{},[2627,310678,310679],{},"Stories and Scenarios:"," Using real-life examples (User Stories) to help users visualize the system. Scenarios are highly structured stories detailing the initial state, normal flow of events, edge cases (what goes wrong), and the final state.",[253,310682,310684],{"id":310683},"_32-requirements-specification","3.2 Requirements Specification",[15,310686,310687,310688,2547],{},"This is the act of translating elicited data into a formal ",[2627,310689,310690],{},"Software Requirements Document (SRS)",[15,310692,310693],{},[2627,310694,310695],{},"Formatting Techniques:",[105,310697,310698,310712,310718],{},[108,310699,310700,310703,310704,310707,310708,310711],{},[2627,310701,310702],{},"Natural Language:"," The most common, but prone to ambiguity. To mitigate this, engineers use strict rules: use ",[2627,310705,310706],{},"\"shall\""," for mandatory requirements and ",[2627,310709,310710],{},"\"should\""," for desirable ones. Always avoid jargon.",[108,310713,310714,310717],{},[2627,310715,310716],{},"Structured Specifications:"," Using standardized templates or forms. This forces the analyst to define inputs, outputs, preconditions, and postconditions for every function.",[108,310719,310720,310723],{},[2627,310721,310722],{},"Graphical Models (UML Use Cases):"," Diagrams that show the interactions between system \"actors\" (users or other systems) and the system itself.",[253,310725,310727],{"id":310726},"_33-requirements-validation","3.3 Requirements Validation",[15,310729,310730,310731,58299,310734,310736],{},"Once the SRS is written, it must be validated. Validation answers the question: ",[71807,310732,310733],{},"\"Are we building the right product?\"",[2627,310735,310050],{}," The cost of fixing a requirements error during the coding or testing phase is exponentially higher than fixing it during the RE phase. If a requirement changes, the design, code, and tests must all be rewritten.",[105,310738,310739],{},[108,310740,310741,310744],{},[2627,310742,310743],{},"Core Checks:",[2230,310745,310746,310752,310758,310764,310770],{},[108,310747,310748,310751],{},[2627,310749,310750],{},"Validity Check:"," Does the system provide the functions that best support the customer's needs?",[108,310753,310754,310757],{},[2627,310755,310756],{},"Consistency Check:"," Are there contradictory requirements?",[108,310759,310760,310763],{},[2627,310761,310762],{},"Completeness Check:"," Are all necessary functions defined?",[108,310765,310766,310769],{},[2627,310767,310768],{},"Realism Check:"," Can this be built with the given budget and technology?",[108,310771,310772,310775],{},[2627,310773,310774],{},"Verifiability Check:"," Can we write a test to prove this requirement was met?",[15,310777,310778],{},[2627,310779,310780],{},"Validation Techniques:",[105,310782,310783,310789,310795],{},[108,310784,310785,310788],{},[2627,310786,310787],{},"Requirements Reviews:"," Systematic analysis by a team of reviewers.",[108,310790,310791,310794],{},[2627,310792,310793],{},"Prototyping:"," Building an executable, scaled-down model for users to experiment with.",[108,310796,310797,310800],{},[2627,310798,310799],{},"Test-Case Generation:"," If you cannot design a test for a requirement, the requirement is inherently flawed and must be rewritten.",[32188,310802],{},[19,310804,310806],{"id":310805},"_4-requirements-change-change-management","4. Requirements Change (Change Management)",[15,310808,310809,310810,310813],{},"The final, sobering reality of Software Engineering is that ",[2627,310811,310812],{},"requirements will always change",". Software often solves \"wicked problems\"—problems so complex they cannot be fully understood until a solution is attempted.",[253,310815,310817],{"id":310816},"_41-why-requirements-evolve","4.1 Why Requirements Evolve",[2230,310819,310820,310826,310832],{},[108,310821,310822,310825],{},[2627,310823,310824],{},"Changing Environments:"," Hardware evolves, business priorities shift, and new laws are passed after the project begins.",[108,310827,310828,310831],{},[2627,310829,310830],{},"Purchaser vs. User Conflict:"," The management team paying for the system has different priorities than the end-users. After deployment, user pushback often forces requirements to change.",[108,310833,310834,310837,310838,310840],{},[2627,310835,310836],{},"Emerging Understanding:"," Seeing the software in action helps users realize what they ",[71807,310839,310668],{}," needed.",[253,310842,310844],{"id":310843},"_42-the-change-management-process","4.2 The Change Management Process",[15,310846,310847,310848,310850],{},"To prevent project chaos, organizations must establish a formal ",[2627,310849,310393],{}," protocol.",[2230,310852,310853,310859],{},[108,310854,310855,310858],{},[2627,310856,310857],{},"Problem Analysis and Change Specification:"," A proposed change is evaluated to see if it is valid and necessary.",[108,310860,310861,310864,310865,310868,310869,310872],{},[2627,310862,310863],{},"Change Analysis and Costing:"," Engineers assess the impact of the change. ",[71807,310866,310867],{},"How much code will this break? How much will it cost?"," 3. ",[2627,310870,310871],{},"Change Implementation:"," The requirements document, system design, and code are sequentially updated.",[105,310874,310875],{},[108,310876,310877,310880,310881,310884],{},[2627,310878,310879],{},"Mastery Application (Traceability):"," To successfully calculate the cost of a change during step 2, engineers rely on ",[2627,310882,310883],{},"Requirements Traceability",". Traceability involves maintaining explicit links (usually via specialized software tools) between a user requirement, its corresponding system requirements, and the actual code. If a manager requests a change to the \"Authentication Requirement,\" traceability tools instantly flag the three system modules and 15 database tables that will be affected by the change, allowing for a precise cost estimate.",{"title":29,"searchDepth":43,"depth":43,"links":310886},[310887,310891,310896,310901],{"id":310399,"depth":43,"text":310400,"children":310888},[310889,310890],{"id":310412,"depth":49,"text":310413},{"id":310447,"depth":49,"text":310448},{"id":310475,"depth":43,"text":310476,"children":310892},[310893,310894,310895],{"id":310482,"depth":49,"text":310483},{"id":310527,"depth":49,"text":310528},{"id":310573,"depth":49,"text":310574},{"id":310604,"depth":43,"text":310605,"children":310897},[310898,310899,310900],{"id":310615,"depth":49,"text":310616},{"id":310683,"depth":49,"text":310684},{"id":310726,"depth":49,"text":310727},{"id":310805,"depth":43,"text":310806,"children":310902},[310903,310904],{"id":310816,"depth":49,"text":310817},{"id":310843,"depth":49,"text":310844},"Requirements Engineering (RE) is the foundational stage of the software development lifecycle. It is the systematic process of discovering, analyzing, documenting, and validating the services that a system should provide and the constraints under which it must operate. Failing to properly engineer requirements leads to systems that do not meet user needs, exceed budgets, and are plagued by constant rework.",{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap4",{"title":310364,"description":310905},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap4","crRbwzPVOoTHsu6O8QQnRYpfqBvmuCSZo-SAWc0QiEU",{"id":310912,"title":310913,"body":310914,"description":29,"extension":85,"meta":311450,"navigation":52,"path":311451,"seo":311452,"stem":311453,"__hash__":311454},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap5.md","Chapter 5: System Modeling",{"type":8,"value":310915,"toc":311441},[310916,310919,310921,310925,310939,310941,310947,310953,310960,310966,310969,310995,311001,311006,311038,311040,311046,311057,311084,311094,311096,311102,311105,311111,311114,311138,311144,311156,311176,311183,311185,311191,311198,311204,311207,311241,311284,311286,311292,311301,311307,311310,311325,311331,311334,311350,311380,311382,311388,311394,311400,311403,311423,311429],[11,310917,310913],{"id":310918},"chapter-5-system-modeling",[32188,310920],{},[253,310922,310923],{"id":309852},[2627,310924,309853],{},[15,310926,310927,310930,310931,310934,310935,310938],{},[2627,310928,310929],{},"System modeling"," is the process of developing abstract models of a system, with each model presenting a different view or perspective of that system. Chapter 5 explores how we use graphical representations—primarily the ",[2627,310932,310933],{},"Unified Modeling Language (UML)","—to design, communicate, and analyze software before and during implementation. This chapter breaks down system modeling into four distinct perspectives (Context, Interaction, Structure, and Behavior) and introduces ",[2627,310936,310937],{},"Model-Driven Engineering (MDE)",", a paradigm where models, rather than raw code, are the primary outputs of the development process.",[32188,310940],{},[253,310942,310944],{"id":310943},"_1-the-foundations-of-system-modeling",[2627,310945,310946],{},"1. The Foundations of System Modeling",[32490,310948,310950],{"id":310949},"what-is-a-model",[2627,310951,310952],{},"What is a Model?",[15,310954,310955,310956,310959],{},"At its core, a model is an ",[2627,310957,310958],{},"abstraction",". It leaves out details to make the system easier to understand. You cannot build a complex system by jumping straight into code; you need blueprints. Models are used during requirements engineering to clarify what the existing system does and during design to explain the proposed system.",[32490,310961,310963],{"id":310962},"the-four-system-perspectives",[2627,310964,310965],{},"The Four System Perspectives",[15,310967,310968],{},"When you look at a building, you can look at its plumbing, its electrical wiring, or its floor plan. Software is identical. We model systems from four perspectives:",[2230,310970,310971,310977,310983,310989],{},[108,310972,310973,310976],{},[2627,310974,310975],{},"An External Perspective (Context):"," Where does the system operate? What is in its environment?",[108,310978,310979,310982],{},[2627,310980,310981],{},"An Interaction Perspective:"," How does the system interact with its environment or how do its internal components interact with each other?",[108,310984,310985,310988],{},[2627,310986,310987],{},"A Structural Perspective:"," How is the data and the architecture organized?",[108,310990,310991,310994],{},[2627,310992,310993],{},"A Behavioral Perspective:"," How does the system react to specific events or data inputs dynamically?",[32490,310996,310998],{"id":310997},"the-role-of-uml",[2627,310999,311000],{},"The Role of UML",[15,311002,240905,311003,311005],{},[2627,311004,310933],{}," is the standard visual language for software design. While UML has over a dozen diagram types, Chapter 5 focuses on the five most critical:",[105,311007,311008,311014,311020,311026,311032],{},[108,311009,311010,311013],{},[2627,311011,311012],{},"Activity Diagrams:"," Show the activities or flow of data\u002Fcontrol.",[108,311015,311016,311019],{},[2627,311017,311018],{},"Use Case Diagrams:"," Show interactions between actors and the system.",[108,311021,311022,311025],{},[2627,311023,311024],{},"Sequence Diagrams:"," Show interactions between actors and the system (or between system objects) over time.",[108,311027,311028,311031],{},[2627,311029,311030],{},"Class Diagrams:"," Show the object classes and their relationships.",[108,311033,311034,311037],{},[2627,311035,311036],{},"State Diagrams:"," Show how the system reacts to internal and external events.",[32188,311039],{},[253,311041,311043],{"id":311042},"_2-context-models-defining-the-boundaries",[2627,311044,311045],{},"2. Context Models: Defining the Boundaries",[15,311047,311048,311049,311052,311053,311056],{},"Before you can build a system, you must define exactly what you are building. ",[2627,311050,311051],{},"Context models"," illustrate the operational context of a system—they show what lies ",[71807,311054,311055],{},"outside"," the system boundaries.",[105,311058,311059,311065,311071],{},[108,311060,311061,311064],{},[2627,311062,311063],{},"System Boundaries:"," Defining the boundary is often a political and social decision as much as a technical one. It determines what features are built into your system and what features are delegated to external systems.",[108,311066,311067,311070],{},[2627,311068,311069],{},"Context Diagrams:"," Simple block diagrams that show the system at the center and the external entities (other systems, databases, hardware) it communicates with.",[108,311072,311073,311076,311077,311079,311080,311083],{},[2627,311074,311075],{},"Process Models:"," Because simple block diagrams don't show ",[71807,311078,310219],{}," the systems interact, we use ",[2627,311081,311082],{},"UML Activity Diagrams"," to show the business processes.",[104833,311085,311086],{},[15,311087,311088,311090,311091,311093],{},[2627,311089,310018],{}," Imagine designing an ATM network. The Context Model would show the ATM itself as the central system, connected to the Bank's Mainframe, a Currency Database, and a Maintenance System. The boundary defines that the ATM does ",[71807,311092,243053],{}," verify account balances itself; it delegates that to the Mainframe.",[32188,311095],{},[253,311097,311099],{"id":311098},"_3-interaction-models-communication-and-use",[2627,311100,311101],{},"3. Interaction Models: Communication and Use",[15,311103,311104],{},"Interaction models help us understand how users interact with the system (vital for user requirements) and how system components interact with each other (vital for system design).",[32490,311106,311108],{"id":311107},"use-case-modeling",[2627,311109,311110],{},"Use Case Modeling",[15,311112,311113],{},"Originally developed for the Object-Oriented Software Engineering (OOSE) method, use cases are a high-level view of interactions.",[105,311115,311116,311122,311128],{},[108,311117,311118,311121],{},[2627,311119,311120],{},"Actors:"," External entities (humans or other systems) that interact with the system. Represented as stick figures.",[108,311123,311124,311127],{},[2627,311125,311126],{},"Use Cases:"," A specific task or interaction (e.g., \"Withdraw Cash\", \"Print Receipt\"). Represented as ovals.",[108,311129,311130,311133,311134,311137],{},[2627,311131,311132],{},"Limitation:"," Use case diagrams are just summaries. They ",[71807,311135,311136],{},"must"," be accompanied by a tabular description or structured text that explains the normal flow, alternative flows, and exceptions.",[32490,311139,311141],{"id":311140},"sequence-diagrams",[2627,311142,311143],{},"Sequence Diagrams",[15,311145,311146,311147,311149,311150,311152,311153,311155],{},"While use cases show ",[71807,311148,310203],{}," happens, ",[2627,311151,311143],{}," show ",[71807,311154,310219],{}," it happens chronologically.",[105,311157,311158,311164,311170],{},[108,311159,311160,311163],{},[2627,311161,311162],{},"Lifelines:"," Vertical dashed lines representing the existence of an object over time.",[108,311165,311166,311169],{},[2627,311167,311168],{},"Messages:"," Horizontal arrows passed between objects. Solid arrows denote synchronous calls (the sender waits for a response); dashed arrows denote return messages.",[108,311171,311172,311175],{},[2627,311173,311174],{},"Activation Boxes:"," Rectangles on the lifeline showing when an object is actively processing data.",[104833,311177,311178],{},[15,311179,311180,311182],{},[2627,311181,310018],{}," In a modern web application (like a login screen), a Sequence Diagram is perfect for visualizing the flow: The User (Actor) sends credentials to the UI -> UI sends an HTTP request to the Auth Controller -> Auth Controller queries the Database. The diagram proves logically whether the controller needs to wait for the database before returning the view to the user.",[32188,311184],{},[253,311186,311188],{"id":311187},"_4-structural-models-the-architecture-of-data",[2627,311189,311190],{},"4. Structural Models: The Architecture of Data",[15,311192,311193,311194,311197],{},"Structural models display the organization of a system in terms of the components that make up that system and their relationships. These are ",[2627,311195,311196],{},"static models"," (they do not show time or behavior).",[32490,311199,311201],{"id":311200},"class-diagrams",[2627,311202,311203],{},"Class Diagrams",[15,311205,311206],{},"The workhorse of object-oriented modeling. A class diagram shows the object classes in the system and the associations between them.",[105,311208,311209,311215,311225,311231],{},[108,311210,311211,311214],{},[2627,311212,311213],{},"Class Box:"," Divided into three compartments: Class Name, Attributes (data), and Operations (methods\u002Ffunctions).",[108,311216,311217,311220,311221,311224],{},[2627,311218,311219],{},"Associations:"," Lines connecting classes, showing a relationship. Multiplicity (e.g., ",[31,311222,311223],{},"1..*"," meaning \"one to many\") is placed on these lines.",[108,311226,311227,311230],{},[2627,311228,311229],{},"Generalization (Inheritance):"," Represented by an empty arrow pointing to the superclass. Used to manage complexity by abstracting common attributes into a single parent class.",[108,311232,311233,311236,311237,311240],{},[2627,311234,311235],{},"Aggregation\u002FComposition:"," Represents a \"has-a\" relationship (e.g., a Library ",[71807,311238,311239],{},"has"," Books). Represented by a diamond on the association line.",[104833,311242,311243],{},[15,311244,311245,311247,311248,311251,311252,195562,311254,311257,311258,195562,311261,311264,311265,311268,311269,311271,311272,311275,311276,311279,311280,311283],{},[2627,311246,310018],{}," Consider a Hospital Management System. A generic ",[31,311249,311250],{},"Employee"," class holds ",[31,311253,2305],{},[31,311255,311256],{},"ID",". ",[31,311259,311260],{},"Doctor",[31,311262,311263],{},"Nurse"," classes ",[2627,311266,311267],{},"inherit"," from ",[31,311270,311250],{},". A ",[31,311273,311274],{},"Ward"," class has an ",[2627,311277,311278],{},"aggregation"," relationship with ",[31,311281,311282],{},"Patient"," (a ward contains many patients, but patients exist independently of the ward). This dictates how you will write your SQL tables and Java\u002FC# classes later.",[32188,311285],{},[253,311287,311289],{"id":311288},"_5-behavioral-models-system-dynamics",[2627,311290,311291],{},"5. Behavioral Models: System Dynamics",[15,311293,311294,311295,195562,311298,2547],{},"Behavioral models are dynamic. They show what happens when the system is running and responding to its environment. There are two primary stimuli a system responds to: ",[2627,311296,311297],{},"Data",[2627,311299,311300],{},"Events",[32490,311302,311304],{"id":311303},"data-driven-modeling",[2627,311305,311306],{},"Data-Driven Modeling",[15,311308,311309],{},"Shows the sequence of actions involved in processing input data and generating an output.",[105,311311,311312,311319],{},[108,311313,311314,311315,311318],{},"Historically done using Data Flow Diagrams (DFDs), but in UML, ",[2627,311316,311317],{},"Activity Diagrams"," are used.",[108,311320,311321,311322],{},"They are excellent for showing end-to-end processing. For example, the flow of an e-commerce order: ",[71807,311323,311324],{},"Receive Order -> Check Inventory -> Process Payment -> Dispatch Goods.",[32490,311326,311328],{"id":311327},"event-driven-modeling-state-machines",[2627,311329,311330],{},"Event-Driven Modeling (State Machines)",[15,311332,311333],{},"Some systems don't just passively process data; they sit in a specific \"state\" waiting for an \"event\" to happen. This is crucial for real-time systems.",[105,311335,311336,311341],{},[108,311337,311338,311340],{},[2627,311339,311036],{}," Show system states as rounded rectangles and events as labeled arrows transitioning between states.",[108,311342,311343,311346,311347,26192],{},[2627,311344,311345],{},"Guards:"," Conditions that must be met for a transition to occur (e.g., ",[31,311348,311349],{},"[balance > 0]",[104833,311351,311352],{},[15,311353,311354,311356,311357,616,311360,616,311363,241592,311365,311368,311369,311371,311372,311375,311376,311379],{},[2627,311355,310018],{}," A Microwave Oven is the classic state machine. Its states are ",[31,311358,311359],{},"Off",[31,311361,311362],{},"Waiting for Input",[31,311364,299745],{},[31,311366,311367],{},"Door Open",". If the microwave is in the ",[31,311370,299745],{}," state and the event ",[31,311373,311374],{},"Door Opened"," occurs, the system must immediately transition to a ",[31,311377,311378],{},"Paused"," state and cut the power. A data-driven model cannot capture this instantaneous safety interrupt; a state diagram captures it perfectly.",[32188,311381],{},[253,311383,311385],{"id":311384},"_6-model-driven-engineering-mde",[2627,311386,311387],{},"6. Model-Driven Engineering (MDE)",[15,311389,311390,311391,311393],{},"Chapter 5 concludes by looking toward the future\u002Falternative paradigms of software engineering. ",[2627,311392,310937],{}," is an approach where models are the focal point of the development process, not the code. In a pure MDE environment, executable code is automatically generated from the models.",[32490,311395,311397],{"id":311396},"model-driven-architecture-mda",[2627,311398,311399],{},"Model-Driven Architecture (MDA)",[15,311401,311402],{},"A specific subset of MDE proposed by the OMG (Object Management Group). It uses a three-tier abstraction:",[2230,311404,311405,311411,311417],{},[108,311406,311407,311410],{},[2627,311408,311409],{},"Computation Independent Model (CIM):"," Models the domain and business requirements (no software details).",[108,311412,311413,311416],{},[2627,311414,311415],{},"Platform Independent Model (PIM):"," A software model that does not reference any specific technology (e.g., a standard UML class diagram).",[108,311418,311419,311422],{},[2627,311420,311421],{},"Platform Specific Model (PSM):"," The PIM is transformed into a model tailored to a specific execution environment (e.g., a Java EE or .NET model).",[32490,311424,311426],{"id":311425},"pros-and-cons-of-mde",[2627,311427,311428],{},"Pros and Cons of MDE",[105,311430,311431,311436],{},[108,311432,311433,311435],{},[2627,311434,241973],{}," Allows systems to be considered at higher levels of abstraction. It is cheaper to adapt a system to a new platform (just generate a new PSM from the existing PIM).",[108,311437,311438,311440],{},[2627,311439,241989],{}," Creating complete, unambiguous models is incredibly difficult. Often, the generated code is bloated or inefficient. Because of this, MDE has seen limited adoption in general-purpose software, though it is highly successful in embedded systems engineering.",{"title":29,"searchDepth":43,"depth":43,"links":311442},[311443,311444,311445,311446,311447,311448,311449],{"id":309852,"depth":49,"text":309853},{"id":310943,"depth":49,"text":310946},{"id":311042,"depth":49,"text":311045},{"id":311098,"depth":49,"text":311101},{"id":311187,"depth":49,"text":311190},{"id":311288,"depth":49,"text":311291},{"id":311384,"depth":49,"text":311387},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap5",{"title":310913,"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap5","AgVkpvM8kqu9J2hblSpJFkwBFjrK1R8GZKyCQfEBsP0",{"id":311456,"title":311457,"body":311458,"description":29,"extension":85,"meta":311912,"navigation":52,"path":311913,"seo":311914,"stem":311915,"__hash__":311916},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap6.md","Chapter 6: Architectural Design",{"type":8,"value":311459,"toc":311886},[311460,311463,311465,311467,311470,311472,311476,311480,311483,311487,311490,311504,311508,311511,311531,311535,311538,311552,311554,311558,311562,311565,311607,311611,311614,311646,311659,311661,311665,311669,311672,311678,311697,311701,311704,311706,311710,311714,311717,311721,311818,311820,311824,311828,311835,311839,311842,311874,311878],[11,311461,311457],{"id":311462},"chapter-6-architectural-design",[32188,311464],{},[19,311466,309853],{"id":309852},[15,311468,311469],{},"Architectural design represents the critical bridge between requirements engineering and software design. It dictates how a software system is fundamentally organized, establishing its major structural components and their interrelationships. This chapter explores how architectural decisions profoundly impact non-functional properties?uch as performance, security, and maintainability. It introduces the concept of modeling architecture from multiple distinct views, using established \"Architectural Patterns\" to reuse proven design solutions, and applying generic \"Application Architectures\" to specific business domains.",[32188,311471],{},[19,311473,311475],{"id":311474},"_1-introduction-to-architectural-design","1. Introduction to Architectural Design",[253,311477,311479],{"id":311478},"_11-what-is-architectural-design","1.1 What is Architectural Design?",[15,311481,311482],{},"Architectural design is the very first stage in the software design process. Because it is the critical link between requirements engineering and design, it identifies the main structural components in a system and defines the relationships between them. The ultimate output of this process is an architectural model that describes how the system is organized as a set of communicating components.",[253,311484,311486],{"id":311485},"_12-architecture-in-the-small-vs-architecture-in-the-large","1.2 Architecture in the Small vs. Architecture in the Large",[15,311488,311489],{},"Software architecture can be designed at two distinct levels of abstraction:",[105,311491,311492,311498],{},[108,311493,311494,311497],{},[2627,311495,311496],{},"Architecture in the small:"," Concerned with the architecture of individual programs and how an individual program is decomposed into components.",[108,311499,311500,311503],{},[2627,311501,311502],{},"Architecture in the large:"," Concerned with the architecture of complex enterprise systems that encompass multiple programs and systems. These are often distributed across different computers and managed by different organizations.",[253,311505,311507],{"id":311506},"_13-why-is-software-architecture-important","1.3 Why is Software Architecture Important?",[15,311509,311510],{},"The architecture of a system is paramount because it exerts the dominant influence on the system's non-functional characteristics?amely its performance, robustness, distributability, and maintainability. Explicitly designing and documenting it yields three major advantages:",[2230,311512,311513,311519,311525],{},[108,311514,311515,311518],{},[2627,311516,311517],{},"Stakeholder Communication:"," It provides a high-level presentation of the system that serves as a focal point for discussion among a wide range of stakeholders (engineers, managers, clients).",[108,311520,311521,311524],{},[2627,311522,311523],{},"System Analysis:"," Making the architecture explicit early allows engineers to analyze whether the system will actually be able to meet critical non-functional requirements.",[108,311526,311527,311530],{},[2627,311528,311529],{},"Large-scale Reuse:"," An architectural model is a compact description of system organization. Systems in similar domains often share the same architecture, which enables large-scale reuse (e.g., product-line architectures).",[253,311532,311534],{"id":311533},"_14-how-architectural-models-are-used","1.4 How Architectural Models are Used",[15,311536,311537],{},"Architectures are often modeled using simple, informal block diagrams. These models generally serve two distinct purposes:",[105,311539,311540,311546],{},[108,311541,311542,311545],{},[2627,311543,311544],{},"To encourage discussion:"," A high-level view facilitates communication and project planning because it is uncluttered by technical details, allowing stakeholders to easily grasp the system.",[108,311547,311548,311551],{},[2627,311549,311550],{},"To document a design:"," A detailed architectural description shows all components, interfaces, and connections, making it easier to understand and evolve the system during implementation.",[32188,311553],{},[19,311555,311557],{"id":311556},"_2-architectural-design-decisions","2. Architectural Design Decisions",[253,311559,311561],{"id":311560},"_21-the-core-decisions","2.1 The Core Decisions",[15,311563,311564],{},"Architectural design is not a rigid sequence of activities, but rather a highly creative process driven by a series of structural decisions. System architects must rely on their expertise to answer several fundamental questions:",[105,311566,311567,311574,311581,311588,311595,311601],{},[108,311568,311569,311570,311573],{},"What ",[2627,311571,311572],{},"Architectural patterns"," or styles might be used?",[108,311575,311576,311577,311580],{},"What fundamental approach will be used to ",[2627,311578,311579],{},"structure"," the system?",[108,311582,311583,311584,311587],{},"How will structural components be ",[2627,311585,311586],{},"decomposed"," into sub-components?",[108,311589,311590,311591,311594],{},"What strategy will be used to ",[2627,311592,311593],{},"control"," the operation of the components?",[108,311596,311597,311598,6701],{},"What organization is best for delivering the ",[2627,311599,311600],{},"non-functional requirements",[108,311602,311603,311604,6701],{},"How should the architecture be ",[2627,311605,311606],{},"documented",[253,311608,311610],{"id":311609},"_22-impact-of-non-functional-requirements-the-trade-offs","2.2 Impact of Non-Functional Requirements (The Trade-offs)",[15,311612,311613],{},"Because non-functional requirements are heavily influenced by the system architecture, architects must structure the system to satisfy its most critical goals.",[105,311615,311616,311622,311628,311634,311640],{},[108,311617,311618,311621],{},[2627,311619,311620],{},"Performance:"," Demands that critical operations be localized within a small number of large components deployed on the same computer to minimize network communication.",[108,311623,311624,311627],{},[2627,311625,311626],{},"Security:"," Demands a layered structure where the most critical assets are protected in the innermost layers, with high-level security validation applied.",[108,311629,311630,311633],{},[2627,311631,311632],{},"Safety:"," Demands that safety-related operations be co-located in a single component (or very few). This simplifies safety validation and enables safe shutdown mechanisms.",[108,311635,311636,311639],{},[2627,311637,311638],{},"Availability:"," Demands redundant components so that parts of the system can be replaced or updated without stopping the entire system.",[108,311641,311642,311645],{},[2627,311643,311644],{},"Maintainability:"," Demands fine-grain, self-contained components where the producers and consumers of data are separated.",[15,311647,311648,311650,311651,311654,311655,311658],{},[2627,311649,310018],{}," Notice the inherent conflict here. If a system requires extremely high ",[71807,311652,311653],{},"Performance",", you must use a few large components. However, if it requires high ",[71807,311656,311657],{},"Maintainability",", you must use many small, fine-grained components. If both are required, the architect must find a compromise, sometimes applying different architectural patterns to different parts of the system to achieve a balance.",[32188,311660],{},[19,311662,311664],{"id":311663},"_3-architectural-views","3. Architectural Views",[253,311666,311668],{"id":311667},"_31-the-necessity-of-multiple-perspectives","3.1 The Necessity of Multiple Perspectives",[15,311670,311671],{},"It is impossible to represent all relevant architectural information in a single diagram; a graphical model can only show one perspective at a time. An architect must present multiple views to show how modules are decomposed, how runtime processes interact, and how components are physically distributed.",[15,311673,311674,311677],{},[2627,311675,311676],{},"Krutchen? 4+1 View Model"," is a widely referenced framework that suggests four fundamental views (linked by common use cases):",[105,311679,311680,311686,311691],{},[108,311681,311682,311685],{},[2627,311683,311684],{},"Logical view:"," Shows the key abstractions in the system as objects or object classes, directly relating to the system requirements.",[108,311687,311688],{},[71807,311689,311690],{},"(The model also includes Process, Development, and Physical views).",[108,311692,311693,311696],{},[2627,311694,311695],{},"Conceptual view:"," (Suggested by Hofmeister et al.) An abstract view that helps decompose high-level requirements and guides decisions about reusable components or product lines.",[253,311698,311700],{"id":311699},"_32-documenting-architectures","3.2 Documenting Architectures",[15,311702,311703],{},"While specialized architectural description languages (ADLs) exist to strictly define components and rules, domain experts often find them too complex. Therefore, informal models and notations?ncluding informal applications of the UML?emain the most commonly used ways of documenting system architectures.",[32188,311705],{},[19,311707,311709],{"id":311708},"_4-architectural-patterns","4. Architectural Patterns",[253,311711,311713],{"id":311712},"_41-what-is-an-architectural-pattern","4.1 What is an Architectural Pattern?",[15,311715,311716],{},"An Architectural pattern (or style) is a stylized, abstract description of good architectural practice that has been tried and tested in different systems and environments. Patterns provide a vocabulary to describe system organization, capturing the essence of an architecture that has proven successful.",[253,311718,311720],{"id":311719},"_42-key-architectural-patterns","4.2 Key Architectural Patterns",[2230,311722,311723,311754,311772,311789,311801],{},[108,311724,311725,311728],{},[2627,311726,311727],{},"Model-View-Controller (MVC)",[105,311729,311730,311748],{},[108,311731,311732,311735,311736,311739,311740,311743,311744,311747],{},[2627,311733,311734],{},"Description:"," Separates presentation and interaction from the system data. The system is structured into three logical components: The ",[71807,311737,311738],{},"Model"," manages the data and operations; the ",[71807,311741,311742],{},"View"," defines how data is presented; and the ",[71807,311745,311746],{},"Controller"," manages user interactions (e.g., clicks) and passes them to the View and Model.",[108,311749,311750,311753],{},[2627,311751,311752],{},"Application:"," Widely used in web-based frameworks to allow independent modification of the data representation and the user interface.",[108,311755,311756,311759],{},[2627,311757,311758],{},"Layered Architecture",[105,311760,311761,311766],{},[108,311762,311763,311765],{},[2627,311764,311734],{}," Organizes system functionality into separate layers. Each layer relies exclusively on the facilities and services offered by the layer immediately beneath it.",[108,311767,311768,311771],{},[2627,311769,311770],{},"Benefits:"," Highly changeable and portable. If an interface is unchanged, a new layer with extended functionality can completely replace an existing layer without affecting the rest of the system. It is also highly suited for enforcing multilevel security.",[108,311773,311774,311777],{},[2627,311775,311776],{},"Repository Architecture",[105,311778,311779,311784],{},[108,311780,311781,311783],{},[2627,311782,311734],{}," All data in a system is managed in a central repository. Components do not interact directly; they communicate exclusively through the shared repository.",[108,311785,311786,311788],{},[2627,311787,311752],{}," Ideal for systems where large volumes of information are generated and stored long-term, such as IDEs, CAD systems, and management information systems.",[108,311790,311791,311794],{},[2627,311792,311793],{},"Client-Server Architecture",[105,311795,311796],{},[108,311797,311798,311800],{},[2627,311799,311734],{}," Models the runtime organization of a distributed system. The system is organized as a set of services and associated servers, along with clients that access and utilize those services over a network.",[108,311802,311803,311806],{},[2627,311804,311805],{},"Pipe and Filter Architecture",[105,311807,311808,311813],{},[108,311809,311810,311812],{},[2627,311811,311734],{}," Models a runtime organization where functional transformations (filters) process their inputs and produce outputs. Data flows (via pipes) from one filter to another, being transformed as it moves through the sequence.",[108,311814,311815,311817],{},[2627,311816,311752],{}," Found heavily in data-processing systems (as batch sequential models) and embedded real-time systems.",[32188,311819],{},[19,311821,311823],{"id":311822},"_5-application-architectures","5. Application Architectures",[253,311825,311827],{"id":311826},"_51-understanding-application-architectures","5.1 Understanding Application Architectures",[15,311829,311830,311831,311834],{},"Application systems designed to meet similar business needs (e.g., all phone companies need to meter calls and issue bills) share significant structural commonalities. ",[2627,311832,311833],{},"Application architectures"," encapsulate these principal characteristics, forming generic models of different system types. This common structure can be heavily reused when developing new systems within the same domain.",[253,311836,311838],{"id":311837},"_52-uses-of-application-architectures","5.2 Uses of Application Architectures",[15,311840,311841],{},"Software designers utilize these generic application architectures in multiple strategic ways:",[2230,311843,311844,311850,311856,311862,311868],{},[108,311845,311846,311849],{},[2627,311847,311848],{},"As a Starting Point:"," If you are unfamiliar with a domain, you can base your initial design on the generic architecture and specialize it.",[108,311851,311852,311855],{},[2627,311853,311854],{},"As a Design Checklist:"," You can evaluate a completed design by comparing it against the generic architecture to ensure consistency.",[108,311857,311858,311861],{},[2627,311859,311860],{},"To Organize Work:"," Because application architectures identify stable structural features, project managers can assign teams to develop different components in parallel.",[108,311863,311864,311867],{},[2627,311865,311866],{},"To Assess Reuse:"," Designers can compare available off-the-shelf components against the generic architecture to see if they fit.",[108,311869,311870,311873],{},[2627,311871,311872],{},"As a Vocabulary:"," It gives engineers a common conceptual vocabulary for discussing and comparing specific applications.",[253,311875,311877],{"id":311876},"_53-common-examples","5.3 Common Examples",[105,311879,311880],{},[108,311881,311882,311885],{},[2627,311883,311884],{},"Language Processing Systems:"," Systems where a user's intentions are expressed in a formal language (like code or XML). The system processes the language into an internal format and interprets it. Compilers are a classic example of this architecture.",{"title":29,"searchDepth":43,"depth":43,"links":311887},[311888,311889,311895,311899,311903,311907],{"id":309852,"depth":43,"text":309853},{"id":311474,"depth":43,"text":311475,"children":311890},[311891,311892,311893,311894],{"id":311478,"depth":49,"text":311479},{"id":311485,"depth":49,"text":311486},{"id":311506,"depth":49,"text":311507},{"id":311533,"depth":49,"text":311534},{"id":311556,"depth":43,"text":311557,"children":311896},[311897,311898],{"id":311560,"depth":49,"text":311561},{"id":311609,"depth":49,"text":311610},{"id":311663,"depth":43,"text":311664,"children":311900},[311901,311902],{"id":311667,"depth":49,"text":311668},{"id":311699,"depth":49,"text":311700},{"id":311708,"depth":43,"text":311709,"children":311904},[311905,311906],{"id":311712,"depth":49,"text":311713},{"id":311719,"depth":49,"text":311720},{"id":311822,"depth":43,"text":311823,"children":311908},[311909,311910,311911],{"id":311826,"depth":49,"text":311827},{"id":311837,"depth":49,"text":311838},{"id":311876,"depth":49,"text":311877},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap6",{"title":311457,"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap6","dAfSxzmKdKPhijplnRv7WBs3NuojSV1_wCdJ9tw8jSA",{"id":311918,"title":311919,"body":311920,"description":29,"extension":85,"meta":312375,"navigation":52,"path":312376,"seo":312377,"stem":312378,"__hash__":312379},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap7.md","Chapter 7: Design and Implementation",{"type":8,"value":311921,"toc":312352},[311922,311925,311927,311929,311943,311945,311949,311960,311963,311965,311969,311980,311983,311987,312021,312025,312028,312032,312035,312055,312075,312079,312082,312108,312112,312115,312132,312134,312138,312148,312152,312155,312181,312185,312222,312224,312228,312231,312235,312238,312264,312268,312271,312285,312289,312292,312306,312308,312312,312315,312319,312322,312326,312329],[11,311923,311919],{"id":311924},"chapter-7-design-and-implementation",[32188,311926],{},[11,311928,309853],{"id":309852},[15,311930,311931,311932,311935,311936,311939,311940,2547],{},"Chapter 7 bridges the gap between requirements engineering and the actual coding of the software. It focuses on the transition from conceptual requirements to an executable system. The chapter emphasizes ",[2627,311933,311934],{},"object-oriented design"," using the Unified Modeling Language (UML), the utilization of ",[2627,311937,311938],{},"design patterns"," to solve recurring problems, and critical implementation issues that software engineers must manage, such as ",[2627,311941,311942],{},"software reuse, configuration management, and open-source development",[32188,311944],{},[19,311946,311948],{"id":311947},"_1-introduction-to-design-and-implementation","1. Introduction to Design and Implementation",[15,311950,311951,311952,311955,311956,311959],{},"Software ",[2627,311953,311954],{},"design"," is a creative process where you identify software components and their relationships based on a customer?�s requirements. ",[2627,311957,311958],{},"Implementation"," is the process of realizing this design as an executable program.",[15,311961,311962],{},"These two activities are inherently linked and invariably interleaved. In agile environments, formal design documentation might be skipped in favor of whiteboard sketches, while plan-driven approaches might mandate detailed UML modeling before any code is written. A critical early decision in this phase is the \"build vs. buy\" dilemma: evaluating whether to build the software from scratch or buy and adapt an off-the-shelf application to save time and money.",[32188,311964],{},[19,311966,311968],{"id":311967},"_2-object-oriented-design-using-the-uml","2. Object-Oriented Design using the UML",[15,311970,311971,311972,311975,311976,311979],{},"An ",[2627,311973,311974],{},"object-oriented (OO) system"," is composed of interacting objects that maintain their own private, local state and provide operations to manipulate that state. This concept of ",[71807,311977,311978],{},"encapsulation"," ensures that the representation of an object's state cannot be accessed directly from outside the object, meaning modifying one object does not unintentionally break others.",[15,311981,311982],{},"The OO design process generally involves five sequential (but often iterative) steps:",[253,311984,311986],{"id":311985},"_21-system-context-and-interactions","2.1 System Context and Interactions",[105,311988,311989,311995],{},[108,311990,311991,311994],{},[2627,311992,311993],{},"The \"Why\"",": Before designing the internal architecture, you must define the system boundaries to understand what lies outside the system.",[108,311996,311997,312000,312001],{},[2627,311998,311999],{},"The \"How\"",":\n",[105,312002,312003,312009],{},[108,312004,241048,312005,312008],{},[2627,312006,312007],{},"System Context Model"," is created to illustrate the external entities (other systems, users, hardware) that interact with the software.",[108,312010,241048,312011,312014,312015,312017,312018,312020],{},[2627,312012,312013],{},"Use Case Model"," is then used to map out ",[71807,312016,310203],{}," interactions occur without detailing ",[71807,312019,310219],{}," they are processed internally.",[253,312022,312024],{"id":312023},"_22-system-architecture","2.2 System Architecture",[15,312026,312027],{},"Once the context and external interactions are clear, engineers design the overarching architecture to handle these interactions. This involves identifying the major components that make up the system and organizing them using architectural patterns (like a layered or client-server model).",[253,312029,312031],{"id":312030},"_23-object-class-identification","2.3 Object Class Identification",[15,312033,312034],{},"Identifying the core objects is one of the most challenging parts of OO design. Engineers use several analytical techniques to discover them:",[2230,312036,312037,312043,312049],{},[108,312038,312039,312042],{},[2627,312040,312041],{},"Grammatical Analysis:"," Parsing natural language requirements. Nouns translate to objects or attributes, while verbs translate to operations or services.",[108,312044,312045,312048],{},[2627,312046,312047],{},"Tangible Entities:"," Identifying physical things (e.g., an aircraft), roles (e.g., manager), or events (e.g., a request) in the application domain.",[108,312050,312051,312054],{},[2627,312052,312053],{},"Scenario-Based Analysis:"," Tracing through specific use cases to see what data and actions are required to fulfill a scenario.",[104833,312056,312057],{},[15,312058,312059,312062,312063,312066,312067,312070,312071,312074],{},[2627,312060,312061],{},"Mastery Application (Object Identification):"," Imagine you are designing software for a wilderness weather station. A grammatical analysis of the requirement ",[71807,312064,312065],{},"\"The system must report weather data\""," reveals a ",[31,312068,312069],{},"WeatherData"," object (from the noun) and a ",[31,312072,312073],{},"report()"," operation (from the verb).",[253,312076,312078],{"id":312077},"_24-design-models","2.4 Design Models",[15,312080,312081],{},"Models act as the bridge between abstract requirements and concrete implementation. In UML, two primary categories of models are used to document a design:",[105,312083,312084,312094],{},[108,312085,312086,312089,312090,312093],{},[2627,312087,312088],{},"Structural (Static) Models:"," Define the system's static structure. Examples include ",[2627,312091,312092],{},"Subsystem Models"," (logical groupings of objects) and Class Diagrams showing inheritance and composition.",[108,312095,312096,312099,312100,312103,312104,312107],{},[2627,312097,312098],{},"Dynamic Models:"," Describe runtime interactions. ",[2627,312101,312102],{},"Sequence Models"," chronologically map out the sequence of object interactions for a specific use case , while ",[2627,312105,312106],{},"State Machine Models"," track how an individual object's state changes in response to specific events.",[253,312109,312111],{"id":312110},"_25-interface-specification","2.5 Interface Specification",[15,312113,312114],{},"Interfaces must be unambiguously defined so that individual objects and subsystems can be developed in parallel by different programmers.",[105,312116,312117,312127],{},[108,312118,312119,312122,312123,312126],{},[2627,312120,312121],{},"The Golden Rule:"," Interface specifications should ",[71807,312124,312125],{},"never"," expose the internal data representation (attributes).",[108,312128,312129,312131],{},[2627,312130,310050],{}," If data representation is hidden, it can be easily changed (e.g., switching from an array to a linked list) without breaking any external objects that rely on that interface. This leads to inherently more maintainable code.",[32188,312133],{},[19,312135,312137],{"id":312136},"_3-design-patterns","3. Design Patterns",[15,312139,241048,312140,312143,312144,312147],{},[2627,312141,312142],{},"design pattern"," is a stylized, abstract description of accumulated wisdom?�a well-tried solution to a common, recurring software design problem. Patterns facilitate high-level \"concept reuse.\" When you try to reuse exact executable code, you are constrained by the original programmer's specific algorithms; patterns allow you to reuse the ",[71807,312145,312146],{},"idea"," but adapt the implementation to your unique system.",[253,312149,312151],{"id":312150},"_31-elements-of-a-design-pattern","3.1 Elements of a Design Pattern",[15,312153,312154],{},"As defined by the \"Gang of Four\" (the pioneers of software design patterns), a pattern includes four essential elements:",[2230,312156,312157,312163,312169,312175],{},[108,312158,312159,312162],{},[2627,312160,312161],{},"Name:"," A meaningful reference to the pattern.",[108,312164,312165,312168],{},[2627,312166,312167],{},"Problem Description:"," Explains the motivation and situations where the pattern is applicable.",[108,312170,312171,312174],{},[2627,312172,312173],{},"Solution Description:"," A template (often graphical) showing the objects, their relationships, and responsibilities.",[108,312176,312177,312180],{},[2627,312178,312179],{},"Consequences:"," The trade-offs and results of applying the pattern.",[253,312182,312184],{"id":312183},"_32-mastery-application-the-observer-pattern","3.2 Mastery Application: The Observer Pattern",[105,312186,312187,312192],{},[108,312188,312189,312191],{},[2627,312190,309896],{}," You need multiple display formats (e.g., a pie chart and a bar graph) for the exact same underlying dataset, and all of them must update instantly when the data changes.",[108,312193,312194,242719,312197,312200,312201,312204,312205,312208,312209,312211,312212,312214,312215,312217,312218,312221],{},[2627,312195,312196],{},"The Solution:",[2627,312198,312199],{},"Observer pattern"," separates the data (the ",[31,312202,312203],{},"Subject",") from its displays (the ",[31,312206,312207],{},"Observers","). The ",[31,312210,312203],{}," maintains the state. When the state changes, the ",[31,312213,312203],{}," issues a notification to all attached ",[31,312216,312207],{},". The ",[31,312219,312220],{},"Observer"," objects then automatically pull the updated data and refresh their specific displays.",[32188,312223],{},[19,312225,312227],{"id":312226},"_4-implementation-issues","4. Implementation Issues",[15,312229,312230],{},"Once the design is mapped out, programmers write the code. However, modern software engineering requires managing three major non-programming implementation issues:",[253,312232,312234],{"id":312233},"_41-software-reuse","4.1 Software Reuse",[15,312236,312237],{},"Developing software entirely from scratch is expensive and risky. Modern software is heavily constructed by reusing existing components.",[105,312239,312240,312246,312252,312258],{},[108,312241,312242,312245],{},[2627,312243,312244],{},"Abstraction Level:"," Reusing successful design patterns and architectural styles.",[108,312247,312248,312251],{},[2627,312249,312250],{},"Object Level:"," Directly reusing specific objects\u002Fmethods from existing libraries (e.g., using a JavaMail library to process emails).",[108,312253,312254,312257],{},[2627,312255,312256],{},"Component Level:"," Reusing entire frameworks or collections of objects (e.g., a User Interface framework that handles window displays and event handling).",[108,312259,312260,312263],{},[2627,312261,312262],{},"The Trade-off:"," While reuse saves time, it involves costs: the time spent searching for and evaluating the software, purchasing licenses, adapting it to your needs, and integrating conflicting components.",[253,312265,312267],{"id":312266},"_42-configuration-management","4.2 Configuration Management",[15,312269,312270],{},"Because software changes continuously during development, change management is essential. If a team doesn't track versions, developers will overwrite each other's code.",[105,312272,312273,312279],{},[108,312274,312275,312278],{},[2627,312276,312277],{},"Version Management:"," Tools that keep track of component versions and coordinate team development to stop code conflicts.",[108,312280,312281,312284],{},[2627,312282,312283],{},"System Integration:"," Tools that help developers define exactly which versions of which components are combined to create a specific system release.",[253,312286,312288],{"id":312287},"_43-host-target-development","4.3 Host-Target Development",[15,312290,312291],{},"Software is rarely executed on the same machine it was coded on.",[105,312293,312294,312300],{},[108,312295,312296,312299],{},[2627,312297,312298],{},"The Host:"," The development platform where the code is written (using an Integrated Development Environment or IDE).",[108,312301,312302,312305],{},[2627,312303,312304],{},"The Target:"," The execution platform where the software will live. Developers must consider the target's specific hardware constraints, physical size, and operating system during implementation.",[32188,312307],{},[19,312309,312311],{"id":312310},"_5-open-source-development","5. Open-Source Development",[15,312313,312314],{},"Open-source development involves publishing the source code of a system and inviting a vast community of volunteers to participate in its development and improvement.",[253,312316,312318],{"id":312317},"_51-business-and-practical-considerations","5.1 Business and Practical Considerations",[15,312320,312321],{},"Making software open-source can reassure customers; if the original company goes out of business, the users know they will still have access to the code and can maintain it themselves. However, open-sourcing a highly specialized application does not guarantee a flood of volunteer developers?�most highly successful open-source projects are broad, foundational platform products.",[253,312323,312325],{"id":312324},"_52-open-source-licensing-models","5.2 Open-Source Licensing Models",[15,312327,312328],{},"While the code is freely available, the developer still legally owns the code and uses licenses to dictate exactly how it can be reused:",[2230,312330,312331,312340,312346],{},[108,312332,312333,312336,312337,312339],{},[2627,312334,312335],{},"GNU General Public License (GPL):"," A \"reciprocal\" license. If you embed GPL-licensed code in your system, your entire system ",[71807,312338,311136],{}," also be made open-source.",[108,312341,312342,312345],{},[2627,312343,312344],{},"GNU Lesser General Public License (LGPL):"," A variant where you can link to LGPL code without open-sourcing your entire system. However, if you modify the LGPL component itself, those specific modifications must be published.",[108,312347,312348,312351],{},[2627,312349,312350],{},"Berkley Standard Distribution (BSD) \u002F MIT:"," Non-reciprocal licenses. You can use the code in proprietary, closed-source commercial systems that you sell, as long as you acknowledge the original creator of the code.",{"title":29,"searchDepth":43,"depth":43,"links":312353},[312354,312355,312362,312366,312371],{"id":311947,"depth":43,"text":311948},{"id":311967,"depth":43,"text":311968,"children":312356},[312357,312358,312359,312360,312361],{"id":311985,"depth":49,"text":311986},{"id":312023,"depth":49,"text":312024},{"id":312030,"depth":49,"text":312031},{"id":312077,"depth":49,"text":312078},{"id":312110,"depth":49,"text":312111},{"id":312136,"depth":43,"text":312137,"children":312363},[312364,312365],{"id":312150,"depth":49,"text":312151},{"id":312183,"depth":49,"text":312184},{"id":312226,"depth":43,"text":312227,"children":312367},[312368,312369,312370],{"id":312233,"depth":49,"text":312234},{"id":312266,"depth":49,"text":312267},{"id":312287,"depth":49,"text":312288},{"id":312310,"depth":43,"text":312311,"children":312372},[312373,312374],{"id":312317,"depth":49,"text":312318},{"id":312324,"depth":49,"text":312325},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap7",{"title":311919,"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap7","wTQNJBO2s0qK0e8ePQtsaGlou0mnbwdiYuaRxfCVcZI",{"id":312381,"title":312382,"body":312383,"description":29,"extension":85,"meta":312661,"navigation":52,"path":312662,"seo":312663,"stem":312664,"__hash__":312665},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap8.md","Chapter 8: Software Testing",{"type":8,"value":312384,"toc":312653},[312385,312388,312390,312394,312397,312399,312405,312408,312411,312425,312431,312434,312448,312450,312456,312459,312465,312468,312471,312485,312491,312498,312504,312507,312509,312515,312521,312526,312546,312551,312577,312579,312585,312588,312591,312619,312621,312627,312630,312633],[11,312386,312382],{"id":312387},"chapter-8-software-testing",[32188,312389],{},[253,312391,312392],{"id":309852},[2627,312393,309853],{},[15,312395,312396],{},"Chapter 8 explores the fundamental concepts and practices of software testing within the software engineering lifecycle. Testing is an execution-based process designed to verify that software meets its intended requirements and to uncover hidden defects before deployment. The chapter breaks down testing into distinct phases?evelopment Testing, Release Testing, and User Testing?hile highlighting modern agile practices like Test-Driven Development (TDD).",[32188,312398],{},[253,312400,312402],{"id":312401},"_1-introduction-to-software-testing",[2627,312403,312404],{},"1. Introduction to Software Testing",[15,312406,312407],{},"At its core, software testing involves executing a program using artificial data. The overarching purpose is to show that the software does what it is intended to do and to discover defects before the product goes live.",[15,312409,312410],{},"Testing generally serves two primary goals:",[105,312412,312413,312419],{},[108,312414,312415,312418],{},[2627,312416,312417],{},"Validation testing:"," Designed to demonstrate to both the developer and the customer that the system meets its requirements and performs correctly under expected use.",[108,312420,312421,312424],{},[2627,312422,312423],{},"Defect testing:"," Designed to intentionally find inputs or sequences where the system behaves incorrectly or fails to conform to its specifications, ultimately exposing hidden bugs.",[15,312426,312427,312430],{},[2627,312428,312429],{},"Verification and Validation (V&V)","\nTesting is a critical, dynamic component of the broader Verification and Validation (V&V) process. Because testing requires executing the software, it is considered dynamic, distinguishing it from static techniques like software inspections that analyze code without running it.",[15,312432,312433],{},"To understand the V&V process, you must separate the two terms conceptually:",[105,312435,312436,312442],{},[108,312437,312438,312441],{},[2627,312439,312440],{},"Validation:"," Asks the question, \"Are we building the right product?\". It ensures the software ultimately aligns with the customer's real-world expectations.",[108,312443,312444,312447],{},[2627,312445,312446],{},"Verification:"," Asks the question, \"Are we building the product right?\". It checks that the software meets its explicitly stated functional and non-functional requirements.",[32188,312449],{},[253,312451,312453],{"id":312452},"_2-development-testing",[2627,312454,312455],{},"2. Development Testing",[15,312457,312458],{},"Development testing encompasses all testing activities performed by the development team. It is generally broken down into three granular stages, moving from the smallest units of code to the full system:",[32490,312460,312462],{"id":312461},"a-unit-testing",[2627,312463,312464],{},"A. Unit Testing",[15,312466,312467],{},"This is the lowest level of testing, focusing on individual program components like specific methods or object classes. It is typically automated using developer frameworks such as JUnit.",[15,312469,312470],{},"To build effective unit tests, developers rely on strategic testing methods:",[105,312472,312473,312479],{},[108,312474,312475,312478],{},[2627,312476,312477],{},"Partition Testing (Equivalence Partitioning):"," This strategy involves identifying groups (or partitions) of inputs that share common characteristics and should, theoretically, be processed by the software in the exact same way. Test cases are carefully selected from the boundaries and midpoints of these input partitions to maximize coverage while minimizing redundant tests.",[108,312480,312481,312484],{},[2627,312482,312483],{},"Guideline-based Testing:"," This method utilizes historical testing experience to purposefully target common programmer errors. Examples include testing single-value sequences, deliberately forcing output errors, or attempting to overload data buffers.",[32490,312486,312488],{"id":312487},"b-component-testing",[2627,312489,312490],{},"B. Component Testing",[15,312492,312493,312494,312497],{},"Once units are tested, several individual units are integrated to create composite components. The focus shifts heavily to ",[2627,312495,312496],{},"interface testing",", ensuring that these distinct objects interact correctly and pass data seamlessly. Common points of failure at this stage include errors within parameter interfaces, shared memory interfaces, or procedural interfaces.",[32490,312499,312501],{"id":312500},"c-system-testing",[2627,312502,312503],{},"C. System Testing",[15,312505,312506],{},"At this stage, all or some composite components are integrated to form a complete version of the system. The primary focus here is evaluating system-wide compatibility, verifying correct interactions on a macro level, and observing emergent behavior that only manifests when the individual parts are brought together.",[32188,312508],{},[253,312510,312512],{"id":312511},"_3-test-driven-development-tdd",[2627,312513,312514],{},"3. Test-Driven Development (TDD)",[15,312516,312517,312518,312520],{},"Test-Driven Development (TDD) is a prominent agile engineering practice where testing and code development are deeply interleaved. Instead of writing code and then testing it, code is developed in small increments, and the test is actually written ",[71807,312519,310072],{}," the code itself.",[15,312522,312523],{},[2627,312524,312525],{},"The TDD Cycle:",[2230,312527,312528,312531,312534,312537,312540,312543],{},[108,312529,312530],{},"Identify new functionality.",[108,312532,312533],{},"Write an automated test for that functionality.",[108,312535,312536],{},"Run the test (it will fail, as the code doesn't exist yet).",[108,312538,312539],{},"Implement the exact functionality needed to pass the test.",[108,312541,312542],{},"Run the test again (it passes).",[108,312544,312545],{},"Move to the next chunk of development.",[15,312547,312548],{},[2627,312549,312550],{},"Core Benefits of TDD:",[105,312552,312553,312559,312565,312571],{},[108,312554,312555,312558],{},[2627,312556,312557],{},"Code Coverage:"," Guarantees that every segment of written code has an associated, passing test.",[108,312560,312561,312564],{},[2627,312562,312563],{},"Regression Testing:"," Naturally builds a robust, incremental suite of tests. This makes it incredibly easy to detect if new code changes accidentally break existing functionality.",[108,312566,312567,312570],{},[2627,312568,312569],{},"Simplified Debugging:"," If a test fails during the cycle, developers know the bug is almost certainly confined to the most recently written chunk of code.",[108,312572,312573,312576],{},[2627,312574,312575],{},"System Documentation:"," The extensive suite of tests acts as a form of living documentation, clearly outlining what the code is supposed to do under specific conditions.",[32188,312578],{},[253,312580,312582],{"id":312581},"_4-release-testing",[2627,312583,312584],{},"4. Release Testing",[15,312586,312587],{},"Once the development team is satisfied, the software moves to release testing. This process tests a version of the system intended for use outside the development team. The primary goal is to prove to external suppliers and customers that the system is fully ready for deployment. This phase relies heavily on \"black-box\" functional testing, meaning the internal code structure is ignored, and only inputs and outputs are evaluated.",[15,312589,312590],{},"Key approaches in release testing include:",[105,312592,312593,312599,312605],{},[108,312594,312595,312598],{},[2627,312596,312597],{},"Requirements-based testing:"," A systematic derivation of test cases designed to explicitly prove that every single specific requirement has been satisfied.",[108,312600,312601,312604],{},[2627,312602,312603],{},"Scenario testing:"," Generating test cases based on realistic, typical stories of how an end-user will actually utilize the system in the real world.",[108,312606,312607,312610,312611],{},[2627,312608,312609],{},"Performance testing:"," Running tests designed specifically to ensure the system can handle its intended operational load.\n",[105,312612,312613],{},[108,312614,312615,312618],{},[71807,312616,312617],{},"Stress Testing:"," A subset of performance testing where the system is intentionally pushed beyond its maximum design limits. This allows engineers to observe how the system fails and to uncover hidden defects that only surface under extreme, full-load conditions.",[32188,312620],{},[253,312622,312624],{"id":312623},"_5-user-testing",[2627,312625,312626],{},"5. User Testing",[15,312628,312629],{},"The final phase involves actual end-users testing the software in their native environments to provide direct input and advice. This is a vital step because development teams can never perfectly replicate the nuanced complexities of a real-world operational environment.",[15,312631,312632],{},"User testing is generally staged in three phases:",[105,312634,312635,312641,312647],{},[108,312636,312637,312640],{},[2627,312638,312639],{},"Alpha testing:"," A small, carefully selected group of users works intimately with the development team to test early, often incomplete releases of the software.",[108,312642,312643,312646],{},[2627,312644,312645],{},"Beta testing:"," An early release is distributed to a much larger, external group of users. These users experiment with the software independently and report any problems or bugs back to the team.",[108,312648,312649,312652],{},[2627,312650,312651],{},"Acceptance testing:"," The final hurdle where the customer formally tests the system utilizing their own real-world data. The results of this test dictate whether the software is officially accepted from the developer and cleared for final deployment.",{"title":29,"searchDepth":43,"depth":43,"links":312654},[312655,312656,312657,312658,312659,312660],{"id":309852,"depth":49,"text":309853},{"id":312401,"depth":49,"text":312404},{"id":312452,"depth":49,"text":312455},{"id":312511,"depth":49,"text":312514},{"id":312581,"depth":49,"text":312584},{"id":312623,"depth":49,"text":312626},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap8",{"title":312382,"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap8","VDgnhVN1LhFyeyOWRLpkfifBCDawFO4lxz-U3juXu6Y",{"id":312667,"title":312668,"body":312669,"description":29,"extension":85,"meta":313064,"navigation":52,"path":313065,"seo":313066,"stem":313067,"__hash__":313068},"notes\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap9.md","Chapter 9: Software Evolution",{"type":8,"value":312670,"toc":313048},[312671,312674,312676,312687,312689,312693,312696,312700,312703,312729,312733,312739,312772,312781,312802,312804,312808,312814,312818,312824,312838,312842,312849,312875,312881,312885,312895,312921,312923,312927,312933,312937,312965,312969,312972,312986,313013,313022,313034],[11,312672,312668],{"id":312673},"chapter-9-software-evolution",[19,312675,309853],{"id":309852},[15,312677,312678,312679,312682,312683,312686],{},"Software development does not end when a system is deployed; in fact, the deployment marks the beginning of the most expensive and prolonged phase of the software lifecycle: ",[2627,312680,312681],{},"Software Evolution",". Historical data indicates that ",[2627,312684,312685],{},"60% to 90% of total software costs are incurred after delivery",". This chapter explores the inevitability of software change, the structured processes used to manage these changes, the complexities of maintaining aging \"legacy\" systems, and the strategic differences between software maintenance, reengineering, and refactoring.",[32188,312688],{},[19,312690,312692],{"id":312691},"_91-the-dynamics-of-software-evolution","9.1 The Dynamics of Software Evolution",[15,312694,312695],{},"Assume you have just launched a brand new software system. Almost immediately, the environment it operates in will begin to shift. Competitors will launch new features, underlying operating systems will be updated, and users will uncover unforeseen bugs or request new functionalities. A system that does not adapt will quickly degrade in usefulness.",[253,312697,312699],{"id":312698},"_911-the-software-lifecycle-model-rajlich-and-bennett","9.1.1 The Software Lifecycle Model (Rajlich and Bennett)",[15,312701,312702],{},"To understand a system's lifespan from the ground up, we rely on a model that divides the software lifecycle into distinct phases:",[2230,312704,312705,312711,312717,312723],{},[108,312706,312707,312710],{},[2627,312708,312709],{},"Evolution:"," The phase immediately following deployment. The software is actively used, and significant changes to the architecture and functionality are frequently made to meet evolving stakeholder requirements.",[108,312712,312713,312716],{},[2627,312714,312715],{},"Servicing:"," Over time, continuous modifications degrade the software's underlying structure, making large architectural changes too expensive or risky. The software enters the servicing phase, where only small, tactical, and essential changes (like minor bug fixes) are made.",[108,312718,312719,312722],{},[2627,312720,312721],{},"Phase-out:"," The system is still in use, but no further changes are made. Users must simply \"work around\" any bugs they encounter.",[108,312724,312725,312728],{},[2627,312726,312727],{},"Retirement:"," The system is taken completely offline, often requiring a complex data migration to a newer replacement system.",[253,312730,312732],{"id":312731},"_912-the-evolution-process","9.1.2 The Evolution Process",[15,312734,312735,312736,5598],{},"The process of evolving software is inherently cyclical and revolves around ",[2627,312737,312738],{},"change management",[105,312740,312741,312747,312753,312759],{},[108,312742,312743,312746],{},[2627,312744,312745],{},"Change Identification:"," Stakeholders (users, management, developers) submit formal or informal change proposals (e.g., bug reports, feature requests).",[108,312748,312749,312752],{},[2627,312750,312751],{},"Impact Analysis:"," Before accepting a change, developers must analyze the existing source code to understand how the proposed change will affect the system's architecture and to estimate the cost.",[108,312754,312755,312758],{},[2627,312756,312757],{},"Release Planning:"," The approved changes are bundled into a planned new release.",[108,312760,312761,312764,312765,2547],{},[2627,312762,312763],{},"Implementation:"," The changes are coded, validated (tested), and deployed. ",[71807,312766,312767,312768,312771],{},"Crucially, if a different team is doing the evolution than did the original development, the first and most difficult step of implementation is ",[2627,312769,312770],{},"program understanding","?iguring out how the original code works",[15,312773,312774,312777,312778,2547],{},[2627,312775,312776],{},"Mastery Application (Emergency Repairs):"," Sometimes, a critical bug (like a security vulnerability) forces developers to bypass the formal evolution process and issue an ",[2627,312779,312780],{},"emergency repair",[105,312782,312783,312788],{},[108,312784,312785,312787],{},[71807,312786,310005],{}," The developer patches the source code directly and immediately deploys it.",[108,312789,312790,312793,312794,312797,312798,312801],{},[71807,312791,312792],{},"The Consequence:"," The requirements and design documentation are not updated to reflect this patch. Over time, these emergency patches cause the system's architecture to severely degrade, accelerating the transition from the ",[71807,312795,312796],{},"Evolution"," phase into the ",[71807,312799,312800],{},"Servicing"," phase.",[32188,312803],{},[19,312805,312807],{"id":312806},"_92-legacy-systems","9.2 Legacy Systems",[15,312809,312810,312813],{},[2627,312811,312812],{},"Legacy systems"," are older software systems that rely on obsolete languages and technologies, but which remain absolutely critical to a business's operations.",[253,312815,312817],{"id":312816},"_921-why-not-just-replace-them","9.2.1 Why Not Just Replace Them?",[15,312819,312820,312821,2547],{},"It seems logical to scrap 20-year-old software, but businesses rarely do. Why? ",[2627,312822,312823],{},"Risk and Cost",[105,312825,312826,312832],{},[108,312827,312828,312831],{},[2627,312829,312830],{},"Embedded Business Rules:"," The legacy code often contains crucial, undocumented business rules (e.g., complex insurance risk-assessment algorithms). If the system is scrapped, that tacit institutional knowledge is lost.",[108,312833,312834,312837],{},[2627,312835,312836],{},"Operational Risk:"," The new replacement system might fail, go over budget, or disrupt daily operations. It is often cheaper to pay the high maintenance costs of an old system than to risk a catastrophic failure of a new one.",[253,312839,312841],{"id":312840},"_922-the-layers-of-a-legacy-system","9.2.2 The Layers of a Legacy System",[15,312843,312844,312845,312848],{},"A legacy system is not just old code; it is a ",[2627,312846,312847],{},"sociotechnical system"," made up of interdependent layers:",[2230,312850,312851,312857,312863,312869],{},[108,312852,312853,312856],{},[2627,312854,312855],{},"Hardware:"," Obsolete mainframe computers.",[108,312858,312859,312862],{},[2627,312860,312861],{},"Support Software:"," Outdated operating systems or compilers.",[108,312864,312865,312868],{},[2627,312866,312867],{},"Application Software:"," The actual legacy code.",[108,312870,312871,312874],{},[2627,312872,312873],{},"Business Processes:"," The organizational workflows built around the quirks of the old software.",[15,312876,312877,312880],{},[71807,312878,312879],{},"Analytical Insight:"," Because these layers are heavily intertwined, you cannot simply swap out the hardware without breaking the application software, which in turn disrupts the human business processes.",[253,312882,312884],{"id":312883},"_923-legacy-system-assessment","9.2.3 Legacy System Assessment",[15,312886,312887,312888,312891,312892,5598],{},"To decide what to do with a legacy system, managers plot the system on a matrix measuring ",[2627,312889,312890],{},"System Quality"," against ",[2627,312893,312894],{},"Business Value",[105,312896,312897,312903,312909,312915],{},[108,312898,312899,312902],{},[2627,312900,312901],{},"Low Quality, Low Business Value:"," Scrap the system.",[108,312904,312905,312908],{},[2627,312906,312907],{},"Low Quality, High Business Value:"," The system is vital but a nightmare to maintain. It should be reengineered or replaced.",[108,312910,312911,312914],{},[2627,312912,312913],{},"High Quality, Low Business Value:"," Replace with off-the-shelf software or maintain it cheaply.",[108,312916,312917,312920],{},[2627,312918,312919],{},"High Quality, High Business Value:"," Leave the system alone and continue normal maintenance.",[32188,312922],{},[19,312924,312926],{"id":312925},"_93-software-maintenance-reengineering","9.3 Software Maintenance & Reengineering",[15,312928,312929,312932],{},[2627,312930,312931],{},"Software Maintenance"," is the general term for changing custom software after it has been delivered, particularly when the maintenance team is different from the original development team.",[253,312934,312936],{"id":312935},"_931-the-three-types-of-maintenance","9.3.1 The Three Types of Maintenance",[2230,312938,312939,312949,312955],{},[108,312940,312941,312944,312945,312948],{},[2627,312942,312943],{},"Fault Repairs (Bug Fixing):"," Correcting coding, design, or requirements errors. (Surprisingly, this usually consumes the ",[71807,312946,312947],{},"least"," amount of maintenance effort).",[108,312950,312951,312954],{},[2627,312952,312953],{},"Environmental Adaptation:"," Modifying the software to work with new hardware, operating systems, or databases.",[108,312956,312957,312960,312961,312964],{},[2627,312958,312959],{},"Functionality Addition:"," Implementing new features to meet changing business demands. (This generally consumes the ",[71807,312962,312963],{},"most"," maintenance effort and budget).",[253,312966,312968],{"id":312967},"_932-software-reengineering-vs-refactoring","9.3.2 Software Reengineering vs. Refactoring",[15,312970,312971],{},"When a legacy system degrades to the point where maintenance is too expensive, the structure must be improved. There are two distinct approaches to this:",[15,312973,312974,312977,312978,312981,312982,312985],{},[2627,312975,312976],{},"1. Software Reengineering (Macro-level):","\nReengineering takes place ",[71807,312979,312980],{},"after"," a system has been maintained for a long time and its structure has decayed. The goal is to rebuild the system to make it maintainable again ",[71807,312983,312984],{},"without"," changing its functionality.",[105,312987,312988],{},[108,312989,312990,312992,312993,312996,312997,313000,313001,313004,313005,313008,313009,313012],{},[2627,312991,309890],{}," 1. ",[71807,312994,312995],{},"Source Code Translation:"," Automatically converting an old language (e.g., COBOL) to a modern one (e.g., Java).\n2. ",[71807,312998,312999],{},"Reverse Engineering:"," Analyzing the code to automatically reconstruct missing design documentation.\n3. ",[71807,313002,313003],{},"Program Structure Improvement:"," Untangling messy \"spaghetti code\".\n4. ",[71807,313006,313007],{},"Program Modularization:"," Grouping related parts of the program together to remove redundancy.\n5. ",[71807,313010,313011],{},"Data Reengineering:"," Cleaning up corrupted databases and redefining schemas.",[15,313014,313015,313018,313019,2547],{},[2627,313016,313017],{},"2. Refactoring (Micro-level \u002F Preventative):","\nUnlike reengineering, which is a massive reactive project, ",[2627,313020,313021],{},"refactoring is a continuous, preventative process",[105,313023,313024,313029],{},[108,313025,313026,313028],{},[71807,313027,310005],{}," As developers add new features (especially in Agile environments), they continuously make small, localized improvements to the code structure (e.g., renaming variables for clarity, breaking a large function into two smaller ones).",[108,313030,313031,313033],{},[71807,313032,310050],{}," Refactoring prevents the structural degradation that eventually necessitates expensive reengineering.",[15,313035,313036,313039,313040,313043,313044,313047],{},[2627,313037,313038],{},"Mastery Application (Refactoring vs. Reengineering):","\nImagine a house. ",[2627,313041,313042],{},"Refactoring"," is like fixing a leaky pipe and repainting the walls every few years so the house stays in great shape. ",[2627,313045,313046],{},"Reengineering"," is what happens when you ignore the house for 40 years, the foundation starts sinking, and you have to hire a crew to lift the entire house and pour a new concrete foundation?ithout changing the floor plan.",{"title":29,"searchDepth":43,"depth":43,"links":313049},[313050,313051,313055,313060],{"id":309852,"depth":43,"text":309853},{"id":312691,"depth":43,"text":312692,"children":313052},[313053,313054],{"id":312698,"depth":49,"text":312699},{"id":312731,"depth":49,"text":312732},{"id":312806,"depth":43,"text":312807,"children":313056},[313057,313058,313059],{"id":312816,"depth":49,"text":312817},{"id":312840,"depth":49,"text":312841},{"id":312883,"depth":49,"text":312884},{"id":312925,"depth":43,"text":312926,"children":313061},[313062,313063],{"id":312935,"depth":49,"text":312936},{"id":312967,"depth":49,"text":312968},{},"\u002Fsource\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap9",{"title":312668,"description":29},"source\u002Fcs\u002Fse\u002Fnotesbychap\u002Fchap9","JdMbuN6OhRg7afJ70uG5pRmJ38AP03PucFxBS8J0hus",{"id":313070,"title":313071,"body":313072,"description":313116,"extension":85,"meta":313117,"navigation":52,"path":313118,"seo":313119,"stem":313120,"__hash__":313121},"notes\u002Fsource\u002Findex.md","Studiorum",{"type":8,"value":313073,"toc":313113},[313074,313077,313080,313083,313105],[11,313075,313071],{"id":313076},"studiorum",[15,313078,313079],{},"知識を、記す。Learning is a journey, not a destination.",[19,313081,313082],{"id":313082},"分類",[105,313084,313085,313092,313099],{},[108,313086,313087,313091],{},[111,313088,313090],{"href":313089},"\u002Fmath\u002Fmath","數學"," — Abstract Algebra \u002F Mathematical Analysis \u002F ODE",[108,313093,313094,313098],{},[111,313095,313097],{"href":313096},"\u002Fcode\u002F","程式設計"," — C++ \u002F Go \u002F Vue.js \u002F Frontend",[108,313100,313101,313104],{},[111,313102,313103],{"href":162016},"計算機科學"," — OS \u002F Algorithm Design & Analysis",[15,313106,313107,313108,313112],{},"這些原文筆記會顯示在 ",[111,313109,313111],{"href":313110},"\u002Flibrary","課程筆記"," 頁面，依課程與章節重新分類、可搜尋。",{"title":29,"searchDepth":43,"depth":43,"links":313114},[313115],{"id":313082,"depth":43,"text":313082},"學習是一段旅程，而非終點。課程原文筆記依科目與章節分類收錄於此。",{},"\u002Fsource",{"title":313071,"description":313116},"source\u002Findex","ZRTyDuUozvl7nQt9XsrII1nA4QNe9rTLdDIz8nx1TNQ",{"id":313123,"title":313124,"body":313125,"description":313134,"extension":85,"meta":313422,"navigation":52,"path":313423,"seo":313424,"stem":313425,"__hash__":313426},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.10.md","2.10. Es gibt, es ist, es sind",{"type":8,"value":313126,"toc":313417},[313127,313130,313135,313149,313155,313169,313174,313194,313198,313218,313220,313226,313229,313319,313321,313327,313330,313380,313385,313387,313393,313399,313405,313411],[11,313128,313124],{"id":313129},"_210-es-gibt-es-ist-es-sind",[15,313131,313132],{},[2627,313133,313134],{},"1. Geben 的原意是“给”，但 es gibt 被习惯用来表示“存在”。根据上下文，它可以翻译为“有”或“存在”（there is \u002F there are）：",[105,313136,313137,313143],{},[108,313138,313139,313142],{},[2627,313140,313141],{},"Es gibt eine Universität in Hamburg."," — 汉堡有一所大学。(There is a university in Hamburg.)",[108,313144,313145,313148],{},[2627,313146,313147],{},"Gibt es viele Studenten auf der Universität?"," — 大学里有很多学生吗？(Are there many students at the university?)",[15,313150,313151,313154],{},[2627,313152,313153],{},"2. Es ist（有一\u002F是）和 es sind（有\u002F是）的使用方式与此类似：","\nEs ist (there is) and es sind (there are) are used in a like manner:",[105,313156,313157,313163],{},[108,313158,313159,313162],{},[2627,313160,313161],{},"Es ist nur ein Mensch in diesem Haus."," — 这个房子里只有一个人。(There is only one person in this house.)",[108,313164,313165,313168],{},[2627,313166,313167],{},"Es sind drei Menschen in dem Volkswagen."," — 大众汽车里有三个人。(There are three people in the Volkswagen.)",[15,313170,313171],{},[2627,313172,313173],{},"3. 这些表达方式的过去时为：",[105,313175,313176,313182,313188],{},[108,313177,313178,313181],{},[2627,313179,313180],{},"es gab"," — 曾有 (there was, there were)",[108,313183,313184,313187],{},[2627,313185,313186],{},"es war"," — 曾是\u002F曾有 (there was)",[108,313189,313190,313193],{},[2627,313191,313192],{},"es waren"," — 曾是\u002F曾有 (there were)",[15,313195,313196],{},[2627,313197,8882],{},[105,313199,313200,313206,313212],{},[108,313201,313202,313205],{},[2627,313203,313204],{},"Ich bin es."," — 是我。(It is I.)",[108,313207,313208,313211],{},[2627,313209,313210],{},"Ist er es?"," — 是他吗？(Is it he?)",[108,313213,313214,313217],{},[2627,313215,313216],{},"Es waren meine Kinder."," — 那是我的孩子们。(It was my children.)",[32188,313219],{},[253,313221,313223],{"id":313222},"练习-exercises",[2627,313224,313225],{},"练习 (Exercises)",[15,313227,313228],{},"请给出下列德语短语的含义：",[2230,313230,313231,313241,313251,313261,313271,313277,313287,313297,313303,313309],{},[108,313232,313233,313236,313237,313240],{},[2627,313234,313235],{},"sehr"," = 非常 (very)；",[2627,313238,313239],{},"Sehr viele Länder umschließen Deutschland."," — 1. 非常多的国家包围着德国。(Very many countries surround Germany.)",[108,313242,313243,313246,313247,313250],{},[2627,313244,313245],{},"die"," = 那些\u002F哪一个 (which)；",[2627,313248,313249],{},"Es gibt neun Länder, die Deutschland umschließen."," — 2. 有九个国家包围着德国。(There are nine countries which surround Germany.)",[108,313252,313253,313256,313257,313260],{},[2627,313254,313255],{},"die Mitte"," = 中间、中心 (middle, center)；",[2627,313258,313259],{},"Es ist in der geographischen Mitte Europas."," — 3. 它位于欧洲的地理中心。(It is in the geographical center of Europe.)",[108,313262,313263,313266,313267,313270],{},[2627,313264,313265],{},"einmal"," = 曾经 (once)；",[2627,313268,313269],{},"Einmal gab es viele Deutsche in der Tschechoslowakei."," — 4. 捷克斯洛伐克曾经有很多德国人。(There were once many Germans in Czechoslovakia.)",[108,313272,313273,313276],{},[2627,313274,313275],{},"Warum ist es ein Durchgangsland?"," — 5. 为什么它是一个通道国家（指他国经过的国家）？(Why is it a country through which others travel?)",[108,313278,313279,313282,313283,313286],{},[2627,313280,313281],{},"da"," = 因为、由于 (since)；",[2627,313284,313285],{},"... da es fast keine natürlichen Grenzen gibt."," — 6. ……因为几乎没有自然边界。(... since there are almost no natural borders.)",[108,313288,313289,313292,313293,313296],{},[2627,313290,313291],{},"eigentlich"," = 真的、实际上 (really, actually)；",[2627,313294,313295],{},"Gab es eigentlich ein Deutschland?"," — 7. 真的曾有一个（统一的）德国吗？(Was there really one Germany?)",[108,313298,313299,313302],{},[2627,313300,313301],{},"Es war sehr schwer, diese Frage zu verstehen."," — 8. 理解这个问题曾经非常困难。(It was very hard to understand this question.)",[108,313304,313305,313308],{},[2627,313306,313307],{},"Es gibt zwei Teile Deutschlands, die zusammen den größten Staat Europas bilden."," — 9. 德国有两个部分，它们共同构成了欧洲最大的国家。(There are two parts of Germany which together form the largest state in Europe.)",[108,313310,313311,313314,313315,313318],{},[2627,313312,313313],{},"das Gebiet"," = 地区 (the area)；",[2627,313316,313317],{},"Es waren auch romanische, slawische und magyarische Völker auf diesem Gebiet."," — 10. 在这个地区也曾有过罗曼、斯拉夫和马扎尔民族。(There were also Romanic, Slavic, and Magyar peoples in this area.)",[32188,313320],{},[253,313322,313324],{"id":313323},"阶段测验-progress-test",[2627,313325,313326],{},"阶段测验 (PROGRESS TEST)",[15,313328,313329],{},"请选择最符合德语含义的选项：",[2230,313331,313332,313342,313352,313361,313370],{},[108,313333,313334,313337,313338,313341],{},[2627,313335,313336],{},"Es sind nur vier Länder auf dieser Landkarte.","\n(a) 他们是 (b) ",[2627,313339,313340],{},"这里有"," (c) 曾经有",[108,313343,313344,313347,313348,313351],{},[2627,313345,313346],{},"Es war nur eine Landkarte an der Wand.","\n(a) ",[2627,313349,313350],{},"曾经有一张"," (b) 它是 (c) 它是（现在时）",[108,313353,313354,313356,313357,313360],{},[2627,313355,313295],{},"\n(a) 有……吗？ (b) ",[2627,313358,313359],{},"曾经有……吗？"," (c) 他给了吗？",[108,313362,313363,313337,313366,313369],{},[2627,313364,313365],{},"Es waren nur vier Länder auf der Landkarte.",[2627,313367,313368],{},"曾经有"," (c) 这里有",[108,313371,313372,313375,313376,313379],{},[2627,313373,313374],{},"Es ist nur eine Person in dem Volkswagen.","\n(a) 它是 (b) ",[2627,313377,313378],{},"这里只有"," (c) 他是",[15,313381,313382],{},[2627,313383,313384],{},"答案：1. (b); 2. (a); 3. (b); 4. (b); 5. (b)。",[32188,313386],{},[253,313388,313390],{"id":313389},"阅读课文德国地理-die-geographie-deutschlands",[2627,313391,313392],{},"阅读课文：德国地理 (Die Geographie Deutschlands)",[15,313394,313395,313398],{},[2627,313396,313397],{},"Wenn wir Deutschland als ein Land verstehen wollen, müssen wir nicht nur Geschichte lernen, sondern auch eine Landkarte anschauen.","\n如果我们想把德国作为一个国家来理解，我们不仅必须学习历史，还必须观察地图。\nIf we want to understand Germany as a country, we must not only learn history, but also look at a map.",[15,313400,313401,313404],{},[2627,313402,313403],{},"Die Geographie Deutschlands kann uns sehr viel sagen. Deutschland liegt in der geographischen Mitte Europas, in dem Teil Europas, den wir Mitteleuropa nennen.","\n德国的地理可以告诉我们很多。德国位于欧洲的地理中心，即我们称之为中欧的那部分欧洲。\nThe geography of Germany can tell us very much. Germany is located in the geographical center of Europe, in the part of Europe which we call Central Europe.",[15,313406,313407,313410],{},[2627,313408,313409],{},"Natürlich liegt nicht nur Deutschland in diesem Gebiet, sondern auch andere Länder. In dieser Zone leben also germanische, slawische, romanische und magyarische Völker.","\n当然，不仅德国位于这个地区，其他国家也是。因此，在这个地带居住着日耳曼、斯拉夫、罗曼和马扎尔民族。\nNaturally, not only Germany is located in this area, but also other countries. In this zone, therefore, live Germanic, Slavic, Romanic, and Magyar peoples.",[15,313412,313413,313416],{},[2627,313414,313415],{},"Zusammen bilden die Bundesrepublik Deutschland (d.h. Westdeutschland) und die Deutsche Demokratische Republik (d.h. Ostdeutschland) den größten Staat Europas, außer Rußland.","\n德意志联邦共和国（即西德）和德意志民主共和国（即东德）共同构成了除俄罗斯外欧洲最大的国家。\nTogether the Federal Republic of Germany (i.e. West Germany) and the German Democratic Republic (i.e. East Germany) form the largest state in Europe, except for Russia.",{"title":29,"searchDepth":43,"depth":43,"links":313418},[313419,313420,313421],{"id":313222,"depth":49,"text":313225},{"id":313323,"depth":49,"text":313326},{"id":313389,"depth":49,"text":313392},{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.10",{"title":313124,"description":313134},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.10","bAOusUcLtqpNAATzA6_rTIQddLLMFMobr2yWudrl3UU",{"id":313428,"title":313429,"body":313430,"description":29,"extension":85,"meta":313639,"navigation":52,"path":313640,"seo":313641,"stem":313642,"__hash__":313643},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.7.md","2.7 动词 - 同源词 (Verbs - Cognates)",{"type":8,"value":313431,"toc":313637},[313432,313435,313519,313524],[11,313433,313429],{"id":313434},"_27-动词-同源词-verbs-cognates",[2230,313436,313437,313466],{},[108,313438,313439,313442],{},[2627,313440,313441],{},"所有德語动词的不定式（infinitive）形式都以 -en 結尾，有时以 -n 結尾：",[105,313443,313444,313454],{},[108,313445,313446,47649,313449],{},[2627,313447,313448],{},"beginnen",[71807,313450,313451],{},[2627,313452,313453],{},"(to begin)",[108,313455,313456,313459,313460,313465],{},[2627,313457,313458],{},"wandern"," ",[71807,313461,313462],{},[2627,313463,313464],{},"(to wander)","\n德语和英语中有许多同源动词，这便于学习和阅读。",[108,313467,313468,313471,313472,313475,313476,2664,313479],{},[2627,313469,313470],{},"不定式只要將动词首字母大写，也可用作名詞。","\n其性別始终為",[2627,313473,313474],{},"中性","。要將用作名詞的不定式翻譯成英語，請在英語不定式後加上 ",[2627,313477,313478],{},"-ing",[105,313480,313481,313489,313499,313509],{},[108,313482,313483,47649,313485],{},[2627,313484,313448],{},[71807,313486,313487],{},[2627,313488,313453],{},[108,313490,313491,47649,313494],{},[2627,313492,313493],{},"das Beginnen",[71807,313495,313496],{},[2627,313497,313498],{},"(the beginning)",[108,313500,313501,47649,313504],{},[2627,313502,313503],{},"trinken",[71807,313505,313506],{},[2627,313507,313508],{},"(to drink)",[108,313510,313511,47649,313514],{},[2627,313512,313513],{},"das Trinken",[71807,313515,313516],{},[2627,313517,313518],{},"(the drinking)",[15,313520,313521],{},[2627,313522,313523],{},"练习：",[2230,313525,313526,313537,313547,313557,313567,313577,313587,313597,313607,313617,313627],{},[108,313527,313528,313531,313532],{},[2627,313529,313530],{},"bringen"," — ",[71807,313533,313534],{},[2627,313535,313536],{},"(to bring)",[108,313538,313539,313531,313542],{},[2627,313540,313541],{},"denken",[71807,313543,313544],{},[2627,313545,313546],{},"(to think)",[108,313548,313549,313531,313552],{},[2627,313550,313551],{},"finden",[71807,313553,313554],{},[2627,313555,313556],{},"(to find)",[108,313558,313559,313531,313562],{},[2627,313560,313561],{},"hindern",[71807,313563,313564],{},[2627,313565,313566],{},"(to hinder) 阻碍",[108,313568,313569,313531,313572],{},[2627,313570,313571],{},"sinken",[71807,313573,313574],{},[2627,313575,313576],{},"(to sink) 下沉",[108,313578,313579,313531,313582],{},[2627,313580,313581],{},"das Sinken",[71807,313583,313584],{},[2627,313585,313586],{},"(the sinking) 下沉（名词）",[108,313588,313589,313531,313592],{},[2627,313590,313591],{},"schlummern",[71807,313593,313594],{},[2627,313595,313596],{},"(to slumber) 打盹",[108,313598,313599,313531,313602],{},[2627,313600,313601],{},"kommen",[71807,313603,313604],{},[2627,313605,313606],{},"(to come)",[108,313608,313609,313531,313612],{},[2627,313610,313611],{},"das Kommen",[71807,313613,313614],{},[2627,313615,313616],{},"(the coming)",[108,313618,313619,313531,313622],{},[2627,313620,313621],{},"das Marschieren",[71807,313623,313624],{},[2627,313625,313626],{},"(the marching)",[108,313628,313629,313531,313632],{},[2627,313630,313631],{},"marschieren",[71807,313633,313634],{},[2627,313635,313636],{},"(to march)",{"title":29,"searchDepth":43,"depth":43,"links":313638},[],{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.7",{"title":313429,"description":29},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.7","65-ztjGmdmSPfKr4QXnMYRsbo2PIoyON1QP9X29tuDI",{"id":313645,"title":313646,"body":313647,"description":29,"extension":85,"meta":313888,"navigation":52,"path":313889,"seo":313890,"stem":313891,"__hash__":313892},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.8.md","2.8 动词-部分同源和虚假同源词 (Verbs - Partial and False Cognates)",{"type":8,"value":313648,"toc":313886},[313649,313652,313754,313757,313855,313861,313881],[11,313650,313646],{"id":313651},"_28-动词-部分同源和虚假同源词-verbs-partial-and-false-cognates",[2230,313653,313654,313709],{},[108,313655,313656,313657],{},"一些部分同源词如下：\n",[105,313658,313659,313669,313679,313689,313699],{},[108,313660,313661,47649,313664],{},[2627,313662,313663],{},"liegen",[71807,313665,313666],{},[2627,313667,313668],{},"(to lie, to be situated)",[108,313670,313671,47649,313674],{},[2627,313672,313673],{},"lernen",[71807,313675,313676],{},[2627,313677,313678],{},"(to learn, to study)",[108,313680,313681,47649,313684],{},[2627,313682,313683],{},"sagen",[71807,313685,313686],{},[2627,313687,313688],{},"(to say, to tell)",[108,313690,313691,47649,313694],{},[2627,313692,313693],{},"scheinen",[71807,313695,313696],{},[2627,313697,313698],{},"(to shine, to seem, to appear)",[108,313700,313701,47649,313704],{},[2627,313702,313703],{},"tun",[71807,313705,313706],{},[2627,313707,313708],{},"(to do, to make, to put)",[108,313710,313711,313712],{},"一些虚假同源词如下（需要背诵）：\n",[105,313713,313714,313724,313734,313744],{},[108,313715,313716,47649,313719],{},[2627,313717,313718],{},"bilden",[71807,313720,313721],{},[2627,313722,313723],{},"(to form, to educate)",[108,313725,313726,47649,313729],{},[2627,313727,313728],{},"bekommen",[71807,313730,313731],{},[2627,313732,313733],{},"(to receive, to get)",[108,313735,313736,47649,313739],{},[2627,313737,313738],{},"handeln",[71807,313740,313741],{},[2627,313742,313743],{},"(to act, to trade, to bargain) 行动，贸易，讨价还价",[108,313745,313746,47649,313749],{},[2627,313747,313748],{},"mangeln",[71807,313750,313751],{},[2627,313752,313753],{},"(to lack, to be missing) 缺乏",[15,313755,313756],{},"下列练习将为您阅读章末的关于德国地理的文章做好准备：",[2230,313758,313759,313775,313781,313791,313801,313815,313825,313835,313845],{},[108,313760,313761,313762,313765,313766,313768,313769,113463,313771,313774],{},"在動詞 ",[2627,313763,313764],{},"schieben"," 中，您可以預期字母 ",[2627,313767,1697],{}," 在英語中會變成 ",[2627,313770,56782],{},[2627,313772,313773],{},"to shove"," (to push)。",[108,313776,313777,313780],{},[2627,313778,313779],{},"den Mann schieben"," — 推那個男人 (to push the man)",[108,313782,313783,313786,313787,313790],{},[2627,313784,313785],{},"die Grenzen"," = the borders；",[2627,313788,313789],{},"die Grenzen verschieben"," — 移動（改變）邊界 (to shift the borders)",[108,313792,313793,313796,313797,313800],{},[2627,313794,313795],{},"mehrfach"," = repeatedly；",[2627,313798,313799],{},"die Grenzen mehrfach verschieben"," — 多次改變邊界 (to shift the borders repeatedly)",[108,313802,313761,313803,313765,313806,313768,313809,113463,313811,313814],{},[2627,313804,313805],{},"suchen",[2627,313807,313808],{},"ch",[2627,313810,33759],{},[2627,313812,313813],{},"to seek"," (to look for, to try)。",[108,313816,313817,313820,313821,313824],{},[2627,313818,313819],{},"versuchen"," = 嘗試 (to try, to attempt)；",[2627,313822,313823],{},"Sie versuchen, Land zu gewinnen."," — 他們試圖獲得土地。",[108,313826,313827,313830,313831,313834],{},[2627,313828,313829],{},"größten"," = largest；",[2627,313832,313833],{},"den größten Staat bilden"," — 形成最大的國家 (to form the largest state)",[108,313836,313837,313840,313841,313844],{},[2627,313838,313839],{},"spielen eine Rolle"," （在英語中，我們說 ",[2627,313842,313843],{},"to play a role","）。 — 扮演一個角色",[108,313846,313847,313850,313851,313854],{},[2627,313848,313849],{},"zusammen"," = together；",[2627,313852,313853],{},"eine Rolle zusammen spielen"," — to play a role together",[15,313856,313857,313860],{},[2627,313858,313859],{},"阶段测试：","\n請選擇最能表達德語斜體字含義的答案。",[2230,313862,313863,313868,313874],{},[108,313864,313865,313867],{},[71807,313866,313819],{},", Land zu gewinnen (a) to forsake 棄絕 (b) to forgive原諒 (c) to seek尋求、嘗試",[108,313869,313870,313871,313873],{},"den größten Staat ",[71807,313872,313718],{}," (a) to educate 教育 (b) to form 形成 (c) to build 建造",[108,313875,313876,313877,313880],{},"die Grenzen ",[71807,313878,313879],{},"verschieben"," (a) to forsake 棄絕 (b) to shovel 鏟除 (c) to shift 改變、移動",[15,313882,313883],{},[2627,313884,313885],{},"答案：1. (c); 2. (b); 3. (c)。",{"title":29,"searchDepth":43,"depth":43,"links":313887},[],{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.8",{"title":313646,"description":29},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.8","UPCBNlpSJzhc5X6AHBa-l-20ZjyesY73y675FoCTHo8",{"id":313894,"title":313895,"body":313896,"description":313905,"extension":85,"meta":314388,"navigation":52,"path":314389,"seo":314390,"stem":314391,"__hash__":314392},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.9.md","2.9 现在时动词词尾；人称代词",{"type":8,"value":313897,"toc":314385},[313898,313901,313906,313919,313924,314029,314034,314039,314064,314069,314084,314089,314213,314221,314223,314227,314230,314285,314290,314340,314345],[11,313899,313895],{"id":313900},"_29-现在时动词词尾人称代词",[15,313902,313903],{},[2627,313904,313905],{},"1. 现在时的词干,通过去掉不定式结尾的 -en 或 -n 得到：",[105,313907,313908,313914],{},[108,313909,313910,313913],{},[2627,313911,313912],{},"singen",": sing-",[108,313915,313916,313918],{},[2627,313917,313458],{},": wander-",[15,313920,313921],{},[2627,313922,313923],{},"2. 德语中大多数动词的现在时可以通过在词干后加上以下词尾来识别：",[13837,313925,313926,313938],{},[13840,313927,313928],{},[13843,313929,313930,313933,313935],{},[13846,313931,313932],{"align":82355},"德语",[13846,313934,99794],{"align":82355},[13846,313936,313937],{"align":82355},"中文",[13853,313939,313940,313953,313966,313979,313992,314004,314017],{},[13843,313941,313942,313947,313950],{},[13858,313943,313944],{"align":82355},[2627,313945,313946],{},"ich singe",[13858,313948,313949],{"align":82355},"I am singing, I sing, I do sing",[13858,313951,313952],{"align":82355},"我唱\u002F正在唱",[13843,313954,313955,313960,313963],{},[13858,313956,313957],{"align":82355},[2627,313958,313959],{},"du singst",[13858,313961,313962],{"align":82355},"you are singing, you sing, you do sing",[13858,313964,313965],{"align":82355},"你唱\u002F正在唱",[13843,313967,313968,313973,313976],{},[13858,313969,313970],{"align":82355},[2627,313971,313972],{},"er (sie, es) singt",[13858,313974,313975],{"align":82355},"he (she, it) is singing, sings, does sing",[13858,313977,313978],{"align":82355},"他（她、它）唱\u002F正在唱",[13843,313980,313981,313986,313989],{},[13858,313982,313983],{"align":82355},[2627,313984,313985],{},"wir singen",[13858,313987,313988],{"align":82355},"we are singing, we sing, we do sing",[13858,313990,313991],{"align":82355},"我们唱\u002F正在唱",[13843,313993,313994,313999,314001],{},[13858,313995,313996],{"align":82355},[2627,313997,313998],{},"ihr singt",[13858,314000,313962],{"align":82355},[13858,314002,314003],{"align":82355},"你们唱\u002F正在唱",[13843,314005,314006,314011,314014],{},[13858,314007,314008],{"align":82355},[2627,314009,314010],{},"sie singen",[13858,314012,314013],{"align":82355},"they are singing, they sing, they do sing",[13858,314015,314016],{"align":82355},"他们唱\u002F正在唱",[13843,314018,314019,314024,314026],{},[13858,314020,314021],{"align":82355},[2627,314022,314023],{},"Sie singen",[13858,314025,313962],{"align":82355},[13858,314027,314028],{"align":82355},"您（们）唱\u002F正在唱",[15,314030,314031],{},[71807,314032,314033],{},"注：德语的一种动词形式可以表达英语现在时的三种含义。德语中没有进行时（I am singing）或强调式（I do sing）。",[15,314035,314036],{},[2627,314037,314038],{},"3. 大写的 Sie 只能翻译为“您\u002F你们”，除非它位于句首。",[105,314040,314041,314051],{},[108,314042,314043,314044,314047,314048,314050],{},"要区分 ",[2627,314045,314046],{},"sie","（她）和 ",[2627,314049,314046],{},"（他们），必须查看动词词尾。",[108,314052,314053,314054,314057,314058,314060,314061,314063],{},"要区分句首的 ",[2627,314055,314056],{},"Sie","（您\u002F你们）、",[2627,314059,314046],{},"（他们）和 ",[2627,314062,314046],{},"（她），必须查看动词词尾或上下文。",[15,314065,314066],{},[2627,314067,314068],{},"4. Du 和 ihr 用于称呼家人、亲密朋友、儿童、动物和神祇。",[105,314070,314071,314076],{},[108,314072,314073,314074,347],{},"对于礼貌和正式的关系，使用 ",[2627,314075,314056],{},[108,314077,314078,314079,258,314081,314083],{},"请注意，",[2627,314080,34599],{},[2627,314082,314046],{}," 既可以指人也可以指物。",[15,314085,314086],{},[2627,314087,314088],{},"5. sein（to be 是）、haben（to have 有）和 werden（to become, to get; will 变成；将要）的变位是不规则的：",[13837,314090,314091,314107],{},[13840,314092,314093],{},[13843,314094,314095,314098,314101,314104],{},[13846,314096,314097],{"align":82355},"人称",[13846,314099,314100],{"align":82355},"sein (be)",[13846,314102,314103],{"align":82355},"haben (have)",[13846,314105,314106],{"align":82355},"werden (become\u002Fwill)",[13853,314108,314109,314125,314141,314157,314173,314189,314201],{},[13843,314110,314111,314116,314119,314122],{},[13858,314112,314113],{"align":82355},[2627,314114,314115],{},"ich",[13858,314117,314118],{"align":82355},"bin",[13858,314120,314121],{"align":82355},"habe",[13858,314123,314124],{"align":82355},"werde",[13843,314126,314127,314132,314135,314138],{},[13858,314128,314129],{"align":82355},[2627,314130,314131],{},"du",[13858,314133,314134],{"align":82355},"bist",[13858,314136,314137],{"align":82355},"hast",[13858,314139,314140],{"align":82355},"wirst",[13843,314142,314143,314148,314151,314154],{},[13858,314144,314145],{"align":82355},[2627,314146,314147],{},"er\u002Fsie\u002Fes",[13858,314149,314150],{"align":82355},"ist",[13858,314152,314153],{"align":82355},"hat",[13858,314155,314156],{"align":82355},"wird",[13843,314158,314159,314164,314167,314170],{},[13858,314160,314161],{"align":82355},[2627,314162,314163],{},"wir",[13858,314165,314166],{"align":82355},"sind",[13858,314168,314169],{"align":82355},"haben",[13858,314171,314172],{"align":82355},"werden",[13843,314174,314175,314180,314183,314186],{},[13858,314176,314177],{"align":82355},[2627,314178,314179],{},"ihr",[13858,314181,314182],{"align":82355},"seid",[13858,314184,314185],{"align":82355},"habt",[13858,314187,314188],{"align":82355},"werdet",[13843,314190,314191,314195,314197,314199],{},[13858,314192,314193],{"align":82355},[2627,314194,314046],{},[13858,314196,314166],{"align":82355},[13858,314198,314169],{"align":82355},[13858,314200,314172],{"align":82355},[13843,314202,314203,314207,314209,314211],{},[13858,314204,314205],{"align":82355},[2627,314206,314056],{},[13858,314208,314166],{"align":82355},[13858,314210,314169],{"align":82355},[13858,314212,314172],{"align":82355},[15,314214,314215],{},[71807,314216,314217,314218,314220],{},"注：",[2627,314219,314172],{}," 只有在与不定式连用构成将来时，才表示“将要（will\u002Fshall）”。",[32188,314222],{},[253,314224,314225],{"id":313222},[2627,314226,313225],{},[15,314228,314229],{},"请选择正确的翻译或代词：",[2230,314231,314232,314238,314247,314256,314261,314266,314272,314277],{},[108,314233,314234,314237],{},[2627,314235,314236],{},"er singt"," — 他 (he)",[108,314239,314240,314243,314244],{},[2627,314241,314242],{},"sie singt"," — 她 (she) — ",[71807,314245,314246],{},"动词词尾 -t 告诉我们 sie 是第三人称单数",[108,314248,314249,314252,314253],{},[2627,314250,314251],{},"Singen Sie?"," — 您唱吗？ (you) — ",[71807,314254,314255],{},"大写的 S 告诉你这是“您”",[108,314257,314258,314260],{},[2627,314259,313946],{}," — 我 (I)",[108,314262,314263,314265],{},[2627,314264,313998],{}," — 你们 (you)",[108,314267,314268,314271],{},[2627,314269,314270],{},"es singt"," — 它 (it)",[108,314273,314274,314276],{},[2627,314275,313959],{}," — 你 (you)",[108,314278,314279,314281,314282],{},[2627,314280,314010],{}," — 他们 (they) — ",[71807,314283,314284],{},"动词词尾 -en 告诉我们 sie 是第三人称复数",[15,314286,314287],{},[2627,314288,314289],{},"根据上下文选择最佳翻译：",[2230,314291,314292,314298,314304,314310,314316,314322,314328,314334],{"start":5576},[108,314293,314294,314297],{},[2627,314295,314296],{},"Das Wasser kocht schon."," — 水已经开了。(is boiling)",[108,314299,314300,314303],{},[2627,314301,314302],{},"Sie geht schon nach Hause."," — 她已经回家了。(is going)",[108,314305,314306,314309],{},[2627,314307,314308],{},"Heute fliegt sie wieder nach Italien."," — 今天她又飞往意大利了。(is flying)",[108,314311,314312,314315],{},[2627,314313,314314],{},"Lernt er auch Deutsch?"," — 他也在学德语吗？(is learning)",[108,314317,314318,314321],{},[2627,314319,314320],{},"Wir gehen oft ins Theater."," — 我们经常去剧院。(go)",[108,314323,314324,314327],{},[2627,314325,314326],{},"Paul tanzt nur mit Margarete."," — 保罗只和玛格丽特跳舞。(dances)",[108,314329,314330,314333],{},[2627,314331,314332],{},"Wo wohnen Sie?"," — 您住在哪儿？(Where do you live?)",[108,314335,314336,314339],{},[2627,314337,314338],{},"Hat er nur heute Zeit zu studieren?"," — 他只有今天有时间学习吗？(Does he only have time...?)",[15,314341,314342],{},[2627,314343,314344],{},"阶段测验 (PROGRESS TEST)：",[2230,314346,314347,314355,314363,314371,314378],{},[108,314348,314349,314352,314353],{},[2627,314350,314351],{},"findet"," — (d) ",[2627,314354,314179],{},[108,314356,314357,314360,314361],{},[2627,314358,314359],{},"lernst"," — (a) ",[2627,314362,314131],{},[108,314364,314365,314367,314368],{},[2627,314366,314115],{}," ___ — (b) ",[2627,314369,314370],{},"koche",[108,314372,314373,314375,314376],{},[2627,314374,34599],{}," ___ — (c) ",[2627,314377,314156],{},[108,314379,314380,314382,314383],{},[2627,314381,314179],{}," ___ — (d) ",[2627,314384,314182],{},{"title":29,"searchDepth":43,"depth":43,"links":314386},[314387],{"id":313222,"depth":49,"text":313225},{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.9",{"title":313895,"description":313905},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap2\u002F2.9","nW9eez9CyiVFJdIyIhQVVJU1MWXGAqjV8XDLl4GM_VI",{"id":314394,"title":314395,"body":314396,"description":29,"extension":85,"meta":314494,"navigation":52,"path":314495,"seo":314496,"stem":314497,"__hash__":314498},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap3\u002F3.11.md","3.11 語序——德語句子的基本结构",{"type":8,"value":314397,"toc":314492},[314398,314401],[11,314399,314395],{"id":314400},"_311-語序德語句子的基本结构",[2230,314402,314403,314419],{},[108,314404,314405,314408],{},[2627,314406,314407],{},"如果知道如何定位动词，阅读德语就会容易不少。一般而言，动词的人称形式（限定动词）在德语中会出现在以下三個位置之一：",[105,314409,314410,314413,314416],{},[108,314411,314412],{},"(a) 緊跟在主詞之後（正常語序 \u002F Normal word order）",[108,314414,314415],{},"(b) 緊接在主詞之前（倒裝語序 \u002F Inverted word order）",[108,314417,314418],{},"(c) 位於句子或從句的末尾（從屬語序 \u002F Dependent word order）",[108,314420,314421,314424],{},[2627,314422,314423],{},"這三種德語語序的基本形式如下：",[105,314425,314426,314440,314466],{},[108,314427,314428,314431,314432],{},[2627,314429,314430],{},"(a) 正常語序 (Normal word order)：","\n主詞及其修飾語 + 動詞 + 限定動詞的修飾語",[105,314433,314434],{},[108,314435,314436,314439],{},[2627,314437,314438],{},"Der alte Mann \u002F geht \u002F langsam in das Haus.","\n（The old man \u002F 慢慢地走 \u002F 進房子裡。）",[108,314441,314442,314445,314446],{},[2627,314443,314444],{},"(b) 倒裝語序 (Inverted word order)：","\n除了連詞或感嘆詞以外的任何單詞 + 動詞 + 主詞及其修飾語 + 其他修飾語",[105,314447,314448,314454,314460],{},[108,314449,314450,314453],{},[2627,314451,314452],{},"Wann \u002F kommt \u002F Karl \u002F nach Hause?","\nWhen is Karl coming home?（卡尔什么时候回家？）",[108,314455,314456,314459],{},[2627,314457,314458],{},"Gehen \u002F Sie \u002F bitte nach Hause!","\nPlease go home!（请回家！）",[108,314461,314462,314465],{},[2627,314463,314464],{},"Den Hund \u002F sehe \u002F ich \u002F auf der Straße.","\n（那隻狗 \u002F 我看見 \u002F 在街道上。\u002F 我在街上看見那隻狗。）",[108,314467,314468,314471,314472],{},[2627,314469,314470],{},"(c) 從屬語序 (Dependent word order)：","\n從屬連詞 + 主詞及其修飾語 + 其他修飾語 + 動詞",[105,314473,314474,314480,314486,314489],{},[108,314475,314476,314479],{},[2627,314477,314478],{},"... weil \u002F Karl \u002F nach Hause \u002F kommt.","\n（……因為 \u002F 卡爾 \u002F 回家。）",[108,314481,314482,314485],{},[2627,314483,314484],{},"... wenn \u002F Sie \u002F nach Hause \u002F gehen.","\n（……如果 \u002F 您 \u002F 回家。）",[108,314487,314488],{},"**... dass \u002F ich \u002F den",[108,314490,314491],{},"Hund \u002F auf der Straße \u002F sehe.**\n（……那隻狗 \u002F 我看見 \u002F 在街道上。\u002F ……我在街上看見那隻狗。）",{"title":29,"searchDepth":43,"depth":43,"links":314493},[],{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap3\u002F3.11",{"title":314395,"description":29},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap3\u002F3.11","VaWQLSmtB969eDPK9BIqGeZr7dSKkDqWWq5EDqj07R8",{"id":314500,"title":314501,"body":314502,"description":29,"extension":85,"meta":314509,"navigation":52,"path":314510,"seo":314511,"stem":314512,"__hash__":314513},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap3\u002F3.12.md","将来时 - werden",{"type":8,"value":314503,"toc":314507},[314504],[11,314505,314501],{"id":314506},"将来时-werden",{"title":29,"searchDepth":43,"depth":43,"links":314508},[],{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap3\u002F3.12",{"title":314501,"description":29},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap3\u002F3.12","XAAf5wW33B_OJAPdahrOhGg6xSyO2Fm76QtG0i_46cw",{"id":314515,"title":229955,"body":314516,"description":29,"extension":85,"meta":314520,"navigation":52,"path":314521,"seo":314522,"stem":314523,"__hash__":314524},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap4\u002F4.15.md",{"type":8,"value":314517,"toc":314518},[],{"title":29,"searchDepth":43,"depth":43,"links":314519},[],{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap4\u002F4.15",{"description":29},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Fchap4\u002F4.15","bNZiHDlPK0kS-_FY3fYEKVf7ueTTXUdvvqK3fN8NgRU",{"id":314526,"title":314527,"body":314528,"description":29,"extension":85,"meta":314543,"navigation":52,"path":314544,"seo":314545,"stem":314546,"__hash__":314547},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Findex.md","German for Reading",{"type":8,"value":314529,"toc":314540},[314530,314533,314537],[11,314531,314527],{"id":314532},"german-for-reading",[19,314534,314536],{"id":314535},"textbook","Textbook",[15,314538,314539],{},"German for Reading by Karl C. Sandberg & John R. Wendel",{"title":29,"searchDepth":43,"depth":43,"links":314541},[314542],{"id":314535,"depth":43,"text":314536},{},"\u002Fsource\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading",{"title":314527,"description":29},"source\u002Flanguage\u002Fdeutsch\u002Fgerman-for-reading\u002Findex","Ke80V_LS99xUaB7BIhpI8leyUe_ICrV7g7Vau-nmNGs",{"id":314549,"title":314550,"body":314551,"description":314558,"extension":85,"meta":314572,"navigation":52,"path":314574,"seo":314575,"stem":314576,"__hash__":314577},"notes\u002Fsource\u002Flanguage\u002Fdeutsch\u002Findex.md","Deutsch（德語）",{"type":8,"value":314552,"toc":314569},[314553,314556,314559,314562],[11,314554,314550],{"id":314555},"deutsch德語",[15,314557,314558],{},"德语学习笔记。",[19,314560,314561],{"id":314561},"教材",[105,314563,314564],{},[108,314565,314566],{},[111,314567,314527],{"href":314568},".\u002Fgerman-for-reading\u002F",{"title":29,"searchDepth":43,"depth":43,"links":314570},[314571],{"id":314561,"depth":43,"text":314561},{"order":314573},0,"\u002Fsource\u002Flanguage\u002Fdeutsch",{"title":314550,"description":314558},"source\u002Flanguage\u002Fdeutsch\u002Findex","rqgJentFKTWu3EfgmkQ5Xu2jDLw3HHaYFzIw1qphYsE",{"id":314579,"title":314580,"body":314581,"description":314588,"extension":85,"meta":314601,"navigation":52,"path":314602,"seo":314603,"stem":314604,"__hash__":314605},"notes\u002Fsource\u002Flanguage\u002Findex.md","語言 · Language",{"type":8,"value":314582,"toc":314598},[314583,314586,314589,314591],[11,314584,314580],{"id":314585},"語言-language",[15,314587,314588],{},"语言学习笔记汇总。",[19,314590,103],{"id":103},[105,314592,314593],{},[108,314594,314595],{},[111,314596,314550],{"href":314597},".\u002Fdeutsch\u002F",{"title":29,"searchDepth":43,"depth":43,"links":314599},[314600],{"id":103,"depth":43,"text":103},{"order":314573},"\u002Fsource\u002Flanguage",{"title":314580,"description":314588},"source\u002Flanguage\u002Findex","wRgcXA3Dp3ELd15BLaUo58fXNIyw4Mb5bRRIdj6YRO8",{"id":314607,"title":314608,"body":314609,"description":29,"extension":85,"meta":330674,"navigation":52,"path":330675,"seo":330676,"stem":330677,"__hash__":330678},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fgroupproblem.md","特殊类型的群 问题详解",{"type":8,"value":314610,"toc":330668},[314611,314614,314616,314620,314624,314986,314991,318389,318391,318395,318455,318459,320990,320992,321136,321334,321407,321630,321780,322512,322585,322718,322948,322950,322954,322958,323289,323294,325425,325427,325552,325695,325700,328955,328957,328961,329082,329086],[11,314612,314608],{"id":314613},"特殊类型的群-问题详解",[32188,314615],{},[253,314617,314619],{"id":314618},"第一部分循环群与生成元基础篇","第一部分：循环群与生成元（基础篇）",[32490,314621,314623],{"id":314622},"题目-2方程解的唯一性","题目 2：方程解的唯一性",[15,314625,314626,314628,314629,314657,314658,314686,314687,314748,314749,314777,314778,314863,314864,314945,314946,314985],{},[2627,314627,101191],{}," 如果群 ",[34,314630,314632,314645],{"className":314631},[31642],[34,314633,314635],{"className":314634},[31646],[31648,314636,314637],{"xmlns":31650},[31652,314638,314639,314643],{},[31655,314640,314641],{},[31658,314642,97709],{},[31665,314644,97709],{"encoding":31667},[34,314646,314648],{"className":314647,"ariaHidden":758},[31672],[34,314649,314651,314654],{"className":314650},[31676],[34,314652],{"className":314653,"style":33218},[31680],[34,314655,97709],{"className":314656},[31685,31686]," 中的元素 ",[34,314659,314661,314674],{"className":314660},[31642],[34,314662,314664],{"className":314663},[31646],[31648,314665,314666],{"xmlns":31650},[31652,314667,314668,314672],{},[31655,314669,314670],{},[31658,314671,31942],{},[31665,314673,31942],{"encoding":31667},[34,314675,314677],{"className":314676,"ariaHidden":758},[31672],[34,314678,314680,314683],{"className":314679},[31676],[34,314681],{"className":314682,"style":38774},[31680],[34,314684,31942],{"className":314685,"style":32062},[31685,31686]," 的阶 ",[34,314688,314690,314712],{"className":314689},[31642],[34,314691,314693],{"className":314692},[31646],[31648,314694,314695],{"xmlns":31650},[31652,314696,314697,314709],{},[31655,314698,314699,314701,314703,314705,314707],{},[31658,314700,162086],{"mathvariant":38079},[31658,314702,31942],{},[31658,314704,162086],{"mathvariant":38079},[31662,314706,202],{},[31658,314708,31660],{},[31665,314710,314711],{"encoding":31667},"|g| = n",[34,314713,314715,314739],{"className":314714,"ariaHidden":758},[31672],[34,314716,314718,314721,314724,314727,314730,314733,314736],{"className":314717},[31676],[34,314719],{"className":314720,"style":31774},[31680],[34,314722,162086],{"className":314723},[31685],[34,314725,31942],{"className":314726,"style":32062},[31685,31686],[34,314728,162086],{"className":314729},[31685],[34,314731],{"className":314732,"style":31846},[31845],[34,314734,202],{"className":314735},[31850],[34,314737],{"className":314738,"style":31846},[31845],[34,314740,314742,314745],{"className":314741},[31676],[34,314743],{"className":314744,"style":32527},[31680],[34,314746,31660],{"className":314747},[31685,31686]," 与正整数 ",[34,314750,314752,314765],{"className":314751},[31642],[34,314753,314755],{"className":314754},[31646],[31648,314756,314757],{"xmlns":31650},[31652,314758,314759,314763],{},[31655,314760,314761],{},[31658,314762,33759],{},[31665,314764,33759],{"encoding":31667},[34,314766,314768],{"className":314767,"ariaHidden":758},[31672],[34,314769,314771,314774],{"className":314770},[31676],[34,314772],{"className":314773,"style":31681},[31680],[34,314775,33759],{"className":314776,"style":33774},[31685,31686]," 互素（即 ",[34,314779,314781,314812],{"className":314780},[31642],[34,314782,314784],{"className":314783},[31646],[31648,314785,314786],{"xmlns":31650},[31652,314787,314788,314809],{},[31655,314789,314790,314793,314795,314797,314799,314801,314803,314805,314807],{},[31658,314791,314792],{},"gcd",[31662,314794,32682],{},[31662,314796,456],{"stretchy":702},[31658,314798,31660],{},[31662,314800,31960],{"separator":758},[31658,314802,33759],{},[31662,314804,2557],{"stretchy":702},[31662,314806,202],{},[31749,314808,640],{},[31665,314810,314811],{"encoding":31667},"\\gcd(n, k) = 1",[34,314813,314815,314854],{"className":314814,"ariaHidden":758},[31672],[34,314816,314818,314821,314827,314830,314833,314836,314839,314842,314845,314848,314851],{"className":314817},[31676],[34,314819],{"className":314820,"style":31774},[31680],[34,314822,314824,314826],{"className":314823},[32767],[34,314825,31942],{"style":38619},"cd",[34,314828,456],{"className":314829},[31782],[34,314831,31660],{"className":314832},[31685,31686],[34,314834,31960],{"className":314835},[32119],[34,314837],{"className":314838,"style":32123},[31845],[34,314840,33759],{"className":314841,"style":33774},[31685,31686],[34,314843,2557],{"className":314844},[31690],[34,314846],{"className":314847,"style":31846},[31845],[34,314849,202],{"className":314850},[31850],[34,314852],{"className":314853,"style":31846},[31845],[34,314855,314857,314860],{"className":314856},[31676],[34,314858],{"className":314859,"style":32617},[31680],[34,314861,640],{"className":314862},[31685],"），证明方程 ",[34,314865,314867,314889],{"className":314866},[31642],[34,314868,314870],{"className":314869},[31646],[31648,314871,314872],{"xmlns":31650},[31652,314873,314874,314886],{},[31655,314875,314876,314882,314884],{},[37624,314877,314878,314880],{},[31658,314879,32936],{},[31658,314881,33759],{},[31662,314883,202],{},[31658,314885,31942],{},[31665,314887,314888],{"encoding":31667},"x^k = g",[34,314890,314892,314936],{"className":314891,"ariaHidden":758},[31672],[34,314893,314895,314898,314927,314930,314933],{"className":314894},[31676],[34,314896],{"className":314897,"style":38486},[31680],[34,314899,314901,314904],{"className":314900},[31685],[34,314902,32936],{"className":314903},[31685,31686],[34,314905,314907],{"className":314906},[31792],[34,314908,314910],{"className":314909},[31796],[34,314911,314913],{"className":314912},[31801],[34,314914,314916],{"className":314915,"style":38486},[31805],[34,314917,314918,314921],{"style":37777},[34,314919],{"className":314920,"style":31814},[31813],[34,314922,314924],{"className":314923},[31818,31819,31820,31821],[34,314925,33759],{"className":314926,"style":33774},[31685,31686,31821],[34,314928],{"className":314929,"style":31846},[31845],[34,314931,202],{"className":314932},[31850],[34,314934],{"className":314935,"style":31846},[31845],[34,314937,314939,314942],{"className":314938},[31676],[34,314940],{"className":314941,"style":38774},[31680],[34,314943,31942],{"className":314944,"style":32062},[31685,31686]," 在 ",[34,314947,314949,314967],{"className":314948},[31642],[34,314950,314952],{"className":314951},[31646],[31648,314953,314954],{"xmlns":31650},[31652,314955,314956,314964],{},[31655,314957,314958,314960,314962],{},[31662,314959,126529],{"stretchy":702},[31658,314961,31942],{},[31662,314963,126544],{"stretchy":702},[31665,314965,314966],{"encoding":31667},"\\langle g \\rangle",[34,314968,314970],{"className":314969,"ariaHidden":758},[31672],[34,314971,314973,314976,314979,314982],{"className":314972},[31676],[34,314974],{"className":314975,"style":31774},[31680],[34,314977,126529],{"className":314978},[31782],[34,314980,31942],{"className":314981,"style":32062},[31685,31686],[34,314983,126544],{"className":314984},[31690]," 内恰有一个解。",[15,314987,314988],{},[2627,314989,314990],{},"【从零解析】",[2230,314992,314993,315428],{},[108,314994,314995,2664,314998],{},[2627,314996,314997],{},"背景知识",[105,314999,315000,315239],{},[108,315001,315002,315040,315041,315069,315070,347],{},[34,315003,315005,315022],{"className":315004},[31642],[34,315006,315008],{"className":315007},[31646],[31648,315009,315010],{"xmlns":31650},[31652,315011,315012,315020],{},[31655,315013,315014,315016,315018],{},[31662,315015,126529],{"stretchy":702},[31658,315017,31942],{},[31662,315019,126544],{"stretchy":702},[31665,315021,314966],{"encoding":31667},[34,315023,315025],{"className":315024,"ariaHidden":758},[31672],[34,315026,315028,315031,315034,315037],{"className":315027},[31676],[34,315029],{"className":315030,"style":31774},[31680],[34,315032,126529],{"className":315033},[31782],[34,315035,31942],{"className":315036,"style":32062},[31685,31686],[34,315038,126544],{"className":315039},[31690]," 是由 ",[34,315042,315044,315057],{"className":315043},[31642],[34,315045,315047],{"className":315046},[31646],[31648,315048,315049],{"xmlns":31650},[31652,315050,315051,315055],{},[31655,315052,315053],{},[31658,315054,31942],{},[31665,315056,31942],{"encoding":31667},[34,315058,315060],{"className":315059,"ariaHidden":758},[31672],[34,315061,315063,315066],{"className":315062},[31676],[34,315064],{"className":315065,"style":38774},[31680],[34,315067,31942],{"className":315068,"style":32062},[31685,31686]," 生成的循环群，里面的元素长这样：",[34,315071,315073,315121],{"className":315072},[31642],[34,315074,315076],{"className":315075},[31646],[31648,315077,315078],{"xmlns":31650},[31652,315079,315080,315118],{},[31655,315081,315082,315084,315086,315088,315090,315092,315098,315100,315102,315104,315116],{},[31662,315083,32685],{"stretchy":702},[31658,315085,33244],{},[31662,315087,31960],{"separator":758},[31658,315089,31942],{},[31662,315091,31960],{"separator":758},[37624,315093,315094,315096],{},[31658,315095,31942],{},[31749,315097,768],{},[31662,315099,31960],{"separator":758},[31662,315101,33934],{},[31662,315103,31960],{"separator":758},[37624,315105,315106,315108],{},[31658,315107,31942],{},[31655,315109,315110,315112,315114],{},[31658,315111,31660],{},[31662,315113,32226],{},[31749,315115,640],{},[31662,315117,32715],{"stretchy":702},[31665,315119,315120],{"encoding":31667},"\\{e, g, g^2, \\dots, g^{n-1}\\}",[34,315122,315124],{"className":315123,"ariaHidden":758},[31672],[34,315125,315127,315130,315133,315136,315139,315142,315145,315148,315151,315180,315183,315186,315189,315192,315195,315198,315236],{"className":315126},[31676],[34,315128],{"className":315129,"style":37755},[31680],[34,315131,32685],{"className":315132},[31782],[34,315134,33244],{"className":315135},[31685,31686],[34,315137,31960],{"className":315138},[32119],[34,315140],{"className":315141,"style":32123},[31845],[34,315143,31942],{"className":315144,"style":32062},[31685,31686],[34,315146,31960],{"className":315147},[32119],[34,315149],{"className":315150,"style":32123},[31845],[34,315152,315154,315157],{"className":315153},[31685],[34,315155,31942],{"className":315156,"style":32062},[31685,31686],[34,315158,315160],{"className":315159},[31792],[34,315161,315163],{"className":315162},[31796],[34,315164,315166],{"className":315165},[31801],[34,315167,315169],{"className":315168,"style":37774},[31805],[34,315170,315171,315174],{"style":37777},[34,315172],{"className":315173,"style":31814},[31813],[34,315175,315177],{"className":315176},[31818,31819,31820,31821],[34,315178,768],{"className":315179},[31685,31821],[34,315181,31960],{"className":315182},[32119],[34,315184],{"className":315185,"style":32123},[31845],[34,315187,33934],{"className":315188},[34006],[34,315190],{"className":315191,"style":32123},[31845],[34,315193,31960],{"className":315194},[32119],[34,315196],{"className":315197,"style":32123},[31845],[34,315199,315201,315204],{"className":315200},[31685],[34,315202,31942],{"className":315203,"style":32062},[31685,31686],[34,315205,315207],{"className":315206},[31792],[34,315208,315210],{"className":315209},[31796],[34,315211,315213],{"className":315212},[31801],[34,315214,315216],{"className":315215,"style":37774},[31805],[34,315217,315218,315221],{"style":37777},[34,315219],{"className":315220,"style":31814},[31813],[34,315222,315224],{"className":315223},[31818,31819,31820,31821],[34,315225,315227,315230,315233],{"className":315226},[31685,31821],[34,315228,31660],{"className":315229},[31685,31686,31821],[34,315231,32226],{"className":315232},[32024,31821],[34,315234,640],{"className":315235},[31685,31821],[34,315237,32715],{"className":315238},[31690],[108,315240,315241,46324,315244,258,315272,315300,315301,315343,315344,315427],{},[2627,315242,315243],{},"互素的威力（Bézout 等式）",[34,315245,315247,315260],{"className":315246},[31642],[34,315248,315250],{"className":315249},[31646],[31648,315251,315252],{"xmlns":31650},[31652,315253,315254,315258],{},[31655,315255,315256],{},[31658,315257,31660],{},[31665,315259,31660],{"encoding":31667},[34,315261,315263],{"className":315262,"ariaHidden":758},[31672],[34,315264,315266,315269],{"className":315265},[31676],[34,315267],{"className":315268,"style":32527},[31680],[34,315270,31660],{"className":315271},[31685,31686],[34,315273,315275,315288],{"className":315274},[31642],[34,315276,315278],{"className":315277},[31646],[31648,315279,315280],{"xmlns":31650},[31652,315281,315282,315286],{},[31655,315283,315284],{},[31658,315285,33759],{},[31665,315287,33759],{"encoding":31667},[34,315289,315291],{"className":315290,"ariaHidden":758},[31672],[34,315292,315294,315297],{"className":315293},[31676],[34,315295],{"className":315296,"style":31681},[31680],[34,315298,33759],{"className":315299,"style":33774},[31685,31686]," 互素，那么一定存在整数 ",[34,315302,315304,315322],{"className":315303},[31642],[34,315305,315307],{"className":315306},[31646],[31648,315308,315309],{"xmlns":31650},[31652,315310,315311,315319],{},[31655,315312,315313,315315,315317],{},[31658,315314,94628],{},[31662,315316,31960],{"separator":758},[31658,315318,56782],{},[31665,315320,315321],{"encoding":31667},"u, v",[34,315323,315325],{"className":315324,"ariaHidden":758},[31672],[34,315326,315328,315331,315334,315337,315340],{"className":315327},[31676],[34,315329],{"className":315330,"style":38774},[31680],[34,315332,94628],{"className":315333},[31685,31686],[34,315335,31960],{"className":315336},[32119],[34,315338],{"className":315339,"style":32123},[31845],[34,315341,56782],{"className":315342,"style":32062},[31685,31686]," 使得 ",[34,315345,315347,315373],{"className":315346},[31642],[34,315348,315350],{"className":315349},[31646],[31648,315351,315352],{"xmlns":31650},[31652,315353,315354,315370],{},[31655,315355,315356,315358,315360,315362,315364,315366,315368],{},[31658,315357,94628],{},[31658,315359,31660],{},[31662,315361,223],{},[31658,315363,56782],{},[31658,315365,33759],{},[31662,315367,202],{},[31749,315369,640],{},[31665,315371,315372],{"encoding":31667},"un + vk = 1",[34,315374,315376,315397,315418],{"className":315375,"ariaHidden":758},[31672],[34,315377,315379,315382,315385,315388,315391,315394],{"className":315378},[31676],[34,315380],{"className":315381,"style":32597},[31680],[34,315383,94628],{"className":315384},[31685,31686],[34,315386,31660],{"className":315387},[31685,31686],[34,315389],{"className":315390,"style":32604},[31845],[34,315392,223],{"className":315393},[32024],[34,315395],{"className":315396,"style":32604},[31845],[34,315398,315400,315403,315406,315409,315412,315415],{"className":315399},[31676],[34,315401],{"className":315402,"style":31681},[31680],[34,315404,56782],{"className":315405,"style":32062},[31685,31686],[34,315407,33759],{"className":315408,"style":33774},[31685,31686],[34,315410],{"className":315411,"style":31846},[31845],[34,315413,202],{"className":315414},[31850],[34,315416],{"className":315417,"style":31846},[31845],[34,315419,315421,315424],{"className":315420},[31676],[34,315422],{"className":315423,"style":32617},[31680],[34,315425,640],{"className":315426},[31685],"。这在代数里是解题的神器。",[108,315429,315430,2664,315433],{},[2627,315431,315432],{},"详细证明步骤",[105,315434,315435,316689,318384],{},[108,315436,315437,315440,315441,315523,315524,315343,315565,315647,315648,315676,315677,315837,315838,315440,316021,11008,316049,316077,316078,316160,316161,316426,316427,316508,316509,316589,316590,316649,316650,316688],{},[2627,315438,315439],{},"第一步：构造一个解（存在性）。","\n因为 ",[34,315442,315444,315473],{"className":315443},[31642],[34,315445,315447],{"className":315446},[31646],[31648,315448,315449],{"xmlns":31650},[31652,315450,315451,315471],{},[31655,315452,315453,315455,315457,315459,315461,315463,315465,315467,315469],{},[31658,315454,314792],{},[31662,315456,32682],{},[31662,315458,456],{"stretchy":702},[31658,315460,31660],{},[31662,315462,31960],{"separator":758},[31658,315464,33759],{},[31662,315466,2557],{"stretchy":702},[31662,315468,202],{},[31749,315470,640],{},[31665,315472,314811],{"encoding":31667},[34,315474,315476,315514],{"className":315475,"ariaHidden":758},[31672],[34,315477,315479,315482,315487,315490,315493,315496,315499,315502,315505,315508,315511],{"className":315478},[31676],[34,315480],{"className":315481,"style":31774},[31680],[34,315483,315485,314826],{"className":315484},[32767],[34,315486,31942],{"style":38619},[34,315488,456],{"className":315489},[31782],[34,315491,31660],{"className":315492},[31685,31686],[34,315494,31960],{"className":315495},[32119],[34,315497],{"className":315498,"style":32123},[31845],[34,315500,33759],{"className":315501,"style":33774},[31685,31686],[34,315503,2557],{"className":315504},[31690],[34,315506],{"className":315507,"style":31846},[31845],[34,315509,202],{"className":315510},[31850],[34,315512],{"className":315513,"style":31846},[31845],[34,315515,315517,315520],{"className":315516},[31676],[34,315518],{"className":315519,"style":32617},[31680],[34,315521,640],{"className":315522},[31685],"，存在整数 ",[34,315525,315527,315544],{"className":315526},[31642],[34,315528,315530],{"className":315529},[31646],[31648,315531,315532],{"xmlns":31650},[31652,315533,315534,315542],{},[31655,315535,315536,315538,315540],{},[31658,315537,94628],{},[31662,315539,31960],{"separator":758},[31658,315541,56782],{},[31665,315543,315321],{"encoding":31667},[34,315545,315547],{"className":315546,"ariaHidden":758},[31672],[34,315548,315550,315553,315556,315559,315562],{"className":315549},[31676],[34,315551],{"className":315552,"style":38774},[31680],[34,315554,94628],{"className":315555},[31685,31686],[34,315557,31960],{"className":315558},[32119],[34,315560],{"className":315561,"style":32123},[31845],[34,315563,56782],{"className":315564,"style":32062},[31685,31686],[34,315566,315568,315593],{"className":315567},[31642],[34,315569,315571],{"className":315570},[31646],[31648,315572,315573],{"xmlns":31650},[31652,315574,315575,315591],{},[31655,315576,315577,315579,315581,315583,315585,315587,315589],{},[31658,315578,94628],{},[31658,315580,31660],{},[31662,315582,223],{},[31658,315584,56782],{},[31658,315586,33759],{},[31662,315588,202],{},[31749,315590,640],{},[31665,315592,315372],{"encoding":31667},[34,315594,315596,315617,315638],{"className":315595,"ariaHidden":758},[31672],[34,315597,315599,315602,315605,315608,315611,315614],{"className":315598},[31676],[34,315600],{"className":315601,"style":32597},[31680],[34,315603,94628],{"className":315604},[31685,31686],[34,315606,31660],{"className":315607},[31685,31686],[34,315609],{"className":315610,"style":32604},[31845],[34,315612,223],{"className":315613},[32024],[34,315615],{"className":315616,"style":32604},[31845],[34,315618,315620,315623,315626,315629,315632,315635],{"className":315619},[31676],[34,315621],{"className":315622,"style":31681},[31680],[34,315624,56782],{"className":315625,"style":32062},[31685,31686],[34,315627,33759],{"className":315628,"style":33774},[31685,31686],[34,315630],{"className":315631,"style":31846},[31845],[34,315633,202],{"className":315634},[31850],[34,315636],{"className":315637,"style":31846},[31845],[34,315639,315641,315644],{"className":315640},[31676],[34,315642],{"className":315643,"style":32617},[31680],[34,315645,640],{"className":315646},[31685],"。\n我们可以把 ",[34,315649,315651,315664],{"className":315650},[31642],[34,315652,315654],{"className":315653},[31646],[31648,315655,315656],{"xmlns":31650},[31652,315657,315658,315662],{},[31655,315659,315660],{},[31658,315661,31942],{},[31665,315663,31942],{"encoding":31667},[34,315665,315667],{"className":315666,"ariaHidden":758},[31672],[34,315668,315670,315673],{"className":315669},[31676],[34,315671],{"className":315672,"style":38774},[31680],[34,315674,31942],{"className":315675,"style":32062},[31685,31686]," 写成：\n",[34,315678,315680,315720],{"className":315679},[31642],[34,315681,315683],{"className":315682},[31646],[31648,315684,315685],{"xmlns":31650},[31652,315686,315687,315717],{},[31655,315688,315689,315691,315693,315699,315701],{},[31658,315690,31942],{},[31662,315692,202],{},[37624,315694,315695,315697],{},[31658,315696,31942],{},[31749,315698,640],{},[31662,315700,202],{},[37624,315702,315703,315705],{},[31658,315704,31942],{},[31655,315706,315707,315709,315711,315713,315715],{},[31658,315708,94628],{},[31658,315710,31660],{},[31662,315712,223],{},[31658,315714,56782],{},[31658,315716,33759],{},[31665,315718,315719],{"encoding":31667},"g = g^1 = g^{un + vk}",[34,315721,315723,315741,315786],{"className":315722,"ariaHidden":758},[31672],[34,315724,315726,315729,315732,315735,315738],{"className":315725},[31676],[34,315727],{"className":315728,"style":38774},[31680],[34,315730,31942],{"className":315731,"style":32062},[31685,31686],[34,315733],{"className":315734,"style":31846},[31845],[34,315736,202],{"className":315737},[31850],[34,315739],{"className":315740,"style":31846},[31845],[34,315742,315744,315748,315777,315780,315783],{"className":315743},[31676],[34,315745],{"className":315746,"style":315747},[31680],"height:1.0085em;vertical-align:-0.1944em;",[34,315749,315751,315754],{"className":315750},[31685],[34,315752,31942],{"className":315753,"style":32062},[31685,31686],[34,315755,315757],{"className":315756},[31792],[34,315758,315760],{"className":315759},[31796],[34,315761,315763],{"className":315762},[31801],[34,315764,315766],{"className":315765,"style":37774},[31805],[34,315767,315768,315771],{"style":37777},[34,315769],{"className":315770,"style":31814},[31813],[34,315772,315774],{"className":315773},[31818,31819,31820,31821],[34,315775,640],{"className":315776},[31685,31821],[34,315778],{"className":315779,"style":31846},[31845],[34,315781,202],{"className":315782},[31850],[34,315784],{"className":315785,"style":31846},[31845],[34,315787,315789,315793],{"className":315788},[31676],[34,315790],{"className":315791,"style":315792},[31680],"height:1.0435em;vertical-align:-0.1944em;",[34,315794,315796,315799],{"className":315795},[31685],[34,315797,31942],{"className":315798,"style":32062},[31685,31686],[34,315800,315802],{"className":315801},[31792],[34,315803,315805],{"className":315804},[31796],[34,315806,315808],{"className":315807},[31801],[34,315809,315811],{"className":315810,"style":38486},[31805],[34,315812,315813,315816],{"style":37777},[34,315814],{"className":315815,"style":31814},[31813],[34,315817,315819],{"className":315818},[31818,31819,31820,31821],[34,315820,315822,315825,315828,315831,315834],{"className":315821},[31685,31821],[34,315823,94628],{"className":315824},[31685,31686,31821],[34,315826,31660],{"className":315827},[31685,31686,31821],[34,315829,223],{"className":315830},[32024,31821],[34,315832,56782],{"className":315833,"style":32062},[31685,31686,31821],[34,315835,33759],{"className":315836,"style":33774},[31685,31686,31821],"\n利用指数法则拆开：\n",[34,315839,315841,315883],{"className":315840},[31642],[34,315842,315844],{"className":315843},[31646],[31648,315845,315846],{"xmlns":31650},[31652,315847,315848,315880],{},[31655,315849,315850,315852,315854,315856,315862,315868,315870],{},[31658,315851,31942],{},[31662,315853,202],{},[31662,315855,456],{"stretchy":702},[37624,315857,315858,315860],{},[31658,315859,31942],{},[31658,315861,31660],{},[37624,315863,315864,315866],{},[31662,315865,2557],{"stretchy":702},[31658,315867,94628],{},[31662,315869,71191],{},[37624,315871,315872,315874],{},[31658,315873,31942],{},[31655,315875,315876,315878],{},[31658,315877,56782],{},[31658,315879,33759],{},[31665,315881,315882],{"encoding":31667},"g = (g^n)^u \\cdot g^{vk}",[34,315884,315886,315904,315980],{"className":315885,"ariaHidden":758},[31672],[34,315887,315889,315892,315895,315898,315901],{"className":315888},[31676],[34,315890],{"className":315891,"style":38774},[31680],[34,315893,31942],{"className":315894,"style":32062},[31685,31686],[34,315896],{"className":315897,"style":31846},[31845],[34,315899,202],{"className":315900},[31850],[34,315902],{"className":315903,"style":31846},[31845],[34,315905,315907,315910,315913,315942,315971,315974,315977],{"className":315906},[31676],[34,315908],{"className":315909,"style":31774},[31680],[34,315911,456],{"className":315912},[31782],[34,315914,315916,315919],{"className":315915},[31685],[34,315917,31942],{"className":315918,"style":32062},[31685,31686],[34,315920,315922],{"className":315921},[31792],[34,315923,315925],{"className":315924},[31796],[34,315926,315928],{"className":315927},[31801],[34,315929,315931],{"className":315930,"style":37959},[31805],[34,315932,315933,315936],{"style":37777},[34,315934],{"className":315935,"style":31814},[31813],[34,315937,315939],{"className":315938},[31818,31819,31820,31821],[34,315940,31660],{"className":315941},[31685,31686,31821],[34,315943,315945,315948],{"className":315944},[31690],[34,315946,2557],{"className":315947},[31690],[34,315949,315951],{"className":315950},[31792],[34,315952,315954],{"className":315953},[31796],[34,315955,315957],{"className":315956},[31801],[34,315958,315960],{"className":315959,"style":37959},[31805],[34,315961,315962,315965],{"style":37777},[34,315963],{"className":315964,"style":31814},[31813],[34,315966,315968],{"className":315967},[31818,31819,31820,31821],[34,315969,94628],{"className":315970},[31685,31686,31821],[34,315972],{"className":315973,"style":32604},[31845],[34,315975,71191],{"className":315976},[32024],[34,315978],{"className":315979,"style":32604},[31845],[34,315981,315983,315986],{"className":315982},[31676],[34,315984],{"className":315985,"style":315792},[31680],[34,315987,315989,315992],{"className":315988},[31685],[34,315990,31942],{"className":315991,"style":32062},[31685,31686],[34,315993,315995],{"className":315994},[31792],[34,315996,315998],{"className":315997},[31796],[34,315999,316001],{"className":316000},[31801],[34,316002,316004],{"className":316003,"style":38486},[31805],[34,316005,316006,316009],{"style":37777},[34,316007],{"className":316008,"style":31814},[31813],[34,316010,316012],{"className":316011},[31818,31819,31820,31821],[34,316013,316015,316018],{"className":316014},[31685,31821],[34,316016,56782],{"className":316017,"style":32062},[31685,31686,31821],[34,316019,33759],{"className":316020,"style":33774},[31685,31686,31821],[34,316022,316024,316037],{"className":316023},[31642],[34,316025,316027],{"className":316026},[31646],[31648,316028,316029],{"xmlns":31650},[31652,316030,316031,316035],{},[31655,316032,316033],{},[31658,316034,31660],{},[31665,316036,31660],{"encoding":31667},[34,316038,316040],{"className":316039,"ariaHidden":758},[31672],[34,316041,316043,316046],{"className":316042},[31676],[34,316044],{"className":316045,"style":32527},[31680],[34,316047,31660],{"className":316048},[31685,31686],[34,316050,316052,316065],{"className":316051},[31642],[34,316053,316055],{"className":316054},[31646],[31648,316056,316057],{"xmlns":31650},[31652,316058,316059,316063],{},[31655,316060,316061],{},[31658,316062,31942],{},[31665,316064,31942],{"encoding":31667},[34,316066,316068],{"className":316067,"ariaHidden":758},[31672],[34,316069,316071,316074],{"className":316070},[31676],[34,316072],{"className":316073,"style":38774},[31680],[34,316075,31942],{"className":316076,"style":32062},[31685,31686]," 的阶，所以 ",[34,316079,316081,316103],{"className":316080},[31642],[34,316082,316084],{"className":316083},[31646],[31648,316085,316086],{"xmlns":31650},[31652,316087,316088,316100],{},[31655,316089,316090,316096,316098],{},[37624,316091,316092,316094],{},[31658,316093,31942],{},[31658,316095,31660],{},[31662,316097,202],{},[31658,316099,33244],{},[31665,316101,316102],{"encoding":31667},"g^n = e",[34,316104,316106,316151],{"className":316105,"ariaHidden":758},[31672],[34,316107,316109,316113,316142,316145,316148],{"className":316108},[31676],[34,316110],{"className":316111,"style":316112},[31680],"height:0.8588em;vertical-align:-0.1944em;",[34,316114,316116,316119],{"className":316115},[31685],[34,316117,31942],{"className":316118,"style":32062},[31685,31686],[34,316120,316122],{"className":316121},[31792],[34,316123,316125],{"className":316124},[31796],[34,316126,316128],{"className":316127},[31801],[34,316129,316131],{"className":316130,"style":37959},[31805],[34,316132,316133,316136],{"style":37777},[34,316134],{"className":316135,"style":31814},[31813],[34,316137,316139],{"className":316138},[31818,31819,31820,31821],[34,316140,31660],{"className":316141},[31685,31686,31821],[34,316143],{"className":316144,"style":31846},[31845],[34,316146,202],{"className":316147},[31850],[34,316149],{"className":316150,"style":31846},[31845],[34,316152,316154,316157],{"className":316153},[31676],[34,316155],{"className":316156,"style":32527},[31680],[34,316158,33244],{"className":316159},[31685,31686],"（单位元）。那么：\n",[34,316162,316164,316218],{"className":316163},[31642],[34,316165,316167],{"className":316166},[31646],[31648,316168,316169],{"xmlns":31650},[31652,316170,316171,316215],{},[31655,316172,316173,316175,316177,316183,316185,316187,316193,316199,316201,316203,316209],{},[31658,316174,31942],{},[31662,316176,202],{},[37624,316178,316179,316181],{},[31658,316180,33244],{},[31658,316182,94628],{},[31662,316184,71191],{},[31662,316186,456],{"stretchy":702},[37624,316188,316189,316191],{},[31658,316190,31942],{},[31658,316192,56782],{},[37624,316194,316195,316197],{},[31662,316196,2557],{"stretchy":702},[31658,316198,33759],{},[31662,316200,202],{},[31662,316202,456],{"stretchy":702},[37624,316204,316205,316207],{},[31658,316206,31942],{},[31658,316208,56782],{},[37624,316210,316211,316213],{},[31662,316212,2557],{"stretchy":702},[31658,316214,33759],{},[31665,316216,316217],{"encoding":31667},"g = e^u \\cdot (g^v)^k = (g^v)^k",[34,316219,316221,316239,316283,316359],{"className":316220,"ariaHidden":758},[31672],[34,316222,316224,316227,316230,316233,316236],{"className":316223},[31676],[34,316225],{"className":316226,"style":38774},[31680],[34,316228,31942],{"className":316229,"style":32062},[31685,31686],[34,316231],{"className":316232,"style":31846},[31845],[34,316234,202],{"className":316235},[31850],[34,316237],{"className":316238,"style":31846},[31845],[34,316240,316242,316245,316274,316277,316280],{"className":316241},[31676],[34,316243],{"className":316244,"style":37959},[31680],[34,316246,316248,316251],{"className":316247},[31685],[34,316249,33244],{"className":316250},[31685,31686],[34,316252,316254],{"className":316253},[31792],[34,316255,316257],{"className":316256},[31796],[34,316258,316260],{"className":316259},[31801],[34,316261,316263],{"className":316262,"style":37959},[31805],[34,316264,316265,316268],{"style":37777},[34,316266],{"className":316267,"style":31814},[31813],[34,316269,316271],{"className":316270},[31818,31819,31820,31821],[34,316272,94628],{"className":316273},[31685,31686,31821],[34,316275],{"className":316276,"style":32604},[31845],[34,316278,71191],{"className":316279},[32024],[34,316281],{"className":316282,"style":32604},[31845],[34,316284,316286,316289,316292,316321,316350,316353,316356],{"className":316285},[31676],[34,316287],{"className":316288,"style":43651},[31680],[34,316290,456],{"className":316291},[31782],[34,316293,316295,316298],{"className":316294},[31685],[34,316296,31942],{"className":316297,"style":32062},[31685,31686],[34,316299,316301],{"className":316300},[31792],[34,316302,316304],{"className":316303},[31796],[34,316305,316307],{"className":316306},[31801],[34,316308,316310],{"className":316309,"style":37959},[31805],[34,316311,316312,316315],{"style":37777},[34,316313],{"className":316314,"style":31814},[31813],[34,316316,316318],{"className":316317},[31818,31819,31820,31821],[34,316319,56782],{"className":316320,"style":32062},[31685,31686,31821],[34,316322,316324,316327],{"className":316323},[31690],[34,316325,2557],{"className":316326},[31690],[34,316328,316330],{"className":316329},[31792],[34,316331,316333],{"className":316332},[31796],[34,316334,316336],{"className":316335},[31801],[34,316337,316339],{"className":316338,"style":38486},[31805],[34,316340,316341,316344],{"style":37777},[34,316342],{"className":316343,"style":31814},[31813],[34,316345,316347],{"className":316346},[31818,31819,31820,31821],[34,316348,33759],{"className":316349,"style":33774},[31685,31686,31821],[34,316351],{"className":316352,"style":31846},[31845],[34,316354,202],{"className":316355},[31850],[34,316357],{"className":316358,"style":31846},[31845],[34,316360,316362,316365,316368,316397],{"className":316361},[31676],[34,316363],{"className":316364,"style":43651},[31680],[34,316366,456],{"className":316367},[31782],[34,316369,316371,316374],{"className":316370},[31685],[34,316372,31942],{"className":316373,"style":32062},[31685,31686],[34,316375,316377],{"className":316376},[31792],[34,316378,316380],{"className":316379},[31796],[34,316381,316383],{"className":316382},[31801],[34,316384,316386],{"className":316385,"style":37959},[31805],[34,316387,316388,316391],{"style":37777},[34,316389],{"className":316390,"style":31814},[31813],[34,316392,316394],{"className":316393},[31818,31819,31820,31821],[34,316395,56782],{"className":316396,"style":32062},[31685,31686,31821],[34,316398,316400,316403],{"className":316399},[31690],[34,316401,2557],{"className":316402},[31690],[34,316404,316406],{"className":316405},[31792],[34,316407,316409],{"className":316408},[31796],[34,316410,316412],{"className":316411},[31801],[34,316413,316415],{"className":316414,"style":38486},[31805],[34,316416,316417,316420],{"style":37777},[34,316418],{"className":316419,"style":31814},[31813],[34,316421,316423],{"className":316422},[31818,31819,31820,31821],[34,316424,33759],{"className":316425,"style":33774},[31685,31686,31821],"\n我们找到了一个元素 ",[34,316428,316430,316452],{"className":316429},[31642],[34,316431,316433],{"className":316432},[31646],[31648,316434,316435],{"xmlns":31650},[31652,316436,316437,316449],{},[31655,316438,316439,316441,316443],{},[31658,316440,32936],{},[31662,316442,202],{},[37624,316444,316445,316447],{},[31658,316446,31942],{},[31658,316448,56782],{},[31665,316450,316451],{"encoding":31667},"x = g^v",[34,316453,316455,316473],{"className":316454,"ariaHidden":758},[31672],[34,316456,316458,316461,316464,316467,316470],{"className":316457},[31676],[34,316459],{"className":316460,"style":32527},[31680],[34,316462,32936],{"className":316463},[31685,31686],[34,316465],{"className":316466,"style":31846},[31845],[34,316468,202],{"className":316469},[31850],[34,316471],{"className":316472,"style":31846},[31845],[34,316474,316476,316479],{"className":316475},[31676],[34,316477],{"className":316478,"style":316112},[31680],[34,316480,316482,316485],{"className":316481},[31685],[34,316483,31942],{"className":316484,"style":32062},[31685,31686],[34,316486,316488],{"className":316487},[31792],[34,316489,316491],{"className":316490},[31796],[34,316492,316494],{"className":316493},[31801],[34,316495,316497],{"className":316496,"style":37959},[31805],[34,316498,316499,316502],{"style":37777},[34,316500],{"className":316501,"style":31814},[31813],[34,316503,316505],{"className":316504},[31818,31819,31820,31821],[34,316506,56782],{"className":316507,"style":32062},[31685,31686,31821],"，它满足 ",[34,316510,316512,316533],{"className":316511},[31642],[34,316513,316515],{"className":316514},[31646],[31648,316516,316517],{"xmlns":31650},[31652,316518,316519,316531],{},[31655,316520,316521,316527,316529],{},[37624,316522,316523,316525],{},[31658,316524,32936],{},[31658,316526,33759],{},[31662,316528,202],{},[31658,316530,31942],{},[31665,316532,314888],{"encoding":31667},[34,316534,316536,316580],{"className":316535,"ariaHidden":758},[31672],[34,316537,316539,316542,316571,316574,316577],{"className":316538},[31676],[34,316540],{"className":316541,"style":38486},[31680],[34,316543,316545,316548],{"className":316544},[31685],[34,316546,32936],{"className":316547},[31685,31686],[34,316549,316551],{"className":316550},[31792],[34,316552,316554],{"className":316553},[31796],[34,316555,316557],{"className":316556},[31801],[34,316558,316560],{"className":316559,"style":38486},[31805],[34,316561,316562,316565],{"style":37777},[34,316563],{"className":316564,"style":31814},[31813],[34,316566,316568],{"className":316567},[31818,31819,31820,31821],[34,316569,33759],{"className":316570,"style":33774},[31685,31686,31821],[34,316572],{"className":316573,"style":31846},[31845],[34,316575,202],{"className":316576},[31850],[34,316578],{"className":316579,"style":31846},[31845],[34,316581,316583,316586],{"className":316582},[31676],[34,316584],{"className":316585,"style":38774},[31680],[34,316587,31942],{"className":316588,"style":32062},[31685,31686],"。而且 ",[34,316591,316593,316611],{"className":316592},[31642],[34,316594,316596],{"className":316595},[31646],[31648,316597,316598],{"xmlns":31650},[31652,316599,316600,316608],{},[31655,316601,316602],{},[37624,316603,316604,316606],{},[31658,316605,31942],{},[31658,316607,56782],{},[31665,316609,316610],{"encoding":31667},"g^v",[34,316612,316614],{"className":316613,"ariaHidden":758},[31672],[34,316615,316617,316620],{"className":316616},[31676],[34,316618],{"className":316619,"style":316112},[31680],[34,316621,316623,316626],{"className":316622},[31685],[34,316624,31942],{"className":316625,"style":32062},[31685,31686],[34,316627,316629],{"className":316628},[31792],[34,316630,316632],{"className":316631},[31796],[34,316633,316635],{"className":316634},[31801],[34,316636,316638],{"className":316637,"style":37959},[31805],[34,316639,316640,316643],{"style":37777},[34,316641],{"className":316642,"style":31814},[31813],[34,316644,316646],{"className":316645},[31818,31819,31820,31821],[34,316647,56782],{"className":316648,"style":32062},[31685,31686,31821]," 显然在 ",[34,316651,316653,316670],{"className":316652},[31642],[34,316654,316656],{"className":316655},[31646],[31648,316657,316658],{"xmlns":31650},[31652,316659,316660,316668],{},[31655,316661,316662,316664,316666],{},[31662,316663,126529],{"stretchy":702},[31658,316665,31942],{},[31662,316667,126544],{"stretchy":702},[31665,316669,314966],{"encoding":31667},[34,316671,316673],{"className":316672,"ariaHidden":758},[31672],[34,316674,316676,316679,316682,316685],{"className":316675},[31676],[34,316677],{"className":316678,"style":31774},[31680],[34,316680,126529],{"className":316681},[31782],[34,316683,31942],{"className":316684,"style":32062},[31685,31686],[34,316686,126544],{"className":316687},[31690]," 里。所以解是存在的。",[108,316690,316691,316694,316695,258,316764,316833,316834,316872,316873,258,316980,317085,317086,317245,317246,317452,317453,317614,317615,317643,317644,316833,317768,317806,317807,317835,317836,317864,317865,258,317893,317921,317922,318004,318005,318033,318034,318085,318086,347],{},[2627,316692,316693],{},"第二步：证明只有一个解（唯一性）。","\n假设有两个解 ",[34,316696,316698,316715],{"className":316697},[31642],[34,316699,316701],{"className":316700},[31646],[31648,316702,316703],{"xmlns":31650},[31652,316704,316705,316713],{},[31655,316706,316707],{},[31743,316708,316709,316711],{},[31658,316710,32936],{},[31749,316712,640],{},[31665,316714,224014],{"encoding":31667},[34,316716,316718],{"className":316717,"ariaHidden":758},[31672],[34,316719,316721,316724],{"className":316720},[31676],[34,316722],{"className":316723,"style":31860},[31680],[34,316725,316727,316730],{"className":316726},[31685],[34,316728,32936],{"className":316729},[31685,31686],[34,316731,316733],{"className":316732},[31792],[34,316734,316736,316756],{"className":316735},[31796,31797],[34,316737,316739,316753],{"className":316738},[31801],[34,316740,316742],{"className":316741,"style":31806},[31805],[34,316743,316744,316747],{"style":31809},[34,316745],{"className":316746,"style":31814},[31813],[34,316748,316750],{"className":316749},[31818,31819,31820,31821],[34,316751,640],{"className":316752},[31685,31821],[34,316754,31829],{"className":316755},[31828],[34,316757,316759],{"className":316758},[31801],[34,316760,316762],{"className":316761,"style":31836},[31805],[34,316763],{},[34,316765,316767,316784],{"className":316766},[31642],[34,316768,316770],{"className":316769},[31646],[31648,316771,316772],{"xmlns":31650},[31652,316773,316774,316782],{},[31655,316775,316776],{},[31743,316777,316778,316780],{},[31658,316779,32936],{},[31749,316781,768],{},[31665,316783,224085],{"encoding":31667},[34,316785,316787],{"className":316786,"ariaHidden":758},[31672],[34,316788,316790,316793],{"className":316789},[31676],[34,316791],{"className":316792,"style":31860},[31680],[34,316794,316796,316799],{"className":316795},[31685],[34,316797,32936],{"className":316798},[31685,31686],[34,316800,316802],{"className":316801},[31792],[34,316803,316805,316825],{"className":316804},[31796,31797],[34,316806,316808,316822],{"className":316807},[31801],[34,316809,316811],{"className":316810,"style":31806},[31805],[34,316812,316813,316816],{"style":31809},[34,316814],{"className":316815,"style":31814},[31813],[34,316817,316819],{"className":316818},[31818,31819,31820,31821],[34,316820,768],{"className":316821},[31685,31821],[34,316823,31829],{"className":316824},[31828],[34,316826,316828],{"className":316827},[31801],[34,316829,316831],{"className":316830,"style":31836},[31805],[34,316832],{}," 都在 ",[34,316835,316837,316854],{"className":316836},[31642],[34,316838,316840],{"className":316839},[31646],[31648,316841,316842],{"xmlns":31650},[31652,316843,316844,316852],{},[31655,316845,316846,316848,316850],{},[31662,316847,126529],{"stretchy":702},[31658,316849,31942],{},[31662,316851,126544],{"stretchy":702},[31665,316853,314966],{"encoding":31667},[34,316855,316857],{"className":316856,"ariaHidden":758},[31672],[34,316858,316860,316863,316866,316869],{"className":316859},[31676],[34,316861],{"className":316862,"style":31774},[31680],[34,316864,126529],{"className":316865},[31782],[34,316867,31942],{"className":316868,"style":32062},[31685,31686],[34,316870,126544],{"className":316871},[31690]," 里，满足 ",[34,316874,316876,316900],{"className":316875},[31642],[34,316877,316879],{"className":316878},[31646],[31648,316880,316881],{"xmlns":31650},[31652,316882,316883,316897],{},[31655,316884,316885,316893,316895],{},[38846,316886,316887,316889,316891],{},[31658,316888,32936],{},[31749,316890,640],{},[31658,316892,33759],{},[31662,316894,202],{},[31658,316896,31942],{},[31665,316898,316899],{"encoding":31667},"x_1^k = g",[34,316901,316903,316971],{"className":316902,"ariaHidden":758},[31672],[34,316904,316906,316910,316962,316965,316968],{"className":316905},[31676],[34,316907],{"className":316908,"style":316909},[31680],"height:1.0972em;vertical-align:-0.2481em;",[34,316911,316913,316916],{"className":316912},[31685],[34,316914,32936],{"className":316915},[31685,31686],[34,316917,316919],{"className":316918},[31792],[34,316920,316922,316954],{"className":316921},[31796,31797],[34,316923,316925,316951],{"className":316924},[31801],[34,316926,316928,316940],{"className":316927,"style":38486},[31805],[34,316929,316931,316934],{"style":316930},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[34,316932],{"className":316933,"style":31814},[31813],[34,316935,316937],{"className":316936},[31818,31819,31820,31821],[34,316938,640],{"className":316939},[31685,31821],[34,316941,316942,316945],{"style":37777},[34,316943],{"className":316944,"style":31814},[31813],[34,316946,316948],{"className":316947},[31818,31819,31820,31821],[34,316949,33759],{"className":316950,"style":33774},[31685,31686,31821],[34,316952,31829],{"className":316953},[31828],[34,316955,316957],{"className":316956},[31801],[34,316958,316960],{"className":316959,"style":39021},[31805],[34,316961],{},[34,316963],{"className":316964,"style":31846},[31845],[34,316966,202],{"className":316967},[31850],[34,316969],{"className":316970,"style":31846},[31845],[34,316972,316974,316977],{"className":316973},[31676],[34,316975],{"className":316976,"style":38774},[31680],[34,316978,31942],{"className":316979,"style":32062},[31685,31686],[34,316981,316983,317007],{"className":316982},[31642],[34,316984,316986],{"className":316985},[31646],[31648,316987,316988],{"xmlns":31650},[31652,316989,316990,317004],{},[31655,316991,316992,317000,317002],{},[38846,316993,316994,316996,316998],{},[31658,316995,32936],{},[31749,316997,768],{},[31658,316999,33759],{},[31662,317001,202],{},[31658,317003,31942],{},[31665,317005,317006],{"encoding":31667},"x_2^k = g",[34,317008,317010,317076],{"className":317009,"ariaHidden":758},[31672],[34,317011,317013,317016,317067,317070,317073],{"className":317012},[31676],[34,317014],{"className":317015,"style":316909},[31680],[34,317017,317019,317022],{"className":317018},[31685],[34,317020,32936],{"className":317021},[31685,31686],[34,317023,317025],{"className":317024},[31792],[34,317026,317028,317059],{"className":317027},[31796,31797],[34,317029,317031,317056],{"className":317030},[31801],[34,317032,317034,317045],{"className":317033,"style":38486},[31805],[34,317035,317036,317039],{"style":316930},[34,317037],{"className":317038,"style":31814},[31813],[34,317040,317042],{"className":317041},[31818,31819,31820,31821],[34,317043,768],{"className":317044},[31685,31821],[34,317046,317047,317050],{"style":37777},[34,317048],{"className":317049,"style":31814},[31813],[34,317051,317053],{"className":317052},[31818,31819,31820,31821],[34,317054,33759],{"className":317055,"style":33774},[31685,31686,31821],[34,317057,31829],{"className":317058},[31828],[34,317060,317062],{"className":317061},[31801],[34,317063,317065],{"className":317064,"style":39021},[31805],[34,317066],{},[34,317068],{"className":317069,"style":31846},[31845],[34,317071,202],{"className":317072},[31850],[34,317074],{"className":317075,"style":31846},[31845],[34,317077,317079,317082],{"className":317078},[31676],[34,317080],{"className":317081,"style":38774},[31680],[34,317083,31942],{"className":317084,"style":32062},[31685,31686],"。\n那么 ",[34,317087,317089,317119],{"className":317088},[31642],[34,317090,317092],{"className":317091},[31646],[31648,317093,317094],{"xmlns":31650},[31652,317095,317096,317116],{},[31655,317097,317098,317106,317108],{},[38846,317099,317100,317102,317104],{},[31658,317101,32936],{},[31749,317103,640],{},[31658,317105,33759],{},[31662,317107,202],{},[38846,317109,317110,317112,317114],{},[31658,317111,32936],{},[31749,317113,768],{},[31658,317115,33759],{},[31665,317117,317118],{"encoding":31667},"x_1^k = x_2^k",[34,317120,317122,317188],{"className":317121,"ariaHidden":758},[31672],[34,317123,317125,317128,317179,317182,317185],{"className":317124},[31676],[34,317126],{"className":317127,"style":316909},[31680],[34,317129,317131,317134],{"className":317130},[31685],[34,317132,32936],{"className":317133},[31685,31686],[34,317135,317137],{"className":317136},[31792],[34,317138,317140,317171],{"className":317139},[31796,31797],[34,317141,317143,317168],{"className":317142},[31801],[34,317144,317146,317157],{"className":317145,"style":38486},[31805],[34,317147,317148,317151],{"style":316930},[34,317149],{"className":317150,"style":31814},[31813],[34,317152,317154],{"className":317153},[31818,31819,31820,31821],[34,317155,640],{"className":317156},[31685,31821],[34,317158,317159,317162],{"style":37777},[34,317160],{"className":317161,"style":31814},[31813],[34,317163,317165],{"className":317164},[31818,31819,31820,31821],[34,317166,33759],{"className":317167,"style":33774},[31685,31686,31821],[34,317169,31829],{"className":317170},[31828],[34,317172,317174],{"className":317173},[31801],[34,317175,317177],{"className":317176,"style":39021},[31805],[34,317178],{},[34,317180],{"className":317181,"style":31846},[31845],[34,317183,202],{"className":317184},[31850],[34,317186],{"className":317187,"style":31846},[31845],[34,317189,317191,317194],{"className":317190},[31676],[34,317192],{"className":317193,"style":316909},[31680],[34,317195,317197,317200],{"className":317196},[31685],[34,317198,32936],{"className":317199},[31685,31686],[34,317201,317203],{"className":317202},[31792],[34,317204,317206,317237],{"className":317205},[31796,31797],[34,317207,317209,317234],{"className":317208},[31801],[34,317210,317212,317223],{"className":317211,"style":38486},[31805],[34,317213,317214,317217],{"style":316930},[34,317215],{"className":317216,"style":31814},[31813],[34,317218,317220],{"className":317219},[31818,31819,31820,31821],[34,317221,768],{"className":317222},[31685,31821],[34,317224,317225,317228],{"style":37777},[34,317226],{"className":317227,"style":31814},[31813],[34,317229,317231],{"className":317230},[31818,31819,31820,31821],[34,317232,33759],{"className":317233,"style":33774},[31685,31686,31821],[34,317235,31829],{"className":317236},[31828],[34,317238,317240],{"className":317239},[31801],[34,317241,317243],{"className":317242,"style":39021},[31805],[34,317244],{},"，也就是 ",[34,317247,317249,317291],{"className":317248},[31642],[34,317250,317252],{"className":317251},[31646],[31648,317253,317254],{"xmlns":31650},[31652,317255,317256,317288],{},[31655,317257,317258,317260,317266,317278,317284,317286],{},[31662,317259,456],{"stretchy":702},[31743,317261,317262,317264],{},[31658,317263,32936],{},[31749,317265,640],{},[38846,317267,317268,317270,317272],{},[31658,317269,32936],{},[31749,317271,768],{},[31655,317273,317274,317276],{},[31662,317275,32226],{},[31749,317277,640],{},[37624,317279,317280,317282],{},[31662,317281,2557],{"stretchy":702},[31658,317283,33759],{},[31662,317285,202],{},[31658,317287,33244],{},[31665,317289,317290],{"encoding":31667},"(x_1 x_2^{-1})^k = e",[34,317292,317294,317443],{"className":317293,"ariaHidden":758},[31672],[34,317295,317297,317301,317304,317344,317405,317434,317437,317440],{"className":317296},[31676],[34,317298],{"className":317299,"style":317300},[31680],"height:1.1205em;vertical-align:-0.2663em;",[34,317302,456],{"className":317303},[31782],[34,317305,317307,317310],{"className":317306},[31685],[34,317308,32936],{"className":317309},[31685,31686],[34,317311,317313],{"className":317312},[31792],[34,317314,317316,317336],{"className":317315},[31796,31797],[34,317317,317319,317333],{"className":317318},[31801],[34,317320,317322],{"className":317321,"style":31806},[31805],[34,317323,317324,317327],{"style":31809},[34,317325],{"className":317326,"style":31814},[31813],[34,317328,317330],{"className":317329},[31818,31819,31820,31821],[34,317331,640],{"className":317332},[31685,31821],[34,317334,31829],{"className":317335},[31828],[34,317337,317339],{"className":317338},[31801],[34,317340,317342],{"className":317341,"style":31836},[31805],[34,317343],{},[34,317345,317347,317350],{"className":317346},[31685],[34,317348,32936],{"className":317349},[31685,31686],[34,317351,317353],{"className":317352},[31792],[34,317354,317356,317396],{"className":317355},[31796,31797],[34,317357,317359,317393],{"className":317358},[31801],[34,317360,317363,317375],{"className":317361,"style":317362},[31805],"height:0.8542em;",[34,317364,317366,317369],{"style":317365},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[34,317367],{"className":317368,"style":31814},[31813],[34,317370,317372],{"className":317371},[31818,31819,31820,31821],[34,317373,768],{"className":317374},[31685,31821],[34,317376,317378,317381],{"style":317377},"top:-3.1031em;margin-right:0.05em;",[34,317379],{"className":317380,"style":31814},[31813],[34,317382,317384],{"className":317383},[31818,31819,31820,31821],[34,317385,317387,317390],{"className":317386},[31685,31821],[34,317388,32226],{"className":317389},[31685,31821],[34,317391,640],{"className":317392},[31685,31821],[34,317394,31829],{"className":317395},[31828],[34,317397,317399],{"className":317398},[31801],[34,317400,317403],{"className":317401,"style":317402},[31805],"height:0.2663em;",[34,317404],{},[34,317406,317408,317411],{"className":317407},[31690],[34,317409,2557],{"className":317410},[31690],[34,317412,317414],{"className":317413},[31792],[34,317415,317417],{"className":317416},[31796],[34,317418,317420],{"className":317419},[31801],[34,317421,317423],{"className":317422,"style":38486},[31805],[34,317424,317425,317428],{"style":37777},[34,317426],{"className":317427,"style":31814},[31813],[34,317429,317431],{"className":317430},[31818,31819,31820,31821],[34,317432,33759],{"className":317433,"style":33774},[31685,31686,31821],[34,317435],{"className":317436,"style":31846},[31845],[34,317438,202],{"className":317439},[31850],[34,317441],{"className":317442,"style":31846},[31845],[34,317444,317446,317449],{"className":317445},[31676],[34,317447],{"className":317448,"style":32527},[31680],[34,317450,33244],{"className":317451},[31685,31686],"。\n这意味着元素 ",[34,317454,317456,317490],{"className":317455},[31642],[34,317457,317459],{"className":317458},[31646],[31648,317460,317461],{"xmlns":31650},[31652,317462,317463,317487],{},[31655,317464,317465,317467,317469,317475],{},[31658,317466,67764],{},[31662,317468,202],{},[31743,317470,317471,317473],{},[31658,317472,32936],{},[31749,317474,640],{},[38846,317476,317477,317479,317481],{},[31658,317478,32936],{},[31749,317480,768],{},[31655,317482,317483,317485],{},[31662,317484,32226],{},[31749,317486,640],{},[31665,317488,317489],{"encoding":31667},"y = x_1 x_2^{-1}",[34,317491,317493,317511],{"className":317492,"ariaHidden":758},[31672],[34,317494,317496,317499,317502,317505,317508],{"className":317495},[31676],[34,317497],{"className":317498,"style":38774},[31680],[34,317500,67764],{"className":317501,"style":32062},[31685,31686],[34,317503],{"className":317504,"style":31846},[31845],[34,317506,202],{"className":317507},[31850],[34,317509],{"className":317510,"style":31846},[31845],[34,317512,317514,317517,317557],{"className":317513},[31676],[34,317515],{"className":317516,"style":317300},[31680],[34,317518,317520,317523],{"className":317519},[31685],[34,317521,32936],{"className":317522},[31685,31686],[34,317524,317526],{"className":317525},[31792],[34,317527,317529,317549],{"className":317528},[31796,31797],[34,317530,317532,317546],{"className":317531},[31801],[34,317533,317535],{"className":317534,"style":31806},[31805],[34,317536,317537,317540],{"style":31809},[34,317538],{"className":317539,"style":31814},[31813],[34,317541,317543],{"className":317542},[31818,31819,31820,31821],[34,317544,640],{"className":317545},[31685,31821],[34,317547,31829],{"className":317548},[31828],[34,317550,317552],{"className":317551},[31801],[34,317553,317555],{"className":317554,"style":31836},[31805],[34,317556],{},[34,317558,317560,317563],{"className":317559},[31685],[34,317561,32936],{"className":317562},[31685,31686],[34,317564,317566],{"className":317565},[31792],[34,317567,317569,317606],{"className":317568},[31796,31797],[34,317570,317572,317603],{"className":317571},[31801],[34,317573,317575,317586],{"className":317574,"style":317362},[31805],[34,317576,317577,317580],{"style":317365},[34,317578],{"className":317579,"style":31814},[31813],[34,317581,317583],{"className":317582},[31818,31819,31820,31821],[34,317584,768],{"className":317585},[31685,31821],[34,317587,317588,317591],{"style":317377},[34,317589],{"className":317590,"style":31814},[31813],[34,317592,317594],{"className":317593},[31818,31819,31820,31821],[34,317595,317597,317600],{"className":317596},[31685,31821],[34,317598,32226],{"className":317599},[31685,31821],[34,317601,640],{"className":317602},[31685,31821],[34,317604,31829],{"className":317605},[31828],[34,317607,317609],{"className":317608},[31801],[34,317610,317612],{"className":317611,"style":317402},[31805],[34,317613],{}," 的阶必须整除 ",[34,317616,317618,317631],{"className":317617},[31642],[34,317619,317621],{"className":317620},[31646],[31648,317622,317623],{"xmlns":31650},[31652,317624,317625,317629],{},[31655,317626,317627],{},[31658,317628,33759],{},[31665,317630,33759],{"encoding":31667},[34,317632,317634],{"className":317633,"ariaHidden":758},[31672],[34,317635,317637,317640],{"className":317636},[31676],[34,317638],{"className":317639,"style":31681},[31680],[34,317641,33759],{"className":317642,"style":33774},[31685,31686],"。\n同时，因为 ",[34,317645,317647,317673],{"className":317646},[31642],[34,317648,317650],{"className":317649},[31646],[31648,317651,317652],{"xmlns":31650},[31652,317653,317654,317670],{},[31655,317655,317656,317662,317664],{},[31743,317657,317658,317660],{},[31658,317659,32936],{},[31749,317661,640],{},[31662,317663,31960],{"separator":758},[31743,317665,317666,317668],{},[31658,317667,32936],{},[31749,317669,768],{},[31665,317671,317672],{"encoding":31667},"x_1, x_2",[34,317674,317676],{"className":317675,"ariaHidden":758},[31672],[34,317677,317679,317682,317722,317725,317728],{"className":317678},[31676],[34,317680],{"className":317681,"style":38774},[31680],[34,317683,317685,317688],{"className":317684},[31685],[34,317686,32936],{"className":317687},[31685,31686],[34,317689,317691],{"className":317690},[31792],[34,317692,317694,317714],{"className":317693},[31796,31797],[34,317695,317697,317711],{"className":317696},[31801],[34,317698,317700],{"className":317699,"style":31806},[31805],[34,317701,317702,317705],{"style":31809},[34,317703],{"className":317704,"style":31814},[31813],[34,317706,317708],{"className":317707},[31818,31819,31820,31821],[34,317709,640],{"className":317710},[31685,31821],[34,317712,31829],{"className":317713},[31828],[34,317715,317717],{"className":317716},[31801],[34,317718,317720],{"className":317719,"style":31836},[31805],[34,317721],{},[34,317723,31960],{"className":317724},[32119],[34,317726],{"className":317727,"style":32123},[31845],[34,317729,317731,317734],{"className":317730},[31685],[34,317732,32936],{"className":317733},[31685,31686],[34,317735,317737],{"className":317736},[31792],[34,317738,317740,317760],{"className":317739},[31796,31797],[34,317741,317743,317757],{"className":317742},[31801],[34,317744,317746],{"className":317745,"style":31806},[31805],[34,317747,317748,317751],{"style":31809},[34,317749],{"className":317750,"style":31814},[31813],[34,317752,317754],{"className":317753},[31818,31819,31820,31821],[34,317755,768],{"className":317756},[31685,31821],[34,317758,31829],{"className":317759},[31828],[34,317761,317763],{"className":317762},[31801],[34,317764,317766],{"className":317765,"style":31836},[31805],[34,317767],{},[34,317769,317771,317788],{"className":317770},[31642],[34,317772,317774],{"className":317773},[31646],[31648,317775,317776],{"xmlns":31650},[31652,317777,317778,317786],{},[31655,317779,317780,317782,317784],{},[31662,317781,126529],{"stretchy":702},[31658,317783,31942],{},[31662,317785,126544],{"stretchy":702},[31665,317787,314966],{"encoding":31667},[34,317789,317791],{"className":317790,"ariaHidden":758},[31672],[34,317792,317794,317797,317800,317803],{"className":317793},[31676],[34,317795],{"className":317796,"style":31774},[31680],[34,317798,126529],{"className":317799},[31782],[34,317801,31942],{"className":317802,"style":32062},[31685,31686],[34,317804,126544],{"className":317805},[31690]," 里，它们的阶也都必须整除 ",[34,317808,317810,317823],{"className":317809},[31642],[34,317811,317813],{"className":317812},[31646],[31648,317814,317815],{"xmlns":31650},[31652,317816,317817,317821],{},[31655,317818,317819],{},[31658,317820,31660],{},[31665,317822,31660],{"encoding":31667},[34,317824,317826],{"className":317825,"ariaHidden":758},[31672],[34,317827,317829,317832],{"className":317828},[31676],[34,317830],{"className":317831,"style":32527},[31680],[34,317833,31660],{"className":317834},[31685,31686],"（群的阶）。\n所以，",[34,317837,317839,317852],{"className":317838},[31642],[34,317840,317842],{"className":317841},[31646],[31648,317843,317844],{"xmlns":31650},[31652,317845,317846,317850],{},[31655,317847,317848],{},[31658,317849,67764],{},[31665,317851,67764],{"encoding":31667},[34,317853,317855],{"className":317854,"ariaHidden":758},[31672],[34,317856,317858,317861],{"className":317857},[31676],[34,317859],{"className":317860,"style":38774},[31680],[34,317862,67764],{"className":317863,"style":32062},[31685,31686]," 的阶必须同时整除 ",[34,317866,317868,317881],{"className":317867},[31642],[34,317869,317871],{"className":317870},[31646],[31648,317872,317873],{"xmlns":31650},[31652,317874,317875,317879],{},[31655,317876,317877],{},[31658,317878,33759],{},[31665,317880,33759],{"encoding":31667},[34,317882,317884],{"className":317883,"ariaHidden":758},[31672],[34,317885,317887,317890],{"className":317886},[31676],[34,317888],{"className":317889,"style":31681},[31680],[34,317891,33759],{"className":317892,"style":33774},[31685,31686],[34,317894,317896,317909],{"className":317895},[31642],[34,317897,317899],{"className":317898},[31646],[31648,317900,317901],{"xmlns":31650},[31652,317902,317903,317907],{},[31655,317904,317905],{},[31658,317906,31660],{},[31665,317908,31660],{"encoding":31667},[34,317910,317912],{"className":317911,"ariaHidden":758},[31672],[34,317913,317915,317918],{"className":317914},[31676],[34,317916],{"className":317917,"style":32527},[31680],[34,317919,31660],{"className":317920},[31685,31686],"。\n但 ",[34,317923,317925,317954],{"className":317924},[31642],[34,317926,317928],{"className":317927},[31646],[31648,317929,317930],{"xmlns":31650},[31652,317931,317932,317952],{},[31655,317933,317934,317936,317938,317940,317942,317944,317946,317948,317950],{},[31658,317935,314792],{},[31662,317937,32682],{},[31662,317939,456],{"stretchy":702},[31658,317941,31660],{},[31662,317943,31960],{"separator":758},[31658,317945,33759],{},[31662,317947,2557],{"stretchy":702},[31662,317949,202],{},[31749,317951,640],{},[31665,317953,314811],{"encoding":31667},[34,317955,317957,317995],{"className":317956,"ariaHidden":758},[31672],[34,317958,317960,317963,317968,317971,317974,317977,317980,317983,317986,317989,317992],{"className":317959},[31676],[34,317961],{"className":317962,"style":31774},[31680],[34,317964,317966,314826],{"className":317965},[32767],[34,317967,31942],{"style":38619},[34,317969,456],{"className":317970},[31782],[34,317972,31660],{"className":317973},[31685,31686],[34,317975,31960],{"className":317976},[32119],[34,317978],{"className":317979,"style":32123},[31845],[34,317981,33759],{"className":317982,"style":33774},[31685,31686],[34,317984,2557],{"className":317985},[31690],[34,317987],{"className":317988,"style":31846},[31845],[34,317990,202],{"className":317991},[31850],[34,317993],{"className":317994,"style":31846},[31845],[34,317996,317998,318001],{"className":317997},[31676],[34,317999],{"className":318000,"style":32617},[31680],[34,318002,640],{"className":318003},[31685],"，它们没有公约数（除了1）。\n所以 ",[34,318006,318008,318021],{"className":318007},[31642],[34,318009,318011],{"className":318010},[31646],[31648,318012,318013],{"xmlns":31650},[31652,318014,318015,318019],{},[31655,318016,318017],{},[31658,318018,67764],{},[31665,318020,67764],{"encoding":31667},[34,318022,318024],{"className":318023,"ariaHidden":758},[31672],[34,318025,318027,318030],{"className":318026},[31676],[34,318028],{"className":318029,"style":38774},[31680],[34,318031,67764],{"className":318032,"style":32062},[31685,31686]," 的阶只能是 1，即 ",[34,318035,318037,318055],{"className":318036},[31642],[34,318038,318040],{"className":318039},[31646],[31648,318041,318042],{"xmlns":31650},[31652,318043,318044,318052],{},[31655,318045,318046,318048,318050],{},[31658,318047,67764],{},[31662,318049,202],{},[31658,318051,33244],{},[31665,318053,318054],{"encoding":31667},"y = e",[34,318056,318058,318076],{"className":318057,"ariaHidden":758},[31672],[34,318059,318061,318064,318067,318070,318073],{"className":318060},[31676],[34,318062],{"className":318063,"style":38774},[31680],[34,318065,67764],{"className":318066,"style":32062},[31685,31686],[34,318068],{"className":318069,"style":31846},[31845],[34,318071,202],{"className":318072},[31850],[34,318074],{"className":318075,"style":31846},[31845],[34,318077,318079,318082],{"className":318078},[31676],[34,318080],{"className":318081,"style":32527},[31680],[34,318083,33244],{"className":318084},[31685,31686],"。\n推导出 ",[34,318087,318089,318143],{"className":318088},[31642],[34,318090,318092],{"className":318091},[31646],[31648,318093,318094],{"xmlns":31650},[31652,318095,318096,318140],{},[31655,318097,318098,318104,318116,318118,318120,318122,318124,318126,318132,318134],{},[31743,318099,318100,318102],{},[31658,318101,32936],{},[31749,318103,640],{},[38846,318105,318106,318108,318110],{},[31658,318107,32936],{},[31749,318109,768],{},[31655,318111,318112,318114],{},[31662,318113,32226],{},[31749,318115,640],{},[31662,318117,202],{},[31658,318119,33244],{},[39361,318121,262622],{},[31662,318123,262625],{},[39361,318125,262622],{},[31743,318127,318128,318130],{},[31658,318129,32936],{},[31749,318131,640],{},[31662,318133,202],{},[31743,318135,318136,318138],{},[31658,318137,32936],{},[31749,318139,768],{},[31665,318141,318142],{"encoding":31667},"x_1 x_2^{-1} = e \\implies x_1 = x_2",[34,318144,318146,318258,318283,318338],{"className":318145,"ariaHidden":758},[31672],[34,318147,318149,318152,318192,318249,318252,318255],{"className":318148},[31676],[34,318150],{"className":318151,"style":317300},[31680],[34,318153,318155,318158],{"className":318154},[31685],[34,318156,32936],{"className":318157},[31685,31686],[34,318159,318161],{"className":318160},[31792],[34,318162,318164,318184],{"className":318163},[31796,31797],[34,318165,318167,318181],{"className":318166},[31801],[34,318168,318170],{"className":318169,"style":31806},[31805],[34,318171,318172,318175],{"style":31809},[34,318173],{"className":318174,"style":31814},[31813],[34,318176,318178],{"className":318177},[31818,31819,31820,31821],[34,318179,640],{"className":318180},[31685,31821],[34,318182,31829],{"className":318183},[31828],[34,318185,318187],{"className":318186},[31801],[34,318188,318190],{"className":318189,"style":31836},[31805],[34,318191],{},[34,318193,318195,318198],{"className":318194},[31685],[34,318196,32936],{"className":318197},[31685,31686],[34,318199,318201],{"className":318200},[31792],[34,318202,318204,318241],{"className":318203},[31796,31797],[34,318205,318207,318238],{"className":318206},[31801],[34,318208,318210,318221],{"className":318209,"style":317362},[31805],[34,318211,318212,318215],{"style":317365},[34,318213],{"className":318214,"style":31814},[31813],[34,318216,318218],{"className":318217},[31818,31819,31820,31821],[34,318219,768],{"className":318220},[31685,31821],[34,318222,318223,318226],{"style":317377},[34,318224],{"className":318225,"style":31814},[31813],[34,318227,318229],{"className":318228},[31818,31819,31820,31821],[34,318230,318232,318235],{"className":318231},[31685,31821],[34,318233,32226],{"className":318234},[31685,31821],[34,318236,640],{"className":318237},[31685,31821],[34,318239,31829],{"className":318240},[31828],[34,318242,318244],{"className":318243},[31801],[34,318245,318247],{"className":318246,"style":317402},[31805],[34,318248],{},[34,318250],{"className":318251,"style":31846},[31845],[34,318253,202],{"className":318254},[31850],[34,318256],{"className":318257,"style":31846},[31845],[34,318259,318261,318265,318268,318271,318274,318277,318280],{"className":318260},[31676],[34,318262],{"className":318263,"style":318264},[31680],"height:0.549em;vertical-align:-0.024em;",[34,318266,33244],{"className":318267},[31685,31686],[34,318269],{"className":318270,"style":31846},[31845],[34,318272],{"className":318273,"style":31846},[31845],[34,318275,262625],{"className":318276},[31850],[34,318278],{"className":318279,"style":31846},[31845],[34,318281],{"className":318282,"style":31846},[31845],[34,318284,318286,318289,318329,318332,318335],{"className":318285},[31676],[34,318287],{"className":318288,"style":31860},[31680],[34,318290,318292,318295],{"className":318291},[31685],[34,318293,32936],{"className":318294},[31685,31686],[34,318296,318298],{"className":318297},[31792],[34,318299,318301,318321],{"className":318300},[31796,31797],[34,318302,318304,318318],{"className":318303},[31801],[34,318305,318307],{"className":318306,"style":31806},[31805],[34,318308,318309,318312],{"style":31809},[34,318310],{"className":318311,"style":31814},[31813],[34,318313,318315],{"className":318314},[31818,31819,31820,31821],[34,318316,640],{"className":318317},[31685,31821],[34,318319,31829],{"className":318320},[31828],[34,318322,318324],{"className":318323},[31801],[34,318325,318327],{"className":318326,"style":31836},[31805],[34,318328],{},[34,318330],{"className":318331,"style":31846},[31845],[34,318333,202],{"className":318334},[31850],[34,318336],{"className":318337,"style":31846},[31845],[34,318339,318341,318344],{"className":318340},[31676],[34,318342],{"className":318343,"style":31860},[31680],[34,318345,318347,318350],{"className":318346},[31685],[34,318348,32936],{"className":318349},[31685,31686],[34,318351,318353],{"className":318352},[31792],[34,318354,318356,318376],{"className":318355},[31796,31797],[34,318357,318359,318373],{"className":318358},[31801],[34,318360,318362],{"className":318361,"style":31806},[31805],[34,318363,318364,318367],{"style":31809},[34,318365],{"className":318366,"style":31814},[31813],[34,318368,318370],{"className":318369},[31818,31819,31820,31821],[34,318371,768],{"className":318372},[31685,31821],[34,318374,31829],{"className":318375},[31828],[34,318377,318379],{"className":318378},[31801],[34,318380,318382],{"className":318381,"style":31836},[31805],[34,318383],{},[108,318385,318386,318388],{},[2627,318387,114043],{},"：解存在且唯一。",[32188,318390],{},[32490,318392,318394],{"id":318393},"题目-3有理数加法群的子群","题目 3：有理数加法群的子群",[15,318396,318397,318399,318400,318454],{},[2627,318398,101191],{}," 证明有理数加法群 ",[34,318401,318403,318426],{"className":318402},[31642],[34,318404,318406],{"className":318405},[31646],[31648,318407,318408],{"xmlns":31650},[31652,318409,318410,318423],{},[31655,318411,318412,318414,318417,318419,318421],{},[31662,318413,456],{"stretchy":702},[31658,318415,116863],{"mathvariant":318416},"double-struck",[31662,318418,31960],{"separator":758},[31662,318420,223],{},[31662,318422,2557],{"stretchy":702},[31665,318424,318425],{"encoding":31667},"(\\mathbb{Q}, +)",[34,318427,318429],{"className":318428,"ariaHidden":758},[31672],[34,318430,318432,318435,318438,318442,318445,318448,318451],{"className":318431},[31676],[34,318433],{"className":318434,"style":31774},[31680],[34,318436,456],{"className":318437},[31782],[34,318439,116863],{"className":318440},[31685,318441],"mathbb",[34,318443,31960],{"className":318444},[32119],[34,318446],{"className":318447,"style":32123},[31845],[34,318449,223],{"className":318450},[31685],[34,318452,2557],{"className":318453},[31690]," 的任一有限生成子群都是循环群。",[15,318456,318457],{},[2627,318458,314990],{},[2230,318460,318461,318467],{},[108,318462,318463,318466],{},[2627,318464,318465],{},"直观理解","：有理数很稠密，但如果你只取“有限个”有理数通过加减法生成一个群，它们其实都“卡”在一个共同分母的网格上。",[108,318468,318469,33183,318472],{},[2627,318470,318471],{},"详细讲解",[105,318473,318474,319067,319245,319645,320371,320644,320785,320791],{},[108,318475,318476,318477,318505,318506,347],{},"设这个子群 ",[34,318478,318480,318493],{"className":318479},[31642],[34,318481,318483],{"className":318482},[31646],[31648,318484,318485],{"xmlns":31650},[31652,318486,318487,318491],{},[31655,318488,318489],{},[31658,318490,118113],{},[31665,318492,118113],{"encoding":31667},[34,318494,318496],{"className":318495,"ariaHidden":758},[31672],[34,318497,318499,318502],{"className":318498},[31676],[34,318500],{"className":318501,"style":33218},[31680],[34,318503,118113],{"className":318504,"style":118235},[31685,31686]," 是由有限个有理数生成的：",[34,318507,318509,318579],{"className":318508},[31642],[34,318510,318512],{"className":318511},[31646],[31648,318513,318514],{"xmlns":31650},[31652,318515,318516,318576],{},[31655,318517,318518,318520,318522,318524,318538,318540,318554,318556,318558,318560,318574],{},[31658,318519,118113],{},[31662,318521,202],{},[31662,318523,126529],{"stretchy":702},[41453,318525,318526,318532],{},[31743,318527,318528,318530],{},[31658,318529,34395],{},[31749,318531,640],{},[31743,318533,318534,318536],{},[31658,318535,31747],{},[31749,318537,640],{},[31662,318539,31960],{"separator":758},[41453,318541,318542,318548],{},[31743,318543,318544,318546],{},[31658,318545,34395],{},[31749,318547,768],{},[31743,318549,318550,318552],{},[31658,318551,31747],{},[31749,318553,768],{},[31662,318555,31960],{"separator":758},[31662,318557,33934],{},[31662,318559,31960],{"separator":758},[41453,318561,318562,318568],{},[31743,318563,318564,318566],{},[31658,318565,34395],{},[31658,318567,31759],{},[31743,318569,318570,318572],{},[31658,318571,31747],{},[31658,318573,31759],{},[31662,318575,126544],{"stretchy":702},[31665,318577,318578],{"encoding":31667},"H = \\langle \\frac{r_1}{s_1}, \\frac{r_2}{s_2}, \\dots, \\frac{r_m}{s_m} \\rangle",[34,318580,318582,318600],{"className":318581,"ariaHidden":758},[31672],[34,318583,318585,318588,318591,318594,318597],{"className":318584},[31676],[34,318586],{"className":318587,"style":33218},[31680],[34,318589,118113],{"className":318590,"style":118235},[31685,31686],[34,318592],{"className":318593,"style":31846},[31845],[34,318595,202],{"className":318596},[31850],[34,318598],{"className":318599,"style":31846},[31845],[34,318601,318603,318607,318610,318756,318759,318762,318904,318907,318910,318913,318916,318919,318922,319064],{"className":318602},[31676],[34,318604],{"className":318605,"style":318606},[31680],"height:1.1951em;vertical-align:-0.4451em;",[34,318608,126529],{"className":318609},[31782],[34,318611,318613,318616,318753],{"className":318612},[31685],[34,318614],{"className":318615},[31782,41488],[34,318617,318619],{"className":318618},[41453],[34,318620,318622,318745],{"className":318621},[31796,31797],[34,318623,318625,318742],{"className":318624},[31801],[34,318626,318629,318682,318690],{"className":318627,"style":318628},[31805],"height:0.7115em;",[34,318630,318631,318634],{"style":41504},[34,318632],{"className":318633,"style":41508},[31813],[34,318635,318637],{"className":318636},[31818,31819,31820,31821],[34,318638,318640],{"className":318639},[31685,31821],[34,318641,318643,318646],{"className":318642},[31685,31821],[34,318644,31747],{"className":318645},[31685,31686,31821],[34,318647,318649],{"className":318648},[31792],[34,318650,318652,318674],{"className":318651},[31796,31797],[34,318653,318655,318671],{"className":318654},[31801],[34,318656,318659],{"className":318657,"style":318658},[31805],"height:0.3173em;",[34,318660,318662,318665],{"style":318661},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[34,318663],{"className":318664,"style":43406},[31813],[34,318666,318668],{"className":318667},[31818,43410,43411,31821],[34,318669,640],{"className":318670},[31685,31821],[34,318672,31829],{"className":318673},[31828],[34,318675,318677],{"className":318676},[31801],[34,318678,318680],{"className":318679,"style":220540},[31805],[34,318681],{},[34,318683,318684,318687],{"style":41520},[34,318685],{"className":318686,"style":41508},[31813],[34,318688],{"className":318689,"style":41528},[41527],[34,318691,318692,318695],{"style":220490},[34,318693],{"className":318694,"style":41508},[31813],[34,318696,318698],{"className":318697},[31818,31819,31820,31821],[34,318699,318701],{"className":318700},[31685,31821],[34,318702,318704,318707],{"className":318703},[31685,31821],[34,318705,34395],{"className":318706,"style":34452},[31685,31686,31821],[34,318708,318710],{"className":318709},[31792],[34,318711,318713,318734],{"className":318712},[31796,31797],[34,318714,318716,318731],{"className":318715},[31801],[34,318717,318719],{"className":318718,"style":318658},[31805],[34,318720,318722,318725],{"style":318721},"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;",[34,318723],{"className":318724,"style":43406},[31813],[34,318726,318728],{"className":318727},[31818,43410,43411,31821],[34,318729,640],{"className":318730},[31685,31821],[34,318732,31829],{"className":318733},[31828],[34,318735,318737],{"className":318736},[31801],[34,318738,318740],{"className":318739,"style":220540},[31805],[34,318741],{},[34,318743,31829],{"className":318744},[31828],[34,318746,318748],{"className":318747},[31801],[34,318749,318751],{"className":318750,"style":231800},[31805],[34,318752],{},[34,318754],{"className":318755},[31690,41488],[34,318757,31960],{"className":318758},[32119],[34,318760],{"className":318761,"style":32123},[31845],[34,318763,318765,318768,318901],{"className":318764},[31685],[34,318766],{"className":318767},[31782,41488],[34,318769,318771],{"className":318770},[41453],[34,318772,318774,318893],{"className":318773},[31796,31797],[34,318775,318777,318890],{"className":318776},[31801],[34,318778,318780,318831,318839],{"className":318779,"style":318628},[31805],[34,318781,318782,318785],{"style":41504},[34,318783],{"className":318784,"style":41508},[31813],[34,318786,318788],{"className":318787},[31818,31819,31820,31821],[34,318789,318791],{"className":318790},[31685,31821],[34,318792,318794,318797],{"className":318793},[31685,31821],[34,318795,31747],{"className":318796},[31685,31686,31821],[34,318798,318800],{"className":318799},[31792],[34,318801,318803,318823],{"className":318802},[31796,31797],[34,318804,318806,318820],{"className":318805},[31801],[34,318807,318809],{"className":318808,"style":318658},[31805],[34,318810,318811,318814],{"style":318661},[34,318812],{"className":318813,"style":43406},[31813],[34,318815,318817],{"className":318816},[31818,43410,43411,31821],[34,318818,768],{"className":318819},[31685,31821],[34,318821,31829],{"className":318822},[31828],[34,318824,318826],{"className":318825},[31801],[34,318827,318829],{"className":318828,"style":220540},[31805],[34,318830],{},[34,318832,318833,318836],{"style":41520},[34,318834],{"className":318835,"style":41508},[31813],[34,318837],{"className":318838,"style":41528},[41527],[34,318840,318841,318844],{"style":220490},[34,318842],{"className":318843,"style":41508},[31813],[34,318845,318847],{"className":318846},[31818,31819,31820,31821],[34,318848,318850],{"className":318849},[31685,31821],[34,318851,318853,318856],{"className":318852},[31685,31821],[34,318854,34395],{"className":318855,"style":34452},[31685,31686,31821],[34,318857,318859],{"className":318858},[31792],[34,318860,318862,318882],{"className":318861},[31796,31797],[34,318863,318865,318879],{"className":318864},[31801],[34,318866,318868],{"className":318867,"style":318658},[31805],[34,318869,318870,318873],{"style":318721},[34,318871],{"className":318872,"style":43406},[31813],[34,318874,318876],{"className":318875},[31818,43410,43411,31821],[34,318877,768],{"className":318878},[31685,31821],[34,318880,31829],{"className":318881},[31828],[34,318883,318885],{"className":318884},[31801],[34,318886,318888],{"className":318887,"style":220540},[31805],[34,318889],{},[34,318891,31829],{"className":318892},[31828],[34,318894,318896],{"className":318895},[31801],[34,318897,318899],{"className":318898,"style":231800},[31805],[34,318900],{},[34,318902],{"className":318903},[31690,41488],[34,318905,31960],{"className":318906},[32119],[34,318908],{"className":318909,"style":32123},[31845],[34,318911,33934],{"className":318912},[34006],[34,318914],{"className":318915,"style":32123},[31845],[34,318917,31960],{"className":318918},[32119],[34,318920],{"className":318921,"style":32123},[31845],[34,318923,318925,318928,319061],{"className":318924},[31685],[34,318926],{"className":318927},[31782,41488],[34,318929,318931],{"className":318930},[41453],[34,318932,318934,319053],{"className":318933},[31796,31797],[34,318935,318937,319050],{"className":318936},[31801],[34,318938,318940,318991,318999],{"className":318939,"style":318628},[31805],[34,318941,318942,318945],{"style":41504},[34,318943],{"className":318944,"style":41508},[31813],[34,318946,318948],{"className":318947},[31818,31819,31820,31821],[34,318949,318951],{"className":318950},[31685,31821],[34,318952,318954,318957],{"className":318953},[31685,31821],[34,318955,31747],{"className":318956},[31685,31686,31821],[34,318958,318960],{"className":318959},[31792],[34,318961,318963,318983],{"className":318962},[31796,31797],[34,318964,318966,318980],{"className":318965},[31801],[34,318967,318969],{"className":318968,"style":220518},[31805],[34,318970,318971,318974],{"style":318661},[34,318972],{"className":318973,"style":43406},[31813],[34,318975,318977],{"className":318976},[31818,43410,43411,31821],[34,318978,31759],{"className":318979},[31685,31686,31821],[34,318981,31829],{"className":318982},[31828],[34,318984,318986],{"className":318985},[31801],[34,318987,318989],{"className":318988,"style":220540},[31805],[34,318990],{},[34,318992,318993,318996],{"style":41520},[34,318994],{"className":318995,"style":41508},[31813],[34,318997],{"className":318998,"style":41528},[41527],[34,319000,319001,319004],{"style":220490},[34,319002],{"className":319003,"style":41508},[31813],[34,319005,319007],{"className":319006},[31818,31819,31820,31821],[34,319008,319010],{"className":319009},[31685,31821],[34,319011,319013,319016],{"className":319012},[31685,31821],[34,319014,34395],{"className":319015,"style":34452},[31685,31686,31821],[34,319017,319019],{"className":319018},[31792],[34,319020,319022,319042],{"className":319021},[31796,31797],[34,319023,319025,319039],{"className":319024},[31801],[34,319026,319028],{"className":319027,"style":220518},[31805],[34,319029,319030,319033],{"style":318721},[34,319031],{"className":319032,"style":43406},[31813],[34,319034,319036],{"className":319035},[31818,43410,43411,31821],[34,319037,31759],{"className":319038},[31685,31686,31821],[34,319040,31829],{"className":319041},[31828],[34,319043,319045],{"className":319044},[31801],[34,319046,319048],{"className":319047,"style":220540},[31805],[34,319049],{},[34,319051,31829],{"className":319052},[31828],[34,319054,319056],{"className":319055},[31801],[34,319057,319059],{"className":319058,"style":231800},[31805],[34,319060],{},[34,319062],{"className":319063},[31690,41488],[34,319065,126544],{"className":319066},[31690],[108,319068,319069,319072,319073,319101,319102,130135,319242,347],{},[2627,319070,319071],{},"通分","：令 ",[34,319074,319076,319089],{"className":319075},[31642],[34,319077,319079],{"className":319078},[31646],[31648,319080,319081],{"xmlns":31650},[31652,319082,319083,319087],{},[31655,319084,319085],{},[31658,319086,94675],{},[31665,319088,94675],{"encoding":31667},[34,319090,319092],{"className":319091,"ariaHidden":758},[31672],[34,319093,319095,319098],{"className":319094},[31676],[34,319096],{"className":319097,"style":33218},[31680],[34,319099,94675],{"className":319100,"style":106455},[31685,31686]," 为所有分母 ",[34,319103,319105,319135],{"className":319104},[31642],[34,319106,319108],{"className":319107},[31646],[31648,319109,319110],{"xmlns":31650},[31652,319111,319112,319132],{},[31655,319113,319114,319120,319122,319124,319126],{},[31743,319115,319116,319118],{},[31658,319117,31747],{},[31749,319119,640],{},[31662,319121,31960],{"separator":758},[31662,319123,33934],{},[31662,319125,31960],{"separator":758},[31743,319127,319128,319130],{},[31658,319129,31747],{},[31658,319131,31759],{},[31665,319133,319134],{"encoding":31667},"s_1, \\dots, s_m",[34,319136,319138],{"className":319137,"ariaHidden":758},[31672],[34,319139,319141,319144,319184,319187,319190,319193,319196,319199,319202],{"className":319140},[31676],[34,319142],{"className":319143,"style":38774},[31680],[34,319145,319147,319150],{"className":319146},[31685],[34,319148,31747],{"className":319149},[31685,31686],[34,319151,319153],{"className":319152},[31792],[34,319154,319156,319176],{"className":319155},[31796,31797],[34,319157,319159,319173],{"className":319158},[31801],[34,319160,319162],{"className":319161,"style":31806},[31805],[34,319163,319164,319167],{"style":31809},[34,319165],{"className":319166,"style":31814},[31813],[34,319168,319170],{"className":319169},[31818,31819,31820,31821],[34,319171,640],{"className":319172},[31685,31821],[34,319174,31829],{"className":319175},[31828],[34,319177,319179],{"className":319178},[31801],[34,319180,319182],{"className":319181,"style":31836},[31805],[34,319183],{},[34,319185,31960],{"className":319186},[32119],[34,319188],{"className":319189,"style":32123},[31845],[34,319191,33934],{"className":319192},[34006],[34,319194],{"className":319195,"style":32123},[31845],[34,319197,31960],{"className":319198},[32119],[34,319200],{"className":319201,"style":32123},[31845],[34,319203,319205,319208],{"className":319204},[31685],[34,319206,31747],{"className":319207},[31685,31686],[34,319209,319211],{"className":319210},[31792],[34,319212,319214,319234],{"className":319213},[31796,31797],[34,319215,319217,319231],{"className":319216},[31801],[34,319218,319220],{"className":319219,"style":32362},[31805],[34,319221,319222,319225],{"style":31809},[34,319223],{"className":319224,"style":31814},[31813],[34,319226,319228],{"className":319227},[31818,31819,31820,31821],[34,319229,31759],{"className":319230},[31685,31686,31821],[34,319232,31829],{"className":319233},[31828],[34,319235,319237],{"className":319236},[31801],[34,319238,319240],{"className":319239,"style":31836},[31805],[34,319241],{},[2627,319243,319244],{},"最小公倍数",[108,319246,319247,319248,319429,319430,319572,319573,319644],{},"那么，每一个生成元 ",[34,319249,319251,319277],{"className":319250},[31642],[34,319252,319254],{"className":319253},[31646],[31648,319255,319256],{"xmlns":31650},[31652,319257,319258,319274],{},[31655,319259,319260],{},[41453,319261,319262,319268],{},[31743,319263,319264,319266],{},[31658,319265,34395],{},[31658,319267,31955],{},[31743,319269,319270,319272],{},[31658,319271,31747],{},[31658,319273,31955],{},[31665,319275,319276],{"encoding":31667},"\\frac{r_i}{s_i}",[34,319278,319280],{"className":319279,"ariaHidden":758},[31672],[34,319281,319283,319287],{"className":319282},[31676],[34,319284],{"className":319285,"style":319286},[31680],"height:1.1566em;vertical-align:-0.4451em;",[34,319288,319290,319293,319426],{"className":319289},[31685],[34,319291],{"className":319292},[31782,41488],[34,319294,319296],{"className":319295},[41453],[34,319297,319299,319418],{"className":319298},[31796,31797],[34,319300,319302,319415],{"className":319301},[31801],[34,319303,319305,319356,319364],{"className":319304,"style":318628},[31805],[34,319306,319307,319310],{"style":41504},[34,319308],{"className":319309,"style":41508},[31813],[34,319311,319313],{"className":319312},[31818,31819,31820,31821],[34,319314,319316],{"className":319315},[31685,31821],[34,319317,319319,319322],{"className":319318},[31685,31821],[34,319320,31747],{"className":319321},[31685,31686,31821],[34,319323,319325],{"className":319324},[31792],[34,319326,319328,319348],{"className":319327},[31796,31797],[34,319329,319331,319345],{"className":319330},[31801],[34,319332,319334],{"className":319333,"style":231708},[31805],[34,319335,319336,319339],{"style":318661},[34,319337],{"className":319338,"style":43406},[31813],[34,319340,319342],{"className":319341},[31818,43410,43411,31821],[34,319343,31955],{"className":319344},[31685,31686,31821],[34,319346,31829],{"className":319347},[31828],[34,319349,319351],{"className":319350},[31801],[34,319352,319354],{"className":319353,"style":220540},[31805],[34,319355],{},[34,319357,319358,319361],{"style":41520},[34,319359],{"className":319360,"style":41508},[31813],[34,319362],{"className":319363,"style":41528},[41527],[34,319365,319366,319369],{"style":220490},[34,319367],{"className":319368,"style":41508},[31813],[34,319370,319372],{"className":319371},[31818,31819,31820,31821],[34,319373,319375],{"className":319374},[31685,31821],[34,319376,319378,319381],{"className":319377},[31685,31821],[34,319379,34395],{"className":319380,"style":34452},[31685,31686,31821],[34,319382,319384],{"className":319383},[31792],[34,319385,319387,319407],{"className":319386},[31796,31797],[34,319388,319390,319404],{"className":319389},[31801],[34,319391,319393],{"className":319392,"style":231708},[31805],[34,319394,319395,319398],{"style":318721},[34,319396],{"className":319397,"style":43406},[31813],[34,319399,319401],{"className":319400},[31818,43410,43411,31821],[34,319402,31955],{"className":319403},[31685,31686,31821],[34,319405,31829],{"className":319406},[31828],[34,319408,319410],{"className":319409},[31801],[34,319411,319413],{"className":319412,"style":220540},[31805],[34,319414],{},[34,319416,31829],{"className":319417},[31828],[34,319419,319421],{"className":319420},[31801],[34,319422,319424],{"className":319423,"style":231800},[31805],[34,319425],{},[34,319427],{"className":319428},[31690,41488]," 都可以写成 ",[34,319431,319433,319455],{"className":319432},[31642],[34,319434,319436],{"className":319435},[31646],[31648,319437,319438],{"xmlns":31650},[31652,319439,319440,319452],{},[31655,319441,319442],{},[41453,319443,319444,319450],{},[31743,319445,319446,319448],{},[31658,319447,33759],{},[31658,319449,31955],{},[31658,319451,94675],{},[31665,319453,319454],{"encoding":31667},"\\frac{k_i}{S}",[34,319456,319458],{"className":319457,"ariaHidden":758},[31672],[34,319459,319461,319465],{"className":319460},[31676],[34,319462],{"className":319463,"style":319464},[31680],"height:1.2412em;vertical-align:-0.345em;",[34,319466,319468,319471,319569],{"className":319467},[31685],[34,319469],{"className":319470},[31782,41488],[34,319472,319474],{"className":319473},[41453],[34,319475,319477,319561],{"className":319476},[31796,31797],[34,319478,319480,319558],{"className":319479},[31801],[34,319481,319484,319498,319506],{"className":319482,"style":319483},[31805],"height:0.8962em;",[34,319485,319486,319489],{"style":41504},[34,319487],{"className":319488,"style":41508},[31813],[34,319490,319492],{"className":319491},[31818,31819,31820,31821],[34,319493,319495],{"className":319494},[31685,31821],[34,319496,94675],{"className":319497,"style":106455},[31685,31686,31821],[34,319499,319500,319503],{"style":41520},[34,319501],{"className":319502,"style":41508},[31813],[34,319504],{"className":319505,"style":41528},[41527],[34,319507,319508,319511],{"style":220490},[34,319509],{"className":319510,"style":41508},[31813],[34,319512,319514],{"className":319513},[31818,31819,31820,31821],[34,319515,319517],{"className":319516},[31685,31821],[34,319518,319520,319523],{"className":319519},[31685,31821],[34,319521,33759],{"className":319522,"style":33774},[31685,31686,31821],[34,319524,319526],{"className":319525},[31792],[34,319527,319529,319550],{"className":319528},[31796,31797],[34,319530,319532,319547],{"className":319531},[31801],[34,319533,319535],{"className":319534,"style":231708},[31805],[34,319536,319538,319541],{"style":319537},"top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;",[34,319539],{"className":319540,"style":43406},[31813],[34,319542,319544],{"className":319543},[31818,43410,43411,31821],[34,319545,31955],{"className":319546},[31685,31686,31821],[34,319548,31829],{"className":319549},[31828],[34,319551,319553],{"className":319552},[31801],[34,319554,319556],{"className":319555,"style":220540},[31805],[34,319557],{},[34,319559,31829],{"className":319560},[31828],[34,319562,319564],{"className":319563},[31801],[34,319565,319567],{"className":319566,"style":41619},[31805],[34,319568],{},[34,319570],{"className":319571},[31690,41488]," 的形式（其中 ",[34,319574,319576,319594],{"className":319575},[31642],[34,319577,319579],{"className":319578},[31646],[31648,319580,319581],{"xmlns":31650},[31652,319582,319583,319591],{},[31655,319584,319585],{},[31743,319586,319587,319589],{},[31658,319588,33759],{},[31658,319590,31955],{},[31665,319592,319593],{"encoding":31667},"k_i",[34,319595,319597],{"className":319596,"ariaHidden":758},[31672],[34,319598,319600,319603],{"className":319599},[31676],[34,319601],{"className":319602,"style":33814},[31680],[34,319604,319606,319609],{"className":319605},[31685],[34,319607,33759],{"className":319608,"style":33774},[31685,31686],[34,319610,319612],{"className":319611},[31792],[34,319613,319615,319636],{"className":319614},[31796,31797],[34,319616,319618,319633],{"className":319617},[31801],[34,319619,319621],{"className":319620,"style":32090},[31805],[34,319622,319624,319627],{"style":319623},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[34,319625],{"className":319626,"style":31814},[31813],[34,319628,319630],{"className":319629},[31818,31819,31820,31821],[34,319631,31955],{"className":319632},[31685,31686,31821],[34,319634,31829],{"className":319635},[31828],[34,319637,319639],{"className":319638},[31801],[34,319640,319642],{"className":319641,"style":31836},[31805],[34,319643],{}," 是整数）。",[108,319646,319647,319648,319676,319677],{},"这意味着，群 ",[34,319649,319651,319664],{"className":319650},[31642],[34,319652,319654],{"className":319653},[31646],[31648,319655,319656],{"xmlns":31650},[31652,319657,319658,319662],{},[31655,319659,319660],{},[31658,319661,118113],{},[31665,319663,118113],{"encoding":31667},[34,319665,319667],{"className":319666,"ariaHidden":758},[31672],[34,319668,319670,319673],{"className":319669},[31676],[34,319671],{"className":319672,"style":33218},[31680],[34,319674,118113],{"className":319675,"style":118235},[31685,31686]," 中的任何元素都可以写成：\n",[34,319678,319680,319772],{"className":319679},[31642],[34,319681,319683],{"className":319682},[31646],[31648,319684,319685],{"xmlns":31650},[31652,319686,319687,319769],{},[31655,319688,319689,319691,319693,319699,319709,319711,319713,319715,319721,319731,319733],{},[31658,319690,32936],{},[31662,319692,202],{},[31743,319694,319695,319697],{},[31658,319696,31660],{},[31749,319698,640],{},[41453,319700,319701,319707],{},[31743,319702,319703,319705],{},[31658,319704,33759],{},[31749,319706,640],{},[31658,319708,94675],{},[31662,319710,223],{},[31662,319712,38874],{},[31662,319714,223],{},[31743,319716,319717,319719],{},[31658,319718,31660],{},[31658,319720,31759],{},[41453,319722,319723,319729],{},[31743,319724,319725,319727],{},[31658,319726,33759],{},[31658,319728,31759],{},[31658,319730,94675],{},[31662,319732,202],{},[41453,319734,319735,319767],{},[31655,319736,319737,319743,319749,319751,319753,319755,319761],{},[31743,319738,319739,319741],{},[31658,319740,31660],{},[31749,319742,640],{},[31743,319744,319745,319747],{},[31658,319746,33759],{},[31749,319748,640],{},[31662,319750,223],{},[31662,319752,38874],{},[31662,319754,223],{},[31743,319756,319757,319759],{},[31658,319758,31660],{},[31658,319760,31759],{},[31743,319762,319763,319765],{},[31658,319764,33759],{},[31658,319766,31759],{},[31658,319768,94675],{},[31665,319770,319771],{"encoding":31667},"x = n_1 \\frac{k_1}{S} + \\dots + n_m \\frac{k_m}{S} = \\frac{n_1 k_1 + \\dots + n_m k_m}{S}",[34,319773,319775,319793,319953,319971,320131],{"className":319774,"ariaHidden":758},[31672],[34,319776,319778,319781,319784,319787,319790],{"className":319777},[31676],[34,319779],{"className":319780,"style":32527},[31680],[34,319782,32936],{"className":319783},[31685,31686],[34,319785],{"className":319786,"style":31846},[31845],[34,319788,202],{"className":319789},[31850],[34,319791],{"className":319792,"style":31846},[31845],[34,319794,319796,319799,319839,319944,319947,319950],{"className":319795},[31676],[34,319797],{"className":319798,"style":319464},[31680],[34,319800,319802,319805],{"className":319801},[31685],[34,319803,31660],{"className":319804},[31685,31686],[34,319806,319808],{"className":319807},[31792],[34,319809,319811,319831],{"className":319810},[31796,31797],[34,319812,319814,319828],{"className":319813},[31801],[34,319815,319817],{"className":319816,"style":31806},[31805],[34,319818,319819,319822],{"style":31809},[34,319820],{"className":319821,"style":31814},[31813],[34,319823,319825],{"className":319824},[31818,31819,31820,31821],[34,319826,640],{"className":319827},[31685,31821],[34,319829,31829],{"className":319830},[31828],[34,319832,319834],{"className":319833},[31801],[34,319835,319837],{"className":319836,"style":31836},[31805],[34,319838],{},[34,319840,319842,319845,319941],{"className":319841},[31685],[34,319843],{"className":319844},[31782,41488],[34,319846,319848],{"className":319847},[41453],[34,319849,319851,319933],{"className":319850},[31796,31797],[34,319852,319854,319930],{"className":319853},[31801],[34,319855,319857,319871,319879],{"className":319856,"style":319483},[31805],[34,319858,319859,319862],{"style":41504},[34,319860],{"className":319861,"style":41508},[31813],[34,319863,319865],{"className":319864},[31818,31819,31820,31821],[34,319866,319868],{"className":319867},[31685,31821],[34,319869,94675],{"className":319870,"style":106455},[31685,31686,31821],[34,319872,319873,319876],{"style":41520},[34,319874],{"className":319875,"style":41508},[31813],[34,319877],{"className":319878,"style":41528},[41527],[34,319880,319881,319884],{"style":220490},[34,319882],{"className":319883,"style":41508},[31813],[34,319885,319887],{"className":319886},[31818,31819,31820,31821],[34,319888,319890],{"className":319889},[31685,31821],[34,319891,319893,319896],{"className":319892},[31685,31821],[34,319894,33759],{"className":319895,"style":33774},[31685,31686,31821],[34,319897,319899],{"className":319898},[31792],[34,319900,319902,319922],{"className":319901},[31796,31797],[34,319903,319905,319919],{"className":319904},[31801],[34,319906,319908],{"className":319907,"style":318658},[31805],[34,319909,319910,319913],{"style":319537},[34,319911],{"className":319912,"style":43406},[31813],[34,319914,319916],{"className":319915},[31818,43410,43411,31821],[34,319917,640],{"className":319918},[31685,31821],[34,319920,31829],{"className":319921},[31828],[34,319923,319925],{"className":319924},[31801],[34,319926,319928],{"className":319927,"style":220540},[31805],[34,319929],{},[34,319931,31829],{"className":319932},[31828],[34,319934,319936],{"className":319935},[31801],[34,319937,319939],{"className":319938,"style":41619},[31805],[34,319940],{},[34,319942],{"className":319943},[31690,41488],[34,319945],{"className":319946,"style":32604},[31845],[34,319948,223],{"className":319949},[32024],[34,319951],{"className":319952,"style":32604},[31845],[34,319954,319956,319959,319962,319965,319968],{"className":319955},[31676],[34,319957],{"className":319958,"style":32597},[31680],[34,319960,38874],{"className":319961},[34006],[34,319963],{"className":319964,"style":32604},[31845],[34,319966,223],{"className":319967},[32024],[34,319969],{"className":319970,"style":32604},[31845],[34,319972,319974,319977,320017,320122,320125,320128],{"className":319973},[31676],[34,319975],{"className":319976,"style":319464},[31680],[34,319978,319980,319983],{"className":319979},[31685],[34,319981,31660],{"className":319982},[31685,31686],[34,319984,319986],{"className":319985},[31792],[34,319987,319989,320009],{"className":319988},[31796,31797],[34,319990,319992,320006],{"className":319991},[31801],[34,319993,319995],{"className":319994,"style":32362},[31805],[34,319996,319997,320000],{"style":31809},[34,319998],{"className":319999,"style":31814},[31813],[34,320001,320003],{"className":320002},[31818,31819,31820,31821],[34,320004,31759],{"className":320005},[31685,31686,31821],[34,320007,31829],{"className":320008},[31828],[34,320010,320012],{"className":320011},[31801],[34,320013,320015],{"className":320014,"style":31836},[31805],[34,320016],{},[34,320018,320020,320023,320119],{"className":320019},[31685],[34,320021],{"className":320022},[31782,41488],[34,320024,320026],{"className":320025},[41453],[34,320027,320029,320111],{"className":320028},[31796,31797],[34,320030,320032,320108],{"className":320031},[31801],[34,320033,320035,320049,320057],{"className":320034,"style":319483},[31805],[34,320036,32003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,"style":43406},[31813],[34,320301,320303],{"className":320302},[31818,43410,43411,31821],[34,320304,31759],{"className":320305},[31685,31686,31821],[34,320307,31829],{"className":320308},[31828],[34,320310,320312],{"className":320311},[31801],[34,320313,320315],{"className":320314,"style":220540},[31805],[34,320316],{},[34,320318,320320,320323],{"className":320319},[31685,31821],[34,320321,33759],{"className":320322,"style":33774},[31685,31686,31821],[34,320324,320326],{"className":320325},[31792],[34,320327,320329,320349],{"className":320328},[31796,31797],[34,320330,320332,320346],{"className":320331},[31801],[34,320333,320335],{"className":320334,"style":220518},[31805],[34,320336,320337,320340],{"style":319537},[34,320338],{"className":320339,"style":43406},[31813],[34,320341,320343],{"className":320342},[31818,43410,43411,31821],[34,320344,31759],{"className":320345},[31685,31686,31821],[34,320347,31829],{"className":320348},[31828],[34,320350,320352],{"className":320351},[31801],[34,320353,320355],{"className":320354,"style":220540},[31805],[34,320356],{},[34,320358,31829],{"className":320359},[31828],[34,320361,320363],{"className":320362},[31801],[34,320364,320366],{"className":320365,"style":41619},[31805],[34,320367],{},[34,320369],{"className":320370},[31690,41488],[108,320372,320373,320374,320643],{},"分子部分 ",[34,320375,320377,320423],{"className":320376},[31642],[34,320378,320380],{"className":320379},[31646],[31648,320381,320382],{"xmlns":31650},[31652,320383,320384,320420],{},[31655,320385,320386,320388,320390,320396,320402,320404,320406,320408,320414],{},[31658,320387,33347],{},[31662,320389,202],{},[31743,320391,320392,320394],{},[31658,320393,31660],{},[31749,320395,640],{},[31743,320397,320398,320400],{},[31658,320399,33759],{},[31749,320401,640],{},[31662,320403,223],{},[31662,320405,38874],{},[31662,320407,223],{},[31743,320409,320410,320412],{},[31658,320411,31660],{},[31658,320413,31759],{},[31743,320415,320416,320418],{},[31658,320417,33759],{},[31658,320419,31759],{},[31665,320421,320422],{"encoding":31667},"N = n_1 k_1 + \\dots + n_m k_m",[34,320424,320426,320444,320539,320557],{"className":320425,"ariaHidden":758},[31672],[34,320427,320429,320432,320435,320438,320441],{"className":320428},[31676],[34,320430],{"className":320431,"style":33218},[31680],[34,320433,33347],{"className":320434,"style":33389},[31685,31686],[34,320436],{"className":320437,"style":31846},[31845],[34,320439,202],{"className":320440},[31850],[34,320442],{"className":320443,"style":31846},[31845],[34,320445,320447,320450,320490,320530,320533,320536],{"className":320446},[31676],[34,320448],{"className":320449,"style":33814},[31680],[34,320451,320453,320456],{"className":320452},[31685],[34,320454,31660],{"className":320455},[31685,31686],[34,320457,320459],{"className":320458},[31792],[34,320460,320462,320482],{"className":320461},[31796,31797],[34,320463,320465,320479],{"className":320464},[31801],[34,320466,320468],{"className":320467,"style":31806},[31805],[34,320469,320470,320473],{"style":31809},[34,320471],{"className":320472,"style":31814},[31813],[34,320474,320476],{"className":320475},[31818,31819,31820,31821],[34,320477,640],{"className":320478},[31685,31821],[34,320480,31829],{"className":320481},[31828],[34,320483,320485],{"className":320484},[31801],[34,320486,320488],{"className":320487,"style":31836},[31805],[34,320489],{},[34,320491,320493,320496],{"className":320492},[31685],[34,320494,33759],{"className":320495,"style":33774},[31685,31686],[34,320497,320499],{"className":320498},[31792],[34,320500,320502,320522],{"className":320501},[31796,31797],[34,320503,320505,320519],{"className":320504},[31801],[34,320506,320508],{"className":320507,"style":31806},[31805],[34,320509,320510,320513],{"style":319623},[34,320511],{"className":320512,"style":31814},[31813],[34,320514,320516],{"className":320515},[31818,31819,31820,31821],[34,320517,640],{"className":320518},[31685,31821],[34,320520,31829],{"className":320521},[31828],[34,320523,320525],{"className":320524},[31801],[34,320526,320528],{"className":320527,"style":31836},[31805],[34,320529],{},[34,320531],{"className":320532,"style":32604},[31845],[34,320534,223],{"className":320535},[32024],[34,320537],{"className":320538,"style":32604},[31845],[34,320540,320542,320545,320548,320551,320554],{"className":320541},[31676],[34,320543],{"className":320544,"style":32597},[31680],[34,320546,38874],{"className":320547},[34006],[34,320549],{"className":320550,"style":32604},[31845],[34,320552,223],{"className":320553},[32024],[34,320555],{"className":320556,"style":32604},[31845],[34,320558,320560,320563,320603],{"className":320559},[31676],[34,320561],{"className":320562,"style":33814},[31680],[34,320564,320566,320569],{"className":320565},[31685],[34,320567,31660],{"className":320568},[31685,31686],[34,320570,320572],{"className":320571},[31792],[34,320573,320575,320595],{"className":320574},[31796,31797],[34,320576,320578,320592],{"className":320577},[31801],[34,320579,320581],{"className":320580,"style":32362},[31805],[34,320582,320583,320586],{"style":31809},[34,320584],{"className":320585,"style":31814},[31813],[34,320587,320589],{"className":320588},[31818,31819,31820,31821],[34,320590,31759],{"className":320591},[31685,31686,31821],[34,320593,31829],{"className":320594},[31828],[34,320596,320598],{"className":320597},[31801],[34,320599,320601],{"className":320600,"style":31836},[31805],[34,320602],{},[34,320604,320606,320609],{"className":320605},[31685],[34,320607,33759],{"className":320608,"style":33774},[31685,31686],[34,320610,320612],{"className":320611},[31792],[34,320613,320615,320635],{"className":320614},[31796,31797],[34,320616,320618,320632],{"className":320617},[31801],[34,320619,320621],{"className":320620,"style":32362},[31805],[34,320622,320623,320626],{"style":319623},[34,320624],{"className":320625,"style":31814},[31813],[34,320627,320629],{"className":320628},[31818,31819,31820,31821],[34,320630,31759],{"className":320631},[31685,31686,31821],[34,320633,31829],{"className":320634},[31828],[34,320636,320638],{"className":320637},[31801],[34,320639,320641],{"className":320640,"style":31836},[31805],[34,320642],{}," 肯定是个整数。",[108,320645,320646,320647,320675,320676,320784],{},"所以，整个子群 ",[34,320648,320650,320663],{"className":320649},[31642],[34,320651,320653],{"className":320652},[31646],[31648,320654,320655],{"xmlns":31650},[31652,320656,320657,320661],{},[31655,320658,320659],{},[31658,320660,118113],{},[31665,320662,118113],{"encoding":31667},[34,320664,320666],{"className":320665,"ariaHidden":758},[31672],[34,320667,320669,320672],{"className":320668},[31676],[34,320670],{"className":320671,"style":33218},[31680],[34,320673,118113],{"className":320674,"style":118235},[31685,31686]," 实际上包含在群 ",[34,320677,320679,320701],{"className":320678},[31642],[34,320680,320682],{"className":320681},[31646],[31648,320683,320684],{"xmlns":31650},[31652,320685,320686,320698],{},[31655,320687,320688,320690,320696],{},[31662,320689,126529],{"stretchy":702},[41453,320691,320692,320694],{},[31749,320693,640],{},[31658,320695,94675],{},[31662,320697,126544],{"stretchy":702},[31665,320699,320700],{"encoding":31667},"\\langle \\frac{1}{S} \\rangle",[34,320702,320704],{"className":320703,"ariaHidden":758},[31672],[34,320705,320707,320710,320713,320781],{"className":320706},[31676],[34,320708],{"className":320709,"style":46727},[31680],[34,320711,126529],{"className":320712},[31782],[34,320714,320716,320719,320778],{"className":320715},[31685],[34,320717],{"className":320718},[31782,41488],[34,320720,320722],{"className":320721},[41453],[34,320723,320725,320770],{"className":320724},[31796,31797],[34,320726,320728,320767],{"className":320727},[31801],[34,320729,320731,320745,320753],{"className":320730,"style":46746},[31805],[34,320732,320733,320736],{"style":41504},[34,320734],{"className":320735,"style":41508},[31813],[34,320737,320739],{"className":320738},[31818,31819,31820,31821],[34,320740,320742],{"className":320741},[31685,31821],[34,320743,94675],{"className":320744,"style":106455},[31685,31686,31821],[34,320746,320747,320750],{"style":41520},[34,320748],{"className":320749,"style":41508},[31813],[34,320751],{"className":320752,"style":41528},[41527],[34,320754,320755,320758],{"style":46771},[34,320756],{"className":320757,"style":41508},[31813],[34,320759,320761],{"className":320760},[31818,31819,31820,31821],[34,320762,320764],{"className":320763},[31685,31821],[34,320765,640],{"className":320766},[31685,31821],[34,320768,31829],{"className":320769},[31828],[34,320771,320773],{"className":320772},[31801],[34,320774,320776],{"className":320775,"style":41619},[31805],[34,320777],{},[34,320779],{"className":320780},[31690,41488],[34,320782,126544],{"className":320783},[31690]," 里面。",[108,320786,320787,320790],{},[2627,320788,320789],{},"关键定理","：循环群的子群必定是循环群。",[108,320792,320793,320900,320901,320931,320932,320960,320961,320989],{},[34,320794,320796,320817],{"className":320795},[31642],[34,320797,320799],{"className":320798},[31646],[31648,320800,320801],{"xmlns":31650},[31652,320802,320803,320815],{},[31655,320804,320805,320807,320813],{},[31662,320806,126529],{"stretchy":702},[41453,320808,320809,320811],{},[31749,320810,640],{},[31658,320812,94675],{},[31662,320814,126544],{"stretchy":702},[31665,320816,320700],{"encoding":31667},[34,320818,320820],{"className":320819,"ariaHidden":758},[31672],[34,320821,320823,320826,320829,320897],{"className":320822},[31676],[34,320824],{"className":320825,"style":46727},[31680],[34,320827,126529],{"className":320828},[31782],[34,320830,320832,320835,320894],{"className":320831},[31685],[34,320833],{"className":320834},[31782,41488],[34,320836,320838],{"className":320837},[41453],[34,320839,320841,320886],{"className":320840},[31796,31797],[34,320842,320844,320883],{"className":320843},[31801],[34,320845,320847,320861,320869],{"className":320846,"style":46746},[31805],[34,320848,320849,320852],{"style":41504},[34,320850],{"className":320851,"style":41508},[31813],[34,320853,320855],{"className":320854},[31818,31819,31820,31821],[34,320856,320858],{"className":320857},[31685,31821],[34,320859,94675],{"className":320860,"style":106455},[31685,31686,31821],[34,320862,320863,320866],{"style":41520},[34,320864],{"className":320865,"style":41508},[31813],[34,320867],{"className":320868,"style":41528},[41527],[34,320870,320871,320874],{"style":46771},[34,320872],{"className":320873,"style":41508},[31813],[34,320875,320877],{"className":320876},[31818,31819,31820,31821],[34,320878,320880],{"className":320879},[31685,31821],[34,320881,640],{"className":320882},[31685,31821],[34,320884,31829],{"className":320885},[31828],[34,320887,320889],{"className":320888},[31801],[34,320890,320892],{"className":320891,"style":41619},[31805],[34,320893],{},[34,320895],{"className":320896},[31690,41488],[34,320898,126544],{"className":320899},[31690]," 同构于整数加法群 ",[34,320902,320904,320918],{"className":320903},[31642],[34,320905,320907],{"className":320906},[31646],[31648,320908,320909],{"xmlns":31650},[31652,320910,320911,320915],{},[31655,320912,320913],{},[31658,320914,97300],{"mathvariant":318416},[31665,320916,320917],{"encoding":31667},"\\mathbb{Z}",[34,320919,320921],{"className":320920,"ariaHidden":758},[31672],[34,320922,320924,320928],{"className":320923},[31676],[34,320925],{"className":320926,"style":320927},[31680],"height:0.6889em;",[34,320929,97300],{"className":320930},[31685,318441],"（它是循环群）。既然 ",[34,320933,320935,320948],{"className":320934},[31642],[34,320936,320938],{"className":320937},[31646],[31648,320939,320940],{"xmlns":31650},[31652,320941,320942,320946],{},[31655,320943,320944],{},[31658,320945,118113],{},[31665,320947,118113],{"encoding":31667},[34,320949,320951],{"className":320950,"ariaHidden":758},[31672],[34,320952,320954,320957],{"className":320953},[31676],[34,320955],{"className":320956,"style":33218},[31680],[34,320958,118113],{"className":320959,"style":118235},[31685,31686]," 是它的子群，那 ",[34,320962,320964,320977],{"className":320963},[31642],[34,320965,320967],{"className":320966},[31646],[31648,320968,320969],{"xmlns":31650},[31652,320970,320971,320975],{},[31655,320972,320973],{},[31658,320974,118113],{},[31665,320976,118113],{"encoding":31667},[34,320978,320980],{"className":320979,"ariaHidden":758},[31672],[34,320981,320983,320986],{"className":320982},[31676],[34,320984],{"className":320985,"style":33218},[31680],[34,320987,118113],{"className":320988,"style":118235},[31685,31686]," 也一定是循环群。",[32188,320991],{},[253,320993,320995,320996,43298,321065,321135],{"id":320994},"第二部分置换群-sns_nsn-与-ana_nan计算篇","第二部分：置换群 ",[34,320997,320999,321016],{"className":320998},[31642],[34,321000,321002],{"className":321001},[31646],[31648,321003,321004],{"xmlns":31650},[31652,321005,321006,321014],{},[31655,321007,321008],{},[31743,321009,321010,321012],{},[31658,321011,94675],{},[31658,321013,31660],{},[31665,321015,218979],{"encoding":31667},[34,321017,321019],{"className":321018,"ariaHidden":758},[31672],[34,321020,321022,321025],{"className":321021},[31676],[34,321023],{"className":321024,"style":70915},[31680],[34,321026,321028,321031],{"className":321027},[31685],[34,321029,94675],{"className":321030,"style":106455},[31685,31686],[34,321032,321034],{"className":321033},[31792],[34,321035,321037,321057],{"className":321036},[31796,31797],[34,321038,321040,321054],{"className":321039},[31801],[34,321041,321043],{"className":321042,"style":32362},[31805],[34,321044,321045,321048],{"style":206422},[34,321046],{"className":321047,"style":31814},[31813],[34,321049,321051],{"className":321050},[31818,31819,31820,31821],[34,321052,31660],{"className":321053},[31685,31686,31821],[34,321055,31829],{"className":321056},[31828],[34,321058,321060],{"className":321059},[31801],[34,321061,321063],{"className":321062,"style":31836},[31805],[34,321064],{},[34,321066,321068,321086],{"className":321067},[31642],[34,321069,321071],{"className":321070},[31646],[31648,321072,321073],{"xmlns":31650},[31652,321074,321075,321083],{},[31655,321076,321077],{},[31743,321078,321079,321081],{},[31658,321080,54151],{},[31658,321082,31660],{},[31665,321084,321085],{"encoding":31667},"A_n",[34,321087,321089],{"className":321088,"ariaHidden":758},[31672],[34,321090,321092,321095],{"className":321091},[31676],[34,321093],{"className":321094,"style":70915},[31680],[34,321096,321098,321101],{"className":321097},[31685],[34,321099,54151],{"className":321100},[31685,31686],[34,321102,321104],{"className":321103},[31792],[34,321105,321107,321127],{"className":321106},[31796,31797],[34,321108,321110,321124],{"className":321109},[31801],[34,321111,321113],{"className":321112,"style":32362},[31805],[34,321114,321115,321118],{"style":31809},[34,321116],{"className":321117,"style":31814},[31813],[34,321119,321121],{"className":321120},[31818,31819,31820,31821],[34,321122,31660],{"className":321123},[31685,31686,31821],[34,321125,31829],{"className":321126},[31828],[34,321128,321130],{"className":321129},[31801],[34,321131,321133],{"className":321132,"style":31836},[31805],[34,321134],{},"（计算篇）",[15,321137,321138,321141,321142,321261,321262,88909,321290,347],{},[2627,321139,321140],{},"核心技巧","：我们要学会用“共轭”来搬运数字。\n公式：",[34,321143,321145,321186],{"className":321144},[31642],[34,321146,321148],{"className":321147},[31646],[31648,321149,321150],{"xmlns":31650},[31652,321151,321152,321183],{},[31655,321153,321154,321157,321159,321161,321163,321165,321167,321169,321171,321173],{},[31658,321155,321156],{},"τ",[31662,321158,456],{"stretchy":702},[31658,321160,31955],{},[39361,321162,130298],{},[31658,321164,31968],{},[39361,321166,130298],{},[31662,321168,33934],{},[39361,321170,39363],{},[31662,321172,2557],{"stretchy":702},[37624,321174,321175,321177],{},[31658,321176,321156],{},[31655,321178,321179,321181],{},[31662,321180,32226],{},[31749,321182,640],{},[31665,321184,321185],{"encoding":31667},"\\tau (i \\ j \\ \\dots) \\tau^{-1}",[34,321187,321189],{"className":321188,"ariaHidden":758},[31672],[34,321190,321192,321195,321199,321202,321205,321208,321211,321214,321217,321220,321223,321226],{"className":321191},[31676],[34,321193],{"className":321194,"style":37755},[31680],[34,321196,321156],{"className":321197,"style":321198},[31685,31686],"margin-right:0.1132em;",[34,321200,456],{"className":321201},[31782],[34,321203,31955],{"className":321204},[31685,31686],[34,321206,130298],{"className":321207},[31845],[34,321209,31968],{"className":321210,"style":32153},[31685,31686],[34,321212,130298],{"className":321213},[31845],[34,321215],{"className":321216,"style":32123},[31845],[34,321218,33934],{"className":321219},[34006],[34,321221],{"className":321222,"style":32123},[31845],[34,321224,2557],{"className":321225},[31690],[34,321227,321229,321232],{"className":321228},[31685],[34,321230,321156],{"className":321231,"style":321198},[31685,31686],[34,321233,321235],{"className":321234},[31792],[34,321236,321238],{"className":321237},[31796],[34,321239,321241],{"className":321240},[31801],[34,321242,321244],{"className":321243,"style":37774},[31805],[34,321245,321246,321249],{"style":37777},[34,321247],{"className":321248,"style":31814},[31813],[34,321250,321252],{"className":321251},[31818,31819,31820,31821],[34,321253,321255,321258],{"className":321254},[31685,31821],[34,321256,32226],{"className":321257},[31685,31821],[34,321259,640],{"className":321260},[31685,31821]," 的结果，就是把括号里的数字 ",[34,321263,321265,321278],{"className":321264},[31642],[34,321266,321268],{"className":321267},[31646],[31648,321269,321270],{"xmlns":31650},[31652,321271,321272,321276],{},[31655,321273,321274],{},[31658,321275,32936],{},[31665,321277,32936],{"encoding":31667},[34,321279,321281],{"className":321280,"ariaHidden":758},[31672],[34,321282,321284,321287],{"className":321283},[31676],[34,321285],{"className":321286,"style":32527},[31680],[34,321288,32936],{"className":321289},[31685,31686],[34,321291,321293,321313],{"className":321292},[31642],[34,321294,321296],{"className":321295},[31646],[31648,321297,321298],{"xmlns":31650},[31652,321299,321300,321310],{},[31655,321301,321302,321304,321306,321308],{},[31658,321303,321156],{},[31662,321305,456],{"stretchy":702},[31658,321307,32936],{},[31662,321309,2557],{"stretchy":702},[31665,321311,321312],{"encoding":31667},"\\tau(x)",[34,321314,321316],{"className":321315,"ariaHidden":758},[31672],[34,321317,321319,321322,321325,321328,321331],{"className":321318},[31676],[34,321320],{"className":321321,"style":31774},[31680],[34,321323,321156],{"className":321324,"style":321198},[31685,31686],[34,321326,456],{"className":321327},[31782],[34,321329,32936],{"className":321330},[31685,31686],[34,321332,2557],{"className":321333},[31690],[32490,321335,118837,321337,321406],{"id":321336},"题目-11sns_nsn-的生成元",[34,321338,321340,321357],{"className":321339},[31642],[34,321341,321343],{"className":321342},[31646],[31648,321344,321345],{"xmlns":31650},[31652,321346,321347,321355],{},[31655,321348,321349],{},[31743,321350,321351,321353],{},[31658,321352,94675],{},[31658,321354,31660],{},[31665,321356,218979],{"encoding":31667},[34,321358,321360],{"className":321359,"ariaHidden":758},[31672],[34,321361,321363,321366],{"className":321362},[31676],[34,321364],{"className":321365,"style":70915},[31680],[34,321367,321369,321372],{"className":321368},[31685],[34,321370,94675],{"className":321371,"style":106455},[31685,31686],[34,321373,321375],{"className":321374},[31792],[34,321376,321378,321398],{"className":321377},[31796,31797],[34,321379,321381,321395],{"className":321380},[31801],[34,321382,321384],{"className":321383,"style":32362},[31805],[34,321385,321386,321389],{"style":206422},[34,321387],{"className":321388,"style":31814},[31813],[34,321390,321392],{"className":321391},[31818,31819,31820,31821],[34,321393,31660],{"className":321394},[31685,31686,31821],[34,321396,31829],{"className":321397},[31828],[34,321399,321401],{"className":321400},[31801],[34,321402,321404],{"className":321403,"style":31836},[31805],[34,321405],{}," 的生成元",[15,321408,321409,321411,321412,321481,321482,321531,321532,321629],{},[2627,321410,101191],{}," 证明 ",[34,321413,321415,321432],{"className":321414},[31642],[34,321416,321418],{"className":321417},[31646],[31648,321419,321420],{"xmlns":31650},[31652,321421,321422,321430],{},[31655,321423,321424],{},[31743,321425,321426,321428],{},[31658,321427,94675],{},[31658,321429,31660],{},[31665,321431,218979],{"encoding":31667},[34,321433,321435],{"className":321434,"ariaHidden":758},[31672],[34,321436,321438,321441],{"className":321437},[31676],[34,321439],{"className":321440,"style":70915},[31680],[34,321442,321444,321447],{"className":321443},[31685],[34,321445,94675],{"className":321446,"style":106455},[31685,31686],[34,321448,321450],{"className":321449},[31792],[34,321451,321453,321473],{"className":321452},[31796,31797],[34,321454,321456,321470],{"className":321455},[31801],[34,321457,321459],{"className":321458,"style":32362},[31805],[34,321460,321461,321464],{"style":206422},[34,321462],{"className":321463,"style":31814},[31813],[34,321465,321467],{"className":321466},[31818,31819,31820,31821],[34,321468,31660],{"className":321469},[31685,31686,31821],[34,321471,31829],{"className":321472},[31828],[34,321474,321476],{"className":321475},[31801],[34,321477,321479],{"className":321478,"style":31836},[31805],[34,321480],{}," 可以由对换 ",[34,321483,321485,321507],{"className":321484},[31642],[34,321486,321488],{"className":321487},[31646],[31648,321489,321490],{"xmlns":31650},[31652,321491,321492,321504],{},[31655,321493,321494,321496,321498,321500,321502],{},[31662,321495,456],{"stretchy":702},[31749,321497,640],{},[39361,321499,130298],{},[31749,321501,768],{},[31662,321503,2557],{"stretchy":702},[31665,321505,321506],{"encoding":31667},"(1\\ 2)",[34,321508,321510],{"className":321509,"ariaHidden":758},[31672],[34,321511,321513,321516,321519,321522,321525,321528],{"className":321512},[31676],[34,321514],{"className":321515,"style":31774},[31680],[34,321517,456],{"className":321518},[31782],[34,321520,640],{"className":321521},[31685],[34,321523,130298],{"className":321524},[31845],[34,321526,768],{"className":321527},[31685],[34,321529,2557],{"className":321530},[31690]," 和轮换 ",[34,321533,321535,321569],{"className":321534},[31642],[34,321536,321538],{"className":321537},[31646],[31648,321539,321540],{"xmlns":31650},[31652,321541,321542,321566],{},[31655,321543,321544,321546,321548,321550,321552,321554,321556,321558,321560,321562,321564],{},[31658,321545,115372],{},[31662,321547,202],{},[31662,321549,456],{"stretchy":702},[31749,321551,640],{},[39361,321553,130298],{},[31749,321555,768],{},[39361,321557,130298],{},[31662,321559,38874],{},[39361,321561,130298],{},[31658,321563,31660],{},[31662,321565,2557],{"stretchy":702},[31665,321567,321568],{"encoding":31667},"\\sigma = (1\\ 2\\ \\cdots\\ n)",[34,321570,321572,321590],{"className":321571,"ariaHidden":758},[31672],[34,321573,321575,321578,321581,321584,321587],{"className":321574},[31676],[34,321576],{"className":321577,"style":32527},[31680],[34,321579,115372],{"className":321580,"style":32062},[31685,31686],[34,321582],{"className":321583,"style":31846},[31845],[34,321585,202],{"className":321586},[31850],[34,321588],{"className":321589,"style":31846},[31845],[34,321591,321593,321596,321599,321602,321605,321608,321611,321614,321617,321620,321623,321626],{"className":321592},[31676],[34,321594],{"className":321595,"style":31774},[31680],[34,321597,456],{"className":321598},[31782],[34,321600,640],{"className":321601},[31685],[34,321603,130298],{"className":321604},[31845],[34,321606,768],{"className":321607},[31685],[34,321609,130298],{"className":321610},[31845],[34,321612],{"className":321613,"style":32123},[31845],[34,321615,38874],{"className":321616},[34006],[34,321618,130298],{"className":321619},[31845],[34,321621],{"className":321622,"style":32123},[31845],[34,321624,31660],{"className":321625},[31685,31686],[34,321627,2557],{"className":321628},[31690]," 生成。",[15,321631,321632,321635,321636,321710,321711,347],{},[2627,321633,321634],{},"【证明思路】","\n只要能造出所有的“相邻对换” ",[34,321637,321639,321665],{"className":321638},[31642],[34,321640,321642],{"className":321641},[31646],[31648,321643,321644],{"xmlns":31650},[31652,321645,321646,321662],{},[31655,321647,321648,321650,321652,321654,321656,321658,321660],{},[31662,321649,456],{"stretchy":702},[31658,321651,33759],{},[31662,321653,31960],{"separator":758},[31658,321655,33759],{},[31662,321657,223],{},[31749,321659,640],{},[31662,321661,2557],{"stretchy":702},[31665,321663,321664],{"encoding":31667},"(k, k+1)",[34,321666,321668,321698],{"className":321667,"ariaHidden":758},[31672],[34,321669,321671,321674,321677,321680,321683,321686,321689,321692,321695],{"className":321670},[31676],[34,321672],{"className":321673,"style":31774},[31680],[34,321675,456],{"className":321676},[31782],[34,321678,33759],{"className":321679,"style":33774},[31685,31686],[34,321681,31960],{"className":321682},[32119],[34,321684],{"className":321685,"style":32123},[31845],[34,321687,33759],{"className":321688,"style":33774},[31685,31686],[34,321690],{"className":321691,"style":32604},[31845],[34,321693,223],{"className":321694},[32024],[34,321696],{"className":321697,"style":32604},[31845],[34,321699,321701,321704,321707],{"className":321700},[31676],[34,321702],{"className":321703,"style":31774},[31680],[34,321705,640],{"className":321706},[31685],[34,321708,2557],{"className":321709},[31690],"，就能生成整个 ",[34,321712,321714,321731],{"className":321713},[31642],[34,321715,321717],{"className":321716},[31646],[31648,321718,321719],{"xmlns":31650},[31652,321720,321721,321729],{},[31655,321722,321723],{},[31743,321724,321725,321727],{},[31658,321726,94675],{},[31658,321728,31660],{},[31665,321730,218979],{"encoding":31667},[34,321732,321734],{"className":321733,"ariaHidden":758},[31672],[34,321735,321737,321740],{"className":321736},[31676],[34,321738],{"className":321739,"style":70915},[31680],[34,321741,321743,321746],{"className":321742},[31685],[34,321744,94675],{"className":321745,"style":106455},[31685,31686],[34,321747,321749],{"className":321748},[31792],[34,321750,321752,321772],{"className":321751},[31796,31797],[34,321753,321755,321769],{"className":321754},[31801],[34,321756,321758],{"className":321757,"style":32362},[31805],[34,321759,321760,321763],{"style":206422},[34,321761],{"className":321762,"style":31814},[31813],[34,321764,321766],{"className":321765},[31818,31819,31820,31821],[34,321767,31660],{"className":321768},[31685,31686,31821],[34,321770,31829],{"className":321771},[31828],[34,321773,321775],{"className":321774},[31801],[34,321776,321778],{"className":321777,"style":31836},[31805],[34,321779],{},[2230,321781,321782,321833,322209,322353,322509],{},[108,321783,321784,321785,347],{},"我们已有 ",[34,321786,321788,321809],{"className":321787},[31642],[34,321789,321791],{"className":321790},[31646],[31648,321792,321793],{"xmlns":31650},[31652,321794,321795,321807],{},[31655,321796,321797,321799,321801,321803,321805],{},[31662,321798,456],{"stretchy":702},[31749,321800,640],{},[39361,321802,130298],{},[31749,321804,768],{},[31662,321806,2557],{"stretchy":702},[31665,321808,321506],{"encoding":31667},[34,321810,321812],{"className":321811,"ariaHidden":758},[31672],[34,321813,321815,321818,321821,321824,321827,321830],{"className":321814},[31676],[34,321816],{"className":321817,"style":31774},[31680],[34,321819,456],{"className":321820},[31782],[34,321822,640],{"className":321823},[31685],[34,321825,130298],{"className":321826},[31845],[34,321828,768],{"className":321829},[31685],[34,321831,2557],{"className":321832},[31690],[108,321834,321835,321836,39984,321935,321963,321964,322015,322016,322067,322068,347],{},"利用共轭作用：",[34,321837,321839,321873],{"className":321838},[31642],[34,321840,321842],{"className":321841},[31646],[31648,321843,321844],{"xmlns":31650},[31652,321845,321846,321870],{},[31655,321847,321848,321850,321852,321854,321856,321858,321860],{},[31658,321849,115372],{},[31662,321851,456],{"stretchy":702},[31749,321853,640],{},[39361,321855,130298],{},[31749,321857,768],{},[31662,321859,2557],{"stretchy":702},[37624,321861,321862,321864],{},[31658,321863,115372],{},[31655,321865,321866,321868],{},[31662,321867,32226],{},[31749,321869,640],{},[31665,321871,321872],{"encoding":31667},"\\sigma (1\\ 2) \\sigma^{-1}",[34,321874,321876],{"className":321875,"ariaHidden":758},[31672],[34,321877,321879,321882,321885,321888,321891,321894,321897,321900],{"className":321878},[31676],[34,321880],{"className":321881,"style":37755},[31680],[34,321883,115372],{"className":321884,"style":32062},[31685,31686],[34,321886,456],{"className":321887},[31782],[34,321889,640],{"className":321890},[31685],[34,321892,130298],{"className":321893},[31845],[34,321895,768],{"className":321896},[31685],[34,321898,2557],{"className":321899},[31690],[34,321901,321903,321906],{"className":321902},[31685],[34,321904,115372],{"className":321905,"style":32062},[31685,31686],[34,321907,321909],{"className":321908},[31792],[34,321910,321912],{"className":321911},[31796],[34,321913,321915],{"className":321914},[31801],[34,321916,321918],{"className":321917,"style":37774},[31805],[34,321919,321920,321923],{"style":37777},[34,321921],{"className":321922,"style":31814},[31813],[34,321924,321926],{"className":321925},[31818,31819,31820,31821],[34,321927,321929,321932],{"className":321928},[31685,31821],[34,321930,32226],{"className":321931},[31685,31821],[34,321933,640],{"className":321934},[31685,31821],[34,321936,321938,321951],{"className":321937},[31642],[34,321939,321941],{"className":321940},[31646],[31648,321942,321943],{"xmlns":31650},[31652,321944,321945,321949],{},[31655,321946,321947],{},[31658,321948,115372],{},[31665,321950,115375],{"encoding":31667},[34,321952,321954],{"className":321953,"ariaHidden":758},[31672],[34,321955,321957,321960],{"className":321956},[31676],[34,321958],{"className":321959,"style":32527},[31680],[34,321961,115372],{"className":321962,"style":32062},[31685,31686]," 把 ",[34,321965,321967,321985],{"className":321966},[31642],[34,321968,321970],{"className":321969},[31646],[31648,321971,321972],{"xmlns":31650},[31652,321973,321974,321982],{},[31655,321975,321976,321978,321980],{},[31749,321977,640],{},[31662,321979,33239],{},[31749,321981,768],{},[31665,321983,321984],{"encoding":31667},"1 \\to 2",[34,321986,321988,322006],{"className":321987,"ariaHidden":758},[31672],[34,321989,321991,321994,321997,322000,322003],{"className":321990},[31676],[34,321992],{"className":321993,"style":32617},[31680],[34,321995,640],{"className":321996},[31685],[34,321998],{"className":321999,"style":31846},[31845],[34,322001,33239],{"className":322002},[31850],[34,322004],{"className":322005,"style":31846},[31845],[34,322007,322009,322012],{"className":322008},[31676],[34,322010],{"className":322011,"style":32617},[31680],[34,322013,768],{"className":322014},[31685],"，把 ",[34,322017,322019,322037],{"className":322018},[31642],[34,322020,322022],{"className":322021},[31646],[31648,322023,322024],{"xmlns":31650},[31652,322025,322026,322034],{},[31655,322027,322028,322030,322032],{},[31749,322029,768],{},[31662,322031,33239],{},[31749,322033,786],{},[31665,322035,322036],{"encoding":31667},"2 \\to 3",[34,322038,322040,322058],{"className":322039,"ariaHidden":758},[31672],[34,322041,322043,322046,322049,322052,322055],{"className":322042},[31676],[34,322044],{"className":322045,"style":32617},[31680],[34,322047,768],{"className":322048},[31685],[34,322050],{"className":322051,"style":31846},[31845],[34,322053,33239],{"className":322054},[31850],[34,322056],{"className":322057,"style":31846},[31845],[34,322059,322061,322064],{"className":322060},[31676],[34,322062],{"className":322063,"style":32617},[31680],[34,322065,786],{"className":322066},[31685],"。所以 ",[34,322069,322071,322117],{"className":322070},[31642],[34,322072,322074],{"className":322073},[31646],[31648,322075,322076],{"xmlns":31650},[31652,322077,322078,322114],{},[31655,322079,322080,322082,322084,322086,322088,322090,322092,322102,322104,322106,322108,322110,322112],{},[31658,322081,115372],{},[31662,322083,456],{"stretchy":702},[31749,322085,640],{},[39361,322087,130298],{},[31749,322089,768],{},[31662,322091,2557],{"stretchy":702},[37624,322093,322094,322096],{},[31658,322095,115372],{},[31655,322097,322098,322100],{},[31662,322099,32226],{},[31749,322101,640],{},[31662,322103,202],{},[31662,322105,456],{"stretchy":702},[31749,322107,768],{},[39361,322109,130298],{},[31749,322111,786],{},[31662,322113,2557],{"stretchy":702},[31665,322115,322116],{"encoding":31667},"\\sigma (1\\ 2) \\sigma^{-1} = (2\\ 3)",[34,322118,322120,322188],{"className":322119,"ariaHidden":758},[31672],[34,322121,322123,322126,322129,322132,322135,322138,322141,322144,322179,322182,322185],{"className":322122},[31676],[34,322124],{"className":322125,"style":37755},[31680],[34,322127,115372],{"className":322128,"style":32062},[31685,31686],[34,322130,456],{"className":322131},[31782],[34,322133,640],{"className":322134},[31685],[34,322136,130298],{"className":322137},[31845],[34,322139,768],{"className":322140},[31685],[34,322142,2557],{"className":322143},[31690],[34,322145,322147,322150],{"className":322146},[31685],[34,322148,115372],{"className":322149,"style":32062},[31685,31686],[34,322151,322153],{"className":322152},[31792],[34,322154,322156],{"className":322155},[31796],[34,322157,322159],{"className":322158},[31801],[34,322160,322162],{"className":322161,"style":37774},[31805],[34,322163,322164,322167],{"style":37777},[34,322165],{"className":322166,"style":31814},[31813],[34,322168,322170],{"className":322169},[31818,31819,31820,31821],[34,322171,322173,322176],{"className":322172},[31685,31821],[34,322174,32226],{"className":322175},[31685,31821],[34,322177,640],{"className":322178},[31685,31821],[34,322180],{"className":322181,"style":31846},[31845],[34,322183,202],{"className":322184},[31850],[34,322186],{"className":322187,"style":31846},[31845],[34,322189,322191,322194,322197,322200,322203,322206],{"className":322190},[31676],[34,322192],{"className":322193,"style":31774},[31680],[34,322195,456],{"className":322196},[31782],[34,322198,768],{"className":322199},[31685],[34,322201,130298],{"className":322202},[31845],[34,322204,786],{"className":322205},[31685],[34,322207,2557],{"className":322208},[31690],[108,322210,322211,322212,347],{},"再搞一次：",[34,322213,322215,322261],{"className":322214},[31642],[34,322216,322218],{"className":322217},[31646],[31648,322219,322220],{"xmlns":31650},[31652,322221,322222,322258],{},[31655,322223,322224,322226,322228,322230,322232,322234,322236,322246,322248,322250,322252,322254,322256],{},[31658,322225,115372],{},[31662,322227,456],{"stretchy":702},[31749,322229,768],{},[39361,322231,130298],{},[31749,322233,786],{},[31662,322235,2557],{"stretchy":702},[37624,322237,322238,322240],{},[31658,322239,115372],{},[31655,322241,322242,322244],{},[31662,322243,32226],{},[31749,322245,640],{},[31662,322247,202],{},[31662,322249,456],{"stretchy":702},[31749,322251,786],{},[39361,322253,130298],{},[31749,322255,15932],{},[31662,322257,2557],{"stretchy":702},[31665,322259,322260],{"encoding":31667},"\\sigma (2\\ 3) \\sigma^{-1} = (3\\ 4)",[34,322262,322264,322332],{"className":322263,"ariaHidden":758},[31672],[34,322265,322267,322270,322273,322276,322279,322282,322285,322288,322323,322326,322329],{"className":322266},[31676],[34,322268],{"className":322269,"style":37755},[31680],[34,322271,115372],{"className":322272,"style":32062},[31685,31686],[34,322274,456],{"className":322275},[31782],[34,322277,768],{"className":322278},[31685],[34,322280,130298],{"className":322281},[31845],[34,322283,786],{"className":322284},[31685],[34,322286,2557],{"className":322287},[31690],[34,322289,322291,322294],{"className":322290},[31685],[34,322292,115372],{"className":322293,"style":32062},[31685,31686],[34,322295,322297],{"className":322296},[31792],[34,322298,322300],{"className":322299},[31796],[34,322301,322303],{"className":322302},[31801],[34,322304,322306],{"className":322305,"style":37774},[31805],[34,322307,322308,322311],{"style":37777},[34,322309],{"className":322310,"style":31814},[31813],[34,322312,322314],{"className":322313},[31818,31819,31820,31821],[34,322315,322317,322320],{"className":322316},[31685,31821],[34,322318,32226],{"className":322319},[31685,31821],[34,322321,640],{"className":322322},[31685,31821],[34,322324],{"className":322325,"style":31846},[31845],[34,322327,202],{"className":322328},[31850],[34,322330],{"className":322331,"style":31846},[31845],[34,322333,322335,322338,322341,322344,322347,322350],{"className":322334},[31676],[34,322336],{"className":322337,"style":31774},[31680],[34,322339,456],{"className":322340},[31782],[34,322342,786],{"className":322343},[31685],[34,322345,130298],{"className":322346},[31845],[34,322348,15932],{"className":322349},[31685],[34,322351,2557],{"className":322352},[31690],[108,322354,322355,322356,347],{},"以此类推，我们可以得到 ",[34,322357,322359,322413],{"className":322358},[31642],[34,322360,322362],{"className":322361},[31646],[31648,322363,322364],{"xmlns":31650},[31652,322365,322366,322410],{},[31655,322367,322368,322370,322372,322374,322376,322378,322380,322382,322384,322386,322388,322390,322392,322394,322396,322398,322400,322402,322404,322406,322408],{},[31662,322369,456],{"stretchy":702},[31749,322371,640],{},[39361,322373,130298],{},[31749,322375,768],{},[31662,322377,2557],{"stretchy":702},[31662,322379,31960],{"separator":758},[31662,322381,456],{"stretchy":702},[31749,322383,768],{},[39361,322385,130298],{},[31749,322387,786],{},[31662,322389,2557],{"stretchy":702},[31662,322391,31960],{"separator":758},[31662,322393,33934],{},[31662,322395,31960],{"separator":758},[31662,322397,456],{"stretchy":702},[31658,322399,31660],{},[31662,322401,32226],{},[31749,322403,640],{},[39361,322405,130298],{},[31658,322407,31660],{},[31662,322409,2557],{"stretchy":702},[31665,322411,322412],{"encoding":31667},"(1\\ 2), (2\\ 3), \\dots, (n-1\\ n)",[34,322414,322416,322491],{"className":322415,"ariaHidden":758},[31672],[34,322417,322419,322422,322425,322428,322431,322434,322437,322440,322443,322446,322449,322452,322455,322458,322461,322464,322467,322470,322473,322476,322479,322482,322485,322488],{"className":322418},[31676],[34,322420],{"className":322421,"style":31774},[31680],[34,322423,456],{"className":322424},[31782],[34,322426,640],{"className":322427},[31685],[34,322429,130298],{"className":322430},[31845],[34,322432,768],{"className":322433},[31685],[34,322435,2557],{"className":322436},[31690],[34,322438,31960],{"className":322439},[32119],[34,322441],{"className":322442,"style":32123},[31845],[34,322444,456],{"className":322445},[31782],[34,322447,768],{"className":322448},[31685],[34,322450,130298],{"className":322451},[31845],[34,322453,786],{"className":322454},[31685],[34,322456,2557],{"className":322457},[31690],[34,322459,31960],{"className":322460},[32119],[34,322462],{"className":322463,"style":32123},[31845],[34,322465,33934],{"className":322466},[34006],[34,322468],{"className":322469,"style":32123},[31845],[34,322471,31960],{"className":322472},[32119],[34,322474],{"className":322475,"style":32123},[31845],[34,322477,456],{"className":322478},[31782],[34,322480,31660],{"className":322481},[31685,31686],[34,322483],{"className":322484,"style":32604},[31845],[34,322486,32226],{"className":322487},[32024],[34,322489],{"className":322490,"style":32604},[31845],[34,322492,322494,322497,322500,322503,322506],{"className":322493},[31676],[34,322495],{"className":322496,"style":31774},[31680],[34,322498,640],{"className":322499},[31685],[34,322501,130298],{"className":322502},[31845],[34,322504,31660],{"className":322505},[31685,31686],[34,322507,2557],{"className":322508},[31690],[108,322510,322511],{},"既然有了所有相邻对换，它们就能生成所有置换。证毕。",[32490,322513,322515,322516,321406],{"id":322514},"题目-12-13ana_nan-的生成元","题目 12 & 13：",[34,322517,322519,322536],{"className":322518},[31642],[34,322520,322522],{"className":322521},[31646],[31648,322523,322524],{"xmlns":31650},[31652,322525,322526,322534],{},[31655,322527,322528],{},[31743,322529,322530,322532],{},[31658,322531,54151],{},[31658,322533,31660],{},[31665,322535,321085],{"encoding":31667},[34,322537,322539],{"className":322538,"ariaHidden":758},[31672],[34,322540,322542,322545],{"className":322541},[31676],[34,322543],{"className":322544,"style":70915},[31680],[34,322546,322548,322551],{"className":322547},[31685],[34,322549,54151],{"className":322550},[31685,31686],[34,322552,322554],{"className":322553},[31792],[34,322555,322557,322577],{"className":322556},[31796,31797],[34,322558,322560,322574],{"className":322559},[31801],[34,322561,322563],{"className":322562,"style":32362},[31805],[34,322564,322565,322568],{"style":31809},[34,322566],{"className":322567,"style":31814},[31813],[34,322569,322571],{"className":322570},[31818,31819,31820,31821],[34,322572,31660],{"className":322573},[31685,31686,31821],[34,322575,31829],{"className":322576},[31828],[34,322578,322580],{"className":322579},[31801],[34,322581,322583],{"className":322582,"style":31836},[31805],[34,322584],{},[15,322586,322587,322588,322657,322658,322717],{},"这道题利用了 ",[34,322589,322591,322608],{"className":322590},[31642],[34,322592,322594],{"className":322593},[31646],[31648,322595,322596],{"xmlns":31650},[31652,322597,322598,322606],{},[31655,322599,322600],{},[31743,322601,322602,322604],{},[31658,322603,54151],{},[31658,322605,31660],{},[31665,322607,321085],{"encoding":31667},[34,322609,322611],{"className":322610,"ariaHidden":758},[31672],[34,322612,322614,322617],{"className":322613},[31676],[34,322615],{"className":322616,"style":70915},[31680],[34,322618,322620,322623],{"className":322619},[31685],[34,322621,54151],{"className":322622},[31685,31686],[34,322624,322626],{"className":322625},[31792],[34,322627,322629,322649],{"className":322628},[31796,31797],[34,322630,322632,322646],{"className":322631},[31801],[34,322633,322635],{"className":322634,"style":32362},[31805],[34,322636,322637,322640],{"style":31809},[34,322638],{"className":322639,"style":31814},[31813],[34,322641,322643],{"className":322642},[31818,31819,31820,31821],[34,322644,31660],{"className":322645},[31685,31686,31821],[34,322647,31829],{"className":322648},[31828],[34,322650,322652],{"className":322651},[31801],[34,322653,322655],{"className":322654,"style":31836},[31805],[34,322656],{},"（偶置换群）是由“3-轮换” ",[34,322659,322661,322687],{"className":322660},[31642],[34,322662,322664],{"className":322663},[31646],[31648,322665,322666],{"xmlns":31650},[31652,322667,322668,322684],{},[31655,322669,322670,322672,322674,322676,322678,322680,322682],{},[31662,322671,456],{"stretchy":702},[31658,322673,31955],{},[39361,322675,130298],{},[31658,322677,31968],{},[39361,322679,130298],{},[31658,322681,33759],{},[31662,322683,2557],{"stretchy":702},[31665,322685,322686],{"encoding":31667},"(i\\ j\\ k)",[34,322688,322690],{"className":322689,"ariaHidden":758},[31672],[34,322691,322693,322696,322699,322702,322705,322708,322711,322714],{"className":322692},[31676],[34,322694],{"className":322695,"style":31774},[31680],[34,322697,456],{"className":322698},[31782],[34,322700,31955],{"className":322701},[31685,31686],[34,322703,130298],{"className":322704},[31845],[34,322706,31968],{"className":322707,"style":32153},[31685,31686],[34,322709,130298],{"className":322710},[31845],[34,322712,33759],{"className":322713,"style":33774},[31685,31686],[34,322715,2557],{"className":322716},[31690]," 生成的事实。",[105,322719,322720,322782],{},[108,322721,322722,322781],{},[34,322723,322725,322751],{"className":322724},[31642],[34,322726,322728],{"className":322727},[31646],[31648,322729,322730],{"xmlns":31650},[31652,322731,322732,322748],{},[31655,322733,322734,322736,322738,322740,322742,322744,322746],{},[31662,322735,456],{"stretchy":702},[31749,322737,640],{},[39361,322739,130298],{},[31749,322741,768],{},[39361,322743,130298],{},[31749,322745,786],{},[31662,322747,2557],{"stretchy":702},[31665,322749,322750],{"encoding":31667},"(1\\ 2\\ 3)",[34,322752,322754],{"className":322753,"ariaHidden":758},[31672],[34,322755,322757,322760,322763,322766,322769,322772,322775,322778],{"className":322756},[31676],[34,322758],{"className":322759,"style":31774},[31680],[34,322761,456],{"className":322762},[31782],[34,322764,640],{"className":322765},[31685],[34,322767,130298],{"className":322768},[31845],[34,322770,768],{"className":322771},[31685],[34,322773,130298],{"className":322774},[31845],[34,322776,786],{"className":322777},[31685],[34,322779,2557],{"className":322780},[31690]," 是一个 3-轮换。",[108,322783,322784,322785,258,322843,322918,322919,322947],{},"如果你有 ",[34,322786,322788,322813],{"className":322787},[31642],[34,322789,322791],{"className":322790},[31646],[31648,322792,322793],{"xmlns":31650},[31652,322794,322795,322811],{},[31655,322796,322797,322799,322801,322803,322805,322807,322809],{},[31662,322798,456],{"stretchy":702},[31749,322800,640],{},[39361,322802,130298],{},[31749,322804,768],{},[39361,322806,130298],{},[31749,322808,786],{},[31662,322810,2557],{"stretchy":702},[31665,322812,322750],{"encoding":31667},[34,322814,322816],{"className":322815,"ariaHidden":758},[31672],[34,322817,322819,322822,322825,322828,322831,322834,322837,322840],{"className":322818},[31676],[34,322820],{"className":322821,"style":31774},[31680],[34,322823,456],{"className":322824},[31782],[34,322826,640],{"className":322827},[31685],[34,322829,130298],{"className":322830},[31845],[34,322832,768],{"className":322833},[31685],[34,322835,130298],{"className":322836},[31845],[34,322838,786],{"className":322839},[31685],[34,322841,2557],{"className":322842},[31690],[34,322844,322846,322876],{"className":322845},[31642],[34,322847,322849],{"className":322848},[31646],[31648,322850,322851],{"xmlns":31650},[31652,322852,322853,322873],{},[31655,322854,322855,322857,322859,322861,322863,322865,322867,322869,322871],{},[31662,322856,456],{"stretchy":702},[31749,322858,640],{},[39361,322860,130298],{},[31749,322862,768],{},[39361,322864,130298],{},[31662,322866,33934],{},[39361,322868,130298],{},[31658,322870,31660],{},[31662,322872,2557],{"stretchy":702},[31665,322874,322875],{"encoding":31667},"(1\\ 2\\ \\dots\\ n)",[34,322877,322879],{"className":322878,"ariaHidden":758},[31672],[34,322880,322882,322885,322888,322891,322894,322897,322900,322903,322906,322909,322912,322915],{"className":322881},[31676],[34,322883],{"className":322884,"style":31774},[31680],[34,322886,456],{"className":322887},[31782],[34,322889,640],{"className":322890},[31685],[34,322892,130298],{"className":322893},[31845],[34,322895,768],{"className":322896},[31685],[34,322898,130298],{"className":322899},[31845],[34,322901],{"className":322902,"style":32123},[31845],[34,322904,33934],{"className":322905},[34006],[34,322907,130298],{"className":322908},[31845],[34,322910],{"className":322911,"style":32123},[31845],[34,322913,31660],{"className":322914},[31685,31686],[34,322916,2557],{"className":322917},[31690],"（当 ",[34,322920,322922,322935],{"className":322921},[31642],[34,322923,322925],{"className":322924},[31646],[31648,322926,322927],{"xmlns":31650},[31652,322928,322929,322933],{},[31655,322930,322931],{},[31658,322932,31660],{},[31665,322934,31660],{"encoding":31667},[34,322936,322938],{"className":322937,"ariaHidden":758},[31672],[34,322939,322941,322944],{"className":322940},[31676],[34,322942],{"className":322943,"style":32527},[31680],[34,322945,31660],{"className":322946},[31685,31686]," 为奇数时，它是偶置换），你就可以像上面一样，通过“推移”造出所有的 3-轮换。",[32188,322949],{},[253,322951,322953],{"id":322952},"第三部分正规子群与中心化子抽象篇","第三部分：正规子群与中心化子（抽象篇）",[32490,322955,322957],{"id":322956},"题目-14中心化子与共轭类","题目 14：中心化子与共轭类",[15,322959,322960,69123,322962,323058,323059,323223,323224,323288],{},[2627,322961,101191],{},[34,322963,322965,322998],{"className":322964},[31642],[34,322966,322968],{"className":322967},[31646],[31648,322969,322970],{"xmlns":31650},[31652,322971,322972,322996],{},[31655,322973,322974,322976,322978,322980,322982,322984,322986,322988,322990,322992,322994],{},[31658,322975,115372],{},[31662,322977,202],{},[31662,322979,456],{"stretchy":702},[31749,322981,640],{},[39361,322983,130298],{},[31749,322985,768],{},[39361,322987,130298],{},[31662,322989,38874],{},[39361,322991,130298],{},[31658,322993,31660],{},[31662,322995,2557],{"stretchy":702},[31665,322997,321568],{"encoding":31667},[34,322999,323001,323019],{"className":323000,"ariaHidden":758},[31672],[34,323002,323004,323007,323010,323013,323016],{"className":323003},[31676],[34,323005],{"className":323006,"style":32527},[31680],[34,323008,115372],{"className":323009,"style":32062},[31685,31686],[34,323011],{"className":323012,"style":31846},[31845],[34,323014,202],{"className":323015},[31850],[34,323017],{"className":323018,"style":31846},[31845],[34,323020,323022,323025,323028,323031,323034,323037,323040,323043,323046,323049,323052,323055],{"className":323021},[31676],[34,323023],{"className":323024,"style":31774},[31680],[34,323026,456],{"className":323027},[31782],[34,323029,640],{"className":323030},[31685],[34,323032,130298],{"className":323033},[31845],[34,323035,768],{"className":323036},[31685],[34,323038,130298],{"className":323039},[31845],[34,323041],{"className":323042,"style":32123},[31845],[34,323044,38874],{"className":323045},[34006],[34,323047,130298],{"className":323048},[31845],[34,323050],{"className":323051,"style":32123},[31845],[34,323053,31660],{"className":323054},[31685,31686],[34,323056,2557],{"className":323057},[31690],"，证明其中心化子 ",[34,323060,323062,323098],{"className":323061},[31642],[34,323063,323065],{"className":323064},[31646],[31648,323066,323067],{"xmlns":31650},[31652,323068,323069,323095],{},[31655,323070,323071,323081,323083,323085,323087,323089,323091,323093],{},[31743,323072,323073,323075],{},[31658,323074,47384],{},[31743,323076,323077,323079],{},[31658,323078,94675],{},[31658,323080,31660],{},[31662,323082,456],{"stretchy":702},[31658,323084,115372],{},[31662,323086,2557],{"stretchy":702},[31662,323088,202],{},[31662,323090,126529],{"stretchy":702},[31658,323092,115372],{},[31662,323094,126544],{"stretchy":702},[31665,323096,323097],{"encoding":31667},"C_{S_n}(\\sigma) = \\langle \\sigma \\rangle",[34,323099,323101,323208],{"className":323100,"ariaHidden":758},[31672],[34,323102,323104,323108,323190,323193,323196,323199,323202,323205],{"className":323103},[31676],[34,323105],{"className":323106,"style":323107},[31680],"height:1.0001em;vertical-align:-0.2501em;",[34,323109,323111,323114],{"className":323110},[31685],[34,323112,47384],{"className":323113,"style":38706},[31685,31686],[34,323115,323117],{"className":323116},[31792],[34,323118,323120,323181],{"className":323119},[31796,31797],[34,323121,323123,323178],{"className":323122},[31801],[34,323124,323126],{"className":323125,"style":116558},[31805],[34,323127,323128,323131],{"style":129511},[34,323129],{"className":323130,"style":31814},[31813],[34,323132,323134],{"className":323133},[31818,31819,31820,31821],[34,323135,323137],{"className":323136},[31685,31821],[34,323138,323140,323143],{"className":323139},[31685,31821],[34,323141,94675],{"className":323142,"style":106455},[31685,31686,31821],[34,323144,323146],{"className":323145},[31792],[34,323147,323149,323170],{"className":323148},[31796,31797],[34,323150,323152,323167],{"className":323151},[31801],[34,323153,323155],{"className":323154,"style":220518},[31805],[34,323156,323158,323161],{"style":323157},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[34,323159],{"className":323160,"style":43406},[31813],[34,323162,323164],{"className":323163},[31818,43410,43411,31821],[34,323165,31660],{"className":323166},[31685,31686,31821],[34,323168,31829],{"className":323169},[31828],[34,323171,323173],{"className":323172},[31801],[34,323174,323176],{"className":323175,"style":220540},[31805],[34,323177],{},[34,323179,31829],{"className":323180},[31828],[34,323182,323184],{"className":323183},[31801],[34,323185,323188],{"className":323186,"style":323187},[31805],"height:0.2501em;",[34,323189],{},[34,323191,456],{"className":323192},[31782],[34,323194,115372],{"className":323195,"style":32062},[31685,31686],[34,323197,2557],{"className":323198},[31690],[34,323200],{"className":323201,"style":31846},[31845],[34,323203,202],{"className":323204},[31850],[34,323206],{"className":323207,"style":31846},[31845],[34,323209,323211,323214,323217,323220],{"className":323210},[31676],[34,323212],{"className":323213,"style":31774},[31680],[34,323215,126529],{"className":323216},[31782],[34,323218,115372],{"className":323219,"style":32062},[31685,31686],[34,323221,126544],{"className":323222},[31690],"，共轭类含 ",[34,323225,323227,323251],{"className":323226},[31642],[34,323228,323230],{"className":323229},[31646],[31648,323231,323232],{"xmlns":31650},[31652,323233,323234,323248],{},[31655,323235,323236,323238,323240,323242,323244,323246],{},[31662,323237,456],{"stretchy":702},[31658,323239,31660],{},[31662,323241,32226],{},[31749,323243,640],{},[31662,323245,2557],{"stretchy":702},[31662,323247,5504],{"stretchy":702},[31665,323249,323250],{"encoding":31667},"(n-1)!",[34,323252,323254,323275],{"className":323253,"ariaHidden":758},[31672],[34,323255,323257,323260,323263,323266,323269,323272],{"className":323256},[31676],[34,323258],{"className":323259,"style":31774},[31680],[34,323261,456],{"className":323262},[31782],[34,323264,31660],{"className":323265},[31685,31686],[34,323267],{"className":323268,"style":32604},[31845],[34,323270,32226],{"className":323271},[32024],[34,323273],{"className":323274,"style":32604},[31845],[34,323276,323278,323281,323284],{"className":323277},[31676],[34,323279],{"className":323280,"style":31774},[31680],[34,323282,640],{"className":323283},[31685],[34,323285,323287],{"className":323286},[31690],")!"," 个元素。",[15,323290,323291],{},[2627,323292,323293],{},"【证明详解】",[2230,323295,323296,324038,324882],{},[108,323297,323298,2664,323301],{},[2627,323299,323300],{},"共轭类的大小",[105,323302,323303,323416,323575,323845],{},[108,323304,2884,323305,323374,323375,323415],{},[34,323306,323308,323325],{"className":323307},[31642],[34,323309,323311],{"className":323310},[31646],[31648,323312,323313],{"xmlns":31650},[31652,323314,323315,323323],{},[31655,323316,323317],{},[31743,323318,323319,323321],{},[31658,323320,94675],{},[31658,323322,31660],{},[31665,323324,218979],{"encoding":31667},[34,323326,323328],{"className":323327,"ariaHidden":758},[31672],[34,323329,323331,323334],{"className":323330},[31676],[34,323332],{"className":323333,"style":70915},[31680],[34,323335,323337,323340],{"className":323336},[31685],[34,323338,94675],{"className":323339,"style":106455},[31685,31686],[34,323341,323343],{"className":323342},[31792],[34,323344,323346,323366],{"className":323345},[31796,31797],[34,323347,323349,323363],{"className":323348},[31801],[34,323350,323352],{"className":323351,"style":32362},[31805],[34,323353,323354,323357],{"style":206422},[34,323355],{"className":323356,"style":31814},[31813],[34,323358,323360],{"className":323359},[31818,31819,31820,31821],[34,323361,31660],{"className":323362},[31685,31686,31821],[34,323364,31829],{"className":323365},[31828],[34,323367,323369],{"className":323368},[31801],[34,323370,323372],{"className":323371,"style":31836},[31805],[34,323373],{}," 中，两个置换共轭 ",[34,323376,323378,323397],{"className":323377},[31642],[34,323379,323381],{"className":323380},[31646],[31648,323382,323383],{"xmlns":31650},[31652,323384,323385,323394],{},[31655,323386,323387,323389,323392],{},[39361,323388,262622],{},[31662,323390,323391],{},"⟺",[39361,323393,262622],{},[31665,323395,323396],{"encoding":31667},"\\iff",[34,323398,323400],{"className":323399,"ariaHidden":758},[31672],[34,323401,323403,323406,323409,323412],{"className":323402},[31676],[34,323404],{"className":323405,"style":318264},[31680],[34,323407],{"className":323408,"style":31846},[31845],[34,323410,323391],{"className":323411},[31850],[34,323413],{"className":323414,"style":31846},[31845]," 它们的循环结构相同。",[108,323417,323418,323446,323447,323475,323476,323545,323546,323574],{},[34,323419,323421,323434],{"className":323420},[31642],[34,323422,323424],{"className":323423},[31646],[31648,323425,323426],{"xmlns":31650},[31652,323427,323428,323432],{},[31655,323429,323430],{},[31658,323431,115372],{},[31665,323433,115375],{"encoding":31667},[34,323435,323437],{"className":323436,"ariaHidden":758},[31672],[34,323438,323440,323443],{"className":323439},[31676],[34,323441],{"className":323442,"style":32527},[31680],[34,323444,115372],{"className":323445,"style":32062},[31685,31686]," 是一个 ",[34,323448,323450,323463],{"className":323449},[31642],[34,323451,323453],{"className":323452},[31646],[31648,323454,323455],{"xmlns":31650},[31652,323456,323457,323461],{},[31655,323458,323459],{},[31658,323460,31660],{},[31665,323462,31660],{"encoding":31667},[34,323464,323466],{"className":323465,"ariaHidden":758},[31672],[34,323467,323469,323472],{"className":323468},[31676],[34,323470],{"className":323471,"style":32527},[31680],[34,323473,31660],{"className":323474},[31685,31686],"-长轮换。在 ",[34,323477,323479,323496],{"className":323478},[31642],[34,323480,323482],{"className":323481},[31646],[31648,323483,323484],{"xmlns":31650},[31652,323485,323486,323494],{},[31655,323487,323488],{},[31743,323489,323490,323492],{},[31658,323491,94675],{},[31658,323493,31660],{},[31665,323495,218979],{"encoding":31667},[34,323497,323499],{"className":323498,"ariaHidden":758},[31672],[34,323500,323502,323505],{"className":323501},[31676],[34,323503],{"className":323504,"style":70915},[31680],[34,323506,323508,323511],{"className":323507},[31685],[34,323509,94675],{"className":323510,"style":106455},[31685,31686],[34,323512,323514],{"className":323513},[31792],[34,323515,323517,323537],{"className":323516},[31796,31797],[34,323518,323520,323534],{"className":323519},[31801],[34,323521,323523],{"className":323522,"style":32362},[31805],[34,323524,323525,323528],{"style":206422},[34,323526],{"className":323527,"style":31814},[31813],[34,323529,323531],{"className":323530},[31818,31819,31820,31821],[34,323532,31660],{"className":323533},[31685,31686,31821],[34,323535,31829],{"className":323536},[31828],[34,323538,323540],{"className":323539},[31801],[34,323541,323543],{"className":323542,"style":31836},[31805],[34,323544],{}," 中，有多少个长为 ",[34,323547,323549,323562],{"className":323548},[31642],[34,323550,323552],{"className":323551},[31646],[31648,323553,323554],{"xmlns":31650},[31652,323555,323556,323560],{},[31655,323557,323558],{},[31658,323559,31660],{},[31665,323561,31660],{"encoding":31667},[34,323563,323565],{"className":323564,"ariaHidden":758},[31672],[34,323566,323568,323571],{"className":323567},[31676],[34,323569],{"className":323570,"style":32527},[31680],[34,323572,31660],{"className":323573},[31685,31686]," 的轮换？",[108,323576,323577,323578,53834,323606,323634,323635,323668,323669,258,323727,323786,323787,323815,323816,323844],{},"排列 ",[34,323579,323581,323594],{"className":323580},[31642],[34,323582,323584],{"className":323583},[31646],[31648,323585,323586],{"xmlns":31650},[31652,323587,323588,323592],{},[31655,323589,323590],{},[31749,323591,640],{},[31665,323593,640],{"encoding":31667},[34,323595,323597],{"className":323596,"ariaHidden":758},[31672],[34,323598,323600,323603],{"className":323599},[31676],[34,323601],{"className":323602,"style":32617},[31680],[34,323604,640],{"className":323605},[31685],[34,323607,323609,323622],{"className":323608},[31642],[34,323610,323612],{"className":323611},[31646],[31648,323613,323614],{"xmlns":31650},[31652,323615,323616,323620],{},[31655,323617,323618],{},[31658,323619,31660],{},[31665,323621,31660],{"encoding":31667},[34,323623,323625],{"className":323624,"ariaHidden":758},[31672],[34,323626,323628,323631],{"className":323627},[31676],[34,323629],{"className":323630,"style":32527},[31680],[34,323632,31660],{"className":323633},[31685,31686]," 有 ",[34,323636,323638,323653],{"className":323637},[31642],[34,323639,323641],{"className":323640},[31646],[31648,323642,323643],{"xmlns":31650},[31652,323644,323645,323651],{},[31655,323646,323647,323649],{},[31658,323648,31660],{},[31662,323650,5504],{"stretchy":702},[31665,323652,31668],{"encoding":31667},[34,323654,323656],{"className":323655,"ariaHidden":758},[31672],[34,323657,323659,323662,323665],{"className":323658},[31676],[34,323660],{"className":323661,"style":31681},[31680],[34,323663,31660],{"className":323664},[31685,31686],[34,323666,5504],{"className":323667},[31690]," 种排法。但是轮换是圆的，",[34,323670,323672,323697],{"className":323671},[31642],[34,323673,323675],{"className":323674},[31646],[31648,323676,323677],{"xmlns":31650},[31652,323678,323679,323695],{},[31655,323680,323681,323683,323685,323687,323689,323691,323693],{},[31662,323682,456],{"stretchy":702},[31749,323684,640],{},[39361,323686,130298],{},[31749,323688,768],{},[39361,323690,130298],{},[31749,323692,786],{},[31662,323694,2557],{"stretchy":702},[31665,323696,322750],{"encoding":31667},[34,323698,323700],{"className":323699,"ariaHidden":758},[31672],[34,323701,323703,323706,323709,323712,323715,323718,323721,323724],{"className":323702},[31676],[34,323704],{"className":323705,"style":31774},[31680],[34,323707,456],{"className":323708},[31782],[34,323710,640],{"className":323711},[31685],[34,323713,130298],{"className":323714},[31845],[34,323716,768],{"className":323717},[31685],[34,323719,130298],{"className":323720},[31845],[34,323722,786],{"className":323723},[31685],[34,323725,2557],{"className":323726},[31690],[34,323728,323730,323756],{"className":323729},[31642],[34,323731,323733],{"className":323732},[31646],[31648,323734,323735],{"xmlns":31650},[31652,323736,323737,323753],{},[31655,323738,323739,323741,323743,323745,323747,323749,323751],{},[31662,323740,456],{"stretchy":702},[31749,323742,768],{},[39361,323744,130298],{},[31749,323746,786],{},[39361,323748,130298],{},[31749,323750,640],{},[31662,323752,2557],{"stretchy":702},[31665,323754,323755],{"encoding":31667},"(2\\ 3\\ 1)",[34,323757,323759],{"className":323758,"ariaHidden":758},[31672],[34,323760,323762,323765,323768,323771,323774,323777,323780,323783],{"className":323761},[31676],[34,323763],{"className":323764,"style":31774},[31680],[34,323766,456],{"className":323767},[31782],[34,323769,768],{"className":323770},[31685],[34,323772,130298],{"className":323773},[31845],[34,323775,786],{"className":323776},[31685],[34,323778,130298],{"className":323779},[31845],[34,323781,640],{"className":323782},[31685],[34,323784,2557],{"className":323785},[31690]," 是一样的。一个 ",[34,323788,323790,323803],{"className":323789},[31642],[34,323791,323793],{"className":323792},[31646],[31648,323794,323795],{"xmlns":31650},[31652,323796,323797,323801],{},[31655,323798,323799],{},[31658,323800,31660],{},[31665,323802,31660],{"encoding":31667},[34,323804,323806],{"className":323805,"ariaHidden":758},[31672],[34,323807,323809,323812],{"className":323808},[31676],[34,323810],{"className":323811,"style":32527},[31680],[34,323813,31660],{"className":323814},[31685,31686],"-轮换有 ",[34,323817,323819,323832],{"className":323818},[31642],[34,323820,323822],{"className":323821},[31646],[31648,323823,323824],{"xmlns":31650},[31652,323825,323826,323830],{},[31655,323827,323828],{},[31658,323829,31660],{},[31665,323831,31660],{"encoding":31667},[34,323833,323835],{"className":323834,"ariaHidden":758},[31672],[34,323836,323838,323841],{"className":323837},[31676],[34,323839],{"className":323840,"style":32527},[31680],[34,323842,31660],{"className":323843},[31685,31686]," 种写法。",[108,323846,323847,323848,323876,323877,347],{},"所以，",[34,323849,323851,323864],{"className":323850},[31642],[34,323852,323854],{"className":323853},[31646],[31648,323855,323856],{"xmlns":31650},[31652,323857,323858,323862],{},[31655,323859,323860],{},[31658,323861,31660],{},[31665,323863,31660],{"encoding":31667},[34,323865,323867],{"className":323866,"ariaHidden":758},[31672],[34,323868,323870,323873],{"className":323869},[31676],[34,323871],{"className":323872,"style":32527},[31680],[34,323874,31660],{"className":323875},[31685,31686],"-轮换的总数（共轭类大小） = ",[34,323878,323880,323916],{"className":323879},[31642],[34,323881,323883],{"className":323882},[31646],[31648,323884,323885],{"xmlns":31650},[31652,323886,323887,323913],{},[31655,323888,323889,323899,323901,323903,323905,323907,323909,323911],{},[41453,323890,323891,323897],{},[31655,323892,323893,323895],{},[31658,323894,31660],{},[31662,323896,5504],{"stretchy":702},[31658,323898,31660],{},[31662,323900,202],{},[31662,323902,456],{"stretchy":702},[31658,323904,31660],{},[31662,323906,32226],{},[31749,323908,640],{},[31662,323910,2557],{"stretchy":702},[31662,323912,5504],{"stretchy":702},[31665,323914,323915],{"encoding":31667},"\\frac{n!}{n} = (n-1)!",[34,323917,323919,324005,324026],{"className":323918,"ariaHidden":758},[31672],[34,323920,323922,323925,323996,323999,324002],{"className":323921},[31676],[34,323923],{"className":323924,"style":232492},[31680],[34,323926,323928,323931,323993],{"className":323927},[31685],[34,323929],{"className":323930},[31782,41488],[34,323932,323934],{"className":323933},[41453],[34,323935,323937,323985],{"className":323936},[31796,31797],[34,323938,323940,323982],{"className":323939},[31801],[34,323941,323943,323957,323965],{"className":323942,"style":232511},[31805],[34,323944,323945,323948],{"style":41504},[34,323946],{"className":323947,"style":41508},[31813],[34,323949,323951],{"className":323950},[31818,31819,31820,31821],[34,323952,323954],{"className":323953},[31685,31821],[34,323955,31660],{"className":323956},[31685,31686,31821],[34,323958,323959,323962],{"style":41520},[34,323960],{"className":323961,"style":41508},[31813],[34,323963],{"className":323964,"style":41528},[41527],[34,323966,323967,323970],{"style":46771},[34,323968],{"className":323969,"style":41508},[31813],[34,323971,323973],{"className":323972},[31818,31819,31820,31821],[34,323974,323976,323979],{"className":323975},[31685,31821],[34,323977,31660],{"className":323978},[31685,31686,31821],[34,323980,5504],{"className":323981},[31690,31821],[34,323983,31829],{"className":323984},[31828],[34,323986,323988],{"className":323987},[31801],[34,323989,323991],{"className":323990,"style":41619},[31805],[34,323992],{},[34,323994],{"className":323995},[31690,41488],[34,323997],{"className":323998,"style":31846},[31845],[34,324000,202],{"className":324001},[31850],[34,324003],{"className":324004,"style":31846},[31845],[34,324006,324008,324011,324014,324017,324020,324023],{"className":324007},[31676],[34,324009],{"className":324010,"style":31774},[31680],[34,324012,456],{"className":324013},[31782],[34,324015,31660],{"className":324016},[31685,31686],[34,324018],{"className":324019,"style":32604},[31845],[34,324021,32226],{"className":324022},[32024],[34,324024],{"className":324025,"style":32604},[31845],[34,324027,324029,324032,324035],{"className":324028},[31676],[34,324030],{"className":324031,"style":31774},[31680],[34,324033,640],{"className":324034},[31685],[34,324036,323287],{"className":324037},[31690],[108,324039,324040,2664,324043],{},[2627,324041,324042],{},"轨道-稳定子定理（Orbit-Stabilizer Theorem）",[105,324044,324045,324204,324369,324594],{},[108,324046,324047,324048,347],{},"这是群论最重要的公式之一：",[34,324049,324051,324107],{"className":324050},[31642],[34,324052,324054],{"className":324053},[31646],[31648,324055,324056],{"xmlns":31650},[31652,324057,324058,324104],{},[31655,324059,324060,324062,324064,324066,324068,324070,324072,324074,324076,324078,324080,324082,324084,324086,324088,324090,324092,324094,324096,324098,324100,324102],{},[31658,324061,162086],{"mathvariant":38079},[31658,324063,97709],{},[31658,324065,162086],{"mathvariant":38079},[31662,324067,202],{},[31658,324069,162086],{"mathvariant":38079},[31658,324071,37890],{},[31658,324073,34395],{},[31658,324075,1697],{},[31662,324077,456],{"stretchy":702},[31658,324079,115372],{},[31662,324081,2557],{"stretchy":702},[31658,324083,162086],{"mathvariant":38079},[31662,324085,71191],{},[31658,324087,162086],{"mathvariant":38079},[31658,324089,94675],{},[31658,324091,33249],{},[31658,324093,111],{},[31658,324095,1697],{},[31662,324097,456],{"stretchy":702},[31658,324099,115372],{},[31662,324101,2557],{"stretchy":702},[31658,324103,162086],{"mathvariant":38079},[31665,324105,324106],{"encoding":31667},"|G| = |Orb(\\sigma)| \\cdot |Stab(\\sigma)|",[34,324108,324110,324134,324173],{"className":324109,"ariaHidden":758},[31672],[34,324111,324113,324116,324119,324122,324125,324128,324131],{"className":324112},[31676],[34,324114],{"className":324115,"style":31774},[31680],[34,324117,162086],{"className":324118},[31685],[34,324120,97709],{"className":324121},[31685,31686],[34,324123,162086],{"className":324124},[31685],[34,324126],{"className":324127,"style":31846},[31845],[34,324129,202],{"className":324130},[31850],[34,324132],{"className":324133,"style":31846},[31845],[34,324135,324137,324140,324143,324146,324149,324152,324155,324158,324161,324164,324167,324170],{"className":324136},[31676],[34,324138],{"className":324139,"style":31774},[31680],[34,324141,162086],{"className":324142},[31685],[34,324144,37890],{"className":324145,"style":34452},[31685,31686],[34,324147,34395],{"className":324148,"style":34452},[31685,31686],[34,324150,1697],{"className":324151},[31685,31686],[34,324153,456],{"className":324154},[31782],[34,324156,115372],{"className":324157,"style":32062},[31685,31686],[34,324159,2557],{"className":324160},[31690],[34,324162,162086],{"className":324163},[31685],[34,324165],{"className":324166,"style":32604},[31845],[34,324168,71191],{"className":324169},[32024],[34,324171],{"className":324172,"style":32604},[31845],[34,324174,324176,324179,324182,324185,324188,324192,324195,324198,324201],{"className":324175},[31676],[34,324177],{"className":324178,"style":31774},[31680],[34,324180,162086],{"className":324181},[31685],[34,324183,94675],{"className":324184,"style":106455},[31685,31686],[34,324186,33249],{"className":324187},[31685,31686],[34,324189,324191],{"className":324190},[31685,31686],"ab",[34,324193,456],{"className":324194},[31782],[34,324196,115372],{"className":324197,"style":32062},[31685,31686],[34,324199,2557],{"className":324200},[31690],[34,324202,162086],{"className":324203},[31685],[108,324205,324206,324207,347],{},"在这里：",[34,324208,324210,324254],{"className":324209},[31642],[34,324211,324213],{"className":324212},[31646],[31648,324214,324215],{"xmlns":31650},[31652,324216,324217,324251],{},[31655,324218,324219,324221,324227,324229,324231,324233,324235,324238,324240,324242,324244,324246,324249],{},[31658,324220,162086],{"mathvariant":38079},[31743,324222,324223,324225],{},[31658,324224,94675],{},[31658,324226,31660],{},[31658,324228,162086],{"mathvariant":38079},[31662,324230,202],{},[31662,324232,456],{"stretchy":702},[31658,324234,115372],{},[39361,324236,324237],{},"的共轭类大小",[31662,324239,2557],{"stretchy":702},[31662,324241,71191],{},[31662,324243,456],{"stretchy":702},[31658,324245,115372],{},[39361,324247,324248],{},"的中心化子阶数",[31662,324250,2557],{"stretchy":702},[31665,324252,324253],{"encoding":31667},"|S_n| = (\\sigma \\text{的共轭类大小}) \\cdot (\\sigma \\text{的中心化子阶数})",[34,324255,324257,324318,324348],{"className":324256,"ariaHidden":758},[31672],[34,324258,324260,324263,324266,324306,324309,324312,324315],{"className":324259},[31676],[34,324261],{"className":324262,"style":31774},[31680],[34,324264,162086],{"className":324265},[31685],[34,324267,324269,324272],{"className":324268},[31685],[34,324270,94675],{"className":324271,"style":106455},[31685,31686],[34,324273,324275],{"className":324274},[31792],[34,324276,324278,324298],{"className":324277},[31796,31797],[34,324279,324281,324295],{"className":324280},[31801],[34,324282,324284],{"className":324283,"style":32362},[31805],[34,324285,324286,324289],{"style":206422},[34,324287],{"className":324288,"style":31814},[31813],[34,324290,324292],{"className":324291},[31818,31819,31820,31821],[34,324293,31660],{"className":324294},[31685,31686,31821],[34,324296,31829],{"className":324297},[31828],[34,324299,324301],{"className":324300},[31801],[34,324302,324304],{"className":324303,"style":31836},[31805],[34,324305],{},[34,324307,162086],{"className":324308},[31685],[34,324310],{"className":324311,"style":31846},[31845],[34,324313,202],{"className":324314},[31850],[34,324316],{"className":324317,"style":31846},[31845],[34,324319,324321,324324,324327,324330,324336,324339,324342,324345],{"className":324320},[31676],[34,324322],{"className":324323,"style":31774},[31680],[34,324325,456],{"className":324326},[31782],[34,324328,115372],{"className":324329,"style":32062},[31685,31686],[34,324331,324333],{"className":324332},[31685,2638],[34,324334,324237],{"className":324335},[31685,105462],[34,324337,2557],{"className":324338},[31690],[34,324340],{"className":324341,"style":32604},[31845],[34,324343,71191],{"className":324344},[32024],[34,324346],{"className":324347,"style":32604},[31845],[34,324349,324351,324354,324357,324360,324366],{"className":324350},[31676],[34,324352],{"className":324353,"style":31774},[31680],[34,324355,456],{"className":324356},[31782],[34,324358,115372],{"className":324359,"style":32062},[31685,31686],[34,324361,324363],{"className":324362},[31685,2638],[34,324364,324248],{"className":324365},[31685,105462],[34,324367,2557],{"className":324368},[31690],[108,324370,324371,324372],{},"代入数值：\n",[34,324373,324375,324427],{"className":324374},[31642],[34,324376,324378],{"className":324377},[31646],[31648,324379,324380],{"xmlns":31650},[31652,324381,324382,324424],{},[31655,324383,324384,324386,324388,324390,324392,324394,324396,324398,324400,324402,324404,324406,324416,324418,324420,324422],{},[31658,324385,31660],{},[31662,324387,5504],{"stretchy":702},[31662,324389,202],{},[31662,324391,456],{"stretchy":702},[31658,324393,31660],{},[31662,324395,32226],{},[31749,324397,640],{},[31662,324399,2557],{"stretchy":702},[31662,324401,5504],{"stretchy":702},[31662,324403,71191],{},[31658,324405,162086],{"mathvariant":38079},[31743,324407,324408,324410],{},[31658,324409,47384],{},[31743,324411,324412,324414],{},[31658,324413,94675],{},[31658,324415,31660],{},[31662,324417,456],{"stretchy":702},[31658,324419,115372],{},[31662,324421,2557],{"stretchy":702},[31658,324423,162086],{"mathvariant":38079},[31665,324425,324426],{"encoding":31667},"n! = (n-1)! \\cdot |C_{S_n}(\\sigma)|",[34,324428,324430,324451,324472,324493],{"className":324429,"ariaHidden":758},[31672],[34,324431,324433,324436,324439,324442,324445,324448],{"className":324432},[31676],[34,324434],{"className":324435,"style":31681},[31680],[34,324437,31660],{"className":324438},[31685,31686],[34,324440,5504],{"className":324441},[31690],[34,324443],{"className":324444,"style":31846},[31845],[34,324446,202],{"className":324447},[31850],[34,324449],{"className":324450,"style":31846},[31845],[34,324452,324454,324457,324460,324463,324466,324469],{"className":324453},[31676],[34,324455],{"className":324456,"style":31774},[31680],[34,324458,456],{"className":324459},[31782],[34,324461,31660],{"className":324462},[31685,31686],[34,324464],{"className":324465,"style":32604},[31845],[34,324467,32226],{"className":324468},[32024],[34,324470],{"className":324471,"style":32604},[31845],[34,324473,324475,324478,324481,324484,324487,324490],{"className":324474},[31676],[34,324476],{"className":324477,"style":31774},[31680],[34,324479,640],{"className":324480},[31685],[34,324482,323287],{"className":324483},[31690],[34,324485],{"className":324486,"style":32604},[31845],[34,324488,71191],{"className":324489},[32024],[34,324491],{"className":324492,"style":32604},[31845],[34,324494,324496,324499,324502,324582,324585,324588,324591],{"className":324495},[31676],[34,324497],{"className":324498,"style":323107},[31680],[34,324500,162086],{"className":324501},[31685],[34,324503,324505,324508],{"className":324504},[31685],[34,324506,47384],{"className":324507,"style":38706},[31685,31686],[34,324509,324511],{"className":324510},[31792],[34,324512,324514,324574],{"className":324513},[31796,31797],[34,324515,324517,324571],{"className":324516},[31801],[34,324518,324520],{"className":324519,"style":116558},[31805],[34,324521,324522,324525],{"style":129511},[34,324523],{"className":324524,"style":31814},[31813],[34,324526,324528],{"className":324527},[31818,31819,31820,31821],[34,324529,324531],{"className":324530},[31685,31821],[34,324532,324534,324537],{"className":324533},[31685,31821],[34,324535,94675],{"className":324536,"style":106455},[31685,31686,31821],[34,324538,324540],{"className":324539},[31792],[34,324541,324543,324563],{"className":324542},[31796,31797],[34,324544,324546,324560],{"className":324545},[31801],[34,324547,324549],{"className":324548,"style":220518},[31805],[34,324550,324551,324554],{"style":323157},[34,324552],{"className":324553,"style":43406},[31813],[34,324555,324557],{"className":324556},[31818,43410,43411,31821],[34,324558,31660],{"className":324559},[31685,31686,31821],[34,324561,31829],{"className":324562},[31828],[34,324564,324566],{"className":324565},[31801],[34,324567,324569],{"className":324568,"style":220540},[31805],[34,324570],{},[34,324572,31829],{"className":324573},[31828],[34,324575,324577],{"className":324576},[31801],[34,324578,324580],{"className":324579,"style":323187},[31805],[34,324581],{},[34,324583,456],{"className":324584},[31782],[34,324586,115372],{"className":324587,"style":32062},[31685,31686],[34,324589,2557],{"className":324590},[31690],[34,324592,162086],{"className":324593},[31685],[108,324595,324596,324597,347],{},"解得：",[34,324598,324600,324660],{"className":324599},[31642],[34,324601,324603],{"className":324602},[31646],[31648,324604,324605],{"xmlns":31650},[31652,324606,324607,324657],{},[31655,324608,324609,324611,324621,324623,324625,324627,324629,324631,324653,324655],{},[31658,324610,162086],{"mathvariant":38079},[31743,324612,324613,324615],{},[31658,324614,47384],{},[31743,324616,324617,324619],{},[31658,324618,94675],{},[31658,324620,31660],{},[31662,324622,456],{"stretchy":702},[31658,324624,115372],{},[31662,324626,2557],{"stretchy":702},[31658,324628,162086],{"mathvariant":38079},[31662,324630,202],{},[41453,324632,324633,324639],{},[31655,324634,324635,324637],{},[31658,324636,31660],{},[31662,324638,5504],{"stretchy":702},[31655,324640,324641,324643,324645,324647,324649,324651],{},[31662,324642,456],{"stretchy":702},[31658,324644,31660],{},[31662,324646,32226],{},[31749,324648,640],{},[31662,324650,2557],{"stretchy":702},[31662,324652,5504],{"stretchy":702},[31662,324654,202],{},[31658,324656,31660],{},[31665,324658,324659],{"encoding":31667},"|C_{S_n}(\\sigma)| = \\frac{n!}{(n-1)!} = n",[34,324661,324663,324773,324873],{"className":324662,"ariaHidden":758},[31672],[34,324664,324666,324669,324672,324752,324755,324758,324761,324764,324767,324770],{"className":324665},[31676],[34,324667],{"className":324668,"style":323107},[31680],[34,324670,162086],{"className":324671},[31685],[34,324673,324675,324678],{"className":324674},[31685],[34,324676,47384],{"className":324677,"style":38706},[31685,31686],[34,324679,324681],{"className":324680},[31792],[34,324682,324684,324744],{"className":324683},[31796,31797],[34,324685,324687,324741],{"className":324686},[31801],[34,324688,324690],{"className":324689,"style":116558},[31805],[34,324691,324692,324695],{"style":129511},[34,324693],{"className":324694,"style":31814},[31813],[34,324696,324698],{"className":324697},[31818,31819,31820,31821],[34,324699,324701],{"className":324700},[31685,31821],[34,324702,324704,324707],{"className":324703},[31685,31821],[34,324705,94675],{"className":324706,"style":106455},[31685,31686,31821],[34,324708,324710],{"className":324709},[31792],[34,324711,324713,324733],{"className":324712},[31796,31797],[34,324714,324716,324730],{"className":324715},[31801],[34,324717,324719],{"className":324718,"style":220518},[31805],[34,324720,324721,324724],{"style":323157},[34,324722],{"className":324723,"style":43406},[31813],[34,324725,324727],{"className":324726},[31818,43410,43411,31821],[34,324728,31660],{"className":324729},[31685,31686,31821],[34,324731,31829],{"className":324732},[31828],[34,324734,324736],{"className":324735},[31801],[34,324737,324739],{"className":324738,"style":220540},[31805],[34,324740],{},[34,324742,31829],{"className":324743},[31828],[34,324745,324747],{"className":324746},[31801],[34,324748,324750],{"className":324749,"style":323187},[31805],[34,324751],{},[34,324753,456],{"className":324754},[31782],[34,324756,115372],{"className":324757,"style":32062},[31685,31686],[34,324759,2557],{"className":324760},[31690],[34,324762,162086],{"className":324763},[31685],[34,324765],{"className":324766,"style":31846},[31845],[34,324768,202],{"className":324769},[31850],[34,324771],{"className":324772,"style":31846},[31845],[34,324774,324776,324780,324864,324867,324870],{"className":324775},[31676],[34,324777],{"className":324778,"style":324779},[31680],"height:1.4001em;vertical-align:-0.52em;",[34,324781,324783,324786,324861],{"className":324782},[31685],[34,324784],{"className":324785},[31782,41488],[34,324787,324789],{"className":324788},[41453],[34,324790,324792,324852],{"className":324791},[31796,31797],[34,324793,324795,324849],{"className":324794},[31801],[34,324796,324798,324824,324832],{"className":324797,"style":232511},[31805],[34,324799,324800,324803],{"style":41504},[34,324801],{"className":324802,"style":41508},[31813],[34,324804,324806],{"className":324805},[31818,31819,31820,31821],[34,324807,324809,324812,324815,324818,324821],{"className":324808},[31685,31821],[34,324810,456],{"className":324811},[31782,31821],[34,324813,31660],{"className":324814},[31685,31686,31821],[34,324816,32226],{"className":324817},[32024,31821],[34,324819,640],{"className":324820},[31685,31821],[34,324822,323287],{"className":324823},[31690,31821],[34,324825,324826,324829],{"style":41520},[34,324827],{"className":324828,"style":41508},[31813],[34,324830],{"className":324831,"style":41528},[41527],[34,324833,324834,324837],{"style":46771},[34,324835],{"className":324836,"style":41508},[31813],[34,324838,324840],{"className":324839},[31818,31819,31820,31821],[34,324841,324843,324846],{"className":324842},[31685,31821],[34,324844,31660],{"className":324845},[31685,31686,31821],[34,324847,5504],{"className":324848},[31690,31821],[34,324850,31829],{"className":324851},[31828],[34,324853,324855],{"className":324854},[31801],[34,324856,324859],{"className":324857,"style":324858},[31805],"height:0.52em;",[34,324860],{},[34,324862],{"className":324863},[31690,41488],[34,324865],{"className":324866,"style":31846},[31845],[34,324868,202],{"className":324869},[31850],[34,324871],{"className":324872,"style":31846},[31845],[34,324874,324876,324879],{"className":324875},[31676],[34,324877],{"className":324878,"style":32527},[31680],[34,324880,31660],{"className":324881},[31685,31686],[108,324883,324884,2664,324887],{},[2627,324885,324886],{},"确定中心化子",[105,324888,324889,325123,325193],{},[108,324890,324891,324892,325093,325094,325122],{},"我们知道 ",[34,324893,324895,324951],{"className":324894},[31642],[34,324896,324898],{"className":324897},[31646],[31648,324899,324900],{"xmlns":31650},[31652,324901,324902,324948],{},[31655,324903,324904,324906,324908,324910,324912,324914,324916,324918,324920,324922,324928,324930,324932,324934,324946],{},[31662,324905,126529],{"stretchy":702},[31658,324907,115372],{},[31662,324909,126544],{"stretchy":702},[31662,324911,202],{},[31662,324913,32685],{"stretchy":702},[31658,324915,33244],{},[31662,324917,31960],{"separator":758},[31658,324919,115372],{},[31662,324921,31960],{"separator":758},[37624,324923,324924,324926],{},[31658,324925,115372],{},[31749,324927,768],{},[31662,324929,31960],{"separator":758},[31662,324931,33934],{},[31662,324933,31960],{"separator":758},[37624,324935,324936,324938],{},[31658,324937,115372],{},[31655,324939,324940,324942,324944],{},[31658,324941,31660],{},[31662,324943,32226],{},[31749,324945,640],{},[31662,324947,32715],{"stretchy":702},[31665,324949,324950],{"encoding":31667},"\\langle \\sigma \\rangle = \\{e, \\sigma, \\sigma^2, \\dots, \\sigma^{n-1}\\}",[34,324952,324954,324978],{"className":324953,"ariaHidden":758},[31672],[34,324955,324957,324960,324963,324966,324969,324972,324975],{"className":324956},[31676],[34,324958],{"className":324959,"style":31774},[31680],[34,324961,126529],{"className":324962},[31782],[34,324964,115372],{"className":324965,"style":32062},[31685,31686],[34,324967,126544],{"className":324968},[31690],[34,324970],{"className":324971,"style":31846},[31845],[34,324973,202],{"className":324974},[31850],[34,324976],{"className":324977,"style":31846},[31845],[34,324979,324981,324984,324987,324990,324993,324996,324999,325002,325005,325034,325037,325040,325043,325046,325049,325052,325090],{"className":324980},[31676],[34,324982],{"className":324983,"style":37755},[31680],[34,324985,32685],{"className":324986},[31782],[34,324988,33244],{"className":324989},[31685,31686],[34,324991,31960],{"className":324992},[32119],[34,324994],{"className":324995,"style":32123},[31845],[34,324997,115372],{"className":324998,"style":32062},[31685,31686],[34,325000,31960],{"className":325001},[32119],[34,325003],{"className":325004,"style":32123},[31845],[34,325006,325008,325011],{"className":325007},[31685],[34,325009,115372],{"className":325010,"style":32062},[31685,31686],[34,325012,325014],{"className":325013},[31792],[34,325015,325017],{"className":325016},[31796],[34,325018,325020],{"className":325019},[31801],[34,325021,325023],{"className":325022,"style":37774},[31805],[34,325024,325025,325028],{"style":37777},[34,325026],{"className":325027,"style":31814},[31813],[34,325029,325031],{"className":325030},[31818,31819,31820,31821],[34,325032,768],{"className":325033},[31685,31821],[34,325035,31960],{"className":325036},[32119],[34,325038],{"className":325039,"style":32123},[31845],[34,325041,33934],{"className":325042},[34006],[34,325044],{"className":325045,"style":32123},[31845],[34,325047,31960],{"className":325048},[32119],[34,325050],{"className":325051,"style":32123},[31845],[34,325053,325055,325058],{"className":325054},[31685],[34,325056,115372],{"className":325057,"style":32062},[31685,31686],[34,325059,325061],{"className":325060},[31792],[34,325062,325064],{"className":325063},[31796],[34,325065,325067],{"className":325066},[31801],[34,325068,325070],{"className":325069,"style":37774},[31805],[34,325071,325072,325075],{"style":37777},[34,325073],{"className":325074,"style":31814},[31813],[34,325076,325078],{"className":325077},[31818,31819,31820,31821],[34,325079,325081,325084,325087],{"className":325080},[31685,31821],[34,325082,31660],{"className":325083},[31685,31686,31821],[34,325085,32226],{"className":325086},[32024,31821],[34,325088,640],{"className":325089},[31685,31821],[34,325091,32715],{"className":325092},[31690]," 里面的元素肯定都和 ",[34,325095,325097,325110],{"className":325096},[31642],[34,325098,325100],{"className":325099},[31646],[31648,325101,325102],{"xmlns":31650},[31652,325103,325104,325108],{},[31655,325105,325106],{},[31658,325107,115372],{},[31665,325109,115375],{"encoding":31667},[34,325111,325113],{"className":325112,"ariaHidden":758},[31672],[34,325114,325116,325119],{"className":325115},[31676],[34,325117],{"className":325118,"style":32527},[31680],[34,325120,115372],{"className":325121,"style":32062},[31685,31686]," 交换（自己跟自己肯定交换）。",[108,325124,325125,325164,325165,347],{},[34,325126,325128,325146],{"className":325127},[31642],[34,325129,325131],{"className":325130},[31646],[31648,325132,325133],{"xmlns":31650},[31652,325134,325135,325143],{},[31655,325136,325137,325139,325141],{},[31662,325138,126529],{"stretchy":702},[31658,325140,115372],{},[31662,325142,126544],{"stretchy":702},[31665,325144,325145],{"encoding":31667},"\\langle \\sigma \\rangle",[34,325147,325149],{"className":325148,"ariaHidden":758},[31672],[34,325150,325152,325155,325158,325161],{"className":325151},[31676],[34,325153],{"className":325154,"style":31774},[31680],[34,325156,126529],{"className":325157},[31782],[34,325159,115372],{"className":325160,"style":32062},[31685,31686],[34,325162,126544],{"className":325163},[31690]," 的阶正好是 ",[34,325166,325168,325181],{"className":325167},[31642],[34,325169,325171],{"className":325170},[31646],[31648,325172,325173],{"xmlns":31650},[31652,325174,325175,325179],{},[31655,325176,325177],{},[31658,325178,31660],{},[31665,325180,31660],{"encoding":31667},[34,325182,325184],{"className":325183,"ariaHidden":758},[31672],[34,325185,325187,325190],{"className":325186},[31676],[34,325188],{"className":325189,"style":32527},[31680],[34,325191,31660],{"className":325192},[31685,31686],[108,325194,325195,325197,325198,97596,325226,325264,325265,347],{},[2627,325196,114043],{},"：既然中心化子的大小是 ",[34,325199,325201,325214],{"className":325200},[31642],[34,325202,325204],{"className":325203},[31646],[31648,325205,325206],{"xmlns":31650},[31652,325207,325208,325212],{},[31655,325209,325210],{},[31658,325211,31660],{},[31665,325213,31660],{"encoding":31667},[34,325215,325217],{"className":325216,"ariaHidden":758},[31672],[34,325218,325220,325223],{"className":325219},[31676],[34,325221],{"className":325222,"style":32527},[31680],[34,325224,31660],{"className":325225},[31685,31686],[34,325227,325229,325246],{"className":325228},[31642],[34,325230,325232],{"className":325231},[31646],[31648,325233,325234],{"xmlns":31650},[31652,325235,325236,325244],{},[31655,325237,325238,325240,325242],{},[31662,325239,126529],{"stretchy":702},[31658,325241,115372],{},[31662,325243,126544],{"stretchy":702},[31665,325245,325145],{"encoding":31667},[34,325247,325249],{"className":325248,"ariaHidden":758},[31672],[34,325250,325252,325255,325258,325261],{"className":325251},[31676],[34,325253],{"className":325254,"style":31774},[31680],[34,325256,126529],{"className":325257},[31782],[34,325259,115372],{"className":325260,"style":32062},[31685,31686],[34,325262,126544],{"className":325263},[31690]," 是它在这个大小下的一个子集，那它们必须相等。即 ",[34,325266,325268,325303],{"className":325267},[31642],[34,325269,325271],{"className":325270},[31646],[31648,325272,325273],{"xmlns":31650},[31652,325274,325275,325301],{},[31655,325276,325277,325287,325289,325291,325293,325295,325297,325299],{},[31743,325278,325279,325281],{},[31658,325280,47384],{},[31743,325282,325283,325285],{},[31658,325284,94675],{},[31658,325286,31660],{},[31662,325288,456],{"stretchy":702},[31658,325290,115372],{},[31662,325292,2557],{"stretchy":702},[31662,325294,202],{},[31662,325296,126529],{"stretchy":702},[31658,325298,115372],{},[31662,325300,126544],{"stretchy":702},[31665,325302,323097],{"encoding":31667},[34,325304,325306,325410],{"className":325305,"ariaHidden":758},[31672],[34,325307,325309,325312,325392,325395,325398,325401,325404,325407],{"className":325308},[31676],[34,325310],{"className":325311,"style":323107},[31680],[34,325313,325315,325318],{"className":325314},[31685],[34,325316,47384],{"className":325317,"style":38706},[31685,31686],[34,325319,325321],{"className":325320},[31792],[34,325322,325324,325384],{"className":325323},[31796,31797],[34,325325,325327,325381],{"className":325326},[31801],[34,325328,325330],{"className":325329,"style":116558},[31805],[34,325331,325332,325335],{"style":129511},[34,325333],{"className":325334,"style":31814},[31813],[34,325336,325338],{"className":325337},[31818,31819,31820,31821],[34,325339,325341],{"className":325340},[31685,31821],[34,325342,325344,325347],{"className":325343},[31685,31821],[34,325345,94675],{"className":325346,"style":106455},[31685,31686,31821],[34,325348,325350],{"className":325349},[31792],[34,325351,325353,325373],{"className":325352},[31796,31797],[34,325354,325356,325370],{"className":325355},[31801],[34,325357,325359],{"className":325358,"style":220518},[31805],[34,325360,325361,325364],{"style":323157},[34,325362],{"className":325363,"style":43406},[31813],[34,325365,325367],{"className":325366},[31818,43410,43411,31821],[34,325368,31660],{"className":325369},[31685,31686,31821],[34,325371,31829],{"className":325372},[31828],[34,325374,325376],{"className":325375},[31801],[34,325377,325379],{"className":325378,"style":220540},[31805],[34,325380],{},[34,325382,31829],{"className":325383},[31828],[34,325385,325387],{"className":325386},[31801],[34,325388,325390],{"className":325389,"style":323187},[31805],[34,325391],{},[34,325393,456],{"className":325394},[31782],[34,325396,115372],{"className":325397,"style":32062},[31685,31686],[34,325399,2557],{"className":325400},[31690],[34,325402],{"className":325403,"style":31846},[31845],[34,325405,202],{"className":325406},[31850],[34,325408],{"className":325409,"style":31846},[31845],[34,325411,325413,325416,325419,325422],{"className":325412},[31676],[34,325414],{"className":325415,"style":31774},[31680],[34,325417,126529],{"className":325418},[31782],[34,325420,115372],{"className":325421,"style":32062},[31685,31686],[34,325423,126544],{"className":325424},[31690],[32188,325426],{},[32490,325428,325430,325431,325500,325501,82346],{"id":325429},"题目-16sns_nsn-的正规子群n5n-ge-5n5","题目 16：",[34,325432,325434,325451],{"className":325433},[31642],[34,325435,325437],{"className":325436},[31646],[31648,325438,325439],{"xmlns":31650},[31652,325440,325441,325449],{},[31655,325442,325443],{},[31743,325444,325445,325447],{},[31658,325446,94675],{},[31658,325448,31660],{},[31665,325450,218979],{"encoding":31667},[34,325452,325454],{"className":325453,"ariaHidden":758},[31672],[34,325455,325457,325460],{"className":325456},[31676],[34,325458],{"className":325459,"style":70915},[31680],[34,325461,325463,325466],{"className":325462},[31685],[34,325464,94675],{"className":325465,"style":106455},[31685,31686],[34,325467,325469],{"className":325468},[31792],[34,325470,325472,325492],{"className":325471},[31796,31797],[34,325473,325475,325489],{"className":325474},[31801],[34,325476,325478],{"className":325477,"style":32362},[31805],[34,325479,325480,325483],{"style":206422},[34,325481],{"className":325482,"style":31814},[31813],[34,325484,325486],{"className":325485},[31818,31819,31820,31821],[34,325487,31660],{"className":325488},[31685,31686,31821],[34,325490,31829],{"className":325491},[31828],[34,325493,325495],{"className":325494},[31801],[34,325496,325498],{"className":325497,"style":31836},[31805],[34,325499],{}," 的正规子群（",[34,325502,325504,325522],{"className":325503},[31642],[34,325505,325507],{"className":325506},[31646],[31648,325508,325509],{"xmlns":31650},[31652,325510,325511,325519],{},[31655,325512,325513,325515,325517],{},[31658,325514,31660],{},[31662,325516,43175],{},[31749,325518,18136],{},[31665,325520,325521],{"encoding":31667},"n \\ge 5",[34,325523,325525,325543],{"className":325524,"ariaHidden":758},[31672],[34,325526,325528,325531,325534,325537,325540],{"className":325527},[31676],[34,325529],{"className":325530,"style":43198},[31680],[34,325532,31660],{"className":325533},[31685,31686],[34,325535],{"className":325536,"style":31846},[31845],[34,325538,43175],{"className":325539},[31850],[34,325541],{"className":325542,"style":31846},[31845],[34,325544,325546,325549],{"className":325545},[31676],[34,325547],{"className":325548,"style":32617},[31680],[34,325550,18136],{"className":325551},[31685],[15,325553,325554,321411,325556,325625,325626,347],{},[2627,325555,101191],{},[34,325557,325559,325576],{"className":325558},[31642],[34,325560,325562],{"className":325561},[31646],[31648,325563,325564],{"xmlns":31650},[31652,325565,325566,325574],{},[31655,325567,325568],{},[31743,325569,325570,325572],{},[31658,325571,94675],{},[31658,325573,31660],{},[31665,325575,218979],{"encoding":31667},[34,325577,325579],{"className":325578,"ariaHidden":758},[31672],[34,325580,325582,325585],{"className":325581},[31676],[34,325583],{"className":325584,"style":70915},[31680],[34,325586,325588,325591],{"className":325587},[31685],[34,325589,94675],{"className":325590,"style":106455},[31685,31686],[34,325592,325594],{"className":325593},[31792],[34,325595,325597,325617],{"className":325596},[31796,31797],[34,325598,325600,325614],{"className":325599},[31801],[34,325601,325603],{"className":325602,"style":32362},[31805],[34,325604,325605,325608],{"style":206422},[34,325606],{"className":325607,"style":31814},[31813],[34,325609,325611],{"className":325610},[31818,31819,31820,31821],[34,325612,31660],{"className":325613},[31685,31686,31821],[34,325615,31829],{"className":325616},[31828],[34,325618,325620],{"className":325619},[31801],[34,325621,325623],{"className":325622,"style":31836},[31805],[34,325624],{}," 只有一个非平凡正规子群 ",[34,325627,325629,325646],{"className":325628},[31642],[34,325630,325632],{"className":325631},[31646],[31648,325633,325634],{"xmlns":31650},[31652,325635,325636,325644],{},[31655,325637,325638],{},[31743,325639,325640,325642],{},[31658,325641,54151],{},[31658,325643,31660],{},[31665,325645,321085],{"encoding":31667},[34,325647,325649],{"className":325648,"ariaHidden":758},[31672],[34,325650,325652,325655],{"className":325651},[31676],[34,325653],{"className":325654,"style":70915},[31680],[34,325656,325658,325661],{"className":325657},[31685],[34,325659,54151],{"className":325660},[31685,31686],[34,325662,325664],{"className":325663},[31792],[34,325665,325667,325687],{"className":325666},[31796,31797],[34,325668,325670,325684],{"className":325669},[31801],[34,325671,325673],{"className":325672,"style":32362},[31805],[34,325674,325675,325678],{"style":31809},[34,325676],{"className":325677,"style":31814},[31813],[34,325679,325681],{"className":325680},[31818,31819,31820,31821],[34,325682,31660],{"className":325683},[31685,31686,31821],[34,325685,31829],{"className":325686},[31828],[34,325688,325690],{"className":325689},[31801],[34,325691,325693],{"className":325692,"style":31836},[31805],[34,325694],{},[15,325696,325697],{},[2627,325698,325699],{},"【证明逻辑重构】",[2230,325701,325702,326123,326323,328858],{},[108,325703,325704,325706,325707,11008,325735,325804,325805,46528,325899,326024,326025,326053,326054,347],{},[2627,325705,233757],{},"：假设 ",[34,325708,325710,325723],{"className":325709},[31642],[34,325711,325713],{"className":325712},[31646],[31648,325714,325715],{"xmlns":31650},[31652,325716,325717,325721],{},[31655,325718,325719],{},[31658,325720,118113],{},[31665,325722,118113],{"encoding":31667},[34,325724,325726],{"className":325725,"ariaHidden":758},[31672],[34,325727,325729,325732],{"className":325728},[31676],[34,325730],{"className":325731,"style":33218},[31680],[34,325733,118113],{"className":325734,"style":118235},[31685,31686],[34,325736,325738,325755],{"className":325737},[31642],[34,325739,325741],{"className":325740},[31646],[31648,325742,325743],{"xmlns":31650},[31652,325744,325745,325753],{},[31655,325746,325747],{},[31743,325748,325749,325751],{},[31658,325750,94675],{},[31658,325752,31660],{},[31665,325754,218979],{"encoding":31667},[34,325756,325758],{"className":325757,"ariaHidden":758},[31672],[34,325759,325761,325764],{"className":325760},[31676],[34,325762],{"className":325763,"style":70915},[31680],[34,325765,325767,325770],{"className":325766},[31685],[34,325768,94675],{"className":325769,"style":106455},[31685,31686],[34,325771,325773],{"className":325772},[31792],[34,325774,325776,325796],{"className":325775},[31796,31797],[34,325777,325779,325793],{"className":325778},[31801],[34,325780,325782],{"className":325781,"style":32362},[31805],[34,325783,325784,325787],{"style":206422},[34,325785],{"className":325786,"style":31814},[31813],[34,325788,325790],{"className":325789},[31818,31819,31820,31821],[34,325791,31660],{"className":325792},[31685,31686,31821],[34,325794,31829],{"className":325795},[31828],[34,325797,325799],{"className":325798},[31801],[34,325800,325802],{"className":325801,"style":31836},[31805],[34,325803],{}," 的一个正规子群，且 ",[34,325806,325808,325830],{"className":325807},[31642],[34,325809,325811],{"className":325810},[31646],[31648,325812,325813],{"xmlns":31650},[31652,325814,325815,325827],{},[31655,325816,325817,325819,325821,325823,325825],{},[31658,325818,118113],{},[31662,325820,54707],{"mathvariant":38079},[31662,325822,32685],{"stretchy":702},[31658,325824,33244],{},[31662,325826,32715],{"stretchy":702},[31665,325828,325829],{"encoding":31667},"H \\neq \\{e\\}",[34,325831,325833,325884],{"className":325832,"ariaHidden":758},[31672],[34,325834,325836,325839,325842,325845,325881],{"className":325835},[31676],[34,325837],{"className":325838,"style":43217},[31680],[34,325840,118113],{"className":325841,"style":118235},[31685,31686],[34,325843],{"className":325844,"style":31846},[31845],[34,325846,325848,325875,325878],{"className":325847},[31850],[34,325849,325851],{"className":325850},[31850],[34,325852,325854],{"className":325853},[31685,54774],[34,325855,325857],{"className":325856},[54778],[34,325858,325860,325863,325872],{"className":325859},[54782],[34,325861],{"className":325862,"style":43217},[31680],[34,325864,325866],{"className":325865},[54789],[34,325867,325869],{"className":325868},[31685],[34,325870,54796],{"className":325871},[31850],[34,325873],{"className":325874},[54800],[34,325876],{"className":325877},[31845,54804],[34,325879,202],{"className":325880},[31850],[34,325882],{"className":325883,"style":31846},[31845],[34,325885,325887,325890,325893,325896],{"className":325886},[31676],[34,325888],{"className":325889,"style":31774},[31680],[34,325891,32685],{"className":325892},[31782],[34,325894,33244],{"className":325895},[31685,31686],[34,325897,32715],{"className":325898},[31690],[34,325900,325902,325924],{"className":325901},[31642],[34,325903,325905],{"className":325904},[31646],[31648,325906,325907],{"xmlns":31650},[31652,325908,325909,325921],{},[31655,325910,325911,325913,325915],{},[31658,325912,118113],{},[31662,325914,54707],{"mathvariant":38079},[31743,325916,325917,325919],{},[31658,325918,94675],{},[31658,325920,31660],{},[31665,325922,325923],{"encoding":31667},"H \\neq S_n",[34,325925,325927,325978],{"className":325926,"ariaHidden":758},[31672],[34,325928,325930,325933,325936,325939,325975],{"className":325929},[31676],[34,325931],{"className":325932,"style":43217},[31680],[34,325934,118113],{"className":325935,"style":118235},[31685,31686],[34,325937],{"className":325938,"style":31846},[31845],[34,325940,325942,325969,325972],{"className":325941},[31850],[34,325943,325945],{"className":325944},[31850],[34,325946,325948],{"className":325947},[31685,54774],[34,325949,325951],{"className":325950},[54778],[34,325952,325954,325957,325966],{"className":325953},[54782],[34,325955],{"className":325956,"style":43217},[31680],[34,325958,325960],{"className":325959},[54789],[34,325961,325963],{"className":325962},[31685],[34,325964,54796],{"className":325965},[31850],[34,325967],{"className":325968},[54800],[34,325970],{"className":325971},[31845,54804],[34,325973,202],{"className":325974},[31850],[34,325976],{"className":325977,"style":31846},[31845],[34,325979,325981,325984],{"className":325980},[31676],[34,325982],{"className":325983,"style":70915},[31680],[34,325985,325987,325990],{"className":325986},[31685],[34,325988,94675],{"className":325989,"style":106455},[31685,31686],[34,325991,325993],{"className":325992},[31792],[34,325994,325996,326016],{"className":325995},[31796,31797],[34,325997,325999,326013],{"className":325998},[31801],[34,326000,326002],{"className":326001,"style":32362},[31805],[34,326003,326004,326007],{"style":206422},[34,326005],{"className":326006,"style":31814},[31813],[34,326008,326010],{"className":326009},[31818,31819,31820,31821],[34,326011,31660],{"className":326012},[31685,31686,31821],[34,326014,31829],{"className":326015},[31828],[34,326017,326019],{"className":326018},[31801],[34,326020,326022],{"className":326021,"style":31836},[31805],[34,326023],{},"。我们要证明 ",[34,326026,326028,326041],{"className":326027},[31642],[34,326029,326031],{"className":326030},[31646],[31648,326032,326033],{"xmlns":31650},[31652,326034,326035,326039],{},[31655,326036,326037],{},[31658,326038,118113],{},[31665,326040,118113],{"encoding":31667},[34,326042,326044],{"className":326043,"ariaHidden":758},[31672],[34,326045,326047,326050],{"className":326046},[31676],[34,326048],{"className":326049,"style":33218},[31680],[34,326051,118113],{"className":326052,"style":118235},[31685,31686]," 只能是 ",[34,326055,326057,326074],{"className":326056},[31642],[34,326058,326060],{"className":326059},[31646],[31648,326061,326062],{"xmlns":31650},[31652,326063,326064,326072],{},[31655,326065,326066],{},[31743,326067,326068,326070],{},[31658,326069,54151],{},[31658,326071,31660],{},[31665,326073,321085],{"encoding":31667},[34,326075,326077],{"className":326076,"ariaHidden":758},[31672],[34,326078,326080,326083],{"className":326079},[31676],[34,326081],{"className":326082,"style":70915},[31680],[34,326084,326086,326089],{"className":326085},[31685],[34,326087,54151],{"className":326088},[31685,31686],[34,326090,326092],{"className":326091},[31792],[34,326093,326095,326115],{"className":326094},[31796,31797],[34,326096,326098,326112],{"className":326097},[31801],[34,326099,326101],{"className":326100,"style":32362},[31805],[34,326102,326103,326106],{"style":31809},[34,326104],{"className":326105,"style":31814},[31813],[34,326107,326109],{"className":326108},[31818,31819,31820,31821],[34,326110,31660],{"className":326111},[31685,31686,31821],[34,326113,31829],{"className":326114},[31828],[34,326116,326118],{"className":326117},[31801],[34,326119,326121],{"className":326120,"style":31836},[31805],[34,326122],{},[108,326124,326125,39984,326198],{},[2627,326126,326127,326128,326197],{},"关键点：",[34,326129,326131,326148],{"className":326130},[31642],[34,326132,326134],{"className":326133},[31646],[31648,326135,326136],{"xmlns":31650},[31652,326137,326138,326146],{},[31655,326139,326140],{},[31743,326141,326142,326144],{},[31658,326143,54151],{},[31658,326145,31660],{},[31665,326147,321085],{"encoding":31667},[34,326149,326151],{"className":326150,"ariaHidden":758},[31672],[34,326152,326154,326157],{"className":326153},[31676],[34,326155],{"className":326156,"style":70915},[31680],[34,326158,326160,326163],{"className":326159},[31685],[34,326161,54151],{"className":326162},[31685,31686],[34,326164,326166],{"className":326165},[31792],[34,326167,326169,326189],{"className":326168},[31796,31797],[34,326170,326172,326186],{"className":326171},[31801],[34,326173,326175],{"className":326174,"style":32362},[31805],[34,326176,326177,326180],{"style":31809},[34,326178],{"className":326179,"style":31814},[31813],[34,326181,326183],{"className":326182},[31818,31819,31820,31821],[34,326184,31660],{"className":326185},[31685,31686,31821],[34,326187,31829],{"className":326188},[31828],[34,326190,326192],{"className":326191},[31801],[34,326193,326195],{"className":326194,"style":31836},[31805],[34,326196],{}," 是单群（Simple Group）",[105,326199,326200],{},[108,326201,59432,326202,326252,326253,326322],{},[34,326203,326205,326222],{"className":326204},[31642],[34,326206,326208],{"className":326207},[31646],[31648,326209,326210],{"xmlns":31650},[31652,326211,326212,326220],{},[31655,326213,326214,326216,326218],{},[31658,326215,31660],{},[31662,326217,43175],{},[31749,326219,18136],{},[31665,326221,325521],{"encoding":31667},[34,326223,326225,326243],{"className":326224,"ariaHidden":758},[31672],[34,326226,326228,326231,326234,326237,326240],{"className":326227},[31676],[34,326229],{"className":326230,"style":43198},[31680],[34,326232,31660],{"className":326233},[31685,31686],[34,326235],{"className":326236,"style":31846},[31845],[34,326238,43175],{"className":326239},[31850],[34,326241],{"className":326242,"style":31846},[31845],[34,326244,326246,326249],{"className":326245},[31676],[34,326247],{"className":326248,"style":32617},[31680],[34,326250,18136],{"className":326251},[31685],"， ",[34,326254,326256,326273],{"className":326255},[31642],[34,326257,326259],{"className":326258},[31646],[31648,326260,326261],{"xmlns":31650},[31652,326262,326263,326271],{},[31655,326264,326265],{},[31743,326266,326267,326269],{},[31658,326268,54151],{},[31658,326270,31660],{},[31665,326272,321085],{"encoding":31667},[34,326274,326276],{"className":326275,"ariaHidden":758},[31672],[34,326277,326279,326282],{"className":326278},[31676],[34,326280],{"className":326281,"style":70915},[31680],[34,326283,326285,326288],{"className":326284},[31685],[34,326286,54151],{"className":326287},[31685,31686],[34,326289,326291],{"className":326290},[31792],[34,326292,326294,326314],{"className":326293},[31796,31797],[34,326295,326297,326311],{"className":326296},[31801],[34,326298,326300],{"className":326299,"style":32362},[31805],[34,326301,326302,326305],{"style":31809},[34,326303],{"className":326304,"style":31814},[31813],[34,326306,326308],{"className":326307},[31818,31819,31820,31821],[34,326309,31660],{"className":326310},[31685,31686,31821],[34,326312,31829],{"className":326313},[31828],[34,326315,326317],{"className":326316},[31801],[34,326318,326320],{"className":326319,"style":31836},[31805],[34,326321],{}," 没有非平凡的正规子群。这是已知的大定理。",[108,326324,326325,33183,326421],{},[2627,326326,326327,326328],{},"分析 ",[34,326329,326331,326354],{"className":326330},[31642],[34,326332,326334],{"className":326333},[31646],[31648,326335,326336],{"xmlns":31650},[31652,326337,326338,326351],{},[31655,326339,326340,326342,326345],{},[31658,326341,118113],{},[31662,326343,326344],{},"∩",[31743,326346,326347,326349],{},[31658,326348,54151],{},[31658,326350,31660],{},[31665,326352,326353],{"encoding":31667},"H \\cap A_n",[34,326355,326357,326375],{"className":326356,"ariaHidden":758},[31672],[34,326358,326360,326363,326366,326369,326372],{"className":326359},[31676],[34,326361],{"className":326362,"style":33218},[31680],[34,326364,118113],{"className":326365,"style":118235},[31685,31686],[34,326367],{"className":326368,"style":32604},[31845],[34,326370,326344],{"className":326371},[32024],[34,326373],{"className":326374,"style":32604},[31845],[34,326376,326378,326381],{"className":326377},[31676],[34,326379],{"className":326380,"style":70915},[31680],[34,326382,326384,326387],{"className":326383},[31685],[34,326385,54151],{"className":326386},[31685,31686],[34,326388,326390],{"className":326389},[31792],[34,326391,326393,326413],{"className":326392},[31796,31797],[34,326394,326396,326410],{"className":326395},[31801],[34,326397,326399],{"className":326398,"style":32362},[31805],[34,326400,326401,326404],{"style":31809},[34,326402],{"className":326403,"style":31814},[31813],[34,326405,326407],{"className":326406},[31818,31819,31820,31821],[34,326408,31660],{"className":326409},[31685,31686,31821],[34,326411,31829],{"className":326412},[31828],[34,326414,326416],{"className":326415},[31801],[34,326417,326419],{"className":326418,"style":31836},[31805],[34,326420],{},[105,326422,326423,326755],{},[108,326424,93310,326425,258,326453,326522,326523,326592,326593,326684,326685,326754],{},[34,326426,326428,326441],{"className":326427},[31642],[34,326429,326431],{"className":326430},[31646],[31648,326432,326433],{"xmlns":31650},[31652,326434,326435,326439],{},[31655,326436,326437],{},[31658,326438,118113],{},[31665,326440,118113],{"encoding":31667},[34,326442,326444],{"className":326443,"ariaHidden":758},[31672],[34,326445,326447,326450],{"className":326446},[31676],[34,326448],{"className":326449,"style":33218},[31680],[34,326451,118113],{"className":326452,"style":118235},[31685,31686],[34,326454,326456,326473],{"className":326455},[31642],[34,326457,326459],{"className":326458},[31646],[31648,326460,326461],{"xmlns":31650},[31652,326462,326463,326471],{},[31655,326464,326465],{},[31743,326466,326467,326469],{},[31658,326468,54151],{},[31658,326470,31660],{},[31665,326472,321085],{"encoding":31667},[34,326474,326476],{"className":326475,"ariaHidden":758},[31672],[34,326477,326479,326482],{"className":326478},[31676],[34,326480],{"className":326481,"style":70915},[31680],[34,326483,326485,326488],{"className":326484},[31685],[34,326486,54151],{"className":326487},[31685,31686],[34,326489,326491],{"className":326490},[31792],[34,326492,326494,326514],{"className":326493},[31796,31797],[34,326495,326497,326511],{"className":326496},[31801],[34,326498,326500],{"className":326499,"style":32362},[31805],[34,326501,326502,326505],{"style":31809},[34,326503],{"className":326504,"style":31814},[31813],[34,326506,326508],{"className":326507},[31818,31819,31820,31821],[34,326509,31660],{"className":326510},[31685,31686,31821],[34,326512,31829],{"className":326513},[31828],[34,326515,326517],{"className":326516},[31801],[34,326518,326520],{"className":326519,"style":31836},[31805],[34,326521],{}," 都是 ",[34,326524,326526,326543],{"className":326525},[31642],[34,326527,326529],{"className":326528},[31646],[31648,326530,326531],{"xmlns":31650},[31652,326532,326533,326541],{},[31655,326534,326535],{},[31743,326536,326537,326539],{},[31658,326538,94675],{},[31658,326540,31660],{},[31665,326542,218979],{"encoding":31667},[34,326544,326546],{"className":326545,"ariaHidden":758},[31672],[34,326547,326549,326552],{"className":326548},[31676],[34,326550],{"className":326551,"style":70915},[31680],[34,326553,326555,326558],{"className":326554},[31685],[34,326556,94675],{"className":326557,"style":106455},[31685,31686],[34,326559,326561],{"className":326560},[31792],[34,326562,326564,326584],{"className":326563},[31796,31797],[34,326565,326567,326581],{"className":326566},[31801],[34,326568,326570],{"className":326569,"style":32362},[31805],[34,326571,326572,326575],{"style":206422},[34,326573],{"className":326574,"style":31814},[31813],[34,326576,326578],{"className":326577},[31818,31819,31820,31821],[34,326579,31660],{"className":326580},[31685,31686,31821],[34,326582,31829],{"className":326583},[31828],[34,326585,326587],{"className":326586},[31801],[34,326588,326590],{"className":326589,"style":31836},[31805],[34,326591],{}," 的正规子群，所以它们的交集 ",[34,326594,326596,326617],{"className":326595},[31642],[34,326597,326599],{"className":326598},[31646],[31648,326600,326601],{"xmlns":31650},[31652,326602,326603,326615],{},[31655,326604,326605,326607,326609],{},[31658,326606,118113],{},[31662,326608,326344],{},[31743,326610,326611,326613],{},[31658,326612,54151],{},[31658,326614,31660],{},[31665,326616,326353],{"encoding":31667},[34,326618,326620,326638],{"className":326619,"ariaHidden":758},[31672],[34,326621,326623,326626,326629,326632,326635],{"className":326622},[31676],[34,326624],{"className":326625,"style":33218},[31680],[34,326627,118113],{"className":326628,"style":118235},[31685,31686],[34,326630],{"className":326631,"style":32604},[31845],[34,326633,326344],{"className":326634},[32024],[34,326636],{"className":326637,"style":32604},[31845],[34,326639,326641,326644],{"className":326640},[31676],[34,326642],{"className":326643,"style":70915},[31680],[34,326645,326647,326650],{"className":326646},[31685],[34,326648,54151],{"className":326649},[31685,31686],[34,326651,326653],{"className":326652},[31792],[34,326654,326656,326676],{"className":326655},[31796,31797],[34,326657,326659,326673],{"className":326658},[31801],[34,326660,326662],{"className":326661,"style":32362},[31805],[34,326663,326664,326667],{"style":31809},[34,326665],{"className":326666,"style":31814},[31813],[34,326668,326670],{"className":326669},[31818,31819,31820,31821],[34,326671,31660],{"className":326672},[31685,31686,31821],[34,326674,31829],{"className":326675},[31828],[34,326677,326679],{"className":326678},[31801],[34,326680,326682],{"className":326681,"style":31836},[31805],[34,326683],{}," 也是 ",[34,326686,326688,326705],{"className":326687},[31642],[34,326689,326691],{"className":326690},[31646],[31648,326692,326693],{"xmlns":31650},[31652,326694,326695,326703],{},[31655,326696,326697],{},[31743,326698,326699,326701],{},[31658,326700,54151],{},[31658,326702,31660],{},[31665,326704,321085],{"encoding":31667},[34,326706,326708],{"className":326707,"ariaHidden":758},[31672],[34,326709,326711,326714],{"className":326710},[31676],[34,326712],{"className":326713,"style":70915},[31680],[34,326715,326717,326720],{"className":326716},[31685],[34,326718,54151],{"className":326719},[31685,31686],[34,326721,326723],{"className":326722},[31792],[34,326724,326726,326746],{"className":326725},[31796,31797],[34,326727,326729,326743],{"className":326728},[31801],[34,326730,326732],{"className":326731,"style":32362},[31805],[34,326733,326734,326737],{"style":31809},[34,326735],{"className":326736,"style":31814},[31813],[34,326738,326740],{"className":326739},[31818,31819,31820,31821],[34,326741,31660],{"className":326742},[31685,31686,31821],[34,326744,31829],{"className":326745},[31828],[34,326747,326749],{"className":326748},[31801],[34,326750,326752],{"className":326751,"style":31836},[31805],[34,326753],{}," 的正规子群。",[108,326756,93310,326757,326826,326827],{},[34,326758,326760,326777],{"className":326759},[31642],[34,326761,326763],{"className":326762},[31646],[31648,326764,326765],{"xmlns":31650},[31652,326766,326767,326775],{},[31655,326768,326769],{},[31743,326770,326771,326773],{},[31658,326772,54151],{},[31658,326774,31660],{},[31665,326776,321085],{"encoding":31667},[34,326778,326780],{"className":326779,"ariaHidden":758},[31672],[34,326781,326783,326786],{"className":326782},[31676],[34,326784],{"className":326785,"style":70915},[31680],[34,326787,326789,326792],{"className":326788},[31685],[34,326790,54151],{"className":326791},[31685,31686],[34,326793,326795],{"className":326794},[31792],[34,326796,326798,326818],{"className":326797},[31796,31797],[34,326799,326801,326815],{"className":326800},[31801],[34,326802,326804],{"className":326803,"style":32362},[31805],[34,326805,326806,326809],{"style":31809},[34,326807],{"className":326808,"style":31814},[31813],[34,326810,326812],{"className":326811},[31818,31819,31820,31821],[34,326813,31660],{"className":326814},[31685,31686,31821],[34,326816,31829],{"className":326817},[31828],[34,326819,326821],{"className":326820},[31801],[34,326822,326824],{"className":326823,"style":31836},[31805],[34,326825],{}," 是单群，这个交集只有两种可能：\n",[105,326828,326829,327547],{},[108,326830,326831,2664,326834,326989,326990,39984,327082],{},[2627,326832,326833],{},"情况 A",[34,326835,326837,326867],{"className":326836},[31642],[34,326838,326840],{"className":326839},[31646],[31648,326841,326842],{"xmlns":31650},[31652,326843,326844,326864],{},[31655,326845,326846,326848,326850,326856,326858],{},[31658,326847,118113],{},[31662,326849,326344],{},[31743,326851,326852,326854],{},[31658,326853,54151],{},[31658,326855,31660],{},[31662,326857,202],{},[31743,326859,326860,326862],{},[31658,326861,54151],{},[31658,326863,31660],{},[31665,326865,326866],{"encoding":31667},"H \\cap A_n = A_n",[34,326868,326870,326888,326943],{"className":326869,"ariaHidden":758},[31672],[34,326871,326873,326876,326879,326882,326885],{"className":326872},[31676],[34,326874],{"className":326875,"style":33218},[31680],[34,326877,118113],{"className":326878,"style":118235},[31685,31686],[34,326880],{"className":326881,"style":32604},[31845],[34,326883,326344],{"className":326884},[32024],[34,326886],{"className":326887,"style":32604},[31845],[34,326889,326891,326894,326934,326937,326940],{"className":326890},[31676],[34,326892],{"className":326893,"style":70915},[31680],[34,326895,326897,326900],{"className":326896},[31685],[34,326898,54151],{"className":326899},[31685,31686],[34,326901,326903],{"className":326902},[31792],[34,326904,326906,326926],{"className":326905},[31796,31797],[34,326907,326909,326923],{"className":326908},[31801],[34,326910,326912],{"className":326911,"style":32362},[31805],[34,326913,326914,326917],{"style":31809},[34,326915],{"className":326916,"style":31814},[31813],[34,326918,326920],{"className":326919},[31818,31819,31820,31821],[34,326921,31660],{"className":326922},[31685,31686,31821],[34,326924,31829],{"className":326925},[31828],[34,326927,326929],{"className":326928},[31801],[34,326930,326932],{"className":326931,"style":31836},[31805],[34,326933],{},[34,326935],{"className":326936,"style":31846},[31845],[34,326938,202],{"className":326939},[31850],[34,326941],{"className":326942,"style":31846},[31845],[34,326944,326946,326949],{"className":326945},[31676],[34,326947],{"className":326948,"style":70915},[31680],[34,326950,326952,326955],{"className":326951},[31685],[34,326953,54151],{"className":326954},[31685,31686],[34,326956,326958],{"className":326957},[31792],[34,326959,326961,326981],{"className":326960},[31796,31797],[34,326962,326964,326978],{"className":326963},[31801],[34,326965,326967],{"className":326966,"style":32362},[31805],[34,326968,326969,326972],{"style":31809},[34,326970],{"className":326971,"style":31814},[31813],[34,326973,326975],{"className":326974},[31818,31819,31820,31821],[34,326976,31660],{"className":326977},[31685,31686,31821],[34,326979,31829],{"className":326980},[31828],[34,326982,326984],{"className":326983},[31801],[34,326985,326987],{"className":326986,"style":31836},[31805],[34,326988],{},"。这意味着 ",[34,326991,326993,327015],{"className":326992},[31642],[34,326994,326996],{"className":326995},[31646],[31648,326997,326998],{"xmlns":31650},[31652,326999,327000,327012],{},[31655,327001,327002,327008,327010],{},[31743,327003,327004,327006],{},[31658,327005,54151],{},[31658,327007,31660],{},[31662,327009,131855],{},[31658,327011,118113],{},[31665,327013,327014],{"encoding":31667},"A_n \\subseteq H",[34,327016,327018,327073],{"className":327017,"ariaHidden":758},[31672],[34,327019,327021,327024,327064,327067,327070],{"className":327020},[31676],[34,327022],{"className":327023,"style":70915},[31680],[34,327025,327027,327030],{"className":327026},[31685],[34,327028,54151],{"className":327029},[31685,31686],[34,327031,327033],{"className":327032},[31792],[34,327034,327036,327056],{"className":327035},[31796,31797],[34,327037,327039,327053],{"className":327038},[31801],[34,327040,327042],{"className":327041,"style":32362},[31805],[34,327043,327044,327047],{"style":31809},[34,327045],{"className":327046,"style":31814},[31813],[34,327048,327050],{"className":327049},[31818,31819,31820,31821],[34,327051,31660],{"className":327052},[31685,31686,31821],[34,327054,31829],{"className":327055},[31828],[34,327057,327059],{"className":327058},[31801],[34,327060,327062],{"className":327061,"style":31836},[31805],[34,327063],{},[34,327065],{"className":327066,"style":31846},[31845],[34,327068,131855],{"className":327069},[31850],[34,327071],{"className":327072,"style":31846},[31845],[34,327074,327076,327079],{"className":327075},[31676],[34,327077],{"className":327078,"style":33218},[31680],[34,327080,118113],{"className":327081,"style":118235},[31685,31686],[105,327083,327084],{},[108,327085,327086,327155,327156,327225,327226,327254,327255,86674,327324,327448,327449,326053,327477,327546],{},[34,327087,327089,327106],{"className":327088},[31642],[34,327090,327092],{"className":327091},[31646],[31648,327093,327094],{"xmlns":31650},[31652,327095,327096,327104],{},[31655,327097,327098],{},[31743,327099,327100,327102],{},[31658,327101,54151],{},[31658,327103,31660],{},[31665,327105,321085],{"encoding":31667},[34,327107,327109],{"className":327108,"ariaHidden":758},[31672],[34,327110,327112,327115],{"className":327111},[31676],[34,327113],{"className":327114,"style":70915},[31680],[34,327116,327118,327121],{"className":327117},[31685],[34,327119,54151],{"className":327120},[31685,31686],[34,327122,327124],{"className":327123},[31792],[34,327125,327127,327147],{"className":327126},[31796,31797],[34,327128,327130,327144],{"className":327129},[31801],[34,327131,327133],{"className":327132,"style":32362},[31805],[34,327134,327135,327138],{"style":31809},[34,327136],{"className":327137,"style":31814},[31813],[34,327139,327141],{"className":327140},[31818,31819,31820,31821],[34,327142,31660],{"className":327143},[31685,31686,31821],[34,327145,31829],{"className":327146},[31828],[34,327148,327150],{"className":327149},[31801],[34,327151,327153],{"className":327152,"style":31836},[31805],[34,327154],{}," 的指数是 2（它占了 ",[34,327157,327159,327176],{"className":327158},[31642],[34,327160,327162],{"className":327161},[31646],[31648,327163,327164],{"xmlns":31650},[31652,327165,327166,327174],{},[31655,327167,327168],{},[31743,327169,327170,327172],{},[31658,327171,94675],{},[31658,327173,31660],{},[31665,327175,218979],{"encoding":31667},[34,327177,327179],{"className":327178,"ariaHidden":758},[31672],[34,327180,327182,327185],{"className":327181},[31676],[34,327183],{"className":327184,"style":70915},[31680],[34,327186,327188,327191],{"className":327187},[31685],[34,327189,94675],{"className":327190,"style":106455},[31685,31686],[34,327192,327194],{"className":327193},[31792],[34,327195,327197,327217],{"className":327196},[31796,31797],[34,327198,327200,327214],{"className":327199},[31801],[34,327201,327203],{"className":327202,"style":32362},[31805],[34,327204,327205,327208],{"style":206422},[34,327206],{"className":327207,"style":31814},[31813],[34,327209,327211],{"className":327210},[31818,31819,31820,31821],[34,327212,31660],{"className":327213},[31685,31686,31821],[34,327215,31829],{"className":327216},[31828],[34,327218,327220],{"className":327219},[31801],[34,327221,327223],{"className":327222,"style":31836},[31805],[34,327224],{}," 的一半）。如果 ",[34,327227,327229,327242],{"className":327228},[31642],[34,327230,327232],{"className":327231},[31646],[31648,327233,327234],{"xmlns":31650},[31652,327235,327236,327240],{},[31655,327237,327238],{},[31658,327239,118113],{},[31665,327241,118113],{"encoding":31667},[34,327243,327245],{"className":327244,"ariaHidden":758},[31672],[34,327246,327248,327251],{"className":327247},[31676],[34,327249],{"className":327250,"style":33218},[31680],[34,327252,118113],{"className":327253,"style":118235},[31685,31686]," 包含 ",[34,327256,327258,327275],{"className":327257},[31642],[34,327259,327261],{"className":327260},[31646],[31648,327262,327263],{"xmlns":31650},[31652,327264,327265,327273],{},[31655,327266,327267],{},[31743,327268,327269,327271],{},[31658,327270,54151],{},[31658,327272,31660],{},[31665,327274,321085],{"encoding":31667},[34,327276,327278],{"className":327277,"ariaHidden":758},[31672],[34,327279,327281,327284],{"className":327280},[31676],[34,327282],{"className":327283,"style":70915},[31680],[34,327285,327287,327290],{"className":327286},[31685],[34,327288,54151],{"className":327289},[31685,31686],[34,327291,327293],{"className":327292},[31792],[34,327294,327296,327316],{"className":327295},[31796,31797],[34,327297,327299,327313],{"className":327298},[31801],[34,327300,327302],{"className":327301,"style":32362},[31805],[34,327303,327304,327307],{"style":31809},[34,327305],{"className":327306,"style":31814},[31813],[34,327308,327310],{"className":327309},[31818,31819,31820,31821],[34,327311,31660],{"className":327312},[31685,31686,31821],[34,327314,31829],{"className":327315},[31828],[34,327317,327319],{"className":327318},[31801],[34,327320,327322],{"className":327321,"style":31836},[31805],[34,327323],{},[34,327325,327327,327348],{"className":327326},[31642],[34,327328,327330],{"className":327329},[31646],[31648,327331,327332],{"xmlns":31650},[31652,327333,327334,327346],{},[31655,327335,327336,327338,327340],{},[31658,327337,118113],{},[31662,327339,54707],{"mathvariant":38079},[31743,327341,327342,327344],{},[31658,327343,94675],{},[31658,327345,31660],{},[31665,327347,325923],{"encoding":31667},[34,327349,327351,327402],{"className":327350,"ariaHidden":758},[31672],[34,327352,327354,327357,327360,327363,327399],{"className":327353},[31676],[34,327355],{"className":327356,"style":43217},[31680],[34,327358,118113],{"className":327359,"style":118235},[31685,31686],[34,327361],{"className":327362,"style":31846},[31845],[34,327364,327366,327393,327396],{"className":327365},[31850],[34,327367,327369],{"className":327368},[31850],[34,327370,327372],{"className":327371},[31685,54774],[34,327373,327375],{"className":327374},[54778],[34,327376,327378,327381,327390],{"className":327377},[54782],[34,327379],{"className":327380,"style":43217},[31680],[34,327382,327384],{"className":327383},[54789],[34,327385,327387],{"className":327386},[31685],[34,327388,54796],{"className":327389},[31850],[34,327391],{"className":327392},[54800],[34,327394],{"className":327395},[31845,54804],[34,327397,202],{"className":327398},[31850],[34,327400],{"className":327401,"style":31846},[31845],[34,327403,327405,327408],{"className":327404},[31676],[34,327406],{"className":327407,"style":70915},[31680],[34,327409,327411,327414],{"className":327410},[31685],[34,327412,94675],{"className":327413,"style":106455},[31685,31686],[34,327415,327417],{"className":327416},[31792],[34,327418,327420,327440],{"className":327419},[31796,31797],[34,327421,327423,327437],{"className":327422},[31801],[34,327424,327426],{"className":327425,"style":32362},[31805],[34,327427,327428,327431],{"style":206422},[34,327429],{"className":327430,"style":31814},[31813],[34,327432,327434],{"className":327433},[31818,31819,31820,31821],[34,327435,31660],{"className":327436},[31685,31686,31821],[34,327438,31829],{"className":327439},[31828],[34,327441,327443],{"className":327442},[31801],[34,327444,327446],{"className":327445,"style":31836},[31805],[34,327447],{},"，那 ",[34,327450,327452,327465],{"className":327451},[31642],[34,327453,327455],{"className":327454},[31646],[31648,327456,327457],{"xmlns":31650},[31652,327458,327459,327463],{},[31655,327460,327461],{},[31658,327462,118113],{},[31665,327464,118113],{"encoding":31667},[34,327466,327468],{"className":327467,"ariaHidden":758},[31672],[34,327469,327471,327474],{"className":327470},[31676],[34,327472],{"className":327473,"style":33218},[31680],[34,327475,118113],{"className":327476,"style":118235},[31685,31686],[34,327478,327480,327497],{"className":327479},[31642],[34,327481,327483],{"className":327482},[31646],[31648,327484,327485],{"xmlns":31650},[31652,327486,327487,327495],{},[31655,327488,327489],{},[31743,327490,327491,327493],{},[31658,327492,54151],{},[31658,327494,31660],{},[31665,327496,321085],{"encoding":31667},[34,327498,327500],{"className":327499,"ariaHidden":758},[31672],[34,327501,327503,327506],{"className":327502},[31676],[34,327504],{"className":327505,"style":70915},[31680],[34,327507,327509,327512],{"className":327508},[31685],[34,327510,54151],{"className":327511},[31685,31686],[34,327513,327515],{"className":327514},[31792],[34,327516,327518,327538],{"className":327517},[31796,31797],[34,327519,327521,327535],{"className":327520},[31801],[34,327522,327524],{"className":327523,"style":32362},[31805],[34,327525,327526,327529],{"style":31809},[34,327527],{"className":327528,"style":31814},[31813],[34,327530,327532],{"className":327531},[31818,31819,31820,31821],[34,327533,31660],{"className":327534},[31685,31686,31821],[34,327536,31829],{"className":327537},[31828],[34,327539,327541],{"className":327540},[31801],[34,327542,327544],{"className":327543,"style":31836},[31805],[34,327545],{},"。得证。",[108,327548,327549,2664,327552,39984,327676],{},[2627,327550,327551],{},"情况 B",[34,327553,327555,327585],{"className":327554},[31642],[34,327556,327558],{"className":327557},[31646],[31648,327559,327560],{"xmlns":31650},[31652,327561,327562,327582],{},[31655,327563,327564,327566,327568,327574,327576,327578,327580],{},[31658,327565,118113],{},[31662,327567,326344],{},[31743,327569,327570,327572],{},[31658,327571,54151],{},[31658,327573,31660],{},[31662,327575,202],{},[31662,327577,32685],{"stretchy":702},[31658,327579,33244],{},[31662,327581,32715],{"stretchy":702},[31665,327583,327584],{"encoding":31667},"H \\cap A_n = \\{e\\}",[34,327586,327588,327606,327661],{"className":327587,"ariaHidden":758},[31672],[34,327589,327591,327594,327597,327600,327603],{"className":327590},[31676],[34,327592],{"className":327593,"style":33218},[31680],[34,327595,118113],{"className":327596,"style":118235},[31685,31686],[34,327598],{"className":327599,"style":32604},[31845],[34,327601,326344],{"className":327602},[32024],[34,327604],{"className":327605,"style":32604},[31845],[34,327607,327609,327612,327652,327655,327658],{"className":327608},[31676],[34,327610],{"className":327611,"style":70915},[31680],[34,327613,327615,327618],{"className":327614},[31685],[34,327616,54151],{"className":327617},[31685,31686],[34,327619,327621],{"className":327620},[31792],[34,327622,327624,327644],{"className":327623},[31796,31797],[34,327625,327627,327641],{"className":327626},[31801],[34,327628,327630],{"className":327629,"style":32362},[31805],[34,327631,327632,327635],{"style":31809},[34,327633],{"className":327634,"style":31814},[31813],[34,327636,327638],{"className":327637},[31818,31819,31820,31821],[34,327639,31660],{"className":327640},[31685,31686,31821],[34,327642,31829],{"className":327643},[31828],[34,327645,327647],{"className":327646},[31801],[34,327648,327650],{"className":327649,"style":31836},[31805],[34,327651],{},[34,327653],{"className":327654,"style":31846},[31845],[34,327656,202],{"className":327657},[31850],[34,327659],{"className":327660,"style":31846},[31845],[34,327662,327664,327667,327670,327673],{"className":327663},[31676],[34,327665],{"className":327666,"style":31774},[31680],[34,327668,32685],{"className":327669},[31782],[34,327671,33244],{"className":327672},[31685,31686],[34,327674,32715],{"className":327675},[31690],[105,327677,327678,328124,328304,328347,328729,328855],{},[108,327679,327680,327681,347],{},"根据同态定理或乘积公式：",[34,327682,327684,327768],{"className":327683},[31642],[34,327685,327687],{"className":327686},[31646],[31648,327688,327689],{"xmlns":31650},[31652,327690,327691,327765],{},[31655,327692,327693,327695,327697,327703,327705,327707,327745,327747,327749,327751,327753,327755],{},[31658,327694,162086],{"mathvariant":38079},[31658,327696,118113],{},[31743,327698,327699,327701],{},[31658,327700,54151],{},[31658,327702,31660],{},[31658,327704,162086],{"mathvariant":38079},[31662,327706,202],{},[41453,327708,327709,327729],{},[31655,327710,327711,327713,327715,327717,327719,327721,327727],{},[31658,327712,162086],{"mathvariant":38079},[31658,327714,118113],{},[31658,327716,162086],{"mathvariant":38079},[31662,327718,71191],{},[31658,327720,162086],{"mathvariant":38079},[31743,327722,327723,327725],{},[31658,327724,54151],{},[31658,327726,31660],{},[31658,327728,162086],{"mathvariant":38079},[31655,327730,327731,327733,327735,327737,327743],{},[31658,327732,162086],{"mathvariant":38079},[31658,327734,118113],{},[31662,327736,326344],{},[31743,327738,327739,327741],{},[31658,327740,54151],{},[31658,327742,31660],{},[31658,327744,162086],{"mathvariant":38079},[31662,327746,202],{},[31658,327748,162086],{"mathvariant":38079},[31658,327750,118113],{},[31658,327752,162086],{"mathvariant":38079},[31662,327754,71191],{},[41453,327756,327757,327763],{},[31655,327758,327759,327761],{},[31658,327760,31660],{},[31662,327762,5504],{"stretchy":702},[31749,327764,768],{},[31665,327766,327767],{"encoding":31667},"|H A_n| = \\frac{|H| \\cdot |A_n|}{|H \\cap A_n|} = |H| \\cdot \\frac{n!}{2}",[34,327769,327771,327835,328023,328047],{"className":327770,"ariaHidden":758},[31672],[34,327772,327774,327777,327780,327783,327823,327826,327829,327832],{"className":327773},[31676],[34,327775],{"className":327776,"style":31774},[31680],[34,327778,162086],{"className":327779},[31685],[34,327781,118113],{"className":327782,"style":118235},[31685,31686],[34,327784,327786,327789],{"className":327785},[31685],[34,327787,54151],{"className":327788},[31685,31686],[34,327790,327792],{"className":327791},[31792],[34,327793,327795,327815],{"className":327794},[31796,31797],[34,327796,327798,327812],{"className":327797},[31801],[34,327799,327801],{"className":327800,"style":32362},[31805],[34,327802,327803,327806],{"style":31809},[34,327804],{"className":327805,"style":31814},[31813],[34,327807,327809],{"className":327808},[31818,31819,31820,31821],[34,327810,31660],{"className":327811},[31685,31686,31821],[34,327813,31829],{"className":327814},[31828],[34,327816,327818],{"className":327817},[31801],[34,327819,327821],{"className":327820,"style":31836},[31805],[34,327822],{},[34,327824,162086],{"className":327825},[31685],[34,327827],{"className":327828,"style":31846},[31845],[34,327830,202],{"className":327831},[31850],[34,327833],{"className":327834,"style":31846},[31845],[34,327836,327838,327842,328014,328017,328020],{"className":327837},[31676],[34,327839],{"className":327840,"style":327841},[31680],"height:1.53em;vertical-align:-0.52em;",[34,327843,327845,327848,328011],{"className":327844},[31685],[34,327846],{"className":327847},[31782,41488],[34,327849,327851],{"className":327850},[41453],[34,327852,327854,328003],{"className":327853},[31796,31797],[34,327855,327857,328000],{"className":327856},[31801],[34,327858,327860,327923,327931],{"className":327859,"style":70504},[31805],[34,327861,327862,327865],{"style":41504},[34,327863],{"className":327864,"style":41508},[31813],[34,327866,327868],{"className":327867},[31818,31819,31820,31821],[34,327869,327871,327874,327877,327880,327920],{"className":327870},[31685,31821],[34,327872,162086],{"className":327873},[31685,31821],[34,327875,118113],{"className":327876,"style":118235},[31685,31686,31821],[34,327878,326344],{"className":327879},[32024,31821],[34,327881,327883,327886],{"className":327882},[31685,31821],[34,327884,54151],{"className":327885},[31685,31686,31821],[34,327887,327889],{"className":327888},[31792],[34,327890,327892,327912],{"className":327891},[31796,31797],[34,327893,327895,327909],{"className":327894},[31801],[34,327896,327898],{"className":327897,"style":220518},[31805],[34,327899,327900,327903],{"style":318661},[34,327901],{"className":327902,"style":43406},[31813],[34,327904,327906],{"className":327905},[31818,43410,43411,31821],[34,327907,31660],{"className":327908},[31685,31686,31821],[34,327910,31829],{"className":327911},[31828],[34,327913,327915],{"className":327914},[31801],[34,327916,327918],{"className":327917,"style":220540},[31805],[34,327919],{},[34,327921,162086],{"className":327922},[31685,31821],[34,327924,327925,327928],{"style":41520},[34,327926],{"className":327927,"style":41508},[31813],[34,327929],{"className":327930,"style":41528},[41527],[34,327932,327933,327936],{"style":70529},[34,327934],{"className":327935,"style":41508},[31813],[34,327937,327939],{"className":327938},[31818,31819,31820,31821],[34,327940,327942,327945,327948,327951,327954,327957,327997],{"className":327941},[31685,31821],[34,327943,162086],{"className":327944},[31685,31821],[34,327946,118113],{"className":327947,"style":118235},[31685,31686,31821],[34,327949,162086],{"className":327950},[31685,31821],[34,327952,71191],{"className":327953},[32024,31821],[34,327955,162086],{"className":327956},[31685,31821],[34,327958,327960,327963],{"className":327959},[31685,31821],[34,327961,54151],{"className":327962},[31685,31686,31821],[34,327964,327966],{"className":327965},[31792],[34,327967,327969,327989],{"className":327968},[31796,31797],[34,327970,327972,327986],{"className":327971},[31801],[34,327973,327975],{"className":327974,"style":220518},[31805],[34,327976,327977,327980],{"style":318661},[34,327978],{"className":327979,"style":43406},[31813],[34,327981,327983],{"className":327982},[31818,43410,43411,31821],[34,327984,31660],{"className":327985},[31685,31686,31821],[34,327987,31829],{"className":327988},[31828],[34,327990,327992],{"className":327991},[31801],[34,327993,327995],{"className":327994,"style":220540},[31805],[34,327996],{},[34,327998,162086],{"className":327999},[31685,31821],[34,328001,31829],{"className":328002},[31828],[34,328004,328006],{"className":328005},[31801],[34,328007,328009],{"className":328008,"style":324858},[31805],[34,328010],{},[34,328012],{"className":328013},[31690,41488],[34,328015],{"className":328016,"style":31846},[31845],[34,328018,202],{"className":328019},[31850],[34,328021],{"className":328022,"style":31846},[31845],[34,328024,328026,328029,328032,328035,328038,328041,328044],{"className":328025},[31676],[34,328027],{"className":328028,"style":31774},[31680],[34,328030,162086],{"className":328031},[31685],[34,328033,118113],{"className":328034,"style":118235},[31685,31686],[34,328036,162086],{"className":328037},[31685],[34,328039],{"className":328040,"style":32604},[31845],[34,328042,71191],{"className":328043},[32024],[34,328045],{"className":328046,"style":32604},[31845],[34,328048,328050,328053],{"className":328049},[31676],[34,328051],{"className":328052,"style":232492},[31680],[34,328054,328056,328059,328121],{"className":328055},[31685],[34,328057],{"className":328058},[31782,41488],[34,328060,328062],{"className":328061},[41453],[34,328063,328065,328113],{"className":328064},[31796,31797],[34,328066,328068,328110],{"className":328067},[31801],[34,328069,328071,328085,328093],{"className":328070,"style":232511},[31805],[34,328072,328073,328076],{"style":41504},[34,328074],{"className":328075,"style":41508},[31813],[34,328077,328079],{"className":328078},[31818,31819,31820,31821],[34,328080,328082],{"className":328081},[31685,31821],[34,328083,768],{"className":328084},[31685,31821],[34,328086,328087,328090],{"style":41520},[34,328088],{"className":328089,"style":41508},[31813],[34,328091],{"className":328092,"style":41528},[41527],[34,328094,328095,328098],{"style":46771},[34,328096],{"className":328097,"style":41508},[31813],[34,328099,328101],{"className":328100},[31818,31819,31820,31821],[34,328102,328104,328107],{"className":328103},[31685,31821],[34,328105,31660],{"className":328106},[31685,31686,31821],[34,328108,5504],{"className":328109},[31690,31821],[34,328111,31829],{"className":328112},[31828],[34,328114,328116],{"className":328115},[31801],[34,328117,328119],{"className":328118,"style":41619},[31805],[34,328120],{},[34,328122],{"className":328123},[31690,41488],[108,328125,93310,328126,11008,328201,328270,328271,347],{},[34,328127,328129,328149],{"className":328128},[31642],[34,328130,328132],{"className":328131},[31646],[31648,328133,328134],{"xmlns":31650},[31652,328135,328136,328146],{},[31655,328137,328138,328140],{},[31658,328139,118113],{},[31743,328141,328142,328144],{},[31658,328143,54151],{},[31658,328145,31660],{},[31665,328147,328148],{"encoding":31667},"H A_n",[34,328150,328152],{"className":328151,"ariaHidden":758},[31672],[34,328153,328155,328158,328161],{"className":328154},[31676],[34,328156],{"className":328157,"style":70915},[31680],[34,328159,118113],{"className":328160,"style":118235},[31685,31686],[34,328162,328164,328167],{"className":328163},[31685],[34,328165,54151],{"className":328166},[31685,31686],[34,328168,328170],{"className":328169},[31792],[34,328171,328173,328193],{"className":328172},[31796,31797],[34,328174,328176,328190],{"className":328175},[31801],[34,328177,328179],{"className":328178,"style":32362},[31805],[34,328180,328181,328184],{"style":31809},[34,328182],{"className":328183,"style":31814},[31813],[34,328185,328187],{"className":328186},[31818,31819,31820,31821],[34,328188,31660],{"className":328189},[31685,31686,31821],[34,328191,31829],{"className":328192},[31828],[34,328194,328196],{"className":328195},[31801],[34,328197,328199],{"className":328198,"style":31836},[31805],[34,328200],{},[34,328202,328204,328221],{"className":328203},[31642],[34,328205,328207],{"className":328206},[31646],[31648,328208,328209],{"xmlns":31650},[31652,328210,328211,328219],{},[31655,328212,328213],{},[31743,328214,328215,328217],{},[31658,328216,94675],{},[31658,328218,31660],{},[31665,328220,218979],{"encoding":31667},[34,328222,328224],{"className":328223,"ariaHidden":758},[31672],[34,328225,328227,328230],{"className":328226},[31676],[34,328228],{"className":328229,"style":70915},[31680],[34,328231,328233,328236],{"className":328232},[31685],[34,328234,94675],{"className":328235,"style":106455},[31685,31686],[34,328237,328239],{"className":328238},[31792],[34,328240,328242,328262],{"className":328241},[31796,31797],[34,328243,328245,328259],{"className":328244},[31801],[34,328246,328248],{"className":328247,"style":32362},[31805],[34,328249,328250,328253],{"style":206422},[34,328251],{"className":328252,"style":31814},[31813],[34,328254,328256],{"className":328255},[31818,31819,31820,31821],[34,328257,31660],{"className":328258},[31685,31686,31821],[34,328260,31829],{"className":328261},[31828],[34,328263,328265],{"className":328264},[31801],[34,328266,328268],{"className":328267,"style":31836},[31805],[34,328269],{}," 的子群，它的大小不能超过 ",[34,328272,328274,328289],{"className":328273},[31642],[34,328275,328277],{"className":328276},[31646],[31648,328278,328279],{"xmlns":31650},[31652,328280,328281,328287],{},[31655,328282,328283,328285],{},[31658,328284,31660],{},[31662,328286,5504],{"stretchy":702},[31665,328288,31668],{"encoding":31667},[34,328290,328292],{"className":328291,"ariaHidden":758},[31672],[34,328293,328295,328298,328301],{"className":328294},[31676],[34,328296],{"className":328297,"style":31681},[31680],[34,328299,31660],{"className":328300},[31685,31686],[34,328302,5504],{"className":328303},[31690],[108,328305,328306,328307,328346],{},"这强迫 ",[34,328308,328310,328328],{"className":328309},[31642],[34,328311,328313],{"className":328312},[31646],[31648,328314,328315],{"xmlns":31650},[31652,328316,328317,328325],{},[31655,328318,328319,328321,328323],{},[31658,328320,162086],{"mathvariant":38079},[31658,328322,118113],{},[31658,328324,162086],{"mathvariant":38079},[31665,328326,328327],{"encoding":31667},"|H|",[34,328329,328331],{"className":328330,"ariaHidden":758},[31672],[34,328332,328334,328337,328340,328343],{"className":328333},[31676],[34,328335],{"className":328336,"style":31774},[31680],[34,328338,162086],{"className":328339},[31685],[34,328341,118113],{"className":328342,"style":118235},[31685,31686],[34,328344,162086],{"className":328345},[31685]," 只能等于 2。",[108,328348,2234,328349,328410,328411,44689,328485,328513,328514,328543,328544,328613,328614,328642,328643,328728],{},[34,328350,328352,328374],{"className":328351},[31642],[34,328353,328355],{"className":328354},[31646],[31648,328356,328357],{"xmlns":31650},[31652,328358,328359,328371],{},[31655,328360,328361,328363,328365,328367,328369],{},[31658,328362,162086],{"mathvariant":38079},[31658,328364,118113],{},[31658,328366,162086],{"mathvariant":38079},[31662,328368,202],{},[31749,328370,768],{},[31665,328372,328373],{"encoding":31667},"|H|=2",[34,328375,328377,328401],{"className":328376,"ariaHidden":758},[31672],[34,328378,328380,328383,328386,328389,328392,328395,328398],{"className":328379},[31676],[34,328381],{"className":328382,"style":31774},[31680],[34,328384,162086],{"className":328385},[31685],[34,328387,118113],{"className":328388,"style":118235},[31685,31686],[34,328390,162086],{"className":328391},[31685],[34,328393],{"className":328394,"style":31846},[31845],[34,328396,202],{"className":328397},[31850],[34,328399],{"className":328400,"style":31846},[31845],[34,328402,328404,328407],{"className":328403},[31676],[34,328405],{"className":328406,"style":32617},[31680],[34,328408,768],{"className":328409},[31685],"，设 ",[34,328412,328414,328440],{"className":328413},[31642],[34,328415,328417],{"className":328416},[31646],[31648,328418,328419],{"xmlns":31650},[31652,328420,328421,328437],{},[31655,328422,328423,328425,328427,328429,328431,328433,328435],{},[31658,328424,118113],{},[31662,328426,202],{},[31662,328428,32685],{"stretchy":702},[31658,328430,33244],{},[31662,328432,31960],{"separator":758},[31658,328434,321156],{},[31662,328436,32715],{"stretchy":702},[31665,328438,328439],{"encoding":31667},"H = \\{e, \\tau\\}",[34,328441,328443,328461],{"className":328442,"ariaHidden":758},[31672],[34,328444,328446,328449,328452,328455,328458],{"className":328445},[31676],[34,328447],{"className":328448,"style":33218},[31680],[34,328450,118113],{"className":328451,"style":118235},[31685,31686],[34,328453],{"className":328454,"style":31846},[31845],[34,328456,202],{"className":328457},[31850],[34,328459],{"className":328460,"style":31846},[31845],[34,328462,328464,328467,328470,328473,328476,328479,328482],{"className":328463},[31676],[34,328465],{"className":328466,"style":31774},[31680],[34,328468,32685],{"className":328469},[31782],[34,328471,33244],{"className":328472},[31685,31686],[34,328474,31960],{"className":328475},[32119],[34,328477],{"className":328478,"style":32123},[31845],[34,328480,321156],{"className":328481,"style":321198},[31685,31686],[34,328483,32715],{"className":328484},[31690],[34,328486,328488,328501],{"className":328487},[31642],[34,328489,328491],{"className":328490},[31646],[31648,328492,328493],{"xmlns":31650},[31652,328494,328495,328499],{},[31655,328496,328497],{},[31658,328498,118113],{},[31665,328500,118113],{"encoding":31667},[34,328502,328504],{"className":328503,"ariaHidden":758},[31672],[34,328505,328507,328510],{"className":328506},[31676],[34,328508],{"className":328509,"style":33218},[31680],[34,328511,118113],{"className":328512,"style":118235},[31685,31686]," 是正规子群，这意味着 ",[34,328515,328517,328531],{"className":328516},[31642],[34,328518,328520],{"className":328519},[31646],[31648,328521,328522],{"xmlns":31650},[31652,328523,328524,328528],{},[31655,328525,328526],{},[31658,328527,321156],{},[31665,328529,328530],{"encoding":31667},"\\tau",[34,328532,328534],{"className":328533,"ariaHidden":758},[31672],[34,328535,328537,328540],{"className":328536},[31676],[34,328538],{"className":328539,"style":32527},[31680],[34,328541,321156],{"className":328542,"style":321198},[31685,31686]," 必须和 ",[34,328545,328547,328564],{"className":328546},[31642],[34,328548,328550],{"className":328549},[31646],[31648,328551,328552],{"xmlns":31650},[31652,328553,328554,328562],{},[31655,328555,328556],{},[31743,328557,328558,328560],{},[31658,328559,94675],{},[31658,328561,31660],{},[31665,328563,218979],{"encoding":31667},[34,328565,328567],{"className":328566,"ariaHidden":758},[31672],[34,328568,328570,328573],{"className":328569},[31676],[34,328571],{"className":328572,"style":70915},[31680],[34,328574,328576,328579],{"className":328575},[31685],[34,328577,94675],{"className":328578,"style":106455},[31685,31686],[34,328580,328582],{"className":328581},[31792],[34,328583,328585,328605],{"className":328584},[31796,31797],[34,328586,328588,328602],{"className":328587},[31801],[34,328589,328591],{"className":328590,"style":32362},[31805],[34,328592,328593,328596],{"style":206422},[34,328594],{"className":328595,"style":31814},[31813],[34,328597,328599],{"className":328598},[31818,31819,31820,31821],[34,328600,31660],{"className":328601},[31685,31686,31821],[34,328603,31829],{"className":328604},[31828],[34,328606,328608],{"className":328607},[31801],[34,328609,328611],{"className":328610,"style":31836},[31805],[34,328612],{}," 中所有元素交换（即 ",[34,328615,328617,328630],{"className":328616},[31642],[34,328618,328620],{"className":328619},[31646],[31648,328621,328622],{"xmlns":31650},[31652,328623,328624,328628],{},[31655,328625,328626],{},[31658,328627,321156],{},[31665,328629,328530],{"encoding":31667},[34,328631,328633],{"className":328632,"ariaHidden":758},[31672],[34,328634,328636,328639],{"className":328635},[31676],[34,328637],{"className":328638,"style":32527},[31680],[34,328640,321156],{"className":328641,"style":321198},[31685,31686]," 在中心 ",[34,328644,328646,328670],{"className":328645},[31642],[34,328647,328649],{"className":328648},[31646],[31648,328650,328651],{"xmlns":31650},[31652,328652,328653,328667],{},[31655,328654,328655,328657,328659,328665],{},[31658,328656,97300],{},[31662,328658,456],{"stretchy":702},[31743,328660,328661,328663],{},[31658,328662,94675],{},[31658,328664,31660],{},[31662,328666,2557],{"stretchy":702},[31665,328668,328669],{"encoding":31667},"Z(S_n)",[34,328671,328673],{"className":328672,"ariaHidden":758},[31672],[34,328674,328676,328679,328682,328685,328725],{"className":328675},[31676],[34,328677],{"className":328678,"style":31774},[31680],[34,328680,97300],{"className":328681,"style":38706},[31685,31686],[34,328683,456],{"className":328684},[31782],[34,328686,328688,328691],{"className":328687},[31685],[34,328689,94675],{"className":328690,"style":106455},[31685,31686],[34,328692,328694],{"className":328693},[31792],[34,328695,328697,328717],{"className":328696},[31796,31797],[34,328698,328700,328714],{"className":328699},[31801],[34,328701,328703],{"className":328702,"style":32362},[31805],[34,328704,328705,328708],{"style":206422},[34,328706],{"className":328707,"style":31814},[31813],[34,328709,328711],{"className":328710},[31818,31819,31820,31821],[34,328712,31660],{"className":328713},[31685,31686,31821],[34,328715,31829],{"className":328716},[31828],[34,328718,328720],{"className":328719},[31801],[34,328721,328723],{"className":328722,"style":31836},[31805],[34,328724],{},[34,328726,2557],{"className":328727},[31690]," 里）。",[108,328730,328731,328732,46528,328783,347],{},"但是！对于 ",[34,328733,328735,328753],{"className":328734},[31642],[34,328736,328738],{"className":328737},[31646],[31648,328739,328740],{"xmlns":31650},[31652,328741,328742,328750],{},[31655,328743,328744,328746,328748],{},[31658,328745,31660],{},[31662,328747,43175],{},[31749,328749,786],{},[31665,328751,328752],{"encoding":31667},"n \\ge 3",[34,328754,328756,328774],{"className":328755,"ariaHidden":758},[31672],[34,328757,328759,328762,328765,328768,328771],{"className":328758},[31676],[34,328760],{"className":328761,"style":43198},[31680],[34,328763,31660],{"className":328764},[31685,31686],[34,328766],{"className":328767,"style":31846},[31845],[34,328769,43175],{"className":328770},[31850],[34,328772],{"className":328773,"style":31846},[31845],[34,328775,328777,328780],{"className":328776},[31676],[34,328778],{"className":328779,"style":32617},[31680],[34,328781,786],{"className":328782},[31685],[2627,328784,328785,328854],{},[34,328786,328788,328805],{"className":328787},[31642],[34,328789,328791],{"className":328790},[31646],[31648,328792,328793],{"xmlns":31650},[31652,328794,328795,328803],{},[31655,328796,328797],{},[31743,328798,328799,328801],{},[31658,328800,94675],{},[31658,328802,31660],{},[31665,328804,218979],{"encoding":31667},[34,328806,328808],{"className":328807,"ariaHidden":758},[31672],[34,328809,328811,328814],{"className":328810},[31676],[34,328812],{"className":328813,"style":70915},[31680],[34,328815,328817,328820],{"className":328816},[31685],[34,328818,94675],{"className":328819,"style":106455},[31685,31686],[34,328821,328823],{"className":328822},[31792],[34,328824,328826,328846],{"className":328825},[31796,31797],[34,328827,328829,328843],{"className":328828},[31801],[34,328830,328832],{"className":328831,"style":32362},[31805],[34,328833,328834,328837],{"style":206422},[34,328835],{"className":328836,"style":31814},[31813],[34,328838,328840],{"className":328839},[31818,31819,31820,31821],[34,328841,31660],{"className":328842},[31685,31686,31821],[34,328844,31829],{"className":328845},[31828],[34,328847,328849],{"className":328848},[31801],[34,328850,328852],{"className":328851,"style":31836},[31805],[34,328853],{}," 的中心是平凡的（只有单位元）",[108,328856,328857],{},"矛盾！所以情况 B 不可能发生。",[108,328859,328860,328862,328863,347],{},[2627,328861,114043],{},"：只能是情况 A，即 ",[34,328864,328866,328888],{"className":328865},[31642],[34,328867,328869],{"className":328868},[31646],[31648,328870,328871],{"xmlns":31650},[31652,328872,328873,328885],{},[31655,328874,328875,328877,328879],{},[31658,328876,118113],{},[31662,328878,202],{},[31743,328880,328881,328883],{},[31658,328882,54151],{},[31658,328884,31660],{},[31665,328886,328887],{"encoding":31667},"H = A_n",[34,328889,328891,328909],{"className":328890,"ariaHidden":758},[31672],[34,328892,328894,328897,328900,328903,328906],{"className":328893},[31676],[34,328895],{"className":328896,"style":33218},[31680],[34,328898,118113],{"className":328899,"style":118235},[31685,31686],[34,328901],{"className":328902,"style":31846},[31845],[34,328904,202],{"className":328905},[31850],[34,328907],{"className":328908,"style":31846},[31845],[34,328910,328912,328915],{"className":328911},[31676],[34,328913],{"className":328914,"style":70915},[31680],[34,328916,328918,328921],{"className":328917},[31685],[34,328919,54151],{"className":328920},[31685,31686],[34,328922,328924],{"className":328923},[31792],[34,328925,328927,328947],{"className":328926},[31796,31797],[34,328928,328930,328944],{"className":328929},[31801],[34,328931,328933],{"className":328932,"style":32362},[31805],[34,328934,328935,328938],{"style":31809},[34,328936],{"className":328937,"style":31814},[31813],[34,328939,328941],{"className":328940},[31818,31819,31820,31821],[34,328942,31660],{"className":328943},[31685,31686,31821],[34,328945,31829],{"className":328946},[31828],[34,328948,328950],{"className":328949},[31801],[34,328951,328953],{"className":328952,"style":31836},[31805],[34,328954],{},[32188,328956],{},[32490,328958,328960],{"id":328959},"题目-20只有两个自同构的群","题目 20：只有两个自同构的群",[15,328962,328963,328965,328966,329052,329053,329081],{},[2627,328964,101191],{}," 如果 ",[34,328967,328969,329001],{"className":328968},[31642],[34,328970,328972],{"className":328971},[31646],[31648,328973,328974],{"xmlns":31650},[31652,328975,328976,328998],{},[31655,328977,328978,328980,328982,328984,328986,328988,328990,328992,328994,328996],{},[31658,328979,162086],{"mathvariant":38079},[31658,328981,54151],{},[31658,328983,94628],{},[31658,328985,33249],{},[31662,328987,456],{"stretchy":702},[31658,328989,97709],{},[31662,328991,2557],{"stretchy":702},[31658,328993,162086],{"mathvariant":38079},[31662,328995,202],{},[31749,328997,768],{},[31665,328999,329000],{"encoding":31667},"|Aut(G)| = 2",[34,329002,329004,329043],{"className":329003,"ariaHidden":758},[31672],[34,329005,329007,329010,329013,329016,329019,329022,329025,329028,329031,329034,329037,329040],{"className":329006},[31676],[34,329008],{"className":329009,"style":31774},[31680],[34,329011,162086],{"className":329012},[31685],[34,329014,54151],{"className":329015},[31685,31686],[34,329017,94628],{"className":329018},[31685,31686],[34,329020,33249],{"className":329021},[31685,31686],[34,329023,456],{"className":329024},[31782],[34,329026,97709],{"className":329027},[31685,31686],[34,329029,2557],{"className":329030},[31690],[34,329032,162086],{"className":329033},[31685],[34,329035],{"className":329036,"style":31846},[31845],[34,329038,202],{"className":329039},[31850],[34,329041],{"className":329042,"style":31846},[31845],[34,329044,329046,329049],{"className":329045},[31676],[34,329047],{"className":329048,"style":32617},[31680],[34,329050,768],{"className":329051},[31685],"，证明 ",[34,329054,329056,329069],{"className":329055},[31642],[34,329057,329059],{"className":329058},[31646],[31648,329060,329061],{"xmlns":31650},[31652,329062,329063,329067],{},[31655,329064,329065],{},[31658,329066,97709],{},[31665,329068,97709],{"encoding":31667},[34,329070,329072],{"className":329071,"ariaHidden":758},[31672],[34,329073,329075,329078],{"className":329074},[31676],[34,329076],{"className":329077,"style":33218},[31680],[34,329079,97709],{"className":329080},[31685,31686]," 是交换群。",[15,329083,329084],{},[2627,329085,314990],{},[2230,329087,329088,329123,329454,329587],{},[108,329089,329090,329093,329094,329122],{},[2627,329091,329092],{},"什么是自同构？"," 群 ",[34,329095,329097,329110],{"className":329096},[31642],[34,329098,329100],{"className":329099},[31646],[31648,329101,329102],{"xmlns":31650},[31652,329103,329104,329108],{},[31655,329105,329106],{},[31658,329107,97709],{},[31665,329109,97709],{"encoding":31667},[34,329111,329113],{"className":329112,"ariaHidden":758},[31672],[34,329114,329116,329119],{"className":329115},[31676],[34,329117],{"className":329118,"style":33218},[31680],[34,329120,97709],{"className":329121},[31685,31686]," 到它自己的同构映射。",[108,329124,329125,329128],{},[2627,329126,329127],{},"什么是内自同构（Inn）？",[105,329129,329130,329344],{},[108,329131,329132,329133,329184,329185,329343],{},"对任意 ",[34,329134,329136,329154],{"className":329135},[31642],[34,329137,329139],{"className":329138},[31646],[31648,329140,329141],{"xmlns":31650},[31652,329142,329143,329151],{},[31655,329144,329145,329147,329149],{},[31658,329146,31942],{},[31662,329148,112296],{},[31658,329150,97709],{},[31665,329152,329153],{"encoding":31667},"g \\in G",[34,329155,329157,329175],{"className":329156,"ariaHidden":758},[31672],[34,329158,329160,329163,329166,329169,329172],{"className":329159},[31676],[34,329161],{"className":329162,"style":86652},[31680],[34,329164,31942],{"className":329165,"style":32062},[31685,31686],[34,329167],{"className":329168,"style":31846},[31845],[34,329170,112296],{"className":329171},[31850],[34,329173],{"className":329174,"style":31846},[31845],[34,329176,329178,329181],{"className":329177},[31676],[34,329179],{"className":329180,"style":33218},[31680],[34,329182,97709],{"className":329183},[31685,31686],"，定义映射 ",[34,329186,329188,329229],{"className":329187},[31642],[34,329189,329191],{"className":329190},[31646],[31648,329192,329193],{"xmlns":31650},[31652,329194,329195,329226],{},[31655,329196,329197,329204,329206,329208,329210,329212,329214,329216],{},[31743,329198,329199,329202],{},[31658,329200,329201],{},"ϕ",[31658,329203,31942],{},[31662,329205,456],{"stretchy":702},[31658,329207,32936],{},[31662,329209,2557],{"stretchy":702},[31662,329211,202],{},[31658,329213,31942],{},[31658,329215,32936],{},[37624,329217,329218,329220],{},[31658,329219,31942],{},[31655,329221,329222,329224],{},[31662,329223,32226],{},[31749,329225,640],{},[31665,329227,329228],{"encoding":31667},"\\phi_g(x) = g x g^{-1}",[34,329230,329232,329296],{"className":329231,"ariaHidden":758},[31672],[34,329233,329235,329238,329278,329281,329284,329287,329290,329293],{"className":329234},[31676],[34,329236],{"className":329237,"style":32058},[31680],[34,329239,329241,329244],{"className":329240},[31685],[34,329242,329201],{"className":329243},[31685,31686],[34,329245,329247],{"className":329246},[31792],[34,329248,329250,329270],{"className":329249},[31796,31797],[34,329251,329253,329267],{"className":329252},[31801],[34,329254,329256],{"className":329255,"style":32362},[31805],[34,329257,329258,329261],{"style":31809},[34,329259],{"className":329260,"style":31814},[31813],[34,329262,329264],{"className":329263},[31818,31819,31820,31821],[34,329265,31942],{"className":329266,"style":32062},[31685,31686,31821],[34,329268,31829],{"className":329269},[31828],[34,329271,329273],{"className":329272},[31801],[34,329274,329276],{"className":329275,"style":32163},[31805],[34,329277],{},[34,329279,456],{"className":329280},[31782],[34,329282,32936],{"className":329283},[31685,31686],[34,329285,2557],{"className":329286},[31690],[34,329288],{"className":329289,"style":31846},[31845],[34,329291,202],{"className":329292},[31850],[34,329294],{"className":329295,"style":31846},[31845],[34,329297,329299,329302,329305,329308],{"className":329298},[31676],[34,329300],{"className":329301,"style":315747},[31680],[34,329303,31942],{"className":329304,"style":32062},[31685,31686],[34,329306,32936],{"className":329307},[31685,31686],[34,329309,329311,329314],{"className":329310},[31685],[34,329312,31942],{"className":329313,"style":32062},[31685,31686],[34,329315,329317],{"className":329316},[31792],[34,329318,329320],{"className":329319},[31796],[34,329321,329323],{"className":329322},[31801],[34,329324,329326],{"className":329325,"style":37774},[31805],[34,329327,329328,329331],{"style":37777},[34,329329],{"className":329330,"style":31814},[31813],[34,329332,329334],{"className":329333},[31818,31819,31820,31821],[34,329335,329337,329340],{"className":329336},[31685,31821],[34,329338,32226],{"className":329339},[31685,31821],[34,329341,640],{"className":329342},[31685,31821],"。这也是一种自同构。",[108,329345,329346,329347,11008,329399,329453],{},"所有的内自同构构成的群 ",[34,329348,329350,329374],{"className":329349},[31642],[34,329351,329353],{"className":329352},[31646],[31648,329354,329355],{"xmlns":31650},[31652,329356,329357,329371],{},[31655,329358,329359,329361,329363,329365,329367,329369],{},[31658,329360,106063],{},[31658,329362,31660],{},[31658,329364,31660],{},[31662,329366,456],{"stretchy":702},[31658,329368,97709],{},[31662,329370,2557],{"stretchy":702},[31665,329372,329373],{"encoding":31667},"Inn(G)",[34,329375,329377],{"className":329376,"ariaHidden":758},[31672],[34,329378,329380,329383,329386,329390,329393,329396],{"className":329379},[31676],[34,329381],{"className":329382,"style":31774},[31680],[34,329384,106063],{"className":329385,"style":67482},[31685,31686],[34,329387,329389],{"className":329388},[31685,31686],"nn",[34,329391,456],{"className":329392},[31782],[34,329394,97709],{"className":329395},[31685,31686],[34,329397,2557],{"className":329398},[31690],[34,329400,329402,329426],{"className":329401},[31642],[34,329403,329405],{"className":329404},[31646],[31648,329406,329407],{"xmlns":31650},[31652,329408,329409,329423],{},[31655,329410,329411,329413,329415,329417,329419,329421],{},[31658,329412,54151],{},[31658,329414,94628],{},[31658,329416,33249],{},[31662,329418,456],{"stretchy":702},[31658,329420,97709],{},[31662,329422,2557],{"stretchy":702},[31665,329424,329425],{"encoding":31667},"Aut(G)",[34,329427,329429],{"className":329428,"ariaHidden":758},[31672],[34,329430,329432,329435,329438,329441,329444,329447,329450],{"className":329431},[31676],[34,329433],{"className":329434,"style":31774},[31680],[34,329436,54151],{"className":329437},[31685,31686],[34,329439,94628],{"className":329440},[31685,31686],[34,329442,33249],{"className":329443},[31685,31686],[34,329445,456],{"className":329446},[31782],[34,329448,97709],{"className":329449},[31685,31686],[34,329451,2557],{"className":329452},[31690]," 的子群。",[108,329455,329456,33183,329459,329557,329558,329586],{},[2627,329457,329458],{},"关键联系",[34,329460,329462,329500],{"className":329461},[31642],[34,329463,329465],{"className":329464},[31646],[31648,329466,329467],{"xmlns":31650},[31652,329468,329469,329497],{},[31655,329470,329471,329473,329475,329477,329479,329481,329483,329485,329487,329489,329491,329493,329495],{},[31658,329472,106063],{},[31658,329474,31660],{},[31658,329476,31660],{},[31662,329478,456],{"stretchy":702},[31658,329480,97709],{},[31662,329482,2557],{"stretchy":702},[31662,329484,134826],{},[31658,329486,97709],{},[31658,329488,38080],{"mathvariant":38079},[31658,329490,97300],{},[31662,329492,456],{"stretchy":702},[31658,329494,97709],{},[31662,329496,2557],{"stretchy":702},[31665,329498,329499],{"encoding":31667},"Inn(G) \\cong G \u002F Z(G)",[34,329501,329503,329533],{"className":329502,"ariaHidden":758},[31672],[34,329504,329506,329509,329512,329515,329518,329521,329524,329527,329530],{"className":329505},[31676],[34,329507],{"className":329508,"style":31774},[31680],[34,329510,106063],{"className":329511,"style":67482},[31685,31686],[34,329513,329389],{"className":329514},[31685,31686],[34,329516,456],{"className":329517},[31782],[34,329519,97709],{"className":329520},[31685,31686],[34,329522,2557],{"className":329523},[31690],[34,329525],{"className":329526,"style":31846},[31845],[34,329528,134826],{"className":329529},[31850],[34,329531],{"className":329532,"style":31846},[31845],[34,329534,329536,329539,329542,329545,329548,329551,329554],{"className":329535},[31676],[34,329537],{"className":329538,"style":31774},[31680],[34,329540,97709],{"className":329541},[31685,31686],[34,329543,38080],{"className":329544},[31685],[34,329546,97300],{"className":329547,"style":38706},[31685,31686],[34,329549,456],{"className":329550},[31782],[34,329552,97709],{"className":329553},[31685,31686],[34,329555,2557],{"className":329556},[31690],"\n即：内自同构群同构于 ",[34,329559,329561,329574],{"className":329560},[31642],[34,329562,329564],{"className":329563},[31646],[31648,329565,329566],{"xmlns":31650},[31652,329567,329568,329572],{},[31655,329569,329570],{},[31658,329571,97709],{},[31665,329573,97709],{"encoding":31667},[34,329575,329577],{"className":329576,"ariaHidden":758},[31672],[34,329578,329580,329583],{"className":329579},[31676],[34,329581],{"className":329582,"style":33218},[31680],[34,329584,97709],{"className":329585},[31685,31686]," 商掉它的中心。",[108,329588,329589,33183,329591],{},[2627,329590,33190],{},[105,329592,329593,329681,329833,330110],{},[108,329594,329595,329596,347],{},"题目说 ",[34,329597,329599,329630],{"className":329598},[31642],[34,329600,329602],{"className":329601},[31646],[31648,329603,329604],{"xmlns":31650},[31652,329605,329606,329628],{},[31655,329607,329608,329610,329612,329614,329616,329618,329620,329622,329624,329626],{},[31658,329609,162086],{"mathvariant":38079},[31658,329611,54151],{},[31658,329613,94628],{},[31658,329615,33249],{},[31662,329617,456],{"stretchy":702},[31658,329619,97709],{},[31662,329621,2557],{"stretchy":702},[31658,329623,162086],{"mathvariant":38079},[31662,329625,202],{},[31749,329627,768],{},[31665,329629,329000],{"encoding":31667},[34,329631,329633,329672],{"className":329632,"ariaHidden":758},[31672],[34,329634,329636,329639,329642,329645,329648,329651,329654,329657,329660,329663,329666,329669],{"className":329635},[31676],[34,329637],{"className":329638,"style":31774},[31680],[34,329640,162086],{"className":329641},[31685],[34,329643,54151],{"className":329644},[31685,31686],[34,329646,94628],{"className":329647},[31685,31686],[34,329649,33249],{"className":329650},[31685,31686],[34,329652,456],{"className":329653},[31782],[34,329655,97709],{"className":329656},[31685,31686],[34,329658,2557],{"className":329659},[31690],[34,329661,162086],{"className":329662},[31685],[34,329664],{"className":329665,"style":31846},[31845],[34,329667,202],{"className":329668},[31850],[34,329670],{"className":329671,"style":31846},[31845],[34,329673,329675,329678],{"className":329674},[31676],[34,329676],{"className":329677,"style":32617},[31680],[34,329679,768],{"className":329680},[31685],[108,329682,93310,329683,329781,329782,329832],{},[34,329684,329686,329724],{"className":329685},[31642],[34,329687,329689],{"className":329688},[31646],[31648,329690,329691],{"xmlns":31650},[31652,329692,329693,329721],{},[31655,329694,329695,329697,329699,329701,329703,329705,329707,329709,329711,329713,329715,329717,329719],{},[31658,329696,106063],{},[31658,329698,31660],{},[31658,329700,31660],{},[31662,329702,456],{"stretchy":702},[31658,329704,97709],{},[31662,329706,2557],{"stretchy":702},[31662,329708,45972],{},[31658,329710,54151],{},[31658,329712,94628],{},[31658,329714,33249],{},[31662,329716,456],{"stretchy":702},[31658,329718,97709],{},[31662,329720,2557],{"stretchy":702},[31665,329722,329723],{"encoding":31667},"Inn(G) \\le Aut(G)",[34,329725,329727,329757],{"className":329726,"ariaHidden":758},[31672],[34,329728,329730,329733,329736,329739,329742,329745,329748,329751,329754],{"className":329729},[31676],[34,329731],{"className":329732,"style":31774},[31680],[34,329734,106063],{"className":329735,"style":67482},[31685,31686],[34,329737,329389],{"className":329738},[31685,31686],[34,329740,456],{"className":329741},[31782],[34,329743,97709],{"className":329744},[31685,31686],[34,329746,2557],{"className":329747},[31690],[34,329749],{"className":329750,"style":31846},[31845],[34,329752,45972],{"className":329753},[31850],[34,329755],{"className":329756,"style":31846},[31845],[34,329758,329760,329763,329766,329769,329772,329775,329778],{"className":329759},[31676],[34,329761],{"className":329762,"style":31774},[31680],[34,329764,54151],{"className":329765},[31685,31686],[34,329767,94628],{"className":329768},[31685,31686],[34,329770,33249],{"className":329771},[31685,31686],[34,329773,456],{"className":329774},[31782],[34,329776,97709],{"className":329777},[31685,31686],[34,329779,2557],{"className":329780},[31690],"，根据拉格朗日定理，",[34,329783,329785,329808],{"className":329784},[31642],[34,329786,329788],{"className":329787},[31646],[31648,329789,329790],{"xmlns":31650},[31652,329791,329792,329806],{},[31655,329793,329794,329796,329798,329800,329802,329804],{},[31658,329795,106063],{},[31658,329797,31660],{},[31658,329799,31660],{},[31662,329801,456],{"stretchy":702},[31658,329803,97709],{},[31662,329805,2557],{"stretchy":702},[31665,329807,329373],{"encoding":31667},[34,329809,329811],{"className":329810,"ariaHidden":758},[31672],[34,329812,329814,329817,329820,329823,329826,329829],{"className":329813},[31676],[34,329815],{"className":329816,"style":31774},[31680],[34,329818,106063],{"className":329819,"style":67482},[31685,31686],[34,329821,329389],{"className":329822},[31685,31686],[34,329824,456],{"className":329825},[31782],[34,329827,97709],{"className":329828},[31685,31686],[34,329830,2557],{"className":329831},[31690]," 的阶只能是 1 或者 2。",[108,329834,329835,2664,329838,39984,329921],{},[2627,329836,329837],{},"情况 1",[34,329839,329841,329873],{"className":329840},[31642],[34,329842,329844],{"className":329843},[31646],[31648,329845,329846],{"xmlns":31650},[31652,329847,329848,329870],{},[31655,329849,329850,329852,329854,329856,329858,329860,329862,329864,329866,329868],{},[31658,329851,162086],{"mathvariant":38079},[31658,329853,106063],{},[31658,329855,31660],{},[31658,329857,31660],{},[31662,329859,456],{"stretchy":702},[31658,329861,97709],{},[31662,329863,2557],{"stretchy":702},[31658,329865,162086],{"mathvariant":38079},[31662,329867,202],{},[31749,329869,640],{},[31665,329871,329872],{"encoding":31667},"|Inn(G)| = 1",[34,329874,329876,329912],{"className":329875,"ariaHidden":758},[31672],[34,329877,329879,329882,329885,329888,329891,329894,329897,329900,329903,329906,329909],{"className":329878},[31676],[34,329880],{"className":329881,"style":31774},[31680],[34,329883,162086],{"className":329884},[31685],[34,329886,106063],{"className":329887,"style":67482},[31685,31686],[34,329889,329389],{"className":329890},[31685,31686],[34,329892,456],{"className":329893},[31782],[34,329895,97709],{"className":329896},[31685,31686],[34,329898,2557],{"className":329899},[31690],[34,329901,162086],{"className":329902},[31685],[34,329904],{"className":329905,"style":31846},[31845],[34,329907,202],{"className":329908},[31850],[34,329910],{"className":329911,"style":31846},[31845],[34,329913,329915,329918],{"className":329914},[31676],[34,329916],{"className":329917,"style":32617},[31680],[34,329919,640],{"className":329920},[31685],[105,329922,329923,330047],{},[108,329924,329925,329926,329980,329981,347],{},"这意味着 ",[34,329927,329929,329953],{"className":329928},[31642],[34,329930,329932],{"className":329931},[31646],[31648,329933,329934],{"xmlns":31650},[31652,329935,329936,329950],{},[31655,329937,329938,329940,329942,329944,329946,329948],{},[31658,329939,97709],{},[31658,329941,38080],{"mathvariant":38079},[31658,329943,97300],{},[31662,329945,456],{"stretchy":702},[31658,329947,97709],{},[31662,329949,2557],{"stretchy":702},[31665,329951,329952],{"encoding":31667},"G\u002FZ(G)",[34,329954,329956],{"className":329955,"ariaHidden":758},[31672],[34,329957,329959,329962,329965,329968,329971,329974,329977],{"className":329958},[31676],[34,329960],{"className":329961,"style":31774},[31680],[34,329963,97709],{"className":329964},[31685,31686],[34,329966,38080],{"className":329967},[31685],[34,329969,97300],{"className":329970,"style":38706},[31685,31686],[34,329972,456],{"className":329973},[31782],[34,329975,97709],{"className":329976},[31685,31686],[34,329978,2557],{"className":329979},[31690]," 只有一个元素。说明 ",[34,329982,329984,330008],{"className":329983},[31642],[34,329985,329987],{"className":329986},[31646],[31648,329988,329989],{"xmlns":31650},[31652,329990,329991,330005],{},[31655,329992,329993,329995,329997,329999,330001,330003],{},[31658,329994,97709],{},[31662,329996,202],{},[31658,329998,97300],{},[31662,330000,456],{"stretchy":702},[31658,330002,97709],{},[31662,330004,2557],{"stretchy":702},[31665,330006,330007],{"encoding":31667},"G = Z(G)",[34,330009,330011,330029],{"className":330010,"ariaHidden":758},[31672],[34,330012,330014,330017,330020,330023,330026],{"className":330013},[31676],[34,330015],{"className":330016,"style":33218},[31680],[34,330018,97709],{"className":330019},[31685,31686],[34,330021],{"className":330022,"style":31846},[31845],[34,330024,202],{"className":330025},[31850],[34,330027],{"className":330028,"style":31846},[31845],[34,330030,330032,330035,330038,330041,330044],{"className":330031},[31676],[34,330033],{"className":330034,"style":31774},[31680],[34,330036,97300],{"className":330037,"style":38706},[31685,31686],[34,330039,456],{"className":330040},[31782],[34,330042,97709],{"className":330043},[31685,31686],[34,330045,2557],{"className":330046},[31690],[108,330048,330049,330050,330078,330079,347],{},"中心等于全集，说明 ",[34,330051,330053,330066],{"className":330052},[31642],[34,330054,330056],{"className":330055},[31646],[31648,330057,330058],{"xmlns":31650},[31652,330059,330060,330064],{},[31655,330061,330062],{},[31658,330063,97709],{},[31665,330065,97709],{"encoding":31667},[34,330067,330069],{"className":330068,"ariaHidden":758},[31672],[34,330070,330072,330075],{"className":330071},[31676],[34,330073],{"className":330074,"style":33218},[31680],[34,330076,97709],{"className":330077},[31685,31686]," 中所有元素都可交换。",[2627,330080,330081,330109],{},[34,330082,330084,330097],{"className":330083},[31642],[34,330085,330087],{"className":330086},[31646],[31648,330088,330089],{"xmlns":31650},[31652,330090,330091,330095],{},[31655,330092,330093],{},[31658,330094,97709],{},[31665,330096,97709],{"encoding":31667},[34,330098,330100],{"className":330099,"ariaHidden":758},[31672],[34,330101,330103,330106],{"className":330102},[31676],[34,330104],{"className":330105,"style":33218},[31680],[34,330107,97709],{"className":330108},[31685,31686]," 是交换群",[108,330111,330112,2664,330115,39984,330198],{},[2627,330113,330114],{},"情况 2",[34,330116,330118,330150],{"className":330117},[31642],[34,330119,330121],{"className":330120},[31646],[31648,330122,330123],{"xmlns":31650},[31652,330124,330125,330147],{},[31655,330126,330127,330129,330131,330133,330135,330137,330139,330141,330143,330145],{},[31658,330128,162086],{"mathvariant":38079},[31658,330130,106063],{},[31658,330132,31660],{},[31658,330134,31660],{},[31662,330136,456],{"stretchy":702},[31658,330138,97709],{},[31662,330140,2557],{"stretchy":702},[31658,330142,162086],{"mathvariant":38079},[31662,330144,202],{},[31749,330146,768],{},[31665,330148,330149],{"encoding":31667},"|Inn(G)| = 2",[34,330151,330153,330189],{"className":330152,"ariaHidden":758},[31672],[34,330154,330156,330159,330162,330165,330168,330171,330174,330177,330180,330183,330186],{"className":330155},[31676],[34,330157],{"className":330158,"style":31774},[31680],[34,330160,162086],{"className":330161},[31685],[34,330163,106063],{"className":330164,"style":67482},[31685,31686],[34,330166,329389],{"className":330167},[31685,31686],[34,330169,456],{"className":330170},[31782],[34,330172,97709],{"className":330173},[31685,31686],[34,330175,2557],{"className":330176},[31690],[34,330178,162086],{"className":330179},[31685],[34,330181],{"className":330182,"style":31846},[31845],[34,330184,202],{"className":330185},[31850],[34,330187],{"className":330188,"style":31846},[31845],[34,330190,330192,330195],{"className":330191},[31676],[34,330193],{"className":330194,"style":32617},[31680],[34,330196,768],{"className":330197},[31685],[105,330199,330200,330288,330294,330380,330454,330484],{},[108,330201,329925,330202,347],{},[34,330203,330205,330237],{"className":330204},[31642],[34,330206,330208],{"className":330207},[31646],[31648,330209,330210],{"xmlns":31650},[31652,330211,330212,330234],{},[31655,330213,330214,330216,330218,330220,330222,330224,330226,330228,330230,330232],{},[31658,330215,162086],{"mathvariant":38079},[31658,330217,97709],{},[31658,330219,38080],{"mathvariant":38079},[31658,330221,97300],{},[31662,330223,456],{"stretchy":702},[31658,330225,97709],{},[31662,330227,2557],{"stretchy":702},[31658,330229,162086],{"mathvariant":38079},[31662,330231,202],{},[31749,330233,768],{},[31665,330235,330236],{"encoding":31667},"|G\u002FZ(G)| = 2",[34,330238,330240,330279],{"className":330239,"ariaHidden":758},[31672],[34,330241,330243,330246,330249,330252,330255,330258,330261,330264,330267,330270,330273,330276],{"className":330242},[31676],[34,330244],{"className":330245,"style":31774},[31680],[34,330247,162086],{"className":330248},[31685],[34,330250,97709],{"className":330251},[31685,31686],[34,330253,38080],{"className":330254},[31685],[34,330256,97300],{"className":330257,"style":38706},[31685,31686],[34,330259,456],{"className":330260},[31782],[34,330262,97709],{"className":330263},[31685,31686],[34,330265,2557],{"className":330266},[31690],[34,330268,162086],{"className":330269},[31685],[34,330271],{"className":330272,"style":31846},[31845],[34,330274,202],{"className":330275},[31850],[34,330277],{"className":330278,"style":31846},[31845],[34,330280,330282,330285],{"className":330281},[31676],[34,330283],{"className":330284,"style":32617},[31680],[34,330286,768],{"className":330287},[31685],[108,330289,330290,330291,347],{},"这里有一个著名的结论：",[2627,330292,330293],{},"“循环群判定法”",[108,330295,330296,330297,330350,330351,330379],{},"如果商群 ",[34,330298,330300,330323],{"className":330299},[31642],[34,330301,330303],{"className":330302},[31646],[31648,330304,330305],{"xmlns":31650},[31652,330306,330307,330321],{},[31655,330308,330309,330311,330313,330315,330317,330319],{},[31658,330310,97709],{},[31658,330312,38080],{"mathvariant":38079},[31658,330314,97300],{},[31662,330316,456],{"stretchy":702},[31658,330318,97709],{},[31662,330320,2557],{"stretchy":702},[31665,330322,329952],{"encoding":31667},[34,330324,330326],{"className":330325,"ariaHidden":758},[31672],[34,330327,330329,330332,330335,330338,330341,330344,330347],{"className":330328},[31676],[34,330330],{"className":330331,"style":31774},[31680],[34,330333,97709],{"className":330334},[31685,31686],[34,330336,38080],{"className":330337},[31685],[34,330339,97300],{"className":330340,"style":38706},[31685,31686],[34,330342,456],{"className":330343},[31782],[34,330345,97709],{"className":330346},[31685,31686],[34,330348,2557],{"className":330349},[31690]," 是循环群，那么 ",[34,330352,330354,330367],{"className":330353},[31642],[34,330355,330357],{"className":330356},[31646],[31648,330358,330359],{"xmlns":31650},[31652,330360,330361,330365],{},[31655,330362,330363],{},[31658,330364,97709],{},[31665,330366,97709],{"encoding":31667},[34,330368,330370],{"className":330369,"ariaHidden":758},[31672],[34,330371,330373,330376],{"className":330372},[31676],[34,330374],{"className":330375,"style":33218},[31680],[34,330377,97709],{"className":330378},[31685,31686]," 一定是交换群。",[108,330381,330382,330383,31901],{},"阶数为 2 的群一定是循环群（同构于 ",[34,330384,330386,330404],{"className":330385},[31642],[34,330387,330389],{"className":330388},[31646],[31648,330390,330391],{"xmlns":31650},[31652,330392,330393,330401],{},[31655,330394,330395],{},[31743,330396,330397,330399],{},[31658,330398,97300],{"mathvariant":318416},[31749,330400,768],{},[31665,330402,330403],{"encoding":31667},"\\mathbb{Z}_2",[34,330405,330407],{"className":330406,"ariaHidden":758},[31672],[34,330408,330410,330414],{"className":330409},[31676],[34,330411],{"className":330412,"style":330413},[31680],"height:0.8389em;vertical-align:-0.15em;",[34,330415,330417,330420],{"className":330416},[31685],[34,330418,97300],{"className":330419},[31685,318441],[34,330421,330423],{"className":330422},[31792],[34,330424,330426,330446],{"className":330425},[31796,31797],[34,330427,330429,330443],{"className":330428},[31801],[34,330430,330432],{"className":330431,"style":31806},[31805],[34,330433,330434,330437],{"style":31809},[34,330435],{"className":330436,"style":31814},[31813],[34,330438,330440],{"className":330439},[31818,31819,31820,31821],[34,330441,768],{"className":330442},[31685,31821],[34,330444,31829],{"className":330445},[31828],[34,330447,330449],{"className":330448},[31801],[34,330450,330452],{"className":330451,"style":31836},[31805],[34,330453],{},[108,330455,175992,330456,329081],{},[34,330457,330459,330472],{"className":330458},[31642],[34,330460,330462],{"className":330461},[31646],[31648,330463,330464],{"xmlns":31650},[31652,330465,330466,330470],{},[31655,330467,330468],{},[31658,330469,97709],{},[31665,330471,97709],{"encoding":31667},[34,330473,330475],{"className":330474,"ariaHidden":758},[31672],[34,330476,330478,330481],{"className":330477},[31676],[34,330479],{"className":330480,"style":33218},[31680],[34,330482,97709],{"className":330483},[31685,31686],[108,330485,330486,330489,330490,330518,330519,330585,330586,330639,330640,329081],{},[71807,330487,330488],{},"进一步补充","：其实如果 ",[34,330491,330493,330506],{"className":330492},[31642],[34,330494,330496],{"className":330495},[31646],[31648,330497,330498],{"xmlns":31650},[31652,330499,330500,330504],{},[31655,330501,330502],{},[31658,330503,97709],{},[31665,330505,97709],{"encoding":31667},[34,330507,330509],{"className":330508,"ariaHidden":758},[31672],[34,330510,330512,330515],{"className":330511},[31676],[34,330513],{"className":330514,"style":33218},[31680],[34,330516,97709],{"className":330517},[31685,31686]," 是交换群，",[34,330520,330522,330546],{"className":330521},[31642],[34,330523,330525],{"className":330524},[31646],[31648,330526,330527],{"xmlns":31650},[31652,330528,330529,330543],{},[31655,330530,330531,330533,330535,330537,330539,330541],{},[31658,330532,97300],{},[31662,330534,456],{"stretchy":702},[31658,330536,97709],{},[31662,330538,2557],{"stretchy":702},[31662,330540,202],{},[31658,330542,97709],{},[31665,330544,330545],{"encoding":31667},"Z(G)=G",[34,330547,330549,330576],{"className":330548,"ariaHidden":758},[31672],[34,330550,330552,330555,330558,330561,330564,330567,330570,330573],{"className":330551},[31676],[34,330553],{"className":330554,"style":31774},[31680],[34,330556,97300],{"className":330557,"style":38706},[31685,31686],[34,330559,456],{"className":330560},[31782],[34,330562,97709],{"className":330563},[31685,31686],[34,330565,2557],{"className":330566},[31690],[34,330568],{"className":330569,"style":31846},[31845],[34,330571,202],{"className":330572},[31850],[34,330574],{"className":330575,"style":31846},[31845],[34,330577,330579,330582],{"className":330578},[31676],[34,330580],{"className":330581,"style":33218},[31680],[34,330583,97709],{"className":330584},[31685,31686],"，那么 ",[34,330587,330589,330612],{"className":330588},[31642],[34,330590,330592],{"className":330591},[31646],[31648,330593,330594],{"xmlns":31650},[31652,330595,330596,330610],{},[31655,330597,330598,330600,330602,330604,330606,330608],{},[31658,330599,97709],{},[31658,330601,38080],{"mathvariant":38079},[31658,330603,97300],{},[31662,330605,456],{"stretchy":702},[31658,330607,97709],{},[31662,330609,2557],{"stretchy":702},[31665,330611,329952],{"encoding":31667},[34,330613,330615],{"className":330614,"ariaHidden":758},[31672],[34,330616,330618,330621,330624,330627,330630,330633,330636],{"className":330617},[31676],[34,330619],{"className":330620,"style":31774},[31680],[34,330622,97709],{"className":330623},[31685,31686],[34,330625,38080],{"className":330626},[31685],[34,330628,97300],{"className":330629,"style":38706},[31685,31686],[34,330631,456],{"className":330632},[31782],[34,330634,97709],{"className":330635},[31685,31686],[34,330637,2557],{"className":330638},[31690]," 只有 1 个元素。所以\"情况2\"实际上是不可能发生的，但无论如何，推导都指向 ",[34,330641,330643,330656],{"className":330642},[31642],[34,330644,330646],{"className":330645},[31646],[31648,330647,330648],{"xmlns":31650},[31652,330649,330650,330654],{},[31655,330651,330652],{},[31658,330653,97709],{},[31665,330655,97709],{"encoding":31667},[34,330657,330659],{"className":330658,"ariaHidden":758},[31672],[34,330660,330662,330665],{"className":330661},[31676],[34,330663],{"className":330664,"style":33218},[31680],[34,330666,97709],{"className":330667},[31685,31686],{"title":29,"searchDepth":43,"depth":43,"links":330669},[330670,330671,330673],{"id":314618,"depth":49,"text":314619},{"id":320994,"depth":49,"text":330672},"第二部分：置换群 SnS_nSn​ 与 AnA_nAn​（计算篇）",{"id":322952,"depth":49,"text":322953},{"order":37},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fgroupproblem",{"title":314608,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002Fgroupproblem","3RDZDVbG7TptQ_QHahOZdABszVwE9KCsjpjqOzR-WEI",{"id":330680,"title":330681,"body":330682,"description":29,"extension":85,"meta":339101,"navigation":52,"path":339102,"seo":339103,"stem":339104,"__hash__":339105},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fprimeandmaximalideal.md","素理想与极大理想",{"type":8,"value":330683,"toc":339086},[330684,330688,330692,330697,330780,330782,330786,330790,331188,331418,331422,331535,331540,331640,332796,332898,333480,333482,333486,333490,333695,333904,333967,333971,334083,334086,334189,334286,334374,334491,334583,334759,334839,335042,335228,335265,335647,335873,335973,336652,336878,336884,337093,337100,337509,337714,337716,337720,337831,337833,337837,338769,338771,338774],[11,330685,330687],{"id":330686},"素理想与极大理想-prime-maximal-ideals","素理想与极大理想 (Prime & Maximal Ideals)",[19,330689,330691],{"id":330690},"_0-核心逻辑图谱","0. 核心逻辑图谱",[15,330693,148881,330694],{},[2627,330695,330696],{},"理想不仅是环的子集，它还是构造新环（商环）的\"模具\"。",[105,330698,330699,330742],{},[108,330700,330701,330702,330705,330706,330737,330738,330741],{},"模具选得好（",[2627,330703,330704],{},"素理想","） ",[34,330707,330709,330724],{"className":330708},[31642],[34,330710,330712],{"className":330711},[31646],[31648,330713,330714],{"xmlns":31650},[31652,330715,330716,330721],{},[31655,330717,330718],{},[31662,330719,330720],{},"⟶",[31665,330722,330723],{"encoding":31667},"\\longrightarrow",[34,330725,330727],{"className":330726,"ariaHidden":758},[31672],[34,330728,330730,330734],{"className":330729},[31676],[34,330731],{"className":330732,"style":330733},[31680],"height:0.522em;vertical-align:-0.011em;",[34,330735,330720],{"className":330736},[31850]," 造出的成品是 ",[2627,330739,330740],{},"整环"," (Integral Domain)。",[108,330743,330744,330745,330705,330748,330737,330776,330779],{},"模具选得最好（",[2627,330746,330747],{},"极大理想",[34,330749,330751,330764],{"className":330750},[31642],[34,330752,330754],{"className":330753},[31646],[31648,330755,330756],{"xmlns":31650},[31652,330757,330758,330762],{},[31655,330759,330760],{},[31662,330761,330720],{},[31665,330763,330723],{"encoding":31667},[34,330765,330767],{"className":330766,"ariaHidden":758},[31672],[34,330768,330770,330773],{"className":330769},[31676],[34,330771],{"className":330772,"style":330733},[31680],[34,330774,330720],{"className":330775},[31850],[2627,330777,330778],{},"域"," (Field)。",[32188,330781],{},[19,330783,330785],{"id":330784},"_1-素理想-prime-ideal","1. 素理想 (Prime Ideal)",[253,330787,330789],{"id":330788},"_11-定义","1.1 定义",[15,330791,133910,330792,330820,330821,330849,330850,330934,330935,330999,331000,331053,331054,331105,331106,331157,331158,49276,331186,347],{},[34,330793,330795,330808],{"className":330794},[31642],[34,330796,330798],{"className":330797},[31646],[31648,330799,330800],{"xmlns":31650},[31652,330801,330802,330806],{},[31655,330803,330804],{},[31658,330805,105852],{},[31665,330807,105852],{"encoding":31667},[34,330809,330811],{"className":330810,"ariaHidden":758},[31672],[34,330812,330814,330817],{"className":330813},[31676],[34,330815],{"className":330816,"style":33218},[31680],[34,330818,105852],{"className":330819,"style":37422},[31685,31686]," 是环 ",[34,330822,330824,330837],{"className":330823},[31642],[34,330825,330827],{"className":330826},[31646],[31648,330828,330829],{"xmlns":31650},[31652,330830,330831,330835],{},[31655,330832,330833],{},[31658,330834,34386],{},[31665,330836,34386],{"encoding":31667},[34,330838,330840],{"className":330839,"ariaHidden":758},[31672],[34,330841,330843,330846],{"className":330842},[31676],[34,330844],{"className":330845,"style":33218},[31680],[34,330847,34386],{"className":330848,"style":34427},[31685,31686]," 的真理想（",[34,330851,330853,330871],{"className":330852},[31642],[34,330854,330856],{"className":330855},[31646],[31648,330857,330858],{"xmlns":31650},[31652,330859,330860,330868],{},[31655,330861,330862,330864,330866],{},[31658,330863,105852],{},[31662,330865,54707],{"mathvariant":38079},[31658,330867,34386],{},[31665,330869,330870],{"encoding":31667},"P \\neq R",[34,330872,330874,330925],{"className":330873,"ariaHidden":758},[31672],[34,330875,330877,330880,330883,330886,330922],{"className":330876},[31676],[34,330878],{"className":330879,"style":43217},[31680],[34,330881,105852],{"className":330882,"style":37422},[31685,31686],[34,330884],{"className":330885,"style":31846},[31845],[34,330887,330889,330916,330919],{"className":330888},[31850],[34,330890,330892],{"className":330891},[31850],[34,330893,330895],{"className":330894},[31685,54774],[34,330896,330898],{"className":330897},[54778],[34,330899,330901,330904,330913],{"className":330900},[54782],[34,330902],{"className":330903,"style":43217},[31680],[34,330905,330907],{"className":330906},[54789],[34,330908,330910],{"className":330909},[31685],[34,330911,54796],{"className":330912},[31850],[34,330914],{"className":330915},[54800],[34,330917],{"className":330918},[31845,54804],[34,330920,202],{"className":330921},[31850],[34,330923],{"className":330924,"style":31846},[31845],[34,330926,330928,330931],{"className":330927},[31676],[34,330929],{"className":330930,"style":33218},[31680],[34,330932,34386],{"className":330933,"style":34427},[31685,31686],"）。\n如果对于任意 ",[34,330936,330938,330960],{"className":330937},[31642],[34,330939,330941],{"className":330940},[31646],[31648,330942,330943],{"xmlns":31650},[31652,330944,330945,330957],{},[31655,330946,330947,330949,330951,330953,330955],{},[31658,330948,111],{},[31662,330950,31960],{"separator":758},[31658,330952,1697],{},[31662,330954,112296],{},[31658,330956,34386],{},[31665,330958,330959],{"encoding":31667},"a, b \\in R",[34,330961,330963,330990],{"className":330962,"ariaHidden":758},[31672],[34,330964,330966,330969,330972,330975,330978,330981,330984,330987],{"className":330965},[31676],[34,330967],{"className":330968,"style":43217},[31680],[34,330970,111],{"className":330971},[31685,31686],[34,330973,31960],{"className":330974},[32119],[34,330976],{"className":330977,"style":32123},[31845],[34,330979,1697],{"className":330980},[31685,31686],[34,330982],{"className":330983,"style":31846},[31845],[34,330985,112296],{"className":330986},[31850],[34,330988],{"className":330989,"style":31846},[31845],[34,330991,330993,330996],{"className":330992},[31676],[34,330994],{"className":330995,"style":33218},[31680],[34,330997,34386],{"className":330998,"style":34427},[31685,31686],"，只要乘积 ",[34,331001,331003,331023],{"className":331002},[31642],[34,331004,331006],{"className":331005},[31646],[31648,331007,331008],{"xmlns":31650},[31652,331009,331010,331020],{},[31655,331011,331012,331014,331016,331018],{},[31658,331013,111],{},[31658,331015,1697],{},[31662,331017,112296],{},[31658,331019,105852],{},[31665,331021,331022],{"encoding":31667},"ab \\in P",[34,331024,331026,331044],{"className":331025,"ariaHidden":758},[31672],[34,331027,331029,331032,331035,331038,331041],{"className":331028},[31676],[34,331030],{"className":331031,"style":46991},[31680],[34,331033,324191],{"className":331034},[31685,31686],[34,331036],{"className":331037,"style":31846},[31845],[34,331039,112296],{"className":331040},[31850],[34,331042],{"className":331043,"style":31846},[31845],[34,331045,331047,331050],{"className":331046},[31676],[34,331048],{"className":331049,"style":33218},[31680],[34,331051,105852],{"className":331052,"style":37422},[31685,31686],"，就一定能推出 ",[34,331055,331057,331075],{"className":331056},[31642],[34,331058,331060],{"className":331059},[31646],[31648,331061,331062],{"xmlns":31650},[31652,331063,331064,331072],{},[31655,331065,331066,331068,331070],{},[31658,331067,111],{},[31662,331069,112296],{},[31658,331071,105852],{},[31665,331073,331074],{"encoding":31667},"a \\in P",[34,331076,331078,331096],{"className":331077,"ariaHidden":758},[31672],[34,331079,331081,331084,331087,331090,331093],{"className":331080},[31676],[34,331082],{"className":331083,"style":34812},[31680],[34,331085,111],{"className":331086},[31685,31686],[34,331088],{"className":331089,"style":31846},[31845],[34,331091,112296],{"className":331092},[31850],[34,331094],{"className":331095,"style":31846},[31845],[34,331097,331099,331102],{"className":331098},[31676],[34,331100],{"className":331101,"style":33218},[31680],[34,331103,105852],{"className":331104,"style":37422},[31685,31686]," 或者 ",[34,331107,331109,331127],{"className":331108},[31642],[34,331110,331112],{"className":331111},[31646],[31648,331113,331114],{"xmlns":31650},[31652,331115,331116,331124],{},[31655,331117,331118,331120,331122],{},[31658,331119,1697],{},[31662,331121,112296],{},[31658,331123,105852],{},[31665,331125,331126],{"encoding":31667},"b \\in P",[34,331128,331130,331148],{"className":331129,"ariaHidden":758},[31672],[34,331131,331133,331136,331139,331142,331145],{"className":331132},[31676],[34,331134],{"className":331135,"style":46991},[31680],[34,331137,1697],{"className":331138},[31685,31686],[34,331140],{"className":331141,"style":31846},[31845],[34,331143,112296],{"className":331144},[31850],[34,331146],{"className":331147,"style":31846},[31845],[34,331149,331151,331154],{"className":331150},[31676],[34,331152],{"className":331153,"style":33218},[31680],[34,331155,105852],{"className":331156,"style":37422},[31685,31686],"。\n则称 ",[34,331159,331161,331174],{"className":331160},[31642],[34,331162,331164],{"className":331163},[31646],[31648,331165,331166],{"xmlns":31650},[31652,331167,331168,331172],{},[31655,331169,331170],{},[31658,331171,105852],{},[31665,331173,105852],{"encoding":31667},[34,331175,331177],{"className":331176,"ariaHidden":758},[31672],[34,331178,331180,331183],{"className":331179},[31676],[34,331181],{"className":331182,"style":33218},[31680],[34,331184,105852],{"className":331185,"style":37422},[31685,31686],[2627,331187,330704],{},[104833,331189,331190],{},[15,331191,331192,331194,331195,93231,331223,331251,331252,327448,331303,331331,331332,4374,331360,331388,331389,331417],{},[2627,331193,318465],{},"：这就像整数里的素数 ",[34,331196,331198,331211],{"className":331197},[31642],[34,331199,331201],{"className":331200},[31646],[31648,331202,331203],{"xmlns":31650},[31652,331204,331205,331209],{},[31655,331206,331207],{},[31658,331208,15],{},[31665,331210,15],{"encoding":31667},[34,331212,331214],{"className":331213,"ariaHidden":758},[31672],[34,331215,331217,331220],{"className":331216},[31676],[34,331218],{"className":331219,"style":38774},[31680],[34,331221,15],{"className":331222},[31685,31686],[34,331224,331226,331239],{"className":331225},[31642],[34,331227,331229],{"className":331228},[31646],[31648,331230,331231],{"xmlns":31650},[31652,331232,331233,331237],{},[31655,331234,331235],{},[31658,331236,15],{},[31665,331238,15],{"encoding":31667},[34,331240,331242],{"className":331241,"ariaHidden":758},[31672],[34,331243,331245,331248],{"className":331244},[31676],[34,331246],{"className":331247,"style":38774},[31680],[34,331249,15],{"className":331250},[31685,31686]," 整除 ",[34,331253,331255,331273],{"className":331254},[31642],[34,331256,331258],{"className":331257},[31646],[31648,331259,331260],{"xmlns":31650},[31652,331261,331262,331270],{},[31655,331263,331264,331266,331268],{},[31658,331265,111],{},[31662,331267,42071],{},[31658,331269,1697],{},[31665,331271,331272],{"encoding":31667},"a \\times b",[34,331274,331276,331294],{"className":331275,"ariaHidden":758},[31672],[34,331277,331279,331282,331285,331288,331291],{"className":331278},[31676],[34,331280],{"className":331281,"style":32597},[31680],[34,331283,111],{"className":331284},[31685,31686],[34,331286],{"className":331287,"style":32604},[31845],[34,331289,42071],{"className":331290},[32024],[34,331292],{"className":331293,"style":32604},[31845],[34,331295,331297,331300],{"className":331296},[31676],[34,331298],{"className":331299,"style":31681},[31680],[34,331301,1697],{"className":331302},[31685,31686],[34,331304,331306,331319],{"className":331305},[31642],[34,331307,331309],{"className":331308},[31646],[31648,331310,331311],{"xmlns":31650},[31652,331312,331313,331317],{},[31655,331314,331315],{},[31658,331316,15],{},[31665,331318,15],{"encoding":31667},[34,331320,331322],{"className":331321,"ariaHidden":758},[31672],[34,331323,331325,331328],{"className":331324},[31676],[34,331326],{"className":331327,"style":38774},[31680],[34,331329,15],{"className":331330},[31685,31686]," 必须整除 ",[34,331333,331335,331348],{"className":331334},[31642],[34,331336,331338],{"className":331337},[31646],[31648,331339,331340],{"xmlns":31650},[31652,331341,331342,331346],{},[31655,331343,331344],{},[31658,331345,111],{},[31665,331347,111],{"encoding":31667},[34,331349,331351],{"className":331350,"ariaHidden":758},[31672],[34,331352,331354,331357],{"className":331353},[31676],[34,331355],{"className":331356,"style":32527},[31680],[34,331358,111],{"className":331359},[31685,31686],[34,331361,331363,331376],{"className":331362},[31642],[34,331364,331366],{"className":331365},[31646],[31648,331367,331368],{"xmlns":31650},[31652,331369,331370,331374],{},[31655,331371,331372],{},[31658,331373,1697],{},[31665,331375,1697],{"encoding":31667},[34,331377,331379],{"className":331378,"ariaHidden":758},[31672],[34,331380,331382,331385],{"className":331381},[31676],[34,331383],{"className":331384,"style":31681},[31680],[34,331386,1697],{"className":331387},[31685,31686],"。素理想 ",[34,331390,331392,331405],{"className":331391},[31642],[34,331393,331395],{"className":331394},[31646],[31648,331396,331397],{"xmlns":31650},[31652,331398,331399,331403],{},[31655,331400,331401],{},[31658,331402,105852],{},[31665,331404,105852],{"encoding":31667},[34,331406,331408],{"className":331407,"ariaHidden":758},[31672],[34,331409,331411,331414],{"className":331410},[31676],[34,331412],{"className":331413,"style":33218},[31680],[34,331415,105852],{"className":331416,"style":37422},[31685,31686]," 就像一个\"过滤器\"，乘积掉进去了，因子肯定至少有一个原本就在里面。",[253,331419,331421],{"id":331420},"_12-核心定理-a-必考","1.2 核心定理 A (必考)",[15,331423,331424,2664,331427,331455,331456,331494,331495,331534],{},[2627,331425,331426],{},"定理 3.5.1",[34,331428,331430,331443],{"className":331429},[31642],[34,331431,331433],{"className":331432},[31646],[31648,331434,331435],{"xmlns":31650},[31652,331436,331437,331441],{},[31655,331438,331439],{},[31658,331440,105852],{},[31665,331442,105852],{"encoding":31667},[34,331444,331446],{"className":331445,"ariaHidden":758},[31672],[34,331447,331449,331452],{"className":331448},[31676],[34,331450],{"className":331451,"style":33218},[31680],[34,331453,105852],{"className":331454,"style":37422},[31685,31686]," 是素理想 ",[34,331457,331459,331476],{"className":331458},[31642],[34,331460,331462],{"className":331461},[31646],[31648,331463,331464],{"xmlns":31650},[31652,331465,331466,331474],{},[31655,331467,331468,331470,331472],{},[39361,331469,262622],{},[31662,331471,323391],{},[39361,331473,262622],{},[31665,331475,323396],{"encoding":31667},[34,331477,331479],{"className":331478,"ariaHidden":758},[31672],[34,331480,331482,331485,331488,331491],{"className":331481},[31676],[34,331483],{"className":331484,"style":318264},[31680],[34,331486],{"className":331487,"style":31846},[31845],[34,331489,323391],{"className":331490},[31850],[34,331492],{"className":331493,"style":31846},[31845]," 商环 ",[34,331496,331498,331516],{"className":331497},[31642],[34,331499,331501],{"className":331500},[31646],[31648,331502,331503],{"xmlns":31650},[31652,331504,331505,331513],{},[31655,331506,331507,331509,331511],{},[31658,331508,34386],{},[31658,331510,38080],{"mathvariant":38079},[31658,331512,105852],{},[31665,331514,331515],{"encoding":31667},"R\u002FP",[34,331517,331519],{"className":331518,"ariaHidden":758},[31672],[34,331520,331522,331525,331528,331531],{"className":331521},[31676],[34,331523],{"className":331524,"style":31774},[31680],[34,331526,34386],{"className":331527,"style":34427},[31685,31686],[34,331529,38080],{"className":331530},[31685],[34,331532,105852],{"className":331533,"style":37422},[31685,31686]," 是整环。",[32490,331536,331538],{"id":331537},"证明详解",[2627,331539,331537],{},[15,331541,331542],{},[2627,331543,456,331544,331572,331573,331601,331602,331534],{},[34,331545,331547,331560],{"className":331546},[31642],[34,331548,331550],{"className":331549},[31646],[31648,331551,331552],{"xmlns":31650},[31652,331553,331554,331558],{},[31655,331555,331556],{},[31662,331557,140094],{},[31665,331559,266872],{"encoding":31667},[34,331561,331563],{"className":331562,"ariaHidden":758},[31672],[34,331564,331566,331569],{"className":331565},[31676],[34,331567],{"className":331568,"style":73089},[31680],[34,331570,140094],{"className":331571},[31850],") 方向：已知 ",[34,331574,331576,331589],{"className":331575},[31642],[34,331577,331579],{"className":331578},[31646],[31648,331580,331581],{"xmlns":31650},[31652,331582,331583,331587],{},[31655,331584,331585],{},[31658,331586,105852],{},[31665,331588,105852],{"encoding":31667},[34,331590,331592],{"className":331591,"ariaHidden":758},[31672],[34,331593,331595,331598],{"className":331594},[31676],[34,331596],{"className":331597,"style":33218},[31680],[34,331599,105852],{"className":331600,"style":37422},[31685,31686]," 是素理想，求证 ",[34,331603,331605,331622],{"className":331604},[31642],[34,331606,331608],{"className":331607},[31646],[31648,331609,331610],{"xmlns":31650},[31652,331611,331612,331620],{},[31655,331613,331614,331616,331618],{},[31658,331615,34386],{},[31658,331617,38080],{"mathvariant":38079},[31658,331619,105852],{},[31665,331621,331515],{"encoding":31667},[34,331623,331625],{"className":331624,"ariaHidden":758},[31672],[34,331626,331628,331631,331634,331637],{"className":331627},[31676],[34,331629],{"className":331630,"style":31774},[31680],[34,331632,34386],{"className":331633,"style":34427},[31685,31686],[34,331635,38080],{"className":331636},[31685],[34,331638,105852],{"className":331639,"style":37422},[31685,31686],[2230,331641,331642,331824,332144,332536,332673,332751],{},[108,331643,331644,331647,331648,331721,331722,4374,331773,347],{},[2627,331645,331646],{},"回顾整环定义","：整环是交换幺环，且没有\"零因子\"。也就是说，如果 ",[34,331649,331651,331673],{"className":331650},[31642],[34,331652,331654],{"className":331653},[31646],[31648,331655,331656],{"xmlns":31650},[31652,331657,331658,331670],{},[31655,331659,331660,331662,331664,331666,331668],{},[31658,331661,67467],{},[31662,331663,71191],{},[31658,331665,67731],{},[31662,331667,202],{},[31749,331669,459],{},[31665,331671,331672],{"encoding":31667},"X \\cdot Y = 0",[34,331674,331676,331694,331712],{"className":331675,"ariaHidden":758},[31672],[34,331677,331679,331682,331685,331688,331691],{"className":331678},[31676],[34,331680],{"className":331681,"style":33218},[31680],[34,331683,67467],{"className":331684,"style":67482},[31685,31686],[34,331686],{"className":331687,"style":32604},[31845],[34,331689,71191],{"className":331690},[32024],[34,331692],{"className":331693,"style":32604},[31845],[34,331695,331697,331700,331703,331706,331709],{"className":331696},[31676],[34,331698],{"className":331699,"style":33218},[31680],[34,331701,67731],{"className":331702,"style":32604},[31685,31686],[34,331704],{"className":331705,"style":31846},[31845],[34,331707,202],{"className":331708},[31850],[34,331710],{"className":331711,"style":31846},[31845],[34,331713,331715,331718],{"className":331714},[31676],[34,331716],{"className":331717,"style":32617},[31680],[34,331719,459],{"className":331720},[31685],"，则必须 ",[34,331723,331725,331743],{"className":331724},[31642],[34,331726,331728],{"className":331727},[31646],[31648,331729,331730],{"xmlns":31650},[31652,331731,331732,331740],{},[31655,331733,331734,331736,331738],{},[31658,331735,67467],{},[31662,331737,202],{},[31749,331739,459],{},[31665,331741,331742],{"encoding":31667},"X=0",[34,331744,331746,331764],{"className":331745,"ariaHidden":758},[31672],[34,331747,331749,331752,331755,331758,331761],{"className":331748},[31676],[34,331750],{"className":331751,"style":33218},[31680],[34,331753,67467],{"className":331754,"style":67482},[31685,31686],[34,331756],{"className":331757,"style":31846},[31845],[34,331759,202],{"className":331760},[31850],[34,331762],{"className":331763,"style":31846},[31845],[34,331765,331767,331770],{"className":331766},[31676],[34,331768],{"className":331769,"style":32617},[31680],[34,331771,459],{"className":331772},[31685],[34,331774,331776,331794],{"className":331775},[31642],[34,331777,331779],{"className":331778},[31646],[31648,331780,331781],{"xmlns":31650},[31652,331782,331783,331791],{},[31655,331784,331785,331787,331789],{},[31658,331786,67731],{},[31662,331788,202],{},[31749,331790,459],{},[31665,331792,331793],{"encoding":31667},"Y=0",[34,331795,331797,331815],{"className":331796,"ariaHidden":758},[31672],[34,331798,331800,331803,331806,331809,331812],{"className":331799},[31676],[34,331801],{"className":331802,"style":33218},[31680],[34,331804,67731],{"className":331805,"style":32604},[31685,31686],[34,331807],{"className":331808,"style":31846},[31845],[34,331810,202],{"className":331811},[31850],[34,331813],{"className":331814,"style":31846},[31845],[34,331816,331818,331821],{"className":331817},[31676],[34,331819],{"className":331820,"style":32617},[31680],[34,331822,459],{"className":331823},[31685],[108,331825,331826,331829,331830,331868,331869,258,331920,39984,331971],{},[2627,331827,331828],{},"设定场景","：在商环 ",[34,331831,331833,331850],{"className":331832},[31642],[34,331834,331836],{"className":331835},[31646],[31648,331837,331838],{"xmlns":31650},[31652,331839,331840,331848],{},[31655,331841,331842,331844,331846],{},[31658,331843,34386],{},[31658,331845,38080],{"mathvariant":38079},[31658,331847,105852],{},[31665,331849,331515],{"encoding":31667},[34,331851,331853],{"className":331852,"ariaHidden":758},[31672],[34,331854,331856,331859,331862,331865],{"className":331855},[31676],[34,331857],{"className":331858,"style":31774},[31680],[34,331860,34386],{"className":331861,"style":34427},[31685,31686],[34,331863,38080],{"className":331864},[31685],[34,331866,105852],{"className":331867,"style":37422},[31685,31686]," 中，取两个非零元素 ",[34,331870,331872,331890],{"className":331871},[31642],[34,331873,331875],{"className":331874},[31646],[31648,331876,331877],{"xmlns":31650},[31652,331878,331879,331887],{},[31655,331880,331881,331883,331885],{},[31658,331882,111],{},[31662,331884,223],{},[31658,331886,105852],{},[31665,331888,331889],{"encoding":31667},"a+P",[34,331891,331893,331911],{"className":331892,"ariaHidden":758},[31672],[34,331894,331896,331899,331902,331905,331908],{"className":331895},[31676],[34,331897],{"className":331898,"style":32597},[31680],[34,331900,111],{"className":331901},[31685,31686],[34,331903],{"className":331904,"style":32604},[31845],[34,331906,223],{"className":331907},[32024],[34,331909],{"className":331910,"style":32604},[31845],[34,331912,331914,331917],{"className":331913},[31676],[34,331915],{"className":331916,"style":33218},[31680],[34,331918,105852],{"className":331919,"style":37422},[31685,31686],[34,331921,331923,331941],{"className":331922},[31642],[34,331924,331926],{"className":331925},[31646],[31648,331927,331928],{"xmlns":31650},[31652,331929,331930,331938],{},[31655,331931,331932,331934,331936],{},[31658,331933,1697],{},[31662,331935,223],{},[31658,331937,105852],{},[31665,331939,331940],{"encoding":31667},"b+P",[34,331942,331944,331962],{"className":331943,"ariaHidden":758},[31672],[34,331945,331947,331950,331953,331956,331959],{"className":331946},[31676],[34,331948],{"className":331949,"style":34299},[31680],[34,331951,1697],{"className":331952},[31685,31686],[34,331954],{"className":331955,"style":32604},[31845],[34,331957,223],{"className":331958},[32024],[34,331960],{"className":331961,"style":32604},[31845],[34,331963,331965,331968],{"className":331964},[31676],[34,331966],{"className":331967,"style":33218},[31680],[34,331969,105852],{"className":331970,"style":37422},[31685,31686],[105,331972,331973],{},[108,331974,331975,331976,86674,332060,347],{},"这里的\"非零\"意味着 ",[34,331977,331979,331997],{"className":331978},[31642],[34,331980,331982],{"className":331981},[31646],[31648,331983,331984],{"xmlns":31650},[31652,331985,331986,331994],{},[31655,331987,331988,331990,331992],{},[31658,331989,111],{},[31662,331991,112266],{"mathvariant":38079},[31658,331993,105852],{},[31665,331995,331996],{"encoding":31667},"a \\notin P",[34,331998,332000,332051],{"className":331999,"ariaHidden":758},[31672],[34,332001,332003,332006,332009,332012,332048],{"className":332002},[31676],[34,332004],{"className":332005,"style":31774},[31680],[34,332007,111],{"className":332008},[31685,31686],[34,332010],{"className":332011,"style":31846},[31845],[34,332013,332015,332021],{"className":332014},[31850],[34,332016,332018],{"className":332017},[31685],[34,332019,112296],{"className":332020},[31850],[34,332022,332024],{"className":332023},[31685,54774],[34,332025,332027],{"className":332026},[54778],[34,332028,332030,332033,332045],{"className":332029},[112306],[34,332031],{"className":332032,"style":31774},[31680],[34,332034,332036],{"className":332035},[54789],[34,332037,332039,332042],{"className":332038},[31685],[34,332040,38080],{"className":332041},[31685],[34,332043],{"className":332044,"style":112322},[31845],[34,332046],{"className":332047},[54800],[34,332049],{"className":332050,"style":31846},[31845],[34,332052,332054,332057],{"className":332053},[31676],[34,332055],{"className":332056,"style":33218},[31680],[34,332058,105852],{"className":332059,"style":37422},[31685,31686],[34,332061,332063,332081],{"className":332062},[31642],[34,332064,332066],{"className":332065},[31646],[31648,332067,332068],{"xmlns":31650},[31652,332069,332070,332078],{},[31655,332071,332072,332074,332076],{},[31658,332073,1697],{},[31662,332075,112266],{"mathvariant":38079},[31658,332077,105852],{},[31665,332079,332080],{"encoding":31667},"b \\notin P",[34,332082,332084,332135],{"className":332083,"ariaHidden":758},[31672],[34,332085,332087,332090,332093,332096,332132],{"className":332086},[31676],[34,332088],{"className":332089,"style":31774},[31680],[34,332091,1697],{"className":332092},[31685,31686],[34,332094],{"className":332095,"style":31846},[31845],[34,332097,332099,332105],{"className":332098},[31850],[34,332100,332102],{"className":332101},[31685],[34,332103,112296],{"className":332104},[31850],[34,332106,332108],{"className":332107},[31685,54774],[34,332109,332111],{"className":332110},[54778],[34,332112,332114,332117,332129],{"className":332113},[112306],[34,332115],{"className":332116,"style":31774},[31680],[34,332118,332120],{"className":332119},[54789],[34,332121,332123,332126],{"className":332122},[31685],[34,332124,38080],{"className":332125},[31685],[34,332127],{"className":332128,"style":112322},[31845],[34,332130],{"className":332131},[54800],[34,332133],{"className":332134,"style":31846},[31845],[34,332136,332138,332141],{"className":332137},[31676],[34,332139],{"className":332140,"style":33218},[31680],[34,332142,105852],{"className":332143,"style":37422},[31685,31686],[108,332145,332146,325706,332149,39984,332329],{},[2627,332147,332148],{},"假设乘积为零",[34,332150,332152,332198],{"className":332151},[31642],[34,332153,332155],{"className":332154},[31646],[31648,332156,332157],{"xmlns":31650},[31652,332158,332159,332195],{},[31655,332160,332161,332163,332165,332167,332169,332171,332173,332175,332177,332179,332181,332183],{},[31662,332162,456],{"stretchy":702},[31658,332164,111],{},[31662,332166,223],{},[31658,332168,105852],{},[31662,332170,2557],{"stretchy":702},[31662,332172,456],{"stretchy":702},[31658,332174,1697],{},[31662,332176,223],{},[31658,332178,105852],{},[31662,332180,2557],{"stretchy":702},[31662,332182,202],{},[31743,332184,332185,332187],{},[31749,332186,459],{},[31655,332188,332189,332191,332193],{},[31658,332190,34386],{},[31658,332192,38080],{"mathvariant":38079},[31658,332194,105852],{},[31665,332196,332197],{"encoding":31667},"(a+P)(b+P) = 0_{R\u002FP}",[34,332199,332201,332222,332249,332270],{"className":332200,"ariaHidden":758},[31672],[34,332202,332204,332207,332210,332213,332216,332219],{"className":332203},[31676],[34,332205],{"className":332206,"style":31774},[31680],[34,332208,456],{"className":332209},[31782],[34,332211,111],{"className":332212},[31685,31686],[34,332214],{"className":332215,"style":32604},[31845],[34,332217,223],{"className":332218},[32024],[34,332220],{"className":332221,"style":32604},[31845],[34,332223,332225,332228,332231,332234,332237,332240,332243,332246],{"className":332224},[31676],[34,332226],{"className":332227,"style":31774},[31680],[34,332229,105852],{"className":332230,"style":37422},[31685,31686],[34,332232,2557],{"className":332233},[31690],[34,332235,456],{"className":332236},[31782],[34,332238,1697],{"className":332239},[31685,31686],[34,332241],{"className":332242,"style":32604},[31845],[34,332244,223],{"className":332245},[32024],[34,332247],{"className":332248,"style":32604},[31845],[34,332250,332252,332255,332258,332261,332264,332267],{"className":332251},[31676],[34,332253],{"className":332254,"style":31774},[31680],[34,332256,105852],{"className":332257,"style":37422},[31685,31686],[34,332259,2557],{"className":332260},[31690],[34,332262],{"className":332263,"style":31846},[31845],[34,332265,202],{"className":332266},[31850],[34,332268],{"className":332269,"style":31846},[31845],[34,332271,332273,332277],{"className":332272},[31676],[34,332274],{"className":332275,"style":332276},[31680],"height:0.9996em;vertical-align:-0.3552em;",[34,332278,332280,332283],{"className":332279},[31685],[34,332281,459],{"className":332282},[31685],[34,332284,332286],{"className":332285},[31792],[34,332287,332289,332320],{"className":332288},[31796,31797],[34,332290,332292,332317],{"className":332291},[31801],[34,332293,332296],{"className":332294,"style":332295},[31805],"height:0.3448em;",[34,332297,332299,332302],{"style":332298},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[34,332300],{"className":332301,"style":31814},[31813],[34,332303,332305],{"className":332304},[31818,31819,31820,31821],[34,332306,332308,332311,332314],{"className":332307},[31685,31821],[34,332309,34386],{"className":332310,"style":34427},[31685,31686,31821],[34,332312,38080],{"className":332313},[31685,31821],[34,332315,105852],{"className":332316,"style":37422},[31685,31686,31821],[34,332318,31829],{"className":332319},[31828],[34,332321,332323],{"className":332322},[31801],[34,332324,332327],{"className":332325,"style":332326},[31805],"height:0.3552em;",[34,332328],{},[105,332330,332331,332407],{},[108,332332,332333,332334,347],{},"商环的零元是 ",[34,332335,332337,332359],{"className":332336},[31642],[34,332338,332340],{"className":332339},[31646],[31648,332341,332342],{"xmlns":31650},[31652,332343,332344,332356],{},[31655,332345,332346,332348,332350,332352,332354],{},[31749,332347,459],{},[31662,332349,223],{},[31658,332351,105852],{},[31662,332353,202],{},[31658,332355,105852],{},[31665,332357,332358],{"encoding":31667},"0+P = P",[34,332360,332362,332380,332398],{"className":332361,"ariaHidden":758},[31672],[34,332363,332365,332368,332371,332374,332377],{"className":332364},[31676],[34,332366],{"className":332367,"style":35942},[31680],[34,332369,459],{"className":332370},[31685],[34,332372],{"className":332373,"style":32604},[31845],[34,332375,223],{"className":332376},[32024],[34,332378],{"className":332379,"style":32604},[31845],[34,332381,332383,332386,332389,332392,332395],{"className":332382},[31676],[34,332384],{"className":332385,"style":33218},[31680],[34,332387,105852],{"className":332388,"style":37422},[31685,31686],[34,332390],{"className":332391,"style":31846},[31845],[34,332393,202],{"className":332394},[31850],[34,332396],{"className":332397,"style":31846},[31845],[34,332399,332401,332404],{"className":332400},[31676],[34,332402],{"className":332403,"style":33218},[31680],[34,332405,105852],{"className":332406,"style":37422},[31685,31686],[108,332408,329925,332409,61605,332484,347],{},[34,332410,332412,332436],{"className":332411},[31642],[34,332413,332415],{"className":332414},[31646],[31648,332416,332417],{"xmlns":31650},[31652,332418,332419,332433],{},[31655,332420,332421,332423,332425,332427,332429,332431],{},[31658,332422,111],{},[31658,332424,1697],{},[31662,332426,223],{},[31658,332428,105852],{},[31662,332430,202],{},[31658,332432,105852],{},[31665,332434,332435],{"encoding":31667},"ab + P = P",[34,332437,332439,332457,332475],{"className":332438,"ariaHidden":758},[31672],[34,332440,332442,332445,332448,332451,332454],{"className":332441},[31676],[34,332443],{"className":332444,"style":34299},[31680],[34,332446,324191],{"className":332447},[31685,31686],[34,332449],{"className":332450,"style":32604},[31845],[34,332452,223],{"className":332453},[32024],[34,332455],{"className":332456,"style":32604},[31845],[34,332458,332460,332463,332466,332469,332472],{"className":332459},[31676],[34,332461],{"className":332462,"style":33218},[31680],[34,332464,105852],{"className":332465,"style":37422},[31685,31686],[34,332467],{"className":332468,"style":31846},[31845],[34,332470,202],{"className":332471},[31850],[34,332473],{"className":332474,"style":31846},[31845],[34,332476,332478,332481],{"className":332477},[31676],[34,332479],{"className":332480,"style":33218},[31680],[34,332482,105852],{"className":332483,"style":37422},[31685,31686],[34,332485,332487,332506],{"className":332486},[31642],[34,332488,332490],{"className":332489},[31646],[31648,332491,332492],{"xmlns":31650},[31652,332493,332494,332504],{},[31655,332495,332496,332498,332500,332502],{},[31658,332497,111],{},[31658,332499,1697],{},[31662,332501,112296],{},[31658,332503,105852],{},[31665,332505,331022],{"encoding":31667},[34,332507,332509,332527],{"className":332508,"ariaHidden":758},[31672],[34,332510,332512,332515,332518,332521,332524],{"className":332511},[31676],[34,332513],{"className":332514,"style":46991},[31680],[34,332516,324191],{"className":332517},[31685,31686],[34,332519],{"className":332520,"style":31846},[31845],[34,332522,112296],{"className":332523},[31850],[34,332525],{"className":332526,"style":31846},[31845],[34,332528,332530,332533],{"className":332529},[31676],[34,332531],{"className":332532,"style":33218},[31680],[34,332534,105852],{"className":332535,"style":37422},[31685,31686],[108,332537,332538,332541,332542,113124,332570,332572,332573,331105,332623,347],{},[2627,332539,332540],{},"利用已知条件","：因为 ",[34,332543,332545,332558],{"className":332544},[31642],[34,332546,332548],{"className":332547},[31646],[31648,332549,332550],{"xmlns":31650},[31652,332551,332552,332556],{},[31655,332553,332554],{},[31658,332555,105852],{},[31665,332557,105852],{"encoding":31667},[34,332559,332561],{"className":332560,"ariaHidden":758},[31672],[34,332562,332564,332567],{"className":332563},[31676],[34,332565],{"className":332566,"style":33218},[31680],[34,332568,105852],{"className":332569,"style":37422},[31685,31686],[2627,332571,330704],{},"，由定义可知：",[34,332574,332576,332593],{"className":332575},[31642],[34,332577,332579],{"className":332578},[31646],[31648,332580,332581],{"xmlns":31650},[31652,332582,332583,332591],{},[31655,332584,332585,332587,332589],{},[31658,332586,111],{},[31662,332588,112296],{},[31658,332590,105852],{},[31665,332592,331074],{"encoding":31667},[34,332594,332596,332614],{"className":332595,"ariaHidden":758},[31672],[34,332597,332599,332602,332605,332608,332611],{"className":332598},[31676],[34,332600],{"className":332601,"style":34812},[31680],[34,332603,111],{"className":332604},[31685,31686],[34,332606],{"className":332607,"style":31846},[31845],[34,332609,112296],{"className":332610},[31850],[34,332612],{"className":332613,"style":31846},[31845],[34,332615,332617,332620],{"className":332616},[31676],[34,332618],{"className":332619,"style":33218},[31680],[34,332621,105852],{"className":332622,"style":37422},[31685,31686],[34,332624,332626,332643],{"className":332625},[31642],[34,332627,332629],{"className":332628},[31646],[31648,332630,332631],{"xmlns":31650},[31652,332632,332633,332641],{},[31655,332634,332635,332637,332639],{},[31658,332636,1697],{},[31662,332638,112296],{},[31658,332640,105852],{},[31665,332642,331126],{"encoding":31667},[34,332644,332646,332664],{"className":332645,"ariaHidden":758},[31672],[34,332647,332649,332652,332655,332658,332661],{"className":332648},[31676],[34,332650],{"className":332651,"style":46991},[31680],[34,332653,1697],{"className":332654},[31685,31686],[34,332656],{"className":332657,"style":31846},[31845],[34,332659,112296],{"className":332660},[31850],[34,332662],{"className":332663,"style":31846},[31845],[34,332665,332667,332670],{"className":332666},[31676],[34,332668],{"className":332669,"style":33218},[31680],[34,332671,105852],{"className":332672,"style":37422},[31685,31686],[108,332674,332675,332678,332679,332721,332722,332750],{},[2627,332676,332677],{},"矛盾\u002F结论","：但这与步骤2中\"",[34,332680,332682,332700],{"className":332681},[31642],[34,332683,332685],{"className":332684},[31646],[31648,332686,332687],{"xmlns":31650},[31652,332688,332689,332697],{},[31655,332690,332691,332693,332695],{},[31658,332692,111],{},[31662,332694,31960],{"separator":758},[31658,332696,1697],{},[31665,332698,332699],{"encoding":31667},"a, b",[34,332701,332703],{"className":332702,"ariaHidden":758},[31672],[34,332704,332706,332709,332712,332715,332718],{"className":332705},[31676],[34,332707],{"className":332708,"style":43217},[31680],[34,332710,111],{"className":332711},[31685,31686],[34,332713,31960],{"className":332714},[32119],[34,332716],{"className":332717,"style":32123},[31845],[34,332719,1697],{"className":332720},[31685,31686]," 都不在 ",[34,332723,332725,332738],{"className":332724},[31642],[34,332726,332728],{"className":332727},[31646],[31648,332729,332730],{"xmlns":31650},[31652,332731,332732,332736],{},[31655,332733,332734],{},[31658,332735,105852],{},[31665,332737,105852],{"encoding":31667},[34,332739,332741],{"className":332740,"ariaHidden":758},[31672],[34,332742,332744,332747],{"className":332743},[31676],[34,332745],{"className":332746,"style":33218},[31680],[34,332748,105852],{"className":332749,"style":37422},[31685,31686]," 中\"矛盾。如果乘积是0，因子必有一个是0。",[108,332752,332753,332756,332757,332795],{},[2627,332754,332755],{},"证毕","：所以 ",[34,332758,332760,332777],{"className":332759},[31642],[34,332761,332763],{"className":332762},[31646],[31648,332764,332765],{"xmlns":31650},[31652,332766,332767,332775],{},[31655,332768,332769,332771,332773],{},[31658,332770,34386],{},[31658,332772,38080],{"mathvariant":38079},[31658,332774,105852],{},[31665,332776,331515],{"encoding":31667},[34,332778,332780],{"className":332779,"ariaHidden":758},[31672],[34,332781,332783,332786,332789,332792],{"className":332782},[31676],[34,332784],{"className":332785,"style":31774},[31680],[34,332787,34386],{"className":332788,"style":34427},[31685,31686],[34,332790,38080],{"className":332791},[31685],[34,332793,105852],{"className":332794,"style":37422},[31685,31686]," 没有零因子，是整环。",[15,332797,332798],{},[2627,332799,456,332800,331572,332830,332868,332869,332897],{},[34,332801,332803,332818],{"className":332802},[31642],[34,332804,332806],{"className":332805},[31646],[31648,332807,332808],{"xmlns":31650},[31652,332809,332810,332815],{},[31655,332811,332812],{},[31662,332813,332814],{},"⇐",[31665,332816,332817],{"encoding":31667},"\\Leftarrow",[34,332819,332821],{"className":332820,"ariaHidden":758},[31672],[34,332822,332824,332827],{"className":332823},[31676],[34,332825],{"className":332826,"style":73089},[31680],[34,332828,332814],{"className":332829},[31850],[34,332831,332833,332850],{"className":332832},[31642],[34,332834,332836],{"className":332835},[31646],[31648,332837,332838],{"xmlns":31650},[31652,332839,332840,332848],{},[31655,332841,332842,332844,332846],{},[31658,332843,34386],{},[31658,332845,38080],{"mathvariant":38079},[31658,332847,105852],{},[31665,332849,331515],{"encoding":31667},[34,332851,332853],{"className":332852,"ariaHidden":758},[31672],[34,332854,332856,332859,332862,332865],{"className":332855},[31676],[34,332857],{"className":332858,"style":31774},[31680],[34,332860,34386],{"className":332861,"style":34427},[31685,31686],[34,332863,38080],{"className":332864},[31685],[34,332866,105852],{"className":332867,"style":37422},[31685,31686]," 是整环，求证 ",[34,332870,332872,332885],{"className":332871},[31642],[34,332873,332875],{"className":332874},[31646],[31648,332876,332877],{"xmlns":31650},[31652,332878,332879,332883],{},[31655,332880,332881],{},[31658,332882,105852],{},[31665,332884,105852],{"encoding":31667},[34,332886,332888],{"className":332887,"ariaHidden":758},[31672],[34,332889,332891,332894],{"className":332890},[31676],[34,332892],{"className":332893,"style":33218},[31680],[34,332895,105852],{"className":332896,"style":37422},[31685,31686]," 是素理想。",[2230,332899,332900,332956,333179,333370,333475],{},[108,332901,332902,325706,332904,347],{},[2627,332903,331828],{},[34,332905,332907,332926],{"className":332906},[31642],[34,332908,332910],{"className":332909},[31646],[31648,332911,332912],{"xmlns":31650},[31652,332913,332914,332924],{},[31655,332915,332916,332918,332920,332922],{},[31658,332917,111],{},[31658,332919,1697],{},[31662,332921,112296],{},[31658,332923,105852],{},[31665,332925,331022],{"encoding":31667},[34,332927,332929,332947],{"className":332928,"ariaHidden":758},[31672],[34,332930,332932,332935,332938,332941,332944],{"className":332931},[31676],[34,332933],{"className":332934,"style":46991},[31680],[34,332936,324191],{"className":332937},[31685,31686],[34,332939],{"className":332940,"style":31846},[31845],[34,332942,112296],{"className":332943},[31850],[34,332945],{"className":332946,"style":31846},[31845],[34,332948,332950,332953],{"className":332949},[31676],[34,332951],{"className":332952,"style":33218},[31680],[34,332954,105852],{"className":332955,"style":37422},[31685,31686],[108,332957,332958,332961,332962,61605,333059,347],{},[2627,332959,332960],{},"翻译到商环","：这意味着在商环里，",[34,332963,332965,332993],{"className":332964},[31642],[34,332966,332968],{"className":332967},[31646],[31648,332969,332970],{"xmlns":31650},[31652,332971,332972,332990],{},[31655,332973,332974,332976,332978,332980,332982,332984,332986,332988],{},[31658,332975,111],{},[31658,332977,1697],{},[31662,332979,223],{},[31658,332981,105852],{},[31662,332983,202],{},[31749,332985,459],{},[31662,332987,223],{},[31658,332989,105852],{},[31665,332991,332992],{"encoding":31667},"ab+P = 0+P",[34,332994,332996,333014,333032,333050],{"className":332995,"ariaHidden":758},[31672],[34,332997,332999,333002,333005,333008,333011],{"className":332998},[31676],[34,333000],{"className":333001,"style":34299},[31680],[34,333003,324191],{"className":333004},[31685,31686],[34,333006],{"className":333007,"style":32604},[31845],[34,333009,223],{"className":333010},[32024],[34,333012],{"className":333013,"style":32604},[31845],[34,333015,333017,333020,333023,333026,333029],{"className":333016},[31676],[34,333018],{"className":333019,"style":33218},[31680],[34,333021,105852],{"className":333022,"style":37422},[31685,31686],[34,333024],{"className":333025,"style":31846},[31845],[34,333027,202],{"className":333028},[31850],[34,333030],{"className":333031,"style":31846},[31845],[34,333033,333035,333038,333041,333044,333047],{"className":333034},[31676],[34,333036],{"className":333037,"style":35942},[31680],[34,333039,459],{"className":333040},[31685],[34,333042],{"className":333043,"style":32604},[31845],[34,333045,223],{"className":333046},[32024],[34,333048],{"className":333049,"style":32604},[31845],[34,333051,333053,333056],{"className":333052},[31676],[34,333054],{"className":333055,"style":33218},[31680],[34,333057,105852],{"className":333058,"style":37422},[31685,31686],[34,333060,333062,333098],{"className":333061},[31642],[34,333063,333065],{"className":333064},[31646],[31648,333066,333067],{"xmlns":31650},[31652,333068,333069,333095],{},[31655,333070,333071,333073,333075,333077,333079,333081,333083,333085,333087,333089,333091,333093],{},[31662,333072,456],{"stretchy":702},[31658,333074,111],{},[31662,333076,223],{},[31658,333078,105852],{},[31662,333080,2557],{"stretchy":702},[31662,333082,456],{"stretchy":702},[31658,333084,1697],{},[31662,333086,223],{},[31658,333088,105852],{},[31662,333090,2557],{"stretchy":702},[31662,333092,202],{},[31749,333094,459],{},[31665,333096,333097],{"encoding":31667},"(a+P)(b+P) = 0",[34,333099,333101,333122,333149,333170],{"className":333100,"ariaHidden":758},[31672],[34,333102,333104,333107,333110,333113,333116,333119],{"className":333103},[31676],[34,333105],{"className":333106,"style":31774},[31680],[34,333108,456],{"className":333109},[31782],[34,333111,111],{"className":333112},[31685,31686],[34,333114],{"className":333115,"style":32604},[31845],[34,333117,223],{"className":333118},[32024],[34,333120],{"className":333121,"style":32604},[31845],[34,333123,333125,333128,333131,333134,333137,333140,333143,333146],{"className":333124},[31676],[34,333126],{"className":333127,"style":31774},[31680],[34,333129,105852],{"className":333130,"style":37422},[31685,31686],[34,333132,2557],{"className":333133},[31690],[34,333135,456],{"className":333136},[31782],[34,333138,1697],{"className":333139},[31685,31686],[34,333141],{"className":333142,"style":32604},[31845],[34,333144,223],{"className":333145},[32024],[34,333147],{"className":333148,"style":32604},[31845],[34,333150,333152,333155,333158,333161,333164,333167],{"className":333151},[31676],[34,333153],{"className":333154,"style":31774},[31680],[34,333156,105852],{"className":333157,"style":37422},[31685,31686],[34,333159,2557],{"className":333160},[31690],[34,333162],{"className":333163,"style":31846},[31845],[34,333165,202],{"className":333166},[31850],[34,333168],{"className":333169,"style":31846},[31845],[34,333171,333173,333176],{"className":333172},[31676],[34,333174],{"className":333175,"style":32617},[31680],[34,333177,459],{"className":333178},[31685],[108,333180,333181,332541,333183,113124,333221,333223,333224,331105,333297,347],{},[2627,333182,332540],{},[34,333184,333186,333203],{"className":333185},[31642],[34,333187,333189],{"className":333188},[31646],[31648,333190,333191],{"xmlns":31650},[31652,333192,333193,333201],{},[31655,333194,333195,333197,333199],{},[31658,333196,34386],{},[31658,333198,38080],{"mathvariant":38079},[31658,333200,105852],{},[31665,333202,331515],{"encoding":31667},[34,333204,333206],{"className":333205,"ariaHidden":758},[31672],[34,333207,333209,333212,333215,333218],{"className":333208},[31676],[34,333210],{"className":333211,"style":31774},[31680],[34,333213,34386],{"className":333214,"style":34427},[31685,31686],[34,333216,38080],{"className":333217},[31685],[34,333219,105852],{"className":333220,"style":37422},[31685,31686],[2627,333222,330740],{},"，没有零因子，所以必须有 ",[34,333225,333227,333249],{"className":333226},[31642],[34,333228,333230],{"className":333229},[31646],[31648,333231,333232],{"xmlns":31650},[31652,333233,333234,333246],{},[31655,333235,333236,333238,333240,333242,333244],{},[31658,333237,111],{},[31662,333239,223],{},[31658,333241,105852],{},[31662,333243,202],{},[31749,333245,459],{},[31665,333247,333248],{"encoding":31667},"a+P=0",[34,333250,333252,333270,333288],{"className":333251,"ariaHidden":758},[31672],[34,333253,333255,333258,333261,333264,333267],{"className":333254},[31676],[34,333256],{"className":333257,"style":32597},[31680],[34,333259,111],{"className":333260},[31685,31686],[34,333262],{"className":333263,"style":32604},[31845],[34,333265,223],{"className":333266},[32024],[34,333268],{"className":333269,"style":32604},[31845],[34,333271,333273,333276,333279,333282,333285],{"className":333272},[31676],[34,333274],{"className":333275,"style":33218},[31680],[34,333277,105852],{"className":333278,"style":37422},[31685,31686],[34,333280],{"className":333281,"style":31846},[31845],[34,333283,202],{"className":333284},[31850],[34,333286],{"className":333287,"style":31846},[31845],[34,333289,333291,333294],{"className":333290},[31676],[34,333292],{"className":333293,"style":32617},[31680],[34,333295,459],{"className":333296},[31685],[34,333298,333300,333322],{"className":333299},[31642],[34,333301,333303],{"className":333302},[31646],[31648,333304,333305],{"xmlns":31650},[31652,333306,333307,333319],{},[31655,333308,333309,333311,333313,333315,333317],{},[31658,333310,1697],{},[31662,333312,223],{},[31658,333314,105852],{},[31662,333316,202],{},[31749,333318,459],{},[31665,333320,333321],{"encoding":31667},"b+P=0",[34,333323,333325,333343,333361],{"className":333324,"ariaHidden":758},[31672],[34,333326,333328,333331,333334,333337,333340],{"className":333327},[31676],[34,333329],{"className":333330,"style":34299},[31680],[34,333332,1697],{"className":333333},[31685,31686],[34,333335],{"className":333336,"style":32604},[31845],[34,333338,223],{"className":333339},[32024],[34,333341],{"className":333342,"style":32604},[31845],[34,333344,333346,333349,333352,333355,333358],{"className":333345},[31676],[34,333347],{"className":333348,"style":33218},[31680],[34,333350,105852],{"className":333351,"style":37422},[31685,31686],[34,333353],{"className":333354,"style":31846},[31845],[34,333356,202],{"className":333357},[31850],[34,333359],{"className":333360,"style":31846},[31845],[34,333362,333364,333367],{"className":333363},[31676],[34,333365],{"className":333366,"style":32617},[31680],[34,333368,459],{"className":333369},[31685],[108,333371,333372,113463,333375,331105,333425,347],{},[2627,333373,333374],{},"还原",[34,333376,333378,333395],{"className":333377},[31642],[34,333379,333381],{"className":333380},[31646],[31648,333382,333383],{"xmlns":31650},[31652,333384,333385,333393],{},[31655,333386,333387,333389,333391],{},[31658,333388,111],{},[31662,333390,112296],{},[31658,333392,105852],{},[31665,333394,331074],{"encoding":31667},[34,333396,333398,333416],{"className":333397,"ariaHidden":758},[31672],[34,333399,333401,333404,333407,333410,333413],{"className":333400},[31676],[34,333402],{"className":333403,"style":34812},[31680],[34,333405,111],{"className":333406},[31685,31686],[34,333408],{"className":333409,"style":31846},[31845],[34,333411,112296],{"className":333412},[31850],[34,333414],{"className":333415,"style":31846},[31845],[34,333417,333419,333422],{"className":333418},[31676],[34,333420],{"className":333421,"style":33218},[31680],[34,333423,105852],{"className":333424,"style":37422},[31685,31686],[34,333426,333428,333445],{"className":333427},[31642],[34,333429,333431],{"className":333430},[31646],[31648,333432,333433],{"xmlns":31650},[31652,333434,333435,333443],{},[31655,333436,333437,333439,333441],{},[31658,333438,1697],{},[31662,333440,112296],{},[31658,333442,105852],{},[31665,333444,331126],{"encoding":31667},[34,333446,333448,333466],{"className":333447,"ariaHidden":758},[31672],[34,333449,333451,333454,333457,333460,333463],{"className":333450},[31676],[34,333452],{"className":333453,"style":46991},[31680],[34,333455,1697],{"className":333456},[31685,31686],[34,333458],{"className":333459,"style":31846},[31845],[34,333461,112296],{"className":333462},[31850],[34,333464],{"className":333465,"style":31846},[31845],[34,333467,333469,333472],{"className":333468},[31676],[34,333470],{"className":333471,"style":33218},[31680],[34,333473,105852],{"className":333474,"style":37422},[31685,31686],[108,333476,333477,333479],{},[2627,333478,332755],{},"：符合素理想定义。",[32188,333481],{},[19,333483,333485],{"id":333484},"_2-极大理想-maximal-ideal","2. 极大理想 (Maximal Ideal)",[253,333487,333489],{"id":333488},"_21-定义","2.1 定义",[15,333491,133910,333492,330820,333520,333548,333549,333608,333609,258,333637,333665,333666,333694],{},[34,333493,333495,333508],{"className":333494},[31642],[34,333496,333498],{"className":333497},[31646],[31648,333499,333500],{"xmlns":31650},[31652,333501,333502,333506],{},[31655,333503,333504],{},[31658,333505,98360],{},[31665,333507,98360],{"encoding":31667},[34,333509,333511],{"className":333510,"ariaHidden":758},[31672],[34,333512,333514,333517],{"className":333513},[31676],[34,333515],{"className":333516,"style":33218},[31680],[34,333518,98360],{"className":333519,"style":33389},[31685,31686],[34,333521,333523,333536],{"className":333522},[31642],[34,333524,333526],{"className":333525},[31646],[31648,333527,333528],{"xmlns":31650},[31652,333529,333530,333534],{},[31655,333531,333532],{},[31658,333533,34386],{},[31665,333535,34386],{"encoding":31667},[34,333537,333539],{"className":333538,"ariaHidden":758},[31672],[34,333540,333542,333545],{"className":333541},[31676],[34,333543],{"className":333544,"style":33218},[31680],[34,333546,34386],{"className":333547,"style":34427},[31685,31686]," 的真理想。如果",[2627,333550,2884,333551,258,333579,333607],{},[34,333552,333554,333567],{"className":333553},[31642],[34,333555,333557],{"className":333556},[31646],[31648,333558,333559],{"xmlns":31650},[31652,333560,333561,333565],{},[31655,333562,333563],{},[31658,333564,98360],{},[31665,333566,98360],{"encoding":31667},[34,333568,333570],{"className":333569,"ariaHidden":758},[31672],[34,333571,333573,333576],{"className":333572},[31676],[34,333574],{"className":333575,"style":33218},[31680],[34,333577,98360],{"className":333578,"style":33389},[31685,31686],[34,333580,333582,333595],{"className":333581},[31642],[34,333583,333585],{"className":333584},[31646],[31648,333586,333587],{"xmlns":31650},[31652,333588,333589,333593],{},[31655,333590,333591],{},[31658,333592,34386],{},[31665,333594,34386],{"encoding":31667},[34,333596,333598],{"className":333597,"ariaHidden":758},[31672],[34,333599,333601,333604],{"className":333600},[31676],[34,333602],{"className":333603,"style":33218},[31680],[34,333605,34386],{"className":333606,"style":34427},[31685,31686]," 之间找不到其他的理想","（除了 ",[34,333610,333612,333625],{"className":333611},[31642],[34,333613,333615],{"className":333614},[31646],[31648,333616,333617],{"xmlns":31650},[31652,333618,333619,333623],{},[31655,333620,333621],{},[31658,333622,98360],{},[31665,333624,98360],{"encoding":31667},[34,333626,333628],{"className":333627,"ariaHidden":758},[31672],[34,333629,333631,333634],{"className":333630},[31676],[34,333632],{"className":333633,"style":33218},[31680],[34,333635,98360],{"className":333636,"style":33389},[31685,31686],[34,333638,333640,333653],{"className":333639},[31642],[34,333641,333643],{"className":333642},[31646],[31648,333644,333645],{"xmlns":31650},[31652,333646,333647,333651],{},[31655,333648,333649],{},[31658,333650,34386],{},[31665,333652,34386],{"encoding":31667},[34,333654,333656],{"className":333655,"ariaHidden":758},[31672],[34,333657,333659,333662],{"className":333658},[31676],[34,333660],{"className":333661,"style":33218},[31680],[34,333663,34386],{"className":333664,"style":34427},[31685,31686]," 自己），则称 ",[34,333667,333669,333682],{"className":333668},[31642],[34,333670,333672],{"className":333671},[31646],[31648,333673,333674],{"xmlns":31650},[31652,333675,333676,333680],{},[31655,333677,333678],{},[31658,333679,98360],{},[31665,333681,98360],{"encoding":31667},[34,333683,333685],{"className":333684,"ariaHidden":758},[31672],[34,333686,333688,333691],{"className":333687},[31676],[34,333689],{"className":333690,"style":33218},[31680],[34,333692,98360],{"className":333693,"style":33389},[31685,31686]," 为极大理想。",[105,333696,333697],{},[108,333698,333699,333700,86674,333773,333801,333802,4374,333853,347],{},"数学语言：若 ",[34,333701,333703,333725],{"className":333702},[31642],[34,333704,333706],{"className":333705},[31646],[31648,333707,333708],{"xmlns":31650},[31652,333709,333710,333722],{},[31655,333711,333712,333714,333716,333718,333720],{},[31658,333713,98360],{},[31662,333715,131855],{},[31658,333717,106063],{},[31662,333719,131855],{},[31658,333721,34386],{},[31665,333723,333724],{"encoding":31667},"M \\subseteq I \\subseteq R",[34,333726,333728,333746,333764],{"className":333727,"ariaHidden":758},[31672],[34,333729,333731,333734,333737,333740,333743],{"className":333730},[31676],[34,333732],{"className":333733,"style":195047},[31680],[34,333735,98360],{"className":333736,"style":33389},[31685,31686],[34,333738],{"className":333739,"style":31846},[31845],[34,333741,131855],{"className":333742},[31850],[34,333744],{"className":333745,"style":31846},[31845],[34,333747,333749,333752,333755,333758,333761],{"className":333748},[31676],[34,333750],{"className":333751,"style":195047},[31680],[34,333753,106063],{"className":333754,"style":67482},[31685,31686],[34,333756],{"className":333757,"style":31846},[31845],[34,333759,131855],{"className":333760},[31850],[34,333762],{"className":333763,"style":31846},[31845],[34,333765,333767,333770],{"className":333766},[31676],[34,333768],{"className":333769,"style":33218},[31680],[34,333771,34386],{"className":333772,"style":34427},[31685,31686],[34,333774,333776,333789],{"className":333775},[31642],[34,333777,333779],{"className":333778},[31646],[31648,333780,333781],{"xmlns":31650},[31652,333782,333783,333787],{},[31655,333784,333785],{},[31658,333786,106063],{},[31665,333788,106063],{"encoding":31667},[34,333790,333792],{"className":333791,"ariaHidden":758},[31672],[34,333793,333795,333798],{"className":333794},[31676],[34,333796],{"className":333797,"style":33218},[31680],[34,333799,106063],{"className":333800,"style":67482},[31685,31686]," 是理想，则必有 ",[34,333803,333805,333823],{"className":333804},[31642],[34,333806,333808],{"className":333807},[31646],[31648,333809,333810],{"xmlns":31650},[31652,333811,333812,333820],{},[31655,333813,333814,333816,333818],{},[31658,333815,106063],{},[31662,333817,202],{},[31658,333819,98360],{},[31665,333821,333822],{"encoding":31667},"I=M",[34,333824,333826,333844],{"className":333825,"ariaHidden":758},[31672],[34,333827,333829,333832,333835,333838,333841],{"className":333828},[31676],[34,333830],{"className":333831,"style":33218},[31680],[34,333833,106063],{"className":333834,"style":67482},[31685,31686],[34,333836],{"className":333837,"style":31846},[31845],[34,333839,202],{"className":333840},[31850],[34,333842],{"className":333843,"style":31846},[31845],[34,333845,333847,333850],{"className":333846},[31676],[34,333848],{"className":333849,"style":33218},[31680],[34,333851,98360],{"className":333852,"style":33389},[31685,31686],[34,333854,333856,333874],{"className":333855},[31642],[34,333857,333859],{"className":333858},[31646],[31648,333860,333861],{"xmlns":31650},[31652,333862,333863,333871],{},[31655,333864,333865,333867,333869],{},[31658,333866,106063],{},[31662,333868,202],{},[31658,333870,34386],{},[31665,333872,333873],{"encoding":31667},"I=R",[34,333875,333877,333895],{"className":333876,"ariaHidden":758},[31672],[34,333878,333880,333883,333886,333889,333892],{"className":333879},[31676],[34,333881],{"className":333882,"style":33218},[31680],[34,333884,106063],{"className":333885,"style":67482},[31685,31686],[34,333887],{"className":333888,"style":31846},[31845],[34,333890,202],{"className":333891},[31850],[34,333893],{"className":333894,"style":31846},[31845],[34,333896,333898,333901],{"className":333897},[31676],[34,333899],{"className":333900,"style":33218},[31680],[34,333902,34386],{"className":333903,"style":34427},[31685,31686],[104833,333905,333906],{},[15,333907,333908,2664,333910,333938,333939,347],{},[2627,333909,318465],{},[34,333911,333913,333926],{"className":333912},[31642],[34,333914,333916],{"className":333915},[31646],[31648,333917,333918],{"xmlns":31650},[31652,333919,333920,333924],{},[31655,333921,333922],{},[31658,333923,98360],{},[31665,333925,98360],{"encoding":31667},[34,333927,333929],{"className":333928,"ariaHidden":758},[31672],[34,333930,333932,333935],{"className":333931},[31676],[34,333933],{"className":333934,"style":33218},[31680],[34,333936,98360],{"className":333937,"style":33389},[31685,31686]," 是\"天花板\"级别的理想。它大到不能再大了，再往里面加任何一个不属于它的元素，它就会瞬间膨胀成整个环 ",[34,333940,333942,333955],{"className":333941},[31642],[34,333943,333945],{"className":333944},[31646],[31648,333946,333947],{"xmlns":31650},[31652,333948,333949,333953],{},[31655,333950,333951],{},[31658,333952,34386],{},[31665,333954,34386],{"encoding":31667},[34,333956,333958],{"className":333957,"ariaHidden":758},[31672],[34,333959,333961,333964],{"className":333960},[31676],[34,333962],{"className":333963,"style":33218},[31680],[34,333965,34386],{"className":333966,"style":34427},[31685,31686],[253,333968,333970],{"id":333969},"_22-核心定理-b-重难点","2.2 核心定理 B (重难点)",[15,333972,333973,2664,333976,334004,334005,331494,334043,334082],{},[2627,333974,333975],{},"定理 3.5.2",[34,333977,333979,333992],{"className":333978},[31642],[34,333980,333982],{"className":333981},[31646],[31648,333983,333984],{"xmlns":31650},[31652,333985,333986,333990],{},[31655,333987,333988],{},[31658,333989,98360],{},[31665,333991,98360],{"encoding":31667},[34,333993,333995],{"className":333994,"ariaHidden":758},[31672],[34,333996,333998,334001],{"className":333997},[31676],[34,333999],{"className":334000,"style":33218},[31680],[34,334002,98360],{"className":334003,"style":33389},[31685,31686]," 是极大理想 ",[34,334006,334008,334025],{"className":334007},[31642],[34,334009,334011],{"className":334010},[31646],[31648,334012,334013],{"xmlns":31650},[31652,334014,334015,334023],{},[31655,334016,334017,334019,334021],{},[39361,334018,262622],{},[31662,334020,323391],{},[39361,334022,262622],{},[31665,334024,323396],{"encoding":31667},[34,334026,334028],{"className":334027,"ariaHidden":758},[31672],[34,334029,334031,334034,334037,334040],{"className":334030},[31676],[34,334032],{"className":334033,"style":318264},[31680],[34,334035],{"className":334036,"style":31846},[31845],[34,334038,323391],{"className":334039},[31850],[34,334041],{"className":334042,"style":31846},[31845],[34,334044,334046,334064],{"className":334045},[31642],[34,334047,334049],{"className":334048},[31646],[31648,334050,334051],{"xmlns":31650},[31652,334052,334053,334061],{},[31655,334054,334055,334057,334059],{},[31658,334056,34386],{},[31658,334058,38080],{"mathvariant":38079},[31658,334060,98360],{},[31665,334062,334063],{"encoding":31667},"R\u002FM",[34,334065,334067],{"className":334066,"ariaHidden":758},[31672],[34,334068,334070,334073,334076,334079],{"className":334069},[31676],[34,334071],{"className":334072,"style":31774},[31680],[34,334074,34386],{"className":334075,"style":34427},[31685,31686],[34,334077,38080],{"className":334078},[31685],[34,334080,98360],{"className":334081,"style":33389},[31685,31686]," 是域。",[32490,334084,331537],{"id":334085},"证明详解-1",[15,334087,334088,334186],{},[2627,334089,83913,334090,334118,334119,334147,334148,334082],{},[34,334091,334093,334106],{"className":334092},[31642],[34,334094,334096],{"className":334095},[31646],[31648,334097,334098],{"xmlns":31650},[31652,334099,334100,334104],{},[31655,334101,334102],{},[31662,334103,140094],{},[31665,334105,266872],{"encoding":31667},[34,334107,334109],{"className":334108,"ariaHidden":758},[31672],[34,334110,334112,334115],{"className":334111},[31676],[34,334113],{"className":334114,"style":73089},[31680],[34,334116,140094],{"className":334117},[31850],"）方向：已知 ",[34,334120,334122,334135],{"className":334121},[31642],[34,334123,334125],{"className":334124},[31646],[31648,334126,334127],{"xmlns":31650},[31652,334128,334129,334133],{},[31655,334130,334131],{},[31658,334132,98360],{},[31665,334134,98360],{"encoding":31667},[34,334136,334138],{"className":334137,"ariaHidden":758},[31672],[34,334139,334141,334144],{"className":334140},[31676],[34,334142],{"className":334143,"style":33218},[31680],[34,334145,98360],{"className":334146,"style":33389},[31685,31686]," 极大，求证 ",[34,334149,334151,334168],{"className":334150},[31642],[34,334152,334154],{"className":334153},[31646],[31648,334155,334156],{"xmlns":31650},[31652,334157,334158,334166],{},[31655,334159,334160,334162,334164],{},[31658,334161,34386],{},[31658,334163,38080],{"mathvariant":38079},[31658,334165,98360],{},[31665,334167,334063],{"encoding":31667},[34,334169,334171],{"className":334170,"ariaHidden":758},[31672],[34,334172,334174,334177,334180,334183],{"className":334173},[31676],[34,334175],{"className":334176,"style":31774},[31680],[34,334178,34386],{"className":334179,"style":34427},[31685,31686],[34,334181,38080],{"className":334182},[31685],[34,334184,98360],{"className":334185,"style":33389},[31685,31686],[71807,334187,334188],{},"关键思路：域就是“每个非零元素都有逆元”的环。我们要造出逆元。",[2230,334190,334191],{},[108,334192,334193,331829,334196,334234,334235,347],{},[2627,334194,334195],{},"取元",[34,334197,334199,334216],{"className":334198},[31642],[34,334200,334202],{"className":334201},[31646],[31648,334203,334204],{"xmlns":31650},[31652,334205,334206,334214],{},[31655,334207,334208,334210,334212],{},[31658,334209,34386],{},[31658,334211,38080],{"mathvariant":38079},[31658,334213,98360],{},[31665,334215,334063],{"encoding":31667},[34,334217,334219],{"className":334218,"ariaHidden":758},[31672],[34,334220,334222,334225,334228,334231],{"className":334221},[31676],[34,334223],{"className":334224,"style":31774},[31680],[34,334226,34386],{"className":334227,"style":34427},[31685,31686],[34,334229,38080],{"className":334230},[31685],[34,334232,98360],{"className":334233,"style":33389},[31685,31686]," 中任取非零元素 ",[34,334236,334238,334256],{"className":334237},[31642],[34,334239,334241],{"className":334240},[31646],[31648,334242,334243],{"xmlns":31650},[31652,334244,334245,334253],{},[31655,334246,334247,334249,334251],{},[31658,334248,111],{},[31662,334250,223],{},[31658,334252,98360],{},[31665,334254,334255],{"encoding":31667},"a+M",[34,334257,334259,334277],{"className":334258,"ariaHidden":758},[31672],[34,334260,334262,334265,334268,334271,334274],{"className":334261},[31676],[34,334263],{"className":334264,"style":32597},[31680],[34,334266,111],{"className":334267},[31685,31686],[34,334269],{"className":334270,"style":32604},[31845],[34,334272,223],{"className":334273},[32024],[34,334275],{"className":334276,"style":32604},[31845],[34,334278,334280,334283],{"className":334279},[31676],[34,334281],{"className":334282,"style":33218},[31680],[34,334284,98360],{"className":334285,"style":33389},[31685,31686],[105,334287,334288],{},[108,334289,329925,334290,347],{},[34,334291,334293,334311],{"className":334292},[31642],[34,334294,334296],{"className":334295},[31646],[31648,334297,334298],{"xmlns":31650},[31652,334299,334300,334308],{},[31655,334301,334302,334304,334306],{},[31658,334303,111],{},[31662,334305,112266],{"mathvariant":38079},[31658,334307,98360],{},[31665,334309,334310],{"encoding":31667},"a \\notin M",[34,334312,334314,334365],{"className":334313,"ariaHidden":758},[31672],[34,334315,334317,334320,334323,334326,334362],{"className":334316},[31676],[34,334318],{"className":334319,"style":31774},[31680],[34,334321,111],{"className":334322},[31685,31686],[34,334324],{"className":334325,"style":31846},[31845],[34,334327,334329,334335],{"className":334328},[31850],[34,334330,334332],{"className":334331},[31685],[34,334333,112296],{"className":334334},[31850],[34,334336,334338],{"className":334337},[31685,54774],[34,334339,334341],{"className":334340},[54778],[34,334342,334344,334347,334359],{"className":334343},[112306],[34,334345],{"className":334346,"style":31774},[31680],[34,334348,334350],{"className":334349},[54789],[34,334351,334353,334356],{"className":334352},[31685],[34,334354,38080],{"className":334355},[31685],[34,334357],{"className":334358,"style":112322},[31845],[34,334360],{"className":334361},[54800],[34,334363],{"className":334364,"style":31846},[31845],[34,334366,334368,334371],{"className":334367},[31676],[34,334369],{"className":334370,"style":33218},[31680],[34,334372,98360],{"className":334373,"style":33389},[31685,31686],[2230,334375,334376],{"start":43},[108,334377,334378,334381,334382,258,334410,334438,334439,347],{},[2627,334379,334380],{},"构造新理想","：考虑由 ",[34,334383,334385,334398],{"className":334384},[31642],[34,334386,334388],{"className":334387},[31646],[31648,334389,334390],{"xmlns":31650},[31652,334391,334392,334396],{},[31655,334393,334394],{},[31658,334395,98360],{},[31665,334397,98360],{"encoding":31667},[34,334399,334401],{"className":334400,"ariaHidden":758},[31672],[34,334402,334404,334407],{"className":334403},[31676],[34,334405],{"className":334406,"style":33218},[31680],[34,334408,98360],{"className":334409,"style":33389},[31685,31686],[34,334411,334413,334426],{"className":334412},[31642],[34,334414,334416],{"className":334415},[31646],[31648,334417,334418],{"xmlns":31650},[31652,334419,334420,334424],{},[31655,334421,334422],{},[31658,334423,111],{},[31665,334425,111],{"encoding":31667},[34,334427,334429],{"className":334428,"ariaHidden":758},[31672],[34,334430,334432,334435],{"className":334431},[31676],[34,334433],{"className":334434,"style":32527},[31680],[34,334436,111],{"className":334437},[31685,31686]," 共同生成的理想，记为 ",[34,334440,334442,334464],{"className":334441},[31642],[34,334443,334445],{"className":334444},[31646],[31648,334446,334447],{"xmlns":31650},[31652,334448,334449,334461],{},[31655,334450,334451,334453,334455,334457,334459],{},[31662,334452,126529],{"stretchy":702},[31658,334454,98360],{},[31662,334456,31960],{"separator":758},[31658,334458,111],{},[31662,334460,126544],{"stretchy":702},[31665,334462,334463],{"encoding":31667},"\\langle M, a \\rangle",[34,334465,334467],{"className":334466,"ariaHidden":758},[31672],[34,334468,334470,334473,334476,334479,334482,334485,334488],{"className":334469},[31676],[34,334471],{"className":334472,"style":31774},[31680],[34,334474,126529],{"className":334475},[31782],[34,334477,98360],{"className":334478,"style":33389},[31685,31686],[34,334480,31960],{"className":334481},[32119],[34,334483],{"className":334484,"style":32123},[31845],[34,334486,111],{"className":334487},[31685,31686],[34,334489,126544],{"className":334490},[31690],[105,334492,334493],{},[108,334494,334495,334496,334524,334525,334553,334554,334582],{},"这个理想包含了 ",[34,334497,334499,334512],{"className":334498},[31642],[34,334500,334502],{"className":334501},[31646],[31648,334503,334504],{"xmlns":31650},[31652,334505,334506,334510],{},[31655,334507,334508],{},[31658,334509,98360],{},[31665,334511,98360],{"encoding":31667},[34,334513,334515],{"className":334514,"ariaHidden":758},[31672],[34,334516,334518,334521],{"className":334517},[31676],[34,334519],{"className":334520,"style":33218},[31680],[34,334522,98360],{"className":334523,"style":33389},[31685,31686],"，而且因为有 ",[34,334526,334528,334541],{"className":334527},[31642],[34,334529,334531],{"className":334530},[31646],[31648,334532,334533],{"xmlns":31650},[31652,334534,334535,334539],{},[31655,334536,334537],{},[31658,334538,111],{},[31665,334540,111],{"encoding":31667},[34,334542,334544],{"className":334543,"ariaHidden":758},[31672],[34,334545,334547,334550],{"className":334546},[31676],[34,334548],{"className":334549,"style":32527},[31680],[34,334551,111],{"className":334552},[31685,31686],"，它比 ",[34,334555,334557,334570],{"className":334556},[31642],[34,334558,334560],{"className":334559},[31646],[31648,334561,334562],{"xmlns":31650},[31652,334563,334564,334568],{},[31655,334565,334566],{},[31658,334567,98360],{},[31665,334569,98360],{"encoding":31667},[34,334571,334573],{"className":334572,"ariaHidden":758},[31672],[34,334574,334576,334579],{"className":334575},[31676],[34,334577],{"className":334578,"style":33218},[31680],[34,334580,98360],{"className":334581,"style":33389},[31685,31686]," 更大。",[2230,334584,334585],{"start":49},[108,334586,334587,332541,334590,334618,334619,334647,334648,44748,334699,334727,334728,347],{},[2627,334588,334589],{},"利用极大性",[34,334591,334593,334606],{"className":334592},[31642],[34,334594,334596],{"className":334595},[31646],[31648,334597,334598],{"xmlns":31650},[31652,334599,334600,334604],{},[31655,334601,334602],{},[31658,334603,98360],{},[31665,334605,98360],{"encoding":31667},[34,334607,334609],{"className":334608,"ariaHidden":758},[31672],[34,334610,334612,334615],{"className":334611},[31676],[34,334613],{"className":334614,"style":33218},[31680],[34,334616,98360],{"className":334617,"style":33389},[31685,31686]," 是极大理想，它上面只有 ",[34,334620,334622,334635],{"className":334621},[31642],[34,334623,334625],{"className":334624},[31646],[31648,334626,334627],{"xmlns":31650},[31652,334628,334629,334633],{},[31655,334630,334631],{},[31658,334632,34386],{},[31665,334634,34386],{"encoding":31667},[34,334636,334638],{"className":334637,"ariaHidden":758},[31672],[34,334639,334641,334644],{"className":334640},[31676],[34,334642],{"className":334643,"style":33218},[31680],[34,334645,34386],{"className":334646,"style":34427},[31685,31686],"。既然 ",[34,334649,334651,334672],{"className":334650},[31642],[34,334652,334654],{"className":334653},[31646],[31648,334655,334656],{"xmlns":31650},[31652,334657,334658,334670],{},[31655,334659,334660,334662,334664,334666,334668],{},[31662,334661,126529],{"stretchy":702},[31658,334663,98360],{},[31662,334665,31960],{"separator":758},[31658,334667,111],{},[31662,334669,126544],{"stretchy":702},[31665,334671,334463],{"encoding":31667},[34,334673,334675],{"className":334674,"ariaHidden":758},[31672],[34,334676,334678,334681,334684,334687,334690,334693,334696],{"className":334677},[31676],[34,334679],{"className":334680,"style":31774},[31680],[34,334682,126529],{"className":334683},[31782],[34,334685,98360],{"className":334686,"style":33389},[31685,31686],[34,334688,31960],{"className":334689},[32119],[34,334691],{"className":334692,"style":32123},[31845],[34,334694,111],{"className":334695},[31685,31686],[34,334697,126544],{"className":334698},[31690],[34,334700,334702,334715],{"className":334701},[31642],[34,334703,334705],{"className":334704},[31646],[31648,334706,334707],{"xmlns":31650},[31652,334708,334709,334713],{},[31655,334710,334711],{},[31658,334712,98360],{},[31665,334714,98360],{"encoding":31667},[34,334716,334718],{"className":334717,"ariaHidden":758},[31672],[34,334719,334721,334724],{"className":334720},[31676],[34,334722],{"className":334723,"style":33218},[31680],[34,334725,98360],{"className":334726,"style":33389},[31685,31686]," 大，那它",[2627,334729,334730,334731],{},"只能是 ",[34,334732,334734,334747],{"className":334733},[31642],[34,334735,334737],{"className":334736},[31646],[31648,334738,334739],{"xmlns":31650},[31652,334740,334741,334745],{},[31655,334742,334743],{},[31658,334744,34386],{},[31665,334746,34386],{"encoding":31667},[34,334748,334750],{"className":334749,"ariaHidden":758},[31672],[34,334751,334753,334756],{"className":334752},[31676],[34,334754],{"className":334755,"style":33218},[31680],[34,334757,34386],{"className":334758,"style":34427},[31685,31686],[105,334760,334761],{},[108,334762,334763,2664,334765,347],{},[2627,334764,114043],{},[34,334766,334768,334794],{"className":334767},[31642],[34,334769,334771],{"className":334770},[31646],[31648,334772,334773],{"xmlns":31650},[31652,334774,334775,334791],{},[31655,334776,334777,334779,334781,334783,334785,334787,334789],{},[31662,334778,126529],{"stretchy":702},[31658,334780,98360],{},[31662,334782,31960],{"separator":758},[31658,334784,111],{},[31662,334786,126544],{"stretchy":702},[31662,334788,202],{},[31658,334790,34386],{},[31665,334792,334793],{"encoding":31667},"\\langle M, a \\rangle = R",[34,334795,334797,334830],{"className":334796,"ariaHidden":758},[31672],[34,334798,334800,334803,334806,334809,334812,334815,334818,334821,334824,334827],{"className":334799},[31676],[34,334801],{"className":334802,"style":31774},[31680],[34,334804,126529],{"className":334805},[31782],[34,334807,98360],{"className":334808,"style":33389},[31685,31686],[34,334810,31960],{"className":334811},[32119],[34,334813],{"className":334814,"style":32123},[31845],[34,334816,111],{"className":334817},[31685,31686],[34,334819,126544],{"className":334820},[31690],[34,334822],{"className":334823,"style":31846},[31845],[34,334825,202],{"className":334826},[31850],[34,334828],{"className":334829,"style":31846},[31845],[34,334831,334833,334836],{"className":334832},[31676],[34,334834],{"className":334835,"style":33218},[31680],[34,334837,34386],{"className":334838,"style":34427},[31685,31686],[2230,334840,334841,334950],{"start":56},[108,334842,334843,334846,334847,334875,334876,347],{},[2627,334844,334845],{},"抓取单位元","：既然整个环 ",[34,334848,334850,334863],{"className":334849},[31642],[34,334851,334853],{"className":334852},[31646],[31648,334854,334855],{"xmlns":31650},[31652,334856,334857,334861],{},[31655,334858,334859],{},[31658,334860,34386],{},[31665,334862,34386],{"encoding":31667},[34,334864,334866],{"className":334865,"ariaHidden":758},[31672],[34,334867,334869,334872],{"className":334868},[31676],[34,334870],{"className":334871,"style":33218},[31680],[34,334873,34386],{"className":334874,"style":34427},[31685,31686]," 都在这个理想里，那么单位元 ",[34,334877,334879,334905],{"className":334878},[31642],[34,334880,334882],{"className":334881},[31646],[31648,334883,334884],{"xmlns":31650},[31652,334885,334886,334902],{},[31655,334887,334888,334890,334892,334894,334896,334898,334900],{},[31749,334889,640],{},[31662,334891,112296],{},[31662,334893,126529],{"stretchy":702},[31658,334895,98360],{},[31662,334897,31960],{"separator":758},[31658,334899,111],{},[31662,334901,126544],{"stretchy":702},[31665,334903,334904],{"encoding":31667},"1 \\in \\langle M, a \\rangle",[34,334906,334908,334926],{"className":334907,"ariaHidden":758},[31672],[34,334909,334911,334914,334917,334920,334923],{"className":334910},[31676],[34,334912],{"className":334913,"style":61814},[31680],[34,334915,640],{"className":334916},[31685],[34,334918],{"className":334919,"style":31846},[31845],[34,334921,112296],{"className":334922},[31850],[34,334924],{"className":334925,"style":31846},[31845],[34,334927,334929,334932,334935,334938,334941,334944,334947],{"className":334928},[31676],[34,334930],{"className":334931,"style":31774},[31680],[34,334933,126529],{"className":334934},[31782],[34,334936,98360],{"className":334937,"style":33389},[31685,31686],[34,334939,31960],{"className":334940},[32119],[34,334942],{"className":334943,"style":32123},[31845],[34,334945,111],{"className":334946},[31685,31686],[34,334948,126544],{"className":334949},[31690],[108,334951,334952,2664,334955,334983,334984,335012,335013,335041],{},[2627,334953,334954],{},"写出表达式",[34,334956,334958,334971],{"className":334957},[31642],[34,334959,334961],{"className":334960},[31646],[31648,334962,334963],{"xmlns":31650},[31652,334964,334965,334969],{},[31655,334966,334967],{},[31749,334968,640],{},[31665,334970,640],{"encoding":31667},[34,334972,334974],{"className":334973,"ariaHidden":758},[31672],[34,334975,334977,334980],{"className":334976},[31676],[34,334978],{"className":334979,"style":32617},[31680],[34,334981,640],{"className":334982},[31685]," 可以写成 ",[34,334985,334987,335000],{"className":334986},[31642],[34,334988,334990],{"className":334989},[31646],[31648,334991,334992],{"xmlns":31650},[31652,334993,334994,334998],{},[31655,334995,334996],{},[31658,334997,98360],{},[31665,334999,98360],{"encoding":31667},[34,335001,335003],{"className":335002,"ariaHidden":758},[31672],[34,335004,335006,335009],{"className":335005},[31676],[34,335007],{"className":335008,"style":33218},[31680],[34,335010,98360],{"className":335011,"style":33389},[31685,31686]," 里的元素和 ",[34,335014,335016,335029],{"className":335015},[31642],[34,335017,335019],{"className":335018},[31646],[31648,335020,335021],{"xmlns":31650},[31652,335022,335023,335027],{},[31655,335024,335025],{},[31658,335026,111],{},[31665,335028,111],{"encoding":31667},[34,335030,335032],{"className":335031,"ariaHidden":758},[31672],[34,335033,335035,335038],{"className":335034},[31676],[34,335036],{"className":335037,"style":32527},[31680],[34,335039,111],{"className":335040},[31685,31686]," 的倍数之和。",[105,335043,335044],{},[108,335045,335046,335047,258,335098,335149,335150,347],{},"存在 ",[34,335048,335050,335068],{"className":335049},[31642],[34,335051,335053],{"className":335052},[31646],[31648,335054,335055],{"xmlns":31650},[31652,335056,335057,335065],{},[31655,335058,335059,335061,335063],{},[31658,335060,31759],{},[31662,335062,112296],{},[31658,335064,98360],{},[31665,335066,335067],{"encoding":31667},"m \\in M",[34,335069,335071,335089],{"className":335070,"ariaHidden":758},[31672],[34,335072,335074,335077,335080,335083,335086],{"className":335073},[31676],[34,335075],{"className":335076,"style":34812},[31680],[34,335078,31759],{"className":335079},[31685,31686],[34,335081],{"className":335082,"style":31846},[31845],[34,335084,112296],{"className":335085},[31850],[34,335087],{"className":335088,"style":31846},[31845],[34,335090,335092,335095],{"className":335091},[31676],[34,335093],{"className":335094,"style":33218},[31680],[34,335096,98360],{"className":335097,"style":33389},[31685,31686],[34,335099,335101,335119],{"className":335100},[31642],[34,335102,335104],{"className":335103},[31646],[31648,335105,335106],{"xmlns":31650},[31652,335107,335108,335116],{},[31655,335109,335110,335112,335114],{},[31658,335111,34395],{},[31662,335113,112296],{},[31658,335115,34386],{},[31665,335117,335118],{"encoding":31667},"r \\in R",[34,335120,335122,335140],{"className":335121,"ariaHidden":758},[31672],[34,335123,335125,335128,335131,335134,335137],{"className":335124},[31676],[34,335126],{"className":335127,"style":34812},[31680],[34,335129,34395],{"className":335130,"style":34452},[31685,31686],[34,335132],{"className":335133,"style":31846},[31845],[34,335135,112296],{"className":335136},[31850],[34,335138],{"className":335139,"style":31846},[31845],[34,335141,335143,335146],{"className":335142},[31676],[34,335144],{"className":335145,"style":33218},[31680],[34,335147,34386],{"className":335148,"style":34427},[31685,31686],"，使得 ",[34,335151,335153,335177],{"className":335152},[31642],[34,335154,335156],{"className":335155},[31646],[31648,335157,335158],{"xmlns":31650},[31652,335159,335160,335174],{},[31655,335161,335162,335164,335166,335168,335170,335172],{},[31749,335163,640],{},[31662,335165,202],{},[31658,335167,31759],{},[31662,335169,223],{},[31658,335171,34395],{},[31658,335173,111],{},[31665,335175,335176],{"encoding":31667},"1 = m + ra",[34,335178,335180,335198,335216],{"className":335179,"ariaHidden":758},[31672],[34,335181,335183,335186,335189,335192,335195],{"className":335182},[31676],[34,335184],{"className":335185,"style":32617},[31680],[34,335187,640],{"className":335188},[31685],[34,335190],{"className":335191,"style":31846},[31845],[34,335193,202],{"className":335194},[31850],[34,335196],{"className":335197,"style":31846},[31845],[34,335199,335201,335204,335207,335210,335213],{"className":335200},[31676],[34,335202],{"className":335203,"style":32597},[31680],[34,335205,31759],{"className":335206},[31685,31686],[34,335208],{"className":335209,"style":32604},[31845],[34,335211,223],{"className":335212},[32024],[34,335214],{"className":335215,"style":32604},[31845],[34,335217,335219,335222,335225],{"className":335218},[31676],[34,335220],{"className":335221,"style":32527},[31680],[34,335223,34395],{"className":335224,"style":34452},[31685,31686],[34,335226,111],{"className":335227},[31685,31686],[2230,335229,335230],{"start":68},[108,335231,335232,335235,335236,335264],{},[2627,335233,335234],{},"回到商环","：两边同时模 ",[34,335237,335239,335252],{"className":335238},[31642],[34,335240,335242],{"className":335241},[31646],[31648,335243,335244],{"xmlns":31650},[31652,335245,335246,335250],{},[31655,335247,335248],{},[31658,335249,98360],{},[31665,335251,98360],{"encoding":31667},[34,335253,335255],{"className":335254,"ariaHidden":758},[31672],[34,335256,335258,335261],{"className":335257},[31676],[34,335259],{"className":335260,"style":33218},[31680],[34,335262,98360],{"className":335263,"style":33389},[31685,31686],"（即在商环中看等式）：",[105,335266,335267,335401,335454],{},[108,335268,335269],{},[34,335270,335272,335308],{"className":335271},[31642],[34,335273,335275],{"className":335274},[31646],[31648,335276,335277],{"xmlns":31650},[31652,335278,335279,335305],{},[31655,335280,335281,335283,335285,335287,335289,335291,335293,335295,335297,335299,335301,335303],{},[31749,335282,640],{},[31662,335284,223],{},[31658,335286,98360],{},[31662,335288,202],{},[31662,335290,456],{"stretchy":702},[31658,335292,31759],{},[31662,335294,223],{},[31658,335296,34395],{},[31658,335298,111],{},[31662,335300,2557],{"stretchy":702},[31662,335302,223],{},[31658,335304,98360],{},[31665,335306,335307],{"encoding":31667},"1 + M = (m + ra) + M",[34,335309,335311,335329,335347,335368,335392],{"className":335310,"ariaHidden":758},[31672],[34,335312,335314,335317,335320,335323,335326],{"className":335313},[31676],[34,335315],{"className":335316,"style":35942},[31680],[34,335318,640],{"className":335319},[31685],[34,335321],{"className":335322,"style":32604},[31845],[34,335324,223],{"className":335325},[32024],[34,335327],{"className":335328,"style":32604},[31845],[34,335330,335332,335335,335338,335341,335344],{"className":335331},[31676],[34,335333],{"className":335334,"style":33218},[31680],[34,335336,98360],{"className":335337,"style":33389},[31685,31686],[34,335339],{"className":335340,"style":31846},[31845],[34,335342,202],{"className":335343},[31850],[34,335345],{"className":335346,"style":31846},[31845],[34,335348,335350,335353,335356,335359,335362,335365],{"className":335349},[31676],[34,335351],{"className":335352,"style":31774},[31680],[34,335354,456],{"className":335355},[31782],[34,335357,31759],{"className":335358},[31685,31686],[34,335360],{"className":335361,"style":32604},[31845],[34,335363,223],{"className":335364},[32024],[34,335366],{"className":335367,"style":32604},[31845],[34,335369,335371,335374,335377,335380,335383,335386,335389],{"className":335370},[31676],[34,335372],{"className":335373,"style":31774},[31680],[34,335375,34395],{"className":335376,"style":34452},[31685,31686],[34,335378,111],{"className":335379},[31685,31686],[34,335381,2557],{"className":335382},[31690],[34,335384],{"className":335385,"style":32604},[31845],[34,335387,223],{"className":335388},[32024],[34,335390],{"className":335391,"style":32604},[31845],[34,335393,335395,335398],{"className":335394},[31676],[34,335396],{"className":335397,"style":33218},[31680],[34,335399,98360],{"className":335400,"style":33389},[31685,31686],[108,335402,93310,335403,335453],{},[34,335404,335406,335423],{"className":335405},[31642],[34,335407,335409],{"className":335408},[31646],[31648,335410,335411],{"xmlns":31650},[31652,335412,335413,335421],{},[31655,335414,335415,335417,335419],{},[31658,335416,31759],{},[31662,335418,112296],{},[31658,335420,98360],{},[31665,335422,335067],{"encoding":31667},[34,335424,335426,335444],{"className":335425,"ariaHidden":758},[31672],[34,335427,335429,335432,335435,335438,335441],{"className":335428},[31676],[34,335430],{"className":335431,"style":34812},[31680],[34,335433,31759],{"className":335434},[31685,31686],[34,335436],{"className":335437,"style":31846},[31845],[34,335439,112296],{"className":335440},[31850],[34,335442],{"className":335443,"style":31846},[31845],[34,335445,335447,335450],{"className":335446},[31676],[34,335448],{"className":335449,"style":33218},[31680],[34,335451,98360],{"className":335452,"style":33389},[31685,31686],"，它在商环里是 0。",[108,335455,175992,335456,347],{},[34,335457,335459,335509],{"className":335458},[31642],[34,335460,335462],{"className":335461},[31646],[31648,335463,335464],{"xmlns":31650},[31652,335465,335466,335506],{},[31655,335467,335468,335470,335472,335474,335476,335478,335480,335482,335484,335486,335488,335490,335492,335494,335496,335498,335500,335502,335504],{},[31749,335469,640],{},[31662,335471,223],{},[31658,335473,98360],{},[31662,335475,202],{},[31658,335477,34395],{},[31658,335479,111],{},[31662,335481,223],{},[31658,335483,98360],{},[31662,335485,202],{},[31662,335487,456],{"stretchy":702},[31658,335489,34395],{},[31662,335491,223],{},[31658,335493,98360],{},[31662,335495,2557],{"stretchy":702},[31662,335497,456],{"stretchy":702},[31658,335499,111],{},[31662,335501,223],{},[31658,335503,98360],{},[31662,335505,2557],{"stretchy":702},[31665,335507,335508],{"encoding":31667},"1 + M = ra + M = (r+M)(a+M)",[34,335510,335512,335530,335548,335569,335587,335608,335635],{"className":335511,"ariaHidden":758},[31672],[34,335513,335515,335518,335521,335524,335527],{"className":335514},[31676],[34,335516],{"className":335517,"style":35942},[31680],[34,335519,640],{"className":335520},[31685],[34,335522],{"className":335523,"style":32604},[31845],[34,335525,223],{"className":335526},[32024],[34,335528],{"className":335529,"style":32604},[31845],[34,335531,335533,335536,335539,335542,335545],{"className":335532},[31676],[34,335534],{"className":335535,"style":33218},[31680],[34,335537,98360],{"className":335538,"style":33389},[31685,31686],[34,335540],{"className":335541,"style":31846},[31845],[34,335543,202],{"className":335544},[31850],[34,335546],{"className":335547,"style":31846},[31845],[34,335549,335551,335554,335557,335560,335563,335566],{"className":335550},[31676],[34,335552],{"className":335553,"style":32597},[31680],[34,335555,34395],{"className":335556,"style":34452},[31685,31686],[34,335558,111],{"className":335559},[31685,31686],[34,335561],{"className":335562,"style":32604},[31845],[34,335564,223],{"className":335565},[32024],[34,335567],{"className":335568,"style":32604},[31845],[34,335570,335572,335575,335578,335581,335584],{"className":335571},[31676],[34,335573],{"className":335574,"style":33218},[31680],[34,335576,98360],{"className":335577,"style":33389},[31685,31686],[34,335579],{"className":335580,"style":31846},[31845],[34,335582,202],{"className":335583},[31850],[34,335585],{"className":335586,"style":31846},[31845],[34,335588,335590,335593,335596,335599,335602,335605],{"className":335589},[31676],[34,335591],{"className":335592,"style":31774},[31680],[34,335594,456],{"className":335595},[31782],[34,335597,34395],{"className":335598,"style":34452},[31685,31686],[34,335600],{"className":335601,"style":32604},[31845],[34,335603,223],{"className":335604},[32024],[34,335606],{"className":335607,"style":32604},[31845],[34,335609,335611,335614,335617,335620,335623,335626,335629,335632],{"className":335610},[31676],[34,335612],{"className":335613,"style":31774},[31680],[34,335615,98360],{"className":335616,"style":33389},[31685,31686],[34,335618,2557],{"className":335619},[31690],[34,335621,456],{"className":335622},[31782],[34,335624,111],{"className":335625},[31685,31686],[34,335627],{"className":335628,"style":32604},[31845],[34,335630,223],{"className":335631},[32024],[34,335633],{"className":335634,"style":32604},[31845],[34,335636,335638,335641,335644],{"className":335637},[31676],[34,335639],{"className":335640,"style":31774},[31680],[34,335642,98360],{"className":335643,"style":33389},[31685,31686],[34,335645,2557],{"className":335646},[31690],[2230,335648,335649],{"start":74},[108,335650,335651,2664,335653,335703,335704,335755,335756,47025,335784,335834,335835,334082],{},[2627,335652,332755],{},[34,335654,335656,335673],{"className":335655},[31642],[34,335657,335659],{"className":335658},[31646],[31648,335660,335661],{"xmlns":31650},[31652,335662,335663,335671],{},[31655,335664,335665,335667,335669],{},[31658,335666,111],{},[31662,335668,223],{},[31658,335670,98360],{},[31665,335672,334255],{"encoding":31667},[34,335674,335676,335694],{"className":335675,"ariaHidden":758},[31672],[34,335677,335679,335682,335685,335688,335691],{"className":335678},[31676],[34,335680],{"className":335681,"style":32597},[31680],[34,335683,111],{"className":335684},[31685,31686],[34,335686],{"className":335687,"style":32604},[31845],[34,335689,223],{"className":335690},[32024],[34,335692],{"className":335693,"style":32604},[31845],[34,335695,335697,335700],{"className":335696},[31676],[34,335698],{"className":335699,"style":33218},[31680],[34,335701,98360],{"className":335702,"style":33389},[31685,31686]," 乘以 ",[34,335705,335707,335725],{"className":335706},[31642],[34,335708,335710],{"className":335709},[31646],[31648,335711,335712],{"xmlns":31650},[31652,335713,335714,335722],{},[31655,335715,335716,335718,335720],{},[31658,335717,34395],{},[31662,335719,223],{},[31658,335721,98360],{},[31665,335723,335724],{"encoding":31667},"r+M",[34,335726,335728,335746],{"className":335727,"ariaHidden":758},[31672],[34,335729,335731,335734,335737,335740,335743],{"className":335730},[31676],[34,335732],{"className":335733,"style":32597},[31680],[34,335735,34395],{"className":335736,"style":34452},[31685,31686],[34,335738],{"className":335739,"style":32604},[31845],[34,335741,223],{"className":335742},[32024],[34,335744],{"className":335745,"style":32604},[31845],[34,335747,335749,335752],{"className":335748},[31676],[34,335750],{"className":335751,"style":33218},[31680],[34,335753,98360],{"className":335754,"style":33389},[31685,31686]," 等于 ",[34,335757,335759,335772],{"className":335758},[31642],[34,335760,335762],{"className":335761},[31646],[31648,335763,335764],{"xmlns":31650},[31652,335765,335766,335770],{},[31655,335767,335768],{},[31749,335769,640],{},[31665,335771,640],{"encoding":31667},[34,335773,335775],{"className":335774,"ariaHidden":758},[31672],[34,335776,335778,335781],{"className":335777},[31676],[34,335779],{"className":335780,"style":32617},[31680],[34,335782,640],{"className":335783},[31685],[34,335785,335787,335804],{"className":335786},[31642],[34,335788,335790],{"className":335789},[31646],[31648,335791,335792],{"xmlns":31650},[31652,335793,335794,335802],{},[31655,335795,335796,335798,335800],{},[31658,335797,111],{},[31662,335799,223],{},[31658,335801,98360],{},[31665,335803,334255],{"encoding":31667},[34,335805,335807,335825],{"className":335806,"ariaHidden":758},[31672],[34,335808,335810,335813,335816,335819,335822],{"className":335809},[31676],[34,335811],{"className":335812,"style":32597},[31680],[34,335814,111],{"className":335815},[31685,31686],[34,335817],{"className":335818,"style":32604},[31845],[34,335820,223],{"className":335821},[32024],[34,335823],{"className":335824,"style":32604},[31845],[34,335826,335828,335831],{"className":335827},[31676],[34,335829],{"className":335830,"style":33218},[31680],[34,335832,98360],{"className":335833,"style":33389},[31685,31686]," 有逆元。所以 ",[34,335836,335838,335855],{"className":335837},[31642],[34,335839,335841],{"className":335840},[31646],[31648,335842,335843],{"xmlns":31650},[31652,335844,335845,335853],{},[31655,335846,335847,335849,335851],{},[31658,335848,34386],{},[31658,335850,38080],{"mathvariant":38079},[31658,335852,98360],{},[31665,335854,334063],{"encoding":31667},[34,335856,335858],{"className":335857,"ariaHidden":758},[31672],[34,335859,335861,335864,335867,335870],{"className":335860},[31676],[34,335862],{"className":335863,"style":31774},[31680],[34,335865,34386],{"className":335866,"style":34427},[31685,31686],[34,335868,38080],{"className":335869},[31685],[34,335871,98360],{"className":335872,"style":33389},[31685,31686],[15,335874,335875],{},[2627,335876,83913,335877,334118,335905,335943,335944,335972],{},[34,335878,335880,335893],{"className":335879},[31642],[34,335881,335883],{"className":335882},[31646],[31648,335884,335885],{"xmlns":31650},[31652,335886,335887,335891],{},[31655,335888,335889],{},[31662,335890,332814],{},[31665,335892,332817],{"encoding":31667},[34,335894,335896],{"className":335895,"ariaHidden":758},[31672],[34,335897,335899,335902],{"className":335898},[31676],[34,335900],{"className":335901,"style":73089},[31680],[34,335903,332814],{"className":335904},[31850],[34,335906,335908,335925],{"className":335907},[31642],[34,335909,335911],{"className":335910},[31646],[31648,335912,335913],{"xmlns":31650},[31652,335914,335915,335923],{},[31655,335916,335917,335919,335921],{},[31658,335918,34386],{},[31658,335920,38080],{"mathvariant":38079},[31658,335922,98360],{},[31665,335924,334063],{"encoding":31667},[34,335926,335928],{"className":335927,"ariaHidden":758},[31672],[34,335929,335931,335934,335937,335940],{"className":335930},[31676],[34,335932],{"className":335933,"style":31774},[31680],[34,335935,34386],{"className":335936,"style":34427},[31685,31686],[34,335938,38080],{"className":335939},[31685],[34,335941,98360],{"className":335942,"style":33389},[31685,31686]," 是域，求证 ",[34,335945,335947,335960],{"className":335946},[31642],[34,335948,335950],{"className":335949},[31646],[31648,335951,335952],{"xmlns":31650},[31652,335953,335954,335958],{},[31655,335955,335956],{},[31658,335957,98360],{},[31665,335959,98360],{"encoding":31667},[34,335961,335963],{"className":335962,"ariaHidden":758},[31672],[34,335964,335966,335969],{"className":335965},[31676],[34,335967],{"className":335968,"style":33218},[31680],[34,335970,98360],{"className":335971,"style":33389},[31685,31686]," 极大。",[2230,335974,335975,336166,336416],{},[108,335976,335977,335980,335981,336009,336010,93535,336082,347],{},[2627,335978,335979],{},"假设中间层","：假设有一个理想 ",[34,335982,335984,335997],{"className":335983},[31642],[34,335985,335987],{"className":335986},[31646],[31648,335988,335989],{"xmlns":31650},[31652,335990,335991,335995],{},[31655,335992,335993],{},[31658,335994,106063],{},[31665,335996,106063],{"encoding":31667},[34,335998,336000],{"className":335999,"ariaHidden":758},[31672],[34,336001,336003,336006],{"className":336002},[31676],[34,336004],{"className":336005,"style":33218},[31680],[34,336007,106063],{"className":336008,"style":67482},[31685,31686]," 夹在中间：",[34,336011,336013,336034],{"className":336012},[31642],[34,336014,336016],{"className":336015},[31646],[31648,336017,336018],{"xmlns":31650},[31652,336019,336020,336032],{},[31655,336021,336022,336024,336026,336028,336030],{},[31658,336023,98360],{},[31662,336025,131855],{},[31658,336027,106063],{},[31662,336029,131855],{},[31658,336031,34386],{},[31665,336033,333724],{"encoding":31667},[34,336035,336037,336055,336073],{"className":336036,"ariaHidden":758},[31672],[34,336038,336040,336043,336046,336049,336052],{"className":336039},[31676],[34,336041],{"className":336042,"style":195047},[31680],[34,336044,98360],{"className":336045,"style":33389},[31685,31686],[34,336047],{"className":336048,"style":31846},[31845],[34,336050,131855],{"className":336051},[31850],[34,336053],{"className":336054,"style":31846},[31845],[34,336056,336058,336061,336064,336067,336070],{"className":336057},[31676],[34,336059],{"className":336060,"style":195047},[31680],[34,336062,106063],{"className":336063,"style":67482},[31685,31686],[34,336065],{"className":336066,"style":31846},[31845],[34,336068,131855],{"className":336069},[31850],[34,336071],{"className":336072,"style":31846},[31845],[34,336074,336076,336079],{"className":336075},[31676],[34,336077],{"className":336078,"style":33218},[31680],[34,336080,34386],{"className":336081,"style":34427},[31685,31686],[34,336083,336085,336103],{"className":336084},[31642],[34,336086,336088],{"className":336087},[31646],[31648,336089,336090],{"xmlns":31650},[31652,336091,336092,336100],{},[31655,336093,336094,336096,336098],{},[31658,336095,98360],{},[31662,336097,54707],{"mathvariant":38079},[31658,336099,106063],{},[31665,336101,336102],{"encoding":31667},"M \\neq I",[34,336104,336106,336157],{"className":336105,"ariaHidden":758},[31672],[34,336107,336109,336112,336115,336118,336154],{"className":336108},[31676],[34,336110],{"className":336111,"style":43217},[31680],[34,336113,98360],{"className":336114,"style":33389},[31685,31686],[34,336116],{"className":336117,"style":31846},[31845],[34,336119,336121,336148,336151],{"className":336120},[31850],[34,336122,336124],{"className":336123},[31850],[34,336125,336127],{"className":336126},[31685,54774],[34,336128,336130],{"className":336129},[54778],[34,336131,336133,336136,336145],{"className":336132},[54782],[34,336134],{"className":336135,"style":43217},[31680],[34,336137,336139],{"className":336138},[54789],[34,336140,336142],{"className":336141},[31685],[34,336143,54796],{"className":336144},[31850],[34,336146],{"className":336147},[54800],[34,336149],{"className":336150},[31845,54804],[34,336152,202],{"className":336153},[31850],[34,336155],{"className":336156,"style":31846},[31845],[34,336158,336160,336163],{"className":336159},[31676],[34,336161],{"className":336162,"style":33218},[31680],[34,336164,106063],{"className":336165,"style":67482},[31685,31686],[108,336167,336168,332541,336171,336254,336255,336283,336284,336312,336313,336341,336342,31901],{},[2627,336169,336170],{},"取差值",[34,336172,336174,336191],{"className":336173},[31642],[34,336175,336177],{"className":336176},[31646],[31648,336178,336179],{"xmlns":31650},[31652,336180,336181,336189],{},[31655,336182,336183,336185,336187],{},[31658,336184,98360],{},[31662,336186,54707],{"mathvariant":38079},[31658,336188,106063],{},[31665,336190,336102],{"encoding":31667},[34,336192,336194,336245],{"className":336193,"ariaHidden":758},[31672],[34,336195,336197,336200,336203,336206,336242],{"className":336196},[31676],[34,336198],{"className":336199,"style":43217},[31680],[34,336201,98360],{"className":336202,"style":33389},[31685,31686],[34,336204],{"className":336205,"style":31846},[31845],[34,336207,336209,336236,336239],{"className":336208},[31850],[34,336210,336212],{"className":336211},[31850],[34,336213,336215],{"className":336214},[31685,54774],[34,336216,336218],{"className":336217},[54778],[34,336219,336221,336224,336233],{"className":336220},[54782],[34,336222],{"className":336223,"style":43217},[31680],[34,336225,336227],{"className":336226},[54789],[34,336228,336230],{"className":336229},[31685],[34,336231,54796],{"className":336232},[31850],[34,336234],{"className":336235},[54800],[34,336237],{"className":336238},[31845,54804],[34,336240,202],{"className":336241},[31850],[34,336243],{"className":336244,"style":31846},[31845],[34,336246,336248,336251],{"className":336247},[31676],[34,336249],{"className":336250,"style":33218},[31680],[34,336252,106063],{"className":336253,"style":67482},[31685,31686],"，必定存在一个元素 ",[34,336256,336258,336271],{"className":336257},[31642],[34,336259,336261],{"className":336260},[31646],[31648,336262,336263],{"xmlns":31650},[31652,336264,336265,336269],{},[31655,336266,336267],{},[31658,336268,111],{},[31665,336270,111],{"encoding":31667},[34,336272,336274],{"className":336273,"ariaHidden":758},[31672],[34,336275,336277,336280],{"className":336276},[31676],[34,336278],{"className":336279,"style":32527},[31680],[34,336281,111],{"className":336282},[31685,31686],"，它在 ",[34,336285,336287,336300],{"className":336286},[31642],[34,336288,336290],{"className":336289},[31646],[31648,336291,336292],{"xmlns":31650},[31652,336293,336294,336298],{},[31655,336295,336296],{},[31658,336297,106063],{},[31665,336299,106063],{"encoding":31667},[34,336301,336303],{"className":336302,"ariaHidden":758},[31672],[34,336304,336306,336309],{"className":336305},[31676],[34,336307],{"className":336308,"style":33218},[31680],[34,336310,106063],{"className":336311,"style":67482},[31685,31686]," 里但不在 ",[34,336314,336316,336329],{"className":336315},[31642],[34,336317,336319],{"className":336318},[31646],[31648,336320,336321],{"xmlns":31650},[31652,336322,336323,336327],{},[31655,336324,336325],{},[31658,336326,98360],{},[31665,336328,98360],{"encoding":31667},[34,336330,336332],{"className":336331,"ariaHidden":758},[31672],[34,336333,336335,336338],{"className":336334},[31676],[34,336336],{"className":336337,"style":33218},[31680],[34,336339,98360],{"className":336340,"style":33389},[31685,31686]," 里（",[34,336343,336345,336368],{"className":336344},[31642],[34,336346,336348],{"className":336347},[31646],[31648,336349,336350],{"xmlns":31650},[31652,336351,336352,336365],{},[31655,336353,336354,336356,336358,336360,336363],{},[31658,336355,111],{},[31662,336357,112296],{},[31658,336359,106063],{},[31662,336361,336362],{},"∖",[31658,336364,98360],{},[31665,336366,336367],{"encoding":31667},"a \\in I \\setminus M",[34,336369,336371,336389,336407],{"className":336370,"ariaHidden":758},[31672],[34,336372,336374,336377,336380,336383,336386],{"className":336373},[31676],[34,336375],{"className":336376,"style":34812},[31680],[34,336378,111],{"className":336379},[31685,31686],[34,336381],{"className":336382,"style":31846},[31845],[34,336384,112296],{"className":336385},[31850],[34,336387],{"className":336388,"style":31846},[31845],[34,336390,336392,336395,336398,336401,336404],{"className":336391},[31676],[34,336393],{"className":336394,"style":31774},[31680],[34,336396,106063],{"className":336397,"style":67482},[31685,31686],[34,336399],{"className":336400,"style":32604},[31845],[34,336402,336362],{"className":336403},[32024],[34,336405],{"className":336406,"style":32604},[31845],[34,336408,336410,336413],{"className":336409},[31676],[34,336411],{"className":336412,"style":33218},[31680],[34,336414,98360],{"className":336415,"style":33389},[31685,31686],[108,336417,336418,331829,336421,336459,336460,336510,336511,336549,336550,336600,336601,347],{},[2627,336419,336420],{},"利用域的性质",[34,336422,336424,336441],{"className":336423},[31642],[34,336425,336427],{"className":336426},[31646],[31648,336428,336429],{"xmlns":31650},[31652,336430,336431,336439],{},[31655,336432,336433,336435,336437],{},[31658,336434,34386],{},[31658,336436,38080],{"mathvariant":38079},[31658,336438,98360],{},[31665,336440,334063],{"encoding":31667},[34,336442,336444],{"className":336443,"ariaHidden":758},[31672],[34,336445,336447,336450,336453,336456],{"className":336446},[31676],[34,336448],{"className":336449,"style":31774},[31680],[34,336451,34386],{"className":336452,"style":34427},[31685,31686],[34,336454,38080],{"className":336455},[31685],[34,336457,98360],{"className":336458,"style":33389},[31685,31686]," 中，",[34,336461,336463,336480],{"className":336462},[31642],[34,336464,336466],{"className":336465},[31646],[31648,336467,336468],{"xmlns":31650},[31652,336469,336470,336478],{},[31655,336471,336472,336474,336476],{},[31658,336473,111],{},[31662,336475,223],{},[31658,336477,98360],{},[31665,336479,334255],{"encoding":31667},[34,336481,336483,336501],{"className":336482,"ariaHidden":758},[31672],[34,336484,336486,336489,336492,336495,336498],{"className":336485},[31676],[34,336487],{"className":336488,"style":32597},[31680],[34,336490,111],{"className":336491},[31685,31686],[34,336493],{"className":336494,"style":32604},[31845],[34,336496,223],{"className":336497},[32024],[34,336499],{"className":336500,"style":32604},[31845],[34,336502,336504,336507],{"className":336503},[31676],[34,336505],{"className":336506,"style":33218},[31680],[34,336508,98360],{"className":336509,"style":33389},[31685,31686]," 是非零的。因为 ",[34,336512,336514,336531],{"className":336513},[31642],[34,336515,336517],{"className":336516},[31646],[31648,336518,336519],{"xmlns":31650},[31652,336520,336521,336529],{},[31655,336522,336523,336525,336527],{},[31658,336524,34386],{},[31658,336526,38080],{"mathvariant":38079},[31658,336528,98360],{},[31665,336530,334063],{"encoding":31667},[34,336532,336534],{"className":336533,"ariaHidden":758},[31672],[34,336535,336537,336540,336543,336546],{"className":336536},[31676],[34,336538],{"className":336539,"style":31774},[31680],[34,336541,34386],{"className":336542,"style":34427},[31685,31686],[34,336544,38080],{"className":336545},[31685],[34,336547,98360],{"className":336548,"style":33389},[31685,31686]," 是域，所以 ",[34,336551,336553,336570],{"className":336552},[31642],[34,336554,336556],{"className":336555},[31646],[31648,336557,336558],{"xmlns":31650},[31652,336559,336560,336568],{},[31655,336561,336562,336564,336566],{},[31658,336563,111],{},[31662,336565,223],{},[31658,336567,98360],{},[31665,336569,334255],{"encoding":31667},[34,336571,336573,336591],{"className":336572,"ariaHidden":758},[31672],[34,336574,336576,336579,336582,336585,336588],{"className":336575},[31676],[34,336577],{"className":336578,"style":32597},[31680],[34,336580,111],{"className":336581},[31685,31686],[34,336583],{"className":336584,"style":32604},[31845],[34,336586,223],{"className":336587},[32024],[34,336589],{"className":336590,"style":32604},[31845],[34,336592,336594,336597],{"className":336593},[31676],[34,336595],{"className":336596,"style":33218},[31680],[34,336598,98360],{"className":336599,"style":33389},[31685,31686]," 有逆元，设为 ",[34,336602,336604,336622],{"className":336603},[31642],[34,336605,336607],{"className":336606},[31646],[31648,336608,336609],{"xmlns":31650},[31652,336610,336611,336619],{},[31655,336612,336613,336615,336617],{},[31658,336614,1697],{},[31662,336616,223],{},[31658,336618,98360],{},[31665,336620,336621],{"encoding":31667},"b+M",[34,336623,336625,336643],{"className":336624,"ariaHidden":758},[31672],[34,336626,336628,336631,336634,336637,336640],{"className":336627},[31676],[34,336629],{"className":336630,"style":34299},[31680],[34,336632,1697],{"className":336633},[31685,31686],[34,336635],{"className":336636,"style":32604},[31845],[34,336638,223],{"className":336639},[32024],[34,336641],{"className":336642,"style":32604},[31845],[34,336644,336646,336649],{"className":336645},[31676],[34,336647],{"className":336648,"style":33218},[31680],[34,336650,98360],{"className":336651,"style":33389},[31685,31686],[105,336653,336654],{},[108,336655,336656,336657,347],{},"即 ",[34,336658,336660,336718],{"className":336659},[31642],[34,336661,336663],{"className":336662},[31646],[31648,336664,336665],{"xmlns":31650},[31652,336666,336667,336715],{},[31655,336668,336669,336671,336673,336675,336677,336679,336681,336683,336685,336687,336689,336691,336693,336695,336697,336699,336701,336703,336705,336707,336709,336711,336713],{},[31662,336670,456],{"stretchy":702},[31658,336672,111],{},[31662,336674,223],{},[31658,336676,98360],{},[31662,336678,2557],{"stretchy":702},[31662,336680,456],{"stretchy":702},[31658,336682,1697],{},[31662,336684,223],{},[31658,336686,98360],{},[31662,336688,2557],{"stretchy":702},[31662,336690,202],{},[31749,336692,640],{},[31662,336694,223],{},[31658,336696,98360],{},[39361,336698,262622],{},[31662,336700,262625],{},[39361,336702,262622],{},[31658,336704,111],{},[31658,336706,1697],{},[31662,336708,32226],{},[31749,336710,640],{},[31662,336712,112296],{},[31658,336714,98360],{},[31665,336716,336717],{"encoding":31667},"(a+M)(b+M) = 1+M \\implies ab - 1 \\in M",[34,336719,336721,336742,336769,336790,336808,336833,336851,336869],{"className":336720,"ariaHidden":758},[31672],[34,336722,336724,336727,336730,336733,336736,336739],{"className":336723},[31676],[34,336725],{"className":336726,"style":31774},[31680],[34,336728,456],{"className":336729},[31782],[34,336731,111],{"className":336732},[31685,31686],[34,336734],{"className":336735,"style":32604},[31845],[34,336737,223],{"className":336738},[32024],[34,336740],{"className":336741,"style":32604},[31845],[34,336743,336745,336748,336751,336754,336757,336760,336763,336766],{"className":336744},[31676],[34,336746],{"className":336747,"style":31774},[31680],[34,336749,98360],{"className":336750,"style":33389},[31685,31686],[34,336752,2557],{"className":336753},[31690],[34,336755,456],{"className":336756},[31782],[34,336758,1697],{"className":336759},[31685,31686],[34,336761],{"className":336762,"style":32604},[31845],[34,336764,223],{"className":336765},[32024],[34,336767],{"className":336768,"style":32604},[31845],[34,336770,336772,336775,336778,336781,336784,336787],{"className":336771},[31676],[34,336773],{"className":336774,"style":31774},[31680],[34,336776,98360],{"className":336777,"style":33389},[31685,31686],[34,336779,2557],{"className":336780},[31690],[34,336782],{"className":336783,"style":31846},[31845],[34,336785,202],{"className":336786},[31850],[34,336788],{"className":336789,"style":31846},[31845],[34,336791,336793,336796,336799,336802,336805],{"className":336792},[31676],[34,336794],{"className":336795,"style":35942},[31680],[34,336797,640],{"className":336798},[31685],[34,336800],{"className":336801,"style":32604},[31845],[34,336803,223],{"className":336804},[32024],[34,336806],{"className":336807,"style":32604},[31845],[34,336809,336811,336815,336818,336821,336824,336827,336830],{"className":336810},[31676],[34,336812],{"className":336813,"style":336814},[31680],"height:0.7073em;vertical-align:-0.024em;",[34,336816,98360],{"className":336817,"style":33389},[31685,31686],[34,336819],{"className":336820,"style":31846},[31845],[34,336822],{"className":336823,"style":31846},[31845],[34,336825,262625],{"className":336826},[31850],[34,336828],{"className":336829,"style":31846},[31845],[34,336831],{"className":336832,"style":31846},[31845],[34,336834,336836,336839,336842,336845,336848],{"className":336835},[31676],[34,336837],{"className":336838,"style":34299},[31680],[34,336840,324191],{"className":336841},[31685,31686],[34,336843],{"className":336844,"style":32604},[31845],[34,336846,32226],{"className":336847},[32024],[34,336849],{"className":336850,"style":32604},[31845],[34,336852,336854,336857,336860,336863,336866],{"className":336853},[31676],[34,336855],{"className":336856,"style":61814},[31680],[34,336858,640],{"className":336859},[31685],[34,336861],{"className":336862,"style":31846},[31845],[34,336864,112296],{"className":336865},[31850],[34,336867],{"className":336868,"style":31846},[31845],[34,336870,336872,336875],{"className":336871},[31676],[34,336873],{"className":336874,"style":33218},[31680],[34,336876,98360],{"className":336877,"style":33389},[31685,31686],[2230,336879,336880],{"start":56},[108,336881,336882,2664],{},[2627,336883,33190],{},[105,336885,336886,337015],{},[108,336887,336888,336889,336964,336965,31901],{},"记 ",[34,336890,336892,336916],{"className":336891},[31642],[34,336893,336895],{"className":336894},[31646],[31648,336896,336897],{"xmlns":31650},[31652,336898,336899,336913],{},[31655,336900,336901,336903,336905,336907,336909,336911],{},[31658,336902,111],{},[31658,336904,1697],{},[31662,336906,32226],{},[31749,336908,640],{},[31662,336910,202],{},[31658,336912,31759],{},[31665,336914,336915],{"encoding":31667},"ab - 1 = m",[34,336917,336919,336937,336955],{"className":336918,"ariaHidden":758},[31672],[34,336920,336922,336925,336928,336931,336934],{"className":336921},[31676],[34,336923],{"className":336924,"style":34299},[31680],[34,336926,324191],{"className":336927},[31685,31686],[34,336929],{"className":336930,"style":32604},[31845],[34,336932,32226],{"className":336933},[32024],[34,336935],{"className":336936,"style":32604},[31845],[34,336938,336940,336943,336946,336949,336952],{"className":336939},[31676],[34,336941],{"className":336942,"style":32617},[31680],[34,336944,640],{"className":336945},[31685],[34,336947],{"className":336948,"style":31846},[31845],[34,336950,202],{"className":336951},[31850],[34,336953],{"className":336954,"style":31846},[31845],[34,336956,336958,336961],{"className":336957},[31676],[34,336959],{"className":336960,"style":32527},[31680],[34,336962,31759],{"className":336963},[31685,31686],"（其中 ",[34,336966,336968,336985],{"className":336967},[31642],[34,336969,336971],{"className":336970},[31646],[31648,336972,336973],{"xmlns":31650},[31652,336974,336975,336983],{},[31655,336976,336977,336979,336981],{},[31658,336978,31759],{},[31662,336980,112296],{},[31658,336982,98360],{},[31665,336984,335067],{"encoding":31667},[34,336986,336988,337006],{"className":336987,"ariaHidden":758},[31672],[34,336989,336991,336994,336997,337000,337003],{"className":336990},[31676],[34,336992],{"className":336993,"style":34812},[31680],[34,336995,31759],{"className":336996},[31685,31686],[34,336998],{"className":336999,"style":31846},[31845],[34,337001,112296],{"className":337002},[31850],[34,337004],{"className":337005,"style":31846},[31845],[34,337007,337009,337012],{"className":337008},[31676],[34,337010],{"className":337011,"style":33218},[31680],[34,337013,98360],{"className":337014,"style":33389},[31685,31686],[108,337016,337017,337018,347],{},"移项得：",[34,337019,337021,337045],{"className":337020},[31642],[34,337022,337024],{"className":337023},[31646],[31648,337025,337026],{"xmlns":31650},[31652,337027,337028,337042],{},[31655,337029,337030,337032,337034,337036,337038,337040],{},[31749,337031,640],{},[31662,337033,202],{},[31658,337035,111],{},[31658,337037,1697],{},[31662,337039,32226],{},[31658,337041,31759],{},[31665,337043,337044],{"encoding":31667},"1 = ab - m",[34,337046,337048,337066,337084],{"className":337047,"ariaHidden":758},[31672],[34,337049,337051,337054,337057,337060,337063],{"className":337050},[31676],[34,337052],{"className":337053,"style":32617},[31680],[34,337055,640],{"className":337056},[31685],[34,337058],{"className":337059,"style":31846},[31845],[34,337061,202],{"className":337062},[31850],[34,337064],{"className":337065,"style":31846},[31845],[34,337067,337069,337072,337075,337078,337081],{"className":337068},[31676],[34,337070],{"className":337071,"style":34299},[31680],[34,337073,324191],{"className":337074},[31685,31686],[34,337076],{"className":337077,"style":32604},[31845],[34,337079,32226],{"className":337080},[32024],[34,337082],{"className":337083,"style":32604},[31845],[34,337085,337087,337090],{"className":337086},[31676],[34,337088],{"className":337089,"style":32527},[31680],[34,337091,31759],{"className":337092},[31685,31686],[2230,337094,337095],{"start":62},[108,337096,337097,2664],{},[2627,337098,337099],{},"判断归属",[105,337101,337102,337212,337341,337456],{},[108,337103,337104,337211],{},[34,337105,337107,337139],{"className":337106},[31642],[34,337108,337110],{"className":337109},[31646],[31648,337111,337112],{"xmlns":31650},[31652,337113,337114,337136],{},[31655,337115,337116,337118,337120,337122,337124,337126,337128,337130,337132,337134],{},[31658,337117,111],{},[31662,337119,112296],{},[31658,337121,106063],{},[39361,337123,262622],{},[31662,337125,262625],{},[39361,337127,262622],{},[31658,337129,111],{},[31658,337131,1697],{},[31662,337133,112296],{},[31658,337135,106063],{},[31665,337137,337138],{"encoding":31667},"a \\in I \\implies ab \\in I",[34,337140,337142,337160,337184,337202],{"className":337141,"ariaHidden":758},[31672],[34,337143,337145,337148,337151,337154,337157],{"className":337144},[31676],[34,337146],{"className":337147,"style":34812},[31680],[34,337149,111],{"className":337150},[31685,31686],[34,337152],{"className":337153,"style":31846},[31845],[34,337155,112296],{"className":337156},[31850],[34,337158],{"className":337159,"style":31846},[31845],[34,337161,337163,337166,337169,337172,337175,337178,337181],{"className":337162},[31676],[34,337164],{"className":337165,"style":336814},[31680],[34,337167,106063],{"className":337168,"style":67482},[31685,31686],[34,337170],{"className":337171,"style":31846},[31845],[34,337173],{"className":337174,"style":31846},[31845],[34,337176,262625],{"className":337177},[31850],[34,337179],{"className":337180,"style":31846},[31845],[34,337182],{"className":337183,"style":31846},[31845],[34,337185,337187,337190,337193,337196,337199],{"className":337186},[31676],[34,337188],{"className":337189,"style":46991},[31680],[34,337191,324191],{"className":337192},[31685,31686],[34,337194],{"className":337195,"style":31846},[31845],[34,337197,112296],{"className":337198},[31850],[34,337200],{"className":337201,"style":31846},[31845],[34,337203,337205,337208],{"className":337204},[31676],[34,337206],{"className":337207,"style":33218},[31680],[34,337209,106063],{"className":337210,"style":67482},[31685,31686],"（理想吸收律）。",[108,337213,337214,347],{},[34,337215,337217,337251],{"className":337216},[31642],[34,337218,337220],{"className":337219},[31646],[31648,337221,337222],{"xmlns":31650},[31652,337223,337224,337248],{},[31655,337225,337226,337228,337230,337232,337234,337236,337238,337240,337242,337244,337246],{},[31658,337227,31759],{},[31662,337229,112296],{},[31658,337231,98360],{},[31662,337233,115762],{},[31658,337235,106063],{},[39361,337237,262622],{},[31662,337239,262625],{},[39361,337241,262622],{},[31658,337243,31759],{},[31662,337245,112296],{},[31658,337247,106063],{},[31665,337249,337250],{"encoding":31667},"m \\in M \\subset I \\implies m \\in I",[34,337252,337254,337272,337290,337314,337332],{"className":337253,"ariaHidden":758},[31672],[34,337255,337257,337260,337263,337266,337269],{"className":337256},[31676],[34,337258],{"className":337259,"style":34812},[31680],[34,337261,31759],{"className":337262},[31685,31686],[34,337264],{"className":337265,"style":31846},[31845],[34,337267,112296],{"className":337268},[31850],[34,337270],{"className":337271,"style":31846},[31845],[34,337273,337275,337278,337281,337284,337287],{"className":337274},[31676],[34,337276],{"className":337277,"style":104901},[31680],[34,337279,98360],{"className":337280,"style":33389},[31685,31686],[34,337282],{"className":337283,"style":31846},[31845],[34,337285,115762],{"className":337286},[31850],[34,337288],{"className":337289,"style":31846},[31845],[34,337291,337293,337296,337299,337302,337305,337308,337311],{"className":337292},[31676],[34,337294],{"className":337295,"style":336814},[31680],[34,337297,106063],{"className":337298,"style":67482},[31685,31686],[34,337300],{"className":337301,"style":31846},[31845],[34,337303],{"className":337304,"style":31846},[31845],[34,337306,262625],{"className":337307},[31850],[34,337309],{"className":337310,"style":31846},[31845],[34,337312],{"className":337313,"style":31846},[31845],[34,337315,337317,337320,337323,337326,337329],{"className":337316},[31676],[34,337318],{"className":337319,"style":34812},[31680],[34,337321,31759],{"className":337322},[31685,31686],[34,337324],{"className":337325,"style":31846},[31845],[34,337327,112296],{"className":337328},[31850],[34,337330],{"className":337331,"style":31846},[31845],[34,337333,337335,337338],{"className":337334},[31676],[34,337336],{"className":337337,"style":33218},[31680],[34,337339,106063],{"className":337340,"style":67482},[31685,31686],[108,337342,337343,337344,337372,337373,337426,337427,337455],{},"两个都在 ",[34,337345,337347,337360],{"className":337346},[31642],[34,337348,337350],{"className":337349},[31646],[31648,337351,337352],{"xmlns":31650},[31652,337353,337354,337358],{},[31655,337355,337356],{},[31658,337357,106063],{},[31665,337359,106063],{"encoding":31667},[34,337361,337363],{"className":337362,"ariaHidden":758},[31672],[34,337364,337366,337369],{"className":337365},[31676],[34,337367],{"className":337368,"style":33218},[31680],[34,337370,106063],{"className":337371,"style":67482},[31685,31686]," 里，它们的差 ",[34,337374,337376,337396],{"className":337375},[31642],[34,337377,337379],{"className":337378},[31646],[31648,337380,337381],{"xmlns":31650},[31652,337382,337383,337393],{},[31655,337384,337385,337387,337389,337391],{},[31658,337386,111],{},[31658,337388,1697],{},[31662,337390,32226],{},[31658,337392,31759],{},[31665,337394,337395],{"encoding":31667},"ab - m",[34,337397,337399,337417],{"className":337398,"ariaHidden":758},[31672],[34,337400,337402,337405,337408,337411,337414],{"className":337401},[31676],[34,337403],{"className":337404,"style":34299},[31680],[34,337406,324191],{"className":337407},[31685,31686],[34,337409],{"className":337410,"style":32604},[31845],[34,337412,32226],{"className":337413},[32024],[34,337415],{"className":337416,"style":32604},[31845],[34,337418,337420,337423],{"className":337419},[31676],[34,337421],{"className":337422,"style":32527},[31680],[34,337424,31759],{"className":337425},[31685,31686]," 也在 ",[34,337428,337430,337443],{"className":337429},[31642],[34,337431,337433],{"className":337432},[31646],[31648,337434,337435],{"xmlns":31650},[31652,337436,337437,337441],{},[31655,337438,337439],{},[31658,337440,106063],{},[31665,337442,106063],{"encoding":31667},[34,337444,337446],{"className":337445,"ariaHidden":758},[31672],[34,337447,337449,337452],{"className":337448},[31676],[34,337450],{"className":337451,"style":33218},[31680],[34,337453,106063],{"className":337454,"style":67482},[31685,31686]," 里。",[108,337457,175992,337458,347],{},[34,337459,337461,337479],{"className":337460},[31642],[34,337462,337464],{"className":337463},[31646],[31648,337465,337466],{"xmlns":31650},[31652,337467,337468,337476],{},[31655,337469,337470,337472,337474],{},[31749,337471,640],{},[31662,337473,112296],{},[31658,337475,106063],{},[31665,337477,337478],{"encoding":31667},"1 \\in I",[34,337480,337482,337500],{"className":337481,"ariaHidden":758},[31672],[34,337483,337485,337488,337491,337494,337497],{"className":337484},[31676],[34,337486],{"className":337487,"style":61814},[31680],[34,337489,640],{"className":337490},[31685],[34,337492],{"className":337493,"style":31846},[31845],[34,337495,112296],{"className":337496},[31850],[34,337498],{"className":337499,"style":31846},[31845],[34,337501,337503,337506],{"className":337502},[31676],[34,337504],{"className":337505,"style":33218},[31680],[34,337507,106063],{"className":337508,"style":67482},[31685,31686],[2230,337510,337511,337624],{"start":68},[108,337512,337513,337515,337516,337544,337545,330585,337573,347],{},[2627,337514,114043],{},"：如果理想 ",[34,337517,337519,337532],{"className":337518},[31642],[34,337520,337522],{"className":337521},[31646],[31648,337523,337524],{"xmlns":31650},[31652,337525,337526,337530],{},[31655,337527,337528],{},[31658,337529,106063],{},[31665,337531,106063],{"encoding":31667},[34,337533,337535],{"className":337534,"ariaHidden":758},[31672],[34,337536,337538,337541],{"className":337537},[31676],[34,337539],{"className":337540,"style":33218},[31680],[34,337542,106063],{"className":337543,"style":67482},[31685,31686]," 含有 ",[34,337546,337548,337561],{"className":337547},[31642],[34,337549,337551],{"className":337550},[31646],[31648,337552,337553],{"xmlns":31650},[31652,337554,337555,337559],{},[31655,337556,337557],{},[31749,337558,640],{},[31665,337560,640],{"encoding":31667},[34,337562,337564],{"className":337563,"ariaHidden":758},[31672],[34,337565,337567,337570],{"className":337566},[31676],[34,337568],{"className":337569,"style":32617},[31680],[34,337571,640],{"className":337572},[31685],[34,337574,337576,337594],{"className":337575},[31642],[34,337577,337579],{"className":337578},[31646],[31648,337580,337581],{"xmlns":31650},[31652,337582,337583,337591],{},[31655,337584,337585,337587,337589],{},[31658,337586,106063],{},[31662,337588,202],{},[31658,337590,34386],{},[31665,337592,337593],{"encoding":31667},"I = R",[34,337595,337597,337615],{"className":337596,"ariaHidden":758},[31672],[34,337598,337600,337603,337606,337609,337612],{"className":337599},[31676],[34,337601],{"className":337602,"style":33218},[31680],[34,337604,106063],{"className":337605,"style":67482},[31685,31686],[34,337607],{"className":337608,"style":31846},[31845],[34,337610,202],{"className":337611},[31850],[34,337613],{"className":337614,"style":31846},[31845],[34,337616,337618,337621],{"className":337617},[31676],[34,337619],{"className":337620,"style":33218},[31680],[34,337622,34386],{"className":337623,"style":34427},[31685,31686],[108,337625,337626,337628,337629,326053,337657,13977,337685,337713],{},[2627,337627,332755],{},"：夹在中间的理想 ",[34,337630,337632,337645],{"className":337631},[31642],[34,337633,337635],{"className":337634},[31646],[31648,337636,337637],{"xmlns":31650},[31652,337638,337639,337643],{},[31655,337640,337641],{},[31658,337642,106063],{},[31665,337644,106063],{"encoding":31667},[34,337646,337648],{"className":337647,"ariaHidden":758},[31672],[34,337649,337651,337654],{"className":337650},[31676],[34,337652],{"className":337653,"style":33218},[31680],[34,337655,106063],{"className":337656,"style":67482},[31685,31686],[34,337658,337660,337673],{"className":337659},[31642],[34,337661,337663],{"className":337662},[31646],[31648,337664,337665],{"xmlns":31650},[31652,337666,337667,337671],{},[31655,337668,337669],{},[31658,337670,34386],{},[31665,337672,34386],{"encoding":31667},[34,337674,337676],{"className":337675,"ariaHidden":758},[31672],[34,337677,337679,337682],{"className":337678},[31676],[34,337680],{"className":337681,"style":33218},[31680],[34,337683,34386],{"className":337684,"style":34427},[31685,31686],[34,337686,337688,337701],{"className":337687},[31642],[34,337689,337691],{"className":337690},[31646],[31648,337692,337693],{"xmlns":31650},[31652,337694,337695,337699],{},[31655,337696,337697],{},[31658,337698,98360],{},[31665,337700,98360],{"encoding":31667},[34,337702,337704],{"className":337703,"ariaHidden":758},[31672],[34,337705,337707,337710],{"className":337706},[31676],[34,337708],{"className":337709,"style":33218},[31680],[34,337711,98360],{"className":337712,"style":33389},[31685,31686]," 是极大的。",[32188,337715],{},[19,337717,337719],{"id":337718},"_3-两者关系与重要推论","3. 两者关系与重要推论",[105,337721,337722,337815],{},[108,337723,337724,337727,337728,337771,337772,337814],{},[2627,337725,337726],{},"逻辑链","：\n因为",[2627,337729,337730,337731,337770],{},"域 ",[34,337732,337734,337752],{"className":337733},[31642],[34,337735,337737],{"className":337736},[31646],[31648,337738,337739],{"xmlns":31650},[31652,337740,337741,337749],{},[31655,337742,337743,337745,337747],{},[39361,337744,262622],{},[31662,337746,262625],{},[39361,337748,262622],{},[31665,337750,337751],{"encoding":31667},"\\implies",[34,337753,337755],{"className":337754,"ariaHidden":758},[31672],[34,337756,337758,337761,337764,337767],{"className":337757},[31676],[34,337759],{"className":337760,"style":318264},[31680],[34,337762],{"className":337763,"style":31846},[31845],[34,337765,262625],{"className":337766},[31850],[34,337768],{"className":337769,"style":31846},[31845]," 整环","，所以",[2627,337773,337774,337775,337813],{},"极大理想 ",[34,337776,337778,337795],{"className":337777},[31642],[34,337779,337781],{"className":337780},[31646],[31648,337782,337783],{"xmlns":31650},[31652,337784,337785,337793],{},[31655,337786,337787,337789,337791],{},[39361,337788,262622],{},[31662,337790,262625],{},[39361,337792,262622],{},[31665,337794,337751],{"encoding":31667},[34,337796,337798],{"className":337797,"ariaHidden":758},[31672],[34,337799,337801,337804,337807,337810],{"className":337800},[31676],[34,337802],{"className":337803,"style":318264},[31680],[34,337805],{"className":337806,"style":31846},[31845],[34,337808,262625],{"className":337809},[31850],[34,337811],{"className":337812,"style":31846},[31845]," 素理想","。\n（口诀：极大的肯定素，素的不一定大。）",[108,337816,337817,33183,337820],{},[2627,337818,337819],{},"存在性（Zorn 引理）",[105,337821,337822,337828],{},[108,337823,337824,337827],{},[2627,337825,337826],{},"推论 3.5.4","：任何带单位元的交换环，一定至少有一个极大理想。",[108,337829,337830],{},"证明用到了佐恩引理，核心思想是\"不断把理想变大，直到撞到天花板\"。",[32188,337832],{},[19,337834,337836],{"id":337835},"_4-例子","4. 例子",[13837,337838,337839,337949],{},[13840,337840,337841],{},[13843,337842,337843,337874,337905,337947],{},[13846,337844,337845,337846],{},"环 ",[34,337847,337849,337862],{"className":337848},[31642],[34,337850,337852],{"className":337851},[31646],[31648,337853,337854],{"xmlns":31650},[31652,337855,337856,337860],{},[31655,337857,337858],{},[31658,337859,34386],{},[31665,337861,34386],{"encoding":31667},[34,337863,337865],{"className":337864,"ariaHidden":758},[31672],[34,337866,337868,337871],{"className":337867},[31676],[34,337869],{"className":337870,"style":33218},[31680],[34,337872,34386],{"className":337873,"style":34427},[31685,31686],[13846,337875,337876,337877],{},"理想 ",[34,337878,337880,337893],{"className":337879},[31642],[34,337881,337883],{"className":337882},[31646],[31648,337884,337885],{"xmlns":31650},[31652,337886,337887,337891],{},[31655,337888,337889],{},[31658,337890,106063],{},[31665,337892,106063],{"encoding":31667},[34,337894,337896],{"className":337895,"ariaHidden":758},[31672],[34,337897,337899,337902],{"className":337898},[31676],[34,337900],{"className":337901,"style":33218},[31680],[34,337903,106063],{"className":337904,"style":67482},[31685,31686],[13846,337906,337907,337946],{},[34,337908,337910,337928],{"className":337909},[31642],[34,337911,337913],{"className":337912},[31646],[31648,337914,337915],{"xmlns":31650},[31652,337916,337917,337925],{},[31655,337918,337919,337921,337923],{},[31658,337920,34386],{},[31658,337922,38080],{"mathvariant":38079},[31658,337924,106063],{},[31665,337926,337927],{"encoding":31667},"R\u002FI",[34,337929,337931],{"className":337930,"ariaHidden":758},[31672],[34,337932,337934,337937,337940,337943],{"className":337933},[31676],[34,337935],{"className":337936,"style":31774},[31680],[34,337938,34386],{"className":337939,"style":34427},[31685,31686],[34,337941,38080],{"className":337942},[31685],[34,337944,106063],{"className":337945,"style":67482},[31685,31686]," 是什么？",[13846,337948,114043],{},[13853,337950,337951,338162,338309,338593],{},[13843,337952,337953,337986,338052,338126],{},[13858,337954,337955],{},[2627,337956,337957,337958],{},"整数环 ",[34,337959,337961,337974],{"className":337960},[31642],[34,337962,337964],{"className":337963},[31646],[31648,337965,337966],{"xmlns":31650},[31652,337967,337968,337972],{},[31655,337969,337970],{},[31658,337971,97300],{"mathvariant":318416},[31665,337973,320917],{"encoding":31667},[34,337975,337977],{"className":337976,"ariaHidden":758},[31672],[34,337978,337980,337983],{"className":337979},[31676],[34,337981],{"className":337982,"style":320927},[31680],[34,337984,97300],{"className":337985},[31685,318441],[13858,337987,337988,83913,338023,338051],{},[34,337989,337991,338007],{"className":337990},[31642],[34,337992,337994],{"className":337993},[31646],[31648,337995,337996],{"xmlns":31650},[31652,337997,337998,338004],{},[31655,337999,338000,338002],{},[31658,338001,15],{},[31658,338003,97300],{"mathvariant":318416},[31665,338005,338006],{"encoding":31667},"p\\mathbb{Z}",[34,338008,338010],{"className":338009,"ariaHidden":758},[31672],[34,338011,338013,338017,338020],{"className":338012},[31676],[34,338014],{"className":338015,"style":338016},[31680],"height:0.8833em;vertical-align:-0.1944em;",[34,338018,15],{"className":338019},[31685,31686],[34,338021,97300],{"className":338022},[31685,318441],[34,338024,338026,338039],{"className":338025},[31642],[34,338027,338029],{"className":338028},[31646],[31648,338030,338031],{"xmlns":31650},[31652,338032,338033,338037],{},[31655,338034,338035],{},[31658,338036,15],{},[31665,338038,15],{"encoding":31667},[34,338040,338042],{"className":338041,"ariaHidden":758},[31672],[34,338043,338045,338048],{"className":338044},[31676],[34,338046],{"className":338047,"style":38774},[31680],[34,338049,15],{"className":338050},[31685,31686]," 是素数）",[13858,338053,338054,338125],{},[34,338055,338057,338075],{"className":338056},[31642],[34,338058,338060],{"className":338059},[31646],[31648,338061,338062],{"xmlns":31650},[31652,338063,338064,338072],{},[31655,338065,338066],{},[31743,338067,338068,338070],{},[31658,338069,97300],{"mathvariant":318416},[31658,338071,15],{},[31665,338073,338074],{"encoding":31667},"\\mathbb{Z}_p",[34,338076,338078],{"className":338077,"ariaHidden":758},[31672],[34,338079,338081,338085],{"className":338080},[31676],[34,338082],{"className":338083,"style":338084},[31680],"height:0.975em;vertical-align:-0.2861em;",[34,338086,338088,338091],{"className":338087},[31685],[34,338089,97300],{"className":338090},[31685,318441],[34,338092,338094],{"className":338093},[31792],[34,338095,338097,338117],{"className":338096},[31796,31797],[34,338098,338100,338114],{"className":338099},[31801],[34,338101,338103],{"className":338102,"style":32362},[31805],[34,338104,338105,338108],{"style":31809},[34,338106],{"className":338107,"style":31814},[31813],[34,338109,338111],{"className":338110},[31818,31819,31820,31821],[34,338112,15],{"className":338113},[31685,31686,31821],[34,338115,31829],{"className":338116},[31828],[34,338118,338120],{"className":338119},[31801],[34,338121,338123],{"className":338122,"style":32163},[31805],[34,338124],{},"（有限域）",[13858,338127,338128,338161],{},[34,338129,338131,338146],{"className":338130},[31642],[34,338132,338134],{"className":338133},[31646],[31648,338135,338136],{"xmlns":31650},[31652,338137,338138,338144],{},[31655,338139,338140,338142],{},[31658,338141,15],{},[31658,338143,97300],{"mathvariant":318416},[31665,338145,338006],{"encoding":31667},[34,338147,338149],{"className":338148,"ariaHidden":758},[31672],[34,338150,338152,338155,338158],{"className":338151},[31676],[34,338153],{"className":338154,"style":338016},[31680],[34,338156,15],{"className":338157},[31685,31686],[34,338159,97300],{"className":338160},[31685,318441]," 是极大理想（也是素理想）",[13843,338163,338164,338196,338237,338268],{},[13858,338165,338166],{},[2627,338167,337957,338168],{},[34,338169,338171,338184],{"className":338170},[31642],[34,338172,338174],{"className":338173},[31646],[31648,338175,338176],{"xmlns":31650},[31652,338177,338178,338182],{},[31655,338179,338180],{},[31658,338181,97300],{"mathvariant":318416},[31665,338183,320917],{"encoding":31667},[34,338185,338187],{"className":338186,"ariaHidden":758},[31672],[34,338188,338190,338193],{"className":338189},[31676],[34,338191],{"className":338192,"style":320927},[31680],[34,338194,97300],{"className":338195},[31685,318441],[13858,338197,338198],{},[34,338199,338201,338219],{"className":338200},[31642],[34,338202,338204],{"className":338203},[31646],[31648,338205,338206],{"xmlns":31650},[31652,338207,338208,338216],{},[31655,338209,338210,338212,338214],{},[31662,338211,32685],{"stretchy":702},[31749,338213,459],{},[31662,338215,32715],{"stretchy":702},[31665,338217,338218],{"encoding":31667},"\\{0\\}",[34,338220,338222],{"className":338221,"ariaHidden":758},[31672],[34,338223,338225,338228,338231,338234],{"className":338224},[31676],[34,338226],{"className":338227,"style":31774},[31680],[34,338229,32685],{"className":338230},[31782],[34,338232,459],{"className":338233},[31685],[34,338235,32715],{"className":338236},[31690],[13858,338238,338239,338267],{},[34,338240,338242,338255],{"className":338241},[31642],[34,338243,338245],{"className":338244},[31646],[31648,338246,338247],{"xmlns":31650},[31652,338248,338249,338253],{},[31655,338250,338251],{},[31658,338252,97300],{"mathvariant":318416},[31665,338254,320917],{"encoding":31667},[34,338256,338258],{"className":338257,"ariaHidden":758},[31672],[34,338259,338261,338264],{"className":338260},[31676],[34,338262],{"className":338263,"style":320927},[31680],[34,338265,97300],{"className":338266},[31685,318441],"（是整环但不是域）",[13858,338269,338270,338308],{},[34,338271,338273,338290],{"className":338272},[31642],[34,338274,338276],{"className":338275},[31646],[31648,338277,338278],{"xmlns":31650},[31652,338279,338280,338288],{},[31655,338281,338282,338284,338286],{},[31662,338283,32685],{"stretchy":702},[31749,338285,459],{},[31662,338287,32715],{"stretchy":702},[31665,338289,338218],{"encoding":31667},[34,338291,338293],{"className":338292,"ariaHidden":758},[31672],[34,338294,338296,338299,338302,338305],{"className":338295},[31676],[34,338297],{"className":338298,"style":31774},[31680],[34,338300,32685],{"className":338301},[31782],[34,338303,459],{"className":338304},[31685],[34,338306,32715],{"className":338307},[31690]," 是素理想，但不是极大理想",[13843,338310,338311,338360,338453,338500],{},[13858,338312,338313],{},[2627,338314,338315,338316],{},"实多项式 ",[34,338317,338319,338339],{"className":338318},[31642],[34,338320,338322],{"className":338321},[31646],[31648,338323,338324],{"xmlns":31650},[31652,338325,338326,338336],{},[31655,338327,338328,338330,338332,338334],{},[31658,338329,34386],{"mathvariant":318416},[31662,338331,15972],{"stretchy":702},[31658,338333,32936],{},[31662,338335,32712],{"stretchy":702},[31665,338337,338338],{"encoding":31667},"\\mathbb{R}[x]",[34,338340,338342],{"className":338341,"ariaHidden":758},[31672],[34,338343,338345,338348,338351,338354,338357],{"className":338344},[31676],[34,338346],{"className":338347,"style":31774},[31680],[34,338349,34386],{"className":338350},[31685,318441],[34,338352,15972],{"className":338353},[31782],[34,338355,32936],{"className":338356},[31685,31686],[34,338358,32712],{"className":338359},[31690],[13858,338361,338362],{},[34,338363,338365,338391],{"className":338364},[31642],[34,338366,338368],{"className":338367},[31646],[31648,338369,338370],{"xmlns":31650},[31652,338371,338372,338388],{},[31655,338373,338374,338376,338382,338384,338386],{},[31662,338375,126529],{"stretchy":702},[37624,338377,338378,338380],{},[31658,338379,32936],{},[31749,338381,768],{},[31662,338383,223],{},[31749,338385,640],{},[31662,338387,126544],{"stretchy":702},[31665,338389,338390],{"encoding":31667},"\\langle x^2+1 \\rangle",[34,338392,338394,338441],{"className":338393,"ariaHidden":758},[31672],[34,338395,338397,338400,338403,338432,338435,338438],{"className":338396},[31676],[34,338398],{"className":338399,"style":37755},[31680],[34,338401,126529],{"className":338402},[31782],[34,338404,338406,338409],{"className":338405},[31685],[34,338407,32936],{"className":338408},[31685,31686],[34,338410,338412],{"className":338411},[31792],[34,338413,338415],{"className":338414},[31796],[34,338416,338418],{"className":338417},[31801],[34,338419,338421],{"className":338420,"style":37774},[31805],[34,338422,338423,338426],{"style":37777},[34,338424],{"className":338425,"style":31814},[31813],[34,338427,338429],{"className":338428},[31818,31819,31820,31821],[34,338430,768],{"className":338431},[31685,31821],[34,338433],{"className":338434,"style":32604},[31845],[34,338436,223],{"className":338437},[32024],[34,338439],{"className":338440,"style":32604},[31845],[34,338442,338444,338447,338450],{"className":338443},[31676],[34,338445],{"className":338446,"style":31774},[31680],[34,338448,640],{"className":338449},[31685],[34,338451,126544],{"className":338452},[31690],[13858,338454,338455,338499],{},[34,338456,338458,338474],{"className":338457},[31642],[34,338459,338461],{"className":338460},[31646],[31648,338462,338463],{"xmlns":31650},[31652,338464,338465,338471],{},[31655,338466,338467,338469],{},[31662,338468,134826],{},[31658,338470,47384],{"mathvariant":318416},[31665,338472,338473],{"encoding":31667},"\\cong \\mathbb{C}",[34,338475,338477,338490],{"className":338476,"ariaHidden":758},[31672],[34,338478,338480,338484,338487],{"className":338479},[31676],[34,338481],{"className":338482,"style":338483},[31680],"height:0.589em;",[34,338485,134826],{"className":338486},[31850],[34,338488],{"className":338489,"style":31846},[31845],[34,338491,338493,338496],{"className":338492},[31676],[34,338494],{"className":338495,"style":320927},[31680],[34,338497,47384],{"className":338498},[31685,318441],"（复数域）",[13858,338501,338502,338592],{},[34,338503,338505,338530],{"className":338504},[31642],[34,338506,338508],{"className":338507},[31646],[31648,338509,338510],{"xmlns":31650},[31652,338511,338512,338528],{},[31655,338513,338514,338516,338522,338524,338526],{},[31662,338515,126529],{"stretchy":702},[37624,338517,338518,338520],{},[31658,338519,32936],{},[31749,338521,768],{},[31662,338523,223],{},[31749,338525,640],{},[31662,338527,126544],{"stretchy":702},[31665,338529,338390],{"encoding":31667},[34,338531,338533,338580],{"className":338532,"ariaHidden":758},[31672],[34,338534,338536,338539,338542,338571,338574,338577],{"className":338535},[31676],[34,338537],{"className":338538,"style":37755},[31680],[34,338540,126529],{"className":338541},[31782],[34,338543,338545,338548],{"className":338544},[31685],[34,338546,32936],{"className":338547},[31685,31686],[34,338549,338551],{"className":338550},[31792],[34,338552,338554],{"className":338553},[31796],[34,338555,338557],{"className":338556},[31801],[34,338558,338560],{"className":338559,"style":37774},[31805],[34,338561,338562,338565],{"style":37777},[34,338563],{"className":338564,"style":31814},[31813],[34,338566,338568],{"className":338567},[31818,31819,31820,31821],[34,338569,768],{"className":338570},[31685,31821],[34,338572],{"className":338573,"style":32604},[31845],[34,338575,223],{"className":338576},[32024],[34,338578],{"className":338579,"style":32604},[31845],[34,338581,338583,338586,338589],{"className":338582},[31676],[34,338584],{"className":338585,"style":31774},[31680],[34,338587,640],{"className":338588},[31685],[34,338590,126544],{"className":338591},[31690]," 是极大理想",[13843,338594,338595,338642,338683,338729],{},[13858,338596,338597],{},[2627,338598,338315,338599],{},[34,338600,338602,338621],{"className":338601},[31642],[34,338603,338605],{"className":338604},[31646],[31648,338606,338607],{"xmlns":31650},[31652,338608,338609,338619],{},[31655,338610,338611,338613,338615,338617],{},[31658,338612,34386],{"mathvariant":318416},[31662,338614,15972],{"stretchy":702},[31658,338616,32936],{},[31662,338618,32712],{"stretchy":702},[31665,338620,338338],{"encoding":31667},[34,338622,338624],{"className":338623,"ariaHidden":758},[31672],[34,338625,338627,338630,338633,338636,338639],{"className":338626},[31676],[34,338628],{"className":338629,"style":31774},[31680],[34,338631,34386],{"className":338632},[31685,318441],[34,338634,15972],{"className":338635},[31782],[34,338637,32936],{"className":338638},[31685,31686],[34,338640,32712],{"className":338641},[31690],[13858,338643,338644],{},[34,338645,338647,338665],{"className":338646},[31642],[34,338648,338650],{"className":338649},[31646],[31648,338651,338652],{"xmlns":31650},[31652,338653,338654,338662],{},[31655,338655,338656,338658,338660],{},[31662,338657,126529],{"stretchy":702},[31658,338659,32936],{},[31662,338661,126544],{"stretchy":702},[31665,338663,338664],{"encoding":31667},"\\langle x \\rangle",[34,338666,338668],{"className":338667,"ariaHidden":758},[31672],[34,338669,338671,338674,338677,338680],{"className":338670},[31676],[34,338672],{"className":338673,"style":31774},[31680],[34,338675,126529],{"className":338676},[31782],[34,338678,32936],{"className":338679},[31685,31686],[34,338681,126544],{"className":338682},[31690],[13858,338684,338685,338728],{},[34,338686,338688,338704],{"className":338687},[31642],[34,338689,338691],{"className":338690},[31646],[31648,338692,338693],{"xmlns":31650},[31652,338694,338695,338701],{},[31655,338696,338697,338699],{},[31662,338698,134826],{},[31658,338700,34386],{"mathvariant":318416},[31665,338702,338703],{"encoding":31667},"\\cong \\mathbb{R}",[34,338705,338707,338719],{"className":338706,"ariaHidden":758},[31672],[34,338708,338710,338713,338716],{"className":338709},[31676],[34,338711],{"className":338712,"style":338483},[31680],[34,338714,134826],{"className":338715},[31850],[34,338717],{"className":338718,"style":31846},[31845],[34,338720,338722,338725],{"className":338721},[31676],[34,338723],{"className":338724,"style":320927},[31680],[34,338726,34386],{"className":338727},[31685,318441],"（实数域）",[13858,338730,338731,338592],{},[34,338732,338734,338751],{"className":338733},[31642],[34,338735,338737],{"className":338736},[31646],[31648,338738,338739],{"xmlns":31650},[31652,338740,338741,338749],{},[31655,338742,338743,338745,338747],{},[31662,338744,126529],{"stretchy":702},[31658,338746,32936],{},[31662,338748,126544],{"stretchy":702},[31665,338750,338664],{"encoding":31667},[34,338752,338754],{"className":338753,"ariaHidden":758},[31672],[34,338755,338757,338760,338763,338766],{"className":338756},[31676],[34,338758],{"className":338759,"style":31774},[31680],[34,338761,126529],{"className":338762},[31782],[34,338764,32936],{"className":338765},[31685,31686],[34,338767,126544],{"className":338768},[31690],[32188,338770],{},[253,338772,338773],{"id":338773},"重点回顾",[2230,338775,338776,338914],{},[108,338777,338778,338781,338782,338810,338811,338884,338885,338913],{},[2627,338779,338780],{},"构造逆元的证明","（定理 3.5.2 的 ",[34,338783,338785,338798],{"className":338784},[31642],[34,338786,338788],{"className":338787},[31646],[31648,338789,338790],{"xmlns":31650},[31652,338791,338792,338796],{},[31655,338793,338794],{},[31662,338795,140094],{},[31665,338797,266872],{"encoding":31667},[34,338799,338801],{"className":338800,"ariaHidden":758},[31672],[34,338802,338804,338807],{"className":338803},[31676],[34,338805],{"className":338806,"style":73089},[31680],[34,338808,140094],{"className":338809},[31850]," 部分）：利用 ",[34,338812,338814,338839],{"className":338813},[31642],[34,338815,338817],{"className":338816},[31646],[31648,338818,338819],{"xmlns":31650},[31652,338820,338821,338837],{},[31655,338822,338823,338825,338827,338829,338831,338833,338835],{},[31749,338824,640],{},[31662,338826,112296],{},[31662,338828,126529],{"stretchy":702},[31658,338830,98360],{},[31662,338832,31960],{"separator":758},[31658,338834,111],{},[31662,338836,126544],{"stretchy":702},[31665,338838,334904],{"encoding":31667},[34,338840,338842,338860],{"className":338841,"ariaHidden":758},[31672],[34,338843,338845,338848,338851,338854,338857],{"className":338844},[31676],[34,338846],{"className":338847,"style":61814},[31680],[34,338849,640],{"className":338850},[31685],[34,338852],{"className":338853,"style":31846},[31845],[34,338855,112296],{"className":338856},[31850],[34,338858],{"className":338859,"style":31846},[31845],[34,338861,338863,338866,338869,338872,338875,338878,338881],{"className":338862},[31676],[34,338864],{"className":338865,"style":31774},[31680],[34,338867,126529],{"className":338868},[31782],[34,338870,98360],{"className":338871,"style":33389},[31685,31686],[34,338873,31960],{"className":338874},[32119],[34,338876],{"className":338877,"style":32123},[31845],[34,338879,111],{"className":338880},[31685,31686],[34,338882,126544],{"className":338883},[31690]," 强行把 ",[34,338886,338888,338901],{"className":338887},[31642],[34,338889,338891],{"className":338890},[31646],[31648,338892,338893],{"xmlns":31650},[31652,338894,338895,338899],{},[31655,338896,338897],{},[31749,338898,640],{},[31665,338900,640],{"encoding":31667},[34,338902,338904],{"className":338903,"ariaHidden":758},[31672],[34,338905,338907,338910],{"className":338906},[31676],[34,338908],{"className":338909,"style":32617},[31680],[34,338911,640],{"className":338912},[31685]," 变出来，是本节最核心的技巧。",[108,338915,338916,339056,339057,339085],{},[2627,338917,338918],{},[34,338919,338921,338961],{"className":338920},[31642],[34,338922,338924],{"className":338923},[31646],[31648,338925,338926],{"xmlns":31650},[31652,338927,338928,338958],{},[31655,338929,338930,338932,338934,338936,338938,338940,338942,338948,338950,338952,338954,338956],{},[31658,338931,34386],{"mathvariant":318416},[31662,338933,15972],{"stretchy":702},[31658,338935,32936],{},[31662,338937,32712],{"stretchy":702},[31658,338939,38080],{"mathvariant":38079},[31662,338941,126529],{"stretchy":702},[37624,338943,338944,338946],{},[31658,338945,32936],{},[31749,338947,768],{},[31662,338949,223],{},[31749,338951,640],{},[31662,338953,126544],{"stretchy":702},[31662,338955,134826],{},[31658,338957,47384],{"mathvariant":318416},[31665,338959,338960],{"encoding":31667},"\\mathbb{R}[x]\u002F\\langle x^2+1 \\rangle \\cong \\mathbb{C}",[34,338962,338964,339026,339047],{"className":338963,"ariaHidden":758},[31672],[34,338965,338967,338970,338973,338976,338979,338982,338985,338988,339017,339020,339023],{"className":338966},[31676],[34,338968],{"className":338969,"style":37755},[31680],[34,338971,34386],{"className":338972},[31685,318441],[34,338974,15972],{"className":338975},[31782],[34,338977,32936],{"className":338978},[31685,31686],[34,338980,32712],{"className":338981},[31690],[34,338983,38080],{"className":338984},[31685],[34,338986,126529],{"className":338987},[31782],[34,338989,338991,338994],{"className":338990},[31685],[34,338992,32936],{"className":338993},[31685,31686],[34,338995,338997],{"className":338996},[31792],[34,338998,339000],{"className":338999},[31796],[34,339001,339003],{"className":339002},[31801],[34,339004,339006],{"className":339005,"style":37774},[31805],[34,339007,339008,339011],{"style":37777},[34,339009],{"className":339010,"style":31814},[31813],[34,339012,339014],{"className":339013},[31818,31819,31820,31821],[34,339015,768],{"className":339016},[31685,31821],[34,339018],{"className":339019,"style":32604},[31845],[34,339021,223],{"className":339022},[32024],[34,339024],{"className":339025,"style":32604},[31845],[34,339027,339029,339032,339035,339038,339041,339044],{"className":339028},[31676],[34,339030],{"className":339031,"style":31774},[31680],[34,339033,640],{"className":339034},[31685],[34,339036,126544],{"className":339037},[31690],[34,339039],{"className":339040,"style":31846},[31845],[34,339042,134826],{"className":339043},[31850],[34,339045],{"className":339046,"style":31846},[31845],[34,339048,339050,339053],{"className":339049},[31676],[34,339051],{"className":339052,"style":320927},[31680],[34,339054,47384],{"className":339055},[31685,318441],"：用商环构造出虚数单位 ",[34,339058,339060,339073],{"className":339059},[31642],[34,339061,339063],{"className":339062},[31646],[31648,339064,339065],{"xmlns":31650},[31652,339066,339067,339071],{},[31655,339068,339069],{},[31658,339070,31955],{},[31665,339072,31955],{"encoding":31667},[34,339074,339076],{"className":339075,"ariaHidden":758},[31672],[34,339077,339079,339082],{"className":339078},[31676],[34,339080],{"className":339081,"style":33104},[31680],[34,339083,31955],{"className":339084},[31685,31686]," 的经典例子。",{"title":29,"searchDepth":43,"depth":43,"links":339087},[339088,339089,339093,339097,339098],{"id":330690,"depth":43,"text":330691},{"id":330784,"depth":43,"text":330785,"children":339090},[339091,339092],{"id":330788,"depth":49,"text":330789},{"id":331420,"depth":49,"text":331421},{"id":333484,"depth":43,"text":333485,"children":339094},[339095,339096],{"id":333488,"depth":49,"text":333489},{"id":333969,"depth":49,"text":333970},{"id":337718,"depth":43,"text":337719},{"id":337835,"depth":43,"text":337836,"children":339099},[339100],{"id":338773,"depth":49,"text":338773},{"order":1985},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fprimeandmaximalideal",{"title":330681,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002Fprimeandmaximalideal","9rOuW7SZ0HQzaMIO2YnVBodMxUEdGfJd8TZxEi-SWEQ",{"id":339107,"title":339108,"body":339109,"description":29,"extension":85,"meta":365350,"navigation":52,"path":365351,"seo":365352,"stem":365353,"__hash__":365354},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002FprimeidealmaximalidealHW.md","素理想与极大理想 习题详解",{"type":8,"value":339110,"toc":365343},[339111,339114,339118,339120,339408,339410,339413,339483,340389,340394,340428,340850,341340,343271,343332,345478,345522,346497,346499,346503,346505,346800,346802,346805,346810,347502,347506,351463,351465,351469,351471,351765,351767,351770,351774,351896,351900,352757,355805,359600,361825,362457,362459,362463,362465,362468,362470,362473,362477,362550,362903,362907,365082,365084,365087],[11,339112,339108],{"id":339113},"素理想与极大理想-习题详解",[253,339115,339117],{"id":339116},"题号-5","题号 5",[32490,339119,162043],{"id":162043},[15,339121,133910,339122,339150,339151,339179,339180,339282,339283,339311,339312,339340,339341,339369,339370,334082],{},[34,339123,339125,339138],{"className":339124},[31642],[34,339126,339128],{"className":339127},[31646],[31648,339129,339130],{"xmlns":31650},[31652,339131,339132,339136],{},[31655,339133,339134],{},[31658,339135,15],{},[31665,339137,15],{"encoding":31667},[34,339139,339141],{"className":339140,"ariaHidden":758},[31672],[34,339142,339144,339147],{"className":339143},[31676],[34,339145],{"className":339146,"style":38774},[31680],[34,339148,15],{"className":339149},[31685,31686]," 是一个素数，",[34,339152,339154,339167],{"className":339153},[31642],[34,339155,339157],{"className":339156},[31646],[31648,339158,339159],{"xmlns":31650},[31652,339160,339161,339165],{},[31655,339162,339163],{},[31658,339164,31660],{},[31665,339166,31660],{"encoding":31667},[34,339168,339170],{"className":339169,"ariaHidden":758},[31672],[34,339171,339173,339176],{"className":339172},[31676],[34,339174],{"className":339175,"style":32527},[31680],[34,339177,31660],{"className":339178},[31685,31686]," 是大于 1 的整数，",[34,339181,339183,339214],{"className":339182},[31642],[34,339184,339186],{"className":339185},[31646],[31648,339187,339188],{"xmlns":31650},[31652,339189,339190,339211],{},[31655,339191,339192,339194,339197,339199,339201,339203,339209],{},[31658,339193,34386],{},[31662,339195,339196],{},"≃",[31658,339198,97300],{"mathvariant":318416},[31658,339200,38080],{"mathvariant":38079},[31662,339202,456],{"stretchy":702},[37624,339204,339205,339207],{},[31658,339206,15],{},[31658,339208,31660],{},[31662,339210,2557],{"stretchy":702},[31665,339212,339213],{"encoding":31667},"R \\simeq \\mathbb{Z}\u002F(p^n)",[34,339215,339217,339235],{"className":339216,"ariaHidden":758},[31672],[34,339218,339220,339223,339226,339229,339232],{"className":339219},[31676],[34,339221],{"className":339222,"style":33218},[31680],[34,339224,34386],{"className":339225,"style":34427},[31685,31686],[34,339227],{"className":339228,"style":31846},[31845],[34,339230,339196],{"className":339231},[31850],[34,339233],{"className":339234,"style":31846},[31845],[34,339236,339238,339241,339244,339247,339250,339279],{"className":339237},[31676],[34,339239],{"className":339240,"style":31774},[31680],[34,339242,97300],{"className":339243},[31685,318441],[34,339245,38080],{"className":339246},[31685],[34,339248,456],{"className":339249},[31782],[34,339251,339253,339256],{"className":339252},[31685],[34,339254,15],{"className":339255},[31685,31686],[34,339257,339259],{"className":339258},[31792],[34,339260,339262],{"className":339261},[31796],[34,339263,339265],{"className":339264},[31801],[34,339266,339268],{"className":339267,"style":37959},[31805],[34,339269,339270,339273],{"style":37777},[34,339271],{"className":339272,"style":31814},[31813],[34,339274,339276],{"className":339275},[31818,31819,31820,31821],[34,339277,31660],{"className":339278},[31685,31686,31821],[34,339280,2557],{"className":339281},[31690],"，证明：\n(1) ",[34,339284,339286,339299],{"className":339285},[31642],[34,339287,339289],{"className":339288},[31646],[31648,339290,339291],{"xmlns":31650},[31652,339292,339293,339297],{},[31655,339294,339295],{},[31658,339296,34386],{},[31665,339298,34386],{"encoding":31667},[34,339300,339302],{"className":339301,"ariaHidden":758},[31672],[34,339303,339305,339308],{"className":339304},[31676],[34,339306],{"className":339307,"style":33218},[31680],[34,339309,34386],{"className":339310,"style":34427},[31685,31686]," 的元素不是可逆元就是幂零元；\n(2) ",[34,339313,339315,339328],{"className":339314},[31642],[34,339316,339318],{"className":339317},[31646],[31648,339319,339320],{"xmlns":31650},[31652,339321,339322,339326],{},[31655,339323,339324],{},[31658,339325,34386],{},[31665,339327,34386],{"encoding":31667},[34,339329,339331],{"className":339330,"ariaHidden":758},[31672],[34,339332,339334,339337],{"className":339333},[31676],[34,339335],{"className":339336,"style":33218},[31680],[34,339338,34386],{"className":339339,"style":34427},[31685,31686]," 只有一个素理想，记做 ",[34,339342,339344,339357],{"className":339343},[31642],[34,339345,339347],{"className":339346},[31646],[31648,339348,339349],{"xmlns":31650},[31652,339350,339351,339355],{},[31655,339352,339353],{},[31658,339354,105852],{},[31665,339356,105852],{"encoding":31667},[34,339358,339360],{"className":339359,"ariaHidden":758},[31672],[34,339361,339363,339366],{"className":339362},[31676],[34,339364],{"className":339365,"style":33218},[31680],[34,339367,105852],{"className":339368,"style":37422},[31685,31686],"；\n(3) 商环 ",[34,339371,339373,339390],{"className":339372},[31642],[34,339374,339376],{"className":339375},[31646],[31648,339377,339378],{"xmlns":31650},[31652,339379,339380,339388],{},[31655,339381,339382,339384,339386],{},[31658,339383,34386],{},[31658,339385,38080],{"mathvariant":38079},[31658,339387,105852],{},[31665,339389,331515],{"encoding":31667},[34,339391,339393],{"className":339392,"ariaHidden":758},[31672],[34,339394,339396,339399,339402,339405],{"className":339395},[31676],[34,339397],{"className":339398,"style":31774},[31680],[34,339400,34386],{"className":339401,"style":34427},[31685,31686],[34,339403,38080],{"className":339404},[31685],[34,339406,105852],{"className":339407,"style":37422},[31685,31686],[32188,339409],{},[32490,339411,339412],{"id":339412},"深度分析与解答",[15,339414,339415,339418,339419,339482],{},[2627,339416,339417],{},"分析框架（Pre-analysis）：","\n这道题探讨的是",[2627,339420,339421,339422,339481],{},"模 ",[34,339423,339425,339443],{"className":339424},[31642],[34,339426,339428],{"className":339427},[31646],[31648,339429,339430],{"xmlns":31650},[31652,339431,339432,339440],{},[31655,339433,339434],{},[37624,339435,339436,339438],{},[31658,339437,15],{},[31658,339439,31660],{},[31665,339441,339442],{"encoding":31667},"p^n",[34,339444,339446],{"className":339445,"ariaHidden":758},[31672],[34,339447,339449,339452],{"className":339448},[31676],[34,339450],{"className":339451,"style":316112},[31680],[34,339453,339455,339458],{"className":339454},[31685],[34,339456,15],{"className":339457},[31685,31686],[34,339459,339461],{"className":339460},[31792],[34,339462,339464],{"className":339463},[31796],[34,339465,339467],{"className":339466},[31801],[34,339468,339470],{"className":339469,"style":37959},[31805],[34,339471,339472,339475],{"style":37777},[34,339473],{"className":339474,"style":31814},[31813],[34,339476,339478],{"className":339477},[31818,31819,31820,31821],[34,339479,31660],{"className":339480},[31685,31686,31821]," 的整数环","的性质。",[105,339484,339485,339787,339791,339989,340133,340383],{},[108,339486,339487,339489,339490,339590,339591,339727,339728,339786],{},[2627,339488,314997],{},"：环 ",[34,339491,339493,339522],{"className":339492},[31642],[34,339494,339496],{"className":339495},[31646],[31648,339497,339498],{"xmlns":31650},[31652,339499,339500,339520],{},[31655,339501,339502,339504,339506,339508,339510,339512,339518],{},[31658,339503,34386],{},[31662,339505,339196],{},[31658,339507,97300],{"mathvariant":318416},[31658,339509,38080],{"mathvariant":38079},[31662,339511,456],{"stretchy":702},[37624,339513,339514,339516],{},[31658,339515,15],{},[31658,339517,31660],{},[31662,339519,2557],{"stretchy":702},[31665,339521,339213],{"encoding":31667},[34,339523,339525,339543],{"className":339524,"ariaHidden":758},[31672],[34,339526,339528,339531,339534,339537,339540],{"className":339527},[31676],[34,339529],{"className":339530,"style":33218},[31680],[34,339532,34386],{"className":339533,"style":34427},[31685,31686],[34,339535],{"className":339536,"style":31846},[31845],[34,339538,339196],{"className":339539},[31850],[34,339541],{"className":339542,"style":31846},[31845],[34,339544,339546,339549,339552,339555,339558,339587],{"className":339545},[31676],[34,339547],{"className":339548,"style":31774},[31680],[34,339550,97300],{"className":339551},[31685,318441],[34,339553,38080],{"className":339554},[31685],[34,339556,456],{"className":339557},[31782],[34,339559,339561,339564],{"className":339560},[31685],[34,339562,15],{"className":339563},[31685,31686],[34,339565,339567],{"className":339566},[31792],[34,339568,339570],{"className":339569},[31796],[34,339571,339573],{"className":339572},[31801],[34,339574,339576],{"className":339575,"style":37959},[31805],[34,339577,339578,339581],{"style":37777},[34,339579],{"className":339580,"style":31814},[31813],[34,339582,339584],{"className":339583},[31818,31819,31820,31821],[34,339585,31660],{"className":339586},[31685,31686,31821],[34,339588,2557],{"className":339589},[31690]," 里的元素实际上就是 ",[34,339592,339594,339632],{"className":339593},[31642],[34,339595,339597],{"className":339596},[31646],[31648,339598,339599],{"xmlns":31650},[31652,339600,339601,339629],{},[31655,339602,339603,339605,339607,339609,339611,339613,339615,339617,339619,339625,339627],{},[31749,339604,459],{},[31662,339606,31960],{"separator":758},[31749,339608,640],{},[31662,339610,31960],{"separator":758},[31749,339612,768],{},[31662,339614,31960],{"separator":758},[31662,339616,33934],{},[31662,339618,31960],{"separator":758},[37624,339620,339621,339623],{},[31658,339622,15],{},[31658,339624,31660],{},[31662,339626,32226],{},[31749,339628,640],{},[31665,339630,339631],{"encoding":31667},"0, 1, 2, \\dots, p^n-1",[34,339633,339635,339718],{"className":339634,"ariaHidden":758},[31672],[34,339636,339638,339641,339644,339647,339650,339653,339656,339659,339662,339665,339668,339671,339674,339677,339680,339709,339712,339715],{"className":339637},[31676],[34,339639],{"className":339640,"style":316112},[31680],[34,339642,459],{"className":339643},[31685],[34,339645,31960],{"className":339646},[32119],[34,339648],{"className":339649,"style":32123},[31845],[34,339651,640],{"className":339652},[31685],[34,339654,31960],{"className":339655},[32119],[34,339657],{"className":339658,"style":32123},[31845],[34,339660,768],{"className":339661},[31685],[34,339663,31960],{"className":339664},[32119],[34,339666],{"className":339667,"style":32123},[31845],[34,339669,33934],{"className":339670},[34006],[34,339672],{"className":339673,"style":32123},[31845],[34,339675,31960],{"className":339676},[32119],[34,339678],{"className":339679,"style":32123},[31845],[34,339681,339683,339686],{"className":339682},[31685],[34,339684,15],{"className":339685},[31685,31686],[34,339687,339689],{"className":339688},[31792],[34,339690,339692],{"className":339691},[31796],[34,339693,339695],{"className":339694},[31801],[34,339696,339698],{"className":339697,"style":37959},[31805],[34,339699,339700,339703],{"style":37777},[34,339701],{"className":339702,"style":31814},[31813],[34,339704,339706],{"className":339705},[31818,31819,31820,31821],[34,339707,31660],{"className":339708},[31685,31686,31821],[34,339710],{"className":339711,"style":32604},[31845],[34,339713,32226],{"className":339714},[32024],[34,339716],{"className":339717,"style":32604},[31845],[34,339719,339721,339724],{"className":339720},[31676],[34,339722],{"className":339723,"style":32617},[31680],[34,339725,640],{"className":339726},[31685]," 这些整数，运算遵循“模 ",[34,339729,339731,339748],{"className":339730},[31642],[34,339732,339734],{"className":339733},[31646],[31648,339735,339736],{"xmlns":31650},[31652,339737,339738,339746],{},[31655,339739,339740],{},[37624,339741,339742,339744],{},[31658,339743,15],{},[31658,339745,31660],{},[31665,339747,339442],{"encoding":31667},[34,339749,339751],{"className":339750,"ariaHidden":758},[31672],[34,339752,339754,339757],{"className":339753},[31676],[34,339755],{"className":339756,"style":316112},[31680],[34,339758,339760,339763],{"className":339759},[31685],[34,339761,15],{"className":339762},[31685,31686],[34,339764,339766],{"className":339765},[31792],[34,339767,339769],{"className":339768},[31796],[34,339770,339772],{"className":339771},[31801],[34,339773,339775],{"className":339774,"style":37959},[31805],[34,339776,339777,339780],{"style":37777},[34,339778],{"className":339779,"style":31814},[31813],[34,339781,339783],{"className":339782},[31818,31819,31820,31821],[34,339784,31660],{"className":339785},[31685,31686,31821],"”的规则。",[108,339788,339789,2664],{},[2627,339790,243605],{},[108,339792,339793,339796,339797,315343,339825,55016,339881,339909,339910,339959,339960,339988],{},[2627,339794,339795],{},"可逆元 (Unit)","：如果存在 ",[34,339798,339800,339813],{"className":339799},[31642],[34,339801,339803],{"className":339802},[31646],[31648,339804,339805],{"xmlns":31650},[31652,339806,339807,339811],{},[31655,339808,339809],{},[31658,339810,67764],{},[31665,339812,67764],{"encoding":31667},[34,339814,339816],{"className":339815,"ariaHidden":758},[31672],[34,339817,339819,339822],{"className":339818},[31676],[34,339820],{"className":339821,"style":38774},[31680],[34,339823,67764],{"className":339824,"style":32062},[31685,31686],[34,339826,339828,339848],{"className":339827},[31642],[34,339829,339831],{"className":339830},[31646],[31648,339832,339833],{"xmlns":31650},[31652,339834,339835,339845],{},[31655,339836,339837,339839,339841,339843],{},[31658,339838,32936],{},[31658,339840,67764],{},[31662,339842,202],{},[31749,339844,640],{},[31665,339846,339847],{"encoding":31667},"xy = 1",[34,339849,339851,339872],{"className":339850,"ariaHidden":758},[31672],[34,339852,339854,339857,339860,339863,339866,339869],{"className":339853},[31676],[34,339855],{"className":339856,"style":38774},[31680],[34,339858,32936],{"className":339859},[31685,31686],[34,339861,67764],{"className":339862,"style":32062},[31685,31686],[34,339864],{"className":339865,"style":31846},[31845],[34,339867,202],{"className":339868},[31850],[34,339870],{"className":339871,"style":31846},[31845],[34,339873,339875,339878],{"className":339874},[31676],[34,339876],{"className":339877,"style":32617},[31680],[34,339879,640],{"className":339880},[31685],[34,339882,339884,339897],{"className":339883},[31642],[34,339885,339887],{"className":339886},[31646],[31648,339888,339889],{"xmlns":31650},[31652,339890,339891,339895],{},[31655,339892,339893],{},[31658,339894,32936],{},[31665,339896,32936],{"encoding":31667},[34,339898,339900],{"className":339899,"ariaHidden":758},[31672],[34,339901,339903,339906],{"className":339902},[31676],[34,339904],{"className":339905,"style":32527},[31680],[34,339907,32936],{"className":339908},[31685,31686]," 是可逆元。在 ",[34,339911,339913,339935],{"className":339912},[31642],[34,339914,339916],{"className":339915},[31646],[31648,339917,339918],{"xmlns":31650},[31652,339919,339920,339932],{},[31655,339921,339922,339924,339926,339928,339930],{},[31658,339923,97300],{"mathvariant":318416},[31658,339925,38080],{"mathvariant":38079},[31662,339927,456],{"stretchy":702},[31658,339929,31759],{},[31662,339931,2557],{"stretchy":702},[31665,339933,339934],{"encoding":31667},"\\mathbb{Z}\u002F(m)",[34,339936,339938],{"className":339937,"ariaHidden":758},[31672],[34,339939,339941,339944,339947,339950,339953,339956],{"className":339940},[31676],[34,339942],{"className":339943,"style":31774},[31680],[34,339945,97300],{"className":339946},[31685,318441],[34,339948,38080],{"className":339949},[31685],[34,339951,456],{"className":339952},[31782],[34,339954,31759],{"className":339955},[31685,31686],[34,339957,2557],{"className":339958},[31690]," 中，与 ",[34,339961,339963,339976],{"className":339962},[31642],[34,339964,339966],{"className":339965},[31646],[31648,339967,339968],{"xmlns":31650},[31652,339969,339970,339974],{},[31655,339971,339972],{},[31658,339973,31759],{},[31665,339975,31759],{"encoding":31667},[34,339977,339979],{"className":339978,"ariaHidden":758},[31672],[34,339980,339982,339985],{"className":339981},[31676],[34,339983],{"className":339984,"style":32527},[31680],[34,339986,31759],{"className":339987},[31685,31686]," 互质的数是可逆元。",[108,339990,339991,339994,339995,315343,340023,55016,340104,340132],{},[2627,339992,339993],{},"幂零元 (Nilpotent)","：如果存在某个正整数 ",[34,339996,339998,340011],{"className":339997},[31642],[34,339999,340001],{"className":340000},[31646],[31648,340002,340003],{"xmlns":31650},[31652,340004,340005,340009],{},[31655,340006,340007],{},[31658,340008,33759],{},[31665,340010,33759],{"encoding":31667},[34,340012,340014],{"className":340013,"ariaHidden":758},[31672],[34,340015,340017,340020],{"className":340016},[31676],[34,340018],{"className":340019,"style":31681},[31680],[34,340021,33759],{"className":340022,"style":33774},[31685,31686],[34,340024,340026,340048],{"className":340025},[31642],[34,340027,340029],{"className":340028},[31646],[31648,340030,340031],{"xmlns":31650},[31652,340032,340033,340045],{},[31655,340034,340035,340041,340043],{},[37624,340036,340037,340039],{},[31658,340038,32936],{},[31658,340040,33759],{},[31662,340042,202],{},[31749,340044,459],{},[31665,340046,340047],{"encoding":31667},"x^k = 0",[34,340049,340051,340095],{"className":340050,"ariaHidden":758},[31672],[34,340052,340054,340057,340086,340089,340092],{"className":340053},[31676],[34,340055],{"className":340056,"style":38486},[31680],[34,340058,340060,340063],{"className":340059},[31685],[34,340061,32936],{"className":340062},[31685,31686],[34,340064,340066],{"className":340065},[31792],[34,340067,340069],{"className":340068},[31796],[34,340070,340072],{"className":340071},[31801],[34,340073,340075],{"className":340074,"style":38486},[31805],[34,340076,340077,340080],{"style":37777},[34,340078],{"className":340079,"style":31814},[31813],[34,340081,340083],{"className":340082},[31818,31819,31820,31821],[34,340084,33759],{"className":340085,"style":33774},[31685,31686,31821],[34,340087],{"className":340088,"style":31846},[31845],[34,340090,202],{"className":340091},[31850],[34,340093],{"className":340094,"style":31846},[31845],[34,340096,340098,340101],{"className":340097},[31676],[34,340099],{"className":340100,"style":32617},[31680],[34,340102,459],{"className":340103},[31685],[34,340105,340107,340120],{"className":340106},[31642],[34,340108,340110],{"className":340109},[31646],[31648,340111,340112],{"xmlns":31650},[31652,340113,340114,340118],{},[31655,340115,340116],{},[31658,340117,32936],{},[31665,340119,32936],{"encoding":31667},[34,340121,340123],{"className":340122,"ariaHidden":758},[31672],[34,340124,340126,340129],{"className":340125},[31676],[34,340127],{"className":340128,"style":32527},[31680],[34,340130,32936],{"className":340131},[31685,31686]," 是幂零元。",[108,340134,340135,340138,340139,340167,340168,340196,340197,4374,340304,340354,340355,332897],{},[2627,340136,340137],{},"素理想 (Prime Ideal)","：如果在理想 ",[34,340140,340142,340155],{"className":340141},[31642],[34,340143,340145],{"className":340144},[31646],[31648,340146,340147],{"xmlns":31650},[31652,340148,340149,340153],{},[31655,340150,340151],{},[31658,340152,105852],{},[31665,340154,105852],{"encoding":31667},[34,340156,340158],{"className":340157,"ariaHidden":758},[31672],[34,340159,340161,340164],{"className":340160},[31676],[34,340162],{"className":340163,"style":33218},[31680],[34,340165,105852],{"className":340166,"style":37422},[31685,31686]," 外找不到两个数的乘积掉进 ",[34,340169,340171,340184],{"className":340170},[31642],[34,340172,340174],{"className":340173},[31646],[31648,340175,340176],{"xmlns":31650},[31652,340177,340178,340182],{},[31655,340179,340180],{},[31658,340181,105852],{},[31665,340183,105852],{"encoding":31667},[34,340185,340187],{"className":340186,"ariaHidden":758},[31672],[34,340188,340190,340193],{"className":340189},[31676],[34,340191],{"className":340192,"style":33218},[31680],[34,340194,105852],{"className":340195,"style":37422},[31685,31686]," 里（即 ",[34,340198,340200,340232],{"className":340199},[31642],[34,340201,340203],{"className":340202},[31646],[31648,340204,340205],{"xmlns":31650},[31652,340206,340207,340229],{},[31655,340208,340209,340211,340213,340215,340217,340219,340221,340223,340225,340227],{},[31658,340210,111],{},[31658,340212,1697],{},[31662,340214,112296],{},[31658,340216,105852],{},[39361,340218,262622],{},[31662,340220,262625],{},[39361,340222,262622],{},[31658,340224,111],{},[31662,340226,112296],{},[31658,340228,105852],{},[31665,340230,340231],{"encoding":31667},"ab \\in P \\implies a \\in P",[34,340233,340235,340253,340277,340295],{"className":340234,"ariaHidden":758},[31672],[34,340236,340238,340241,340244,340247,340250],{"className":340237},[31676],[34,340239],{"className":340240,"style":46991},[31680],[34,340242,324191],{"className":340243},[31685,31686],[34,340245],{"className":340246,"style":31846},[31845],[34,340248,112296],{"className":340249},[31850],[34,340251],{"className":340252,"style":31846},[31845],[34,340254,340256,340259,340262,340265,340268,340271,340274],{"className":340255},[31676],[34,340257],{"className":340258,"style":336814},[31680],[34,340260,105852],{"className":340261,"style":37422},[31685,31686],[34,340263],{"className":340264,"style":31846},[31845],[34,340266],{"className":340267,"style":31846},[31845],[34,340269,262625],{"className":340270},[31850],[34,340272],{"className":340273,"style":31846},[31845],[34,340275],{"className":340276,"style":31846},[31845],[34,340278,340280,340283,340286,340289,340292],{"className":340279},[31676],[34,340281],{"className":340282,"style":34812},[31680],[34,340284,111],{"className":340285},[31685,31686],[34,340287],{"className":340288,"style":31846},[31845],[34,340290,112296],{"className":340291},[31850],[34,340293],{"className":340294,"style":31846},[31845],[34,340296,340298,340301],{"className":340297},[31676],[34,340299],{"className":340300,"style":33218},[31680],[34,340302,105852],{"className":340303,"style":37422},[31685,31686],[34,340305,340307,340324],{"className":340306},[31642],[34,340308,340310],{"className":340309},[31646],[31648,340311,340312],{"xmlns":31650},[31652,340313,340314,340322],{},[31655,340315,340316,340318,340320],{},[31658,340317,1697],{},[31662,340319,112296],{},[31658,340321,105852],{},[31665,340323,331126],{"encoding":31667},[34,340325,340327,340345],{"className":340326,"ariaHidden":758},[31672],[34,340328,340330,340333,340336,340339,340342],{"className":340329},[31676],[34,340331],{"className":340332,"style":46991},[31680],[34,340334,1697],{"className":340335},[31685,31686],[34,340337],{"className":340338,"style":31846},[31845],[34,340340,112296],{"className":340341},[31850],[34,340343],{"className":340344,"style":31846},[31845],[34,340346,340348,340351],{"className":340347},[31676],[34,340349],{"className":340350,"style":33218},[31680],[34,340352,105852],{"className":340353,"style":37422},[31685,31686],"），则 ",[34,340356,340358,340371],{"className":340357},[31642],[34,340359,340361],{"className":340360},[31646],[31648,340362,340363],{"xmlns":31650},[31652,340364,340365,340369],{},[31655,340366,340367],{},[31658,340368,105852],{},[31665,340370,105852],{"encoding":31667},[34,340372,340374],{"className":340373,"ariaHidden":758},[31672],[34,340375,340377,340380],{"className":340376},[31676],[34,340378],{"className":340379,"style":33218},[31680],[34,340381,105852],{"className":340382,"style":37422},[31685,31686],[108,340384,340385,340388],{},[2627,340386,340387],{},"域 (Field)","：非零元素全是可逆元的环。",[15,340390,340391],{},[2627,340392,340393],{},"详细解答步骤：",[15,340395,340396],{},[2627,340397,340398,340399,340427],{},"证明 (1)：",[34,340400,340402,340415],{"className":340401},[31642],[34,340403,340405],{"className":340404},[31646],[31648,340406,340407],{"xmlns":31650},[31652,340408,340409,340413],{},[31655,340410,340411],{},[31658,340412,34386],{},[31665,340414,34386],{"encoding":31667},[34,340416,340418],{"className":340417,"ariaHidden":758},[31672],[34,340419,340421,340424],{"className":340420},[31676],[34,340422],{"className":340423,"style":33218},[31680],[34,340425,34386],{"className":340426,"style":34427},[31685,31686]," 的元素不是可逆元就是幂零元",[2230,340429,340430,340667],{},[108,340431,340432,340433,340534,340535,340563,340564,31901],{},"在环 ",[34,340434,340436,340466],{"className":340435},[31642],[34,340437,340439],{"className":340438},[31646],[31648,340440,340441],{"xmlns":31650},[31652,340442,340443,340463],{},[31655,340444,340445,340447,340449,340451,340453,340455,340461],{},[31658,340446,34386],{},[31662,340448,202],{},[31658,340450,97300],{"mathvariant":318416},[31658,340452,38080],{"mathvariant":38079},[31662,340454,456],{"stretchy":702},[37624,340456,340457,340459],{},[31658,340458,15],{},[31658,340460,31660],{},[31662,340462,2557],{"stretchy":702},[31665,340464,340465],{"encoding":31667},"R = \\mathbb{Z}\u002F(p^n)",[34,340467,340469,340487],{"className":340468,"ariaHidden":758},[31672],[34,340470,340472,340475,340478,340481,340484],{"className":340471},[31676],[34,340473],{"className":340474,"style":33218},[31680],[34,340476,34386],{"className":340477,"style":34427},[31685,31686],[34,340479],{"className":340480,"style":31846},[31845],[34,340482,202],{"className":340483},[31850],[34,340485],{"className":340486,"style":31846},[31845],[34,340488,340490,340493,340496,340499,340502,340531],{"className":340489},[31676],[34,340491],{"className":340492,"style":31774},[31680],[34,340494,97300],{"className":340495},[31685,318441],[34,340497,38080],{"className":340498},[31685],[34,340500,456],{"className":340501},[31782],[34,340503,340505,340508],{"className":340504},[31685],[34,340506,15],{"className":340507},[31685,31686],[34,340509,340511],{"className":340510},[31792],[34,340512,340514],{"className":340513},[31796],[34,340515,340517],{"className":340516},[31801],[34,340518,340520],{"className":340519,"style":37959},[31805],[34,340521,340522,340525],{"style":37777},[34,340523],{"className":340524,"style":31814},[31813],[34,340526,340528],{"className":340527},[31818,31819,31820,31821],[34,340529,31660],{"className":340530},[31685,31686,31821],[34,340532,2557],{"className":340533},[31690]," 中，任意元素都可以用一个整数 ",[34,340536,340538,340551],{"className":340537},[31642],[34,340539,340541],{"className":340540},[31646],[31648,340542,340543],{"xmlns":31650},[31652,340544,340545,340549],{},[31655,340546,340547],{},[31658,340548,111],{},[31665,340550,111],{"encoding":31667},[34,340552,340554],{"className":340553,"ariaHidden":758},[31672],[34,340555,340557,340560],{"className":340556},[31676],[34,340558],{"className":340559,"style":32527},[31680],[34,340561,111],{"className":340562},[31685,31686]," 来表示（",[34,340565,340567,340593],{"className":340566},[31642],[34,340568,340570],{"className":340569},[31646],[31648,340571,340572],{"xmlns":31650},[31652,340573,340574,340590],{},[31655,340575,340576,340578,340580,340582,340584],{},[31749,340577,459],{},[31662,340579,45972],{},[31658,340581,111],{},[31662,340583,932],{},[37624,340585,340586,340588],{},[31658,340587,15],{},[31658,340589,31660],{},[31665,340591,340592],{"encoding":31667},"0 \\le a \u003C p^n",[34,340594,340596,340614,340632],{"className":340595,"ariaHidden":758},[31672],[34,340597,340599,340602,340605,340608,340611],{"className":340598},[31676],[34,340600],{"className":340601,"style":55382},[31680],[34,340603,459],{"className":340604},[31685],[34,340606],{"className":340607,"style":31846},[31845],[34,340609,45972],{"className":340610},[31850],[34,340612],{"className":340613,"style":31846},[31845],[34,340615,340617,340620,340623,340626,340629],{"className":340616},[31676],[34,340618],{"className":340619,"style":34812},[31680],[34,340621,111],{"className":340622},[31685,31686],[34,340624],{"className":340625,"style":31846},[31845],[34,340627,932],{"className":340628},[31850],[34,340630],{"className":340631,"style":31846},[31845],[34,340633,340635,340638],{"className":340634},[31676],[34,340636],{"className":340637,"style":316112},[31680],[34,340639,340641,340644],{"className":340640},[31685],[34,340642,15],{"className":340643},[31685,31686],[34,340645,340647],{"className":340646},[31792],[34,340648,340650],{"className":340649},[31796],[34,340651,340653],{"className":340652},[31801],[34,340654,340656],{"className":340655,"style":37959},[31805],[34,340657,340658,340661],{"style":37777},[34,340659],{"className":340660,"style":31814},[31813],[34,340662,340664],{"className":340663},[31818,31819,31820,31821],[34,340665,31660],{"className":340666},[31685,31686,31821],[108,340668,340669,340670,340698,340699,340757,340758,340849],{},"对于任意整数 ",[34,340671,340673,340686],{"className":340672},[31642],[34,340674,340676],{"className":340675},[31646],[31648,340677,340678],{"xmlns":31650},[31652,340679,340680,340684],{},[31655,340681,340682],{},[31658,340683,111],{},[31665,340685,111],{"encoding":31667},[34,340687,340689],{"className":340688,"ariaHidden":758},[31672],[34,340690,340692,340695],{"className":340691},[31676],[34,340693],{"className":340694,"style":32527},[31680],[34,340696,111],{"className":340697},[31685,31686],"，它与模数 ",[34,340700,340702,340719],{"className":340701},[31642],[34,340703,340705],{"className":340704},[31646],[31648,340706,340707],{"xmlns":31650},[31652,340708,340709,340717],{},[31655,340710,340711],{},[37624,340712,340713,340715],{},[31658,340714,15],{},[31658,340716,31660],{},[31665,340718,339442],{"encoding":31667},[34,340720,340722],{"className":340721,"ariaHidden":758},[31672],[34,340723,340725,340728],{"className":340724},[31676],[34,340726],{"className":340727,"style":316112},[31680],[34,340729,340731,340734],{"className":340730},[31685],[34,340732,15],{"className":340733},[31685,31686],[34,340735,340737],{"className":340736},[31792],[34,340738,340740],{"className":340739},[31796],[34,340741,340743],{"className":340742},[31801],[34,340744,340746],{"className":340745,"style":37959},[31805],[34,340747,340748,340751],{"style":37777},[34,340749],{"className":340750,"style":31814},[31813],[34,340752,340754],{"className":340753},[31818,31819,31820,31821],[34,340755,31660],{"className":340756},[31685,31686,31821]," 的最大公约数 ",[34,340759,340761,340791],{"className":340760},[31642],[34,340762,340764],{"className":340763},[31646],[31648,340765,340766],{"xmlns":31650},[31652,340767,340768,340788],{},[31655,340769,340770,340772,340774,340776,340778,340780,340786],{},[31658,340771,314792],{},[31662,340773,32682],{},[31662,340775,456],{"stretchy":702},[31658,340777,111],{},[31662,340779,31960],{"separator":758},[37624,340781,340782,340784],{},[31658,340783,15],{},[31658,340785,31660],{},[31662,340787,2557],{"stretchy":702},[31665,340789,340790],{"encoding":31667},"\\gcd(a, p^n)",[34,340792,340794],{"className":340793,"ariaHidden":758},[31672],[34,340795,340797,340800,340805,340808,340811,340814,340817,340846],{"className":340796},[31676],[34,340798],{"className":340799,"style":31774},[31680],[34,340801,340803,314826],{"className":340802},[32767],[34,340804,31942],{"style":38619},[34,340806,456],{"className":340807},[31782],[34,340809,111],{"className":340810},[31685,31686],[34,340812,31960],{"className":340813},[32119],[34,340815],{"className":340816,"style":32123},[31845],[34,340818,340820,340823],{"className":340819},[31685],[34,340821,15],{"className":340822},[31685,31686],[34,340824,340826],{"className":340825},[31792],[34,340827,340829],{"className":340828},[31796],[34,340830,340832],{"className":340831},[31801],[34,340833,340835],{"className":340834,"style":37959},[31805],[34,340836,340837,340840],{"style":37777},[34,340838],{"className":340839,"style":31814},[31813],[34,340841,340843],{"className":340842},[31818,31819,31820,31821],[34,340844,31660],{"className":340845},[31685,31686,31821],[34,340847,2557],{"className":340848},[31690]," 只有两种情况：",[105,340851,340852,341026],{},[108,340853,340854,2664,340856,326989,340969,340997,340998,347],{},[2627,340855,326833],{},[34,340857,340859,340893],{"className":340858},[31642],[34,340860,340862],{"className":340861},[31646],[31648,340863,340864],{"xmlns":31650},[31652,340865,340866,340890],{},[31655,340867,340868,340870,340872,340874,340876,340878,340884,340886,340888],{},[31658,340869,314792],{},[31662,340871,32682],{},[31662,340873,456],{"stretchy":702},[31658,340875,111],{},[31662,340877,31960],{"separator":758},[37624,340879,340880,340882],{},[31658,340881,15],{},[31658,340883,31660],{},[31662,340885,2557],{"stretchy":702},[31662,340887,202],{},[31749,340889,640],{},[31665,340891,340892],{"encoding":31667},"\\gcd(a, p^n) = 1",[34,340894,340896,340960],{"className":340895,"ariaHidden":758},[31672],[34,340897,340899,340902,340907,340910,340913,340916,340919,340948,340951,340954,340957],{"className":340898},[31676],[34,340900],{"className":340901,"style":31774},[31680],[34,340903,340905,314826],{"className":340904},[32767],[34,340906,31942],{"style":38619},[34,340908,456],{"className":340909},[31782],[34,340911,111],{"className":340912},[31685,31686],[34,340914,31960],{"className":340915},[32119],[34,340917],{"className":340918,"style":32123},[31845],[34,340920,340922,340925],{"className":340921},[31685],[34,340923,15],{"className":340924},[31685,31686],[34,340926,340928],{"className":340927},[31792],[34,340929,340931],{"className":340930},[31796],[34,340932,340934],{"className":340933},[31801],[34,340935,340937],{"className":340936,"style":37959},[31805],[34,340938,340939,340942],{"style":37777},[34,340940],{"className":340941,"style":31814},[31813],[34,340943,340945],{"className":340944},[31818,31819,31820,31821],[34,340946,31660],{"className":340947},[31685,31686,31821],[34,340949,2557],{"className":340950},[31690],[34,340952],{"className":340953,"style":31846},[31845],[34,340955,202],{"className":340956},[31850],[34,340958],{"className":340959,"style":31846},[31845],[34,340961,340963,340966],{"className":340962},[31676],[34,340964],{"className":340965,"style":32617},[31680],[34,340967,640],{"className":340968},[31685],[34,340970,340972,340985],{"className":340971},[31642],[34,340973,340975],{"className":340974},[31646],[31648,340976,340977],{"xmlns":31650},[31652,340978,340979,340983],{},[31655,340980,340981],{},[31658,340982,111],{},[31665,340984,111],{"encoding":31667},[34,340986,340988],{"className":340987,"ariaHidden":758},[31672],[34,340989,340991,340994],{"className":340990},[31676],[34,340992],{"className":340993,"style":32527},[31680],[34,340995,111],{"className":340996},[31685,31686]," 不包含因子 ",[34,340999,341001,341014],{"className":341000},[31642],[34,341002,341004],{"className":341003},[31646],[31648,341005,341006],{"xmlns":31650},[31652,341007,341008,341012],{},[31655,341009,341010],{},[31658,341011,15],{},[31665,341013,15],{"encoding":31667},[34,341015,341017],{"className":341016,"ariaHidden":758},[31672],[34,341018,341020,341023],{"className":341019},[31676],[34,341021],{"className":341022,"style":38774},[31680],[34,341024,15],{"className":341025},[31685,31686],[108,341027,341028,2664,341030,44689,341143,341201,341202,341230,341231,341259,341260,341288,341289,347],{},[2627,341029,327551],{},[34,341031,341033,341067],{"className":341032},[31642],[34,341034,341036],{"className":341035},[31646],[31648,341037,341038],{"xmlns":31650},[31652,341039,341040,341064],{},[31655,341041,341042,341044,341046,341048,341050,341052,341058,341060,341062],{},[31658,341043,314792],{},[31662,341045,32682],{},[31662,341047,456],{"stretchy":702},[31658,341049,111],{},[31662,341051,31960],{"separator":758},[37624,341053,341054,341056],{},[31658,341055,15],{},[31658,341057,31660],{},[31662,341059,2557],{"stretchy":702},[31662,341061,29622],{},[31749,341063,640],{},[31665,341065,341066],{"encoding":31667},"\\gcd(a, p^n) > 1",[34,341068,341070,341134],{"className":341069,"ariaHidden":758},[31672],[34,341071,341073,341076,341081,341084,341087,341090,341093,341122,341125,341128,341131],{"className":341072},[31676],[34,341074],{"className":341075,"style":31774},[31680],[34,341077,341079,314826],{"className":341078},[32767],[34,341080,31942],{"style":38619},[34,341082,456],{"className":341083},[31782],[34,341085,111],{"className":341086},[31685,31686],[34,341088,31960],{"className":341089},[32119],[34,341091],{"className":341092,"style":32123},[31845],[34,341094,341096,341099],{"className":341095},[31685],[34,341097,15],{"className":341098},[31685,31686],[34,341100,341102],{"className":341101},[31792],[34,341103,341105],{"className":341104},[31796],[34,341106,341108],{"className":341107},[31801],[34,341109,341111],{"className":341110,"style":37959},[31805],[34,341112,341113,341116],{"style":37777},[34,341114],{"className":341115,"style":31814},[31813],[34,341117,341119],{"className":341118},[31818,31819,31820,31821],[34,341120,31660],{"className":341121},[31685,31686,31821],[34,341123,2557],{"className":341124},[31690],[34,341126],{"className":341127,"style":31846},[31845],[34,341129,29622],{"className":341130},[31850],[34,341132],{"className":341133,"style":31846},[31845],[34,341135,341137,341140],{"className":341136},[31676],[34,341138],{"className":341139,"style":32617},[31680],[34,341141,640],{"className":341142},[31685],[34,341144,341146,341163],{"className":341145},[31642],[34,341147,341149],{"className":341148},[31646],[31648,341150,341151],{"xmlns":31650},[31652,341152,341153,341161],{},[31655,341154,341155],{},[37624,341156,341157,341159],{},[31658,341158,15],{},[31658,341160,31660],{},[31665,341162,339442],{"encoding":31667},[34,341164,341166],{"className":341165,"ariaHidden":758},[31672],[34,341167,341169,341172],{"className":341168},[31676],[34,341170],{"className":341171,"style":316112},[31680],[34,341173,341175,341178],{"className":341174},[31685],[34,341176,15],{"className":341177},[31685,31686],[34,341179,341181],{"className":341180},[31792],[34,341182,341184],{"className":341183},[31796],[34,341185,341187],{"className":341186},[31801],[34,341188,341190],{"className":341189,"style":37959},[31805],[34,341191,341192,341195],{"style":37777},[34,341193],{"className":341194,"style":31814},[31813],[34,341196,341198],{"className":341197},[31818,31819,31820,31821],[34,341199,31660],{"className":341200},[31685,31686,31821]," 只有素因子 ",[34,341203,341205,341218],{"className":341204},[31642],[34,341206,341208],{"className":341207},[31646],[31648,341209,341210],{"xmlns":31650},[31652,341211,341212,341216],{},[31655,341213,341214],{},[31658,341215,15],{},[31665,341217,15],{"encoding":31667},[34,341219,341221],{"className":341220,"ariaHidden":758},[31672],[34,341222,341224,341227],{"className":341223},[31676],[34,341225],{"className":341226,"style":38774},[31680],[34,341228,15],{"className":341229},[31685,31686],"，这意味着 ",[34,341232,341234,341247],{"className":341233},[31642],[34,341235,341237],{"className":341236},[31646],[31648,341238,341239],{"xmlns":31650},[31652,341240,341241,341245],{},[31655,341242,341243],{},[31658,341244,111],{},[31665,341246,111],{"encoding":31667},[34,341248,341250],{"className":341249,"ariaHidden":758},[31672],[34,341251,341253,341256],{"className":341252},[31676],[34,341254],{"className":341255,"style":32527},[31680],[34,341257,111],{"className":341258},[31685,31686]," 必须是 ",[34,341261,341263,341276],{"className":341262},[31642],[34,341264,341266],{"className":341265},[31646],[31648,341267,341268],{"xmlns":31650},[31652,341269,341270,341274],{},[31655,341271,341272],{},[31658,341273,15],{},[31665,341275,15],{"encoding":31667},[34,341277,341279],{"className":341278,"ariaHidden":758},[31672],[34,341280,341282,341285],{"className":341281},[31676],[34,341283],{"className":341284,"style":38774},[31680],[34,341286,15],{"className":341287},[31685,31686]," 的倍数，即 ",[34,341290,341292,341310],{"className":341291},[31642],[34,341293,341295],{"className":341294},[31646],[31648,341296,341297],{"xmlns":31650},[31652,341298,341299,341307],{},[31655,341300,341301,341303,341305],{},[31658,341302,15],{},[31662,341304,162086],{},[31658,341306,111],{},[31665,341308,341309],{"encoding":31667},"p \\mid a",[34,341311,341313,341331],{"className":341312,"ariaHidden":758},[31672],[34,341314,341316,341319,341322,341325,341328],{"className":341315},[31676],[34,341317],{"className":341318,"style":31774},[31680],[34,341320,15],{"className":341321},[31685,31686],[34,341323],{"className":341324,"style":31846},[31845],[34,341326,162086],{"className":341327},[31850],[34,341329],{"className":341330,"style":31846},[31845],[34,341332,341334,341337],{"className":341333},[31676],[34,341335],{"className":341336,"style":32527},[31680],[34,341338,111],{"className":341339},[31685,31686],[2230,341341,341342,341937,343180],{"start":49},[108,341343,341344,341347,341348,341460,341461,315523,341464,315343,341505,341618,341619,341677,341678,341819,341820,314945,341848,341876,341877,341905,341906,113124,341934,347],{},[2627,341345,341346],{},"分析情况 A（可逆元）","：\n如果 ",[34,341349,341351,341384],{"className":341350},[31642],[34,341352,341354],{"className":341353},[31646],[31648,341355,341356],{"xmlns":31650},[31652,341357,341358,341382],{},[31655,341359,341360,341362,341364,341366,341368,341370,341376,341378,341380],{},[31658,341361,314792],{},[31662,341363,32682],{},[31662,341365,456],{"stretchy":702},[31658,341367,111],{},[31662,341369,31960],{"separator":758},[37624,341371,341372,341374],{},[31658,341373,15],{},[31658,341375,31660],{},[31662,341377,2557],{"stretchy":702},[31662,341379,202],{},[31749,341381,640],{},[31665,341383,340892],{"encoding":31667},[34,341385,341387,341451],{"className":341386,"ariaHidden":758},[31672],[34,341388,341390,341393,341398,341401,341404,341407,341410,341439,341442,341445,341448],{"className":341389},[31676],[34,341391],{"className":341392,"style":31774},[31680],[34,341394,341396,314826],{"className":341395},[32767],[34,341397,31942],{"style":38619},[34,341399,456],{"className":341400},[31782],[34,341402,111],{"className":341403},[31685,31686],[34,341405,31960],{"className":341406},[32119],[34,341408],{"className":341409,"style":32123},[31845],[34,341411,341413,341416],{"className":341412},[31685],[34,341414,15],{"className":341415},[31685,31686],[34,341417,341419],{"className":341418},[31792],[34,341420,341422],{"className":341421},[31796],[34,341423,341425],{"className":341424},[31801],[34,341426,341428],{"className":341427,"style":37959},[31805],[34,341429,341430,341433],{"style":37777},[34,341431],{"className":341432,"style":31814},[31813],[34,341434,341436],{"className":341435},[31818,31819,31820,31821],[34,341437,31660],{"className":341438},[31685,31686,31821],[34,341440,2557],{"className":341441},[31690],[34,341443],{"className":341444,"style":31846},[31845],[34,341446,202],{"className":341447},[31850],[34,341449],{"className":341450,"style":31846},[31845],[34,341452,341454,341457],{"className":341453},[31676],[34,341455],{"className":341456,"style":32617},[31680],[34,341458,640],{"className":341459},[31685],"，根据数论中的",[2627,341462,341463],{},"贝祖定理 (Bézout's identity)",[34,341465,341467,341484],{"className":341466},[31642],[34,341468,341470],{"className":341469},[31646],[31648,341471,341472],{"xmlns":31650},[31652,341473,341474,341482],{},[31655,341475,341476,341478,341480],{},[31658,341477,94628],{},[31662,341479,31960],{"separator":758},[31658,341481,56782],{},[31665,341483,315321],{"encoding":31667},[34,341485,341487],{"className":341486,"ariaHidden":758},[31672],[34,341488,341490,341493,341496,341499,341502],{"className":341489},[31676],[34,341491],{"className":341492,"style":38774},[31680],[34,341494,94628],{"className":341495},[31685,31686],[34,341497,31960],{"className":341498},[32119],[34,341500],{"className":341501,"style":32123},[31845],[34,341503,56782],{"className":341504,"style":32062},[31685,31686],[34,341506,341508,341538],{"className":341507},[31642],[34,341509,341511],{"className":341510},[31646],[31648,341512,341513],{"xmlns":31650},[31652,341514,341515,341535],{},[31655,341516,341517,341519,341521,341523,341529,341531,341533],{},[31658,341518,111],{},[31658,341520,94628],{},[31662,341522,223],{},[37624,341524,341525,341527],{},[31658,341526,15],{},[31658,341528,31660],{},[31658,341530,56782],{},[31662,341532,202],{},[31749,341534,640],{},[31665,341536,341537],{"encoding":31667},"au + p^n v = 1",[34,341539,341541,341562,341609],{"className":341540,"ariaHidden":758},[31672],[34,341542,341544,341547,341550,341553,341556,341559],{"className":341543},[31676],[34,341545],{"className":341546,"style":32597},[31680],[34,341548,111],{"className":341549},[31685,31686],[34,341551,94628],{"className":341552},[31685,31686],[34,341554],{"className":341555,"style":32604},[31845],[34,341557,223],{"className":341558},[32024],[34,341560],{"className":341561,"style":32604},[31845],[34,341563,341565,341568,341597,341600,341603,341606],{"className":341564},[31676],[34,341566],{"className":341567,"style":316112},[31680],[34,341569,341571,341574],{"className":341570},[31685],[34,341572,15],{"className":341573},[31685,31686],[34,341575,341577],{"className":341576},[31792],[34,341578,341580],{"className":341579},[31796],[34,341581,341583],{"className":341582},[31801],[34,341584,341586],{"className":341585,"style":37959},[31805],[34,341587,341588,341591],{"style":37777},[34,341589],{"className":341590,"style":31814},[31813],[34,341592,341594],{"className":341593},[31818,31819,31820,31821],[34,341595,31660],{"className":341596},[31685,31686,31821],[34,341598,56782],{"className":341599,"style":32062},[31685,31686],[34,341601],{"className":341602,"style":31846},[31845],[34,341604,202],{"className":341605},[31850],[34,341607],{"className":341608,"style":31846},[31845],[34,341610,341612,341615],{"className":341611},[31676],[34,341613],{"className":341614,"style":32617},[31680],[34,341616,640],{"className":341617},[31685],"。\n在模 ",[34,341620,341622,341639],{"className":341621},[31642],[34,341623,341625],{"className":341624},[31646],[31648,341626,341627],{"xmlns":31650},[31652,341628,341629,341637],{},[31655,341630,341631],{},[37624,341632,341633,341635],{},[31658,341634,15],{},[31658,341636,31660],{},[31665,341638,339442],{"encoding":31667},[34,341640,341642],{"className":341641,"ariaHidden":758},[31672],[34,341643,341645,341648],{"className":341644},[31676],[34,341646],{"className":341647,"style":316112},[31680],[34,341649,341651,341654],{"className":341650},[31685],[34,341652,15],{"className":341653},[31685,31686],[34,341655,341657],{"className":341656},[31792],[34,341658,341660],{"className":341659},[31796],[34,341661,341663],{"className":341662},[31801],[34,341664,341666],{"className":341665,"style":37959},[31805],[34,341667,341668,341671],{"style":37777},[34,341669],{"className":341670,"style":31814},[31813],[34,341672,341674],{"className":341673},[31818,31819,31820,31821],[34,341675,31660],{"className":341676},[31685,31686,31821]," 的意义下，这个等式变为：\n",[34,341679,341681,341726],{"className":341680},[31642],[34,341682,341684],{"className":341683},[31646],[31648,341685,341686],{"xmlns":31650},[31652,341687,341688,341723],{},[31655,341689,341690,341692,341694,341697,341699,341701,341703,341705,341713,341715,341721],{},[31658,341691,111],{},[31658,341693,94628],{},[31662,341695,341696],{},"≡",[31749,341698,640],{},[31845,341700],{},[31845,341702],{"width":208910},[31662,341704,456],{"stretchy":702},[31655,341706,341707,341709,341711],{},[31658,341708,31759],{"mathvariant":38079},[31658,341710,43383],{"mathvariant":38079},[31658,341712,46900],{"mathvariant":38079},[31845,341714],{"width":208923},[37624,341716,341717,341719],{},[31658,341718,15],{},[31658,341720,31660],{},[31662,341722,2557],{"stretchy":702},[31665,341724,341725],{"encoding":31667},"au \\equiv 1 \\pmod{p^n}",[34,341727,341729,341751,341766],{"className":341728,"ariaHidden":758},[31672],[34,341730,341732,341736,341739,341742,341745,341748],{"className":341731},[31676],[34,341733],{"className":341734,"style":341735},[31680],"height:0.4637em;",[34,341737,111],{"className":341738},[31685,31686],[34,341740,94628],{"className":341741},[31685,31686],[34,341743],{"className":341744,"style":31846},[31845],[34,341746,341696],{"className":341747},[31850],[34,341749],{"className":341750,"style":31846},[31845],[34,341752,341754,341757,341760,341763],{"className":341753},[31676],[34,341755],{"className":341756,"style":32617},[31680],[34,341758,640],{"className":341759},[31685],[34,341761],{"className":341762},[31845,208951],[34,341764],{"className":341765,"style":208955},[31845],[34,341767,341769,341772,341775,341784,341787,341816],{"className":341768},[31676],[34,341770],{"className":341771,"style":31774},[31680],[34,341773,456],{"className":341774},[31782],[34,341776,341778],{"className":341777},[31685],[34,341779,341781],{"className":341780},[31685],[34,341782,207924],{"className":341783},[31685,207923],[34,341785],{"className":341786,"style":208977},[31845],[34,341788,341790,341793],{"className":341789},[31685],[34,341791,15],{"className":341792},[31685,31686],[34,341794,341796],{"className":341795},[31792],[34,341797,341799],{"className":341798},[31796],[34,341800,341802],{"className":341801},[31801],[34,341803,341805],{"className":341804,"style":37959},[31805],[34,341806,341807,341810],{"style":37777},[34,341808],{"className":341809,"style":31814},[31813],[34,341811,341813],{"className":341812},[31818,31819,31820,31821],[34,341814,31660],{"className":341815},[31685,31686,31821],[34,341817,2557],{"className":341818},[31690],"\n这意味着 ",[34,341821,341823,341836],{"className":341822},[31642],[34,341824,341826],{"className":341825},[31646],[31648,341827,341828],{"xmlns":31650},[31652,341829,341830,341834],{},[31655,341831,341832],{},[31658,341833,111],{},[31665,341835,111],{"encoding":31667},[34,341837,341839],{"className":341838,"ariaHidden":758},[31672],[34,341840,341842,341845],{"className":341841},[31676],[34,341843],{"className":341844,"style":32527},[31680],[34,341846,111],{"className":341847},[31685,31686],[34,341849,341851,341864],{"className":341850},[31642],[34,341852,341854],{"className":341853},[31646],[31648,341855,341856],{"xmlns":31650},[31652,341857,341858,341862],{},[31655,341859,341860],{},[31658,341861,34386],{},[31665,341863,34386],{"encoding":31667},[34,341865,341867],{"className":341866,"ariaHidden":758},[31672],[34,341868,341870,341873],{"className":341869},[31676],[34,341871],{"className":341872,"style":33218},[31680],[34,341874,34386],{"className":341875,"style":34427},[31685,31686]," 中有乘法逆元 ",[34,341878,341880,341893],{"className":341879},[31642],[34,341881,341883],{"className":341882},[31646],[31648,341884,341885],{"xmlns":31650},[31652,341886,341887,341891],{},[31655,341888,341889],{},[31658,341890,94628],{},[31665,341892,94628],{"encoding":31667},[34,341894,341896],{"className":341895,"ariaHidden":758},[31672],[34,341897,341899,341902],{"className":341898},[31676],[34,341900],{"className":341901,"style":32527},[31680],[34,341903,94628],{"className":341904},[31685,31686],"。所以，",[34,341907,341909,341922],{"className":341908},[31642],[34,341910,341912],{"className":341911},[31646],[31648,341913,341914],{"xmlns":31650},[31652,341915,341916,341920],{},[31655,341917,341918],{},[31658,341919,111],{},[31665,341921,111],{"encoding":31667},[34,341923,341925],{"className":341924,"ariaHidden":758},[31672],[34,341926,341928,341931],{"className":341927},[31676],[34,341929],{"className":341930,"style":32527},[31680],[34,341932,111],{"className":341933},[31685,31686],[2627,341935,341936],{},"可逆元",[108,341938,341939,341347,341942,55016,342054,342104,342105,342133,342134,342207,342208,342236,342237,67452,342265,342293,342294,342541,342542,342600,342601,11008,342712,342770,342771,343039,343040,315343,343068,13977,343149,113124,343177,347],{},[2627,341940,341941],{},"分析情况 B（幂零元）",[34,341943,341945,341978],{"className":341944},[31642],[34,341946,341948],{"className":341947},[31646],[31648,341949,341950],{"xmlns":31650},[31652,341951,341952,341976],{},[31655,341953,341954,341956,341958,341960,341962,341964,341970,341972,341974],{},[31658,341955,314792],{},[31662,341957,32682],{},[31662,341959,456],{"stretchy":702},[31658,341961,111],{},[31662,341963,31960],{"separator":758},[37624,341965,341966,341968],{},[31658,341967,15],{},[31658,341969,31660],{},[31662,341971,2557],{"stretchy":702},[31662,341973,29622],{},[31749,341975,640],{},[31665,341977,341066],{"encoding":31667},[34,341979,341981,342045],{"className":341980,"ariaHidden":758},[31672],[34,341982,341984,341987,341992,341995,341998,342001,342004,342033,342036,342039,342042],{"className":341983},[31676],[34,341985],{"className":341986,"style":31774},[31680],[34,341988,341990,314826],{"className":341989},[32767],[34,341991,31942],{"style":38619},[34,341993,456],{"className":341994},[31782],[34,341996,111],{"className":341997},[31685,31686],[34,341999,31960],{"className":342000},[32119],[34,342002],{"className":342003,"style":32123},[31845],[34,342005,342007,342010],{"className":342006},[31685],[34,342008,15],{"className":342009},[31685,31686],[34,342011,342013],{"className":342012},[31792],[34,342014,342016],{"className":342015},[31796],[34,342017,342019],{"className":342018},[31801],[34,342020,342022],{"className":342021,"style":37959},[31805],[34,342023,342024,342027],{"style":37777},[34,342025],{"className":342026,"style":31814},[31813],[34,342028,342030],{"className":342029},[31818,31819,31820,31821],[34,342031,31660],{"className":342032},[31685,31686,31821],[34,342034,2557],{"className":342035},[31690],[34,342037],{"className":342038,"style":31846},[31845],[34,342040,29622],{"className":342041},[31850],[34,342043],{"className":342044,"style":31846},[31845],[34,342046,342048,342051],{"className":342047},[31676],[34,342049],{"className":342050,"style":32617},[31680],[34,342052,640],{"className":342053},[31685],[34,342055,342057,342074],{"className":342056},[31642],[34,342058,342060],{"className":342059},[31646],[31648,342061,342062],{"xmlns":31650},[31652,342063,342064,342072],{},[31655,342065,342066,342068,342070],{},[31658,342067,15],{},[31662,342069,162086],{},[31658,342071,111],{},[31665,342073,341309],{"encoding":31667},[34,342075,342077,342095],{"className":342076,"ariaHidden":758},[31672],[34,342078,342080,342083,342086,342089,342092],{"className":342079},[31676],[34,342081],{"className":342082,"style":31774},[31680],[34,342084,15],{"className":342085},[31685,31686],[34,342087],{"className":342088,"style":31846},[31845],[34,342090,162086],{"className":342091},[31850],[34,342093],{"className":342094,"style":31846},[31845],[34,342096,342098,342101],{"className":342097},[31676],[34,342099],{"className":342100,"style":32527},[31680],[34,342102,111],{"className":342103},[31685,31686],"。我们可以把 ",[34,342106,342108,342121],{"className":342107},[31642],[34,342109,342111],{"className":342110},[31646],[31648,342112,342113],{"xmlns":31650},[31652,342114,342115,342119],{},[31655,342116,342117],{},[31658,342118,111],{},[31665,342120,111],{"encoding":31667},[34,342122,342124],{"className":342123,"ariaHidden":758},[31672],[34,342125,342127,342130],{"className":342126},[31676],[34,342128],{"className":342129,"style":32527},[31680],[34,342131,111],{"className":342132},[31685,31686]," 写成 ",[34,342135,342137,342159],{"className":342136},[31642],[34,342138,342140],{"className":342139},[31646],[31648,342141,342142],{"xmlns":31650},[31652,342143,342144,342156],{},[31655,342145,342146,342148,342150,342152,342154],{},[31658,342147,111],{},[31662,342149,202],{},[31658,342151,33759],{},[31662,342153,71191],{},[31658,342155,15],{},[31665,342157,342158],{"encoding":31667},"a = k \\cdot p",[34,342160,342162,342180,342198],{"className":342161,"ariaHidden":758},[31672],[34,342163,342165,342168,342171,342174,342177],{"className":342164},[31676],[34,342166],{"className":342167,"style":32527},[31680],[34,342169,111],{"className":342170},[31685,31686],[34,342172],{"className":342173,"style":31846},[31845],[34,342175,202],{"className":342176},[31850],[34,342178],{"className":342179,"style":31846},[31845],[34,342181,342183,342186,342189,342192,342195],{"className":342182},[31676],[34,342184],{"className":342185,"style":31681},[31680],[34,342187,33759],{"className":342188,"style":33774},[31685,31686],[34,342190],{"className":342191,"style":32604},[31845],[34,342193,71191],{"className":342194},[32024],[34,342196],{"className":342197,"style":32604},[31845],[34,342199,342201,342204],{"className":342200},[31676],[34,342202],{"className":342203,"style":38774},[31680],[34,342205,15],{"className":342206},[31685,31686]," 的形式，其中 ",[34,342209,342211,342224],{"className":342210},[31642],[34,342212,342214],{"className":342213},[31646],[31648,342215,342216],{"xmlns":31650},[31652,342217,342218,342222],{},[31655,342219,342220],{},[31658,342221,33759],{},[31665,342223,33759],{"encoding":31667},[34,342225,342227],{"className":342226,"ariaHidden":758},[31672],[34,342228,342230,342233],{"className":342229},[31676],[34,342231],{"className":342232,"style":31681},[31680],[34,342234,33759],{"className":342235,"style":33774},[31685,31686]," 是某个整数。\n让我们计算 ",[34,342238,342240,342253],{"className":342239},[31642],[34,342241,342243],{"className":342242},[31646],[31648,342244,342245],{"xmlns":31650},[31652,342246,342247,342251],{},[31655,342248,342249],{},[31658,342250,111],{},[31665,342252,111],{"encoding":31667},[34,342254,342256],{"className":342255,"ariaHidden":758},[31672],[34,342257,342259,342262],{"className":342258},[31676],[34,342260],{"className":342261,"style":32527},[31680],[34,342263,111],{"className":342264},[31685,31686],[34,342266,342268,342281],{"className":342267},[31642],[34,342269,342271],{"className":342270},[31646],[31648,342272,342273],{"xmlns":31650},[31652,342274,342275,342279],{},[31655,342276,342277],{},[31658,342278,31660],{},[31665,342280,31660],{"encoding":31667},[34,342282,342284],{"className":342283,"ariaHidden":758},[31672],[34,342285,342287,342290],{"className":342286},[31676],[34,342288],{"className":342289,"style":32527},[31680],[34,342291,31660],{"className":342292},[31685,31686]," 次方：\n",[34,342295,342297,342347],{"className":342296},[31642],[34,342298,342300],{"className":342299},[31646],[31648,342301,342302],{"xmlns":31650},[31652,342303,342304,342344],{},[31655,342305,342306,342312,342314,342316,342318,342320,342322,342328,342330,342336,342338],{},[37624,342307,342308,342310],{},[31658,342309,111],{},[31658,342311,31660],{},[31662,342313,202],{},[31662,342315,456],{"stretchy":702},[31658,342317,33759],{},[31662,342319,71191],{},[31658,342321,15],{},[37624,342323,342324,342326],{},[31662,342325,2557],{"stretchy":702},[31658,342327,31660],{},[31662,342329,202],{},[37624,342331,342332,342334],{},[31658,342333,33759],{},[31658,342335,31660],{},[31662,342337,71191],{},[37624,342339,342340,342342],{},[31658,342341,15],{},[31658,342343,31660],{},[31665,342345,342346],{"encoding":31667},"a^n = (k \\cdot p)^n = k^n \\cdot p^n",[34,342348,342350,342394,342415,342462,342506],{"className":342349,"ariaHidden":758},[31672],[34,342351,342353,342356,342385,342388,342391],{"className":342352},[31676],[34,342354],{"className":342355,"style":37959},[31680],[34,342357,342359,342362],{"className":342358},[31685],[34,342360,111],{"className":342361},[31685,31686],[34,342363,342365],{"className":342364},[31792],[34,342366,342368],{"className":342367},[31796],[34,342369,342371],{"className":342370},[31801],[34,342372,342374],{"className":342373,"style":37959},[31805],[34,342375,342376,342379],{"style":37777},[34,342377],{"className":342378,"style":31814},[31813],[34,342380,342382],{"className":342381},[31818,31819,31820,31821],[34,342383,31660],{"className":342384},[31685,31686,31821],[34,342386],{"className":342387,"style":31846},[31845],[34,342389,202],{"className":342390},[31850],[34,342392],{"className":342393,"style":31846},[31845],[34,342395,342397,342400,342403,342406,342409,342412],{"className":342396},[31676],[34,342398],{"className":342399,"style":31774},[31680],[34,342401,456],{"className":342402},[31782],[34,342404,33759],{"className":342405,"style":33774},[31685,31686],[34,342407],{"className":342408,"style":32604},[31845],[34,342410,71191],{"className":342411},[32024],[34,342413],{"className":342414,"style":32604},[31845],[34,342416,342418,342421,342424,342453,342456,342459],{"className":342417},[31676],[34,342419],{"className":342420,"style":31774},[31680],[34,342422,15],{"className":342423},[31685,31686],[34,342425,342427,342430],{"className":342426},[31690],[34,342428,2557],{"className":342429},[31690],[34,342431,342433],{"className":342432},[31792],[34,342434,342436],{"className":342435},[31796],[34,342437,342439],{"className":342438},[31801],[34,342440,342442],{"className":342441,"style":37959},[31805],[34,342443,342444,342447],{"style":37777},[34,342445],{"className":342446,"style":31814},[31813],[34,342448,342450],{"className":342449},[31818,31819,31820,31821],[34,342451,31660],{"className":342452},[31685,31686,31821],[34,342454],{"className":342455,"style":31846},[31845],[34,342457,202],{"className":342458},[31850],[34,342460],{"className":342461,"style":31846},[31845],[34,342463,342465,342468,342497,342500,342503],{"className":342464},[31676],[34,342466],{"className":342467,"style":31681},[31680],[34,342469,342471,342474],{"className":342470},[31685],[34,342472,33759],{"className":342473,"style":33774},[31685,31686],[34,342475,342477],{"className":342476},[31792],[34,342478,342480],{"className":342479},[31796],[34,342481,342483],{"className":342482},[31801],[34,342484,342486],{"className":342485,"style":37959},[31805],[34,342487,342488,342491],{"style":37777},[34,342489],{"className":342490,"style":31814},[31813],[34,342492,342494],{"className":342493},[31818,31819,31820,31821],[34,342495,31660],{"className":342496},[31685,31686,31821],[34,342498],{"className":342499,"style":32604},[31845],[34,342501,71191],{"className":342502},[32024],[34,342504],{"className":342505,"style":32604},[31845],[34,342507,342509,342512],{"className":342508},[31676],[34,342510],{"className":342511,"style":316112},[31680],[34,342513,342515,342518],{"className":342514},[31685],[34,342516,15],{"className":342517},[31685,31686],[34,342519,342521],{"className":342520},[31792],[34,342522,342524],{"className":342523},[31796],[34,342525,342527],{"className":342526},[31801],[34,342528,342530],{"className":342529,"style":37959},[31805],[34,342531,342532,342535],{"style":37777},[34,342533],{"className":342534,"style":31814},[31813],[34,342536,342538],{"className":342537},[31818,31819,31820,31821],[34,342539,31660],{"className":342540},[31685,31686,31821],"\n因为我们要模 ",[34,342543,342545,342562],{"className":342544},[31642],[34,342546,342548],{"className":342547},[31646],[31648,342549,342550],{"xmlns":31650},[31652,342551,342552,342560],{},[31655,342553,342554],{},[37624,342555,342556,342558],{},[31658,342557,15],{},[31658,342559,31660],{},[31665,342561,339442],{"encoding":31667},[34,342563,342565],{"className":342564,"ariaHidden":758},[31672],[34,342566,342568,342571],{"className":342567},[31676],[34,342569],{"className":342570,"style":316112},[31680],[34,342572,342574,342577],{"className":342573},[31685],[34,342575,15],{"className":342576},[31685,31686],[34,342578,342580],{"className":342579},[31792],[34,342581,342583],{"className":342582},[31796],[34,342584,342586],{"className":342585},[31801],[34,342587,342589],{"className":342588,"style":37959},[31805],[34,342590,342591,342594],{"style":37777},[34,342592],{"className":342593,"style":31814},[31813],[34,342595,342597],{"className":342596},[31818,31819,31820,31821],[34,342598,31660],{"className":342599},[31685,31686,31821],"，显然 ",[34,342602,342604,342630],{"className":342603},[31642],[34,342605,342607],{"className":342606},[31646],[31648,342608,342609],{"xmlns":31650},[31652,342610,342611,342627],{},[31655,342612,342613,342619,342621],{},[37624,342614,342615,342617],{},[31658,342616,33759],{},[31658,342618,31660],{},[31662,342620,71191],{},[37624,342622,342623,342625],{},[31658,342624,15],{},[31658,342626,31660],{},[31665,342628,342629],{"encoding":31667},"k^n \\cdot p^n",[34,342631,342633,342677],{"className":342632,"ariaHidden":758},[31672],[34,342634,342636,342639,342668,342671,342674],{"className":342635},[31676],[34,342637],{"className":342638,"style":31681},[31680],[34,342640,342642,342645],{"className":342641},[31685],[34,342643,33759],{"className":342644,"style":33774},[31685,31686],[34,342646,342648],{"className":342647},[31792],[34,342649,342651],{"className":342650},[31796],[34,342652,342654],{"className":342653},[31801],[34,342655,342657],{"className":342656,"style":37959},[31805],[34,342658,342659,342662],{"style":37777},[34,342660],{"className":342661,"style":31814},[31813],[34,342663,342665],{"className":342664},[31818,31819,31820,31821],[34,342666,31660],{"className":342667},[31685,31686,31821],[34,342669],{"className":342670,"style":32604},[31845],[34,342672,71191],{"className":342673},[32024],[34,342675],{"className":342676,"style":32604},[31845],[34,342678,342680,342683],{"className":342679},[31676],[34,342681],{"className":342682,"style":316112},[31680],[34,342684,342686,342689],{"className":342685},[31685],[34,342687,15],{"className":342688},[31685,31686],[34,342690,342692],{"className":342691},[31792],[34,342693,342695],{"className":342694},[31796],[34,342696,342698],{"className":342697},[31801],[34,342699,342701],{"className":342700,"style":37959},[31805],[34,342702,342703,342706],{"style":37777},[34,342704],{"className":342705,"style":31814},[31813],[34,342707,342709],{"className":342708},[31818,31819,31820,31821],[34,342710,31660],{"className":342711},[31685,31686,31821],[34,342713,342715,342732],{"className":342714},[31642],[34,342716,342718],{"className":342717},[31646],[31648,342719,342720],{"xmlns":31650},[31652,342721,342722,342730],{},[31655,342723,342724],{},[37624,342725,342726,342728],{},[31658,342727,15],{},[31658,342729,31660],{},[31665,342731,339442],{"encoding":31667},[34,342733,342735],{"className":342734,"ariaHidden":758},[31672],[34,342736,342738,342741],{"className":342737},[31676],[34,342739],{"className":342740,"style":316112},[31680],[34,342742,342744,342747],{"className":342743},[31685],[34,342745,15],{"className":342746},[31685,31686],[34,342748,342750],{"className":342749},[31792],[34,342751,342753],{"className":342752},[31796],[34,342754,342756],{"className":342755},[31801],[34,342757,342759],{"className":342758,"style":37959},[31805],[34,342760,342761,342764],{"style":37777},[34,342762],{"className":342763,"style":31814},[31813],[34,342765,342767],{"className":342766},[31818,31819,31820,31821],[34,342768,31660],{"className":342769},[31685,31686,31821]," 的倍数。\n",[34,342772,342774,342836],{"className":342773},[31642],[34,342775,342777],{"className":342776},[31646],[31648,342778,342779],{"xmlns":31650},[31652,342780,342781,342833],{},[31655,342782,342783,342789,342791,342797,342799,342805,342807,342809,342811,342813,342815,342823,342825,342831],{},[37624,342784,342785,342787],{},[31658,342786,111],{},[31658,342788,31660],{},[31662,342790,202],{},[37624,342792,342793,342795],{},[31658,342794,33759],{},[31658,342796,31660],{},[31662,342798,71191],{},[37624,342800,342801,342803],{},[31658,342802,15],{},[31658,342804,31660],{},[31662,342806,341696],{},[31749,342808,459],{},[31845,342810],{},[31845,342812],{"width":208910},[31662,342814,456],{"stretchy":702},[31655,342816,342817,342819,342821],{},[31658,342818,31759],{"mathvariant":38079},[31658,342820,43383],{"mathvariant":38079},[31658,342822,46900],{"mathvariant":38079},[31845,342824],{"width":208923},[37624,342826,342827,342829],{},[31658,342828,15],{},[31658,342830,31660],{},[31662,342832,2557],{"stretchy":702},[31665,342834,342835],{"encoding":31667},"a^n = k^n \\cdot p^n \\equiv 0 \\pmod{p^n}",[34,342837,342839,342883,342927,342971,342986],{"className":342838,"ariaHidden":758},[31672],[34,342840,342842,342845,342874,342877,342880],{"className":342841},[31676],[34,342843],{"className":342844,"style":37959},[31680],[34,342846,342848,342851],{"className":342847},[31685],[34,342849,111],{"className":342850},[31685,31686],[34,342852,342854],{"className":342853},[31792],[34,342855,342857],{"className":342856},[31796],[34,342858,342860],{"className":342859},[31801],[34,342861,342863],{"className":342862,"style":37959},[31805],[34,342864,342865,342868],{"style":37777},[34,342866],{"className":342867,"style":31814},[31813],[34,342869,342871],{"className":342870},[31818,31819,31820,31821],[34,342872,31660],{"className":342873},[31685,31686,31821],[34,342875],{"className":342876,"style":31846},[31845],[34,342878,202],{"className":342879},[31850],[34,342881],{"className":342882,"style":31846},[31845],[34,342884,342886,342889,342918,342921,342924],{"className":342885},[31676],[34,342887],{"className":342888,"style":31681},[31680],[34,342890,342892,342895],{"className":342891},[31685],[34,342893,33759],{"className":342894,"style":33774},[31685,31686],[34,342896,342898],{"className":342897},[31792],[34,342899,342901],{"className":342900},[31796],[34,342902,342904],{"className":342903},[31801],[34,342905,342907],{"className":342906,"style":37959},[31805],[34,342908,342909,342912],{"style":37777},[34,342910],{"className":342911,"style":31814},[31813],[34,342913,342915],{"className":342914},[31818,31819,31820,31821],[34,342916,31660],{"className":342917},[31685,31686,31821],[34,342919],{"className":342920,"style":32604},[31845],[34,342922,71191],{"className":342923},[32024],[34,342925],{"className":342926,"style":32604},[31845],[34,342928,342930,342933,342962,342965,342968],{"className":342929},[31676],[34,342931],{"className":342932,"style":316112},[31680],[34,342934,342936,342939],{"className":342935},[31685],[34,342937,15],{"className":342938},[31685,31686],[34,342940,342942],{"className":342941},[31792],[34,342943,342945],{"className":342944},[31796],[34,342946,342948],{"className":342947},[31801],[34,342949,342951],{"className":342950,"style":37959},[31805],[34,342952,342953,342956],{"style":37777},[34,342954],{"className":342955,"style":31814},[31813],[34,342957,342959],{"className":342958},[31818,31819,31820,31821],[34,342960,31660],{"className":342961},[31685,31686,31821],[34,342963],{"className":342964,"style":31846},[31845],[34,342966,341696],{"className":342967},[31850],[34,342969],{"className":342970,"style":31846},[31845],[34,342972,342974,342977,342980,342983],{"className":342973},[31676],[34,342975],{"className":342976,"style":32617},[31680],[34,342978,459],{"className":342979},[31685],[34,342981],{"className":342982},[31845,208951],[34,342984],{"className":342985,"style":208955},[31845],[34,342987,342989,342992,342995,343004,343007,343036],{"className":342988},[31676],[34,342990],{"className":342991,"style":31774},[31680],[34,342993,456],{"className":342994},[31782],[34,342996,342998],{"className":342997},[31685],[34,342999,343001],{"className":343000},[31685],[34,343002,207924],{"className":343003},[31685,207923],[34,343005],{"className":343006,"style":208977},[31845],[34,343008,343010,343013],{"className":343009},[31685],[34,343011,15],{"className":343012},[31685,31686],[34,343014,343016],{"className":343015},[31792],[34,343017,343019],{"className":343018},[31796],[34,343020,343022],{"className":343021},[31801],[34,343023,343025],{"className":343024,"style":37959},[31805],[34,343026,343027,343030],{"style":37777},[34,343028],{"className":343029,"style":31814},[31813],[34,343031,343033],{"className":343032},[31818,31819,31820,31821],[34,343034,31660],{"className":343035},[31685,31686,31821],[34,343037,2557],{"className":343038},[31690],"\n根据定义，存在正整数 ",[34,343041,343043,343056],{"className":343042},[31642],[34,343044,343046],{"className":343045},[31646],[31648,343047,343048],{"xmlns":31650},[31652,343049,343050,343054],{},[31655,343051,343052],{},[31658,343053,31660],{},[31665,343055,31660],{"encoding":31667},[34,343057,343059],{"className":343058,"ariaHidden":758},[31672],[34,343060,343062,343065],{"className":343061},[31676],[34,343063],{"className":343064,"style":32527},[31680],[34,343066,31660],{"className":343067},[31685,31686],[34,343069,343071,343093],{"className":343070},[31642],[34,343072,343074],{"className":343073},[31646],[31648,343075,343076],{"xmlns":31650},[31652,343077,343078,343090],{},[31655,343079,343080,343086,343088],{},[37624,343081,343082,343084],{},[31658,343083,111],{},[31658,343085,31660],{},[31662,343087,202],{},[31749,343089,459],{},[31665,343091,343092],{"encoding":31667},"a^n = 0",[34,343094,343096,343140],{"className":343095,"ariaHidden":758},[31672],[34,343097,343099,343102,343131,343134,343137],{"className":343098},[31676],[34,343100],{"className":343101,"style":37959},[31680],[34,343103,343105,343108],{"className":343104},[31685],[34,343106,111],{"className":343107},[31685,31686],[34,343109,343111],{"className":343110},[31792],[34,343112,343114],{"className":343113},[31796],[34,343115,343117],{"className":343116},[31801],[34,343118,343120],{"className":343119,"style":37959},[31805],[34,343121,343122,343125],{"style":37777},[34,343123],{"className":343124,"style":31814},[31813],[34,343126,343128],{"className":343127},[31818,31819,31820,31821],[34,343129,31660],{"className":343130},[31685,31686,31821],[34,343132],{"className":343133,"style":31846},[31845],[34,343135,202],{"className":343136},[31850],[34,343138],{"className":343139,"style":31846},[31845],[34,343141,343143,343146],{"className":343142},[31676],[34,343144],{"className":343145,"style":32617},[31680],[34,343147,459],{"className":343148},[31685],[34,343150,343152,343165],{"className":343151},[31642],[34,343153,343155],{"className":343154},[31646],[31648,343156,343157],{"xmlns":31650},[31652,343158,343159,343163],{},[31655,343160,343161],{},[31658,343162,111],{},[31665,343164,111],{"encoding":31667},[34,343166,343168],{"className":343167,"ariaHidden":758},[31672],[34,343169,343171,343174],{"className":343170},[31676],[34,343172],{"className":343173,"style":32527},[31680],[34,343175,111],{"className":343176},[31685,31686],[2627,343178,343179],{},"幂零元",[108,343181,343182,2664,343184,343212,343213,343241,343242,343270],{},[2627,343183,114043],{},[34,343185,343187,343200],{"className":343186},[31642],[34,343188,343190],{"className":343189},[31646],[31648,343191,343192],{"xmlns":31650},[31652,343193,343194,343198],{},[31655,343195,343196],{},[31658,343197,34386],{},[31665,343199,34386],{"encoding":31667},[34,343201,343203],{"className":343202,"ariaHidden":758},[31672],[34,343204,343206,343209],{"className":343205},[31676],[34,343207],{"className":343208,"style":33218},[31680],[34,343210,34386],{"className":343211,"style":34427},[31685,31686]," 中的任何元素要么与 ",[34,343214,343216,343229],{"className":343215},[31642],[34,343217,343219],{"className":343218},[31646],[31648,343220,343221],{"xmlns":31650},[31652,343222,343223,343227],{},[31655,343224,343225],{},[31658,343226,15],{},[31665,343228,15],{"encoding":31667},[34,343230,343232],{"className":343231,"ariaHidden":758},[31672],[34,343233,343235,343238],{"className":343234},[31676],[34,343236],{"className":343237,"style":38774},[31680],[34,343239,15],{"className":343240},[31685,31686]," 互质（可逆元），要么是 ",[34,343243,343245,343258],{"className":343244},[31642],[34,343246,343248],{"className":343247},[31646],[31648,343249,343250],{"xmlns":31650},[31652,343251,343252,343256],{},[31655,343253,343254],{},[31658,343255,15],{},[31665,343257,15],{"encoding":31667},[34,343259,343261],{"className":343260,"ariaHidden":758},[31672],[34,343262,343264,343267],{"className":343263},[31676],[34,343265],{"className":343266,"style":38774},[31680],[34,343268,15],{"className":343269},[31685,31686]," 的倍数（幂零元）。不存在第三种状态。证毕。",[15,343272,343273],{},[2627,343274,343275,343276,339340,343304],{},"证明 (2)：",[34,343277,343279,343292],{"className":343278},[31642],[34,343280,343282],{"className":343281},[31646],[31648,343283,343284],{"xmlns":31650},[31652,343285,343286,343290],{},[31655,343287,343288],{},[31658,343289,34386],{},[31665,343291,34386],{"encoding":31667},[34,343293,343295],{"className":343294,"ariaHidden":758},[31672],[34,343296,343298,343301],{"className":343297},[31676],[34,343299],{"className":343300,"style":33218},[31680],[34,343302,34386],{"className":343303,"style":34427},[31685,31686],[34,343305,343307,343320],{"className":343306},[31642],[34,343308,343310],{"className":343309},[31646],[31648,343311,343312],{"xmlns":31650},[31652,343313,343314,343318],{},[31655,343315,343316],{},[31658,343317,105852],{},[31665,343319,105852],{"encoding":31667},[34,343321,343323],{"className":343322,"ariaHidden":758},[31672],[34,343324,343326,343329],{"className":343325},[31676],[34,343327],{"className":343328,"style":33218},[31680],[34,343330,105852],{"className":343331,"style":37422},[31685,31686],[2230,343333,343334,343527,343967,344489,345385],{},[108,343335,343336,343339,343340,343368,343369,330585,343425,4374,343476,347],{},[2627,343337,343338],{},"回顾素理想定义","：理想 ",[34,343341,343343,343356],{"className":343342},[31642],[34,343344,343346],{"className":343345},[31646],[31648,343347,343348],{"xmlns":31650},[31652,343349,343350,343354],{},[31655,343351,343352],{},[31658,343353,105852],{},[31665,343355,105852],{"encoding":31667},[34,343357,343359],{"className":343358,"ariaHidden":758},[31672],[34,343360,343362,343365],{"className":343361},[31676],[34,343363],{"className":343364,"style":33218},[31680],[34,343366,105852],{"className":343367,"style":37422},[31685,31686]," 是素理想，意味着如果 ",[34,343370,343372,343392],{"className":343371},[31642],[34,343373,343375],{"className":343374},[31646],[31648,343376,343377],{"xmlns":31650},[31652,343378,343379,343389],{},[31655,343380,343381,343383,343385,343387],{},[31658,343382,32936],{},[31658,343384,67764],{},[31662,343386,112296],{},[31658,343388,105852],{},[31665,343390,343391],{"encoding":31667},"xy \\in P",[34,343393,343395,343416],{"className":343394,"ariaHidden":758},[31672],[34,343396,343398,343401,343404,343407,343410,343413],{"className":343397},[31676],[34,343399],{"className":343400,"style":86652},[31680],[34,343402,32936],{"className":343403},[31685,31686],[34,343405,67764],{"className":343406,"style":32062},[31685,31686],[34,343408],{"className":343409,"style":31846},[31845],[34,343411,112296],{"className":343412},[31850],[34,343414],{"className":343415,"style":31846},[31845],[34,343417,343419,343422],{"className":343418},[31676],[34,343420],{"className":343421,"style":33218},[31680],[34,343423,105852],{"className":343424,"style":37422},[31685,31686],[34,343426,343428,343446],{"className":343427},[31642],[34,343429,343431],{"className":343430},[31646],[31648,343432,343433],{"xmlns":31650},[31652,343434,343435,343443],{},[31655,343436,343437,343439,343441],{},[31658,343438,32936],{},[31662,343440,112296],{},[31658,343442,105852],{},[31665,343444,343445],{"encoding":31667},"x \\in P",[34,343447,343449,343467],{"className":343448,"ariaHidden":758},[31672],[34,343450,343452,343455,343458,343461,343464],{"className":343451},[31676],[34,343453],{"className":343454,"style":34812},[31680],[34,343456,32936],{"className":343457},[31685,31686],[34,343459],{"className":343460,"style":31846},[31845],[34,343462,112296],{"className":343463},[31850],[34,343465],{"className":343466,"style":31846},[31845],[34,343468,343470,343473],{"className":343469},[31676],[34,343471],{"className":343472,"style":33218},[31680],[34,343474,105852],{"className":343475,"style":37422},[31685,31686],[34,343477,343479,343497],{"className":343478},[31642],[34,343480,343482],{"className":343481},[31646],[31648,343483,343484],{"xmlns":31650},[31652,343485,343486,343494],{},[31655,343487,343488,343490,343492],{},[31658,343489,67764],{},[31662,343491,112296],{},[31658,343493,105852],{},[31665,343495,343496],{"encoding":31667},"y \\in P",[34,343498,343500,343518],{"className":343499,"ariaHidden":758},[31672],[34,343501,343503,343506,343509,343512,343515],{"className":343502},[31676],[34,343504],{"className":343505,"style":86652},[31680],[34,343507,67764],{"className":343508,"style":32062},[31685,31686],[34,343510],{"className":343511,"style":31846},[31845],[34,343513,112296],{"className":343514},[31850],[34,343516],{"className":343517,"style":31846},[31845],[34,343519,343521,343524],{"className":343520},[31676],[34,343522],{"className":343523,"style":33218},[31680],[34,343525,105852],{"className":343526,"style":37422},[31685,31686],[108,343528,343529,343532,343533,343612,343613,343671,343672,343700,343701,39984,343762,343904,343905,343933,343934,343962,343963,343966],{},[2627,343530,343531],{},"寻找候选理想","：在 ",[34,343534,343536,343562],{"className":343535},[31642],[34,343537,343539],{"className":343538},[31646],[31648,343540,343541],{"xmlns":31650},[31652,343542,343543,343559],{},[31655,343544,343545,343547,343549,343551,343557],{},[31658,343546,97300],{"mathvariant":318416},[31658,343548,38080],{"mathvariant":38079},[31662,343550,456],{"stretchy":702},[37624,343552,343553,343555],{},[31658,343554,15],{},[31658,343556,31660],{},[31662,343558,2557],{"stretchy":702},[31665,343560,343561],{"encoding":31667},"\\mathbb{Z}\u002F(p^n)",[34,343563,343565],{"className":343564,"ariaHidden":758},[31672],[34,343566,343568,343571,343574,343577,343580,343609],{"className":343567},[31676],[34,343569],{"className":343570,"style":31774},[31680],[34,343572,97300],{"className":343573},[31685,318441],[34,343575,38080],{"className":343576},[31685],[34,343578,456],{"className":343579},[31782],[34,343581,343583,343586],{"className":343582},[31685],[34,343584,15],{"className":343585},[31685,31686],[34,343587,343589],{"className":343588},[31792],[34,343590,343592],{"className":343591},[31796],[34,343593,343595],{"className":343594},[31801],[34,343596,343598],{"className":343597,"style":37959},[31805],[34,343599,343600,343603],{"style":37777},[34,343601],{"className":343602,"style":31814},[31813],[34,343604,343606],{"className":343605},[31818,31819,31820,31821],[34,343607,31660],{"className":343608},[31685,31686,31821],[34,343610,2557],{"className":343611},[31690]," 中，理想是由 ",[34,343614,343616,343633],{"className":343615},[31642],[34,343617,343619],{"className":343618},[31646],[31648,343620,343621],{"xmlns":31650},[31652,343622,343623,343631],{},[31655,343624,343625],{},[37624,343626,343627,343629],{},[31658,343628,15],{},[31658,343630,31660],{},[31665,343632,339442],{"encoding":31667},[34,343634,343636],{"className":343635,"ariaHidden":758},[31672],[34,343637,343639,343642],{"className":343638},[31676],[34,343640],{"className":343641,"style":316112},[31680],[34,343643,343645,343648],{"className":343644},[31685],[34,343646,15],{"className":343647},[31685,31686],[34,343649,343651],{"className":343650},[31792],[34,343652,343654],{"className":343653},[31796],[34,343655,343657],{"className":343656},[31801],[34,343658,343660],{"className":343659,"style":37959},[31805],[34,343661,343662,343665],{"style":37777},[34,343663],{"className":343664,"style":31814},[31813],[34,343666,343668],{"className":343667},[31818,31819,31820,31821],[34,343669,31660],{"className":343670},[31685,31686,31821]," 的因子的倍数构成的。考虑由元素 ",[34,343673,343675,343688],{"className":343674},[31642],[34,343676,343678],{"className":343677},[31646],[31648,343679,343680],{"xmlns":31650},[31652,343681,343682,343686],{},[31655,343683,343684],{},[31658,343685,15],{},[31665,343687,15],{"encoding":31667},[34,343689,343691],{"className":343690,"ariaHidden":758},[31672],[34,343692,343694,343697],{"className":343693},[31676],[34,343695],{"className":343696,"style":38774},[31680],[34,343698,15],{"className":343699},[31685,31686]," 生成的主理想，记为 ",[34,343702,343704,343726],{"className":343703},[31642],[34,343705,343707],{"className":343706},[31646],[31648,343708,343709],{"xmlns":31650},[31652,343710,343711,343723],{},[31655,343712,343713,343715,343717,343719,343721],{},[31658,343714,105852],{},[31662,343716,202],{},[31662,343718,456],{"stretchy":702},[31658,343720,15],{},[31662,343722,2557],{"stretchy":702},[31665,343724,343725],{"encoding":31667},"P = (p)",[34,343727,343729,343747],{"className":343728,"ariaHidden":758},[31672],[34,343730,343732,343735,343738,343741,343744],{"className":343731},[31676],[34,343733],{"className":343734,"style":33218},[31680],[34,343736,105852],{"className":343737,"style":37422},[31685,31686],[34,343739],{"className":343740,"style":31846},[31845],[34,343742,202],{"className":343743},[31850],[34,343745],{"className":343746,"style":31846},[31845],[34,343748,343750,343753,343756,343759],{"className":343749},[31676],[34,343751],{"className":343752,"style":31774},[31680],[34,343754,456],{"className":343755},[31782],[34,343757,15],{"className":343758},[31685,31686],[34,343760,2557],{"className":343761},[31690],[34,343763,343765,343805],{"className":343764},[31642],[34,343766,343768],{"className":343767},[31646],[31648,343769,343770],{"xmlns":31650},[31652,343771,343772,343802],{},[31655,343773,343774,343776,343778,343780,343782,343784,343786,343788,343790,343792,343794,343796,343798,343800],{},[31658,343775,105852],{},[31662,343777,202],{},[31662,343779,456],{"stretchy":702},[31658,343781,15],{},[31662,343783,2557],{"stretchy":702},[31662,343785,202],{},[31662,343787,32685],{"stretchy":702},[31658,343789,34395],{},[31658,343791,15],{},[31662,343793,162086],{},[31658,343795,34395],{},[31662,343797,112296],{},[31658,343799,34386],{},[31662,343801,32715],{"stretchy":702},[31665,343803,343804],{"encoding":31667},"P = (p) = \\{ rp \\mid r \\in R \\}",[34,343806,343808,343826,343850,343874,343892],{"className":343807,"ariaHidden":758},[31672],[34,343809,343811,343814,343817,343820,343823],{"className":343810},[31676],[34,343812],{"className":343813,"style":33218},[31680],[34,343815,105852],{"className":343816,"style":37422},[31685,31686],[34,343818],{"className":343819,"style":31846},[31845],[34,343821,202],{"className":343822},[31850],[34,343824],{"className":343825,"style":31846},[31845],[34,343827,343829,343832,343835,343838,343841,343844,343847],{"className":343828},[31676],[34,343830],{"className":343831,"style":31774},[31680],[34,343833,456],{"className":343834},[31782],[34,343836,15],{"className":343837},[31685,31686],[34,343839,2557],{"className":343840},[31690],[34,343842],{"className":343843,"style":31846},[31845],[34,343845,202],{"className":343846},[31850],[34,343848],{"className":343849,"style":31846},[31845],[34,343851,343853,343856,343859,343862,343865,343868,343871],{"className":343852},[31676],[34,343854],{"className":343855,"style":31774},[31680],[34,343857,32685],{"className":343858},[31782],[34,343860,34395],{"className":343861,"style":34452},[31685,31686],[34,343863,15],{"className":343864},[31685,31686],[34,343866],{"className":343867,"style":31846},[31845],[34,343869,162086],{"className":343870},[31850],[34,343872],{"className":343873,"style":31846},[31845],[34,343875,343877,343880,343883,343886,343889],{"className":343876},[31676],[34,343878],{"className":343879,"style":34812},[31680],[34,343881,34395],{"className":343882,"style":34452},[31685,31686],[34,343884],{"className":343885,"style":31846},[31845],[34,343887,112296],{"className":343888},[31850],[34,343890],{"className":343891,"style":31846},[31845],[34,343893,343895,343898,343901],{"className":343894},[31676],[34,343896],{"className":343897,"style":31774},[31680],[34,343899,34386],{"className":343900,"style":34427},[31685,31686],[34,343902,32715],{"className":343903},[31690],"\n这个理想包含了所有 ",[34,343906,343908,343921],{"className":343907},[31642],[34,343909,343911],{"className":343910},[31646],[31648,343912,343913],{"xmlns":31650},[31652,343914,343915,343919],{},[31655,343916,343917],{},[31658,343918,15],{},[31665,343920,15],{"encoding":31667},[34,343922,343924],{"className":343923,"ariaHidden":758},[31672],[34,343925,343927,343930],{"className":343926},[31676],[34,343928],{"className":343929,"style":38774},[31680],[34,343931,15],{"className":343932},[31685,31686]," 的倍数，也就是包含了 ",[34,343935,343937,343950],{"className":343936},[31642],[34,343938,343940],{"className":343939},[31646],[31648,343941,343942],{"xmlns":31650},[31652,343943,343944,343948],{},[31655,343945,343946],{},[31658,343947,34386],{},[31665,343949,34386],{"encoding":31667},[34,343951,343953],{"className":343952,"ariaHidden":758},[31672],[34,343954,343956,343959],{"className":343955},[31676],[34,343957],{"className":343958,"style":33218},[31680],[34,343960,34386],{"className":343961,"style":34427},[31685,31686]," 中所有的",[2627,343964,343965],{},"不可逆元素","（即所有的幂零元）。",[108,343968,343969,344001,344002,326989,344054,11008,344084,341288,344112,86621,344165,344193,344194,344246,344247,4374,344297,344348,344349,4374,344399,322067,344449,344488],{},[2627,343970,343971,343972,344000],{},"验证 ",[34,343973,343975,343988],{"className":343974},[31642],[34,343976,343978],{"className":343977},[31646],[31648,343979,343980],{"xmlns":31650},[31652,343981,343982,343986],{},[31655,343983,343984],{},[31658,343985,105852],{},[31665,343987,105852],{"encoding":31667},[34,343989,343991],{"className":343990,"ariaHidden":758},[31672],[34,343992,343994,343997],{"className":343993},[31676],[34,343995],{"className":343996,"style":33218},[31680],[34,343998,105852],{"className":343999,"style":37422},[31685,31686]," 是素理想","：\n假设 ",[34,344003,344005,344024],{"className":344004},[31642],[34,344006,344008],{"className":344007},[31646],[31648,344009,344010],{"xmlns":31650},[31652,344011,344012,344022],{},[31655,344013,344014,344016,344018,344020],{},[31658,344015,111],{},[31658,344017,1697],{},[31662,344019,112296],{},[31658,344021,105852],{},[31665,344023,331022],{"encoding":31667},[34,344025,344027,344045],{"className":344026,"ariaHidden":758},[31672],[34,344028,344030,344033,344036,344039,344042],{"className":344029},[31676],[34,344031],{"className":344032,"style":46991},[31680],[34,344034,324191],{"className":344035},[31685,31686],[34,344037],{"className":344038,"style":31846},[31845],[34,344040,112296],{"className":344041},[31850],[34,344043],{"className":344044,"style":31846},[31845],[34,344046,344048,344051],{"className":344047},[31676],[34,344049],{"className":344050,"style":33218},[31680],[34,344052,105852],{"className":344053,"style":37422},[31685,31686],[34,344055,344057,344072],{"className":344056},[31642],[34,344058,344060],{"className":344059},[31646],[31648,344061,344062],{"xmlns":31650},[31652,344063,344064,344070],{},[31655,344065,344066,344068],{},[31658,344067,111],{},[31658,344069,1697],{},[31665,344071,324191],{"encoding":31667},[34,344073,344075],{"className":344074,"ariaHidden":758},[31672],[34,344076,344078,344081],{"className":344077},[31676],[34,344079],{"className":344080,"style":31681},[31680],[34,344082,324191],{"className":344083},[31685,31686],[34,344085,344087,344100],{"className":344086},[31642],[34,344088,344090],{"className":344089},[31646],[31648,344091,344092],{"xmlns":31650},[31652,344093,344094,344098],{},[31655,344095,344096],{},[31658,344097,15],{},[31665,344099,15],{"encoding":31667},[34,344101,344103],{"className":344102,"ariaHidden":758},[31672],[34,344104,344106,344109],{"className":344105},[31676],[34,344107],{"className":344108,"style":38774},[31680],[34,344110,15],{"className":344111},[31685,31686],[34,344113,344115,344135],{"className":344114},[31642],[34,344116,344118],{"className":344117},[31646],[31648,344119,344120],{"xmlns":31650},[31652,344121,344122,344132],{},[31655,344123,344124,344126,344128,344130],{},[31658,344125,15],{},[31662,344127,162086],{},[31658,344129,111],{},[31658,344131,1697],{},[31665,344133,344134],{"encoding":31667},"p \\mid ab",[34,344136,344138,344156],{"className":344137,"ariaHidden":758},[31672],[34,344139,344141,344144,344147,344150,344153],{"className":344140},[31676],[34,344142],{"className":344143,"style":31774},[31680],[34,344145,15],{"className":344146},[31685,31686],[34,344148],{"className":344149,"style":31846},[31845],[34,344151,162086],{"className":344152},[31850],[34,344154],{"className":344155,"style":31846},[31845],[34,344157,344159,344162],{"className":344158},[31676],[34,344160],{"className":344161,"style":31681},[31680],[34,344163,324191],{"className":344164},[31685,31686],[34,344166,344168,344181],{"className":344167},[31642],[34,344169,344171],{"className":344170},[31646],[31648,344172,344173],{"xmlns":31650},[31652,344174,344175,344179],{},[31655,344176,344177],{},[31658,344178,15],{},[31665,344180,15],{"encoding":31667},[34,344182,344184],{"className":344183,"ariaHidden":758},[31672],[34,344185,344187,344190],{"className":344186},[31676],[34,344188],{"className":344189,"style":38774},[31680],[34,344191,15],{"className":344192},[31685,31686]," 是素数，若 ",[34,344195,344197,344216],{"className":344196},[31642],[34,344198,344200],{"className":344199},[31646],[31648,344201,344202],{"xmlns":31650},[31652,344203,344204,344214],{},[31655,344205,344206,344208,344210,344212],{},[31658,344207,15],{},[31662,344209,162086],{},[31658,344211,111],{},[31658,344213,1697],{},[31665,344215,344134],{"encoding":31667},[34,344217,344219,344237],{"className":344218,"ariaHidden":758},[31672],[34,344220,344222,344225,344228,344231,344234],{"className":344221},[31676],[34,344223],{"className":344224,"style":31774},[31680],[34,344226,15],{"className":344227},[31685,31686],[34,344229],{"className":344230,"style":31846},[31845],[34,344232,162086],{"className":344233},[31850],[34,344235],{"className":344236,"style":31846},[31845],[34,344238,344240,344243],{"className":344239},[31676],[34,344241],{"className":344242,"style":31681},[31680],[34,344244,324191],{"className":344245},[31685,31686],"，则必有 ",[34,344248,344250,344267],{"className":344249},[31642],[34,344251,344253],{"className":344252},[31646],[31648,344254,344255],{"xmlns":31650},[31652,344256,344257,344265],{},[31655,344258,344259,344261,344263],{},[31658,344260,15],{},[31662,344262,162086],{},[31658,344264,111],{},[31665,344266,341309],{"encoding":31667},[34,344268,344270,344288],{"className":344269,"ariaHidden":758},[31672],[34,344271,344273,344276,344279,344282,344285],{"className":344272},[31676],[34,344274],{"className":344275,"style":31774},[31680],[34,344277,15],{"className":344278},[31685,31686],[34,344280],{"className":344281,"style":31846},[31845],[34,344283,162086],{"className":344284},[31850],[34,344286],{"className":344287,"style":31846},[31845],[34,344289,344291,344294],{"className":344290},[31676],[34,344292],{"className":344293,"style":32527},[31680],[34,344295,111],{"className":344296},[31685,31686],[34,344298,344300,344318],{"className":344299},[31642],[34,344301,344303],{"className":344302},[31646],[31648,344304,344305],{"xmlns":31650},[31652,344306,344307,344315],{},[31655,344308,344309,344311,344313],{},[31658,344310,15],{},[31662,344312,162086],{},[31658,344314,1697],{},[31665,344316,344317],{"encoding":31667},"p \\mid b",[34,344319,344321,344339],{"className":344320,"ariaHidden":758},[31672],[34,344322,344324,344327,344330,344333,344336],{"className":344323},[31676],[34,344325],{"className":344326,"style":31774},[31680],[34,344328,15],{"className":344329},[31685,31686],[34,344331],{"className":344332,"style":31846},[31845],[34,344334,162086],{"className":344335},[31850],[34,344337],{"className":344338,"style":31846},[31845],[34,344340,344342,344345],{"className":344341},[31676],[34,344343],{"className":344344,"style":31681},[31680],[34,344346,1697],{"className":344347},[31685,31686],"。\n这等价于 ",[34,344350,344352,344369],{"className":344351},[31642],[34,344353,344355],{"className":344354},[31646],[31648,344356,344357],{"xmlns":31650},[31652,344358,344359,344367],{},[31655,344360,344361,344363,344365],{},[31658,344362,111],{},[31662,344364,112296],{},[31658,344366,105852],{},[31665,344368,331074],{"encoding":31667},[34,344370,344372,344390],{"className":344371,"ariaHidden":758},[31672],[34,344373,344375,344378,344381,344384,344387],{"className":344374},[31676],[34,344376],{"className":344377,"style":34812},[31680],[34,344379,111],{"className":344380},[31685,31686],[34,344382],{"className":344383,"style":31846},[31845],[34,344385,112296],{"className":344386},[31850],[34,344388],{"className":344389,"style":31846},[31845],[34,344391,344393,344396],{"className":344392},[31676],[34,344394],{"className":344395,"style":33218},[31680],[34,344397,105852],{"className":344398,"style":37422},[31685,31686],[34,344400,344402,344419],{"className":344401},[31642],[34,344403,344405],{"className":344404},[31646],[31648,344406,344407],{"xmlns":31650},[31652,344408,344409,344417],{},[31655,344410,344411,344413,344415],{},[31658,344412,1697],{},[31662,344414,112296],{},[31658,344416,105852],{},[31665,344418,331126],{"encoding":31667},[34,344420,344422,344440],{"className":344421,"ariaHidden":758},[31672],[34,344423,344425,344428,344431,344434,344437],{"className":344424},[31676],[34,344426],{"className":344427,"style":46991},[31680],[34,344429,1697],{"className":344430},[31685,31686],[34,344432],{"className":344433,"style":31846},[31845],[34,344435,112296],{"className":344436},[31850],[34,344438],{"className":344439,"style":31846},[31845],[34,344441,344443,344446],{"className":344442},[31676],[34,344444],{"className":344445,"style":33218},[31680],[34,344447,105852],{"className":344448,"style":37422},[31685,31686],[34,344450,344452,344470],{"className":344451},[31642],[34,344453,344455],{"className":344454},[31646],[31648,344456,344457],{"xmlns":31650},[31652,344458,344459,344467],{},[31655,344460,344461,344463,344465],{},[31662,344462,456],{"stretchy":702},[31658,344464,15],{},[31662,344466,2557],{"stretchy":702},[31665,344468,344469],{"encoding":31667},"(p)",[34,344471,344473],{"className":344472,"ariaHidden":758},[31672],[34,344474,344476,344479,344482,344485],{"className":344475},[31676],[34,344477],{"className":344478,"style":31774},[31680],[34,344480,456],{"className":344481},[31782],[34,344483,15],{"className":344484},[31685,31686],[34,344486,2557],{"className":344487},[31690]," 是一个素理想。",[108,344490,344491,344001,344494,11008,344522,344550,344551,314945,344632,344660,344661,344712,344713,1840,344808,344836,344837,1330,344865,344908,344909,344937,344938,344966,344967,344995,344996,345047,345048,345098,345099,345127,345128,345156,345157,345218,345219,11008,345247,345275,345276,345314,345315,345317,345318,345346,345347,347],{},[2627,344492,344493],{},"证明唯一性",[34,344495,344497,344510],{"className":344496},[31642],[34,344498,344500],{"className":344499},[31646],[31648,344501,344502],{"xmlns":31650},[31652,344503,344504,344508],{},[31655,344505,344506],{},[31658,344507,116863],{},[31665,344509,116863],{"encoding":31667},[34,344511,344513],{"className":344512,"ariaHidden":758},[31672],[34,344514,344516,344519],{"className":344515},[31676],[34,344517],{"className":344518,"style":70739},[31680],[34,344520,116863],{"className":344521},[31685,31686],[34,344523,344525,344538],{"className":344524},[31642],[34,344526,344528],{"className":344527},[31646],[31648,344529,344530],{"xmlns":31650},[31652,344531,344532,344536],{},[31655,344533,344534],{},[31658,344535,34386],{},[31665,344537,34386],{"encoding":31667},[34,344539,344541],{"className":344540,"ariaHidden":758},[31672],[34,344542,344544,344547],{"className":344543},[31676],[34,344545],{"className":344546,"style":33218},[31680],[34,344548,34386],{"className":344549,"style":34427},[31685,31686]," 中的任意一个素理想。\n我们知道 ",[34,344552,344554,344576],{"className":344553},[31642],[34,344555,344557],{"className":344556},[31646],[31648,344558,344559],{"xmlns":31650},[31652,344560,344561,344573],{},[31655,344562,344563,344569,344571],{},[37624,344564,344565,344567],{},[31658,344566,15],{},[31658,344568,31660],{},[31662,344570,202],{},[31749,344572,459],{},[31665,344574,344575],{"encoding":31667},"p^n = 0",[34,344577,344579,344623],{"className":344578,"ariaHidden":758},[31672],[34,344580,344582,344585,344614,344617,344620],{"className":344581},[31676],[34,344583],{"className":344584,"style":316112},[31680],[34,344586,344588,344591],{"className":344587},[31685],[34,344589,15],{"className":344590},[31685,31686],[34,344592,344594],{"className":344593},[31792],[34,344595,344597],{"className":344596},[31796],[34,344598,344600],{"className":344599},[31801],[34,344601,344603],{"className":344602,"style":37959},[31805],[34,344604,344605,344608],{"style":37777},[34,344606],{"className":344607,"style":31814},[31813],[34,344609,344611],{"className":344610},[31818,31819,31820,31821],[34,344612,31660],{"className":344613},[31685,31686,31821],[34,344615],{"className":344616,"style":31846},[31845],[34,344618,202],{"className":344619},[31850],[34,344621],{"className":344622,"style":31846},[31845],[34,344624,344626,344629],{"className":344625},[31676],[34,344627],{"className":344628,"style":32617},[31680],[34,344630,459],{"className":344631},[31685],[34,344633,344635,344648],{"className":344634},[31642],[34,344636,344638],{"className":344637},[31646],[31648,344639,344640],{"xmlns":31650},[31652,344641,344642,344646],{},[31655,344643,344644],{},[31658,344645,34386],{},[31665,344647,34386],{"encoding":31667},[34,344649,344651],{"className":344650,"ariaHidden":758},[31672],[34,344652,344654,344657],{"className":344653},[31676],[34,344655],{"className":344656,"style":33218},[31680],[34,344658,34386],{"className":344659,"style":34427},[31685,31686]," 中。显然 ",[34,344662,344664,344682],{"className":344663},[31642],[34,344665,344667],{"className":344666},[31646],[31648,344668,344669],{"xmlns":31650},[31652,344670,344671,344679],{},[31655,344672,344673,344675,344677],{},[31749,344674,459],{},[31662,344676,112296],{},[31658,344678,116863],{},[31665,344680,344681],{"encoding":31667},"0 \\in Q",[34,344683,344685,344703],{"className":344684,"ariaHidden":758},[31672],[34,344686,344688,344691,344694,344697,344700],{"className":344687},[31676],[34,344689],{"className":344690,"style":61814},[31680],[34,344692,459],{"className":344693},[31685],[34,344695],{"className":344696,"style":31846},[31845],[34,344698,112296],{"className":344699},[31850],[34,344701],{"className":344702,"style":31846},[31845],[34,344704,344706,344709],{"className":344705},[31676],[34,344707],{"className":344708,"style":70739},[31680],[34,344710,116863],{"className":344711},[31685,31686],"（理想包含零元）。\n即 ",[34,344714,344716,344742],{"className":344715},[31642],[34,344717,344719],{"className":344718},[31646],[31648,344720,344721],{"xmlns":31650},[31652,344722,344723,344739],{},[31655,344724,344725,344727,344729,344731,344733,344735,344737],{},[31658,344726,15],{},[31662,344728,71191],{},[31658,344730,15],{},[31662,344732,71191],{},[31662,344734,38874],{},[31662,344736,71191],{},[31658,344738,15],{},[31665,344740,344741],{"encoding":31667},"p \\cdot p \\cdot \\dots \\cdot p",[34,344743,344745,344763,344781,344799],{"className":344744,"ariaHidden":758},[31672],[34,344746,344748,344751,344754,344757,344760],{"className":344747},[31676],[34,344749],{"className":344750,"style":73503},[31680],[34,344752,15],{"className":344753},[31685,31686],[34,344755],{"className":344756,"style":32604},[31845],[34,344758,71191],{"className":344759},[32024],[34,344761],{"className":344762,"style":32604},[31845],[34,344764,344766,344769,344772,344775,344778],{"className":344765},[31676],[34,344767],{"className":344768,"style":73503},[31680],[34,344770,15],{"className":344771},[31685,31686],[34,344773],{"className":344774,"style":32604},[31845],[34,344776,71191],{"className":344777},[32024],[34,344779],{"className":344780,"style":32604},[31845],[34,344782,344784,344787,344790,344793,344796],{"className":344783},[31676],[34,344785],{"className":344786,"style":224869},[31680],[34,344788,38874],{"className":344789},[34006],[34,344791],{"className":344792,"style":32604},[31845],[34,344794,71191],{"className":344795},[32024],[34,344797],{"className":344798,"style":32604},[31845],[34,344800,344802,344805],{"className":344801},[31676],[34,344803],{"className":344804,"style":38774},[31680],[34,344806,15],{"className":344807},[31685,31686],[34,344809,344811,344824],{"className":344810},[31642],[34,344812,344814],{"className":344813},[31646],[31648,344815,344816],{"xmlns":31650},[31652,344817,344818,344822],{},[31655,344819,344820],{},[31658,344821,31660],{},[31665,344823,31660],{"encoding":31667},[34,344825,344827],{"className":344826,"ariaHidden":758},[31672],[34,344828,344830,344833],{"className":344829},[31676],[34,344831],{"className":344832,"style":32527},[31680],[34,344834,31660],{"className":344835},[31685,31686]," 个 ",[34,344838,344840,344853],{"className":344839},[31642],[34,344841,344843],{"className":344842},[31646],[31648,344844,344845],{"xmlns":31650},[31652,344846,344847,344851],{},[31655,344848,344849],{},[31658,344850,15],{},[31665,344852,15],{"encoding":31667},[34,344854,344856],{"className":344855,"ariaHidden":758},[31672],[34,344857,344859,344862],{"className":344858},[31676],[34,344860],{"className":344861,"style":38774},[31680],[34,344863,15],{"className":344864},[31685,31686],[34,344866,344868,344884],{"className":344867},[31642],[34,344869,344871],{"className":344870},[31646],[31648,344872,344873],{"xmlns":31650},[31652,344874,344875,344881],{},[31655,344876,344877,344879],{},[31662,344878,112296],{},[31658,344880,116863],{},[31665,344882,344883],{"encoding":31667},"\\in Q",[34,344885,344887,344899],{"className":344886,"ariaHidden":758},[31672],[34,344888,344890,344893,344896],{"className":344889},[31676],[34,344891],{"className":344892,"style":34812},[31680],[34,344894,112296],{"className":344895},[31850],[34,344897],{"className":344898,"style":31846},[31845],[34,344900,344902,344905],{"className":344901},[31676],[34,344903],{"className":344904,"style":70739},[31680],[34,344906,116863],{"className":344907},[31685,31686],"。\n根据素理想的定义，如果乘积在 ",[34,344910,344912,344925],{"className":344911},[31642],[34,344913,344915],{"className":344914},[31646],[31648,344916,344917],{"xmlns":31650},[31652,344918,344919,344923],{},[31655,344920,344921],{},[31658,344922,116863],{},[31665,344924,116863],{"encoding":31667},[34,344926,344928],{"className":344927,"ariaHidden":758},[31672],[34,344929,344931,344934],{"className":344930},[31676],[34,344932],{"className":344933,"style":70739},[31680],[34,344935,116863],{"className":344936},[31685,31686]," 中，因子必须有一个在 ",[34,344939,344941,344954],{"className":344940},[31642],[34,344942,344944],{"className":344943},[31646],[31648,344945,344946],{"xmlns":31650},[31652,344947,344948,344952],{},[31655,344949,344950],{},[31658,344951,116863],{},[31665,344953,116863],{"encoding":31667},[34,344955,344957],{"className":344956,"ariaHidden":758},[31672],[34,344958,344960,344963],{"className":344959},[31676],[34,344961],{"className":344962,"style":70739},[31680],[34,344964,116863],{"className":344965},[31685,31686]," 中。\n既然因子都是 ",[34,344968,344970,344983],{"className":344969},[31642],[34,344971,344973],{"className":344972},[31646],[31648,344974,344975],{"xmlns":31650},[31652,344976,344977,344981],{},[31655,344978,344979],{},[31658,344980,15],{},[31665,344982,15],{"encoding":31667},[34,344984,344986],{"className":344985,"ariaHidden":758},[31672],[34,344987,344989,344992],{"className":344988},[31676],[34,344990],{"className":344991,"style":38774},[31680],[34,344993,15],{"className":344994},[31685,31686],"，那么必须有 ",[34,344997,344999,345017],{"className":344998},[31642],[34,345000,345002],{"className":345001},[31646],[31648,345003,345004],{"xmlns":31650},[31652,345005,345006,345014],{},[31655,345007,345008,345010,345012],{},[31658,345009,15],{},[31662,345011,112296],{},[31658,345013,116863],{},[31665,345015,345016],{"encoding":31667},"p \\in Q",[34,345018,345020,345038],{"className":345019,"ariaHidden":758},[31672],[34,345021,345023,345026,345029,345032,345035],{"className":345022},[31676],[34,345024],{"className":345025,"style":86652},[31680],[34,345027,15],{"className":345028},[31685,31686],[34,345030],{"className":345031,"style":31846},[31845],[34,345033,112296],{"className":345034},[31850],[34,345036],{"className":345037,"style":31846},[31845],[34,345039,345041,345044],{"className":345040},[31676],[34,345042],{"className":345043,"style":70739},[31680],[34,345045,116863],{"className":345046},[31685,31686],"。\n一旦 ",[34,345049,345051,345068],{"className":345050},[31642],[34,345052,345054],{"className":345053},[31646],[31648,345055,345056],{"xmlns":31650},[31652,345057,345058,345066],{},[31655,345059,345060,345062,345064],{},[31658,345061,15],{},[31662,345063,112296],{},[31658,345065,116863],{},[31665,345067,345016],{"encoding":31667},[34,345069,345071,345089],{"className":345070,"ariaHidden":758},[31672],[34,345072,345074,345077,345080,345083,345086],{"className":345073},[31676],[34,345075],{"className":345076,"style":86652},[31680],[34,345078,15],{"className":345079},[31685,31686],[34,345081],{"className":345082,"style":31846},[31845],[34,345084,112296],{"className":345085},[31850],[34,345087],{"className":345088,"style":31846},[31845],[34,345090,345092,345095],{"className":345091},[31676],[34,345093],{"className":345094,"style":70739},[31680],[34,345096,116863],{"className":345097},[31685,31686],"，由于理想对乘法封闭，所有 ",[34,345100,345102,345115],{"className":345101},[31642],[34,345103,345105],{"className":345104},[31646],[31648,345106,345107],{"xmlns":31650},[31652,345108,345109,345113],{},[31655,345110,345111],{},[31658,345112,15],{},[31665,345114,15],{"encoding":31667},[34,345116,345118],{"className":345117,"ariaHidden":758},[31672],[34,345119,345121,345124],{"className":345120},[31676],[34,345122],{"className":345123,"style":38774},[31680],[34,345125,15],{"className":345126},[31685,31686]," 的倍数都在 ",[34,345129,345131,345144],{"className":345130},[31642],[34,345132,345134],{"className":345133},[31646],[31648,345135,345136],{"xmlns":31650},[31652,345137,345138,345142],{},[31655,345139,345140],{},[31658,345141,116863],{},[31665,345143,116863],{"encoding":31667},[34,345145,345147],{"className":345146,"ariaHidden":758},[31672],[34,345148,345150,345153],{"className":345149},[31676],[34,345151],{"className":345152,"style":70739},[31680],[34,345154,116863],{"className":345155},[31685,31686]," 中，即 ",[34,345158,345160,345182],{"className":345159},[31642],[34,345161,345163],{"className":345162},[31646],[31648,345164,345165],{"xmlns":31650},[31652,345166,345167,345179],{},[31655,345168,345169,345171,345173,345175,345177],{},[31662,345170,456],{"stretchy":702},[31658,345172,15],{},[31662,345174,2557],{"stretchy":702},[31662,345176,131855],{},[31658,345178,116863],{},[31665,345180,345181],{"encoding":31667},"(p) \\subseteq Q",[34,345183,345185,345209],{"className":345184,"ariaHidden":758},[31672],[34,345186,345188,345191,345194,345197,345200,345203,345206],{"className":345187},[31676],[34,345189],{"className":345190,"style":31774},[31680],[34,345192,456],{"className":345193},[31782],[34,345195,15],{"className":345196},[31685,31686],[34,345198,2557],{"className":345199},[31690],[34,345201],{"className":345202,"style":31846},[31845],[34,345204,131855],{"className":345205},[31850],[34,345207],{"className":345208,"style":31846},[31845],[34,345210,345212,345215],{"className":345211},[31676],[34,345213],{"className":345214,"style":70739},[31680],[34,345216,116863],{"className":345217},[31685,31686],"。\n又因为 ",[34,345220,345222,345235],{"className":345221},[31642],[34,345223,345225],{"className":345224},[31646],[31648,345226,345227],{"xmlns":31650},[31652,345228,345229,345233],{},[31655,345230,345231],{},[31658,345232,116863],{},[31665,345234,116863],{"encoding":31667},[34,345236,345238],{"className":345237,"ariaHidden":758},[31672],[34,345239,345241,345244],{"className":345240},[31676],[34,345242],{"className":345243,"style":70739},[31680],[34,345245,116863],{"className":345246},[31685,31686],[34,345248,345250,345263],{"className":345249},[31642],[34,345251,345253],{"className":345252},[31646],[31648,345254,345255],{"xmlns":31650},[31652,345256,345257,345261],{},[31655,345258,345259],{},[31658,345260,34386],{},[31665,345262,34386],{"encoding":31667},[34,345264,345266],{"className":345265,"ariaHidden":758},[31672],[34,345267,345269,345272],{"className":345268},[31676],[34,345270],{"className":345271,"style":33218},[31680],[34,345273,34386],{"className":345274,"style":34427},[31685,31686]," 的真理想（素理想定义要求是真理想），而 ",[34,345277,345279,345296],{"className":345278},[31642],[34,345280,345282],{"className":345281},[31646],[31648,345283,345284],{"xmlns":31650},[31652,345285,345286,345294],{},[31655,345287,345288,345290,345292],{},[31662,345289,456],{"stretchy":702},[31658,345291,15],{},[31662,345293,2557],{"stretchy":702},[31665,345295,344469],{"encoding":31667},[34,345297,345299],{"className":345298,"ariaHidden":758},[31672],[34,345300,345302,345305,345308,345311],{"className":345301},[31676],[34,345303],{"className":345304,"style":31774},[31680],[34,345306,456],{"className":345307},[31782],[34,345309,15],{"className":345310},[31685,31686],[34,345312,2557],{"className":345313},[31690]," 实际上是",[2627,345316,330747],{},"（我们将在第3问看到），所以 ",[34,345319,345321,345334],{"className":345320},[31642],[34,345322,345324],{"className":345323},[31646],[31648,345325,345326],{"xmlns":31650},[31652,345327,345328,345332],{},[31655,345329,345330],{},[31658,345331,116863],{},[31665,345333,116863],{"encoding":31667},[34,345335,345337],{"className":345336,"ariaHidden":758},[31672],[34,345338,345340,345343],{"className":345339},[31676],[34,345341],{"className":345342,"style":70739},[31680],[34,345344,116863],{"className":345345},[31685,31686]," 只能等于 ",[34,345348,345350,345367],{"className":345349},[31642],[34,345351,345353],{"className":345352},[31646],[31648,345354,345355],{"xmlns":31650},[31652,345356,345357,345365],{},[31655,345358,345359,345361,345363],{},[31662,345360,456],{"stretchy":702},[31658,345362,15],{},[31662,345364,2557],{"stretchy":702},[31665,345366,344469],{"encoding":31667},[34,345368,345370],{"className":345369,"ariaHidden":758},[31672],[34,345371,345373,345376,345379,345382],{"className":345372},[31676],[34,345374],{"className":345375,"style":31774},[31680],[34,345377,456],{"className":345378},[31782],[34,345380,15],{"className":345381},[31685,31686],[34,345383,2557],{"className":345384},[31690],[108,345386,345387,2664,345389,345417,345418,347],{},[2627,345388,114043],{},[34,345390,345392,345405],{"className":345391},[31642],[34,345393,345395],{"className":345394},[31646],[31648,345396,345397],{"xmlns":31650},[31652,345398,345399,345403],{},[31655,345400,345401],{},[31658,345402,34386],{},[31665,345404,34386],{"encoding":31667},[34,345406,345408],{"className":345407,"ariaHidden":758},[31672],[34,345409,345411,345414],{"className":345410},[31676],[34,345412],{"className":345413,"style":33218},[31680],[34,345415,34386],{"className":345416,"style":34427},[31685,31686]," 只有一个素理想 ",[34,345419,345421,345442],{"className":345420},[31642],[34,345422,345424],{"className":345423},[31646],[31648,345425,345426],{"xmlns":31650},[31652,345427,345428,345440],{},[31655,345429,345430,345432,345434,345436,345438],{},[31658,345431,105852],{},[31662,345433,202],{},[31662,345435,456],{"stretchy":702},[31658,345437,15],{},[31662,345439,2557],{"stretchy":702},[31665,345441,343725],{"encoding":31667},[34,345443,345445,345463],{"className":345444,"ariaHidden":758},[31672],[34,345446,345448,345451,345454,345457,345460],{"className":345447},[31676],[34,345449],{"className":345450,"style":33218},[31680],[34,345452,105852],{"className":345453,"style":37422},[31685,31686],[34,345455],{"className":345456,"style":31846},[31845],[34,345458,202],{"className":345459},[31850],[34,345461],{"className":345462,"style":31846},[31845],[34,345464,345466,345469,345472,345475],{"className":345465},[31676],[34,345467],{"className":345468,"style":31774},[31680],[34,345470,456],{"className":345471},[31782],[34,345473,15],{"className":345474},[31685,31686],[34,345476,2557],{"className":345477},[31690],[15,345479,345480],{},[2627,345481,345482,345483,345521],{},"证明 (3)：商环 ",[34,345484,345486,345503],{"className":345485},[31642],[34,345487,345489],{"className":345488},[31646],[31648,345490,345491],{"xmlns":31650},[31652,345492,345493,345501],{},[31655,345494,345495,345497,345499],{},[31658,345496,34386],{},[31658,345498,38080],{"mathvariant":38079},[31658,345500,105852],{},[31665,345502,331515],{"encoding":31667},[34,345504,345506],{"className":345505,"ariaHidden":758},[31672],[34,345507,345509,345512,345515,345518],{"className":345508},[31676],[34,345510],{"className":345511,"style":31774},[31680],[34,345513,34386],{"className":345514,"style":34427},[31685,31686],[34,345516,38080],{"className":345517},[31685],[34,345519,105852],{"className":345520,"style":37422},[31685,31686]," 是域",[2230,345523,345524,345730,345993,346406],{},[108,345525,345526,345529,345530,345630,345631,345691,345692,347],{},[2627,345527,345528],{},"构造商环","：\n我们已经知道 ",[34,345531,345533,345562],{"className":345532},[31642],[34,345534,345536],{"className":345535},[31646],[31648,345537,345538],{"xmlns":31650},[31652,345539,345540,345560],{},[31655,345541,345542,345544,345546,345548,345550,345552,345558],{},[31658,345543,34386],{},[31662,345545,202],{},[31658,345547,97300],{"mathvariant":318416},[31658,345549,38080],{"mathvariant":38079},[31662,345551,456],{"stretchy":702},[37624,345553,345554,345556],{},[31658,345555,15],{},[31658,345557,31660],{},[31662,345559,2557],{"stretchy":702},[31665,345561,340465],{"encoding":31667},[34,345563,345565,345583],{"className":345564,"ariaHidden":758},[31672],[34,345566,345568,345571,345574,345577,345580],{"className":345567},[31676],[34,345569],{"className":345570,"style":33218},[31680],[34,345572,34386],{"className":345573,"style":34427},[31685,31686],[34,345575],{"className":345576,"style":31846},[31845],[34,345578,202],{"className":345579},[31850],[34,345581],{"className":345582,"style":31846},[31845],[34,345584,345586,345589,345592,345595,345598,345627],{"className":345585},[31676],[34,345587],{"className":345588,"style":31774},[31680],[34,345590,97300],{"className":345591},[31685,318441],[34,345593,38080],{"className":345594},[31685],[34,345596,456],{"className":345597},[31782],[34,345599,345601,345604],{"className":345600},[31685],[34,345602,15],{"className":345603},[31685,31686],[34,345605,345607],{"className":345606},[31792],[34,345608,345610],{"className":345609},[31796],[34,345611,345613],{"className":345612},[31801],[34,345614,345616],{"className":345615,"style":37959},[31805],[34,345617,345618,345621],{"style":37777},[34,345619],{"className":345620,"style":31814},[31813],[34,345622,345624],{"className":345623},[31818,31819,31820,31821],[34,345625,31660],{"className":345626},[31685,31686,31821],[34,345628,2557],{"className":345629},[31690],"，且素理想 ",[34,345632,345634,345655],{"className":345633},[31642],[34,345635,345637],{"className":345636},[31646],[31648,345638,345639],{"xmlns":31650},[31652,345640,345641,345653],{},[31655,345642,345643,345645,345647,345649,345651],{},[31658,345644,105852],{},[31662,345646,202],{},[31662,345648,456],{"stretchy":702},[31658,345650,15],{},[31662,345652,2557],{"stretchy":702},[31665,345654,343725],{"encoding":31667},[34,345656,345658,345676],{"className":345657,"ariaHidden":758},[31672],[34,345659,345661,345664,345667,345670,345673],{"className":345660},[31676],[34,345662],{"className":345663,"style":33218},[31680],[34,345665,105852],{"className":345666,"style":37422},[31685,31686],[34,345668],{"className":345669,"style":31846},[31845],[34,345671,202],{"className":345672},[31850],[34,345674],{"className":345675,"style":31846},[31845],[34,345677,345679,345682,345685,345688],{"className":345678},[31676],[34,345680],{"className":345681,"style":31774},[31680],[34,345683,456],{"className":345684},[31782],[34,345686,15],{"className":345687},[31685,31686],[34,345689,2557],{"className":345690},[31690],"。\n我们要考察的结构是 ",[34,345693,345695,345712],{"className":345694},[31642],[34,345696,345698],{"className":345697},[31646],[31648,345699,345700],{"xmlns":31650},[31652,345701,345702,345710],{},[31655,345703,345704,345706,345708],{},[31658,345705,34386],{},[31658,345707,38080],{"mathvariant":38079},[31658,345709,105852],{},[31665,345711,331515],{"encoding":31667},[34,345713,345715],{"className":345714,"ariaHidden":758},[31672],[34,345716,345718,345721,345724,345727],{"className":345717},[31676],[34,345719],{"className":345720,"style":31774},[31680],[34,345722,34386],{"className":345723,"style":34427},[31685,31686],[34,345725,38080],{"className":345726},[31685],[34,345728,105852],{"className":345729,"style":37422},[31685,31686],[108,345731,345732,345735,345736,345764,345765],{},[2627,345733,345734],{},"利用同构定理化简","：\n根据环的第三同构定理，这等价于直接在整数环 ",[34,345737,345739,345752],{"className":345738},[31642],[34,345740,345742],{"className":345741},[31646],[31648,345743,345744],{"xmlns":31650},[31652,345745,345746,345750],{},[31655,345747,345748],{},[31658,345749,97300],{"mathvariant":318416},[31665,345751,320917],{"encoding":31667},[34,345753,345755],{"className":345754,"ariaHidden":758},[31672],[34,345756,345758,345761],{"className":345757},[31676],[34,345759],{"className":345760,"style":320927},[31680],[34,345762,97300],{"className":345763},[31685,318441]," 上操作：\n",[34,345766,345768,345835],{"className":345767},[31642],[34,345769,345771],{"className":345770},[31646],[31648,345772,345773],{"xmlns":31650},[31652,345774,345775,345832],{},[31655,345776,345777,345779,345781,345783,345785,345787,345789,345791,345797,345799,345802,345804,345806,345808,345810,345812,345818,345820,345822,345824,345826,345828,345830],{},[31658,345778,34386],{},[31658,345780,38080],{"mathvariant":38079},[31658,345782,105852],{},[31662,345784,202],{},[31658,345786,97300],{"mathvariant":318416},[31658,345788,38080],{"mathvariant":38079},[31662,345790,456],{"stretchy":702},[37624,345792,345793,345795],{},[31658,345794,15],{},[31658,345796,31660],{},[31662,345798,2557],{"stretchy":702},[31662,345800,38080],{"fence":702,"stretchy":758,"minsize":345801,"maxsize":345801},"1.2em",[31662,345803,456],{"stretchy":702},[31658,345805,15],{},[31662,345807,2557],{"stretchy":702},[31658,345809,38080],{"mathvariant":38079},[31662,345811,456],{"stretchy":702},[37624,345813,345814,345816],{},[31658,345815,15],{},[31658,345817,31660],{},[31662,345819,2557],{"stretchy":702},[31662,345821,339196],{},[31658,345823,97300],{"mathvariant":318416},[31658,345825,38080],{"mathvariant":38079},[31662,345827,456],{"stretchy":702},[31658,345829,15],{},[31662,345831,2557],{"stretchy":702},[31665,345833,345834],{"encoding":31667},"R\u002FP = \\mathbb{Z}\u002F(p^n) \\big\u002F (p)\u002F(p^n) \\simeq \\mathbb{Z}\u002F(p)",[34,345836,345838,345862,345972],{"className":345837,"ariaHidden":758},[31672],[34,345839,345841,345844,345847,345850,345853,345856,345859],{"className":345840},[31676],[34,345842],{"className":345843,"style":31774},[31680],[34,345845,34386],{"className":345846,"style":34427},[31685,31686],[34,345848,38080],{"className":345849},[31685],[34,345851,105852],{"className":345852,"style":37422},[31685,31686],[34,345854],{"className":345855,"style":31846},[31845],[34,345857,202],{"className":345858},[31850],[34,345860],{"className":345861,"style":31846},[31845],[34,345863,345865,345869,345872,345875,345878,345907,345910,345916,345919,345922,345925,345928,345931,345960,345963,345966,345969],{"className":345864},[31676],[34,345866],{"className":345867,"style":345868},[31680],"height:1.2em;vertical-align:-0.35em;",[34,345870,97300],{"className":345871},[31685,318441],[34,345873,38080],{"className":345874},[31685],[34,345876,456],{"className":345877},[31782],[34,345879,345881,345884],{"className":345880},[31685],[34,345882,15],{"className":345883},[31685,31686],[34,345885,345887],{"className":345886},[31792],[34,345888,345890],{"className":345889},[31796],[34,345891,345893],{"className":345892},[31801],[34,345894,345896],{"className":345895,"style":37959},[31805],[34,345897,345898,345901],{"style":37777},[34,345899],{"className":345900,"style":31814},[31813],[34,345902,345904],{"className":345903},[31818,31819,31820,31821],[34,345905,31660],{"className":345906},[31685,31686,31821],[34,345908,2557],{"className":345909},[31690],[34,345911,345913],{"className":345912},[31685],[34,345914,38080],{"className":345915},[71281,43411],[34,345917,456],{"className":345918},[31782],[34,345920,15],{"className":345921},[31685,31686],[34,345923,2557],{"className":345924},[31690],[34,345926,38080],{"className":345927},[31685],[34,345929,456],{"className":345930},[31782],[34,345932,345934,345937],{"className":345933},[31685],[34,345935,15],{"className":345936},[31685,31686],[34,345938,345940],{"className":345939},[31792],[34,345941,345943],{"className":345942},[31796],[34,345944,345946],{"className":345945},[31801],[34,345947,345949],{"className":345948,"style":37959},[31805],[34,345950,345951,345954],{"style":37777},[34,345952],{"className":345953,"style":31814},[31813],[34,345955,345957],{"className":345956},[31818,31819,31820,31821],[34,345958,31660],{"className":345959},[31685,31686,31821],[34,345961,2557],{"className":345962},[31690],[34,345964],{"className":345965,"style":31846},[31845],[34,345967,339196],{"className":345968},[31850],[34,345970],{"className":345971,"style":31846},[31845],[34,345973,345975,345978,345981,345984,345987,345990],{"className":345974},[31676],[34,345976],{"className":345977,"style":31774},[31680],[34,345979,97300],{"className":345980},[31685,318441],[34,345982,38080],{"className":345983},[31685],[34,345985,456],{"className":345986},[31782],[34,345988,15],{"className":345989},[31685,31686],[34,345991,2557],{"className":345992},[31690],[108,345994,345995,33183,346046,346094,346095,346123,346124,93389,346233,346261,346262,346345,346346,346374,346375,346403,346404,347],{},[2627,345996,326327,345997],{},[34,345998,346000,346022],{"className":345999},[31642],[34,346001,346003],{"className":346002},[31646],[31648,346004,346005],{"xmlns":31650},[31652,346006,346007,346019],{},[31655,346008,346009,346011,346013,346015,346017],{},[31658,346010,97300],{"mathvariant":318416},[31658,346012,38080],{"mathvariant":38079},[31662,346014,456],{"stretchy":702},[31658,346016,15],{},[31662,346018,2557],{"stretchy":702},[31665,346020,346021],{"encoding":31667},"\\mathbb{Z}\u002F(p)",[34,346023,346025],{"className":346024,"ariaHidden":758},[31672],[34,346026,346028,346031,346034,346037,346040,346043],{"className":346027},[31676],[34,346029],{"className":346030,"style":31774},[31680],[34,346032,97300],{"className":346033},[31685,318441],[34,346035,38080],{"className":346036},[31685],[34,346038,456],{"className":346039},[31782],[34,346041,15],{"className":346042},[31685,31686],[34,346044,2557],{"className":346045},[31690],[34,346047,346049,346070],{"className":346048},[31642],[34,346050,346052],{"className":346051},[31646],[31648,346053,346054],{"xmlns":31650},[31652,346055,346056,346068],{},[31655,346057,346058,346060,346062,346064,346066],{},[31658,346059,97300],{"mathvariant":318416},[31658,346061,38080],{"mathvariant":38079},[31662,346063,456],{"stretchy":702},[31658,346065,15],{},[31662,346067,2557],{"stretchy":702},[31665,346069,346021],{"encoding":31667},[34,346071,346073],{"className":346072,"ariaHidden":758},[31672],[34,346074,346076,346079,346082,346085,346088,346091],{"className":346075},[31676],[34,346077],{"className":346078,"style":31774},[31680],[34,346080,97300],{"className":346081},[31685,318441],[34,346083,38080],{"className":346084},[31685],[34,346086,456],{"className":346087},[31782],[34,346089,15],{"className":346090},[31685,31686],[34,346092,2557],{"className":346093},[31690]," 是模素数 ",[34,346096,346098,346111],{"className":346097},[31642],[34,346099,346101],{"className":346100},[31646],[31648,346102,346103],{"xmlns":31650},[31652,346104,346105,346109],{},[31655,346106,346107],{},[31658,346108,15],{},[31665,346110,15],{"encoding":31667},[34,346112,346114],{"className":346113,"ariaHidden":758},[31672],[34,346115,346117,346120],{"className":346116},[31676],[34,346118],{"className":346119,"style":38774},[31680],[34,346121,15],{"className":346122},[31685,31686]," 的整数环。\n对于任意非零元素 ",[34,346125,346127,346158],{"className":346126},[31642],[34,346128,346130],{"className":346129},[31646],[31648,346131,346132],{"xmlns":31650},[31652,346133,346134,346155],{},[31655,346135,346136,346143,346145,346147,346149,346151,346153],{},[134014,346137,346138,346140],{"accent":758},[31658,346139,111],{},[31662,346141,346142],{},"ˉ",[31662,346144,112296],{},[31658,346146,97300],{"mathvariant":318416},[31658,346148,38080],{"mathvariant":38079},[31662,346150,456],{"stretchy":702},[31658,346152,15],{},[31662,346154,2557],{"stretchy":702},[31665,346156,346157],{"encoding":31667},"\\bar{a} \\in \\mathbb{Z}\u002F(p)",[34,346159,346161,346212],{"className":346160,"ariaHidden":758},[31672],[34,346162,346164,346168,346203,346206,346209],{"className":346163},[31676],[34,346165],{"className":346166,"style":346167},[31680],"height:0.6069em;vertical-align:-0.0391em;",[34,346169,346172],{"className":346170},[31685,346171],"accent",[34,346173,346175],{"className":346174},[31796],[34,346176,346178],{"className":346177},[31801],[34,346179,346182,346190],{"className":346180,"style":346181},[31805],"height:0.5678em;",[34,346183,346184,346187],{"style":41565},[34,346185],{"className":346186,"style":41508},[31813],[34,346188,111],{"className":346189},[31685,31686],[34,346191,346192,346195],{"style":41565},[34,346193],{"className":346194,"style":41508},[31813],[34,346196,346200],{"className":346197,"style":346199},[346198],"accent-body","left:-0.25em;",[34,346201,346142],{"className":346202},[31685],[34,346204],{"className":346205,"style":31846},[31845],[34,346207,112296],{"className":346208},[31850],[34,346210],{"className":346211,"style":31846},[31845],[34,346213,346215,346218,346221,346224,346227,346230],{"className":346214},[31676],[34,346216],{"className":346217,"style":31774},[31680],[34,346219,97300],{"className":346220},[31685,318441],[34,346222,38080],{"className":346223},[31685],[34,346225,456],{"className":346226},[31782],[34,346228,15],{"className":346229},[31685,31686],[34,346231,2557],{"className":346232},[31690],[34,346234,346236,346249],{"className":346235},[31642],[34,346237,346239],{"className":346238},[31646],[31648,346240,346241],{"xmlns":31650},[31652,346242,346243,346247],{},[31655,346244,346245],{},[31658,346246,15],{},[31665,346248,15],{"encoding":31667},[34,346250,346252],{"className":346251,"ariaHidden":758},[31672],[34,346253,346255,346258],{"className":346254},[31676],[34,346256],{"className":346257,"style":38774},[31680],[34,346259,15],{"className":346260},[31685,31686]," 是素数，",[34,346263,346265,346295],{"className":346264},[31642],[34,346266,346268],{"className":346267},[31646],[31648,346269,346270],{"xmlns":31650},[31652,346271,346272,346292],{},[31655,346273,346274,346276,346278,346280,346282,346284,346286,346288,346290],{},[31658,346275,314792],{},[31662,346277,32682],{},[31662,346279,456],{"stretchy":702},[31658,346281,111],{},[31662,346283,31960],{"separator":758},[31658,346285,15],{},[31662,346287,2557],{"stretchy":702},[31662,346289,202],{},[31749,346291,640],{},[31665,346293,346294],{"encoding":31667},"\\gcd(a, p) = 1",[34,346296,346298,346336],{"className":346297,"ariaHidden":758},[31672],[34,346299,346301,346304,346309,346312,346315,346318,346321,346324,346327,346330,346333],{"className":346300},[31676],[34,346302],{"className":346303,"style":31774},[31680],[34,346305,346307,314826],{"className":346306},[32767],[34,346308,31942],{"style":38619},[34,346310,456],{"className":346311},[31782],[34,346313,111],{"className":346314},[31685,31686],[34,346316,31960],{"className":346317},[32119],[34,346319],{"className":346320,"style":32123},[31845],[34,346322,15],{"className":346323},[31685,31686],[34,346325,2557],{"className":346326},[31690],[34,346328],{"className":346329,"style":31846},[31845],[34,346331,202],{"className":346332},[31850],[34,346334],{"className":346335,"style":31846},[31845],[34,346337,346339,346342],{"className":346338},[31676],[34,346340],{"className":346341,"style":32617},[31680],[34,346343,640],{"className":346344},[31685],"。\n这意味着 ",[34,346347,346349,346362],{"className":346348},[31642],[34,346350,346352],{"className":346351},[31646],[31648,346353,346354],{"xmlns":31650},[31652,346355,346356,346360],{},[31655,346357,346358],{},[31658,346359,111],{},[31665,346361,111],{"encoding":31667},[34,346363,346365],{"className":346364,"ariaHidden":758},[31672],[34,346366,346368,346371],{"className":346367},[31676],[34,346369],{"className":346370,"style":32527},[31680],[34,346372,111],{"className":346373},[31685,31686]," 在模 ",[34,346376,346378,346391],{"className":346377},[31642],[34,346379,346381],{"className":346380},[31646],[31648,346382,346383],{"xmlns":31650},[31652,346384,346385,346389],{},[31655,346386,346387],{},[31658,346388,15],{},[31665,346390,15],{"encoding":31667},[34,346392,346394],{"className":346393,"ariaHidden":758},[31672],[34,346395,346397,346400],{"className":346396},[31676],[34,346398],{"className":346399,"style":38774},[31680],[34,346401,15],{"className":346402},[31685,31686]," 下一定有逆元。\n一个所有非零元素都有逆元的交换环，定义为",[2627,346405,340387],{},[108,346407,346408,332541,346410,346458,346459,334082],{},[2627,346409,114043],{},[34,346411,346413,346434],{"className":346412},[31642],[34,346414,346416],{"className":346415},[31646],[31648,346417,346418],{"xmlns":31650},[31652,346419,346420,346432],{},[31655,346421,346422,346424,346426,346428,346430],{},[31658,346423,97300],{"mathvariant":318416},[31658,346425,38080],{"mathvariant":38079},[31662,346427,456],{"stretchy":702},[31658,346429,15],{},[31662,346431,2557],{"stretchy":702},[31665,346433,346021],{"encoding":31667},[34,346435,346437],{"className":346436,"ariaHidden":758},[31672],[34,346438,346440,346443,346446,346449,346452,346455],{"className":346439},[31676],[34,346441],{"className":346442,"style":31774},[31680],[34,346444,97300],{"className":346445},[31685,318441],[34,346447,38080],{"className":346448},[31685],[34,346450,456],{"className":346451},[31782],[34,346453,15],{"className":346454},[31685,31686],[34,346456,2557],{"className":346457},[31690]," 是域，所以商环 ",[34,346460,346462,346479],{"className":346461},[31642],[34,346463,346465],{"className":346464},[31646],[31648,346466,346467],{"xmlns":31650},[31652,346468,346469,346477],{},[31655,346470,346471,346473,346475],{},[31658,346472,34386],{},[31658,346474,38080],{"mathvariant":38079},[31658,346476,105852],{},[31665,346478,331515],{"encoding":31667},[34,346480,346482],{"className":346481,"ariaHidden":758},[31672],[34,346483,346485,346488,346491,346494],{"className":346484},[31676],[34,346486],{"className":346487,"style":31774},[31680],[34,346489,34386],{"className":346490,"style":34427},[31685,31686],[34,346492,38080],{"className":346493},[31685],[34,346495,105852],{"className":346496,"style":37422},[31685,31686],[32188,346498],{},[253,346500,346502],{"id":346501},"题号-8","题号 8",[32490,346504,162043],{"id":167236},[15,346506,346507,346508,346536,346537,346565,346566,346700,346701,346729,346730,347],{},"若环 ",[34,346509,346511,346524],{"className":346510},[31642],[34,346512,346514],{"className":346513},[31646],[31648,346515,346516],{"xmlns":31650},[31652,346517,346518,346522],{},[31655,346519,346520],{},[31658,346521,34386],{},[31665,346523,34386],{"encoding":31667},[34,346525,346527],{"className":346526,"ariaHidden":758},[31672],[34,346528,346530,346533],{"className":346529},[31676],[34,346531],{"className":346532,"style":33218},[31680],[34,346534,34386],{"className":346535,"style":34427},[31685,31686]," 的一个素理想 ",[34,346538,346540,346553],{"className":346539},[31642],[34,346541,346543],{"className":346542},[31646],[31648,346544,346545],{"xmlns":31650},[31652,346546,346547,346551],{},[31655,346548,346549],{},[31658,346550,105852],{},[31665,346552,105852],{"encoding":31667},[34,346554,346556],{"className":346555,"ariaHidden":758},[31672],[34,346557,346559,346562],{"className":346558},[31676],[34,346560],{"className":346561,"style":33218},[31680],[34,346563,105852],{"className":346564,"style":37422},[31685,31686]," 包含有限多个理想 ",[34,346567,346569,346603],{"className":346568},[31642],[34,346570,346572],{"className":346571},[31646],[31648,346573,346574],{"xmlns":31650},[31652,346575,346576,346600],{},[31655,346577,346578,346584,346586,346588,346590,346592,346594,346596,346598],{},[31743,346579,346580,346582],{},[31658,346581,106063],{},[31658,346583,31955],{},[39361,346585,39363],{},[31662,346587,456],{"stretchy":702},[31749,346589,640],{},[31662,346591,45972],{},[31658,346593,31955],{},[31662,346595,45972],{},[31658,346597,31660],{},[31662,346599,2557],{"stretchy":702},[31665,346601,346602],{"encoding":31667},"I_i \\, (1 \\le i \\le n)",[34,346604,346606,346670,346688],{"className":346605,"ariaHidden":758},[31672],[34,346607,346609,346612,346652,346655,346658,346661,346664,346667],{"className":346608},[31676],[34,346610],{"className":346611,"style":31774},[31680],[34,346613,346615,346618],{"className":346614},[31685],[34,346616,106063],{"className":346617,"style":67482},[31685,31686],[34,346619,346621],{"className":346620},[31792],[34,346622,346624,346644],{"className":346623},[31796,31797],[34,346625,346627,346641],{"className":346626},[31801],[34,346628,346630],{"className":346629,"style":32090},[31805],[34,346631,346632,346635],{"style":217742},[34,346633],{"className":346634,"style":31814},[31813],[34,346636,346638],{"className":346637},[31818,31819,31820,31821],[34,346639,31955],{"className":346640},[31685,31686,31821],[34,346642,31829],{"className":346643},[31828],[34,346645,346647],{"className":346646},[31801],[34,346648,346650],{"className":346649,"style":31836},[31805],[34,346651],{},[34,346653],{"className":346654,"style":32123},[31845],[34,346656,456],{"className":346657},[31782],[34,346659,640],{"className":346660},[31685],[34,346662],{"className":346663,"style":31846},[31845],[34,346665,45972],{"className":346666},[31850],[34,346668],{"className":346669,"style":31846},[31845],[34,346671,346673,346676,346679,346682,346685],{"className":346672},[31676],[34,346674],{"className":346675,"style":55401},[31680],[34,346677,31955],{"className":346678},[31685,31686],[34,346680],{"className":346681,"style":31846},[31845],[34,346683,45972],{"className":346684},[31850],[34,346686],{"className":346687,"style":31846},[31845],[34,346689,346691,346694,346697],{"className":346690},[31676],[34,346692],{"className":346693,"style":31774},[31680],[34,346695,31660],{"className":346696},[31685,31686],[34,346698,2557],{"className":346699},[31690]," 的交，证明 ",[34,346702,346704,346717],{"className":346703},[31642],[34,346705,346707],{"className":346706},[31646],[31648,346708,346709],{"xmlns":31650},[31652,346710,346711,346715],{},[31655,346712,346713],{},[31658,346714,105852],{},[31665,346716,105852],{"encoding":31667},[34,346718,346720],{"className":346719,"ariaHidden":758},[31672],[34,346721,346723,346726],{"className":346722},[31676],[34,346724],{"className":346725,"style":33218},[31680],[34,346727,105852],{"className":346728,"style":37422},[31685,31686]," 包含某个 ",[34,346731,346733,346751],{"className":346732},[31642],[34,346734,346736],{"className":346735},[31646],[31648,346737,346738],{"xmlns":31650},[31652,346739,346740,346748],{},[31655,346741,346742],{},[31743,346743,346744,346746],{},[31658,346745,106063],{},[31658,346747,31955],{},[31665,346749,346750],{"encoding":31667},"I_i",[34,346752,346754],{"className":346753,"ariaHidden":758},[31672],[34,346755,346757,346760],{"className":346756},[31676],[34,346758],{"className":346759,"style":70915},[31680],[34,346761,346763,346766],{"className":346762},[31685],[34,346764,106063],{"className":346765,"style":67482},[31685,31686],[34,346767,346769],{"className":346768},[31792],[34,346770,346772,346792],{"className":346771},[31796,31797],[34,346773,346775,346789],{"className":346774},[31801],[34,346776,346778],{"className":346777,"style":32090},[31805],[34,346779,346780,346783],{"style":217742},[34,346781],{"className":346782,"style":31814},[31813],[34,346784,346786],{"className":346785},[31818,31819,31820,31821],[34,346787,31955],{"className":346788},[31685,31686,31821],[34,346790,31829],{"className":346791},[31828],[34,346793,346795],{"className":346794},[31801],[34,346796,346798],{"className":346797,"style":31836},[31805],[34,346799],{},[32188,346801],{},[32490,346803,339412],{"id":346804},"深度分析与解答-1",[15,346806,346807],{},[2627,346808,346809],{},"分析框架：",[105,346811,346812,347179,347496],{},[108,346813,346814,346816,346817,347058,347059,335149,347087,347],{},[2627,346815,233757],{},"：已知 ",[34,346818,346820,346863],{"className":346819},[31642],[34,346821,346823],{"className":346822},[31646],[31648,346824,346825],{"xmlns":31650},[31652,346826,346827,346860],{},[31655,346828,346829,346831,346834,346840,346842,346848,346850,346852,346854],{},[31658,346830,105852],{},[31662,346832,346833],{},"⊇",[31743,346835,346836,346838],{},[31658,346837,106063],{},[31749,346839,640],{},[31662,346841,326344],{},[31743,346843,346844,346846],{},[31658,346845,106063],{},[31749,346847,768],{},[31662,346849,326344],{},[31662,346851,38874],{},[31662,346853,326344],{},[31743,346855,346856,346858],{},[31658,346857,106063],{},[31658,346859,31660],{},[31665,346861,346862],{"encoding":31667},"P \\supseteq I_1 \\cap I_2 \\cap \\dots \\cap I_n",[34,346864,346866,346884,346939,346994,347012],{"className":346865,"ariaHidden":758},[31672],[34,346867,346869,346872,346875,346878,346881],{"className":346868},[31676],[34,346870],{"className":346871,"style":195047},[31680],[34,346873,105852],{"className":346874,"style":37422},[31685,31686],[34,346876],{"className":346877,"style":31846},[31845],[34,346879,346833],{"className":346880},[31850],[34,346882],{"className":346883,"style":31846},[31845],[34,346885,346887,346890,346930,346933,346936],{"className":346886},[31676],[34,346888],{"className":346889,"style":70915},[31680],[34,346891,346893,346896],{"className":346892},[31685],[34,346894,106063],{"className":346895,"style":67482},[31685,31686],[34,346897,346899],{"className":346898},[31792],[34,346900,346902,346922],{"className":346901},[31796,31797],[34,346903,346905,346919],{"className":346904},[31801],[34,346906,346908],{"className":346907,"style":31806},[31805],[34,346909,346910,346913],{"style":217742},[34,346911],{"className":346912,"style":31814},[31813],[34,346914,346916],{"className":346915},[31818,31819,31820,31821],[34,346917,640],{"className":346918},[31685,31821],[34,346920,31829],{"className":346921},[31828],[34,346923,346925],{"className":346924},[31801],[34,346926,346928],{"className":346927,"style":31836},[31805],[34,346929],{},[34,346931],{"className":346932,"style":32604},[31845],[34,346934,326344],{"className":346935},[32024],[34,346937],{"className":346938,"style":32604},[31845],[34,346940,346942,346945,346985,346988,346991],{"className":346941},[31676],[34,346943],{"className":346944,"style":70915},[31680],[34,346946,346948,346951],{"className":346947},[31685],[34,346949,106063],{"className":346950,"style":67482},[31685,31686],[34,346952,346954],{"className":346953},[31792],[34,346955,346957,346977],{"className":346956},[31796,31797],[34,346958,346960,346974],{"className":346959},[31801],[34,346961,346963],{"className":346962,"style":31806},[31805],[34,346964,346965,346968],{"style":217742},[34,346966],{"className":346967,"style":31814},[31813],[34,346969,346971],{"className":346970},[31818,31819,31820,31821],[34,346972,768],{"className":346973},[31685,31821],[34,346975,31829],{"className":346976},[31828],[34,346978,346980],{"className":346979},[31801],[34,346981,346983],{"className":346982,"style":31836},[31805],[34,346984],{},[34,346986],{"className":346987,"style":32604},[31845],[34,346989,326344],{"className":346990},[32024],[34,346992],{"className":346993,"style":32604},[31845],[34,346995,346997,347000,347003,347006,347009],{"className":346996},[31676],[34,346998],{"className":346999,"style":147334},[31680],[34,347001,38874],{"className":347002},[34006],[34,347004],{"className":347005,"style":32604},[31845],[34,347007,326344],{"className":347008},[32024],[34,347010],{"className":347011,"style":32604},[31845],[34,347013,347015,347018],{"className":347014},[31676],[34,347016],{"className":347017,"style":70915},[31680],[34,347019,347021,347024],{"className":347020},[31685],[34,347022,106063],{"className":347023,"style":67482},[31685,31686],[34,347025,347027],{"className":347026},[31792],[34,347028,347030,347050],{"className":347029},[31796,31797],[34,347031,347033,347047],{"className":347032},[31801],[34,347034,347036],{"className":347035,"style":32362},[31805],[34,347037,347038,347041],{"style":217742},[34,347039],{"className":347040,"style":31814},[31813],[34,347042,347044],{"className":347043},[31818,31819,31820,31821],[34,347045,31660],{"className":347046},[31685,31686,31821],[34,347048,31829],{"className":347049},[31828],[34,347051,347053],{"className":347052},[31801],[34,347054,347056],{"className":347055,"style":31836},[31805],[34,347057],{},"，求证存在某个 ",[34,347060,347062,347075],{"className":347061},[31642],[34,347063,347065],{"className":347064},[31646],[31648,347066,347067],{"xmlns":31650},[31652,347068,347069,347073],{},[31655,347070,347071],{},[31658,347072,33759],{},[31665,347074,33759],{"encoding":31667},[34,347076,347078],{"className":347077,"ariaHidden":758},[31672],[34,347079,347081,347084],{"className":347080},[31676],[34,347082],{"className":347083,"style":31681},[31680],[34,347085,33759],{"className":347086,"style":33774},[31685,31686],[34,347088,347090,347112],{"className":347089},[31642],[34,347091,347093],{"className":347092},[31646],[31648,347094,347095],{"xmlns":31650},[31652,347096,347097,347109],{},[31655,347098,347099,347101,347103],{},[31658,347100,105852],{},[31662,347102,346833],{},[31743,347104,347105,347107],{},[31658,347106,106063],{},[31658,347108,33759],{},[31665,347110,347111],{"encoding":31667},"P \\supseteq I_k",[34,347113,347115,347133],{"className":347114,"ariaHidden":758},[31672],[34,347116,347118,347121,347124,347127,347130],{"className":347117},[31676],[34,347119],{"className":347120,"style":195047},[31680],[34,347122,105852],{"className":347123,"style":37422},[31685,31686],[34,347125],{"className":347126,"style":31846},[31845],[34,347128,346833],{"className":347129},[31850],[34,347131],{"className":347132,"style":31846},[31845],[34,347134,347136,347139],{"className":347135},[31676],[34,347137],{"className":347138,"style":70915},[31680],[34,347140,347142,347145],{"className":347141},[31685],[34,347143,106063],{"className":347144,"style":67482},[31685,31686],[34,347146,347148],{"className":347147},[31792],[34,347149,347151,347171],{"className":347150},[31796,31797],[34,347152,347154,347168],{"className":347153},[31801],[34,347155,347157],{"className":347156,"style":33833},[31805],[34,347158,347159,347162],{"style":217742},[34,347160],{"className":347161,"style":31814},[31813],[34,347163,347165],{"className":347164},[31818,31819,31820,31821],[34,347166,33759],{"className":347167,"style":33774},[31685,31686,31821],[34,347169,31829],{"className":347170},[31828],[34,347172,347174],{"className":347173},[31801],[34,347175,347177],{"className":347176,"style":31836},[31805],[34,347178],{},[108,347180,347181,347184,347185,55016,347237,4374,347287,347337,347338,55016,347394,4374,347445,347],{},[2627,347182,347183],{},"关键性质","：这是素理想定义的推广。素理想最基本的定义是：若 ",[34,347186,347188,347207],{"className":347187},[31642],[34,347189,347191],{"className":347190},[31646],[31648,347192,347193],{"xmlns":31650},[31652,347194,347195,347205],{},[31655,347196,347197,347199,347201,347203],{},[31658,347198,111],{},[31658,347200,1697],{},[31662,347202,112296],{},[31658,347204,105852],{},[31665,347206,331022],{"encoding":31667},[34,347208,347210,347228],{"className":347209,"ariaHidden":758},[31672],[34,347211,347213,347216,347219,347222,347225],{"className":347212},[31676],[34,347214],{"className":347215,"style":46991},[31680],[34,347217,324191],{"className":347218},[31685,31686],[34,347220],{"className":347221,"style":31846},[31845],[34,347223,112296],{"className":347224},[31850],[34,347226],{"className":347227,"style":31846},[31845],[34,347229,347231,347234],{"className":347230},[31676],[34,347232],{"className":347233,"style":33218},[31680],[34,347235,105852],{"className":347236,"style":37422},[31685,31686],[34,347238,347240,347257],{"className":347239},[31642],[34,347241,347243],{"className":347242},[31646],[31648,347244,347245],{"xmlns":31650},[31652,347246,347247,347255],{},[31655,347248,347249,347251,347253],{},[31658,347250,111],{},[31662,347252,112296],{},[31658,347254,105852],{},[31665,347256,331074],{"encoding":31667},[34,347258,347260,347278],{"className":347259,"ariaHidden":758},[31672],[34,347261,347263,347266,347269,347272,347275],{"className":347262},[31676],[34,347264],{"className":347265,"style":34812},[31680],[34,347267,111],{"className":347268},[31685,31686],[34,347270],{"className":347271,"style":31846},[31845],[34,347273,112296],{"className":347274},[31850],[34,347276],{"className":347277,"style":31846},[31845],[34,347279,347281,347284],{"className":347280},[31676],[34,347282],{"className":347283,"style":33218},[31680],[34,347285,105852],{"className":347286,"style":37422},[31685,31686],[34,347288,347290,347307],{"className":347289},[31642],[34,347291,347293],{"className":347292},[31646],[31648,347294,347295],{"xmlns":31650},[31652,347296,347297,347305],{},[31655,347298,347299,347301,347303],{},[31658,347300,1697],{},[31662,347302,112296],{},[31658,347304,105852],{},[31665,347306,331126],{"encoding":31667},[34,347308,347310,347328],{"className":347309,"ariaHidden":758},[31672],[34,347311,347313,347316,347319,347322,347325],{"className":347312},[31676],[34,347314],{"className":347315,"style":46991},[31680],[34,347317,1697],{"className":347318},[31685,31686],[34,347320],{"className":347321,"style":31846},[31845],[34,347323,112296],{"className":347324},[31850],[34,347326],{"className":347327,"style":31846},[31845],[34,347329,347331,347334],{"className":347330},[31676],[34,347332],{"className":347333,"style":33218},[31680],[34,347335,105852],{"className":347336,"style":37422},[31685,31686],"。对于理想而言，若乘积 ",[34,347339,347341,347361],{"className":347340},[31642],[34,347342,347344],{"className":347343},[31646],[31648,347345,347346],{"xmlns":31650},[31652,347347,347348,347358],{},[31655,347349,347350,347352,347354,347356],{},[31658,347351,54151],{},[31658,347353,54224],{},[31662,347355,131855],{},[31658,347357,105852],{},[31665,347359,347360],{"encoding":31667},"AB \\subseteq P",[34,347362,347364,347385],{"className":347363,"ariaHidden":758},[31672],[34,347365,347367,347370,347373,347376,347379,347382],{"className":347366},[31676],[34,347368],{"className":347369,"style":195047},[31680],[34,347371,54151],{"className":347372},[31685,31686],[34,347374,54224],{"className":347375,"style":54252},[31685,31686],[34,347377],{"className":347378,"style":31846},[31845],[34,347380,131855],{"className":347381},[31850],[34,347383],{"className":347384,"style":31846},[31845],[34,347386,347388,347391],{"className":347387},[31676],[34,347389],{"className":347390,"style":33218},[31680],[34,347392,105852],{"className":347393,"style":37422},[31685,31686],[34,347395,347397,347415],{"className":347396},[31642],[34,347398,347400],{"className":347399},[31646],[31648,347401,347402],{"xmlns":31650},[31652,347403,347404,347412],{},[31655,347405,347406,347408,347410],{},[31658,347407,54151],{},[31662,347409,131855],{},[31658,347411,105852],{},[31665,347413,347414],{"encoding":31667},"A \\subseteq P",[34,347416,347418,347436],{"className":347417,"ariaHidden":758},[31672],[34,347419,347421,347424,347427,347430,347433],{"className":347420},[31676],[34,347422],{"className":347423,"style":195047},[31680],[34,347425,54151],{"className":347426},[31685,31686],[34,347428],{"className":347429,"style":31846},[31845],[34,347431,131855],{"className":347432},[31850],[34,347434],{"className":347435,"style":31846},[31845],[34,347437,347439,347442],{"className":347438},[31676],[34,347440],{"className":347441,"style":33218},[31680],[34,347443,105852],{"className":347444,"style":37422},[31685,31686],[34,347446,347448,347466],{"className":347447},[31642],[34,347449,347451],{"className":347450},[31646],[31648,347452,347453],{"xmlns":31650},[31652,347454,347455,347463],{},[31655,347456,347457,347459,347461],{},[31658,347458,54224],{},[31662,347460,131855],{},[31658,347462,105852],{},[31665,347464,347465],{"encoding":31667},"B \\subseteq P",[34,347467,347469,347487],{"className":347468,"ariaHidden":758},[31672],[34,347470,347472,347475,347478,347481,347484],{"className":347471},[31676],[34,347473],{"className":347474,"style":195047},[31680],[34,347476,54224],{"className":347477,"style":54252},[31685,31686],[34,347479],{"className":347480,"style":31846},[31845],[34,347482,131855],{"className":347483},[31850],[34,347485],{"className":347486,"style":31846},[31845],[34,347488,347490,347493],{"className":347489},[31676],[34,347491],{"className":347492,"style":33218},[31680],[34,347494,105852],{"className":347495,"style":37422},[31685,31686],[108,347497,347498,347501],{},[2627,347499,347500],{},"技巧","：利用“交”与“积”的关系。理想的乘积总是包含在理想的交集中的。",[15,347503,347504],{},[2627,347505,340393],{},[2230,347507,347508,348937,349603,351275],{},[108,347509,347510,347513,347514,347542,347543,347716,347717,347890,347891,348024,348025,348095,348096,348165,348166,348338,348339,348409,348410,348480,348481,348551,348552,348936],{},[2627,347511,347512],{},"建立理想的积与交的关系","：\n考虑这 ",[34,347515,347517,347530],{"className":347516},[31642],[34,347518,347520],{"className":347519},[31646],[31648,347521,347522],{"xmlns":31650},[31652,347523,347524,347528],{},[31655,347525,347526],{},[31658,347527,31660],{},[31665,347529,31660],{"encoding":31667},[34,347531,347533],{"className":347532,"ariaHidden":758},[31672],[34,347534,347536,347539],{"className":347535},[31676],[34,347537],{"className":347538,"style":32527},[31680],[34,347540,31660],{"className":347541},[31685,31686]," 个理想的乘积 ",[34,347544,347546,347578],{"className":347545},[31642],[34,347547,347549],{"className":347548},[31646],[31648,347550,347551],{"xmlns":31650},[31652,347552,347553,347575],{},[31655,347554,347555,347561,347567,347569],{},[31743,347556,347557,347559],{},[31658,347558,106063],{},[31749,347560,640],{},[31743,347562,347563,347565],{},[31658,347564,106063],{},[31749,347566,768],{},[31662,347568,33934],{},[31743,347570,347571,347573],{},[31658,347572,106063],{},[31658,347574,31660],{},[31665,347576,347577],{"encoding":31667},"I_1 I_2 \\dots I_n",[34,347579,347581],{"className":347580,"ariaHidden":758},[31672],[34,347582,347584,347587,347627,347667,347670,347673,347676],{"className":347583},[31676],[34,347585],{"className":347586,"style":70915},[31680],[34,347588,347590,347593],{"className":347589},[31685],[34,347591,106063],{"className":347592,"style":67482},[31685,31686],[34,347594,347596],{"className":347595},[31792],[34,347597,347599,347619],{"className":347598},[31796,31797],[34,347600,347602,347616],{"className":347601},[31801],[34,347603,347605],{"className":347604,"style":31806},[31805],[34,347606,347607,347610],{"style":217742},[34,347608],{"className":347609,"style":31814},[31813],[34,347611,347613],{"className":347612},[31818,31819,31820,31821],[34,347614,640],{"className":347615},[31685,31821],[34,347617,31829],{"className":347618},[31828],[34,347620,347622],{"className":347621},[31801],[34,347623,347625],{"className":347624,"style":31836},[31805],[34,347626],{},[34,347628,347630,347633],{"className":347629},[31685],[34,347631,106063],{"className":347632,"style":67482},[31685,31686],[34,347634,347636],{"className":347635},[31792],[34,347637,347639,347659],{"className":347638},[31796,31797],[34,347640,347642,347656],{"className":347641},[31801],[34,347643,347645],{"className":347644,"style":31806},[31805],[34,347646,347647,347650],{"style":217742},[34,347648],{"className":347649,"style":31814},[31813],[34,347651,347653],{"className":347652},[31818,31819,31820,31821],[34,347654,768],{"className":347655},[31685,31821],[34,347657,31829],{"className":347658},[31828],[34,347660,347662],{"className":347661},[31801],[34,347663,347665],{"className":347664,"style":31836},[31805],[34,347666],{},[34,347668],{"className":347669,"style":32123},[31845],[34,347671,33934],{"className":347672},[34006],[34,347674],{"className":347675,"style":32123},[31845],[34,347677,347679,347682],{"className":347678},[31685],[34,347680,106063],{"className":347681,"style":67482},[31685,31686],[34,347683,347685],{"className":347684},[31792],[34,347686,347688,347708],{"className":347687},[31796,31797],[34,347689,347691,347705],{"className":347690},[31801],[34,347692,347694],{"className":347693,"style":32362},[31805],[34,347695,347696,347699],{"style":217742},[34,347697],{"className":347698,"style":31814},[31813],[34,347700,347702],{"className":347701},[31818,31819,31820,31821],[34,347703,31660],{"className":347704},[31685,31686,31821],[34,347706,31829],{"className":347707},[31828],[34,347709,347711],{"className":347710},[31801],[34,347712,347714],{"className":347713,"style":31836},[31805],[34,347715],{},"。\n理想的乘积定义为有限个形式为 ",[34,347718,347720,347752],{"className":347719},[31642],[34,347721,347723],{"className":347722},[31646],[31648,347724,347725],{"xmlns":31650},[31652,347726,347727,347749],{},[31655,347728,347729,347735,347741,347743],{},[31743,347730,347731,347733],{},[31658,347732,32936],{},[31749,347734,640],{},[31743,347736,347737,347739],{},[31658,347738,32936],{},[31749,347740,768],{},[31662,347742,33934],{},[31743,347744,347745,347747],{},[31658,347746,32936],{},[31658,347748,31660],{},[31665,347750,347751],{"encoding":31667},"x_1 x_2 \\dots x_n",[34,347753,347755],{"className":347754,"ariaHidden":758},[31672],[34,347756,347758,347761,347801,347841,347844,347847,347850],{"className":347757},[31676],[34,347759],{"className":347760,"style":31860},[31680],[34,347762,347764,347767],{"className":347763},[31685],[34,347765,32936],{"className":347766},[31685,31686],[34,347768,347770],{"className":347769},[31792],[34,347771,347773,347793],{"className":347772},[31796,31797],[34,347774,347776,347790],{"className":347775},[31801],[34,347777,347779],{"className":347778,"style":31806},[31805],[34,347780,347781,347784],{"style":31809},[34,347782],{"className":347783,"style":31814},[31813],[34,347785,347787],{"className":347786},[31818,31819,31820,31821],[34,347788,640],{"className":347789},[31685,31821],[34,347791,31829],{"className":347792},[31828],[34,347794,347796],{"className":347795},[31801],[34,347797,347799],{"className":347798,"style":31836},[31805],[34,347800],{},[34,347802,347804,347807],{"className":347803},[31685],[34,347805,32936],{"className":347806},[31685,31686],[34,347808,347810],{"className":347809},[31792],[34,347811,347813,347833],{"className":347812},[31796,31797],[34,347814,347816,347830],{"className":347815},[31801],[34,347817,347819],{"className":347818,"style":31806},[31805],[34,347820,347821,347824],{"style":31809},[34,347822],{"className":347823,"style":31814},[31813],[34,347825,347827],{"className":347826},[31818,31819,31820,31821],[34,347828,768],{"className":347829},[31685,31821],[34,347831,31829],{"className":347832},[31828],[34,347834,347836],{"className":347835},[31801],[34,347837,347839],{"className":347838,"style":31836},[31805],[34,347840],{},[34,347842],{"className":347843,"style":32123},[31845],[34,347845,33934],{"className":347846},[34006],[34,347848],{"className":347849,"style":32123},[31845],[34,347851,347853,347856],{"className":347852},[31685],[34,347854,32936],{"className":347855},[31685,31686],[34,347857,347859],{"className":347858},[31792],[34,347860,347862,347882],{"className":347861},[31796,31797],[34,347863,347865,347879],{"className":347864},[31801],[34,347866,347868],{"className":347867,"style":32362},[31805],[34,347869,347870,347873],{"style":31809},[34,347871],{"className":347872,"style":31814},[31813],[34,347874,347876],{"className":347875},[31818,31819,31820,31821],[34,347877,31660],{"className":347878},[31685,31686,31821],[34,347880,31829],{"className":347881},[31828],[34,347883,347885],{"className":347884},[31801],[34,347886,347888],{"className":347887,"style":31836},[31805],[34,347889],{}," (其中 ",[34,347892,347894,347920],{"className":347893},[31642],[34,347895,347897],{"className":347896},[31646],[31648,347898,347899],{"xmlns":31650},[31652,347900,347901,347917],{},[31655,347902,347903,347909,347911],{},[31743,347904,347905,347907],{},[31658,347906,32936],{},[31658,347908,33759],{},[31662,347910,112296],{},[31743,347912,347913,347915],{},[31658,347914,106063],{},[31658,347916,33759],{},[31665,347918,347919],{"encoding":31667},"x_k \\in I_k",[34,347921,347923,347978],{"className":347922,"ariaHidden":758},[31672],[34,347924,347926,347929,347969,347972,347975],{"className":347925},[31676],[34,347927],{"className":347928,"style":225773},[31680],[34,347930,347932,347935],{"className":347931},[31685],[34,347933,32936],{"className":347934},[31685,31686],[34,347936,347938],{"className":347937},[31792],[34,347939,347941,347961],{"className":347940},[31796,31797],[34,347942,347944,347958],{"className":347943},[31801],[34,347945,347947],{"className":347946,"style":33833},[31805],[34,347948,347949,347952],{"style":31809},[34,347950],{"className":347951,"style":31814},[31813],[34,347953,347955],{"className":347954},[31818,31819,31820,31821],[34,347956,33759],{"className":347957,"style":33774},[31685,31686,31821],[34,347959,31829],{"className":347960},[31828],[34,347962,347964],{"className":347963},[31801],[34,347965,347967],{"className":347966,"style":31836},[31805],[34,347968],{},[34,347970],{"className":347971,"style":31846},[31845],[34,347973,112296],{"className":347974},[31850],[34,347976],{"className":347977,"style":31846},[31845],[34,347979,347981,347984],{"className":347980},[31676],[34,347982],{"className":347983,"style":70915},[31680],[34,347985,347987,347990],{"className":347986},[31685],[34,347988,106063],{"className":347989,"style":67482},[31685,31686],[34,347991,347993],{"className":347992},[31792],[34,347994,347996,348016],{"className":347995},[31796,31797],[34,347997,347999,348013],{"className":347998},[31801],[34,348000,348002],{"className":348001,"style":33833},[31805],[34,348003,348004,348007],{"style":217742},[34,348005],{"className":348006,"style":31814},[31813],[34,348008,348010],{"className":348009},[31818,31819,31820,31821],[34,348011,33759],{"className":348012,"style":33774},[31685,31686,31821],[34,348014,31829],{"className":348015},[31828],[34,348017,348019],{"className":348018},[31801],[34,348020,348022],{"className":348021,"style":31836},[31805],[34,348023],{},") 的元素之和。\n因为理想具有吸收性（吸附性），如果一个元素属于 ",[34,348026,348028,348046],{"className":348027},[31642],[34,348029,348031],{"className":348030},[31646],[31648,348032,348033],{"xmlns":31650},[31652,348034,348035,348043],{},[31655,348036,348037],{},[31743,348038,348039,348041],{},[31658,348040,106063],{},[31658,348042,33759],{},[31665,348044,348045],{"encoding":31667},"I_k",[34,348047,348049],{"className":348048,"ariaHidden":758},[31672],[34,348050,348052,348055],{"className":348051},[31676],[34,348053],{"className":348054,"style":70915},[31680],[34,348056,348058,348061],{"className":348057},[31685],[34,348059,106063],{"className":348060,"style":67482},[31685,31686],[34,348062,348064],{"className":348063},[31792],[34,348065,348067,348087],{"className":348066},[31796,31797],[34,348068,348070,348084],{"className":348069},[31801],[34,348071,348073],{"className":348072,"style":33833},[31805],[34,348074,348075,348078],{"style":217742},[34,348076],{"className":348077,"style":31814},[31813],[34,348079,348081],{"className":348080},[31818,31819,31820,31821],[34,348082,33759],{"className":348083,"style":33774},[31685,31686,31821],[34,348085,31829],{"className":348086},[31828],[34,348088,348090],{"className":348089},[31801],[34,348091,348093],{"className":348092,"style":31836},[31805],[34,348094],{},"，那么它乘以任何其他元素还在 ",[34,348097,348099,348116],{"className":348098},[31642],[34,348100,348102],{"className":348101},[31646],[31648,348103,348104],{"xmlns":31650},[31652,348105,348106,348114],{},[31655,348107,348108],{},[31743,348109,348110,348112],{},[31658,348111,106063],{},[31658,348113,33759],{},[31665,348115,348045],{"encoding":31667},[34,348117,348119],{"className":348118,"ariaHidden":758},[31672],[34,348120,348122,348125],{"className":348121},[31676],[34,348123],{"className":348124,"style":70915},[31680],[34,348126,348128,348131],{"className":348127},[31685],[34,348129,106063],{"className":348130,"style":67482},[31685,31686],[34,348132,348134],{"className":348133},[31792],[34,348135,348137,348157],{"className":348136},[31796,31797],[34,348138,348140,348154],{"className":348139},[31801],[34,348141,348143],{"className":348142,"style":33833},[31805],[34,348144,348145,348148],{"style":217742},[34,348146],{"className":348147,"style":31814},[31813],[34,348149,348151],{"className":348150},[31818,31819,31820,31821],[34,348152,33759],{"className":348153,"style":33774},[31685,31686,31821],[34,348155,31829],{"className":348156},[31828],[34,348158,348160],{"className":348159},[31801],[34,348161,348163],{"className":348162,"style":31836},[31805],[34,348164],{}," 中。\n所以，乘积 ",[34,348167,348169,348200],{"className":348168},[31642],[34,348170,348172],{"className":348171},[31646],[31648,348173,348174],{"xmlns":31650},[31652,348175,348176,348198],{},[31655,348177,348178,348184,348190,348192],{},[31743,348179,348180,348182],{},[31658,348181,106063],{},[31749,348183,640],{},[31743,348185,348186,348188],{},[31658,348187,106063],{},[31749,348189,768],{},[31662,348191,33934],{},[31743,348193,348194,348196],{},[31658,348195,106063],{},[31658,348197,31660],{},[31665,348199,347577],{"encoding":31667},[34,348201,348203],{"className":348202,"ariaHidden":758},[31672],[34,348204,348206,348209,348249,348289,348292,348295,348298],{"className":348205},[31676],[34,348207],{"className":348208,"style":70915},[31680],[34,348210,348212,348215],{"className":348211},[31685],[34,348213,106063],{"className":348214,"style":67482},[31685,31686],[34,348216,348218],{"className":348217},[31792],[34,348219,348221,348241],{"className":348220},[31796,31797],[34,348222,348224,348238],{"className":348223},[31801],[34,348225,348227],{"className":348226,"style":31806},[31805],[34,348228,348229,348232],{"style":217742},[34,348230],{"className":348231,"style":31814},[31813],[34,348233,348235],{"className":348234},[31818,31819,31820,31821],[34,348236,640],{"className":348237},[31685,31821],[34,348239,31829],{"className":348240},[31828],[34,348242,348244],{"className":348243},[31801],[34,348245,348247],{"className":348246,"style":31836},[31805],[34,348248],{},[34,348250,348252,348255],{"className":348251},[31685],[34,348253,106063],{"className":348254,"style":67482},[31685,31686],[34,348256,348258],{"className":348257},[31792],[34,348259,348261,348281],{"className":348260},[31796,31797],[34,348262,348264,348278],{"className":348263},[31801],[34,348265,348267],{"className":348266,"style":31806},[31805],[34,348268,348269,348272],{"style":217742},[34,348270],{"className":348271,"style":31814},[31813],[34,348273,348275],{"className":348274},[31818,31819,31820,31821],[34,348276,768],{"className":348277},[31685,31821],[34,348279,31829],{"className":348280},[31828],[34,348282,348284],{"className":348283},[31801],[34,348285,348287],{"className":348286,"style":31836},[31805],[34,348288],{},[34,348290],{"className":348291,"style":32123},[31845],[34,348293,33934],{"className":348294},[34006],[34,348296],{"className":348297,"style":32123},[31845],[34,348299,348301,348304],{"className":348300},[31685],[34,348302,106063],{"className":348303,"style":67482},[31685,31686],[34,348305,348307],{"className":348306},[31792],[34,348308,348310,348330],{"className":348309},[31796,31797],[34,348311,348313,348327],{"className":348312},[31801],[34,348314,348316],{"className":348315,"style":32362},[31805],[34,348317,348318,348321],{"style":217742},[34,348319],{"className":348320,"style":31814},[31813],[34,348322,348324],{"className":348323},[31818,31819,31820,31821],[34,348325,31660],{"className":348326},[31685,31686,31821],[34,348328,31829],{"className":348329},[31828],[34,348331,348333],{"className":348332},[31801],[34,348334,348336],{"className":348335,"style":31836},[31805],[34,348337],{}," 中的元素一定同时属于 ",[34,348340,348342,348360],{"className":348341},[31642],[34,348343,348345],{"className":348344},[31646],[31648,348346,348347],{"xmlns":31650},[31652,348348,348349,348357],{},[31655,348350,348351],{},[31743,348352,348353,348355],{},[31658,348354,106063],{},[31749,348356,640],{},[31665,348358,348359],{"encoding":31667},"I_1",[34,348361,348363],{"className":348362,"ariaHidden":758},[31672],[34,348364,348366,348369],{"className":348365},[31676],[34,348367],{"className":348368,"style":70915},[31680],[34,348370,348372,348375],{"className":348371},[31685],[34,348373,106063],{"className":348374,"style":67482},[31685,31686],[34,348376,348378],{"className":348377},[31792],[34,348379,348381,348401],{"className":348380},[31796,31797],[34,348382,348384,348398],{"className":348383},[31801],[34,348385,348387],{"className":348386,"style":31806},[31805],[34,348388,348389,348392],{"style":217742},[34,348390],{"className":348391,"style":31814},[31813],[34,348393,348395],{"className":348394},[31818,31819,31820,31821],[34,348396,640],{"className":348397},[31685,31821],[34,348399,31829],{"className":348400},[31828],[34,348402,348404],{"className":348403},[31801],[34,348405,348407],{"className":348406,"style":31836},[31805],[34,348408],{},"，属于 ",[34,348411,348413,348431],{"className":348412},[31642],[34,348414,348416],{"className":348415},[31646],[31648,348417,348418],{"xmlns":31650},[31652,348419,348420,348428],{},[31655,348421,348422],{},[31743,348423,348424,348426],{},[31658,348425,106063],{},[31749,348427,768],{},[31665,348429,348430],{"encoding":31667},"I_2",[34,348432,348434],{"className":348433,"ariaHidden":758},[31672],[34,348435,348437,348440],{"className":348436},[31676],[34,348438],{"className":348439,"style":70915},[31680],[34,348441,348443,348446],{"className":348442},[31685],[34,348444,106063],{"className":348445,"style":67482},[31685,31686],[34,348447,348449],{"className":348448},[31792],[34,348450,348452,348472],{"className":348451},[31796,31797],[34,348453,348455,348469],{"className":348454},[31801],[34,348456,348458],{"className":348457,"style":31806},[31805],[34,348459,348460,348463],{"style":217742},[34,348461],{"className":348462,"style":31814},[31813],[34,348464,348466],{"className":348465},[31818,31819,31820,31821],[34,348467,768],{"className":348468},[31685,31821],[34,348470,31829],{"className":348471},[31828],[34,348473,348475],{"className":348474},[31801],[34,348476,348478],{"className":348477,"style":31836},[31805],[34,348479],{},"，……，属于 ",[34,348482,348484,348502],{"className":348483},[31642],[34,348485,348487],{"className":348486},[31646],[31648,348488,348489],{"xmlns":31650},[31652,348490,348491,348499],{},[31655,348492,348493],{},[31743,348494,348495,348497],{},[31658,348496,106063],{},[31658,348498,31660],{},[31665,348500,348501],{"encoding":31667},"I_n",[34,348503,348505],{"className":348504,"ariaHidden":758},[31672],[34,348506,348508,348511],{"className":348507},[31676],[34,348509],{"className":348510,"style":70915},[31680],[34,348512,348514,348517],{"className":348513},[31685],[34,348515,106063],{"className":348516,"style":67482},[31685,31686],[34,348518,348520],{"className":348519},[31792],[34,348521,348523,348543],{"className":348522},[31796,31797],[34,348524,348526,348540],{"className":348525},[31801],[34,348527,348529],{"className":348528,"style":32362},[31805],[34,348530,348531,348534],{"style":217742},[34,348532],{"className":348533,"style":31814},[31813],[34,348535,348537],{"className":348536},[31818,31819,31820,31821],[34,348538,31660],{"className":348539},[31685,31686,31821],[34,348541,31829],{"className":348542},[31828],[34,348544,348546],{"className":348545},[31801],[34,348547,348549],{"className":348548,"style":31836},[31805],[34,348550],{},"。\n因此，我们有一个极其重要的包含关系：\n",[34,348553,348555,348615],{"className":348554},[31642],[34,348556,348558],{"className":348557},[31646],[31648,348559,348560],{"xmlns":31650},[31652,348561,348562,348612],{},[31655,348563,348564,348570,348576,348578,348584,348586,348592,348594,348600,348602,348604,348606],{},[31743,348565,348566,348568],{},[31658,348567,106063],{},[31749,348569,640],{},[31743,348571,348572,348574],{},[31658,348573,106063],{},[31749,348575,768],{},[31662,348577,33934],{},[31743,348579,348580,348582],{},[31658,348581,106063],{},[31658,348583,31660],{},[31662,348585,131855],{},[31743,348587,348588,348590],{},[31658,348589,106063],{},[31749,348591,640],{},[31662,348593,326344],{},[31743,348595,348596,348598],{},[31658,348597,106063],{},[31749,348599,768],{},[31662,348601,326344],{},[31662,348603,38874],{},[31662,348605,326344],{},[31743,348607,348608,348610],{},[31658,348609,106063],{},[31658,348611,31660],{},[31665,348613,348614],{"encoding":31667},"I_1 I_2 \\dots I_n \\subseteq I_1 \\cap I_2 \\cap \\dots \\cap I_n",[34,348616,348618,348762,348817,348872,348890],{"className":348617,"ariaHidden":758},[31672],[34,348619,348621,348624,348664,348704,348707,348710,348713,348753,348756,348759],{"className":348620},[31676],[34,348622],{"className":348623,"style":70915},[31680],[34,348625,348627,348630],{"className":348626},[31685],[34,348628,106063],{"className":348629,"style":67482},[31685,31686],[34,348631,348633],{"className":348632},[31792],[34,348634,348636,348656],{"className":348635},[31796,31797],[34,348637,348639,348653],{"className":348638},[31801],[34,348640,348642],{"className":348641,"style":31806},[31805],[34,348643,348644,348647],{"style":217742},[34,348645],{"className":348646,"style":31814},[31813],[34,348648,348650],{"className":348649},[31818,31819,31820,31821],[34,348651,640],{"className":348652},[31685,31821],[34,348654,31829],{"className":348655},[31828],[34,348657,348659],{"className":348658},[31801],[34,348660,348662],{"className":348661,"style":31836},[31805],[34,348663],{},[34,348665,348667,348670],{"className":348666},[31685],[34,348668,106063],{"className":348669,"style":67482},[31685,31686],[34,348671,348673],{"className":348672},[31792],[34,348674,348676,348696],{"className":348675},[31796,31797],[34,348677,348679,348693],{"className":348678},[31801],[34,348680,348682],{"className":348681,"style":31806},[31805],[34,348683,348684,348687],{"style":217742},[34,348685],{"className":348686,"style":31814},[31813],[34,348688,348690],{"className":348689},[31818,31819,31820,31821],[34,348691,768],{"className":348692},[31685,31821],[34,348694,31829],{"className":348695},[31828],[34,348697,348699],{"className":348698},[31801],[34,348700,348702],{"className":348701,"style":31836},[31805],[34,348703],{},[34,348705],{"className":348706,"style":32123},[31845],[34,348708,33934],{"className":348709},[34006],[34,348711],{"className":348712,"style":32123},[31845],[34,348714,348716,348719],{"className":348715},[31685],[34,348717,106063],{"className":348718,"style":67482},[31685,31686],[34,348720,348722],{"className":348721},[31792],[34,348723,348725,348745],{"className":348724},[31796,31797],[34,348726,348728,348742],{"className":348727},[31801],[34,348729,348731],{"className":348730,"style":32362},[31805],[34,348732,348733,348736],{"style":217742},[34,348734],{"className":348735,"style":31814},[31813],[34,348737,348739],{"className":348738},[31818,31819,31820,31821],[34,348740,31660],{"className":348741},[31685,31686,31821],[34,348743,31829],{"className":348744},[31828],[34,348746,348748],{"className":348747},[31801],[34,348749,348751],{"className":348750,"style":31836},[31805],[34,348752],{},[34,348754],{"className":348755,"style":31846},[31845],[34,348757,131855],{"className":348758},[31850],[34,348760],{"className":348761,"style":31846},[31845],[34,348763,348765,348768,348808,348811,348814],{"className":348764},[31676],[34,348766],{"className":348767,"style":70915},[31680],[34,348769,348771,348774],{"className":348770},[31685],[34,348772,106063],{"className":348773,"style":67482},[31685,31686],[34,348775,348777],{"className":348776},[31792],[34,348778,348780,348800],{"className":348779},[31796,31797],[34,348781,348783,348797],{"className":348782},[31801],[34,348784,348786],{"className":348785,"style":31806},[31805],[34,348787,348788,348791],{"style":217742},[34,348789],{"className":348790,"style":31814},[31813],[34,348792,348794],{"className":348793},[31818,31819,31820,31821],[34,348795,640],{"className":348796},[31685,31821],[34,348798,31829],{"className":348799},[31828],[34,348801,348803],{"className":348802},[31801],[34,348804,348806],{"className":348805,"style":31836},[31805],[34,348807],{},[34,348809],{"className":348810,"style":32604},[31845],[34,348812,326344],{"className":348813},[32024],[34,348815],{"className":348816,"style":32604},[31845],[34,348818,348820,348823,348863,348866,348869],{"className":348819},[31676],[34,348821],{"className":348822,"style":70915},[31680],[34,348824,348826,348829],{"className":348825},[31685],[34,348827,106063],{"className":348828,"style":67482},[31685,31686],[34,348830,348832],{"className":348831},[31792],[34,348833,348835,348855],{"className":348834},[31796,31797],[34,348836,348838,348852],{"className":348837},[31801],[34,348839,348841],{"className":348840,"style":31806},[31805],[34,348842,348843,348846],{"style":217742},[34,348844],{"className":348845,"style":31814},[31813],[34,348847,348849],{"className":348848},[31818,31819,31820,31821],[34,348850,768],{"className":348851},[31685,31821],[34,348853,31829],{"className":348854},[31828],[34,348856,348858],{"className":348857},[31801],[34,348859,348861],{"className":348860,"style":31836},[31805],[34,348862],{},[34,348864],{"className":348865,"style":32604},[31845],[34,348867,326344],{"className":348868},[32024],[34,348870],{"className":348871,"style":32604},[31845],[34,348873,348875,348878,348881,348884,348887],{"className":348874},[31676],[34,348876],{"className":348877,"style":147334},[31680],[34,348879,38874],{"className":348880},[34006],[34,348882],{"className":348883,"style":32604},[31845],[34,348885,326344],{"className":348886},[32024],[34,348888],{"className":348889,"style":32604},[31845],[34,348891,348893,348896],{"className":348892},[31676],[34,348894],{"className":348895,"style":70915},[31680],[34,348897,348899,348902],{"className":348898},[31685],[34,348900,106063],{"className":348901,"style":67482},[31685,31686],[34,348903,348905],{"className":348904},[31792],[34,348906,348908,348928],{"className":348907},[31796,31797],[34,348909,348911,348925],{"className":348910},[31801],[34,348912,348914],{"className":348913,"style":32362},[31805],[34,348915,348916,348919],{"style":217742},[34,348917],{"className":348918,"style":31814},[31813],[34,348920,348922],{"className":348921},[31818,31819,31820,31821],[34,348923,31660],{"className":348924},[31685,31686,31821],[34,348926,31829],{"className":348927},[31828],[34,348929,348931],{"className":348930},[31801],[34,348932,348934],{"className":348933,"style":31836},[31805],[34,348935],{},"\n即：理想的乘积包含在理想的交集中。",[108,348938,348939,348942,348943,348971,348972,349211,349212,349407,349408],{},[2627,348940,348941],{},"利用题目已知条件","：\n题目告诉我们素理想 ",[34,348944,348946,348959],{"className":348945},[31642],[34,348947,348949],{"className":348948},[31646],[31648,348950,348951],{"xmlns":31650},[31652,348952,348953,348957],{},[31655,348954,348955],{},[31658,348956,105852],{},[31665,348958,105852],{"encoding":31667},[34,348960,348962],{"className":348961,"ariaHidden":758},[31672],[34,348963,348965,348968],{"className":348964},[31676],[34,348966],{"className":348967,"style":33218},[31680],[34,348969,105852],{"className":348970,"style":37422},[31685,31686]," 包含了这个交集：\n",[34,348973,348975,349016],{"className":348974},[31642],[34,348976,348978],{"className":348977},[31646],[31648,348979,348980],{"xmlns":31650},[31652,348981,348982,349014],{},[31655,348983,348984,348986,348988,348994,348996,349002,349004,349006,349008],{},[31658,348985,105852],{},[31662,348987,346833],{},[31743,348989,348990,348992],{},[31658,348991,106063],{},[31749,348993,640],{},[31662,348995,326344],{},[31743,348997,348998,349000],{},[31658,348999,106063],{},[31749,349001,768],{},[31662,349003,326344],{},[31662,349005,38874],{},[31662,349007,326344],{},[31743,349009,349010,349012],{},[31658,349011,106063],{},[31658,349013,31660],{},[31665,349015,346862],{"encoding":31667},[34,349017,349019,349037,349092,349147,349165],{"className":349018,"ariaHidden":758},[31672],[34,349020,349022,349025,349028,349031,349034],{"className":349021},[31676],[34,349023],{"className":349024,"style":195047},[31680],[34,349026,105852],{"className":349027,"style":37422},[31685,31686],[34,349029],{"className":349030,"style":31846},[31845],[34,349032,346833],{"className":349033},[31850],[34,349035],{"className":349036,"style":31846},[31845],[34,349038,349040,349043,349083,349086,349089],{"className":349039},[31676],[34,349041],{"className":349042,"style":70915},[31680],[34,349044,349046,349049],{"className":349045},[31685],[34,349047,106063],{"className":349048,"style":67482},[31685,31686],[34,349050,349052],{"className":349051},[31792],[34,349053,349055,349075],{"className":349054},[31796,31797],[34,349056,349058,349072],{"className":349057},[31801],[34,349059,349061],{"className":349060,"style":31806},[31805],[34,349062,349063,349066],{"style":217742},[34,349064],{"className":349065,"style":31814},[31813],[34,349067,349069],{"className":349068},[31818,31819,31820,31821],[34,349070,640],{"className":349071},[31685,31821],[34,349073,31829],{"className":349074},[31828],[34,349076,349078],{"className":349077},[31801],[34,349079,349081],{"className":349080,"style":31836},[31805],[34,349082],{},[34,349084],{"className":349085,"style":32604},[31845],[34,349087,326344],{"className":349088},[32024],[34,349090],{"className":349091,"style":32604},[31845],[34,349093,349095,349098,349138,349141,349144],{"className":349094},[31676],[34,349096],{"className":349097,"style":70915},[31680],[34,349099,349101,349104],{"className":349100},[31685],[34,349102,106063],{"className":349103,"style":67482},[31685,31686],[34,349105,349107],{"className":349106},[31792],[34,349108,349110,349130],{"className":349109},[31796,31797],[34,349111,349113,349127],{"className":349112},[31801],[34,349114,349116],{"className":349115,"style":31806},[31805],[34,349117,349118,349121],{"style":217742},[34,349119],{"className":349120,"style":31814},[31813],[34,349122,349124],{"className":349123},[31818,31819,31820,31821],[34,349125,768],{"className":349126},[31685,31821],[34,349128,31829],{"className":349129},[31828],[34,349131,349133],{"className":349132},[31801],[34,349134,349136],{"className":349135,"style":31836},[31805],[34,349137],{},[34,349139],{"className":349140,"style":32604},[31845],[34,349142,326344],{"className":349143},[32024],[34,349145],{"className":349146,"style":32604},[31845],[34,349148,349150,349153,349156,349159,349162],{"className":349149},[31676],[34,349151],{"className":349152,"style":147334},[31680],[34,349154,38874],{"className":349155},[34006],[34,349157],{"className":349158,"style":32604},[31845],[34,349160,326344],{"className":349161},[32024],[34,349163],{"className":349164,"style":32604},[31845],[34,349166,349168,349171],{"className":349167},[31676],[34,349169],{"className":349170,"style":70915},[31680],[34,349172,349174,349177],{"className":349173},[31685],[34,349175,106063],{"className":349176,"style":67482},[31685,31686],[34,349178,349180],{"className":349179},[31792],[34,349181,349183,349203],{"className":349182},[31796,31797],[34,349184,349186,349200],{"className":349185},[31801],[34,349187,349189],{"className":349188,"style":32362},[31805],[34,349190,349191,349194],{"style":217742},[34,349192],{"className":349193,"style":31814},[31813],[34,349195,349197],{"className":349196},[31818,31819,31820,31821],[34,349198,31660],{"className":349199},[31685,31686,31821],[34,349201,31829],{"className":349202},[31828],[34,349204,349206],{"className":349205},[31801],[34,349207,349209],{"className":349208,"style":31836},[31805],[34,349210],{},"\n结合第1步，我们得到：\n",[34,349213,349215,349251],{"className":349214},[31642],[34,349216,349218],{"className":349217},[31646],[31648,349219,349220],{"xmlns":31650},[31652,349221,349222,349248],{},[31655,349223,349224,349226,349228,349234,349240,349242],{},[31658,349225,105852],{},[31662,349227,346833],{},[31743,349229,349230,349232],{},[31658,349231,106063],{},[31749,349233,640],{},[31743,349235,349236,349238],{},[31658,349237,106063],{},[31749,349239,768],{},[31662,349241,33934],{},[31743,349243,349244,349246],{},[31658,349245,106063],{},[31658,349247,31660],{},[31665,349249,349250],{"encoding":31667},"P \\supseteq I_1 I_2 \\dots I_n",[34,349252,349254,349272],{"className":349253,"ariaHidden":758},[31672],[34,349255,349257,349260,349263,349266,349269],{"className":349256},[31676],[34,349258],{"className":349259,"style":195047},[31680],[34,349261,105852],{"className":349262,"style":37422},[31685,31686],[34,349264],{"className":349265,"style":31846},[31845],[34,349267,346833],{"className":349268},[31850],[34,349270],{"className":349271,"style":31846},[31845],[34,349273,349275,349278,349318,349358,349361,349364,349367],{"className":349274},[31676],[34,349276],{"className":349277,"style":70915},[31680],[34,349279,349281,349284],{"className":349280},[31685],[34,349282,106063],{"className":349283,"style":67482},[31685,31686],[34,349285,349287],{"className":349286},[31792],[34,349288,349290,349310],{"className":349289},[31796,31797],[34,349291,349293,349307],{"className":349292},[31801],[34,349294,349296],{"className":349295,"style":31806},[31805],[34,349297,349298,349301],{"style":217742},[34,349299],{"className":349300,"style":31814},[31813],[34,349302,349304],{"className":349303},[31818,31819,31820,31821],[34,349305,640],{"className":349306},[31685,31821],[34,349308,31829],{"className":349309},[31828],[34,349311,349313],{"className":349312},[31801],[34,349314,349316],{"className":349315,"style":31836},[31805],[34,349317],{},[34,349319,349321,349324],{"className":349320},[31685],[34,349322,106063],{"className":349323,"style":67482},[31685,31686],[34,349325,349327],{"className":349326},[31792],[34,349328,349330,349350],{"className":349329},[31796,31797],[34,349331,349333,349347],{"className":349332},[31801],[34,349334,349336],{"className":349335,"style":31806},[31805],[34,349337,349338,349341],{"style":217742},[34,349339],{"className":349340,"style":31814},[31813],[34,349342,349344],{"className":349343},[31818,31819,31820,31821],[34,349345,768],{"className":349346},[31685,31821],[34,349348,31829],{"className":349349},[31828],[34,349351,349353],{"className":349352},[31801],[34,349354,349356],{"className":349355,"style":31836},[31805],[34,349357],{},[34,349359],{"className":349360,"style":32123},[31845],[34,349362,33934],{"className":349363},[34006],[34,349365],{"className":349366,"style":32123},[31845],[34,349368,349370,349373],{"className":349369},[31685],[34,349371,106063],{"className":349372,"style":67482},[31685,31686],[34,349374,349376],{"className":349375},[31792],[34,349377,349379,349399],{"className":349378},[31796,31797],[34,349380,349382,349396],{"className":349381},[31801],[34,349383,349385],{"className":349384,"style":32362},[31805],[34,349386,349387,349390],{"style":217742},[34,349388],{"className":349389,"style":31814},[31813],[34,349391,349393],{"className":349392},[31818,31819,31820,31821],[34,349394,31660],{"className":349395},[31685,31686,31821],[34,349397,31829],{"className":349398},[31828],[34,349400,349402],{"className":349401},[31801],[34,349403,349405],{"className":349404,"style":31836},[31805],[34,349406],{},"\n所以：\n",[34,349409,349411,349447],{"className":349410},[31642],[34,349412,349414],{"className":349413},[31646],[31648,349415,349416],{"xmlns":31650},[31652,349417,349418,349444],{},[31655,349419,349420,349426,349432,349434,349440,349442],{},[31743,349421,349422,349424],{},[31658,349423,106063],{},[31749,349425,640],{},[31743,349427,349428,349430],{},[31658,349429,106063],{},[31749,349431,768],{},[31662,349433,33934],{},[31743,349435,349436,349438],{},[31658,349437,106063],{},[31658,349439,31660],{},[31662,349441,131855],{},[31658,349443,105852],{},[31665,349445,349446],{"encoding":31667},"I_1 I_2 \\dots I_n \\subseteq P",[34,349448,349450,349594],{"className":349449,"ariaHidden":758},[31672],[34,349451,349453,349456,349496,349536,349539,349542,349545,349585,349588,349591],{"className":349452},[31676],[34,349454],{"className":349455,"style":70915},[31680],[34,349457,349459,349462],{"className":349458},[31685],[34,349460,106063],{"className":349461,"style":67482},[31685,31686],[34,349463,349465],{"className":349464},[31792],[34,349466,349468,349488],{"className":349467},[31796,31797],[34,349469,349471,349485],{"className":349470},[31801],[34,349472,349474],{"className":349473,"style":31806},[31805],[34,349475,349476,349479],{"style":217742},[34,349477],{"className":349478,"style":31814},[31813],[34,349480,349482],{"className":349481},[31818,31819,31820,31821],[34,349483,640],{"className":349484},[31685,31821],[34,349486,31829],{"className":349487},[31828],[34,349489,349491],{"className":349490},[31801],[34,349492,349494],{"className":349493,"style":31836},[31805],[34,349495],{},[34,349497,349499,349502],{"className":349498},[31685],[34,349500,106063],{"className":349501,"style":67482},[31685,31686],[34,349503,349505],{"className":349504},[31792],[34,349506,349508,349528],{"className":349507},[31796,31797],[34,349509,349511,349525],{"className":349510},[31801],[34,349512,349514],{"className":349513,"style":31806},[31805],[34,349515,349516,349519],{"style":217742},[34,349517],{"className":349518,"style":31814},[31813],[34,349520,349522],{"className":349521},[31818,31819,31820,31821],[34,349523,768],{"className":349524},[31685,31821],[34,349526,31829],{"className":349527},[31828],[34,349529,349531],{"className":349530},[31801],[34,349532,349534],{"className":349533,"style":31836},[31805],[34,349535],{},[34,349537],{"className":349538,"style":32123},[31845],[34,349540,33934],{"className":349541},[34006],[34,349543],{"className":349544,"style":32123},[31845],[34,349546,349548,349551],{"className":349547},[31685],[34,349549,106063],{"className":349550,"style":67482},[31685,31686],[34,349552,349554],{"className":349553},[31792],[34,349555,349557,349577],{"className":349556},[31796,31797],[34,349558,349560,349574],{"className":349559},[31801],[34,349561,349563],{"className":349562,"style":32362},[31805],[34,349564,349565,349568],{"style":217742},[34,349566],{"className":349567,"style":31814},[31813],[34,349569,349571],{"className":349570},[31818,31819,31820,31821],[34,349572,31660],{"className":349573},[31685,31686,31821],[34,349575,31829],{"className":349576},[31828],[34,349578,349580],{"className":349579},[31801],[34,349581,349583],{"className":349582,"style":31836},[31805],[34,349584],{},[34,349586],{"className":349587,"style":31846},[31845],[34,349589,131855],{"className":349590},[31850],[34,349592],{"className":349593,"style":31846},[31845],[34,349595,349597,349600],{"className":349596},[31676],[34,349598],{"className":349599,"style":33218},[31680],[34,349601,105852],{"className":349602,"style":37422},[31685,31686],[108,349604,349605,349608,349609,349637,349638,349698,349701,349702,349730,349731,349800,349801,349829,349830,315343,349963,350088,350089,350284,350285,350313,350314,86621,350486,13977,350680,350730,350731,350759,350760,350955,350956,351048,351049,351119,351120,351148,351149,351274],{},[2627,349606,349607],{},"利用素理想的性质","：\n对于素理想 ",[34,349610,349612,349625],{"className":349611},[31642],[34,349613,349615],{"className":349614},[31646],[31648,349616,349617],{"xmlns":31650},[31652,349618,349619,349623],{},[31655,349620,349621],{},[31658,349622,105852],{},[31665,349624,105852],{"encoding":31667},[34,349626,349628],{"className":349627,"ariaHidden":758},[31672],[34,349629,349631,349634],{"className":349630},[31676],[34,349632],{"className":349633,"style":33218},[31680],[34,349635,105852],{"className":349636,"style":37422},[31685,31686],"，有一个等价定义或基本性质：\n",[2627,349639,349640,349641,349669,349670,50492],{},"如果理想的乘积包含在 ",[34,349642,349644,349657],{"className":349643},[31642],[34,349645,349647],{"className":349646},[31646],[31648,349648,349649],{"xmlns":31650},[31652,349650,349651,349655],{},[31655,349652,349653],{},[31658,349654,105852],{},[31665,349656,105852],{"encoding":31667},[34,349658,349660],{"className":349659,"ariaHidden":758},[31672],[34,349661,349663,349666],{"className":349662},[31676],[34,349664],{"className":349665,"style":33218},[31680],[34,349667,105852],{"className":349668,"style":37422},[31685,31686]," 中，那么至少有一个因子理想包含在 ",[34,349671,349673,349686],{"className":349672},[31642],[34,349674,349676],{"className":349675},[31646],[31648,349677,349678],{"xmlns":31650},[31652,349679,349680,349684],{},[31655,349681,349682],{},[31658,349683,105852],{},[31665,349685,105852],{"encoding":31667},[34,349687,349689],{"className":349688,"ariaHidden":758},[31672],[34,349690,349692,349695],{"className":349691},[31676],[34,349693],{"className":349694,"style":33218},[31680],[34,349696,105852],{"className":349697,"style":37422},[31685,31686],[71807,349699,349700],{},"为了严谨，如果你没学过这个推论，我们在这里简单反证一下：","\n假设 ",[34,349703,349705,349718],{"className":349704},[31642],[34,349706,349708],{"className":349707},[31646],[31648,349709,349710],{"xmlns":31650},[31652,349711,349712,349716],{},[31655,349713,349714],{},[31658,349715,105852],{},[31665,349717,105852],{"encoding":31667},[34,349719,349721],{"className":349720,"ariaHidden":758},[31672],[34,349722,349724,349727],{"className":349723},[31676],[34,349725],{"className":349726,"style":33218},[31680],[34,349728,105852],{"className":349729,"style":37422},[31685,31686]," 不包含任何一个 ",[34,349732,349734,349751],{"className":349733},[31642],[34,349735,349737],{"className":349736},[31646],[31648,349738,349739],{"xmlns":31650},[31652,349740,349741,349749],{},[31655,349742,349743],{},[31743,349744,349745,349747],{},[31658,349746,106063],{},[31658,349748,31955],{},[31665,349750,346750],{"encoding":31667},[34,349752,349754],{"className":349753,"ariaHidden":758},[31672],[34,349755,349757,349760],{"className":349756},[31676],[34,349758],{"className":349759,"style":70915},[31680],[34,349761,349763,349766],{"className":349762},[31685],[34,349764,106063],{"className":349765,"style":67482},[31685,31686],[34,349767,349769],{"className":349768},[31792],[34,349770,349772,349792],{"className":349771},[31796,31797],[34,349773,349775,349789],{"className":349774},[31801],[34,349776,349778],{"className":349777,"style":32090},[31805],[34,349779,349780,349783],{"style":217742},[34,349781],{"className":349782,"style":31814},[31813],[34,349784,349786],{"className":349785},[31818,31819,31820,31821],[34,349787,31955],{"className":349788},[31685,31686,31821],[34,349790,31829],{"className":349791},[31828],[34,349793,349795],{"className":349794},[31801],[34,349796,349798],{"className":349797,"style":31836},[31805],[34,349799],{},"。\n这意味着对于每一个 ",[34,349802,349804,349817],{"className":349803},[31642],[34,349805,349807],{"className":349806},[31646],[31648,349808,349809],{"xmlns":31650},[31652,349810,349811,349815],{},[31655,349812,349813],{},[31658,349814,31955],{},[31665,349816,31955],{"encoding":31667},[34,349818,349820],{"className":349819,"ariaHidden":758},[31672],[34,349821,349823,349826],{"className":349822},[31676],[34,349824],{"className":349825,"style":33104},[31680],[34,349827,31955],{"className":349828},[31685,31686],"，都存在一个元素 ",[34,349831,349833,349859],{"className":349832},[31642],[34,349834,349836],{"className":349835},[31646],[31648,349837,349838],{"xmlns":31650},[31652,349839,349840,349856],{},[31655,349841,349842,349848,349850],{},[31743,349843,349844,349846],{},[31658,349845,111],{},[31658,349847,31955],{},[31662,349849,112296],{},[31743,349851,349852,349854],{},[31658,349853,106063],{},[31658,349855,31955],{},[31665,349857,349858],{"encoding":31667},"a_i \\in I_i",[34,349860,349862,349917],{"className":349861,"ariaHidden":758},[31672],[34,349863,349865,349868,349908,349911,349914],{"className":349864},[31676],[34,349866],{"className":349867,"style":225773},[31680],[34,349869,349871,349874],{"className":349870},[31685],[34,349872,111],{"className":349873},[31685,31686],[34,349875,349877],{"className":349876},[31792],[34,349878,349880,349900],{"className":349879},[31796,31797],[34,349881,349883,349897],{"className":349882},[31801],[34,349884,349886],{"className":349885,"style":32090},[31805],[34,349887,349888,349891],{"style":31809},[34,349889],{"className":349890,"style":31814},[31813],[34,349892,349894],{"className":349893},[31818,31819,31820,31821],[34,349895,31955],{"className":349896},[31685,31686,31821],[34,349898,31829],{"className":349899},[31828],[34,349901,349903],{"className":349902},[31801],[34,349904,349906],{"className":349905,"style":31836},[31805],[34,349907],{},[34,349909],{"className":349910,"style":31846},[31845],[34,349912,112296],{"className":349913},[31850],[34,349915],{"className":349916,"style":31846},[31845],[34,349918,349920,349923],{"className":349919},[31676],[34,349921],{"className":349922,"style":70915},[31680],[34,349924,349926,349929],{"className":349925},[31685],[34,349927,106063],{"className":349928,"style":67482},[31685,31686],[34,349930,349932],{"className":349931},[31792],[34,349933,349935,349955],{"className":349934},[31796,31797],[34,349936,349938,349952],{"className":349937},[31801],[34,349939,349941],{"className":349940,"style":32090},[31805],[34,349942,349943,349946],{"style":217742},[34,349944],{"className":349945,"style":31814},[31813],[34,349947,349949],{"className":349948},[31818,31819,31820,31821],[34,349950,31955],{"className":349951},[31685,31686,31821],[34,349953,31829],{"className":349954},[31828],[34,349956,349958],{"className":349957},[31801],[34,349959,349961],{"className":349960,"style":31836},[31805],[34,349962],{},[34,349964,349966,349988],{"className":349965},[31642],[34,349967,349969],{"className":349968},[31646],[31648,349970,349971],{"xmlns":31650},[31652,349972,349973,349985],{},[31655,349974,349975,349981,349983],{},[31743,349976,349977,349979],{},[31658,349978,111],{},[31658,349980,31955],{},[31662,349982,112266],{"mathvariant":38079},[31658,349984,105852],{},[31665,349986,349987],{"encoding":31667},"a_i \\notin P",[34,349989,349991,350079],{"className":349990,"ariaHidden":758},[31672],[34,349992,349994,349997,350037,350040,350076],{"className":349993},[31676],[34,349995],{"className":349996,"style":31774},[31680],[34,349998,350000,350003],{"className":349999},[31685],[34,350001,111],{"className":350002},[31685,31686],[34,350004,350006],{"className":350005},[31792],[34,350007,350009,350029],{"className":350008},[31796,31797],[34,350010,350012,350026],{"className":350011},[31801],[34,350013,350015],{"className":350014,"style":32090},[31805],[34,350016,350017,350020],{"style":31809},[34,350018],{"className":350019,"style":31814},[31813],[34,350021,350023],{"className":350022},[31818,31819,31820,31821],[34,350024,31955],{"className":350025},[31685,31686,31821],[34,350027,31829],{"className":350028},[31828],[34,350030,350032],{"className":350031},[31801],[34,350033,350035],{"className":350034,"style":31836},[31805],[34,350036],{},[34,350038],{"className":350039,"style":31846},[31845],[34,350041,350043,350049],{"className":350042},[31850],[34,350044,350046],{"className":350045},[31685],[34,350047,112296],{"className":350048},[31850],[34,350050,350052],{"className":350051},[31685,54774],[34,350053,350055],{"className":350054},[54778],[34,350056,350058,350061,350073],{"className":350057},[112306],[34,350059],{"className":350060,"style":31774},[31680],[34,350062,350064],{"className":350063},[54789],[34,350065,350067,350070],{"className":350066},[31685],[34,350068,38080],{"className":350069},[31685],[34,350071],{"className":350072,"style":112322},[31845],[34,350074],{"className":350075},[54800],[34,350077],{"className":350078,"style":31846},[31845],[34,350080,350082,350085],{"className":350081},[31676],[34,350083],{"className":350084,"style":33218},[31680],[34,350086,105852],{"className":350087,"style":37422},[31685,31686],"。\n考虑元素 ",[34,350090,350092,350128],{"className":350091},[31642],[34,350093,350095],{"className":350094},[31646],[31648,350096,350097],{"xmlns":31650},[31652,350098,350099,350125],{},[31655,350100,350101,350103,350105,350111,350117,350119],{},[31658,350102,32936],{},[31662,350104,202],{},[31743,350106,350107,350109],{},[31658,350108,111],{},[31749,350110,640],{},[31743,350112,350113,350115],{},[31658,350114,111],{},[31749,350116,768],{},[31662,350118,33934],{},[31743,350120,350121,350123],{},[31658,350122,111],{},[31658,350124,31660],{},[31665,350126,350127],{"encoding":31667},"x = a_1 a_2 \\dots a_n",[34,350129,350131,350149],{"className":350130,"ariaHidden":758},[31672],[34,350132,350134,350137,350140,350143,350146],{"className":350133},[31676],[34,350135],{"className":350136,"style":32527},[31680],[34,350138,32936],{"className":350139},[31685,31686],[34,350141],{"className":350142,"style":31846},[31845],[34,350144,202],{"className":350145},[31850],[34,350147],{"className":350148,"style":31846},[31845],[34,350150,350152,350155,350195,350235,350238,350241,350244],{"className":350151},[31676],[34,350153],{"className":350154,"style":31860},[31680],[34,350156,350158,350161],{"className":350157},[31685],[34,350159,111],{"className":350160},[31685,31686],[34,350162,350164],{"className":350163},[31792],[34,350165,350167,350187],{"className":350166},[31796,31797],[34,350168,350170,350184],{"className":350169},[31801],[34,350171,350173],{"className":350172,"style":31806},[31805],[34,350174,350175,350178],{"style":31809},[34,350176],{"className":350177,"style":31814},[31813],[34,350179,350181],{"className":350180},[31818,31819,31820,31821],[34,350182,640],{"className":350183},[31685,31821],[34,350185,31829],{"className":350186},[31828],[34,350188,350190],{"className":350189},[31801],[34,350191,350193],{"className":350192,"style":31836},[31805],[34,350194],{},[34,350196,350198,350201],{"className":350197},[31685],[34,350199,111],{"className":350200},[31685,31686],[34,350202,350204],{"className":350203},[31792],[34,350205,350207,350227],{"className":350206},[31796,31797],[34,350208,350210,350224],{"className":350209},[31801],[34,350211,350213],{"className":350212,"style":31806},[31805],[34,350214,350215,350218],{"style":31809},[34,350216],{"className":350217,"style":31814},[31813],[34,350219,350221],{"className":350220},[31818,31819,31820,31821],[34,350222,768],{"className":350223},[31685,31821],[34,350225,31829],{"className":350226},[31828],[34,350228,350230],{"className":350229},[31801],[34,350231,350233],{"className":350232,"style":31836},[31805],[34,350234],{},[34,350236],{"className":350237,"style":32123},[31845],[34,350239,33934],{"className":350240},[34006],[34,350242],{"className":350243,"style":32123},[31845],[34,350245,350247,350250],{"className":350246},[31685],[34,350248,111],{"className":350249},[31685,31686],[34,350251,350253],{"className":350252},[31792],[34,350254,350256,350276],{"className":350255},[31796,31797],[34,350257,350259,350273],{"className":350258},[31801],[34,350260,350262],{"className":350261,"style":32362},[31805],[34,350263,350264,350267],{"style":31809},[34,350265],{"className":350266,"style":31814},[31813],[34,350268,350270],{"className":350269},[31818,31819,31820,31821],[34,350271,31660],{"className":350272},[31685,31686,31821],[34,350274,31829],{"className":350275},[31828],[34,350277,350279],{"className":350278},[31801],[34,350280,350282],{"className":350281,"style":31836},[31805],[34,350283],{},"。\n这个 ",[34,350286,350288,350301],{"className":350287},[31642],[34,350289,350291],{"className":350290},[31646],[31648,350292,350293],{"xmlns":31650},[31652,350294,350295,350299],{},[31655,350296,350297],{},[31658,350298,32936],{},[31665,350300,32936],{"encoding":31667},[34,350302,350304],{"className":350303,"ariaHidden":758},[31672],[34,350305,350307,350310],{"className":350306},[31676],[34,350308],{"className":350309,"style":32527},[31680],[34,350311,32936],{"className":350312},[31685,31686]," 显然属于乘积理想 ",[34,350315,350317,350348],{"className":350316},[31642],[34,350318,350320],{"className":350319},[31646],[31648,350321,350322],{"xmlns":31650},[31652,350323,350324,350346],{},[31655,350325,350326,350332,350338,350340],{},[31743,350327,350328,350330],{},[31658,350329,106063],{},[31749,350331,640],{},[31743,350333,350334,350336],{},[31658,350335,106063],{},[31749,350337,768],{},[31662,350339,33934],{},[31743,350341,350342,350344],{},[31658,350343,106063],{},[31658,350345,31660],{},[31665,350347,347577],{"encoding":31667},[34,350349,350351],{"className":350350,"ariaHidden":758},[31672],[34,350352,350354,350357,350397,350437,350440,350443,350446],{"className":350353},[31676],[34,350355],{"className":350356,"style":70915},[31680],[34,350358,350360,350363],{"className":350359},[31685],[34,350361,106063],{"className":350362,"style":67482},[31685,31686],[34,350364,350366],{"className":350365},[31792],[34,350367,350369,350389],{"className":350368},[31796,31797],[34,350370,350372,350386],{"className":350371},[31801],[34,350373,350375],{"className":350374,"style":31806},[31805],[34,350376,350377,350380],{"style":217742},[34,350378],{"className":350379,"style":31814},[31813],[34,350381,350383],{"className":350382},[31818,31819,31820,31821],[34,350384,640],{"className":350385},[31685,31821],[34,350387,31829],{"className":350388},[31828],[34,350390,350392],{"className":350391},[31801],[34,350393,350395],{"className":350394,"style":31836},[31805],[34,350396],{},[34,350398,350400,350403],{"className":350399},[31685],[34,350401,106063],{"className":350402,"style":67482},[31685,31686],[34,350404,350406],{"className":350405},[31792],[34,350407,350409,350429],{"className":350408},[31796,31797],[34,350410,350412,350426],{"className":350411},[31801],[34,350413,350415],{"className":350414,"style":31806},[31805],[34,350416,350417,350420],{"style":217742},[34,350418],{"className":350419,"style":31814},[31813],[34,350421,350423],{"className":350422},[31818,31819,31820,31821],[34,350424,768],{"className":350425},[31685,31821],[34,350427,31829],{"className":350428},[31828],[34,350430,350432],{"className":350431},[31801],[34,350433,350435],{"className":350434,"style":31836},[31805],[34,350436],{},[34,350438],{"className":350439,"style":32123},[31845],[34,350441,33934],{"className":350442},[34006],[34,350444],{"className":350445,"style":32123},[31845],[34,350447,350449,350452],{"className":350448},[31685],[34,350450,106063],{"className":350451,"style":67482},[31685,31686],[34,350453,350455],{"className":350454},[31792],[34,350456,350458,350478],{"className":350457},[31796,31797],[34,350459,350461,350475],{"className":350460},[31801],[34,350462,350464],{"className":350463,"style":32362},[31805],[34,350465,350466,350469],{"style":217742},[34,350467],{"className":350468,"style":31814},[31813],[34,350470,350472],{"className":350471},[31818,31819,31820,31821],[34,350473,31660],{"className":350474},[31685,31686,31821],[34,350476,31829],{"className":350477},[31828],[34,350479,350481],{"className":350480},[31801],[34,350482,350484],{"className":350483,"style":31836},[31805],[34,350485],{},[34,350487,350489,350524],{"className":350488},[31642],[34,350490,350492],{"className":350491},[31646],[31648,350493,350494],{"xmlns":31650},[31652,350495,350496,350522],{},[31655,350497,350498,350504,350510,350512,350518,350520],{},[31743,350499,350500,350502],{},[31658,350501,106063],{},[31749,350503,640],{},[31743,350505,350506,350508],{},[31658,350507,106063],{},[31749,350509,768],{},[31662,350511,33934],{},[31743,350513,350514,350516],{},[31658,350515,106063],{},[31658,350517,31660],{},[31662,350519,131855],{},[31658,350521,105852],{},[31665,350523,349446],{"encoding":31667},[34,350525,350527,350671],{"className":350526,"ariaHidden":758},[31672],[34,350528,350530,350533,350573,350613,350616,350619,350622,350662,350665,350668],{"className":350529},[31676],[34,350531],{"className":350532,"style":70915},[31680],[34,350534,350536,350539],{"className":350535},[31685],[34,350537,106063],{"className":350538,"style":67482},[31685,31686],[34,350540,350542],{"className":350541},[31792],[34,350543,350545,350565],{"className":350544},[31796,31797],[34,350546,350548,350562],{"className":350547},[31801],[34,350549,350551],{"className":350550,"style":31806},[31805],[34,350552,350553,350556],{"style":217742},[34,350554],{"className":350555,"style":31814},[31813],[34,350557,350559],{"className":350558},[31818,31819,31820,31821],[34,350560,640],{"className":350561},[31685,31821],[34,350563,31829],{"className":350564},[31828],[34,350566,350568],{"className":350567},[31801],[34,350569,350571],{"className":350570,"style":31836},[31805],[34,350572],{},[34,350574,350576,350579],{"className":350575},[31685],[34,350577,106063],{"className":350578,"style":67482},[31685,31686],[34,350580,350582],{"className":350581},[31792],[34,350583,350585,350605],{"className":350584},[31796,31797],[34,350586,350588,350602],{"className":350587},[31801],[34,350589,350591],{"className":350590,"style":31806},[31805],[34,350592,350593,350596],{"style":217742},[34,350594],{"className":350595,"style":31814},[31813],[34,350597,350599],{"className":350598},[31818,31819,31820,31821],[34,350600,768],{"className":350601},[31685,31821],[34,350603,31829],{"className":350604},[31828],[34,350606,350608],{"className":350607},[31801],[34,350609,350611],{"className":350610,"style":31836},[31805],[34,350612],{},[34,350614],{"className":350615,"style":32123},[31845],[34,350617,33934],{"className":350618},[34006],[34,350620],{"className":350621,"style":32123},[31845],[34,350623,350625,350628],{"className":350624},[31685],[34,350626,106063],{"className":350627,"style":67482},[31685,31686],[34,350629,350631],{"className":350630},[31792],[34,350632,350634,350654],{"className":350633},[31796,31797],[34,350635,350637,350651],{"className":350636},[31801],[34,350638,350640],{"className":350639,"style":32362},[31805],[34,350641,350642,350645],{"style":217742},[34,350643],{"className":350644,"style":31814},[31813],[34,350646,350648],{"className":350647},[31818,31819,31820,31821],[34,350649,31660],{"className":350650},[31685,31686,31821],[34,350652,31829],{"className":350653},[31828],[34,350655,350657],{"className":350656},[31801],[34,350658,350660],{"className":350659,"style":31836},[31805],[34,350661],{},[34,350663],{"className":350664,"style":31846},[31845],[34,350666,131855],{"className":350667},[31850],[34,350669],{"className":350670,"style":31846},[31845],[34,350672,350674,350677],{"className":350673},[31676],[34,350675],{"className":350676,"style":33218},[31680],[34,350678,105852],{"className":350679,"style":37422},[31685,31686],[34,350681,350683,350700],{"className":350682},[31642],[34,350684,350686],{"className":350685},[31646],[31648,350687,350688],{"xmlns":31650},[31652,350689,350690,350698],{},[31655,350691,350692,350694,350696],{},[31658,350693,32936],{},[31662,350695,112296],{},[31658,350697,105852],{},[31665,350699,343445],{"encoding":31667},[34,350701,350703,350721],{"className":350702,"ariaHidden":758},[31672],[34,350704,350706,350709,350712,350715,350718],{"className":350705},[31676],[34,350707],{"className":350708,"style":34812},[31680],[34,350710,32936],{"className":350711},[31685,31686],[34,350713],{"className":350714,"style":31846},[31845],[34,350716,112296],{"className":350717},[31850],[34,350719],{"className":350720,"style":31846},[31845],[34,350722,350724,350727],{"className":350723},[31676],[34,350725],{"className":350726,"style":33218},[31680],[34,350728,105852],{"className":350729,"style":37422},[31685,31686],"。\n但是 ",[34,350732,350734,350747],{"className":350733},[31642],[34,350735,350737],{"className":350736},[31646],[31648,350738,350739],{"xmlns":31650},[31652,350740,350741,350745],{},[31655,350742,350743],{},[31658,350744,105852],{},[31665,350746,105852],{"encoding":31667},[34,350748,350750],{"className":350749,"ariaHidden":758},[31672],[34,350751,350753,350756],{"className":350752},[31676],[34,350754],{"className":350755,"style":33218},[31680],[34,350757,105852],{"className":350758,"style":37422},[31685,31686]," 是素理想，如果 ",[34,350761,350763,350799],{"className":350762},[31642],[34,350764,350766],{"className":350765},[31646],[31648,350767,350768],{"xmlns":31650},[31652,350769,350770,350796],{},[31655,350771,350772,350778,350784,350786,350792,350794],{},[31743,350773,350774,350776],{},[31658,350775,111],{},[31749,350777,640],{},[31743,350779,350780,350782],{},[31658,350781,111],{},[31749,350783,768],{},[31662,350785,33934],{},[31743,350787,350788,350790],{},[31658,350789,111],{},[31658,350791,31660],{},[31662,350793,112296],{},[31658,350795,105852],{},[31665,350797,350798],{"encoding":31667},"a_1 a_2 \\dots a_n \\in P",[34,350800,350802,350946],{"className":350801,"ariaHidden":758},[31672],[34,350803,350805,350808,350848,350888,350891,350894,350897,350937,350940,350943],{"className":350804},[31676],[34,350806],{"className":350807,"style":225773},[31680],[34,350809,350811,350814],{"className":350810},[31685],[34,350812,111],{"className":350813},[31685,31686],[34,350815,350817],{"className":350816},[31792],[34,350818,350820,350840],{"className":350819},[31796,31797],[34,350821,350823,350837],{"className":350822},[31801],[34,350824,350826],{"className":350825,"style":31806},[31805],[34,350827,350828,350831],{"style":31809},[34,350829],{"className":350830,"style":31814},[31813],[34,350832,350834],{"className":350833},[31818,31819,31820,31821],[34,350835,640],{"className":350836},[31685,31821],[34,350838,31829],{"className":350839},[31828],[34,350841,350843],{"className":350842},[31801],[34,350844,350846],{"className":350845,"style":31836},[31805],[34,350847],{},[34,350849,350851,350854],{"className":350850},[31685],[34,350852,111],{"className":350853},[31685,31686],[34,350855,350857],{"className":350856},[31792],[34,350858,350860,350880],{"className":350859},[31796,31797],[34,350861,350863,350877],{"className":350862},[31801],[34,350864,350866],{"className":350865,"style":31806},[31805],[34,350867,350868,350871],{"style":31809},[34,350869],{"className":350870,"style":31814},[31813],[34,350872,350874],{"className":350873},[31818,31819,31820,31821],[34,350875,768],{"className":350876},[31685,31821],[34,350878,31829],{"className":350879},[31828],[34,350881,350883],{"className":350882},[31801],[34,350884,350886],{"className":350885,"style":31836},[31805],[34,350887],{},[34,350889],{"className":350890,"style":32123},[31845],[34,350892,33934],{"className":350893},[34006],[34,350895],{"className":350896,"style":32123},[31845],[34,350898,350900,350903],{"className":350899},[31685],[34,350901,111],{"className":350902},[31685,31686],[34,350904,350906],{"className":350905},[31792],[34,350907,350909,350929],{"className":350908},[31796,31797],[34,350910,350912,350926],{"className":350911},[31801],[34,350913,350915],{"className":350914,"style":32362},[31805],[34,350916,350917,350920],{"style":31809},[34,350918],{"className":350919,"style":31814},[31813],[34,350921,350923],{"className":350922},[31818,31819,31820,31821],[34,350924,31660],{"className":350925},[31685,31686,31821],[34,350927,31829],{"className":350928},[31828],[34,350930,350932],{"className":350931},[31801],[34,350933,350935],{"className":350934,"style":31836},[31805],[34,350936],{},[34,350938],{"className":350939,"style":31846},[31845],[34,350941,112296],{"className":350942},[31850],[34,350944],{"className":350945,"style":31846},[31845],[34,350947,350949,350952],{"className":350948},[31676],[34,350950],{"className":350951,"style":33218},[31680],[34,350953,105852],{"className":350954,"style":37422},[31685,31686],"，则必有某个 ",[34,350957,350959,350981],{"className":350958},[31642],[34,350960,350962],{"className":350961},[31646],[31648,350963,350964],{"xmlns":31650},[31652,350965,350966,350978],{},[31655,350967,350968,350974,350976],{},[31743,350969,350970,350972],{},[31658,350971,111],{},[31658,350973,33759],{},[31662,350975,112296],{},[31658,350977,105852],{},[31665,350979,350980],{"encoding":31667},"a_k \\in P",[34,350982,350984,351039],{"className":350983,"ariaHidden":758},[31672],[34,350985,350987,350990,351030,351033,351036],{"className":350986},[31676],[34,350988],{"className":350989,"style":225773},[31680],[34,350991,350993,350996],{"className":350992},[31685],[34,350994,111],{"className":350995},[31685,31686],[34,350997,350999],{"className":350998},[31792],[34,351000,351002,351022],{"className":351001},[31796,31797],[34,351003,351005,351019],{"className":351004},[31801],[34,351006,351008],{"className":351007,"style":33833},[31805],[34,351009,351010,351013],{"style":31809},[34,351011],{"className":351012,"style":31814},[31813],[34,351014,351016],{"className":351015},[31818,31819,31820,31821],[34,351017,33759],{"className":351018,"style":33774},[31685,31686,31821],[34,351020,31829],{"className":351021},[31828],[34,351023,351025],{"className":351024},[31801],[34,351026,351028],{"className":351027,"style":31836},[31805],[34,351029],{},[34,351031],{"className":351032,"style":31846},[31845],[34,351034,112296],{"className":351035},[31850],[34,351037],{"className":351038,"style":31846},[31845],[34,351040,351042,351045],{"className":351041},[31676],[34,351043],{"className":351044,"style":33218},[31680],[34,351046,105852],{"className":351047,"style":37422},[31685,31686],"。\n这与我们假设的“",[34,351050,351052,351070],{"className":351051},[31642],[34,351053,351055],{"className":351054},[31646],[31648,351056,351057],{"xmlns":31650},[31652,351058,351059,351067],{},[31655,351060,351061],{},[31743,351062,351063,351065],{},[31658,351064,111],{},[31658,351066,33759],{},[31665,351068,351069],{"encoding":31667},"a_k",[34,351071,351073],{"className":351072,"ariaHidden":758},[31672],[34,351074,351076,351079],{"className":351075},[31676],[34,351077],{"className":351078,"style":31860},[31680],[34,351080,351082,351085],{"className":351081},[31685],[34,351083,111],{"className":351084},[31685,31686],[34,351086,351088],{"className":351087},[31792],[34,351089,351091,351111],{"className":351090},[31796,31797],[34,351092,351094,351108],{"className":351093},[31801],[34,351095,351097],{"className":351096,"style":33833},[31805],[34,351098,351099,351102],{"style":31809},[34,351100],{"className":351101,"style":31814},[31813],[34,351103,351105],{"className":351104},[31818,31819,31820,31821],[34,351106,33759],{"className":351107,"style":33774},[31685,31686,31821],[34,351109,31829],{"className":351110},[31828],[34,351112,351114],{"className":351113},[31801],[34,351115,351117],{"className":351116,"style":31836},[31805],[34,351118],{}," 是专门从 ",[34,351121,351123,351136],{"className":351122},[31642],[34,351124,351126],{"className":351125},[31646],[31648,351127,351128],{"xmlns":31650},[31652,351129,351130,351134],{},[31655,351131,351132],{},[31658,351133,105852],{},[31665,351135,105852],{"encoding":31667},[34,351137,351139],{"className":351138,"ariaHidden":758},[31672],[34,351140,351142,351145],{"className":351141},[31676],[34,351143],{"className":351144,"style":33218},[31680],[34,351146,105852],{"className":351147,"style":37422},[31685,31686]," 外面挑出来的 (",[34,351150,351152,351174],{"className":351151},[31642],[34,351153,351155],{"className":351154},[31646],[31648,351156,351157],{"xmlns":31650},[31652,351158,351159,351171],{},[31655,351160,351161,351167,351169],{},[31743,351162,351163,351165],{},[31658,351164,111],{},[31658,351166,33759],{},[31662,351168,112266],{"mathvariant":38079},[31658,351170,105852],{},[31665,351172,351173],{"encoding":31667},"a_k \\notin P",[34,351175,351177,351265],{"className":351176,"ariaHidden":758},[31672],[34,351178,351180,351183,351223,351226,351262],{"className":351179},[31676],[34,351181],{"className":351182,"style":31774},[31680],[34,351184,351186,351189],{"className":351185},[31685],[34,351187,111],{"className":351188},[31685,31686],[34,351190,351192],{"className":351191},[31792],[34,351193,351195,351215],{"className":351194},[31796,31797],[34,351196,351198,351212],{"className":351197},[31801],[34,351199,351201],{"className":351200,"style":33833},[31805],[34,351202,351203,351206],{"style":31809},[34,351204],{"className":351205,"style":31814},[31813],[34,351207,351209],{"className":351208},[31818,31819,31820,31821],[34,351210,33759],{"className":351211,"style":33774},[31685,31686,31821],[34,351213,31829],{"className":351214},[31828],[34,351216,351218],{"className":351217},[31801],[34,351219,351221],{"className":351220,"style":31836},[31805],[34,351222],{},[34,351224],{"className":351225,"style":31846},[31845],[34,351227,351229,351235],{"className":351228},[31850],[34,351230,351232],{"className":351231},[31685],[34,351233,112296],{"className":351234},[31850],[34,351236,351238],{"className":351237},[31685,54774],[34,351239,351241],{"className":351240},[54778],[34,351242,351244,351247,351259],{"className":351243},[112306],[34,351245],{"className":351246,"style":31774},[31680],[34,351248,351250],{"className":351249},[54789],[34,351251,351253,351256],{"className":351252},[31685],[34,351254,38080],{"className":351255},[31685],[34,351257],{"className":351258,"style":112322},[31845],[34,351260],{"className":351261},[54800],[34,351263],{"className":351264,"style":31846},[31845],[34,351266,351268,351271],{"className":351267},[31676],[34,351269],{"className":351270,"style":33218},[31680],[34,351272,105852],{"className":351273,"style":37422},[31685,31686],")”相矛盾。",[108,351276,351277,351279,351280,335149,351370,351462],{},[2627,351278,114043],{},"：\n假设不成立。因此，必然存在某个 ",[34,351281,351283,351313],{"className":351282},[31642],[34,351284,351286],{"className":351285},[31646],[31648,351287,351288],{"xmlns":31650},[31652,351289,351290,351310],{},[31655,351291,351292,351294,351296,351298,351300,351302,351304,351306,351308],{},[31658,351293,33759],{},[31662,351295,112296],{},[31662,351297,32685],{"stretchy":702},[31749,351299,640],{},[31662,351301,31960],{"separator":758},[31662,351303,33934],{},[31662,351305,31960],{"separator":758},[31658,351307,31660],{},[31662,351309,32715],{"stretchy":702},[31665,351311,351312],{"encoding":31667},"k \\in \\{1, \\dots, n\\}",[34,351314,351316,351334],{"className":351315,"ariaHidden":758},[31672],[34,351317,351319,351322,351325,351328,351331],{"className":351318},[31676],[34,351320],{"className":351321,"style":46991},[31680],[34,351323,33759],{"className":351324,"style":33774},[31685,31686],[34,351326],{"className":351327,"style":31846},[31845],[34,351329,112296],{"className":351330},[31850],[34,351332],{"className":351333,"style":31846},[31845],[34,351335,351337,351340,351343,351346,351349,351352,351355,351358,351361,351364,351367],{"className":351336},[31676],[34,351338],{"className":351339,"style":31774},[31680],[34,351341,32685],{"className":351342},[31782],[34,351344,640],{"className":351345},[31685],[34,351347,31960],{"className":351348},[32119],[34,351350],{"className":351351,"style":32123},[31845],[34,351353,33934],{"className":351354},[34006],[34,351356],{"className":351357,"style":32123},[31845],[34,351359,31960],{"className":351360},[32119],[34,351362],{"className":351363,"style":32123},[31845],[34,351365,31660],{"className":351366},[31685,31686],[34,351368,32715],{"className":351369},[31690],[34,351371,351373,351395],{"className":351372},[31642],[34,351374,351376],{"className":351375},[31646],[31648,351377,351378],{"xmlns":31650},[31652,351379,351380,351392],{},[31655,351381,351382,351388,351390],{},[31743,351383,351384,351386],{},[31658,351385,106063],{},[31658,351387,33759],{},[31662,351389,131855],{},[31658,351391,105852],{},[31665,351393,351394],{"encoding":31667},"I_k \\subseteq P",[34,351396,351398,351453],{"className":351397,"ariaHidden":758},[31672],[34,351399,351401,351404,351444,351447,351450],{"className":351400},[31676],[34,351402],{"className":351403,"style":70915},[31680],[34,351405,351407,351410],{"className":351406},[31685],[34,351408,106063],{"className":351409,"style":67482},[31685,31686],[34,351411,351413],{"className":351412},[31792],[34,351414,351416,351436],{"className":351415},[31796,31797],[34,351417,351419,351433],{"className":351418},[31801],[34,351420,351422],{"className":351421,"style":33833},[31805],[34,351423,351424,351427],{"style":217742},[34,351425],{"className":351426,"style":31814},[31813],[34,351428,351430],{"className":351429},[31818,31819,31820,31821],[34,351431,33759],{"className":351432,"style":33774},[31685,31686,31821],[34,351434,31829],{"className":351435},[31828],[34,351437,351439],{"className":351438},[31801],[34,351440,351442],{"className":351441,"style":31836},[31805],[34,351443],{},[34,351445],{"className":351446,"style":31846},[31845],[34,351448,131855],{"className":351449},[31850],[34,351451],{"className":351452,"style":31846},[31845],[34,351454,351456,351459],{"className":351455},[31676],[34,351457],{"className":351458,"style":33218},[31680],[34,351460,105852],{"className":351461,"style":37422},[31685,31686],"。证毕。",[32188,351464],{},[253,351466,351468],{"id":351467},"题号-9","题号 9",[32490,351470,162043],{"id":168779},[15,351472,346507,351473,351501,351502,351530,351531,351665,351666,351694,351695,347],{},[34,351474,351476,351489],{"className":351475},[31642],[34,351477,351479],{"className":351478},[31646],[31648,351480,351481],{"xmlns":31650},[31652,351482,351483,351487],{},[31655,351484,351485],{},[31658,351486,34386],{},[31665,351488,34386],{"encoding":31667},[34,351490,351492],{"className":351491,"ariaHidden":758},[31672],[34,351493,351495,351498],{"className":351494},[31676],[34,351496],{"className":351497,"style":33218},[31680],[34,351499,34386],{"className":351500,"style":34427},[31685,31686]," 的一个理想 ",[34,351503,351505,351518],{"className":351504},[31642],[34,351506,351508],{"className":351507},[31646],[31648,351509,351510],{"xmlns":31650},[31652,351511,351512,351516],{},[31655,351513,351514],{},[31658,351515,106063],{},[31665,351517,106063],{"encoding":31667},[34,351519,351521],{"className":351520,"ariaHidden":758},[31672],[34,351522,351524,351527],{"className":351523},[31676],[34,351525],{"className":351526,"style":33218},[31680],[34,351528,106063],{"className":351529,"style":67482},[31685,31686]," 含于有限多个素理想 ",[34,351532,351534,351568],{"className":351533},[31642],[34,351535,351537],{"className":351536},[31646],[31648,351538,351539],{"xmlns":31650},[31652,351540,351541,351565],{},[31655,351542,351543,351549,351551,351553,351555,351557,351559,351561,351563],{},[31743,351544,351545,351547],{},[31658,351546,105852],{},[31658,351548,31955],{},[39361,351550,39363],{},[31662,351552,456],{"stretchy":702},[31749,351554,640],{},[31662,351556,45972],{},[31658,351558,31955],{},[31662,351560,45972],{},[31658,351562,31660],{},[31662,351564,2557],{"stretchy":702},[31665,351566,351567],{"encoding":31667},"P_i \\, (1 \\le i \\le n)",[34,351569,351571,351635,351653],{"className":351570,"ariaHidden":758},[31672],[34,351572,351574,351577,351617,351620,351623,351626,351629,351632],{"className":351573},[31676],[34,351575],{"className":351576,"style":31774},[31680],[34,351578,351580,351583],{"className":351579},[31685],[34,351581,105852],{"className":351582,"style":37422},[31685,31686],[34,351584,351586],{"className":351585},[31792],[34,351587,351589,351609],{"className":351588},[31796,31797],[34,351590,351592,351606],{"className":351591},[31801],[34,351593,351595],{"className":351594,"style":32090},[31805],[34,351596,351597,351600],{"style":214424},[34,351598],{"className":351599,"style":31814},[31813],[34,351601,351603],{"className":351602},[31818,31819,31820,31821],[34,351604,31955],{"className":351605},[31685,31686,31821],[34,351607,31829],{"className":351608},[31828],[34,351610,351612],{"className":351611},[31801],[34,351613,351615],{"className":351614,"style":31836},[31805],[34,351616],{},[34,351618],{"className":351619,"style":32123},[31845],[34,351621,456],{"className":351622},[31782],[34,351624,640],{"className":351625},[31685],[34,351627],{"className":351628,"style":31846},[31845],[34,351630,45972],{"className":351631},[31850],[34,351633],{"className":351634,"style":31846},[31845],[34,351636,351638,351641,351644,351647,351650],{"className":351637},[31676],[34,351639],{"className":351640,"style":55401},[31680],[34,351642,31955],{"className":351643},[31685,31686],[34,351645],{"className":351646,"style":31846},[31845],[34,351648,45972],{"className":351649},[31850],[34,351651],{"className":351652,"style":31846},[31845],[34,351654,351656,351659,351662],{"className":351655},[31676],[34,351657],{"className":351658,"style":31774},[31680],[34,351660,31660],{"className":351661},[31685,31686],[34,351663,2557],{"className":351664},[31690]," 的并，证明 ",[34,351667,351669,351682],{"className":351668},[31642],[34,351670,351672],{"className":351671},[31646],[31648,351673,351674],{"xmlns":31650},[31652,351675,351676,351680],{},[31655,351677,351678],{},[31658,351679,106063],{},[31665,351681,106063],{"encoding":31667},[34,351683,351685],{"className":351684,"ariaHidden":758},[31672],[34,351686,351688,351691],{"className":351687},[31676],[34,351689],{"className":351690,"style":33218},[31680],[34,351692,106063],{"className":351693,"style":67482},[31685,31686]," 含于某个 ",[34,351696,351698,351716],{"className":351697},[31642],[34,351699,351701],{"className":351700},[31646],[31648,351702,351703],{"xmlns":31650},[31652,351704,351705,351713],{},[31655,351706,351707],{},[31743,351708,351709,351711],{},[31658,351710,105852],{},[31658,351712,31955],{},[31665,351714,351715],{"encoding":31667},"P_i",[34,351717,351719],{"className":351718,"ariaHidden":758},[31672],[34,351720,351722,351725],{"className":351721},[31676],[34,351723],{"className":351724,"style":70915},[31680],[34,351726,351728,351731],{"className":351727},[31685],[34,351729,105852],{"className":351730,"style":37422},[31685,31686],[34,351732,351734],{"className":351733},[31792],[34,351735,351737,351757],{"className":351736},[31796,31797],[34,351738,351740,351754],{"className":351739},[31801],[34,351741,351743],{"className":351742,"style":32090},[31805],[34,351744,351745,351748],{"style":214424},[34,351746],{"className":351747,"style":31814},[31813],[34,351749,351751],{"className":351750},[31818,31819,31820,31821],[34,351752,31955],{"className":351753},[31685,31686,31821],[34,351755,31829],{"className":351756},[31828],[34,351758,351760],{"className":351759},[31801],[34,351761,351763],{"className":351762,"style":31836},[31805],[34,351764],{},[32188,351766],{},[32490,351768,339412],{"id":351769},"深度分析与解答-2",[15,351771,351772],{},[2627,351773,346809],{},[105,351775,351776,351785,351791],{},[108,351777,351778,351781,351782,347],{},[2627,351779,351780],{},"这道题的名字","：这在代数中非常著名，叫做",[2627,351783,351784],{},"素理想规避引理 (Prime Avoidance Lemma)",[108,351786,351787,351790],{},[2627,351788,351789],{},"直觉","：理想（作为加法群）是一个很“结实”的整体，它很难被几个素理想“拼凑”着覆盖住。如果它被覆盖了，那它肯定完全躺在某一个里面。",[108,351792,351793,351796,351797,351825,351826,351895],{},[2627,351794,351795],{},"证明方法","：反证法 + 归纳法。我们要构造一个元素，它属于 ",[34,351798,351800,351813],{"className":351799},[31642],[34,351801,351803],{"className":351802},[31646],[31648,351804,351805],{"xmlns":31650},[31652,351806,351807,351811],{},[31655,351808,351809],{},[31658,351810,106063],{},[31665,351812,106063],{"encoding":31667},[34,351814,351816],{"className":351815,"ariaHidden":758},[31672],[34,351817,351819,351822],{"className":351818},[31676],[34,351820],{"className":351821,"style":33218},[31680],[34,351823,106063],{"className":351824,"style":67482},[31685,31686],"，但不属于任何 ",[34,351827,351829,351846],{"className":351828},[31642],[34,351830,351832],{"className":351831},[31646],[31648,351833,351834],{"xmlns":31650},[31652,351835,351836,351844],{},[31655,351837,351838],{},[31743,351839,351840,351842],{},[31658,351841,105852],{},[31658,351843,31955],{},[31665,351845,351715],{"encoding":31667},[34,351847,351849],{"className":351848,"ariaHidden":758},[31672],[34,351850,351852,351855],{"className":351851},[31676],[34,351853],{"className":351854,"style":70915},[31680],[34,351856,351858,351861],{"className":351857},[31685],[34,351859,105852],{"className":351860,"style":37422},[31685,31686],[34,351862,351864],{"className":351863},[31792],[34,351865,351867,351887],{"className":351866},[31796,31797],[34,351868,351870,351884],{"className":351869},[31801],[34,351871,351873],{"className":351872,"style":32090},[31805],[34,351874,351875,351878],{"style":214424},[34,351876],{"className":351877,"style":31814},[31813],[34,351879,351881],{"className":351880},[31818,31819,31820,31821],[34,351882,31955],{"className":351883},[31685,31686,31821],[34,351885,31829],{"className":351886},[31828],[34,351888,351890],{"className":351889},[31801],[34,351891,351893],{"className":351892,"style":31836},[31805],[34,351894],{},"，从而导出矛盾。",[15,351897,351898],{},[2627,351899,340393],{},[2230,351901,351902,352388],{},[108,351903,351904,344001,351907,352081,352082,47649,352110,352113,352114,352183,352184,352264,352265,347],{},[2627,351905,351906],{},"反证法假设",[34,351908,351910,351947],{"className":351909},[31642],[34,351911,351913],{"className":351912},[31646],[31648,351914,351915],{"xmlns":31650},[31652,351916,351917,351944],{},[31655,351918,351919,351921,351923,351938],{},[31658,351920,106063],{},[31662,351922,131855],{},[38846,351924,351925,351928,351936],{},[31662,351926,351927],{},"⋃",[31655,351929,351930,351932,351934],{},[31658,351931,31955],{},[31662,351933,202],{},[31749,351935,640],{},[31658,351937,31660],{},[31743,351939,351940,351942],{},[31658,351941,105852],{},[31658,351943,31955],{},[31665,351945,351946],{"encoding":31667},"I \\subseteq \\bigcup_{i=1}^n P_i",[34,351948,351950,351968],{"className":351949,"ariaHidden":758},[31672],[34,351951,351953,351956,351959,351962,351965],{"className":351952},[31676],[34,351954],{"className":351955,"style":195047},[31680],[34,351957,106063],{"className":351958,"style":67482},[31685,31686],[34,351960],{"className":351961,"style":31846},[31845],[34,351963,131855],{"className":351964},[31850],[34,351966],{"className":351967,"style":31846},[31845],[34,351969,351971,351975,352038,352041],{"className":351970},[31676],[34,351972],{"className":351973,"style":351974},[31680],"height:1.104em;vertical-align:-0.2997em;",[34,351976,351978,351981],{"className":351977},[32767],[34,351979,351927],{"className":351980,"style":156561},[32767,131249,156560],[34,351982,351984],{"className":351983},[31792],[34,351985,351987,352029],{"className":351986},[31796,31797],[34,351988,351990,352026],{"className":351989},[31801],[34,351991,351994,352014],{"className":351992,"style":351993},[31805],"height:0.8043em;",[34,351995,351996,351999],{"style":156577},[34,351997],{"className":351998,"style":31814},[31813],[34,352000,352002],{"className":352001},[31818,31819,31820,31821],[34,352003,352005,352008,352011],{"className":352004},[31685,31821],[34,352006,31955],{"className":352007},[31685,31686,31821],[34,352009,202],{"className":352010},[31850,31821],[34,352012,640],{"className":352013},[31685,31821],[34,352015,352017,352020],{"style":352016},"top:-3.2029em;margin-right:0.05em;",[34,352018],{"className":352019,"style":31814},[31813],[34,352021,352023],{"className":352022},[31818,31819,31820,31821],[34,352024,31660],{"className":352025},[31685,31686,31821],[34,352027,31829],{"className":352028},[31828],[34,352030,352032],{"className":352031},[31801],[34,352033,352036],{"className":352034,"style":352035},[31805],"height:0.2997em;",[34,352037],{},[34,352039],{"className":352040,"style":32123},[31845],[34,352042,352044,352047],{"className":352043},[31685],[34,352045,105852],{"className":352046,"style":37422},[31685,31686],[34,352048,352050],{"className":352049},[31792],[34,352051,352053,352073],{"className":352052},[31796,31797],[34,352054,352056,352070],{"className":352055},[31801],[34,352057,352059],{"className":352058,"style":32090},[31805],[34,352060,352061,352064],{"style":214424},[34,352062],{"className":352063,"style":31814},[31813],[34,352065,352067],{"className":352066},[31818,31819,31820,31821],[34,352068,31955],{"className":352069},[31685,31686,31821],[34,352071,31829],{"className":352072},[31828],[34,352074,352076],{"className":352075},[31801],[34,352077,352079],{"className":352078,"style":31836},[31805],[34,352080],{},"，但是 ",[34,352083,352085,352098],{"className":352084},[31642],[34,352086,352088],{"className":352087},[31646],[31648,352089,352090],{"xmlns":31650},[31652,352091,352092,352096],{},[31655,352093,352094],{},[31658,352095,106063],{},[31665,352097,106063],{"encoding":31667},[34,352099,352101],{"className":352100,"ariaHidden":758},[31672],[34,352102,352104,352107],{"className":352103},[31676],[34,352105],{"className":352106,"style":33218},[31680],[34,352108,106063],{"className":352109,"style":67482},[31685,31686],[2627,352111,352112],{},"不含于","任何一个单独的 ",[34,352115,352117,352134],{"className":352116},[31642],[34,352118,352120],{"className":352119},[31646],[31648,352121,352122],{"xmlns":31650},[31652,352123,352124,352132],{},[31655,352125,352126],{},[31743,352127,352128,352130],{},[31658,352129,105852],{},[31658,352131,31955],{},[31665,352133,351715],{"encoding":31667},[34,352135,352137],{"className":352136,"ariaHidden":758},[31672],[34,352138,352140,352143],{"className":352139},[31676],[34,352141],{"className":352142,"style":70915},[31680],[34,352144,352146,352149],{"className":352145},[31685],[34,352147,105852],{"className":352148,"style":37422},[31685,31686],[34,352150,352152],{"className":352151},[31792],[34,352153,352155,352175],{"className":352154},[31796,31797],[34,352156,352158,352172],{"className":352157},[31801],[34,352159,352161],{"className":352160,"style":32090},[31805],[34,352162,352163,352166],{"style":214424},[34,352164],{"className":352165,"style":31814},[31813],[34,352167,352169],{"className":352168},[31818,31819,31820,31821],[34,352170,31955],{"className":352171},[31685,31686,31821],[34,352173,31829],{"className":352174},[31828],[34,352176,352178],{"className":352177},[31801],[34,352179,352181],{"className":352180,"style":31836},[31805],[34,352182],{},"。\n即对于所有的 ",[34,352185,352187,352213],{"className":352186},[31642],[34,352188,352190],{"className":352189},[31646],[31648,352191,352192],{"xmlns":31650},[31652,352193,352194,352210],{},[31655,352195,352196,352198,352200,352202,352204,352206,352208],{},[31658,352197,33759],{},[31662,352199,202],{},[31749,352201,640],{},[31662,352203,31960],{"separator":758},[31662,352205,33934],{},[31662,352207,31960],{"separator":758},[31658,352209,31660],{},[31665,352211,352212],{"encoding":31667},"k = 1, \\dots, n",[34,352214,352216,352234],{"className":352215,"ariaHidden":758},[31672],[34,352217,352219,352222,352225,352228,352231],{"className":352218},[31676],[34,352220],{"className":352221,"style":31681},[31680],[34,352223,33759],{"className":352224,"style":33774},[31685,31686],[34,352226],{"className":352227,"style":31846},[31845],[34,352229,202],{"className":352230},[31850],[34,352232],{"className":352233,"style":31846},[31845],[34,352235,352237,352240,352243,352246,352249,352252,352255,352258,352261],{"className":352236},[31676],[34,352238],{"className":352239,"style":40292},[31680],[34,352241,640],{"className":352242},[31685],[34,352244,31960],{"className":352245},[32119],[34,352247],{"className":352248,"style":32123},[31845],[34,352250,33934],{"className":352251},[34006],[34,352253],{"className":352254,"style":32123},[31845],[34,352256,31960],{"className":352257},[32119],[34,352259],{"className":352260,"style":32123},[31845],[34,352262,31660],{"className":352263},[31685,31686],"，都有 ",[34,352266,352268,352291],{"className":352267},[31642],[34,352269,352271],{"className":352270},[31646],[31648,352272,352273],{"xmlns":31650},[31652,352274,352275,352288],{},[31655,352276,352277,352279,352282],{},[31658,352278,106063],{},[31662,352280,352281],{},"⊈",[31743,352283,352284,352286],{},[31658,352285,105852],{},[31658,352287,33759],{},[31665,352289,352290],{"encoding":31667},"I \\not\\subseteq P_k",[34,352292,352294,352342],{"className":352293,"ariaHidden":758},[31672],[34,352295,352297,352300,352303,352306,352333,352336,352339],{"className":352296},[31676],[34,352298],{"className":352299,"style":43217},[31680],[34,352301,106063],{"className":352302,"style":67482},[31685,31686],[34,352304],{"className":352305,"style":31846},[31845],[34,352307,352309],{"className":352308},[31850],[34,352310,352312],{"className":352311},[31685,54774],[34,352313,352315],{"className":352314},[54778],[34,352316,352318,352321,352330],{"className":352317},[54782],[34,352319],{"className":352320,"style":43217},[31680],[34,352322,352324],{"className":352323},[54789],[34,352325,352327],{"className":352326},[31685],[34,352328,54796],{"className":352329},[31850],[34,352331],{"className":352332},[54800],[34,352334],{"className":352335},[31845,54804],[34,352337,131855],{"className":352338},[31850],[34,352340],{"className":352341,"style":31846},[31845],[34,352343,352345,352348],{"className":352344},[31676],[34,352346],{"className":352347,"style":70915},[31680],[34,352349,352351,352354],{"className":352350},[31685],[34,352352,105852],{"className":352353,"style":37422},[31685,31686],[34,352355,352357],{"className":352356},[31792],[34,352358,352360,352380],{"className":352359},[31796,31797],[34,352361,352363,352377],{"className":352362},[31801],[34,352364,352366],{"className":352365,"style":33833},[31805],[34,352367,352368,352371],{"style":214424},[34,352369],{"className":352370,"style":31814},[31813],[34,352372,352374],{"className":352373},[31818,31819,31820,31821],[34,352375,33759],{"className":352376,"style":33774},[31685,31686,31821],[34,352378,31829],{"className":352379},[31828],[34,352381,352383],{"className":352382},[31801],[34,352384,352386],{"className":352385,"style":31836},[31805],[34,352387],{},[108,352389,352390,352393,352394,335149,352445,352570,352571,352599,352600,352628,352629,352727,352728,352756],{},[2627,352391,352392],{},"归纳法策略","：\n我们要证明存在一个元素 ",[34,352395,352397,352415],{"className":352396},[31642],[34,352398,352400],{"className":352399},[31646],[31648,352401,352402],{"xmlns":31650},[31652,352403,352404,352412],{},[31655,352405,352406,352408,352410],{},[31658,352407,32936],{},[31662,352409,112296],{},[31658,352411,106063],{},[31665,352413,352414],{"encoding":31667},"x \\in I",[34,352416,352418,352436],{"className":352417,"ariaHidden":758},[31672],[34,352419,352421,352424,352427,352430,352433],{"className":352420},[31676],[34,352422],{"className":352423,"style":34812},[31680],[34,352425,32936],{"className":352426},[31685,31686],[34,352428],{"className":352429,"style":31846},[31845],[34,352431,112296],{"className":352432},[31850],[34,352434],{"className":352435,"style":31846},[31845],[34,352437,352439,352442],{"className":352438},[31676],[34,352440],{"className":352441,"style":33218},[31680],[34,352443,106063],{"className":352444,"style":67482},[31685,31686],[34,352446,352448,352470],{"className":352447},[31642],[34,352449,352451],{"className":352450},[31646],[31648,352452,352453],{"xmlns":31650},[31652,352454,352455,352467],{},[31655,352456,352457,352459,352461],{},[31658,352458,32936],{},[31662,352460,112266],{"mathvariant":38079},[31743,352462,352463,352465],{},[31658,352464,105852],{},[31658,352466,33759],{},[31665,352468,352469],{"encoding":31667},"x \\notin P_k",[34,352471,352473,352524],{"className":352472,"ariaHidden":758},[31672],[34,352474,352476,352479,352482,352485,352521],{"className":352475},[31676],[34,352477],{"className":352478,"style":31774},[31680],[34,352480,32936],{"className":352481},[31685,31686],[34,352483],{"className":352484,"style":31846},[31845],[34,352486,352488,352494],{"className":352487},[31850],[34,352489,352491],{"className":352490},[31685],[34,352492,112296],{"className":352493},[31850],[34,352495,352497],{"className":352496},[31685,54774],[34,352498,352500],{"className":352499},[54778],[34,352501,352503,352506,352518],{"className":352502},[112306],[34,352504],{"className":352505,"style":31774},[31680],[34,352507,352509],{"className":352508},[54789],[34,352510,352512,352515],{"className":352511},[31685],[34,352513,38080],{"className":352514},[31685],[34,352516],{"className":352517,"style":112322},[31845],[34,352519],{"className":352520},[54800],[34,352522],{"className":352523,"style":31846},[31845],[34,352525,352527,352530],{"className":352526},[31676],[34,352528],{"className":352529,"style":70915},[31680],[34,352531,352533,352536],{"className":352532},[31685],[34,352534,105852],{"className":352535,"style":37422},[31685,31686],[34,352537,352539],{"className":352538},[31792],[34,352540,352542,352562],{"className":352541},[31796,31797],[34,352543,352545,352559],{"className":352544},[31801],[34,352546,352548],{"className":352547,"style":33833},[31805],[34,352549,352550,352553],{"style":214424},[34,352551],{"className":352552,"style":31814},[31813],[34,352554,352556],{"className":352555},[31818,31819,31820,31821],[34,352557,33759],{"className":352558,"style":33774},[31685,31686,31821],[34,352560,31829],{"className":352561},[31828],[34,352563,352565],{"className":352564},[31801],[34,352566,352568],{"className":352567,"style":31836},[31805],[34,352569],{}," 对所有 ",[34,352572,352574,352587],{"className":352573},[31642],[34,352575,352577],{"className":352576},[31646],[31648,352578,352579],{"xmlns":31650},[31652,352580,352581,352585],{},[31655,352582,352583],{},[31658,352584,33759],{},[31665,352586,33759],{"encoding":31667},[34,352588,352590],{"className":352589,"ariaHidden":758},[31672],[34,352591,352593,352596],{"className":352592},[31676],[34,352594],{"className":352595,"style":31681},[31680],[34,352597,33759],{"className":352598,"style":33774},[31685,31686]," 成立（这说明 ",[34,352601,352603,352616],{"className":352602},[31642],[34,352604,352606],{"className":352605},[31646],[31648,352607,352608],{"xmlns":31650},[31652,352609,352610,352614],{},[31655,352611,352612],{},[31658,352613,32936],{},[31665,352615,32936],{"encoding":31667},[34,352617,352619],{"className":352618,"ariaHidden":758},[31672],[34,352620,352622,352625],{"className":352621},[31676],[34,352623],{"className":352624,"style":32527},[31680],[34,352626,32936],{"className":352627},[31685,31686]," 不在并集中，与 ",[34,352630,352632,352657],{"className":352631},[31642],[34,352633,352635],{"className":352634},[31646],[31648,352636,352637],{"xmlns":31650},[31652,352638,352639,352654],{},[31655,352640,352641,352643,352645,352648],{},[31658,352642,106063],{},[31662,352644,131855],{},[31662,352646,352647],{},"∪",[31743,352649,352650,352652],{},[31658,352651,105852],{},[31658,352653,31955],{},[31665,352655,352656],{"encoding":31667},"I \\subseteq \\cup P_i",[34,352658,352660,352678],{"className":352659,"ariaHidden":758},[31672],[34,352661,352663,352666,352669,352672,352675],{"className":352662},[31676],[34,352664],{"className":352665,"style":195047},[31680],[34,352667,106063],{"className":352668,"style":67482},[31685,31686],[34,352670],{"className":352671,"style":31846},[31845],[34,352673,131855],{"className":352674},[31850],[34,352676],{"className":352677,"style":31846},[31845],[34,352679,352681,352684,352687],{"className":352680},[31676],[34,352682],{"className":352683,"style":70915},[31680],[34,352685,352647],{"className":352686},[31685],[34,352688,352690,352693],{"className":352689},[31685],[34,352691,105852],{"className":352692,"style":37422},[31685,31686],[34,352694,352696],{"className":352695},[31792],[34,352697,352699,352719],{"className":352698},[31796,31797],[34,352700,352702,352716],{"className":352701},[31801],[34,352703,352705],{"className":352704,"style":32090},[31805],[34,352706,352707,352710],{"style":214424},[34,352708],{"className":352709,"style":31814},[31813],[34,352711,352713],{"className":352712},[31818,31819,31820,31821],[34,352714,31955],{"className":352715},[31685,31686,31821],[34,352717,31829],{"className":352718},[31828],[34,352720,352722],{"className":352721},[31801],[34,352723,352725],{"className":352724,"style":31836},[31805],[34,352726],{}," 矛盾）。\n我们对素理想的个数 ",[34,352729,352731,352744],{"className":352730},[31642],[34,352732,352734],{"className":352733},[31646],[31648,352735,352736],{"xmlns":31650},[31652,352737,352738,352742],{},[31655,352739,352740],{},[31658,352741,31660],{},[31665,352743,31660],{"encoding":31667},[34,352745,352747],{"className":352746,"ariaHidden":758},[31672],[34,352748,352750,352753],{"className":352749},[31676],[34,352751],{"className":352752,"style":32527},[31680],[34,352754,31660],{"className":352755},[31685,31686]," 进行归纳。",[105,352758,352759,352908,354277,354360,354884],{},[108,352760,352761,352814,352815,352907],{},[2627,352762,352763,352813],{},[34,352764,352766,352783],{"className":352765},[31642],[34,352767,352769],{"className":352768},[31646],[31648,352770,352771],{"xmlns":31650},[31652,352772,352773,352781],{},[31655,352774,352775,352777,352779],{},[31658,352776,31660],{},[31662,352778,202],{},[31749,352780,640],{},[31665,352782,32850],{"encoding":31667},[34,352784,352786,352804],{"className":352785,"ariaHidden":758},[31672],[34,352787,352789,352792,352795,352798,352801],{"className":352788},[31676],[34,352790],{"className":352791,"style":32527},[31680],[34,352793,31660],{"className":352794},[31685,31686],[34,352796],{"className":352797,"style":31846},[31845],[34,352799,202],{"className":352800},[31850],[34,352802],{"className":352803,"style":31846},[31845],[34,352805,352807,352810],{"className":352806},[31676],[34,352808],{"className":352809,"style":32617},[31680],[34,352811,640],{"className":352812},[31685]," 时","：显然 ",[34,352816,352818,352840],{"className":352817},[31642],[34,352819,352821],{"className":352820},[31646],[31648,352822,352823],{"xmlns":31650},[31652,352824,352825,352837],{},[31655,352826,352827,352829,352831],{},[31658,352828,106063],{},[31662,352830,131855],{},[31743,352832,352833,352835],{},[31658,352834,105852],{},[31749,352836,640],{},[31665,352838,352839],{"encoding":31667},"I \\subseteq P_1",[34,352841,352843,352861],{"className":352842,"ariaHidden":758},[31672],[34,352844,352846,352849,352852,352855,352858],{"className":352845},[31676],[34,352847],{"className":352848,"style":195047},[31680],[34,352850,106063],{"className":352851,"style":67482},[31685,31686],[34,352853],{"className":352854,"style":31846},[31845],[34,352856,131855],{"className":352857},[31850],[34,352859],{"className":352860,"style":31846},[31845],[34,352862,352864,352867],{"className":352863},[31676],[34,352865],{"className":352866,"style":70915},[31680],[34,352868,352870,352873],{"className":352869},[31685],[34,352871,105852],{"className":352872,"style":37422},[31685,31686],[34,352874,352876],{"className":352875},[31792],[34,352877,352879,352899],{"className":352878},[31796,31797],[34,352880,352882,352896],{"className":352881},[31801],[34,352883,352885],{"className":352884,"style":31806},[31805],[34,352886,352887,352890],{"style":214424},[34,352888],{"className":352889,"style":31814},[31813],[34,352891,352893],{"className":352892},[31818,31819,31820,31821],[34,352894,640],{"className":352895},[31685,31821],[34,352897,31829],{"className":352898},[31828],[34,352900,352902],{"className":352901},[31801],[34,352903,352905],{"className":352904,"style":31836},[31805],[34,352906],{},"，不需要证明。",[108,352909,352910,325706,352963,86674,353085,353207,353208,353363,353364,353392,353393,353463,353464,89494,353619,327448,353785,353854,353855,86674,353988,354121,354122,347],{},[2627,352911,352912,352813],{},[34,352913,352915,352933],{"className":352914},[31642],[34,352916,352918],{"className":352917},[31646],[31648,352919,352920],{"xmlns":31650},[31652,352921,352922,352930],{},[31655,352923,352924,352926,352928],{},[31658,352925,31660],{},[31662,352927,202],{},[31749,352929,768],{},[31665,352931,352932],{"encoding":31667},"n=2",[34,352934,352936,352954],{"className":352935,"ariaHidden":758},[31672],[34,352937,352939,352942,352945,352948,352951],{"className":352938},[31676],[34,352940],{"className":352941,"style":32527},[31680],[34,352943,31660],{"className":352944},[31685,31686],[34,352946],{"className":352947,"style":31846},[31845],[34,352949,202],{"className":352950},[31850],[34,352952],{"className":352953,"style":31846},[31845],[34,352955,352957,352960],{"className":352956},[31676],[34,352958],{"className":352959,"style":32617},[31680],[34,352961,768],{"className":352962},[31685],[34,352964,352966,352988],{"className":352965},[31642],[34,352967,352969],{"className":352968},[31646],[31648,352970,352971],{"xmlns":31650},[31652,352972,352973,352985],{},[31655,352974,352975,352977,352979],{},[31658,352976,106063],{},[31662,352978,352281],{},[31743,352980,352981,352983],{},[31658,352982,105852],{},[31749,352984,640],{},[31665,352986,352987],{"encoding":31667},"I \\not\\subseteq P_1",[34,352989,352991,353039],{"className":352990,"ariaHidden":758},[31672],[34,352992,352994,352997,353000,353003,353030,353033,353036],{"className":352993},[31676],[34,352995],{"className":352996,"style":43217},[31680],[34,352998,106063],{"className":352999,"style":67482},[31685,31686],[34,353001],{"className":353002,"style":31846},[31845],[34,353004,353006],{"className":353005},[31850],[34,353007,353009],{"className":353008},[31685,54774],[34,353010,353012],{"className":353011},[54778],[34,353013,353015,353018,353027],{"className":353014},[54782],[34,353016],{"className":353017,"style":43217},[31680],[34,353019,353021],{"className":353020},[54789],[34,353022,353024],{"className":353023},[31685],[34,353025,54796],{"className":353026},[31850],[34,353028],{"className":353029},[54800],[34,353031],{"className":353032},[31845,54804],[34,353034,131855],{"className":353035},[31850],[34,353037],{"className":353038,"style":31846},[31845],[34,353040,353042,353045],{"className":353041},[31676],[34,353043],{"className":353044,"style":70915},[31680],[34,353046,353048,353051],{"className":353047},[31685],[34,353049,105852],{"className":353050,"style":37422},[31685,31686],[34,353052,353054],{"className":353053},[31792],[34,353055,353057,353077],{"className":353056},[31796,31797],[34,353058,353060,353074],{"className":353059},[31801],[34,353061,353063],{"className":353062,"style":31806},[31805],[34,353064,353065,353068],{"style":214424},[34,353066],{"className":353067,"style":31814},[31813],[34,353069,353071],{"className":353070},[31818,31819,31820,31821],[34,353072,640],{"className":353073},[31685,31821],[34,353075,31829],{"className":353076},[31828],[34,353078,353080],{"className":353079},[31801],[34,353081,353083],{"className":353082,"style":31836},[31805],[34,353084],{},[34,353086,353088,353110],{"className":353087},[31642],[34,353089,353091],{"className":353090},[31646],[31648,353092,353093],{"xmlns":31650},[31652,353094,353095,353107],{},[31655,353096,353097,353099,353101],{},[31658,353098,106063],{},[31662,353100,352281],{},[31743,353102,353103,353105],{},[31658,353104,105852],{},[31749,353106,768],{},[31665,353108,353109],{"encoding":31667},"I \\not\\subseteq P_2",[34,353111,353113,353161],{"className":353112,"ariaHidden":758},[31672],[34,353114,353116,353119,353122,353125,353152,353155,353158],{"className":353115},[31676],[34,353117],{"className":353118,"style":43217},[31680],[34,353120,106063],{"className":353121,"style":67482},[31685,31686],[34,353123],{"className":353124,"style":31846},[31845],[34,353126,353128],{"className":353127},[31850],[34,353129,353131],{"className":353130},[31685,54774],[34,353132,353134],{"className":353133},[54778],[34,353135,353137,353140,353149],{"className":353136},[54782],[34,353138],{"className":353139,"style":43217},[31680],[34,353141,353143],{"className":353142},[54789],[34,353144,353146],{"className":353145},[31685],[34,353147,54796],{"className":353148},[31850],[34,353150],{"className":353151},[54800],[34,353153],{"className":353154},[31845,54804],[34,353156,131855],{"className":353157},[31850],[34,353159],{"className":353160,"style":31846},[31845],[34,353162,353164,353167],{"className":353163},[31676],[34,353165],{"className":353166,"style":70915},[31680],[34,353168,353170,353173],{"className":353169},[31685],[34,353171,105852],{"className":353172,"style":37422},[31685,31686],[34,353174,353176],{"className":353175},[31792],[34,353177,353179,353199],{"className":353178},[31796,31797],[34,353180,353182,353196],{"className":353181},[31801],[34,353183,353185],{"className":353184,"style":31806},[31805],[34,353186,353187,353190],{"style":214424},[34,353188],{"className":353189,"style":31814},[31813],[34,353191,353193],{"className":353192},[31818,31819,31820,31821],[34,353194,768],{"className":353195},[31685,31821],[34,353197,31829],{"className":353198},[31828],[34,353200,353202],{"className":353201},[31801],[34,353203,353205],{"className":353204,"style":31836},[31805],[34,353206],{},"。\n选取 ",[34,353209,353211,353241],{"className":353210},[31642],[34,353212,353214],{"className":353213},[31646],[31648,353215,353216],{"xmlns":31650},[31652,353217,353218,353238],{},[31655,353219,353220,353226,353228,353230,353232],{},[31743,353221,353222,353224],{},[31658,353223,32936],{},[31749,353225,640],{},[31662,353227,112296],{},[31658,353229,106063],{},[31662,353231,336362],{},[31743,353233,353234,353236],{},[31658,353235,105852],{},[31749,353237,640],{},[31665,353239,353240],{"encoding":31667},"x_1 \\in I \\setminus P_1",[34,353242,353244,353299,353317],{"className":353243,"ariaHidden":758},[31672],[34,353245,353247,353250,353290,353293,353296],{"className":353246},[31676],[34,353248],{"className":353249,"style":225773},[31680],[34,353251,353253,353256],{"className":353252},[31685],[34,353254,32936],{"className":353255},[31685,31686],[34,353257,353259],{"className":353258},[31792],[34,353260,353262,353282],{"className":353261},[31796,31797],[34,353263,353265,353279],{"className":353264},[31801],[34,353266,353268],{"className":353267,"style":31806},[31805],[34,353269,353270,353273],{"style":31809},[34,353271],{"className":353272,"style":31814},[31813],[34,353274,353276],{"className":353275},[31818,31819,31820,31821],[34,353277,640],{"className":353278},[31685,31821],[34,353280,31829],{"className":353281},[31828],[34,353283,353285],{"className":353284},[31801],[34,353286,353288],{"className":353287,"style":31836},[31805],[34,353289],{},[34,353291],{"className":353292,"style":31846},[31845],[34,353294,112296],{"className":353295},[31850],[34,353297],{"className":353298,"style":31846},[31845],[34,353300,353302,353305,353308,353311,353314],{"className":353301},[31676],[34,353303],{"className":353304,"style":31774},[31680],[34,353306,106063],{"className":353307,"style":67482},[31685,31686],[34,353309],{"className":353310,"style":32604},[31845],[34,353312,336362],{"className":353313},[32024],[34,353315],{"className":353316,"style":32604},[31845],[34,353318,353320,353323],{"className":353319},[31676],[34,353321],{"className":353322,"style":70915},[31680],[34,353324,353326,353329],{"className":353325},[31685],[34,353327,105852],{"className":353328,"style":37422},[31685,31686],[34,353330,353332],{"className":353331},[31792],[34,353333,353335,353355],{"className":353334},[31796,31797],[34,353336,353338,353352],{"className":353337},[31801],[34,353339,353341],{"className":353340,"style":31806},[31805],[34,353342,353343,353346],{"style":214424},[34,353344],{"className":353345,"style":31814},[31813],[34,353347,353349],{"className":353348},[31818,31819,31820,31821],[34,353350,640],{"className":353351},[31685,31821],[34,353353,31829],{"className":353354},[31828],[34,353356,353358],{"className":353357},[31801],[34,353359,353361],{"className":353360,"style":31836},[31805],[34,353362],{},"（在 ",[34,353365,353367,353380],{"className":353366},[31642],[34,353368,353370],{"className":353369},[31646],[31648,353371,353372],{"xmlns":31650},[31652,353373,353374,353378],{},[31655,353375,353376],{},[31658,353377,106063],{},[31665,353379,106063],{"encoding":31667},[34,353381,353383],{"className":353382,"ariaHidden":758},[31672],[34,353384,353386,353389],{"className":353385},[31676],[34,353387],{"className":353388,"style":33218},[31680],[34,353390,106063],{"className":353391,"style":67482},[31685,31686]," 中但不在 ",[34,353394,353396,353414],{"className":353395},[31642],[34,353397,353399],{"className":353398},[31646],[31648,353400,353401],{"xmlns":31650},[31652,353402,353403,353411],{},[31655,353404,353405],{},[31743,353406,353407,353409],{},[31658,353408,105852],{},[31749,353410,640],{},[31665,353412,353413],{"encoding":31667},"P_1",[34,353415,353417],{"className":353416,"ariaHidden":758},[31672],[34,353418,353420,353423],{"className":353419},[31676],[34,353421],{"className":353422,"style":70915},[31680],[34,353424,353426,353429],{"className":353425},[31685],[34,353427,105852],{"className":353428,"style":37422},[31685,31686],[34,353430,353432],{"className":353431},[31792],[34,353433,353435,353455],{"className":353434},[31796,31797],[34,353436,353438,353452],{"className":353437},[31801],[34,353439,353441],{"className":353440,"style":31806},[31805],[34,353442,353443,353446],{"style":214424},[34,353444],{"className":353445,"style":31814},[31813],[34,353447,353449],{"className":353448},[31818,31819,31820,31821],[34,353450,640],{"className":353451},[31685,31821],[34,353453,31829],{"className":353454},[31828],[34,353456,353458],{"className":353457},[31801],[34,353459,353461],{"className":353460,"style":31836},[31805],[34,353462],{}," 中）和 ",[34,353465,353467,353497],{"className":353466},[31642],[34,353468,353470],{"className":353469},[31646],[31648,353471,353472],{"xmlns":31650},[31652,353473,353474,353494],{},[31655,353475,353476,353482,353484,353486,353488],{},[31743,353477,353478,353480],{},[31658,353479,32936],{},[31749,353481,768],{},[31662,353483,112296],{},[31658,353485,106063],{},[31662,353487,336362],{},[31743,353489,353490,353492],{},[31658,353491,105852],{},[31749,353493,768],{},[31665,353495,353496],{"encoding":31667},"x_2 \\in I \\setminus P_2",[34,353498,353500,353555,353573],{"className":353499,"ariaHidden":758},[31672],[34,353501,353503,353506,353546,353549,353552],{"className":353502},[31676],[34,353504],{"className":353505,"style":225773},[31680],[34,353507,353509,353512],{"className":353508},[31685],[34,353510,32936],{"className":353511},[31685,31686],[34,353513,353515],{"className":353514},[31792],[34,353516,353518,353538],{"className":353517},[31796,31797],[34,353519,353521,353535],{"className":353520},[31801],[34,353522,353524],{"className":353523,"style":31806},[31805],[34,353525,353526,353529],{"style":31809},[34,353527],{"className":353528,"style":31814},[31813],[34,353530,353532],{"className":353531},[31818,31819,31820,31821],[34,353533,768],{"className":353534},[31685,31821],[34,353536,31829],{"className":353537},[31828],[34,353539,353541],{"className":353540},[31801],[34,353542,353544],{"className":353543,"style":31836},[31805],[34,353545],{},[34,353547],{"className":353548,"style":31846},[31845],[34,353550,112296],{"className":353551},[31850],[34,353553],{"className":353554,"style":31846},[31845],[34,353556,353558,353561,353564,353567,353570],{"className":353557},[31676],[34,353559],{"className":353560,"style":31774},[31680],[34,353562,106063],{"className":353563,"style":67482},[31685,31686],[34,353565],{"className":353566,"style":32604},[31845],[34,353568,336362],{"className":353569},[32024],[34,353571],{"className":353572,"style":32604},[31845],[34,353574,353576,353579],{"className":353575},[31676],[34,353577],{"className":353578,"style":70915},[31680],[34,353580,353582,353585],{"className":353581},[31685],[34,353583,105852],{"className":353584,"style":37422},[31685,31686],[34,353586,353588],{"className":353587},[31792],[34,353589,353591,353611],{"className":353590},[31796,31797],[34,353592,353594,353608],{"className":353593},[31801],[34,353595,353597],{"className":353596,"style":31806},[31805],[34,353598,353599,353602],{"style":214424},[34,353600],{"className":353601,"style":31814},[31813],[34,353603,353605],{"className":353604},[31818,31819,31820,31821],[34,353606,768],{"className":353607},[31685,31821],[34,353609,31829],{"className":353610},[31828],[34,353612,353614],{"className":353613},[31801],[34,353615,353617],{"className":353616,"style":31836},[31805],[34,353618],{},[34,353620,353622,353648],{"className":353621},[31642],[34,353623,353625],{"className":353624},[31646],[31648,353626,353627],{"xmlns":31650},[31652,353628,353629,353645],{},[31655,353630,353631,353637,353639],{},[31743,353632,353633,353635],{},[31658,353634,32936],{},[31749,353636,640],{},[31662,353638,112266],{"mathvariant":38079},[31743,353640,353641,353643],{},[31658,353642,105852],{},[31749,353644,768],{},[31665,353646,353647],{"encoding":31667},"x_1 \\notin P_2",[34,353649,353651,353739],{"className":353650,"ariaHidden":758},[31672],[34,353652,353654,353657,353697,353700,353736],{"className":353653},[31676],[34,353655],{"className":353656,"style":31774},[31680],[34,353658,353660,353663],{"className":353659},[31685],[34,353661,32936],{"className":353662},[31685,31686],[34,353664,353666],{"className":353665},[31792],[34,353667,353669,353689],{"className":353668},[31796,31797],[34,353670,353672,353686],{"className":353671},[31801],[34,353673,353675],{"className":353674,"style":31806},[31805],[34,353676,353677,353680],{"style":31809},[34,353678],{"className":353679,"style":31814},[31813],[34,353681,353683],{"className":353682},[31818,31819,31820,31821],[34,353684,640],{"className":353685},[31685,31821],[34,353687,31829],{"className":353688},[31828],[34,353690,353692],{"className":353691},[31801],[34,353693,353695],{"className":353694,"style":31836},[31805],[34,353696],{},[34,353698],{"className":353699,"style":31846},[31845],[34,353701,353703,353709],{"className":353702},[31850],[34,353704,353706],{"className":353705},[31685],[34,353707,112296],{"className":353708},[31850],[34,353710,353712],{"className":353711},[31685,54774],[34,353713,353715],{"className":353714},[54778],[34,353716,353718,353721,353733],{"className":353717},[112306],[34,353719],{"className":353720,"style":31774},[31680],[34,353722,353724],{"className":353723},[54789],[34,353725,353727,353730],{"className":353726},[31685],[34,353728,38080],{"className":353729},[31685],[34,353731],{"className":353732,"style":112322},[31845],[34,353734],{"className":353735},[54800],[34,353737],{"className":353738,"style":31846},[31845],[34,353740,353742,353745],{"className":353741},[31676],[34,353743],{"className":353744,"style":70915},[31680],[34,353746,353748,353751],{"className":353747},[31685],[34,353749,105852],{"className":353750,"style":37422},[31685,31686],[34,353752,353754],{"className":353753},[31792],[34,353755,353757,353777],{"className":353756},[31796,31797],[34,353758,353760,353774],{"className":353759},[31801],[34,353761,353763],{"className":353762,"style":31806},[31805],[34,353764,353765,353768],{"style":214424},[34,353766],{"className":353767,"style":31814},[31813],[34,353769,353771],{"className":353770},[31818,31819,31820,31821],[34,353772,768],{"className":353773},[31685,31821],[34,353775,31829],{"className":353776},[31828],[34,353778,353780],{"className":353779},[31801],[34,353781,353783],{"className":353782,"style":31836},[31805],[34,353784],{},[34,353786,353788,353805],{"className":353787},[31642],[34,353789,353791],{"className":353790},[31646],[31648,353792,353793],{"xmlns":31650},[31652,353794,353795,353803],{},[31655,353796,353797],{},[31743,353798,353799,353801],{},[31658,353800,32936],{},[31749,353802,640],{},[31665,353804,224014],{"encoding":31667},[34,353806,353808],{"className":353807,"ariaHidden":758},[31672],[34,353809,353811,353814],{"className":353810},[31676],[34,353812],{"className":353813,"style":31860},[31680],[34,353815,353817,353820],{"className":353816},[31685],[34,353818,32936],{"className":353819},[31685,31686],[34,353821,353823],{"className":353822},[31792],[34,353824,353826,353846],{"className":353825},[31796,31797],[34,353827,353829,353843],{"className":353828},[31801],[34,353830,353832],{"className":353831,"style":31806},[31805],[34,353833,353834,353837],{"style":31809},[34,353835],{"className":353836,"style":31814},[31813],[34,353838,353840],{"className":353839},[31818,31819,31820,31821],[34,353841,640],{"className":353842},[31685,31821],[34,353844,31829],{"className":353845},[31828],[34,353847,353849],{"className":353848},[31801],[34,353850,353852],{"className":353851,"style":31836},[31805],[34,353853],{}," 就是我们要找的元素，矛盾。\n如果 ",[34,353856,353858,353884],{"className":353857},[31642],[34,353859,353861],{"className":353860},[31646],[31648,353862,353863],{"xmlns":31650},[31652,353864,353865,353881],{},[31655,353866,353867,353873,353875],{},[31743,353868,353869,353871],{},[31658,353870,32936],{},[31749,353872,640],{},[31662,353874,112296],{},[31743,353876,353877,353879],{},[31658,353878,105852],{},[31749,353880,768],{},[31665,353882,353883],{"encoding":31667},"x_1 \\in P_2",[34,353885,353887,353942],{"className":353886,"ariaHidden":758},[31672],[34,353888,353890,353893,353933,353936,353939],{"className":353889},[31676],[34,353891],{"className":353892,"style":225773},[31680],[34,353894,353896,353899],{"className":353895},[31685],[34,353897,32936],{"className":353898},[31685,31686],[34,353900,353902],{"className":353901},[31792],[34,353903,353905,353925],{"className":353904},[31796,31797],[34,353906,353908,353922],{"className":353907},[31801],[34,353909,353911],{"className":353910,"style":31806},[31805],[34,353912,353913,353916],{"style":31809},[34,353914],{"className":353915,"style":31814},[31813],[34,353917,353919],{"className":353918},[31818,31819,31820,31821],[34,353920,640],{"className":353921},[31685,31821],[34,353923,31829],{"className":353924},[31828],[34,353926,353928],{"className":353927},[31801],[34,353929,353931],{"className":353930,"style":31836},[31805],[34,353932],{},[34,353934],{"className":353935,"style":31846},[31845],[34,353937,112296],{"className":353938},[31850],[34,353940],{"className":353941,"style":31846},[31845],[34,353943,353945,353948],{"className":353944},[31676],[34,353946],{"className":353947,"style":70915},[31680],[34,353949,353951,353954],{"className":353950},[31685],[34,353952,105852],{"className":353953,"style":37422},[31685,31686],[34,353955,353957],{"className":353956},[31792],[34,353958,353960,353980],{"className":353959},[31796,31797],[34,353961,353963,353977],{"className":353962},[31801],[34,353964,353966],{"className":353965,"style":31806},[31805],[34,353967,353968,353971],{"style":214424},[34,353969],{"className":353970,"style":31814},[31813],[34,353972,353974],{"className":353973},[31818,31819,31820,31821],[34,353975,768],{"className":353976},[31685,31821],[34,353978,31829],{"className":353979},[31828],[34,353981,353983],{"className":353982},[31801],[34,353984,353986],{"className":353985,"style":31836},[31805],[34,353987],{},[34,353989,353991,354017],{"className":353990},[31642],[34,353992,353994],{"className":353993},[31646],[31648,353995,353996],{"xmlns":31650},[31652,353997,353998,354014],{},[31655,353999,354000,354006,354008],{},[31743,354001,354002,354004],{},[31658,354003,32936],{},[31749,354005,768],{},[31662,354007,112296],{},[31743,354009,354010,354012],{},[31658,354011,105852],{},[31749,354013,640],{},[31665,354015,354016],{"encoding":31667},"x_2 \\in P_1",[34,354018,354020,354075],{"className":354019,"ariaHidden":758},[31672],[34,354021,354023,354026,354066,354069,354072],{"className":354022},[31676],[34,354024],{"className":354025,"style":225773},[31680],[34,354027,354029,354032],{"className":354028},[31685],[34,354030,32936],{"className":354031},[31685,31686],[34,354033,354035],{"className":354034},[31792],[34,354036,354038,354058],{"className":354037},[31796,31797],[34,354039,354041,354055],{"className":354040},[31801],[34,354042,354044],{"className":354043,"style":31806},[31805],[34,354045,354046,354049],{"style":31809},[34,354047],{"className":354048,"style":31814},[31813],[34,354050,354052],{"className":354051},[31818,31819,31820,31821],[34,354053,768],{"className":354054},[31685,31821],[34,354056,31829],{"className":354057},[31828],[34,354059,354061],{"className":354060},[31801],[34,354062,354064],{"className":354063,"style":31836},[31805],[34,354065],{},[34,354067],{"className":354068,"style":31846},[31845],[34,354070,112296],{"className":354071},[31850],[34,354073],{"className":354074,"style":31846},[31845],[34,354076,354078,354081],{"className":354077},[31676],[34,354079],{"className":354080,"style":70915},[31680],[34,354082,354084,354087],{"className":354083},[31685],[34,354085,105852],{"className":354086,"style":37422},[31685,31686],[34,354088,354090],{"className":354089},[31792],[34,354091,354093,354113],{"className":354092},[31796,31797],[34,354094,354096,354110],{"className":354095},[31801],[34,354097,354099],{"className":354098,"style":31806},[31805],[34,354100,354101,354104],{"style":214424},[34,354102],{"className":354103,"style":31814},[31813],[34,354105,354107],{"className":354106},[31818,31819,31820,31821],[34,354108,640],{"className":354109},[31685,31821],[34,354111,31829],{"className":354112},[31828],[34,354114,354116],{"className":354115},[31801],[34,354117,354119],{"className":354118,"style":31836},[31805],[34,354120],{},"，我们考虑元素 ",[34,354123,354125,354155],{"className":354124},[31642],[34,354126,354128],{"className":354127},[31646],[31648,354129,354130],{"xmlns":31650},[31652,354131,354132,354152],{},[31655,354133,354134,354136,354138,354144,354146],{},[31658,354135,67764],{},[31662,354137,202],{},[31743,354139,354140,354142],{},[31658,354141,32936],{},[31749,354143,640],{},[31662,354145,223],{},[31743,354147,354148,354150],{},[31658,354149,32936],{},[31749,354151,768],{},[31665,354153,354154],{"encoding":31667},"y = x_1 + x_2",[34,354156,354158,354176,354231],{"className":354157,"ariaHidden":758},[31672],[34,354159,354161,354164,354167,354170,354173],{"className":354160},[31676],[34,354162],{"className":354163,"style":38774},[31680],[34,354165,67764],{"className":354166,"style":32062},[31685,31686],[34,354168],{"className":354169,"style":31846},[31845],[34,354171,202],{"className":354172},[31850],[34,354174],{"className":354175,"style":31846},[31845],[34,354177,354179,354182,354222,354225,354228],{"className":354178},[31676],[34,354180],{"className":354181,"style":40522},[31680],[34,354183,354185,354188],{"className":354184},[31685],[34,354186,32936],{"className":354187},[31685,31686],[34,354189,354191],{"className":354190},[31792],[34,354192,354194,354214],{"className":354193},[31796,31797],[34,354195,354197,354211],{"className":354196},[31801],[34,354198,354200],{"className":354199,"style":31806},[31805],[34,354201,354202,354205],{"style":31809},[34,354203],{"className":354204,"style":31814},[31813],[34,354206,354208],{"className":354207},[31818,31819,31820,31821],[34,354209,640],{"className":354210},[31685,31821],[34,354212,31829],{"className":354213},[31828],[34,354215,354217],{"className":354216},[31801],[34,354218,354220],{"className":354219,"style":31836},[31805],[34,354221],{},[34,354223],{"className":354224,"style":32604},[31845],[34,354226,223],{"className":354227},[32024],[34,354229],{"className":354230,"style":32604},[31845],[34,354232,354234,354237],{"className":354233},[31676],[34,354235],{"className":354236,"style":31860},[31680],[34,354238,354240,354243],{"className":354239},[31685],[34,354241,32936],{"className":354242},[31685,31686],[34,354244,354246],{"className":354245},[31792],[34,354247,354249,354269],{"className":354248},[31796,31797],[34,354250,354252,354266],{"className":354251},[31801],[34,354253,354255],{"className":354254,"style":31806},[31805],[34,354256,354257,354260],{"style":31809},[34,354258],{"className":354259,"style":31814},[31813],[34,354261,354263],{"className":354262},[31818,31819,31820,31821],[34,354264,768],{"className":354265},[31685,31821],[34,354267,31829],{"className":354268},[31828],[34,354270,354272],{"className":354271},[31801],[34,354273,354275],{"className":354274,"style":31836},[31805],[34,354276],{},[108,354278,354279,354330,354331,354359],{},[34,354280,354282,354300],{"className":354281},[31642],[34,354283,354285],{"className":354284},[31646],[31648,354286,354287],{"xmlns":31650},[31652,354288,354289,354297],{},[31655,354290,354291,354293,354295],{},[31658,354292,67764],{},[31662,354294,112296],{},[31658,354296,106063],{},[31665,354298,354299],{"encoding":31667},"y \\in I",[34,354301,354303,354321],{"className":354302,"ariaHidden":758},[31672],[34,354304,354306,354309,354312,354315,354318],{"className":354305},[31676],[34,354307],{"className":354308,"style":86652},[31680],[34,354310,67764],{"className":354311,"style":32062},[31685,31686],[34,354313],{"className":354314,"style":31846},[31845],[34,354316,112296],{"className":354317},[31850],[34,354319],{"className":354320,"style":31846},[31845],[34,354322,354324,354327],{"className":354323},[31676],[34,354325],{"className":354326,"style":33218},[31680],[34,354328,106063],{"className":354329,"style":67482},[31685,31686],"（因为 ",[34,354332,354334,354347],{"className":354333},[31642],[34,354335,354337],{"className":354336},[31646],[31648,354338,354339],{"xmlns":31650},[31652,354340,354341,354345],{},[31655,354342,354343],{},[31658,354344,106063],{},[31665,354346,106063],{"encoding":31667},[34,354348,354350],{"className":354349,"ariaHidden":758},[31672],[34,354351,354353,354356],{"className":354352},[31676],[34,354354],{"className":354355,"style":33218},[31680],[34,354357,106063],{"className":354358,"style":67482},[31685,31686]," 是理想，对加法封闭）。",[108,354361,354362,314945,354390,354459,354460,112501,354592,354758,354759,347],{},[34,354363,354365,354378],{"className":354364},[31642],[34,354366,354368],{"className":354367},[31646],[31648,354369,354370],{"xmlns":31650},[31652,354371,354372,354376],{},[31655,354373,354374],{},[31658,354375,67764],{},[31665,354377,67764],{"encoding":31667},[34,354379,354381],{"className":354380,"ariaHidden":758},[31672],[34,354382,354384,354387],{"className":354383},[31676],[34,354385],{"className":354386,"style":38774},[31680],[34,354388,67764],{"className":354389,"style":32062},[31685,31686],[34,354391,354393,354410],{"className":354392},[31642],[34,354394,354396],{"className":354395},[31646],[31648,354397,354398],{"xmlns":31650},[31652,354399,354400,354408],{},[31655,354401,354402],{},[31743,354403,354404,354406],{},[31658,354405,105852],{},[31749,354407,640],{},[31665,354409,353413],{"encoding":31667},[34,354411,354413],{"className":354412,"ariaHidden":758},[31672],[34,354414,354416,354419],{"className":354415},[31676],[34,354417],{"className":354418,"style":70915},[31680],[34,354420,354422,354425],{"className":354421},[31685],[34,354423,105852],{"className":354424,"style":37422},[31685,31686],[34,354426,354428],{"className":354427},[31792],[34,354429,354431,354451],{"className":354430},[31796,31797],[34,354432,354434,354448],{"className":354433},[31801],[34,354435,354437],{"className":354436,"style":31806},[31805],[34,354438,354439,354442],{"style":214424},[34,354440],{"className":354441,"style":31814},[31813],[34,354443,354445],{"className":354444},[31818,31819,31820,31821],[34,354446,640],{"className":354447},[31685,31821],[34,354449,31829],{"className":354450},[31828],[34,354452,354454],{"className":354453},[31801],[34,354455,354457],{"className":354456,"style":31836},[31805],[34,354458],{}," 吗？",[34,354461,354463,354488],{"className":354462},[31642],[34,354464,354466],{"className":354465},[31646],[31648,354467,354468],{"xmlns":31650},[31652,354469,354470,354486],{},[31655,354471,354472,354478,354480],{},[31743,354473,354474,354476],{},[31658,354475,32936],{},[31749,354477,768],{},[31662,354479,112296],{},[31743,354481,354482,354484],{},[31658,354483,105852],{},[31749,354485,640],{},[31665,354487,354016],{"encoding":31667},[34,354489,354491,354546],{"className":354490,"ariaHidden":758},[31672],[34,354492,354494,354497,354537,354540,354543],{"className":354493},[31676],[34,354495],{"className":354496,"style":225773},[31680],[34,354498,354500,354503],{"className":354499},[31685],[34,354501,32936],{"className":354502},[31685,31686],[34,354504,354506],{"className":354505},[31792],[34,354507,354509,354529],{"className":354508},[31796,31797],[34,354510,354512,354526],{"className":354511},[31801],[34,354513,354515],{"className":354514,"style":31806},[31805],[34,354516,354517,354520],{"style":31809},[34,354518],{"className":354519,"style":31814},[31813],[34,354521,354523],{"className":354522},[31818,31819,31820,31821],[34,354524,768],{"className":354525},[31685,31821],[34,354527,31829],{"className":354528},[31828],[34,354530,354532],{"className":354531},[31801],[34,354533,354535],{"className":354534,"style":31836},[31805],[34,354536],{},[34,354538],{"className":354539,"style":31846},[31845],[34,354541,112296],{"className":354542},[31850],[34,354544],{"className":354545,"style":31846},[31845],[34,354547,354549,354552],{"className":354548},[31676],[34,354550],{"className":354551,"style":70915},[31680],[34,354553,354555,354558],{"className":354554},[31685],[34,354556,105852],{"className":354557,"style":37422},[31685,31686],[34,354559,354561],{"className":354560},[31792],[34,354562,354564,354584],{"className":354563},[31796,31797],[34,354565,354567,354581],{"className":354566},[31801],[34,354568,354570],{"className":354569,"style":31806},[31805],[34,354571,354572,354575],{"style":214424},[34,354573],{"className":354574,"style":31814},[31813],[34,354576,354578],{"className":354577},[31818,31819,31820,31821],[34,354579,640],{"className":354580},[31685,31821],[34,354582,31829],{"className":354583},[31828],[34,354585,354587],{"className":354586},[31801],[34,354588,354590],{"className":354589,"style":31836},[31805],[34,354591],{},[34,354593,354595,354621],{"className":354594},[31642],[34,354596,354598],{"className":354597},[31646],[31648,354599,354600],{"xmlns":31650},[31652,354601,354602,354618],{},[31655,354603,354604,354610,354612],{},[31743,354605,354606,354608],{},[31658,354607,32936],{},[31749,354609,640],{},[31662,354611,112266],{"mathvariant":38079},[31743,354613,354614,354616],{},[31658,354615,105852],{},[31749,354617,640],{},[31665,354619,354620],{"encoding":31667},"x_1 \\notin P_1",[34,354622,354624,354712],{"className":354623,"ariaHidden":758},[31672],[34,354625,354627,354630,354670,354673,354709],{"className":354626},[31676],[34,354628],{"className":354629,"style":31774},[31680],[34,354631,354633,354636],{"className":354632},[31685],[34,354634,32936],{"className":354635},[31685,31686],[34,354637,354639],{"className":354638},[31792],[34,354640,354642,354662],{"className":354641},[31796,31797],[34,354643,354645,354659],{"className":354644},[31801],[34,354646,354648],{"className":354647,"style":31806},[31805],[34,354649,354650,354653],{"style":31809},[34,354651],{"className":354652,"style":31814},[31813],[34,354654,354656],{"className":354655},[31818,31819,31820,31821],[34,354657,640],{"className":354658},[31685,31821],[34,354660,31829],{"className":354661},[31828],[34,354663,354665],{"className":354664},[31801],[34,354666,354668],{"className":354667,"style":31836},[31805],[34,354669],{},[34,354671],{"className":354672,"style":31846},[31845],[34,354674,354676,354682],{"className":354675},[31850],[34,354677,354679],{"className":354678},[31685],[34,354680,112296],{"className":354681},[31850],[34,354683,354685],{"className":354684},[31685,54774],[34,354686,354688],{"className":354687},[54778],[34,354689,354691,354694,354706],{"className":354690},[112306],[34,354692],{"className":354693,"style":31774},[31680],[34,354695,354697],{"className":354696},[54789],[34,354698,354700,354703],{"className":354699},[31685],[34,354701,38080],{"className":354702},[31685],[34,354704],{"className":354705,"style":112322},[31845],[34,354707],{"className":354708},[54800],[34,354710],{"className":354711,"style":31846},[31845],[34,354713,354715,354718],{"className":354714},[31676],[34,354716],{"className":354717,"style":70915},[31680],[34,354719,354721,354724],{"className":354720},[31685],[34,354722,105852],{"className":354723,"style":37422},[31685,31686],[34,354725,354727],{"className":354726},[31792],[34,354728,354730,354750],{"className":354729},[31796,31797],[34,354731,354733,354747],{"className":354732},[31801],[34,354734,354736],{"className":354735,"style":31806},[31805],[34,354737,354738,354741],{"style":214424},[34,354739],{"className":354740,"style":31814},[31813],[34,354742,354744],{"className":354743},[31818,31819,31820,31821],[34,354745,640],{"className":354746},[31685,31821],[34,354748,31829],{"className":354749},[31828],[34,354751,354753],{"className":354752},[31801],[34,354754,354756],{"className":354755,"style":31836},[31805],[34,354757],{},"。理想加非理想，结果不在理想中。所以 ",[34,354760,354762,354784],{"className":354761},[31642],[34,354763,354765],{"className":354764},[31646],[31648,354766,354767],{"xmlns":31650},[31652,354768,354769,354781],{},[31655,354770,354771,354773,354775],{},[31658,354772,67764],{},[31662,354774,112266],{"mathvariant":38079},[31743,354776,354777,354779],{},[31658,354778,105852],{},[31749,354780,640],{},[31665,354782,354783],{"encoding":31667},"y \\notin P_1",[34,354785,354787,354838],{"className":354786,"ariaHidden":758},[31672],[34,354788,354790,354793,354796,354799,354835],{"className":354789},[31676],[34,354791],{"className":354792,"style":31774},[31680],[34,354794,67764],{"className":354795,"style":32062},[31685,31686],[34,354797],{"className":354798,"style":31846},[31845],[34,354800,354802,354808],{"className":354801},[31850],[34,354803,354805],{"className":354804},[31685],[34,354806,112296],{"className":354807},[31850],[34,354809,354811],{"className":354810},[31685,54774],[34,354812,354814],{"className":354813},[54778],[34,354815,354817,354820,354832],{"className":354816},[112306],[34,354818],{"className":354819,"style":31774},[31680],[34,354821,354823],{"className":354822},[54789],[34,354824,354826,354829],{"className":354825},[31685],[34,354827,38080],{"className":354828},[31685],[34,354830],{"className":354831,"style":112322},[31845],[34,354833],{"className":354834},[54800],[34,354836],{"className":354837,"style":31846},[31845],[34,354839,354841,354844],{"className":354840},[31676],[34,354842],{"className":354843,"style":70915},[31680],[34,354845,354847,354850],{"className":354846},[31685],[34,354848,105852],{"className":354849,"style":37422},[31685,31686],[34,354851,354853],{"className":354852},[31792],[34,354854,354856,354876],{"className":354855},[31796,31797],[34,354857,354859,354873],{"className":354858},[31801],[34,354860,354862],{"className":354861,"style":31806},[31805],[34,354863,354864,354867],{"style":214424},[34,354865],{"className":354866,"style":31814},[31813],[34,354868,354870],{"className":354869},[31818,31819,31820,31821],[34,354871,640],{"className":354872},[31685,31821],[34,354874,31829],{"className":354875},[31828],[34,354877,354879],{"className":354878},[31801],[34,354880,354882],{"className":354881,"style":31836},[31805],[34,354883],{},[108,354885,354886,314945,354914,354459,354984,112501,355116,355282,355283,355408,355409,355459,355460,355648,355649,355804],{},[34,354887,354889,354902],{"className":354888},[31642],[34,354890,354892],{"className":354891},[31646],[31648,354893,354894],{"xmlns":31650},[31652,354895,354896,354900],{},[31655,354897,354898],{},[31658,354899,67764],{},[31665,354901,67764],{"encoding":31667},[34,354903,354905],{"className":354904,"ariaHidden":758},[31672],[34,354906,354908,354911],{"className":354907},[31676],[34,354909],{"className":354910,"style":38774},[31680],[34,354912,67764],{"className":354913,"style":32062},[31685,31686],[34,354915,354917,354935],{"className":354916},[31642],[34,354918,354920],{"className":354919},[31646],[31648,354921,354922],{"xmlns":31650},[31652,354923,354924,354932],{},[31655,354925,354926],{},[31743,354927,354928,354930],{},[31658,354929,105852],{},[31749,354931,768],{},[31665,354933,354934],{"encoding":31667},"P_2",[34,354936,354938],{"className":354937,"ariaHidden":758},[31672],[34,354939,354941,354944],{"className":354940},[31676],[34,354942],{"className":354943,"style":70915},[31680],[34,354945,354947,354950],{"className":354946},[31685],[34,354948,105852],{"className":354949,"style":37422},[31685,31686],[34,354951,354953],{"className":354952},[31792],[34,354954,354956,354976],{"className":354955},[31796,31797],[34,354957,354959,354973],{"className":354958},[31801],[34,354960,354962],{"className":354961,"style":31806},[31805],[34,354963,354964,354967],{"style":214424},[34,354965],{"className":354966,"style":31814},[31813],[34,354968,354970],{"className":354969},[31818,31819,31820,31821],[34,354971,768],{"className":354972},[31685,31821],[34,354974,31829],{"className":354975},[31828],[34,354977,354979],{"className":354978},[31801],[34,354980,354982],{"className":354981,"style":31836},[31805],[34,354983],{},[34,354985,354987,355012],{"className":354986},[31642],[34,354988,354990],{"className":354989},[31646],[31648,354991,354992],{"xmlns":31650},[31652,354993,354994,355010],{},[31655,354995,354996,355002,355004],{},[31743,354997,354998,355000],{},[31658,354999,32936],{},[31749,355001,640],{},[31662,355003,112296],{},[31743,355005,355006,355008],{},[31658,355007,105852],{},[31749,355009,768],{},[31665,355011,353883],{"encoding":31667},[34,355013,355015,355070],{"className":355014,"ariaHidden":758},[31672],[34,355016,355018,355021,355061,355064,355067],{"className":355017},[31676],[34,355019],{"className":355020,"style":225773},[31680],[34,355022,355024,355027],{"className":355023},[31685],[34,355025,32936],{"className":355026},[31685,31686],[34,355028,355030],{"className":355029},[31792],[34,355031,355033,355053],{"className":355032},[31796,31797],[34,355034,355036,355050],{"className":355035},[31801],[34,355037,355039],{"className":355038,"style":31806},[31805],[34,355040,355041,355044],{"style":31809},[34,355042],{"className":355043,"style":31814},[31813],[34,355045,355047],{"className":355046},[31818,31819,31820,31821],[34,355048,640],{"className":355049},[31685,31821],[34,355051,31829],{"className":355052},[31828],[34,355054,355056],{"className":355055},[31801],[34,355057,355059],{"className":355058,"style":31836},[31805],[34,355060],{},[34,355062],{"className":355063,"style":31846},[31845],[34,355065,112296],{"className":355066},[31850],[34,355068],{"className":355069,"style":31846},[31845],[34,355071,355073,355076],{"className":355072},[31676],[34,355074],{"className":355075,"style":70915},[31680],[34,355077,355079,355082],{"className":355078},[31685],[34,355080,105852],{"className":355081,"style":37422},[31685,31686],[34,355083,355085],{"className":355084},[31792],[34,355086,355088,355108],{"className":355087},[31796,31797],[34,355089,355091,355105],{"className":355090},[31801],[34,355092,355094],{"className":355093,"style":31806},[31805],[34,355095,355096,355099],{"style":214424},[34,355097],{"className":355098,"style":31814},[31813],[34,355100,355102],{"className":355101},[31818,31819,31820,31821],[34,355103,768],{"className":355104},[31685,31821],[34,355106,31829],{"className":355107},[31828],[34,355109,355111],{"className":355110},[31801],[34,355112,355114],{"className":355113,"style":31836},[31805],[34,355115],{},[34,355117,355119,355145],{"className":355118},[31642],[34,355120,355122],{"className":355121},[31646],[31648,355123,355124],{"xmlns":31650},[31652,355125,355126,355142],{},[31655,355127,355128,355134,355136],{},[31743,355129,355130,355132],{},[31658,355131,32936],{},[31749,355133,768],{},[31662,355135,112266],{"mathvariant":38079},[31743,355137,355138,355140],{},[31658,355139,105852],{},[31749,355141,768],{},[31665,355143,355144],{"encoding":31667},"x_2 \\notin P_2",[34,355146,355148,355236],{"className":355147,"ariaHidden":758},[31672],[34,355149,355151,355154,355194,355197,355233],{"className":355150},[31676],[34,355152],{"className":355153,"style":31774},[31680],[34,355155,355157,355160],{"className":355156},[31685],[34,355158,32936],{"className":355159},[31685,31686],[34,355161,355163],{"className":355162},[31792],[34,355164,355166,355186],{"className":355165},[31796,31797],[34,355167,355169,355183],{"className":355168},[31801],[34,355170,355172],{"className":355171,"style":31806},[31805],[34,355173,355174,355177],{"style":31809},[34,355175],{"className":355176,"style":31814},[31813],[34,355178,355180],{"className":355179},[31818,31819,31820,31821],[34,355181,768],{"className":355182},[31685,31821],[34,355184,31829],{"className":355185},[31828],[34,355187,355189],{"className":355188},[31801],[34,355190,355192],{"className":355191,"style":31836},[31805],[34,355193],{},[34,355195],{"className":355196,"style":31846},[31845],[34,355198,355200,355206],{"className":355199},[31850],[34,355201,355203],{"className":355202},[31685],[34,355204,112296],{"className":355205},[31850],[34,355207,355209],{"className":355208},[31685,54774],[34,355210,355212],{"className":355211},[54778],[34,355213,355215,355218,355230],{"className":355214},[112306],[34,355216],{"className":355217,"style":31774},[31680],[34,355219,355221],{"className":355220},[54789],[34,355222,355224,355227],{"className":355223},[31685],[34,355225,38080],{"className":355226},[31685],[34,355228],{"className":355229,"style":112322},[31845],[34,355231],{"className":355232},[54800],[34,355234],{"className":355235,"style":31846},[31845],[34,355237,355239,355242],{"className":355238},[31676],[34,355240],{"className":355241,"style":70915},[31680],[34,355243,355245,355248],{"className":355244},[31685],[34,355246,105852],{"className":355247,"style":37422},[31685,31686],[34,355249,355251],{"className":355250},[31792],[34,355252,355254,355274],{"className":355253},[31796,31797],[34,355255,355257,355271],{"className":355256},[31801],[34,355258,355260],{"className":355259,"style":31806},[31805],[34,355261,355262,355265],{"style":214424},[34,355263],{"className":355264,"style":31814},[31813],[34,355266,355268],{"className":355267},[31818,31819,31820,31821],[34,355269,768],{"className":355270},[31685,31821],[34,355272,31829],{"className":355273},[31828],[34,355275,355277],{"className":355276},[31801],[34,355278,355280],{"className":355279,"style":31836},[31805],[34,355281],{},"。同理， ",[34,355284,355286,355308],{"className":355285},[31642],[34,355287,355289],{"className":355288},[31646],[31648,355290,355291],{"xmlns":31650},[31652,355292,355293,355305],{},[31655,355294,355295,355297,355299],{},[31658,355296,67764],{},[31662,355298,112266],{"mathvariant":38079},[31743,355300,355301,355303],{},[31658,355302,105852],{},[31749,355304,768],{},[31665,355306,355307],{"encoding":31667},"y \\notin P_2",[34,355309,355311,355362],{"className":355310,"ariaHidden":758},[31672],[34,355312,355314,355317,355320,355323,355359],{"className":355313},[31676],[34,355315],{"className":355316,"style":31774},[31680],[34,355318,67764],{"className":355319,"style":32062},[31685,31686],[34,355321],{"className":355322,"style":31846},[31845],[34,355324,355326,355332],{"className":355325},[31850],[34,355327,355329],{"className":355328},[31685],[34,355330,112296],{"className":355331},[31850],[34,355333,355335],{"className":355334},[31685,54774],[34,355336,355338],{"className":355337},[54778],[34,355339,355341,355344,355356],{"className":355340},[112306],[34,355342],{"className":355343,"style":31774},[31680],[34,355345,355347],{"className":355346},[54789],[34,355348,355350,355353],{"className":355349},[31685],[34,355351,38080],{"className":355352},[31685],[34,355354],{"className":355355,"style":112322},[31845],[34,355357],{"className":355358},[54800],[34,355360],{"className":355361,"style":31846},[31845],[34,355363,355365,355368],{"className":355364},[31676],[34,355366],{"className":355367,"style":70915},[31680],[34,355369,355371,355374],{"className":355370},[31685],[34,355372,105852],{"className":355373,"style":37422},[31685,31686],[34,355375,355377],{"className":355376},[31792],[34,355378,355380,355400],{"className":355379},[31796,31797],[34,355381,355383,355397],{"className":355382},[31801],[34,355384,355386],{"className":355385,"style":31806},[31805],[34,355387,355388,355391],{"style":214424},[34,355389],{"className":355390,"style":31814},[31813],[34,355392,355394],{"className":355393},[31818,31819,31820,31821],[34,355395,768],{"className":355396},[31685,31821],[34,355398,31829],{"className":355399},[31828],[34,355401,355403],{"className":355402},[31801],[34,355404,355406],{"className":355405,"style":31836},[31805],[34,355407],{},"。\n所以 ",[34,355410,355412,355429],{"className":355411},[31642],[34,355413,355415],{"className":355414},[31646],[31648,355416,355417],{"xmlns":31650},[31652,355418,355419,355427],{},[31655,355420,355421,355423,355425],{},[31658,355422,67764],{},[31662,355424,112296],{},[31658,355426,106063],{},[31665,355428,354299],{"encoding":31667},[34,355430,355432,355450],{"className":355431,"ariaHidden":758},[31672],[34,355433,355435,355438,355441,355444,355447],{"className":355434},[31676],[34,355436],{"className":355437,"style":86652},[31680],[34,355439,67764],{"className":355440,"style":32062},[31685,31686],[34,355442],{"className":355443,"style":31846},[31845],[34,355445,112296],{"className":355446},[31850],[34,355448],{"className":355449,"style":31846},[31845],[34,355451,355453,355456],{"className":355452},[31676],[34,355454],{"className":355455,"style":33218},[31680],[34,355457,106063],{"className":355458,"style":67482},[31685,31686]," 但 ",[34,355461,355463,355493],{"className":355462},[31642],[34,355464,355466],{"className":355465},[31646],[31648,355467,355468],{"xmlns":31650},[31652,355469,355470,355490],{},[31655,355471,355472,355474,355476,355482,355484],{},[31658,355473,67764],{},[31662,355475,112266],{"mathvariant":38079},[31743,355477,355478,355480],{},[31658,355479,105852],{},[31749,355481,640],{},[31662,355483,352647],{},[31743,355485,355486,355488],{},[31658,355487,105852],{},[31749,355489,768],{},[31665,355491,355492],{"encoding":31667},"y \\notin P_1 \\cup P_2",[34,355494,355496,355547,355602],{"className":355495,"ariaHidden":758},[31672],[34,355497,355499,355502,355505,355508,355544],{"className":355498},[31676],[34,355500],{"className":355501,"style":31774},[31680],[34,355503,67764],{"className":355504,"style":32062},[31685,31686],[34,355506],{"className":355507,"style":31846},[31845],[34,355509,355511,355517],{"className":355510},[31850],[34,355512,355514],{"className":355513},[31685],[34,355515,112296],{"className":355516},[31850],[34,355518,355520],{"className":355519},[31685,54774],[34,355521,355523],{"className":355522},[54778],[34,355524,355526,355529,355541],{"className":355525},[112306],[34,355527],{"className":355528,"style":31774},[31680],[34,355530,355532],{"className":355531},[54789],[34,355533,355535,355538],{"className":355534},[31685],[34,355536,38080],{"className":355537},[31685],[34,355539],{"className":355540,"style":112322},[31845],[34,355542],{"className":355543},[54800],[34,355545],{"className":355546,"style":31846},[31845],[34,355548,355550,355553,355593,355596,355599],{"className":355549},[31676],[34,355551],{"className":355552,"style":70915},[31680],[34,355554,355556,355559],{"className":355555},[31685],[34,355557,105852],{"className":355558,"style":37422},[31685,31686],[34,355560,355562],{"className":355561},[31792],[34,355563,355565,355585],{"className":355564},[31796,31797],[34,355566,355568,355582],{"className":355567},[31801],[34,355569,355571],{"className":355570,"style":31806},[31805],[34,355572,355573,355576],{"style":214424},[34,355574],{"className":355575,"style":31814},[31813],[34,355577,355579],{"className":355578},[31818,31819,31820,31821],[34,355580,640],{"className":355581},[31685,31821],[34,355583,31829],{"className":355584},[31828],[34,355586,355588],{"className":355587},[31801],[34,355589,355591],{"className":355590,"style":31836},[31805],[34,355592],{},[34,355594],{"className":355595,"style":32604},[31845],[34,355597,352647],{"className":355598},[32024],[34,355600],{"className":355601,"style":32604},[31845],[34,355603,355605,355608],{"className":355604},[31676],[34,355606],{"className":355607,"style":70915},[31680],[34,355609,355611,355614],{"className":355610},[31685],[34,355612,105852],{"className":355613,"style":37422},[31685,31686],[34,355615,355617],{"className":355616},[31792],[34,355618,355620,355640],{"className":355619},[31796,31797],[34,355621,355623,355637],{"className":355622},[31801],[34,355624,355626],{"className":355625,"style":31806},[31805],[34,355627,355628,355631],{"style":214424},[34,355629],{"className":355630,"style":31814},[31813],[34,355632,355634],{"className":355633},[31818,31819,31820,31821],[34,355635,768],{"className":355636},[31685,31821],[34,355638,31829],{"className":355639},[31828],[34,355641,355643],{"className":355642},[31801],[34,355644,355646],{"className":355645,"style":31836},[31805],[34,355647],{},"，这与 ",[34,355650,355652,355682],{"className":355651},[31642],[34,355653,355655],{"className":355654},[31646],[31648,355656,355657],{"xmlns":31650},[31652,355658,355659,355679],{},[31655,355660,355661,355663,355665,355671,355673],{},[31658,355662,106063],{},[31662,355664,131855],{},[31743,355666,355667,355669],{},[31658,355668,105852],{},[31749,355670,640],{},[31662,355672,352647],{},[31743,355674,355675,355677],{},[31658,355676,105852],{},[31749,355678,768],{},[31665,355680,355681],{"encoding":31667},"I \\subseteq P_1 \\cup P_2",[34,355683,355685,355703,355758],{"className":355684,"ariaHidden":758},[31672],[34,355686,355688,355691,355694,355697,355700],{"className":355687},[31676],[34,355689],{"className":355690,"style":195047},[31680],[34,355692,106063],{"className":355693,"style":67482},[31685,31686],[34,355695],{"className":355696,"style":31846},[31845],[34,355698,131855],{"className":355699},[31850],[34,355701],{"className":355702,"style":31846},[31845],[34,355704,355706,355709,355749,355752,355755],{"className":355705},[31676],[34,355707],{"className":355708,"style":70915},[31680],[34,355710,355712,355715],{"className":355711},[31685],[34,355713,105852],{"className":355714,"style":37422},[31685,31686],[34,355716,355718],{"className":355717},[31792],[34,355719,355721,355741],{"className":355720},[31796,31797],[34,355722,355724,355738],{"className":355723},[31801],[34,355725,355727],{"className":355726,"style":31806},[31805],[34,355728,355729,355732],{"style":214424},[34,355730],{"className":355731,"style":31814},[31813],[34,355733,355735],{"className":355734},[31818,31819,31820,31821],[34,355736,640],{"className":355737},[31685,31821],[34,355739,31829],{"className":355740},[31828],[34,355742,355744],{"className":355743},[31801],[34,355745,355747],{"className":355746,"style":31836},[31805],[34,355748],{},[34,355750],{"className":355751,"style":32604},[31845],[34,355753,352647],{"className":355754},[32024],[34,355756],{"className":355757,"style":32604},[31845],[34,355759,355761,355764],{"className":355760},[31676],[34,355762],{"className":355763,"style":70915},[31680],[34,355765,355767,355770],{"className":355766},[31685],[34,355768,105852],{"className":355769,"style":37422},[31685,31686],[34,355771,355773],{"className":355772},[31792],[34,355774,355776,355796],{"className":355775},[31796,31797],[34,355777,355779,355793],{"className":355778},[31801],[34,355780,355782],{"className":355781,"style":31806},[31805],[34,355783,355784,355787],{"style":214424},[34,355785],{"className":355786,"style":31814},[31813],[34,355788,355790],{"className":355789},[31818,31819,31820,31821],[34,355791,768],{"className":355792},[31685,31821],[34,355794,31829],{"className":355795},[31828],[34,355797,355799],{"className":355798},[31801],[34,355800,355802],{"className":355801,"style":31836},[31805],[34,355803],{}," 矛盾。",[2230,355806,355807],{"start":49},[108,355808,355809,355841,355842,355892,355893,355921,355922,356062,356063,356091,356092,356162,356163,356191,356192,356242,356243,356293,356294,83913,356322,66331,356395,356423,356424,356592,356593,356663,356664,356916,356917,357083,357084,357168,357169,314945,357238,357307,357308,357377,357378,357448,357449,357452,357453,357481,357482,357510,357511,357580,357581,93535,357673,357742,357743,357812,357813,358071,358072,358141,358142,358211,358212,358215,358218,358219,358247,358248,358467,358468,315343,358560,352570,358726,358810,358811,358839,358840,358910,358911,358980,358981,359050,359051,359079,359080,352570,359213,359241,359242,359500,359501,359529,359530,359599],{},[2627,355810,355811,355812,355840],{},"推广到 ",[34,355813,355815,355828],{"className":355814},[31642],[34,355816,355818],{"className":355817},[31646],[31648,355819,355820],{"xmlns":31650},[31652,355821,355822,355826],{},[31655,355823,355824],{},[31658,355825,31660],{},[31665,355827,31660],{"encoding":31667},[34,355829,355831],{"className":355830,"ariaHidden":758},[31672],[34,355832,355834,355837],{"className":355833},[31676],[34,355835],{"className":355836,"style":32527},[31680],[34,355838,31660],{"className":355839},[31685,31686]," 个素理想","：\n假设对于 ",[34,355843,355845,355862],{"className":355844},[31642],[34,355846,355848],{"className":355847},[31646],[31648,355849,355850],{"xmlns":31650},[31652,355851,355852,355860],{},[31655,355853,355854,355856,355858],{},[31658,355855,31660],{},[31662,355857,32226],{},[31749,355859,640],{},[31665,355861,32587],{"encoding":31667},[34,355863,355865,355883],{"className":355864,"ariaHidden":758},[31672],[34,355866,355868,355871,355874,355877,355880],{"className":355867},[31676],[34,355869],{"className":355870,"style":32597},[31680],[34,355872,31660],{"className":355873},[31685,31686],[34,355875],{"className":355876,"style":32604},[31845],[34,355878,32226],{"className":355879},[32024],[34,355881],{"className":355882,"style":32604},[31845],[34,355884,355886,355889],{"className":355885},[31676],[34,355887],{"className":355888,"style":32617},[31680],[34,355890,640],{"className":355891},[31685]," 个素理想结论成立。\n现在考虑 ",[34,355894,355896,355909],{"className":355895},[31642],[34,355897,355899],{"className":355898},[31646],[31648,355900,355901],{"xmlns":31650},[31652,355902,355903,355907],{},[31655,355904,355905],{},[31658,355906,31660],{},[31665,355908,31660],{"encoding":31667},[34,355910,355912],{"className":355911,"ariaHidden":758},[31672],[34,355913,355915,355918],{"className":355914},[31676],[34,355916],{"className":355917,"style":32527},[31680],[34,355919,31660],{"className":355920},[31685,31686]," 个素理想 ",[34,355923,355925,355955],{"className":355924},[31642],[34,355926,355928],{"className":355927},[31646],[31648,355929,355930],{"xmlns":31650},[31652,355931,355932,355952],{},[31655,355933,355934,355940,355942,355944,355946],{},[31743,355935,355936,355938],{},[31658,355937,105852],{},[31749,355939,640],{},[31662,355941,31960],{"separator":758},[31662,355943,33934],{},[31662,355945,31960],{"separator":758},[31743,355947,355948,355950],{},[31658,355949,105852],{},[31658,355951,31660],{},[31665,355953,355954],{"encoding":31667},"P_1, \\dots, P_n",[34,355956,355958],{"className":355957,"ariaHidden":758},[31672],[34,355959,355961,355964,356004,356007,356010,356013,356016,356019,356022],{"className":355960},[31676],[34,355962],{"className":355963,"style":70739},[31680],[34,355965,355967,355970],{"className":355966},[31685],[34,355968,105852],{"className":355969,"style":37422},[31685,31686],[34,355971,355973],{"className":355972},[31792],[34,355974,355976,355996],{"className":355975},[31796,31797],[34,355977,355979,355993],{"className":355978},[31801],[34,355980,355982],{"className":355981,"style":31806},[31805],[34,355983,355984,355987],{"style":214424},[34,355985],{"className":355986,"style":31814},[31813],[34,355988,355990],{"className":355989},[31818,31819,31820,31821],[34,355991,640],{"className":355992},[31685,31821],[34,355994,31829],{"className":355995},[31828],[34,355997,355999],{"className":355998},[31801],[34,356000,356002],{"className":356001,"style":31836},[31805],[34,356003],{},[34,356005,31960],{"className":356006},[32119],[34,356008],{"className":356009,"style":32123},[31845],[34,356011,33934],{"className":356012},[34006],[34,356014],{"className":356015,"style":32123},[31845],[34,356017,31960],{"className":356018},[32119],[34,356020],{"className":356021,"style":32123},[31845],[34,356023,356025,356028],{"className":356024},[31685],[34,356026,105852],{"className":356027,"style":37422},[31685,31686],[34,356029,356031],{"className":356030},[31792],[34,356032,356034,356054],{"className":356033},[31796,31797],[34,356035,356037,356051],{"className":356036},[31801],[34,356038,356040],{"className":356039,"style":32362},[31805],[34,356041,356042,356045],{"style":214424},[34,356043],{"className":356044,"style":31814},[31813],[34,356046,356048],{"className":356047},[31818,31819,31820,31821],[34,356049,31660],{"className":356050},[31685,31686,31821],[34,356052,31829],{"className":356053},[31828],[34,356055,356057],{"className":356056},[31801],[34,356058,356060],{"className":356059,"style":31836},[31805],[34,356061],{},"。\n根据假设，",[34,356064,356066,356079],{"className":356065},[31642],[34,356067,356069],{"className":356068},[31646],[31648,356070,356071],{"xmlns":31650},[31652,356072,356073,356077],{},[31655,356074,356075],{},[31658,356076,106063],{},[31665,356078,106063],{"encoding":31667},[34,356080,356082],{"className":356081,"ariaHidden":758},[31672],[34,356083,356085,356088],{"className":356084},[31676],[34,356086],{"className":356087,"style":33218},[31680],[34,356089,106063],{"className":356090,"style":67482},[31685,31686]," 不含于任何单个 ",[34,356093,356095,356113],{"className":356094},[31642],[34,356096,356098],{"className":356097},[31646],[31648,356099,356100],{"xmlns":31650},[31652,356101,356102,356110],{},[31655,356103,356104],{},[31743,356105,356106,356108],{},[31658,356107,105852],{},[31658,356109,33759],{},[31665,356111,356112],{"encoding":31667},"P_k",[34,356114,356116],{"className":356115,"ariaHidden":758},[31672],[34,356117,356119,356122],{"className":356118},[31676],[34,356120],{"className":356121,"style":70915},[31680],[34,356123,356125,356128],{"className":356124},[31685],[34,356126,105852],{"className":356127,"style":37422},[31685,31686],[34,356129,356131],{"className":356130},[31792],[34,356132,356134,356154],{"className":356133},[31796,31797],[34,356135,356137,356151],{"className":356136},[31801],[34,356138,356140],{"className":356139,"style":33833},[31805],[34,356141,356142,356145],{"style":214424},[34,356143],{"className":356144,"style":31814},[31813],[34,356146,356148],{"className":356147},[31818,31819,31820,31821],[34,356149,33759],{"className":356150,"style":33774},[31685,31686,31821],[34,356152,31829],{"className":356153},[31828],[34,356155,356157],{"className":356156},[31801],[34,356158,356160],{"className":356159,"style":31836},[31805],[34,356161],{},"。\n更进一步，我们可以假设 ",[34,356164,356166,356179],{"className":356165},[31642],[34,356167,356169],{"className":356168},[31646],[31648,356170,356171],{"xmlns":31650},[31652,356172,356173,356177],{},[31655,356174,356175],{},[31658,356176,106063],{},[31665,356178,106063],{"encoding":31667},[34,356180,356182],{"className":356181,"ariaHidden":758},[31672],[34,356183,356185,356188],{"className":356184},[31676],[34,356186],{"className":356187,"style":33218},[31680],[34,356189,106063],{"className":356190,"style":67482},[31685,31686]," 也不含于任何 ",[34,356193,356195,356212],{"className":356194},[31642],[34,356196,356198],{"className":356197},[31646],[31648,356199,356200],{"xmlns":31650},[31652,356201,356202,356210],{},[31655,356203,356204,356206,356208],{},[31658,356205,31660],{},[31662,356207,32226],{},[31749,356209,640],{},[31665,356211,32587],{"encoding":31667},[34,356213,356215,356233],{"className":356214,"ariaHidden":758},[31672],[34,356216,356218,356221,356224,356227,356230],{"className":356217},[31676],[34,356219],{"className":356220,"style":32597},[31680],[34,356222,31660],{"className":356223},[31685,31686],[34,356225],{"className":356226,"style":32604},[31845],[34,356228,32226],{"className":356229},[32024],[34,356231],{"className":356232,"style":32604},[31845],[34,356234,356236,356239],{"className":356235},[31676],[34,356237],{"className":356238,"style":32617},[31680],[34,356240,640],{"className":356241},[31685]," 个素理想的并集（否则就归结为 ",[34,356244,356246,356263],{"className":356245},[31642],[34,356247,356249],{"className":356248},[31646],[31648,356250,356251],{"xmlns":31650},[31652,356252,356253,356261],{},[31655,356254,356255,356257,356259],{},[31658,356256,31660],{},[31662,356258,32226],{},[31749,356260,640],{},[31665,356262,32587],{"encoding":31667},[34,356264,356266,356284],{"className":356265,"ariaHidden":758},[31672],[34,356267,356269,356272,356275,356278,356281],{"className":356268},[31676],[34,356270],{"className":356271,"style":32597},[31680],[34,356273,31660],{"className":356274},[31685,31686],[34,356276],{"className":356277,"style":32604},[31845],[34,356279,32226],{"className":356280},[32024],[34,356282],{"className":356283,"style":32604},[31845],[34,356285,356287,356290],{"className":356286},[31676],[34,356288],{"className":356289,"style":32617},[31680],[34,356291,640],{"className":356292},[31685]," 的情况，直接得证）。\n这意味着，对于每一个固定索引 ",[34,356295,356297,356310],{"className":356296},[31642],[34,356298,356300],{"className":356299},[31646],[31648,356301,356302],{"xmlns":31650},[31652,356303,356304,356308],{},[31655,356305,356306],{},[31658,356307,33759],{},[31665,356309,33759],{"encoding":31667},[34,356311,356313],{"className":356312,"ariaHidden":758},[31672],[34,356314,356316,356319],{"className":356315},[31676],[34,356317],{"className":356318,"style":31681},[31680],[34,356320,33759],{"className":356321,"style":33774},[31685,31686],[34,356323,356325,356347],{"className":356324},[31642],[34,356326,356328],{"className":356327},[31646],[31648,356329,356330],{"xmlns":31650},[31652,356331,356332,356344],{},[31655,356333,356334,356336,356338,356340,356342],{},[31749,356335,640],{},[31662,356337,45972],{},[31658,356339,33759],{},[31662,356341,45972],{},[31658,356343,31660],{},[31665,356345,356346],{"encoding":31667},"1 \\le k \\le n",[34,356348,356350,356368,356386],{"className":356349,"ariaHidden":758},[31672],[34,356351,356353,356356,356359,356362,356365],{"className":356352},[31676],[34,356354],{"className":356355,"style":55382},[31680],[34,356357,640],{"className":356358},[31685],[34,356360],{"className":356361,"style":31846},[31845],[34,356363,45972],{"className":356364},[31850],[34,356366],{"className":356367,"style":31846},[31845],[34,356369,356371,356374,356377,356380,356383],{"className":356370},[31676],[34,356372],{"className":356373,"style":47709},[31680],[34,356375,33759],{"className":356376,"style":33774},[31685,31686],[34,356378],{"className":356379,"style":31846},[31845],[34,356381,45972],{"className":356382},[31850],[34,356384],{"className":356385,"style":31846},[31845],[34,356387,356389,356392],{"className":356388},[31676],[34,356390],{"className":356391,"style":32527},[31680],[34,356393,31660],{"className":356394},[31685,31686],[34,356396,356398,356411],{"className":356397},[31642],[34,356399,356401],{"className":356400},[31646],[31648,356402,356403],{"xmlns":31650},[31652,356404,356405,356409],{},[31655,356406,356407],{},[31658,356408,106063],{},[31665,356410,106063],{"encoding":31667},[34,356412,356414],{"className":356413,"ariaHidden":758},[31672],[34,356415,356417,356420],{"className":356416},[31676],[34,356418],{"className":356419,"style":33218},[31680],[34,356421,106063],{"className":356422,"style":67482},[31685,31686]," 都不含于 ",[34,356425,356427,356457],{"className":356426},[31642],[34,356428,356430],{"className":356429},[31646],[31648,356431,356432],{"xmlns":31650},[31652,356433,356434,356454],{},[31655,356435,356436,356448],{},[31743,356437,356438,356440],{},[31662,356439,351927],{},[31655,356441,356442,356444,356446],{},[31658,356443,31968],{},[31662,356445,54707],{"mathvariant":38079},[31658,356447,33759],{},[31743,356449,356450,356452],{},[31658,356451,105852],{},[31658,356453,31968],{},[31665,356455,356456],{"encoding":31667},"\\bigcup_{j \\neq k} P_j",[34,356458,356460],{"className":356459,"ariaHidden":758},[31672],[34,356461,356463,356466,356549,356552],{"className":356462},[31676],[34,356464],{"className":356465,"style":156901},[31680],[34,356467,356469,356472],{"className":356468},[32767],[34,356470,351927],{"className":356471,"style":156561},[32767,131249,156560],[34,356473,356475],{"className":356474},[31792],[34,356476,356478,356541],{"className":356477},[31796,31797],[34,356479,356481,356538],{"className":356480},[31801],[34,356482,356485],{"className":356483,"style":356484},[31805],"height:0.1864em;",[34,356486,356487,356490],{"style":156577},[34,356488],{"className":356489,"style":31814},[31813],[34,356491,356493],{"className":356492},[31818,31819,31820,31821],[34,356494,356496,356499,356535],{"className":356495},[31685,31821],[34,356497,31968],{"className":356498,"style":32153},[31685,31686,31821],[34,356500,356502,356529,356532],{"className":356501},[31850,31821],[34,356503,356505],{"className":356504},[31850,31821],[34,356506,356508],{"className":356507},[31685,54774,31821],[34,356509,356511],{"className":356510},[54778,31821],[34,356512,356514,356517,356526],{"className":356513},[54782,31821],[34,356515],{"className":356516,"style":43217},[31680],[34,356518,356520],{"className":356519},[54789],[34,356521,356523],{"className":356522},[31685,31821],[34,356524,54796],{"className":356525},[31850,31821],[34,356527],{"className":356528},[54800],[34,356530],{"className":356531},[31845,54804,31821],[34,356533,202],{"className":356534},[31850,31821],[34,356536,33759],{"className":356537,"style":33774},[31685,31686,31821],[34,356539,31829],{"className":356540},[31828],[34,356542,356544],{"className":356543},[31801],[34,356545,356547],{"className":356546,"style":156941},[31805],[34,356548],{},[34,356550],{"className":356551,"style":32123},[31845],[34,356553,356555,356558],{"className":356554},[31685],[34,356556,105852],{"className":356557,"style":37422},[31685,31686],[34,356559,356561],{"className":356560},[31792],[34,356562,356564,356584],{"className":356563},[31796,31797],[34,356565,356567,356581],{"className":356566},[31801],[34,356568,356570],{"className":356569,"style":32090},[31805],[34,356571,356572,356575],{"style":214424},[34,356573],{"className":356574,"style":31814},[31813],[34,356576,356578],{"className":356577},[31818,31819,31820,31821],[34,356579,31968],{"className":356580,"style":32153},[31685,31686,31821],[34,356582,31829],{"className":356583},[31828],[34,356585,356587],{"className":356586},[31801],[34,356588,356590],{"className":356589,"style":32163},[31805],[34,356591],{},"。\n所以，我们可以找到一个元素 ",[34,356594,356596,356614],{"className":356595},[31642],[34,356597,356599],{"className":356598},[31646],[31648,356600,356601],{"xmlns":31650},[31652,356602,356603,356611],{},[31655,356604,356605],{},[31743,356606,356607,356609],{},[31658,356608,32936],{},[31658,356610,33759],{},[31665,356612,356613],{"encoding":31667},"x_k",[34,356615,356617],{"className":356616,"ariaHidden":758},[31672],[34,356618,356620,356623],{"className":356619},[31676],[34,356621],{"className":356622,"style":31860},[31680],[34,356624,356626,356629],{"className":356625},[31685],[34,356627,32936],{"className":356628},[31685,31686],[34,356630,356632],{"className":356631},[31792],[34,356633,356635,356655],{"className":356634},[31796,31797],[34,356636,356638,356652],{"className":356637},[31801],[34,356639,356641],{"className":356640,"style":33833},[31805],[34,356642,356643,356646],{"style":31809},[34,356644],{"className":356645,"style":31814},[31813],[34,356647,356649],{"className":356648},[31818,31819,31820,31821],[34,356650,33759],{"className":356651,"style":33774},[31685,31686,31821],[34,356653,31829],{"className":356654},[31828],[34,356656,356658],{"className":356657},[31801],[34,356659,356661],{"className":356660,"style":31836},[31805],[34,356662],{},"，满足：\n",[34,356665,356667,356709],{"className":356666},[31642],[34,356668,356670],{"className":356669},[31646],[31648,356671,356672],{"xmlns":31650},[31652,356673,356674,356706],{},[31655,356675,356676,356682,356684,356686,356688,356700],{},[31743,356677,356678,356680],{},[31658,356679,32936],{},[31658,356681,33759],{},[31662,356683,112296],{},[31658,356685,106063],{},[31662,356687,336362],{},[31743,356689,356690,356692],{},[31662,356691,351927],{},[31655,356693,356694,356696,356698],{},[31658,356695,31968],{},[31662,356697,54707],{"mathvariant":38079},[31658,356699,33759],{},[31743,356701,356702,356704],{},[31658,356703,105852],{},[31658,356705,31968],{},[31665,356707,356708],{"encoding":31667},"x_k \\in I \\setminus \\bigcup_{j \\neq k} P_j",[34,356710,356712,356767,356785],{"className":356711,"ariaHidden":758},[31672],[34,356713,356715,356718,356758,356761,356764],{"className":356714},[31676],[34,356716],{"className":356717,"style":225773},[31680],[34,356719,356721,356724],{"className":356720},[31685],[34,356722,32936],{"className":356723},[31685,31686],[34,356725,356727],{"className":356726},[31792],[34,356728,356730,356750],{"className":356729},[31796,31797],[34,356731,356733,356747],{"className":356732},[31801],[34,356734,356736],{"className":356735,"style":33833},[31805],[34,356737,356738,356741],{"style":31809},[34,356739],{"className":356740,"style":31814},[31813],[34,356742,356744],{"className":356743},[31818,31819,31820,31821],[34,356745,33759],{"className":356746,"style":33774},[31685,31686,31821],[34,356748,31829],{"className":356749},[31828],[34,356751,356753],{"className":356752},[31801],[34,356754,356756],{"className":356755,"style":31836},[31805],[34,356757],{},[34,356759],{"className":356760,"style":31846},[31845],[34,356762,112296],{"className":356763},[31850],[34,356765],{"className":356766,"style":31846},[31845],[34,356768,356770,356773,356776,356779,356782],{"className":356769},[31676],[34,356771],{"className":356772,"style":31774},[31680],[34,356774,106063],{"className":356775,"style":67482},[31685,31686],[34,356777],{"className":356778,"style":32604},[31845],[34,356780,336362],{"className":356781},[32024],[34,356783],{"className":356784,"style":32604},[31845],[34,356786,356788,356791,356873,356876],{"className":356787},[31676],[34,356789],{"className":356790,"style":156901},[31680],[34,356792,356794,356797],{"className":356793},[32767],[34,356795,351927],{"className":356796,"style":156561},[32767,131249,156560],[34,356798,356800],{"className":356799},[31792],[34,356801,356803,356865],{"className":356802},[31796,31797],[34,356804,356806,356862],{"className":356805},[31801],[34,356807,356809],{"className":356808,"style":356484},[31805],[34,356810,356811,356814],{"style":156577},[34,356812],{"className":356813,"style":31814},[31813],[34,356815,356817],{"className":356816},[31818,31819,31820,31821],[34,356818,356820,356823,356859],{"className":356819},[31685,31821],[34,356821,31968],{"className":356822,"style":32153},[31685,31686,31821],[34,356824,356826,356853,356856],{"className":356825},[31850,31821],[34,356827,356829],{"className":356828},[31850,31821],[34,356830,356832],{"className":356831},[31685,54774,31821],[34,356833,356835],{"className":356834},[54778,31821],[34,356836,356838,356841,356850],{"className":356837},[54782,31821],[34,356839],{"className":356840,"style":43217},[31680],[34,356842,356844],{"className":356843},[54789],[34,356845,356847],{"className":356846},[31685,31821],[34,356848,54796],{"className":356849},[31850,31821],[34,356851],{"className":356852},[54800],[34,356854],{"className":356855},[31845,54804,31821],[34,356857,202],{"className":356858},[31850,31821],[34,356860,33759],{"className":356861,"style":33774},[31685,31686,31821],[34,356863,31829],{"className":356864},[31828],[34,356866,356868],{"className":356867},[31801],[34,356869,356871],{"className":356870,"style":156941},[31805],[34,356872],{},[34,356874],{"className":356875,"style":32123},[31845],[34,356877,356879,356882],{"className":356878},[31685],[34,356880,105852],{"className":356881,"style":37422},[31685,31686],[34,356883,356885],{"className":356884},[31792],[34,356886,356888,356908],{"className":356887},[31796,31797],[34,356889,356891,356905],{"className":356890},[31801],[34,356892,356894],{"className":356893,"style":32090},[31805],[34,356895,356896,356899],{"style":214424},[34,356897],{"className":356898,"style":31814},[31813],[34,356900,356902],{"className":356901},[31818,31819,31820,31821],[34,356903,31968],{"className":356904,"style":32153},[31685,31686,31821],[34,356906,31829],{"className":356907},[31828],[34,356909,356911],{"className":356910},[31801],[34,356912,356914],{"className":356913,"style":32163},[31805],[34,356915],{},"\n这实际上意味着 ",[34,356918,356920,356946],{"className":356919},[31642],[34,356921,356923],{"className":356922},[31646],[31648,356924,356925],{"xmlns":31650},[31652,356926,356927,356943],{},[31655,356928,356929,356935,356937],{},[31743,356930,356931,356933],{},[31658,356932,32936],{},[31658,356934,33759],{},[31662,356936,112266],{"mathvariant":38079},[31743,356938,356939,356941],{},[31658,356940,105852],{},[31658,356942,31968],{},[31665,356944,356945],{"encoding":31667},"x_k \\notin P_j",[34,356947,356949,357037],{"className":356948,"ariaHidden":758},[31672],[34,356950,356952,356955,356995,356998,357034],{"className":356951},[31676],[34,356953],{"className":356954,"style":31774},[31680],[34,356956,356958,356961],{"className":356957},[31685],[34,356959,32936],{"className":356960},[31685,31686],[34,356962,356964],{"className":356963},[31792],[34,356965,356967,356987],{"className":356966},[31796,31797],[34,356968,356970,356984],{"className":356969},[31801],[34,356971,356973],{"className":356972,"style":33833},[31805],[34,356974,356975,356978],{"style":31809},[34,356976],{"className":356977,"style":31814},[31813],[34,356979,356981],{"className":356980},[31818,31819,31820,31821],[34,356982,33759],{"className":356983,"style":33774},[31685,31686,31821],[34,356985,31829],{"className":356986},[31828],[34,356988,356990],{"className":356989},[31801],[34,356991,356993],{"className":356992,"style":31836},[31805],[34,356994],{},[34,356996],{"className":356997,"style":31846},[31845],[34,356999,357001,357007],{"className":357000},[31850],[34,357002,357004],{"className":357003},[31685],[34,357005,112296],{"className":357006},[31850],[34,357008,357010],{"className":357009},[31685,54774],[34,357011,357013],{"className":357012},[54778],[34,357014,357016,357019,357031],{"className":357015},[112306],[34,357017],{"className":357018,"style":31774},[31680],[34,357020,357022],{"className":357021},[54789],[34,357023,357025,357028],{"className":357024},[31685],[34,357026,38080],{"className":357027},[31685],[34,357029],{"className":357030,"style":112322},[31845],[34,357032],{"className":357033},[54800],[34,357035],{"className":357036,"style":31846},[31845],[34,357038,357040,357043],{"className":357039},[31676],[34,357041],{"className":357042,"style":55484},[31680],[34,357044,357046,357049],{"className":357045},[31685],[34,357047,105852],{"className":357048,"style":37422},[31685,31686],[34,357050,357052],{"className":357051},[31792],[34,357053,357055,357075],{"className":357054},[31796,31797],[34,357056,357058,357072],{"className":357057},[31801],[34,357059,357061],{"className":357060,"style":32090},[31805],[34,357062,357063,357066],{"style":214424},[34,357064],{"className":357065,"style":31814},[31813],[34,357067,357069],{"className":357068},[31818,31819,31820,31821],[34,357070,31968],{"className":357071,"style":32153},[31685,31686,31821],[34,357073,31829],{"className":357074},[31828],[34,357076,357078],{"className":357077},[31801],[34,357079,357081],{"className":357080,"style":32163},[31805],[34,357082],{}," (对于所有 ",[34,357085,357087,357105],{"className":357086},[31642],[34,357088,357090],{"className":357089},[31646],[31648,357091,357092],{"xmlns":31650},[31652,357093,357094,357102],{},[31655,357095,357096,357098,357100],{},[31658,357097,31968],{},[31662,357099,54707],{"mathvariant":38079},[31658,357101,33759],{},[31665,357103,357104],{"encoding":31667},"j \\ne k",[34,357106,357108,357159],{"className":357107,"ariaHidden":758},[31672],[34,357109,357111,357114,357117,357120,357156],{"className":357110},[31676],[34,357112],{"className":357113,"style":43217},[31680],[34,357115,31968],{"className":357116,"style":32153},[31685,31686],[34,357118],{"className":357119,"style":31846},[31845],[34,357121,357123,357150,357153],{"className":357122},[31850],[34,357124,357126],{"className":357125},[31850],[34,357127,357129],{"className":357128},[31685,54774],[34,357130,357132],{"className":357131},[54778],[34,357133,357135,357138,357147],{"className":357134},[54782],[34,357136],{"className":357137,"style":43217},[31680],[34,357139,357141],{"className":357140},[54789],[34,357142,357144],{"className":357143},[31685],[34,357145,54796],{"className":357146},[31850],[34,357148],{"className":357149},[54800],[34,357151],{"className":357152},[31845,54804],[34,357154,202],{"className":357155},[31850],[34,357157],{"className":357158,"style":31846},[31845],[34,357160,357162,357165],{"className":357161},[31676],[34,357163],{"className":357164,"style":31681},[31680],[34,357166,33759],{"className":357167,"style":33774},[31685,31686],")。那 ",[34,357170,357172,357189],{"className":357171},[31642],[34,357173,357175],{"className":357174},[31646],[31648,357176,357177],{"xmlns":31650},[31652,357178,357179,357187],{},[31655,357180,357181],{},[31743,357182,357183,357185],{},[31658,357184,32936],{},[31658,357186,33759],{},[31665,357188,356613],{"encoding":31667},[34,357190,357192],{"className":357191,"ariaHidden":758},[31672],[34,357193,357195,357198],{"className":357194},[31676],[34,357196],{"className":357197,"style":31860},[31680],[34,357199,357201,357204],{"className":357200},[31685],[34,357202,32936],{"className":357203},[31685,31686],[34,357205,357207],{"className":357206},[31792],[34,357208,357210,357230],{"className":357209},[31796,31797],[34,357211,357213,357227],{"className":357212},[31801],[34,357214,357216],{"className":357215,"style":33833},[31805],[34,357217,357218,357221],{"style":31809},[34,357219],{"className":357220,"style":31814},[31813],[34,357222,357224],{"className":357223},[31818,31819,31820,31821],[34,357225,33759],{"className":357226,"style":33774},[31685,31686,31821],[34,357228,31829],{"className":357229},[31828],[34,357231,357233],{"className":357232},[31801],[34,357234,357236],{"className":357235,"style":31836},[31805],[34,357237],{},[34,357239,357241,357258],{"className":357240},[31642],[34,357242,357244],{"className":357243},[31646],[31648,357245,357246],{"xmlns":31650},[31652,357247,357248,357256],{},[31655,357249,357250],{},[31743,357251,357252,357254],{},[31658,357253,105852],{},[31658,357255,33759],{},[31665,357257,356112],{"encoding":31667},[34,357259,357261],{"className":357260,"ariaHidden":758},[31672],[34,357262,357264,357267],{"className":357263},[31676],[34,357265],{"className":357266,"style":70915},[31680],[34,357268,357270,357273],{"className":357269},[31685],[34,357271,105852],{"className":357272,"style":37422},[31685,31686],[34,357274,357276],{"className":357275},[31792],[34,357277,357279,357299],{"className":357278},[31796,31797],[34,357280,357282,357296],{"className":357281},[31801],[34,357283,357285],{"className":357284,"style":33833},[31805],[34,357286,357287,357290],{"style":214424},[34,357288],{"className":357289,"style":31814},[31813],[34,357291,357293],{"className":357292},[31818,31819,31820,31821],[34,357294,33759],{"className":357295,"style":33774},[31685,31686,31821],[34,357297,31829],{"className":357298},[31828],[34,357300,357302],{"className":357301},[31801],[34,357303,357305],{"className":357304,"style":31836},[31805],[34,357306],{}," 里吗？通常是的（否则我们就直接找到了既不在 ",[34,357309,357311,357328],{"className":357310},[31642],[34,357312,357314],{"className":357313},[31646],[31648,357315,357316],{"xmlns":31650},[31652,357317,357318,357326],{},[31655,357319,357320],{},[31743,357321,357322,357324],{},[31658,357323,105852],{},[31658,357325,33759],{},[31665,357327,356112],{"encoding":31667},[34,357329,357331],{"className":357330,"ariaHidden":758},[31672],[34,357332,357334,357337],{"className":357333},[31676],[34,357335],{"className":357336,"style":70915},[31680],[34,357338,357340,357343],{"className":357339},[31685],[34,357341,105852],{"className":357342,"style":37422},[31685,31686],[34,357344,357346],{"className":357345},[31792],[34,357347,357349,357369],{"className":357348},[31796,31797],[34,357350,357352,357366],{"className":357351},[31801],[34,357353,357355],{"className":357354,"style":33833},[31805],[34,357356,357357,357360],{"style":214424},[34,357358],{"className":357359,"style":31814},[31813],[34,357361,357363],{"className":357362},[31818,31819,31820,31821],[34,357364,33759],{"className":357365,"style":33774},[31685,31686,31821],[34,357367,31829],{"className":357368},[31828],[34,357370,357372],{"className":357371},[31801],[34,357373,357375],{"className":357374,"style":31836},[31805],[34,357376],{}," 也不在其他 ",[34,357379,357381,357399],{"className":357380},[31642],[34,357382,357384],{"className":357383},[31646],[31648,357385,357386],{"xmlns":31650},[31652,357387,357388,357396],{},[31655,357389,357390],{},[31743,357391,357392,357394],{},[31658,357393,105852],{},[31658,357395,31968],{},[31665,357397,357398],{"encoding":31667},"P_j",[34,357400,357402],{"className":357401,"ariaHidden":758},[31672],[34,357403,357405,357408],{"className":357404},[31676],[34,357406],{"className":357407,"style":55484},[31680],[34,357409,357411,357414],{"className":357410},[31685],[34,357412,105852],{"className":357413,"style":37422},[31685,31686],[34,357415,357417],{"className":357416},[31792],[34,357418,357420,357440],{"className":357419},[31796,31797],[34,357421,357423,357437],{"className":357422},[31801],[34,357424,357426],{"className":357425,"style":32090},[31805],[34,357427,357428,357431],{"style":214424},[34,357429],{"className":357430,"style":31814},[31813],[34,357432,357434],{"className":357433},[31818,31819,31820,31821],[34,357435,31968],{"className":357436,"style":32153},[31685,31686,31821],[34,357438,31829],{"className":357439},[31828],[34,357441,357443],{"className":357442},[31801],[34,357444,357446],{"className":357445,"style":32163},[31805],[34,357447],{}," 的元素）。\n",[2627,357450,357451],{},"关键构造","：\n我们要构造一个“终极元素” ",[34,357454,357456,357469],{"className":357455},[31642],[34,357457,357459],{"className":357458},[31646],[31648,357460,357461],{"xmlns":31650},[31652,357462,357463,357467],{},[31655,357464,357465],{},[31658,357466,67764],{},[31665,357468,67764],{"encoding":31667},[34,357470,357472],{"className":357471,"ariaHidden":758},[31672],[34,357473,357475,357478],{"className":357474},[31676],[34,357476],{"className":357477,"style":38774},[31680],[34,357479,67764],{"className":357480,"style":32062},[31685,31686],"。\n对于每个 ",[34,357483,357485,357498],{"className":357484},[31642],[34,357486,357488],{"className":357487},[31646],[31648,357489,357490],{"xmlns":31650},[31652,357491,357492,357496],{},[31655,357493,357494],{},[31658,357495,33759],{},[31665,357497,33759],{"encoding":31667},[34,357499,357501],{"className":357500,"ariaHidden":758},[31672],[34,357502,357504,357507],{"className":357503},[31676],[34,357505],{"className":357506,"style":31681},[31680],[34,357508,33759],{"className":357509,"style":33774},[31685,31686],"，考虑元素 ",[34,357512,357514,357531],{"className":357513},[31642],[34,357515,357517],{"className":357516},[31646],[31648,357518,357519],{"xmlns":31650},[31652,357520,357521,357529],{},[31655,357522,357523],{},[31743,357524,357525,357527],{},[31658,357526,32936],{},[31658,357528,33759],{},[31665,357530,356613],{"encoding":31667},[34,357532,357534],{"className":357533,"ariaHidden":758},[31672],[34,357535,357537,357540],{"className":357536},[31676],[34,357538],{"className":357539,"style":31860},[31680],[34,357541,357543,357546],{"className":357542},[31685],[34,357544,32936],{"className":357545},[31685,31686],[34,357547,357549],{"className":357548},[31792],[34,357550,357552,357572],{"className":357551},[31796,31797],[34,357553,357555,357569],{"className":357554},[31801],[34,357556,357558],{"className":357557,"style":33833},[31805],[34,357559,357560,357563],{"style":31809},[34,357561],{"className":357562,"style":31814},[31813],[34,357564,357566],{"className":357565},[31818,31819,31820,31821],[34,357567,33759],{"className":357568,"style":33774},[31685,31686,31821],[34,357570,31829],{"className":357571},[31828],[34,357573,357575],{"className":357574},[31801],[34,357576,357578],{"className":357577,"style":31836},[31805],[34,357579],{},"。我们知道 ",[34,357582,357584,357606],{"className":357583},[31642],[34,357585,357587],{"className":357586},[31646],[31648,357588,357589],{"xmlns":31650},[31652,357590,357591,357603],{},[31655,357592,357593,357599,357601],{},[31743,357594,357595,357597],{},[31658,357596,32936],{},[31658,357598,33759],{},[31662,357600,112296],{},[31658,357602,106063],{},[31665,357604,357605],{"encoding":31667},"x_k \\in I",[34,357607,357609,357664],{"className":357608,"ariaHidden":758},[31672],[34,357610,357612,357615,357655,357658,357661],{"className":357611},[31676],[34,357613],{"className":357614,"style":225773},[31680],[34,357616,357618,357621],{"className":357617},[31685],[34,357619,32936],{"className":357620},[31685,31686],[34,357622,357624],{"className":357623},[31792],[34,357625,357627,357647],{"className":357626},[31796,31797],[34,357628,357630,357644],{"className":357629},[31801],[34,357631,357633],{"className":357632,"style":33833},[31805],[34,357634,357635,357638],{"style":31809},[34,357636],{"className":357637,"style":31814},[31813],[34,357639,357641],{"className":357640},[31818,31819,31820,31821],[34,357642,33759],{"className":357643,"style":33774},[31685,31686,31821],[34,357645,31829],{"className":357646},[31828],[34,357648,357650],{"className":357649},[31801],[34,357651,357653],{"className":357652,"style":31836},[31805],[34,357654],{},[34,357656],{"className":357657,"style":31846},[31845],[34,357659,112296],{"className":357660},[31850],[34,357662],{"className":357663,"style":31846},[31845],[34,357665,357667,357670],{"className":357666},[31676],[34,357668],{"className":357669,"style":33218},[31680],[34,357671,106063],{"className":357672,"style":67482},[31685,31686],[34,357674,357676,357693],{"className":357675},[31642],[34,357677,357679],{"className":357678},[31646],[31648,357680,357681],{"xmlns":31650},[31652,357682,357683,357691],{},[31655,357684,357685],{},[31743,357686,357687,357689],{},[31658,357688,32936],{},[31658,357690,33759],{},[31665,357692,356613],{"encoding":31667},[34,357694,357696],{"className":357695,"ariaHidden":758},[31672],[34,357697,357699,357702],{"className":357698},[31676],[34,357700],{"className":357701,"style":31860},[31680],[34,357703,357705,357708],{"className":357704},[31685],[34,357706,32936],{"className":357707},[31685,31686],[34,357709,357711],{"className":357710},[31792],[34,357712,357714,357734],{"className":357713},[31796,31797],[34,357715,357717,357731],{"className":357716},[31801],[34,357718,357720],{"className":357719,"style":33833},[31805],[34,357721,357722,357725],{"style":31809},[34,357723],{"className":357724,"style":31814},[31813],[34,357726,357728],{"className":357727},[31818,31819,31820,31821],[34,357729,33759],{"className":357730,"style":33774},[31685,31686,31821],[34,357732,31829],{"className":357733},[31828],[34,357735,357737],{"className":357736},[31801],[34,357738,357740],{"className":357739,"style":31836},[31805],[34,357741],{}," 避开了除了 ",[34,357744,357746,357763],{"className":357745},[31642],[34,357747,357749],{"className":357748},[31646],[31648,357750,357751],{"xmlns":31650},[31652,357752,357753,357761],{},[31655,357754,357755],{},[31743,357756,357757,357759],{},[31658,357758,105852],{},[31658,357760,33759],{},[31665,357762,356112],{"encoding":31667},[34,357764,357766],{"className":357765,"ariaHidden":758},[31672],[34,357767,357769,357772],{"className":357768},[31676],[34,357770],{"className":357771,"style":70915},[31680],[34,357773,357775,357778],{"className":357774},[31685],[34,357776,105852],{"className":357777,"style":37422},[31685,31686],[34,357779,357781],{"className":357780},[31792],[34,357782,357784,357804],{"className":357783},[31796,31797],[34,357785,357787,357801],{"className":357786},[31801],[34,357788,357790],{"className":357789,"style":33833},[31805],[34,357791,357792,357795],{"style":214424},[34,357793],{"className":357794,"style":31814},[31813],[34,357796,357798],{"className":357797},[31818,31819,31820,31821],[34,357799,33759],{"className":357800,"style":33774},[31685,31686,31821],[34,357802,31829],{"className":357803},[31828],[34,357805,357807],{"className":357806},[31801],[34,357808,357810],{"className":357809,"style":31836},[31805],[34,357811],{}," 以外的所有素理想。\n令 ",[34,357814,357816,357860],{"className":357815},[31642],[34,357817,357819],{"className":357818},[31646],[31648,357820,357821],{"xmlns":31650},[31652,357822,357823,357857],{},[31655,357824,357825,357827,357829,357835,357837,357843,357849,357851],{},[31658,357826,67764],{},[31662,357828,202],{},[31743,357830,357831,357833],{},[31658,357832,32936],{},[31749,357834,640],{},[31662,357836,223],{},[31743,357838,357839,357841],{},[31658,357840,32936],{},[31749,357842,768],{},[31743,357844,357845,357847],{},[31658,357846,32936],{},[31749,357848,786],{},[31662,357850,33934],{},[31743,357852,357853,357855],{},[31658,357854,32936],{},[31658,357856,31660],{},[31665,357858,357859],{"encoding":31667},"y = x_1 + x_2 x_3 \\dots x_n",[34,357861,357863,357881,357936],{"className":357862,"ariaHidden":758},[31672],[34,357864,357866,357869,357872,357875,357878],{"className":357865},[31676],[34,357867],{"className":357868,"style":38774},[31680],[34,357870,67764],{"className":357871,"style":32062},[31685,31686],[34,357873],{"className":357874,"style":31846},[31845],[34,357876,202],{"className":357877},[31850],[34,357879],{"className":357880,"style":31846},[31845],[34,357882,357884,357887,357927,357930,357933],{"className":357883},[31676],[34,357885],{"className":357886,"style":40522},[31680],[34,357888,357890,357893],{"className":357889},[31685],[34,357891,32936],{"className":357892},[31685,31686],[34,357894,357896],{"className":357895},[31792],[34,357897,357899,357919],{"className":357898},[31796,31797],[34,357900,357902,357916],{"className":357901},[31801],[34,357903,357905],{"className":357904,"style":31806},[31805],[34,357906,357907,357910],{"style":31809},[34,357908],{"className":357909,"style":31814},[31813],[34,357911,357913],{"className":357912},[31818,31819,31820,31821],[34,357914,640],{"className":357915},[31685,31821],[34,357917,31829],{"className":357918},[31828],[34,357920,357922],{"className":357921},[31801],[34,357923,357925],{"className":357924,"style":31836},[31805],[34,357926],{},[34,357928],{"className":357929,"style":32604},[31845],[34,357931,223],{"className":357932},[32024],[34,357934],{"className":357935,"style":32604},[31845],[34,357937,357939,357942,357982,358022,358025,358028,358031],{"className":357938},[31676],[34,357940],{"className":357941,"style":31860},[31680],[34,357943,357945,357948],{"className":357944},[31685],[34,357946,32936],{"className":357947},[31685,31686],[34,357949,357951],{"className":357950},[31792],[34,357952,357954,357974],{"className":357953},[31796,31797],[34,357955,357957,357971],{"className":357956},[31801],[34,357958,357960],{"className":357959,"style":31806},[31805],[34,357961,357962,357965],{"style":31809},[34,357963],{"className":357964,"style":31814},[31813],[34,357966,357968],{"className":357967},[31818,31819,31820,31821],[34,357969,768],{"className":357970},[31685,31821],[34,357972,31829],{"className":357973},[31828],[34,357975,357977],{"className":357976},[31801],[34,357978,357980],{"className":357979,"style":31836},[31805],[34,357981],{},[34,357983,357985,357988],{"className":357984},[31685],[34,357986,32936],{"className":357987},[31685,31686],[34,357989,357991],{"className":357990},[31792],[34,357992,357994,358014],{"className":357993},[31796,31797],[34,357995,357997,358011],{"className":357996},[31801],[34,357998,358000],{"className":357999,"style":31806},[31805],[34,358001,358002,358005],{"style":31809},[34,358003],{"className":358004,"style":31814},[31813],[34,358006,358008],{"className":358007},[31818,31819,31820,31821],[34,358009,786],{"className":358010},[31685,31821],[34,358012,31829],{"className":358013},[31828],[34,358015,358017],{"className":358016},[31801],[34,358018,358020],{"className":358019,"style":31836},[31805],[34,358021],{},[34,358023],{"className":358024,"style":32123},[31845],[34,358026,33934],{"className":358027},[34006],[34,358029],{"className":358030,"style":32123},[31845],[34,358032,358034,358037],{"className":358033},[31685],[34,358035,32936],{"className":358036},[31685,31686],[34,358038,358040],{"className":358039},[31792],[34,358041,358043,358063],{"className":358042},[31796,31797],[34,358044,358046,358060],{"className":358045},[31801],[34,358047,358049],{"className":358048,"style":32362},[31805],[34,358050,358051,358054],{"style":31809},[34,358052],{"className":358053,"style":31814},[31813],[34,358055,358057],{"className":358056},[31818,31819,31820,31821],[34,358058,31660],{"className":358059},[31685,31686,31821],[34,358061,31829],{"className":358062},[31828],[34,358064,358066],{"className":358065},[31801],[34,358067,358069],{"className":358068,"style":31836},[31805],[34,358070],{},"（注意：第一项是 ",[34,358073,358075,358092],{"className":358074},[31642],[34,358076,358078],{"className":358077},[31646],[31648,358079,358080],{"xmlns":31650},[31652,358081,358082,358090],{},[31655,358083,358084],{},[31743,358085,358086,358088],{},[31658,358087,32936],{},[31749,358089,640],{},[31665,358091,224014],{"encoding":31667},[34,358093,358095],{"className":358094,"ariaHidden":758},[31672],[34,358096,358098,358101],{"className":358097},[31676],[34,358099],{"className":358100,"style":31860},[31680],[34,358102,358104,358107],{"className":358103},[31685],[34,358105,32936],{"className":358106},[31685,31686],[34,358108,358110],{"className":358109},[31792],[34,358111,358113,358133],{"className":358112},[31796,31797],[34,358114,358116,358130],{"className":358115},[31801],[34,358117,358119],{"className":358118,"style":31806},[31805],[34,358120,358121,358124],{"style":31809},[34,358122],{"className":358123,"style":31814},[31813],[34,358125,358127],{"className":358126},[31818,31819,31820,31821],[34,358128,640],{"className":358129},[31685,31821],[34,358131,31829],{"className":358132},[31828],[34,358134,358136],{"className":358135},[31801],[34,358137,358139],{"className":358138,"style":31836},[31805],[34,358140],{},"，第二项是除了 ",[34,358143,358145,358162],{"className":358144},[31642],[34,358146,358148],{"className":358147},[31646],[31648,358149,358150],{"xmlns":31650},[31652,358151,358152,358160],{},[31655,358153,358154],{},[31743,358155,358156,358158],{},[31658,358157,32936],{},[31749,358159,640],{},[31665,358161,224014],{"encoding":31667},[34,358163,358165],{"className":358164,"ariaHidden":758},[31672],[34,358166,358168,358171],{"className":358167},[31676],[34,358169],{"className":358170,"style":31860},[31680],[34,358172,358174,358177],{"className":358173},[31685],[34,358175,32936],{"className":358176},[31685,31686],[34,358178,358180],{"className":358179},[31792],[34,358181,358183,358203],{"className":358182},[31796,31797],[34,358184,358186,358200],{"className":358185},[31801],[34,358187,358189],{"className":358188,"style":31806},[31805],[34,358190,358191,358194],{"style":31809},[34,358192],{"className":358193,"style":31814},[31813],[34,358195,358197],{"className":358196},[31818,31819,31820,31821],[34,358198,640],{"className":358199},[31685,31821],[34,358201,31829],{"className":358202},[31828],[34,358204,358206],{"className":358205},[31801],[34,358207,358209],{"className":358208,"style":31836},[31805],[34,358210],{}," 以外所有元素的积）。\n",[71807,358213,358214],{},"这里我们需要调整一下构造以确保更严谨（经典的 Prime Avoidance 构造）：",[2627,358216,358217],{},"更严谨的构造","：\n对每个 ",[34,358220,358222,358235],{"className":358221},[31642],[34,358223,358225],{"className":358224},[31646],[31648,358226,358227],{"xmlns":31650},[31652,358228,358229,358233],{},[31655,358230,358231],{},[31658,358232,31955],{},[31665,358234,31955],{"encoding":31667},[34,358236,358238],{"className":358237,"ariaHidden":758},[31672],[34,358239,358241,358244],{"className":358240},[31676],[34,358242],{"className":358243,"style":33104},[31680],[34,358245,31955],{"className":358246},[31685,31686],"，我们知道 ",[34,358249,358251,358285],{"className":358250},[31642],[34,358252,358254],{"className":358253},[31646],[31648,358255,358256],{"xmlns":31650},[31652,358257,358258,358282],{},[31655,358259,358260,358262,358264,358276],{},[31658,358261,106063],{},[31662,358263,352281],{},[31743,358265,358266,358268],{},[31662,358267,351927],{},[31655,358269,358270,358272,358274],{},[31658,358271,31968],{},[31662,358273,54707],{"mathvariant":38079},[31658,358275,31955],{},[31743,358277,358278,358280],{},[31658,358279,105852],{},[31658,358281,31968],{},[31665,358283,358284],{"encoding":31667},"I \\not\\subseteq \\bigcup_{j \\ne i} P_j",[34,358286,358288,358336],{"className":358287,"ariaHidden":758},[31672],[34,358289,358291,358294,358297,358300,358327,358330,358333],{"className":358290},[31676],[34,358292],{"className":358293,"style":43217},[31680],[34,358295,106063],{"className":358296,"style":67482},[31685,31686],[34,358298],{"className":358299,"style":31846},[31845],[34,358301,358303],{"className":358302},[31850],[34,358304,358306],{"className":358305},[31685,54774],[34,358307,358309],{"className":358308},[54778],[34,358310,358312,358315,358324],{"className":358311},[54782],[34,358313],{"className":358314,"style":43217},[31680],[34,358316,358318],{"className":358317},[54789],[34,358319,358321],{"className":358320},[31685],[34,358322,54796],{"className":358323},[31850],[34,358325],{"className":358326},[54800],[34,358328],{"className":358329},[31845,54804],[34,358331,131855],{"className":358332},[31850],[34,358334],{"className":358335,"style":31846},[31845],[34,358337,358339,358342,358424,358427],{"className":358338},[31676],[34,358340],{"className":358341,"style":156901},[31680],[34,358343,358345,358348],{"className":358344},[32767],[34,358346,351927],{"className":358347,"style":156561},[32767,131249,156560],[34,358349,358351],{"className":358350},[31792],[34,358352,358354,358416],{"className":358353},[31796,31797],[34,358355,358357,358413],{"className":358356},[31801],[34,358358,358360],{"className":358359,"style":356484},[31805],[34,358361,358362,358365],{"style":156577},[34,358363],{"className":358364,"style":31814},[31813],[34,358366,358368],{"className":358367},[31818,31819,31820,31821],[34,358369,358371,358374,358410],{"className":358370},[31685,31821],[34,358372,31968],{"className":358373,"style":32153},[31685,31686,31821],[34,358375,358377,358404,358407],{"className":358376},[31850,31821],[34,358378,358380],{"className":358379},[31850,31821],[34,358381,358383],{"className":358382},[31685,54774,31821],[34,358384,358386],{"className":358385},[54778,31821],[34,358387,358389,358392,358401],{"className":358388},[54782,31821],[34,358390],{"className":358391,"style":43217},[31680],[34,358393,358395],{"className":358394},[54789],[34,358396,358398],{"className":358397},[31685,31821],[34,358399,54796],{"className":358400},[31850,31821],[34,358402],{"className":358403},[54800],[34,358405],{"className":358406},[31845,54804,31821],[34,358408,202],{"className":358409},[31850,31821],[34,358411,31955],{"className":358412},[31685,31686,31821],[34,358414,31829],{"className":358415},[31828],[34,358417,358419],{"className":358418},[31801],[34,358420,358422],{"className":358421,"style":156941},[31805],[34,358423],{},[34,358425],{"className":358426,"style":32123},[31845],[34,358428,358430,358433],{"className":358429},[31685],[34,358431,105852],{"className":358432,"style":37422},[31685,31686],[34,358434,358436],{"className":358435},[31792],[34,358437,358439,358459],{"className":358438},[31796,31797],[34,358440,358442,358456],{"className":358441},[31801],[34,358443,358445],{"className":358444,"style":32090},[31805],[34,358446,358447,358450],{"style":214424},[34,358448],{"className":358449,"style":31814},[31813],[34,358451,358453],{"className":358452},[31818,31819,31820,31821],[34,358454,31968],{"className":358455,"style":32153},[31685,31686,31821],[34,358457,31829],{"className":358458},[31828],[34,358460,358462],{"className":358461},[31801],[34,358463,358465],{"className":358464,"style":32163},[31805],[34,358466],{},"。所以存在 ",[34,358469,358471,358493],{"className":358470},[31642],[34,358472,358474],{"className":358473},[31646],[31648,358475,358476],{"xmlns":31650},[31652,358477,358478,358490],{},[31655,358479,358480,358486,358488],{},[31743,358481,358482,358484],{},[31658,358483,111],{},[31658,358485,31955],{},[31662,358487,112296],{},[31658,358489,106063],{},[31665,358491,358492],{"encoding":31667},"a_i \\in I",[34,358494,358496,358551],{"className":358495,"ariaHidden":758},[31672],[34,358497,358499,358502,358542,358545,358548],{"className":358498},[31676],[34,358500],{"className":358501,"style":225773},[31680],[34,358503,358505,358508],{"className":358504},[31685],[34,358506,111],{"className":358507},[31685,31686],[34,358509,358511],{"className":358510},[31792],[34,358512,358514,358534],{"className":358513},[31796,31797],[34,358515,358517,358531],{"className":358516},[31801],[34,358518,358520],{"className":358519,"style":32090},[31805],[34,358521,358522,358525],{"style":31809},[34,358523],{"className":358524,"style":31814},[31813],[34,358526,358528],{"className":358527},[31818,31819,31820,31821],[34,358529,31955],{"className":358530},[31685,31686,31821],[34,358532,31829],{"className":358533},[31828],[34,358535,358537],{"className":358536},[31801],[34,358538,358540],{"className":358539,"style":31836},[31805],[34,358541],{},[34,358543],{"className":358544,"style":31846},[31845],[34,358546,112296],{"className":358547},[31850],[34,358549],{"className":358550,"style":31846},[31845],[34,358552,358554,358557],{"className":358553},[31676],[34,358555],{"className":358556,"style":33218},[31680],[34,358558,106063],{"className":358559,"style":67482},[31685,31686],[34,358561,358563,358589],{"className":358562},[31642],[34,358564,358566],{"className":358565},[31646],[31648,358567,358568],{"xmlns":31650},[31652,358569,358570,358586],{},[31655,358571,358572,358578,358580],{},[31743,358573,358574,358576],{},[31658,358575,111],{},[31658,358577,31955],{},[31662,358579,112266],{"mathvariant":38079},[31743,358581,358582,358584],{},[31658,358583,105852],{},[31658,358585,31968],{},[31665,358587,358588],{"encoding":31667},"a_i \\notin P_j",[34,358590,358592,358680],{"className":358591,"ariaHidden":758},[31672],[34,358593,358595,358598,358638,358641,358677],{"className":358594},[31676],[34,358596],{"className":358597,"style":31774},[31680],[34,358599,358601,358604],{"className":358600},[31685],[34,358602,111],{"className":358603},[31685,31686],[34,358605,358607],{"className":358606},[31792],[34,358608,358610,358630],{"className":358609},[31796,31797],[34,358611,358613,358627],{"className":358612},[31801],[34,358614,358616],{"className":358615,"style":32090},[31805],[34,358617,358618,358621],{"style":31809},[34,358619],{"className":358620,"style":31814},[31813],[34,358622,358624],{"className":358623},[31818,31819,31820,31821],[34,358625,31955],{"className":358626},[31685,31686,31821],[34,358628,31829],{"className":358629},[31828],[34,358631,358633],{"className":358632},[31801],[34,358634,358636],{"className":358635,"style":31836},[31805],[34,358637],{},[34,358639],{"className":358640,"style":31846},[31845],[34,358642,358644,358650],{"className":358643},[31850],[34,358645,358647],{"className":358646},[31685],[34,358648,112296],{"className":358649},[31850],[34,358651,358653],{"className":358652},[31685,54774],[34,358654,358656],{"className":358655},[54778],[34,358657,358659,358662,358674],{"className":358658},[112306],[34,358660],{"className":358661,"style":31774},[31680],[34,358663,358665],{"className":358664},[54789],[34,358666,358668,358671],{"className":358667},[31685],[34,358669,38080],{"className":358670},[31685],[34,358672],{"className":358673,"style":112322},[31845],[34,358675],{"className":358676},[54800],[34,358678],{"className":358679,"style":31846},[31845],[34,358681,358683,358686],{"className":358682},[31676],[34,358684],{"className":358685,"style":55484},[31680],[34,358687,358689,358692],{"className":358688},[31685],[34,358690,105852],{"className":358691,"style":37422},[31685,31686],[34,358693,358695],{"className":358694},[31792],[34,358696,358698,358718],{"className":358697},[31796,31797],[34,358699,358701,358715],{"className":358700},[31801],[34,358702,358704],{"className":358703,"style":32090},[31805],[34,358705,358706,358709],{"style":214424},[34,358707],{"className":358708,"style":31814},[31813],[34,358710,358712],{"className":358711},[31818,31819,31820,31821],[34,358713,31968],{"className":358714,"style":32153},[31685,31686,31821],[34,358716,31829],{"className":358717},[31828],[34,358719,358721],{"className":358720},[31801],[34,358722,358724],{"className":358723,"style":32163},[31805],[34,358725],{},[34,358727,358729,358747],{"className":358728},[31642],[34,358730,358732],{"className":358731},[31646],[31648,358733,358734],{"xmlns":31650},[31652,358735,358736,358744],{},[31655,358737,358738,358740,358742],{},[31658,358739,31968],{},[31662,358741,54707],{"mathvariant":38079},[31658,358743,31955],{},[31665,358745,358746],{"encoding":31667},"j \\ne i",[34,358748,358750,358801],{"className":358749,"ariaHidden":758},[31672],[34,358751,358753,358756,358759,358762,358798],{"className":358752},[31676],[34,358754],{"className":358755,"style":43217},[31680],[34,358757,31968],{"className":358758,"style":32153},[31685,31686],[34,358760],{"className":358761,"style":31846},[31845],[34,358763,358765,358792,358795],{"className":358764},[31850],[34,358766,358768],{"className":358767},[31850],[34,358769,358771],{"className":358770},[31685,54774],[34,358772,358774],{"className":358773},[54778],[34,358775,358777,358780,358789],{"className":358776},[54782],[34,358778],{"className":358779,"style":43217},[31680],[34,358781,358783],{"className":358782},[54789],[34,358784,358786],{"className":358785},[31685],[34,358787,54796],{"className":358788},[31850],[34,358790],{"className":358791},[54800],[34,358793],{"className":358794},[31845,54804],[34,358796,202],{"className":358797},[31850],[34,358799],{"className":358800,"style":31846},[31845],[34,358802,358804,358807],{"className":358803},[31676],[34,358805],{"className":358806,"style":33104},[31680],[34,358808,31955],{"className":358809},[31685,31686]," 成立。\n如果对于某个 ",[34,358812,358814,358827],{"className":358813},[31642],[34,358815,358817],{"className":358816},[31646],[31648,358818,358819],{"xmlns":31650},[31652,358820,358821,358825],{},[31655,358822,358823],{},[31658,358824,31955],{},[31665,358826,31955],{"encoding":31667},[34,358828,358830],{"className":358829,"ariaHidden":758},[31672],[34,358831,358833,358836],{"className":358832},[31676],[34,358834],{"className":358835,"style":33104},[31680],[34,358837,31955],{"className":358838},[31685,31686],"，这个 ",[34,358841,358843,358861],{"className":358842},[31642],[34,358844,358846],{"className":358845},[31646],[31648,358847,358848],{"xmlns":31650},[31652,358849,358850,358858],{},[31655,358851,358852],{},[31743,358853,358854,358856],{},[31658,358855,111],{},[31658,358857,31955],{},[31665,358859,358860],{"encoding":31667},"a_i",[34,358862,358864],{"className":358863,"ariaHidden":758},[31672],[34,358865,358867,358870],{"className":358866},[31676],[34,358868],{"className":358869,"style":31860},[31680],[34,358871,358873,358876],{"className":358872},[31685],[34,358874,111],{"className":358875},[31685,31686],[34,358877,358879],{"className":358878},[31792],[34,358880,358882,358902],{"className":358881},[31796,31797],[34,358883,358885,358899],{"className":358884},[31801],[34,358886,358888],{"className":358887,"style":32090},[31805],[34,358889,358890,358893],{"style":31809},[34,358891],{"className":358892,"style":31814},[31813],[34,358894,358896],{"className":358895},[31818,31819,31820,31821],[34,358897,31955],{"className":358898},[31685,31686,31821],[34,358900,31829],{"className":358901},[31828],[34,358903,358905],{"className":358904},[31801],[34,358906,358908],{"className":358907,"style":31836},[31805],[34,358909],{}," 恰好也不在 ",[34,358912,358914,358931],{"className":358913},[31642],[34,358915,358917],{"className":358916},[31646],[31648,358918,358919],{"xmlns":31650},[31652,358920,358921,358929],{},[31655,358922,358923],{},[31743,358924,358925,358927],{},[31658,358926,105852],{},[31658,358928,31955],{},[31665,358930,351715],{"encoding":31667},[34,358932,358934],{"className":358933,"ariaHidden":758},[31672],[34,358935,358937,358940],{"className":358936},[31676],[34,358938],{"className":358939,"style":70915},[31680],[34,358941,358943,358946],{"className":358942},[31685],[34,358944,105852],{"className":358945,"style":37422},[31685,31686],[34,358947,358949],{"className":358948},[31792],[34,358950,358952,358972],{"className":358951},[31796,31797],[34,358953,358955,358969],{"className":358954},[31801],[34,358956,358958],{"className":358957,"style":32090},[31805],[34,358959,358960,358963],{"style":214424},[34,358961],{"className":358962,"style":31814},[31813],[34,358964,358966],{"className":358965},[31818,31819,31820,31821],[34,358967,31955],{"className":358968},[31685,31686,31821],[34,358970,31829],{"className":358971},[31828],[34,358973,358975],{"className":358974},[31801],[34,358976,358978],{"className":358977,"style":31836},[31805],[34,358979],{}," 里，那 ",[34,358982,358984,359001],{"className":358983},[31642],[34,358985,358987],{"className":358986},[31646],[31648,358988,358989],{"xmlns":31650},[31652,358990,358991,358999],{},[31655,358992,358993],{},[31743,358994,358995,358997],{},[31658,358996,111],{},[31658,358998,31955],{},[31665,359000,358860],{"encoding":31667},[34,359002,359004],{"className":359003,"ariaHidden":758},[31672],[34,359005,359007,359010],{"className":359006},[31676],[34,359008],{"className":359009,"style":31860},[31680],[34,359011,359013,359016],{"className":359012},[31685],[34,359014,111],{"className":359015},[31685,31686],[34,359017,359019],{"className":359018},[31792],[34,359020,359022,359042],{"className":359021},[31796,31797],[34,359023,359025,359039],{"className":359024},[31801],[34,359026,359028],{"className":359027,"style":32090},[31805],[34,359029,359030,359033],{"style":31809},[34,359031],{"className":359032,"style":31814},[31813],[34,359034,359036],{"className":359035},[31818,31819,31820,31821],[34,359037,31955],{"className":359038},[31685,31686,31821],[34,359040,31829],{"className":359041},[31828],[34,359043,359045],{"className":359044},[31801],[34,359046,359048],{"className":359047,"style":31836},[31805],[34,359049],{}," 就是我们要找的元素（它不在任何 ",[34,359052,359054,359067],{"className":359053},[31642],[34,359055,359057],{"className":359056},[31646],[31648,359058,359059],{"xmlns":31650},[31652,359060,359061,359065],{},[31655,359062,359063],{},[31658,359064,105852],{},[31665,359066,105852],{"encoding":31667},[34,359068,359070],{"className":359069,"ariaHidden":758},[31672],[34,359071,359073,359076],{"className":359072},[31676],[34,359074],{"className":359075,"style":33218},[31680],[34,359077,105852],{"className":359078,"style":37422},[31685,31686]," 里），矛盾得证。\n所以，我们假设 ",[34,359081,359083,359109],{"className":359082},[31642],[34,359084,359086],{"className":359085},[31646],[31648,359087,359088],{"xmlns":31650},[31652,359089,359090,359106],{},[31655,359091,359092,359098,359100],{},[31743,359093,359094,359096],{},[31658,359095,111],{},[31658,359097,31955],{},[31662,359099,112296],{},[31743,359101,359102,359104],{},[31658,359103,105852],{},[31658,359105,31955],{},[31665,359107,359108],{"encoding":31667},"a_i \\in P_i",[34,359110,359112,359167],{"className":359111,"ariaHidden":758},[31672],[34,359113,359115,359118,359158,359161,359164],{"className":359114},[31676],[34,359116],{"className":359117,"style":225773},[31680],[34,359119,359121,359124],{"className":359120},[31685],[34,359122,111],{"className":359123},[31685,31686],[34,359125,359127],{"className":359126},[31792],[34,359128,359130,359150],{"className":359129},[31796,31797],[34,359131,359133,359147],{"className":359132},[31801],[34,359134,359136],{"className":359135,"style":32090},[31805],[34,359137,359138,359141],{"style":31809},[34,359139],{"className":359140,"style":31814},[31813],[34,359142,359144],{"className":359143},[31818,31819,31820,31821],[34,359145,31955],{"className":359146},[31685,31686,31821],[34,359148,31829],{"className":359149},[31828],[34,359151,359153],{"className":359152},[31801],[34,359154,359156],{"className":359155,"style":31836},[31805],[34,359157],{},[34,359159],{"className":359160,"style":31846},[31845],[34,359162,112296],{"className":359163},[31850],[34,359165],{"className":359166,"style":31846},[31845],[34,359168,359170,359173],{"className":359169},[31676],[34,359171],{"className":359172,"style":70915},[31680],[34,359174,359176,359179],{"className":359175},[31685],[34,359177,105852],{"className":359178,"style":37422},[31685,31686],[34,359180,359182],{"className":359181},[31792],[34,359183,359185,359205],{"className":359184},[31796,31797],[34,359186,359188,359202],{"className":359187},[31801],[34,359189,359191],{"className":359190,"style":32090},[31805],[34,359192,359193,359196],{"style":214424},[34,359194],{"className":359195,"style":31814},[31813],[34,359197,359199],{"className":359198},[31818,31819,31820,31821],[34,359200,31955],{"className":359201},[31685,31686,31821],[34,359203,31829],{"className":359204},[31828],[34,359206,359208],{"className":359207},[31801],[34,359209,359211],{"className":359210,"style":31836},[31805],[34,359212],{},[34,359214,359216,359229],{"className":359215},[31642],[34,359217,359219],{"className":359218},[31646],[31648,359220,359221],{"xmlns":31650},[31652,359222,359223,359227],{},[31655,359224,359225],{},[31658,359226,31955],{},[31665,359228,31955],{"encoding":31667},[34,359230,359232],{"className":359231,"ariaHidden":758},[31672],[34,359233,359235,359238],{"className":359234},[31676],[34,359236],{"className":359237,"style":33104},[31680],[34,359239,31955],{"className":359240},[31685,31686]," 成立。\n构造元素：\n",[34,359243,359245,359289],{"className":359244},[31642],[34,359246,359248],{"className":359247},[31646],[31648,359249,359250],{"xmlns":31650},[31652,359251,359252,359286],{},[31655,359253,359254,359256,359258,359264,359266,359272,359278,359280],{},[31658,359255,67764],{},[31662,359257,202],{},[31743,359259,359260,359262],{},[31658,359261,111],{},[31749,359263,640],{},[31662,359265,223],{},[31743,359267,359268,359270],{},[31658,359269,111],{},[31749,359271,768],{},[31743,359273,359274,359276],{},[31658,359275,111],{},[31749,359277,786],{},[31662,359279,33934],{},[31743,359281,359282,359284],{},[31658,359283,111],{},[31658,359285,31660],{},[31665,359287,359288],{"encoding":31667},"y = a_1 + a_2 a_3 \\dots a_n",[34,359290,359292,359310,359365],{"className":359291,"ariaHidden":758},[31672],[34,359293,359295,359298,359301,359304,359307],{"className":359294},[31676],[34,359296],{"className":359297,"style":38774},[31680],[34,359299,67764],{"className":359300,"style":32062},[31685,31686],[34,359302],{"className":359303,"style":31846},[31845],[34,359305,202],{"className":359306},[31850],[34,359308],{"className":359309,"style":31846},[31845],[34,359311,359313,359316,359356,359359,359362],{"className":359312},[31676],[34,359314],{"className":359315,"style":40522},[31680],[34,359317,359319,359322],{"className":359318},[31685],[34,359320,111],{"className":359321},[31685,31686],[34,359323,359325],{"className":359324},[31792],[34,359326,359328,359348],{"className":359327},[31796,31797],[34,359329,359331,359345],{"className":359330},[31801],[34,359332,359334],{"className":359333,"style":31806},[31805],[34,359335,359336,359339],{"style":31809},[34,359337],{"className":359338,"style":31814},[31813],[34,359340,359342],{"className":359341},[31818,31819,31820,31821],[34,359343,640],{"className":359344},[31685,31821],[34,359346,31829],{"className":359347},[31828],[34,359349,359351],{"className":359350},[31801],[34,359352,359354],{"className":359353,"style":31836},[31805],[34,359355],{},[34,359357],{"className":359358,"style":32604},[31845],[34,359360,223],{"className":359361},[32024],[34,359363],{"className":359364,"style":32604},[31845],[34,359366,359368,359371,359411,359451,359454,359457,359460],{"className":359367},[31676],[34,359369],{"className":359370,"style":31860},[31680],[34,359372,359374,359377],{"className":359373},[31685],[34,359375,111],{"className":359376},[31685,31686],[34,359378,359380],{"className":359379},[31792],[34,359381,359383,359403],{"className":359382},[31796,31797],[34,359384,359386,359400],{"className":359385},[31801],[34,359387,359389],{"className":359388,"style":31806},[31805],[34,359390,359391,359394],{"style":31809},[34,359392],{"className":359393,"style":31814},[31813],[34,359395,359397],{"className":359396},[31818,31819,31820,31821],[34,359398,768],{"className":359399},[31685,31821],[34,359401,31829],{"className":359402},[31828],[34,359404,359406],{"className":359405},[31801],[34,359407,359409],{"className":359408,"style":31836},[31805],[34,359410],{},[34,359412,359414,359417],{"className":359413},[31685],[34,359415,111],{"className":359416},[31685,31686],[34,359418,359420],{"className":359419},[31792],[34,359421,359423,359443],{"className":359422},[31796,31797],[34,359424,359426,359440],{"className":359425},[31801],[34,359427,359429],{"className":359428,"style":31806},[31805],[34,359430,359431,359434],{"style":31809},[34,359432],{"className":359433,"style":31814},[31813],[34,359435,359437],{"className":359436},[31818,31819,31820,31821],[34,359438,786],{"className":359439},[31685,31821],[34,359441,31829],{"className":359442},[31828],[34,359444,359446],{"className":359445},[31801],[34,359447,359449],{"className":359448,"style":31836},[31805],[34,359450],{},[34,359452],{"className":359453,"style":32123},[31845],[34,359455,33934],{"className":359456},[34006],[34,359458],{"className":359459,"style":32123},[31845],[34,359461,359463,359466],{"className":359462},[31685],[34,359464,111],{"className":359465},[31685,31686],[34,359467,359469],{"className":359468},[31792],[34,359470,359472,359492],{"className":359471},[31796,31797],[34,359473,359475,359489],{"className":359474},[31801],[34,359476,359478],{"className":359477,"style":32362},[31805],[34,359479,359480,359483],{"style":31809},[34,359481],{"className":359482,"style":31814},[31813],[34,359484,359486],{"className":359485},[31818,31819,31820,31821],[34,359487,31660],{"className":359488},[31685,31686,31821],[34,359490,31829],{"className":359491},[31828],[34,359493,359495],{"className":359494},[31801],[34,359496,359498],{"className":359497,"style":31836},[31805],[34,359499],{},"\n让我们检查 ",[34,359502,359504,359517],{"className":359503},[31642],[34,359505,359507],{"className":359506},[31646],[31648,359508,359509],{"xmlns":31650},[31652,359510,359511,359515],{},[31655,359512,359513],{},[31658,359514,67764],{},[31665,359516,67764],{"encoding":31667},[34,359518,359520],{"className":359519,"ariaHidden":758},[31672],[34,359521,359523,359526],{"className":359522},[31676],[34,359524],{"className":359525,"style":38774},[31680],[34,359527,67764],{"className":359528,"style":32062},[31685,31686]," 是否在某个 ",[34,359531,359533,359550],{"className":359532},[31642],[34,359534,359536],{"className":359535},[31646],[31648,359537,359538],{"xmlns":31650},[31652,359539,359540,359548],{},[31655,359541,359542],{},[31743,359543,359544,359546],{},[31658,359545,105852],{},[31658,359547,33759],{},[31665,359549,356112],{"encoding":31667},[34,359551,359553],{"className":359552,"ariaHidden":758},[31672],[34,359554,359556,359559],{"className":359555},[31676],[34,359557],{"className":359558,"style":70915},[31680],[34,359560,359562,359565],{"className":359561},[31685],[34,359563,105852],{"className":359564,"style":37422},[31685,31686],[34,359566,359568],{"className":359567},[31792],[34,359569,359571,359591],{"className":359570},[31796,31797],[34,359572,359574,359588],{"className":359573},[31801],[34,359575,359577],{"className":359576,"style":33833},[31805],[34,359578,359579,359582],{"style":214424},[34,359580],{"className":359581,"style":31814},[31813],[34,359583,359585],{"className":359584},[31818,31819,31820,31821],[34,359586,33759],{"className":359587,"style":33774},[31685,31686,31821],[34,359589,31829],{"className":359590},[31828],[34,359592,359594],{"className":359593},[31801],[34,359595,359597],{"className":359596,"style":31836},[31805],[34,359598],{}," 中：",[105,359601,359602,360668],{},[108,359603,359604,33183,359676,359809,359810,44689,360046,360115,360116,355408,360339,347],{},[2627,359605,359606,359607],{},"检查 ",[34,359608,359610,359627],{"className":359609},[31642],[34,359611,359613],{"className":359612},[31646],[31648,359614,359615],{"xmlns":31650},[31652,359616,359617,359625],{},[31655,359618,359619],{},[31743,359620,359621,359623],{},[31658,359622,105852],{},[31749,359624,640],{},[31665,359626,353413],{"encoding":31667},[34,359628,359630],{"className":359629,"ariaHidden":758},[31672],[34,359631,359633,359636],{"className":359632},[31676],[34,359634],{"className":359635,"style":70915},[31680],[34,359637,359639,359642],{"className":359638},[31685],[34,359640,105852],{"className":359641,"style":37422},[31685,31686],[34,359643,359645],{"className":359644},[31792],[34,359646,359648,359668],{"className":359647},[31796,31797],[34,359649,359651,359665],{"className":359650},[31801],[34,359652,359654],{"className":359653,"style":31806},[31805],[34,359655,359656,359659],{"style":214424},[34,359657],{"className":359658,"style":31814},[31813],[34,359660,359662],{"className":359661},[31818,31819,31820,31821],[34,359663,640],{"className":359664},[31685,31821],[34,359666,31829],{"className":359667},[31828],[34,359669,359671],{"className":359670},[31801],[34,359672,359674],{"className":359673,"style":31836},[31805],[34,359675],{},[34,359677,359679,359705],{"className":359678},[31642],[34,359680,359682],{"className":359681},[31646],[31648,359683,359684],{"xmlns":31650},[31652,359685,359686,359702],{},[31655,359687,359688,359694,359696],{},[31743,359689,359690,359692],{},[31658,359691,111],{},[31749,359693,640],{},[31662,359695,112296],{},[31743,359697,359698,359700],{},[31658,359699,105852],{},[31749,359701,640],{},[31665,359703,359704],{"encoding":31667},"a_1 \\in P_1",[34,359706,359708,359763],{"className":359707,"ariaHidden":758},[31672],[34,359709,359711,359714,359754,359757,359760],{"className":359710},[31676],[34,359712],{"className":359713,"style":225773},[31680],[34,359715,359717,359720],{"className":359716},[31685],[34,359718,111],{"className":359719},[31685,31686],[34,359721,359723],{"className":359722},[31792],[34,359724,359726,359746],{"className":359725},[31796,31797],[34,359727,359729,359743],{"className":359728},[31801],[34,359730,359732],{"className":359731,"style":31806},[31805],[34,359733,359734,359737],{"style":31809},[34,359735],{"className":359736,"style":31814},[31813],[34,359738,359740],{"className":359739},[31818,31819,31820,31821],[34,359741,640],{"className":359742},[31685,31821],[34,359744,31829],{"className":359745},[31828],[34,359747,359749],{"className":359748},[31801],[34,359750,359752],{"className":359751,"style":31836},[31805],[34,359753],{},[34,359755],{"className":359756,"style":31846},[31845],[34,359758,112296],{"className":359759},[31850],[34,359761],{"className":359762,"style":31846},[31845],[34,359764,359766,359769],{"className":359765},[31676],[34,359767],{"className":359768,"style":70915},[31680],[34,359770,359772,359775],{"className":359771},[31685],[34,359773,105852],{"className":359774,"style":37422},[31685,31686],[34,359776,359778],{"className":359777},[31792],[34,359779,359781,359801],{"className":359780},[31796,31797],[34,359782,359784,359798],{"className":359783},[31801],[34,359785,359787],{"className":359786,"style":31806},[31805],[34,359788,359789,359792],{"style":214424},[34,359790],{"className":359791,"style":31814},[31813],[34,359793,359795],{"className":359794},[31818,31819,31820,31821],[34,359796,640],{"className":359797},[31685,31821],[34,359799,31829],{"className":359800},[31828],[34,359802,359804],{"className":359803},[31801],[34,359805,359807],{"className":359806,"style":31836},[31805],[34,359808],{}," (根据上述假设)。\n",[34,359811,359813,359851],{"className":359812},[31642],[34,359814,359816],{"className":359815},[31646],[31648,359817,359818],{"xmlns":31650},[31652,359819,359820,359848],{},[31655,359821,359822,359828,359830,359832,359834,359840,359842],{},[31743,359823,359824,359826],{},[31658,359825,111],{},[31749,359827,768],{},[31662,359829,31960],{"separator":758},[31662,359831,33934],{},[31662,359833,31960],{"separator":758},[31743,359835,359836,359838],{},[31658,359837,111],{},[31658,359839,31660],{},[31662,359841,112266],{"mathvariant":38079},[31743,359843,359844,359846],{},[31658,359845,105852],{},[31749,359847,640],{},[31665,359849,359850],{"encoding":31667},"a_2, \\dots, a_n \\notin P_1",[34,359852,359854,360000],{"className":359853,"ariaHidden":758},[31672],[34,359855,359857,359860,359900,359903,359906,359909,359912,359915,359918,359958,359961,359997],{"className":359856},[31676],[34,359858],{"className":359859,"style":31774},[31680],[34,359861,359863,359866],{"className":359862},[31685],[34,359864,111],{"className":359865},[31685,31686],[34,359867,359869],{"className":359868},[31792],[34,359870,359872,359892],{"className":359871},[31796,31797],[34,359873,359875,359889],{"className":359874},[31801],[34,359876,359878],{"className":359877,"style":31806},[31805],[34,359879,359880,359883],{"style":31809},[34,359881],{"className":359882,"style":31814},[31813],[34,359884,359886],{"className":359885},[31818,31819,31820,31821],[34,359887,768],{"className":359888},[31685,31821],[34,359890,31829],{"className":359891},[31828],[34,359893,359895],{"className":359894},[31801],[34,359896,359898],{"className":359897,"style":31836},[31805],[34,359899],{},[34,359901,31960],{"className":359902},[32119],[34,359904],{"className":359905,"style":32123},[31845],[34,359907,33934],{"className":359908},[34006],[34,359910],{"className":359911,"style":32123},[31845],[34,359913,31960],{"className":359914},[32119],[34,359916],{"className":359917,"style":32123},[31845],[34,359919,359921,359924],{"className":359920},[31685],[34,359922,111],{"className":359923},[31685,31686],[34,359925,359927],{"className":359926},[31792],[34,359928,359930,359950],{"className":359929},[31796,31797],[34,359931,359933,359947],{"className":359932},[31801],[34,359934,359936],{"className":359935,"style":32362},[31805],[34,359937,359938,359941],{"style":31809},[34,359939],{"className":359940,"style":31814},[31813],[34,359942,359944],{"className":359943},[31818,31819,31820,31821],[34,359945,31660],{"className":359946},[31685,31686,31821],[34,359948,31829],{"className":359949},[31828],[34,359951,359953],{"className":359952},[31801],[34,359954,359956],{"className":359955,"style":31836},[31805],[34,359957],{},[34,359959],{"className":359960,"style":31846},[31845],[34,359962,359964,359970],{"className":359963},[31850],[34,359965,359967],{"className":359966},[31685],[34,359968,112296],{"className":359969},[31850],[34,359971,359973],{"className":359972},[31685,54774],[34,359974,359976],{"className":359975},[54778],[34,359977,359979,359982,359994],{"className":359978},[112306],[34,359980],{"className":359981,"style":31774},[31680],[34,359983,359985],{"className":359984},[54789],[34,359986,359988,359991],{"className":359987},[31685],[34,359989,38080],{"className":359990},[31685],[34,359992],{"className":359993,"style":112322},[31845],[34,359995],{"className":359996},[54800],[34,359998],{"className":359999,"style":31846},[31845],[34,360001,360003,360006],{"className":360002},[31676],[34,360004],{"className":360005,"style":70915},[31680],[34,360007,360009,360012],{"className":360008},[31685],[34,360010,105852],{"className":360011,"style":37422},[31685,31686],[34,360013,360015],{"className":360014},[31792],[34,360016,360018,360038],{"className":360017},[31796,31797],[34,360019,360021,360035],{"className":360020},[31801],[34,360022,360024],{"className":360023,"style":31806},[31805],[34,360025,360026,360029],{"style":214424},[34,360027],{"className":360028,"style":31814},[31813],[34,360030,360032],{"className":360031},[31818,31819,31820,31821],[34,360033,640],{"className":360034},[31685,31821],[34,360036,31829],{"className":360037},[31828],[34,360039,360041],{"className":360040},[31801],[34,360042,360044],{"className":360043,"style":31836},[31805],[34,360045],{},[34,360047,360049,360066],{"className":360048},[31642],[34,360050,360052],{"className":360051},[31646],[31648,360053,360054],{"xmlns":31650},[31652,360055,360056,360064],{},[31655,360057,360058],{},[31743,360059,360060,360062],{},[31658,360061,105852],{},[31749,360063,640],{},[31665,360065,353413],{"encoding":31667},[34,360067,360069],{"className":360068,"ariaHidden":758},[31672],[34,360070,360072,360075],{"className":360071},[31676],[34,360073],{"className":360074,"style":70915},[31680],[34,360076,360078,360081],{"className":360077},[31685],[34,360079,105852],{"className":360080,"style":37422},[31685,31686],[34,360082,360084],{"className":360083},[31792],[34,360085,360087,360107],{"className":360086},[31796,31797],[34,360088,360090,360104],{"className":360089},[31801],[34,360091,360093],{"className":360092,"style":31806},[31805],[34,360094,360095,360098],{"style":214424},[34,360096],{"className":360097,"style":31814},[31813],[34,360099,360101],{"className":360100},[31818,31819,31820,31821],[34,360102,640],{"className":360103},[31685,31821],[34,360105,31829],{"className":360106},[31828],[34,360108,360110],{"className":360109},[31801],[34,360111,360113],{"className":360112,"style":31836},[31805],[34,360114],{}," 是素理想，所以乘积 ",[34,360117,360119,360153],{"className":360118},[31642],[34,360120,360122],{"className":360121},[31646],[31648,360123,360124],{"xmlns":31650},[31652,360125,360126,360150],{},[31655,360127,360128,360134,360136,360142,360144],{},[31743,360129,360130,360132],{},[31658,360131,111],{},[31749,360133,768],{},[31662,360135,33934],{},[31743,360137,360138,360140],{},[31658,360139,111],{},[31658,360141,31660],{},[31662,360143,112266],{"mathvariant":38079},[31743,360145,360146,360148],{},[31658,360147,105852],{},[31749,360149,640],{},[31665,360151,360152],{"encoding":31667},"a_2 \\dots a_n \\notin P_1",[34,360154,360156,360293],{"className":360155,"ariaHidden":758},[31672],[34,360157,360159,360162,360202,360205,360208,360211,360251,360254,360290],{"className":360158},[31676],[34,360160],{"className":360161,"style":31774},[31680],[34,360163,360165,360168],{"className":360164},[31685],[34,360166,111],{"className":360167},[31685,31686],[34,360169,360171],{"className":360170},[31792],[34,360172,360174,360194],{"className":360173},[31796,31797],[34,360175,360177,360191],{"className":360176},[31801],[34,360178,360180],{"className":360179,"style":31806},[31805],[34,360181,360182,360185],{"style":31809},[34,360183],{"className":360184,"style":31814},[31813],[34,360186,360188],{"className":360187},[31818,31819,31820,31821],[34,360189,768],{"className":360190},[31685,31821],[34,360192,31829],{"className":360193},[31828],[34,360195,360197],{"className":360196},[31801],[34,360198,360200],{"className":360199,"style":31836},[31805],[34,360201],{},[34,360203],{"className":360204,"style":32123},[31845],[34,360206,33934],{"className":360207},[34006],[34,360209],{"className":360210,"style":32123},[31845],[34,360212,360214,360217],{"className":360213},[31685],[34,360215,111],{"className":360216},[31685,31686],[34,360218,360220],{"className":360219},[31792],[34,360221,360223,360243],{"className":360222},[31796,31797],[34,360224,360226,360240],{"className":360225},[31801],[34,360227,360229],{"className":360228,"style":32362},[31805],[34,360230,360231,360234],{"style":31809},[34,360232],{"className":360233,"style":31814},[31813],[34,360235,360237],{"className":360236},[31818,31819,31820,31821],[34,360238,31660],{"className":360239},[31685,31686,31821],[34,360241,31829],{"className":360242},[31828],[34,360244,360246],{"className":360245},[31801],[34,360247,360249],{"className":360248,"style":31836},[31805],[34,360250],{},[34,360252],{"className":360253,"style":31846},[31845],[34,360255,360257,360263],{"className":360256},[31850],[34,360258,360260],{"className":360259},[31685],[34,360261,112296],{"className":360262},[31850],[34,360264,360266],{"className":360265},[31685,54774],[34,360267,360269],{"className":360268},[54778],[34,360270,360272,360275,360287],{"className":360271},[112306],[34,360273],{"className":360274,"style":31774},[31680],[34,360276,360278],{"className":360277},[54789],[34,360279,360281,360284],{"className":360280},[31685],[34,360282,38080],{"className":360283},[31685],[34,360285],{"className":360286,"style":112322},[31845],[34,360288],{"className":360289},[54800],[34,360291],{"className":360292,"style":31846},[31845],[34,360294,360296,360299],{"className":360295},[31676],[34,360297],{"className":360298,"style":70915},[31680],[34,360300,360302,360305],{"className":360301},[31685],[34,360303,105852],{"className":360304,"style":37422},[31685,31686],[34,360306,360308],{"className":360307},[31792],[34,360309,360311,360331],{"className":360310},[31796,31797],[34,360312,360314,360328],{"className":360313},[31801],[34,360315,360317],{"className":360316,"style":31806},[31805],[34,360318,360319,360322],{"style":214424},[34,360320],{"className":360321,"style":31814},[31813],[34,360323,360325],{"className":360324},[31818,31819,31820,31821],[34,360326,640],{"className":360327},[31685,31821],[34,360329,31829],{"className":360330},[31828],[34,360332,360334],{"className":360333},[31801],[34,360335,360337],{"className":360336,"style":31836},[31805],[34,360338],{},[34,360340,360342,360402],{"className":360341},[31642],[34,360343,360345],{"className":360344},[31646],[31648,360346,360347],{"xmlns":31650},[31652,360348,360349,360399],{},[31655,360350,360351,360353,360355,360357,360360,360366,360368,360370,360372,360375,360381,360383,360385,360387,360389,360391,360393],{},[31658,360352,67764],{},[31662,360354,202],{},[31662,360356,456],{"stretchy":702},[39361,360358,360359],{},"属于 ",[31743,360361,360362,360364],{},[31658,360363,105852],{},[31749,360365,640],{},[31662,360367,2557],{"stretchy":702},[31662,360369,223],{},[31662,360371,456],{"stretchy":702},[39361,360373,360374],{},"不属于 ",[31743,360376,360377,360379],{},[31658,360378,105852],{},[31749,360380,640],{},[31662,360382,2557],{"stretchy":702},[39361,360384,262622],{},[31662,360386,262625],{},[39361,360388,262622],{},[31658,360390,67764],{},[31662,360392,112266],{"mathvariant":38079},[31743,360394,360395,360397],{},[31658,360396,105852],{},[31749,360398,640],{},[31665,360400,360401],{"encoding":31667},"y = (\\text{属于 } P_1) + (\\text{不属于 } P_1) \\implies y \\notin P_1",[34,360403,360405,360423,360494,360571,360622],{"className":360404,"ariaHidden":758},[31672],[34,360406,360408,360411,360414,360417,360420],{"className":360407},[31676],[34,360409],{"className":360410,"style":38774},[31680],[34,360412,67764],{"className":360413,"style":32062},[31685,31686],[34,360415],{"className":360416,"style":31846},[31845],[34,360418,202],{"className":360419},[31850],[34,360421],{"className":360422,"style":31846},[31845],[34,360424,360426,360429,360432,360442,360482,360485,360488,360491],{"className":360425},[31676],[34,360427],{"className":360428,"style":31774},[31680],[34,360430,456],{"className":360431},[31782],[34,360433,360435,360439],{"className":360434},[31685,2638],[34,360436,360438],{"className":360437},[31685,105462],"属于",[34,360440,130298],{"className":360441},[31685],[34,360443,360445,360448],{"className":360444},[31685],[34,360446,105852],{"className":360447,"style":37422},[31685,31686],[34,360449,360451],{"className":360450},[31792],[34,360452,360454,360474],{"className":360453},[31796,31797],[34,360455,360457,360471],{"className":360456},[31801],[34,360458,360460],{"className":360459,"style":31806},[31805],[34,360461,360462,360465],{"style":214424},[34,360463],{"className":360464,"style":31814},[31813],[34,360466,360468],{"className":360467},[31818,31819,31820,31821],[34,360469,640],{"className":360470},[31685,31821],[34,360472,31829],{"className":360473},[31828],[34,360475,360477],{"className":360476},[31801],[34,360478,360480],{"className":360479,"style":31836},[31805],[34,360481],{},[34,360483,2557],{"className":360484},[31690],[34,360486],{"className":360487,"style":32604},[31845],[34,360489,223],{"className":360490},[32024],[34,360492],{"className":360493,"style":32604},[31845],[34,360495,360497,360500,360503,360513,360553,360556,360559,360562,360565,360568],{"className":360496},[31676],[34,360498],{"className":360499,"style":31774},[31680],[34,360501,456],{"className":360502},[31782],[34,360504,360506,360510],{"className":360505},[31685,2638],[34,360507,360509],{"className":360508},[31685,105462],"不属于",[34,360511,130298],{"className":360512},[31685],[34,360514,360516,360519],{"className":360515},[31685],[34,360517,105852],{"className":360518,"style":37422},[31685,31686],[34,360520,360522],{"className":360521},[31792],[34,360523,360525,360545],{"className":360524},[31796,31797],[34,360526,360528,360542],{"className":360527},[31801],[34,360529,360531],{"className":360530,"style":31806},[31805],[34,360532,360533,360536],{"style":214424},[34,360534],{"className":360535,"style":31814},[31813],[34,360537,360539],{"className":360538},[31818,31819,31820,31821],[34,360540,640],{"className":360541},[31685,31821],[34,360543,31829],{"className":360544},[31828],[34,360546,360548],{"className":360547},[31801],[34,360549,360551],{"className":360550,"style":31836},[31805],[34,360552],{},[34,360554,2557],{"className":360555},[31690],[34,360557],{"className":360558,"style":31846},[31845],[34,360560],{"className":360561,"style":31846},[31845],[34,360563,262625],{"className":360564},[31850],[34,360566],{"className":360567,"style":31846},[31845],[34,360569],{"className":360570,"style":31846},[31845],[34,360572,360574,360577,360580,360583,360619],{"className":360573},[31676],[34,360575],{"className":360576,"style":31774},[31680],[34,360578,67764],{"className":360579,"style":32062},[31685,31686],[34,360581],{"className":360582,"style":31846},[31845],[34,360584,360586,360592],{"className":360585},[31850],[34,360587,360589],{"className":360588},[31685],[34,360590,112296],{"className":360591},[31850],[34,360593,360595],{"className":360594},[31685,54774],[34,360596,360598],{"className":360597},[54778],[34,360599,360601,360604,360616],{"className":360600},[112306],[34,360602],{"className":360603,"style":31774},[31680],[34,360605,360607],{"className":360606},[54789],[34,360608,360610,360613],{"className":360609},[31685],[34,360611,38080],{"className":360612},[31685],[34,360614],{"className":360615,"style":112322},[31845],[34,360617],{"className":360618},[54800],[34,360620],{"className":360621,"style":31846},[31845],[34,360623,360625,360628],{"className":360624},[31676],[34,360626],{"className":360627,"style":70915},[31680],[34,360629,360631,360634],{"className":360630},[31685],[34,360632,105852],{"className":360633,"style":37422},[31685,31686],[34,360635,360637],{"className":360636},[31792],[34,360638,360640,360660],{"className":360639},[31796,31797],[34,360641,360643,360657],{"className":360642},[31801],[34,360644,360646],{"className":360645,"style":31806},[31805],[34,360647,360648,360651],{"style":214424},[34,360649],{"className":360650,"style":31814},[31813],[34,360652,360654],{"className":360653},[31818,31819,31820,31821],[34,360655,640],{"className":360656},[31685,31821],[34,360658,31829],{"className":360659},[31828],[34,360661,360663],{"className":360662},[31801],[34,360664,360666],{"className":360665,"style":31836},[31805],[34,360667],{},[108,360669,360670,360792,360793,360966,360967,361036,361037,361121,361122,361288,361289,361359,361360,361429,361430,361499,361500,347],{},[2627,360671,359606,360672,347890,360741,2557],{},[34,360673,360675,360692],{"className":360674},[31642],[34,360676,360678],{"className":360677},[31646],[31648,360679,360680],{"xmlns":31650},[31652,360681,360682,360690],{},[31655,360683,360684],{},[31743,360685,360686,360688],{},[31658,360687,105852],{},[31658,360689,33759],{},[31665,360691,356112],{"encoding":31667},[34,360693,360695],{"className":360694,"ariaHidden":758},[31672],[34,360696,360698,360701],{"className":360697},[31676],[34,360699],{"className":360700,"style":70915},[31680],[34,360702,360704,360707],{"className":360703},[31685],[34,360705,105852],{"className":360706,"style":37422},[31685,31686],[34,360708,360710],{"className":360709},[31792],[34,360711,360713,360733],{"className":360712},[31796,31797],[34,360714,360716,360730],{"className":360715},[31801],[34,360717,360719],{"className":360718,"style":33833},[31805],[34,360720,360721,360724],{"style":214424},[34,360722],{"className":360723,"style":31814},[31813],[34,360725,360727],{"className":360726},[31818,31819,31820,31821],[34,360728,33759],{"className":360729,"style":33774},[31685,31686,31821],[34,360731,31829],{"className":360732},[31828],[34,360734,360736],{"className":360735},[31801],[34,360737,360739],{"className":360738,"style":31836},[31805],[34,360740],{},[34,360742,360744,360762],{"className":360743},[31642],[34,360745,360747],{"className":360746},[31646],[31648,360748,360749],{"xmlns":31650},[31652,360750,360751,360759],{},[31655,360752,360753,360755,360757],{},[31658,360754,33759],{},[31662,360756,43175],{},[31749,360758,768],{},[31665,360760,360761],{"encoding":31667},"k \\ge 2",[34,360763,360765,360783],{"className":360764,"ariaHidden":758},[31672],[34,360766,360768,360771,360774,360777,360780],{"className":360767},[31676],[34,360769],{"className":360770,"style":47709},[31680],[34,360772,33759],{"className":360773,"style":33774},[31685,31686],[34,360775],{"className":360776,"style":31846},[31845],[34,360778,43175],{"className":360779},[31850],[34,360781],{"className":360782,"style":31846},[31845],[34,360784,360786,360789],{"className":360785},[31676],[34,360787],{"className":360788,"style":32617},[31680],[34,360790,768],{"className":360791},[31685],"：\n乘积项 ",[34,360794,360796,360828],{"className":360795},[31642],[34,360797,360799],{"className":360798},[31646],[31648,360800,360801],{"xmlns":31650},[31652,360802,360803,360825],{},[31655,360804,360805,360811,360817,360819],{},[31743,360806,360807,360809],{},[31658,360808,111],{},[31749,360810,768],{},[31743,360812,360813,360815],{},[31658,360814,111],{},[31749,360816,786],{},[31662,360818,33934],{},[31743,360820,360821,360823],{},[31658,360822,111],{},[31658,360824,31660],{},[31665,360826,360827],{"encoding":31667},"a_2 a_3 \\dots a_n",[34,360829,360831],{"className":360830,"ariaHidden":758},[31672],[34,360832,360834,360837,360877,360917,360920,360923,360926],{"className":360833},[31676],[34,360835],{"className":360836,"style":31860},[31680],[34,360838,360840,360843],{"className":360839},[31685],[34,360841,111],{"className":360842},[31685,31686],[34,360844,360846],{"className":360845},[31792],[34,360847,360849,360869],{"className":360848},[31796,31797],[34,360850,360852,360866],{"className":360851},[31801],[34,360853,360855],{"className":360854,"style":31806},[31805],[34,360856,360857,360860],{"style":31809},[34,360858],{"className":360859,"style":31814},[31813],[34,360861,360863],{"className":360862},[31818,31819,31820,31821],[34,360864,768],{"className":360865},[31685,31821],[34,360867,31829],{"className":360868},[31828],[34,360870,360872],{"className":360871},[31801],[34,360873,360875],{"className":360874,"style":31836},[31805],[34,360876],{},[34,360878,360880,360883],{"className":360879},[31685],[34,360881,111],{"className":360882},[31685,31686],[34,360884,360886],{"className":360885},[31792],[34,360887,360889,360909],{"className":360888},[31796,31797],[34,360890,360892,360906],{"className":360891},[31801],[34,360893,360895],{"className":360894,"style":31806},[31805],[34,360896,360897,360900],{"style":31809},[34,360898],{"className":360899,"style":31814},[31813],[34,360901,360903],{"className":360902},[31818,31819,31820,31821],[34,360904,786],{"className":360905},[31685,31821],[34,360907,31829],{"className":360908},[31828],[34,360910,360912],{"className":360911},[31801],[34,360913,360915],{"className":360914,"style":31836},[31805],[34,360916],{},[34,360918],{"className":360919,"style":32123},[31845],[34,360921,33934],{"className":360922},[34006],[34,360924],{"className":360925,"style":32123},[31845],[34,360927,360929,360932],{"className":360928},[31685],[34,360930,111],{"className":360931},[31685,31686],[34,360933,360935],{"className":360934},[31792],[34,360936,360938,360958],{"className":360937},[31796,31797],[34,360939,360941,360955],{"className":360940},[31801],[34,360942,360944],{"className":360943,"style":32362},[31805],[34,360945,360946,360949],{"style":31809},[34,360947],{"className":360948,"style":31814},[31813],[34,360950,360952],{"className":360951},[31818,31819,31820,31821],[34,360953,31660],{"className":360954},[31685,31686,31821],[34,360956,31829],{"className":360957},[31828],[34,360959,360961],{"className":360960},[31801],[34,360962,360964],{"className":360963,"style":31836},[31805],[34,360965],{}," 中包含了 ",[34,360968,360970,360987],{"className":360969},[31642],[34,360971,360973],{"className":360972},[31646],[31648,360974,360975],{"xmlns":31650},[31652,360976,360977,360985],{},[31655,360978,360979],{},[31743,360980,360981,360983],{},[31658,360982,111],{},[31658,360984,33759],{},[31665,360986,351069],{"encoding":31667},[34,360988,360990],{"className":360989,"ariaHidden":758},[31672],[34,360991,360993,360996],{"className":360992},[31676],[34,360994],{"className":360995,"style":31860},[31680],[34,360997,360999,361002],{"className":360998},[31685],[34,361000,111],{"className":361001},[31685,31686],[34,361003,361005],{"className":361004},[31792],[34,361006,361008,361028],{"className":361007},[31796,31797],[34,361009,361011,361025],{"className":361010},[31801],[34,361012,361014],{"className":361013,"style":33833},[31805],[34,361015,361016,361019],{"style":31809},[34,361017],{"className":361018,"style":31814},[31813],[34,361020,361022],{"className":361021},[31818,31819,31820,31821],[34,361023,33759],{"className":361024,"style":33774},[31685,31686,31821],[34,361026,31829],{"className":361027},[31828],[34,361029,361031],{"className":361030},[31801],[34,361032,361034],{"className":361033,"style":31836},[31805],[34,361035],{},"，所以乘积项 ",[34,361038,361040,361060],{"className":361039},[31642],[34,361041,361043],{"className":361042},[31646],[31648,361044,361045],{"xmlns":31650},[31652,361046,361047,361057],{},[31655,361048,361049,361051],{},[31662,361050,112296],{},[31743,361052,361053,361055],{},[31658,361054,105852],{},[31658,361056,33759],{},[31665,361058,361059],{"encoding":31667},"\\in P_k",[34,361061,361063,361075],{"className":361062,"ariaHidden":758},[31672],[34,361064,361066,361069,361072],{"className":361065},[31676],[34,361067],{"className":361068,"style":34812},[31680],[34,361070,112296],{"className":361071},[31850],[34,361073],{"className":361074,"style":31846},[31845],[34,361076,361078,361081],{"className":361077},[31676],[34,361079],{"className":361080,"style":70915},[31680],[34,361082,361084,361087],{"className":361083},[31685],[34,361085,105852],{"className":361086,"style":37422},[31685,31686],[34,361088,361090],{"className":361089},[31792],[34,361091,361093,361113],{"className":361092},[31796,31797],[34,361094,361096,361110],{"className":361095},[31801],[34,361097,361099],{"className":361098,"style":33833},[31805],[34,361100,361101,361104],{"style":214424},[34,361102],{"className":361103,"style":31814},[31813],[34,361105,361107],{"className":361106},[31818,31819,31820,31821],[34,361108,33759],{"className":361109,"style":33774},[31685,31686,31821],[34,361111,31829],{"className":361112},[31828],[34,361114,361116],{"className":361115},[31801],[34,361117,361119],{"className":361118,"style":31836},[31805],[34,361120],{},"。\n第一项 ",[34,361123,361125,361151],{"className":361124},[31642],[34,361126,361128],{"className":361127},[31646],[31648,361129,361130],{"xmlns":31650},[31652,361131,361132,361148],{},[31655,361133,361134,361140,361142],{},[31743,361135,361136,361138],{},[31658,361137,111],{},[31749,361139,640],{},[31662,361141,112266],{"mathvariant":38079},[31743,361143,361144,361146],{},[31658,361145,105852],{},[31658,361147,33759],{},[31665,361149,361150],{"encoding":31667},"a_1 \\notin P_k",[34,361152,361154,361242],{"className":361153,"ariaHidden":758},[31672],[34,361155,361157,361160,361200,361203,361239],{"className":361156},[31676],[34,361158],{"className":361159,"style":31774},[31680],[34,361161,361163,361166],{"className":361162},[31685],[34,361164,111],{"className":361165},[31685,31686],[34,361167,361169],{"className":361168},[31792],[34,361170,361172,361192],{"className":361171},[31796,31797],[34,361173,361175,361189],{"className":361174},[31801],[34,361176,361178],{"className":361177,"style":31806},[31805],[34,361179,361180,361183],{"style":31809},[34,361181],{"className":361182,"style":31814},[31813],[34,361184,361186],{"className":361185},[31818,31819,31820,31821],[34,361187,640],{"className":361188},[31685,31821],[34,361190,31829],{"className":361191},[31828],[34,361193,361195],{"className":361194},[31801],[34,361196,361198],{"className":361197,"style":31836},[31805],[34,361199],{},[34,361201],{"className":361202,"style":31846},[31845],[34,361204,361206,361212],{"className":361205},[31850],[34,361207,361209],{"className":361208},[31685],[34,361210,112296],{"className":361211},[31850],[34,361213,361215],{"className":361214},[31685,54774],[34,361216,361218],{"className":361217},[54778],[34,361219,361221,361224,361236],{"className":361220},[112306],[34,361222],{"className":361223,"style":31774},[31680],[34,361225,361227],{"className":361226},[54789],[34,361228,361230,361233],{"className":361229},[31685],[34,361231,38080],{"className":361232},[31685],[34,361234],{"className":361235,"style":112322},[31845],[34,361237],{"className":361238},[54800],[34,361240],{"className":361241,"style":31846},[31845],[34,361243,361245,361248],{"className":361244},[31676],[34,361246],{"className":361247,"style":70915},[31680],[34,361249,361251,361254],{"className":361250},[31685],[34,361252,105852],{"className":361253,"style":37422},[31685,31686],[34,361255,361257],{"className":361256},[31792],[34,361258,361260,361280],{"className":361259},[31796,31797],[34,361261,361263,361277],{"className":361262},[31801],[34,361264,361266],{"className":361265,"style":33833},[31805],[34,361267,361268,361271],{"style":214424},[34,361269],{"className":361270,"style":31814},[31813],[34,361272,361274],{"className":361273},[31818,31819,31820,31821],[34,361275,33759],{"className":361276,"style":33774},[31685,31686,31821],[34,361278,31829],{"className":361279},[31828],[34,361281,361283],{"className":361282},[31801],[34,361284,361286],{"className":361285,"style":31836},[31805],[34,361287],{}," (因为 ",[34,361290,361292,361310],{"className":361291},[31642],[34,361293,361295],{"className":361294},[31646],[31648,361296,361297],{"xmlns":31650},[31652,361298,361299,361307],{},[31655,361300,361301],{},[31743,361302,361303,361305],{},[31658,361304,111],{},[31749,361306,640],{},[31665,361308,361309],{"encoding":31667},"a_1",[34,361311,361313],{"className":361312,"ariaHidden":758},[31672],[34,361314,361316,361319],{"className":361315},[31676],[34,361317],{"className":361318,"style":31860},[31680],[34,361320,361322,361325],{"className":361321},[31685],[34,361323,111],{"className":361324},[31685,31686],[34,361326,361328],{"className":361327},[31792],[34,361329,361331,361351],{"className":361330},[31796,31797],[34,361332,361334,361348],{"className":361333},[31801],[34,361335,361337],{"className":361336,"style":31806},[31805],[34,361338,361339,361342],{"style":31809},[34,361340],{"className":361341,"style":31814},[31813],[34,361343,361345],{"className":361344},[31818,31819,31820,31821],[34,361346,640],{"className":361347},[31685,31821],[34,361349,31829],{"className":361350},[31828],[34,361352,361354],{"className":361353},[31801],[34,361355,361357],{"className":361356,"style":31836},[31805],[34,361358],{}," 只在 ",[34,361361,361363,361380],{"className":361362},[31642],[34,361364,361366],{"className":361365},[31646],[31648,361367,361368],{"xmlns":31650},[31652,361369,361370,361378],{},[31655,361371,361372],{},[31743,361373,361374,361376],{},[31658,361375,105852],{},[31749,361377,640],{},[31665,361379,353413],{"encoding":31667},[34,361381,361383],{"className":361382,"ariaHidden":758},[31672],[34,361384,361386,361389],{"className":361385},[31676],[34,361387],{"className":361388,"style":70915},[31680],[34,361390,361392,361395],{"className":361391},[31685],[34,361393,105852],{"className":361394,"style":37422},[31685,31686],[34,361396,361398],{"className":361397},[31792],[34,361399,361401,361421],{"className":361400},[31796,31797],[34,361402,361404,361418],{"className":361403},[31801],[34,361405,361407],{"className":361406,"style":31806},[31805],[34,361408,361409,361412],{"style":214424},[34,361410],{"className":361411,"style":31814},[31813],[34,361413,361415],{"className":361414},[31818,31819,31820,31821],[34,361416,640],{"className":361417},[31685,31821],[34,361419,31829],{"className":361420},[31828],[34,361422,361424],{"className":361423},[31801],[34,361425,361427],{"className":361426,"style":31836},[31805],[34,361428],{}," 里，避开了其他所有 ",[34,361431,361433,361450],{"className":361432},[31642],[34,361434,361436],{"className":361435},[31646],[31648,361437,361438],{"xmlns":31650},[31652,361439,361440,361448],{},[31655,361441,361442],{},[31743,361443,361444,361446],{},[31658,361445,105852],{},[31658,361447,31968],{},[31665,361449,357398],{"encoding":31667},[34,361451,361453],{"className":361452,"ariaHidden":758},[31672],[34,361454,361456,361459],{"className":361455},[31676],[34,361457],{"className":361458,"style":55484},[31680],[34,361460,361462,361465],{"className":361461},[31685],[34,361463,105852],{"className":361464,"style":37422},[31685,31686],[34,361466,361468],{"className":361467},[31792],[34,361469,361471,361491],{"className":361470},[31796,31797],[34,361472,361474,361488],{"className":361473},[31801],[34,361475,361477],{"className":361476,"style":32090},[31805],[34,361478,361479,361482],{"style":214424},[34,361480],{"className":361481,"style":31814},[31813],[34,361483,361485],{"className":361484},[31818,31819,31820,31821],[34,361486,31968],{"className":361487,"style":32153},[31685,31686,31821],[34,361489,31829],{"className":361490},[31828],[34,361492,361494],{"className":361493},[31801],[34,361495,361497],{"className":361496,"style":32163},[31805],[34,361498],{},")。\n所以 ",[34,361501,361503,361561],{"className":361502},[31642],[34,361504,361506],{"className":361505},[31646],[31648,361507,361508],{"xmlns":31650},[31652,361509,361510,361558],{},[31655,361511,361512,361514,361516,361518,361520,361526,361528,361530,361532,361534,361540,361542,361544,361546,361548,361550,361552],{},[31658,361513,67764],{},[31662,361515,202],{},[31662,361517,456],{"stretchy":702},[39361,361519,360374],{},[31743,361521,361522,361524],{},[31658,361523,105852],{},[31658,361525,33759],{},[31662,361527,2557],{"stretchy":702},[31662,361529,223],{},[31662,361531,456],{"stretchy":702},[39361,361533,360359],{},[31743,361535,361536,361538],{},[31658,361537,105852],{},[31658,361539,33759],{},[31662,361541,2557],{"stretchy":702},[39361,361543,262622],{},[31662,361545,262625],{},[39361,361547,262622],{},[31658,361549,67764],{},[31662,361551,112266],{"mathvariant":38079},[31743,361553,361554,361556],{},[31658,361555,105852],{},[31658,361557,33759],{},[31665,361559,361560],{"encoding":31667},"y = (\\text{不属于 } P_k) + (\\text{属于 } P_k) \\implies y \\notin P_k",[34,361562,361564,361582,361652,361728,361779],{"className":361563,"ariaHidden":758},[31672],[34,361565,361567,361570,361573,361576,361579],{"className":361566},[31676],[34,361568],{"className":361569,"style":38774},[31680],[34,361571,67764],{"className":361572,"style":32062},[31685,31686],[34,361574],{"className":361575,"style":31846},[31845],[34,361577,202],{"className":361578},[31850],[34,361580],{"className":361581,"style":31846},[31845],[34,361583,361585,361588,361591,361600,361640,361643,361646,361649],{"className":361584},[31676],[34,361586],{"className":361587,"style":31774},[31680],[34,361589,456],{"className":361590},[31782],[34,361592,361594,361597],{"className":361593},[31685,2638],[34,361595,360509],{"className":361596},[31685,105462],[34,361598,130298],{"className":361599},[31685],[34,361601,361603,361606],{"className":361602},[31685],[34,361604,105852],{"className":361605,"style":37422},[31685,31686],[34,361607,361609],{"className":361608},[31792],[34,361610,361612,361632],{"className":361611},[31796,31797],[34,361613,361615,361629],{"className":361614},[31801],[34,361616,361618],{"className":361617,"style":33833},[31805],[34,361619,361620,361623],{"style":214424},[34,361621],{"className":361622,"style":31814},[31813],[34,361624,361626],{"className":361625},[31818,31819,31820,31821],[34,361627,33759],{"className":361628,"style":33774},[31685,31686,31821],[34,361630,31829],{"className":361631},[31828],[34,361633,361635],{"className":361634},[31801],[34,361636,361638],{"className":361637,"style":31836},[31805],[34,361639],{},[34,361641,2557],{"className":361642},[31690],[34,361644],{"className":361645,"style":32604},[31845],[34,361647,223],{"className":361648},[32024],[34,361650],{"className":361651,"style":32604},[31845],[34,361653,361655,361658,361661,361670,361710,361713,361716,361719,361722,361725],{"className":361654},[31676],[34,361656],{"className":361657,"style":31774},[31680],[34,361659,456],{"className":361660},[31782],[34,361662,361664,361667],{"className":361663},[31685,2638],[34,361665,360438],{"className":361666},[31685,105462],[34,361668,130298],{"className":361669},[31685],[34,361671,361673,361676],{"className":361672},[31685],[34,361674,105852],{"className":361675,"style":37422},[31685,31686],[34,361677,361679],{"className":361678},[31792],[34,361680,361682,361702],{"className":361681},[31796,31797],[34,361683,361685,361699],{"className":361684},[31801],[34,361686,361688],{"className":361687,"style":33833},[31805],[34,361689,361690,361693],{"style":214424},[34,361691],{"className":361692,"style":31814},[31813],[34,361694,361696],{"className":361695},[31818,31819,31820,31821],[34,361697,33759],{"className":361698,"style":33774},[31685,31686,31821],[34,361700,31829],{"className":361701},[31828],[34,361703,361705],{"className":361704},[31801],[34,361706,361708],{"className":361707,"style":31836},[31805],[34,361709],{},[34,361711,2557],{"className":361712},[31690],[34,361714],{"className":361715,"style":31846},[31845],[34,361717],{"className":361718,"style":31846},[31845],[34,361720,262625],{"className":361721},[31850],[34,361723],{"className":361724,"style":31846},[31845],[34,361726],{"className":361727,"style":31846},[31845],[34,361729,361731,361734,361737,361740,361776],{"className":361730},[31676],[34,361732],{"className":361733,"style":31774},[31680],[34,361735,67764],{"className":361736,"style":32062},[31685,31686],[34,361738],{"className":361739,"style":31846},[31845],[34,361741,361743,361749],{"className":361742},[31850],[34,361744,361746],{"className":361745},[31685],[34,361747,112296],{"className":361748},[31850],[34,361750,361752],{"className":361751},[31685,54774],[34,361753,361755],{"className":361754},[54778],[34,361756,361758,361761,361773],{"className":361757},[112306],[34,361759],{"className":361760,"style":31774},[31680],[34,361762,361764],{"className":361763},[54789],[34,361765,361767,361770],{"className":361766},[31685],[34,361768,38080],{"className":361769},[31685],[34,361771],{"className":361772,"style":112322},[31845],[34,361774],{"className":361775},[54800],[34,361777],{"className":361778,"style":31846},[31845],[34,361780,361782,361785],{"className":361781},[31676],[34,361783],{"className":361784,"style":70915},[31680],[34,361786,361788,361791],{"className":361787},[31685],[34,361789,105852],{"className":361790,"style":37422},[31685,31686],[34,361792,361794],{"className":361793},[31792],[34,361795,361797,361817],{"className":361796},[31796,31797],[34,361798,361800,361814],{"className":361799},[31801],[34,361801,361803],{"className":361802,"style":33833},[31805],[34,361804,361805,361808],{"style":214424},[34,361806],{"className":361807,"style":31814},[31813],[34,361809,361811],{"className":361810},[31818,31819,31820,31821],[34,361812,33759],{"className":361813,"style":33774},[31685,31686,31821],[34,361815,31829],{"className":361816},[31828],[34,361818,361820],{"className":361819},[31801],[34,361821,361823],{"className":361822,"style":31836},[31805],[34,361824],{},[2230,361826,361827],{"start":56},[108,361828,361829,361831,361832,361860,361861,112501,361889,361917,361918,346345,361987,362189,362190,362358,362359,362387,362388,351462],{},[2627,361830,114043],{},"：\n构造出的 ",[34,361833,361835,361848],{"className":361834},[31642],[34,361836,361838],{"className":361837},[31646],[31648,361839,361840],{"xmlns":31650},[31652,361841,361842,361846],{},[31655,361843,361844],{},[31658,361845,67764],{},[31665,361847,67764],{"encoding":31667},[34,361849,361851],{"className":361850,"ariaHidden":758},[31672],[34,361852,361854,361857],{"className":361853},[31676],[34,361855],{"className":361856,"style":38774},[31680],[34,361858,67764],{"className":361859,"style":32062},[31685,31686]," 属于 ",[34,361862,361864,361877],{"className":361863},[31642],[34,361865,361867],{"className":361866},[31646],[31648,361868,361869],{"xmlns":31650},[31652,361870,361871,361875],{},[31655,361872,361873],{},[31658,361874,106063],{},[31665,361876,106063],{"encoding":31667},[34,361878,361880],{"className":361879,"ariaHidden":758},[31672],[34,361881,361883,361886],{"className":361882},[31676],[34,361884],{"className":361885,"style":33218},[31680],[34,361887,106063],{"className":361888,"style":67482},[31685,31686],[34,361890,361892,361905],{"className":361891},[31642],[34,361893,361895],{"className":361894},[31646],[31648,361896,361897],{"xmlns":31650},[31652,361898,361899,361903],{},[31655,361900,361901],{},[31658,361902,67764],{},[31665,361904,67764],{"encoding":31667},[34,361906,361908],{"className":361907,"ariaHidden":758},[31672],[34,361909,361911,361914],{"className":361910},[31676],[34,361912],{"className":361913,"style":38774},[31680],[34,361915,67764],{"className":361916,"style":32062},[31685,31686]," 不属于任何一个 ",[34,361919,361921,361938],{"className":361920},[31642],[34,361922,361924],{"className":361923},[31646],[31648,361925,361926],{"xmlns":31650},[31652,361927,361928,361936],{},[31655,361929,361930],{},[31743,361931,361932,361934],{},[31658,361933,105852],{},[31658,361935,31955],{},[31665,361937,351715],{"encoding":31667},[34,361939,361941],{"className":361940,"ariaHidden":758},[31672],[34,361942,361944,361947],{"className":361943},[31676],[34,361945],{"className":361946,"style":70915},[31680],[34,361948,361950,361953],{"className":361949},[31685],[34,361951,105852],{"className":361952,"style":37422},[31685,31686],[34,361954,361956],{"className":361955},[31792],[34,361957,361959,361979],{"className":361958},[31796,31797],[34,361960,361962,361976],{"className":361961},[31801],[34,361963,361965],{"className":361964,"style":32090},[31805],[34,361966,361967,361970],{"style":214424},[34,361968],{"className":361969,"style":31814},[31813],[34,361971,361973],{"className":361972},[31818,31819,31820,31821],[34,361974,31955],{"className":361975},[31685,31686,31821],[34,361977,31829],{"className":361978},[31828],[34,361980,361982],{"className":361981},[31801],[34,361983,361985],{"className":361984,"style":31836},[31805],[34,361986],{},[34,361988,361990,362026],{"className":361989},[31642],[34,361991,361993],{"className":361992},[31646],[31648,361994,361995],{"xmlns":31650},[31652,361996,361997,362023],{},[31655,361998,361999,362001,362003,362017],{},[31658,362000,67764],{},[31662,362002,112266],{"mathvariant":38079},[38846,362004,362005,362007,362015],{},[31662,362006,351927],{},[31655,362008,362009,362011,362013],{},[31658,362010,31955],{},[31662,362012,202],{},[31749,362014,640],{},[31658,362016,31660],{},[31743,362018,362019,362021],{},[31658,362020,105852],{},[31658,362022,31955],{},[31665,362024,362025],{"encoding":31667},"y \\notin \\bigcup_{i=1}^n P_i",[34,362027,362029,362080],{"className":362028,"ariaHidden":758},[31672],[34,362030,362032,362035,362038,362041,362077],{"className":362031},[31676],[34,362033],{"className":362034,"style":31774},[31680],[34,362036,67764],{"className":362037,"style":32062},[31685,31686],[34,362039],{"className":362040,"style":31846},[31845],[34,362042,362044,362050],{"className":362043},[31850],[34,362045,362047],{"className":362046},[31685],[34,362048,112296],{"className":362049},[31850],[34,362051,362053],{"className":362052},[31685,54774],[34,362054,362056],{"className":362055},[54778],[34,362057,362059,362062,362074],{"className":362058},[112306],[34,362060],{"className":362061,"style":31774},[31680],[34,362063,362065],{"className":362064},[54789],[34,362066,362068,362071],{"className":362067},[31685],[34,362069,38080],{"className":362070},[31685],[34,362072],{"className":362073,"style":112322},[31845],[34,362075],{"className":362076},[54800],[34,362078],{"className":362079,"style":31846},[31845],[34,362081,362083,362086,362146,362149],{"className":362082},[31676],[34,362084],{"className":362085,"style":351974},[31680],[34,362087,362089,362092],{"className":362088},[32767],[34,362090,351927],{"className":362091,"style":156561},[32767,131249,156560],[34,362093,362095],{"className":362094},[31792],[34,362096,362098,362138],{"className":362097},[31796,31797],[34,362099,362101,362135],{"className":362100},[31801],[34,362102,362104,362124],{"className":362103,"style":351993},[31805],[34,362105,362106,362109],{"style":156577},[34,362107],{"className":362108,"style":31814},[31813],[34,362110,362112],{"className":362111},[31818,31819,31820,31821],[34,362113,362115,362118,362121],{"className":362114},[31685,31821],[34,362116,31955],{"className":362117},[31685,31686,31821],[34,362119,202],{"className":362120},[31850,31821],[34,362122,640],{"className":362123},[31685,31821],[34,362125,362126,362129],{"style":352016},[34,362127],{"className":362128,"style":31814},[31813],[34,362130,362132],{"className":362131},[31818,31819,31820,31821],[34,362133,31660],{"className":362134},[31685,31686,31821],[34,362136,31829],{"className":362137},[31828],[34,362139,362141],{"className":362140},[31801],[34,362142,362144],{"className":362143,"style":352035},[31805],[34,362145],{},[34,362147],{"className":362148,"style":32123},[31845],[34,362150,362152,362155],{"className":362151},[31685],[34,362153,105852],{"className":362154,"style":37422},[31685,31686],[34,362156,362158],{"className":362157},[31792],[34,362159,362161,362181],{"className":362160},[31796,31797],[34,362162,362164,362178],{"className":362163},[31801],[34,362165,362167],{"className":362166,"style":32090},[31805],[34,362168,362169,362172],{"style":214424},[34,362170],{"className":362171,"style":31814},[31813],[34,362173,362175],{"className":362174},[31818,31819,31820,31821],[34,362176,31955],{"className":362177},[31685,31686,31821],[34,362179,31829],{"className":362180},[31828],[34,362182,362184],{"className":362183},[31801],[34,362185,362187],{"className":362186,"style":31836},[31805],[34,362188],{},"。\n这与已知条件 ",[34,362191,362193,362228],{"className":362192},[31642],[34,362194,362196],{"className":362195},[31646],[31648,362197,362198],{"xmlns":31650},[31652,362199,362200,362226],{},[31655,362201,362202,362204,362206,362220],{},[31658,362203,106063],{},[31662,362205,131855],{},[38846,362207,362208,362210,362218],{},[31662,362209,351927],{},[31655,362211,362212,362214,362216],{},[31658,362213,31955],{},[31662,362215,202],{},[31749,362217,640],{},[31658,362219,31660],{},[31743,362221,362222,362224],{},[31658,362223,105852],{},[31658,362225,31955],{},[31665,362227,351946],{"encoding":31667},[34,362229,362231,362249],{"className":362230,"ariaHidden":758},[31672],[34,362232,362234,362237,362240,362243,362246],{"className":362233},[31676],[34,362235],{"className":362236,"style":195047},[31680],[34,362238,106063],{"className":362239,"style":67482},[31685,31686],[34,362241],{"className":362242,"style":31846},[31845],[34,362244,131855],{"className":362245},[31850],[34,362247],{"className":362248,"style":31846},[31845],[34,362250,362252,362255,362315,362318],{"className":362251},[31676],[34,362253],{"className":362254,"style":351974},[31680],[34,362256,362258,362261],{"className":362257},[32767],[34,362259,351927],{"className":362260,"style":156561},[32767,131249,156560],[34,362262,362264],{"className":362263},[31792],[34,362265,362267,362307],{"className":362266},[31796,31797],[34,362268,362270,362304],{"className":362269},[31801],[34,362271,362273,362293],{"className":362272,"style":351993},[31805],[34,362274,362275,362278],{"style":156577},[34,362276],{"className":362277,"style":31814},[31813],[34,362279,362281],{"className":362280},[31818,31819,31820,31821],[34,362282,362284,362287,362290],{"className":362283},[31685,31821],[34,362285,31955],{"className":362286},[31685,31686,31821],[34,362288,202],{"className":362289},[31850,31821],[34,362291,640],{"className":362292},[31685,31821],[34,362294,362295,362298],{"style":352016},[34,362296],{"className":362297,"style":31814},[31813],[34,362299,362301],{"className":362300},[31818,31819,31820,31821],[34,362302,31660],{"className":362303},[31685,31686,31821],[34,362305,31829],{"className":362306},[31828],[34,362308,362310],{"className":362309},[31801],[34,362311,362313],{"className":362312,"style":352035},[31805],[34,362314],{},[34,362316],{"className":362317,"style":32123},[31845],[34,362319,362321,362324],{"className":362320},[31685],[34,362322,105852],{"className":362323,"style":37422},[31685,31686],[34,362325,362327],{"className":362326},[31792],[34,362328,362330,362350],{"className":362329},[31796,31797],[34,362331,362333,362347],{"className":362332},[31801],[34,362334,362336],{"className":362335,"style":32090},[31805],[34,362337,362338,362341],{"style":214424},[34,362339],{"className":362340,"style":31814},[31813],[34,362342,362344],{"className":362343},[31818,31819,31820,31821],[34,362345,31955],{"className":362346},[31685,31686,31821],[34,362348,31829],{"className":362349},[31828],[34,362351,362353],{"className":362352},[31801],[34,362354,362356],{"className":362355,"style":31836},[31805],[34,362357],{}," 矛盾。\n所以，假设错误，",[34,362360,362362,362375],{"className":362361},[31642],[34,362363,362365],{"className":362364},[31646],[31648,362366,362367],{"xmlns":31650},[31652,362368,362369,362373],{},[31655,362370,362371],{},[31658,362372,106063],{},[31665,362374,106063],{"encoding":31667},[34,362376,362378],{"className":362377,"ariaHidden":758},[31672],[34,362379,362381,362384],{"className":362380},[31676],[34,362382],{"className":362383,"style":33218},[31680],[34,362385,106063],{"className":362386,"style":67482},[31685,31686]," 必须含于某个 ",[34,362389,362391,362408],{"className":362390},[31642],[34,362392,362394],{"className":362393},[31646],[31648,362395,362396],{"xmlns":31650},[31652,362397,362398,362406],{},[31655,362399,362400],{},[31743,362401,362402,362404],{},[31658,362403,105852],{},[31658,362405,31955],{},[31665,362407,351715],{"encoding":31667},[34,362409,362411],{"className":362410,"ariaHidden":758},[31672],[34,362412,362414,362417],{"className":362413},[31676],[34,362415],{"className":362416,"style":70915},[31680],[34,362418,362420,362423],{"className":362419},[31685],[34,362421,105852],{"className":362422,"style":37422},[31685,31686],[34,362424,362426],{"className":362425},[31792],[34,362427,362429,362449],{"className":362428},[31796,31797],[34,362430,362432,362446],{"className":362431},[31801],[34,362433,362435],{"className":362434,"style":32090},[31805],[34,362436,362437,362440],{"style":214424},[34,362438],{"className":362439,"style":31814},[31813],[34,362441,362443],{"className":362442},[31818,31819,31820,31821],[34,362444,31955],{"className":362445},[31685,31686,31821],[34,362447,31829],{"className":362448},[31828],[34,362450,362452],{"className":362451},[31801],[34,362453,362455],{"className":362454,"style":31836},[31805],[34,362456],{},[32188,362458],{},[253,362460,362462],{"id":362461},"题号-10","题号 10",[32490,362464,162043],{"id":170039},[15,362466,362467],{},"证明有限环的素理想都是极大理想。",[32188,362469],{},[32490,362471,339412],{"id":362472},"深度分析与解答-3",[15,362474,362475],{},[2627,362476,346809],{},[105,362478,362479,362545],{},[108,362480,362481,362484,362485,113124,362513,362516,362517,31901],{},[2627,362482,362483],{},"假设","：这里通常假设环 ",[34,362486,362488,362501],{"className":362487},[31642],[34,362489,362491],{"className":362490},[31646],[31648,362492,362493],{"xmlns":31650},[31652,362494,362495,362499],{},[31655,362496,362497],{},[31658,362498,34386],{},[31665,362500,34386],{"encoding":31667},[34,362502,362504],{"className":362503,"ariaHidden":758},[31672],[34,362505,362507,362510],{"className":362506},[31676],[34,362508],{"className":362509,"style":33218},[31680],[34,362511,34386],{"className":362512,"style":34427},[31685,31686],[2627,362514,362515],{},"交换幺环","（有乘法单位元 ",[34,362518,362520,362533],{"className":362519},[31642],[34,362521,362523],{"className":362522},[31646],[31648,362524,362525],{"xmlns":31650},[31652,362526,362527,362531],{},[31655,362528,362529],{},[31749,362530,640],{},[31665,362532,640],{"encoding":31667},[34,362534,362536],{"className":362535,"ariaHidden":758},[31672],[34,362537,362539,362542],{"className":362538},[31676],[34,362540],{"className":362541,"style":32617},[31680],[34,362543,640],{"className":362544},[31685],[108,362546,362547,2664],{},[2627,362548,362549],{},"逻辑链条",[2230,362551,362552,362690,362827,362897],{},[108,362553,362554,11008,362582,362610,362611,331494,362649,113124,362687,347],{},[34,362555,362557,362570],{"className":362556},[31642],[34,362558,362560],{"className":362559},[31646],[31648,362561,362562],{"xmlns":31650},[31652,362563,362564,362568],{},[31655,362565,362566],{},[31658,362567,105852],{},[31665,362569,105852],{"encoding":31667},[34,362571,362573],{"className":362572,"ariaHidden":758},[31672],[34,362574,362576,362579],{"className":362575},[31676],[34,362577],{"className":362578,"style":33218},[31680],[34,362580,105852],{"className":362581,"style":37422},[31685,31686],[34,362583,362585,362598],{"className":362584},[31642],[34,362586,362588],{"className":362587},[31646],[31648,362589,362590],{"xmlns":31650},[31652,362591,362592,362596],{},[31655,362593,362594],{},[31658,362595,34386],{},[31665,362597,34386],{"encoding":31667},[34,362599,362601],{"className":362600,"ariaHidden":758},[31672],[34,362602,362604,362607],{"className":362603},[31676],[34,362605],{"className":362606,"style":33218},[31680],[34,362608,34386],{"className":362609,"style":34427},[31685,31686]," 的素理想 ",[34,362612,362614,362631],{"className":362613},[31642],[34,362615,362617],{"className":362616},[31646],[31648,362618,362619],{"xmlns":31650},[31652,362620,362621,362629],{},[31655,362622,362623,362625,362627],{},[39361,362624,262622],{},[31662,362626,323391],{},[39361,362628,262622],{},[31665,362630,323396],{"encoding":31667},[34,362632,362634],{"className":362633,"ariaHidden":758},[31672],[34,362635,362637,362640,362643,362646],{"className":362636},[31676],[34,362638],{"className":362639,"style":318264},[31680],[34,362641],{"className":362642,"style":31846},[31845],[34,362644,323391],{"className":362645},[31850],[34,362647],{"className":362648,"style":31846},[31845],[34,362650,362652,362669],{"className":362651},[31642],[34,362653,362655],{"className":362654},[31646],[31648,362656,362657],{"xmlns":31650},[31652,362658,362659,362667],{},[31655,362660,362661,362663,362665],{},[31658,362662,34386],{},[31658,362664,38080],{"mathvariant":38079},[31658,362666,105852],{},[31665,362668,331515],{"encoding":31667},[34,362670,362672],{"className":362671,"ariaHidden":758},[31672],[34,362673,362675,362678,362681,362684],{"className":362674},[31676],[34,362676],{"className":362677,"style":31774},[31680],[34,362679,34386],{"className":362680,"style":34427},[31685,31686],[34,362682,38080],{"className":362683},[31685],[34,362685,105852],{"className":362686,"style":37422},[31685,31686],[2627,362688,362689],{},"整环 (Integral Domain)",[108,362691,362692,11008,362720,362748,362749,331494,362787,113124,362825,347],{},[34,362693,362695,362708],{"className":362694},[31642],[34,362696,362698],{"className":362697},[31646],[31648,362699,362700],{"xmlns":31650},[31652,362701,362702,362706],{},[31655,362703,362704],{},[31658,362705,105852],{},[31665,362707,105852],{"encoding":31667},[34,362709,362711],{"className":362710,"ariaHidden":758},[31672],[34,362712,362714,362717],{"className":362713},[31676],[34,362715],{"className":362716,"style":33218},[31680],[34,362718,105852],{"className":362719,"style":37422},[31685,31686],[34,362721,362723,362736],{"className":362722},[31642],[34,362724,362726],{"className":362725},[31646],[31648,362727,362728],{"xmlns":31650},[31652,362729,362730,362734],{},[31655,362731,362732],{},[31658,362733,34386],{},[31665,362735,34386],{"encoding":31667},[34,362737,362739],{"className":362738,"ariaHidden":758},[31672],[34,362740,362742,362745],{"className":362741},[31676],[34,362743],{"className":362744,"style":33218},[31680],[34,362746,34386],{"className":362747,"style":34427},[31685,31686]," 的极大理想 ",[34,362750,362752,362769],{"className":362751},[31642],[34,362753,362755],{"className":362754},[31646],[31648,362756,362757],{"xmlns":31650},[31652,362758,362759,362767],{},[31655,362760,362761,362763,362765],{},[39361,362762,262622],{},[31662,362764,323391],{},[39361,362766,262622],{},[31665,362768,323396],{"encoding":31667},[34,362770,362772],{"className":362771,"ariaHidden":758},[31672],[34,362773,362775,362778,362781,362784],{"className":362774},[31676],[34,362776],{"className":362777,"style":318264},[31680],[34,362779],{"className":362780,"style":31846},[31845],[34,362782,323391],{"className":362783},[31850],[34,362785],{"className":362786,"style":31846},[31845],[34,362788,362790,362807],{"className":362789},[31642],[34,362791,362793],{"className":362792},[31646],[31648,362794,362795],{"xmlns":31650},[31652,362796,362797,362805],{},[31655,362798,362799,362801,362803],{},[31658,362800,34386],{},[31658,362802,38080],{"mathvariant":38079},[31658,362804,105852],{},[31665,362806,331515],{"encoding":31667},[34,362808,362810],{"className":362809,"ariaHidden":758},[31672],[34,362811,362813,362816,362819,362822],{"className":362812},[31676],[34,362814],{"className":362815,"style":31774},[31680],[34,362817,34386],{"className":362818,"style":34427},[31685,31686],[34,362820,38080],{"className":362821},[31685],[34,362823,105852],{"className":362824,"style":37422},[31685,31686],[2627,362826,340387],{},[108,362828,93310,362829,362857,362858,362896],{},[34,362830,362832,362845],{"className":362831},[31642],[34,362833,362835],{"className":362834},[31646],[31648,362836,362837],{"xmlns":31650},[31652,362838,362839,362843],{},[31655,362840,362841],{},[31658,362842,34386],{},[31665,362844,34386],{"encoding":31667},[34,362846,362848],{"className":362847,"ariaHidden":758},[31672],[34,362849,362851,362854],{"className":362850},[31676],[34,362852],{"className":362853,"style":33218},[31680],[34,362855,34386],{"className":362856,"style":34427},[31685,31686]," 是有限环，商环 ",[34,362859,362861,362878],{"className":362860},[31642],[34,362862,362864],{"className":362863},[31646],[31648,362865,362866],{"xmlns":31650},[31652,362867,362868,362876],{},[31655,362869,362870,362872,362874],{},[31658,362871,34386],{},[31658,362873,38080],{"mathvariant":38079},[31658,362875,105852],{},[31665,362877,331515],{"encoding":31667},[34,362879,362881],{"className":362880,"ariaHidden":758},[31672],[34,362882,362884,362887,362890,362893],{"className":362883},[31676],[34,362885],{"className":362886,"style":31774},[31680],[34,362888,34386],{"className":362889,"style":34427},[31685,31686],[34,362891,38080],{"className":362892},[31685],[34,362894,105852],{"className":362895,"style":37422},[31685,31686]," 也是有限的。",[108,362898,362899,362900,347],{},"题目转化为证明：",[2627,362901,362902],{},"有限整环必定是域",[15,362904,362905],{},[2627,362906,340393],{},[2230,362908,362909,363153,363744,364501,364783],{},[108,362910,362911,362914,362915,362943,362944,11008,362972,363000,363001,363062,363063,363091,363092,363120,363121,363149,363150,347],{},[2627,362912,362913],{},"第一步：利用素理想性质转化问题","\n设 ",[34,362916,362918,362931],{"className":362917},[31642],[34,362919,362921],{"className":362920},[31646],[31648,362922,362923],{"xmlns":31650},[31652,362924,362925,362929],{},[31655,362926,362927],{},[31658,362928,34386],{},[31665,362930,34386],{"encoding":31667},[34,362932,362934],{"className":362933,"ariaHidden":758},[31672],[34,362935,362937,362940],{"className":362936},[31676],[34,362938],{"className":362939,"style":33218},[31680],[34,362941,34386],{"className":362942,"style":34427},[31685,31686]," 是一个有限交换幺环，",[34,362945,362947,362960],{"className":362946},[31642],[34,362948,362950],{"className":362949},[31646],[31648,362951,362952],{"xmlns":31650},[31652,362953,362954,362958],{},[31655,362955,362956],{},[31658,362957,105852],{},[31665,362959,105852],{"encoding":31667},[34,362961,362963],{"className":362962,"ariaHidden":758},[31672],[34,362964,362966,362969],{"className":362965},[31676],[34,362967],{"className":362968,"style":33218},[31680],[34,362970,105852],{"className":362971,"style":37422},[31685,31686],[34,362973,362975,362988],{"className":362974},[31642],[34,362976,362978],{"className":362977},[31646],[31648,362979,362980],{"xmlns":31650},[31652,362981,362982,362986],{},[31655,362983,362984],{},[31658,362985,34386],{},[31665,362987,34386],{"encoding":31667},[34,362989,362991],{"className":362990,"ariaHidden":758},[31672],[34,362992,362994,362997],{"className":362993},[31676],[34,362995],{"className":362996,"style":33218},[31680],[34,362998,34386],{"className":362999,"style":34427},[31685,31686]," 的一个素理想。\n根据素理想的定义，商环 ",[34,363002,363004,363026],{"className":363003},[31642],[34,363005,363007],{"className":363006},[31646],[31648,363008,363009],{"xmlns":31650},[31652,363010,363011,363023],{},[31655,363012,363013,363015,363017,363019,363021],{},[31658,363014,91994],{},[31662,363016,202],{},[31658,363018,34386],{},[31658,363020,38080],{"mathvariant":38079},[31658,363022,105852],{},[31665,363024,363025],{"encoding":31667},"D = R\u002FP",[34,363027,363029,363047],{"className":363028,"ariaHidden":758},[31672],[34,363030,363032,363035,363038,363041,363044],{"className":363031},[31676],[34,363033],{"className":363034,"style":33218},[31680],[34,363036,91994],{"className":363037,"style":34452},[31685,31686],[34,363039],{"className":363040,"style":31846},[31845],[34,363042,202],{"className":363043},[31850],[34,363045],{"className":363046,"style":31846},[31845],[34,363048,363050,363053,363056,363059],{"className":363049},[31676],[34,363051],{"className":363052,"style":31774},[31680],[34,363054,34386],{"className":363055,"style":34427},[31685,31686],[34,363057,38080],{"className":363058},[31685],[34,363060,105852],{"className":363061,"style":37422},[31685,31686]," 是一个整环。\n由于 ",[34,363064,363066,363079],{"className":363065},[31642],[34,363067,363069],{"className":363068},[31646],[31648,363070,363071],{"xmlns":31650},[31652,363072,363073,363077],{},[31655,363074,363075],{},[31658,363076,34386],{},[31665,363078,34386],{"encoding":31667},[34,363080,363082],{"className":363081,"ariaHidden":758},[31672],[34,363083,363085,363088],{"className":363084},[31676],[34,363086],{"className":363087,"style":33218},[31680],[34,363089,34386],{"className":363090,"style":34427},[31685,31686]," 是有限集合，其商集 ",[34,363093,363095,363108],{"className":363094},[31642],[34,363096,363098],{"className":363097},[31646],[31648,363099,363100],{"xmlns":31650},[31652,363101,363102,363106],{},[31655,363103,363104],{},[31658,363105,91994],{},[31665,363107,91994],{"encoding":31667},[34,363109,363111],{"className":363110,"ariaHidden":758},[31672],[34,363112,363114,363117],{"className":363113},[31676],[34,363115],{"className":363116,"style":33218},[31680],[34,363118,91994],{"className":363119,"style":34452},[31685,31686]," 显然也是有限集合。\n因此，",[34,363122,363124,363137],{"className":363123},[31642],[34,363125,363127],{"className":363126},[31646],[31648,363128,363129],{"xmlns":31650},[31652,363130,363131,363135],{},[31655,363132,363133],{},[31658,363134,91994],{},[31665,363136,91994],{"encoding":31667},[34,363138,363140],{"className":363139,"ariaHidden":758},[31672],[34,363141,363143,363146],{"className":363142},[31676],[34,363144],{"className":363145,"style":33218},[31680],[34,363147,91994],{"className":363148,"style":34452},[31685,31686]," 是一个",[2627,363151,363152],{},"有限整环",[108,363154,363155,363158,363159,363187,363188,363216,363217,363415,363416,1840,363467,363551,363552,363685,363686,363714,363715,363743],{},[2627,363156,363157],{},"第二步：证明有限整环是域","\n要证明 ",[34,363160,363162,363175],{"className":363161},[31642],[34,363163,363165],{"className":363164},[31646],[31648,363166,363167],{"xmlns":31650},[31652,363168,363169,363173],{},[31655,363170,363171],{},[31658,363172,91994],{},[31665,363174,91994],{"encoding":31667},[34,363176,363178],{"className":363177,"ariaHidden":758},[31672],[34,363179,363181,363184],{"className":363180},[31676],[34,363182],{"className":363183,"style":33218},[31680],[34,363185,91994],{"className":363186,"style":34452},[31685,31686]," 是域，我们需要证明 ",[34,363189,363191,363204],{"className":363190},[31642],[34,363192,363194],{"className":363193},[31646],[31648,363195,363196],{"xmlns":31650},[31652,363197,363198,363202],{},[31655,363199,363200],{},[31658,363201,91994],{},[31665,363203,91994],{"encoding":31667},[34,363205,363207],{"className":363206,"ariaHidden":758},[31672],[34,363208,363210,363213],{"className":363209},[31676],[34,363211],{"className":363212,"style":33218},[31680],[34,363214,91994],{"className":363215,"style":34452},[31685,31686]," 中的每一个非零元素都是可逆的。\n设 ",[34,363218,363220,363266],{"className":363219},[31642],[34,363221,363223],{"className":363222},[31646],[31648,363224,363225],{"xmlns":31650},[31652,363226,363227,363263],{},[31655,363228,363229,363231,363233,363235,363237,363239,363241,363243,363249,363251,363253,363255,363261],{},[31658,363230,91994],{},[31662,363232,202],{},[31662,363234,32685],{"stretchy":702},[31749,363236,459],{},[31662,363238,31960],{"separator":758},[31749,363240,640],{},[31662,363242,31960],{"separator":758},[31743,363244,363245,363247],{},[31658,363246,46900],{},[31749,363248,786],{},[31662,363250,31960],{"separator":758},[31662,363252,33934],{},[31662,363254,31960],{"separator":758},[31743,363256,363257,363259],{},[31658,363258,46900],{},[31658,363260,31759],{},[31662,363262,32715],{"stretchy":702},[31665,363264,363265],{"encoding":31667},"D = \\{0, 1, d_3, \\dots, d_m\\}",[34,363267,363269,363287],{"className":363268,"ariaHidden":758},[31672],[34,363270,363272,363275,363278,363281,363284],{"className":363271},[31676],[34,363273],{"className":363274,"style":33218},[31680],[34,363276,91994],{"className":363277,"style":34452},[31685,31686],[34,363279],{"className":363280,"style":31846},[31845],[34,363282,202],{"className":363283},[31850],[34,363285],{"className":363286,"style":31846},[31845],[34,363288,363290,363293,363296,363299,363302,363305,363308,363311,363314,363354,363357,363360,363363,363366,363369,363372,363412],{"className":363289},[31676],[34,363291],{"className":363292,"style":31774},[31680],[34,363294,32685],{"className":363295},[31782],[34,363297,459],{"className":363298},[31685],[34,363300,31960],{"className":363301},[32119],[34,363303],{"className":363304,"style":32123},[31845],[34,363306,640],{"className":363307},[31685],[34,363309,31960],{"className":363310},[32119],[34,363312],{"className":363313,"style":32123},[31845],[34,363315,363317,363320],{"className":363316},[31685],[34,363318,46900],{"className":363319},[31685,31686],[34,363321,363323],{"className":363322},[31792],[34,363324,363326,363346],{"className":363325},[31796,31797],[34,363327,363329,363343],{"className":363328},[31801],[34,363330,363332],{"className":363331,"style":31806},[31805],[34,363333,363334,363337],{"style":31809},[34,363335],{"className":363336,"style":31814},[31813],[34,363338,363340],{"className":363339},[31818,31819,31820,31821],[34,363341,786],{"className":363342},[31685,31821],[34,363344,31829],{"className":363345},[31828],[34,363347,363349],{"className":363348},[31801],[34,363350,363352],{"className":363351,"style":31836},[31805],[34,363353],{},[34,363355,31960],{"className":363356},[32119],[34,363358],{"className":363359,"style":32123},[31845],[34,363361,33934],{"className":363362},[34006],[34,363364],{"className":363365,"style":32123},[31845],[34,363367,31960],{"className":363368},[32119],[34,363370],{"className":363371,"style":32123},[31845],[34,363373,363375,363378],{"className":363374},[31685],[34,363376,46900],{"className":363377},[31685,31686],[34,363379,363381],{"className":363380},[31792],[34,363382,363384,363404],{"className":363383},[31796,31797],[34,363385,363387,363401],{"className":363386},[31801],[34,363388,363390],{"className":363389,"style":32362},[31805],[34,363391,363392,363395],{"style":31809},[34,363393],{"className":363394,"style":31814},[31813],[34,363396,363398],{"className":363397},[31818,31819,31820,31821],[34,363399,31759],{"className":363400},[31685,31686,31821],[34,363402,31829],{"className":363403},[31828],[34,363405,363407],{"className":363406},[31801],[34,363408,363410],{"className":363409,"style":31836},[31805],[34,363411],{},[34,363413,32715],{"className":363414},[31690]," 是有限整环的元素集合。\n取任意非零元素 ",[34,363417,363419,363437],{"className":363418},[31642],[34,363420,363422],{"className":363421},[31646],[31648,363423,363424],{"xmlns":31650},[31652,363425,363426,363434],{},[31655,363427,363428,363430,363432],{},[31658,363429,111],{},[31662,363431,112296],{},[31658,363433,91994],{},[31665,363435,363436],{"encoding":31667},"a \\in D",[34,363438,363440,363458],{"className":363439,"ariaHidden":758},[31672],[34,363441,363443,363446,363449,363452,363455],{"className":363442},[31676],[34,363444],{"className":363445,"style":34812},[31680],[34,363447,111],{"className":363448},[31685,31686],[34,363450],{"className":363451,"style":31846},[31845],[34,363453,112296],{"className":363454},[31850],[34,363456],{"className":363457,"style":31846},[31845],[34,363459,363461,363464],{"className":363460},[31676],[34,363462],{"className":363463,"style":33218},[31680],[34,363465,91994],{"className":363466,"style":34452},[31685,31686],[34,363468,363470,363488],{"className":363469},[31642],[34,363471,363473],{"className":363472},[31646],[31648,363474,363475],{"xmlns":31650},[31652,363476,363477,363485],{},[31655,363478,363479,363481,363483],{},[31658,363480,111],{},[31662,363482,54707],{"mathvariant":38079},[31749,363484,459],{},[31665,363486,363487],{"encoding":31667},"a \\neq 0",[34,363489,363491,363542],{"className":363490,"ariaHidden":758},[31672],[34,363492,363494,363497,363500,363503,363539],{"className":363493},[31676],[34,363495],{"className":363496,"style":43217},[31680],[34,363498,111],{"className":363499},[31685,31686],[34,363501],{"className":363502,"style":31846},[31845],[34,363504,363506,363533,363536],{"className":363505},[31850],[34,363507,363509],{"className":363508},[31850],[34,363510,363512],{"className":363511},[31685,54774],[34,363513,363515],{"className":363514},[54778],[34,363516,363518,363521,363530],{"className":363517},[54782],[34,363519],{"className":363520,"style":43217},[31680],[34,363522,363524],{"className":363523},[54789],[34,363525,363527],{"className":363526},[31685],[34,363528,54796],{"className":363529},[31850],[34,363531],{"className":363532},[54800],[34,363534],{"className":363535},[31845,54804],[34,363537,202],{"className":363538},[31850],[34,363540],{"className":363541,"style":31846},[31845],[34,363543,363545,363548],{"className":363544},[31676],[34,363546],{"className":363547,"style":32617},[31680],[34,363549,459],{"className":363550},[31685],")。\n考虑构造一个映射（或者说是做乘法变换）：\n",[34,363553,363555,363595],{"className":363554},[31642],[34,363556,363558],{"className":363557},[31646],[31648,363559,363560],{"xmlns":31650},[31652,363561,363562,363592],{},[31655,363563,363564,363566,363568,363570,363572,363574,363576,363578,363580,363582,363584,363586,363588,363590],{},[31658,363565,329201],{},[31662,363567,5598],{},[31658,363569,91994],{},[31662,363571,33239],{},[31658,363573,91994],{},[31662,363575,31960],{"separator":758},[31845,363577],{"width":71082},[31658,363579,329201],{},[31662,363581,456],{"stretchy":702},[31658,363583,32936],{},[31662,363585,2557],{"stretchy":702},[31662,363587,202],{},[31658,363589,111],{},[31658,363591,32936],{},[31665,363593,363594],{"encoding":31667},"\\phi: D \\to D, \\quad \\phi(x) = ax",[34,363596,363598,363616,363634,363673],{"className":363597,"ariaHidden":758},[31672],[34,363599,363601,363604,363607,363610,363613],{"className":363600},[31676],[34,363602],{"className":363603,"style":43217},[31680],[34,363605,329201],{"className":363606},[31685,31686],[34,363608],{"className":363609,"style":31846},[31845],[34,363611,5598],{"className":363612},[31850],[34,363614],{"className":363615,"style":31846},[31845],[34,363617,363619,363622,363625,363628,363631],{"className":363618},[31676],[34,363620],{"className":363621,"style":33218},[31680],[34,363623,91994],{"className":363624,"style":34452},[31685,31686],[34,363626],{"className":363627,"style":31846},[31845],[34,363629,33239],{"className":363630},[31850],[34,363632],{"className":363633,"style":31846},[31845],[34,363635,363637,363640,363643,363646,363649,363652,363655,363658,363661,363664,363667,363670],{"className":363636},[31676],[34,363638],{"className":363639,"style":31774},[31680],[34,363641,91994],{"className":363642,"style":34452},[31685,31686],[34,363644,31960],{"className":363645},[32119],[34,363647],{"className":363648,"style":128438},[31845],[34,363650],{"className":363651,"style":32123},[31845],[34,363653,329201],{"className":363654},[31685,31686],[34,363656,456],{"className":363657},[31782],[34,363659,32936],{"className":363660},[31685,31686],[34,363662,2557],{"className":363663},[31690],[34,363665],{"className":363666,"style":31846},[31845],[34,363668,202],{"className":363669},[31850],[34,363671],{"className":363672,"style":31846},[31845],[34,363674,363676,363679,363682],{"className":363675},[31676],[34,363677],{"className":363678,"style":32527},[31680],[34,363680,111],{"className":363681},[31685,31686],[34,363683,32936],{"className":363684},[31685,31686],"\n即用 ",[34,363687,363689,363702],{"className":363688},[31642],[34,363690,363692],{"className":363691},[31646],[31648,363693,363694],{"xmlns":31650},[31652,363695,363696,363700],{},[31655,363697,363698],{},[31658,363699,111],{},[31665,363701,111],{"encoding":31667},[34,363703,363705],{"className":363704,"ariaHidden":758},[31672],[34,363706,363708,363711],{"className":363707},[31676],[34,363709],{"className":363710,"style":32527},[31680],[34,363712,111],{"className":363713},[31685,31686]," 去乘 ",[34,363716,363718,363731],{"className":363717},[31642],[34,363719,363721],{"className":363720},[31646],[31648,363722,363723],{"xmlns":31650},[31652,363724,363725,363729],{},[31655,363726,363727],{},[31658,363728,91994],{},[31665,363730,91994],{"encoding":31667},[34,363732,363734],{"className":363733,"ariaHidden":758},[31672],[34,363735,363737,363740],{"className":363736},[31676],[34,363738],{"className":363739,"style":33218},[31680],[34,363741,91994],{"className":363742,"style":34452},[31685,31686]," 中的所有元素。",[108,363745,363746,315440,363749,363777,363778,363781,363782,55016,363925,86621,364095,364178,364179,61605,364334,364467,364468,113124,364497,364500],{},[2627,363747,363748],{},"第三步：利用消去律证明单射",[34,363750,363752,363765],{"className":363751},[31642],[34,363753,363755],{"className":363754},[31646],[31648,363756,363757],{"xmlns":31650},[31652,363758,363759,363763],{},[31655,363760,363761],{},[31658,363762,91994],{},[31665,363764,91994],{"encoding":31667},[34,363766,363768],{"className":363767,"ariaHidden":758},[31672],[34,363769,363771,363774],{"className":363770},[31676],[34,363772],{"className":363773,"style":33218},[31680],[34,363775,91994],{"className":363776,"style":34452},[31685,31686]," 是整环，整环满足",[2627,363779,363780],{},"消去律","（无零因子）。\n如果 ",[34,363783,363785,363815],{"className":363784},[31642],[34,363786,363788],{"className":363787},[31646],[31648,363789,363790],{"xmlns":31650},[31652,363791,363792,363812],{},[31655,363793,363794,363796,363802,363804,363806],{},[31658,363795,111],{},[31743,363797,363798,363800],{},[31658,363799,32936],{},[31749,363801,640],{},[31662,363803,202],{},[31658,363805,111],{},[31743,363807,363808,363810],{},[31658,363809,32936],{},[31749,363811,768],{},[31665,363813,363814],{"encoding":31667},"ax_1 = ax_2",[34,363816,363818,363876],{"className":363817,"ariaHidden":758},[31672],[34,363819,363821,363824,363827,363867,363870,363873],{"className":363820},[31676],[34,363822],{"className":363823,"style":31860},[31680],[34,363825,111],{"className":363826},[31685,31686],[34,363828,363830,363833],{"className":363829},[31685],[34,363831,32936],{"className":363832},[31685,31686],[34,363834,363836],{"className":363835},[31792],[34,363837,363839,363859],{"className":363838},[31796,31797],[34,363840,363842,363856],{"className":363841},[31801],[34,363843,363845],{"className":363844,"style":31806},[31805],[34,363846,363847,363850],{"style":31809},[34,363848],{"className":363849,"style":31814},[31813],[34,363851,363853],{"className":363852},[31818,31819,31820,31821],[34,363854,640],{"className":363855},[31685,31821],[34,363857,31829],{"className":363858},[31828],[34,363860,363862],{"className":363861},[31801],[34,363863,363865],{"className":363864,"style":31836},[31805],[34,363866],{},[34,363868],{"className":363869,"style":31846},[31845],[34,363871,202],{"className":363872},[31850],[34,363874],{"className":363875,"style":31846},[31845],[34,363877,363879,363882,363885],{"className":363878},[31676],[34,363880],{"className":363881,"style":31860},[31680],[34,363883,111],{"className":363884},[31685,31686],[34,363886,363888,363891],{"className":363887},[31685],[34,363889,32936],{"className":363890},[31685,31686],[34,363892,363894],{"className":363893},[31792],[34,363895,363897,363917],{"className":363896},[31796,31797],[34,363898,363900,363914],{"className":363899},[31801],[34,363901,363903],{"className":363902,"style":31806},[31805],[34,363904,363905,363908],{"style":31809},[34,363906],{"className":363907,"style":31814},[31813],[34,363909,363911],{"className":363910},[31818,31819,31820,31821],[34,363912,768],{"className":363913},[31685,31821],[34,363915,31829],{"className":363916},[31828],[34,363918,363920],{"className":363919},[31801],[34,363921,363923],{"className":363922,"style":31836},[31805],[34,363924],{},[34,363926,363928,363964],{"className":363927},[31642],[34,363929,363931],{"className":363930},[31646],[31648,363932,363933],{"xmlns":31650},[31652,363934,363935,363961],{},[31655,363936,363937,363939,363941,363947,363949,363955,363957,363959],{},[31658,363938,111],{},[31662,363940,456],{"stretchy":702},[31743,363942,363943,363945],{},[31658,363944,32936],{},[31749,363946,640],{},[31662,363948,32226],{},[31743,363950,363951,363953],{},[31658,363952,32936],{},[31749,363954,768],{},[31662,363956,2557],{"stretchy":702},[31662,363958,202],{},[31749,363960,459],{},[31665,363962,363963],{"encoding":31667},"a(x_1 - x_2) = 0",[34,363965,363967,364028,364086],{"className":363966,"ariaHidden":758},[31672],[34,363968,363970,363973,363976,363979,364019,364022,364025],{"className":363969},[31676],[34,363971],{"className":363972,"style":31774},[31680],[34,363974,111],{"className":363975},[31685,31686],[34,363977,456],{"className":363978},[31782],[34,363980,363982,363985],{"className":363981},[31685],[34,363983,32936],{"className":363984},[31685,31686],[34,363986,363988],{"className":363987},[31792],[34,363989,363991,364011],{"className":363990},[31796,31797],[34,363992,363994,364008],{"className":363993},[31801],[34,363995,363997],{"className":363996,"style":31806},[31805],[34,363998,363999,364002],{"style":31809},[34,364000],{"className":364001,"style":31814},[31813],[34,364003,364005],{"className":364004},[31818,31819,31820,31821],[34,364006,640],{"className":364007},[31685,31821],[34,364009,31829],{"className":364010},[31828],[34,364012,364014],{"className":364013},[31801],[34,364015,364017],{"className":364016,"style":31836},[31805],[34,364018],{},[34,364020],{"className":364021,"style":32604},[31845],[34,364023,32226],{"className":364024},[32024],[34,364026],{"className":364027,"style":32604},[31845],[34,364029,364031,364034,364074,364077,364080,364083],{"className":364030},[31676],[34,364032],{"className":364033,"style":31774},[31680],[34,364035,364037,364040],{"className":364036},[31685],[34,364038,32936],{"className":364039},[31685,31686],[34,364041,364043],{"className":364042},[31792],[34,364044,364046,364066],{"className":364045},[31796,31797],[34,364047,364049,364063],{"className":364048},[31801],[34,364050,364052],{"className":364051,"style":31806},[31805],[34,364053,364054,364057],{"style":31809},[34,364055],{"className":364056,"style":31814},[31813],[34,364058,364060],{"className":364059},[31818,31819,31820,31821],[34,364061,768],{"className":364062},[31685,31821],[34,364064,31829],{"className":364065},[31828],[34,364067,364069],{"className":364068},[31801],[34,364070,364072],{"className":364071,"style":31836},[31805],[34,364073],{},[34,364075,2557],{"className":364076},[31690],[34,364078],{"className":364079,"style":31846},[31845],[34,364081,202],{"className":364082},[31850],[34,364084],{"className":364085,"style":31846},[31845],[34,364087,364089,364092],{"className":364088},[31676],[34,364090],{"className":364091,"style":32617},[31680],[34,364093,459],{"className":364094},[31685],[34,364096,364098,364115],{"className":364097},[31642],[34,364099,364101],{"className":364100},[31646],[31648,364102,364103],{"xmlns":31650},[31652,364104,364105,364113],{},[31655,364106,364107,364109,364111],{},[31658,364108,111],{},[31662,364110,54707],{"mathvariant":38079},[31749,364112,459],{},[31665,364114,363487],{"encoding":31667},[34,364116,364118,364169],{"className":364117,"ariaHidden":758},[31672],[34,364119,364121,364124,364127,364130,364166],{"className":364120},[31676],[34,364122],{"className":364123,"style":43217},[31680],[34,364125,111],{"className":364126},[31685,31686],[34,364128],{"className":364129,"style":31846},[31845],[34,364131,364133,364160,364163],{"className":364132},[31850],[34,364134,364136],{"className":364135},[31850],[34,364137,364139],{"className":364138},[31685,54774],[34,364140,364142],{"className":364141},[54778],[34,364143,364145,364148,364157],{"className":364144},[54782],[34,364146],{"className":364147,"style":43217},[31680],[34,364149,364151],{"className":364150},[54789],[34,364152,364154],{"className":364153},[31685],[34,364155,54796],{"className":364156},[31850],[34,364158],{"className":364159},[54800],[34,364161],{"className":364162},[31845,54804],[34,364164,202],{"className":364165},[31850],[34,364167],{"className":364168,"style":31846},[31845],[34,364170,364172,364175],{"className":364171},[31676],[34,364173],{"className":364174,"style":32617},[31680],[34,364176,459],{"className":364177},[31685]," 且整环无零因子，必然有 ",[34,364180,364182,364212],{"className":364181},[31642],[34,364183,364185],{"className":364184},[31646],[31648,364186,364187],{"xmlns":31650},[31652,364188,364189,364209],{},[31655,364190,364191,364197,364199,364205,364207],{},[31743,364192,364193,364195],{},[31658,364194,32936],{},[31749,364196,640],{},[31662,364198,32226],{},[31743,364200,364201,364203],{},[31658,364202,32936],{},[31749,364204,768],{},[31662,364206,202],{},[31749,364208,459],{},[31665,364210,364211],{"encoding":31667},"x_1 - x_2 = 0",[34,364213,364215,364270,364325],{"className":364214,"ariaHidden":758},[31672],[34,364216,364218,364221,364261,364264,364267],{"className":364217},[31676],[34,364219],{"className":364220,"style":40522},[31680],[34,364222,364224,364227],{"className":364223},[31685],[34,364225,32936],{"className":364226},[31685,31686],[34,364228,364230],{"className":364229},[31792],[34,364231,364233,364253],{"className":364232},[31796,31797],[34,364234,364236,364250],{"className":364235},[31801],[34,364237,364239],{"className":364238,"style":31806},[31805],[34,364240,364241,364244],{"style":31809},[34,364242],{"className":364243,"style":31814},[31813],[34,364245,364247],{"className":364246},[31818,31819,31820,31821],[34,364248,640],{"className":364249},[31685,31821],[34,364251,31829],{"className":364252},[31828],[34,364254,364256],{"className":364255},[31801],[34,364257,364259],{"className":364258,"style":31836},[31805],[34,364260],{},[34,364262],{"className":364263,"style":32604},[31845],[34,364265,32226],{"className":364266},[32024],[34,364268],{"className":364269,"style":32604},[31845],[34,364271,364273,364276,364316,364319,364322],{"className":364272},[31676],[34,364274],{"className":364275,"style":31860},[31680],[34,364277,364279,364282],{"className":364278},[31685],[34,364280,32936],{"className":364281},[31685,31686],[34,364283,364285],{"className":364284},[31792],[34,364286,364288,364308],{"className":364287},[31796,31797],[34,364289,364291,364305],{"className":364290},[31801],[34,364292,364294],{"className":364293,"style":31806},[31805],[34,364295,364296,364299],{"style":31809},[34,364297],{"className":364298,"style":31814},[31813],[34,364300,364302],{"className":364301},[31818,31819,31820,31821],[34,364303,768],{"className":364304},[31685,31821],[34,364306,31829],{"className":364307},[31828],[34,364309,364311],{"className":364310},[31801],[34,364312,364314],{"className":364313,"style":31836},[31805],[34,364315],{},[34,364317],{"className":364318,"style":31846},[31845],[34,364320,202],{"className":364321},[31850],[34,364323],{"className":364324,"style":31846},[31845],[34,364326,364328,364331],{"className":364327},[31676],[34,364329],{"className":364330,"style":32617},[31680],[34,364332,459],{"className":364333},[31685],[34,364335,364337,364363],{"className":364336},[31642],[34,364338,364340],{"className":364339},[31646],[31648,364341,364342],{"xmlns":31650},[31652,364343,364344,364360],{},[31655,364345,364346,364352,364354],{},[31743,364347,364348,364350],{},[31658,364349,32936],{},[31749,364351,640],{},[31662,364353,202],{},[31743,364355,364356,364358],{},[31658,364357,32936],{},[31749,364359,768],{},[31665,364361,364362],{"encoding":31667},"x_1 = x_2",[34,364364,364366,364421],{"className":364365,"ariaHidden":758},[31672],[34,364367,364369,364372,364412,364415,364418],{"className":364368},[31676],[34,364370],{"className":364371,"style":31860},[31680],[34,364373,364375,364378],{"className":364374},[31685],[34,364376,32936],{"className":364377},[31685,31686],[34,364379,364381],{"className":364380},[31792],[34,364382,364384,364404],{"className":364383},[31796,31797],[34,364385,364387,364401],{"className":364386},[31801],[34,364388,364390],{"className":364389,"style":31806},[31805],[34,364391,364392,364395],{"style":31809},[34,364393],{"className":364394,"style":31814},[31813],[34,364396,364398],{"className":364397},[31818,31819,31820,31821],[34,364399,640],{"className":364400},[31685,31821],[34,364402,31829],{"className":364403},[31828],[34,364405,364407],{"className":364406},[31801],[34,364408,364410],{"className":364409,"style":31836},[31805],[34,364411],{},[34,364413],{"className":364414,"style":31846},[31845],[34,364416,202],{"className":364417},[31850],[34,364419],{"className":364420,"style":31846},[31845],[34,364422,364424,364427],{"className":364423},[31676],[34,364425],{"className":364426,"style":31860},[31680],[34,364428,364430,364433],{"className":364429},[31685],[34,364431,32936],{"className":364432},[31685,31686],[34,364434,364436],{"className":364435},[31792],[34,364437,364439,364459],{"className":364438},[31796,31797],[34,364440,364442,364456],{"className":364441},[31801],[34,364443,364445],{"className":364444,"style":31806},[31805],[34,364446,364447,364450],{"style":31809},[34,364448],{"className":364449,"style":31814},[31813],[34,364451,364453],{"className":364452},[31818,31819,31820,31821],[34,364454,768],{"className":364455},[31685,31821],[34,364457,31829],{"className":364458},[31828],[34,364460,364462],{"className":364461},[31801],[34,364463,364465],{"className":364464,"style":31836},[31805],[34,364466],{},"。\n这说明映射 ",[34,364469,364471,364485],{"className":364470},[31642],[34,364472,364474],{"className":364473},[31646],[31648,364475,364476],{"xmlns":31650},[31652,364477,364478,364482],{},[31655,364479,364480],{},[31658,364481,329201],{},[31665,364483,364484],{"encoding":31667},"\\phi",[34,364486,364488],{"className":364487,"ariaHidden":758},[31672],[34,364489,364491,364494],{"className":364490},[31676],[34,364492],{"className":364493,"style":43217},[31680],[34,364495,329201],{"className":364496},[31685,31686],[2627,364498,364499],{},"单射","（Injective，即一对一）。",[108,364502,364503,364506,364534,364535,364538,364539,364567,364568,364596,364597,364648,364649,364677,364678,364729,364730],{},[2627,364504,364505],{},"第四步：利用有限性证明满射（鸽巢原理）",[34,364507,364509,364522],{"className":364508},[31642],[34,364510,364512],{"className":364511},[31646],[31648,364513,364514],{"xmlns":31650},[31652,364515,364516,364520],{},[31655,364517,364518],{},[31658,364519,91994],{},[31665,364521,91994],{"encoding":31667},[34,364523,364525],{"className":364524,"ariaHidden":758},[31672],[34,364526,364528,364531],{"className":364527},[31676],[34,364529],{"className":364530,"style":33218},[31680],[34,364532,91994],{"className":364533,"style":34452},[31685,31686]," 是一个有限集合。对于从有限集合到自身的映射，如果是单射，那么它必然也是",[2627,364536,364537],{},"满射","（Surjective）。\n这意味着，",[34,364540,364542,364555],{"className":364541},[31642],[34,364543,364545],{"className":364544},[31646],[31648,364546,364547],{"xmlns":31650},[31652,364548,364549,364553],{},[31655,364550,364551],{},[31658,364552,91994],{},[31665,364554,91994],{"encoding":31667},[34,364556,364558],{"className":364557,"ariaHidden":758},[31672],[34,364559,364561,364564],{"className":364560},[31676],[34,364562],{"className":364563,"style":33218},[31680],[34,364565,91994],{"className":364566,"style":34452},[31685,31686]," 中的每一个元素都可以被写成 ",[34,364569,364571,364584],{"className":364570},[31642],[34,364572,364574],{"className":364573},[31646],[31648,364575,364576],{"xmlns":31650},[31652,364577,364578,364582],{},[31655,364579,364580],{},[31658,364581,111],{},[31665,364583,111],{"encoding":31667},[34,364585,364587],{"className":364586,"ariaHidden":758},[31672],[34,364588,364590,364593],{"className":364589},[31676],[34,364591],{"className":364592,"style":32527},[31680],[34,364594,111],{"className":364595},[31685,31686]," 乘以某数的形式。\n特别地，单位元 ",[34,364598,364600,364618],{"className":364599},[31642],[34,364601,364603],{"className":364602},[31646],[31648,364604,364605],{"xmlns":31650},[31652,364606,364607,364615],{},[31655,364608,364609,364611,364613],{},[31749,364610,640],{},[31662,364612,112296],{},[31658,364614,91994],{},[31665,364616,364617],{"encoding":31667},"1 \\in D",[34,364619,364621,364639],{"className":364620,"ariaHidden":758},[31672],[34,364622,364624,364627,364630,364633,364636],{"className":364623},[31676],[34,364625],{"className":364626,"style":61814},[31680],[34,364628,640],{"className":364629},[31685],[34,364631],{"className":364632,"style":31846},[31845],[34,364634,112296],{"className":364635},[31850],[34,364637],{"className":364638,"style":31846},[31845],[34,364640,364642,364645],{"className":364641},[31676],[34,364643],{"className":364644,"style":33218},[31680],[34,364646,91994],{"className":364647,"style":34452},[31685,31686]," 必定在 ",[34,364650,364652,364665],{"className":364651},[31642],[34,364653,364655],{"className":364654},[31646],[31648,364656,364657],{"xmlns":31650},[31652,364658,364659,364663],{},[31655,364660,364661],{},[31658,364662,329201],{},[31665,364664,364484],{"encoding":31667},[34,364666,364668],{"className":364667,"ariaHidden":758},[31672],[34,364669,364671,364674],{"className":364670},[31676],[34,364672],{"className":364673,"style":43217},[31680],[34,364675,329201],{"className":364676},[31685,31686]," 的值域中。\n所以，存在某个 ",[34,364679,364681,364699],{"className":364680},[31642],[34,364682,364684],{"className":364683},[31646],[31648,364685,364686],{"xmlns":31650},[31652,364687,364688,364696],{},[31655,364689,364690,364692,364694],{},[31658,364691,1697],{},[31662,364693,112296],{},[31658,364695,91994],{},[31665,364697,364698],{"encoding":31667},"b \\in D",[34,364700,364702,364720],{"className":364701,"ariaHidden":758},[31672],[34,364703,364705,364708,364711,364714,364717],{"className":364704},[31676],[34,364706],{"className":364707,"style":46991},[31680],[34,364709,1697],{"className":364710},[31685,31686],[34,364712],{"className":364713,"style":31846},[31845],[34,364715,112296],{"className":364716},[31850],[34,364718],{"className":364719,"style":31846},[31845],[34,364721,364723,364726],{"className":364722},[31676],[34,364724],{"className":364725,"style":33218},[31680],[34,364727,91994],{"className":364728,"style":34452},[31685,31686],"，使得：\n",[34,364731,364733,364753],{"className":364732},[31642],[34,364734,364736],{"className":364735},[31646],[31648,364737,364738],{"xmlns":31650},[31652,364739,364740,364750],{},[31655,364741,364742,364744,364746,364748],{},[31658,364743,111],{},[31658,364745,1697],{},[31662,364747,202],{},[31749,364749,640],{},[31665,364751,364752],{"encoding":31667},"ab = 1",[34,364754,364756,364774],{"className":364755,"ariaHidden":758},[31672],[34,364757,364759,364762,364765,364768,364771],{"className":364758},[31676],[34,364760],{"className":364761,"style":31681},[31680],[34,364763,324191],{"className":364764},[31685,31686],[34,364766],{"className":364767,"style":31846},[31845],[34,364769,202],{"className":364770},[31850],[34,364772],{"className":364773,"style":31846},[31845],[34,364775,364777,364780],{"className":364776},[31676],[34,364778],{"className":364779,"style":32617},[31680],[34,364781,640],{"className":364782},[31685],[108,364784,364785,364788,364789,364839,364840,315343,364868,47025,364921,364949,364950,365010,365011,365049,365050,365078,365079,365081],{},[2627,364786,364787],{},"第五步：得出结论","\n既然对于任意非零 ",[34,364790,364792,364809],{"className":364791},[31642],[34,364793,364795],{"className":364794},[31646],[31648,364796,364797],{"xmlns":31650},[31652,364798,364799,364807],{},[31655,364800,364801,364803,364805],{},[31658,364802,111],{},[31662,364804,112296],{},[31658,364806,91994],{},[31665,364808,363436],{"encoding":31667},[34,364810,364812,364830],{"className":364811,"ariaHidden":758},[31672],[34,364813,364815,364818,364821,364824,364827],{"className":364814},[31676],[34,364816],{"className":364817,"style":34812},[31680],[34,364819,111],{"className":364820},[31685,31686],[34,364822],{"className":364823,"style":31846},[31845],[34,364825,112296],{"className":364826},[31850],[34,364828],{"className":364829,"style":31846},[31845],[34,364831,364833,364836],{"className":364832},[31676],[34,364834],{"className":364835,"style":33218},[31680],[34,364837,91994],{"className":364838,"style":34452},[31685,31686],"，都存在 ",[34,364841,364843,364856],{"className":364842},[31642],[34,364844,364846],{"className":364845},[31646],[31648,364847,364848],{"xmlns":31650},[31652,364849,364850,364854],{},[31655,364851,364852],{},[31658,364853,1697],{},[31665,364855,1697],{"encoding":31667},[34,364857,364859],{"className":364858,"ariaHidden":758},[31672],[34,364860,364862,364865],{"className":364861},[31676],[34,364863],{"className":364864,"style":31681},[31680],[34,364866,1697],{"className":364867},[31685,31686],[34,364869,364871,364891],{"className":364870},[31642],[34,364872,364874],{"className":364873},[31646],[31648,364875,364876],{"xmlns":31650},[31652,364877,364878,364888],{},[31655,364879,364880,364882,364884,364886],{},[31658,364881,111],{},[31658,364883,1697],{},[31662,364885,202],{},[31749,364887,640],{},[31665,364889,364890],{"encoding":31667},"ab=1",[34,364892,364894,364912],{"className":364893,"ariaHidden":758},[31672],[34,364895,364897,364900,364903,364906,364909],{"className":364896},[31676],[34,364898],{"className":364899,"style":31681},[31680],[34,364901,324191],{"className":364902},[31685,31686],[34,364904],{"className":364905,"style":31846},[31845],[34,364907,202],{"className":364908},[31850],[34,364910],{"className":364911,"style":31846},[31845],[34,364913,364915,364918],{"className":364914},[31676],[34,364916],{"className":364917,"style":32617},[31680],[34,364919,640],{"className":364920},[31685],[34,364922,364924,364937],{"className":364923},[31642],[34,364925,364927],{"className":364926},[31646],[31648,364928,364929],{"xmlns":31650},[31652,364930,364931,364935],{},[31655,364932,364933],{},[31658,364934,111],{},[31665,364936,111],{"encoding":31667},[34,364938,364940],{"className":364939,"ariaHidden":758},[31672],[34,364941,364943,364946],{"className":364942},[31676],[34,364944],{"className":364945,"style":32527},[31680],[34,364947,111],{"className":364948},[31685,31686]," 是可逆的。\n所以 ",[34,364951,364953,364974],{"className":364952},[31642],[34,364954,364956],{"className":364955},[31646],[31648,364957,364958],{"xmlns":31650},[31652,364959,364960,364972],{},[31655,364961,364962,364964,364966,364968,364970],{},[31658,364963,91994],{},[31662,364965,202],{},[31658,364967,34386],{},[31658,364969,38080],{"mathvariant":38079},[31658,364971,105852],{},[31665,364973,363025],{"encoding":31667},[34,364975,364977,364995],{"className":364976,"ariaHidden":758},[31672],[34,364978,364980,364983,364986,364989,364992],{"className":364979},[31676],[34,364981],{"className":364982,"style":33218},[31680],[34,364984,91994],{"className":364985,"style":34452},[31685,31686],[34,364987],{"className":364988,"style":31846},[31845],[34,364990,202],{"className":364991},[31850],[34,364993],{"className":364994,"style":31846},[31845],[34,364996,364998,365001,365004,365007],{"className":364997},[31676],[34,364999],{"className":365000,"style":31774},[31680],[34,365002,34386],{"className":365003,"style":34427},[31685,31686],[34,365005,38080],{"className":365006},[31685],[34,365008,105852],{"className":365009,"style":37422},[31685,31686]," 是一个域。\n当商环 ",[34,365012,365014,365031],{"className":365013},[31642],[34,365015,365017],{"className":365016},[31646],[31648,365018,365019],{"xmlns":31650},[31652,365020,365021,365029],{},[31655,365022,365023,365025,365027],{},[31658,365024,34386],{},[31658,365026,38080],{"mathvariant":38079},[31658,365028,105852],{},[31665,365030,331515],{"encoding":31667},[34,365032,365034],{"className":365033,"ariaHidden":758},[31672],[34,365035,365037,365040,365043,365046],{"className":365036},[31676],[34,365038],{"className":365039,"style":31774},[31680],[34,365041,34386],{"className":365042,"style":34427},[31685,31686],[34,365044,38080],{"className":365045},[31685],[34,365047,105852],{"className":365048,"style":37422},[31685,31686]," 是域时，对应的理想 ",[34,365051,365053,365066],{"className":365052},[31642],[34,365054,365056],{"className":365055},[31646],[31648,365057,365058],{"xmlns":31650},[31652,365059,365060,365064],{},[31655,365061,365062],{},[31658,365063,105852],{},[31665,365065,105852],{"encoding":31667},[34,365067,365069],{"className":365068,"ariaHidden":758},[31672],[34,365070,365072,365075],{"className":365071},[31676],[34,365073],{"className":365074,"style":33218},[31680],[34,365076,105852],{"className":365077,"style":37422},[31685,31686]," 就是",[2627,365080,330747],{},"。\n综上所述，有限环的素理想一定是极大理想。证毕。",[32188,365083],{},[253,365085,365086],{"id":365086},"总结",[105,365088,365089,365129,365331,365337],{},[108,365090,365091,365094,365095,365128],{},[2627,365092,365093],{},"题5"," 展示了模运算环的基本结构，利用 ",[34,365096,365098,365114],{"className":365097},[31642],[34,365099,365101],{"className":365100},[31646],[31648,365102,365103],{"xmlns":31650},[31652,365104,365105,365111],{},[31655,365106,365107,365109],{},[31658,365108,314792],{},[31662,365110,32682],{},[31665,365112,365113],{"encoding":31667},"\\gcd",[34,365115,365117],{"className":365116,"ariaHidden":758},[31672],[34,365118,365120,365123],{"className":365119},[31676],[34,365121],{"className":365122,"style":43217},[31680],[34,365124,365126,314826],{"className":365125},[32767],[34,365127,31942],{"style":38619}," 区分了可逆元和幂零元。",[108,365130,365131,365134,365135,365330],{},[2627,365132,365133],{},"题8"," 利用了理想乘积与交集的关系 (",[34,365136,365138,365174],{"className":365137},[31642],[34,365139,365141],{"className":365140},[31646],[31648,365142,365143],{"xmlns":31650},[31652,365144,365145,365171],{},[31655,365146,365147,365153,365155,365161,365163,365165],{},[31743,365148,365149,365151],{},[31658,365150,106063],{},[31749,365152,640],{},[31662,365154,33934],{},[31743,365156,365157,365159],{},[31658,365158,106063],{},[31658,365160,31660],{},[31662,365162,131855],{},[31662,365164,326344],{},[31743,365166,365167,365169],{},[31658,365168,106063],{},[31658,365170,31955],{},[31665,365172,365173],{"encoding":31667},"I_1 \\dots I_n \\subseteq \\cap I_i",[34,365175,365177,365281],{"className":365176,"ariaHidden":758},[31672],[34,365178,365180,365183,365223,365226,365229,365232,365272,365275,365278],{"className":365179},[31676],[34,365181],{"className":365182,"style":70915},[31680],[34,365184,365186,365189],{"className":365185},[31685],[34,365187,106063],{"className":365188,"style":67482},[31685,31686],[34,365190,365192],{"className":365191},[31792],[34,365193,365195,365215],{"className":365194},[31796,31797],[34,365196,365198,365212],{"className":365197},[31801],[34,365199,365201],{"className":365200,"style":31806},[31805],[34,365202,365203,365206],{"style":217742},[34,365204],{"className":365205,"style":31814},[31813],[34,365207,365209],{"className":365208},[31818,31819,31820,31821],[34,365210,640],{"className":365211},[31685,31821],[34,365213,31829],{"className":365214},[31828],[34,365216,365218],{"className":365217},[31801],[34,365219,365221],{"className":365220,"style":31836},[31805],[34,365222],{},[34,365224],{"className":365225,"style":32123},[31845],[34,365227,33934],{"className":365228},[34006],[34,365230],{"className":365231,"style":32123},[31845],[34,365233,365235,365238],{"className":365234},[31685],[34,365236,106063],{"className":365237,"style":67482},[31685,31686],[34,365239,365241],{"className":365240},[31792],[34,365242,365244,365264],{"className":365243},[31796,31797],[34,365245,365247,365261],{"className":365246},[31801],[34,365248,365250],{"className":365249,"style":32362},[31805],[34,365251,365252,365255],{"style":217742},[34,365253],{"className":365254,"style":31814},[31813],[34,365256,365258],{"className":365257},[31818,31819,31820,31821],[34,365259,31660],{"className":365260},[31685,31686,31821],[34,365262,31829],{"className":365263},[31828],[34,365265,365267],{"className":365266},[31801],[34,365268,365270],{"className":365269,"style":31836},[31805],[34,365271],{},[34,365273],{"className":365274,"style":31846},[31845],[34,365276,131855],{"className":365277},[31850],[34,365279],{"className":365280,"style":31846},[31845],[34,365282,365284,365287,365290],{"className":365283},[31676],[34,365285],{"className":365286,"style":70915},[31680],[34,365288,326344],{"className":365289},[31685],[34,365291,365293,365296],{"className":365292},[31685],[34,365294,106063],{"className":365295,"style":67482},[31685,31686],[34,365297,365299],{"className":365298},[31792],[34,365300,365302,365322],{"className":365301},[31796,31797],[34,365303,365305,365319],{"className":365304},[31801],[34,365306,365308],{"className":365307,"style":32090},[31805],[34,365309,365310,365313],{"style":217742},[34,365311],{"className":365312,"style":31814},[31813],[34,365314,365316],{"className":365315},[31818,31819,31820,31821],[34,365317,31955],{"className":365318},[31685,31686,31821],[34,365320,31829],{"className":365321},[31828],[34,365323,365325],{"className":365324},[31801],[34,365326,365328],{"className":365327,"style":31836},[31805],[34,365329],{},") 证明了素理想的吸附性。",[108,365332,365333,365336],{},[2627,365334,365335],{},"题9"," 使用了精妙的构造法（反证+归纳），证明了理想无法躲在有限个素理想的并集里（除非它已经在某一个里面）。",[108,365338,365339,365342],{},[2627,365340,365341],{},"题10"," 将环论问题转化为“有限整环即为域”这一经典定理，利用了有限集合上单射即满射的原理。",{"title":29,"searchDepth":43,"depth":43,"links":365344},[365345,365346,365347,365348,365349],{"id":339116,"depth":49,"text":339117},{"id":346501,"depth":49,"text":346502},{"id":351467,"depth":49,"text":351468},{"id":362461,"depth":49,"text":362462},{"id":365086,"depth":49,"text":365086},{"order":1990},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fprimeidealmaximalidealhw",{"title":339108,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002FprimeidealmaximalidealHW","ea73z76H2sMvitWX_V388bRZ91ktzog7by4c7qUeF2o",{"id":365356,"title":365357,"body":365358,"description":29,"extension":85,"meta":371597,"navigation":52,"path":371598,"seo":371599,"stem":371600,"__hash__":371601},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fring.md","环的基本理论",{"type":8,"value":365359,"toc":371583},[365360,365362,365366,365369,365404,365406,365410,365413,365417,365597,367174,367178,367181,367657,367850,367852,367856,367859,367863,368127,368132,368737,368741,368744,369721,369723,369727,369730,370420,370424],[11,365361,365357],{"id":365357},[19,365363,365365],{"id":365364},"_1-开篇总结-executive-summary","1. 开篇总结 (Executive Summary)",[15,365367,365368],{},"本课程的核心目标是将代数结构从单一运算（群）推广到双运算（环）。",[15,365370,365371,365374,365375,365403],{},[2627,365372,365373],{},"核心洞察："," 环（Ring）是对整数集合 ",[34,365376,365378,365391],{"className":365377},[31642],[34,365379,365381],{"className":365380},[31646],[31648,365382,365383],{"xmlns":31650},[31652,365384,365385,365389],{},[31655,365386,365387],{},[31658,365388,97300],{"mathvariant":318416},[31665,365390,320917],{"encoding":31667},[34,365392,365394],{"className":365393,"ariaHidden":758},[31672],[34,365395,365397,365400],{"className":365396},[31676],[34,365398],{"className":365399,"style":320927},[31680],[34,365401,97300],{"className":365402},[31685,318441]," 的抽象化，旨在研究“加法”与“乘法”共存时的代数性质。相比于群，环增加了结构的复杂性（两个运算），但也因为分配律的存在，引入了更丰富的代数工具。课程的难点在于理解**零因子（Zero Divisors）**的存在如何破坏了初等代数中的“消去律”。",[32188,365405],{},[19,365407,365409],{"id":365408},"_2-环的公理化定义体系-axiomatic-definitions","2. 环的公理化定义体系 (Axiomatic Definitions)",[15,365411,365412],{},"首先给出环的通用定义，随后通过增加公理（Axioms）不断缩小范围，构建了特殊的环类。",[253,365414,365416],{"id":365415},"_21-环-ring-的基础定义","2.1 环 (Ring) 的基础定义",[15,365418,133910,365419,365447,365448,365487,365488,365527,365528,365593,365594,2664],{},[34,365420,365422,365435],{"className":365421},[31642],[34,365423,365425],{"className":365424},[31646],[31648,365426,365427],{"xmlns":31650},[31652,365428,365429,365433],{},[31655,365430,365431],{},[31658,365432,34386],{},[31665,365434,34386],{"encoding":31667},[34,365436,365438],{"className":365437,"ariaHidden":758},[31672],[34,365439,365441,365444],{"className":365440},[31676],[34,365442],{"className":365443,"style":33218},[31680],[34,365445,34386],{"className":365446,"style":34427},[31685,31686]," 是一个非空集合，配备两个二元运算：加法 ",[34,365449,365451,365469],{"className":365450},[31642],[34,365452,365454],{"className":365453},[31646],[31648,365455,365456],{"xmlns":31650},[31652,365457,365458,365466],{},[31655,365459,365460,365462,365464],{},[31662,365461,456],{"stretchy":702},[31662,365463,223],{},[31662,365465,2557],{"stretchy":702},[31665,365467,365468],{"encoding":31667},"(+)",[34,365470,365472],{"className":365471,"ariaHidden":758},[31672],[34,365473,365475,365478,365481,365484],{"className":365474},[31676],[34,365476],{"className":365477,"style":31774},[31680],[34,365479,456],{"className":365480},[31782],[34,365482,223],{"className":365483},[31685],[34,365485,2557],{"className":365486},[31690]," 和乘法 ",[34,365489,365491,365509],{"className":365490},[31642],[34,365492,365494],{"className":365493},[31646],[31648,365495,365496],{"xmlns":31650},[31652,365497,365498,365506],{},[31655,365499,365500,365502,365504],{},[31662,365501,456],{"stretchy":702},[31662,365503,71191],{},[31662,365505,2557],{"stretchy":702},[31665,365507,365508],{"encoding":31667},"(\\cdot)",[34,365510,365512],{"className":365511,"ariaHidden":758},[31672],[34,365513,365515,365518,365521,365524],{"className":365514},[31676],[34,365516],{"className":365517,"style":31774},[31680],[34,365519,456],{"className":365520},[31782],[34,365522,71191],{"className":365523},[31685],[34,365525,2557],{"className":365526},[31690],"。若满足以下三组公理，则称 ",[34,365529,365531,365557],{"className":365530},[31642],[34,365532,365534],{"className":365533},[31646],[31648,365535,365536],{"xmlns":31650},[31652,365537,365538,365554],{},[31655,365539,365540,365542,365544,365546,365548,365550,365552],{},[31662,365541,456],{"stretchy":702},[31658,365543,34386],{},[31662,365545,31960],{"separator":758},[31662,365547,223],{},[31662,365549,31960],{"separator":758},[31662,365551,71191],{},[31662,365553,2557],{"stretchy":702},[31665,365555,365556],{"encoding":31667},"(R, +, \\cdot)",[34,365558,365560],{"className":365559,"ariaHidden":758},[31672],[34,365561,365563,365566,365569,365572,365575,365578,365581,365584,365587,365590],{"className":365562},[31676],[34,365564],{"className":365565,"style":31774},[31680],[34,365567,456],{"className":365568},[31782],[34,365570,34386],{"className":365571,"style":34427},[31685,31686],[34,365573,31960],{"className":365574},[32119],[34,365576],{"className":365577,"style":32123},[31845],[34,365579,223],{"className":365580},[31685],[34,365582,31960],{"className":365583},[32119],[34,365585],{"className":365586,"style":32123},[31845],[34,365588,71191],{"className":365589},[31685],[34,365591,2557],{"className":365592},[31690]," 为一个",[2627,365595,365596],{},"环",[2230,365598,365599,366448,366818],{},[108,365600,365601,2664,365604,365656,365657,39984,365660],{},[2627,365602,365603],{},"加法群公理",[34,365605,365607,365629],{"className":365606},[31642],[34,365608,365610],{"className":365609},[31646],[31648,365611,365612],{"xmlns":31650},[31652,365613,365614,365626],{},[31655,365615,365616,365618,365620,365622,365624],{},[31662,365617,456],{"stretchy":702},[31658,365619,34386],{},[31662,365621,31960],{"separator":758},[31662,365623,223],{},[31662,365625,2557],{"stretchy":702},[31665,365627,365628],{"encoding":31667},"(R, +)",[34,365630,365632],{"className":365631,"ariaHidden":758},[31672],[34,365633,365635,365638,365641,365644,365647,365650,365653],{"className":365634},[31676],[34,365636],{"className":365637,"style":31774},[31680],[34,365639,456],{"className":365640},[31782],[34,365642,34386],{"className":365643,"style":34427},[31685,31686],[34,365645,31960],{"className":365646},[32119],[34,365648],{"className":365649,"style":32123},[31845],[34,365651,223],{"className":365652},[31685],[34,365654,2557],{"className":365655},[31690]," 构成",[2627,365658,365659],{},"阿贝尔群（Abelian Group）",[105,365661,365662,365811,365972,366147,366351],{},[108,365663,365664,365665],{},"封闭性：",[34,365666,365668,365709],{"className":365667},[31642],[34,365669,365671],{"className":365670},[31646],[31648,365672,365673],{"xmlns":31650},[31652,365674,365675,365706],{},[31655,365676,365677,365680,365682,365684,365686,365688,365690,365692,365694,365696,365698,365700,365702,365704],{},[31658,365678,365679],{"mathvariant":38079},"∀",[31658,365681,111],{},[31662,365683,31960],{"separator":758},[31658,365685,1697],{},[31662,365687,112296],{},[31658,365689,34386],{},[39361,365691,262622],{},[31662,365693,262625],{},[39361,365695,262622],{},[31658,365697,111],{},[31662,365699,223],{},[31658,365701,1697],{},[31662,365703,112296],{},[31658,365705,34386],{},[31665,365707,365708],{"encoding":31667},"\\forall a, b \\in R \\implies a+b \\in R",[34,365710,365712,365742,365766,365784,365802],{"className":365711,"ariaHidden":758},[31672],[34,365713,365715,365718,365721,365724,365727,365730,365733,365736,365739],{"className":365714},[31676],[34,365716],{"className":365717,"style":43217},[31680],[34,365719,365679],{"className":365720},[31685],[34,365722,111],{"className":365723},[31685,31686],[34,365725,31960],{"className":365726},[32119],[34,365728],{"className":365729,"style":32123},[31845],[34,365731,1697],{"className":365732},[31685,31686],[34,365734],{"className":365735,"style":31846},[31845],[34,365737,112296],{"className":365738},[31850],[34,365740],{"className":365741,"style":31846},[31845],[34,365743,365745,365748,365751,365754,365757,365760,365763],{"className":365744},[31676],[34,365746],{"className":365747,"style":336814},[31680],[34,365749,34386],{"className":365750,"style":34427},[31685,31686],[34,365752],{"className":365753,"style":31846},[31845],[34,365755],{"className":365756,"style":31846},[31845],[34,365758,262625],{"className":365759},[31850],[34,365761],{"className":365762,"style":31846},[31845],[34,365764],{"className":365765,"style":31846},[31845],[34,365767,365769,365772,365775,365778,365781],{"className":365768},[31676],[34,365770],{"className":365771,"style":32597},[31680],[34,365773,111],{"className":365774},[31685,31686],[34,365776],{"className":365777,"style":32604},[31845],[34,365779,223],{"className":365780},[32024],[34,365782],{"className":365783,"style":32604},[31845],[34,365785,365787,365790,365793,365796,365799],{"className":365786},[31676],[34,365788],{"className":365789,"style":46991},[31680],[34,365791,1697],{"className":365792},[31685,31686],[34,365794],{"className":365795,"style":31846},[31845],[34,365797,112296],{"className":365798},[31850],[34,365800],{"className":365801,"style":31846},[31845],[34,365803,365805,365808],{"className":365804},[31676],[34,365806],{"className":365807,"style":33218},[31680],[34,365809,34386],{"className":365810,"style":34427},[31685,31686],[108,365812,124156,365813],{},[34,365814,365816,365858],{"className":365815},[31642],[34,365817,365819],{"className":365818},[31646],[31648,365820,365821],{"xmlns":31650},[31652,365822,365823,365855],{},[31655,365824,365825,365827,365829,365831,365833,365835,365837,365839,365841,365843,365845,365847,365849,365851,365853],{},[31662,365826,456],{"stretchy":702},[31658,365828,111],{},[31662,365830,223],{},[31658,365832,1697],{},[31662,365834,2557],{"stretchy":702},[31662,365836,223],{},[31658,365838,31965],{},[31662,365840,202],{},[31658,365842,111],{},[31662,365844,223],{},[31662,365846,456],{"stretchy":702},[31658,365848,1697],{},[31662,365850,223],{},[31658,365852,31965],{},[31662,365854,2557],{"stretchy":702},[31665,365856,365857],{"encoding":31667},"(a+b)+c = a+(b+c)",[34,365859,365861,365882,365903,365921,365939,365960],{"className":365860,"ariaHidden":758},[31672],[34,365862,365864,365867,365870,365873,365876,365879],{"className":365863},[31676],[34,365865],{"className":365866,"style":31774},[31680],[34,365868,456],{"className":365869},[31782],[34,365871,111],{"className":365872},[31685,31686],[34,365874],{"className":365875,"style":32604},[31845],[34,365877,223],{"className":365878},[32024],[34,365880],{"className":365881,"style":32604},[31845],[34,365883,365885,365888,365891,365894,365897,365900],{"className":365884},[31676],[34,365886],{"className":365887,"style":31774},[31680],[34,365889,1697],{"className":365890},[31685,31686],[34,365892,2557],{"className":365893},[31690],[34,365895],{"className":365896,"style":32604},[31845],[34,365898,223],{"className":365899},[32024],[34,365901],{"className":365902,"style":32604},[31845],[34,365904,365906,365909,365912,365915,365918],{"className":365905},[31676],[34,365907],{"className":365908,"style":32527},[31680],[34,365910,31965],{"className":365911},[31685,31686],[34,365913],{"className":365914,"style":31846},[31845],[34,365916,202],{"className":365917},[31850],[34,365919],{"className":365920,"style":31846},[31845],[34,365922,365924,365927,365930,365933,365936],{"className":365923},[31676],[34,365925],{"className":365926,"style":32597},[31680],[34,365928,111],{"className":365929},[31685,31686],[34,365931],{"className":365932,"style":32604},[31845],[34,365934,223],{"className":365935},[32024],[34,365937],{"className":365938,"style":32604},[31845],[34,365940,365942,365945,365948,365951,365954,365957],{"className":365941},[31676],[34,365943],{"className":365944,"style":31774},[31680],[34,365946,456],{"className":365947},[31782],[34,365949,1697],{"className":365950},[31685,31686],[34,365952],{"className":365953,"style":32604},[31845],[34,365955,223],{"className":365956},[32024],[34,365958],{"className":365959,"style":32604},[31845],[34,365961,365963,365966,365969],{"className":365962},[31676],[34,365964],{"className":365965,"style":31774},[31680],[34,365967,31965],{"className":365968},[31685,31686],[34,365970,2557],{"className":365971},[31690],[108,365973,365974,365975,335149,366030],{},"单位元（零元）：",[34,365976,365978,365999],{"className":365977},[31642],[34,365979,365981],{"className":365980},[31646],[31648,365982,365983],{"xmlns":31650},[31652,365984,365985,365996],{},[31655,365986,365987,365990,365992,365994],{},[31658,365988,365989],{"mathvariant":38079},"∃",[31749,365991,459],{},[31662,365993,112296],{},[31658,365995,34386],{},[31665,365997,365998],{"encoding":31667},"\\exists 0 \\in R",[34,366000,366002,366021],{"className":366001,"ariaHidden":758},[31672],[34,366003,366005,366008,366012,366015,366018],{"className":366004},[31676],[34,366006],{"className":366007,"style":46991},[31680],[34,366009,366011],{"className":366010},[31685],"∃0",[34,366013],{"className":366014,"style":31846},[31845],[34,366016,112296],{"className":366017},[31850],[34,366019],{"className":366020,"style":31846},[31845],[34,366022,366024,366027],{"className":366023},[31676],[34,366025],{"className":366026,"style":33218},[31680],[34,366028,34386],{"className":366029,"style":34427},[31685,31686],[34,366031,366033,366063],{"className":366032},[31642],[34,366034,366036],{"className":366035},[31646],[31648,366037,366038],{"xmlns":31650},[31652,366039,366040,366060],{},[31655,366041,366042,366044,366046,366048,366050,366052,366054,366056,366058],{},[31658,366043,111],{},[31662,366045,223],{},[31749,366047,459],{},[31662,366049,202],{},[31749,366051,459],{},[31662,366053,223],{},[31658,366055,111],{},[31662,366057,202],{},[31658,366059,111],{},[31665,366061,366062],{"encoding":31667},"a+0 = 0+a = a",[34,366064,366066,366084,366102,366120,366138],{"className":366065,"ariaHidden":758},[31672],[34,366067,366069,366072,366075,366078,366081],{"className":366068},[31676],[34,366070],{"className":366071,"style":32597},[31680],[34,366073,111],{"className":366074},[31685,31686],[34,366076],{"className":366077,"style":32604},[31845],[34,366079,223],{"className":366080},[32024],[34,366082],{"className":366083,"style":32604},[31845],[34,366085,366087,366090,366093,366096,366099],{"className":366086},[31676],[34,366088],{"className":366089,"style":32617},[31680],[34,366091,459],{"className":366092},[31685],[34,366094],{"className":366095,"style":31846},[31845],[34,366097,202],{"className":366098},[31850],[34,366100],{"className":366101,"style":31846},[31845],[34,366103,366105,366108,366111,366114,366117],{"className":366104},[31676],[34,366106],{"className":366107,"style":35942},[31680],[34,366109,459],{"className":366110},[31685],[34,366112],{"className":366113,"style":32604},[31845],[34,366115,223],{"className":366116},[32024],[34,366118],{"className":366119,"style":32604},[31845],[34,366121,366123,366126,366129,366132,366135],{"className":366122},[31676],[34,366124],{"className":366125,"style":32527},[31680],[34,366127,111],{"className":366128},[31685,31686],[34,366130],{"className":366131,"style":31846},[31845],[34,366133,202],{"className":366134},[31850],[34,366136],{"className":366137,"style":31846},[31845],[34,366139,366141,366144],{"className":366140},[31676],[34,366142],{"className":366143,"style":32527},[31680],[34,366145,111],{"className":366146},[31685,31686],[108,366148,366149,366150,335149,366263],{},"逆元：",[34,366151,366153,366185],{"className":366152},[31642],[34,366154,366156],{"className":366155},[31646],[31648,366157,366158],{"xmlns":31650},[31652,366159,366160,366182],{},[31655,366161,366162,366164,366166,366168,366170,366172,366174,366176,366178,366180],{},[31658,366163,365679],{"mathvariant":38079},[31658,366165,111],{},[31662,366167,112296],{},[31658,366169,34386],{},[31662,366171,31960],{"separator":758},[31658,366173,365989],{"mathvariant":38079},[31662,366175,32226],{},[31658,366177,111],{},[31662,366179,112296],{},[31658,366181,34386],{},[31665,366183,366184],{"encoding":31667},"\\forall a \\in R, \\exists -a \\in R",[34,366186,366188,366209,366236,366254],{"className":366187,"ariaHidden":758},[31672],[34,366189,366191,366194,366197,366200,366203,366206],{"className":366190},[31676],[34,366192],{"className":366193,"style":46991},[31680],[34,366195,365679],{"className":366196},[31685],[34,366198,111],{"className":366199},[31685,31686],[34,366201],{"className":366202,"style":31846},[31845],[34,366204,112296],{"className":366205},[31850],[34,366207],{"className":366208,"style":31846},[31845],[34,366210,366212,366215,366218,366221,366224,366227,366230,366233],{"className":366211},[31676],[34,366213],{"className":366214,"style":43217},[31680],[34,366216,34386],{"className":366217,"style":34427},[31685,31686],[34,366219,31960],{"className":366220},[32119],[34,366222],{"className":366223,"style":32123},[31845],[34,366225,365989],{"className":366226},[31685],[34,366228],{"className":366229,"style":32604},[31845],[34,366231,32226],{"className":366232},[32024],[34,366234],{"className":366235,"style":32604},[31845],[34,366237,366239,366242,366245,366248,366251],{"className":366238},[31676],[34,366240],{"className":366241,"style":34812},[31680],[34,366243,111],{"className":366244},[31685,31686],[34,366246],{"className":366247,"style":31846},[31845],[34,366249,112296],{"className":366250},[31850],[34,366252],{"className":366253,"style":31846},[31845],[34,366255,366257,366260],{"className":366256},[31676],[34,366258],{"className":366259,"style":33218},[31680],[34,366261,34386],{"className":366262,"style":34427},[31685,31686],[34,366264,366266,366294],{"className":366265},[31642],[34,366267,366269],{"className":366268},[31646],[31648,366270,366271],{"xmlns":31650},[31652,366272,366273,366291],{},[31655,366274,366275,366277,366279,366281,366283,366285,366287,366289],{},[31658,366276,111],{},[31662,366278,223],{},[31662,366280,456],{"stretchy":702},[31662,366282,32226],{},[31658,366284,111],{},[31662,366286,2557],{"stretchy":702},[31662,366288,202],{},[31749,366290,459],{},[31665,366292,366293],{"encoding":31667},"a+(-a) = 0",[34,366295,366297,366315,366342],{"className":366296,"ariaHidden":758},[31672],[34,366298,366300,366303,366306,366309,366312],{"className":366299},[31676],[34,366301],{"className":366302,"style":32597},[31680],[34,366304,111],{"className":366305},[31685,31686],[34,366307],{"className":366308,"style":32604},[31845],[34,366310,223],{"className":366311},[32024],[34,366313],{"className":366314,"style":32604},[31845],[34,366316,366318,366321,366324,366327,366330,366333,366336,366339],{"className":366317},[31676],[34,366319],{"className":366320,"style":31774},[31680],[34,366322,456],{"className":366323},[31782],[34,366325,32226],{"className":366326},[31685],[34,366328,111],{"className":366329},[31685,31686],[34,366331,2557],{"className":366332},[31690],[34,366334],{"className":366335,"style":31846},[31845],[34,366337,202],{"className":366338},[31850],[34,366340],{"className":366341,"style":31846},[31845],[34,366343,366345,366348],{"className":366344},[31676],[34,366346],{"className":366347,"style":32617},[31680],[34,366349,459],{"className":366350},[31685],[108,366352,124147,366353],{},[34,366354,366356,366382],{"className":366355},[31642],[34,366357,366359],{"className":366358},[31646],[31648,366360,366361],{"xmlns":31650},[31652,366362,366363,366379],{},[31655,366364,366365,366367,366369,366371,366373,366375,366377],{},[31658,366366,111],{},[31662,366368,223],{},[31658,366370,1697],{},[31662,366372,202],{},[31658,366374,1697],{},[31662,366376,223],{},[31658,366378,111],{},[31665,366380,366381],{"encoding":31667},"a+b = b+a",[34,366383,366385,366403,366421,366439],{"className":366384,"ariaHidden":758},[31672],[34,366386,366388,366391,366394,366397,366400],{"className":366387},[31676],[34,366389],{"className":366390,"style":32597},[31680],[34,366392,111],{"className":366393},[31685,31686],[34,366395],{"className":366396,"style":32604},[31845],[34,366398,223],{"className":366399},[32024],[34,366401],{"className":366402,"style":32604},[31845],[34,366404,366406,366409,366412,366415,366418],{"className":366405},[31676],[34,366407],{"className":366408,"style":31681},[31680],[34,366410,1697],{"className":366411},[31685,31686],[34,366413],{"className":366414,"style":31846},[31845],[34,366416,202],{"className":366417},[31850],[34,366419],{"className":366420,"style":31846},[31845],[34,366422,366424,366427,366430,366433,366436],{"className":366423},[31676],[34,366425],{"className":366426,"style":34299},[31680],[34,366428,1697],{"className":366429},[31685,31686],[34,366431],{"className":366432,"style":32604},[31845],[34,366434,223],{"className":366435},[32024],[34,366437],{"className":366438,"style":32604},[31845],[34,366440,366442,366445],{"className":366441},[31676],[34,366443],{"className":366444,"style":32527},[31680],[34,366446,111],{"className":366447},[31685,31686],[108,366449,366450,2664,366453,365656,366505,39984,366508],{},[2627,366451,366452],{},"乘法半群公理",[34,366454,366456,366478],{"className":366455},[31642],[34,366457,366459],{"className":366458},[31646],[31648,366460,366461],{"xmlns":31650},[31652,366462,366463,366475],{},[31655,366464,366465,366467,366469,366471,366473],{},[31662,366466,456],{"stretchy":702},[31658,366468,34386],{},[31662,366470,31960],{"separator":758},[31662,366472,71191],{},[31662,366474,2557],{"stretchy":702},[31665,366476,366477],{"encoding":31667},"(R, \\cdot)",[34,366479,366481],{"className":366480,"ariaHidden":758},[31672],[34,366482,366484,366487,366490,366493,366496,366499,366502],{"className":366483},[31676],[34,366485],{"className":366486,"style":31774},[31680],[34,366488,456],{"className":366489},[31782],[34,366491,34386],{"className":366492,"style":34427},[31685,31686],[34,366494,31960],{"className":366495},[32119],[34,366497],{"className":366498,"style":32123},[31845],[34,366500,71191],{"className":366501},[31685],[34,366503,2557],{"className":366504},[31690],[2627,366506,366507],{},"半群（Semigroup）",[105,366509,366510,366657],{},[108,366511,365664,366512],{},[34,366513,366515,366555],{"className":366514},[31642],[34,366516,366518],{"className":366517},[31646],[31648,366519,366520],{"xmlns":31650},[31652,366521,366522,366552],{},[31655,366523,366524,366526,366528,366530,366532,366534,366536,366538,366540,366542,366544,366546,366548,366550],{},[31658,366525,365679],{"mathvariant":38079},[31658,366527,111],{},[31662,366529,31960],{"separator":758},[31658,366531,1697],{},[31662,366533,112296],{},[31658,366535,34386],{},[39361,366537,262622],{},[31662,366539,262625],{},[39361,366541,262622],{},[31658,366543,111],{},[31662,366545,71191],{},[31658,366547,1697],{},[31662,366549,112296],{},[31658,366551,34386],{},[31665,366553,366554],{"encoding":31667},"\\forall a, b \\in R \\implies a \\cdot b \\in R",[34,366556,366558,366588,366612,366630,366648],{"className":366557,"ariaHidden":758},[31672],[34,366559,366561,366564,366567,366570,366573,366576,366579,366582,366585],{"className":366560},[31676],[34,366562],{"className":366563,"style":43217},[31680],[34,366565,365679],{"className":366566},[31685],[34,366568,111],{"className":366569},[31685,31686],[34,366571,31960],{"className":366572},[32119],[34,366574],{"className":366575,"style":32123},[31845],[34,366577,1697],{"className":366578},[31685,31686],[34,366580],{"className":366581,"style":31846},[31845],[34,366583,112296],{"className":366584},[31850],[34,366586],{"className":366587,"style":31846},[31845],[34,366589,366591,366594,366597,366600,366603,366606,366609],{"className":366590},[31676],[34,366592],{"className":366593,"style":336814},[31680],[34,366595,34386],{"className":366596,"style":34427},[31685,31686],[34,366598],{"className":366599,"style":31846},[31845],[34,366601],{"className":366602,"style":31846},[31845],[34,366604,262625],{"className":366605},[31850],[34,366607],{"className":366608,"style":31846},[31845],[34,366610],{"className":366611,"style":31846},[31845],[34,366613,366615,366618,366621,366624,366627],{"className":366614},[31676],[34,366616],{"className":366617,"style":224869},[31680],[34,366619,111],{"className":366620},[31685,31686],[34,366622],{"className":366623,"style":32604},[31845],[34,366625,71191],{"className":366626},[32024],[34,366628],{"className":366629,"style":32604},[31845],[34,366631,366633,366636,366639,366642,366645],{"className":366632},[31676],[34,366634],{"className":366635,"style":46991},[31680],[34,366637,1697],{"className":366638},[31685,31686],[34,366640],{"className":366641,"style":31846},[31845],[34,366643,112296],{"className":366644},[31850],[34,366646],{"className":366647,"style":31846},[31845],[34,366649,366651,366654],{"className":366650},[31676],[34,366652],{"className":366653,"style":33218},[31680],[34,366655,34386],{"className":366656,"style":34427},[31685,31686],[108,366658,124156,366659],{},[34,366660,366662,366704],{"className":366661},[31642],[34,366663,366665],{"className":366664},[31646],[31648,366666,366667],{"xmlns":31650},[31652,366668,366669,366701],{},[31655,366670,366671,366673,366675,366677,366679,366681,366683,366685,366687,366689,366691,366693,366695,366697,366699],{},[31662,366672,456],{"stretchy":702},[31658,366674,111],{},[31662,366676,71191],{},[31658,366678,1697],{},[31662,366680,2557],{"stretchy":702},[31662,366682,71191],{},[31658,366684,31965],{},[31662,366686,202],{},[31658,366688,111],{},[31662,366690,71191],{},[31662,366692,456],{"stretchy":702},[31658,366694,1697],{},[31662,366696,71191],{},[31658,366698,31965],{},[31662,366700,2557],{"stretchy":702},[31665,366702,366703],{"encoding":31667},"(a \\cdot b) \\cdot c = a \\cdot (b \\cdot c)",[34,366705,366707,366728,366749,366767,366785,366806],{"className":366706,"ariaHidden":758},[31672],[34,366708,366710,366713,366716,366719,366722,366725],{"className":366709},[31676],[34,366711],{"className":366712,"style":31774},[31680],[34,366714,456],{"className":366715},[31782],[34,366717,111],{"className":366718},[31685,31686],[34,366720],{"className":366721,"style":32604},[31845],[34,366723,71191],{"className":366724},[32024],[34,366726],{"className":366727,"style":32604},[31845],[34,366729,366731,366734,366737,366740,366743,366746],{"className":366730},[31676],[34,366732],{"className":366733,"style":31774},[31680],[34,366735,1697],{"className":366736},[31685,31686],[34,366738,2557],{"className":366739},[31690],[34,366741],{"className":366742,"style":32604},[31845],[34,366744,71191],{"className":366745},[32024],[34,366747],{"className":366748,"style":32604},[31845],[34,366750,366752,366755,366758,366761,366764],{"className":366751},[31676],[34,366753],{"className":366754,"style":32527},[31680],[34,366756,31965],{"className":366757},[31685,31686],[34,366759],{"className":366760,"style":31846},[31845],[34,366762,202],{"className":366763},[31850],[34,366765],{"className":366766,"style":31846},[31845],[34,366768,366770,366773,366776,366779,366782],{"className":366769},[31676],[34,366771],{"className":366772,"style":224869},[31680],[34,366774,111],{"className":366775},[31685,31686],[34,366777],{"className":366778,"style":32604},[31845],[34,366780,71191],{"className":366781},[32024],[34,366783],{"className":366784,"style":32604},[31845],[34,366786,366788,366791,366794,366797,366800,366803],{"className":366787},[31676],[34,366789],{"className":366790,"style":31774},[31680],[34,366792,456],{"className":366793},[31782],[34,366795,1697],{"className":366796},[31685,31686],[34,366798],{"className":366799,"style":32604},[31845],[34,366801,71191],{"className":366802},[32024],[34,366804],{"className":366805,"style":32604},[31845],[34,366807,366809,366812,366815],{"className":366808},[31676],[34,366810],{"className":366811,"style":31774},[31680],[34,366813,31965],{"className":366814},[31685,31686],[34,366816,2557],{"className":366817},[31690],[108,366819,366820,366823,366824],{},[2627,366821,366822],{},"相容性公理（分配律）","：乘法对加法满足分配律。\n",[105,366825,366826,367000],{},[108,366827,366828,366829],{},"左分配律：",[34,366830,366832,366874],{"className":366831},[31642],[34,366833,366835],{"className":366834},[31646],[31648,366836,366837],{"xmlns":31650},[31652,366838,366839,366871],{},[31655,366840,366841,366843,366845,366847,366849,366851,366853,366855,366857,366859,366861,366863,366865,366867,366869],{},[31658,366842,111],{},[31662,366844,71191],{},[31662,366846,456],{"stretchy":702},[31658,366848,1697],{},[31662,366850,223],{},[31658,366852,31965],{},[31662,366854,2557],{"stretchy":702},[31662,366856,202],{},[31658,366858,111],{},[31662,366860,71191],{},[31658,366862,1697],{},[31662,366864,223],{},[31658,366866,111],{},[31662,366868,71191],{},[31658,366870,31965],{},[31665,366872,366873],{"encoding":31667},"a \\cdot (b+c) = a \\cdot b + a \\cdot c",[34,366875,366877,366895,366916,366937,366955,366973,366991],{"className":366876,"ariaHidden":758},[31672],[34,366878,366880,366883,366886,366889,366892],{"className":366879},[31676],[34,366881],{"className":366882,"style":224869},[31680],[34,366884,111],{"className":366885},[31685,31686],[34,366887],{"className":366888,"style":32604},[31845],[34,366890,71191],{"className":366891},[32024],[34,366893],{"className":366894,"style":32604},[31845],[34,366896,366898,366901,366904,366907,366910,366913],{"className":366897},[31676],[34,366899],{"className":366900,"style":31774},[31680],[34,366902,456],{"className":366903},[31782],[34,366905,1697],{"className":366906},[31685,31686],[34,366908],{"className":366909,"style":32604},[31845],[34,366911,223],{"className":366912},[32024],[34,366914],{"className":366915,"style":32604},[31845],[34,366917,366919,366922,366925,366928,366931,366934],{"className":366918},[31676],[34,366920],{"className":366921,"style":31774},[31680],[34,366923,31965],{"className":366924},[31685,31686],[34,366926,2557],{"className":366927},[31690],[34,366929],{"className":366930,"style":31846},[31845],[34,366932,202],{"className":366933},[31850],[34,366935],{"className":366936,"style":31846},[31845],[34,366938,366940,366943,366946,366949,366952],{"className":366939},[31676],[34,366941],{"className":366942,"style":224869},[31680],[34,366944,111],{"className":366945},[31685,31686],[34,366947],{"className":366948,"style":32604},[31845],[34,366950,71191],{"className":366951},[32024],[34,366953],{"className":366954,"style":32604},[31845],[34,366956,366958,366961,366964,366967,366970],{"className":366957},[31676],[34,366959],{"className":366960,"style":34299},[31680],[34,366962,1697],{"className":366963},[31685,31686],[34,366965],{"className":366966,"style":32604},[31845],[34,366968,223],{"className":366969},[32024],[34,366971],{"className":366972,"style":32604},[31845],[34,366974,366976,366979,366982,366985,366988],{"className":366975},[31676],[34,366977],{"className":366978,"style":224869},[31680],[34,366980,111],{"className":366981},[31685,31686],[34,366983],{"className":366984,"style":32604},[31845],[34,366986,71191],{"className":366987},[32024],[34,366989],{"className":366990,"style":32604},[31845],[34,366992,366994,366997],{"className":366993},[31676],[34,366995],{"className":366996,"style":32527},[31680],[34,366998,31965],{"className":366999},[31685,31686],[108,367001,367002,367003],{},"右分配律：",[34,367004,367006,367048],{"className":367005},[31642],[34,367007,367009],{"className":367008},[31646],[31648,367010,367011],{"xmlns":31650},[31652,367012,367013,367045],{},[31655,367014,367015,367017,367019,367021,367023,367025,367027,367029,367031,367033,367035,367037,367039,367041,367043],{},[31662,367016,456],{"stretchy":702},[31658,367018,111],{},[31662,367020,223],{},[31658,367022,1697],{},[31662,367024,2557],{"stretchy":702},[31662,367026,71191],{},[31658,367028,31965],{},[31662,367030,202],{},[31658,367032,111],{},[31662,367034,71191],{},[31658,367036,31965],{},[31662,367038,223],{},[31658,367040,1697],{},[31662,367042,71191],{},[31658,367044,31965],{},[31665,367046,367047],{"encoding":31667},"(a+b) \\cdot c = a \\cdot c + b \\cdot c",[34,367049,367051,367072,367093,367111,367129,367147,367165],{"className":367050,"ariaHidden":758},[31672],[34,367052,367054,367057,367060,367063,367066,367069],{"className":367053},[31676],[34,367055],{"className":367056,"style":31774},[31680],[34,367058,456],{"className":367059},[31782],[34,367061,111],{"className":367062},[31685,31686],[34,367064],{"className":367065,"style":32604},[31845],[34,367067,223],{"className":367068},[32024],[34,367070],{"className":367071,"style":32604},[31845],[34,367073,367075,367078,367081,367084,367087,367090],{"className":367074},[31676],[34,367076],{"className":367077,"style":31774},[31680],[34,367079,1697],{"className":367080},[31685,31686],[34,367082,2557],{"className":367083},[31690],[34,367085],{"className":367086,"style":32604},[31845],[34,367088,71191],{"className":367089},[32024],[34,367091],{"className":367092,"style":32604},[31845],[34,367094,367096,367099,367102,367105,367108],{"className":367095},[31676],[34,367097],{"className":367098,"style":32527},[31680],[34,367100,31965],{"className":367101},[31685,31686],[34,367103],{"className":367104,"style":31846},[31845],[34,367106,202],{"className":367107},[31850],[34,367109],{"className":367110,"style":31846},[31845],[34,367112,367114,367117,367120,367123,367126],{"className":367113},[31676],[34,367115],{"className":367116,"style":224869},[31680],[34,367118,111],{"className":367119},[31685,31686],[34,367121],{"className":367122,"style":32604},[31845],[34,367124,71191],{"className":367125},[32024],[34,367127],{"className":367128,"style":32604},[31845],[34,367130,367132,367135,367138,367141,367144],{"className":367131},[31676],[34,367133],{"className":367134,"style":32597},[31680],[34,367136,31965],{"className":367137},[31685,31686],[34,367139],{"className":367140,"style":32604},[31845],[34,367142,223],{"className":367143},[32024],[34,367145],{"className":367146,"style":32604},[31845],[34,367148,367150,367153,367156,367159,367162],{"className":367149},[31676],[34,367151],{"className":367152,"style":31681},[31680],[34,367154,1697],{"className":367155},[31685,31686],[34,367157],{"className":367158,"style":32604},[31845],[34,367160,71191],{"className":367161},[32024],[34,367163],{"className":367164,"style":32604},[31845],[34,367166,367168,367171],{"className":367167},[31676],[34,367169],{"className":367170,"style":32527},[31680],[34,367172,31965],{"className":367173},[31685,31686],[253,367175,367177],{"id":367176},"_22-环的分类与层级结构","2.2 环的分类与层级结构",[15,367179,367180],{},"讲课中口语表达较为琐碎，现将其整理为严谨的层级框架：",[13837,367182,367183,367196],{},[13840,367184,367185],{},[13843,367186,367187,367190,367193],{},[13846,367188,367189],{"align":82355},"术语 (Term)",[13846,367191,367192],{"align":82355},"定义特征 (Defining Characteristics)",[13846,367194,367195],{"align":82355},"备注",[13853,367197,367198,367211,367389,367458,367556,367602],{},[13843,367199,367200,367205,367208],{},[13858,367201,367202],{"align":82355},[2627,367203,367204],{},"环 (Ring)",[13858,367206,367207],{"align":82355},"基础定义（如上）",[13858,367209,367210],{"align":82355},"最一般的结构",[13843,367212,367213,367218,367301],{},[13858,367214,367215],{"align":82355},[2627,367216,367217],{},"幺环 (Unital Ring)",[13858,367219,367220,367221,33775,367249,367300],{"align":82355},"乘法有单位元 ",[34,367222,367224,367237],{"className":367223},[31642],[34,367225,367227],{"className":367226},[31646],[31648,367228,367229],{"xmlns":31650},[31652,367230,367231,367235],{},[31655,367232,367233],{},[31749,367234,640],{},[31665,367236,640],{"encoding":31667},[34,367238,367240],{"className":367239,"ariaHidden":758},[31672],[34,367241,367243,367246],{"className":367242},[31676],[34,367244],{"className":367245,"style":32617},[31680],[34,367247,640],{"className":367248},[31685],[34,367250,367252,367273],{"className":367251},[31642],[34,367253,367255],{"className":367254},[31646],[31648,367256,367257],{"xmlns":31650},[31652,367258,367259,367271],{},[31655,367260,367261,367263,367265,367267,367269],{},[31662,367262,456],{"stretchy":702},[31658,367264,34386],{},[31662,367266,31960],{"separator":758},[31662,367268,71191],{},[31662,367270,2557],{"stretchy":702},[31665,367272,366477],{"encoding":31667},[34,367274,367276],{"className":367275,"ariaHidden":758},[31672],[34,367277,367279,367282,367285,367288,367291,367294,367297],{"className":367278},[31676],[34,367280],{"className":367281,"style":31774},[31680],[34,367283,456],{"className":367284},[31782],[34,367286,34386],{"className":367287,"style":34427},[31685,31686],[34,367289,31960],{"className":367290},[32119],[34,367292],{"className":367293,"style":32123},[31845],[34,367295,71191],{"className":367296},[31685],[34,367298,2557],{"className":367299},[31690]," 是幺半群）",[13858,367302,367303,367304,367388],{"align":82355},"要求 ",[34,367305,367307,367325],{"className":367306},[31642],[34,367308,367310],{"className":367309},[31646],[31648,367311,367312],{"xmlns":31650},[31652,367313,367314,367322],{},[31655,367315,367316,367318,367320],{},[31749,367317,640],{},[31662,367319,54707],{"mathvariant":38079},[31749,367321,459],{},[31665,367323,367324],{"encoding":31667},"1 \\neq 0",[34,367326,367328,367379],{"className":367327,"ariaHidden":758},[31672],[34,367329,367331,367334,367337,367340,367376],{"className":367330},[31676],[34,367332],{"className":367333,"style":43217},[31680],[34,367335,640],{"className":367336},[31685],[34,367338],{"className":367339,"style":31846},[31845],[34,367341,367343,367370,367373],{"className":367342},[31850],[34,367344,367346],{"className":367345},[31850],[34,367347,367349],{"className":367348},[31685,54774],[34,367350,367352],{"className":367351},[54778],[34,367353,367355,367358,367367],{"className":367354},[54782],[34,367356],{"className":367357,"style":43217},[31680],[34,367359,367361],{"className":367360},[54789],[34,367362,367364],{"className":367363},[31685],[34,367365,54796],{"className":367366},[31850],[34,367368],{"className":367369},[54800],[34,367371],{"className":367372},[31845,54804],[34,367374,202],{"className":367375},[31850],[34,367377],{"className":367378,"style":31846},[31845],[34,367380,367382,367385],{"className":367381},[31676],[34,367383],{"className":367384,"style":32617},[31680],[34,367386,459],{"className":367387},[31685]," (通常)",[13843,367390,367391,367396,367455],{},[13858,367392,367393],{"align":82355},[2627,367394,367395],{},"交换环 (Commutative Ring)",[13858,367397,367398,367399,2557],{"align":82355},"乘法满足交换律 (",[34,367400,367402,367424],{"className":367401},[31642],[34,367403,367405],{"className":367404},[31646],[31648,367406,367407],{"xmlns":31650},[31652,367408,367409,367421],{},[31655,367410,367411,367413,367415,367417,367419],{},[31658,367412,111],{},[31658,367414,1697],{},[31662,367416,202],{},[31658,367418,1697],{},[31658,367420,111],{},[31665,367422,367423],{"encoding":31667},"ab=ba",[34,367425,367427,367445],{"className":367426,"ariaHidden":758},[31672],[34,367428,367430,367433,367436,367439,367442],{"className":367429},[31676],[34,367431],{"className":367432,"style":31681},[31680],[34,367434,324191],{"className":367435},[31685,31686],[34,367437],{"className":367438,"style":31846},[31845],[34,367440,202],{"className":367441},[31850],[34,367443],{"className":367444,"style":31846},[31845],[34,367446,367448,367451],{"className":367447},[31676],[34,367449],{"className":367450,"style":31681},[31680],[34,367452,367454],{"className":367453},[31685,31686],"ba",[13858,367456,367457],{"align":82355},"如多项式环",[13843,367459,367460,367465,367553],{},[13858,367461,367462],{"align":82355},[2627,367463,367464],{},"除环 (Division Ring)",[13858,367466,367467,367468,367552],{"align":82355},"幺环，且非零元素均可逆（",[34,367469,367471,367501],{"className":367470},[31642],[34,367472,367474],{"className":367473},[31646],[31648,367475,367476],{"xmlns":31650},[31652,367477,367478,367498],{},[31655,367479,367480,367482,367484,367486,367488,367490,367492,367494,367496],{},[31662,367481,456],{"stretchy":702},[31658,367483,34386],{},[31662,367485,336362],{},[31662,367487,32685],{"stretchy":702},[31749,367489,459],{},[31662,367491,32715],{"stretchy":702},[31662,367493,31960],{"separator":758},[31662,367495,71191],{},[31662,367497,2557],{"stretchy":702},[31665,367499,367500],{"encoding":31667},"(R\\setminus\\{0\\}, \\cdot)",[34,367502,367504,367525],{"className":367503,"ariaHidden":758},[31672],[34,367505,367507,367510,367513,367516,367519,367522],{"className":367506},[31676],[34,367508],{"className":367509,"style":31774},[31680],[34,367511,456],{"className":367512},[31782],[34,367514,34386],{"className":367515,"style":34427},[31685,31686],[34,367517],{"className":367518,"style":32604},[31845],[34,367520,336362],{"className":367521},[32024],[34,367523],{"className":367524,"style":32604},[31845],[34,367526,367528,367531,367534,367537,367540,367543,367546,367549],{"className":367527},[31676],[34,367529],{"className":367530,"style":31774},[31680],[34,367532,32685],{"className":367533},[31782],[34,367535,459],{"className":367536},[31685],[34,367538,32715],{"className":367539},[31690],[34,367541,31960],{"className":367542},[32119],[34,367544],{"className":367545,"style":32123},[31845],[34,367547,71191],{"className":367548},[31685],[34,367550,2557],{"className":367551},[31690]," 是群）",[13858,367554,367555],{"align":82355},"也称斜域 (Skew Field)",[13843,367557,367558,367562,367570],{},[13858,367559,367560],{"align":82355},[2627,367561,362689],{},[13858,367563,367564,367566,367567],{"align":82355},[2627,367565,362515],{},"，且",[2627,367568,367569],{},"无零因子",[13858,367571,367572,367573,367601],{"align":82355},"类似 ",[34,367574,367576,367589],{"className":367575},[31642],[34,367577,367579],{"className":367578},[31646],[31648,367580,367581],{"xmlns":31650},[31652,367582,367583,367587],{},[31655,367584,367585],{},[31658,367586,97300],{"mathvariant":318416},[31665,367588,320917],{"encoding":31667},[34,367590,367592],{"className":367591,"ariaHidden":758},[31672],[34,367593,367595,367598],{"className":367594},[31676],[34,367596],{"className":367597,"style":320927},[31680],[34,367599,97300],{"className":367600},[31685,318441]," 的结构",[13843,367603,367604,367608,367613],{},[13858,367605,367606],{"align":82355},[2627,367607,340387],{},[13858,367609,367610],{"align":82355},[2627,367611,367612],{},"交换除环",[13858,367614,367572,367615,367601],{"align":82355},[34,367616,367618,367636],{"className":367617},[31642],[34,367619,367621],{"className":367620},[31646],[31648,367622,367623],{"xmlns":31650},[31652,367624,367625,367633],{},[31655,367626,367627,367629,367631],{},[31658,367628,116863],{"mathvariant":318416},[31662,367630,31960],{"separator":758},[31658,367632,34386],{"mathvariant":318416},[31665,367634,367635],{"encoding":31667},"\\mathbb{Q}, \\mathbb{R}",[34,367637,367639],{"className":367638,"ariaHidden":758},[31672],[34,367640,367642,367645,367648,367651,367654],{"className":367641},[31676],[34,367643],{"className":367644,"style":338016},[31680],[34,367646,116863],{"className":367647},[31685,318441],[34,367649,31960],{"className":367650},[32119],[34,367652],{"className":367653,"style":32123},[31845],[34,367655,34386],{"className":367656},[31685,318441],[104833,367658,367659,367665],{},[15,367660,367661,367664],{},[2627,367662,367663],{},"关键辨析："," 讲稿中对于“单位元”和“单位”做了区分，这是代数中的易混点。",[105,367666,367667,367701,367844],{},[108,367668,367669,367672,367673,347],{},[2627,367670,367671],{},"单位元 (Identity\u002FUnity):"," 指乘法幺元 ",[34,367674,367676,367689],{"className":367675},[31642],[34,367677,367679],{"className":367678},[31646],[31648,367680,367681],{"xmlns":31650},[31652,367682,367683,367687],{},[31655,367684,367685],{},[31749,367686,640],{},[31665,367688,640],{"encoding":31667},[34,367690,367692],{"className":367691,"ariaHidden":758},[31672],[34,367693,367695,367698],{"className":367694},[31676],[34,367696],{"className":367697,"style":32617},[31680],[34,367699,640],{"className":367700},[31685],[108,367702,367703,367706,367707,367735,367736,315343,367764,347],{},[2627,367704,367705],{},"单位 (Unit\u002FInvertible Element):"," 指乘法可逆的元素 ",[34,367708,367710,367723],{"className":367709},[31642],[34,367711,367713],{"className":367712},[31646],[31648,367714,367715],{"xmlns":31650},[31652,367716,367717,367721],{},[31655,367718,367719],{},[31658,367720,94628],{},[31665,367722,94628],{"encoding":31667},[34,367724,367726],{"className":367725,"ariaHidden":758},[31672],[34,367727,367729,367732],{"className":367728},[31676],[34,367730],{"className":367731,"style":32527},[31680],[34,367733,94628],{"className":367734},[31685,31686],"，即存在 ",[34,367737,367739,367752],{"className":367738},[31642],[34,367740,367742],{"className":367741},[31646],[31648,367743,367744],{"xmlns":31650},[31652,367745,367746,367750],{},[31655,367747,367748],{},[31658,367749,56782],{},[31665,367751,56782],{"encoding":31667},[34,367753,367755],{"className":367754,"ariaHidden":758},[31672],[34,367756,367758,367761],{"className":367757},[31676],[34,367759],{"className":367760,"style":32527},[31680],[34,367762,56782],{"className":367763,"style":32062},[31685,31686],[34,367765,367767,367793],{"className":367766},[31642],[34,367768,367770],{"className":367769},[31646],[31648,367771,367772],{"xmlns":31650},[31652,367773,367774,367790],{},[31655,367775,367776,367778,367780,367782,367784,367786,367788],{},[31658,367777,94628],{},[31658,367779,56782],{},[31662,367781,202],{},[31658,367783,56782],{},[31658,367785,94628],{},[31662,367787,202],{},[31749,367789,640],{},[31665,367791,367792],{"encoding":31667},"uv=vu=1",[34,367794,367796,367814,367835],{"className":367795,"ariaHidden":758},[31672],[34,367797,367799,367802,367805,367808,367811],{"className":367798},[31676],[34,367800],{"className":367801,"style":32527},[31680],[34,367803,134150],{"className":367804,"style":32062},[31685,31686],[34,367806],{"className":367807,"style":31846},[31845],[34,367809,202],{"className":367810},[31850],[34,367812],{"className":367813,"style":31846},[31845],[34,367815,367817,367820,367823,367826,367829,367832],{"className":367816},[31676],[34,367818],{"className":367819,"style":32527},[31680],[34,367821,56782],{"className":367822,"style":32062},[31685,31686],[34,367824,94628],{"className":367825},[31685,31686],[34,367827],{"className":367828,"style":31846},[31845],[34,367830,202],{"className":367831},[31850],[34,367833],{"className":367834,"style":31846},[31845],[34,367836,367838,367841],{"className":367837},[31676],[34,367839],{"className":367840,"style":32617},[31680],[34,367842,640],{"className":367843},[31685],[108,367845,367846,367849],{},[2627,367847,367848],{},"域的定义："," 域既是整环（无零因子），又是除环（非零元可逆）。",[32188,367851],{},[19,367853,367855],{"id":367854},"_3-核心概念深入分析零因子与整环","3. 核心概念深入分析：零因子与整环",[15,367857,367858],{},"零因子是理解环论与初等算术差异的关键。",[253,367860,367862],{"id":367861},"_31-零因子-zero-divisors","3.1 零因子 (Zero Divisors)",[15,367864,367865,367868,367869,367897,367898,335149,367995,368068,368069,368097,368098,368126],{},[2627,367866,367867],{},"定义："," 在环 ",[34,367870,367872,367885],{"className":367871},[31642],[34,367873,367875],{"className":367874},[31646],[31648,367876,367877],{"xmlns":31650},[31652,367878,367879,367883],{},[31655,367880,367881],{},[31658,367882,34386],{},[31665,367884,34386],{"encoding":31667},[34,367886,367888],{"className":367887,"ariaHidden":758},[31672],[34,367889,367891,367894],{"className":367890},[31676],[34,367892],{"className":367893,"style":33218},[31680],[34,367895,34386],{"className":367896,"style":34427},[31685,31686]," 中，若存在非零元素 ",[34,367899,367901,367923],{"className":367900},[31642],[34,367902,367904],{"className":367903},[31646],[31648,367905,367906],{"xmlns":31650},[31652,367907,367908,367920],{},[31655,367909,367910,367912,367914,367916,367918],{},[31658,367911,111],{},[31662,367913,31960],{"separator":758},[31658,367915,1697],{},[31662,367917,54707],{"mathvariant":38079},[31749,367919,459],{},[31665,367921,367922],{"encoding":31667},"a, b \\neq 0",[34,367924,367926,367986],{"className":367925,"ariaHidden":758},[31672],[34,367927,367929,367932,367935,367938,367941,367944,367947,367983],{"className":367928},[31676],[34,367930],{"className":367931,"style":43217},[31680],[34,367933,111],{"className":367934},[31685,31686],[34,367936,31960],{"className":367937},[32119],[34,367939],{"className":367940,"style":32123},[31845],[34,367942,1697],{"className":367943},[31685,31686],[34,367945],{"className":367946,"style":31846},[31845],[34,367948,367950,367977,367980],{"className":367949},[31850],[34,367951,367953],{"className":367952},[31850],[34,367954,367956],{"className":367955},[31685,54774],[34,367957,367959],{"className":367958},[54778],[34,367960,367962,367965,367974],{"className":367961},[54782],[34,367963],{"className":367964,"style":43217},[31680],[34,367966,367968],{"className":367967},[54789],[34,367969,367971],{"className":367970},[31685],[34,367972,54796],{"className":367973},[31850],[34,367975],{"className":367976},[54800],[34,367978],{"className":367979},[31845,54804],[34,367981,202],{"className":367982},[31850],[34,367984],{"className":367985,"style":31846},[31845],[34,367987,367989,367992],{"className":367988},[31676],[34,367990],{"className":367991,"style":32617},[31680],[34,367993,459],{"className":367994},[31685],[34,367996,367998,368020],{"className":367997},[31642],[34,367999,368001],{"className":368000},[31646],[31648,368002,368003],{"xmlns":31650},[31652,368004,368005,368017],{},[31655,368006,368007,368009,368011,368013,368015],{},[31658,368008,111],{},[31662,368010,71191],{},[31658,368012,1697],{},[31662,368014,202],{},[31749,368016,459],{},[31665,368018,368019],{"encoding":31667},"a \\cdot b = 0",[34,368021,368023,368041,368059],{"className":368022,"ariaHidden":758},[31672],[34,368024,368026,368029,368032,368035,368038],{"className":368025},[31676],[34,368027],{"className":368028,"style":224869},[31680],[34,368030,111],{"className":368031},[31685,31686],[34,368033],{"className":368034,"style":32604},[31845],[34,368036,71191],{"className":368037},[32024],[34,368039],{"className":368040,"style":32604},[31845],[34,368042,368044,368047,368050,368053,368056],{"className":368043},[31676],[34,368045],{"className":368046,"style":31681},[31680],[34,368048,1697],{"className":368049},[31685,31686],[34,368051],{"className":368052,"style":31846},[31845],[34,368054,202],{"className":368055},[31850],[34,368057],{"className":368058,"style":31846},[31845],[34,368060,368062,368065],{"className":368061},[31676],[34,368063],{"className":368064,"style":32617},[31680],[34,368066,459],{"className":368067},[31685],"，则称 ",[34,368070,368072,368085],{"className":368071},[31642],[34,368073,368075],{"className":368074},[31646],[31648,368076,368077],{"xmlns":31650},[31652,368078,368079,368083],{},[31655,368080,368081],{},[31658,368082,111],{},[31665,368084,111],{"encoding":31667},[34,368086,368088],{"className":368087,"ariaHidden":758},[31672],[34,368089,368091,368094],{"className":368090},[31676],[34,368092],{"className":368093,"style":32527},[31680],[34,368095,111],{"className":368096},[31685,31686]," 为左零因子，",[34,368099,368101,368114],{"className":368100},[31642],[34,368102,368104],{"className":368103},[31646],[31648,368105,368106],{"xmlns":31650},[31652,368107,368108,368112],{},[31655,368109,368110],{},[31658,368111,1697],{},[31665,368113,1697],{"encoding":31667},[34,368115,368117],{"className":368116,"ariaHidden":758},[31672],[34,368118,368120,368123],{"className":368119},[31676],[34,368121],{"className":368122,"style":31681},[31680],[34,368124,1697],{"className":368125},[31685,31686]," 为右零因子。",[15,368128,368129],{},[2627,368130,368131],{},"分析与洞察：",[105,368133,368134,368625],{},[108,368135,368136,368139,368140,86674,368198,368281,368282,368333,368334,368362,368363,39984,368366],{},[2627,368137,368138],{},"消去律的失效："," 在小学算术中，若 ",[34,368141,368143,368165],{"className":368142},[31642],[34,368144,368146],{"className":368145},[31646],[31648,368147,368148],{"xmlns":31650},[31652,368149,368150,368162],{},[31655,368151,368152,368154,368156,368158,368160],{},[31658,368153,111],{},[31658,368155,1697],{},[31662,368157,202],{},[31658,368159,111],{},[31658,368161,31965],{},[31665,368163,368164],{"encoding":31667},"ab=ac",[34,368166,368168,368186],{"className":368167,"ariaHidden":758},[31672],[34,368169,368171,368174,368177,368180,368183],{"className":368170},[31676],[34,368172],{"className":368173,"style":31681},[31680],[34,368175,324191],{"className":368176},[31685,31686],[34,368178],{"className":368179,"style":31846},[31845],[34,368181,202],{"className":368182},[31850],[34,368184],{"className":368185,"style":31846},[31845],[34,368187,368189,368192,368195],{"className":368188},[31676],[34,368190],{"className":368191,"style":32527},[31680],[34,368193,111],{"className":368194},[31685,31686],[34,368196,31965],{"className":368197},[31685,31686],[34,368199,368201,368218],{"className":368200},[31642],[34,368202,368204],{"className":368203},[31646],[31648,368205,368206],{"xmlns":31650},[31652,368207,368208,368216],{},[31655,368209,368210,368212,368214],{},[31658,368211,111],{},[31662,368213,54707],{"mathvariant":38079},[31749,368215,459],{},[31665,368217,363487],{"encoding":31667},[34,368219,368221,368272],{"className":368220,"ariaHidden":758},[31672],[34,368222,368224,368227,368230,368233,368269],{"className":368223},[31676],[34,368225],{"className":368226,"style":43217},[31680],[34,368228,111],{"className":368229},[31685,31686],[34,368231],{"className":368232,"style":31846},[31845],[34,368234,368236,368263,368266],{"className":368235},[31850],[34,368237,368239],{"className":368238},[31850],[34,368240,368242],{"className":368241},[31685,54774],[34,368243,368245],{"className":368244},[54778],[34,368246,368248,368251,368260],{"className":368247},[54782],[34,368249],{"className":368250,"style":43217},[31680],[34,368252,368254],{"className":368253},[54789],[34,368255,368257],{"className":368256},[31685],[34,368258,54796],{"className":368259},[31850],[34,368261],{"className":368262},[54800],[34,368264],{"className":368265},[31845,54804],[34,368267,202],{"className":368268},[31850],[34,368270],{"className":368271,"style":31846},[31845],[34,368273,368275,368278],{"className":368274},[31676],[34,368276],{"className":368277,"style":32617},[31680],[34,368279,459],{"className":368280},[31685],"，我们可以推出 ",[34,368283,368285,368303],{"className":368284},[31642],[34,368286,368288],{"className":368287},[31646],[31648,368289,368290],{"xmlns":31650},[31652,368291,368292,368300],{},[31655,368293,368294,368296,368298],{},[31658,368295,1697],{},[31662,368297,202],{},[31658,368299,31965],{},[31665,368301,368302],{"encoding":31667},"b=c",[34,368304,368306,368324],{"className":368305,"ariaHidden":758},[31672],[34,368307,368309,368312,368315,368318,368321],{"className":368308},[31676],[34,368310],{"className":368311,"style":31681},[31680],[34,368313,1697],{"className":368314},[31685,31686],[34,368316],{"className":368317,"style":31846},[31845],[34,368319,202],{"className":368320},[31850],[34,368322],{"className":368323,"style":31846},[31845],[34,368325,368327,368330],{"className":368326},[31676],[34,368328],{"className":368329,"style":32527},[31680],[34,368331,31965],{"className":368332},[31685,31686],"。但在环论中，如果 ",[34,368335,368337,368350],{"className":368336},[31642],[34,368338,368340],{"className":368339},[31646],[31648,368341,368342],{"xmlns":31650},[31652,368343,368344,368348],{},[31655,368345,368346],{},[31658,368347,111],{},[31665,368349,111],{"encoding":31667},[34,368351,368353],{"className":368352,"ariaHidden":758},[31672],[34,368354,368356,368359],{"className":368355},[31676],[34,368357],{"className":368358,"style":32527},[31680],[34,368360,111],{"className":368361},[31685,31686]," 是零因子，",[2627,368364,368365],{},"消去律失效",[105,368367,368368],{},[108,368369,368370,47649,368373,368522,368523,368551,368552,347],{},[71807,368371,368372],{},"推导：",[34,368374,368376,368420],{"className":368375},[31642],[34,368377,368379],{"className":368378},[31646],[31648,368380,368381],{"xmlns":31650},[31652,368382,368383,368417],{},[31655,368384,368385,368387,368389,368391,368393,368395,368397,368399,368401,368403,368405,368407,368409,368411,368413,368415],{},[31658,368386,111],{},[31658,368388,1697],{},[31662,368390,202],{},[31658,368392,111],{},[31658,368394,31965],{},[39361,368396,262622],{},[31662,368398,262625],{},[39361,368400,262622],{},[31658,368402,111],{},[31662,368404,456],{"stretchy":702},[31658,368406,1697],{},[31662,368408,32226],{},[31658,368410,31965],{},[31662,368412,2557],{"stretchy":702},[31662,368414,202],{},[31749,368416,459],{},[31665,368418,368419],{"encoding":31667},"ab=ac \\implies a(b-c)=0",[34,368421,368423,368441,368468,368492,368513],{"className":368422,"ariaHidden":758},[31672],[34,368424,368426,368429,368432,368435,368438],{"className":368425},[31676],[34,368427],{"className":368428,"style":31681},[31680],[34,368430,324191],{"className":368431},[31685,31686],[34,368433],{"className":368434,"style":31846},[31845],[34,368436,202],{"className":368437},[31850],[34,368439],{"className":368440,"style":31846},[31845],[34,368442,368444,368447,368450,368453,368456,368459,368462,368465],{"className":368443},[31676],[34,368445],{"className":368446,"style":318264},[31680],[34,368448,111],{"className":368449},[31685,31686],[34,368451,31965],{"className":368452},[31685,31686],[34,368454],{"className":368455,"style":31846},[31845],[34,368457],{"className":368458,"style":31846},[31845],[34,368460,262625],{"className":368461},[31850],[34,368463],{"className":368464,"style":31846},[31845],[34,368466],{"className":368467,"style":31846},[31845],[34,368469,368471,368474,368477,368480,368483,368486,368489],{"className":368470},[31676],[34,368472],{"className":368473,"style":31774},[31680],[34,368475,111],{"className":368476},[31685,31686],[34,368478,456],{"className":368479},[31782],[34,368481,1697],{"className":368482},[31685,31686],[34,368484],{"className":368485,"style":32604},[31845],[34,368487,32226],{"className":368488},[32024],[34,368490],{"className":368491,"style":32604},[31845],[34,368493,368495,368498,368501,368504,368507,368510],{"className":368494},[31676],[34,368496],{"className":368497,"style":31774},[31680],[34,368499,31965],{"className":368500},[31685,31686],[34,368502,2557],{"className":368503},[31690],[34,368505],{"className":368506,"style":31846},[31845],[34,368508,202],{"className":368509},[31850],[34,368511],{"className":368512,"style":31846},[31845],[34,368514,368516,368519],{"className":368515},[31676],[34,368517],{"className":368518,"style":32617},[31680],[34,368520,459],{"className":368521},[31685],"。若 ",[34,368524,368526,368539],{"className":368525},[31642],[34,368527,368529],{"className":368528},[31646],[31648,368530,368531],{"xmlns":31650},[31652,368532,368533,368537],{},[31655,368534,368535],{},[31658,368536,111],{},[31665,368538,111],{"encoding":31667},[34,368540,368542],{"className":368541,"ariaHidden":758},[31672],[34,368543,368545,368548],{"className":368544},[31676],[34,368546],{"className":368547,"style":32527},[31680],[34,368549,111],{"className":368550},[31685,31686]," 是零因子，不能断定 ",[34,368553,368555,368577],{"className":368554},[31642],[34,368556,368558],{"className":368557},[31646],[31648,368559,368560],{"xmlns":31650},[31652,368561,368562,368574],{},[31655,368563,368564,368566,368568,368570,368572],{},[31658,368565,1697],{},[31662,368567,32226],{},[31658,368569,31965],{},[31662,368571,202],{},[31749,368573,459],{},[31665,368575,368576],{"encoding":31667},"b-c=0",[34,368578,368580,368598,368616],{"className":368579,"ariaHidden":758},[31672],[34,368581,368583,368586,368589,368592,368595],{"className":368582},[31676],[34,368584],{"className":368585,"style":34299},[31680],[34,368587,1697],{"className":368588},[31685,31686],[34,368590],{"className":368591,"style":32604},[31845],[34,368593,32226],{"className":368594},[32024],[34,368596],{"className":368597,"style":32604},[31845],[34,368599,368601,368604,368607,368610,368613],{"className":368600},[31676],[34,368602],{"className":368603,"style":32527},[31680],[34,368605,31965],{"className":368606},[31685,31686],[34,368608],{"className":368609,"style":31846},[31845],[34,368611,202],{"className":368612},[31850],[34,368614],{"className":368615,"style":31846},[31845],[34,368617,368619,368622],{"className":368618},[31676],[34,368620],{"className":368621,"style":32617},[31680],[34,368623,459],{"className":368624},[31685],[108,368626,368627,368630,368631,368659,368660,368733,368734,368736],{},[2627,368628,368629],{},"平凡零因子："," 元素 ",[34,368632,368634,368647],{"className":368633},[31642],[34,368635,368637],{"className":368636},[31646],[31648,368638,368639],{"xmlns":31650},[31652,368640,368641,368645],{},[31655,368642,368643],{},[31749,368644,459],{},[31665,368646,459],{"encoding":31667},[34,368648,368650],{"className":368649,"ariaHidden":758},[31672],[34,368651,368653,368656],{"className":368652},[31676],[34,368654],{"className":368655,"style":32617},[31680],[34,368657,459],{"className":368658},[31685]," 总是满足 ",[34,368661,368663,368685],{"className":368662},[31642],[34,368664,368666],{"className":368665},[31646],[31648,368667,368668],{"xmlns":31650},[31652,368669,368670,368682],{},[31655,368671,368672,368674,368676,368678,368680],{},[31749,368673,459],{},[31662,368675,71191],{},[31658,368677,32936],{},[31662,368679,202],{},[31749,368681,459],{},[31665,368683,368684],{"encoding":31667},"0 \\cdot x = 0",[34,368686,368688,368706,368724],{"className":368687,"ariaHidden":758},[31672],[34,368689,368691,368694,368697,368700,368703],{"className":368690},[31676],[34,368692],{"className":368693,"style":32617},[31680],[34,368695,459],{"className":368696},[31685],[34,368698],{"className":368699,"style":32604},[31845],[34,368701,71191],{"className":368702},[32024],[34,368704],{"className":368705,"style":32604},[31845],[34,368707,368709,368712,368715,368718,368721],{"className":368708},[31676],[34,368710],{"className":368711,"style":32527},[31680],[34,368713,32936],{"className":368714},[31685,31686],[34,368716],{"className":368717,"style":31846},[31845],[34,368719,202],{"className":368720},[31850],[34,368722],{"className":368723,"style":31846},[31845],[34,368725,368727,368730],{"className":368726},[31676],[34,368728],{"className":368729,"style":32617},[31680],[34,368731,459],{"className":368732},[31685],"，被称为平凡零因子。",[2627,368735,330740],{},"的精髓在于它排除了“非平凡零因子”，从而挽救了消去律。",[253,368738,368740],{"id":368739},"_32-基础性质推导","3.2 基础性质推导",[15,368742,368743],{},"讲稿中提及了三个命题，利用公理化定义可以严谨证明：",[2230,368745,368746,369340],{},[108,368747,368748,368824],{},[2627,368749,368750,368823],{},[34,368751,368753,368775],{"className":368752},[31642],[34,368754,368756],{"className":368755},[31646],[31648,368757,368758],{"xmlns":31650},[31652,368759,368760,368772],{},[31655,368761,368762,368764,368766,368768,368770],{},[31749,368763,459],{},[31662,368765,71191],{},[31658,368767,111],{},[31662,368769,202],{},[31749,368771,459],{},[31665,368773,368774],{"encoding":31667},"0 \\cdot a = 0",[34,368776,368778,368796,368814],{"className":368777,"ariaHidden":758},[31672],[34,368779,368781,368784,368787,368790,368793],{"className":368780},[31676],[34,368782],{"className":368783,"style":32617},[31680],[34,368785,459],{"className":368786},[31685],[34,368788],{"className":368789,"style":32604},[31845],[34,368791,71191],{"className":368792},[32024],[34,368794],{"className":368795,"style":32604},[31845],[34,368797,368799,368802,368805,368808,368811],{"className":368798},[31676],[34,368800],{"className":368801,"style":32527},[31680],[34,368803,111],{"className":368804},[31685,31686],[34,368806],{"className":368807,"style":31846},[31845],[34,368809,202],{"className":368810},[31850],[34,368812],{"className":368813,"style":31846},[31845],[34,368815,368817,368820],{"className":368816},[31676],[34,368818],{"className":368819,"style":32617},[31680],[34,368821,459],{"className":368822},[31685]," 的证明：",[34,368825,368827],{"className":368826},[126385],[34,368828,368830,368984],{"className":368829},[31642],[34,368831,368833],{"className":368832},[31646],[31648,368834,368835],{"xmlns":31650,"display":126394},[31652,368836,368837,368981],{},[71078,368838,368842,368893,368944],{"rowspacing":368839,"columnalign":368840,"columnspacing":368841},"0.25em","right left right","0em 1em",[71084,368843,368844,368856,368880],{},[71087,368845,368846],{},[71090,368847,368848],{"scriptlevel":459,"displaystyle":758},[31655,368849,368850,368852,368854],{},[31749,368851,459],{},[31662,368853,71191],{},[31658,368855,111],{},[71087,368857,368858],{},[71090,368859,368860],{"scriptlevel":459,"displaystyle":758},[31655,368861,368862,368864,368866,368868,368870,368872,368874,368876,368878],{},[31655,368863],{},[31662,368865,202],{},[31662,368867,456],{"stretchy":702},[31749,368869,459],{},[31662,368871,223],{},[31749,368873,459],{},[31662,368875,2557],{"stretchy":702},[31662,368877,71191],{},[31658,368879,111],{},[71087,368881,368882],{},[71090,368883,368884],{"scriptlevel":459,"displaystyle":758},[31655,368885,368886,368888,368891],{},[31662,368887,456],{"stretchy":702},[39361,368889,368890],{},"加法单位元定义",[31662,368892,2557],{"stretchy":702},[71084,368894,368895,368907,368931],{},[71087,368896,368897],{},[71090,368898,368899],{"scriptlevel":459,"displaystyle":758},[31655,368900,368901,368903,368905],{},[31749,368902,459],{},[31662,368904,71191],{},[31658,368906,111],{},[71087,368908,368909],{},[71090,368910,368911],{"scriptlevel":459,"displaystyle":758},[31655,368912,368913,368915,368917,368919,368921,368923,368925,368927,368929],{},[31655,368914],{},[31662,368916,202],{},[31749,368918,459],{},[31662,368920,71191],{},[31658,368922,111],{},[31662,368924,223],{},[31749,368926,459],{},[31662,368928,71191],{},[31658,368930,111],{},[71087,368932,368933],{},[71090,368934,368935],{"scriptlevel":459,"displaystyle":758},[31655,368936,368937,368939,368942],{},[31662,368938,456],{"stretchy":702},[39361,368940,368941],{},"分配律",[31662,368943,2557],{"stretchy":702},[71084,368945,368946,368952,368968],{},[71087,368947,368948],{},[71090,368949,368950],{"scriptlevel":459,"displaystyle":758},[31749,368951,459],{},[71087,368953,368954],{},[71090,368955,368956],{"scriptlevel":459,"displaystyle":758},[31655,368957,368958,368960,368962,368964,368966],{},[31655,368959],{},[31662,368961,202],{},[31749,368963,459],{},[31662,368965,71191],{},[31658,368967,111],{},[71087,368969,368970],{},[71090,368971,368972],{"scriptlevel":459,"displaystyle":758},[31655,368973,368974,368976,368979],{},[31662,368975,456],{"stretchy":702},[39361,368977,368978],{},"加法群消去律",[31662,368980,2557],{"stretchy":702},[31665,368982,368983],{"encoding":31667},"\\begin{aligned}\n0 \\cdot a &= (0+0) \\cdot a & (\\text{加法单位元定义}) \\\\\n0 \\cdot a &= 0 \\cdot a + 0 \\cdot a & (\\text{分配律}) \\\\\n0 &= 0 \\cdot a & (\\text{加法群消去律})\n\\end{aligned}",[34,368985,368987],{"className":368986,"ariaHidden":758},[31672],[34,368988,368990,368994],{"className":368989},[31676],[34,368991],{"className":368992,"style":368993},[31680],"height:4.2em;vertical-align:-1.85em;",[34,368995,368997],{"className":368996},[31685],[34,368998,369000,369084,369254,369257],{"className":368999},[71078],[34,369001,369004],{"className":369002},[369003],"col-align-r",[34,369005,369007,369076],{"className":369006},[31796,31797],[34,369008,369010,369073],{"className":369009},[31801],[34,369011,369013,369037,369061],{"className":369012,"style":77995},[31805],[34,369014,369016,369019],{"style":369015},"top:-4.51em;",[34,369017],{"className":369018,"style":41508},[31813],[34,369020,369022,369025,369028,369031,369034],{"className":369021},[31685],[34,369023,459],{"className":369024},[31685],[34,369026],{"className":369027,"style":32604},[31845],[34,369029,71191],{"className":369030},[32024],[34,369032],{"className":369033,"style":32604},[31845],[34,369035,111],{"className":369036},[31685,31686],[34,369038,369040,369043],{"style":369039},"top:-3.01em;",[34,369041],{"className":369042,"style":41508},[31813],[34,369044,369046,369049,369052,369055,369058],{"className":369045},[31685],[34,369047,459],{"className":369048},[31685],[34,369050],{"className":369051,"style":32604},[31845],[34,369053,71191],{"className":369054},[32024],[34,369056],{"className":369057,"style":32604},[31845],[34,369059,111],{"className":369060},[31685,31686],[34,369062,369064,369067],{"style":369063},"top:-1.51em;",[34,369065],{"className":369066,"style":41508},[31813],[34,369068,369070],{"className":369069},[31685],[34,369071,459],{"className":369072},[31685],[34,369074,31829],{"className":369075},[31828],[34,369077,369079],{"className":369078},[31801],[34,369080,369082],{"className":369081,"style":78064},[31805],[34,369083],{},[34,369085,369087],{"className":369086},[71384],[34,369088,369090,369246],{"className":369089},[31796,31797],[34,369091,369093,369243],{"className":369092},[31801],[34,369094,369096,369149,369208],{"className":369095,"style":77995},[31805],[34,369097,369098,369101],{"style":369015},[34,369099],{"className":369100,"style":41508},[31813],[34,369102,369104,369107,369110,369113,369116,369119,369122,369125,369128,369131,369134,369137,369140,369143,369146],{"className":369103},[31685],[34,369105],{"className":369106},[31685],[34,369108],{"className":369109,"style":31846},[31845],[34,369111,202],{"className":369112},[31850],[34,369114],{"className":369115,"style":31846},[31845],[34,369117,456],{"className":369118},[31782],[34,369120,459],{"className":369121},[31685],[34,369123],{"className":369124,"style":32604},[31845],[34,369126,223],{"className":369127},[32024],[34,369129],{"className":369130,"style":32604},[31845],[34,369132,459],{"className":369133},[31685],[34,369135,2557],{"className":369136},[31690],[34,369138],{"className":369139,"style":32604},[31845],[34,369141,71191],{"className":369142},[32024],[34,369144],{"className":369145,"style":32604},[31845],[34,369147,111],{"className":369148},[31685,31686],[34,369150,369151,369154],{"style":369039},[34,369152],{"className":369153,"style":41508},[31813],[34,369155,369157,369160,369163,369166,369169,369172,369175,369178,369181,369184,369187,369190,369193,369196,369199,369202,369205],{"className":369156},[31685],[34,369158],{"className":369159},[31685],[34,369161],{"className":369162,"style":31846},[31845],[34,369164,202],{"className":369165},[31850],[34,369167],{"className":369168,"style":31846},[31845],[34,369170,459],{"className":369171},[31685],[34,369173],{"className":369174,"style":32604},[31845],[34,369176,71191],{"className":369177},[32024],[34,369179],{"className":369180,"style":32604},[31845],[34,369182,111],{"className":369183},[31685,31686],[34,369185],{"className":369186,"style":32604},[31845],[34,369188,223],{"className":369189},[32024],[34,369191],{"className":369192,"style":32604},[31845],[34,369194,459],{"className":369195},[31685],[34,369197],{"className":369198,"style":32604},[31845],[34,369200,71191],{"className":369201},[32024],[34,369203],{"className":369204,"style":32604},[31845],[34,369206,111],{"className":369207},[31685,31686],[34,369209,369210,369213],{"style":369063},[34,369211],{"className":369212,"style":41508},[31813],[34,369214,369216,369219,369222,369225,369228,369231,369234,369237,369240],{"className":369215},[31685],[34,369217],{"className":369218},[31685],[34,369220],{"className":369221,"style":31846},[31845],[34,369223,202],{"className":369224},[31850],[34,369226],{"className":369227,"style":31846},[31845],[34,369229,459],{"className":369230},[31685],[34,369232],{"className":369233,"style":32604},[31845],[34,369235,71191],{"className":369236},[32024],[34,369238],{"className":369239,"style":32604},[31845],[34,369241,111],{"className":369242},[31685,31686],[34,369244,31829],{"className":369245},[31828],[34,369247,369249],{"className":369248},[31801],[34,369250,369252],{"className":369251,"style":78064},[31805],[34,369253],{},[34,369255],{"className":369256,"style":71733},[71732],[34,369258,369260],{"className":369259},[369003],[34,369261,369263,369332],{"className":369262},[31796,31797],[34,369264,369266,369329],{"className":369265},[31801],[34,369267,369269,369289,369309],{"className":369268,"style":77995},[31805],[34,369270,369271,369274],{"style":369015},[34,369272],{"className":369273,"style":41508},[31813],[34,369275,369277,369280,369286],{"className":369276},[31685],[34,369278,456],{"className":369279},[31782],[34,369281,369283],{"className":369282},[31685,2638],[34,369284,368890],{"className":369285},[31685,105462],[34,369287,2557],{"className":369288},[31690],[34,369290,369291,369294],{"style":369039},[34,369292],{"className":369293,"style":41508},[31813],[34,369295,369297,369300,369306],{"className":369296},[31685],[34,369298,456],{"className":369299},[31782],[34,369301,369303],{"className":369302},[31685,2638],[34,369304,368941],{"className":369305},[31685,105462],[34,369307,2557],{"className":369308},[31690],[34,369310,369311,369314],{"style":369063},[34,369312],{"className":369313,"style":41508},[31813],[34,369315,369317,369320,369326],{"className":369316},[31685],[34,369318,456],{"className":369319},[31782],[34,369321,369323],{"className":369322},[31685,2638],[34,369324,368978],{"className":369325},[31685,105462],[34,369327,2557],{"className":369328},[31690],[34,369330,31829],{"className":369331},[31828],[34,369333,369335],{"className":369334},[31801],[34,369336,369338],{"className":369337,"style":78064},[31805],[34,369339],{},[108,369341,369342,369433,369434,369522,369523,347],{},[2627,369343,369344,369345,2664],{},"符号规则 ",[34,369346,369348,369382],{"className":369347},[31642],[34,369349,369351],{"className":369350},[31646],[31648,369352,369353],{"xmlns":31650},[31652,369354,369355,369379],{},[31655,369356,369357,369359,369361,369363,369365,369367,369369,369371,369373,369375,369377],{},[31662,369358,456],{"stretchy":702},[31662,369360,32226],{},[31658,369362,111],{},[31662,369364,2557],{"stretchy":702},[31658,369366,1697],{},[31662,369368,202],{},[31662,369370,32226],{},[31662,369372,456],{"stretchy":702},[31658,369374,111],{},[31658,369376,1697],{},[31662,369378,2557],{"stretchy":702},[31665,369380,369381],{"encoding":31667},"(-a)b = -(ab)",[34,369383,369385,369415],{"className":369384,"ariaHidden":758},[31672],[34,369386,369388,369391,369394,369397,369400,369403,369406,369409,369412],{"className":369387},[31676],[34,369389],{"className":369390,"style":31774},[31680],[34,369392,456],{"className":369393},[31782],[34,369395,32226],{"className":369396},[31685],[34,369398,111],{"className":369399},[31685,31686],[34,369401,2557],{"className":369402},[31690],[34,369404,1697],{"className":369405},[31685,31686],[34,369407],{"className":369408,"style":31846},[31845],[34,369410,202],{"className":369411},[31850],[34,369413],{"className":369414,"style":31846},[31845],[34,369416,369418,369421,369424,369427,369430],{"className":369417},[31676],[34,369419],{"className":369420,"style":31774},[31680],[34,369422,32226],{"className":369423},[31685],[34,369425,456],{"className":369426},[31782],[34,369428,324191],{"className":369429},[31685,31686],[34,369431,2557],{"className":369432},[31690],"\n这说明“负号”可以像我们直觉中那样在乘法中移动。它建立在 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0",[34,369466,369468,369486,369513],{"className":369467,"ariaHidden":758},[31672],[34,369469,369471,369474,369477,369480,369483],{"className":369470},[31676],[34,369472],{"className":369473,"style":32597},[31680],[34,369475,111],{"className":369476},[31685,31686],[34,369478],{"className":369479,"style":32604},[31845],[34,369481,223],{"className":369482},[32024],[34,369484],{"className":369485,"style":32604},[31845],[34,369487,369489,369492,369495,369498,369501,369504,369507,369510],{"className":369488},[31676],[34,369490],{"className":369491,"style":31774},[31680],[34,369493,456],{"className":369494},[31782],[34,369496,32226],{"className":369497},[31685],[34,369499,111],{"className":369500},[31685,31686],[34,369502,2557],{"className":369503},[31690],[34,369505],{"className":369506,"style":31846},[31845],[34,369508,202],{"className":369509},[31850],[34,369511],{"className":369512,"style":31846},[31845],[34,369514,369516,369519],{"className":369515},[31676],[34,369517],{"className":369518,"style":32617},[31680],[34,369520,459],{"className":369521},[31685]," 的基础上，通过分配律导出 ",[34,369524,369526,369580],{"className":369525},[31642],[34,369527,369529],{"className":369528},[31646],[31648,369530,369531],{"xmlns":31650},[31652,369532,369533,369577],{},[31655,369534,369535,369537,369539,369541,369543,369545,369547,369549,369551,369553,369555,369557,369559,369561,369563,369565,369567,369569,369571,369573,369575],{},[31658,369536,111],{},[31658,369538,1697],{},[31662,369540,223],{},[31662,369542,456],{"stretchy":702},[31662,369544,32226],{},[31658,369546,111],{},[31662,369548,2557],{"stretchy":702},[31658,369550,1697],{},[31662,369552,202],{},[31662,369554,456],{"stretchy":702},[31658,369556,111],{},[31662,369558,32226],{},[31658,369560,111],{},[31662,369562,2557],{"stretchy":702},[31658,369564,1697],{},[31662,369566,202],{},[31749,369568,459],{},[31662,369570,71191],{},[31658,369572,1697],{},[31662,369574,202],{},[31749,369576,459],{},[31665,369578,369579],{"encoding":31667},"ab + (-a)b = (a-a)b = 0 \\cdot b = 0",[34,369581,369583,369601,369631,369652,369676,369694,369712],{"className":369582,"ariaHidden":758},[31672],[34,369584,369586,369589,369592,369595,369598],{"className":369585},[31676],[34,369587],{"className":369588,"style":34299},[31680],[34,369590,324191],{"className":369591},[31685,31686],[34,369593],{"className":369594,"style":32604},[31845],[34,369596,223],{"className":369597},[32024],[34,369599],{"className":369600,"style":32604},[31845],[34,369602,369604,369607,369610,369613,369616,369619,369622,369625,369628],{"className":369603},[31676],[34,369605],{"className":369606,"style":31774},[31680],[34,369608,456],{"className":369609},[31782],[34,369611,32226],{"className":369612},[31685],[34,369614,111],{"className":369615},[31685,31686],[34,369617,2557],{"className":369618},[31690],[34,369620,1697],{"className":369621},[31685,31686],[34,369623],{"className":369624,"style":31846},[31845],[34,369626,202],{"className":369627},[31850],[34,369629],{"className":369630,"style":31846},[31845],[34,369632,369634,369637,369640,369643,369646,369649],{"className":369633},[31676],[34,369635],{"className":369636,"style":31774},[31680],[34,369638,456],{"className":369639},[31782],[34,369641,111],{"className":369642},[31685,31686],[34,369644],{"className":369645,"style":32604},[31845],[34,369647,32226],{"className":369648},[32024],[34,369650],{"className":369651,"style":32604},[31845],[34,369653,369655,369658,369661,369664,369667,369670,369673],{"className":369654},[31676],[34,369656],{"className":369657,"style":31774},[31680],[34,369659,111],{"className":369660},[31685,31686],[34,369662,2557],{"className":369663},[31690],[34,369665,1697],{"className":369666},[31685,31686],[34,369668],{"className":369669,"style":31846},[31845],[34,369671,202],{"className":369672},[31850],[34,369674],{"className":369675,"style":31846},[31845],[34,369677,369679,369682,369685,369688,369691],{"className":369678},[31676],[34,369680],{"className":369681,"style":32617},[31680],[34,369683,459],{"className":369684},[31685],[34,369686],{"className":369687,"style":32604},[31845],[34,369689,71191],{"className":369690},[32024],[34,369692],{"className":369693,"style":32604},[31845],[34,369695,369697,369700,369703,369706,369709],{"className":369696},[31676],[34,369698],{"className":369699,"style":31681},[31680],[34,369701,1697],{"className":369702},[31685,31686],[34,369704],{"className":369705,"style":31846},[31845],[34,369707,202],{"className":369708},[31850],[34,369710],{"className":369711,"style":31846},[31845],[34,369713,369715,369718],{"className":369714},[31676],[34,369716],{"className":369717,"style":32617},[31680],[34,369719,459],{"className":369720},[31685],[32188,369722],{},[19,369724,369726],{"id":369725},"_4-实例分析与反思-case-studies","4. 实例分析与反思 (Case Studies)",[15,369728,369729],{},"根据讲课内容，我对提及的例子进行了结构化对比分析，以展示不同环的特性。",[13837,369731,369732,369754],{},[13840,369733,369734],{},[13843,369735,369736,369738,369742,369745,369748,369751],{},[13846,369737,233206],{"align":82355},[13846,369739,369741],{"align":369740},"center","交换性?",[13846,369743,369744],{"align":369740},"含幺?",[13846,369746,369747],{"align":369740},"无零因子?",[13846,369749,369750],{"align":369740},"可逆性 (非零元)?",[13846,369752,369753],{"align":82355},"结论类型",[13853,369755,369756,369839,369889,369993,370227],{},[13843,369757,369758,369791,369793,369795,369797,369834],{},[13858,369759,369760],{"align":82355},[2627,369761,369762,369763],{},"整数 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1",[34,369820,369822],{"className":369821,"ariaHidden":758},[31672],[34,369823,369825,369828,369831],{"className":369824},[31676],[34,369826],{"className":369827,"style":35942},[31680],[34,369829,41461],{"className":369830},[31685],[34,369832,640],{"className":369833},[31685],[13858,369835,369836,369838],{"align":82355},[2627,369837,330740],{}," (Integral Domain)",[13843,369840,369841,369876,369878,369880,369882,369884],{},[13858,369842,369843],{"align":82355},[2627,369844,369845,369846],{},"有理数 ",[34,369847,369849,369863],{"className":369848},[31642],[34,369850,369852],{"className":369851},[31646],[31648,369853,369854],{"xmlns":31650},[31652,369855,369856,369860],{},[31655,369857,369858],{},[31658,369859,116863],{"mathvariant":318416},[31665,369861,369862],{"encoding":31667},"\\mathbb{Q}",[34,369864,369866],{"className":369865,"ariaHidden":758},[31672],[34,369867,369869,369873],{"className":369868},[31676],[34,369870],{"className":369871,"style":369872},[31680],"height:0.8556em;vertical-align:-0.1667em;",[34,369874,116863],{"className":369875},[31685,318441],[13858,369877,289512],{"align":369740},[13858,369879,289512],{"align":369740},[13858,369881,289512],{"align":369740},[13858,369883,289512],{"align":369740},[13858,369885,369886,369888],{"align":82355},[2627,369887,330778],{}," (Field)",[13843,369890,369891,369981,369983,369985,369987,369990],{},[13858,369892,369893],{"align":82355},[2627,369894,369895,369896],{},"矩阵 ",[34,369897,369899,369923],{"className":369898},[31642],[34,369900,369902],{"className":369901},[31646],[31648,369903,369904],{"xmlns":31650},[31652,369905,369906,369920],{},[31655,369907,369908,369914,369916,369918],{},[31743,369909,369910,369912],{},[31658,369911,98360],{},[31658,369913,31660],{},[31662,369915,456],{"stretchy":702},[31658,369917,34386],{"mathvariant":318416},[31662,369919,2557],{"stretchy":702},[31665,369921,369922],{"encoding":31667},"M_n(\\mathbb{R})",[34,369924,369926],{"className":369925,"ariaHidden":758},[31672],[34,369927,369929,369932,369972,369975,369978],{"className":369928},[31676],[34,369930],{"className":369931,"style":31774},[31680],[34,369933,369935,369938],{"className":369934},[31685],[34,369936,98360],{"className":369937,"style":33389},[31685,31686],[34,369939,369941],{"className":369940},[31792],[34,369942,369944,369964],{"className":369943},[31796,31797],[34,369945,369947,369961],{"className":369946},[31801],[34,369948,369950],{"className":369949,"style":32362},[31805],[34,369951,369952,369955],{"style":231884},[34,369953],{"className":369954,"style":31814},[31813],[34,369956,369958],{"className":369957},[31818,31819,31820,31821],[34,369959,31660],{"className":369960},[31685,31686,31821],[34,369962,31829],{"className":369963},[31828],[34,369965,369967],{"className":369966},[31801],[34,369968,369970],{"className":369969,"style":31836},[31805],[34,369971],{},[34,369973,456],{"className":369974},[31782],[34,369976,34386],{"className":369977},[31685,318441],[34,369979,2557],{"className":369980},[31690],[13858,369982,289557],{"align":369740},[13858,369984,289512],{"align":369740},[13858,369986,289557],{"align":369740},[13858,369988,369989],{"align":369740},"否 (仅非奇异矩阵)",[13858,369991,369992],{"align":82355},"非交换幺环 (有零因子)",[13843,369994,369995,370098,370100,370102,370133,370163],{},[13858,369996,369997],{"align":82355},[2627,369998,339421,369999,370027,370028],{},[34,370000,370002,370015],{"className":370001},[31642],[34,370003,370005],{"className":370004},[31646],[31648,370006,370007],{"xmlns":31650},[31652,370008,370009,370013],{},[31655,370010,370011],{},[31658,370012,31660],{},[31665,370014,31660],{"encoding":31667},[34,370016,370018],{"className":370017,"ariaHidden":758},[31672],[34,370019,370021,370024],{"className":370020},[31676],[34,370022],{"className":370023,"style":32527},[31680],[34,370025,31660],{"className":370026},[31685,31686]," 整数 ",[34,370029,370031,370049],{"className":370030},[31642],[34,370032,370034],{"className":370033},[31646],[31648,370035,370036],{"xmlns":31650},[31652,370037,370038,370046],{},[31655,370039,370040],{},[31743,370041,370042,370044],{},[31658,370043,97300],{"mathvariant":318416},[31658,370045,31660],{},[31665,370047,370048],{"encoding":31667},"\\mathbb{Z}_n",[34,370050,370052],{"className":370051,"ariaHidden":758},[31672],[34,370053,370055,370058],{"className":370054},[31676],[34,370056],{"className":370057,"style":330413},[31680],[34,370059,370061,370064],{"className":370060},[31685],[34,370062,97300],{"className":370063},[31685,318441],[34,370065,370067],{"className":370066},[31792],[34,370068,370070,370090],{"className":370069},[31796,31797],[34,370071,370073,370087],{"className":370072},[31801],[34,370074,370076],{"className":370075,"style":32362},[31805],[34,370077,370078,370081],{"style":31809},[34,370079],{"className":370080,"style":31814},[31813],[34,370082,370084],{"className":370083},[31818,31819,31820,31821],[34,370085,31660],{"className":370086},[31685,31686,31821],[34,370088,31829],{"className":370089},[31828],[34,370091,370093],{"className":370092},[31801],[34,370094,370096],{"className":370095,"style":31836},[31805],[34,370097],{},[13858,370099,289512],{"align":369740},[13858,370101,289512],{"align":369740},[13858,370103,370104,370105],{"align":369740},"取决于 ",[34,370106,370108,370121],{"className":370107},[31642],[34,370109,370111],{"className":370110},[31646],[31648,370112,370113],{"xmlns":31650},[31652,370114,370115,370119],{},[31655,370116,370117],{},[31658,370118,31660],{},[31665,370120,31660],{"encoding":31667},[34,370122,370124],{"className":370123,"ariaHidden":758},[31672],[34,370125,370127,370130],{"className":370126},[31676],[34,370128],{"className":370129,"style":32527},[31680],[34,370131,31660],{"className":370132},[31685,31686],[13858,370134,370104,370135],{"align":369740},[34,370136,370138,370151],{"className":370137},[31642],[34,370139,370141],{"className":370140},[31646],[31648,370142,370143],{"xmlns":31650},[31652,370144,370145,370149],{},[31655,370146,370147],{},[31658,370148,31660],{},[31665,370150,31660],{"encoding":31667},[34,370152,370154],{"className":370153,"ariaHidden":758},[31672],[34,370155,370157,370160],{"className":370156},[31676],[34,370158],{"className":370159,"style":32527},[31680],[34,370161,31660],{"className":370162},[31685,31686],[13858,370164,43439,370165,370193,370194,43439,370196,370224,370225],{"align":82355},[34,370166,370168,370181],{"className":370167},[31642],[34,370169,370171],{"className":370170},[31646],[31648,370172,370173],{"xmlns":31650},[31652,370174,370175,370179],{},[31655,370176,370177],{},[31658,370178,31660],{},[31665,370180,31660],{"encoding":31667},[34,370182,370184],{"className":370183,"ariaHidden":758},[31672],[34,370185,370187,370190],{"className":370186},[31676],[34,370188],{"className":370189,"style":32527},[31680],[34,370191,31660],{"className":370192},[31685,31686]," 为合数，则有零因子；",[58531,370195],{},[34,370197,370199,370212],{"className":370198},[31642],[34,370200,370202],{"className":370201},[31646],[31648,370203,370204],{"xmlns":31650},[31652,370205,370206,370210],{},[31655,370207,370208],{},[31658,370209,31660],{},[31665,370211,31660],{"encoding":31667},[34,370213,370215],{"className":370214,"ariaHidden":758},[31672],[34,370216,370218,370221],{"className":370217},[31676],[34,370219],{"className":370220,"style":32527},[31680],[34,370222,31660],{"className":370223},[31685,31686]," 为素数，则是",[2627,370226,330778],{},[13843,370228,370229,370278,370309,370311,370342,370345],{},[13858,370230,370231],{"align":82355},[2627,370232,370233,370234],{},"多项式 ",[34,370235,370237,370257],{"className":370236},[31642],[34,370238,370240],{"className":370239},[31646],[31648,370241,370242],{"xmlns":31650},[31652,370243,370244,370254],{},[31655,370245,370246,370248,370250,370252],{},[31658,370247,34386],{},[31662,370249,15972],{"stretchy":702},[31658,370251,32936],{},[31662,370253,32712],{"stretchy":702},[31665,370255,370256],{"encoding":31667},"R[x]",[34,370258,370260],{"className":370259,"ariaHidden":758},[31672],[34,370261,370263,370266,370269,370272,370275],{"className":370262},[31676],[34,370264],{"className":370265,"style":31774},[31680],[34,370267,34386],{"className":370268,"style":34427},[31685,31686],[34,370270,15972],{"className":370271},[31782],[34,370273,32936],{"className":370274},[31685,31686],[34,370276,32712],{"className":370277},[31690],[13858,370279,43439,370280,370308],{"align":369740},[34,370281,370283,370296],{"className":370282},[31642],[34,370284,370286],{"className":370285},[31646],[31648,370287,370288],{"xmlns":31650},[31652,370289,370290,370294],{},[31655,370291,370292],{},[31658,370293,34386],{},[31665,370295,34386],{"encoding":31667},[34,370297,370299],{"className":370298,"ariaHidden":758},[31672],[34,370300,370302,370305],{"className":370301},[31676],[34,370303],{"className":370304,"style":33218},[31680],[34,370306,34386],{"className":370307,"style":34427},[31685,31686]," 交换",[13858,370310,289512],{"align":369740},[13858,370312,43439,370313,370341],{"align":369740},[34,370314,370316,370329],{"className":370315},[31642],[34,370317,370319],{"className":370318},[31646],[31648,370320,370321],{"xmlns":31650},[31652,370322,370323,370327],{},[31655,370324,370325],{},[31658,370326,34386],{},[31665,370328,34386],{"encoding":31667},[34,370330,370332],{"className":370331,"ariaHidden":758},[31672],[34,370333,370335,370338],{"className":370334},[31676],[34,370336],{"className":370337,"style":33218},[31680],[34,370339,34386],{"className":370340,"style":34427},[31685,31686]," 无零因子",[13858,370343,370344],{"align":369740},"否 (仅常数项可逆)",[13858,370346,43439,370347,370375,370376,370419],{"align":82355},[34,370348,370350,370363],{"className":370349},[31642],[34,370351,370353],{"className":370352},[31646],[31648,370354,370355],{"xmlns":31650},[31652,370356,370357,370361],{},[31655,370358,370359],{},[31658,370360,34386],{},[31665,370362,34386],{"encoding":31667},[34,370364,370366],{"className":370365,"ariaHidden":758},[31672],[34,370367,370369,370372],{"className":370368},[31676],[34,370370],{"className":370371,"style":33218},[31680],[34,370373,34386],{"className":370374,"style":34427},[31685,31686]," 是整环，则 ",[34,370377,370379,370398],{"className":370378},[31642],[34,370380,370382],{"className":370381},[31646],[31648,370383,370384],{"xmlns":31650},[31652,370385,370386,370396],{},[31655,370387,370388,370390,370392,370394],{},[31658,370389,34386],{},[31662,370391,15972],{"stretchy":702},[31658,370393,32936],{},[31662,370395,32712],{"stretchy":702},[31665,370397,370256],{"encoding":31667},[34,370399,370401],{"className":370400,"ariaHidden":758},[31672],[34,370402,370404,370407,370410,370413,370416],{"className":370403},[31676],[34,370405],{"className":370406,"style":31774},[31680],[34,370408,34386],{"className":370409,"style":34427},[31685,31686],[34,370411,15972],{"className":370412},[31782],[34,370414,32936],{"className":370415},[31685,31686],[34,370417,32712],{"className":370418},[31690]," 是整环",[253,370421,370423],{"id":370422},"关键案例深度剖析","关键案例深度剖析：",[2230,370425,370426,371048],{},[108,370427,370428,370517,370518],{},[2627,370429,370430,370431,370516],{},"矩阵环 ",[34,370432,370434,370458],{"className":370433},[31642],[34,370435,370437],{"className":370436},[31646],[31648,370438,370439],{"xmlns":31650},[31652,370440,370441,370455],{},[31655,370442,370443,370449,370451,370453],{},[31743,370444,370445,370447],{},[31658,370446,98360],{},[31658,370448,31660],{},[31662,370450,456],{"stretchy":702},[31658,370452,34386],{},[31662,370454,2557],{"stretchy":702},[31665,370456,370457],{"encoding":31667},"M_n(R)",[34,370459,370461],{"className":370460,"ariaHidden":758},[31672],[34,370462,370464,370467,370507,370510,370513],{"className":370463},[31676],[34,370465],{"className":370466,"style":31774},[31680],[34,370468,370470,370473],{"className":370469},[31685],[34,370471,98360],{"className":370472,"style":33389},[31685,31686],[34,370474,370476],{"className":370475},[31792],[34,370477,370479,370499],{"className":370478},[31796,31797],[34,370480,370482,370496],{"className":370481},[31801],[34,370483,370485],{"className":370484,"style":32362},[31805],[34,370486,370487,370490],{"style":231884},[34,370488],{"className":370489,"style":31814},[31813],[34,370491,370493],{"className":370492},[31818,31819,31820,31821],[34,370494,31660],{"className":370495},[31685,31686,31821],[34,370497,31829],{"className":370498},[31828],[34,370500,370502],{"className":370501},[31801],[34,370503,370505],{"className":370504,"style":31836},[31805],[34,370506],{},[34,370508,456],{"className":370509},[31782],[34,370511,34386],{"className":370512,"style":34427},[31685,31686],[34,370514,2557],{"className":370515},[31690]," 的非交换性与零因子：","\n这是最典型的“非良态”例子。\n",[105,370519,370520],{},[108,370521,370522,370523,371047],{},"举例：",[34,370524,370526,370652],{"className":370525},[31642],[34,370527,370529],{"className":370528},[31646],[31648,370530,370531],{"xmlns":31650},[31652,370532,370533,370649],{},[31655,370534,370535,370573,370575,370611,370613],{},[31655,370536,370537,370539,370571],{},[31662,370538,456],{"fence":758},[71078,370540,370543,370557],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},"0.16em","center center",[71084,370544,370545,370551],{},[71087,370546,370547],{},[71090,370548,370549],{"scriptlevel":459,"displaystyle":702},[31749,370550,459],{},[71087,370552,370553],{},[71090,370554,370555],{"scriptlevel":459,"displaystyle":702},[31749,370556,640],{},[71084,370558,370559,370565],{},[71087,370560,370561],{},[71090,370562,370563],{"scriptlevel":459,"displaystyle":702},[31749,370564,459],{},[71087,370566,370567],{},[71090,370568,370569],{"scriptlevel":459,"displaystyle":702},[31749,370570,459],{},[31662,370572,2557],{"fence":758},[31662,370574,71191],{},[31655,370576,370577,370579,370609],{},[31662,370578,456],{"fence":758},[71078,370580,370581,370595],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,370582,370583,370589],{},[71087,370584,370585],{},[71090,370586,370587],{"scriptlevel":459,"displaystyle":702},[31749,370588,640],{},[71087,370590,370591],{},[71090,370592,370593],{"scriptlevel":459,"displaystyle":702},[31749,370594,459],{},[71084,370596,370597,370603],{},[71087,370598,370599],{},[71090,370600,370601],{"scriptlevel":459,"displaystyle":702},[31749,370602,459],{},[71087,370604,370605],{},[71090,370606,370607],{"scriptlevel":459,"displaystyle":702},[31749,370608,459],{},[31662,370610,2557],{"fence":758},[31662,370612,202],{},[31655,370614,370615,370617,370647],{},[31662,370616,456],{"fence":758},[71078,370618,370619,370633],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,370620,370621,370627],{},[71087,370622,370623],{},[71090,370624,370625],{"scriptlevel":459,"displaystyle":702},[31749,370626,459],{},[71087,370628,370629],{},[71090,370630,370631],{"scriptlevel":459,"displaystyle":702},[31749,370632,459],{},[71084,370634,370635,370641],{},[71087,370636,370637],{},[71090,370638,370639],{"scriptlevel":459,"displaystyle":702},[31749,370640,459],{},[71087,370642,370643],{},[71090,370644,370645],{"scriptlevel":459,"displaystyle":702},[31749,370646,459],{},[31662,370648,2557],{"fence":758},[31665,370650,370651],{"encoding":31667},"\\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} = \\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}",[34,370653,370655,370792,370924],{"className":370654,"ariaHidden":758},[31672],[34,370656,370658,370662,370783,370786,370789],{"className":370657},[31676],[34,370659],{"className":370660,"style":370661},[31680],"height:2.4em;vertical-align:-0.95em;",[34,370663,370665,370671,370777],{"className":370664},[34006],[34,370666,370668],{"className":370667,"style":75811},[31782,75810],[34,370669,456],{"className":370670},[71281,31820],[34,370672,370674],{"className":370673},[31685],[34,370675,370677,370726,370729,370732],{"className":370676},[71078],[34,370678,370680],{"className":370679},[141481],[34,370681,370683,370717],{"className":370682},[31796,31797],[34,370684,370686,370714],{"className":370685},[31801],[34,370687,370690,370702],{"className":370688,"style":370689},[31805],"height:1.45em;",[34,370691,370693,370696],{"style":370692},"top:-3.61em;",[34,370694],{"className":370695,"style":41508},[31813],[34,370697,370699],{"className":370698},[31685],[34,370700,459],{"className":370701},[31685],[34,370703,370705,370708],{"style":370704},"top:-2.41em;",[34,370706],{"className":370707,"style":41508},[31813],[34,370709,370711],{"className":370710},[31685],[34,370712,459],{"className":370713},[31685],[34,370715,31829],{"className":370716},[31828],[34,370718,370720],{"className":370719},[31801],[34,370721,370724],{"className":370722,"style":370723},[31805],"height:0.95em;",[34,370725],{},[34,370727],{"className":370728,"style":141477},[71732],[34,370730],{"className":370731,"style":141477},[71732],[34,370733,370735],{"className":370734},[141481],[34,370736,370738,370769],{"className":370737},[31796,31797],[34,370739,370741,370766],{"className":370740},[31801],[34,370742,370744,370755],{"className":370743,"style":370689},[31805],[34,370745,370746,370749],{"style":370692},[34,370747],{"className":370748,"style":41508},[31813],[34,370750,370752],{"className":370751},[31685],[34,370753,640],{"className":370754},[31685],[34,370756,370757,370760],{"style":370704},[34,370758],{"className":370759,"style":41508},[31813],[34,370761,370763],{"className":370762},[31685],[34,370764,459],{"className":370765},[31685],[34,370767,31829],{"className":370768},[31828],[34,370770,370772],{"className":370771},[31801],[34,370773,370775],{"className":370774,"style":370723},[31805],[34,370776],{},[34,370778,370780],{"className":370779,"style":75811},[31690,75810],[34,370781,2557],{"className":370782},[71281,31820],[34,370784],{"className":370785,"style":32604},[31845],[34,370787,71191],{"className":370788},[32024],[34,370790],{"className":370791,"style":32604},[31845],[34,370793,370795,370798,370915,370918,370921],{"className":370794},[31676],[34,370796],{"className":370797,"style":370661},[31680],[34,370799,370801,370807,370909],{"className":370800},[34006],[34,370802,370804],{"className":370803,"style":75811},[31782,75810],[34,370805,456],{"className":370806},[71281,31820],[34,370808,370810],{"className":370809},[31685],[34,370811,370813,370858,370861,370864],{"className":370812},[71078],[34,370814,370816],{"className":370815},[141481],[34,370817,370819,370850],{"className":370818},[31796,31797],[34,370820,370822,370847],{"className":370821},[31801],[34,370823,370825,370836],{"className":370824,"style":370689},[31805],[34,370826,370827,370830],{"style":370692},[34,370828],{"className":370829,"style":41508},[31813],[34,370831,370833],{"className":370832},[31685],[34,370834,640],{"className":370835},[31685],[34,370837,370838,370841],{"style":370704},[34,370839],{"className":370840,"style":41508},[31813],[34,370842,370844],{"className":370843},[31685],[34,370845,459],{"className":370846},[31685],[34,370848,31829],{"className":370849},[31828],[34,370851,370853],{"className":370852},[31801],[34,370854,370856],{"className":370855,"style":370723},[31805],[34,370857],{},[34,370859],{"className":370860,"style":141477},[71732],[34,370862],{"className":370863,"style":141477},[71732],[34,370865,370867],{"className":370866},[141481],[34,370868,370870,370901],{"className":370869},[31796,31797],[34,370871,370873,370898],{"className":370872},[31801],[34,370874,370876,370887],{"className":370875,"style":370689},[31805],[34,370877,370878,370881],{"style":370692},[34,370879],{"className":370880,"style":41508},[31813],[34,370882,370884],{"className":370883},[31685],[34,370885,459],{"className":370886},[31685],[34,370888,370889,370892],{"style":370704},[34,370890],{"className":370891,"style":41508},[31813],[34,370893,370895],{"className":370894},[31685],[34,370896,459],{"className":370897},[31685],[34,370899,31829],{"className":370900},[31828],[34,370902,370904],{"className":370903},[31801],[34,370905,370907],{"className":370906,"style":370723},[31805],[34,370908],{},[34,370910,370912],{"className":370911,"style":75811},[31690,75810],[34,370913,2557],{"className":370914},[71281,31820],[34,370916],{"className":370917,"style":31846},[31845],[34,370919,202],{"className":370920},[31850],[34,370922],{"className":370923,"style":31846},[31845],[34,37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的二象性：",[105,371123,371124,371483],{},[108,371125,371126,336459,371196,13977,371421,371482],{},[34,371127,371129,371147],{"className":371128},[31642],[34,371130,371132],{"className":371131},[31646],[31648,371133,371134],{"xmlns":31650},[31652,371135,371136,371144],{},[31655,371137,371138],{},[31743,371139,371140,371142],{},[31658,371141,97300],{"mathvariant":318416},[31749,371143,15932],{},[31665,371145,371146],{"encoding":31667},"\\mathbb{Z}_4",[34,371148,371150],{"className":371149,"ariaHidden":758},[31672],[34,371151,371153,371156],{"className":371152},[31676],[34,371154],{"className":371155,"style":330413},[31680],[34,371157,371159,371162],{"className":371158},[31685],[34,371160,97300],{"className":371161},[31685,318441],[34,371163,371165],{"className":371164},[31792],[34,371166,371168,371188],{"className":371167},[31796,31797],[34,371169,371171,371185],{"className":371170},[31801],[34,371172,371174],{"className":371173,"style":31806},[31805],[34,371175,371176,371179],{"style":31809},[34,371177],{"className":371178,"style":31814},[31813],[34,371180,371182],{"className":371181},[31818,31819,31820,31821],[34,371183,15932],{"className":371184},[31685,31821],[34,371186,31829],{"className":371187},[31828],[34,371189,371191],{"className":371190},[31801],[34,371192,371194],{"className":371193,"style":31836},[31805],[34,371195],{},[34,371197,371199,371241],{"className":371198},[31642],[34,371200,371202],{"className":371201},[31646],[31648,371203,371204],{"xmlns":31650},[31652,371205,371206,371238],{},[31655,371207,371208,371214,371216,371222,371224,371230,371232],{},[134014,371209,371210,371212],{"accent":758},[31749,371211,768],{},[31662,371213,346142],{},[31662,371215,71191],{},[134014,371217,371218,371220],{"accent":758},[31749,371219,768],{},[31662,371221,346142],{},[31662,371223,202],{},[134014,371225,371226,371228],{"accent":758},[31749,371227,15932],{},[31662,371229,346142],{},[31662,371231,202],{},[134014,371233,371234,371236],{"accent":758},[31749,371235,459],{},[31662,371237,346142],{},[31665,371239,371240],{"encoding":31667},"\\bar{2} \\cdot \\bar{2} = \\bar{4} = \\bar{0}",[34,371242,371244,371292,371338,371384],{"className":371243,"ariaHidden":758},[31672],[34,371245,371247,371251,371283,371286,371289],{"className":371246},[31676],[34,371248],{"className":371249,"style":371250},[31680],"height:0.7812em;",[34,371252,371254],{"className":371253},[31685,346171],[34,371255,371257],{"className":371256},[31796],[34,371258,371260],{"className":371259},[31801],[34,371261,371263,371271],{"className":371262,"style":371250},[31805],[34,371264,371265,371268],{"style":41565},[34,371266],{"className":371267,"style":41508},[31813],[34,371269,768],{"className":371270},[31685],[34,371272,371274,371277],{"style":371273},"top:-3.2134em;",[34,371275],{"className":371276,"style":41508},[31813],[34,371278,371280],{"className":371279,"style":346199},[346198],[34,371281,346142],{"className":371282},[31685],[34,371284],{"className":371285,"style":32604},[31845],[34,371287,71191],{"className":371288},[32024],[34,371290],{"className":371291,"style":32604},[31845],[34,371293,371295,371298,371329,371332,371335],{"className":371294},[31676],[34,371296],{"className":371297,"style":371250},[31680],[34,371299,371301],{"className":371300},[31685,346171],[34,371302,371304],{"className":371303},[31796],[34,371305,371307],{"className":371306},[31801],[34,371308,371310,371318],{"className":371309,"style":371250},[31805],[34,371311,371312,371315],{"style":41565},[34,371313],{"className":371314,"style":41508},[31813],[34,371316,768],{"className":371317},[31685],[34,371319,371320,371323],{"style":371273},[34,371321],{"className":371322,"style":41508},[31813],[34,371324,371326],{"className":371325,"style":346199},[346198],[34,371327,346142],{"className":371328},[31685],[34,371330],{"className":371331,"style":31846},[31845],[34,371333,202],{"className":371334},[31850],[34,371336],{"className":371337,"style":31846},[31845],[34,371339,371341,371344,371375,371378,371381],{"className":371340},[31676],[34,371342],{"className":371343,"style":371250},[31680],[34,371345,371347],{"className":371346},[31685,346171],[34,371348,371350],{"className":371349},[31796],[34,371351,371353],{"className":371352},[31801],[34,371354,371356,371364],{"className":371355,"style":371250},[31805],[34,371357,371358,371361],{"style":41565},[34,371359],{"className":371360,"style":41508},[31813],[34,371362,15932],{"className":371363},[31685],[34,371365,371366,371369],{"style":371273},[34,371367],{"className":371368,"style":41508},[31813],[34,371370,371372],{"className":371371,"style":346199},[346198],[34,371373,346142],{"className":371374},[31685],[34,371376],{"className":371377,"style":31846},[31845],[34,371379,202],{"className":371380},[31850],[34,371382],{"className":371383,"style":31846},[31845],[34,371385,371387,371390],{"className":371386},[31676],[34,371388],{"className":371389,"style":371250},[31680],[34,371391,371393],{"className":371392},[31685,346171],[34,371394,371396],{"className":371395},[31796],[34,371397,371399],{"className":371398},[31801],[34,371400,371402,371410],{"className":371401,"style":371250},[31805],[34,371403,371404,371407],{"style":41565},[34,371405],{"className":371406,"style":41508},[31813],[34,371408,459],{"className":371409},[31685],[34,371411,371412,371415],{"style":371273},[34,371413],{"className":371414,"style":41508},[31813],[34,371416,371418],{"className":371417,"style":346199},[346198],[34,371419,346142],{"className":371420},[31685],[34,371422,371424,371442],{"className":371423},[31642],[34,371425,371427],{"className":371426},[31646],[31648,371428,371429],{"xmlns":31650},[31652,371430,371431,371439],{},[31655,371432,371433],{},[134014,371434,371435,371437],{"accent":758},[31749,371436,768],{},[31662,371438,346142],{},[31665,371440,371441],{"encoding":31667},"\\bar{2}",[34,371443,371445],{"className":371444,"ariaHidden":758},[31672],[34,371446,371448,371451],{"className":371447},[31676],[34,371449],{"className":371450,"style":371250},[31680],[34,371452,371454],{"className":371453},[31685,346171],[34,371455,371457],{"className":371456},[31796],[34,371458,371460],{"className":371459},[31801],[34,371461,371463,371471],{"className":371462,"style":371250},[31805],[34,371464,371465,371468],{"style":41565},[34,371466],{"className":371467,"style":41508},[31813],[34,371469,768],{"className":371470},[31685],[34,371472,371473,371476],{"style":371273},[34,371474],{"className":371475,"style":41508},[31813],[34,371477,371479],{"className":371478,"style":346199},[346198],[34,371480,346142],{"className":371481},[31685]," 是零因子。",[108,371484,371485,1840,371554,371582],{},[34,371486,371488,371505],{"className":371487},[31642],[34,371489,371491],{"className":371490},[31646],[31648,371492,371493],{"xmlns":31650},[31652,371494,371495,371503],{},[31655,371496,371497],{},[31743,371498,371499,371501],{},[31658,371500,97300],{"mathvariant":318416},[31658,371502,15],{},[31665,371504,338074],{"encoding":31667},[34,371506,371508],{"className":371507,"ariaHidden":758},[31672],[34,371509,371511,371514],{"className":371510},[31676],[34,371512],{"className":371513,"style":338084},[31680],[34,371515,371517,371520],{"className":371516},[31685],[34,371518,97300],{"className":371519},[31685,318441],[34,371521,371523],{"className":371522},[31792],[34,371524,371526,371546],{"className":371525},[31796,31797],[34,371527,371529,371543],{"className":371528},[31801],[34,371530,371532],{"className":371531,"style":32362},[31805],[34,371533,371534,371537],{"style":31809},[34,371535],{"className":371536,"style":31814},[31813],[34,371538,371540],{"className":371539},[31818,31819,31820,31821],[34,371541,15],{"className":371542},[31685,31686,31821],[34,371544,31829],{"className":371545},[31828],[34,371547,371549],{"className":371548},[31801],[34,371550,371552],{"className":371551,"style":32163},[31805],[34,371553],{},[34,371555,371557,371570],{"className":371556},[31642],[34,371558,371560],{"className":371559},[31646],[31648,371561,371562],{"xmlns":31650},[31652,371563,371564,371568],{},[31655,371565,371566],{},[31658,371567,15],{},[31665,371569,15],{"encoding":31667},[34,371571,371573],{"className":371572,"ariaHidden":758},[31672],[34,371574,371576,371579],{"className":371575},[31676],[34,371577],{"className":371578,"style":38774},[31680],[34,371580,15],{"className":371581},[31685,31686]," 为素数) 中，利用贝祖等式（Bezout's Identity）可证所有非零元均有逆元，从而构成域。这是数论与环论的交叉点。",{"title":29,"searchDepth":43,"depth":43,"links":371584},[371585,371586,371590,371594],{"id":365364,"depth":43,"text":365365},{"id":365408,"depth":43,"text":365409,"children":371587},[371588,371589],{"id":365415,"depth":49,"text":365416},{"id":367176,"depth":49,"text":367177},{"id":367854,"depth":43,"text":367855,"children":371591},[371592,371593],{"id":367861,"depth":49,"text":367862},{"id":368739,"depth":49,"text":368740},{"id":369725,"depth":43,"text":369726,"children":371595},[371596],{"id":370422,"depth":49,"text":370423},{"order":43},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fring",{"title":365357,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002Fring","kL6aeXKSqHwuc-yjLmwdIZ1VKF750jFhk8EQsUs_TCU",{"id":371603,"title":371604,"body":371605,"description":29,"extension":85,"meta":402619,"navigation":52,"path":402620,"seo":402621,"stem":402622,"__hash__":402623},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002FringHW.md","环的基本理论 习题详解",{"type":8,"value":371606,"toc":402602},[371607,371610,371752,371754,371786,371889,372258,372491,373720,373722,373754,373857,373996,375975,375977,375980,376086,376543,377524,377526,377529,377888,377890,377893,378229,378231,378315,378351,378386,379345,379384,379389,379391,379394,379472,379666,379880,379949,379952,380554,380560,382485,382543,382545,382548,382644,382769,382896,382963,382966,383026,383060,384604,384639,386117,386314,387433,387495,387497,387500,387561,387564,387571,387634,388190,388253,389056,389118,389837,389899,390263,390265,390268,390271,390274,390280,390622,390666,391615,391658,393485,393529,395606,395608,395611,395894,395897,396167,396319,397252,397286,397288,397291,397427,397430,397544,397668,398919,399120,399876,399878,399881,400016,400019,400464,400882,401905,402271],[253,371608,371609],{"id":371609},"证明习题",[15,371611,371612,2664,371615,371684,371685,47649,371723,371751],{},[2627,371613,371614],{},"命题",[34,371616,371618,371635],{"className":371617},[31642],[34,371619,371621],{"className":371620},[31646],[31648,371622,371623],{"xmlns":31650},[31652,371624,371625,371633],{},[31655,371626,371627],{},[31743,371628,371629,371631],{},[31658,371630,97300],{"mathvariant":318416},[31658,371632,15],{},[31665,371634,338074],{"encoding":31667},[34,371636,371638],{"className":371637,"ariaHidden":758},[31672],[34,371639,371641,371644],{"className":371640},[31676],[34,371642],{"className":371643,"style":338084},[31680],[34,371645,371647,371650],{"className":371646},[31685],[34,371648,97300],{"className":371649},[31685,318441],[34,371651,371653],{"className":371652},[31792],[34,371654,371656,371676],{"className":371655},[31796,31797],[34,371657,371659,371673],{"className":371658},[31801],[34,371660,371662],{"className":371661,"style":32362},[31805],[34,371663,371664,371667],{"style":31809},[34,371665],{"className":371666,"style":31814},[31813],[34,371668,371670],{"className":371669},[31818,31819,31820,31821],[34,371671,15],{"className":371672},[31685,31686,31821],[34,371674,31829],{"className":371675},[31828],[34,371677,371679],{"className":371678},[31801],[34,371680,371682],{"className":371681,"style":32163},[31805],[34,371683],{}," 是整环 ",[34,371686,371688,371705],{"className":371687},[31642],[34,371689,371691],{"className":371690},[31646],[31648,371692,371693],{"xmlns":31650},[31652,371694,371695,371703],{},[31655,371696,371697,371699,371701],{},[39361,371698,262622],{},[31662,371700,323391],{},[39361,371702,262622],{},[31665,371704,323396],{"encoding":31667},[34,371706,371708],{"className":371707,"ariaHidden":758},[31672],[34,371709,371711,371714,371717,371720],{"className":371710},[31676],[34,371712],{"className":371713,"style":318264},[31680],[34,371715],{"className":371716,"style":31846},[31845],[34,371718,323391],{"className":371719},[31850],[34,371721],{"className":371722,"style":31846},[31845],[34,371724,371726,371739],{"className":371725},[31642],[34,371727,371729],{"className":371728},[31646],[31648,371730,371731],{"xmlns":31650},[31652,371732,371733,371737],{},[31655,371734,371735],{},[31658,371736,15],{},[31665,371738,15],{"encoding":31667},[34,371740,371742],{"className":371741,"ariaHidden":758},[31672],[34,371743,371745,371748],{"className":371744},[31676],[34,371746],{"className":371747,"style":38774},[31680],[34,371749,15],{"className":371750},[31685,31686]," 是素数。",[32188,371753],{},[253,371755,371757,371758,2557],{"id":371756},"第一部分充分性证明-leftarrow","第一部分：充分性证明 (",[34,371759,371761,371774],{"className":371760},[31642],[34,371762,371764],{"className":371763},[31646],[31648,371765,371766],{"xmlns":31650},[31652,371767,371768,371772],{},[31655,371769,371770],{},[31662,371771,332814],{},[31665,371773,332817],{"encoding":31667},[34,371775,371777],{"className":371776,"ariaHidden":758},[31672],[34,371778,371780,371783],{"className":371779},[31676],[34,371781],{"className":371782,"style":73089},[31680],[34,371784,332814],{"className":371785},[31850],[15,371787,371788,346816,371790,371818,371819,371888],{},[2627,371789,233757],{},[34,371791,371793,371806],{"className":371792},[31642],[34,371794,371796],{"className":371795},[31646],[31648,371797,371798],{"xmlns":31650},[31652,371799,371800,371804],{},[31655,371801,371802],{},[31658,371803,15],{},[31665,371805,15],{"encoding":31667},[34,371807,371809],{"className":371808,"ariaHidden":758},[31672],[34,371810,371812,371815],{"className":371811},[31676],[34,371813],{"className":371814,"style":38774},[31680],[34,371816,15],{"className":371817},[31685,31686]," 是素数，证明 ",[34,371820,371822,371839],{"className":371821},[31642],[34,371823,371825],{"className":371824},[31646],[31648,371826,371827],{"xmlns":31650},[31652,371828,371829,371837],{},[31655,371830,371831],{},[31743,371832,371833,371835],{},[31658,371834,97300],{"mathvariant":318416},[31658,371836,15],{},[31665,371838,338074],{"encoding":31667},[34,371840,371842],{"className":371841,"ariaHidden":758},[31672],[34,371843,371845,371848],{"className":371844},[31676],[34,371846],{"className":371847,"style":338084},[31680],[34,371849,371851,371854],{"className":371850},[31685],[34,371852,97300],{"className":371853},[31685,318441],[34,371855,371857],{"className":371856},[31792],[34,371858,371860,371880],{"className":371859},[31796,31797],[34,371861,371863,371877],{"className":371862},[31801],[34,371864,371866],{"className":371865,"style":32362},[31805],[34,371867,371868,371871],{"style":31809},[34,371869],{"className":371870,"style":31814},[31813],[34,371872,371874],{"className":371873},[31818,31819,31820,31821],[34,371875,15],{"className":371876},[31685,31686,31821],[34,371878,31829],{"className":371879},[31828],[34,371881,371883],{"className":371882},[31801],[34,371884,371886],{"className":371885,"style":32163},[31805],[34,371887],{}," 里没有零因子。",[15,371890,371891,371894,371895,371964,371965,258,372026,356663,372089],{},[2627,371892,371893],{},"证明逻辑","：\n假设在 ",[34,371896,371898,371915],{"className":371897},[31642],[34,371899,371901],{"className":371900},[31646],[31648,371902,371903],{"xmlns":31650},[31652,371904,371905,371913],{},[31655,371906,371907],{},[31743,371908,371909,371911],{},[31658,371910,97300],{"mathvariant":318416},[31658,371912,15],{},[31665,371914,338074],{"encoding":31667},[34,371916,371918],{"className":371917,"ariaHidden":758},[31672],[34,371919,371921,371924],{"className":371920},[31676],[34,371922],{"className":371923,"style":338084},[31680],[34,371925,371927,371930],{"className":371926},[31685],[34,371928,97300],{"className":371929},[31685,318441],[34,371931,371933],{"className":371932},[31792],[34,371934,371936,371956],{"className":371935},[31796,31797],[34,371937,371939,371953],{"className":371938},[31801],[34,371940,371942],{"className":371941,"style":32362},[31805],[34,371943,371944,371947],{"style":31809},[34,371945],{"className":371946,"style":31814},[31813],[34,371948,371950],{"className":371949},[31818,31819,31820,31821],[34,371951,15],{"className":371952},[31685,31686,31821],[34,371954,31829],{"className":371955},[31828],[34,371957,371959],{"className":371958},[31801],[34,371960,371962],{"className":371961,"style":32163},[31805],[34,371963],{}," 中有两个元素 ",[34,371966,371968,371986],{"className":371967},[31642],[34,371969,371971],{"className":371970},[31646],[31648,371972,371973],{"xmlns":31650},[31652,371974,371975,371983],{},[31655,371976,371977],{},[134014,371978,371979,371981],{"accent":758},[31658,371980,111],{},[31662,371982,346142],{},[31665,371984,371985],{"encoding":31667},"\\bar{a}",[34,371987,371989],{"className":371988,"ariaHidden":758},[31672],[34,371990,371992,371995],{"className":371991},[31676],[34,371993],{"className":371994,"style":346181},[31680],[34,371996,371998],{"className":371997},[31685,346171],[34,371999,372001],{"className":372000},[31796],[34,372002,372004],{"className":372003},[31801],[34,372005,372007,372015],{"className":372006,"style":346181},[31805],[34,372008,372009,372012],{"style":41565},[34,372010],{"className":372011,"style":41508},[31813],[34,372013,111],{"className":372014},[31685,31686],[34,372016,372017,372020],{"style":41565},[34,372018],{"className":372019,"style":41508},[31813],[34,372021,372023],{"className":372022,"style":346199},[346198],[34,372024,346142],{"className":372025},[31685],[34,372027,372029,372047],{"className":372028},[31642],[34,372030,372032],{"className":372031},[31646],[31648,372033,372034],{"xmlns":31650},[31652,372035,372036,372044],{},[31655,372037,372038],{},[134014,372039,372040,372042],{"accent":758},[31658,372041,1697],{},[31662,372043,346142],{},[31665,372045,372046],{"encoding":31667},"\\bar{b}",[34,372048,372050],{"className":372049,"ariaHidden":758},[31672],[34,372051,372053,372057],{"className":372052},[31676],[34,372054],{"className":372055,"style":372056},[31680],"height:0.8312em;",[34,372058,372060],{"className":372059},[31685,346171],[34,372061,372063],{"className":372062},[31796],[34,372064,372066],{"className":372065},[31801],[34,372067,372069,372077],{"className":372068,"style":372056},[31805],[34,372070,372071,372074],{"style":41565},[34,372072],{"className":372073,"style":41508},[31813],[34,372075,1697],{"className":372076},[31685,31686],[34,372078,372080,372083],{"style":372079},"top:-3.2634em;",[34,372081],{"className":372082,"style":41508},[31813],[34,372084,372086],{"className":372085,"style":346199},[346198],[34,372087,346142],{"className":372088},[31685],[34,372090,372092,372126],{"className":372091},[31642],[34,372093,372095],{"className":372094},[31646],[31648,372096,372097],{"xmlns":31650},[31652,372098,372099,372123],{},[31655,372100,372101,372107,372109,372115,372117],{},[134014,372102,372103,372105],{"accent":758},[31658,372104,111],{},[31662,372106,346142],{},[31662,372108,71191],{},[134014,372110,372111,372113],{"accent":758},[31658,372112,1697],{},[31662,372114,346142],{},[31662,372116,202],{},[134014,372118,372119,372121],{"accent":758},[31749,372120,459],{},[31662,372122,346142],{},[31665,372124,372125],{"encoding":31667},"\\bar{a} \\cdot \\bar{b} = \\bar{0}",[34,372127,372129,372175,372221],{"className":372128,"ariaHidden":758},[31672],[34,372130,372132,372135,372166,372169,372172],{"className":372131},[31676],[34,372133],{"className":372134,"style":346181},[31680],[34,372136,372138],{"className":372137},[31685,346171],[34,372139,372141],{"className":372140},[31796],[34,372142,372144],{"className":372143},[31801],[34,372145,372147,372155],{"className":372146,"style":346181},[31805],[34,372148,372149,372152],{"style":41565},[34,372150],{"className":372151,"style":41508},[31813],[34,372153,111],{"className":372154},[31685,31686],[34,372156,372157,372160],{"style":41565},[34,372158],{"className":372159,"style":41508},[31813],[34,372161,372163],{"className":372162,"style":346199},[346198],[34,372164,346142],{"className":372165},[31685],[34,372167],{"className":372168,"style":32604},[31845],[34,372170,71191],{"className":372171},[32024],[34,372173],{"className":372174,"style":32604},[31845],[34,372176,372178,372181,372212,372215,372218],{"className":372177},[31676],[34,372179],{"className":372180,"style":372056},[31680],[34,372182,372184],{"className":372183},[31685,346171],[34,372185,372187],{"className":372186},[31796],[34,372188,372190],{"className":372189},[31801],[34,372191,372193,372201],{"className":372192,"style":372056},[31805],[34,372194,372195,372198],{"style":41565},[34,372196],{"className":372197,"style":41508},[31813],[34,372199,1697],{"className":372200},[31685,31686],[34,372202,372203,372206],{"style":372079},[34,372204],{"className":372205,"style":41508},[31813],[34,372207,372209],{"className":372208,"style":346199},[346198],[34,372210,346142],{"className":372211},[31685],[34,372213],{"className":372214,"style":31846},[31845],[34,372216,202],{"className":372217},[31850],[34,372219],{"className":372220,"style":31846},[31845],[34,372222,372224,372227],{"className":372223},[31676],[34,372225],{"className":372226,"style":371250},[31680],[34,372228,372230],{"className":372229},[31685,346171],[34,372231,372233],{"className":372232},[31796],[34,372234,372236],{"className":372235},[31801],[34,372237,372239,372247],{"className":372238,"style":371250},[31805],[34,372240,372241,372244],{"style":41565},[34,372242],{"className":372243,"style":41508},[31813],[34,372245,459],{"className":372246},[31685],[34,372248,372249,372252],{"style":371273},[34,372250],{"className":372251,"style":41508},[31813],[34,372253,372255],{"className":372254,"style":346199},[346198],[34,372256,346142],{"className":372257},[31685],[15,372259,372260,372261,331105,372376,347],{},"我们要证明 ",[34,372262,372264,372290],{"className":372263},[31642],[34,372265,372267],{"className":372266},[31646],[31648,372268,372269],{"xmlns":31650},[31652,372270,372271,372287],{},[31655,372272,372273,372279,372281],{},[134014,372274,372275,372277],{"accent":758},[31658,372276,111],{},[31662,372278,346142],{},[31662,372280,202],{},[134014,372282,372283,372285],{"accent":758},[31749,372284,459],{},[31662,372286,346142],{},[31665,372288,372289],{"encoding":31667},"\\bar{a} = \\bar{0}",[34,372291,372293,372339],{"className":372292,"ariaHidden":758},[31672],[34,372294,372296,372299,372330,372333,372336],{"className":372295},[31676],[34,372297],{"className":372298,"style":346181},[31680],[34,372300,372302],{"className":372301},[31685,346171],[34,372303,372305],{"className":372304},[31796],[34,372306,372308],{"className":372307},[31801],[34,372309,372311,372319],{"className":372310,"style":346181},[31805],[34,372312,372313,372316],{"style":41565},[34,372314],{"className":372315,"style":41508},[31813],[34,372317,111],{"className":372318},[31685,31686],[34,372320,372321,372324],{"style":41565},[34,372322],{"className":372323,"style":41508},[31813],[34,372325,372327],{"className":372326,"style":346199},[346198],[34,372328,346142],{"className":372329},[31685],[34,372331],{"className":372332,"style":31846},[31845],[34,372334,202],{"className":372335},[31850],[34,372337],{"className":372338,"style":31846},[31845],[34,372340,372342,372345],{"className":372341},[31676],[34,372343],{"className":372344,"style":371250},[31680],[34,372346,372348],{"className":372347},[31685,346171],[34,372349,372351],{"className":372350},[31796],[34,372352,372354],{"className":372353},[31801],[34,372355,372357,372365],{"className":372356,"style":371250},[31805],[34,372358,372359,372362],{"style":41565},[34,372360],{"className":372361,"style":41508},[31813],[34,372363,459],{"className":372364},[31685],[34,372366,372367,372370],{"style":371273},[34,372368],{"className":372369,"style":41508},[31813],[34,372371,372373],{"className":372372,"style":346199},[346198],[34,372374,346142],{"className":372375},[31685],[34,372377,372379,372405],{"className":372378},[31642],[34,372380,372382],{"className":372381},[31646],[31648,372383,372384],{"xmlns":31650},[31652,372385,372386,372402],{},[31655,372387,372388,372394,372396],{},[134014,372389,372390,372392],{"accent":758},[31658,372391,1697],{},[31662,372393,346142],{},[31662,372395,202],{},[134014,372397,372398,372400],{"accent":758},[31749,372399,459],{},[31662,372401,346142],{},[31665,372403,372404],{"encoding":31667},"\\bar{b} = \\bar{0}",[34,372406,372408,372454],{"className":372407,"ariaHidden":758},[31672],[34,372409,372411,372414,372445,372448,372451],{"className":372410},[31676],[34,372412],{"className":372413,"style":372056},[31680],[34,372415,372417],{"className":372416},[31685,346171],[34,372418,372420],{"className":372419},[31796],[34,372421,372423],{"className":372422},[31801],[34,372424,372426,372434],{"className":372425,"style":372056},[31805],[34,372427,372428,372431],{"style":41565},[34,372429],{"className":372430,"style":41508},[31813],[34,372432,1697],{"className":372433},[31685,31686],[34,372435,372436,372439],{"style":372079},[34,372437],{"className":372438,"style":41508},[31813],[34,372440,372442],{"className":372441,"style":346199},[346198],[34,372443,346142],{"className":372444},[31685],[34,372446],{"className":372447,"style":31846},[31845],[34,372449,202],{"className":372450},[31850],[34,372452],{"className":372453,"style":31846},[31845],[34,372455,372457,372460],{"className":372456},[31676],[34,372458],{"className":372459,"style":371250},[31680],[34,372461,372463],{"className":372462},[31685,346171],[34,372464,372466],{"className":372465},[31796],[34,372467,372469],{"className":372468},[31801],[34,372470,372472,372480],{"className":372471,"style":371250},[31805],[34,372473,372474,372477],{"style":41565},[34,372475],{"className":372476,"style":41508},[31813],[34,372478,459],{"className":372479},[31685],[34,372481,372482,372485],{"style":371273},[34,372483],{"className":372484,"style":41508},[31813],[34,372486,372488],{"className":372487,"style":346199},[346198],[34,372489,346142],{"className":372490},[31685],[2230,372492,372493,372880,373090,373543],{},[108,372494,372495,33183,372498,314945,372666,372735,372736,372787,372788,372816,372817],{},[2627,372496,372497],{},"翻译成数论语言",[34,372499,372501,372534],{"className":372500},[31642],[34,372502,372504],{"className":372503},[31646],[31648,372505,372506],{"xmlns":31650},[31652,372507,372508,372532],{},[31655,372509,372510,372516,372518,372524,372526],{},[134014,372511,372512,372514],{"accent":758},[31658,372513,111],{},[31662,372515,346142],{},[31662,372517,71191],{},[134014,372519,372520,372522],{"accent":758},[31658,372521,1697],{},[31662,372523,346142],{},[31662,372525,202],{},[134014,372527,372528,372530],{"accent":758},[31749,372529,459],{},[31662,372531,346142],{},[31665,372533,372125],{"encoding":31667},[34,372535,372537,372583,372629],{"className":372536,"ariaHidden":758},[31672],[34,372538,372540,372543,372574,372577,372580],{"className":372539},[31676],[34,372541],{"className":372542,"style":346181},[31680],[34,372544,372546],{"className":372545},[31685,346171],[34,372547,372549],{"className":372548},[31796],[34,372550,372552],{"className":372551},[31801],[34,372553,372555,372563],{"className":372554,"style":346181},[31805],[34,372556,372557,372560],{"style":41565},[34,372558],{"className":372559,"style":41508},[31813],[34,372561,111],{"className":372562},[31685,31686],[34,372564,372565,372568],{"style":41565},[34,372566],{"className":372567,"style":41508},[31813],[34,372569,372571],{"className":372570,"style":346199},[346198],[34,372572,346142],{"className":372573},[31685],[34,372575],{"className":372576,"style":32604},[31845],[34,372578,71191],{"className":372579},[32024],[34,372581],{"className":372582,"style":32604},[31845],[34,372584,372586,372589,372620,372623,372626],{"className":372585},[31676],[34,372587],{"className":372588,"style":372056},[31680],[34,372590,372592],{"className":372591},[31685,346171],[34,372593,372595],{"className":372594},[31796],[34,372596,372598],{"className":372597},[31801],[34,372599,372601,372609],{"className":372600,"style":372056},[31805],[34,372602,372603,372606],{"style":41565},[34,372604],{"className":372605,"style":41508},[31813],[34,372607,1697],{"className":372608},[31685,31686],[34,372610,372611,372614],{"style":372079},[34,372612],{"className":372613,"style":41508},[31813],[34,372615,372617],{"className":372616,"style":346199},[346198],[34,372618,346142],{"className":372619},[31685],[34,372621],{"className":372622,"style":31846},[31845],[34,372624,202],{"className":372625},[31850],[34,372627],{"className":372628,"style":31846},[31845],[34,372630,372632,372635],{"className":372631},[31676],[34,372633],{"className":372634,"style":371250},[31680],[34,372636,372638],{"className":372637},[31685,346171],[34,372639,372641],{"className":372640},[31796],[34,372642,372644],{"className":372643},[31801],[34,372645,372647,372655],{"className":372646,"style":371250},[31805],[34,372648,372649,372652],{"style":41565},[34,372650],{"className":372651,"style":41508},[31813],[34,372653,459],{"className":372654},[31685],[34,372656,372657,372660],{"style":371273},[34,372658],{"className":372659,"style":41508},[31813],[34,372661,372663],{"className":372662,"style":346199},[346198],[34,372664,346142],{"className":372665},[31685],[34,372667,372669,372686],{"className":372668},[31642],[34,372670,372672],{"className":372671},[31646],[31648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中的意思就是：整数乘积 ",[34,372737,372739,372757],{"className":372738},[31642],[34,372740,372742],{"className":372741},[31646],[31648,372743,372744],{"xmlns":31650},[31652,372745,372746,372754],{},[31655,372747,372748,372750,372752],{},[31658,372749,111],{},[31662,372751,71191],{},[31658,372753,1697],{},[31665,372755,372756],{"encoding":31667},"a \\cdot b",[34,372758,372760,372778],{"className":372759,"ariaHidden":758},[31672],[34,372761,372763,372766,372769,372772,372775],{"className":372762},[31676],[34,372764],{"className":372765,"style":224869},[31680],[34,372767,111],{"className":372768},[31685,31686],[34,372770],{"className":372771,"style":32604},[31845],[34,372773,71191],{"className":372774},[32024],[34,372776],{"className":372777,"style":32604},[31845],[34,372779,372781,372784],{"className":372780},[31676],[34,372782],{"className":372783,"style":31681},[31680],[34,372785,1697],{"className":372786},[31685,31686]," 被 ",[34,372789,372791,372804],{"className":372790},[31642],[34,372792,372794],{"className":372793},[31646],[31648,372795,372796],{"xmlns":31650},[31652,372797,372798,372802],{},[31655,372799,372800],{},[31658,372801,15],{},[31665,372803,15],{"encoding":31667},[34,372805,372807],{"className":372806,"ariaHidden":758},[31672],[34,372808,372810,372813],{"className":372809},[31676],[34,372811],{"className":372812,"style":38774},[31680],[34,372814,15],{"className":372815},[31685,31686]," 整除。\n记作：\n",[34,372818,372820,372844],{"className":372819},[31642],[34,372821,372823],{"className":372822},[31646],[31648,372824,372825],{"xmlns":31650},[31652,372826,372827,372841],{},[31655,372828,372829,372831,372833,372835,372837,372839],{},[31658,372830,15],{},[31662,372832,162086],{},[31662,372834,456],{"stretchy":702},[31658,372836,111],{},[31658,372838,1697],{},[31662,372840,2557],{"stretchy":702},[31665,372842,372843],{"encoding":31667},"p \\mid (ab)",[34,372845,372847,372865],{"className":372846,"ariaHidden":758},[31672],[34,372848,372850,372853,372856,372859,372862],{"className":372849},[31676],[34,372851],{"className":372852,"style":31774},[31680],[34,372854,15],{"className":372855},[31685,31686],[34,372857],{"className":372858,"style":31846},[31845],[34,372860,162086],{"className":372861},[31850],[34,372863],{"className":372864,"style":31846},[31845],[34,372866,372868,372871,372874,372877],{"className":372867},[31676],[34,372869],{"className":372870,"style":31774},[31680],[34,372872,456],{"className":372873},[31782],[34,372875,324191],{"className":372876},[31685,31686],[34,372878,2557],{"className":372879},[31690],[108,372881,372882,2664,372885],{},[2627,372883,372884],{},"利用素数的性质 (欧几里得引理)",[104833,372886,372887],{},[15,372888,372889],{},[2627,372890,2234,372891,372919,372920,331251,372948,330585,372978,331331,373006,331105,373034,331331,373062,347],{},[34,372892,372894,372907],{"className":372893},[31642],[34,372895,372897],{"className":372896},[31646],[31648,372898,372899],{"xmlns":31650},[31652,372900,372901,372905],{},[31655,372902,372903],{},[31658,372904,15],{},[31665,372906,15],{"encoding":31667},[34,372908,372910],{"className":372909,"ariaHidden":758},[31672],[34,372911,372913,372916],{"className":372912},[31676],[34,372914],{"className":372915,"style":38774},[31680],[34,372917,15],{"className":372918},[31685,31686]," 是素数，且 ",[34,372921,372923,372936],{"className":372922},[31642],[34,372924,372926],{"className":372925},[31646],[31648,372927,372928],{"xmlns":31650},[31652,372929,372930,372934],{},[31655,372931,372932],{},[31658,372933,15],{},[31665,372935,15],{"encoding":31667},[34,372937,372939],{"className":372938,"ariaHidden":758},[31672],[34,372940,372942,372945],{"className":372941},[31676],[34,372943],{"className":372944,"style":38774},[31680],[34,372946,15],{"className":372947},[31685,31686],[34,372949,372951,372966],{"className":372950},[31642],[34,372952,372954],{"className":372953},[31646],[31648,372955,372956],{"xmlns":31650},[31652,372957,372958,372964],{},[31655,372959,372960,372962],{},[31658,372961,111],{},[31658,372963,1697],{},[31665,372965,324191],{"encoding":31667},[34,372967,372969],{"className":372968,"ariaHidden":758},[31672],[34,372970,372972,372975],{"className":372971},[31676],[34,372973],{"className":372974,"style":31681},[31680],[34,372976,324191],{"className":372977},[31685,31686],[34,372979,372981,372994],{"className":372980},[31642],[34,372982,372984],{"className":372983},[31646],[31648,372985,372986],{"xmlns":31650},[31652,372987,372988,372992],{},[31655,372989,372990],{},[31658,372991,15],{},[31665,372993,15],{"encoding":31667},[34,372995,372997],{"className":372996,"ariaHidden":758},[31672],[34,372998,373000,373003],{"className":372999},[31676],[34,373001],{"className":373002,"style":38774},[31680],[34,373004,15],{"className":373005},[31685,31686],[34,373007,373009,373022],{"className":373008},[31642],[34,373010,373012],{"className":373011},[31646],[31648,373013,373014],{"xmlns":31650},[31652,373015,373016,373020],{},[31655,373017,373018],{},[31658,373019,111],{},[31665,373021,111],{"encoding":31667},[34,373023,373025],{"className":373024,"ariaHidden":758},[31672],[34,373026,373028,373031],{"className":373027},[31676],[34,373029],{"className":373030,"style":32527},[31680],[34,373032,111],{"className":373033},[31685,31686],[34,373035,373037,373050],{"className":373036},[31642],[34,373038,373040],{"className":373039},[31646],[31648,373041,373042],{"xmlns":31650},[31652,373043,373044,373048],{},[31655,373045,373046],{},[31658,373047,15],{},[31665,373049,15],{"encoding":31667},[34,373051,373053],{"className":373052,"ariaHidden":758},[31672],[34,373054,373056,373059],{"className":373055},[31676],[34,373057],{"className":373058,"style":38774},[31680],[34,373060,15],{"className":373061},[31685,31686],[34,373063,373065,373078],{"className":373064},[31642],[34,373066,373068],{"className":373067},[31646],[31648,373069,373070],{"xmlns":31650},[31652,373071,373072,373076],{},[31655,373073,373074],{},[31658,373075,1697],{},[31665,373077,1697],{"encoding":31667},[34,373079,373081],{"className":373080,"ariaHidden":758},[31672],[34,373082,373084,373087],{"className":373083},[31676],[34,373085],{"className":373086,"style":31681},[31680],[34,373088,1697],{"className":373089},[31685,31686],[108,373091,373092,2664,373095],{},[2627,373093,373094],{},"回到环论语言",[105,373096,373097,373321],{},[108,373098,2234,373099,341230,373149,373177,373178,373206,373207,347],{},[34,373100,373102,373119],{"className":373101},[31642],[34,373103,373105],{"className":373104},[31646],[31648,373106,373107],{"xmlns":31650},[31652,373108,373109,373117],{},[31655,373110,373111,373113,373115],{},[31658,373112,15],{},[31662,373114,162086],{},[31658,373116,111],{},[31665,373118,341309],{"encoding":31667},[34,373120,373122,373140],{"className":373121,"ariaHidden":758},[31672],[34,373123,373125,373128,373131,373134,373137],{"className":373124},[31676],[34,373126],{"className":373127,"style":31774},[31680],[34,373129,15],{"className":373130},[31685,31686],[34,373132],{"className":373133,"style":31846},[31845],[34,373135,162086],{"className":373136},[31850],[34,373138],{"className":373139,"style":31846},[31845],[34,373141,373143,373146],{"className":373142},[31676],[34,373144],{"className":373145,"style":32527},[31680],[34,373147,111],{"className":373148},[31685,31686],[34,373150,373152,373165],{"className":373151},[31642],[34,373153,373155],{"className":373154},[31646],[31648,373156,373157],{"xmlns":31650},[31652,373158,373159,373163],{},[31655,373160,373161],{},[31658,373162,111],{},[31665,373164,111],{"encoding":31667},[34,373166,373168],{"className":373167,"ariaHidden":758},[31672],[34,373169,373171,373174],{"className":373170},[31676],[34,373172],{"className":373173,"style":32527},[31680],[34,373175,111],{"className":373176},[31685,31686]," 除以 ",[34,373179,373181,373194],{"className":373180},[31642],[34,373182,373184],{"className":373183},[31646],[31648,373185,373186],{"xmlns":31650},[31652,373187,373188,373192],{},[31655,373189,373190],{},[31658,373191,15],{},[31665,373193,15],{"encoding":31667},[34,373195,373197],{"className":373196,"ariaHidden":758},[31672],[34,373198,373200,373203],{"className":373199},[31676],[34,373201],{"className":373202,"style":38774},[31680],[34,373204,15],{"className":373205},[31685,31686]," 余数为 0，即 ",[34,373208,373210,373235],{"className":373209},[31642],[34,373211,373213],{"className":373212},[31646],[31648,373214,373215],{"xmlns":31650},[31652,373216,373217,373233],{},[31655,373218,373219,373225,373227],{},[134014,373220,373221,373223],{"accent":758},[31658,373222,111],{},[31662,373224,346142],{},[31662,373226,202],{},[134014,373228,373229,373231],{"accent":758},[31749,373230,459],{},[31662,373232,346142],{},[31665,373234,372289],{"encoding":31667},[34,373236,373238,373284],{"className":373237,"ariaHidden":758},[31672],[34,373239,373241,373244,373275,373278,373281],{"className":373240},[31676],[34,373242],{"className":373243,"style":346181},[31680],[34,373245,373247],{"className":373246},[31685,346171],[34,373248,373250],{"className":373249},[31796],[34,373251,373253],{"className":373252},[31801],[34,373254,373256,373264],{"className":373255,"style":346181},[31805],[34,373257,373258,373261],{"style":41565},[34,373259],{"className":373260,"style":41508},[31813],[34,373262,111],{"className":373263},[31685,31686],[34,373265,373266,373269],{"style":41565},[34,373267],{"className":373268,"style":41508},[31813],[34,373270,373272],{"className":373271,"style":346199},[346198],[34,373273,346142],{"className":373274},[31685],[34,373276],{"className":373277,"style":31846},[31845],[34,373279,202],{"className":373280},[31850],[34,373282],{"className":373283,"style":31846},[31845],[34,373285,373287,373290],{"className":373286},[31676],[34,373288],{"className":373289,"style":371250},[31680],[34,373291,373293],{"className":373292},[31685,346171],[34,373294,373296],{"className":373295},[31796],[34,373297,373299],{"className":373298},[31801],[34,373300,373302,373310],{"className":373301,"style":371250},[31805],[34,373303,373304,373307],{"style":41565},[34,373305],{"className":373306,"style":41508},[31813],[34,373308,459],{"className":373309},[31685],[34,373311,373312,373315],{"style":371273},[34,373313],{"className":373314,"style":41508},[31813],[34,373316,373318],{"className":373317,"style":346199},[346198],[34,373319,346142],{"className":373320},[31685],[108,373322,2234,373323,341230,373373,373177,373401,373206,373429,347],{},[34,373324,373326,373343],{"className":373325},[31642],[34,373327,373329],{"className":373328},[31646],[31648,373330,373331],{"xmlns":31650},[31652,373332,373333,373341],{},[31655,373334,373335,373337,373339],{},[31658,373336,15],{},[31662,373338,162086],{},[31658,373340,1697],{},[31665,373342,344317],{"encoding":31667},[34,373344,373346,373364],{"className":373345,"ariaHidden":758},[31672],[34,373347,373349,373352,373355,373358,373361],{"className":373348},[31676],[34,373350],{"className":373351,"style":31774},[31680],[34,373353,15],{"className":373354},[31685,31686],[34,373356],{"className":373357,"style":31846},[31845],[34,373359,162086],{"className":373360},[31850],[34,373362],{"className":373363,"style":31846},[31845],[34,373365,373367,373370],{"className":373366},[31676],[34,373368],{"className":373369,"style":31681},[31680],[34,373371,1697],{"className":373372},[31685,31686],[34,373374,373376,373389],{"className":373375},[31642],[34,373377,373379],{"className":373378},[31646],[31648,373380,373381],{"xmlns":31650},[31652,373382,373383,373387],{},[31655,373384,373385],{},[31658,373386,1697],{},[31665,373388,1697],{"encoding":31667},[34,373390,373392],{"className":373391,"ariaHidden":758},[31672],[34,373393,373395,373398],{"className":373394},[31676],[34,373396],{"className":373397,"style":31681},[31680],[34,373399,1697],{"className":373400},[31685,31686],[34,373402,373404,373417],{"className":373403},[31642],[34,373405,373407],{"className":373406},[31646],[31648,373408,373409],{"xmlns":31650},[31652,373410,373411,373415],{},[31655,373412,373413],{},[31658,373414,15],{},[31665,373416,15],{"encoding":31667},[34,373418,373420],{"className":373419,"ariaHidden":758},[31672],[34,373421,373423,373426],{"className":373422},[31676],[34,373424],{"className":373425,"style":38774},[31680],[34,373427,15],{"className":373428},[31685,31686],[34,373430,373432,373457],{"className":373431},[31642],[34,373433,373435],{"className":373434},[31646],[31648,373436,373437],{"xmlns":31650},[31652,373438,373439,373455],{},[31655,373440,373441,373447,373449],{},[134014,373442,373443,373445],{"accent":758},[31658,373444,1697],{},[31662,373446,346142],{},[31662,373448,202],{},[134014,373450,373451,373453],{"accent":758},[31749,373452,459],{},[31662,373454,346142],{},[31665,373456,372404],{"encoding":31667},[34,373458,373460,373506],{"className":373459,"ariaHidden":758},[31672],[34,373461,373463,373466,373497,373500,373503],{"className":373462},[31676],[34,373464],{"className":373465,"style":372056},[31680],[34,373467,373469],{"className":373468},[31685,346171],[34,373470,373472],{"className":373471},[31796],[34,373473,373475],{"className":373474},[31801],[34,373476,373478,373486],{"className":373477,"style":372056},[31805],[34,373479,373480,373483],{"style":41565},[34,373481],{"className":373482,"style":41508},[31813],[34,373484,1697],{"className":373485},[31685,31686],[34,373487,373488,373491],{"style":372079},[34,373489],{"className":373490,"style":41508},[31813],[34,373492,373494],{"className":373493,"style":346199},[346198],[34,373495,346142],{"className":373496},[31685],[34,373498],{"className":373499,"style":31846},[31845],[34,373501,202],{"className":373502},[31850],[34,373504],{"className":373505,"style":31846},[31845],[34,373507,373509,373512],{"className":373508},[31676],[34,373510],{"className":373511,"style":371250},[31680],[34,373513,373515],{"className":373514},[31685,346171],[34,373516,373518],{"className":373517},[31796],[34,373519,373521],{"className":373520},[31801],[34,373522,373524,373532],{"className":373523,"style":371250},[31805],[34,373525,373526,373529],{"style":41565},[34,373527],{"className":373528,"style":41508},[31813],[34,373530,459],{"className":373531},[31685],[34,373533,373534,373537],{"style":371273},[34,373535],{"className":373536,"style":41508},[31813],[34,373538,373540],{"className":373539,"style":346199},[346198],[34,373541,346142],{"className":373542},[31685],[108,373544,373545,373547,373548,373617,373618,373621,373622,373650,373651,331534],{},[2627,373546,114043],{},"：\n既然乘积为 0 推出了因子至少有一个为 0，说明 ",[34,373549,373551,373568],{"className":373550},[31642],[34,373552,373554],{"className":373553},[31646],[31648,373555,373556],{"xmlns":31650},[31652,373557,373558,373566],{},[31655,373559,373560],{},[31743,373561,373562,373564],{},[31658,373563,97300],{"mathvariant":318416},[31658,373565,15],{},[31665,373567,338074],{"encoding":31667},[34,373569,373571],{"className":373570,"ariaHidden":758},[31672],[34,373572,373574,373577],{"className":373573},[31676],[34,373575],{"className":373576,"style":338084},[31680],[34,373578,373580,373583],{"className":373579},[31685],[34,373581,97300],{"className":373582},[31685,318441],[34,373584,373586],{"className":373585},[31792],[34,373587,373589,373609],{"className":373588},[31796,31797],[34,373590,373592,373606],{"className":373591},[31801],[34,373593,373595],{"className":373594,"style":32362},[31805],[34,373596,373597,373600],{"style":31809},[34,373598],{"className":373599,"style":31814},[31813],[34,373601,373603],{"className":373602},[31818,31819,31820,31821],[34,373604,15],{"className":373605},[31685,31686,31821],[34,373607,31829],{"className":373608},[31828],[34,373610,373612],{"className":373611},[31801],[34,373613,373615],{"className":373614,"style":32163},[31805],[34,373616],{}," 中",[2627,373619,373620],{},"不存在","非零零因子。\n因此，当 ",[34,373623,373625,373638],{"className":373624},[31642],[34,373626,373628],{"className":373627},[31646],[31648,373629,373630],{"xmlns":31650},[31652,373631,373632,373636],{},[31655,373633,373634],{},[31658,373635,15],{},[31665,373637,15],{"encoding":31667},[34,373639,373641],{"className":373640,"ariaHidden":758},[31672],[34,373642,373644,373647],{"className":373643},[31676],[34,373645],{"className":373646,"style":38774},[31680],[34,373648,15],{"className":373649},[31685,31686]," 是素数时，",[34,373652,373654,373671],{"className":373653},[31642],[34,373655,373657],{"className":373656},[31646],[31648,373658,373659],{"xmlns":31650},[31652,373660,373661,373669],{},[31655,373662,373663],{},[31743,373664,373665,373667],{},[31658,373666,97300],{"mathvariant":318416},[31658,373668,15],{},[31665,373670,338074],{"encoding":31667},[34,373672,373674],{"className":373673,"ariaHidden":758},[31672],[34,373675,373677,373680],{"className":373676},[31676],[34,373678],{"className":373679,"style":338084},[31680],[34,373681,373683,373686],{"className":373682},[31685],[34,373684,97300],{"className":373685},[31685,318441],[34,373687,373689],{"className":373688},[31792],[34,373690,373692,373712],{"className":373691},[31796,31797],[34,373693,373695,373709],{"className":373694},[31801],[34,373696,373698],{"className":373697,"style":32362},[31805],[34,373699,373700,373703],{"style":31809},[34,373701],{"className":373702,"style":31814},[31813],[34,373704,373706],{"className":373705},[31818,31819,31820,31821],[34,373707,15],{"className":373708},[31685,31686,31821],[34,373710,31829],{"className":373711},[31828],[34,373713,373715],{"className":373714},[31801],[34,373716,373718],{"className":373717,"style":32163},[31805],[34,373719],{},[32188,373721],{},[253,373723,373725,373726,2557],{"id":373724},"第二部分必要性证明-rightarrow","第二部分：必要性证明 (",[34,373727,373729,373742],{"className":373728},[31642],[34,373730,373732],{"className":373731},[31646],[31648,373733,373734],{"xmlns":31650},[31652,373735,373736,373740],{},[31655,373737,373738],{},[31662,373739,140094],{},[31665,373741,266872],{"encoding":31667},[34,373743,373745],{"className":373744,"ariaHidden":758},[31672],[34,373746,373748,373751],{"className":373747},[31676],[34,373749],{"className":373750,"style":73089},[31680],[34,373752,140094],{"className":373753},[31850],[15,373755,373756,346816,373758,373827,373828,373856],{},[2627,373757,233757],{},[34,373759,373761,373778],{"className":373760},[31642],[34,373762,373764],{"className":373763},[31646],[31648,373765,373766],{"xmlns":31650},[31652,373767,373768,373776],{},[31655,373769,373770],{},[31743,373771,373772,373774],{},[31658,373773,97300],{"mathvariant":318416},[31658,373775,15],{},[31665,373777,338074],{"encoding":31667},[34,373779,373781],{"className":373780,"ariaHidden":758},[31672],[34,373782,373784,373787],{"className":373783},[31676],[34,373785],{"className":373786,"style":338084},[31680],[34,373788,373790,373793],{"className":373789},[31685],[34,373791,97300],{"className":373792},[31685,318441],[34,373794,373796],{"className":373795},[31792],[34,373797,373799,373819],{"className":373798},[31796,31797],[34,373800,373802,373816],{"className":373801},[31801],[34,373803,373805],{"className":373804,"style":32362},[31805],[34,373806,373807,373810],{"style":31809},[34,373808],{"className":373809,"style":31814},[31813],[34,373811,373813],{"className":373812},[31818,31819,31820,31821],[34,373814,15],{"className":373815},[31685,31686,31821],[34,373817,31829],{"className":373818},[31828],[34,373820,373822],{"className":373821},[31801],[34,373823,373825],{"className":373824,"style":32163},[31805],[34,373826],{}," 是整环，证明 ",[34,373829,373831,373844],{"className":373830},[31642],[34,373832,373834],{"className":373833},[31646],[31648,373835,373836],{"xmlns":31650},[31652,373837,373838,373842],{},[31655,373839,373840],{},[31658,373841,15],{},[31665,373843,15],{"encoding":31667},[34,373845,373847],{"className":373846,"ariaHidden":758},[31672],[34,373848,373850,373853],{"className":373849},[31676],[34,373851],{"className":373852,"style":38774},[31680],[34,373854,15],{"className":373855},[31685,31686]," 只能是素数。",[15,373858,373859,373861,373862,373865,373866,373935,373936,47649,373964,373966,373967,373995],{},[2627,373860,371893],{},"：我们要用",[2627,373863,373864],{},"反证法","。\n假设 ",[34,373867,373869,373886],{"className":373868},[31642],[34,373870,373872],{"className":373871},[31646],[31648,373873,373874],{"xmlns":31650},[31652,373875,373876,373884],{},[31655,373877,373878],{},[31743,373879,373880,373882],{},[31658,373881,97300],{"mathvariant":318416},[31658,373883,15],{},[31665,373885,338074],{"encoding":31667},[34,373887,373889],{"className":373888,"ariaHidden":758},[31672],[34,373890,373892,373895],{"className":373891},[31676],[34,373893],{"className":373894,"style":338084},[31680],[34,373896,373898,373901],{"className":373897},[31685],[34,373899,97300],{"className":373900},[31685,318441],[34,373902,373904],{"className":373903},[31792],[34,373905,373907,373927],{"className":373906},[31796,31797],[34,373908,373910,373924],{"className":373909},[31801],[34,373911,373913],{"className":373912,"style":32362},[31805],[34,373914,373915,373918],{"style":31809},[34,373916],{"className":373917,"style":31814},[31813],[34,373919,373921],{"className":373920},[31818,31819,31820,31821],[34,373922,15],{"className":373923},[31685,31686,31821],[34,373925,31829],{"className":373926},[31828],[34,373928,373930],{"className":373929},[31801],[34,373931,373933],{"className":373932,"style":32163},[31805],[34,373934],{}," 是整环，但 ",[34,373937,373939,373952],{"className":373938},[31642],[34,373940,373942],{"className":373941},[31646],[31648,373943,373944],{"xmlns":31650},[31652,373945,373946,373950],{},[31655,373947,373948],{},[31658,373949,15],{},[31665,373951,15],{"encoding":31667},[34,373953,373955],{"className":373954,"ariaHidden":758},[31672],[34,373956,373958,373961],{"className":373957},[31676],[34,373959],{"className":373960,"style":38774},[31680],[34,373962,15],{"className":373963},[31685,31686],[2627,373965,243550],{},"素数（即 ",[34,373968,373970,373983],{"className":373969},[31642],[34,373971,373973],{"className":373972},[31646],[31648,373974,373975],{"xmlns":31650},[31652,373976,373977,373981],{},[31655,373978,373979],{},[31658,373980,15],{},[31665,373982,15],{"encoding":31667},[34,373984,373986],{"className":373985,"ariaHidden":758},[31672],[34,373987,373989,373992],{"className":373988},[31676],[34,373990],{"className":373991,"style":38774},[31680],[34,373993,15],{"className":373994},[31685,31686]," 是合数），看看会发生什么矛盾。",[2230,373997,373998,374281,375042,375523,375912],{},[108,373999,374000,341347,374003,374031,374032,374060,374061,374134,374135,86674,374208,347],{},[2627,374001,374002],{},"利用合数的定义",[34,374004,374006,374019],{"className":374005},[31642],[34,374007,374009],{"className":374008},[31646],[31648,374010,374011],{"xmlns":31650},[31652,374012,374013,374017],{},[31655,374014,374015],{},[31658,374016,15],{},[31665,374018,15],{"encoding":31667},[34,374020,374022],{"className":374021,"ariaHidden":758},[31672],[34,374023,374025,374028],{"className":374024},[31676],[34,374026],{"className":374027,"style":38774},[31680],[34,374029,15],{"className":374030},[31685,31686]," 是合数，那么 ",[34,374033,374035,374048],{"className":374034},[31642],[34,374036,374038],{"className":374037},[31646],[31648,374039,374040],{"xmlns":31650},[31652,374041,374042,374046],{},[31655,374043,374044],{},[31658,374045,15],{},[31665,374047,15],{"encoding":31667},[34,374049,374051],{"className":374050,"ariaHidden":758},[31672],[34,374052,374054,374057],{"className":374053},[31676],[34,374055],{"className":374056,"style":38774},[31680],[34,374058,15],{"className":374059},[31685,31686]," 可以分解成两个比它小的整数相乘：\n",[34,374062,374064,374086],{"className":374063},[31642],[34,374065,374067],{"className":374066},[31646],[31648,374068,374069],{"xmlns":31650},[31652,374070,374071,374083],{},[31655,374072,374073,374075,374077,374079,374081],{},[31658,374074,15],{},[31662,374076,202],{},[31658,374078,111],{},[31662,374080,71191],{},[31658,374082,1697],{},[31665,374084,374085],{"encoding":31667},"p = a \\cdot b",[34,374087,374089,374107,374125],{"className":374088,"ariaHidden":758},[31672],[34,374090,374092,374095,374098,374101,374104],{"className":374091},[31676],[34,374093],{"className":374094,"style":38774},[31680],[34,374096,15],{"className":374097},[31685,31686],[34,374099],{"className":374100,"style":31846},[31845],[34,374102,202],{"className":374103},[31850],[34,374105],{"className":374106,"style":31846},[31845],[34,374108,374110,374113,374116,374119,374122],{"className":374109},[31676],[34,374111],{"className":374112,"style":224869},[31680],[34,374114,111],{"className":374115},[31685,31686],[34,374117],{"className":374118,"style":32604},[31845],[34,374120,71191],{"className":374121},[32024],[34,374123],{"className":374124,"style":32604},[31845],[34,374126,374128,374131],{"className":374127},[31676],[34,374129],{"className":374130,"style":31681},[31680],[34,374132,1697],{"className":374133},[31685,31686],"\n其中 ",[34,374136,374138,374160],{"className":374137},[31642],[34,374139,374141],{"className":374140},[31646],[31648,374142,374143],{"xmlns":31650},[31652,374144,374145,374157],{},[31655,374146,374147,374149,374151,374153,374155],{},[31749,374148,640],{},[31662,374150,932],{},[31658,374152,111],{},[31662,374154,932],{},[31658,374156,15],{},[31665,374158,374159],{"encoding":31667},"1 \u003C a \u003C p",[34,374161,374163,374181,374199],{"className":374162,"ariaHidden":758},[31672],[34,374164,374166,374169,374172,374175,374178],{"className":374165},[31676],[34,374167],{"className":374168,"style":61814},[31680],[34,374170,640],{"className":374171},[31685],[34,374173],{"className":374174,"style":31846},[31845],[34,374176,932],{"className":374177},[31850],[34,374179],{"className":374180,"style":31846},[31845],[34,374182,374184,374187,374190,374193,374196],{"className":374183},[31676],[34,374185],{"className":374186,"style":34812},[31680],[34,374188,111],{"className":374189},[31685,31686],[34,374191],{"className":374192,"style":31846},[31845],[34,374194,932],{"className":374195},[31850],[34,374197],{"className":374198,"style":31846},[31845],[34,374200,374202,374205],{"className":374201},[31676],[34,374203],{"className":374204,"style":38774},[31680],[34,374206,15],{"className":374207},[31685,31686],[34,374209,374211,374233],{"className":374210},[31642],[34,374212,374214],{"className":374213},[31646],[31648,374215,374216],{"xmlns":31650},[31652,374217,374218,374230],{},[31655,374219,374220,374222,374224,374226,374228],{},[31749,374221,640],{},[31662,374223,932],{},[31658,374225,1697],{},[31662,374227,932],{},[31658,374229,15],{},[31665,374231,374232],{"encoding":31667},"1 \u003C b \u003C p",[34,374234,374236,374254,374272],{"className":374235,"ariaHidden":758},[31672],[34,374237,374239,374242,374245,374248,374251],{"className":374238},[31676],[34,374240],{"className":374241,"style":61814},[31680],[34,374243,640],{"className":374244},[31685],[34,374246],{"className":374247,"style":31846},[31845],[34,374249,932],{"className":374250},[31850],[34,374252],{"className":374253,"style":31846},[31845],[34,374255,374257,374260,374263,374266,374269],{"className":374256},[31676],[34,374258],{"className":374259,"style":46991},[31680],[34,374261,1697],{"className":374262},[31685,31686],[34,374264],{"className":374265,"style":31846},[31845],[34,374267,932],{"className":374268},[31850],[34,374270],{"className":374271,"style":31846},[31845],[34,374273,374275,374278],{"className":374274},[31676],[34,374276],{"className":374277,"style":38774},[31680],[34,374279,15],{"className":374280},[31685,31686],[108,374282,374283,374355,374356,258,374384,314945,374412,374481,374482],{},[2627,374284,374285,374286,373617],{},"翻译到 ",[34,374287,374289,374306],{"className":374288},[31642],[34,374290,374292],{"className":374291},[31646],[31648,374293,374294],{"xmlns":31650},[31652,374295,374296,374304],{},[31655,374297,374298],{},[31743,374299,374300,374302],{},[31658,374301,97300],{"mathvariant":318416},[31658,374303,15],{},[31665,374305,338074],{"encoding":31667},[34,374307,374309],{"className":374308,"ariaHidden":758},[31672],[34,374310,374312,374315],{"className":374311},[31676],[34,374313],{"className":374314,"style":338084},[31680],[34,374316,374318,374321],{"className":374317},[31685],[34,374319,97300],{"className":374320},[31685,318441],[34,374322,374324],{"className":374323},[31792],[34,374325,374327,374347],{"className":374326},[31796,31797],[34,374328,374330,374344],{"className":374329},[31801],[34,374331,374333],{"className":374332,"style":32362},[31805],[34,374334,374335,374338],{"style":31809},[34,374336],{"className":374337,"style":31814},[31813],[34,374339,374341],{"className":374340},[31818,31819,31820,31821],[34,374342,15],{"className":374343},[31685,31686,31821],[34,374345,31829],{"className":374346},[31828],[34,374348,374350],{"className":374349},[31801],[34,374351,374353],{"className":374352,"style":32163},[31805],[34,374354],{},"：\n这两个数 ",[34,374357,374359,374372],{"className":374358},[31642],[34,374360,374362],{"className":374361},[31646],[31648,374363,374364],{"xmlns":31650},[31652,374365,374366,374370],{},[31655,374367,374368],{},[31658,374369,111],{},[31665,374371,111],{"encoding":31667},[34,374373,374375],{"className":374374,"ariaHidden":758},[31672],[34,374376,374378,374381],{"className":374377},[31676],[34,374379],{"className":374380,"style":32527},[31680],[34,374382,111],{"className":374383},[31685,31686],[34,374385,374387,374400],{"className":374386},[31642],[34,374388,374390],{"className":374389},[31646],[31648,374391,374392],{"xmlns":31650},[31652,374393,374394,374398],{},[31655,374395,374396],{},[31658,374397,1697],{},[31665,374399,1697],{"encoding":31667},[34,374401,374403],{"className":374402,"ariaHidden":758},[31672],[34,374404,374406,374409],{"className":374405},[31676],[34,374407],{"className":374408,"style":31681},[31680],[34,374410,1697],{"className":374411},[31685,31686],[34,374413,374415,374432],{"className":374414},[31642],[34,374416,374418],{"className":374417},[31646],[31648,374419,374420],{"xmlns":31650},[31652,374421,374422,374430],{},[31655,374423,374424],{},[31743,374425,374426,374428],{},[31658,374427,97300],{"mathvariant":318416},[31658,374429,15],{},[31665,374431,338074],{"encoding":31667},[34,374433,374435],{"className":374434,"ariaHidden":758},[31672],[34,374436,374438,374441],{"className":374437},[31676],[34,374439],{"className":374440,"style":338084},[31680],[34,374442,374444,374447],{"className":374443},[31685],[34,374445,97300],{"className":374446},[31685,318441],[34,374448,374450],{"className":374449},[31792],[34,374451,374453,374473],{"className":374452},[31796,31797],[34,374454,374456,374470],{"className":374455},[31801],[34,374457,374459],{"className":374458,"style":32362},[31805],[34,374460,374461,374464],{"style":31809},[34,374462],{"className":374463,"style":31814},[31813],[34,374465,374467],{"className":374466},[31818,31819,31820,31821],[34,374468,15],{"className":374469},[31685,31686,31821],[34,374471,31829],{"className":374472},[31828],[34,374474,374476],{"className":374475},[31801],[34,374477,374479],{"className":374478,"style":32163},[31805],[34,374480],{}," 里：",[105,374483,374484,374763],{},[108,374485,93310,374486,13977,374558,372787,374586,374614,374615,347],{},[34,374487,374489,374510],{"className":374488},[31642],[34,374490,374492],{"className":374491},[31646],[31648,374493,374494],{"xmlns":31650},[31652,374495,374496,374508],{},[31655,374497,374498,374500,374502,374504,374506],{},[31749,374499,640],{},[31662,374501,932],{},[31658,374503,111],{},[31662,374505,932],{},[31658,374507,15],{},[31665,374509,374159],{"encoding":31667},[34,374511,374513,374531,374549],{"className":374512,"ariaHidden":758},[31672],[34,374514,374516,374519,374522,374525,374528],{"className":374515},[31676],[34,374517],{"className":374518,"style":61814},[31680],[34,374520,640],{"className":374521},[31685],[34,374523],{"className":374524,"style":31846},[31845],[34,374526,932],{"className":374527},[31850],[34,374529],{"className":374530,"style":31846},[31845],[34,374532,374534,374537,374540,374543,374546],{"className":374533},[31676],[34,374535],{"className":374536,"style":34812},[31680],[34,374538,111],{"className":374539},[31685,31686],[34,374541],{"className":374542,"style":31846},[31845],[34,374544,932],{"className":374545},[31850],[34,374547],{"className":374548,"style":31846},[31845],[34,374550,374552,374555],{"className":374551},[31676],[34,374553],{"className":374554,"style":38774},[31680],[34,374556,15],{"className":374557},[31685,31686],[34,374559,374561,374574],{"className":374560},[31642],[34,374562,374564],{"className":374563},[31646],[31648,374565,374566],{"xmlns":31650},[31652,374567,374568,374572],{},[31655,374569,374570],{},[31658,374571,111],{},[31665,374573,111],{"encoding":31667},[34,374575,374577],{"className":374576,"ariaHidden":758},[31672],[34,374578,374580,374583],{"className":374579},[31676],[34,374581],{"className":374582,"style":32527},[31680],[34,374584,111],{"className":374585},[31685,31686],[34,374587,374589,374602],{"className":374588},[31642],[34,374590,374592],{"className":374591},[31646],[31648,374593,374594],{"xmlns":31650},[31652,374595,374596,374600],{},[31655,374597,374598],{},[31658,374599,15],{},[31665,374601,15],{"encoding":31667},[34,374603,374605],{"className":374604,"ariaHidden":758},[31672],[34,374606,374608,374611],{"className":374607},[31676],[34,374609],{"className":374610,"style":38774},[31680],[34,374612,15],{"className":374613},[31685,31686]," 除的余数不是 0，即 ",[34,374616,374618,374644],{"className":374617},[31642],[34,374619,374621],{"className":374620},[31646],[31648,374622,374623],{"xmlns":31650},[31652,374624,374625,374641],{},[31655,374626,374627,374633,374635],{},[134014,374628,374629,374631],{"accent":758},[31658,374630,111],{},[31662,374632,346142],{},[31662,374634,54707],{"mathvariant":38079},[134014,374636,374637,374639],{"accent":758},[31749,374638,459],{},[31662,374640,346142],{},[31665,374642,374643],{"encoding":31667},"\\bar{a} \\neq \\bar{0}",[34,374645,374647,374726],{"className":374646,"ariaHidden":758},[31672],[34,374648,374650,374653,374684,374687,374723],{"className":374649},[31676],[34,374651],{"className":374652,"style":43217},[31680],[34,374654,374656],{"className":374655},[31685,346171],[34,374657,374659],{"className":374658},[31796],[34,374660,374662],{"className":374661},[31801],[34,374663,374665,374673],{"className":374664,"style":346181},[31805],[34,374666,374667,374670],{"style":41565},[34,374668],{"className":374669,"style":41508},[31813],[34,374671,111],{"className":374672},[31685,31686],[34,374674,374675,374678],{"style":41565},[34,374676],{"className":374677,"style":41508},[31813],[34,374679,374681],{"className":374680,"style":346199},[346198],[34,374682,346142],{"className":374683},[31685],[34,374685],{"className":374686,"style":31846},[31845],[34,374688,374690,374717,374720],{"className":374689},[31850],[34,374691,374693],{"className":374692},[31850],[34,374694,374696],{"className":374695},[31685,54774],[34,374697,374699],{"className":374698},[54778],[34,374700,374702,374705,374714],{"className":374701},[54782],[34,374703],{"className":374704,"style":43217},[31680],[34,374706,374708],{"className":374707},[54789],[34,374709,374711],{"className":374710},[31685],[34,374712,54796],{"className":374713},[31850],[34,374715],{"className":374716},[54800],[34,374718],{"className":374719},[31845,54804],[34,374721,202],{"className":374722},[31850],[34,374724],{"className":374725,"style":31846},[31845],[34,374727,374729,374732],{"className":374728},[31676],[34,374730],{"className":374731,"style":371250},[31680],[34,374733,374735],{"className":374734},[31685,346171],[34,374736,374738],{"className":374737},[31796],[34,374739,374741],{"className":374740},[31801],[34,374742,374744,374752],{"className":374743,"style":371250},[31805],[34,374745,374746,374749],{"style":41565},[34,374747],{"className":374748,"style":41508},[31813],[34,374750,459],{"className":374751},[31685],[34,374753,374754,374757],{"style":371273},[34,374755],{"className":374756,"style":41508},[31813],[34,374758,374760],{"className":374759,"style":346199},[346198],[34,374761,346142],{"className":374762},[31685],[108,374764,93310,374765,13977,374837,372787,374865,374614,374893,347],{},[34,374766,374768,374789],{"className":374767},[31642],[34,374769,374771],{"className":374770},[31646],[31648,374772,374773],{"xmlns":31650},[31652,374774,374775,374787],{},[31655,374776,374777,374779,374781,374783,374785],{},[31749,374778,640],{},[31662,374780,932],{},[31658,374782,1697],{},[31662,374784,932],{},[31658,374786,15],{},[31665,374788,374232],{"encoding":31667},[34,374790,374792,374810,374828],{"className":374791,"ariaHidden":758},[31672],[34,374793,374795,374798,374801,374804,374807],{"className":374794},[31676],[34,374796],{"className":374797,"style":61814},[31680],[34,374799,640],{"className":374800},[31685],[34,374802],{"className":374803,"style":31846},[31845],[34,374805,932],{"className":374806},[31850],[34,374808],{"className":374809,"style":31846},[31845],[34,374811,374813,374816,374819,374822,374825],{"className":374812},[31676],[34,374814],{"className":374815,"style":46991},[31680],[34,374817,1697],{"className":374818},[31685,31686],[34,374820],{"className":374821,"style":31846},[31845],[34,374823,932],{"className":374824},[31850],[34,374826],{"className":374827,"style":31846},[31845],[34,374829,374831,374834],{"className":374830},[31676],[34,374832],{"className":374833,"style":38774},[31680],[34,374835,15],{"className":374836},[31685,31686],[34,374838,374840,374853],{"className":374839},[31642],[34,374841,374843],{"className":374842},[31646],[31648,374844,374845],{"xmlns":31650},[31652,374846,374847,374851],{},[31655,374848,374849],{},[31658,374850,1697],{},[31665,374852,1697],{"encoding":31667},[34,374854,374856],{"className":374855,"ariaHidden":758},[31672],[34,374857,374859,374862],{"className":374858},[31676],[34,374860],{"className":374861,"style":31681},[31680],[34,374863,1697],{"className":374864},[31685,31686],[34,374866,374868,374881],{"className":374867},[31642],[34,374869,374871],{"className":374870},[31646],[31648,374872,374873],{"xmlns":31650},[31652,374874,374875,374879],{},[31655,374876,374877],{},[31658,374878,15],{},[31665,374880,15],{"encoding":31667},[34,374882,374884],{"className":374883,"ariaHidden":758},[31672],[34,374885,374887,374890],{"className":374886},[31676],[34,374888],{"className":374889,"style":38774},[31680],[34,374891,15],{"className":374892},[31685,31686],[34,374894,374896,374922],{"className":374895},[31642],[34,374897,374899],{"className":374898},[31646],[31648,374900,374901],{"xmlns":31650},[31652,374902,374903,374919],{},[31655,374904,374905,374911,374913],{},[134014,374906,374907,374909],{"accent":758},[31658,374908,1697],{},[31662,374910,346142],{},[31662,374912,54707],{"mathvariant":38079},[134014,374914,374915,374917],{"accent":758},[31749,374916,459],{},[31662,374918,346142],{},[31665,374920,374921],{"encoding":31667},"\\bar{b} \\neq \\bar{0}",[34,374923,374925,375005],{"className":374924,"ariaHidden":758},[31672],[34,374926,374928,374932,374963,374966,375002],{"className":374927},[31676],[34,374929],{"className":374930,"style":374931},[31680],"height:1.0257em;vertical-align:-0.1944em;",[34,374933,374935],{"className":374934},[31685,346171],[34,374936,374938],{"className":374937},[31796],[34,374939,374941],{"className":374940},[31801],[34,374942,374944,374952],{"className":374943,"style":372056},[31805],[34,374945,374946,374949],{"style":41565},[34,374947],{"className":374948,"style":41508},[31813],[34,374950,1697],{"className":374951},[31685,31686],[34,374953,374954,374957],{"style":372079},[34,374955],{"className":374956,"style":41508},[31813],[34,374958,374960],{"className":374959,"style":346199},[346198],[34,374961,346142],{"className":374962},[31685],[34,374964],{"className":374965,"style":31846},[31845],[34,374967,374969,374996,374999],{"className":374968},[31850],[34,374970,374972],{"className":374971},[31850],[34,374973,374975],{"className":374974},[31685,54774],[34,374976,374978],{"className":374977},[54778],[34,374979,374981,374984,374993],{"className":374980},[54782],[34,374982],{"className":374983,"style":43217},[31680],[34,374985,374987],{"className":374986},[54789],[34,374988,374990],{"className":374989},[31685],[34,374991,54796],{"className":374992},[31850],[34,374994],{"className":374995},[54800],[34,374997],{"className":374998},[31845,54804],[34,375000,202],{"className":375001},[31850],[34,375003],{"className":375004,"style":31846},[31845],[34,375006,375008,375011],{"className":375007},[31676],[34,375009],{"className":375010,"style":371250},[31680],[34,375012,375014],{"className":375013},[31685,346171],[34,375015,375017],{"className":375016},[31796],[34,375018,375020],{"className":375019},[31801],[34,375021,375023,375031],{"className":375022,"style":371250},[31805],[34,375024,375025,375028],{"style":41565},[34,375026],{"className":375027,"style":41508},[31813],[34,375029,459],{"className":375030},[31685],[34,375032,375033,375036],{"style":371273},[34,375034],{"className":375035,"style":41508},[31813],[34,375037,375039],{"className":375038,"style":346199},[346198],[34,375040,346142],{"className":375041},[31685],[108,375043,375044,375047,375048,375051,375052,315440,375227,97596,375280,346374,375308,375336,375337],{},[2627,375045,375046],{},"计算乘积","：\n现在我们把这两个",[2627,375049,375050],{},"非零","元素乘起来：\n",[34,375053,375055,375093],{"className":375054},[31642],[34,375056,375058],{"className":375057},[31646],[31648,375059,375060],{"xmlns":31650},[31652,375061,375062,375090],{},[31655,375063,375064,375070,375072,375078,375080],{},[134014,375065,375066,375068],{"accent":758},[31658,375067,111],{},[31662,375069,346142],{},[31662,375071,71191],{},[134014,375073,375074,375076],{"accent":758},[31658,375075,1697],{},[31662,375077,346142],{},[31662,375079,202],{},[134014,375081,375082,375088],{"accent":758},[31655,375083,375084,375086],{},[31658,375085,111],{},[31658,375087,1697],{},[31662,375089,105837],{"stretchy":758},[31665,375091,375092],{"encoding":31667},"\\bar{a} \\cdot \\bar{b} = \\overline{ab}",[34,375094,375096,375142,375188],{"className":375095,"ariaHidden":758},[31672],[34,375097,375099,375102,375133,375136,375139],{"className":375098},[31676],[34,375100],{"className":375101,"style":346181},[31680],[34,375103,375105],{"className":375104},[31685,346171],[34,375106,375108],{"className":375107},[31796],[34,375109,375111],{"className":375110},[31801],[34,375112,375114,375122],{"className":375113,"style":346181},[31805],[34,375115,375116,375119],{"style":41565},[34,375117],{"className":375118,"style":41508},[31813],[34,375120,111],{"className":375121},[31685,31686],[34,375123,375124,375127],{"style":41565},[34,375125],{"className":375126,"style":41508},[31813],[34,375128,375130],{"className":375129,"style":346199},[346198],[34,375131,346142],{"className":375132},[31685],[34,375134],{"className":375135,"style":32604},[31845],[34,375137,71191],{"className":375138},[32024],[34,375140],{"className":375141,"style":32604},[31845],[34,375143,375145,375148,375179,375182,375185],{"className":375144},[31676],[34,375146],{"className":375147,"style":372056},[31680],[34,375149,375151],{"className":375150},[31685,346171],[34,375152,375154],{"className":375153},[31796],[34,375155,375157],{"className":375156},[31801],[34,375158,375160,375168],{"className":375159,"style":372056},[31805],[34,375161,375162,375165],{"style":41565},[34,375163],{"className":375164,"style":41508},[31813],[34,375166,1697],{"className":375167},[31685,31686],[34,375169,375170,375173],{"style":372079},[34,375171],{"className":375172,"style":41508},[31813],[34,375174,375176],{"className":375175,"style":346199},[346198],[34,375177,346142],{"className":375178},[31685],[34,375180],{"className":375181,"style":31846},[31845],[34,375183,202],{"className":375184},[31850],[34,375186],{"className":375187,"style":31846},[31845],[34,375189,375191,375195],{"className":375190},[31676],[34,375192],{"className":375193,"style":375194},[31680],"height:0.8944em;",[34,375196,375198],{"className":375197},[31685,134036],[34,375199,375201],{"className":375200},[31796],[34,375202,375204],{"className":375203},[31801],[34,375205,375207,375218],{"className":375206,"style":375194},[31805],[34,375208,375209,375212],{"style":41565},[34,375210],{"className":375211,"style":41508},[31813],[34,375213,375215],{"className":375214},[31685],[34,375216,324191],{"className":375217},[31685,31686],[34,375219,375221,375224],{"style":375220},"top:-3.8144em;",[34,375222],{"className":375223,"style":41508},[31813],[34,375225],{"className":375226,"style":41528},[134066],[34,375228,375230,375250],{"className":375229},[31642],[34,375231,375233],{"className":375232},[31646],[31648,375234,375235],{"xmlns":31650},[31652,375236,375237,375247],{},[31655,375238,375239,375241,375243,375245],{},[31658,375240,111],{},[31658,375242,1697],{},[31662,375244,202],{},[31658,375246,15],{},[31665,375248,375249],{"encoding":31667},"ab = p",[34,375251,375253,375271],{"className":375252,"ariaHidden":758},[31672],[34,375254,375256,375259,375262,375265,375268],{"className":375255},[31676],[34,375257],{"className":375258,"style":31681},[31680],[34,375260,324191],{"className":375261},[31685,31686],[34,375263],{"className":375264,"style":31846},[31845],[34,375266,202],{"className":375267},[31850],[34,375269],{"className":375270,"style":31846},[31845],[34,375272,375274,375277],{"className":375273},[31676],[34,375275],{"className":375276,"style":38774},[31680],[34,375278,15],{"className":375279},[31685,31686],[34,375281,375283,375296],{"className":375282},[31642],[34,375284,375286],{"className":375285},[31646],[31648,375287,375288],{"xmlns":31650},[31652,375289,375290,375294],{},[31655,375291,375292],{},[31658,375293,15],{},[31665,375295,15],{"encoding":31667},[34,375297,375299],{"className":375298,"ariaHidden":758},[31672],[34,375300,375302,375305],{"className":375301},[31676],[34,375303],{"className":375304,"style":38774},[31680],[34,375306,15],{"className":375307},[31685,31686],[34,375309,375311,375324],{"className":375310},[31642],[34,375312,375314],{"className":375313},[31646],[31648,375315,375316],{"xmlns":31650},[31652,375317,375318,375322],{},[31655,375319,375320],{},[31658,375321,15],{},[31665,375323,15],{"encoding":31667},[34,375325,375327],{"className":375326,"ariaHidden":758},[31672],[34,375328,375330,375333],{"className":375329},[31676],[34,375331],{"className":375332,"style":38774},[31680],[34,375334,15],{"className":375335},[31685,31686]," 意义下就是 0。\n",[34,375338,375340,375378],{"className":375339},[31642],[34,375341,375343],{"className":375342},[31646],[31648,375344,375345],{"xmlns":31650},[31652,375346,375347,375375],{},[31655,375348,375349,375359,375361,375367,375369],{},[134014,375350,375351,375357],{"accent":758},[31655,375352,375353,375355],{},[31658,375354,111],{},[31658,375356,1697],{},[31662,375358,105837],{"stretchy":758},[31662,375360,202],{},[134014,375362,375363,375365],{"accent":758},[31658,375364,15],{},[31662,375366,346142],{},[31662,375368,202],{},[134014,375370,375371,375373],{"accent":758},[31749,375372,459],{},[31662,375374,346142],{},[31665,375376,375377],{"encoding":31667},"\\overline{ab} = \\bar{p} = \\bar{0}",[34,375379,375381,375427,375486],{"className":375380,"ariaHidden":758},[31672],[34,375382,375384,375387,375418,375421,375424],{"className":375383},[31676],[34,375385],{"className":375386,"style":375194},[31680],[34,375388,375390],{"className":375389},[31685,134036],[34,375391,375393],{"className":375392},[31796],[34,375394,375396],{"className":375395},[31801],[34,375397,375399,375410],{"className":375398,"style":375194},[31805],[34,375400,375401,375404],{"style":41565},[34,375402],{"className":375403,"style":41508},[31813],[34,375405,375407],{"className":375406},[31685],[34,375408,324191],{"className":375409},[31685,31686],[34,375411,375412,375415],{"style":375220},[34,375413],{"className":375414,"style":41508},[31813],[34,375416],{"className":375417,"style":41528},[134066],[34,375419],{"className":375420,"style":31846},[31845],[34,375422,202],{"className":375423},[31850],[34,375425],{"className":375426,"style":31846},[31845],[34,375428,375430,375434,375477,375480,375483],{"className":375429},[31676],[34,375431],{"className":375432,"style":375433},[31680],"height:0.7622em;vertical-align:-0.1944em;",[34,375435,375437],{"className":375436},[31685,346171],[34,375438,375440,375469],{"className":375439},[31796,31797],[34,375441,375443,375466],{"className":375442},[31801],[34,375444,375446,375454],{"className":375445,"style":346181},[31805],[34,375447,375448,375451],{"style":41565},[34,375449],{"className":375450,"style":41508},[31813],[34,375452,15],{"className":375453},[31685,31686],[34,375455,375456,375459],{"style":41565},[34,375457],{"className":375458,"style":41508},[31813],[34,375460,375463],{"className":375461,"style":375462},[346198],"left:-0.1667em;",[34,375464,346142],{"className":375465},[31685],[34,375467,31829],{"className":375468},[31828],[34,375470,375472],{"className":375471},[31801],[34,375473,375475],{"className":375474,"style":44616},[31805],[34,375476],{},[34,375478],{"className":375479,"style":31846},[31845],[34,375481,202],{"className":375482},[31850],[34,375484],{"className":375485,"style":31846},[31845],[34,375487,375489,375492],{"className":375488},[31676],[34,375490],{"className":375491,"style":371250},[31680],[34,375493,375495],{"className":375494},[31685,346171],[34,375496,375498],{"className":375497},[31796],[34,375499,375501],{"className":375500},[31801],[34,375502,375504,375512],{"className":375503,"style":371250},[31805],[34,375505,375506,375509],{"style":41565},[34,375507],{"className":375508,"style":41508},[31813],[34,375510,459],{"className":375511},[31685],[34,375513,375514,375517],{"style":371273},[34,375515],{"className":375516,"style":41508},[31813],[34,375518,375520],{"className":375519,"style":346199},[346198],[34,375521,346142],{"className":375522},[31685],[108,375524,375525,375528,375529,375531,375532,258,375592,375652,375653,346345,375714,258,375774,113124,375834,375837,375838,375907,375908,375911],{},[2627,375526,375527],{},"发现矛盾","：\n我们找到了两个",[2627,375530,375050],{},"的元素 ",[34,375533,375535,375552],{"className":375534},[31642],[34,375536,375538],{"className":375537},[31646],[31648,375539,375540],{"xmlns":31650},[31652,375541,375542,375550],{},[31655,375543,375544],{},[134014,375545,375546,375548],{"accent":758},[31658,375547,111],{},[31662,375549,346142],{},[31665,375551,371985],{"encoding":31667},[34,375553,375555],{"className":375554,"ariaHidden":758},[31672],[34,375556,375558,375561],{"className":375557},[31676],[34,375559],{"className":375560,"style":346181},[31680],[34,375562,375564],{"className":375563},[31685,346171],[34,375565,375567],{"className":375566},[31796],[34,375568,375570],{"className":375569},[31801],[34,375571,375573,375581],{"className":375572,"style":346181},[31805],[34,375574,375575,375578],{"style":41565},[34,375576],{"className":375577,"style":41508},[31813],[34,375579,111],{"className":375580},[31685,31686],[34,375582,375583,375586],{"style":41565},[34,375584],{"className":375585,"style":41508},[31813],[34,375587,375589],{"className":375588,"style":346199},[346198],[34,375590,346142],{"className":375591},[31685],[34,375593,375595,375612],{"className":375594},[31642],[34,375596,375598],{"className":375597},[31646],[31648,375599,375600],{"xmlns":31650},[31652,375601,375602,375610],{},[31655,375603,375604],{},[134014,375605,375606,375608],{"accent":758},[31658,375607,1697],{},[31662,375609,346142],{},[31665,375611,372046],{"encoding":31667},[34,375613,375615],{"className":375614,"ariaHidden":758},[31672],[34,375616,375618,375621],{"className":375617},[31676],[34,375619],{"className":375620,"style":372056},[31680],[34,375622,375624],{"className":375623},[31685,346171],[34,375625,375627],{"className":375626},[31796],[34,375628,375630],{"className":375629},[31801],[34,375631,375633,375641],{"className":375632,"style":372056},[31805],[34,375634,375635,375638],{"style":41565},[34,375636],{"className":375637,"style":41508},[31813],[34,375639,1697],{"className":375640},[31685,31686],[34,375642,375643,375646],{"style":372079},[34,375644],{"className":375645,"style":41508},[31813],[34,375647,375649],{"className":375648,"style":346199},[346198],[34,375650,346142],{"className":375651},[31685],"，它们的乘积却是 ",[34,375654,375656,375674],{"className":375655},[31642],[34,375657,375659],{"className":375658},[31646],[31648,375660,375661],{"xmlns":31650},[31652,375662,375663,375671],{},[31655,375664,375665],{},[134014,375666,375667,375669],{"accent":758},[31749,375668,459],{},[31662,375670,346142],{},[31665,375672,375673],{"encoding":31667},"\\bar{0}",[34,375675,375677],{"className":375676,"ariaHidden":758},[31672],[34,375678,375680,375683],{"className":375679},[31676],[34,375681],{"className":375682,"style":371250},[31680],[34,375684,375686],{"className":375685},[31685,346171],[34,375687,375689],{"className":375688},[31796],[34,375690,375692],{"className":375691},[31801],[34,375693,375695,375703],{"className":375694,"style":371250},[31805],[34,375696,375697,375700],{"style":41565},[34,375698],{"className":375699,"style":41508},[31813],[34,375701,459],{"className":375702},[31685],[34,375704,375705,375708],{"style":371273},[34,375706],{"className":375707,"style":41508},[31813],[34,375709,375711],{"className":375710,"style":346199},[346198],[34,375712,346142],{"className":375713},[31685],[34,375715,375717,375734],{"className":375716},[31642],[34,375718,375720],{"className":375719},[31646],[31648,375721,375722],{"xmlns":31650},[31652,375723,375724,375732],{},[31655,375725,375726],{},[134014,375727,375728,375730],{"accent":758},[31658,375729,111],{},[31662,375731,346142],{},[31665,375733,371985],{"encoding":31667},[34,375735,375737],{"className":375736,"ariaHidden":758},[31672],[34,375738,375740,375743],{"className":375739},[31676],[34,375741],{"className":375742,"style":346181},[31680],[34,375744,375746],{"className":375745},[31685,346171],[34,375747,375749],{"className":375748},[31796],[34,375750,375752],{"className":375751},[31801],[34,375753,375755,375763],{"className":375754,"style":346181},[31805],[34,375756,375757,375760],{"style":41565},[34,375758],{"className":375759,"style":41508},[31813],[34,375761,111],{"className":375762},[31685,31686],[34,375764,375765,375768],{"style":41565},[34,375766],{"className":375767,"style":41508},[31813],[34,375769,375771],{"className":375770,"style":346199},[346198],[34,375772,346142],{"className":375773},[31685],[34,375775,375777,375794],{"className":375776},[31642],[34,375778,375780],{"className":375779},[31646],[31648,375781,375782],{"xmlns":31650},[31652,375783,375784,375792],{},[31655,375785,375786],{},[134014,375787,375788,375790],{"accent":758},[31658,375789,1697],{},[31662,375791,346142],{},[31665,375793,372046],{"encoding":31667},[34,375795,375797],{"className":375796,"ariaHidden":758},[31672],[34,375798,375800,375803],{"className":375799},[31676],[34,375801],{"className":375802,"style":372056},[31680],[34,375804,375806],{"className":375805},[31685,346171],[34,375807,375809],{"className":375808},[31796],[34,375810,375812],{"className":375811},[31801],[34,375813,375815,375823],{"className":375814,"style":372056},[31805],[34,375816,375817,375820],{"style":41565},[34,375818],{"className":375819,"style":41508},[31813],[34,375821,1697],{"className":375822},[31685,31686],[34,375824,375825,375828],{"style":372079},[34,375826],{"className":375827,"style":41508},[31813],[34,375829,375831],{"className":375830,"style":346199},[346198],[34,375832,346142],{"className":375833},[31685],[2627,375835,375836],{},"零因子","。\n这与前提“",[34,375839,375841,375858],{"className":375840},[31642],[34,375842,375844],{"className":375843},[31646],[31648,375845,375846],{"xmlns":31650},[31652,375847,375848,375856],{},[31655,375849,375850],{},[31743,375851,375852,375854],{},[31658,375853,97300],{"mathvariant":318416},[31658,375855,15],{},[31665,375857,338074],{"encoding":31667},[34,375859,375861],{"className":375860,"ariaHidden":758},[31672],[34,375862,375864,375867],{"className":375863},[31676],[34,375865],{"className":375866,"style":338084},[31680],[34,375868,375870,375873],{"className":375869},[31685],[34,375871,97300],{"className":375872},[31685,318441],[34,375874,375876],{"className":375875},[31792],[34,375877,375879,375899],{"className":375878},[31796,31797],[34,375880,375882,375896],{"className":375881},[31801],[34,375883,375885],{"className":375884,"style":32362},[31805],[34,375886,375887,375890],{"style":31809},[34,375888],{"className":375889,"style":31814},[31813],[34,375891,375893],{"className":375892},[31818,31819,31820,31821],[34,375894,15],{"className":375895},[31685,31686,31821],[34,375897,31829],{"className":375898},[31828],[34,375900,375902],{"className":375901},[31801],[34,375903,375905],{"className":375904,"style":32163},[31805],[34,375906],{}," 是整环（无零因子）”",[2627,375909,375910],{},"矛盾","！",[108,375913,375914,375916,375917,375945,375946,375974],{},[2627,375915,114043],{},"：\n假设不成立。所以 ",[34,375918,375920,375933],{"className":375919},[31642],[34,375921,375923],{"className":375922},[31646],[31648,375924,375925],{"xmlns":31650},[31652,375926,375927,375931],{},[31655,375928,375929],{},[31658,375930,15],{},[31665,375932,15],{"encoding":31667},[34,375934,375936],{"className":375935,"ariaHidden":758},[31672],[34,375937,375939,375942],{"className":375938},[31676],[34,375940],{"className":375941,"style":38774},[31680],[34,375943,15],{"className":375944},[31685,31686]," 不能是合数，",[34,375947,375949,375962],{"className":375948},[31642],[34,375950,375952],{"className":375951},[31646],[31648,375953,375954],{"xmlns":31650},[31652,375955,375956,375960],{},[31655,375957,375958],{},[31658,375959,15],{},[31665,375961,15],{"encoding":31667},[34,375963,375965],{"className":375964,"ariaHidden":758},[31672],[34,375966,375968,375971],{"className":375967},[31676],[34,375969],{"className":375970,"style":38774},[31680],[34,375972,15],{"className":375973},[31685,31686]," 必须是素数。",[32188,375976],{},[253,375978,375979],{"id":375979},"直观的例子对比",[15,375981,375982,375983,376034,376035,376085],{},"对比一下 ",[34,375984,375986,376004],{"className":375985},[31642],[34,375987,375989],{"className":375988},[31646],[31648,375990,375991],{"xmlns":31650},[31652,375992,375993,376001],{},[31655,375994,375995,375997,375999],{},[31658,375996,15],{},[31662,375998,202],{},[31749,376000,18136],{},[31665,376002,376003],{"encoding":31667},"p=5",[34,376005,376007,376025],{"className":376006,"ariaHidden":758},[31672],[34,376008,376010,376013,376016,376019,376022],{"className":376009},[31676],[34,376011],{"className":376012,"style":38774},[31680],[34,376014,15],{"className":376015},[31685,31686],[34,376017],{"className":376018,"style":31846},[31845],[34,376020,202],{"className":376021},[31850],[34,376023],{"className":376024,"style":31846},[31845],[34,376026,376028,376031],{"className":376027},[31676],[34,376029],{"className":376030,"style":32617},[31680],[34,376032,18136],{"className":376033},[31685],"（素数）和 ",[34,376036,376038,376055],{"className":376037},[31642],[34,376039,376041],{"className":376040},[31646],[31648,376042,376043],{"xmlns":31650},[31652,376044,376045,376053],{},[31655,376046,376047,376049,376051],{},[31658,376048,31660],{},[31662,376050,202],{},[31749,376052,36656],{},[31665,376054,72457],{"encoding":31667},[34,376056,376058,376076],{"className":376057,"ariaHidden":758},[31672],[34,376059,376061,376064,376067,376070,376073],{"className":376060},[31676],[34,376062],{"className":376063,"style":32527},[31680],[34,376065,31660],{"className":376066},[31685,31686],[34,376068],{"className":376069,"style":31846},[31845],[34,376071,202],{"className":376072},[31850],[34,376074],{"className":376075,"style":31846},[31845],[34,376077,376079,376082],{"className":376078},[31676],[34,376080],{"className":376081,"style":32617},[31680],[34,376083,36656],{"className":376084},[31685],"（合数）的情况。",[15,376087,376088,376162,376163,376254,376255,376350,376445,376540,376541,347],{},[2627,376089,376090,376091,376161],{},"案例 A：",[34,376092,376094,376112],{"className":376093},[31642],[34,376095,376097],{"className":376096},[31646],[31648,376098,376099],{"xmlns":31650},[31652,376100,376101,376109],{},[31655,376102,376103],{},[31743,376104,376105,376107],{},[31658,376106,97300],{"mathvariant":318416},[31749,376108,18136],{},[31665,376110,376111],{"encoding":31667},"\\mathbb{Z}_5",[34,376113,376115],{"className":376114,"ariaHidden":758},[31672],[34,376116,376118,376121],{"className":376117},[31676],[34,376119],{"className":376120,"style":330413},[31680],[34,376122,376124,376127],{"className":376123},[31685],[34,376125,97300],{"className":376126},[31685,318441],[34,376128,376130],{"className":376129},[31792],[34,376131,376133,376153],{"className":376132},[31796,31797],[34,376134,376136,376150],{"className":376135},[31801],[34,376137,376139],{"className":376138,"style":31806},[31805],[34,376140,376141,376144],{"style":31809},[34,376142],{"className":376143,"style":31814},[31813],[34,376145,376147],{"className":376146},[31818,31819,31820,31821],[34,376148,18136],{"className":376149},[31685,31821],[34,376151,31829],{"className":376152},[31828],[34,376154,376156],{"className":376155},[31801],[34,376157,376159],{"className":376158,"style":31836},[31805],[34,376160],{}," (素数)","\n元素：",[34,376164,376166,376200],{"className":376165},[31642],[34,376167,376169],{"className":376168},[31646],[31648,376170,376171],{"xmlns":31650},[31652,376172,376173,376197],{},[31655,376174,376175,376177,376179,376181,376183,376185,376187,376189,376191,376193,376195],{},[31662,376176,32685],{"stretchy":702},[31749,376178,459],{},[31662,376180,31960],{"separator":758},[31749,376182,640],{},[31662,376184,31960],{"separator":758},[31749,376186,768],{},[31662,376188,31960],{"separator":758},[31749,376190,786],{},[31662,376192,31960],{"separator":758},[31749,376194,15932],{},[31662,376196,32715],{"stretchy":702},[31665,376198,376199],{"encoding":31667},"\\{0, 1, 2, 3, 4\\}",[34,376201,376203],{"className":376202,"ariaHidden":758},[31672],[34,376204,376206,376209,376212,376215,376218,376221,376224,376227,376230,376233,376236,376239,376242,376245,376248,376251],{"className":376205},[31676],[34,376207],{"className":376208,"style":31774},[31680],[34,376210,32685],{"className":376211},[31782],[34,376213,459],{"className":376214},[31685],[34,376216,31960],{"className":376217},[32119],[34,376219],{"className":376220,"style":32123},[31845],[34,376222,640],{"className":376223},[31685],[34,376225,31960],{"className":376226},[32119],[34,376228],{"className":376229,"style":32123},[31845],[34,376231,768],{"className":376232},[31685],[34,376234,31960],{"className":376235},[32119],[34,376237],{"className":376238,"style":32123},[31845],[34,376240,786],{"className":376241},[31685],[34,376243,31960],{"className":376244},[32119],[34,376246],{"className":376247,"style":32123},[31845],[34,376249,15932],{"className":376250},[31685],[34,376252,32715],{"className":376253},[31690],"。\n试着找两个非零数乘起来等于 5 的倍数：\n",[34,376256,376258,376284],{"className":376257},[31642],[34,376259,376261],{"className":376260},[31646],[31648,376262,376263],{"xmlns":31650},[31652,376264,376265,376281],{},[31655,376266,376267,376269,376271,376273,376275,376277,376279],{},[31749,376268,768],{},[31662,376270,42071],{},[31749,376272,786],{},[31662,376274,202],{},[31749,376276,36656],{},[31662,376278,341696],{},[31749,376280,640],{},[31665,376282,376283],{"encoding":31667},"2 \\times 3 = 6 \\equiv 1",[34,376285,376287,376305,376323,376341],{"className":376286,"ariaHidden":758},[31672],[34,376288,376290,376293,376296,376299,376302],{"className":376289},[31676],[34,376291],{"className":376292,"style":35942},[31680],[34,376294,768],{"className":376295},[31685],[34,376297],{"className":376298,"style":32604},[31845],[34,376300,42071],{"className":376301},[32024],[34,376303],{"className":376304,"style":32604},[31845],[34,376306,376308,376311,376314,376317,376320],{"className":376307},[31676],[34,376309],{"className":376310,"style":32617},[31680],[34,376312,786],{"className":376313},[31685],[34,376315],{"className":376316,"style":31846},[31845],[34,376318,202],{"className":376319},[31850],[34,376321],{"className":376322,"style":31846},[31845],[34,376324,376326,376329,376332,376335,376338],{"className":376325},[31676],[34,376327],{"className":376328,"style":32617},[31680],[34,376330,36656],{"className":376331},[31685],[34,376333],{"className":376334,"style":31846},[31845],[34,376336,341696],{"className":376337},[31850],[34,376339],{"className":376340,"style":31846},[31845],[34,376342,376344,376347],{"className":376343},[31676],[34,376345],{"className":376346,"style":32617},[31680],[34,376348,640],{"className":376349},[31685],[34,376351,376353,376379],{"className":376352},[31642],[34,376354,376356],{"className":376355},[31646],[31648,376357,376358],{"xmlns":31650},[31652,376359,376360,376376],{},[31655,376361,376362,376364,376366,376368,376370,376372,376374],{},[31749,376363,768],{},[31662,376365,42071],{},[31749,376367,15932],{},[31662,376369,202],{},[31749,376371,60525],{},[31662,376373,341696],{},[31749,376375,786],{},[31665,376377,376378],{"encoding":31667},"2 \\times 4 = 8 \\equiv 3",[34,376380,376382,376400,376418,376436],{"className":376381,"ariaHidden":758},[31672],[34,376383,376385,376388,376391,376394,376397],{"className":376384},[31676],[34,376386],{"className":376387,"style":35942},[31680],[34,376389,768],{"className":376390},[31685],[34,376392],{"className":376393,"style":32604},[31845],[34,376395,42071],{"className":376396},[32024],[34,376398],{"className":376399,"style":32604},[31845],[34,376401,376403,376406,376409,376412,376415],{"className":376402},[31676],[34,376404],{"className":376405,"style":32617},[31680],[34,376407,15932],{"className":376408},[31685],[34,376410],{"className":376411,"style":31846},[31845],[34,376413,202],{"className":376414},[31850],[34,376416],{"className":376417,"style":31846},[31845],[34,376419,376421,376424,376427,376430,376433],{"className":376420},[31676],[34,376422],{"className":376423,"style":32617},[31680],[34,376425,60525],{"className":376426},[31685],[34,376428],{"className":376429,"style":31846},[31845],[34,376431,341696],{"className":376432},[31850],[34,376434],{"className":376435,"style":31846},[31845],[34,376437,376439,376442],{"className":376438},[31676],[34,376440],{"className":376441,"style":32617},[31680],[34,376443,786],{"className":376444},[31685],[34,376446,376448,376474],{"className":376447},[31642],[34,376449,376451],{"className":376450},[31646],[31648,376452,376453],{"xmlns":31650},[31652,376454,376455,376471],{},[31655,376456,376457,376459,376461,376463,376465,376467,376469],{},[31749,376458,786],{},[31662,376460,42071],{},[31749,376462,15932],{},[31662,376464,202],{},[31749,376466,71869],{},[31662,376468,341696],{},[31749,376470,768],{},[31665,376472,376473],{"encoding":31667},"3 \\times 4 = 12 \\equiv 2",[34,376475,376477,376495,376513,376531],{"className":376476,"ariaHidden":758},[31672],[34,376478,376480,376483,376486,376489,376492],{"className":376479},[31676],[34,376481],{"className":376482,"style":35942},[31680],[34,376484,786],{"className":376485},[31685],[34,376487],{"className":376488,"style":32604},[31845],[34,376490,42071],{"className":376491},[32024],[34,376493],{"className":376494,"style":32604},[31845],[34,376496,376498,376501,376504,376507,376510],{"className":376497},[31676],[34,376499],{"className":376500,"style":32617},[31680],[34,376502,15932],{"className":376503},[31685],[34,376505],{"className":376506,"style":31846},[31845],[34,376508,202],{"className":376509},[31850],[34,376511],{"className":376512,"style":31846},[31845],[34,376514,376516,376519,376522,376525,376528],{"className":376515},[31676],[34,376517],{"className":376518,"style":32617},[31680],[34,376520,71869],{"className":376521},[31685],[34,376523],{"className":376524,"style":31846},[31845],[34,376526,341696],{"className":376527},[31850],[34,376529],{"className":376530,"style":31846},[31845],[34,376532,376534,376537],{"className":376533},[31676],[34,376535],{"className":376536,"style":32617},[31680],[34,376538,768],{"className":376539},[31685],"\n...\n你会发现，怎么乘都不可能得到 0。所以它是",[2627,376542,330740],{},[15,376544,376545,376162,376619,86621,376723,376751,376752,376803,376804,376873,376874,377023,377171,377172,377395,377396,258,377456,377517,377518,377520,377521,347],{},[2627,376546,376547,376548,376618],{},"案例 B：",[34,376549,376551,376569],{"className":376550},[31642],[34,376552,376554],{"className":376553},[31646],[31648,376555,376556],{"xmlns":31650},[31652,376557,376558,376566],{},[31655,376559,376560],{},[31743,376561,376562,376564],{},[31658,376563,97300],{"mathvariant":318416},[31749,376565,36656],{},[31665,376567,376568],{"encoding":31667},"\\mathbb{Z}_6",[34,376570,376572],{"className":376571,"ariaHidden":758},[31672],[34,376573,376575,376578],{"className":376574},[31676],[34,376576],{"className":376577,"style":330413},[31680],[34,376579,376581,376584],{"className":376580},[31685],[34,376582,97300],{"className":376583},[31685,318441],[34,376585,376587],{"className":376586},[31792],[34,376588,376590,376610],{"className":376589},[31796,31797],[34,376591,376593,376607],{"className":376592},[31801],[34,376594,376596],{"className":376595,"style":31806},[31805],[34,376597,376598,376601],{"style":31809},[34,376599],{"className":376600,"style":31814},[31813],[34,376602,376604],{"className":376603},[31818,31819,31820,31821],[34,376605,36656],{"className":376606},[31685,31821],[34,376608,31829],{"className":376609},[31828],[34,376611,376613],{"className":376612},[31801],[34,376614,376616],{"className":376615,"style":31836},[31805],[34,376617],{}," (合数)",[34,376620,376622,376660],{"className":376621},[31642],[34,376623,376625],{"className":376624},[31646],[31648,376626,376627],{"xmlns":31650},[31652,376628,376629,376657],{},[31655,376630,376631,376633,376635,376637,376639,376641,376643,376645,376647,376649,376651,376653,376655],{},[31662,376632,32685],{"stretchy":702},[31749,376634,459],{},[31662,376636,31960],{"separator":758},[31749,376638,640],{},[31662,376640,31960],{"separator":758},[31749,376642,768],{},[31662,376644,31960],{"separator":758},[31749,376646,786],{},[31662,376648,31960],{"separator":758},[31749,376650,15932],{},[31662,376652,31960],{"separator":758},[31749,376654,18136],{},[31662,376656,32715],{"stretchy":702},[31665,376658,376659],{"encoding":31667},"\\{0, 1, 2, 3, 4, 5\\}",[34,376661,376663],{"className":376662,"ariaHidden":758},[31672],[34,376664,376666,376669,376672,376675,376678,376681,376684,376687,376690,376693,376696,376699,376702,376705,376708,376711,376714,376717,376720],{"className":376665},[31676],[34,376667],{"className":376668,"style":31774},[31680],[34,376670,32685],{"className":376671},[31782],[34,376673,459],{"className":376674},[31685],[34,376676,31960],{"className":376677},[32119],[34,376679],{"className":376680,"style":32123},[31845],[34,376682,640],{"className":376683},[31685],[34,376685,31960],{"className":376686},[32119],[34,376688],{"className":376689,"style":32123},[31845],[34,376691,768],{"className":376692},[31685],[34,376694,31960],{"className":376695},[32119],[34,376697],{"className":376698,"style":32123},[31845],[34,376700,786],{"className":376701},[31685],[34,376703,31960],{"className":376704},[32119],[34,376706],{"className":376707,"style":32123},[31845],[34,376709,15932],{"className":376710},[31685],[34,376712,31960],{"className":376713},[32119],[34,376715],{"className":376716,"style":32123},[31845],[34,376718,18136],{"className":376719},[31685],[34,376721,32715],{"className":376722},[31690],[34,376724,376726,376739],{"className":376725},[31642],[34,376727,376729],{"className":376728},[31646],[31648,376730,376731],{"xmlns":31650},[31652,376732,376733,376737],{},[31655,376734,376735],{},[31749,376736,36656],{},[31665,376738,36656],{"encoding":31667},[34,376740,376742],{"className":376741,"ariaHidden":758},[31672],[34,376743,376745,376748],{"className":376744},[31676],[34,376746],{"className":376747,"style":32617},[31680],[34,376749,36656],{"className":376750},[31685]," 是合数，可以写成 ",[34,376753,376755,376773],{"className":376754},[31642],[34,376756,376758],{"className":376757},[31646],[31648,376759,376760],{"xmlns":31650},[31652,376761,376762,376770],{},[31655,376763,376764,376766,376768],{},[31749,376765,768],{},[31662,376767,42071],{},[31749,376769,786],{},[31665,376771,376772],{"encoding":31667},"2 \\times 3",[34,376774,376776,376794],{"className":376775,"ariaHidden":758},[31672],[34,376777,376779,376782,376785,376788,376791],{"className":376778},[31676],[34,376780],{"className":376781,"style":35942},[31680],[34,376783,768],{"className":376784},[31685],[34,376786],{"className":376787,"style":32604},[31845],[34,376789,42071],{"className":376790},[32024],[34,376792],{"className":376793,"style":32604},[31845],[34,376795,376797,376800],{"className":376796},[31676],[34,376798],{"className":376799,"style":32617},[31680],[34,376801,786],{"className":376802},[31685],"。\n在 ",[34,376805,376807,376824],{"className":376806},[31642],[34,376808,376810],{"className":376809},[31646],[31648,376811,376812],{"xmlns":31650},[31652,376813,376814,376822],{},[31655,376815,376816],{},[31743,376817,376818,376820],{},[31658,376819,97300],{"mathvariant":318416},[31749,376821,36656],{},[31665,376823,376568],{"encoding":31667},[34,376825,376827],{"className":376826,"ariaHidden":758},[31672],[34,376828,376830,376833],{"className":376829},[31676],[34,376831],{"className":376832,"style":330413},[31680],[34,376834,376836,376839],{"className":376835},[31685],[34,376837,97300],{"className":376838},[31685,318441],[34,376840,376842],{"className":376841},[31792],[34,376843,376845,376865],{"className":376844},[31796,31797],[34,376846,376848,376862],{"className":376847},[31801],[34,376849,376851],{"className":376850,"style":31806},[31805],[34,376852,376853,376856],{"style":31809},[34,376854],{"className":376855,"style":31814},[31813],[34,376857,376859],{"className":376858},[31818,31819,31820,31821],[34,376860,36656],{"className":376861},[31685,31821],[34,376863,31829],{"className":376864},[31828],[34,376866,376868],{"className":376867},[31801],[34,376869,376871],{"className":376870,"style":31836},[31805],[34,376872],{}," 里：\n",[34,376875,376877,376903],{"className":376876},[31642],[34,376878,376880],{"className":376879},[31646],[31648,376881,376882],{"xmlns":31650},[31652,376883,376884,376900],{},[31655,376885,376886,376892,376894],{},[134014,376887,376888,376890],{"accent":758},[31749,376889,768],{},[31662,376891,346142],{},[31662,376893,54707],{"mathvariant":38079},[134014,376895,376896,376898],{"accent":758},[31749,376897,459],{},[31662,376899,346142],{},[31665,376901,376902],{"encoding":31667},"\\bar{2} \\neq \\bar{0}",[34,376904,376906,376986],{"className":376905,"ariaHidden":758},[31672],[34,376907,376909,376913,376944,376947,376983],{"className":376908},[31676],[34,376910],{"className":376911,"style":376912},[31680],"height:0.9757em;vertical-align:-0.1944em;",[34,376914,376916],{"className":376915},[31685,346171],[34,376917,376919],{"className":376918},[31796],[34,376920,376922],{"className":376921},[31801],[34,376923,376925,376933],{"className":376924,"style":371250},[31805],[34,376926,376927,376930],{"style":41565},[34,376928],{"className":376929,"style":41508},[31813],[34,376931,768],{"className":376932},[31685],[34,376934,376935,376938],{"style":371273},[34,376936],{"className":376937,"style":41508},[31813],[34,376939,376941],{"className":376940,"style":346199},[346198],[34,376942,346142],{"className":376943},[31685],[34,376945],{"className":376946,"style":31846},[31845],[34,376948,376950,376977,376980],{"className":376949},[31850],[34,376951,376953],{"className":376952},[31850],[34,376954,376956],{"className":376955},[31685,54774],[34,376957,376959],{"className":376958},[54778],[34,376960,376962,376965,376974],{"className":376961},[54782],[34,376963],{"className":376964,"style":43217},[31680],[34,376966,376968],{"className":376967},[54789],[34,376969,376971],{"className":376970},[31685],[34,376972,54796],{"className":376973},[31850],[34,376975],{"className":376976},[54800],[34,376978],{"className":376979},[31845,54804],[34,376981,202],{"className":376982},[31850],[34,376984],{"className":376985,"style":31846},[31845],[34,376987,376989,376992],{"className":376988},[31676],[34,376990],{"className":376991,"style":371250},[31680],[34,376993,376995],{"className":376994},[31685,346171],[34,376996,376998],{"className":376997},[31796],[34,376999,377001],{"className":377000},[31801],[34,377002,377004,377012],{"className":377003,"style":371250},[31805],[34,377005,377006,377009],{"style":41565},[34,377007],{"className":377008,"style":41508},[31813],[34,377010,459],{"className":377011},[31685],[34,377013,377014,377017],{"style":371273},[34,377015],{"className":377016,"style":41508},[31813],[34,377018,377020],{"className":377019,"style":346199},[346198],[34,377021,346142],{"className":377022},[31685],[34,377024,377026,377052],{"className":377025},[31642],[34,377027,377029],{"className":377028},[31646],[31648,377030,377031],{"xmlns":31650},[31652,377032,377033,377049],{},[31655,377034,377035,377041,377043],{},[134014,377036,377037,377039],{"accent":758},[31749,377038,786],{},[31662,377040,346142],{},[31662,377042,54707],{"mathvariant":38079},[134014,377044,377045,377047],{"accent":758},[31749,377046,459],{},[31662,377048,346142],{},[31665,377050,377051],{"encoding":31667},"\\bar{3} \\neq \\bar{0}",[34,377053,377055,377134],{"className":377054,"ariaHidden":758},[31672],[34,377056,377058,377061,377092,377095,377131],{"className":377057},[31676],[34,377059],{"className":377060,"style":376912},[31680],[34,377062,377064],{"className":377063},[31685,346171],[34,377065,377067],{"className":377066},[31796],[34,377068,377070],{"className":377069},[31801],[34,377071,377073,377081],{"className":377072,"style":371250},[31805],[34,377074,377075,377078],{"style":41565},[34,377076],{"className":377077,"style":41508},[31813],[34,377079,786],{"className":377080},[31685],[34,377082,377083,377086],{"style":371273},[34,377084],{"className":377085,"style":41508},[31813],[34,377087,377089],{"className":377088,"style":346199},[346198],[34,377090,346142],{"className":377091},[31685],[34,377093],{"className":377094,"style":31846},[31845],[34,377096,377098,377125,377128],{"className":377097},[31850],[34,377099,377101],{"className":377100},[31850],[34,377102,377104],{"className":377103},[31685,54774],[34,377105,377107],{"className":377106},[54778],[34,377108,377110,377113,377122],{"className":377109},[54782],[34,377111],{"className":377112,"style":43217},[31680],[34,377114,377116],{"className":377115},[54789],[34,377117,377119],{"className":377118},[31685],[34,377120,54796],{"className":377121},[31850],[34,377123],{"className":377124},[54800],[34,377126],{"className":377127},[31845,54804],[34,377129,202],{"className":377130},[31850],[34,377132],{"className":377133,"style":31846},[31845],[34,377135,377137,377140],{"className":377136},[31676],[34,377138],{"className":377139,"style":371250},[31680],[34,377141,377143],{"className":377142},[31685,346171],[34,377144,377146],{"className":377145},[31796],[34,377147,377149],{"className":377148},[31801],[34,377150,377152,377160],{"className":377151,"style":371250},[31805],[34,377153,377154,377157],{"style":41565},[34,377155],{"className":377156,"style":41508},[31813],[34,377158,459],{"className":377159},[31685],[34,377161,377162,377165],{"style":371273},[34,377163],{"className":377164,"style":41508},[31813],[34,377166,377168],{"className":377167,"style":346199},[346198],[34,377169,346142],{"className":377170},[31685],"\n但是：\n",[34,377173,377175,377217],{"className":377174},[31642],[34,377176,377178],{"className":377177},[31646],[31648,377179,377180],{"xmlns":31650},[31652,377181,377182,377214],{},[31655,377183,377184,377190,377192,377198,377200,377206,377208],{},[134014,377185,377186,377188],{"accent":758},[31749,377187,768],{},[31662,377189,346142],{},[31662,377191,71191],{},[134014,377193,377194,377196],{"accent":758},[31749,377195,786],{},[31662,377197,346142],{},[31662,377199,202],{},[134014,377201,377202,377204],{"accent":758},[31749,377203,36656],{},[31662,377205,346142],{},[31662,377207,202],{},[134014,377209,377210,377212],{"accent":758},[31749,377211,459],{},[31662,377213,346142],{},[31665,377215,377216],{"encoding":31667},"\\bar{2} \\cdot \\bar{3} = \\bar{6} = \\bar{0}",[34,377218,377220,377266,377312,377358],{"className":377219,"ariaHidden":758},[31672],[34,377221,377223,377226,377257,377260,377263],{"className":377222},[31676],[34,377224],{"className":377225,"style":371250},[31680],[34,377227,377229],{"className":377228},[31685,346171],[34,377230,377232],{"className":377231},[31796],[34,377233,377235],{"className":377234},[31801],[34,377236,377238,377246],{"className":377237,"style":371250},[31805],[34,377239,377240,377243],{"style":41565},[34,377241],{"className":377242,"style":41508},[31813],[34,377244,768],{"className":377245},[31685],[34,377247,377248,377251],{"style":371273},[34,377249],{"className":377250,"style":41508},[31813],[34,377252,377254],{"className":377253,"style":346199},[346198],[34,377255,346142],{"className":377256},[31685],[34,377258],{"className":377259,"style":32604},[31845],[34,377261,71191],{"className":377262},[32024],[34,377264],{"className":377265,"style":32604},[31845],[34,377267,377269,377272,377303,377306,377309],{"className":377268},[31676],[34,377270],{"className":377271,"style":371250},[31680],[34,377273,377275],{"className":377274},[31685,346171],[34,377276,377278],{"className":377277},[31796],[34,377279,377281],{"className":377280},[31801],[34,377282,377284,377292],{"className":377283,"style":371250},[31805],[34,377285,377286,377289],{"style":41565},[34,377287],{"className":377288,"style":41508},[31813],[34,377290,786],{"className":377291},[31685],[34,377293,377294,377297],{"style":371273},[34,377295],{"className":377296,"style":41508},[31813],[34,377298,377300],{"className":377299,"style":346199},[346198],[34,377301,346142],{"className":377302},[31685],[34,377304],{"className":377305,"style":31846},[31845],[34,377307,202],{"className":377308},[31850],[34,377310],{"className":377311,"style":31846},[31845],[34,377313,377315,377318,377349,377352,377355],{"className":377314},[31676],[34,377316],{"className":377317,"style":371250},[31680],[34,377319,377321],{"className":377320},[31685,346171],[34,377322,377324],{"className":377323},[31796],[34,377325,377327],{"className":377326},[31801],[34,377328,377330,377338],{"className":377329,"style":371250},[31805],[34,377331,377332,377335],{"style":41565},[34,377333],{"className":377334,"style":41508},[31813],[34,377336,36656],{"className":377337},[31685],[34,377339,377340,377343],{"style":371273},[34,377341],{"className":377342,"style":41508},[31813],[34,377344,377346],{"className":377345,"style":346199},[346198],[34,377347,346142],{"className":377348},[31685],[34,377350],{"className":377351,"style":31846},[31845],[34,377353,202],{"className":377354},[31850],[34,377356],{"className":377357,"style":31846},[31845],[34,377359,377361,377364],{"className":377360},[31676],[34,377362],{"className":377363,"style":371250},[31680],[34,377365,377367],{"className":377366},[31685,346171],[34,377368,377370],{"className":377369},[31796],[34,377371,377373],{"className":377372},[31801],[34,377374,377376,377384],{"className":377375,"style":371250},[31805],[34,377377,377378,377381],{"style":41565},[34,377379],{"className":377380,"style":41508},[31813],[34,377382,459],{"className":377383},[31685],[34,377385,377386,377389],{"style":371273},[34,377387],{"className":377388,"style":41508},[31813],[34,377390,377392],{"className":377391,"style":346199},[346198],[34,377393,346142],{"className":377394},[31685],"\n这里 ",[34,377397,377399,377416],{"className":377398},[31642],[34,377400,377402],{"className":377401},[31646],[31648,377403,377404],{"xmlns":31650},[31652,377405,377406,377414],{},[31655,377407,377408],{},[134014,377409,377410,377412],{"accent":758},[31749,377411,768],{},[31662,377413,346142],{},[31665,377415,371441],{"encoding":31667},[34,377417,377419],{"className":377418,"ariaHidden":758},[31672],[34,377420,377422,377425],{"className":377421},[31676],[34,377423],{"className":377424,"style":371250},[31680],[34,377426,377428],{"className":377427},[31685,346171],[34,377429,377431],{"className":377430},[31796],[34,377432,377434],{"className":377433},[31801],[34,377435,377437,377445],{"className":377436,"style":371250},[31805],[34,377438,377439,377442],{"style":41565},[34,377440],{"className":377441,"style":41508},[31813],[34,377443,768],{"className":377444},[31685],[34,377446,377447,377450],{"style":371273},[34,377448],{"className":377449,"style":41508},[31813],[34,377451,377453],{"className":377452,"style":346199},[346198],[34,377454,346142],{"className":377455},[31685],[34,377457,377459,377477],{"className":377458},[31642],[34,377460,377462],{"className":377461},[31646],[31648,377463,377464],{"xmlns":31650},[31652,377465,377466,377474],{},[31655,377467,377468],{},[134014,377469,377470,377472],{"accent":758},[31749,377471,786],{},[31662,377473,346142],{},[31665,377475,377476],{"encoding":31667},"\\bar{3}",[34,377478,377480],{"className":377479,"ariaHidden":758},[31672],[34,377481,377483,377486],{"className":377482},[31676],[34,377484],{"className":377485,"style":371250},[31680],[34,377487,377489],{"className":377488},[31685,346171],[34,377490,377492],{"className":377491},[31796],[34,377493,377495],{"className":377494},[31801],[34,377496,377498,377506],{"className":377497,"style":371250},[31805],[34,377499,377500,377503],{"style":41565},[34,377501],{"className":377502,"style":41508},[31813],[34,377504,786],{"className":377505},[31685],[34,377507,377508,377511],{"style":371273},[34,377509],{"className":377510,"style":41508},[31813],[34,377512,377514],{"className":377513,"style":346199},[346198],[34,377515,346142],{"className":377516},[31685]," 即使不是 0，乘起来却是 0。这就有了",[2627,377519,375836],{},"，所以它",[2627,377522,377523],{},"不是整环",[253,377525,365086],{"id":365086},[15,377527,377528],{},"证明的核心就在于连接这两个概念：",[105,377530,377531,377571],{},[108,377532,377533,377536,377537,47649,377567,377570],{},[2627,377534,377535],{},"环论","里的“零因子” ",[34,377538,377540,377555],{"className":377539},[31642],[34,377541,377543],{"className":377542},[31646],[31648,377544,377545],{"xmlns":31650},[31652,377546,377547,377552],{},[31655,377548,377549],{},[31662,377550,377551],{},"⟷",[31665,377553,377554],{"encoding":31667},"\\longleftrightarrow",[34,377556,377558],{"className":377557,"ariaHidden":758},[31672],[34,377559,377561,377564],{"className":377560},[31676],[34,377562],{"className":377563,"style":330733},[31680],[34,377565,377551],{"className":377566},[31850],[2627,377568,377569],{},"数论","里的“因子分解”。",[108,377572,377573,355459,377626,47649,377778,47649,377806,377834,377835,347],{},[34,377574,377576,377596],{"className":377575},[31642],[34,377577,377579],{"className":377578},[31646],[31648,377580,377581],{"xmlns":31650},[31652,377582,377583,377593],{},[31655,377584,377585,377587,377589,377591],{},[31658,377586,111],{},[31658,377588,1697],{},[31662,377590,202],{},[31749,377592,459],{},[31665,377594,377595],{"encoding":31667},"ab=0",[34,377597,377599,377617],{"className":377598,"ariaHidden":758},[31672],[34,377600,377602,377605,377608,377611,377614],{"className":377601},[31676],[34,377603],{"className":377604,"style":31681},[31680],[34,377606,324191],{"className":377607},[31685,31686],[34,377609],{"className":377610,"style":31846},[31845],[34,377612,202],{"className":377613},[31850],[34,377615],{"className":377616,"style":31846},[31845],[34,377618,377620,377623],{"className":377619},[31676],[34,377621],{"className":377622,"style":32617},[31680],[34,377624,459],{"className":377625},[31685],[34,377627,377629,377655],{"className":377628},[31642],[34,377630,377632],{"className":377631},[31646],[31648,377633,377634],{"xmlns":31650},[31652,377635,377636,377652],{},[31655,377637,377638,377640,377642,377644,377646,377648,377650],{},[31658,377639,111],{},[31662,377641,54707],{"mathvariant":38079},[31749,377643,459],{},[31662,377645,31960],{"separator":758},[31658,377647,1697],{},[31662,377649,54707],{"mathvariant":38079},[31749,377651,459],{},[31665,377653,377654],{"encoding":31667},"a \\neq 0, b \\neq 0",[34,377656,377658,377709,377769],{"className":377657,"ariaHidden":758},[31672],[34,377659,377661,377664,377667,377670,377706],{"className":377660},[31676],[34,377662],{"className":377663,"style":43217},[31680],[34,377665,111],{"className":377666},[31685,31686],[34,377668],{"className":377669,"style":31846},[31845],[34,377671,377673,377700,377703],{"className":377672},[31850],[34,377674,377676],{"className":377675},[31850],[34,377677,377679],{"className":377678},[31685,54774],[34,377680,377682],{"className":377681},[54778],[34,377683,377685,377688,377697],{"className":377684},[54782],[34,377686],{"className":377687,"style":43217},[31680],[34,377689,377691],{"className":377690},[54789],[34,377692,377694],{"className":377693},[31685],[34,377695,54796],{"className":377696},[31850],[34,377698],{"className":377699},[54800],[34,377701],{"className":377702},[31845,54804],[34,377704,202],{"className":377705},[31850],[34,377707],{"className":377708,"style":31846},[31845],[34,377710,377712,377715,377718,377721,377724,377727,377730,377766],{"className":377711},[31676],[34,377713],{"className":377714,"style":43217},[31680],[34,377716,459],{"className":377717},[31685],[34,377719,31960],{"className":377720},[32119],[34,377722],{"className":377723,"style":32123},[31845],[34,377725,1697],{"className":377726},[31685,31686],[34,377728],{"className":377729,"style":31846},[31845],[34,377731,377733,377760,377763],{"className":377732},[31850],[34,377734,377736],{"className":377735},[31850],[34,377737,377739],{"className":377738},[31685,54774],[34,377740,377742],{"className":377741},[54778],[34,377743,377745,377748,377757],{"className":377744},[54782],[34,377746],{"className":377747,"style":43217},[31680],[34,377749,377751],{"className":377750},[54789],[34,377752,377754],{"className":377753},[31685],[34,377755,54796],{"className":377756},[31850],[34,377758],{"className":377759},[54800],[34,377761],{"className":377762},[31845,54804],[34,377764,202],{"className":377765},[31850],[34,377767],{"className":377768,"style":31846},[31845],[34,377770,377772,377775],{"className":377771},[31676],[34,377773],{"className":377774,"style":32617},[31680],[34,377776,459],{"className":377777},[31685],[34,377779,377781,377794],{"className":377780},[31642],[34,377782,377784],{"className":377783},[31646],[31648,377785,377786],{"xmlns":31650},[31652,377787,377788,377792],{},[31655,377789,377790],{},[31662,377791,377551],{},[31665,377793,377554],{"encoding":31667},[34,377795,377797],{"className":377796,"ariaHidden":758},[31672],[34,377798,377800,377803],{"className":377799},[31676],[34,377801],{"className":377802,"style":330733},[31680],[34,377804,377551],{"className":377805},[31850],[34,377807,377809,377822],{"className":377808},[31642],[34,377810,377812],{"className":377811},[31646],[31648,377813,377814],{"xmlns":31650},[31652,377815,377816,377820],{},[31655,377817,377818],{},[31658,377819,31660],{},[31665,377821,31660],{"encoding":31667},[34,377823,377825],{"className":377824,"ariaHidden":758},[31672],[34,377826,377828,377831],{"className":377827},[31676],[34,377829],{"className":377830,"style":32527},[31680],[34,377832,31660],{"className":377833},[31685,31686]," 可以分解为真因子 ",[34,377836,377838,377858],{"className":377837},[31642],[34,377839,377841],{"className":377840},[31646],[31648,377842,377843],{"xmlns":31650},[31652,377844,377845,377855],{},[31655,377846,377847,377849,377851,377853],{},[31658,377848,111],{},[31658,377850,1697],{},[31662,377852,202],{},[31658,377854,31660],{},[31665,377856,377857],{"encoding":31667},"ab=n",[34,377859,377861,377879],{"className":377860,"ariaHidden":758},[31672],[34,377862,377864,377867,377870,377873,377876],{"className":377863},[31676],[34,377865],{"className":377866,"style":31681},[31680],[34,377868,324191],{"className":377869},[31685,31686],[34,377871],{"className":377872,"style":31846},[31845],[34,377874,202],{"className":377875},[31850],[34,377877],{"className":377878,"style":31846},[31845],[34,377880,377882,377885],{"className":377881},[31676],[34,377883],{"className":377884,"style":32527},[31680],[34,377886,31660],{"className":377887},[31685,31686],[32188,377889],{},[253,377891,377892],{"id":377892},"题目1",[15,377894,377895,377896,377924,377925,377989,377990,326252,378066,378162,378163,378228],{},"如果把整数环 ",[34,377897,377899,377912],{"className":377898},[31642],[34,377900,377902],{"className":377901},[31646],[31648,377903,377904],{"xmlns":31650},[31652,377905,377906,377910],{},[31655,377907,377908],{},[31658,377909,97300],{"mathvariant":318416},[31665,377911,320917],{"encoding":31667},[34,377913,377915],{"className":377914,"ariaHidden":758},[31672],[34,377916,377918,377921],{"className":377917},[31676],[34,377919],{"className":377920,"style":320927},[31680],[34,377922,97300],{"className":377923},[31685,318441]," 中的加法和乘法的定义互换，即对于 ",[34,377926,377928,377950],{"className":377927},[31642],[34,377929,377931],{"className":377930},[31646],[31648,377932,377933],{"xmlns":31650},[31652,377934,377935,377947],{},[31655,377936,377937,377939,377941,377943,377945],{},[31658,377938,111],{},[31662,377940,31960],{"separator":758},[31658,377942,1697],{},[31662,377944,112296],{},[31658,377946,97300],{"mathvariant":318416},[31665,377948,377949],{"encoding":31667},"a,b \\in \\mathbb{Z}",[34,377951,377953,377980],{"className":377952,"ariaHidden":758},[31672],[34,377954,377956,377959,377962,377965,377968,377971,377974,377977],{"className":377955},[31676],[34,377957],{"className":377958,"style":43217},[31680],[34,377960,111],{"className":377961},[31685,31686],[34,377963,31960],{"className":377964},[32119],[34,377966],{"className":377967,"style":32123},[31845],[34,377969,1697],{"className":377970},[31685,31686],[34,377972],{"className":377973,"style":31846},[31845],[34,377975,112296],{"className":377976},[31850],[34,377978],{"className":377979,"style":31846},[31845],[34,377981,377983,377986],{"className":377982},[31676],[34,377984],{"className":377985,"style":320927},[31680],[34,377987,97300],{"className":377988},[31685,318441],"，定义 ",[34,377991,377993,378018],{"className":377992},[31642],[34,377994,377996],{"className":377995},[31646],[31648,377997,377998],{"xmlns":31650},[31652,377999,378000,378015],{},[31655,378001,378002,378004,378007,378009,378011,378013],{},[31658,378003,111],{},[31662,378005,378006],{},"⊕",[31658,378008,1697],{},[31662,378010,202],{},[31658,378012,111],{},[31658,378014,1697],{},[31665,378016,378017],{"encoding":31667},"a \\oplus b = ab",[34,378019,378021,378039,378057],{"className":378020,"ariaHidden":758},[31672],[34,378022,378024,378027,378030,378033,378036],{"className":378023},[31676],[34,378025],{"className":378026,"style":32597},[31680],[34,378028,111],{"className":378029},[31685,31686],[34,378031],{"className":378032,"style":32604},[31845],[34,378034,378006],{"className":378035},[32024],[34,378037],{"className":378038,"style":32604},[31845],[34,378040,378042,378045,378048,378051,378054],{"className":378041},[31676],[34,378043],{"className":378044,"style":31681},[31680],[34,378046,1697],{"className":378047},[31685,31686],[34,378049],{"className":378050,"style":31846},[31845],[34,378052,202],{"className":378053},[31850],[34,378055],{"className":378056,"style":31846},[31845],[34,378058,378060,378063],{"className":378059},[31676],[34,378061],{"className":378062,"style":31681},[31680],[34,378064,324191],{"className":378065},[31685,31686],[34,378067,378069,378096],{"className":378068},[31642],[34,378070,378072],{"className":378071},[31646],[31648,378073,378074],{"xmlns":31650},[31652,378075,378076,378093],{},[31655,378077,378078,378080,378083,378085,378087,378089,378091],{},[31658,378079,111],{},[31662,378081,378082],{},"⊙",[31658,378084,1697],{},[31662,378086,202],{},[31658,378088,111],{},[31662,378090,223],{},[31658,378092,1697],{},[31665,378094,378095],{"encoding":31667},"a \\odot b = a + b",[34,378097,378099,378117,378135,378153],{"className":378098,"ariaHidden":758},[31672],[34,378100,378102,378105,378108,378111,378114],{"className":378101},[31676],[34,378103],{"className":378104,"style":32597},[31680],[34,378106,111],{"className":378107},[31685,31686],[34,378109],{"className":378110,"style":32604},[31845],[34,378112,378082],{"className":378113},[32024],[34,378115],{"className":378116,"style":32604},[31845],[34,378118,378120,378123,378126,378129,378132],{"className":378119},[31676],[34,378121],{"className":378122,"style":31681},[31680],[34,378124,1697],{"className":378125},[31685,31686],[34,378127],{"className":378128,"style":31846},[31845],[34,378130,202],{"className":378131},[31850],[34,378133],{"className":378134,"style":31846},[31845],[34,378136,378138,378141,378144,378147,378150],{"className":378137},[31676],[34,378139],{"className":378140,"style":32597},[31680],[34,378142,111],{"className":378143},[31685,31686],[34,378145],{"className":378146,"style":32604},[31845],[34,378148,223],{"className":378149},[32024],[34,378151],{"className":378152,"style":32604},[31845],[34,378154,378156,378159],{"className":378155},[31676],[34,378157],{"className":378158,"style":31681},[31680],[34,378160,1697],{"className":378161},[31685,31686],"，试问 ",[34,378164,378166,378192],{"className":378165},[31642],[34,378167,378169],{"className":378168},[31646],[31648,378170,378171],{"xmlns":31650},[31652,378172,378173,378189],{},[31655,378174,378175,378177,378179,378181,378183,378185,378187],{},[31662,378176,456],{"stretchy":702},[31658,378178,97300],{"mathvariant":318416},[31662,378180,199477],{"separator":758},[31662,378182,378006],{},[31662,378184,31960],{"separator":758},[31662,378186,378082],{},[31662,378188,2557],{"stretchy":702},[31665,378190,378191],{"encoding":31667},"(\\mathbb{Z}; \\oplus, \\odot)",[34,378193,378195],{"className":378194,"ariaHidden":758},[31672],[34,378196,378198,378201,378204,378207,378210,378213,378216,378219,378222,378225],{"className":378197},[31676],[34,378199],{"className":378200,"style":31774},[31680],[34,378202,456],{"className":378203},[31782],[34,378205,97300],{"className":378206},[31685,318441],[34,378208,199477],{"className":378209},[32119],[34,378211],{"className":378212,"style":32123},[31845],[34,378214,378006],{"className":378215},[31685],[34,378217,31960],{"className":378218},[32119],[34,378220],{"className":378221,"style":32123},[31845],[34,378223,378082],{"className":378224},[31685],[34,378226,2557],{"className":378227},[31690]," 是否构成环？",[32490,378230,297792],{"id":297792},[15,378232,378233,378236,378237,378310,378311,378314],{},[2627,378234,378235],{},"1. 分析意图","\n这道题是想考你对“环的定义”是否死板。环定义里的“加法”不一定非要是我们小学学的 ",[34,378238,378240,378262],{"className":378239},[31642],[34,378241,378243],{"className":378242},[31646],[31648,378244,378245],{"xmlns":31650},[31652,378246,378247,378259],{},[31655,378248,378249,378251,378253,378255,378257],{},[31749,378250,640],{},[31662,378252,223],{},[31749,378254,640],{},[31662,378256,202],{},[31749,378258,768],{},[31665,378260,378261],{"encoding":31667},"1+1=2",[34,378263,378265,378283,378301],{"className":378264,"ariaHidden":758},[31672],[34,378266,378268,378271,378274,378277,378280],{"className":378267},[31676],[34,378269],{"className":378270,"style":35942},[31680],[34,378272,640],{"className":378273},[31685],[34,378275],{"className":378276,"style":32604},[31845],[34,378278,223],{"className":378279},[32024],[34,378281],{"className":378282,"style":32604},[31845],[34,378284,378286,378289,378292,378295,378298],{"className":378285},[31676],[34,378287],{"className":378288,"style":32617},[31680],[34,378290,640],{"className":378291},[31685],[34,378293],{"className":378294,"style":31846},[31845],[34,378296,202],{"className":378297},[31850],[34,378299],{"className":378300,"style":31846},[31845],[34,378302,378304,378307],{"className":378303},[31676],[34,378305],{"className":378306,"style":32617},[31680],[34,378308,768],{"className":378309},[31685],"，它只是一个",[2627,378312,378313],{},"第一运算","的名字。这道题把小学乘法变成了“第一运算”，把小学加法变成了“第二运算”。",[15,378316,378317,378318,378347,378348,347],{},"我们要验证它是不是环，首先要验证第一运算（这里叫 ",[34,378319,378321,378335],{"className":378320},[31642],[34,378322,378324],{"className":378323},[31646],[31648,378325,378326],{"xmlns":31650},[31652,378327,378328,378332],{},[31655,378329,378330],{},[31662,378331,378006],{},[31665,378333,378334],{"encoding":31667},"\\oplus",[34,378336,378338],{"className":378337,"ariaHidden":758},[31672],[34,378339,378341,378344],{"className":378340},[31676],[34,378342],{"className":378343,"style":32597},[31680],[34,378345,378006],{"className":378346},[31685],"）是否构成",[2627,378349,378350],{},"阿贝尔群",[15,378352,378353,378385],{},[2627,378354,378355,378356,378384],{},"2. 验证第一运算 ",[34,378357,378359,378372],{"className":378358},[31642],[34,378360,378362],{"className":378361},[31646],[31648,378363,378364],{"xmlns":31650},[31652,378365,378366,378370],{},[31655,378367,378368],{},[31662,378369,378006],{},[31665,378371,378334],{"encoding":31667},[34,378373,378375],{"className":378374,"ariaHidden":758},[31672],[34,378376,378378,378381],{"className":378377},[31676],[34,378379],{"className":378380,"style":32597},[31680],[34,378382,378006],{"className":378383},[31685]," (也就是原来的乘法)","\n要构成群，必须满足四个条件：封闭性、结合律、单位元、逆元。",[105,378387,378388,378394,378488,378762],{},[108,378389,378390,378393],{},[2627,378391,378392],{},"封闭性","：整数乘整数还是整数。满足。",[108,378395,378396,2664,378399,378487],{},[2627,378397,378398],{},"结合律",[34,378400,378402,378436],{"className":378401},[31642],[34,378403,378405],{"className":378404},[31646],[31648,378406,378407],{"xmlns":31650},[31652,378408,378409,378433],{},[31655,378410,378411,378413,378415,378417,378419,378421,378423,378425,378427,378429,378431],{},[31662,378412,456],{"stretchy":702},[31658,378414,111],{},[31658,378416,1697],{},[31662,378418,2557],{"stretchy":702},[31658,378420,31965],{},[31662,378422,202],{},[31658,378424,111],{},[31662,378426,456],{"stretchy":702},[31658,378428,1697],{},[31658,378430,31965],{},[31662,378432,2557],{"stretchy":702},[31665,378434,378435],{"encoding":31667},"(ab)c = a(bc)",[34,378437,378439,378466],{"className":378438,"ariaHidden":758},[31672],[34,378440,378442,378445,378448,378451,378454,378457,378460,378463],{"className":378441},[31676],[34,378443],{"className":378444,"style":31774},[31680],[34,378446,456],{"className":378447},[31782],[34,378449,324191],{"className":378450},[31685,31686],[34,378452,2557],{"className":378453},[31690],[34,378455,31965],{"className":378456},[31685,31686],[34,378458],{"className":378459,"style":31846},[31845],[34,378461,202],{"className":378462},[31850],[34,378464],{"className":378465,"style":31846},[31845],[34,378467,378469,378472,378475,378478,378481,378484],{"className":378468},[31676],[34,378470],{"className":378471,"style":31774},[31680],[34,378473,111],{"className":378474},[31685,31686],[34,378476,456],{"className":378477},[31782],[34,378479,1697],{"className":378480},[31685,31686],[34,378482,31965],{"className":378483},[31685,31686],[34,378485,2557],{"className":378486},[31690],"。满足。",[108,378489,378490,378493,378494,378522,378523,352264,378551,39984,378624],{},[2627,378491,378492],{},"单位元","：我们需要找一个元素 ",[34,378495,378497,378510],{"className":378496},[31642],[34,378498,378500],{"className":378499},[31646],[31648,378501,378502],{"xmlns":31650},[31652,378503,378504,378508],{},[31655,378505,378506],{},[31658,378507,33244],{},[31665,378509,33244],{"encoding":31667},[34,378511,378513],{"className":378512,"ariaHidden":758},[31672],[34,378514,378516,378519],{"className":378515},[31676],[34,378517],{"className":378518,"style":32527},[31680],[34,378520,33244],{"className":378521},[31685,31686],"，使得对于任意整数 ",[34,378524,378526,378539],{"className":378525},[31642],[34,378527,378529],{"className":378528},[31646],[31648,378530,378531],{"xmlns":31650},[31652,378532,378533,378537],{},[31655,378534,378535],{},[31658,378536,111],{},[31665,378538,111],{"encoding":31667},[34,378540,378542],{"className":378541,"ariaHidden":758},[31672],[34,378543,378545,378548],{"className":378544},[31676],[34,378546],{"className":378547,"style":32527},[31680],[34,378549,111],{"className":378550},[31685,31686],[34,378552,378554,378576],{"className":378553},[31642],[34,378555,378557],{"className":378556},[31646],[31648,378558,378559],{"xmlns":31650},[31652,378560,378561,378573],{},[31655,378562,378563,378565,378567,378569,378571],{},[31658,378564,111],{},[31662,378566,378006],{},[31658,378568,33244],{},[31662,378570,202],{},[31658,378572,111],{},[31665,378574,378575],{"encoding":31667},"a \\oplus e = a",[34,378577,378579,378597,378615],{"className":378578,"ariaHidden":758},[31672],[34,378580,378582,378585,378588,378591,378594],{"className":378581},[31676],[34,378583],{"className":378584,"style":32597},[31680],[34,378586,111],{"className":378587},[31685,31686],[34,378589],{"className":378590,"style":32604},[31845],[34,378592,378006],{"className":378593},[32024],[34,378595],{"className":378596,"style":32604},[31845],[34,378598,378600,378603,378606,378609,378612],{"className":378599},[31676],[34,378601],{"className":378602,"style":32527},[31680],[34,378604,33244],{"className":378605},[31685,31686],[34,378607],{"className":378608,"style":31846},[31845],[34,378610,202],{"className":378611},[31850],[34,378613],{"className":378614,"style":31846},[31845],[34,378616,378618,378621],{"className":378617},[31676],[34,378619],{"className":378620,"style":32527},[31680],[34,378622,111],{"className":378623},[31685,31686],[105,378625,378626,378702],{},[108,378627,378628,378629,347],{},"翻译成原来的运算就是：",[34,378630,378632,378654],{"className":378631},[31642],[34,378633,378635],{"className":378634},[31646],[31648,378636,378637],{"xmlns":31650},[31652,378638,378639,378651],{},[31655,378640,378641,378643,378645,378647,378649],{},[31658,378642,111],{},[31662,378644,71191],{},[31658,378646,33244],{},[31662,378648,202],{},[31658,378650,111],{},[31665,378652,378653],{"encoding":31667},"a \\cdot e = a",[34,378655,378657,378675,378693],{"className":378656,"ariaHidden":758},[31672],[34,378658,378660,378663,378666,378669,378672],{"className":378659},[31676],[34,378661],{"className":378662,"style":224869},[31680],[34,378664,111],{"className":378665},[31685,31686],[34,378667],{"className":378668,"style":32604},[31845],[34,378670,71191],{"className":378671},[32024],[34,378673],{"className":378674,"style":32604},[31845],[34,378676,378678,378681,378684,378687,378690],{"className":378677},[31676],[34,378679],{"className":378680,"style":32527},[31680],[34,378682,33244],{"className":378683},[31685,31686],[34,378685],{"className":378686,"style":31846},[31845],[34,378688,202],{"className":378689},[31850],[34,378691],{"className":378692,"style":31846},[31845],[34,378694,378696,378699],{"className":378695},[31676],[34,378697],{"className":378698,"style":32527},[31680],[34,378700,111],{"className":378701},[31685,31686],[108,378703,378704,378705,378733,378734,347],{},"显然，整数 ",[34,378706,378708,378721],{"className":378707},[31642],[34,378709,378711],{"className":378710},[31646],[31648,378712,378713],{"xmlns":31650},[31652,378714,378715,378719],{},[31655,378716,378717],{},[31749,378718,640],{},[31665,378720,640],{"encoding":31667},[34,378722,378724],{"className":378723,"ariaHidden":758},[31672],[34,378725,378727,378730],{"className":378726},[31676],[34,378728],{"className":378729,"style":32617},[31680],[34,378731,640],{"className":378732},[31685]," 就是这个单位元。所以，这套新系统的“零元”其实是 ",[34,378735,378737,378750],{"className":378736},[31642],[34,378738,378740],{"className":378739},[31646],[31648,378741,378742],{"xmlns":31650},[31652,378743,378744,378748],{},[31655,378745,378746],{},[31749,378747,640],{},[31665,378749,640],{"encoding":31667},[34,378751,378753],{"className":378752,"ariaHidden":758},[31672],[34,378754,378756,378759],{"className":378755},[31676],[34,378757],{"className":378758,"style":32617},[31680],[34,378760,640],{"className":378761},[31685],[108,378763,378764,378767,378768,378796,378797,335149,378825,378898,378899,68913,378927],{},[2627,378765,378766],{},"逆元 (关键点！)","：对于任意整数 ",[34,378769,378771,378784],{"className":378770},[31642],[34,378772,378774],{"className":378773},[31646],[31648,378775,378776],{"xmlns":31650},[31652,378777,378778,378782],{},[31655,378779,378780],{},[31658,378781,111],{},[31665,378783,111],{"encoding":31667},[34,378785,378787],{"className":378786,"ariaHidden":758},[31672],[34,378788,378790,378793],{"className":378789},[31676],[34,378791],{"className":378792,"style":32527},[31680],[34,378794,111],{"className":378795},[31685,31686],"，我们需要找一个 ",[34,378798,378800,378813],{"className":378799},[31642],[34,378801,378803],{"className":378802},[31646],[31648,378804,378805],{"xmlns":31650},[31652,378806,378807,378811],{},[31655,378808,378809],{},[31658,378810,32936],{},[31665,378812,32936],{"encoding":31667},[34,378814,378816],{"className":378815,"ariaHidden":758},[31672],[34,378817,378819,378822],{"className":378818},[31676],[34,378820],{"className":378821,"style":32527},[31680],[34,378823,32936],{"className":378824},[31685,31686],[34,378826,378828,378850],{"className":378827},[31642],[34,378829,378831],{"className":378830},[31646],[31648,378832,378833],{"xmlns":31650},[31652,378834,378835,378847],{},[31655,378836,378837,378839,378841,378843,378845],{},[31658,378838,111],{},[31662,378840,378006],{},[31658,378842,32936],{},[31662,378844,202],{},[31658,378846,33244],{},[31665,378848,378849],{"encoding":31667},"a \\oplus x = e",[34,378851,378853,378871,378889],{"className":378852,"ariaHidden":758},[31672],[34,378854,378856,378859,378862,378865,378868],{"className":378855},[31676],[34,378857],{"className":378858,"style":32597},[31680],[34,378860,111],{"className":378861},[31685,31686],[34,378863],{"className":378864,"style":32604},[31845],[34,378866,378006],{"className":378867},[32024],[34,378869],{"className":378870,"style":32604},[31845],[34,378872,378874,378877,378880,378883,378886],{"className":378873},[31676],[34,378875],{"className":378876,"style":32527},[31680],[34,378878,32936],{"className":378879},[31685,31686],[34,378881],{"className":378882,"style":31846},[31845],[34,378884,202],{"className":378885},[31850],[34,378887],{"className":378888,"style":31846},[31845],[34,378890,378892,378895],{"className":378891},[31676],[34,378893],{"className":378894,"style":32527},[31680],[34,378896,33244],{"className":378897},[31685,31686],"（也就是刚才找到的单位元 ",[34,378900,378902,378915],{"className":378901},[31642],[34,378903,378905],{"className":378904},[31646],[31648,378906,378907],{"xmlns":31650},[31652,378908,378909,378913],{},[31655,378910,378911],{},[31749,378912,640],{},[31665,378914,640],{"encoding":31667},[34,378916,378918],{"className":378917,"ariaHidden":758},[31672],[34,378919,378921,378924],{"className":378920},[31676],[34,378922],{"className":378923,"style":32617},[31680],[34,378925,640],{"className":378926},[31685],[105,378928,378929,379005,379216,379251],{},[108,378930,378931,378932,347],{},"翻译成原来的运算：",[34,378933,378935,378957],{"className":378934},[31642],[34,378936,378938],{"className":378937},[31646],[31648,378939,378940],{"xmlns":31650},[31652,378941,378942,378954],{},[31655,378943,378944,378946,378948,378950,378952],{},[31658,378945,111],{},[31662,378947,71191],{},[31658,378949,32936],{},[31662,378951,202],{},[31749,378953,640],{},[31665,378955,378956],{"encoding":31667},"a \\cdot x = 1",[34,378958,378960,378978,378996],{"className":378959,"ariaHidden":758},[31672],[34,378961,378963,378966,378969,378972,378975],{"className":378962},[31676],[34,378964],{"className":378965,"style":224869},[31680],[34,378967,111],{"className":378968},[31685,31686],[34,378970],{"className":378971,"style":32604},[31845],[34,378973,71191],{"className":378974},[32024],[34,378976],{"className":378977,"style":32604},[31845],[34,378979,378981,378984,378987,378990,378993],{"className":378980},[31676],[34,378982],{"className":378983,"style":32527},[31680],[34,378985,32936],{"className":378986},[31685,31686],[34,378988],{"className":378989,"style":31846},[31845],[34,378991,202],{"className":378992},[31850],[34,378994],{"className":378995,"style":31846},[31845],[34,378997,378999,379002],{"className":378998},[31676],[34,379000],{"className":379001,"style":32617},[31680],[34,379003,640],{"className":379004},[31685],[108,379006,379007,379010,379011,379039,379040,330585,379091,379164,379165,347],{},[2627,379008,379009],{},"出问题了！"," 在整数集合 ",[34,379012,379014,379027],{"className":379013},[31642],[34,379015,379017],{"className":379016},[31646],[31648,379018,379019],{"xmlns":31650},[31652,379020,379021,379025],{},[31655,379022,379023],{},[31658,379024,97300],{"mathvariant":318416},[31665,379026,320917],{"encoding":31667},[34,379028,379030],{"className":379029,"ariaHidden":758},[31672],[34,379031,379033,379036],{"className":379032},[31676],[34,379034],{"className":379035,"style":320927},[31680],[34,379037,97300],{"className":379038},[31685,318441]," 里，如果我们取 ",[34,379041,379043,379061],{"className":379042},[31642],[34,379044,379046],{"className":379045},[31646],[31648,379047,379048],{"xmlns":31650},[31652,379049,379050,379058],{},[31655,379051,379052,379054,379056],{},[31658,379053,111],{},[31662,379055,202],{},[31749,379057,768],{},[31665,379059,379060],{"encoding":31667},"a=2",[34,379062,379064,379082],{"className":379063,"ariaHidden":758},[31672],[34,379065,379067,379070,379073,379076,379079],{"className":379066},[31676],[34,379068],{"className":379069,"style":32527},[31680],[34,379071,111],{"className":379072},[31685,31686],[34,379074],{"className":379075,"style":31846},[31845],[34,379077,202],{"className":379078},[31850],[34,379080],{"className":379081,"style":31846},[31845],[34,379083,379085,379088],{"className":379084},[31676],[34,379086],{"className":379087,"style":32617},[31680],[34,379089,768],{"className":379090},[31685],[34,379092,379094,379116],{"className":379093},[31642],[34,379095,379097],{"className":379096},[31646],[31648,379098,379099],{"xmlns":31650},[31652,379100,379101,379113],{},[31655,379102,379103,379105,379107,379109,379111],{},[31749,379104,768],{},[31662,379106,71191],{},[31658,379108,32936],{},[31662,379110,202],{},[31749,379112,640],{},[31665,379114,379115],{"encoding":31667},"2 \\cdot x = 1",[34,379117,379119,379137,379155],{"className":379118,"ariaHidden":758},[31672],[34,379120,379122,379125,379128,379131,379134],{"className":379121},[31676],[34,379123],{"className":379124,"style":32617},[31680],[34,379126,768],{"className":379127},[31685],[34,379129],{"className":379130,"style":32604},[31845],[34,379132,71191],{"className":379133},[32024],[34,379135],{"className":379136,"style":32604},[31845],[34,379138,379140,379143,379146,379149,379152],{"className":379139},[31676],[34,379141],{"className":379142,"style":32527},[31680],[34,379144,32936],{"className":379145},[31685,31686],[34,379147],{"className":379148,"style":31846},[31845],[34,379150,202],{"className":379151},[31850],[34,379153],{"className":379154,"style":31846},[31845],[34,379156,379158,379161],{"className":379157},[31676],[34,379159],{"className":379160,"style":32617},[31680],[34,379162,640],{"className":379163},[31685]," 的解是 ",[34,379166,379168,379186],{"className":379167},[31642],[34,379169,379171],{"className":379170},[31646],[31648,379172,379173],{"xmlns":31650},[31652,379174,379175,379183],{},[31655,379176,379177,379179,379181],{},[31658,379178,32936],{},[31662,379180,202],{},[31749,379182,65507],{},[31665,379184,379185],{"encoding":31667},"x = 0.5",[34,379187,379189,379207],{"className":379188,"ariaHidden":758},[31672],[34,379190,379192,379195,379198,379201,379204],{"className":379191},[31676],[34,379193],{"className":379194,"style":32527},[31680],[34,379196,32936],{"className":379197},[31685,31686],[34,379199],{"className":379200,"style":31846},[31845],[34,379202,202],{"className":379203},[31850],[34,379205],{"className":379206,"style":31846},[31845],[34,379208,379210,379213],{"className":379209},[31676],[34,379211],{"className":379212,"style":32617},[31680],[34,379214,65507],{"className":379215},[31685],[108,379217,379218,379219,47649,379247,379250],{},"但是 ",[34,379220,379222,379235],{"className":379221},[31642],[34,379223,379225],{"className":379224},[31646],[31648,379226,379227],{"xmlns":31650},[31652,379228,379229,379233],{},[31655,379230,379231],{},[31749,379232,65507],{},[31665,379234,65507],{"encoding":31667},[34,379236,379238],{"className":379237,"ariaHidden":758},[31672],[34,379239,379241,379244],{"className":379240},[31676],[34,379242],{"className":379243,"style":32617},[31680],[34,379245,65507],{"className":379246},[31685],[2627,379248,379249],{},"不是整数","！它跑出这个集合了。",[108,379252,379253,379254,379282,379283,258,379311,379344],{},"这意味着，在 ",[34,379255,379257,379270],{"className":379256},[31642],[34,379258,379260],{"className":379259},[31646],[31648,379261,379262],{"xmlns":31650},[31652,379263,379264,379268],{},[31655,379265,379266],{},[31658,379267,97300],{"mathvariant":318416},[31665,379269,320917],{"encoding":31667},[34,379271,379273],{"className":379272,"ariaHidden":758},[31672],[34,379274,379276,379279],{"className":379275},[31676],[34,379277],{"className":379278,"style":320927},[31680],[34,379280,97300],{"className":379281},[31685,318441]," 里，除了 ",[34,379284,379286,379299],{"className":379285},[31642],[34,379287,379289],{"className":379288},[31646],[31648,379290,379291],{"xmlns":31650},[31652,379292,379293,379297],{},[31655,379294,379295],{},[31749,379296,640],{},[31665,379298,640],{"encoding":31667},[34,379300,379302],{"className":379301,"ariaHidden":758},[31672],[34,379303,379305,379308],{"className":379304},[31676],[34,379306],{"className":379307,"style":32617},[31680],[34,379309,640],{"className":379310},[31685],[34,379312,379314,379329],{"className":379313},[31642],[34,379315,379317],{"className":379316},[31646],[31648,379318,379319],{"xmlns":31650},[31652,379320,379321,379327],{},[31655,379322,379323,379325],{},[31662,379324,32226],{},[31749,379326,640],{},[31665,379328,259653],{"encoding":31667},[34,379330,379332],{"className":379331,"ariaHidden":758},[31672],[34,379333,379335,379338,379341],{"className":379334},[31676],[34,379336],{"className":379337,"style":35942},[31680],[34,379339,32226],{"className":379340},[31685],[34,379342,640],{"className":379343},[31685]," 以外，其他元素在乘法下都没有逆元。",[15,379346,379347,379350,379351,379379,379380,379383],{},[2627,379348,379349],{},"3. 结论","\n因为第一运算 ",[34,379352,379354,379367],{"className":379353},[31642],[34,379355,379357],{"className":379356},[31646],[31648,379358,379359],{"xmlns":31650},[31652,379360,379361,379365],{},[31655,379362,379363],{},[31662,379364,378006],{},[31665,379366,378334],{"encoding":31667},[34,379368,379370],{"className":379369,"ariaHidden":758},[31672],[34,379371,379373,379376],{"className":379372},[31676],[34,379374],{"className":379375,"style":32597},[31680],[34,379377,378006],{"className":379378},[31685],"（原乘法）甚至连",[2627,379381,379382],{},"群","都构不成（因为缺逆元），所以它绝对构不成环。",[15,379385,379386,379388],{},[2627,379387,118575],{},"：不构成环。",[32188,379390],{},[253,379392,379393],{"id":379393},"题目2",[15,379395,379396,379397,379471],{},"在集合 ",[34,379398,379400,379422],{"className":379399},[31642],[34,379401,379403],{"className":379402},[31646],[31648,379404,379405],{"xmlns":31650},[31652,379406,379407,379419],{},[31655,379408,379409,379411,379413,379415,379417],{},[31658,379410,94675],{},[31662,379412,202],{},[31658,379414,97300],{"mathvariant":318416},[31662,379416,42071],{},[31658,379418,97300],{"mathvariant":318416},[31665,379420,379421],{"encoding":31667},"S = \\mathbb{Z} \\times \\mathbb{Z}",[34,379423,379425,379443,379462],{"className":379424,"ariaHidden":758},[31672],[34,379426,379428,379431,379434,379437,379440],{"className":379427},[31676],[34,379429],{"className":379430,"style":33218},[31680],[34,379432,94675],{"className":379433,"style":106455},[31685,31686],[34,379435],{"className":379436,"style":31846},[31845],[34,379438,202],{"className":379439},[31850],[34,379441],{"className":379442,"style":31846},[31845],[34,379444,379446,379450,379453,379456,379459],{"className":379445},[31676],[34,379447],{"className":379448,"style":379449},[31680],"height:0.7722em;vertical-align:-0.0833em;",[34,379451,97300],{"className":379452},[31685,318441],[34,379454],{"className":379455,"style":32604},[31845],[34,379457,42071],{"className":379458},[32024],[34,379460],{"className":379461,"style":32604},[31845],[34,379463,379465,379468],{"className":379464},[31676],[34,379466],{"className":379467,"style":320927},[31680],[34,379469,97300],{"className":379470},[31685,318441]," 上定义",[34,379473,379475],{"className":379474},[126385],[34,379476,379478,379534],{"className":379477},[31642],[34,379479,379481],{"className":379480},[31646],[31648,379482,379483],{"xmlns":31650,"display":126394},[31652,379484,379485,379531],{},[31655,379486,379487,379489,379491,379493,379495,379497,379499,379501,379503,379505,379507,379509,379511,379513,379515,379517,379519,379521,379523,379525,379527,379529],{},[31662,379488,456],{"stretchy":702},[31658,379490,111],{},[31662,379492,31960],{"separator":758},[31658,379494,1697],{},[31662,379496,2557],{"stretchy":702},[31662,379498,223],{},[31662,379500,456],{"stretchy":702},[31658,379502,31965],{},[31662,379504,31960],{"separator":758},[31658,379506,46900],{},[31662,379508,2557],{"stretchy":702},[31662,379510,202],{},[31662,379512,456],{"stretchy":702},[31658,379514,111],{},[31662,379516,223],{},[31658,379518,31965],{},[31662,379520,31960],{"separator":758},[31658,379522,1697],{},[31662,379524,223],{},[31658,379526,46900],{},[31662,379528,2557],{"stretchy":702},[31662,379530,31960],{"separator":758},[31665,379532,379533],{"encoding":31667},"(a,b) + (c,d) = (a + c, b + d),",[34,379535,379537,379570,379603,379624,379651],{"className":379536,"ariaHidden":758},[31672],[34,379538,379540,379543,379546,379549,379552,379555,379558,379561,379564,379567],{"className":379539},[31676],[34,379541],{"className":379542,"style":31774},[31680],[34,379544,456],{"className":379545},[31782],[34,379547,111],{"className":379548},[31685,31686],[34,379550,31960],{"className":379551},[32119],[34,379553],{"className":379554,"style":32123},[31845],[34,379556,1697],{"className":379557},[31685,31686],[34,379559,2557],{"className":379560},[31690],[34,379562],{"className":379563,"style":32604},[31845],[34,379565,223],{"className":379566},[32024],[34,379568],{"className":379569,"style":32604},[31845],[34,379571,379573,379576,379579,379582,379585,379588,379591,379594,379597,379600],{"className":379572},[31676],[34,379574],{"className":379575,"style":31774},[31680],[34,379577,456],{"className":379578},[31782],[34,379580,31965],{"className":379581},[31685,31686],[34,379583,31960],{"className":379584},[32119],[34,379586],{"className":379587,"style":32123},[31845],[34,379589,46900],{"className":379590},[31685,31686],[34,379592,2557],{"className":379593},[31690],[34,379595],{"className":379596,"style":31846},[31845],[34,379598,202],{"className":379599},[31850],[34,379601],{"className":379602,"style":31846},[31845],[34,379604,379606,379609,379612,379615,379618,379621],{"className":379605},[31676],[34,379607],{"className":379608,"style":31774},[31680],[34,379610,456],{"className":379611},[31782],[34,379613,111],{"className":379614},[31685,31686],[34,379616],{"className":379617,"style":32604},[31845],[34,379619,223],{"className":379620},[32024],[34,379622],{"className":379623,"style":32604},[31845],[34,379625,379627,379630,379633,379636,379639,379642,379645,379648],{"className":379626},[31676],[34,379628],{"className":379629,"style":43217},[31680],[34,379631,31965],{"className":379632},[31685,31686],[34,379634,31960],{"className":379635},[32119],[34,379637],{"className":379638,"style":32123},[31845],[34,379640,1697],{"className":379641},[31685,31686],[34,379643],{"className":379644,"style":32604},[31845],[34,379646,223],{"className":379647},[32024],[34,379649],{"className":379650,"style":32604},[31845],[34,379652,379654,379657,379660,379663],{"className":379653},[31676],[34,379655],{"className":379656,"style":31774},[31680],[34,379658,46900],{"className":379659},[31685,31686],[34,379661,2557],{"className":379662},[31690],[34,379664,31960],{"className":379665},[32119],[34,379667,379669],{"className":379668},[126385],[34,379670,379672,379736],{"className":379671},[31642],[34,379673,379675],{"className":379674},[31646],[31648,379676,379677],{"xmlns":31650,"display":126394},[31652,379678,379679,379733],{},[31655,379680,379681,379683,379685,379687,379689,379691,379693,379695,379697,379699,379701,379703,379705,379707,379709,379711,379713,379715,379717,379719,379721,379723,379725,379727,379729,379731],{},[31662,379682,456],{"stretchy":702},[31658,379684,111],{},[31662,379686,31960],{"separator":758},[31658,379688,1697],{},[31662,379690,2557],{"stretchy":702},[31662,379692,71191],{},[31662,379694,456],{"stretchy":702},[31658,379696,31965],{},[31662,379698,31960],{"separator":758},[31658,379700,46900],{},[31662,379702,2557],{"stretchy":702},[31662,379704,202],{},[31662,379706,456],{"stretchy":702},[31658,379708,111],{},[31658,379710,31965],{},[31662,379712,32226],{},[31658,379714,1697],{},[31658,379716,46900],{},[31662,379718,31960],{"separator":758},[31658,379720,111],{},[31658,379722,46900],{},[31662,379724,223],{},[31658,379726,1697],{},[31658,379728,31965],{},[31662,379730,2557],{"stretchy":702},[31662,379732,31960],{"separator":758},[31665,379734,379735],{"encoding":31667},"(a,b) \\cdot (c,d) = (ac - bd, ad + bc),",[34,379737,379739,379772,379805,379829,379862],{"className":379738,"ariaHidden":758},[31672],[34,379740,379742,379745,379748,379751,379754,379757,379760,379763,379766,379769],{"className":379741},[31676],[34,379743],{"className":379744,"style":31774},[31680],[34,379746,456],{"className":379747},[31782],[34,379749,111],{"className":379750},[31685,31686],[34,379752,31960],{"className":379753},[32119],[34,379755],{"className":379756,"style":32123},[31845],[34,379758,1697],{"className":379759},[31685,31686],[34,379761,2557],{"className":379762},[31690],[34,379764],{"className":379765,"style":32604},[31845],[34,379767,71191],{"className":379768},[32024],[34,379770],{"className":379771,"style":32604},[31845],[34,379773,379775,379778,379781,379784,379787,379790,379793,379796,379799,379802],{"className":379774},[31676],[34,379776],{"className":379777,"style":31774},[31680],[34,379779,456],{"className":379780},[31782],[34,379782,31965],{"className":379783},[31685,31686],[34,379785,31960],{"className":379786},[32119],[34,379788],{"className":379789,"style":32123},[31845],[34,379791,46900],{"className":379792},[31685,31686],[34,379794,2557],{"className":379795},[31690],[34,379797],{"className":379798,"style":31846},[31845],[34,379800,202],{"className":379801},[31850],[34,379803],{"className":379804,"style":31846},[31845],[34,379806,379808,379811,379814,379817,379820,379823,379826],{"className":379807},[31676],[34,379809],{"className":379810,"style":31774},[31680],[34,379812,456],{"className":379813},[31782],[34,379815,111],{"className":379816},[31685,31686],[34,379818,31965],{"className":379819},[31685,31686],[34,379821],{"className":379822,"style":32604},[31845],[34,379824,32226],{"className":379825},[32024],[34,379827],{"className":379828,"style":32604},[31845],[34,379830,379832,379835,379838,379841,379844,379847,379850,379853,379856,379859],{"className":379831},[31676],[34,379833],{"className":379834,"style":43217},[31680],[34,379836,1697],{"className":379837},[31685,31686],[34,379839,46900],{"className":379840},[31685,31686],[34,379842,31960],{"className":379843},[32119],[34,379845],{"className":379846,"style":32123},[31845],[34,379848,111],{"className":379849},[31685,31686],[34,379851,46900],{"className":379852},[31685,31686],[34,379854],{"className":379855,"style":32604},[31845],[34,379857,223],{"className":379858},[32024],[34,379860],{"className":379861,"style":32604},[31845],[34,379863,379865,379868,379871,379874,379877],{"className":379864},[31676],[34,379866],{"className":379867,"style":31774},[31680],[34,379869,1697],{"className":379870},[31685,31686],[34,379872,31965],{"className":379873},[31685,31686],[34,379875,2557],{"className":379876},[31690],[34,379878,31960],{"className":379879},[32119],[15,379881,379882,379883,379948],{},"证明 ",[34,379884,379886,379912],{"className":379885},[31642],[34,379887,379889],{"className":379888},[31646],[31648,379890,379891],{"xmlns":31650},[31652,379892,379893,379909],{},[31655,379894,379895,379897,379899,379901,379903,379905,379907],{},[31662,379896,456],{"stretchy":702},[31658,379898,94675],{},[31662,379900,199477],{"separator":758},[31662,379902,223],{},[31662,379904,31960],{"separator":758},[31662,379906,71191],{},[31662,379908,2557],{"stretchy":702},[31665,379910,379911],{"encoding":31667},"(S; +, \\cdot)",[34,379913,379915],{"className":379914,"ariaHidden":758},[31672],[34,379916,379918,379921,379924,379927,379930,379933,379936,379939,379942,379945],{"className":379917},[31676],[34,379919],{"className":379920,"style":31774},[31680],[34,379922,456],{"className":379923},[31782],[34,379925,94675],{"className":379926,"style":106455},[31685,31686],[34,379928,199477],{"className":379929},[32119],[34,379931],{"className":379932,"style":32123},[31845],[34,379934,223],{"className":379935},[31685],[34,379937,31960],{"className":379938},[32119],[34,379940],{"className":379941,"style":32123},[31845],[34,379943,71191],{"className":379944},[31685],[34,379946,2557],{"className":379947},[31690]," 是交换幺环。",[32490,379950,297792],{"id":379951},"详细解析-1",[15,379953,379954,379957,379958,258,380019,380080,380081,380084,380085,380476,380477,380549,380550,380553],{},[2627,379955,379956],{},"1. 基础知识铺垫：这是什么？","\n看着这个奇怪的乘法定义：",[34,379959,379961,379983],{"className":379960},[31642],[34,379962,379964],{"className":379963},[31646],[31648,379965,379966],{"xmlns":31650},[31652,379967,379968,379980],{},[31655,379969,379970,379972,379974,379976,379978],{},[31658,379971,111],{},[31658,379973,31965],{},[31662,379975,32226],{},[31658,379977,1697],{},[31658,379979,46900],{},[31665,379981,379982],{"encoding":31667},"ac - bd",[34,379984,379986,380007],{"className":379985,"ariaHidden":758},[31672],[34,379987,379989,379992,379995,379998,380001,380004],{"className":379988},[31676],[34,379990],{"className":379991,"style":32597},[31680],[34,379993,111],{"className":379994},[31685,31686],[34,379996,31965],{"className":379997},[31685,31686],[34,379999],{"className":380000,"style":32604},[31845],[34,380002,32226],{"className":380003},[32024],[34,380005],{"className":380006,"style":32604},[31845],[34,380008,380010,380013,380016],{"className":380009},[31676],[34,380011],{"className":380012,"style":31681},[31680],[34,380014,1697],{"className":380015},[31685,31686],[34,380017,46900],{"className":380018},[31685,31686],[34,380020,380022,380044],{"className":380021},[31642],[34,380023,380025],{"className":380024},[31646],[31648,380026,380027],{"xmlns":31650},[31652,380028,380029,380041],{},[31655,380030,380031,380033,380035,380037,380039],{},[31658,380032,111],{},[31658,380034,46900],{},[31662,380036,223],{},[31658,380038,1697],{},[31658,380040,31965],{},[31665,380042,380043],{"encoding":31667},"ad + bc",[34,380045,380047,380068],{"className":380046,"ariaHidden":758},[31672],[34,380048,380050,380053,380056,380059,380062,380065],{"className":380049},[31676],[34,380051],{"className":380052,"style":34299},[31680],[34,380054,111],{"className":380055},[31685,31686],[34,380057,46900],{"className":380058},[31685,31686],[34,380060],{"className":380061,"style":32604},[31845],[34,380063,223],{"className":380064},[32024],[34,380066],{"className":380067,"style":32604},[31845],[34,380069,380071,380074,380077],{"className":380070},[31676],[34,380072],{"className":380073,"style":31681},[31680],[34,380075,1697],{"className":380076},[31685,31686],[34,380078,31965],{"className":380079},[31685,31686],"。你有没有觉得眼熟？\n如果你学过",[2627,380082,380083],{},"复数","，回忆一下复数乘法：\n",[34,380086,380088,380192],{"className":380087},[31642],[34,380089,380091],{"className":380090},[31646],[31648,380092,380093],{"xmlns":31650},[31652,380094,380095,380189],{},[31655,380096,380097,380099,380101,380103,380105,380107,380109,380111,380113,380115,380117,380119,380121,380123,380125,380127,380129,380131,380133,380135,380137,380139,380141,380143,380145,380147,380149,380155,380157,380159,380161,380163,380165,380167,380169,380171,380173,380175,380177,380179,380181,380183,380185,380187],{},[31662,380098,456],{"stretchy":702},[31658,380100,111],{},[31662,380102,223],{},[31658,380104,1697],{},[31658,380106,31955],{},[31662,380108,2557],{"stretchy":702},[31662,380110,456],{"stretchy":702},[31658,380112,31965],{},[31662,380114,223],{},[31658,380116,46900],{},[31658,380118,31955],{},[31662,380120,2557],{"stretchy":702},[31662,380122,202],{},[31658,380124,111],{},[31658,380126,31965],{},[31662,380128,223],{},[31658,380130,111],{},[31658,380132,46900],{},[31658,380134,31955],{},[31662,380136,223],{},[31658,380138,1697],{},[31658,380140,31965],{},[31658,380142,31955],{},[31662,380144,223],{},[31658,380146,1697],{},[31658,380148,46900],{},[37624,380150,380151,380153],{},[31658,380152,31955],{},[31749,380154,768],{},[31662,380156,202],{},[31662,380158,456],{"stretchy":702},[31658,380160,111],{},[31658,380162,31965],{},[31662,380164,32226],{},[31658,380166,1697],{},[31658,380168,46900],{},[31662,380170,2557],{"stretchy":702},[31662,380172,223],{},[31662,380174,456],{"stretchy":702},[31658,380176,111],{},[31658,380178,46900],{},[31662,380180,223],{},[31658,380182,1697],{},[31658,380184,31965],{},[31662,380186,2557],{"stretchy":702},[31658,380188,31955],{},[31665,380190,380191],{"encoding":31667},"(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i",[34,380193,380195,380216,380243,380267,380288,380312,380336,380386,380410,380434,380458],{"className":380194,"ariaHidden":758},[31672],[34,380196,380198,380201,380204,380207,380210,380213],{"className":380197},[31676],[34,380199],{"className":380200,"style":31774},[31680],[34,380202,456],{"className":380203},[31782],[34,380205,111],{"className":380206},[31685,31686],[34,380208],{"className":380209,"style":32604},[31845],[34,380211,223],{"className":380212},[32024],[34,380214],{"className":380215,"style":32604},[31845],[34,380217,380219,380222,380225,380228,380231,380234,380237,380240],{"className":380218},[31676],[34,380220],{"className":380221,"style":31774},[31680],[34,380223,177253],{"className":380224},[31685,31686],[34,380226,2557],{"className":380227},[31690],[34,380229,456],{"className":380230},[31782],[34,380232,31965],{"className":380233},[31685,31686],[34,380235],{"className":380236,"style":32604},[31845],[34,380238,223],{"className":380239},[32024],[34,380241],{"className":380242,"style":32604},[31845],[34,380244,380246,380249,380252,380255,380258,380261,380264],{"className":380245},[31676],[34,380247],{"className":380248,"style":31774},[31680],[34,380250,46900],{"className":380251},[31685,31686],[34,380253,31955],{"className":380254},[31685,31686],[34,380256,2557],{"className":380257},[31690],[34,380259],{"className":380260,"style":31846},[31845],[34,380262,202],{"className":380263},[31850],[34,380265],{"className":380266,"style":31846},[31845],[34,380268,380270,380273,380276,380279,380282,380285],{"className":380269},[31676],[34,380271],{"className":380272,"style":32597},[31680],[34,380274,111],{"className":380275},[31685,31686],[34,380277,31965],{"className":380278},[31685,31686],[34,380280],{"className":380281,"style":32604},[31845],[34,380283,223],{"className":380284},[32024],[34,380286],{"className":380287,"style":32604},[31845],[34,380289,380291,380294,380297,380300,380303,380306,380309],{"className":380290},[31676],[34,380292],{"className":380293,"style":34299},[31680],[34,380295,111],{"className":380296},[31685,31686],[34,380298,46900],{"className":380299},[31685,31686],[34,380301,31955],{"className":380302},[31685,31686],[34,380304],{"className":380305,"style":32604},[31845],[34,380307,223],{"className":380308},[32024],[34,380310],{"className":380311,"style":32604},[31845],[34,380313,380315,380318,380321,380324,380327,380330,380333],{"className":380314},[31676],[34,380316],{"className":380317,"style":34299},[31680],[34,380319,1697],{"className":380320},[31685,31686],[34,380322,31965],{"className":380323},[31685,31686],[34,380325,31955],{"className":380326},[31685,31686],[34,380328],{"className":380329,"style":32604},[31845],[34,380331,223],{"className":380332},[32024],[34,380334],{"className":380335,"style":32604},[31845],[34,380337,380339,380342,380345,380348,380377,380380,380383],{"className":380338},[31676],[34,380340],{"className":380341,"style":37774},[31680],[34,380343,1697],{"className":380344},[31685,31686],[34,380346,46900],{"className":380347},[31685,31686],[34,380349,380351,380354],{"className":380350},[31685],[34,380352,31955],{"className":380353},[31685,31686],[34,380355,380357],{"className":380356},[31792],[34,380358,380360],{"className":380359},[31796],[34,380361,380363],{"className":380362},[31801],[34,380364,380366],{"className":380365,"style":37774},[31805],[34,380367,380368,380371],{"style":37777},[34,380369],{"className":380370,"style":31814},[31813],[34,380372,380374],{"className":380373},[31818,31819,31820,31821],[34,380375,768],{"className":380376},[31685,31821],[34,380378],{"className":380379,"style":31846},[31845],[34,380381,202],{"className":380382},[31850],[34,380384],{"className":380385,"style":31846},[31845],[34,380387,380389,380392,380395,380398,380401,380404,380407],{"className":380388},[31676],[34,380390],{"className":380391,"style":31774},[31680],[34,380393,456],{"className":380394},[31782],[34,380396,111],{"className":380397},[31685,31686],[34,380399,31965],{"className":380400},[31685,31686],[34,380402],{"className":380403,"style":32604},[31845],[34,380405,32226],{"className":380406},[32024],[34,380408],{"className":380409,"style":32604},[31845],[34,380411,380413,380416,380419,380422,380425,380428,380431],{"className":380412},[31676],[34,380414],{"className":380415,"style":31774},[31680],[34,380417,1697],{"className":380418},[31685,31686],[34,380420,46900],{"className":380421},[31685,31686],[34,380423,2557],{"className":380424},[31690],[34,380426],{"className":380427,"style":32604},[31845],[34,380429,223],{"className":380430},[32024],[34,380432],{"className":380433,"style":32604},[31845],[34,380435,380437,380440,380443,380446,380449,380452,380455],{"className":380436},[31676],[34,380438],{"className":380439,"style":31774},[31680],[34,380441,456],{"className":380442},[31782],[34,380444,111],{"className":380445},[31685,31686],[34,380447,46900],{"className":380448},[31685,31686],[34,380450],{"className":380451,"style":32604},[31845],[34,380453,223],{"className":380454},[32024],[34,380456],{"className":380457,"style":32604},[31845],[34,380459,380461,380464,380467,380470,380473],{"className":380460},[31676],[34,380462],{"className":380463,"style":31774},[31680],[34,380465,1697],{"className":380466},[31685,31686],[34,380468,31965],{"className":380469},[31685,31686],[34,380471,2557],{"className":380472},[31690],[34,380474,31955],{"className":380475},[31685,31686],"。\n这两个定义是一模一样的。\n题目中的集合 ",[34,380478,380480,380501],{"className":380479},[31642],[34,380481,380483],{"className":380482},[31646],[31648,380484,380485],{"xmlns":31650},[31652,380486,380487,380499],{},[31655,380488,380489,380491,380493,380495,380497],{},[31658,380490,94675],{},[31662,380492,202],{},[31658,380494,97300],{"mathvariant":318416},[31662,380496,42071],{},[31658,380498,97300],{"mathvariant":318416},[31665,380500,379421],{"encoding":31667},[34,380502,380504,380522,380540],{"className":380503,"ariaHidden":758},[31672],[34,380505,380507,380510,380513,380516,380519],{"className":380506},[31676],[34,380508],{"className":380509,"style":33218},[31680],[34,380511,94675],{"className":380512,"style":106455},[31685,31686],[34,380514],{"className":380515,"style":31846},[31845],[34,380517,202],{"className":380518},[31850],[34,380520],{"className":380521,"style":31846},[31845],[34,380523,380525,380528,380531,380534,380537],{"className":380524},[31676],[34,380526],{"className":380527,"style":379449},[31680],[34,380529,97300],{"className":380530},[31685,318441],[34,380532],{"className":380533,"style":32604},[31845],[34,380535,42071],{"className":380536},[32024],[34,380538],{"className":380539,"style":32604},[31845],[34,380541,380543,380546],{"className":380542},[31676],[34,380544],{"className":380545,"style":320927},[31680],[34,380547,97300],{"className":380548},[31685,318441]," 其实就是",[2627,380551,380552],{},"高斯整数环","（所有实部和虚部都是整数的复数）。",[15,380555,380556,380559],{},[2627,380557,380558],{},"2. 证明步骤","\n我们要证明它是“交换幺环”，需要验证三个大项：",[105,380561,380562,380894,382299],{},[108,380563,380564,380567],{},[2627,380565,380566],{},"Step 1: 加法构成阿贝尔群",[105,380568,380569,380572,380771,380891],{},[108,380570,380571],{},"显然，向量加法满足交换律、结合律。",[108,380573,380574,2664,380577,44689,380629,347],{},[2627,380575,380576],{},"零元",[34,380578,380580,380602],{"className":380579},[31642],[34,380581,380583],{"className":380582},[31646],[31648,380584,380585],{"xmlns":31650},[31652,380586,380587,380599],{},[31655,380588,380589,380591,380593,380595,380597],{},[31662,380590,456],{"stretchy":702},[31749,380592,459],{},[31662,380594,31960],{"separator":758},[31749,380596,459],{},[31662,380598,2557],{"stretchy":702},[31665,380600,380601],{"encoding":31667},"(0,0)",[34,380603,380605],{"className":380604,"ariaHidden":758},[31672],[34,380606,380608,380611,380614,380617,380620,380623,380626],{"className":380607},[31676],[34,380609],{"className":380610,"style":31774},[31680],[34,380612,456],{"className":380613},[31782],[34,380615,459],{"className":380616},[31685],[34,380618,31960],{"className":380619},[32119],[34,380621],{"className":380622,"style":32123},[31845],[34,380624,459],{"className":380625},[31685],[34,380627,2557],{"className":380628},[31690],[34,380630,380632,380678],{"className":380631},[31642],[34,380633,380635],{"className":380634},[31646],[31648,380636,380637],{"xmlns":31650},[31652,380638,380639,380675],{},[31655,380640,380641,380643,380645,380647,380649,380651,380653,380655,380657,380659,380661,380663,380665,380667,380669,380671,380673],{},[31662,380642,456],{"stretchy":702},[31658,380644,111],{},[31662,380646,31960],{"separator":758},[31658,380648,1697],{},[31662,380650,2557],{"stretchy":702},[31662,380652,223],{},[31662,380654,456],{"stretchy":702},[31749,380656,459],{},[31662,380658,31960],{"separator":758},[31749,380660,459],{},[31662,380662,2557],{"stretchy":702},[31662,380664,202],{},[31662,380666,456],{"stretchy":702},[31658,380668,111],{},[31662,380670,31960],{"separator":758},[31658,380672,1697],{},[31662,380674,2557],{"stretchy":702},[31665,380676,380677],{"encoding":31667},"(a,b) + (0,0) = (a,b)",[34,380679,380681,380714,380747],{"className":380680,"ariaHidden":758},[31672],[34,380682,380684,380687,380690,380693,380696,380699,380702,380705,380708,380711],{"className":380683},[31676],[34,380685],{"className":380686,"style":31774},[31680],[34,380688,456],{"className":380689},[31782],[34,380691,111],{"className":380692},[31685,31686],[34,380694,31960],{"className":380695},[32119],[34,380697],{"className":380698,"style":32123},[31845],[34,380700,1697],{"className":380701},[31685,31686],[34,380703,2557],{"className":380704},[31690],[34,380706],{"className":380707,"style":32604},[31845],[34,380709,223],{"className":380710},[32024],[34,380712],{"className":380713,"style":32604},[31845],[34,380715,380717,380720,380723,380726,380729,380732,380735,380738,380741,380744],{"className":380716},[31676],[34,380718],{"className":380719,"style":31774},[31680],[34,380721,456],{"className":380722},[31782],[34,380724,459],{"className":380725},[31685],[34,380727,31960],{"className":380728},[32119],[34,380730],{"className":380731,"style":32123},[31845],[34,380733,459],{"className":380734},[31685],[34,380736,2557],{"className":380737},[31690],[34,380739],{"className":380740,"style":31846},[31845],[34,380742,202],{"className":380743},[31850],[34,380745],{"className":380746,"style":31846},[31845],[34,380748,380750,380753,380756,380759,380762,380765,380768],{"className":380749},[31676],[34,380751],{"className":380752,"style":31774},[31680],[34,380754,456],{"className":380755},[31782],[34,380757,111],{"className":380758},[31685,31686],[34,380760,31960],{"className":380761},[32119],[34,380763],{"className":380764,"style":32123},[31845],[34,380766,1697],{"className":380767},[31685,31686],[34,380769,2557],{"className":380770},[31690],[108,380772,380773,2664,380776,380828,380829,347],{},[2627,380774,380775],{},"负元",[34,380777,380779,380801],{"className":380778},[31642],[34,380780,380782],{"className":380781},[31646],[31648,380783,380784],{"xmlns":31650},[31652,380785,380786,380798],{},[31655,380787,380788,380790,380792,380794,380796],{},[31662,380789,456],{"stretchy":702},[31658,380791,111],{},[31662,380793,31960],{"separator":758},[31658,380795,1697],{},[31662,380797,2557],{"stretchy":702},[31665,380799,380800],{"encoding":31667},"(a,b)",[34,380802,380804],{"className":380803,"ariaHidden":758},[31672],[34,380805,380807,380810,380813,380816,380819,380822,380825],{"className":380806},[31676],[34,380808],{"className":380809,"style":31774},[31680],[34,380811,456],{"className":380812},[31782],[34,380814,111],{"className":380815},[31685,31686],[34,380817,31960],{"className":380818},[32119],[34,380820],{"className":380821,"style":32123},[31845],[34,380823,1697],{"className":380824},[31685,31686],[34,380826,2557],{"className":380827},[31690]," 的负元是 ",[34,380830,380832,380858],{"className":380831},[31642],[34,380833,380835],{"className":380834},[31646],[31648,380836,380837],{"xmlns":31650},[31652,380838,380839,380855],{},[31655,380840,380841,380843,380845,380847,380849,380851,380853],{},[31662,380842,456],{"stretchy":702},[31662,380844,32226],{},[31658,380846,111],{},[31662,380848,31960],{"separator":758},[31662,380850,32226],{},[31658,380852,1697],{},[31662,380854,2557],{"stretchy":702},[31665,380856,380857],{"encoding":31667},"(-a, -b)",[34,380859,380861],{"className":380860,"ariaHidden":758},[31672],[34,380862,380864,380867,380870,380873,380876,380879,380882,380885,380888],{"className":380863},[31676],[34,380865],{"className":380866,"style":31774},[31680],[34,380868,456],{"className":380869},[31782],[34,380871,32226],{"className":380872},[31685],[34,380874,111],{"className":380875},[31685,31686],[34,380877,31960],{"className":380878},[32119],[34,380880],{"className":380881,"style":32123},[31845],[34,380883,32226],{"className":380884},[31685],[34,380886,1697],{"className":380887},[31685,31686],[34,380889,2557],{"className":380890},[31690],[108,380892,380893],{},"这步通常很显然，简单写一下即可。",[108,380895,380896,380899],{},[2627,380897,380898],{},"Step 2: 乘法构成交换幺半群",[105,380900,380901,380934,380939,381422],{},[108,380902,380903,380905,380906,337455],{},[2627,380904,378392],{},"：整数加减乘还是整数，所以结果还在 ",[34,380907,380909,380922],{"className":380908},[31642],[34,380910,380912],{"className":380911},[31646],[31648,380913,380914],{"xmlns":31650},[31652,380915,380916,380920],{},[31655,380917,380918],{},[31658,380919,94675],{},[31665,380921,94675],{"encoding":31667},[34,380923,380925],{"className":380924,"ariaHidden":758},[31672],[34,380926,380928,380931],{"className":380927},[31676],[34,380929],{"className":380930,"style":33218},[31680],[34,380932,94675],{"className":380933,"style":106455},[31685,31686],[108,380935,380936,380938],{},[2627,380937,378398],{},"：复数乘法本身满足结合律，这里也一定满足（硬算很麻烦，可以直接引用复数性质）。",[108,380940,380941,33183,380944,381150,381356,381357,381418,381419,347],{},[2627,380942,380943],{},"交换律",[34,380945,380947,381009],{"className":380946},[31642],[34,380948,380950],{"className":380949},[31646],[31648,380951,380952],{"xmlns":31650},[31652,380953,380954,381006],{},[31655,380955,380956,380958,380960,380962,380964,380966,380968,380970,380972,380974,380976,380978,380980,380982,380984,380986,380988,380990,380992,380994,380996,380998,381000,381002,381004],{},[31662,380957,456],{"stretchy":702},[31658,380959,111],{},[31662,380961,31960],{"separator":758},[31658,380963,1697],{},[31662,380965,2557],{"stretchy":702},[31662,380967,71191],{},[31662,380969,456],{"stretchy":702},[31658,380971,31965],{},[31662,380973,31960],{"separator":758},[31658,380975,46900],{},[31662,380977,2557],{"stretchy":702},[31662,380979,202],{},[31662,380981,456],{"stretchy":702},[31658,380983,111],{},[31658,380985,31965],{},[31662,380987,32226],{},[31658,380989,1697],{},[31658,380991,46900],{},[31662,380993,31960],{"separator":758},[31658,380995,111],{},[31658,380997,46900],{},[31662,380999,223],{},[31658,381001,1697],{},[31658,381003,31965],{},[31662,381005,2557],{"stretchy":702},[31665,381007,381008],{"encoding":31667},"(a,b)\\cdot(c,d) = (ac-bd, ad+bc)",[34,381010,381012,381045,381078,381102,381135],{"className":381011,"ariaHidden":758},[31672],[34,381013,381015,381018,381021,381024,381027,381030,381033,381036,381039,381042],{"className":381014},[31676],[34,381016],{"className":381017,"style":31774},[31680],[34,381019,456],{"className":381020},[31782],[34,381022,111],{"className":381023},[31685,31686],[34,381025,31960],{"className":381026},[32119],[34,381028],{"className":381029,"style":32123},[31845],[34,381031,1697],{"className":381032},[31685,31686],[34,381034,2557],{"className":381035},[31690],[34,381037],{"className":381038,"style":32604},[31845],[34,381040,71191],{"className":381041},[32024],[34,381043],{"className":381044,"style":32604},[31845],[34,381046,381048,381051,381054,381057,381060,381063,381066,381069,381072,381075],{"className":381047},[31676],[34,381049],{"className":381050,"style":31774},[31680],[34,381052,456],{"className":381053},[31782],[34,381055,31965],{"className":381056},[31685,31686],[34,381058,31960],{"className":381059},[32119],[34,381061],{"className":381062,"style":32123},[31845],[34,381064,46900],{"className":381065},[31685,31686],[34,381067,2557],{"className":381068},[31690],[34,381070],{"className":381071,"style":31846},[31845],[34,381073,202],{"className":381074},[31850],[34,381076],{"className":381077,"style":31846},[31845],[34,381079,381081,381084,381087,381090,381093,381096,381099],{"className":381080},[31676],[34,381082],{"className":381083,"style":31774},[31680],[34,381085,456],{"className":381086},[31782],[34,381088,111],{"className":381089},[31685,31686],[34,381091,31965],{"className":381092},[31685,31686],[34,381094],{"className":381095,"style":32604},[31845],[34,381097,32226],{"className":381098},[32024],[34,381100],{"className":381101,"style":32604},[31845],[34,381103,381105,381108,381111,381114,381117,381120,381123,381126,381129,381132],{"className":381104},[31676],[34,381106],{"className":381107,"style":43217},[31680],[34,381109,1697],{"className":381110},[31685,31686],[34,381112,46900],{"className":381113},[31685,31686],[34,381115,31960],{"className":381116},[32119],[34,381118],{"className":381119,"style":32123},[31845],[34,381121,111],{"className":381122},[31685,31686],[34,381124,46900],{"className":381125},[31685,31686],[34,381127],{"className":381128,"style":32604},[31845],[34,381130,223],{"className":381131},[32024],[34,381133],{"className":381134,"style":32604},[31845],[34,381136,381138,381141,381144,381147],{"className":381137},[31676],[34,381139],{"className":381140,"style":31774},[31680],[34,381142,1697],{"className":381143},[31685,31686],[34,381145,31965],{"className":381146},[31685,31686],[34,381148,2557],{"className":381149},[31690],[34,381151,381153,381215],{"className":381152},[31642],[34,381154,381156],{"className":381155},[31646],[31648,381157,381158],{"xmlns":31650},[31652,381159,381160,381212],{},[31655,381161,381162,381164,381166,381168,381170,381172,381174,381176,381178,381180,381182,381184,381186,381188,381190,381192,381194,381196,381198,381200,381202,381204,381206,381208,381210],{},[31662,381163,456],{"stretchy":702},[31658,381165,31965],{},[31662,381167,31960],{"separator":758},[31658,381169,46900],{},[31662,381171,2557],{"stretchy":702},[31662,381173,71191],{},[31662,381175,456],{"stretchy":702},[31658,381177,111],{},[31662,381179,31960],{"separator":758},[31658,381181,1697],{},[31662,381183,2557],{"stretchy":702},[31662,381185,202],{},[31662,381187,456],{"stretchy":702},[31658,381189,31965],{},[31658,381191,111],{},[31662,381193,32226],{},[31658,381195,46900],{},[31658,381197,1697],{},[31662,381199,31960],{"separator":758},[31658,381201,31965],{},[31658,381203,1697],{},[31662,381205,223],{},[31658,381207,46900],{},[31658,381209,111],{},[31662,381211,2557],{"stretchy":702},[31665,381213,381214],{"encoding":31667},"(c,d)\\cdot(a,b) = (ca-db, cb+da)",[34,381216,381218,381251,381284,381308,381341],{"className":381217,"ariaHidden":758},[31672],[34,381219,381221,381224,381227,381230,381233,381236,381239,381242,381245,381248],{"className":381220},[31676],[34,381222],{"className":381223,"style":31774},[31680],[34,381225,456],{"className":381226},[31782],[34,381228,31965],{"className":381229},[31685,31686],[34,381231,31960],{"className":381232},[32119],[34,381234],{"className":381235,"style":32123},[31845],[34,381237,46900],{"className":381238},[31685,31686],[34,381240,2557],{"className":381241},[31690],[34,381243],{"className":381244,"style":32604},[31845],[34,381246,71191],{"className":381247},[32024],[34,381249],{"className":381250,"style":32604},[31845],[34,381252,381254,381257,381260,381263,381266,381269,381272,381275,381278,381281],{"className":381253},[31676],[34,381255],{"className":381256,"style":31774},[31680],[34,381258,456],{"className":381259},[31782],[34,381261,111],{"className":381262},[31685,31686],[34,381264,31960],{"className":381265},[32119],[34,381267],{"className":381268,"style":32123},[31845],[34,381270,1697],{"className":381271},[31685,31686],[34,381273,2557],{"className":381274},[31690],[34,381276],{"className":381277,"style":31846},[31845],[34,381279,202],{"className":381280},[31850],[34,381282],{"className":381283,"style":31846},[31845],[34,381285,381287,381290,381293,381296,381299,381302,381305],{"className":381286},[31676],[34,381288],{"className":381289,"style":31774},[31680],[34,381291,456],{"className":381292},[31782],[34,381294,31965],{"className":381295},[31685,31686],[34,381297,111],{"className":381298},[31685,31686],[34,381300],{"className":381301,"style":32604},[31845],[34,381303,32226],{"className":381304},[32024],[34,381306],{"className":381307,"style":32604},[31845],[34,381309,381311,381314,381317,381320,381323,381326,381329,381332,381335,381338],{"className":381310},[31676],[34,381312],{"className":381313,"style":43217},[31680],[34,381315,46900],{"className":381316},[31685,31686],[34,381318,1697],{"className":381319},[31685,31686],[34,381321,31960],{"className":381322},[32119],[34,381324],{"className":381325,"style":32123},[31845],[34,381327,31965],{"className":381328},[31685,31686],[34,381330,1697],{"className":381331},[31685,31686],[34,381333],{"className":381334,"style":32604},[31845],[34,381336,223],{"className":381337},[32024],[34,381339],{"className":381340,"style":32604},[31845],[34,381342,381344,381347,381350,381353],{"className":381343},[31676],[34,381345],{"className":381346,"style":31774},[31680],[34,381348,46900],{"className":381349},[31685,31686],[34,381351,111],{"className":381352},[31685,31686],[34,381354,2557],{"className":381355},[31690],"\n因为整数乘法 ",[34,381358,381360,381382],{"className":381359},[31642],[34,381361,381363],{"className":381362},[31646],[31648,381364,381365],{"xmlns":31650},[31652,381366,381367,381379],{},[31655,381368,381369,381371,381373,381375,381377],{},[31658,381370,111],{},[31658,381372,31965],{},[31662,381374,202],{},[31658,381376,31965],{},[31658,381378,111],{},[31665,381380,381381],{"encoding":31667},"ac=ca",[34,381383,381385,381406],{"className":381384,"ariaHidden":758},[31672],[34,381386,381388,381391,381394,381397,381400,381403],{"className":381387},[31676],[34,381389],{"className":381390,"style":32527},[31680],[34,381392,111],{"className":381393},[31685,31686],[34,381395,31965],{"className":381396},[31685,31686],[34,381398],{"className":381399,"style":31846},[31845],[34,381401,202],{"className":381402},[31850],[34,381404],{"className":381405,"style":31846},[31845],[34,381407,381409,381412,381415],{"className":381408},[31676],[34,381410],{"className":381411,"style":32527},[31680],[34,381413,31965],{"className":381414},[31685,31686],[34,381416,111],{"className":381417},[31685,31686],"，所以两者相等。",[2627,381420,381421],{},"满足交换律",[108,381423,381424,381427,381428,315343,381562,381786,381787,317245,381815,381871,381872,381924,381925,382244,382245,347],{},[2627,381425,381426],{},"幺元（单位元）","：我们需要找一个 ",[34,381429,381431,381461],{"className":381430},[31642],[34,381432,381434],{"className":381433},[31646],[31648,381435,381436],{"xmlns":31650},[31652,381437,381438,381458],{},[31655,381439,381440,381442,381448,381450,381456],{},[31662,381441,456],{"stretchy":702},[31743,381443,381444,381446],{},[31658,381445,33244],{},[31749,381447,640],{},[31662,381449,31960],{"separator":758},[31743,381451,381452,381454],{},[31658,381453,33244],{},[31749,381455,768],{},[31662,381457,2557],{"stretchy":702},[31665,381459,381460],{"encoding":31667},"(e_1, e_2)",[34,381462,381464],{"className":381463,"ariaHidden":758},[31672],[34,381465,381467,381470,381473,381513,381516,381519,381559],{"className":381466},[31676],[34,381468],{"className":381469,"style":31774},[31680],[34,381471,456],{"className":381472},[31782],[34,381474,381476,381479],{"className":381475},[31685],[34,381477,33244],{"className":381478},[31685,31686],[34,381480,381482],{"className":381481},[31792],[34,381483,381485,381505],{"className":381484},[31796,31797],[34,381486,381488,381502],{"className":381487},[31801],[34,381489,381491],{"className":381490,"style":31806},[31805],[34,381492,381493,381496],{"style":31809},[34,381494],{"className":381495,"style":31814},[31813],[34,381497,381499],{"className":381498},[31818,31819,31820,31821],[34,381500,640],{"className":381501},[31685,31821],[34,381503,31829],{"className":381504},[31828],[34,381506,381508],{"className":381507},[31801],[34,381509,381511],{"className":381510,"style":31836},[31805],[34,381512],{},[34,381514,31960],{"className":381515},[32119],[34,381517],{"className":381518,"style":32123},[31845],[34,381520,381522,381525],{"className":381521},[31685],[34,381523,33244],{"className":381524},[31685,31686],[34,381526,381528],{"className":381527},[31792],[34,381529,381531,381551],{"className":381530},[31796,31797],[34,381532,381534,381548],{"className":381533},[31801],[34,381535,381537],{"className":381536,"style":31806},[31805],[34,381538,381539,381542],{"style":31809},[34,381540],{"className":381541,"style":31814},[31813],[34,381543,381545],{"className":381544},[31818,31819,31820,31821],[34,381546,768],{"className":381547},[31685,31821],[34,381549,31829],{"className":381550},[31828],[34,381552,381554],{"className":381553},[31801],[34,381555,381557],{"className":381556,"style":31836},[31805],[34,381558],{},[34,381560,2557],{"className":381561},[31690],[34,381563,381565,381619],{"className":381564},[31642],[34,381566,381568],{"className":381567},[31646],[31648,381569,381570],{"xmlns":31650},[31652,381571,381572,381616],{},[31655,381573,381574,381576,381578,381580,381582,381584,381586,381588,381594,381596,381602,381604,381606,381608,381610,381612,381614],{},[31662,381575,456],{"stretchy":702},[31658,381577,111],{},[31662,381579,31960],{"separator":758},[31658,381581,1697],{},[31662,381583,2557],{"stretchy":702},[31662,381585,71191],{},[31662,381587,456],{"stretchy":702},[31743,381589,381590,381592],{},[31658,381591,33244],{},[31749,381593,640],{},[31662,381595,31960],{"separator":758},[31743,381597,381598,381600],{},[31658,381599,33244],{},[31749,381601,768],{},[31662,381603,2557],{"stretchy":702},[31662,381605,202],{},[31662,381607,456],{"stretchy":702},[31658,381609,111],{},[31662,381611,31960],{"separator":758},[31658,381613,1697],{},[31662,381615,2557],{"stretchy":702},[31665,381617,381618],{"encoding":31667},"(a,b) \\cdot (e_1, e_2) = (a,b)",[34,381620,381622,381655,381762],{"className":381621,"ariaHidden":758},[31672],[34,381623,381625,381628,381631,381634,381637,381640,381643,381646,381649,381652],{"className":381624},[31676],[34,381626],{"className":381627,"style":31774},[31680],[34,381629,456],{"className":381630},[31782],[34,381632,111],{"className":381633},[31685,31686],[34,381635,31960],{"className":381636},[32119],[34,381638],{"className":381639,"style":32123},[31845],[34,381641,1697],{"className":381642},[31685,31686],[34,381644,2557],{"className":381645},[31690],[34,381647],{"className":381648,"style":32604},[31845],[34,381650,71191],{"className":381651},[32024],[34,381653],{"className":381654,"style":32604},[31845],[34,381656,381658,381661,381664,381704,381707,381710,381750,381753,381756,381759],{"className":381657},[31676],[34,381659],{"className":381660,"style":31774},[31680],[34,381662,456],{"className":381663},[31782],[34,381665,381667,381670],{"className":381666},[31685],[34,381668,33244],{"className":381669},[31685,31686],[34,381671,381673],{"className":381672},[31792],[34,381674,381676,381696],{"className":381675},[31796,31797],[34,381677,381679,381693],{"className":381678},[31801],[34,381680,381682],{"className":381681,"style":31806},[31805],[34,381683,381684,381687],{"style":31809},[34,381685],{"className":381686,"style":31814},[31813],[34,381688,381690],{"className":381689},[31818,31819,31820,31821],[34,381691,640],{"className":381692},[31685,31821],[34,381694,31829],{"className":381695},[31828],[34,381697,381699],{"className":381698},[31801],[34,381700,381702],{"className":381701,"style":31836},[31805],[34,381703],{},[34,381705,31960],{"className":381706},[32119],[34,381708],{"className":381709,"style":32123},[31845],[34,381711,381713,381716],{"className":381712},[31685],[34,381714,33244],{"className":381715},[31685,31686],[34,381717,381719],{"className":381718},[31792],[34,381720,381722,381742],{"className":381721},[31796,31797],[34,381723,381725,381739],{"className":381724},[31801],[34,381726,381728],{"className":381727,"style":31806},[31805],[34,381729,381730,381733],{"style":31809},[34,381731],{"className":381732,"style":31814},[31813],[34,381734,381736],{"className":381735},[31818,31819,31820,31821],[34,381737,768],{"className":381738},[31685,31821],[34,381740,31829],{"className":381741},[31828],[34,381743,381745],{"className":381744},[31801],[34,381746,381748],{"className":381747,"style":31836},[31805],[34,381749],{},[34,381751,2557],{"className":381752},[31690],[34,381754],{"className":381755,"style":31846},[31845],[34,381757,202],{"className":381758},[31850],[34,381760],{"className":381761,"style":31846},[31845],[34,381763,381765,381768,381771,381774,381777,381780,381783],{"className":381764},[31676],[34,381766],{"className":381767,"style":31774},[31680],[34,381769,456],{"className":381770},[31782],[34,381772,111],{"className":381773},[31685,31686],[34,381775,31960],{"className":381776},[32119],[34,381778],{"className":381779,"style":32123},[31845],[34,381781,1697],{"className":381782},[31685,31686],[34,381784,2557],{"className":381785},[31690],"。\n对应复数的 ",[34,381788,381790,381803],{"className":381789},[31642],[34,381791,381793],{"className":381792},[31646],[31648,381794,381795],{"xmlns":31650},[31652,381796,381797,381801],{},[31655,381798,381799],{},[31749,381800,640],{},[31665,381802,640],{"encoding":31667},[34,381804,381806],{"className":381805,"ariaHidden":758},[31672],[34,381807,381809,381812],{"className":381808},[31676],[34,381810],{"className":381811,"style":32617},[31680],[34,381813,640],{"className":381814},[31685],[34,381816,381818,381838],{"className":381817},[31642],[34,381819,381821],{"className":381820},[31646],[31648,381822,381823],{"xmlns":31650},[31652,381824,381825,381835],{},[31655,381826,381827,381829,381831,381833],{},[31749,381828,640],{},[31662,381830,223],{},[31749,381832,459],{},[31658,381834,31955],{},[31665,381836,381837],{"encoding":31667},"1 + 0i",[34,381839,381841,381859],{"className":381840,"ariaHidden":758},[31672],[34,381842,381844,381847,381850,381853,381856],{"className":381843},[31676],[34,381845],{"className":381846,"style":35942},[31680],[34,381848,640],{"className":381849},[31685],[34,381851],{"className":381852,"style":32604},[31845],[34,381854,223],{"className":381855},[32024],[34,381857],{"className":381858,"style":32604},[31845],[34,381860,381862,381865,381868],{"className":381861},[31676],[34,381863],{"className":381864,"style":33104},[31680],[34,381866,459],{"className":381867},[31685],[34,381869,31955],{"className":381870},[31685,31686],"。我们要猜单位元是 ",[34,381873,381875,381897],{"className":381874},[31642],[34,381876,381878],{"className":381877},[31646],[31648,381879,381880],{"xmlns":31650},[31652,381881,381882,381894],{},[31655,381883,381884,381886,381888,381890,381892],{},[31662,381885,456],{"stretchy":702},[31749,381887,640],{},[31662,381889,31960],{"separator":758},[31749,381891,459],{},[31662,381893,2557],{"stretchy":702},[31665,381895,381896],{"encoding":31667},"(1,0)",[34,381898,381900],{"className":381899,"ariaHidden":758},[31672],[34,381901,381903,381906,381909,381912,381915,381918,381921],{"className":381902},[31676],[34,381904],{"className":381905,"style":31774},[31680],[34,381907,456],{"className":381908},[31782],[34,381910,640],{"className":381911},[31685],[34,381913,31960],{"className":381914},[32119],[34,381916],{"className":381917,"style":32123},[31845],[34,381919,459],{"className":381920},[31685],[34,381922,2557],{"className":381923},[31690],"。\n验证：",[34,381926,381928,382010],{"className":381927},[31642],[34,381929,381931],{"className":381930},[31646],[31648,381932,381933],{"xmlns":31650},[31652,381934,381935,382007],{},[31655,381936,381937,381939,381941,381943,381945,381947,381949,381951,381953,381955,381957,381959,381961,381963,381965,381967,381969,381971,381973,381975,381977,381979,381981,381983,381985,381987,381989,381991,381993,381995,381997,381999,382001,382003,382005],{},[31662,381938,456],{"stretchy":702},[31658,381940,111],{},[31662,381942,31960],{"separator":758},[31658,381944,1697],{},[31662,381946,2557],{"stretchy":702},[31662,381948,71191],{},[31662,381950,456],{"stretchy":702},[31749,381952,640],{},[31662,381954,31960],{"separator":758},[31749,381956,459],{},[31662,381958,2557],{"stretchy":702},[31662,381960,202],{},[31662,381962,456],{"stretchy":702},[31658,381964,111],{},[31662,381966,71191],{},[31749,381968,640],{},[31662,381970,32226],{},[31658,381972,1697],{},[31662,381974,71191],{},[31749,381976,459],{},[31662,381978,31960],{"separator":758},[31658,381980,111],{},[31662,381982,71191],{},[31749,381984,459],{},[31662,381986,223],{},[31658,381988,1697],{},[31662,381990,71191],{},[31749,381992,640],{},[31662,381994,2557],{"stretchy":702},[31662,381996,202],{},[31662,381998,456],{"stretchy":702},[31658,382000,111],{},[31662,382002,31960],{"separator":758},[31658,382004,1697],{},[31662,382006,2557],{"stretchy":702},[31665,382008,382009],{"encoding":31667},"(a,b) \\cdot (1,0) = (a\\cdot1 - b\\cdot0, a\\cdot0 + b\\cdot1) = (a, b)",[34,382011,382013,382046,382079,382100,382118,382136,382163,382181,382199,382220],{"className":382012,"ariaHidden":758},[31672],[34,382014,382016,382019,382022,382025,382028,382031,382034,382037,382040,382043],{"className":382015},[31676],[34,382017],{"className":382018,"style":31774},[31680],[34,382020,456],{"className":382021},[31782],[34,382023,111],{"className":382024},[31685,31686],[34,382026,31960],{"className":382027},[32119],[34,382029],{"className":382030,"style":32123},[31845],[34,382032,1697],{"className":382033},[31685,31686],[34,382035,2557],{"className":382036},[31690],[34,382038],{"className":382039,"style":32604},[31845],[34,382041,71191],{"className":382042},[32024],[34,382044],{"className":382045,"style":32604},[31845],[34,382047,382049,382052,382055,382058,382061,382064,382067,382070,382073,382076],{"className":382048},[31676],[34,382050],{"className":382051,"style":31774},[31680],[34,382053,456],{"className":382054},[31782],[34,382056,640],{"className":382057},[31685],[34,382059,31960],{"className":382060},[32119],[34,382062],{"className":382063,"style":32123},[31845],[34,382065,459],{"className":382066},[31685],[34,382068,2557],{"className":382069},[31690],[34,382071],{"className":382072,"style":31846},[31845],[34,382074,202],{"className":382075},[31850],[34,382077],{"className":382078,"style":31846},[31845],[34,382080,382082,382085,382088,382091,382094,382097],{"className":382081},[31676],[34,382083],{"className":382084,"style":31774},[31680],[34,382086,456],{"className":382087},[31782],[34,382089,111],{"className":382090},[31685,31686],[34,382092],{"className":382093,"style":32604},[31845],[34,382095,71191],{"className":382096},[32024],[34,382098],{"className":382099,"style":32604},[31845],[34,382101,382103,382106,382109,382112,382115],{"className":382102},[31676],[34,382104],{"className":382105,"style":35942},[31680],[34,382107,640],{"className":382108},[31685],[34,382110],{"className":382111,"style":32604},[31845],[34,382113,32226],{"className":382114},[32024],[34,382116],{"className":382117,"style":32604},[31845],[34,382119,382121,382124,382127,382130,382133],{"className":382120},[31676],[34,382122],{"className":382123,"style":31681},[31680],[34,382125,1697],{"className":382126},[31685,31686],[34,382128],{"className":382129,"style":32604},[31845],[34,382131,71191],{"className":382132},[32024],[34,382134],{"className":382135,"style":32604},[31845],[34,382137,382139,382142,382145,382148,382151,382154,382157,382160],{"className":382138},[31676],[34,382140],{"className":382141,"style":40292},[31680],[34,382143,459],{"className":382144},[31685],[34,382146,31960],{"className":382147},[32119],[34,382149],{"className":382150,"style":32123},[31845],[34,382152,111],{"className":382153},[31685,31686],[34,382155],{"className":382156,"style":32604},[31845],[34,382158,71191],{"className":382159},[32024],[34,382161],{"className":382162,"style":32604},[31845],[34,382164,382166,382169,382172,382175,382178],{"className":382165},[31676],[34,382167],{"className":382168,"style":35942},[31680],[34,382170,459],{"className":382171},[31685],[34,382173],{"className":382174,"style":32604},[31845],[34,382176,223],{"className":382177},[32024],[34,382179],{"className":382180,"style":32604},[31845],[34,382182,382184,382187,382190,382193,382196],{"className":382183},[31676],[34,382185],{"className":382186,"style":31681},[31680],[34,382188,1697],{"className":382189},[31685,31686],[34,382191],{"className":382192,"style":32604},[31845],[34,382194,71191],{"className":382195},[32024],[34,382197],{"className":382198,"style":32604},[31845],[34,382200,382202,382205,382208,382211,382214,382217],{"className":382201},[31676],[34,382203],{"className":382204,"style":31774},[31680],[34,382206,640],{"className":382207},[31685],[34,382209,2557],{"className":382210},[31690],[34,382212],{"className":382213,"style":31846},[31845],[34,382215,202],{"className":382216},[31850],[34,382218],{"className":382219,"style":31846},[31845],[34,382221,382223,382226,382229,382232,382235,382238,382241],{"className":382222},[31676],[34,382224],{"className":382225,"style":31774},[31680],[34,382227,456],{"className":382228},[31782],[34,382230,111],{"className":382231},[31685,31686],[34,382233,31960],{"className":382234},[32119],[34,382236],{"className":382237,"style":32123},[31845],[34,382239,1697],{"className":382240},[31685,31686],[34,382242,2557],{"className":382243},[31690],"。\n所以",[2627,382246,382247,382248],{},"有单位元 ",[34,382249,382251,382272],{"className":382250},[31642],[34,382252,382254],{"className":382253},[31646],[31648,382255,382256],{"xmlns":31650},[31652,382257,382258,382270],{},[31655,382259,382260,382262,382264,382266,382268],{},[31662,382261,456],{"stretchy":702},[31749,382263,640],{},[31662,382265,31960],{"separator":758},[31749,382267,459],{},[31662,382269,2557],{"stretchy":702},[31665,382271,381896],{"encoding":31667},[34,382273,382275],{"className":382274,"ariaHidden":758},[31672],[34,382276,382278,382281,382284,382287,382290,382293,382296],{"className":382277},[31676],[34,382279],{"className":382280,"style":31774},[31680],[34,382282,456],{"className":382283},[31782],[34,382285,640],{"className":382286},[31685],[34,382288,31960],{"className":382289},[32119],[34,382291],{"className":382292,"style":32123},[31845],[34,382294,459],{"className":382295},[31685],[34,382297,2557],{"className":382298},[31690],[108,382300,382301,382304],{},[2627,382302,382303],{},"Step 3: 分配律",[105,382305,382306,382482],{},[108,382307,382308,382309,347],{},"我们需要验证 ",[34,382310,382312,382355],{"className":382311},[31642],[34,382313,382315],{"className":382314},[31646],[31648,382316,382317],{"xmlns":31650},[31652,382318,382319,382352],{},[31655,382320,382321,382323,382325,382327,382329,382331,382334,382336,382338,382340,382342,382344,382346,382348,382350],{},[31658,382322,32936],{},[31662,382324,71191],{},[31662,382326,456],{"stretchy":702},[31658,382328,67764],{},[31662,382330,223],{},[31658,382332,382333],{},"z",[31662,382335,2557],{"stretchy":702},[31662,382337,202],{},[31658,382339,32936],{},[31662,382341,71191],{},[31658,382343,67764],{},[31662,382345,223],{},[31658,382347,32936],{},[31662,382349,71191],{},[31658,382351,382333],{},[31665,382353,382354],{"encoding":31667},"x \\cdot (y + z) = x \\cdot y + x \\cdot z",[34,382356,382358,382376,382397,382419,382437,382455,382473],{"className":382357,"ariaHidden":758},[31672],[34,382359,382361,382364,382367,382370,382373],{"className":382360},[31676],[34,382362],{"className":382363,"style":224869},[31680],[34,382365,32936],{"className":382366},[31685,31686],[34,382368],{"className":382369,"style":32604},[31845],[34,382371,71191],{"className":382372},[32024],[34,382374],{"className":382375,"style":32604},[31845],[34,382377,382379,382382,382385,382388,382391,382394],{"className":382378},[31676],[34,382380],{"className":382381,"style":31774},[31680],[34,382383,456],{"className":382384},[31782],[34,382386,67764],{"className":382387,"style":32062},[31685,31686],[34,382389],{"className":382390,"style":32604},[31845],[34,382392,223],{"className":382393},[32024],[34,382395],{"className":382396,"style":32604},[31845],[34,382398,382400,382403,382407,382410,382413,382416],{"className":382399},[31676],[34,382401],{"className":382402,"style":31774},[31680],[34,382404,382333],{"className":382405,"style":382406},[31685,31686],"margin-right:0.044em;",[34,382408,2557],{"className":382409},[31690],[34,382411],{"className":382412,"style":31846},[31845],[34,382414,202],{"className":382415},[31850],[34,382417],{"className":382418,"style":31846},[31845],[34,382420,382422,382425,382428,382431,382434],{"className":382421},[31676],[34,382423],{"className":382424,"style":224869},[31680],[34,382426,32936],{"className":382427},[31685,31686],[34,382429],{"className":382430,"style":32604},[31845],[34,382432,71191],{"className":382433},[32024],[34,382435],{"className":382436,"style":32604},[31845],[34,382438,382440,382443,382446,382449,382452],{"className":382439},[31676],[34,382441],{"className":382442,"style":55949},[31680],[34,382444,67764],{"className":382445,"style":32062},[31685,31686],[34,382447],{"className":382448,"style":32604},[31845],[34,382450,223],{"className":382451},[32024],[34,382453],{"className":382454,"style":32604},[31845],[34,382456,382458,382461,382464,382467,382470],{"className":382457},[31676],[34,382459],{"className":382460,"style":224869},[31680],[34,382462,32936],{"className":382463},[31685,31686],[34,382465],{"className":382466,"style":32604},[31845],[34,382468,71191],{"className":382469},[32024],[34,382471],{"className":382472,"style":32604},[31845],[34,382474,382476,382479],{"className":382475},[31676],[34,382477],{"className":382478,"style":32527},[31680],[34,382480,382333],{"className":382481,"style":382406},[31685,31686],[108,382483,382484],{},"由于这就是复数乘法和加法，复数是满足分配律的，所以这里也一定满足。",[15,382486,382487,382490,382491,382542],{},[2627,382488,382489],{},"3. 总结","\n这个系统具备了环的所有性质，乘法可交换，且有单位元 ",[34,382492,382494,382515],{"className":382493},[31642],[34,382495,382497],{"className":382496},[31646],[31648,382498,382499],{"xmlns":31650},[31652,382500,382501,382513],{},[31655,382502,382503,382505,382507,382509,382511],{},[31662,382504,456],{"stretchy":702},[31749,382506,640],{},[31662,382508,31960],{"separator":758},[31749,382510,459],{},[31662,382512,2557],{"stretchy":702},[31665,382514,381896],{"encoding":31667},[34,382516,382518],{"className":382517,"ariaHidden":758},[31672],[34,382519,382521,382524,382527,382530,382533,382536,382539],{"className":382520},[31676],[34,382522],{"className":382523,"style":31774},[31680],[34,382525,456],{"className":382526},[31782],[34,382528,640],{"className":382529},[31685],[34,382531,31960],{"className":382532},[32119],[34,382534],{"className":382535,"style":32123},[31845],[34,382537,459],{"className":382538},[31685],[34,382540,2557],{"className":382541},[31690],"。所以它是交换幺环。",[32188,382544],{},[253,382546,382547],{"id":382547},"题目3",[15,382549,133910,382550,382578,382579,382643],{},[34,382551,382553,382566],{"className":382552},[31642],[34,382554,382556],{"className":382555},[31646],[31648,382557,382558],{"xmlns":31650},[31652,382559,382560,382564],{},[31655,382561,382562],{},[31658,382563,34386],{},[31665,382565,34386],{"encoding":31667},[34,382567,382569],{"className":382568,"ariaHidden":758},[31672],[34,382570,382572,382575],{"className":382571},[31676],[34,382573],{"className":382574,"style":33218},[31680],[34,382576,34386],{"className":382577,"style":34427},[31685,31686]," 是交换幺环。对于 ",[34,382580,382582,382604],{"className":382581},[31642],[34,382583,382585],{"className":382584},[31646],[31648,382586,382587],{"xmlns":31650},[31652,382588,382589,382601],{},[31655,382590,382591,382593,382595,382597,382599],{},[31658,382592,111],{},[31662,382594,31960],{"separator":758},[31658,382596,1697],{},[31662,382598,112296],{},[31658,382600,34386],{},[31665,382602,382603],{"encoding":31667},"a,b \\in R",[34,382605,382607,382634],{"className":382606,"ariaHidden":758},[31672],[34,382608,382610,382613,382616,382619,382622,382625,382628,382631],{"className":382609},[31676],[34,382611],{"className":382612,"style":43217},[31680],[34,382614,111],{"className":382615},[31685,31686],[34,382617,31960],{"className":382618},[32119],[34,382620],{"className":382621,"style":32123},[31845],[34,382623,1697],{"className":382624},[31685,31686],[34,382626],{"className":382627,"style":31846},[31845],[34,382629,112296],{"className":382630},[31850],[34,382632],{"className":382633,"style":31846},[31845],[34,382635,382637,382640],{"className":382636},[31676],[34,382638],{"className":382639,"style":33218},[31680],[34,382641,34386],{"className":382642,"style":34427},[31685,31686],"，定义",[34,382645,382647],{"className":382646},[126385],[34,382648,382650,382682],{"className":382649},[31642],[34,382651,382653],{"className":382652},[31646],[31648,382654,382655],{"xmlns":31650,"display":126394},[31652,382656,382657,382679],{},[31655,382658,382659,382661,382663,382665,382667,382669,382671,382673,382675,382677],{},[31658,382660,111],{},[31662,382662,378006],{},[31658,382664,1697],{},[31662,382666,202],{},[31658,382668,111],{},[31662,382670,223],{},[31658,382672,1697],{},[31662,382674,32226],{},[31749,382676,640],{},[31662,382678,31960],{"separator":758},[31665,382680,382681],{"encoding":31667},"a \\oplus b = a + b - 1,",[34,382683,382685,382703,382721,382739,382757],{"className":382684,"ariaHidden":758},[31672],[34,382686,382688,382691,382694,382697,382700],{"className":382687},[31676],[34,382689],{"className":382690,"style":32597},[31680],[34,382692,111],{"className":382693},[31685,31686],[34,382695],{"className":382696,"style":32604},[31845],[34,382698,378006],{"className":382699},[32024],[34,382701],{"className":382702,"style":32604},[31845],[34,382704,382706,382709,382712,382715,382718],{"className":382705},[31676],[34,382707],{"className":382708,"style":31681},[31680],[34,382710,1697],{"className":382711},[31685,31686],[34,382713],{"className":382714,"style":31846},[31845],[34,382716,202],{"className":382717},[31850],[34,382719],{"className":382720,"style":31846},[31845],[34,382722,382724,382727,382730,382733,382736],{"className":382723},[31676],[34,382725],{"className":382726,"style":32597},[31680],[34,382728,111],{"className":382729},[31685,31686],[34,382731],{"className":382732,"style":32604},[31845],[34,382734,223],{"className":382735},[32024],[34,382737],{"className":382738,"style":32604},[31845],[34,382740,382742,382745,382748,382751,382754],{"className":382741},[31676],[34,382743],{"className":382744,"style":34299},[31680],[34,382746,1697],{"className":382747},[31685,31686],[34,382749],{"className":382750,"style":32604},[31845],[34,382752,32226],{"className":382753},[32024],[34,382755],{"className":382756,"style":32604},[31845],[34,382758,382760,382763,382766],{"className":382759},[31676],[34,382761],{"className":382762,"style":40292},[31680],[34,382764,640],{"className":382765},[31685],[34,382767,31960],{"className":382768},[32119],[34,382770,382772],{"className":382771},[126385],[34,382773,382775,382809],{"className":382774},[31642],[34,382776,382778],{"className":382777},[31646],[31648,382779,382780],{"xmlns":31650,"display":126394},[31652,382781,382782,382806],{},[31655,382783,382784,382786,382788,382790,382792,382794,382796,382798,382800,382802,382804],{},[31658,382785,111],{},[31662,382787,378082],{},[31658,382789,1697],{},[31662,382791,202],{},[31658,382793,111],{},[31662,382795,223],{},[31658,382797,1697],{},[31662,382799,32226],{},[31658,382801,111],{},[31658,382803,1697],{},[31662,382805,31960],{"separator":758},[31665,382807,382808],{"encoding":31667},"a \\odot b = a + b - ab,",[34,382810,382812,382830,382848,382866,382884],{"className":382811,"ariaHidden":758},[31672],[34,382813,382815,382818,382821,382824,382827],{"className":382814},[31676],[34,382816],{"className":382817,"style":32597},[31680],[34,382819,111],{"className":382820},[31685,31686],[34,382822],{"className":382823,"style":32604},[31845],[34,382825,378082],{"className":382826},[32024],[34,382828],{"className":382829,"style":32604},[31845],[34,382831,382833,382836,382839,382842,382845],{"className":382832},[31676],[34,382834],{"className":382835,"style":31681},[31680],[34,382837,1697],{"className":382838},[31685,31686],[34,382840],{"className":382841,"style":31846},[31845],[34,382843,202],{"className":382844},[31850],[34,382846],{"className":382847,"style":31846},[31845],[34,382849,382851,382854,382857,382860,382863],{"className":382850},[31676],[34,382852],{"className":382853,"style":32597},[31680],[34,382855,111],{"className":382856},[31685,31686],[34,382858],{"className":382859,"style":32604},[31845],[34,382861,223],{"className":382862},[32024],[34,382864],{"className":382865,"style":32604},[31845],[34,382867,382869,382872,382875,382878,382881],{"className":382868},[31676],[34,382870],{"className":382871,"style":34299},[31680],[34,382873,1697],{"className":382874},[31685,31686],[34,382876],{"className":382877,"style":32604},[31845],[34,382879,32226],{"className":382880},[32024],[34,382882],{"className":382883,"style":32604},[31845],[34,382885,382887,382890,382893],{"className":382886},[31676],[34,382888],{"className":382889,"style":43217},[31680],[34,382891,324191],{"className":382892},[31685,31686],[34,382894,31960],{"className":382895},[32119],[15,382897,379882,382898,379948],{},[34,382899,382901,382927],{"className":382900},[31642],[34,382902,382904],{"className":382903},[31646],[31648,382905,382906],{"xmlns":31650},[31652,382907,382908,382924],{},[31655,382909,382910,382912,382914,382916,382918,382920,382922],{},[31662,382911,456],{"stretchy":702},[31658,382913,34386],{},[31662,382915,199477],{"separator":758},[31662,382917,378006],{},[31662,382919,31960],{"separator":758},[31662,382921,378082],{},[31662,382923,2557],{"stretchy":702},[31665,382925,382926],{"encoding":31667},"(R; \\oplus, \\odot)",[34,382928,382930],{"className":382929,"ariaHidden":758},[31672],[34,382931,382933,382936,382939,382942,382945,382948,382951,382954,382957,382960],{"className":382932},[31676],[34,382934],{"className":382935,"style":31774},[31680],[34,382937,456],{"className":382938},[31782],[34,382940,34386],{"className":382941,"style":34427},[31685,31686],[34,382943,199477],{"className":382944},[32119],[34,382946],{"className":382947,"style":32123},[31845],[34,382949,378006],{"className":382950},[31685],[34,382952,31960],{"className":382953},[32119],[34,382955],{"className":382956,"style":32123},[31845],[34,382958,378082],{"className":382959},[31685],[34,382961,2557],{"className":382962},[31690],[32490,382964,297792],{"id":382965},"详细解析-2",[15,382967,382968,382969,258,382997,383025],{},"这道题属于“换元法”或者“同构”类的题目。它把原来的 ",[34,382970,382972,382985],{"className":382971},[31642],[34,382973,382975],{"className":382974},[31646],[31648,382976,382977],{"xmlns":31650},[31652,382978,382979,382983],{},[31655,382980,382981],{},[31749,382982,459],{},[31665,382984,459],{"encoding":31667},[34,382986,382988],{"className":382987,"ariaHidden":758},[31672],[34,382989,382991,382994],{"className":382990},[31676],[34,382992],{"className":382993,"style":32617},[31680],[34,382995,459],{"className":382996},[31685],[34,382998,383000,383013],{"className":382999},[31642],[34,383001,383003],{"className":383002},[31646],[31648,383004,383005],{"xmlns":31650},[31652,383006,383007,383011],{},[31655,383008,383009],{},[31749,383010,640],{},[31665,383012,640],{"encoding":31667},[34,383014,383016],{"className":383015,"ariaHidden":758},[31672],[34,383017,383019,383022],{"className":383018},[31676],[34,383020],{"className":383021,"style":32617},[31680],[34,383023,640],{"className":383024},[31685]," 的位置挪动了。",[15,383027,383028],{},[2627,383029,383030,383031,383059],{},"1. 验证加法 ",[34,383032,383034,383047],{"className":383033},[31642],[34,383035,383037],{"className":383036},[31646],[31648,383038,383039],{"xmlns":31650},[31652,383040,383041,383045],{},[31655,383042,383043],{},[31662,383044,378006],{},[31665,383046,378334],{"encoding":31667},[34,383048,383050],{"className":383049,"ariaHidden":758},[31672],[34,383051,383053,383056],{"className":383052},[31676],[34,383054],{"className":383055,"style":32597},[31680],[34,383057,378006],{"className":383058},[31685]," 构成阿贝尔群",[105,383061,383062,383649,383792,384082],{},[108,383063,383064,2664,383066,39984,383357,383648],{},[2627,383065,378398],{},[34,383067,383069,383135],{"className":383068},[31642],[34,383070,383072],{"className":383071},[31646],[31648,383073,383074],{"xmlns":31650},[31652,383075,383076,383132],{},[31655,383077,383078,383080,383082,383084,383086,383088,383090,383092,383094,383096,383098,383100,383102,383104,383106,383108,383110,383112,383114,383116,383118,383120,383122,383124,383126,383128,383130],{},[31662,383079,456],{"stretchy":702},[31658,383081,111],{},[31662,383083,378006],{},[31658,383085,1697],{},[31662,383087,2557],{"stretchy":702},[31662,383089,378006],{},[31658,383091,31965],{},[31662,383093,202],{},[31662,383095,456],{"stretchy":702},[31658,383097,111],{},[31662,383099,223],{},[31658,383101,1697],{},[31662,383103,32226],{},[31749,383105,640],{},[31662,383107,2557],{"stretchy":702},[31662,383109,223],{},[31658,383111,31965],{},[31662,383113,32226],{},[31749,383115,640],{},[31662,383117,202],{},[31658,383119,111],{},[31662,383121,223],{},[31658,383123,1697],{},[31662,383125,223],{},[31658,383127,31965],{},[31662,383129,32226],{},[31749,383131,768],{},[31665,383133,383134],{"encoding":31667},"(a \\oplus b) \\oplus c = (a+b-1) + c - 1 = a+b+c-2",[34,383136,383138,383159,383180,383198,383219,383237,383258,383276,383294,383312,383330,383348],{"className":383137,"ariaHidden":758},[31672],[34,383139,383141,383144,383147,383150,383153,383156],{"className":383140},[31676],[34,383142],{"className":383143,"style":31774},[31680],[34,383145,456],{"className":383146},[31782],[34,383148,111],{"className":383149},[31685,31686],[34,383151],{"className":383152,"style":32604},[31845],[34,383154,378006],{"className":383155},[32024],[34,383157],{"className":383158,"style":32604},[31845],[34,383160,383162,383165,383168,383171,383174,383177],{"className":383161},[31676],[34,383163],{"className":383164,"style":31774},[31680],[34,383166,1697],{"className":383167},[31685,31686],[34,383169,2557],{"className":383170},[31690],[34,383172],{"className":383173,"style":32604},[31845],[34,383175,378006],{"className":383176},[32024],[34,383178],{"className":383179,"style":32604},[31845],[34,383181,383183,383186,383189,383192,383195],{"className":383182},[31676],[34,383184],{"className":383185,"style":32527},[31680],[34,383187,31965],{"className":383188},[31685,31686],[34,383190],{"className":383191,"style":31846},[31845],[34,383193,202],{"className":383194},[31850],[34,383196],{"className":383197,"style":31846},[31845],[34,383199,383201,383204,383207,383210,383213,383216],{"className":383200},[31676],[34,383202],{"className":383203,"style":31774},[31680],[34,383205,456],{"className":383206},[31782],[34,383208,111],{"className":383209},[31685,31686],[34,383211],{"className":383212,"style":32604},[31845],[34,383214,223],{"className":383215},[32024],[34,383217],{"className":383218,"style":32604},[31845],[34,383220,383222,383225,383228,383231,383234],{"className":383221},[31676],[34,383223],{"className":383224,"style":34299},[31680],[34,383226,1697],{"className":383227},[31685,31686],[34,383229],{"className":383230,"style":32604},[31845],[34,383232,32226],{"className":383233},[32024],[34,383235],{"className":383236,"style":32604},[31845],[34,383238,383240,383243,383246,383249,383252,383255],{"className":383239},[31676],[34,383241],{"className":383242,"style":31774},[31680],[34,383244,640],{"className":383245},[31685],[34,383247,2557],{"className":383248},[31690],[34,383250],{"className":383251,"style":32604},[31845],[34,383253,223],{"className":383254},[32024],[34,383256],{"className":383257,"style":32604},[31845],[34,383259,383261,383264,383267,383270,383273],{"className":383260},[31676],[34,383262],{"className":383263,"style":32597},[31680],[34,383265,31965],{"className":383266},[31685,31686],[34,383268],{"className":383269,"style":32604},[31845],[34,383271,32226],{"className":383272},[32024],[34,383274],{"className":383275,"style":32604},[31845],[34,383277,383279,383282,383285,383288,383291],{"className":383278},[31676],[34,383280],{"className":383281,"style":32617},[31680],[34,383283,640],{"className":383284},[31685],[34,383286],{"className":383287,"style":31846},[31845],[34,383289,202],{"className":383290},[31850],[34,383292],{"className":383293,"style":31846},[31845],[34,383295,383297,383300,383303,383306,383309],{"className":383296},[31676],[34,383298],{"className":383299,"style":32597},[31680],[34,383301,111],{"className":383302},[31685,31686],[34,383304],{"className":383305,"style":32604},[31845],[34,383307,223],{"className":383308},[32024],[34,383310],{"className":383311,"style":32604},[31845],[34,383313,383315,383318,383321,383324,383327],{"className":383314},[31676],[34,383316],{"className":383317,"style":34299},[31680],[34,383319,1697],{"className":383320},[31685,31686],[34,383322],{"className":383323,"style":32604},[31845],[34,383325,223],{"className":383326},[32024],[34,383328],{"className":383329,"style":32604},[31845],[34,383331,383333,383336,383339,383342,383345],{"className":383332},[31676],[34,383334],{"className":383335,"style":32597},[31680],[34,383337,31965],{"className":383338},[31685,31686],[34,383340],{"className":383341,"style":32604},[31845],[34,383343,32226],{"className":383344},[32024],[34,383346],{"className":383347,"style":32604},[31845],[34,383349,383351,383354],{"className":383350},[31676],[34,383352],{"className":383353,"style":32617},[31680],[34,383355,768],{"className":383356},[31685],[34,383358,383360,383426],{"className":383359},[31642],[34,383361,383363],{"className":383362},[31646],[31648,383364,383365],{"xmlns":31650},[31652,383366,383367,383423],{},[31655,383368,383369,383371,383373,383375,383377,383379,383381,383383,383385,383387,383389,383391,383393,383395,383397,383399,383401,383403,383405,383407,383409,383411,383413,383415,383417,383419,383421],{},[31658,383370,111],{},[31662,383372,378006],{},[31662,383374,456],{"stretchy":702},[31658,383376,1697],{},[31662,383378,378006],{},[31658,383380,31965],{},[31662,383382,2557],{"stretchy":702},[31662,383384,202],{},[31658,383386,111],{},[31662,383388,223],{},[31662,383390,456],{"stretchy":702},[31658,383392,1697],{},[31662,383394,223],{},[31658,383396,31965],{},[31662,383398,32226],{},[31749,383400,640],{},[31662,383402,2557],{"stretchy":702},[31662,383404,32226],{},[31749,383406,640],{},[31662,383408,202],{},[31658,383410,111],{},[31662,383412,223],{},[31658,383414,1697],{},[31662,383416,223],{},[31658,383418,31965],{},[31662,383420,32226],{},[31749,383422,768],{},[31665,383424,383425],{"encoding":31667},"a \\oplus (b \\oplus c) = a + (b+c-1) - 1 = a+b+c-2",[34,383427,383429,383447,383468,383489,383507,383528,383546,383567,383585,383603,383621,383639],{"className":383428,"ariaHidden":758},[31672],[34,383430,383432,383435,383438,383441,383444],{"className":383431},[31676],[34,383433],{"className":383434,"style":32597},[31680],[34,383436,111],{"className":383437},[31685,31686],[34,383439],{"className":383440,"style":32604},[31845],[34,383442,378006],{"className":383443},[32024],[34,383445],{"className":383446,"style":32604},[31845],[34,383448,383450,383453,383456,383459,383462,383465],{"className":383449},[31676],[34,383451],{"className":383452,"style":31774},[31680],[34,383454,456],{"className":383455},[31782],[34,383457,1697],{"className":383458},[31685,31686],[34,383460],{"className":383461,"style":32604},[31845],[34,383463,378006],{"className":383464},[32024],[34,383466],{"className":383467,"style":32604},[31845],[34,383469,383471,383474,383477,383480,383483,383486],{"className":383470},[31676],[34,383472],{"className":383473,"style":31774},[31680],[34,383475,31965],{"className":383476},[31685,31686],[34,383478,2557],{"className":383479},[31690],[34,383481],{"className":383482,"style":31846},[31845],[34,383484,202],{"className":383485},[31850],[34,383487],{"className":383488,"style":31846},[31845],[34,383490,383492,383495,383498,383501,383504],{"className":383491},[31676],[34,383493],{"className":383494,"style":32597},[31680],[34,383496,111],{"className":383497},[31685,31686],[34,383499],{"className":383500,"style":32604},[31845],[34,383502,223],{"className":383503},[32024],[34,383505],{"className":383506,"style":32604},[31845],[34,383508,383510,383513,383516,383519,383522,383525],{"className":383509},[31676],[34,383511],{"className":383512,"style":31774},[31680],[34,383514,456],{"className":383515},[31782],[34,383517,1697],{"className":383518},[31685,31686],[34,383520],{"className":383521,"style":32604},[31845],[34,383523,223],{"className":383524},[32024],[34,383526],{"className":383527,"style":32604},[31845],[34,383529,383531,383534,383537,383540,383543],{"className":383530},[31676],[34,383532],{"className":383533,"style":32597},[31680],[34,383535,31965],{"className":383536},[31685,31686],[34,383538],{"className":383539,"style":32604},[31845],[34,383541,32226],{"className":383542},[32024],[34,383544],{"className":383545,"style":32604},[31845],[34,383547,383549,383552,383555,383558,383561,383564],{"className":383548},[31676],[34,383550],{"className":383551,"style":31774},[31680],[34,383553,640],{"className":383554},[31685],[34,383556,2557],{"className":383557},[31690],[34,383559],{"className":383560,"style":32604},[31845],[34,383562,32226],{"className":383563},[32024],[34,383565],{"className":383566,"style":32604},[31845],[34,383568,383570,383573,383576,383579,383582],{"className":383569},[31676],[34,383571],{"className":383572,"style":32617},[31680],[34,383574,640],{"className":383575},[31685],[34,383577],{"className":383578,"style":31846},[31845],[34,383580,202],{"className":383581},[31850],[34,383583],{"className":383584,"style":31846},[31845],[34,383586,383588,383591,383594,383597,383600],{"className":383587},[31676],[34,383589],{"className":383590,"style":32597},[31680],[34,383592,111],{"className":383593},[31685,31686],[34,383595],{"className":383596,"style":32604},[31845],[34,383598,223],{"className":383599},[32024],[34,383601],{"className":383602,"style":32604},[31845],[34,383604,383606,383609,383612,383615,383618],{"className":383605},[31676],[34,383607],{"className":383608,"style":34299},[31680],[34,383610,1697],{"className":383611},[31685,31686],[34,383613],{"className":383614,"style":32604},[31845],[34,383616,223],{"className":383617},[32024],[34,383619],{"className":383620,"style":32604},[31845],[34,383622,383624,383627,383630,383633,383636],{"className":383623},[31676],[34,383625],{"className":383626,"style":32597},[31680],[34,383628,31965],{"className":383629},[31685,31686],[34,383631],{"className":383632,"style":32604},[31845],[34,383634,32226],{"className":383635},[32024],[34,383637],{"className":383638,"style":32604},[31845],[34,383640,383642,383645],{"className":383641},[31676],[34,383643],{"className":383644,"style":32617},[31680],[34,383646,768],{"className":383647},[31685],"。相等，满足。",[108,383650,383651,352814,383653,378487],{},[2627,383652,380943],{},[34,383654,383656,383690],{"className":383655},[31642],[34,383657,383659],{"className":383658},[31646],[31648,383660,383661],{"xmlns":31650},[31652,383662,383663,383687],{},[31655,383664,383665,383667,383669,383671,383673,383675,383677,383679,383681,383683,383685],{},[31658,383666,111],{},[31662,383668,223],{},[31658,383670,1697],{},[31662,383672,32226],{},[31749,383674,640],{},[31662,383676,202],{},[31658,383678,1697],{},[31662,383680,223],{},[31658,383682,111],{},[31662,383684,32226],{},[31749,383686,640],{},[31665,383688,383689],{"encoding":31667},"a+b-1 = b+a-1",[34,383691,383693,383711,383729,383747,383765,383783],{"className":383692,"ariaHidden":758},[31672],[34,383694,383696,383699,383702,383705,383708],{"className":383695},[31676],[34,383697],{"className":383698,"style":32597},[31680],[34,383700,111],{"className":383701},[31685,31686],[34,383703],{"className":383704,"style":32604},[31845],[34,383706,223],{"className":383707},[32024],[34,383709],{"className":383710,"style":32604},[31845],[34,383712,383714,383717,383720,383723,383726],{"className":383713},[31676],[34,383715],{"className":383716,"style":34299},[31680],[34,383718,1697],{"className":383719},[31685,31686],[34,383721],{"className":383722,"style":32604},[31845],[34,383724,32226],{"className":383725},[32024],[34,383727],{"className":383728,"style":32604},[31845],[34,383730,383732,383735,383738,383741,383744],{"className":383731},[31676],[34,383733],{"className":383734,"style":32617},[31680],[34,383736,640],{"className":383737},[31685],[34,383739],{"className":383740,"style":31846},[31845],[34,383742,202],{"className":383743},[31850],[34,383745],{"className":383746,"style":31846},[31845],[34,383748,383750,383753,383756,383759,383762],{"className":383749},[31676],[34,383751],{"className":383752,"style":34299},[31680],[34,383754,1697],{"className":383755},[31685,31686],[34,383757],{"className":383758,"style":32604},[31845],[34,383760,223],{"className":383761},[32024],[34,383763],{"className":383764,"style":32604},[31845],[34,383766,383768,383771,383774,383777,383780],{"className":383767},[31676],[34,383769],{"className":383770,"style":32597},[31680],[34,383772,111],{"className":383773},[31685,31686],[34,383775],{"className":383776,"style":32604},[31845],[34,383778,32226],{"className":383779},[32024],[34,383781],{"className":383782,"style":32604},[31845],[34,383784,383786,383789],{"className":383785},[31676],[34,383787],{"className":383788,"style":32617},[31680],[34,383790,640],{"className":383791},[31685],[108,383793,383794,383797,383798,383826,383827,383900,383901,384050,384051,347],{},[2627,383795,383796],{},"零元（关键！）","：设新零元为 ",[34,383799,383801,383814],{"className":383800},[31642],[34,383802,383804],{"className":383803},[31646],[31648,383805,383806],{"xmlns":31650},[31652,383807,383808,383812],{},[31655,383809,383810],{},[31658,383811,382333],{},[31665,383813,382333],{"encoding":31667},[34,383815,383817],{"className":383816,"ariaHidden":758},[31672],[34,383818,383820,383823],{"className":383819},[31676],[34,383821],{"className":383822,"style":32527},[31680],[34,383824,382333],{"className":383825,"style":382406},[31685,31686],"。\n需满足 ",[34,383828,383830,383852],{"className":383829},[31642],[34,383831,383833],{"className":383832},[31646],[31648,383834,383835],{"xmlns":31650},[31652,383836,383837,383849],{},[31655,383838,383839,383841,383843,383845,383847],{},[31658,383840,111],{},[31662,383842,378006],{},[31658,383844,382333],{},[31662,383846,202],{},[31658,383848,111],{},[31665,383850,383851],{"encoding":31667},"a \\oplus z = a",[34,383853,383855,383873,383891],{"className":383854,"ariaHidden":758},[31672],[34,383856,383858,383861,383864,383867,383870],{"className":383857},[31676],[34,383859],{"className":383860,"style":32597},[31680],[34,383862,111],{"className":383863},[31685,31686],[34,383865],{"className":383866,"style":32604},[31845],[34,383868,378006],{"className":383869},[32024],[34,383871],{"className":383872,"style":32604},[31845],[34,383874,383876,383879,383882,383885,383888],{"className":383875},[31676],[34,383877],{"className":383878,"style":32527},[31680],[34,383880,382333],{"className":383881,"style":382406},[31685,31686],[34,383883],{"className":383884,"style":31846},[31845],[34,383886,202],{"className":383887},[31850],[34,383889],{"className":383890,"style":31846},[31845],[34,383892,383894,383897],{"className":383893},[31676],[34,383895],{"className":383896,"style":32527},[31680],[34,383898,111],{"className":383899},[31685,31686],"。\n代入定义：",[34,383902,383904,383942],{"className":383903},[31642],[34,383905,383907],{"className":383906},[31646],[31648,383908,383909],{"xmlns":31650},[31652,383910,383911,383939],{},[31655,383912,383913,383915,383917,383919,383921,383923,383925,383927,383929,383931,383933,383935,383937],{},[31658,383914,111],{},[31662,383916,223],{},[31658,383918,382333],{},[31662,383920,32226],{},[31749,383922,640],{},[31662,383924,202],{},[31658,383926,111],{},[39361,383928,262622],{},[31662,383930,262625],{},[39361,383932,262622],{},[31658,383934,382333],{},[31662,383936,202],{},[31749,383938,640],{},[31665,383940,383941],{"encoding":31667},"a + z - 1 = a \\implies z = 1",[34,383943,383945,383963,383981,383999,384023,384041],{"className":383944,"ariaHidden":758},[31672],[34,383946,383948,383951,383954,383957,383960],{"className":383947},[31676],[34,383949],{"className":383950,"style":32597},[31680],[34,383952,111],{"className":383953},[31685,31686],[34,383955],{"className":383956,"style":32604},[31845],[34,383958,223],{"className":383959},[32024],[34,383961],{"className":383962,"style":32604},[31845],[34,383964,383966,383969,383972,383975,383978],{"className":383965},[31676],[34,383967],{"className":383968,"style":32597},[31680],[34,383970,382333],{"className":383971,"style":382406},[31685,31686],[34,383973],{"className":383974,"style":32604},[31845],[34,383976,32226],{"className":383977},[32024],[34,383979],{"className":383980,"style":32604},[31845],[34,383982,383984,383987,383990,383993,383996],{"className":383983},[31676],[34,383985],{"className":383986,"style":32617},[31680],[34,383988,640],{"className":383989},[31685],[34,383991],{"className":383992,"style":31846},[31845],[34,383994,202],{"className":383995},[31850],[34,383997],{"className":383998,"style":31846},[31845],[34,384000,384002,384005,384008,384011,384014,384017,384020],{"className":384001},[31676],[34,384003],{"className":384004,"style":318264},[31680],[34,384006,111],{"className":384007},[31685,31686],[34,384009],{"className":384010,"style":31846},[31845],[34,384012],{"className":384013,"style":31846},[31845],[34,384015,262625],{"className":384016},[31850],[34,384018],{"className":384019,"style":31846},[31845],[34,384021],{"className":384022,"style":31846},[31845],[34,384024,384026,384029,384032,384035,384038],{"className":384025},[31676],[34,384027],{"className":384028,"style":32527},[31680],[34,384030,382333],{"className":384031,"style":382406},[31685,31686],[34,384033],{"className":384034,"style":31846},[31845],[34,384036,202],{"className":384037},[31850],[34,384039],{"className":384040,"style":31846},[31845],[34,384042,384044,384047],{"className":384043},[31676],[34,384045],{"className":384046,"style":32617},[31680],[34,384048,640],{"className":384049},[31685],"。\n所以，",[2627,384052,384053,384054],{},"这个新环的“零”其实是原来的 ",[34,384055,384057,384070],{"className":384056},[31642],[34,384058,384060],{"className":384059},[31646],[31648,384061,384062],{"xmlns":31650},[31652,384063,384064,384068],{},[31655,384065,384066],{},[31749,384067,640],{},[31665,384069,640],{"encoding":31667},[34,384071,384073],{"className":384072,"ariaHidden":758},[31672],[34,384074,384076,384079],{"className":384075},[31676],[34,384077],{"className":384078,"style":32617},[31680],[34,384080,640],{"className":384081},[31685],[108,384083,384084,34554,384086,384114,384115,383826,384177,384283,384284,384522,384523,384551,384552,384603],{},[2627,384085,380775],{},[34,384087,384089,384102],{"className":384088},[31642],[34,384090,384092],{"className":384091},[31646],[31648,384093,384094],{"xmlns":31650},[31652,384095,384096,384100],{},[31655,384097,384098],{},[31658,384099,111],{},[31665,384101,111],{"encoding":31667},[34,384103,384105],{"className":384104,"ariaHidden":758},[31672],[34,384106,384108,384111],{"className":384107},[31676],[34,384109],{"className":384110,"style":32527},[31680],[34,384112,111],{"className":384113},[31685,31686]," 的负元为 ",[34,384116,384118,384136],{"className":384117},[31642],[34,384119,384121],{"className":384120},[31646],[31648,384122,384123],{"xmlns":31650},[31652,384124,384125,384133],{},[31655,384126,384127],{},[37624,384128,384129,384131],{},[31658,384130,111],{},[31662,384132,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,384134,384135],{"encoding":31667},"a'",[34,384137,384139],{"className":384138,"ariaHidden":758},[31672],[34,384140,384142,384145],{"className":384141},[31676],[34,384143],{"className":384144,"style":132027},[31680],[34,384146,384148,384151],{"className":384147},[31685],[34,384149,111],{"className":384150},[31685,31686],[34,384152,384154],{"className":384153},[31792],[34,384155,384157],{"className":384156},[31796],[34,384158,384160],{"className":384159},[31801],[34,384161,384163],{"className":384162,"style":132027},[31805],[34,384164,384165,384168],{"style":37777},[34,384166],{"className":384167,"style":31814},[31813],[34,384169,384171],{"className":384170},[31818,31819,31820,31821],[34,384172,384174],{"className":384173},[31685,31821],[34,384175,116495],{"className":384176},[31685,31821],[34,384178,384180,384206],{"className":384179},[31642],[34,384181,384183],{"className":384182},[31646],[31648,384184,384185],{"xmlns":31650},[31652,384186,384187,384203],{},[31655,384188,384189,384191,384193,384199,384201],{},[31658,384190,111],{},[31662,384192,378006],{},[37624,384194,384195,384197],{},[31658,384196,111],{},[31662,384198,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,384200,202],{},[31658,384202,382333],{},[31665,384204,384205],{"encoding":31667},"a \\oplus a' = z",[34,384207,384209,384227,384274],{"className":384208,"ariaHidden":758},[31672],[34,384210,384212,384215,384218,384221,384224],{"className":384211},[31676],[34,384213],{"className":384214,"style":32597},[31680],[34,384216,111],{"className":384217},[31685,31686],[34,384219],{"className":384220,"style":32604},[31845],[34,384222,378006],{"className":384223},[32024],[34,384225],{"className":384226,"style":32604},[31845],[34,384228,384230,384233,384265,384268,384271],{"className":384229},[31676],[34,384231],{"className":384232,"style":132027},[31680],[34,384234,384236,384239],{"className":384235},[31685],[34,384237,111],{"className":384238},[31685,31686],[34,384240,384242],{"className":384241},[31792],[34,384243,384245],{"className":384244},[31796],[34,384246,384248],{"className":384247},[31801],[34,384249,384251],{"className":384250,"style":132027},[31805],[34,384252,384253,384256],{"style":37777},[34,384254],{"className":384255,"style":31814},[31813],[34,384257,384259],{"className":384258},[31818,31819,31820,31821],[34,384260,384262],{"className":384261},[31685,31821],[34,384263,116495],{"className":384264},[31685,31821],[34,384266],{"className":384267,"style":31846},[31845],[34,384269,202],{"className":384270},[31850],[34,384272],{"className":384273,"style":31846},[31845],[34,384275,384277,384280],{"className":384276},[31676],[34,384278],{"className":384279,"style":32527},[31680],[34,384281,382333],{"className":384282,"style":382406},[31685,31686],"（新零元）。\n即 ",[34,384285,384287,384337],{"className":384286},[31642],[34,384288,384290],{"className":384289},[31646],[31648,384291,384292],{"xmlns":31650},[31652,384293,384294,384334],{},[31655,384295,384296,384298,384300,384306,384308,384310,384312,384314,384316,384318,384320,384326,384328,384330,384332],{},[31658,384297,111],{},[31662,384299,223],{},[37624,384301,384302,384304],{},[31658,384303,111],{},[31662,384305,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,384307,32226],{},[31749,384309,640],{},[31662,384311,202],{},[31749,384313,640],{},[39361,384315,262622],{},[31662,384317,262625],{},[39361,384319,262622],{},[37624,384321,384322,384324],{},[31658,384323,111],{},[31662,384325,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,384327,202],{},[31749,384329,768],{},[31662,384331,32226],{},[31658,384333,111],{},[31665,384335,384336],{"encoding":31667},"a + a' - 1 = 1 \\implies a' = 2 - a",[34,384338,384340,384358,384406,384424,384448,384495,384513],{"className":384339,"ariaHidden":758},[31672],[34,384341,384343,384346,384349,384352,384355],{"className":384342},[31676],[34,384344],{"className":384345,"style":32597},[31680],[34,384347,111],{"className":384348},[31685,31686],[34,384350],{"className":384351,"style":32604},[31845],[34,384353,223],{"className":384354},[32024],[34,384356],{"className":384357,"style":32604},[31845],[34,384359,384361,384365,384397,384400,384403],{"className":384360},[31676],[34,384362],{"className":384363,"style":384364},[31680],"height:0.8352em;vertical-align:-0.0833em;",[34,384366,384368,384371],{"className":384367},[31685],[34,384369,111],{"className":384370},[31685,31686],[34,384372,384374],{"className":384373},[31792],[34,384375,384377],{"className":384376},[31796],[34,384378,384380],{"className":384379},[31801],[34,384381,384383],{"className":384382,"style":132027},[31805],[34,384384,384385,384388],{"style":37777},[34,384386],{"className":384387,"style":31814},[31813],[34,384389,384391],{"className":384390},[31818,31819,31820,31821],[34,384392,384394],{"className":384393},[31685,31821],[34,384395,116495],{"className":384396},[31685,31821],[34,384398],{"className":384399,"style":32604},[31845],[34,384401,32226],{"className":384402},[32024],[34,384404],{"className":384405,"style":32604},[31845],[34,384407,384409,384412,384415,384418,384421],{"className":384408},[31676],[34,384410],{"className":384411,"style":32617},[31680],[34,384413,640],{"className":384414},[31685],[34,384416],{"className":384417,"style":31846},[31845],[34,384419,202],{"className":384420},[31850],[34,384422],{"className":384423,"style":31846},[31845],[34,384425,384427,384430,384433,384436,384439,384442,384445],{"className":384426},[31676],[34,384428],{"className":384429,"style":262700},[31680],[34,384431,640],{"className":384432},[31685],[34,384434],{"className":384435,"style":31846},[31845],[34,384437],{"className":384438,"style":31846},[31845],[34,384440,262625],{"className":384441},[31850],[34,384443],{"className":384444,"style":31846},[31845],[34,384446],{"className":384447,"style":31846},[31845],[34,384449,384451,384454,384486,384489,384492],{"className":384450},[31676],[34,384452],{"className":384453,"style":132027},[31680],[34,384455,384457,384460],{"className":384456},[31685],[34,384458,111],{"className":384459},[31685,31686],[34,384461,384463],{"className":384462},[31792],[34,384464,384466],{"className":384465},[31796],[34,384467,384469],{"className":384468},[31801],[34,384470,384472],{"className":384471,"style":132027},[31805],[34,384473,384474,384477],{"style":37777},[34,384475],{"className":384476,"style":31814},[31813],[34,384478,384480],{"className":384479},[31818,31819,31820,31821],[34,384481,384483],{"className":384482},[31685,31821],[34,384484,116495],{"className":384485},[31685,31821],[34,384487],{"className":384488,"style":31846},[31845],[34,384490,202],{"className":384491},[31850],[34,384493],{"className":384494,"style":31846},[31845],[34,384496,384498,384501,384504,384507,384510],{"className":384497},[31676],[34,384499],{"className":384500,"style":35942},[31680],[34,384502,768],{"className":384503},[31685],[34,384505],{"className":384506,"style":32604},[31845],[34,384508,32226],{"className":384509},[32024],[34,384511],{"className":384512,"style":32604},[31845],[34,384514,384516,384519],{"className":384515},[31676],[34,384517],{"className":384518,"style":32527},[31680],[34,384520,111],{"className":384521},[31685,31686],"。\n在环里，",[34,384524,384526,384539],{"className":384525},[31642],[34,384527,384529],{"className":384528},[31646],[31648,384530,384531],{"xmlns":31650},[31652,384532,384533,384537],{},[31655,384534,384535],{},[31749,384536,768],{},[31665,384538,768],{"encoding":31667},[34,384540,384542],{"className":384541,"ariaHidden":758},[31672],[34,384543,384545,384548],{"className":384544},[31676],[34,384546],{"className":384547,"style":32617},[31680],[34,384549,768],{"className":384550},[31685]," 就是 ",[34,384553,384555,384573],{"className":384554},[31642],[34,384556,384558],{"className":384557},[31646],[31648,384559,384560],{"xmlns":31650},[31652,384561,384562,384570],{},[31655,384563,384564,384566,384568],{},[31749,384565,640],{},[31662,384567,223],{},[31749,384569,640],{},[31665,384571,384572],{"encoding":31667},"1+1",[34,384574,384576,384594],{"className":384575,"ariaHidden":758},[31672],[34,384577,384579,384582,384585,384588,384591],{"className":384578},[31676],[34,384580],{"className":384581,"style":35942},[31680],[34,384583,640],{"className":384584},[31685],[34,384586],{"className":384587,"style":32604},[31845],[34,384589,223],{"className":384590},[32024],[34,384592],{"className":384593,"style":32604},[31845],[34,384595,384597,384600],{"className":384596},[31676],[34,384598],{"className":384599,"style":32617},[31680],[34,384601,640],{"className":384602},[31685],"。存在，满足。",[15,384605,384606],{},[2627,384607,384608,384609,384638],{},"2. 验证乘法 ",[34,384610,384612,384626],{"className":384611},[31642],[34,384613,384615],{"className":384614},[31646],[31648,384616,384617],{"xmlns":31650},[31652,384618,384619,384623],{},[31655,384620,384621],{},[31662,384622,378082],{},[31665,384624,384625],{"encoding":31667},"\\odot",[34,384627,384629],{"className":384628,"ariaHidden":758},[31672],[34,384630,384632,384635],{"className":384631},[31676],[34,384633],{"className":384634,"style":32597},[31680],[34,384636,378082],{"className":384637},[31685]," 构成交换幺半群",[105,384640,384641,385269,385424],{},[108,384642,384643,384645,384646,39984,385185,385268],{},[2627,384644,378398],{},"：需要耐心计算。\n",[34,384647,384649,384777],{"className":384648},[31642],[34,384650,384652],{"className":384651},[31646],[31648,384653,384654],{"xmlns":31650},[31652,384655,384656,384774],{},[31655,384657,384658,384660,384662,384664,384666,384668,384670,384672,384674,384676,384678,384680,384682,384684,384686,384688,384690,384692,384694,384696,384698,384700,384702,384704,384706,384708,384710,384712,384714,384716,384718,384720,384722,384724,384726,384728,384730,384732,384734,384736,384738,384740,384742,384744,384746,384748,384750,384752,384754,384756,384758,384760,384762,384764,384766,384768,384770,384772],{},[31662,384659,456],{"stretchy":702},[31658,384661,111],{},[31662,384663,378082],{},[31658,384665,1697],{},[31662,384667,2557],{"stretchy":702},[31662,384669,378082],{},[31658,384671,31965],{},[31662,384673,202],{},[31662,384675,456],{"stretchy":702},[31658,384677,111],{},[31662,384679,223],{},[31658,384681,1697],{},[31662,384683,32226],{},[31658,384685,111],{},[31658,384687,1697],{},[31662,384689,2557],{"stretchy":702},[31662,384691,378082],{},[31658,384693,31965],{},[31662,384695,202],{},[31662,384697,456],{"stretchy":702},[31658,384699,111],{},[31662,384701,223],{},[31658,384703,1697],{},[31662,384705,32226],{},[31658,384707,111],{},[31658,384709,1697],{},[31662,384711,2557],{"stretchy":702},[31662,384713,223],{},[31658,384715,31965],{},[31662,384717,32226],{},[31662,384719,456],{"stretchy":702},[31658,384721,111],{},[31662,384723,223],{},[31658,384725,1697],{},[31662,384727,32226],{},[31658,384729,111],{},[31658,384731,1697],{},[31662,384733,2557],{"stretchy":702},[31658,384735,31965],{},[31662,384737,202],{},[31658,384739,111],{},[31662,384741,223],{},[31658,384743,1697],{},[31662,384745,223],{},[31658,384747,31965],{},[31662,384749,32226],{},[31658,384751,111],{},[31658,384753,1697],{},[31662,384755,32226],{},[31658,384757,111],{},[31658,384759,31965],{},[31662,384761,32226],{},[31658,384763,1697],{},[31658,384765,31965],{},[31662,384767,223],{},[31658,384769,111],{},[31658,384771,1697],{},[31658,384773,31965],{},[31665,384775,384776],{"encoding":31667},"(a \\odot b) \\odot c = (a+b-ab) \\odot c = (a+b-ab) + c - (a+b-ab)c = a+b+c - ab - ac - bc + abc",[34,384778,384780,384801,384822,384840,384861,384879,384900,384918,384939,384957,384978,384996,385017,385035,385059,385077,385095,385113,385131,385152,385173],{"className":384779,"ariaHidden":758},[31672],[34,384781,384783,384786,384789,384792,384795,384798],{"className":384782},[31676],[34,384784],{"className":384785,"style":31774},[31680],[34,384787,456],{"className":384788},[31782],[34,384790,111],{"className":384791},[31685,31686],[34,384793],{"className":384794,"style":32604},[31845],[34,384796,378082],{"className":384797},[32024],[34,384799],{"className":384800,"style":32604},[31845],[34,384802,384804,384807,384810,384813,384816,384819],{"className":384803},[31676],[34,384805],{"className":384806,"style":31774},[31680],[34,384808,1697],{"className":384809},[31685,31686],[34,384811,2557],{"className":384812},[31690],[34,384814],{"className":384815,"style":32604},[31845],[34,384817,378082],{"className":384818},[32024],[34,384820],{"className":384821,"style":32604},[31845],[34,384823,384825,384828,384831,384834,384837],{"className":384824},[31676],[34,384826],{"className":384827,"style":32527},[31680],[34,384829,31965],{"className":384830},[31685,31686],[34,384832],{"className":384833,"style":31846},[31845],[34,384835,202],{"className":384836},[31850],[34,384838],{"className":384839,"style":31846},[31845],[34,384841,384843,384846,384849,384852,384855,384858],{"className":384842},[31676],[34,384844],{"className":384845,"style":31774},[31680],[34,384847,456],{"className":384848},[31782],[34,384850,111],{"className":384851},[31685,31686],[34,384853],{"className":384854,"style":32604},[31845],[34,384856,223],{"className":384857},[32024],[34,384859],{"className":384860,"style":32604},[31845],[34,384862,384864,384867,384870,384873,384876],{"className":384863},[31676],[34,384865],{"className":384866,"style":34299},[31680],[34,384868,1697],{"className":384869},[31685,31686],[34,384871],{"className":384872,"style":32604},[31845],[34,384874,32226],{"className":384875},[32024],[34,384877],{"className":384878,"style":32604},[31845],[34,384880,384882,384885,384888,384891,384894,384897],{"className":384881},[31676],[34,384883],{"className":384884,"style":31774},[31680],[34,384886,324191],{"className":384887},[31685,31686],[34,384889,2557],{"className":384890},[31690],[34,384892],{"className":384893,"style":32604},[31845],[34,384895,378082],{"className":384896},[32024],[34,384898],{"className":384899,"style":32604},[31845],[34,384901,384903,384906,384909,384912,384915],{"className":384902},[31676],[34,384904],{"className":384905,"style":32527},[31680],[34,384907,31965],{"className":384908},[31685,31686],[34,384910],{"className":384911,"style":31846},[31845],[34,384913,202],{"className":384914},[31850],[34,384916],{"className":384917,"style":31846},[31845],[34,384919,384921,384924,384927,384930,384933,384936],{"className":384920},[31676],[34,384922],{"className":384923,"style":31774},[31680],[34,384925,456],{"className":384926},[31782],[34,384928,111],{"className":384929},[31685,31686],[34,384931],{"className":384932,"style":32604},[31845],[34,384934,223],{"className":384935},[32024],[34,384937],{"className":384938,"style":32604},[31845],[34,384940,384942,384945,384948,384951,384954],{"className":384941},[31676],[34,384943],{"className":384944,"style":34299},[31680],[34,384946,1697],{"className":384947},[31685,31686],[34,384949],{"className":384950,"style":32604},[31845],[34,384952,32226],{"className":384953},[32024],[34,384955],{"className":384956,"style":32604},[31845],[34,384958,384960,384963,384966,384969,384972,384975],{"className":384959},[31676],[34,384961],{"className":384962,"style":31774},[31680],[34,384964,324191],{"className":384965},[31685,31686],[34,384967,2557],{"className":384968},[31690],[34,384970],{"className":384971,"style":32604},[31845],[34,384973,223],{"className":384974},[32024],[34,384976],{"className":384977,"style":32604},[31845],[34,384979,384981,384984,384987,384990,384993],{"className":384980},[31676],[34,384982],{"className":384983,"style":32597},[31680],[34,384985,31965],{"className":384986},[31685,31686],[34,384988],{"className":384989,"style":32604},[31845],[34,384991,32226],{"className":384992},[32024],[34,384994],{"className":384995,"style":32604},[31845],[34,384997,384999,385002,385005,385008,385011,385014],{"className":384998},[31676],[34,385000],{"className":385001,"style":31774},[31680],[34,385003,456],{"className":385004},[31782],[34,385006,111],{"className":385007},[31685,31686],[34,385009],{"className":385010,"style":32604},[31845],[34,385012,223],{"className":385013},[32024],[34,385015],{"className":385016,"style":32604},[31845],[34,385018,385020,385023,385026,385029,385032],{"className":385019},[31676],[34,385021],{"className":385022,"style":34299},[31680],[34,385024,1697],{"className":385025},[31685,31686],[34,385027],{"className":385028,"style":32604},[31845],[34,385030,32226],{"className":385031},[32024],[34,385033],{"className":385034,"style":32604},[31845],[34,38503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\\odot (b \\odot c)",[34,385215,385217,385235,385256],{"className":385216,"ariaHidden":758},[31672],[34,385218,385220,385223,385226,385229,385232],{"className":385219},[31676],[34,385221],{"className":385222,"style":32597},[31680],[34,385224,111],{"className":385225},[31685,31686],[34,385227],{"className":385228,"style":32604},[31845],[34,385230,378082],{"className":385231},[32024],[34,385233],{"className":385234,"style":32604},[31845],[34,385236,385238,385241,385244,385247,385250,385253],{"className":385237},[31676],[34,385239],{"className":385240,"style":31774},[31680],[34,385242,456],{"className":385243},[31782],[34,385245,1697],{"className":385246},[31685,31686],[34,385248],{"className":385249,"style":32604},[31845],[34,385251,378082],{"className":385252},[32024],[34,385254],{"className":385255,"style":32604},[31845],[34,385257,385259,385262,385265],{"className":385258},[31676],[34,385260],{"className":385261,"style":31774},[31680],[34,385263,31965],{"className":385264},[31685,31686],[34,385266,2557],{"className":385267},[31690]," 算出来也是这一长串。满足。",[108,385270,385271,2664,385273,385348,385349,378487],{},[2627,385272,380943],{},[34,385274,385276,385300],{"className":385275},[31642],[34,385277,385279],{"className":385278},[31646],[31648,385280,385281],{"xmlns":31650},[31652,385282,385283,385297],{},[31655,385284,385285,385287,385289,385291,385293,385295],{},[31658,385286,111],{},[31662,385288,223],{},[31658,385290,1697],{},[31662,385292,32226],{},[31658,385294,111],{},[31658,385296,1697],{},[31665,385298,385299],{"encoding":31667},"a+b-ab",[34,385301,385303,385321,385339],{"className":385302,"ariaHidden":758},[31672],[34,385304,385306,385309,385312,385315,385318],{"className":385305},[31676],[34,385307],{"className":385308,"style":32597},[31680],[34,385310,111],{"className":385311},[31685,31686],[34,385313],{"className":385314,"style":32604},[31845],[34,385316,223],{"className":385317},[32024],[34,385319],{"className":385320,"style":32604},[31845],[34,385322,385324,385327,385330,385333,385336],{"className":385323},[31676],[34,385325],{"className":385326,"style":34299},[31680],[34,385328,1697],{"className":385329},[31685,31686],[34,385331],{"className":385332,"style":32604},[31845],[34,385334,32226],{"className":385335},[32024],[34,385337],{"className":385338,"style":32604},[31845],[34,385340,385342,385345],{"className":385341},[31676],[34,385343],{"className":385344,"style":31681},[31680],[34,385346,324191],{"className":385347},[31685,31686]," 显然等于 ",[34,385350,385352,385376],{"className":385351},[31642],[34,385353,385355],{"className":385354},[31646],[31648,385356,385357],{"xmlns":31650},[31652,385358,385359,385373],{},[31655,385360,385361,385363,385365,385367,385369,385371],{},[31658,385362,1697],{},[31662,385364,223],{},[31658,385366,111],{},[31662,385368,32226],{},[31658,385370,1697],{},[31658,385372,111],{},[31665,385374,385375],{"encoding":31667},"b+a-ba",[34,385377,385379,385397,385415],{"className":385378,"ariaHidden":758},[31672],[34,385380,385382,385385,385388,385391,385394],{"className":385381},[31676],[34,385383],{"className":385384,"style":34299},[31680],[34,385386,1697],{"className":385387},[31685,31686],[34,385389],{"className":385390,"style":32604},[31845],[34,385392,223],{"className":385393},[32024],[34,385395],{"className":385396,"style":32604},[31845],[34,385398,385400,385403,385406,385409,385412],{"className":385399},[31676],[34,385401],{"className":385402,"style":32597},[31680],[34,385404,111],{"className":385405},[31685,31686],[34,385407],{"className":385408,"style":32604},[31845],[34,385410,32226],{"className":385411},[32024],[34,385413],{"className":385414,"style":32604},[31845],[34,385416,385418,385421],{"className":385417},[31676],[34,385419],{"className":385420,"style":31681},[31680],[34,385422,367454],{"className":385423},[31685,31686],[108,385425,385426,385428,385429,383826,385457,383900,385530,385630,385732,385844,385845,385873,385874,381924,385925,384050,386086,347],{},[2627,385427,381426],{},"：设新单位元为 ",[34,385430,385432,385445],{"className":385431},[31642],[34,385433,385435],{"className":385434},[31646],[31648,385436,385437],{"xmlns":31650},[31652,385438,385439,385443],{},[31655,385440,385441],{},[31658,385442,33244],{},[31665,385444,33244],{"encoding":31667},[34,385446,385448],{"className":385447,"ariaHidden":758},[31672],[34,385449,385451,385454],{"className":385450},[31676],[34,385452],{"className":385453,"style":32527},[31680],[34,385455,33244],{"className":385456},[31685,31686],[34,385458,385460,385482],{"className":385459},[31642],[34,385461,385463],{"className":385462},[31646],[31648,385464,385465],{"xmlns":31650},[31652,385466,385467,385479],{},[31655,385468,385469,385471,385473,385475,385477],{},[31658,385470,111],{},[31662,385472,378082],{},[31658,385474,33244],{},[31662,385476,202],{},[31658,385478,111],{},[31665,385480,385481],{"encoding":31667},"a \\odot e = a",[34,385483,385485,385503,385521],{"className":385484,"ariaHidden":758},[31672],[34,385486,385488,385491,385494,385497,385500],{"className":385487},[31676],[34,385489],{"className":385490,"style":32597},[31680],[34,385492,111],{"className":385493},[31685,31686],[34,385495],{"className":385496,"style":32604},[31845],[34,385498,378082],{"className":385499},[32024],[34,385501],{"className":385502,"style":32604},[31845],[34,385504,385506,385509,385512,385515,385518],{"className":385505},[31676],[34,385507],{"className":385508,"style":32527},[31680],[34,385510,33244],{"className":385511},[31685,31686],[34,385513],{"className":385514,"style":31846},[31845],[34,385516,202],{"className":385517},[31850],[34,385519],{"className":385520,"style":31846},[31845],[34,385522,385524,385527],{"className":385523},[31676],[34,385525],{"className":385526,"style":32527},[31680],[34,385528,111],{"className":385529},[31685,31686],[34,385531,385533,385561],{"className":385532},[31642],[34,385534,385536],{"className":385535},[31646],[31648,385537,385538],{"xmlns":31650},[31652,385539,385540,385558],{},[31655,385541,385542,385544,385546,385548,385550,385552,385554,385556],{},[31658,385543,111],{},[31662,385545,223],{},[31658,385547,33244],{},[31662,385549,32226],{},[31658,385551,111],{},[31658,385553,33244],{},[31662,385555,202],{},[31658,385557,111],{},[31665,385559,385560],{"encoding":31667},"a + e - ae = a",[34,385562,385564,385582,385600,385621],{"className":385563,"ariaHidden":758},[31672],[34,385565,385567,385570,385573,385576,385579],{"className":385566},[31676],[34,385568],{"className":385569,"style":32597},[31680],[34,385571,111],{"className":385572},[31685,31686],[34,385574],{"className":385575,"style":32604},[31845],[34,385577,223],{"className":385578},[32024],[34,385580],{"className":385581,"style":32604},[31845],[34,385583,385585,385588,385591,385594,385597],{"className":385584},[31676],[34,385586],{"className":385587,"style":32597},[31680],[34,385589,33244],{"className":385590},[31685,31686],[34,385592],{"className":385593,"style":32604},[31845],[34,385595,32226],{"className":385596},[32024],[34,385598],{"className":385599,"style":32604},[31845],[34,385601,385603,385606,385609,385612,385615,385618],{"className":385602},[31676],[34,385604],{"className":385605,"style":32527},[31680],[34,385607,111],{"className":385608},[31685,31686],[34,385610,33244],{"className":385611},[31685,31686],[34,385613],{"className":385614,"style":31846},[31845],[34,385616,202],{"className":385617},[31850],[34,385619],{"className":385620,"style":31846},[31845],[34,385622,385624,385627],{"className":385623},[31676],[34,385625],{"className":385626,"style":32527},[31680],[34,385628,111],{"className":385629},[31685,31686],[34,385631,385633,385663],{"className":385632},[31642],[34,385634,385636],{"className":385635},[31646],[31648,385637,385638],{"xmlns":31650},[31652,385639,385640,385660],{},[31655,385641,385642,385644,385646,385648,385650,385652,385654,385656,385658],{},[39361,385643,262622],{},[31662,385645,262625],{},[39361,385647,262622],{},[31658,385649,33244],{},[31662,385651,32226],{},[31658,385653,111],{},[31658,385655,33244],{},[31662,385657,202],{},[31749,385659,459],{},[31665,385661,385662],{"encoding":31667},"\\implies e - ae = 0",[34,385664,385666,385684,385702,385723],{"className":385665,"ariaHidden":758},[31672],[34,385667,385669,385672,385675,385678,385681],{"className":385668},[31676],[34,385670],{"className":385671,"style":318264},[31680],[34,385673],{"className":385674,"style":31846},[31845],[34,385676,262625],{"className":385677},[31850],[34,385679],{"className":385680,"style":31846},[31845],[34,385682],{"className":385683,"style":31846},[31845],[34,385685,385687,385690,385693,385696,385699],{"className":385686},[31676],[34,385688],{"className":385689,"style":32597},[31680],[34,385691,33244],{"className":385692},[31685,31686],[34,385694],{"className":385695,"style":32604},[31845],[34,385697,32226],{"className":385698},[32024],[34,385700],{"className":385701,"style":32604},[31845],[34,385703,385705,385708,385711,385714,385717,385720],{"className":385704},[31676],[34,385706],{"className":385707,"style":32527},[31680],[34,385709,111],{"className":385710},[31685,31686],[34,385712,33244],{"className":385713},[31685,31686],[34,385715],{"className":385716,"style":31846},[31845],[34,385718,202],{"className":385719},[31850],[34,385721],{"className":385722,"style":31846},[31845],[34,385724,385726,385729],{"className":385725},[31676],[34,385727],{"className":385728,"style":32617},[31680],[34,385730,459],{"className":385731},[31685],[34,385733,385735,385769],{"className":385734},[31642],[34,385736,385738],{"className":385737},[31646],[31648,385739,385740],{"xmlns":31650},[31652,385741,385742,385766],{},[31655,385743,385744,385746,385748,385750,385752,385754,385756,385758,385760,385762,385764],{},[39361,385745,262622],{},[31662,385747,262625],{},[39361,385749,262622],{},[31658,385751,33244],{},[31662,385753,456],{"stretchy":702},[31749,385755,640],{},[31662,385757,32226],{},[31658,385759,111],{},[31662,385761,2557],{"stretchy":702},[31662,385763,202],{},[31749,385765,459],{},[31665,385767,385768],{"encoding":31667},"\\implies e(1-a) = 0",[34,385770,385772,385790,385814,385835],{"className":385771,"ariaHidden":758},[31672],[34,385773,385775,385778,385781,385784,385787],{"className":385774},[31676],[34,385776],{"className":385777,"style":318264},[31680],[34,385779],{"className":385780,"style":31846},[31845],[34,385782,262625],{"className":385783},[31850],[34,385785],{"className":385786,"style":31846},[31845],[34,385788],{"className":385789,"style":31846},[31845],[34,385791,385793,385796,385799,385802,385805,385808,385811],{"className":385792},[31676],[34,385794],{"className":385795,"style":31774},[31680],[34,385797,33244],{"className":385798},[31685,31686],[34,385800,456],{"className":385801},[31782],[34,385803,640],{"className":385804},[31685],[34,385806],{"className":385807,"style":32604},[31845],[34,385809,32226],{"className":385810},[32024],[34,385812],{"className":385813,"style":32604},[31845],[34,385815,385817,385820,385823,385826,385829,385832],{"className":385816},[31676],[34,385818],{"className":385819,"style":31774},[31680],[34,385821,111],{"className":385822},[31685,31686],[34,385824,2557],{"className":385825},[31690],[34,385827],{"className":385828,"style":31846},[31845],[34,385830,202],{"className":385831},[31850],[34,385833],{"className":385834,"style":31846},[31845],[34,385836,385838,385841],{"className":385837},[31676],[34,385839],{"className":385840,"style":32617},[31680],[34,385842,459],{"className":385843},[31685],"。\n如果要对所有 ",[34,385846,385848,385861],{"className":385847},[31642],[34,385849,385851],{"className":385850},[31646],[31648,385852,385853],{"xmlns":31650},[31652,385854,385855,385859],{},[31655,385856,385857],{},[31658,385858,111],{},[31665,385860,111],{"encoding":31667},[34,385862,385864],{"className":385863,"ariaHidden":758},[31672],[34,385865,385867,385870],{"className":385866},[31676],[34,385868],{"className":385869,"style":32527},[31680],[34,385871,111],{"className":385872},[31685,31686]," 成立，唯有 ",[34,385875,385877,385895],{"className":385876},[31642],[34,385878,385880],{"className":385879},[31646],[31648,385881,385882],{"xmlns":31650},[31652,385883,385884,385892],{},[31655,385885,385886,385888,385890],{},[31658,385887,33244],{},[31662,385889,202],{},[31749,385891,459],{},[31665,385893,385894],{"encoding":31667},"e = 0",[34,385896,385898,385916],{"className":385897,"ariaHidden":758},[31672],[34,385899,385901,385904,385907,385910,385913],{"className":385900},[31676],[34,385902],{"className":385903,"style":32527},[31680],[34,385905,33244],{"className":385906},[31685,31686],[34,385908],{"className":385909,"style":31846},[31845],[34,385911,202],{"className":385912},[31850],[34,385914],{"className":385915,"style":31846},[31845],[34,385917,385919,385922],{"className":385918},[31676],[34,385920],{"className":385921,"style":32617},[31680],[34,385923,459],{"className":385924},[31685],[34,385926,385928,385966],{"className":385927},[31642],[34,385929,385931],{"className":385930},[31646],[31648,385932,385933],{"xmlns":31650},[31652,385934,385935,385963],{},[31655,385936,385937,385939,385941,385943,385945,385947,385949,385951,385953,385955,385957,385959,385961],{},[31658,385938,111],{},[31662,385940,378082],{},[31749,385942,459],{},[31662,385944,202],{},[31658,385946,111],{},[31662,385948,223],{},[31749,385950,459],{},[31662,385952,32226],{},[31658,385954,111],{},[31662,385956,71191],{},[31749,385958,459],{},[31662,385960,202],{},[31658,385962,111],{},[31665,385964,385965],{"encoding":31667},"a \\odot 0 = a + 0 - a\\cdot 0 = a",[34,385967,385969,385987,386005,386023,386041,386059,386077],{"className":385968,"ariaHidden":758},[31672],[34,385970,385972,385975,385978,385981,385984],{"className":385971},[31676],[34,385973],{"className":385974,"style":32597},[31680],[34,385976,111],{"className":385977},[31685,31686],[34,385979],{"className":385980,"style":32604},[31845],[34,385982,378082],{"className":385983},[32024],[34,385985],{"className":385986,"style":32604},[31845],[34,385988,385990,385993,385996,385999,386002],{"className":385989},[31676],[34,385991],{"className":385992,"style":32617},[31680],[34,385994,459],{"className":385995},[31685],[34,385997],{"className":385998,"style":31846},[31845],[34,386000,202],{"className":386001},[31850],[34,386003],{"className":386004,"style":31846},[31845],[34,386006,386008,386011,386014,386017,386020],{"className":386007},[31676],[34,386009],{"className":386010,"style":32597},[31680],[34,386012,111],{"className":386013},[31685,31686],[34,386015],{"className":386016,"style":32604},[31845],[34,386018,223],{"className":386019},[32024],[34,386021],{"className":386022,"style":32604},[31845],[34,386024,386026,386029,386032,386035,386038],{"className":386025},[31676],[34,386027],{"className":386028,"style":35942},[31680],[34,386030,459],{"className":386031},[31685],[34,386033],{"className":386034,"style":32604},[31845],[34,386036,32226],{"className":386037},[32024],[34,386039],{"className":386040,"style":32604},[31845],[34,386042,386044,386047,386050,386053,386056],{"className":386043},[31676],[34,386045],{"className":386046,"style":224869},[31680],[34,386048,111],{"className":386049},[31685,31686],[34,386051],{"className":386052,"style":32604},[31845],[34,386054,71191],{"className":386055},[32024],[34,386057],{"className":386058,"style":32604},[31845],[34,386060,386062,386065,386068,386071,386074],{"className":386061},[31676],[34,386063],{"className":386064,"style":32617},[31680],[34,386066,459],{"className":386067},[31685],[34,386069],{"className":386070,"style":31846},[31845],[34,386072,202],{"className":386073},[31850],[34,386075],{"className":386076,"style":31846},[31845],[34,386078,386080,386083],{"className":386079},[31676],[34,386081],{"className":386082,"style":32527},[31680],[34,386084,111],{"className":386085},[31685,31686],[2627,386087,386088,386089],{},"这个新环的“单位元”其实是原来的 ",[34,386090,386092,386105],{"className":386091},[31642],[34,386093,386095],{"className":386094},[31646],[31648,386096,386097],{"xmlns":31650},[31652,386098,386099,386103],{},[31655,386100,386101],{},[31749,386102,459],{},[31665,386104,459],{"encoding":31667},[34,386106,386108],{"className":386107,"ariaHidden":758},[31672],[34,386109,386111,386114],{"className":386110},[31676],[34,386112],{"className":386113,"style":32617},[31680],[34,386115,459],{"className":386116},[31685],[15,386118,386119,386122,386123,347],{},[2627,386120,386121],{},"3. 验证分配律","\n我们要证 ",[34,386124,386126,386176],{"className":386125},[31642],[34,386127,386129],{"className":386128},[31646],[31648,386130,386131],{"xmlns":31650},[31652,386132,386133,386173],{},[31655,386134,386135,386137,386139,386141,386143,386145,386147,386149,386151,386153,386155,386157,386159,386161,386163,386165,386167,386169,386171],{},[31658,386136,111],{},[31662,386138,378082],{},[31662,386140,456],{"stretchy":702},[31658,386142,1697],{},[31662,386144,378006],{},[31658,386146,31965],{},[31662,386148,2557],{"stretchy":702},[31662,386150,202],{},[31662,386152,456],{"stretchy":702},[31658,386154,111],{},[31662,386156,378082],{},[31658,386158,1697],{},[31662,386160,2557],{"stretchy":702},[31662,386162,378006],{},[31662,386164,456],{"stretchy":702},[31658,386166,111],{},[31662,386168,378082],{},[31658,386170,31965],{},[31662,386172,2557],{"stretchy":702},[31665,386174,386175],{"encoding":31667},"a \\odot (b \\oplus c) = (a \\odot b) \\oplus (a \\odot c)",[34,386177,386179,386197,386218,386239,386260,386281,386302],{"className":386178,"ariaHidden":758},[31672],[34,386180,386182,386185,386188,386191,386194],{"className":386181},[31676],[34,386183],{"className":386184,"style":32597},[31680],[34,386186,111],{"className":386187},[31685,31686],[34,386189],{"className":386190,"style":32604},[31845],[34,386192,378082],{"className":386193},[32024],[34,386195],{"className":386196,"style":32604},[31845],[34,386198,386200,386203,386206,386209,386212,386215],{"className":386199},[31676],[34,386201],{"className":386202,"style":31774},[31680],[34,386204,456],{"className":386205},[31782],[34,386207,1697],{"className":386208},[31685,31686],[34,386210],{"className":386211,"style":32604},[31845],[34,386213,378006],{"className":386214},[32024],[34,386216],{"className":386217,"style":32604},[31845],[34,386219,386221,386224,386227,386230,386233,386236],{"className":386220},[31676],[34,386222],{"className":386223,"style":31774},[31680],[34,386225,31965],{"className":386226},[31685,31686],[34,386228,2557],{"className":386229},[31690],[34,386231],{"className":386232,"style":31846},[31845],[34,386234,202],{"className":386235},[31850],[34,386237],{"className":386238,"style":31846},[31845],[34,386240,386242,386245,386248,386251,386254,386257],{"className":386241},[31676],[34,386243],{"className":386244,"style":31774},[31680],[34,386246,456],{"className":386247},[31782],[34,386249,111],{"className":386250},[31685,31686],[34,386252],{"className":386253,"style":32604},[31845],[34,386255,378082],{"className":386256},[32024],[34,386258],{"className":386259,"style":32604},[31845],[34,386261,386263,386266,386269,386272,386275,386278],{"className":386262},[31676],[34,386264],{"className":386265,"style":31774},[31680],[34,386267,1697],{"className":386268},[31685,31686],[34,386270,2557],{"className":386271},[31690],[34,386273],{"className":386274,"style":32604},[31845],[34,386276,378006],{"className":386277},[32024],[34,386279],{"className":386280,"style":32604},[31845],[34,386282,386284,386287,386290,386293,386296,386299],{"className":386283},[31676],[34,386285],{"className":386286,"style":31774},[31680],[34,386288,456],{"className":386289},[31782],[34,386291,111],{"className":386292},[31685,31686],[34,386294],{"className":386295,"style":32604},[31845],[34,386297,378082],{"className":386298},[32024],[34,386300],{"className":386301,"style":32604},[31845],[34,386303,386305,386308,386311],{"className":386304},[31676],[34,386306],{"className":386307,"style":31774},[31680],[34,386309,31965],{"className":386310},[31685,31686],[34,386312,2557],{"className":386313},[31690],[105,386315,386316,386922,387430],{},[108,386317,386318,386319,347],{},"左边 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a \\odot (b+c-1) = a + (b+c-1) - a(b+c-1) = a+b+c-1-ab-ac+a = 2a+b+c-1-ab-ac",[34,386455,386457,386469,386487,386508,386526,386547,386565,386586,386604,386625,386649,386667,386688,386706,386724,386742,386760,386778,386799,386817,386838,386856,386874,386892,386910],{"className":386456,"ariaHidden":758},[31672],[34,386458,386460,386463,386466],{"className":386459},[31676],[34,386461],{"className":386462,"style":73089},[31680],[34,386464,202],{"className":386465},[31850],[34,386467],{"className":386468,"style":31846},[31845],[34,386470,386472,386475,386478,386481,386484],{"className":386471},[31676],[34,386473],{"className":386474,"style":32597},[31680],[34,386476,111],{"className":386477},[31685,31686],[34,386479],{"className":386480,"style":32604},[31845],[34,386482,378082],{"className":386483},[32024],[34,386485],{"className":386486,"style":32604},[31845],[34,386488,386490,386493,386496,386499,386502,386505],{"className":386489},[31676],[34,386491],{"className":386492,"style":31774},[31680],[34,386494,456],{"className":386495},[31782],[34,386497,1697],{"className":386498},[31685,31686],[34,386500],{"className":386501,"style":32604},[31845],[34,386503,223],{"className":386504},[32024],[34,386506],{"className":386507,"style":32604},[31845],[34,386509,386511,386514,386517,386520,386523],{"className":386510},[31676],[34,386512],{"className":386513,"style":32597},[31680],[34,386515,31965],{"className":386516},[31685,31686],[34,386518],{"className":386519,"style":32604},[31845],[34,386521,32226],{"className":386522},[32024],[34,386524],{"className":386525,"style":32604},[31845],[34,386527,386529,386532,386535,386538,386541,386544],{"className":386528},[31676],[34,386530],{"className":386531,"style":31774},[31680],[34,386533,640],{"className":386534},[31685],[34,386536,2557],{"className":386537},[31690],[34,386539],{"className":386540,"style":31846},[31845],[34,386542,202],{"className":386543},[31850],[34,386545],{"className":386546,"style":31846},[31845],[34,386548,386550,386553,386556,386559,386562],{"className":386549},[31676],[34,386551],{"className":386552,"style":32597},[31680],[34,386554,111],{"className":386555},[31685,31686],[34,386557],{"className":386558,"style":32604},[31845],[34,386560,223],{"className":386561},[32024],[34,386563],{"className":386564,"style":32604},[31845],[34,386566,386568,386571,386574,386577,386580,386583],{"className":386567},[31676],[34,386569],{"className":386570,"style":31774},[31680],[34,386572,456],{"className":386573},[31782],[34,386575,1697],{"className":386576},[31685,31686],[34,386578],{"className":386579,"style":32604},[31845],[34,386581,223],{"className":386582},[32024],[34,386584],{"className":386585,"style":32604},[31845],[34,386587,386589,386592,386595,386598,386601],{"className":386588},[31676],[34,386590],{"className":386591,"style":32597},[31680],[34,386593,31965],{"className":386594},[31685,31686],[34,386596],{"className":386597,"style":32604},[31845],[34,386599,32226],{"className":386600},[32024],[34,386602],{"className":386603,"style":32604},[31845],[34,386605,386607,386610,386613,386616,386619,386622],{"className":386606},[31676],[34,386608],{"className":386609,"style":31774},[31680],[34,386611,640],{"className":386612},[31685],[34,386614,2557],{"className":386615},[31690],[34,386617],{"className":386618,"style":32604},[31845],[34,386620,32226],{"className":386621},[32024],[34,386623],{"className":386624,"style":32604},[31845],[34,386626,386628,386631,386634,386637,386640,386643,386646],{"className":386627},[31676],[34,386629],{"className":386630,"style":31774},[31680],[34,386632,111],{"className":386633},[31685,31686],[34,386635,456],{"className":386636},[31782],[34,386638,1697],{"className":386639},[31685,31686],[34,386641],{"className":386642,"style":32604},[31845],[34,386644,223],{"className":386645},[32024],[34,386647],{"className":386648,"style":32604},[31845],[34,386650,386652,386655,386658,386661,386664],{"className":386651},[31676],[34,386653],{"className":386654,"style":32597},[31680],[34,386656,31965],{"className":386657},[31685,31686],[34,386659],{"className":386660,"style":32604},[31845],[34,386662,32226],{"className":386663},[32024],[34,386665],{"className":386666,"style":32604},[31845],[34,386668,386670,386673,386676,386679,386682,386685],{"className":386669},[31676],[34,386671],{"className":386672,"style":31774},[31680],[34,386674,640],{"className":386675},[31685],[34,386677,2557],{"className":386678},[31690],[34,386680],{"className":386681,"style":31846},[31845],[34,386683,202],{"className":386684},[31850],[34,386686],{"className":386687,"style":31846},[31845],[34,386689,386691,386694,386697,386700,386703],{"className":386690},[31676],[34,386692],{"className":386693,"style":32597},[31680],[34,386695,111],{"className":386696},[31685,31686],[34,386698],{"className":386699,"style":32604},[31845],[34,386701,223],{"className":386702},[32024],[34,386704],{"className":386705,"style":32604},[31845],[34,386707,386709,386712,386715,386718,386721],{"className":386708},[31676],[34,386710],{"className":386711,"style"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",[34,386926,386928,387046],{"className":386927},[31642],[34,386929,386931],{"className":386930},[31646],[31648,386932,386933],{"xmlns":31650},[31652,386934,386935,387043],{},[31655,386936,386937,386939,386941,386943,386945,386947,386949,386951,386953,386955,386957,386959,386961,386963,386965,386967,386969,386971,386973,386975,386977,386979,386981,386983,386985,386987,386989,386991,386993,386995,386997,386999,387001,387003,387005,387007,387009,387011,387013,387015,387017,387019,387021,387023,387025,387027,387029,387031,387033,387035,387037,387039,387041],{},[31662,386938,202],{},[31662,386940,456],{"stretchy":702},[31658,386942,111],{},[31662,386944,223],{},[31658,386946,1697],{},[31662,386948,32226],{},[31658,386950,111],{},[31658,386952,1697],{},[31662,386954,2557],{"stretchy":702},[31662,386956,378006],{},[31662,386958,456],{"stretchy":702},[31658,386960,111],{},[31662,386962,223],{},[31658,386964,31965],{},[31662,386966,32226],{},[31658,386968,111],{},[31658,386970,31965],{},[31662,386972,2557],{"stretchy":702},[31662,386974,202],{},[31662,386976,456],{"stretchy":702},[31658,386978,111],{},[31662,386980,223],{},[31658,386982,1697],{},[31662,386984,32226],{},[31658,386986,111],{},[31658,386988,1697],{},[31662,386990,2557],{"stretchy":702},[31662,386992,223],{},[31662,386994,456],{"stretchy":702},[31658,386996,111],{},[31662,386998,223],{},[31658,387000,31965],{},[31662,387002,32226],{},[31658,387004,111],{},[31658,387006,31965],{},[31662,387008,2557],{"stretchy":702},[31662,387010,32226],{},[31749,387012,640],{},[31662,387014,202],{},[31749,387016,768],{},[31658,387018,111],{},[31662,387020,223],{},[31658,387022,1697],{},[31662,387024,223],{},[31658,387026,31965],{},[31662,387028,32226],{},[31749,387030,640],{},[31662,387032,32226],{},[31658,387034,111],{},[31658,387036,1697],{},[31662,387038,32226],{},[31658,387040,111],{},[31658,387042,31965],{},[31665,387044,387045],{"encoding":31667},"= (a+b-ab) \\oplus (a+c-ac) = (a+b-ab) + (a+c-ac) - 1 = 2a+b+c-1-ab-ac",[34,387047,387049,387061,387082,387100,387121,387142,387160,387184,387205,387223,387244,387265,387283,387307,387325,387346,387364,387382,387400,387418],{"className":387048,"ariaHidden":758},[31672],[34,387050,387052,387055,387058],{"className":387051},[31676],[34,387053],{"className":387054,"style":73089},[31680],[34,387056,202],{"className":387057},[31850],[34,387059],{"className":387060,"style":31846},[31845],[34,387062,387064,387067,387070,387073,387076,387079],{"className":387063},[31676],[34,387065],{"className":387066,"style":31774},[31680],[34,387068,456],{"className":387069},[31782],[34,387071,111],{"className":387072},[31685,31686],[34,387074],{"className":387075,"style":32604},[31845],[34,387077,223],{"className":387078},[32024],[34,387080],{"className":387081,"style":32604},[31845],[34,387083,387085,387088,387091,387094,387097],{"className":387084},[31676],[34,387086],{"className":387087,"style":34299},[31680],[34,387089,1697],{"className":387090},[31685,31686],[34,387092],{"className":387093,"style":32604},[31845],[34,387095,32226],{"className":387096},[32024],[34,387098],{"className":387099,"style":32604},[31845],[34,387101,387103,387106,387109,387112,387115,387118],{"className":387102},[31676],[34,387104],{"className":387105,"style":31774},[31680],[34,387107,324191],{"className":387108},[31685,31686],[34,387110,2557],{"className":387111},[31690],[34,387113],{"className":387114,"style":32604},[31845],[34,387116,378006],{"className":387117},[32024],[34,387119],{"className":387120,"style":32604},[31845],[34,387122,387124,387127,387130,387133,387136,387139],{"className":387123},[31676],[34,387125],{"className":387126,"style":31774},[31680],[34,387128,456],{"className":387129},[31782],[34,387131,111],{"className":387132},[31685,31686],[34,387134],{"className":387135,"style":32604},[31845],[34,387137,223],{"className":387138},[32024],[34,387140],{"className":387141,"style":32604},[31845],[34,387143,387145,387148,387151,387154,387157],{"className":387144},[31676],[34,387146],{"className":387147,"style":32597},[31680],[34,387149,31965],{"className":387150},[31685,31686],[34,387152],{"className":387153,"style":32604},[31845],[34,387155,32226],{"className":387156},[32024],[34,387158],{"className":387159,"style":32604},[31845],[34,387161,387163,387166,387169,387172,387175,387178,387181],{"className":387162},[31676],[34,387164],{"className":387165,"style":31774},[31680],[34,387167,111],{"className":387168},[31685,31686],[34,387170,31965],{"className":387171},[31685,31686],[34,387173,2557],{"className":387174},[31690],[34,387176],{"className":387177,"style":31846},[31845],[34,387179,202],{"className":387180},[31850],[34,387182],{"className":387183,"style":31846},[31845],[34,387185,387187,387190,387193,387196,387199,387202],{"className":387186},[31676],[34,387188],{"className":387189,"style":31774},[31680],[34,387191,456],{"className":387192},[31782],[34,387194,111],{"className":387195},[31685,31686],[34,387197],{"className":387198,"style":32604},[31845],[34,387200,223],{"className":387201},[32024],[34,387203],{"className":387204,"style":32604},[31845],[34,387206,387208,387211,387214,387217,387220],{"className":387207},[31676],[34,387209],{"className":387210,"style":34299},[31680],[34,387212,1697],{"className":387213},[31685,31686],[34,387215],{"className":387216,"style":32604},[31845],[34,387218,32226],{"className":387219},[32024],[34,387221],{"className":387222,"style":32604},[31845],[34,387224,387226,387229,387232,387235,387238,387241],{"className":387225},[31676],[34,387227],{"className":387228,"style":31774},[31680],[34,387230,324191],{"className":387231},[31685,31686],[34,387233,2557],{"className":387234},[31690],[34,387236],{"className":387237,"style":32604},[31845],[34,387239,223],{"className":387240},[32024],[34,387242],{"className":387243,"style":32604},[31845],[34,387245,387247,387250,387253,387256,387259,387262],{"className":387246},[31676],[34,387248],{"className":387249,"style":31774},[31680],[34,387251,456],{"className":387252},[31782],[34,387254,111],{"className":387255},[31685,31686],[34,387257],{"className":387258,"style":32604},[31845],[34,387260,223],{"className":387261},[32024],[34,387263],{"className":387264,"style":32604},[31845],[34,387266,387268,387271,387274,387277,387280],{"className":387267},[31676],[34,387269],{"className":387270,"style":32597},[31680],[34,387272,31965],{"className":387273},[31685,31686],[34,387275],{"className":387276,"style":32604},[31845],[34,387278,32226],{"className":387279},[32024],[34,387281],{"className":387282,"style":32604},[31845],[34,387284,387286,387289,387292,387295,387298,387301,387304],{"className":387285},[31676],[34,387287],{"className":387288,"style":31774},[31680],[34,387290,111],{"className":387291},[31685,31686],[34,387293,31965],{"className":387294},[31685,31686],[34,387296,2557],{"className":387297},[31690],[34,387299],{"className":387300,"style":32604},[31845],[34,387302,32226],{"className":387303},[32024],[34,387305],{"className":387306,"style":32604},[31845],[34,387308,387310,387313,387316,387319,387322],{"className":387309},[31676],[34,387311],{"className":387312,"style":32617},[31680],[34,387314,640],{"className":387315},[31685],[34,387317],{"className":387318,"style":31846},[31845],[34,387320,202],{"className":387321},[31850],[34,387323],{"className":387324,"style":31846},[31845],[34,387326,387328,387331,387334,387337,387340,387343],{"className":387327},[31676],[34,387329],{"className":387330,"style":35942},[31680],[34,387332,768],{"className":387333},[31685],[34,387335,111],{"className":387336},[31685,31686],[34,387338],{"className":387339,"style":32604},[31845],[34,387341,223],{"className":387342},[32024],[34,387344],{"className":387345,"style":32604},[31845],[34,387347,387349,387352,387355,387358,387361],{"className":387348},[31676],[34,387350],{"className":387351,"style":34299},[31680],[34,387353,1697],{"className":387354},[31685,31686],[34,387356],{"className":387357,"style":32604},[31845],[34,387359,223],{"className":387360},[32024],[34,387362],{"className":387363,"style":32604},[31845],[34,387365,387367,387370,387373,387376,387379],{"className":387366},[31676],[34,387368],{"className":387369,"style":32597},[31680],[34,387371,31965],{"className":387372},[31685,31686],[34,387374],{"className":387375,"style":32604},[31845],[34,387377,32226],{"className":387378},[32024],[34,387380],{"className":387381,"style":32604},[31845],[34,387383,387385,387388,387391,387394,387397],{"className":387384},[31676],[34,387386],{"className":387387,"style":35942},[31680],[34,387389,640],{"className":387390},[31685],[34,387392],{"className":387393,"style":32604},[31845],[34,387395,32226],{"className":387396},[32024],[34,387398],{"className":387399,"style":32604},[31845],[34,387401,387403,387406,387409,387412,387415],{"className":387402},[31676],[34,387404],{"className":387405,"style":34299},[31680],[34,387407,324191],{"className":387408},[31685,31686],[34,387410],{"className":387411,"style":32604},[31845],[34,387413,32226],{"className":387414},[32024],[34,387416],{"className":387417,"style":32604},[31845],[34,387419,387421,387424,387427],{"className":387420},[31676],[34,387422],{"className":387423,"style":32527},[31680],[34,387425,111],{"className":387426},[31685,31686],[34,387428,31965],{"className":387429},[31685,31686],[108,387431,387432],{},"左右相等，得证。",[15,387434,387435,387437,387438,387466,387467,347],{},[2627,387436,114043],{},"：这是一个交换幺环，其零元是旧 ",[34,387439,387441,387454],{"className":387440},[31642],[34,387442,387444],{"className":387443},[31646],[31648,387445,387446],{"xmlns":31650},[31652,387447,387448,387452],{},[31655,387449,387450],{},[31749,387451,640],{},[31665,387453,640],{"encoding":31667},[34,387455,387457],{"className":387456,"ariaHidden":758},[31672],[34,387458,387460,387463],{"className":387459},[31676],[34,387461],{"className":387462,"style":32617},[31680],[34,387464,640],{"className":387465},[31685],"，单位元是旧 ",[34,387468,387470,387483],{"className":387469},[31642],[34,387471,387473],{"className":387472},[31646],[31648,387474,387475],{"xmlns":31650},[31652,387476,387477,387481],{},[31655,387478,387479],{},[31749,387480,459],{},[31665,387482,459],{"encoding":31667},[34,387484,387486],{"className":387485,"ariaHidden":758},[31672],[34,387487,387489,387492],{"className":387488},[31676],[34,387490],{"className":387491,"style":32617},[31680],[34,387493,459],{"className":387494},[31685],[32188,387496],{},[253,387498,387499],{"id":387499},"题目8",[15,387501,387502,387503,387531,387532,387560],{},"给出环 ",[34,387504,387506,387519],{"className":387505},[31642],[34,387507,387509],{"className":387508},[31646],[31648,387510,387511],{"xmlns":31650},[31652,387512,387513,387517],{},[31655,387514,387515],{},[31658,387516,34386],{},[31665,387518,34386],{"encoding":31667},[34,387520,387522],{"className":387521,"ariaHidden":758},[31672],[34,387523,387525,387528],{"className":387524},[31676],[34,387526],{"className":387527,"style":33218},[31680],[34,387529,34386],{"className":387530,"style":34427},[31685,31686]," 和它的子环 ",[34,387533,387535,387548],{"className":387534},[31642],[34,387536,387538],{"className":387537},[31646],[31648,387539,387540],{"xmlns":31650},[31652,387541,387542,387546],{},[31655,387543,387544],{},[31658,387545,94675],{},[31665,387547,94675],{"encoding":31667},[34,387549,387551],{"className":387550,"ariaHidden":758},[31672],[34,387552,387554,387557],{"className":387553},[31676],[34,387555],{"className":387556,"style":33218},[31680],[34,387558,94675],{"className":387559,"style":106455},[31685,31686]," 的例子，使得它们满足以下条件...",[32490,387562,297792],{"id":387563},"详细解析-3",[15,387565,387566,387567,387570],{},"这道题考的是对",[2627,387568,387569],{},"子环","概念的辨析。注意：子环的单位元不一定要和父环一样，甚至子环可以没有单位元。",[15,387572,387573],{},[2627,387574,387575,387576,387604,387605,387633],{},"(1) ",[34,387577,387579,387592],{"className":387578},[31642],[34,387580,387582],{"className":387581},[31646],[31648,387583,387584],{"xmlns":31650},[31652,387585,387586,387590],{},[31655,387587,387588],{},[31658,387589,34386],{},[31665,387591,34386],{"encoding":31667},[34,387593,387595],{"className":387594,"ariaHidden":758},[31672],[34,387596,387598,387601],{"className":387597},[31676],[34,387599],{"className":387600,"style":33218},[31680],[34,387602,34386],{"className":387603,"style":34427},[31685,31686]," 有1，但 ",[34,387606,387608,387621],{"className":387607},[31642],[34,387609,387611],{"className":387610},[31646],[31648,387612,387613],{"xmlns":31650},[31652,387614,387615,387619],{},[31655,387616,387617],{},[31658,387618,94675],{},[31665,387620,94675],{"encoding":31667},[34,387622,387624],{"className":387623,"ariaHidden":758},[31672],[34,387625,387627,387630],{"className":387626},[31676],[34,387628],{"className":387629,"style":33218},[31680],[34,387631,94675],{"className":387632,"style":106455},[31685,31686]," 没有1",[105,387635,387636,387859],{},[108,387637,387638,387640,387641,387692,387693,347],{},[2627,387639,233206],{},"：整数环 ",[34,387642,387644,387662],{"className":387643},[31642],[34,387645,387647],{"className":387646},[31646],[31648,387648,387649],{"xmlns":31650},[31652,387650,387651,387659],{},[31655,387652,387653,387655,387657],{},[31658,387654,34386],{},[31662,387656,202],{},[31658,387658,97300],{"mathvariant":318416},[31665,387660,387661],{"encoding":31667},"R = \\mathbb{Z}",[34,387663,387665,387683],{"className":387664,"ariaHidden":758},[31672],[34,387666,387668,387671,387674,387677,387680],{"className":387667},[31676],[34,387669],{"className":387670,"style":33218},[31680],[34,387672,34386],{"className":387673,"style":34427},[31685,31686],[34,387675],{"className":387676,"style":31846},[31845],[34,387678,202],{"className":387679},[31850],[34,387681],{"className":387682,"style":31846},[31845],[34,387684,387686,387689],{"className":387685},[31676],[34,387687],{"className":387688,"style":320927},[31680],[34,387690,97300],{"className":387691},[31685,318441],"，偶数子环 ",[34,387694,387696,387748],{"className":387695},[31642],[34,387697,387699],{"className":387698},[31646],[31648,387700,387701],{"xmlns":31650},[31652,387702,387703,387745],{},[31655,387704,387705,387707,387709,387711,387713,387715,387717,387719,387721,387723,387725,387727,387729,387731,387733,387735,387737,387739,387741,387743],{},[31658,387706,94675],{},[31662,387708,202],{},[31749,387710,768],{},[31658,387712,97300],{"mathvariant":318416},[31662,387714,202],{},[31662,387716,32685],{"stretchy":702},[31662,387718,33934],{},[31662,387720,31960],{"separator":758},[31662,387722,32226],{},[31749,387724,768],{},[31662,387726,31960],{"separator":758},[31749,387728,459],{},[31662,387730,31960],{"separator":758},[31749,387732,768],{},[31662,387734,31960],{"separator":758},[31749,387736,15932],{},[31662,387738,31960],{"separator":758},[31662,387740,33934],{},[39361,387742,39363],{},[31662,387744,32715],{"stretchy":702},[31665,387746,387747],{"encoding":31667},"S = 2\\mathbb{Z} = \\{\\dots, -2, 0, 2, 4, \\dots\\}",[34,387749,387751,387769,387790],{"className":387750,"ariaHidden":758},[31672],[34,387752,387754,387757,387760,387763,387766],{"className":387753},[31676],[34,387755],{"className":387756,"style":33218},[31680],[34,387758,94675],{"className":387759,"style":106455},[31685,31686],[34,387761],{"className":387762,"style":31846},[31845],[34,387764,202],{"className":387765},[31850],[34,387767],{"className":387768,"style":31846},[31845],[34,387770,387772,387775,387778,387781,387784,387787],{"className":387771},[31676],[34,387773],{"className":387774,"style":320927},[31680],[34,387776,768],{"className":387777},[31685],[34,387779,97300],{"className":387780},[31685,318441],[34,387782],{"className":387783,"style":31846},[31845],[34,387785,202],{"className":387786},[31850],[34,387788],{"className":387789,"style":31846},[31845],[34,387791,387793,387796,387799,387802,387805,387808,387811,387814,387817,387820,387823,387826,387829,387832,387835,387838,387841,387844,387847,387850,387853,387856],{"className":387792},[31676],[34,387794],{"className":387795,"style":31774},[31680],[34,387797,32685],{"className":387798},[31782],[34,387800,33934],{"className":387801},[34006],[34,387803],{"className":387804,"style":32123},[31845],[34,387806,31960],{"className":387807},[32119],[34,387809],{"className":387810,"style":32123},[31845],[34,387812,32226],{"className":387813},[31685],[34,387815,768],{"className":387816},[31685],[34,387818,31960],{"className":387819},[32119],[34,387821],{"className":387822,"style":32123},[31845],[34,387824,459],{"className":387825},[31685],[34,387827,31960],{"className":387828},[32119],[34,387830],{"className":387831,"style":32123},[31845],[34,387833,768],{"className":387834},[31685],[34,387836,31960],{"className":387837},[32119],[34,387839],{"className":387840,"style":32123},[31845],[34,387842,15932],{"className":387843},[31685],[34,387845,31960],{"className":387846},[32119],[34,387848],{"className":387849,"style":32123},[31845],[34,387851,33934],{"className":387852},[34006],[34,387854],{"className":387855,"style":32123},[31845],[34,387857,32715],{"className":387858},[31690],[108,387860,387861,2664,387863,387891,387892,387920,387921,387949,387950,337455,387978,388006,388007,388080,388081,388132,388133,388161,388162,328728],{},[2627,387862,118582],{},[34,387864,387866,387879],{"className":387865},[31642],[34,387867,387869],{"className":387868},[31646],[31648,387870,387871],{"xmlns":31650},[31652,387872,387873,387877],{},[31655,387874,387875],{},[31658,387876,97300],{"mathvariant":318416},[31665,387878,320917],{"encoding":31667},[34,387880,387882],{"className":387881,"ariaHidden":758},[31672],[34,387883,387885,387888],{"className":387884},[31676],[34,387886],{"className":387887,"style":320927},[31680],[34,387889,97300],{"className":387890},[31685,318441]," 的单位元是 ",[34,387893,387895,387908],{"className":387894},[31642],[34,387896,387898],{"className":387897},[31646],[31648,387899,387900],{"xmlns":31650},[31652,387901,387902,387906],{},[31655,387903,387904],{},[31749,387905,640],{},[31665,387907,640],{"encoding":31667},[34,387909,387911],{"className":387910,"ariaHidden":758},[31672],[34,387912,387914,387917],{"className":387913},[31676],[34,387915],{"className":387916,"style":32617},[31680],[34,387918,640],{"className":387919},[31685],"。但 ",[34,387922,387924,387937],{"className":387923},[31642],[34,387925,387927],{"className":387926},[31646],[31648,387928,387929],{"xmlns":31650},[31652,387930,387931,387935],{},[31655,387932,387933],{},[31749,387934,640],{},[31665,387936,640],{"encoding":31667},[34,387938,387940],{"className":387939,"ariaHidden":758},[31672],[34,387941,387943,387946],{"className":387942},[31676],[34,387944],{"className":387945,"style":32617},[31680],[34,387947,640],{"className":387948},[31685]," 不是偶数，所以不在 ",[34,387951,387953,387966],{"className":387952},[31642],[34,387954,387956],{"className":387955},[31646],[31648,387957,387958],{"xmlns":31650},[31652,387959,387960,387964],{},[31655,387961,387962],{},[31658,387963,94675],{},[31665,387965,94675],{"encoding":31667},[34,387967,387969],{"className":387968,"ariaHidden":758},[31672],[34,387970,387972,387975],{"className":387971},[31676],[34,387973],{"className":387974,"style":33218},[31680],[34,387976,94675],{"className":387977,"style":106455},[31685,31686],[34,387979,387981,387994],{"className":387980},[31642],[34,387982,387984],{"className":387983},[31646],[31648,387985,387986],{"xmlns":31650},[31652,387987,387988,387992],{},[31655,387989,387990],{},[31658,387991,94675],{},[31665,387993,94675],{"encoding":31667},[34,387995,387997],{"className":387996,"ariaHidden":758},[31672],[34,387998,388000,388003],{"className":387999},[31676],[34,388001],{"className":388002,"style":33218},[31680],[34,388004,94675],{"className":388005,"style":106455},[31685,31686]," 里没有任何元素能起到单位元的作用（比如 ",[34,388008,388010,388032],{"className":388009},[31642],[34,388011,388013],{"className":388012},[31646],[31648,388014,388015],{"xmlns":31650},[31652,388016,388017,388029],{},[31655,388018,388019,388021,388023,388025,388027],{},[31749,388020,768],{},[31662,388022,42071],{},[31658,388024,32936],{},[31662,388026,202],{},[31749,388028,768],{},[31665,388030,388031],{"encoding":31667},"2 \\times x = 2",[34,388033,388035,388053,388071],{"className":388034,"ariaHidden":758},[31672],[34,388036,388038,388041,388044,388047,388050],{"className":388037},[31676],[34,388039],{"className":388040,"style":35942},[31680],[34,388042,768],{"className":388043},[31685],[34,388045],{"className":388046,"style":32604},[31845],[34,388048,42071],{"className":388049},[32024],[34,388051],{"className":388052,"style":32604},[31845],[34,388054,388056,388059,388062,388065,388068],{"className":388055},[31676],[34,388057],{"className":388058,"style":32527},[31680],[34,388060,32936],{"className":388061},[31685,31686],[34,388063],{"className":388064,"style":31846},[31845],[34,388066,202],{"className":388067},[31850],[34,388069],{"className":388070,"style":31846},[31845],[34,388072,388074,388077],{"className":388073},[31676],[34,388075],{"className":388076,"style":32617},[31680],[34,388078,768],{"className":388079},[31685]," 只有 ",[34,388082,388084,388102],{"className":388083},[31642],[34,388085,388087],{"className":388086},[31646],[31648,388088,388089],{"xmlns":31650},[31652,388090,388091,388099],{},[31655,388092,388093,388095,388097],{},[31658,388094,32936],{},[31662,388096,202],{},[31749,388098,640],{},[31665,388100,388101],{"encoding":31667},"x=1",[34,388103,388105,388123],{"className":388104,"ariaHidden":758},[31672],[34,388106,388108,388111,388114,388117,388120],{"className":388107},[31676],[34,388109],{"className":388110,"style":32527},[31680],[34,388112,32936],{"className":388113},[31685,31686],[34,388115],{"className":388116,"style":31846},[31845],[34,388118,202],{"className":388119},[31850],[34,388121],{"className":388122,"style":31846},[31845],[34,388124,388126,388129],{"className":388125},[31676],[34,388127],{"className":388128,"style":32617},[31680],[34,388130,640],{"className":388131},[31685]," 才行，但 ",[34,388134,388136,388149],{"className":388135},[31642],[34,388137,388139],{"className":388138},[31646],[31648,388140,388141],{"xmlns":31650},[31652,388142,388143,388147],{},[31655,388144,388145],{},[31749,388146,640],{},[31665,388148,640],{"encoding":31667},[34,388150,388152],{"className":388151,"ariaHidden":758},[31672],[34,388153,388155,388158],{"className":388154},[31676],[34,388156],{"className":388157,"style":32617},[31680],[34,388159,640],{"className":388160},[31685]," 不在 ",[34,388163,388165,388178],{"className":388164},[31642],[34,388166,388168],{"className":388167},[31646],[31648,388169,388170],{"xmlns":31650},[31652,388171,388172,388176],{},[31655,388173,388174],{},[31658,388175,94675],{},[31665,388177,94675],{"encoding":31667},[34,388179,388181],{"className":388180,"ariaHidden":758},[31672],[34,388182,388184,388187],{"className":388183},[31676],[34,388185],{"className":388186,"style":33218},[31680],[34,388188,94675],{"className":388189,"style":106455},[31685,31686],[15,388191,388192],{},[2627,388193,388194,388195,388223,388224,388252],{},"(2) ",[34,388196,388198,388211],{"className":388197},[31642],[34,388199,388201],{"className":388200},[31646],[31648,388202,388203],{"xmlns":31650},[31652,388204,388205,388209],{},[31655,388206,388207],{},[31658,388208,34386],{},[31665,388210,34386],{"encoding":31667},[34,388212,388214],{"className":388213,"ariaHidden":758},[31672],[34,388215,388217,388220],{"className":388216},[31676],[34,388218],{"className":388219,"style":33218},[31680],[34,388221,34386],{"className":388222,"style":34427},[31685,31686]," 没有1，但 ",[34,388225,388227,388240],{"className":388226},[31642],[34,388228,388230],{"className":388229},[31646],[31648,388231,388232],{"xmlns":31650},[31652,388233,388234,388238],{},[31655,388235,388236],{},[31658,388237,94675],{},[31665,388239,94675],{"encoding":31667},[34,388241,388243],{"className":388242,"ariaHidden":758},[31672],[34,388244,388246,388249],{"className":388245},[31676],[34,388247],{"className":388248,"style":33218},[31680],[34,388250,94675],{"className":388251,"style":106455},[31685,31686]," 有1",[105,388254,388255,388608],{},[108,388256,388257,2664,388259,388440,388441,347],{},[2627,388258,233206],{},[34,388260,388262,388316],{"className":388261},[31642],[34,388263,388265],{"className":388264},[31646],[31648,388266,388267],{"xmlns":31650},[31652,388268,388269,388313],{},[31655,388270,388271,388273,388275,388277,388279,388281,388283,388285,388287,388289,388291,388293,388295,388297,388299,388301,388303,388305,388307,388309,388311],{},[31658,388272,34386],{},[31662,388274,202],{},[31658,388276,97300],{"mathvariant":318416},[31662,388278,378006],{},[31749,388280,768],{},[31658,388282,97300],{"mathvariant":318416},[31662,388284,202],{},[31662,388286,32685],{"stretchy":702},[31662,388288,456],{"stretchy":702},[31658,388290,111],{},[31662,388292,31960],{"separator":758},[31749,388294,768],{},[31658,388296,1697],{},[31662,388298,2557],{"stretchy":702},[31658,388300,162086],{"mathvariant":38079},[31658,388302,111],{},[31662,388304,31960],{"separator":758},[31658,388306,1697],{},[31662,388308,112296],{},[31658,388310,97300],{"mathvariant":318416},[31662,388312,32715],{"stretchy":702},[31665,388314,388315],{"encoding":31667},"R = \\mathbb{Z} \\oplus 2\\mathbb{Z} = \\{(a, 2b) | a,b \\in \\mathbb{Z}\\}",[34,388317,388319,388337,388355,388376,388428],{"className":388318,"ariaHidden":758},[31672],[34,388320,388322,388325,388328,388331,388334],{"className":388321},[31676],[34,388323],{"className":388324,"style":33218},[31680],[34,388326,34386],{"className":388327,"style":34427},[31685,31686],[34,388329],{"className":388330,"style":31846},[31845],[34,388332,202],{"className":388333},[31850],[34,388335],{"className":388336,"style":31846},[31845],[34,388338,388340,388343,388346,388349,388352],{"className":388339},[31676],[34,388341],{"className":388342,"style":379449},[31680],[34,388344,97300],{"className":388345},[31685,318441],[34,388347],{"className":388348,"style":32604},[31845],[34,388350,378006],{"className":388351},[32024],[34,388353],{"className":388354,"style":32604},[31845],[34,388356,388358,388361,388364,388367,388370,388373],{"className":388357},[31676],[34,388359],{"className":388360,"style":320927},[31680],[34,388362,768],{"className":388363},[31685],[34,388365,97300],{"className":388366},[31685,318441],[34,388368],{"className":388369,"style":31846},[31845],[34,388371,202],{"className":388372},[31850],[34,388374],{"className":388375,"style":31846},[31845],[34,388377,388379,388382,388386,388389,388392,388395,388398,388401,388404,388407,388410,388413,388416,388419,388422,388425],{"className":388378},[31676],[34,388380],{"className":388381,"style":31774},[31680],[34,388383,388385],{"className":388384},[31782],"{(",[34,388387,111],{"className":388388},[31685,31686],[34,388390,31960],{"className":388391},[32119],[34,388393],{"className":388394,"style":32123},[31845],[34,388396,768],{"className":388397},[31685],[34,388399,1697],{"className":388400},[31685,31686],[34,388402,2557],{"className":388403},[31690],[34,388405,162086],{"className":388406},[31685],[34,388408,111],{"className":388409},[31685,31686],[34,388411,31960],{"className":388412},[32119],[34,388414],{"className":388415,"style":32123},[31845],[34,388417,1697],{"className":388418},[31685,31686],[34,388420],{"className":388421,"style":31846},[31845],[34,388423,112296],{"className":388424},[31850],[34,388426],{"className":388427,"style":31846},[31845],[34,388429,388431,388434,388437],{"className":388430},[31676],[34,388432],{"className":388433,"style":31774},[31680],[34,388435,97300],{"className":388436},[31685,318441],[34,388438,32715],{"className":388439},[31690],"，乘法是分量相乘。\n",[34,388442,388444,388494],{"className":388443},[31642],[34,388445,388447],{"className":388446},[31646],[31648,388448,388449],{"xmlns":31650},[31652,388450,388451,388491],{},[31655,388452,388453,388455,388457,388459,388461,388463,388465,388467,388469,388471,388473,388475,388477,388479,388481,388483,388485,388487,388489],{},[31658,388454,94675],{},[31662,388456,202],{},[31658,388458,97300],{"mathvariant":318416},[31662,388460,378006],{},[31662,388462,32685],{"stretchy":702},[31749,388464,459],{},[31662,388466,32715],{"stretchy":702},[31662,388468,202],{},[31662,388470,32685],{"stretchy":702},[31662,388472,456],{"stretchy":702},[31658,388474,111],{},[31662,388476,31960],{"separator":758},[31749,388478,459],{},[31662,388480,2557],{"stretchy":702},[31658,388482,162086],{"mathvariant":38079},[31658,388484,111],{},[31662,388486,112296],{},[31658,388488,97300],{"mathvariant":318416},[31662,388490,32715],{"stretchy":702},[31665,388492,388493],{"encoding":31667},"S = \\mathbb{Z} \\oplus \\{0\\} = \\{(a, 0) | a \\in \\mathbb{Z}\\}",[34,388495,388497,388515,388533,388557,388596],{"className":388496,"ariaHidden":758},[31672],[34,388498,388500,388503,388506,388509,388512],{"className":388499},[31676],[34,388501],{"className":388502,"style":33218},[31680],[34,388504,94675],{"className":388505,"style":106455},[31685,31686],[34,388507],{"className":388508,"style":31846},[31845],[34,388510,202],{"className":388511},[31850],[34,388513],{"className":388514,"style":31846},[31845],[34,388516,388518,388521,388524,388527,388530],{"className":388517},[31676],[34,388519],{"className":388520,"style":379449},[31680],[34,388522,97300],{"className":388523},[31685,318441],[34,388525],{"className":388526,"style":32604},[31845],[34,388528,378006],{"className":388529},[32024],[34,388531],{"className":388532,"style":32604},[31845],[34,388534,388536,388539,388542,388545,388548,388551,388554],{"className":388535},[31676],[34,388537],{"className":388538,"style":31774},[31680],[34,388540,32685],{"className":388541},[31782],[34,388543,459],{"className":388544},[31685],[34,388546,32715],{"className":388547},[31690],[34,388549],{"className":388550,"style":31846},[31845],[34,388552,202],{"className":388553},[31850],[34,388555],{"className":388556,"style":31846},[31845],[34,388558,388560,388563,388566,388569,388572,388575,388578,388581,388584,388587,388590,388593],{"className":388559},[31676],[34,388561],{"className":388562,"style":31774},[31680],[34,388564,388385],{"className":388565},[31782],[34,388567,111],{"className":388568},[31685,31686],[34,388570,31960],{"className":388571},[32119],[34,388573],{"className":388574,"style":32123},[31845],[34,388576,459],{"className":388577},[31685],[34,388579,2557],{"className":388580},[31690],[34,388582,162086],{"className":388583},[31685],[34,388585,111],{"className":388586},[31685,31686],[34,388588],{"className":388589,"style":31846},[31845],[34,388591,112296],{"className":388592},[31850],[34,388594],{"className":388595,"style":31846},[31845],[34,388597,388599,388602,388605],{"className":388598},[31676],[34,388600],{"className":388601,"style":31774},[31680],[34,388603,97300],{"className":388604},[31685,318441],[34,388606,32715],{"className":388607},[31690],[108,388609,388610,33183,388612],{},[2627,388611,118582],{},[105,388613,388614,388673,388881],{},[108,388615,388616,11008,388644,388672],{},[34,388617,388619,388632],{"className":388618},[31642],[34,388620,388622],{"className":388621},[31646],[31648,388623,388624],{"xmlns":31650},[31652,388625,388626,388630],{},[31655,388627,388628],{},[31658,388629,94675],{},[31665,388631,94675],{"encoding":31667},[34,388633,388635],{"className":388634,"ariaHidden":758},[31672],[34,388636,388638,388641],{"className":388637},[31676],[34,388639],{"className":388640,"style":33218},[31680],[34,388642,94675],{"className":388643,"style":106455},[31685,31686],[34,388645,388647,388660],{"className":388646},[31642],[34,388648,388650],{"className":388649},[31646],[31648,388651,388652],{"xmlns":31650},[31652,388653,388654,388658],{},[31655,388655,388656],{},[31658,388657,34386],{},[31665,388659,34386],{"encoding":31667},[34,388661,388663],{"className":388662,"ariaHidden":758},[31672],[34,388664,388666,388669],{"className":388665},[31676],[34,388667],{"className":388668,"style":33218},[31680],[34,388670,34386],{"className":388671,"style":34427},[31685,31686]," 的子环。",[108,388674,2884,388675,388703,388704,388755,388756,347],{},[34,388676,388678,388691],{"className":388677},[31642],[34,388679,388681],{"className":388680},[31646],[31648,388682,388683],{"xmlns":31650},[31652,388684,388685,388689],{},[31655,388686,388687],{},[31658,388688,94675],{},[31665,388690,94675],{"encoding":31667},[34,388692,388694],{"className":388693,"ariaHidden":758},[31672],[34,388695,388697,388700],{"className":388696},[31676],[34,388698],{"className":388699,"style":33218},[31680],[34,388701,94675],{"className":388702,"style":106455},[31685,31686]," 里，元素 ",[34,388705,388707,388728],{"className":388706},[31642],[34,388708,388710],{"className":388709},[31646],[31648,388711,388712],{"xmlns":31650},[31652,388713,388714,388726],{},[31655,388715,388716,388718,388720,388722,388724],{},[31662,388717,456],{"stretchy":702},[31749,388719,640],{},[31662,388721,31960],{"separator":758},[31749,388723,459],{},[31662,388725,2557],{"stretchy":702},[31665,388727,381896],{"encoding":31667},[34,388729,388731],{"className":388730,"ariaHidden":758},[31672],[34,388732,388734,388737,388740,388743,388746,388749,388752],{"className":388733},[31676],[34,388735],{"className":388736,"style":31774},[31680],[34,388738,456],{"className":388739},[31782],[34,388741,640],{"className":388742},[31685],[34,388744,31960],{"className":388745},[32119],[34,388747],{"className":388748,"style":32123},[31845],[34,388750,459],{"className":388751},[31685],[34,388753,2557],{"className":388754},[31690]," 就是单位元，因为 ",[34,388757,388759,388803],{"className":388758},[31642],[34,388760,388762],{"className":388761},[31646],[31648,388763,388764],{"xmlns":31650},[31652,388765,388766,388800],{},[31655,388767,388768,388770,388772,388774,388776,388778,388780,388782,388784,388786,388788,388790,388792,388794,388796,388798],{},[31662,388769,456],{"stretchy":702},[31658,388771,111],{},[31662,388773,31960],{"separator":758},[31749,388775,459],{},[31662,388777,2557],{"stretchy":702},[31662,388779,456],{"stretchy":702},[31749,388781,640],{},[31662,388783,31960],{"separator":758},[31749,388785,459],{},[31662,388787,2557],{"stretchy":702},[31662,388789,202],{},[31662,388791,456],{"stretchy":702},[31658,388793,111],{},[31662,388795,31960],{"separator":758},[31749,388797,459],{},[31662,388799,2557],{"stretchy":702},[31665,388801,388802],{"encoding":31667},"(a,0)(1,0) = (a,0)",[34,388804,388806,388857],{"className":388805,"ariaHidden":758},[31672],[34,388807,388809,388812,388815,388818,388821,388824,388827,388830,388833,388836,388839,388842,388845,388848,388851,388854],{"className":388808},[31676],[34,388810],{"className":388811,"style":31774},[31680],[34,388813,456],{"className":388814},[31782],[34,388816,111],{"className":388817},[31685,31686],[34,388819,31960],{"className":388820},[32119],[34,388822],{"className":388823,"style":32123},[31845],[34,388825,459],{"className":388826},[31685],[34,388828,2557],{"className":388829},[31690],[34,388831,456],{"className":388832},[31782],[34,388834,640],{"className":388835},[31685],[34,388837,31960],{"className":388838},[32119],[34,388840],{"className":388841,"style":32123},[31845],[34,388843,459],{"className":388844},[31685],[34,388846,2557],{"className":388847},[31690],[34,388849],{"className":388850,"style":31846},[31845],[34,388852,202],{"className":388853},[31850],[34,388855],{"className":388856,"style":31846},[31845],[34,388858,388860,388863,388866,388869,388872,388875,388878],{"className":388859},[31676],[34,388861],{"className":388862,"style":31774},[31680],[34,388864,456],{"className":388865},[31782],[34,388867,111],{"className":388868},[31685,31686],[34,388870,31960],{"className":388871},[32119],[34,388873],{"className":388874,"style":32123},[31845],[34,388876,459],{"className":388877},[31685],[34,388879,2557],{"className":388880},[31690],[108,388882,2884,388883,388911,388912,388968,388969,388997,388998,389026,389027,389055],{},[34,388884,388886,388899],{"className":388885},[31642],[34,388887,388889],{"className":388888},[31646],[31648,388890,388891],{"xmlns":31650},[31652,388892,388893,388897],{},[31655,388894,388895],{},[31658,388896,34386],{},[31665,388898,34386],{"encoding":31667},[34,388900,388902],{"className":388901,"ariaHidden":758},[31672],[34,388903,388905,388908],{"className":388904},[31676],[34,388906],{"className":388907,"style":33218},[31680],[34,388909,34386],{"className":388910,"style":34427},[31685,31686]," 里，要想有单位元，必须是 ",[34,388913,388915,388941],{"className":388914},[31642],[34,388916,388918],{"className":388917},[31646],[31648,388919,388920],{"xmlns":31650},[31652,388921,388922,388938],{},[31655,388923,388924,388926,388928,388930,388932,388934,388936],{},[31662,388925,456],{"stretchy":702},[31749,388927,640],{},[31662,388929,31960],{"separator":758},[31749,388931,640],{},[31658,388933,38080],{"mathvariant":38079},[31749,388935,768],{},[31662,388937,2557],{"stretchy":702},[31665,388939,388940],{"encoding":31667},"(1, 1\u002F2)",[34,388942,388944],{"className":388943,"ariaHidden":758},[31672],[34,388945,388947,388950,388953,388956,388959,388962,388965],{"className":388946},[31676],[34,388948],{"className":388949,"style":31774},[31680],[34,388951,456],{"className":388952},[31782],[34,388954,640],{"className":388955},[31685],[34,388957,31960],{"className":388958},[32119],[34,388960],{"className":388961,"style":32123},[31845],[34,388963,42407],{"className":388964},[31685],[34,388966,2557],{"className":388967},[31690]," 这种形式，但这不在 ",[34,388970,388972,388985],{"className":388971},[31642],[34,388973,388975],{"className":388974},[31646],[31648,388976,388977],{"xmlns":31650},[31652,388978,388979,388983],{},[31655,388980,388981],{},[31658,388982,34386],{},[31665,388984,34386],{"encoding":31667},[34,388986,388988],{"className":388987,"ariaHidden":758},[31672],[34,388989,388991,388994],{"className":388990},[31676],[34,388992],{"className":388993,"style":33218},[31680],[34,388995,34386],{"className":388996,"style":34427},[31685,31686]," 里。所以 ",[34,388999,389001,389014],{"className":389000},[31642],[34,389002,389004],{"className":389003},[31646],[31648,389005,389006],{"xmlns":31650},[31652,389007,389008,389012],{},[31655,389009,389010],{},[31658,389011,34386],{},[31665,389013,34386],{"encoding":31667},[34,389015,389017],{"className":389016,"ariaHidden":758},[31672],[34,389018,389020,389023],{"className":389019},[31676],[34,389021],{"className":389022,"style":33218},[31680],[34,389024,34386],{"className":389025,"style":34427},[31685,31686]," 整体没有单位元，但它的局部 ",[34,389028,389030,389043],{"className":389029},[31642],[34,389031,389033],{"className":389032},[31646],[31648,389034,389035],{"xmlns":31650},[31652,389036,389037,389041],{},[31655,389038,389039],{},[31658,389040,94675],{},[31665,389042,94675],{"encoding":31667},[34,389044,389046],{"className":389045,"ariaHidden":758},[31672],[34,389047,389049,389052],{"className":389048},[31676],[34,389050],{"className":389051,"style":33218},[31680],[34,389053,94675],{"className":389054,"style":106455},[31685,31686]," 却有。",[15,389057,389058],{},[2627,389059,389060,389061,43298,389089,389117],{},"(3) ",[34,389062,389064,389077],{"className":389063},[31642],[34,389065,389067],{"className":389066},[31646],[31648,389068,389069],{"xmlns":31650},[31652,389070,389071,389075],{},[31655,389072,389073],{},[31658,389074,34386],{},[31665,389076,34386],{"encoding":31667},[34,389078,389080],{"className":389079,"ariaHidden":758},[31672],[34,389081,389083,389086],{"className":389082},[31676],[34,389084],{"className":389085,"style":33218},[31680],[34,389087,34386],{"className":389088,"style":34427},[31685,31686],[34,389090,389092,389105],{"className":389091},[31642],[34,389093,389095],{"className":389094},[31646],[31648,389096,389097],{"xmlns":31650},[31652,389098,389099,389103],{},[31655,389100,389101],{},[31658,389102,94675],{},[31665,389104,94675],{"encoding":31667},[34,389106,389108],{"className":389107,"ariaHidden":758},[31672],[34,389109,389111,389114],{"className":389110},[31676],[34,389112],{"className":389113,"style":33218},[31680],[34,389115,94675],{"className":389116,"style":106455},[31685,31686]," 都有1",[105,389119,389120,389667],{},[108,389121,389122,39984,389154],{},[2627,389123,389124,389125,389153],{},"情况A：",[34,389126,389128,389141],{"className":389127},[31642],[34,389129,389131],{"className":389130},[31646],[31648,389132,389133],{"xmlns":31650},[31652,389134,389135,389139],{},[31655,389136,389137],{},[31749,389138,640],{},[31665,389140,640],{"encoding":31667},[34,389142,389144],{"className":389143,"ariaHidden":758},[31672],[34,389145,389147,389150],{"className":389146},[31676],[34,389148],{"className":389149,"style":32617},[31680],[34,389151,640],{"className":389152},[31685]," 不一样",[105,389155,389156,389284,389420],{},[108,389157,389158,389231,389232,347],{},[34,389159,389161,389183],{"className":389160},[31642],[34,389162,389164],{"className":389163},[31646],[31648,389165,389166],{"xmlns":31650},[31652,389167,389168,389180],{},[31655,389169,389170,389172,389174,389176,389178],{},[31658,389171,34386],{},[31662,389173,202],{},[31658,389175,97300],{"mathvariant":318416},[31662,389177,42071],{},[31658,389179,97300],{"mathvariant":318416},[31665,389181,389182],{"encoding":31667},"R = \\mathbb{Z} \\times \\mathbb{Z}",[34,389184,389186,389204,389222],{"className":389185,"ariaHidden":758},[31672],[34,389187,389189,389192,389195,389198,389201],{"className":389188},[31676],[34,389190],{"className":389191,"style":33218},[31680],[34,389193,34386],{"className":389194,"style":34427},[31685,31686],[34,389196],{"className":389197,"style":31846},[31845],[34,389199,202],{"className":389200},[31850],[34,389202],{"className":389203,"style":31846},[31845],[34,389205,389207,389210,389213,389216,389219],{"className":389206},[31676],[34,389208],{"className":389209,"style":379449},[31680],[34,389211,97300],{"className":389212},[31685,318441],[34,389214],{"className":389215,"style":32604},[31845],[34,389217,42071],{"className":389218},[32024],[34,389220],{"className":389221,"style":32604},[31845],[34,389223,389225,389228],{"className":389224},[31676],[34,389226],{"className":389227,"style":320927},[31680],[34,389229,97300],{"className":389230},[31685,318441],"，单位元是 ",[34,389233,389235,389257],{"className":389234},[31642],[34,389236,389238],{"className":389237},[31646],[31648,389239,389240],{"xmlns":31650},[31652,389241,389242,389254],{},[31655,389243,389244,389246,389248,389250,389252],{},[31662,389245,456],{"stretchy":702},[31749,389247,640],{},[31662,389249,31960],{"separator":758},[31749,389251,640],{},[31662,389253,2557],{"stretchy":702},[31665,389255,389256],{"encoding":31667},"(1,1)",[34,389258,389260],{"className":389259,"ariaHidden":758},[31672],[34,389261,389263,389266,389269,389272,389275,389278,389281],{"className":389262},[31676],[34,389264],{"className":389265,"style":31774},[31680],[34,389267,456],{"className":389268},[31782],[34,389270,640],{"className":389271},[31685],[34,389273,31960],{"className":389274},[32119],[34,389276],{"className":389277,"style":32123},[31845],[34,389279,640],{"className":389280},[31685],[34,389282,2557],{"className":389283},[31690],[108,389285,389286,389231,389369,347],{},[34,389287,389289,389315],{"className":389288},[31642],[34,389290,389292],{"className":389291},[31646],[31648,389293,389294],{"xmlns":31650},[31652,389295,389296,389312],{},[31655,389297,389298,389300,389302,389304,389306,389308,389310],{},[31658,389299,94675],{},[31662,389301,202],{},[31658,389303,97300],{"mathvariant":318416},[31662,389305,42071],{},[31662,389307,32685],{"stretchy":702},[31749,389309,459],{},[31662,389311,32715],{"stretchy":702},[31665,389313,389314],{"encoding":31667},"S = \\mathbb{Z} \\times \\{0\\}",[34,389316,389318,389336,389354],{"className":389317,"ariaHidden":758},[31672],[34,389319,389321,389324,389327,389330,389333],{"className":389320},[31676],[34,389322],{"className":389323,"style":33218},[31680],[34,389325,94675],{"className":389326,"style":106455},[31685,31686],[34,389328],{"className":389329,"style":31846},[31845],[34,389331,202],{"className":389332},[31850],[34,389334],{"className":389335,"style":31846},[31845],[34,389337,389339,389342,389345,389348,389351],{"className":389338},[31676],[34,389340],{"className":389341,"style":379449},[31680],[34,389343,97300],{"className":389344},[31685,318441],[34,389346],{"className":389347,"style":32604},[31845],[34,389349,42071],{"className":389350},[32024],[34,389352],{"className":389353,"style":32604},[31845],[34,389355,389357,389360,389363,389366],{"className":389356},[31676],[34,389358],{"className":389359,"style":31774},[31680],[34,389361,32685],{"className":389362},[31782],[34,389364,459],{"className":389365},[31685],[34,389367,32715],{"className":389368},[31690],[34,389370,389372,389393],{"className":389371},[31642],[34,389373,389375],{"className":389374},[31646],[31648,389376,389377],{"xmlns":31650},[31652,389378,389379,389391],{},[31655,389380,389381,389383,389385,389387,389389],{},[31662,389382,456],{"stretchy":702},[31749,389384,640],{},[31662,389386,31960],{"separator":758},[31749,389388,459],{},[31662,389390,2557],{"stretchy":702},[31665,389392,381896],{"encoding":31667},[34,389394,389396],{"className":389395,"ariaHidden":758},[31672],[34,389397,389399,389402,389405,389408,389411,389414,389417],{"className":389398},[31676],[34,389400],{"className":389401,"style":31774},[31680],[34,389403,456],{"className":389404},[31782],[34,389406,640],{"className":389407},[31685],[34,389409,31960],{"className":389410},[32119],[34,389412],{"className":389413,"style":32123},[31845],[34,389415,459],{"className":389416},[31685],[34,389418,2557],{"className":389419},[31690],[108,389421,389422,389423,314945,389474,389502,389503,389531,389532,389584,389585,389637,389638,389666],{},"注意 ",[34,389424,389426,389447],{"className":389425},[31642],[34,389427,389429],{"className":389428},[31646],[31648,389430,389431],{"xmlns":31650},[31652,389432,389433,389445],{},[31655,389434,389435,389437,389439,389441,389443],{},[31662,389436,456],{"stretchy":702},[31749,389438,640],{},[31662,389440,31960],{"separator":758},[31749,389442,459],{},[31662,389444,2557],{"stretchy":702},[31665,389446,381896],{"encoding":31667},[34,389448,389450],{"className":389449,"ariaHidden":758},[31672],[34,389451,389453,389456,389459,389462,389465,389468,389471],{"className":389452},[31676],[34,389454],{"className":389455,"style":31774},[31680],[34,389457,456],{"className":389458},[31782],[34,389460,640],{"className":389461},[31685],[34,389463,31960],{"className":389464},[32119],[34,389466],{"className":389467,"style":32123},[31845],[34,389469,459],{"className":389470},[31685],[34,389472,2557],{"className":389473},[31690],[34,389475,389477,389490],{"className":389476},[31642],[34,389478,389480],{"className":389479},[31646],[31648,389481,389482],{"xmlns":31650},[31652,389483,389484,389488],{},[31655,389485,389486],{},[31658,389487,94675],{},[31665,389489,94675],{"encoding":31667},[34,389491,389493],{"className":389492,"ariaHidden":758},[31672],[34,389494,389496,389499],{"className":389495},[31676],[34,389497],{"className":389498,"style":33218},[31680],[34,389500,94675],{"className":389501,"style":106455},[31685,31686]," 内部好使，但放到 ",[34,389504,389506,389519],{"className":389505},[31642],[34,389507,389509],{"className":389508},[31646],[31648,389510,389511],{"xmlns":31650},[31652,389512,389513,389517],{},[31655,389514,389515],{},[31658,389516,34386],{},[31665,389518,34386],{"encoding":31667},[34,389520,389522],{"className":389521,"ariaHidden":758},[31672],[34,389523,389525,389528],{"className":389524},[31676],[34,389526],{"className":389527,"style":33218},[31680],[34,389529,34386],{"className":389530,"style":34427},[31685,31686]," 里乘上 ",[34,389533,389535,389557],{"className":389534},[31642],[34,389536,389538],{"className":389537},[31646],[31648,389539,389540],{"xmlns":31650},[31652,389541,389542,389554],{},[31655,389543,389544,389546,389548,389550,389552],{},[31662,389545,456],{"stretchy":702},[31749,389547,768],{},[31662,389549,31960],{"separator":758},[31749,389551,786],{},[31662,389553,2557],{"stretchy":702},[31665,389555,389556],{"encoding":31667},"(2,3)",[34,389558,389560],{"className":389559,"ariaHidden":758},[31672],[34,389561,389563,389566,389569,389572,389575,389578,389581],{"className":389562},[31676],[34,389564],{"className":389565,"style":31774},[31680],[34,389567,456],{"className":389568},[31782],[34,389570,768],{"className":389571},[31685],[34,389573,31960],{"className":389574},[32119],[34,389576],{"className":389577,"style":32123},[31845],[34,389579,786],{"className":389580},[31685],[34,389582,2557],{"className":389583},[31690]," 就变成了 ",[34,389586,389588,389610],{"className":389587},[31642],[34,389589,389591],{"className":389590},[31646],[31648,389592,389593],{"xmlns":31650},[31652,389594,389595,389607],{},[31655,389596,389597,389599,389601,389603,389605],{},[31662,389598,456],{"stretchy":702},[31749,389600,768],{},[31662,389602,31960],{"separator":758},[31749,389604,459],{},[31662,389606,2557],{"stretchy":702},[31665,389608,389609],{"encoding":31667},"(2,0)",[34,389611,389613],{"className":389612,"ariaHidden":758},[31672],[34,389614,389616,389619,389622,389625,389628,389631,389634],{"className":389615},[31676],[34,389617],{"className":389618,"style":31774},[31680],[34,389620,456],{"className":389621},[31782],[34,389623,768],{"className":389624},[31685],[34,389626,31960],{"className":389627},[32119],[34,389629],{"className":389630,"style":32123},[31845],[34,389632,459],{"className":389633},[31685],[34,389635,2557],{"className":389636},[31690],"，所以它不是 ",[34,389639,389641,389654],{"className":389640},[31642],[34,389642,389644],{"className":389643},[31646],[31648,389645,389646],{"xmlns":31650},[31652,389647,389648,389652],{},[31655,389649,389650],{},[31658,389651,34386],{},[31665,389653,34386],{"encoding":31667},[34,389655,389657],{"className":389656,"ariaHidden":758},[31672],[34,389658,389660,389663],{"className":389659},[31676],[34,389661],{"className":389662,"style":33218},[31680],[34,389664,34386],{"className":389665,"style":34427},[31685,31686]," 的单位元。",[108,389668,389669,39984,389701],{},[2627,389670,389671,389672,389700],{},"情况B：",[34,389673,389675,389688],{"className":389674},[31642],[34,389676,389678],{"className":389677},[31646],[31648,389679,389680],{"xmlns":31650},[31652,389681,389682,389686],{},[31655,389683,389684],{},[31749,389685,640],{},[31665,389687,640],{"encoding":31667},[34,389689,389691],{"className":389690,"ariaHidden":758},[31672],[34,389692,389694,389697],{"className":389693},[31676],[34,389695],{"className":389696,"style":32617},[31680],[34,389698,640],{"className":389699},[31685]," 一样",[105,389702,389703],{},[108,389704,389705,389756,389757,389808,389809,347],{},[34,389706,389708,389726],{"className":389707},[31642],[34,389709,389711],{"className":389710},[31646],[31648,389712,389713],{"xmlns":31650},[31652,389714,389715,389723],{},[31655,389716,389717,389719,389721],{},[31658,389718,34386],{},[31662,389720,202],{},[31658,389722,116863],{"mathvariant":318416},[31665,389724,389725],{"encoding":31667},"R = \\mathbb{Q}",[34,389727,389729,389747],{"className":389728,"ariaHidden":758},[31672],[34,389730,389732,389735,389738,389741,389744],{"className":389731},[31676],[34,389733],{"className":389734,"style":33218},[31680],[34,389736,34386],{"className":389737,"style":34427},[31685,31686],[34,389739],{"className":389740,"style":31846},[31845],[34,389742,202],{"className":389743},[31850],[34,389745],{"className":389746,"style":31846},[31845],[34,389748,389750,389753],{"className":389749},[31676],[34,389751],{"className":389752,"style":369872},[31680],[34,389754,116863],{"className":389755},[31685,318441]," (有理数)，",[34,389758,389760,389778],{"className":389759},[31642],[34,389761,389763],{"className":389762},[31646],[31648,389764,389765],{"xmlns":31650},[31652,389766,389767,389775],{},[31655,389768,389769,389771,389773],{},[31658,389770,94675],{},[31662,389772,202],{},[31658,389774,97300],{"mathvariant":318416},[31665,389776,389777],{"encoding":31667},"S = \\mathbb{Z}",[34,389779,389781,389799],{"className":389780,"ariaHidden":758},[31672],[34,389782,389784,389787,389790,389793,389796],{"className":389783},[31676],[34,389785],{"className":389786,"style":33218},[31680],[34,389788,94675],{"className":389789,"style":106455},[31685,31686],[34,389791],{"className":389792,"style":31846},[31845],[34,389794,202],{"className":389795},[31850],[34,389797],{"className":389798,"style":31846},[31845],[34,389800,389802,389805],{"className":389801},[31676],[34,389803],{"className":389804,"style":320927},[31680],[34,389806,97300],{"className":389807},[31685,318441]," (整数)。大家共用同一个 ",[34,389810,389812,389825],{"className":389811},[31642],[34,389813,389815],{"className":389814},[31646],[31648,389816,389817],{"xmlns":31650},[31652,389818,389819,389823],{},[31655,389820,389821],{},[31749,389822,640],{},[31665,389824,640],{"encoding":31667},[34,389826,389828],{"className":389827,"ariaHidden":758},[31672],[34,389829,389831,389834],{"className":389830},[31676],[34,389832],{"className":389833,"style":32617},[31680],[34,389835,640],{"className":389836},[31685],[15,389838,389839],{},[2627,389840,389841,389842,389870,389871,370308],{},"(4) ",[34,389843,389845,389858],{"className":389844},[31642],[34,389846,389848],{"className":389847},[31646],[31648,389849,389850],{"xmlns":31650},[31652,389851,389852,389856],{},[31655,389853,389854],{},[31658,389855,34386],{},[31665,389857,34386],{"encoding":31667},[34,389859,389861],{"className":389860,"ariaHidden":758},[31672],[34,389862,389864,389867],{"className":389863},[31676],[34,389865],{"className":389866,"style":33218},[31680],[34,389868,34386],{"className":389869,"style":34427},[31685,31686]," 不交换，但 ",[34,389872,389874,389887],{"className":389873},[31642],[34,389875,389877],{"className":389876},[31646],[31648,389878,389879],{"xmlns":31650},[31652,389880,389881,389885],{},[31655,389882,389883],{},[31658,389884,94675],{},[31665,389886,94675],{"encoding":31667},[34,389888,389890],{"className":389889,"ariaHidden":758},[31672],[34,389891,389893,389896],{"className":389892},[31676],[34,389894],{"className":389895,"style":33218},[31680],[34,389897,94675],{"className":389898,"style":106455},[31685,31686],[105,389900,389901,390046],{},[108,389902,389903,2664,389905,390012,390013,390041,390042,390045],{},[2627,389904,233206],{},[34,389906,389908,389936],{"className":389907},[31642],[34,389909,389911],{"className":389910},[31646],[31648,389912,389913],{"xmlns":31650},[31652,389914,389915,389933],{},[31655,389916,389917,389919,389921,389927,389929,389931],{},[31658,389918,34386],{},[31662,389920,202],{},[31743,389922,389923,389925],{},[31658,389924,98360],{},[31749,389926,768],{},[31662,389928,456],{"stretchy":702},[31658,389930,34386],{"mathvariant":318416},[31662,389932,2557],{"stretchy":702},[31665,389934,389935],{"encoding":31667},"R = M_2(\\mathbb{R})",[34,389937,389939,389957],{"className":389938,"ariaHidden":758},[31672],[34,389940,389942,389945,389948,389951,389954],{"className":389941},[31676],[34,389943],{"className":389944,"style":33218},[31680],[34,389946,34386],{"className":389947,"style":34427},[31685,31686],[34,389949],{"className":389950,"style":31846},[31845],[34,389952,202],{"className":389953},[31850],[34,389955],{"className":389956,"style":31846},[31845],[34,389958,389960,389963,390003,390006,390009],{"className":389959},[31676],[34,389961],{"className":389962,"style":31774},[31680],[34,389964,389966,389969],{"className":389965},[31685],[34,389967,98360],{"className":389968,"style":33389},[31685,31686],[34,389970,389972],{"className":389971},[31792],[34,389973,389975,389995],{"className":389974},[31796,31797],[34,389976,389978,389992],{"className":389977},[31801],[34,389979,389981],{"className":389980,"style":31806},[31805],[34,389982,389983,389986],{"style":231884},[34,389984],{"className":389985,"style":31814},[31813],[34,389987,389989],{"className":389988},[31818,31819,31820,31821],[34,389990,768],{"className":389991},[31685,31821],[34,389993,31829],{"className":389994},[31828],[34,389996,389998],{"className":389997},[31801],[34,389999,390001],{"className":390000,"style":31836},[31805],[34,390002],{},[34,390004,456],{"className":390005},[31782],[34,390007,34386],{"className":390008},[31685,318441],[34,390010,2557],{"className":390011},[31690]," (二阶实矩阵环)，",[34,390014,390016,390029],{"className":390015},[31642],[34,390017,390019],{"className":390018},[31646],[31648,390020,390021],{"xmlns":31650},[31652,390022,390023,390027],{},[31655,390024,390025],{},[31658,390026,94675],{},[31665,390028,94675],{"encoding":31667},[34,390030,390032],{"className":390031,"ariaHidden":758},[31672],[34,390033,390035,390038],{"className":390034},[31676],[34,390036],{"className":390037,"style":33218},[31680],[34,390039,94675],{"className":390040,"style":106455},[31685,31686]," 是所有",[2627,390043,390044],{},"对角矩阵","的集合。",[108,390047,390048,33183,390050],{},[2627,390049,118582],{},[105,390051,390052,390084],{},[108,390053,390054,390055,390083],{},"矩阵乘法一般不交换，所以 ",[34,390056,390058,390071],{"className":390057},[31642],[34,390059,390061],{"className":390060},[31646],[31648,390062,390063],{"xmlns":31650},[31652,390064,390065,390069],{},[31655,390066,390067],{},[31658,390068,34386],{},[31665,390070,34386],{"encoding":31667},[34,390072,390074],{"className":390073,"ariaHidden":758},[31672],[34,390075,390077,390080],{"className":390076},[31676],[34,390078],{"className":390079,"style":33218},[31680],[34,390081,34386],{"className":390082,"style":34427},[31685,31686]," 不交换。",[108,390085,390086,390087,390262],{},"但是对角矩阵 ",[34,390088,390090,390136],{"className":390089},[31642],[34,390091,390093],{"className":390092},[31646],[31648,390094,390095],{"xmlns":31650},[31652,390096,390097,390133],{},[31655,390098,390099,390101,390131],{},[31662,390100,456],{"fence":758},[71078,390102,390103,390117],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,390104,390105,390111],{},[71087,390106,390107],{},[71090,390108,390109],{"scriptlevel":459,"displaystyle":702},[31658,390110,111],{},[71087,390112,390113],{},[71090,390114,390115],{"scriptlevel":459,"displaystyle":702},[31749,390116,459],{},[71084,390118,390119,390125],{},[71087,390120,390121],{},[71090,390122,390123],{"scriptlevel":459,"displaystyle":702},[31749,390124,459],{},[71087,390126,390127],{},[71090,390128,390129],{"scriptlevel":459,"displaystyle":702},[31658,390130,1697],{},[31662,390132,2557],{"fence":758},[31665,390134,390135],{"encoding":31667},"\\begin{pmatrix} a & 0 \\\\ 0 & b \\end{pmatrix}",[34,390137,390139],{"className":390138,"ariaHidden":758},[31672],[34,390140,390142,390145],{"className":390141},[31676],[34,390143],{"className":390144,"style":370661},[31680],[34,390146,390148,390154,390256],{"className":390147},[34006],[34,390149,390151],{"className":390150,"style":75811},[31782,75810],[34,390152,456],{"className":390153},[71281,31820],[34,390155,390157],{"className":390156},[31685],[34,390158,390160,390205,390208,390211],{"className":390159},[71078],[34,390161,390163],{"className":390162},[141481],[34,390164,390166,390197],{"className":390165},[31796,31797],[34,390167,390169,390194],{"className":390168},[31801],[34,390170,390172,390183],{"className":390171,"style":370689},[31805],[34,390173,390174,390177],{"style":370692},[34,390175],{"className":390176,"style":41508},[31813],[34,390178,390180],{"className":390179},[31685],[34,390181,111],{"className":390182},[31685,31686],[34,390184,390185,390188],{"style":370704},[34,390186],{"className":390187,"style":41508},[31813],[34,390189,390191],{"className":390190},[31685],[34,390192,459],{"className":390193},[31685],[34,390195,31829],{"className":390196},[31828],[34,390198,390200],{"className":390199},[31801],[34,390201,390203],{"className":390202,"style":370723},[31805],[34,390204],{},[34,390206],{"className":390207,"style":141477},[71732],[34,390209],{"className":390210,"style":141477},[71732],[34,390212,390214],{"className":390213},[141481],[34,390215,390217,390248],{"className":390216},[31796,31797],[34,390218,390220,390245],{"className":390219},[31801],[34,390221,390223,390234],{"className":390222,"style":370689},[31805],[34,390224,390225,390228],{"style":370692},[34,390226],{"className":390227,"style":41508},[31813],[34,390229,390231],{"className":390230},[31685],[34,390232,459],{"className":390233},[31685],[34,390235,390236,390239],{"style":370704},[34,390237],{"className":390238,"style":41508},[31813],[34,390240,390242],{"className":390241},[31685],[34,390243,1697],{"className":390244},[31685,31686],[34,390246,31829],{"className":390247},[31828],[34,390249,390251],{"className":390250},[31801],[34,390252,390254],{"className":390253,"style":370723},[31805],[34,390255],{},[34,390257,390259],{"className":390258,"style":75811},[31690,75810],[34,390260,2557],{"className":390261},[71281,31820]," 乘起来只看对角线，就像实数乘法一样，是可交换的。",[32188,390264],{},[253,390266,390267],{"id":390267},"题目9",[15,390269,390270],{},"关于左幺元和右幺元的证明。",[32490,390272,297792],{"id":390273},"详细解析-4",[15,390275,390276,390277,2664],{},"这是一个纯逻辑推导题。\n",[2627,390278,390279],{},"定义回顾",[105,390281,390282,390452],{},[108,390283,390284,390354,390355,347],{},[34,390285,390287,390305],{"className":390286},[31642],[34,390288,390290],{"className":390289},[31646],[31648,390291,390292],{"xmlns":31650},[31652,390293,390294,390302],{},[31655,390295,390296],{},[31743,390297,390298,390300],{},[31658,390299,33244],{},[31658,390301,43379],{},[31665,390303,390304],{"encoding":31667},"e_l",[34,390306,390308],{"className":390307,"ariaHidden":758},[31672],[34,390309,390311,390314],{"className":390310},[31676],[34,390312],{"className":390313,"style":31860},[31680],[34,390315,390317,390320],{"className":390316},[31685],[34,390318,33244],{"className":390319},[31685,31686],[34,390321,390323],{"className":390322},[31792],[34,390324,390326,390346],{"className":390325},[31796,31797],[34,390327,390329,390343],{"className":390328},[31801],[34,390330,390332],{"className":390331,"style":33833},[31805],[34,390333,390334,390337],{"style":31809},[34,390335],{"className":390336,"style":31814},[31813],[34,390338,390340],{"className":390339},[31818,31819,31820,31821],[34,390341,43379],{"className":390342,"style":47113},[31685,31686,31821],[34,390344,31829],{"className":390345},[31828],[34,390347,390349],{"className":390348},[31801],[34,390350,390352],{"className":390351,"style":31836},[31805],[34,390353],{}," (Left) 是左幺元：",[34,390356,390358,390382],{"className":390357},[31642],[34,390359,390361],{"className":390360},[31646],[31648,390362,390363],{"xmlns":31650},[31652,390364,390365,390379],{},[31655,390366,390367,390373,390375,390377],{},[31743,390368,390369,390371],{},[31658,390370,33244],{},[31658,390372,43379],{},[31658,390374,111],{},[31662,390376,202],{},[31658,390378,111],{},[31665,390380,390381],{"encoding":31667},"e_l a = a",[34,390383,390385,390443],{"className":390384,"ariaHidden":758},[31672],[34,390386,390388,390391,390431,390434,390437,390440],{"className":390387},[31676],[34,390389],{"className":390390,"style":31860},[31680],[34,390392,390394,390397],{"className":390393},[31685],[34,390395,33244],{"className":390396},[31685,31686],[34,390398,390400],{"className":390399},[31792],[34,390401,390403,390423],{"className":390402},[31796,31797],[34,390404,390406,390420],{"className":390405},[31801],[34,390407,390409],{"className":390408,"style":33833},[31805],[34,390410,390411,390414],{"style":31809},[34,390412],{"className":390413,"style":31814},[31813],[34,390415,390417],{"className":390416},[31818,31819,31820,31821],[34,390418,43379],{"className":390419,"style":47113},[31685,31686,31821],[34,390421,31829],{"className":390422},[31828],[34,390424,390426],{"className":390425},[31801],[34,390427,390429],{"className":390428,"style":31836},[31805],[34,390430],{},[34,390432,111],{"className":390433},[31685,31686],[34,390435],{"className":390436,"style":31846},[31845],[34,390438,202],{"className":390439},[31850],[34,390441],{"className":390442,"style":31846},[31845],[34,390444,390446,390449],{"className":390445},[31676],[34,390447],{"className":390448,"style":32527},[31680],[34,390450,111],{"className":390451},[31685,31686],[108,390453,390454,390524,390525,347],{},[34,390455,390457,390475],{"className":390456},[31642],[34,390458,390460],{"className":390459},[31646],[31648,390461,390462],{"xmlns":31650},[31652,390463,390464,390472],{},[31655,390465,390466],{},[31743,390467,390468,390470],{},[31658,390469,33244],{},[31658,390471,34395],{},[31665,390473,390474],{"encoding":31667},"e_r",[34,390476,390478],{"className":390477,"ariaHidden":758},[31672],[34,390479,390481,390484],{"className":390480},[31676],[34,390482],{"className":390483,"style":31860},[31680],[34,390485,390487,390490],{"className":390486},[31685],[34,390488,33244],{"className":390489},[31685,31686],[34,390491,390493],{"className":390492},[31792],[34,390494,390496,390516],{"className":390495},[31796,31797],[34,390497,390499,390513],{"className":390498},[31801],[34,390500,390502],{"className":390501,"style":32362},[31805],[34,390503,390504,390507],{"style":31809},[34,390505],{"className":390506,"style":31814},[31813],[34,390508,390510],{"className":390509},[31818,31819,31820,31821],[34,390511,34395],{"className":390512,"style":34452},[31685,31686,31821],[34,390514,31829],{"className":390515},[31828],[34,390517,390519],{"className":390518},[31801],[34,390520,390522],{"className":390521,"style":31836},[31805],[34,390523],{}," (Right) 是右幺元：",[34,390526,390528,390552],{"className":390527},[31642],[34,390529,390531],{"className":390530},[31646],[31648,390532,390533],{"xmlns":31650},[31652,390534,390535,390549],{},[31655,390536,390537,390539,390545,390547],{},[31658,390538,111],{},[31743,390540,390541,390543],{},[31658,390542,33244],{},[31658,390544,34395],{},[31662,390546,202],{},[31658,390548,111],{},[31665,390550,390551],{"encoding":31667},"a e_r = a",[34,390553,390555,390613],{"className":390554,"ariaHidden":758},[31672],[34,390556,390558,390561,390564,390604,390607,390610],{"className":390557},[31676],[34,390559],{"className":390560,"style":31860},[31680],[34,390562,111],{"className":390563},[31685,31686],[34,390565,390567,390570],{"className":390566},[31685],[34,390568,33244],{"className":390569},[31685,31686],[34,390571,390573],{"className":390572},[31792],[34,390574,390576,390596],{"className":390575},[31796,31797],[34,390577,390579,390593],{"className":390578},[31801],[34,390580,390582],{"className":390581,"style":32362},[31805],[34,390583,390584,390587],{"style":31809},[34,390585],{"className":390586,"style":31814},[31813],[34,390588,390590],{"className":390589},[31818,31819,31820,31821],[34,390591,34395],{"className":390592,"style":34452},[31685,31686,31821],[34,390594,31829],{"className":390595},[31828],[34,390597,390599],{"className":390598},[31801],[34,390600,390602],{"className":390601,"style":31836},[31805],[34,390603],{},[34,390605],{"className":390606,"style":31846},[31845],[34,390608,202],{"className":390609},[31850],[34,390611],{"className":390612,"style":31846},[31845],[34,390614,390616,390619],{"className":390615},[31676],[34,390617],{"className":390618,"style":32527},[31680],[34,390620,111],{"className":390621},[31685,31686],[15,390623,390624],{},[2627,390625,390626,390627,390665],{},"(1) 既有左又有右 ",[34,390628,390630,390647],{"className":390629},[31642],[34,390631,390633],{"className":390632},[31646],[31648,390634,390635],{"xmlns":31650},[31652,390636,390637,390645],{},[31655,390638,390639,390641,390643],{},[39361,390640,262622],{},[31662,390642,262625],{},[39361,390644,262622],{},[31665,390646,337751],{"encoding":31667},[34,390648,390650],{"className":390649,"ariaHidden":758},[31672],[34,390651,390653,390656,390659,390662],{"className":390652},[31676],[34,390654],{"className":390655,"style":318264},[31680],[34,390657],{"className":390658,"style":31846},[31845],[34,390660,262625],{"className":390661},[31850],[34,390663],{"className":390664,"style":31846},[31845]," 有幺元",[105,390667,390668],{},[108,390669,390670,390673,390674,390807,390808,390924,390925],{},[2627,390671,390672],{},"证明","：我们要证明 ",[34,390675,390677,390703],{"className":390676},[31642],[34,390678,390680],{"className":390679},[31646],[31648,390681,390682],{"xmlns":31650},[31652,390683,390684,390700],{},[31655,390685,390686,390692,390694],{},[31743,390687,390688,390690],{},[31658,390689,33244],{},[31658,390691,43379],{},[31662,390693,202],{},[31743,390695,390696,390698],{},[31658,390697,33244],{},[31658,390699,34395],{},[31665,390701,390702],{"encoding":31667},"e_l = e_r",[34,390704,390706,390761],{"className":390705,"ariaHidden":758},[31672],[34,390707,390709,390712,390752,390755,390758],{"className":390708},[31676],[34,390710],{"className":390711,"style":31860},[31680],[34,390713,390715,390718],{"className":390714},[31685],[34,390716,33244],{"className":390717},[31685,31686],[34,390719,390721],{"className":390720},[31792],[34,390722,390724,390744],{"className":390723},[31796,31797],[34,390725,390727,390741],{"className":390726},[31801],[34,390728,390730],{"className":390729,"style":33833},[31805],[34,390731,390732,390735],{"style":31809},[34,390733],{"className":390734,"style":31814},[31813],[34,390736,390738],{"className":390737},[31818,31819,31820,31821],[34,390739,43379],{"className":390740,"style":47113},[31685,31686,31821],[34,390742,31829],{"className":390743},[31828],[34,390745,390747],{"className":390746},[31801],[34,390748,390750],{"className":390749,"style":31836},[31805],[34,390751],{},[34,390753],{"className":390754,"style":31846},[31845],[34,390756,202],{"className":390757},[31850],[34,390759],{"className":390760,"style":31846},[31845],[34,390762,390764,390767],{"className":390763},[31676],[34,390765],{"className":390766,"style":31860},[31680],[34,390768,390770,390773],{"className":390769},[31685],[34,390771,33244],{"className":390772},[31685,31686],[34,390774,390776],{"className":390775},[31792],[34,390777,390779,390799],{"className":390778},[31796,31797],[34,390780,390782,390796],{"className":390781},[31801],[34,390783,390785],{"className":390784,"style":32362},[31805],[34,390786,390787,390790],{"style":31809},[34,390788],{"className":390789,"style":31814},[31813],[34,390791,390793],{"className":390792},[31818,31819,31820,31821],[34,390794,34395],{"className":390795,"style":34452},[31685,31686,31821],[34,390797,31829],{"className":390798},[31828],[34,390800,390802],{"className":390801},[31801],[34,390803,390805],{"className":390804,"style":31836},[31805],[34,390806],{},"，这样它就是双边幺元了。\n考虑计算 ",[34,390809,390811,390835],{"className":390810},[31642],[34,390812,390814],{"className":390813},[31646],[31648,390815,390816],{"xmlns":31650},[31652,390817,390818,390832],{},[31655,390819,390820,390826],{},[31743,390821,390822,390824],{},[31658,390823,33244],{},[31658,390825,43379],{},[31743,390827,390828,390830],{},[31658,390829,33244],{},[31658,390831,34395],{},[31665,390833,390834],{"encoding":31667},"e_l e_r",[34,390836,390838],{"className":390837,"ariaHidden":758},[31672],[34,390839,390841,390844,390884],{"className":390840},[31676],[34,390842],{"className":390843,"style":31860},[31680],[34,390845,390847,390850],{"className":390846},[31685],[34,390848,33244],{"className":390849},[31685,31686],[34,390851,390853],{"className":390852},[31792],[34,390854,390856,390876],{"className":390855},[31796,31797],[34,390857,390859,390873],{"className":390858},[31801],[34,390860,390862],{"className":390861,"style":33833},[31805],[34,390863,390864,390867],{"style":31809},[34,390865],{"className":390866,"style":31814},[31813],[34,390868,390870],{"className":390869},[31818,31819,31820,31821],[34,390871,43379],{"className":390872,"style":47113},[31685,31686,31821],[34,390874,31829],{"className":390875},[31828],[34,390877,390879],{"className":390878},[31801],[34,390880,390882],{"className":390881,"style":31836},[31805],[34,390883],{},[34,390885,390887,390890],{"className":390886},[31685],[34,390888,33244],{"className":390889},[31685,31686],[34,390891,390893],{"className":390892},[31792],[34,390894,390896,390916],{"className":390895},[31796,31797],[34,390897,390899,390913],{"className":390898},[31801],[34,390900,390902],{"className":390901,"style":32362},[31805],[34,390903,390904,390907],{"style":31809},[34,390905],{"className":390906,"style":31814},[31813],[34,390908,390910],{"className":390909},[31818,31819,31820,31821],[34,390911,34395],{"className":390912,"style":34452},[31685,31686,31821],[34,390914,31829],{"className":390915},[31828],[34,390917,390919],{"className":390918},[31801],[34,390920,390922],{"className":390921,"style":31836},[31805],[34,390923],{}," 这个值。\n",[105,390926,390927,391188,391449,391583],{},[108,390928,2892,390929,390998,390999,347],{},[34,390930,390932,390949],{"className":390931},[31642],[34,390933,390935],{"className":390934},[31646],[31648,390936,390937],{"xmlns":31650},[31652,390938,390939,390947],{},[31655,390940,390941],{},[31743,390942,390943,390945],{},[31658,390944,33244],{},[31658,390946,43379],{},[31665,390948,390304],{"encoding":31667},[34,390950,390952],{"className":390951,"ariaHidden":758},[31672],[34,390953,390955,390958],{"className":390954},[31676],[34,390956],{"className":390957,"style":31860},[31680],[34,390959,390961,390964],{"className":390960},[31685],[34,390962,33244],{"className":390963},[31685,31686],[34,390965,390967],{"className":390966},[31792],[34,390968,390970,390990],{"className":390969},[31796,31797],[34,390971,390973,390987],{"className":390972},[31801],[34,390974,390976],{"className":390975,"style":33833},[31805],[34,390977,390978,390981],{"style":31809},[34,390979],{"className":390980,"style":31814},[31813],[34,390982,390984],{"className":390983},[31818,31819,31820,31821],[34,390985,43379],{"className":390986,"style":47113},[31685,31686,31821],[34,390988,31829],{"className":390989},[31828],[34,390991,390993],{"className":390992},[31801],[34,390994,390996],{"className":390995,"style":31836},[31805],[34,390997],{}," 看作左幺元：",[34,391000,391002,391038],{"className":391001},[31642],[34,391003,391005],{"className":391004},[31646],[31648,391006,391007],{"xmlns":31650},[31652,391008,391009,391035],{},[31655,391010,391011,391017,391019,391025,391027,391029],{},[31743,391012,391013,391015],{},[31658,391014,33244],{},[31658,391016,43379],{},[31662,391018,456],{"stretchy":702},[31743,391020,391021,391023],{},[31658,391022,33244],{},[31658,391024,34395],{},[31662,391026,2557],{"stretchy":702},[31662,391028,202],{},[31743,391030,391031,391033],{},[31658,391032,33244],{},[31658,391034,34395],{},[31665,391036,391037],{"encoding":31667},"e_l (e_r) = e_r",[34,391039,391041,391142],{"className":391040,"ariaHidden":758},[31672],[34,391042,391044,391047,391087,391090,391130,391133,391136,391139],{"className":391043},[31676],[34,391045],{"className":391046,"style":31774},[31680],[34,391048,391050,391053],{"className":391049},[31685],[34,391051,33244],{"className":391052},[31685,31686],[34,391054,391056],{"className":391055},[31792],[34,391057,391059,391079],{"className":391058},[31796,31797],[34,391060,391062,391076],{"className":391061},[31801],[34,391063,391065],{"className":391064,"style":33833},[31805],[34,391066,391067,391070],{"style":31809},[34,391068],{"className":391069,"style":31814},[31813],[34,391071,391073],{"className":391072},[31818,31819,31820,31821],[34,391074,43379],{"className":391075,"style":47113},[31685,31686,31821],[34,391077,31829],{"className":391078},[31828],[34,391080,391082],{"className":391081},[31801],[34,391083,391085],{"className":391084,"style":31836},[31805],[34,391086],{},[34,391088,456],{"className":391089},[31782],[34,391091,391093,391096],{"className":391092},[31685],[34,391094,33244],{"className":391095},[31685,31686],[34,391097,391099],{"className":391098},[31792],[34,391100,391102,391122],{"className":391101},[31796,31797],[34,391103,391105,391119],{"className":391104},[31801],[34,391106,391108],{"className":391107,"style":32362},[31805],[34,391109,391110,391113],{"style":31809},[34,391111],{"className":391112,"style":31814},[31813],[34,391114,391116],{"className":391115},[31818,31819,31820,31821],[34,391117,34395],{"className":391118,"style":34452},[31685,31686,31821],[34,391120,31829],{"className":391121},[31828],[34,391123,391125],{"className":391124},[31801],[34,391126,391128],{"className":391127,"style":31836},[31805],[34,391129],{},[34,391131,2557],{"className":391132},[31690],[34,391134],{"className":391135,"style":31846},[31845],[34,391137,202],{"className":391138},[31850],[34,391140],{"className":391141,"style":31846},[31845],[34,391143,391145,391148],{"className":391144},[31676],[34,391146],{"className":391147,"style":31860},[31680],[34,391149,391151,391154],{"className":391150},[31685],[34,391152,33244],{"className":391153},[31685,31686],[34,391155,391157],{"className":391156},[31792],[34,391158,391160,391180],{"className":391159},[31796,31797],[34,391161,391163,391177],{"className":391162},[31801],[34,391164,391166],{"className":391165,"style":32362},[31805],[34,391167,391168,391171],{"style":31809},[34,391169],{"className":391170,"style":31814},[31813],[34,391172,391174],{"className":391173},[31818,31819,31820,31821],[34,391175,34395],{"className":391176,"style":34452},[31685,31686,31821],[34,391178,31829],{"className":391179},[31828],[34,391181,391183],{"className":391182},[31801],[34,391184,391186],{"className":391185,"style":31836},[31805],[34,391187],{},[108,391189,2892,391190,391259,391260,347],{},[34,391191,391193,391210],{"className":391192},[31642],[34,391194,391196],{"className":391195},[31646],[31648,391197,391198],{"xmlns":31650},[31652,391199,391200,391208],{},[31655,391201,391202],{},[31743,391203,391204,391206],{},[31658,391205,33244],{},[31658,391207,34395],{},[31665,391209,390474],{"encoding":31667},[34,391211,391213],{"className":391212,"ariaHidden":758},[31672],[34,391214,391216,391219],{"className":391215},[31676],[34,391217],{"className":391218,"style":31860},[31680],[34,391220,391222,391225],{"className":391221},[31685],[34,391223,33244],{"className":391224},[31685,31686],[34,391226,391228],{"className":391227},[31792],[34,391229,391231,391251],{"className":391230},[31796,31797],[34,391232,391234,391248],{"className":391233},[31801],[34,391235,391237],{"className":391236,"style":32362},[31805],[34,391238,391239,391242],{"style":31809},[34,391240],{"className":391241,"style":31814},[31813],[34,391243,391245],{"className":391244},[31818,31819,31820,31821],[34,391246,34395],{"className":391247,"style":34452},[31685,31686,31821],[34,391249,31829],{"className":391250},[31828],[34,391252,391254],{"className":391253},[31801],[34,391255,391257],{"className":391256,"style":31836},[31805],[34,391258],{}," 看作右幺元：",[34,391261,391263,391299],{"className":391262},[31642],[34,391264,391266],{"className":391265},[31646],[31648,391267,391268],{"xmlns":31650},[31652,391269,391270,391296],{},[31655,391271,391272,391274,391280,391282,391288,391290],{},[31662,391273,456],{"stretchy":702},[31743,391275,391276,391278],{},[31658,391277,33244],{},[31658,391279,43379],{},[31662,391281,2557],{"stretchy":702},[31743,391283,391284,391286],{},[31658,391285,33244],{},[31658,391287,34395],{},[31662,391289,202],{},[31743,391291,391292,391294],{},[31658,391293,33244],{},[31658,391295,43379],{},[31665,391297,391298],{"encoding":31667},"(e_l) e_r = e_l",[34,391300,391302,391403],{"className":391301,"ariaHidden":758},[31672],[34,391303,391305,391308,391311,391351,391354,391394,391397,391400],{"className":391304},[31676],[34,391306],{"className":391307,"style":31774},[31680],[34,391309,456],{"className":391310},[31782],[34,391312,391314,391317],{"className":391313},[31685],[34,391315,33244],{"className":391316},[31685,31686],[34,391318,391320],{"className":391319},[31792],[34,391321,391323,391343],{"className":391322},[31796,31797],[34,391324,391326,391340],{"className":391325},[31801],[34,391327,391329],{"className":391328,"style":33833},[31805],[34,391330,391331,391334],{"style":31809},[34,391332],{"className":391333,"style":31814},[31813],[34,391335,391337],{"className":391336},[31818,31819,31820,31821],[34,391338,43379],{"className":391339,"style":47113},[31685,31686,31821],[34,391341,31829],{"className":391342},[31828],[34,391344,391346],{"className":391345},[31801],[34,391347,391349],{"className":391348,"style":31836},[31805],[34,391350],{},[34,391352,2557],{"className":391353},[31690],[34,391355,391357,391360],{"className":391356},[31685],[34,391358,33244],{"className":391359},[31685,31686],[34,391361,391363],{"className":391362},[31792],[34,391364,391366,391386],{"className":391365},[31796,31797],[34,391367,391369,391383],{"className":391368},[31801],[34,391370,391372],{"className":391371,"style":32362},[31805],[34,391373,391374,391377],{"style":31809},[34,391375],{"className":391376,"style":31814},[31813],[34,391378,391380],{"className":391379},[31818,31819,31820,31821],[34,391381,34395],{"className":391382,"style":34452},[31685,31686,31821],[34,391384,31829],{"className":391385},[31828],[34,391387,391389],{"className":391388},[31801],[34,391390,391392],{"className":391391,"style":31836},[31805],[34,391393],{},[34,391395],{"className":391396,"style":31846},[31845],[34,391398,202],{"className":391399},[31850],[34,391401],{"className":391402,"style":31846},[31845],[34,391404,391406,391409],{"className":391405},[31676],[34,391407],{"className":391408,"style":31860},[31680],[34,391410,391412,391415],{"className":391411},[31685],[34,391413,33244],{"className":391414},[31685,31686],[34,391416,391418],{"className":391417},[31792],[34,391419,391421,391441],{"className":391420},[31796,31797],[34,391422,391424,391438],{"className":391423},[31801],[34,391425,391427],{"className":391426,"style":33833},[31805],[34,391428,391429,391432],{"style":31809},[34,391430],{"className":391431,"style":31814},[31813],[34,391433,391435],{"className":391434},[31818,31819,31820,31821],[34,391436,43379],{"className":391437,"style":47113},[31685,31686,31821],[34,391439,31829],{"className":391440},[31828],[34,391442,391444],{"className":391443},[31801],[34,391445,391447],{"className":391446,"style":31836},[31805],[34,391448],{},[108,391450,175992,391451,347],{},[34,391452,391454,391479],{"className":391453},[31642],[34,391455,391457],{"className":391456},[31646],[31648,391458,391459],{"xmlns":31650},[31652,391460,391461,391477],{},[31655,391462,391463,391469,391471],{},[31743,391464,391465,391467],{},[31658,391466,33244],{},[31658,391468,43379],{},[31662,391470,202],{},[31743,391472,391473,391475],{},[31658,391474,33244],{},[31658,391476,34395],{},[31665,391478,390702],{"encoding":31667},[34,391480,391482,391537],{"className":391481,"ariaHidden":758},[31672],[34,391483,391485,391488,391528,391531,391534],{"className":391484},[31676],[34,391486],{"className":391487,"style":31860},[31680],[34,391489,391491,391494],{"className":391490},[31685],[34,391492,33244],{"className":391493},[31685,31686],[34,391495,391497],{"className":391496},[31792],[34,391498,391500,391520],{"className":391499},[31796,31797],[34,391501,391503,391517],{"className":391502},[31801],[34,391504,391506],{"className":391505,"style":33833},[31805],[34,391507,391508,391511],{"style":31809},[34,391509],{"className":391510,"style":31814},[31813],[34,391512,391514],{"className":391513},[31818,31819,31820,31821],[34,391515,43379],{"className":391516,"style":47113},[31685,31686,31821],[34,391518,31829],{"className":391519},[31828],[34,391521,391523],{"className":391522},[31801],[34,391524,391526],{"className":391525,"style":31836},[31805],[34,391527],{},[34,391529],{"className":391530,"style":31846},[31845],[34,391532,202],{"className":391533},[31850],[34,391535],{"className":391536,"style":31846},[31845],[34,391538,391540,391543],{"className":391539},[31676],[34,391541],{"className":391542,"style":31860},[31680],[34,391544,391546,391549],{"className":391545},[31685],[34,391547,33244],{"className":391548},[31685,31686],[34,391550,391552],{"className":391551},[31792],[34,391553,391555,391575],{"className":391554},[31796,31797],[34,391556,391558,391572],{"className":391557},[31801],[34,391559,391561],{"className":391560,"style":32362},[31805],[34,391562,391563,391566],{"style":31809},[34,391564],{"className":391565,"style":31814},[31813],[34,391567,391569],{"className":391568},[31818,31819,31820,31821],[34,391570,34395],{"className":391571,"style":34452},[31685,31686,31821],[34,391573,31829],{"className":391574},[31828],[34,391576,391578],{"className":391577},[31801],[34,391579,391581],{"className":391580,"style":31836},[31805],[34,391582],{},[108,391584,391585,391586,391614],{},"既然左右相等，记为 ",[34,391587,391589,391602],{"className":391588},[31642],[34,391590,391592],{"className":391591},[31646],[31648,391593,391594],{"xmlns":31650},[31652,391595,391596,391600],{},[31655,391597,391598],{},[31658,391599,33244],{},[31665,391601,33244],{"encoding":31667},[34,391603,391605],{"className":391604,"ariaHidden":758},[31672],[34,391606,391608,391611],{"className":391607},[31676],[34,391609],{"className":391610,"style":32527},[31680],[34,391612,33244],{"className":391613},[31685,31686],"，它既能左乘也能右乘，就是幺元。",[15,391616,391617],{},[2627,391618,391619,391620,390665],{},"(2) 有左幺元 + 无零因子 ",[34,391621,391623,391640],{"className":391622},[31642],[34,391624,391626],{"className":391625},[31646],[31648,391627,391628],{"xmlns":31650},[31652,391629,391630,391638],{},[31655,391631,391632,391634,391636],{},[39361,391633,262622],{},[31662,391635,262625],{},[39361,391637,262622],{},[31665,391639,337751],{"encoding":31667},[34,391641,391643],{"className":391642,"ariaHidden":758},[31672],[34,391644,391646,391649,391652,391655],{"className":391645},[31676],[34,391647],{"className":391648,"style":318264},[31680],[34,391650],{"className":391651,"style":31846},[31845],[34,391653,262625],{"className":391654},[31850],[34,391656],{"className":391657,"style":31846},[31845],[105,391659,391660,391930],{},[108,391661,391662,391665,391666,391735,391736,391832,391833,347],{},[2627,391663,391664],{},"思路","：我们手里有 ",[34,391667,391669,391686],{"className":391668},[31642],[34,391670,391672],{"className":391671},[31646],[31648,391673,391674],{"xmlns":31650},[31652,391675,391676,391684],{},[31655,391677,391678],{},[31743,391679,391680,391682],{},[31658,391681,33244],{},[31658,391683,43379],{},[31665,391685,390304],{"encoding":31667},[34,391687,391689],{"className":391688,"ariaHidden":758},[31672],[34,391690,391692,391695],{"className":391691},[31676],[34,391693],{"className":391694,"style":31860},[31680],[34,391696,391698,391701],{"className":391697},[31685],[34,391699,33244],{"className":391700},[31685,31686],[34,391702,391704],{"className":391703},[31792],[34,391705,391707,391727],{"className":391706},[31796,31797],[34,391708,391710,391724],{"className":391709},[31801],[34,391711,391713],{"className":391712,"style":33833},[31805],[34,391714,391715,391718],{"style":31809},[34,391716],{"className":391717,"style":31814},[31813],[34,391719,391721],{"className":391720},[31818,31819,31820,31821],[34,391722,43379],{"className":391723,"style":47113},[31685,31686,31821],[34,391725,31829],{"className":391726},[31828],[34,391728,391730],{"className":391729},[31801],[34,391731,391733],{"className":391732,"style":31836},[31805],[34,391734],{},"，满足 ",[34,391737,391739,391762],{"className":391738},[31642],[34,391740,391742],{"className":391741},[31646],[31648,391743,391744],{"xmlns":31650},[31652,391745,391746,391760],{},[31655,391747,391748,391754,391756,391758],{},[31743,391749,391750,391752],{},[31658,391751,33244],{},[31658,391753,43379],{},[31658,391755,111],{},[31662,391757,202],{},[31658,391759,111],{},[31665,391761,390381],{"encoding":31667},[34,391763,391765,391823],{"className":391764,"ariaHidden":758},[31672],[34,391766,391768,391771,391811,391814,391817,391820],{"className":391767},[31676],[34,391769],{"className":391770,"style":31860},[31680],[34,391772,391774,391777],{"className":391773},[31685],[34,391775,33244],{"className":391776},[31685,31686],[34,391778,391780],{"className":391779},[31792],[34,391781,391783,391803],{"className":391782},[31796,31797],[34,391784,391786,391800],{"className":391785},[31801],[34,391787,391789],{"className":391788,"style":33833},[31805],[34,391790,391791,391794],{"style":31809},[34,391792],{"className":391793,"style":31814},[31813],[34,391795,391797],{"className":391796},[31818,31819,31820,31821],[34,391798,43379],{"className":391799,"style":47113},[31685,31686,31821],[34,391801,31829],{"className":391802},[31828],[34,391804,391806],{"className":391805},[31801],[34,391807,391809],{"className":391808,"style":31836},[31805],[34,391810],{},[34,391812,111],{"className":391813},[31685,31686],[34,391815],{"className":391816,"style":31846},[31845],[34,391818,202],{"className":391819},[31850],[34,391821],{"className":391822,"style":31846},[31845],[34,391824,391826,391829],{"className":391825},[31676],[34,391827],{"className":391828,"style":32527},[31680],[34,391830,111],{"className":391831},[31685,31686],"。我们要证明它也是右幺元，即证明 ",[34,391834,391836,391860],{"className":391835},[31642],[34,391837,391839],{"className":391838},[31646],[31648,391840,391841],{"xmlns":31650},[31652,391842,391843,391857],{},[31655,391844,391845,391847,391853,391855],{},[31658,391846,111],{},[31743,391848,391849,391851],{},[31658,391850,33244],{},[31658,391852,43379],{},[31662,391854,202],{},[31658,391856,111],{},[31665,391858,391859],{"encoding":31667},"a e_l = a",[34,391861,391863,391921],{"className":391862,"ariaHidden":758},[31672],[34,391864,391866,391869,391872,391912,391915,391918],{"className":391865},[31676],[34,391867],{"className":391868,"style":31860},[31680],[34,391870,111],{"className":391871},[31685,31686],[34,391873,391875,391878],{"className":391874},[31685],[34,391876,33244],{"className":391877},[31685,31686],[34,391879,391881],{"className":391880},[31792],[34,391882,391884,391904],{"className":391883},[31796,31797],[34,391885,391887,391901],{"className":391886},[31801],[34,391888,391890],{"className":391889,"style":33833},[31805],[34,391891,391892,391895],{"style":31809},[34,391893],{"className":391894,"style":31814},[31813],[34,391896,391898],{"className":391897},[31818,31819,31820,31821],[34,391899,43379],{"className":391900,"style":47113},[31685,31686,31821],[34,391902,31829],{"className":391903},[31828],[34,391905,391907],{"className":391906},[31801],[34,391908,391910],{"className":391909,"style":31836},[31805],[34,391911],{},[34,391913],{"className":391914,"style":31846},[31845],[34,391916,202],{"className":391917},[31850],[34,391919],{"className":391920,"style":31846},[31845],[34,391922,391924,391927],{"className":391923},[31676],[34,391925],{"className":391926,"style":32527},[31680],[34,391928,111],{"className":391929},[31685,31686],[108,391931,391932,391934,391935,392032,392033,33183,392162,392308,392309,13977,392416,392485,392486,39984,392598,392940,392941,393092,393093,86621,393096,393124,393125,393208,393209,47649,393316,393318,393319,355408,393415,393484],{},[2627,391933,390672],{},"：\n考察 ",[34,391936,391938,391962],{"className":391937},[31642],[34,391939,391941],{"className":391940},[31646],[31648,391942,391943],{"xmlns":31650},[31652,391944,391945,391959],{},[31655,391946,391947,391949,391955,391957],{},[31658,391948,111],{},[31743,391950,391951,391953],{},[31658,391952,33244],{},[31658,391954,43379],{},[31662,391956,32226],{},[31658,391958,111],{},[31665,391960,391961],{"encoding":31667},"a e_l - a",[34,391963,391965,392023],{"className":391964,"ariaHidden":758},[31672],[34,391966,391968,391971,391974,392014,392017,392020],{"className":391967},[31676],[34,391969],{"className":391970,"style":40522},[31680],[34,391972,111],{"className":391973},[31685,31686],[34,391975,391977,391980],{"className":391976},[31685],[34,391978,33244],{"className":391979},[31685,31686],[34,391981,391983],{"className":391982},[31792],[34,391984,391986,392006],{"className":391985},[31796,31797],[34,391987,391989,392003],{"className":391988},[31801],[34,391990,391992],{"className":391991,"style":33833},[31805],[34,391993,391994,391997],{"style":31809},[34,391995],{"className":391996,"style":31814},[31813],[34,391998,392000],{"className":391999},[31818,31819,31820,31821],[34,392001,43379],{"className":392002,"style":47113},[31685,31686,31821],[34,392004,31829],{"className":392005},[31828],[34,392007,392009],{"className":392008},[31801],[34,392010,392012],{"className":392011,"style":31836},[31805],[34,392013],{},[34,392015],{"className":392016,"style":32604},[31845],[34,392018,32226],{"className":392019},[32024],[34,392021],{"className":392022,"style":32604},[31845],[34,392024,392026,392029],{"className":392025},[31676],[34,392027],{"className":392028,"style":32527},[31680],[34,392030,111],{"className":392031},[31685,31686]," 这个式子，如果它等于 0，我们就赢了。\n我们想办法构造出“零因子”的形式。\n计算 ",[34,392034,392036,392068],{"className":392035},[31642],[34,392037,392039],{"className":392038},[31646],[31648,392040,392041],{"xmlns":31650},[31652,392042,392043,392065],{},[31655,392044,392045,392047,392049,392055,392057,392059,392061,392063],{},[31662,392046,456],{"stretchy":702},[31658,392048,111],{},[31743,392050,392051,392053],{},[31658,392052,33244],{},[31658,392054,43379],{},[31662,392056,32226],{},[31658,392058,111],{},[31662,392060,2557],{"stretchy":702},[31662,392062,71191],{},[31658,392064,111],{},[31665,392066,392067],{"encoding":31667},"(a e_l - a) \\cdot a",[34,392069,392071,392132,392153],{"className":392070,"ariaHidden":758},[31672],[34,392072,392074,392077,392080,392083,392123,392126,392129],{"className":392073},[31676],[34,392075],{"className":392076,"style":31774},[31680],[34,392078,456],{"className":392079},[31782],[34,392081,111],{"className":392082},[31685,31686],[34,392084,392086,392089],{"className":392085},[31685],[34,392087,33244],{"className":392088},[31685,31686],[34,392090,392092],{"className":392091},[31792],[34,392093,392095,392115],{"className":392094},[31796,31797],[34,392096,392098,392112],{"className":392097},[31801],[34,392099,392101],{"className":392100,"style":33833},[31805],[34,392102,392103,392106],{"style":31809},[34,392104],{"className":392105,"style":31814},[31813],[34,392107,392109],{"className":392108},[31818,31819,31820,31821],[34,392110,43379],{"className":392111,"style":47113},[31685,31686,31821],[34,392113,31829],{"className":392114},[31828],[34,392116,392118],{"className":392117},[31801],[34,392119,392121],{"className":392120,"style":31836},[31805],[34,392122],{},[34,392124],{"className":392125,"style":32604},[31845],[34,392127,32226],{"className":392128},[32024],[34,392130],{"className":392131,"style":32604},[31845],[34,392133,392135,392138,392141,392144,392147,392150],{"className":392134},[31676],[34,392136],{"className":392137,"style":31774},[31680],[34,392139,111],{"className":392140},[31685,31686],[34,392142,2557],{"className":392143},[31690],[34,392145],{"className":392146,"style":32604},[31845],[34,392148,71191],{"className":392149},[32024],[34,392151],{"className":392152,"style":32604},[31845],[34,392154,392156,392159],{"className":392155},[31676],[34,392157],{"className":392158,"style":32527},[31680],[34,392160,111],{"className":392161},[31685,31686],[34,392163,392165,392197],{"className":392164},[31642],[34,392166,392168],{"className":392167},[31646],[31648,392169,392170],{"xmlns":31650},[31652,392171,392172,392194],{},[31655,392173,392174,392176,392178,392184,392186,392188],{},[31662,392175,202],{},[31658,392177,111],{},[31743,392179,392180,392182],{},[31658,392181,33244],{},[31658,392183,43379],{},[31658,392185,111],{},[31662,392187,32226],{},[37624,392189,392190,392192],{},[31658,392191,111],{},[31749,392193,768],{},[31665,392195,392196],{"encoding":31667},"= a e_l a - a^2",[34,392198,392200,392212,392273],{"className":392199,"ariaHidden":758},[31672],[34,392201,392203,392206,392209],{"className":392202},[31676],[34,392204],{"className":392205,"style":73089},[31680],[34,392207,202],{"className":392208},[31850],[34,392210],{"className":392211,"style":31846},[31845],[34,392213,392215,392218,392221,392261,392264,392267,392270],{"className":392214},[31676],[34,392216],{"className":392217,"style":40522},[31680],[34,392219,111],{"className":392220},[31685,31686],[34,392222,392224,392227],{"className":392223},[31685],[34,392225,33244],{"className":392226},[31685,31686],[34,392228,392230],{"className":392229},[31792],[34,392231,392233,392253],{"className":392232},[31796,31797],[34,392234,392236,392250],{"className":392235},[31801],[34,392237,392239],{"className":392238,"style":33833},[31805],[34,392240,392241,392244],{"style":31809},[34,392242],{"className":392243,"style":31814},[31813],[34,392245,392247],{"className":392246},[31818,31819,31820,31821],[34,392248,43379],{"className":392249,"style":47113},[31685,31686,31821],[34,392251,31829],{"className":392252},[31828],[34,392254,392256],{"className":392255},[31801],[34,392257,392259],{"className":392258,"style":31836},[31805],[34,392260],{},[34,392262,111],{"className":392263},[31685,31686],[34,392265],{"className":392266,"style":32604},[31845],[34,392268,32226],{"className":392269},[32024],[34,392271],{"className":392272,"style":32604},[31845],[34,392274,392276,392279],{"className":392275},[31676],[34,392277],{"className":392278,"style":37774},[31680],[34,392280,392282,392285],{"className":392281},[31685],[34,392283,111],{"className":392284},[31685,31686],[34,392286,392288],{"className":392287},[31792],[34,392289,392291],{"className":392290},[31796],[34,392292,392294],{"className":392293},[31801],[34,392295,392297],{"className":392296,"style":37774},[31805],[34,392298,392299,392302],{"style":37777},[34,392300],{"className":392301,"style":31814},[31813],[34,392303,392305],{"className":392304},[31818,31819,31820,31821],[34,392306,768],{"className":392307},[31685,31821],"\n利用左幺元性质：因为 ",[34,392310,392312,392340],{"className":392311},[31642],[34,392313,392315],{"className":392314},[31646],[31648,392316,392317],{"xmlns":31650},[31652,392318,392319,392337],{},[31655,392320,392321,392327,392329,392331,392333,392335],{},[31743,392322,392323,392325],{},[31658,392324,33244],{},[31658,392326,43379],{},[31662,392328,456],{"stretchy":702},[31658,392330,111],{},[31662,392332,2557],{"stretchy":702},[31662,392334,202],{},[31658,392336,111],{},[31665,392338,392339],{"encoding":31667},"e_l (a) = a",[34,392341,392343,392407],{"className":392342,"ariaHidden":758},[31672],[34,392344,392346,392349,392389,392392,392395,392398,392401,392404],{"className":392345},[31676],[34,392347],{"className":392348,"style":31774},[31680],[34,392350,392352,392355],{"className":392351},[31685],[34,392353,33244],{"className":392354},[31685,31686],[34,392356,392358],{"className":392357},[31792],[34,392359,392361,392381],{"className":392360},[31796,31797],[34,392362,392364,392378],{"className":392363},[31801],[34,392365,392367],{"className":392366,"style":33833},[31805],[34,392368,392369,392372],{"style":31809},[34,392370],{"className":392371,"style":31814},[31813],[34,392373,392375],{"className":392374},[31818,31819,31820,31821],[34,392376,43379],{"className":392377,"style":47113},[31685,31686,31821],[34,392379,31829],{"className":392380},[31828],[34,392382,392384],{"className":392383},[31801],[34,392385,392387],{"className":392386,"style":31836},[31805],[34,392388],{},[34,392390,456],{"className":392391},[31782],[34,392393,111],{"className":392394},[31685,31686],[34,392396,2557],{"className":392397},[31690],[34,392399],{"className":392400,"style":31846},[31845],[34,392402,202],{"className":392403},[31850],[34,392405],{"className":392406,"style":31846},[31845],[34,392408,392410,392413],{"className":392409},[31676],[34,392411],{"className":392412,"style":32527},[31680],[34,392414,111],{"className":392415},[31685,31686],[34,392417,392419,392436],{"className":392418},[31642],[34,392420,392422],{"className":392421},[31646],[31648,392423,392424],{"xmlns":31650},[31652,392425,392426,392434],{},[31655,392427,392428],{},[31743,392429,392430,392432],{},[31658,392431,33244],{},[31658,392433,43379],{},[31665,392435,390304],{"encoding":31667},[34,392437,392439],{"className":392438,"ariaHidden":758},[31672],[34,392440,392442,392445],{"className":392441},[31676],[34,392443],{"className":392444,"style":31860},[31680],[34,392446,392448,392451],{"className":392447},[31685],[34,392449,33244],{"className":392450},[31685,31686],[34,392452,392454],{"className":392453},[31792],[34,392455,392457,392477],{"className":392456},[31796,31797],[34,392458,392460,392474],{"className":392459},[31801],[34,392461,392463],{"className":392462,"style":33833},[31805],[34,392464,392465,392468],{"style":31809},[34,392466],{"className":392467,"style":31814},[31813],[34,392469,392471],{"className":392470},[31818,31819,31820,31821],[34,392472,43379],{"className":392473,"style":47113},[31685,31686,31821],[34,392475,31829],{"className":392476},[31828],[34,392478,392480],{"className":392479},[31801],[34,392481,392483],{"className":392482,"style":31836},[31805],[34,392484],{}," 也可以作用在任何元素上。\n哎，这里要用一个小技巧。我们反过来算 ",[34,392487,392489,392519],{"className":392488},[31642],[34,392490,392492],{"className":392491},[31646],[31648,392493,392494],{"xmlns":31650},[31652,392495,392496,392516],{},[31655,392497,392498,392500,392502,392508,392510,392512,392514],{},[31658,392499,111],{},[31662,392501,456],{"stretchy":702},[31743,392503,392504,392506],{},[31658,392505,33244],{},[31658,392507,43379],{},[31658,392509,111],{},[31662,392511,32226],{},[31658,392513,111],{},[31662,392515,2557],{"stretchy":702},[31665,392517,392518],{"encoding":31667},"a (e_l a - a)",[34,392520,392522,392586],{"className":392521,"ariaHidden":758},[31672],[34,392523,392525,392528,392531,392534,392574,392577,392580,392583],{"className":392524},[31676],[34,392526],{"className":392527,"style":31774},[31680],[34,392529,111],{"className":392530},[31685,31686],[34,392532,456],{"className":392533},[31782],[34,392535,392537,392540],{"className":392536},[31685],[34,392538,33244],{"className":392539},[31685,31686],[34,392541,392543],{"className":392542},[31792],[34,392544,392546,392566],{"className":392545},[31796,31797],[34,392547,392549,392563],{"className":392548},[31801],[34,392550,392552],{"className":392551,"style":33833},[31805],[34,392553,392554,392557],{"style":31809},[34,392555],{"className":392556,"style":31814},[31813],[34,392558,392560],{"className":392559},[31818,31819,31820,31821],[34,392561,43379],{"className":392562,"style":47113},[31685,31686,31821],[34,392564,31829],{"className":392565},[31828],[34,392567,392569],{"className":392568},[31801],[34,392570,392572],{"className":392571,"style":31836},[31805],[34,392573],{},[34,392575,111],{"className":392576},[31685,31686],[34,392578],{"className":392579,"style":32604},[31845],[34,392581,32226],{"className":392582},[32024],[34,392584],{"className":392585,"style":32604},[31845],[34,392587,392589,392592,392595],{"className":392588},[31676],[34,392590],{"className":392591,"style":31774},[31680],[34,392593,111],{"className":392594},[31685,31686],[34,392596,2557],{"className":392597},[31690],[34,392599,392601,392675],{"className":392600},[31642],[34,392602,392604],{"className":392603},[31646],[31648,392605,392606],{"xmlns":31650},[31652,392607,392608,392672],{},[31655,392609,392610,392612,392614,392620,392622,392624,392626,392628,392630,392632,392634,392636,392638,392640,392642,392644,392646,392652,392654,392660,392662,392668,392670],{},[31658,392611,111],{},[31662,392613,456],{"stretchy":702},[31743,392615,392616,392618],{},[31658,392617,33244],{},[31658,392619,43379],{},[31658,392621,111],{},[31662,392623,2557],{"stretchy":702},[31662,392625,32226],{},[31658,392627,111],{},[31662,392629,456],{"stretchy":702},[31658,392631,111],{},[31662,392633,2557],{"stretchy":702},[31662,392635,202],{},[31658,392637,111],{},[31662,392639,456],{"stretchy":702},[31658,392641,111],{},[31662,392643,2557],{"stretchy":702},[31662,392645,32226],{},[37624,392647,392648,392650],{},[31658,392649,111],{},[31749,392651,768],{},[31662,392653,202],{},[37624,392655,392656,392658],{},[31658,392657,111],{},[31749,392659,768],{},[31662,392661,32226],{},[37624,392663,392664,392666],{},[31658,392665,111],{},[31749,392667,768],{},[31662,392669,202],{},[31749,392671,459],{},[31665,392673,392674],{"encoding":31667},"a (e_l a) - a(a) = a(a) - a^2 = a^2 - a^2 = 0",[34,392676,392678,392745,392772,392799,392843,392887,392931],{"className":392677,"ariaHidden":758},[31672],[34,392679,392681,392684,392687,392690,392730,392733,392736,392739,392742],{"className":392680},[31676],[34,392682],{"className":392683,"style":31774},[31680],[34,392685,111],{"className":392686},[31685,31686],[34,392688,456],{"className":392689},[31782],[34,392691,392693,392696],{"className":392692},[31685],[34,392694,33244],{"className":392695},[31685,31686],[34,392697,392699],{"className":392698},[31792],[34,392700,392702,392722],{"className":392701},[31796,31797],[34,392703,392705,392719],{"className":392704},[31801],[34,392706,392708],{"className":392707,"style":33833},[31805],[34,392709,392710,392713],{"style":31809},[34,392711],{"className":392712,"style":31814},[31813],[34,392714,392716],{"className":392715},[31818,31819,31820,31821],[34,392717,43379],{"className":392718,"style":47113},[31685,31686,31821],[34,392720,31829],{"className":392721},[31828],[34,392723,392725],{"className":392724},[31801],[34,392726,392728],{"className":392727,"style":31836},[31805],[34,392729],{},[34,392731,111],{"className":392732},[31685,31686],[34,392734,2557],{"className":392735},[31690],[34,392737],{"className":392738,"style":32604},[31845],[34,392740,32226],{"className":392741},[32024],[34,392743],{"className":392744,"style":32604},[31845],[34,392746,392748,392751,392754,392757,392760,392763,392766,392769],{"className":392747},[31676],[34,392749],{"className":392750,"style":31774},[31680],[34,392752,111],{"className":392753},[31685,31686],[34,392755,456],{"className":392756},[31782],[34,392758,111],{"className":392759},[31685,31686],[34,392761,2557],{"className":392762},[31690],[34,392764],{"className":392765,"style":31846},[31845],[34,392767,202],{"className":392768},[31850],[34,392770],{"className":392771,"style":31846},[31845],[34,392773,392775,392778,392781,392784,392787,392790,392793,392796],{"className":392774},[31676],[34,392776],{"className":392777,"style":31774},[31680],[34,392779,111],{"className":392780},[31685,31686],[34,392782,456],{"className":392783},[31782],[34,392785,111],{"className":392786},[31685,31686],[34,392788,2557],{"className":392789},[31690],[34,392791],{"className":392792,"style":32604},[31845],[34,392794,32226],{"className":392795},[32024],[34,392797],{"className":392798,"style":32604},[31845],[34,392800,392802,392805,392834,392837,392840],{"className":392801},[31676],[34,392803],{"className":392804,"style":37774},[31680],[34,392806,392808,392811],{"className":392807},[31685],[34,392809,111],{"className":392810},[31685,31686],[34,392812,392814],{"className":392813},[31792],[34,392815,392817],{"className":392816},[31796],[34,392818,392820],{"className":392819},[31801],[34,392821,392823],{"className":392822,"style":37774},[31805],[34,392824,392825,392828],{"style":37777},[34,392826],{"className":392827,"style":31814},[31813],[34,392829,392831],{"className":392830},[31818,31819,31820,31821],[34,392832,768],{"className":392833},[31685,31821],[34,392835],{"className":392836,"style":31846},[31845],[34,392838,202],{"className":392839},[31850],[34,392841],{"className":392842,"style":31846},[31845],[34,392844,392846,392849,392878,392881,392884],{"className":392845},[31676],[34,392847],{"className":392848,"style":40044},[31680],[34,392850,392852,392855],{"className":392851},[31685],[34,392853,111],{"className":392854},[31685,31686],[34,392856,392858],{"className":392857},[31792],[34,392859,392861],{"className":392860},[31796],[34,392862,392864],{"className":392863},[31801],[34,392865,392867],{"className":392866,"style":37774},[31805],[34,392868,392869,392872],{"style":37777},[34,392870],{"className":392871,"style":31814},[31813],[34,392873,392875],{"className":392874},[31818,31819,31820,31821],[34,392876,768],{"className":392877},[31685,31821],[34,392879],{"className":392880,"style":32604},[31845],[34,392882,32226],{"className":392883},[32024],[34,392885],{"className":392886,"style":32604},[31845],[34,392888,392890,392893,392922,392925,392928],{"className":392889},[31676],[34,392891],{"className":392892,"style":37774},[31680],[34,392894,392896,392899],{"className":392895},[31685],[34,392897,111],{"className":392898},[31685,31686],[34,392900,392902],{"className":392901},[31792],[34,392903,392905],{"className":392904},[31796],[34,392906,392908],{"className":392907},[31801],[34,392909,392911],{"className":392910,"style":37774},[31805],[34,392912,392913,392916],{"style":37777},[34,392914],{"className":392915,"style":31814},[31813],[34,392917,392919],{"className":392918},[31818,31819,31820,31821],[34,392920,768],{"className":392921},[31685,31821],[34,392923],{"className":392924,"style":31846},[31845],[34,392926,202],{"className":392927},[31850],[34,392929],{"className":392930,"style":31846},[31845],[34,392932,392934,392937],{"className":392933},[31676],[34,392935],{"className":392936,"style":32617},[31680],[34,392938,459],{"className":392939},[31685],"。\n现在我们得到了 ",[34,392942,392944,392980],{"className":392943},[31642],[34,392945,392947],{"className":392946},[31646],[31648,392948,392949],{"xmlns":31650},[31652,392950,392951,392977],{},[31655,392952,392953,392955,392957,392959,392965,392967,392969,392971,392973,392975],{},[31658,392954,111],{},[31662,392956,71191],{},[31662,392958,456],{"stretchy":702},[31743,392960,392961,392963],{},[31658,392962,33244],{},[31658,392964,43379],{},[31658,392966,111],{},[31662,392968,32226],{},[31658,392970,111],{},[31662,392972,2557],{"stretchy":702},[31662,392974,202],{},[31749,392976,459],{},[31665,392978,392979],{"encoding":31667},"a \\cdot (e_l a - a) = 0",[34,392981,392983,393001,393062,393083],{"className":392982,"ariaHidden":758},[31672],[34,392984,392986,392989,392992,392995,392998],{"className":392985},[31676],[34,392987],{"className":392988,"style":224869},[31680],[34,392990,111],{"className":392991},[31685,31686],[34,392993],{"className":392994,"style":32604},[31845],[34,392996,71191],{"className":392997},[32024],[34,392999],{"className":393000,"style":32604},[31845],[34,393002,393004,393007,393010,393050,393053,393056,393059],{"className":393003},[31676],[34,393005],{"className":393006,"style":31774},[31680],[34,393008,456],{"className":393009},[31782],[34,393011,393013,393016],{"className":393012},[31685],[34,393014,33244],{"className":393015},[31685,31686],[34,393017,393019],{"className":393018},[31792],[34,393020,393022,393042],{"className":393021},[31796,31797],[34,393023,393025,393039],{"className":393024},[31801],[34,393026,393028],{"className":393027,"style":33833},[31805],[34,393029,393030,393033],{"style":31809},[34,393031],{"className":393032,"style":31814},[31813],[34,393034,393036],{"className":393035},[31818,31819,31820,31821],[34,393037,43379],{"className":393038,"style":47113},[31685,31686,31821],[34,393040,31829],{"className":393041},[31828],[34,393043,393045],{"className":393044},[31801],[34,393046,393048],{"className":393047,"style":31836},[31805],[34,393049],{},[34,393051,111],{"className":393052},[31685,31686],[34,393054],{"className":393055,"style":32604},[31845],[34,393057,32226],{"className":393058},[32024],[34,393060],{"className":393061,"style":32604},[31845],[34,393063,393065,393068,393071,393074,393077,393080],{"className":393064},[31676],[34,393066],{"className":393067,"style":31774},[31680],[34,393069,111],{"className":393070},[31685,31686],[34,393072,2557],{"className":393073},[31690],[34,393075],{"className":393076,"style":31846},[31845],[34,393078,202],{"className":393079},[31850],[34,393081],{"className":393082,"style":31846},[31845],[34,393084,393086,393089],{"className":393085},[31676],[34,393087],{"className":393088,"style":32617},[31680],[34,393090,459],{"className":393091},[31685],"。\n题目说",[2627,393094,393095],{},"没有非零零因子",[34,393097,393099,393112],{"className":393098},[31642],[34,393100,393102],{"className":393101},[31646],[31648,393103,393104],{"xmlns":31650},[31652,393105,393106,393110],{},[31655,393107,393108],{},[31658,393109,111],{},[31665,393111,111],{"encoding":31667},[34,393113,393115],{"className":393114,"ariaHidden":758},[31672],[34,393116,393118,393121],{"className":393117},[31676],[34,393119],{"className":393120,"style":32527},[31680],[34,393122,111],{"className":393123},[31685,31686]," 可以取非零元素，既然 ",[34,393126,393128,393145],{"className":393127},[31642],[34,393129,393131],{"className":393130},[31646],[31648,393132,393133],{"xmlns":31650},[31652,393134,393135,393143],{},[31655,393136,393137,393139,393141],{},[31658,393138,111],{},[31662,393140,54707],{"mathvariant":38079},[31749,393142,459],{},[31665,393144,363487],{"encoding":31667},[34,393146,393148,393199],{"className":393147,"ariaHidden":758},[31672],[34,393149,393151,393154,393157,393160,393196],{"className":393150},[31676],[34,393152],{"className":393153,"style":43217},[31680],[34,393155,111],{"className":393156},[31685,31686],[34,393158],{"className":393159,"style":31846},[31845],[34,393161,393163,393190,393193],{"className":393162},[31850],[34,393164,393166],{"className":393165},[31850],[34,393167,393169],{"className":393168},[31685,54774],[34,393170,393172],{"className":393171},[54778],[34,393173,393175,393178,393187],{"className":393174},[54782],[34,393176],{"className":393177,"style":43217},[31680],[34,393179,393181],{"className":393180},[54789],[34,393182,393184],{"className":393183},[31685],[34,393185,54796],{"className":393186},[31850],[34,393188],{"className":393189},[54800],[34,393191],{"className":393192},[31845,54804],[34,393194,202],{"className":393195},[31850],[34,393197],{"className":393198,"style":31846},[31845],[34,393200,393202,393205],{"className":393201},[31676],[34,393203],{"className":393204,"style":32617},[31680],[34,393206,459],{"className":393207},[31685]," 且乘积为 0，那么另一项必须为 0。\n所以 ",[34,393210,393212,393240],{"className":393211},[31642],[34,393213,393215],{"className":393214},[31646],[31648,393216,393217],{"xmlns":31650},[31652,393218,393219,393237],{},[31655,393220,393221,393223,393229,393231,393233,393235],{},[31662,393222,456],{"stretchy":702},[31743,393224,393225,393227],{},[31658,393226,33244],{},[31658,393228,43379],{},[31658,393230,111],{},[31662,393232,32226],{},[31658,393234,111],{},[31662,393236,2557],{"stretchy":702},[31665,393238,393239],{"encoding":31667},"(e_l a - a)",[34,393241,393243,393304],{"className":393242,"ariaHidden":758},[31672],[34,393244,393246,393249,393252,393292,393295,393298,393301],{"className":393245},[31676],[34,393247],{"className":393248,"style":31774},[31680],[34,393250,456],{"className":393251},[31782],[34,393253,393255,393258],{"className":393254},[31685],[34,393256,33244],{"className":393257},[31685,31686],[34,393259,393261],{"className":393260},[31792],[34,393262,393264,393284],{"className":393263},[31796,31797],[34,393265,393267,393281],{"className":393266},[31801],[34,393268,393270],{"className":393269,"style":33833},[31805],[34,393271,393272,393275],{"style":31809},[34,393273],{"className":393274,"style":31814},[31813],[34,393276,393278],{"className":393277},[31818,31819,31820,31821],[34,393279,43379],{"className":393280,"style":47113},[31685,31686,31821],[34,393282,31829],{"className":393283},[31828],[34,393285,393287],{"className":393286},[31801],[34,393288,393290],{"className":393289,"style":31836},[31805],[34,393291],{},[34,393293,111],{"className":393294},[31685,31686],[34,393296],{"className":393297,"style":32604},[31845],[34,393299,32226],{"className":393300},[32024],[34,393302],{"className":393303,"style":32604},[31845],[34,393305,393307,393310,393313],{"className":393306},[31676],[34,393308],{"className":393309,"style":31774},[31680],[34,393311,111],{"className":393312},[31685,31686],[34,393314,2557],{"className":393315},[31690],[2627,393317,307037],{},"等于 0。\n这就意味着 ",[34,393320,393322,393345],{"className":393321},[31642],[34,393323,393325],{"className":393324},[31646],[31648,393326,393327],{"xmlns":31650},[31652,393328,393329,393343],{},[31655,393330,393331,393337,393339,393341],{},[31743,393332,393333,393335],{},[31658,393334,33244],{},[31658,393336,43379],{},[31658,393338,111],{},[31662,393340,202],{},[31658,393342,111],{},[31665,393344,390381],{"encoding":31667},[34,393346,393348,393406],{"className":393347,"ariaHidden":758},[31672],[34,393349,393351,393354,393394,393397,393400,393403],{"className":393350},[31676],[34,393352],{"className":393353,"style":31860},[31680],[34,393355,393357,393360],{"className":393356},[31685],[34,393358,33244],{"className":393359},[31685,31686],[34,393361,393363],{"className":393362},[31792],[34,393364,393366,393386],{"className":393365},[31796,31797],[34,393367,393369,393383],{"className":393368},[31801],[34,393370,393372],{"className":393371,"style":33833},[31805],[34,393373,393374,393377],{"style":31809},[34,393375],{"className":393376,"style":31814},[31813],[34,393378,393380],{"className":393379},[31818,31819,31820,31821],[34,393381,43379],{"className":393382,"style":47113},[31685,31686,31821],[34,393384,31829],{"className":393385},[31828],[34,393387,393389],{"className":393388},[31801],[34,393390,393392],{"className":393391,"style":31836},[31805],[34,393393],{},[34,393395,111],{"className":393396},[31685,31686],[34,393398],{"className":393399,"style":31846},[31845],[34,393401,202],{"className":393402},[31850],[34,393404],{"className":393405,"style":31846},[31845],[34,393407,393409,393412],{"className":393408},[31676],[34,393410],{"className":393411,"style":32527},[31680],[34,393413,111],{"className":393414},[31685,31686],[34,393416,393418,393435],{"className":393417},[31642],[34,393419,393421],{"className":393420},[31646],[31648,393422,393423],{"xmlns":31650},[31652,393424,393425,393433],{},[31655,393426,393427],{},[31743,393428,393429,393431],{},[31658,393430,33244],{},[31658,393432,43379],{},[31665,393434,390304],{"encoding":31667},[34,393436,393438],{"className":393437,"ariaHidden":758},[31672],[34,393439,393441,393444],{"className":393440},[31676],[34,393442],{"className":393443,"style":31860},[31680],[34,393445,393447,393450],{"className":393446},[31685],[34,393448,33244],{"className":393449},[31685,31686],[34,393451,393453],{"className":393452},[31792],[34,393454,393456,393476],{"className":393455},[31796,31797],[34,393457,393459,393473],{"className":393458},[31801],[34,393460,393462],{"className":393461,"style":33833},[31805],[34,393463,393464,393467],{"style":31809},[34,393465],{"className":393466,"style":31814},[31813],[34,393468,393470],{"className":393469},[31818,31819,31820,31821],[34,393471,43379],{"className":393472,"style":47113},[31685,31686,31821],[34,393474,31829],{"className":393475},[31828],[34,393477,393479],{"className":393478},[31801],[34,393480,393482],{"className":393481,"style":31836},[31805],[34,393483],{}," 也是右幺元。结合(1)，它就是幺元。",[15,393486,393487],{},[2627,393488,393489,393490,393528],{},"(3) 有左无右 ",[34,393491,393493,393510],{"className":393492},[31642],[34,393494,393496],{"className":393495},[31646],[31648,393497,393498],{"xmlns":31650},[31652,393499,393500,393508],{},[31655,393501,393502,393504,393506],{},[39361,393503,262622],{},[31662,393505,262625],{},[39361,393507,262622],{},[31665,393509,337751],{"encoding":31667},[34,393511,393513],{"className":393512,"ariaHidden":758},[31672],[34,393514,393516,393519,393522,393525],{"className":393515},[31676],[34,393517],{"className":393518,"style":318264},[31680],[34,393520],{"className":393521,"style":31846},[31845],[34,393523,262625],{"className":393524},[31850],[34,393526],{"className":393527,"style":31846},[31845]," 至少两个左幺元",[105,393530,393531,394185],{},[108,393532,393533,393535,393536,393605,393606,335149,393634,393764,393765,393917,393918,394037,394038,347],{},[2627,393534,391664],{},"：既然 ",[34,393537,393539,393556],{"className":393538},[31642],[34,393540,393542],{"className":393541},[31646],[31648,393543,393544],{"xmlns":31650},[31652,393545,393546,393554],{},[31655,393547,393548],{},[31743,393549,393550,393552],{},[31658,393551,33244],{},[31658,393553,43379],{},[31665,393555,390304],{"encoding":31667},[34,393557,393559],{"className":393558,"ariaHidden":758},[31672],[34,393560,393562,393565],{"className":393561},[31676],[34,393563],{"className":393564,"style":31860},[31680],[34,393566,393568,393571],{"className":393567},[31685],[34,393569,33244],{"className":393570},[31685,31686],[34,393572,393574],{"className":393573},[31792],[34,393575,393577,393597],{"className":393576},[31796,31797],[34,393578,393580,393594],{"className":393579},[31801],[34,393581,393583],{"className":393582,"style":33833},[31805],[34,393584,393585,393588],{"style":31809},[34,393586],{"className":393587,"style":31814},[31813],[34,393589,393591],{"className":393590},[31818,31819,31820,31821],[34,393592,43379],{"className":393593,"style":47113},[31685,31686,31821],[34,393595,31829],{"className":393596},[31828],[34,393598,393600],{"className":393599},[31801],[34,393601,393603],{"className":393602,"style":31836},[31805],[34,393604],{}," 只是左不是右，说明存在某个坏孩子 ",[34,393607,393609,393622],{"className":393608},[31642],[34,393610,393612],{"className":393611},[31646],[31648,393613,393614],{"xmlns":31650},[31652,393615,393616,393620],{},[31655,393617,393618],{},[31658,393619,111],{},[31665,393621,111],{"encoding":31667},[34,393623,393625],{"className":393624,"ariaHidden":758},[31672],[34,393626,393628,393631],{"className":393627},[31676],[34,393629],{"className":393630,"style":32527},[31680],[34,393632,111],{"className":393633},[31685,31686],[34,393635,393637,393661],{"className":393636},[31642],[34,393638,393640],{"className":393639},[31646],[31648,393641,393642],{"xmlns":31650},[31652,393643,393644,393658],{},[31655,393645,393646,393648,393654,393656],{},[31658,393647,111],{},[31743,393649,393650,393652],{},[31658,393651,33244],{},[31658,393653,43379],{},[31662,393655,54707],{"mathvariant":38079},[31658,393657,111],{},[31665,393659,393660],{"encoding":31667},"a e_l \\neq a",[34,393662,393664,393755],{"className":393663,"ariaHidden":758},[31672],[34,393665,393667,393670,393673,393713,393716,393752],{"className":393666},[31676],[34,393668],{"className":393669,"style":43217},[31680],[34,393671,111],{"className":393672},[31685,31686],[34,393674,393676,393679],{"className":393675},[31685],[34,393677,33244],{"className":393678},[31685,31686],[34,393680,393682],{"className":393681},[31792],[34,393683,393685,393705],{"className":393684},[31796,31797],[34,393686,393688,393702],{"className":393687},[31801],[34,393689,393691],{"className":393690,"style":33833},[31805],[34,393692,393693,393696],{"style":31809},[34,393694],{"className":393695,"style":31814},[31813],[34,393697,393699],{"className":393698},[31818,31819,31820,31821],[34,393700,43379],{"className":393701,"style":47113},[31685,31686,31821],[34,393703,31829],{"className":393704},[31828],[34,393706,393708],{"className":393707},[31801],[34,393709,393711],{"className":393710,"style":31836},[31805],[34,393712],{},[34,393714],{"className":393715,"style":31846},[31845],[34,393717,393719,393746,393749],{"className":393718},[31850],[34,393720,393722],{"className":393721},[31850],[34,393723,393725],{"className":393724},[31685,54774],[34,393726,393728],{"className":393727},[54778],[34,393729,393731,393734,393743],{"className":393730},[54782],[34,393732],{"className":393733,"style":43217},[31680],[34,393735,393737],{"className":393736},[54789],[34,393738,393740],{"className":393739},[31685],[34,393741,54796],{"className":393742},[31850],[34,393744],{"className":393745},[54800],[34,393747],{"className":393748},[31845,54804],[34,393750,202],{"className":393751},[31850],[34,393753],{"className":393754,"style":31846},[31845],[34,393756,393758,393761],{"className":393757},[31676],[34,393759],{"className":393760,"style":32527},[31680],[34,393762,111],{"className":393763},[31685,31686],"。也就是 ",[34,393766,393768,393796],{"className":393767},[31642],[34,393769,393771],{"className":393770},[31646],[31648,393772,393773],{"xmlns":31650},[31652,393774,393775,393793],{},[31655,393776,393777,393779,393785,393787,393789,393791],{},[31658,393778,111],{},[31743,393780,393781,393783],{},[31658,393782,33244],{},[31658,393784,43379],{},[31662,393786,32226],{},[31658,393788,111],{},[31662,393790,54707],{"mathvariant":38079},[31749,393792,459],{},[31665,393794,393795],{"encoding":31667},"a e_l - a \\neq 0",[34,393797,393799,393857,393908],{"className":393798,"ariaHidden":758},[31672],[34,393800,393802,393805,393808,393848,393851,393854],{"className":393801},[31676],[34,393803],{"className":393804,"style":40522},[31680],[34,393806,111],{"className":393807},[31685,31686],[34,393809,393811,393814],{"className":393810},[31685],[34,393812,33244],{"className":393813},[31685,31686],[34,393815,393817],{"className":393816},[31792],[34,393818,393820,393840],{"className":393819},[31796,31797],[34,393821,393823,393837],{"className":393822},[31801],[34,393824,393826],{"className":393825,"style":33833},[31805],[34,393827,393828,393831],{"style":31809},[34,393829],{"className":393830,"style":31814},[31813],[34,393832,393834],{"className":393833},[31818,31819,31820,31821],[34,393835,43379],{"className":393836,"style":47113},[31685,31686,31821],[34,393838,31829],{"className":393839},[31828],[34,393841,393843],{"className":393842},[31801],[34,393844,393846],{"className":393845,"style":31836},[31805],[34,393847],{},[34,393849],{"className":393850,"style":32604},[31845],[34,393852,32226],{"className":393853},[32024],[34,393855],{"className":393856,"style":32604},[31845],[34,393858,393860,393863,393866,393869,393905],{"className":393859},[31676],[34,393861],{"className":393862,"style":43217},[31680],[34,393864,111],{"className":393865},[31685,31686],[34,393867],{"className":393868,"style":31846},[31845],[34,393870,393872,393899,393902],{"className":393871},[31850],[34,393873,393875],{"className":393874},[31850],[34,393876,393878],{"className":393877},[31685,54774],[34,393879,393881],{"className":393880},[54778],[34,393882,393884,393887,393896],{"className":393883},[54782],[34,393885],{"className":393886,"style":43217},[31680],[34,393888,393890],{"className":393889},[54789],[34,393891,393893],{"className":393892},[31685],[34,393894,54796],{"className":393895},[31850],[34,393897],{"className":393898},[54800],[34,393900],{"className":393901},[31845,54804],[34,393903,202],{"className":393904},[31850],[34,393906],{"className":393907,"style":31846},[31845],[34,393909,393911,393914],{"className":393910},[31676],[34,393912],{"className":393913,"style":32617},[31680],[34,393915,459],{"className":393916},[31685],"。\n我们要利用这个非零的差值，构造一个新的左幺元。\n设 ",[34,393919,393921,393949],{"className":393920},[31642],[34,393922,393924],{"className":393923},[31646],[31648,393925,393926],{"xmlns":31650},[31652,393927,393928,393946],{},[31655,393929,393930,393932,393934,393936,393942,393944],{},[31658,393931,94628],{},[31662,393933,202],{},[31658,393935,111],{},[31743,393937,393938,393940],{},[31658,393939,33244],{},[31658,393941,43379],{},[31662,393943,32226],{},[31658,393945,111],{},[31665,393947,393948],{"encoding":31667},"u = a e_l - a",[34,393950,393952,393970,394028],{"className":393951,"ariaHidden":758},[31672],[34,393953,393955,393958,393961,393964,393967],{"className":393954},[31676],[34,393956],{"className":393957,"style":32527},[31680],[34,393959,94628],{"className":393960},[31685,31686],[34,393962],{"className":393963,"style":31846},[31845],[34,393965,202],{"className":393966},[31850],[34,393968],{"className":393969,"style":31846},[31845],[34,393971,393973,393976,393979,394019,394022,394025],{"className":393972},[31676],[34,393974],{"className":393975,"style":40522},[31680],[34,393977,111],{"className":393978},[31685,31686],[34,393980,393982,393985],{"className":393981},[31685],[34,393983,33244],{"className":393984},[31685,31686],[34,393986,393988],{"className":393987},[31792],[34,393989,393991,394011],{"className":393990},[31796,31797],[34,393992,393994,394008],{"className":393993},[31801],[34,393995,393997],{"className":393996,"style":33833},[31805],[34,393998,393999,394002],{"style":31809},[34,394000],{"className":394001,"style":31814},[31813],[34,394003,394005],{"className":394004},[31818,31819,31820,31821],[34,394006,43379],{"className":394007,"style":47113},[31685,31686,31821],[34,394009,31829],{"className":394010},[31828],[34,394012,394014],{"className":394013},[31801],[34,394015,394017],{"className":394016,"style":31836},[31805],[34,394018],{},[34,394020],{"className":394021,"style":32604},[31845],[34,394023,32226],{"className":394024},[32024],[34,394026],{"className":394027,"style":32604},[31845],[34,394029,394031,394034],{"className":394030},[31676],[34,394032],{"className":394033,"style":32527},[31680],[34,394035,111],{"className":394036},[31685,31686],"。\n构造 ",[34,394039,394041,394071],{"className":394040},[31642],[34,394042,394044],{"className":394043},[31646],[31648,394045,394046],{"xmlns":31650},[31652,394047,394048,394068],{},[31655,394049,394050,394056,394058,394064,394066],{},[37624,394051,394052,394054],{},[31658,394053,33244],{},[31662,394055,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,394057,202],{},[31743,394059,394060,394062],{},[31658,394061,33244],{},[31658,394063,43379],{},[31662,394065,223],{},[31658,394067,94628],{},[31665,394069,394070],{"encoding":31667},"e' = e_l + u",[34,394072,394074,394121,394176],{"className":394073,"ariaHidden":758},[31672],[34,394075,394077,394080,394112,394115,394118],{"className":394076},[31676],[34,394078],{"className":394079,"style":132027},[31680],[34,394081,394083,394086],{"className":394082},[31685],[34,394084,33244],{"className":394085},[31685,31686],[34,394087,394089],{"className":394088},[31792],[34,394090,394092],{"className":394091},[31796],[34,394093,394095],{"className":394094},[31801],[34,394096,394098],{"className":394097,"style":132027},[31805],[34,394099,394100,394103],{"style":37777},[34,394101],{"className":394102,"style":31814},[31813],[34,394104,394106],{"className":394105},[31818,31819,31820,31821],[34,394107,394109],{"className":394108},[31685,31821],[34,394110,116495],{"className":394111},[31685,31821],[34,394113],{"className":394114,"style":31846},[31845],[34,394116,202],{"className":394117},[31850],[34,394119],{"className":394120,"style":31846},[31845],[34,394122,394124,394127,394167,394170,394173],{"className":394123},[31676],[34,394125],{"className":394126,"style":40522},[31680],[34,394128,394130,394133],{"className":394129},[31685],[34,394131,33244],{"className":394132},[31685,31686],[34,394134,394136],{"className":394135},[31792],[34,394137,394139,394159],{"className":394138},[31796,31797],[34,394140,394142,394156],{"className":394141},[31801],[34,394143,394145],{"className":394144,"style":33833},[31805],[34,394146,394147,394150],{"style":31809},[34,394148],{"className":394149,"style":31814},[31813],[34,394151,394153],{"className":394152},[31818,31819,31820,31821],[34,394154,43379],{"className":394155,"style":47113},[31685,31686,31821],[34,394157,31829],{"className":394158},[31828],[34,394160,394162],{"className":394161},[31801],[34,394163,394165],{"className":394164,"style":31836},[31805],[34,394166],{},[34,394168],{"className":394169,"style":32604},[31845],[34,394171,223],{"className":394172},[32024],[34,394174],{"className":394175,"style":32604},[31845],[34,394177,394179,394182],{"className":394178},[31676],[34,394180],{"className":394181,"style":32527},[31680],[34,394183,94628],{"className":394184},[31685,31686],[108,394186,394187,394252,394253,394304,394305,86621,394705,394774,394775,355408,394872,355408,394968,395082,395083,355408,395182,395271,395272,395300,395301,395362,395363,13977,395447,327546],{},[2627,394188,343971,394189,394251],{},[34,394190,394192,394210],{"className":394191},[31642],[34,394193,394195],{"className":394194},[31646],[31648,394196,394197],{"xmlns":31650},[31652,394198,394199,394207],{},[31655,394200,394201],{},[37624,394202,394203,394205],{},[31658,394204,33244],{},[31662,394206,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,394208,394209],{"encoding":31667},"e'",[34,394211,394213],{"className":394212,"ariaHidden":758},[31672],[34,394214,394216,394219],{"className":394215},[31676],[34,394217],{"className":394218,"style":132027},[31680],[34,394220,394222,394225],{"className":394221},[31685],[34,394223,33244],{"className":394224},[31685,31686],[34,394226,394228],{"className":394227},[31792],[34,394229,394231],{"className":394230},[31796],[34,394232,394234],{"className":394233},[31801],[34,394235,394237],{"className":394236,"style":132027},[31805],[34,394238,394239,394242],{"style":37777},[34,394240],{"className":394241,"style":31814},[31813],[34,394243,394245],{"className":394244},[31818,31819,31820,31821],[34,394246,394248],{"className":394247},[31685,31821],[34,394249,116495],{"className":394250},[31685,31821]," 是左幺元","：\n对于任意 ",[34,394254,394256,394274],{"className":394255},[31642],[34,394257,394259],{"className":394258},[31646],[31648,394260,394261],{"xmlns":31650},[31652,394262,394263,394271],{},[31655,394264,394265,394267,394269],{},[31658,394266,1697],{},[31662,394268,112296],{},[31658,394270,34386],{},[31665,394272,394273],{"encoding":31667},"b \\in R",[34,394275,394277,394295],{"className":394276,"ariaHidden":758},[31672],[34,394278,394280,394283,394286,394289,394292],{"className":394279},[31676],[34,394281],{"className":394282,"style":46991},[31680],[34,394284,1697],{"className":394285},[31685,31686],[34,394287],{"className":394288,"style":31846},[31845],[34,394290,112296],{"className":394291},[31850],[34,394293],{"className":394294,"style":31846},[31845],[34,394296,394298,394301],{"className":394297},[31676],[34,394299],{"className":394300,"style":33218},[31680],[34,394302,34386],{"className":394303,"style":34427},[31685,31686],"，计算 ",[34,394306,394308,394384],{"className":394307},[31642],[34,394309,394311],{"className":394310},[31646],[31648,394312,394313],{"xmlns":31650},[31652,394314,394315,394381],{},[31655,394316,394317,394323,394325,394327,394329,394335,394337,394339,394345,394347,394349,394351,394353,394355,394361,394363,394365,394367,394373,394375,394377,394379],{},[37624,394318,394319,394321],{},[31658,394320,33244],{},[31662,394322,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,394324,1697],{},[31662,394326,202],{},[31662,394328,456],{"stretchy":702},[31743,394330,394331,394333],{},[31658,394332,33244],{},[31658,394334,43379],{},[31662,394336,223],{},[31658,394338,111],{},[31743,394340,394341,394343],{},[31658,394342,33244],{},[31658,394344,43379],{},[31662,394346,32226],{},[31658,394348,111],{},[31662,394350,2557],{"stretchy":702},[31658,394352,1697],{},[31662,394354,202],{},[31743,394356,394357,394359],{},[31658,394358,33244],{},[31658,394360,43379],{},[31658,394362,1697],{},[31662,394364,223],{},[31658,394366,111],{},[31743,394368,394369,394371],{},[31658,394370,33244],{},[31658,394372,43379],{},[31658,394374,1697],{},[31662,394376,32226],{},[31658,394378,111],{},[31658,394380,1697],{},[31665,394382,394383],{"encoding":31667},"e' b = (e_l + a e_l - a) b = e_l b + a e_l b - a b",[34,394385,394387,394437,394495,394553,394577,394635,394696],{"className":394386,"ariaHidden":758},[31672],[34,394388,394390,394393,394425,394428,394431,394434],{"className":394389},[31676],[34,394391],{"className":394392,"style":132027},[31680],[34,394394,394396,394399],{"className":394395},[31685],[34,394397,33244],{"className":394398},[31685,31686],[34,394400,394402],{"className":394401},[31792],[34,394403,394405],{"className":394404},[31796],[34,394406,394408],{"className":394407},[31801],[34,394409,394411],{"className":394410,"style":132027},[31805],[34,394412,394413,394416],{"style":37777},[34,394414],{"className":394415,"style":31814},[31813],[34,394417,394419],{"className":394418},[31818,31819,31820,31821],[34,394420,394422],{"className":394421},[31685,31821],[34,394423,116495],{"className":394424},[31685,31821],[34,394426,1697],{"className":394427},[31685,31686],[34,394429],{"className":394430,"style":31846},[31845],[34,394432,202],{"className":394433},[31850],[34,394435],{"className":394436,"style":31846},[31845],[34,394438,394440,394443,394446,394486,394489,394492],{"className":394439},[31676],[34,394441],{"className":394442,"style":31774},[31680],[34,394444,456],{"className":394445},[31782],[34,394447,394449,394452],{"className":394448},[31685],[34,394450,33244],{"className":394451},[31685,31686],[34,394453,394455],{"className":394454},[31792],[34,394456,394458,394478],{"className":394457},[31796,31797],[34,394459,394461,394475],{"className":394460},[31801],[34,394462,394464],{"className":394463,"style":33833},[31805],[34,394465,394466,394469],{"style":31809},[34,394467],{"className":394468,"style":31814},[31813],[34,394470,394472],{"className":394471},[31818,31819,31820,31821],[34,394473,43379],{"className":394474,"style":47113},[31685,31686,31821],[34,394476,31829],{"className":394477},[31828],[34,394479,394481],{"className":394480},[31801],[34,394482,394484],{"className":394483,"style":31836},[31805],[34,394485],{},[34,394487],{"className":394488,"style":32604},[31845],[34,394490,223],{"className":394491},[32024],[34,394493],{"className":394494,"style":32604},[31845],[34,394496,394498,394501,394504,394544,394547,394550],{"className":394497},[31676],[34,394499],{"className":394500,"style":40522},[31680],[34,394502,111],{"className":394503},[31685,31686],[34,394505,394507,394510],{"className":394506},[31685],[34,394508,33244],{"className":394509},[31685,31686],[34,394511,394513],{"className":394512},[31792],[34,394514,394516,394536],{"className":394515},[31796,31797],[34,394517,394519,394533],{"className":394518},[31801],[34,394520,394522],{"className":394521,"style":33833},[31805],[34,394523,394524,394527],{"style":31809},[34,394525],{"className":394526,"style":31814},[31813],[34,394528,394530],{"className":394529},[31818,31819,31820,31821],[34,394531,43379],{"className":394532,"style":47113},[31685,31686,31821],[34,394534,31829],{"className":394535},[31828],[34,394537,394539],{"className":394538},[31801],[34,394540,394542],{"className":394541,"style":31836},[31805],[34,394543],{},[34,394545],{"className":394546,"style":32604},[31845],[34,394548,32226],{"className":394549},[32024],[34,394551],{"className":394552,"style":32604},[31845],[34,394554,394556,394559,394562,394565,394568,394571,394574],{"className":394555},[31676],[34,394557],{"className":394558,"style":31774},[31680],[34,394560,111],{"className":394561},[31685,31686],[34,394563,2557],{"className":394564},[31690],[34,394566,1697],{"className":394567},[31685,31686],[34,394569],{"className":394570,"style":31846},[31845],[34,394572,202],{"className":394573},[31850],[34,394575],{"className":394576,"style":31846},[31845],[34,394578,394580,394583,394623,394626,394629,394632],{"className":394579},[31676],[34,394581],{"className":394582,"style":33814},[31680],[34,394584,394586,394589],{"className":394585},[31685],[34,394587,33244],{"className":394588},[31685,31686],[34,394590,394592],{"className":394591},[31792],[34,394593,394595,394615],{"className":394594},[31796,31797],[34,394596,394598,394612],{"className":394597},[31801],[34,394599,394601],{"className":394600,"style":33833},[31805],[34,394602,394603,394606],{"style":31809},[34,394604],{"className":394605,"style":31814},[31813],[34,394607,394609],{"className":394608},[31818,31819,31820,31821],[34,394610,43379],{"className":394611,"style":47113},[31685,31686,31821],[34,394613,31829],{"className":394614},[31828],[34,394616,394618],{"className":394617},[31801],[34,394619,394621],{"className":394620,"style":31836},[31805],[34,394622],{},[34,394624,1697],{"className":394625},[31685,31686],[34,394627],{"className":394628,"style":32604},[31845],[34,394630,223],{"className":394631},[32024],[34,394633],{"className":394634,"style":32604},[31845],[34,394636,394638,394641,394644,394684,394687,394690,394693],{"className":394637},[31676],[34,394639],{"className":394640,"style":33814},[31680],[34,394642,111],{"className":394643},[31685,31686],[34,394645,394647,394650],{"className":394646},[31685],[34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是左幺元，",[34,394776,394778,394802],{"className":394777},[31642],[34,394779,394781],{"className":394780},[31646],[31648,394782,394783],{"xmlns":31650},[31652,394784,394785,394799],{},[31655,394786,394787,394793,394795,394797],{},[31743,394788,394789,394791],{},[31658,394790,33244],{},[31658,394792,43379],{},[31658,394794,1697],{},[31662,394796,202],{},[31658,394798,1697],{},[31665,394800,394801],{"encoding":31667},"e_l b = b",[34,394803,394805,394863],{"className":394804,"ariaHidden":758},[31672],[34,394806,394808,394811,394851,394854,394857,394860],{"className":394807},[31676],[34,394809],{"className":394810,"style":33814},[31680],[34,394812,394814,394817],{"className":394813},[31685],[34,394815,33244],{"className":394816},[31685,31686],[34,394818,394820],{"className":394819},[31792],[34,394821,394823,394843],{"className":394822},[31796,31797],[34,394824,394826,394840],{"className":394825},[31801],[34,394827,394829],{"className":394828,"style":33833},[31805],[34,394830,394831,394834],{"style":31809},[34,394832],{"className":394833,"style":31814},[31813],[34,394835,394837],{"className":394836},[31818,31819,31820,31821],[34,394838,43379],{"className":394839,"style":47113},[31685,31686,31821],[34,394841,31829],{"className":394842},[31828],[34,394844,394846],{"className":394845},[31801],[34,394847,394849],{"className":394848,"style":31836},[31805],[34,394850],{},[34,394852,1697],{"className":394853},[31685,31686],[34,394855],{"className":394856,"style":31846},[31845],[34,394858,202],{"className":394859},[31850],[34,394861],{"className":394862,"style":31846},[31845],[34,394864,394866,394869],{"className":394865},[31676],[34,394867],{"className":394868,"style":31681},[31680],[34,394870,1697],{"className":394871},[31685,31686],[34,394873,394875,394898],{"className":394874},[31642],[34,394876,394878],{"className":394877},[31646],[31648,394879,394880],{"xmlns":31650},[31652,394881,394882,394896],{},[31655,394883,394884,394890,394892,394894],{},[31743,394885,394886,394888],{},[31658,394887,33244],{},[31658,394889,43379],{},[31658,394891,1697],{},[31662,394893,202],{},[31658,394895,1697],{},[31665,394897,394801],{"encoding":31667},[34,394899,394901,394959],{"className":394900,"ariaHidden":758},[31672],[34,394902,394904,394907,394947,394950,394953,394956],{"className":394903},[31676],[34,394905],{"className":394906,"style":33814},[31680],[34,394908,394910,394913],{"className":394909},[31685],[34,394911,33244],{"className":394912},[31685,31686],[34,394914,394916],{"className":394915},[31792],[34,394917,394919,394939],{"className":394918},[31796,31797],[34,394920,394922,394936],{"className":394921},[31801],[34,394923,394925],{"className":394924,"style":33833},[31805],[34,394926,394927,394930],{"style":31809},[34,394928],{"className":394929,"style":31814},[31813],[34,394931,394933],{"className":394932},[31818,31819,31820,31821],[34,394934,43379],{"className":394935,"style":47113},[31685,31686,31821],[34,394937,31829],{"className":394938},[31828],[34,394940,394942],{"className":394941},[31801],[34,394943,394945],{"className":394944,"style":31836},[31805],[34,394946],{},[34,394948,1697],{"className":394949},[31685,31686],[34,394951],{"className":394952,"style":31846},[31845],[34,394954,202],{"className":394955},[31850],[34,394957],{"className":394958,"style":31846},[31845],[34,394960,394962,394965],{"className":394961},[31676],[34,394963],{"className":394964,"style":31681},[31680],[34,394966,1697],{"className":394967},[31685,31686],[34,394969,394971,395003],{"className":394970},[31642],[34,394972,394974],{"className":394973},[31646],[31648,394975,394976],{"xmlns":31650},[31652,394977,394978,395000],{},[31655,394979,394980,394982,394984,394990,394992,394994,394996,394998],{},[31658,394981,111],{},[31662,394983,456],{"stretchy":702},[31743,394985,394986,394988],{},[31658,394987,33244],{},[31658,394989,43379],{},[31658,394991,1697],{},[31662,394993,2557],{"stretchy":702},[31662,394995,202],{},[31658,394997,111],{},[31658,394999,1697],{},[31665,395001,395002],{"encoding":31667},"a (e_l b) = ab",[34,395004,395006,395073],{"className":395005,"ariaHidden":758},[31672],[34,395007,395009,395012,395015,395018,395058,395061,395064,395067,395070],{"className":395008},[31676],[34,395010],{"className":395011,"style":31774},[31680],[34,395013,111],{"className":395014},[31685,31686],[34,395016,456],{"className":395017},[31782],[34,395019,395021,395024],{"className":395020},[31685],[34,395022,33244],{"className":395023},[31685,31686],[34,395025,395027],{"className":395026},[31792],[34,395028,395030,395050],{"className":395029},[31796,31797],[34,395031,395033,395047],{"className":395032},[31801],[34,395034,395036],{"className":395035,"style":33833},[31805],[34,395037,395038,395041],{"style":31809},[34,395039],{"className":395040,"style":31814},[31813],[34,395042,395044],{"className":395043},[31818,31819,31820,31821],[34,395045,43379],{"className":395046,"style":47113},[31685,31686,31821],[34,395048,31829],{"className":395049},[31828],[34,395051,395053],{"className":395052},[31801],[34,395054,395056],{"className":395055,"style":31836},[31805],[34,395057],{},[34,395059,1697],{"className":395060},[31685,31686],[34,395062,2557],{"className":395063},[31690],[34,395065],{"className":395066,"style":31846},[31845],[34,395068,202],{"className":395069},[31850],[34,395071],{"className":395072,"style":31846},[31845],[34,395074,395076,395079],{"className":395075},[31676],[34,395077],{"className":395078,"style":31681},[31680],[34,395080,324191],{"className":395081},[31685,31686],"。\n代入得：",[34,395084,395086,395116],{"className":395085},[31642],[34,395087,395089],{"className":395088},[31646],[31648,395090,395091],{"xmlns":31650},[31652,395092,395093,395113],{},[31655,395094,395095,395097,395099,395101,395103,395105,395107,395109,395111],{},[31658,395096,1697],{},[31662,395098,223],{},[31658,395100,111],{},[31658,395102,1697],{},[31662,395104,32226],{},[31658,395106,111],{},[31658,395108,1697],{},[31662,395110,202],{},[31658,395112,1697],{},[31665,395114,395115],{"encoding":31667},"b + ab - ab = b",[34,395117,395119,395137,395155,395173],{"className":395118,"ariaHidden":758},[31672],[34,395120,395122,395125,395128,395131,395134],{"className":395121},[31676],[34,395123],{"className":395124,"style":34299},[31680],[34,395126,1697],{"className":395127},[31685,31686],[34,395129],{"className":395130,"style":32604},[31845],[34,395132,223],{"className":395133},[32024],[34,395135],{"className":395136,"style":32604},[31845],[34,395138,395140,395143,395146,395149,395152],{"className":395139},[31676],[34,395141],{"className":395142,"style":34299},[31680],[34,395144,324191],{"className":395145},[31685,31686],[34,395147],{"className":395148,"style":32604},[31845],[34,395150,32226],{"className":395151},[32024],[34,395153],{"className":395154,"style":32604},[31845],[34,395156,395158,395161,395164,395167,395170],{"className":395157},[31676],[34,395159],{"className":395160,"style":31681},[31680],[34,395162,324191],{"className":395163},[31685,31686],[34,395165],{"className":395166,"style":31846},[31845],[34,395168,202],{"className":395169},[31850],[34,395171],{"className":395172,"style":31846},[31845],[34,395174,395176,395179],{"className":395175},[31676],[34,395177],{"className":395178,"style":31681},[31680],[34,395180,1697],{"className":395181},[31685,31686],[34,395183,395185,395209],{"className":395184},[31642],[34,395186,395188],{"className":395187},[31646],[31648,395189,395190],{"xmlns":31650},[31652,395191,395192,395206],{},[31655,395193,395194,395200,395202,395204],{},[37624,395195,395196,395198],{},[31658,395197,33244],{},[31662,395199,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,395201,1697],{},[31662,395203,202],{},[31658,395205,1697],{},[31665,395207,395208],{"encoding":31667},"e' b = b",[34,395210,395212,395262],{"className":395211,"ariaHidden":758},[31672],[34,395213,395215,395218,395250,395253,395256,395259],{"className":395214},[31676],[34,395216],{"className":395217,"style":132027},[31680],[34,395219,395221,395224],{"className":395220},[31685],[34,395222,33244],{"className":395223},[31685,31686],[34,395225,395227],{"className":395226},[31792],[34,395228,395230],{"className":395229},[31796],[34,395231,395233],{"className":395232},[31801],[34,395234,395236],{"className":395235,"style":132027},[31805],[34,395237,395238,395241],{"style":37777},[34,395239],{"className":395240,"style":31814},[31813],[34,395242,395244],{"className":395243},[31818,31819,31820,31821],[34,395245,395247],{"className":395246},[31685,31821],[34,395248,116495],{"className":395249},[31685,31821],[34,395251,1697],{"className":395252},[31685,31686],[34,395254],{"className":395255,"style":31846},[31845],[34,395257,202],{"className":395258},[31850],[34,395260],{"className":395261,"style":31846},[31845],[34,395263,395265,395268],{"className":395264},[31676],[34,395266],{"className":395267,"style":31681},[31680],[34,395269,1697],{"className":395270},[31685,31686]," 对任意 ",[34,395273,395275,395288],{"className":395274},[31642],[34,395276,395278],{"className":395277},[31646],[31648,395279,395280],{"xmlns":31650},[31652,395281,395282,395286],{},[31655,395283,395284],{},[31658,395285,1697],{},[31665,395287,1697],{"encoding":31667},[34,395289,395291],{"className":395290,"ariaHidden":758},[31672],[34,395292,395294,395297],{"className":395293},[31676],[34,395295],{"className":395296,"style":31681},[31680],[34,395298,1697],{"className":395299},[31685,31686]," 成立。",[34,395302,395304,395321],{"className":395303},[31642],[34,395305,395307],{"className":395306},[31646],[31648,395308,395309],{"xmlns":31650},[31652,395310,395311,395319],{},[31655,395312,395313],{},[37624,395314,395315,395317],{},[31658,395316,33244],{},[31662,395318,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,395320,394209],{"encoding":31667},[34,395322,395324],{"className":395323,"ariaHidden":758},[31672],[34,395325,395327,395330],{"className":395326},[31676],[34,395328],{"className":395329,"style":132027},[31680],[34,395331,395333,395336],{"className":395332},[31685],[34,395334,33244],{"className":395335},[31685,31686],[34,395337,395339],{"className":395338},[31792],[34,395340,395342],{"className":395341},[31796],[34,395343,395345],{"className":395344},[31801],[34,395346,395348],{"className":395347,"style":132027},[31805],[34,395349,395350,395353],{"style":37777},[34,395351],{"className":395352,"style":31814},[31813],[34,395354,395356],{"className":395355},[31818,31819,31820,31821],[34,395357,395359],{"className":395358},[31685,31821],[34,395360,116495],{"className":395361},[31685,31821]," 确实是左幺元。\n而且 ",[34,395364,395366,395384],{"className":395365},[31642],[34,395367,395369],{"className":395368},[31646],[31648,395370,395371],{"xmlns":31650},[31652,395372,395373,395381],{},[31655,395374,395375,395377,395379],{},[31658,395376,94628],{},[31662,395378,54707],{"mathvariant":38079},[31749,395380,459],{},[31665,395382,395383],{"encoding":31667},"u \\neq 0",[34,395385,395387,395438],{"className":395386,"ariaHidden":758},[31672],[34,395388,395390,395393,395396,395399,395435],{"className":395389},[31676],[34,395391],{"className":395392,"style":43217},[31680],[34,395394,94628],{"className":395395},[31685,31686],[34,395397],{"className":395398,"style":31846},[31845],[34,395400,395402,395429,395432],{"className":395401},[31850],[34,395403,395405],{"className":395404},[31850],[34,395406,395408],{"className":395407},[31685,54774],[34,395409,395411],{"className":395410},[54778],[34,395412,395414,395417,395426],{"className":395413},[54782],[34,395415],{"className":395416,"style":43217},[31680],[34,395418,395420],{"className":395419},[54789],[34,395421,395423],{"className":395422},[31685],[34,395424,54796],{"className":395425},[31850],[34,395427],{"className":395428},[54800],[34,395430],{"className":395431},[31845,54804],[34,395433,202],{"className":395434},[31850],[34,395436],{"className":395437,"style":31846},[31845],[34,395439,395441,395444],{"className":395440},[31676],[34,395442],{"className":395443,"style":32617},[31680],[34,395445,459],{"className":395446},[31685],[34,395448,395450,395476],{"className":395449},[31642],[34,395451,395453],{"className":395452},[31646],[31648,395454,395455],{"xmlns":31650},[31652,395456,395457,395473],{},[31655,395458,395459,395465,395467],{},[37624,395460,395461,395463],{},[31658,395462,33244],{},[31662,395464,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,395466,54707],{"mathvariant":38079},[31743,395468,395469,395471],{},[31658,395470,33244],{},[31658,395472,43379],{},[31665,395474,395475],{"encoding":31667},"e' \\neq e_l",[34,395477,395479,395560],{"className":395478,"ariaHidden":758},[31672],[34,395480,395482,395486,395518,395521,395557],{"className":395481},[31676],[34,395483],{"className":395484,"style":395485},[31680],"height:0.9463em;vertical-align:-0.1944em;",[34,395487,395489,395492],{"className":395488},[31685],[34,395490,33244],{"className":395491},[31685,31686],[34,395493,395495],{"className":395494},[31792],[34,395496,395498],{"className":395497},[31796],[34,395499,395501],{"className":395500},[31801],[34,395502,395504],{"className":395503,"style":132027},[31805],[34,395505,395506,395509],{"style":37777},[34,395507],{"className":395508,"style":31814},[31813],[34,395510,395512],{"className":395511},[31818,31819,31820,31821],[34,395513,395515],{"className":395514},[31685,31821],[34,395516,116495],{"className":395517},[31685,31821],[34,395519],{"className":395520,"style":31846},[31845],[34,395522,395524,395551,395554],{"className":395523},[31850],[34,395525,395527],{"className":395526},[31850],[34,395528,395530],{"className":395529},[31685,54774],[34,395531,395533],{"className":395532},[54778],[34,395534,395536,395539,395548],{"className":395535},[54782],[34,395537],{"className":395538,"style":43217},[31680],[34,395540,395542],{"className":395541},[54789],[34,395543,395545],{"className":395544},[31685],[34,395546,54796],{"className":395547},[31850],[34,395549],{"className":395550},[54800],[34,395552],{"className":395553},[31845,54804],[34,395555,202],{"className":395556},[31850],[34,395558],{"className":395559,"style":31846},[31845],[34,395561,395563,395566],{"className":395562},[31676],[34,395564],{"className":395565,"style":31860},[31680],[34,395567,395569,395572],{"className":395568},[31685],[34,395570,33244],{"className":395571},[31685,31686],[34,395573,395575],{"className":395574},[31792],[34,395576,395578,395598],{"className":395577},[31796,31797],[34,395579,395581,395595],{"className":395580},[31801],[34,395582,395584],{"className":395583,"style":33833},[31805],[34,395585,395586,395589],{"style":31809},[34,395587],{"className":395588,"style":31814},[31813],[34,395590,395592],{"className":395591},[31818,31819,31820,31821],[34,395593,43379],{"className":395594,"style":47113},[31685,31686,31821],[34,395596,31829],{"className":395597},[31828],[34,395599,395601],{"className":395600},[31801],[34,395602,395604],{"className":395603,"style":31836},[31805],[34,395605],{},[32188,395607],{},[253,395609,395610],{"id":395610},"题目10",[15,395612,133910,395613,395641,395642,395725,395726,315343,395810,329052,395866,371482],{},[34,395614,395616,395629],{"className":395615},[31642],[34,395617,395619],{"className":395618},[31646],[31648,395620,395621],{"xmlns":31650},[31652,395622,395623,395627],{},[31655,395624,395625],{},[31658,395626,34386],{},[31665,395628,34386],{"encoding":31667},[34,395630,395632],{"className":395631,"ariaHidden":758},[31672],[34,395633,395635,395638],{"className":395634},[31676],[34,395636],{"className":395637,"style":33218},[31680],[34,395639,34386],{"className":395640,"style":34427},[31685,31686]," 是环， ",[34,395643,395645,395662],{"className":395644},[31642],[34,395646,395648],{"className":395647},[31646],[31648,395649,395650],{"xmlns":31650},[31652,395651,395652,395660],{},[31655,395653,395654,395656,395658],{},[31658,395655,111],{},[31662,395657,54707],{"mathvariant":38079},[31749,395659,459],{},[31665,395661,363487],{"encoding":31667},[34,395663,395665,395716],{"className":395664,"ariaHidden":758},[31672],[34,395666,395668,395671,395674,395677,395713],{"className":395667},[31676],[34,395669],{"className":395670,"style":43217},[31680],[34,395672,111],{"className":395673},[31685,31686],[34,395675],{"className":395676,"style":31846},[31845],[34,395678,395680,395707,395710],{"className":395679},[31850],[34,395681,395683],{"className":395682},[31850],[34,395684,395686],{"className":395685},[31685,54774],[34,395687,395689],{"className":395688},[54778],[34,395690,395692,395695,395704],{"className":395691},[54782],[34,395693],{"className":395694,"style":43217},[31680],[34,395696,395698],{"className":395697},[54789],[34,395699,395701],{"className":395700},[31685],[34,395702,54796],{"className":395703},[31850],[34,395705],{"className":395706},[54800],[34,395708],{"className":395709},[31845,54804],[34,395711,202],{"className":395712},[31850],[34,395714],{"className":395715,"style":31846},[31845],[34,395717,395719,395722],{"className":395718},[31676],[34,395720],{"className":395721,"style":32617},[31680],[34,395723,459],{"className":395724},[31685],"，存在 ",[34,395727,395729,395747],{"className":395728},[31642],[34,395730,395732],{"className":395731},[31646],[31648,395733,395734],{"xmlns":31650},[31652,395735,395736,395744],{},[31655,395737,395738,395740,395742],{},[31658,395739,1697],{},[31662,395741,54707],{"mathvariant":38079},[31749,395743,459],{},[31665,395745,395746],{"encoding":31667},"b \\neq 0",[34,395748,395750,395801],{"className":395749,"ariaHidden":758},[31672],[34,395751,395753,395756,395759,395762,395798],{"className":395752},[31676],[34,395754],{"className":395755,"style":43217},[31680],[34,395757,1697],{"className":395758},[31685,31686],[34,395760],{"className":395761,"style":31846},[31845],[34,395763,395765,395792,395795],{"className":395764},[31850],[34,395766,395768],{"className":395767},[31850],[34,395769,395771],{"className":395770},[31685,54774],[34,395772,395774],{"className":395773},[54778],[34,395775,395777,395780,395789],{"className":395776},[54782],[34,395778],{"className":395779,"style":43217},[31680],[34,395781,395783],{"className":395782},[54789],[34,395784,395786],{"className":395785},[31685],[34,395787,54796],{"className":395788},[31850],[34,395790],{"className":395791},[54800],[34,395793],{"className":395794},[31845,54804],[34,395796,202],{"className":395797},[31850],[34,395799],{"className":395800,"style":31846},[31845],[34,395802,395804,395807],{"className":395803},[31676],[34,395805],{"className":395806,"style":32617},[31680],[34,395808,459],{"className":395809},[31685],[34,395811,395813,395835],{"className":395812},[31642],[34,395814,395816],{"className":395815},[31646],[31648,395817,395818],{"xmlns":31650},[31652,395819,395820,395832],{},[31655,395821,395822,395824,395826,395828,395830],{},[31658,395823,111],{},[31658,395825,1697],{},[31658,395827,111],{},[31662,395829,202],{},[31749,395831,459],{},[31665,395833,395834],{"encoding":31667},"aba = 0",[34,395836,395838,395857],{"className":395837,"ariaHidden":758},[31672],[34,395839,395841,395844,395848,395851,395854],{"className":395840},[31676],[34,395842],{"className":395843,"style":31681},[31680],[34,395845,395847],{"className":395846},[31685,31686],"aba",[34,395849],{"className":395850,"style":31846},[31845],[34,395852,202],{"className":395853},[31850],[34,395855],{"className":395856,"style":31846},[31845],[34,395858,395860,395863],{"className":395859},[31676],[34,395861],{"className":395862,"style":32617},[31680],[34,395864,459],{"className":395865},[31685],[34,395867,395869,395882],{"className":395868},[31642],[34,395870,395872],{"className":395871},[31646],[31648,395873,395874],{"xmlns":31650},[31652,395875,395876,395880],{},[31655,395877,395878],{},[31658,395879,111],{},[31665,395881,111],{"encoding":31667},[34,395883,395885],{"className":395884,"ariaHidden":758},[31672],[34,395886,395888,395891],{"className":395887},[31676],[34,395889],{"className":395890,"style":32527},[31680],[34,395892,111],{"className":395893},[31685,31686],[32490,395895,297792],{"id":395896},"详细解析-5",[15,395898,395899,89374,395902,355459,396054,327448,396110,258,396138,396166],{},[2627,395900,395901],{},"什么是零因子？",[34,395903,395905,395931],{"className":395904},[31642],[34,395906,395908],{"className":395907},[31646],[31648,395909,395910],{"xmlns":31650},[31652,395911,395912,395928],{},[31655,395913,395914,395916,395918,395920,395922,395924,395926],{},[31658,395915,32936],{},[31662,395917,54707],{"mathvariant":38079},[31749,395919,459],{},[31662,395921,31960],{"separator":758},[31658,395923,67764],{},[31662,395925,54707],{"mathvariant":38079},[31749,395927,459],{},[31665,395929,395930],{"encoding":31667},"x \\neq 0, y \\neq 0",[34,395932,395934,395985,396045],{"className":395933,"ariaHidden":758},[31672],[34,395935,395937,395940,395943,395946,395982],{"className":395936},[31676],[34,395938],{"className":395939,"style":43217},[31680],[34,395941,32936],{"className":395942},[31685,31686],[34,395944],{"className":395945,"style":31846},[31845],[34,395947,395949,395976,395979],{"className":395948},[31850],[34,395950,395952],{"className":395951},[31850],[34,395953,395955],{"className":395954},[31685,54774],[34,395956,395958],{"className":395957},[54778],[34,395959,395961,395964,395973],{"className":395960},[54782],[34,395962],{"className":395963,"style":43217},[31680],[34,395965,395967],{"className":395966},[54789],[34,395968,395970],{"className":395969},[31685],[34,395971,54796],{"className":395972},[31850],[34,395974],{"className":395975},[54800],[34,395977],{"className":395978},[31845,54804],[34,395980,202],{"className":395981},[31850],[34,395983],{"className":395984,"style":31846},[31845],[34,395986,395988,395991,395994,395997,396000,396003,396006,396042],{"className":395987},[31676],[34,395989],{"className":395990,"style":43217},[31680],[34,395992,459],{"className":395993},[31685],[34,395995,31960],{"className":395996},[32119],[34,395998],{"className":395999,"style":32123},[31845],[34,396001,67764],{"className":396002,"style":32062},[31685,31686],[34,396004],{"className":396005,"style":31846},[31845],[34,396007,396009,396036,396039],{"className":396008},[31850],[34,396010,396012],{"className":396011},[31850],[34,396013,396015],{"className":396014},[31685,54774],[34,396016,396018],{"className":396017},[54778],[34,396019,396021,396024,396033],{"className":396020},[54782],[34,396022],{"className":396023,"style":43217},[31680],[34,396025,396027],{"className":396026},[54789],[34,396028,396030],{"className":396029},[31685],[34,396031,54796],{"className":396032},[31850],[34,396034],{"className":396035},[54800],[34,396037],{"className":396038},[31845,54804],[34,396040,202],{"className":396041},[31850],[34,396043],{"className":396044,"style":31846},[31845],[34,396046,396048,396051],{"className":396047},[31676],[34,396049],{"className":396050,"style":32617},[31680],[34,396052,459],{"className":396053},[31685],[34,396055,396057,396077],{"className":396056},[31642],[34,396058,396060],{"className":396059},[31646],[31648,396061,396062],{"xmlns":31650},[31652,396063,396064,396074],{},[31655,396065,396066,396068,396070,396072],{},[31658,396067,32936],{},[31658,396069,67764],{},[31662,396071,202],{},[31749,396073,459],{},[31665,396075,396076],{"encoding":31667},"xy = 0",[34,396078,396080,396101],{"className":396079,"ariaHidden":758},[31672],[34,396081,396083,396086,396089,396092,396095,396098],{"className":396082},[31676],[34,396084],{"className":396085,"style":38774},[31680],[34,396087,32936],{"className":396088},[31685,31686],[34,396090,67764],{"className":396091,"style":32062},[31685,31686],[34,396093],{"className":396094,"style":31846},[31845],[34,396096,202],{"className":396097},[31850],[34,396099],{"className":396100,"style":31846},[31845],[34,396102,396104,396107],{"className":396103},[31676],[34,396105],{"className":396106,"style":32617},[31680],[34,396108,459],{"className":396109},[31685],[34,396111,396113,396126],{"className":396112},[31642],[34,396114,396116],{"className":396115},[31646],[31648,396117,396118],{"xmlns":31650},[31652,396119,396120,396124],{},[31655,396121,396122],{},[31658,396123,32936],{},[31665,396125,32936],{"encoding":31667},[34,396127,396129],{"className":396128,"ariaHidden":758},[31672],[34,396130,396132,396135],{"className":396131},[31676],[34,396133],{"className":396134,"style":32527},[31680],[34,396136,32936],{"className":396137},[31685,31686],[34,396139,396141,396154],{"className":396140},[31642],[34,396142,396144],{"className":396143},[31646],[31648,396145,396146],{"xmlns":31650},[31652,396147,396148,396152],{},[31655,396149,396150],{},[31658,396151,67764],{},[31665,396153,67764],{"encoding":31667},[34,396155,396157],{"className":396156,"ariaHidden":758},[31672],[34,396158,396160,396163],{"className":396159},[31676],[34,396161],{"className":396162,"style":38774},[31680],[34,396164,67764],{"className":396165,"style":32062},[31685,31686]," 都是零因子。",[15,396168,396169,396171,396172,396267,396268,396318],{},[2627,396170,390672],{},"\n题目给了等式：",[34,396173,396175,396201],{"className":396174},[31642],[34,396176,396178],{"className":396177},[31646],[31648,396179,396180],{"xmlns":31650},[31652,396181,396182,396198],{},[31655,396183,396184,396186,396188,396190,396192,396194,396196],{},[31658,396185,111],{},[31662,396187,71191],{},[31658,396189,1697],{},[31662,396191,71191],{},[31658,396193,111],{},[31662,396195,202],{},[31749,396197,459],{},[31665,396199,396200],{"encoding":31667},"a \\cdot b \\cdot a = 0",[34,396202,396204,396222,396240,396258],{"className":396203,"ariaHidden":758},[31672],[34,396205,396207,396210,396213,396216,396219],{"className":396206},[31676],[34,396208],{"className":396209,"style":224869},[31680],[34,396211,111],{"className":396212},[31685,31686],[34,396214],{"className":396215,"style":32604},[31845],[34,396217,71191],{"className":396218},[32024],[34,396220],{"className":396221,"style":32604},[31845],[34,396223,396225,396228,396231,396234,396237],{"className":396224},[31676],[34,396226],{"className":396227,"style":31681},[31680],[34,396229,1697],{"className":396230},[31685,31686],[34,396232],{"className":396233,"style":32604},[31845],[34,396235,71191],{"className":396236},[32024],[34,396238],{"className":396239,"style":32604},[31845],[34,396241,396243,396246,396249,396252,396255],{"className":396242},[31676],[34,396244],{"className":396245,"style":32527},[31680],[34,396247,111],{"className":396248},[31685,31686],[34,396250],{"className":396251,"style":31846},[31845],[34,396253,202],{"className":396254},[31850],[34,396256],{"className":396257,"style":31846},[31845],[34,396259,396261,396264],{"className":396260},[31676],[34,396262],{"className":396263,"style":32617},[31680],[34,396265,459],{"className":396266},[31685],"。\n我们要讨论中间那个乘积 ",[34,396269,396271,396288],{"className":396270},[31642],[34,396272,396274],{"className":396273},[31646],[31648,396275,396276],{"xmlns":31650},[31652,396277,396278,396286],{},[31655,396279,396280,396282,396284],{},[31658,396281,111],{},[31662,396283,71191],{},[31658,396285,1697],{},[31665,396287,372756],{"encoding":31667},[34,396289,396291,396309],{"className":396290,"ariaHidden":758},[31672],[34,396292,396294,396297,396300,396303,396306],{"className":396293},[31676],[34,396295],{"className":396296,"style":224869},[31680],[34,396298,111],{"className":396299},[31685,31686],[34,396301],{"className":396302,"style":32604},[31845],[34,396304,71191],{"className":396305},[32024],[34,396307],{"className":396308,"style":32604},[31845],[34,396310,396312,396315],{"className":396311},[31676],[34,396313],{"className":396314,"style":31681},[31680],[34,396316,1697],{"className":396317},[31685,31686]," 的状态。",[105,396320,396321,396600],{},[108,396322,396323,46324,396326,39984,396398],{},[2627,396324,396325],{},"情形 1",[34,396327,396329,396350],{"className":396328},[31642],[34,396330,396332],{"className":396331},[31646],[31648,396333,396334],{"xmlns":31650},[31652,396335,396336,396348],{},[31655,396337,396338,396340,396342,396344,396346],{},[31658,396339,111],{},[31662,396341,71191],{},[31658,396343,1697],{},[31662,396345,202],{},[31749,396347,459],{},[31665,396349,368019],{"encoding":31667},[34,396351,396353,396371,396389],{"className":396352,"ariaHidden":758},[31672],[34,396354,396356,396359,396362,396365,396368],{"className":396355},[31676],[34,396357],{"className":396358,"style":224869},[31680],[34,396360,111],{"className":396361},[31685,31686],[34,396363],{"className":396364,"style":32604},[31845],[34,396366,71191],{"className":396367},[32024],[34,396369],{"className":396370,"style":32604},[31845],[34,396372,396374,396377,396380,396383,396386],{"className":396373},[31676],[34,396375],{"className":396376,"style":31681},[31680],[34,396378,1697],{"className":396379},[31685,31686],[34,396381],{"className":396382,"style":31846},[31845],[34,396384,202],{"className":396385},[31850],[34,396387],{"className":396388,"style":31846},[31845],[34,396390,396392,396395],{"className":396391},[31676],[34,396393],{"className":396394,"style":32617},[31680],[34,396396,459],{"className":396397},[31685],[105,396399,396400,396568],{},[108,396401,93310,396402,86674,396485,347],{},[34,396403,396405,396422],{"className":396404},[31642],[34,396406,396408],{"className":396407},[31646],[31648,396409,396410],{"xmlns":31650},[31652,396411,396412,396420],{},[31655,396413,396414,396416,396418],{},[31658,396415,111],{},[31662,396417,54707],{"mathvariant":38079},[31749,396419,459],{},[31665,396421,363487],{"encoding":31667},[34,396423,396425,396476],{"className":396424,"ariaHidden":758},[31672],[34,396426,396428,396431,396434,396437,396473],{"className":396427},[31676],[34,396429],{"className":396430,"style":43217},[31680],[34,396432,111],{"className":396433},[31685,31686],[34,396435],{"className":396436,"style":31846},[31845],[34,396438,396440,396467,396470],{"className":396439},[31850],[34,396441,396443],{"className":396442},[31850],[34,396444,396446],{"className":396445},[31685,54774],[34,396447,396449],{"className":396448},[54778],[34,396450,396452,396455,396464],{"className":396451},[54782],[34,396453],{"className":396454,"style":43217},[31680],[34,396456,396458],{"className":396457},[54789],[34,396459,396461],{"className":396460},[31685],[34,396462,54796],{"className":396463},[31850],[34,396465],{"className":396466},[54800],[34,396468],{"className":396469},[31845,54804],[34,396471,202],{"className":396472},[31850],[34,396474],{"className":396475,"style":31846},[31845],[34,396477,396479,396482],{"className":396478},[31676],[34,396480],{"className":396481,"style":32617},[31680],[34,396483,459],{"className":396484},[31685],[34,396486,396488,396505],{"className":396487},[31642],[34,396489,396491],{"className":396490},[31646],[31648,396492,396493],{"xmlns":31650},[31652,396494,396495,396503],{},[31655,396496,396497,396499,396501],{},[31658,396498,1697],{},[31662,396500,54707],{"mathvariant":38079},[31749,396502,459],{},[31665,396504,395746],{"encoding":31667},[34,396506,396508,396559],{"className":396507,"ariaHidden":758},[31672],[34,396509,396511,396514,396517,396520,396556],{"className":396510},[31676],[34,396512],{"className":396513,"style":43217},[31680],[34,396515,1697],{"className":396516},[31685,31686],[34,396518],{"className":396519,"style":31846},[31845],[34,396521,396523,396550,396553],{"className":396522},[31850],[34,396524,396526],{"className":396525},[31850],[34,396527,396529],{"className":396528},[31685,54774],[34,396530,396532],{"className":396531},[54778],[34,396533,396535,396538,396547],{"className":396534},[54782],[34,396536],{"className":396537,"style":43217},[31680],[34,396539,396541],{"className":396540},[54789],[34,396542,396544],{"className":396543},[31685],[34,396545,54796],{"className":396546},[31850],[34,396548],{"className":396549},[54800],[34,396551],{"className":396552},[31845,54804],[34,396554,202],{"className":396555},[31850],[34,396557],{"className":396558,"style":31846},[31845],[34,396560,396562,396565],{"className":396561},[31676],[34,396563],{"className":396564,"style":32617},[31680],[34,396566,459],{"className":396567},[31685],[108,396569,396570,396571,396599],{},"根据定义，直接得证：",[34,396572,396574,396587],{"className":396573},[31642],[34,396575,396577],{"className":396576},[31646],[31648,396578,396579],{"xmlns":31650},[31652,396580,396581,396585],{},[31655,396582,396583],{},[31658,396584,111],{},[31665,396586,111],{"encoding":31667},[34,396588,396590],{"className":396589,"ariaHidden":758},[31672],[34,396591,396593,396596],{"className":396592},[31676],[34,396594],{"className":396595,"style":32527},[31680],[34,396597,111],{"className":396598},[31685,31686]," 是左零因子。",[108,396601,396602,46324,396605,39984,396711],{},[2627,396603,396604],{},"情形 2",[34,396606,396608,396630],{"className":396607},[31642],[34,396609,396611],{"className":396610},[31646],[31648,396612,396613],{"xmlns":31650},[31652,396614,396615,396627],{},[31655,396616,396617,396619,396621,396623,396625],{},[31658,396618,111],{},[31662,396620,71191],{},[31658,396622,1697],{},[31662,396624,54707],{"mathvariant":38079},[31749,396626,459],{},[31665,396628,396629],{"encoding":31667},"a \\cdot b \\neq 0",[34,396631,396633,396651,396702],{"className":396632,"ariaHidden":758},[31672],[34,396634,396636,396639,396642,396645,396648],{"className":396635},[31676],[34,396637],{"className":396638,"style":224869},[31680],[34,396640,111],{"className":396641},[31685,31686],[34,396643],{"className":396644,"style":32604},[31845],[34,396646,71191],{"className":396647},[32024],[34,396649],{"className":396650,"style":32604},[31845],[34,396652,396654,396657,396660,396663,396699],{"className":396653},[31676],[34,396655],{"className":396656,"style":43217},[31680],[34,396658,1697],{"className":396659},[31685,31686],[34,396661],{"className":396662,"style":31846},[31845],[34,396664,396666,396693,396696],{"className":396665},[31850],[34,396667,396669],{"className":396668},[31850],[34,396670,396672],{"className":396671},[31685,54774],[34,396673,396675],{"className":396674},[54778],[34,396676,396678,396681,396690],{"className":396677},[54782],[34,396679],{"className":396680,"style":43217},[31680],[34,396682,396684],{"className":396683},[54789],[34,396685,396687],{"className":396686},[31685],[34,396688,54796],{"className":396689},[31850],[34,396691],{"className":396692},[54800],[34,396694],{"className":396695},[31845,54804],[34,396697,202],{"className":396698},[31850],[34,396700],{"className":396701,"style":31846},[31845],[34,396703,396705,396708],{"className":396704},[31676],[34,396706],{"className":396707,"style":32617},[31680],[34,396709,459],{"className":396710},[31685],[105,396712,396713,396857,396914,396990,397159,397220],{},[108,396714,396715,396716,396771,396772,347],{},"我们将等式 ",[34,396717,396719,396741],{"className":396718},[31642],[34,396720,396722],{"className":396721},[31646],[31648,396723,396724],{"xmlns":31650},[31652,396725,396726,396738],{},[31655,396727,396728,396730,396732,396734,396736],{},[31658,396729,111],{},[31658,396731,1697],{},[31658,396733,111],{},[31662,396735,202],{},[31749,396737,459],{},[31665,396739,396740],{"encoding":31667},"aba=0",[34,396742,396744,396762],{"className":396743,"ariaHidden":758},[31672],[34,396745,396747,396750,396753,396756,396759],{"className":396746},[31676],[34,396748],{"className":396749,"style":31681},[31680],[34,396751,395847],{"className":396752},[31685,31686],[34,396754],{"className":396755,"style":31846},[31845],[34,396757,202],{"className":396758},[31850],[34,396760],{"className":396761,"style":31846},[31845],[34,396763,396765,396768],{"className":396764},[31676],[34,396766],{"className":396767,"style":32617},[31680],[34,396769,459],{"className":396770},[31685]," 结合一下：",[34,396773,396775,396803],{"className":396774},[31642],[34,396776,396778],{"className":396777},[31646],[31648,396779,396780],{"xmlns":31650},[31652,396781,396782,396800],{},[31655,396783,396784,396786,396788,396790,396792,396794,396796,396798],{},[31662,396785,456],{"stretchy":702},[31658,396787,111],{},[31658,396789,1697],{},[31662,396791,2557],{"stretchy":702},[31662,396793,71191],{},[31658,396795,111],{},[31662,396797,202],{},[31749,396799,459],{},[31665,396801,396802],{"encoding":31667},"(ab) \\cdot a = 0",[34,396804,396806,396830,396848],{"className":396805,"ariaHidden":758},[31672],[34,396807,396809,396812,396815,396818,396821,396824,396827],{"className":396808},[31676],[34,396810],{"className":396811,"style":31774},[31680],[34,396813,456],{"className":396814},[31782],[34,396816,324191],{"className":396817},[31685,31686],[34,396819,2557],{"className":396820},[31690],[34,396822],{"className":396823,"style":32604},[31845],[34,396825,71191],{"className":396826},[32024],[34,396828],{"className":396829,"style":32604},[31845],[34,396831,396833,396836,396839,396842,396845],{"className":396832},[31676],[34,396834],{"className":396835,"style":32527},[31680],[34,396837,111],{"className":396838},[31685,31686],[34,396840],{"className":396841,"style":31846},[31845],[34,396843,202],{"className":396844},[31850],[34,396846],{"className":396847,"style":31846},[31845],[34,396849,396851,396854],{"className":396850},[31676],[34,396852],{"className":396853,"style":32617},[31680],[34,396855,459],{"className":396856},[31685],[108,396858,396859,396860,396913],{},"令 ",[34,396861,396863,396883],{"className":396862},[31642],[34,396864,396866],{"className":396865},[31646],[31648,396867,396868],{"xmlns":31650},[31652,396869,396870,396880],{},[31655,396871,396872,396874,396876,396878],{},[31658,396873,47384],{},[31662,396875,202],{},[31658,396877,111],{},[31658,396879,1697],{},[31665,396881,396882],{"encoding":31667},"C = ab",[34,396884,396886,396904],{"className":396885,"ariaHidden":758},[31672],[34,396887,396889,396892,396895,396898,396901],{"className":396888},[31676],[34,396890],{"className":396891,"style":33218},[31680],[34,396893,47384],{"className":396894,"style":38706},[31685,31686],[34,396896],{"className":396897,"style":31846},[31845],[34,396899,202],{"className":396900},[31850],[34,396902],{"className":396903,"style":31846},[31845],[34,396905,396907,396910],{"className":396906},[31676],[34,396908],{"className":396909,"style":31681},[31680],[34,396911,324191],{"className":396912},[31685,31686],"。这是一个非零元素（根据当前假设）。",[108,396915,396916,396917,347],{},"等式变成了 ",[34,396918,396920,396942],{"className":396919},[31642],[34,396921,396923],{"className":396922},[31646],[31648,396924,396925],{"xmlns":31650},[31652,396926,396927,396939],{},[31655,396928,396929,396931,396933,396935,396937],{},[31658,396930,47384],{},[31662,396932,71191],{},[31658,396934,111],{},[31662,396936,202],{},[31749,396938,459],{},[31665,396940,396941],{"encoding":31667},"C \\cdot a = 0",[34,396943,396945,396963,396981],{"className":396944,"ariaHidden":758},[31672],[34,396946,396948,396951,396954,396957,396960],{"className":396947},[31676],[34,396949],{"className":396950,"style":33218},[31680],[34,396952,47384],{"className":396953,"style":38706},[31685,31686],[34,396955],{"className":396956,"style":32604},[31845],[34,396958,71191],{"className":396959},[32024],[34,396961],{"className":396962,"style":32604},[31845],[34,396964,396966,396969,396972,396975,396978],{"className":396965},[31676],[34,396967],{"className":396968,"style":32527},[31680],[34,396970,111],{"className":396971},[31685,31686],[34,396973],{"className":396974,"style":31846},[31845],[34,396976,202],{"className":396977},[31850],[34,396979],{"className":396980,"style":31846},[31845],[34,396982,396984,396987],{"className":396983},[31676],[34,396985],{"className":396986,"style":32617},[31680],[34,396988,459],{"className":396989},[31685],[108,396991,25496,396992,93535,397076,347],{},[34,396993,396995,397013],{"className":396994},[31642],[34,396996,396998],{"className":396997},[31646],[31648,396999,397000],{"xmlns":31650},[31652,397001,397002,397010],{},[31655,397003,397004,397006,397008],{},[31658,397005,47384],{},[31662,397007,54707],{"mathvariant":38079},[31749,397009,459],{},[31665,397011,397012],{"encoding":31667},"C \\neq 0",[34,397014,397016,397067],{"className":397015,"ariaHidden":758},[31672],[34,397017,397019,397022,397025,397028,397064],{"className":397018},[31676],[34,397020],{"className":397021,"style":43217},[31680],[34,397023,47384],{"className":397024,"style":38706},[31685,31686],[34,397026],{"className":397027,"style":31846},[31845],[34,397029,397031,397058,397061],{"className":397030},[31850],[34,397032,397034],{"className":397033},[31850],[34,397035,397037],{"className":397036},[31685,54774],[34,397038,397040],{"className":397039},[54778],[34,397041,397043,397046,397055],{"className":397042},[54782],[34,397044],{"className":397045,"style":43217},[31680],[34,397047,397049],{"className":397048},[54789],[34,397050,397052],{"className":397051},[31685],[34,397053,54796],{"className":397054},[31850],[34,397056],{"className":397057},[54800],[34,397059],{"className":397060},[31845,54804],[34,397062,202],{"className":397063},[31850],[34,397065],{"className":397066,"style":31846},[31845],[34,397068,397070,397073],{"className":397069},[31676],[34,397071],{"className":397072,"style":32617},[31680],[34,397074,459],{"className":397075},[31685],[34,397077,397079,397096],{"className":397078},[31642],[34,397080,397082],{"className":397081},[31646],[31648,397083,397084],{"xmlns":31650},[31652,397085,397086,397094],{},[31655,397087,397088,397090,397092],{},[31658,397089,111],{},[31662,397091,54707],{"mathvariant":38079},[31749,397093,459],{},[31665,397095,363487],{"encoding":31667},[34,397097,397099,397150],{"className":397098,"ariaHidden":758},[31672],[34,397100,397102,397105,397108,397111,397147],{"className":397101},[31676],[34,397103],{"className":397104,"style":43217},[31680],[34,397106,111],{"className":397107},[31685,31686],[34,397109],{"className":397110,"style":31846},[31845],[34,397112,397114,397141,397144],{"className":397113},[31850],[34,397115,397117],{"className":397116},[31850],[34,397118,397120],{"className":397119},[31685,54774],[34,397121,397123],{"className":397122},[54778],[34,397124,397126,397129,397138],{"className":397125},[54782],[34,397127],{"className":397128,"style":43217},[31680],[34,397130,397132],{"className":397131},[54789],[34,397133,397135],{"className":397134},[31685],[34,397136,54796],{"className":397137},[31850],[34,397139],{"className":397140},[54800],[34,397142],{"className":397143},[31845,54804],[34,397145,202],{"className":397146},[31850],[34,397148],{"className":397149,"style":31846},[31845],[34,397151,397153,397156],{"className":397152},[31676],[34,397154],{"className":397155,"style":32617},[31680],[34,397157,459],{"className":397158},[31685],[108,397160,397161,397162,397190,397191,397219],{},"这说明 ",[34,397163,397165,397178],{"className":397164},[31642],[34,397166,397168],{"className":397167},[31646],[31648,397169,397170],{"xmlns":31650},[31652,397171,397172,397176],{},[31655,397173,397174],{},[31658,397175,111],{},[31665,397177,111],{"encoding":31667},[34,397179,397181],{"className":397180,"ariaHidden":758},[31672],[34,397182,397184,397187],{"className":397183},[31676],[34,397185],{"className":397186,"style":32527},[31680],[34,397188,111],{"className":397189},[31685,31686]," 能把非零的 ",[34,397192,397194,397207],{"className":397193},[31642],[34,397195,397197],{"className":397196},[31646],[31648,397198,397199],{"xmlns":31650},[31652,397200,397201,397205],{},[31655,397202,397203],{},[31658,397204,47384],{},[31665,397206,47384],{"encoding":31667},[34,397208,397210],{"className":397209,"ariaHidden":758},[31672],[34,397211,397213,397216],{"className":397212},[31676],[34,397214],{"className":397215,"style":33218},[31680],[34,397217,47384],{"className":397218,"style":38706},[31685,31686]," 右乘变成 0。",[108,397221,397222,397223,397251],{},"根据定义，",[34,397224,397226,397239],{"className":397225},[31642],[34,397227,397229],{"className":397228},[31646],[31648,397230,397231],{"xmlns":31650},[31652,397232,397233,397237],{},[31655,397234,397235],{},[31658,397236,111],{},[31665,397238,111],{"encoding":31667},[34,397240,397242],{"className":397241,"ariaHidden":758},[31672],[34,397243,397245,397248],{"className":397244},[31676],[34,397246],{"className":397247,"style":32527},[31680],[34,397249,111],{"className":397250},[31685,31686]," 是右零因子。",[15,397253,397254,397256,397257,397285],{},[2627,397255,365086],{},"：不管哪种情况，",[34,397258,397260,397273],{"className":397259},[31642],[34,397261,397263],{"className":397262},[31646],[31648,397264,397265],{"xmlns":31650},[31652,397266,397267,397271],{},[31655,397268,397269],{},[31658,397270,111],{},[31665,397272,111],{"encoding":31667},[34,397274,397276],{"className":397275,"ariaHidden":758},[31672],[34,397277,397279,397282],{"className":397278},[31676],[34,397280],{"className":397281,"style":32527},[31680],[34,397283,111],{"className":397284},[31685,31686]," 只要不是把所有非零数都变成非零，它就得是零因子。",[32188,397287],{},[253,397289,397290],{"id":397290},"题目11",[15,397292,133910,397293,397321,397322,329052,397374,347],{},[34,397294,397296,397309],{"className":397295},[31642],[34,397297,397299],{"className":397298},[31646],[31648,397300,397301],{"xmlns":31650},[31652,397302,397303,397307],{},[31655,397304,397305],{},[31658,397306,34386],{},[31665,397308,34386],{"encoding":31667},[34,397310,397312],{"className":397311,"ariaHidden":758},[31672],[34,397313,397315,397318],{"className":397314},[31676],[34,397316],{"className":397317,"style":33218},[31680],[34,397319,34386],{"className":397320,"style":34427},[31685,31686]," 是有限幺环， ",[34,397323,397325,397344],{"className":397324},[31642],[34,397326,397328],{"className":397327},[31646],[31648,397329,397330],{"xmlns":31650},[31652,397331,397332,397342],{},[31655,397333,397334,397336,397338,397340],{},[31658,397335,111],{},[31658,397337,1697],{},[31662,397339,202],{},[31749,397341,640],{},[31665,397343,364752],{"encoding":31667},[34,397345,397347,397365],{"className":397346,"ariaHidden":758},[31672],[34,397348,397350,397353,397356,397359,397362],{"className":397349},[31676],[34,397351],{"className":397352,"style":31681},[31680],[34,397354,324191],{"className":397355},[31685,31686],[34,397357],{"className":397358,"style":31846},[31845],[34,397360,202],{"className":397361},[31850],[34,397363],{"className":397364,"style":31846},[31845],[34,397366,397368,397371],{"className":397367},[31676],[34,397369],{"className":397370,"style":32617},[31680],[34,397372,640],{"className":397373},[31685],[34,397375,397377,397397],{"className":397376},[31642],[34,397378,397380],{"className":397379},[31646],[31648,397381,397382],{"xmlns":31650},[31652,397383,397384,397394],{},[31655,397385,397386,397388,397390,397392],{},[31658,397387,1697],{},[31658,397389,111],{},[31662,397391,202],{},[31749,397393,640],{},[31665,397395,397396],{"encoding":31667},"ba = 1",[34,397398,397400,397418],{"className":397399,"ariaHidden":758},[31672],[34,397401,397403,397406,397409,397412,397415],{"className":397402},[31676],[34,397404],{"className":397405,"style":31681},[31680],[34,397407,367454],{"className":397408},[31685,31686],[34,397410],{"className":397411,"style":31846},[31845],[34,397413,202],{"className":397414},[31850],[34,397416],{"className":397417,"style":31846},[31845],[34,397419,397421,397424],{"className":397420},[31676],[34,397422],{"className":397423,"style":32617},[31680],[34,397425,640],{"className":397426},[31685],[32490,397428,297792],{"id":397429},"详细解析-6",[15,397431,397432,397433,397436,397437,397489,397490,397543],{},"这道题利用了",[2627,397434,397435],{},"有限集合","的特性。在无限集合里（比如线性算子），",[34,397438,397440,397459],{"className":397439},[31642],[34,397441,397443],{"className":397442},[31646],[31648,397444,397445],{"xmlns":31650},[31652,397446,397447,397457],{},[31655,397448,397449,397451,397453,397455],{},[31658,397450,111],{},[31658,397452,1697],{},[31662,397454,202],{},[31749,397456,640],{},[31665,397458,364890],{"encoding":31667},[34,397460,397462,397480],{"className":397461,"ariaHidden":758},[31672],[34,397463,397465,397468,397471,397474,397477],{"className":397464},[31676],[34,397466],{"className":397467,"style":31681},[31680],[34,397469,324191],{"className":397470},[31685,31686],[34,397472],{"className":397473,"style":31846},[31845],[34,397475,202],{"className":397476},[31850],[34,397478],{"className":397479,"style":31846},[31845],[34,397481,397483,397486],{"className":397482},[31676],[34,397484],{"className":397485,"style":32617},[31680],[34,397487,640],{"className":397488},[31685]," 推不出 ",[34,397491,397493,397513],{"className":397492},[31642],[34,397494,397496],{"className":397495},[31646],[31648,397497,397498],{"xmlns":31650},[31652,397499,397500,397510],{},[31655,397501,397502,397504,397506,397508],{},[31658,397503,1697],{},[31658,397505,111],{},[31662,397507,202],{},[31749,397509,640],{},[31665,397511,397512],{"encoding":31667},"ba=1",[34,397514,397516,397534],{"className":397515,"ariaHidden":758},[31672],[34,397517,397519,397522,397525,397528,397531],{"className":397518},[31676],[34,397520],{"className":397521,"style":31681},[31680],[34,397523,367454],{"className":397524},[31685,31686],[34,397526],{"className":397527,"style":31846},[31845],[34,397529,202],{"className":397530},[31850],[34,397532],{"className":397533,"style":31846},[31845],[34,397535,397537,397540],{"className":397536},[31676],[34,397538],{"className":397539,"style":32617},[31680],[34,397541,640],{"className":397542},[31685],"。但在有限环里可以。",[15,397545,397546,397549,397550,397638,397639,397667],{},[2627,397547,397548],{},"1. 构造映射","\n想象一个函数 ",[34,397551,397553,397581],{"className":397552},[31642],[34,397554,397556],{"className":397555},[31646],[31648,397557,397558],{"xmlns":31650},[31652,397559,397560,397578],{},[31655,397561,397562,397564,397566,397568,397570,397572,397574,397576],{},[31658,397563,31739],{},[31662,397565,456],{"stretchy":702},[31658,397567,32936],{},[31662,397569,2557],{"stretchy":702},[31662,397571,202],{},[31658,397573,1697],{},[31662,397575,71191],{},[31658,397577,32936],{},[31665,397579,397580],{"encoding":31667},"f(x) = b \\cdot x",[34,397582,397584,397611,397629],{"className":397583,"ariaHidden":758},[31672],[34,397585,397587,397590,397593,397596,397599,397602,397605,397608],{"className":397586},[31676],[34,397588],{"className":397589,"style":31774},[31680],[34,397591,31739],{"className":397592,"style":31778},[31685,31686],[34,397594,456],{"className":397595},[31782],[34,397597,32936],{"className":397598},[31685,31686],[34,397600,2557],{"className":397601},[31690],[34,397603],{"className":397604,"style":31846},[31845],[34,397606,202],{"className":397607},[31850],[34,397609],{"className":397610,"style":31846},[31845],[34,397612,397614,397617,397620,397623,397626],{"className":397613},[31676],[34,397615],{"className":397616,"style":31681},[31680],[34,397618,1697],{"className":397619},[31685,31686],[34,397621],{"className":397622,"style":32604},[31845],[34,397624,71191],{"className":397625},[32024],[34,397627],{"className":397628,"style":32604},[31845],[34,397630,397632,397635],{"className":397631},[31676],[34,397633],{"className":397634,"style":32527},[31680],[34,397636,32936],{"className":397637},[31685,31686],"。这个函数把环里的元素 ",[34,397640,397642,397655],{"className":397641},[31642],[34,397643,397645],{"className":397644},[31646],[31648,397646,397647],{"xmlns":31650},[31652,397648,397649,397653],{},[31655,397650,397651],{},[31658,397652,32936],{},[31665,397654,32936],{"encoding":31667},[34,397656,397658],{"className":397657,"ariaHidden":758},[31672],[34,397659,397661,397664],{"className":397660},[31676],[34,397662],{"className":397663,"style":32527},[31680],[34,397665,32936],{"className":397666},[31685,31686]," 映射到另一个元素。",[15,397669,397670,397673,397674,397797,397798,317085,397941,398111,398112,47649,398115,33183,398143,398357,398539,398540,398592,398593,398916,398917,347],{},[2627,397671,397672],{},"2. 证明单射 (One-to-One)","\n假设有两个不同的元素 ",[34,397675,397677,397702],{"className":397676},[31642],[34,397678,397680],{"className":397679},[31646],[31648,397681,397682],{"xmlns":31650},[31652,397683,397684,397700],{},[31655,397685,397686,397692,397694],{},[31743,397687,397688,397690],{},[31658,397689,32936],{},[31749,397691,640],{},[31662,397693,31960],{"separator":758},[31743,397695,397696,397698],{},[31658,397697,32936],{},[31749,397699,768],{},[31665,397701,317672],{"encoding":31667},[34,397703,397705],{"className":397704,"ariaHidden":758},[31672],[34,397706,397708,397711,397751,397754,397757],{"className":397707},[31676],[34,397709],{"className":397710,"style":38774},[31680],[34,397712,397714,397717],{"className":397713},[31685],[34,397715,32936],{"className":397716},[31685,31686],[34,397718,397720],{"className":397719},[31792],[34,397721,397723,397743],{"className":397722},[31796,31797],[34,397724,397726,397740],{"className":397725},[31801],[34,397727,397729],{"className":397728,"style":31806},[31805],[34,397730,397731,397734],{"style":31809},[34,397732],{"className":397733,"style":31814},[31813],[34,397735,397737],{"className":397736},[31818,31819,31820,31821],[34,397738,640],{"className":397739},[31685,31821],[34,397741,31829],{"className":397742},[31828],[34,397744,397746],{"className":397745},[31801],[34,397747,397749],{"className":397748,"style":31836},[31805],[34,397750],{},[34,397752,31960],{"className":397753},[32119],[34,397755],{"className":397756,"style":32123},[31845],[34,397758,397760,397763],{"className":397759},[31685],[34,397761,32936],{"className":397762},[31685,31686],[34,397764,397766],{"className":397765},[31792],[34,397767,397769,397789],{"className":397768},[31796,31797],[34,397770,397772,397786],{"className":397771},[31801],[34,397773,397775],{"className":397774,"style":31806},[31805],[34,397776,397777,397780],{"style":31809},[34,397778],{"className":397779,"style":31814},[31813],[34,397781,397783],{"className":397782},[31818,31819,31820,31821],[34,397784,768],{"className":397785},[31685,31821],[34,397787,31829],{"className":397788},[31828],[34,397790,397792],{"className":397791},[31801],[34,397793,397795],{"className":397794,"style":31836},[31805],[34,397796],{}," 被映射到了同一个地方，即 ",[34,397799,397801,397831],{"className":397800},[31642],[34,397802,397804],{"className":397803},[31646],[31648,397805,397806],{"xmlns":31650},[31652,397807,397808,397828],{},[31655,397809,397810,397812,397818,397820,397822],{},[31658,397811,1697],{},[31743,397813,397814,397816],{},[31658,397815,32936],{},[31749,397817,640],{},[31662,397819,202],{},[31658,397821,1697],{},[31743,397823,397824,397826],{},[31658,397825,32936],{},[31749,397827,768],{},[31665,397829,397830],{"encoding":31667},"b x_1 = b x_2",[34,397832,397834,397892],{"className":397833,"ariaHidden":758},[31672],[34,397835,397837,397840,397843,397883,397886,397889],{"className":397836},[31676],[34,397838],{"className":397839,"style":33814},[31680],[34,397841,1697],{"className":397842},[31685,31686],[34,397844,397846,397849],{"className":397845},[31685],[34,397847,32936],{"className":397848},[31685,31686],[34,397850,397852],{"className":397851},[31792],[34,397853,397855,397875],{"className":397854},[31796,31797],[34,397856,397858,397872],{"className":397857},[31801],[34,397859,397861],{"className":397860,"style":31806},[31805],[34,397862,397863,397866],{"style":31809},[34,397864],{"className":397865,"style":31814},[31813],[34,397867,397869],{"className":397868},[31818,31819,31820,31821],[34,397870,640],{"className":397871},[31685,31821],[34,397873,31829],{"className":397874},[31828],[34,397876,397878],{"className":397877},[31801],[34,397879,397881],{"className":397880,"style":31836},[31805],[34,397882],{},[34,397884],{"className":397885,"style":31846},[31845],[34,397887,202],{"className":397888},[31850],[34,397890],{"className":397891,"style":31846},[31845],[34,397893,397895,397898,397901],{"className":397894},[31676],[34,397896],{"className":397897,"style":33814},[31680],[34,397899,1697],{"className":397900},[31685,31686],[34,397902,397904,397907],{"className":397903},[31685],[34,397905,32936],{"className":397906},[31685,31686],[34,397908,397910],{"className":397909},[31792],[34,397911,397913,397933],{"className":397912},[31796,31797],[34,397914,397916,397930],{"className":397915},[31801],[34,397917,397919],{"className":397918,"style":31806},[31805],[34,397920,397921,397924],{"style":31809},[34,397922],{"className":397923,"style":31814},[31813],[34,397925,397927],{"className":397926},[31818,31819,31820,31821],[34,397928,768],{"className":397929},[31685,31821],[34,397931,31829],{"className":397932},[31828],[34,397934,397936],{"className":397935},[31801],[34,397937,397939],{"className":397938,"style":31836},[31805],[34,397940],{},[34,397942,397944,397980],{"className":397943},[31642],[34,397945,397947],{"className":397946},[31646],[31648,397948,397949],{"xmlns":31650},[31652,397950,397951,397977],{},[31655,397952,397953,397955,397957,397963,397965,397971,397973,397975],{},[31658,397954,1697],{},[31662,397956,456],{"stretchy":702},[31743,397958,397959,397961],{},[31658,397960,32936],{},[31749,397962,640],{},[31662,397964,32226],{},[31743,397966,397967,397969],{},[31658,397968,32936],{},[31749,397970,768],{},[31662,397972,2557],{"stretchy":702},[31662,397974,202],{},[31749,397976,459],{},[31665,397978,397979],{"encoding":31667},"b(x_1 - x_2) = 0",[34,397981,397983,398044,398102],{"className":397982,"ariaHidden":758},[31672],[34,397984,397986,397989,397992,397995,398035,398038,398041],{"className":397985},[31676],[34,397987],{"className":397988,"style":31774},[31680],[34,397990,1697],{"className":397991},[31685,31686],[34,397993,456],{"className":397994},[31782],[34,397996,397998,398001],{"className":397997},[31685],[34,397999,32936],{"className":398000},[31685,31686],[34,398002,398004],{"className":398003},[31792],[34,398005,398007,398027],{"className":398006},[31796,31797],[34,398008,398010,398024],{"className":398009},[31801],[34,398011,398013],{"className":398012,"style":31806},[31805],[34,398014,398015,398018],{"style":31809},[34,398016],{"className":398017,"style":31814},[31813],[34,398019,398021],{"className":398020},[31818,31819,31820,31821],[34,398022,640],{"className":398023},[31685,31821],[34,398025,31829],{"className":398026},[31828],[34,398028,398030],{"className":398029},[31801],[34,398031,398033],{"className":398032,"style":31836},[31805],[34,398034],{},[34,398036],{"className":398037,"style":32604},[31845],[34,398039,32226],{"className":398040},[32024],[34,398042],{"className":398043,"style":32604},[31845],[34,398045,398047,398050,398090,398093,398096,398099],{"className":398046},[31676],[34,398048],{"className":398049,"style":31774},[31680],[34,398051,398053,398056],{"className":398052},[31685],[34,398054,32936],{"className":398055},[31685,31686],[34,398057,398059],{"className":398058},[31792],[34,398060,398062,398082],{"className":398061},[31796,31797],[34,398063,398065,398079],{"className":398064},[31801],[34,398066,398068],{"className":398067,"style":31806},[31805],[34,398069,398070,398073],{"style":31809},[34,398071],{"className":398072,"style":31814},[31813],[34,398074,398076],{"className":398075},[31818,31819,31820,31821],[34,398077,768],{"className":398078},[31685,31821],[34,398080,31829],{"className":398081},[31828],[34,398083,398085],{"className":398084},[31801],[34,398086,398088],{"className":398087,"style":31836},[31805],[34,398089],{},[34,398091,2557],{"className":398092},[31690],[34,398094],{"className":398095,"style":31846},[31845],[34,398097,202],{"className":398098},[31850],[34,398100],{"className":398101,"style":31846},[31845],[34,398103,398105,398108],{"className":398104},[31676],[34,398106],{"className":398107,"style":32617},[31680],[34,398109,459],{"className":398110},[31685],"。\n我们在两边同时",[2627,398113,398114],{},"左乘",[34,398116,398118,398131],{"className":398117},[31642],[34,398119,398121],{"className":398120},[31646],[31648,398122,398123],{"xmlns":31650},[31652,398124,398125,398129],{},[31655,398126,398127],{},[31658,398128,111],{},[31665,398130,111],{"encoding":31667},[34,398132,398134],{"className":398133,"ariaHidden":758},[31672],[34,398135,398137,398140],{"className":398136},[31676],[34,398138],{"className":398139,"style":32527},[31680],[34,398141,111],{"className":398142},[31685,31686],[34,398144,398146,398190],{"className":398145},[31642],[34,398147,398149],{"className":398148},[31646],[31648,398150,398151],{"xmlns":31650},[31652,398152,398153,398187],{},[31655,398154,398155,398157,398159,398161,398163,398169,398171,398177,398179,398181,398183,398185],{},[31658,398156,111],{},[31662,398158,71191],{},[31658,398160,1697],{},[31662,398162,456],{"stretchy":702},[31743,398164,398165,398167],{},[31658,398166,32936],{},[31749,398168,640],{},[31662,398170,32226],{},[31743,398172,398173,398175],{},[31658,398174,32936],{},[31749,398176,768],{},[31662,398178,2557],{"stretchy":702},[31662,398180,202],{},[31658,398182,111],{},[31662,398184,71191],{},[31749,398186,459],{},[31665,398188,398189],{"encoding":31667},"a \\cdot b(x_1 - x_2) = a \\cdot 0",[34,398191,398193,398211,398272,398330,398348],{"className":398192,"ariaHidden":758},[31672],[34,398194,398196,398199,398202,398205,398208],{"className":398195},[31676],[34,398197],{"className":398198,"style":224869},[31680],[34,398200,111],{"className":398201},[31685,31686],[34,398203],{"className":398204,"style":32604},[31845],[34,398206,71191],{"className":398207},[32024],[34,398209],{"className":398210,"style":32604},[31845],[34,398212,398214,398217,398220,398223,398263,398266,398269],{"className":398213},[31676],[34,398215],{"className":398216,"style":31774},[31680],[34,398218,1697],{"className":398219},[31685,31686],[34,398221,456],{"className":398222},[31782],[34,398224,398226,398229],{"className":398225},[31685],[34,398227,32936],{"className":398228},[31685,31686],[34,398230,398232],{"className":398231},[31792],[34,398233,398235,398255],{"className":398234},[31796,31797],[34,398236,398238,398252],{"className":398237},[31801],[34,398239,398241],{"className":398240,"style":31806},[31805],[34,398242,398243,398246],{"style":31809},[34,398244],{"className":398245,"style":31814},[31813],[34,398247,398249],{"className":398248},[31818,31819,31820,31821],[34,398250,640],{"className":398251},[31685,31821],[34,398253,31829],{"className":398254},[31828],[34,398256,398258],{"className":398257},[31801],[34,398259,398261],{"className":398260,"style":31836},[31805],[34,398262],{},[34,398264],{"className":398265,"style":32604},[31845],[34,398267,32226],{"className":398268},[32024],[34,398270],{"className":398271,"style":32604},[31845],[34,398273,398275,398278,398318,398321,398324,398327],{"className":398274},[31676],[34,398276],{"className":398277,"style":31774},[31680],[34,398279,398281,398284],{"className":398280},[31685],[34,398282,32936],{"className":398283},[31685,31686],[34,398285,398287],{"className":398286},[31792],[34,398288,398290,398310],{"className":398289},[31796,31797],[34,398291,398293,398307],{"className":398292},[31801],[34,398294,398296],{"className":398295,"style":31806},[31805],[34,398297,398298,398301],{"style":31809},[34,398299],{"className":398300,"style":31814},[31813],[34,398302,398304],{"className":398303},[31818,31819,31820,31821],[34,398305,768],{"className":398306},[31685,31821],[34,398308,31829],{"className":398309},[31828],[34,398311,398313],{"className":398312},[31801],[34,398314,398316],{"className":398315,"style":31836},[31805],[34,398317],{},[34,398319,2557],{"className":398320},[31690],[34,398322],{"className":398323,"style":31846},[31845],[34,398325,202],{"className":398326},[31850],[34,398328],{"className":398329,"style":31846},[31845],[34,398331,398333,398336,398339,398342,398345],{"className":398332},[31676],[34,398334],{"className":398335,"style":224869},[31680],[34,398337,111],{"className":398338},[31685,31686],[34,398340],{"className":398341,"style":32604},[31845],[34,398343,71191],{"className":398344},[32024],[34,398346],{"className":398347,"style":32604},[31845],[34,398349,398351,398354],{"className":398350},[31676],[34,398352],{"className":398353,"style":32617},[31680],[34,398355,459],{"className":398356},[31685],[34,398358,398360,398402],{"className":398359},[31642],[34,398361,398363],{"className":398362},[31646],[31648,398364,398365],{"xmlns":31650},[31652,398366,398367,398399],{},[31655,398368,398369,398371,398373,398375,398377,398379,398385,398387,398393,398395,398397],{},[31662,398370,456],{"stretchy":702},[31658,398372,111],{},[31658,398374,1697],{},[31662,398376,2557],{"stretchy":702},[31662,398378,456],{"stretchy":702},[31743,398380,398381,398383],{},[31658,398382,32936],{},[31749,398384,640],{},[31662,398386,32226],{},[31743,398388,398389,398391],{},[31658,398390,32936],{},[31749,398392,768],{},[31662,398394,2557],{"stretchy":702},[31662,398396,202],{},[31749,398398,459],{},[31665,398400,398401],{"encoding":31667},"(ab)(x_1 - x_2) = 0",[34,398403,398405,398472,398530],{"className":398404,"ariaHidden":758},[31672],[34,398406,398408,398411,398414,398417,398420,398423,398463,398466,398469],{"className":398407},[31676],[34,398409],{"className":398410,"style":31774},[31680],[34,398412,456],{"className":398413},[31782],[34,398415,324191],{"className":398416},[31685,31686],[34,398418,2557],{"className":398419},[31690],[34,398421,456],{"className":398422},[31782],[34,398424,398426,398429],{"className":398425},[31685],[34,398427,32936],{"className":398428},[31685,31686],[34,398430,398432],{"className":398431},[31792],[34,398433,398435,398455],{"className":398434},[31796,31797],[34,398436,398438,398452],{"className":398437},[31801],[34,398439,398441],{"className":398440,"style":31806},[31805],[34,398442,398443,398446],{"style":31809},[34,398444],{"className":398445,"style":31814},[31813],[34,398447,398449],{"className":398448},[31818,31819,31820,31821],[34,398450,640],{"className":398451},[31685,31821],[34,398453,31829],{"className":398454},[31828],[34,398456,398458],{"className":398457},[31801],[34,398459,398461],{"className":398460,"style":31836},[31805],[34,398462],{},[34,398464],{"className":398465,"style":32604},[31845],[34,398467,32226],{"className":398468},[32024],[34,398470],{"className":398471,"style":32604},[31845],[34,398473,398475,398478,398518,398521,398524,398527],{"className":398474},[31676],[34,398476],{"className":398477,"style":31774},[31680],[34,398479,398481,398484],{"className":398480},[31685],[34,398482,32936],{"className":398483},[31685,31686],[34,398485,398487],{"className":398486},[31792],[34,398488,398490,398510],{"className":398489},[31796,31797],[34,398491,398493,398507],{"className":398492},[31801],[34,398494,398496],{"className":398495,"style":31806},[31805],[34,398497,398498,398501],{"style":31809},[34,398499],{"className":398500,"style":31814},[31813],[34,398502,398504],{"className":398503},[31818,31819,31820,31821],[34,398505,768],{"className":398506},[31685,31821],[34,398508,31829],{"className":398509},[31828],[34,398511,398513],{"className":398512},[31801],[34,398514,398516],{"className":398515,"style":31836},[31805],[34,398517],{},[34,398519,2557],{"className":398520},[31690],[34,398522],{"className":398523,"style":31846},[31845],[34,398525,202],{"className":398526},[31850],[34,398528],{"className":398529,"style":31846},[31845],[34,398531,398533,398536],{"className":398532},[31676],[34,398534],{"className":398535,"style":32617},[31680],[34,398537,459],{"className":398538},[31685],"\n因为题目已知 ",[34,398541,398543,398562],{"className":398542},[31642],[34,398544,398546],{"className":398545},[31646],[31648,398547,398548],{"xmlns":31650},[31652,398549,398550,398560],{},[31655,398551,398552,398554,398556,398558],{},[31658,398553,111],{},[31658,398555,1697],{},[31662,398557,202],{},[31749,398559,640],{},[31665,398561,364752],{"encoding":31667},[34,398563,398565,398583],{"className":398564,"ariaHidden":758},[31672],[34,398566,398568,398571,398574,398577,398580],{"className":398567},[31676],[34,398569],{"className":398570,"style":31681},[31680],[34,398572,324191],{"className":398573},[31685,31686],[34,398575],{"className":398576,"style":31846},[31845],[34,398578,202],{"className":398579},[31850],[34,398581],{"className":398582,"style":31846},[31845],[34,398584,398586,398589],{"className":398585},[31676],[34,398587],{"className":398588,"style":32617},[31680],[34,398590,640],{"className":398591},[31685],"，所以：\n",[34,398594,398596,398654],{"className":398595},[31642],[34,398597,398599],{"className":398598},[31646],[31648,398600,398601],{"xmlns":31650},[31652,398602,398603,398651],{},[31655,398604,398605,398607,398609,398611,398617,398619,398625,398627,398629,398631,398633,398635,398637,398643,398645],{},[31749,398606,640],{},[31662,398608,71191],{},[31662,398610,456],{"stretchy":702},[31743,398612,398613,398615],{},[31658,398614,32936],{},[31749,398616,640],{},[31662,398618,32226],{},[31743,398620,398621,398623],{},[31658,398622,32936],{},[31749,398624,768],{},[31662,398626,2557],{"stretchy":702},[31662,398628,202],{},[31749,398630,459],{},[39361,398632,262622],{},[31662,398634,262625],{},[39361,398636,262622],{},[31743,398638,398639,398641],{},[31658,398640,32936],{},[31749,398642,640],{},[31662,398644,202],{},[31743,398646,398647,398649],{},[31658,398648,32936],{},[31749,398650,768],{},[31665,398652,398653],{"encoding":31667},"1 \\cdot (x_1 - x_2) = 0 \\implies x_1 = x_2",[34,398655,398657,398675,398733,398791,398815,398870],{"className":398656,"ariaHidden":758},[31672],[34,398658,398660,398663,398666,398669,398672],{"className":398659},[31676],[34,398661],{"className":398662,"style":32617},[31680],[34,398664,640],{"className":398665},[31685],[34,398667],{"className":398668,"style":32604},[31845],[34,398670,71191],{"className":398671},[32024],[34,398673],{"className":398674,"style":32604},[31845],[34,398676,398678,398681,398684,398724,398727,398730],{"className":398677},[31676],[34,398679],{"className":398680,"style":31774},[31680],[34,398682,456],{"className":398683},[31782],[34,398685,398687,398690],{"className":398686},[31685],[34,398688,32936],{"className":398689},[31685,31686],[34,398691,398693],{"className":398692},[31792],[34,398694,398696,398716],{"className":398695},[31796,31797],[34,398697,398699,398713],{"className":398698},[31801],[34,398700,398702],{"className":398701,"style":31806},[31805],[34,398703,398704,398707],{"style":31809},[34,398705],{"className":398706,"style":31814},[31813],[34,398708,398710],{"className":398709},[31818,31819,31820,31821],[34,398711,640],{"className":398712},[31685,31821],[34,398714,31829],{"className":398715},[31828],[34,398717,398719],{"className":398718},[31801],[34,398720,398722],{"className":398721,"style":31836},[31805],[34,398723],{},[34,398725],{"className":398726,"style":32604},[31845],[34,398728,32226],{"className":398729},[32024],[34,398731],{"className":398732,"style":32604},[31845],[34,398734,398736,398739,398779,398782,398785,398788],{"className":398735},[31676],[34,398737],{"className":398738,"style":31774},[31680],[34,398740,398742,398745],{"className":398741},[31685],[34,398743,32936],{"className":398744},[31685,31686],[34,398746,398748],{"className":398747},[31792],[34,398749,398751,398771],{"className":398750},[31796,31797],[34,398752,398754,398768],{"className":398753},[31801],[34,398755,398757],{"className":398756,"style":31806},[31805],[34,398758,398759,398762],{"style":31809},[34,398760],{"className":398761,"style":31814},[31813],[34,398763,398765],{"className":398764},[31818,31819,31820,31821],[34,398766,768],{"className":398767},[31685,31821],[34,398769,31829],{"className":398770},[31828],[34,398772,398774],{"className":398773},[31801],[34,398775,398777],{"className":398776,"style":31836},[31805],[34,398778],{},[34,398780,2557],{"className":398781},[31690],[34,398783],{"className":398784,"style":31846},[31845],[34,398786,202],{"className":398787},[31850],[34,398789],{"className":398790,"style":31846},[31845],[34,398792,398794,398797,398800,398803,398806,398809,398812],{"className":398793},[31676],[34,398795],{"className":398796,"style":262700},[31680],[34,398798,459],{"className":398799},[31685],[34,398801],{"className":398802,"style":31846},[31845],[34,398804],{"className":398805,"style":31846},[31845],[34,398807,262625],{"className":398808},[31850],[34,398810],{"className":398811,"style":31846},[31845],[34,398813],{"className":398814,"style":31846},[31845],[34,398816,398818,398821,398861,398864,398867],{"className":398817},[31676],[34,398819],{"className":398820,"style":31860},[31680],[34,398822,398824,398827],{"className":398823},[31685],[34,398825,32936],{"className":398826},[31685,31686],[34,398828,398830],{"className":398829},[31792],[34,398831,398833,398853],{"className":398832},[31796,31797],[34,398834,398836,398850],{"className":398835},[31801],[34,398837,398839],{"className":398838,"style":31806},[31805],[34,398840,398841,398844],{"style":31809},[34,398842],{"className":398843,"style":31814},[31813],[34,398845,398847],{"className":398846},[31818,31819,31820,31821],[34,398848,640],{"className":398849},[31685,31821],[34,398851,31829],{"className":398852},[31828],[34,398854,398856],{"className":398855},[31801],[34,398857,398859],{"className":398858,"style":31836},[31805],[34,398860],{},[34,398862],{"className":398863,"style":31846},[31845],[34,398865,202],{"className":398866},[31850],[34,398868],{"className":398869,"style":31846},[31845],[34,398871,398873,398876],{"className":398872},[31676],[34,398874],{"className":398875,"style":31860},[31680],[34,398877,398879,398882],{"className":398878},[31685],[34,398880,32936],{"className":398881},[31685,31686],[34,398883,398885],{"className":398884},[31792],[34,398886,398888,398908],{"className":398887},[31796,31797],[34,398889,398891,398905],{"className":398890},[31801],[34,398892,398894],{"className":398893,"style":31806},[31805],[34,398895,398896,398899],{"style":31809},[34,398897],{"className":398898,"style":31814},[31813],[34,398900,398902],{"className":398901},[31818,31819,31820,31821],[34,398903,768],{"className":398904},[31685,31821],[34,398906,31829],{"className":398907},[31828],[34,398909,398911],{"className":398910},[31801],[34,398912,398914],{"className":398913,"style":31836},[31805],[34,398915],{},"。\n这说明：只有相同的输入才有相同的输出。这个映射是",[2627,398918,364499],{},[15,398920,398921,398924,398925,398953,398954,398982,398983,398986,398987,399015,399016,399087,399088,399090,399091,399119],{},[2627,398922,398923],{},"3. 利用有限性 (Pigeonhole Principle)","\n如果你有 ",[34,398926,398928,398941],{"className":398927},[31642],[34,398929,398931],{"className":398930},[31646],[31648,398932,398933],{"xmlns":31650},[31652,398934,398935,398939],{},[31655,398936,398937],{},[31658,398938,33347],{},[31665,398940,33347],{"encoding":31667},[34,398942,398944],{"className":398943,"ariaHidden":758},[31672],[34,398945,398947,398950],{"className":398946},[31676],[34,398948],{"className":398949,"style":33218},[31680],[34,398951,33347],{"className":398952,"style":33389},[31685,31686]," 个座位，有 ",[34,398955,398957,398970],{"className":398956},[31642],[34,398958,398960],{"className":398959},[31646],[31648,398961,398962],{"xmlns":31650},[31652,398963,398964,398968],{},[31655,398965,398966],{},[31658,398967,33347],{},[31665,398969,33347],{"encoding":31667},[34,398971,398973],{"className":398972,"ariaHidden":758},[31672],[34,398974,398976,398979],{"className":398975},[31676],[34,398977],{"className":398978,"style":33218},[31680],[34,398980,33347],{"className":398981,"style":33389},[31685,31686]," 个人。如果你保证每个人都坐不同的座位（单射），那么结果一定是",[2627,398984,398985],{},"所有座位都被坐满了","（满射）。\n因为 ",[34,398988,398990,399003],{"className":398989},[31642],[34,398991,398993],{"className":398992},[31646],[31648,398994,398995],{"xmlns":31650},[31652,398996,398997,399001],{},[31655,398998,398999],{},[31658,399000,34386],{},[31665,399002,34386],{"encoding":31667},[34,399004,399006],{"className":399005,"ariaHidden":758},[31672],[34,399007,399009,399012],{"className":399008},[31676],[34,399010],{"className":399011,"style":33218},[31680],[34,399013,34386],{"className":399014,"style":34427},[31685,31686]," 是有限的，所以 ",[34,399017,399019,399045],{"className":399018},[31642],[34,399020,399022],{"className":399021},[31646],[31648,399023,399024],{"xmlns":31650},[31652,399025,399026,399042],{},[31655,399027,399028,399030,399032,399034,399036,399038,399040],{},[31658,399029,31739],{},[31662,399031,456],{"stretchy":702},[31658,399033,32936],{},[31662,399035,2557],{"stretchy":702},[31662,399037,202],{},[31658,399039,1697],{},[31658,399041,32936],{},[31665,399043,399044],{"encoding":31667},"f(x) = bx",[34,399046,399048,399075],{"className":399047,"ariaHidden":758},[31672],[34,399049,399051,399054,399057,399060,399063,399066,399069,399072],{"className":399050},[31676],[34,399052],{"className":399053,"style":31774},[31680],[34,399055,31739],{"className":399056,"style":31778},[31685,31686],[34,399058,456],{"className":399059},[31782],[34,399061,32936],{"className":399062},[31685,31686],[34,399064,2557],{"className":399065},[31690],[34,399067],{"className":399068,"style":31846},[31845],[34,399070,202],{"className":399071},[31850],[34,399073],{"className":399074,"style":31846},[31845],[34,399076,399078,399081,399084],{"className":399077},[31676],[34,399079],{"className":399080,"style":31681},[31680],[34,399082,1697],{"className":399083},[31685,31686],[34,399085,32936],{"className":399086},[31685,31686]," 这个映射一定是",[2627,399089,364537],{},"（Onto）。\n这意味着，环里的每一个元素，都可以被写成 ",[34,399092,399094,399107],{"className":399093},[31642],[34,399095,399097],{"className":399096},[31646],[31648,399098,399099],{"xmlns":31650},[31652,399100,399101,399105],{},[31655,399102,399103],{},[31658,399104,1697],{},[31665,399106,1697],{"encoding":31667},[34,399108,399110],{"className":399109,"ariaHidden":758},[31672],[34,399111,399113,399116],{"className":399112},[31676],[34,399114],{"className":399115,"style":31681},[31680],[34,399117,1697],{"className":399118},[31685,31686]," 乘以某样东西。",[15,399121,399122,399125,399126,399154,399155,335149,399206,61605,399272,399328,399329,86674,399381,399437,399438,399466,399467,39984,399495,355408,399715,399766,399767,399823,399824,347],{},[2627,399123,399124],{},"4. 寻找逆元","\n既然是满射，那么单位元 ",[34,399127,399129,399142],{"className":399128},[31642],[34,399130,399132],{"className":399131},[31646],[31648,399133,399134],{"xmlns":31650},[31652,399135,399136,399140],{},[31655,399137,399138],{},[31749,399139,640],{},[31665,399141,640],{"encoding":31667},[34,399143,399145],{"className":399144,"ariaHidden":758},[31672],[34,399146,399148,399151],{"className":399147},[31676],[34,399149],{"className":399150,"style":32617},[31680],[34,399152,640],{"className":399153},[31685]," 也一定能被映射到。\n也就是说，存在某个元素 ",[34,399156,399158,399176],{"className":399157},[31642],[34,399159,399161],{"className":399160},[31646],[31648,399162,399163],{"xmlns":31650},[31652,399164,399165,399173],{},[31655,399166,399167,399169,399171],{},[31658,399168,31965],{},[31662,399170,112296],{},[31658,399172,34386],{},[31665,399174,399175],{"encoding":31667},"c \\in R",[34,399177,399179,399197],{"className":399178,"ariaHidden":758},[31672],[34,399180,399182,399185,399188,399191,399194],{"className":399181},[31676],[34,399183],{"className":399184,"style":34812},[31680],[34,399186,31965],{"className":399187},[31685,31686],[34,399189],{"className":399190,"style":31846},[31845],[34,399192,112296],{"className":399193},[31850],[34,399195],{"className":399196,"style":31846},[31845],[34,399198,399200,399203],{"className":399199},[31676],[34,399201],{"className":399202,"style":33218},[31680],[34,399204,34386],{"className":399205,"style":34427},[31685,31686],[34,399207,399209,399233],{"className":399208},[31642],[34,399210,399212],{"className":399211},[31646],[31648,399213,399214],{"xmlns":31650},[31652,399215,399216,399230],{},[31655,399217,399218,399220,399222,399224,399226,399228],{},[31658,399219,31739],{},[31662,399221,456],{"stretchy":702},[31658,399223,31965],{},[31662,399225,2557],{"stretchy":702},[31662,399227,202],{},[31749,399229,640],{},[31665,399231,399232],{"encoding":31667},"f(c) = 1",[34,399234,399236,399263],{"className":399235,"ariaHidden":758},[31672],[34,399237,399239,399242,399245,399248,399251,399254,399257,399260],{"className":399238},[31676],[34,399240],{"className":399241,"style":31774},[31680],[34,399243,31739],{"className":399244,"style":31778},[31685,31686],[34,399246,456],{"className":399247},[31782],[34,399249,31965],{"className":399250},[31685,31686],[34,399252,2557],{"className":399253},[31690],[34,399255],{"className":399256,"style":31846},[31845],[34,399258,202],{"className":399259},[31850],[34,399261],{"className":399262,"style":31846},[31845],[34,399264,399266,399269],{"className":399265},[31676],[34,399267],{"className":399268,"style":32617},[31680],[34,399270,640],{"className":399271},[31685],[34,399273,399275,399295],{"className":399274},[31642],[34,399276,399278],{"className":399277},[31646],[31648,399279,399280],{"xmlns":31650},[31652,399281,399282,399292],{},[31655,399283,399284,399286,399288,399290],{},[31658,399285,1697],{},[31658,399287,31965],{},[31662,399289,202],{},[31749,399291,640],{},[31665,399293,399294],{"encoding":31667},"b c = 1",[34,399296,399298,399319],{"className":399297,"ariaHidden":758},[31672],[34,399299,399301,399304,399307,399310,399313,399316],{"className":399300},[31676],[34,399302],{"className":399303,"style":31681},[31680],[34,399305,1697],{"className":399306},[31685,31686],[34,399308,31965],{"className":399309},[31685,31686],[34,399311],{"className":399312,"style":31846},[31845],[34,399314,202],{"className":399315},[31850],[34,399317],{"className":399318,"style":31846},[31845],[34,399320,399322,399325],{"className":399321},[31676],[34,399323],{"className":399324,"style":32617},[31680],[34,399326,640],{"className":399327},[31685],"。\n现在的局面是：",[34,399330,399332,399351],{"className":399331},[31642],[34,399333,399335],{"className":399334},[31646],[31648,399336,399337],{"xmlns":31650},[31652,399338,399339,399349],{},[31655,399340,399341,399343,399345,399347],{},[31658,399342,111],{},[31658,399344,1697],{},[31662,399346,202],{},[31749,399348,640],{},[31665,399350,364752],{"encoding":31667},[34,399352,399354,399372],{"className":399353,"ariaHidden":758},[31672],[34,399355,399357,399360,399363,399366,399369],{"className":399356},[31676],[34,399358],{"className":399359,"style":31681},[31680],[34,399361,324191],{"className":399362},[31685,31686],[34,399364],{"className":399365,"style":31846},[31845],[34,399367,202],{"className":399368},[31850],[34,399370],{"className":399371,"style":31846},[31845],[34,399373,399375,399378],{"className":399374},[31676],[34,399376],{"className":399377,"style":32617},[31680],[34,399379,640],{"className":399380},[31685],[34,399382,399384,399404],{"className":399383},[31642],[34,399385,399387],{"className":399386},[31646],[31648,399388,399389],{"xmlns":31650},[31652,399390,399391,399401],{},[31655,399392,399393,399395,399397,399399],{},[31658,399394,1697],{},[31658,399396,31965],{},[31662,399398,202],{},[31749,399400,640],{},[31665,399402,399403],{"encoding":31667},"bc = 1",[34,399405,399407,399428],{"className":399406,"ariaHidden":758},[31672],[34,399408,399410,399413,399416,399419,399422,399425],{"className":399409},[31676],[34,399411],{"className":399412,"style":31681},[31680],[34,399414,1697],{"className":399415},[31685,31686],[34,399417,31965],{"className":399418},[31685,31686],[34,399420],{"className":399421,"style":31846},[31845],[34,399423,202],{"className":399424},[31850],[34,399426],{"className":399427,"style":31846},[31845],[34,399429,399431,399434],{"className":399430},[31676],[34,399432],{"className":399433,"style":32617},[31680],[34,399435,640],{"className":399436},[31685],"。\n我们要证明 ",[34,399439,399441,399454],{"className":399440},[31642],[34,399442,399444],{"className":399443},[31646],[31648,399445,399446],{"xmlns":31650},[31652,399447,399448,399452],{},[31655,399449,399450],{},[31658,399451,31965],{},[31665,399453,31965],{"encoding":31667},[34,399455,399457],{"className":399456,"ariaHidden":758},[31672],[34,399458,399460,399463],{"className":399459},[31676],[34,399461],{"className":399462,"style":32527},[31680],[34,399464,31965],{"className":399465},[31685,31686]," 其实就是 ",[34,399468,399470,399483],{"className":399469},[31642],[34,399471,399473],{"className":399472},[31646],[31648,399474,399475],{"xmlns":31650},[31652,399476,399477,399481],{},[31655,399478,399479],{},[31658,399480,111],{},[31665,399482,111],{"encoding":31667},[34,399484,399486],{"className":399485,"ariaHidden":758},[31672],[34,399487,399489,399492],{"className":399488},[31676],[34,399490],{"className":399491,"style":32527},[31680],[34,399493,111],{"className":399494},[31685,31686],[34,399496,399498,399556],{"className":399497},[31642],[34,399499,399501],{"className":399500},[31646],[31648,399502,399503],{"xmlns":31650},[31652,399504,399505,399553],{},[31655,399506,399507,399509,399511,399513,399515,399517,399519,399521,399523,399525,399527,399529,399531,399533,399535,399537,399539,399541,399543,399545,399547,399549,399551],{},[31658,399508,111],{},[31662,399510,202],{},[31658,399512,111],{},[31662,399514,71191],{},[31749,399516,640],{},[31662,399518,202],{},[31658,399520,111],{},[31662,399522,456],{"stretchy":702},[31658,399524,1697],{},[31658,399526,31965],{},[31662,399528,2557],{"stretchy":702},[31662,399530,202],{},[31662,399532,456],{"stretchy":702},[31658,399534,111],{},[31658,399536,1697],{},[31662,399538,2557],{"stretchy":702},[31658,399540,31965],{},[31662,399542,202],{},[31749,399544,640],{},[31662,399546,71191],{},[31658,399548,31965],{},[31662,399550,202],{},[31658,399552,31965],{},[31665,399554,399555],{"encoding":31667},"a = a \\cdot 1 = a(bc) = (ab)c = 1 \\cdot c = c",[34,399557,399559,399577,399595,399613,399643,399670,399688,399706],{"className":399558,"ariaHidden":758},[31672],[34,399560,399562,399565,399568,399571,399574],{"className":399561},[31676],[34,399563],{"className":399564,"style":32527},[31680],[34,399566,111],{"className":399567},[31685,31686],[34,399569],{"className":399570,"style":31846},[31845],[34,399572,202],{"className":399573},[31850],[34,399575],{"className":399576,"style":31846},[31845],[34,399578,399580,399583,399586,399589,399592],{"className":399579},[31676],[34,399581],{"className":399582,"style":224869},[31680],[34,399584,111],{"className":399585},[31685,31686],[34,399587],{"className":399588,"style":32604},[31845],[34,399590,71191],{"className":399591},[32024],[34,399593],{"className":399594,"style":32604},[31845],[34,399596,399598,399601,399604,399607,399610],{"className":399597},[31676],[34,399599],{"className":399600,"style":32617},[31680],[34,399602,640],{"className":399603},[31685],[34,399605],{"className":399606,"style":31846},[31845],[34,399608,202],{"className":399609},[31850],[34,399611],{"className":399612,"style":31846},[31845],[34,399614,399616,399619,399622,399625,399628,399631,399634,399637,399640],{"className":399615},[31676],[34,399617],{"className":399618,"style":31774},[31680],[34,399620,111],{"className":399621},[31685,31686],[34,399623,456],{"className":399624},[31782],[34,399626,1697],{"className":399627},[31685,31686],[34,399629,31965],{"className":399630},[31685,31686],[34,399632,2557],{"className":399633},[31690],[34,399635],{"className":399636,"style":31846},[31845],[34,399638,202],{"className":399639},[31850],[34,399641],{"className":399642,"style":31846},[31845],[34,399644,399646,399649,399652,399655,399658,399661,399664,399667],{"className":399645},[31676],[34,399647],{"className":399648,"style":31774},[31680],[34,399650,456],{"className":399651},[31782],[34,399653,324191],{"className":399654},[31685,31686],[34,399656,2557],{"className":399657},[31690],[34,399659,31965],{"className":399660},[31685,31686],[34,399662],{"className":399663,"style":31846},[31845],[34,399665,202],{"className":399666},[31850],[34,399668],{"className":399669,"style":31846},[31845],[34,399671,399673,399676,399679,399682,399685],{"className":399672},[31676],[34,399674],{"className":399675,"style":32617},[31680],[34,399677,640],{"className":399678},[31685],[34,399680],{"className":399681,"style":32604},[31845],[34,399683,71191],{"className":399684},[32024],[34,399686],{"className":399687,"style":32604},[31845],[34,399689,399691,399694,399697,399700,399703],{"className":399690},[31676],[34,399692],{"className":399693,"style":32527},[31680],[34,399695,31965],{"className":399696},[31685,31686],[34,399698],{"className":399699,"style":31846},[31845],[34,399701,202],{"className":399702},[31850],[34,399704],{"className":399705,"style":31846},[31845],[34,399707,399709,399712],{"className":399708},[31676],[34,399710],{"className":399711,"style":32527},[31680],[34,399713,31965],{"className":399714},[31685,31686],[34,399716,399718,399736],{"className":399717},[31642],[34,399719,399721],{"className":399720},[31646],[31648,399722,399723],{"xmlns":31650},[31652,399724,399725,399733],{},[31655,399726,399727,399729,399731],{},[31658,399728,111],{},[31662,399730,202],{},[31658,399732,31965],{},[31665,399734,399735],{"encoding":31667},"a = c",[34,399737,399739,399757],{"className":399738,"ariaHidden":758},[31672],[34,399740,399742,399745,399748,399751,399754],{"className":399741},[31676],[34,399743],{"className":399744,"style":32527},[31680],[34,399746,111],{"className":399747},[31685,31686],[34,399749],{"className":399750,"style":31846},[31845],[34,399752,202],{"className":399753},[31850],[34,399755],{"className":399756,"style":31846},[31845],[34,399758,399760,399763],{"className":399759},[31676],[34,399761],{"className":399762,"style":32527},[31680],[34,399764,31965],{"className":399765},[31685,31686],"。\n代回 ",[34,399768,399770,399790],{"className":399769},[31642],[34,399771,399773],{"className":399772},[31646],[31648,399774,399775],{"xmlns":31650},[31652,399776,399777,399787],{},[31655,399778,399779,399781,399783,399785],{},[31658,399780,1697],{},[31658,399782,31965],{},[31662,399784,202],{},[31749,399786,640],{},[31665,399788,399789],{"encoding":31667},"bc=1",[34,399791,399793,399814],{"className":399792,"ariaHidden":758},[31672],[34,399794,399796,399799,399802,399805,399808,399811],{"className":399795},[31676],[34,399797],{"className":399798,"style":31681},[31680],[34,399800,1697],{"className":399801},[31685,31686],[34,399803,31965],{"className":399804},[31685,31686],[34,399806],{"className":399807,"style":31846},[31845],[34,399809,202],{"className":399810},[31850],[34,399812],{"className":399813,"style":31846},[31845],[34,399815,399817,399820],{"className":399816},[31676],[34,399818],{"className":399819,"style":32617},[31680],[34,399821,640],{"className":399822},[31685],"，就得到了 ",[34,399825,399827,399846],{"className":399826},[31642],[34,399828,399830],{"className":399829},[31646],[31648,399831,399832],{"xmlns":31650},[31652,399833,399834,399844],{},[31655,399835,399836,399838,399840,399842],{},[31658,399837,1697],{},[31658,399839,111],{},[31662,399841,202],{},[31749,399843,640],{},[31665,399845,397396],{"encoding":31667},[34,399847,399849,399867],{"className":399848,"ariaHidden":758},[31672],[34,399850,399852,399855,399858,399861,399864],{"className":399851},[31676],[34,399853],{"className":399854,"style":31681},[31680],[34,399856,367454],{"className":399857},[31685,31686],[34,399859],{"className":399860,"style":31846},[31845],[34,399862,202],{"className":399863},[31850],[34,399865],{"className":399866,"style":31846},[31845],[34,399868,399870,399873],{"className":399869},[31676],[34,399871],{"className":399872,"style":32617},[31680],[34,399874,640],{"className":399875},[31685],[32188,399877],{},[253,399879,399880],{"id":399880},"题目13",[15,399882,2234,399883,399963,399964,400015],{},[34,399884,399886,399907],{"className":399885},[31642],[34,399887,399889],{"className":399888},[31646],[31648,399890,399891],{"xmlns":31650},[31652,399892,399893,399905],{},[31655,399894,399895,399901,399903],{},[37624,399896,399897,399899],{},[31658,399898,111],{},[31658,399900,31660],{},[31662,399902,202],{},[31749,399904,459],{},[31665,399906,343092],{"encoding":31667},[34,399908,399910,399954],{"className":399909,"ariaHidden":758},[31672],[34,399911,399913,399916,399945,399948,399951],{"className":399912},[31676],[34,399914],{"className":399915,"style":37959},[31680],[34,399917,399919,399922],{"className":399918},[31685],[34,399920,111],{"className":399921},[31685,31686],[34,399923,399925],{"className":399924},[31792],[34,399926,399928],{"className":399927},[31796],[34,399929,399931],{"className":399930},[31801],[34,399932,399934],{"className":399933,"style":37959},[31805],[34,399935,399936,399939],{"style":37777},[34,399937],{"className":399938,"style":31814},[31813],[34,399940,399942],{"className":399941},[31818,31819,31820,31821],[34,399943,31660],{"className":399944},[31685,31686,31821],[34,399946],{"className":399947,"style":31846},[31845],[34,399949,202],{"className":399950},[31850],[34,399952],{"className":399953,"style":31846},[31845],[34,399955,399957,399960],{"className":399956},[31676],[34,399958],{"className":399959,"style":32617},[31680],[34,399961,459],{"className":399962},[31685],"（幂零元），证明 ",[34,399965,399967,399985],{"className":399966},[31642],[34,399968,399970],{"className":399969},[31646],[31648,399971,399972],{"xmlns":31650},[31652,399973,399974,399982],{},[31655,399975,399976,399978,399980],{},[31749,399977,640],{},[31662,399979,32226],{},[31658,399981,111],{},[31665,399983,399984],{"encoding":31667},"1 - a",[34,399986,399988,400006],{"className":399987,"ariaHidden":758},[31672],[34,399989,399991,399994,399997,400000,400003],{"className":399990},[31676],[34,399992],{"className":399993,"style":35942},[31680],[34,399995,640],{"className":399996},[31685],[34,399998],{"className":399999,"style":32604},[31845],[34,400001,32226],{"className":400002},[32024],[34,400004],{"className":400005,"style":32604},[31845],[34,400007,400009,400012],{"className":400008},[31676],[34,400010],{"className":400011,"style":32527},[31680],[34,400013,111],{"className":400014},[31685,31686]," 可逆。",[32490,400017,297792],{"id":400018},"详细解析-7",[15,400020,400021,400024,400025,400028,400029,400312,400313,400374,400375,400403,400404,400463],{},[2627,400022,400023],{},"1. 直觉来源","\n这道题的灵感来源于微积分里的",[2627,400026,400027],{},"几何级数","。\n回忆一下：",[34,400030,400032,400084],{"className":400031},[31642],[34,400033,400035],{"className":400034},[31646],[31648,400036,400037],{"xmlns":31650},[31652,400038,400039,400081],{},[31655,400040,400041,400053,400055,400057,400059,400061,400063,400069,400071,400077,400079],{},[41453,400042,400043,400045],{},[31749,400044,640],{},[31655,400046,400047,400049,400051],{},[31749,400048,640],{},[31662,400050,32226],{},[31658,400052,32936],{},[31662,400054,202],{},[31749,400056,640],{},[31662,400058,223],{},[31658,400060,32936],{},[31662,400062,223],{},[37624,400064,400065,400067],{},[31658,400066,32936],{},[31749,400068,768],{},[31662,400070,223],{},[37624,400072,400073,400075],{},[31658,400074,32936],{},[31749,400076,786],{},[31662,400078,223],{},[31662,400080,33934],{},[31665,400082,400083],{"encoding":31667},"\\frac{1}{1-x} = 1 + x + x^2 + x^3 + \\dots",[34,400085,400087,400178,400196,400214,400258,400302],{"className":400086,"ariaHidden":758},[31672],[34,400088,400090,400094,400169,400172,400175],{"className":400089},[31676],[34,400091],{"className":400092,"style":400093},[31680],"height:1.2484em;vertical-align:-0.4033em;",[34,400095,400097,400100,400166],{"className":400096},[31685],[34,400098],{"className":400099},[31782,41488],[34,400101,400103],{"className":400102},[41453],[34,400104,400106,400157],{"className":400105},[31796,31797],[34,400107,400109,400154],{"className":400108},[31801],[34,400110,400112,400132,400140],{"className":400111,"style":46746},[31805],[34,400113,400114,400117],{"style":41504},[34,400115],{"className":400116,"style":41508},[31813],[34,400118,400120],{"className":400119},[31818,31819,31820,31821],[34,400121,400123,400126,400129],{"className":400122},[31685,31821],[34,400124,640],{"className":400125},[31685,31821],[34,400127,32226],{"className":400128},[32024,31821],[34,400130,32936],{"className":400131},[31685,31686,31821],[34,400133,400134,400137],{"style":41520},[34,400135],{"className":400136,"style":41508},[31813],[34,400138],{"className":400139,"style":41528},[41527],[34,400141,400142,400145],{"style":46771},[34,400143],{"className":400144,"style":41508},[31813],[34,400146,400148],{"className":400147},[31818,31819,31820,31821],[34,400149,400151],{"className":400150},[31685,31821],[34,400152,640],{"className":400153},[31685,31821],[34,400155,31829],{"className":400156},[31828],[34,400158,400160],{"className":400159},[31801],[34,400161,400164],{"className":400162,"style":400163},[31805],"height:0.4033em;",[34,400165],{},[34,400167],{"className":400168},[31690,41488],[34,400170],{"className":400171,"style":31846},[31845],[34,400173,202],{"className":400174},[31850],[34,400176],{"className":400177,"style":31846},[31845],[34,400179,400181,400184,400187,400190,400193],{"className":400180},[31676],[34,400182],{"className":400183,"style":35942},[31680],[34,400185,640],{"className":400186},[31685],[34,400188],{"className":400189,"style":32604},[31845],[34,400191,223],{"className":400192},[32024],[34,400194],{"className":400195,"style":32604},[31845],[34,400197,400199,400202,400205,400208,400211],{"className":400198},[31676],[34,400200],{"className":400201,"style":32597},[31680],[34,400203,32936],{"className":400204},[31685,31686],[34,400206],{"className":400207,"style":32604},[31845],[34,400209,223],{"className":400210},[32024],[34,400212],{"className":400213,"style":32604},[31845],[34,400215,400217,400220,400249,400252,400255],{"className":400216},[31676],[34,400218],{"className":400219,"style":40044},[31680],[34,400221,400223,400226],{"className":400222},[31685],[34,400224,32936],{"className":400225},[31685,31686],[34,400227,400229],{"className":400228},[31792],[34,400230,400232],{"className":400231},[31796],[34,400233,400235],{"className":400234},[31801],[34,400236,400238],{"className":400237,"style":37774},[31805],[34,400239,400240,400243],{"style":37777},[34,400241],{"className":400242,"style":31814},[31813],[34,400244,400246],{"className":400245},[31818,31819,31820,31821],[34,400247,768],{"className":400248},[31685,31821],[34,400250],{"className":400251,"style":32604},[31845],[34,400253,223],{"className":400254},[32024],[34,400256],{"className":400257,"style":32604},[31845],[34,400259,400261,400264,400293,400296,400299],{"className":400260},[31676],[34,400262],{"className":400263,"style":40044},[31680],[34,400265,400267,400270],{"className":400266},[31685],[34,400268,32936],{"className":400269},[31685,31686],[34,400271,400273],{"className":400272},[31792],[34,400274,400276],{"className":400275},[31796],[34,400277,400279],{"className":400278},[31801],[34,400280,400282],{"className":400281,"style":37774},[31805],[34,400283,400284,400287],{"style":37777},[34,400285],{"className":400286,"style":31814},[31813],[34,400288,400290],{"className":400289},[31818,31819,31820,31821],[34,400291,786],{"className":400292},[31685,31821],[34,400294],{"className":400295,"style":32604},[31845],[34,400297,223],{"className":400298},[32024],[34,400300],{"className":400301,"style":32604},[31845],[34,400303,400305,400309],{"className":400304},[31676],[34,400306],{"className":400307,"style":400308},[31680],"height:0.123em;",[34,400310,33934],{"className":400311},[34006]," （当 ",[34,400314,400316,400338],{"className":400315},[31642],[34,400317,400319],{"className":400318},[31646],[31648,400320,400321],{"xmlns":31650},[31652,400322,400323,400335],{},[31655,400324,400325,400327,400329,400331,400333],{},[31658,400326,162086],{"mathvariant":38079},[31658,400328,32936],{},[31658,400330,162086],{"mathvariant":38079},[31662,400332,932],{},[31749,400334,640],{},[31665,400336,400337],{"encoding":31667},"|x|\u003C1",[34,400339,400341,400365],{"className":400340,"ariaHidden":758},[31672],[34,400342,400344,400347,400350,400353,400356,400359,400362],{"className":400343},[31676],[34,400345],{"className":400346,"style":31774},[31680],[34,400348,162086],{"className":400349},[31685],[34,400351,32936],{"className":400352},[31685,31686],[34,400354,162086],{"className":400355},[31685],[34,400357],{"className":400358,"style":31846},[31845],[34,400360,932],{"className":400361},[31850],[34,400363],{"className":400364,"style":31846},[31845],[34,400366,400368,400371],{"className":400367},[31676],[34,400369],{"className":400370,"style":32617},[31680],[34,400372,640],{"className":400373},[31685]," 时）。\n在环论里，虽然没有收敛的概念，但如果 ",[34,400376,400378,400391],{"className":400377},[31642],[34,400379,400381],{"className":400380},[31646],[31648,400382,400383],{"xmlns":31650},[31652,400384,400385,400389],{},[31655,400386,400387],{},[31658,400388,32936],{},[31665,400390,32936],{"encoding":31667},[34,400392,400394],{"className":400393,"ariaHidden":758},[31672],[34,400395,400397,400400],{"className":400396},[31676],[34,400398],{"className":400399,"style":32527},[31680],[34,400401,32936],{"className":400402},[31685,31686]," 到某一项 ",[34,400405,400407,400425],{"className":400406},[31642],[34,400408,400410],{"className":400409},[31646],[31648,400411,400412],{"xmlns":31650},[31652,400413,400414,400422],{},[31655,400415,400416],{},[37624,400417,400418,400420],{},[31658,400419,32936],{},[31658,400421,31660],{},[31665,400423,400424],{"encoding":31667},"x^n",[34,400426,400428],{"className":400427,"ariaHidden":758},[31672],[34,400429,400431,400434],{"className":400430},[31676],[34,400432],{"className":400433,"style":37959},[31680],[34,400435,400437,400440],{"className":400436},[31685],[34,400438,32936],{"className":400439},[31685,31686],[34,400441,400443],{"className":400442},[31792],[34,400444,400446],{"className":400445},[31796],[34,400447,400449],{"className":400448},[31801],[34,400450,400452],{"className":400451,"style":37959},[31805],[34,400453,400454,400457],{"style":37777},[34,400455],{"className":400456,"style":31814},[31813],[34,400458,400460],{"className":400459},[31818,31819,31820,31821],[34,400461,31660],{"className":400462},[31685,31686,31821]," 直接变成 0 了，后面无穷多项就不用加了，变成了一个有限的多项式！",[15,400465,400466,400469,400470,400521,400522,335149,400550,400638,400639,400667,400668,347],{},[2627,400467,400468],{},"2. 猜想逆元","\n我们要证明 ",[34,400471,400473,400491],{"className":400472},[31642],[34,400474,400476],{"className":400475},[31646],[31648,400477,400478],{"xmlns":31650},[31652,400479,400480,400488],{},[31655,400481,400482,400484,400486],{},[31749,400483,640],{},[31662,400485,32226],{},[31658,400487,111],{},[31665,400489,400490],{"encoding":31667},"1-a",[34,400492,400494,400512],{"className":400493,"ariaHidden":758},[31672],[34,400495,400497,400500,400503,400506,400509],{"className":400496},[31676],[34,400498],{"className":400499,"style":35942},[31680],[34,400501,640],{"className":400502},[31685],[34,400504],{"className":400505,"style":32604},[31845],[34,400507,32226],{"className":400508},[32024],[34,400510],{"className":400511,"style":32604},[31845],[34,400513,400515,400518],{"className":400514},[31676],[34,400516],{"className":400517,"style":32527},[31680],[34,400519,111],{"className":400520},[31685,31686]," 可逆，就是要找到一个 ",[34,400523,400525,400538],{"className":400524},[31642],[34,400526,400528],{"className":400527},[31646],[31648,400529,400530],{"xmlns":31650},[31652,400531,400532,400536],{},[31655,400533,400534],{},[31658,400535,54224],{},[31665,400537,54224],{"encoding":31667},[34,400539,400541],{"className":400540,"ariaHidden":758},[31672],[34,400542,400544,400547],{"className":400543},[31676],[34,400545],{"className":400546,"style":33218},[31680],[34,400548,54224],{"className":400549,"style":54252},[31685,31686],[34,400551,400553,400581],{"className":400552},[31642],[34,400554,400556],{"className":400555},[31646],[31648,400557,400558],{"xmlns":31650},[31652,400559,400560,400578],{},[31655,400561,400562,400564,400566,400568,400570,400572,400574,400576],{},[31662,400563,456],{"stretchy":702},[31749,400565,640],{},[31662,400567,32226],{},[31658,400569,111],{},[31662,400571,2557],{"stretchy":702},[31658,400573,54224],{},[31662,400575,202],{},[31749,400577,640],{},[31665,400579,400580],{"encoding":31667},"(1-a)B = 1",[34,400582,400584,400605,400629],{"className":400583,"ariaHidden":758},[31672],[34,400585,400587,400590,400593,400596,400599,400602],{"className":400586},[31676],[34,400588],{"className":400589,"style":31774},[31680],[34,400591,456],{"className":400592},[31782],[34,400594,640],{"className":400595},[31685],[34,400597],{"className":400598,"style":32604},[31845],[34,400600,32226],{"className":400601},[32024],[34,400603],{"className":400604,"style":32604},[31845],[34,400606,400608,400611,400614,400617,400620,400623,400626],{"className":400607},[31676],[34,400609],{"className":400610,"style":31774},[31680],[34,400612,111],{"className":400613},[31685,31686],[34,400615,2557],{"className":400616},[31690],[34,400618,54224],{"className":400619,"style":54252},[31685,31686],[34,400621],{"className":400622,"style":31846},[31845],[34,400624,202],{"className":400625},[31850],[34,400627],{"className":400628,"style":31846},[31845],[34,400630,400632,400635],{"className":400631},[31676],[34,400633],{"className":400634,"style":32617},[31680],[34,400636,640],{"className":400637},[31685],"。\n根据上面的灵感，我们猜这个 ",[34,400640,400642,400655],{"className":400641},[31642],[34,400643,400645],{"className":400644},[31646],[31648,400646,400647],{"xmlns":31650},[31652,400648,400649,400653],{},[31655,400650,400651],{},[31658,400652,54224],{},[31665,400654,54224],{"encoding":31667},[34,400656,400658],{"className":400657,"ariaHidden":758},[31672],[34,400659,400661,400664],{"className":400660},[31676],[34,400662],{"className":400663,"style":33218},[31680],[34,400665,54224],{"className":400666,"style":54252},[31685,31686]," 就是：\n",[34,400669,400671,400719],{"className":400670},[31642],[34,400672,400674],{"className":400673},[31646],[31648,400675,400676],{"xmlns":31650},[31652,400677,400678,400716],{},[31655,400679,400680,400682,400684,400686,400688,400690,400692,400698,400700,400702,400704],{},[31658,400681,54224],{},[31662,400683,202],{},[31749,400685,640],{},[31662,400687,223],{},[31658,400689,111],{},[31662,400691,223],{},[37624,400693,400694,400696],{},[31658,400695,111],{},[31749,400697,768],{},[31662,400699,223],{},[31662,400701,38874],{},[31662,400703,223],{},[37624,400705,400706,400708],{},[31658,400707,111],{},[31655,400709,400710,400712,400714],{},[31658,400711,31660],{},[31662,400713,32226],{},[31749,400715,640],{},[31665,400717,400718],{"encoding":31667},"B = 1 + a + a^2 + \\dots + a^{n-1}",[34,400720,400722,400740,400758,400776,400820,400838],{"className":400721,"ariaHidden":758},[31672],[34,400723,400725,400728,400731,400734,400737],{"className":400724},[31676],[34,400726],{"className":400727,"style":33218},[31680],[34,400729,54224],{"className":400730,"style":54252},[31685,31686],[34,400732],{"className":400733,"style":31846},[31845],[34,400735,202],{"className":400736},[31850],[34,400738],{"className":400739,"style":31846},[31845],[34,400741,400743,400746,400749,400752,400755],{"className":400742},[31676],[34,400744],{"className":400745,"style":35942},[31680],[34,400747,640],{"className":400748},[31685],[34,400750],{"className":400751,"style":32604},[31845],[34,400753,223],{"className":400754},[32024],[34,400756],{"className":400757,"style":32604},[31845],[34,400759,400761,400764,400767,400770,400773],{"className":400760},[31676],[34,400762],{"className":400763,"style":32597},[31680],[34,400765,111],{"className":400766},[31685,31686],[34,400768],{"className":400769,"style":32604},[31845],[34,400771,223],{"className":400772},[32024],[34,400774],{"className":400775,"style":32604},[31845],[34,400777,400779,400782,400811,400814,400817],{"className":400778},[31676],[34,400780],{"className":400781,"style":40044},[31680],[34,400783,400785,400788],{"className":400784},[31685],[34,400786,111],{"className":400787},[31685,31686],[34,400789,400791],{"className":400790},[31792],[34,400792,400794],{"className":400793},[31796],[34,400795,400797],{"className":400796},[31801],[34,400798,400800],{"className":400799,"style":37774},[31805],[34,400801,400802,400805],{"style":37777},[34,400803],{"className":400804,"style":31814},[31813],[34,400806,400808],{"className":400807},[31818,31819,31820,31821],[34,400809,768],{"className":400810},[31685,31821],[34,400812],{"className":400813,"style":32604},[31845],[34,400815,223],{"className":400816},[32024],[34,400818],{"className":400819,"style":32604},[31845],[34,400821,400823,400826,400829,400832,400835],{"className":400822},[31676],[34,400824],{"className":400825,"style":32597},[31680],[34,400827,38874],{"className":400828},[34006],[34,400830],{"className":400831,"style":32604},[31845],[34,400833,223],{"className":400834},[32024],[34,400836],{"className":400837,"style":32604},[31845],[34,400839,400841,400844],{"className":400840},[31676],[34,400842],{"className":400843,"style":37774},[31680],[34,400845,400847,400850],{"className":400846},[31685],[34,400848,111],{"className":400849},[31685,31686],[34,400851,400853],{"className":400852},[31792],[34,400854,400856],{"className":400855},[31796],[34,400857,400859],{"className":400858},[31801],[34,400860,400862],{"className":400861,"style":37774},[31805],[34,400863,400864,400867],{"style":37777},[34,400865],{"className":400866,"style":31814},[31813],[34,400868,400870],{"className":400869},[31818,31819,31820,31821],[34,400871,400873,400876,400879],{"className":400872},[31685,31821],[34,400874,31660],{"className":400875},[31685,31686,31821],[34,400877,32226],{"className":400878},[32024,31821],[34,400880,640],{"className":400881},[31685,31821],[15,400883,400884,400887,400888,401127,401128,401479],{},[2627,400885,400886],{},"3. 验证","\n我们来做乘法：\n",[34,400889,400891,400949],{"className":400890},[31642],[34,400892,400894],{"className":400893},[31646],[31648,400895,400896],{"xmlns":31650},[31652,400897,400898,400946],{},[31655,400899,400900,400902,400904,400906,400908,400910,400912,400914,400916,400918,400920,400926,400928,400930,400932,400944],{},[31662,400901,456],{"stretchy":702},[31749,400903,640],{},[31662,400905,32226],{},[31658,400907,111],{},[31662,400909,2557],{"stretchy":702},[31662,400911,456],{"stretchy":702},[31749,400913,640],{},[31662,400915,223],{},[31658,400917,111],{},[31662,400919,223],{},[37624,400921,400922,400924],{},[31658,400923,111],{},[31749,400925,768],{},[31662,400927,223],{},[31662,400929,38874],{},[31662,400931,223],{},[37624,400933,400934,400936],{},[31658,400935,111],{},[31655,400937,400938,400940,400942],{},[31658,400939,31660],{},[31662,400941,32226],{},[31749,400943,640],{},[31662,400945,2557],{"stretchy":702},[31665,400947,400948],{"encoding":31667},"(1 - a)(1 + a + a^2 + \\dots + a^{n-1})",[34,400950,400952,400973,401000,401018,401062,401080],{"className":400951,"ariaHidden":758},[31672],[34,400953,400955,400958,400961,400964,400967,400970],{"className":400954},[31676],[34,400956],{"className":400957,"style":31774},[31680],[34,400959,456],{"className":400960},[31782],[34,400962,640],{"className":400963},[31685],[34,400965],{"className":400966,"style":32604},[31845],[34,400968,32226],{"className":400969},[32024],[34,400971],{"className":400972,"style":32604},[31845],[34,400974,400976,400979,400982,400985,400988,400991,400994,400997],{"className":400975},[31676],[34,400977],{"className":400978,"style":31774},[31680],[34,400980,111],{"className":400981},[31685,31686],[34,400983,2557],{"className":400984},[31690],[34,400986,456],{"className":400987},[31782],[34,400989,640],{"className":400990},[31685],[34,400992],{"className":400993,"style":32604},[31845],[34,400995,223],{"className":400996},[32024],[34,400998],{"className":400999,"style":32604},[31845],[34,401001,401003,401006,401009,401012,401015],{"className":401002},[31676],[34,401004],{"className":401005,"style":32597},[31680],[34,401007,111],{"className":401008},[31685,31686],[34,401010],{"className":401011,"style":32604},[31845],[34,401013,223],{"className":401014},[32024],[34,401016],{"className":401017,"style":32604},[31845],[34,401019,401021,401024,401053,401056,401059],{"className":401020},[31676],[34,401022],{"className":401023,"style":40044},[31680],[34,401025,401027,401030],{"className":401026},[31685],[34,401028,111],{"className":401029},[31685,31686],[34,401031,401033],{"className":401032},[31792],[34,401034,401036],{"className":401035},[31796],[34,401037,401039],{"className":401038},[31801],[34,401040,401042],{"className":401041,"style":37774},[31805],[34,401043,401044,401047],{"style":37777},[34,401045],{"className":401046,"style":31814},[31813],[34,401048,401050],{"className":401049},[31818,31819,31820,31821],[34,401051,768],{"className":401052},[31685,31821],[34,401054],{"className":401055,"style":32604},[31845],[34,401057,223],{"className":401058},[32024],[34,401060],{"className":401061,"style":32604},[31845],[34,401063,401065,401068,401071,401074,401077],{"className":401064},[31676],[34,401066],{"className":401067,"style":32597},[31680],[34,401069,38874],{"className":401070},[34006],[34,401072],{"className":401073,"style":32604},[31845],[34,401075,223],{"className":401076},[32024],[34,401078],{"className":401079,"style":32604},[31845],[34,401081,401083,401086,401124],{"className":401082},[31676],[34,401084],{"className":401085,"style":37755},[31680],[34,401087,401089,401092],{"className":401088},[31685],[34,401090,111],{"className":401091},[31685,31686],[34,401093,401095],{"className":401094},[31792],[34,401096,401098],{"className":401097},[31796],[34,401099,401101],{"className":401100},[31801],[34,401102,401104],{"className":401103,"style":37774},[31805],[34,401105,401106,401109],{"style":37777},[34,401107],{"className":401108,"style":31814},[31813],[34,401110,401112],{"className":401111},[31818,31819,31820,31821],[34,401113,401115,401118,401121],{"className":401114},[31685,31821],[34,401116,31660],{"className":401117},[31685,31686,31821],[34,401119,32226],{"className":401120},[32024,31821],[34,401122,640],{"className":401123},[31685,31821],[34,401125,2557],{"className":401126},[31690],"\n逐项展开：\n",[34,401129,401131,401211],{"className":401130},[31642],[34,401132,401134],{"className":401133},[31646],[31648,401135,401136],{"xmlns":31650},[31652,401137,401138,401208],{},[31655,401139,401140,401142,401144,401146,401148,401150,401152,401154,401156,401158,401160,401172,401174,401176,401178,401180,401182,401184,401186,401188,401190,401192,401194,401206],{},[31662,401141,202],{},[31749,401143,640],{},[31662,401145,71191],{},[31662,401147,456],{"stretchy":702},[31749,401149,640],{},[31662,401151,223],{},[31658,401153,111],{},[31662,401155,223],{},[31662,401157,38874],{},[31662,401159,223],{},[37624,401161,401162,401164],{},[31658,401163,111],{},[31655,401165,401166,401168,401170],{},[31658,401167,31660],{},[31662,401169,32226],{},[31749,401171,640],{},[31662,401173,2557],{"stretchy":702},[31662,401175,32226],{},[31658,401177,111],{},[31662,401179,71191],{},[31662,401181,456],{"stretchy":702},[31749,401183,640],{},[31662,401185,223],{},[31658,401187,111],{},[31662,401189,223],{},[31662,401191,38874],{},[31662,401193,223],{},[37624,401195,401196,401198],{},[31658,401197,111],{},[31655,401199,401200,401202,401204],{},[31658,401201,31660],{},[31662,401203,32226],{},[31749,401205,640],{},[31662,401207,2557],{"stretchy":702},[31665,401209,401210],{"encoding":31667},"= 1 \\cdot (1 + a + \\dots + a^{n-1}) - a \\cdot (1 + a + \\dots + a^{n-1})",[34,401212,401214,401226,401244,401265,401283,401301,401357,401375,401396,401414,401432],{"className":401213,"ariaHidden":758},[31672],[34,401215,401217,401220,401223],{"className":401216},[31676],[34,401218],{"className":401219,"style":73089},[31680],[34,401221,202],{"className":401222},[31850],[34,401224],{"className":401225,"style":31846},[31845],[34,401227,401229,401232,401235,401238,401241],{"className":401228},[31676],[34,401230],{"className":401231,"style":32617},[31680],[34,401233,640],{"className":401234},[31685],[34,401236],{"className":401237,"style":32604},[31845],[34,401239,71191],{"className":401240},[32024],[34,401242],{"className":401243,"style":32604},[31845],[34,401245,401247,401250,401253,401256,401259,401262],{"className":401246},[31676],[34,401248],{"className":401249,"style":31774},[31680],[34,401251,456],{"className":401252},[31782],[34,401254,640],{"className":401255},[31685],[34,401257],{"className":401258,"style":32604},[31845],[34,401260,223],{"className":401261},[32024],[34,401263],{"className":401264,"style":32604},[31845],[34,401266,401268,401271,401274,401277,401280],{"className":401267},[31676],[34,401269],{"className":401270,"style":32597},[31680],[34,401272,111],{"className":401273},[31685,31686],[34,401275],{"className":401276,"style":32604},[31845],[34,401278,223],{"className":401279},[32024],[34,401281],{"className":401282,"style":32604},[31845],[34,401284,401286,401289,401292,401295,401298],{"className":401285},[31676],[34,401287],{"className":401288,"style":32597},[31680],[34,401290,38874],{"className":401291},[34006],[34,401293],{"className":401294,"style":32604},[31845],[34,401296,223],{"className":401297},[32024],[34,401299],{"className":401300,"style":32604},[31845],[34,401302,401304,401307,401345,401348,401351,401354],{"className":401303},[31676],[34,401305],{"className":401306,"style":37755},[31680],[34,401308,401310,401313],{"className":401309},[31685],[34,401311,111],{"className":401312},[31685,31686],[34,401314,401316],{"className":401315},[31792],[34,401317,401319],{"className":401318},[31796],[34,401320,401322],{"className":401321},[31801],[34,401323,401325],{"className":401324,"style":37774},[31805],[34,401326,401327,401330],{"style":37777},[34,401328],{"className":401329,"style":31814},[31813],[34,401331,401333],{"className":401332},[31818,31819,31820,31821],[34,401334,401336,401339,401342],{"className":401335},[31685,31821],[34,401337,31660],{"className":401338},[31685,31686,31821],[34,401340,32226],{"className":401341},[32024,31821],[34,401343,640],{"className":401344},[31685,31821],[34,401346,2557],{"className":401347},[31690],[34,401349],{"className":401350,"style":32604},[31845],[34,401352,32226],{"className":401353},[32024],[34,401355],{"className":401356,"style":32604},[31845],[34,401358,401360,401363,401366,401369,401372],{"className":401359},[31676],[34,401361],{"className":401362,"style":224869},[31680],[34,401364,111],{"className":401365},[31685,31686],[34,401367],{"className":401368,"style":32604},[31845],[34,401370,71191],{"className":401371},[32024],[34,401373],{"className":401374,"style":32604},[31845],[34,401376,401378,401381,401384,401387,401390,401393],{"className":401377},[31676],[34,401379],{"className":401380,"style":31774},[31680],[34,401382,456],{"className":401383},[31782],[34,401385,640],{"className":401386},[31685],[34,401388],{"className":401389,"style":32604},[31845],[34,401391,223],{"className":401392},[32024],[34,401394],{"className":401395,"style":32604},[31845],[34,401397,401399,401402,401405,401408,401411],{"className":401398},[31676],[34,401400],{"className":401401,"style":32597},[31680],[34,401403,111],{"className":401404},[31685,31686],[34,401406],{"className":401407,"style":32604},[31845],[34,401409,223],{"className":401410},[32024],[34,401412],{"className":401413,"style":32604},[31845],[34,401415,401417,401420,401423,401426,401429],{"className":401416},[31676],[34,401418],{"className":401419,"style":32597},[31680],[34,401421,38874],{"className":401422},[34006],[34,401424],{"className":401425,"style":32604},[31845],[34,401427,223],{"className":401428},[32024],[34,401430],{"className":401431,"style":32604},[31845],[34,401433,401435,401438,401476],{"className":401434},[31676],[34,401436],{"className":401437,"style":37755},[31680],[34,401439,401441,401444],{"className":401440},[31685],[34,401442,111],{"className":401443},[31685,31686],[34,401445,401447],{"className":401446},[31792],[34,401448,401450],{"className":401449},[31796],[34,401451,401453],{"className":401452},[31801],[34,401454,401456],{"className":401455,"style":37774},[31805],[34,401457,401458,401461],{"style":37777},[34,401459],{"className":401460,"style":31814},[31813],[34,401462,401464],{"className":401463},[31818,31819,31820,31821],[34,401465,401467,401470,401473],{"className":401466},[31685,31821],[34,401468,31660],{"className":401469},[31685,31686,31821],[34,401471,32226],{"className":401472},[32024,31821],[34,401474,640],{"className":401475},[31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(1 + a + a^2 + \\dots + a^{n-1}) - (a + a^2 + a^3 + \\dots + a^n)",[34,401569,401571,401583,401604,401622,401666,401684,401740,401761,401805,401849,401867],{"className":401570,"ariaHidden":758},[31672],[34,401572,401574,401577,401580],{"className":401573},[31676],[34,401575],{"className":401576,"style":73089},[31680],[34,401578,202],{"className":401579},[31850],[34,401581],{"className":401582,"style":31846},[31845],[34,401584,401586,401589,401592,401595,401598,401601],{"className":401585},[31676],[34,401587],{"className":401588,"style":31774},[31680],[34,401590,456],{"className":401591},[31782],[34,401593,640],{"className":401594},[31685],[34,401596],{"className":401597,"style":32604},[31845],[34,401599,223],{"className":401600},[32024],[34,401602],{"className":401603,"style":32604},[31845],[34,401605,401607,401610,401613,401616,401619],{"className":401606},[31676],[34,401608],{"className":401609,"style":32597},[31680],[34,401611,111],{"className":401612},[31685,31686],[34,401614],{"className":401615,"style":32604},[31845],[34,401617,223],{"className":401618},[32024],[34,401620],{"className":401621,"style":32604},[31845],[34,401623,401625,401628,401657,401660,401663],{"className":401624},[31676],[34,401626],{"className":401627,"style":40044},[31680],[34,401629,401631,401634],{"className":401630},[31685],[34,401632,111],{"className":401633},[31685,31686],[34,401635,401637],{"className":401636},[31792],[34,401638,401640],{"className":401639},[31796],[34,401641,401643],{"className":401642},[31801],[34,401644,401646],{"className":401645,"style":37774},[31805],[34,401647,401648,401651],{"style":37777},[34,401649],{"className":401650,"style":31814},[31813],[34,401652,401654],{"className":401653},[31818,31819,31820,31821],[34,401655,768],{"className":401656},[31685,31821],[34,401658],{"className":401659,"style":32604},[31845],[34,401661,223],{"className":401662},[32024],[34,401664],{"className":401665,"style":32604},[31845],[34,401667,401669,401672,401675,401678,401681],{"className":401668},[31676],[34,401670],{"className":401671,"style":32597},[31680],[34,401673,38874],{"className":401674},[34006],[34,401676],{"className":401677,"style":32604},[31845],[34,401679,223],{"className":401680},[32024],[34,401682],{"className":401683,"style":32604},[31845],[34,401685,401687,401690,401728,401731,401734,401737],{"className":401686},[31676],[34,401688],{"className":401689,"style":37755},[31680],[34,401691,401693,401696],{"className":401692},[31685],[34,401694,111],{"className":401695},[31685,31686],[34,401697,401699],{"className":401698},[31792],[34,401700,401702],{"className":401701},[31796],[34,401703,401705],{"className":401704},[31801],[34,401706,401708],{"className":401707,"style":37774},[31805],[34,401709,401710,401713],{"style":37777},[34,401711],{"className":401712,"style":31814},[31813],[34,401714,401716],{"className":401715},[31818,31819,31820,31821],[34,401717,401719,401722,401725],{"className":401718},[31685,31821],[34,401720,31660],{"className":401721},[31685,31686,31821],[34,401723,32226],{"className":401724},[32024,31821],[34,401726,640],{"className":401727},[31685,31821],[34,401729,2557],{"className":401730},[31690],[34,401732],{"className":401733,"style":32604},[31845],[34,401735,32226],{"className":401736},[32024],[34,401738],{"className":401739,"style":32604},[31845],[34,401741,401743,401746,401749,401752,401755,401758],{"className":401742},[31676],[34,401744],{"className":401745,"style":31774},[31680],[34,401747,456],{"className":401748},[31782],[34,401750,111],{"className":401751},[31685,31686],[34,401753],{"className":401754,"style":32604},[31845],[34,401756,223],{"className":401757},[32024],[34,401759],{"className":401760,"style":32604},[31845],[34,401762,401764,401767,401796,401799,401802],{"className":401763},[31676],[34,401765],{"className":401766,"style":40044},[31680],[34,401768,401770,401773],{"className":401769},[31685],[34,401771,111],{"className":401772},[31685,31686],[34,401774,401776],{"className":401775},[31792],[34,401777,401779],{"className":401778},[31796],[34,401780,401782],{"className":401781},[31801],[34,401783,401785],{"className":401784,"style":37774},[31805],[34,401786,401787,401790],{"style":37777},[34,401788],{"className":401789,"style":31814},[31813],[34,401791,401793],{"className":401792},[31818,31819,31820,31821],[34,401794,768],{"className":401795},[31685,31821],[34,401797],{"className":401798,"style":32604},[31845],[34,401800,223],{"className":401801},[32024],[34,401803],{"className":401804,"style":32604},[31845],[34,401806,401808,401811,401840,401843,401846],{"className":401807},[31676],[34,401809],{"className":401810,"style":40044},[31680],[34,401812,401814,401817],{"className":401813},[31685],[34,401815,111],{"className":401816},[31685,31686],[34,401818,401820],{"className":401819},[31792],[34,401821,401823],{"className":401822},[31796],[34,401824,401826],{"className":401825},[31801],[34,401827,401829],{"className":401828,"style":37774},[31805],[34,401830,401831,401834],{"style":37777},[34,401832],{"className":401833,"style":31814},[31813],[34,401835,401837],{"className":401836},[31818,31819,31820,31821],[34,401838,786],{"className":401839},[31685,31821],[34,401841],{"className":401842,"style":32604},[31845],[34,401844,223],{"className":401845},[32024],[34,401847],{"className":401848,"style":32604},[31845],[34,401850,401852,401855,401858,401861,401864],{"className":401851},[31676],[34,401853],{"className":401854,"style":32597},[31680],[34,401856,38874],{"className":401857},[34006],[34,401859],{"className":401860,"style":32604},[31845],[34,401862,223],{"className":401863},[32024],[34,401865],{"className":401866,"style":32604},[31845],[34,401868,401870,401873,401902],{"className":401869},[31676],[34,401871],{"className":401872,"style":31774},[31680],[34,401874,401876,401879],{"className":401875},[31685],[34,401877,111],{"className":401878},[31685,31686],[34,401880,401882],{"className":401881},[31792],[34,401883,401885],{"className":401884},[31796],[34,401886,401888],{"className":401887},[31801],[34,401889,401891],{"className":401890,"style":37959},[31805],[34,401892,401893,401896],{"style":37777},[34,401894],{"className":401895,"style":31814},[31813],[34,401897,401899],{"className":401898},[31818,31819,31820,31821],[34,401900,31660],{"className":401901},[31685,31686,31821],[34,401903,2557],{"className":401904},[31690],[15,401906,401907,401908,401936,401937,326252,401965,401936,402024,402082,402083,402111,402112,39984,402176],{},"仔细看，中间的项全抵消了！\n",[34,401909,401911,401924],{"className":401910},[31642],[34,401912,401914],{"className":401913},[31646],[31648,401915,401916],{"xmlns":31650},[31652,401917,401918,401922],{},[31655,401919,401920],{},[31658,401921,111],{},[31665,401923,111],{"encoding":31667},[34,401925,401927],{"className":401926,"ariaHidden":758},[31672],[34,401928,401930,401933],{"className":401929},[31676],[34,401931],{"className":401932,"style":32527},[31680],[34,401934,111],{"className":401935},[31685,31686]," 减 ",[34,401938,401940,401953],{"className":401939},[31642],[34,401941,401943],{"className":401942},[31646],[31648,401944,401945],{"xmlns":31650},[31652,401946,401947,401951],{},[31655,401948,401949],{},[31658,401950,111],{},[31665,401952,111],{"encoding":31667},[34,401954,401956],{"className":401955,"ariaHidden":758},[31672],[34,401957,401959,401962],{"className":401958},[31676],[34,401960],{"className":401961,"style":32527},[31680],[34,401963,111],{"className":401964},[31685,31686],[34,401966,401968,401986],{"className":401967},[31642],[34,401969,401971],{"className":401970},[31646],[31648,401972,401973],{"xmlns":31650},[31652,401974,401975,401983],{},[31655,401976,401977],{},[37624,401978,401979,401981],{},[31658,401980,111],{},[31749,401982,768],{},[31665,401984,401985],{"encoding":31667},"a^2",[34,401987,401989],{"className":401988,"ariaHidden":758},[31672],[34,401990,401992,401995],{"className":401991},[31676],[34,401993],{"className":401994,"style":37774},[31680],[34,401996,401998,402001],{"className":401997},[31685],[34,401999,111],{"className":402000},[31685,31686],[34,402002,402004],{"className":402003},[31792],[34,402005,402007],{"className":402006},[31796],[34,402008,402010],{"className":402009},[31801],[34,402011,402013],{"className":402012,"style":37774},[31805],[34,402014,402015,402018],{"style":37777},[34,402016],{"className":402017,"style":31814},[31813],[34,402019,402021],{"className":402020},[31818,31819,31820,31821],[34,402022,768],{"className":402023},[31685,31821],[34,402025,402027,402044],{"className":402026},[31642],[34,402028,402030],{"className":402029},[31646],[31648,402031,402032],{"xmlns":31650},[31652,402033,402034,402042],{},[31655,402035,402036],{},[37624,402037,402038,402040],{},[31658,402039,111],{},[31749,402041,768],{},[31665,402043,401985],{"encoding":31667},[34,402045,402047],{"className":402046,"ariaHidden":758},[31672],[34,402048,402050,402053],{"className":402049},[31676],[34,402051],{"className":402052,"style":37774},[31680],[34,402054,402056,402059],{"className":402055},[31685],[34,402057,111],{"className":402058},[31685,31686],[34,402060,402062],{"className":402061},[31792],[34,402063,402065],{"className":402064},[31796],[34,402066,402068],{"className":402067},[31801],[34,402069,402071],{"className":402070,"style":37774},[31805],[34,402072,402073,402076],{"style":37777},[34,402074],{"className":402075,"style":31814},[31813],[34,402077,402079],{"className":402078},[31818,31819,31820,31821],[34,402080,768],{"className":402081},[31685,31821],"……\n剩下的只有最前面的 ",[34,402084,402086,402099],{"className":402085},[31642],[34,402087,402089],{"className":402088},[31646],[31648,402090,402091],{"xmlns":31650},[31652,402092,402093,402097],{},[31655,402094,402095],{},[31749,402096,640],{},[31665,402098,640],{"encoding":31667},[34,402100,402102],{"className":402101,"ariaHidden":758},[31672],[34,402103,402105,402108],{"className":402104},[31676],[34,402106],{"className":402107,"style":32617},[31680],[34,402109,640],{"className":402110},[31685]," 和最后面的 ",[34,402113,402115,402135],{"className":402114},[31642],[34,402116,402118],{"className":402117},[31646],[31648,402119,402120],{"xmlns":31650},[31652,402121,402122,402132],{},[31655,402123,402124,402126],{},[31662,402125,32226],{},[37624,402127,402128,402130],{},[31658,402129,111],{},[31658,402131,31660],{},[31665,402133,402134],{"encoding":31667},"-a^n",[34,402136,402138],{"className":402137,"ariaHidden":758},[31672],[34,402139,402141,402144,402147],{"className":402140},[31676],[34,402142],{"className":402143,"style":37940},[31680],[34,402145,32226],{"className":402146},[31685],[34,402148,402150,402153],{"className":402149},[31685],[34,402151,111],{"className":402152},[31685,31686],[34,402154,402156],{"className":402155},[31792],[34,402157,402159],{"className":402158},[31796],[34,402160,402162],{"className":402161},[31801],[34,402163,402165],{"className":402164,"style":37959},[31805],[34,402166,402167,402170],{"style":37777},[34,402168],{"className":402169,"style":31814},[31813],[34,402171,402173],{"className":402172},[31818,31819,31820,31821],[34,402174,31660],{"className":402175},[31685,31686,31821],[34,402177,402179,402203],{"className":402178},[31642],[34,402180,402182],{"className":402181},[31646],[31648,402183,402184],{"xmlns":31650},[31652,402185,402186,402200],{},[31655,402187,402188,402190,402192,402194],{},[31662,402189,202],{},[31749,402191,640],{},[31662,402193,32226],{},[37624,402195,402196,402198],{},[31658,402197,111],{},[31658,402199,31660],{},[31665,402201,402202],{"encoding":31667},"= 1 - a^n",[34,402204,402206,402218,402236],{"className":402205,"ariaHidden":758},[31672],[34,402207,402209,402212,402215],{"className":402208},[31676],[34,402210],{"className":402211,"style":73089},[31680],[34,402213,202],{"className":402214},[31850],[34,402216],{"className":402217,"style":31846},[31845],[34,402219,402221,402224,402227,402230,402233],{"className":402220},[31676],[34,402222],{"className":402223,"style":35942},[31680],[34,402225,640],{"className":402226},[31685],[34,402228],{"className":402229,"style":32604},[31845],[34,402231,32226],{"className":402232},[32024],[34,402234],{"className":402235,"style":32604},[31845],[34,402237,402239,402242],{"className":402238},[31676],[34,402240],{"className":402241,"style":37959},[31680],[34,402243,402245,402248],{"className":402244},[31685],[34,402246,111],{"className":402247},[31685,31686],[34,402249,402251],{"className":402250},[31792],[34,402252,402254],{"className":402253},[31796],[34,402255,402257],{"className":402256},[31801],[34,402258,402260],{"className":402259,"style":37959},[31805],[34,402261,402262,402265],{"style":37777},[34,402263],{"className":402264,"style":31814},[31813],[34,402266,402268],{"className":402267},[31818,31819,31820,31821],[34,402269,31660],{"className":402270},[31685,31686,31821],[15,402272,402273,402276,402277,402357,402358,402445,402446,330585,402551,402601],{},[2627,402274,402275],{},"4. 结论","\n题目已知 ",[34,402278,402280,402301],{"className":402279},[31642],[34,402281,402283],{"className":402282},[31646],[31648,402284,402285],{"xmlns":31650},[31652,402286,402287,402299],{},[31655,402288,402289,402295,402297],{},[37624,402290,402291,402293],{},[31658,402292,111],{},[31658,402294,31660],{},[31662,402296,202],{},[31749,402298,459],{},[31665,402300,343092],{"encoding":31667},[34,402302,402304,402348],{"className":402303,"ariaHidden":758},[31672],[34,402305,402307,402310,402339,402342,402345],{"className":402306},[31676],[34,402308],{"className":402309,"style":37959},[31680],[34,402311,402313,402316],{"className":402312},[31685],[34,402314,111],{"className":402315},[31685,31686],[34,402317,402319],{"className":402318},[31792],[34,402320,402322],{"className":402321},[31796],[34,402323,402325],{"className":402324},[31801],[34,402326,402328],{"className":402327,"style":37959},[31805],[34,402329,402330,402333],{"style":37777},[34,402331],{"className":402332,"style":31814},[31813],[34,402334,402336],{"className":402335},[31818,31819,31820,31821],[34,402337,31660],{"className":402338},[31685,31686,31821],[34,402340],{"className":402341,"style":31846},[31845],[34,402343,202],{"className":402344},[31850],[34,402346],{"className":402347,"style":31846},[31845],[34,402349,402351,402354],{"className":402350},[31676],[34,402352],{"className":402353,"style":32617},[31680],[34,402355,459],{"className":402356},[31685],"。\n所以结果 ",[34,402359,402361,402385],{"className":402360},[31642],[34,402362,402364],{"className":402363},[31646],[31648,402365,402366],{"xmlns":31650},[31652,402367,402368,402382],{},[31655,402369,402370,402372,402374,402376,402378,402380],{},[31662,402371,202],{},[31749,402373,640],{},[31662,402375,32226],{},[31749,402377,459],{},[31662,402379,202],{},[31749,402381,640],{},[31665,402383,402384],{"encoding":31667},"= 1 - 0 = 1",[34,402386,402388,402400,402418,402436],{"className":402387,"ariaHidden":758},[31672],[34,402389,402391,402394,402397],{"className":402390},[31676],[34,402392],{"className":402393,"style":73089},[31680],[34,402395,202],{"className":402396},[31850],[34,402398],{"className":402399,"style":31846},[31845],[34,402401,402403,402406,402409,402412,402415],{"className":402402},[31676],[34,402404],{"className":402405,"style":35942},[31680],[34,402407,640],{"className":402408},[31685],[34,402410],{"className":402411,"style":32604},[31845],[34,402413,32226],{"className":402414},[32024],[34,402416],{"className":402417,"style":32604},[31845],[34,402419,402421,402424,402427,402430,402433],{"className":402420},[31676],[34,402422],{"className":402423,"style":32617},[31680],[34,402425,459],{"className":402426},[31685],[34,402428],{"className":402429,"style":31846},[31845],[34,402431,202],{"className":402432},[31850],[34,402434],{"className":402435,"style":31846},[31845],[34,402437,402439,402442],{"className":402438},[31676],[34,402440],{"className":402441,"style":32617},[31680],[34,402443,640],{"className":402444},[31685],"。\n既然 ",[34,402447,402449,402479],{"className":402448},[31642],[34,402450,402452],{"className":402451},[31646],[31648,402453,402454],{"xmlns":31650},[31652,402455,402456,402476],{},[31655,402457,402458,402460,402462,402464,402466,402468,402470,402472,402474],{},[31662,402459,456],{"stretchy":702},[31749,402461,640],{},[31662,402463,32226],{},[31658,402465,111],{},[31662,402467,2557],{"stretchy":702},[31662,402469,71191],{},[31658,402471,54224],{},[31662,402473,202],{},[31749,402475,640],{},[31665,402477,402478],{"encoding":31667},"(1-a) \\cdot B = 1",[34,402480,402482,402503,402524,402542],{"className":402481,"ariaHidden":758},[31672],[34,402483,402485,402488,402491,402494,402497,402500],{"className":402484},[31676],[34,402486],{"className":402487,"style":31774},[31680],[34,402489,456],{"className":402490},[31782],[34,402492,640],{"className":402493},[31685],[34,402495],{"className":402496,"style":32604},[31845],[34,402498,32226],{"className":402499},[32024],[34,402501],{"className":402502,"style":32604},[31845],[34,402504,402506,402509,402512,402515,402518,402521],{"className":402505},[31676],[34,402507],{"className":402508,"style":31774},[31680],[34,402510,111],{"className":402511},[31685,31686],[34,402513,2557],{"className":402514},[31690],[34,402516],{"className":402517,"style":32604},[31845],[34,402519,71191],{"className":402520},[32024],[34,402522],{"className":402523,"style":32604},[31845],[34,402525,402527,402530,402533,402536,402539],{"className":402526},[31676],[34,402528],{"className":402529,"style":33218},[31680],[34,402531,54224],{"className":402532,"style":54252},[31685,31686],[34,402534],{"className":402535,"style":31846},[31845],[34,402537,202],{"className":402538},[31850],[34,402540],{"className":402541,"style":31846},[31845],[34,402543,402545,402548],{"className":402544},[31676],[34,402546],{"className":402547,"style":32617},[31680],[34,402549,640],{"className":402550},[31685],[34,402552,402554,402571],{"className":402553},[31642],[34,402555,402557],{"className":402556},[31646],[31648,402558,402559],{"xmlns":31650},[31652,402560,402561,402569],{},[31655,402562,402563,402565,402567],{},[31749,402564,640],{},[31662,402566,32226],{},[31658,402568,111],{},[31665,402570,400490],{"encoding":31667},[34,402572,402574,402592],{"className":402573,"ariaHidden":758},[31672],[34,402575,402577,402580,402583,402586,402589],{"className":402576},[31676],[34,402578],{"className":402579,"style":35942},[31680],[34,402581,640],{"className":402582},[31685],[34,402584],{"className":402585,"style":32604},[31845],[34,402587,32226],{"className":402588},[32024],[34,402590],{"className":402591,"style":32604},[31845],[34,402593,402595,402598],{"className":402594},[31676],[34,402596],{"className":402597,"style":32527},[31680],[34,402599,111],{"className":402600},[31685,31686]," 就是可逆的，且逆元就是那个多项式。",{"title":29,"searchDepth":43,"depth":43,"links":402603},[402604,402605,402607,402609,402610,402611,402612,402613,402614,402615,402616,402617,402618],{"id":371609,"depth":49,"text":371609},{"id":371756,"depth":49,"text":402606},"第一部分：充分性证明 (⇐\\Leftarrow⇐)",{"id":373724,"depth":49,"text":402608},"第二部分：必要性证明 (⇒\\Rightarrow⇒)",{"id":375979,"depth":49,"text":375979},{"id":365086,"depth":49,"text":365086},{"id":377892,"depth":49,"text":377892},{"id":379393,"depth":49,"text":379393},{"id":382547,"depth":49,"text":382547},{"id":387499,"depth":49,"text":387499},{"id":390267,"depth":49,"text":390267},{"id":395610,"depth":49,"text":395610},{"id":397290,"depth":49,"text":397290},{"id":399880,"depth":49,"text":399880},{"order":49},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringhw",{"title":371604,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002FringHW","j-4JrnLGYQdmZU5HiDB6jTFlP5EiT-eW-0764ZIg-BM",{"id":402625,"title":402626,"body":402627,"description":402634,"extension":85,"meta":417965,"navigation":52,"path":417966,"seo":417967,"stem":417968,"__hash__":417969},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringdirectprod.md","环的直积与中国剩余定理",{"type":8,"value":402628,"toc":417959},[402629,402632,402635,402649,402651,402655,402658,402662,402749,404507,404511,404676,404845,404847,404851,404854,405350,405662,405666,405669,406111,406312,406317,407231,407325,407327,407331,407517,407522,415634,415636,415640,415701,417353,417355,417358],[11,402630,402631],{"id":402631},"环的直积直和与中国剩余定理",[15,402633,402634],{},"核心在于讲清楚两件事：",[2230,402636,402637,402643],{},[108,402638,402639,402642],{},[2627,402640,402641],{},"定义工具","：什么是直积和直和（把小环拼成大环）。",[108,402644,402645,402648],{},[2627,402646,402647],{},"核心定理","：中国剩余定理（CRT）—— 如何把一个大环的问题，拆解成几个小环的问题来解，并且证明它们是等价的。",[32188,402650],{},[253,402652,402654],{"id":402653},"第一阶段弄清楚概念直积与直和","第一阶段：弄清楚概念（直积与直和）",[15,402656,402657],{},"在进入复杂的证明前，我们必须先看懂符号。",[32490,402659,402661],{"id":402660},"_1-定义-341直积-direct-product","1. 定义 3.4.1：直积 (Direct Product)",[15,402663,402664,2664,402666,402745,402748],{},[2627,402665,126612],{},[34,402667,402669,402690],{"className":402668},[31642],[34,402670,402672],{"className":402671},[31646],[31648,402673,402674],{"xmlns":31650},[31652,402675,402676,402687],{},[31655,402677,402678,402681],{},[31662,402679,402680],{},"∏",[31743,402682,402683,402685],{},[31658,402684,34386],{},[31658,402686,31955],{},[31665,402688,402689],{"encoding":31667},"\\prod R_i",[34,402691,402693],{"className":402692,"ariaHidden":758},[31672],[34,402694,402696,402699,402702,402705],{"className":402695},[31676],[34,402697],{"className":402698,"style":31774},[31680],[34,402700,402680],{"className":402701,"style":156561},[32767,131249,156560],[34,402703],{"className":402704,"style":32123},[31845],[34,402706,402708,402711],{"className":402707},[31685],[34,402709,34386],{"className":402710,"style":34427},[31685,31686],[34,402712,402714],{"className":402713},[31792],[34,402715,402717,402737],{"className":402716},[31796,31797],[34,402718,402720,402734],{"className":402719},[31801],[34,402721,402723],{"className":402722,"style":32090},[31805],[34,402724,402725,402728],{"style":104694},[34,402726],{"className":402727,"style":31814},[31813],[34,402729,402731],{"className":402730},[31818,31819,31820,31821],[34,402732,31955],{"className":402733},[31685,31686,31821],[34,402735,31829],{"className":402736},[31828],[34,402738,402740],{"className":402739},[31801],[34,402741,402743],{"className":402742,"style":31836},[31805],[34,402744],{},[2627,402746,402747],{},"这是什么","：这是一个“大集合”，里面的元素是由很多小元素组成的“长列表”（数组）。",[105,402750,402751,403012,403499],{},[108,402752,402753,402754,402824,402825,31901],{},"假设你有一族环 ",[34,402755,402757,402775],{"className":402756},[31642],[34,402758,402760],{"className":402759},[31646],[31648,402761,402762],{"xmlns":31650},[31652,402763,402764,402772],{},[31655,402765,402766],{},[31743,402767,402768,402770],{},[31658,402769,34386],{},[31658,402771,31955],{},[31665,402773,402774],{"encoding":31667},"R_i",[34,402776,402778],{"className":402777,"ariaHidden":758},[31672],[34,402779,402781,402784],{"className":402780},[31676],[34,402782],{"className":402783,"style":70915},[31680],[34,402785,402787,402790],{"className":402786},[31685],[34,402788,34386],{"className":402789,"style":34427},[31685,31686],[34,402791,402793],{"className":402792},[31792],[34,402794,402796,402816],{"className":402795},[31796,31797],[34,402797,402799,402813],{"className":402798},[31801],[34,402800,402802],{"className":402801,"style":32090},[31805],[34,402803,402804,402807],{"style":104694},[34,402805],{"className":402806,"style":31814},[31813],[34,402808,402810],{"className":402809},[31818,31819,31820,31821],[34,402811,31955],{"className":402812},[31685,31686,31821],[34,402814,31829],{"className":402815},[31828],[34,402817,402819],{"className":402818},[31801],[34,402820,402822],{"className":402821,"style":31836},[31805],[34,402823],{},"（比如 ",[34,402826,402828,402868],{"className":402827},[31642],[34,402829,402831],{"className":402830},[31646],[31648,402832,402833],{"xmlns":31650},[31652,402834,402835,402865],{},[31655,402836,402837,402843,402845,402851,402853,402859,402861,402863],{},[31743,402838,402839,402841],{},[31658,402840,34386],{},[31749,402842,640],{},[31662,402844,31960],{"separator":758},[31743,402846,402847,402849],{},[31658,402848,34386],{},[31749,402850,768],{},[31662,402852,31960],{"separator":758},[31743,402854,402855,402857],{},[31658,402856,34386],{},[31749,402858,786],{},[31658,402860,2547],{"mathvariant":38079},[31658,402862,2547],{"mathvariant":38079},[31658,402864,2547],{"mathvariant":38079},[31665,402866,402867],{"encoding":31667},"R_1, R_2, R_3...",[34,402869,402871],{"className":402870,"ariaHidden":758},[31672],[34,402872,402874,402877,402917,402920,402923,402963,402966,402969,403009],{"className":402873},[31676],[34,402875],{"className":402876,"style":70739},[31680],[34,402878,402880,402883],{"className":402879},[31685],[34,402881,34386],{"className":402882,"style":34427},[31685,31686],[34,402884,402886],{"className":402885},[31792],[34,402887,402889,402909],{"className":402888},[31796,31797],[34,402890,402892,402906],{"className":402891},[31801],[34,402893,402895],{"className":402894,"style":31806},[31805],[34,402896,402897,402900],{"style":104694},[34,402898],{"className":402899,"style":31814},[31813],[34,402901,402903],{"className":402902},[31818,31819,31820,31821],[34,402904,640],{"className":402905},[31685,31821],[34,402907,31829],{"className":402908},[31828],[34,402910,402912],{"className":402911},[31801],[34,402913,402915],{"className":402914,"style":31836},[31805],[34,402916],{},[34,402918,31960],{"className":402919},[32119],[34,402921],{"className":402922,"style":32123},[31845],[34,402924,402926,402929],{"className":402925},[31685],[34,402927,34386],{"className":402928,"style":34427},[31685,31686],[34,402930,402932],{"className":402931},[31792],[34,402933,402935,402955],{"className":402934},[31796,31797],[34,402936,402938,402952],{"className":402937},[31801],[34,402939,402941],{"className":402940,"style":31806},[31805],[34,402942,402943,402946],{"style":104694},[34,402944],{"className":402945,"style":31814},[31813],[34,402947,402949],{"className":402948},[31818,31819,31820,31821],[34,402950,768],{"className":402951},[31685,31821],[34,402953,31829],{"className":402954},[31828],[34,402956,402958],{"className":402957},[31801],[34,402959,402961],{"className":402960,"style":31836},[31805],[34,402962],{},[34,402964,31960],{"className":402965},[32119],[34,402967],{"className":402968,"style":32123},[31845],[34,402970,402972,402975],{"className":402971},[31685],[34,402973,34386],{"className":402974,"style":34427},[31685,31686],[34,402976,402978],{"className":402977},[31792],[34,402979,402981,403001],{"className":402980},[31796,31797],[34,402982,402984,402998],{"className":402983},[31801],[34,402985,402987],{"className":402986,"style":31806},[31805],[34,402988,402989,402992],{"style":104694},[34,402990],{"className":402991,"style":31814},[31813],[34,402993,402995],{"className":402994},[31818,31819,31820,31821],[34,402996,786],{"className":402997},[31685,31821],[34,402999,31829],{"className":403000},[31828],[34,403002,403004],{"className":403003},[31801],[34,403005,403007],{"className":403006,"style":31836},[31805],[34,403008],{},[34,403010,1167],{"className":403011},[31685],[108,403013,403014,403015,70884,403221,361860,403290,326252,403359,361860,403429,403498],{},"直积里的一个元素长这样：",[34,403016,403018,403062],{"className":403017},[31642],[34,403019,403021],{"className":403020},[31646],[31648,403022,403023],{"xmlns":31650},[31652,403024,403025,403059],{},[31655,403026,403027,403029,403035,403037,403043,403045,403051,403053,403055,403057],{},[31662,403028,456],{"stretchy":702},[31743,403030,403031,403033],{},[31658,403032,111],{},[31749,403034,640],{},[31662,403036,31960],{"separator":758},[31743,403038,403039,403041],{},[31658,403040,111],{},[31749,403042,768],{},[31662,403044,31960],{"separator":758},[31743,403046,403047,403049],{},[31658,403048,111],{},[31749,403050,786],{},[31662,403052,31960],{"separator":758},[31662,403054,33934],{},[39361,403056,39363],{},[31662,403058,2557],{"stretchy":702},[31665,403060,403061],{"encoding":31667},"(a_1, a_2, a_3, \\dots)",[34,403063,403065],{"className":403064,"ariaHidden":758},[31672],[34,403066,403068,403071,403074,403114,403117,403120,403160,403163,403166,403206,403209,403212,403215,403218],{"className":403067},[31676],[34,403069],{"className":403070,"style":31774},[31680],[34,403072,456],{"className":403073},[31782],[34,403075,403077,403080],{"className":403076},[31685],[34,403078,111],{"className":403079},[31685,31686],[34,403081,403083],{"className":403082},[31792],[34,403084,403086,403106],{"className":403085},[31796,31797],[34,403087,403089,403103],{"className":403088},[31801],[34,403090,403092],{"className":403091,"style":31806},[31805],[34,403093,403094,403097],{"style":31809},[34,403095],{"className":403096,"style":31814},[31813],[34,403098,403100],{"className":403099},[31818,31819,31820,31821],[34,403101,640],{"className":403102},[31685,31821],[34,403104,31829],{"className":403105},[31828],[34,403107,403109],{"className":403108},[31801],[34,403110,403112],{"className":403111,"style":31836},[31805],[34,403113],{},[34,403115,31960],{"className":403116},[32119],[34,403118],{"className":403119,"style":32123},[31845],[34,403121,403123,403126],{"className":403122},[31685],[34,403124,111],{"className":403125},[31685,31686],[34,403127,403129],{"className":403128},[31792],[34,403130,403132,403152],{"className":403131},[31796,31797],[34,403133,403135,403149],{"className":403134},[31801],[34,403136,403138],{"className":403137,"style":31806},[31805],[34,403139,403140,403143],{"style":31809},[34,403141],{"className":403142,"style":31814},[31813],[34,403144,403146],{"className":403145},[31818,31819,31820,31821],[34,403147,768],{"className":403148},[31685,31821],[34,403150,31829],{"className":403151},[31828],[34,403153,403155],{"className":403154},[31801],[34,403156,403158],{"className":403157,"style":31836},[31805],[34,403159],{},[34,403161,31960],{"className":403162},[32119],[34,403164],{"className":403165,"style":32123},[31845],[34,403167,403169,403172],{"className":403168},[31685],[34,403170,111],{"className":403171},[31685,31686],[34,403173,403175],{"className":403174},[31792],[34,403176,403178,403198],{"className":403177},[31796,31797],[34,403179,403181,403195],{"className":403180},[31801],[34,403182,403184],{"className":403183,"style":31806},[31805],[34,403185,403186,403189],{"style":31809},[34,403187],{"className":403188,"style":31814},[31813],[34,403190,403192],{"className":403191},[31818,31819,31820,31821],[34,403193,786],{"className":403194},[31685,31821],[34,403196,31829],{"className":403197},[31828],[34,403199,403201],{"className":403200},[31801],[34,403202,403204],{"className":403203,"style":31836},[31805],[34,403205],{},[34,403207,31960],{"className":403208},[32119],[34,403210],{"className":403211,"style":32123},[31845],[34,403213,33934],{"className":403214},[34006],[34,403216],{"className":403217,"style":32123},[31845],[34,403219,2557],{"className":403220},[31690],[34,403222,403224,403241],{"className":403223},[31642],[34,403225,403227],{"className":403226},[31646],[31648,403228,403229],{"xmlns":31650},[31652,403230,403231,403239],{},[31655,403232,403233],{},[31743,403234,403235,403237],{},[31658,403236,111],{},[31749,403238,640],{},[31665,403240,361309],{"encoding":31667},[34,403242,403244],{"className":403243,"ariaHidden":758},[31672],[34,403245,403247,403250],{"className":403246},[31676],[34,403248],{"className":403249,"style":31860},[31680],[34,403251,403253,403256],{"className":403252},[31685],[34,403254,111],{"className":403255},[31685,31686],[34,403257,403259],{"className":403258},[31792],[34,403260,403262,403282],{"className":403261},[31796,31797],[34,403263,403265,403279],{"className":403264},[31801],[34,403266,403268],{"className":403267,"style":31806},[31805],[34,403269,403270,403273],{"style":31809},[34,403271],{"className":403272,"style":31814},[31813],[34,403274,403276],{"className":403275},[31818,31819,31820,31821],[34,403277,640],{"className":403278},[31685,31821],[34,403280,31829],{"className":403281},[31828],[34,403283,403285],{"className":403284},[31801],[34,403286,403288],{"className":403287,"style":31836},[31805],[34,403289],{},[34,403291,403293,403310],{"className":403292},[31642],[34,403294,403296],{"className":403295},[31646],[31648,403297,403298],{"xmlns":31650},[31652,403299,403300,403308],{},[31655,403301,403302],{},[31743,403303,403304,403306],{},[31658,403305,34386],{},[31749,403307,640],{},[31665,403309,107510],{"encoding":31667},[34,403311,403313],{"className":403312,"ariaHidden":758},[31672],[34,403314,403316,403319],{"className":403315},[31676],[34,403317],{"className":403318,"style":70915},[31680],[34,403320,403322,403325],{"className":403321},[31685],[34,403323,34386],{"className":403324,"style":34427},[31685,31686],[34,403326,403328],{"className":403327},[31792],[34,403329,403331,403351],{"className":403330},[31796,31797],[34,403332,403334,403348],{"className":403333},[31801],[34,403335,403337],{"className":403336,"style":31806},[31805],[34,403338,403339,403342],{"style":104694},[34,403340],{"className":403341,"style":31814},[31813],[34,403343,403345],{"className":403344},[31818,31819,31820,31821],[34,403346,640],{"className":403347},[31685,31821],[34,403349,31829],{"className":403350},[31828],[34,403352,403354],{"className":403353},[31801],[34,403355,403357],{"className":403356,"style":31836},[31805],[34,403358],{},[34,403360,403362,403380],{"className":403361},[31642],[34,403363,403365],{"className":403364},[31646],[31648,403366,403367],{"xmlns":31650},[31652,403368,403369,403377],{},[31655,403370,403371],{},[31743,403372,403373,403375],{},[31658,403374,111],{},[31749,403376,768],{},[31665,403378,403379],{"encoding":31667},"a_2",[34,403381,403383],{"className":403382,"ariaHidden":758},[31672],[34,403384,403386,403389],{"className":403385},[31676],[34,403387],{"className":403388,"style":31860},[31680],[34,403390,403392,403395],{"className":403391},[31685],[34,403393,111],{"className":403394},[31685,31686],[34,403396,403398],{"className":403397},[31792],[34,403399,403401,403421],{"className":403400},[31796,31797],[34,403402,403404,403418],{"className":403403},[31801],[34,403405,403407],{"className":403406,"style":31806},[31805],[34,403408,403409,403412],{"style":31809},[34,403410],{"className":403411,"style":31814},[31813],[34,403413,403415],{"className":403414},[31818,31819,31820,31821],[34,403416,768],{"className":403417},[31685,31821],[34,403419,31829],{"className":403420},[31828],[34,403422,403424],{"className":403423},[31801],[34,403425,403427],{"className":403426,"style":31836},[31805],[34,403428],{},[34,403430,403432,403449],{"className":403431},[31642],[34,403433,403435],{"className":403434},[31646],[31648,403436,403437],{"xmlns":31650},[31652,403438,403439,403447],{},[31655,403440,403441],{},[31743,403442,403443,403445],{},[31658,403444,34386],{},[31749,403446,768],{},[31665,403448,107707],{"encoding":31667},[34,403450,403452],{"className":403451,"ariaHidden":758},[31672],[34,403453,403455,403458],{"className":403454},[31676],[34,403456],{"className":403457,"style":70915},[31680],[34,403459,403461,403464],{"className":403460},[31685],[34,403462,34386],{"className":403463,"style":34427},[31685,31686],[34,403465,403467],{"className":403466},[31792],[34,403468,403470,403490],{"className":403469},[31796,31797],[34,403471,403473,403487],{"className":403472},[31801],[34,403474,403476],{"className":403475,"style":31806},[31805],[34,403477,403478,403481],{"style":104694},[34,403479],{"className":403480,"style":31814},[31813],[34,403482,403484],{"className":403483},[31818,31819,31820,31821],[34,403485,768],{"className":403486},[31685,31821],[34,403488,31829],{"className":403489},[31828],[34,403491,403493],{"className":403492},[31801],[34,403494,403496],{"className":403495,"style":31836},[31805],[34,403497],{},"，以此类推。",[108,403500,403501,403504,403505],{},[2627,403502,403503],{},"运算规则","：就是“各算各的”。\n",[105,403506,403507,404024],{},[108,403508,403509,403510],{},"加法：",[34,403511,403513,403599],{"className":403512},[31642],[34,403514,403516],{"className":403515},[31646],[31648,403517,403518],{"xmlns":31650},[31652,403519,403520,403596],{},[31655,403521,403522,403524,403530,403532,403538,403540,403542,403544,403550,403552,403558,403560,403562,403564,403570,403572,403578,403580,403586,403588,403594],{},[31662,403523,456],{"stretchy":702},[31743,403525,403526,403528],{},[31658,403527,111],{},[31749,403529,640],{},[31662,403531,31960],{"separator":758},[31743,403533,403534,403536],{},[31658,403535,111],{},[31749,403537,768],{},[31662,403539,2557],{"stretchy":702},[31662,403541,223],{},[31662,403543,456],{"stretchy":702},[31743,403545,403546,403548],{},[31658,403547,1697],{},[31749,403549,640],{},[31662,403551,31960],{"separator":758},[31743,403553,403554,403556],{},[31658,403555,1697],{},[31749,403557,768],{},[31662,403559,2557],{"stretchy":702},[31662,403561,202],{},[31662,403563,456],{"stretchy":702},[31743,403565,403566,403568],{},[31658,403567,111],{},[31749,403569,640],{},[31662,403571,223],{},[31743,403573,403574,403576],{},[31658,403575,1697],{},[31749,403577,640],{},[31662,403579,31960],{"separator":758},[31743,403581,403582,403584],{},[31658,403583,111],{},[31749,403585,768],{},[31662,403587,223],{},[31743,403589,403590,403592],{},[31658,403591,1697],{},[31749,403593,768],{},[31662,403595,2557],{"stretchy":702},[31665,403597,403598],{"encoding":31667},"(a_1, a_2) + (b_1, b_2) = (a_1+b_1, a_2+b_2)",[34,403600,403602,403709,403816,403874,403975],{"className":403601,"ariaHidden":758},[31672],[34,403603,403605,403608,403611,403651,403654,403657,403697,403700,403703,403706],{"className":403604},[31676],[34,403606],{"className":403607,"style":31774},[31680],[34,403609,456],{"className":403610},[31782],[34,403612,403614,403617],{"className":403613},[31685],[34,403615,111],{"className":403616},[31685,31686],[34,403618,403620],{"className":403619},[31792],[34,403621,403623,403643],{"className":403622},[31796,31797],[34,403624,403626,403640],{"className":403625},[31801],[34,403627,403629],{"className":403628,"style":31806},[31805],[34,403630,403631,403634],{"style":31809},[34,403632],{"className":403633,"style":31814},[31813],[34,403635,403637],{"className":403636},[31818,31819,31820,31821],[34,403638,640],{"className":403639},[31685,31821],[34,403641,31829],{"className":403642},[31828],[34,403644,403646],{"className":403645},[31801],[34,403647,403649],{"className":403648,"style":31836},[31805],[34,403650],{},[34,403652,31960],{"className":403653},[32119],[34,403655],{"className":403656,"style":32123},[31845],[34,403658,403660,403663],{"className":403659},[31685],[34,403661,111],{"className":403662},[31685,31686],[34,403664,403666],{"className":403665},[31792],[34,403667,403669,403689],{"className":403668},[31796,31797],[34,403670,403672,403686],{"className":403671},[31801],[34,403673,403675],{"className":403674,"style":31806},[31805],[34,403676,403677,403680],{"style":31809},[34,403678],{"className":403679,"style":31814},[31813],[34,403681,403683],{"className":403682},[31818,31819,31820,31821],[34,403684,768],{"className":403685},[31685,31821],[34,403687,31829],{"className":403688},[31828],[34,403690,403692],{"className":403691},[31801],[34,403693,403695],{"className":403694,"style":31836},[31805],[34,403696],{},[34,403698,2557],{"className":403699},[31690],[34,403701],{"className":403702,"style":32604},[31845],[34,403704,223],{"className":403705},[32024],[34,403707],{"className":403708,"style":32604},[31845],[34,403710,403712,403715,403718,403758,403761,403764,403804,403807,403810,403813],{"className":403711},[31676],[34,403713],{"className":403714,"style":31774},[31680],[34,403716,456],{"className":403717},[31782],[34,403719,403721,403724],{"className":403720},[31685],[34,403722,1697],{"className":403723},[31685,31686],[34,403725,403727],{"className":403726},[31792],[34,403728,403730,403750],{"className":403729},[31796,31797],[34,403731,403733,403747],{"className":403732},[31801],[34,403734,403736],{"className":403735,"style":31806},[31805],[34,403737,403738,403741],{"style":31809},[34,403739],{"className":403740,"style":31814},[31813],[34,403742,403744],{"className":403743},[31818,31819,31820,31821],[34,403745,640],{"className":403746},[31685,31821],[34,403748,31829],{"className":403749},[31828],[34,403751,403753],{"className":403752},[31801],[34,403754,403756],{"className":403755,"style":31836},[31805],[34,403757],{},[34,403759,31960],{"className":403760},[32119],[34,403762],{"className":403763,"style":32123},[31845],[34,403765,403767,403770],{"className":403766},[31685],[34,403768,1697],{"className":403769},[31685,31686],[34,403771,403773],{"className":403772},[31792],[34,403774,403776,403796],{"className":403775},[31796,31797],[34,403777,403779,403793],{"className":403778},[31801],[34,403780,403782],{"className":403781,"style":31806},[31805],[34,403783,403784,403787],{"style":31809},[34,403785],{"className":403786,"style":31814},[31813],[34,403788,403790],{"className":403789},[31818,31819,31820,31821],[34,403791,768],{"className":403792},[31685,31821],[34,403794,31829],{"className":403795},[31828],[34,403797,403799],{"className":403798},[31801],[34,403800,403802],{"className":403801,"style":31836},[31805],[34,403803],{},[34,403805,2557],{"className":403806},[31690],[34,403808],{"className":403809,"style":31846},[31845],[34,403811,202],{"className":403812},[31850],[34,403814],{"className":403815,"style":31846},[31845],[34,403817,403819,403822,403825,403865,403868,403871],{"className":403818},[31676],[34,403820],{"className":403821,"style":31774},[31680],[34,403823,456],{"className":403824},[31782],[34,403826,403828,403831],{"className":403827},[31685],[34,403829,111],{"className":403830},[31685,31686],[34,403832,403834],{"className":403833},[31792],[34,403835,403837,403857],{"className":403836},[31796,31797],[34,403838,403840,403854],{"className":403839},[31801],[34,403841,403843],{"className":403842,"style":31806},[31805],[34,403844,403845,403848],{"style":31809},[34,403846],{"className":403847,"style":31814},[31813],[34,403849,403851],{"className":403850},[31818,31819,31820,31821],[34,403852,640],{"className":403853},[31685,31821],[34,403855,31829],{"className":403856},[31828],[34,403858,403860],{"className":403859},[31801],[34,403861,403863],{"className":403862,"style":31836},[31805],[34,403864],{},[34,403866],{"className":403867,"style":32604},[31845],[34,403869,223],{"className":403870},[32024],[34,403872],{"className":403873,"style":32604},[31845],[34,403875,403877,403880,403920,403923,403926,403966,403969,403972],{"className":403876},[31676],[34,403878],{"className":403879,"style":43217},[31680],[34,403881,403883,403886],{"className":403882},[31685],[34,403884,1697],{"className":403885},[31685,31686],[34,403887,403889],{"className":403888},[31792],[34,403890,403892,403912],{"className":403891},[31796,31797],[34,403893,403895,403909],{"className":403894},[31801],[34,403896,403898],{"className":403897,"style":31806},[31805],[34,403899,403900,403903],{"style":31809},[34,403901],{"className":403902,"style":31814},[31813],[34,403904,403906],{"className":403905},[31818,31819,31820,31821],[34,403907,640],{"className":403908},[31685,31821],[34,403910,31829],{"className":403911},[31828],[34,403913,403915],{"className":403914},[31801],[34,403916,403918],{"className":403917,"style":31836},[31805],[34,403919],{},[34,403921,31960],{"className":403922},[32119],[34,403924],{"className":403925,"style":32123},[31845],[34,403927,403929,403932],{"className":403928},[31685],[34,403930,111],{"className":403931},[31685,31686],[34,403933,403935],{"className":403934},[31792],[34,403936,403938,403958],{"className":403937},[31796,31797],[34,403939,403941,403955],{"className":403940},[31801],[34,403942,403944],{"className":403943,"style":31806},[31805],[34,403945,403946,403949],{"style":31809},[34,403947],{"className":403948,"style":31814},[31813],[34,403950,403952],{"className":403951},[31818,31819,31820,31821],[34,403953,768],{"className":403954},[31685,31821],[34,403956,31829],{"className":403957},[31828],[34,403959,403961],{"className":403960},[31801],[34,403962,403964],{"className":403963,"style":31836},[31805],[34,403965],{},[34,403967],{"className":403968,"style":32604},[31845],[34,403970,223],{"className":403971},[32024],[34,403973],{"className":403974,"style":32604},[31845],[34,403976,403978,403981,404021],{"className":403977},[31676],[34,403979],{"className":403980,"style":31774},[31680],[34,403982,403984,403987],{"className":403983},[31685],[34,403985,1697],{"className":403986},[31685,31686],[34,403988,403990],{"className":403989},[31792],[34,403991,403993,404013],{"className":403992},[31796,31797],[34,403994,403996,404010],{"className":403995},[31801],[34,403997,403999],{"className":403998,"style":31806},[31805],[34,404000,404001,404004],{"style":31809},[34,404002],{"className":404003,"style":31814},[31813],[34,404005,404007],{"className":404006},[31818,31819,31820,31821],[34,404008,768],{"className":404009},[31685,31821],[34,404011,31829],{"className":404012},[31828],[34,404014,404016],{"className":404015},[31801],[34,404017,404019],{"className":404018,"style":31836},[31805],[34,404020],{},[34,404022,2557],{"className":404023},[31690],[108,404025,404026,404027],{},"乘法：",[34,404028,404030,404112],{"className":404029},[31642],[34,404031,404033],{"className":404032},[31646],[31648,404034,404035],{"xmlns":31650},[31652,404036,404037,404109],{},[31655,404038,404039,404041,404047,404049,404055,404057,404059,404061,404067,404069,404075,404077,404079,404081,404087,404093,404095,404101,404107],{},[31662,404040,456],{"stretchy":702},[31743,404042,404043,404045],{},[31658,404044,111],{},[31749,404046,640],{},[31662,404048,31960],{"separator":758},[31743,404050,404051,404053],{},[31658,404052,111],{},[31749,404054,768],{},[31662,404056,2557],{"stretchy":702},[31662,404058,71191],{},[31662,404060,456],{"stretchy":702},[31743,404062,404063,404065],{},[31658,404064,1697],{},[31749,404066,640],{},[31662,404068,31960],{"separator":758},[31743,404070,404071,404073],{},[31658,404072,1697],{},[31749,404074,768],{},[31662,404076,2557],{"stretchy":702},[31662,404078,202],{},[31662,404080,456],{"stretchy":702},[31743,404082,404083,404085],{},[31658,404084,111],{},[31749,404086,640],{},[31743,404088,404089,404091],{},[31658,404090,1697],{},[31749,404092,640],{},[31662,404094,31960],{"separator":758},[31743,404096,404097,404099],{},[31658,404098,111],{},[31749,404100,768],{},[31743,404102,404103,404105],{},[31658,404104,1697],{},[31749,404106,768],{},[31662,404108,2557],{"stretchy":702},[31665,404110,404111],{"encoding":31667},"(a_1, a_2) \\cdot (b_1, b_2) = (a_1 b_1, a_2 b_2)",[34,404113,404115,404222,404329],{"className":404114,"ariaHidden":758},[31672],[34,404116,404118,404121,404124,404164,404167,404170,404210,404213,404216,404219],{"className":404117},[31676],[34,404119],{"className":404120,"style":31774},[31680],[34,404122,456],{"className":404123},[31782],[34,404125,404127,404130],{"className":404126},[31685],[34,404128,111],{"className":404129},[31685,31686],[34,404131,404133],{"className":404132},[31792],[34,404134,404136,404156],{"className":404135},[31796,31797],[34,404137,404139,404153],{"className":404138},[31801],[34,404140,404142],{"className":404141,"style":31806},[31805],[34,404143,404144,404147],{"style":31809},[34,404145],{"className":404146,"style":31814},[31813],[34,404148,404150],{"className":404149},[31818,31819,31820,31821],[34,404151,640],{"className":404152},[31685,31821],[34,404154,31829],{"className":404155},[31828],[34,404157,404159],{"className":404158},[31801],[34,404160,404162],{"className":404161,"style":31836},[31805],[34,404163],{},[34,404165,31960],{"className":404166},[32119],[34,404168],{"className":404169,"style":32123},[31845],[34,404171,404173,404176],{"className":404172},[31685],[34,404174,111],{"className":404175},[31685,31686],[34,404177,404179],{"className":404178},[31792],[34,404180,404182,404202],{"className":404181},[31796,31797],[34,404183,404185,404199],{"className":404184},[31801],[34,404186,404188],{"className":404187,"style":31806},[31805],[34,404189,404190,404193],{"style":31809},[34,404191],{"className":404192,"style":31814},[31813],[34,404194,404196],{"className":404195},[31818,31819,31820,31821],[34,404197,768],{"className":404198},[31685,31821],[34,404200,31829],{"className":404201},[31828],[34,404203,404205],{"className":404204},[31801],[34,404206,404208],{"className":404207,"style":31836},[31805],[34,404209],{},[34,404211,2557],{"className":404212},[31690],[34,404214],{"className":404215,"style":32604},[31845],[34,404217,71191],{"className":404218},[32024],[34,404220],{"className":404221,"style":32604},[31845],[34,404223,404225,404228,404231,404271,404274,404277,404317,404320,404323,404326],{"className":404224},[31676],[34,404226],{"className":404227,"style":31774},[31680],[34,404229,456],{"className":404230},[31782],[34,404232,404234,404237],{"className":404233},[31685],[34,404235,1697],{"className":404236},[31685,31686],[34,404238,404240],{"className":404239},[31792],[34,404241,404243,404263],{"className":404242},[31796,31797],[34,404244,404246,404260],{"className":404245},[31801],[34,404247,404249],{"className":404248,"style":31806},[31805],[34,404250,404251,404254],{"style":31809},[34,404252],{"className":404253,"style":31814},[31813],[34,404255,404257],{"className":404256},[31818,31819,31820,31821],[34,404258,640],{"className":404259},[31685,31821],[34,404261,31829],{"className":404262},[31828],[34,404264,404266],{"className":404265},[31801],[34,404267,404269],{"className":404268,"style":31836},[31805],[34,404270],{},[34,404272,31960],{"className":404273},[32119],[34,404275],{"className":404276,"style":32123},[31845],[34,404278,404280,404283],{"className":404279},[31685],[34,404281,1697],{"className":404282},[31685,31686],[34,404284,404286],{"className":404285},[31792],[34,404287,404289,404309],{"className":404288},[31796,31797],[34,404290,404292,404306],{"className":404291},[31801],[34,404293,404295],{"className":404294,"style":31806},[31805],[34,404296,404297,404300],{"style":31809},[34,404298],{"className":404299,"style":31814},[31813],[34,404301,404303],{"className":404302},[31818,31819,31820,31821],[34,404304,768],{"className":404305},[31685,31821],[34,404307,31829],{"className":404308},[31828],[34,404310,404312],{"className":404311},[31801],[34,404313,404315],{"className":404314,"style":31836},[31805],[34,404316],{},[34,404318,2557],{"className":404319},[31690],[34,404321],{"className":404322,"style":31846},[31845],[34,404324,202],{"className":404325},[31850],[34,404327],{"className":404328,"style":31846},[31845],[34,404330,404332,404335,404338,404378,404418,404421,404424,404464,404504],{"className":404331},[31676],[34,404333],{"className":404334,"style":31774},[31680],[34,404336,456],{"className":404337},[31782],[34,404339,404341,404344],{"className":404340},[31685],[34,404342,111],{"className":404343},[31685,31686],[34,404345,404347],{"className":404346},[31792],[34,404348,404350,404370],{"className":404349},[31796,31797],[34,404351,404353,404367],{"className":404352},[31801],[34,404354,404356],{"className":404355,"style":31806},[31805],[34,404357,404358,404361],{"style":31809},[34,404359],{"className":404360,"style":31814},[31813],[34,404362,404364],{"className":404363},[31818,31819,31820,31821],[34,404365,640],{"className":404366},[31685,31821],[34,404368,31829],{"className":404369},[31828],[34,404371,404373],{"className":404372},[31801],[34,404374,404376],{"className":404375,"style":31836},[31805],[34,404377],{},[34,404379,404381,404384],{"className":404380},[31685],[34,404382,1697],{"className":4043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定义 3.4.1(ii)：直和 (Direct Sum)",[15,404512,404513,2664,404515,4374,404594,404673,2664],{},[2627,404514,126612],{},[34,404516,404518,404539],{"className":404517},[31642],[34,404519,404521],{"className":404520},[31646],[31648,404522,404523],{"xmlns":31650},[31652,404524,404525,404536],{},[31655,404526,404527,404530],{},[31662,404528,404529],{},"∐",[31743,404531,404532,404534],{},[31658,404533,34386],{},[31658,404535,31955],{},[31665,404537,404538],{"encoding":31667},"\\coprod R_i",[34,404540,404542],{"className":404541,"ariaHidden":758},[31672],[34,404543,404545,404548,404551,404554],{"className":404544},[31676],[34,404546],{"className":404547,"style":31774},[31680],[34,404549,404529],{"className":404550,"style":156561},[32767,131249,156560],[34,404552],{"className":404553,"style":32123},[31845],[34,404555,404557,404560],{"className":404556},[31685],[34,404558,34386],{"className":404559,"style":34427},[31685,31686],[34,404561,404563],{"className":404562},[31792],[34,404564,404566,404586],{"className":404565},[31796,31797],[34,404567,404569,404583],{"className":404568},[31801],[34,404570,404572],{"className":404571,"style":32090},[31805],[34,404573,404574,404577],{"style":104694},[34,404575],{"className":404576,"style":31814},[31813],[34,404578,404580],{"className":404579},[31818,31819,31820,31821],[34,404581,31955],{"className":404582},[31685,31686,31821],[34,404584,31829],{"className":404585},[31828],[34,404587,404589],{"className":404588},[31801],[34,404590,404592],{"className":404591,"style":31836},[31805],[34,404593],{},[34,404595,404597,404618],{"className":404596},[31642],[34,404598,404600],{"className":404599},[31646],[31648,404601,404602],{"xmlns":31650},[31652,404603,404604,404615],{},[31655,404605,404606,404609],{},[31662,404607,404608],{},"⨁",[31743,404610,404611,404613],{},[31658,404612,34386],{},[31658,404614,31955],{},[31665,404616,404617],{"encoding":31667},"\\bigoplus R_i",[34,404619,404621],{"className":404620,"ariaHidden":758},[31672],[34,404622,404624,404627,404630,404633],{"className":404623},[31676],[34,404625],{"className":404626,"style":31774},[31680],[34,404628,404608],{"className":404629,"style":156561},[32767,131249,156560],[34,404631],{"className":404632,"style":32123},[31845],[34,404634,404636,404639],{"className":404635},[31685],[34,404637,34386],{"className":404638,"style":34427},[31685,31686],[34,404640,404642],{"className":404641},[31792],[34,404643,404645,404665],{"className":404644},[31796,31797],[34,404646,404648,404662],{"className":404647},[31801],[34,404649,404651],{"className":404650,"style":32090},[31805],[34,404652,404653,404656],{"style":104694},[34,404654],{"className":404655,"style":31814},[31813],[34,404657,404659],{"className":404658},[31818,31819,31820,31821],[34,404660,31955],{"className":404661},[31685,31686,31821],[34,404663,31829],{"className":404664},[31828],[34,404666,404668],{"className":404667},[31801],[34,404669,404671],{"className":404670,"style":31836},[31805],[34,404672],{},[2627,404674,404675],{},"它和直积的区别",[105,404677,404678,404684,404694],{},[108,404679,404680,404683],{},[2627,404681,404682],{},"直积","：列表是无限长的，每个位置都可以是非零数。",[108,404685,404686,404689,404690,404693],{},[2627,404687,404688],{},"直和","：列表虽然可能无限长，但要求",[2627,404691,404692],{},"只有有限个位置是非零的","，剩下的全是 0。",[108,404695,404696,404699,404700,404840,404841,404844],{},[2627,404697,404698],{},"特别注意","：如果只有有限个环（比如 ",[34,404701,404703,404733],{"className":404702},[31642],[34,404704,404706],{"className":404705},[31646],[31648,404707,404708],{"xmlns":31650},[31652,404709,404710,404730],{},[31655,404711,404712,404718,404720,404722,404724],{},[31743,404713,404714,404716],{},[31658,404715,34386],{},[31749,404717,640],{},[31662,404719,31960],{"separator":758},[31662,404721,33934],{},[31662,404723,31960],{"separator":758},[31743,404725,404726,404728],{},[31658,404727,34386],{},[31658,404729,31660],{},[31665,404731,404732],{"encoding":31667},"R_1, \\dots, R_n",[34,404734,404736],{"className":404735,"ariaHidden":758},[31672],[34,404737,404739,404742,404782,404785,404788,404791,404794,404797,404800],{"className":404738},[31676],[34,404740],{"className":404741,"style":70739},[31680],[34,404743,404745,404748],{"className":404744},[31685],[34,404746,34386],{"className":404747,"style":34427},[31685,31686],[34,404749,404751],{"className":404750},[31792],[34,404752,404754,404774],{"className":404753},[31796,31797],[34,404755,404757,404771],{"className":404756},[31801],[34,404758,404760],{"className":404759,"style":31806},[31805],[34,404761,404762,404765],{"style":104694},[34,404763],{"className":404764,"style":31814},[31813],[34,404766,404768],{"className":404767},[31818,31819,31820,31821],[34,404769,640],{"className":404770},[31685,31821],[34,404772,31829],{"className":404773},[31828],[34,404775,404777],{"className":404776},[31801],[34,404778,404780],{"className":404779,"style":31836},[31805],[34,404781],{},[34,404783,31960],{"className":404784},[32119],[34,404786],{"className":404787,"style":32123},[31845],[34,404789,33934],{"className":404790},[34006],[34,404792],{"className":404793,"style":32123},[31845],[34,404795,31960],{"className":404796},[32119],[34,404798],{"className":404799,"style":32123},[31845],[34,404801,404803,404806],{"className":404802},[31685],[34,404804,34386],{"className":404805,"style":34427},[31685,31686],[34,404807,404809],{"className":404808},[31792],[34,404810,404812,404832],{"className":404811},[31796,31797],[34,404813,404815,404829],{"className":404814},[31801],[34,404816,404818],{"className":404817,"style":32362},[31805],[34,404819,404820,404823],{"style":104694},[34,404821],{"className":404822,"style":31814},[31813],[34,404824,404826],{"className":404825},[31818,31819,31820,31821],[34,404827,31660],{"className":404828},[31685,31686,31821],[34,404830,31829],{"className":404831},[31828],[34,404833,404835],{"className":404834},[31801],[34,404836,404838],{"className":404837,"style":31836},[31805],[34,404839],{},"），那么",[2627,404842,404843],{},"直积 = 直和","。这就是为什么后面的定理里，这两个符号经常混用，因为我们处理的通常是有限个理想的情况。",[32188,404846],{},[253,404848,404850],{"id":404849},"第二阶段重头戏中国剩余定理证明详解","第二阶段：重头戏——中国剩余定理（证明详解）",[15,404852,404853],{},"这是最难的部分。",[15,404855,404856,362914,404859,330820,404999,405027,405028,405031,405032],{},[2627,404857,404858],{},"定理 3.4.2 (中国剩余定理 CRT)",[34,404860,404862,404892],{"className":404861},[31642],[34,404863,404865],{"className":404864},[31646],[31648,404866,404867],{"xmlns":31650},[31652,404868,404869,404889],{},[31655,404870,404871,404877,404879,404881,404883],{},[31743,404872,404873,404875],{},[31658,404874,106063],{},[31749,404876,640],{},[31662,404878,31960],{"separator":758},[31662,404880,33934],{},[31662,404882,31960],{"separator":758},[31743,404884,404885,404887],{},[31658,404886,106063],{},[31658,404888,31660],{},[31665,404890,404891],{"encoding":31667},"I_1, \\dots, I_n",[34,404893,404895],{"className":404894,"ariaHidden":758},[31672],[34,404896,404898,404901,404941,404944,404947,404950,404953,404956,404959],{"className":404897},[31676],[34,404899],{"className":404900,"style":70739},[31680],[34,404902,404904,404907],{"className":404903},[31685],[34,404905,106063],{"className":404906,"style":67482},[31685,31686],[34,404908,404910],{"className":404909},[31792],[34,404911,404913,404933],{"className":404912},[31796,31797],[34,404914,404916,404930],{"className":404915},[31801],[34,404917,404919],{"className":404918,"style":31806},[31805],[34,404920,404921,404924],{"style":217742},[34,404922],{"className":404923,"style":31814},[31813],[34,404925,404927],{"className":404926},[31818,31819,31820,31821],[34,404928,640],{"className":404929},[31685,31821],[34,404931,31829],{"className":404932},[31828],[34,404934,404936],{"className":404935},[31801],[34,404937,404939],{"className":404938,"style":31836},[31805],[34,404940],{},[34,404942,31960],{"className":404943},[32119],[34,404945],{"className":404946,"style":32123},[31845],[34,404948,33934],{"className":404949},[34006],[34,404951],{"className":404952,"style":32123},[31845],[34,404954,31960],{"className":404955},[32119],[34,404957],{"className":404958,"style":32123},[31845],[34,404960,404962,404965],{"className":404961},[31685],[34,404963,106063],{"className":404964,"style":67482},[31685,31686],[34,404966,404968],{"className":404967},[31792],[34,404969,404971,404991],{"className":404970},[31796,31797],[34,404972,404974,404988],{"className":404973},[31801],[34,404975,404977],{"className":404976,"style":32362},[31805],[34,404978,404979,404982],{"style":217742},[34,404980],{"className":404981,"style":31814},[31813],[34,404983,404985],{"className":404984},[31818,31819,31820,31821],[34,404986,31660],{"className":404987},[31685,31686,31821],[34,404989,31829],{"className":404990},[31828],[34,404992,404994],{"className":404993},[31801],[34,404995,404997],{"className":404996,"style":31836},[31805],[34,404998],{},[34,405000,405002,405015],{"className":405001},[31642],[34,405003,405005],{"className":405004},[31646],[31648,405006,405007],{"xmlns":31650},[31652,405008,405009,405013],{},[31655,405010,405011],{},[31658,405012,34386],{},[31665,405014,34386],{"encoding":31667},[34,405016,405018],{"className":405017,"ariaHidden":758},[31672],[34,405019,405021,405024],{"className":405020},[31676],[34,405022],{"className":405023,"style":33218},[31680],[34,405025,34386],{"className":405026,"style":34427},[31685,31686]," 的理想，且",[2627,405029,405030],{},"两两互素","。则：\n",[34,405033,405035,405102],{"className":405034},[31642],[34,405036,405038],{"className":405037},[31646],[31648,405039,405040],{"xmlns":31650},[31652,405041,405042,405099],{},[31655,405043,405044,405046,405048,405050,405065,405071,405073,405075,405089,405091,405093],{},[31658,405045,34386],{},[31658,405047,38080],{"mathvariant":38079},[31662,405049,456],{"stretchy":702},[38846,405051,405052,405055,405063],{},[31662,405053,405054],{},"⋂",[31655,405056,405057,405059,405061],{},[31658,405058,33759],{},[31662,405060,202],{},[31749,405062,640],{},[31658,405064,31660],{},[31743,405066,405067,405069],{},[31658,405068,106063],{},[31658,405070,33759],{},[31662,405072,2557],{"stretchy":702},[31662,405074,134826],{},[38846,405076,405077,405079,405087],{},[31662,405078,404608],{},[31655,405080,405081,405083,405085],{},[31658,405082,33759],{},[31662,405084,202],{},[31749,405086,640],{},[31658,405088,31660],{},[31658,405090,34386],{},[31658,405092,38080],{"mathvariant":38079},[31743,405094,405095,405097],{},[31658,405096,106063],{},[31658,405098,33759],{},[31665,405100,405101],{"encoding":31667},"R \u002F (\\bigcap_{k=1}^n I_k) \\cong \\bigoplus_{k=1}^n R\u002FI_k",[34,405103,405105,405235],{"className":405104,"ariaHidden":758},[31672],[34,405106,405108,405111,405114,405117,405120,405180,405183,405223,405226,405229,405232],{"className":405107},[31676],[34,405109],{"className":405110,"style":351974},[31680],[34,405112,34386],{"className":405113,"style":34427},[31685,31686],[34,405115,38080],{"className":405116},[31685],[34,405118,456],{"className":405119},[31782],[34,405121,405123,405126],{"className":405122},[32767],[34,405124,405054],{"className":405125,"style":156561},[32767,131249,156560],[34,405127,405129],{"className":405128},[31792],[34,405130,405132,405172],{"className":405131},[31796,31797],[34,405133,405135,405169],{"className":405134},[31801],[34,405136,405138,405158],{"className":405137,"style":351993},[31805],[34,405139,405140,405143],{"style":156577},[34,405141],{"className":405142,"style":31814},[31813],[34,405144,405146],{"className":405145},[31818,31819,31820,31821],[34,405147,405149,405152,405155],{"className":405148},[31685,31821],[34,405150,33759],{"className":405151,"style":33774},[31685,31686,31821],[34,405153,202],{"className":405154},[31850,31821],[34,405156,640],{"className":405157},[31685,31821],[34,405159,405160,405163],{"style":352016},[34,405161],{"className":405162,"style":31814},[31813],[34,405164,405166],{"className":405165},[31818,31819,31820,31821],[34,405167,31660],{"className":405168},[31685,31686,31821],[34,405170,31829],{"className":405171},[31828],[34,405173,405175],{"className":405174},[31801],[34,405176,405178],{"className":405177,"style":352035},[31805],[34,405179],{},[34,405181],{"className":405182,"style":32123},[31845],[34,405184,405186,405189],{"className":405185},[31685],[34,405187,106063],{"className":405188,"style":67482},[31685,31686],[34,405190,405192],{"className":405191},[31792],[34,405193,405195,405215],{"className":405194},[31796,31797],[34,405196,405198,405212],{"className":405197},[31801],[34,405199,405201],{"className":405200,"style":33833},[31805],[34,405202,405203,405206],{"style":217742},[34,405204],{"className":405205,"style":31814},[31813],[34,405207,405209],{"className":405208},[31818,31819,31820,31821],[34,405210,33759],{"className":405211,"style":33774},[31685,31686,31821],[34,405213,31829],{"className":405214},[31828],[34,405216,405218],{"className":405217},[31801],[34,405219,405221],{"className":405220,"style":31836},[31805],[34,405222],{},[34,405224,2557],{"className":405225},[31690],[34,405227],{"className":405228,"style":31846},[31845],[34,405230,134826],{"className":405231},[31850],[34,405233],{"className":405234,"style":31846},[31845],[34,405236,405238,405241,405301,405304,405307,405310],{"className":405237},[31676],[34,405239],{"className":405240,"style":351974},[31680],[34,405242,405244,405247],{"className":405243},[32767],[34,405245,404608],{"className":405246,"style":156561},[32767,131249,156560],[34,405248,405250],{"className":405249},[31792],[34,405251,405253,405293],{"className":405252},[31796,31797],[34,405254,405256,405290],{"className":405255},[31801],[34,405257,405259,405279],{"className":405258,"style":351993},[31805],[34,405260,405261,405264],{"style":156577},[34,405262],{"className":405263,"style":31814},[31813],[34,405265,405267],{"className":405266},[31818,31819,31820,31821],[34,405268,405270,405273,405276],{"className":405269},[31685,31821],[34,405271,33759],{"className":405272,"style":33774},[31685,31686,31821],[34,405274,202],{"className":405275},[31850,31821],[34,405277,640],{"className":405278},[31685,31821],[34,405280,405281,405284],{"style":352016},[34,405282],{"className":405283,"style":31814},[31813],[34,405285,405287],{"className":405286},[31818,31819,31820,31821],[34,405288,31660],{"className":405289},[31685,31686,31821],[34,405291,31829],{"className":405292},[31828],[34,405294,405296],{"className":405295},[31801],[34,405297,405299],{"className":405298,"style":352035},[31805],[34,405300],{},[34,405302],{"className":405303,"style":32123},[31845],[34,405305,34386],{"className":405306,"style":34427},[31685,31686],[34,405308,38080],{"className":405309},[31685],[34,405311,405313,405316],{"className":405312},[31685],[34,405314,106063],{"className":405315,"style":67482},[31685,31686],[34,405317,405319],{"className":405318},[31792],[34,405320,405322,405342],{"className":405321},[31796,31797],[34,405323,405325,405339],{"className":405324},[31801],[34,405326,405328],{"className":405327,"style":33833},[31805],[34,405329,405330,405333],{"style":217742},[34,405331],{"className":405332,"style":31814},[31813],[34,405334,405336],{"className":405335},[31818,31819,31820,31821],[34,405337,33759],{"className":405338,"style":33774},[31685,31686,31821],[34,405340,31829],{"className":405341},[31828],[34,405343,405345],{"className":405344},[31801],[34,405346,405348],{"className":405347,"style":31836},[31805],[34,405349],{},[15,405351,405352,405355,405356,258,405384,405412,405413,405486,405487,258,405537,335149,405588,405661],{},[2627,405353,405354],{},"前提知识补充：什么是“两两互素”？","\n在环论里，理想 ",[34,405357,405359,405372],{"className":405358},[31642],[34,405360,405362],{"className":405361},[31646],[31648,405363,405364],{"xmlns":31650},[31652,405365,405366,405370],{},[31655,405367,405368],{},[31658,405369,106063],{},[31665,405371,106063],{"encoding":31667},[34,405373,405375],{"className":405374,"ariaHidden":758},[31672],[34,405376,405378,405381],{"className":405377},[31676],[34,405379],{"className":405380,"style":33218},[31680],[34,405382,106063],{"className":405383,"style":67482},[31685,31686],[34,405385,405387,405400],{"className":405386},[31642],[34,405388,405390],{"className":405389},[31646],[31648,405391,405392],{"xmlns":31650},[31652,405393,405394,405398],{},[31655,405395,405396],{},[31658,405397,223320],{},[31665,405399,223320],{"encoding":31667},[34,405401,405403],{"className":405402,"ariaHidden":758},[31672],[34,405404,405406,405409],{"className":405405},[31676],[34,405407],{"className":405408,"style":33218},[31680],[34,405410,223320],{"className":405411,"style":223361},[31685,31686]," 互素意味着 ",[34,405414,405416,405438],{"className":405415},[31642],[34,405417,405419],{"className":405418},[31646],[31648,405420,405421],{"xmlns":31650},[31652,405422,405423,405435],{},[31655,405424,405425,405427,405429,405431,405433],{},[31658,405426,106063],{},[31662,405428,223],{},[31658,405430,223320],{},[31662,405432,202],{},[31658,405434,34386],{},[31665,405436,405437],{"encoding":31667},"I + J = R",[34,405439,405441,405459,405477],{"className":405440,"ariaHidden":758},[31672],[34,405442,405444,405447,405450,405453,405456],{"className":405443},[31676],[34,405445],{"className":405446,"style":34944},[31680],[34,405448,106063],{"className":405449,"style":67482},[31685,31686],[34,405451],{"className":405452,"style":32604},[31845],[34,405454,223],{"className":405455},[32024],[34,405457],{"className":405458,"style":32604},[31845],[34,405460,405462,405465,405468,405471,405474],{"className":405461},[31676],[34,405463],{"className":405464,"style":33218},[31680],[34,405466,223320],{"className":405467,"style":223361},[31685,31686],[34,405469],{"className":405470,"style":31846},[31845],[34,405472,202],{"className":405473},[31850],[34,405475],{"className":405476,"style":31846},[31845],[34,405478,405480,405483],{"className":405479},[31676],[34,405481],{"className":405482,"style":33218},[31680],[34,405484,34386],{"className":405485,"style":34427},[31685,31686],"。\n这意味着，我们可以找到 ",[34,405488,405490,405507],{"className":405489},[31642],[34,405491,405493],{"className":405492},[31646],[31648,405494,405495],{"xmlns":31650},[31652,405496,405497,405505],{},[31655,405498,405499,405501,405503],{},[31658,405500,32936],{},[31662,405502,112296],{},[31658,405504,106063],{},[31665,405506,352414],{"encoding":31667},[34,405508,405510,405528],{"className":405509,"ariaHidden":758},[31672],[34,405511,405513,405516,405519,405522,405525],{"className":405512},[31676],[34,405514],{"className":405515,"style":34812},[31680],[34,405517,32936],{"className":405518},[31685,31686],[34,405520],{"className":405521,"style":31846},[31845],[34,405523,112296],{"className":405524},[31850],[34,405526],{"className":405527,"style":31846},[31845],[34,405529,405531,405534],{"className":405530},[31676],[34,405532],{"className":405533,"style":33218},[31680],[34,405535,106063],{"className":405536,"style":67482},[31685,31686],[34,405538,405540,405558],{"className":405539},[31642],[34,405541,405543],{"className":405542},[31646],[31648,405544,405545],{"xmlns":31650},[31652,405546,405547,405555],{},[31655,405548,405549,405551,405553],{},[31658,405550,67764],{},[31662,405552,112296],{},[31658,405554,223320],{},[31665,405556,405557],{"encoding":31667},"y \\in J",[34,405559,405561,405579],{"className":405560,"ariaHidden":758},[31672],[34,405562,405564,405567,405570,405573,405576],{"className":405563},[31676],[34,405565],{"className":405566,"style":86652},[31680],[34,405568,67764],{"className":405569,"style":32062},[31685,31686],[34,405571],{"className":405572,"style":31846},[31845],[34,405574,112296],{"className":405575},[31850],[34,405577],{"className":405578,"style":31846},[31845],[34,405580,405582,405585],{"className":405581},[31676],[34,405583],{"className":405584,"style":33218},[31680],[34,405586,223320],{"className":405587,"style":223361},[31685,31686],[34,405589,405591,405613],{"className":405590},[31642],[34,405592,405594],{"className":405593},[31646],[31648,405595,405596],{"xmlns":31650},[31652,405597,405598,405610],{},[31655,405599,405600,405602,405604,405606,405608],{},[31658,405601,32936],{},[31662,405603,223],{},[31658,405605,67764],{},[31662,405607,202],{},[31749,405609,640],{},[31665,405611,405612],{"encoding":31667},"x + y = 1",[34,405614,405616,405634,405652],{"className":405615,"ariaHidden":758},[31672],[34,405617,405619,405622,405625,405628,405631],{"className":405618},[31676],[34,405620],{"className":405621,"style":32597},[31680],[34,405623,32936],{"className":405624},[31685,31686],[34,405626],{"className":405627,"style":32604},[31845],[34,405629,223],{"className":405630},[32024],[34,405632],{"className":405633,"style":32604},[31845],[34,405635,405637,405640,405643,405646,405649],{"className":405636},[31676],[34,405638],{"className":405639,"style":38774},[31680],[34,405641,67764],{"className":405642,"style":32062},[31685,31686],[34,405644],{"className":405645,"style":31846},[31845],[34,405647,202],{"className":405648},[31850],[34,405650],{"className":405651,"style":31846},[31845],[34,405653,405655,405658],{"className":405654},[31676],[34,405656],{"className":405657,"style":32617},[31680],[34,405659,640],{"className":405660},[31685],"。这一点在后面的证明中至关重要（用来制造“1”）。",[32490,405663,405665],{"id":405664},"证明第一步构造映射","证明第一步：构造映射",[15,405667,405668],{},"我们想证明左边（大环模交集）和右边（小环直和）是一样的（同构），首先得架一座桥。",[15,405670,405671,405672,405804],{},"定义映射 ",[34,405673,405675,405707],{"className":405674},[31642],[34,405676,405678],{"className":405677},[31646],[31648,405679,405680],{"xmlns":31650},[31652,405681,405682,405704],{},[31655,405683,405684,405686,405688,405690,405692,405694,405696,405698],{},[31658,405685,329201],{},[31662,405687,5598],{},[31658,405689,34386],{},[31662,405691,33239],{},[31662,405693,404608],{},[31658,405695,34386],{},[31658,405697,38080],{"mathvariant":38079},[31743,405699,405700,405702],{},[31658,405701,106063],{},[31658,405703,33759],{},[31665,405705,405706],{"encoding":31667},"\\phi: R \\to \\bigoplus R\u002FI_k",[34,405708,405710,405728,405746],{"className":405709,"ariaHidden":758},[31672],[34,405711,405713,405716,405719,405722,405725],{"className":405712},[31676],[34,405714],{"className":405715,"style":43217},[31680],[34,405717,329201],{"className":405718},[31685,31686],[34,405720],{"className":405721,"style":31846},[31845],[34,405723,5598],{"className":405724},[31850],[34,405726],{"className":405727,"style":31846},[31845],[34,405729,405731,405734,405737,405740,405743],{"className":405730},[31676],[34,405732],{"className":405733,"style":33218},[31680],[34,405735,34386],{"className":405736,"style":34427},[31685,31686],[34,405738],{"className":405739,"style":31846},[31845],[34,405741,33239],{"className":405742},[31850],[34,405744],{"className":405745,"style":31846},[31845],[34,405747,405749,405752,405755,405758,405761,405764],{"className":405748},[31676],[34,405750],{"className":405751,"style":31774},[31680],[34,405753,404608],{"className":405754,"style":156561},[32767,131249,156560],[34,405756],{"className":405757,"style":32123},[31845],[34,405759,34386],{"className":405760,"style":34427},[31685,31686],[34,405762,38080],{"className":405763},[31685],[34,405765,405767,405770],{"className":405766},[31685],[34,405768,106063],{"className":405769,"style":67482},[31685,31686],[34,405771,405773],{"className":405772},[31792],[34,405774,405776,405796],{"className":405775},[31796,31797],[34,405777,405779,405793],{"className":405778},[31801],[34,405780,405782],{"className":405781,"style":33833},[31805],[34,405783,405784,405787],{"style":217742},[34,405785],{"className":405786,"style":31814},[31813],[34,405788,405790],{"className":405789},[31818,31819,31820,31821],[34,405791,33759],{"className":405792,"style":33774},[31685,31686,31821],[34,405794,31829],{"className":405795},[31828],[34,405797,405799],{"className":405798},[31801],[34,405800,405802],{"className":405801,"style":31836},[31805],[34,405803],{},[34,405805,405807,405871],{"className":405806},[31642],[34,405808,405810],{"className":405809},[31646],[31648,405811,405812],{"xmlns":31650},[31652,405813,405814,405868],{},[31655,405815,405816,405818,405820,405822,405824,405826,405828,405830,405832,405838,405840,405842,405844,405850,405852,405854,405856,405858,405860,405866],{},[31658,405817,329201],{},[31662,405819,456],{"stretchy":702},[31658,405821,111],{},[31662,405823,2557],{"stretchy":702},[31662,405825,202],{},[31662,405827,456],{"stretchy":702},[31658,405829,111],{},[31662,405831,223],{},[31743,405833,405834,405836],{},[31658,405835,106063],{},[31749,405837,640],{},[31662,405839,31960],{"separator":758},[31658,405841,111],{},[31662,405843,223],{},[31743,405845,405846,405848],{},[31658,405847,106063],{},[31749,405849,768],{},[31662,405851,31960],{"separator":758},[31662,405853,33934],{},[31662,405855,31960],{"separator":758},[31658,405857,111],{},[31662,405859,223],{},[31743,405861,405862,405864],{},[31658,405863,106063],{},[31658,405865,31660],{},[31662,405867,2557],{"stretchy":702},[31665,405869,405870],{"encoding":31667},"\\phi(a) = (a + I_1, a + I_2, \\dots, a + I_n)",[34,405872,405874,405901,405922,405986,406062],{"className":405873,"ariaHidden":758},[31672],[34,405875,405877,405880,405883,405886,405889,405892,405895,405898],{"className":405876},[31676],[34,405878],{"className":405879,"style":31774},[31680],[34,405881,329201],{"className":405882},[31685,31686],[34,405884,456],{"className":405885},[31782],[34,405887,111],{"className":405888},[31685,31686],[34,405890,2557],{"className":405891},[31690],[34,405893],{"className":405894,"style":31846},[31845],[34,405896,202],{"className":405897},[31850],[34,405899],{"className":405900,"style":31846},[31845],[34,405902,405904,405907,405910,405913,405916,405919],{"className":405903},[31676],[34,405905],{"className":405906,"style":31774},[31680],[34,405908,456],{"className":405909},[31782],[34,405911,111],{"className":405912},[31685,31686],[34,405914],{"className":405915,"style":32604},[31845],[34,405917,223],{"className":405918},[32024],[34,405920],{"className":405921,"style":32604},[31845],[34,405923,405925,405928,405968,405971,405974,405977,405980,405983],{"className":405924},[31676],[34,405926],{"className":405927,"style":70739},[31680],[34,405929,405931,405934],{"className":405930},[31685],[34,405932,106063],{"className":405933,"style":67482},[31685,31686],[34,405935,405937],{"className":405936},[31792],[34,405938,405940,405960],{"className":405939},[31796,31797],[34,405941,405943,405957],{"className":405942},[31801],[34,405944,405946],{"className":405945,"style":31806},[31805],[34,405947,405948,405951],{"style":217742},[34,405949],{"className":405950,"style":31814},[31813],[34,405952,405954],{"className":405953},[31818,31819,31820,31821],[34,405955,640],{"className":405956},[31685,31821],[34,405958,31829],{"className":405959},[31828],[34,405961,405963],{"className":405962},[31801],[34,405964,405966],{"className":405965,"style":31836},[31805],[34,405967],{},[34,405969,31960],{"className":405970},[32119],[34,405972],{"className":405973,"style":32123},[31845],[34,405975,111],{"className":405976},[31685,31686],[34,405978],{"className":405979,"style":32604},[31845],[34,405981,223],{"className":405982},[32024],[34,405984],{"className":405985,"style":32604},[31845],[34,405987,405989,405992,406032,406035,406038,406041,406044,406047,406050,406053,406056,406059],{"className":405988},[31676],[34,405990],{"className":405991,"style":70739},[31680],[34,405993,405995,405998],{"className":405994},[31685],[34,405996,106063],{"className":405997,"style":67482},[31685,31686],[34,405999,406001],{"className":406000},[31792],[34,406002,406004,406024],{"className":406003},[31796,31797],[34,406005,406007,406021],{"className":406006},[31801],[34,406008,406010],{"className":406009,"style":31806},[31805],[34,406011,406012,406015],{"style":217742},[34,406013],{"className":406014,"style":31814},[31813],[34,406016,406018],{"className":406017},[31818,31819,31820,31821],[34,406019,768],{"className":406020},[31685,31821],[34,406022,31829],{"className":406023},[31828],[34,406025,406027],{"className":406026},[31801],[34,406028,406030],{"className":406029,"style":31836},[31805],[34,406031],{},[34,406033,31960],{"className":406034},[32119],[34,406036],{"className":406037,"style":32123},[31845],[34,406039,33934],{"className":406040},[34006],[34,406042],{"className":406043,"style":32123},[31845],[34,406045,31960],{"className":406046},[32119],[34,406048],{"className":406049,"style":32123},[31845],[34,406051,111],{"className":406052},[31685,31686],[34,406054],{"className":406055,"style":32604},[31845],[34,406057,223],{"className":406058},[32024],[34,406060],{"className":406061,"style":32604},[31845],[34,406063,406065,406068,406108],{"className":406064},[31676],[34,406066],{"className":406067,"style":31774},[31680],[34,406069,406071,406074],{"className":406070},[31685],[34,406072,106063],{"className":406073,"style":67482},[31685,31686],[34,406075,406077],{"className":406076},[31792],[34,406078,406080,406100],{"className":406079},[31796,31797],[34,406081,406083,406097],{"className":406082},[31801],[34,406084,406086],{"className":406085,"style":32362},[31805],[34,406087,406088,406091],{"style":217742},[34,406089],{"className":406090,"style":31814},[31813],[34,406092,406094],{"className":406093},[31818,31819,31820,31821],[34,406095,31660],{"className":406096},[31685,31686,31821],[34,406098,31829],{"className":406099},[31828],[34,406101,406103],{"className":406102},[31801],[34,406104,406106],{"className":406105,"style":31836},[31805],[34,406107],{},[34,406109,2557],{"className":406110},[31690],[105,406112,406113,406309],{},[108,406114,406115,406117,406118,406146,406147,406175,406176,406308],{},[2627,406116,233260],{},"：把 ",[34,406119,406121,406134],{"className":406120},[31642],[34,406122,406124],{"className":406123},[31646],[31648,406125,406126],{"xmlns":31650},[31652,406127,406128,406132],{},[31655,406129,406130],{},[31658,406131,34386],{},[31665,406133,34386],{"encoding":31667},[34,406135,406137],{"className":406136,"ariaHidden":758},[31672],[34,406138,406140,406143],{"className":406139},[31676],[34,406141],{"className":406142,"style":33218},[31680],[34,406144,34386],{"className":406145,"style":34427},[31685,31686]," 里的一个元素 ",[34,406148,406150,406163],{"className":406149},[31642],[34,406151,406153],{"className":406152},[31646],[31648,406154,406155],{"xmlns":31650},[31652,406156,406157,406161],{},[31655,406158,406159],{},[31658,406160,111],{},[31665,406162,111],{"encoding":31667},[34,406164,406166],{"className":406165,"ariaHidden":758},[31672],[34,406167,406169,406172],{"className":406168},[31676],[34,406170],{"className":406171,"style":32527},[31680],[34,406173,111],{"className":406174},[31685,31686],"，分别拿到 ",[34,406177,406179,406207],{"className":406178},[31642],[34,406180,406182],{"className":406181},[31646],[31648,406183,406184],{"xmlns":31650},[31652,406185,406186,406204],{},[31655,406187,406188,406194,406196,406202],{},[31743,406189,406190,406192],{},[31658,406191,106063],{},[31749,406193,640],{},[31662,406195,31960],{"separator":758},[31743,406197,406198,406200],{},[31658,406199,106063],{},[31749,406201,768],{},[31662,406203,33934],{},[31665,406205,406206],{"encoding":31667},"I_1, I_2 \\dots",[34,406208,406210],{"className":406209,"ariaHidden":758},[31672],[34,406211,406213,406216,406256,406259,406262,406302,406305],{"className":406212},[31676],[34,406214],{"className":406215,"style":70739},[31680],[34,406217,406219,406222],{"className":406218},[31685],[34,406220,106063],{"className":406221,"style":67482},[31685,31686],[34,406223,406225],{"className":406224},[31792],[34,406226,406228,406248],{"className":406227},[31796,31797],[34,406229,406231,406245],{"className":406230},[31801],[34,406232,406234],{"className":406233,"style":31806},[31805],[34,406235,406236,406239],{"style":217742},[34,406237],{"className":406238,"style":31814},[31813],[34,406240,406242],{"className":406241},[31818,31819,31820,31821],[34,406243,640],{"className":406244},[31685,31821],[34,406246,31829],{"className":406247},[31828],[34,406249,406251],{"className":406250},[31801],[34,406252,406254],{"className":406253,"style":31836},[31805],[34,406255],{},[34,406257,31960],{"className":406258},[32119],[34,406260],{"className":406261,"style":32123},[31845],[34,406263,406265,406268],{"className":406264},[31685],[34,406266,106063],{"className":406267,"style":67482},[31685,31686],[34,406269,406271],{"className":406270},[31792],[34,406272,406274,406294],{"className":406273},[31796,31797],[34,406275,406277,406291],{"className":406276},[31801],[34,406278,406280],{"className":406279,"style":31806},[31805],[34,406281,406282,406285],{"style":217742},[34,406283],{"className":406284,"style":31814},[31813],[34,406286,406288],{"className":406287},[31818,31819,31820,31821],[34,406289,768],{"className":406290},[31685,31821],[34,406292,31829],{"className":406293},[31828],[34,406295,406297],{"className":406296},[31801],[34,406298,406300],{"className":406299,"style":31836},[31805],[34,406301],{},[34,406303],{"className":406304,"style":32123},[31845],[34,406306,33934],{"className":406307},[34006]," 里去取余数，组成一个列表。",[108,406310,406311],{},"显然这是一个同态（保持加法和乘法）。",[15,406313,406314],{},[2627,406315,406316],{},"我们要找它的核 (Kernel)：",[105,406318,406319,406390,406682,406908],{},[108,406320,406321,406361,406362,347],{},[34,406322,406324,406343],{"className":406323},[31642],[34,406325,406327],{"className":406326},[31646],[31648,406328,406329],{"xmlns":31650},[31652,406330,406331,406340],{},[31655,406332,406333,406336,406338],{},[31658,406334,406335],{},"ker",[31662,406337,32682],{},[31658,406339,329201],{},[31665,406341,406342],{"encoding":31667},"\\ker \\phi",[34,406344,406346],{"className":406345,"ariaHidden":758},[31672],[34,406347,406349,406352,406355,406358],{"className":406348},[31676],[34,406350],{"className":406351,"style":43217},[31680],[34,406353,406335],{"className":406354},[32767],[34,406356],{"className":406357,"style":32123},[31845],[34,406359,329201],{"className":406360},[31685,31686]," 就是那些被映射成全是 0 的元素 ",[34,406363,406365,406378],{"className":406364},[31642],[34,406366,406368],{"className":406367},[31646],[31648,406369,406370],{"xmlns":31650},[31652,406371,406372,406376],{},[31655,406373,406374],{},[31658,406375,111],{},[31665,406377,111],{"encoding":31667},[34,406379,406381],{"className":406380,"ariaHidden":758},[31672],[34,406382,406384,406387],{"className":406383},[31676],[34,406385],{"className":406386,"style":32527},[31680],[34,406388,111],{"className":406389},[31685,31686],[108,406391,329925,406392,93535,406537,104831],{},[34,406393,406395,406437],{"className":406394},[31642],[34,406396,406398],{"className":406397},[31646],[31648,406399,406400],{"xmlns":31650},[31652,406401,406402,406434],{},[31655,406403,406404,406406,406408,406410,406412,406414,406416,406424,406426,406432],{},[31658,406405,111],{},[31662,406407,341696],{},[31749,406409,459],{},[31845,406411],{},[31845,406413],{"width":208910},[31662,406415,456],{"stretchy":702},[31655,406417,406418,406420,406422],{},[31658,406419,31759],{"mathvariant":38079},[31658,406421,43383],{"mathvariant":38079},[31658,406423,46900],{"mathvariant":38079},[31845,406425],{"width":208923},[31743,406427,406428,406430],{},[31658,406429,106063],{},[31749,406431,640],{},[31662,406433,2557],{"stretchy":702},[31665,406435,406436],{"encoding":31667},"a \\equiv 0 \\pmod{I_1}",[34,406438,406440,406458,406473],{"className":406439,"ariaHidden":758},[31672],[34,406441,406443,406446,406449,406452,406455],{"className":406442},[31676],[34,406444],{"className":406445,"style":341735},[31680],[34,406447,111],{"className":406448},[31685,31686],[34,406450],{"className":406451,"style":31846},[31845],[34,406453,341696],{"className":406454},[31850],[34,406456],{"className":406457,"style":31846},[31845],[34,406459,406461,406464,406467,406470],{"className":406460},[31676],[34,406462],{"className":406463,"style":32617},[31680],[34,406465,459],{"className":406466},[31685],[34,406468],{"className":406469},[31845,208951],[34,406471],{"className":406472,"style":208955},[31845],[34,406474,406476,406479,406482,406491,406494,406534],{"className":406475},[31676],[34,406477],{"className":406478,"style":31774},[31680],[34,406480,456],{"className":406481},[31782],[34,406483,406485],{"className":406484},[31685],[34,406486,406488],{"className":406487},[31685],[34,406489,207924],{"className":406490},[31685,207923],[34,406492],{"className":406493,"style":208977},[31845],[34,406495,406497,406500],{"className":406496},[31685],[34,406498,106063],{"className":406499,"style":67482},[31685,31686],[34,406501,406503],{"className":406502},[31792],[34,406504,406506,406526],{"className":406505},[31796,31797],[34,406507,406509,406523],{"className":406508},[31801],[34,406510,406512],{"className":406511,"style":31806},[31805],[34,406513,406514,406517],{"style":217742},[34,406515],{"className":406516,"style":31814},[31813],[34,406518,406520],{"className":406519},[31818,31819,31820,31821],[34,406521,640],{"className":406522},[31685,31821],[34,406524,31829],{"className":406525},[31828],[34,406527,406529],{"className":406528},[31801],[34,406530,406532],{"className":406531,"style":31836},[31805],[34,406533],{},[34,406535,2557],{"className":406536},[31690],[34,406538,406540,406582],{"className":406539},[31642],[34,406541,406543],{"className":406542},[31646],[31648,406544,406545],{"xmlns":31650},[31652,406546,406547,406579],{},[31655,406548,406549,406551,406553,406555,406557,406559,406561,406569,406571,406577],{},[31658,406550,111],{},[31662,406552,341696],{},[31749,406554,459],{},[31845,406556],{},[31845,406558],{"width":208910},[31662,406560,456],{"stretchy":702},[31655,406562,406563,406565,406567],{},[31658,406564,31759],{"mathvariant":38079},[31658,406566,43383],{"mathvariant":38079},[31658,406568,46900],{"mathvariant":38079},[31845,406570],{"width":208923},[31743,406572,406573,406575],{},[31658,406574,106063],{},[31749,406576,768],{},[31662,406578,2557],{"stretchy":702},[31665,406580,406581],{"encoding":31667},"a \\equiv 0 \\pmod{I_2}",[34,406583,406585,406603,406618],{"className":406584,"ariaHidden":758},[31672],[34,406586,406588,406591,406594,406597,406600],{"className":406587},[31676],[34,406589],{"className":406590,"style":341735},[31680],[34,406592,111],{"className":406593},[31685,31686],[34,406595],{"className":406596,"style":31846},[31845],[34,406598,341696],{"className":406599},[31850],[34,406601],{"className":406602,"style":31846},[31845],[34,406604,406606,406609,406612,406615],{"className":406605},[31676],[34,406607],{"className":406608,"style":32617},[31680],[34,406610,459],{"className":406611},[31685],[34,406613],{"className":406614},[31845,208951],[34,406616],{"className":406617,"style":208955},[31845],[34,406619,406621,406624,406627,406636,406639,406679],{"className":406620},[31676],[34,406622],{"className":406623,"style":31774},[31680],[34,406625,456],{"className":406626},[31782],[34,406628,406630],{"className":406629},[31685],[34,406631,406633],{"className":406632},[31685],[34,406634,207924],{"className":406635},[31685,207923],[34,406637],{"className":406638,"style":208977},[31845],[34,406640,406642,406645],{"className":406641},[31685],[34,406643,106063],{"className":406644,"style":67482},[31685,31686],[34,406646,406648],{"className":406647},[31792],[34,406649,406651,406671],{"className":406650},[31796,31797],[34,406652,406654,406668],{"className":406653},[31801],[34,406655,406657],{"className":406656,"style":31806},[31805],[34,406658,406659,406662],{"style":217742},[34,406660],{"className":406661,"style":31814},[31813],[34,406663,406665],{"className":406664},[31818,31819,31820,31821],[34,406666,768],{"className":406667},[31685,31821],[34,406669,31829],{"className":406670},[31828],[34,406672,406674],{"className":406673},[31801],[34,406675,406677],{"className":406676,"style":31836},[31805],[34,406678],{},[34,406680,2557],{"className":406681},[31690],[108,406683,406684,406685,406713,406714,347],{},"也就是说 ",[34,406686,406688,406701],{"className":406687},[31642],[34,406689,406691],{"className":406690},[31646],[31648,406692,406693],{"xmlns":31650},[31652,406694,406695,406699],{},[31655,406696,406697],{},[31658,406698,111],{},[31665,406700,111],{"encoding":31667},[34,406702,406704],{"className":406703,"ariaHidden":758},[31672],[34,406705,406707,406710],{"className":406706},[31676],[34,406708],{"className":406709,"style":32527},[31680],[34,406711,111],{"className":406712},[31685,31686]," 必须同时属于 ",[34,406715,406717,406755],{"className":406716},[31642],[34,406718,406720],{"className":406719},[31646],[31648,406721,406722],{"xmlns":31650},[31652,406723,406724,406752],{},[31655,406725,406726,406732,406734,406740,406742,406744,406746],{},[31743,406727,406728,406730],{},[31658,406729,106063],{},[31749,406731,640],{},[31662,406733,31960],{"separator":758},[31743,406735,406736,406738],{},[31658,406737,106063],{},[31749,406739,768],{},[31662,406741,31960],{"separator":758},[31662,406743,33934],{},[31662,406745,31960],{"separator":758},[31743,406747,406748,406750],{},[31658,406749,106063],{},[31658,406751,31660],{},[31665,406753,406754],{"encoding":31667},"I_1, I_2, \\dots, I_n",[34,406756,406758],{"className":406757,"ariaHidden":758},[31672],[34,406759,406761,406764,406804,406807,406810,406850,406853,406856,406859,406862,406865,406868],{"className":406760},[31676],[34,406762],{"className":406763,"style":70739},[31680],[34,406765,406767,406770],{"className":406766},[31685],[34,406768,106063],{"className":406769,"style":67482},[31685,31686],[34,406771,406773],{"className":406772},[31792],[34,406774,406776,406796],{"className":406775},[31796,31797],[34,406777,406779,406793],{"className":406778},[31801],[34,406780,406782],{"className":406781,"style":31806},[31805],[34,406783,406784,406787],{"style":217742},[34,406785],{"className":406786,"style":31814},[31813],[34,406788,406790],{"className":406789},[31818,31819,31820,31821],[34,406791,640],{"className":406792},[31685,31821],[34,406794,31829],{"className":406795},[31828],[34,406797,406799],{"className":406798},[31801],[34,406800,406802],{"className":406801,"style":31836},[31805],[34,406803],{},[34,406805,31960],{"className":406806},[32119],[34,406808],{"className":406809,"style":32123},[31845],[34,406811,406813,406816],{"className":406812},[31685],[34,406814,106063],{"className":406815,"style":67482},[31685,31686],[34,406817,406819],{"className":406818},[31792],[34,406820,406822,406842],{"className":406821},[31796,31797],[34,406823,406825,406839],{"className":406824},[31801],[34,406826,406828],{"className":406827,"style":31806},[31805],[34,406829,406830,406833],{"style":217742},[34,406831],{"className":406832,"style":31814},[31813],[34,406834,406836],{"className":406835},[31818,31819,31820,31821],[34,406837,768],{"className":406838},[31685,31821],[34,406840,31829],{"className":406841},[31828],[34,406843,406845],{"className":406844},[31801],[34,406846,406848],{"className":406847,"style":31836},[31805],[34,406849],{},[34,406851,31960],{"className":406852},[32119],[34,406854],{"className":406855,"style":32123},[31845],[34,406857,33934],{"className":406858},[34006],[34,406860],{"className":406861,"style":32123},[31845],[34,406863,31960],{"className":406864},[32119],[34,406866],{"className":406867,"style":32123},[31845],[34,406869,406871,406874],{"className":406870},[31685],[34,406872,106063],{"className":406873,"style":67482},[31685,31686],[34,406875,406877],{"className":406876},[31792],[34,406878,406880,406900],{"className":406879},[31796,31797],[34,406881,406883,406897],{"className":406882},[31801],[34,406884,406886],{"className":406885,"style":32362},[31805],[34,406887,406888,406891],{"style":217742},[34,406889],{"className":406890,"style":31814},[31813],[34,406892,406894],{"className":406893},[31818,31819,31820,31821],[34,406895,31660],{"className":406896},[31685,31686,31821],[34,406898,31829],{"className":406899},[31828],[34,406901,406903],{"className":406902},[31801],[34,406904,406906],{"className":406905,"style":31836},[31805],[34,406907],{},[108,406909,323847,406910,347],{},[34,406911,406913,406969],{"className":406912},[31642],[34,406914,406916],{"className":406915},[31646],[31648,406917,406918],{"xmlns":31650},[31652,406919,406920,406966],{},[31655,406921,406922,406924,406926,406928,406930,406936,406938,406944,406946,406948,406950,406956,406958,406960],{},[31658,406923,406335],{},[31662,406925,32682],{},[31658,406927,329201],{},[31662,406929,202],{},[31743,406931,406932,406934],{},[31658,406933,106063],{},[31749,406935,640],{},[31662,406937,326344],{},[31743,406939,406940,406942],{},[31658,406941,106063],{},[31749,406943,768],{},[31662,406945,326344],{},[31662,406947,38874],{},[31662,406949,326344],{},[31743,406951,406952,406954],{},[31658,406953,106063],{},[31658,406955,31660],{},[31662,406957,202],{},[31662,406959,405054],{},[31743,406961,406962,406964],{},[31658,406963,106063],{},[31658,406965,33759],{},[31665,406967,406968],{"encoding":31667},"\\ker \\phi = I_1 \\cap I_2 \\cap \\dots \\cap I_n = \\bigcap I_k",[34,406970,406972,406996,407051,407106,407124,407179],{"className":406971,"ariaHidden":758},[31672],[34,406973,406975,406978,406981,406984,406987,406990,406993],{"className":406974},[31676],[34,406976],{"className":406977,"style":43217},[31680],[34,406979,406335],{"className":406980},[32767],[34,406982],{"className":406983,"style":32123},[31845],[34,406985,329201],{"className":406986},[31685,31686],[34,406988],{"className":406989,"style":31846},[31845],[34,406991,202],{"className":406992},[31850],[34,406994],{"className":406995,"style":31846},[31845],[34,406997,406999,407002,407042,407045,407048],{"className":406998},[31676],[34,407000],{"className":407001,"style":70915},[31680],[34,407003,407005,407008],{"className":407004},[31685],[34,407006,106063],{"className":407007,"style":67482},[31685,31686],[34,407009,407011],{"className":407010},[31792],[34,407012,407014,407034],{"className":407013},[31796,31797],[34,407015,407017,407031],{"className":407016},[31801],[34,407018,407020],{"className":407019,"style":31806},[31805],[34,407021,407022,407025],{"style":217742},[34,407023],{"className":407024,"style":31814},[31813],[34,407026,407028],{"className":407027},[31818,31819,31820,31821],[34,407029,640],{"className":407030},[31685,31821],[34,407032,31829],{"className":407033},[31828],[34,407035,407037],{"className":407036},[31801],[34,407038,407040],{"className":407039,"style":31836},[31805],[34,407041],{},[34,407043],{"className":407044,"style":32604},[31845],[34,407046,326344],{"className":407047},[32024],[34,407049],{"className":407050,"style":32604},[31845],[34,407052,407054,407057,407097,407100,407103],{"className":407053},[31676],[34,407055],{"className":407056,"style":70915},[31680],[34,407058,407060,407063],{"className":407059},[31685],[34,407061,106063],{"className":407062,"style":67482},[31685,31686],[34,407064,407066],{"className":407065},[31792],[34,407067,407069,407089],{"className":407068},[31796,31797],[34,407070,407072,407086],{"className":407071},[31801],[34,407073,407075],{"className":407074,"style":31806},[31805],[34,407076,407077,407080],{"style":217742},[34,407078],{"className":407079,"style":31814},[31813],[34,407081,407083],{"className":407082},[31818,31819,31820,31821],[34,407084,768],{"className":407085},[31685,31821],[34,407087,31829],{"className":407088},[31828],[34,407090,407092],{"className":407091},[31801],[34,407093,407095],{"className":407094,"style":31836},[31805],[34,407096],{},[34,407098],{"className":407099,"style":32604},[31845],[34,407101,326344],{"className":407102},[32024],[34,407104],{"className":407105,"style":32604},[31845],[34,407107,407109,407112,407115,407118,407121],{"className":407108},[31676],[34,407110],{"className":407111,"style":147334},[31680],[34,407113,38874],{"className":407114},[34006],[34,407116],{"className":407117,"style":32604},[31845],[34,407119,326344],{"className":407120},[32024],[34,407122],{"className":407123,"style":32604},[31845],[34,407125,407127,407130,407170,407173,407176],{"className":407126},[31676],[34,407128],{"className":407129,"style":70915},[31680],[34,407131,407133,407136],{"className":407132},[31685],[34,407134,106063],{"className":407135,"style":67482},[31685,31686],[34,407137,407139],{"className":407138},[31792],[34,407140,407142,407162],{"className":407141},[31796,31797],[34,407143,407145,407159],{"className":407144},[31801],[34,407146,407148],{"className":407147,"style":32362},[31805],[34,407149,407150,407153],{"style":217742},[34,407151],{"className":407152,"style":31814},[31813],[34,407154,407156],{"className":407155},[31818,31819,31820,31821],[34,407157,31660],{"className":407158},[31685,31686,31821],[34,407160,31829],{"className":407161},[31828],[34,407163,407165],{"className":407164},[31801],[34,407166,407168],{"className":407167,"style":31836},[31805],[34,407169],{},[34,407171],{"className":407172,"style":31846},[31845],[34,407174,202],{"className":407175},[31850],[34,407177],{"className":407178,"style":31846},[31845],[34,407180,407182,407185,407188,407191],{"className":407181},[31676],[34,407183],{"className":407184,"style":31774},[31680],[34,407186,405054],{"className":407187,"style":156561},[32767,131249,156560],[34,407189],{"className":407190,"style":32123},[31845],[34,407192,407194,407197],{"className":407193},[31685],[34,407195,106063],{"className":407196,"style":67482},[31685,31686],[34,407198,407200],{"className":407199},[31792],[34,407201,407203,407223],{"className":407202},[31796,31797],[34,407204,407206,407220],{"className":407205},[31801],[34,407207,407209],{"className":407208,"style":33833},[31805],[34,407210,407211,407214],{"style":217742},[34,407212],{"className":407213,"style":31814},[31813],[34,407215,407217],{"className":407216},[31818,31819,31820,31821],[34,407218,33759],{"className":407219,"style":33774},[31685,31686,31821],[34,407221,31829],{"className":407222},[31828],[34,407224,407226],{"className":407225},[31801],[34,407227,407229],{"className":407228,"style":31836},[31805],[34,407230],{},[15,407232,407233,407234,2664,407237,407320,407321,407324],{},"根据",[2627,407235,407236],{},"同态基本定理",[34,407238,407240,407269],{"className":407239},[31642],[34,407241,407243],{"className":407242},[31646],[31648,407244,407245],{"xmlns":31650},[31652,407246,407247,407266],{},[31655,407248,407249,407251,407253,407255,407257,407259,407261,407264],{},[31658,407250,34386],{},[31658,407252,38080],{"mathvariant":38079},[31658,407254,406335],{},[31662,407256,32682],{},[31658,407258,329201],{},[31662,407260,134826],{},[39361,407262,407263],{},"Im ",[31658,407265,329201],{},[31665,407267,407268],{"encoding":31667},"R \u002F \\ker \\phi \\cong \\text{Im } \\phi",[34,407270,407272,407305],{"className":407271,"ariaHidden":758},[31672],[34,407273,407275,407278,407281,407284,407287,407290,407293,407296,407299,407302],{"className":407274},[31676],[34,407276],{"className":407277,"style":31774},[31680],[34,407279,34386],{"className":407280,"style":34427},[31685,31686],[34,407282,38080],{"className":407283},[31685],[34,407285],{"className":407286,"style":32123},[31845],[34,407288,406335],{"className":407289},[32767],[34,407291],{"className":407292,"style":32123},[31845],[34,407294,329201],{"className":407295},[31685,31686],[34,407297],{"className":407298,"style":31846},[31845],[34,407300,134826],{"className":407301},[31850],[34,407303],{"className":407304,"style":31846},[31845],[34,407306,407308,407311,407317],{"className":407307},[31676],[34,407309],{"className":407310,"style":43217},[31680],[34,407312,407314],{"className":407313},[31685,2638],[34,407315,407263],{"className":407316},[31685],[34,407318,329201],{"className":407319},[31685,31686],"。\n现在左边有了，我们只差最后一步：",[2627,407322,407323],{},"证明右边是满射 (Surjective)","。\n也就是要证明：对于右边任意一个列表，我们都能在左边找到对应的原像。",[32188,407326],{},[32490,407328,407330],{"id":407329},"证明第二步证明满射最难的归纳法部分","证明第二步：证明满射（最难的归纳法部分）",[15,407332,407333,407334,407417,407418,407446,407447,407516],{},"我们要证明满射，本质上是要构造一组“开关”。\n比如我想构造列表 ",[34,407335,407337,407369],{"className":407336},[31642],[34,407338,407340],{"className":407339},[31646],[31648,407341,407342],{"xmlns":31650},[31652,407343,407344,407366],{},[31655,407345,407346,407348,407350,407352,407354,407356,407358,407360,407362,407364],{},[31662,407347,456],{"stretchy":702},[31749,407349,640],{},[31662,407351,31960],{"separator":758},[31749,407353,459],{},[31662,407355,31960],{"separator":758},[31749,407357,459],{},[31662,407359,31960],{"separator":758},[31662,407361,33934],{},[39361,407363,39363],{},[31662,407365,2557],{"stretchy":702},[31665,407367,407368],{"encoding":31667},"(1, 0, 0, \\dots)",[34,407370,407372],{"className":407371,"ariaHidden":758},[31672],[34,407373,407375,407378,407381,407384,407387,407390,407393,407396,407399,407402,407405,407408,407411,407414],{"className":407374},[31676],[34,407376],{"className":407377,"style":31774},[31680],[34,407379,456],{"className":407380},[31782],[34,407382,640],{"className":407383},[31685],[34,407385,31960],{"className":407386},[32119],[34,407388],{"className":407389,"style":32123},[31845],[34,407391,459],{"className":407392},[31685],[34,407394,31960],{"className":407395},[32119],[34,407397],{"className":407398,"style":32123},[31845],[34,407400,459],{"className":407401},[31685],[34,407403,31960],{"className":407404},[32119],[34,407406],{"className":407407,"style":32123},[31845],[34,407409,33934],{"className":407410},[34006],[34,407412],{"className":407413,"style":32123},[31845],[34,407415,2557],{"className":407416},[31690],"，我需要找到一个 ",[34,407419,407421,407434],{"className":407420},[31642],[34,407422,407424],{"className":407423},[31646],[31648,407425,407426],{"xmlns":31650},[31652,407427,407428,407432],{},[31655,407429,407430],{},[31658,407431,32936],{},[31665,407433,32936],{"encoding":31667},[34,407435,407437],{"className":407436,"ariaHidden":758},[31672],[34,407438,407440,407443],{"className":407439},[31676],[34,407441],{"className":407442,"style":32527},[31680],[34,407444,32936],{"className":407445},[31685,31686],"，它模 ",[34,407448,407450,407467],{"className":407449},[31642],[34,407451,407453],{"className":407452},[31646],[31648,407454,407455],{"xmlns":31650},[31652,407456,407457,407465],{},[31655,407458,407459],{},[31743,407460,407461,407463],{},[31658,407462,106063],{},[31749,407464,640],{},[31665,407466,348359],{"encoding":31667},[34,407468,407470],{"className":407469,"ariaHidden":758},[31672],[34,407471,407473,407476],{"className":407472},[31676],[34,407474],{"className":407475,"style":70915},[31680],[34,407477,407479,407482],{"className":407478},[31685],[34,407480,106063],{"className":407481,"style":67482},[31685,31686],[34,407483,407485],{"className":407484},[31792],[34,407486,407488,407508],{"className":407487},[31796,31797],[34,407489,407491,407505],{"className":407490},[31801],[34,407492,407494],{"className":407493,"style":31806},[31805],[34,407495,407496,407499],{"style":217742},[34,407497],{"className":407498,"style":31814},[31813],[34,407500,407502],{"className":407501},[31818,31819,31820,31821],[34,407503,640],{"className":407504},[31685,31821],[34,407506,31829],{"className":407507},[31828],[34,407509,407511],{"className":407510},[31801],[34,407512,407514],{"className":407513,"style":31836},[31805],[34,407515],{}," 是 1，模其他都是 0。",[15,407518,407519],{},[2627,407520,407521],{},"课件里的构造逻辑：",[2230,407523,407524,408864,411109,413641],{},[108,407525,407526,33183,407599,407829],{},[2627,407527,407528,407529],{},"定义 ",[34,407530,407532,407550],{"className":407531},[31642],[34,407533,407535],{"className":407534},[31646],[31648,407536,407537],{"xmlns":31650},[31652,407538,407539,407547],{},[31655,407540,407541],{},[31743,407542,407543,407545],{},[31658,407544,98360],{},[31658,407546,31955],{},[31665,407548,407549],{"encoding":31667},"M_i",[34,407551,407553],{"className":407552,"ariaHidden":758},[31672],[34,407554,407556,407559],{"className":407555},[31676],[34,407557],{"className":407558,"style":70915},[31680],[34,407560,407562,407565],{"className":407561},[31685],[34,407563,98360],{"className":407564,"style":33389},[31685,31686],[34,407566,407568],{"className":407567},[31792],[34,407569,407571,407591],{"className":407570},[31796,31797],[34,407572,407574,407588],{"className":407573},[31801],[34,407575,407577],{"className":407576,"style":32090},[31805],[34,407578,407579,407582],{"style":231884},[34,407580],{"className":407581,"style":31814},[31813],[34,407583,407585],{"className":407584},[31818,31819,31820,31821],[34,407586,31955],{"className":407587},[31685,31686,31821],[34,407589,31829],{"className":407590},[31828],[34,407592,407594],{"className":407593},[31801],[34,407595,407597],{"className":407596,"style":31836},[31805],[34,407598],{},[34,407600,407602,407640],{"className":407601},[31642],[34,407603,407605],{"className":407604},[31646],[31648,407606,407607],{"xmlns":31650},[31652,407608,407609,407637],{},[31655,407610,407611,407617,407619,407631],{},[31743,407612,407613,407615],{},[31658,407614,98360],{},[31658,407616,31955],{},[31662,407618,202],{},[31743,407620,407621,407623],{},[31662,407622,402680],{},[31655,407624,407625,407627,407629],{},[31658,407626,31968],{},[31662,407628,54707],{"mathvariant":38079},[31658,407630,31955],{},[31743,407632,407633,407635],{},[31658,407634,106063],{},[31658,407636,31968],{},[31665,407638,407639],{"encoding":31667},"M_i = \\prod_{j \\neq i} I_j",[34,407641,407643,407698],{"className":407642,"ariaHidden":758},[31672],[34,407644,407646,407649,407689,407692,407695],{"className":407645},[31676],[34,407647],{"className":407648,"style":70915},[31680],[34,407650,407652,407655],{"className":407651},[31685],[34,407653,98360],{"className":407654,"style":33389},[31685,31686],[34,407656,407658],{"className":407657},[31792],[34,407659,407661,407681],{"className":407660},[31796,31797],[34,407662,407664,407678],{"className":407663},[31801],[34,407665,407667],{"className":407666,"style":32090},[31805],[34,407668,407669,407672],{"style":231884},[34,407670],{"className":407671,"style":31814},[31813],[34,407673,407675],{"className":407674},[31818,31819,31820,31821],[34,407676,31955],{"className":407677},[31685,31686,31821],[34,407679,31829],{"className":407680},[31828],[34,407682,407684],{"className":407683},[31801],[34,407685,407687],{"className":407686,"style":31836},[31805],[34,407688],{},[34,407690],{"className":407691,"style":31846},[31845],[34,407693,202],{"className":407694},[31850],[34,407696],{"className":407697,"style":31846},[31845],[34,407699,407701,407704,407786,407789],{"className":407700},[31676],[34,407702],{"className":407703,"style":156901},[31680],[34,407705,407707,407710],{"className":407706},[32767],[34,407708,402680],{"className":407709,"style":156561},[32767,131249,156560],[34,407711,407713],{"className":407712},[31792],[34,407714,407716,407778],{"className":407715},[31796,31797],[34,407717,407719,407775],{"className":407718},[31801],[34,407720,407722],{"className":407721,"style":356484},[31805],[34,407723,407724,407727],{"style":156577},[34,407725],{"className":407726,"style":31814},[31813],[34,407728,407730],{"className":407729},[31818,31819,31820,31821],[34,407731,407733,407736,407772],{"className":407732},[31685,31821],[34,407734,31968],{"className":407735,"style":32153},[31685,31686,31821],[34,407737,407739,407766,407769],{"className":407738},[31850,31821],[34,407740,407742],{"className":407741},[31850,31821],[34,407743,407745],{"className":407744},[31685,54774,31821],[34,407746,407748],{"className":407747},[54778,31821],[34,407749,407751,407754,407763],{"className":407750},[54782,31821],[34,407752],{"className":407753,"style":43217},[31680],[34,407755,407757],{"className":407756},[54789],[34,407758,407760],{"className":407759},[31685,31821],[34,407761,54796],{"className":407762},[31850,31821],[34,407764],{"className":407765},[54800],[34,407767],{"className":407768},[31845,54804,31821],[34,407770,202],{"className":407771},[31850,31821],[34,407773,31955],{"className":407774},[31685,31686,31821],[34,407776,31829],{"className":407777},[31828],[34,407779,407781],{"className":407780},[31801],[34,407782,407784],{"className":407783,"style":156941},[31805],[34,407785],{},[34,407787],{"className":407788,"style":32123},[31845],[34,407790,407792,407795],{"className":407791},[31685],[34,407793,106063],{"className":407794,"style":67482},[31685,31686],[34,407796,407798],{"className":407797},[31792],[34,407799,407801,407821],{"className":407800},[31796,31797],[34,407802,407804,407818],{"className":407803},[31801],[34,407805,407807],{"className":407806,"style":32090},[31805],[34,407808,407809,407812],{"style":217742},[34,407810],{"className":407811,"style":31814},[31813],[34,407813,407815],{"className":407814},[31818,31819,31820,31821],[34,407816,31968],{"className":407817,"style":32153},[31685,31686,31821],[34,407819,31829],{"className":407820},[31828],[34,407822,407824],{"className":407823},[31801],[34,407825,407827],{"className":407826,"style":32163},[31805],[34,407828],{},[105,407830,407831,408165,408572],{},[108,407832,407833,2664,407835,11008,407905,333608,408095,408164],{},[2627,407834,233260],{},[34,407836,407838,407856],{"className":407837},[31642],[34,407839,407841],{"className":407840},[31646],[31648,407842,407843],{"xmlns":31650},[31652,407844,407845,407853],{},[31655,407846,407847],{},[31743,407848,407849,407851],{},[31658,407850,98360],{},[31749,407852,640],{},[31665,407854,407855],{"encoding":31667},"M_1",[34,407857,407859],{"className":407858,"ariaHidden":758},[31672],[34,407860,407862,407865],{"className":407861},[31676],[34,407863],{"className":407864,"style":70915},[31680],[34,407866,407868,407871],{"className":407867},[31685],[34,407869,98360],{"className":407870,"style":33389},[31685,31686],[34,407872,407874],{"className":407873},[31792],[34,407875,407877,407897],{"className":407876},[31796,31797],[34,407878,407880,407894],{"className":407879},[31801],[34,407881,407883],{"className":407882,"style":31806},[31805],[34,407884,407885,407888],{"style":231884},[34,407886],{"className":407887,"style":31814},[31813],[34,407889,407891],{"className":407890},[31818,31819,31820,31821],[34,407892,640],{"className":407893},[31685,31821],[34,407895,31829],{"className":407896},[31828],[34,407898,407900],{"className":407899},[31801],[34,407901,407903],{"className":407902,"style":31836},[31805],[34,407904],{},[34,407906,407908,407942],{"className"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\\cdot I_3 \\cdots I_n",[34,407943,407945,408000],{"className":407944,"ariaHidden":758},[31672],[34,407946,407948,407951,407991,407994,407997],{"className":407947},[31676],[34,407949],{"className":407950,"style":70915},[31680],[34,407952,407954,407957],{"className":407953},[31685],[34,407955,106063],{"className":407956,"style":67482},[31685,31686],[34,407958,407960],{"className":407959},[31792],[34,407961,407963,407983],{"className":407962},[31796,31797],[34,407964,407966,407980],{"className":407965},[31801],[34,407967,407969],{"className":407968,"style":31806},[31805],[34,407970,407971,407974],{"style":217742},[34,407972],{"className":407973,"style":31814},[31813],[34,407975,407977],{"className":407976},[31818,31819,31820,31821],[34,407978,768],{"className":407979},[31685,31821],[34,407981,31829],{"className":407982},[31828],[34,407984,407986],{"className":407985},[31801],[34,407987,407989],{"className":407988,"style":31836},[31805],[34,407990],{},[34,407992],{"className":407993,"style":32604},[31845],[34,407995,71191],{"className":407996},[32024],[34,407998],{"className":407999,"style":32604},[31845],[34,408001,408003,408006,408046,408049,408052,408055],{"className":408002},[31676],[34,408004],{"className":408005,"style":70915},[31680],[34,408007,408009,408012],{"className":408008},[31685],[34,408010,106063],{"className":408011,"style":67482},[31685,31686],[34,408013,408015],{"className":408014},[31792],[34,408016,408018,408038],{"className":408017},[31796,31797],[34,408019,408021,408035],{"className":408020},[31801],[34,408022,408024],{"className":408023,"style":31806},[31805],[34,408025,408026,408029],{"style":217742},[34,408027],{"className":408028,"style":31814},[31813],[34,408030,408032],{"className":408031},[31818,31819,31820,31821],[34,408033,786],{"className":408034},[31685,31821],[34,408036,31829],{"className":408037},[31828],[34,408039,408041],{"className":408040},[31801],[34,408042,408044],{"className":408043,"style":31836},[31805],[34,408045],{},[34,408047],{"className":408048,"style":32123},[31845],[34,408050,38874],{"className":408051},[34006],[34,408053],{"className":408054,"style":32123},[31845],[34,408056,408058,408061],{"className":408057},[31685],[34,408059,106063],{"className":408060,"style":67482},[31685,31686],[34,408062,408064],{"className":408063},[31792],[34,408065,408067,408087],{"className":408066},[31796,31797],[34,408068,408070,408084],{"className":408069},[31801],[34,408071,408073],{"className":408072,"style":32362},[31805],[34,408074,408075,408078],{"style":217742},[34,408076],{"className":408077,"style":31814},[31813],[34,408079,408081],{"className":408080},[31818,31819,31820,31821],[34,408082,31660],{"className":408083},[31685,31686,31821],[34,408085,31829],{"className":408086},[31828],[34,408088,408090],{"className":408089},[31801],[34,408091,408093],{"className":408092,"style":31836},[31805],[34,408094],{},[34,408096,408098,408115],{"className":408097},[31642],[34,408099,408101],{"className":408100},[31646],[31648,408102,408103],{"xmlns":31650},[31652,408104,408105,408113],{},[31655,408106,408107],{},[31743,408108,408109,408111],{},[31658,408110,106063],{},[31749,408112,640],{},[31665,408114,348359],{"encoding":31667},[34,408116,408118],{"className":408117,"ariaHidden":758},[31672],[34,408119,408121,408124],{"className":408120},[31676],[34,408122],{"className":408123,"style":70915},[31680],[34,408125,408127,408130],{"className":408126},[31685],[34,408128,106063],{"className":408129,"style":67482},[31685,31686],[34,408131,408133],{"className":408132},[31792],[34,408134,408136,408156],{"className":408135},[31796,31797],[34,408137,408139,408153],{"className":408138},[31801],[34,408140,408142],{"className":408141,"style":31806},[31805],[34,408143,408144,408147],{"style":217742},[34,408145],{"className":408146,"style":31814},[31813],[34,408148,408150],{"className":408149},[31818,31819,31820,31821],[34,408151,640],{"className":408152},[31685,31821],[34,408154,31829],{"className":408155},[31828],[34,408157,408159],{"className":408158},[31801],[34,408160,408162],{"className":408161,"style":31836},[31805],[34,408163],{}," 以外所有理想的乘积）。",[108,408166,408167,332541,408169,327254,408238,408370,408371,408440,408441,320784],{},[2627,408168,347183],{},[34,408170,408172,408189],{"className":408171},[31642],[34,408173,408175],{"className":408174},[31646],[31648,408176,408177],{"xmlns":31650},[31652,408178,408179,408187],{},[31655,408180,408181],{},[31743,408182,408183,408185],{},[31658,408184,98360],{},[31749,408186,640],{},[31665,408188,407855],{"encoding":31667},[34,408190,408192],{"className":408191,"ariaHidden":758},[31672],[34,408193,408195,408198],{"className":408194},[31676],[34,408196],{"className":408197,"style":70915},[31680],[34,408199,408201,408204],{"className":408200},[31685],[34,408202,98360],{"className":408203,"style":33389},[31685,31686],[34,408205,408207],{"className":408206},[31792],[34,408208,408210,408230],{"className":408209},[31796,31797],[34,408211,408213,408227],{"className":408212},[31801],[34,408214,408216],{"className":408215,"style":31806},[31805],[34,408217,408218,408221],{"style":231884},[34,408219],{"className":408220,"style":31814},[31813],[34,408222,408224],{"className":408223},[31818,31819,31820,31821],[34,408225,640],{"className":408226},[31685,31821],[34,408228,31829],{"className":408229},[31828],[34,408231,408233],{"className":408232},[31801],[34,408234,408236],{"className":408235,"style":31836},[31805],[34,408237],{},[34,408239,408241,408269],{"className":408240},[31642],[34,408242,408244],{"className":408243},[31646],[31648,408245,408246],{"xmlns":31650},[31652,408247,408248,408266],{},[31655,408249,408250,408256,408258,408264],{},[31743,408251,408252,408254],{},[31658,408253,106063],{},[31749,408255,768],{},[31662,408257,31960],{"separator":758},[31743,408259,408260,408262],{},[31658,408261,106063],{},[31749,408263,786],{},[31662,408265,33934],{},[31665,408267,408268],{"encoding":31667},"I_2, I_3 \\dots",[34,408270,408272],{"className":408271,"ariaHidden":758},[31672],[34,408273,408275,408278,408318,408321,408324,408364,408367],{"className":408274},[31676],[34,408276],{"className":408277,"style":70739},[31680],[34,408279,408281,408284],{"className":408280},[31685],[34,408282,106063],{"className":408283,"style":67482},[31685,31686],[34,408285,408287],{"className":408286},[31792],[34,408288,408290,408310],{"className":408289},[31796,31797],[34,408291,408293,408307],{"className":408292},[31801],[34,408294,408296],{"className":408295,"style":31806},[31805],[34,408297,408298,408301],{"style":217742},[34,408299],{"className":408300,"style":31814},[31813],[34,408302,408304],{"className":408303},[31818,31819,31820,31821],[34,408305,768],{"className":408306},[31685,31821],[34,408308,31829],{"className":408309},[31828],[34,408311,408313],{"className":408312},[31801],[34,408314,408316],{"className":408315,"style":31836},[31805],[34,408317],{},[34,408319,31960],{"className":408320},[32119],[34,408322],{"className":408323,"style":32123},[31845],[34,408325,408327,408330],{"className":408326},[31685],[34,408328,106063],{"className":408329,"style":67482},[31685,31686],[34,408331,408333],{"className":408332},[31792],[34,408334,408336,408356],{"className":408335},[31796,31797],[34,408337,408339,408353],{"className":408338},[31801],[34,408340,408342],{"className":408341,"style":31806},[31805],[34,408343,408344,408347],{"style":217742},[34,408345],{"className":408346,"style":31814},[31813],[34,408348,408350],{"className":408349},[31818,31819,31820,31821],[34,408351,786],{"className":408352},[31685,31821],[34,408354,31829],{"className":408355},[31828],[34,408357,408359],{"className":408358},[31801],[34,408360,408362],{"className":408361,"style":31836},[31805],[34,408363],{},[34,408365],{"className":408366,"style":32123},[31845],[34,408368,33934],{"className":408369},[34006]," 的因子，所以 ",[34,408372,408374,408391],{"className":408373},[31642],[34,408375,408377],{"className":408376},[31646],[31648,408378,408379],{"xmlns":31650},[31652,408380,408381,408389],{},[31655,408382,408383],{},[31743,408384,408385,408387],{},[31658,408386,98360],{},[31749,408388,640],{},[31665,408390,407855],{"encoding":31667},[34,408392,408394],{"className":408393,"ariaHidden":758},[31672],[34,408395,408397,408400],{"className":408396},[31676],[34,408398],{"className":408399,"style":70915},[31680],[34,408401,408403,408406],{"className":408402},[31685],[34,408404,98360],{"className":408405,"style":33389},[31685,31686],[34,408407,408409],{"className":408408},[31792],[34,408410,408412,408432],{"className":408411},[31796,31797],[34,408413,408415,408429],{"className":408414},[31801],[34,408416,408418],{"className":408417,"style":31806},[31805],[34,408419,408420,408423],{"style":231884},[34,408421],{"className":408422,"style":31814},[31813],[34,408424,408426],{"className":408425},[31818,31819,31820,31821],[34,408427,640],{"className":408428},[31685,31821],[34,408430,31829],{"className":408431},[31828],[34,408433,408435],{"className":408434},[31801],[34,408436,408438],{"className":408437,"style":31836},[31805],[34,408439],{}," 落在 ",[34,408442,408444,408471],{"className":408443},[31642],[34,408445,408447],{"className":408446},[31646],[31648,408448,408449],{"xmlns":31650},[31652,408450,408451,408469],{},[31655,408452,408453,408459,408461,408467],{},[31743,408454,408455,408457],{},[31658,408456,106063],{},[31749,408458,768],{},[31662,408460,31960],{"separator":758},[31743,408462,408463,408465],{},[31658,408464,106063],{},[31749,408466,786],{},[31662,408468,33934],{},[31665,408470,408268],{"encoding":31667},[34,408472,408474],{"className":408473,"ariaHidden":758},[31672],[34,408475,408477,408480,408520,408523,408526,408566,408569],{"className":408476},[31676],[34,408478],{"className":408479,"style":70739},[31680],[34,408481,408483,408486],{"className":408482},[31685],[34,408484,106063],{"className":408485,"style":67482},[31685,31686],[34,408487,408489],{"className":408488},[31792],[34,408490,408492,408512],{"className":408491},[31796,31797],[34,408493,408495,408509],{"className":408494},[31801],[34,408496,408498],{"className":408497,"style":31806},[31805],[34,408499,408500,408503],{"style":217742},[34,408501],{"className":408502,"style":31814},[31813],[34,408504,408506],{"className":408505},[31818,31819,31820,31821],[34,408507,768],{"className":408508},[31685,31821],[34,408510,31829],{"className":408511},[31828],[34,408513,408515],{"className":408514},[31801],[34,408516,408518],{"className":408517,"style":31836},[31805],[34,408519],{},[34,408521,31960],{"className":408522},[32119],[34,408524],{"className":408525,"style":32123},[31845],[34,408527,408529,408532],{"className":408528},[31685],[34,408530,106063],{"className":408531,"style":67482},[31685,31686],[34,408533,408535],{"className":408534},[31792],[34,408536,408538,408558],{"className":408537},[31796,31797],[34,408539,408541,408555],{"className":408540},[31801],[34,408542,408544],{"className":408543,"style":31806},[31805],[34,408545,408546,408549],{"style":217742},[34,408547],{"className":408548,"style":31814},[31813],[34,408550,408552],{"className":408551},[31818,31819,31820,31821],[34,408553,786],{"className":408554},[31685,31821],[34,408556,31829],{"className":408557},[31828],[34,408559,408561],{"className":408560},[31801],[34,408562,408564],{"className":408563,"style":31836},[31805],[34,408565],{},[34,408567],{"className":408568,"style":32123},[31845],[34,408570,33934],{"className":408571},[34006],[108,408573,408574,408575,352264,408659,408792,408793,408863],{},"也就是说：对于任何 ",[34,408576,408578,408596],{"className":408577},[31642],[34,408579,408581],{"className":408580},[31646],[31648,408582,408583],{"xmlns":31650},[31652,408584,408585,408593],{},[31655,408586,408587,408589,408591],{},[31658,408588,31968],{},[31662,408590,54707],{"mathvariant":38079},[31749,408592,640],{},[31665,408594,408595],{"encoding":31667},"j \\neq 1",[34,408597,408599,408650],{"className":408598,"ariaHidden":758},[31672],[34,408600,408602,408605,408608,408611,408647],{"className":408601},[31676],[34,408603],{"className":408604,"style":43217},[31680],[34,408606,31968],{"className":408607,"style":32153},[31685,31686],[34,408609],{"className":408610,"style":31846},[31845],[34,408612,408614,408641,408644],{"className":408613},[31850],[34,408615,408617],{"className":408616},[31850],[34,408618,408620],{"className":408619},[31685,54774],[34,408621,408623],{"className":408622},[54778],[34,408624,408626,408629,408638],{"className":408625},[54782],[34,408627],{"className":408628,"style":43217},[31680],[34,408630,408632],{"className":408631},[54789],[34,408633,408635],{"className":408634},[31685],[34,408636,54796],{"className":408637},[31850],[34,408639],{"className":408640},[54800],[34,408642],{"className":408643},[31845,54804],[34,408645,202],{"className":408646},[31850],[34,408648],{"className":408649,"style":31846},[31845],[34,408651,408653,408656],{"className":408652},[31676],[34,408654],{"className":408655,"style":32617},[31680],[34,408657,640],{"className":408658},[31685],[34,408660,408662,408688],{"className":408661},[31642],[34,408663,408665],{"className":408664},[31646],[31648,408666,408667],{"xmlns":31650},[31652,408668,408669,408685],{},[31655,408670,408671,408677,408679],{},[31743,408672,408673,408675],{},[31658,408674,98360],{},[31749,408676,640],{},[31662,408678,131855],{},[31743,408680,408681,408683],{},[31658,408682,106063],{},[31658,408684,31968],{},[31665,408686,408687],{"encoding":31667},"M_1 \\subseteq I_j",[34,408689,408691,408746],{"className":408690,"ariaHidden":758},[31672],[34,408692,408694,408697,408737,408740,408743],{"className":408693},[31676],[34,408695],{"className":408696,"style":70915},[31680],[34,408698,408700,408703],{"className":408699},[31685],[34,408701,98360],{"className":408702,"style":33389},[31685,31686],[34,408704,408706],{"className":408705},[31792],[34,408707,408709,408729],{"className":408708},[31796,31797],[34,408710,408712,408726],{"className":408711},[31801],[34,408713,408715],{"className":408714,"style":31806},[31805],[34,408716,408717,408720],{"style":231884},[34,408718],{"className":408719,"style":31814},[31813],[34,408721,408723],{"className":408722},[31818,31819,31820,31821],[34,408724,640],{"className":408725},[31685,31821],[34,408727,31829],{"className":408728},[31828],[34,408730,408732],{"className":408731},[31801],[34,408733,408735],{"className":408734,"style":31836},[31805],[34,408736],{},[34,408738],{"className":408739,"style":31846},[31845],[34,408741,131855],{"className":408742},[31850],[34,408744],{"className":408745,"style":31846},[31845],[34,408747,408749,408752],{"className":408748},[31676],[34,408750],{"className":408751,"style":55484},[31680],[34,408753,408755,408758],{"className":408754},[31685],[34,408756,106063],{"className":408757,"style":67482},[31685,31686],[34,408759,408761],{"className":408760},[31792],[34,408762,408764,408784],{"className":408763},[31796,31797],[34,408765,408767,408781],{"className":408766},[31801],[34,408768,408770],{"className":408769,"style":32090},[31805],[34,408771,408772,408775],{"style":217742},[34,408773],{"className":408774,"style":31814},[31813],[34,408776,408778],{"className":408777},[31818,31819,31820,31821],[34,408779,31968],{"className":408780,"style":32153},[31685,31686,31821],[34,408782,31829],{"className":408783},[31828],[34,408785,408787],{"className":408786},[31801],[34,408788,408790],{"className":408789,"style":32163},[31805],[34,408791],{},"（即模 ",[34,408794,408796,408814],{"className":408795},[31642],[34,408797,408799],{"className":408798},[31646],[31648,408800,408801],{"xmlns":31650},[31652,408802,408803,408811],{},[31655,408804,408805],{},[31743,408806,408807,408809],{},[31658,408808,106063],{},[31658,408810,31968],{},[31665,408812,408813],{"encoding":31667},"I_j",[34,408815,408817],{"className":408816,"ariaHidden":758},[31672],[34,408818,408820,408823],{"className":408819},[31676],[34,408821],{"className":408822,"style":55484},[31680],[34,408824,408826,408829],{"className":408825},[31685],[34,408827,106063],{"className":408828,"style":67482},[31685,31686],[34,408830,408832],{"className":408831},[31792],[34,408833,408835,408855],{"className":408834},[31796,31797],[34,408836,408838,408852],{"className":408837},[31801],[34,408839,408841],{"className":408840,"style":32090},[31805],[34,408842,408843,408846],{"style":217742},[34,408844],{"className":408845,"style":31814},[31813],[34,408847,408849],{"className":408848},[31818,31819,31820,31821],[34,408850,31968],{"className":408851,"style":32153},[31685,31686,31821],[34,408853,31829],{"className":408854},[31828],[34,408856,408858],{"className":408857},[31801],[34,408859,408861],{"className":408860,"style":32163},[31805],[34,408862],{}," 为 0）。",[108,408865,408866,2664,408869,409109,409110,347,409112,409114,409117],{},[2627,408867,408868],{},"断言",[34,408870,408872,408914],{"className":408871},[31642],[34,408873,408875],{"className":408874},[31646],[31648,408876,408877],{"xmlns":31650},[31652,408878,408879,408911],{},[31655,408880,408881,408887,408889,408895,408897,408899,408901,408907,408909],{},[31743,408882,408883,408885],{},[31658,408884,98360],{},[31749,408886,640],{},[31662,408888,223],{},[31743,408890,408891,408893],{},[31658,408892,98360],{},[31749,408894,768],{},[31662,408896,223],{},[31662,408898,38874],{},[31662,408900,223],{},[31743,408902,408903,408905],{},[31658,408904,98360],{},[31658,408906,31660],{},[31662,408908,202],{},[31658,408910,34386],{},[31665,408912,408913],{"encoding":31667},"M_1 + M_2 + \\dots + M_n = R",[34,408915,408917,408972,409027,409045,409100],{"className":408916,"ariaHidden":758},[31672],[34,408918,408920,408923,408963,408966,408969],{"className":408919},[31676],[34,408921],{"className":408922,"style":70915},[31680],[34,408924,408926,408929],{"className":408925},[31685],[34,408927,98360],{"className":408928,"style":33389},[31685,31686],[34,408930,408932],{"className":408931},[31792],[34,408933,408935,408955],{"className":408934},[31796,31797],[34,408936,408938,408952],{"className":408937},[31801],[34,408939,408941],{"className":408940,"style":31806},[31805],[34,408942,408943,408946],{"style":231884},[34,408944],{"className":408945,"style":31814},[31813],[34,408947,408949],{"className":408948},[31818,31819,31820,31821],[34,408950,640],{"className":408951},[31685,31821],[34,408953,31829],{"className":408954},[31828],[34,408956,408958],{"className":408957},[31801],[34,408959,408961],{"className":408960,"style":31836},[31805],[34,408962],{},[34,408964],{"className":408965,"style":32604},[31845],[34,408967,223],{"className":408968},[32024],[34,408970],{"className":408971,"style":32604},[31845],[34,408973,408975,408978,409018,409021,409024],{"className":408974},[31676],[34,408976],{"className":408977,"style":70915},[31680],[34,408979,408981,408984],{"className":408980},[31685],[34,408982,98360],{"className":408983,"style":33389},[31685,31686],[34,408985,408987],{"className":408986},[31792],[34,408988,408990,409010],{"className":408989},[31796,31797],[34,408991,408993,409007],{"className":408992},[31801],[34,408994,408996],{"className":408995,"style":31806},[31805],[34,408997,408998,409001],{"style":231884},[34,408999],{"className":409000,"style":31814},[31813],[34,409002,409004],{"className":409003},[31818,31819,31820,31821],[34,409005,768],{"className":409006},[31685,31821],[34,409008,31829],{"className":409009},[31828],[34,409011,409013],{"className":409012},[31801],[34,409014,409016],{"className":409015,"style":31836},[31805],[34,409017],{},[34,409019],{"className":409020,"style":32604},[31845],[34,409022,223],{"className":409023},[32024],[34,409025],{"className":409026,"style":32604},[31845],[34,409028,409030,409033,409036,409039,409042],{"className":409029},[31676],[34,409031],{"className":409032,"style":32597},[31680],[34,409034,38874],{"className":409035},[34006],[34,409037],{"className":409038,"style":32604},[31845],[34,409040,223],{"className":409041},[32024],[34,409043],{"className":409044,"style":32604},[31845],[34,409046,409048,409051,409091,409094,409097],{"className":409047},[31676],[34,409049],{"className":409050,"style":70915},[31680],[34,409052,409054,409057],{"className":409053},[31685],[34,409055,98360],{"className":409056,"style":33389},[31685,31686],[34,409058,409060],{"className":409059},[31792],[34,409061,409063,409083],{"className":409062},[31796,31797],[34,409064,409066,409080],{"className":409065},[31801],[34,409067,409069],{"className":409068,"style":32362},[31805],[34,409070,409071,409074],{"style":231884},[34,409072],{"className":409073,"style":31814},[31813],[34,409075,409077],{"className":409076},[31818,31819,31820,31821],[34,409078,31660],{"className":409079},[31685,31686,31821],[34,409081,31829],{"className":409082},[31828],[34,409084,409086],{"className":409085},[31801],[34,409087,409089],{"className":409088,"style":31836},[31805],[34,409090],{},[34,409092],{"className":409093,"style":31846},[31845],[34,409095,202],{"className":409096},[31850],[34,409098],{"className":409099,"style":31846},[31845],[34,409101,409103,409106],{"className":409102},[31676],[34,409104],{"className":409105,"style":33218},[31680],[34,409107,34386],{"className":409108,"style":34427},[31685,31686],"。\n课件用了一个复杂的归纳法来证明这一点。目的是为了说明这些“乘积”加起来能覆盖整个环，特别是能造出 ",[2627,409111,640],{},[58531,409113],{},[2627,409115,409116],{},"归纳法拆解：",[105,409118,409119,409415,409645,409970,410864],{},[108,409120,324891,409121,258,409190,409259,409260,31901],{},[34,409122,409124,409141],{"className":409123},[31642],[34,409125,409127],{"className":409126},[31646],[31648,409128,409129],{"xmlns":31650},[31652,409130,409131,409139],{},[31655,409132,409133],{},[31743,409134,409135,409137],{},[31658,409136,106063],{},[31658,409138,31955],{},[31665,409140,346750],{"encoding":31667},[34,409142,409144],{"className":409143,"ariaHidden":758},[31672],[34,409145,409147,409150],{"className":409146},[31676],[34,409148],{"className":409149,"style":70915},[31680],[34,409151,409153,409156],{"className":409152},[31685],[34,409154,106063],{"className":409155,"style":67482},[31685,31686],[34,409157,409159],{"className":409158},[31792],[34,409160,409162,409182],{"className":409161},[31796,31797],[34,409163,409165,409179],{"className":409164},[31801],[34,409166,409168],{"className":409167,"style":32090},[31805],[34,409169,409170,409173],{"style":217742},[34,409171],{"className":409172,"style":31814},[31813],[34,409174,409176],{"className":409175},[31818,31819,31820,31821],[34,409177,31955],{"className":409178},[31685,31686,31821],[34,409180,31829],{"className":409181},[31828],[34,409183,409185],{"className":409184},[31801],[34,409186,409188],{"className":409187,"style":31836},[31805],[34,409189],{},[34,409191,409193,409210],{"className":409192},[31642],[34,409194,409196],{"className":409195},[31646],[31648,409197,409198],{"xmlns":31650},[31652,409199,409200,409208],{},[31655,409201,409202],{},[31743,409203,409204,409206],{},[31658,409205,106063],{},[31658,409207,31968],{},[31665,409209,408813],{"encoding":31667},[34,409211,409213],{"className":409212,"ariaHidden":758},[31672],[34,409214,409216,409219],{"className":409215},[31676],[34,409217],{"className":409218,"style":55484},[31680],[34,409220,409222,409225],{"className":409221},[31685],[34,409223,106063],{"className":409224,"style":67482},[31685,31686],[34,409226,409228],{"className":409227},[31792],[34,409229,409231,409251],{"className":409230},[31796,31797],[34,409232,409234,409248],{"className":409233},[31801],[34,409235,409237],{"className":409236,"style":32090},[31805],[34,409238,409239,409242],{"style":217742},[34,409240],{"className":409241,"style":31814},[31813],[34,409243,409245],{"className":409244},[31818,31819,31820,31821],[34,409246,31968],{"className":409247,"style":32153},[31685,31686,31821],[34,409249,31829],{"className":409250},[31828],[34,409252,409254],{"className":409253},[31801],[34,409255,409257],{"className":409256,"style":32163},[31805],[34,409258],{}," 是互素的（",[34,409261,409263,409293],{"className":409262},[31642],[34,409264,409266],{"className":409265},[31646],[31648,409267,409268],{"xmlns":31650},[31652,409269,409270,409290],{},[31655,409271,409272,409278,409280,409286,409288],{},[31743,409273,409274,409276],{},[31658,409275,106063],{},[31658,409277,31955],{},[31662,409279,223],{},[31743,409281,409282,409284],{},[31658,409283,106063],{},[31658,409285,31968],{},[31662,409287,202],{},[31658,409289,34386],{},[31665,409291,409292],{"encoding":31667},"I_i + I_j = R",[34,409294,409296,409351,409406],{"className":409295,"ariaHidden":758},[31672],[34,409297,409299,409302,409342,409345,409348],{"className":409298},[31676],[34,409300],{"className":409301,"style":70915},[31680],[34,409303,409305,409308],{"className":409304},[31685],[34,409306,106063],{"className":409307,"style":67482},[31685,31686],[34,409309,409311],{"className":409310},[31792],[34,409312,409314,409334],{"className":409313},[31796,31797],[34,409315,409317,409331],{"className":409316},[31801],[34,409318,409320],{"className":409319,"style":32090},[31805],[34,409321,409322,409325],{"style":217742},[34,409323],{"className":409324,"style":31814},[31813],[34,409326,409328],{"className":409327},[31818,31819,31820,31821],[34,409329,31955],{"className":409330},[31685,31686,31821],[34,409332,31829],{"className":409333},[31828],[34,409335,409337],{"className":409336},[31801],[34,409338,409340],{"className":409339,"style":31836},[31805],[34,409341],{},[34,409343],{"className":409344,"style":32604},[31845],[34,409346,223],{"className":409347},[32024],[34,409349],{"className":409350,"style":32604},[31845],[34,409352,409354,409357,409397,409400,409403],{"className":409353},[31676],[34,409355],{"className":409356,"style":55484},[31680],[34,409358,409360,409363],{"className":409359},[31685],[34,409361,106063],{"className":409362,"style":67482},[31685,31686],[34,409364,409366],{"className":409365},[31792],[34,409367,409369,409389],{"className":409368},[31796,31797],[34,409370,409372,409386],{"className":409371},[31801],[34,409373,409375],{"className":409374,"style":32090},[31805],[34,409376,409377,409380],{"style":217742},[34,409378],{"className":409379,"style":31814},[31813],[34,409381,409383],{"className":409382},[31818,31819,31820,31821],[34,409384,31968],{"className":409385,"style":32153},[31685,31686,31821],[34,409387,31829],{"className":409388},[31828],[34,409390,409392],{"className":409391},[31801],[34,409393,409395],{"className":409394,"style":32163},[31805],[34,409396],{},[34,409398],{"className":409399,"style":31846},[31845],[34,409401,202],{"className":409402},[31850],[34,409404],{"className":409405,"style":31846},[31845],[34,409407,409409,409412],{"className":409408},[31676],[34,409410],{"className":409411,"style":33218},[31680],[34,409413,34386],{"className":409414,"style":34427},[31685,31686],[108,409416,409417,409420,409421,86674,409494,330585,409567,347],{},[2627,409418,409419],{},"课件引用的命题 3.3.5"," 其实是一个性质：如果 ",[34,409422,409424,409446],{"className":409423},[31642],[34,409425,409427],{"className":409426},[31646],[31648,409428,409429],{"xmlns":31650},[31652,409430,409431,409443],{},[31655,409432,409433,409435,409437,409439,409441],{},[31658,409434,54151],{},[31662,409436,223],{},[31658,409438,54224],{},[31662,409440,202],{},[31658,409442,34386],{},[31665,409444,409445],{"encoding":31667},"A+B=R",[34,409447,409449,409467,409485],{"className":409448,"ariaHidden":758},[31672],[34,409450,409452,409455,409458,409461,409464],{"className":409451},[31676],[34,409453],{"className":409454,"style":34944},[31680],[34,409456,54151],{"className":409457},[31685,31686],[34,409459],{"className":409460,"style":32604},[31845],[34,409462,223],{"className":409463},[32024],[34,409465],{"className":409466,"style":32604},[31845],[34,409468,409470,409473,409476,409479,409482],{"className":409469},[31676],[34,409471],{"className":409472,"style":33218},[31680],[34,409474,54224],{"className":409475,"style":54252},[31685,31686],[34,409477],{"className":409478,"style":31846},[31845],[34,409480,202],{"className":409481},[31850],[34,409483],{"className":409484,"style":31846},[31845],[34,409486,409488,409491],{"className":409487},[31676],[34,409489],{"className":409490,"style":33218},[31680],[34,409492,34386],{"className":409493,"style":34427},[31685,31686],[34,409495,409497,409519],{"className":409496},[31642],[34,409498,409500],{"className":409499},[31646],[31648,409501,409502],{"xmlns":31650},[31652,409503,409504,409516],{},[31655,409505,409506,409508,409510,409512,409514],{},[31658,409507,54151],{},[31662,409509,223],{},[31658,409511,47384],{},[31662,409513,202],{},[31658,409515,34386],{},[31665,409517,409518],{"encoding":31667},"A+C=R",[34,409520,409522,409540,409558],{"className":409521,"ariaHidden":758},[31672],[34,409523,409525,409528,409531,409534,409537],{"className":409524},[31676],[34,409526],{"className":409527,"style":34944},[31680],[34,409529,54151],{"className":409530},[31685,31686],[34,409532],{"className":409533,"style":32604},[31845],[34,409535,223],{"className":409536},[32024],[34,409538],{"className":409539,"style":32604},[31845],[34,409541,409543,409546,409549,409552,409555],{"className":409542},[31676],[34,409544],{"className":409545,"style":33218},[31680],[34,409547,47384],{"className":409548,"style":38706},[31685,31686],[34,409550],{"className":409551,"style":31846},[31845],[34,409553,202],{"className":409554},[31850],[34,409556],{"className":409557,"style":31846},[31845],[34,409559,409561,409564],{"className":409560},[31676],[34,409562],{"className":409563,"style":33218},[31680],[34,409565,34386],{"className":409566,"style":34427},[31685,31686],[34,409568,409570,409594],{"className":409569},[31642],[34,409571,409573],{"className":409572},[31646],[31648,409574,409575],{"xmlns":31650},[31652,409576,409577,409591],{},[31655,409578,409579,409581,409583,409585,409587,409589],{},[31658,409580,54151],{},[31662,409582,223],{},[31658,409584,54224],{},[31658,409586,47384],{},[31662,409588,202],{},[31658,409590,34386],{},[31665,409592,409593],{"encoding":31667},"A+BC=R",[34,409595,409597,409615,409636],{"className":409596,"ariaHidden":758},[31672],[34,409598,409600,409603,409606,409609,409612],{"className":409599},[31676],[34,409601],{"className":409602,"style":34944},[31680],[34,409604,54151],{"className":409605},[31685,31686],[34,409607],{"className":409608,"style":32604},[31845],[34,409610,223],{"className":409611},[32024],[34,409613],{"className":409614,"style":32604},[31845],[34,409616,409618,409621,409624,409627,409630,409633],{"className":409617},[31676],[34,409619],{"className":409620,"style":33218},[31680],[34,409622,54224],{"className":409623,"style":54252},[31685,31686],[34,409625,47384],{"className":409626,"style":38706},[31685,31686],[34,409628],{"className":409629,"style":31846},[31845],[34,409631,202],{"className":409632},[31850],[34,409634],{"className":409635,"style":31846},[31845],[34,409637,409639,409642],{"className":409638},[31676],[34,409640],{"className":409641,"style":33218},[31680],[34,409643,34386],{"className":409644,"style":34427},[31685,31686],[108,409646,409647,325706,409650,409940,409941,409969],{},[2627,409648,409649],{},"归纳假设",[34,409651,409653,409705],{"className":409652},[31642],[34,409654,409656],{"className":409655},[31646],[31648,409657,409658],{"xmlns":31650},[31652,409659,409660,409702],{},[31655,409661,409662,409668,409670,409672,409674,409680,409682,409694,409696],{},[31743,409663,409664,409666],{},[31658,409665,98360],{},[31749,409667,640],{},[31662,409669,223],{},[31662,409671,38874],{},[31662,409673,223],{},[31743,409675,409676,409678],{},[31658,409677,98360],{},[31658,409679,33759],{},[31662,409681,202],{},[31743,409683,409684,409686],{},[31658,409685,106063],{},[31655,409687,409688,409690,409692],{},[31658,409689,33759],{},[31662,409691,223],{},[31749,409693,640],{},[31662,409695,38874],{},[31743,409697,409698,409700],{},[31658,409699,106063],{},[31658,409701,31660],{},[31665,409703,4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+ \\dots + M_k = I_{k+1} \\cdots I_n",[34,409706,409708,409763,409781,409836],{"className":409707,"ariaHidden":758},[31672],[34,409709,409711,409714,409754,409757,409760],{"className":409710},[31676],[34,409712],{"className":409713,"style":70915},[31680],[34,409715,409717,409720],{"className":409716},[31685],[34,409718,98360],{"className":409719,"style":33389},[31685,31686],[34,409721,409723],{"className":409722},[31792],[34,409724,409726,409746],{"className":409725},[31796,31797],[34,409727,409729,409743],{"className":409728},[31801],[34,409730,409732],{"className":409731,"style":31806},[31805],[34,409733,409734,409737],{"style":231884},[34,409735],{"className":409736,"style":31814},[31813],[34,409738,409740],{"className":409739},[31818,31819,31820,31821],[34,409741,640],{"className":409742},[31685,31821],[34,409744,31829],{"className":409745},[31828],[34,409747,409749],{"className":409748},[31801],[34,409750,409752],{"className":409751,"style":31836},[31805],[34,409753],{},[34,409755],{"className":409756,"style":32604},[31845],[34,409758,223],{"className":409759},[32024],[34,409761],{"className":409762,"style":32604},[31845],[34,409764,409766,409769,409772,409775,409778],{"className":409765},[31676],[34,409767],{"className":409768,"style":32597},[31680],[34,409770,38874],{"className":409771},[34006],[34,409773],{"className":409774,"style":32604},[31845],[34,409776,223],{"className":409777},[32024],[34,409779],{"className":409780,"style":32604},[31845],[34,409782,409784,409787,409827,409830,409833],{"className":409783},[31676],[34,409785],{"className":409786,"style":70915},[31680],[34,409788,409790,409793],{"className":409789},[31685],[34,409791,98360],{"className":409792,"style":33389},[31685,31686],[34,409794,409796],{"className":409795},[31792],[34,409797,409799,409819],{"className":409798},[31796,31797],[34,409800,409802,409816],{"className":409801},[31801],[34,409803,409805],{"className":409804,"style":33833},[31805],[34,409806,409807,409810],{"style":231884},[34,409808],{"className":409809,"style":31814},[31813],[34,409811,409813],{"className":409812},[31818,31819,31820,31821],[34,409814,33759],{"className":409815,"style":33774},[31685,31686,31821],[34,409817,31829],{"className":409818},[31828],[34,409820,409822],{"className":409821},[31801],[34,409823,409825],{"className":409824,"style":31836},[31805],[34,409826],{},[34,409828],{"className":409829,"style":31846},[31845],[34,409831,202],{"className":409832},[31850],[34,409834],{"className":409835,"style":31846},[31845],[34,409837,409839,409842,409891,409894,409897,409900],{"className":409838},[31676],[34,409840],{"className":409841,"style":70992},[31680],[34,409843,409845,409848],{"className":409844},[31685],[34,409846,106063],{"className":409847,"style":67482},[31685,31686],[34,409849,409851],{"className":409850},[31792],[34,409852,409854,409883],{"className":409853},[31796,31797],[34,409855,409857,409880],{"className":409856},[31801],[34,409858,409860],{"className":409859,"style":33833},[31805],[34,409861,409862,409865],{"style":217742},[34,409863],{"className":409864,"style":31814},[31813],[34,409866,409868],{"className":409867},[31818,31819,31820,31821],[34,409869,409871,409874,409877],{"className":409870},[31685,31821],[34,409872,33759],{"className":409873,"style":33774},[31685,31686,31821],[34,409875,223],{"className":409876},[32024,31821],[34,409878,640],{"className":409879},[31685,31821],[34,409881,31829],{"className":409882},[31828],[34,409884,409886],{"className":409885},[31801],[34,409887,409889],{"className":409888,"style":32037},[31805],[34,409890],{},[34,409892],{"className":409893,"style":32123},[31845],[34,409895,38874],{"className":409896},[34006],[34,409898],{"className":409899,"style":32123},[31845],[34,409901,409903,409906],{"className":409902},[31685],[34,409904,106063],{"className":409905,"style":67482},[31685,31686],[34,409907,409909],{"className":409908},[31792],[34,409910,409912,409932],{"className":409911},[31796,31797],[34,409913,409915,409929],{"className":409914},[31801],[34,409916,409918],{"className":409917,"style":32362},[31805],[34,409919,409920,409923],{"style":217742},[34,409921],{"className":409922,"style":31814},[31813],[34,409924,409926],{"className":409925},[31818,31819,31820,31821],[34,409927,31660],{"className":409928},[31685,31686,31821],[34,409930,31829],{"className":409931},[31828],[34,409933,409935],{"className":409934},[31801],[34,409936,409938],{"className":409937,"style":31836},[31805],[34,409939],{}," （课件这里写得非常简略，其实意思是前 ",[34,409942,409944,409957],{"className":409943},[31642],[34,409945,409947],{"className":409946},[31646],[31648,409948,409949],{"xmlns":31650},[31652,409950,409951,409955],{},[31655,409952,409953],{},[31658,409954,33759],{},[31665,409956,33759],{"encoding":31667},[34,409958,409960],{"className":409959,"ariaHidden":758},[31672],[34,409961,409963,409966],{"className":409962},[31676],[34,409964],{"className":409965,"style":31681},[31680],[34,409967,33759],{"className":409968,"style":33774},[31685,31686]," 项的和，包含了后面剩余项的乘积，这里有笔误或者是符号借用，我们直接看它的核心推导）。",[108,409971,409972,33183,409975],{},[2627,409973,409974],{},"核心推导",[34,409976,409978],{"className":409977},[126385],[34,409979,409981,410143],{"className":409980},[31642],[34,409982,409984],{"className":409983},[31646],[31648,409985,409986],{"xmlns":31650,"display":126394},[31652,409987,409988,410140],{},[71078,409989,409991,410070],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},"right 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+ \\dots + M_k &= I_{k} \\cdots I_n + \\prod_{j \\neq k} I_j \\quad (\\text{这一步是在拆分}) \\\\\n&= (I_k + I_1 \\cdots I_{k-1}) I_{k+1} \\cdots I_n\n\\end{aligned}",[34,410144,410146],{"className":410145,"ariaHidden":758},[31672],[34,410147,410149,410153],{"className":410148},[31676],[34,410150],{"className":410151,"style":410152},[31680],"height:3.9882em;vertical-align:-1.7441em;",[34,410154,410156],{"className":410155},[31685],[34,410157,410159,410303],{"className":410158},[71078],[34,410160,410162],{"className":410161},[369003],[34,410163,410165,410294],{"className":410164},[31796,31797],[34,410166,410168,410291],{"className":410167},[31801],[34,410169,410172,410282],{"className":410170,"style":410171},[31805],"height:2.2441em;",[34,410173,410175,410178],{"style":410174},"top:-4.2441em;",[34,410176],{"className":410177,"style":131222},[31813],[34,410179,410181,410221,410224,410227,410230,410233,410236,410239,410242],{"className":410180},[31685],[34,410182,410184,410187],{"className":410183},[31685],[34,410185,98360],{"className":410186,"style":33389},[31685,31686],[34,410188,410190],{"className":410189},[31792],[34,410191,410193,410213],{"className":410192},[31796,31797],[34,410194,410196,410210],{"className":410195},[31801],[34,410197,410199],{"className":410198,"style":31806},[31805],[34,410200,410201,410204],{"style":231884},[34,410202],{"className":410203,"style":31814},[31813],[34,410205,410207],{"className":410206},[31818,31819,31820,31821],[34,410208,640],{"className":410209},[31685,31821],[34,410211,31829],{"className":410212},[31828],[34,410214,410216],{"className":410215},[31801],[34,410217,410219],{"className":410218,"style":31836},[31805],[34,410220],{},[34,410222],{"className":410223,"style":32604},[31845],[34,410225,223],{"className":410226},[32024],[34,410228],{"className":410229,"style":32604},[31845],[34,410231,38874],{"className":410232},[34006],[34,410234],{"className":410235,"style":32604},[31845],[34,410237,223],{"className":410238},[32024],[34,410240],{"className":410241,"style":32604},[31845],[34,410243,410245,410248],{"className":410244},[31685],[34,410246,98360],{"className":410247,"style":33389},[31685,31686],[34,410249,410251],{"className":410250},[31792],[34,410252,410254,410274],{"className":410253},[31796,31797],[34,410255,410257,410271],{"className":410256},[31801],[34,410258,410260],{"className":410259,"style":33833},[31805],[34,410261,410262,410265],{"style":231884},[34,410263],{"className":410264,"style":31814},[31813],[34,410266,410268],{"className":410267},[31818,31819,31820,31821],[34,410269,33759],{"className":410270,"style":33774},[31685,31686,31821],[34,410272,31829],{"className":410273},[31828],[34,410275,410277],{"className":410276},[31801],[34,410278,410280],{"className":410279,"style":31836},[31805],[34,410281],{},[34,410283,410285,410288],{"style":410284},"top:-1.6659em;",[34,410286],{"className":410287,"style":131222},[31813],[34,410289],{"className":410290},[31685],[34,410292,31829],{"className":410293},[31828],[34,410295,410297],{"className":410296},[31801],[34,410298,410301],{"className":410299,"style":410300},[31805],"height:1.7441em;",[34,410302],{},[34,410304,410306],{"className":410305},[71384],[34,410307,410309,410856],{"className":410308},[31796,31797],[34,410310,410312,410853],{"className":410311},[31801],[34,410313,410315,410582],{"className":410314,"style":410171},[31805],[34,410316,410317,410320],{"style":410174},[34,410318],{"className":410319,"style":131222},[31813],[34,410321,410323,410326,410329,410332,410335,410378,410381,410384,410387,410427,410430,410433,410436,410524,410527,410567,410570,410573,410579],{"className":410322},[31685],[34,410324],{"className":410325},[31685],[34,410327],{"className":410328,"style":31846},[31845],[34,410330,202],{"className":410331},[31850],[34,410333],{"className":410334,"style":31846},[31845],[34,410336,410338,410341],{"className":410337},[31685],[34,410339,106063],{"className":410340,"style":67482},[31685,31686],[34,410342,410344],{"className":410343},[31792],[34,410345,410347,410370],{"className":410346},[31796,31797],[34,410348,410350,410367],{"className":410349},[31801],[34,410351,410353],{"className":410352,"style":33833},[31805],[34,410354,410355,410358],{"style":217742},[34,410356],{"className":410357,"style":31814},[31813],[34,410359,410361],{"className":410360},[31818,31819,31820,31821],[34,410362,410364],{"className":410363},[31685,31821],[34,410365,33759],{"className":410366,"style":33774},[31685,31686,31821],[34,410368,31829],{"className":410369},[31828],[34,410371,410373],{"className":410372},[31801],[34,410374,410376],{"className":410375,"style":31836},[31805],[34,410377],{},[34,410379],{"className":410380,"style":32123},[31845],[34,410382,38874],{"className":410383},[34006],[34,410385],{"className":410386,"style":32123},[31845],[34,410388,410390,410393],{"className":410389},[31685],[34,410391,106063],{"className":410392,"style":67482},[31685,31686],[34,410394,410396],{"className":410395},[31792],[34,410397,410399,410419],{"className":410398},[31796,31797],[34,410400,410402,410416],{"className":410401},[31801],[34,410403,410405],{"clas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",[34,411116,411118,411142],{"className":411117},[31642],[34,411119,411121],{"className":411120},[31646],[31648,411122,411123],{"xmlns":31650},[31652,411124,411125,411139],{},[31655,411126,411127,411129,411135,411137],{},[31662,411128,131138],{},[31743,411130,411131,411133],{},[31658,411132,98360],{},[31658,411134,31955],{},[31662,411136,202],{},[31658,411138,34386],{},[31665,411140,411141],{"encoding":31667},"\\sum M_i = R",[34,411143,411145,411206],{"className":411144,"ariaHidden":758},[31672],[34,411146,411148,411151,411154,411157,411197,411200,411203],{"className":411147},[31676],[34,411149],{"className":411150,"style":31774},[31680],[34,411152,131138],{"className":411153,"style":156561},[32767,131249,156560],[34,411155],{"className":411156,"style":32123},[31845],[34,411158,411160,411163],{"className":411159},[31685],[34,411161,98360],{"className":411162,"style":33389},[31685,31686],[34,411164,411166],{"className":411165},[31792],[34,411167,411169,411189],{"className":411168},[31796,31797],[34,411170,411172,411186],{"className":411171},[31801],[34,411173,411175],{"className":411174,"style":32090},[31805],[34,411176,411177,411180],{"style":231884},[34,411178],{"className":411179,"style":31814},[31813],[34,411181,411183],{"className":411182},[31818,31819,31820,31821],[34,411184,31955],{"className":411185},[31685,31686,31821],[34,411187,31829],{"className":411188},[31828],[34,411190,411192],{"className":411191},[31801],[34,411193,411195],{"className":411194,"style":31836},[31805],[34,411196],{},[34,411198],{"className":411199,"style":31846},[31845],[34,411201,202],{"className":411202},[31850],[34,411204],{"className":411205,"style":31846},[31845],[34,411207,411209,411212],{"className":411208},[31676],[34,411210],{"className":411211,"style":33218},[31680],[34,411213,34386],{"className":411214,"style":34427},[31685,31686],"，那么单位元 ",[2627,411217,640],{}," 也可以写成它们的和。\n存在 ",[34,411220,411222,411248],{"className":411221},[31642],[34,411223,411225],{"className":411224},[31646],[31648,411226,411227],{"xmlns":31650},[31652,411228,411229,411245],{},[31655,411230,411231,411237,411239],{},[31743,411232,411233,411235],{},[31658,411234,31759],{},[31658,411236,31955],{},[31662,411238,112296],{},[31743,411240,411241,411243],{},[31658,411242,98360],{},[31658,411244,31955],{},[31665,411246,411247],{"encoding":31667},"m_i \\in M_i",[34,411249,411251,411306],{"className":411250,"ariaHidden":758},[31672],[34,411252,411254,411257,411297,411300,411303],{"className":411253},[31676],[34,411255],{"className":411256,"style":225773},[31680],[34,411258,411260,411263],{"className":411259},[31685],[34,411261,31759],{"className":411262},[31685,31686],[34,411264,411266],{"className":411265},[31792],[34,411267,411269,411289],{"className":411268},[31796,31797],[34,411270,411272,411286],{"className":411271},[31801],[34,411273,411275],{"className":411274,"style":32090},[31805],[34,411276,411277,411280],{"style":31809},[34,411278],{"className":411279,"style":31814},[31813],[34,411281,411283],{"className":411282},[31818,31819,31820,31821],[34,411284,31955],{"className":411285},[31685,31686,31821],[34,411287,31829],{"className":411288},[31828],[34,411290,411292],{"className":411291},[31801],[34,411293,411295],{"className":411294,"style":31836},[31805],[34,411296],{},[34,411298],{"className":411299,"style":31846},[31845],[34,411301,112296],{"className":411302},[31850],[34,411304],{"className":411305,"style":31846},[31845],[34,411307,411309,411312],{"className":411308},[31676],[34,411310],{"className":411311,"style":70915},[31680],[34,411313,411315,411318],{"className":411314},[31685],[34,411316,98360],{"className":411317,"style":33389},[31685,31686],[34,411319,411321],{"className":411320},[31792],[34,411322,411324,411344],{"className":411323},[31796,31797],[34,411325,411327,411341],{"className":411326},[31801],[34,411328,411330],{"className":411329,"style":32090},[31805],[34,411331,411332,411335],{"style":231884},[34,411333],{"className":411334,"style":31814},[31813],[34,411336,411338],{"className":411337},[31818,31819,31820,31821],[34,411339,31955],{"className":411340},[31685,31686,31821],[34,411342,31829],{"className":411343},[31828],[34,411345,411347],{"className":411346},[31801],[34,411348,411350],{"className":411349,"style":31836},[31805],[34,411351],{},[34,411353,411355,411397],{"className":411354},[31642],[34,411356,411358],{"className":411357},[31646],[31648,411359,411360],{"xmlns":31650},[31652,411361,411362,411394],{},[31655,411363,411364,411366,411368,411374,411376,411382,411384,411386,411388],{},[31749,411365,640],{},[31662,411367,202],{},[31743,411369,411370,411372],{},[31658,411371,31759],{},[31749,411373,640],{},[31662,411375,223],{},[31743,411377,411378,411380],{},[31658,411379,31759],{},[31749,411381,768],{},[31662,411383,223],{},[31662,411385,38874],{},[31662,411387,223],{},[31743,411389,411390,411392],{},[31658,411391,31759],{},[31658,411393,31660],{},[31665,411395,411396],{"encoding":31667},"1 = m_1 + m_2 + \\dots + m_n",[34,411398,411400,411418,411473,411528,411546],{"className":411399,"ariaHidden":758},[31672],[34,411401,411403,411406,411409,411412,411415],{"className":411402},[31676],[34,411404],{"className":411405,"style":32617},[31680],[34,411407,640],{"className":411408},[31685],[34,411410],{"className":411411,"style":31846},[31845],[34,411413,202],{"className":411414},[31850],[34,411416],{"className":411417,"style":31846},[31845],[34,411419,411421,411424,411464,411467,411470],{"className":411420},[31676],[34,411422],{"className":411423,"style":40522},[31680],[34,411425,411427,411430],{"className":411426},[31685],[34,411428,31759],{"className":411429},[31685,31686],[34,411431,411433],{"className":411432},[31792],[34,411434,411436,411456],{"className":411435},[31796,31797],[34,411437,411439,411453],{"className":411438},[31801],[34,411440,411442],{"className":411441,"style":31806},[31805],[34,411443,411444,411447],{"style":31809},[34,411445],{"className":411446,"style":31814},[31813],[34,411448,411450],{"className":411449},[31818,31819,31820,31821],[34,411451,640],{"className":411452},[31685,31821],[34,411454,31829],{"className":411455},[31828],[34,411457,411459],{"className":411458},[31801],[34,411460,411462],{"className":411461,"style":31836},[31805],[34,411463],{},[34,411465],{"className":411466,"style":32604},[31845],[34,411468,223],{"className":411469},[32024],[34,411471],{"className":411472,"style":32604},[31845],[34,411474,411476,411479,411519,411522,411525],{"className":411475},[31676],[34,411477],{"className":411478,"style":40522},[31680],[34,411480,411482,411485],{"className":411481},[31685],[34,411483,31759],{"className":411484},[31685,31686],[34,411486,411488],{"className":411487},[31792],[34,411489,411491,411511],{"className":411490},[31796,31797],[34,411492,411494,411508],{"className":411493},[31801],[34,411495,411497],{"className":411496,"style":31806},[31805],[34,411498,411499,411502],{"style":31809},[34,411500],{"className":411501,"style":31814},[31813],[34,411503,411505],{"className":411504},[31818,31819,31820,31821],[34,411506,768],{"className":411507},[31685,31821],[34,411509,31829],{"className":411510},[31828],[34,411512,411514],{"className":411513},[31801],[34,411515,411517],{"className":411516,"style":31836},[31805],[34,411518],{},[34,411520],{"className":411521,"style":32604},[31845],[34,411523,223],{"className":411524},[32024],[34,411526],{"className":411527,"style":32604},[31845],[34,411529,411531,411534,411537,411540,411543],{"className":411530},[31676],[34,411532],{"className":411533,"style":32597},[31680],[34,411535,38874],{"className":411536},[34006],[34,411538],{"className":411539,"style":32604},[31845],[34,411541,223],{"className":411542},[32024],[34,411544],{"className":411545,"style":32604},[31845],[34,411547,411549,411552],{"className":411548},[31676],[34,411550],{"className":411551,"style":31860},[31680],[34,411553,411555,411558],{"className":411554},[31685],[34,411556,31759],{"className":411557},[31685,31686],[34,411559,411561],{"className":411560},[31792],[34,411562,411564,411584],{"className":411563},[31796,31797],[34,411565,411567,411581],{"className":411566},[31801],[34,411568,411570],{"className":411569,"style":32362},[31805],[34,411571,411572,411575],{"style":31809},[34,411573],{"className":411574,"style":31814},[31813],[34,411576,411578],{"className":411577},[31818,31819,31820,31821],[34,411579,31660],{"className":411580},[31685,31686,31821],[34,411582,31829],{"className":411583},[31828],[34,411585,411587],{"className":411586},[31801],[34,411588,411590],{"className":411589,"style":31836},[31805],[34,411591],{},[58531,411593],{},[2627,411595,411596],{},"请盯着这个式子看，这是证明的灵魂：",[105,411598,411599,412145],{},[108,411600,411601,411602,411672,411673,411742,411743,11008,411812,411943,411944,314945,412013,412144],{},"考察 ",[34,411603,411605,411623],{"className":411604},[31642],[34,411606,411608],{"className":411607},[31646],[31648,411609,411610],{"xmlns":31650},[31652,411611,411612,411620],{},[31655,411613,411614],{},[31743,411615,411616,411618],{},[31658,411617,31759],{},[31749,411619,640],{},[31665,411621,411622],{"encoding":31667},"m_1",[34,411624,411626],{"className":411625,"ariaHidden":758},[31672],[34,411627,411629,411632],{"className":411628},[31676],[34,411630],{"className":411631,"style":31860},[31680],[34,411633,411635,411638],{"className":411634},[31685],[34,411636,31759],{"className":411637},[31685,31686],[34,411639,411641],{"className":411640},[31792],[34,411642,411644,411664],{"className":411643},[31796,31797],[34,411645,411647,411661],{"className":411646},[31801],[34,411648,411650],{"className":411649,"style":31806},[31805],[34,411651,411652,411655],{"style":31809},[34,411653],{"className":411654,"style":31814},[31813],[34,411656,411658],{"className":411657},[31818,31819,31820,31821],[34,411659,640],{"className":411660},[31685,31821],[34,411662,31829],{"className":411663},[31828],[34,411665,411667],{"className":411666},[31801],[34,411668,411670],{"className":411669,"style":31836},[31805],[34,411671],{},"：它属于 ",[34,411674,411676,411693],{"className":411675},[31642],[34,411677,411679],{"className":411678},[31646],[31648,411680,411681],{"xmlns":31650},[31652,411682,411683,411691],{},[31655,411684,411685],{},[31743,411686,411687,411689],{},[31658,411688,98360],{},[31749,411690,640],{},[31665,411692,407855],{"encoding":31667},[34,411694,411696],{"className":411695,"ariaHidden":758},[31672],[34,411697,411699,411702],{"className":411698},[31676],[34,411700],{"className":411701,"style":70915},[31680],[34,411703,411705,411708],{"className":411704},[31685],[34,411706,98360],{"className":411707,"style":33389},[31685,31686],[34,411709,411711],{"className":411710},[31792],[34,411712,411714,411734],{"className":411713},[31796,31797],[34,411715,411717,411731],{"className":411716},[31801],[34,411718,411720],{"className":411719,"style":31806},[31805],[34,411721,411722,411725],{"style":231884},[34,411723],{"className":411724,"style":31814},[31813],[34,411726,411728],{"className":411727},[31818,31819,31820,31821],[34,411729,640],{"className":411730},[31685,31821],[34,411732,31829],{"className":411733},[31828],[34,411735,411737],{"className":411736},[31801],[34,411738,411740],{"className":411739,"style":31836},[31805],[34,411741],{},"。由定义，",[34,411744,411746,411763],{"className":411745},[31642],[34,411747,411749],{"className":411748},[31646],[31648,411750,411751],{"xmlns":31650},[31652,411752,411753,411761],{},[31655,411754,411755],{},[31743,411756,411757,411759],{},[31658,411758,98360],{},[31749,411760,640],{},[31665,411762,407855],{"encoding":31667},[34,411764,411766],{"className":411765,"ariaHidden":758},[31672],[34,411767,411769,411772],{"className":411768},[31676],[34,411770],{"className":411771,"style":70915},[31680],[34,411773,411775,411778],{"className":411774},[31685],[34,411776,98360],{"className":411777,"style":33389},[31685,31686],[34,411779,411781],{"className":411780},[31792],[34,411782,411784,411804],{"className":411783},[31796,31797],[34,411785,411787,411801],{"className":411786},[31801],[34,411788,411790],{"className":411789,"style":31806},[31805],[34,411791,411792,411795],{"style":231884},[34,411793],{"className":411794,"style":31814},[31813],[34,411796,411798],{"className":411797},[31818,31819,31820,31821],[34,411799,640],{"className":411800},[31685,31821],[34,411802,31829],{"className":411803},[31828],[34,411805,411807],{"className":411806},[31801],[34,411808,411810],{"className":411809,"style":31836},[31805],[34,411811],{},[34,411813,411815,411842],{"className":411814},[31642],[34,411816,411818],{"className":411817},[31646],[31648,411819,411820],{"xmlns":31650},[31652,411821,411822,411840],{},[31655,411823,411824,411830,411832,411838],{},[31743,411825,411826,411828],{},[31658,411827,106063],{},[31749,411829,768],{},[31662,411831,31960],{"separator":758},[31743,411833,411834,411836],{},[31658,411835,106063],{},[31749,411837,786],{},[31662,411839,33934],{},[31665,411841,408268],{"encoding":31667},[34,411843,411845],{"className":411844,"ariaHidden":758},[31672],[34,411846,411848,411851,411891,411894,411897,411937,411940],{"className":411847},[31676],[34,411849],{"className":411850,"style":70739},[31680],[34,411852,411854,411857],{"className":411853},[31685],[34,411855,106063],{"className":411856,"style":67482},[31685,31686],[34,411858,411860],{"className":411859},[31792],[34,411861,411863,411883],{"className":411862},[31796,31797],[34,411864,411866,411880],{"className":411865},[31801],[34,411867,411869],{"className":411868,"style":31806},[31805],[34,411870,411871,411874],{"style":217742},[34,411872],{"className":411873,"style":31814},[31813],[34,411875,411877],{"className":411876},[31818,31819,31820,31821],[34,411878,768],{"className":411879},[31685,31821],[34,411881,31829],{"className":411882},[31828],[34,411884,411886],{"className":411885},[31801],[34,411887,411889],{"className":411888,"style":31836},[31805],[34,411890],{},[34,411892,31960],{"className":411893},[32119],[34,411895],{"className":411896,"style":32123},[31845],[34,411898,411900,411903],{"className":411899},[31685],[34,411901,106063],{"className":411902,"style":67482},[31685,31686],[34,411904,411906],{"className":411905},[31792],[34,411907,411909,411929],{"className":411908},[31796,31797],[34,411910,411912,411926],{"className":411911},[31801],[34,411913,411915],{"className":411914,"style":31806},[31805],[34,411916,411917,411920],{"style":217742},[34,411918],{"className":411919,"style":31814},[31813],[34,411921,411923],{"className":411922},[31818,31819,31820,31821],[34,411924,786],{"className":411925},[31685,31821],[34,411927,31829],{"className":411928},[31828],[34,411930,411932],{"className":411931},[31801],[34,411933,411935],{"className":411934,"style":31836},[31805],[34,411936],{},[34,411938],{"className":411939,"style":32123},[31845],[34,411941,33934],{"className":411942},[34006]," 的乘积。所以 ",[34,411945,411947,411964],{"className":411946},[31642],[34,411948,411950],{"className":411949},[31646],[31648,411951,411952],{"xmlns":31650},[31652,411953,411954,411962],{},[31655,411955,411956],{},[31743,411957,411958,411960],{},[31658,411959,31759],{},[31749,411961,640],{},[31665,411963,411622],{"encoding":31667},[34,411965,411967],{"className":411966,"ariaHidden":758},[31672],[34,411968,411970,411973],{"className":411969},[31676],[34,411971],{"className":411972,"style":31860},[31680],[34,411974,411976,411979],{"className":411975},[31685],[34,411977,31759],{"className":411978},[31685,31686],[34,411980,411982],{"className":411981},[31792],[34,411983,411985,412005],{"className":411984},[31796,31797],[34,411986,411988,412002],{"className":411987},[31801],[34,411989,411991],{"className":411990,"style":31806},[31805],[34,411992,411993,411996],{"style":31809},[34,411994],{"className":411995,"style":31814},[31813],[34,411997,411999],{"className":411998},[31818,31819,31820,31821],[34,412000,640],{"className":412001},[31685,31821],[34,412003,31829],{"className":412004},[31828],[34,412006,412008],{"className":412007},[31801],[34,412009,412011],{"className":412010,"style":31836},[31805],[34,412012],{},[34,412014,412016,412043],{"className":412015},[31642],[34,412017,412019],{"className":412018},[31646],[31648,412020,412021],{"xmlns":31650},[31652,412022,412023,412041],{},[31655,412024,412025,412031,412033,412039],{},[31743,412026,412027,412029],{},[31658,412028,106063],{},[31749,412030,768],{},[31662,412032,31960],{"separator":758},[31743,412034,412035,412037],{},[31658,412036,106063],{},[31749,412038,786],{},[31662,412040,33934],{},[31665,412042,408268],{"encoding":31667},[34,412044,412046],{"className":412045,"ariaHidden":758},[31672],[34,412047,412049,412052,412092,412095,412098,412138,412141],{"className":412048},[31676],[34,412050],{"className":412051,"style":70739},[31680],[34,412053,412055,412058],{"className":412054},[31685],[34,412056,106063],{"className":412057,"style":67482},[31685,31686],[34,412059,412061],{"className":412060},[31792],[34,412062,412064,412084],{"className":412063},[31796,31797],[34,412065,412067,412081],{"className":412066},[31801],[34,412068,412070],{"className":412069,"style":31806},[31805],[34,412071,412072,412075],{"style":217742},[34,412073],{"className":412074,"style":31814},[31813],[34,412076,412078],{"className":412077},[31818,31819,31820,31821],[34,412079,768],{"className":412080},[31685,31821],[34,412082,31829],{"className":412083},[31828],[34,412085,412087],{"className":412086},[31801],[34,412088,412090],{"className":412089,"style":31836},[31805],[34,412091],{},[34,412093,31960],{"className":412094},[32119],[34,412096],{"className":412097,"style":32123},[31845],[34,412099,412101,412104],{"className":412100},[31685],[34,412102,106063],{"className":412103,"style":67482},[31685,31686],[34,412105,412107],{"className":412106},[31792],[34,412108,412110,412130],{"className":412109},[31796,31797],[34,412111,412113,412127],{"className":412112},[31801],[34,412114,412116],{"className":412115,"style":31806},[31805],[34,412117,412118,412121],{"style":217742},[34,412119],{"className":412120,"style":31814},[31813],[34,412122,412124],{"className":412123},[31818,31819,31820,31821],[34,412125,786],{"className":412126},[31685,31821],[34,412128,31829],{"className":412129},[31828],[34,412131,412133],{"className":412132},[31801],[34,412134,412136],{"className":412135,"style":31836},[31805],[34,412137],{},[34,412139],{"className":412140,"style":32123},[31845],[34,412142,33934],{"className":412143},[34006]," 里都是 0。",[108,412146,412147,412148,33183,412217,315440,412551,316833,412683,412807,412808,412877,412878,412947,412948],{},"考察上面的等式两边同时模 ",[34,412149,412151,412168],{"className":412150},[31642],[34,412152,412154],{"className":412153},[31646],[31648,412155,412156],{"xmlns":31650},[31652,412157,412158,412166],{},[31655,412159,412160],{},[31743,412161,412162,412164],{},[31658,412163,106063],{},[31749,412165,640],{},[31665,412167,348359],{"encoding":31667},[34,412169,412171],{"className":412170,"ariaHidden":758},[31672],[34,412172,412174,412177],{"className":412173},[31676],[34,412175],{"className":412176,"style":70915},[31680],[34,412178,412180,412183],{"className":412179},[31685],[34,412181,106063],{"className":412182,"style":67482},[31685,31686],[34,412184,412186],{"className":412185},[31792],[34,412187,412189,412209],{"className":412188},[31796,31797],[34,412190,412192,412206],{"className":412191},[31801],[34,412193,412195],{"className":412194,"style":31806},[31805],[34,412196,412197,412200],{"style":217742},[34,412198],{"className":412199,"style":31814},[31813],[34,412201,412203],{"className":412202},[31818,31819,31820,31821],[34,412204,640],{"className":412205},[31685,31821],[34,412207,31829],{"className":412208},[31828],[34,412210,412212],{"className":412211},[31801],[34,412213,412215],{"className":412214,"style":31836},[31805],[34,412216],{},[34,412218,412220,412286],{"className":412219},[31642],[34,412221,412223],{"className":412222},[31646],[31648,412224,412225],{"xmlns":31650},[31652,412226,412227,412283],{},[31655,412228,412229,412231,412233,412239,412241,412247,412249,412251,412253,412259,412261,412263,412265,412273,412275,412281],{},[31749,412230,640],{},[31662,412232,341696],{},[31743,412234,412235,412237],{},[31658,412236,31759],{},[31749,412238,640],{},[31662,412240,223],{},[31743,412242,412243,412245],{},[31658,412244,31759],{},[31749,412246,768],{},[31662,412248,223],{},[31662,412250,38874],{},[31662,412252,223],{},[31743,412254,412255,412257],{},[31658,412256,31759],{},[31658,412258,31660],{},[31845,412260],{},[31845,412262],{"width":208910},[31662,412264,456],{"stretchy":702},[31655,412266,412267,412269,412271],{},[31658,412268,31759],{"mathvariant":38079},[31658,412270,43383],{"mathvariant":38079},[31658,412272,46900],{"mathvariant":38079},[31845,412274],{"width":208923},[31743,412276,412277,412279],{},[31658,412278,106063],{},[31749,412280,640],{},[31662,412282,2557],{"stretchy":702},[31665,412284,412285],{"encoding":31667},"1 \\equiv m_1 + m_2 + \\dots + m_n \\pmod{I_1}",[34,412287,412289,412307,412362,412417,412435,412487],{"className":412288,"ariaHidden":758},[31672],[34,412290,412292,412295,412298,412301,412304],{"className":412291},[31676],[34,412293],{"className":412294,"style":32617},[31680],[34,412296,640],{"className":412297},[31685],[34,412299],{"className":412300,"style":31846},[31845],[34,412302,341696],{"className":412303},[31850],[34,412305],{"className":412306,"style":31846},[31845],[34,412308,412310,412313,412353,412356,412359],{"className":412309},[31676],[34,412311],{"className":412312,"style":40522},[31680],[34,412314,412316,412319],{"className":412315},[31685],[34,412317,31759],{"className":412318},[31685,31686],[34,412320,412322],{"className":412321},[31792],[34,412323,412325,412345],{"className":412324},[31796,31797],[34,412326,412328,412342],{"className":412327},[31801],[34,412329,412331],{"className":412330,"style":31806},[31805],[34,412332,412333,412336],{"style":31809},[34,412334],{"className":412335,"style":31814},[31813],[34,412337,412339],{"className":412338},[31818,31819,31820,31821],[34,412340,640],{"className":412341},[31685,31821],[34,412343,31829],{"className":412344},[31828],[34,412346,412348],{"className":412347},[31801],[34,412349,412351],{"className":412350,"style":31836},[31805],[34,412352],{},[34,412354],{"className":412355,"style":32604},[31845],[34,412357,223],{"className":412358},[32024],[34,412360],{"className":412361,"style":32604},[31845],[34,412363,412365,412368,412408,412411,412414],{"className":412364},[31676],[34,412366],{"className":412367,"style":40522},[31680],[34,412369,412371,412374],{"className":412370},[31685],[34,412372,31759],{"className":412373},[31685,31686],[34,412375,412377],{"className":412376},[31792],[34,412378,412380,412400],{"className":412379},[31796,31797],[34,412381,412383,412397],{"className":412382},[31801],[34,412384,412386],{"className":412385,"style":31806},[31805],[34,412387,412388,412391],{"style":31809},[34,412389],{"className":412390,"style":31814},[31813],[34,412392,412394],{"className":412393},[31818,31819,31820,31821],[34,412395,768],{"className":412396},[31685,31821],[34,412398,31829],{"className":412399},[31828],[34,412401,412403],{"className":412402},[31801],[34,412404,412406],{"className":412405,"style":31836},[31805],[34,412407],{},[34,412409],{"className":412410,"style":32604},[31845],[34,412412,223],{"className":412413},[32024],[34,412415],{"className":412416,"style":32604},[31845],[34,412418,412420,412423,412426,412429,412432],{"className":412419},[31676],[34,412421],{"className":412422,"style":32597},[31680],[34,412424,38874],{"className":412425},[34006],[34,412427],{"className":412428,"style":32604},[31845],[34,412430,223],{"className":412431},[32024],[34,412433],{"className":412434,"style":32604},[31845],[34,412436,412438,412441,412481,412484],{"className":412437},[31676],[34,412439],{"className":412440,"style":31860},[31680],[34,412442,412444,412447],{"className":412443},[31685],[34,412445,31759],{"className":412446},[31685,31686],[34,412448,412450],{"className":412449},[31792],[34,412451,412453,412473],{"className":412452},[31796,31797],[34,412454,412456,412470],{"className":412455},[31801],[34,412457,412459],{"className":412458,"style":32362},[31805],[34,412460,412461,412464],{"style":31809},[34,412462],{"className":412463,"style":31814},[31813],[34,412465,412467],{"className":412466},[31818,31819,31820,31821],[34,412468,31660],{"className":412469},[31685,31686,31821],[34,412471,31829],{"className":412472},[31828],[34,412474,412476],{"className":412475},[31801],[34,412477,412479],{"className":412478,"style":31836},[31805],[34,412480],{},[34,412482],{"className":412483},[31845,208951],[34,412485],{"className":412486,"style":208955},[31845],[34,412488,412490,412493,412496,412505,412508,412548],{"className":412489},[31676],[34,412491],{"className":412492,"style":31774},[31680],[34,412494,456],{"className":412495},[31782],[34,412497,412499],{"className":412498},[31685],[34,412500,412502],{"className":412501},[31685],[34,412503,207924],{"className":412504},[31685,207923],[34,412506],{"className":412507,"style":208977},[31845],[34,412509,412511,412514],{"className":412510},[31685],[34,412512,106063],{"className":412513,"style":67482},[31685,31686],[34,412515,412517],{"className":412516},[31792],[34,412518,412520,412540],{"className":412519},[31796,31797],[34,412521,412523,412537],{"className":412522},[31801],[34,412524,412526],{"className":412525,"style":31806},[31805],[34,412527,412528,412531],{"style":217742},[34,412529],{"className":412530,"style":31814},[31813],[34,412532,412534],{"className":412533},[31818,31819,31820,31821],[34,412535,640],{"className":412536},[31685,31821],[34,412538,31829],{"className":412539},[31828],[34,412541,412543],{"className":412542},[31801],[34,412544,412546],{"className":412545,"style":31836},[31805],[34,412547],{},[34,412549,2557],{"className":412550},[31690],[34,412552,412554,412582],{"className":412553},[31642],[34,412555,412557],{"className":412556},[31646],[31648,412558,412559],{"xmlns":31650},[31652,412560,412561,412579],{},[31655,412562,412563,412569,412571,412577],{},[31743,412564,412565,412567],{},[31658,412566,31759],{},[31749,412568,768],{},[31662,412570,31960],{"separator":758},[31743,412572,412573,412575],{},[31658,412574,31759],{},[31749,412576,786],{},[31662,412578,33934],{},[31665,412580,412581],{"encoding":31667},"m_2, m_3 \\dots",[34,412583,412585],{"className":412584,"ariaHidden":758},[31672],[34,412586,412588,412591,412631,412634,412637,412677,412680],{"className":412587},[31676],[34,412589],{"className":412590,"style":38774},[31680],[34,412592,412594,412597],{"className":412593},[31685],[34,412595,31759],{"className":412596},[31685,31686],[34,412598,412600],{"className":412599},[31792],[34,412601,412603,412623],{"className":412602},[31796,31797],[34,412604,412606,412620],{"className":412605},[31801],[34,412607,412609],{"className":412608,"style":31806},[31805],[34,412610,412611,412614],{"style":31809},[34,412612],{"className":412613,"style":31814},[31813],[34,412615,412617],{"className":412616},[31818,31819,31820,31821],[34,412618,768],{"className":412619},[31685,31821],[34,412621,31829],{"className":412622},[31828],[34,412624,412626],{"className":412625},[31801],[34,412627,412629],{"className":412628,"style":31836},[31805],[34,412630],{},[34,412632,31960],{"className":412633},[32119],[34,412635],{"className":412636,"style":32123},[31845],[34,412638,412640,412643],{"className":412639},[31685],[34,412641,31759],{"className":412642},[31685,31686],[34,412644,412646],{"className":412645},[31792],[34,412647,412649,412669],{"className":412648},[31796,31797],[34,412650,412652,412666],{"className":412651},[31801],[34,412653,412655],{"className":412654,"style":31806},[31805],[34,412656,412657,412660],{"style":31809},[34,412658],{"className":412659,"style":31814},[31813],[34,412661,412663],{"className":412662},[31818,31819,31820,31821],[34,412664,786],{"className":412665},[31685,31821],[34,412667,31829],{"className":412668},[31828],[34,412670,412672],{"className":412671},[31801],[34,412673,412675],{"className":412674,"style":31836},[31805],[34,412676],{},[34,412678],{"className":412679,"style":32123},[31845],[34,412681,33934],{"className":412682},[34006],[34,412684,412686,412712],{"className":412685},[31642],[34,412687,412689],{"className":412688},[31646],[31648,412690,412691],{"xmlns":31650},[31652,412692,412693,412709],{},[31655,412694,412695,412701,412703],{},[31743,412696,412697,412699],{},[31658,412698,98360],{},[31749,412700,768],{},[31662,412702,31960],{"separator":758},[31743,412704,412705,412707],{},[31658,412706,98360],{},[31749,412708,786],{},[31665,412710,412711],{"encoding":31667},"M_2, M_3",[34,412713,412715],{"className":412714,"ariaHidden":758},[31672],[34,412716,412718,412721,412761,412764,412767],{"className":412717},[31676],[34,412719],{"className":412720,"style":70739},[31680],[34,412722,412724,412727],{"className":412723},[31685],[34,412725,98360],{"className":412726,"style":33389},[31685,31686],[34,412728,412730],{"className":412729},[31792],[34,412731,412733,412753],{"className":412732},[31796,31797],[34,412734,412736,412750],{"className":412735},[31801],[34,412737,412739],{"className":412738,"style":31806},[31805],[34,412740,412741,412744],{"style":231884},[34,412742],{"className":412743,"style":31814},[31813],[34,412745,412747],{"className":412746},[31818,31819,31820,31821],[34,412748,768],{"className":412749},[31685,31821],[34,412751,31829],{"className":412752},[31828],[34,412754,412756],{"className":412755},[31801],[34,412757,412759],{"className":412758,"style":31836},[31805],[34,412760],{},[34,412762,31960],{"className":412763},[32119],[34,412765],{"className":412766,"style":32123},[31845],[34,412768,412770,412773],{"className":412769},[31685],[34,412771,98360],{"className":412772,"style":33389},[31685,31686],[34,412774,412776],{"className":412775},[31792],[34,412777,412779,412799],{"className":412778},[31796,31797],[34,412780,412782,412796],{"className":412781},[31801],[34,412783,412785],{"className":412784,"style":31806},[31805],[34,412786,412787,412790],{"style":231884},[34,412788],{"className":412789,"style":31814},[31813],[34,412791,412793],{"className":412792},[31818,31819,31820,31821],[34,412794,786],{"className":412795},[31685,31821],[34,412797,31829],{"className":412798},[31828],[34,412800,412802],{"className":412801},[31801],[34,412803,412805],{"className":412804,"style":31836},[31805],[34,412806],{}," 里（都包含 ",[34,412809,412811,412828],{"className":412810},[31642],[34,412812,412814],{"className":412813},[31646],[31648,412815,412816],{"xmlns":31650},[31652,412817,412818,412826],{},[31655,412819,412820],{},[31743,412821,412822,412824],{},[31658,412823,106063],{},[31749,412825,640],{},[31665,412827,348359],{"encoding":31667},[34,412829,412831],{"className":412830,"ariaHidden":758},[31672],[34,412832,412834,412837],{"className":412833},[31676],[34,412835],{"className":412836,"style":70915},[31680],[34,412838,412840,412843],{"className":412839},[31685],[34,412841,106063],{"className":412842,"style":67482},[31685,31686],[34,412844,412846],{"className":412845},[31792],[34,412847,412849,412869],{"className":412848},[31796,31797],[34,412850,412852,412866],{"className":412851},[31801],[34,412853,412855],{"className":412854,"style":31806},[31805],[34,412856,412857,412860],{"style":217742},[34,412858],{"className":412859,"style":31814},[31813],[34,412861,412863],{"className":412862},[31818,31819,31820,31821],[34,412864,640],{"className":412865},[31685,31821],[34,412867,31829],{"className":412868},[31828],[34,412870,412872],{"className":412871},[31801],[34,412873,412875],{"className":412874,"style":31836},[31805],[34,412876],{}," 这个因子），所以它们模 ",[34,412879,412881,412898],{"className":412880},[31642],[34,412882,412884],{"className":412883},[31646],[31648,412885,412886],{"xmlns":31650},[31652,412887,412888,412896],{},[31655,412889,412890],{},[31743,412891,412892,412894],{},[31658,412893,106063],{},[31749,412895,640],{},[31665,412897,348359],{"encoding":31667},[34,412899,412901],{"className":412900,"ariaHidden":758},[31672],[34,412902,412904,412907],{"className":412903},[31676],[34,412905],{"className":412906,"style":70915},[31680],[34,412908,412910,412913],{"className":412909},[31685],[34,412911,106063],{"className":412912,"style":67482},[31685,31686],[34,412914,412916],{"className":412915},[31792],[34,412917,412919,412939],{"className":412918},[31796,31797],[34,412920,412922,412936],{"className":412921},[31801],[34,412923,412925],{"className":412924,"style":31806},[31805],[34,412926,412927,412930],{"style":217742},[34,412928],{"className":412929,"style":31814},[31813],[34,412931,412933],{"className":412932},[31818,31819,31820,31821],[34,412934,640],{"className":412935},[31685,31821],[34,412937,31829],{"className":412938},[31828],[34,412940,412942],{"className":412941},[31801],[34,412943,412945],{"className":412944,"style":31836},[31805],[34,412946],{}," 都是 0。\n",[34,412949,412951,413049],{"className":412950},[31642],[34,412952,412954],{"className":412953},[31646],[31648,412955,412956],{"xmlns":31650},[31652,412957,412958,413046],{},[31655,412959,412960,412962,412964,412970,412972,412974,412976,412978,412980,412982,412984,412986,412988,412996,412998,413004,413006,413008,413010,413012,413018,413020,413022,413024,413026,413028,413036,413038,413044],{},[31749,412961,640],{},[31662,412963,341696],{},[31743,412965,412966,412968],{},[31658,412967,31759],{},[31749,412969,640],{},[31662,412971,223],{},[31749,412973,459],{},[31662,412975,223],{},[31662,412977,38874],{},[31662,412979,223],{},[31749,412981,459],{},[31845,412983],{},[31845,412985],{"width":208910},[31662,412987,456],{"stretchy":702},[31655,412989,412990,412992,412994],{},[31658,412991,31759],{"mathvariant":38079},[31658,412993,43383],{"mathvariant":38079},[31658,412995,46900],{"mathvariant":38079},[31845,412997],{"width":208923},[31743,412999,413000,413002],{},[31658,413001,106063],{},[31749,413003,640],{},[31662,413005,2557],{"stretchy":702},[39361,413007,262622],{},[31662,413009,262625],{},[39361,413011,262622],{},[31743,413013,413014,413016],{},[31658,413015,31759],{},[31749,413017,640],{},[31662,413019,341696],{},[31749,413021,640],{},[31845,413023],{},[31845,413025],{"width":208910},[31662,413027,456],{"stretchy":702},[31655,413029,413030,413032,413034],{},[31658,413031,31759],{"mathvariant":38079},[31658,413033,43383],{"mathvariant":38079},[31658,413035,46900],{"mathvariant":38079},[31845,413037],{"width":208923},[31743,413039,413040,413042],{},[31658,413041,106063],{},[31749,413043,640],{},[31662,413045,2557],{"stretchy":702},[31665,413047,413048],{"encoding":31667},"1 \\equiv m_1 + 0 + \\dots + 0 \\pmod{I_1} \\implies m_1 \\equiv 1 \\pmod{I_1}",[34,413050,413052,413070,413125,413143,413161,413176,413255,413311,413326],{"className":413051,"ariaHidden":758},[31672],[34,413053,413055,413058,413061,413064,413067],{"className":413054},[31676],[34,413056],{"className":413057,"style":32617},[31680],[34,413059,640],{"className":413060},[31685],[34,413062],{"className":413063,"style":31846},[31845],[34,413065,341696],{"className":413066},[31850],[34,413068],{"className":413069,"style":31846},[31845],[34,413071,413073,413076,413116,413119,413122],{"className":413072},[31676],[34,413074],{"className":413075,"style":40522},[31680],[34,413077,413079,413082],{"className":413078},[31685],[34,413080,31759],{"className":413081},[31685,31686],[34,413083,413085],{"className":413084},[31792],[34,413086,413088,413108],{"className":413087},[31796,31797],[34,413089,413091,413105],{"className":413090},[31801],[34,413092,413094],{"className":413093,"style":31806},[31805],[34,413095,413096,413099],{"style":31809},[34,413097],{"className":413098,"style":31814},[31813],[34,413100,413102],{"className":413101},[31818,31819,31820,31821],[34,413103,640],{"className":413104},[31685,31821],[34,413106,31829],{"className":413107},[31828],[34,413109,413111],{"className":413110},[31801],[34,413112,413114],{"className":413113,"style":31836},[31805],[34,413115],{},[34,413117],{"className":413118,"style":32604},[31845],[34,413120,223],{"className":413121},[32024],[34,413123],{"className":413124,"style":32604},[31845],[34,413126,413128,413131,413134,413137,413140],{"className":413127},[31676],[34,413129],{"className":413130,"style":35942},[31680],[34,413132,459],{"className":413133},[31685],[34,413135],{"className":413136,"style":32604},[31845],[34,413138,223],{"className":413139},[32024],[34,413141],{"className":413142,"style":32604},[31845],[34,413144,413146,413149,413152,413155,413158],{"className":413145},[31676],[34,413147],{"className":413148,"style":32597},[31680],[34,413150,38874],{"className":413151},[34006],[34,413153],{"className":413154,"style":32604},[31845],[34,413156,223],{"className":413157},[32024],[34,413159],{"className":413160,"style":32604},[31845],[34,413162,413164,413167,413170,413173],{"className":413163},[31676],[34,413165],{"className":413166,"style":32617},[31680],[34,413168,459],{"className":413169},[31685],[34,413171],{"className":413172},[31845,208951],[34,413174],{"className":413175,"style":208955},[31845],[34,413177,413179,413182,413185,413194,413197,413237,413240,413243,413246,413249,413252],{"className":413178},[31676],[34,413180],{"className":413181,"style":31774},[31680],[34,413183,456],{"className":413184},[31782],[34,413186,413188],{"className":413187},[31685],[34,413189,413191],{"className":413190},[31685],[34,413192,207924],{"className":413193},[31685,207923],[34,413195],{"className":413196,"style":208977},[31845],[34,413198,413200,413203],{"className":413199},[31685],[34,413201,106063],{"className":413202,"style":67482},[31685,31686],[34,413204,413206],{"className":413205},[31792],[34,413207,413209,413229],{"className":413208},[31796,31797],[34,413210,413212,413226],{"className":413211},[31801],[34,413213,413215],{"className":413214,"style":31806},[31805],[34,413216,413217,413220],{"style":217742},[34,413218],{"className":413219,"style":31814},[31813],[34,413221,413223],{"className":413222},[31818,31819,31820,31821],[34,413224,640],{"className":413225},[31685,31821],[34,413227,31829],{"className":413228},[31828],[34,413230,413232],{"className":413231},[31801],[34,413233,413235],{"className":413234,"style":31836},[31805],[34,413236],{},[34,413238,2557],{"className":413239},[31690],[34,413241],{"className":413242,"style":31846},[31845],[34,413244],{"className":413245,"style":31846},[31845],[34,413247,262625],{"className":413248},[31850],[34,413250],{"className":413251,"style":31846},[31845],[34,413253],{"className":413254,"style":31846},[31845],[34,413256,413258,413262,413302,413305,413308],{"className":413257},[31676],[34,413259],{"className":413260,"style":413261},[31680],"height:0.6138em;vertical-align:-0.15em;",[34,413263,413265,413268],{"className":413264},[31685],[34,413266,31759],{"className":413267},[31685,31686],[34,413269,413271],{"className":413270},[31792],[34,413272,413274,413294],{"className":413273},[31796,31797],[34,413275,413277,413291],{"className":413276},[31801],[34,413278,413280],{"className":413279,"style":31806},[31805],[34,413281,413282,413285],{"style":31809},[34,413283],{"className":413284,"style":31814},[31813],[34,413286,413288],{"className":413287},[31818,31819,31820,31821],[34,413289,640],{"className":413290},[31685,31821],[34,413292,31829],{"className":413293},[31828],[34,413295,413297],{"className":413296},[31801],[34,413298,413300],{"className":413299,"style":31836},[31805],[34,413301],{},[34,413303],{"className":413304,"style":31846},[31845],[34,413306,341696],{"className":413307},[31850],[34,413309],{"className":413310,"style":31846},[31845],[34,413312,413314,413317,413320,413323],{"className":413313},[31676],[34,413315],{"className":413316,"style":32617},[31680],[34,413318,640],{"className":413319},[31685],[34,413321],{"className":413322},[31845,208951],[34,413324],{"className":413325,"style":208955},[31845],[34,413327,413329,413332,413335,413344,413347,413387],{"className":413328},[31676],[34,413330],{"className":413331,"style":31774},[31680],[34,413333,456],{"className":413334},[31782],[34,413336,413338],{"className":413337},[31685],[34,413339,413341],{"className":413340},[31685],[34,413342,207924],{"className":413343},[31685,207923],[34,413345],{"className":413346,"style":208977},[31845],[34,413348,413350,413353],{"className":413349},[31685],[34,413351,106063],{"className":413352,"style":67482},[31685,31686],[34,413354,413356],{"className":413355},[31792],[34,413357,413359,413379],{"className":413358},[31796,31797],[34,413360,413362,413376],{"className":413361},[31801],[34,413363,413365],{"className":413364,"style":31806},[31805],[34,413366,413367,413370],{"style":217742},[34,413368],{"className":413369,"style":31814},[31813],[34,413371,413373],{"className":413372},[31818,31819,31820,31821],[34,413374,640],{"className":413375},[31685,31821],[34,413377,31829],{"className":413378},[31828],[34,413380,413382],{"className":413381},[31801],[34,413383,413385],{"className":413384,"style":31836},[31805],[34,413386],{},[34,413388,2557],{"className":413389},[31690],[58531,413391],{},[2627,413393,413394,413395,413465],{},"总结 ",[34,413396,413398,413416],{"className":413397},[31642],[34,413399,413401],{"className":413400},[31646],[31648,413402,413403],{"xmlns":31650},[31652,413404,413405,413413],{},[31655,413406,413407],{},[31743,413408,413409,413411],{},[31658,413410,31759],{},[31658,413412,31955],{},[31665,413414,413415],{"encoding":31667},"m_i",[34,413417,413419],{"className":413418,"ariaHidden":758},[31672],[34,413420,413422,413425],{"className":413421},[31676],[34,413423],{"className":413424,"style":31860},[31680],[34,413426,413428,413431],{"className":413427},[31685],[34,413429,31759],{"className":413430},[31685,31686],[34,413432,413434],{"className":413433},[31792],[34,413435,413437,413457],{"className":413436},[31796,31797],[34,413438,413440,413454],{"className":413439},[31801],[34,413441,413443],{"className":413442,"style":32090},[31805],[34,413444,413445,413448],{"style":31809},[34,413446],{"className":413447,"style":31814},[31813],[34,413449,413451],{"className":413450},[31818,31819,31820,31821],[34,413452,31955],{"className":413453},[31685,31686,31821],[34,413455,31829],{"className":413456},[31828],[34,413458,413460],{"className":413459},[31801],[34,413461,413463],{"className":413462,"style":31836},[31805],[34,413464],{}," 的性质",[105,413467,413468,413569],{},[108,413469,413470,413539,413540,413568],{},[34,413471,413473,413490],{"className":413472},[31642],[34,413474,413476],{"className":413475},[31646],[31648,413477,413478],{"xmlns":31650},[31652,413479,413480,413488],{},[31655,413481,413482],{},[31743,413483,413484,413486],{},[31658,413485,31759],{},[31658,413487,31955],{},[31665,413489,413415],{"encoding":31667},[34,413491,413493],{"className":413492,"ariaHidden":758},[31672],[34,413494,413496,413499],{"className":413495},[31676],[34,413497],{"className":413498,"style":31860},[31680],[34,413500,413502,413505],{"className":413501},[31685],[34,413503,31759],{"className":413504},[31685,31686],[34,413506,413508],{"className":413507},[31792],[34,413509,413511,413531],{"className":413510},[31796,31797],[34,413512,413514,413528],{"className":413513},[31801],[34,413515,413517],{"className":413516,"style":32090},[31805],[34,413518,413519,413522],{"style":31809},[34,413520],{"className":413521,"style":31814},[31813],[34,413523,413525],{"className":413524},[31818,31819,31820,31821],[34,413526,31955],{"className":413527},[31685,31686,31821],[34,413529,31829],{"className":413530},[31828],[34,413532,413534],{"className":413533},[31801],[34,413535,413537],{"className":413536,"style":31836},[31805],[34,413538],{}," 在第 ",[34,413541,413543,413556],{"className":413542},[31642],[34,413544,413546],{"className":413545},[31646],[31648,413547,413548],{"xmlns":31650},[31652,413549,413550,413554],{},[31655,413551,413552],{},[31658,413553,31955],{},[31665,413555,31955],{"encoding":31667},[34,413557,413559],{"className":413558,"ariaHidden":758},[31672],[34,413560,413562,413565],{"className":413561},[31676],[34,413563],{"className":413564,"style":33104},[31680],[34,413566,31955],{"className":413567},[31685,31686]," 个位置看是 1。",[108,413570,413571,413640],{},[34,413572,413574,413591],{"className":413573},[31642],[34,413575,413577],{"className":413576},[31646],[31648,413578,413579],{"xmlns":31650},[31652,413580,413581,413589],{},[31655,413582,413583],{},[31743,413584,413585,413587],{},[31658,413586,31759],{},[31658,413588,31955],{},[31665,413590,413415],{"encoding":31667},[34,413592,413594],{"className":413593,"ariaHidden":758},[31672],[34,413595,413597,413600],{"className":413596},[31676],[34,413598],{"className":413599,"style":31860},[31680],[34,413601,413603,413606],{"className":413602},[31685],[34,413604,31759],{"className":413605},[31685,31686],[34,413607,413609],{"className":413608},[31792],[34,413610,413612,413632],{"className":413611},[31796,31797],[34,413613,413615,413629],{"className":413614},[31801],[34,413616,413618],{"className":413617,"style":32090},[31805],[34,413619,413620,413623],{"style":31809},[34,413621],{"className":413622,"style":31814},[31813],[34,413624,413626],{"className":413625},[31818,31819,31820,31821],[34,413627,31955],{"className":413628},[31685,31686,31821],[34,413630,31829],{"className":413631},[31828],[34,413633,413635],{"className":413634},[31801],[34,413636,413638],{"className":413637,"style":31836},[31805],[34,413639],{}," 在别的位置看是 0。",[108,413642,413643,413646,413647,413851,413852,414230,414232,414233,414277,414278,414428,414429,414457,414458,414527,414528,415628,415630,415633],{},[2627,413644,413645],{},"最后合成解：","\n现在对于任意的目标列表 ",[34,413648,413650,413692],{"className":413649},[31642],[34,413651,413653],{"className":413652},[31646],[31648,413654,413655],{"xmlns":31650},[31652,413656,413657,413689],{},[31655,413658,413659,413661,413667,413669,413675,413677,413679,413681,413687],{},[31662,413660,456],{"stretchy":702},[31743,413662,413663,413665],{},[31658,413664,32936],{},[31749,413666,640],{},[31662,413668,31960],{"separator":758},[31743,413670,413671,413673],{},[31658,413672,32936],{},[31749,413674,768],{},[31662,413676,31960],{"separator":758},[31662,413678,33934],{},[31662,413680,31960],{"separator":758},[31743,413682,413683,413685],{},[31658,413684,32936],{},[31658,413686,31660],{},[31662,413688,2557],{"stretchy":702},[31665,413690,413691],{"encoding":31667},"(x_1, x_2, \\dots, x_n)",[34,413693,413695],{"className":413694,"ariaHidden":758},[31672],[34,413696,413698,413701,413704,413744,413747,413750,413790,413793,413796,413799,413802,413805,413808,413848],{"className":413697},[31676],[34,413699],{"className":413700,"style":31774},[31680],[34,413702,456],{"className":413703},[31782],[34,413705,413707,413710],{"className":413706},[31685],[34,413708,32936],{"className":413709},[31685,31686],[34,413711,413713],{"className":413712},[31792],[34,413714,413716,413736],{"className":413715},[31796,31797],[34,413717,413719,413733],{"className":413718},[31801],[34,413720,413722],{"className":413721,"style":31806},[31805],[34,413723,413724,413727],{"style":31809},[34,413725],{"className":413726,"style":31814},[31813],[34,413728,413730],{"className":413729},[31818,31819,31820,31821],[34,413731,640],{"className":413732},[31685,31821],[34,413734,31829],{"className":413735},[31828],[34,413737,413739],{"className":413738},[31801],[34,413740,413742],{"className":413741,"style":31836},[31805],[34,413743],{},[34,413745,31960],{"className":413746},[32119],[34,413748],{"className":413749,"style":32123},[31845],[34,413751,413753,413756],{"className":413752},[31685],[34,413754,32936],{"className":413755},[31685,31686],[34,413757,413759],{"className":413758},[31792],[34,413760,413762,413782],{"className":413761},[31796,31797],[34,413763,413765,413779],{"className":413764},[31801],[34,413766,413768],{"className":413767,"style":31806},[31805],[34,413769,413770,413773],{"style":31809},[34,413771],{"className":413772,"style":31814},[31813],[34,413774,413776],{"className":413775},[31818,31819,31820,31821],[34,413777,768],{"className":413778},[31685,31821],[34,413780,31829],{"className":413781},[31828],[34,413783,413785],{"className":413784},[31801],[34,413786,413788],{"className":413787,"style":31836},[31805],[34,413789],{},[34,413791,31960],{"className":413792},[32119],[34,413794],{"className":413795,"style":32123},[31845],[34,413797,33934],{"className":413798},[34006],[34,413800],{"className":413801,"style":32123},[31845],[34,413803,31960],{"className":413804},[32119],[34,413806],{"className":413807,"style":32123},[31845],[34,413809,413811,413814],{"className":413810},[31685],[34,413812,32936],{"className":413813},[31685,31686],[34,413815,413817],{"className":413816},[31792],[34,413818,413820,413840],{"className":413819},[31796,31797],[34,413821,413823,413837],{"className":413822},[31801],[34,413824,413826],{"className":413825,"style":32362},[31805],[34,413827,413828,413831],{"style":31809},[34,413829],{"className":413830,"style":31814},[31813],[34,413832,413834],{"className":413833},[31818,31819,31820,31821],[34,413835,31660],{"className":413836},[31685,31686,31821],[34,413838,31829],{"className":413839},[31828],[34,413841,413843],{"className":413842},[31801],[34,413844,413846],{"className":413845,"style":31836},[31805],[34,413847],{},[34,413849,2557],{"className":413850},[31690],"，我们令：\n",[34,413853,413855,413915],{"className":413854},[31642],[34,413856,413858],{"className":413857},[31646],[31648,413859,413860],{"xmlns":31650},[31652,413861,413862,413912],{},[31655,413863,413864,413866,413868,413874,413880,413882,413888,413894,413896,413898,413900,413906],{},[31658,413865,32936],{},[31662,413867,202],{},[31743,413869,413870,413872],{},[31658,413871,31759],{},[31749,413873,640],{},[31743,413875,413876,413878],{},[31658,413877,32936],{},[31749,413879,640],{},[31662,413881,223],{},[31743,413883,413884,413886],{},[31658,413885,31759],{},[31749,413887,768],{},[31743,413889,413890,413892],{},[31658,413891,32936],{},[31749,413893,768],{},[31662,413895,223],{},[31662,413897,38874],{},[31662,413899,223],{},[31743,413901,413902,413904],{},[31658,413903,31759],{},[31658,413905,31660],{},[31743,413907,413908,413910],{},[31658,413909,32936],{},[31658,413911,31660],{},[31665,413913,413914],{"encoding":31667},"x = m_1 x_1 + m_2 x_2 + \\dots + m_n x_n",[34,413916,413918,413936,414031,414126,414144],{"className":413917,"ariaHidden":758},[31672],[34,413919,413921,413924,413927,413930,413933],{"className":413920},[31676],[34,413922],{"className":413923,"style":32527},[31680],[34,413925,32936],{"className":413926},[31685,31686],[34,413928],{"className":413929,"style":31846},[31845],[34,413931,202],{"className":413932},[31850],[34,413934],{"className":413935,"style":31846},[31845],[34,413937,413939,413942,413982,414022,414025,414028],{"className":413938},[31676],[34,413940],{"className":413941,"style":40522},[31680],[34,413943,413945,413948],{"className":413944},[31685],[34,413946,31759],{"className":413947},[31685,31686],[34,413949,413951],{"className":413950},[31792],[34,413952,413954,413974],{"className":413953},[31796,31797],[34,413955,413957,413971],{"className":413956},[31801],[34,413958,413960],{"className":413959,"style":31806},[31805],[34,413961,413962,413965],{"style":31809},[34,413963],{"className":413964,"style":31814},[31813],[34,413966,413968],{"className":413967},[31818,31819,31820,31821],[34,413969,640],{"className":413970},[31685,31821],[34,413972,31829],{"className":413973},[31828],[34,413975,413977],{"className":413976},[31801],[34,413978,413980],{"className":413979,"style":31836},[31805],[34,413981],{},[34,413983,413985,413988],{"className":413984},[31685],[34,413986,32936],{"className":413987},[31685,31686],[34,413989,413991],{"className":413990},[31792],[34,413992,413994,414014],{"className":413993},[31796,31797],[34,413995,413997,414011],{"className":413996},[31801],[34,413998,414000],{"className":413999,"style":31806},[31805],[34,414001,414002,414005],{"style":31809},[34,414003],{"className":414004,"style":31814},[31813],[34,414006,414008],{"className":414007},[31818,31819,31820,31821],[34,414009,640],{"className":414010},[31685,31821],[34,414012,31829],{"className":414013},[31828],[34,414015,414017],{"className":414016},[31801],[34,414018,414020],{"className":414019,"style":31836},[31805],[34,414021],{},[34,414023],{"className":414024,"style":32604},[31845],[34,414026,223],{"className":414027},[32024],[34,414029],{"className":414030,"style":32604},[31845],[34,414032,414034,414037,414077,414117,414120,414123],{"className":414033},[31676],[34,414035],{"className":414036,"style":40522},[31680],[34,414038,414040,414043],{"className":414039},[31685],[34,414041,31759],{"className":414042},[31685,31686],[34,414044,414046],{"className":414045},[31792],[34,414047,414049,414069],{"className":414048},[31796,31797],[34,414050,414052,414066],{"className":414051},[31801],[34,414053,414055],{"className":414054,"style":31806},[31805],[34,414056,414057,414060],{"style":31809},[34,414058],{"className":414059,"style":31814},[31813],[34,414061,414063],{"className":414062},[31818,31819,31820,31821],[34,414064,768],{"className":414065},[31685,31821],[34,414067,31829],{"className":414068},[31828],[34,414070,414072],{"className":414071},[31801],[34,414073,414075],{"className":414074,"style":31836},[31805],[34,414076],{},[34,414078,414080,414083],{"className":414079},[31685],[34,414081,32936],{"className":414082},[31685,31686],[34,414084,414086],{"className":414085},[31792],[34,414087,414089,414109],{"className":414088},[31796,31797],[34,414090,414092,414106],{"className":414091},[31801],[34,414093,414095],{"className":414094,"style":31806},[31805],[34,414096,414097,414100],{"style":31809},[34,414098],{"className":414099,"style":31814},[31813],[34,414101,414103],{"className":414102},[31818,31819,31820,31821],[34,414104,768],{"className":414105},[31685,31821],[34,414107,31829],{"className":414108},[31828],[34,414110,414112],{"className":414111},[31801],[34,414113,414115],{"className":414114,"style":31836},[31805],[34,414116],{},[34,414118],{"className":414119,"style":32604},[31845],[34,414121,223],{"className":414122},[32024],[34,414124],{"className":414125,"style":32604},[31845],[34,414127,414129,414132,414135,414138,414141],{"className":414128},[31676],[34,414130],{"className":414131,"style":32597},[31680],[34,414133,38874],{"className":414134},[34006],[34,414136],{"className":414137,"style":32604},[31845],[34,414139,223],{"className":414140},[32024],[34,414142],{"className":414143,"style":32604},[31845],[34,414145,414147,414150,414190],{"className":414146},[31676],[34,414148],{"className":414149,"style":31860},[31680],[34,414151,414153,414156],{"className":414152},[31685],[34,414154,31759],{"className":414155},[31685,31686],[34,414157,414159],{"className":414158},[31792],[34,414160,414162,414182],{"className":414161},[31796,31797],[34,414163,414165,414179],{"className":414164},[31801],[34,414166,414168],{"className":414167,"style":32362},[31805],[34,414169,414170,414173],{"style":31809},[34,414171],{"className":414172,"style":31814},[31813],[34,414174,414176],{"className":414175},[31818,31819,31820,31821],[34,414177,31660],{"className":414178},[31685,31686,31821],[34,414180,31829],{"className":414181},[31828],[34,414183,414185],{"className":414184},[318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",[34,414234,414236,414256],{"className":414235},[31642],[34,414237,414239],{"className":414238},[31646],[31648,414240,414241],{"xmlns":31650},[31652,414242,414243,414253],{},[31655,414244,414245,414247,414249,414251],{},[31658,414246,329201],{},[31662,414248,456],{"stretchy":702},[31658,414250,32936],{},[31662,414252,2557],{"stretchy":702},[31665,414254,414255],{"encoding":31667},"\\phi(x)",[34,414257,414259],{"className":414258,"ariaHidden":758},[31672],[34,414260,414262,414265,414268,414271,414274],{"className":414261},[31676],[34,414263],{"className":414264,"style":31774},[31680],[34,414266,329201],{"className":414267},[31685,31686],[34,414269,456],{"className":414270},[31782],[34,414272,32936],{"className":414273},[31685,31686],[34,414275,2557],{"className":414276},[31690]," 是不是等于 ",[34,414279,414281,414315],{"className":414280},[31642],[34,414282,414284],{"className":414283},[31646],[31648,414285,414286],{"xmlns":31650},[31652,414287,414288,414312],{},[31655,414289,414290,414292,414298,414300,414302,414304,414310],{},[31662,414291,456],{"stretchy":702},[31743,414293,414294,414296],{},[31658,414295,32936],{},[31749,414297,640],{},[31662,414299,31960],{"separator":758},[31662,414301,33934],{},[31662,414303,31960],{"separator":758},[31743,414305,414306,414308],{},[31658,414307,32936],{},[31658,414309,31660],{},[31662,414311,2557],{"stretchy":702},[31665,414313,414314],{"encoding":31667},"(x_1, \\dots, x_n)",[34,414316,414318],{"className":414317,"ariaHidden":758},[31672],[34,414319,414321,414324,414327,414367,414370,414373,414376,414379,414382,414385,414425],{"className":414320},[31676],[34,414322],{"className":414323,"style":31774},[31680],[34,414325,456],{"className":414326},[31782],[34,414328,414330,414333],{"className":414329},[31685],[34,414331,32936],{"className":414332},[31685,31686],[34,414334,414336],{"className":414335},[31792],[34,414337,414339,414359],{"className":414338},[31796,31797],[34,414340,414342,414356],{"className":414341},[31801],[34,414343,414345],{"className":414344,"style":31806},[31805],[34,414346,414347,414350],{"style":31809},[34,414348],{"className":414349,"style":31814},[31813],[34,414351,414353],{"className":414352},[31818,31819,31820,31821],[34,414354,640],{"className":414355},[31685,31821],[34,414357,31829],{"className":414358},[31828],[34,414360,414362],{"className":414361},[31801],[34,414363,414365],{"className":414364,"style":31836},[31805],[34,414366],{},[34,414368,31960],{"className":414369},[32119],[34,414371],{"className":414372,"style":32123},[31845],[34,414374,33934],{"className":414375},[34006],[34,414377],{"className":414378,"style":32123},[31845],[34,414380,31960],{"className":414381},[32119],[34,414383],{"className":414384,"style":32123},[31845],[34,414386,414388,414391],{"className":414387},[31685],[34,414389,32936],{"className":414390},[31685,31686],[34,414392,414394],{"className":414393},[31792],[34,414395,414397,414417],{"className":414396},[31796,31797],[34,414398,414400,414414],{"className":414399},[31801],[34,414401,414403],{"className":414402,"style":32362},[31805],[34,414404,414405,414408],{"style":31809},[34,414406],{"className":414407,"style":31814},[31813],[34,414409,414411],{"className":414410},[31818,31819,31820,31821],[34,414412,31660],{"className":414413},[31685,31686,31821],[34,414415,31829],{"className":414416},[31828],[34,414418,414420],{"className":414419},[31801],[34,414421,414423],{"className":414422,"style":31836},[31805],[34,414424],{},[34,414426,2557],{"className":414427},[31690],"。\n只看第 ",[34,414430,414432,414445],{"className":414431},[31642],[34,414433,414435],{"className":414434},[31646],[31648,414436,414437],{"xmlns":31650},[31652,414438,414439,414443],{},[31655,414440,414441],{},[31658,414442,33759],{},[31665,414444,33759],{"encoding":31667},[34,414446,414448],{"className":414447,"ariaHidden":758},[31672],[34,414449,414451,414454],{"className":414450},[31676],[34,414452],{"className":414453,"style":31681},[31680],[34,414455,33759],{"className":414456,"style":33774},[31685,31686]," 个分量（模 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每一个分量都完美对上了。所以是满射。",[32188,415635],{},[253,415637,415639],{"id":415638},"第三阶段应用到整数推论-341","第三阶段：应用到整数（推论 3.4.1）",[15,415641,415642,415643,415671,415672,415700],{},"现在我们把抽象的“环 ",[34,415644,415646,415659],{"className":415645},[31642],[34,415647,415649],{"className":415648},[31646],[31648,415650,415651],{"xmlns":31650},[31652,415652,415653,415657],{},[31655,415654,415655],{},[31658,415656,34386],{},[31665,415658,34386],{"encoding":31667},[34,415660,415662],{"className":415661,"ariaHidden":758},[31672],[34,415663,415665,415668],{"className":415664},[31676],[34,415666],{"className":415667,"style":33218},[31680],[34,415669,34386],{"className":415670,"style":34427},[31685,31686],"”换成我们熟悉的“整数 ",[34,415673,415675,415688],{"className":415674},[31642],[34,415676,415678],{"className":415677},[31646],[31648,415679,415680],{"xmlns":31650},[31652,415681,415682,415686],{},[31655,415683,415684],{},[31658,415685,97300],{"mathvariant":318416},[31665,415687,320917],{"encoding":31667},[34,415689,415691],{"className":415690,"ariaHidden":758},[31672],[34,415692,415694,415697],{"className":415693},[31676],[34,415695],{"className":415696,"style":320927},[31680],[34,415698,97300],{"className":415699},[31685,318441],"”。",[2230,415702,415703,415849,416483,416883],{},[108,415704,415705,2664,415708,415848],{},[2627,415706,415707],{},"设定",[34,415709,415711,415741],{"className":415710},[31642],[34,415712,415714],{"className":415713},[31646],[31648,415715,415716],{"xmlns":31650},[31652,415717,415718,415738],{},[31655,415719,415720,415726,415728,415730,415732],{},[31743,415721,415722,415724],{},[31658,415723,15],{},[31749,415725,640],{},[31662,415727,31960],{"separator":758},[31662,415729,33934],{},[31662,415731,31960],{"separator":758},[31743,415733,415734,415736],{},[31658,415735,15],{},[31658,415737,31660],{},[31665,415739,415740],{"encoding":31667},"p_1, \\dots, p_n",[34,415742,415744],{"className":415743,"ariaHidden":758},[31672],[34,415745,415747,415750,415790,415793,415796,415799,415802,415805,415808],{"className":415746},[31676],[34,415748],{"className":415749,"style":38774},[31680],[34,415751,415753,415756],{"className":415752},[31685],[34,415754,15],{"className":415755},[31685,31686],[34,415757,415759],{"className":415758},[31792],[34,415760,415762,415782],{"className":415761},[31796,31797],[34,415763,415765,415779],{"className":415764},[31801],[34,415766,415768],{"className":415767,"style":31806},[31805],[34,415769,415770,415773],{"style":31809},[34,415771],{"className":415772,"style":31814},[31813],[34,415774,415776],{"className":415775},[31818,31819,31820,31821],[34,415777,640],{"className":415778},[31685,31821],[34,415780,31829],{"className":415781},[31828],[34,415783,415785],{"className":415784},[31801],[34,415786,415788],{"className":415787,"style":31836},[31805],[34,415789],{},[34,415791,31960],{"className":415792},[32119],[34,415794],{"className":415795,"style":32123},[31845],[34,415797,33934],{"className":415798},[34006],[34,415800],{"className":415801,"style":32123},[31845],[34,415803,31960],{"className":415804},[32119],[34,415806],{"className":415807,"style":32123},[31845],[34,415809,415811,415814],{"className":415810},[31685],[34,415812,15],{"className":415813},[31685,31686],[34,415815,415817],{"className":415816},[31792],[34,415818,415820,415840],{"className":415819},[31796,31797],[34,415821,415823,415837],{"className":415822},[31801],[34,415824,415826],{"className":415825,"style":32362},[31805],[34,415827,415828,415831],{"style":31809},[34,415829],{"className":415830,"style":31814},[31813],[34,415832,415834],{"className":415833},[31818,31819,31820,31821],[34,415835,31660],{"className":415836},[31685,31686,31821],[34,415838,31829],{"className":415839},[31828],[34,415841,415843],{"className":415842},[31801],[34,415844,415846],{"className":415845,"style":31836},[31805],[34,415847],{}," 是两两互素的整数（比如 3, 5, 7，不一定是质数，只要最大公约数是1即可）。",[108,415850,415851,33183,415854],{},[2627,415852,415853],{},"对应关系",[105,415855,415856,415943,416188],{},[108,415857,337845,415858,47649,415886,415914,415915,347],{},[34,415859,415861,415874],{"className":415860},[31642],[34,415862,415864],{"className":415863},[31646],[31648,415865,415866],{"xmlns":31650},[31652,415867,415868,415872],{},[31655,415869,415870],{},[31658,415871,34386],{},[31665,415873,34386],{"encoding":31667},[34,415875,415877],{"className":415876,"ariaHidden":758},[31672],[34,415878,415880,415883],{"className":415879},[31676],[34,415881],{"className":415882,"style":33218},[31680],[34,415884,34386],{"className":415885,"style":34427},[31685,31686],[34,415887,415889,415902],{"className":415888},[31642],[34,415890,415892],{"className":415891},[31646],[31648,415893,415894],{"xmlns":31650},[31652,415895,415896,415900],{},[31655,415897,415898],{},[31662,415899,33239],{},[31665,415901,119689],{"encoding":31667},[34,415903,415905],{"className":415904,"ariaHidden":758},[31672],[34,415906,415908,415911],{"className":415907},[31676],[34,415909],{"className":415910,"style":73089},[31680],[34,415912,33239],{"className":415913},[31850]," 整数集 ",[34,415916,415918,415931],{"className":415917},[31642],[34,415919,415921],{"className":415920},[31646],[31648,415922,415923],{"xmlns":31650},[31652,415924,415925,415929],{},[31655,415926,415927],{},[31658,415928,97300],{"mathvariant":318416},[31665,415930,320917],{"encoding":31667},[34,415932,415934],{"className":415933,"ariaHidden":758},[31672],[34,415935,415937,415940],{"className":415936},[31676],[34,415938],{"className":415939,"style":320927},[31680],[34,415941,97300],{"className":415942},[31685,318441],[108,415944,337876,415945,47649,416014,47649,416042,83913,416117,416187],{},[34,415946,415948,415965],{"className":415947},[31642],[34,415949,415951],{"className":415950},[31646],[31648,415952,415953],{"xmlns":31650},[31652,415954,415955,415963],{},[31655,415956,415957],{},[31743,415958,415959,415961],{},[31658,415960,106063],{},[31658,415962,33759],{},[31665,415964,348045],{"encoding":31667},[34,415966,415968],{"className":415967,"ariaHidden":758},[31672],[34,415969,415971,415974],{"className":415970},[31676],[34,415972],{"className":415973,"style":70915},[31680],[34,415975,415977,415980],{"className":415976},[31685],[34,415978,106063],{"className":415979,"style":67482},[31685,31686],[34,415981,415983],{"className":415982},[31792],[34,415984,415986,416006],{"className":415985},[31796,31797],[34,415987,415989,416003],{"className":415988},[31801],[34,415990,415992],{"className":415991,"style":33833},[31805],[34,415993,415994,415997],{"style":217742},[34,415995],{"className":415996,"style":31814},[31813],[34,415998,416000],{"className":415999},[31818,31819,31820,31821],[34,416001,33759],{"className":416002,"style":33774},[31685,31686,31821],[34,416004,31829],{"className":416005},[31828],[34,416007,416009],{"className":416008},[31801],[34,416010,416012],{"className":416011,"style":31836},[31805],[34,416013],{},[34,416015,416017,416030],{"className":416016},[31642],[34,416018,416020],{"className":416019},[31646],[31648,416021,416022],{"xmlns":31650},[31652,416023,416024,416028],{},[31655,416025,416026],{},[31662,416027,33239],{},[31665,416029,119689],{"encoding":31667},[34,416031,416033],{"className":416032,"ariaHidden":758},[31672],[34,416034,416036,416039],{"className":416035},[31676],[34,416037],{"className":416038,"style":73089},[31680],[34,416040,33239],{"className":416041},[31850],[34,416043,416045,416065],{"className":416044},[31642],[34,416046,416048],{"className":416047},[31646],[31648,416049,416050],{"xmlns":31650},[31652,416051,416052,416062],{},[31655,416053,416054,416060],{},[31743,416055,416056,416058],{},[31658,416057,15],{},[31658,416059,33759],{},[31658,416061,97300],{"mathvariant":318416},[31665,416063,416064],{"encoding":31667},"p_k \\mathbb{Z}",[34,416066,416068],{"className":416067,"ariaHidden":758},[31672],[34,416069,416071,416074,416114],{"className":416070},[31676],[34,416072],{"className":416073,"style":338016},[31680],[34,416075,416077,416080],{"className":416076},[31685],[34,416078,15],{"className":416079},[31685,31686],[34,416081,416083],{"className":416082},[31792],[34,416084,416086,416106],{"className":416085},[31796,31797],[34,416087,416089,416103],{"className":416088},[31801],[34,416090,416092],{"className":416091,"style":33833},[31805],[34,416093,416094,416097],{"style":31809},[34,416095],{"className":416096,"style":31814},[31813],[34,416098,416100],{"className":416099},[31818,31819,31820,31821],[34,416101,33759],{"className":416102,"style":33774},[31685,31686,31821],[34,416104,31829],{"className":416105},[31828],[34,416107,416109],{"className":416108},[31801],[34,416110,416112],{"className":416111,"style":31836},[31805],[34,416113],{},[34,416115,97300],{"className":416116},[31685,318441],[34,416118,416120,416138],{"className":416119},[31642],[34,416121,416123],{"className":416122},[31646],[31648,416124,416125],{"xmlns":31650},[31652,416126,416127,416135],{},[31655,416128,416129],{},[31743,416130,416131,416133],{},[31658,416132,15],{},[31658,416134,33759],{},[31665,416136,416137],{"encoding":31667},"p_k",[34,416139,416141],{"className":416140,"ariaHidden":758},[31672],[34,416142,416144,416147],{"className":416143},[31676],[34,416145],{"className":416146,"style":38774},[31680],[34,416148,416150,416153],{"className":416149},[31685],[34,416151,15],{"className":416152},[31685,31686],[34,416154,416156],{"className":416155},[31792],[34,416157,416159,416179],{"className":416158},[31796,31797],[34,416160,416162,416176],{"className":416161},[31801],[34,416163,416165],{"className":416164,"style":33833},[31805],[34,416166,416167,416170],{"style":31809},[34,416168],{"className":416169,"style":31814},[31813],[34,416171,416173],{"className":416172},[31818,31819,31820,31821],[34,416174,33759],{"className":416175,"style":33774},[31685,31686,31821],[34,416177,31829],{"className":416178},[31828],[34,416180,416182],{"className":416181},[31801],[34,416183,416185],{"className":416184,"style":31836},[31805],[34,416186],{}," 的倍数）。",[108,416189,416190,416191,47649,416266,416294,416295,416389,416390,347],{},"交集 ",[34,416192,416194,416214],{"className":416193},[31642],[34,416195,416197],{"className":416196},[31646],[31648,416198,416199],{"xmlns":31650},[31652,416200,416201,416211],{},[31655,416202,416203,416205],{},[31662,416204,326344],{},[31743,416206,416207,416209],{},[31658,416208,106063],{},[31658,416210,33759],{},[31665,416212,416213],{"encoding":31667},"\\cap I_k",[34,416215,416217],{"className":416216,"ariaHidden":758},[31672],[34,416218,416220,416223,416226],{"className":416219},[31676],[34,416221],{"className":416222,"style":70915},[31680],[34,416224,326344],{"className":416225},[31685],[34,416227,416229,416232],{"className":416228},[31685],[34,416230,106063],{"className":416231,"style":67482},[31685,31686],[34,416233,416235],{"className":416234},[31792],[34,416236,416238,416258],{"className":416237},[31796,31797],[34,416239,416241,416255],{"className":416240},[31801],[34,416242,416244],{"className":416243,"style":33833},[31805],[34,416245,416246,416249],{"style":217742},[34,416247],{"className":416248,"style":31814},[31813],[34,416250,416252],{"className":416251},[31818,31819,31820,31821],[34,416253,33759],{"className":416254,"style":33774},[31685,31686,31821],[34,416256,31829],{"className":416257},[31828],[34,416259,416261],{"className":416260},[31801],[34,416262,416264],{"className":416263,"style":31836},[31805],[34,416265],{},[34,416267,416269,416282],{"className":416268},[31642],[34,416270,416272],{"className":416271},[31646],[31648,416273,416274],{"xmlns":31650},[31652,416275,416276,416280],{},[31655,416277,416278],{},[31662,416279,33239],{},[31665,416281,119689],{"encoding":31667},[34,416283,416285],{"className":416284,"ariaHidden":758},[31672],[34,416286,416288,416291],{"className":416287},[31676],[34,416289],{"className":416290,"style":73089},[31680],[34,416292,33239],{"className":416293},[31850]," 最小公倍数 ",[34,416296,416298,416325],{"className":416297},[31642],[34,416299,416301],{"className":416300},[31646],[31648,416302,416303],{"xmlns":31650},[31652,416304,416305,416322],{},[31655,416306,416307,416310,416312,416318,416320],{},[39361,416308,416309],{},"lcm",[31662,416311,456],{"stretchy":702},[31743,416313,416314,416316],{},[31658,416315,15],{},[31658,416317,31955],{},[31662,416319,2557],{"stretchy":702},[31658,416321,97300],{"mathvariant":318416},[31665,416323,416324],{"encoding":31667},"\\text{lcm}(p_i)\\mathbb{Z}",[34,416326,416328],{"className":416327,"ariaHidden":758},[31672],[34,416329,416331,416334,416340,416343,416383,416386],{"className":416330},[31676],[34,416332],{"className":416333,"style":31774},[31680],[34,416335,416337],{"className":416336},[31685,2638],[34,416338,416309],{"className":416339},[31685],[34,416341,456],{"className":416342},[31782],[34,416344,416346,416349],{"className":416345},[31685],[34,416347,15],{"className":416348},[31685,31686],[34,416350,416352],{"className":416351},[31792],[34,416353,416355,416375],{"className":416354},[31796,31797],[34,416356,416358,416372],{"className":416357},[31801],[34,416359,416361],{"className":416360,"style":32090},[31805],[34,416362,416363,416366],{"style":31809},[34,416364],{"className":416365,"style":31814},[31813],[34,416367,416369],{"className":416368},[31818,31819,31820,31821],[34,416370,31955],{"className":416371},[31685,31686,31821],[34,416373,31829],{"className":416374},[31828],[34,416376,416378],{"className":416377},[31801],[34,416379,416381],{"className":416380,"style":31836},[31805],[34,416382],{},[34,416384,2557],{"className":416385},[31690],[34,416387,97300],{"className":416388},[31685,318441],"。因为互素，这就是乘积 ",[34,416391,416393,416419],{"className":416392},[31642],[34,416394,416396],{"className":416395},[31646],[31648,416397,416398],{"xmlns":31650},[31652,416399,416400,416416],{},[31655,416401,416402,416404,416406,416412,416414],{},[31662,416403,456],{"stretchy":702},[31662,416405,402680],{},[31743,416407,416408,416410],{},[31658,416409,15],{},[31658,416411,31955],{},[31662,416413,2557],{"stretchy":702},[31658,416415,97300],{"mathvariant":318416},[31665,416417,416418],{"encoding":31667},"(\\prod p_i)\\mathbb{Z}",[34,416420,416422],{"className":416421,"ariaHidden":758},[31672],[34,416423,416425,416428,416431,416434,416437,416477,416480],{"className":416424},[31676],[34,416426],{"className":416427,"style":31774},[31680],[34,416429,456],{"className":416430},[31782],[34,416432,402680],{"className":416433,"style":156561},[32767,131249,156560],[34,416435],{"className":416436,"style":32123},[31845],[34,416438,416440,416443],{"className":416439},[31685],[34,416441,15],{"className":416442},[31685,31686],[34,416444,416446],{"className":416445},[31792],[34,416447,416449,416469],{"className":416448},[31796,31797],[34,416450,416452,416466],{"className":416451},[31801],[34,416453,416455],{"className":416454,"style":32090},[31805],[34,416456,416457,416460],{"style":31809},[34,416458],{"className":416459,"style":31814},[31813],[34,416461,416463],{"className":416462},[31818,31819,31820,31821],[34,416464,31955],{"className":416465},[31685,31686,31821],[34,416467,31829],{"className":416468},[31828],[34,416470,416472],{"className":416471},[31801],[34,416473,416475],{"className":416474,"style":31836},[31805],[34,416476],{},[34,416478,2557],{"className":416479},[31690],[34,416481,97300],{"className":416482},[31685,318441],[108,416484,416485,33183,416488,416674,416675,416703,416704,416732,416733,5916],{},[2627,416486,416487],{},"方程组",[34,416489,416491,416537],{"className":416490},[31642],[34,416492,416494],{"className":416493},[31646],[31648,416495,416496],{"xmlns":31650},[31652,416497,416498,416534],{},[31655,416499,416500,416502,416504,416510,416512,416514,416516,416524,416526,416532],{},[31658,416501,32936],{},[31662,416503,341696],{},[31743,416505,416506,416508],{},[31658,416507,111],{},[31658,416509,31955],{},[31845,416511],{},[31845,416513],{"width":208910},[31662,416515,456],{"stretchy":702},[31655,416517,416518,416520,416522],{},[31658,416519,31759],{"mathvariant":38079},[31658,416521,43383],{"mathvariant":38079},[31658,416523,46900],{"mathvariant":38079},[31845,416525],{"width":208923},[31743,416527,416528,416530],{},[31658,416529,15],{},[31658,416531,31955],{},[31662,416533,2557],{"stretchy":702},[31665,416535,416536],{"encoding":31667},"x \\equiv a_i \\pmod{p_i}",[34,416538,416540,416558,416610],{"className":416539,"ariaHidden":758},[31672],[34,416541,416543,416546,416549,416552,416555],{"className":416542},[31676],[34,416544],{"className":416545,"style":341735},[31680],[34,416547,32936],{"className":416548},[31685,31686],[34,416550],{"className":416551,"style":31846},[31845],[34,416553,341696],{"className":416554},[31850],[34,416556],{"className":416557,"style":31846},[31845],[34,416559,416561,416564,416604,416607],{"className":416560},[31676],[34,416562],{"className":416563,"style":31860},[31680],[34,416565,416567,416570],{"className":416566},[31685],[34,416568,111],{"className":416569},[31685,31686],[34,416571,416573],{"className":416572},[31792],[34,416574,416576,416596],{"className":416575},[31796,31797],[34,416577,416579,416593],{"className":416578},[31801],[34,416580,416582],{"className":416581,"style":32090},[31805],[34,416583,416584,416587],{"style":31809},[34,416585],{"className":416586,"style":31814},[31813],[34,416588,416590],{"className":416589},[31818,31819,31820,31821],[34,416591,31955],{"className":416592},[31685,31686,31821],[34,416594,31829],{"className":416595},[31828],[34,416597,416599],{"className":416598},[31801],[34,416600,416602],{"className":416601,"style":31836},[31805],[34,416603],{},[34,416605],{"className":416606},[31845,208951],[34,416608],{"className":416609,"style":208955},[31845],[34,416611,416613,416616,416619,416628,416631,416671],{"className":416612},[31676],[34,416614],{"className":416615,"style":31774},[31680],[34,416617,456],{"className":416618},[31782],[34,416620,416622],{"className":416621},[31685],[34,416623,416625],{"className":416624},[31685],[34,416626,207924],{"className":416627},[31685,207923],[34,416629],{"className":416630,"style":208977},[31845],[34,416632,416634,416637],{"className":416633},[31685],[34,416635,15],{"className":416636},[31685,31686],[34,416638,416640],{"className":416639},[31792],[34,416641,416643,416663],{"className":416642},[31796,31797],[34,416644,416646,416660],{"className":416645},[31801],[34,416647,416649],{"className":416648,"style":32090},[31805],[34,416650,416651,416654],{"style":31809},[34,416652],{"className":416653,"style":31814},[31813],[34,416655,416657],{"className":416656},[31818,31819,31820,31821],[34,416658,31955],{"className":416659},[31685,31686,31821],[34,416661,31829],{"className":416662},[31828],[34,416664,416666],{"className":416665},[31801],[34,416667,416669],{"className":416668,"style":31836},[31805],[34,416670],{},[34,416672,2557],{"className":416673},[31690],"\n这其实就是在问：有没有一个整数 ",[34,416676,416678,416691],{"className":416677},[31642],[34,416679,416681],{"className":416680},[31646],[31648,416682,416683],{"xmlns":31650},[31652,416684,416685,416689],{},[31655,416686,416687],{},[31658,416688,32936],{},[31665,416690,32936],{"encoding":31667},[34,416692,416694],{"className":416693,"ariaHidden":758},[31672],[34,416695,416697,416700],{"className":416696},[31676],[34,416698],{"className":416699,"style":32527},[31680],[34,416701,32936],{"className":416702},[31685,31686],"，经过映射 ",[34,416705,416707,416720],{"className":416706},[31642],[34,416708,416710],{"className":416709},[31646],[31648,416711,416712],{"xmlns":31650},[31652,416713,416714,416718],{},[31655,416715,416716],{},[31658,416717,329201],{},[31665,416719,364484],{"encoding":31667},[34,416721,416723],{"className":416722,"ariaHidden":758},[31672],[34,416724,416726,416729],{"className":416725},[31676],[34,416727],{"className":416728,"style":43217},[31680],[34,416730,329201],{"className":416731},[31685,31686]," 后，变成了 ",[34,416734,416736,416770],{"className":416735},[31642],[34,416737,416739],{"className":416738},[31646],[31648,416740,416741],{"xmlns":31650},[31652,416742,416743,416767],{},[31655,416744,416745,416747,416753,416755,416757,416759,416765],{},[31662,416746,456],{"stretchy":702},[31743,416748,416749,416751],{},[31658,416750,111],{},[31749,416752,640],{},[31662,416754,31960],{"separator":758},[31662,416756,33934],{},[31662,416758,31960],{"separator":758},[31743,416760,416761,416763],{},[31658,416762,111],{},[31658,416764,31660],{},[31662,416766,2557],{"stretchy":702},[31665,416768,416769],{"encoding":31667},"(a_1, \\dots, a_n)",[34,416771,416773],{"className":416772,"ariaHidden":758},[31672],[34,416774,416776,416779,416782,416822,416825,416828,416831,416834,416837,416840,416880],{"className":416775},[31676],[34,416777],{"className":416778,"style":31774},[31680],[34,416780,456],{"className":416781},[31782],[34,416783,416785,416788],{"className":416784},[31685],[34,416786,111],{"className":416787},[31685,31686],[34,416789,416791],{"className":416790},[31792],[34,416792,416794,416814],{"className":416793},[31796,31797],[34,416795,416797,416811],{"className":416796},[31801],[34,416798,416800],{"className":416799,"style":31806},[31805],[34,416801,416802,416805],{"style":31809},[34,416803],{"className":416804,"style":31814},[31813],[34,416806,416808],{"className":416807},[31818,31819,31820,31821],[34,416809,640],{"className":416810},[31685,31821],[34,416812,31829],{"className":416813},[31828],[34,416815,416817],{"className":416816},[31801],[34,416818,416820],{"className":416819,"style":31836},[31805],[34,416821],{},[34,416823,31960],{"className":416824},[32119],[34,416826],{"className":416827,"style":32123},[31845],[34,416829,33934],{"className":416830},[34006],[34,416832],{"className":416833,"style":32123},[31845],[34,416835,31960],{"className":416836},[32119],[34,416838],{"className":416839,"style":32123},[31845],[34,416841,416843,416846],{"className":416842},[31685],[34,416844,111],{"className":416845},[31685,31686],[34,416847,416849],{"className":416848},[31792],[34,416850,416852,416872],{"className":416851},[31796,31797],[34,416853,416855,416869],{"className":416854},[31801],[34,416856,416858],{"className":416857,"style":32362},[31805],[34,416859,416860,416863],{"style":31809},[34,416861],{"className":416862,"style":31814},[31813],[34,416864,416866],{"className":416865},[31818,31819,31820,31821],[34,416867,31660],{"className":416868},[31685,31686,31821],[34,416870,31829],{"className":416871},[31828],[34,416873,416875],{"className":416874},[31801],[34,416876,416878],{"className":416877,"style":31836},[31805],[34,416879],{},[34,416881,2557],{"className":416882},[31690],[108,416884,416885,416887,416888,416890,416891],{},[2627,416886,114043],{},"：\n根据上面的定理，映射是",[2627,416889,364537],{},"（Surjective）。\n",[105,416892,416893,417000,417172],{},[108,416894,416895,416897,416898,416901,416902,416971,416972,347],{},[2627,416896,364537],{},"意味着：",[2627,416899,416900],{},"总有解","。无论 ",[34,416903,416905,416922],{"className":416904},[31642],[34,416906,416908],{"className":416907},[31646],[31648,416909,416910],{"xmlns":31650},[31652,416911,416912,416920],{},[31655,416913,416914],{},[31743,416915,416916,416918],{},[31658,416917,111],{},[31658,416919,31955],{},[31665,416921,358860],{"encoding":31667},[34,416923,416925],{"className":416924,"ariaHidden":758},[31672],[34,416926,416928,416931],{"className":416927},[31676],[34,416929],{"className":416930,"style":31860},[31680],[34,416932,416934,416937],{"className":416933},[31685],[34,416935,111],{"className":416936},[31685,31686],[34,416938,416940],{"className":416939},[31792],[34,416941,416943,416963],{"className":416942},[31796,31797],[34,416944,416946,416960],{"className":416945},[31801],[34,416947,416949],{"className":416948,"style":32090},[31805],[34,416950,416951,416954],{"style":31809},[34,416952],{"className":416953,"style":31814},[31813],[34,416955,416957],{"className":416956},[31818,31819,31820,31821],[34,416958,31955],{"className":416959},[31685,31686,31821],[34,416961,31829],{"className":416962},[31828],[34,416964,416966],{"className":416965},[31801],[34,416967,416969],{"className":416968,"style":31836},[31805],[34,416970],{}," 是多少，都能找到 ",[34,416973,416975,416988],{"className":416974},[31642],[34,416976,416978],{"className":416977},[31646],[31648,416979,416980],{"xmlns":31650},[31652,416981,416982,416986],{},[31655,416983,416984],{},[31658,416985,32936],{},[31665,416987,32936],{"encoding":31667},[34,416989,416991],{"className":416990,"ariaHidden":758},[31672],[34,416992,416994,416997],{"className":416993},[31676],[34,416995],{"className":416996,"style":32527},[31680],[34,416998,32936],{"className":416999},[31685,31686],[108,417001,417002,417005,417006,417089,417090,417168,417169,31618],{},[2627,417003,417004],{},"核","是 ",[34,417007,417009,417031],{"className":417008},[31642],[34,417010,417012],{"className":417011},[31646],[31648,417013,417014],{"xmlns":31650},[31652,417015,417016,417028],{},[31655,417017,417018,417020,417026],{},[31662,417019,402680],{},[31743,417021,417022,417024],{},[31658,417023,15],{},[31658,417025,31955],{},[31658,417027,97300],{"mathvariant":318416},[31665,417029,417030],{"encoding":31667},"\\prod p_i \\mathbb{Z}",[34,417032,417034],{"className":417033,"ariaHidden":758},[31672],[34,417035,417037,417040,417043,417046,417086],{"className":417036},[31676],[34,417038],{"className":417039,"style":31774},[31680],[34,417041,402680],{"className":417042,"style":156561},[32767,131249,156560],[34,417044],{"className":417045,"style":32123},[31845],[34,417047,417049,417052],{"className":417048},[31685],[34,417050,15],{"className":417051},[31685,31686],[34,417053,417055],{"className":417054},[31792],[34,417056,417058,417078],{"className":417057},[31796,31797],[34,417059,417061,417075],{"className":417060},[31801],[34,417062,417064],{"className":417063,"style":32090},[31805],[34,417065,417066,417069],{"style":31809},[34,417067],{"className":417068,"style":31814},[31813],[34,417070,417072],{"className":417071},[31818,31819,31820,31821],[34,417073,31955],{"className":417074},[31685,31686,31821],[34,417076,31829],{"className":417077},[31828],[34,417079,417081],{"className":417080},[31801],[34,417082,417084],{"className":417083,"style":31836},[31805],[34,417085],{},[34,417087,97300],{"className":417088},[31685,318441],"：这意味着解在模 ",[34,417091,417093,417113],{"className":417092},[31642],[34,417094,417096],{"className":417095},[31646],[31648,417097,417098],{"xmlns":31650},[31652,417099,417100,417110],{},[31655,417101,417102,417104],{},[31662,417103,402680],{},[31743,417105,417106,417108],{},[31658,417107,15],{},[31658,417109,31955],{},[31665,417111,417112],{"encoding":31667},"\\prod p_i",[34,417114,417116],{"className":417115,"ariaHidden":758},[31672],[34,417117,417119,417122,417125,417128],{"className":417118},[31676],[34,417120],{"className":417121,"style":31774},[31680],[34,417123,402680],{"className":417124,"style":156561},[32767,131249,156560],[34,417126],{"className":417127,"style":32123},[31845],[34,417129,417131,417134],{"className":417130},[31685],[34,417132,15],{"className":417133},[31685,31686],[34,417135,417137],{"className":417136},[31792],[34,417138,417140,417160],{"className":417139},[31796,31797],[34,417141,417143,417157],{"className":417142},[31801],[34,417144,417146],{"className":417145,"style":32090},[31805],[34,417147,417148,417151],{"style":31809},[34,417149],{"className":417150,"style":31814},[31813],[34,417152,417154],{"className":417153},[31818,31819,31820,31821],[34,417155,31955],{"className":417156},[31685,31686,31821],[34,417158,31829],{"className":417159},[31828],[34,417161,417163],{"className":417162},[31801],[34,417164,417166],{"className":417165,"style":31836},[31805],[34,417167],{}," 的意义下是",[2627,417170,417171],{},"唯一",[108,417173,417174,417175,417295,417296,258,417324,417352],{},"推论里写的 ",[34,417176,417178,417210],{"className":417177},[31642],[34,417179,417181],{"className":417180},[31646],[31648,417182,417183],{"xmlns":31650},[31652,417184,417185,417207],{},[31655,417186,417187,417189,417195,417197,417199,417201,417203,417205],{},[31662,417188,402680],{},[31743,417190,417191,417193],{},[31658,417192,15],{},[31658,417194,31955],{},[31658,417196,162086],{"mathvariant":38079},[31662,417198,456],{"stretchy":702},[31658,417200,32936],{},[31662,417202,32226],{},[31658,417204,67764],{},[31662,417206,2557],{"stretchy":702},[31665,417208,417209],{"encoding":31667},"\\prod p_i | (x-y)",[34,417211,417213,417283],{"className":417212,"ariaHidden":758},[31672],[34,417214,417216,417219,417222,417225,417265,417268,417271,417274,417277,417280],{"className":417215},[31676],[34,417217],{"className":417218,"style":31774},[31680],[34,417220,402680],{"className":417221,"style":156561},[32767,131249,156560],[34,417223],{"className":417224,"style":32123},[31845],[34,417226,417228,417231],{"className":417227},[31685],[34,417229,15],{"className":417230},[31685,31686],[34,417232,417234],{"className":417233},[31792],[34,417235,417237,417257],{"className":417236},[31796,31797],[34,417238,417240,417254],{"className":417239},[31801],[34,417241,417243],{"className":417242,"style":32090},[31805],[34,417244,417245,417248],{"style":31809},[34,417246],{"className":417247,"style":31814},[31813],[34,417249,417251],{"className":417250},[31818,31819,31820,31821],[34,417252,31955],{"className":417253},[31685,31686,31821],[34,417255,31829],{"className":417256},[31828],[34,417258,417260],{"className":417259},[31801],[34,417261,417263],{"className":417262,"style":31836},[31805],[34,417264],{},[34,417266,162086],{"className":417267},[31685],[34,417269,456],{"className":417270},[31782],[34,417272,32936],{"className":417273},[31685,31686],[34,417275],{"className":417276,"style":32604},[31845],[34,417278,32226],{"className":417279},[32024],[34,417281],{"className":417282,"style":32604},[31845],[34,417284,417286,417289,417292],{"className":417285},[31676],[34,417287],{"className":417288,"style":31774},[31680],[34,417290,67764],{"className":417291,"style":32062},[31685,31686],[34,417293,2557],{"className":417294},[31690]," 就是指：如果有两个解 ",[34,417297,417299,417312],{"className":417298},[31642],[34,417300,417302],{"className":417301},[31646],[31648,417303,417304],{"xmlns":31650},[31652,417305,417306,417310],{},[31655,417307,417308],{},[31658,417309,32936],{},[31665,417311,32936],{"encoding":31667},[34,417313,417315],{"className":417314,"ariaHidden":758},[31672],[34,417316,417318,417321],{"className":417317},[31676],[34,417319],{"className":417320,"style":32527},[31680],[34,417322,32936],{"className":417323},[31685,31686],[34,417325,417327,417340],{"className":417326},[31642],[34,417328,417330],{"className":417329},[31646],[31648,417331,417332],{"xmlns":31650},[31652,417333,417334,417338],{},[31655,417335,417336],{},[31658,417337,67764],{},[31665,417339,67764],{"encoding":31667},[34,417341,417343],{"className":417342,"ariaHidden":758},[31672],[34,417344,417346,417349],{"className":417345},[31676],[34,417347],{"className":417348,"style":38774},[31680],[34,417350,67764],{"className":417351,"style":32062},[31685,31686],"，它们的差必须能被总乘积整除（也就是它们其实是同一个解）。",[32188,417354],{},[253,417356,417357],{"id":417357},"全局总结",[2230,417359,417360,417366,417372],{},[108,417361,417362,417365],{},[2627,417363,417364],{},"直积\u002F直和","：就是把环拼起来的容器。有限个时它俩一样。",[108,417367,417368,417371],{},[2627,417369,417370],{},"中国剩余定理 (CRT)","：告诉我们，如果几个理想互不干扰（互素），那么“整体模交集”就完全等同于“分开模各自分量”。",[108,417373,417374,417377,417378],{},[2627,417375,417376],{},"证明的核心技巧","：如何证明你能控制每一个分量？\n",[105,417379,417380,417526,417668,417809],{},[108,417381,417382,417383,47649,417386,347],{},"你需要找到一组",[2627,417384,417385],{},"开关",[34,417387,417389,417419],{"className":417388},[31642],[34,417390,417392],{"className":417391},[31646],[31648,417393,417394],{"xmlns":31650},[31652,417395,417396,417416],{},[31655,417397,417398,417404,417406,417408,417410],{},[31743,417399,417400,417402],{},[31658,417401,31759],{},[31749,417403,640],{},[31662,417405,31960],{"separator":758},[31662,417407,33934],{},[31662,417409,31960],{"separator":758},[31743,417411,417412,417414],{},[31658,417413,31759],{},[31658,417415,31660],{},[31665,417417,417418],{"encoding":31667},"m_1, \\dots, m_n",[34,417420,417422],{"className":417421,"ariaHidden":758},[31672],[34,417423,417425,417428,417468,417471,417474,417477,417480,417483,417486],{"className":417424},[31676],[34,417426],{"className":417427,"style":38774},[31680],[34,417429,417431,417434],{"className":417430},[31685],[34,417432,31759],{"className":417433},[31685,31686],[34,417435,417437],{"className":417436},[31792],[34,417438,417440,417460],{"className":417439},[31796,31797],[34,417441,417443,417457],{"className":417442},[31801],[34,417444,417446],{"className":417445,"style":31806},[31805],[34,417447,417448,417451],{"style":31809},[34,417449],{"className":417450,"style":31814},[31813],[34,417452,417454],{"className":417453},[31818,31819,31820,31821],[34,417455,640],{"className":417456},[31685,31821],[34,417458,31829],{"className":417459},[31828],[34,417461,417463],{"className":417462},[31801],[34,417464,417466],{"className":417465,"style":31836},[31805],[34,417467],{},[34,417469,31960],{"className":417470},[32119],[34,417472],{"className":417473,"style":32123},[31845],[34,417475,33934],{"className":417476},[34006],[34,417478],{"className":417479,"style":32123},[31845],[34,417481,31960],{"className":417482},[32119],[34,417484],{"className":417485,"style":32123},[31845],[34,417487,417489,417492],{"className":417488},[31685],[34,417490,31759],{"className":417491},[31685,31686],[34,417493,417495],{"className":417494},[31792],[34,417496,417498,417518],{"className":417497},[31796,31797],[34,417499,417501,417515],{"className":417500},[31801],[34,417502,417504],{"className":417503,"style":32362},[31805],[34,417505,417506,417509],{"style":31809},[34,417507],{"className":417508,"style":31814},[31813],[34,417510,417512],{"className":417511},[31818,31819,31820,31821],[34,417513,31660],{"className":417514},[31685,31686,31821],[34,417516,31829],{"className":417517},[31828],[34,417519,417521],{"className":417520},[31801],[34,417522,417524],{"className":417523,"style":31836},[31805],[34,417525],{},[108,417527,417528,417597,417598,417667],{},[34,417529,417531,417548],{"className":417530},[31642],[34,417532,417534],{"className":417533},[31646],[31648,417535,417536],{"xmlns":31650},[31652,417537,417538,417546],{},[31655,417539,417540],{},[31743,417541,417542,417544],{},[31658,417543,31759],{},[31749,417545,640],{},[31665,417547,411622],{"encoding":31667},[34,417549,417551],{"className":417550,"ariaHidden":758},[31672],[34,417552,417554,417557],{"className":417553},[31676],[34,417555],{"className":417556,"style":31860},[31680],[34,417558,417560,417563],{"className":417559},[31685],[34,417561,31759],{"className":417562},[31685,31686],[34,417564,417566],{"className":417565},[31792],[34,417567,417569,417589],{"className":417568},[31796,31797],[34,417570,417572,417586],{"className":417571},[31801],[34,417573,417575],{"className":417574,"style":31806},[31805],[34,417576,417577,417580],{"style":31809},[34,417578],{"className":417579,"style":31814},[31813],[34,417581,417583],{"className":417582},[31818,31819,31820,31821],[34,417584,640],{"className":417585},[31685,31821],[34,417587,31829],{"className":417588},[31828],[34,417590,417592],{"className":417591},[31801],[34,417593,417595],{"className":417594,"style":31836},[31805],[34,417596],{}," 负责 ",[34,417599,417601,417618],{"className":417600},[31642],[34,417602,417604],{"className":417603},[31646],[31648,417605,417606],{"xmlns":31650},[31652,417607,417608,417616],{},[31655,417609,417610],{},[31743,417611,417612,417614],{},[31658,417613,106063],{},[31749,417615,640],{},[31665,417617,348359],{"encoding":31667},[34,417619,417621],{"className":417620,"ariaHidden":758},[31672],[34,417622,417624,417627],{"className":417623},[31676],[34,417625],{"className":417626,"style":70915},[31680],[34,417628,417630,417633],{"className":417629},[31685],[34,417631,106063],{"className":417632,"style":67482},[31685,31686],[34,417634,417636],{"className":417635},[31792],[34,417637,417639,417659],{"className":417638},[31796,31797],[34,417640,417642,417656],{"className":417641},[31801],[34,417643,417645],{"className":417644,"style":31806},[31805],[34,417646,417647,417650],{"style":217742},[34,417648],{"className":417649,"style":31814},[31813],[34,417651,417653],{"className":417652},[31818,31819,31820,31821],[34,417654,640],{"className":417655},[31685,31821],[34,417657,31829],{"className":417658},[31828],[34,417660,417662],{"className":417661},[31801],[34,417663,417665],{"className":417664,"style":31836},[31805],[34,417666],{},"（在这里是1，在别处是0）。",[108,417669,417670,417597,417740,417667],{},[34,417671,417673,417691],{"className":417672},[31642],[34,417674,417676],{"className":417675},[31646],[31648,417677,417678],{"xmlns":31650},[31652,417679,417680,417688],{},[31655,417681,417682],{},[31743,417683,417684,417686],{},[31658,417685,31759],{},[31749,417687,768],{},[31665,417689,417690],{"encoding":31667},"m_2",[34,417692,417694],{"className":417693,"ariaHidden":758},[31672],[34,417695,417697,417700],{"className":417696},[31676],[34,417698],{"className":417699,"style":31860},[31680],[34,417701,417703,417706],{"className":417702},[31685],[34,417704,31759],{"className":417705},[31685,31686],[34,417707,417709],{"className":417708},[31792],[34,417710,417712,417732],{"className":417711},[31796,31797],[34,417713,417715,417729],{"className":417714},[31801],[34,417716,417718],{"className":417717,"style":31806},[31805],[34,417719,417720,417723],{"style":31809},[34,417721],{"className":417722,"style":31814},[31813],[34,417724,417726],{"className":417725},[31818,31819,31820,31821],[34,417727,768],{"className":417728},[31685,31821],[34,417730,31829],{"className":417731},[31828],[34,417733,417735],{"className":417734},[31801],[34,417736,417738],{"className":417737,"style":31836},[31805],[34,417739],{},[34,417741,417743,417760],{"className":417742},[31642],[34,417744,417746],{"className":417745},[31646],[31648,417747,417748],{"xmlns":31650},[31652,417749,417750,417758],{},[31655,417751,417752],{},[31743,417753,417754,417756],{},[31658,417755,106063],{},[31749,417757,768],{},[31665,417759,348430],{"encoding":31667},[34,417761,417763],{"className":417762,"ariaHidden":758},[31672],[34,417764,417766,417769],{"className":417765},[31676],[34,417767],{"className":417768,"style":70915},[31680],[34,417770,417772,417775],{"className":417771},[31685],[34,417773,106063],{"className":417774,"style":67482},[31685,31686],[34,417776,417778],{"className":417777},[31792],[34,417779,417781,417801],{"className":417780},[31796,31797],[34,417782,417784,417798],{"className":417783},[31801],[34,417785,417787],{"className":417786,"style":31806},[31805],[34,417788,417789,417792],{"style":217742},[34,417790],{"className":417791,"style":31814},[31813],[34,417793,417795],{"className":417794},[31818,31819,31820,31821],[34,417796,768],{"className":417797},[31685,31821],[34,417799,31829],{"className":417800},[31828],[34,417802,417804],{"className":417803},[31801],[34,417805,417807],{"className":417806,"style":31836},[31805],[34,417808],{},[108,417810,417811,417812,417958],{},"然后用 ",[34,417813,417815,417845],{"className":417814},[31642],[34,417816,417818],{"className":417817},[31646],[31648,417819,417820],{"xmlns":31650},[31652,417821,417822,417842],{},[31655,417823,417824,417826,417828,417830,417836],{},[31658,417825,32936],{},[31662,417827,202],{},[31662,417829,131138],{},[31743,417831,417832,417834],{},[31658,417833,31759],{},[31658,417835,31955],{},[31743,417837,417838,417840],{},[31658,417839,32936],{},[31658,417841,31955],{},[31665,417843,417844],{"encoding":31667},"x = \\sum m_i x_i",[34,417846,417848,417866],{"className":417847,"ariaHidden":758},[31672],[34,417849,417851,417854,417857,417860,417863],{"className":417850},[31676],[34,417852],{"className":417853,"style":32527},[31680],[34,417855,32936],{"className":417856},[31685,31686],[34,417858],{"className":417859,"style":31846},[31845],[34,417861,202],{"className":417862},[31850],[34,417864],{"className":417865,"style":31846},[31845],[34,417867,417869,417872,417875,417878,417918],{"className":417868},[31676],[34,417870],{"className":417871,"style":31774},[31680],[34,417873,131138],{"className":417874,"style":156561},[32767,131249,156560],[34,417876],{"className":417877,"style":32123},[31845],[34,417879,417881,417884],{"className":417880},[31685],[34,417882,31759],{"className":417883},[31685,31686],[34,417885,417887],{"className":417886},[31792],[34,417888,417890,417910],{"className":417889},[31796,31797],[34,417891,417893,417907],{"className":417892},[31801],[34,417894,417896],{"className":417895,"style":32090},[31805],[34,417897,417898,417901],{"style":31809},[34,417899],{"className":417900,"style":31814},[31813],[34,417902,417904],{"className":417903},[31818,31819,31820,31821],[34,417905,31955],{"className":417906},[31685,31686,31821],[34,417908,31829],{"className":417909},[31828],[34,417911,417913],{"className":417912},[31801],[34,417914,417916],{"className":417915,"style":31836},[31805],[34,417917],{},[34,417919,417921,417924],{"className":417920},[31685],[34,417922,32936],{"className":417923},[31685,31686],[34,417925,417927],{"className":417926},[31792],[34,417928,417930,417950],{"className":417929},[31796,31797],[34,417931,417933,417947],{"className":417932},[31801],[34,417934,417936],{"className":417935,"style":32090},[31805],[34,417937,417938,417941],{"style":31809},[34,417939],{"className":417940,"style":31814},[31813],[34,417942,417944],{"className":417943},[31818,31819,31820,31821],[34,417945,31955],{"className":417946},[31685,31686,31821],[34,417948,31829],{"className":417949},[31828],[34,417951,417953],{"className":417952},[31801],[34,417954,417956],{"className":417955,"style":31836},[31805],[34,417957],{}," 就能拼出任何你要的结果。",{"title":29,"searchDepth":43,"depth":43,"links":417960},[417961,417962,417963,417964],{"id":402653,"depth":49,"text":402654},{"id":404849,"depth":49,"text":404850},{"id":415638,"depth":49,"text":415639},{"id":417357,"depth":49,"text":417357},{"order":1971},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringdirectprod",{"title":402626,"description":402634},"source\u002Fmath\u002Fabstract_algebra\u002Fringdirectprod","6bDlENKflFZpDVo48Tow-JKqhLL9N-4tZKSoXLp7Z8s",{"id":417971,"title":417972,"body":417973,"description":29,"extension":85,"meta":438825,"navigation":52,"path":438826,"seo":438827,"stem":438828,"__hash__":438829},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002FringdirectprodHW.md","环的直积与中国剩余定理 习题详解",{"type":8,"value":417974,"toc":438819},[417975,417979,417981,417987,418567,419635,419640,419822,419828,421393,421399,421756,421762,424069,424075,426486,426490,426492,426498,427217,427221,427290,427357,427363,427506,429308,429314,429422,431082,431117,431590,432070,432132,432136,432138,432173,432570,432575,432783,432945,432999,433687,434128,434202,434262,435256,435581,436052,436087,436256,436287,437342,437484,437517,438081,438332,438366,438370],[11,417976,417978],{"id":417977},"环的直积直和与中国剩余定理-习题详解","环的直积直和与中国剩余定理 习题详解",[32188,417980],{},[253,417982,417984],{"id":417983},"题目-1环的内直和判别准则定理-341",[2627,417985,417986],{},"题目 1：环的内直和判别准则（定理 3.4.1）",[15,417988,417989,349701,417991,418019,418020,11008,418157,418185,418186,33775,418352,418380,418381,418409,418410,418566],{},[2627,417990,214361],{},[34,417992,417994,418007],{"className":417993},[31642],[34,417995,417997],{"className":417996},[31646],[31648,417998,417999],{"xmlns":31650},[31652,418000,418001,418005],{},[31655,418002,418003],{},[31658,418004,34386],{},[31665,418006,34386],{"encoding":31667},[34,418008,418010],{"className":418009,"ariaHidden":758},[31672],[34,418011,418013,418016],{"className":418012},[31676],[34,418014],{"className":418015,"style":33218},[31680],[34,418017,34386],{"className":418018,"style":34427},[31685,31686]," 是一个环，且 ",[34,418021,418023,418055],{"className":418022},[31642],[34,418024,418026],{"className":418025},[31646],[31648,418027,418028],{"xmlns":31650},[31652,418029,418030,418052],{},[31655,418031,418032,418034,418040],{},[31662,418033,32685],{"stretchy":702},[31743,418035,418036,418038],{},[31658,418037,34386],{},[31658,418039,31955],{},[31743,418041,418042,418044],{},[31662,418043,32715],{"stretchy":702},[31655,418045,418046,418048,418050],{},[31658,418047,31955],{},[31662,418049,112296],{},[31658,418051,106063],{},[31665,418053,418054],{"encoding":31667},"\\{R_i\\}_{i \\in I}",[34,418056,418058],{"className":418057,"ariaHidden":758},[31672],[34,418059,418061,418064,418067,418107],{"className":418060},[31676],[34,418062],{"className":418063,"style":31774},[31680],[34,418065,32685],{"className":418066},[31782],[34,418068,418070,418073],{"className":418069},[31685],[34,418071,34386],{"className":418072,"style":34427},[31685,31686],[34,418074,418076],{"className":418075},[31792],[34,418077,418079,418099],{"className":418078},[31796,31797],[34,418080,418082,418096],{"className":418081},[31801],[34,418083,418085],{"className":418084,"style":32090},[31805],[34,418086,418087,418090],{"style":104694},[34,418088],{"className":418089,"style":31814},[31813],[34,418091,418093],{"className":418092},[31818,31819,31820,31821],[34,418094,31955],{"className":418095},[31685,31686,31821],[34,418097,31829],{"className":418098},[31828],[34,418100,418102],{"className":418101},[31801],[34,418103,418105],{"className":418104,"style":31836},[31805],[34,418106],{},[34,418108,418110,418113],{"className":418109},[31690],[34,418111,32715],{"className":418112},[31690],[34,418114,418116],{"className":418115},[31792],[34,418117,418119,418148],{"className":418118},[31796,31797],[34,418120,418122,418145],{"className":418121},[31801],[34,418123,418125],{"className":418124,"style":116558},[31805],[34,418126,418127,418130],{"style":31809},[34,418128],{"className":418129,"style":31814},[31813],[34,418131,418133],{"className":418132},[31818,31819,31820,31821],[34,418134,418136,418139,418142],{"className":418135},[31685,31821],[34,418137,31955],{"className":418138},[31685,31686,31821],[34,418140,112296],{"className":418141},[31850,31821],[34,418143,106063],{"className":418144,"style":67482},[31685,31686,31821],[34,418146,31829],{"className":418147},[31828],[34,418149,418151],{"className":418150},[31801],[34,418152,418155],{"className":418153,"style":418154},[31805],"height:0.1774em;",[34,418156],{},[34,418158,418160,418173],{"className":418159},[31642],[34,418161,418163],{"className":418162},[31646],[31648,418164,418165],{"xmlns":31650},[31652,418166,418167,418171],{},[31655,418168,418169],{},[31658,418170,34386],{},[31665,418172,34386],{"encoding":31667},[34,418174,418176],{"className":418175,"ariaHidden":758},[31672],[34,418177,418179,418182],{"className":418178},[31676],[34,418180],{"className":418181,"style":33218},[31680],[34,418183,34386],{"className":418184,"style":34427},[31685,31686]," 的一族理想，使得 ",[34,418187,418189,418227],{"className":418188},[31642],[34,418190,418192],{"className":418191},[31646],[31648,418193,418194],{"xmlns":31650},[31652,418195,418196,418224],{},[31655,418197,418198,418200,418202,418204,418216,418222],{},[31658,418199,34386],{},[31662,418201,202],{},[31662,418203,126529],{"stretchy":702},[31743,418205,418206,418208],{},[31662,418207,351927],{},[31655,418209,418210,418212,418214],{},[31658,418211,31955],{},[31662,418213,112296],{},[31658,418215,106063],{},[31743,418217,418218,418220],{},[31658,418219,34386],{},[31658,418221,31955],{},[31662,418223,126544],{"stretchy":702},[31665,418225,418226],{"encoding":31667},"R = \\langle \\bigcup_{i \\in I} R_i \\rangle",[34,418228,418230,418248],{"className":418229,"ariaHidden":758},[31672],[34,418231,418233,418236,418239,418242,418245],{"className":418232},[31676],[34,418234],{"className":418235,"style":33218},[31680],[34,418237,34386],{"className":418238,"style":34427},[31685,31686],[34,418240],{"className":418241,"style":31846},[31845],[34,418243,202],{"className":418244},[31850],[34,418246],{"className":418247,"style":31846},[31845],[34,418249,418251,418254,418257,418306,418309,418349],{"className":418250},[31676],[34,418252],{"className":418253,"style":156553},[31680],[34,418255,126529],{"className":418256},[31782],[34,418258,418260,418263],{"className":418259},[32767],[34,418261,351927],{"className":418262,"style":156561},[32767,131249,156560],[34,418264,418266],{"className":418265},[31792],[34,418267,418269,418298],{"className":418268},[31796,31797],[34,418270,418272,418295],{"className":418271},[31801],[34,418273,418275],{"className":418274,"style":156574},[31805],[34,418276,418277,418280],{"style":156577},[34,418278],{"className":418279,"style":31814},[31813],[34,418281,418283],{"className":418282},[31818,31819,31820,31821],[34,418284,418286,418289,418292],{"className":418285},[31685,31821],[34,418287,31955],{"className":418288},[31685,31686,31821],[34,418290,112296],{"className":418291},[31850,31821],[34,418293,106063],{"className":418294,"style":67482},[31685,31686,31821],[34,418296,31829],{"className":418297},[31828],[34,418299,418301],{"className":418300},[31801],[34,418302,418304],{"className":418303,"style":156605},[31805],[34,418305],{},[34,418307],{"className":418308,"style":32123},[31845],[34,418310,418312,418315],{"className":418311},[31685],[34,418313,34386],{"className":418314,"style":34427},[31685,31686],[34,418316,418318],{"className":418317},[31792],[34,418319,418321,418341],{"className":418320},[31796,31797],[34,418322,418324,418338],{"className":418323},[31801],[34,418325,418327],{"className":418326,"style":32090},[31805],[34,418328,418329,418332],{"style":104694},[34,418330],{"className":418331,"style":31814},[31813],[34,418333,418335],{"className":418334},[31818,31819,31820,31821],[34,418336,31955],{"className":418337},[31685,31686,31821],[34,418339,31829],{"className":418340},[31828],[34,418342,418344],{"className":418343},[31801],[34,418345,418347],{"className":418346,"style":31836},[31805],[34,418348],{},[34,418350,126544],{"className":418351},[31690],[34,418353,418355,418368],{"className":418354},[31642],[34,418356,418358],{"className":418357},[31646],[31648,418359,418360],{"xmlns":31650},[31652,418361,418362,418366],{},[31655,418363,418364],{},[31658,418365,34386],{},[31665,418367,34386],{"encoding":31667},[34,418369,418371],{"className":418370,"ariaHidden":758},[31672],[34,418372,418374,418377],{"className":418373},[31676],[34,418375],{"className":418376,"style":33218},[31680],[34,418378,34386],{"className":418379,"style":34427},[31685,31686]," 由这些理想的并集生成，或者说 ",[34,418382,418384,418397],{"className":418383},[31642],[34,418385,418387],{"className":418386},[31646],[31648,418388,418389],{"xmlns":31650},[31652,418390,418391,418395],{},[31655,418392,418393],{},[31658,418394,34386],{},[31665,418396,34386],{"encoding":31667},[34,418398,418400],{"className":418399,"ariaHidden":758},[31672],[34,418401,418403,418406],{"className":418402},[31676],[34,418404],{"className":418405,"style":33218},[31680],[34,418407,34386],{"className":418408,"style":34427},[31685,31686]," 是这些理想的和 ",[34,418411,418413,418447],{"className":418412},[31642],[34,418414,418416],{"className":418415},[31646],[31648,418417,418418],{"xmlns":31650},[31652,418419,418420,418444],{},[31655,418421,418422,418424,418426,418438],{},[31658,418423,34386],{},[31662,418425,202],{},[31743,418427,418428,418430],{},[31662,418429,131138],{},[31655,418431,418432,418434,418436],{},[31658,418433,31955],{},[31662,418435,112296],{},[31658,418437,106063],{},[31743,418439,418440,418442],{},[31658,418441,34386],{},[31658,418443,31955],{},[31665,418445,418446],{"encoding":31667},"R = \\sum_{i \\in I} R_i",[34,418448,418450,418468],{"className":418449,"ariaHidden":758},[31672],[34,418451,418453,418456,418459,418462,418465],{"className":418452},[31676],[34,418454],{"className":418455,"style":33218},[31680],[34,418457,34386],{"className":418458,"style":34427},[31685,31686],[34,418460],{"className":418461,"style":31846},[31845],[34,418463,202],{"className":418464},[31850],[34,418466],{"className":418467,"style":31846},[31845],[34,418469,418471,418474,418523,418526],{"className":418470},[31676],[34,418472],{"className":418473,"style":156553},[31680],[34,418475,418477,418480],{"className":418476},[32767],[34,418478,131138],{"className":418479,"style":156561},[32767,131249,156560],[34,418481,418483],{"className":418482},[31792],[34,418484,418486,418515],{"className":418485},[31796,31797],[34,418487,418489,418512],{"className":418488},[31801],[34,418490,418492],{"className":418491,"style":156574},[31805],[34,418493,418494,418497],{"style":156577},[34,418495],{"className":418496,"style":31814},[31813],[34,418498,418500],{"className":418499},[31818,31819,31820,31821],[34,418501,418503,418506,418509],{"className":418502},[31685,31821],[34,418504,31955],{"className":418505},[31685,31686,31821],[34,418507,112296],{"className":418508},[31850,31821],[34,418510,106063],{"className":418511,"style":67482},[31685,31686,31821],[34,418513,31829],{"className":418514},[31828],[34,418516,418518],{"className":418517},[31801],[34,418519,418521],{"className":418520,"style":156605},[31805],[34,418522],{},[34,418524],{"className":418525,"style":32123},[31845],[34,418527,418529,418532],{"className":418528},[31685],[34,418530,34386],{"className":418531,"style":34427},[31685,31686],[34,418533,418535],{"className":418534},[31792],[34,418536,418538,418558],{"className":418537},[31796,31797],[34,418539,418541,418555],{"className":418540},[31801],[34,418542,418544],{"className":418543,"style":32090},[31805],[34,418545,418546,418549],{"style":104694},[34,418547],{"className":418548,"style":31814},[31813],[34,418550,418552],{"className":418551},[31818,31819,31820,31821],[34,418553,31955],{"className":418554},[31685,31686,31821],[34,418556,31829],{"className":418557},[31828],[34,418559,418561],{"className":418560},[31801],[34,418562,418564],{"className":418563,"style":31836},[31805],[34,418565],{},"）。证明以下命题是等价的：",[2230,418568,418569,419044,419177,419304],{},[108,418570,405671,418571,418749,418750,419014,419015,419043],{},[34,418572,418574,418612],{"className":418573},[31642],[34,418575,418577],{"className":418576},[31646],[31648,418578,418579],{"xmlns":31650},[31652,418580,418581,418609],{},[31655,418582,418583,418585,418587,418599,418605,418607],{},[31658,418584,115372],{},[31662,418586,5598],{},[31743,418588,418589,418591],{},[31662,418590,404529],{},[31655,418592,418593,418595,418597],{},[31658,418594,31955],{},[31662,418596,112296],{},[31658,418598,106063],{},[31743,418600,418601,418603],{},[31658,418602,34386],{},[31658,418604,31955],{},[31662,418606,33239],{},[31658,418608,34386],{},[31665,418610,418611],{"encoding":31667},"\\sigma: \\coprod_{i \\in I} R_i \\to R",[34,418613,418615,418633,418740],{"className":418614,"ariaHidden":758},[31672],[34,418616,418618,418621,418624,418627,418630],{"className":418617},[31676],[34,418619],{"className":418620,"style":32527},[31680],[34,418622,115372],{"className":418623,"style":32062},[31685,31686],[34,418625],{"className":418626,"style":31846},[31845],[34,418628,5598],{"className":418629},[31850],[34,418631],{"className":418632,"style":31846},[31845],[34,418634,418636,418639,418688,418691,418731,418734,418737],{"className":418635},[31676],[34,418637],{"className":418638,"style":156553},[31680],[34,418640,418642,418645],{"className":418641},[32767],[34,418643,404529],{"className":418644,"style":156561},[32767,131249,156560],[34,418646,418648],{"className":418647},[31792],[34,418649,418651,418680],{"className":418650},[31796,31797],[34,418652,418654,418677],{"className":418653},[31801],[34,418655,418657],{"className":418656,"style":156574},[31805],[34,418658,418659,418662],{"style":156577},[34,418660],{"className":418661,"style":31814},[31813],[34,418663,418665],{"className":418664},[31818,31819,31820,31821],[34,418666,418668,418671,418674],{"className":418667},[31685,31821],[34,418669,31955],{"className":418670},[31685,31686,31821],[34,418672,112296],{"className":418673},[31850,31821],[34,418675,106063],{"className":418676,"style":67482},[31685,31686,31821],[34,418678,31829],{"className":418679},[31828],[34,418681,418683],{"className":418682},[31801],[34,418684,418686],{"className":418685,"style":156605},[31805],[34,418687],{},[34,418689],{"className":418690,"style":32123},[31845],[34,418692,418694,418697],{"className":418693},[31685],[34,418695,34386],{"className":418696,"style":34427},[31685,31686],[34,418698,418700],{"className":418699},[31792],[34,418701,418703,418723],{"className":418702},[31796,31797],[34,418704,418706,418720],{"className":418705},[31801],[34,418707,418709],{"className":418708,"style":32090},[31805],[34,418710,418711,418714],{"style":104694},[34,418712],{"className":418713,"style":31814},[31813],[34,418715,418717],{"className":418716},[31818,31819,31820,31821],[34,418718,31955],{"className":418719},[31685,31686,31821],[34,418721,31829],{"className":418722},[31828],[34,418724,418726],{"className":418725},[31801],[34,418727,418729],{"className":418728,"style":31836},[31805],[34,418730],{},[34,418732],{"className":418733,"style":31846},[31845],[34,418735,33239],{"className":418736},[31850],[34,418738],{"className":418739,"style":31846},[31845],[34,418741,418743,418746],{"className":418742},[31676],[34,418744],{"className":418745,"style":33218},[31680],[34,418747,34386],{"className":418748,"style":34427},[31685,31686]," 为 ",[34,418751,418753,418806],{"className":418752},[31642],[34,418754,418756],{"className":418755},[31646],[31648,418757,418758],{"xmlns":31650},[31652,418759,418760,418803],{},[31655,418761,418762,418764,418770,418782,418785,418797],{},[31662,418763,456],{"stretchy":702},[31743,418765,418766,418768],{},[31658,418767,34395],{},[31658,418769,31955],{},[31743,418771,418772,418774],{},[31662,418773,2557],{"stretchy":702},[31655,418775,418776,418778,418780],{},[31658,418777,31955],{},[31662,418779,112296],{},[31658,418781,106063],{},[31662,418783,418784],{},"↦",[31743,418786,418787,418789],{},[31662,418788,131138],{},[31655,418790,418791,418793,418795],{},[31658,418792,31955],{},[31662,418794,112296],{},[31658,418796,106063],{},[31743,418798,418799,418801],{},[31658,418800,34395],{},[31658,418802,31955],{},[31665,418804,418805],{"encoding":31667},"(r_i)_{i \\in I} \\mapsto \\sum_{i \\in I} r_i",[34,418807,418809,418916],{"className":418808,"ariaHidden":758},[31672],[34,418810,418812,418815,418818,418858,418907,418910,418913],{"className":418811},[31676],[34,418813],{"className":418814,"style":31774},[31680],[34,418816,456],{"className":418817},[31782],[34,418819,418821,418824],{"className":418820},[31685],[34,418822,34395],{"className":418823,"style":34452},[31685,31686],[34,418825,418827],{"className":418826},[31792],[34,418828,418830,418850],{"className":418829},[31796,31797],[34,418831,418833,418847],{"className":418832},[31801],[34,418834,418836],{"className":418835,"style":32090},[31805],[34,418837,418838,418841],{"style":34467},[34,418839],{"className":418840,"style":31814},[31813],[34,418842,418844],{"className":418843},[31818,31819,31820,31821],[34,418845,31955],{"className":418846},[31685,31686,31821],[34,418848,31829],{"className":418849},[31828],[34,418851,418853],{"className":418852},[31801],[34,418854,418856],{"className":418855,"style":31836},[31805],[34,418857],{},[34,418859,418861,418864],{"className":418860},[31690],[34,418862,2557],{"className":418863},[31690],[34,418865,418867],{"className":418866},[31792],[34,418868,418870,418899],{"className":418869},[31796,31797],[34,418871,418873,418896],{"className":418872},[31801],[34,418874,418876],{"className":418875,"style":116558},[31805],[34,418877,418878,418881],{"style":31809},[34,418879],{"className":418880,"style":31814},[31813],[34,418882,418884],{"className":418883},[31818,31819,31820,31821],[34,418885,418887,418890,418893],{"className":418886},[31685,31821],[34,418888,31955],{"className":418889},[31685,31686,31821],[34,418891,112296],{"className":418892},[31850,31821],[34,418894,106063],{"className":418895,"style":67482},[31685,31686,31821],[34,418897,31829],{"className":418898},[31828],[34,418900,418902],{"className":418901},[31801],[34,418903,418905],{"className":418904,"style":418154},[31805],[34,418906],{},[34,418908],{"className":418909,"style":31846},[31845],[34,418911,418784],{"className":418912},[31850],[34,418914],{"className":418915,"style":31846},[31845],[34,418917,418919,418922,418971,418974],{"className":418918},[31676],[34,418920],{"className":418921,"style":156553},[31680],[34,418923,418925,418928],{"className":418924},[32767],[34,418926,131138],{"className":418927,"style":156561},[32767,131249,156560],[34,418929,418931],{"className":418930},[31792],[34,418932,418934,418963],{"className":418933},[31796,31797],[34,418935,418937,418960],{"className":418936},[31801],[34,418938,418940],{"className":418939,"style":156574},[31805],[34,418941,418942,418945],{"style":156577},[34,418943],{"className":418944,"style":31814},[31813],[34,418946,418948],{"className":418947},[31818,31819,31820,31821],[34,418949,418951,418954,418957],{"className":418950},[31685,31821],[34,418952,31955],{"className":418953},[31685,31686,31821],[34,418955,112296],{"className":418956},[31850,31821],[34,418958,106063],{"className":418959,"style":67482},[31685,31686,31821],[34,418961,31829],{"className":418962},[31828],[34,418964,418966],{"className":418965},[31801],[34,418967,418969],{"className":418968,"style":156605},[31805],[34,418970],{},[34,418972],{"className":418973,"style":32123},[31845],[34,418975,418977,418980],{"className":418976},[31685],[34,418978,34395],{"className":418979,"style":34452},[31685,31686],[34,418981,418983],{"className":418982},[31792],[34,418984,418986,419006],{"className":418985},[31796,31797],[34,418987,418989,419003],{"className":418988},[31801],[34,418990,418992],{"className":418991,"style":32090},[31805],[34,418993,418994,418997],{"style":34467},[34,418995],{"className":418996,"style":31814},[31813],[34,418998,419000],{"className":418999},[31818,31819,31820,31821],[34,419001,31955],{"className":419002},[31685,31686,31821],[34,419004,31829],{"className":419005},[31828],[34,419007,419009],{"className":419008},[31801],[34,419010,419012],{"className":419011,"style":31836},[31805],[34,419013],{},"，该映射 ",[34,419016,419018,419031],{"className":419017},[31642],[34,419019,419021],{"className":419020},[31646],[31648,419022,419023],{"xmlns":31650},[31652,419024,419025,419029],{},[31655,419026,419027],{},[31658,419028,115372],{},[31665,419030,115375],{"encoding":31667},[34,419032,419034],{"className":419033,"ariaHidden":758},[31672],[34,419035,419037,419040],{"className":419036},[31676],[34,419038],{"className":419039,"style":32527},[31680],[34,419041,115372],{"className":419042,"style":32062},[31685,31686]," 是一个同构 (Isomorphism)。",[108,419045,419046,419074,419075,419103,419104,419106,419107,419176],{},[34,419047,419049,419062],{"className":419048},[31642],[34,419050,419052],{"className":419051},[31646],[31648,419053,419054],{"xmlns":31650},[31652,419055,419056,419060],{},[31655,419057,419058],{},[31658,419059,34386],{},[31665,419061,34386],{"encoding":31667},[34,419063,419065],{"className":419064,"ariaHidden":758},[31672],[34,419066,419068,419071],{"className":419067},[31676],[34,419069],{"className":419070,"style":33218},[31680],[34,419072,34386],{"className":419073,"style":34427},[31685,31686]," 中的任意元素 ",[34,419076,419078,419091],{"className":419077},[31642],[34,419079,419081],{"className":419080},[31646],[31648,419082,419083],{"xmlns":31650},[31652,419084,419085,419089],{},[31655,419086,419087],{},[31658,419088,34395],{},[31665,419090,34395],{"encoding":31667},[34,419092,419094],{"className":419093,"ariaHidden":758},[31672],[34,419095,419097,419100],{"className":419096},[31676],[34,419098],{"className":419099,"style":32527},[31680],[34,419101,34395],{"className":419102,"style":34452},[31685,31686]," 都可以",[2627,419105,417171],{},"地表示为来自 ",[34,419108,419110,419127],{"className":419109},[31642],[34,419111,419113],{"className":419112},[31646],[31648,419114,419115],{"xmlns":31650},[31652,419116,419117,419125],{},[31655,419118,419119],{},[31743,419120,419121,419123],{},[31658,419122,34386],{},[31658,419124,31955],{},[31665,419126,402774],{"encoding":31667},[34,419128,419130],{"className":419129,"ariaHidden":758},[31672],[34,419131,419133,419136],{"className":419132},[31676],[34,419134],{"className":419135,"style":70915},[31680],[34,419137,419139,419142],{"className":419138},[31685],[34,419140,34386],{"className":419141,"style":34427},[31685,31686],[34,419143,419145],{"className":419144},[31792],[34,419146,419148,419168],{"className":419147},[31796,31797],[34,419149,419151,419165],{"className":419150},[31801],[34,419152,419154],{"className":419153,"style":32090},[31805],[34,419155,419156,419159],{"style":104694},[34,419157],{"className":419158,"style":31814},[31813],[34,419160,419162],{"className":419161},[31818,31819,31820,31821],[34,419163,31955],{"className":419164},[31685,31686,31821],[34,419166,31829],{"className":419167},[31828],[34,419169,419171],{"className":419170},[31801],[34,419172,419174],{"className":419173,"style":31836},[31805],[34,419175],{}," 的元素之和。",[108,419178,419179,419180,419231,419232,419106,419234,419303],{},"零元素 ",[34,419181,419183,419201],{"className":419182},[31642],[34,419184,419186],{"className":419185},[31646],[31648,419187,419188],{"xmlns":31650},[31652,419189,419190,419198],{},[31655,419191,419192,419194,419196],{},[31749,419193,459],{},[31662,419195,112296],{},[31658,419197,34386],{},[31665,419199,419200],{"encoding":31667},"0 \\in R",[34,419202,419204,419222],{"className":419203,"ariaHidden":758},[31672],[34,419205,419207,419210,419213,419216,419219],{"className":419206},[31676],[34,419208],{"className":419209,"style":61814},[31680],[34,419211,459],{"className":419212},[31685],[34,419214],{"className":419215,"style":31846},[31845],[34,419217,112296],{"className":419218},[31850],[34,419220],{"className":419221,"style":31846},[31845],[34,419223,419225,419228],{"className":419224},[31676],[34,419226],{"className":419227,"style":33218},[31680],[34,419229,34386],{"className":419230,"style":34427},[31685,31686]," 可以",[2627,419233,417171],{},[34,419235,419237,419254],{"className":419236},[31642],[34,419238,419240],{"className":419239},[31646],[31648,419241,419242],{"xmlns":31650},[31652,419243,419244,419252],{},[31655,419245,419246],{},[31743,419247,419248,419250],{},[31658,419249,34386],{},[31658,419251,31955],{},[31665,419253,402774],{"encoding":31667},[34,419255,419257],{"className":419256,"ariaHidden":758},[31672],[34,419258,419260,419263],{"className":419259},[31676],[34,419261],{"className":419262,"style":70915},[31680],[34,419264,419266,419269],{"className":419265},[31685],[34,419267,34386],{"className":419268,"style":34427},[31685,31686],[34,419270,419272],{"className":419271},[31792],[34,419273,419275,419295],{"className":419274},[31796,31797],[34,419276,419278,419292],{"className":419277},[31801],[34,419279,419281],{"className":419280,"style":32090},[31805],[34,419282,419283,419286],{"style":104694},[34,419284],{"className":419285,"style":31814},[31813],[34,419287,419289],{"className":419288},[31818,31819,31820,31821],[34,419290,31955],{"className":419291},[31685,31686,31821],[34,419293,31829],{"className":419294},[31828],[34,419296,419298],{"className":419297},[31801],[34,419299,419301],{"className":419300,"style":31836},[31805],[34,419302],{}," 的元素之和（即全为0的表示）。",[108,419305,419306,419307,352264,419358,347],{},"对于任意 ",[34,419308,419310,419328],{"className":419309},[31642],[34,419311,419313],{"className":419312},[31646],[31648,419314,419315],{"xmlns":31650},[31652,419316,419317,419325],{},[31655,419318,419319,419321,419323],{},[31658,419320,33759],{},[31662,419322,112296],{},[31658,419324,106063],{},[31665,419326,419327],{"encoding":31667},"k \\in I",[34,419329,419331,419349],{"className":419330,"ariaHidden":758},[31672],[34,419332,419334,419337,419340,419343,419346],{"className":419333},[31676],[34,419335],{"className":419336,"style":46991},[31680],[34,419338,33759],{"className":419339,"style":33774},[31685,31686],[34,419341],{"className":419342,"style":31846},[31845],[34,419344,112296],{"className":419345},[31850],[34,419347],{"className":419348,"style":31846},[31845],[34,419350,419352,419355],{"className":419351},[31676],[34,419353],{"className":419354,"style":33218},[31680],[34,419356,106063],{"className":419357,"style":67482},[31685,31686],[34,419359,419361,419413],{"className":419360},[31642],[34,419362,419364],{"className":419363},[31646],[31648,419365,419366],{"xmlns":31650},[31652,419367,419368,419410],{},[31655,419369,419370,419376,419378,419402,419404,419406,419408],{},[31743,419371,419372,419374],{},[31658,419373,34386],{},[31658,419375,33759],{},[31662,419377,405054],{},[31655,419379,419380,419382,419394,419400],{},[31662,419381,456],{"fence":758},[31743,419383,419384,419386],{},[31662,419385,131138],{},[31655,419387,419388,419390,419392],{},[31658,419389,31968],{},[31662,419391,54707],{"mathvariant":38079},[31658,419393,33759],{},[31743,419395,419396,419398],{},[31658,419397,34386],{},[31658,419399,31968],{},[31662,419401,2557],{"fence":758},[31662,419403,202],{},[31662,419405,32685],{"stretchy":702},[31749,419407,459],{},[31662,419409,32715],{"stretchy":702},[31665,419411,419412],{"encoding":31667},"R_k \\bigcap \\left( \\sum_{j \\neq k} R_j \\right) = \\{0\\}",[34,419414,419416,419620],{"className":419415,"ariaHidden":758},[31672],[34,419417,419419,419422,419462,419465,419468,419471,419611,419614,419617],{"className":419418},[31676],[34,419420],{"className":419421,"style":141282},[31680],[34,419423,419425,419428],{"className":419424},[31685],[34,419426,34386],{"className":419427,"style":34427},[31685,31686],[34,419429,419431],{"className":419430},[31792],[34,419432,419434,419454],{"className":419433},[31796,31797],[34,419435,419437,419451],{"className":419436},[31801],[34,419438,419440],{"className":419439,"style":33833},[31805],[34,419441,419442,419445],{"style":104694},[34,419443],{"className":419444,"style":31814},[31813],[34,419446,419448],{"className":419447},[31818,31819,31820,31821],[34,419449,33759],{"className":419450,"style":33774},[31685,31686,31821],[34,419452,31829],{"className":419453},[31828],[34,419455,419457],{"className":419456},[31801],[34,419458,419460],{"className":419459,"style":31836},[31805],[34,419461],{},[34,419463],{"className":419464,"style":32123},[31845],[34,419466,405054],{"className":419467,"style":156561},[32767,131249,156560],[34,419469],{"className":419470,"style":32123},[31845],[34,419472,419474,419480,419562,419565,419605],{"className":419473},[34006],[34,419475,419477],{"className":419476,"style":75811},[31782,75810],[34,419478,456],{"className":419479},[71281,141292],[34,419481,419483,419486],{"className":419482},[32767],[34,419484,131138],{"className":419485,"style":156561},[32767,131249,156560],[34,419487,419489],{"className":419488},[31792],[34,419490,419492,419554],{"className":419491},[31796,31797],[34,419493,419495,419551],{"className":419494},[31801],[34,419496,419498],{"className":419497,"style":356484},[31805],[34,419499,419500,419503],{"style":156577},[34,419501],{"className":419502,"style":31814},[31813],[34,419504,419506],{"className":419505},[31818,31819,31820,31821],[34,419507,419509,419512,419548],{"className":419508},[31685,31821],[34,419510,31968],{"className":419511,"style":32153},[31685,31686,31821],[34,419513,419515,419542,419545],{"className":419514},[31850,31821],[34,419516,419518],{"className":419517},[31850,31821],[34,419519,419521],{"className":419520},[31685,54774,31821],[34,419522,419524],{"className":419523},[54778,31821],[34,419525,419527,419530,419539],{"className":419526},[54782,31821],[34,419528],{"className":419529,"style":43217},[31680],[34,419531,419533],{"className":419532},[54789],[34,419534,419536],{"className":419535},[31685,31821],[34,419537,54796],{"className":419538},[31850,31821],[34,419540],{"className":419541},[54800],[34,419543],{"className":419544},[31845,54804,31821],[34,419546,202],{"className":419547},[31850,31821],[34,419549,33759],{"className":419550,"style":33774},[31685,31686,31821],[34,419552,31829],{"className":419553},[31828],[34,419555,419557],{"className":419556},[31801],[34,419558,419560],{"className":419559,"style":156941},[31805],[34,419561],{},[34,419563],{"className":419564,"style":32123},[31845],[34,419566,419568,419571],{"className":419567},[31685],[34,419569,34386],{"className":419570,"style":34427},[31685,31686],[34,419572,419574],{"className":419573},[31792],[34,419575,419577,419597],{"className":419576},[31796,31797],[34,419578,419580,419594],{"className":419579},[31801],[34,419581,419583],{"className":419582,"style":32090},[31805],[34,419584,419585,419588],{"style":104694},[34,419586],{"className":419587,"style":31814},[31813],[34,419589,419591],{"className":419590},[31818,31819,31820,31821],[34,419592,31968],{"className":419593,"style":32153},[31685,31686,31821],[34,419595,31829],{"className":419596},[31828],[34,419598,419600],{"className":419599},[31801],[34,419601,419603],{"className":419602,"style":32163},[31805],[34,419604],{},[34,419606,419608],{"className":419607,"style":75811},[31690,75810],[34,419609,2557],{"className":419610},[71281,141292],[34,419612],{"className":419613,"style":31846},[31845],[34,419615,202],{"className":419616},[31850],[34,419618],{"className":419619,"style":31846},[31845],[34,419621,419623,419626,419629,419632],{"className":419622},[31676],[34,419624],{"className":419625,"style":31774},[31680],[34,419627,32685],{"className":419628},[31782],[34,419630,459],{"className":419631},[31685],[34,419633,32715],{"className":419634},[31690],[15,419636,419637],{},[2627,419638,419639],{},"深度解析与证明：",[15,419641,419642,419643,347],{},"我们要证明四个命题等价，最清晰的逻辑链条通常是循环证明：",[34,419644,419646,419704],{"className":419645},[31642],[34,419647,419649],{"className":419648},[31646],[31648,419650,419651],{"xmlns":31650},[31652,419652,419653,419701],{},[31655,419654,419655,419657,419659,419661,419663,419665,419667,419669,419671,419673,419675,419677,419679,419681,419683,419685,419687,419689,419691,419693,419695,419697,419699],{},[31662,419656,456],{"stretchy":702},[31658,419658,31955],{},[31662,419660,2557],{"stretchy":702},[31662,419662,140094],{},[31662,419664,456],{"stretchy":702},[31658,419666,31955],{},[31658,419668,31955],{},[31662,419670,2557],{"stretchy":702},[31662,419672,140094],{},[31662,419674,456],{"stretchy":702},[31658,419676,31955],{},[31658,419678,31955],{},[31658,419680,31955],{},[31662,419682,2557],{"stretchy":702},[31662,419684,140094],{},[31662,419686,456],{"stretchy":702},[31658,419688,31955],{},[31658,419690,56782],{},[31662,419692,2557],{"stretchy":702},[31662,419694,140094],{},[31662,419696,456],{"stretchy":702},[31658,419698,31955],{},[31662,419700,2557],{"stretchy":702},[31665,419702,419703],{"encoding":31667},"(i) \\Rightarrow (ii) \\Rightarrow (iii) \\Rightarrow (iv) \\Rightarrow (i)",[34,419705,419707,419731,419755,419780,419807],{"className":419706,"ariaHidden":758},[31672],[34,419708,419710,419713,419716,419719,419722,419725,419728],{"className":419709},[31676],[34,419711],{"className":419712,"style":31774},[31680],[34,419714,456],{"className":419715},[31782],[34,419717,31955],{"className":419718},[31685,31686],[34,419720,2557],{"className":419721},[31690],[34,419723],{"className":419724,"style":31846},[31845],[34,419726,140094],{"className":419727},[31850],[34,419729],{"className":419730,"style":31846},[31845],[34,419732,419734,419737,419740,419743,419746,419749,419752],{"className":419733},[31676],[34,419735],{"className":419736,"style":31774},[31680],[34,419738,456],{"className":419739},[31782],[34,419741,148811],{"className":419742},[31685,31686],[34,419744,2557],{"className":419745},[31690],[34,419747],{"className":419748,"style":31846},[31845],[34,419750,140094],{"className":419751},[31850],[34,419753],{"className":419754,"style":31846},[31845],[34,419756,419758,419761,419764,419768,419771,419774,419777],{"className":419757},[31676],[34,419759],{"className":419760,"style":31774},[31680],[34,419762,456],{"className":419763},[31782],[34,419765,419767],{"className":419766},[31685,31686],"iii",[34,419769,2557],{"className":419770},[31690],[34,419772],{"className":419773,"style":31846},[31845],[34,419775,140094],{"className":419776},[31850],[34,419778],{"className":419779,"style":31846},[31845],[34,419781,419783,419786,419789,419792,419795,419798,419801,419804],{"className":419782},[31676],[34,419784],{"className":419785,"style":31774},[31680],[34,419787,456],{"className":419788},[31782],[34,419790,31955],{"className":419791},[31685,31686],[34,419793,56782],{"className":419794,"style":32062},[31685,31686],[34,419796,2557],{"className":419797},[31690],[34,419799],{"className":419800,"style":31846},[31845],[34,419802,140094],{"className":419803},[31850],[34,419805],{"className":419806,"style":31846},[31845],[34,419808,419810,419813,419816,419819],{"className":419809},[31676],[34,419811],{"className":419812,"style":31774},[31680],[34,419814,456],{"className":419815},[31782],[34,419817,31955],{"className":419818},[31685,31686],[34,419820,2557],{"className":419821},[31690],[32490,419823,419825],{"id":419824},"证明步骤-1由-i-推导-ii",[2627,419826,419827],{},"证明步骤 1：由 (i) 推导 (ii)",[105,419829,419830,419959,420247,421388],{},[108,419831,419832,419835,419836,419958],{},[2627,419833,419834],{},"假设 (i) 成立","：映射 ",[34,419837,419839,419867],{"className":419838},[31642],[34,419840,419842],{"className":419841},[31646],[31648,419843,419844],{"xmlns":31650},[31652,419845,419846,419864],{},[31655,419847,419848,419850,419852,419854,419860,419862],{},[31658,419849,115372],{},[31662,419851,5598],{},[31662,419853,404529],{},[31743,419855,419856,419858],{},[31658,419857,34386],{},[31658,419859,31955],{},[31662,419861,33239],{},[31658,419863,34386],{},[31665,419865,419866],{"encoding":31667},"\\sigma: \\coprod R_i \\to R",[34,419868,419870,419888,419949],{"className":419869,"ariaHidden":758},[31672],[34,419871,419873,419876,419879,419882,419885],{"className":419872},[31676],[34,419874],{"className":419875,"style":32527},[31680],[34,419877,115372],{"className":419878,"style":32062},[31685,31686],[34,419880],{"className":419881,"style":31846},[31845],[34,419883,5598],{"className":419884},[31850],[34,419886],{"className":419887,"style":31846},[31845],[34,419889,419891,419894,419897,419900,419940,419943,419946],{"className":419890},[31676],[34,419892],{"className":419893,"style":31774},[31680],[34,419895,404529],{"className":419896,"style":156561},[32767,131249,156560],[34,419898],{"className":419899,"style":32123},[31845],[34,419901,419903,419906],{"className":419902},[31685],[34,419904,34386],{"className":419905,"style":34427},[31685,31686],[34,419907,419909],{"className":419908},[31792],[34,419910,419912,419932],{"className":419911},[31796,31797],[34,419913,419915,419929],{"className":419914},[31801],[34,419916,419918],{"className":419917,"style":32090},[31805],[34,419919,419920,419923],{"style":104694},[34,419921],{"className":419922,"style":31814},[31813],[34,419924,419926],{"className":419925},[31818,31819,31820,31821],[34,419927,31955],{"className":419928},[31685,31686,31821],[34,419930,31829],{"className":419931},[31828],[34,419933,419935],{"className":419934},[31801],[34,419936,419938],{"className":419937,"style":31836},[31805],[34,419939],{},[34,419941],{"className":419942,"style":31846},[31845],[34,419944,33239],{"className":419945},[31850],[34,419947],{"className":419948,"style":31846},[31845],[34,419950,419952,419955],{"className":419951},[31676],[34,419953],{"className":419954,"style":33218},[31680],[34,419956,34386],{"className":419957,"style":34427},[31685,31686]," 是一个同构（即既是满射又是单射）。",[108,419960,419961,33183,419964],{},[2627,419962,419963],{},"分析",[105,419965,419966,420212],{},[108,419967,419968,419996,419997,420133,420134,347],{},[34,419969,419971,419984],{"className":419970},[31642],[34,419972,419974],{"className":419973},[31646],[31648,419975,419976],{"xmlns":31650},[31652,419977,419978,419982],{},[31655,419979,419980],{},[31658,419981,115372],{},[31665,419983,115375],{"encoding":31667},[34,419985,419987],{"className":419986,"ariaHidden":758},[31672],[34,419988,419990,419993],{"className":419989},[31676],[34,419991],{"className":419992,"style":32527},[31680],[34,419994,115372],{"className":419995,"style":32062},[31685,31686]," 的定义是将直和中的元素序列 ",[34,419998,420000,420032],{"className":419999},[31642],[34,420001,420003],{"className":420002},[31646],[31648,420004,420005],{"xmlns":31650},[31652,420006,420007,420029],{},[31655,420008,420009,420011,420017],{},[31662,420010,456],{"stretchy":702},[31743,420012,420013,420015],{},[31658,420014,34395],{},[31658,420016,31955],{},[31743,420018,420019,420021],{},[31662,420020,2557],{"stretchy":702},[31655,420022,420023,420025,420027],{},[31658,420024,31955],{},[31662,420026,112296],{},[31658,420028,106063],{},[31665,420030,420031],{"encoding":31667},"(r_i)_{i \\in I}",[34,420033,420035],{"className":420034,"ariaHidden":758},[31672],[34,420036,420038,420041,420044,420084],{"className":420037},[31676],[34,420039],{"className":420040,"style":31774},[31680],[34,420042,456],{"className":420043},[31782],[34,420045,420047,420050],{"className":420046},[31685],[34,420048,34395],{"className":420049,"style":34452},[31685,31686],[34,420051,420053],{"className":420052},[31792],[34,420054,420056,420076],{"className":420055},[31796,31797],[34,420057,420059,420073],{"className":420058},[31801],[34,420060,420062],{"className":420061,"style":32090},[31805],[34,420063,420064,420067],{"style":34467},[34,420065],{"className":420066,"style":31814},[31813],[34,420068,420070],{"className":420069},[31818,31819,31820,31821],[34,420071,31955],{"className":420072},[31685,31686,31821],[34,420074,31829],{"className":420075},[31828],[34,420077,420079],{"className":420078},[31801],[34,420080,420082],{"className":420081,"style":31836},[31805],[34,420083],{},[34,420085,420087,420090],{"className":420086},[31690],[34,420088,2557],{"className":420089},[31690],[34,420091,420093],{"className":420092},[31792],[34,420094,420096,420125],{"className":420095},[31796,31797],[34,420097,420099,420122],{"className":420098},[31801],[34,420100,420102],{"className":420101,"style":116558},[31805],[34,420103,420104,420107],{"style":31809},[34,420105],{"className":420106,"style":31814},[31813],[34,420108,420110],{"className":420109},[31818,31819,31820,31821],[34,420111,420113,420116,420119],{"className":420112},[31685,31821],[34,420114,31955],{"className":420115},[31685,31686,31821],[34,420117,112296],{"className":420118},[31850,31821],[34,420120,106063],{"className":420121,"style":67482},[31685,31686,31821],[34,420123,31829],{"className":420124},[31828],[34,420126,420128],{"className":420127},[31801],[34,420129,420131],{"className":420130,"style":418154},[31805],[34,420132],{}," 映射为它们的和 ",[34,420135,420137,420157],{"className":420136},[31642],[34,420138,420140],{"className":420139},[31646],[31648,420141,420142],{"xmlns":31650},[31652,420143,420144,420154],{},[31655,420145,420146,420148],{},[31662,420147,131138],{},[31743,420149,420150,420152],{},[31658,420151,34395],{},[31658,420153,31955],{},[31665,420155,420156],{"encoding":31667},"\\sum r_i",[34,420158,420160],{"className":420159,"ariaHidden":758},[31672],[34,420161,420163,420166,420169,420172],{"className":420162},[31676],[34,420164],{"className":420165,"style":31774},[31680],[34,420167,131138],{"className":420168,"style":156561},[32767,131249,156560],[34,420170],{"className":420171,"style":32123},[31845],[34,420173,420175,420178],{"className":420174},[31685],[34,420176,34395],{"className":420177,"style":34452},[31685,31686],[34,420179,420181],{"className":420180},[31792],[34,420182,420184,420204],{"className":420183},[31796,31797],[34,420185,420187,420201],{"className":420186},[31801],[34,420188,420190],{"className":420189,"style":32090},[31805],[34,420191,420192,420195],{"style":34467},[34,420193],{"className":420194,"style":31814},[31813],[34,420196,420198],{"className":420197},[31818,31819,31820,31821],[34,420199,31955],{"className":420200},[31685,31686,31821],[34,420202,31829],{"className":420203},[31828],[34,420205,420207],{"className":420206},[31801],[34,420208,420210],{"className":420209,"style":31836},[31805],[34,420211],{},[108,420213,93310,420214,420242,420243,420246],{},[34,420215,420217,420230],{"className":420216},[31642],[34,420218,420220],{"className":420219},[31646],[31648,420221,420222],{"xmlns":31650},[31652,420223,420224,420228],{},[31655,420225,420226],{},[31658,420227,115372],{},[31665,420229,115375],{"encoding":31667},[34,420231,420233],{"className":420232,"ariaHidden":758},[31672],[34,420234,420236,420239],{"className":420235},[31676],[34,420237],{"className":420238,"style":32527},[31680],[34,420240,115372],{"className":420241,"style":32062},[31685,31686]," 是同构，所以它是",[2627,420244,420245],{},"双射","（一一对应）。",[108,420248,420249,33183,420252],{},[2627,420250,420251],{},"推导 (ii)",[105,420253,420254,420707],{},[108,420255,59432,420256,419074,420284,93389,420312,420340,420341,315343,420492,61605,420603,420703,420704,347],{},[34,420257,420259,420272],{"className":420258},[31642],[34,420260,420262],{"className":420261},[31646],[31648,420263,420264],{"xmlns":31650},[31652,420265,420266,420270],{},[31655,420267,420268],{},[31658,420269,34386],{},[31665,420271,34386],{"encoding":31667},[34,420273,420275],{"className":420274,"ariaHidden":758},[31672],[34,420276,420278,420281],{"className":420277},[31676],[34,420279],{"className":420280,"style":33218},[31680],[34,420282,34386],{"className":420283,"style":34427},[31685,31686],[34,420285,420287,420300],{"className":420286},[31642],[34,420288,420290],{"className":420289},[31646],[31648,420291,420292],{"xmlns":31650},[31652,420293,420294,420298],{},[31655,420295,420296],{},[31658,420297,34395],{},[31665,420299,34395],{"encoding":31667},[34,420301,420303],{"className":420302,"ariaHidden":758},[31672],[34,420304,420306,420309],{"className":420305},[31676],[34,420307],{"className":420308,"style":32527},[31680],[34,420310,34395],{"className":420311,"style":34452},[31685,31686],[34,420313,420315,420328],{"className":420314},[31642],[34,420316,420318],{"className":420317},[31646],[31648,420319,420320],{"xmlns":31650},[31652,420321,420322,420326],{},[31655,420323,420324],{},[31658,420325,115372],{},[31665,420327,115375],{"encoding":31667},[34,420329,420331],{"className":420330,"ariaHidden":758},[31672],[34,420332,420334,420337],{"className":420333},[31676],[34,420335],{"className":420336,"style":32527},[31680],[34,420338,115372],{"className":420339,"style":32062},[31685,31686]," 是满射，所以必然存在一个序列 ",[34,420342,420344,420376],{"className":420343},[31642],[34,420345,420347],{"className":420346},[31646],[31648,420348,420349],{"xmlns":31650},[31652,420350,420351,420373],{},[31655,420352,420353,420355,420361,420363,420365,420367],{},[31662,420354,456],{"stretchy":702},[31743,420356,420357,420359],{},[31658,420358,34395],{},[31658,420360,31955],{},[31662,420362,2557],{"stretchy":702},[31662,420364,112296],{},[31662,420366,404529],{},[31743,420368,420369,420371],{},[31658,420370,34386],{},[31658,420372,31955],{},[31665,420374,420375],{"encoding":31667},"(r_i) \\in \\coprod R_i",[34,420377,420379,420440],{"className":420378,"ariaHidden":758},[31672],[34,420380,420382,420385,420388,420428,420431,420434,420437],{"className":420381},[31676],[34,420383],{"className":420384,"style":31774},[31680],[34,420386,456],{"className":420387},[31782],[34,420389,420391,420394],{"className":420390},[31685],[34,420392,34395],{"className":420393,"style":34452},[31685,31686],[34,420395,420397],{"className":420396},[31792],[34,420398,420400,420420],{"className":420399},[31796,31797],[34,420401,420403,420417],{"className":420402},[31801],[34,420404,420406],{"className":420405,"style":32090},[31805],[34,420407,420408,420411],{"style":34467},[34,420409],{"className":420410,"style":31814},[31813],[34,420412,420414],{"className":420413},[31818,31819,31820,31821],[34,420415,31955],{"className":420416},[31685,31686,31821],[34,420418,31829],{"className":420419},[31828],[34,420421,420423],{"className":420422},[31801],[34,420424,420426],{"className":420425,"style":31836},[31805],[34,420427],{},[34,420429,2557],{"className":420430},[31690],[34,420432],{"className":420433,"style":31846},[31845],[34,420435,112296],{"className":420436},[31850],[34,420438],{"className":420439,"style":31846},[31845],[34,420441,420443,420446,420449,420452],{"className":420442},[31676],[34,420444],{"className":420445,"style":31774},[31680],[34,420447,404529],{"className":420448,"style":156561},[32767,131249,156560],[34,420450],{"className":420451,"style":32123},[31845],[34,420453,420455,420458],{"className":420454},[31685],[34,420456,34386],{"className":420457,"style":34427},[31685,31686],[34,420459,420461],{"className":420460},[31792],[34,420462,420464,420484],{"className":420463},[31796,31797],[34,420465,420467,420481],{"className":420466},[31801],[34,420468,420470],{"className":420469,"style":32090},[31805],[34,420471,420472,420475],{"style":104694},[34,420473],{"className":420474,"style":31814},[31813],[34,420476,420478],{"className":420477},[31818,31819,31820,31821],[34,420479,31955],{"className":420480},[31685,31686,31821],[34,420482,31829],{"className":420483},[31828],[34,420485,420487],{"className":420486},[31801],[34,420488,420490],{"className":420489,"style":31836},[31805],[34,420491],{},[34,420493,420495,420527],{"className":420494},[31642],[34,420496,420498],{"className":420497},[31646],[31648,420499,420500],{"xmlns":31650},[31652,420501,420502,420524],{},[31655,420503,420504,420506,420508,420510,420516,420518,420520,420522],{},[31658,420505,115372],{},[31662,420507,456],{"stretchy":702},[31662,420509,456],{"stretchy":702},[31743,420511,420512,420514],{},[31658,420513,34395],{},[31658,420515,31955],{},[31662,420517,2557],{"stretchy":702},[31662,420519,2557],{"stretchy":702},[31662,420521,202],{},[31658,420523,34395],{},[31665,420525,420526],{"encoding":31667},"\\sigma((r_i)) = r",[34,420528,420530,420594],{"className":420529,"ariaHidden":758},[31672],[34,420531,420533,420536,420539,420542,420582,420585,420588,420591],{"className":420532},[31676],[34,420534],{"className":420535,"style":31774},[31680],[34,420537,115372],{"className":420538,"style":32062},[31685,31686],[34,420540,1323],{"className":420541},[31782],[34,420543,420545,420548],{"className":420544},[31685],[34,420546,34395],{"className":420547,"style":34452},[31685,31686],[34,420549,420551],{"className":420550},[31792],[34,420552,420554,420574],{"className":420553},[31796,31797],[34,420555,420557,420571],{"className":420556},[31801],[34,420558,420560],{"className":420559,"style":32090},[31805],[34,420561,420562,420565],{"style":34467},[34,420563],{"className":420564,"style":31814},[31813],[34,420566,420568],{"className":420567},[31818,31819,31820,31821],[34,420569,31955],{"className":420570},[31685,31686,31821],[34,420572,31829],{"className":420573},[31828],[34,420575,420577],{"className":420576},[31801],[34,420578,420580],{"className":420579,"style":31836},[31805],[34,420581],{},[34,420583,44253],{"className":420584},[31690],[34,420586],{"className":420587,"style":31846},[31845],[34,420589,202],{"className":420590},[31850],[34,420592],{"className":420593,"style":31846},[31845],[34,420595,420597,420600],{"className":420596},[31676],[34,420598],{"className":420599,"style":32527},[31680],[34,420601,34395],{"className":420602,"style":34452},[31685,31686],[34,420604,420606,420630],{"className":420605},[31642],[34,420607,420609],{"className":420608},[31646],[31648,420610,420611],{"xmlns":31650},[31652,420612,420613,420627],{},[31655,420614,420615,420617,420623,420625],{},[31662,420616,131138],{},[31743,420618,420619,420621],{},[31658,420620,34395],{},[31658,420622,31955],{},[31662,420624,202],{},[31658,420626,34395],{},[31665,420628,420629],{"encoding":31667},"\\sum r_i = r",[34,420631,420633,420694],{"className":420632,"ariaHidden":758},[31672],[34,420634,420636,420639,420642,420645,420685,420688,420691],{"className":420635},[31676],[34,420637],{"className":420638,"style":31774},[31680],[34,420640,131138],{"className":420641,"style":156561},[32767,131249,156560],[34,420643],{"className":420644,"style":32123},[31845],[34,420646,420648,420651],{"className":420647},[31685],[34,420649,34395],{"className":420650,"style":34452},[31685,31686],[34,420652,420654],{"className":420653},[31792],[34,420655,420657,420677],{"className":420656},[31796,31797],[34,420658,420660,420674],{"className":420659},[31801],[34,420661,420663],{"className":420662,"style":32090},[31805],[34,420664,420665,420668],{"style":34467},[34,420666],{"className":420667,"style":31814},[31813],[34,420669,420671],{"className":420670},[31818,31819,31820,31821],[34,420672,31955],{"className":420673},[31685,31686,31821],[34,420675,31829],{"className":420676},[31828],[34,420678,420680],{"className":420679},[31801],[34,420681,420683],{"className":420682,"style":31836},[31805],[34,420684],{},[34,420686],{"className":420687,"style":31846},[31845],[34,420689,202],{"className":420690},[31850],[34,420692],{"className":420693,"style":31846},[31845],[34,420695,420697,420700],{"className":420696},[31676],[34,420698],{"className":420699,"style":32527},[31680],[34,420701,34395],{"className":420702,"style":34452},[31685,31686],"。这说明表示形式",[2627,420705,420706],{},"存在",[108,420708,93310,420709,420737,420738,420818,420819,420937,420938,421065,421066,421235,421236,421385,421386,31618],{},[34,420710,420712,420725],{"className":420711},[31642],[34,420713,420715],{"className":420714},[31646],[31648,420716,420717],{"xmlns":31650},[31652,420718,420719,420723],{},[31655,420720,420721],{},[31658,420722,115372],{},[31665,420724,115375],{"encoding":31667},[34,420726,420728],{"className":420727,"ariaHidden":758},[31672],[34,420729,420731,420734],{"className":420730},[31676],[34,420732],{"className":420733,"style":32527},[31680],[34,420735,115372],{"className":420736,"style":32062},[31685,31686]," 是单射，所以这个序列 ",[34,420739,420741,420763],{"className":420740},[31642],[34,420742,420744],{"className":420743},[31646],[31648,420745,420746],{"xmlns":31650},[31652,420747,420748,420760],{},[31655,420749,420750,420752,420758],{},[31662,420751,456],{"stretchy":702},[31743,420753,420754,420756],{},[31658,420755,34395],{},[31658,420757,31955],{},[31662,420759,2557],{"stretchy":702},[31665,420761,420762],{"encoding":31667},"(r_i)",[34,420764,420766],{"className":420765,"ariaHidden":758},[31672],[34,420767,420769,420772,420775,420815],{"className":420768},[31676],[34,420770],{"className":420771,"style":31774},[31680],[34,420773,456],{"className":420774},[31782],[34,420776,420778,420781],{"className":420777},[31685],[34,420779,34395],{"className":420780,"style":34452},[31685,31686],[34,420782,420784],{"className":420783},[31792],[34,420785,420787,420807],{"className":420786},[31796,31797],[34,420788,420790,420804],{"className":420789},[31801],[34,420791,420793],{"className":420792,"style":32090},[31805],[34,420794,420795,420798],{"style":34467},[34,420796],{"className":420797,"style":31814},[31813],[34,420799,420801],{"className":420800},[31818,31819,31820,31821],[34,420802,31955],{"className":420803},[31685,31686,31821],[34,420805,31829],{"className":420806},[31828],[34,420808,420810],{"className":420809},[31801],[34,420811,420813],{"className":420812,"style":31836},[31805],[34,420814],{},[34,420816,2557],{"className":420817},[31690]," 是唯一的。如果有另一种表示 ",[34,420820,420822,420848],{"className":420821},[31642],[34,420823,420825],{"className":420824},[31646],[31648,420826,420827],{"xmlns":31650},[31652,420828,420829,420845],{},[31655,420830,420831,420833,420835,420837],{},[31658,420832,34395],{},[31662,420834,202],{},[31662,420836,131138],{},[38846,420838,420839,420841,420843],{},[31658,420840,34395],{},[31658,420842,31955],{},[31662,420844,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,420846,420847],{"encoding":31667},"r = \\sum r'_i",[34,420849,420851,420869],{"className":420850,"ariaHidden":758},[31672],[34,420852,420854,420857,420860,420863,420866],{"className":420853},[31676],[34,420855],{"className":420856,"style":32527},[31680],[34,420858,34395],{"className":420859,"style":34452},[31685,31686],[34,420861],{"className":420862,"style":31846},[31845],[34,420864,202],{"className":420865},[31850],[34,420867],{"className":420868,"style":31846},[31845],[34,420870,420872,420876,420879,420882],{"className":420871},[31676],[34,420873],{"className":420874,"style":420875},[31680],"height:1.0106em;vertical-align:-0.2587em;",[34,420877,131138],{"className":420878,"style":156561},[32767,131249,156560],[34,420880],{"className":420881,"style":32123},[31845],[34,420883,420885,420888],{"className":420884},[31685],[34,420886,34395],{"className":420887,"style":34452},[31685,31686],[34,420889,420891],{"className":420890},[31792],[34,420892,420894,420929],{"className":420893},[31796,31797],[34,420895,420897,420926],{"className":420896},[31801],[34,420898,420900,420912],{"className":420899,"style":132027},[31805],[34,420901,420903,420906],{"style":420902},"top:-2.4413em;margin-left:-0.0278em;margin-right:0.05em;",[34,420904],{"className":420905,"style":31814},[31813],[34,420907,420909],{"className":420908},[31818,31819,31820,31821],[34,420910,31955],{"className":420911},[31685,31686,31821],[34,420913,420914,420917],{"style":37777},[34,420915],{"className":420916,"style":31814},[31813],[34,420918,420920],{"className":420919},[31818,31819,31820,31821],[34,420921,420923],{"className":420922},[31685,31821],[34,420924,116495],{"className":420925},[31685,31821],[34,420927,31829],{"className":420928},[31828],[34,420930,420932],{"className":420931},[31801],[34,420933,420935],{"className":420934,"style":39572},[31805],[34,420936],{},"，则意味着 ",[34,420939,420941,420975],{"className":420940},[31642],[34,420942,420944],{"className":420943},[31646],[31648,420945,420946],{"xmlns":31650},[31652,420947,420948,420972],{},[31655,420949,420950,420952,420954,420956,420964,420966,420968,420970],{},[31658,420951,115372],{},[31662,420953,456],{"stretchy":702},[31662,420955,456],{"stretchy":702},[38846,420957,420958,420960,420962],{},[31658,420959,34395],{},[31658,420961,31955],{},[31662,420963,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,420965,2557],{"stretchy":702},[31662,420967,2557],{"stretchy":702},[31662,420969,202],{},[31658,420971,34395],{},[31665,420973,420974],{"encoding":31667},"\\sigma((r'_i)) = r",[34,420976,420978,421056],{"className":420977,"ariaHidden":758},[31672],[34,420979,420981,420984,420987,420990,421044,421047,421050,421053],{"className":420980},[31676],[34,420982],{"className":420983,"style":420875},[31680],[34,420985,115372],{"className":420986,"style":32062},[31685,31686],[34,420988,1323],{"className":420989},[31782],[34,420991,420993,420996],{"className":420992},[31685],[34,420994,34395],{"className":420995,"style":34452},[31685,31686],[34,420997,420999],{"className":420998},[31792],[34,421000,421002,421036],{"className":421001},[31796,31797],[34,421003,421005,421033],{"className":421004},[31801],[34,421006,421008,421019],{"className":421007,"style":132027},[31805],[34,421009,421010,421013],{"style":420902},[34,421011],{"className":421012,"style":31814},[31813],[34,421014,421016],{"className":421015},[31818,31819,31820,31821],[34,421017,31955],{"className":421018},[31685,31686,31821],[34,421020,421021,421024],{"style":37777},[34,421022],{"className":421023,"style":31814},[31813],[34,421025,421027],{"className":421026},[31818,31819,31820,31821],[34,421028,421030],{"className":421029},[31685,31821],[34,421031,116495],{"className":421032},[31685,31821],[34,421034,31829],{"className":421035},[31828],[34,421037,421039],{"className":421038},[31801],[34,421040,421042],{"className":421041,"style":39572},[31805],[34,421043],{},[34,421045,44253],{"className":421046},[31690],[34,421048],{"className":421049,"style":31846},[31845],[34,421051,202],{"className":421052},[31850],[34,421054],{"className":421055,"style":31846},[31845],[34,421057,421059,421062],{"className":421058},[31676],[34,421060],{"className":421061,"style":32527},[31680],[34,421063,34395],{"className":421064,"style":34452},[31685,31686],"。由单射性，",[34,421067,421069,421105],{"className":421068},[31642],[34,421070,421072],{"className":421071},[31646],[31648,421073,421074],{"xmlns":31650},[31652,421075,421076,421102],{},[31655,421077,421078,421080,421086,421088,421090,421092,421100],{},[31662,421079,456],{"stretchy":702},[31743,421081,421082,421084],{},[31658,421083,34395],{},[31658,421085,31955],{},[31662,421087,2557],{"stretchy":702},[31662,421089,202],{},[31662,421091,456],{"stretchy":702},[38846,421093,421094,421096,421098],{},[31658,421095,34395],{},[31658,421097,31955],{},[31662,421099,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,421101,2557],{"stretchy":702},[31665,421103,421104],{"encoding":31667},"(r_i) = (r'_i)",[34,421106,421108,421169],{"className":421107,"ariaHidden":758},[31672],[34,421109,421111,421114,421117,421157,421160,421163,421166],{"className":421110},[31676],[34,421112],{"className":421113,"style":31774},[31680],[34,421115,456],{"className":421116},[31782],[34,421118,421120,421123],{"className":421119},[31685],[34,421121,34395],{"className":421122,"style":34452},[31685,31686],[34,421124,421126],{"className":421125},[31792],[34,421127,421129,421149],{"className":421128},[31796,31797],[34,421130,421132,421146],{"className":421131},[31801],[34,421133,421135],{"className":421134,"style":32090},[31805],[34,421136,421137,421140],{"style":34467},[34,421138],{"className":421139,"style":31814},[31813],[34,421141,421143],{"className":421142},[31818,31819,31820,31821],[34,421144,31955],{"className":421145},[31685,31686,31821],[34,421147,31829],{"className":421148},[31828],[34,421150,421152],{"className":421151},[31801],[34,421153,421155],{"className":421154,"style":31836},[31805],[34,421156],{},[34,421158,2557],{"className":421159},[31690],[34,421161],{"className":421162,"style":31846},[31845],[34,421164,202],{"className":421165},[31850],[34,421167],{"className":421168,"style":31846},[31845],[34,421170,421172,421175,421178,421232],{"className":421171},[31676],[34,421173],{"className":421174,"style":420875},[31680],[34,421176,456],{"className":421177},[31782],[34,421179,421181,421184],{"className":421180},[31685],[34,421182,34395],{"className":421183,"style":34452},[31685,31686],[34,421185,421187],{"className":421186},[31792],[34,421188,421190,421224],{"className":421189},[31796,31797],[34,421191,421193,421221],{"className":421192},[31801],[34,421194,421196,421207],{"className":421195,"style":132027},[31805],[34,421197,421198,421201],{"style":420902},[34,421199],{"className":421200,"style":31814},[31813],[34,421202,421204],{"className":421203},[31818,31819,31820,31821],[34,421205,31955],{"className":421206},[31685,31686,31821],[34,421208,421209,421212],{"style":37777},[34,421210],{"className":421211,"style":31814},[31813],[34,421213,421215],{"className":421214},[31818,31819,31820,31821],[34,421216,421218],{"className":421217},[31685,31821],[34,421219,116495],{"className":421220},[31685,31821],[34,421222,31829],{"className":421223},[31828],[34,421225,421227],{"className":421226},[31801],[34,421228,421230],{"className":421229,"style":39572},[31805],[34,421231],{},[34,421233,2557],{"className":421234},[31690],"，即每一项 ",[34,421237,421239,421267],{"className":421238},[31642],[34,421240,421242],{"className":421241},[31646],[31648,421243,421244],{"xmlns":31650},[31652,421245,421246,421264],{},[31655,421247,421248,421254,421256],{},[31743,421249,421250,421252],{},[31658,421251,34395],{},[31658,421253,31955],{},[31662,421255,202],{},[38846,421257,421258,421260,421262],{},[31658,421259,34395],{},[31658,421261,31955],{},[31662,421263,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,421265,421266],{"encoding":31667},"r_i = r'_i",[34,421268,421270,421325],{"className":421269,"ariaHidden":758},[31672],[34,421271,421273,421276,421316,421319,421322],{"className":421272},[31676],[34,421274],{"className":421275,"style":31860},[31680],[34,421277,421279,421282],{"className":421278},[31685],[34,421280,34395],{"className":421281,"style":34452},[31685,31686],[34,421283,421285],{"className":421284},[31792],[34,421286,421288,421308],{"className":421287},[31796,31797],[34,421289,421291,421305],{"className":421290},[31801],[34,421292,421294],{"className":421293,"style":32090},[31805],[34,421295,421296,421299],{"style":34467},[34,421297],{"className":421298,"style":31814},[31813],[34,421300,421302],{"className":421301},[31818,31819,31820,31821],[34,421303,31955],{"className":421304},[31685,31686,31821],[34,421306,31829],{"className":421307},[31828],[34,421309,421311],{"className":421310},[31801],[34,421312,421314],{"className":421313,"style":31836},[31805],[34,421315],{},[34,421317],{"className":421318,"style":31846},[31845],[34,421320,202],{"className":421321},[31850],[34,421323],{"className":421324,"style":31846},[31845],[34,421326,421328,421331],{"className":421327},[31676],[34,421329],{"className":421330,"style":420875},[31680],[34,421332,421334,421337],{"className":421333},[31685],[34,421335,34395],{"className":421336,"style":34452},[31685,31686],[34,421338,421340],{"className":421339},[31792],[34,421341,421343,421377],{"className":421342},[31796,31797],[34,421344,421346,421374],{"className":421345},[31801],[34,421347,421349,421360],{"className":421348,"style":132027},[31805],[34,421350,421351,421354],{"style":420902},[34,421352],{"className":421353,"style":31814},[31813],[34,421355,421357],{"className":421356},[31818,31819,31820,31821],[34,421358,31955],{"className":421359},[31685,31686,31821],[34,421361,421362,421365],{"style":37777},[34,421363],{"className":421364,"style":31814},[31813],[34,421366,421368],{"className":421367},[31818,31819,31820,31821],[34,421369,421371],{"className":421370},[31685,31821],[34,421372,116495],{"className":421373},[31685,31821],[34,421375,31829],{"className":421376},[31828],[34,421378,421380],{"className":421379},[31801],[34,421381,421383],{"className":421382,"style":39572},[31805],[34,421384],{},"。这说明表示形式是",[2627,421387,417171],{},[108,421389,421390,421392],{},[2627,421391,114043],{},"：(ii) 成立。",[32490,421394,421396],{"id":421395},"证明步骤-2由-ii-推导-iii",[2627,421397,421398],{},"证明步骤 2：由 (ii) 推导 (iii)",[105,421400,421401,421438,421450,421751],{},[108,421402,421403,2664,421406,373617,421434,421437],{},[2627,421404,421405],{},"假设 (ii) 成立",[34,421407,421409,421422],{"className":421408},[31642],[34,421410,421412],{"className":421411},[31646],[31648,421413,421414],{"xmlns":31650},[31652,421415,421416,421420],{},[31655,421417,421418],{},[31658,421419,34386],{},[31665,421421,34386],{"encoding":31667},[34,421423,421425],{"className":421424,"ariaHidden":758},[31672],[34,421426,421428,421431],{"className":421427},[31676],[34,421429],{"className":421430,"style":33218},[31680],[34,421432,34386],{"className":421433,"style":34427},[31685,31686],[2627,421435,421436],{},"任意","元素都有唯一的和表示。",[108,421439,421440,33183,421442],{},[2627,421441,419963],{},[105,421443,421444],{},[108,421445,421446,421447,421449],{},"既然是“任意元素”，那么对于特殊元素 ",[2627,421448,459],{}," (零元) 自然也成立。",[108,421451,421452,33183,421455],{},[2627,421453,421454],{},"推导 (iii)",[105,421456,421457,421576],{},[108,421458,421459,421460,421575],{},"显然，",[34,421461,421463,421493],{"className":421462},[31642],[34,421464,421466],{"className":421465},[31646],[31648,421467,421468],{"xmlns":31650},[31652,421469,421470,421490],{},[31655,421471,421472,421474,421476,421488],{},[31749,421473,459],{},[31662,421475,202],{},[31743,421477,421478,421480],{},[31662,421479,131138],{},[31655,421481,421482,421484,421486],{},[31658,421483,31955],{},[31662,421485,112296],{},[31658,421487,106063],{},[31749,421489,459],{},[31665,421491,421492],{"encoding":31667},"0 = \\sum_{i \\in I} 0",[34,421494,421496,421514],{"className":421495,"ariaHidden":758},[31672],[34,421497,421499,421502,421505,421508,421511],{"className":421498},[31676],[34,421500],{"className":421501,"style":32617},[31680],[34,421503,459],{"className":421504},[31685],[34,421506],{"className":421507,"style":31846},[31845],[34,421509,202],{"className":421510},[31850],[34,421512],{"className":421513,"style":31846},[31845],[34,421515,421517,421520,421569,421572],{"className":421516},[31676],[34,421518],{"className":421519,"style":156553},[31680],[34,421521,421523,421526],{"className":421522},[32767],[34,421524,131138],{"className":421525,"style":156561},[32767,131249,156560],[34,421527,421529],{"className":421528},[31792],[34,421530,421532,421561],{"className":421531},[31796,31797],[34,421533,421535,421558],{"className":421534},[31801],[34,421536,421538],{"className":421537,"style":156574},[31805],[34,421539,421540,421543],{"style":156577},[34,421541],{"className":421542,"style":31814},[31813],[34,421544,421546],{"className":421545},[31818,31819,31820,31821],[34,421547,421549,421552,421555],{"className":421548},[31685,31821],[34,421550,31955],{"className":421551},[31685,31686,31821],[34,421553,112296],{"className":421554},[31850,31821],[34,421556,106063],{"className":421557,"style":67482},[31685,31686,31821],[34,421559,31829],{"className":421560},[31828],[34,421562,421564],{"className":421563},[31801],[34,421565,421567],{"className":421566,"style":156605},[31805],[34,421568],{},[34,421570],{"className":421571,"style":32123},[31845],[34,421573,459],{"className":421574},[31685]," 是 0 的一种表示法（每一项都取 0）。",[108,421577,421578,421579,421581,421582,315343,421651,347],{},"根据 (ii) 的唯一性，这必须是 0 的",[2627,421580,417171],{},"表示法。也就是说，不可能存在不全为 0 的 ",[34,421583,421585,421602],{"className":421584},[31642],[34,421586,421588],{"className":421587},[31646],[31648,421589,421590],{"xmlns":31650},[31652,421591,421592,421600],{},[31655,421593,421594],{},[31743,421595,421596,421598],{},[31658,421597,34395],{},[31658,421599,31955],{},[31665,421601,34855],{"encoding":31667},[34,421603,421605],{"className":421604,"ariaHidden":758},[31672],[34,421606,421608,421611],{"className":421607},[31676],[34,421609],{"className":421610,"style":31860},[31680],[34,421612,421614,421617],{"className":421613},[31685],[34,421615,34395],{"className":421616,"style":34452},[31685,31686],[34,421618,421620],{"className":421619},[31792],[34,421621,421623,421643],{"className":421622},[31796,31797],[34,421624,421626,421640],{"className":421625},[31801],[34,421627,421629],{"className":421628,"style":32090},[31805],[34,421630,421631,421634],{"style":34467},[34,421632],{"className":421633,"style":31814},[31813],[34,421635,421637],{"className":421636},[31818,31819,31820,31821],[34,421638,31955],{"className":421639},[31685,31686,31821],[34,421641,31829],{"className":421642},[31828],[34,421644,421646],{"className":421645},[31801],[34,421647,421649],{"className":421648,"style":31836},[31805],[34,421650],{},[34,421652,421654,421678],{"className":421653},[31642],[34,421655,421657],{"className":421656},[31646],[31648,421658,421659],{"xmlns":31650},[31652,421660,421661,421675],{},[31655,421662,421663,421665,421671,421673],{},[31662,421664,131138],{},[31743,421666,421667,421669],{},[31658,421668,34395],{},[31658,421670,31955],{},[31662,421672,202],{},[31749,421674,459],{},[31665,421676,421677],{"encoding":31667},"\\sum r_i = 0",[34,421679,421681,421742],{"className":421680,"ariaHidden":758},[31672],[34,421682,421684,421687,421690,421693,421733,421736,421739],{"className":421683},[31676],[34,421685],{"className":421686,"style":31774},[31680],[34,421688,131138],{"className":421689,"style":156561},[32767,131249,156560],[34,421691],{"className":421692,"style":32123},[31845],[34,421694,421696,421699],{"className":421695},[31685],[34,421697,34395],{"className":421698,"style":34452},[31685,31686],[34,421700,421702],{"className":421701},[31792],[34,421703,421705,421725],{"className":421704},[31796,31797],[34,421706,421708,421722],{"className":421707},[31801],[34,421709,421711],{"className":421710,"style":32090},[31805],[34,421712,421713,421716],{"style":34467},[34,421714],{"className":421715,"style":31814},[31813],[34,421717,421719],{"className":421718},[31818,31819,31820,31821],[34,421720,31955],{"className":421721},[31685,31686,31821],[34,421723,31829],{"className":421724},[31828],[34,421726,421728],{"className":421727},[31801],[34,421729,421731],{"className":421730,"style":31836},[31805],[34,421732],{},[34,421734],{"className":421735,"style":31846},[31845],[34,421737,202],{"className":421738},[31850],[34,421740],{"className":421741,"style":31846},[31845],[34,421743,421745,421748],{"className":421744},[31676],[34,421746],{"className":421747,"style":32617},[31680],[34,421749,459],{"className":421750},[31685],[108,421752,421753,421755],{},[2627,421754,114043],{},"：(iii) 成立。",[32490,421757,421759],{"id":421758},"证明步骤-3由-iii-推导-iv",[2627,421760,421761],{},"证明步骤 3：由 (iii) 推导 (iv)",[105,421763,421764,421798,422075,423792],{},[108,421765,421766,2664,421769,421797],{},[2627,421767,421768],{},"假设 (iii) 成立",[34,421770,421772,421785],{"className":421771},[31642],[34,421773,421775],{"className":421774},[31646],[31648,421776,421777],{"xmlns":31650},[31652,421778,421779,421783],{},[31655,421780,421781],{},[31749,421782,459],{},[31665,421784,459],{"encoding":31667},[34,421786,421788],{"className":421787,"ariaHidden":758},[31672],[34,421789,421791,421794],{"className":421790},[31676],[34,421792],{"className":421793,"style":32617},[31680],[34,421795,459],{"className":421796},[31685]," 只能唯一地表示为所有项均为 0 的和。",[108,421799,421800,421802,421803,347],{},[2627,421801,233757],{},"：证明 ",[34,421804,421806,421856],{"className":421805},[31642],[34,421807,421809],{"className":421808},[31646],[31648,421810,421811],{"xmlns":31650},[31652,421812,421813,421853],{},[31655,421814,421815,421821,421823,421825,421837,421843,421845,421847,421849,421851],{},[31743,421816,421817,421819],{},[31658,421818,34386],{},[31658,421820,33759],{},[31662,421822,326344],{},[31662,421824,456],{"stretchy":702},[31743,421826,421827,421829],{},[31662,421828,131138],{},[31655,421830,421831,421833,421835],{},[31658,421832,31968],{},[31662,421834,54707],{"mathvariant":38079},[31658,421836,33759],{},[31743,421838,421839,421841],{},[31658,421840,34386],{},[31658,421842,31968],{},[31662,421844,2557],{"stretchy":702},[31662,421846,202],{},[31662,421848,32685],{"stretchy":702},[31749,421850,459],{},[31662,421852,32715],{"stretchy":702},[31665,421854,421855],{"encoding":31667},"R_k \\cap (\\sum_{j \\neq k} R_j) = \\{0\\}",[34,421857,421859,421914,422060],{"className":421858,"ariaHidden":758},[31672],[34,421860,421862,421865,421905,421908,421911],{"className":421861},[31676],[34,421863],{"className":421864,"style":70915},[31680],[34,421866,421868,421871],{"className":421867},[31685],[34,421869,34386],{"className":421870,"style":34427},[31685,31686],[34,421872,421874],{"className":421873},[31792],[34,421875,421877,421897],{"className":421876},[31796,31797],[34,421878,421880,421894],{"className":421879},[31801],[34,421881,421883],{"className":421882,"style":33833},[31805],[34,421884,421885,421888],{"style":104694},[34,421886],{"className":421887,"style":31814},[31813],[34,421889,421891],{"className":421890},[31818,31819,31820,31821],[34,421892,33759],{"className":421893,"style":33774},[31685,31686,31821],[34,421895,31829],{"className":421896},[31828],[34,421898,421900],{"className":421899},[31801],[34,421901,421903],{"className":421902,"style":31836},[31805],[34,421904],{},[34,421906],{"className":421907,"style":32604},[31845],[34,421909,326344],{"className":421910},[32024],[34,421912],{"className":421913,"style":32604},[31845],[34,421915,421917,421920,421923,422005,422008,422048,422051,422054,422057],{"className":421916},[31676],[34,421918],{"className":421919,"style":156901},[31680],[34,421921,456],{"className":421922},[31782],[34,421924,421926,421929],{"className":421925},[32767],[34,421927,131138],{"className":421928,"style":156561},[32767,131249,156560],[34,421930,421932],{"className":421931},[31792],[34,421933,421935,421997],{"className":421934},[31796,31797],[34,421936,421938,421994],{"className":421937},[31801],[34,421939,421941],{"className":421940,"style":356484},[31805],[34,421942,421943,421946],{"style":156577},[34,421944],{"className":421945,"style":31814},[31813],[34,421947,421949],{"className":421948},[31818,31819,31820,31821],[34,421950,421952,421955,421991],{"className":421951},[31685,31821],[34,421953,31968],{"className":421954,"style":32153},[31685,31686,31821],[34,421956,421958,421985,421988],{"className":421957},[31850,31821],[34,421959,421961],{"className":421960},[31850,31821],[34,421962,421964],{"className":421963},[31685,54774,31821],[34,421965,421967],{"className":421966},[54778,31821],[34,421968,421970,421973,421982],{"className":421969},[54782,31821],[34,421971],{"className":421972,"style":43217},[31680],[34,421974,421976],{"className":421975},[54789],[34,421977,421979],{"className":421978},[31685,31821],[34,421980,54796],{"className":421981},[31850,31821],[34,421983],{"className":421984},[54800],[34,421986],{"className":421987},[31845,54804,31821],[34,421989,202],{"className":421990},[31850,31821],[34,421992,33759],{"className":421993,"style":33774},[31685,31686,31821],[34,421995,31829],{"className":421996},[31828],[34,421998,422000],{"className":421999},[31801],[34,422001,422003],{"className":422002,"style":156941},[31805],[34,422004],{},[34,422006],{"className":422007,"style":32123},[31845],[34,422009,422011,422014],{"className":422010},[31685],[34,422012,34386],{"className":422013,"style":34427},[31685,31686],[34,422015,422017],{"className":422016},[31792],[34,422018,422020,422040],{"className":422019},[31796,31797],[34,422021,422023,422037],{"className":422022},[31801],[34,422024,422026],{"className":422025,"style":32090},[31805],[34,422027,422028,422031],{"style":104694},[34,422029],{"className":422030,"style":31814},[31813],[34,422032,422034],{"className":422033},[31818,31819,31820,31821],[34,422035,31968],{"className":422036,"style":32153},[31685,31686,31821],[34,422038,31829],{"className":422039},[31828],[34,422041,422043],{"className":422042},[31801],[34,422044,422046],{"className":422045,"style":32163},[31805],[34,422047],{},[34,422049,2557],{"className":422050},[31690],[34,422052],{"className":422053,"style":31846},[31845],[34,422055,202],{"className":422056},[31850],[34,422058],{"className":422059,"style":31846},[31845],[34,422061,422063,422066,422069,422072],{"className":422062},[31676],[34,422064],{"className":422065,"style":31774},[31680],[34,422067,32685],{"className":422068},[31782],[34,422070,459],{"className":422071},[31685],[34,422073,32715],{"className":422074},[31690],[108,422076,422077,33183,422079],{},[2627,422078,33190],{},[105,422080,422081,422112,422266,422810,423251,423650],{},[108,422082,422083,422084,347],{},"取交集中的任意一个元素 ",[34,422085,422087,422100],{"className":422086},[31642],[34,422088,422090],{"className":422089},[31646],[31648,422091,422092],{"xmlns":31650},[31652,422093,422094,422098],{},[31655,422095,422096],{},[31658,422097,32936],{},[31665,422099,32936],{"encoding":31667},[34,422101,422103],{"className":422102,"ariaHidden":758},[31672],[34,422104,422106,422109],{"className":422105},[31676],[34,422107],{"className":422108,"style":32527},[31680],[34,422110,32936],{"className":422111},[31685,31686],[108,422113,422114,422115,422207,422208,422236,422237,422265],{},"一方面，",[34,422116,422118,422140],{"className":422117},[31642],[34,422119,422121],{"className":422120},[31646],[31648,422122,422123],{"xmlns":31650},[31652,422124,422125,422137],{},[31655,422126,422127,422129,422131],{},[31658,422128,32936],{},[31662,422130,112296],{},[31743,422132,422133,422135],{},[31658,422134,34386],{},[31658,422136,33759],{},[31665,422138,422139],{"encoding":31667},"x \\in R_k",[34,422141,422143,422161],{"className":422142,"ariaHidden":758},[31672],[34,422144,422146,422149,422152,422155,422158],{"className":422145},[31676],[34,422147],{"className":422148,"style":34812},[31680],[34,422150,32936],{"className":422151},[31685,31686],[34,422153],{"className":422154,"style":31846},[31845],[34,422156,112296],{"className":422157},[31850],[34,422159],{"className":422160,"style":31846},[31845],[34,422162,422164,422167],{"className":422163},[31676],[34,422165],{"className":422166,"style":70915},[31680],[34,422168,422170,422173],{"className":422169},[31685],[34,422171,34386],{"className":422172,"style":34427},[31685,31686],[34,422174,422176],{"className":422175},[31792],[34,422177,422179,422199],{"className":422178},[31796,31797],[34,422180,422182,422196],{"className":422181},[31801],[34,422183,422185],{"className":422184,"style":33833},[31805],[34,422186,422187,422190],{"style":104694},[34,422188],{"className":422189,"style":31814},[31813],[34,422191,422193],{"className":422192},[31818,31819,31820,31821],[34,422194,33759],{"className":422195,"style":33774},[31685,31686,31821],[34,422197,31829],{"className":422198},[31828],[34,422200,422202],{"className":422201},[31801],[34,422203,422205],{"className":422204,"style":31836},[31805],[34,422206],{},"，所以我们可以把 ",[34,422209,422211,422224],{"className":422210},[31642],[34,422212,422214],{"className":422213},[31646],[31648,422215,422216],{"xmlns":31650},[31652,422217,422218,422222],{},[31655,422219,422220],{},[31658,422221,32936],{},[31665,422223,32936],{"encoding":31667},[34,422225,422227],{"className":422226,"ariaHidden":758},[31672],[34,422228,422230,422233],{"className":422229},[31676],[34,422231],{"className":422232,"style":32527},[31680],[34,422234,32936],{"className":422235},[31685,31686]," 看作第 ",[34,422238,422240,422253],{"className":422239},[31642],[34,422241,422243],{"className":422242},[31646],[31648,422244,422245],{"xmlns":31650},[31652,422246,422247,422251],{},[31655,422248,422249],{},[31658,422250,33759],{},[31665,422252,33759],{"encoding":31667},[34,422254,422256],{"className":422255,"ariaHidden":758},[31672],[34,422257,422259,422262],{"className":422258},[31676],[34,422260],{"className":422261,"style":31681},[31680],[34,422263,33759],{"className":422264,"style":33774},[31685,31686]," 项。",[108,422267,422268,422269,341230,422458,422486,422487,70884,422676,347],{},"另一方面，",[34,422270,422272,422306],{"className":422271},[31642],[34,422273,422275],{"className":422274},[31646],[31648,422276,422277],{"xmlns":31650},[31652,422278,422279,422303],{},[31655,422280,422281,422283,422285,422297],{},[31658,422282,32936],{},[31662,422284,112296],{},[31743,422286,422287,422289],{},[31662,422288,131138],{},[31655,422290,422291,422293,422295],{},[31658,422292,31968],{},[31662,422294,54707],{"mathvariant":38079},[31658,422296,33759],{},[31743,422298,422299,422301],{},[31658,422300,34386],{},[31658,422302,31968],{},[31665,422304,422305],{"encoding":31667},"x \\in \\sum_{j \\neq k} R_j",[34,422307,422309,422327],{"className":422308,"ariaHidden":758},[31672],[34,422310,422312,422315,422318,422321,422324],{"className":422311},[31676],[34,422313],{"className":422314,"style":34812},[31680],[34,422316,32936],{"className":422317},[31685,31686],[34,422319],{"className":422320,"style":31846},[31845],[34,422322,112296],{"className":422323},[31850],[34,422325],{"className":422326,"style":31846},[31845],[34,422328,422330,422333,422415,422418],{"className":422329},[31676],[34,422331],{"className":422332,"style":156901},[31680],[34,422334,422336,422339],{"className":422335},[32767],[34,422337,131138],{"className":422338,"style":156561},[32767,131249,156560],[34,422340,422342],{"className":422341},[31792],[34,422343,422345,422407],{"className":422344},[31796,31797],[34,422346,422348,422404],{"className":422347},[31801],[34,422349,422351],{"className":422350,"style":356484},[31805],[34,422352,422353,422356],{"style":156577},[34,422354],{"className":422355,"style":31814},[31813],[34,422357,422359],{"className":422358},[31818,31819,31820,31821],[34,422360,422362,422365,422401],{"className":422361},[31685,31821],[34,422363,31968],{"className":422364,"style":32153},[31685,31686,31821],[34,422366,422368,422395,422398],{"className":422367},[31850,31821],[34,422369,422371],{"className":422370},[31850,31821],[34,422372,422374],{"className":422373},[31685,54774,31821],[34,422375,422377],{"className":422376},[54778,31821],[34,422378,422380,422383,422392],{"className":422379},[54782,31821],[34,422381],{"className":422382,"style":43217},[31680],[34,422384,422386],{"className":422385},[54789],[34,422387,422389],{"className":422388},[31685,31821],[34,422390,54796],{"className":422391},[31850,31821],[34,422393],{"className":422394},[54800],[34,422396],{"className":422397},[31845,54804,31821],[34,422399,202],{"className":422400},[31850,31821],[34,422402,33759],{"className":422403,"style":33774},[31685,31686,31821],[34,422405,31829],{"className":422406},[31828],[34,422408,422410],{"className":422409},[31801],[34,422411,422413],{"className":422412,"style":156941},[31805],[34,422414],{},[34,422416],{"className":422417,"style":32123},[31845],[34,422419,422421,422424],{"className":422420},[31685],[34,422422,34386],{"className":422423,"style":34427},[31685,31686],[34,422425,422427],{"className":422426},[31792],[34,422428,422430,422450],{"className":422429},[31796,31797],[34,422431,422433,422447],{"className":422432},[31801],[34,422434,422436],{"className":422435,"style":32090},[31805],[34,422437,422438,422441],{"style":104694},[34,422439],{"className":422440,"style":31814},[31813],[34,422442,422444],{"className":422443},[31818,31819,31820,31821],[34,422445,31968],{"className":422446,"style":32153},[31685,31686,31821],[34,422448,31829],{"className":422449},[31828],[34,422451,422453],{"className":422452},[31801],[34,422454,422456],{"className":422455,"style":32163},[31805],[34,422457],{},[34,422459,422461,422474],{"className":422460},[31642],[34,422462,422464],{"className":422463},[31646],[31648,422465,422466],{"xmlns":31650},[31652,422467,422468,422472],{},[31655,422469,422470],{},[31658,422471,32936],{},[31665,422473,32936],{"encoding":31667},[34,422475,422477],{"className":422476,"ariaHidden":758},[31672],[34,422478,422480,422483],{"className":422479},[31676],[34,422481],{"className":422482,"style":32527},[31680],[34,422484,32936],{"className":422485},[31685,31686]," 可以写成其他理想中元素的和：",[34,422488,422490,422524],{"className":422489},[31642],[34,422491,422493],{"className":422492},[31646],[31648,422494,422495],{"xmlns":31650},[31652,422496,422497,422521],{},[31655,422498,422499,422501,422503,422515],{},[31658,422500,32936],{},[31662,422502,202],{},[31743,422504,422505,422507],{},[31662,422506,131138],{},[31655,422508,422509,422511,422513],{},[31658,422510,31968],{},[31662,422512,54707],{"mathvariant":38079},[31658,422514,33759],{},[31743,422516,422517,422519],{},[31658,422518,34395],{},[31658,422520,31968],{},[31665,422522,422523],{"encoding":31667},"x = \\sum_{j \\neq k} r_j",[34,422525,422527,422545],{"className":422526,"ariaHidden":758},[31672],[34,422528,422530,422533,422536,422539,422542],{"className":422529},[31676],[34,422531],{"className":422532,"style":32527},[31680],[34,422534,32936],{"className":422535},[31685,31686],[34,422537],{"className":422538,"style":31846},[31845],[34,422540,202],{"className":422541},[31850],[34,422543],{"className":422544,"style":31846},[31845],[34,422546,422548,422551,422633,422636],{"className":422547},[31676],[34,422549],{"className":422550,"style":156901},[31680],[34,422552,422554,422557],{"className":422553},[32767],[34,422555,131138],{"className":422556,"style":156561},[32767,131249,156560],[34,422558,422560],{"className":422559},[31792],[34,422561,422563,422625],{"className":422562},[31796,31797],[34,422564,422566,422622],{"className":422565},[31801],[34,422567,422569],{"className":422568,"style":356484},[31805],[34,422570,422571,422574],{"style":156577},[34,422572],{"className":422573,"style":31814},[31813],[34,422575,422577],{"className":422576},[31818,31819,31820,31821],[34,422578,422580,422583,422619],{"className":422579},[31685,31821],[34,422581,31968],{"className":422582,"style":32153},[31685,31686,31821],[34,422584,422586,422613,422616],{"className":422585},[31850,31821],[34,422587,422589],{"className":422588},[31850,31821],[34,422590,422592],{"className":422591},[31685,54774,31821],[34,422593,422595],{"className":422594},[54778,31821],[34,422596,422598,422601,422610],{"className":422597},[54782,31821],[34,422599],{"className":422600,"style":43217},[31680],[34,422602,422604],{"className":422603},[54789],[34,422605,422607],{"className":422606},[31685,31821],[34,422608,54796],{"className":422609},[31850,31821],[34,422611],{"className":422612},[54800],[34,422614],{"className":422615},[31845,54804,31821],[34,422617,202],{"className":422618},[31850,31821],[34,422620,33759],{"className":422621,"style":33774},[31685,31686,31821],[34,422623,31829],{"className":422624},[31828],[34,422626,422628],{"className":422627},[31801],[34,422629,422631],{"className":422630,"style":156941},[31805],[34,422632],{},[34,422634],{"className":422635,"style":32123},[31845],[34,422637,422639,422642],{"className":422638},[31685],[34,422640,34395],{"className":422641,"style":34452},[31685,31686],[34,422643,422645],{"className":422644},[31792],[34,422646,422648,422668],{"className":422647},[31796,31797],[34,422649,422651,422665],{"className":422650},[31801],[34,422652,422654],{"className":422653,"style":32090},[31805],[34,422655,422656,422659],{"style":34467},[34,422657],{"className":422658,"style":31814},[31813],[34,422660,422662],{"className":422661},[31818,31819,31820,31821],[34,422663,31968],{"className":422664,"style":32153},[31685,31686,31821],[34,422666,31829],{"className":422667},[31828],[34,422669,422671],{"className":422670},[31801],[34,422672,422674],{"className":422673,"style":32163},[31805],[34,422675],{},[34,422677,422679,422705],{"className":422678},[31642],[34,422680,422682],{"className":422681},[31646],[31648,422683,422684],{"xmlns":31650},[31652,422685,422686,422702],{},[31655,422687,422688,422694,422696],{},[31743,422689,422690,422692],{},[31658,422691,34395],{},[31658,422693,31968],{},[31662,422695,112296],{},[31743,422697,422698,422700],{},[31658,422699,34386],{},[31658,422701,31968],{},[31665,422703,422704],{"encoding":31667},"r_j \\in R_j",[34,422706,422708,422764],{"className":422707,"ariaHidden":758},[31672],[34,422709,422711,422715,422755,422758,422761],{"className":422710},[31676],[34,422712],{"className":422713,"style":422714},[31680],"height:0.8252em;vertical-align:-0.2861em;",[34,422716,422718,422721],{"className":422717},[31685],[34,422719,34395],{"className":422720,"style":34452},[31685,31686],[34,422722,422724],{"className":422723},[31792],[34,422725,422727,422747],{"className":422726},[31796,31797],[34,422728,422730,422744],{"className":422729},[31801],[34,422731,422733],{"className":422732,"style":32090},[31805],[34,422734,422735,422738],{"style":34467},[34,422736],{"className":422737,"style":31814},[31813],[34,422739,422741],{"className":422740},[31818,31819,31820,31821],[34,422742,31968],{"className":422743,"style":32153},[31685,31686,31821],[34,422745,31829],{"className":422746},[31828],[34,422748,422750],{"className":422749},[31801],[34,422751,422753],{"className":422752,"style":32163},[31805],[34,422754],{},[34,422756],{"className":422757,"style":31846},[31845],[34,422759,112296],{"className":422760},[31850],[34,422762],{"className":422763,"style":31846},[31845],[34,422765,422767,422770],{"className":422766},[31676],[34,422768],{"className":422769,"style":55484},[31680],[34,422771,422773,422776],{"className":422772},[31685],[34,422774,34386],{"className":422775,"style":34427},[31685,31686],[34,422777,422779],{"className":422778},[31792],[34,422780,422782,422802],{"className":422781},[31796,31797],[34,422783,422785,422799],{"className":422784},[31801],[34,422786,422788],{"className":422787,"style":32090},[31805],[34,422789,422790,422793],{"style":104694},[34,422791],{"className":422792,"style":31814},[31813],[34,422794,422796],{"className":422795},[31818,31819,31820,31821],[34,422797,31968],{"className":422798,"style":32153},[31685,31686,31821],[34,422800,31829],{"className":422801},[31828],[34,422803,422805],{"className":422804},[31801],[34,422806,422808],{"className":422807,"style":32163},[31805],[34,422809],{},[108,422811,422812,422813,423024,423025],{},"现在我们将两边相减（移项），构造一个等于 0 的式子：\n",[34,422814,422816,422854],{"className":422815},[31642],[34,422817,422819],{"className":422818},[31646],[31648,422820,422821],{"xmlns":31650},[31652,422822,422823,422851],{},[31655,422824,422825,422827,422829,422841,422847,422849],{},[31658,422826,32936],{},[31662,422828,32226],{},[31743,422830,422831,422833],{},[31662,422832,131138],{},[31655,422834,422835,422837,422839],{},[31658,422836,31968],{},[31662,422838,54707],{"mathvariant":38079},[31658,422840,33759],{},[31743,422842,422843,422845],{},[31658,422844,34395],{},[31658,422846,31968],{},[31662,422848,202],{},[31749,422850,459],{},[31665,422852,422853],{"encoding":31667},"x - \\sum_{j \\neq k} r_j = 0",[34,422855,422857,422875,423015],{"className":422856,"ariaHidden":758},[31672],[34,422858,422860,422863,422866,422869,422872],{"className":422859},[31676],[34,422861],{"className":422862,"style":32597},[31680],[34,422864,32936],{"className":422865},[31685,31686],[34,422867],{"className":422868,"style":32604},[31845],[34,422870,32226],{"className":422871},[32024],[34,422873],{"className":422874,"style":32604},[31845],[34,422876,422878,422881,422963,422966,423006,423009,423012],{"className":422877},[31676],[34,422879],{"className":422880,"style":156901},[31680],[34,422882,422884,422887],{"className":422883},[32767],[34,422885,131138],{"className":422886,"style":156561},[32767,131249,156560],[34,422888,422890],{"className":422889},[31792],[34,422891,422893,422955],{"className":422892},[31796,31797],[34,422894,422896,422952],{"className":422895},[31801],[34,422897,422899],{"className":422898,"style":356484},[31805],[34,422900,422901,422904],{"style":156577},[34,422902],{"className":422903,"style":31814},[31813],[34,422905,422907],{"className":422906},[31818,31819,31820,31821],[34,422908,422910,422913,422949],{"className":422909},[31685,31821],[34,422911,31968],{"className":422912,"style":32153},[31685,31686,31821],[34,422914,422916,422943,422946],{"className":422915},[31850,31821],[34,422917,422919],{"className":422918},[31850,31821],[34,422920,422922],{"className":422921},[31685,54774,31821],[34,422923,422925],{"className":422924},[54778,31821],[34,422926,422928,422931,422940],{"className":422927},[54782,31821],[34,422929],{"className":422930,"style":43217},[31680],[34,422932,422934],{"className":422933},[54789],[34,422935,422937],{"className":422936},[31685,31821],[34,422938,54796],{"className":422939},[31850,31821],[34,422941],{"className":422942},[54800],[34,422944],{"className":422945},[31845,54804,31821],[34,422947,202],{"className":422948},[31850,31821],[34,422950,33759],{"className":422951,"style":33774},[31685,31686,31821],[34,422953,31829],{"className":422954},[31828],[34,422956,422958],{"className":422957},[31801],[34,422959,422961],{"className":422960,"style":156941},[31805],[34,422962],{},[34,422964],{"className":422965,"style":32123},[31845],[34,422967,422969,422972],{"className":422968},[31685],[34,422970,34395],{"className":422971,"style":34452},[31685,31686],[34,422973,422975],{"className":422974},[31792],[34,422976,422978,422998],{"className":422977},[31796,31797],[34,422979,422981,422995],{"className":422980},[31801],[34,422982,422984],{"className":422983,"style":32090},[31805],[34,422985,422986,422989],{"style":34467},[34,422987],{"className":422988,"style":31814},[31813],[34,422990,422992],{"className":422991},[31818,31819,31820,31821],[34,422993,31968],{"className":422994,"style":32153},[31685,31686,31821],[34,422996,31829],{"className":422997},[31828],[34,422999,423001],{"className":423000},[31801],[34,423002,423004],{"className":423003,"style":32163},[31805],[34,423005],{},[34,423007],{"className":423008,"style":31846},[31845],[34,423010,202],{"className":423011},[31850],[34,423013],{"className":423014,"style":31846},[31845],[34,423016,423018,423021],{"className":423017},[31676],[34,423019],{"className":423020,"style":32617},[31680],[34,423022,459],{"className":423023},[31685],"\n或者写成规范的和的形式：\n",[34,423026,423028,423072],{"className":423027},[31642],[34,423029,423031],{"className":423030},[31646],[31648,423032,423033],{"xmlns":31650},[31652,423034,423035,423069],{},[31655,423036,423037,423039,423041,423043,423045,423047,423059,423065,423067],{},[31662,423038,456],{"stretchy":702},[31662,423040,32226],{},[31658,423042,32936],{},[31662,423044,2557],{"stretchy":702},[31662,423046,223],{},[31743,423048,423049,423051],{},[31662,423050,131138],{},[31655,423052,423053,423055,423057],{},[31658,423054,31968],{},[31662,423056,54707],{"mathvariant":38079},[31658,423058,33759],{},[31743,423060,423061,423063],{},[31658,423062,34395],{},[31658,423064,31968],{},[31662,423066,202],{},[31749,423068,459],{},[31665,423070,423071],{"encoding":31667},"(-x) + \\sum_{j \\neq k} r_j = 0",[34,423073,423075,423102,423242],{"className":423074,"ariaHidden":758},[31672],[34,423076,423078,423081,423084,423087,423090,423093,423096,423099],{"className":423077},[31676],[34,423079],{"className":423080,"style":31774},[31680],[34,423082,456],{"className":423083},[31782],[34,423085,32226],{"className":423086},[31685],[34,423088,32936],{"className":423089},[31685,31686],[34,423091,2557],{"className":423092},[31690],[34,423094],{"className":423095,"style":32604},[31845],[34,423097,223],{"className":423098},[32024],[34,423100],{"className":423101,"style":32604},[31845],[34,423103,423105,423108,423190,423193,423233,423236,423239],{"className":423104},[31676],[34,423106],{"className":423107,"style":156901},[31680],[34,423109,423111,423114],{"className":423110},[32767],[34,423112,131138],{"className":423113,"style":156561},[32767,131249,156560],[34,423115,423117],{"className":423116},[31792],[34,423118,423120,423182],{"className":423119},[31796,31797],[34,423121,423123,423179],{"className":423122},[31801],[34,423124,423126],{"className":423125,"style":356484},[31805],[34,423127,423128,423131],{"style":156577},[34,423129],{"className":423130,"style":31814},[31813],[34,423132,423134],{"className":423133},[31818,31819,31820,31821],[34,423135,423137,423140,423176],{"className":423136},[31685,31821],[34,423138,31968],{"className":423139,"style":32153},[31685,31686,31821],[34,423141,423143,423170,423173],{"className":423142},[31850,31821],[34,423144,423146],{"className":423145},[31850,31821],[34,423147,423149],{"className":423148},[31685,54774,31821],[34,423150,423152],{"className":423151},[54778,31821],[34,423153,423155,423158,423167],{"className":423154},[54782,31821],[34,423156],{"className":423157,"style":43217},[31680],[34,423159,423161],{"className":423160},[54789],[34,423162,423164],{"className":423163},[31685,31821],[34,423165,54796],{"className":423166},[31850,31821],[34,423168],{"className":423169},[54800],[34,423171],{"className":423172},[31845,54804,31821],[34,423174,202],{"className":423175},[31850,31821],[34,423177,33759],{"className":423178,"style":33774},[31685,31686,31821],[34,423180,31829],{"className":423181},[31828],[34,423183,423185],{"className":423184},[31801],[34,423186,423188],{"className":423187,"style":156941},[31805],[34,423189],{},[34,423191],{"className":423192,"style":32123},[31845],[34,423194,423196,423199],{"className":423195},[31685],[34,423197,34395],{"className":423198,"style":34452},[31685,31686],[34,423200,423202],{"className":423201},[31792],[34,423203,423205,423225],{"className":423204},[31796,31797],[34,423206,423208,423222],{"className":423207},[31801],[34,423209,423211],{"className":423210,"style":32090},[31805],[34,423212,423213,423216],{"style":34467},[34,423214],{"className":423215,"style":31814},[31813],[34,423217,423219],{"className":423218},[31818,31819,31820,31821],[34,423220,31968],{"className":423221,"style":32153},[31685,31686,31821],[34,423223,31829],{"className":423224},[31828],[34,423226,423228],{"className":423227},[31801],[34,423229,423231],{"className":423230,"style":32163},[31805],[34,423232],{},[34,423234],{"className":423235,"style":31846},[31845],[34,423237,202],{"className":423238},[31850],[34,423240],{"className":423241,"style":31846},[31845],[34,423243,423245,423248],{"className":423244},[31676],[34,423246],{"className":423247,"style":32617},[31680],[34,423249,459],{"className":423250},[31685],[108,423252,423253,423254],{},"注意看这个式子：\n",[105,423255,423256,423392,423647],{},[108,423257,178926,423258,423286,423287,423321,423322,31901],{},[34,423259,423261,423274],{"className":423260},[31642],[34,423262,423264],{"className":423263},[31646],[31648,423265,423266],{"xmlns":31650},[31652,423267,423268,423272],{},[31655,423269,423270],{},[31658,423271,33759],{},[31665,423273,33759],{"encoding":31667},[34,423275,423277],{"className":423276,"ariaHidden":758},[31672],[34,423278,423280,423283],{"className":423279},[31676],[34,423281],{"className":423282,"style":31681},[31680],[34,423284,33759],{"className":423285,"style":33774},[31685,31686]," 个位置的元素是 ",[34,423288,423290,423306],{"className":423289},[31642],[34,423291,423293],{"className":423292},[31646],[31648,423294,423295],{"xmlns":31650},[31652,423296,423297,423303],{},[31655,423298,423299,423301],{},[31662,423300,32226],{},[31658,423302,32936],{},[31665,423304,423305],{"encoding":31667},"-x",[34,423307,423309],{"className":423308,"ariaHidden":758},[31672],[34,423310,423312,423315,423318],{"className":423311},[31676],[34,423313],{"className":423314,"style":32597},[31680],[34,423316,32226],{"className":423317},[31685],[34,423319,32936],{"className":423320},[31685,31686],"（属于 ",[34,423323,423325,423343],{"className":423324},[31642],[34,423326,423328],{"className":423327},[31646],[31648,423329,423330],{"xmlns":31650},[31652,423331,423332,423340],{},[31655,423333,423334],{},[31743,423335,423336,423338],{},[31658,423337,34386],{},[31658,423339,33759],{},[31665,423341,423342],{"encoding":31667},"R_k",[34,423344,423346],{"className":423345,"ariaHidden":758},[31672],[34,423347,423349,423352],{"className":423348},[31676],[34,423350],{"className":423351,"style":70915},[31680],[34,423353,423355,423358],{"className":423354},[31685],[34,423356,34386],{"className":423357,"style":34427},[31685,31686],[34,423359,423361],{"className":423360},[31792],[34,423362,423364,423384],{"className":423363},[31796,31797],[34,423365,423367,423381],{"className":423366},[31801],[34,423368,423370],{"className":423369,"style":33833},[31805],[34,423371,423372,423375],{"style":104694},[34,423373],{"className":423374,"style":31814},[31813],[34,423376,423378],{"className":423377},[31818,31819,31820,31821],[34,423379,33759],{"className":423380,"style":33774},[31685,31686,31821],[34,423382,31829],{"className":423383},[31828],[34,423385,423387],{"className":423386},[31801],[34,423388,423390],{"className":423389,"style":31836},[31805],[34,423391],{},[108,423393,178926,423394,1840,423422,423506,423507,423321,423577,31901],{},[34,423395,423397,423410],{"className":423396},[31642],[34,423398,423400],{"className":423399},[31646],[31648,423401,423402],{"xmlns":31650},[31652,423403,423404,423408],{},[31655,423405,423406],{},[31658,423407,31968],{},[31665,423409,31968],{"encoding":31667},[34,423411,423413],{"className":423412,"ariaHidden":758},[31672],[34,423414,423416,423419],{"className":423415},[31676],[34,423417],{"className":423418,"style":33123},[31680],[34,423420,31968],{"className":423421,"style":32153},[31685,31686],[34,423423,423425,423443],{"className":423424},[31642],[34,423426,423428],{"className":423427},[31646],[31648,423429,423430],{"xmlns":31650},[31652,423431,423432,423440],{},[31655,423433,423434,423436,423438],{},[31658,423435,31968],{},[31662,423437,54707],{"mathvariant":38079},[31658,423439,33759],{},[31665,423441,423442],{"encoding":31667},"j \\neq k",[34,423444,423446,423497],{"className":423445,"ariaHidden":758},[31672],[34,423447,423449,423452,423455,423458,423494],{"className":423448},[31676],[34,423450],{"className":423451,"style":43217},[31680],[34,423453,31968],{"className":423454,"style":32153},[31685,31686],[34,423456],{"className":423457,"style":31846},[31845],[34,423459,423461,423488,423491],{"className":423460},[31850],[34,423462,423464],{"className":423463},[31850],[34,423465,423467],{"className":423466},[31685,54774],[34,423468,423470],{"className":423469},[54778],[34,423471,423473,423476,423485],{"className":423472},[54782],[34,423474],{"className":423475,"style":43217},[31680],[34,423477,423479],{"className":423478},[54789],[34,423480,423482],{"className":423481},[31685],[34,423483,54796],{"className":423484},[31850],[34,423486],{"className":423487},[54800],[34,423489],{"className":423490},[31845,54804],[34,423492,202],{"className":423493},[31850],[34,423495],{"className":423496,"style":31846},[31845],[34,423498,423500,423503],{"className":423499},[31676],[34,423501],{"className":423502,"style":31681},[31680],[34,423504,33759],{"className":423505,"style":33774},[31685,31686],") 个位置的元素是 ",[34,423508,423510,423528],{"className":423509},[31642],[34,423511,423513],{"className":423512},[31646],[31648,423514,423515],{"xmlns":31650},[31652,423516,423517,423525],{},[31655,423518,423519],{},[31743,423520,423521,423523],{},[31658,423522,34395],{},[31658,423524,31968],{},[31665,423526,423527],{"encoding":31667},"r_j",[34,423529,423531],{"className":423530,"ariaHidden":758},[31672],[34,423532,423534,423537],{"className":423533},[31676],[34,423535],{"className":423536,"style":56005},[31680],[34,423538,423540,423543],{"className":423539},[31685],[34,423541,34395],{"className":423542,"style":34452},[31685,31686],[34,423544,423546],{"className":423545},[31792],[34,423547,423549,423569],{"className":423548},[31796,31797],[34,423550,423552,423566],{"className":423551},[31801],[34,423553,423555],{"className":423554,"style":32090},[31805],[34,423556,423557,423560],{"style":34467},[34,423558],{"className":423559,"style":31814},[31813],[34,423561,423563],{"className":423562},[31818,31819,31820,31821],[34,423564,31968],{"className":423565,"style":32153},[31685,31686,31821],[34,423567,31829],{"className":423568},[31828],[34,423570,423572],{"className":423571},[31801],[34,423573,423575],{"className":423574,"style":32163},[31805],[34,423576],{},[34,423578,423580,423598],{"className":423579},[31642],[34,423581,423583],{"className":423582},[31646],[31648,423584,423585],{"xmlns":31650},[31652,423586,423587,423595],{},[31655,423588,423589],{},[31743,423590,423591,423593],{},[31658,423592,34386],{},[31658,423594,31968],{},[31665,423596,423597],{"encoding":31667},"R_j",[34,423599,423601],{"className":423600,"ariaHidden":758},[31672],[34,423602,423604,423607],{"className":423603},[31676],[34,423605],{"className":423606,"style":55484},[31680],[34,423608,423610,423613],{"className":423609},[31685],[34,423611,34386],{"className":423612,"style":34427},[31685,31686],[34,423614,423616],{"className":423615},[31792],[34,423617,423619,423639],{"className":423618},[31796,31797],[34,423620,423622,423636],{"className":423621},[31801],[34,423623,423625],{"className":423624,"style":32090},[31805],[34,423626,423627,423630],{"style":104694},[34,423628],{"className":423629,"style":31814},[31813],[34,423631,423633],{"className":423632},[31818,31819,31820,31821],[34,423634,31968],{"className":423635,"style":32153},[31685,31686,31821],[34,423637,31829],{"className":423638},[31828],[34,423640,423642],{"className":423641},[31801],[34,423643,423645],{"className":423644,"style":32163},[31805],[34,423646],{},[108,423648,423649],{},"它们的总和是 0。",[108,423651,423652,423653,423681,423682,347],{},"利用假设 (iii)“0 的表示是唯一的”：\n这就意味着上述表达式中的每一项都必须必须是 0。\n特别是第 ",[34,423654,423656,423669],{"className":423655},[31642],[34,423657,423659],{"className":423658},[31646],[31648,423660,423661],{"xmlns":31650},[31652,423662,423663,423667],{},[31655,423664,423665],{},[31658,423666,33759],{},[31665,423668,33759],{"encoding":31667},[34,423670,423672],{"className":423671,"ariaHidden":758},[31672],[34,423673,423675,423678],{"className":423674},[31676],[34,423676],{"className":423677,"style":31681},[31680],[34,423679,33759],{"className":423680,"style":33774},[31685,31686]," 项：",[34,423683,423685,423717],{"className":423684},[31642],[34,423686,423688],{"className":423687},[31646],[31648,423689,423690],{"xmlns":31650},[31652,423691,423692,423714],{},[31655,423693,423694,423696,423698,423700,423702,423704,423706,423708,423710,423712],{},[31662,423695,32226],{},[31658,423697,32936],{},[31662,423699,202],{},[31749,423701,459],{},[39361,423703,262622],{},[31662,423705,262625],{},[39361,423707,262622],{},[31658,423709,32936],{},[31662,423711,202],{},[31749,423713,459],{},[31665,423715,423716],{"encoding":31667},"-x = 0 \\implies x = 0",[34,423718,423720,423741,423765,423783],{"className":423719,"ariaHidden":758},[31672],[34,423721,423723,423726,423729,423732,423735,423738],{"className":423722},[31676],[34,423724],{"className":423725,"style":32597},[31680],[34,423727,32226],{"className":423728},[31685],[34,423730,32936],{"className":423731},[31685,31686],[34,423733],{"className":423734,"style":31846},[31845],[34,423736,202],{"className":423737},[31850],[34,423739],{"className":423740,"style":31846},[31845],[34,423742,423744,423747,423750,423753,423756,423759,423762],{"className":423743},[31676],[34,423745],{"className":423746,"style":262700},[31680],[34,423748,459],{"className":423749},[31685],[34,423751],{"className":423752,"style":31846},[31845],[34,423754],{"className":423755,"style":31846},[31845],[34,423757,262625],{"className":423758},[31850],[34,423760],{"className":423761,"style":31846},[31845],[34,423763],{"className":423764,"style":31846},[31845],[34,423766,423768,423771,423774,423777,423780],{"className":423767},[31676],[34,423769],{"className":423770,"style":32527},[31680],[34,423772,32936],{"className":423773},[31685,31686],[34,423775],{"className":423776,"style":31846},[31845],[34,423778,202],{"className":423779},[31850],[34,423781],{"className":423782,"style":31846},[31845],[34,423784,423786,423789],{"className":423785},[31676],[34,423787],{"className":423788,"style":32617},[31680],[34,423790,459],{"className":423791},[31685],[108,423793,423794,423796,423797,424068],{},[2627,423795,114043],{},"：交集里只有 0 元素，所以 ",[34,423798,423800,423849],{"className":423799},[31642],[34,423801,423803],{"className":423802},[31646],[31648,423804,423805],{"xmlns":31650},[31652,423806,423807,423847],{},[31655,423808,423809,423815,423817,423819,423831,423837,423839,423841,423843,423845],{},[31743,423810,423811,423813],{},[31658,423812,34386],{},[31658,423814,33759],{},[31662,423816,326344],{},[31662,423818,456],{"stretchy":702},[31743,423820,423821,423823],{},[31662,423822,131138],{},[31655,423824,423825,423827,423829],{},[31658,423826,31968],{},[31662,423828,54707],{"mathvariant":38079},[31658,423830,33759],{},[31743,423832,423833,423835],{},[31658,423834,34386],{},[31658,423836,31968],{},[31662,423838,2557],{"stretchy":702},[31662,423840,202],{},[31662,423842,32685],{"stretchy":702},[31749,423844,459],{},[31662,423846,32715],{"stretchy":702},[31665,423848,421855],{"encoding":31667},[34,423850,423852,423907,424053],{"className":423851,"ariaHidden":758},[31672],[34,423853,423855,423858,423898,423901,423904],{"className":423854},[31676],[34,423856],{"className":423857,"style":70915},[31680],[34,423859,423861,423864],{"className":423860},[31685],[34,423862,34386],{"className":423863,"style":34427},[31685,31686],[34,423865,423867],{"className":423866},[31792],[34,423868,423870,423890],{"className":423869},[31796,31797],[34,423871,423873,423887],{"className":423872},[31801],[34,423874,423876],{"className":423875,"style":33833},[31805],[34,423877,423878,423881],{"style":104694},[34,423879],{"className":423880,"style":31814},[31813],[34,423882,423884],{"className":423883},[31818,31819,31820,31821],[34,423885,33759],{"className":423886,"style":33774},[31685,31686,31821],[34,423888,31829],{"className":423889},[31828],[34,423891,423893],{"className":423892},[31801],[34,423894,423896],{"className":423895,"style":31836},[31805],[34,423897],{},[34,423899],{"className":423900,"style":32604},[31845],[34,423902,326344],{"className":423903},[32024],[34,423905],{"className":423906,"style":32604},[31845],[34,423908,423910,423913,423916,423998,424001,424041,424044,424047,424050],{"className":423909},[31676],[34,423911],{"className":423912,"style":156901},[31680],[34,423914,456],{"className":423915},[31782],[34,423917,423919,423922],{"className":423918},[32767],[34,423920,131138],{"className":423921,"style":156561},[32767,131249,156560],[34,423923,423925],{"className":423924},[31792],[34,423926,423928,423990],{"className":423927},[31796,31797],[34,423929,423931,423987],{"className":423930},[31801],[34,423932,423934],{"className":423933,"style":356484},[31805],[34,423935,423936,423939],{"style":156577},[34,423937],{"className":423938,"style":31814},[31813],[34,423940,423942],{"className":423941},[31818,31819,31820,31821],[34,423943,423945,423948,423984],{"className":423944},[31685,31821],[34,423946,31968],{"className":423947,"style":32153},[31685,31686,31821],[34,423949,423951,423978,423981],{"className":423950},[31850,31821],[34,423952,423954],{"className":423953},[31850,31821],[34,423955,423957],{"className":423956},[31685,54774,31821],[34,423958,423960],{"className":423959},[54778,31821],[34,423961,423963,423966,423975],{"className":423962},[54782,31821],[34,423964],{"className":423965,"style":43217},[31680],[34,423967,423969],{"className":423968},[54789],[34,423970,423972],{"className":423971},[31685,31821],[34,423973,54796],{"className":423974},[31850,31821],[34,423976],{"className":423977},[54800],[34,423979],{"className":423980},[31845,54804,31821],[34,423982,202],{"className":423983},[31850,31821],[34,423985,33759],{"className":423986,"style":33774},[31685,31686,31821],[34,423988,31829],{"className":423989},[31828],[34,423991,423993],{"className":423992},[31801],[34,423994,423996],{"className":423995,"style":156941},[31805],[34,423997],{},[34,423999],{"className":424000,"style":32123},[31845],[34,424002,424004,424007],{"className":424003},[31685],[34,424005,34386],{"className":424006,"style":34427},[31685,31686],[34,424008,424010],{"className":424009},[31792],[34,424011,424013,424033],{"className":424012},[31796,31797],[34,424014,424016,424030],{"className":424015},[31801],[34,424017,424019],{"className":424018,"style":32090},[31805],[34,424020,424021,424024],{"style":104694},[34,424022],{"className":424023,"style":31814},[31813],[34,424025,424027],{"className":424026},[31818,31819,31820,31821],[34,424028,31968],{"className":424029,"style":32153},[31685,31686,31821],[34,424031,31829],{"className":424032},[31828],[34,424034,424036],{"className":424035},[31801],[34,424037,424039],{"className":424038,"style":32163},[31805],[34,424040],{},[34,424042,2557],{"className":424043},[31690],[34,424045],{"className":424046,"style":31846},[31845],[34,424048,202],{"className":424049},[31850],[34,424051],{"className":424052,"style":31846},[31845],[34,424054,424056,424059,424062,424065],{"className":424055},[31676],[34,424057],{"className":424058,"style":31774},[31680],[34,424060,32685],{"className":424061},[31782],[34,424063,459],{"className":424064},[31685],[34,424066,32715],{"className":424067},[31690],"，(iv) 成立。",[32490,424070,424072],{"id":424071},"证明步骤-4由-iv-推导-i",[2627,424073,424074],{},"证明步骤 4：由 (iv) 推导 (i)",[105,424076,424077,424153,424186,426452],{},[108,424078,424079,424082,424083,424152],{},[2627,424080,424081],{},"假设 (iv) 成立","：每个 ",[34,424084,424086,424103],{"className":424085},[31642],[34,424087,424089],{"className":424088},[31646],[31648,424090,424091],{"xmlns":31650},[31652,424092,424093,424101],{},[31655,424094,424095],{},[31743,424096,424097,424099],{},[31658,424098,34386],{},[31658,424100,33759],{},[31665,424102,423342],{"encoding":31667},[34,424104,424106],{"className":424105,"ariaHidden":758},[31672],[34,424107,424109,424112],{"className":424108},[31676],[34,424110],{"className":424111,"style":70915},[31680],[34,424113,424115,424118],{"className":424114},[31685],[34,424116,34386],{"className":424117,"style":34427},[31685,31686],[34,424119,424121],{"className":424120},[31792],[34,424122,424124,424144],{"className":424123},[31796,31797],[34,424125,424127,424141],{"className":424126},[31801],[34,424128,424130],{"className":424129,"style":33833},[31805],[34,424131,424132,424135],{"style":104694},[34,424133],{"className":424134,"style":31814},[31813],[34,424136,424138],{"className":424137},[31818,31819,31820,31821],[34,424139,33759],{"className":424140,"style":33774},[31685,31686,31821],[34,424142,31829],{"className":424143},[31828],[34,424145,424147],{"className":424146},[31801],[34,424148,424150],{"className":424149,"style":31836},[31805],[34,424151],{}," 都与其余理想的和互不干扰（交集为 0）。",[108,424154,424155,421802,424157,424185],{},[2627,424156,233757],{},[34,424158,424160,424173],{"className":424159},[31642],[34,424161,424163],{"className":424162},[31646],[31648,424164,424165],{"xmlns":31650},[31652,424166,424167,424171],{},[31655,424168,424169],{},[31658,424170,115372],{},[31665,424172,115375],{"encoding":31667},[34,424174,424176],{"className":424175,"ariaHidden":758},[31672],[34,424177,424179,424182],{"className":424178},[31676],[34,424180],{"className":424181,"style":32527},[31680],[34,424183,115372],{"className":424184,"style":32062},[31685,31686]," 是同构。",[108,424187,424188,33183,424190],{},[2627,424189,33190],{},[105,424191,424192,424426,424988,425329,425807,426116,426249,426351],{},[108,424193,424194,424197,424198,341230,424298,424326,424327,424396,424397,424425],{},[2627,424195,424196],{},"满射性","：题目已知 ",[34,424199,424201,424225],{"className":424200},[31642],[34,424202,424204],{"className":424203},[31646],[31648,424205,424206],{"xmlns":31650},[31652,424207,424208,424222],{},[31655,424209,424210,424212,424214,424216],{},[31658,424211,34386],{},[31662,424213,202],{},[31662,424215,131138],{},[31743,424217,424218,424220],{},[31658,424219,34386],{},[31658,424221,31955],{},[31665,424223,424224],{"encoding":31667},"R = \\sum R_i",[34,424226,424228,424246],{"className":424227,"ariaHidden":758},[31672],[34,424229,424231,424234,424237,424240,424243],{"className":424230},[31676],[34,424232],{"className":424233,"style":33218},[31680],[34,424235,34386],{"className":424236,"style":34427},[31685,31686],[34,424238],{"className":424239,"style":31846},[31845],[34,424241,202],{"className":424242},[31850],[34,424244],{"className":424245,"style":31846},[31845],[34,424247,424249,424252,424255,424258],{"className":424248},[31676],[34,424250],{"className":424251,"style":31774},[31680],[34,424253,131138],{"className":424254,"style":156561},[32767,131249,156560],[34,424256],{"className":424257,"style":32123},[31845],[34,424259,424261,424264],{"className":424260},[31685],[34,424262,34386],{"className":424263,"style":34427},[31685,31686],[34,424265,424267],{"className":424266},[31792],[34,424268,424270,424290],{"className":424269},[31796,31797],[34,424271,424273,424287],{"className":424272},[31801],[34,424274,424276],{"className":424275,"style":32090},[31805],[34,424277,424278,424281],{"style":104694},[34,424279],{"className":424280,"style":31814},[31813],[34,424282,424284],{"className":424283},[31818,31819,31820,31821],[34,424285,31955],{"className":424286},[31685,31686,31821],[34,424288,31829],{"className":424289},[31828],[34,424291,424293],{"className":424292},[31801],[34,424294,424296],{"className":424295,"style":31836},[31805],[34,424297],{},[34,424299,424301,424314],{"className":424300},[31642],[34,424302,424304],{"className":424303},[31646],[31648,424305,424306],{"xmlns":31650},[31652,424307,424308,424312],{},[31655,424309,424310],{},[31658,424311,34386],{},[31665,424313,34386],{"encoding":31667},[34,424315,424317],{"className":424316,"ariaHidden":758},[31672],[34,424318,424320,424323],{"className":424319},[31676],[34,424321],{"className":424322,"style":33218},[31680],[34,424324,34386],{"className":424325,"style":34427},[31685,31686]," 中任何元素都能写成有限个 ",[34,424328,424330,424347],{"className":424329},[31642],[34,424331,424333],{"className":424332},[31646],[31648,424334,424335],{"xmlns":31650},[31652,424336,424337,424345],{},[31655,424338,424339],{},[31743,424340,424341,424343],{},[31658,424342,34386],{},[31658,424344,31955],{},[31665,424346,402774],{"encoding":31667},[34,424348,424350],{"className":424349,"ariaHidden":758},[31672],[34,424351,424353,424356],{"className":424352},[31676],[34,424354],{"className":424355,"style":70915},[31680],[34,424357,424359,424362],{"className":424358},[31685],[34,424360,34386],{"className":424361,"style":34427},[31685,31686],[34,424363,424365],{"className":424364},[31792],[34,424366,424368,424388],{"className":424367},[31796,31797],[34,424369,424371,424385],{"className":424370},[31801],[34,424372,424374],{"className":424373,"style":32090},[31805],[34,424375,424376,424379],{"style":104694},[34,424377],{"className":424378,"style":31814},[31813],[34,424380,424382],{"className":424381},[31818,31819,31820,31821],[34,424383,31955],{"className":424384},[31685,31686,31821],[34,424386,31829],{"className":424387},[31828],[34,424389,424391],{"className":424390},[31801],[34,424392,424394],{"className":424393,"style":31836},[31805],[34,424395],{}," 元素的和。所以 ",[34,424398,424400,424413],{"className":424399},[31642],[34,424401,424403],{"className":424402},[31646],[31648,424404,424405],{"xmlns":31650},[31652,424406,424407,424411],{},[31655,424408,424409],{},[31658,424410,115372],{},[31665,424412,115375],{"encoding":31667},[34,424414,424416],{"className":424415,"ariaHidden":758},[31672],[34,424417,424419,424422],{"className":424418},[31676],[34,424420],{"className":424421,"style":32527},[31680],[34,424423,115372],{"className":424424,"style":32062},[31685,31686]," 天然是满射。",[108,424427,424428,424431,424432,424503,424504,11008,424639,424678,424679,424917,424918,424987],{},[2627,424429,424430],{},"单射性","：我们需要证明 ",[34,424433,424435,424461],{"className":424434},[31642],[34,424436,424438],{"className":424437},[31646],[31648,424439,424440],{"xmlns":31650},[31652,424441,424442,424458],{},[31655,424443,424444,424446,424448,424450,424452,424454,424456],{},[31658,424445,406335],{},[31662,424447,32682],{},[31658,424449,115372],{},[31662,424451,202],{},[31662,424453,32685],{"stretchy":702},[31749,424455,459],{},[31662,424457,32715],{"stretchy":702},[31665,424459,424460],{"encoding":31667},"\\ker \\sigma = \\{0\\}",[34,424462,424464,424488],{"className":424463,"ariaHidden":758},[31672],[34,424465,424467,424470,424473,424476,424479,424482,424485],{"className":424466},[31676],[34,424468],{"className":424469,"style":31681},[31680],[34,424471,406335],{"className":424472},[32767],[34,424474],{"className":424475,"style":32123},[31845],[34,424477,115372],{"className":424478,"style":32062},[31685,31686],[34,424480],{"className":424481,"style":31846},[31845],[34,424483,202],{"className":424484},[31850],[34,424486],{"className":424487,"style":31846},[31845],[34,424489,424491,424494,424497,424500],{"className":424490},[31676],[34,424492],{"className":424493,"style":31774},[31680],[34,424495,32685],{"className":424496},[31782],[34,424498,459],{"className":424499},[31685],[34,424501,32715],{"className":424502},[31690],"。\n设 ",[34,424505,424507,424538],{"className":424506},[31642],[34,424508,424510],{"className":424509},[31646],[31648,424511,424512],{"xmlns":31650},[31652,424513,424514,424536],{},[31655,424515,424516,424518,424524],{},[31662,424517,456],{"stretchy":702},[31743,424519,424520,424522],{},[31658,424521,34395],{},[31658,424523,31955],{},[31743,424525,424526,424528],{},[31662,424527,2557],{"stretchy":702},[31655,424529,424530,424532,424534],{},[31658,424531,31955],{},[31662,424533,112296],{},[31658,424535,106063],{},[31665,424537,420031],{"encoding":31667},[34,424539,424541],{"className":424540,"ariaHidden":758},[31672],[34,424542,424544,424547,424550,424590],{"className":424543},[31676],[34,424545],{"className":424546,"style":31774},[31680],[34,424548,456],{"className":424549},[31782],[34,424551,424553,424556],{"className":424552},[31685],[34,424554,34395],{"className":424555,"style":34452},[31685,31686],[34,424557,424559],{"className":424558},[31792],[34,424560,424562,424582],{"className":424561},[31796,31797],[34,424563,424565,424579],{"className":424564},[31801],[34,424566,424568],{"className":424567,"style":32090},[31805],[34,424569,424570,424573],{"style":34467},[34,424571],{"className":424572,"style":31814},[31813],[34,424574,424576],{"className":424575},[31818,31819,31820,31821],[34,424577,31955],{"className":424578},[31685,31686,31821],[34,424580,31829],{"className":424581},[31828],[34,424583,424585],{"className":424584},[31801],[34,424586,424588],{"className":424587,"style":31836},[31805],[34,424589],{},[34,424591,424593,424596],{"className":424592},[31690],[34,424594,2557],{"className":424595},[31690],[34,424597,424599],{"className":424598},[31792],[34,424600,424602,424631],{"className":424601},[31796,31797],[34,424603,424605,424628],{"className":424604},[31801],[34,424606,424608],{"className":424607,"style":116558},[31805],[34,424609,424610,424613],{"style":31809},[34,424611],{"className":424612,"style":31814},[31813],[34,424614,424616],{"className":424615},[31818,31819,31820,31821],[34,424617,424619,424622,424625],{"className":424618},[31685,31821],[34,424620,31955],{"className":424621},[31685,31686,31821],[34,424623,112296],{"className":424624},[31850,31821],[34,424626,106063],{"className":424627,"style":67482},[31685,31686,31821],[34,424629,31829],{"className":424630},[31828],[34,424632,424634],{"className":424633},[31801],[34,424635,424637],{"className":424636,"style":418154},[31805],[34,424638],{},[34,424640,424642,424660],{"className":424641},[31642],[34,424643,424645],{"className":424644},[31646],[31648,424646,424647],{"xmlns":31650},[31652,424648,424649,424657],{},[31655,424650,424651,424653,424655],{},[31658,424652,406335],{},[31662,424654,32682],{},[31658,424656,115372],{},[31665,424658,424659],{"encoding":31667},"\\ker \\sigma",[34,424661,424663],{"className":424662,"ariaHidden":758},[31672],[34,424664,424666,424669,424672,424675],{"className":424665},[31676],[34,424667],{"className":424668,"style":31681},[31680],[34,424670,406335],{"className":424671},[32767],[34,424673],{"className":424674,"style":32123},[31845],[34,424676,115372],{"className":424677,"style":32062},[31685,31686]," 中的一个元素。这意味着：\n",[34,424680,424682,424734],{"className":424681},[31642],[34,424683,424685],{"className":424684},[31646],[31648,424686,424687],{"xmlns":31650},[31652,424688,424689,424731],{},[31655,424690,424691,424693,424695,424697,424703,424705,424707,424709,424721,424727,424729],{},[31658,424692,115372],{},[31662,424694,456],{"stretchy":702},[31662,424696,456],{"stretchy":702},[31743,424698,424699,424701],{},[31658,424700,34395],{},[31658,424702,31955],{},[31662,424704,2557],{"stretchy":702},[31662,424706,2557],{"stretchy":702},[31662,424708,202],{},[31743,424710,424711,424713],{},[31662,424712,131138],{},[31655,424714,424715,424717,424719],{},[31658,424716,31955],{},[31662,424718,112296],{},[31658,424720,106063],{},[31743,424722,424723,424725],{},[31658,424724,34395],{},[31658,424726,31955],{},[31662,424728,202],{},[31749,424730,459],{},[31665,424732,424733],{"encoding":31667},"\\sigma((r_i)) = \\sum_{i \\in I} r_i = 0",[34,424735,424737,424801,424908],{"className":424736,"ariaHidden":758},[31672],[34,424738,424740,424743,424746,424749,424789,424792,424795,424798],{"className":424739},[31676],[34,424741],{"className":424742,"style":31774},[31680],[34,424744,115372],{"className":424745,"style":32062},[31685,31686],[34,424747,1323],{"className":424748},[31782],[34,424750,424752,424755],{"className":424751},[31685],[34,424753,34395],{"className":424754,"style":34452},[31685,31686],[34,424756,424758],{"className":424757},[31792],[34,424759,424761,424781],{"className":424760},[31796,31797],[34,424762,424764,424778],{"className":424763},[31801],[34,424765,424767],{"className":424766,"style":32090},[31805],[34,424768,424769,424772],{"style":34467},[34,424770],{"className":424771,"style":31814},[31813],[34,424773,424775],{"className":424774},[31818,31819,31820,31821],[34,424776,31955],{"className":424777},[31685,31686,31821],[34,424779,31829],{"className":424780},[31828],[34,424782,424784],{"className":424783},[31801],[34,424785,424787],{"className":424786,"style":31836},[31805],[34,424788],{},[34,424790,44253],{"className":424791},[31690],[34,424793],{"className":424794,"style":31846},[31845],[34,424796,202],{"className":424797},[31850],[34,424799],{"className":424800,"style":31846},[31845],[34,424802,424804,424807,424856,424859,424899,424902,424905],{"className":424803},[31676],[34,424805],{"className":424806,"style":156553},[31680],[34,424808,424810,424813],{"className":424809},[32767],[34,424811,131138],{"className":424812,"style":156561},[32767,131249,156560],[34,424814,424816],{"className":424815},[31792],[34,424817,424819,424848],{"className":424818},[31796,31797],[34,424820,424822,424845],{"className":424821},[31801],[34,424823,424825],{"className":424824,"style":156574},[31805],[34,424826,424827,424830],{"style":156577},[34,424828],{"className":424829,"style":31814},[31813],[34,424831,424833],{"className":424832},[31818,31819,31820,31821],[34,424834,424836,424839,424842],{"className":424835},[31685,31821],[34,424837,31955],{"className":424838},[31685,31686,31821],[34,424840,112296],{"className":424841},[31850,31821],[34,424843,106063],{"className":424844,"style":67482},[31685,31686,31821],[34,424846,31829],{"className":424847},[31828],[34,424849,424851],{"className":424850},[31801],[34,424852,424854],{"className":424853,"style":156605},[31805],[34,424855],{},[34,424857],{"className":424858,"style":32123},[31845],[34,424860,424862,424865],{"className":424861},[31685],[34,424863,34395],{"className":424864,"style":34452},[31685,31686],[34,424866,424868],{"className":424867},[31792],[34,424869,424871,424891],{"className":424870},[31796,31797],[34,424872,424874,424888],{"className":424873},[31801],[34,424875,424877],{"className":424876,"style":32090},[31805],[34,424878,424879,424882],{"style":34467},[34,424880],{"className":424881,"style":31814},[31813],[34,424883,424885],{"className":424884},[31818,31819,31820,31821],[34,424886,31955],{"className":424887},[31685,31686,31821],[34,424889,31829],{"className":424890},[31828],[34,424892,424894],{"className":424893},[31801],[34,424895,424897],{"className":424896,"style":31836},[31805],[34,424898],{},[34,424900],{"className":424901,"style":31846},[31845],[34,424903,202],{"className":424904},[31850],[34,424906],{"className":424907,"style":31846},[31845],[34,424909,424911,424914],{"className":424910},[31676],[34,424912],{"className":424913,"style":32617},[31680],[34,424915,459],{"className":424916},[31685],"\n我们要证明所有的 ",[34,424919,424921,424938],{"className":424920},[31642],[34,424922,424924],{"className":424923},[31646],[31648,424925,424926],{"xmlns":31650},[31652,424927,424928,424936],{},[31655,424929,424930],{},[31743,424931,424932,424934],{},[31658,424933,34395],{},[31658,424935,31955],{},[31665,424937,34855],{"encoding":31667},[34,424939,424941],{"className":424940,"ariaHidden":758},[31672],[34,424942,424944,424947],{"className":424943},[31676],[34,424945],{"className":424946,"style":31860},[31680],[34,424948,424950,424953],{"className":424949},[31685],[34,424951,34395],{"className":424952,"style":34452},[31685,31686],[34,424954,424956],{"className":424955},[31792],[34,424957,424959,424979],{"className":424958},[31796,31797],[34,424960,424962,424976],{"className":424961},[31801],[34,424963,424965],{"className":424964,"style":32090},[31805],[34,424966,424967,424970],{"style":34467},[34,424968],{"className":424969,"style":31814},[31813],[34,424971,424973],{"className":424972},[31818,31819,31820,31821],[34,424974,31955],{"className":424975},[31685,31686,31821],[34,424977,31829],{"className":424978},[31828],[34,424980,424982],{"className":424981},[31801],[34,424983,424985],{"className":424984,"style":31836},[31805],[34,424986],{}," 都必须是 0。",[108,424989,424990,424991,425019,425020,425090,425091],{},"对于任意的下标 ",[34,424992,424994,425007],{"className":424993},[31642],[34,424995,424997],{"className":424996},[31646],[31648,424998,424999],{"xmlns":31650},[31652,425000,425001,425005],{},[31655,425002,425003],{},[31658,425004,33759],{},[31665,425006,33759],{"encoding":31667},[34,425008,425010],{"className":425009,"ariaHidden":758},[31672],[34,425011,425013,425016],{"className":425012},[31676],[34,425014],{"className":425015,"style":31681},[31680],[34,425017,33759],{"className":425018,"style":33774},[31685,31686],"，我们将上式中的 ",[34,425021,425023,425041],{"className":425022},[31642],[34,425024,425026],{"className":425025},[31646],[31648,425027,425028],{"xmlns":31650},[31652,425029,425030,425038],{},[31655,425031,425032],{},[31743,425033,425034,425036],{},[31658,425035,34395],{},[31658,425037,33759],{},[31665,425039,425040],{"encoding":31667},"r_k",[34,425042,425044],{"className":425043,"ariaHidden":758},[31672],[34,425045,425047,425050],{"className":425046},[31676],[34,425048],{"className":425049,"style":31860},[31680],[34,425051,425053,425056],{"className":425052},[31685],[34,425054,34395],{"className":425055,"style":34452},[31685,31686],[34,425057,425059],{"className":425058},[31792],[34,425060,425062,425082],{"className":425061},[31796,31797],[34,425063,425065,425079],{"className":425064},[31801],[34,425066,425068],{"className":425067,"style":33833},[31805],[34,425069,425070,425073],{"style":34467},[34,425071],{"className":425072,"style":31814},[31813],[34,425074,425076],{"className":425075},[31818,31819,31820,31821],[34,425077,33759],{"className":425078,"style":33774},[31685,31686,31821],[34,425080,31829],{"className":425081},[31828],[34,425083,425085],{"className":425084},[31801],[34,425086,425088],{"className":425087,"style":31836},[31805],[34,425089],{}," 单独留在一边，其他的移到另一边：\n",[34,425092,425094,425134],{"className":425093},[31642],[34,425095,425097],{"className":425096},[31646],[31648,425098,425099],{"xmlns":31650},[31652,425100,425101,425131],{},[31655,425102,425103,425109,425111,425113,425125],{},[31743,425104,425105,425107],{},[31658,425106,34395],{},[31658,425108,33759],{},[31662,425110,202],{},[31662,425112,32226],{},[31743,425114,425115,425117],{},[31662,425116,131138],{},[31655,425118,425119,425121,425123],{},[31658,425120,31968],{},[31662,425122,54707],{"mathvariant":38079},[31658,425124,33759],{},[31743,425126,425127,425129],{},[31658,425128,34395],{},[31658,425130,31968],{},[31665,425132,425133],{"encoding":31667},"r_k = - \\sum_{j \\neq k} r_j",[34,425135,425137,425192],{"className":425136,"ariaHidden":758},[31672],[34,425138,425140,425143,425183,425186,425189],{"className":425139},[31676],[34,425141],{"className":425142,"style":31860},[31680],[34,425144,425146,425149],{"className":425145},[31685],[34,425147,34395],{"className":425148,"style":34452},[31685,31686],[34,425150,425152],{"className":425151},[31792],[34,425153,425155,425175],{"className":425154},[31796,31797],[34,425156,425158,425172],{"className":425157},[31801],[34,425159,425161],{"className":425160,"style":33833},[31805],[34,425162,425163,425166],{"style":34467},[34,425164],{"className":425165,"style":31814},[31813],[34,425167,425169],{"className":425168},[31818,31819,31820,31821],[34,425170,33759],{"className":425171,"style":33774},[31685,31686,31821],[34,425173,31829],{"className":425174},[31828],[34,425176,425178],{"className":425177},[31801],[34,425179,425181],{"className":425180,"style":31836},[31805],[34,425182],{},[34,425184],{"className":425185,"style":31846},[31845],[34,425187,202],{"className":425188},[31850],[34,425190],{"className":425191,"style":31846},[31845],[34,425193,425195,425198,425201,425204,425286,425289],{"className":425194},[31676],[34,425196],{"className":425197,"style":156901},[31680],[34,425199,32226],{"className":425200},[31685],[34,425202],{"className":425203,"style":32123},[31845],[34,425205,425207,425210],{"className":425206},[32767],[34,425208,131138],{"className":425209,"style":156561},[32767,131249,156560],[34,425211,425213],{"className":425212},[31792],[34,425214,425216,425278],{"className":425215},[31796,31797],[34,425217,425219,425275],{"className":425218},[31801],[34,425220,425222],{"className":425221,"style":356484},[31805],[34,425223,425224,425227],{"style":156577},[34,425225],{"className":425226,"style":31814},[31813],[34,425228,425230],{"className":425229},[31818,31819,31820,31821],[34,425231,425233,425236,425272],{"className":425232},[31685,31821],[34,425234,31968],{"className":425235,"style":32153},[31685,31686,31821],[34,425237,425239,425266,425269],{"className":425238},[31850,31821],[34,425240,425242],{"className":425241},[31850,31821],[34,425243,425245],{"className":425244},[31685,54774,31821],[34,425246,425248],{"className":425247},[54778,31821],[34,425249,425251,425254,425263],{"className":425250},[54782,31821],[34,425252],{"className":425253,"style":43217},[31680],[34,425255,425257],{"className":425256},[54789],[34,425258,425260],{"className":425259},[31685,31821],[34,425261,54796],{"className":425262},[31850,31821],[34,425264],{"className":425265},[54800],[34,425267],{"className":425268},[31845,54804,31821],[34,425270,202],{"className":425271},[31850,31821],[34,425273,33759],{"className":425274,"style":33774},[31685,31686,31821],[34,425276,31829],{"className":425277},[31828],[34,425279,425281],{"className":425280},[31801],[34,425282,425284],{"className":425283,"style":156941},[31805],[34,425285],{},[34,425287],{"className":425288,"style":32123},[31845],[34,425290,425292,425295],{"className":425291},[31685],[34,425293,34395],{"className":425294,"style":34452},[31685,31686],[34,425296,425298],{"className":425297},[31792],[34,425299,425301,425321],{"className":425300},[31796,31797],[34,425302,425304,425318],{"className":425303},[31801],[34,425305,425307],{"className":425306,"style":32090},[31805],[34,425308,425309,425312],{"style":34467},[34,425310],{"className":425311,"style":31814},[31813],[34,425313,425315],{"className":425314},[31818,31819,31820,31821],[34,425316,31968],{"className":425317,"style":32153},[31685,31686,31821],[34,425319,31829],{"className":425320},[31828],[34,425322,425324],{"className":425323},[31801],[34,425325,425327],{"className":425326,"style":32163},[31805],[34,425328],{},[108,425330,425331,425332],{},"观察等式两边：\n",[105,425333,425334,425469],{},[108,425335,386318,425336,347],{},[34,425337,425339,425365],{"className":425338},[31642],[34,425340,425342],{"className":425341},[31646],[31648,425343,425344],{"xmlns":31650},[31652,425345,425346,425362],{},[31655,425347,425348,425354,425356],{},[31743,425349,425350,425352],{},[31658,425351,34395],{},[31658,425353,33759],{},[31662,425355,112296],{},[31743,425357,425358,425360],{},[31658,425359,34386],{},[31658,425361,33759],{},[31665,425363,425364],{"encoding":31667},"r_k \\in R_k",[34,425366,425368,425423],{"className":425367,"ariaHidden":758},[31672],[34,425369,425371,425374,425414,425417,425420],{"className":425370},[31676],[34,425372],{"className":425373,"style":225773},[31680],[34,425375,425377,425380],{"className":425376},[31685],[34,425378,34395],{"className":425379,"style":34452},[31685,31686],[34,425381,425383],{"className":425382},[31792],[34,425384,425386,425406],{"className":425385},[31796,31797],[34,425387,425389,425403],{"className":425388},[31801],[34,425390,425392],{"className":425391,"style":33833},[31805],[34,425393,425394,425397],{"style":34467},[34,425395],{"className":425396,"style":31814},[31813],[34,425398,425400],{"className":425399},[31818,31819,31820,31821],[34,425401,33759],{"className":425402,"style":33774},[31685,31686,31821],[34,425404,31829],{"className":425405},[31828],[34,425407,425409],{"className":425408},[31801],[34,425410,425412],{"className":425411,"style":31836},[31805],[34,425413],{},[34,425415],{"className":425416,"style":31846},[31845],[34,425418,112296],{"className":425419},[31850],[34,425421],{"className":425422,"style":31846},[31845],[34,425424,425426,425429],{"className":425425},[31676],[34,425427],{"className":425428,"style":70915},[31680],[34,425430,425432,425435],{"className":425431},[31685],[34,425433,34386],{"className":425434,"style":34427},[31685,31686],[34,425436,425438],{"className":425437},[31792],[34,425439,425441,425461],{"className":425440},[31796,31797],[34,425442,425444,425458],{"className":425443},[31801],[34,425445,425447],{"className":425446,"style":33833},[31805],[34,425448,425449,425452],{"style":104694},[34,425450],{"className":425451,"style":31814},[31813],[34,425453,425455],{"className":425454},[31818,31819,31820,31821],[34,425456,33759],{"className":425457,"style":33774},[31685,31686,31821],[34,425459,31829],{"className":425460},[31828],[34,425462,425464],{"className":425463},[31801],[34,425465,425467],{"className":425466,"style":31836},[31805],[34,425468],{},[108,425470,386924,425471,425806],{},[34,425472,425474,425526],{"className":425473},[31642],[34,425475,425477],{"className":425476},[31646],[31648,425478,425479],{"xmlns":31650},[31652,425480,425481,425523],{},[31655,425482,425483,425485,425497,425503,425505,425517],{},[31662,425484,32226],{},[31743,425486,425487,425489],{},[31662,425488,131138],{},[31655,425490,425491,425493,425495],{},[31658,425492,31968],{},[31662,425494,54707],{"mathvariant":38079},[31658,425496,33759],{},[31743,425498,425499,425501],{},[31658,425500,34395],{},[31658,425502,31968],{},[31662,425504,112296],{},[31743,425506,425507,425509],{},[31662,425508,131138],{},[31655,425510,425511,425513,425515],{},[31658,425512,31968],{},[31662,425514,54707],{"mathvariant":38079},[31658,425516,33759],{},[31743,425518,425519,425521],{},[31658,425520,34386],{},[31658,425522,31968],{},[31665,425524,425525],{"encoding":31667},"- \\sum_{j \\neq k} r_j \\in \\sum_{j \\neq k} R_j",[34,425527,425529,425675],{"className":425528,"ariaHidden":758},[31672],[34,425530,425532,425535,425538,425541,425623,425626,425666,425669,425672],{"className":425531},[31676],[34,425533],{"className":425534,"style":156901},[31680],[34,425536,32226],{"className":425537},[31685],[34,425539],{"className":425540,"style":32123},[31845],[34,425542,425544,425547],{"className":425543},[32767],[34,425545,131138],{"className":425546,"style":156561},[32767,131249,156560],[34,425548,425550],{"className":425549},[31792],[34,425551,425553,425615],{"className":425552},[31796,31797],[34,425554,425556,425612],{"className":425555},[31801],[34,425557,425559],{"className":425558,"style":356484},[31805],[34,425560,425561,425564],{"style":156577},[34,425562],{"className":425563,"style":31814},[31813],[34,425565,425567],{"className":425566},[31818,31819,31820,31821],[34,425568,425570,425573,425609],{"className":425569},[31685,31821],[34,425571,31968],{"className":425572,"style":32153},[31685,31686,31821],[34,425574,425576,425603,425606],{"className":425575},[31850,31821],[34,425577,425579],{"className":425578},[31850,31821],[34,425580,425582],{"className":425581},[31685,54774,31821],[34,425583,425585],{"className":425584},[54778,31821],[34,425586,425588,425591,425600],{"className":425587},[54782,31821],[34,425589],{"className":425590,"style":43217},[31680],[34,425592,425594],{"className":425593},[54789],[34,425595,425597],{"className":425596},[31685,31821],[34,425598,54796],{"className":425599},[31850,31821],[34,425601],{"className":425602},[54800],[34,425604],{"className":425605},[31845,54804,31821],[34,425607,202],{"className":425608},[31850,31821],[34,425610,33759],{"className":425611,"style":33774},[31685,31686,31821],[34,425613,31829],{"className":425614},[31828],[34,425616,425618],{"className":425617},[31801],[34,425619,425621],{"className":425620,"style":156941},[31805],[34,425622],{},[34,425624],{"className":425625,"style":32123},[31845],[34,425627,425629,425632],{"className":425628},[31685],[34,425630,34395],{"className":425631,"style":34452},[31685,31686],[34,425633,425635],{"className":425634},[31792],[34,425636,425638,425658],{"className":425637},[31796,31797],[34,425639,425641,425655],{"className":425640},[31801],[34,425642,425644],{"className":425643,"style":32090},[31805],[34,425645,425646,425649],{"style":34467},[34,425647],{"className":425648,"style":31814},[31813],[34,425650,425652],{"className":425651},[31818,31819,31820,31821],[34,425653,31968],{"className":425654,"style":32153},[31685,31686,31821],[34,425656,31829],{"className":425657},[31828],[34,425659,425661],{"className":425660},[31801],[34,425662,425664],{"className":425663,"style":32163},[31805],[34,425665],{},[34,425667],{"className":425668,"style":31846},[31845],[34,425670,112296],{"className":425671},[31850],[34,425673],{"className":425674,"style":31846},[31845],[34,425676,425678,425681,425763,425766],{"className":425677},[31676],[34,425679],{"className":425680,"style":156901},[31680],[34,425682,425684,425687],{"className":425683},[32767],[34,425685,131138],{"className":425686,"style":156561},[32767,131249,156560],[34,425688,425690],{"className":425689},[31792],[34,425691,425693,425755],{"className":425692},[31796,31797],[34,425694,425696,425752],{"className":425695},[31801],[34,425697,425699],{"className":425698,"style":356484},[31805],[34,425700,425701,425704],{"style":156577},[34,425702],{"className":425703,"style":31814},[31813],[34,425705,425707],{"className":425706},[31818,31819,31820,31821],[34,425708,425710,425713,425749],{"className":425709},[31685,31821],[34,425711,31968],{"className":425712,"style":32153},[31685,31686,31821],[34,425714,425716,425743,425746],{"className":425715},[31850,31821],[34,425717,425719],{"className":425718},[31850,31821],[34,425720,425722],{"className":425721},[31685,54774,31821],[34,425723,425725],{"className":425724},[54778,31821],[34,425726,425728,425731,425740],{"className":425727},[54782,31821],[34,425729],{"className":425730,"style":43217},[31680],[34,425732,425734],{"className":425733},[54789],[34,425735,425737],{"className":425736},[31685,31821],[34,425738,54796],{"className":425739},[31850,31821],[34,425741],{"className":425742},[54800],[34,425744],{"className":425745},[31845,54804,31821],[34,425747,202],{"className":425748},[31850,31821],[34,425750,33759],{"className":425751,"style":33774},[31685,31686,31821],[34,425753,31829],{"className":425754},[31828],[34,425756,425758],{"className":425757},[31801],[34,425759,425761],{"className":425760,"style":156941},[31805],[34,425762],{},[34,425764],{"className":425765,"style":32123},[31845],[34,425767,425769,425772],{"className":425768},[31685],[34,425770,34386],{"className":425771,"style":34427},[31685,31686],[34,425773,425775],{"className":425774},[31792],[34,425776,425778,425798],{"className":425777},[31796,31797],[34,425779,425781,425795],{"className":425780},[31801],[34,425782,425784],{"className":425783,"style":32090},[31805],[34,425785,425786,425789],{"style":104694},[34,425787],{"className":425788,"style":31814},[31813],[34,425790,425792],{"className":425791},[31818,31819,31820,31821],[34,425793,31968],{"className":425794,"style":32153},[31685,31686,31821],[34,425796,31829],{"className":425797},[31828],[34,425799,425801],{"className":425800},[31801],[34,425802,425804],{"className":425803,"style":32163},[31805],[34,425805],{},"（因为理想对加法封闭）。",[108,425808,323847,425809,425878,425879,425948,425949,347],{},[34,425810,425812,425829],{"className":425811},[31642],[34,425813,425815],{"className":425814},[31646],[31648,425816,425817],{"xmlns":31650},[31652,425818,425819,425827],{},[31655,425820,425821],{},[31743,425822,425823,425825],{},[31658,425824,34395],{},[31658,425826,33759],{},[31665,425828,425040],{"encoding":31667},[34,425830,425832],{"className":425831,"ariaHidden":758},[31672],[34,425833,425835,425838],{"className":425834},[31676],[34,425836],{"className":425837,"style":31860},[31680],[34,425839,425841,425844],{"className":425840},[31685],[34,425842,34395],{"className":425843,"style":34452},[31685,31686],[34,425845,425847],{"className":425846},[31792],[34,425848,425850,425870],{"className":425849},[31796,31797],[34,425851,425853,425867],{"className":425852},[31801],[34,425854,425856],{"className":425855,"style":33833},[31805],[34,425857,425858,425861],{"style":34467},[34,425859],{"className":425860,"style":31814},[31813],[34,425862,425864],{"className":425863},[31818,31819,31820,31821],[34,425865,33759],{"className":425866,"style":33774},[31685,31686,31821],[34,425868,31829],{"className":425869},[31828],[34,425871,425873],{"className":425872},[31801],[34,425874,425876],{"className":425875,"style":31836},[31805],[34,425877],{}," 既属于 ",[34,425880,425882,425899],{"className":425881},[31642],[34,425883,425885],{"className":425884},[31646],[31648,425886,425887],{"xmlns":31650},[31652,425888,425889,425897],{},[31655,425890,425891],{},[31743,425892,425893,425895],{},[31658,425894,34386],{},[31658,425896,33759],{},[31665,425898,423342],{"encoding":31667},[34,425900,425902],{"className":425901,"ariaHidden":758},[31672],[34,425903,425905,425908],{"className":425904},[31676],[34,425906],{"className":425907,"style":70915},[31680],[34,425909,425911,425914],{"className":425910},[31685],[34,425912,34386],{"className":425913,"style":34427},[31685,31686],[34,425915,425917],{"className":425916},[31792],[34,425918,425920,425940],{"className":425919},[31796,31797],[34,425921,425923,425937],{"className":425922},[31801],[34,425924,425926],{"className":425925,"style":33833},[31805],[34,425927,425928,425931],{"style":104694},[34,425929],{"className":425930,"style":31814},[31813],[34,425932,425934],{"className":425933},[31818,31819,31820,31821],[34,425935,33759],{"className":425936,"style":33774},[31685,31686,31821],[34,425938,31829],{"className":425939},[31828],[34,425941,425943],{"className":425942},[31801],[34,425944,425946],{"className":425945,"style":31836},[31805],[34,425947],{},"，又属于 ",[34,425950,425952,425982],{"className":425951},[31642],[34,425953,425955],{"className":425954},[31646],[31648,425956,425957],{"xmlns":31650},[31652,425958,425959,425979],{},[31655,425960,425961,425973],{},[31743,425962,425963,425965],{},[31662,425964,131138],{},[31655,425966,425967,425969,425971],{},[31658,425968,31968],{},[31662,425970,54707],{"mathvariant":38079},[31658,425972,33759],{},[31743,425974,425975,425977],{},[31658,425976,34386],{},[31658,425978,31968],{},[31665,425980,425981],{"encoding":31667},"\\sum_{j \\neq k} R_j",[34,425983,425985],{"className":425984,"ariaHidden":758},[31672],[34,425986,425988,425991,426073,426076],{"className":425987},[31676],[34,425989],{"className":425990,"style":156901},[31680],[34,425992,425994,425997],{"className":425993},[32767],[34,425995,131138],{"className":425996,"style":156561},[32767,131249,156560],[34,425998,426000],{"className":425999},[31792],[34,426001,426003,426065],{"className":426002},[31796,31797],[34,426004,426006,426062],{"className":426005},[31801],[34,426007,426009],{"className":426008,"style":356484},[31805],[34,426010,426011,426014],{"style":156577},[34,426012],{"className":426013,"style":31814},[31813],[34,426015,426017],{"className":426016},[31818,31819,31820,31821],[34,426018,426020,426023,426059],{"className":426019},[31685,31821],[34,426021,31968],{"className":426022,"style":32153},[31685,31686,31821],[34,426024,426026,426053,426056],{"className":426025},[31850,31821],[34,426027,426029],{"className":426028},[31850,31821],[34,426030,426032],{"className":426031},[31685,54774,31821],[34,426033,426035],{"className":426034},[54778,31821],[34,426036,426038,426041,426050],{"className":426037},[54782,31821],[34,426039],{"className":426040,"style":43217},[31680],[34,426042,426044],{"className":426043},[54789],[34,426045,426047],{"className":426046},[31685,31821],[34,426048,54796],{"className":426049},[31850,31821],[34,426051],{"className":426052},[54800],[34,426054],{"className":426055},[31845,54804,31821],[34,426057,202],{"className":426058},[31850,31821],[34,426060,33759],{"className":426061,"style":33774},[31685,31686,31821],[34,426063,31829],{"className":426064},[31828],[34,426066,426068],{"className":426067},[31801],[34,426069,426071],{"className":426070,"style":156941},[31805],[34,426072],{},[34,426074],{"className":426075,"style":32123},[31845],[34,426077,426079,426082],{"className":426078},[31685],[34,426080,34386],{"className":426081,"style":34427},[31685,31686],[34,426083,426085],{"className":426084},[31792],[34,426086,426088,426108],{"className":426087},[31796,31797],[34,426089,426091,426105],{"className":426090},[31801],[34,426092,426094],{"className":426093,"style":32090},[31805],[34,426095,426096,426099],{"style":104694},[34,426097],{"className":426098,"style":31814},[31813],[34,426100,426102],{"className":426101},[31818,31819,31820,31821],[34,426103,31968],{"className":426104,"style":32153},[31685,31686,31821],[34,426106,31829],{"className":426107},[31828],[34,426109,426111],{"className":426110},[31801],[34,426112,426114],{"className":426113,"style":32163},[31805],[34,426115],{},[108,426117,426118,426119,13977,426157,347],{},"根据假设 (iv)，这个交集只有 ",[34,426120,426122,426139],{"className":426121},[31642],[34,426123,426125],{"className":426124},[31646],[31648,426126,426127],{"xmlns":31650},[31652,426128,426129,426137],{},[31655,426130,426131,426133,426135],{},[31662,426132,32685],{"stretchy":702},[31749,426134,459],{},[31662,426136,32715],{"stretchy":702},[31665,426138,338218],{"encoding":31667},[34,426140,426142],{"className":426141,"ariaHidden":758},[31672],[34,426143,426145,426148,426151,426154],{"className":426144},[31676],[34,426146],{"className":426147,"style":31774},[31680],[34,426149,32685],{"className":426150},[31782],[34,426152,459],{"className":426153},[31685],[34,426155,32715],{"className":426156},[31690],[34,426158,426160,426182],{"className":426159},[31642],[34,426161,426163],{"className":426162},[31646],[31648,426164,426165],{"xmlns":31650},[31652,426166,426167,426179],{},[31655,426168,426169,426175,426177],{},[31743,426170,426171,426173],{},[31658,426172,34395],{},[31658,426174,33759],{},[31662,426176,202],{},[31749,426178,459],{},[31665,426180,426181],{"encoding":31667},"r_k = 0",[34,426183,426185,426240],{"className":426184,"ariaHidden":758},[31672],[34,426186,426188,426191,426231,426234,426237],{"className":426187},[31676],[34,426189],{"className":426190,"style":31860},[31680],[34,426192,426194,426197],{"className":426193},[31685],[34,426195,34395],{"className":426196,"style":34452},[31685,31686],[34,426198,426200],{"className":426199},[31792],[34,426201,426203,426223],{"className":426202},[31796,31797],[34,426204,426206,426220],{"className":426205},[31801],[34,426207,426209],{"className":426208,"style":33833},[31805],[34,426210,426211,426214],{"style":34467},[34,426212],{"className":426213,"style":31814},[31813],[34,426215,426217],{"className":426216},[31818,31819,31820,31821],[34,426218,33759],{"className":426219,"style":33774},[31685,31686,31821],[34,426221,31829],{"className":426222},[31828],[34,426224,426226],{"className":426225},[31801],[34,426227,426229],{"className":426228,"style":31836},[31805],[34,426230],{},[34,426232],{"className":426233,"style":31846},[31845],[34,426235,202],{"className":426236},[31850],[34,426238],{"className":426239,"style":31846},[31845],[34,426241,426243,426246],{"className":426242},[31676],[34,426244],{"className":426245,"style":32617},[31680],[34,426247,459],{"className":426248},[31685],[108,426250,426251,426252,426280,426281,426350],{},"既然 ",[34,426253,426255,426268],{"className":426254},[31642],[34,426256,426258],{"className":426257},[31646],[31648,426259,426260],{"xmlns":31650},[31652,426261,426262,426266],{},[31655,426263,426264],{},[31658,426265,33759],{},[31665,426267,33759],{"encoding":31667},[34,426269,426271],{"className":426270,"ariaHidden":758},[31672],[34,426272,426274,426277],{"className":426273},[31676],[34,426275],{"className":426276,"style":31681},[31680],[34,426278,33759],{"className":426279,"style":33774},[31685,31686]," 是任意的，那么所有的 ",[34,426282,426284,426301],{"className":426283},[31642],[34,426285,426287],{"className":426286},[31646],[31648,426288,426289],{"xmlns":31650},[31652,426290,426291,426299],{},[31655,426292,426293],{},[31743,426294,426295,426297],{},[31658,426296,34395],{},[31658,426298,31955],{},[31665,426300,34855],{"encoding":31667},[34,426302,426304],{"className":426303,"ariaHidden":758},[31672],[34,426305,426307,426310],{"className":426306},[31676],[34,426308],{"className":426309,"style":31860},[31680],[34,426311,426313,426316],{"className":426312},[31685],[34,426314,34395],{"className":426315,"style":34452},[31685,31686],[34,426317,426319],{"className":426318},[31792],[34,426320,426322,426342],{"className":426321},[31796,31797],[34,426323,426325,426339],{"className":426324},[31801],[34,426326,426328],{"className":426327,"style":32090},[31805],[34,426329,426330,426333],{"style":34467},[34,426331],{"className":426332,"style":31814},[31813],[34,426334,426336],{"className":426335},[31818,31819,31820,31821],[34,426337,31955],{"className":426338},[31685,31686,31821],[34,426340,31829],{"className":426341},[31828],[34,426343,426345],{"className":426344},[31801],[34,426346,426348],{"className":426347,"style":31836},[31805],[34,426349],{}," 都是 0。",[108,426352,397161,426353,61605,426423,426451],{},[34,426354,426356,426381],{"className":426355},[31642],[34,426357,426359],{"className":426358},[31646],[31648,426360,426361],{"xmlns":31650},[31652,426362,426363,426379],{},[31655,426364,426365,426367,426369,426371,426373,426375,426377],{},[31658,426366,406335],{},[31662,426368,32682],{},[31658,426370,115372],{},[31662,426372,202],{},[31662,426374,32685],{"stretchy":702},[31749,426376,459],{},[31662,426378,32715],{"stretchy":702},[31665,426380,424460],{"encoding":31667},[34,426382,426384,426408],{"className":426383,"ariaHidden":758},[31672],[34,426385,426387,426390,426393,426396,426399,426402,426405],{"className":426386},[31676],[34,426388],{"className":426389,"style":31681},[31680],[34,426391,406335],{"className":426392},[32767],[34,426394],{"className":426395,"style":32123},[31845],[34,426397,115372],{"className":426398,"style":32062},[31685,31686],[34,426400],{"className":426401,"style":31846},[31845],[34,426403,202],{"className":426404},[31850],[34,426406],{"className":426407,"style":31846},[31845],[34,426409,426411,426414,426417,426420],{"className":426410},[31676],[34,426412],{"className":426413,"style":31774},[31680],[34,426415,32685],{"className":426416},[31782],[34,426418,459],{"className":426419},[31685],[34,426421,32715],{"className":426422},[31690],[34,426424,426426,426439],{"className":426425},[31642],[34,426427,426429],{"className":426428},[31646],[31648,426430,426431],{"xmlns":31650},[31652,426432,426433,426437],{},[31655,426434,426435],{},[31658,426436,115372],{},[31665,426438,115375],{"encoding":31667},[34,426440,426442],{"className":426441,"ariaHidden":758},[31672],[34,426443,426445,426448],{"className":426444},[31676],[34,426446],{"className":426447,"style":32527},[31680],[34,426449,115372],{"className":426450,"style":32062},[31685,31686]," 是单射。",[108,426453,426454,426456,426457,426485],{},[2627,426455,114043],{},"：既是单射又是满射，",[34,426458,426460,426473],{"className":426459},[31642],[34,426461,426463],{"className":426462},[31646],[31648,426464,426465],{"xmlns":31650},[31652,426466,426467,426471],{},[31655,426468,426469],{},[31658,426470,115372],{},[31665,426472,115375],{"encoding":31667},[34,426474,426476],{"className":426475,"ariaHidden":758},[31672],[34,426477,426479,426482],{"className":426478},[31676],[34,426480],{"className":426481,"style":32527},[31680],[34,426483,115372],{"className":426484,"style":32062},[31685,31686]," 是同构，(i) 成立。",[15,426487,426488],{},[2627,426489,225949],{},[32188,426491],{},[253,426493,426495],{"id":426494},"题目-2验证中国剩余定理中的映射是同态",[2627,426496,426497],{},"题目 2：验证中国剩余定理中的映射是同态",[15,426499,426500,349701,426502,426530,426531,426670,426671,426879,426880,427187,427188,427216],{},[2627,426501,214361],{},[34,426503,426505,426518],{"className":426504},[31642],[34,426506,426508],{"className":426507},[31646],[31648,426509,426510],{"xmlns":31650},[31652,426511,426512,426516],{},[31655,426513,426514],{},[31658,426515,34386],{},[31665,426517,34386],{"encoding":31667},[34,426519,426521],{"className":426520,"ariaHidden":758},[31672],[34,426522,426524,426527],{"className":426523},[31676],[34,426525],{"className":426526,"style":33218},[31680],[34,426528,34386],{"className":426529,"style":34427},[31685,31686]," 是一个幺环，且 ",[34,426532,426534,426563],{"className":426533},[31642],[34,426535,426537],{"className":426536},[31646],[31648,426538,426539],{"xmlns":31650},[31652,426540,426541,426561],{},[31655,426542,426543,426549,426551,426553,426555],{},[31743,426544,426545,426547],{},[31658,426546,106063],{},[31749,426548,640],{},[31662,426550,31960],{"separator":758},[31662,426552,33934],{},[31662,426554,31960],{"separator":758},[31743,426556,426557,426559],{},[31658,426558,106063],{},[31658,426560,31660],{},[31665,426562,404891],{"encoding":31667},[34,426564,426566],{"className":426565,"ariaHidden":758},[31672],[34,426567,426569,426572,426612,426615,426618,426621,426624,426627,426630],{"className":426568},[31676],[34,426570],{"className":426571,"style":70739},[31680],[34,426573,426575,426578],{"className":426574},[31685],[34,426576,106063],{"className":426577,"style":67482},[31685,31686],[34,426579,426581],{"className":426580},[31792],[34,426582,426584,426604],{"className":426583},[31796,31797],[34,426585,426587,426601],{"className":426586},[31801],[34,426588,426590],{"className":426589,"style":31806},[31805],[34,426591,426592,426595],{"style":217742},[34,426593],{"className":426594,"style":31814},[31813],[34,426596,426598],{"className":426597},[31818,31819,31820,31821],[34,426599,640],{"className":426600},[31685,31821],[34,426602,31829],{"className":426603},[31828],[34,426605,426607],{"className":426606},[31801],[34,426608,426610],{"className":426609,"style":31836},[31805],[34,426611],{},[34,426613,31960],{"className":426614},[32119],[34,426616],{"className":426617,"style":32123},[31845],[34,426619,33934],{"className":426620},[34006],[34,426622],{"className":426623,"style":32123},[31845],[34,426625,31960],{"className":426626},[32119],[34,426628],{"className":426629,"style":32123},[31845],[34,426631,426633,426636],{"className":426632},[31685],[34,426634,106063],{"className":426635,"style":67482},[31685,31686],[34,426637,426639],{"className":426638},[31792],[34,426640,426642,426662],{"className":426641},[31796,31797],[34,426643,426645,426659],{"className":426644},[31801],[34,426646,426648],{"className":426647,"style":32362},[31805],[34,426649,426650,426653],{"style":217742},[34,426651],{"className":426652,"style":31814},[31813],[34,426654,426656],{"className":426655},[31818,31819,31820,31821],[34,426657,31660],{"className":426658},[31685,31686,31821],[34,426660,31829],{"className":426661},[31828],[34,426663,426665],{"className":426664},[31801],[34,426666,426668],{"className":426667,"style":31836},[31805],[34,426669],{}," 是理想。定义映射 ",[34,426672,426674,426722],{"className":426673},[31642],[34,426675,426677],{"className":426676},[31646],[31648,426678,426679],{"xmlns":31650},[31652,426680,426681,426719],{},[31655,426682,426683,426685,426687,426689,426691,426705,426707,426709,426711,426717],{},[31658,426684,329201],{},[31662,426686,5598],{},[31658,426688,34386],{},[31662,426690,33239],{},[38846,426692,426693,426695,426703],{},[31662,426694,404608],{},[31655,426696,426697,426699,426701],{},[31658,426698,33759],{},[31662,426700,202],{},[31749,426702,640],{},[31658,426704,31660],{},[31662,426706,456],{"stretchy":702},[31658,426708,34386],{},[31658,426710,38080],{"mathvariant":38079},[31743,426712,426713,426715],{},[31658,426714,106063],{},[31658,426716,33759],{},[31662,426718,2557],{"stretchy":702},[31665,426720,426721],{"encoding":31667},"\\phi: R \\to \\bigoplus_{k=1}^n (R\u002FI_k)",[34,426723,426725,426743,426761],{"className":426724,"ariaHidden":758},[31672],[34,426726,426728,426731,426734,426737,426740],{"className":426727},[31676],[34,426729],{"className":426730,"style":43217},[31680],[34,426732,329201],{"className":426733},[31685,31686],[34,426735],{"className":426736,"style":31846},[31845],[34,426738,5598],{"className":426739},[31850],[34,426741],{"className":426742,"style":31846},[31845],[34,426744,426746,426749,426752,426755,426758],{"className":426745},[31676],[34,426747],{"className":426748,"style":33218},[31680],[34,426750,34386],{"className":426751,"style":34427},[31685,31686],[34,426753],{"className":426754,"style":31846},[31845],[34,426756,33239],{"className":426757},[31850],[34,426759],{"className":426760,"style":31846},[31845],[34,426762,426764,426767,426827,426830,426833,426836,426876],{"className":426763},[31676],[34,426765],{"className":426766,"style":351974},[31680],[34,426768,426770,426773],{"className":426769},[32767],[34,426771,404608],{"className":426772,"style":156561},[32767,131249,156560],[34,426774,426776],{"className":426775},[31792],[34,426777,426779,426819],{"className":426778},[31796,31797],[34,426780,426782,426816],{"className":426781},[31801],[34,426783,426785,426805],{"className":426784,"style":351993},[31805],[34,426786,426787,426790],{"style":156577},[34,426788],{"className":426789,"style":31814},[31813],[34,426791,426793],{"className":426792},[31818,31819,31820,31821],[34,426794,426796,426799,426802],{"className":426795},[31685,31821],[34,426797,33759],{"className":426798,"style":33774},[31685,31686,31821],[34,426800,202],{"className":426801},[31850,31821],[34,426803,640],{"className":426804},[31685,31821],[34,426806,426807,426810],{"style":352016},[34,426808],{"className":426809,"style":31814},[31813],[34,426811,426813],{"className":426812},[31818,31819,31820,31821],[34,426814,31660],{"className":426815},[31685,31686,31821],[34,426817,31829],{"className":426818},[31828],[34,426820,426822],{"className":426821},[31801],[34,426823,426825],{"className":426824,"style":352035},[31805],[34,426826],{},[34,426828,456],{"className":426829},[31782],[34,426831,34386],{"className":426832,"style":34427},[31685,31686],[34,426834,38080],{"className":426835},[31685],[34,426837,426839,426842],{"className":426838},[31685],[34,426840,106063],{"className":426841,"style":67482},[31685,31686],[34,426843,426845],{"className":426844},[31792],[34,426846,426848,426868],{"className":426847},[31796,31797],[34,426849,426851,426865],{"className":426850},[31801],[34,426852,426854],{"className":426853,"style":33833},[31805],[34,426855,426856,426859],{"style":217742},[34,426857],{"className":426858,"style":31814},[31813],[34,426860,426862],{"className":426861},[31818,31819,31820,31821],[34,426863,33759],{"className":426864,"style":33774},[31685,31686,31821],[34,426866,31829],{"className":426867},[31828],[34,426869,426871],{"className":426870},[31801],[34,426872,426874],{"className":426873,"style":31836},[31805],[34,426875],{},[34,426877,2557],{"className":426878},[31690]," 如下：\n",[34,426881,426883,426947],{"className":426882},[31642],[34,426884,426886],{"className":426885},[31646],[31648,426887,426888],{"xmlns":31650},[31652,426889,426890,426944],{},[31655,426891,426892,426894,426896,426898,426900,426902,426904,426906,426908,426914,426916,426918,426920,426926,426928,426930,426932,426934,426936,426942],{},[31658,426893,329201],{},[31662,426895,456],{"stretchy":702},[31658,426897,32936],{},[31662,426899,2557],{"stretchy":702},[31662,426901,202],{},[31662,426903,456],{"stretchy":702},[31658,426905,32936],{},[31662,426907,223],{},[31743,426909,426910,426912],{},[31658,426911,106063],{},[31749,426913,640],{},[31662,426915,31960],{"separator":758},[31658,426917,32936],{},[31662,426919,223],{},[31743,426921,426922,426924],{},[31658,426923,106063],{},[31749,426925,768],{},[31662,426927,31960],{"separator":758},[31662,426929,33934],{},[31662,426931,31960],{"separator":758},[31658,426933,32936],{},[31662,426935,223],{},[31743,426937,426938,426940],{},[31658,426939,106063],{},[31658,426941,31660],{},[31662,426943,2557],{"stretchy":702},[31665,426945,426946],{"encoding":31667},"\\phi(x) = (x+I_1, x+I_2, \\dots, x+I_n)",[34,426948,426950,426977,426998,427062,427138],{"className":426949,"ariaHidden":758},[31672],[34,426951,426953,426956,426959,426962,426965,426968,426971,426974],{"className":426952},[31676],[34,426954],{"className":426955,"style":31774},[31680],[34,426957,329201],{"className":426958},[31685,31686],[34,426960,456],{"className":426961},[31782],[34,426963,32936],{"className":426964},[31685,31686],[34,426966,2557],{"className":426967},[31690],[34,426969],{"className":426970,"style":31846},[31845],[34,426972,202],{"className":426973},[31850],[34,426975],{"className":426976,"style":31846},[31845],[34,426978,426980,426983,426986,426989,426992,426995],{"className":426979},[31676],[34,426981],{"className":426982,"style":31774},[31680],[34,426984,456],{"className":426985},[31782],[34,426987,32936],{"className":426988},[31685,31686],[34,426990],{"className":426991,"style":32604},[31845],[34,426993,223],{"className":426994},[32024],[34,426996],{"className":426997,"style":32604},[31845],[34,426999,427001,427004,427044,427047,427050,427053,427056,427059],{"className":427000},[31676],[34,427002],{"className":427003,"style":70739},[31680],[34,427005,427007,427010],{"className":427006},[31685],[34,427008,106063],{"className":427009,"style":67482},[31685,31686],[34,427011,427013],{"className":427012},[31792],[34,427014,427016,427036],{"className":427015},[31796,31797],[34,427017,427019,427033],{"className":427018},[31801],[34,427020,427022],{"className":427021,"style":31806},[31805],[34,427023,427024,427027],{"style":217742},[34,427025],{"className":427026,"style":31814},[31813],[34,427028,427030],{"className":427029},[31818,31819,31820,31821],[34,427031,640],{"className":427032},[31685,31821],[34,427034,31829],{"className":427035},[31828],[34,427037,427039],{"className":427038},[31801],[34,427040,427042],{"className":427041,"style":31836},[31805],[34,427043],{},[34,427045,31960],{"className":427046},[32119],[34,427048],{"className":427049,"style":32123},[31845],[34,427051,32936],{"className":427052},[31685,31686],[34,427054],{"className":427055,"style":32604},[31845],[34,427057,223],{"className":427058},[32024],[34,427060],{"className":427061,"style":32604},[31845],[34,427063,427065,427068,427108,427111,427114,427117,427120,427123,427126,427129,427132,427135],{"className":427064},[31676],[34,427066],{"className":427067,"style":70739},[31680],[34,427069,427071,427074],{"className":427070},[31685],[34,427072,106063],{"className":427073,"style":67482},[31685,31686],[34,427075,427077],{"className":427076},[31792],[34,427078,427080,427100],{"className":427079},[31796,31797],[34,427081,427083,427097],{"className":427082},[31801],[34,427084,427086],{"className":427085,"style":31806},[31805],[34,427087,427088,427091],{"style":217742},[34,427089],{"className":427090,"style":31814},[31813],[34,427092,427094],{"className":427093},[31818,31819,31820,31821],[34,427095,768],{"className":427096},[31685,31821],[34,427098,31829],{"className":427099},[31828],[34,427101,427103],{"className":427102},[31801],[34,427104,427106],{"className":427105,"style":31836},[31805],[34,427107],{},[34,427109,31960],{"className":427110},[32119],[34,427112],{"className":427113,"style":32123},[31845],[34,427115,33934],{"className":427116},[34006],[34,427118],{"className":427119,"style":32123},[31845],[34,427121,31960],{"className":427122},[32119],[34,427124],{"className":427125,"style":32123},[31845],[34,427127,32936],{"className":427128},[31685,31686],[34,427130],{"className":427131,"style":32604},[31845],[34,427133,223],{"className":427134},[32024],[34,427136],{"className":427137,"style":32604},[31845],[34,427139,427141,427144,427184],{"className":427140},[31676],[34,427142],{"className":427143,"style":31774},[31680],[34,427145,427147,427150],{"className":427146},[31685],[34,427148,106063],{"className":427149,"style":67482},[31685,31686],[34,427151,427153],{"className":427152},[31792],[34,427154,427156,427176],{"className":427155},[31796,31797],[34,427157,427159,427173],{"className":427158},[31801],[34,427160,427162],{"className":427161,"style":32362},[31805],[34,427163,427164,427167],{"style":217742},[34,427165],{"className":427166,"style":31814},[31813],[34,427168,427170],{"className":427169},[31818,31819,31820,31821],[34,427171,31660],{"className":427172},[31685,31686,31821],[34,427174,31829],{"className":427175},[31828],[34,427177,427179],{"className":427178},[31801],[34,427180,427182],{"className":427181,"style":31836},[31805],[34,427183],{},[34,427185,2557],{"className":427186},[31690],"\n请证明 ",[34,427189,427191,427204],{"className":427190},[31642],[34,427192,427194],{"className":427193},[31646],[31648,427195,427196],{"xmlns":31650},[31652,427197,427198,427202],{},[31655,427199,427200],{},[31658,427201,329201],{},[31665,427203,364484],{"encoding":31667},[34,427205,427207],{"className":427206,"ariaHidden":758},[31672],[34,427208,427210,427213],{"className":427209},[31676],[34,427211],{"className":427212,"style":43217},[31680],[34,427214,329201],{"className":427215},[31685,31686]," 是一个同态 (Homomorphism)。",[15,427218,427219],{},[2627,427220,419639],{},[15,427222,427223,427224,427252,427253,119123,427256,93231,427259,427287,427288,347],{},"要证明一个映射 ",[34,427225,427227,427240],{"className":427226},[31642],[34,427228,427230],{"className":427229},[31646],[31648,427231,427232],{"xmlns":31650},[31652,427233,427234,427238],{},[31655,427235,427236],{},[31658,427237,329201],{},[31665,427239,364484],{"encoding":31667},[34,427241,427243],{"className":427242,"ariaHidden":758},[31672],[34,427244,427246,427249],{"className":427245},[31676],[34,427247],{"className":427248,"style":43217},[31680],[34,427250,329201],{"className":427251},[31685,31686]," 是环同态，我们需要验证它保持环的两个基本运算：",[2627,427254,427255],{},"加法",[2627,427257,427258],{},"乘法",[34,427260,427262,427275],{"className":427261},[31642],[34,427263,427265],{"className":427264},[31646],[31648,427266,427267],{"xmlns":31650},[31652,427268,427269,427273],{},[31655,427270,427271],{},[31658,427272,34386],{},[31665,427274,34386],{"encoding":31667},[34,427276,427278],{"className":427277,"ariaHidden":758},[31672],[34,427279,427281,427284],{"className":427280},[31676],[34,427282],{"className":427283,"style":33218},[31680],[34,427285,34386],{"className":427286,"style":34427},[31685,31686]," 是幺环（有单位元 1），通常还需要验证它保持",[2627,427289,378492],{},[15,427291,133910,427292,427356],{},[34,427293,427295,427317],{"className":427294},[31642],[34,427296,427298],{"className":427297},[31646],[31648,427299,427300],{"xmlns":31650},[31652,427301,427302,427314],{},[31655,427303,427304,427306,427308,427310,427312],{},[31658,427305,32936],{},[31662,427307,31960],{"separator":758},[31658,427309,67764],{},[31662,427311,112296],{},[31658,427313,34386],{},[31665,427315,427316],{"encoding":31667},"x, y \\in R",[34,427318,427320,427347],{"className":427319,"ariaHidden":758},[31672],[34,427321,427323,427326,427329,427332,427335,427338,427341,427344],{"className":427322},[31676],[34,427324],{"className":427325,"style":86652},[31680],[34,427327,32936],{"className":427328},[31685,31686],[34,427330,31960],{"className":427331},[32119],[34,427333],{"className":427334,"style":32123},[31845],[34,427336,67764],{"className":427337,"style":32062},[31685,31686],[34,427339],{"className":427340,"style":31846},[31845],[34,427342,112296],{"className":427343},[31850],[34,427345],{"className":427346,"style":31846},[31845],[34,427348,427350,427353],{"className":427349},[31676],[34,427351],{"className":427352,"style":33218},[31680],[34,427354,34386],{"className":427355,"style":34427},[31685,31686]," 是环中的任意两个元素。",[32490,427358,427360],{"id":427359},"_1-保持加法",[2627,427361,427362],{},"1. 保持加法",[15,427364,427365,427366,347],{},"我们需要证明 ",[34,427367,427369,427413],{"className":427368},[31642],[34,427370,427372],{"className":427371},[31646],[31648,427373,427374],{"xmlns":31650},[31652,427375,427376,427410],{},[31655,427377,427378,427380,427382,427384,427386,427388,427390,427392,427394,427396,427398,427400,427402,427404,427406,427408],{},[31658,427379,329201],{},[31662,427381,456],{"stretchy":702},[31658,427383,32936],{},[31662,427385,223],{},[31658,427387,67764],{},[31662,427389,2557],{"stretchy":702},[31662,427391,202],{},[31658,427393,329201],{},[31662,427395,456],{"stretchy":702},[31658,427397,32936],{},[31662,427399,2557],{"stretchy":702},[31662,427401,223],{},[31658,427403,329201],{},[31662,427405,456],{"stretchy":702},[31658,427407,67764],{},[31662,427409,2557],{"stretchy":702},[31665,427411,427412],{"encoding":31667},"\\phi(x+y) = \\phi(x) + \\phi(y)",[34,427414,427416,427440,427461,427488],{"className":427415,"ariaHidden":758},[31672],[34,427417,427419,427422,427425,427428,427431,427434,427437],{"className":427418},[31676],[34,427420],{"className":427421,"style":31774},[31680],[34,427423,329201],{"className":427424},[31685,31686],[34,427426,456],{"className":427427},[31782],[34,427429,32936],{"className":427430},[31685,31686],[34,427432],{"className":427433,"style":32604},[31845],[34,427435,223],{"className":427436},[32024],[34,427438],{"className":427439,"style":32604},[31845],[34,427441,427443,427446,427449,427452,427455,427458],{"className":427442},[31676],[34,427444],{"className":427445,"style":31774},[31680],[34,427447,67764],{"className":427448,"style":32062},[31685,31686],[34,427450,2557],{"className":427451},[31690],[34,427453],{"className":427454,"style":31846},[31845],[34,427456,202],{"className":427457},[31850],[34,427459],{"className":427460,"style":31846},[31845],[34,427462,427464,427467,427470,427473,427476,427479,427482,427485],{"className":427463},[31676],[34,427465],{"className":427466,"style":31774},[31680],[34,427468,329201],{"className":427469},[31685,31686],[34,427471,456],{"className":427472},[31782],[34,427474,32936],{"className":427475},[31685,31686],[34,427477,2557],{"className":427478},[31690],[34,427480],{"className":427481,"style":32604},[31845],[34,427483,223],{"className":427484},[32024],[34,427486],{"className":427487,"style":32604},[31845],[34,427489,427491,427494,427497,427500,427503],{"className":427490},[31676],[34,427492],{"className":427493,"style":31774},[31680],[34,427495,329201],{"className":427496},[31685,31686],[34,427498,456],{"className":427499},[31782],[34,427501,67764],{"className":427502,"style":32062},[31685,31686],[34,427504,2557],{"className":427505},[31690],[105,427507,427508,428128,429273],{},[108,427509,427510,33183,427513,427935,427936,428127],{},[2627,427511,427512],{},"计算左边",[34,427514,427516,427608],{"className":427515},[31642],[34,427517,427519],{"className":427518},[31646],[31648,427520,427521],{"xmlns":31650},[31652,427522,427523,427605],{},[31655,427524,427525,427527,427529,427531,427533,427535,427537,427539,427541,427543,427545,427547,427549,427551,427553,427559,427561,427563,427565,427567,427569,427571,427573,427579,427581,427583,427585,427587,427589,427591,427593,427595,427597,427603],{},[31658,427526,329201],{},[31662,427528,456],{"stretchy":702},[31658,427530,32936],{},[31662,427532,223],{},[31658,427534,67764],{},[31662,427536,2557],{"stretchy":702},[31662,427538,202],{},[31662,427540,456],{"stretchy":702},[31662,427542,456],{"stretchy":702},[31658,427544,32936],{},[31662,427546,223],{},[31658,427548,67764],{},[31662,427550,2557],{"stretchy":702},[31662,427552,223],{},[31743,427554,427555,427557],{},[31658,427556,106063],{},[31749,427558,640],{},[31662,427560,31960],{"separator":758},[31662,427562,456],{"stretchy":702},[31658,427564,32936],{},[31662,427566,223],{},[31658,427568,67764],{},[31662,427570,2557],{"stretchy":702},[31662,427572,223],{},[31743,427574,427575,427577],{},[31658,427576,106063],{},[31749,427578,768],{},[31662,427580,31960],{"separator":758},[31662,427582,33934],{},[31662,427584,31960],{"separator":758},[31662,427586,456],{"stretchy":702},[31658,427588,32936],{},[31662,427590,223],{},[31658,427592,67764],{},[31662,427594,2557],{"stretchy":702},[31662,427596,223],{},[31743,427598,427599,427601],{},[31658,427600,106063],{},[31658,427602,31660],{},[31662,427604,2557],{"stretchy":702},[31665,427606,427607],{"encoding":31667},"\\phi(x+y) = ((x+y)+I_1, (x+y)+I_2, \\dots, (x+y)+I_n)",[34,427609,427611,427635,427656,427677,427698,427765,427786,427865,427886],{"className":427610,"ariaHidden":758},[31672],[34,427612,427614,427617,427620,427623,427626,427629,427632],{"className":427613},[31676],[34,427615],{"className":427616,"style":31774},[31680],[34,427618,329201],{"className":427619},[31685,31686],[34,427621,456],{"className":427622},[31782],[34,427624,32936],{"className":427625},[31685,31686],[34,427627],{"className":427628,"style":32604},[31845],[34,427630,223],{"className":427631},[32024],[34,427633],{"className":427634,"style":32604},[31845],[34,427636,427638,427641,427644,427647,427650,427653],{"className":427637},[31676],[34,427639],{"className":427640,"style":31774},[31680],[34,427642,67764],{"className":427643,"style":32062},[31685,31686],[34,427645,2557],{"className":427646},[31690],[34,427648],{"className":427649,"style":31846},[31845],[34,427651,202],{"className":427652},[31850],[34,427654],{"className":427655,"style":31846},[31845],[34,427657,427659,427662,427665,427668,427671,427674],{"className":427658},[31676],[34,427660],{"className":427661,"style":31774},[31680],[34,427663,1323],{"className":427664},[31782],[34,427666,32936],{"className":427667},[31685,31686],[34,427669],{"className":427670,"style":32604},[31845],[34,427672,223],{"className":427673},[32024],[34,427675],{"className":427676,"style":32604},[31845],[34,427678,427680,427683,427686,427689,427692,427695],{"className":427679},[31676],[34,427681],{"className":427682,"style":31774},[31680],[34,427684,67764],{"className":427685,"style":32062},[31685,31686],[34,427687,2557],{"className":427688},[31690],[34,427690],{"className":427691,"style":32604},[31845],[34,427693,223],{"className":427694},[32024],[34,427696],{"className":427697,"style":32604},[31845],[34,427699,427701,427704,427744,427747,427750,427753,427756,427759,427762],{"className":427700},[31676],[34,427702],{"className":427703,"style":31774},[31680],[34,427705,427707,427710],{"className":427706},[31685],[34,427708,106063],{"className":427709,"style":67482},[31685,31686],[34,427711,427713],{"className":427712},[31792],[34,427714,427716,427736],{"className":427715},[31796,31797],[34,427717,427719,427733],{"className":427718},[31801],[34,427720,427722],{"className":427721,"style":31806},[31805],[34,427723,427724,427727],{"style":217742},[34,427725],{"className":427726,"style":31814},[31813],[34,427728,427730],{"className":427729},[31818,31819,31820,31821],[34,427731,640],{"className":427732},[31685,31821],[34,427734,31829],{"className":427735},[31828],[34,427737,427739],{"className":427738},[31801],[34,427740,427742],{"className":427741,"style":31836},[31805],[34,427743],{},[34,427745,31960],{"className":427746},[32119],[34,427748],{"className":427749,"style":32123},[31845],[34,427751,456],{"className":427752},[31782],[34,427754,32936],{"className":427755},[31685,31686],[34,427757],{"className":427758,"style":32604},[31845],[34,427760,223],{"className":427761},[32024],[34,427763],{"className":427764,"style":32604},[31845],[34,427766,427768,427771,427774,427777,427780,427783],{"className":427767},[31676],[34,427769],{"className":427770,"style":31774},[31680],[34,427772,67764],{"className":427773,"style":32062},[31685,31686],[34,427775,2557],{"className":427776},[31690],[34,427778],{"className":427779,"style":32604},[31845],[34,427781,223],{"className":427782},[32024],[34,427784],{"className":427785,"style":32604},[31845],[34,427787,427789,427792,427832,427835,427838,427841,427844,427847,427850,427853,427856,427859,427862],{"className":427788},[31676],[34,427790],{"className":427791,"style":31774},[31680],[34,427793,427795,427798],{"className":427794},[31685],[34,427796,106063],{"className":427797,"style":67482},[31685,31686],[34,427799,427801],{"className":427800},[31792],[34,427802,427804,427824],{"className":427803},[31796,31797],[34,427805,427807,427821],{"className":427806},[31801],[34,427808,427810],{"className":427809,"style":31806},[31805],[34,427811,427812,427815],{"style":217742},[34,427813],{"className":427814,"style":31814},[31813],[34,427816,427818],{"className":427817},[31818,31819,31820,31821],[34,427819,768],{"className":427820},[31685,31821],[34,427822,31829],{"className":427823},[31828],[34,427825,427827],{"className":427826},[31801],[34,427828,427830],{"className":427829,"style":31836},[31805],[34,427831],{},[34,427833,31960],{"className":427834},[32119],[34,427836],{"className":427837,"style":32123},[31845],[34,427839,33934],{"className":427840},[34006],[34,427842],{"className":427843,"style":32123},[31845],[34,427845,31960],{"className":427846},[32119],[34,427848],{"className":427849,"style":32123},[31845],[34,427851,456],{"className":427852},[31782],[34,427854,32936],{"className":427855},[31685,31686],[34,427857],{"className":427858,"style":32604},[31845],[34,427860,223],{"className":427861},[32024],[34,427863],{"className":427864,"style":32604},[31845],[34,427866,427868,427871,427874,427877,427880,427883],{"className":427867},[31676],[34,427869],{"className":427870,"style":31774},[31680],[34,427872,67764],{"className":427873,"style":32062},[31685,31686],[34,427875,2557],{"className":427876},[31690],[34,427878],{"className":427879,"style":32604},[31845],[34,427881,223],{"className":427882},[32024],[34,427884],{"className":427885,"style":32604},[31845],[34,427887,427889,427892,427932],{"className":427888},[31676],[34,427890],{"className":427891,"style":31774},[31680],[34,427893,427895,427898],{"className":427894},[31685],[34,427896,106063],{"className":427897,"style":67482},[31685,31686],[34,427899,427901],{"className":427900},[31792],[34,427902,427904,427924],{"className":427903},[31796,31797],[34,427905,427907,427921],{"className":427906},[31801],[34,427908,427910],{"className":427909,"style":32362},[31805],[34,427911,427912,427915],{"style":217742},[34,427913],{"className":427914,"style":31814},[31813],[34,427916,427918],{"className":427917},[31818,31819,31820,31821],[34,427919,31660],{"className":427920},[31685,31686,31821],[34,427922,31829],{"className":427923},[31828],[34,427925,427927],{"className":427926},[31801],[34,427928,427930],{"className":427929,"style":31836},[31805],[34,427931],{},[34,427933,2557],{"className":427934},[31690],"\n根据商环加法的定义 ",[34,427937,427939,427989],{"className":427938},[31642],[34,427940,427942],{"className":427941},[31646],[31648,427943,427944],{"xmlns":31650},[31652,427945,427946,427986],{},[31655,427947,427948,427950,427952,427954,427956,427958,427960,427962,427964,427966,427968,427970,427972,427974,427976,427978,427980,427982,427984],{},[31662,427949,456],{"stretchy":702},[31658,427951,111],{},[31662,427953,223],{},[31658,427955,106063],{},[31662,427957,2557],{"stretchy":702},[31662,427959,223],{},[31662,427961,456],{"stretchy":702},[31658,427963,1697],{},[31662,427965,223],{},[31658,427967,106063],{},[31662,427969,2557],{"stretchy":702},[31662,427971,202],{},[31662,427973,456],{"stretchy":702},[31658,427975,111],{},[31662,427977,223],{},[31658,427979,1697],{},[31662,427981,2557],{"stretchy":702},[31662,427983,223],{},[31658,427985,106063],{},[31665,427987,427988],{"encoding":31667},"(a+I) + (b+I) = (a+b)+I",[34,427990,427992,428013,428034,428055,428076,428097,428118],{"className":427991,"ariaHidden":758},[31672],[34,427993,427995,427998,428001,428004,428007,428010],{"className":427994},[31676],[34,427996],{"className":427997,"style":31774},[31680],[34,427999,456],{"className":428000},[31782],[34,428002,111],{"className":428003},[31685,31686],[34,428005],{"className":428006,"style":32604},[31845],[34,428008,223],{"className":428009},[32024],[34,428011],{"className":428012,"style":32604},[31845],[34,428014,428016,428019,428022,428025,428028,428031],{"className":428015},[31676],[34,428017],{"className":428018,"style":31774},[31680],[34,428020,106063],{"className":428021,"style":67482},[31685,31686],[34,428023,2557],{"className":428024},[31690],[34,428026],{"className":428027,"style":32604},[31845],[34,428029,223],{"className":428030},[32024],[34,428032],{"className":428033,"style":32604},[31845],[34,428035,428037,428040,428043,428046,428049,428052],{"className":428036},[31676],[34,428038],{"className":428039,"style":31774},[31680],[34,428041,456],{"className":428042},[31782],[34,428044,1697],{"className":428045},[31685,31686],[34,428047],{"className":428048,"style":32604},[31845],[34,428050,223],{"className":428051},[32024],[34,428053],{"className":428054,"style":32604},[31845],[34,428056,428058,428061,428064,428067,428070,428073],{"className":428057},[31676],[34,428059],{"className":428060,"style":31774},[31680],[34,428062,106063],{"className":428063,"style":67482},[31685,31686],[34,428065,2557],{"className":428066},[31690],[34,428068],{"className":428069,"style":31846},[31845],[34,428071,202],{"className":428072},[31850],[34,428074],{"className":428075,"style":31846},[31845],[34,428077,428079,428082,428085,428088,428091,428094],{"className":428078},[31676],[34,428080],{"className":428081,"style":31774},[31680],[34,428083,456],{"className":428084},[31782],[34,428086,111],{"className":428087},[31685,31686],[34,428089],{"className":428090,"style":32604},[31845],[34,428092,223],{"className":428093},[32024],[34,428095],{"className":428096,"style":32604},[31845],[34,428098,428100,428103,428106,428109,428112,428115],{"className":428099},[31676],[34,428101],{"className":428102,"style":31774},[31680],[34,428104,1697],{"className":428105},[31685,31686],[34,428107,2557],{"className":428108},[31690],[34,428110],{"className":428111,"style":32604},[31845],[34,428113,223],{"className":428114},[32024],[34,428116],{"className":428117,"style":32604},[31845],[34,428119,428121,428124],{"className":428120},[31676],[34,428122],{"className":428123,"style":33218},[31680],[34,428125,106063],{"className":428126,"style":67482},[31685,31686],"，我们可以把每一项拆开。",[108,428129,428130,33183,428133],{},[2627,428131,428132],{},"计算右边",[34,428134,428136],{"className":428135},[126385],[34,428137,428139,428401],{"className":428138},[31642],[34,428140,428142],{"className":428141},[31646],[31648,428143,428144],{"xmlns":31650,"display":126394},[31652,428145,428146,428398],{},[71078,428147,428148,428246,428334],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,428149,428150,428174],{},[71087,428151,428152],{},[71090,428153,428154],{"scriptlevel":459,"displaystyle":758},[31655,428155,428156,428158,428160,428162,428164,428166,428168,428170,428172],{},[31658,428157,329201],{},[31662,428159,456],{"stretchy":702},[31658,428161,32936],{},[31662,428163,2557],{"stretchy":702},[31662,428165,223],{},[31658,428167,329201],{},[31662,428169,456],{"stretchy":702},[31658,428171,67764],{},[31662,428173,2557],{"stretchy":702},[71087,428175,428176],{},[71090,428177,428178],{"scriptlevel":459,"displaystyle":758},[31655,428179,428180,428182,428184,428186,428188,428190,428196,428198,428200,428202,428204,428206,428212,428214,428216,428218,428220,428222,428228,428230,428232,428234,428236,428238,428244],{},[31655,428181],{},[31662,428183,202],{},[31662,428185,456],{"stretchy":702},[31658,428187,32936],{},[31662,428189,223],{},[31743,428191,428192,428194],{},[31658,428193,106063],{},[31749,428195,640],{},[31662,428197,31960],{"separator":758},[31662,428199,33934],{},[31662,428201,31960],{"separator":758},[31658,428203,32936],{},[31662,428205,223],{},[31743,428207,428208,428210],{},[31658,428209,106063],{},[31658,428211,31660],{},[31662,428213,2557],{"stretchy":702},[31662,428215,223],{},[31662,428217,456],{"stretchy":702},[31658,428219,67764],{},[31662,428221,223],{},[31743,428223,428224,428226],{},[31658,428225,106063],{},[31749,428227,640],{},[31662,428229,31960],{"separator":758},[31662,428231,33934],{},[31662,428233,31960],{"separator":758},[31658,428235,67764],{},[31662,428237,223],{},[31743,428239,428240,428242],{},[31658,428241,106063],{},[31658,428243,31660],{},[31662,428245,2557],{"stretchy":702},[71084,428247,428248,428254],{},[71087,428249,428250],{},[71090,428251,428252],{"scriptlevel":459,"displaystyle":758},[31655,428253],{},[71087,428255,428256],{},[71090,428257,428258],{"scriptlevel":459,"displaystyle":758},[31655,428259,428260,428262,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+ \\phi(y) &= (x+I_1, \\dots, x+I_n) + (y+I_1, \\dots, y+I_n) \\\\\n&= ((x+I_1) + (y+I_1), \\dots, (x+I_n) + (y+I_n)) \\\\\n&= ((x+y)+I_1, \\dots, (x+y)+I_n)\n\\end{aligned}",[34,428402,428404],{"className":428403,"ariaHidden":758},[31672],[34,428405,428407,428410],{"className":428406},[31676],[34,428408],{"className":428409,"style":368993},[31680],[34,428411,428413],{"className":428412},[31685],[34,428414,428416,428496],{"className":428415},[71078],[34,428417,428419],{"className":428418},[369003],[34,428420,428422,428488],{"className":428421},[31796,31797],[34,428423,428425,428485],{"className":428424},[31801],[34,428426,428428,428469,428477],{"className":428427,"style":77995},[31805],[34,428429,428430,428433],{"style":369015},[34,428431],{"className":428432,"style":41508},[31813],[34,428434,428436,428439,428442,428445,428448,428451,428454,428457,428460,428463,428466],{"className":428435},[31685],[34,428437,329201],{"className":428438},[31685,31686],[34,428440,456],{"className":428441},[31782],[34,428443,32936],{"className":428444},[31685,31686],[34,428446,2557],{"className":428447},[31690],[34,428449],{"className":428450,"style":32604},[31845],[34,428452,223],{"className":428453},[32024],[34,428455],{"className":428456,"style":32604},[31845],[34,428458,329201],{"className":428459},[31685,31686],[34,428461,456],{"className":428462},[31782],[34,428464,67764],{"className":428465,"style":32062},[31685,31686],[34,428467,2557],{"className":428468},[31690],[34,428470,428471,428474],{"style":369039},[34,428472],{"className":428473,"style":41508},[31813],[34,428475],{"className":428476},[31685],[34,428478,428479,428482],{"style":369063},[34,428480],{"className":428481,"style":41508},[31813],[34,428483],{"className":428484},[31685],[34,428486,31829],{"className":428487},[31828],[34,428489,428491],{"className":428490},[31801],[34,428492,428494],{"className":428493,"style":78064},[31805],[34,428495],{},[34,428497,428499],{"className":428498},[71384],[34,428500,428502,429265],{"className":428501},[31796,31797],[34,428503,428505,429262],{"className":428504},[31801],[34,428506,428508,428793,429081],{"className":428507,"style":77995},[31805],[34,428509,428510,428513],{"style":369015},[34,428511],{"className":428512,"style":41508},[31813],[34,428514,428516,428519,428522,428525,428528,428531,428534,428537,428540,428543,428583,428586,428589,428592,428595,428598,428601,428604,428607,428610,428613,428653,428656,428659,428662,428665,428668,428671,428674,428677,428680,428720,428723,428726,428729,428732,428735,428738,428741,428744,428747,428750,428790],{"className":428515},[31685],[34,428517],{"className":428518},[31685],[34,428520],{"className":428521,"style":31846},[31845],[34,428523,202],{"className":428524},[31850],[34,428526],{"className":428527,"style":31846},[31845],[34,428529,456],{"className":428530},[31782],[34,428532,32936],{"className":428533},[31685,31686],[34,428535],{"className":428536,"style":32604},[31845],[34,428538,223],{"className":428539},[32024],[34,428541],{"className":428542,"style":32604},[31845],[34,428544,428546,428549],{"className":428545},[31685],[34,428547,106063],{"className":428548,"style":67482},[31685,31686],[34,428550,428552],{"className":428551},[31792],[34,428553,428555,428575],{"className":428554},[31796,31797],[34,428556,428558,428572],{"className":428557},[31801],[34,428559,428561],{"className":428560,"style":31806},[31805],[34,428562,428563,428566],{"style":217742},[34,428564],{"className":428565,"style":31814},[31813],[34,428567,428569],{"className":428568},[31818,31819,31820,31821],[34,428570,640],{"className":428571},[31685,31821],[34,428573,31829],{"className":428574},[31828],[34,428576,428578],{"className":428577},[31801],[34,428579,428581],{"className":428580,"style":31836},[31805],[34,428582],{},[34,428584,31960],{"className":428585},[32119],[34,428587],{"className":428588,"style":32123},[31845],[34,428590,33934],{"className":428591},[34006],[34,428593],{"className":428594,"style":32123},[31845],[34,428596,31960],{"className":428597},[32119],[34,428599],{"className":428600,"style":32123},[31845],[34,428602,32936],{"className":428603},[31685,31686],[34,428605],{"className":428606,"style":32604},[31845],[34,428608,223],{"className":428609},[32024],[34,428611],{"className":428612,"style":32604},[31845],[34,428614,428616,428619],{"className":428615},[31685],[34,428617,106063],{"className":428618,"style":67482},[31685,31686],[34,428620,428622],{"className":428621},[31792],[34,428623,428625,428645],{"className":428624},[31796,31797],[34,428626,428628,428642],{"className":428627},[31801],[34,428629,428631],{"className":428630,"style":32362},[31805],[34,428632,428633,428636],{"style":217742},[34,428634],{"className":428635,"style":31814},[31813],[34,428637,428639],{"className":428638},[31818,31819,31820,31821],[34,428640,31660],{"className":428641},[31685,31686,31821],[34,428643,31829],{"className":428644},[31828],[34,428646,428648],{"className":428647},[31801],[34,428649,428651],{"className":428650,"style":31836},[31805],[34,428652],{},[34,428654,2557],{"className":428655},[31690],[34,428657],{"className":428658,"style":32604},[3184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34,429190,456],{"className":429191},[31782],[34,429193,32936],{"className":429194},[31685,31686],[34,429196],{"className":429197,"style":32604},[31845],[34,429199,223],{"className":429200},[32024],[34,429202],{"className":429203,"style":32604},[31845],[34,429205,67764],{"className":429206,"style":32062},[31685,31686],[34,429208,2557],{"className":429209},[31690],[34,429211],{"className":429212,"style":32604},[31845],[34,429214,223],{"className":429215},[32024],[34,429217],{"className":429218,"style":32604},[31845],[34,429220,429222,429225],{"className":429221},[31685],[34,429223,106063],{"className":429224,"style":67482},[31685,31686],[34,429226,429228],{"className":429227},[31792],[34,429229,429231,429251],{"className":429230},[31796,31797],[34,429232,429234,429248],{"className":429233},[31801],[34,429235,429237],{"className":429236,"style":32362},[31805],[34,429238,429239,429242],{"style":217742},[34,429240],{"className":429241,"style":31814},[31813],[34,429243,429245],{"className":429244},[31818,31819,31820,31821],[34,429246,31660],{"className":429247},[31685,31686,31821],[34,429249,31829],{"className":429250},[31828],[34,429252,429254],{"className":429253},[31801],[34,429255,429257],{"className":429256,"style":31836},[31805],[34,429258],{},[34,429260,2557],{"className":429261},[31690],[34,429263,31829],{"className":429264},[31828],[34,429266,429268],{"className":429267},[31801],[34,429269,429271],{"className":429270,"style":78064},[31805],[34,429272],{},[108,429274,429275,429278,429279,429307],{},[2627,429276,429277],{},"比较","：左边 = 右边。所以 ",[34,429280,429282,429295],{"className":429281},[31642],[34,429283,429285],{"className":429284},[31646],[31648,429286,429287],{"xmlns":31650},[31652,429288,429289,429293],{},[31655,429290,429291],{},[31658,429292,329201],{},[31665,429294,364484],{"encoding":31667},[34,429296,429298],{"className":429297,"ariaHidden":758},[31672],[34,429299,429301,429304],{"className":429300},[31676],[34,429302],{"className":429303,"style":43217},[31680],[34,429305,329201],{"className":429306},[31685,31686]," 保持加法。",[32490,429309,429311],{"id":429310},"_2-保持乘法",[2627,429312,429313],{},"2. 保持乘法",[15,429315,427365,429316,347],{},[34,429317,429319,429359],{"className":429318},[31642],[34,429320,429322],{"className":429321},[31646],[31648,429323,429324],{"xmlns":31650},[31652,429325,429326,429356],{},[31655,429327,429328,429330,429332,429334,429336,429338,429340,429342,429344,429346,429348,429350,429352,429354],{},[31658,429329,329201],{},[31662,429331,456],{"stretchy":702},[31658,429333,32936],{},[31658,429335,67764],{},[31662,429337,2557],{"stretchy":702},[31662,429339,202],{},[31658,429341,329201],{},[31662,429343,456],{"stretchy":702},[31658,429345,32936],{},[31662,429347,2557],{"stretchy":702},[31658,429349,329201],{},[31662,429351,456],{"stretchy":702},[31658,429353,67764],{},[31662,429355,2557],{"stretchy":702},[31665,429357,429358],{"encoding":31667},"\\phi(xy) = \\phi(x)\\phi(y)",[34,429360,429362,429392],{"className":429361,"ariaHidden":758},[31672],[34,429363,429365,429368,429371,429374,429377,429380,429383,429386,429389],{"className":429364},[31676],[34,429366],{"className":429367,"style":31774},[31680],[34,429369,329201],{"className":429370},[31685,31686],[34,429372,456],{"className":429373},[31782],[34,429375,32936],{"className":429376},[31685,31686],[34,429378,67764],{"className":429379,"style":32062},[31685,31686],[34,429381,2557],{"className":429382},[31690],[34,429384],{"className":429385,"style":31846},[31845],[34,429387,202],{"className":429388},[31850],[34,429390],{"className":429391,"style":31846},[31845],[34,429393,429395,429398,429401,429404,429407,429410,429413,429416,429419],{"className":429394},[31676],[34,429396],{"className":429397,"style":31774},[31680],[34,429399,329201],{"className":429400},[31685,31686],[34,429402,456],{"className":429403},[31782],[34,429405,32936],{"className":429406},[31685,31686],[34,429408,2557],{"className":429409},[31690],[34,429411,329201],{"className":429412},[31685,31686],[34,429414,456],{"className":429415},[31782],[34,429417,67764],{"className":429418,"style":32062},[31685,31686],[34,429420,2557],{"className":429421},[31690],[105,429423,429424,429755,431049],{},[108,429425,429426,33183,429428],{},[2627,429427,427512],{},[34,429429,429431,429503],{"className":429430},[31642],[34,429432,429434],{"className":429433},[31646],[31648,429435,429436],{"xmlns":31650},[31652,429437,429438,429500],{},[31655,429439,429440,429442,429444,429446,429448,429450,429452,429454,429456,429458,429460,429466,429468,429470,429472,429474,429480,429482,429484,429486,429488,429490,429492,429498],{},[31658,429441,329201],{},[31662,429443,456],{"stretchy":702},[31658,429445,32936],{},[31658,429447,67764],{},[31662,429449,2557],{"stretchy":702},[31662,429451,202],{},[31662,429453,456],{"stretchy":702},[31658,429455,32936],{},[31658,429457,67764],{},[31662,429459,223],{},[31743,429461,429462,429464],{},[31658,429463,106063],{},[31749,429465,640],{},[31662,429467,31960],{"separator":758},[31658,429469,32936],{},[31658,429471,67764],{},[31662,429473,223],{},[31743,429475,429476,429478],{},[31658,429477,106063],{},[31749,429479,768],{},[31662,429481,31960],{"separator":758},[31662,429483,33934],{},[31662,429485,31960],{"separator":758},[31658,429487,32936],{},[31658,429489,67764],{},[31662,429491,223],{},[31743,429493,429494,429496],{},[31658,429495,106063],{},[31658,429497,31660],{},[31662,429499,2557],{"stretchy":702},[31665,429501,429502],{"encoding":31667},"\\phi(xy) = (xy+I_1, xy+I_2, \\dots, xy+I_n)",[34,429504,429506,429536,429560,429627,429706],{"className":429505,"ariaHidden":758},[31672],[34,429507,429509,429512,429515,429518,429521,429524,429527,429530,429533],{"className":429508},[31676],[34,429510],{"className":429511,"style":31774},[31680],[34,429513,329201],{"className":429514},[31685,31686],[34,429516,456],{"className":429517},[31782],[34,429519,32936],{"className":429520},[31685,31686],[34,429522,67764],{"className":429523,"style":32062},[31685,31686],[34,429525,2557],{"className":429526},[31690],[34,429528],{"className":429529,"style":31846},[31845],[34,429531,202],{"className":429532},[31850],[34,429534],{"className":429535,"style":31846},[31845],[34,429537,429539,429542,429545,429548,429551,429554,429557],{"className":429538},[31676],[34,429540],{"className":429541,"style":31774},[31680],[34,429543,456],{"className":429544},[31782],[34,429546,32936],{"className":429547},[31685,31686],[34,429549,67764],{"className":429550,"style":32062},[31685,31686],[34,429552],{"className":429553,"style":32604},[31845],[34,429555,223],{"className":429556},[32024],[34,429558],{"className":429559,"style":32604},[31845],[34,429561,429563,429566,429606,429609,429612,429615,429618,429621,429624],{"className":429562},[31676],[34,429564],{"className":429565,"style":70739},[31680],[34,429567,429569,429572],{"className":429568},[31685],[34,429570,106063],{"className":429571,"style":67482},[31685,31686],[34,429573,429575],{"className":429574},[31792],[34,429576,429578,429598],{"className":429577},[31796,31797],[34,429579,429581,429595],{"className":429580},[31801],[34,429582,429584],{"className":429583,"style":31806},[31805],[34,429585,429586,429589],{"style":217742},[34,429587],{"className":429588,"style":31814},[31813],[34,429590,429592],{"className":429591},[31818,31819,31820,31821],[34,429593,640],{"className":429594},[31685,31821],[34,429596,31829],{"className":429597},[31828],[34,429599,429601],{"className":429600},[31801],[34,429602,429604],{"className":429603,"style":31836},[31805],[34,429605],{},[34,429607,31960],{"className":429608},[32119],[34,429610],{"className":429611,"style":32123},[31845],[34,429613,32936],{"className":429614},[31685,31686],[34,429616,67764],{"className":429617,"style":32062},[31685,31686],[34,429619],{"className":429620,"style":32604},[31845],[34,429622,223],{"className":429623},[32024],[34,429625],{"className":429626,"style":32604},[31845],[34,429628,429630,429633,429673,429676,429679,429682,429685,429688,429691,429694,429697,429700,429703],{"className":429629},[31676],[34,429631],{"className":429632,"style":70739},[31680],[34,429634,429636,429639],{"className":429635},[31685],[34,429637,106063],{"className":429638,"style":67482},[31685,31686],[34,429640,429642],{"className":429641},[31792],[34,429643,429645,429665],{"className":429644},[31796,31797],[34,429646,429648,429662],{"className":429647},[31801],[34,429649,429651],{"className":429650,"style":31806},[31805],[34,429652,429653,429656],{"style":217742},[34,429654],{"className":429655,"style":31814},[31813],[34,429657,429659],{"className":429658},[31818,31819,31820,31821],[34,429660,768],{"className":429661},[31685,31821],[34,429663,31829],{"className":429664},[31828],[34,429666,429668],{"className":429667},[31801],[34,429669,429671],{"className":429670,"style":31836},[31805],[34,429672],{},[34,429674,31960],{"className":429675},[32119],[34,429677],{"className":429678,"style":32123},[31845],[34,429680,33934],{"className":429681},[34006],[34,429683],{"className":429684,"style":32123},[31845],[34,429686,31960],{"className":429687},[32119],[34,429689],{"className":429690,"style":32123},[31845],[34,429692,32936],{"className":429693},[31685,31686],[34,429695,67764],{"className":429696,"style":32062},[31685,31686],[34,429698],{"className":429699,"style":32604},[31845],[34,429701,223],{"className":429702},[32024],[34,429704],{"className":429705,"style":32604},[31845],[34,429707,429709,429712,429752],{"className":429708},[31676],[34,429710],{"className":429711,"style":31774},[31680],[34,429713,429715,429718],{"className":429714},[31685],[34,429716,106063],{"className":429717,"style":67482},[31685,31686],[34,429719,429721],{"className":429720},[31792],[34,429722,429724,429744],{"className":429723},[31796,31797],[34,429725,429727,429741],{"className":429726},[31801],[34,429728,429730],{"className":429729,"style":32362},[31805],[34,429731,429732,429735],{"style":217742},[34,429733],{"className":429734,"style":31814},[31813],[34,429736,429738],{"className":429737},[31818,31819,31820,31821],[34,429739,31660],{"className":429740},[31685,31686,31821],[34,429742,31829],{"className":429743},[31828],[34,429745,429747],{"className":429746},[31801],[34,429748,429750],{"className":429749,"style":31836},[31805],[34,429751],{},[34,429753,2557],{"className":429754},[31690],[108,429756,429757,33183,429759],{},[2627,429758,428132],{},[34,429760,429762],{"className":429761},[126385],[34,429763,429765,430083],{"className":429764},[31642],[34,429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658,430051,67764],{},[31662,430053,223],{},[31743,430055,430056,430058],{},[31658,430057,106063],{},[31749,430059,640],{},[31662,430061,31960],{"separator":758},[31662,430063,33934],{},[31662,430065,31960],{"separator":758},[31658,430067,32936],{},[31658,430069,67764],{},[31662,430071,223],{},[31743,430073,430074,430076],{},[31658,430075,106063],{},[31658,430077,31660],{},[31662,430079,2557],{"stretchy":702},[31665,430081,430082],{"encoding":31667},"\\begin{aligned}\n\\phi(x)\\phi(y) &= (x+I_1, \\dots, x+I_n) \\cdot (y+I_1, \\dots, y+I_n) \\\\\n&\\quad \\text{直和(积)中的乘法是按分量相乘：} \\\\\n&= ((x+I_1)(y+I_1), \\dots, (x+I_n)(y+I_n)) \\\\\n&\\quad \\text{根据商环乘法定义 } (a+I)(b+I) = ab+I \\text{：} \\\\\n&= (xy+I_1, \\dots, xy+I_n)\n\\end{aligned}",[34,430084,430086],{"className":430085,"ariaHidden":758},[31672],[34,430087,430089,430092],{"className":430088},[31676],[34,430090],{"className":430091,"style":81591},[31680],[34,430093,430095],{"className":430094},[31685],[34,430096,430098,430187],{"className":430097},[71078],[34,430099,430101],{"className":430100},[369003],[34,430102,430104,430179],{"className":430103},[31796,31797],[34,430105,430107,430176],{"className":430106},[31801],[34,430108,430110,430143,430151,430159,430167],{"className":430109,"style":81616},[31805],[34,430111,430113,430116],{"style":430112},"top:-6.01em;",[34,430114],{"className":430115,"style":41508},[31813],[34,430117,430119,430122,430125,430128,430131,430134,430137,430140],{"className":430118},[31685],[34,430120,329201],{"className":430121},[31685,31686],[34,430123,456],{"className":430124},[31782],[34,430126,32936],{"className":430127},[31685,31686],[34,430129,2557],{"className":430130},[31690],[34,430132,329201],{"className":430133},[31685,31686],[34,430135,456],{"className":430136},[31782],[34,430138,67764],{"className":430139,"style":32062},[31685,31686],[34,430141,2557],{"className":430142},[31690],[34,430144,430145,430148],{"style":369015},[34,430146],{"className":430147,"style":41508},[31813],[34,430149],{"className":430150},[31685],[34,430152,430153,430156],{"style":369039},[34,430154],{"className":430155,"style":41508},[31813],[34,430157],{"className":430158},[31685],[34,430160,430161,430164],{"style":369063},[34,430162],{"className":430163,"style":41508},[31813],[34,430165],{"className":430166},[31685],[34,430168,430170,430173],{"style":430169},"top:-0.01em;",[34,430171],{"className":430172,"style":41508},[31813],[34,430174],{"className":430175},[31685],[34,430177,31829],{"className":430178},[31828],[34,430180,430182],{"className":430181},[31801],[34,430183,430185],{"className":430184,"style":81688},[31805],[34,430186],{},[34,430188,430190],{"className":430189},[71384],[34,430191,430193,431041],{"className":430192},[31796,31797],[34,430194,430196,431038],{"className":430195},[31801],[34,430197,430199,430484,430518,430788,430884],{"className":430198,"style":81616},[31805],[34,430200,430201,430204],{"style":430112},[34,430202],{"className":430203,"style":41508},[31813],[34,430205,430207,430210,430213,430216,430219,430222,430225,430228,430231,430234,430274,430277,430280,430283,430286,430289,430292,430295,430298,430301,430304,430344,430347,430350,430353,430356,430359,430362,430365,430368,430371,430411,430414,430417,430420,430423,430426,430429,430432,430435,430438,430441,430481],{"className":430206},[31685],[34,430208],{"className":430209},[31685],[34,430211],{"className":430212,"style":31846},[31845],[34,430214,202],{"className":430215},[31850],[34,430217],{"className":430218,"style":31846},[31845],[34,430220,456],{"className":430221},[31782],[34,430223,32936],{"className":430224},[31685,31686],[34,430226],{"className":430227,"style":32604},[31845],[34,430229,223],{"className":430230},[32024],[34,430232],{"className":430233,"style":32604},[31845],[34,430235,430237,430240],{"className":430236},[31685],[34,430238,106063],{"className":430239,"style":67482},[31685,31686],[34,430241,430243],{"className":430242},[31792],[34,430244,430246,430266],{"className":430245},[31796,31797],[34,430247,430249,430263],{"className":430248},[31801],[34,430250,430252],{"className":430251,"style":31806},[31805],[34,430253,430254,430257],{"style":217742},[34,430255],{"className":430256,"style":31814},[31813],[34,430258,430260],{"className":430259},[31818,31819,31820,31821],[34,430261,640],{"className":430262},[31685,31821],[34,430264,31829],{"className":430265},[31828],[34,430267,430269],{"className":430268},[31801],[34,430270,430272],{"className":430271,"style":31836},[31805],[34,430273],{},[34,430275,31960],{"className":430276},[32119],[34,430278],{"className":430279,"style":32123},[31845],[34,430281,33934],{"className":430282},[34006],[34,430284],{"className":430285,"style":32123},[31845],[34,430287,31960],{"className":430288},[32119],[34,430290],{"className":430291,"style":32123},[31845],[34,430293,32936],{"className":430294},[31685,31686],[34,430296],{"className":430297,"style":32604},[31845],[34,430299,223],{"className":430300},[32024],[34,430302],{"className":430303,"style":32604},[31845],[34,430305,430307,430310],{"className":430306},[31685],[34,430308,106063],{"className":430309,"style":67482},[31685,31686],[34,430311,430313],{"className":430312},[31792],[34,430314,430316,430336],{"className":430315},[31796,31797],[34,430317,430319,430333],{"className":430318},[31801],[34,430320,430322],{"className":430321,"style":32362},[31805],[34,430323,430324,430327],{"style":217742},[34,430325],{"className":430326,"style":31814},[31813],[34,430328,430330],{"className":430329},[31818,31819,31820,31821],[34,430331,31660],{"className":430332},[31685,31686,31821],[34,430334,31829],{"className":430335},[31828],[34,430337,430339],{"className":430338},[31801],[34,430340,430342],{"className":430341,"style":31836},[31805],[34,430343],{},[34,430345,2557],{"className":430346},[31690],[34,430348],{"className":430349,"style":32604},[31845],[34,430351,71191],{"className":430352},[32024],[34,430354],{"className":430355,"style":32604},[31845],[34,430357,456],{"className":430358},[31782],[34,430360,67764],{"className":430361,"style":32062},[31685,31686],[34,430363],{"className":430364,"style":32604},[31845],[34,430366,223],{"className":430367},[32024],[34,430369],{"className":430370,"style":32604},[31845],[34,430372,430374,430377],{"className":430373},[31685],[34,430375,106063],{"className":430376,"style":67482},[31685,31686],[34,430378,430380],{"className":430379},[31792],[34,430381,430383,430403],{"className":430382},[31796,31797],[34,430384,430386,430400],{"className":430385},[31801],[34,430387,430389],{"className":430388,"style":31806},[31805],[34,430390,430391,430394],{"style":217742},[34,430392],{"className":430393,"style":31814},[31813],[34,430395,430397],{"className":430396},[31818,31819,31820,31821],[34,430398,640],{"className":430399},[31685,31821],[34,430401,31829],{"className":430402},[31828],[34,430404,430406],{"className":430405},[31801],[34,430407,430409],{"className":430408,"style":31836},[31805],[34,430410],{},[34,430412,31960],{"className":430413},[32119],[34,430415],{"className":430416,"style":32123},[31845],[34,430418,33934],{"className":430419},[34006],[34,430421],{"className":430422,"style":32123},[31845],[34,430424,31960],{"className":430425},[32119],[34,430427],{"className":430428,"style":32123},[31845],[34,430430,67764],{"className":430431,"style":32062},[31685,31686],[34,430433],{"className":430434,"style":32604},[31845],[34,430436,223],{"className":430437},[32024],[34,430439],{"className":430440,"style":32604},[31845],[34,430442,430444,430447],{"className":430443},[31685],[34,430445,106063],{"className":430446,"style":67482},[31685,31686],[34,430448,430450],{"className":430449},[31792],[34,430451,430453,430473],{"className":430452},[31796,31797],[34,430454,430456,430470],{"className":430455},[31801],[34,430457,430459],{"className":430458,"style":32362},[31805],[34,430460,430461,430464],{"style":217742},[34,430462],{"className":430463,"style":31814},[31813],[34,430465,430467],{"className":430466},[31818,31819,31820,31821],[34,430468,31660],{"className":430469},[31685,31686,31821],[34,430471,31829],{"className":430472},[31828],[34,430474,430476],{"className":430475},[31801],[34,430477,430479],{"className":430478,"style":31836},[31805],[34,430480],{},[34,430482,2557],{"className":430483},[31690],[34,430485,430486,430489],{"style":369015},[34,430487],{"className":430488,"style":41508},[31813],[34,430490,430492,430495,430498],{"className":430491},[31685],[34,430493],{"className":430494},[31685],[34,430496],{"className":430497,"style":128438},[31845],[34,430499,430501,430504,430507,430511,430514],{"className":430500},[31685,2638],[34,430502,404688],{"className":430503},[31685,105462],[34,430505,456],{"className":430506},[31685],[34,430508,430510],{"className":430509},[31685,105462],"积",[34,430512,2557],{"className":430513},[31685],[34,430515,430517],{"className":430516},[31685,105462],"中的乘法是按分量相乘：",[34,430519,430520,430523],{"style":369039},[34,430521],{"className":430522,"style":41508},[31813],[34,430524,430526,430529,430532,430535,430538,430541,430544,430547,430550,430553,430593,430596,430599,430602,430605,430608,430611,430651,430654,430657,430660,430663,430666,430669,430672,430675,430678,430681,430684,430687,430727,430730,430733,430736,430739,430742,430745,430785],{"className":430525},[31685],[34,430527],{"className":430528},[31685],[34,430530],{"className":430531,"style":31846},[31845],[34,430533,202],{"className":430534},[31850],[34,430536],{"className":430537,"style":31846},[31845],[34,430539,1323],{"className":430540},[31782],[34,430542,32936],{"className":430543},[31685,31686],[34,430545],{"className":430546,"style":32604},[31845],[34,430548,223],{"className":430549},[32024],[34,430551],{"className":430552,"style":32604},[31845],[34,430554,430556,430559],{"className":430555},[31685],[34,430557,106063],{"className":430558,"style":67482},[31685,31686],[34,430560,430562],{"className":430561},[31792],[34,430563,430565,430585],{"className":430564},[31796,31797],[34,430566,430568,430582],{"className":430567},[31801],[34,430569,430571],{"className":430570,"style":31806},[31805],[34,430572,430573,430576],{"style":217742},[34,430574],{"className":430575,"style":31814},[31813],[34,430577,430579],{"className":430578},[31818,31819,31820,31821],[34,430580,640],{"className":430581},[31685,31821],[34,430583,31829],{"className":430584},[31828],[34,430586,430588],{"className":430587},[31801],[34,430589,430591],{"className":430590,"style":31836},[31805],[34,430592],{},[34,430594,2557],{"className":430595},[31690],[34,430597,456],{"className":430598},[31782],[34,430600,67764],{"className":430601,"style":32062},[31685,31686],[34,430603],{"className":430604,"style":32604},[31845],[3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Name":430877,"style":67482},[31685,31686],[34,430879,430881],{"className":430880},[31685,2638],[34,430882,2664],{"className":430883},[31685,105462],[34,430885,430886,430889],{"style":430169},[34,430887],{"className":430888,"style":41508},[31813],[34,430890,430892,430895,430898,430901,430904,430907,430910,430913,430916,430919,430922,430962,430965,430968,430971,430974,430977,430980,430983,430986,430989,430992,430995,431035],{"className":430891},[31685],[34,430893],{"className":430894},[31685],[34,430896],{"className":430897,"style":31846},[31845],[34,430899,202],{"className":430900},[31850],[34,430902],{"className":430903,"style":31846},[31845],[34,430905,456],{"className":430906},[31782],[34,430908,32936],{"className":430909},[31685,31686],[34,430911,67764],{"className":430912,"style":32062},[31685,31686],[34,430914],{"className":430915,"style":32604},[31845],[34,430917,223],{"className":430918},[32024],[34,430920],{"className":430921,"style":32604},[31845],[34,430923,430925,430928],{"className":430924},[31685],[34,430926,106063],{"className":430927,"style":67482},[31685,31686],[34,430929,430931],{"className":430930},[31792],[34,430932,430934,430954],{"className":430933},[31796,31797],[34,430935,430937,430951],{"className":430936},[31801],[34,430938,430940],{"className":430939,"style":31806},[31805],[34,430941,430942,430945],{"style":217742},[34,430943],{"className":430944,"style":31814},[31813],[34,430946,430948],{"className":430947},[31818,31819,31820,31821],[34,430949,640],{"className":430950},[31685,31821],[34,430952,31829],{"className":430953},[31828],[34,430955,430957],{"className":430956},[31801],[34,430958,430960],{"className":430959,"style":31836},[31805],[34,430961],{},[34,430963,31960],{"className":430964},[32119],[34,430966],{"className":430967,"style":32123},[31845],[34,430969,33934],{"className":430970},[34006],[34,430972],{"className":430973,"style":32123},[31845],[34,430975,31960],{"className":430976},[32119],[34,430978],{"className":430979,"style":32123},[31845],[34,430981,32936],{"className":430982},[31685,31686],[34,430984,67764],{"className":430985,"style":32062},[31685,31686],[34,430987],{"className":430988,"style":32604},[31845],[34,430990,223],{"className":430991},[32024],[34,430993],{"className":430994,"style":32604},[31845],[34,430996,430998,431001],{"className":430997},[31685],[34,430999,106063],{"className":431000,"style":67482},[31685,31686],[34,431002,431004],{"className":431003},[31792],[34,431005,431007,431027],{"className":431006},[31796,31797],[34,431008,431010,431024],{"className":431009},[31801],[34,431011,431013],{"className":431012,"style":32362},[31805],[34,431014,431015,431018],{"style":217742},[34,431016],{"className":431017,"style":31814},[31813],[34,431019,431021],{"className":431020},[31818,31819,31820,31821],[34,431022,31660],{"className":431023},[31685,31686,31821],[34,431025,31829],{"className":431026},[31828],[34,431028,431030],{"className":431029},[31801],[34,431031,431033],{"className":431032,"style":31836},[31805],[34,431034],{},[34,431036,2557],{"className":431037},[31690],[34,431039,31829],{"className":431040},[31828],[34,431042,431044],{"className":431043},[31801],[34,431045,431047],{"className":431046,"style":81688},[31805],[34,431048],{},[108,431050,431051,429278,431053,431081],{},[2627,431052,429277],{},[34,431054,431056,431069],{"className":431055},[31642],[34,431057,431059],{"className":431058},[31646],[31648,431060,431061],{"xmlns":31650},[31652,431062,431063,431067],{},[31655,431064,431065],{},[31658,431066,329201],{},[31665,431068,364484],{"encoding":31667},[34,431070,431072],{"className":431071,"ariaHidden":758},[31672],[34,431073,431075,431078],{"className":431074},[31676],[34,431076],{"className":431077,"style":43217},[31680],[34,431079,329201],{"className":431080},[31685,31686]," 保持乘法。",[32490,431083,431085],{"id":431084},"_3-保持单位元如果-rrr-是幺环",[2627,431086,431087,431088,431116],{},"3. 保持单位元（如果 ",[34,431089,431091,431104],{"className":431090},[31642],[34,431092,431094],{"className":431093},[31646],[31648,431095,431096],{"xmlns":31650},[31652,431097,431098,431102],{},[31655,431099,431100],{},[31658,431101,34386],{},[31665,431103,34386],{"encoding":31667},[34,431105,431107],{"className":431106,"ariaHidden":758},[31672],[34,431108,431110,431113],{"className":431109},[31676],[34,431111],{"className":431112,"style":33218},[31680],[34,431114,34386],{"className":431115,"style":34427},[31685,31686]," 是幺环）",[15,431118,133910,431119,11008,431189,431217,431218,431313,431314,347],{},[34,431120,431122,431140],{"className":431121},[31642],[34,431123,431125],{"className":431124},[31646],[31648,431126,431127],{"xmlns":31650},[31652,431128,431129,431137],{},[31655,431130,431131],{},[31743,431132,431133,431135],{},[31749,431134,640],{},[31658,431136,34386],{},[31665,431138,431139],{"encoding":31667},"1_R",[34,431141,431143],{"className":431142,"ariaHidden":758},[31672],[34,431144,431146,431149],{"className":431145},[31676],[34,431147],{"className":431148,"style":38357},[31680],[34,431150,431152,431155],{"className":431151},[31685],[34,431153,640],{"className":431154},[31685],[34,431156,431158],{"className":431157},[31792],[34,431159,431161,431181],{"className":431160},[31796,31797],[34,431162,431164,431178],{"className":431163},[31801],[34,431165,431167],{"className":431166,"style":116558},[31805],[34,431168,431169,431172],{"style":31809},[34,431170],{"className":431171,"style":31814},[31813],[34,431173,431175],{"className":431174},[31818,31819,31820,31821],[34,431176,34386],{"className":431177,"style":34427},[31685,31686,31821],[34,431179,31829],{"className":431180},[31828],[34,431182,431184],{"className":431183},[31801],[34,431185,431187],{"className":431186,"style":31836},[31805],[34,431188],{},[34,431190,431192,431205],{"className":431191},[31642],[34,431193,431195],{"className":431194},[31646],[31648,431196,431197],{"xmlns":31650},[31652,431198,431199,431203],{},[31655,431200,431201],{},[31658,431202,34386],{},[31665,431204,34386],{"encoding":31667},[34,431206,431208],{"className":431207,"ariaHidden":758},[31672],[34,431209,431211,431214],{"className":431210},[31676],[34,431212],{"className":431213,"style":33218},[31680],[34,431215,34386],{"className":431216,"style":34427},[31685,31686]," 的单位元。直和环 ",[34,431219,431221,431249],{"className":431220},[31642],[34,431222,431224],{"className":431223},[31646],[31648,431225,431226],{"xmlns":31650},[31652,431227,431228,431246],{},[31655,431229,431230,431232,431234,431236,431238,431244],{},[31662,431231,404608],{},[31662,431233,456],{"stretchy":702},[31658,431235,34386],{},[31658,431237,38080],{"mathvariant":38079},[31743,431239,431240,431242],{},[31658,431241,106063],{},[31658,431243,33759],{},[31662,431245,2557],{"stretchy":702},[31665,431247,431248],{"encoding":31667},"\\bigoplus (R\u002FI_k)",[34,431250,431252],{"className":431251,"ariaHidden":758},[31672],[34,431253,431255,431258,431261,431264,431267,431270,431310],{"className":431254},[31676],[34,431256],{"className":431257,"style":31774},[31680],[34,431259,404608],{"className":431260,"style":156561},[32767,131249,156560],[34,431262,456],{"className":431263},[31782],[34,431265,34386],{"className":431266,"style":34427},[31685,31686],[34,431268,38080],{"className":431269},[31685],[34,431271,431273,431276],{"className":431272},[31685],[34,431274,106063],{"className":431275,"style":67482},[31685,31686],[34,431277,431279],{"className":431278},[31792],[34,431280,431282,431302],{"className":431281},[31796,31797],[34,431283,431285,431299],{"className":431284},[31801],[34,431286,431288],{"className":431287,"style":33833},[31805],[34,431289,431290,431293],{"style":217742},[34,431291],{"className":431292,"style":31814},[31813],[34,431294,431296],{"className":431295},[31818,31819,31820,31821],[34,431297,33759],{"className":431298,"style":33774},[31685,31686,31821],[34,431300,31829],{"className":431301},[31828],[34,431303,431305],{"className":431304},[31801],[34,431306,431308],{"className":431307,"style":31836},[31805],[34,431309],{},[34,431311,2557],{"className":431312},[31690]," 的单位元是每个分量的单位元组成的序列，即 ",[34,431315,431317,431367],{"className":431316},[31642],[34,431318,431320],{"className":431319},[31646],[31648,431321,431322],{"xmlns":31650},[31652,431323,431324,431364],{},[31655,431325,431326,431328,431334,431336,431342,431344,431346,431348,431354,431356,431362],{},[31662,431327,456],{"stretchy":702},[31743,431329,431330,431332],{},[31749,431331,640],{},[31658,431333,34386],{},[31662,431335,223],{},[31743,431337,431338,431340],{},[31658,431339,106063],{},[31749,431341,640],{},[31662,431343,31960],{"separator":758},[31662,431345,33934],{},[31662,431347,31960],{"separator":758},[31743,431349,431350,431352],{},[31749,431351,640],{},[31658,431353,34386],{},[31662,431355,223],{},[31743,431357,431358,431360],{},[31658,431359,106063],{},[31658,431361,31660],{},[31662,431363,2557],{"stretchy":702},[31665,431365,431366],{"encoding":31667},"(1_R+I_1, \\dots, 1_R+I_n)",[34,431368,431370,431428,431541],{"className":431369,"ariaHidden":758},[31672],[34,431371,431373,431376,431379,431419,431422,431425],{"className":431372},[31676],[34,431374],{"className":431375,"style":31774},[31680],[34,431377,456],{"className":431378},[31782],[34,431380,431382,431385],{"className":431381},[31685],[34,431383,640],{"className":431384},[31685],[34,431386,431388],{"className":431387},[31792],[34,431389,431391,431411],{"className":431390},[31796,31797],[34,431392,431394,431408],{"className":431393},[31801],[34,431395,431397],{"className":431396,"style":116558},[31805],[34,431398,431399,431402],{"style":31809},[34,431400],{"className":431401,"style":31814},[31813],[34,431403,431405],{"className":431404},[31818,31819,31820,31821],[34,431406,34386],{"className":431407,"style":34427},[31685,31686,31821],[34,431409,31829],{"className":431410},[31828],[34,431412,431414],{"className":431413},[31801],[34,431415,431417],{"className":431416,"style":31836},[31805],[34,431418],{},[34,431420],{"className":431421,"style":32604},[31845],[34,431423,223],{"className":431424},[32024],[34,431426],{"className":431427,"style":32604},[31845],[34,431429,431431,431434,431474,431477,431480,431483,431486,431489,431492,431532,431535,431538],{"className":431430},[31676],[34,431432],{"className":431433,"style":70739},[31680],[34,431435,431437,431440],{"className":431436},[31685],[34,431438,106063],{"className":431439,"style":67482},[31685,31686],[34,431441,431443],{"className":431442},[31792],[34,431444,431446,431466],{"className":431445},[31796,31797],[34,431447,431449,431463],{"className":431448},[31801],[34,431450,431452],{"className":431451,"style":31806},[31805],[34,431453,431454,431457],{"style":217742},[34,431455],{"className":431456,"style":31814},[31813],[34,431458,431460],{"className":431459},[31818,31819,31820,31821],[34,431461,640],{"className":431462},[31685,31821],[34,431464,31829],{"className":431465},[31828],[34,431467,431469],{"className":431468},[31801],[34,431470,431472],{"className":431471,"style":31836},[31805],[34,431473],{},[34,431475,31960],{"className":431476},[32119],[34,431478],{"className":431479,"style":32123},[31845],[34,431481,33934],{"className":431482},[34006],[34,431484],{"className":431485,"style":32123},[31845],[34,431487,31960],{"className":431488},[32119],[34,431490],{"className":431491,"style":32123},[31845],[34,431493,431495,431498],{"className":431494},[31685],[34,431496,640],{"className":431497},[31685],[34,431499,431501],{"className":431500},[31792],[34,431502,431504,431524],{"className":431503},[31796,31797],[34,431505,431507,431521],{"className":431506},[31801],[34,431508,431510],{"className":431509,"style":116558},[31805],[34,431511,431512,431515],{"style":31809},[34,431513],{"className":431514,"style":31814},[31813],[34,431516,431518],{"className":431517},[31818,31819,31820,31821],[34,431519,34386],{"className":431520,"style":34427},[31685,31686,31821],[34,431522,31829],{"className":431523},[31828],[34,431525,431527],{"className":431526},[31801],[34,431528,431530],{"className":431529,"style":31836},[31805],[34,431531],{},[34,431533],{"className":431534,"style":32604},[31845],[34,431536,223],{"className":431537},[32024],[34,431539],{"className":431540,"style":32604},[31845],[34,431542,431544,431547,431587],{"className":431543},[31676],[34,431545],{"className":431546,"style":31774},[31680],[34,431548,431550,431553],{"className":431549},[31685],[34,431551,106063],{"className":431552,"style":67482},[31685,31686],[34,431554,431556],{"className":431555},[31792],[34,431557,431559,431579],{"className":431558},[31796,31797],[34,431560,431562,431576],{"className":431561},[31801],[34,431563,431565],{"className":431564,"style":32362},[31805],[34,431566,431567,431570],{"style":217742},[34,431568],{"className":431569,"style":31814},[31813],[34,431571,431573],{"className":431572},[31818,31819,31820,31821],[34,431574,31660],{"className":431575},[31685,31686,31821],[34,431577,31829],{"className":431578},[31828],[34,431580,431582],{"className":431581},[31801],[34,431583,431585],{"className":431584,"style":31836},[31805],[34,431586],{},[34,431588,2557],{"className":431589},[31690],[105,431591,431592,432067],{},[108,431593,431594,33183,431596],{},[2627,431595,207541],{},[34,431597,431599,431679],{"className":431598},[31642],[34,431600,431602],{"className":431601},[31646],[31648,431603,431604],{"xmlns":31650},[31652,431605,431606,431676],{},[31655,431607,431608,431610,431612,431618,431620,431622,431624,431630,431632,431638,431640,431646,431648,431654,431656,431658,431660,431666,431668,431674],{},[31658,431609,329201],{},[31662,431611,456],{"stretchy":702},[31743,431613,431614,431616],{},[31749,431615,640],{},[31658,431617,34386],{},[31662,431619,2557],{"stretchy":702},[31662,431621,202],{},[31662,431623,456],{"stretchy":702},[31743,431625,431626,431628],{},[31749,431627,640],{},[31658,431629,34386],{},[31662,431631,223],{},[31743,431633,431634,431636],{},[31658,431635,106063],{},[31749,431637,640],{},[31662,431639,31960],{"separator":758},[31743,431641,431642,431644],{},[31749,431643,640],{},[31658,431645,34386],{},[31662,431647,223],{},[31743,431649,431650,431652],{},[31658,431651,106063],{},[31749,431653,768],{},[31662,431655,31960],{"separator":758},[31662,431657,33934],{},[31662,431659,31960],{"separator":758},[31743,431661,431662,431664],{},[31749,431663,640],{},[31658,431665,34386],{},[31662,431667,223],{},[31743,431669,431670,431672],{},[31658,431671,106063],{},[31658,431673,31660],{},[31662,431675,2557],{"stretchy":702},[31665,431677,431678],{"encoding":31667},"\\phi(1_R) = (1_R+I_1, 1_R+I_2, \\dots, 1_R+I_n)",[34,431680,431682,431746,431804,431905,432018],{"className":431681,"ariaHidden":758},[31672],[34,431683,431685,431688,431691,431694,431734,431737,431740,431743],{"className":431684},[31676],[34,431686],{"className":431687,"style":31774},[31680],[34,431689,329201],{"className":431690},[31685,31686],[34,431692,456],{"className":431693},[31782],[34,431695,431697,431700],{"className":431696},[31685],[34,431698,640],{"className":431699},[31685],[34,431701,431703],{"className":431702},[31792],[34,431704,431706,431726],{"className":431705},[31796,31797],[34,431707,431709,431723],{"className":431708},[31801],[34,431710,431712],{"className":431711,"style":116558},[31805],[34,431713,431714,431717],{"style":31809},[34,431715],{"className":431716,"style":31814},[31813],[34,431718,431720],{"className":431719},[31818,31819,31820,31821],[34,431721,34386],{"className":431722,"style":34427},[31685,31686,31821],[34,431724,31829],{"className":431725},[31828],[34,431727,431729],{"className":431728},[31801],[34,431730,431732],{"className":431731,"style":31836},[31805],[34,431733],{},[34,431735,2557],{"className":431736},[31690],[34,431738],{"className":431739,"style":31846},[31845],[34,431741,202],{"className":431742},[31850],[34,431744],{"className":431745,"style":31846},[31845],[34,431747,431749,431752,431755,431795,431798,431801],{"className":431748},[31676],[34,431750],{"className":431751,"style":31774},[31680],[34,431753,456],{"className":431754},[31782],[34,431756,431758,431761],{"className":431757},[31685],[34,431759,640],{"className":431760},[31685],[34,431762,431764],{"className":431763},[31792],[34,431765,431767,431787],{"className":431766},[31796,31797],[34,431768,431770,431784],{"className":431769},[31801],[34,431771,431773],{"className":431772,"style":116558},[31805],[34,431774,431775,431778],{"style":31809},[34,431776],{"className":431777,"style":31814},[31813],[34,431779,431781],{"className":431780},[31818,31819,31820,31821],[34,431782,34386],{"className":431783,"style":34427},[31685,31686,31821],[34,431785,31829],{"className":431786},[31828],[34,431788,431790],{"className":431789},[31801],[34,431791,431793],{"className":431792,"style":31836},[31805],[34,431794],{},[34,431796],{"className":431797,"style":32604},[31845],[34,431799,223],{"className":431800},[32024],[34,431802],{"className":431803,"style":32604},[31845],[34,431805,431807,431810,431850,431853,431856,431896,431899,431902],{"className":431806},[31676],[34,431808],{"className":431809,"style":70739},[31680],[34,431811,431813,431816],{"className":431812},[31685],[34,431814,106063],{"className":431815,"style":67482},[31685,31686],[34,431817,431819],{"className":431818},[31792],[34,431820,431822,431842],{"className":431821},[31796,31797],[34,431823,431825,431839],{"className":431824},[31801],[34,431826,431828],{"className":431827,"style":31806},[31805],[34,431829,431830,431833],{"style":217742},[34,431831],{"className":431832,"style":31814},[31813],[34,431834,431836],{"className":431835},[31818,31819,31820,31821],[34,431837,640],{"className":431838},[31685,31821],[34,431840,31829],{"className":431841},[31828],[34,431843,431845],{"className":431844},[31801],[34,431846,431848],{"className":431847,"style":31836},[31805],[34,431849],{},[34,431851,31960],{"className":431852},[32119],[34,431854],{"className":431855,"style":32123},[31845],[34,431857,431859,431862],{"className":431858},[31685],[34,431860,640],{"className":431861},[31685],[34,431863,431865],{"className":431864},[31792],[34,431866,431868,431888],{"className":431867},[31796,31797],[34,431869,431871,431885],{"className":431870},[31801],[34,431872,431874],{"className":431873,"style":116558},[31805],[34,431875,431876,431879],{"style":31809},[34,431877],{"className":431878,"style":31814},[31813],[34,431880,431882],{"className":431881},[31818,31819,31820,31821],[34,431883,34386],{"className":431884,"style":34427},[31685,31686,31821],[34,431886,31829],{"className":431887},[31828],[34,431889,431891],{"className":431890},[31801],[34,431892,431894],{"className":431893,"style":31836},[31805],[34,431895],{},[34,431897],{"className":431898,"style":32604},[31845],[34,431900,223],{"className":431901},[32024],[34,431903],{"className":431904,"style":32604},[31845],[34,431906,431908,431911,431951,431954,431957,431960,431963,431966,431969,432009,432012,432015],{"className":431907},[31676],[34,431909],{"className":431910,"style":70739},[31680],[34,431912,431914,431917],{"className":431913},[31685],[34,431915,106063],{"className":431916,"style":67482},[31685,31686],[34,431918,431920],{"className":431919},[31792],[34,431921,431923,431943],{"className":431922},[31796,31797],[34,431924,431926,431940],{"className":431925},[31801],[34,431927,431929],{"className":431928,"style":31806},[31805],[34,431930,431931,431934],{"style":217742},[34,431932],{"className":431933,"style":31814},[31813],[34,431935,431937],{"className":431936},[31818,31819,31820,31821],[34,431938,768],{"className":431939},[31685,31821],[34,431941,31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保持加法、乘法和单位元，因此 ",[34,432104,432106,432119],{"className":432105},[31642],[34,432107,432109],{"className":432108},[31646],[31648,432110,432111],{"xmlns":31650},[31652,432112,432113,432117],{},[31655,432114,432115],{},[31658,432116,329201],{},[31665,432118,364484],{"encoding":31667},[34,432120,432122],{"className":432121,"ariaHidden":758},[31672],[34,432123,432125,432128],{"className":432124},[31676],[34,432126],{"className":432127,"style":43217},[31680],[34,432129,329201],{"className":432130},[31685,31686]," 是一个环同态。",[15,432133,432134],{},[2627,432135,225949],{},[32188,432137],{},[253,432139,432141],{"id":432140},"题目-3寻找一般线性群的-sylow-ppp-子群-p64-第21题",[2627,432142,432143,432144,432172],{},"题目 3：寻找一般线性群的 Sylow ",[34,432145,432147,432160],{"className":432146},[31642],[34,432148,432150],{"className":432149},[31646],[31648,432151,432152],{"xmlns":31650},[31652,432153,432154,432158],{},[31655,432155,432156],{},[31658,432157,15],{},[31665,432159,15],{"encoding":31667},[34,432161,432163],{"className":432162,"ariaHidden":758},[31672],[34,432164,432166,432169],{"className":432165},[31676],[34,432167],{"className":432168,"style":38774},[31680],[34,432170,15],{"className":432171},[31685,31686],"-子群 (P64 第21题)",[15,432174,432175,362914,432177,432205,432206,432272,432273,432343,432344,432459,432460,432511,432512,432540,432541,432569],{},[2627,432176,214361],{},[34,432178,432180,432193],{"className":432179},[31642],[34,432181,432183],{"className":432182},[31646],[31648,432184,432185],{"xmlns":31650},[31652,432186,432187,432191],{},[31655,432188,432189],{},[31658,432190,15],{},[31665,432192,15],{"encoding":31667},[34,432194,432196],{"className":432195,"ariaHidden":758},[31672],[34,432197,432199,432202],{"className":432198},[31676],[34,432200],{"className":432201,"style":38774},[31680],[34,432203,15],{"className":432204},[31685,31686]," 是素数, ",[34,432207,432209,432233],{"className":432208},[31642],[34,432210,432212],{"className":432211},[31646],[31648,432213,432214],{"xmlns":31650},[31652,432215,432216,432230],{},[31655,432217,432218,432220,432222,432224,432226,432228],{},[31658,432219,51834],{},[31662,432221,202],{},[31658,432223,97300],{"mathvariant":318416},[31658,432225,38080],{"mathvariant":38079},[31658,432227,15],{},[31658,432229,97300],{"mathvariant":318416},[31665,432231,432232],{"encoding":31667},"F = \\mathbb{Z}\u002Fp\\mathbb{Z}",[34,432234,432236,432254],{"className":432235,"ariaHidden":758},[31672],[34,432237,432239,432242,432245,432248,432251],{"className":432238},[31676],[34,432240],{"className":432241,"style":33218},[31680],[34,432243,51834],{"className":432244,"style":37422},[31685,31686],[34,432246],{"className":432247,"style":31846},[31845],[34,432249,202],{"className":432250},[31850],[34,432252],{"className":432253,"style":31846},[31845],[34,432255,432257,432260,432263,432266,432269],{"className":432256},[31676],[34,432258],{"className":432259,"style":31774},[31680],[34,432261,97300],{"className":432262},[31685,318441],[34,432264,38080],{"className":432265},[31685],[34,432267,15],{"className":432268},[31685,31686],[34,432270,97300],{"className":432271},[31685,318441]," (即有限域 ",[34,432274,432276,432294],{"className":432275},[31642],[34,432277,432279],{"className":432278},[31646],[31648,432280,432281],{"xmlns":31650},[31652,432282,432283,432291],{},[31655,432284,432285],{},[31743,432286,432287,432289],{},[31658,432288,51834],{"mathvariant":318416},[31658,432290,15],{},[31665,432292,432293],{"encoding":31667},"\\mathbb{F}_p",[34,432295,432297],{"className":432296,"ariaHidden":758},[31672],[34,432298,432300,432303],{"className":432299},[31676],[34,432301],{"className":432302,"style":338084},[31680],[34,432304,432306,432309],{"className":432305},[31685],[34,432307,51834],{"className":432308},[31685,318441],[34,432310,432312],{"className":432311},[31792],[34,432313,432315,432335],{"className":432314},[31796,31797],[34,432316,432318,432332],{"className":432317},[31801],[34,432319,432321],{"className":432320,"style":32362},[31805],[34,432322,432323,432326],{"style":31809},[34,432324],{"className":432325,"style":31814},[31813],[34,432327,432329],{"className":432328},[31818,31819,31820,31821],[34,432330,15],{"className":432331},[31685,31686,31821],[34,432333,31829],{"className":432334},[31828],[34,432336,432338],{"className":432337},[31801],[34,432339,432341],{"className":432340,"style":32163},[31805],[34,432342],{},"), ",[34,432345,432347,432379],{"className":432346},[31642],[34,432348,432350],{"className":432349},[31646],[31648,432351,432352],{"xmlns":31650},[31652,432353,432354,432376],{},[31655,432355,432356,432358,432360,432370,432372,432374],{},[31658,432357,97709],{},[31662,432359,202],{},[31743,432361,432362,432368],{},[31655,432363,432364,432366],{},[31658,432365,97709],{"mathvariant":38079},[31658,432367,33206],{"mathvariant":38079},[31658,432369,31660],{},[31662,432371,456],{"stretchy":702},[31658,432373,51834],{},[31662,432375,2557],{"stretchy":702},[31665,432377,432378],{"encoding":31667},"G = \\mathrm{GL}_n(F)",[34,432380,432382,432400],{"className":432381,"ariaHidden":758},[31672],[34,432383,432385,432388,432391,432394,432397],{"className":432384},[31676],[34,432386],{"className":432387,"style":33218},[31680],[34,432389,97709],{"className":432390},[31685,31686],[34,432392],{"className":432393,"style":31846},[31845],[34,432395,202],{"className":432396},[31850],[34,432398],{"className":432399,"style":31846},[31845],[34,432401,432403,432406,432450,432453,432456],{"className":432402},[31676],[34,432404],{"className":432405,"style":31774},[31680],[34,432407,432409,432416],{"className":432408},[31685],[34,432410,432412],{"className":432411},[31685],[34,432413,432415],{"className":432414},[31685,207923],"GL",[34,432417,432419],{"className":432418},[31792],[34,432420,432422,432442],{"className":432421},[31796,31797],[34,432423,432425,432439],{"className":432424},[31801],[34,432426,432428],{"className":432427,"style":32362},[31805],[34,432429,432430,432433],{"style":56235},[34,432431],{"className":432432,"style":31814},[31813],[34,432434,432436],{"className":432435},[31818,31819,31820,31821],[34,432437,31660],{"className":432438},[31685,31686,31821],[34,432440,31829],{"className":432441},[31828],[34,432443,432445],{"className":432444},[31801],[34,432446,432448],{"className":432447,"style":31836},[31805],[34,432449],{},[34,432451,456],{"className":432452},[31782],[34,432454,51834],{"className":432455,"style":37422},[31685,31686],[34,432457,2557],{"className":432458},[31690]," (即 ",[34,432461,432463,432481],{"className":432462},[31642],[34,432464,432466],{"className":432465},[31646],[31648,432467,432468],{"xmlns":31650},[31652,432469,432470,432478],{},[31655,432471,432472,432474,432476],{},[31658,432473,31660],{},[31662,432475,42071],{},[31658,432477,31660],{},[31665,432479,432480],{"encoding":31667},"n \\times n",[34,432482,432484,432502],{"className":432483,"ariaHidden":758},[31672],[34,432485,432487,432490,432493,432496,432499],{"className":432486},[31676],[34,432488],{"className":432489,"style":32597},[31680],[34,432491,31660],{"className":432492},[31685,31686],[34,432494],{"className":432495,"style":32604},[31845],[34,432497,42071],{"className":432498},[32024],[34,432500],{"className":432501,"style":32604},[31845],[34,432503,432505,432508],{"className":432504},[31676],[34,432506],{"className":432507,"style":32527},[31680],[34,432509,31660],{"className":432510},[31685,31686]," 可逆矩阵构成的群)。具体写出 ",[34,432513,432515,432528],{"className":432514},[31642],[34,432516,432518],{"className":432517},[31646],[31648,432519,432520],{"xmlns":31650},[31652,432521,432522,432526],{},[31655,432523,432524],{},[31658,432525,97709],{},[31665,432527,97709],{"encoding":31667},[34,432529,432531],{"className":432530,"ariaHidden":758},[31672],[34,432532,432534,432537],{"className":432533},[31676],[34,432535],{"className":432536,"style":33218},[31680],[34,432538,97709],{"className":432539},[31685,31686]," 的一个 Sylow ",[34,432542,432544,432557],{"className":432543},[31642],[34,432545,432547],{"className":432546},[31646],[31648,432548,432549],{"xmlns":31650},[31652,432550,432551,432555],{},[31655,432552,432553],{},[31658,432554,15],{},[31665,432556,15],{"encoding":31667},[34,432558,432560],{"className":432559,"ariaHidden":758},[31672],[34,432561,432563,432566],{"className":432562},[31676],[34,432564],{"className":432565,"style":38774},[31680],[34,432567,15],{"className":432568},[31685,31686]," 子群。",[15,432571,432572],{},[2627,432573,432574],{},"深度解析与解答：",[15,432576,432577,432578,432606,432607,432635,432636,432664,432665,432724,432725,347],{},"要找到 Sylow ",[34,432579,432581,432594],{"className":432580},[31642],[34,432582,432584],{"className":432583},[31646],[31648,432585,432586],{"xmlns":31650},[31652,432587,432588,432592],{},[31655,432589,432590],{},[31658,432591,15],{},[31665,432593,15],{"encoding":31667},[34,432595,432597],{"className":432596,"ariaHidden":758},[31672],[34,432598,432600,432603],{"className":432599},[31676],[34,432601],{"className":432602,"style":38774},[31680],[34,432604,15],{"className":432605},[31685,31686],"-子群，我们需要先算出群 ",[34,432608,432610,432623],{"className":432609},[31642],[34,432611,432613],{"className":432612},[31646],[31648,432614,432615],{"xmlns":31650},[31652,432616,432617,432621],{},[31655,432618,432619],{},[31658,432620,97709],{},[31665,432622,97709],{"encoding":31667},[34,432624,432626],{"className":432625,"ariaHidden":758},[31672],[34,432627,432629,432632],{"className":432628},[31676],[34,432630],{"className":432631,"style":33218},[31680],[34,432633,97709],{"className":432634},[31685,31686]," 的总元素个数（阶），找出其中包含的 ",[34,432637,432639,432652],{"className":432638},[31642],[34,432640,432642],{"className":432641},[31646],[31648,432643,432644],{"xmlns":31650},[31652,432645,432646,432650],{},[31655,432647,432648],{},[31658,432649,15],{},[31665,432651,15],{"encoding":31667},[34,432653,432655],{"className":432654,"ariaHidden":758},[31672],[34,432656,432658,432661],{"className":432657},[31676],[34,432659],{"className":432660,"style":38774},[31680],[34,432662,15],{"className":432663},[31685,31686]," 的最高次幂 ",[34,432666,432668,432686],{"className":432667},[31642],[34,432669,432671],{"className":432670},[31646],[31648,432672,432673],{"xmlns":31650},[31652,432674,432675,432683],{},[31655,432676,432677],{},[37624,432678,432679,432681],{},[31658,432680,15],{},[31658,432682,31759],{},[31665,432684,432685],{"encoding":31667},"p^m",[34,432687,432689],{"className":432688,"ariaHidden":758},[31672],[34,432690,432692,432695],{"className":432691},[31676],[34,432693],{"className":432694,"style":316112},[31680],[34,432696,432698,432701],{"className":432697},[31685],[34,432699,15],{"className":432700},[31685,31686],[34,432702,432704],{"className":432703},[31792],[34,432705,432707],{"className":432706},[31796],[34,432708,432710],{"className":432709},[31801],[34,432711,432713],{"className":432712,"style":37959},[31805],[34,432714,432715,432718],{"style":37777},[34,432716],{"className":432717,"style":31814},[31813],[34,432719,432721],{"className":432720},[31818,31819,31820,31821],[34,432722,31759],{"className":432723},[31685,31686,31821],"，然后找到一个子群，其大小恰好是 ",[34,432726,432728,432745],{"className":432727},[31642],[34,432729,432731],{"className":432730},[31646],[31648,432732,432733],{"xmlns":31650},[31652,432734,432735,432743],{},[31655,432736,432737],{},[37624,432738,432739,432741],{},[31658,432740,15],{},[31658,432742,31759],{},[31665,432744,432685],{"encoding":31667},[34,432746,432748],{"className":432747,"ariaHidden":758},[31672],[34,432749,432751,432754],{"className":432750},[31676],[34,432752],{"className":432753,"style":316112},[31680],[34,432755,432757,432760],{"className":432756},[31685],[34,432758,15],{"className":432759},[31685,31686],[34,432761,432763],{"className":432762},[31792],[34,432764,432766],{"className":432765},[31796],[34,432767,432769],{"className":432768},[31801],[34,432770,432772],{"className":432771,"style":37959},[31805],[34,432773,432774,432777],{"style":37777},[34,432775],{"className":432776,"style":31814},[31813],[34,432778,432780],{"className":432779},[31818,31819,31820,31821],[34,432781,31759],{"className":432782},[31685,31686,31821],[32490,432784,432786],{"id":432785},"步骤-1计算-gglnfpg-mathrmgl_nmathbbf_pgglnfp-的阶",[2627,432787,432788,432789,432944],{},"步骤 1：计算 ",[34,432790,432792,432828],{"className":432791},[31642],[34,432793,432795],{"className":432794},[31646],[31648,432796,432797],{"xmlns":31650},[31652,432798,432799,432825],{},[31655,432800,432801,432803,432805,432815,432817,432823],{},[31658,432802,97709],{},[31662,432804,202],{},[31743,432806,432807,432813],{},[31655,432808,432809,432811],{},[31658,432810,97709],{"mathvariant":38079},[31658,432812,33206],{"mathvariant":38079},[31658,432814,31660],{},[31662,432816,456],{"stretchy":702},[31743,432818,432819,432821],{},[31658,432820,51834],{"mathvariant":318416},[31658,432822,15],{},[31662,432824,2557],{"stretchy":702},[31665,432826,432827],{"encoding":31667},"G = \\mathrm{GL}_n(\\mathbb{F}_p)",[34,432829,432831,432849],{"className":432830,"ariaHidden":758},[31672],[34,432832,432834,432837,432840,432843,432846],{"className":432833},[31676],[34,432835],{"className":432836,"style":33218},[31680],[34,432838,97709],{"className":432839},[31685,31686],[34,432841],{"className":432842,"style":31846},[31845],[34,432844,202],{"className":432845},[31850],[34,432847],{"className":432848,"style":31846},[31845],[34,432850,432852,432855,432898,432901,432941],{"className":432851},[31676],[34,432853],{"className":432854,"style":32058},[31680],[34,432856,432858,432864],{"className":432857},[31685],[34,432859,432861],{"className":432860},[31685],[34,432862,432415],{"className":432863},[31685,207923],[34,432865,432867],{"className":432866},[31792],[34,432868,432870,432890],{"className":432869},[31796,31797],[34,432871,432873,432887],{"className":432872},[31801],[34,432874,432876],{"className":432875,"style":32362},[31805],[34,432877,432878,432881],{"style":56235},[34,432879],{"className":432880,"style":31814},[31813],[34,432882,432884],{"className":432883},[31818,31819,31820,31821],[34,432885,31660],{"className":432886},[31685,31686,31821],[34,432888,31829],{"className":432889},[31828],[34,432891,432893],{"className":432892},[31801],[34,432894,432896],{"className":432895,"style":31836},[31805],[34,432897],{},[34,432899,456],{"className":432900},[31782],[34,432902,432904,432907],{"className":432903},[31685],[34,432905,51834],{"className":432906},[31685,318441],[34,432908,432910],{"className":432909},[31792],[34,432911,432913,432933],{"className":432912},[31796,31797],[34,432914,432916,432930],{"className":432915},[31801],[34,432917,432919],{"className":432918,"style":32362},[31805],[34,432920,432921,432924],{"style":31809},[34,432922],{"className":432923,"style":31814},[31813],[34,432925,432927],{"className":432926},[31818,31819,31820,31821],[34,432928,15],{"className":432929},[31685,31686,31821],[34,432931,31829],{"className":432932},[31828],[34,432934,432936],{"className":432935},[31801],[34,432937,432939],{"className":432938,"style":32163},[31805],[34,432940],{},[34,432942,2557],{"className":432943},[31690]," 的阶",[15,432946,432947,432948,432998],{},"一个 ",[34,432949,432951,432968],{"className":432950},[31642],[34,432952,432954],{"className":432953},[31646],[31648,432955,432956],{"xmlns":31650},[31652,432957,432958,432966],{},[31655,432959,432960,432962,432964],{},[31658,432961,31660],{},[31662,432963,42071],{},[31658,432965,31660],{},[31665,432967,432480],{"encoding":31667},[34,432969,432971,432989],{"className":432970,"ariaHidden":758},[31672],[34,432972,432974,432977,432980,432983,432986],{"className":432973},[31676],[34,432975],{"className":432976,"style":32597},[31680],[34,432978,31660],{"className":432979},[31685,31686],[34,432981],{"className":432982,"style":32604},[31845],[34,432984,42071],{"className":432985},[32024],[34,432987],{"className":432988,"style":32604},[31845],[34,432990,432992,432995],{"className":432991},[31676],[34,432993],{"className":432994,"style":32527},[31680],[34,432996,31660],{"className":432997},[31685,31686]," 矩阵是可逆的，当且仅当它的行向量是线性无关的。我们可以通过逐行构造的方式来计数：",[2230,433000,433001,433222,433343,433524,433527],{},[108,433002,433003,433006,433007,433075,433076],{},[2627,433004,433005],{},"第一行","：可以是除了零向量 ",[34,433008,433010,433036],{"className":433009},[31642],[34,433011,433013],{"className":433012},[31646],[31648,433014,433015],{"xmlns":31650},[31652,433016,433017,433033],{},[31655,433018,433019,433021,433023,433025,433027,433029,433031],{},[31662,433020,456],{"stretchy":702},[31749,433022,459],{},[31662,433024,31960],{"separator":758},[31662,433026,33934],{},[31662,433028,31960],{"separator":758},[31749,433030,459],{},[31662,433032,2557],{"stretchy":702},[31665,433034,433035],{"encoding":31667},"(0, \\dots, 0)",[34,433037,433039],{"className":433038,"ariaHidden":758},[31672],[34,433040,433042,433045,433048,433051,433054,433057,433060,433063,433066,433069,433072],{"className":433041},[31676],[34,433043],{"className":433044,"style":31774},[31680],[34,433046,456],{"className":433047},[31782],[34,433049,459],{"className":433050},[31685],[34,433052,31960],{"className":433053},[32119],[34,433055],{"className":433056,"style":32123},[31845],[34,433058,33934],{"className":433059},[34006],[34,433061],{"className":433062,"style":32123},[31845],[34,433064,31960],{"className":433065},[32119],[34,433067],{"className":433068,"style":32123},[31845],[34,433070,459],{"className":433071},[31685],[34,433073,2557],{"className":433074},[31690]," 以外的任意向量。\n",[105,433077,433078],{},[108,433079,433080,433081,433139,433140,433221],{},"总向量数是 ",[34,433082,433084,433101],{"className":433083},[31642],[34,433085,433087],{"className":433086},[31646],[31648,433088,433089],{"xmlns":31650},[31652,433090,433091,433099],{},[31655,433092,433093],{},[37624,433094,433095,433097],{},[31658,433096,15],{},[31658,433098,31660],{},[31665,433100,339442],{"encoding":31667},[34,433102,433104],{"className":433103,"ariaHidden":758},[31672],[34,433105,433107,433110],{"className":433106},[31676],[34,433108],{"className":433109,"style":316112},[31680],[34,433111,433113,433116],{"className":433112},[31685],[34,433114,15],{"className":433115},[31685,31686],[34,433117,433119],{"className":433118},[31792],[34,433120,433122],{"className":433121},[31796],[34,433123,433125],{"className":433124},[31801],[34,433126,433128],{"className":433127,"style":37959},[31805],[34,433129,433130,433133],{"style":37777},[34,433131],{"className":433132,"style":31814},[31813],[34,433134,433136],{"className":433135},[31818,31819,31820,31821],[34,433137,31660],{"className":433138},[31685,31686,31821],"，排除零向量，所以有 ",[34,433141,433143,433165],{"className":433142},[31642],[34,433144,433146],{"className":433145},[31646],[31648,433147,433148],{"xmlns":31650},[31652,433149,433150,433162],{},[31655,433151,433152,433158,433160],{},[37624,433153,433154,433156],{},[31658,433155,15],{},[31658,433157,31660],{},[31662,433159,32226],{},[31749,433161,640],{},[31665,433163,433164],{"encoding":31667},"p^n - 1",[34,433166,433168,433212],{"className":433167,"ariaHidden":758},[31672],[34,433169,433171,433174,433203,433206,433209],{"className":433170},[31676],[34,433172],{"className":433173,"style":316112},[31680],[34,433175,433177,433180],{"className":433176},[31685],[34,433178,15],{"className":433179},[31685,31686],[34,433181,433183],{"className":433182},[31792],[34,433184,433186],{"className":433185},[31796],[34,433187,433189],{"className":433188},[31801],[34,433190,433192],{"className":433191,"style":37959},[31805],[34,433193,433194,433197],{"style":37777},[34,433195],{"className":433196,"style":31814},[31813],[34,433198,433200],{"className":433199},[31818,31819,31820,31821],[34,433201,31660],{"className":433202},[31685,31686,31821],[34,433204],{"className":433205,"style":32604},[31845],[34,433207,32226],{"className":433208},[32024],[34,433210],{"className":433211,"style":32604},[31845],[34,433213,433215,433218],{"className":433214},[31676],[34,433216],{"className":433217,"style":32617},[31680],[34,433219,640],{"className":433220},[31685]," 种选择。",[108,433223,433224,433227,433228],{},[2627,433225,433226],{},"第二行","：可以是除了第一行的倍数以外的任意向量。\n",[105,433229,433230],{},[108,433231,433232,433233,433261,433262,433221],{},"第一行生成的子空间有 ",[34,433234,433236,433249],{"className":433235},[31642],[34,433237,433239],{"className":433238},[31646],[31648,433240,433241],{"xmlns":31650},[31652,433242,433243,433247],{},[31655,433244,433245],{},[31658,433246,15],{},[31665,433248,15],{"encoding":31667},[34,433250,433252],{"className":433251,"ariaHidden":758},[31672],[34,433253,433255,433258],{"className":433254},[31676],[34,433256],{"className":433257,"style":38774},[31680],[34,433259,15],{"className":433260},[31685,31686]," 个元素。所以有 ",[34,433263,433265,433287],{"className":433264},[31642],[34,433266,433268],{"className":433267},[31646],[31648,433269,433270],{"xmlns":31650},[31652,433271,433272,433284],{},[31655,433273,433274,433280,433282],{},[37624,433275,433276,433278],{},[31658,433277,15],{},[31658,433279,31660],{},[31662,433281,32226],{},[31658,433283,15],{},[31665,433285,433286],{"encoding":31667},"p^n - p",[34,433288,433290,433334],{"className":433289,"ariaHidden":758},[31672],[34,433291,433293,433296,433325,433328,433331],{"className":433292},[31676],[34,433294],{"className":433295,"style":316112},[31680],[34,433297,433299,433302],{"className":433298},[31685],[34,433300,15],{"className":433301},[31685,31686],[34,433303,433305],{"className":433304},[31792],[34,433306,433308],{"className":433307},[31796],[34,433309,433311],{"className":433310},[31801],[34,433312,433314],{"className":433313,"style":37959},[31805],[34,433315,433316,433319],{"style":37777},[34,433317],{"className":433318,"style":31814},[31813],[34,433320,433322],{"className":433321},[31818,31819,31820,31821],[34,433323,31660],{"className":433324},[31685,31686,31821],[34,433326],{"className":433327,"style":32604},[31845],[34,433329,32226],{"className":433330},[32024],[34,433332],{"className":433333,"style":32604},[31845],[34,433335,433337,433340],{"className":433336},[31676],[34,433338],{"className":433339,"style":38774},[31680],[34,433341,15],{"className":433342},[31685,31686],[108,433344,433345,433348,433349],{},[2627,433346,433347],{},"第三行","：不能在前两行生成的平面内。\n",[105,433350,433351],{},[108,433352,433353,433354,433261,433413,433221],{},"前两行生成的子空间有 ",[34,433355,433357,433375],{"className":433356},[31642],[34,433358,433360],{"className":433359},[31646],[31648,433361,433362],{"xmlns":31650},[31652,433363,433364,433372],{},[31655,433365,433366],{},[37624,433367,433368,433370],{},[31658,433369,15],{},[31749,433371,768],{},[31665,433373,433374],{"encoding":31667},"p^2",[34,433376,433378],{"className":433377,"ariaHidden":758},[31672],[34,433379,433381,433384],{"className":433380},[31676],[34,433382],{"className":433383,"style":315747},[31680],[34,433385,433387,433390],{"className":433386},[31685],[34,433388,15],{"className":433389},[31685,31686],[34,433391,433393],{"className":433392},[31792],[34,433394,433396],{"className":433395},[31796],[34,433397,433399],{"className":433398},[31801],[34,433400,433402],{"className":433401,"style":37774},[31805],[34,433403,433404,433407],{"style":37777},[34,433405],{"className":433406,"style":31814},[31813],[34,433408,433410],{"className":433409},[31818,31819,31820,31821],[34,433411,768],{"className":433412},[31685,31821],[34,433414,433416,433442],{"className":433415},[31642],[34,433417,433419],{"className":433418},[31646],[31648,433420,433421],{"xmlns":31650},[31652,433422,433423,433439],{},[31655,433424,433425,433431,433433],{},[37624,433426,433427,433429],{},[31658,433428,15],{},[31658,433430,31660],{},[31662,433432,32226],{},[37624,433434,433435,433437],{},[31658,433436,15],{},[31749,433438,768],{},[31665,433440,433441],{"encoding":31667},"p^n - p^2",[34,433443,433445,433489],{"className":433444,"ariaHidden":758},[31672],[34,433446,433448,433451,433480,433483,433486],{"className":433447},[31676],[34,433449],{"className":433450,"style":316112},[31680],[34,433452,433454,433457],{"className":433453},[31685],[34,433455,15],{"className":433456},[31685,31686],[34,433458,433460],{"className":433459},[31792],[34,433461,433463],{"className":433462},[31796],[34,433464,433466],{"className":433465},[31801],[34,433467,433469],{"className":433468,"style":37959},[31805],[34,433470,433471,433474],{"style":37777},[34,433472],{"className":433473,"style":31814},[31813],[34,433475,433477],{"className":433476},[31818,31819,31820,31821],[34,433478,31660],{"className":433479},[31685,31686,31821],[34,433481],{"className":433482,"style":32604},[31845],[34,433484,32226],{"className":433485},[32024],[34,433487],{"className":433488,"style":32604},[31845],[34,433490,433492,433495],{"className":433491},[31676],[34,433493],{"className":433494,"style":315747},[31680],[34,433496,433498,433501],{"className":433497},[31685],[34,433499,15],{"className":433500},[31685,31686],[34,433502,433504],{"className":433503},[31792],[34,433505,433507],{"className":433506},[31796],[34,433508,433510],{"className":433509},[31801],[34,433511,433513],{"className":433512,"style":37774},[31805],[34,433514,433515,433518],{"style":37777},[34,433516],{"className":433517,"style":31814},[31813],[34,433519,433521],{"className":433520},[31818,31819,31820,31821],[34,433522,768],{"className":433523},[31685,31821],[108,433525,433526],{},"...以此类推...",[108,433528,433529,433560,433561,433221],{},[2627,433530,178926,433531,433559],{},[34,433532,433534,433547],{"className":433533},[31642],[34,433535,433537],{"className":433536},[31646],[31648,433538,433539],{"xmlns":31650},[31652,433540,433541,433545],{},[31655,433542,433543],{},[31658,433544,33759],{},[31665,433546,33759],{"encoding":31667},[34,433548,433550],{"className":433549,"ariaHidden":758},[31672],[34,433551,433553,433556],{"className":433552},[31676],[34,433554],{"className":433555,"style":31681},[31680],[34,433557,33759],{"className":433558,"style":33774},[31685,31686]," 行","：有 ",[34,433562,433564,433596],{"className":433563},[31642],[34,433565,433567],{"className":433566},[31646],[31648,433568,433569],{"xmlns":31650},[31652,433570,433571,433593],{},[31655,433572,433573,433579,433581],{},[37624,433574,433575,433577],{},[31658,433576,15],{},[31658,433578,31660],{},[31662,433580,32226],{},[37624,433582,433583,433585],{},[31658,433584,15],{},[31655,433586,433587,433589,433591],{},[31658,433588,33759],{},[31662,433590,32226],{},[31749,433592,640],{},[31665,433594,433595],{"encoding":31667},"p^n - p^{k-1}",[34,433597,433599,433643],{"className":433598,"ariaHidden":758},[31672],[34,433600,433602,433605,433634,433637,433640],{"className":433601},[31676],[34,433603],{"className":433604,"style":316112},[31680],[34,433606,433608,433611],{"className":433607},[31685],[34,433609,15],{"className":433610},[31685,31686],[34,433612,433614],{"className":433613},[31792],[34,433615,433617],{"className":433616},[31796],[34,433618,433620],{"className":433619},[31801],[34,433621,433623],{"className":433622,"style":37959},[31805],[34,433624,433625,433628],{"style":37777},[34,433626],{"className":433627,"style":31814},[31813],[34,433629,433631],{"className":433630},[31818,31819,31820,31821],[34,433632,31660],{"className":433633},[31685,31686,31821],[34,433635],{"className":433636,"style":32604},[31845],[34,433638,32226],{"className":433639},[32024],[34,433641],{"className":433642,"style":32604},[31845],[34,433644,433646,433649],{"className":433645},[31676],[34,433647],{"className":433648,"style":315792},[31680],[34,433650,433652,433655],{"className":433651},[31685],[34,433653,15],{"className":433654},[31685,31686],[34,433656,433658],{"className":433657},[31792],[34,433659,433661],{"className":433660},[31796],[34,433662,433664],{"className":433663},[31801],[34,433665,433667],{"className":433666,"style":38486},[31805],[34,433668,433669,433672],{"style":37777},[34,433670],{"className":433671,"style":31814},[31813],[34,433673,433675],{"className":433674},[31818,31819,31820,31821],[34,433676,433678,433681,433684],{"className":433677},[31685,31821],[34,433679,33759],{"className":433680,"style":33774},[31685,31686,31821],[34,433682,32226],{"className":433683},[32024,31821],[34,433685,640],{"className":433686},[31685,31821],[15,433688,323847,433689,433717,433718],{},[34,433690,433692,433705],{"className":433691},[31642],[34,433693,433695],{"className":433694},[31646],[31648,433696,433697],{"xmlns":31650},[31652,433698,433699,433703],{},[31655,433700,433701],{},[31658,433702,97709],{},[31665,433704,97709],{"encoding":31667},[34,433706,433708],{"className":433707,"ariaHidden":758},[31672],[34,433709,433711,433714],{"className":433710},[31676],[34,433712],{"className":433713,"style":33218},[31680],[34,433715,97709],{"className":433716},[31685,31686]," 的阶为：\n",[34,433719,433721,433813],{"className":433720},[31642],[34,433722,433724],{"className":433723},[31646],[31648,433725,433726],{"xmlns":31650},[31652,433727,433728,433810],{},[31655,433729,433730,433732,433734,433736,433738,433740,433746,433748,433750,433752,433754,433760,433762,433764,433766,433768,433774,433776,433782,433784,433786,433788,433794,433796,433808],{},[31658,433731,162086],{"mathvariant":38079},[31658,433733,97709],{},[31658,433735,162086],{"mathvariant":38079},[31662,433737,202],{},[31662,433739,456],{"stretchy":702},[37624,433741,433742,433744],{},[31658,433743,15],{},[31658,433745,31660],{},[31662,433747,32226],{},[31749,433749,640],{},[31662,433751,2557],{"stretchy":702},[31662,433753,456],{"stretchy":702},[37624,433755,433756,433758],{},[31658,433757,15],{},[31658,433759,31660],{},[31662,433761,32226],{},[31658,433763,15],{},[31662,433765,2557],{"stretchy":702},[31662,433767,456],{"stretchy":702},[37624,433769,433770,433772],{},[31658,433771,15],{},[31658,433773,31660],{},[31662,433775,32226],{},[37624,433777,433778,433780],{},[31658,433779,15],{},[31749,433781,768],{},[31662,433783,2557],{"stretchy":702},[31662,433785,38874],{},[31662,433787,456],{"stretchy":702},[37624,433789,433790,433792],{},[31658,433791,15],{},[31658,433793,31660],{},[31662,433795,32226],{},[37624,433797,433798,433800],{},[31658,433799,15],{},[31655,433801,433802,433804,433806],{},[31658,433803,31660],{},[31662,433805,32226],{},[31749,433807,640],{},[31662,433809,2557],{"stretchy":702},[31665,433811,433812],{"encoding":31667},"|G| = (p^n - 1)(p^n - p)(p^n - p^2) \\cdots (p^n - p^{n-1})",[34,433814,433816,433840,433887,433940,433993,434081],{"className":433815,"ariaHidden":758},[31672],[34,433817,433819,433822,433825,433828,433831,433834,433837],{"className":433818},[31676],[34,433820],{"className":433821,"style":31774},[31680],[34,433823,162086],{"className":433824},[31685],[34,433826,97709],{"className":433827},[31685,31686],[34,433829,162086],{"className":433830},[31685],[34,433832],{"className":433833,"style":31846},[31845],[34,433835,202],{"className":433836},[31850],[34,433838],{"className":433839,"style":31846},[31845],[34,433841,433843,433846,433849,433878,433881,433884],{"className":433842},[31676],[34,433844],{"className":433845,"style":31774},[31680],[34,433847,456],{"className":433848},[31782],[34,433850,433852,433855],{"className":433851},[31685],[34,433853,15],{"className":433854},[31685,31686],[34,433856,433858],{"className":433857},[31792],[34,433859,433861],{"className":433860},[31796],[34,433862,433864],{"className":433863},[31801],[34,433865,433867],{"className":433866,"style":37959},[31805],[34,433868,433869,433872],{"style":37777},[34,433870],{"className":433871,"style":31814},[31813],[34,433873,433875],{"className":433874},[31818,31819,31820,31821],[34,433876,31660],{"className":433877},[31685,31686,31821],[34,433879],{"className":433880,"style":32604},[31845],[34,433882,32226],{"className":433883},[32024],[34,433885],{"className":433886,"style":32604},[31845],[34,433888,433890,433893,433896,433899,433902,433931,433934,433937],{"className":433889},[31676],[34,433891],{"className":433892,"style":31774},[31680],[34,433894,640],{"className":433895},[31685],[34,433897,2557],{"className":433898},[31690],[34,433900,456],{"className":433901},[31782],[34,433903,433905,433908],{"className":433904},[31685],[34,433906,15],{"className":433907},[31685,31686],[34,433909,433911],{"className":433910},[31792],[34,433912,433914],{"className":433913},[31796],[34,433915,433917],{"className":433916},[31801],[34,433918,433920],{"className":433919,"style":37959},[31805],[34,433921,433922,433925],{"style":37777},[34,433923],{"className":433924,"style":31814},[31813],[34,433926,433928],{"className":433927},[31818,31819,31820,31821],[34,433929,31660],{"className":433930},[31685,31686,31821],[34,433932],{"className":433933,"style":32604},[31845],[34,433935,32226],{"className":433936},[32024],[34,433938],{"className":433939,"style":32604},[31845],[34,433941,433943,433946,433949,433952,433955,433984,433987,433990],{"className":433942},[31676],[34,433944],{"className":433945,"style":31774},[31680],[34,433947,15],{"className":433948},[31685,31686],[34,433950,2557],{"className":433951},[31690],[34,433953,456],{"className":433954},[31782],[34,433956,433958,433961],{"className":433957},[31685],[34,433959,15],{"className":433960},[31685,31686],[34,433962,433964],{"className":433963},[31792],[34,433965,433967],{"className":433966},[31796],[34,433968,433970],{"className":433969},[31801],[34,433971,433973],{"className":433972,"style":37959},[31805],[34,433974,433975,433978],{"style":37777},[34,433976],{"className":433977,"style":31814},[31813],[34,433979,433981],{"className":433980},[31818,31819,31820,31821],[34,433982,31660],{"className":433983},[31685,31686,31821],[34,433985],{"className":433986,"style":32604},[31845],[34,433988,32226],{"className":433989},[32024],[34,433991],{"className":433992,"style":32604},[31845],[34,433994,433996,433999,434028,434031,434034,434037,434040,434043,434072,434075,434078],{"className":433995},[31676],[34,433997],{"className":433998,"style":37755},[31680],[34,434000,434002,434005],{"className":434001},[31685],[34,434003,15],{"className":434004},[31685,31686],[34,434006,434008],{"className":434007},[31792],[34,434009,434011],{"className":434010},[31796],[34,434012,434014],{"className":434013},[31801],[34,434015,434017],{"className":434016,"style":37774},[31805],[34,434018,434019,434022],{"style":37777},[34,434020],{"className":434021,"style":31814},[31813],[34,434023,434025],{"className":434024},[31818,31819,31820,31821],[34,434026,768],{"className":434027},[31685,31821],[34,434029,2557],{"className":434030},[31690],[34,434032],{"className":434033,"style":32123},[31845],[34,434035,38874],{"className":434036},[34006],[34,434038],{"className":434039,"style":32123},[31845],[34,434041,456],{"className":434042},[31782],[34,434044,434046,434049],{"className":434045},[31685],[34,434047,15],{"className":434048},[31685,31686],[34,434050,434052],{"className":434051},[31792],[34,434053,434055],{"className":434054},[31796],[34,434056,434058],{"className":434057},[31801],[34,434059,434061],{"className":434060,"style":37959},[31805],[34,434062,434063,434066],{"style":37777},[34,434064],{"className":434065,"style":31814},[31813],[34,434067,434069],{"className":434068},[31818,31819,31820,31821],[34,434070,31660],{"className":434071},[31685,31686,31821],[34,434073],{"className":434074,"style":32604},[31845],[34,434076,32226],{"className":434077},[32024],[34,434079],{"className":434080,"style":32604},[31845],[34,434082,434084,434087,434125],{"className":434083},[31676],[34,434085],{"className":434086,"style":37755},[31680],[34,434088,434090,434093],{"className":434089},[31685],[34,434091,15],{"className":434092},[31685,31686],[34,434094,434096],{"className":434095},[31792],[34,434097,434099],{"className":434098},[31796],[34,434100,434102],{"className":434101},[31801],[34,434103,434105],{"className":434104,"style":37774},[31805],[34,434106,434107,434110],{"style":37777},[34,434108],{"className":434109,"style":31814},[31813],[34,434111,434113],{"className":434112},[31818,31819,31820,31821],[34,434114,434116,434119,434122],{"className":434115},[31685,31821],[34,434117,31660],{"className":434118},[31685,31686,31821],[34,434120,32226],{"className":434121},[32024,31821],[34,434123,640],{"className":434124},[31685,31821],[34,434126,2557],{"className":434127},[31690],[32490,434129,434131],{"id":434130},"步骤-2确定-ggg-中-ppp-的最高次幂",[2627,434132,434133,434134,93436,434173,434201],{},"步骤 2：确定 ",[34,434135,434137,434155],{"className":434136},[31642],[34,434138,434140],{"className":434139},[31646],[31648,434141,434142],{"xmlns":31650},[31652,434143,434144,434152],{},[31655,434145,434146,434148,434150],{},[31658,434147,162086],{"mathvariant":38079},[31658,434149,97709],{},[31658,434151,162086],{"mathvariant":38079},[31665,434153,434154],{"encoding":31667},"|G|",[34,434156,434158],{"className":434157,"ariaHidden":758},[31672],[34,434159,434161,434164,434167,434170],{"className":434160},[31676],[34,434162],{"className":434163,"style":31774},[31680],[34,434165,162086],{"className":434166},[31685],[34,434168,97709],{"className":434169},[31685,31686],[34,434171,162086],{"className":434172},[31685],[34,434174,434176,434189],{"className":434175},[31642],[34,434177,434179],{"className":434178},[31646],[31648,434180,434181],{"xmlns":31650},[31652,434182,434183,434187],{},[31655,434184,434185],{},[31658,434186,15],{},[31665,434188,15],{"encoding":31667},[34,434190,434192],{"className":434191,"ariaHidden":758},[31672],[34,434193,434195,434198],{"className":434194},[31676],[34,434196],{"className":434197,"style":38774},[31680],[34,434199,15],{"className":434200},[31685,31686]," 的最高次幂",[15,434203,434204,434205,434233,434234,2664],{},"我们需要从上面的乘积中提取出所有的因子 ",[34,434206,434208,434221],{"className":434207},[31642],[34,434209,434211],{"className":434210},[31646],[31648,434212,434213],{"xmlns":31650},[31652,434214,434215,434219],{},[31655,434216,434217],{},[31658,434218,15],{},[31665,434220,15],{"encoding":31667},[34,434222,434224],{"className":434223,"ariaHidden":758},[31672],[34,434225,434227,434230],{"className":434226},[31676],[34,434228],{"className":434229,"style":38774},[31680],[34,434231,15],{"className":434232},[31685,31686],"。\n我们将每一项提取公因子 ",[34,434235,434237,434250],{"className":434236},[31642],[34,434238,434240],{"className":434239},[31646],[31648,434241,434242],{"xmlns":31650},[31652,434243,434244,434248],{},[31655,434245,434246],{},[31658,434247,15],{},[31665,434249,15],{"encoding":31667},[34,434251,434253],{"className":434252,"ariaHidden":758},[31672],[34,434254,434256,434259],{"className":434255},[31676],[34,434257],{"className":434258,"style":38774},[31680],[34,434260,15],{"className":434261},[31685,31686],[105,434263,434264,434387,434616,434904,434906],{},[108,434265,434266,434267,434358,434359,347],{},"第 1 项：",[34,434268,434270,434296],{"className":434269},[31642],[34,434271,434273],{"className":434272},[31646],[31648,434274,434275],{"xmlns":31650},[31652,434276,434277,434293],{},[31655,434278,434279,434281,434287,434289,434291],{},[31662,434280,456],{"stretchy":702},[37624,434282,434283,434285],{},[31658,434284,15],{},[31658,434286,31660],{},[31662,434288,32226],{},[31749,434290,640],{},[31662,434292,2557],{"stretchy":702},[31665,434294,434295],{"encoding":31667},"(p^n - 1)",[34,434297,434299,434346],{"className":434298,"ariaHidden":758},[31672],[34,434300,434302,434305,434308,434337,434340,434343],{"className":434301},[31676],[34,434303],{"className":434304,"style":31774},[31680],[34,434306,456],{"className":434307},[31782],[34,434309,434311,434314],{"className":434310},[31685],[34,434312,15],{"className":434313},[31685,31686],[34,434315,434317],{"className":434316},[31792],[34,434318,434320],{"className":434319},[31796],[34,434321,434323],{"className":434322},[31801],[34,434324,434326],{"className":434325,"style":37959},[31805],[34,434327,434328,434331],{"style":37777},[34,434329],{"className":434330,"style":31814},[31813],[34,434332,434334],{"className":434333},[31818,31819,31820,31821],[34,434335,31660],{"className":434336},[31685,31686,31821],[34,434338],{"className":434339,"style":32604},[31845],[34,434341,32226],{"className":434342},[32024],[34,434344],{"className":434345,"style":32604},[31845],[34,434347,434349,434352,434355],{"className":434348},[31676],[34,434350],{"className":434351,"style":31774},[31680],[34,434353,640],{"className":434354},[31685],[34,434356,2557],{"className":434357},[31690]," —— 没有因子 ",[34,434360,434362,434375],{"className":434361},[31642],[34,434363,434365],{"className":434364},[31646],[31648,434366,434367],{"xmlns":31650},[31652,434368,434369,434373],{},[31655,434370,434371],{},[31658,434372,15],{},[31665,434374,15],{"encoding":31667},[34,434376,434378],{"className":434377,"ariaHidden":758},[31672],[34,434379,434381,434384],{"className":434380},[31676],[34,434382],{"className":434383,"style":38774},[31680],[34,434385,15],{"className":434386},[31685,31686],[108,434388,434389,434390,434585,434586,344836,434588,347],{},"第 2 项：",[34,434391,434393,434443],{"className":434392},[31642],[34,434394,434396],{"className":434395},[31646],[31648,434397,434398],{"xmlns":31650},[31652,434399,434400,434440],{},[31655,434401,434402,434404,434410,434412,434414,434416,434418,434420,434422,434434,434436,434438],{},[31662,434403,456],{"stretchy":702},[37624,434405,434406,434408],{},[31658,434407,15],{},[31658,434409,31660],{},[31662,434411,32226],{},[31658,434413,15],{},[31662,434415,2557],{"stretchy":702},[31662,434417,202],{},[31658,434419,15],{},[31662,434421,456],{"stretchy":702},[37624,434423,434424,434426],{},[31658,434425,15],{},[31655,434427,434428,434430,434432],{},[31658,434429,31660],{},[31662,434431,32226],{},[31749,434433,640],{},[31662,434435,32226],{},[31749,434437,640],{},[31662,434439,2557],{"stretchy":702},[31665,434441,434442],{"encoding":31667},"(p^n - p) = p(p^{n-1} - 1)",[34,434444,434446,434493,434514,434573],{"className":434445,"ariaHidden":758},[31672],[34,434447,434449,434452,434455,434484,434487,434490],{"className":434448},[31676],[34,434450],{"className":434451,"style":31774},[31680],[34,434453,456],{"className":434454},[31782],[34,434456,434458,434461],{"className":434457},[31685],[34,434459,15],{"className":434460},[31685,31686],[34,434462,434464],{"className":434463},[31792],[34,434465,434467],{"className":434466},[31796],[34,434468,434470],{"className":434469},[31801],[34,434471,434473],{"className":434472,"style":37959},[31805],[34,434474,434475,434478],{"style":37777},[34,434476],{"className":434477,"style":31814},[31813],[34,434479,434481],{"className":434480},[31818,31819,31820,31821],[34,434482,31660],{"className":434483},[31685,31686,31821],[34,434485],{"className":434486,"style":32604},[31845],[34,434488,32226],{"className":434489},[32024],[34,434491],{"className":434492,"style":32604},[31845],[34,434494,434496,434499,434502,434505,434508,434511],{"className":434495},[31676],[34,434497],{"className":434498,"style":31774},[31680],[34,434500,15],{"className":434501},[31685,31686],[34,434503,2557],{"className":434504},[31690],[34,434506],{"className":434507,"style":31846},[31845],[34,434509,202],{"className":434510},[31850],[34,434512],{"className":434513,"style":31846},[31845],[34,434515,434517,434520,434523,434526,434564,434567,434570],{"className":434516},[31676],[34,434518],{"className":434519,"style":37755},[31680],[34,434521,15],{"className":434522},[31685,31686],[34,434524,456],{"className":434525},[31782],[34,434527,434529,434532],{"className":434528},[31685],[34,434530,15],{"className":434531},[31685,31686],[34,434533,434535],{"className":434534},[31792],[34,434536,434538],{"className":434537},[31796],[34,434539,434541],{"className":434540},[31801],[34,434542,434544],{"className":434543,"style":37774},[31805],[34,434545,434546,434549],{"style":37777},[34,434547],{"className":434548,"style":31814},[31813],[34,434550,434552],{"className":434551},[31818,31819,31820,31821],[34,434553,434555,434558,434561],{"className":434554},[31685,31821],[34,434556,31660],{"className":434557},[31685,31686,31821],[34,434559,32226],{"className":434560},[32024,31821],[34,434562,640],{"className":434563},[31685,31821],[34,434565],{"className":434566,"style":32604},[31845],[34,434568,32226],{"className":434569},[32024],[34,434571],{"className":434572,"style":32604},[31845],[34,434574,434576,434579,434582],{"className":434575},[31676],[34,434577],{"className":434578,"style":31774},[31680],[34,434580,640],{"className":434581},[31685],[34,434583,2557],{"className":434584},[31690]," —— 有 ",[2627,434587,640],{},[34,434589,434591,434604],{"className":434590},[31642],[34,434592,434594],{"className":434593},[31646],[31648,434595,434596],{"xmlns":31650},[31652,434597,434598,434602],{},[31655,434599,434600],{},[31658,434601,15],{},[31665,434603,15],{"encoding":31667},[34,434605,434607],{"className":434606,"ariaHidden":758},[31672],[34,434608,434610,434613],{"className":434609},[31676],[34,434611],{"className":434612,"style":38774},[31680],[34,434614,15],{"className":434615},[31685,31686],[108,434617,434618,434619,434585,434874,344836,434876,347],{},"第 3 项：",[34,434620,434622,434680],{"className":434621},[31642],[34,434623,434625],{"className":434624},[31646],[31648,434626,434627],{"xmlns":31650},[31652,434628,434629,434677],{},[31655,434630,434631,434633,434639,434641,434647,434649,434651,434657,434659,434671,434673,434675],{},[31662,434632,456],{"stretchy":702},[37624,434634,434635,434637],{},[31658,434636,15],{},[31658,434638,31660],{},[31662,434640,32226],{},[37624,434642,434643,434645],{},[31658,434644,15],{},[31749,434646,768],{},[31662,434648,2557],{"stretchy":702},[31662,434650,202],{},[37624,434652,434653,434655],{},[31658,434654,15],{},[31749,434656,768],{},[31662,434658,456],{"stretchy":702},[37624,434660,434661,434663],{},[31658,434662,15],{},[31655,434664,434665,434667,434669],{},[31658,434666,31660],{},[31662,434668,32226],{},[31749,434670,768],{},[31662,434672,32226],{},[31749,434674,640],{},[31662,434676,2557],{"stretchy":702},[31665,434678,434679],{"encoding":31667},"(p^n - p^2) = p^2(p^{n-2} - 1)",[34,434681,434683,434730,434777,434862],{"className":434682,"ariaHidden":758},[31672],[34,434684,434686,434689,434692,434721,434724,434727],{"className":434685},[31676],[34,434687],{"className":434688,"style":31774},[31680],[34,434690,456],{"className":434691},[31782],[34,434693,434695,434698],{"className":434694},[31685],[34,434696,15],{"className":434697},[31685,31686],[34,434699,434701],{"className":434700},[31792],[34,434702,434704],{"className":434703},[31796],[34,434705,434707],{"className":434706},[31801],[34,434708,434710],{"className":434709,"style":37959},[31805],[34,434711,434712,434715],{"style":37777},[34,434713],{"className":434714,"style":31814},[31813],[34,434716,434718],{"className":434717},[31818,31819,31820,31821],[34,434719,31660],{"className":434720},[31685,31686,31821],[34,434722],{"className":434723,"style":32604},[31845],[34,434725,32226],{"className":434726},[32024],[34,434728],{"className":434729,"style":32604},[31845],[34,434731,434733,434736,434765,434768,434771,434774],{"className":434732},[31676],[34,434734],{"className":434735,"style":37755},[31680],[34,434737,434739,434742],{"className":434738},[31685],[34,434740,15],{"className":434741},[31685,31686],[34,434743,434745],{"className":434744},[31792],[34,434746,434748],{"className":434747},[31796],[34,434749,434751],{"className":434750},[31801],[34,434752,434754],{"className":434753,"style":37774},[31805],[34,434755,434756,434759],{"style":37777},[34,434757],{"className":434758,"style":31814},[31813],[34,434760,434762],{"className":434761},[31818,31819,31820,31821],[34,434763,768],{"className":434764},[31685,31821],[34,434766,2557],{"className":434767},[31690],[34,434769],{"className":434770,"style":31846},[31845],[34,434772,202],{"className":434773},[31850],[34,434775],{"className":434776,"style":31846},[31845],[34,434778,434780,434783,434812,434815,434853,434856,434859],{"className":434779},[31676],[34,434781],{"className":434782,"style":37755},[31680],[34,434784,434786,434789],{"className":434785},[31685],[34,434787,15],{"className":434788},[31685,31686],[34,434790,434792],{"className":434791},[31792],[34,434793,434795],{"className":434794},[31796],[34,434796,434798],{"className":434797},[31801],[34,434799,434801],{"className":434800,"style":37774},[31805],[34,434802,434803,434806],{"style":37777},[34,434804],{"className":434805,"style":31814},[31813],[34,434807,434809],{"className":434808},[31818,31819,31820,31821],[34,434810,768],{"className":434811},[31685,31821],[34,434813,456],{"className":434814},[31782],[34,434816,434818,434821],{"className":434817},[31685],[34,434819,15],{"className":434820},[31685,31686],[34,434822,434824],{"className":434823},[31792],[34,434825,434827],{"className":434826},[31796],[34,434828,434830],{"className":434829},[31801],[34,434831,434833],{"className":434832,"style":37774},[31805],[34,434834,434835,434838],{"style":37777},[34,434836],{"className":434837,"style":31814},[31813],[34,434839,434841],{"className":434840},[31818,31819,31820,31821],[34,434842,434844,434847,434850],{"className":434843},[31685,31821],[34,434845,31660],{"className":434846},[31685,31686,31821],[34,434848,32226],{"className":434849},[32024,31821],[34,434851,768],{"className":434852},[31685,31821],[34,434854],{"className":434855,"style":32604},[31845],[34,434857,32226],{"className":434858},[32024],[34,434860],{"className":434861,"style":32604},[31845],[34,434863,434865,434868,434871],{"className":434864},[31676],[34,434866],{"className":434867,"style":31774},[31680],[34,434869,640],{"className":434870},[31685],[34,434872,2557],{"className":434873},[31690],[2627,434875,768],{},[34,434877,434879,434892],{"className":434878},[31642],[34,434880,434882],{"className":434881},[31646],[31648,434883,434884],{"xmlns":31650},[31652,434885,434886,434890],{},[31655,434887,434888],{},[31658,434889,15],{},[31665,434891,15],{"encoding":31667},[34,434893,434895],{"className":434894,"ariaHidden":758},[31672],[34,434896,434898,434901],{"className":434897},[31676],[34,434899],{"className":434900,"style":38774},[31680],[34,434902,15],{"className":434903},[31685,31686],[108,434905,1167],{},[108,434907,178926,434908,423681,434936,434585,435176,344836,435228,347],{},[34,434909,434911,434924],{"className":434910},[31642],[34,434912,434914],{"className":434913},[31646],[31648,434915,434916],{"xmlns":31650},[31652,434917,434918,434922],{},[31655,434919,434920],{},[31658,434921,31660],{},[31665,434923,31660],{"encoding":31667},[34,434925,434927],{"className":434926,"ariaHidden":758},[31672],[34,434928,434930,434933],{"className":434929},[31676],[34,434931],{"className":434932,"style":32527},[31680],[34,434934,31660],{"className":434935},[31685,31686],[34,434937,434939,434999],{"className":434938},[31642],[34,434940,434942],{"className":434941},[31646],[31648,434943,434944],{"xmlns":31650},[31652,434945,434946,434996],{},[31655,434947,434948,434950,434956,434958,434970,434972,434974,434986,434988,434990,434992,434994],{},[31662,434949,456],{"stretchy":702},[37624,434951,434952,434954],{},[31658,434953,15],{},[31658,434955,31660],{},[31662,434957,32226],{},[37624,434959,434960,434962],{},[31658,434961,15],{},[31655,434963,434964,434966,434968],{},[31658,434965,31660],{},[31662,434967,32226],{},[31749,434969,640],{},[31662,434971,2557],{"stretchy":702},[31662,434973,202],{},[37624,434975,434976,434978],{},[31658,434977,15],{},[31655,434979,434980,434982,434984],{},[31658,434981,31660],{},[31662,434983,32226],{},[31749,434985,640],{},[31662,434987,456],{"stretchy":702},[31658,434989,15],{},[31662,434991,32226],{},[31749,434993,640],{},[31662,434995,2557],{"stretchy":702},[31665,434997,434998],{"encoding":31667},"(p^n - p^{n-1}) = p^{n-1}(p - 1)",[34,435000,435002,435049,435105,435164],{"className":435001,"ariaHidden":758},[31672],[34,435003,435005,435008,435011,435040,435043,435046],{"className":435004},[31676],[34,435006],{"className":435007,"style":31774},[31680],[34,435009,456],{"className":435010},[31782],[34,435012,435014,435017],{"className":435013},[31685],[34,435015,15],{"className":435016},[31685,31686],[34,435018,435020],{"className":435019},[31792],[34,435021,435023],{"className":435022},[31796],[34,435024,435026],{"className":435025},[31801],[34,435027,435029],{"className":435028,"style":37959},[31805],[34,435030,435031,435034],{"style":37777},[34,435032],{"className":435033,"style":31814},[31813],[34,435035,435037],{"className":435036},[31818,31819,31820,31821],[34,435038,31660],{"className":435039},[31685,31686,31821],[34,435041],{"className":435042,"style":32604},[31845],[34,435044,32226],{"className":435045},[32024],[34,435047],{"className":435048,"style":32604},[31845],[34,435050,435052,435055,435093,435096,435099,435102],{"className":435051},[31676],[34,435053],{"className":435054,"style":37755},[31680],[34,435056,435058,435061],{"className":435057},[31685],[34,435059,15],{"className":435060},[31685,31686],[34,435062,435064],{"className":435063},[31792],[34,435065,435067],{"className":435066},[31796],[34,435068,435070],{"className":435069},[31801],[34,435071,435073],{"className":435072,"style":37774},[31805],[34,435074,435075,435078],{"style":37777},[34,435076],{"className":435077,"style":31814},[31813],[34,435079,435081],{"className":435080},[31818,31819,31820,31821],[34,435082,435084,435087,435090],{"className":435083},[31685,31821],[34,435085,31660],{"className":435086},[31685,31686,31821],[34,435088,32226],{"className":435089},[32024,31821],[34,435091,640],{"className":435092},[31685,31821],[34,435094,2557],{"className":435095},[31690],[34,435097],{"className":435098,"style":31846},[31845],[34,435100,202],{"className":435101},[31850],[34,435103],{"className":435104,"style":31846},[31845],[34,435106,435108,435111,435149,435152,435155,435158,435161],{"className":435107},[31676],[34,435109],{"className":435110,"style":37755},[31680],[34,435112,435114,435117],{"className":435113},[31685],[34,435115,15],{"className":435116},[31685,31686],[34,435118,435120],{"className":435119},[31792],[34,435121,435123],{"className":435122},[31796],[34,435124,435126],{"className":435125},[31801],[34,435127,435129],{"className":435128,"style":37774},[31805],[34,435130,435131,435134],{"style":37777},[34,435132],{"className":435133,"style":31814},[31813],[34,435135,435137],{"className":435136},[31818,31819,31820,31821],[34,435138,435140,435143,435146],{"className":435139},[31685,31821],[34,435141,31660],{"className":435142},[31685,31686,31821],[34,435144,32226],{"className":435145},[32024,31821],[34,435147,640],{"className":435148},[31685,31821],[34,435150,456],{"className":435151},[31782],[34,435153,15],{"className":435154},[31685,31686],[34,435156],{"className":435157,"style":32604},[31845],[34,435159,32226],{"className":435160},[32024],[34,435162],{"className":435163,"style":32604},[31845],[34,435165,435167,435170,435173],{"className":435166},[31676],[34,435168],{"className":435169,"style":31774},[31680],[34,435171,640],{"className":435172},[31685],[34,435174,2557],{"className":435175},[31690],[2627,435177,435178],{},[34,435179,435181,435198],{"className":435180},[31642],[34,435182,435184],{"className":435183},[31646],[31648,435185,435186],{"xmlns":31650},[31652,435187,435188,435196],{},[31655,435189,435190,435192,435194],{},[31658,435191,31660],{},[31662,435193,32226],{},[31749,435195,640],{},[31665,435197,32587],{"encoding":31667},[34,435199,435201,435219],{"className":435200,"ariaHidden":758},[31672],[34,435202,435204,435207,435210,435213,435216],{"className":435203},[31676],[34,435205],{"className":435206,"style":32597},[31680],[34,435208,31660],{"className":435209},[31685,31686],[34,435211],{"className":435212,"style":32604},[31845],[34,435214,32226],{"className":435215},[32024],[34,435217],{"className":435218,"style":32604},[31845],[34,435220,435222,435225],{"className":435221},[31676],[34,435223],{"className":435224,"style":32617},[31680],[34,435226,640],{"className":435227},[31685],[34,435229,435231,435244],{"className":435230},[31642],[34,435232,435234],{"className":435233},[31646],[31648,435235,435236],{"xmlns":31650},[31652,435237,435238,435242],{},[31655,435239,435240],{},[31658,435241,15],{},[31665,435243,15],{"encoding":31667},[34,435245,435247],{"className":435246,"ariaHidden":758},[31672],[34,435248,435250,435253],{"className":435249},[31676],[34,435251],{"className":435252,"style":38774},[31680],[34,435254,15],{"className":435255},[31685,31686],[15,435257,435258,435259,435287,435288],{},"将这些 ",[34,435260,435262,435275],{"className":435261},[31642],[34,435263,435265],{"className":435264},[31646],[31648,435266,435267],{"xmlns":31650},[31652,435268,435269,435273],{},[31655,435270,435271],{},[31658,435272,15],{},[31665,435274,15],{"encoding":31667},[34,435276,435278],{"className":435277,"ariaHidden":758},[31672],[34,435279,435281,435284],{"className":435280},[31676],[34,435282],{"className":435283,"style":38774},[31680],[34,435285,15],{"className":435286},[31685,31686]," 的幂次加起来：\n",[34,435289,435291,435354],{"className":435290},[31642],[34,435292,435294],{"className":435293},[31646],[31648,435295,435296],{"xmlns":31650},[31652,435297,435298,435351],{},[31655,435299,435300,435303,435305,435307,435309,435311,435313,435315,435317,435319,435321,435323,435325,435327,435329,435331,435333],{},[39361,435301,435302],{},"总幂次",[31662,435304,202],{},[31749,435306,459],{},[31662,435308,223],{},[31749,435310,640],{},[31662,435312,223],{},[31749,435314,768],{},[31662,435316,223],{},[31662,435318,38874],{},[31662,435320,223],{},[31662,435322,456],{"stretchy":702},[31658,435324,31660],{},[31662,435326,32226],{},[31749,435328,640],{},[31662,435330,2557],{"stretchy":702},[31662,435332,202],{},[41453,435334,435335,435349],{},[31655,435336,435337,435339,435341,435343,435345,435347],{},[31658,435338,31660],{},[31662,435340,456],{"stretchy":702},[31658,435342,31660],{},[31662,435344,32226],{},[31749,435346,640],{},[31662,435348,2557],{"stretchy":702},[31749,435350,768],{},[31665,435352,435353],{"encoding":31667},"\\text{总幂次} = 0 + 1 + 2 + \\cdots + (n-1) = \\frac{n(n-1)}{2}",[34,435355,435357,435378,435396,435414,435432,435450,435471,435492],{"className":435356,"ariaHidden":758},[31672],[34,435358,435360,435363,435369,435372,435375],{"className":435359},[31676],[34,435361],{"className":435362,"style":33218},[31680],[34,435364,435366],{"className":435365},[31685,2638],[34,435367,435302],{"className":435368},[31685,105462],[34,435370],{"className":435371,"style":31846},[31845],[34,435373,202],{"className":435374},[31850],[34,435376],{"className":435377,"style":31846},[31845],[34,435379,435381,435384,435387,435390,435393],{"className":435380},[31676],[34,435382],{"className":435383,"style":35942},[31680],[34,435385,459],{"className":435386},[31685],[34,435388],{"className":435389,"style":32604},[31845],[34,435391,223],{"className":435392},[32024],[34,435394],{"className":435395,"style":32604},[31845],[34,435397,435399,435402,435405,435408,435411],{"className":435398},[31676],[34,435400],{"className":435401,"style":35942},[31680],[34,435403,640],{"className":435404},[31685],[34,435406],{"className":435407,"style":32604},[31845],[34,435409,223],{"className":435410},[32024],[34,435412],{"className":435413,"style":32604},[31845],[34,435415,435417,435420,435423,435426,435429],{"className":435416},[31676],[34,435418],{"className":435419,"style":35942},[31680],[34,435421,768],{"className":435422},[31685],[34,435424],{"className":435425,"style":32604},[31845],[34,435427,223],{"className":435428},[32024],[34,435430],{"className":435431,"style":32604},[31845],[34,435433,435435,435438,435441,435444,435447],{"className":435434},[31676],[34,435436],{"className":435437,"style":32597},[31680],[34,435439,38874],{"className":435440},[34006],[34,435442],{"className":435443,"style":32604},[31845],[34,435445,223],{"className":435446},[32024],[34,435448],{"className":435449,"style":32604},[31845],[34,435451,435453,435456,435459,435462,435465,435468],{"className":435452},[31676],[34,435454],{"className":435455,"style":31774},[31680],[34,435457,456],{"className":435458},[31782],[34,435460,31660],{"className":435461},[31685,31686],[34,435463],{"className":435464,"style":32604},[31845],[34,435466,32226],{"className":435467},[32024],[34,435469],{"className":435470,"style":32604},[31845],[34,435472,435474,435477,435480,435483,435486,435489],{"className":435473},[31676],[34,435475],{"className":435476,"style":31774},[31680],[34,435478,640],{"className":435479},[31685],[34,435481,2557],{"className":435482},[31690],[34,435484],{"className":435485,"style":31846},[31845],[34,435487,202],{"className":435488},[31850],[34,435490],{"className":435491,"style":31846},[31845],[34,435493,435495,435498],{"className":435494},[31676],[34,435496],{"className":435497,"style":70485},[31680],[34,435499,435501,435504,435578],{"className":435500},[31685],[34,435502],{"className":435503},[31782,41488],[34,435505,435507],{"className":435506},[41453],[34,435508,435510,435570],{"className":435509},[31796,31797],[34,435511,435513,435567],{"className":435512},[31801],[34,435514,435516,435530,435538],{"className":435515,"style":70504},[31805],[34,435517,435518,435521],{"style":41504},[34,435519],{"className":435520,"style":41508},[31813],[34,435522,435524],{"className":435523},[31818,31819,31820,31821],[34,435525,435527],{"className":435526},[31685,31821],[34,435528,768],{"className":435529},[31685,31821],[34,435531,435532,435535],{"style":41520},[34,435533],{"className":435534,"style":41508},[31813],[34,435536],{"className":435537,"style":41528},[41527],[34,435539,435540,435543],{"style":70529},[34,435541],{"className":435542,"style":41508},[31813],[34,435544,435546],{"className":435545},[31818,31819,31820,31821],[34,435547,435549,435552,435555,435558,435561,435564],{"className":435548},[31685,31821],[34,435550,31660],{"className":435551},[31685,31686,31821],[34,435553,456],{"className":435554},[31782,31821],[34,435556,31660],{"className":435557},[31685,31686,31821],[34,435559,32226],{"className":435560},[32024,31821],[34,435562,640],{"className":435563},[31685,31821],[34,435565,2557],{"className":435566},[31690,31821],[34,435568,31829],{"className":435569},[31828],[34,435571,435573],{"className":435572},[31801],[34,435574,435576],{"className":435575,"style":41619},[31805],[34,435577],{},[34,435579],{"className":435580},[31690,41488],[15,435582,435583,435584,70884,435806,43298,435834,435862,435863,435891,435892,347],{},"因此，",[34,435585,435587,435633],{"className":435586},[31642],[34,435588,435590],{"className":435589},[31646],[31648,435591,435592],{"xmlns":31650},[31652,435593,435594,435630],{},[31655,435595,435596,435598,435600,435602,435604,435626,435628],{},[31658,435597,162086],{"mathvariant":38079},[31658,435599,97709],{},[31658,435601,162086],{"mathvariant":38079},[31662,435603,202],{},[37624,435605,435606,435608],{},[31658,435607,15],{},[41453,435609,435610,435624],{},[31655,435611,435612,435614,435616,435618,435620,435622],{},[31658,435613,31660],{},[31662,435615,456],{"stretchy":702},[31658,435617,31660],{},[31662,435619,32226],{},[31749,435621,640],{},[31662,435623,2557],{"stretchy":702},[31749,435625,768],{},[31662,435627,71191],{},[31658,435629,31759],{},[31665,435631,435632],{"encoding":31667},"|G| = p^{\\frac{n(n-1)}{2}} \\cdot m",[34,435634,435636,435660,435797],{"className":435635,"ariaHidden":758},[31672],[34,435637,435639,435642,435645,435648,435651,435654,435657],{"className":435638},[31676],[34,435640],{"className":435641,"style":31774},[31680],[34,435643,162086],{"className":435644},[31685],[34,435646,97709],{"className":435647},[31685,31686],[34,435649,162086],{"className":435650},[31685],[34,435652],{"className":435653,"style":31846},[31845],[34,435655,202],{"className":435656},[31850],[34,435658],{"className":435659,"style":31846},[31845],[34,435661,435663,435667,435788,435791,435794],{"className":435662},[31676],[34,435664],{"className":435665,"style":435666},[31680],"height:1.2839em;vertical-align:-0.1944em;",[34,435668,435670,435673],{"className":435669},[31685],[34,435671,15],{"className":435672},[31685,31686],[34,435674,435676],{"className":435675},[31792],[34,435677,435679],{"className":435678},[31796],[34,435680,435682],{"className":435681},[31801],[34,435683,435686],{"className":435684,"style":435685},[31805],"height:1.0895em;",[34,435687,435689,435692],{"style":435688},"top:-3.363em;margin-right:0.05em;",[34,435690],{"className":435691,"style":41508},[31813],[34,435693,435695],{"className":435694},[31818,31819,31820,31821],[34,435696,435698],{"className":435697},[31685,31821],[34,435699,435701,435705,435785],{"className":435700},[31685,31821],[34,435702],{"className":435703},[31782,41488,31818,43410,435704],"size6",[34,435706,435708],{"className":435707},[41453],[34,435709,435711,435776],{"className":435710},[31796,31797],[34,435712,435714,435773],{"className":435713},[31801],[34,435715,435718,435733,435743],{"className":435716,"style":435717},[31805],"height:1.0378em;",[34,435719,435721,435724],{"style":435720},"top:-2.656em;",[34,435722],{"className":435723,"style":41508},[31813],[34,435725,435727],{"className":435726},[31818,43410,43411,31821],[34,435728,435730],{"className":435729},[31685,31821],[34,435731,768],{"className":435732},[31685,31821],[34,435734,435736,435739],{"style":435735},"top:-3.2255em;",[34,435737],{"className":435738,"style":41508},[31813],[34,435740],{"className":435741,"style":435742},[41527,31821],"border-bottom-width:0.049em;",[34,435744,435746,435749],{"style":435745},"top:-3.5021em;",[34,435747],{"className":435748,"style":41508},[31813],[34,435750,435752],{"className":435751},[31818,43410,43411,31821],[34,435753,435755,435758,435761,435764,435767,435770],{"className":435754},[31685,31821],[34,435756,31660],{"className":435757},[31685,31686,31821],[34,435759,456],{"className":435760},[31782,31821],[34,435762,31660],{"className":435763},[31685,31686,31821],[34,435765,32226],{"className":435766},[32024,31821],[34,435768,640],{"className":435769},[31685,31821],[34,435771,2557],{"className":435772},[31690,31821],[34,435774,31829],{"className":435775},[31828],[34,435777,435779],{"className":435778},[31801],[34,435780,435783],{"className":435781,"style":435782},[31805],"height:0.344em;",[34,435784],{},[34,435786],{"className":435787},[31690,41488,31818,43410,435704],[34,435789],{"className":435790,"style":32604},[31845],[34,435792,71191],{"className":435793},[32024],[34,435795],{"className":435796,"style":32604},[31845],[34,435798,435800,435803],{"className":435799},[31676],[34,435801],{"className":435802,"style":32527},[31680],[34,435804,31759],{"className":435805},[31685,31686],[34,435807,435809,435822],{"className":435808},[31642],[34,435810,435812],{"className":435811},[31646],[31648,435813,435814],{"xmlns":31650},[31652,435815,435816,435820],{},[31655,435817,435818],{},[31658,435819,31759],{},[31665,435821,31759],{"encoding":31667},[34,435823,435825],{"className":435824,"ariaHidden":758},[31672],[34,435826,435828,435831],{"className":435827},[31676],[34,435829],{"className":435830,"style":32527},[31680],[34,435832,31759],{"className":435833},[31685,31686],[34,435835,435837,435850],{"className":435836},[31642],[34,435838,435840],{"className":435839},[31646],[31648,435841,435842],{"xmlns":31650},[31652,435843,435844,435848],{},[31655,435845,435846],{},[31658,435847,15],{},[31665,435849,15],{"encoding":31667},[34,435851,435853],{"className":435852,"ariaHidden":758},[31672],[34,435854,435856,435859],{"className":435855},[31676],[34,435857],{"className":435858,"style":38774},[31680],[34,435860,15],{"className":435861},[31685,31686]," 互素。\n这意味着 Sylow ",[34,435864,435866,435879],{"className":435865},[31642],[34,435867,435869],{"className":435868},[31646],[31648,435870,435871],{"xmlns":31650},[31652,435872,435873,435877],{},[31655,435874,435875],{},[31658,435876,15],{},[31665,435878,15],{"encoding":31667},[34,435880,435882],{"className":435881,"ariaHidden":758},[31672],[34,435883,435885,435888],{"className":435884},[31676],[34,435886],{"className":435887,"style":38774},[31680],[34,435889,15],{"className":435890},[31685,31686],"-子群的阶必须严格等于 ",[2627,435893,435894],{},[34,435895,435897,435931],{"className":435896},[31642],[34,435898,435900],{"className":435899},[31646],[31648,435901,435902],{"xmlns":31650},[31652,435903,435904,435928],{},[31655,435905,435906],{},[37624,435907,435908,435910],{},[31658,435909,15],{},[41453,435911,435912,435926],{},[31655,435913,435914,435916,435918,435920,435922,435924],{},[31658,435915,31660],{},[31662,435917,456],{"stretchy":702},[31658,435919,31660],{},[31662,435921,32226],{},[31749,435923,640],{},[31662,435925,2557],{"stretchy":702},[31749,435927,768],{},[31665,435929,435930],{"encoding":31667},"p^{\\frac{n(n-1)}{2}}",[34,435932,435934],{"className":435933,"ariaHidden":758},[31672],[34,435935,435937,435940],{"className":435936},[31676],[34,435938],{"className":435939,"style":435666},[31680],[34,435941,435943,435946],{"className":435942},[31685],[34,435944,15],{"className":435945},[31685,31686],[34,435947,435949],{"className":435948},[31792],[34,435950,435952],{"className":435951},[31796],[34,435953,435955],{"className":435954},[31801],[34,435956,435958],{"className":435957,"style":435685},[31805],[34,435959,435960,435963],{"style":435688},[34,435961],{"className":435962,"style":41508},[31813],[34,435964,435966],{"className":435965},[31818,31819,31820,31821],[34,435967,435969],{"className":435968},[31685,31821],[34,435970,435972,435975,436049],{"className":435971},[31685,31821],[34,435973],{"className":435974},[31782,41488,31818,43410,435704],[34,435976,435978],{"className":435977},[41453],[34,435979,435981,436041],{"className":435980},[31796,31797],[34,435982,435984,436038],{"className":435983},[31801],[34,435985,435987,436001,436009],{"className":435986,"style":435717},[31805],[34,435988,435989,435992],{"style":435720},[34,435990],{"className":435991,"style":41508},[31813],[34,435993,435995],{"className":435994},[31818,43410,43411,31821],[34,435996,435998],{"className":435997},[31685,31821],[34,435999,768],{"className":436000},[31685,31821],[34,436002,436003,436006],{"style":435735},[34,436004],{"className":436005,"style":41508},[31813],[34,436007],{"className":436008,"style":435742},[41527,31821],[34,436010,436011,436014],{"style":435745},[34,436012],{"className":436013,"style":41508},[31813],[34,436015,436017],{"className":436016},[31818,43410,43411,31821],[34,436018,436020,436023,436026,436029,436032,436035],{"className":436019},[31685,31821],[34,436021,31660],{"className":436022},[31685,31686,31821],[34,436024,456],{"className":436025},[31782,31821],[34,436027,31660],{"className":436028},[31685,31686,31821],[34,436030,32226],{"className":436031},[32024,31821],[34,436033,640],{"className":436034},[31685,31821],[34,436036,2557],{"className":436037},[31690,31821],[34,436039,31829],{"className":436040},[31828],[34,436042,436044],{"className":436043},[31801],[34,436045,436047],{"className":436046,"style":435782},[31805],[34,436048],{},[34,436050],{"className":436051},[31690,41488,31818,43410,435704],[32490,436053,436055],{"id":436054},"步骤-3构造-sylow-ppp-子群",[2627,436056,436057,436058,436086],{},"步骤 3：构造 Sylow ",[34,436059,436061,436074],{"className":436060},[31642],[34,436062,436064],{"className":436063},[31646],[31648,436065,436066],{"xmlns":31650},[31652,436067,436068,436072],{},[31655,436069,436070],{},[31658,436071,15],{},[31665,436073,15],{"encoding":31667},[34,436075,436077],{"className":436076,"ariaHidden":758},[31672],[34,436078,436080,436083],{"className":436079},[31676],[34,436081],{"className":436082,"style":38774},[31680],[34,436084,15],{"className":436085},[31685,31686],"-子群",[15,436088,436089,436090,436247,436248,436251,436252,436255],{},"我们需要寻找一个子群，其结构简单且大小正好是 ",[34,436091,436093,436126],{"className":436092},[31642],[34,436094,436096],{"className":436095},[31646],[31648,436097,436098],{"xmlns":31650},[31652,436099,436100,436124],{},[31655,436101,436102],{},[37624,436103,436104,436106],{},[31658,436105,15],{},[41453,436107,436108,436122],{},[31655,436109,436110,436112,436114,436116,436118,436120],{},[31658,436111,31660],{},[31662,436113,456],{"stretchy":702},[31658,436115,31660],{},[31662,436117,32226],{},[31749,436119,640],{},[31662,436121,2557],{"stretchy":702},[31749,436123,768],{},[31665,436125,435930],{"encoding":31667},[34,436127,436129],{"className":436128,"ariaHidden":758},[31672],[34,436130,436132,436135],{"className":436131},[31676],[34,436133],{"className":436134,"style":435666},[31680],[34,436136,436138,436141],{"className":436137},[31685],[34,436139,15],{"className":436140},[31685,31686],[34,436142,436144],{"className":436143},[31792],[34,436145,436147],{"className":436146},[31796],[34,436148,436150],{"className":436149},[31801],[34,436151,436153],{"className":436152,"style":435685},[31805],[34,436154,436155,436158],{"style":435688},[34,436156],{"className":436157,"style":41508},[31813],[34,436159,436161],{"className":436160},[31818,31819,31820,31821],[34,436162,436164],{"className":436163},[31685,31821],[34,436165,436167,436170,436244],{"className":436166},[31685,31821],[34,436168],{"className":436169},[31782,41488,31818,43410,435704],[34,436171,436173],{"className":436172},[41453],[34,436174,436176,436236],{"className":436175},[31796,31797],[34,436177,436179,436233],{"className":436178},[31801],[34,436180,436182,436196,436204],{"className":436181,"style":435717},[31805],[34,436183,436184,436187],{"style":435720},[34,436185],{"className":436186,"style":41508},[31813],[34,436188,436190],{"className":436189},[31818,43410,43411,31821],[34,436191,436193],{"className":436192},[31685,31821],[34,436194,768],{"className":436195},[31685,31821],[34,436197,436198,436201],{"style":435735},[34,436199],{"className":436200,"style":41508},[31813],[34,436202],{"className":436203,"style":435742},[41527,31821],[34,436205,436206,436209],{"style":435745},[34,436207],{"className":436208,"style":41508},[31813],[34,436210,436212],{"className":436211},[31818,43410,43411,31821],[34,436213,436215,436218,436221,436224,436227,436230],{"className":436214},[31685,31821],[34,436216,31660],{"className":436217},[31685,31686,31821],[34,436219,456],{"className":436220},[31782,31821],[34,436222,31660],{"className":436223},[31685,31686,31821],[34,436225,32226],{"className":436226},[32024,31821],[34,436228,640],{"className":436229},[31685,31821],[34,436231,2557],{"className":436232},[31690,31821],[34,436234,31829],{"className":436235},[31828],[34,436237,436239],{"className":436238},[31801],[34,436240,436242],{"className":436241,"style":435782},[31805],[34,436243],{},[34,436245],{"className":436246},[31690,41488,31818,43410,435704],"。\n考虑",[2627,436249,436250],{},"上三角矩阵","的一个特殊子类：",[2627,436253,436254],{},"对角线上全为 1 的上三角矩阵","（Unitriangular matrices）。",[15,436257,133910,436258,436286],{},[34,436259,436261,436274],{"className":436260},[31642],[34,436262,436264],{"className":436263},[31646],[31648,436265,436266],{"xmlns":31650},[31652,436267,436268,436272],{},[31655,436269,436270],{},[31658,436271,33350],{},[31665,436273,33350],{"encoding":31667},[34,436275,436277],{"className":436276,"ariaHidden":758},[31672],[34,436278,436280,436283],{"className":436279},[31676],[34,436281],{"className":436282,"style":33218},[31680],[34,436284,33350],{"className":436285,"style":33389},[31685,31686]," 为所有形式如下的矩阵集合：",[34,436288,436290],{"className":436289},[126385],[34,436291,436293,436537],{"className":436292},[31642],[34,436294,436296],{"className":436295},[31646],[31648,436297,436298],{"xmlns":31650,"display":126394},[31652,436299,436300,436534],{},[31655,436301,436302,436304,436532],{},[31662,436303,456],{"fence":758},[71078,436305,436307,436355,436399,436439,436500],{"rowspacing":370541,"columnalign":436306,"columnspacing":71082},"center center center center center",[71084,436308,436309,436315,436325,436335,436341],{},[71087,436310,436311],{},[71090,436312,436313],{"scriptlevel":459,"displaystyle":702},[31749,436314,640],{},[71087,436316,436317],{},[71090,436318,436319],{"scriptlevel":459,"displaystyle":702},[31743,436320,436321,436323],{},[31658,436322,111],{},[31749,436324,71869],{},[71087,436326,436327],{},[71090,436328,436329],{"scriptlevel":459,"displaystyle":702},[31743,436330,436331,436333],{},[31658,436332,111],{},[31749,436334,205234],{},[71087,436336,436337],{},[71090,436338,436339],{"scriptlevel":459,"displaystyle":702},[31662,436340,38874],{"lspace":116494,"rspace":116494},[71087,436342,436343],{},[71090,436344,436345],{"scriptlevel":459,"displaystyle":702},[31743,436346,436347,436349],{},[31658,436348,111],{},[31655,436350,436351,436353],{},[31749,436352,640],{},[31658,436354,31660],{},[71084,436356,436357,436363,436369,436379,436385],{},[71087,436358,436359],{},[71090,436360,436361],{"scriptlevel":459,"displaystyle":702},[31749,436362,459],{},[71087,436364,436365],{},[71090,436366,436367],{"scriptlevel":459,"displaystyle":702},[31749,436368,640],{},[71087,436370,436371],{},[71090,436372,436373],{"scriptlevel":459,"displaystyle":702},[31743,436374,436375,436377],{},[31658,436376,111],{},[31749,436378,94703],{},[71087,436380,436381],{},[71090,436382,436383],{"scriptlevel":459,"displaystyle":702},[31662,436384,38874],{"lspace":116494,"rspace":116494},[71087,436386,436387],{},[71090,436388,436389],{"scriptlevel":459,"displaystyle":702},[31743,436390,436391,436393],{},[31658,436392,111],{},[31655,436394,436395,436397],{},[31749,436396,768],{},[31658,436398,31660],{},[71084,436400,436401,436407,436413,436419,436425],{},[71087,436402,436403],{},[71090,436404,436405],{"scriptlevel":459,"displaystyle":702},[31749,436406,459],{},[71087,436408,436409],{},[71090,436410,436411],{"scriptlevel":459,"displaystyle":702},[31749,436412,459],{},[71087,436414,436415],{},[71090,436416,436417],{"scriptlevel":459,"displaystyle":702},[31749,436418,640],{},[71087,436420,436421],{},[71090,436422,436423],{"scriptlevel":459,"displaystyle":702},[31662,436424,38874],{"lspace":116494,"rspace":116494},[71087,436426,436427],{},[71090,436428,436429],{"scriptlevel":459,"displaystyle":702},[31743,436430,436431,436433],{},[31658,436432,111],{},[31655,436434,436435,436437],{},[31749,436436,786],{},[31658,436438,31660],{},[71084,436440,436441,436457,436469,436481,436488],{},[71087,436442,436443],{},[71090,436444,436445],{"scriptlevel":459,"displaystyle":702},[31655,436446,436447,436450],{},[31658,436448,436449],{"mathvariant":38079},"⋮",[436451,436452,436453],"mpadded",{"height":116494,"voffset":116494},[31845,436454],{"mathbackground":436455,"width":116494,"height":436456},"black","1.5em",[71087,436458,436459],{},[71090,436460,436461],{"scriptlevel":459,"displaystyle":702},[31655,436462,436463,436465],{},[31658,436464,436449],{"mathvariant":38079},[436451,436466,436467],{"height":116494,"voffset":116494},[31845,436468],{"mathbackground":436455,"width":116494,"height":436456},[71087,436470,436471],{},[71090,436472,436473],{"scriptlevel":459,"displaystyle":702},[31655,436474,436475,436477],{},[31658,436476,436449],{"mathvariant":38079},[436451,436478,436479],{"height":116494,"voffset":116494},[31845,436480],{"mathbackground":436455,"width":116494,"height":436456},[71087,436482,436483],{},[71090,436484,436485],{"scriptlevel":459,"displaystyle":702},[31662,436486,436487],{"lspace":116494,"rspace":116494},"⋱",[71087,436489,436490],{},[71090,436491,436492],{"scriptlevel":459,"displaystyle":702},[31655,436493,436494,436496],{},[31658,436495,436449],{"mathvariant":38079},[436451,436497,436498],{"height":116494,"voffset":116494},[31845,436499],{"mathbackground":436455,"width":116494,"height":436456},[71084,436501,436502,436508,436514,436520,436526],{},[71087,436503,436504],{},[71090,436505,436506],{"scriptlevel":459,"displaystyle":702},[31749,436507,459],{},[71087,436509,436510],{},[71090,436511,436512],{"scriptlevel":459,"displaystyle":702},[31749,436513,459],{},[71087,436515,436516],{},[71090,436517,436518],{"scriptlevel":459,"displaystyle":702},[31749,436519,459],{},[71087,436521,436522],{},[71090,436523,436524],{"scriptlevel":459,"displaystyle":702},[31662,436525,38874],{"lspace":116494,"rspace":116494},[71087,436527,436528],{},[71090,436529,436530],{"scriptlevel":459,"displaystyle":702},[31749,436531,640],{},[31662,436533,2557],{"fence":758},[31665,436535,436536],{"encoding":31667},"\\begin{pmatrix}\n1 & a_{12} & a_{13} & \\cdots & a_{1n} \\\\\n0 & 1 & a_{23} & \\cdots & a_{2n} \\\\\n0 & 0 & 1 & \\cdots & a_{3n} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 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\\in \\mathbb{F}_p",[34,437378,437380,437438],{"className":437379,"ariaHidden":758},[31672],[34,437381,437383,437386,437429,437432,437435],{"className":437382},[31676],[34,437384],{"className":437385,"style":422714},[31680],[34,437387,437389,437392],{"className":437388},[31685],[34,437390,111],{"className":437391},[31685,31686],[34,437393,437395],{"className":437394},[31792],[34,437396,437398,437421],{"className":437397},[31796,31797],[34,437399,437401,437418],{"className":437400},[31801],[34,437402,437404],{"className":437403,"style":32090},[31805],[34,437405,437406,437409],{"style":31809},[34,437407],{"className":437408,"style":31814},[31813],[34,437410,437412],{"className":437411},[31818,31819,31820,31821],[34,437413,437415],{"className":437414},[31685,31821],[34,437416,141027],{"className":437417,"style":32153},[31685,31686,31821],[34,437419,31829],{"className":437420},[31828],[34,437422,437424],{"className":437423},[31801],[34,437425,437427],{"className":437426,"style":32163},[31805],[34,437428],{},[34,437430],{"className":437431,"style":31846},[31845],[34,437433,112296],{"className":437434},[31850],[34,437436],{"className":437437,"style":31846},[31845],[34,437439,437441,437444],{"className":437440},[31676],[34,437442],{"className":437443,"style":338084},[31680],[34,437445,437447,437450],{"className":437446},[31685],[34,437448,51834],{"className":437449},[31685,318441],[34,437451,437453],{"className":437452},[31792],[34,437454,437456,437476],{"className":437455},[31796,31797],[34,437457,437459,437473],{"className":437458},[31801],[34,437460,437462],{"className":437461,"style":32362},[31805],[34,437463,437464,437467],{"style":31809},[34,437465],{"className":437466,"style":31814},[31813],[34,437468,437470],{"className":437469},[31818,31819,31820,31821],[34,437471,15],{"className":437472},[31685,31686,31821],[34,437474,31829],{"className":437475},[31828],[34,437477,437479],{"className":437478},[31801],[34,437480,437482],{"className":437481,"style":32163},[31805],[34,437483],{},[15,437485,437486],{},[2627,437487,343971,437488,437516],{},[34,437489,437491,437504],{"className":437490},[31642],[34,437492,437494],{"className":437493},[31646],[31648,437495,437496],{"xmlns":31650},[31652,437497,437498,437502],{},[31655,437499,437500],{},[31658,437501,33350],{},[31665,437503,33350],{"encoding":31667},[34,437505,437507],{"className":437506,"ariaHidden":758},[31672],[34,437508,437510,437513],{"className":437509},[31676],[34,437511],{"className":437512,"style":33218},[31680],[34,437514,33350],{"className":437515,"style":33389},[31685,31686]," 的大小：",[105,437518,437519,437522,437525,437606],{},[108,437520,437521],{},"对角线元素固定为 1（无选择）。",[108,437523,437524],{},"下三角元素固定为 0（无选择）。",[108,437526,437527,437528,437605],{},"上三角区域（对角线右上方）的元素 ",[34,437529,437531,437553],{"className":437530},[31642],[34,437532,437534],{"className":437533},[31646],[31648,437535,437536],{"xmlns":31650},[31652,437537,437538,437550],{},[31655,437539,437540],{},[31743,437541,437542,437544],{},[31658,437543,111],{},[31655,437545,437546,437548],{},[31658,437547,31955],{},[31658,437549,31968],{},[31665,437551,437552],{"encoding":31667},"a_{ij}",[34,437554,437556],{"className":437555,"ariaHidden":758},[31672],[34,437557,437559,437562],{"className":437558},[31676],[34,437560],{"className":437561,"style":56005},[31680],[34,437563,437565,437568],{"className":437564},[31685],[34,437566,111],{"className":437567},[31685,31686],[34,437569,437571],{"className":437570},[31792],[34,437572,437574,437597],{"className":437573},[31796,31797],[34,437575,437577,437594],{"className":437576},[31801],[34,437578,437580],{"className":437579,"style":32090},[31805],[34,437581,437582,437585],{"style":31809},[34,437583],{"className":437584,"style":31814},[31813],[34,437586,437588],{"className":437587},[31818,31819,31820,31821],[34,437589,437591],{"className":437590},[31685,31821],[34,437592,141027],{"className":437593,"style":32153},[31685,31686,31821],[34,437595,31829],{"className":437596},[31828],[34,437598,437600],{"className":437599},[31801],[34,437601,437603],{"className":437602,"style":32163},[31805],[34,437604],{}," 可以任意选择。",[108,437607,437608,437609],{},"上三角区域的元素个数有多少个？\n",[105,437610,437611,437665,437718,437720,437801],{},[108,437612,437613,437614,437664],{},"第 1 行有 ",[34,437615,437617,437634],{"className":437616},[31642],[34,437618,437620],{"className":437619},[31646],[31648,437621,437622],{"xmlns":31650},[31652,437623,437624,437632],{},[31655,437625,437626,437628,437630],{},[31658,437627,31660],{},[31662,437629,32226],{},[31749,437631,640],{},[31665,437633,32587],{"encoding":31667},[34,437635,437637,437655],{"className":437636,"ariaHidden":758},[31672],[34,437638,437640,437643,437646,437649,437652],{"className":437639},[31676],[34,437641],{"className":437642,"style":32597},[31680],[34,437644,31660],{"className":437645},[31685,31686],[34,437647],{"className":437648,"style":32604},[31845],[34,437650,32226],{"className":437651},[32024],[34,437653],{"className":437654,"style":32604},[31845],[34,437656,437658,437661],{"className":437657},[31676],[34,437659],{"className":437660,"style":32617},[31680],[34,437662,640],{"className":437663},[31685]," 个自由位置。",[108,437666,437667,437668,437664],{},"第 2 行有 ",[34,437669,437671,437688],{"className":437670},[31642],[34,437672,437674],{"className":437673},[31646],[31648,437675,437676],{"xmlns":31650},[31652,437677,437678,437686],{},[31655,437679,437680,437682,437684],{},[31658,437681,31660],{},[31662,437683,32226],{},[31749,437685,768],{},[31665,437687,85119],{"encoding":31667},[34,437689,437691,437709],{"className":437690,"ariaHidden":758},[31672],[34,437692,437694,437697,437700,437703,437706],{"className":437693},[31676],[34,437695],{"className":437696,"style":32597},[31680],[34,437698,31660],{"className":437699},[31685,31686],[34,437701],{"className":437702,"style":32604},[31845],[34,437704,32226],{"className":437705},[32024],[34,437707],{"className":437708,"style":32604},[31845],[34,437710,437712,437715],{"className":437711},[31676],[34,437713],{"className":437714,"style":32617},[31680],[34,437716,768],{"className":437717},[31685],[108,437719,1167],{},[108,437721,178926,437722,437772,437773,437664],{},[34,437723,437725,437742],{"className":437724},[31642],[34,437726,437728],{"className":437727},[31646],[31648,437729,437730],{"xmlns":31650},[31652,437731,437732,437740],{},[31655,437733,437734,437736,437738],{},[31658,437735,31660],{},[31662,437737,32226],{},[31749,437739,640],{},[31665,437741,32587],{"encoding":31667},[34,437743,437745,437763],{"className":437744,"ariaHidden":758},[31672],[34,437746,437748,437751,437754,437757,437760],{"className":437747},[31676],[34,437749],{"className":437750,"style":32597},[31680],[34,437752,31660],{"className":437753},[31685,31686],[34,437755],{"className":437756,"style":32604},[31845],[34,437758,32226],{"className":437759},[32024],[34,437761],{"className":437762,"style":32604},[31845],[34,437764,437766,437769],{"className":437765},[31676],[34,437767],{"className":437768,"style":32617},[31680],[34,437770,640],{"className":437771},[31685]," 行有 ",[34,437774,437776,437789],{"className":437775},[31642],[34,437777,437779],{"className":437778},[31646],[31648,437780,437781],{"xmlns":31650},[31652,437782,437783,437787],{},[31655,437784,437785],{},[31749,437786,640],{},[31665,437788,640],{"encoding":31667},[34,437790,437792],{"className":437791,"ariaHidden":758},[31672],[34,437793,437795,437798],{"className":437794},[31676],[34,437796],{"className":437797,"style":32617},[31680],[34,437799,640],{"className":437800},[31685],[108,437802,437803,437804,347],{},"总自由位置数 = ",[34,437805,437807,437869],{"className":437806},[31642],[34,437808,437810],{"className":437809},[31646],[31648,437811,437812],{"xmlns":31650},[31652,437813,437814,437866],{},[31655,437815,437816,437818,437820,437822,437824,437826,437828,437830,437832,437834,437836,437838,437840,437842,437844,437846,437848],{},[31662,437817,456],{"stretchy":702},[31658,437819,31660],{},[31662,437821,32226],{},[31749,437823,640],{},[31662,437825,2557],{"stretchy":702},[31662,437827,223],{},[31662,437829,456],{"stretchy":702},[31658,437831,31660],{},[31662,437833,32226],{},[31749,437835,768],{},[31662,437837,2557],{"stretchy":702},[31662,437839,223],{},[31662,437841,38874],{},[31662,437843,223],{},[31749,437845,640],{},[31662,437847,202],{},[41453,437849,437850,437864],{},[31655,437851,437852,437854,437856,437858,437860,437862],{},[31658,437853,31660],{},[31662,437855,456],{"stretchy":702},[31658,437857,31660],{},[31662,437859,32226],{},[31749,437861,640],{},[31662,437863,2557],{"stretchy":702},[31749,437865,768],{},[31665,437867,437868],{"encoding":31667},"(n-1) + (n-2) + \\dots + 1 = \\frac{n(n-1)}{2}",[34,437870,437872,437893,437914,437935,437956,437974,437992],{"className":437871,"ariaHidden":758},[31672],[34,437873,437875,437878,437881,437884,437887,437890],{"className":437874},[31676],[34,437876],{"className":437877,"style":31774},[31680],[34,437879,456],{"className":437880},[31782],[34,437882,31660],{"className":437883},[31685,31686],[34,437885],{"className":437886,"style":32604},[31845],[34,437888,32226],{"className":437889},[32024],[34,437891],{"className":437892,"style":32604},[31845],[34,437894,437896,437899,437902,437905,437908,437911],{"className":437895},[31676],[34,437897],{"className":437898,"style":31774},[31680],[34,437900,640],{"className":437901},[31685],[34,437903,2557],{"className":437904},[31690],[34,437906],{"className":437907,"style":32604},[31845],[34,437909,223],{"className":437910},[32024],[34,437912],{"className":437913,"style":32604},[31845],[34,437915,437917,437920,437923,437926,437929,437932],{"className":437916},[31676],[34,437918],{"className":437919,"style":31774},[31680],[34,437921,456],{"className":437922},[31782],[34,437924,31660],{"className":437925},[31685,31686],[34,437927],{"className":437928,"style":32604},[31845],[34,437930,32226],{"className":437931},[32024],[34,437933],{"className":437934,"style":32604},[31845],[34,437936,437938,437941,437944,437947,437950,437953],{"className":437937},[31676],[34,437939],{"className":437940,"style":31774},[31680],[34,437942,768],{"className":437943},[31685],[34,437945,2557],{"className":437946},[31690],[34,437948],{"className":437949,"style":32604},[31845],[34,437951,223],{"className":437952},[32024],[34,437954],{"className":437955,"style":32604},[31845],[34,437957,437959,437962,437965,437968,437971],{"className":437958},[31676],[34,437960],{"className":437961,"style":32597},[31680],[34,437963,38874],{"className":437964},[34006],[34,437966],{"className":437967,"style":32604},[31845],[34,437969,223],{"className":437970},[32024],[34,437972],{"className":437973,"style":32604},[31845],[34,437975,437977,437980,437983,437986,437989],{"className":437976},[31676],[34,437978],{"className":437979,"style":32617},[31680],[34,437981,640],{"className":437982},[31685],[34,437984],{"className":437985,"style":31846},[31845],[34,437987,202],{"className":437988},[31850],[34,437990],{"className":437991,"style":31846},[31845],[34,437993,437995,437998],{"className":437994},[31676],[34,437996],{"className":437997,"style":70485},[31680],[34,437999,438001,438004,438078],{"className":438000},[31685],[34,438002],{"className":438003},[31782,41488],[34,438005,438007],{"className":438006},[41453],[34,438008,438010,438070],{"className":438009},[31796,31797],[34,438011,438013,438067],{"className":438012},[31801],[34,438014,438016,438030,438038],{"className":438015,"style":70504},[31805],[34,438017,438018,438021],{"style":41504},[34,438019],{"className":438020,"style":41508},[31813],[34,438022,438024],{"className":438023},[31818,31819,31820,31821],[34,438025,438027],{"className":438026},[31685,31821],[34,438028,768],{"className":438029},[31685,31821],[34,438031,438032,438035],{"style":41520},[34,438033],{"className":438034,"style":41508},[31813],[34,438036],{"className":438037,"style":41528},[41527],[34,438039,438040,438043],{"style":70529},[34,438041],{"className":438042,"style":41508},[31813],[34,438044,438046],{"className":438045},[31818,31819,31820,31821],[34,438047,438049,438052,438055,438058,438061,438064],{"className":438048},[31685,31821],[34,438050,31660],{"className":438051},[31685,31686,31821],[34,438053,456],{"className":438054},[31782,31821],[34,438056,31660],{"className":438057},[31685,31686,31821],[34,438059,32226],{"className":438060},[32024,31821],[34,438062,640],{"className":438063},[31685,31821],[34,438065,2557],{"className":438066},[31690,31821],[34,438068,31829],{"className":438069},[31828],[34,438071,438073],{"className":438072},[31801],[34,438074,438076],{"className":438075,"style":41619},[31805],[34,438077],{},[34,438079],{"className":438080},[31690,41488],[15,438082,438083,438084,438112,438113,438141,438142],{},"因为每个位置有 ",[34,438085,438087,438100],{"className":438086},[31642],[34,438088,438090],{"className":438089},[31646],[31648,438091,438092],{"xmlns":31650},[31652,438093,438094,438098],{},[31655,438095,438096],{},[31658,438097,15],{},[31665,438099,15],{"encoding":31667},[34,438101,438103],{"className":438102,"ariaHidden":758},[31672],[34,438104,438106,438109],{"className":438105},[31676],[34,438107],{"className":438108,"style":38774},[31680],[34,438110,15],{"className":438111},[31685,31686]," 种选择，所以 ",[34,438114,438116,438129],{"className":438115},[31642],[34,438117,438119],{"className":438118},[31646],[31648,438120,438121],{"xmlns":31650},[31652,438122,438123,438127],{},[31655,438124,438125],{},[31658,438126,33350],{},[31665,438128,33350],{"encoding":31667},[34,438130,438132],{"className":438131,"ariaHidden":758},[31672],[34,438133,438135,438138],{"className":438134},[31676],[34,438136],{"className":438137,"style":33218},[31680],[34,438139,33350],{"className":438140,"style":33389},[31685,31686]," 的总元素个数为：\n",[34,438143,438145,438187],{"className":438144},[31642],[34,438146,438148],{"className":438147},[31646],[31648,438149,438150],{"xmlns":31650},[31652,438151,438152,438184],{},[31655,438153,438154,438156,438158,438160,438162],{},[31658,438155,162086],{"mathvariant":38079},[31658,438157,33350],{},[31658,438159,162086],{"mathvariant":38079},[31662,438161,202],{},[37624,438163,438164,438166],{},[31658,438165,15],{},[41453,438167,438168,438182],{},[31655,438169,438170,438172,438174,438176,438178,438180],{},[31658,438171,31660],{},[31662,438173,456],{"stretchy":702},[31658,438175,31660],{},[31662,438177,32226],{},[31749,438179,640],{},[31662,438181,2557],{"stretchy":702},[31749,438183,768],{},[31665,438185,438186],{"encoding":31667},"|U| = p^{\\frac{n(n-1)}{2}}",[34,438188,438190,438214],{"className":438189,"ariaHidden":758},[31672],[34,438191,438193,438196,438199,438202,438205,438208,438211],{"className":438192},[31676],[34,438194],{"className":438195,"style":31774},[31680],[34,438197,162086],{"className":438198},[31685],[34,438200,33350],{"className":438201,"style":33389},[31685,31686],[34,438203,162086],{"className":438204},[31685],[34,438206],{"className":438207,"style":31846},[31845],[34,438209,202],{"className":438210},[31850],[34,438212],{"className":438213,"style":31846},[31845],[34,438215,438217,438220],{"className":438216},[31676],[34,438218],{"className":438219,"style":435666},[31680],[34,438221,438223,438226],{"className":438222},[31685],[34,438224,15],{"className":438225},[31685,31686],[34,438227,438229],{"className":438228},[31792],[34,438230,438232],{"className":438231},[31796],[34,438233,438235],{"className":438234},[31801],[34,438236,438238],{"className":438237,"style":435685},[31805],[34,438239,438240,438243],{"style":435688},[34,438241],{"className":438242,"style":41508},[31813],[34,438244,438246],{"className":438245},[31818,31819,31820,31821],[34,438247,438249],{"className":438248},[31685,31821],[34,438250,438252,438255,438329],{"className":438251},[31685,31821],[34,438253],{"className":438254},[31782,41488,31818,43410,435704],[34,438256,438258],{"className":438257},[41453],[34,438259,438261,438321],{"className":438260},[31796,31797],[34,438262,438264,438318],{"className":438263},[31801],[34,438265,438267,438281,438289],{"className":438266,"style":435717},[31805],[34,438268,438269,438272],{"style":435720},[34,438270],{"className":438271,"style":41508},[31813],[34,438273,438275],{"className":438274},[31818,43410,43411,31821],[34,438276,438278],{"className":438277},[31685,31821],[34,438279,768],{"className":438280},[31685,31821],[34,438282,438283,438286],{"style":435735},[34,438284],{"className":438285,"style":41508},[31813],[34,438287],{"className":438288,"style":435742},[41527,31821],[34,438290,438291,438294],{"style":435745},[34,438292],{"className":438293,"style":41508},[31813],[34,438295,438297],{"className":438296},[31818,43410,43411,31821],[34,438298,438300,438303,438306,438309,438312,438315],{"className":438299},[31685,31821],[34,438301,31660],{"className":438302},[31685,31686,31821],[34,438304,456],{"className":438305},[31782,31821],[34,438307,31660],{"className":438308},[31685,31686,31821],[34,438310,32226],{"className":438311},[32024,31821],[34,438313,640],{"className":438314},[31685,31821],[34,438316,2557],{"className":438317},[31690,31821],[34,438319,31829],{"className":438320},[31828],[34,438322,438324],{"className":438323},[31801],[34,438325,438327],{"className":438326,"style":435782},[31805],[34,438328],{},[34,438330],{"className":438331},[31690,41488,31818,43410,435704],[15,438333,438334,438365],{},[2627,438335,343971,438336,438364],{},[34,438337,438339,438352],{"className":438338},[31642],[34,438340,438342],{"className":438341},[31646],[31648,438343,438344],{"xmlns":31650},[31652,438345,438346,438350],{},[31655,438347,438348],{},[31658,438349,33350],{},[31665,438351,33350],{"encoding":31667},[34,438353,438355],{"className":438354,"ariaHidden":758},[31672],[34,438356,438358,438361],{"className":438357},[31676],[34,438359],{"className":438360,"style":33218},[31680],[34,438362,33350],{"className":438363,"style":33389},[31685,31686]," 是子群：","\n两个上三角且对角线为1的矩阵相乘，结果仍然是上三角且对角线为1；其逆矩阵也是同种形式。（这是一个经典结论）。",[32490,438367,438368],{"id":114043},[2627,438369,114043],{},[15,438371,438372,432540,438526,438554,438555,438558,438559,4374,438685,31901],{},[34,438373,438375,438410],{"className":438374},[31642],[34,438376,438378],{"className":438377},[31646],[31648,438379,438380],{"xmlns":31650},[31652,438381,438382,438408],{},[31655,438383,438384,438386,438388,438398,438400,438406],{},[31658,438385,97709],{},[31662,438387,202],{},[31743,438389,438390,438396],{},[31655,438391,438392,438394],{},[31658,438393,97709],{"mathvariant":38079},[31658,438395,33206],{"mathvariant":38079},[31658,438397,31660],{},[31662,438399,456],{"stretchy":702},[31743,438401,438402,438404],{},[31658,438403,51834],{"mathvariant":318416},[31658,438405,15],{},[31662,438407,2557],{"stretchy":702},[31665,438409,432827],{"encoding":31667},[34,438411,438413,438431],{"className":438412,"ariaHidden":758},[31672],[34,438414,438416,438419,438422,438425,438428],{"className":438415},[31676],[34,438417],{"className":438418,"style":33218},[31680],[34,438420,97709],{"className":438421},[31685,31686],[34,438423],{"className":438424,"style":31846},[31845],[34,438426,202],{"className":438427},[31850],[34,438429],{"className":438430,"style":31846},[31845],[34,438432,438434,438437,438480,438483,438523],{"className":438433},[31676],[34,438435],{"className":438436,"style":32058},[31680],[34,438438,438440,438446],{"className":438439},[31685],[34,438441,438443],{"className":438442},[31685],[34,438444,432415],{"className":438445},[31685,207923],[34,438447,438449],{"className":438448},[31792],[34,438450,438452,438472],{"className":438451},[31796,31797],[34,438453,438455,438469],{"className":438454},[31801],[34,438456,438458],{"className":438457,"style":32362},[31805],[34,438459,438460,438463],{"style":56235},[34,438461],{"className":438462,"style":31814},[31813],[34,438464,438466],{"className":438465},[31818,31819,31820,31821],[34,438467,31660],{"className":438468},[31685,31686,31821],[34,438470,31829],{"className":438471},[31828],[34,438473,438475],{"className":438474},[31801],[34,438476,438478],{"className":438477,"style":31836},[31805],[34,438479],{},[34,438481,456],{"className":438482},[31782],[34,438484,438486,438489],{"className":438485},[31685],[34,438487,51834],{"className":438488},[31685,318441],[34,438490,438492],{"className":438491},[31792],[34,438493,438495,438515],{"className":438494},[31796,31797],[34,438496,438498,438512],{"className":438497},[31801],[34,438499,438501],{"className":438500,"style":32362},[31805],[34,438502,438503,438506],{"style":31809},[34,438504],{"className":438505,"style":31814},[31813],[34,438507,438509],{"className":438508},[31818,31819,31820,31821],[34,438510,15],{"className":438511},[31685,31686,31821],[34,438513,31829],{"className":438514},[31828],[34,438516,438518],{"className":438517},[31801],[34,438519,438521],{"className":438520,"style":32163},[31805],[34,438522],{},[34,438524,2557],{"className":438525},[31690],[34,438527,438529,438542],{"className":438528},[31642],[34,438530,438532],{"className":438531},[31646],[31648,438533,438534],{"xmlns":31650},[31652,438535,438536,438540],{},[31655,438537,438538],{},[31658,438539,15],{},[31665,438541,15],{"encoding":31667},[34,438543,438545],{"className":438544,"ariaHidden":758},[31672],[34,438546,438548,438551],{"className":438547},[31676],[34,438549],{"className":438550,"style":38774},[31680],[34,438552,15],{"className":438553},[31685,31686],"-子群是：\n",[2627,438556,438557],{},"由所有对角线元素为 1 的上三角矩阵构成的子群。","\n（通常记作 ",[34,438560,438562,438590],{"className":438561},[31642],[34,438563,438565],{"className":438564},[31646],[31648,438566,438567],{"xmlns":31650},[31652,438568,438569,438587],{},[31655,438570,438571,438577,438579,438585],{},[31743,438572,438573,438575],{},[31658,438574,33350],{},[31658,438576,31660],{},[31662,438578,456],{"stretchy":702},[31743,438580,438581,438583],{},[31658,438582,51834],{"mathvariant":318416},[31658,438584,15],{},[31662,438586,2557],{"stretchy":702},[31665,438588,438589],{"encoding":31667},"U_n(\\mathbb{F}_p)",[34,438591,438593],{"className":438592,"ariaHidden":758},[31672],[34,438594,438596,438599,438639,438642,438682],{"className":438595},[31676],[34,438597],{"className":438598,"style":32058},[31680],[34,438600,438602,438605],{"className":438601},[31685],[34,438603,33350],{"className":438604,"style":33389},[31685,31686],[34,438606,438608],{"className":438607},[31792],[34,438609,438611,438631],{"className":438610},[31796,31797],[34,438612,438614,438628],{"className":438613},[31801],[34,438615,438617],{"className":438616,"style":32362},[31805],[34,438618,438619,438622],{"style":231884},[34,438620],{"className":438621,"style":31814},[31813],[34,438623,438625],{"className":438624},[31818,31819,31820,31821],[34,438626,31660],{"className":438627},[31685,31686,31821],[34,438629,31829],{"className":438630},[31828],[34,438632,438634],{"className":438633},[31801],[34,438635,438637],{"className":438636,"style":31836},[31805],[34,438638],{},[34,438640,456],{"className":438641},[31782],[34,438643,438645,438648],{"className":438644},[31685],[34,438646,51834],{"className":438647},[31685,318441],[34,438649,438651],{"className":438650},[31792],[34,438652,438654,438674],{"className":438653},[31796,31797],[34,438655,438657,438671],{"className":438656},[31801],[34,438658,438660],{"className":438659,"style":32362},[31805],[34,438661,438662,438665],{"style":31809},[34,438663],{"className":438664,"style":31814},[31813],[34,438666,438668],{"className":438667},[31818,31819,31820,31821],[34,438669,15],{"className":438670},[31685,31686,31821],[34,438672,31829],{"className":438673},[31828],[34,438675,438677],{"className":438676},[31801],[34,438678,438680],{"className":438679,"style":32163},[31805],[34,438681],{},[34,438683,2557],{"className":438684},[31690],[34,438686,438688,438720],{"className":438687},[31642],[34,438689,438691],{"className":438690},[31646],[31648,438692,438693],{"xmlns":31650},[31652,438694,438695,438717],{},[31655,438696,438697,438707,438709,438715],{},[31743,438698,438699,438705],{},[31655,438700,438701,438703],{},[31658,438702,33350],{"mathvariant":38079},[31658,438704,37400],{"mathvariant":38079},[31658,438706,31660],{},[31662,438708,456],{"stretchy":702},[31743,438710,438711,438713],{},[31658,438712,51834],{"mathvariant":318416},[31658,438714,15],{},[31662,438716,2557],{"stretchy":702},[31665,438718,438719],{"encoding":31667},"\\mathrm{UT}_n(\\mathbb{F}_p)",[34,438721,438723],{"className":438722,"ariaHidden":758},[31672],[34,438724,438726,438729,438773,438776,438816],{"className":438725},[31676],[34,438727],{"className":438728,"style":32058},[31680],[34,438730,438732,438739],{"className":438731},[31685],[34,438733,438735],{"className":438734},[31685],[34,438736,438738],{"className":438737},[31685,207923],"UT",[34,438740,438742],{"className":438741},[31792],[34,438743,438745,438765],{"className":438744},[31796,31797],[34,438746,438748,438762],{"className":438747},[31801],[34,438749,438751],{"className":438750,"style":32362},[31805],[34,438752,438753,438756],{"style":56235},[34,438754],{"className":438755,"style":31814},[31813],[34,438757,438759],{"className":438758},[31818,31819,31820,31821],[34,438760,31660],{"className":438761},[31685,31686,31821],[34,438763,31829],{"className":438764},[31828],[34,438766,438768],{"className":438767},[31801],[34,438769,438771],{"className":438770,"style":31836},[31805],[34,438772],{},[34,438774,456],{"className":438775},[31782],[34,438777,438779,438782],{"className":438778},[31685],[34,438780,51834],{"className":438781},[31685,318441],[34,438783,438785],{"className":438784},[31792],[34,438786,438788,438808],{"className":438787},[31796,31797],[34,438789,438791,438805],{"className":438790},[31801],[34,438792,438794],{"className":438793,"style":32362},[31805],[34,438795,438796,438799],{"style":31809},[34,438797],{"className":438798,"style":31814},[31813],[34,438800,438802],{"className":438801},[31818,31819,31820,31821],[34,438803,15],{"className":438804},[31685,31686,31821],[34,438806,31829],{"className":438807},[31828],[34,438809,438811],{"className":438810},[31801],[34,438812,438814],{"className":438813,"style":32163},[31805],[34,438815],{},[34,438817,2557],{"className":438818},[31690],{"title":29,"searchDepth":43,"depth":43,"links":438820},[438821,438822,438823],{"id":417983,"depth":49,"text":417986},{"id":426494,"depth":49,"text":426497},{"id":432140,"depth":49,"text":438824},"题目 3：寻找一般线性群的 Sylow ppp-子群 (P64 第21题)",{"order":1976},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringdirectprodhw",{"title":417972,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002FringdirectprodHW","WIk1hc0FjQJprgrGrVGAIIjd-QRCBuidntYgeSQkus4",{"id":438831,"title":438832,"body":438833,"description":29,"extension":85,"meta":463158,"navigation":52,"path":463159,"seo":463160,"stem":463161,"__hash__":463162},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringhomomorphism.md","环同态基本定理",{"type":8,"value":438834,"toc":463151},[438835,438839,438843,439303,439306,440348,440426,440724,440726,440730,441094,441099,441227,443898,444027,445221,446272,446274,446278,446508,446917,447726,448272,448440,448812,449149,449511,449616,452986,452988,452992,454157,454159,454163,454352,455731,456554,458038,458656,458691,460982],[11,438836,438838],{"id":438837},"_33-同态基本定理-完整讲义版","3.3 同态基本定理 (完整讲义版)",[253,438840,438842],{"id":438841},"_1-环同态的定义与分类","1. 环同态的定义与分类",[15,438844,438845,69123,438848,438890,438891,85658,438964,439033,439034,439271,439272,363149,439300,347],{},[2627,438846,438847],{},"定义 3.3.1.",[34,438849,438851,438869],{"className":438850},[31642],[34,438852,438854],{"className":438853},[31646],[31648,438855,438856],{"xmlns":31650},[31652,438857,438858,438866],{},[31655,438859,438860,438862,438864],{},[31658,438861,34386],{},[31662,438863,31960],{"separator":758},[31658,438865,94675],{},[31665,438867,438868],{"encoding":31667},"R, S",[34,438870,438872],{"className":438871,"ariaHidden":758},[31672],[34,438873,438875,438878,438881,438884,438887],{"className":438874},[31676],[34,438876],{"className":438877,"style":70739},[31680],[34,438879,34386],{"className":438880,"style":34427},[31685,31686],[34,438882,31960],{"className":438883},[32119],[34,438885],{"className":438886,"style":32123},[31845],[34,438888,94675],{"className":438889,"style":106455},[31685,31686]," 是两个环。如果映射 ",[34,438892,438894,438916],{"className":438893},[31642],[34,438895,438897],{"className":438896},[31646],[31648,438898,438899],{"xmlns":31650},[31652,438900,438901,438913],{},[31655,438902,438903,438905,438907,438909,438911],{},[31658,438904,329201],{},[31662,438906,5598],{},[31658,438908,34386],{},[31662,438910,33239],{},[31658,438912,94675],{},[31665,438914,438915],{"encoding":31667},"\\phi: R \\to S",[34,438917,438919,438937,438955],{"className":438918,"ariaHidden":758},[31672],[34,438920,438922,438925,438928,438931,438934],{"className":438921},[31676],[34,438923],{"className":438924,"style":43217},[31680],[34,438926,329201],{"className":438927},[31685,31686],[34,438929],{"className":438930,"style":31846},[31845],[34,438932,5598],{"className":438933},[31850],[34,438935],{"className":438936,"style":31846},[31845],[34,438938,438940,438943,438946,438949,438952],{"className":438939},[31676],[34,438941],{"className":438942,"style":33218},[31680],[34,438944,34386],{"className":438945,"style":34427},[31685,31686],[34,438947],{"className":438948,"style":31846},[31845],[34,438950,33239],{"className":438951},[31850],[34,438953],{"className":438954,"style":31846},[31845],[34,438956,438958,438961],{"className":438957},[31676],[34,438959],{"className":438960,"style":33218},[31680],[34,438962,94675],{"className":438963,"style":106455},[31685,31686],[34,438965,438967,438991],{"className":438966},[31642],[34,438968,438970],{"className":438969},[31646],[31648,438971,438972],{"xmlns":31650},[31652,438973,438974,438988],{},[31655,438975,438976,438978,438980,438982,438984,438986],{},[31658,438977,365679],{"mathvariant":38079},[31658,438979,111],{},[31662,438981,31960],{"separator":758},[31658,438983,1697],{},[31662,438985,112296],{},[31658,438987,34386],{},[31665,438989,438990],{"encoding":31667},"\\forall a, b \\in R",[34,438992,438994,439024],{"className":438993,"ariaHidden":758},[31672],[34,438995,438997,439000,439003,439006,439009,439012,439015,439018,439021],{"className":438996},[31676],[34,438998],{"className":438999,"style":43217},[31680],[34,439001,365679],{"className":439002},[31685],[34,439004,111],{"className":439005},[31685,31686],[34,439007,31960],{"className":439008},[32119],[34,439010],{"className":439011,"style":32123},[31845],[34,439013,1697],{"className":439014},[31685,31686],[34,439016],{"className":439017,"style":31846},[31845],[34,439019,112296],{"className":439020},[31850],[34,439022],{"className":439023,"style":31846},[31845],[34,439025,439027,439030],{"className":439026},[31676],[34,439028],{"className":439029,"style":33218},[31680],[34,439031,34386],{"className":439032,"style":34427},[31685,31686],"，都有\n",[34,439035,439037,439115],{"className":439036},[31642],[34,439038,439040],{"className":439039},[31646],[31648,439041,439042],{"xmlns":31650},[31652,439043,439044,439112],{},[31655,439045,439046,439048,439050,439052,439054,439056,439058,439060,439062,439064,439066,439068,439070,439072,439074,439076,439078,439080,439082,439084,439086,439088,439090,439092,439094,439096,439098,439100,439102,439104,439106,439108,439110],{},[31658,439047,329201],{},[31662,439049,456],{"stretchy":702},[31658,439051,111],{},[31662,439053,223],{},[31658,439055,1697],{},[31662,439057,2557],{"stretchy":702},[31662,439059,202],{},[31658,439061,329201],{},[31662,439063,456],{"stretchy":702},[31658,439065,111],{},[31662,439067,2557],{"stretchy":702},[31662,439069,223],{},[31658,439071,329201],{},[31662,439073,456],{"stretchy":702},[31658,439075,1697],{},[31662,439077,2557],{"stretchy":702},[31662,439079,31960],{"separator":758},[31845,439081],{"width":71082},[31658,439083,329201],{},[31662,439085,456],{"stretchy":702},[31658,439087,111],{},[31658,439089,1697],{},[31662,439091,2557],{"stretchy":702},[31662,439093,202],{},[31658,439095,329201],{},[31662,439097,456],{"stretchy":702},[31658,439099,111],{},[31662,439101,2557],{"stretchy":702},[31658,439103,329201],{},[31662,439105,456],{"stretchy":702},[31658,439107,1697],{},[31662,439109,2557],{"stretchy":702},[31658,439111,2547],{"mathvariant":38079},[31665,439113,439114],{"encoding":31667},"\\phi(a+b) = \\phi(a) + \\phi(b), \\quad \\phi(ab) = \\phi(a)\\phi(b).",[34,439116,439118,439142,439163,439190,439238],{"className":439117,"ariaHidden":758},[31672],[34,439119,439121,439124,439127,439130,439133,439136,439139],{"className":439120},[31676],[34,439122],{"className":439123,"style":31774},[31680],[34,439125,329201],{"className":439126},[31685,31686],[34,439128,456],{"className":439129},[31782],[34,439131,111],{"className":439132},[31685,31686],[34,439134],{"className":439135,"style":32604},[31845],[34,439137,223],{"className":439138},[32024],[34,439140],{"className":439141,"style":32604},[31845],[34,439143,439145,439148,439151,439154,439157,439160],{"className":439144},[31676],[34,439146],{"className":439147,"style":31774},[31680],[34,439149,1697],{"className":439150},[31685,31686],[34,439152,2557],{"className":439153},[31690],[34,439155],{"className":439156,"style":31846},[31845],[34,439158,202],{"className":439159},[31850],[34,439161],{"className":439162,"style":31846},[31845],[34,439164,439166,439169,439172,439175,439178,439181,439184,439187],{"className":439165},[31676],[34,439167],{"className":439168,"style":31774},[31680],[34,439170,329201],{"className":439171},[31685,31686],[34,439173,456],{"className":439174},[31782],[34,439176,111],{"className":439177},[31685,31686],[34,439179,2557],{"className":439180},[31690],[34,439182],{"className":439183,"style":32604},[31845],[34,439185,223],{"className":439186},[32024],[34,439188],{"className":439189,"style":32604},[31845],[34,439191,439193,439196,439199,439202,439205,439208,439211,439214,439217,439220,439223,439226,439229,439232,439235],{"className":439192},[31676],[34,439194],{"className":439195,"style":31774},[31680],[34,439197,329201],{"className":439198},[31685,31686],[34,439200,456],{"className":439201},[31782],[34,439203,1697],{"className":439204},[31685,31686],[34,439206,2557],{"className":439207},[31690],[34,439209,31960],{"className":439210},[32119],[34,439212],{"className":439213,"style":128438},[31845],[34,439215],{"className":439216,"style":32123},[31845],[34,439218,329201],{"className":439219},[31685,31686],[34,439221,456],{"className":439222},[31782],[34,439224,324191],{"className":439225},[31685,31686],[34,439227,2557],{"className":439228},[31690],[34,439230],{"className":439231,"style":31846},[31845],[34,439233,202],{"className":439234},[31850],[34,439236],{"className":439237,"style":31846},[31845],[34,439239,439241,439244,439247,439250,439253,439256,439259,439262,439265,439268],{"className":439240},[31676],[34,439242],{"className":439243,"style":31774},[31680],[34,439245,329201],{"className":439246},[31685,31686],[34,439248,456],{"className":439249},[31782],[34,439251,111],{"className":439252},[31685,31686],[34,439254,2557],{"className":439255},[31690],[34,439257,329201],{"className":439258},[31685,31686],[34,439260,456],{"className":439261},[31782],[34,439263,1697],{"className":439264},[31685,31686],[34,439266,2557],{"className":439267},[31690],[34,439269,2547],{"className":439270},[31685],"\n则称 ",[34,439273,439275,439288],{"className":439274},[31642],[34,439276,439278],{"className":439277},[31646],[31648,439279,439280],{"xmlns":31650},[31652,439281,439282,439286],{},[31655,439283,439284],{},[31658,439285,329201],{},[31665,439287,364484],{"encoding":31667},[34,439289,439291],{"className":439290,"ariaHidden":758},[31672],[34,439292,439294,439297],{"className":439293},[31676],[34,439295],{"className":439296,"style":43217},[31680],[34,439298,329201],{"className":439299},[31685,31686],[2627,439301,439302],{},"环同态",[15,439304,439305],{},"在此基础上，我们补充以下具体的同态分类定义（原文省略部分）：",[2230,439307,439308,439574,439799,439955,440149],{},[108,439309,439310,439313,439314,439342,439343,439411,439412,68309,439493,439544,439545,439573],{},[2627,439311,439312],{},"单同态 (Monomorphism)",": 若 ",[34,439315,439317,439330],{"className":439316},[31642],[34,439318,439320],{"className":439319},[31646],[31648,439321,439322],{"xmlns":31650},[31652,439323,439324,439328],{},[31655,439325,439326],{},[31658,439327,329201],{},[31665,439329,364484],{"encoding":31667},[34,439331,439333],{"className":439332,"ariaHidden":758},[31672],[34,439334,439336,439339],{"className":439335},[31676],[34,439337],{"className":439338,"style":43217},[31680],[34,439340,329201],{"className":439341},[31685,31686]," 是单射（即 ",[34,439344,439346,439369],{"className":439345},[31642],[34,439347,439349],{"className":439348},[31646],[31648,439350,439351],{"xmlns":31650},[31652,439352,439353,439367],{},[31655,439354,439355,439357,439359,439361,439363,439365],{},[31658,439356,365679],{"mathvariant":38079},[31658,439358,111],{},[31662,439360,31960],{"separator":758},[31658,439362,1697],{},[31662,439364,112296],{},[31658,439366,34386],{},[31665,439368,438990],{"encoding":31667},[34,439370,439372,439402],{"className":439371,"ariaHidden":758},[31672],[34,439373,439375,439378,439381,439384,439387,439390,439393,439396,439399],{"className":439374},[31676],[34,439376],{"className":439377,"style":43217},[31680],[34,439379,365679],{"className":439380},[31685],[34,439382,111],{"className":439383},[31685,31686],[34,439385,31960],{"className":439386},[32119],[34,439388],{"className":439389,"style":32123},[31845],[34,439391,1697],{"className":439392},[31685,31686],[34,439394],{"className":439395,"style":31846},[31845],[34,439397,112296],{"className":439398},[31850],[34,439400],{"className":439401,"style":31846},[31845],[34,439403,439405,439408],{"className":439404},[31676],[34,439406],{"className":439407,"style":33218},[31680],[34,439409,34386],{"className":439410,"style":34427},[31685,31686],"，若 ",[34,439413,439415,439445],{"className":439414},[31642],[34,439416,439418],{"className":439417},[31646],[31648,439419,439420],{"xmlns":31650},[31652,439421,439422,439442],{},[31655,439423,439424,439426,439428,439430,439432,439434,439436,439438,439440],{},[31658,439425,329201],{},[31662,439427,456],{"stretchy":702},[31658,439429,111],{},[31662,439431,2557],{"stretchy":702},[31662,439433,202],{},[31658,439435,329201],{},[31662,439437,456],{"stretchy":702},[31658,439439,1697],{},[31662,439441,2557],{"stretchy":702},[31665,439443,439444],{"encoding":31667},"\\phi(a) = \\phi(b)",[34,439446,439448,439475],{"className":439447,"ariaHidden":758},[31672],[34,439449,439451,439454,439457,439460,439463,439466,439469,439472],{"className":439450},[31676],[34,439452],{"className":439453,"style":31774},[31680],[34,439455,329201],{"className":439456},[31685,31686],[34,439458,456],{"className":439459},[31782],[34,439461,111],{"className":439462},[31685,31686],[34,439464,2557],{"className":439465},[31690],[34,439467],{"className":439468,"style":31846},[31845],[34,439470,202],{"className":439471},[31850],[34,439473],{"className":439474,"style":31846},[31845],[34,439476,439478,439481,439484,439487,439490],{"className":439477},[31676],[34,439479],{"className":439480,"style":31774},[31680],[34,439482,329201],{"className":439483},[31685,31686],[34,439485,456],{"className":439486},[31782],[34,439488,1697],{"className":439489},[31685,31686],[34,439491,2557],{"className":439492},[31690],[34,439494,439496,439514],{"className":439495},[31642],[34,439497,439499],{"className":439498},[31646],[31648,439500,439501],{"xmlns":31650},[31652,439502,439503,439511],{},[31655,439504,439505,439507,439509],{},[31658,439506,111],{},[31662,439508,202],{},[31658,439510,1697],{},[31665,439512,439513],{"encoding":31667},"a=b",[34,439515,439517,439535],{"className":439516,"ariaHidden":758},[31672],[34,439518,439520,439523,439526,439529,439532],{"className":439519},[31676],[34,439521],{"className":439522,"style":32527},[31680],[34,439524,111],{"className":439525},[31685,31686],[34,439527],{"className":439528,"style":31846},[31845],[34,439530,202],{"className":439531},[31850],[34,439533],{"className":439534,"style":31846},[31845],[34,439536,439538,439541],{"className":439537},[31676],[34,439539],{"className":439540,"style":31681},[31680],[34,439542,1697],{"className":439543},[31685,31686],"），称 ",[34,439546,439548,439561],{"className":439547},[31642],[34,439549,439551],{"className":439550},[31646],[31648,439552,439553],{"xmlns":31650},[31652,439554,439555,439559],{},[31655,439556,439557],{},[31658,439558,329201],{},[31665,439560,364484],{"encoding":31667},[34,439562,439564],{"className":439563,"ariaHidden":758},[31672],[34,439565,439567,439570],{"className":439566},[31676],[34,439568],{"className":439569,"style":43217},[31680],[34,439571,329201],{"className":439572},[31685,31686]," 为单同态。",[108,439575,439576,439313,439579,439607,439608,315343,439704,439544,439770,439798],{},[2627,439577,439578],{},"满同态 (Epimorphism)",[34,439580,439582,439595],{"className":439581},[31642],[34,439583,439585],{"className":439584},[31646],[31648,439586,439587],{"xmlns":31650},[31652,439588,439589,439593],{},[31655,439590,439591],{},[31658,439592,329201],{},[31665,439594,364484],{"encoding":31667},[34,439596,439598],{"className":439597,"ariaHidden":758},[31672],[34,439599,439601,439604],{"className":439600},[31676],[34,439602],{"className":439603,"style":43217},[31680],[34,439605,329201],{"className":439606},[31685,31686]," 是满射（即 ",[34,439609,439611,439641],{"className":439610},[31642],[34,439612,439614],{"className":439613},[31646],[31648,439615,439616],{"xmlns":31650},[31652,439617,439618,439638],{},[31655,439619,439620,439622,439624,439626,439628,439630,439632,439634,439636],{},[31658,439621,365679],{"mathvariant":38079},[31658,439623,31747],{},[31662,439625,112296],{},[31658,439627,94675],{},[31662,439629,31960],{"separator":758},[31658,439631,365989],{"mathvariant":38079},[31658,439633,34395],{},[31662,439635,112296],{},[31658,439637,34386],{},[31665,439639,439640],{"encoding":31667},"\\forall s \\in S, \\exists r \\in R",[34,439642,439644,439665,439695],{"className":439643,"ariaHidden":758},[31672],[34,439645,439647,439650,439653,439656,439659,439662],{"className":439646},[31676],[34,439648],{"className":439649,"style":46991},[31680],[34,439651,365679],{"className":439652},[31685],[34,439654,31747],{"className":439655},[31685,31686],[34,439657],{"className":439658,"style":31846},[31845],[34,439660,112296],{"className":439661},[31850],[34,439663],{"className":439664,"style":31846},[31845],[34,439666,439668,439671,439674,439677,439680,439683,439686,439689,439692],{"className":439667},[31676],[34,439669],{"className":439670,"style":43217},[31680],[34,439672,94675],{"className":439673,"style":106455},[31685,31686],[34,439675,31960],{"className":439676},[32119],[34,439678],{"className":439679,"style":32123},[31845],[34,439681,365989],{"className":439682},[31685],[34,439684,34395],{"className":439685,"style":34452},[31685,31686],[34,439687],{"className":439688,"style":31846},[31845],[34,439690,112296],{"className":439691},[31850],[34,439693],{"className":439694,"style":31846},[31845],[34,439696,439698,439701],{"className":439697},[31676],[34,439699],{"className":439700,"style":33218},[31680],[34,439702,34386],{"className":439703,"style":34427},[31685,31686],[34,439705,439707,439731],{"className":439706},[31642],[34,439708,439710],{"className":439709},[31646],[31648,439711,439712],{"xmlns":31650},[31652,439713,439714,439728],{},[31655,439715,439716,439718,439720,439722,439724,439726],{},[31658,439717,329201],{},[31662,439719,456],{"stretchy":702},[31658,439721,34395],{},[31662,439723,2557],{"stretchy":702},[31662,439725,202],{},[31658,439727,31747],{},[31665,439729,439730],{"encoding":31667},"\\phi(r)=s",[34,439732,439734,439761],{"className":439733,"ariaHidden":758},[31672],[34,439735,439737,439740,439743,439746,439749,439752,439755,439758],{"className":439736},[31676],[34,439738],{"className":439739,"style":31774},[31680],[34,439741,329201],{"className":439742},[31685,31686],[34,439744,456],{"className":439745},[31782],[34,439747,34395],{"className":439748,"style":34452},[31685,31686],[34,439750,2557],{"className":439751},[31690],[34,439753],{"className":439754,"style":31846},[31845],[34,439756,202],{"className":439757},[31850],[34,439759],{"className":439760,"style":31846},[31845],[34,439762,439764,439767],{"className":439763},[31676],[34,439765],{"className":439766,"style":32527},[31680],[34,439768,31747],{"className":439769},[31685,31686],[34,439771,439773,439786],{"className":439772},[31642],[34,439774,439776],{"className":439775},[31646],[31648,439777,439778],{"xmlns":31650},[31652,439779,439780,439784],{},[31655,439781,439782],{},[31658,439783,329201],{},[31665,439785,364484],{"encoding":31667},[34,439787,439789],{"className":439788,"ariaHidden":758},[31672],[34,439790,439792,439795],{"className":439791},[31676],[34,439793],{"className":439794,"style":43217},[31680],[34,439796,329201],{"className":439797},[31685,31686]," 为满同态。",[108,439800,439801,439313,439804,439832,439833,439861,439862,439903,439904,347],{},[2627,439802,439803],{},"同构 (Isomorphism)",[34,439805,439807,439820],{"className":439806},[31642],[34,439808,439810],{"className":439809},[31646],[31648,439811,439812],{"xmlns":31650},[31652,439813,439814,439818],{},[31655,439815,439816],{},[31658,439817,329201],{},[31665,439819,364484],{"encoding":31667},[34,439821,439823],{"className":439822,"ariaHidden":758},[31672],[34,439824,439826,439829],{"className":439825},[31676],[34,439827],{"className":439828,"style":43217},[31680],[34,439830,329201],{"className":439831},[31685,31686]," 既是单射又是满射（即双射），称 ",[34,439834,439836,439849],{"className":439835},[31642],[34,439837,439839],{"className":439838},[31646],[31648,439840,439841],{"xmlns":31650},[31652,439842,439843,439847],{},[31655,439844,439845],{},[31658,439846,329201],{},[31665,439848,364484],{"encoding":31667},[34,439850,439852],{"className":439851,"ariaHidden":758},[31672],[34,439853,439855,439858],{"className":439854},[31676],[34,439856],{"className":439857,"style":43217},[31680],[34,439859,329201],{"className":439860},[31685,31686]," 为同构。若 ",[34,439863,439865,439882],{"className":439864},[31642],[34,439866,439868],{"className":439867},[31646],[31648,439869,439870],{"xmlns":31650},[31652,439871,439872,439880],{},[31655,439873,439874,439876,439878],{},[31658,439875,34386],{},[31662,439877,31960],{"separator":758},[31658,439879,94675],{},[31665,439881,438868],{"encoding":31667},[34,439883,439885],{"className":439884,"ariaHidden":758},[31672],[34,439886,439888,439891,439894,439897,439900],{"className":439887},[31676],[34,439889],{"className":439890,"style":70739},[31680],[34,439892,34386],{"className":439893,"style":34427},[31685,31686],[34,439895,31960],{"className":439896},[32119],[34,439898],{"className":439899,"style":32123},[31845],[34,439901,94675],{"className":439902,"style":106455},[31685,31686]," 之间存在同构，记为 ",[34,439905,439907,439925],{"className":439906},[31642],[34,439908,439910],{"className":439909},[31646],[31648,439911,439912],{"xmlns":31650},[31652,439913,439914,439922],{},[31655,439915,439916,439918,439920],{},[31658,439917,34386],{},[31662,439919,134826],{},[31658,439921,94675],{},[31665,439923,439924],{"encoding":31667},"R \\cong S",[34,439926,439928,439946],{"className":439927,"ariaHidden":758},[31672],[34,439929,439931,439934,439937,439940,439943],{"className":439930},[31676],[34,439932],{"className":439933,"style":33218},[31680],[34,439935,34386],{"className":439936,"style":34427},[31685,31686],[34,439938],{"className":439939,"style":31846},[31845],[34,439941,134826],{"className":439942},[31850],[34,439944],{"className":439945,"style":31846},[31845],[34,439947,439949,439952],{"className":439948},[31676],[34,439950],{"className":439951,"style":33218},[31680],[34,439953,94675],{"className":439954,"style":106455},[31685,31686],[108,439956,439957,439313,439960,440011,440012,440040,440041,440069,440070,440098,440099,347],{},[2627,439958,439959],{},"自同态 (Endomorphism)",[34,439961,439963,439981],{"className":439962},[31642],[34,439964,439966],{"className":439965},[31646],[31648,439967,439968],{"xmlns":31650},[31652,439969,439970,439978],{},[31655,439971,439972,439974,439976],{},[31658,439973,94675],{},[31662,439975,202],{},[31658,439977,34386],{},[31665,439979,439980],{"encoding":31667},"S=R",[34,439982,439984,440002],{"className":439983,"ariaHidden":758},[31672],[34,439985,439987,439990,439993,439996,439999],{"className":439986},[31676],[34,439988],{"className":439989,"style":33218},[31680],[34,439991,94675],{"className":439992,"style":106455},[31685,31686],[34,439994],{"className":439995,"style":31846},[31845],[34,439997,202],{"className":439998},[31850],[34,440000],{"className":440001,"style":31846},[31845],[34,440003,440005,440008],{"className":440004},[31676],[34,440006],{"className":440007,"style":33218},[31680],[34,440009,34386],{"className":440010,"style":34427},[31685,31686],"，即从环 ",[34,440013,440015,440028],{"className":440014},[31642],[34,440016,440018],{"className":440017},[31646],[31648,440019,440020],{"xmlns":31650},[31652,440021,440022,440026],{},[31655,440023,440024],{},[31658,440025,34386],{},[31665,440027,34386],{"encoding":31667},[34,440029,440031],{"className":440030,"ariaHidden":758},[31672],[34,440032,440034,440037],{"className":440033},[31676],[34,440035],{"className":440036,"style":33218},[31680],[34,440038,34386],{"className":440039,"style":34427},[31685,31686]," 到自身的同态，称 ",[34,440042,440044,440057],{"className":440043},[31642],[34,440045,440047],{"className":440046},[31646],[31648,440048,440049],{"xmlns":31650},[31652,440050,440051,440055],{},[31655,440052,440053],{},[31658,440054,329201],{},[31665,440056,364484],{"encoding":31667},[34,440058,440060],{"className":440059,"ariaHidden":758},[31672],[34,440061,440063,440066],{"className":440062},[31676],[34,440064],{"className":440065,"style":43217},[31680],[34,440067,329201],{"className":440068},[31685,31686]," 为自同态。所有 ",[34,440071,440073,440086],{"className":440072},[31642],[34,440074,440076],{"className":440075},[31646],[31648,440077,440078],{"xmlns":31650},[31652,440079,440080,440084],{},[31655,440081,440082],{},[31658,440083,34386],{},[31665,440085,34386],{"encoding":31667},[34,440087,440089],{"className":440088,"ariaHidden":758},[31672],[34,440090,440092,440095],{"className":440091},[31676],[34,440093],{"className":440094,"style":33218},[31680],[34,440096,34386],{"className":440097,"style":34427},[31685,31686]," 的自同态构成的集合记为 ",[34,440100,440102,440125],{"className":440101},[31642],[34,440103,440105],{"className":440104},[31646],[31648,440106,440107],{"xmlns":31650},[31652,440108,440109,440122],{},[31655,440110,440111,440114,440116,440118,440120],{},[31658,440112,440113],{"mathvariant":38079},"End",[31662,440115,32682],{},[31662,440117,456],{"stretchy":702},[31658,440119,34386],{},[31662,440121,2557],{"stretchy":702},[31665,440123,440124],{"encoding":31667},"\\operatorname{End}(R)",[34,440126,440128],{"className":440127,"ariaHidden":758},[31672],[34,440129,440131,440134,440140,440143,440146],{"className":440130},[31676],[34,440132],{"className":440133,"style":31774},[31680],[34,440135,440137],{"className":440136},[32767],[34,440138,440113],{"className":440139},[31685,207923],[34,440141,456],{"className":440142},[31782],[34,440144,34386],{"className":440145,"style":34427},[31685,31686],[34,440147,2557],{"className":440148},[31690],[108,440150,440151,439313,440154,11008,440182,53834,440210,440238,440239,440267,440268,440296,440297,440347],{},[2627,440152,440153],{},"自同构 (Automorphism)",[34,440155,440157,440170],{"className":440156},[31642],[34,440158,440160],{"className":440159},[31646],[31648,440161,440162],{"xmlns":31650},[31652,440163,440164,440168],{},[31655,440165,440166],{},[31658,440167,329201],{},[31665,440169,364484],{"encoding":31667},[34,440171,440173],{"className":440172,"ariaHidden":758},[31672],[34,440174,440176,440179],{"className":440175},[31676],[34,440177],{"className":440178,"style":43217},[31680],[34,440180,329201],{"className":440181},[31685,31686],[34,440183,440185,440198],{"className":440184},[31642],[34,440186,440188],{"className":440187},[31646],[31648,440189,440190],{"xmlns":31650},[31652,440191,440192,440196],{},[31655,440193,440194],{},[31658,440195,34386],{},[31665,440197,34386],{"encoding":31667},[34,440199,440201],{"className":440200,"ariaHidden":758},[31672],[34,440202,440204,440207],{"className":440203},[31676],[34,440205],{"className":440206,"style":33218},[31680],[34,440208,34386],{"className":440209,"style":34427},[31685,31686],[34,440211,440213,440226],{"className":440212},[31642],[34,440214,440216],{"className":440215},[31646],[31648,440217,440218],{"xmlns":31650},[31652,440219,440220,440224],{},[31655,440221,440222],{},[31658,440223,34386],{},[31665,440225,34386],{"encoding":31667},[34,440227,440229],{"className":440228,"ariaHidden":758},[31672],[34,440230,440232,440235],{"className":440231},[31676],[34,440233],{"className":440234,"style":33218},[31680],[34,440236,34386],{"className":440237,"style":34427},[31685,31686]," 的同构，称 ",[34,440240,440242,440255],{"className":440241},[31642],[34,440243,440245],{"className":440244},[31646],[31648,440246,440247],{"xmlns":31650},[31652,440248,440249,440253],{},[31655,440250,440251],{},[31658,440252,329201],{},[31665,440254,364484],{"encoding":31667},[34,440256,440258],{"className":440257,"ariaHidden":758},[31672],[34,440259,440261,440264],{"className":440260},[31676],[34,440262],{"className":440263,"style":43217},[31680],[34,440265,329201],{"className":440266},[31685,31686]," 为自同构。所有 ",[34,440269,440271,440284],{"className":440270},[31642],[34,440272,440274],{"className":440273},[31646],[31648,440275,440276],{"xmlns":31650},[31652,440277,440278,440282],{},[31655,440279,440280],{},[31658,440281,34386],{},[31665,440283,34386],{"encoding":31667},[34,440285,440287],{"className":440286,"ariaHidden":758},[31672],[34,440288,440290,440293],{"className":440289},[31676],[34,440291],{"className":440292,"style":33218},[31680],[34,440294,34386],{"className":440295,"style":34427},[31685,31686]," 的自同构构成的集合记为 ",[34,440298,440300,440323],{"className":440299},[31642],[34,440301,440303],{"className":440302},[31646],[31648,440304,440305],{"xmlns":31650},[31652,440306,440307,440320],{},[31655,440308,440309,440312,440314,440316,440318],{},[31658,440310,440311],{"mathvariant":38079},"Aut",[31662,440313,32682],{},[31662,440315,456],{"stretchy":702},[31658,440317,34386],{},[31662,440319,2557],{"stretchy":702},[31665,440321,440322],{"encoding":31667},"\\operatorname{Aut}(R)",[34,440324,440326],{"className":440325,"ariaHidden":758},[31672],[34,440327,440329,440332,440338,440341,440344],{"className":440328},[31676],[34,440330],{"className":440331,"style":31774},[31680],[34,440333,440335],{"className":440334},[32767],[34,440336,440311],{"className":440337},[31685,207923],[34,440339,456],{"className":440340},[31782],[34,440342,34386],{"className":440343,"style":34427},[31685,31686],[34,440345,2557],{"className":440346},[31690],"（它关于映射复合构成一个群）。",[15,440349,440350,69123,440353,440425],{},[2627,440351,440352],{},"定义.",[34,440354,440356,440377],{"className":440355},[31642],[34,440357,440359],{"className":440358},[31646],[31648,440360,440361],{"xmlns":31650},[31652,440362,440363,440375],{},[31655,440364,440365,440367,440369,440371,440373],{},[31658,440366,329201],{},[31662,440368,5598],{},[31658,440370,34386],{},[31662,440372,33239],{},[31658,440374,94675],{},[31665,440376,438915],{"encoding":31667},[34,440378,440380,440398,440416],{"className":440379,"ariaHidden":758},[31672],[34,440381,440383,440386,440389,440392,440395],{"className":440382},[31676],[34,440384],{"className":440385,"style":43217},[31680],[34,440387,329201],{"className":440388},[31685,31686],[34,440390],{"className":440391,"style":31846},[31845],[34,440393,5598],{"className":440394},[31850],[34,440396],{"className":440397,"style":31846},[31845],[34,440399,440401,440404,440407,440410,440413],{"className":440400},[31676],[34,440402],{"className":440403,"style":33218},[31680],[34,440405,34386],{"className":440406,"style":34427},[31685,31686],[34,440408],{"className":440409,"style":31846},[31845],[34,440411,33239],{"className":440412},[31850],[34,440414],{"className":440415,"style":31846},[31845],[34,440417,440419,440422],{"className":440418},[31676],[34,440420],{"className":440421,"style":33218},[31680],[34,440423,94675],{"className":440424,"style":106455},[31685,31686]," 是环同态。",[105,440427,440428,440585],{},[108,440429,440430,8471,440433,347],{},[2627,440431,440432],{},"核 (Kernel)",[34,440434,440436,440480],{"className":440435},[31642],[34,440437,440439],{"className":440438},[31646],[31648,440440,440441],{"xmlns":31650},[31652,440442,440443,440477],{},[31655,440444,440445,440447,440449,440451,440453,440455,440457,440459,440461,440463,440465,440467,440469,440471,440473,440475],{},[31658,440446,406335],{},[31662,440448,32682],{},[31658,440450,329201],{},[31662,440452,202],{},[31662,440454,32685],{"stretchy":702},[31658,440456,111],{},[31662,440458,112296],{},[31658,440460,34386],{},[31662,440462,162086],{},[31658,440464,329201],{},[31662,440466,456],{"stretchy":702},[31658,440468,111],{},[31662,440470,2557],{"stretchy":702},[31662,440472,202],{},[31749,440474,459],{},[31662,440476,32715],{"stretchy":702},[31665,440478,440479],{"encoding":31667},"\\ker \\phi = \\{a \\in R \\mid \\phi(a) = 0\\}",[34,440481,440483,440507,440528,440546,440573],{"className":440482,"ariaHidden":758},[31672],[34,440484,440486,440489,440492,440495,440498,440501,440504],{"className":440485},[31676],[34,440487],{"className":440488,"style":43217},[31680],[34,440490,406335],{"className":440491},[32767],[34,440493],{"className":440494,"style":32123},[31845],[34,440496,329201],{"className":440497},[31685,31686],[34,440499],{"className":440500,"style":31846},[31845],[34,440502,202],{"className":440503},[31850],[34,440505],{"className":440506,"style":31846},[31845],[34,440508,440510,440513,440516,440519,440522,440525],{"className":440509},[31676],[34,440511],{"className":440512,"style":31774},[31680],[34,440514,32685],{"className":440515},[31782],[34,440517,111],{"className":440518},[31685,31686],[34,440520],{"className":440521,"style":31846},[31845],[34,440523,112296],{"className":440524},[31850],[34,440526],{"className":440527,"style":31846},[31845],[34,440529,440531,440534,440537,440540,440543],{"className":440530},[31676],[34,440532],{"className":440533,"style":31774},[31680],[34,440535,34386],{"className":440536,"style":34427},[31685,31686],[34,440538],{"className":440539,"style":31846},[31845],[34,440541,162086],{"className":440542},[31850],[34,440544],{"className":440545,"style":31846},[31845],[34,440547,440549,440552,440555,440558,440561,440564,440567,440570],{"className":440548},[31676],[34,440550],{"className":440551,"style":31774},[31680],[34,440553,329201],{"className":440554},[31685,31686],[34,440556,456],{"className":440557},[31782],[34,440559,111],{"className":440560},[31685,31686],[34,440562,2557],{"className":440563},[31690],[34,440565],{"className":440566,"style":31846},[31845],[34,440568,202],{"className":440569},[31850],[34,440571],{"className":440572,"style":31846},[31845],[34,440574,440576,440579,440582],{"className":440575},[31676],[34,440577],{"className":440578,"style":31774},[31680],[34,440580,459],{"className":440581},[31685],[34,440583,32715],{"className":440584},[31690],[108,440586,440587,8471,440590,347],{},[2627,440588,440589],{},"像 (Image)",[34,440591,440593,440634],{"className":440592},[31642],[34,440594,440596],{"className":440595},[31646],[31648,440597,440598],{"xmlns":31650},[31652,440599,440600,440631],{},[31655,440601,440602,440605,440607,440609,440611,440613,440615,440617,440619,440621,440623,440625,440627,440629],{},[31658,440603,440604],{"mathvariant":38079},"Im",[31662,440606,32682],{},[31658,440608,329201],{},[31662,440610,202],{},[31662,440612,32685],{"stretchy":702},[31658,440614,329201],{},[31662,440616,456],{"stretchy":702},[31658,440618,111],{},[31662,440620,2557],{"stretchy":702},[31662,440622,162086],{},[31658,440624,111],{},[31662,440626,112296],{},[31658,440628,34386],{},[31662,440630,32715],{"stretchy":702},[31665,440632,440633],{"encoding":31667},"\\operatorname{Im} \\phi = \\{\\phi(a) \\mid a \\in R\\}",[34,440635,440637,440664,440694,440712],{"className":440636,"ariaHidden":758},[31672],[34,440638,440640,440643,440649,440652,440655,440658,440661],{"className":440639},[31676],[34,440641],{"className":440642,"style":43217},[31680],[34,440644,440646],{"className":440645},[32767],[34,440647,440604],{"className":440648},[31685,207923],[34,440650],{"className":440651,"style":32123},[31845],[34,440653,329201],{"className":440654},[31685,31686],[34,440656],{"className":440657,"style":31846},[31845],[34,440659,202],{"className":440660},[31850],[34,440662],{"className":440663,"style":31846},[31845],[34,440665,440667,440670,440673,440676,440679,440682,440685,440688,440691],{"className":440666},[31676],[34,440668],{"className":440669,"style":31774},[31680],[34,440671,32685],{"className":440672},[31782],[34,440674,329201],{"className":440675},[31685,31686],[34,440677,456],{"className":440678},[31782],[34,440680,111],{"className":440681},[31685,31686],[34,440683,2557],{"className":440684},[31690],[34,440686],{"className":440687,"style":31846},[31845],[34,440689,162086],{"className":440690},[31850],[34,440692],{"className":440693,"style":31846},[31845],[34,440695,440697,440700,440703,440706,440709],{"className":440696},[31676],[34,440698],{"className":440699,"style":34812},[31680],[34,440701,111],{"className":440702},[31685,31686],[34,440704],{"className":440705,"style":31846},[31845],[34,440707,112296],{"className":440708},[31850],[34,440710],{"className":440711,"style":31846},[31845],[34,440713,440715,440718,440721],{"className":440714},[31676],[34,440716],{"className":440717,"style":31774},[31680],[34,440719,34386],{"className":440720,"style":34427},[31685,31686],[34,440722,32715],{"className":440723},[31690],[32188,440725],{},[253,440727,440729],{"id":440728},"_2-同态的基本性质","2. 同态的基本性质",[15,440731,440732,69123,440735,440807,440808,440870,440871,440899,440900,440964,440965,440993,440994,441022,441023,2547],{},[2627,440733,440734],{},"命题 3.3.1.",[34,440736,440738,440759],{"className":440737},[31642],[34,440739,440741],{"className":440740},[31646],[31648,440742,440743],{"xmlns":31650},[31652,440744,440745,440757],{},[31655,440746,440747,440749,440751,440753,440755],{},[31658,440748,329201],{},[31662,440750,5598],{},[31658,440752,34386],{},[31662,440754,33239],{},[31658,440756,94675],{},[31665,440758,438915],{"encoding":31667},[34,440760,440762,440780,440798],{"className":440761,"ariaHidden":758},[31672],[34,440763,440765,440768,440771,440774,440777],{"className":440764},[31676],[34,440766],{"className":440767,"style":43217},[31680],[34,440769,329201],{"className":440770},[31685,31686],[34,440772],{"className":440773,"style":31846},[31845],[34,440775,5598],{"className":440776},[31850],[34,440778],{"className":440779,"style":31846},[31845],[34,440781,440783,440786,440789,440792,440795],{"className":440782},[31676],[34,440784],{"className":440785,"style":33218},[31680],[34,440787,34386],{"className":440788,"style":34427},[31685,31686],[34,440790],{"className":440791,"style":31846},[31845],[34,440793,33239],{"className":440794},[31850],[34,440796],{"className":440797,"style":31846},[31845],[34,440799,440801,440804],{"className":440800},[31676],[34,440802],{"className":440803,"style":33218},[31680],[34,440805,94675],{"className":440806,"style":106455},[31685,31686]," 是一个环同态。\n(i) ",[34,440809,440811,440834],{"className":440810},[31642],[34,440812,440814],{"className":440813},[31646],[31648,440815,440816],{"xmlns":31650},[31652,440817,440818,440831],{},[31655,440819,440820,440822,440824,440826,440829],{},[31658,440821,406335],{},[31662,440823,32682],{},[31658,440825,329201],{},[31662,440827,440828],{},"⊲",[31658,440830,34386],{},[31665,440832,440833],{"encoding":31667},"\\ker \\phi \\lhd R",[34,440835,440837,440861],{"className":440836,"ariaHidden":758},[31672],[34,440838,440840,440843,440846,440849,440852,440855,440858],{"className":440839},[31676],[34,440841],{"className":440842,"style":43217},[31680],[34,440844,406335],{"className":440845},[32767],[34,440847],{"className":440848,"style":32123},[31845],[34,440850,329201],{"className":440851},[31685,31686],[34,440853],{"className":440854,"style":32604},[31845],[34,440856,440828],{"className":440857},[32024,129261],[34,440859],{"className":440860,"style":32604},[31845],[34,440862,440864,440867],{"className":440863},[31676],[34,440865],{"className":440866,"style":33218},[31680],[34,440868,34386],{"className":440869,"style":34427},[31685,31686]," (核是 ",[34,440872,440874,440887],{"className":440873},[31642],[34,440875,440877],{"className":440876},[31646],[31648,440878,440879],{"xmlns":31650},[31652,440880,440881,440885],{},[31655,440882,440883],{},[31658,440884,34386],{},[31665,440886,34386],{"encoding":31667},[34,440888,440890],{"className":440889,"ariaHidden":758},[31672],[34,440891,440893,440896],{"className":440892},[31676],[34,440894],{"className":440895,"style":33218},[31680],[34,440897,34386],{"className":440898,"style":34427},[31685,31686]," 的理想), ",[34,440901,440903,440925],{"className":440902},[31642],[34,440904,440906],{"className":440905},[31646],[31648,440907,440908],{"xmlns":31650},[31652,440909,440910,440922],{},[31655,440911,440912,440914,440916,440918,440920],{},[31658,440913,440604],{"mathvariant":38079},[31662,440915,32682],{},[31658,440917,329201],{},[31662,440919,45972],{},[31658,440921,94675],{},[31665,440923,440924],{"encoding":31667},"\\operatorname{Im} \\phi \\le S",[34,440926,440928,440955],{"className":440927,"ariaHidden":758},[31672],[34,440929,440931,440934,440940,440943,440946,440949,440952],{"className":440930},[31676],[34,440932],{"className":440933,"style":43217},[31680],[34,440935,440937],{"className":440936},[32767],[34,440938,440604],{"className":440939},[31685,207923],[34,440941],{"className":440942,"style":32123},[31845],[34,440944,329201],{"className":440945},[31685,31686],[34,440947],{"className":440948,"style":31846},[31845],[34,440950,45972],{"className":440951},[31850],[34,440953],{"className":440954,"style":31846},[31845],[34,440956,440958,440961],{"className":440957},[31676],[34,440959],{"className":440960,"style":33218},[31680],[34,440962,94675],{"className":440963,"style":106455},[31685,31686]," (像是 ",[34,440966,440968,440981],{"className":440967},[31642],[34,440969,440971],{"className":440970},[31646],[31648,440972,440973],{"xmlns":31650},[31652,440974,440975,440979],{},[31655,440976,440977],{},[31658,440978,94675],{},[31665,440980,94675],{"encoding":31667},[34,440982,440984],{"className":440983,"ariaHidden":758},[31672],[34,440985,440987,440990],{"className":440986},[31676],[34,440988],{"className":440989,"style":33218},[31680],[34,440991,94675],{"className":440992,"style":106455},[31685,31686]," 的子环).\n(ii) ",[34,440995,440997,441010],{"className":440996},[31642],[34,440998,441000],{"className":440999},[31646],[31648,441001,441002],{"xmlns":31650},[31652,441003,441004,441008],{},[31655,441005,441006],{},[31658,441007,329201],{},[31665,441009,364484],{"encoding":31667},[34,441011,441013],{"className":441012,"ariaHidden":758},[31672],[34,441014,441016,441019],{"className":441015},[31676],[34,441017],{"className":441018,"style":43217},[31680],[34,441020,329201],{"className":441021},[31685,31686]," 是单射，当且仅当 ",[34,441024,441026,441052],{"className":441025},[31642],[34,441027,441029],{"className":441028},[31646],[31648,441030,441031],{"xmlns":31650},[31652,441032,441033,441049],{},[31655,441034,441035,441037,441039,441041,441043,441045,441047],{},[31658,441036,406335],{},[31662,441038,32682],{},[31658,441040,329201],{},[31662,441042,202],{},[31662,441044,32685],{"stretchy":702},[31749,441046,459],{},[31662,441048,32715],{"stretchy":702},[31665,441050,441051],{"encoding":31667},"\\ker \\phi = \\{0\\}",[34,441053,441055,441079],{"className":441054,"ariaHidden":758},[31672],[34,441056,441058,441061,441064,441067,441070,441073,441076],{"className":441057},[31676],[34,441059],{"className":441060,"style":43217},[31680],[34,441062,406335],{"className":441063},[32767],[34,441065],{"className":441066,"style":32123},[31845],[34,441068,329201],{"className":441069},[31685,31686],[34,441071],{"className":441072,"style":31846},[31845],[34,441074,202],{"className":441075},[31850],[34,441077],{"className":441078,"style":31846},[31845],[34,441080,441082,441085,441088,441091],{"className":441081},[31676],[34,441083],{"className":441084,"style":31774},[31680],[34,441086,32685],{"className":441087},[31782],[34,441089,459],{"className":441090},[31685],[34,441092,32715],{"className":441093},[31690],[15,441095,441096],{},[2627,441097,441098],{},"证明 (补全作业):",[15,441100,441101],{},[2627,441102,441103,441104,258,441164],{},"(i) 证明 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证明 ",[34,441235,441237,441254],{"className":441236},[31642],[34,441238,441240],{"className":441239},[31646],[31648,441241,441242],{"xmlns":31650},[31652,441243,441244,441252],{},[31655,441245,441246,441248,441250],{},[31658,441247,406335],{},[31662,441249,32682],{},[31658,441251,329201],{},[31665,441253,406342],{"encoding":31667},[34,441255,441257],{"className":441256,"ariaHidden":758},[31672],[34,441258,441260,441263,441266,441269],{"className":441259},[31676],[34,441261],{"className":441262,"style":43217},[31680],[34,441264,406335],{"className":441265},[32767],[34,441267],{"className":441268,"style":32123},[31845],[34,441270,329201],{"className":441271},[31685,31686],[34,441273,441275,441288],{"className":441274},[31642],[34,441276,441278],{"className":441277},[31646],[31648,441279,441280],{"xmlns":31650},[31652,441281,441282,441286],{},[31655,441283,441284],{},[31658,441285,34386],{},[31665,441287,34386],{"encoding":31667},[34,441289,441291],{"className":441290,"ariaHidden":758},[31672],[34,441292,441294,441297],{"className":441293},[31676],[34,441295],{"className":441296,"style":33218},[31680],[34,441298,34386],{"className":441299,"style":34427},[31685,31686]," 的理想:",[105,441302,441303,441682,442179],{},[108,441304,441305,441308,441309,441337,441338,13977,441486,61605,441588,347],{},[2627,441306,441307],{},"非空性",": 因为 ",[34,441310,441312,441325],{"className":441311},[31642],[34,441313,441315],{"className":441314},[31646],[31648,441316,441317],{"xmlns":31650},[31652,441318,441319,441323],{},[31655,441320,441321],{},[31658,441322,329201],{},[31665,441324,364484],{"encoding":31667},[34,441326,441328],{"className":441327,"ariaHidden":758},[31672],[34,441329,441331,441334],{"className":441330},[31676],[34,441332],{"className":441333,"style":43217},[31680],[34,441335,329201],{"className":441336},[31685,31686]," 是加法群同态，故 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0_S",[34,441374,441376,441440],{"className":441375,"ariaHidden":758},[31672],[34,441377,441379,441382,441385,441388,441428,441431,441434,441437],{"className":441378},[31676],[34,441380],{"className":441381,"style":31774},[31680],[34,441383,329201],{"className":441384},[31685,31686],[34,441386,456],{"className":441387},[31782],[34,441389,441391,441394],{"className":441390},[31685],[34,441392,459],{"className":441393},[31685],[34,441395,441397],{"className":441396},[31792],[34,441398,441400,441420],{"className":441399},[31796,31797],[34,441401,441403,441417],{"className":441402},[31801],[34,441404,441406],{"className":441405,"style":116558},[31805],[34,441407,441408,441411],{"style":31809},[34,441409],{"className":441410,"style":31814},[31813],[34,441412,441414],{"className":441413},[31818,31819,31820,31821],[34,441415,34386],{"className":441416,"style":34427},[31685,31686,31821],[34,441418,31829],{"className":441419},[31828],[34,441421,441423],{"className":441422},[31801],[34,441424,441426],{"className":441425,"style":31836},[31805],[34,441427],{},[34,441429,2557],{"className":441430},[31690],[34,441432],{"className":441433,"style":31846},[31845],[34,441435,202],{"className":441436},[31850],[34,441438],{"className":441439,"style":31846},[31845],[34,441441,441443,441446],{"className":441442},[31676],[34,441444],{"className":441445,"style":38357},[31680],[34,441447,441449,441452],{"className":441448},[31685],[34,441450,459],{"className":441451},[31685],[34,441453,441455],{"className":441454},[31792],[34,441456,441458,441478],{"className":441457},[31796,31797],[34,441459,441461,441475],{"className":441460},[31801],[34,441462,441464],{"className":441463,"style":116558},[31805],[34,441465,441466,441469],{"style":31809},[34,441467],{"className":441468,"style":31814},[31813],[34,441470,441472],{"className":441471},[31818,31819,31820,31821],[34,441473,94675],{"className":441474,"style":106455},[31685,31686,31821],[34,441476,31829],{"className":441477},[31828],[34,441479,441481],{"className":441480},[31801],[34,441482,441484],{"className":441483,"style":31836},[31805],[34,441485],{},[34,441487,441489,441515],{"className":441488},[31642],[34,441490,441492],{"className":441491},[31646],[31648,441493,441494],{"xmlns":31650},[31652,441495,441496,441512],{},[31655,441497,441498,441504,441506,441508,441510],{},[31743,441499,441500,441502],{},[31749,441501,459],{},[31658,441503,34386],{},[31662,441505,112296],{},[31658,441507,406335],{},[31662,441509,32682],{},[31658,441511,329201],{},[31665,441513,441514],{"encoding":31667},"0_R \\in \\ker \\phi",[34,441516,441518,441573],{"className":441517,"ariaHidden":758},[31672],[34,441519,441521,441524,441564,441567,441570],{"className":441520},[31676],[34,441522],{"className":441523,"style":38357},[31680],[34,441525,441527,441530],{"className":441526},[31685],[34,441528,459],{"className":441529},[31685],[34,441531,441533],{"className":441532},[31792],[34,441534,441536,441556],{"className":441535},[31796,31797],[34,441537,441539,441553],{"className":441538},[31801],[34,441540,441542],{"className":441541,"style":116558},[31805],[34,441543,441544,441547],{"style":31809},[34,441545],{"className":441546,"style":31814},[31813],[34,441548,441550],{"className":441549},[31818,31819,31820,31821],[34,441551,34386],{"className":441552,"style":34427},[31685,31686,31821],[34,441554,31829],{"className":441555},[31828],[34,441557,441559],{"className":441558},[31801],[34,441560,441562],{"className":441561,"style":31836},[31805],[34,441563],{},[34,441565],{"className":441566,"style":31846},[31845],[34,441568,112296],{"className":441569},[31850],[34,441571],{"className":441572,"style":31846},[31845],[34,441574,441576,441579,441582,441585],{"className":441575},[31676],[34,441577],{"className":441578,"style":43217},[31680],[34,441580,406335],{"className":441581},[32767],[34,441583],{"className":441584,"style":32123},[31845],[34,441586,329201],{"className":441587},[31685,31686],[34,441589,441591,441613],{"className":441590},[31642],[34,441592,441594],{"className":441593},[31646],[31648,441595,441596],{"xmlns":31650},[31652,441597,441598,441610],{},[31655,441599,441600,441602,441604,441606,441608],{},[31658,441601,406335],{},[31662,441603,32682],{},[31658,441605,329201],{},[31662,441607,54707],{"mathvariant":38079},[31658,441609,129226],{"mathvariant":38079},[31665,441611,441612],{"encoding":31667},"\\ker \\phi \\neq \\emptyset",[34,441614,441616,441673],{"className":441615,"ariaHidden":758},[31672],[34,441617,441619,441622,441625,441628,441631,441634,441670],{"className":441618},[31676],[34,441620],{"className":441621,"style":43217},[31680],[34,441623,406335],{"className":441624},[32767],[34,441626],{"className":441627,"style":32123},[31845],[34,441629,329201],{"className":441630},[31685,31686],[34,441632],{"className":441633,"style":31846},[31845],[34,441635,441637,441664,441667],{"className":441636},[31850],[34,441638,441640],{"className":441639},[31850],[34,441641,441643],{"className":441642},[31685,54774],[34,441644,441646],{"className":441645},[54778],[34,441647,441649,441652,441661],{"className":441648},[54782],[34,441650],{"className":441651,"style":43217},[31680],[34,441653,441655],{"className":441654},[54789],[34,441656,441658],{"className":441657},[31685],[34,441659,54796],{"className":441660},[31850],[34,441662],{"className":441663},[54800],[34,441665],{"className":441666},[31845,54804],[34,441668,202],{"className":441669},[31850],[34,441671],{"className":441672,"style":31846},[31845],[34,441674,441676,441679],{"className":441675},[31676],[34,441677],{"className":441678,"style":168117},[31680],[34,441680,129226],{"className":441681},[31685],[108,441683,441684,8471,441687,64615,441766,39984,441882],{},[2627,441685,441686],{},"加法封闭性 (对减法封闭)",[34,441688,441690,441718],{"className":441689},[31642],[34,441691,441693],{"className":441692},[31646],[31648,441694,441695],{"xmlns":31650},[31652,441696,441697,441715],{},[31655,441698,441699,441701,441703,441705,441707,441709,441711,441713],{},[31658,441700,365679],{"mathvariant":38079},[31658,441702,111],{},[31662,441704,31960],{"separator":758},[31658,441706,1697],{},[31662,441708,112296],{},[31658,441710,406335],{},[31662,441712,32682],{},[31658,441714,329201],{},[31665,441716,441717],{"encoding":31667},"\\forall a, b \\in \\ker \\phi",[34,441719,441721,441751],{"className":441720,"ariaHidden":758},[31672],[34,441722,441724,441727,441730,441733,441736,441739,441742,441745,441748],{"className":441723},[31676],[34,441725],{"className":441726,"style":43217},[31680],[34,441728,365679],{"className":441729},[31685],[34,441731,111],{"className":441732},[31685,31686],[34,441734,31960],{"className":441735},[32119],[34,441737],{"className":441738,"style":32123},[31845],[34,441740,1697],{"className":441741},[31685,31686],[34,441743],{"className":441744,"style":31846},[31845],[34,441746,112296],{"className":441747},[31850],[34,441749],{"className":441750,"style":31846},[31845],[34,441752,441754,441757,441760,441763],{"className":441753},[31676],[34,441755],{"className":441756,"style":43217},[31680],[34,441758,406335],{"className":441759},[32767],[34,441761],{"className":441762,"style":32123},[31845],[34,441764,329201],{"className":441765},[31685,31686],[34,441767,441769,441807],{"className":441768},[31642],[34,441770,441772],{"className":441771},[31646],[31648,441773,441774],{"xmlns":31650},[31652,441775,441776,441804],{},[31655,441777,441778,441780,441782,441784,441786,441788,441790,441792,441794,441796,441798,441800,441802],{},[31658,441779,329201],{},[31662,441781,456],{"stretchy":702},[31658,441783,111],{},[31662,441785,2557],{"stretchy":702},[31662,441787,202],{},[31749,441789,459],{},[31662,441791,31960],{"separator":758},[31658,441793,329201],{},[31662,441795,456],{"stretchy":702},[31658,441797,1697],{},[31662,441799,2557],{"stretchy":702},[31662,441801,202],{},[31749,441803,459],{},[31665,441805,441806],{"encoding":31667},"\\phi(a)=0, \\phi(b)=0",[34,441808,441810,441837,441873],{"className":441809,"ariaHidden":758},[31672],[34,441811,441813,441816,441819,441822,441825,441828,441831,441834],{"className":441812},[31676],[34,441814],{"className":441815,"style":31774},[31680],[34,441817,329201],{"className":441818},[31685,31686],[34,441820,456],{"className":441821},[31782],[34,441823,111],{"className":441824},[31685,31686],[34,441826,2557],{"className":441827},[31690],[34,441829],{"className":441830,"style":31846},[31845],[34,441832,202],{"className":441833},[31850],[34,441835],{"className":441836,"style":31846},[31845],[34,441838,441840,441843,441846,441849,441852,441855,441858,441861,441864,441867,441870],{"className":441839},[31676],[34,441841],{"className":441842,"style":31774},[31680],[34,441844,459],{"className":441845},[31685],[34,441847,31960],{"className":441848},[32119],[34,441850],{"className":441851,"style":32123},[31845],[34,441853,329201],{"className":441854},[31685,31686],[34,441856,456],{"className":441857},[31782],[34,441859,1697],{"className":441860},[31685,31686],[34,441862,2557],{"className":441863},[31690],[34,441865],{"className":441866,"style":31846},[31845],[34,441868,202],{"className":441869},[31850],[34,441871],{"className":441872,"style":31846},[31845],[34,441874,441876,441879],{"className":441875},[31676],[34,441877],{"className":441878,"style":32617},[31680],[34,441880,459],{"className":441881},[31685],[34,441883,441885,441963],{"className":441884},[31642],[34,441886,441888],{"className":441887},[31646],[31648,441889,441890],{"xmlns":31650},[31652,441891,441892,441960],{},[31655,441893,441894,441896,441898,441900,441902,441904,441906,441908,441910,441912,441914,441916,441918,441920,441922,441924,441926,441928,441930,441932,441934,441936,441938,441940,441942,441944,441946,441948,441950,441952,441954,441956,441958],{},[31658,441895,329201],{},[31662,441897,456],{"stretchy":702},[31658,441899,111],{},[31662,441901,32226],{},[31658,441903,1697],{},[31662,441905,2557],{"stretchy":702},[31662,441907,202],{},[31658,441909,329201],{},[31662,441911,456],{"stretchy":702},[31658,441913,111],{},[31662,441915,2557],{"stretchy":702},[31662,441917,32226],{},[31658,441919,329201],{},[31662,441921,456],{"stretchy":702},[31658,441923,1697],{},[31662,441925,2557],{"stretchy":702},[31662,441927,202],{},[31749,441929,459],{},[31662,441931,32226],{},[31749,441933,459],{},[31662,441935,202],{},[31749,441937,459],{},[39361,441939,262622],{},[31662,441941,262625],{},[39361,441943,262622],{},[31658,441945,111],{},[31662,441947,32226],{},[31658,441949,1697],{},[31662,441951,112296],{},[31658,441953,406335],{},[31662,441955,32682],{},[31658,441957,329201],{},[31658,441959,2547],{"mathvariant":38079},[31665,441961,441962],{"encoding":31667},"\\phi(a-b) = \\phi(a) - \\phi(b) = 0 - 0 = 0 \\implies a-b \\in \\ker \\phi.",[34,441964,441966,441990,442011,442038,442065,442083,442101,442125,442143,442161],{"className":441965,"ariaHidden":758},[31672],[34,441967,441969,441972,441975,441978,441981,441984,441987],{"className":441968},[31676],[34,441970],{"className":441971,"style":31774},[31680],[34,441973,329201],{"className":441974},[31685,31686],[34,441976,456],{"className":441977},[31782],[34,441979,111],{"className":441980},[31685,31686],[34,441982],{"className":441983,"style":32604},[31845],[34,441985,32226],{"className":441986},[32024],[34,441988],{"className":441989,"style":32604},[31845],[34,441991,441993,441996,441999,442002,442005,442008],{"className":441992},[31676],[34,441994],{"className":441995,"style":31774},[31680],[34,441997,1697],{"className":441998},[31685,31686],[34,442000,2557],{"className":442001},[31690],[34,442003],{"className":442004,"style":31846},[31845],[34,442006,202],{"className":442007},[31850],[34,442009],{"className":442010,"style":31846},[31845],[34,442012,442014,442017,442020,442023,442026,442029,442032,442035],{"className":442013},[31676],[34,442015],{"className":442016,"style":31774},[31680],[34,442018,329201],{"className":442019},[31685,31686],[34,442021,456],{"className":442022},[31782],[34,442024,111],{"className":442025},[31685,31686],[34,442027,2557],{"className":442028},[31690],[34,442030],{"className":442031,"style":32604},[31845],[34,442033,32226],{"className":442034},[32024],[34,442036],{"className":442037,"style":32604},[31845],[34,442039,442041,442044,442047,442050,442053,442056,442059,442062],{"className":442040},[31676],[34,442042],{"className":442043,"style":31774},[31680],[34,442045,329201],{"className":442046},[31685,31686],[34,442048,456],{"className":442049},[31782],[34,442051,1697],{"className":442052},[31685,31686],[34,442054,2557],{"className":442055},[31690],[34,442057],{"className":442058,"style":31846},[31845],[34,442060,202],{"className":442061},[31850],[34,442063],{"className":442064,"style":31846},[31845],[34,442066,442068,442071,442074,442077,442080],{"className":442067},[31676],[34,442069],{"className":442070,"style":35942},[31680],[34,442072,459],{"className":442073},[31685],[34,442075],{"className":442076,"style":32604},[31845],[34,442078,32226],{"className":442079},[32024],[34,442081],{"className":442082,"style":32604},[31845],[34,442084,442086,442089,442092,442095,442098],{"className":442085},[31676],[34,442087],{"className":442088,"style":32617},[31680],[34,442090,459],{"className":442091},[31685],[34,442093],{"className":442094,"style":31846},[31845],[34,442096,202],{"className":442097},[31850],[34,442099],{"className":442100,"style":31846},[31845],[34,442102,442104,442107,442110,442113,442116,442119,442122],{"className":442103},[31676],[34,442105],{"className":442106,"style":262700},[31680],[34,442108,459],{"className":442109},[31685],[34,442111],{"className":442112,"style":31846},[31845],[34,442114],{"className":442115,"style":31846},[31845],[34,442117,262625],{"className":442118},[31850],[34,442120],{"className":442121,"style":31846},[31845],[34,442123],{"className":442124,"style":31846},[31845],[34,442126,442128,442131,442134,442137,442140],{"className":442127},[31676],[34,442129],{"className":442130,"style":32597},[31680],[34,442132,111],{"className":442133},[31685,31686],[34,442135],{"className":442136,"style":32604},[31845],[34,442138,32226],{"className":442139},[32024],[34,442141],{"className":442142,"style":32604},[31845],[34,442144,442146,442149,442152,442155,442158],{"className":442145},[31676],[34,442147],{"className":442148,"style":46991},[31680],[34,442150,1697],{"className":442151},[31685,31686],[34,442153],{"className":442154,"style":31846},[31845],[34,442156,112296],{"className":442157},[31850],[34,442159],{"className":442160,"style":31846},[31845],[34,442162,442164,442167,442170,442173,442176],{"className":442163},[31676],[34,442165],{"className":442166,"style":43217},[31680],[34,442168,406335],{"className":442169},[32767],[34,442171],{"className":442172,"style":32123},[31845],[34,442174,329201],{"className":442175},[31685,31686],[34,442177,2547],{"className":442178},[31685],[108,442180,442181,8471,442184,432459,442250,432343,442316,39984,442372,442633,442894,442895,11008,442933,442961,442962,347],{},[2627,442182,442183],{},"吸收律",[34,442185,442187,442211],{"className":442186},[31642],[34,442188,442190],{"className":442189},[31646],[31648,442191,442192],{"xmlns":31650},[31652,442193,442194,442208],{},[31655,442195,442196,442198,442200,442202,442204,442206],{},[31658,442197,365679],{"mathvariant":38079},[31658,442199,111],{},[31662,442201,112296],{},[31658,442203,406335],{},[31662,442205,32682],{},[31658,442207,329201],{},[31665,442209,442210],{"encoding":31667},"\\forall a \\in \\ker \\phi",[34,442212,442214,442235],{"className":442213,"ariaHidden":758},[31672],[34,442215,442217,442220,442223,442226,442229,442232],{"className":442216},[31676],[34,442218],{"className":442219,"style":46991},[31680],[34,442221,365679],{"className":442222},[31685],[34,442224,111],{"className":442225},[31685,31686],[34,442227],{"className":442228,"style":31846},[31845],[34,442230,112296],{"className":442231},[31850],[34,442233],{"className":442234,"style":31846},[31845],[34,442236,442238,442241,442244,442247],{"className":442237},[31676],[34,442239],{"className":442240,"style":43217},[31680],[34,442242,406335],{"className":442243},[32767],[34,442245],{"className":442246,"style":32123},[31845],[34,442248,329201],{"className":442249},[31685,31686],[34,442251,442253,442277],{"className":442252},[31642],[34,442254,442256],{"className":442255},[31646],[31648,442257,442258],{"xmlns":31650},[31652,442259,442260,442274],{},[31655,442261,442262,442264,442266,442268,442270,442272],{},[31658,442263,329201],{},[31662,442265,456],{"stretchy":702},[31658,442267,111],{},[31662,442269,2557],{"stretchy":702},[31662,442271,202],{},[31749,442273,459],{},[31665,442275,442276],{"encoding":31667},"\\phi(a)=0",[34,442278,442280,442307],{"className":442279,"ariaHidden":758},[31672],[34,442281,442283,442286,442289,442292,442295,442298,442301,442304],{"className":442282},[31676],[34,442284],{"className":442285,"style":31774},[31680],[34,442287,329201],{"className":442288},[31685,31686],[34,442290,456],{"className":442291},[31782],[34,442293,111],{"className":442294},[31685,31686],[34,442296,2557],{"className":442297},[31690],[34,442299],{"className":442300,"style":31846},[31845],[34,442302,202],{"className":442303},[31850],[34,442305],{"className":442306,"style":31846},[31845],[34,442308,442310,442313],{"className":442309},[31676],[34,442311],{"className":442312,"style":32617},[31680],[34,442314,459],{"className":442315},[31685],[34,442317,442319,442339],{"className":442318},[31642],[34,442320,442322],{"className":442321},[31646],[31648,442323,442324],{"xmlns":31650},[31652,442325,442326,442336],{},[31655,442327,442328,442330,442332,442334],{},[31658,442329,365679],{"mathvariant":38079},[31658,442331,34395],{},[31662,442333,112296],{},[31658,442335,34386],{},[31665,442337,442338],{"encoding":31667},"\\forall r \\in R",[34,442340,442342,442363],{"className":442341,"ariaHidden":758},[31672],[34,442343,442345,442348,442351,442354,442357,442360],{"className":442344},[31676],[34,442346],{"className":442347,"style":46991},[31680],[34,442349,365679],{"className":442350},[31685],[34,442352,34395],{"className":442353,"style":34452},[31685,31686],[34,442355],{"className":442356,"style":31846},[31845],[34,442358,112296],{"className":442359},[31850],[34,442361],{"className":442362,"style":31846},[31845],[34,442364,442366,442369],{"className":442365},[31676],[34,442367],{"className":442368,"style":33218},[31680],[34,442370,34386],{"className":442371,"style":34427},[31685,31686],[34,442373,442375,442453],{"className":442374},[31642],[34,442376,442378],{"className":442377},[31646],[31648,442379,442380],{"xmlns":31650},[31652,442381,442382,442450],{},[31655,442383,442384,442386,442388,442390,442392,442394,442396,442398,442400,442402,442404,442406,442408,442410,442412,442414,442416,442418,442420,442422,442424,442426,442428,442430,442432,442434,442436,442438,442440,442442,442444,442446,442448],{},[31658,442385,329201],{},[31662,442387,456],{"stretchy":702},[31658,442389,34395],{},[31658,442391,111],{},[31662,442393,2557],{"stretchy":702},[31662,442395,202],{},[31658,442397,329201],{},[31662,442399,456],{"stretchy":702},[31658,442401,34395],{},[31662,442403,2557],{"stretchy":702},[31658,442405,329201],{},[31662,442407,456],{"stretchy":702},[31658,442409,111],{},[31662,442411,2557],{"stretchy":702},[31662,442413,202],{},[31658,442415,329201],{},[31662,442417,456],{"stretchy":702},[31658,442419,34395],{},[31662,442421,2557],{"stretchy":702},[31662,442423,71191],{},[31749,442425,459],{},[31662,442427,202],{},[31749,442429,459],{},[39361,442431,262622],{},[31662,442433,262625],{},[39361,442435,262622],{},[31658,442437,34395],{},[31658,442439,111],{},[31662,442441,112296],{},[31658,442443,406335],{},[31662,442445,32682],{},[31658,442447,329201],{},[31658,442449,2547],{"mathvariant":38079},[31665,442451,442452],{"encoding":31667},"\\phi(ra) = \\phi(r)\\phi(a) = \\phi(r) \\cdot 0 = 0 \\implies ra \\in \\ker \\phi.",[34,442454,442456,442486,442525,442552,442570,442594,442615],{"className":442455,"ariaHidden":758},[31672],[34,442457,442459,442462,442465,442468,442471,442474,442477,442480,442483],{"className":442458},[31676],[34,442460],{"className":442461,"style":31774},[31680],[34,442463,329201],{"className":442464},[31685,31686],[34,442466,456],{"className":442467},[31782],[34,442469,34395],{"className":442470,"style":34452},[31685,31686],[34,442472,111],{"className":442473},[31685,31686],[34,442475,2557],{"className":442476},[31690],[34,442478],{"className":442479,"style":31846},[31845],[34,442481,202],{"className":442482},[31850],[34,442484],{"className":442485,"style":31846},[31845],[34,442487,442489,442492,442495,442498,442501,442504,442507,442510,442513,442516,442519,442522],{"className":442488},[31676],[34,442490],{"className":442491,"style":31774},[31680],[34,442493,329201],{"className":442494},[31685,31686],[34,442496,456],{"className":442497},[31782],[34,442499,34395],{"className":442500,"style":34452},[31685,31686],[34,442502,2557],{"className":442503},[31690],[34,442505,329201],{"className":442506},[31685,31686],[34,442508,456],{"className":442509},[31782],[34,442511,111],{"className":442512},[31685,31686],[34,442514,2557],{"className":442515},[31690],[34,442517],{"className":442518,"style":31846},[31845],[34,442520,202],{"className":442521},[31850],[34,442523],{"className":442524,"style":31846},[31845],[34,442526,442528,442531,442534,442537,442540,442543,442546,442549],{"className":442527},[31676],[34,442529],{"className":442530,"style":31774},[31680],[34,442532,329201],{"className":442533},[31685,31686],[34,442535,456],{"className":442536},[31782],[34,442538,34395],{"className":442539,"style":34452},[31685,31686],[34,442541,2557],{"className":442542},[31690],[34,442544],{"className":442545,"style":32604},[31845],[34,442547,71191],{"className":442548},[32024],[34,442550],{"className":442551,"style":32604},[31845],[34,442553,442555,442558,442561,442564,442567],{"className":442554},[31676],[34,442556],{"className":442557,"style":32617},[31680],[34,442559,459],{"className":442560},[31685],[34,442562],{"className":442563,"style":31846},[31845],[34,442565,202],{"className":442566},[31850],[34,442568],{"className":442569,"style":31846},[31845],[34,442571,442573,442576,442579,442582,442585,442588,442591],{"className":442572},[31676],[34,442574],{"className":442575,"style":262700},[31680],[34,442577,459],{"className":442578},[31685],[34,442580],{"className":442581,"style":31846},[31845],[34,442583],{"className":442584,"style":31846},[31845],[34,442586,262625],{"className":442587},[31850],[34,442589],{"className":442590,"style":31846},[31845],[34,442592],{"className":442593,"style":31846},[31845],[34,442595,442597,442600,442603,442606,442609,442612],{"className":442596},[31676],[34,442598],{"className":442599,"style":34812},[31680],[34,442601,34395],{"className":442602,"style":34452},[31685,31686],[34,442604,111],{"className":442605},[31685,31686],[34,442607],{"className":442608,"style":31846},[31845],[34,442610,112296],{"className":442611},[31850],[34,442613],{"className":442614,"style":31846},[31845],[34,442616,442618,442621,442624,442627,442630],{"className":442617},[31676],[34,442619],{"className":442620,"style":43217},[31680],[34,442622,406335],{"className":442623},[32767],[34,442625],{"className":442626,"style":32123},[31845],[34,442628,329201],{"className":442629},[31685,31686],[34,442631,2547],{"className":442632},[31685],[34,442634,442636,442714],{"className":442635},[31642],[34,442637,442639],{"className":442638},[31646],[31648,442640,442641],{"xmlns":31650},[31652,442642,442643,442711],{},[31655,442644,442645,442647,442649,442651,442653,442655,442657,442659,442661,442663,442665,442667,442669,442671,442673,442675,442677,442679,442681,442683,442685,442687,442689,442691,442693,442695,442697,442699,442701,442703,442705,442707,442709],{},[31658,442646,329201],{},[31662,442648,456],{"stretchy":702},[31658,442650,111],{},[31658,442652,34395],{},[31662,442654,2557],{"stretchy":702},[31662,442656,202],{},[31658,442658,329201],{},[31662,442660,456],{"stretchy":702},[31658,442662,111],{},[31662,442664,2557],{"stretchy":702},[31658,442666,329201],{},[31662,442668,456],{"stretchy":702},[31658,442670,34395],{},[31662,442672,2557],{"stretchy":702},[31662,442674,202],{},[31749,442676,459],{},[31662,442678,71191],{},[31658,442680,329201],{},[31662,442682,456],{"stretchy":702},[31658,442684,34395],{},[31662,442686,2557],{"stretchy":702},[31662,442688,202],{},[31749,442690,459],{},[39361,442692,262622],{},[31662,442694,262625],{},[39361,442696,262622],{},[31658,442698,111],{},[31658,442700,34395],{},[31662,442702,112296],{},[31658,442704,406335],{},[31662,442706,32682],{},[31658,442708,329201],{},[31658,442710,2547],{"mathvariant":38079},[31665,442712,442713],{"encoding":31667},"\\phi(ar) = \\phi(a)\\phi(r) = 0 \\cdot \\phi(r) = 0 \\implies ar \\in \\ker \\phi.",[34,442715,442717,442747,442786,442804,442831,442855,442876],{"className":442716,"ariaHidden":758},[31672],[34,442718,442720,442723,442726,442729,442732,442735,442738,442741,442744],{"className":442719},[31676],[34,442721],{"className":442722,"style":31774},[31680],[34,442724,329201],{"className":442725},[31685,31686],[34,442727,456],{"className":442728},[31782],[34,442730,111],{"className":442731},[31685,31686],[34,442733,34395],{"className":442734,"style":34452},[31685,31686],[34,442736,2557],{"className":442737},[31690],[34,442739],{"className":442740,"style":31846},[31845],[34,442742,202],{"className":442743},[31850],[34,442745],{"className":442746,"style":31846},[31845],[34,442748,442750,442753,442756,442759,442762,442765,442768,442771,442774,442777,442780,442783],{"className":442749},[31676],[34,442751],{"className":442752,"style":31774},[31680],[34,442754,329201],{"className":442755},[31685,31686],[34,442757,456],{"className":442758},[31782],[34,442760,111],{"className":442761},[31685,31686],[34,442763,2557],{"className":442764},[31690],[34,442766,329201],{"className":442767},[31685,31686],[34,442769,456],{"className":442770},[31782],[34,442772,34395],{"className":442773,"style":34452},[31685,31686],[34,442775,2557],{"className":442776},[31690],[34,442778],{"className":442779,"style":31846},[31845],[34,442781,202],{"className":442782},[31850],[34,442784],{"className":442785,"style":31846},[31845],[34,442787,442789,442792,442795,442798,442801],{"className":442788},[31676],[34,442790],{"className":442791,"style":32617},[31680],[34,442793,459],{"className":442794},[31685],[34,442796],{"className":442797,"style":32604},[31845],[34,442799,71191],{"className":442800},[32024],[34,442802],{"className":442803,"style":32604},[31845],[34,442805,442807,442810,442813,442816,442819,442822,442825,442828],{"className":442806},[31676],[34,442808],{"className":442809,"style":31774},[31680],[34,442811,329201],{"className":442812},[31685,31686],[34,442814,456],{"className":442815},[31782],[34,442817,34395],{"className":442818,"style":34452},[31685,31686],[34,442820,2557],{"className":442821},[31690],[34,442823],{"className":442824,"style":31846},[31845],[34,442826,202],{"className":442827},[31850],[34,442829],{"className":442830,"style":31846},[31845],[34,442832,442834,442837,442840,442843,442846,442849,442852],{"className":442833},[31676],[34,442835],{"className":442836,"style":262700},[31680],[34,442838,459],{"className":442839},[31685],[34,442841],{"className":442842,"style":31846},[31845],[34,442844],{"className":442845,"style":31846},[31845],[34,442847,262625],{"className":442848},[31850],[34,442850],{"className":442851,"style":31846},[31845],[34,442853],{"className":442854,"style":31846},[31845],[34,442856,442858,442861,442864,442867,442870,442873],{"className":442857},[31676],[34,442859],{"className":442860,"style":34812},[31680],[34,442862,111],{"className":442863},[31685,31686],[34,442865,34395],{"className":442866,"style":34452},[31685,31686],[34,442868],{"className":442869,"style":31846},[31845],[34,442871,112296],{"className":442872},[31850],[34,442874],{"className":442875,"style":31846},[31845],[34,442877,442879,442882,442885,442888,442891],{"className":442878},[31676],[34,442880],{"className":442881,"style":43217},[31680],[34,442883,406335],{"className":442884},[32767],[34,442886],{"className":442887,"style":32123},[31845],[34,442889,329201],{"className":442890},[31685,31686],[34,442892,2547],{"className":442893},[31685],"\n综上，",[34,442896,442898,442915],{"className":442897},[31642],[34,442899,442901],{"className":442900},[31646],[31648,442902,442903],{"xmlns":31650},[31652,442904,442905,442913],{},[31655,442906,442907,442909,442911],{},[31658,442908,406335],{},[31662,442910,32682],{},[31658,442912,329201],{},[31665,442914,406342],{"encoding":31667},[34,442916,442918],{"className":442917,"ariaHidden":758},[31672],[34,442919,442921,442924,442927,442930],{"className":442920},[31676],[34,442922],{"className":442923,"style":43217},[31680],[34,442925,406335],{"className":442926},[32767],[34,442928],{"className":442929,"style":32123},[31845],[34,442931,329201],{"className":442932},[31685,31686],[34,442934,442936,442949],{"className":442935},[31642],[34,442937,442939],{"className":442938},[31646],[31648,442940,442941],{"xmlns":31650},[31652,442942,442943,442947],{},[31655,442944,442945],{},[31658,442946,34386],{},[31665,442948,34386],{"encoding":31667},[34,442950,442952],{"className":442951,"ariaHidden":758},[31672],[34,442953,442955,442958],{"className":442954},[31676],[34,442956],{"className":442957,"style":33218},[31680],[34,442959,34386],{"className":442960,"style":34427},[31685,31686]," 的理想，即 ",[34,442963,442965,442986],{"className":442964},[31642],[34,442966,442968],{"className":442967},[31646],[31648,442969,442970],{"xmlns":31650},[31652,442971,442972,442984],{},[31655,442973,442974,442976,442978,442980,442982],{},[31658,442975,406335],{},[31662,442977,32682],{},[31658,442979,329201],{},[31662,442981,440828],{},[31658,442983,34386],{},[31665,442985,440833],{"encoding":31667},[34,442987,442989,443013],{"className":442988,"ariaHidden":758},[31672],[34,442990,442992,442995,442998,443001,443004,443007,443010],{"className":442991},[31676],[34,442993],{"className":442994,"style":43217},[31680],[34,442996,406335],{"className":442997},[32767],[34,442999],{"className":443000,"style":32123},[31845],[34,443002,329201],{"className":443003},[31685,31686],[34,443005],{"className":443006,"style":32604},[31845],[34,443008,440828],{"className":443009},[32024,129261],[34,443011],{"className":443012,"style":32604},[31845],[34,443014,443016,443019],{"className":443015},[31676],[34,443017],{"className":443018,"style":33218},[31680],[34,443020,34386],{"className":443021,"style":34427},[31685,31686],[108,443023,443024,443098],{},[2627,443025,443026,443027,11008,443069,443097],{},"2. 证明 ",[34,443028,443030,443048],{"className":443029},[31642],[34,443031,443033],{"className":443032},[31646],[31648,443034,443035],{"xmlns":31650},[31652,443036,443037,443045],{},[31655,443038,443039,443041,443043],{},[31658,443040,440604],{"mathvariant":38079},[31662,443042,32682],{},[31658,443044,329201],{},[31665,443046,443047],{"encoding":31667},"\\operatorname{Im} \\phi",[34,443049,443051],{"className":443050,"ariaHidden":758},[31672],[34,443052,443054,443057,443063,443066],{"className":443053},[31676],[34,443055],{"className":443056,"style":43217},[31680],[34,443058,443060],{"className":443059},[32767],[34,443061,440604],{"className":443062},[31685,207923],[34,443064],{"className":443065,"style":32123},[31845],[34,443067,329201],{"className":443068},[31685,31686],[34,443070,443072,443085],{"className":443071},[31642],[34,443073,443075],{"className":443074},[31646],[31648,443076,443077],{"xmlns":31650},[31652,443078,443079,443083],{},[31655,443080,443081],{},[31658,443082,94675],{},[31665,443084,94675],{"encoding":31667},[34,443086,443088],{"className":443087,"ariaHidden":758},[31672],[34,443089,443091,443094],{"className":443090},[31676],[34,443092],{"className":443093,"style":33218},[31680],[34,443095,94675],{"className":443096,"style":106455},[31685,31686]," 的子环:",[105,443099,443100,443590],{},[108,443101,443102,8471,443105,395725,443187,315343,443250,39984,443366],{},[2627,443103,443104],{},"减法封闭",[34,443106,443108,443136],{"className":443107},[31642],[34,443109,443111],{"className":443110},[31646],[31648,443112,443113],{"xmlns":31650},[31652,443114,443115,443133],{},[31655,443116,443117,443119,443121,443123,443125,443127,443129,443131],{},[31658,443118,365679],{"mathvariant":38079},[31658,443120,32936],{},[31662,443122,31960],{"separator":758},[31658,443124,67764],{},[31662,443126,112296],{},[31658,443128,440604],{"mathvariant":38079},[31662,443130,32682],{},[31658,443132,329201],{},[31665,443134,443135],{"encoding":31667},"\\forall x, y \\in \\operatorname{Im} \\phi",[34,443137,443139,443169],{"className":443138,"ariaHidden":758},[31672],[34,443140,443142,443145,443148,443151,443154,443157,443160,443163,443166],{"className":443141},[31676],[34,443143],{"className":443144,"style":43217},[31680],[34,443146,365679],{"className":443147},[31685],[34,443149,32936],{"className":443150},[31685,31686],[34,443152,31960],{"className":443153},[32119],[34,443155],{"className":443156,"style":32123},[31845],[34,443158,67764],{"className":443159,"style":32062},[31685,31686],[34,443161],{"className":443162,"style":31846},[31845],[34,443164,112296],{"className":443165},[31850],[34,443167],{"className":443168,"style":31846},[31845],[34,443170,443172,443175,443181,443184],{"className":443171},[31676],[34,443173],{"className":443174,"style":43217},[31680],[34,443176,443178],{"className":443177},[32767],[34,443179,440604],{"className":443180},[31685,207923],[34,443182],{"className":443183,"style":32123},[31845],[34,443185,329201],{"className":443186},[31685,31686],[34,443188,443190,443211],{"className":443189},[31642],[34,443191,443193],{"className":443192},[31646],[31648,443194,443195],{"xmlns":31650},[31652,443196,443197,443209],{},[31655,443198,443199,443201,443203,443205,443207],{},[31658,443200,111],{},[31662,443202,31960],{"separator":758},[31658,443204,1697],{},[31662,443206,112296],{},[31658,443208,34386],{},[31665,443210,330959],{"encoding":31667},[34,443212,443214,443241],{"className":443213,"ariaHidden":758},[31672],[34,443215,443217,443220,443223,443226,443229,443232,443235,443238],{"className":443216},[31676],[34,443218],{"className":443219,"style":43217},[31680],[34,443221,111],{"className":443222},[31685,31686],[34,443224,31960],{"className":443225},[32119],[34,443227],{"className":443228,"style":32123},[31845],[34,443230,1697],{"className":443231},[31685,31686],[34,443233],{"className":443234,"style":31846},[31845],[34,443236,112296],{"className":443237},[31850],[34,443239],{"className":443240,"style":31846},[31845],[34,443242,443244,443247],{"className":443243},[31676],[34,443245],{"className":443246,"style":33218},[31680],[34,443248,34386],{"className":443249,"style":34427},[31685,31686],[34,443251,443253,443291],{"className":443252},[31642],[34,443254,443256],{"className":443255},[31646],[31648,443257,443258],{"xmlns":31650},[31652,443259,443260,443288],{},[31655,443261,443262,443264,443266,443268,443270,443272,443274,443276,443278,443280,443282,443284,443286],{},[31658,443263,32936],{},[31662,443265,202],{},[31658,443267,329201],{},[31662,443269,456],{"stretchy":702},[31658,443271,111],{},[31662,443273,2557],{"stretchy":702},[31662,443275,31960],{"separator":758},[31658,443277,67764],{},[31662,443279,202],{},[31658,443281,329201],{},[31662,443283,456],{"stretchy":702},[31658,443285,1697],{},[31662,443287,2557],{"stretchy":702},[31665,443289,443290],{"encoding":31667},"x=\\phi(a), y=\\phi(b)",[34,443292,443294,443312,443348],{"className":443293,"ariaHidden":758},[31672],[34,443295,443297,443300,443303,443306,443309],{"className":443296},[31676],[34,443298],{"className":443299,"style":32527},[31680],[34,443301,32936],{"className":443302},[31685,31686],[34,443304],{"className":443305,"style":31846},[31845],[34,443307,202],{"className":443308},[31850],[34,443310],{"className":443311,"style":31846},[31845],[34,443313,443315,443318,443321,443324,443327,443330,443333,443336,443339,443342,443345],{"className":443314},[31676],[34,443316],{"className":443317,"style":31774},[31680],[34,443319,329201],{"className":443320},[31685,31686],[34,443322,456],{"className":443323},[31782],[34,443325,111],{"className":443326},[31685,31686],[34,443328,2557],{"className":443329},[31690],[34,443331,31960],{"className":443332},[32119],[34,443334],{"className":443335,"style":32123},[31845],[34,443337,67764],{"className":443338,"style":32062},[31685,31686],[34,443340],{"className":443341,"style":31846},[31845],[34,443343,202],{"className":443344},[31850],[34,443346],{"className":443347,"style":31846},[31845],[34,443349,443351,443354,443357,443360,443363],{"className":443350},[31676],[34,443352],{"className":443353,"style":31774},[31680],[34,443355,329201],{"className":443356},[31685,31686],[34,443358,456],{"className":443359},[31782],[34,443361,1697],{"className":443362},[31685,31686],[34,443364,2557],{"className":443365},[31690],[34,443367,443369,443431],{"className":443368},[31642],[34,443370,443372],{"className":443371},[31646],[31648,443373,443374],{"xmlns":31650},[31652,443375,443376,443428],{},[31655,443377,443378,443380,443382,443384,443386,443388,443390,443392,443394,443396,443398,443400,443402,443404,443406,443408,443410,443412,443414,443416,443418,443420,443422,443424,443426],{},[31658,443379,32936],{},[31662,443381,32226],{},[31658,443383,67764],{},[31662,443385,202],{},[31658,443387,329201],{},[31662,443389,456],{"stretchy":702},[31658,443391,111],{},[31662,443393,2557],{"stretchy":702},[31662,443395,32226],{},[31658,443397,329201],{},[31662,443399,456],{"stretchy":702},[31658,443401,1697],{},[31662,443403,2557],{"stretchy":702},[31662,443405,202],{},[31658,443407,329201],{},[31662,443409,456],{"stretchy":702},[31658,443411,111],{},[31662,443413,32226],{},[31658,443415,1697],{},[31662,443417,2557],{"stretchy":702},[31662,443419,112296],{},[31658,443421,440604],{"mathvariant":38079},[31662,443423,32682],{},[31658,443425,329201],{},[31658,443427,2547],{"mathvariant":38079},[31665,443429,443430],{"encoding":31667},"x - y = \\phi(a) - \\phi(b) = \\phi(a-b) \\in \\operatorname{Im} \\phi.",[34,443432,443434,443452,443470,443497,443524,443548,443569],{"className":443433,"ariaHidden":758},[31672],[34,443435,443437,443440,443443,443446,443449],{"className":443436},[31676],[34,443438],{"className":443439,"style":32597},[31680],[34,443441,32936],{"className":443442},[31685,31686],[34,443444],{"className":443445,"style":32604},[31845],[34,443447,32226],{"className":443448},[32024],[34,443450],{"className":443451,"style":32604},[31845],[34,443453,443455,443458,443461,443464,443467],{"className":443454},[31676],[34,443456],{"className":443457,"style":38774},[31680],[34,443459,67764],{"className":443460,"style":32062},[31685,31686],[34,443462],{"className":443463,"style":31846},[31845],[34,443465,202],{"className":443466},[31850],[34,443468],{"className":443469,"style":31846},[31845],[34,443471,443473,443476,443479,443482,443485,443488,443491,443494],{"className":443472},[31676],[34,443474],{"className":443475,"style":31774},[31680],[34,443477,329201],{"className":443478},[31685,31686],[34,443480,456],{"className":443481},[31782],[34,443483,111],{"className":443484},[31685,31686],[34,443486,2557],{"className":443487},[31690],[34,443489],{"className":443490,"style":32604},[31845],[34,443492,32226],{"className":443493},[32024],[34,443495],{"className":443496,"style":32604},[31845],[34,443498,443500,443503,443506,443509,443512,443515,443518,443521],{"className":443499},[31676],[34,443501],{"className":443502,"style":31774},[31680],[34,443504,329201],{"className":443505},[31685,31686],[34,443507,456],{"className":443508},[31782],[34,443510,1697],{"className":443511},[31685,31686],[34,443513,2557],{"className":443514},[31690],[34,443516],{"className":443517,"style":31846},[31845],[34,443519,202],{"className":443520},[31850],[34,443522],{"className":443523,"style":31846},[31845],[34,443525,443527,443530,443533,443536,443539,443542,443545],{"className":443526},[31676],[34,443528],{"className":443529,"style":31774},[31680],[34,443531,329201],{"className":443532},[31685,31686],[34,443534,456],{"className":443535},[31782],[34,443537,111],{"className":443538},[31685,31686],[34,443540],{"className":443541,"style":32604},[31845],[34,443543,32226],{"className":443544},[32024],[34,443546],{"className":443547,"style":32604},[31845],[34,443549,443551,443554,443557,443560,443563,443566],{"className":443550},[31676],[34,443552],{"className":443553,"style":31774},[31680],[34,443555,1697],{"className":443556},[31685,31686],[34,443558,2557],{"className":443559},[31690],[34,443561],{"className":443562,"style":31846},[31845],[34,443564,112296],{"className":443565},[31850],[34,443567],{"className":443568,"style":31846},[31845],[34,443570,443572,443575,443581,443584,443587],{"className":443571},[31676],[34,443573],{"className":443574,"style":43217},[31680],[34,443576,443578],{"className":443577},[32767],[34,443579,440604],{"className":443580},[31685,207923],[34,443582],{"className":443583,"style":32123},[31845],[34,443585,329201],{"className":443586},[31685,31686],[34,443588,2547],{"className":443589},[31685],[108,443591,443592,312000,443595,442894,443765,11008,443806,443834,443835,347],{},[2627,443593,443594],{},"乘法封闭",[34,443596,443598,443654],{"className":443597},[31642],[34,443599,443601],{"className":443600},[31646],[31648,443602,443603],{"xmlns":31650},[31652,443604,443605,443651],{},[31655,443606,443607,443609,443611,443613,443615,443617,443619,443621,443623,443625,443627,443629,443631,443633,443635,443637,443639,443641,443643,443645,443647,443649],{},[31658,443608,32936],{},[31658,443610,67764],{},[31662,443612,202],{},[31658,443614,329201],{},[31662,443616,456],{"stretchy":702},[31658,443618,111],{},[31662,443620,2557],{"stretchy":702},[31658,443622,329201],{},[31662,443624,456],{"stretchy":702},[31658,443626,1697],{},[31662,443628,2557],{"stretchy":702},[31662,443630,202],{},[31658,443632,329201],{},[31662,443634,456],{"stretchy":702},[31658,443636,111],{},[31658,443638,1697],{},[31662,443640,2557],{"stretchy":702},[31662,443642,112296],{},[31658,443644,440604],{"mathvariant":38079},[31662,443646,32682],{},[31658,443648,329201],{},[31658,443650,2547],{"mathvariant":38079},[31665,443652,443653],{"encoding":31667},"xy = \\phi(a)\\phi(b) = \\phi(ab) \\in \\operatorname{Im} \\phi.",[34,443655,443657,443678,443717,443744],{"className":443656,"ariaHidden":758},[31672],[34,443658,443660,443663,443666,443669,443672,443675],{"className":443659},[31676],[34,443661],{"className":443662,"style":38774},[31680],[34,443664,32936],{"className":443665},[31685,31686],[34,443667,67764],{"className":443668,"style":32062},[31685,31686],[34,443670],{"className":443671,"style":31846},[31845],[34,443673,202],{"className":443674},[31850],[34,443676],{"className":443677,"style":31846},[31845],[34,443679,443681,443684,443687,443690,443693,443696,443699,443702,443705,443708,443711,443714],{"className":443680},[31676],[34,443682],{"className":443683,"style":31774},[31680],[34,443685,329201],{"className":443686},[31685,31686],[34,443688,456],{"className":443689},[31782],[34,443691,111],{"className":443692},[31685,31686],[34,443694,2557],{"className":443695},[31690],[34,443697,329201],{"className":443698},[31685,31686],[34,443700,456],{"className":443701},[31782],[34,443703,1697],{"className":443704},[31685,31686],[34,443706,2557],{"className":443707},[31690],[34,443709],{"className":443710,"style":31846},[31845],[34,443712,202],{"className":443713},[31850],[34,443715],{"className":443716,"style":31846},[31845],[34,443718,443720,443723,443726,443729,443732,443735,443738,443741],{"className":443719},[31676],[34,443721],{"className":443722,"style":31774},[31680],[34,443724,329201],{"className":443725},[31685,31686],[34,443727,456],{"className":443728},[31782],[34,443730,324191],{"className":443731},[31685,31686],[34,443733,2557],{"className":443734},[31690],[34,443736],{"className":443737,"style":31846},[31845],[34,443739,112296],{"className":443740},[31850],[34,443742],{"className":443743,"style":31846},[31845],[34,443745,443747,443750,443756,443759,443762],{"className":443746},[31676],[34,443748],{"className":443749,"style":43217},[31680],[34,443751,443753],{"className":443752},[32767],[34,443754,440604],{"className":443755},[31685,207923],[34,443757],{"className":443758,"style":32123},[31845],[34,443760,329201],{"className":443761},[31685,31686],[34,443763,2547],{"className":443764},[31685],[34,443766,443768,443785],{"className":443767},[31642],[34,443769,443771],{"className":443770},[31646],[31648,443772,443773],{"xmlns":31650},[31652,443774,443775,443783],{},[31655,443776,443777,443779,443781],{},[31658,443778,440604],{"mathvariant":38079},[31662,443780,32682],{},[31658,443782,329201],{},[31665,443784,443047],{"encoding":31667},[34,443786,443788],{"className":443787,"ariaHidden":758},[31672],[34,443789,443791,443794,443800,443803],{"className":443790},[31676],[34,443792],{"className":443793,"style":43217},[31680],[34,443795,443797],{"className":443796},[32767],[34,443798,440604],{"className":443799},[31685,207923],[34,443801],{"className":443802,"style":32123},[31845],[34,443804,329201],{"className":443805},[31685,31686],[34,443807,443809,443822],{"className":443808},[31642],[34,443810,443812],{"className":443811},[31646],[31648,443813,443814],{"xmlns":31650},[31652,443815,443816,443820],{},[31655,443817,443818],{},[31658,443819,94675],{},[31665,443821,94675],{"encoding":31667},[34,443823,443825],{"className":443824,"ariaHidden":758},[31672],[34,443826,443828,443831],{"className":443827},[31676],[34,443829],{"className":443830,"style":33218},[31680],[34,443832,94675],{"className":443833,"style":106455},[31685,31686]," 的子环，即 ",[34,443836,443838,443859],{"className":443837},[31642],[34,443839,443841],{"className":443840},[31646],[31648,443842,443843],{"xmlns":31650},[31652,443844,443845,443857],{},[31655,443846,443847,443849,443851,443853,443855],{},[31658,443848,440604],{"mathvariant":38079},[31662,443850,32682],{},[31658,443852,329201],{},[31662,443854,45972],{},[31658,443856,94675],{},[31665,443858,440924],{"encoding":31667},[34,443860,443862,443889],{"className":443861,"ariaHidden":758},[31672],[34,443863,443865,443868,443874,443877,443880,443883,443886],{"className":443864},[31676],[34,443866],{"className":443867,"style":43217},[31680],[34,443869,443871],{"className":443870},[32767],[34,443872,440604],{"className":443873},[31685,207923],[34,443875],{"className":443876,"style":32123},[31845],[34,443878,329201],{"className":443879},[31685,31686],[34,443881],{"className":443882,"style":31846},[31845],[34,443884,45972],{"className":443885},[31850],[34,443887],{"className":443888,"style":31846},[31845],[34,443890,443892,443895],{"className":443891},[31676],[34,443893],{"className":443894,"style":33218},[31680],[34,443896,94675],{"className":443897,"style":106455},[31685,31686],[15,443899,443900],{},[2627,443901,443902,443903,443931,443932],{},"(ii) 证明 ",[34,443904,443906,443919],{"className":443905},[31642],[34,443907,443909],{"className":443908},[31646],[31648,443910,443911],{"xmlns":31650},[31652,443912,443913,443917],{},[31655,443914,443915],{},[31658,443916,329201],{},[31665,443918,364484],{"encoding":31667},[34,443920,443922],{"className":443921,"ariaHidden":758},[31672],[34,443923,443925,443928],{"className":443924},[31676],[34,443926],{"className":443927,"style":43217},[31680],[34,443929,329201],{"className":443930},[31685,31686]," 是单射 ",[34,443933,443935,443967],{"className":443934},[31642],[34,443936,443938],{"className":443937},[31646],[31648,443939,443940],{"xmlns":31650},[31652,443941,443942,443964],{},[31655,443943,443944,443946,443948,443950,443952,443954,443956,443958,443960,443962],{},[39361,443945,262622],{},[31662,443947,323391],{},[39361,443949,262622],{},[31658,443951,406335],{},[31662,443953,32682],{},[31658,443955,329201],{},[31662,443957,202],{},[31662,443959,32685],{"stretchy":702},[31749,443961,459],{},[31662,443963,32715],{"stretchy":702},[31665,443965,443966],{"encoding":31667},"\\iff \\ker \\phi = \\{0\\}",[34,443968,443970,443988,444012],{"className":443969,"ariaHidden":758},[31672],[34,443971,443973,443976,443979,443982,443985],{"className":443972},[31676],[34,443974],{"className":443975,"style":318264},[31680],[34,443977],{"className":443978,"style":31846},[31845],[34,443980,323391],{"className":443981},[31850],[34,443983],{"className":443984,"style":31846},[31845],[34,443986],{"className":443987,"style":31846},[31845],[34,443989,443991,443994,443997,444000,444003,444006,444009],{"className":443990},[31676],[34,443992],{"className":443993,"style":43217},[31680],[34,443995,406335],{"className":443996},[32767],[34,443998],{"className":443999,"style":32123},[31845],[34,444001,329201],{"className":444002},[31685,31686],[34,444004],{"className":444005,"style":31846},[31845],[34,444007,202],{"className":444008},[31850],[34,444010],{"className":444011,"style":31846},[31845],[34,444013,444015,444018,444021,444024],{"className":444014},[31676],[34,444016],{"className":444017,"style":31774},[31680],[34,444019,32685],{"className":444020},[31782],[34,444022,459],{"className":444023},[31685],[34,444025,32715],{"className":444026},[31690],[105,444028,444029,444441],{},[108,444030,456,444031,444059,444060,444088,444089,368522,444155,55016,444216,444319,444320,35847,444371,347],{},[34,444032,444034,444047],{"className":444033},[31642],[34,444035,444037],{"className":444036},[31646],[31648,444038,444039],{"xmlns":31650},[31652,444040,444041,444045],{},[31655,444042,444043],{},[31662,444044,140094],{},[31665,444046,266872],{"encoding":31667},[34,444048,444050],{"className":444049,"ariaHidden":758},[31672],[34,444051,444053,444056],{"className":444052},[31676],[34,444054],{"className":444055,"style":73089},[31680],[34,444057,140094],{"className":444058},[31850],") 设 ",[34,444061,444063,444076],{"className":444062},[31642],[34,444064,444066],{"className":444065},[31646],[31648,444067,444068],{"xmlns":31650},[31652,444069,444070,444074],{},[31655,444071,444072],{},[31658,444073,329201],{},[31665,444075,364484],{"encoding":31667},[34,444077,444079],{"className":444078,"ariaHidden":758},[31672],[34,444080,444082,444085],{"className":444081},[31676],[34,444083],{"className":444084,"style":43217},[31680],[34,444086,329201],{"className":444087},[31685,31686]," 是单射。已知 ",[34,444090,444092,444116],{"className":444091},[31642],[34,444093,444095],{"className":444094},[31646],[31648,444096,444097],{"xmlns":31650},[31652,444098,444099,444113],{},[31655,444100,444101,444103,444105,444107,444109,444111],{},[31658,444102,329201],{},[31662,444104,456],{"stretchy":702},[31749,444106,459],{},[31662,444108,2557],{"stretchy":702},[31662,444110,202],{},[31749,444112,459],{},[31665,444114,444115],{"encoding":31667},"\\phi(0)=0",[34,444117,444119,444146],{"className":444118,"ariaHidden":758},[31672],[34,444120,444122,444125,444128,444131,444134,444137,444140,444143],{"className":444121},[31676],[34,444123],{"className":444124,"style":31774},[31680],[34,444126,329201],{"className":444127},[31685,31686],[34,444129,456],{"className":444130},[31782],[34,444132,459],{"className":444133},[31685],[34,444135,2557],{"className":444136},[31690],[34,444138],{"className":444139,"style":31846},[31845],[34,444141,202],{"className":444142},[31850],[34,444144],{"className":444145,"style":31846},[31845],[34,444147,444149,444152],{"className":444148},[31676],[34,444150],{"className":444151,"style":32617},[31680],[34,444153,459],{"className":444154},[31685],[34,444156,444158,444180],{"className":444157},[31642],[34,444159,444161],{"className":444160},[31646],[31648,444162,444163],{"xmlns":31650},[31652,444164,444165,444177],{},[31655,444166,444167,444169,444171,444173,444175],{},[31658,444168,111],{},[31662,444170,112296],{},[31658,444172,406335],{},[31662,444174,32682],{},[31658,444176,329201],{},[31665,444178,444179],{"encoding":31667},"a \\in \\ker \\phi",[34,444181,444183,444201],{"className":444182,"ariaHidden":758},[31672],[34,444184,444186,444189,444192,444195,444198],{"className":444185},[31676],[34,444187],{"className":444188,"style":34812},[31680],[34,444190,111],{"className":444191},[31685,31686],[34,444193],{"className":444194,"style":31846},[31845],[34,444196,112296],{"className":444197},[31850],[34,444199],{"className":444200,"style":31846},[31845],[34,444202,444204,444207,444210,444213],{"className":444203},[31676],[34,444205],{"className":444206,"style":43217},[31680],[34,444208,406335],{"className":444209},[32767],[34,444211],{"className":444212,"style":32123},[31845],[34,444214,329201],{"className":444215},[31685,31686],[34,444217,444219,444253],{"className":444218},[31642],[34,444220,444222],{"className":444221},[31646],[31648,444223,444224],{"xmlns":31650},[31652,444225,444226,444250],{},[31655,444227,444228,444230,444232,444234,444236,444238,444240,444242,444244,444246,444248],{},[31658,444229,329201],{},[31662,444231,456],{"stretchy":702},[31658,444233,111],{},[31662,444235,2557],{"stretchy":702},[31662,444237,202],{},[31749,444239,459],{},[31662,444241,202],{},[31658,444243,329201],{},[31662,444245,456],{"stretchy":702},[31749,444247,459],{},[31662,444249,2557],{"stretchy":702},[31665,444251,444252],{"encoding":31667},"\\phi(a)=0=\\phi(0)",[34,444254,444256,444283,444301],{"className":444255,"ariaHidden":758},[31672],[34,444257,444259,444262,444265,444268,444271,444274,444277,444280],{"className":444258},[31676],[34,444260],{"className":444261,"style":31774},[31680],[34,444263,329201],{"className":444264},[31685,31686],[34,444266,456],{"className":444267},[31782],[34,444269,111],{"className":444270},[31685,31686],[34,444272,2557],{"className":444273},[31690],[34,444275],{"className":444276,"style":31846},[31845],[34,444278,202],{"className":444279},[31850],[34,444281],{"className":444282,"style":31846},[31845],[34,444284,444286,444289,444292,444295,444298],{"className":444285},[31676],[34,444287],{"className":444288,"style":32617},[31680],[34,444290,459],{"className":444291},[31685],[34,444293],{"className":444294,"style":31846},[31845],[34,444296,202],{"className":444297},[31850],[34,444299],{"className":444300,"style":31846},[31845],[34,444302,444304,444307,444310,444313,444316],{"className":444303},[31676],[34,444305],{"className":444306,"style":31774},[31680],[34,444308,329201],{"className":444309},[31685,31686],[34,444311,456],{"className":444312},[31782],[34,444314,459],{"className":444315},[31685],[34,444317,2557],{"className":444318},[31690],"。由单射定义知 ",[34,444321,444323,444341],{"className":444322},[31642],[34,444324,444326],{"className":444325},[31646],[31648,444327,444328],{"xmlns":31650},[31652,444329,444330,444338],{},[31655,444331,444332,444334,444336],{},[31658,444333,111],{},[31662,444335,202],{},[31749,444337,459],{},[31665,444339,444340],{"encoding":31667},"a=0",[34,444342,444344,444362],{"className":444343,"ariaHidden":758},[31672],[34,444345,444347,444350,444353,444356,444359],{"className":444346},[31676],[34,444348],{"className":444349,"style":32527},[31680],[34,444351,111],{"className":444352},[31685,31686],[34,444354],{"className":444355,"style":31846},[31845],[34,444357,202],{"className":444358},[31850],[34,444360],{"className":444361,"style":31846},[31845],[34,444363,444365,444368],{"className":444364},[31676],[34,444366],{"className":444367,"style":32617},[31680],[34,444369,459],{"className":444370},[31685],[34,444372,444374,444399],{"className":444373},[31642],[34,444375,444377],{"className":444376},[31646],[31648,444378,444379],{"xmlns":31650},[31652,444380,444381,444397],{},[31655,444382,444383,444385,444387,444389,444391,444393,444395],{},[31658,444384,406335],{},[31662,444386,32682],{},[31658,444388,329201],{},[31662,444390,202],{},[31662,444392,32685],{"stretchy":702},[31749,444394,459],{},[31662,444396,32715],{"stretchy":702},[31665,444398,441051],{"encoding":31667},[34,444400,444402,444426],{"className":444401,"ariaHidden":758},[31672],[34,444403,444405,444408,444411,444414,444417,444420,444423],{"className":444404},[31676],[34,444406],{"className":444407,"style":43217},[31680],[34,444409,406335],{"className":444410},[32767],[34,444412],{"className":444413,"style":32123},[31845],[34,444415,329201],{"className":444416},[31685,31686],[34,444418],{"className":444419,"style":31846},[31845],[34,444421,202],{"className":444422},[31850],[34,444424],{"className":444425,"style":31846},[31845],[34,444427,444429,444432,444435,444438],{"className":444428},[31676],[34,444430],{"className":444431,"style":31774},[31680],[34,444433,32685],{"className":444434},[31782],[34,444436,459],{"className":444437},[31685],[34,444439,32715],{"className":444440},[31690],[108,444442,456,444443,444059,444471,368522,444541,85658,444609,444689,444690,341819,444885,44689,444968,13977,445038,61605,445111,35847,445161,445189,445190],{},[34,444444,444446,444459],{"className":444445},[31642],[34,444447,444449],{"className":444448},[31646],[31648,444450,444451],{"xmlns":31650},[31652,444452,444453,444457],{},[31655,444454,444455],{},[31662,444456,332814],{},[31665,444458,332817],{"encoding":31667},[34,444460,444462],{"className":444461,"ariaHidden":758},[31672],[34,444463,444465,444468],{"className":444464},[31676],[34,444466],{"className":444467,"style":73089},[31680],[34,444469,332814],{"className":444470},[31850],[34,444472,444474,444499],{"className":444473},[31642],[34,444475,444477],{"className":444476},[31646],[31648,444478,444479],{"xmlns":31650},[31652,444480,444481,444497],{},[31655,444482,444483,444485,444487,444489,444491,444493,444495],{},[31658,444484,406335],{},[31662,444486,32682],{},[31658,444488,329201],{},[31662,444490,202],{},[31662,444492,32685],{"stretchy":702},[31749,444494,459],{},[31662,444496,32715],{"stretchy":702},[31665,444498,441051],{"encoding":31667},[34,444500,444502,444526],{"className":444501,"ariaHidden":758},[31672],[34,444503,444505,444508,444511,444514,444517,444520,444523],{"className":444504},[31676],[34,444506],{"className":444507,"style":43217},[31680],[34,444509,406335],{"className":444510},[32767],[34,444512],{"className":444513,"style":32123},[31845],[34,444515,329201],{"className":444516},[31685,31686],[34,444518],{"className":444519,"style":31846},[31845],[34,444521,202],{"className":444522},[31850],[34,444524],{"className":444525,"style":31846},[31845],[34,444527,444529,444532,444535,444538],{"className":444528},[31676],[34,444530],{"className":444531,"style":31774},[31680],[34,444533,32685],{"className":444534},[31782],[34,444536,459],{"className":444537},[31685],[34,444539,32715],{"className":444540},[31690],[34,444542,444544,444567],{"className":444543},[31642],[34,444545,444547],{"className":444546},[31646],[31648,444548,444549],{"xmlns":31650},[31652,444550,444551,444565],{},[31655,444552,444553,444555,444557,444559,444561,444563],{},[31658,444554,365679],{"mathvariant":38079},[31658,444556,111],{},[31662,444558,31960],{"separator":758},[31658,444560,1697],{},[31662,444562,112296],{},[31658,444564,34386],{},[31665,444566,438990],{"encoding":31667},[34,444568,444570,444600],{"className":444569,"ariaHidden":758},[31672],[34,444571,444573,444576,444579,444582,444585,444588,444591,444594,444597],{"className":444572},[31676],[34,444574],{"className":444575,"style":43217},[31680],[34,444577,365679],{"className":444578},[31685],[34,444580,111],{"className":444581},[31685,31686],[34,444583,31960],{"className":444584},[32119],[34,444586],{"className":444587,"style":32123},[31845],[34,444589,1697],{"className":444590},[31685,31686],[34,444592],{"className":444593,"style":31846},[31845],[34,444595,112296],{"className":444596},[31850],[34,444598],{"className":444599,"style":31846},[31845],[34,444601,444603,444606],{"className":444602},[31676],[34,444604],{"className":444605,"style":33218},[31680],[34,444607,34386],{"className":444608,"style":34427},[31685,31686],[34,444610,444612,444641],{"className":444611},[31642],[34,444613,444615],{"className":444614},[31646],[31648,444616,444617],{"xmlns":31650},[31652,444618,444619,444639],{},[31655,444620,444621,444623,444625,444627,444629,444631,444633,444635,444637],{},[31658,444622,329201],{},[31662,444624,456],{"stretchy":702},[31658,444626,111],{},[31662,444628,2557],{"stretchy":702},[31662,444630,202],{},[31658,444632,329201],{},[31662,444634,456],{"stretchy":702},[31658,444636,1697],{},[31662,444638,2557],{"stretchy":702},[31665,444640,439444],{"encoding":31667},[34,444642,444644,444671],{"className":444643,"ariaHidden":758},[31672],[34,444645,444647,444650,444653,444656,444659,444662,444665,444668],{"className":444646},[31676],[34,444648],{"className":444649,"style":31774},[31680],[34,444651,329201],{"className":444652},[31685,31686],[34,444654,456],{"className":444655},[31782],[34,444657,111],{"className":444658},[31685,31686],[34,444660,2557],{"className":444661},[31690],[34,444663],{"className":444664,"style":31846},[31845],[34,444666,202],{"className":444667},[31850],[34,444669],{"className":444670,"style":31846},[31845],[34,444672,444674,444677,444680,444683,444686],{"className":444673},[31676],[34,444675],{"className":444676,"style":31774},[31680],[34,444678,329201],{"className":444679},[31685,31686],[34,444681,456],{"className":444682},[31782],[34,444684,1697],{"className":444685},[31685,31686],[34,444687,2557],{"className":444688},[31690],"，则：\n",[34,444691,444693,444750],{"className":444692},[31642],[34,444694,444696],{"className":444695},[31646],[31648,444697,444698],{"xmlns":31650},[31652,444699,444700,444747],{},[31655,444701,444702,444704,444706,444708,444710,444712,444714,444716,444718,444720,444722,444724,444726,444728,444730,444732,444734,444736,444738,444740,444742,444744],{},[31658,444703,329201],{},[31662,444705,456],{"stretchy":702},[31658,444707,111],{},[31662,444709,2557],{"stretchy":702},[31662,444711,32226],{},[31658,444713,329201],{},[31662,444715,456],{"stretchy":702},[31658,444717,1697],{},[31662,444719,2557],{"stretchy":702},[31662,444721,202],{},[31749,444723,459],{},[39361,444725,262622],{},[31662,444727,262625],{},[39361,444729,262622],{},[31658,444731,329201],{},[31662,444733,456],{"stretchy":702},[31658,444735,111],{},[31662,444737,32226],{},[31658,444739,1697],{},[31662,444741,2557],{"stretchy":702},[31662,444743,202],{},[31749,444745,444746],{},"0.",[31665,444748,444749],{"encoding":31667},"\\phi(a) - \\phi(b) = 0 \\implies \\phi(a-b) = 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利用命题 3.2.3, ",[34,445577,445579,445625],{"className":445578},[31642],[34,445580,445582],{"className":445581},[31646],[31648,445583,445584],{"xmlns":31650},[31652,445585,445586,445622],{},[31655,445587,445588,445590,445592,445594,445596,445598,445608,445610,445612,445614,445616,445618,445620],{},[31658,445589,365679],{"mathvariant":38079},[31658,445591,111],{},[31662,445593,31960],{"separator":758},[31658,445595,1697],{},[31662,445597,112296],{},[37624,445599,445600,445602],{},[31658,445601,329201],{},[31655,445603,445604,445606],{},[31662,445605,32226],{},[31749,445607,640],{},[31662,445609,456],{"stretchy":702},[31658,445611,223320],{},[31662,445613,2557],{"stretchy":702},[31662,445615,31960],{"separator":758},[31658,445617,34395],{},[31662,445619,112296],{},[31658,445621,34386],{},[31665,445623,445624],{"encoding":31667},"\\forall a, b \\in \\phi^{-1}(J), r \\in 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都有\n",[34,445737,445739,445861],{"className":445738},[31642],[34,445740,445742],{"className":445741},[31646],[31648,445743,445744],{"xmlns":31650},[31652,445745,445746,445858],{},[31655,445747,445748,445750,445752,445754,445756,445758,445760,445762,445764,445766,445768,445770,445772,445774,445776,445778,445780,445782,445784,445786,445788,445790,445792,445794,445796,445798,445800,445802,445804,445806,445808,445810,445812,445814,445816,445818,445820,445822,445824,445826,445828,445830,445832,445834,445836,445838,445840,445842,445844,445846,445848,445850,445852,445854,445856],{},[31658,445749,329201],{},[31662,445751,456],{"stretchy":702},[31658,445753,111],{},[31662,445755,32226],{},[31658,445757,1697],{},[31662,445759,2557],{"stretchy":702},[31662,445761,202],{},[31658,445763,329201],{},[31662,445765,456],{"stretchy":702},[31658,445767,111],{},[31662,445769,2557],{"stretchy":702},[31662,445771,32226],{},[31658,445773,329201],{},[31662,445775,456],{"stretchy":702},[31658,445777,1697],{},[31662,445779,2557],{"stretchy":702},[31662,445781,112296],{},[31658,445783,223320],{},[31662,445785,31960],{"separator":758},[31845,445787],{"width":71082},[31658,445789,329201],{},[31662,445791,456],{"stretchy":702},[31658,445793,34395],{},[31658,445795,111],{},[31662,445797,2557],{"stretchy":702},[31662,445799,202],{},[31658,445801,329201],{},[31662,445803,456],{"stretchy":702},[31658,445805,34395],{},[31662,445807,2557],{"stretchy":702},[31658,445809,329201],{},[31662,445811,456],{"stretchy":702},[31658,445813,111],{},[31662,445815,2557],{"stretchy":702},[31662,445817,112296],{},[31658,445819,223320],{},[31662,445821,31960],{"separator":758},[31845,445823],{"width":71082},[31658,445825,329201],{},[31662,445827,456],{"stretchy":702},[31658,445829,111],{},[31658,445831,34395],{},[31662,445833,2557],{"stretchy":702},[31662,445835,202],{},[31658,445837,329201],{},[31662,445839,456],{"stretchy":702},[31658,445841,111],{},[31662,445843,2557],{"stretchy":702},[31658,445845,329201],{},[31662,445847,456],{"stretchy":702},[31658,445849,34395],{},[31662,445851,2557],{"stretchy":702},[31662,445853,112296],{},[31658,445855,223320],{},[31658,445857,2547],{"mathvariant":38079},[31665,445859,445860],{"encoding":31667},"\\phi(a-b) = \\phi(a) - \\phi(b) \\in J, \\quad \\phi(ra) = \\phi(r)\\phi(a) \\in J, \\quad \\phi(ar) = \\phi(a)\\phi(r) \\in J.",[34,445862,445864,445888,445909,445936,445963,446005,446044,446086,446125],{"className":445863,"ariaHidden":758},[31672],[34,445865,445867,445870,445873,445876,445879,445882,445885],{"className":445866},[31676],[34,445868],{"className":445869,"style":31774},[31680],[34,445871,329201],{"className":445872},[31685,31686],[34,445874,456],{"className":445875},[31782],[34,445877,111],{"className":445878},[31685,31686],[34,445880],{"className":445881,"style":32604},[31845],[34,445883,32226],{"className":445884},[32024],[34,445886],{"className":445887,"style":32604},[31845],[34,445889,445891,445894,445897,445900,445903,445906],{"className":445890},[31676],[34,445892],{"className":445893,"style":31774},[31680],[34,445895,1697],{"className":445896},[31685,31686],[34,445898,2557],{"className":445899},[31690],[34,445901],{"className":445902,"style":31846},[31845],[34,445904,202],{"className":445905},[31850],[34,445907],{"className":445908,"style":31846},[31845],[34,445910,445912,445915,445918,445921,445924,445927,445930,445933],{"className":445911},[31676],[34,445913],{"className":445914,"style":31774},[31680],[34,445916,329201],{"className":445917},[31685,31686],[34,445919,456],{"className":445920},[31782],[34,445922,111],{"className":445923},[31685,31686],[34,445925,2557],{"className":445926},[31690],[34,445928],{"className":445929,"style":32604},[31845],[34,445931,32226],{"className":445932},[32024],[34,445934],{"className":445935,"style":32604},[31845],[34,445937,445939,445942,445945,445948,445951,445954,445957,445960],{"className":445938},[31676],[34,445940],{"className":445941,"style":31774},[31680],[34,445943,329201],{"className":445944},[31685,31686],[34,445946,456],{"className":445947},[31782],[34,445949,1697],{"className":445950},[31685,31686],[34,445952,2557],{"className":445953},[31690],[34,445955],{"className":445956,"style":31846},[31845],[34,445958,112296],{"className":445959},[31850],[34,445961],{"className":445962,"style":31846},[31845],[34,445964,445966,445969,445972,445975,445978,445981,445984,445987,445990,445993,445996,445999,446002],{"className":445965},[31676],[34,445967],{"className":445968,"style":31774},[31680],[34,445970,223320],{"className":445971,"style":223361},[31685,31686],[34,445973,31960],{"className":445974},[32119],[34,445976],{"className":445977,"style":128438},[31845],[34,445979],{"className":445980,"style":32123},[31845],[34,445982,329201],{"className":445983},[31685,31686],[34,445985,456],{"className":445986},[31782],[34,445988,34395],{"className":445989,"style":34452},[31685,31686],[34,445991,111],{"className":445992},[31685,31686],[34,445994,2557],{"className":445995},[31690],[34,445997],{"className":445998,"style":31846},[31845],[34,446000,202],{"className":446001},[31850],[34,446003],{"className":446004,"style":31846},[31845],[34,446006,446008,446011,446014,446017,446020,446023,446026,446029,446032,446035,446038,446041],{"className":446007},[31676],[34,446009],{"className":446010,"style":31774},[31680],[34,446012,329201],{"className":446013},[31685,31686],[34,446015,456],{"className":446016},[31782],[34,446018,34395],{"className":446019,"style":34452},[31685,31686],[34,446021,2557],{"className":446022},[31690],[34,446024,329201],{"className":446025},[31685,31686],[34,446027,456],{"className":446028},[31782],[34,446030,111],{"className":446031},[31685,31686],[34,446033,2557],{"className":446034},[31690],[34,446036],{"className":446037,"style":31846},[31845],[34,446039,112296],{"className":446040},[31850],[34,446042],{"className":446043,"style":31846},[31845],[34,446045,446047,446050,446053,446056,446059,446062,446065,446068,446071,446074,446077,446080,446083],{"className":446046},[31676],[34,446048],{"className":446049,"style":31774},[31680],[34,446051,223320],{"className":446052,"style":223361},[31685,31686],[34,446054,31960],{"className":446055},[32119],[34,446057],{"className":446058,"style":128438},[31845],[34,446060],{"className":446061,"style":32123},[31845],[34,446063,329201],{"className":446064},[31685,31686],[34,446066,456],{"className":446067},[31782],[34,446069,111],{"className":446070},[31685,31686],[34,446072,34395],{"className":446073,"style":34452},[31685,31686],[34,446075,2557],{"className":446076},[31690],[34,446078],{"className":446079,"style":31846},[31845],[34,446081,202],{"className":446082},[31850],[34,446084],{"className":446085,"style":31846},[31845],[34,446087,446089,446092,446095,446098,446101,446104,446107,446110,446113,446116,446119,446122],{"className":446088},[31676],[34,446090],{"className":446091,"style":31774},[31680],[34,446093,329201],{"className":446094},[31685,31686],[34,446096,456],{"className":446097},[31782],[34,446099,111],{"className":446100},[31685,31686],[34,446102,2557],{"className":446103},[31690],[34,446105,329201],{"className":446106},[31685,31686],[34,446108,456],{"className":446109},[31782],[34,446111,34395],{"className":446112,"style":34452},[31685,31686],[34,446114,2557],{"className":446115},[31690],[34,446117],{"className":446118,"style":31846},[31845],[34,446120,112296],{"className":446121},[31850],[34,446123],{"className":446124,"style":31846},[31845],[34,446126,446128,446131,446134],{"className":446127},[31676],[34,446129],{"className":446130,"style":33218},[31680],[34,446132,223320],{"className":446133,"style":223361},[31685,31686],[34,446135,2547],{"className":446136},[31685],"\n因此 ",[34,446139,446141,446173],{"className":446140},[31642],[34,446142,446144],{"className":446143},[31646],[31648,446145,446146],{"xmlns":31650},[31652,446147,446148,446170],{},[31655,446149,446150,446160,446162,446164,446166,446168],{},[37624,446151,446152,446154],{},[31658,446153,329201],{},[31655,446155,446156,446158],{},[31662,446157,32226],{},[31749,446159,640],{},[31662,446161,456],{"stretchy":702},[31658,446163,223320],{},[31662,446165,2557],{"stretchy":702},[31662,446167,440828],{},[31658,446169,34386],{},[31665,446171,446172],{"encoding":31667},"\\phi^{-1}(J) \\lhd R",[34,446174,446176,446235],{"className":446175,"ariaHidden":758},[31672],[34,446177,446179,446182,446217,446220,446223,446226,446229,446232],{"className":446178},[31676],[34,446180],{"className":446181,"style":37755},[31680],[34,446183,446185,446188],{"className":446184},[31685],[34,446186,329201],{"className":446187},[31685,31686],[34,446189,446191],{"className":446190},[31792],[34,446192,446194],{"className":446193},[31796],[34,446195,446197],{"className":446196},[31801],[34,446198,446200],{"className":446199,"style":37774},[31805],[34,446201,446202,446205],{"style":37777},[34,446203],{"className":446204,"style":31814},[31813],[34,446206,446208],{"className":446207},[31818,31819,31820,31821],[34,446209,446211,446214],{"className":446210},[31685,31821],[34,446212,32226],{"className":446213},[31685,31821],[34,446215,640],{"className":446216},[31685,31821],[34,446218,456],{"className":446219},[31782],[34,446221,223320],{"className":446222,"style":223361},[31685,31686],[34,446224,2557],{"className":446225},[31690],[34,446227],{"className":446228,"style":32604},[31845],[34,446230,440828],{"className":446231},[32024,129261],[34,446233],{"className":446234,"style":32604},[31845],[34,446236,446238,446241],{"className":446237},[31676],[34,446239],{"className":446240,"style":33218},[31680],[34,446242,34386],{"className":446243,"style":34427},[31685,31686],[34,446245,446247,446260],{"className":446246},[31642],[34,446248,446250],{"className":446249},[31646],[31648,446251,446252],{"xmlns":31650},[31652,446253,446254,446258],{},[31655,446255,446256],{},[31658,446257,445204],{"mathvariant":38079},[31665,446259,445207],{"encoding":31667},[34,446261,446263],{"className":446262,"ariaHidden":758},[31672],[34,446264,446266,446269],{"className":446265},[31676],[34,446267],{"className":446268,"style":445217},[31680],[34,446270,445204],{"className":446271},[31685,129261],[32188,446273],{},[253,446275,446277],{"id":446276},"_3-环同态定理","3. 环同态定理",[15,446279,446280,69123,446283,446334,446335,446418,446419,446507],{},[2627,446281,446282],{},"定理 3.3.1 (典范同态).",[34,446284,446286,446304],{"className":446285},[31642],[34,446287,446289],{"className":446288},[31646],[31648,446290,446291],{"xmlns":31650},[31652,446292,446293,446301],{},[31655,446294,446295,446297,446299],{},[31658,446296,106063],{},[31662,446298,440828],{},[31658,446300,34386],{},[31665,446302,446303],{"encoding":31667},"I \\lhd R",[34,446305,446307,446325],{"className":446306,"ariaHidden":758},[31672],[34,446308,446310,446313,446316,446319,446322],{"className":446309},[31676],[34,446311],{"className":446312,"style":445329},[31680],[34,446314,106063],{"className":446315,"style":67482},[31685,31686],[34,446317],{"className":446318,"style":32604},[31845],[34,446320,440828],{"className":446321},[32024,129261],[34,446323],{"className":446324,"style":32604},[31845],[34,446326,446328,446331],{"className":446327},[31676],[34,446329],{"className":446330,"style":33218},[31680],[34,446332,34386],{"className":446333,"style":34427},[31685,31686],". 则映射 ",[34,446336,446338,446364],{"className":446337},[31642],[34,446339,446341],{"className":446340},[31646],[31648,446342,446343],{"xmlns":31650},[31652,446344,446345,446361],{},[31655,446346,446347,446349,446351,446353,446355,446357,446359],{},[31658,446348,58442],{},[31662,446350,5598],{},[31658,446352,34386],{},[31662,446354,33239],{},[31658,446356,34386],{},[31658,446358,38080],{"mathvariant":38079},[31658,446360,106063],{},[31665,446362,446363],{"encoding":31667},"\\pi: R \\to R\u002FI",[34,446365,446367,446385,446403],{"className":446366,"ariaHidden":758},[31672],[34,446368,446370,446373,446376,446379,446382],{"className":446369},[31676],[34,446371],{"className":446372,"style":32527},[31680],[34,446374,58442],{"className":446375,"style":32062},[31685,31686],[34,446377],{"className":446378,"style":31846},[31845],[34,446380,5598],{"className":446381},[31850],[34,446383],{"className":446384,"style":31846},[31845],[34,446386,446388,446391,446394,446397,446400],{"className":446387},[31676],[34,446389],{"className":446390,"style":33218},[31680],[34,446392,34386],{"className":446393,"style":34427},[31685,31686],[34,446395],{"className":446396,"style":31846},[31845],[34,446398,33239],{"className":446399},[31850],[34,446401],{"className":446402,"style":31846},[31845],[34,446404,446406,446409,446412,446415],{"className":446405},[31676],[34,446407],{"className":446408,"style":31774},[31680],[34,446410,34386],{"className":446411,"style":34427},[31685,31686],[34,446413,38080],{"className":446414},[31685],[34,446416,106063],{"className":446417,"style":67482},[31685,31686]," 定义为 ",[34,446420,446422,446450],{"className":446421},[31642],[34,446423,446425],{"className":446424},[31646],[31648,446426,446427],{"xmlns":31650},[31652,446428,446429,446447],{},[31655,446430,446431,446433,446435,446437,446439,446441,446443,446445],{},[31658,446432,58442],{},[31662,446434,456],{"stretchy":702},[31658,446436,34395],{},[31662,446438,2557],{"stretchy":702},[31662,446440,202],{},[31658,446442,34395],{},[31662,446444,223],{},[31658,446446,106063],{},[31665,446448,446449],{"encoding":31667},"\\pi(r) = r+I",[34,446451,446453,446480,446498],{"className":446452,"ariaHidden":758},[31672],[34,446454,446456,446459,446462,446465,446468,446471,446474,446477],{"className":446455},[31676],[34,446457],{"className":446458,"style":31774},[31680],[34,446460,58442],{"className":446461,"style":32062},[31685,31686],[34,446463,456],{"className":446464},[31782],[34,446466,34395],{"className":446467,"style":34452},[31685,31686],[34,446469,2557],{"className":446470},[31690],[34,446472],{"className":446473,"style":31846},[31845],[34,446475,202],{"className":446476},[31850],[34,446478],{"className":446479,"style":31846},[31845],[34,446481,446483,446486,446489,446492,446495],{"className":446482},[31676],[34,446484],{"className":446485,"style":32597},[31680],[34,446487,34395],{"className":446488,"style":34452},[31685,31686],[34,446490],{"className":446491,"style":32604},[31845],[34,446493,223],{"className":446494},[32024],[34,446496],{"className":446497,"style":32604},[31845],[34,446499,446501,446504],{"className":446500},[31676],[34,446502],{"className":446503,"style":33218},[31680],[34,446505,106063],{"className":446506,"style":67482},[31685,31686]," 是一个环同态，称为典范同态。",[15,446509,446510,446512,446513,446541,446542,446580,446581,446772,446773,347],{},[2627,446511,441098],{},"\n我们需要验证 ",[34,446514,446516,446529],{"className":446515},[31642],[34,446517,446519],{"className":446518},[31646],[31648,446520,446521],{"xmlns":31650},[31652,446522,446523,446527],{},[31655,446524,446525],{},[31658,446526,58442],{},[31665,446528,58445],{"encoding":31667},[34,446530,446532],{"className":446531,"ariaHidden":758},[31672],[34,446533,446535,446538],{"className":446534},[31676],[34,446536],{"className":446537,"style":32527},[31680],[34,446539,58442],{"className":446540,"style":32062},[31685,31686]," 保持加法和乘法运算。回顾商环 ",[34,446543,446545,446562],{"className":446544},[31642],[34,446546,446548],{"className":446547},[31646],[31648,446549,446550],{"xmlns":31650},[31652,446551,446552,446560],{},[31655,446553,446554,446556,446558],{},[31658,446555,34386],{},[31658,446557,38080],{"mathvariant":38079},[31658,446559,106063],{},[31665,446561,337927],{"encoding":31667},[34,446563,446565],{"className":446564,"ariaHidden":758},[31672],[34,446566,446568,446571,446574,446577],{"className":446567},[31676],[34,446569],{"className":446570,"style":31774},[31680],[34,446572,34386],{"className":446573,"style":34427},[31685,31686],[34,446575,38080],{"className":446576},[31685],[34,446578,106063],{"className":446579,"style":67482},[31685,31686]," 的运算定义：",[34,446582,446584,446634],{"className":446583},[31642],[34,446585,446587],{"className":446586},[31646],[31648,446588,446589],{"xmlns":31650},[31652,446590,446591,446631],{},[31655,446592,446593,446595,446597,446599,446601,446603,446605,446607,446609,446611,446613,446615,446617,446619,446621,446623,446625,446627,446629],{},[31662,446594,456],{"stretchy":702},[31658,446596,111],{},[31662,446598,223],{},[31658,446600,106063],{},[31662,446602,2557],{"stretchy":702},[31662,446604,223],{},[31662,446606,456],{"stretchy":702},[31658,446608,1697],{},[31662,446610,223],{},[31658,446612,106063],{},[31662,446614,2557],{"stretchy":702},[31662,446616,202],{},[31662,446618,456],{"stretchy":702},[31658,446620,111],{},[31662,446622,223],{},[31658,446624,1697],{},[31662,446626,2557],{"stretchy":702},[31662,446628,223],{},[31658,446630,106063],{},[31665,446632,446633],{"encoding":31667},"(a+I)+(b+I) = (a+b)+I",[34,446635,446637,446658,446679,446700,446721,446742,446763],{"className":446636,"ariaHidden":758},[31672],[34,446638,446640,446643,446646,446649,446652,446655],{"className":446639},[31676],[34,446641],{"className":446642,"style":31774},[31680],[34,446644,456],{"className":446645},[31782],[34,446647,111],{"className":446648},[31685,31686],[34,446650],{"className":446651,"style":32604},[31845],[34,446653,223],{"className":446654},[32024],[34,446656],{"className":446657,"style":32604},[31845],[34,446659,446661,446664,446667,446670,446673,446676],{"className":446660},[31676],[34,446662],{"className":446663,"style":31774},[31680],[34,446665,106063],{"className":446666,"style":67482},[31685,31686],[34,446668,2557],{"className":446669},[31690],[34,446671],{"className":446672,"style":32604},[31845],[34,446674,223],{"className":446675},[32024],[34,446677],{"className":446678,"style":32604},[31845],[34,446680,446682,446685,446688,446691,446694,446697],{"className":446681},[31676],[34,446683],{"className":446684,"style":31774},[31680],[34,446686,456],{"className":446687},[31782],[34,446689,1697],{"className":446690},[31685,31686],[34,446692],{"className":446693,"style":32604},[31845],[34,446695,223],{"className":446696},[32024],[34,446698],{"className":446699,"style":32604},[31845],[34,446701,446703,446706,446709,446712,446715,446718],{"className":446702},[31676],[34,446704],{"className":446705,"style":31774},[31680],[34,446707,106063],{"className":446708,"style":67482},[31685,31686],[34,446710,2557],{"className":446711},[31690],[34,446713],{"className":446714,"style":31846},[31845],[34,446716,202],{"className":446717},[31850],[34,446719],{"className":446720,"style":31846},[31845],[34,446722,446724,446727,446730,446733,446736,446739],{"className":446723},[31676],[34,446725],{"className":446726,"style":31774},[31680],[34,446728,456],{"className":446729},[31782],[34,446731,111],{"className":446732},[31685,31686],[34,446734],{"className":446735,"style":32604},[31845],[34,446737,223],{"className":446738},[32024],[34,446740],{"className":446741,"style":32604},[31845],[34,446743,446745,446748,446751,446754,446757,446760],{"className":446744},[31676],[34,446746],{"className":446747,"style":31774},[31680],[34,446749,1697],{"className":446750},[31685,31686],[34,446752,2557],{"className":446753},[31690],[34,446755],{"className":446756,"style":32604},[31845],[34,446758,223],{"className":446759},[32024],[34,446761],{"className":446762,"style":32604},[31845],[34,446764,446766,446769],{"className":446765},[31676],[34,446767],{"className":446768,"style":33218},[31680],[34,446770,106063],{"className":446771,"style":67482},[31685,31686]," 以及 ",[34,446774,446776,446818],{"className":446775},[31642],[34,446777,446779],{"className":446778},[31646],[31648,446780,446781],{"xmlns":31650},[31652,446782,446783,446815],{},[31655,446784,446785,446787,446789,446791,446793,446795,446797,446799,446801,446803,446805,446807,446809,446811,446813],{},[31662,446786,456],{"stretchy":702},[31658,446788,111],{},[31662,446790,223],{},[31658,446792,106063],{},[31662,446794,2557],{"stretchy":702},[31662,446796,456],{"stretchy":702},[31658,446798,1697],{},[31662,446800,223],{},[31658,446802,106063],{},[31662,446804,2557],{"stretchy":702},[31662,446806,202],{},[31658,446808,111],{},[31658,446810,1697],{},[31662,446812,223],{},[31658,446814,106063],{},[31665,446816,446817],{"encoding":31667},"(a+I)(b+I) = ab+I",[34,446819,446821,446842,446869,446890,446908],{"className":446820,"ariaHidden":758},[31672],[34,446822,446824,446827,446830,446833,446836,446839],{"className":446823},[31676],[34,446825],{"className":446826,"style":31774},[31680],[34,446828,456],{"className":446829},[31782],[34,446831,111],{"className":446832},[31685,31686],[34,446834],{"className":446835,"style":32604},[31845],[34,446837,223],{"className":446838},[32024],[34,446840],{"className":446841,"style":32604},[31845],[34,446843,446845,446848,446851,446854,446857,446860,446863,446866],{"className":446844},[31676],[34,446846],{"className":446847,"style":31774},[31680],[34,446849,106063],{"className":446850,"style":67482},[31685,31686],[34,446852,2557],{"className":446853},[31690],[34,446855,456],{"className":446856},[31782],[34,446858,1697],{"className":446859},[31685,31686],[34,446861],{"className":446862,"style":32604},[31845],[34,446864,223],{"className":446865},[32024],[34,446867],{"className":446868,"style":32604},[31845],[34,446870,446872,446875,446878,446881,446884,446887],{"className":446871},[31676],[34,446873],{"className":446874,"style":31774},[31680],[34,446876,106063],{"className":446877,"style":67482},[31685,31686],[34,446879,2557],{"className":446880},[31690],[34,446882],{"className":446883,"style":31846},[31845],[34,446885,202],{"className":446886},[31850],[34,446888],{"className":446889,"style":31846},[31845],[34,446891,446893,446896,446899,446902,446905],{"className":446892},[31676],[34,446894],{"className":446895,"style":34299},[31680],[34,446897,324191],{"className":446898},[31685,31686],[34,446900],{"className":446901,"style":32604},[31845],[34,446903,223],{"className":446904},[32024],[34,446906],{"className":446907,"style":32604},[31845],[34,446909,446911,446914],{"className":446910},[31676],[34,446912],{"className":446913,"style":33218},[31680],[34,446915,106063],{"className":446916,"style":67482},[31685,31686],[2230,446918,446919,447321],{},[108,446920,446921,8471,446924,684,446992],{},[2627,446922,446923],{},"保持加法",[34,446925,446927,446950],{"className":446926},[31642],[34,446928,446930],{"className":446929},[31646],[31648,446931,446932],{"xmlns":31650},[31652,446933,446934,446948],{},[31655,446935,446936,446938,446940,446942,446944,446946],{},[31658,446937,365679],{"mathvariant":38079},[31658,446939,111],{},[31662,446941,31960],{"separator":758},[31658,446943,1697],{},[31662,446945,112296],{},[31658,446947,34386],{},[31665,446949,438990],{"encoding":31667},[34,446951,446953,446983],{"className":446952,"ariaHidden":758},[31672],[34,446954,446956,446959,446962,446965,446968,446971,446974,446977,446980],{"className":446955},[31676],[34,446957],{"className":446958,"style":43217},[31680],[34,446960,365679],{"className":446961},[31685],[34,446963,111],{"className":446964},[31685,31686],[34,446966,31960],{"className":446967},[32119],[34,446969],{"className":446970,"style":32123},[31845],[34,446972,1697],{"className":446973},[31685,31686],[34,446975],{"className":446976,"style":31846},[31845],[34,446978,112296],{"className":446979},[31850],[34,446981],{"className":446982,"style":31846},[31845],[34,446984,446986,446989],{"className":446985},[31676],[34,446987],{"className":446988,"style":33218},[31680],[34,446990,34386],{"className":446991,"style":34427},[31685,31686],[34,446993,446995,447081],{"className":446994},[31642],[34,446996,446998],{"className":446997},[31646],[31648,446999,447000],{"xmlns":31650},[31652,447001,447002,447078],{},[31655,447003,447004,447006,447008,447010,447012,447014,447016,447018,447020,447022,447024,447026,447028,447030,447032,447034,447036,447038,447040,447042,447044,447046,447048,447050,447052,447054,447056,447058,447060,447062,447064,447066,447068,447070,447072,447074,447076],{},[31658,447005,58442],{},[31662,447007,456],{"stretchy":702},[31658,447009,111],{},[31662,447011,223],{},[31658,447013,1697],{},[31662,447015,2557],{"stretchy":702},[31662,447017,202],{},[31662,447019,456],{"stretchy":702},[31658,447021,111],{},[31662,447023,223],{},[31658,447025,1697],{},[31662,447027,2557],{"stretchy":702},[31662,447029,223],{},[31658,447031,106063],{},[31662,447033,202],{},[31662,447035,456],{"stretchy":702},[31658,447037,111],{},[31662,447039,223],{},[31658,447041,106063],{},[31662,447043,2557],{"stretchy":702},[31662,447045,223],{},[31662,447047,456],{"stretchy":702},[31658,447049,1697],{},[31662,447051,223],{},[31658,447053,106063],{},[31662,447055,2557],{"stretchy":702},[31662,447057,202],{},[31658,447059,58442],{},[31662,447061,456],{"stretchy":702},[31658,447063,111],{},[31662,447065,2557],{"stretchy":702},[31662,447067,223],{},[31658,447069,58442],{},[31662,447071,456],{"stretchy":702},[31658,447073,1697],{},[31662,447075,2557],{"stretchy":702},[31658,447077,2547],{"mathvariant":38079},[31665,447079,447080],{"encoding":31667},"\\pi(a+b) = (a+b) + I = (a+I) + (b+I) = \\pi(a) + \\pi(b).",[34,447082,447084,447108,447129,447150,447171,447189,447210,447231,447252,447273,447300],{"className":447083,"ariaHidden":758},[31672],[34,447085,447087,447090,447093,447096,447099,447102,447105],{"className":447086},[31676],[34,447088],{"className":447089,"style":31774},[31680],[34,447091,58442],{"className":447092,"style":32062},[31685,31686],[34,447094,456],{"className":447095},[31782],[34,447097,111],{"className":447098},[31685,31686],[34,447100],{"className":447101,"style":32604},[31845],[34,447103,223],{"className":447104},[32024],[34,447106],{"className":447107,"style":32604},[31845],[34,447109,447111,447114,447117,447120,447123,447126],{"className":447110},[31676],[34,447112],{"className":447113,"style":31774},[31680],[34,447115,1697],{"className":447116},[31685,31686],[34,447118,2557],{"className":447119},[31690],[34,447121],{"className":447122,"style":31846},[31845],[34,447124,202],{"className":447125},[31850],[34,447127],{"className":447128,"style":31846},[31845],[34,447130,447132,447135,447138,447141,447144,447147],{"className":447131},[31676],[34,447133],{"className":447134,"style":31774},[31680],[34,447136,456],{"className":447137},[31782],[34,447139,111],{"className":447140},[31685,31686],[34,447142],{"className":447143,"style":32604},[31845],[34,447145,223],{"className":447146},[32024],[34,447148],{"className":447149,"style":32604},[31845],[34,447151,447153,447156,447159,447162,447165,447168],{"className":447152},[31676],[34,447154],{"className":447155,"style":31774},[31680],[34,447157,1697],{"className":447158},[31685,31686],[34,447160,2557],{"className":447161},[31690],[34,447163],{"className":447164,"style":32604},[31845],[34,447166,223],{"className":447167},[32024],[34,447169],{"className":447170,"style":32604},[31845],[34,447172,447174,447177,447180,447183,447186],{"className":447173},[31676],[34,447175],{"className":447176,"style":33218},[31680],[34,447178,106063],{"className":447179,"style":67482},[31685,31686],[34,447181],{"className":447182,"style":31846},[31845],[34,447184,202],{"className":447185},[31850],[34,447187],{"className":447188,"style":31846},[31845],[34,447190,447192,447195,447198,447201,447204,447207],{"className":447191},[31676],[34,447193],{"className":447194,"style":31774},[31680],[34,447196,456],{"className":447197},[31782],[34,447199,111],{"className":447200},[31685,31686],[34,447202],{"className":447203,"style":32604},[31845],[34,447205,223],{"className":447206},[32024],[34,447208],{"className":447209,"style":32604},[31845],[34,447211,447213,447216,447219,447222,447225,447228],{"className":447212},[31676],[34,447214],{"className":447215,"style":31774},[31680],[34,447217,106063],{"className":447218,"style":67482},[31685,31686],[34,447220,2557],{"className":447221},[31690],[34,447223],{"className":447224,"style":32604},[31845],[34,447226,223],{"className":447227},[32024],[34,447229],{"className":447230,"style":32604},[31845],[34,447232,447234,447237,447240,447243,447246,447249],{"className":447233},[31676],[34,447235],{"className":447236,"style":31774},[31680],[34,447238,456],{"className":447239},[31782],[34,447241,1697],{"className":447242},[31685,31686],[34,447244],{"className":447245,"style":32604},[31845],[34,447247,223],{"className":447248},[32024],[34,447250],{"className":447251,"style":32604},[31845],[34,447253,447255,447258,447261,447264,447267,447270],{"className":447254},[31676],[34,447256],{"className":447257,"style":31774},[31680],[34,447259,106063],{"className":447260,"style":67482},[31685,31686],[34,447262,2557],{"className":447263},[31690],[34,447265],{"className":447266,"style":31846},[31845],[34,447268,202],{"className":447269},[31850],[34,447271],{"className":447272,"style":31846},[31845],[34,447274,447276,447279,447282,447285,447288,447291,447294,447297],{"className":447275},[31676],[34,447277],{"className":447278,"style":31774},[31680],[34,447280,58442],{"className":447281,"style":32062},[31685,31686],[34,447283,456],{"className":447284},[31782],[34,447286,111],{"className":447287},[31685,31686],[34,447289,2557],{"className":447290},[31690],[34,447292],{"className":447293,"style":32604},[31845],[34,447295,223],{"className":447296},[32024],[34,447298],{"className":447299,"style":32604},[31845],[34,447301,447303,447306,447309,447312,447315,447318],{"className":447302},[31676],[34,447304],{"className":447305,"style":31774},[31680],[34,447307,58442],{"className":447308,"style":32062},[31685,31686],[34,447310,456],{"className":447311},[31782],[34,447313,1697],{"className":447314},[31685,31686],[34,447316,2557],{"className":447317},[31690],[34,447319,2547],{"className":447320},[31685],[108,447322,447323,8471,447326,684,447394,447639,447640,447668,447669,447697,447698],{},[2627,447324,447325],{},"保持乘法",[34,447327,447329,447352],{"className":447328},[31642],[34,447330,447332],{"className":447331},[31646],[31648,447333,447334],{"xmlns":31650},[31652,447335,447336,447350],{},[31655,447337,447338,447340,447342,447344,447346,447348],{},[31658,44733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= ab + I = (a+I)(b+I) = \\pi(a)\\pi(b).",[34,447472,447474,447501,447519,447537,447558,447585,447606],{"className":447473,"ariaHidden":758},[31672],[34,447475,447477,447480,447483,447486,447489,447492,447495,447498],{"className":447476},[31676],[34,447478],{"className":447479,"style":31774},[31680],[34,447481,58442],{"className":447482,"style":32062},[31685,31686],[34,447484,456],{"className":447485},[31782],[34,447487,324191],{"className":447488},[31685,31686],[34,447490,2557],{"className":447491},[31690],[34,447493],{"className":447494,"style":31846},[31845],[34,447496,202],{"className":447497},[31850],[34,447499],{"className":447500,"style":31846},[31845],[34,447502,447504,447507,447510,447513,447516],{"className":447503},[31676],[34,447505],{"className":447506,"style":34299},[31680],[34,447508,324191],{"className":447509},[31685,31686],[34,447511],{"className":447512,"style":32604},[31845],[34,447514,223],{"className":447515},[32024],[34,447517],{"className":447518,"style":32604},[31845],[34,447520,447522,447525,447528,447531,447534],{"className":447521},[31676],[34,447523],{"className":447524,"style":33218},[31680],[34,447526,106063],{"className":447527,"style":67482},[31685,31686],[34,447529],{"className":447530,"style":31846},[31845],[34,447532,202],{"className":447533},[31850],[34,447535],{"className":447536,"style":31846},[31845],[34,447538,447540,447543,447546,447549,447552,447555],{"className":447539},[31676],[34,447541],{"className":447542,"style":31774},[31680],[34,447544,456],{"className":447545},[31782],[34,447547,111],{"className":447548},[31685,31686],[34,447550],{"className":447551,"style":32604},[31845],[34,447553,223],{"className":447554},[32024],[34,447556],{"className":447557,"style":32604},[31845],[34,447559,447561,447564,447567,447570,447573,447576,447579,447582],{"className":447560},[31676],[34,447562],{"className":447563,"style":31774},[31680],[34,447565,106063],{"className":447566,"style":67482},[31685,31686],[34,447568,2557],{"className":447569},[31690],[34,447571,456],{"className":447572},[31782],[34,447574,1697],{"className":447575},[31685,31686],[34,447577],{"className":447578,"style":32604},[31845],[34,447580,223],{"className":447581},[32024],[34,447583],{"className":447584,"style":32604},[31845],[34,447586,447588,447591,447594,447597,447600,447603],{"className":447587},[31676],[34,447589],{"className":447590,"style":31774},[31680],[34,447592,106063],{"className":447593,"style":67482},[31685,31686],[34,447595,2557],{"className":447596},[31690],[34,447598],{"className":447599,"style":31846},[31845],[34,447601,202],{"className":447602},[31850],[34,447604],{"className":447605,"style":31846},[31845],[34,447607,447609,447612,447615,447618,447621,447624,447627,447630,447633,447636],{"className":447608},[31676],[34,447610],{"className":447611,"style":31774},[31680],[34,447613,58442],{"className":447614,"style":32062},[31685,31686],[34,447616,456],{"className":447617},[31782],[34,447619,111],{"className":447620},[31685,31686],[34,447622,2557],{"className":447623},[31690],[34,447625,58442],{"className":447626,"style":32062},[31685,31686],[34,447628,456],{"className":447629},[31782],[34,447631,1697],{"className":447632},[31685,31686],[34,447634,2557],{"className":447635},[31690],[34,447637,2547],{"className":447638},[31685],"\n显见 ",[34,447641,447643,447656],{"className":447642},[31642],[34,447644,447646],{"className":447645},[31646],[31648,447647,447648],{"xmlns":31650},[31652,447649,447650,447654],{},[31655,447651,447652],{},[31658,447653,58442],{},[31665,447655,58445],{"encoding":31667},[34,447657,447659],{"className":447658,"ariaHidden":758},[31672],[34,447660,447662,447665],{"className":447661},[31676],[34,447663],{"className":447664,"style":32527},[31680],[34,447666,58442],{"className":447667,"style":32062},[31685,31686]," 是满射。故 ",[34,447670,447672,447685],{"className":447671},[31642],[34,447673,447675],{"className":447674},[31646],[31648,447676,447677],{"xmlns":31650},[31652,447678,447679,447683],{},[31655,447680,447681],{},[31658,447682,58442],{},[31665,447684,58445],{"encoding":31667},[34,447686,447688],{"className":447687,"ariaHidden":758},[31672],[34,447689,447691,447694],{"className":447690},[31676],[34,447692],{"className":447693,"style":32527},[31680],[34,447695,58442],{"className":447696,"style":32062},[31685,31686]," 是一个满的环同态。 ",[34,447699,447701,447714],{"className":447700},[31642],[34,447702,447704],{"className":447703},[31646],[31648,447705,447706],{"xmlns":31650},[31652,447707,447708,447712],{},[31655,447709,447710],{},[31658,447711,445204],{"mathvariant":38079},[31665,447713,445207],{"encoding":31667},[34,447715,447717],{"className":447716,"ariaHidden":758},[31672],[34,447718,447720,447723],{"className":447719},[31676],[34,447721],{"className":447722,"style":445217},[31680],[34,447724,445204],{"className":447725},[31685,129261],[15,447727,447728,69123,447731,616,447781,447853,447854,33775,447915,447943,447944,448070,448071,448188,448189,448271],{},[2627,447729,447730],{},"定理 3.3.2 (第零同态定理 \u002F 通用性质).",[34,447732,447734,447751],{"className":447733},[31642],[34,447735,447737],{"className":447736},[31646],[31648,447738,447739],{"xmlns":31650},[31652,447740,447741,447749],{},[31655,447742,447743,447745,447747],{},[31658,447744,106063],{},[31662,447746,440828],{},[31658,447748,34386],{},[31665,447750,446303],{"encoding":31667},[34,447752,447754,447772],{"className":447753,"ariaHidden":758},[31672],[34,447755,447757,447760,447763,447766,447769],{"className":447756},[31676],[34,447758],{"className":447759,"style":445329},[31680],[34,447761,106063],{"className":447762,"style":67482},[31685,31686],[34,447764],{"className":447765,"style":32604},[31845],[34,447767,440828],{"className":447768},[32024,129261],[34,447770],{"className":447771,"style":32604},[31845],[34,447773,447775,447778],{"className":447774},[31676],[34,447776],{"className":447777,"style":33218},[31680],[34,447779,34386],{"className":447780,"style":34427},[31685,31686],[34,447782,447784,447805],{"className":447783},[31642],[34,447785,447787],{"className":447786},[31646],[31648,447788,447789],{"xmlns":31650},[31652,447790,447791,447803],{},[31655,447792,447793,447795,447797,447799,447801],{},[31658,447794,329201],{},[31662,447796,5598],{},[31658,447798,34386],{},[31662,447800,33239],{},[31658,447802,94675],{},[31665,447804,438915],{"encoding":31667},[34,447806,447808,447826,447844],{"className":447807,"ariaHidden":758},[31672],[34,447809,447811,447814,447817,447820,447823],{"className":447810},[31676],[34,447812],{"className":447813,"style":43217},[31680],[34,447815,329201],{"className":447816},[31685,31686],[34,447818],{"className":447819,"style":31846},[31845],[34,447821,5598],{"className":447822},[31850],[34,447824],{"className":447825,"style":31846},[31845],[34,447827,447829,447832,447835,447838,447841],{"className":447828},[31676],[34,447830],{"className":447831,"style":33218},[31680],[34,447833,34386],{"className":447834,"style":34427},[31685,31686],[34,447836],{"className":447837,"style":31846},[31845],[34,447839,33239],{"className":447840},[31850],[34,447842],{"className":447843,"style":31846},[31845],[34,447845,447847,447850],{"className":447846},[31676],[34,447848],{"className":447849,"style":33218},[31680],[34,447851,94675],{"className":447852,"style":106455},[31685,31686]," 是环同态，且 ",[34,447855,447857,447879],{"className":447856},[31642],[34,447858,447860],{"className":447859},[31646],[31648,447861,447862],{"xmlns":31650},[31652,447863,447864,447876],{},[31655,447865,447866,447868,447870,447872,447874],{},[31658,447867,106063],{},[31662,447869,45972],{},[31658,447871,406335],{},[31662,447873,32682],{},[31658,447875,329201],{},[31665,447877,447878],{"encoding":31667},"I \\le \\ker \\phi",[34,447880,447882,447900],{"className":447881,"ariaHidden":758},[31672],[34,447883,447885,447888,447891,447894,447897],{"className":447884},[31676],[34,447886],{"className":447887,"style":195047},[31680],[34,447889,106063],{"className":447890,"style":67482},[31685,31686],[34,447892],{"className":447893,"style":31846},[31845],[34,447895,45972],{"className":447896},[31850],[34,447898],{"className":447899,"style":31846},[31845],[34,447901,447903,447906,447909,447912],{"className":447902},[31676],[34,447904],{"className":447905,"style":43217},[31680],[34,447907,406335],{"className":447908},[32767],[34,447910],{"className":447911,"style":32123},[31845],[34,447913,329201],{"className":447914},[31685,31686],[34,447916,447918,447931],{"className":447917},[31642],[34,447919,447921],{"className":447920},[31646],[31648,447922,447923],{"xmlns":31650},[31652,447924,447925,447929],{},[31655,447926,447927],{},[31658,447928,106063],{},[31665,447930,106063],{"encoding":31667},[34,447932,447934],{"className":447933,"ariaHidden":758},[31672],[34,447935,447937,447940],{"className":447936},[31676],[34,447938],{"className":447939,"style":33218},[31680],[34,447941,106063],{"className":447942,"style":67482},[31685,31686]," 包含在核内）。则存在唯一的环同态 ",[34,447945,447947,447977],{"className":447946},[31642],[34,447948,447950],{"className":447949},[31646],[31648,447951,447952],{"xmlns":31650},[31652,447953,447954,447974],{},[31655,447955,447956,447962,447964,447966,447968,447970,447972],{},[134014,447957,447958,447960],{"accent":758},[31658,447959,329201],{},[31662,447961,346142],{},[31662,447963,5598],{},[31658,447965,34386],{},[31658,447967,38080],{"mathvariant":38079},[31658,447969,106063],{},[31662,447971,33239],{},[31658,447973,94675],{},[31665,447975,447976],{"encoding":31667},"\\bar{\\phi}: R\u002FI \\to S",[34,447978,447980,448037,448061],{"className":447979,"ariaHidden":758},[31672],[34,447981,447983,447986,448028,448031,448034],{"className":447982},[31676],[34,447984],{"className":447985,"style":374931},[31680],[34,447987,447989],{"className":447988},[31685,346171],[34,447990,447992,448020],{"className":447991},[31796,31797],[34,447993,447995,448017],{"className":447994},[31801],[34,447996,447998,448006],{"className":447997,"style":372056},[31805],[34,447999,448000,448003],{"style":41565},[34,448001],{"className":448002,"style":41508},[31813],[34,448004,329201],{"className":448005},[31685,31686],[34,448007,448008,448011],{"style":372079},[34,448009],{"className":448010,"style":41508},[31813],[34,448012,448014],{"className":448013,"style":375462},[346198],[34,448015,346142],{"className":448016},[31685],[34,448018,31829],{"className":448019},[31828],[34,448021,448023],{"className":448022},[31801],[34,448024,448026],{"className":448025,"style":44616},[31805],[34,448027],{},[34,448029],{"className":448030,"style":31846},[31845],[34,448032,5598],{"className":448033},[31850],[34,448035],{"className":448036,"style":31846},[31845],[34,448038,448040,448043,448046,448049,448052,448055,448058],{"className":448039},[31676],[34,448041],{"className":448042,"style":31774},[31680],[34,448044,34386],{"className":448045,"style":34427},[31685,31686],[34,448047,38080],{"className":448048},[31685],[34,448050,106063],{"className":448051,"style":67482},[31685,31686],[34,448053],{"className":448054,"style":31846},[31845],[34,448056,33239],{"className":448057},[31850],[34,448059],{"className":448060,"style":31846},[31845],[34,448062,448064,448067],{"className":448063},[31676],[34,448065],{"className":448066,"style":33218},[31680],[34,448068,94675],{"className":448069,"style":106455},[31685,31686],", 满足 ",[34,448072,448074,448101],{"className":448073},[31642],[34,448075,448077],{"className":448076},[31646],[31648,448078,448079],{"xmlns":31650},[31652,448080,448081,448098],{},[31655,448082,448083,448085,448087,448093,448096],{},[31658,448084,329201],{},[31662,448086,202],{},[134014,448088,448089,448091],{"accent":758},[31658,448090,329201],{},[31662,448092,346142],{},[31662,448094,448095],{},"∘",[31658,448097,58442],{},[31665,448099,448100],{"encoding":31667},"\\phi = \\bar{\\phi} \\circ \\pi",[34,448102,448104,448122,448179],{"className":448103,"ariaHidden":758},[31672],[34,448105,448107,448110,448113,448116,448119],{"className":448106},[31676],[34,448108],{"className":448109,"style":43217},[31680],[34,448111,329201],{"className":448112},[31685,31686],[34,448114],{"className":448115,"style":31846},[31845],[34,448117,202],{"className":448118},[31850],[34,448120],{"className":448121,"style":31846},[31845],[34,448123,448125,448128,448170,448173,448176],{"className":448124},[31676],[34,448126],{"className":448127,"style":374931},[31680],[34,448129,448131],{"className":448130},[31685,346171],[34,448132,448134,448162],{"className":448133},[31796,31797],[34,448135,448137,448159],{"className":448136},[31801],[34,448138,448140,448148],{"className":448139,"style":372056},[31805],[34,448141,448142,448145],{"style":41565},[34,448143],{"className":448144,"style":41508},[31813],[34,448146,329201],{"className":448147},[31685,31686],[34,448149,448150,448153],{"style":372079},[34,448151],{"className":448152,"style":41508},[31813],[34,448154,448156],{"className":448155,"style":375462},[346198],[34,448157,346142],{"className":448158},[31685],[34,448160,31829],{"className":448161},[31828],[34,448163,448165],{"className":448164},[31801],[34,448166,448168],{"className":448167,"style":44616},[31805],[34,448169],{},[34,448171],{"className":448172,"style":32604},[31845],[34,448174,448095],{"className":448175},[32024],[34,448177],{"className":448178,"style":32604},[31845],[34,448180,448182,448185],{"className":448181},[31676],[34,448183],{"className":448184,"style":32527},[31680],[34,448186,58442],{"className":448187,"style":32062},[31685,31686],", 其中 ",[34,448190,448192,448217],{"className":448191},[31642],[34,448193,448195],{"className":448194},[31646],[31648,448196,448197],{"xmlns":31650},[31652,448198,448199,448215],{},[31655,448200,448201,448203,448205,448207,448209,448211,448213],{},[31658,448202,58442],{},[31662,448204,5598],{},[31658,448206,34386],{},[31662,448208,33239],{},[31658,448210,34386],{},[31658,448212,38080],{"mathvariant":38079},[31658,448214,106063],{},[31665,448216,446363],{"encoding":31667},[34,448218,448220,448238,448256],{"className":448219,"ariaHidden":758},[31672],[34,448221,448223,448226,448229,448232,448235],{"className":448222},[31676],[34,448224],{"className":448225,"style":32527},[31680],[34,448227,58442],{"className":448228,"style":32062},[31685,31686],[34,448230],{"className":448231,"style":31846},[31845],[34,448233,5598],{"className":448234},[31850],[34,448236],{"className":448237,"style":31846},[31845],[34,448239,448241,448244,448247,448250,448253],{"className":448240},[31676],[34,448242],{"className":448243,"style":33218},[31680],[34,448245,34386],{"className":448246,"style":34427},[31685,31686],[34,448248],{"className":448249,"style":31846},[31845],[34,448251,33239],{"className":448252},[31850],[34,448254],{"className":448255,"style":31846},[31845],[34,448257,448259,448262,448265,448268],{"className":448258},[31676],[34,448260],{"className":448261,"style":31774},[31680],[34,448263,34386],{"className":448264,"style":34427},[31685,31686],[34,448266,38080],{"className":448267},[31685],[34,448269,106063],{"className":448270,"style":67482},[31685,31686]," 是典范同态。",[15,448273,448274,69123,448277,448349,448350,445571,448437],{},[2627,448275,448276],{},"定理 3.3.3 (第一同态定理).",[34,448278,448280,448301],{"className":448279},[31642],[34,448281,448283],{"className":448282},[31646],[31648,448284,448285],{"xmlns":31650},[31652,448286,448287,448299],{},[31655,448288,448289,448291,448293,448295,448297],{},[31658,448290,329201],{},[31662,448292,5598],{},[31658,448294,34386],{},[31662,448296,33239],{},[31658,448298,94675],{},[31665,448300,438915],{"encoding":31667},[34,448302,448304,448322,448340],{"className":448303,"ariaHidden":758},[31672],[34,448305,448307,448310,448313,448316,448319],{"className":448306},[31676],[34,448308],{"className":448309,"style":43217},[31680],[34,448311,329201],{"className":448312},[31685,31686],[34,448314],{"className":448315,"style":31846},[31845],[34,448317,5598],{"className":448318},[31850],[34,448320],{"className":448321,"style":31846},[31845],[34,448323,448325,448328,448331,448334,448337],{"className":448324},[31676],[34,448326],{"className":448327,"style":33218},[31680],[34,448329,34386],{"className":448330,"style":34427},[31685,31686],[34,448332],{"className":448333,"style":31846},[31845],[34,448335,33239],{"className":448336},[31850],[34,448338],{"className":448339,"style":31846},[31845],[34,448341,448343,448346],{"className":448342},[31676],[34,448344],{"className":448345,"style":33218},[31680],[34,448347,94675],{"className":448348,"style":106455},[31685,31686]," 是环同态。则 ",[34,448351,448353,448383],{"className":448352},[31642],[34,448354,448356],{"className":448355},[31646],[31648,448357,448358],{"xmlns":31650},[31652,448359,448360,448380],{},[31655,448361,448362,448364,448366,448368,448370,448372,448374,448376,448378],{},[31658,448363,34386],{},[31658,448365,38080],{"mathvariant":38079},[31658,448367,406335],{},[31662,448369,32682],{},[31658,448371,329201],{},[31662,448373,134826],{},[31658,448375,440604],{"mathvariant":38079},[31662,448377,32682],{},[31658,448379,329201],{},[31665,448381,448382],{"encoding":31667},"R\u002F\\ker \\phi \\cong \\operatorname{Im} \\phi",[34,448384,448386,448419],{"className":448385,"ariaHidden":758},[31672],[34,448387,448389,448392,448395,448398,448401,448404,448407,448410,448413,448416],{"className":448388},[31676],[34,448390],{"className":448391,"style":31774},[31680],[34,448393,34386],{"className":448394,"style":34427},[31685,31686],[34,448396,38080],{"className":448397},[31685],[34,448399],{"className":448400,"style":32123},[31845],[34,448402,406335],{"className":448403},[32767],[34,448405],{"className":448406,"style":32123},[31845],[34,448408,329201],{"className":448409},[31685,31686],[34,448411],{"className":448412,"style":31846},[31845],[34,448414,134826],{"className":448415},[31850],[34,448417],{"className":448418,"style":31846},[31845],[34,448420,448422,448425,448431,448434],{"className":448421},[31676],[34,448423],{"className":448424,"style":43217},[31680],[34,448426,448428],{"className":448427},[32767],[34,448429,440604],{"className":448430},[31685,207923],[34,448432],{"className":448433,"style":32123},[31845],[34,448435,329201],{"className":448436},[31685,31686],[71807,448438,448439],{},"(注：这是环论中最核心的定理，它建立了同态像与商环之间的代数同构关系。)",[15,448441,448442,448445,448446,448519,448520,448606,448607,448679,448680,448754,448755,11008,448783,448811],{},[2627,448443,448444],{},"推论 3.3.1.","\n(i) 如果 ",[34,448447,448449,448471],{"className":448448},[31642],[34,448450,448452],{"className":448451},[31646],[31648,448453,448454],{"xmlns":31650},[31652,448455,448456,448468],{},[31655,448457,448458,448460,448462,448464,448466],{},[31658,448459,329201],{},[31662,448461,5598],{},[31658,448463,34386],{},[31662,448465,33239],{},[31658,448467,34386],{},[31665,448469,448470],{"encoding":31667},"\\phi: R \\to R",[34,448472,448474,448492,448510],{"className":448473,"ariaHidden":758},[31672],[34,448475,448477,448480,448483,448486,448489],{"className":448476},[31676],[34,448478],{"className":448479,"style":43217},[31680],[34,448481,329201],{"className":448482},[31685,31686],[34,448484],{"className":448485,"style":31846},[31845],[34,448487,5598],{"className":448488},[31850],[34,448490],{"className":448491,"style":31846},[31845],[34,448493,448495,448498,448501,448504,448507],{"className":448494},[31676],[34,448496],{"className":448497,"style":33218},[31680],[34,448499,34386],{"className":448500,"style":34427},[31685,31686],[34,448502],{"className":448503,"style":31846},[31845],[34,448505,33239],{"className":448506},[31850],[34,448508],{"className":448509,"style":31846},[31845],[34,448511,448513,448516],{"className":448512},[31676],[34,448514],{"className":448515,"style":33218},[31680],[34,448517,34386],{"className":448518,"style":34427},[31685,31686]," 是单的环同态。则 ",[34,448521,448523,448549],{"className":448522},[31642],[34,448524,448526],{"className":448525},[31646],[31648,448527,448528],{"xmlns":31650},[31652,448529,448530,448546],{},[31655,448531,448532,448534,448536,448538,448540,448542,448544],{},[31658,448533,34386],{},[31662,448535,134826],{},[31658,448537,440604],{"mathvariant":38079},[31662,448539,32682],{},[31658,448541,329201],{},[31662,448543,45972],{},[31658,448545,94675],{},[31665,448547,448548],{"encoding":31667},"R \\cong \\operatorname{Im} \\phi \\le S",[34,448550,448552,448570,448597],{"className":448551,"ariaHidden":758},[31672],[34,448553,448555,448558,448561,448564,448567],{"className":448554},[31676],[34,448556],{"className":448557,"style":33218},[31680],[34,448559,34386],{"className":448560,"style":34427},[31685,31686],[34,448562],{"className":448563,"style":31846},[31845],[34,448565,134826],{"className":448566},[31850],[34,448568],{"className":448569,"style":31846},[31845],[34,448571,448573,448576,448582,448585,448588,448591,448594],{"className":448572},[31676],[34,448574],{"className":448575,"style":43217},[31680],[34,448577,448579],{"className":448578},[32767],[34,448580,440604],{"className":448581},[31685,207923],[34,448583],{"className":448584,"style":32123},[31845],[34,448586,329201],{"className":448587},[31685,31686],[34,448589],{"className":448590,"style":31846},[31845],[34,448592,45972],{"className":448593},[31850],[34,448595],{"className":448596,"style":31846},[31845],[34,448598,448600,448603],{"className":448599},[31676],[34,448601],{"className":448602,"style":33218},[31680],[34,448604,94675],{"className":448605,"style":106455},[31685,31686],".\n(ii) 设 ",[34,448608,448610,448631],{"className":448609},[31642],[34,448611,448613],{"className":448612},[31646],[31648,448614,448615],{"xmlns":31650},[31652,448616,448617,448629],{},[31655,448618,448619,448621,448623,448625,448627],{},[31658,448620,329201],{},[31662,448622,5598],{},[31658,448624,34386],{},[31662,448626,33239],{},[31658,448628,94675],{},[31665,448630,438915],{"encoding":31667},[34,448632,448634,448652,448670],{"className":448633,"ariaHidden":758},[31672],[34,448635,448637,448640,448643,448646,448649],{"className":448636},[31676],[34,448638],{"className":448639,"style":43217},[31680],[34,448641,329201],{"className":448642},[31685,31686],[34,448644],{"className":448645,"style":31846},[31845],[34,448647,5598],{"className":448648},[31850],[34,448650],{"className":448651,"style":31846},[31845],[34,448653,448655,448658,448661,448664,448667],{"className":448654},[31676],[34,448656],{"className":448657,"style":33218},[31680],[34,448659,34386],{"className":448660,"style":34427},[31685,31686],[34,448662],{"className":448663,"style":31846},[31845],[34,448665,33239],{"className":448666},[31850],[34,448668],{"className":448669,"style":31846},[31845],[34,448671,448673,448676],{"className":448672},[31676],[34,448674],{"className":448675,"style":33218},[31680],[34,448677,94675],{"className":448678,"style":106455},[31685,31686]," 是满的环同态。则 ",[34,448681,448683,448709],{"className":448682},[31642],[34,448684,448686],{"className":448685},[31646],[31648,448687,448688],{"xmlns":31650},[31652,448689,448690,448706],{},[31655,448691,448692,448694,448696,448698,448700,448702,448704],{},[31658,448693,34386],{},[31658,448695,38080],{"mathvariant":38079},[31658,448697,406335],{},[31662,448699,32682],{},[31658,448701,329201],{},[31662,448703,134826],{},[31658,448705,94675],{},[31665,448707,448708],{"encoding":31667},"R\u002F\\ker \\phi \\cong S",[34,448710,448712,448745],{"className":448711,"ariaHidden":758},[31672],[34,448713,448715,448718,448721,448724,448727,448730,448733,448736,448739,448742],{"className":448714},[31676],[34,448716],{"className":448717,"style":31774},[31680],[34,448719,34386],{"className":448720,"style":34427},[31685,31686],[34,448722,38080],{"className":448723},[31685],[34,448725],{"className":448726,"style":32123},[31845],[34,448728,406335],{"className":448729},[32767],[34,448731],{"className":448732,"style":32123},[31845],[34,448734,329201],{"className":448735},[31685,31686],[34,448737],{"className":448738,"style":31846},[31845],[34,448740,134826],{"className":448741},[31850],[34,448743],{"className":448744,"style":31846},[31845],[34,448746,448748,448751],{"className":448747},[31676],[34,448749],{"className":448750,"style":33218},[31680],[34,448752,94675],{"className":448753,"style":106455},[31685,31686],", 称 ",[34,448756,448758,448771],{"className":448757},[31642],[34,448759,448761],{"className":448760},[31646],[31648,448762,448763],{"xmlns":31650},[31652,448764,448765,448769],{},[31655,448766,448767],{},[31658,448768,94675],{},[31665,448770,94675],{"encoding":31667},[34,448772,448774],{"className":448773,"ariaHidden":758},[31672],[34,448775,448777,448780],{"className":448776},[31676],[34,448778],{"className":448779,"style":33218},[31680],[34,448781,94675],{"className":448782,"style":106455},[31685,31686],[34,448784,448786,448799],{"className":448785},[31642],[34,448787,448789],{"className":448788},[31646],[31648,448790,448791],{"xmlns":31650},[31652,448792,448793,448797],{},[31655,448794,448795],{},[31658,448796,34386],{},[31665,448798,34386],{"encoding":31667},[34,448800,448802],{"className":448801,"ariaHidden":758},[31672],[34,448803,448805,448808],{"className":448804},[31676],[34,448806],{"className":448807,"style":33218},[31680],[34,448809,34386],{"className":448810,"style":34427},[31685,31686]," 的一个同态像。",[15,448813,448814,69123,448817,445351,448867,448905,448906,448934,448935,448963,448964,449037,449038,2547],{},[2627,448815,448816],{},"定理 3.3.4 (第二同态定理 \u002F 对应定理).",[34,448818,448820,448837],{"className":448819},[31642],[34,448821,448823],{"className":448822},[31646],[31648,448824,448825],{"xmlns":31650},[31652,448826,448827,448835],{},[31655,448828,448829,448831,448833],{},[31658,448830,106063],{},[31662,448832,440828],{},[31658,448834,34386],{},[31665,448836,446303],{"encoding":31667},[34,448838,448840,448858],{"className":448839,"ariaHidden":758},[31672],[34,448841,448843,448846,448849,448852,448855],{"className":448842},[31676],[34,448844],{"className":448845,"style":445329},[31680],[34,448847,106063],{"className":448848,"style":67482},[31685,31686],[34,448850],{"className":448851,"style":32604},[31845],[34,448853,440828],{"className":448854},[32024,129261],[34,448856],{"className":448857,"style":32604},[31845],[34,448859,448861,448864],{"className":448860},[31676],[34,448862],{"className":448863,"style":33218},[31680],[34,448865,34386],{"className":448866,"style":34427},[31685,31686],[34,448868,448870,448887],{"className":448869},[31642],[34,448871,448873],{"className":448872},[31646],[31648,448874,448875],{"xmlns":31650},[31652,448876,448877,448885],{},[31655,448878,448879,448881,448883],{},[31658,448880,34386],{},[31658,448882,38080],{"mathvariant":38079},[31658,448884,106063],{},[31665,448886,337927],{"encoding":31667},[34,448888,448890],{"className":448889,"ariaHidden":758},[31672],[34,448891,448893,448896,448899,448902],{"className":448892},[31676],[34,448894],{"className":448895,"style":31774},[31680],[34,448897,34386],{"className":448898,"style":34427},[31685,31686],[34,448900,38080],{"className":448901},[31685],[34,448903,106063],{"className":448904,"style":67482},[31685,31686]," 的子环(理想)与 ",[34,448907,448909,448922],{"className":448908},[31642],[34,448910,448912],{"className":448911},[31646],[31648,448913,448914],{"xmlns":31650},[31652,448915,448916,448920],{},[31655,448917,448918],{},[31658,448919,34386],{},[31665,448921,34386],{"encoding":31667},[34,448923,448925],{"className":448924,"ariaHidden":758},[31672],[34,448926,448928,448931],{"className":448927},[31676],[34,448929],{"className":448930,"style":33218},[31680],[34,448932,34386],{"className":448933,"style":34427},[31685,31686]," 的包含 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R",[34,448990,448992,449010,449028],{"className":448991,"ariaHidden":758},[31672],[34,448993,448995,448998,449001,449004,449007],{"className":448994},[31676],[34,448996],{"className":448997,"style":445329},[31680],[34,448999,106063],{"className":449000,"style":67482},[31685,31686],[34,449002],{"className":449003,"style":32604},[31845],[34,449005,440828],{"className":449006},[32024,129261],[34,449008],{"className":449009,"style":32604},[31845],[34,449011,449013,449016,449019,449022,449025],{"className":449012},[31676],[34,449014],{"className":449015,"style":445329},[31680],[34,449017,223320],{"className":449018,"style":223361},[31685,31686],[34,449020],{"className":449021,"style":32604},[31845],[34,449023,440828],{"className":449024},[32024,129261],[34,449026],{"className":449027,"style":32604},[31845],[34,449029,449031,449034],{"className":449030},[31676],[34,449032],{"className":449033,"style":33218},[31680],[34,449035,34386],{"className":449036,"style":34427},[31685,31686],", 则有 ",[34,449039,449041,449083],{"className":449040},[31642],[34,449042,449044],{"className":449043},[31646],[31648,449045,449046],{"xmlns":31650},[31652,449047,449048,449080],{},[31655,449049,449050,449052,449054,449056,449058,449060,449062,449064,449066,449068,449070,449072,449074,449076,449078],{},[31658,449051,34386],{},[31658,449053,38080],{"mathvariant":38079},[31658,449055,223320],{},[31662,449057,134826],{},[31662,449059,456],{"stretchy":702},[31658,449061,34386],{},[31658,449063,38080],{"mathvariant":38079},[31658,449065,106063],{},[31662,449067,2557],{"stretchy":702},[31658,449069,38080],{"mathvariant":38079},[31662,449071,456],{"stretchy":702},[31658,449073,223320],{},[31658,449075,38080],{"mathvariant":38079},[31658,449077,106063],{},[31662,449079,2557],{"stretchy":702},[31665,449081,449082],{"encoding":31667},"R\u002FJ \\cong (R\u002FI)\u002F(J\u002FI)",[34,449084,449086,449110],{"className":449085,"ariaHidden":758},[31672],[34,449087,449089,449092,449095,449098,449101,449104,449107],{"className":449088},[31676],[34,449090],{"className":449091,"style":31774},[31680],[34,449093,34386],{"className":449094,"style":34427},[31685,31686],[34,449096,38080],{"className":449097},[31685],[34,449099,223320],{"className":449100,"style":223361},[31685,31686],[34,449102],{"className":449103,"style":31846},[31845],[34,449105,134826],{"className":449106},[31850],[34,449108],{"className":449109,"style":31846},[31845],[34,449111,449113,449116,449119,449122,449125,449128,449131,449134,449137,449140,449143,449146],{"className":449112},[31676],[34,449114],{"className":449115,"style":31774},[31680],[34,449117,456],{"className":449118},[31782],[34,449120,34386],{"className":449121,"style":34427},[31685,31686],[34,449123,38080],{"className":449124},[31685],[34,449126,106063],{"className":449127,"style":67482},[31685,31686],[34,449129,2557],{"className":449130},[31690],[34,449132,38080],{"className":449133},[31685],[34,449135,456],{"className":449136},[31782],[34,449138,223320],{"className":449139,"style":223361},[31685,31686],[34,449141,38080],{"className":449142},[31685],[34,449144,106063],{"className":449145,"style":67482},[31685,31686],[34,449147,2557],{"className":449148},[31690],[15,449150,449151,69123,449154,445351,449240,449370,449371],{},[2627,449152,449153],{},"定理 3.3.5 (第三同态定理).",[34,449155,449157,449183],{"className":449156},[31642],[34,449158,449160],{"className":449159},[31646],[31648,449161,449162],{"xmlns":31650},[31652,449163,449164,449180],{},[31655,449165,449166,449168,449170,449172,449174,449176,449178],{},[31658,449167,106063],{},[31662,449169,440828],{},[31658,449171,34386],{},[31662,449173,31960],{"separator":758},[31658,449175,38691],{},[31662,449177,45972],{},[31658,449179,34386],{},[31665,449181,449182],{"encoding":31667},"I \\lhd R, K \\le R",[34,449184,449186,449204,449231],{"className":449185,"ariaHidden":758},[31672],[34,449187,449189,449192,449195,449198,449201],{"className":449188},[31676],[34,449190],{"className":449191,"style":445329},[31680],[34,449193,106063],{"className":449194,"style":67482},[31685,31686],[34,449196],{"className":449197,"style":32604},[31845],[34,449199,440828],{"className":449200},[32024,129261],[34,449202],{"className":449203,"style":32604},[31845],[34,449205,449207,449210,449213,449216,449219,449222,449225,449228],{"className":449206},[31676],[34,449208],{"className":449209,"style":70739},[31680],[34,449211,34386],{"className":449212,"style":34427},[31685,31686],[34,449214,31960],{"className":449215},[32119],[34,449217],{"className":449218,"style":32123},[31845],[34,449220,38691],{"className":449221,"style":38706},[31685,31686],[34,449223],{"className":449224,"style":31846},[31845],[34,449226,45972],{"className":449227},[31850],[34,449229],{"className":449230,"style":31846},[31845],[34,449232,449234,449237],{"className":449233},[31676],[34,449235],{"className":449236,"style":33218},[31680],[34,449238,34386],{"className":449239,"style":34427},[31685,31686],[34,449241,449243,449277],{"className":449242},[31642],[34,449244,449246],{"className":449245},[31646],[31648,449247,449248],{"xmlns":31650},[31652,449249,449250,449274],{},[31655,449251,449252,449254,449256,449258,449260,449262,449264,449266,449268,449270,449272],{},[31658,449253,106063],{},[31662,449255,223],{},[31658,449257,38691],{},[31662,449259,45972],{},[31658,449261,34386],{},[31662,449263,31960],{"separator":758},[31658,449265,106063],{},[31662,449267,326344],{},[31658,449269,38691],{},[31662,449271,440828],{},[31658,449273,38691],{},[31665,449275,449276],{"encoding":31667},"I+K \\le R, I \\cap K \\lhd K",[34,449278,449280,449298,449316,449343,449361],{"className":449279,"ariaHidden":758},[31672],[34,449281,449283,449286,449289,449292,449295],{"className":449282},[31676],[34,449284],{"className":449285,"style":34944},[31680],[34,449287,106063],{"className":449288,"style":67482},[31685,31686],[34,449290],{"className":449291,"style":32604},[31845],[34,449293,223],{"className":449294},[32024],[34,449296],{"className":449297,"style":32604},[31845],[34,449299,449301,449304,449307,449310,449313],{"className":449300},[31676],[34,449302],{"className":449303,"style":195047},[31680],[34,449305,38691],{"className":449306,"style":38706},[31685,31686],[34,449308],{"className":449309,"style":31846},[31845],[34,449311,45972],{"className":449312},[31850],[34,449314],{"className":449315,"style":31846},[31845],[34,449317,449319,449322,449325,449328,449331,449334,449337,449340],{"className":449318},[31676],[34,449320],{"className":449321,"style":70739},[31680],[34,449323,34386],{"className":449324,"style":34427},[31685,31686],[34,449326,31960],{"className":449327},[32119],[34,449329],{"className":449330,"style":32123},[31845],[34,449332,106063],{"className":449333,"style":67482},[31685,31686],[34,449335],{"className":449336,"style":32604},[31845],[34,449338,326344],{"className":449339},[32024],[34,449341],{"className":449342,"style":32604},[31845],[34,449344,449346,449349,449352,449355,449358],{"className":449345},[31676],[34,449347],{"className":449348,"style":445329},[31680],[34,449350,38691],{"className":449351,"style":38706},[31685,31686],[34,449353],{"className":449354,"style":32604},[31845],[34,449356,440828],{"className":449357},[32024,129261],[34,449359],{"className":449360,"style":32604},[31845],[34,449362,449364,449367],{"className":449363},[31676],[34,449365],{"className":449366,"style":33218},[31680],[34,449368,38691],{"className":449369,"style":38706},[31685,31686],", 且\n",[34,449372,449374,449418],{"className":449373},[31642],[34,449375,449377],{"className":449376},[31646],[31648,449378,449379],{"xmlns":31650},[31652,449380,449381,449415],{},[31655,449382,449383,449385,449387,449389,449391,449393,449395,449397,449399,449401,449403,449405,449407,449409,449411,449413],{},[31662,449384,456],{"stretchy":702},[31658,449386,106063],{},[31662,449388,223],{},[31658,449390,38691],{},[31662,449392,2557],{"stretchy":702},[31658,449394,38080],{"mathvariant":38079},[31658,449396,106063],{},[31662,449398,134826],{},[31658,449400,38691],{},[31658,449402,38080],{"mathvariant":38079},[31662,449404,456],{"stretchy":702},[31658,449406,106063],{},[31662,449408,326344],{},[31658,449410,38691],{},[31662,449412,2557],{"stretchy":702},[31658,449414,2547],{"mathvariant":38079},[31665,449416,449417],{"encoding":31667},"(I+K)\u002FI \\cong K\u002F(I \\cap K).",[34,449419,449421,449442,449469,449496],{"className":449420,"ariaHidden":758},[31672],[34,449422,449424,449427,449430,449433,449436,449439],{"className":449423},[31676],[34,449425],{"className":449426,"style":31774},[31680],[34,449428,456],{"className":449429},[31782],[34,449431,106063],{"className":449432,"style":67482},[31685,31686],[34,449434],{"className":449435,"style":32604},[31845],[34,449437,223],{"className":449438},[32024],[34,449440],{"className":449441,"style":32604},[31845],[34,449443,449445,449448,449451,449454,449457,449460,449463,449466],{"className":449444},[31676],[34,449446],{"className":449447,"style":31774},[31680],[34,449449,38691],{"className":449450,"style":38706},[31685,31686],[34,449452,2557],{"className":449453},[31690],[34,449455,38080],{"className":449456},[31685],[34,449458,106063],{"className":449459,"style":67482},[31685,31686],[34,449461],{"className":449462,"style":31846},[31845],[34,449464,134826],{"className":449465},[31850],[34,449467],{"className":449468,"style":31846},[31845],[34,449470,449472,449475,449478,449481,449484,449487,449490,449493],{"className":449471},[31676],[34,449473],{"className":449474,"style":31774},[31680],[34,449476,38691],{"className":449477,"style":38706},[31685,31686],[34,449479,38080],{"className":449480},[31685],[34,449482,456],{"className":449483},[31782],[34,449485,106063],{"className":449486,"style":67482},[31685,31686],[34,449488],{"className":449489,"style":32604},[31845],[34,449491,326344],{"className":449492},[32024],[34,449494],{"className":449495,"style":32604},[31845],[34,449497,449499,449502,449505,449508],{"className":449498},[31676],[34,449500],{"className":449501,"style":31774},[31680],[34,449503,38691],{"className":449504,"style":38706},[31685,31686],[34,449506,2557],{"className":449507},[31690],[34,449509,2547],{"className":449510},[31685],[15,449512,449513,449515,449516,53834,449544,449615],{},[2627,449514,441098],{},"\n此定理描述了子环与理想之和的商结构。证明思路是构造一个从 ",[34,449517,449519,449532],{"className":449518},[31642],[34,449520,449522],{"className":449521},[31646],[31648,449523,449524],{"xmlns":31650},[31652,449525,449526,449530],{},[31655,449527,449528],{},[31658,449529,38691],{},[31665,449531,38691],{"encoding":31667},[34,449533,449535],{"className":449534,"ariaHidden":758},[31672],[34,449536,449538,449541],{"className":449537},[31676],[34,449539],{"className":449540,"style":33218},[31680],[34,449542,38691],{"className":449543,"style":38706},[31685,31686],[34,449545,449547,449573],{"className":449546},[31642],[34,449548,449550],{"className":449549},[31646],[31648,449551,449552],{"xmlns":31650},[31652,449553,449554,449570],{},[31655,449555,449556,449558,449560,449562,449564,449566,449568],{},[31662,449557,456],{"stretchy":702},[31658,449559,106063],{},[31662,449561,223],{},[31658,449563,38691],{},[31662,449565,2557],{"stretchy":702},[31658,449567,38080],{"mathvariant":38079},[31658,449569,106063],{},[31665,449571,449572],{"encoding":31667},"(I+K)\u002FI",[34,449574,449576,449597],{"className":449575,"ariaHidden":758},[31672],[34,449577,449579,449582,449585,449588,449591,449594],{"className":449578},[31676],[34,449580],{"className":449581,"style":31774},[31680],[34,449583,456],{"className":449584},[31782],[34,449586,106063],{"className":449587,"style":67482},[31685,31686],[34,449589],{"className":449590,"style":32604},[31845],[34,449592,223],{"className":449593},[32024],[34,449595],{"className":449596,"style":32604},[31845],[34,449598,449600,449603,449606,449609,449612],{"className":449599},[31676],[34,449601],{"className":449602,"style":31774},[31680],[34,449604,38691],{"className":449605,"style":38706},[31685,31686],[34,449607,2557],{"className":449608},[31690],[34,449610,38080],{"className":449611},[31685],[34,449613,106063],{"className":449614,"style":67482},[31685,31686]," 的满同态，并利用第一同态定理。",[2230,449617,449618,450048,451476,452052,452700],{},[108,449619,449620,449623,449624,449740,449741,449829,449830,13977,449925,449976,449977,450047],{},[2627,449621,449622],{},"构造映射",": 定义 ",[34,449625,449627,449662],{"className":449626},[31642],[34,449628,449630],{"className":449629},[31646],[31648,449631,449632],{"xmlns":31650},[31652,449633,449634,449659],{},[31655,449635,449636,449639,449641,449643,449645,449647,449649,449651,449653,449655,449657],{},[31658,449637,449638],{},"ψ",[31662,449640,5598],{},[31658,449642,38691],{},[31662,449644,33239],{},[31662,449646,456],{"stretchy":702},[31658,449648,106063],{},[31662,449650,223],{},[31658,449652,38691],{},[31662,449654,2557],{"stretchy":702},[31658,449656,38080],{"mathvariant":38079},[31658,449658,106063],{},[31665,449660,449661],{"encoding":31667},"\\psi: K \\to (I+K)\u002FI",[34,449663,449665,449683,449701,449722],{"className":449664,"ariaHidden":758},[31672],[34,449666,449668,449671,449674,449677,449680],{"className":449667},[31676],[34,449669],{"className":449670,"style":43217},[31680],[34,449672,449638],{"className":449673,"style":32062},[31685,31686],[34,449675],{"className":449676,"style":31846},[31845],[34,449678,5598],{"className":449679},[31850],[34,449681],{"className":449682,"style":31846},[31845],[34,449684,449686,449689,449692,449695,449698],{"className":449685},[31676],[34,449687],{"className":449688,"style":33218},[31680],[34,449690,38691],{"className":449691,"style":38706},[31685,31686],[34,449693],{"className":449694,"style":31846},[31845],[34,449696,33239],{"className":449697},[31850],[34,449699],{"className":449700,"style":31846},[31845],[34,449702,449704,449707,449710,449713,449716,449719],{"className":449703},[31676],[34,449705],{"className":449706,"style":31774},[31680],[34,449708,456],{"className":449709},[31782],[34,449711,106063],{"className":449712,"style":67482},[31685,31686],[34,449714],{"className":449715,"style":32604},[31845],[34,449717,223],{"className":449718},[32024],[34,449720],{"className":449721,"style":32604},[31845],[34,449723,449725,449728,449731,449734,449737],{"className":449724},[31676],[34,449726],{"className":449727,"style":31774},[31680],[34,449729,38691],{"className":449730,"style":38706},[31685,31686],[34,449732,2557],{"className":449733},[31690],[34,449735,38080],{"className":449736},[31685],[34,449738,106063],{"className":449739,"style":67482},[31685,31686],"，规则为 ",[34,449742,449744,449772],{"className":449743},[31642],[34,449745,449747],{"className":449746},[31646],[31648,449748,449749],{"xmlns":31650},[31652,449750,449751,449769],{},[31655,449752,449753,449755,449757,449759,449761,449763,449765,449767],{},[31658,449754,449638],{},[31662,449756,456],{"stretchy":702},[31658,449758,33759],{},[31662,449760,2557],{"stretchy":702},[31662,449762,202],{},[31658,449764,33759],{},[31662,449766,223],{},[31658,449768,106063],{},[31665,449770,449771],{"encoding":31667},"\\psi(k) = k + I",[34,449773,449775,449802,449820],{"className":449774,"ariaHidden":758},[31672],[34,449776,449778,449781,449784,449787,449790,449793,449796,449799],{"className":449777},[31676],[34,449779],{"className":449780,"style":31774},[31680],[34,449782,449638],{"className":449783,"style":32062},[31685,31686],[34,449785,456],{"className":449786},[31782],[34,449788,33759],{"className":449789,"style":33774},[31685,31686],[34,449791,2557],{"className":449792},[31690],[34,449794],{"className":449795,"style":31846},[31845],[34,449797,202],{"className":449798},[31850],[34,449800],{"className":449801,"style":31846},[31845],[34,449803,449805,449808,449811,449814,449817],{"className":449804},[31676],[34,449806],{"className":449807,"style":34299},[31680],[34,449809,33759],{"className":449810,"style":33774},[31685,31686],[34,449812],{"className":449813,"style":32604},[31845],[34,449815,223],{"className":449816},[32024],[34,449818],{"className":449819,"style":32604},[31845],[34,449821,449823,449826],{"className":449822},[31676],[34,449824],{"className":449825,"style":33218},[31680],[34,449827,106063],{"className":449828,"style":67482},[31685,31686],"。\n注意：这里 ",[34,449831,449833,449859],{"className":449832},[31642],[34,449834,449836],{"className":449835},[31646],[31648,449837,449838],{"xmlns":31650},[31652,449839,449840,449856],{},[31655,449841,449842,449844,449846,449848,449850,449852,449854],{},[31658,449843,33759],{},[31662,449845,112296],{},[31658,449847,38691],{},[31662,449849,131855],{},[31658,449851,106063],{},[31662,449853,223],{},[31658,449855,38691],{},[31665,449857,449858],{"encoding":31667},"k \\in K \\subseteq I+K",[34,449860,449862,449880,449898,449916],{"className":449861,"ariaHidden":758},[31672],[34,449863,449865,449868,449871,449874,449877],{"className":449864},[31676],[34,449866],{"className":449867,"style":46991},[31680],[34,449869,33759],{"className":449870,"style":33774},[31685,31686],[34,449872],{"className":449873,"style":31846},[31845],[34,449875,112296],{"className":449876},[31850],[34,449878],{"className":449879,"style":31846},[31845],[34,449881,449883,449886,449889,449892,449895],{"className":449882},[31676],[34,449884],{"className":449885,"style":195047},[31680],[34,449887,38691],{"className":449888,"style":38706},[31685,31686],[34,449890],{"className":449891,"style":31846},[31845],[34,449893,131855],{"className":449894},[31850],[34,449896],{"className":449897,"style":31846},[31845],[34,449899,449901,449904,449907,449910,449913],{"className":449900},[31676],[34,449902],{"className":449903,"style":34944},[31680],[34,449905,106063],{"className":449906,"style":67482},[31685,31686],[34,449908],{"className":449909,"style":32604},[31845],[34,449911,223],{"className":449912},[32024],[34,449914],{"className":449915,"style":32604},[31845],[34,449917,449919,449922],{"className":449918},[31676],[34,449920],{"className":449921,"style":33218},[31680],[34,449923,38691],{"className":449924,"style":38706},[31685,31686],[34,449926,449928,449946],{"className":449927},[31642],[34,449929,449931],{"className":449930},[31646],[31648,449932,449933],{"xmlns":31650},[31652,449934,449935,449943],{},[31655,449936,449937,449939,449941],{},[31658,449938,33759],{},[31662,449940,223],{},[31658,449942,106063],{},[31665,449944,449945],{"encoding":31667},"k+I",[34,449947,449949,449967],{"className":449948,"ariaHidden":758},[31672],[34,449950,449952,449955,449958,449961,449964],{"className":449951},[31676],[34,449953],{"className":449954,"style":34299},[31680],[34,449956,33759],{"className":449957,"style":33774},[31685,31686],[34,449959],{"className":449960,"style":32604},[31845],[34,449962,223],{"className":449963},[32024],[34,449965],{"className":449966,"style":32604},[31845],[34,449968,449970,449973],{"className":449969},[31676],[34,449971],{"className":449972,"style":33218},[31680],[34,449974,106063],{"className":449975,"style":67482},[31685,31686]," 确实是商环 ",[34,449978,449980,450005],{"className":449979},[31642],[34,449981,449983],{"className":449982},[31646],[31648,449984,449985],{"xmlns":31650},[31652,449986,449987,450003],{},[31655,449988,449989,449991,449993,449995,449997,449999,450001],{},[31662,449990,456],{"stretchy":702},[31658,449992,106063],{},[31662,449994,223],{},[31658,449996,38691],{},[31662,449998,2557],{"stretchy":702},[31658,450000,38080],{"mathvariant":38079},[31658,450002,106063],{},[31665,450004,449572],{"encoding":31667},[34,450006,450008,450029],{"className":450007,"ariaHidden":758},[31672],[34,450009,450011,450014,450017,450020,450023,450026],{"className":450010},[31676],[34,450012],{"className":450013,"style":31774},[31680],[34,450015,456],{"className":450016},[31782],[34,450018,106063],{"className":450019,"style":67482},[31685,31686],[34,450021],{"className":450022,"style":32604},[31845],[34,450024,223],{"className":450025},[32024],[34,450027],{"className":450028,"style":32604},[31845],[34,450030,450032,450035,450038,450041,450044],{"className":450031},[31676],[34,450033],{"className":450034,"style":31774},[31680],[34,450036,38691],{"className":450037,"style":38706},[31685,31686],[34,450039,2557],{"className":450040},[31690],[34,450042,38080],{"className":450043},[31685],[34,450045,106063],{"className":450046,"style":67482},[31685,31686]," 中的元素。",[108,450049,450050,8471,450053,684,450204],{},[2627,450051,450052],{},"验证同态",[34,450054,450056,450088],{"className":450055},[31642],[34,450057,450059],{"className":450058},[31646],[31648,450060,450061],{"xmlns":31650},[31652,450062,450063,450085],{},[31655,450064,450065,450067,450073,450075,450081,450083],{},[31658,450066,365679],{"mathvariant":38079},[31743,450068,450069,450071],{},[31658,450070,33759],{},[31749,450072,640],{},[31662,450074,31960],{"separator":758},[31743,450076,450077,450079],{},[31658,450078,33759],{},[31749,450080,768],{},[31662,450082,112296],{},[31658,450084,38691],{},[31665,450086,450087],{"encoding":31667},"\\forall k_1, k_2 \\in K",[34,450089,450091,450195],{"className":450090,"ariaHidden":758},[31672],[34,450092,450094,450097,450100,450140,450143,450146,450186,450189,450192],{"className":450093},[31676],[34,450095],{"className":450096,"style":43217},[31680],[34,450098,365679],{"className":450099},[31685],[34,450101,450103,450106],{"className":450102},[31685],[34,450104,33759],{"className":450105,"style":33774},[31685,31686],[34,450107,450109],{"className":450108},[31792],[34,450110,450112,450132],{"className":450111},[31796,31797],[34,450113,450115,450129],{"className":450114},[31801],[34,450116,450118],{"className":450117,"style":31806},[31805],[34,450119,450120,450123],{"style":319623},[34,450121],{"className":450122,"style":31814},[31813],[34,450124,450126],{"className":450125},[31818,31819,31820,31821],[34,450127,640],{"className":450128},[31685,31821],[34,450130,31829],{"className":450131},[31828],[34,450133,450135],{"className":450134},[31801],[34,450136,450138],{"className":450137,"style":31836},[31805],[34,450139],{},[34,450141,31960],{"className":450142},[32119],[34,450144],{"className":450145,"style":32123},[31845],[34,450147,450149,450152],{"className":450148},[31685],[34,450150,33759],{"className":450151,"style":33774},[31685,31686],[34,450153,450155],{"className":450154},[31792],[34,450156,450158,450178],{"className":450157},[31796,31797],[34,450159,450161,450175],{"className":450160},[31801],[34,450162,450164],{"className":450163,"style":31806},[31805],[34,450165,450166,450169],{"style":319623},[34,450167],{"className":450168,"style":31814},[31813],[34,450170,450172],{"className":450171},[31818,31819,31820,31821],[34,450173,768],{"className":450174},[31685,31821],[34,450176,31829],{"className":450177},[31828],[34,450179,450181],{"className":450180},[31801],[34,450182,450184],{"className":450183,"style":31836},[31805],[34,450185],{},[34,450187],{"className":450188,"style":31846},[31845],[34,450190,112296],{"className":450191},[31850],[34,450193],{"className":450194,"style":31846},[31845],[34,450196,450198,450201],{"className":450197},[31676],[34,450199],{"className":450200,"style":33218},[31680],[34,450202,38691],{"className":450203,"style":38706},[31685,31686],[105,450205,450206,450860],{},[108,450207,450208,2547],{},[34,450209,450211,450327],{"className":450210},[31642],[34,450212,450214],{"className":450213},[31646],[31648,450215,450216],{"xmlns":31650},[31652,450217,450218,450324],{},[31655,450219,450220,450222,450224,450230,450232,450238,450240,450242,450244,450250,450252,450258,450260,450262,450264,450266,450268,450274,450276,450278,450280,450282,450284,450290,450292,450294,450296,450298,450300,450302,450308,450310,450312,450314,450316,450322],{},[31658,450221,449638],{},[31662,450223,456],{"stretchy":702},[31743,450225,450226,450228],{},[31658,450227,33759],{},[31749,450229,640],{},[31662,450231,223],{},[31743,450233,450234,450236],{},[31658,450235,33759],{},[31749,450237,768],{},[31662,450239,2557],{"stretchy":702},[31662,450241,202],{},[31662,450243,456],{"stretchy":702},[31743,450245,450246,450248],{},[31658,450247,33759],{},[31749,450249,640],{},[31662,450251,223],{},[31743,450253,450254,450256],{},[31658,450255,33759],{},[31749,450257,768],{},[31662,450259,2557],{"stretchy":702},[31662,450261,223],{},[31658,450263,106063],{},[31662,450265,202],{},[31662,450267,456],{"stretchy":702},[31743,450269,450270,450272],{},[31658,450271,33759],{},[31749,450273,640],{},[31662,450275,223],{},[31658,450277,106063],{},[31662,450279,2557],{"stretchy":702},[31662,450281,223],{},[31662,450283,456],{"stretchy":702},[31743,450285,450286,450288],{},[31658,450287,33759],{},[31749,450289,768],{},[31662,450291,223],{},[31658,450293,106063],{},[31662,450295,2557],{"stretchy":702},[31662,450297,202],{},[31658,450299,449638],{},[31662,450301,456],{"stretchy":702},[31743,450303,450304,450306],{},[31658,450305,33759],{},[31749,450307,640],{},[31662,450309,2557],{"stretchy":702},[31662,450311,223],{},[31658,450313,449638],{},[31662,450315,456],{"stretchy":702},[31743,450317,450318,450320],{},[31658,450319,33759],{},[31749,450321,768],{},[31662,450323,2557],{"stretchy":702},[31665,450325,450326],{"encoding":31667},"\\psi(k_1+k_2) = (k_1+k_2)+I = (k_1+I)+(k_2+I) = \\psi(k_1)+\\psi(k_2)",[34,450328,450330,450391,450449,450507,450565,450583,450641,450662,450720,450741,450805],{"className":450329,"ariaHidden":758},[31672],[34,450331,450333,450336,450339,450342,450382,450385,450388],{"className":450332},[31676],[34,450334],{"className":450335,"style":31774},[31680],[34,450337,449638],{"className":450338,"style":32062},[31685,31686],[34,450340,456],{"className":450341},[31782],[34,450343,450345,450348],{"className":450344},[31685],[34,450346,33759],{"className":450347,"style":33774},[31685,31686],[34,450349,450351],{"className":450350},[31792],[34,450352,450354,450374],{"className":450353},[31796,31797],[34,450355,450357,450371],{"className":450356},[31801],[34,450358,450360],{"className":450359,"style":31806},[31805],[34,450361,450362,450365],{"style":319623},[34,450363],{"className":450364,"style":31814},[31813],[34,450366,450368],{"className":450367},[31818,31819,31820,31821],[34,450369,640],{"className":450370},[31685,31821],[34,450372,31829],{"className":450373},[31828],[34,450375,450377],{"className":450376},[31801],[34,450378,450380],{"className":450379,"style":31836},[31805],[34,450381],{},[34,450383],{"className":450384,"style":32604},[31845],[34,450386,223],{"className":450387},[32024],[34,450389],{"className":450390,"style":32604},[31845],[34,450392,450394,450397,450437,450440,450443,450446],{"className":450393},[31676],[34,450395],{"className":450396,"style":31774},[31680],[34,450398,450400,450403],{"className":450399},[31685],[34,450401,33759],{"className":450402,"style":33774},[31685,31686],[34,450404,450406],{"className":450405},[31792],[34,450407,450409,450429],{"className":450408},[31796,31797],[34,450410,450412,450426],{"className":450411},[31801],[34,450413,450415],{"className":450414,"style":31806},[31805],[34,450416,450417,450420],{"style":319623},[34,450418],{"className":450419,"style":31814},[31813],[34,450421,450423],{"className":450422},[31818,31819,31820,31821],[34,450424,768],{"className":450425},[31685,31821],[34,450427,31829],{"className":450428},[31828],[34,450430,450432],{"className":450431},[31801],[34,450433,450435],{"className":450434,"style":31836},[31805],[34,450436],{},[34,450438,2557],{"className":450439},[31690],[34,450441],{"className":450442,"style":31846},[31845],[34,450444,202],{"className":450445},[31850],[34,450447],{"className":450448,"style":31846},[31845],[34,450450,450452,450455,450458,450498,450501,450504],{"className":450451},[31676],[34,450453],{"className":450454,"style":31774},[31680],[34,450456,456],{"className":450457},[31782],[34,450459,450461,450464],{"className":450460},[31685],[34,450462,33759],{"className":450463,"style":33774},[31685,31686],[34,450465,450467],{"className":450466},[31792],[34,450468,450470,450490],{"className":450469},[31796,31797],[34,450471,450473,450487],{"className":450472},[31801],[34,450474,450476],{"className":450475,"style":31806},[31805],[34,450477,450478,450481],{"style":319623},[34,450479],{"className":450480,"style":31814},[31813],[34,450482,450484],{"className":450483},[31818,31819,31820,31821],[34,450485,640],{"className":450486},[31685,31821],[34,450488,31829],{"className":450489},[31828],[34,450491,450493],{"className":450492},[31801],[34,450494,450496],{"className":450495,"style":31836},[31805],[34,450497],{},[34,450499],{"className":450500,"style":32604},[31845],[34,450502,223],{"className":450503},[32024],[34,450505],{"className":450506,"style":32604},[31845],[34,450508,450510,450513,450553,450556,450559,450562],{"className":450509},[31676],[34,450511],{"className":450512,"style":31774},[31680],[34,450514,450516,450519],{"className":450515},[31685],[34,450517,33759],{"className":450518,"style":33774},[31685,31686],[34,450520,450522],{"className":450521},[31792],[34,450523,450525,450545],{"className":450524},[31796,31797],[34,450526,450528,450542],{"className":450527},[31801],[34,450529,450531],{"className":450530,"style":31806},[31805],[34,450532,450533,450536],{"style":319623},[34,450534],{"className":450535,"style":31814},[31813],[34,450537,450539],{"className":450538},[31818,31819,31820,31821],[34,450540,768],{"className":450541},[31685,31821],[34,450543,31829],{"className":450544},[31828],[34,450546,450548],{"className":450547},[31801],[34,450549,450551],{"className":450550,"style":31836},[31805],[34,450552],{},[34,450554,2557],{"className":450555},[31690],[34,450557],{"className":450558,"style":32604},[31845],[34,450560,223],{"className":450561},[32024],[34,450563],{"className":450564,"style":32604},[31845],[34,450566,450568,450571,450574,450577,450580],{"className":450567},[31676],[34,450569],{"className":450570,"style":33218},[31680],[34,450572,106063],{"className":450573,"style":67482},[31685,31686],[34,450575],{"className":450576,"style":31846},[31845],[34,450578,202],{"className":450579},[31850],[34,450581],{"className":450582,"style":31846},[31845],[34,450584,450586,450589,450592,450632,450635,450638],{"className":450585},[31676],[34,450587],{"className":450588,"style":31774},[31680],[34,450590,456],{"className":450591},[31782],[34,450593,450595,450598],{"className":450594},[31685],[34,450596,33759],{"className":450597,"style":33774},[31685,31686],[34,450599,450601],{"className":450600},[31792],[34,450602,450604,450624],{"className":450603},[31796,31797],[34,450605,450607,450621],{"className":450606},[31801],[34,450608,450610],{"className":450609,"style":31806},[31805],[34,450611,450612,450615],{"style":319623},[34,450613],{"className":450614,"style":31814},[31813],[34,450616,450618],{"className":450617},[31818,31819,31820,31821],[34,450619,640],{"className":450620},[31685,31821],[34,450622,31829],{"className":450623},[31828],[34,450625,450627],{"className":450626},[31801],[34,450628,450630],{"className":450629,"style":31836},[31805],[34,450631],{},[34,450633],{"className":450634,"style":32604},[31845],[34,450636,223],{"className":450637},[32024],[34,450639],{"className":450640,"style":32604},[31845],[34,450642,450644,450647,450650,450653,450656,450659],{"className":450643},[31676],[34,450645],{"className":450646,"style":31774},[31680],[34,450648,106063],{"className":450649,"style":67482},[31685,31686],[34,450651,2557],{"className":450652},[31690],[34,450654],{"className":450655,"style":32604},[31845],[34,450657,223],{"className":450658},[32024],[34,450660],{"className":450661,"style":32604},[31845],[34,450663,450665,450668,450671,450711,450714,450717],{"className":450664},[31676],[34,450666],{"className":450667,"style":31774},[31680],[34,450669,456],{"className":450670},[31782],[34,450672,450674,450677],{"className":450673},[31685],[34,450675,33759],{"className":450676,"style":33774},[31685,31686],[34,450678,450680],{"className":450679},[31792],[34,450681,450683,450703],{"className":450682},[31796,31797],[34,450684,450686,450700],{"className":450685},[31801],[34,450687,450689],{"className":450688,"style":31806},[31805],[34,450690,450691,450694],{"style":319623},[34,450692],{"className":450693,"style":31814},[31813],[34,450695,450697],{"className":450696},[31818,31819,31820,31821],[34,450698,768],{"className":450699},[31685,31821],[34,450701,31829],{"className":450702},[31828],[34,450704,450706],{"className":450705},[31801],[34,450707,450709],{"className":450708,"style":31836},[31805],[34,450710],{},[34,450712],{"className":450713,"style":32604},[31845],[34,450715,223],{"className":450716},[32024],[34,450718],{"className":450719,"style":32604},[31845],[34,450721,450723,450726,450729,450732,450735,450738],{"className":450722},[31676],[34,450724],{"className":450725,"style":31774},[31680],[34,450727,106063],{"className":450728,"style":67482},[31685,31686],[34,450730,2557],{"className":450731},[31690],[34,450733],{"className":450734,"style":31846},[31845],[34,450736,202],{"className":450737},[31850],[34,450739],{"className":450740,"style":31846},[31845],[34,450742,450744,450747,450750,450753,450793,450796,450799,450802],{"className":450743},[31676],[34,450745],{"className":450746,"style":31774},[31680],[34,450748,449638],{"className":450749,"style":32062},[31685,31686],[34,450751,456],{"className":450752},[31782],[34,450754,450756,450759],{"className":450755},[31685],[34,450757,33759],{"className":450758,"style":33774},[31685,31686],[34,450760,450762],{"className":450761},[31792],[34,450763,450765,450785],{"className":450764},[31796,31797],[34,450766,450768,450782],{"className":450767},[31801],[34,450769,450771],{"className":450770,"style":31806},[31805],[34,450772,450773,450776],{"style":319623},[34,450774],{"className":450775,"style":31814},[31813],[34,450777,450779],{"className":450778},[31818,31819,31820,31821],[34,450780,640],{"className":450781},[31685,31821],[34,450783,31829],{"className":450784},[31828],[34,450786,450788],{"className":450787},[31801],[34,450789,450791],{"className":450790,"style":31836},[31805],[34,450792],{},[34,450794,2557],{"className":450795},[31690],[34,450797],{"className":450798,"style":32604},[31845],[34,450800,223],{"className":450801},[32024],[34,450803],{"className":450804,"style":32604},[31845],[34,450806,450808,450811,450814,450817,450857],{"className":450807},[31676],[34,450809],{"className":450810,"style":31774},[31680],[34,450812,449638],{"className":450813,"style":32062},[31685,31686],[34,450815,456],{"className":450816},[31782],[34,450818,450820,450823],{"className":450819},[31685],[34,450821,33759],{"className":450822,"style":33774},[31685,31686],[34,450824,450826],{"className":450825},[31792],[34,450827,450829,450849],{"className":450828},[31796,31797],[34,450830,450832,450846],{"className":450831},[31801],[34,450833,450835],{"className":450834,"style":31806},[31805],[34,450836,450837,450840],{"style":319623},[34,450838],{"className":450839,"style":31814},[31813],[34,450841,450843],{"className":450842},[31818,31819,31820,31821],[34,450844,768],{"className":450845},[31685,31821],[34,450847,31829],{"className":450848},[31828],[34,450850,450852],{"className":450851},[31801],[34,450853,450855],{"className":450854,"style":31836},[31805],[34,450856],{},[34,450858,2557],{"className":450859},[31690],[108,450861,450862,451446,451447,440425],{},[34,450863,450865,450973],{"className":450864},[31642],[34,450866,450868],{"className":450867},[31646],[31648,450869,450870],{"xmlns":31650},[31652,450871,450872,450970],{},[31655,450873,450874,450876,450878,450884,450890,450892,450894,450896,450902,450908,450910,450912,450914,450916,450918,450924,450926,450928,450930,450932,450938,450940,450942,450944,450946,450948,450950,450956,450958,450960,450962,450968],{},[31658,450875,449638],{},[31662,450877,456],{"stretchy":702},[31743,450879,450880,450882],{},[31658,450881,33759],{},[31749,450883,640],{},[31743,450885,450886,450888],{},[31658,450887,33759],{},[31749,450889,768],{},[31662,450891,2557],{"stretchy":702},[31662,450893,202],{},[31662,450895,456],{"stretchy":702},[31743,450897,450898,450900],{},[31658,450899,33759],{},[31749,450901,640],{},[31743,450903,450904,450906],{},[31658,450905,33759],{},[31749,450907,768],{},[31662,450909,2557],{"stretchy":702},[31662,450911,223],{},[31658,450913,106063],{},[31662,450915,202],{},[31662,450917,456],{"stretchy":702},[31743,450919,450920,450922],{},[31658,450921,33759],{},[31749,450923,640],{},[31662,450925,223],{},[31658,450927,106063],{},[31662,450929,2557],{"stretchy":702},[31662,450931,456],{"stretchy":702},[31743,450933,450934,450936],{},[31658,450935,33759],{},[31749,450937,768],{},[31662,450939,223],{},[31658,450941,106063],{},[31662,450943,2557],{"stretchy":702},[31662,450945,202],{},[31658,450947,449638],{},[31662,450949,456],{"stretchy":702},[31743,450951,450952,450954],{},[31658,450953,33759],{},[31749,450955,640],{},[31662,450957,2557],{"stretchy":702},[31658,450959,449638],{},[31662,450961,456],{"stretchy":702},[31743,450963,450964,450966],{},[31658,450965,33759],{},[31749,450967,768],{},[31662,450969,2557],{"stretchy":702},[31665,450971,450972],{"encoding":31667},"\\psi(k_1 k_2) = (k_1 k_2)+I = (k_1+I)(k_2+I) = \\psi(k_1)\\psi(k_2)",[34,450974,450976,451080,451181,451199,451257,451321,451342],{"className":450975,"ariaHidden":758},[31672],[34,450977,450979,450982,450985,450988,451028,451068,451071,451074,451077],{"className":450978},[31676],[34,450980],{"className":450981,"style":31774},[31680],[34,450983,449638],{"className":450984,"style":32062},[31685,31686],[34,450986,456],{"className":450987},[31782],[34,450989,450991,450994],{"className":450990},[31685],[34,450992,33759],{"className":450993,"style":33774},[31685,31686],[34,450995,450997],{"className":450996},[31792],[34,450998,451000,451020],{"className":450999},[31796,31797],[34,451001,451003,451017],{"className":451002},[31801],[34,451004,451006],{"className":451005,"style":31806},[31805],[34,451007,451008,451011],{"style":319623},[34,451009],{"className":451010,"style":31814},[31813],[34,451012,451014],{"className":451013},[31818,31819,31820,31821],[34,451015,640],{"className":451016},[31685,31821],[34,451018,31829],{"className":451019},[31828],[34,451021,451023],{"className":451022},[31801],[34,451024,451026],{"className":451025,"style":31836},[31805],[34,451027],{},[34,451029,451031,451034],{"className":451030},[31685],[34,451032,33759],{"className":451033,"style":33774},[31685,31686],[34,451035,451037],{"className":451036},[31792],[34,451038,451040,451060],{"className":451039},[31796,31797],[34,451041,451043,451057],{"className":451042},[31801],[34,451044,451046],{"className":451045,"style":31806},[31805],[34,451047,451048,451051],{"style":319623},[34,451049],{"className":451050,"style":31814},[31813],[34,451052,451054],{"className":451053},[31818,31819,31820,31821],[34,451055,768],{"className":451056},[31685,31821],[34,451058,31829],{"className":451059},[31828],[34,451061,451063],{"className":451062},[31801],[34,451064,451066],{"className":451065,"style":31836},[31805],[34,451067],{},[34,451069,2557],{"className":451070},[31690],[34,451072],{"className":451073,"style":31846},[31845],[34,451075,202],{"className":451076},[31850],[34,451078],{"className":451079,"style":31846},[31845],[34,451081,451083,451086,451089,451129,451169,451172,451175,451178],{"className":451082},[31676],[34,451084],{"className":451085,"style":31774},[31680],[34,451087,456],{"className":451088},[31782],[34,451090,451092,451095],{"className":451091},[31685],[34,451093,33759],{"className":451094,"style":33774},[31685,31686],[34,451096,451098],{"className":451097},[31792],[34,451099,451101,451121],{"className":451100},[31796,31797],[34,451102,451104,451118],{"className":451103},[31801],[34,451105,451107],{"className":451106,"style":31806},[31805],[34,451108,451109,451112],{"style":319623},[34,451110],{"className":451111,"style":31814},[31813],[34,451113,451115],{"className":451114},[31818,31819,31820,31821],[34,451116,640],{"className":451117},[31685,31821],[34,451119,31829],{"className":451120},[31828],[34,451122,451124],{"className":451123},[31801],[34,451125,451127],{"className":451126,"style":31836},[31805],[34,451128],{},[34,451130,451132,451135],{"className":451131},[31685],[34,451133,33759],{"className":451134,"style":33774},[31685,31686],[34,451136,451138],{"className":451137},[31792],[34,451139,451141,451161],{"className":451140},[31796,31797],[34,451142,451144,451158],{"className":451143},[31801],[34,451145,451147],{"className":451146,"style":31806},[31805],[34,451148,451149,451152],{"style":319623},[34,451150],{"className":451151,"style":31814},[31813],[34,451153,451155],{"className":451154},[31818,31819,31820,31821],[34,451156,768],{"className":451157},[31685,31821],[34,451159,31829],{"className":451160},[31828],[34,451162,451164],{"className":451163},[31801],[34,451165,451167],{"className":451166,"style":31836},[31805],[34,451168],{},[34,451170,2557],{"className":451171},[31690],[34,451173],{"className":451174,"style":32604},[31845],[34,451176,223],{"className":451177},[32024],[34,451179],{"className":451180,"style":32604},[31845],[34,451182,451184,451187,451190,451193,451196],{"className":451183},[31676],[34,451185],{"className":451186,"style":33218},[31680],[34,451188,106063],{"className":451189,"style":67482},[31685,31686],[34,451191],{"className":451192,"style":31846},[31845],[34,451194,202],{"className":451195},[31850],[34,451197],{"className":451198,"style":31846},[31845],[34,451200,451202,451205,451208,451248,451251,451254],{"className":451201},[31676],[34,451203],{"className":451204,"style":31774},[31680],[34,451206,456],{"className":451207},[31782],[34,451209,451211,451214],{"className":451210},[31685],[34,451212,33759],{"className":451213,"style":33774},[31685,31686],[34,451215,451217],{"className":451216},[31792],[34,451218,451220,451240],{"className":451219},[31796,31797],[34,451221,451223,451237],{"className":451222},[31801],[34,451224,451226],{"className":451225,"style":31806},[31805],[34,451227,451228,451231],{"style":319623},[34,451229],{"className":451230,"style":31814},[31813],[34,451232,451234],{"className":451233},[31818,31819,31820,31821],[34,451235,640],{"className":451236},[31685,31821],[34,451238,31829],{"className":451239},[31828],[34,451241,451243],{"className":451242},[31801],[34,451244,451246],{"className":451245,"style":31836},[31805],[34,451247],{},[34,451249],{"className":451250,"style":32604},[31845],[34,451252,223],{"className":451253},[32024],[34,451255],{"className":451256,"style":32604},[31845],[34,451258,451260,451263,451266,451269,451272,451312,451315,451318],{"className":451259},[31676],[34,451261],{"className":451262,"style":31774},[31680],[34,451264,106063],{"className":451265,"style":67482},[31685,31686],[34,451267,2557],{"className":451268},[31690],[34,451270,456],{"className":451271},[31782],[34,451273,451275,451278],{"className":451274},[31685],[34,451276,33759],{"className":451277,"style":33774},[31685,31686],[34,451279,451281],{"className":451280},[31792],[34,451282,451284,451304],{"className":451283},[31796,31797],[34,451285,451287,451301],{"className":451286},[31801],[34,451288,451290],{"className":451289,"style":31806},[31805],[34,451291,451292,451295],{"style":319623},[34,451293],{"className":451294,"style":31814},[31813],[34,451296,451298],{"className":451297},[31818,31819,31820,31821],[34,451299,768],{"className":451300},[31685,31821],[34,451302,31829],{"className":451303},[31828],[34,451305,451307],{"className":451306},[31801],[34,451308,451310],{"className":451309,"style":31836},[31805],[34,451311],{},[34,451313],{"className":451314,"style":32604},[31845],[34,451316,223],{"className":451317},[32024],[34,451319],{"className":451320,"style":32604},[31845],[34,451322,451324,451327,451330,451333,451336,451339],{"className":451323},[31676],[34,451325],{"className":451326,"style":31774},[31680],[34,451328,106063],{"className":451329,"style":67482},[31685,31686],[34,451331,2557],{"className":451332},[31690],[34,451334],{"className":451335,"style":31846},[31845],[34,451337,202],{"className":451338},[31850],[34,451340],{"className":451341,"style":31846},[31845],[34,451343,451345,451348,451351,451354,451394,451397,451400,451403,451443],{"className":451344},[31676],[34,451346],{"className":451347,"style":31774},[31680],[34,451349,449638],{"className":451350,"style":32062},[31685,31686],[34,451352,456],{"className":451353},[31782],[34,451355,451357,451360],{"className":451356},[31685],[34,451358,33759],{"className":451359,"style":33774},[31685,31686],[34,451361,451363],{"className":451362},[31792],[34,451364,451366,451386],{"className":451365},[31796,31797],[34,451367,451369,451383],{"className":451368},[31801],[34,451370,451372],{"className":451371,"style":31806},[31805],[34,451373,451374,451377],{"style":319623},[34,451375],{"className":451376,"style":31814},[31813],[34,451378,451380],{"className":451379},[31818,31819,31820,31821],[34,451381,640],{"className":451382},[31685,31821],[34,451384,31829],{"className":451385},[31828],[34,451387,451389],{"className":451388},[31801],[34,451390,451392],{"className":451391,"style":31836},[31805],[34,451393],{},[34,451395,2557],{"className":451396},[31690],[34,451398,449638],{"className":451399,"style":32062},[31685,31686],[34,451401,456],{"className":451402},[31782],[34,451404,451406,451409],{"className":451405},[31685],[34,451407,33759],{"className":451408,"style":33774},[31685,31686],[34,451410,451412],{"className":451411},[31792],[34,451413,451415,451435],{"className":451414},[31796,31797],[34,451416,451418,451432],{"className":451417},[31801],[34,451419,451421],{"className":451420,"style":31806},[31805],[34,451422,451423,451426],{"style":319623},[34,451424],{"className":451425,"style":31814},[31813],[34,451427,451429],{"className":451428},[31818,31819,31820,31821],[34,451430,768],{"className":451431},[31685,31821],[34,451433,31829],{"className":451434},[31828],[34,451436,451438],{"className":451437},[31801],[34,451439,451441],{"className":451440,"style":31836},[31805],[34,451442],{},[34,451444,2557],{"className":451445},[31690],".\n所以 ",[34,451448,451450,451464],{"className":451449},[31642],[34,451451,451453],{"className":451452},[31646],[31648,451454,451455],{"xmlns":31650},[31652,451456,451457,451461],{},[31655,451458,451459],{},[31658,451460,449638],{},[31665,451462,451463],{"encoding":31667},"\\psi",[34,451465,451467],{"className":451466,"ariaHidden":758},[31672],[34,451468,451470,451473],{"className":451469},[31676],[34,451471],{"className":451472,"style":43217},[31680],[34,451474,449638],{"className":451475,"style":32062},[31685,31686],[108,451477,451478,312000,451481,451551,451552,70884,451635,451722,451723,354330,451926,451977,451978,452022,452023,452051],{},[2627,451479,451480],{},"验证满射",[34,451482,451484,451509],{"className":451483},[31642],[34,451485,451487],{"className":451486},[31646],[31648,451488,451489],{"xmlns":31650},[31652,451490,451491,451507],{},[31655,451492,451493,451495,451497,451499,451501,451503,451505],{},[31662,451494,456],{"stretchy":702},[31658,451496,106063],{},[31662,451498,223],{},[31658,451500,38691],{},[31662,451502,2557],{"stretchy":702},[31658,451504,38080],{"mathvariant":38079},[31658,451506,106063],{},[31665,451508,449572],{"encoding":31667},[34,451510,451512,451533],{"className":451511,"ariaHidden":758},[31672],[34,451513,451515,451518,451521,451524,451527,451530],{"className":451514},[31676],[34,451516],{"className":451517,"style":31774},[31680],[34,451519,456],{"className":451520},[31782],[34,451522,106063],{"className":451523,"style":67482},[31685,31686],[34,451525],{"className":451526,"style":32604},[31845],[34,451528,223],{"className":451529},[32024],[34,451531],{"className":451532,"style":32604},[31845],[34,451534,451536,451539,451542,451545,451548],{"className":451535},[31676],[34,451537],{"className":451538,"style":31774},[31680],[34,451540,38691],{"className":451541,"style":38706},[31685,31686],[34,451543,2557],{"className":451544},[31690],[34,451546,38080],{"className":451547},[31685],[34,451549,106063],{"className":451550,"style":67482},[31685,31686]," 中的任意元素形式为 ",[34,451553,451555,451581],{"className":451554},[31642],[34,451556,451558],{"className":451557},[31646],[31648,451559,451560],{"xmlns":31650},[31652,451561,451562,451578],{},[31655,451563,451564,451566,451568,451570,451572,451574,451576],{},[31662,451565,456],{"stretchy":702},[31658,451567,31955],{},[31662,451569,223],{},[31658,451571,33759],{},[31662,451573,2557],{"stretchy":702},[31662,451575,223],{},[31658,451577,106063],{},[31665,451579,451580],{"encoding":31667},"(i+k)+I",[34,451582,451584,451605,451626],{"className":451583,"ariaHidden":758},[31672],[34,451585,451587,451590,451593,451596,451599,451602],{"className":451586},[31676],[34,451588],{"className":451589,"style":31774},[31680],[34,451591,456],{"className":451592},[31782],[34,451594,31955],{"className":451595},[31685,31686],[34,451597],{"className":451598,"style":32604},[31845],[34,451600,223],{"className":451601},[32024],[34,451603],{"className":451604,"style":32604},[31845],[34,451606,451608,451611,451614,451617,451620,451623],{"className":451607},[31676],[34,451609],{"className":451610,"style":31774},[31680],[34,451612,33759],{"className":451613,"style":33774},[31685,31686],[34,451615,2557],{"className":451616},[31690],[34,451618],{"className":451619,"style":32604},[31845],[34,451621,223],{"className":451622},[32024],[34,451624],{"className":451625,"style":32604},[31845],[34,451627,451629,451632],{"className":451628},[31676],[34,451630],{"className":451631,"style":33218},[31680],[34,451633,106063],{"className":451634,"style":67482},[31685,31686],[34,451636,451638,451664],{"className":451637},[31642],[34,451639,451641],{"className":451640},[31646],[31648,451642,451643],{"xmlns":31650},[31652,451644,451645,451661],{},[31655,451646,451647,451649,451651,451653,451655,451657,451659],{},[31658,451648,31955],{},[31662,451650,112296],{},[31658,451652,106063],{},[31662,451654,31960],{"separator":758},[31658,451656,33759],{},[31662,451658,112296],{},[31658,451660,38691],{},[31665,451662,451663],{"encoding":31667},"i \\in I, k \\in K",[34,451665,451667,451686,451713],{"className":451666,"ariaHidden":758},[31672],[34,451668,451670,451674,451677,451680,451683],{"className":451669},[31676],[34,451671],{"className":451672,"style":451673},[31680],"height:0.6986em;vertical-align:-0.0391em;",[34,451675,31955],{"className":451676},[31685,31686],[34,451678],{"className":451679,"style":31846},[31845],[34,451681,112296],{"className":451682},[31850],[34,451684],{"className":451685,"style":31846},[31845],[34,451687,451689,451692,451695,451698,451701,451704,451707,451710],{"className":451688},[31676],[34,451690],{"className":451691,"style":43217},[31680],[34,451693,106063],{"className":451694,"style":67482},[31685,31686],[34,451696,31960],{"className":451697},[32119],[34,451699],{"className":451700,"style":32123},[31845],[34,451702,33759],{"className":451703,"style":33774},[31685,31686],[34,451705],{"className":451706,"style":31846},[31845],[34,451708,112296],{"className":451709},[31850],[34,451711],{"className":451712,"style":31846},[31845],[34,451714,451716,451719],{"className":451715},[31676],[34,451717],{"className":451718,"style":33218},[31680],[34,451720,38691],{"className":451721,"style":38706},[31685,31686],"。\n根据商环陪集的性质，",[34,451724,451726,451776],{"className":451725},[31642],[34,451727,451729],{"className":451728},[31646],[31648,451730,451731],{"xmlns":31650},[31652,451732,451733,451773],{},[31655,451734,451735,451737,451739,451741,451743,451745,451747,451749,451751,451753,451755,451757,451759,451761,451763,451765,451767,451769,451771],{},[31662,451736,456],{"stretchy":702},[31658,451738,31955],{},[31662,451740,223],{},[31658,451742,33759],{},[31662,451744,2557],{"stretchy":702},[31662,451746,223],{},[31658,451748,106063],{},[31662,451750,202],{},[31658,451752,33759],{},[31662,451754,223],{},[31662,451756,456],{"stretchy":702},[31658,451758,31955],{},[31662,451760,223],{},[31658,451762,106063],{},[31662,451764,2557],{"stretchy":702},[31662,451766,202],{},[31658,451768,33759],{},[31662,451770,223],{},[31658,451772,106063],{},[31665,451774,451775],{"encoding":31667},"(i+k)+I = k+(i+I) = k+I",[34,451777,451779,451800,451821,451839,451857,451878,451899,451917],{"className":451778,"ariaHidden":758},[31672],[34,451780,451782,451785,451788,451791,451794,451797],{"className":451781},[31676],[34,451783],{"className":451784,"style":31774},[31680],[34,451786,456],{"className":451787},[31782],[34,451789,31955],{"className":451790},[31685,31686],[34,451792],{"className":451793,"style":32604},[31845],[34,451795,223],{"className":451796},[32024],[34,451798],{"className":451799,"style":32604},[31845],[34,451801,451803,451806,451809,451812,451815,451818],{"className":451802},[31676],[34,451804],{"className":451805,"style":31774},[31680],[34,451807,33759],{"className":451808,"style":33774},[31685,31686],[34,451810,2557],{"className":451811},[31690],[34,451813],{"className":451814,"style":32604},[31845],[34,451816,223],{"className":451817},[32024],[34,451819],{"className":451820,"style":32604},[31845],[34,451822,451824,451827,451830,451833,451836],{"className":451823},[31676],[34,451825],{"className":451826,"style":33218},[31680],[34,451828,106063],{"className":451829,"style":67482},[31685,31686],[34,451831],{"className":451832,"style":31846},[31845],[34,451834,202],{"className":451835},[31850],[34,451837],{"className":451838,"style":31846},[31845],[34,451840,451842,451845,451848,451851,451854],{"className":451841},[31676],[34,451843],{"className":451844,"style":34299},[31680],[34,451846,33759],{"className":451847,"style":33774},[31685,31686],[34,451849],{"className":451850,"style":32604},[31845],[34,451852,223],{"className":451853},[32024],[34,451855],{"className":451856,"style":32604},[31845],[34,451858,451860,451863,451866,451869,451872,451875],{"className":451859},[31676],[34,451861],{"className":451862,"style":31774},[31680],[34,451864,456],{"className":451865},[31782],[34,451867,31955],{"className":451868},[31685,31686],[34,451870],{"className":451871,"style":32604},[31845],[34,451873,223],{"className":451874},[32024],[34,451876],{"className":451877,"style":32604},[31845],[34,451879,451881,451884,451887,451890,451893,451896],{"className":451880},[31676],[34,451882],{"className":451883,"style":31774},[31680],[34,451885,106063],{"className":451886,"style":67482},[31685,31686],[34,451888,2557],{"className":451889},[31690],[34,451891],{"className":451892,"style":31846},[31845],[34,451894,202],{"className":451895},[31850],[34,451897],{"className":451898,"style":31846},[31845],[34,451900,451902,451905,451908,451911,451914],{"className":451901},[31676],[34,451903],{"className":451904,"style":34299},[31680],[34,451906,33759],{"className":451907,"style":33774},[31685,31686],[34,451909],{"className":451910,"style":32604},[31845],[34,451912,223],{"className":451913},[32024],[34,451915],{"className":451916,"style":32604},[31845],[34,451918,451920,451923],{"className":451919},[31676],[34,451921],{"className":451922,"style":33218},[31680],[34,451924,106063],{"className":451925,"style":67482},[31685,31686],[34,451927,451929,451947],{"className":451928},[31642],[34,451930,451932],{"className":451931},[31646],[31648,451933,451934],{"xmlns":31650},[31652,451935,451936,451944],{},[31655,451937,451938,451940,451942],{},[31658,451939,31955],{},[31662,451941,112296],{},[31658,451943,106063],{},[31665,451945,451946],{"encoding":31667},"i \\in I",[34,451948,451950,451968],{"className":451949,"ariaHidden":758},[31672],[34,451951,451953,451956,451959,451962,451965],{"className":451952},[31676],[34,451954],{"className":451955,"style":451673},[31680],[34,451957,31955],{"className":451958},[31685,31686],[34,451960],{"className":451961,"style":31846},[31845],[34,451963,112296],{"className":451964},[31850],[34,451966],{"className":451967,"style":31846},[31845],[34,451969,451971,451974],{"className":451970},[31676],[34,451972],{"className":451973,"style":33218},[31680],[34,451975,106063],{"className":451976,"style":67482},[31685,31686],"）。\n即任意元素均可写成 ",[34,451979,451981,452001],{"className":451980},[31642],[34,451982,451984],{"className":451983},[31646],[31648,451985,451986],{"xmlns":31650},[31652,451987,451988,451998],{},[31655,451989,451990,451992,451994,451996],{},[31658,451991,449638],{},[31662,451993,456],{"stretchy":702},[31658,451995,33759],{},[31662,451997,2557],{"stretchy":702},[31665,451999,452000],{"encoding":31667},"\\psi(k)",[34,452002,452004],{"className":452003,"ariaHidden":758},[31672],[34,452005,452007,452010,452013,452016,452019],{"className":452006},[31676],[34,452008],{"className":452009,"style":31774},[31680],[34,452011,449638],{"className":452012,"style":32062},[31685,31686],[34,452014,456],{"className":452015},[31782],[34,452017,33759],{"className":452018,"style":33774},[31685,31686],[34,452020,2557],{"className":452021},[31690]," 的形式，故 ",[34,452024,452026,452039],{"className":452025},[31642],[34,452027,452029],{"className":452028},[31646],[31648,452030,452031],{"xmlns":31650},[31652,452032,452033,452037],{},[31655,452034,452035],{},[31658,452036,449638],{},[31665,452038,451463],{"encoding":31667},[34,452040,452042],{"className":452041,"ariaHidden":758},[31672],[34,452043,452045,452048],{"className":452044},[31676],[34,452046],{"className":452047,"style":43217},[31680],[34,452049,449638],{"className":452050,"style":32062},[31685,31686]," 是满同态。",[108,452053,452054,312000,452096],{},[2627,452055,452056,452057,2557],{},"计算核 (",[34,452058,452060,452078],{"className":452059},[31642],[34,452061,452063],{"className":452062},[31646],[31648,452064,452065],{"xmlns":31650},[31652,452066,452067,452075],{},[31655,452068,452069,452071,452073],{},[31658,452070,406335],{},[31662,452072,32682],{},[31658,452074,449638],{},[31665,452076,452077],{"encoding":31667},"\\ker \\psi",[34,452079,452081],{"className":452080,"ariaHidden":758},[31672],[34,452082,452084,452087,452090,452093],{"className":452083},[31676],[34,452085],{"className":452086,"style":43217},[31680],[34,452088,406335],{"className":452089},[32767],[34,452091],{"className":452092,"style":32123},[31845],[34,452094,449638],{"className":452095,"style":32062},[31685,31686],[34,452097,452099,452279],{"className":452098},[31642],[34,452100,452102],{"className":452101},[31646],[31648,452103,452104],{"xmlns":31650},[31652,452105,452106,452276],{},[71078,452107,452108,452174,452214,452250],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,452109,452110,452122],{},[71087,452111,452112],{},[71090,452113,452114],{"scriptlevel":459,"displaystyle":758},[31655,452115,452116,452118,452120],{},[31658,452117,406335],{},[31662,452119,32682],{},[31658,452121,449638],{},[71087,452123,452124],{},[71090,452125,452126],{"scriptlevel":459,"displaystyle":758},[31655,452127,452128,452130,452132,452134,452136,452138,452140,452142,452144,452146,452148,452150,452152,452172],{},[31655,452129],{},[31662,452131,202],{},[31662,452133,32685],{"stretchy":702},[31658,452135,33759],{},[31662,452137,112296],{},[31658,452139,38691],{},[31662,452141,162086],{},[31658,452143,449638],{},[31662,452145,456],{"stretchy":702},[31658,452147,33759],{},[31662,452149,2557],{"stretchy":702},[31662,452151,202],{},[31743,452153,452154,452156],{},[31749,452155,459],{},[31655,452157,452158,452160,452162,452164,452166,452168,452170],{},[31662,452159,456],{"stretchy":702},[31658,452161,106063],{},[31662,452163,223],{},[31658,452165,38691],{},[31662,452167,2557],{"stretchy":702},[31658,452169,38080],{"mathvariant":38079},[31658,452171,106063],{},[31662,452173,32715],{"stretchy":702},[71084,452175,452176,452182],{},[71087,452177,452178],{},[71090,452179,452180],{"scriptlevel":459,"displaystyle":758},[31655,452181],{},[71087,452183,452184],{},[71090,452185,452186],{"scriptlevel":459,"displaystyle":758},[31655,452187,452188,452190,452192,452194,452196,452198,452200,452202,452204,452206,452208,452210,452212],{},[31655,452189],{},[31662,452191,202],{},[31662,452193,32685],{"stretchy":702},[31658,452195,33759],{},[31662,452197,112296],{},[31658,452199,38691],{},[31662,452201,162086],{},[31658,452203,33759],{},[31662,452205,223],{},[31658,452207,106063],{},[31662,452209,202],{},[31658,452211,106063],{},[31662,452213,32715],{"stretchy":702},[71084,452215,452216,452222],{},[71087,452217,452218],{},[71090,452219,452220],{"scriptlevel":459,"displaystyle":758},[31655,452221],{},[71087,452223,452224],{},[71090,452225,452226],{"scriptlevel":459,"displaystyle":758},[31655,452227,452228,452230,452232,452234,452236,452238,452240,452242,452244,452246,452248],{},[31655,452229],{},[31662,452231,202],{},[31662,452233,32685],{"stretchy":702},[31658,452235,33759],{},[31662,452237,112296],{},[31658,452239,38691],{},[31662,452241,162086],{},[31658,452243,33759],{},[31662,452245,112296],{},[31658,452247,106063],{},[31662,452249,32715],{"stretchy":702},[71084,452251,452252,452258],{},[71087,452253,452254],{},[71090,452255,452256],{"scriptlevel":459,"displaystyle":758},[31655,452257],{},[71087,452259,452260],{},[71090,452261,452262],{"scriptlevel":459,"displaystyle":758},[31655,452263,452264,452266,452268,452270,452272,452274],{},[31655,452265],{},[31662,452267,202],{},[31658,452269,38691],{},[31662,452271,326344],{},[31658,452273,106063],{},[31658,452275,2547],{"mathvariant":38079},[31665,452277,452278],{"encoding":31667},"\\begin{aligned} \\ker \\psi &= \\{k \\in K \\mid \\psi(k) = 0_{(I+K)\u002FI}\\} \\\\ &= \\{k \\in K \\mid k+I = I\\} \\\\ &= \\{k \\in K \\mid k \\in I\\} \\\\ &= K \\cap I. \\end{aligned}",[34,452280,452282],{"className":452281,"ariaHidden":758},[31672],[34,452283,452285,452289],{"className":452284},[31676],[34,452286],{"className":452287,"style":452288},[31680],"height:5.7em;vertical-align:-2.6em;",[34,452290,452292],{"className":452291},[31685],[34,452293,452295,452365],{"className":452294},[71078],[34,452296,452298],{"className":452297},[369003],[34,452299,452301,452356],{"className":452300},[31796,31797],[34,452302,452304,452353],{"className":452303},[31801],[34,452305,452308,452326,452335,452344],{"className":452306,"style":452307},[31805],"height:3.1em;",[34,452309,452311,452314],{"style":452310},"top:-5.26em;",[34,452312],{"className":452313,"style":41508},[31813],[34,452315,452317,452320,452323],{"className":452316},[31685],[34,452318,406335],{"className":452319},[32767],[34,452321],{"className":452322,"style":32123},[31845],[34,452324,449638],{"className":452325,"style":32062},[31685,31686],[34,452327,452329,452332],{"style":452328},"top:-3.76em;",[34,452330],{"className":452331,"style":41508},[31813],[34,452333],{"className":452334},[31685],[34,452336,452338,452341],{"style":452337},"top:-2.26em;",[34,452339],{"className":452340,"style":41508},[31813],[34,452342],{"className":452343},[31685],[34,452345,452347,452350],{"style":452346},"top:-0.76em;",[34,452348],{"className":452349,"style":41508},[31813],[34,452351],{"className":452352},[31685],[34,452354,31829],{"className":452355},[31828],[34,452357,452359],{"className":452358},[31801],[34,452360,452363],{"className":452361,"style":452362},[31805],"height:2.6em;",[34,452364],{},[34,452366,452368],{"className":452367},[71384],[34,452369,452371,452692],{"className":452370},[31796,31797],[34,452372,452374,452689],{"className":452373},[31801],[34,452375,452377,452509,452586,452651],{"className":452376,"style":452307},[31805],[34,452378,452379,452382],{"style":452310},[34,452380],{"className":452381,"style":41508},[31813],[34,452383,452385,452388,452391,452394,452397,452400,452403,452406,452409,452412,452415,452418,452421,452424,452427,452430,452433,452436,452439,452442,452445,452506],{"className":452384},[31685],[34,452386],{"className":452387},[31685],[34,452389],{"className":452390,"style":31846},[31845],[34,452392,202],{"className":452393},[31850],[34,452395],{"className":452396,"style":31846},[31845],[34,452398,32685],{"className":452399},[31782],[34,452401,33759],{"className":452402,"style":33774},[31685,31686],[34,452404],{"className":452405,"style":31846},[31845],[34,452407,112296],{"className":452408},[31850],[34,452410],{"className":452411,"style":31846},[31845],[34,452413,38691],{"className":452414,"style":38706},[31685,31686],[34,452416],{"className":452417,"style":31846},[31845],[34,452419,162086],{"className":452420},[31850],[34,452422],{"className":452423,"style":31846},[31845],[34,452425,449638],{"className":452426,"style":32062},[31685,31686],[34,452428,456],{"className":452429},[31782],[34,452431,33759],{"className":452432,"style":33774},[31685,31686],[34,452434,2557],{"className":452435},[31690],[34,452437],{"className":452438,"style":31846},[31845],[34,452440,202],{"className":452441},[31850],[34,452443],{"className":452444,"style":31846},[31845],[34,452446,452448,452451],{"className":452447},[31685],[34,452449,459],{"className":452450},[31685],[34,452452,452454],{"className":452453},[31792],[34,452455,452457,452498],{"className":452456},[31796,31797],[34,452458,452460,452495],{"className":452459},[31801],[34,452461,452463],{"className":452462,"style":332295},[31805],[34,452464,452465,452468],{"style":332298},[34,452466],{"className":452467,"style":31814},[31813],[34,452469,452471],{"className":452470},[31818,31819,31820,31821],[34,452472,452474,452477,452480,452483,452486,452489,452492],{"className":452473},[31685,31821],[34,452475,456],{"className":452476},[31782,31821],[34,452478,106063],{"className":452479,"style":67482},[31685,31686,31821],[34,452481,223],{"className":452482},[32024,31821],[34,452484,38691],{"className":452485,"style":38706},[31685,31686,31821],[34,452487,2557],{"className":452488},[31690,31821],[34,452490,38080],{"className":452491},[31685,31821],[34,452493,106063],{"className":452494,"style":67482},[31685,31686,31821],[34,452496,31829],{"className":452497},[31828],[34,452499,452501],{"className":452500},[31801],[34,452502,452504],{"className":452503,"style":332326},[31805],[34,452505],{},[34,452507,32715],{"className":452508},[31690],[34,452510,452511,452514],{"style":452328},[34,452512],{"className":452513,"style":41508},[31813],[34,452515,452517,452520,452523,452526,452529,452532,452535,452538,452541,452544,452547,452550,452553,452556,452559,452562,452565,452568,452571,452574,452577,452580,452583],{"className":452516},[31685],[34,452518],{"className":452519},[31685],[34,452521],{"className":452522,"style":31846},[31845],[34,452524,202],{"className":452525},[31850],[34,452527],{"className":452528,"style":31846},[31845],[34,452530,32685],{"className":452531},[31782],[34,452533,33759],{"className":452534,"style":33774},[31685,31686],[34,452536],{"className":452537,"style":31846},[31845],[34,452539,112296],{"className":452540},[31850],[34,452542],{"className":452543,"style":31846},[31845],[34,452545,38691],{"className":452546,"style":38706},[31685,31686],[34,452548],{"className":452549,"style":31846},[31845],[34,452551,162086],{"className":452552},[31850],[34,452554],{"className":452555,"style":31846},[31845],[34,452557,33759],{"className":452558,"style":33774},[31685,31686],[34,452560],{"className":452561,"style":32604},[31845],[34,452563,223],{"className":452564},[32024],[34,452566],{"className":452567,"style":32604},[31845],[34,452569,106063],{"className":452570,"style":67482},[31685,31686],[34,452572],{"className":452573,"style":31846},[31845],[34,452575,202],{"className":452576},[31850],[34,452578],{"className":452579,"style":31846},[31845],[34,452581,106063],{"className":452582,"style":67482},[31685,31686],[34,452584,32715],{"className":452585},[31690],[34,452587,452588,452591],{"style":452337},[34,452589],{"className":452590,"style":41508},[31813],[34,452592,452594,452597,452600,452603,452606,452609,452612,452615,452618,452621,452624,452627,452630,452633,452636,452639,452642,452645,452648],{"className":452593},[31685],[34,452595],{"className":452596},[31685],[34,452598],{"className":452599,"style":31846},[31845],[34,452601,202],{"className":452602},[31850],[34,452604],{"className":452605,"style":31846},[31845],[34,452607,32685],{"className":452608},[31782],[34,452610,33759],{"className":452611,"style":33774},[31685,31686],[34,452613],{"className":452614,"style":31846},[31845],[34,452616,112296],{"className":452617},[31850],[34,452619],{"className":452620,"style":31846},[31845],[34,452622,38691],{"className":452623,"style":38706},[31685,31686],[34,452625],{"className":452626,"style":31846},[31845],[34,452628,162086],{"className":452629},[31850],[34,452631],{"className":452632,"style":31846},[31845],[34,452634,33759],{"className":452635,"style":33774},[31685,31686],[34,452637],{"className":452638,"style":31846},[31845],[34,452640,112296],{"className":452641},[31850],[34,452643],{"className":452644,"style":31846},[31845],[34,452646,106063],{"className":452647,"style":67482},[31685,31686],[34,452649,32715],{"className":452650},[31690],[34,452652,452653,452656],{"style":452346},[34,452654],{"className":452655,"style":41508},[31813],[34,452657,452659,452662,452665,452668,452671,452674,452677,452680,452683,452686],{"className":452658},[31685],[34,452660],{"className":452661},[31685],[34,452663],{"className":452664,"style":31846},[31845],[34,452666,202],{"className":452667},[31850],[34,452669],{"className":452670,"style":31846},[31845],[34,452672,38691],{"className":452673,"style":38706},[31685,31686],[34,452675],{"className":452676,"style":32604},[31845],[34,452678,326344],{"className":452679},[32024],[34,452681],{"className":452682,"style":32604},[31845],[34,452684,106063],{"className":452685,"style":67482},[31685,31686],[34,452687,2547],{"className":452688},[31685],[34,452690,31829],{"className":452691},[31828],[34,452693,452695],{"className":452694},[31801],[34,452696,452698],{"className":452697,"style":452362},[31805],[34,452699],{},[108,452701,452702,452705,452706,64615,452734,452821,452822,452957,452958],{},[2627,452703,452704],{},"应用第一同态定理",":\n根据定理 3.3.3，对于满同态 ",[34,452707,452709,452722],{"className":452708},[31642],[34,452710,452712],{"className":452711},[31646],[31648,452713,452714],{"xmlns":31650},[31652,452715,452716,452720],{},[31655,452717,452718],{},[31658,452719,449638],{},[31665,452721,451463],{"encoding":31667},[34,452723,452725],{"className":452724,"ariaHidden":758},[31672],[34,452726,452728,452731],{"className":452727},[31676],[34,452729],{"className":452730,"style":43217},[31680],[34,452732,449638],{"className":452733,"style":32062},[31685,31686],[34,452735,452737,452767],{"className":452736},[31642],[34,452738,452740],{"className":452739},[31646],[31648,452741,452742],{"xmlns":31650},[31652,452743,452744,452764],{},[31655,452745,452746,452748,452750,452752,452754,452756,452758,452760,452762],{},[31658,452747,38691],{},[31658,452749,38080],{"mathvariant":38079},[31658,452751,406335],{},[31662,452753,32682],{},[31658,452755,449638],{},[31662,452757,134826],{},[31658,452759,440604],{"mathvariant":38079},[31662,452761,32682],{},[31658,452763,449638],{},[31665,452765,452766],{"encoding":31667},"K \u002F \\ker \\psi \\cong \\operatorname{Im} \\psi",[34,452768,452770,452803],{"className":452769,"ariaHidden":758},[31672],[34,452771,452773,452776,452779,452782,452785,452788,452791,452794,452797,452800],{"className":452772},[31676],[34,452774],{"className":452775,"style":31774},[31680],[34,452777,38691],{"className":452778,"style":38706},[31685,31686],[34,452780,38080],{"className":452781},[31685],[34,452783],{"className":452784,"style":32123},[31845],[34,452786,406335],{"className":452787},[32767],[34,452789],{"className":452790,"style":32123},[31845],[34,452792,449638],{"className":452793,"style":32062},[31685,31686],[34,452795],{"className":452796,"style":31846},[31845],[34,452798,134826],{"className":452799},[31850],[34,452801],{"className":452802,"style":31846},[31845],[34,452804,452806,452809,452815,452818],{"className":452805},[31676],[34,452807],{"className":452808,"style":43217},[31680],[34,452810,452812],{"className":452811},[32767],[34,452813,440604],{"className":452814},[31685,207923],[34,452816],{"className":452817,"style":32123},[31845],[34,452819,449638],{"className":452820,"style":32062},[31685,31686],"。\n即 ",[34,452823,452825,452867],{"className":452824},[31642],[34,452826,452828],{"className":452827},[31646],[31648,452829,452830],{"xmlns":31650},[31652,452831,452832,452864],{},[31655,452833,452834,452836,452838,452840,452842,452844,452846,452848,452850,452852,452854,452856,452858,452860,452862],{},[31658,452835,38691],{},[31658,452837,38080],{"mathvariant":38079},[31662,452839,456],{"stretchy":702},[31658,452841,106063],{},[31662,452843,326344],{},[31658,452845,38691],{},[31662,452847,2557],{"stretchy":702},[31662,452849,134826],{},[31662,452851,456],{"stretchy":702},[31658,452853,106063],{},[31662,452855,223],{},[31658,452857,38691],{},[31662,452859,2557],{"stretchy":702},[31658,452861,38080],{"mathvariant":38079},[31658,452863,106063],{},[31665,452865,452866],{"encoding":31667},"K \u002F (I \\cap K) \\cong (I+K)\u002FI",[34,452868,452870,452897,452918,452939],{"className":452869,"ariaHidden":758},[31672],[34,452871,452873,452876,452879,452882,452885,452888,452891,452894],{"className":452872},[31676],[34,452874],{"className":452875,"style":31774},[31680],[34,452877,38691],{"className":452878,"style":38706},[31685,31686],[34,452880,38080],{"className":452881},[31685],[34,452883,456],{"className":452884},[31782],[34,452886,106063],{"className":452887,"style":67482},[31685,31686],[34,452889],{"className":452890,"style":32604},[31845],[34,452892,326344],{"className":452893},[32024],[34,452895],{"className":452896,"style":32604},[31845],[34,452898,452900,452903,452906,452909,452912,452915],{"className":452899},[31676],[34,452901],{"className":452902,"style":31774},[31680],[34,452904,38691],{"className":452905,"style":38706},[31685,31686],[34,452907,2557],{"className":452908},[31690],[34,452910],{"className":452911,"style":31846},[31845],[34,452913,134826],{"className":452914},[31850],[34,452916],{"className":452917,"style":31846},[31845],[34,452919,452921,452924,452927,452930,452933,452936],{"className":452920},[31676],[34,452922],{"className":452923,"style":31774},[31680],[34,452925,456],{"className":452926},[31782],[34,452928,106063],{"className":452929,"style":67482},[31685,31686],[34,452931],{"className":452932,"style":32604},[31845],[34,452934,223],{"className":452935},[32024],[34,452937],{"className":452938,"style":32604},[31845],[34,452940,452942,452945,452948,452951,452954],{"className":452941},[31676],[34,452943],{"className":452944,"style":31774},[31680],[34,452946,38691],{"className":452947,"style":38706},[31685,31686],[34,452949,2557],{"className":452950},[31690],[34,452952,38080],{"className":452953},[31685],[34,452955,106063],{"className":452956,"style":67482},[31685,31686],"。 ",[34,452959,452961,452974],{"className":452960},[31642],[34,452962,452964],{"className":452963},[31646],[31648,452965,452966],{"xmlns":31650},[31652,452967,452968,452972],{},[31655,452969,452970],{},[31658,452971,445204],{"mathvariant":38079},[31665,452973,445207],{"encoding":31667},[34,452975,452977],{"className":452976,"ariaHidden":758},[31672],[34,452978,452980,452983],{"className":452979},[31676],[34,452981],{"className":452982,"style":445217},[31680],[34,452984,445204],{"className":452985},[31685,129261],[32188,452987],{},[253,452989,452991],{"id":452990},"_4-典型例子","4. 典型例子",[15,452993,452994,452997,452998,453112,453113,453192,453193,453422,453423,11008,453451,453503,453504,311257,453555,447697,453693,453867,453868,684,453947],{},[2627,452995,452996],{},"例 3.3.1.","\n(i) 典范态射 ",[34,452999,453001,453027],{"className":453000},[31642],[34,453002,453004],{"className":453003},[31646],[31648,453005,453006],{"xmlns":31650},[31652,453007,453008,453024],{},[31655,453009,453010,453012,453014,453016,453018],{},[31658,453011,58442],{},[31662,453013,5598],{},[31658,453015,97300],{"mathvariant":318416},[31662,453017,33239],{},[31743,453019,453020,453022],{},[31658,453021,97300],{"mathvariant":318416},[31658,453023,31660],{},[31665,453025,453026],{"encoding":31667},"\\pi: \\mathbb{Z} \\to \\mathbb{Z}_n",[34,453028,453030,453048,453066],{"className":453029,"ariaHidden":758},[31672],[34,453031,453033,453036,453039,453042,453045],{"className":453032},[31676],[34,453034],{"className":453035,"style":32527},[31680],[34,453037,58442],{"className":453038,"style":32062},[31685,31686],[34,453040],{"className":453041,"style":31846},[31845],[34,453043,5598],{"className":453044},[31850],[34,453046],{"className":453047,"style":31846},[31845],[34,453049,453051,453054,453057,453060,453063],{"className":453050},[31676],[34,453052],{"className":453053,"style":320927},[31680],[34,453055,97300],{"className":453056},[31685,318441],[34,453058],{"className":453059,"style":31846},[31845],[34,453061,33239],{"className":453062},[31850],[34,453064],{"className":453065,"style":31846},[31845],[34,453067,453069,453072],{"className":453068},[31676],[34,453070],{"className":453071,"style":330413},[31680],[34,453073,453075,453078],{"className":453074},[31685],[34,453076,97300],{"className":453077},[31685,318441],[34,453079,453081],{"className":453080},[31792],[34,453082,453084,453104],{"className":453083},[31796,31797],[34,453085,453087,453101],{"className":453086},[31801],[34,453088,453090],{"className":453089,"style":32362},[31805],[34,453091,453092,453095],{"style":31809},[34,453093],{"className":453094,"style":31814},[31813],[34,453096,453098],{"className":453097},[31818,31819,31820,31821],[34,453099,31660],{"className":453100},[31685,31686,31821],[34,453102,31829],{"className":453103},[31828],[34,453105,453107],{"className":453106},[31801],[34,453108,453110],{"className":453109,"style":31836},[31805],[34,453111],{}," 是一个满的环同态。\n(ii) 设 ",[34,453114,453116,453144],{"className":453115},[31642],[34,453117,453119],{"className":453118},[31646],[31648,453120,453121],{"xmlns":31650},[31652,453122,453123,453141],{},[31655,453124,453125,453127,453129,453131,453133,453135,453137,453139],{},[31658,453126,54151],{},[31662,453128,112296],{},[31658,453130,98360],{},[31662,453132,456],{"stretchy":702},[31658,453134,31660],{},[31662,453136,31960],{"separator":758},[31658,453138,34386],{"mathvariant":318416},[31662,453140,2557],{"stretchy":702},[31665,453142,453143],{"encoding":31667},"A \\in M(n, \\mathbb{R})",[34,453145,453147,453165],{"className":453146,"ariaHidden":758},[31672],[34,453148,453150,453153,453156,453159,453162],{"className":453149},[31676],[34,453151],{"className":453152,"style":104901},[31680],[34,453154,54151],{"className":453155},[31685,31686],[34,453157],{"className":453158,"style":31846},[31845],[34,453160,112296],{"className":453161},[31850],[34,453163],{"className":453164,"style":31846},[31845],[34,453166,453168,453171,453174,453177,453180,453183,453186,453189],{"className":453167},[31676],[34,453169],{"className":453170,"style":31774},[31680],[34,453172,98360],{"className":453173,"style":33389},[31685,31686],[34,453175,456],{"className":453176},[31782],[34,453178,31660],{"className":453179},[31685,31686],[34,453181,31960],{"className":453182},[32119],[34,453184],{"className":453185,"style":32123},[31845],[34,453187,34386],{"className":453188},[31685,318441],[34,453190,2557],{"className":453191},[31690]," 是可逆矩阵。则 ",[34,453194,453196,453264],{"className":453195},[31642],[34,453197,453199],{"className":453198},[31646],[31648,453200,453201],{"xmlns":31650},[31652,453202,453203,453261],{},[31655,453204,453205,453207,453209,453211,453213,453215,453217,453219,453221,453223,453225,453227,453229,453231,453233,453235,453237,453239,453241,453243,453245,453247,453249,453251],{},[31658,453206,329201],{},[31662,453208,5598],{},[31658,453210,98360],{},[31662,453212,456],{"stretchy":702},[31658,453214,31660],{},[31662,453216,31960],{"separator":758},[31658,453218,34386],{"mathvariant":318416},[31662,453220,2557],{"stretchy":702},[31662,453222,33239],{},[31658,453224,98360],{},[31662,453226,456],{"stretchy":702},[31658,453228,31660],{},[31662,453230,31960],{"separator":758},[31658,453232,34386],{"mathvariant":318416},[31662,453234,2557],{"stretchy":702},[31662,453236,31960],{"separator":758},[31658,453238,329201],{},[31662,453240,456],{"stretchy":702},[31658,453242,67467],{},[31662,453244,2557],{"stretchy":702},[31662,453246,202],{},[31658,453248,54151],{},[31658,453250,67467],{},[37624,453252,453253,453255],{},[31658,453254,54151],{},[31655,453256,453257,453259],{},[31662,453258,32226],{},[31749,453260,640],{},[31665,453262,453263],{"encoding":31667},"\\phi: M(n, \\mathbb{R}) \\to M(n, \\mathbb{R}), \\phi(X) = AXA^{-1}",[34,453265,453267,453285,453321,453375],{"className":453266,"ariaHidden":758},[31672],[34,453268,453270,453273,453276,453279,453282],{"className":453269},[31676],[34,453271],{"className":453272,"style":43217},[31680],[34,453274,329201],{"className":453275},[31685,31686],[34,453277],{"className":453278,"style":31846},[31845],[34,453280,5598],{"className":453281},[31850],[34,453283],{"className":453284,"style":31846},[31845],[34,453286,453288,453291,453294,453297,453300,453303,453306,453309,453312,453315,453318],{"className":453287},[31676],[34,453289],{"className":453290,"style":31774},[31680],[34,453292,98360],{"className":453293,"style":33389},[31685,31686],[34,453295,456],{"className":453296},[31782],[34,453298,31660],{"className":453299},[31685,31686],[34,453301,31960],{"className":453302},[32119],[34,453304],{"className":453305,"style":32123},[31845],[34,453307,34386],{"className":453308},[31685,318441],[34,453310,2557],{"className":453311},[31690],[34,453313],{"className":453314,"style":31846},[31845],[34,453316,33239],{"className":453317},[31850],[34,453319],{"className":453320,"style":31846},[31845],[34,453322,453324,453327,453330,453333,453336,453339,453342,453345,453348,453351,453354,453357,453360,453363,453366,453369,453372],{"className":453323},[31676],[34,453325],{"className":453326,"style":31774},[31680],[34,453328,98360],{"className":453329,"style":33389},[31685,31686],[34,453331,456],{"className":453332},[31782],[34,453334,31660],{"className":453335},[31685,31686],[34,453337,31960],{"className":453338},[32119],[34,453340],{"className":453341,"style":32123},[31845],[34,453343,34386],{"className":453344},[31685,318441],[34,453346,2557],{"className":453347},[31690],[34,453349,31960],{"className":453350},[32119],[34,453352],{"className":453353,"style":32123},[31845],[34,453355,329201],{"className":453356},[31685,31686],[34,453358,456],{"className":453359},[31782],[34,453361,67467],{"className":453362,"style":67482},[31685,31686],[34,453364,2557],{"className":453365},[31690],[34,453367],{"className":453368,"style":31846},[31845],[34,453370,202],{"className":453371},[31850],[34,453373],{"className":453374,"style":31846},[31845],[34,453376,453378,453381,453384,453387],{"className":453377},[31676],[34,453379],{"className":453380,"style":37774},[31680],[34,453382,54151],{"className":453383},[31685,31686],[34,453385,67467],{"className":453386,"style":67482},[31685,31686],[34,453388,453390,453393],{"className":453389},[31685],[34,453391,54151],{"className":453392},[31685,31686],[34,453394,453396],{"className":453395},[31792],[34,453397,453399],{"className":453398},[31796],[34,453400,453402],{"className":453401},[31801],[34,453403,453405],{"className":453404,"style":37774},[31805],[34,453406,453407,453410],{"style":37777},[34,453408],{"className":453409,"style":31814},[31813],[34,453411,453413],{"className":453412},[31818,31819,31820,31821],[34,453414,453416,453419],{"className":453415},[31685,31821],[34,453417,32226],{"className":453418},[31685,31821],[34,453420,640],{"className":453421},[31685,31821]," 是一个环同构。称为内自同构。\n(iii) 设 ",[34,453424,453426,453439],{"className":453425},[31642],[34,453427,453429],{"className":453428},[31646],[31648,453430,453431],{"xmlns":31650},[31652,453432,453433,453437],{},[31655,453434,453435],{},[31658,453436,34386],{},[31665,453438,34386],{"encoding":31667},[34,453440,453442],{"className":453441,"ariaHidden":758},[31672],[34,453443,453445,453448],{"className":453444},[31676],[34,453446],{"className":453447,"style":33218},[31680],[34,453449,34386],{"className":453450,"style":34427},[31685,31686],[34,453452,453454,453476],{"className":453453},[31642],[34,453455,453457],{"className":453456},[31646],[31648,453458,453459],{"xmlns":31650},[31652,453460,453461,453473],{},[31655,453462,453463,453465,453467,453469,453471],{},[31662,453464,15972],{"stretchy":702},[31749,453466,459],{},[31662,453468,31960],{"separator":758},[31749,453470,640],{},[31662,453472,32712],{"stretchy":702},[31665,453474,453475],{"encoding":31667},"[0, 1]",[34,453477,453479],{"className":453478,"ariaHidden":758},[31672],[34,453480,453482,453485,453488,453491,453494,453497,453500],{"className":453481},[31676],[34,453483],{"className":453484,"style":31774},[31680],[34,453486,15972],{"className":453487},[31782],[34,453489,459],{"className":453490},[31685],[34,453492,31960],{"className":453493},[32119],[34,453495],{"className":453496,"style":32123},[31845],[34,453498,640],{"className":453499},[31685],[34,453501,32712],{"className":453502},[31690]," 上的连续实函数环， ",[34,453505,453507,453525],{"className":453506},[31642],[34,453508,453510],{"className":453509},[31646],[31648,453511,453512],{"xmlns":31650},[31652,453513,453514,453522],{},[31655,453515,453516,453518,453520],{},[31658,453517,111],{},[31662,453519,112296],{},[31658,453521,34386],{"mathvariant":318416},[31665,453523,453524],{"encoding":31667},"a \\in \\mathbb{R}",[34,453526,453528,453546],{"className":453527,"ariaHidden":758},[31672],[34,453529,453531,453534,453537,453540,453543],{"className":453530},[31676],[34,453532],{"className":453533,"style":34812},[31680],[34,453535,111],{"className":453536},[31685,31686],[34,453538],{"className":453539,"style":31846},[31845],[34,453541,112296],{"className":453542},[31850],[34,453544],{"className":453545,"style":31846},[31845],[34,453547,453549,453552],{"className":453548},[31676],[34,453550],{"className":453551,"style":320927},[31680],[34,453553,34386],{"className":453554},[31685,318441],[34,453556,453558,453600],{"className":453557},[31642],[34,453559,453561],{"className":453560},[31646],[31648,453562,453563],{"xmlns":31650},[31652,453564,453565,453597],{},[31655,453566,453567,453569,453571,453573,453575,453577,453579,453581,453583,453585,453587,453589,453591,453593,453595],{},[31658,453568,329201],{},[31662,453570,5598],{},[31658,453572,34386],{},[31662,453574,33239],{},[31658,453576,34386],{"mathvariant":318416},[31662,453578,31960],{"separator":758},[31658,453580,329201],{},[31662,453582,456],{"stretchy":702},[31658,453584,31739],{},[31662,453586,2557],{"stretchy":702},[31662,453588,202],{},[31658,453590,31739],{},[31662,453592,456],{"stretchy":702},[31658,453594,111],{},[31662,453596,2557],{"stretchy":702},[31665,453598,453599],{"encoding":31667},"\\phi: R \\to \\mathbb{R}, \\phi(f) = f(a)",[34,453601,453603,453621,453639,453675],{"className":453602,"ariaHidden":758},[31672],[34,453604,453606,453609,453612,453615,453618],{"className":453605},[31676],[34,453607],{"className":453608,"style":43217},[31680],[34,453610,329201],{"className":453611},[31685,31686],[34,453613],{"className":453614,"style":31846},[31845],[34,453616,5598],{"className":453617},[31850],[34,453619],{"className":453620,"style":31846},[31845],[34,453622,453624,453627,453630,453633,453636],{"className":453623},[31676],[34,453625],{"className":453626,"style":33218},[31680],[34,453628,34386],{"className":453629,"style":34427},[31685,31686],[34,453631],{"className":453632,"style":31846},[31845],[34,453634,33239],{"className":453635},[31850],[34,453637],{"className":453638,"style":31846},[31845],[34,453640,453642,453645,453648,453651,453654,453657,453660,453663,453666,453669,453672],{"className":453641},[31676],[34,453643],{"className":453644,"style":31774},[31680],[34,453646,34386],{"className":453647},[31685,318441],[34,453649,31960],{"className":453650},[32119],[34,453652],{"className":453653,"style":32123},[31845],[34,453655,329201],{"className":453656},[31685,31686],[34,453658,456],{"className":453659},[31782],[34,453661,31739],{"className":453662,"style":31778},[31685,31686],[34,453664,2557],{"className":453665},[31690],[34,453667],{"className":453668,"style":31846},[31845],[34,453670,202],{"className":453671},[31850],[34,453673],{"className":453674,"style":31846},[31845],[34,453676,453678,453681,453684,453687,453690],{"className":453677},[31676],[34,453679],{"className":453680,"style":31774},[31680],[34,453682,31739],{"className":453683,"style":31778},[31685,31686],[34,453685,456],{"className":453686},[31782],[34,453688,111],{"className":453689},[31685,31686],[34,453691,2557],{"className":453692},[31690],[34,453694,453696,453744],{"className":453695},[31642],[34,453697,453699],{"className":453698},[31646],[31648,453700,453701],{"xmlns":31650},[31652,453702,453703,453741],{},[31655,453704,453705,453707,453709,453711,453713,453715,453717,453719,453721,453723,453725,453727,453729,453731,453733,453735,453737,453739],{},[31658,453706,406335],{},[31662,453708,32682],{},[31658,453710,329201],{},[31662,453712,202],{},[31662,453714,32685],{"stretchy":702},[31658,453716,31739],{},[31662,453718,112296],{},[31658,453720,34386],{},[31662,453722,162086],{},[31658,453724,31739],{},[31662,453726,456],{"stretchy":702},[31658,453728,111],{},[31662,453730,2557],{"stretchy":702},[31662,453732,202],{},[31749,453734,459],{},[31662,453736,32715],{"stretchy":702},[31662,453738,440828],{},[31658,453740,34386],{},[31665,453742,453743],{"encoding":31667},"\\ker \\phi = \\{f \\in R \\mid f(a) = 0\\} \\lhd R",[34,453745,453747,453771,453792,453810,453837,453858],{"className":453746,"ariaHidden":758},[31672],[34,453748,453750,453753,453756,453759,453762,453765,453768],{"className":453749},[31676],[34,453751],{"className":453752,"style":43217},[31680],[34,453754,406335],{"className":453755},[32767],[34,453757],{"className":453758,"style":32123},[31845],[34,453760,329201],{"className":453761},[31685,31686],[34,453763],{"className":453764,"style":31846},[31845],[34,453766,202],{"className":453767},[31850],[34,453769],{"className":453770,"style":31846},[31845],[34,453772,453774,453777,453780,453783,453786,453789],{"className":453773},[31676],[34,453775],{"className":453776,"style":31774},[31680],[34,453778,32685],{"className":453779},[31782],[34,453781,31739],{"className":453782,"style":31778},[31685,31686],[34,453784],{"className":453785,"style":31846},[31845],[34,453787,112296],{"className":453788},[31850],[34,453790],{"className":453791,"style":31846},[31845],[34,453793,453795,453798,453801,453804,453807],{"className":453794},[31676],[34,453796],{"className":453797,"style":31774},[31680],[34,453799,34386],{"className":453800,"style":34427},[31685,31686],[34,453802],{"className":453803,"style":31846},[31845],[34,453805,162086],{"className":453806},[31850],[34,453808],{"className":453809,"style":31846},[31845],[34,453811,453813,453816,453819,453822,453825,453828,453831,453834],{"className":453812},[31676],[34,453814],{"className":453815,"style":31774},[31680],[34,453817,31739],{"className":453818,"style":31778},[31685,31686],[34,453820,456],{"className":453821},[31782],[34,453823,111],{"className":453824},[31685,31686],[34,453826,2557],{"className":453827},[31690],[34,453829],{"className":453830,"style":31846},[31845],[34,453832,202],{"className":453833},[31850],[34,453835],{"className":453836,"style":31846},[31845],[34,453838,453840,453843,453846,453849,453852,453855],{"className":453839},[31676],[34,453841],{"className":453842,"style":31774},[31680],[34,453844,459],{"className":453845},[31685],[34,453847,32715],{"className":453848},[31690],[34,453850],{"className":453851,"style":32604},[31845],[34,453853,440828],{"className":453854},[32024,129261],[34,453856],{"className":453857,"style":32604},[31845],[34,453859,453861,453864],{"className":453860},[31676],[34,453862],{"className":453863,"style":33218},[31680],[34,453865,34386],{"className":453866,"style":34427},[31685,31686],". 特别的， ",[34,453869,453871,453899],{"className":453870},[31642],[34,453872,453874],{"className":453873},[31646],[31648,453875,453876],{"xmlns":31650},[31652,453877,453878,453896],{},[31655,453879,453880,453882,453884,453886,453888,453890,453892,453894],{},[31658,453881,365679],{"mathvariant":38079},[31658,453883,94675],{},[31662,453885,131855],{},[31662,453887,15972],{"stretchy":702},[31749,453889,459],{},[31662,453891,31960],{"separator":758},[31749,453893,640],{},[31662,453895,32712],{"stretchy":702},[31665,453897,453898],{"encoding":31667},"\\forall S \\subseteq [0, 1]",[34,453900,453902,453923],{"className":453901,"ariaHidden":758},[31672],[34,453903,453905,453908,453911,453914,453917,453920],{"className":453904},[31676],[34,453906],{"className":453907,"style":47709},[31680],[34,453909,365679],{"className":453910},[31685],[34,453912,94675],{"className":453913,"style":106455},[31685,31686],[34,453915],{"className":453916,"style":31846},[31845],[34,453918,131855],{"className":453919},[31850],[34,453921],{"className":453922,"style":31846},[31845],[34,453924,453926,453929,453932,453935,453938,453941,453944],{"className":453925},[31676],[34,453927],{"className":453928,"style":31774},[31680],[34,453930,15972],{"className":453931},[31782],[34,453933,459],{"className":453934},[31685],[34,453936,31960],{"className":453937},[32119],[34,453939],{"className":453940,"style":32123},[31845],[34,453942,640],{"className":453943},[31685],[34,453945,32712],{"className":453946},[31690],[34,453948,453950,454000],{"className":453949},[31642],[34,453951,453953],{"className":453952},[31646],[31648,453954,453955],{"xmlns":31650},[31652,453956,453957,453997],{},[31655,453958,453959,453965,453967,453969,453971,453973,453975,453977,453979,453981,453983,453985,453987,453989,453991,453993,453995],{},[31743,453960,453961,453963],{},[31658,453962,106063],{},[31658,453964,94675],{},[31662,453966,202],{},[31662,453968,32685],{"stretchy":702},[31658,453970,31739],{},[31662,453972,112296],{},[31658,453974,34386],{},[31662,453976,162086],{},[31658,453978,31739],{},[31662,453980,456],{"stretchy":702},[31658,453982,94675],{},[31662,453984,2557],{"stretchy":702},[31662,453986,202],{},[31749,453988,459],{},[31662,453990,32715],{"stretchy":702},[31662,453992,440828],{},[31658,453994,34386],{},[31658,453996,2547],{"mathvariant":38079},[31665,453998,453999],{"encoding":31667},"I_S = \\{f \\in R \\mid f(S) = 0\\} \\lhd R.",[34,454001,454003,454058,454079,454097,454124,454145],{"className":454002,"ariaHidden":758},[31672],[34,454004,454006,454009,454049,454052,454055],{"className":454005},[31676],[34,454007],{"className":454008,"style":70915},[31680],[34,454010,454012,454015],{"className":454011},[31685],[34,454013,106063],{"className":454014,"style":67482},[31685,31686],[34,454016,454018],{"className":454017},[31792],[34,454019,454021,454041],{"className":454020},[31796,31797],[34,454022,454024,454038],{"className":454023},[31801],[34,454025,454027],{"className":454026,"style":116558},[31805],[34,454028,454029,454032],{"style":217742},[34,454030],{"className":454031,"style":31814},[31813],[34,454033,454035],{"className":454034},[31818,31819,31820,31821],[34,454036,94675],{"className":454037,"style":106455},[31685,31686,31821],[34,454039,31829],{"className":454040},[31828],[34,454042,454044],{"className":454043},[31801],[34,454045,454047],{"className":454046,"style":31836},[31805],[34,454048],{},[34,454050],{"className":454051,"style":31846},[31845],[34,454053,202],{"className":454054},[31850],[34,454056],{"className":454057,"style":31846},[31845],[34,454059,454061,454064,454067,454070,454073,454076],{"className":454060},[31676],[34,454062],{"className":454063,"style":31774},[31680],[34,454065,32685],{"className":454066},[31782],[34,454068,31739],{"className":454069,"style":31778},[31685,31686],[34,454071],{"className":454072,"style":31846},[31845],[34,454074,112296],{"className":454075},[31850],[34,454077],{"className":454078,"style":31846},[31845],[34,454080,454082,454085,454088,454091,454094],{"className":454081},[31676],[34,454083],{"className":454084,"style":31774},[31680],[34,454086,34386],{"className":454087,"style":34427},[31685,31686],[34,454089],{"className":454090,"style":31846},[31845],[34,454092,162086],{"className":454093},[31850],[34,454095],{"className":454096,"style":31846},[31845],[34,454098,454100,454103,454106,454109,454112,454115,454118,454121],{"className":454099},[31676],[34,454101],{"className":454102,"style":31774},[31680],[34,454104,31739],{"className":454105,"style":31778},[31685,31686],[34,454107,456],{"className":454108},[31782],[34,454110,94675],{"className":454111,"style":106455},[31685,31686],[34,454113,2557],{"className":454114},[31690],[34,454116],{"className":454117,"style":31846},[31845],[34,454119,202],{"className":454120},[31850],[34,454122],{"className":454123,"style":31846},[31845],[34,454125,454127,454130,454133,454136,454139,454142],{"className":454126},[31676],[34,454128],{"className":454129,"style":31774},[31680],[34,454131,459],{"className":454132},[31685],[34,454134,32715],{"className":454135},[31690],[34,454137],{"className":454138,"style":32604},[31845],[34,454140,440828],{"className":454141},[32024,129261],[34,454143],{"className":454144,"style":32604},[31845],[34,454146,454148,454151,454154],{"className":454147},[31676],[34,454149],{"className":454150,"style":33218},[31680],[34,454152,34386],{"className":454153,"style":34427},[31685,31686],[34,454155,2547],{"className":454156},[31685],[32188,454158],{},[253,454160,454162],{"id":454161},"_5-理想的互素与积","5. 理想的互素与积",[15,454164,454165,69123,454168,454232,454233,454306,454307,47649,454349,347],{},[2627,454166,454167],{},"定义 3.3.2.",[34,454169,454171,454193],{"className":454170},[31642],[34,454172,454174],{"className":454173},[31646],[31648,454175,454176],{"xmlns":31650},[31652,454177,454178,454190],{},[31655,454179,454180,454182,454184,454186,454188],{},[31658,454181,106063],{},[31662,454183,31960],{"separator":758},[31658,454185,223320],{},[31662,454187,440828],{},[31658,454189,34386],{},[31665,454191,454192],{"encoding":31667},"I, J \\lhd R",[34,454194,454196,454223],{"className":454195,"ariaHidden":758},[31672],[34,454197,454199,454202,454205,454208,454211,454214,454217,454220],{"className":454198},[31676],[34,454200],{"className":454201,"style":70739},[31680],[34,454203,106063],{"className":454204,"style":67482},[31685,31686],[34,454206,31960],{"className":454207},[32119],[34,454209],{"className":454210,"style":32123},[31845],[34,454212,223320],{"className":454213,"style":223361},[31685,31686],[34,454215],{"className":454216,"style":32604},[31845],[34,454218,440828],{"className":454219},[32024,129261],[34,454221],{"className":454222,"style":32604},[31845],[34,454224,454226,454229],{"className":454225},[31676],[34,454227],{"className":454228,"style":33218},[31680],[34,454230,34386],{"className":454231,"style":34427},[31685,31686],". 如果 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则称 ",[34,454308,454310,454328],{"className":454309},[31642],[34,454311,454313],{"className":454312},[31646],[31648,454314,454315],{"xmlns":31650},[31652,454316,454317,454325],{},[31655,454318,454319,454321,454323],{},[31658,454320,106063],{},[31662,454322,31960],{"separator":758},[31658,454324,223320],{},[31665,454326,454327],{"encoding":31667},"I, J",[34,454329,454331],{"className":454330,"ariaHidden":758},[31672],[34,454332,454334,454337,454340,454343,454346],{"className":454333},[31676],[34,454335],{"className":454336,"style":70739},[31680],[34,454338,106063],{"className":454339,"style":67482},[31685,31686],[34,454341,31960],{"className":454342},[32119],[34,454344],{"className":454345,"style":32123},[31845],[34,454347,223320],{"className":454348,"style":223361},[31685,31686],[2627,454350,454351],{},"互素",[15,454353,454354,69123,454357,454385,454386,445351,454449,454490,454491,454577,454578,445571,454651,454653,439313,454681,93389,454753,454781,454782,454877,454878,315343,454963,39984,455036,455064,455065,454577,455150,445351,455222,455295,455296,64615,455346,354330,455463,455514,455515,53141,455588,455660,455661,455702,455703],{},[2627,454355,454356],{},"命题 3.3.3.",[34,454358,454360,454373],{"className":454359},[31642],[34,454361,454363],{"className":454362},[31646],[31648,454364,454365],{"xmlns":31650},[31652,454366,454367,454371],{},[31655,454368,454369],{},[31658,454370,34386],{},[31665,454372,34386],{"encoding":31667},[34,454374,454376],{"className":454375,"ariaHidden":758},[31672],[34,454377,454379,454382],{"className":454378},[31676],[34,454380],{"className":454381,"style":33218},[31680],[34,454383,34386],{"className":454384,"style":34427},[31685,31686]," 是幺环, ",[34,454387,454389,454410],{"className":454388},[31642],[34,454390,454392],{"className":454391},[31646],[31648,454393,454394],{"xmlns":31650},[31652,454395,454396,454408],{},[31655,454397,454398,454400,454402,454404,454406],{},[31658,454399,106063],{},[31662,454401,31960],{"separator":758},[31658,454403,223320],{},[31662,454405,440828],{},[31658,454407,34386],{},[31665,454409,454192],{"encoding":31667},[34,454411,454413,454440],{"className":454412,"ariaHidden":758},[31672],[34,454414,454416,454419,454422,454425,454428,454431,454434,454437],{"className":454415},[31676],[34,454417],{"className":454418,"style":70739},[31680],[34,454420,106063],{"className":454421,"style":67482},[31685,31686],[34,454423,31960],{"className":454424},[32119],[34,454426],{"className":454427,"style":32123},[31845],[34,454429,223320],{"className":454430,"style":223361},[31685,31686],[34,454432],{"className":454433,"style":32604},[31845],[34,454435,440828],{"className":454436},[32024,129261],[34,454438],{"className":454439,"style":32604},[31845],[34,454441,454443,454446],{"className":454442},[31676],[34,454444],{"className":454445,"style":33218},[31680],[34,454447,34386],{"className":454448,"style":34427},[31685,31686],[34,454450,454452,454469],{"className":454451},[31642],[34,454453,454455],{"className":454454},[31646],[31648,454456,454457],{"xmlns":31650},[31652,454458,454459,454467],{},[31655,454460,454461,454463,454465],{},[31658,454462,106063],{},[31662,454464,31960],{"separator":758},[31658,454466,223320],{},[31665,454468,454327],{"encoding":31667},[34,454470,454472],{"className":454471,"ariaHidden":758},[31672],[34,454473,454475,454478,454481,454484,454487],{"className":454474},[31676],[34,454476],{"className":454477,"style":70739},[31680],[34,454479,106063],{"className":454480,"style":67482},[31685,31686],[34,454482,31960],{"className":454483},[32119],[34,454485],{"className":454486,"style":32123},[31845],[34,454488,223320],{"className":454489,"style":223361},[31685,31686]," 互素，当且仅当存在 ",[34,454492,454494,454520],{"className":454493},[31642],[34,454495,454497],{"className":454496},[31646],[31648,454498,454499],{"xmlns":31650},[31652,454500,454501,454517],{},[31655,454502,454503,454505,454507,454509,454511,454513,454515],{},[31658,454504,31955],{},[31662,454506,112296],{},[31658,454508,106063],{},[31662,454510,31960],{"separator":758},[31658,454512,31968],{},[31662,454514,112296],{},[31658,454516,223320],{},[31665,454518,454519],{"encoding":31667},"i \\in I, j \\in J",[34,454521,454523,454541,454568],{"className":454522,"ariaHidden":758},[31672],[34,454524,454526,454529,454532,454535,454538],{"className":454525},[31676],[34,454527],{"className":454528,"style":451673},[31680],[34,454530,31955],{"className":454531},[31685,31686],[34,454533],{"className":454534,"style":31846},[31845],[34,454536,112296],{"className":454537},[31850],[34,454539],{"className":454540,"style":31846},[31845],[34,454542,454544,454547,454550,454553,454556,454559,454562,454565],{"className":454543},[31676],[34,454545],{"className":454546,"style":70739},[31680],[34,454548,106063],{"className":454549,"style":67482},[31685,31686],[34,454551,31960],{"className":454552},[32119],[34,454554],{"className":454555,"style":32123},[31845],[34,454557,31968],{"className":454558,"style":32153},[31685,31686],[34,454560],{"className":454561,"style":31846},[31845],[34,454563,112296],{"className":454564},[31850],[34,454566],{"className":454567,"style":31846},[31845],[34,454569,454571,454574],{"className":454570},[31676],[34,454572],{"className":454573,"style":33218},[31680],[34,454575,223320],{"className":454576,"style":223361},[31685,31686],", 使得 ",[34,454579,454581,454603],{"className":454580},[31642],[34,454582,454584],{"className":454583},[31646],[31648,454585,454586],{"xmlns":31650},[31652,454587,454588,454600],{},[31655,454589,454590,454592,454594,454596,454598],{},[31749,454591,640],{},[31662,454593,202],{},[31658,454595,31955],{},[31662,454597,223],{},[31658,454599,31968],{},[31665,454601,454602],{"encoding":31667},"1 = 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是幺环，",[34,454783,454785,454811],{"className":454784},[31642],[34,454786,454788],{"className":454787},[31646],[31648,454789,454790],{"xmlns":31650},[31652,454791,454792,454808],{},[31655,454793,454794,454796,454798,454800,454802,454804,454806],{},[31749,454795,640],{},[31662,454797,112296],{},[31658,454799,34386],{},[31662,454801,202],{},[31658,454803,106063],{},[31662,454805,223],{},[31658,454807,223320],{},[31665,454809,454810],{"encoding":31667},"1 \\in R = I+J",[34,454812,454814,454832,454850,454868],{"className":454813,"ariaHidden":758},[31672],[34,454815,454817,454820,454823,454826,454829],{"className":454816},[31676],[34,454818],{"className":454819,"style":61814},[31680],[34,454821,640],{"className":454822},[31685],[34,454824],{"className":454825,"style":31846},[31845],[34,454827,112296],{"className":454828},[31850],[34,454830],{"className":454831,"style":31846},[31845],[34,454833,454835,454838,454841,454844,454847],{"className":454834},[31676],[34,454836],{"className":454837,"style":33218},[31680],[34,454839,34386],{"className":454840,"style":34427},[31685,31686],[34,454842],{"className":454843,"style":31846},[31845],[34,454845,202],{"className":454846},[31850],[34,454848],{"className":454849,"style":31846},[31845],[34,454851,454853,454856,454859,454862,454865],{"className":454852},[31676],[34,454854],{"className":454855,"style":34944},[31680],[34,454857,106063],{"className":454858,"style":67482},[31685,31686],[34,454860],{"className":454861,"style":32604},[31845],[34,454863,223],{"className":454864},[32024],[34,454866],{"className":454867,"style":32604},[31845],[34,454869,454871,454874],{"className":454870},[31676],[34,454872],{"className":454873,"style":33218},[31680],[34,454875,223320],{"className":454876,"style":223361},[31685,31686],"。故存在 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若存在 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\\in I+J",[34,455248,455250,455268,455286],{"className":455249,"ariaHidden":758},[31672],[34,455251,455253,455256,455259,455262,455265],{"className":455252},[31676],[34,455254],{"className":455255,"style":61814},[31680],[34,455257,640],{"className":455258},[31685],[34,455260],{"className":455261,"style":31846},[31845],[34,455263,112296],{"className":455264},[31850],[34,455266],{"className":455267,"style":31846},[31845],[34,455269,455271,455274,455277,455280,455283],{"className":455270},[31676],[34,455272],{"className":455273,"style":34944},[31680],[34,455275,106063],{"className":455276,"style":67482},[31685,31686],[34,455278],{"className":455279,"style":32604},[31845],[34,455281,223],{"className":455282},[32024],[34,455284],{"className":455285,"style":32604},[31845],[34,455287,455289,455292],{"className":455288},[31676],[34,455290],{"className":455291,"style":33218},[31680],[34,455293,223320],{"className":455294,"style":223361},[31685,31686],". 对于任意 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= r \\cdot 1 \\in 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是理想）。因此 ",[34,455516,455518,455540],{"className":455517},[31642],[34,455519,455521],{"className":455520},[31646],[31648,455522,455523],{"xmlns":31650},[31652,455524,455525,455537],{},[31655,455526,455527,455529,455531,455533,455535],{},[31658,455528,34386],{},[31662,455530,131855],{},[31658,455532,106063],{},[31662,455534,223],{},[31658,455536,223320],{},[31665,455538,455539],{"encoding":31667},"R \\subseteq I+J",[34,455541,455543,455561,455579],{"className":455542,"ariaHidden":758},[31672],[34,455544,455546,455549,455552,455555,455558],{"className":455545},[31676],[34,455547],{"className":455548,"style":195047},[31680],[34,455550,34386],{"className":455551,"style":34427},[31685,31686],[34,455553],{"className":455554,"style":31846},[31845],[34,455556,131855],{"className":455557},[31850],[34,455559],{"className":455560,"style":31846},[31845],[34,455562,455564,455567,455570,455573,455576],{"className":455563},[31676],[34,455565],{"className":455566,"style":34944},[31680],[34,455568,106063],{"className":455569,"style":67482},[31685,31686],[34,455571],{"className":455572,"style":32604},[31845],[34,455574,223],{"className":455575},[32024],[34,455577],{"className":455578,"style":32604},[31845],[34,455580,455582,455585],{"className":455581},[31676],[34,455583],{"className":455584,"style":33218},[31680],[34,455586,223320],{"className":455587,"style":223361},[31685,31686],[34,455589,455591,455612],{"className":455590},[31642],[34,455592,455594],{"className":455593},[31646],[31648,455595,455596],{"xmlns":31650},[31652,455597,455598,455610],{},[31655,455599,455600,455602,455604,455606,455608],{},[31658,455601,106063],{},[31662,455603,223],{},[31658,455605,223320],{},[31662,455607,202],{},[31658,455609,34386],{},[31665,455611,454257],{"encoding":31667},[34,455613,455615,455633,455651],{"className":455614,"ariaHidden":758},[31672],[34,455616,455618,455621,455624,455627,455630],{"className":455617},[31676],[34,455619],{"className":455620,"style":34944},[31680],[34,455622,106063],{"className":455623,"style":67482},[31685,31686],[34,455625],{"className":455626,"style":32604},[31845],[34,455628,223],{"className":455629},[32024],[34,455631],{"className":455632,"style":32604},[31845],[34,455634,455636,455639,455642,455645,455648],{"className":455635},[31676],[34,455637],{"className":455638,"style":33218},[31680],[34,455640,223320],{"className":455641,"style":223361},[31685,31686],[34,455643],{"className":455644,"style":31846},[31845],[34,455646,202],{"className":455647},[31850],[34,455649],{"className":455650,"style":31846},[31845],[34,455652,455654,455657],{"className":455653},[31676],[34,455655],{"className":455656,"style":33218},[31680],[34,455658,34386],{"className":455659,"style":34427},[31685,31686],", 即 ",[34,455662,455664,455681],{"className":455663},[31642],[34,455665,455667],{"className":455666},[31646],[31648,455668,455669],{"xmlns":31650},[31652,455670,455671,455679],{},[31655,455672,455673,455675,455677],{},[31658,455674,106063],{},[31662,455676,31960],{"separator":758},[31658,455678,223320],{},[31665,455680,454327],{"encoding":31667},[34,455682,455684],{"className":455683,"ariaHidden":758},[31672],[34,455685,455687,455690,455693,455696,455699],{"className":455686},[31676],[34,455688],{"className":455689,"style":70739},[31680],[34,455691,106063],{"className":455692,"style":67482},[31685,31686],[34,455694,31960],{"className":455695},[32119],[34,455697],{"className":455698,"style":32123},[31845],[34,455700,223320],{"className":455701,"style":223361},[31685,31686]," 互素。 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R",[34,455775,455777,455890],{"className":455776,"ariaHidden":758},[31672],[34,455778,455780,455783,455823,455826,455829,455832,455835,455838,455841,455881,455884,455887],{"className":455779},[31676],[34,455781],{"className":455782,"style":70739},[31680],[34,455784,455786,455789],{"className":455785},[31685],[34,455787,106063],{"className":455788,"style":67482},[31685,31686],[34,455790,455792],{"className":455791},[31792],[34,455793,455795,455815],{"className":455794},[31796,31797],[34,455796,455798,455812],{"className":455797},[31801],[34,455799,455801],{"className":455800,"style":31806},[31805],[34,455802,455803,455806],{"style":217742},[34,455804],{"className":455805,"style":31814},[31813],[34,455807,455809],{"className":455808},[31818,31819,31820,31821],[34,455810,640],{"className":455811},[31685,31821],[34,455813,31829],{"className":455814},[31828],[34,455816,455818],{"className":455817},[31801],[34,455819,455821],{"className":455820,"style":31836},[31805],[34,455822],{},[34,455824,31960],{"className":455825},[32119],[34,455827],{"className":455828,"style":32123},[31845],[34,455830,33934],{"className":455831},[34006],[34,455833],{"className":455834,"style":32123},[31845],[34,455836,31960],{"className":455837},[32119],[34,455839],{"className":455840,"style":32123},[31845],[34,455842,455844,455847],{"className":455843},[31685],[34,455845,106063],{"className":455846,"style":67482},[31685,31686],[34,455848,455850],{"className":455849},[31792],[34,455851,455853,455873],{"className":455852},[31796,31797],[34,455854,455856,455870],{"className":455855},[31801],[34,455857,455859],{"className":455858,"style":33833},[31805],[34,455860,455861,455864],{"style":217742},[34,455862],{"className":455863,"style":31814},[31813],[34,455865,455867],{"className":455866},[31818,31819,31820,31821],[34,455868,33759],{"className":455869,"style":33774},[31685,31686,31821],[34,455871,31829],{"className":455872},[31828],[34,455874,455876],{"className":455875},[31801],[34,455877,455879],{"className":455878,"style":31836},[31805],[34,455880],{},[34,455882],{"className":455883,"style":32604},[31845],[34,455885,440828],{"className":455886},[32024,129261],[34,455888],{"className":455889,"style":32604},[31845],[34,455891,455893,455896],{"className":455892},[31676],[34,455894],{"className":455895,"style":33218},[31680],[34,455897,34386],{"className":455898,"style":34427},[31685,31686],", 定义\n",[34,455901,455903,455963],{"className":455902},[31642],[34,455904,455906],{"className":455905},[31646],[31648,455907,455908],{"xmlns":31650},[31652,455909,455910,455960],{},[31655,455911,455912,455918,455920,455926,455928,455930,455936,455938,455950,455952,455958],{},[31743,455913,455914,455916],{},[31658,455915,106063],{},[31749,455917,640],{},[31662,455919,38874],{},[31743,455921,455922,455924],{},[31658,455923,106063],{},[31658,455925,33759],{},[31662,455927,202],{},[31662,455929,456],{"stretchy":702},[31743,455931,455932,455934],{},[31658,455933,106063],{},[31749,455935,640],{},[31662,455937,38874],{},[31743,455939,455940,455942],{},[31658,455941,106063],{},[31655,455943,455944,455946,455948],{},[31658,455945,33759],{},[31662,455947,32226],{},[31749,455949,640],{},[31662,455951,2557],{"stretchy":702},[31743,455953,455954,455956],{},[31658,455955,106063],{},[31658,455957,33759],{},[31658,455959,2547],{"mathvariant":38079},[31665,455961,455962],{"encoding":31667},"I_1 \\cdots I_k = (I_1 \\cdots I_{k-1})I_k.",[34,455964,455966,456070],{"className":455965,"ariaHidden":758},[31672],[34,455967,455969,455972,456012,456015,456018,456021,456061,456064,456067],{"className":455968},[31676],[34,455970],{"className":455971,"style":70915},[31680],[34,455973,455975,455978],{"className":455974},[31685],[34,455976,106063],{"className":455977,"style":67482},[31685,31686],[34,455979,455981],{"className":455980},[31792],[34,455982,455984,456004],{"className":455983},[31796,31797],[34,455985,455987,456001],{"className":455986},[31801],[34,455988,455990],{"className":455989,"style":31806},[31805],[34,455991,455992,455995],{"style":217742},[34,455993],{"className":455994,"style":31814},[31813],[34,455996,455998],{"className":455997},[31818,31819,31820,31821],[34,455999,640],{"className":456000},[31685,31821],[34,456002,31829],{"className":456003},[31828],[34,456005,456007],{"className":456006},[31801],[34,456008,456010],{"className":456009,"style":31836},[31805],[34,456011],{},[34,456013],{"className":456014,"style":32123},[31845],[34,456016,38874],{"className":456017},[34006],[34,456019],{"className":456020,"style":32123},[31845],[34,456022,456024,456027],{"className":456023},[31685],[34,456025,106063],{"className":456026,"style":67482},[31685,31686],[34,456028,456030],{"className":456029},[31792],[34,456031,456033,456053],{"className":456032},[31796,31797],[34,456034,456036,456050],{"className":456035},[31801],[34,456037,456039],{"className":456038,"style":33833},[31805],[34,456040,456041,456044],{"style":217742},[34,456042],{"className":456043,"style":31814},[31813],[34,456045,456047],{"className":456046},[31818,31819,31820,31821],[34,456048,33759],{"className":456049,"style":33774},[31685,31686,31821],[34,456051,31829],{"className":456052},[31828],[34,456054,456056],{"className":456055},[31801],[34,456057,456059],{"className":456058,"style":31836},[31805],[34,456060],{},[34,456062],{"className":456063,"style":31846},[31845],[34,456065,202],{"className":456066},[31850],[34,456068],{"className":456069,"style":31846},[31845],[34,456071,456073,456076,456079,456119,456122,456125,456128,456177,456180,456220],{"className":456072},[31676],[34,456074],{"className":456075,"style":31774},[31680],[34,456077,456],{"className":456078},[31782],[34,456080,456082,456085],{"className":456081},[31685],[34,456083,106063],{"className":456084,"style":67482},[31685,31686],[34,456086,456088],{"className":456087},[31792],[34,456089,456091,456111],{"className":456090},[31796,31797],[34,456092,456094,456108],{"className":456093},[31801],[34,456095,456097],{"className":456096,"style":31806},[31805],[34,456098,456099,456102],{"style":217742},[34,456100],{"className":456101,"style":31814},[31813],[34,456103,456105],{"className":456104},[31818,31819,31820,31821],[34,456106,640],{"className":456107},[31685,31821],[34,456109,31829],{"className":456110},[31828],[34,456112,456114],{"className":456113},[31801],[34,456115,456117],{"className":456116,"style":31836},[31805],[34,456118],{},[34,456120],{"className":456121,"style":32123},[31845],[34,456123,38874],{"className":456124},[34006],[34,456126],{"className":456127,"style":32123},[31845],[34,456129,456131,456134],{"className":456130},[31685],[34,456132,106063],{"className":456133,"style":67482},[31685,31686],[34,456135,456137],{"className":456136},[31792],[34,456138,456140,456169],{"className":456139},[31796,31797],[34,456141,456143,456166],{"className":456142},[31801],[34,456144,456146],{"className":456145,"style":33833},[31805],[34,456147,456148,456151],{"style":217742},[34,456149],{"className":456150,"style":31814},[31813],[34,456152,456154],{"className":456153},[31818,31819,31820,31821],[34,456155,456157,456160,456163],{"className":456156},[31685,31821],[34,456158,33759],{"className":456159,"style":33774},[31685,31686,31821],[34,456161,32226],{"className":456162},[32024,31821],[34,456164,640],{"className":456165},[31685,31821],[34,456167,31829],{"className":456168},[31828],[34,456170,456172],{"className":456171},[31801],[34,456173,456175],{"className":456174,"style":32037},[31805],[34,456176],{},[34,456178,2557],{"className":456179},[31690],[34,456181,456183,456186],{"className":456182},[31685],[34,456184,106063],{"className":456185,"style":67482},[31685,31686],[34,456187,456189],{"className":456188},[31792],[34,456190,456192,456212],{"className":456191},[31796,31797],[34,456193,456195,456209],{"className":456194},[31801],[34,456196,456198],{"className":456197,"style":33833},[31805],[34,456199,456200,456203],{"style":217742},[34,456201],{"className":456202,"style":31814},[31813],[34,456204,456206],{"className":456205},[31818,31819,31820,31821],[34,456207,33759],{"className":456208,"style":33774},[31685,31686,31821],[34,456210,31829],{"className":456211},[31828],[34,456213,456215],{"className":456214},[31801],[34,456216,456218],{"className":456217,"style":31836},[31805],[34,456219],{},[34,456221,2547],{"className":456222},[31685],[71807,456224,456225,456226,456260,456261,1840,456385,456553],{},"(注：理想的积 ",[34,456227,456229,456245],{"className":456228},[31642],[34,456230,456232],{"className":456231},[31646],[31648,456233,456234],{"xmlns":31650},[31652,456235,456236,456242],{},[31655,456237,456238,456240],{},[31658,456239,106063],{},[31658,456241,223320],{},[31665,456243,456244],{"encoding":31667},"IJ",[34,456246,456248],{"className":456247,"ariaHidden":758},[31672],[34,456249,456251,456254,456257],{"className":456250},[31676],[34,456252],{"className":456253,"style":33218},[31680],[34,456255,106063],{"className":456256,"style":67482},[31685,31686],[34,456258,223320],{"className":456259,"style":223361},[31685,31686]," 是由所有形式为 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b_k",[34,456291,456293],{"className":456292,"ariaHidden":758},[31672],[34,456294,456296,456299,456302,456305,456345],{"className":456295},[31676],[34,456297],{"className":456298,"style":31774},[31680],[34,456300,131138],{"className":456301,"style":156561},[32767,131249,156560],[34,456303],{"className":456304,"style":32123},[31845],[34,456306,456308,456311],{"className":456307},[31685],[34,456309,111],{"className":456310},[31685,31686],[34,456312,456314],{"className":456313},[31792],[34,456315,456317,456337],{"className":456316},[31796,31797],[34,456318,456320,456334],{"className":456319},[31801],[34,456321,456323],{"className":456322,"style":33833},[31805],[34,456324,456325,456328],{"style":31809},[34,456326],{"className":456327,"style":31814},[31813],[34,456329,456331],{"className":456330},[31818,31819,31820,31821],[34,456332,33759],{"className":456333,"style":33774},[31685,31686,31821],[34,456335,31829],{"className":456336},[31828],[34,456338,456340],{"className":456339},[31801],[34,456341,456343],{"className":456342,"style":31836},[31805],[34,456344],{},[34,456346,456348,456351],{"className":456347},[31685],[34,456349,1697],{"className":456350},[31685,31686],[34,456352,456354],{"className":456353},[31792],[34,456355,456357,456377],{"className":456356},[31796,31797],[34,456358,456360,456374],{"className":456359},[31801],[34,456361,456363],{"className":456362,"style":33833},[31805],[34,456364,456365,456368],{"style":31809},[34,456366],{"className":456367,"style":31814},[31813],[34,456369,456371],{"className":456370},[31818,31819,31820,31821],[34,456372,33759],{"className":456373,"style":33774},[31685,31686,31821],[34,456375,31829],{"className":456376},[31828],[34,456378,456380],{"className":456379},[31801],[34,456381,456383],{"className":456382,"style":31836},[31805],[34,456384],{},[34,456386,456388,456422],{"className":456387},[31642],[34,456389,456391],{"className":456390},[31646],[31648,456392,456393],{"xmlns":31650},[31652,456394,456395,456419],{},[31655,456396,456397,456403,456405,456407,456409,456415,456417],{},[31743,456398,456399,456401],{},[31658,456400,111],{},[31658,456402,33759],{},[31662,456404,112296],{},[31658,456406,106063],{},[31662,456408,31960],{"separator":758},[31743,456410,456411,456413],{},[31658,456412,1697],{},[31658,456414,33759],{},[31662,456416,112296],{},[31658,456418,223320],{},[31665,456420,456421],{"encoding":31667},"a_k 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的有限和构成的集合，而不仅仅是元素乘积的集合。)",[15,456555,456556,69123,456559,456587,456588,456651,456652,456693,456694,445571,456799,456801,456802,456880,456881,456693,456922,456994,456995,315343,457080,445571,457152,457230,457231,315440,457422,457537,457538,457680,457681,361499,457742,457852,457853,457931,457932,452957,458010],{},[2627,456557,456558],{},"命题 3.3.4.",[34,456560,456562,456575],{"className":456561},[31642],[34,456563,456565],{"className":456564},[31646],[31648,456566,456567],{"xmlns":31650},[31652,456568,456569,456573],{},[31655,456570,456571],{},[31658,456572,34386],{},[31665,456574,34386],{"encoding":31667},[34,456576,456578],{"className":456577,"ariaHidden":758},[31672],[34,456579,456581,456584],{"className":456580},[31676],[34,456582],{"className":456583,"style":33218},[31680],[34,456585,34386],{"className":456586,"style":34427},[31685,31686]," 是交换幺环, 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若 ",[34,456653,456655,456672],{"className":456654},[31642],[34,456656,456658],{"className":456657},[31646],[31648,456659,456660],{"xmlns":31650},[31652,456661,456662,456670],{},[31655,456663,456664,456666,456668],{},[31658,456665,106063],{},[31662,456667,31960],{"separator":758},[31658,456669,223320],{},[31665,456671,454327],{"encoding":31667},[34,456673,456675],{"className":456674,"ariaHidden":758},[31672],[34,456676,456678,456681,456684,456687,456690],{"className":456677},[31676],[34,456679],{"className":456680,"style":70739},[31680],[34,456682,106063],{"className":456683,"style":67482},[31685,31686],[34,456685,31960],{"className":456686},[32119],[34,456688],{"className":456689,"style":32123},[31845],[34,456691,223320],{"className":456692,"style":223361},[31685,31686]," 互素，则 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JI",[34,456728,456730,456751,456769,456787],{"className":456729,"ariaHidden":758},[31672],[34,456731,456733,456736,456739,456742,456745,456748],{"className":456732},[31676],[34,456734],{"className":456735,"style":33218},[31680],[34,456737,106063],{"className":456738,"style":67482},[31685,31686],[34,456740,223320],{"className":456741,"style":223361},[31685,31686],[34,456743],{"className":456744,"style":31846},[31845],[34,456746,202],{"className":456747},[31850],[34,456749],{"className":456750,"style":31846},[31845],[34,456752,456754,456757,456760,456763,456766],{"className":456753},[31676],[34,456755],{"className":456756,"style":33218},[31680],[34,456758,106063],{"className":456759,"style":67482},[31685,31686],[34,456761],{"className":456762,"style":32604},[31845],[34,456764,326344],{"className":456765},[32024],[34,456767],{"className":456768,"style":32604},[31845],[34,456770,456772,456775,456778,456781,456784],{"className":456771},[31676],[34,456773],{"className":456774,"style":33218},[31680],[34,456776,223320],{"className":456777,"style":223361},[31685,31686],[34,456779],{"className":456780,"style":31846},[31845],[34,456782,202],{"className":456783},[31850],[34,456785],{"className":456786,"style":31846},[31845],[34,456788,456790,456793,456796],{"className":456789},[31676],[34,456791],{"className":456792,"style":33218},[31680],[34,456794,223320],{"className":456795,"style":223361},[31685,31686],[34,456797,106063],{"className":456798,"style":67482},[31685,31686],[2627,456800,445574],{},"\n一方面，显然有 ",[34,456803,456805,456829],{"className":456804},[31642],[34,456806,456808],{"className":456807},[31646],[31648,456809,456810],{"xmlns":31650},[31652,456811,456812,456826],{},[31655,456813,456814,456816,456818,456820,456822,456824],{},[31658,456815,106063],{},[31658,456817,223320],{},[31662,456819,131855],{},[31658,456821,106063],{},[31662,456823,326344],{},[31658,456825,223320],{},[31665,456827,456828],{"encoding":31667},"IJ \\subseteq I \\cap J",[34,456830,456832,456853,456871],{"className":456831,"ariaHidden":758},[31672],[34,456833,456835,456838,456841,456844,456847,456850],{"className":456834},[31676],[34,456836],{"className":456837,"style":195047},[31680],[34,456839,106063],{"className":456840,"style":67482},[31685,31686],[34,456842,223320],{"className":456843,"style":223361},[31685,31686],[34,456845],{"className":456846,"style":31846},[31845],[34,456848,131855],{"className":456849},[31850],[34,456851],{"className":456852,"style":31846},[31845],[34,456854,456856,456859,456862,456865,456868],{"className":456855},[31676],[34,456857],{"className":456858,"style":33218},[31680],[34,456860,106063],{"className":456861,"style":67482},[31685,31686],[34,456863],{"className":456864,"style":32604},[31845],[34,456866,326344],{"className":456867},[32024],[34,456869],{"className":456870,"style":32604},[31845],[34,456872,456874,456877],{"className":456873},[31676],[34,456875],{"className":456876,"style":33218},[31680],[34,456878,223320],{"className":456879,"style":223361},[31685,31686],"（这是理想积的一般性质）。\n反过来，如果 ",[34,456882,456884,456901],{"className":456883},[31642],[34,456885,456887],{"className":456886},[31646],[31648,456888,456889],{"xmlns":31650},[31652,456890,456891,456899],{},[31655,456892,456893,456895,456897],{},[31658,456894,106063],{},[31662,456896,31960],{"separator":758},[31658,456898,223320],{},[31665,456900,454327],{"encoding":31667},[34,456902,456904],{"className":456903,"ariaHidden":758},[31672],[34,456905,456907,456910,456913,456916,456919],{"className":456906},[31676],[34,456908],{"className":456909,"style":70739},[31680],[34,456911,106063],{"className":456912,"style":67482},[31685,31686],[34,456914,31960],{"className":456915},[32119],[34,456917],{"className":456918,"style":32123},[31845],[34,456920,223320],{"className":456921,"style":223361},[31685,31686],[34,456923,456925,456946],{"className":456924},[31642],[34,456926,456928],{"className":456927},[31646],[31648,456929,456930],{"xmlns":31650},[31652,456931,456932,456944],{},[31655,456933,456934,456936,456938,456940,456942],{},[31658,456935,106063],{},[31662,456937,223],{},[31658,456939,223320],{},[31662,456941,202],{},[31658,456943,34386],{},[31665,456945,454257],{"encoding":31667},[34,456947,456949,456967,456985],{"className":456948,"ariaHidden":758},[31672],[34,456950,456952,456955,456958,456961,456964],{"className":456951},[31676],[34,456953],{"className":456954,"style":34944},[31680],[34,456956,106063],{"className":456957,"style":67482},[31685,31686],[34,456959],{"className":456960,"style":32604},[31845],[34,456962,223],{"className":456963},[32024],[34,456965],{"className":456966,"style":32604},[31845],[34,456968,456970,456973,456976,456979,456982],{"className":456969},[31676],[34,456971],{"className":456972,"style":33218},[31680],[34,456974,223320],{"className":456975,"style":223361},[31685,31686],[34,456977],{"className":456978,"style":31846},[31845],[34,456980,202],{"className":456981},[31850],[34,456983],{"className":456984,"style":31846},[31845],[34,456986,456988,456991],{"className":456987},[31676],[34,456989],{"className":456990,"style":33218},[31680],[34,456992,34386],{"className":456993,"style":34427},[31685,31686],". 由命题 3.3.3，存在 ",[34,456996,456998,457023],{"className":456997},[31642],[34,456999,457001],{"className":457000},[31646],[31648,457002,457003],{"xmlns":31650},[31652,457004,457005,457021],{},[31655,457006,457007,457009,457011,457013,457015,457017,457019],{},[31658,457008,31955],{},[31662,457010,112296],{},[31658,457012,106063],{},[31662,457014,31960],{"separator":758},[31658,457016,31968],{},[31662,457018,112296],{},[31658,457020,223320],{},[31665,457022,454519],{"encoding":31667},[34,457024,457026,457044,457071],{"className":457025,"ariaHidden":758},[31672],[34,457027,457029,457032,457035,457038,457041],{"className":457028},[31676],[34,457030],{"className":457031,"style":451673},[31680],[34,457033,31955],{"className":457034},[31685,31686],[34,457036],{"className":457037,"style":31846},[31845],[34,457039,112296],{"className":457040},[31850],[34,457042],{"className":457043,"style":31846},[31845],[34,457045,457047,457050,457053,457056,457059,457062,457065,457068],{"className":457046},[31676],[34,457048],{"className":457049,"style":70739},[31680],[34,457051,106063],{"className":457052,"style":67482},[31685,31686],[34,457054,31960],{"className":457055},[32119],[34,457057],{"className":457058,"style":32123},[31845],[34,457060,31968],{"className":457061,"style":32153},[31685,31686],[34,457063],{"className":457064,"style":31846},[31845],[34,457066,112296],{"className":457067},[31850],[34,457069],{"className":457070,"style":31846},[31845],[34,457072,457074,457077],{"className":457073},[31676],[34,457075],{"className":457076,"style":33218},[31680],[34,457078,223320],{"className":457079,"style":223361},[31685,31686],[34,457081,457083,457104],{"className":457082},[31642],[34,457084,457086],{"className":457085},[31646],[31648,457087,457088],{"xmlns":31650},[31652,457089,457090,457102],{},[31655,457091,457092,457094,457096,457098,457100],{},[31749,457093,640],{},[31662,457095,202],{},[31658,457097,31955],{},[31662,457099,223],{},[31658,457101,31968],{},[31665,457103,454602],{"encoding":31667},[34,457105,457107,457125,457143],{"className":457106,"ariaHidden":758},[31672],[34,457108,457110,457113,457116,457119,457122],{"className":457109},[31676],[34,457111],{"className":457112,"style":32617},[31680],[34,457114,640],{"className":457115},[31685],[34,457117],{"className":457118,"style":31846},[31845],[34,457120,202],{"className":457121},[31850],[34,457123],{"className":457124,"style":31846},[31845],[34,457126,457128,457131,457134,457137,457140],{"className":457127},[31676],[34,457129],{"className":457130,"style":37322},[31680],[34,457132,31955],{"className":457133},[31685,31686],[34,457135],{"className":457136,"style":32604},[31845],[34,457138,223],{"className":457139},[32024],[34,457141],{"className":457142,"style":32604},[31845],[34,457144,457146,457149],{"className":457145},[31676],[34,457147],{"className":457148,"style":33123},[31680],[34,457150,31968],{"className":457151,"style":32153},[31685,31686],[34,457153,457155,457179],{"className":457154},[31642],[34,457156,457158],{"className":457157},[31646],[31648,457159,457160],{"xmlns":31650},[31652,457161,457162,457176],{},[31655,457163,457164,457166,457168,457170,457172,457174],{},[31658,457165,365679],{"mathvariant":38079},[31658,457167,32936],{},[31662,457169,112296],{},[31658,457171,106063],{},[31662,457173,326344],{},[31658,457175,223320],{},[31665,457177,457178],{"encoding":31667},"\\forall x \\in I \\cap J",[34,457180,457182,457203,457221],{"className":457181,"ariaHidden":758},[31672],[34,457183,457185,457188,457191,457194,457197,457200],{"className":457184},[31676],[34,457186],{"className":457187,"style":46991},[31680],[34,457189,365679],{"className":457190},[31685],[34,457192,32936],{"className":457193},[31685,31686],[34,457195],{"className":457196,"style":31846},[31845],[34,457198,112296],{"className":457199},[31850],[34,457201],{"className":457202,"style":31846},[31845],[34,457204,457206,457209,457212,457215,457218],{"className":457205},[31676],[34,457207],{"className":457208,"style":33218},[31680],[34,457210,106063],{"className":457211,"style":67482},[31685,31686],[34,457213],{"className":457214,"style":32604},[31845],[34,457216,326344],{"className":457217},[32024],[34,457219],{"className":457220,"style":32604},[31845],[34,457222,457224,457227],{"className":457223},[31676],[34,457225],{"className":457226,"style":33218},[31680],[34,457228,223320],{"className":457229,"style":223361},[31685,31686],", 有\n",[34,457232,457234,457284],{"className":457233},[31642],[34,457235,457237],{"className":457236},[31646],[31648,457238,457239],{"xmlns":31650},[31652,457240,457241,457281],{},[31655,457242,457243,457245,457247,457249,457251,457253,457255,457257,457259,457261,457263,457265,457267,457269,457271,457273,457275,457277,457279],{},[31658,457244,32936],{},[31662,457246,202],{},[31749,457248,640],{},[31662,457250,71191],{},[31658,457252,32936],{},[31662,457254,202],{},[31662,457256,456],{"stretchy":702},[31658,457258,31955],{},[31662,457260,223],{},[31658,457262,31968],{},[31662,457264,2557],{"stretchy":702},[31658,457266,32936],{},[31662,457268,202],{},[31658,457270,31955],{},[31658,457272,32936],{},[31662,457274,223],{},[31658,457276,31968],{},[31658,457278,32936],{},[31658,457280,2547],{"mathvariant":38079},[31665,457282,457283],{"encoding":31667},"x = 1 \\cdot x = (i+j)x = ix + jx.",[34,457285,457287,457305,457323,457341,457362,457386,457407],{"className":457286,"ariaHidden":758},[31672],[34,457288,457290,457293,457296,457299,457302],{"className":457289},[31676],[34,457291],{"className":457292,"style":32527},[31680],[34,457294,32936],{"className":457295},[31685,31686],[34,457297],{"className":457298,"style":31846},[31845],[34,457300,202],{"className":457301},[31850],[34,457303],{"className":457304,"style":31846},[31845],[34,457306,457308,457311,457314,457317,457320],{"className":457307},[31676],[34,457309],{"className":457310,"style":32617},[31680],[34,457312,640],{"className":457313},[31685],[34,457315],{"className":457316,"style":32604},[31845],[34,457318,71191],{"className":457319},[32024],[34,457321],{"className":457322,"style":32604},[31845],[34,457324,457326,457329,457332,457335,457338],{"className":457325},[31676],[34,457327],{"className":457328,"style":32527},[31680],[34,457330,32936],{"className":457331},[31685,31686],[34,457333],{"className":457334,"style":31846},[31845],[34,457336,202],{"className":457337},[31850],[34,457339],{"className":457340,"style":31846},[31845],[34,457342,457344,457347,457350,457353,457356,457359],{"className":457343},[31676],[34,457345],{"className":457346,"style":31774},[31680],[34,457348,456],{"className":457349},[31782],[34,457351,31955],{"className":457352},[31685,31686],[34,457354],{"className":457355,"style":32604},[31845],[34,457357,223],{"className":457358},[32024],[34,457360],{"className":457361,"style":32604},[31845],[34,457363,457365,457368,457371,457374,457377,457380,457383],{"className":457364},[31676],[34,457366],{"className":457367,"style":31774},[31680],[34,457369,31968],{"className":457370,"style":32153},[31685,31686],[34,457372,2557],{"className":457373},[31690],[34,457375,32936],{"className":457376},[31685,31686],[34,457378],{"className":457379,"style":31846},[31845],[34,457381,202],{"className":457382},[31850],[34,457384],{"className":457385,"style":31846},[31845],[34,457387,457389,457392,457395,457398,457401,457404],{"className":457388},[31676],[34,457390],{"className":457391,"style":37322},[31680],[34,457393,31955],{"className":457394},[31685,31686],[34,457396,32936],{"className":457397},[31685,31686],[34,457399],{"className":457400,"style":32604},[31845],[34,457402,223],{"className":457403},[32024],[34,457405],{"className":457406,"style":32604},[31845],[34,457408,457410,457413,457416,457419],{"className":457409},[31676],[34,457411],{"className":457412,"style":33123},[31680],[34,457414,31968],{"className":457415,"style":32153},[31685,31686],[34,457417,32936],{"className":457418},[31685,31686],[34,457420,2547],{"className":457421},[31685],[34,457423,457425,457459],{"className":457424},[31642],[34,457426,457428],{"className":457427},[31646],[31648,457429,457430],{"xmlns":31650},[31652,457431,457432,457456],{},[31655,457433,457434,457436,457438,457440,457442,457444,457446,457448,457450,457452,457454],{},[31658,457435,32936],{},[31662,457437,112296],{},[31658,457439,223320],{},[39361,457441,262622],{},[31662,457443,262625],{},[39361,457445,262622],{},[31658,457447,31955],{},[31658,457449,32936],{},[31662,457451,112296],{},[31658,457453,106063],{},[31658,457455,223320],{},[31665,457457,457458],{"encoding":31667},"x \\in J \\implies ix \\in IJ",[34,457460,457462,457480,457504,457525],{"className":457461,"ariaHidden":758},[31672],[34,457463,457465,457468,457471,457474,457477],{"className":457464},[31676],[34,457466],{"className":457467,"style":34812},[31680],[34,457469,32936],{"className":457470},[31685,31686],[34,457472],{"className":457473,"style":31846},[31845],[34,457475,112296],{"className":457476},[31850],[34,457478],{"className":457479,"style":31846},[31845],[34,457481,457483,457486,457489,457492,457495,457498,457501],{"className":457482},[31676],[34,457484],{"className":457485,"style":336814},[31680],[34,457487,223320],{"className":457488,"style":223361},[31685,31686],[34,457490],{"className":457491,"style":31846},[31845],[34,457493],{"className":457494,"style":31846},[31845],[34,457496,262625],{"className":457497},[31850],[34,457499],{"className":457500,"style":31846},[31845],[34,457502],{"className":457503,"style":31846},[31845],[34,457505,457507,457510,457513,457516,457519,457522],{"className":457506},[31676],[34,457508],{"className":457509,"style":451673},[31680],[34,457511,31955],{"className":457512},[31685,31686],[34,457514,32936],{"className":457515},[31685,31686],[34,457517],{"className":457518,"style":31846},[31845],[34,457520,112296],{"className":457521},[31850],[34,457523],{"className":457524,"style":31846},[31845],[34,457526,457528,457531,457534],{"className":457527},[31676],[34,457529],{"className":457530,"style":33218},[31680],[34,457532,106063],{"className":457533,"style":67482},[31685,31686],[34,457535,223320],{"className":457536,"style":223361},[31685,31686]," (理想性质)，且 ",[34,457539,457541,457581],{"className":457540},[31642],[34,457542,457544],{"className":457543},[31646],[31648,457545,457546],{"xmlns":31650},[31652,457547,457548,457578],{},[31655,457549,457550,457552,457554,457556,457558,457560,457562,457564,457566,457568,457570,457572,457574,457576],{},[31658,457551,32936],{},[31662,457553,112296],{},[31658,457555,106063],{},[39361,457557,262622],{},[31662,457559,262625],{},[39361,457561,262622],{},[31658,457563,31968],{},[31658,457565,32936],{},[31662,457567,112296],{},[31658,457569,223320],{},[31658,457571,106063],{},[31662,457573,202],{},[31658,457575,106063],{},[31658,457577,223320],{},[31665,457579,457580],{"encoding":31667},"x \\in I \\implies jx \\in JI = IJ",[34,457582,457584,457602,457626,457647,457668],{"className":457583,"ariaHidden":758},[31672],[34,457585,457587,457590,457593,457596,457599],{"className":457586},[31676],[34,457588],{"className":457589,"style":34812},[31680],[34,457591,32936],{"className":457592},[31685,31686],[34,457594],{"className":457595,"style":31846},[31845],[34,457597,112296],{"className":457598},[31850],[34,457600],{"className":457601,"style":31846},[31845],[34,457603,457605,457608,457611,457614,457617,457620,457623],{"className":457604},[31676],[34,457606],{"className":457607,"style":336814},[31680],[34,457609,106063],{"className":457610,"style":67482},[31685,31686],[34,457612],{"className":457613,"style":31846},[31845],[34,457615],{"className":457616,"style":31846},[31845],[34,457618,262625],{"className":457619},[31850],[34,457621],{"className":457622,"style":31846},[31845],[34,457624],{"className":457625,"style":31846},[31845],[34,457627,457629,457632,457635,457638,457641,457644],{"className":457628},[31676],[34,457630],{"className":457631,"style":33123},[31680],[34,457633,31968],{"className":457634,"style":32153},[31685,31686],[34,457636,32936],{"className":457637},[31685,31686],[34,457639],{"className":457640,"style":31846},[31845],[34,457642,112296],{"className":457643},[31850],[34,457645],{"className":457646,"style":31846},[31845],[34,457648,457650,457653,457656,457659,457662,457665],{"className":457649},[31676],[34,457651],{"className":457652,"style":33218},[31680],[34,457654,223320],{"className":457655,"style":223361},[31685,31686],[34,457657,106063],{"className":457658,"style":67482},[31685,31686],[34,457660],{"className":457661,"style":31846},[31845],[34,457663,202],{"className":457664},[31850],[34,457666],{"className":457667,"style":31846},[31845],[34,457669,457671,457674,457677],{"className":457670},[31676],[34,457672],{"className":457673,"style":33218},[31680],[34,457675,106063],{"className":457676,"style":67482},[31685,31686],[34,457678,223320],{"className":457679,"style":223361},[31685,31686]," (交换环中 ",[34,457682,457684,457706],{"className":457683},[31642],[34,457685,457687],{"className":457686},[31646],[31648,457688,457689],{"xmlns":31650},[31652,457690,457691,457703],{},[31655,457692,457693,457695,457697,457699,457701],{},[31658,457694,223320],{},[31658,457696,106063],{},[31662,457698,202],{},[31658,457700,106063],{},[31658,457702,223320],{},[31665,457704,457705],{"encoding":31667},"JI=IJ",[34,457707,457709,457730],{"className":457708,"ariaHidden":758},[31672],[34,457710,457712,457715,457718,457721,457724,457727],{"className":457711},[31676],[34,457713],{"className":457714,"style":33218},[31680],[34,457716,223320],{"className":457717,"style":223361},[31685,31686],[34,457719,106063],{"className":457720,"style":67482},[31685,31686],[34,457722],{"className":457723,"style":31846},[31845],[34,457725,202],{"className":457726},[31850],[34,457728],{"className":457729,"style":31846},[31845],[34,457731,457733,457736,457739],{"className":457732},[31676],[34,457734],{"className":457735,"style":33218},[31680],[34,457737,106063],{"className":457738,"style":67482},[31685,31686],[34,457740,223320],{"className":457741,"style":223361},[31685,31686],[34,457743,457745,457777],{"className":457744},[31642],[34,457746,457748],{"className":457747},[31646],[31648,457749,457750],{"xmlns":31650},[31652,457751,457752,457774],{},[31655,457753,457754,457756,457758,457760,457762,457764,457766,457768,457770,457772],{},[31658,457755,32936],{},[31662,457757,112296],{},[31658,457759,106063],{},[31658,457761,223320],{},[31662,457763,223],{},[31658,457765,106063],{},[31658,457767,223320],{},[31662,457769,202],{},[31658,457771,106063],{},[31658,457773,223320],{},[31665,457775,457776],{"encoding":31667},"x \\in IJ + IJ = IJ",[34,457778,457780,457798,457819,457840],{"className":457779,"ariaHidden":758},[31672],[34,457781,457783,457786,457789,457792,457795],{"className":457782},[31676],[34,457784],{"className":457785,"style":34812},[31680],[34,457787,32936],{"className":457788},[31685,31686],[34,457790],{"className":457791,"style":31846},[31845],[34,457793,112296],{"className":457794},[31850],[34,457796],{"className":457797,"style":31846},[31845],[34,457799,457801,457804,457807,457810,457813,457816],{"className":457800},[31676],[34,457802],{"className":457803,"style":34944},[31680],[34,457805,106063],{"className":457806,"style":67482},[31685,31686],[34,457808,223320],{"className":457809,"style":223361},[31685,31686],[34,457811],{"className":457812,"style":32604},[31845],[34,457814,223],{"className":457815},[32024],[34,457817],{"className":457818,"style":32604},[31845],[34,457820,457822,457825,457828,457831,457834,457837],{"className":457821},[31676],[34,457823],{"className":457824,"style":33218},[31680],[34,457826,106063],{"className":457827,"style":67482},[31685,31686],[34,457829,223320],{"className":457830,"style":223361},[31685,31686],[34,457832],{"className":457833,"style":31846},[31845],[34,457835,202],{"className":457836},[31850],[34,457838],{"className":457839,"style":31846},[31845],[34,457841,457843,457846,457849],{"className":457842},[31676],[34,457844],{"className":457845,"style":33218},[31680],[34,457847,106063],{"className":457848,"style":67482},[31685,31686],[34,457850,223320],{"className":457851,"style":223361},[31685,31686],"。\n因此 ",[34,457854,457856,457880],{"className":457855},[31642],[34,457857,457859],{"className":457858},[31646],[31648,457860,457861],{"xmlns":31650},[31652,457862,457863,457877],{},[31655,457864,457865,457867,457869,457871,457873,457875],{},[31658,457866,106063],{},[31662,457868,326344],{},[31658,457870,223320],{},[31662,457872,131855],{},[31658,457874,106063],{},[31658,457876,223320],{},[31665,457878,457879],{"encoding":31667},"I \\cap J \\subseteq IJ",[34,457881,457883,457901,457919],{"className":457882,"ariaHidden":758},[31672],[34,457884,457886,457889,457892,457895,457898],{"className":457885},[31676],[34,457887],{"className":457888,"style":33218},[31680],[34,457890,106063],{"className":457891,"style":67482},[31685,31686],[34,457893],{"className":457894,"style":32604},[31845],[34,457896,326344],{"className":457897},[32024],[34,457899],{"className":457900,"style":32604},[31845],[34,457902,457904,457907,457910,457913,457916],{"className":457903},[31676],[34,457905],{"className":457906,"style":195047},[31680],[34,457908,223320],{"className":457909,"style":223361},[31685,31686],[34,457911],{"className":457912,"style":31846},[31845],[34,457914,131855],{"className":457915},[31850],[34,457917],{"className":457918,"style":31846},[31845],[34,457920,457922,457925,457928],{"className":457921},[31676],[34,457923],{"className":457924,"style":33218},[31680],[34,457926,106063],{"className":457927,"style":67482},[31685,31686],[34,457929,223320],{"className":457930,"style":223361},[31685,31686],". 综上 ",[34,457933,457935,457959],{"className":457934},[31642],[34,457936,457938],{"className":457937},[31646],[31648,457939,457940],{"xmlns":31650},[31652,457941,457942,457956],{},[31655,457943,457944,457946,457948,457950,457952,457954],{},[31658,457945,106063],{},[31658,457947,223320],{},[31662,457949,202],{},[31658,457951,106063],{},[31662,457953,326344],{},[31658,457955,223320],{},[31665,457957,457958],{"encoding":31667},"IJ = I \\cap J",[34,457960,457962,457983,458001],{"className":457961,"ariaHidden":758},[31672],[34,457963,457965,457968,457971,457974,457977,457980],{"className":457964},[31676],[34,457966],{"className":457967,"style":33218},[31680],[34,457969,106063],{"className":457970,"style":67482},[31685,31686],[34,457972,223320],{"className":457973,"style":223361},[31685,31686],[34,457975],{"className":457976,"style":31846},[31845],[34,457978,202],{"className":457979},[31850],[34,457981],{"className":457982,"style":31846},[31845],[34,457984,457986,457989,457992,457995,457998],{"className":457985},[31676],[34,457987],{"className":457988,"style":33218},[31680],[34,457990,106063],{"className":457991,"style":67482},[31685,31686],[34,457993],{"className":457994,"style":32604},[31845],[34,457996,326344],{"className":457997},[32024],[34,457999],{"className":458000,"style":32604},[31845],[34,458002,458004,458007],{"className":458003},[31676],[34,458005],{"className":458006,"style":33218},[31680],[34,458008,223320],{"className":458009,"style":223361},[31685,31686],[34,458011,458013,458026],{"className":458012},[31642],[34,458014,458016],{"className":458015},[31646],[31648,458017,458018],{"xmlns":31650},[31652,458019,458020,458024],{},[31655,458021,458022],{},[31658,458023,445204],{"mathvariant":38079},[31665,458025,445207],{"encoding":31667},[34,458027,458029],{"className":458028,"ariaHidden":758},[31672],[34,458030,458032,458035],{"className":458031},[31676],[34,458033],{"className":458034,"style":445217},[31680],[34,458036,445204],{"className":458037},[31685,129261],[15,458039,458040,69123,458043,454385,458071,456651,458246,458329,458330,616,458358,340354,458472,43298,458627,458655],{},[2627,458041,458042],{},"命题 3.3.5.",[34,458044,458046,458059],{"className":458045},[31642],[34,458047,458049],{"className":458048},[31646],[31648,458050,458051],{"xmlns":31650},[31652,458052,458053,458057],{},[31655,458054,458055],{},[31658,458056,34386],{},[31665,458058,34386],{"encoding":31667},[34,458060,458062],{"className":458061,"ariaHidden":758},[31672],[34,458063,458065,458068],{"className":458064},[31676],[34,458066],{"className":458067,"style":33218},[31680],[34,458069,34386],{"className":458070,"style":34427},[31685,31686],[34,458072,458074,458112],{"className":458073},[31642],[34,458075,458077],{"className":458076},[31646],[31648,458078,458079],{"xmlns":31650},[31652,458080,458081,458109],{},[31655,458082,458083,458089,458091,458093,458095,458101,458103,458105,458107],{},[31743,458084,458085,458087],{},[31658,458086,106063],{},[31749,458088,640],{},[31662,458090,31960],{"separator":758},[31662,458092,33934],{},[31662,458094,31960],{"separator":758},[31743,458096,458097,458099],{},[31658,458098,106063],{},[31658,458100,33759],{},[31662,458102,31960],{"separator":758},[31658,458104,223320],{},[31662,458106,440828],{},[31658,458108,34386],{},[31665,458110,458111],{"encoding":31667},"I_1, \\dots, I_k, J \\lhd R",[34,458113,458115,458237],{"className":458114,"ariaHidden":758},[31672],[34,458116,458118,458121,458161,458164,458167,458170,458173,458176,458179,458219,458222,458225,458228,458231,458234],{"className":458117},[31676],[34,458119],{"className":458120,"style":70739},[31680],[34,458122,458124,458127],{"className":458123},[31685],[34,458125,106063],{"className":458126,"style":67482},[31685,31686],[34,458128,458130],{"className":458129},[31792],[34,458131,458133,458153],{"className":458132},[31796,31797],[34,458134,458136,458150],{"className":458135},[31801],[34,458137,458139],{"className":458138,"style":31806},[31805],[34,458140,458141,458144],{"style":217742},[34,458142],{"className":458143,"style":31814},[31813],[34,458145,458147],{"className":458146},[31818,31819,31820,31821],[34,458148,640],{"className":458149},[31685,31821],[34,458151,31829],{"className":458152},[31828],[34,458154,458156],{"className":458155},[31801],[34,458157,458159],{"className":458158,"style":31836},[31805],[34,458160],{},[34,458162,31960],{"className":458163},[32119],[34,458165],{"className":458166,"style":32123},[31845],[34,458168,33934],{"className":458169},[34006],[34,458171],{"className":458172,"style":32123},[31845],[34,458174,31960],{"className":458175},[32119],[34,458177],{"className":458178,"style":32123},[31845],[34,458180,458182,458185],{"className":458181},[31685],[34,458183,106063],{"className":458184,"style":67482},[31685,31686],[34,458186,458188],{"className":458187},[31792],[34,458189,458191,458211],{"className":458190},[31796,31797],[34,458192,458194,458208],{"className":458193},[31801],[34,458195,458197],{"className":458196,"style":33833},[31805],[34,458198,458199,458202],{"style":217742},[34,458200],{"className":458201,"style":31814},[31813],[34,458203,458205],{"className":458204},[31818,31819,31820,31821],[34,458206,33759],{"className":458207,"style":33774},[31685,31686,31821],[34,458209,31829],{"className":458210},[31828],[34,458212,458214],{"className":458213},[31801],[34,458215,458217],{"className":458216,"style":31836},[31805],[34,458218],{},[34,458220,31960],{"className":458221},[32119],[34,458223],{"className":458224,"style":32123},[31845],[34,458226,223320],{"className":458227,"style":223361},[31685,31686],[34,458229],{"className":458230,"style":32604},[31845],[34,458232,440828],{"className":458233},[32024,129261],[34,458235],{"className":458236,"style":32604},[31845],[34,458238,458240,458243],{"className":458239},[31676],[34,458241],{"className":458242,"style":33218},[31680],[34,458244,34386],{"className":458245,"style":34427},[31685,31686],[34,458247,458249,458271],{"className":458248},[31642],[34,458250,458252],{"className":458251},[31646],[31648,458253,458254],{"xmlns":31650},[31652,458255,458256,458268],{},[31655,458257,458258,458264,458266],{},[31743,458259,458260,458262],{},[31658,458261,106063],{},[31658,458263,31747],{},[31662,458265,31960],{"separator":758},[31658,458267,223320],{},[31665,458269,458270],{"encoding":31667},"I_s, J",[34,458272,458274],{"className":458273,"ariaHidden":758},[31672],[34,458275,458277,458280,458320,458323,458326],{"className":458276},[31676],[34,458278],{"className":458279,"style":70739},[31680],[34,458281,458283,458286],{"className":458282},[31685],[34,458284,106063],{"className":458285,"style":67482},[31685,31686],[34,458287,458289],{"className":458288},[31792],[34,458290,458292,458312],{"className":458291},[31796,31797],[34,458293,458295,458309],{"className":458294},[31801],[34,458296,458298],{"className":458297,"style":32362},[31805],[34,458299,458300,458303],{"style":217742},[34,458301],{"className":458302,"style":31814},[31813],[34,458304,458306],{"className":458305},[31818,31819,31820,31821],[34,458307,31747],{"className":458308},[31685,31686,31821],[34,458310,31829],{"className":458311},[31828],[34,458313,458315],{"className":458314},[31801],[34,458316,458318],{"className":458317,"style":31836},[31805],[34,458319],{},[34,458321,31960],{"className":458322},[32119],[34,458324],{"className":458325,"style":32123},[31845],[34,458327,223320],{"className":458328,"style":223361},[31685,31686]," 都互素（即对于每个 ",[34,458331,458333,458346],{"className":458332},[31642],[34,458334,458336],{"className":458335},[31646],[31648,458337,458338],{"xmlns":31650},[31652,458339,458340,458344],{},[31655,458341,458342],{},[31658,458343,31747],{},[31665,458345,31747],{"encoding":31667},[34,458347,458349],{"className":458348,"ariaHidden":758},[31672],[34,458350,458352,458355],{"className":458351},[31676],[34,458353],{"className":458354,"style":32527},[31680],[34,458356,31747],{"className":458357},[31685,31686],[34,458359,458361,458387],{"className":458360},[31642],[34,458362,458364],{"className":458363},[31646],[31648,458365,458366],{"xmlns":31650},[31652,458367,458368,458384],{},[31655,458369,458370,458376,458378,458380,458382],{},[31743,458371,458372,458374],{},[31658,458373,106063],{},[31658,458375,31747],{},[31662,458377,223],{},[31658,458379,223320],{},[31662,458381,202],{},[31658,458383,34386],{},[31665,458385,458386],{"encoding":31667},"I_s+J=R",[34,458388,458390,458445,458463],{"className":458389,"ariaHidden":758},[31672],[34,458391,458393,458396,458436,458439,458442],{"className":458392},[31676],[34,458394],{"className":458395,"style":70915},[31680],[34,458397,458399,458402],{"className":458398},[31685],[34,458400,106063],{"className":458401,"style":67482},[31685,31686],[34,458403,458405],{"className":458404},[31792],[34,458406,458408,458428],{"className":458407},[31796,31797],[34,458409,458411,458425],{"className":458410},[31801],[34,458412,458414],{"className":458413,"style":32362},[31805],[34,458415,458416,458419],{"style":217742},[34,458417],{"className":458418,"style":31814},[31813],[34,458420,458422],{"className":458421},[31818,31819,31820,31821],[34,458423,31747],{"className":458424},[31685,31686,31821],[34,458426,31829],{"className":458427},[31828],[34,458429,458431],{"className":458430},[31801],[34,458432,458434],{"className":458433,"style":31836},[31805],[34,458435],{},[34,458437],{"className":458438,"style":32604},[31845],[34,458440,223],{"className":458441},[32024],[34,458443],{"className":458444,"style":32604},[31845],[34,458446,458448,458451,458454,458457,458460],{"className":458447},[31676],[34,458449],{"className":458450,"style":33218},[31680],[34,458452,223320],{"className":458453,"style":223361},[31685,31686],[34,458455],{"className":458456,"style":31846},[31845],[34,458458,202],{"className":458459},[31850],[34,458461],{"className":458462,"style":31846},[31845],[34,458464,458466,458469],{"className":458465},[31676],[34,458467],{"className":458468,"style":33218},[31680],[34,458470,34386],{"className":458471,"style":34427},[31685,31686],[34,458473,458475,458505],{"className":458474},[31642],[34,458476,458478],{"className":458477},[31646],[31648,458479,458480],{"xmlns":31650},[31652,458481,458482,458502],{},[31655,458483,458484,458490,458492,458494,458496],{},[31743,458485,458486,458488],{},[31658,458487,106063],{},[31749,458489,640],{},[31662,458491,326344],{},[31662,458493,38874],{},[31662,458495,326344],{},[31743,458497,458498,458500],{},[31658,458499,106063],{},[31658,458501,33759],{},[31665,458503,458504],{"encoding":31667},"I_1 \\cap \\dots \\cap I_k",[34,458506,458508,458563,458581],{"className":458507,"ariaHidden":758},[31672],[34,458509,458511,458514,458554,458557,458560],{"className":458510},[31676],[34,458512],{"className":458513,"style":70915},[31680],[34,458515,458517,458520],{"className":458516},[31685],[34,458518,106063],{"className":458519,"style":67482},[31685,31686],[34,458521,458523],{"className":458522},[31792],[34,458524,458526,458546],{"className":458525},[31796,31797],[34,458527,458529,458543],{"className":458528},[31801],[34,458530,458532],{"className":458531,"style":31806},[31805],[34,458533,458534,458537],{"style":217742},[34,458535],{"className":458536,"style":31814},[31813],[34,458538,458540],{"className":458539},[31818,31819,31820,31821],[34,458541,640],{"className":458542},[31685,31821],[34,458544,31829],{"className":458545},[31828],[34,458547,458549],{"className":458548},[31801],[34,458550,458552],{"className":458551,"style":31836},[31805],[34,458553],{},[34,458555],{"className":458556,"style":32604},[31845],[34,458558,326344],{"className":458559},[32024],[34,458561],{"className":458562,"style":32604},[31845],[34,458564,458566,458569,458572,458575,458578],{"className":458565},[31676],[34,458567],{"className":458568,"style":147334},[31680],[34,458570,38874],{"className":458571},[34006],[34,458573],{"className":458574,"style":32604},[31845],[34,458576,326344],{"className":458577},[32024],[34,458579],{"className":458580,"style":32604},[31845],[34,458582,458584,458587],{"className":458583},[31676],[34,458585],{"className":458586,"style":70915},[31680],[34,458588,458590,458593],{"className":458589},[31685],[34,458591,106063],{"className":458592,"style":67482},[31685,31686],[34,458594,458596],{"className":458595},[31792],[34,458597,458599,458619],{"className":458598},[31796,31797],[34,458600,458602,458616],{"className":458601},[31801],[34,458603,458605],{"className":458604,"style":33833},[31805],[34,458606,458607,458610],{"style":217742},[34,458608],{"className":458609,"style":31814},[31813],[34,458611,458613],{"className":458612},[31818,31819,31820,31821],[34,458614,33759],{"className":458615,"style":33774},[31685,31686,31821],[34,458617,31829],{"className":458618},[31828],[34,458620,458622],{"className":458621},[31801],[34,458623,458625],{"className":458624,"style":31836},[31805],[34,458626],{},[34,458628,458630,458643],{"className":458629},[31642],[34,458631,458633],{"className":458632},[31646],[31648,458634,458635],{"xmlns":31650},[31652,458636,458637,458641],{},[31655,458638,458639],{},[31658,458640,223320],{},[31665,458642,223320],{"encoding":31667},[34,458644,458646],{"className":458645,"ariaHidden":758},[31672],[34,458647,458649,458652],{"className":458648},[31676],[34,458650],{"className":458651,"style":33218},[31680],[34,458653,223320],{"className":458654,"style":223361},[31685,31686]," 也互素。",[15,458657,458658,458661,458662,458690],{},[2627,458659,458660],{},"证明 (详细展开):","\n对 ",[34,458663,458665,458678],{"className":458664},[31642],[34,458666,458668],{"className":458667},[31646],[31648,458669,458670],{"xmlns":31650},[31652,458671,458672,458676],{},[31655,458673,458674],{},[31658,458675,33759],{},[31665,458677,33759],{"encoding":31667},[34,458679,458681],{"className":458680,"ariaHidden":758},[31672],[34,458682,458684,458687],{"className":458683},[31676],[34,458685],{"className":458686,"style":31681},[31680],[34,458688,33759],{"className":458689,"style":33774},[31685,31686]," 用数学归纳法。",[105,458692,458693,458746,459066],{},[108,458694,458695,458745],{},[34,458696,458698,458715],{"className":458697},[31642],[34,458699,458701],{"className":458700},[31646],[31648,458702,458703],{"xmlns":31650},[31652,458704,458705,458713],{},[31655,458706,458707,458709,458711],{},[31658,458708,33759],{},[31662,458710,202],{},[31749,458712,640],{},[31665,458714,45814],{"encoding":31667},[34,458716,458718,458736],{"className":458717,"ariaHidden":758},[31672],[34,458719,458721,458724,458727,458730,458733],{"className":458720},[31676],[34,458722],{"className":458723,"style":31681},[31680],[34,458725,33759],{"className":458726,"style":33774},[31685,31686],[34,458728],{"className":458729,"style":31846},[31845],[34,458731,202],{"className":458732},[31850],[34,458734],{"className":458735,"style":31846},[31845],[34,458737,458739,458742],{"className":458738},[31676],[34,458740],{"className":458741,"style":32617},[31680],[34,458743,640],{"className":458744},[31685]," 时，由已知条件显然成立。",[108,458747,458748,458749,458800,458801,64615,458993,347],{},"假设 ",[34,458750,458752,458770],{"className":458751},[31642],[34,458753,458755],{"className":458754},[31646],[31648,458756,458757],{"xmlns":31650},[31652,458758,458759,458767],{},[31655,458760,458761,458763,458765],{},[31658,458762,33759],{},[31662,458764,32226],{},[31749,458766,640],{},[31665,458768,458769],{"encoding":31667},"k-1",[34,458771,458773,458791],{"className":458772,"ariaHidden":758},[31672],[34,458774,458776,458779,458782,458785,458788],{"className":458775},[31676],[34,458777],{"className":458778,"style":34299},[31680],[34,458780,33759],{"className":458781,"style":33774},[31685,31686],[34,458783],{"className":458784,"style":32604},[31845],[34,458786,32226],{"className":458787},[32024],[34,458789],{"className":458790,"style":32604},[31845],[34,458792,458794,458797],{"className":458793},[31676],[34,458795],{"className":458796,"style":32617},[31680],[34,458798,640],{"className":458799},[31685]," 时结论成立，即设 ",[34,458802,458804,458844],{"className":458803},[31642],[34,458805,458807],{"className":458806},[31646],[31648,458808,458809],{"xmlns":31650},[31652,458810,458811,458841],{},[31655,458812,458813,458815,458817,458823,458825,458827,458829],{},[31658,458814,54151],{},[31662,458816,202],{},[31743,458818,458819,458821],{},[31658,458820,106063],{},[31749,458822,640],{},[31662,458824,326344],{},[31662,458826,38874],{},[31662,458828,326344],{},[31743,458830,458831,458833],{},[31658,458832,106063],{},[31655,458834,458835,458837,458839],{},[31658,458836,33759],{},[31662,458838,32226],{},[31749,458840,640],{},[31665,458842,458843],{"encoding":31667},"A = I_1 \\cap \\dots \\cap I_{k-1}",[34,458845,458847,458865,458920,458938],{"className":458846,"ariaHidden":758},[31672],[34,458848,458850,458853,458856,458859,458862],{"className":458849},[31676],[34,458851],{"className":458852,"style":33218},[31680],[34,458854,54151],{"className":458855},[31685,31686],[34,458857],{"className":458858,"style":31846},[31845],[34,458860,202],{"className":458861},[31850],[34,458863],{"className":458864,"style":31846},[31845],[34,458866,458868,458871,458911,458914,458917],{"className":458867},[31676],[34,458869],{"className":458870,"style":70915},[31680],[34,458872,458874,458877],{"className":458873},[31685],[34,458875,106063],{"className":458876,"style":67482},[31685,31686],[34,458878,458880],{"className":458879},[31792],[34,458881,458883,458903],{"className":458882},[31796,31797],[34,458884,458886,458900],{"className":458885},[31801],[34,458887,458889],{"className":458888,"style":31806},[31805],[34,458890,458891,458894],{"style":217742},[34,458892],{"className":458893,"style":31814},[31813],[34,458895,458897],{"className":458896},[31818,31819,31820,31821],[34,458898,640],{"className":458899},[31685,31821],[34,458901,31829],{"className":458902},[31828],[34,458904,458906],{"className":458905},[31801],[34,458907,458909],{"className":458908,"style":31836},[31805],[34,458910],{},[34,458912],{"className":458913,"style":32604},[31845],[34,458915,326344],{"className":458916},[32024],[34,458918],{"className":458919,"style":32604},[31845],[34,458921,458923,458926,458929,458932,458935],{"className":458922},[31676],[34,458924],{"className":458925,"style":147334},[31680],[34,458927,38874],{"className":458928},[34006],[34,458930],{"className":458931,"style":32604},[31845],[34,458933,326344],{"className":458934},[32024],[34,458936],{"className":458937,"style":32604},[31845],[34,458939,458941,458944],{"className":458940},[31676],[34,458942],{"className":458943,"style":70992},[31680],[34,458945,458947,458950],{"className":458946},[31685],[34,458948,106063],{"className":458949,"style":67482},[31685,31686],[34,458951,458953],{"className":458952},[31792],[34,458954,458956,458985],{"className":458955},[31796,31797],[34,458957,458959,458982],{"className":458958},[31801],[34,458960,458962],{"className":458961,"style":33833},[31805],[34,458963,458964,458967],{"style":217742},[34,458965],{"className":458966,"style":31814},[31813],[34,458968,458970],{"className":458969},[31818,31819,31820,31821],[34,458971,458973,458976,458979],{"className":458972},[31685,31821],[34,458974,33759],{"className":458975,"style":33774},[31685,31686,31821],[34,458977,32226],{"className":458978},[32024,31821],[34,458980,640],{"className":458981},[31685,31821],[34,458983,31829],{"className":458984},[31828],[34,458986,458988],{"className":458987},[31801],[34,458989,458991],{"className":458990,"style":32037},[31805],[34,458992],{},[34,458994,458996,459018],{"className":458995},[31642],[34,458997,458999],{"className":458998},[31646],[31648,459000,459001],{"xmlns":31650},[31652,459002,459003,459015],{},[31655,459004,459005,459007,459009,459011,459013],{},[31658,459006,54151],{},[31662,459008,223],{},[31658,459010,223320],{},[31662,459012,202],{},[31658,459014,34386],{},[31665,459016,459017],{"encoding":31667},"A+J=R",[34,459019,459021,459039,459057],{"className":459020,"ariaHidden":758},[31672],[34,459022,459024,459027,459030,459033,459036],{"className":459023},[31676],[34,459025],{"className":459026,"style":34944},[31680],[34,459028,54151],{"className":459029},[31685,31686],[34,459031],{"className":459032,"style":32604},[31845],[34,459034,223],{"className":459035},[32024],[34,459037],{"className":459038,"style":32604},[31845],[34,459040,459042,459045,459048,459051,459054],{"className":459041},[31676],[34,459043],{"className":459044,"style":33218},[31680],[34,459046,223320],{"className":459047,"style":223361},[31685,31686],[34,459049],{"className":459050,"style":31846},[31845],[34,459052,202],{"className":459053},[31850],[34,459055],{"className":459056,"style":31846},[31845],[34,459058,459060,459063],{"className":459059},[31676],[34,459061],{"className":459062,"style":33218},[31680],[34,459064,34386],{"className":459065,"style":34427},[31685,31686],[108,459067,459068,459069,459097,459098,459190,459191,86674,459263,459336,459337,347,459442,459444,459445,459543,459544,459572,459573,459742,459743,459910,459911,460195,460197,460198,460561,460562,460667,460668,452821,460772,43298,460926,455702,460954],{},"现在考虑 ",[34,459070,459072,459085],{"className":459071},[31642],[34,459073,459075],{"className":459074},[31646],[31648,459076,459077],{"xmlns":31650},[31652,459078,459079,459083],{},[31655,459080,459081],{},[31658,459082,33759],{},[31665,459084,33759],{"encoding":31667},[34,459086,459088],{"className":459087,"ariaHidden":758},[31672],[34,459089,459091,459094],{"className":459090},[31676],[34,459092],{"className":459093,"style":31681},[31680],[34,459095,33759],{"className":459096,"style":33774},[31685,31686]," 的情况。令 ",[34,459099,459101,459123],{"className":459100},[31642],[34,459102,459104],{"className":459103},[31646],[31648,459105,459106],{"xmlns":31650},[31652,459107,459108,459120],{},[31655,459109,459110,459112,459114],{},[31658,459111,54224],{},[31662,459113,202],{},[31743,459115,459116,459118],{},[31658,459117,106063],{},[31658,459119,33759],{},[31665,459121,459122],{"encoding":31667},"B = I_k",[34,459124,459126,459144],{"className":459125,"ariaHidden":758},[31672],[34,459127,459129,459132,459135,459138,459141],{"className":459128},[31676],[34,459130],{"className":459131,"style":33218},[31680],[34,459133,54224],{"className":459134,"style":54252},[31685,31686],[34,459136],{"className":459137,"style":31846},[31845],[34,459139,202],{"className":459140},[31850],[34,459142],{"className":459143,"style":31846},[31845],[34,459145,459147,459150],{"className":459146},[31676],[34,459148],{"className":459149,"style":70915},[31680],[34,459151,459153,459156],{"className":459152},[31685],[34,459154,106063],{"className":459155,"style":67482},[31685,31686],[34,459157,459159],{"className":459158},[31792],[34,459160,459162,459182],{"className":459161},[31796,31797],[34,459163,459165,459179],{"className":459164},[31801],[34,459166,459168],{"className":459167,"style":33833},[31805],[34,459169,459170,459173],{"style":217742},[34,459171],{"className":459172,"style":31814},[31813],[34,459174,459176],{"className":459175},[31818,31819,31820,31821],[34,459177,33759],{"className":459178,"style":33774},[31685,31686,31821],[34,459180,31829],{"className":459181},[31828],[34,459183,459185],{"className":459184},[31801],[34,459186,459188],{"className":459187,"style":31836},[31805],[34,459189],{},"。已知 ",[34,459192,459194,459215],{"className":459193},[31642],[34,459195,459197],{"className":459196},[31646],[31648,459198,459199],{"xmlns":31650},[31652,459200,459201,459213],{},[31655,459202,459203,459205,459207,459209,459211],{},[31658,459204,54151],{},[31662,459206,223],{},[31658,459208,223320],{},[31662,459210,202],{},[31658,459212,34386],{},[31665,459214,459017],{"encoding":31667},[34,459216,459218,459236,459254],{"className":459217,"ariaHidden":758},[31672],[34,459219,459221,459224,459227,459230,459233],{"className":459220},[31676],[34,459222],{"className":459223,"style":34944},[31680],[34,459225,54151],{"className":459226},[31685,31686],[34,459228],{"className":459229,"style":32604},[31845],[34,459231,223],{"className":459232},[32024],[34,459234],{"className":459235,"style":32604},[31845],[34,459237,459239,459242,459245,459248,459251],{"className":459238},[31676],[34,459240],{"className":459241,"style":33218},[31680],[34,459243,223320],{"className":459244,"style":223361},[31685,31686],[34,459246],{"className":459247,"style":31846},[31845],[34,459249,202],{"className":459250},[31850],[34,459252],{"className":459253,"style":31846},[31845],[34,459255,459257,459260],{"className":459256},[31676],[34,459258],{"className":459259,"style":33218},[31680],[34,459261,34386],{"className":459262,"style":34427},[31685,31686],[34,459264,459266,459288],{"className":459265},[31642],[34,459267,459269],{"className":459268},[31646],[31648,459270,459271],{"xmlns":31650},[31652,459272,459273,459285],{},[31655,459274,459275,459277,459279,459281,459283],{},[31658,459276,54224],{},[31662,459278,223],{},[31658,459280,223320],{},[31662,459282,202],{},[31658,459284,34386],{},[31665,459286,459287],{"encoding":31667},"B+J=R",[34,459289,459291,459309,459327],{"className":459290,"ariaHidden":758},[31672],[34,459292,459294,459297,459300,459303,459306],{"className":459293},[31676],[34,459295],{"className":459296,"style":34944},[31680],[34,459298,54224],{"className":459299,"style":54252},[31685,31686],[34,459301],{"className":459302,"style":32604},[31845],[34,459304,223],{"className":459305},[32024],[34,459307],{"className":459308,"style":32604},[31845],[34,459310,459312,459315,459318,459321,459324],{"className":459311},[31676],[34,459313],{"className":459314,"style":33218},[31680],[34,459316,223320],{"className":459317,"style":223361},[31685,31686],[34,459319],{"className":459320,"style":31846},[31845],[34,459322,202],{"className":459323},[31850],[34,459325],{"className":459326,"style":31846},[31845],[34,459328,459330,459333],{"className":459329},[31676],[34,459331],{"className":459332,"style":33218},[31680],[34,459334,34386],{"className":459335,"style":34427},[31685,31686],"。我们只需证明 ",[34,459338,459340,459370],{"className":459339},[31642],[34,459341,459343],{"className":459342},[31646],[31648,459344,459345],{"xmlns":31650},[31652,459346,459347,459367],{},[31655,459348,459349,459351,459353,459355,459357,459359,459361,459363,459365],{},[31662,459350,456],{"stretchy":702},[31658,459352,54151],{},[31662,459354,326344],{},[31658,459356,54224],{},[31662,459358,2557],{"stretchy":702},[31662,459360,223],{},[31658,459362,223320],{},[31662,459364,202],{},[31658,459366,34386],{},[31665,459368,459369],{"encoding":31667},"(A \\cap B) + J = R",[34,459371,459373,459394,459415,459433],{"className":459372,"ariaHidden":758},[31672],[34,459374,459376,459379,459382,459385,459388,459391],{"className":459375},[31676],[34,459377],{"className":459378,"style":31774},[31680],[34,459380,456],{"className":459381},[31782],[34,459383,54151],{"className":459384},[31685,31686],[34,459386],{"className":459387,"style":32604},[31845],[34,459389,326344],{"className":459390},[32024],[34,459392],{"className":459393,"style":32604},[31845],[34,459395,459397,459400,459403,459406,459409,459412],{"className":459396},[31676],[34,459398],{"className":459399,"style":31774},[31680],[34,459401,54224],{"className":459402,"style":54252},[31685,31686],[34,459404,2557],{"className":459405},[31690],[34,459407],{"className":459408,"style":32604},[31845],[34,459410,223],{"className":459411},[32024],[34,459413],{"className":459414,"style":32604},[31845],[34,459416,459418,459421,459424,459427,459430],{"className":459417},[31676],[34,459419],{"className":459420,"style":33218},[31680],[34,459422,223320],{"className":459423,"style":223361},[31685,31686],[34,459425],{"className":459426,"style":31846},[31845],[34,459428,202],{"className":459429},[31850],[34,459431],{"className":459432,"style":31846},[31845],[34,459434,459436,459439],{"className":459435},[31676],[34,459437],{"className":459438,"style":33218},[31680],[34,459440,34386],{"className":459441,"style":34427},[31685,31686],[58531,459443],{},"考察乘积 ",[34,459446,459448,459480],{"className":459447},[31642],[34,459449,459451],{"className":459450},[31646],[31648,459452,459453],{"xmlns":31650},[31652,459454,459455,459477],{},[31655,459456,459457,459459,459461,459463,459465,459467,459469,459471,459473,459475],{},[31662,459458,456],{"stretchy":702},[31658,459460,54151],{},[31662,459462,223],{},[31658,459464,223320],{},[31662,459466,2557],{"stretchy":702},[31662,459468,456],{"stretchy":702},[31658,459470,54224],{},[31662,459472,223],{},[31658,459474,223320],{},[31662,459476,2557],{"stretchy":702},[31665,459478,459479],{"encoding":31667},"(A+J)(B+J)",[34,459481,459483,459504,459531],{"className":459482,"ariaHidden":758},[31672],[34,459484,459486,459489,459492,459495,459498,459501],{"className":459485},[31676],[34,459487],{"className":459488,"style":31774},[31680],[34,459490,456],{"className":459491},[31782],[34,459493,54151],{"className":459494},[31685,31686],[34,459496],{"className":459497,"style":32604},[31845],[34,459499,223],{"className":459500},[32024],[34,459502],{"className":459503,"style":32604},[31845],[34,459505,459507,459510,459513,459516,459519,459522,459525,459528],{"className":459506},[31676],[34,459508],{"className":459509,"style":31774},[31680],[34,459511,223320],{"className":459512,"style":223361},[31685,31686],[34,459514,2557],{"className":459515},[31690],[34,459517,456],{"className":459518},[31782],[34,459520,54224],{"className":459521,"style":54252},[31685,31686],[34,459523],{"className":459524,"style":32604},[31845],[34,459526,223],{"className":459527},[32024],[34,459529],{"className":459530,"style":32604},[31845],[34,459532,459534,459537,459540],{"className":459533},[31676],[34,459535],{"className":459536,"style":31774},[31680],[34,459538,223320],{"className":459539,"style":223361},[31685,31686],[34,459541,2557],{"className":459542},[31690],"。\n由于 ",[34,459545,459547,459560],{"className":459546},[31642],[34,459548,459550],{"className":459549},[31646],[31648,459551,459552],{"xmlns":31650},[31652,459553,459554,459558],{},[31655,459555,459556],{},[31658,459557,34386],{},[31665,459559,34386],{"encoding":31667},[34,459561,459563],{"className":459562,"ariaHidden":758},[31672],[34,459564,459566,459569],{"className":459565},[31676],[34,459567],{"className":459568,"style":33218},[31680],[34,459570,34386],{"className":459571,"style":34427},[31685,31686]," 是幺环，\n",[34,459574,459576,459622],{"className":459575},[31642],[34,459577,459579],{"className":459578},[31646],[31648,459580,459581],{"xmlns":31650},[31652,459582,459583,459619],{},[31655,459584,459585,459587,459589,459591,459593,459595,459597,459599,459601,459603,459605,459607,459609,459611,459613,459615,459617],{},[31658,459586,34386],{},[31662,459588,202],{},[31658,459590,34386],{},[31662,459592,71191],{},[31658,459594,34386],{},[31662,459596,202],{},[31662,459598,456],{"stretchy":702},[31658,459600,54151],{},[31662,459602,223],{},[31658,459604,223320],{},[31662,459606,2557],{"stretchy":702},[31662,459608,456],{"stretchy":702},[31658,459610,54224],{},[31662,459612,223],{},[31658,459614,223320],{},[31662,459616,2557],{"stretchy":702},[31658,459618,2547],{"mathvariant":38079},[31665,459620,459621],{"encoding":31667},"R = R \\cdot R = (A+J)(B+J).",[34,459623,459625,459643,459661,459679,459700,459727],{"className":459624,"ariaHidden":758},[31672],[34,459626,459628,459631,459634,459637,459640],{"className":459627},[31676],[34,459629],{"className":459630,"style":33218},[31680],[34,459632,34386],{"className":459633,"style":34427},[31685,31686],[34,459635],{"className":459636,"style":31846},[31845],[34,459638,202],{"className":459639},[31850],[34,459641],{"className":459642,"style":31846},[31845],[34,459644,459646,459649,459652,459655,459658],{"className":459645},[31676],[34,459647],{"className":459648,"style":33218},[31680],[34,459650,34386],{"className":459651,"style":34427},[31685,31686],[34,459653],{"className":459654,"style":32604},[31845],[34,459656,71191],{"className":459657},[32024],[34,459659],{"className":459660,"style":32604},[31845],[34,459662,459664,459667,459670,459673,459676],{"className":459663},[31676],[34,459665],{"className":459666,"style":33218},[31680],[34,459668,34386],{"className":459669,"style":34427},[31685,31686],[34,459671],{"className":459672,"style":31846},[31845],[34,459674,202],{"className":459675},[31850],[34,459677],{"className":459678,"style":31846},[31845],[34,459680,459682,459685,459688,459691,459694,459697],{"className":459681},[31676],[34,459683],{"className":459684,"style":31774},[31680],[34,459686,456],{"className":459687},[31782],[34,459689,54151],{"className":459690},[31685,31686],[34,459692],{"className":459693,"style":32604},[31845],[34,459695,223],{"className":459696},[32024],[34,459698],{"className":459699,"style":32604},[31845],[34,459701,459703,459706,459709,459712,459715,459718,459721,459724],{"className":459702},[31676],[34,459704],{"className":459705,"style":31774},[31680],[34,459707,223320],{"className":459708,"style":223361},[31685,31686],[34,459710,2557],{"className":459711},[31690],[34,459713,456],{"className":459714},[31782],[34,459716,54224],{"className":459717,"style":54252},[31685,31686],[34,459719],{"className":459720,"style":32604},[31845],[34,459722,223],{"className":459723},[32024],[34,459725],{"className":459726,"style":32604},[31845],[34,459728,459730,459733,459736,459739],{"className":459729},[31676],[34,459731],{"className":459732,"style":31774},[31680],[34,459734,223320],{"className":459735,"style":223361},[31685,31686],[34,459737,2557],{"className":459738},[31690],[34,459740,2547],{"className":459741},[31685],"\n展开上式（利用理想加法和乘法的分配律）：\n",[34,459744,459746,459788],{"className":459745},[31642],[34,459747,459749],{"className":459748},[31646],[31648,459750,459751],{"xmlns":31650},[31652,459752,459753,459785],{},[31655,459754,459755,459757,459759,459761,459763,459765,459767,459769,459771,459773,459775,459777,459783],{},[31658,459756,34386],{},[31662,459758,202],{},[31658,459760,54151],{},[31658,459762,54224],{},[31662,459764,223],{},[31658,459766,54151],{},[31658,459768,223320],{},[31662,459770,223],{},[31658,459772,223320],{},[31658,459774,54224],{},[31662,459776,223],{},[37624,459778,459779,459781],{},[31658,459780,223320],{},[31749,459782,768],{},[31658,459784,2547],{"mathvariant":38079},[31665,459786,459787],{"encoding":31667},"R = AB + AJ + JB + J^2.",[34,459789,459791,459809,459830,459851,459872],{"className":459790,"ariaHidden":758},[31672],[34,459792,459794,459797,459800,459803,459806],{"className":459793},[31676],[34,459795],{"className":459796,"style":33218},[31680],[34,459798,34386],{"className":459799,"style":34427},[31685,31686],[34,459801],{"className":459802,"style":31846},[31845],[34,459804,202],{"className":459805},[31850],[34,459807],{"className":459808,"style":31846},[31845],[34,459810,459812,459815,459818,459821,459824,459827],{"className":459811},[31676],[34,459813],{"className":459814,"style":34944},[31680],[34,459816,54151],{"className":459817},[31685,31686],[34,459819,54224],{"className":459820,"style":54252},[31685,31686],[34,459822],{"className":459823,"style":32604},[31845],[34,459825,223],{"className":459826},[32024],[34,459828],{"className":459829,"style":32604},[31845],[34,459831,459833,459836,459839,459842,459845,459848],{"className":459832},[31676],[34,459834],{"className":459835,"style":34944},[31680],[34,459837,54151],{"className":459838},[31685,31686],[34,459840,223320],{"className":459841,"style":223361},[31685,31686],[34,459843],{"className":459844,"style":32604},[31845],[34,459846,223],{"className":459847},[32024],[34,459849],{"className":459850,"style":32604},[31845],[34,459852,459854,459857,459860,459863,459866,459869],{"className":459853},[31676],[34,459855],{"className":459856,"style":34944},[31680],[34,459858,223320],{"className":459859,"style":223361},[31685,31686],[34,459861,54224],{"className":459862,"style":54252},[31685,31686],[34,459864],{"className":459865,"style":32604},[31845],[34,459867,223],{"className":459868},[32024],[34,459870],{"className":459871,"style":32604},[31845],[34,459873,459875,459878,459907],{"className":459874},[31676],[34,459876],{"className":459877,"style":37774},[31680],[34,459879,459881,459884],{"className":459880},[31685],[34,459882,223320],{"className":459883,"style":223361},[31685,31686],[34,459885,459887],{"className":459886},[31792],[34,459888,459890],{"className":459889},[31796],[34,459891,459893],{"className":459892},[31801],[34,459894,459896],{"className":459895,"style":37774},[31805],[34,459897,459898,459901],{"style":37777},[34,459899],{"className":459900,"style":31814},[31813],[34,459902,459904],{"className":459903},[31818,31819,31820,31821],[34,459905,768],{"className":459906},[31685,31821],[34,459908,2547],{"className":459909},[31685],"\n分析各项的包含关系：",[2230,459912,459913,459994,460053,460111],{},[108,459914,459915,459993],{},[34,459916,459918,459942],{"className":459917},[31642],[34,459919,459921],{"className":459920},[31646],[31648,459922,459923],{"xmlns":31650},[31652,459924,459925,459939],{},[31655,459926,459927,459929,459931,459933,459935,459937],{},[31658,459928,54151],{},[31658,459930,54224],{},[31662,459932,131855],{},[31658,459934,54151],{},[31662,459936,326344],{},[31658,459938,54224],{},[31665,459940,459941],{"encoding":31667},"AB \\subseteq A \\cap B",[34,459943,459945,459966,459984],{"className":459944,"ariaHidden":758},[31672],[34,459946,459948,459951,459954,459957,459960,459963],{"className":459947},[31676],[34,459949],{"className":459950,"style":195047},[31680],[34,459952,54151],{"className":459953},[31685,31686],[34,459955,54224],{"className":459956,"style":54252},[31685,31686],[34,459958],{"className":459959,"style":31846},[31845],[34,459961,131855],{"className":459962},[31850],[34,459964],{"className":459965,"style":31846},[31845],[34,459967,459969,459972,459975,459978,459981],{"className":459968},[31676],[34,459970],{"className":459971,"style":33218},[31680],[34,459973,54151],{"className":459974},[31685,31686],[34,459976],{"className":459977,"style":32604},[31845],[34,459979,326344],{"className":459980},[32024],[34,459982],{"className":459983,"style":32604},[31845],[34,459985,459987,459990],{"className":459986},[31676],[34,459988],{"className":459989,"style":33218},[31680],[34,459991,54224],{"className":459992,"style":54252},[31685,31686]," (两个理想的积总是包含在交集中)。",[108,459995,459996,460052],{},[34,459997,459999,460019],{"className":459998},[31642],[34,460000,460002],{"className":460001},[31646],[31648,460003,460004],{"xmlns":31650},[31652,460005,460006,460016],{},[31655,460007,460008,460010,460012,460014],{},[31658,460009,54151],{},[31658,460011,223320],{},[31662,460013,131855],{},[31658,460015,223320],{},[31665,460017,460018],{"encoding":31667},"AJ \\subseteq J",[34,460020,460022,460043],{"className":460021,"ariaHidden":758},[31672],[34,460023,460025,460028,460031,460034,460037,460040],{"className":460024},[31676],[34,460026],{"className":460027,"style":195047},[31680],[34,460029,54151],{"className":460030},[31685,31686],[34,460032,223320],{"className":460033,"style":223361},[31685,31686],[34,460035],{"className":460036,"style":31846},[31845],[34,460038,131855],{"className":460039},[31850],[34,460041],{"className":460042,"style":31846},[31845],[34,460044,460046,460049],{"className":460045},[31676],[34,460047],{"className":460048,"style":33218},[31680],[34,460050,223320],{"className":460051,"style":223361},[31685,31686]," (理想的定义)。",[108,460054,460055,347],{},[34,460056,460058,460078],{"className":460057},[31642],[34,460059,460061],{"className":460060},[31646],[31648,460062,460063],{"xmlns":31650},[31652,460064,460065,460075],{},[31655,460066,460067,460069,460071,460073],{},[31658,460068,223320],{},[31658,460070,54224],{},[31662,460072,131855],{},[31658,460074,223320],{},[31665,460076,460077],{"encoding":31667},"JB \\subseteq J",[34,460079,460081,460102],{"className":460080,"ariaHidden":758},[31672],[34,460082,460084,460087,460090,460093,460096,460099],{"className":460083},[31676],[34,460085],{"className":460086,"style":195047},[31680],[34,460088,223320],{"className":460089,"style":223361},[31685,31686],[34,460091,54224],{"className":460092,"style":54252},[31685,31686],[34,460094],{"className":460095,"style":31846},[31845],[34,460097,131855],{"className":460098},[31850],[34,460100],{"className":460101,"style":31846},[31845],[34,460103,460105,460108],{"className":460104},[31676],[34,460106],{"className":460107,"style":33218},[31680],[34,460109,223320],{"className":460110,"style":223361},[31685,31686],[108,460112,460113,347],{},[34,460114,460116,460138],{"className":460115},[31642],[34,460117,460119],{"className":460118},[31646],[31648,460120,460121],{"xmlns":31650},[31652,460122,460123,460135],{},[31655,460124,460125,460131,460133],{},[37624,460126,460127,460129],{},[31658,460128,223320],{},[31749,460130,768],{},[31662,460132,131855],{},[31658,460134,223320],{},[31665,460136,460137],{"encoding":31667},"J^2 \\subseteq J",[34,460139,460141,460186],{"className":460140,"ariaHidden":758},[31672],[34,460142,460144,460148,460177,460180,460183],{"className":460143},[31676],[34,460145],{"className":460146,"style":460147},[31680],"height:0.9501em;vertical-align:-0.136em;",[34,460149,460151,460154],{"className":460150},[31685],[34,460152,223320],{"className":460153,"style":223361},[31685,31686],[34,460155,460157],{"className":460156},[31792],[34,460158,460160],{"className":460159},[31796],[34,460161,460163],{"className":460162},[31801],[34,460164,460166],{"className":460165,"style":37774},[31805],[34,460167,460168,460171],{"style":37777},[34,460169],{"className":460170,"style":31814},[31813],[34,460172,460174],{"className":460173},[31818,31819,31820,31821],[34,460175,768],{"className":460176},[31685,31821],[34,460178],{"className":460179,"style":31846},[31845],[34,460181,131855],{"className":460182},[31850],[34,460184],{"className":460185,"style":31846},[31845],[34,460187,460189,460192],{"className":460188},[31676],[34,460190],{"className":460191,"style":33218},[31680],[34,460193,223320],{"className":460194,"style":223361},[31685,31686],[58531,460196],{},"因此，\n",[34,460199,460201,460283],{"className":460200},[31642],[34,460202,460204],{"className":460203},[31646],[31648,460205,460206],{"xmlns":31650},[31652,460207,460208,460280],{},[31655,460209,460210,460212,460214,460216,460218,460220,460222,460224,460226,460228,460230,460232,460238,460240,460242,460244,460246,460248,460250,460252,460254,460256,460258,460260,460262,460264,460266,460268,460270,460272,460274,460276,460278],{},[31658,460211,34386],{},[31662,460213,202],{},[31658,460215,54151],{},[31658,460217,54224],{},[31662,460219,223],{},[31658,460221,54151],{},[31658,460223,223320],{},[31662,460225,223],{},[31658,460227,223320],{},[31658,460229,54224],{},[31662,460231,223],{},[37624,460233,460234,460236],{},[31658,460235,223320],{},[31749,460237,768],{},[31662,460239,131855],{},[31662,460241,456],{"stretchy":702},[31658,460243,54151],{},[31662,460245,326344],{},[31658,460247,54224],{},[31662,460249,2557],{"stretchy":702},[31662,460251,223],{},[31658,460253,223320],{},[31662,460255,223],{},[31658,460257,223320],{},[31662,460259,223],{},[31658,460261,223320],{},[31662,460263,202],{},[31662,460265,456],{"stretchy":702},[31658,460267,54151],{},[31662,460269,326344],{},[31658,460271,54224],{},[31662,460273,2557],{"stretchy":702},[31662,460275,223],{},[31658,460277,223320],{},[31658,460279,2547],{"mathvariant":38079},[31665,460281,460282],{"encoding":31667},"R = AB + AJ + JB + J^2 \\subseteq (A \\cap B) + J + J + J = (A \\cap B) + J.",[34,460284,460286,460304,460325,460346,460367,460411,460432,460453,460471,460489,460507,460528,460549],{"className":460285,"ariaHidden":758},[31672],[34,460287,460289,460292,460295,460298,460301],{"className":460288},[31676],[34,460290],{"className":460291,"style":33218},[31680],[34,460293,34386],{"className":460294,"style":34427},[31685,31686],[34,460296],{"className":460297,"style":31846},[31845],[34,460299,202],{"className":460300},[31850],[34,460302],{"className":460303,"style":31846},[31845],[34,460305,460307,460310,460313,460316,460319,460322],{"className":460306},[31676],[34,460308],{"className":460309,"style":34944},[31680],[34,460311,54151],{"className":460312},[31685,31686],[34,460314,54224],{"className":460315,"style":54252},[31685,31686],[34,460317],{"className":460318,"style":32604},[31845],[34,460320,223],{"className":460321},[32024],[34,460323],{"className":460324,"style":32604},[31845],[34,460326,460328,460331,460334,460337,460340,460343],{"className":460327},[31676],[34,460329],{"className":460330,"style":34944},[31680],[34,460332,54151],{"className":460333},[31685,31686],[34,460335,223320],{"className":460336,"style":223361},[31685,31686],[34,460338],{"className":460339,"style":32604},[31845],[34,460341,223],{"className":460342},[32024],[34,460344],{"className":460345,"style":32604},[31845],[34,460347,460349,460352,460355,460358,460361,460364],{"className":460348},[31676],[34,460350],{"className":460351,"style":34944},[31680],[34,460353,223320],{"className":460354,"style":223361},[31685,31686],[34,460356,54224],{"className":460357,"style":54252},[31685,31686],[34,460359],{"className":460360,"style":32604},[31845],[34,460362,223],{"className":460363},[32024],[34,460365],{"className":460366,"style":32604},[31845],[34,460368,460370,460373,460402,460405,460408],{"className":460369},[31676],[34,460371],{"className":460372,"style":460147},[31680],[34,460374,460376,460379],{"className":460375},[31685],[34,460377,223320],{"className":460378,"style":223361},[31685,31686],[34,460380,460382],{"className":460381},[31792],[34,460383,460385],{"className":460384},[31796],[34,460386,460388],{"className":460387},[31801],[34,460389,460391],{"className":460390,"style":37774},[31805],[34,460392,460393,460396],{"style":37777},[34,460394],{"className":460395,"style":31814},[31813],[34,460397,460399],{"className":460398},[31818,31819,31820,31821],[34,460400,768],{"className":460401},[31685,31821],[34,460403],{"className":460404,"style":31846},[31845],[34,460406,131855],{"className":460407},[31850],[34,460409],{"className":460410,"style":31846},[31845],[34,460412,460414,460417,460420,460423,460426,460429],{"className":460413},[31676],[34,460415],{"className":460416,"style":31774},[31680],[34,460418,456],{"className":460419},[31782],[34,460421,54151],{"className":460422},[31685,31686],[34,460424],{"className":460425,"style":32604},[31845],[34,460427,326344],{"className":460428},[32024],[34,460430],{"className":460431,"style":32604},[31845],[34,460433,460435,460438,460441,460444,460447,460450],{"className":460434},[31676],[34,460436],{"className":460437,"style":31774},[31680],[34,460439,54224],{"className":460440,"style":54252},[31685,31686],[34,460442,2557],{"className":460443},[31690],[34,460445],{"className":460446,"style":32604},[31845],[34,460448,223],{"className":460449},[32024],[34,460451],{"className":460452,"style":32604},[31845],[34,460454,460456,460459,460462,460465,460468],{"className":460455},[31676],[34,460457],{"className":460458,"style":34944},[31680],[34,460460,223320],{"className":460461,"style":223361},[31685,31686],[34,460463],{"className":460464,"style":32604},[31845],[34,460466,223],{"className":460467},[32024],[34,460469],{"className":460470,"style":32604},[31845],[34,460472,460474,460477,460480,460483,460486],{"className":460473},[31676],[34,460475],{"className":460476,"style":34944},[31680],[34,460478,223320],{"className":460479,"style":223361},[31685,31686],[34,460481],{"className":460482,"style":32604},[31845],[34,460484,223],{"className":460485},[32024],[34,460487],{"className":460488,"style":32604},[31845],[34,460490,460492,460495,460498,460501,460504],{"className":460491},[31676],[34,460493],{"className":460494,"style":33218},[31680],[34,460496,223320],{"className":460497,"style":223361},[31685,31686],[34,460499],{"className":460500,"style":31846},[31845],[34,460502,202],{"className":460503},[31850],[34,460505],{"className":460506,"style":31846},[31845],[34,460508,460510,460513,460516,460519,460522,460525],{"className":460509},[31676],[34,460511],{"className":460512,"style":31774},[31680],[34,460514,456],{"className":460515},[31782],[34,460517,54151],{"className":460518},[31685,31686],[34,460520],{"className":460521,"style":32604},[31845],[34,460523,326344],{"className":460524},[32024],[34,460526],{"className":460527,"style":32604},[31845],[34,460529,460531,460534,460537,460540,460543,460546],{"className":460530},[31676],[34,460532],{"className":460533,"style":31774},[31680],[34,460535,54224],{"className":460536,"style":54252},[31685,31686],[34,460538,2557],{"className":460539},[31690],[34,460541],{"className":460542,"style":32604},[31845],[34,460544,223],{"className":460545},[32024],[34,460547],{"className":460548,"style":32604},[31845],[34,460550,460552,460555,460558],{"className":460551},[31676],[34,460553],{"className":460554,"style":33218},[31680],[34,460556,223320],{"className":460557,"style":223361},[31685,31686],[34,460559,2547],{"className":460560},[31685],"\n显然 ",[34,460563,460565,460595],{"className":460564},[31642],[34,460566,460568],{"className":460567},[31646],[31648,460569,460570],{"xmlns":31650},[31652,460571,460572,460592],{},[31655,460573,460574,460576,460578,460580,460582,460584,460586,460588,460590],{},[31662,460575,456],{"stretchy":702},[31658,460577,54151],{},[31662,460579,326344],{},[31658,460581,54224],{},[31662,460583,2557],{"stretchy":702},[31662,460585,223],{},[31658,460587,223320],{},[31662,460589,131855],{},[31658,460591,34386],{},[31665,460593,460594],{"encoding":31667},"(A \\cap B) + J \\subseteq R",[34,460596,460598,460619,460640,460658],{"className":460597,"ariaHidden":758},[31672],[34,460599,460601,460604,460607,460610,460613,460616],{"className":460600},[31676],[34,460602],{"className":460603,"style":31774},[31680],[34,460605,456],{"className":460606},[31782],[34,460608,54151],{"className":460609},[31685,31686],[34,460611],{"className":460612,"style":32604},[31845],[34,460614,326344],{"className":460615},[32024],[34,460617],{"className":460618,"style":32604},[31845],[34,460620,460622,460625,460628,460631,460634,460637],{"className":460621},[31676],[34,460623],{"className":460624,"style":31774},[31680],[34,460626,54224],{"className":460627,"style":54252},[31685,31686],[34,460629,2557],{"className":460630},[31690],[34,460632],{"className":460633,"style":32604},[31845],[34,460635,223],{"className":460636},[32024],[34,460638],{"className":460639,"style":32604},[31845],[34,460641,460643,460646,460649,460652,460655],{"className":460642},[31676],[34,460644],{"className":460645,"style":195047},[31680],[34,460647,223320],{"className":460648,"style":223361},[31685,31686],[34,460650],{"className":460651,"style":31846},[31845],[34,460653,131855],{"className":460654},[31850],[34,460656],{"className":460657,"style":31846},[31845],[34,460659,460661,460664],{"className":460660},[31676],[34,460662],{"className":460663,"style":33218},[31680],[34,460665,34386],{"className":460666,"style":34427},[31685,31686],"。\n故 ",[34,460669,460671,460700],{"className":460670},[31642],[34,460672,460674],{"className":460673},[31646],[31648,460675,460676],{"xmlns":31650},[31652,460677,460678,460698],{},[31655,460679,460680,460682,460684,460686,460688,460690,460692,460694,460696],{},[31662,460681,456],{"stretchy":702},[31658,460683,54151],{},[31662,460685,326344],{},[31658,460687,54224],{},[31662,460689,2557],{"stretchy":702},[31662,460691,223],{},[31658,460693,223320],{},[31662,460695,202],{},[31658,460697,34386],{},[31665,460699,459369],{"encoding":31667},[34,460701,460703,460724,460745,460763],{"className":460702,"ariaHidden":758},[31672],[34,460704,460706,460709,460712,460715,460718,460721],{"className":460705},[31676],[34,460707],{"className":460708,"style":31774},[31680],[34,460710,456],{"className":460711},[31782],[34,460713,54151],{"className":460714},[31685,31686],[34,460716],{"className":460717,"style":32604},[31845],[34,460719,326344],{"className":460720},[32024],[34,460722],{"className":460723,"style":32604},[31845],[34,460725,460727,460730,460733,460736,460739,460742],{"className":460726},[31676],[34,460728],{"className":460729,"style":31774},[31680],[34,460731,54224],{"className":460732,"style":54252},[31685,31686],[34,460734,2557],{"className":460735},[31690],[34,460737],{"className":460738,"style":32604},[31845],[34,460740,223],{"className":460741},[32024],[34,460743],{"className":460744,"style":32604},[31845],[34,460746,460748,460751,460754,460757,460760],{"className":460747},[31676],[34,460749],{"className":460750,"style":33218},[31680],[34,460752,223320],{"className":460753,"style":223361},[31685,31686],[34,460755],{"className":460756,"style":31846},[31845],[34,460758,202],{"className":460759},[31850],[34,460761],{"className":460762,"style":31846},[31845],[34,460764,460766,460769],{"className":460765},[31676],[34,460767],{"className":460768,"style":33218},[31680],[34,460770,34386],{"className":460771,"style":34427},[31685,31686],[34,460773,460775,460804],{"className":460774},[31642],[34,460776,460778],{"className":460777},[31646],[31648,460779,460780],{"xmlns":31650},[31652,460781,460782,460802],{},[31655,460783,460784,460790,460792,460794,460796],{},[31743,460785,460786,460788],{},[31658,460787,106063],{},[31749,460789,640],{},[31662,460791,326344],{},[31662,460793,38874],{},[31662,460795,326344],{},[31743,460797,460798,460800],{},[31658,460799,106063],{},[31658,460801,33759],{},[31665,460803,458504],{"encoding":31667},[34,460805,460807,460862,460880],{"className":460806,"ariaHidden":758},[31672],[34,460808,460810,460813,460853,460856,460859],{"className":460809},[31676],[34,460811],{"className":460812,"style":70915},[31680],[34,460814,460816,460819],{"className":460815},[31685],[34,460817,106063],{"className":460818,"style":67482},[31685,31686],[34,460820,460822],{"className":460821},[31792],[34,460823,460825,460845],{"className":460824},[31796,31797],[34,460826,460828,460842],{"className":460827},[31801],[34,460829,460831],{"className":460830,"style":31806},[31805],[34,460832,460833,460836],{"style":217742},[34,460834],{"className":460835,"style":31814},[31813],[34,460837,460839],{"className":460838},[31818,31819,31820,31821],[34,460840,640],{"className":460841},[31685,31821],[34,460843,31829],{"className":460844},[31828],[34,460846,460848],{"className":460847},[31801],[34,460849,460851],{"className":460850,"style":31836},[31805],[34,460852],{},[34,460854],{"className":460855,"style":32604},[31845],[34,460857,326344],{"className":460858},[32024],[34,460860],{"className":460861,"style":32604},[31845],[34,460863,460865,460868,460871,460874,460877],{"className":460864},[31676],[34,460866],{"className":460867,"style":147334},[31680],[34,460869,38874],{"className":460870},[34006],[34,460872],{"className":460873,"style":32604},[31845],[34,460875,326344],{"className":460876},[32024],[34,460878],{"className":460879,"style":32604},[31845],[34,460881,460883,460886],{"className":460882},[31676],[34,460884],{"className":460885,"style":70915},[31680],[34,460887,460889,460892],{"className":460888},[31685],[34,460890,106063],{"className":460891,"style":67482},[31685,31686],[34,460893,460895],{"className":460894},[31792],[34,460896,460898,460918],{"className":460897},[31796,31797],[34,460899,460901,460915],{"className":460900},[31801],[34,460902,460904],{"className":460903,"style":33833},[31805],[34,460905,460906,460909],{"style":217742},[34,460907],{"className":460908,"style":31814},[31813],[34,460910,460912],{"className":460911},[31818,31819,31820,31821],[34,460913,33759],{"className":460914,"style":33774},[31685,31686,31821],[34,460916,31829],{"className":460917},[31828],[34,460919,460921],{"className":460920},[31801],[34,460922,460924],{"className":460923,"style":31836},[31805],[34,460925],{},[34,460927,460929,460942],{"className":460928},[31642],[34,460930,460932],{"className":460931},[31646],[31648,460933,460934],{"xmlns":31650},[31652,460935,460936,460940],{},[31655,460937,460938],{},[31658,460939,223320],{},[31665,460941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3.3.2.",[34,460988,460990,461003],{"className":460989},[31642],[34,460991,460993],{"className":460992},[31646],[31648,460994,460995],{"xmlns":31650},[31652,460996,460997,461001],{},[31655,460998,460999],{},[31658,461000,34386],{},[31665,461002,34386],{"encoding":31667},[34,461004,461006],{"className":461005,"ariaHidden":758},[31672],[34,461007,461009,461012],{"className":461008},[31676],[34,461010],{"className":461011,"style":33218},[31680],[34,461013,34386],{"className":461014,"style":34427},[31685,31686],[34,461016,461018,461055],{"className":461017},[31642],[34,461019,461021],{"className":461020},[31646],[31648,461022,461023],{"xmlns":31650},[31652,461024,461025,461053],{},[31655,461026,461027,461033,461035,461037,461039,461045,461047,461049,461051],{},[31743,461028,461029,461031],{},[31658,461030,106063],{},[31749,461032,640],{},[31662,461034,31960],{"separator":758},[31662,461036,33934],{},[31662,461038,31960],{"separator":758},[31743,461040,461041,461043],{},[31658,461042,106063],{},[31658,461044,33759],{},[31662,461046,31960],{"separator":758},[31658,461048,223320],{},[31662,461050,440828],{},[31658,461052,34386],{},[31665,461054,458111],{"encoding":31667},[34,461056,461058,461180],{"className":461057,"ariaHidden":758},[31672],[34,461059,461061,461064,461104,461107,461110,461113,461116,461119,461122,461162,461165,461168,461171,461174,461177],{"className":461060},[31676],[34,461062],{"className":461063,"style":70739},[31680],[34,461065,461067,461070],{"className":461066},[31685],[34,461068,106063],{"className":461069,"style":67482},[31685,31686],[34,461071,461073],{"className":461072},[31792],[34,461074,461076,461096],{"className":461075},[31796,31797],[34,461077,461079,461093],{"className":461078},[31801],[34,461080,461082],{"className":461081,"style":31806},[31805],[34,461083,461084,461087],{"style":217742},[34,461085],{"className":461086,"style":31814},[31813],[34,461088,461090],{"className":461089},[31818,31819,31820,31821],[34,461091,640],{"className":461092},[31685,31821],[34,461094,31829],{"className":461095},[31828],[34,461097,461099],{"className":461098},[31801],[34,461100,461102],{"className":461101,"style":31836},[31805],[34,461103],{},[34,461105,31960],{"className":461106},[32119],[34,461108],{"className":461109,"style":32123},[31845],[34,461111,33934],{"className":461112},[34006],[34,461114],{"className":461115,"style":32123},[31845],[34,461117,31960],{"className":461118},[32119],[34,461120],{"className":461121,"style":32123},[31845],[34,461123,461125,461128],{"className":461124},[31685],[34,461126,106063],{"className":461127,"style":67482},[31685,31686],[34,461129,461131],{"className":461130},[31792],[34,461132,461134,461154],{"className":461133},[31796,31797],[34,461135,461137,461151],{"className":461136},[31801],[34,461138,461140],{"className":461139,"style":33833},[31805],[34,461141,461142,461145],{"style":217742},[34,461143],{"className":461144,"style":31814},[31813],[34,461146,461148],{"className":461147},[31818,31819,31820,31821],[34,461149,33759],{"className":461150,"style":33774},[31685,31686,31821],[34,461152,31829],{"className":461153},[31828],[34,461155,461157],{"className":461156},[31801],[34,461158,461160],{"className":461159,"style":31836},[31805],[34,461161],{},[34,461163,31960],{"className":461164},[32119],[34,461166],{"className":461167,"style":32123},[31845],[34,461169,223320],{"className":461170,"style":223361},[31685,31686],[34,461172],{"className":461173,"style":32604},[31845],[34,461175,440828],{"className":461176},[32024,129261],[34,461178],{"className":461179,"style":32604},[31845],[34,461181,461183,461186],{"className":461182},[31676],[34,461184],{"className":461185,"style":33218},[31680],[34,461187,34386],{"className":461188,"style":34427},[31685,31686],[34,461190,461192,461213],{"className":461191},[31642],[34,461193,461195],{"className":461194},[31646],[31648,461196,461197],{"xmlns":31650},[31652,461198,461199,461211],{},[31655,461200,461201,461207,461209],{},[31743,461202,461203,461205],{},[31658,461204,106063],{},[31658,461206,31747],{},[31662,461208,31960],{"separator":758},[31658,461210,223320],{},[31665,461212,458270],{"encoding":31667},[34,461214,461216],{"className":461215,"ariaHidden":758},[31672],[34,461217,461219,461222,461262,461265,461268],{"className":461218},[31676],[34,461220],{"className":461221,"style":70739},[31680],[34,461223,461225,461228],{"className":461224},[31685],[34,461226,106063],{"className":461227,"style":67482},[31685,31686],[34,461229,461231],{"className":461230},[31792],[34,461232,461234,461254],{"className":461233},[31796,31797],[34,461235,461237,461251],{"className":461236},[31801],[34,461238,461240],{"className":461239,"style":32362},[31805],[34,461241,461242,461245],{"style":217742},[34,461243],{"className":461244,"style":31814},[31813],[34,461246,461248],{"className":461247},[31818,31819,31820,31821],[34,461249,31747],{"className":461250},[31685,31686,31821],[34,461252,31829],{"className":461253},[31828],[34,461255,461257],{"className":461256},[31801],[34,461258,461260],{"className":461259,"style":31836},[31805],[34,461261],{},[34,461263,31960],{"className":461264},[32119],[34,461266],{"className":461267,"style":32123},[31845],[34,461269,223320],{"className":461270,"style":223361},[31685,31686]," 都互素，则 ",[34,461273,461275,461301],{"className":461274},[31642],[34,461276,461278],{"className":461277},[31646],[31648,461279,461280],{"xmlns":31650},[31652,461281,461282,461298],{},[31655,461283,461284,461290,461292],{},[31743,461285,461286,461288],{},[31658,461287,106063],{},[31749,461289,640],{},[31662,461291,38874],{},[31743,461293,461294,461296],{},[31658,461295,106063],{},[31658,461297,33759],{},[31665,461299,461300],{"encoding":31667},"I_1 \\cdots I_k",[34,461302,461304],{"className":461303,"ariaHidden":758},[31672],[34,461305,461307,461310,461350,461353,461356,461359],{"className":461306},[31676],[34,461308],{"className":461309,"style":70915},[31680],[34,461311,461313,461316],{"className":461312},[31685],[34,461314,106063],{"className":461315,"style":67482},[31685,31686],[34,461317,461319],{"className":461318},[31792],[34,461320,461322,461342],{"className":461321},[31796,31797],[34,461323,461325,461339],{"className":461324},[31801],[34,461326,461328],{"className":461327,"style":31806},[31805],[34,461329,461330,461333],{"style":217742},[34,461331],{"className":461332,"style":31814},[31813],[34,461334,461336],{"className":461335},[31818,31819,31820,31821],[34,461337,640],{"className":461338},[31685,31821],[34,461340,31829],{"className":461341},[31828],[34,461343,461345],{"className":461344},[31801],[34,461346,461348],{"className":461347,"style":31836},[31805],[34,461349],{},[34,461351],{"className":461352,"style":32123},[31845],[34,461354,38874],{"className":461355},[34006],[34,461357],{"className":461358,"style":32123},[31845],[34,461360,461362,461365],{"className":461361},[31685],[34,461363,106063],{"className":461364,"style":67482},[31685,31686],[34,461366,461368],{"className":461367},[31792],[34,461369,461371,461391],{"className":461370},[31796,31797],[34,461372,461374,461388],{"className":461373},[31801],[34,461375,461377],{"className":461376,"style":33833},[31805],[34,461378,461379,461382],{"style":217742},[34,461380],{"className":461381,"style":31814},[31813],[34,461383,461385],{"className":461384},[31818,31819,31820,31821],[34,461386,33759],{"className":461387,"style":33774},[31685,31686,31821],[34,461389,31829],{"className":461390},[31828],[34,461392,461394],{"className":461393},[31801],[34,461395,461397],{"className":461396,"style":31836},[31805],[34,461398],{},[34,461400,461402,461415],{"className":461401},[31642],[34,461403,461405],{"className":461404},[31646],[31648,461406,461407],{"xmlns":31650},[31652,461408,461409,461413],{},[31655,461410,461411],{},[31658,461412,223320],{},[31665,461414,223320],{"encoding":31667},[34,461416,461418],{"className":461417,"ariaHidden":758},[31672],[34,461419,461421,461424],{"className":461420},[31676],[34,461422],{"className":461423,"style":33218},[31680],[34,461425,223320],{"className":461426,"style":223361},[31685,31686]," 也互素。\n",[71807,461429,461430,461431,461513,461514,461640,461641,43298,461795,461823,461824,462099,462100,462291,462292,462567,462568,462571,462572,462794,462795,463150],{},"(注：这是因为在交换幺环中，若 ",[34,461432,461434,461455],{"className":461433},[31642],[34,461435,461437],{"className":461436},[31646],[31648,461438,461439],{"xmlns":31650},[31652,461440,461441,461453],{},[31655,461442,461443,461449,461451],{},[31743,461444,461445,461447],{},[31658,461446,106063],{},[31658,461448,31747],{},[31662,461450,31960],{"separator":758},[31658,461452,223320],{},[31665,461454,458270],{"encoding":31667},[34,461456,461458],{"className":461457,"ariaHidden":758},[31672],[34,461459,461461,461464,461504,461507,461510],{"className":461460},[31676],[34,461462],{"className":461463,"style":70739},[31680],[34,461465,461467,461470],{"className":461466},[31685],[34,461468,106063],{"className":461469,"style":67482},[31685,31686],[34,461471,461473],{"className":461472},[31792],[34,461474,461476,461496],{"className":461475},[31796,31797],[34,461477,461479,461493],{"className":461478},[31801],[34,461480,461482],{"className":461481,"style":32362},[31805],[34,461483,461484,461487],{"style":217742},[34,461485],{"className":461486,"style":31814},[31813],[34,461488,461490],{"className":461489},[31818,31819,31820,31821],[34,461491,31747],{"className":461492},[31685,31686,31821],[34,461494,31829],{"className":461495},[31828],[34,461497,461499],{"className":461498},[31801],[34,461500,461502],{"className":461501,"style":31836},[31805],[34,461503],{},[34,461505,31960],{"className":461506},[32119],[34,461508],{"className":461509,"style":32123},[31845],[34,461511,223320],{"className":461512,"style":223361},[31685,31686]," 互素，则积 ",[34,461515,461517,461542],{"className":461516},[31642],[34,461518,461520],{"className":461519},[31646],[31648,461521,461522],{"xmlns":31650},[31652,461523,461524,461540],{},[31655,461525,461526,461532,461534],{},[31743,461527,461528,461530],{},[31658,461529,106063],{},[31749,461531,640],{},[31662,461533,38874],{},[31743,461535,461536,461538],{},[31658,461537,106063],{},[31658,461539,33759],{},[31665,461541,461300],{"encoding":31667},[34,461543,461545],{"className":461544,"ariaHidden":758},[31672],[34,461546,461548,461551,461591,461594,461597,461600],{"className":461547},[31676],[34,461549],{"className":461550,"style":70915},[31680],[34,461552,461554,461557],{"className":461553},[31685],[34,461555,106063],{"className":461556,"style":67482},[31685,31686],[34,461558,461560],{"className":461559},[31792],[34,461561,461563,461583],{"className":461562},[31796,31797],[34,461564,461566,461580],{"className":461565},[31801],[34,461567,461569],{"className":461568,"style":31806},[31805],[34,461570,461571,461574],{"style":217742},[34,461572],{"className":461573,"style":31814},[31813],[34,461575,461577],{"className":461576},[31818,31819,31820,31821],[34,461578,640],{"className":461579},[31685,31821],[34,461581,31829],{"className":461582},[31828],[34,461584,461586],{"className":461585},[31801],[34,461587,461589],{"className":461588,"style":31836},[31805],[34,461590],{},[34,461592],{"className":461593,"style":32123},[31845],[34,461595,38874],{"className":461596},[34006],[34,461598],{"className":461599,"style":32123},[31845],[34,461601,461603,461606],{"className":461602},[31685],[34,461604,106063],{"className":461605,"style":67482},[31685,31686],[34,461607,461609],{"className":461608},[31792],[34,461610,461612,461632],{"className":461611},[31796,31797],[34,461613,461615,461629],{"className":461614},[31801],[34,461616,461618],{"className":461617,"style":33833},[31805],[34,461619,461620,461623],{"style":217742},[34,461621],{"className":461622,"style":31814},[31813],[34,461624,461626],{"className":461625},[31818,31819,31820,31821],[34,461627,33759],{"className":461628,"style":33774},[31685,31686,31821],[34,461630,31829],{"className":461631},[31828],[34,461633,461635],{"className":461634},[31801],[34,461636,461638],{"className":461637,"style":31836},[31805],[34,461639],{}," 包含了交集 ",[34,461642,461644,461673],{"className":461643},[31642],[34,461645,461647],{"className":461646},[31646],[31648,461648,461649],{"xmlns":31650},[31652,461650,461651,461671],{},[31655,461652,461653,461659,461661,461663,461665],{},[31743,461654,461655,461657],{},[31658,461656,106063],{},[31749,461658,640],{},[31662,461660,326344],{},[31662,461662,38874],{},[31662,461664,326344],{},[31743,461666,461667,461669],{},[31658,461668,106063],{},[31658,461670,33759],{},[31665,461672,458504],{"encoding":31667},[34,461674,461676,461731,461749],{"className":461675,"ariaHidden":758},[31672],[34,461677,461679,461682,461722,461725,461728],{"className":461678},[31676],[34,461680],{"className":461681,"style":70915},[31680],[34,461683,461685,461688],{"className":461684},[31685],[34,461686,106063],{"className":461687,"style":67482},[31685,31686],[34,461689,461691],{"className":461690},[31792],[34,461692,461694,461714],{"className":461693},[31796,31797],[34,461695,461697,461711],{"className":461696},[31801],[34,461698,461700],{"className":461699,"style":31806},[31805],[34,461701,461702,461705],{"style":217742},[34,461703],{"className":461704,"style":31814},[31813],[34,461706,461708],{"className":461707},[31818,31819,31820,31821],[34,461709,640],{"className":461710},[31685,31821],[34,461712,31829],{"className":461713},[31828],[34,461715,461717],{"className":461716},[31801],[34,461718,461720],{"className":461719,"style":31836},[31805],[34,461721],{},[34,461723],{"className":461724,"style":32604},[31845],[34,461726,326344],{"className":461727},[32024],[34,461729],{"className":461730,"style":32604},[31845],[34,461732,461734,461737,461740,461743,461746],{"className":461733},[31676],[34,461735],{"className":461736,"style":147334},[31680],[34,461738,38874],{"className":461739},[34006],[34,461741],{"className":461742,"style":32604},[31845],[34,461744,326344],{"className":461745},[32024],[34,461747],{"className":461748,"style":32604},[31845],[34,461750,461752,461755],{"className":461751},[31676],[34,461753],{"className":461754,"style":70915},[31680],[34,461756,461758,461761],{"className":461757},[31685],[34,461759,106063],{"className":461760,"style":67482},[31685,31686],[34,461762,461764],{"className":461763},[31792],[34,461765,461767,461787],{"className":461766},[31796,31797],[34,461768,461770,461784],{"className":461769},[31801],[34,461771,461773],{"className":461772,"style":33833},[31805],[34,461774,461775,461778],{"style":217742},[34,461776],{"className":461777,"style":31814},[31813],[34,461779,461781],{"className":461780},[31818,31819,31820,31821],[34,461782,33759],{"className":461783,"style":33774},[31685,31686,31821],[34,461785,31829],{"className":461786},[31828],[34,461788,461790],{"className":461789},[31801],[34,461791,461793],{"className":461792,"style":31836},[31805],[34,461794],{},[34,461796,461798,461811],{"className":461797},[31642],[34,461799,461801],{"className":461800},[31646],[31648,461802,461803],{"xmlns":31650},[31652,461804,461805,461809],{},[31655,461806,461807],{},[31658,461808,223320],{},[31665,461810,223320],{"encoding":31667},[34,461812,461814],{"className":461813,"ariaHidden":758},[31672],[34,461815,461817,461820],{"className":461816},[31676],[34,461818],{"className":461819,"style":33218},[31680],[34,461821,223320],{"className":461822,"style":223361},[31685,31686]," 的某种组合关系，或者直接利用命题 3.3.5 证明交集互素后，再利用包含关系 ",[34,461825,461827,461873],{"className":461826},[31642],[34,461828,461830],{"className":461829},[31646],[31648,461831,461832],{"xmlns":31650},[31652,461833,461834,461870],{},[31655,461835,461836,461842,461844,461850,461852,461858,461860,461862,461864],{},[31743,461837,461838,461840],{},[31658,461839,106063],{},[31749,461841,640],{},[31662,461843,38874],{},[31743,461845,461846,461848],{},[31658,461847,106063],{},[31658,461849,33759],{},[31662,461851,131855],{},[31743,461853,461854,461856],{},[31658,461855,106063],{},[31749,461857,640],{},[31662,461859,326344],{},[31662,461861,38874],{},[31662,461863,326344],{},[31743,461865,461866,461868],{},[31658,461867,106063],{},[31658,461869,33759],{},[31665,461871,461872],{"encoding":31667},"I_1 \\cdots I_k \\subseteq I_1 \\cap \\dots \\cap I_k",[34,461874,461876,461980,462035,462053],{"className":461875,"ariaHidden":758},[31672],[34,461877,461879,461882,461922,461925,461928,461931,461971,461974,461977],{"className":461878},[31676],[34,461880],{"className":461881,"style":70915},[31680],[34,461883,461885,461888],{"className":461884},[31685],[34,461886,106063],{"className":461887,"style":67482},[31685,31686],[34,461889,461891],{"className":461890},[31792],[34,461892,461894,461914],{"className":461893},[31796,31797],[34,461895,461897,461911],{"className":461896},[31801],[34,461898,461900],{"className":461899,"style":31806},[31805],[34,461901,461902,461905],{"style":217742},[34,461903],{"className":461904,"style":31814},[31813],[34,461906,461908],{"className":461907},[31818,31819,31820,31821],[34,461909,640],{"className":461910},[31685,31821],[34,461912,31829],{"className":461913},[31828],[34,461915,461917],{"className":461916},[31801],[34,461918,461920],{"className":461919,"style":31836},[31805],[34,461921],{},[34,461923],{"className":461924,"style":32123},[31845],[34,461926,38874],{"className":461927},[34006],[34,461929],{"className":461930,"style":32123},[31845],[34,461932,461934,461937],{"className":461933},[31685],[34,461935,106063],{"className":461936,"style":67482},[31685,31686],[34,461938,461940],{"className":461939},[31792],[34,461941,461943,461963],{"className":461942},[31796,31797],[34,461944,461946,461960],{"className":461945},[31801],[34,461947,461949],{"className":461948,"style":33833},[31805],[34,461950,461951,461954],{"style":217742},[34,461952],{"className":461953,"style":31814},[31813],[34,461955,461957],{"className":461956},[31818,31819,31820,31821],[34,461958,33759],{"className":461959,"style":33774},[31685,31686,31821],[34,461961,31829],{"className":461962},[31828],[34,461964,461966],{"className":461965},[31801],[34,461967,461969],{"className":461968,"style":31836},[31805],[34,461970],{},[34,461972],{"className":461973,"style":31846},[31845],[34,461975,131855],{"className":461976},[31850],[34,461978],{"className":461979,"style":31846},[31845],[34,461981,461983,461986,462026,462029,462032],{"className":461982},[31676],[34,461984],{"className":461985,"style":70915},[31680],[34,461987,461989,461992],{"className":461988},[31685],[34,461990,106063],{"className":461991,"style":67482},[31685,31686],[34,461993,461995],{"className":461994},[31792],[34,461996,461998,462018],{"className":461997},[31796,31797],[34,461999,462001,462015],{"className":462000},[31801],[34,462002,462004],{"className":462003,"style":31806},[31805],[34,462005,462006,462009],{"style":217742},[34,462007],{"className":462008,"style":31814},[31813],[34,462010,462012],{"className":462011},[31818,31819,31820,31821],[34,462013,640],{"className":462014},[31685,31821],[34,462016,31829],{"className":462017},[31828],[34,462019,462021],{"className":462020},[31801],[34,462022,462024],{"className":462023,"style":31836},[31805],[34,462025],{},[34,462027],{"className":462028,"style":32604},[31845],[34,462030,326344],{"className":462031},[32024],[34,462033],{"className":462034,"style":32604},[31845],[34,462036,462038,462041,462044,462047,462050],{"className":462037},[31676],[34,462039],{"className":462040,"style":147334},[31680],[34,462042,38874],{"className":462043},[34006],[34,462045],{"className":462046,"style":32604},[31845],[34,462048,326344],{"className":462049},[32024],[34,462051],{"className":462052,"style":32604},[31845],[34,462054,462056,462059],{"className":462055},[31676],[34,462057],{"className":462058,"style":70915},[31680],[34,462060,462062,462065],{"className":462061},[31685],[34,462063,106063],{"className":462064,"style":67482},[31685,31686],[34,462066,462068],{"className":462067},[31792],[34,462069,462071,462091],{"className":462070},[31796,31797],[34,462072,462074,462088],{"className":462073},[31801],[34,462075,462077],{"className":462076,"style":33833},[31805],[34,462078,462079,462082],{"style":217742},[34,462080],{"className":462081,"style":31814},[31813],[34,462083,462085],{"className":462084},[31818,31819,31820,31821],[34,462086,33759],{"className":462087,"style":33774},[31685,31686,31821],[34,462089,31829],{"className":462090},[31828],[34,462092,462094],{"className":462093},[31801],[34,462095,462097],{"className":462096,"style":31836},[31805],[34,462098],{}," 并不足以直接推出互素，这里通常需要重新利用 ",[34,462101,462103,462145],{"className":462102},[31642],[34,462104,462106],{"className":462105},[31646],[31648,462107,462108],{"xmlns":31650},[31652,462109,462110,462142],{},[31655,462111,462112,462114,462120,462122,462124,462126,462128,462130,462136,462138,462140],{},[31662,462113,456],{"stretchy":702},[31743,462115,462116,462118],{},[31658,462117,106063],{},[31749,462119,640],{},[31662,462121,223],{},[31658,462123,223320],{},[31662,462125,2557],{"stretchy":702},[31662,462127,38874],{},[31662,462129,456],{"stretchy":702},[31743,462131,462132,462134],{},[31658,462133,106063],{},[31658,462135,33759],{},[31662,462137,223],{},[31658,462139,223320],{},[31662,462141,2557],{"stretchy":702},[31665,462143,462144],{"encoding":31667},"(I_1+J)\\cdots(I_k+J)",[34,462146,462148,462206,462279],{"className":462147,"ariaHidden":758},[31672],[34,462149,462151,462154,462157,462197,462200,462203],{"className":462150},[31676],[34,462152],{"className":462153,"style":31774},[31680],[34,462155,456],{"className":462156},[31782],[34,462158,462160,462163],{"className":462159},[31685],[34,462161,106063],{"className":462162,"style":67482},[31685,31686],[34,462164,462166],{"className":462165},[31792],[34,462167,462169,462189],{"className":462168},[31796,31797],[34,462170,462172,462186],{"className":462171},[31801],[34,462173,462175],{"className":462174,"style":31806},[31805],[34,462176,462177,462180],{"style":217742},[34,462178],{"className":462179,"style":31814},[31813],[34,462181,462183],{"className":462182},[31818,31819,31820,31821],[34,462184,640],{"className":462185},[31685,31821],[34,462187,31829],{"className":462188},[31828],[34,462190,462192],{"className":462191},[31801],[34,462193,462195],{"className":462194,"style":31836},[31805],[34,462196],{},[34,462198],{"className":462199,"style":32604},[31845],[34,462201,223],{"className":462202},[32024],[34,462204],{"className":462205,"style":32604},[31845],[34,462207,462209,462212,462215,462218,462221,462224,462227,462230,462270,462273,462276],{"className":462208},[31676],[34,462210],{"className":462211,"style":31774},[31680],[34,462213,223320],{"className":462214,"style":223361},[31685,31686],[34,462216,2557],{"className":462217},[31690],[34,462219],{"className":462220,"style":32123},[31845],[34,462222,38874],{"className":462223},[34006],[34,462225],{"className":462226,"style":32123},[31845],[34,462228,456],{"className":462229},[31782],[34,462231,462233,462236],{"className":462232},[31685],[34,462234,106063],{"className":462235,"style":67482},[31685,31686],[34,462237,462239],{"className":462238},[31792],[34,462240,462242,462262],{"className":462241},[31796,31797],[34,462243,462245,462259],{"className":462244},[31801],[34,462246,462248],{"className":462247,"style":33833},[31805],[34,462249,462250,462253],{"style":217742},[34,462251],{"className":462252,"style":31814},[31813],[34,462254,462256],{"className":462255},[31818,31819,31820,31821],[34,462257,33759],{"className":462258,"style":33774},[31685,31686,31821],[34,462260,31829],{"className":462261},[31828],[34,462263,462265],{"className":462264},[31801],[34,462266,462268],{"className":462267,"style":31836},[31805],[34,462269],{},[34,462271],{"className":462272,"style":32604},[31845],[34,462274,223],{"className":462275},[32024],[34,462277],{"className":462278,"style":32604},[31845],[34,462280,462282,462285,462288],{"className":462281},[31676],[34,462283],{"className":462284,"style":31774},[31680],[34,462286,223320],{"className":462287,"style":223361},[31685,31686],[34,462289,2557],{"className":462290},[31690]," 的展开式或者利用根理想性质。但在本讲义的逻辑流中，通常是利用 ",[34,462293,462295,462341],{"className":462294},[31642],[34,462296,462298],{"className":462297},[31646],[31648,462299,462300],{"xmlns":31650},[31652,462301,462302,462338],{},[31655,462303,462304,462310,462312,462314,462316,462322,462324,462330,462332],{},[31743,462305,462306,462308],{},[31658,462307,106063],{},[31749,462309,640],{},[31662,462311,326344],{},[31662,462313,38874],{},[31662,462315,326344],{},[31743,462317,462318,462320],{},[31658,462319,106063],{},[31658,462321,33759],{},[31662,462323,346833],{},[31743,462325,462326,462328],{},[31658,462327,106063],{},[31749,462329,640],{},[31662,462331,38874],{},[31743,462333,462334,462336],{},[31658,462335,106063],{},[31658,462337,33759],{},[31665,462339,462340],{"encoding":31667},"I_1 \\cap \\dots \\cap I_k \\supseteq I_1 \\cdots I_k",[34,462342,462344,462399,462417,462472],{"className":462343,"ariaHidden":758},[31672],[34,462345,462347,462350,462390,462393,462396],{"className":462346},[31676],[34,462348],{"className":462349,"style":70915},[31680],[34,462351,462353,462356],{"className":462352},[31685],[34,462354,106063],{"className":462355,"style":67482},[31685,31686],[34,462357,462359],{"className":462358},[31792],[34,462360,462362,462382],{"className":462361},[31796,31797],[34,462363,462365,462379],{"className":462364},[31801],[34,462366,462368],{"className":462367,"style":31806},[31805],[34,462369,462370,462373],{"style":217742},[34,462371],{"className":462372,"style":31814},[31813],[34,462374,462376],{"className":462375},[31818,31819,31820,31821],[34,462377,640],{"className":462378},[31685,31821],[34,462380,31829],{"className":462381},[31828],[34,462383,462385],{"className":462384},[31801],[34,462386,462388],{"className":462387,"style":31836},[31805],[34,462389],{},[34,462391],{"className":462392,"style":32604},[31845],[34,462394,326344],{"className":462395},[32024],[34,462397],{"className":462398,"style":32604},[31845],[34,462400,462402,462405,462408,462411,462414],{"className":462401},[31676],[34,462403],{"className":462404,"style":147334},[31680],[34,462406,38874],{"className":462407},[34006],[34,462409],{"className":462410,"style":32604},[31845],[34,462412,326344],{"className":462413},[32024],[34,462415],{"className":462416,"style":32604},[31845],[34,462418,462420,462423,462463,462466,462469],{"className":462419},[31676],[34,462421],{"className":462422,"style":70915},[31680],[34,462424,462426,462429],{"className":462425},[31685],[34,462427,106063],{"className":462428,"style":67482},[31685,31686],[34,462430,462432],{"className":462431},[31792],[34,462433,462435,462455],{"className":462434},[31796,31797],[34,462436,462438,462452],{"className":462437},[31801],[34,462439,462441],{"className":462440,"style":33833},[31805],[34,462442,462443,462446],{"style":217742},[34,462444],{"className":462445,"style":31814},[31813],[34,462447,462449],{"className":462448},[31818,31819,31820,31821],[34,462450,33759],{"className":462451,"style":33774},[31685,31686,31821],[34,462453,31829],{"className":462454},[31828],[34,462456,462458],{"className":462457},[31801],[34,462459,462461],{"className":462460,"style":31836},[31805],[34,462462],{},[34,462464],{"className":462465,"style":31846},[31845],[34,462467,346833],{"className":462468},[31850],[34,462470],{"className":462471,"style":31846},[31845],[34,462473,462475,462478,462518,462521,462524,462527],{"className":462474},[31676],[34,462476],{"className":462477,"style":70915},[31680],[34,462479,462481,462484],{"className":462480},[31685],[34,462482,106063],{"className":462483,"style":67482},[31685,31686],[34,462485,462487],{"className":462486},[31792],[34,462488,462490,462510],{"className":462489},[31796,31797],[34,462491,462493,462507],{"className":462492},[31801],[34,462494,462496],{"className":462495,"style":31806},[31805],[34,462497,462498,462501],{"style":217742},[34,462499],{"className":462500,"style":31814},[31813],[34,462502,462504],{"className":462503},[31818,31819,31820,31821],[34,462505,640],{"className":462506},[31685,31821],[34,462508,31829],{"className":462509},[31828],[34,462511,462513],{"className":462512},[31801],[34,462514,462516],{"className":462515,"style":31836},[31805],[34,462517],{},[34,462519],{"className":462520,"style":32123},[31845],[34,462522,38874],{"className":462523},[34006],[34,462525],{"className":462526,"style":32123},[31845],[34,462528,462530,462533],{"className":462529},[31685],[34,462531,106063],{"className":462532,"style":67482},[31685,31686],[34,462534,462536],{"className":462535},[31792],[34,462537,462539,462559],{"className":462538},[31796,31797],[34,462540,462542,462556],{"className":462541},[31801],[34,462543,462545],{"className":462544,"style":33833},[31805],[34,462546,462547,462550],{"style":217742},[34,462548],{"className":462549,"style":31814},[31813],[34,462551,462553],{"className":462552},[31818,31819,31820,31821],[34,462554,33759],{"className":462555,"style":33774},[31685,31686,31821],[34,462557,31829],{"className":462558},[31828],[34,462560,462562],{"className":462561},[31801],[34,462563,462565],{"className":462564,"style":31836},[31805],[34,462566],{},"，如果交集与 J 互素，我们需要更强的条件。",[2627,462569,462570],{},"更正","：实际上，推论通常是由命题 3.3.5 结合 ",[34,462573,462575,462621],{"className":462574},[31642],[34,462576,462578],{"className":462577},[31646],[31648,462579,462580],{"xmlns":31650},[31652,462581,462582,462618],{},[31655,462583,462584,462586,462588,462590,462592,462598,462600,462602,462604,462606,462608,462614,462616],{},[31658,462585,34386],{},[31662,462587,202],{},[31662,462589,402680],{},[31662,462591,456],{"stretchy":702},[31743,462593,462594,462596],{},[31658,462595,106063],{},[31658,462597,31747],{},[31662,462599,223],{},[31658,462601,223320],{},[31662,462603,2557],{"stretchy":702},[31662,462605,131855],{},[31662,462607,402680],{},[31743,462609,462610,462612],{},[31658,462611,106063],{},[31658,462613,31747],{},[31662,462615,223],{},[31658,462617,223320],{},[31665,462619,462620],{"encoding":31667},"R = \\prod (I_s+J) \\subseteq \\prod I_s + J",[34,462622,462624,462642,462703,462724,462785],{"className":462623,"ariaHidden":758},[31672],[34,462625,462627,462630,462633,462636,462639],{"className":462626},[31676],[34,462628],{"className":462629,"style":33218},[31680],[34,462631,34386],{"className":462632,"style":34427},[31685,31686],[34,462634],{"className":462635,"style":31846},[31845],[34,462637,202],{"className":462638},[31850],[34,462640],{"className":462641,"style":31846},[31845],[34,462643,462645,462648,462651,462654,462694,462697,462700],{"className":462644},[31676],[34,462646],{"className":462647,"style":31774},[31680],[34,462649,402680],{"className":462650,"style":156561},[32767,131249,156560],[34,462652,456],{"className":462653},[31782],[34,462655,462657,462660],{"className":462656},[31685],[34,462658,106063],{"className":462659,"style":67482},[31685,31686],[34,462661,462663],{"className":462662},[31792],[34,462664,462666,462686],{"className":462665},[31796,31797],[34,462667,462669,462683],{"className":462668},[31801],[34,462670,462672],{"className":462671,"style":32362},[31805],[34,462673,462674,462677],{"style":217742},[34,462675],{"className":462676,"style":31814},[31813],[34,462678,462680],{"className":462679},[31818,31819,31820,31821],[34,462681,31747],{"className":462682},[31685,31686,31821],[34,462684,31829],{"className":462685},[31828],[34,462687,462689],{"className":462688},[31801],[34,462690,462692],{"className":462691,"style":31836},[31805],[34,462693],{},[34,462695],{"className":462696,"style":32604},[31845],[34,462698,223],{"className":462699},[32024],[34,462701],{"className":462702,"style":32604},[31845],[34,462704,462706,462709,462712,462715,462718,462721],{"className":462705},[31676],[34,462707],{"className":462708,"style":31774},[31680],[34,462710,223320],{"className":462711,"style":223361},[31685,31686],[34,462713,2557],{"className":462714},[31690],[34,462716],{"className":462717,"style":31846},[31845],[34,462719,131855],{"className":462720},[31850],[34,462722],{"className":462723,"style":31846},[31845],[34,462725,462727,462730,462733,462736,462776,462779,462782],{"className":462726},[31676],[34,462728],{"className":462729,"style":31774},[31680],[34,462731,402680],{"className":462732,"style":156561},[32767,131249,156560],[34,462734],{"className":462735,"style":32123},[31845],[34,462737,462739,462742],{"className":462738},[31685],[34,462740,106063],{"className":462741,"style":67482},[31685,31686],[34,462743,462745],{"className":462744},[31792],[34,462746,462748,462768],{"className":462747},[31796,31797],[34,462749,462751,462765],{"className":462750},[31801],[34,462752,462754],{"className":462753,"style":32362},[31805],[34,462755,462756,462759],{"style":217742},[34,462757],{"className":462758,"style":31814},[31813],[34,462760,462762],{"className":462761},[31818,31819,31820,31821],[34,462763,31747],{"className":462764},[31685,31686,31821],[34,462766,31829],{"className":462767},[31828],[34,462769,462771],{"className":462770},[31801],[34,462772,462774],{"className":462773,"style":31836},[31805],[34,462775],{},[34,462777],{"className":462778,"style":32604},[31845],[34,462780,223],{"className":462781},[32024],[34,462783],{"className":462784,"style":32604},[31845],[34,462786,462788,462791],{"className":462787},[31676],[34,462789],{"className":462790,"style":33218},[31680],[34,462792,223320],{"className":462793,"style":223361},[31685,31686]," 得到的。即 ",[34,462796,462798,462864],{"className":462797},[31642],[34,462799,462801],{"className":462800},[31646],[31648,462802,462803],{"xmlns":31650},[31652,462804,462805,462861],{},[31655,462806,462807,462809,462811,462813,462819,462821,462823,462825,462827,462829,462835,462837,462839,462841,462843,462849,462851,462857,462859],{},[31658,462808,34386],{},[31662,462810,202],{},[31662,462812,456],{"stretchy":702},[31743,462814,462815,462817],{},[31658,462816,106063],{},[31749,462818,640],{},[31662,462820,223],{},[31658,462822,223320],{},[31662,462824,2557],{"stretchy":702},[31662,462826,33934],{},[31662,462828,456],{"stretchy":702},[31743,462830,462831,462833],{},[31658,462832,106063],{},[31658,462834,33759],{},[31662,462836,223],{},[31658,462838,223320],{},[31662,462840,2557],{"stretchy":702},[31662,462842,131855],{},[31743,462844,462845,462847],{},[31658,462846,106063],{},[31749,462848,640],{},[31662,462850,33934],{},[31743,462852,462853,462855],{},[31658,462854,106063],{},[31658,462856,33759],{},[31662,462858,223],{},[31658,462860,223320],{},[31665,462862,462863],{"encoding":31667},"R = (I_1+J)\\dots(I_k+J) \\subseteq I_1\\dots I_k + J",[34,462865,462867,462885,462943,463016,463037,463141],{"className":462866,"ariaHidden":758},[31672],[34,462868,462870,462873,462876,462879,462882],{"className":462869},[31676],[34,462871],{"className":462872,"style":33218},[31680],[34,462874,34386],{"className":462875,"style":34427},[31685,31686],[34,462877],{"className":462878,"style":31846},[31845],[34,462880,202],{"className":462881},[31850],[34,462883],{"className":462884,"style":31846},[31845],[34,462886,462888,462891,462894,462934,462937,462940],{"className":462887},[31676],[34,462889],{"className":462890,"style":31774},[31680],[34,462892,456],{"className":462893},[31782],[34,462895,462897,462900],{"className":462896},[31685],[34,462898,106063],{"className":462899,"style":67482},[31685,31686],[34,462901,462903],{"className":462902},[31792],[34,462904,462906,462926],{"className":462905},[31796,31797],[34,462907,462909,462923],{"className":462908},[31801],[34,462910,462912],{"className":462911,"style":31806},[31805],[34,462913,462914,462917],{"style":217742},[34,462915],{"className":462916,"style":31814},[31813],[34,462918,462920],{"className":462919},[31818,31819,31820,31821],[34,462921,640],{"className":462922},[31685,31821],[34,462924,31829],{"className":462925},[31828],[34,462927,462929],{"className":462928},[31801],[34,462930,462932],{"className":462931,"style":31836},[31805],[34,462933],{},[34,462935],{"className":462936,"style":32604},[31845],[34,462938,223],{"className":462939},[32024],[34,462941],{"className":462942,"style":32604},[31845],[34,462944,462946,462949,462952,462955,462958,462961,462964,462967,463007,463010,463013],{"className":462945},[31676],[34,462947],{"className":462948,"style":31774},[31680],[34,462950,223320],{"className":462951,"style":223361},[31685,31686],[34,462953,2557],{"className":462954},[31690],[34,462956],{"className":462957,"style":32123},[31845],[34,462959,33934],{"className":462960},[34006],[34,462962],{"className":462963,"style":32123},[31845],[34,462965,456],{"className":462966},[31782],[34,462968,462970,462973],{"className":462969},[31685],[34,462971,106063],{"className":462972,"style":67482},[31685,31686],[34,462974,462976],{"className":462975},[31792],[34,462977,462979,462999],{"className":462978},[31796,31797],[34,462980,462982,462996],{"className":462981},[31801],[34,462983,462985],{"className":462984,"style":33833},[31805],[34,462986,462987,462990],{"style":217742},[34,462988],{"className":462989,"style":31814},[31813],[34,462991,462993],{"className":462992},[31818,31819,31820,31821],[34,462994,33759],{"className":462995,"style":33774},[31685,31686,31821],[34,462997,31829],{"className":462998},[31828],[34,463000,463002],{"className":463001},[31801],[34,463003,463005],{"className":463004,"style":31836},[31805],[34,463006],{},[34,463008],{"className":463009,"style":32604},[31845],[34,463011,223],{"className":463012},[32024],[34,463014],{"className":463015,"style":32604},[31845],[34,463017,463019,463022,463025,463028,463031,463034],{"className":463018},[31676],[34,463020],{"className":463021,"style":31774},[31680],[34,463023,223320],{"className":463024,"style":223361},[31685,31686],[34,463026,2557],{"className":463027},[31690],[34,463029],{"className":463030,"style":31846},[31845],[34,463032,131855],{"className":463033},[31850],[34,463035],{"className":463036,"style":31846},[31845],[34,463038,463040,463043,463083,463086,463089,463092,463132,463135,463138],{"className":463039},[31676],[34,463041],{"className":463042,"style":70915},[31680],[34,463044,463046,463049],{"className":463045},[31685],[34,463047,106063],{"className":463048,"style":67482},[31685,31686],[34,463050,463052],{"className":463051},[31792],[34,463053,463055,463075],{"className":463054},[31796,31797],[34,463056,463058,463072],{"className":463057},[31801],[34,463059,463061],{"className":463060,"style":31806},[31805],[34,463062,463063,463066],{"style":217742},[34,463064],{"className":463065,"style":31814},[31813],[34,463067,463069],{"className":463068},[31818,31819,31820,31821],[34,463070,640],{"className":463071},[31685,31821],[34,463073,31829],{"className":463074},[31828],[34,463076,463078],{"className":463077},[31801],[34,463079,463081],{"className":463080,"style":31836},[31805],[34,463082],{},[34,463084],{"className":463085,"style":32123},[31845],[34,463087,33934],{"className":463088},[34006],[34,463090],{"className":463091,"style":32123},[31845],[34,463093,463095,463098],{"className":463094},[31685],[34,463096,106063],{"className":463097,"style":67482},[31685,31686],[34,463099,463101],{"className":463100},[31792],[34,463102,463104,463124],{"className":463103},[31796,31797],[34,463105,463107,463121],{"className":463106},[31801],[34,463108,463110],{"className":463109,"style":33833},[31805],[34,463111,463112,463115],{"style":217742},[34,463113],{"className":463114,"style":31814},[31813],[34,463116,463118],{"className":463117},[31818,31819,31820,31821],[34,463119,33759],{"className":463120,"style":33774},[31685,31686,31821],[34,463122,31829],{"className":463123},[31828],[34,463125,463127],{"className":463126},[31801],[34,463128,463130],{"className":463129,"style":31836},[31805],[34,463131],{},[34,463133],{"className":463134,"style":32604},[31845],[34,463136,223],{"className":463137},[32024],[34,463139],{"className":463140,"style":32604},[31845],[34,463142,463144,463147],{"className":463143},[31676],[34,463145],{"className":463146,"style":33218},[31680],[34,463148,223320],{"className":463149,"style":223361},[31685,31686],", 从而得证。)",{"title":29,"searchDepth":43,"depth":43,"links":463152},[463153,463154,463155,463156,463157],{"id":438841,"depth":49,"text":438842},{"id":440728,"depth":49,"text":440729},{"id":446276,"depth":49,"text":446277},{"id":452990,"depth":49,"text":452991},{"id":454161,"depth":49,"text":454162},{"order":68},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringhomomorphism",{"title":438832,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002Fringhomomorphism","IqswxktgDKTOmFTcCPHBq7vlhvt2XwaLUCUUbQ-STkA",{"id":463164,"title":463165,"body":463166,"description":29,"extension":85,"meta":495542,"navigation":52,"path":495543,"seo":495544,"stem":495545,"__hash__":495546},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002FringhomomorphismHW.md","环同态基本定理 习题详解",{"type":8,"value":463167,"toc":495537},[463168,463171,463173,463177,463180,463182,463458,463463,463775,463780,463854,466590,466663,466734,468940,469039,469041,469043,469250,469254,469385,469546,472270,472373,472375,472377,472644,472648,473004,473008,474523,474525,474529,475059,475063,475647,476797,476800,477687,479320,479322,479326,479330,479420,479424,479671,479823,482554,482635,482637,482641,482868,482901,483756,483789,484896,484929,485209,485271,485484,485518,486019,486148,486690,486692,486696,486979,486983,486991,489058,489060,489064,489437,489441,489833,490000,491421,491542,491544,491548,491552,491720,491724,492030,492034,493300,493302,493305,493485,493489,493860,493864,495450],[11,463169,463165],{"id":463170},"环同态基本定理-习题详解",[32188,463172],{},[253,463174,463176],{"id":463175},"第一部分基础同态性质证明","第一部分：基础同态性质证明",[15,463178,463179],{},"这部分题目考察的是环同态最基本的“保结构”性质。",[32490,463181,214292],{"id":214289},[15,463183,463184,463259],{},[2627,463185,133910,463186,432131],{},[34,463187,463189,463211],{"className":463188},[31642],[34,463190,463192],{"className":463191},[31646],[31648,463193,463194],{"xmlns":31650},[31652,463195,463196,463208],{},[31655,463197,463198,463200,463202,463204,463206],{},[31658,463199,31739],{},[31662,463201,5598],{},[31658,463203,34386],{},[31662,463205,33239],{},[31658,463207,94675],{},[31665,463209,463210],{"encoding":31667},"f: R \\to S",[34,463212,463214,463232,463250],{"className":463213,"ariaHidden":758},[31672],[34,463215,463217,463220,463223,463226,463229],{"className":463216},[31676],[34,463218],{"className":463219,"style":43217},[31680],[34,463221,31739],{"className":463222,"style":31778},[31685,31686],[34,463224],{"className":463225,"style":31846},[31845],[34,463227,5598],{"className":463228},[31850],[34,463230],{"className":463231,"style":31846},[31845],[34,463233,463235,463238,463241,463244,463247],{"className":463234},[31676],[34,463236],{"className":463237,"style":33218},[31680],[34,463239,34386],{"className":463240,"style":34427},[31685,31686],[34,463242],{"className":463243,"style":31846},[31845],[34,463245,33239],{"className":463246},[31850],[34,463248],{"className":463249,"style":31846},[31845],[34,463251,463253,463256],{"className":463252},[31676],[34,463254],{"className":463255,"style":33218},[31680],[34,463257,94675],{"className":463258,"style":106455},[31685,31686],[2627,463260,225289,463261,463289,463290,11008,463332,463360,463361,463389,463390,11008,463429,463457],{},[34,463262,463264,463277],{"className":463263},[31642],[34,463265,463267],{"className":463266},[31646],[31648,463268,463269],{"xmlns":31650},[31652,463270,463271,463275],{},[31655,463272,463273],{},[31658,463274,31739],{},[31665,463276,31739],{"encoding":31667},[34,463278,463280],{"className":463279,"ariaHidden":758},[31672],[34,463281,463283,463286],{"className":463282},[31676],[34,463284],{"className":463285,"style":43217},[31680],[34,463287,31739],{"className":463288,"style":31778},[31685,31686]," 的像 ",[34,463291,463293,463311],{"className":463292},[31642],[34,463294,463296],{"className":463295},[31646],[31648,463297,463298],{"xmlns":31650},[31652,463299,463300,463308],{},[31655,463301,463302,463304,463306],{},[31658,463303,440604],{"mathvariant":38079},[31662,463305,32682],{},[31658,463307,31739],{},[31665,463309,463310],{"encoding":31667},"\\operatorname{Im} f",[34,463312,463314],{"className":463313,"ariaHidden":758},[31672],[34,463315,463317,463320,463326,463329],{"className":463316},[31676],[34,463318],{"className":463319,"style":43217},[31680],[34,463321,463323],{"className":463322},[32767],[34,463324,440604],{"className":463325},[31685,207923],[34,463327],{"className":463328,"style":32123},[31845],[34,463330,31739],{"className":463331,"style":31778},[31685,31686],[34,463333,463335,463348],{"className":463334},[31642],[34,463336,463338],{"className":463337},[31646],[31648,463339,463340],{"xmlns":31650},[31652,463341,463342,463346],{},[31655,463343,463344],{},[31658,463345,94675],{},[31665,463347,94675],{"encoding":31667},[34,463349,463351],{"className":463350,"ariaHidden":758},[31672],[34,463352,463354,463357],{"className":463353},[31676],[34,463355],{"className":463356,"style":33218},[31680],[34,463358,94675],{"className":463359,"style":106455},[31685,31686]," 的一个子环，且 ",[34,463362,463364,463377],{"className":463363},[31642],[34,463365,463367],{"className":463366},[31646],[31648,463368,463369],{"xmlns":31650},[31652,463370,463371,463375],{},[31655,463372,463373],{},[31658,463374,31739],{},[31665,463376,31739],{"encoding":31667},[34,463378,463380],{"className":463379,"ariaHidden":758},[31672],[34,463381,463383,463386],{"className":463382},[31676],[34,463384],{"className":463385,"style":43217},[31680],[34,463387,31739],{"className":463388,"style":31778},[31685,31686]," 的核 ",[34,463391,463393,463411],{"className":463392},[31642],[34,463394,463396],{"className":463395},[31646],[31648,463397,463398],{"xmlns":31650},[31652,463399,463400,463408],{},[31655,463401,463402,463404,463406],{},[31658,463403,406335],{},[31662,463405,32682],{},[31658,463407,31739],{},[31665,463409,463410],{"encoding":31667},"\\ker f",[34,463412,463414],{"className":463413,"ariaHidden":758},[31672],[34,463415,463417,463420,463423,463426],{"className":463416},[31676],[34,463418],{"className":463419,"style":43217},[31680],[34,463421,406335],{"className":463422},[32767],[34,463424],{"className":463425,"style":32123},[31845],[34,463427,31739],{"className":463428,"style":31778},[31685,31686],[34,463430,463432,463445],{"className":463431},[31642],[34,463433,463435],{"className":463434},[31646],[31648,463436,463437],{"xmlns":31650},[31652,463438,463439,463443],{},[31655,463440,463441],{},[31658,463442,34386],{},[31665,463444,34386],{"encoding":31667},[34,463446,463448],{"className":463447,"ariaHidden":758},[31672],[34,463449,463451,463454],{"className":463450},[31676],[34,463452],{"className":463453,"style":33218},[31680],[34,463455,34386],{"className":463456,"style":34427},[31685,31686]," 的一个理想。",[15,463459,463460],{},[2627,463461,463462],{},"【深度解析】",[105,463464,463465,463600],{},[108,463466,463467,463512,463513,463541,463542,463570,463571,463599],{},[2627,463468,463469,463470,463511],{},"什么是像 (",[34,463471,463473,463490],{"className":463472},[31642],[34,463474,463476],{"className":463475},[31646],[31648,463477,463478],{"xmlns":31650},[31652,463479,463480,463488],{},[31655,463481,463482,463484,463486],{},[31658,463483,440604],{"mathvariant":38079},[31662,463485,32682],{},[31658,463487,31739],{},[31665,463489,463310],{"encoding":31667},[34,463491,463493],{"className":463492,"ariaHidden":758},[31672],[34,463494,463496,463499,463505,463508],{"className":463495},[31676],[34,463497],{"className":463498,"style":43217},[31680],[34,463500,463502],{"className":463501},[32767],[34,463503,440604],{"className":463504},[31685,207923],[34,463506],{"className":463507,"style":32123},[31845],[34,463509,31739],{"className":463510,"style":31778},[31685,31686],")？"," 它是 ",[34,463514,463516,463529],{"className":463515},[31642],[34,463517,463519],{"className":463518},[31646],[31648,463520,463521],{"xmlns":31650},[31652,463522,463523,463527],{},[31655,463524,463525],{},[31658,463526,34386],{},[31665,463528,34386],{"encoding":31667},[34,463530,463532],{"className":463531,"ariaHidden":758},[31672],[34,463533,463535,463538],{"className":463534},[31676],[34,463536],{"className":463537,"style":33218},[31680],[34,463539,34386],{"className":463540,"style":34427},[31685,31686]," 中所有元素通过 ",[34,463543,463545,463558],{"className":463544},[31642],[34,463546,463548],{"className":463547},[31646],[31648,463549,463550],{"xmlns":31650},[31652,463551,463552,463556],{},[31655,463553,463554],{},[31658,463555,31739],{},[31665,463557,31739],{"encoding":31667},[34,463559,463561],{"className":463560,"ariaHidden":758},[31672],[34,463562,463564,463567],{"className":463563},[31676],[34,463565],{"className":463566,"style":43217},[31680],[34,463568,31739],{"className":463569,"style":31778},[31685,31686]," 映射过去后，在 ",[34,463572,463574,463587],{"className":463573},[31642],[34,463575,463577],{"className":463576},[31646],[31648,463578,463579],{"xmlns":31650},[31652,463580,463581,463585],{},[31655,463582,463583],{},[31658,463584,94675],{},[31665,463586,94675],{"encoding":31667},[34,463588,463590],{"className":463589,"ariaHidden":758},[31672],[34,463591,463593,463596],{"className":463592},[31676],[34,463594],{"className":463595,"style":33218},[31680],[34,463597,94675],{"className":463598,"style":106455},[31685,31686]," 中形成的集合。证明它是子环，要验证：做减法不出圈，做乘法不出圈。",[108,463601,463602,463512,463643,463671,463672,463700,463701,463771,463772,347],{},[2627,463603,463604,463605,463511],{},"什么是核 (",[34,463606,463608,463625],{"className":463607},[31642],[34,463609,463611],{"className":463610},[31646],[31648,463612,463613],{"xmlns":31650},[31652,463614,463615,463623],{},[31655,463616,463617,463619,463621],{},[31658,463618,406335],{},[31662,463620,32682],{},[31658,463622,31739],{},[31665,463624,463410],{"encoding":31667},[34,463626,463628],{"className":463627,"ariaHidden":758},[31672],[34,463629,463631,463634,463637,463640],{"className":463630},[31676],[34,463632],{"className":463633,"style":43217},[31680],[34,463635,406335],{"className":463636},[32767],[34,463638],{"className":463639,"style":32123},[31845],[34,463641,31739],{"className":463642,"style":31778},[31685,31686],[34,463644,463646,463659],{"className":463645},[31642],[34,463647,463649],{"className":463648},[31646],[31648,463650,463651],{"xmlns":31650},[31652,463652,463653,463657],{},[31655,463654,463655],{},[31658,463656,34386],{},[31665,463658,34386],{"encoding":31667},[34,463660,463662],{"className":463661,"ariaHidden":758},[31672],[34,463663,463665,463668],{"className":463664},[31676],[34,463666],{"className":463667,"style":33218},[31680],[34,463669,34386],{"className":463670,"style":34427},[31685,31686]," 中所有被映射成 ",[34,463673,463675,463688],{"className":463674},[31642],[34,463676,463678],{"className":463677},[31646],[31648,463679,463680],{"xmlns":31650},[31652,463681,463682,463686],{},[31655,463683,463684],{},[31658,463685,94675],{},[31665,463687,94675],{"encoding":31667},[34,463689,463691],{"className":463690,"ariaHidden":758},[31672],[34,463692,463694,463697],{"className":463693},[31676],[34,463695],{"className":463696,"style":33218},[31680],[34,463698,94675],{"className":463699,"style":106455},[31685,31686]," 中零元 (",[34,463702,463704,463722],{"className":463703},[31642],[34,463705,463707],{"className":463706},[31646],[31648,463708,463709],{"xmlns":31650},[31652,463710,463711,463719],{},[31655,463712,463713],{},[31743,463714,463715,463717],{},[31749,463716,459],{},[31658,463718,94675],{},[31665,463720,463721],{"encoding":31667},"0_S",[34,463723,463725],{"className":463724,"ariaHidden":758},[31672],[34,463726,463728,463731],{"className":463727},[31676],[34,463729],{"className":463730,"style":38357},[31680],[34,463732,463734,463737],{"className":463733},[31685],[34,463735,459],{"className":463736},[31685],[34,463738,463740],{"className":463739},[31792],[34,463741,463743,463763],{"className":463742},[31796,31797],[34,463744,463746,463760],{"className":463745},[31801],[34,463747,463749],{"className":463748,"style":116558},[31805],[34,463750,463751,463754],{"style":31809},[34,463752],{"className":463753,"style":31814},[31813],[34,463755,463757],{"className":463756},[31818,31819,31820,31821],[34,463758,94675],{"className":463759,"style":106455},[31685,31686,31821],[34,463761,31829],{"className":463762},[31828],[34,463764,463766],{"className":463765},[31801],[34,463767,463769],{"className":463768,"style":31836},[31805],[34,463770],{},") 的元素集合。证明它是理想，要验证：做减法不出圈，",[2627,463773,463774],{},"被外界元素乘（吸收律）也不出圈",[15,463776,463777],{},[2627,463778,463779],{},"【详细证明】",[15,463781,463782],{},[2627,463783,441233,463784,11008,463825,463853],{},[34,463785,463787,463804],{"className":463786},[31642],[34,463788,463790],{"className":463789},[31646],[31648,463791,463792],{"xmlns":31650},[31652,463793,463794,463802],{},[31655,463795,463796,463798,463800],{},[31658,463797,440604],{"mathvariant":38079},[31662,463799,32682],{},[31658,463801,31739],{},[31665,463803,463310],{"encoding":31667},[34,463805,463807],{"className":463806,"ariaHidden":758},[31672],[34,463808,463810,463813,463819,463822],{"className":463809},[31676],[34,463811],{"className":463812,"style":43217},[31680],[34,463814,463816],{"className":463815},[32767],[34,463817,440604],{"className":463818},[31685,207923],[34,463820],{"className":463821,"style":32123},[31845],[34,463823,31739],{"className":463824,"style":31778},[31685,31686],[34,463826,463828,463841],{"className":463827},[31642],[34,463829,463831],{"className":463830},[31646],[31648,463832,463833],{"xmlns":31650},[31652,463834,463835,463839],{},[31655,463836,463837],{},[31658,463838,94675],{},[31665,463840,94675],{"encoding":31667},[34,463842,463844],{"className":463843,"ariaHidden":758},[31672],[34,463845,463847,463850],{"className":463846},[31676],[34,463848],{"className":463849,"style":33218},[31680],[34,463851,94675],{"className":463852,"style":106455},[31685,31686]," 的子环",[15,463855,336888,463856,463989,463990,332541,463992,464020,464021,53141,464169,464274,464275,34554,464277,464436,464437,315343,464583,46528,464731,39984,464879,465309,465310,465450,465451,13977,465606,465789,465790,465958,465959,33183,465961,466391,466392,466548,466549,50492],{},[34,463857,463859,463899],{"className":463858},[31642],[34,463860,463862],{"className":463861},[31646],[31648,463863,463864],{"xmlns":31650},[31652,463865,463866,463896],{},[31655,463867,463868,463870,463872,463874,463876,463878,463880,463882,463884,463886,463888,463890,463892,463894],{},[31658,463869,440604],{"mathvariant":38079},[31662,463871,32682],{},[31658,463873,31739],{},[31662,463875,202],{},[31662,463877,32685],{"stretchy":702},[31658,463879,31739],{},[31662,463881,456],{"stretchy":702},[31658,463883,34395],{},[31662,463885,2557],{"stretchy":702},[31662,463887,162086],{},[31658,463889,34395],{},[31662,463891,112296],{},[31658,463893,34386],{},[31662,463895,32715],{"stretchy":702},[31665,463897,463898],{"encoding":31667},"\\operatorname{Im} f = \\{f(r) \\mid r \\in R\\}",[34,463900,463902,463929,463959,463977],{"className":463901,"ariaHidden":758},[31672],[34,463903,463905,463908,463914,463917,463920,463923,463926],{"className":463904},[31676],[34,463906],{"className":463907,"style":43217},[31680],[34,463909,463911],{"className":463910},[32767],[34,463912,440604],{"className":463913},[31685,207923],[34,463915],{"className":463916,"style":32123},[31845],[34,463918,31739],{"className":463919,"style":31778},[31685,31686],[34,463921],{"className":463922,"style":31846},[31845],[34,463924,202],{"className":463925},[31850],[34,463927],{"className":463928,"style":31846},[31845],[34,463930,463932,463935,463938,463941,463944,463947,463950,463953,463956],{"className":463931},[31676],[34,463933],{"className":463934,"style":31774},[31680],[34,463936,32685],{"className":463937},[31782],[34,463939,31739],{"className":463940,"style":31778},[31685,31686],[34,463942,456],{"className":463943},[31782],[34,463945,34395],{"className":463946,"style":34452},[31685,31686],[34,463948,2557],{"className":463949},[31690],[34,463951],{"className":463952,"style":31846},[31845],[34,463954,162086],{"className":463955},[31850],[34,463957],{"className":463958,"style":31846},[31845],[34,463960,463962,463965,463968,463971,463974],{"className":463961},[31676],[34,463963],{"className":463964,"style":34812},[31680],[34,463966,34395],{"className":463967,"style":34452},[31685,31686],[34,463969],{"className":463970,"style":31846},[31845],[34,463972,112296],{"className":463973},[31850],[34,463975],{"className":463976,"style":31846},[31845],[34,463978,463980,463983,463986],{"className":463979},[31676],[34,463981],{"className":463982,"style":31774},[31680],[34,463984,34386],{"className":463985,"style":34427},[31685,31686],[34,463987,32715],{"className":463988},[31690],"。\n(1) ",[2627,463991,441307],{},[34,463993,463995,464008],{"className":463994},[31642],[34,463996,463998],{"className":463997},[31646],[31648,463999,464000],{"xmlns":31650},[31652,464001,464002,464006],{},[31655,464003,464004],{},[31658,464005,31739],{},[31665,464007,31739],{"encoding":31667},[34,464009,464011],{"className":464010,"ariaHidden":758},[31672],[34,464012,464014,464017],{"className":464013},[31676],[34,464015],{"className":464016,"style":43217},[31680],[34,464018,31739],{"className":464019,"style":31778},[31685,31686]," 是同态，所以 ",[34,464022,464024,464056],{"className":464023},[31642],[34,464025,464027],{"className":464026},[31646],[31648,464028,464029],{"xmlns":31650},[31652,464030,464031,464053],{},[31655,464032,464033,464035,464037,464043,464045,464047],{},[31658,464034,31739],{},[31662,464036,456],{"stretchy":702},[31743,464038,464039,464041],{},[31749,464040,459],{},[31658,464042,34386],{},[31662,464044,2557],{"stretchy":702},[31662,464046,202],{},[31743,464048,464049,464051],{},[31749,464050,459],{},[31658,464052,94675],{},[31665,464054,464055],{"encoding":31667},"f(0_R) = 0_S",[34,464057,464059,464123],{"className":464058,"ariaHidden":758},[31672],[34,464060,464062,464065,464068,464071,464111,464114,464117,464120],{"className":464061},[31676],[34,464063],{"className":464064,"style":31774},[31680],[34,464066,31739],{"className":464067,"style":31778},[31685,31686],[34,464069,456],{"className":464070},[31782],[34,464072,464074,464077],{"className":464073},[31685],[34,464075,459],{"className":464076},[31685],[34,464078,464080],{"className":464079},[31792],[34,464081,464083,464103],{"className":464082},[31796,31797],[34,464084,464086,464100],{"className":464085},[31801],[34,464087,464089],{"className":464088,"style":116558},[31805],[34,464090,464091,464094],{"style":31809},[34,464092],{"className":464093,"style":31814},[31813],[34,464095,464097],{"className":464096},[31818,31819,31820,31821],[34,464098,34386],{"className":464099,"style":34427},[31685,31686,31821],[34,464101,31829],{"className":464102},[31828],[34,464104,464106],{"className":464105},[31801],[34,464107,464109],{"className":464108,"style":31836},[31805],[34,464110],{},[34,464112,2557],{"className":464113},[31690],[34,464115],{"className":464116,"style":31846},[31845],[34,464118,202],{"className":464119},[31850],[34,464121],{"className":464122,"style":31846},[31845],[34,464124,464126,464129],{"className":464125},[31676],[34,464127],{"className":464128,"style":38357},[31680],[34,464130,464132,464135],{"className":464131},[31685],[34,464133,459],{"className":464134},[31685],[34,464136,464138],{"className":464137},[31792],[34,464139,464141,464161],{"className":464140},[31796,31797],[34,464142,464144,464158],{"className":464143},[31801],[34,464145,464147],{"className":464146,"style":116558},[31805],[34,464148,464149,464152],{"style":31809},[34,464150],{"className":464151,"style":31814},[31813],[34,464153,464155],{"className":464154},[31818,31819,31820,31821],[34,464156,94675],{"className":464157,"style":106455},[31685,31686,31821],[34,464159,31829],{"className":464160},[31828],[34,464162,464164],{"className":464163},[31801],[34,464165,464167],{"className":464166,"style":31836},[31805],[34,464168],{},[34,464170,464172,464198],{"className":464171},[31642],[34,464173,464175],{"className":464174},[31646],[31648,464176,464177],{"xmlns":31650},[31652,464178,464179,464195],{},[31655,464180,464181,464187,464189,464191,464193],{},[31743,464182,464183,464185],{},[31749,464184,459],{},[31658,464186,94675],{},[31662,464188,112296],{},[31658,464190,440604],{"mathvariant":38079},[31662,464192,32682],{},[31658,464194,31739],{},[31665,464196,464197],{"encoding":31667},"0_S \\in \\operatorname{Im} f",[34,464199,464201,464256],{"className":464200,"ariaHidden":758},[31672],[34,464202,464204,464207,464247,464250,464253],{"className":464203},[31676],[34,464205],{"className":464206,"style":38357},[31680],[34,464208,464210,464213],{"className":464209},[31685],[34,464211,459],{"className":464212},[31685],[34,464214,464216],{"className":464215},[31792],[34,464217,464219,464239],{"className":464218},[31796,31797],[34,464220,464222,464236],{"className":464221},[31801],[34,464223,464225],{"className":464224,"style":116558},[31805],[34,464226,464227,464230],{"style":31809},[34,464228],{"className":464229,"style":31814},[31813],[34,464231,464233],{"className":464232},[31818,31819,31820,31821],[34,464234,94675],{"className":464235,"style":106455},[31685,31686,31821],[34,464237,31829],{"className":464238},[31828],[34,464240,464242],{"className":464241},[31801],[34,464243,464245],{"className":464244,"style":31836},[31805],[34,464246],{},[34,464248],{"className":464249,"style":31846},[31845],[34,464251,112296],{"className":464252},[31850],[34,464254],{"className":464255,"style":31846},[31845],[34,464257,464259,464262,464268,464271],{"className":464258},[31676],[34,464260],{"className":464261,"style":43217},[31680],[34,464263,464265],{"className":464264},[32767],[34,464266,440604],{"className":464267},[31685,207923],[34,464269],{"className":464270,"style":32123},[31845],[34,464272,31739],{"className":464273,"style":31778},[31685,31686],"，集合非空。\n(2) ",[2627,464276,443104],{},[34,464278,464280,464314],{"className":464279},[31642],[34,464281,464283],{"className":464282},[31646],[31648,464284,464285],{"xmlns":31650},[31652,464286,464287,464311],{},[31655,464288,464289,464295,464297,464303,464305,464307,464309],{},[31743,464290,464291,464293],{},[31658,464292,67764],{},[31749,464294,640],{},[31662,464296,31960],{"separator":758},[31743,464298,464299,464301],{},[31658,464300,67764],{},[31749,464302,768],{},[31662,464304,112296],{},[31658,464306,440604],{"mathvariant":38079},[31662,464308,32682],{},[31658,464310,31739],{},[31665,464312,464313],{"encoding":31667},"y_1, y_2 \\in \\operatorname{Im} f",[34,464315,464317,464418],{"className":464316,"ariaHidden":758},[31672],[34,464318,464320,464323,464363,464366,464369,464409,464412,464415],{"className":464319},[31676],[34,464321],{"className":464322,"style":86652},[31680],[34,464324,464326,464329],{"className":464325},[31685],[34,464327,67764],{"className":464328,"style":32062},[31685,31686],[34,464330,464332],{"className":464331},[31792],[34,464333,464335,464355],{"className":464334},[31796,31797],[34,464336,464338,464352],{"className":464337},[31801],[34,464339,464341],{"className":464340,"style":31806},[31805],[34,464342,464343,464346],{"style":38795},[34,464344],{"className":464345,"style":31814},[31813],[34,464347,464349],{"className":464348},[31818,31819,31820,31821],[34,464350,640],{"className":464351},[31685,31821],[34,464353,31829],{"className":464354},[31828],[34,464356,464358],{"className":464357},[31801],[34,464359,464361],{"className":464360,"style":31836},[31805],[34,464362],{},[34,464364,31960],{"className":464365},[32119],[34,464367],{"className":464368,"style":32123},[31845],[34,464370,464372,464375],{"className":464371},[31685],[34,464373,67764],{"className":464374,"style":32062},[31685,31686],[34,464376,464378],{"className":464377},[31792],[34,464379,464381,464401],{"className":464380},[31796,31797],[34,464382,464384,464398],{"className":464383},[31801],[34,464385,464387],{"className":464386,"style":31806},[31805],[34,464388,464389,464392],{"style":38795},[34,464390],{"className":464391,"style":31814},[31813],[34,464393,464395],{"className":464394},[31818,31819,31820,31821],[34,464396,768],{"className":464397},[31685,31821],[34,464399,31829],{"className":464400},[31828],[34,464402,464404],{"className":464403},[31801],[34,464405,464407],{"className":464406,"style":31836},[31805],[34,464408],{},[34,464410],{"className":464411,"style":31846},[31845],[34,464413,112296],{"className":464414},[31850],[34,464416],{"className":464417,"style":31846},[31845],[34,464419,464421,464424,464430,464433],{"className":464420},[31676],[34,464422],{"className":464423,"style":43217},[31680],[34,464425,464427],{"className":464426},[32767],[34,464428,440604],{"className":464429},[31685,207923],[34,464431],{"className":464432,"style":32123},[31845],[34,464434,31739],{"className":464435,"style":31778},[31685,31686],"。\n根据定义，存在 ",[34,464438,464440,464470],{"className":464439},[31642],[34,464441,464443],{"className":464442},[31646],[31648,464444,464445],{"xmlns":31650},[31652,464446,464447,464467],{},[31655,464448,464449,464455,464457,464463,464465],{},[31743,464450,464451,464453],{},[31658,464452,32936],{},[31749,464454,640],{},[31662,464456,31960],{"separator":758},[31743,464458,464459,464461],{},[31658,464460,32936],{},[31749,464462,768],{},[31662,464464,112296],{},[31658,464466,34386],{},[31665,464468,464469],{"encoding":31667},"x_1, x_2 \\in R",[34,464471,464473,464574],{"className":464472,"ariaHidden":758},[31672],[34,464474,464476,464479,464519,464522,464525,464565,464568,464571],{"className":464475},[31676],[34,464477],{"className":464478,"style":86652},[31680],[34,464480,464482,464485],{"className":464481},[31685],[34,464483,32936],{"className":464484},[31685,31686],[34,464486,464488],{"className":464487},[31792],[34,464489,464491,464511],{"className":464490},[31796,31797],[34,464492,464494,464508],{"className":464493},[31801],[34,464495,464497],{"className":464496,"style":31806},[31805],[34,464498,464499,464502],{"style":31809},[34,464500],{"className":464501,"style":31814},[31813],[34,464503,464505],{"className":464504},[31818,31819,31820,31821],[34,464506,640],{"className":464507},[31685,31821],[34,464509,31829],{"className":464510},[31828],[34,464512,464514],{"className":464513},[31801],[34,464515,464517],{"className":464516,"style":31836},[31805],[34,464518],{},[34,464520,31960],{"className":464521},[32119],[34,464523],{"className":464524,"style":32123},[31845],[34,464526,464528,464531],{"className":464527},[31685],[34,464529,32936],{"className":464530},[31685,31686],[34,464532,464534],{"className":464533},[31792],[34,464535,464537,464557],{"className":464536},[31796,31797],[34,464538,464540,464554],{"className":464539},[31801],[34,464541,464543],{"className":464542,"style":31806},[31805],[34,464544,464545,464548],{"style":31809},[34,464546],{"className":464547,"style":31814},[31813],[34,464549,464551],{"className":464550},[31818,31819,31820,31821],[34,464552,768],{"className":464553},[31685,31821],[34,464555,31829],{"className":464556},[31828],[34,464558,464560],{"className":464559},[31801],[34,464561,464563],{"className":464562,"style":31836},[31805],[34,464564],{},[34,464566],{"className":464567,"style":31846},[31845],[34,464569,112296],{"className":464570},[31850],[34,464572],{"className":464573,"style":31846},[31845],[34,464575,464577,464580],{"className":464576},[31676],[34,464578],{"className":464579,"style":33218},[31680],[34,464581,34386],{"className":464582,"style":34427},[31685,31686],[34,464584,464586,464618],{"className":464585},[31642],[34,464587,464589],{"className":464588},[31646],[31648,464590,464591],{"xmlns":31650},[31652,464592,464593,464615],{},[31655,464594,464595,464597,464599,464605,464607,464609],{},[31658,464596,31739],{},[31662,464598,456],{"stretchy":702},[31743,464600,464601,464603],{},[31658,464602,32936],{},[31749,464604,640],{},[31662,464606,2557],{"stretchy":702},[31662,464608,202],{},[31743,464610,464611,464613],{},[31658,464612,67764],{},[31749,464614,640],{},[31665,464616,464617],{"encoding":31667},"f(x_1) = y_1",[34,464619,464621,464685],{"className":464620,"ariaHidden":758},[31672],[34,464622,464624,464627,464630,464633,464673,464676,464679,464682],{"className":464623},[31676],[34,464625],{"className":464626,"style":31774},[31680],[34,464628,31739],{"className":464629,"style":31778},[31685,31686],[34,464631,456],{"className":464632},[31782],[34,464634,464636,464639],{"className":464635},[31685],[34,464637,32936],{"className":464638},[31685,31686],[34,464640,464642],{"className":464641},[31792],[34,464643,464645,464665],{"className":464644},[31796,31797],[34,464646,464648,464662],{"className":464647},[31801],[34,464649,464651],{"className":464650,"style":31806},[31805],[34,464652,464653,464656],{"style":31809},[34,464654],{"className":464655,"style":31814},[31813],[34,464657,464659],{"className":464658},[31818,31819,31820,31821],[34,464660,640],{"className":464661},[31685,31821],[34,464663,31829],{"className":464664},[31828],[34,464666,464668],{"className":464667},[31801],[34,464669,464671],{"className":464670,"style":31836},[31805],[34,464672],{},[34,464674,2557],{"className":464675},[31690],[34,464677],{"className":464678,"style":31846},[31845],[34,464680,202],{"className":464681},[31850],[34,464683],{"className":464684,"style":31846},[31845],[34,464686,464688,464691],{"className":464687},[31676],[34,464689],{"className":464690,"style":38774},[31680],[34,464692,464694,464697],{"className":464693},[31685],[34,464695,67764],{"className":464696,"style":32062},[31685,31686],[34,464698,464700],{"className":464699},[31792],[34,464701,464703,464723],{"className":464702},[31796,31797],[34,464704,464706,464720],{"className":464705},[31801],[34,464707,464709],{"className":464708,"style":31806},[31805],[34,464710,464711,464714],{"style":38795},[34,464712],{"className":464713,"style":31814},[31813],[34,464715,464717],{"className":464716},[31818,31819,31820,31821],[34,464718,640],{"className":464719},[31685,31821],[34,464721,31829],{"className":464722},[31828],[34,464724,464726],{"className":464725},[31801],[34,464727,464729],{"className":464728,"style":31836},[31805],[34,464730],{},[34,464732,464734,464766],{"className":464733},[31642],[34,464735,464737],{"className":464736},[31646],[31648,464738,464739],{"xmlns":31650},[31652,464740,464741,464763],{},[31655,464742,464743,464745,464747,464753,464755,464757],{},[31658,464744,31739],{},[31662,464746,456],{"stretchy":702},[31743,464748,464749,464751],{},[31658,464750,32936],{},[31749,464752,768],{},[31662,464754,2557],{"stretchy":702},[31662,464756,202],{},[31743,464758,464759,464761],{},[31658,464760,67764],{},[31749,464762,768],{},[31665,464764,464765],{"encoding":31667},"f(x_2) = y_2",[34,464767,464769,464833],{"className":464768,"ariaHidden":758},[31672],[34,464770,464772,464775,464778,464781,464821,464824,464827,464830],{"className":464771},[31676],[34,464773],{"className":464774,"style":31774},[31680],[34,464776,31739],{"className":464777,"style":31778},[31685,31686],[34,464779,456],{"className":464780},[31782],[34,464782,464784,464787],{"className":464783},[31685],[34,464785,32936],{"className":464786},[31685,31686],[34,464788,464790],{"className":464789},[31792],[34,464791,464793,464813],{"className":464792},[31796,31797],[34,464794,464796,464810],{"className":464795},[31801],[34,464797,464799],{"className":464798,"style":31806},[31805],[34,464800,464801,464804],{"style":31809},[34,464802],{"className":464803,"style":31814},[31813],[34,464805,464807],{"className":464806},[31818,31819,31820,31821],[34,464808,768],{"className":464809},[31685,31821],[34,464811,31829],{"className":464812},[31828],[34,464814,464816],{"className":464815},[31801],[34,464817,464819],{"className":464818,"style":31836},[31805],[34,464820],{},[34,464822,2557],{"className":464823},[31690],[34,464825],{"className":464826,"style":31846},[31845],[34,464828,202],{"className":464829},[31850],[34,464831],{"className":464832,"style":31846},[31845],[34,464834,464836,464839],{"className":464835},[31676],[34,464837],{"className":464838,"style":38774},[31680],[34,464840,464842,464845],{"className":464841},[31685],[34,464843,67764],{"className":464844,"style":32062},[31685,31686],[34,464846,464848],{"className":464847},[31792],[34,464849,464851,464871],{"className":464850},[31796,31797],[34,464852,464854,464868],{"className":464853},[31801],[34,464855,464857],{"className":464856,"style":31806},[31805],[34,464858,464859,464862],{"style":38795},[34,464860],{"className":464861,"style":31814},[31813],[34,464863,464865],{"className":464864},[31818,31819,31820,31821],[34,464866,768],{"className":464867},[31685,31821],[34,464869,31829],{"className":464870},[31828],[34,464872,464874],{"className":464873},[31801],[34,464875,464877],{"className":464876,"style":31836},[31805],[34,464878],{},[34,464880,464882,464958],{"className":464881},[31642],[34,464883,464885],{"className":464884},[31646],[31648,464886,464887],{"xmlns":31650},[31652,464888,464889,464955],{},[31655,464890,464891,464897,464899,464905,464907,464909,464911,464917,464919,464921,464923,464925,464931,464933,464935,464937,464939,464945,464947,464953],{},[31743,464892,464893,464895],{},[31658,464894,67764],{},[31749,464896,640],{},[31662,464898,32226],{},[31743,464900,464901,464903],{},[31658,464902,67764],{},[31749,464904,768],{},[31662,464906,202],{},[31658,464908,31739],{},[31662,464910,456],{"stretchy":702},[31743,464912,464913,464915],{},[31658,464914,32936],{},[31749,464916,640],{},[31662,464918,2557],{"stretchy":702},[31662,464920,32226],{},[31658,464922,31739],{},[31662,464924,456],{"stretchy":702},[31743,464926,464927,464929],{},[31658,464928,32936],{},[31749,464930,768],{},[31662,464932,2557],{"stretchy":702},[31662,464934,202],{},[31658,464936,31739],{},[31662,464938,456],{"stretchy":702},[31743,464940,464941,464943],{},[31658,464942,32936],{},[31749,464944,640],{},[31662,464946,32226],{},[31743,464948,464949,464951],{},[31658,464950,32936],{},[31749,464952,768],{},[31662,464954,2557],{"stretchy":702},[31665,464956,464957],{"encoding":31667},"y_1 - y_2 = f(x_1) - f(x_2) = f(x_1 - x_2)",[34,464959,464961,465016,465071,465135,465199,465260],{"className":464960,"ariaHidden":758},[31672],[34,464962,464964,464967,465007,465010,465013],{"className":464963},[31676],[34,464965],{"className":464966,"style":55949},[31680],[34,464968,464970,464973],{"className":464969},[31685],[34,464971,67764],{"className":464972,"style":32062},[31685,31686],[34,464974,464976],{"className":464975},[31792],[34,464977,464979,464999],{"className":464978},[31796,31797],[34,464980,464982,464996],{"className":464981},[31801],[34,464983,464985],{"className":464984,"style":31806},[31805],[34,464986,464987,464990],{"style":38795},[34,464988],{"className":464989,"style":31814},[31813],[34,464991,464993],{"className":464992},[31818,31819,31820,31821],[34,464994,640],{"className":464995},[31685,31821],[34,464997,31829],{"className":464998},[31828],[34,465000,465002],{"className":465001},[31801],[34,465003,465005],{"className":465004,"style":31836},[31805],[34,465006],{},[34,465008],{"className":465009,"style":32604},[31845],[34,465011,32226],{"className":465012},[32024],[34,465014],{"className":465015,"style":32604},[31845],[34,465017,465019,465022,465062,465065,465068],{"className":465018},[31676],[34,465020],{"className":465021,"style":38774},[31680],[34,465023,465025,465028],{"className":465024},[31685],[34,465026,67764],{"className":465027,"style":32062},[31685,31686],[34,465029,465031],{"className":465030},[31792],[34,465032,465034,465054],{"className":465033},[31796,31797],[34,465035,465037,465051],{"className":465036},[31801],[34,465038,465040],{"className":465039,"style":31806},[31805],[34,465041,465042,465045],{"style":38795},[34,465043],{"className":465044,"style":31814},[31813],[34,465046,465048],{"className":465047},[31818,31819,31820,31821],[34,465049,768],{"className":465050},[31685,31821],[34,465052,31829],{"className":465053},[31828],[34,465055,465057],{"className":465056},[31801],[34,465058,465060],{"className":465059,"style":31836},[31805],[34,465061],{},[34,465063],{"className":465064,"style":31846},[31845],[34,465066,202],{"className":465067},[31850],[34,465069],{"className":465070,"style":31846},[31845],[34,465072,465074,465077,465080,465083,465123,465126,465129,465132],{"className":465073},[31676],[34,465075],{"className":465076,"style":31774},[31680],[34,465078,31739],{"className":465079,"style":31778},[31685,31686],[34,465081,456],{"className":465082},[31782],[34,465084,465086,465089],{"className":465085},[31685],[34,465087,32936],{"className":465088},[31685,31686],[34,465090,465092],{"className":465091},[31792],[34,465093,465095,465115],{"className":465094},[31796,31797],[34,465096,465098,465112],{"className":465097},[31801],[34,465099,465101],{"className":465100,"style":31806},[31805],[34,465102,465103,465106],{"style":31809},[34,465104],{"className":465105,"style":31814},[31813],[34,465107,465109],{"className":465108},[31818,31819,31820,31821],[34,465110,640],{"className":465111},[31685,31821],[34,465113,31829],{"className":465114},[31828],[34,465116,465118],{"className":465117},[31801],[34,465119,465121],{"className":465120,"style":31836},[31805],[34,465122],{},[34,465124,2557],{"className":465125},[31690],[34,465127],{"className":465128,"style":32604},[31845],[34,465130,32226],{"className":465131},[32024],[34,465133],{"className":465134,"style":32604},[31845],[34,465136,465138,465141,465144,465147,465187,465190,465193,465196],{"className":465137},[31676],[34,465139],{"className":465140,"style":31774},[31680],[34,465142,31739],{"className":465143,"style":31778},[31685,31686],[34,465145,456],{"className":465146},[31782],[34,465148,465150,465153],{"className":465149},[31685],[34,465151,32936],{"className":465152},[31685,31686],[34,465154,465156],{"className":465155},[31792],[34,465157,465159,465179],{"className":465158},[31796,31797],[34,465160,465162,465176],{"className":465161},[31801],[34,465163,465165],{"className":465164,"style":31806},[31805],[34,465166,465167,465170],{"style":31809},[34,465168],{"className":465169,"style":31814},[31813],[34,465171,465173],{"className":465172},[31818,31819,31820,31821],[34,465174,768],{"className":465175},[31685,31821],[34,465177,31829],{"className":465178},[31828],[34,465180,465182],{"className":465181},[31801],[34,465183,465185],{"className":465184,"style":31836},[31805],[34,465186],{},[34,465188,2557],{"className":465189},[31690],[34,465191],{"className":465192,"style":31846},[31845],[34,465194,202],{"className":465195},[31850],[34,465197],{"className":465198,"style":31846},[31845],[34,465200,465202,465205,465208,465211,465251,465254,465257],{"className":465201},[31676],[34,465203],{"className":465204,"style":31774},[31680],[34,465206,31739],{"className":465207,"style":31778},[31685,31686],[34,465209,456],{"className":465210},[31782],[34,465212,465214,465217],{"className":465213},[31685],[34,465215,32936],{"className":465216},[31685,31686],[34,465218,465220],{"className":465219},[31792],[34,465221,465223,465243],{"className":465222},[31796,31797],[34,465224,465226,465240],{"className":465225},[31801],[34,465227,465229],{"className":465228,"style":31806},[31805],[34,465230,465231,465234],{"style":31809},[34,465232],{"className":465233,"style":31814},[31813],[34,465235,465237],{"className":465236},[31818,31819,31820,31821],[34,465238,640],{"className":465239},[31685,31821],[34,465241,31829],{"className":465242},[31828],[34,465244,465246],{"className":465245},[31801],[34,465247,465249],{"className":465248,"style":31836},[31805],[34,465250],{},[34,465252],{"className":465253,"style":32604},[31845],[34,465255,32226],{"className":465256},[32024],[34,465258],{"className":465259,"style":32604},[31845],[34,465261,465263,465266,465306],{"className":465262},[31676],[34,465264],{"className":465265,"style":31774},[31680],[34,465267,465269,465272],{"className":465268},[31685],[34,465270,32936],{"className":465271},[31685,31686],[34,465273,465275],{"className":465274},[31792],[34,465276,465278,465298],{"className":465277},[31796,31797],[34,465279,465281,465295],{"className":465280},[31801],[34,465282,465284],{"className":465283,"style":31806},[31805],[34,465285,465286,465289],{"style":31809},[34,465287],{"className":465288,"style":31814},[31813],[34,465290,465292],{"className":465291},[31818,31819,31820,31821],[34,465293,768],{"className":465294},[31685,31821],[34,465296,31829],{"className":465297},[31828],[34,465299,465301],{"className":465300},[31801],[34,465302,465304],{"className":465303,"style":31836},[31805],[34,465305],{},[34,465307,2557],{"className":465308},[31690],"\n（注：同态保持加减法，",[34,465311,465313,465357],{"className":465312},[31642],[34,465314,465316],{"className":465315},[31646],[31648,465317,465318],{"xmlns":31650},[31652,465319,465320,465354],{},[31655,465321,465322,465324,465326,465328,465330,465332,465334,465336,465338,465340,465342,465344,465346,465348,465350,465352],{},[31658,465323,31739],{},[31662,465325,456],{"stretchy":702},[31658,465327,111],{},[31662,465329,2557],{"stretchy":702},[31662,465331,32226],{},[31658,465333,31739],{},[31662,465335,456],{"stretchy":702},[31658,465337,1697],{},[31662,465339,2557],{"stretchy":702},[31662,465341,202],{},[31658,465343,31739],{},[31662,465345,456],{"stretchy":702},[31658,465347,111],{},[31662,465349,32226],{},[31658,465351,1697],{},[31662,465353,2557],{"stretchy":702},[31665,465355,465356],{"encoding":31667},"f(a)-f(b)=f(a-b)",[34,465358,465360,465387,465414,465438],{"className":465359,"ariaHidden":758},[31672],[34,465361,465363,465366,465369,465372,465375,465378,465381,465384],{"className":465362},[31676],[34,465364],{"className":465365,"style":31774},[31680],[34,465367,31739],{"className":465368,"style":31778},[31685,31686],[34,465370,456],{"className":465371},[31782],[34,465373,111],{"className":465374},[31685,31686],[34,465376,2557],{"className":465377},[31690],[34,465379],{"className":465380,"style":32604},[31845],[34,465382,32226],{"className":465383},[32024],[34,465385],{"className":465386,"style":32604},[31845],[34,465388,465390,465393,465396,465399,465402,465405,465408,465411],{"className":465389},[31676],[34,465391],{"className":465392,"style":31774},[31680],[34,465394,31739],{"className":465395,"style":31778},[31685,31686],[34,465397,456],{"className":465398},[31782],[34,465400,1697],{"className":465401},[31685,31686],[34,465403,2557],{"className":465404},[31690],[34,465406],{"className":465407,"style":31846},[31845],[34,465409,202],{"className":465410},[31850],[34,465412],{"className":465413,"style":31846},[31845],[34,465415,465417,465420,465423,465426,465429,465432,465435],{"className":465416},[31676],[34,465418],{"className":465419,"style":31774},[31680],[34,465421,31739],{"className":465422,"style":31778},[31685,31686],[34,465424,456],{"className":465425},[31782],[34,465427,111],{"className":465428},[31685,31686],[34,465430],{"className":465431,"style":32604},[31845],[34,465433,32226],{"className":465434},[32024],[34,465436],{"className":465437,"style":32604},[31845],[34,465439,465441,465444,465447],{"className":465440},[31676],[34,465442],{"className":465443,"style":31774},[31680],[34,465445,1697],{"className":465446},[31685,31686],[34,465448,2557],{"className":465449},[31690],"）\n因为 ",[34,465452,465454,465484],{"className":465453},[31642],[34,465455,465457],{"className":465456},[31646],[31648,465458,465459],{"xmlns":31650},[31652,465460,465461,465481],{},[31655,465462,465463,465469,465471,465477,465479],{},[31743,465464,465465,465467],{},[31658,465466,32936],{},[31749,465468,640],{},[31662,465470,32226],{},[31743,465472,465473,465475],{},[31658,465474,32936],{},[31749,465476,768],{},[31662,465478,112296],{},[31658,465480,34386],{},[31665,465482,465483],{"encoding":31667},"x_1 - x_2 \\in R",[34,465485,465487,465542,465597],{"className":465486,"ariaHidden":758},[31672],[34,465488,465490,465493,465533,465536,465539],{"className":465489},[31676],[34,465491],{"className":465492,"style":40522},[31680],[34,465494,465496,465499],{"className":465495},[31685],[34,465497,32936],{"className":465498},[31685,31686],[34,465500,465502],{"className":465501},[31792],[34,465503,465505,465525],{"className":465504},[31796,31797],[34,465506,465508,465522],{"className":465507},[31801],[34,465509,465511],{"className":465510,"style":31806},[31805],[34,465512,465513,465516],{"style":31809},[34,465514],{"className":465515,"style":31814},[31813],[34,465517,465519],{"className":465518},[31818,31819,31820,31821],[34,465520,640],{"className":465521},[31685,31821],[34,465523,31829],{"className":465524},[31828],[34,465526,465528],{"className":465527},[31801],[34,465529,465531],{"className":465530,"style":31836},[31805],[34,465532],{},[34,465534],{"className":465535,"style":32604},[31845],[34,465537,32226],{"className":465538},[32024],[34,465540],{"className":465541,"style":32604},[31845],[34,465543,465545,465548,465588,465591,465594],{"className":465544},[31676],[34,465546],{"className":465547,"style":225773},[31680],[34,465549,465551,465554],{"className":465550},[31685],[34,465552,32936],{"className":465553},[31685,31686],[34,465555,465557],{"className":465556},[31792],[34,465558,465560,465580],{"className":465559},[31796,31797],[34,465561,465563,465577],{"className":465562},[31801],[34,465564,465566],{"className":465565,"style":31806},[31805],[34,465567,465568,465571],{"style":31809},[34,465569],{"className":465570,"style":31814},[31813],[34,465572,465574],{"className":465573},[31818,31819,31820,31821],[34,465575,768],{"className":465576},[31685,31821],[34,465578,31829],{"className":465579},[31828],[34,465581,465583],{"className":465582},[31801],[34,465584,465586],{"className":465585,"style":31836},[31805],[34,465587],{},[34,465589],{"className":465590,"style":31846},[31845],[34,465592,112296],{"className":465593},[31850],[34,465595],{"className":465596,"style":31846},[31845],[34,465598,465600,465603],{"className":465599},[31676],[34,465601],{"className":465602,"style":33218},[31680],[34,465604,34386],{"className":465605,"style":34427},[31685,31686],[34,465607,465609,465649],{"className":465608},[31642],[34,465610,465612],{"className":465611},[31646],[31648,465613,465614],{"xmlns":31650},[31652,465615,465616,465646],{},[31655,465617,465618,465620,465622,465628,465630,465636,465638,465640,465642,465644],{},[31658,465619,31739],{},[31662,465621,456],{"stretchy":702},[31743,465623,465624,465626],{},[31658,465625,32936],{},[31749,465627,640],{},[31662,465629,32226],{},[31743,465631,465632,465634],{},[31658,465633,32936],{},[31749,465635,768],{},[31662,465637,2557],{"stretchy":702},[31662,465639,112296],{},[31658,465641,440604],{"mathvariant":38079},[31662,465643,32682],{},[31658,465645,31739],{},[31665,465647,465648],{"encoding":31667},"f(x_1 - x_2) \\in \\operatorname{Im} f",[34,465650,465652,465713,465771],{"className":465651,"ariaHidden":758},[31672],[34,465653,465655,465658,465661,465664,465704,465707,465710],{"className":465654},[31676],[34,465656],{"className":465657,"style":31774},[31680],[34,465659,31739],{"className":465660,"style":31778},[31685,31686],[34,465662,456],{"className":465663},[31782],[34,465665,465667,465670],{"className":465666},[31685],[34,465668,32936],{"className":465669},[31685,31686],[34,465671,465673],{"className":465672},[31792],[34,465674,465676,465696],{"className":465675},[31796,31797],[34,465677,465679,465693],{"className":465678},[31801],[34,465680,465682],{"className":465681,"style":31806},[31805],[34,465683,465684,465687],{"style":31809},[34,465685],{"className":465686,"style":31814},[31813],[34,465688,465690],{"className":465689},[31818,31819,31820,31821],[34,465691,640],{"className":465692},[31685,31821],[34,465694,31829],{"className":465695},[31828],[34,465697,465699],{"className":465698},[31801],[34,465700,465702],{"className":465701,"style":31836},[31805],[34,465703],{},[34,465705],{"className":465706,"style":32604},[31845],[34,465708,32226],{"className":465709},[32024],[34,465711],{"className":465712,"style":32604},[31845],[34,465714,465716,465719,465759,465762,465765,465768],{"className":465715},[31676],[34,465717],{"className":465718,"style":31774},[31680],[34,465720,465722,465725],{"className":465721},[31685],[34,465723,32936],{"className":465724},[31685,31686],[34,465726,465728],{"className":465727},[31792],[34,465729,465731,465751],{"className":465730},[31796,31797],[34,465732,465734,465748],{"className":465733},[31801],[34,465735,465737],{"className":465736,"style":31806},[31805],[34,465738,465739,465742],{"style":31809},[34,465740],{"className":465741,"style":31814},[31813],[34,465743,465745],{"className":465744},[31818,31819,31820,31821],[34,465746,768],{"className":465747},[31685,31821],[34,465749,31829],{"className":465750},[31828],[34,465752,465754],{"className":465753},[31801],[34,465755,465757],{"className":465756,"style":31836},[31805],[34,465758],{},[34,465760,2557],{"className":465761},[31690],[34,465763],{"className":465764,"style":31846},[31845],[34,465766,112296],{"className":465767},[31850],[34,465769],{"className":465770,"style":31846},[31845],[34,465772,465774,465777,465783,465786],{"className":465773},[31676],[34,465775],{"className":465776,"style":43217},[31680],[34,465778,465780],{"className":465779},[32767],[34,465781,440604],{"className":465782},[31685,207923],[34,465784],{"className":465785,"style":32123},[31845],[34,465787,31739],{"className":465788,"style":31778},[31685,31686],"。即 ",[34,465791,465793,465827],{"className":465792},[31642],[34,465794,465796],{"className":465795},[31646],[31648,465797,465798],{"xmlns":31650},[31652,465799,465800,465824],{},[31655,465801,465802,465808,465810,465816,465818,465820,465822],{},[31743,465803,465804,465806],{},[31658,465805,67764],{},[31749,465807,640],{},[31662,465809,32226],{},[31743,465811,465812,465814],{},[31658,465813,67764],{},[31749,465815,768],{},[31662,465817,112296],{},[31658,465819,440604],{"mathvariant":38079},[31662,465821,32682],{},[31658,465823,31739],{},[31665,465825,465826],{"encoding":31667},"y_1 - y_2 \\in \\operatorname{Im} f",[34,465828,465830,465885,465940],{"className":465829,"ariaHidden":758},[31672],[34,465831,465833,465836,465876,465879,465882],{"className":465832},[31676],[34,465834],{"className":465835,"style":55949},[31680],[34,465837,465839,465842],{"className":465838},[31685],[34,465840,67764],{"className":465841,"style":32062},[31685,31686],[34,465843,465845],{"className":465844},[31792],[34,465846,465848,465868],{"className":465847},[31796,31797],[34,465849,465851,465865],{"className":465850},[31801],[34,465852,465854],{"className":465853,"style":31806},[31805],[34,465855,465856,465859],{"style":38795},[34,465857],{"className":465858,"style":31814},[31813],[34,465860,465862],{"className":465861},[31818,31819,31820,31821],[34,465863,640],{"className":465864},[31685,31821],[34,465866,31829],{"className":465867},[31828],[34,465869,465871],{"className":465870},[31801],[34,465872,465874],{"className":465873,"style":31836},[31805],[34,465875],{},[34,465877],{"className":465878,"style":32604},[31845],[34,465880,32226],{"className":465881},[32024],[34,465883],{"className":465884,"style":32604},[31845],[34,465886,465888,465891,465931,465934,465937],{"className":465887},[31676],[34,465889],{"className":465890,"style":86652},[31680],[34,465892,465894,465897],{"className":465893},[31685],[34,465895,67764],{"className":465896,"style":32062},[31685,31686],[34,465898,465900],{"className":465899},[31792],[34,465901,465903,465923],{"className":465902},[31796,31797],[34,465904,465906,465920],{"className":465905},[31801],[34,465907,465909],{"className":465908,"style":31806},[31805],[34,465910,465911,465914],{"style":38795},[34,465912],{"className":465913,"style":31814},[31813],[34,465915,465917],{"className":465916},[31818,31819,31820,31821],[34,465918,768],{"className":465919},[31685,31821],[34,465921,31829],{"className":465922},[31828],[34,465924,465926],{"className":465925},[31801],[34,465927,465929],{"className":465928,"style":31836},[31805],[34,465930],{},[34,465932],{"className":465933,"style":31846},[31845],[34,465935,112296],{"className":465936},[31850],[34,465938],{"className":465939,"style":31846},[31845],[34,465941,465943,465946,465952,465955],{"className":465942},[31676],[34,465944],{"className":465945,"style":43217},[31680],[34,465947,465949],{"className":465948},[32767],[34,465950,440604],{"className":465951},[31685,207923],[34,465953],{"className":465954,"style":32123},[31845],[34,465956,31739],{"className":465957,"style":31778},[31685,31686],"。\n(3) ",[2627,465960,443594],{},[34,465962,465964,466040],{"className":465963},[31642],[34,465965,465967],{"className":465966},[31646],[31648,465968,465969],{"xmlns":31650},[31652,465970,465971,466037],{},[31655,465972,465973,465979,465981,465987,465989,465991,465993,465999,466001,466003,466005,466007,466013,466015,466017,466019,466021,466027,466029,466035],{},[31743,465974,465975,465977],{},[31658,465976,67764],{},[31749,465978,640],{},[31662,465980,71191],{},[31743,465982,465983,465985],{},[31658,465984,67764],{},[31749,465986,768],{},[31662,465988,202],{},[31658,465990,31739],{},[31662,465992,456],{"stretchy":702},[31743,465994,465995,465997],{},[31658,465996,32936],{},[31749,465998,640],{},[31662,466000,2557],{"stretchy":702},[31662,466002,71191],{},[31658,466004,31739],{},[31662,466006,456],{"stretchy":702},[31743,466008,466009,466011],{},[31658,466010,32936],{},[31749,466012,768],{},[31662,466014,2557],{"stretchy":702},[31662,466016,202],{},[31658,466018,31739],{},[31662,466020,456],{"stretchy":702},[31743,466022,466023,466025],{},[31658,466024,32936],{},[31749,466026,640],{},[31662,466028,71191],{},[31743,466030,466031,466033],{},[31658,466032,32936],{},[31749,466034,768],{},[31662,466036,2557],{"stretchy":702},[31665,466038,466039],{"encoding":31667},"y_1 \\cdot y_2 = f(x_1) \\cdot f(x_2) = f(x_1 \\cdot x_2)",[34,466041,466043,466098,466153,466217,466281,466342],{"className":466042,"ariaHidden":758},[31672],[34,466044,466046,466049,466089,466092,466095],{"className":466045},[31676],[34,466047],{"className":466048,"style":73503},[31680],[34,466050,466052,466055],{"className":466051},[31685],[34,466053,67764],{"className":466054,"style":32062},[31685,31686],[34,466056,466058],{"className":466057},[31792],[34,466059,466061,466081],{"className":466060},[31796,31797],[34,466062,466064,466078],{"className":466063},[31801],[34,466065,466067],{"className":466066,"style":31806},[31805],[34,466068,466069,466072],{"style":38795},[34,466070],{"className":466071,"style":31814},[31813],[34,466073,466075],{"className":466074},[31818,31819,31820,31821],[34,466076,640],{"className":466077},[31685,31821],[34,466079,31829],{"className":466080},[31828],[34,466082,466084],{"className":466083},[31801],[34,466085,466087],{"className":466086,"style":31836},[31805],[34,466088],{},[34,466090],{"className":466091,"style":32604},[31845],[34,466093,71191],{"className":466094},[32024],[34,466096],{"className":466097,"style":32604},[31845],[34,466099,466101,466104,466144,466147,466150],{"className":466100},[31676],[34,466102],{"className":466103,"style":38774},[31680],[34,466105,466107,466110],{"className":466106},[31685],[34,466108,67764],{"className":466109,"style":32062},[31685,31686],[34,466111,466113],{"className":466112},[31792],[34,466114,466116,466136],{"className":466115},[31796,31797],[34,466117,466119,466133],{"className":466118},[31801],[34,466120,466122],{"className":466121,"style":31806},[31805],[34,466123,466124,466127],{"style":38795},[34,466125],{"className":466126,"style":31814},[31813],[34,466128,466130],{"className":466129},[31818,31819,31820,31821],[34,466131,768],{"className":466132},[31685,31821],[34,466134,31829],{"className":466135},[31828],[34,466137,466139],{"className":466138},[31801],[34,466140,466142],{"className":466141,"style":31836},[31805],[34,466143],{},[34,466145],{"className":466146,"style":31846},[31845],[34,466148,202],{"className":466149},[31850],[34,466151],{"className":466152,"style":31846},[31845],[34,466154,466156,466159,466162,466165,466205,466208,466211,466214],{"className":466155},[31676],[34,466157],{"className":466158,"style":31774},[31680],[34,466160,31739],{"className":466161,"style":31778},[31685,31686],[34,466163,456],{"className":466164},[31782],[34,466166,466168,466171],{"className":466167},[31685],[34,466169,32936],{"className":466170},[31685,31686],[34,466172,466174],{"className":466173},[31792],[34,466175,466177,466197],{"className":466176},[31796,31797],[34,466178,466180,466194],{"className":466179},[31801],[34,466181,466183],{"className":466182,"style":31806},[31805],[34,466184,466185,466188],{"style":31809},[34,466186],{"className":466187,"style":31814},[31813],[34,466189,466191],{"className":466190},[31818,31819,31820,31821],[34,466192,640],{"className":466193},[31685,31821],[34,466195,31829],{"className":466196},[31828],[34,466198,466200],{"className":466199},[31801],[34,466201,466203],{"className":466202,"style":31836},[31805],[34,466204],{},[34,466206,2557],{"className":466207},[31690],[34,466209],{"className":466210,"style":32604},[31845],[34,466212,71191],{"className":466213},[32024],[34,466215],{"className":466216,"style":32604},[31845],[34,466218,466220,466223,466226,466229,466269,466272,466275,466278],{"className":466219},[31676],[34,466221],{"className":466222,"style":31774},[31680],[34,466224,31739],{"className":466225,"style":31778},[31685,31686],[34,466227,456],{"className":466228},[31782],[34,466230,466232,466235],{"className":466231},[31685],[34,466233,32936],{"className":466234},[31685,31686],[34,466236,466238],{"className":466237},[31792],[34,466239,466241,466261],{"className":466240},[31796,31797],[34,466242,466244,466258],{"className":466243},[31801],[34,466245,466247],{"className":466246,"style":31806},[31805],[34,466248,466249,466252],{"style":31809},[34,466250],{"className":466251,"style":31814},[31813],[34,466253,466255],{"className":466254},[31818,31819,31820,31821],[34,466256,768],{"className":466257},[31685,31821],[34,466259,31829],{"className":466260},[31828],[34,466262,466264],{"className":466263},[31801],[34,466265,466267],{"className":466266,"style":31836},[31805],[34,466268],{},[34,466270,2557],{"className":466271},[31690],[34,466273],{"className":466274,"style":31846},[31845],[34,466276,202],{"className":466277},[31850],[34,466279],{"className":466280,"style":31846},[31845],[34,466282,466284,466287,466290,466293,466333,466336,466339],{"className":466283},[31676],[34,466285],{"className":466286,"style":31774},[31680],[34,466288,31739],{"className":466289,"style":31778},[31685,31686],[34,466291,456],{"className":466292},[31782],[34,466294,466296,466299],{"className":466295},[31685],[34,466297,32936],{"className":466298},[31685,31686],[34,466300,466302],{"className":466301},[31792],[34,466303,466305,466325],{"className":466304},[31796,31797],[34,466306,466308,466322],{"className":466307},[31801],[34,466309,466311],{"className":466310,"style":31806},[31805],[34,466312,466313,466316],{"style":31809},[34,466314],{"className":466315,"style":31814},[31813],[34,466317,466319],{"className":466318},[31818,31819,31820,31821],[34,466320,640],{"className":466321},[31685,31821],[34,466323,31829],{"className":466324},[31828],[34,466326,466328],{"className":466327},[31801],[34,466329,466331],{"className":466330,"style":31836},[31805],[34,466332],{},[34,466334],{"className":466335,"style":32604},[31845],[34,466337,71191],{"className":466338},[32024],[34,466340],{"className":466341,"style":32604},[31845],[34,466343,466345,466348,466388],{"className":466344},[31676],[34,466346],{"className":466347,"style":31774},[31680],[34,466349,466351,466354],{"className":466350},[31685],[34,466352,32936],{"className":466353},[31685,31686],[34,466355,466357],{"className":466356},[31792],[34,466358,466360,466380],{"className":466359},[31796,31797],[34,466361,466363,466377],{"className":466362},[31801],[34,466364,466366],{"className":466365,"style":31806},[31805],[34,466367,466368,466371],{"style":31809},[34,466369],{"className":466370,"style":31814},[31813],[34,466372,466374],{"className":466373},[31818,31819,31820,31821],[34,466375,768],{"className":466376},[31685,31821],[34,466378,31829],{"className":466379},[31828],[34,466381,466383],{"className":466382},[31801],[34,466384,466386],{"className":466385,"style":31836},[31805],[34,466387],{},[34,466389,2557],{"className":466390},[31690],"\n（注：同态保持乘法）\n因为 ",[34,466393,466395,466425],{"className":466394},[31642],[34,466396,466398],{"className":466397},[31646],[31648,466399,466400],{"xmlns":31650},[31652,466401,466402,466422],{},[31655,466403,466404,466410,466412,466418,466420],{},[31743,466405,466406,466408],{},[31658,466407,32936],{},[31749,466409,640],{},[31662,466411,71191],{},[31743,466413,466414,466416],{},[31658,466415,32936],{},[31749,466417,768],{},[31662,466419,112296],{},[31658,466421,34386],{},[31665,466423,466424],{"encoding":31667},"x_1 \\cdot x_2 \\in R",[34,466426,466428,466484,466539],{"className":466427,"ariaHidden":758},[31672],[34,466429,466431,466435,466475,466478,466481],{"className":466430},[31676],[34,466432],{"className":466433,"style":466434},[31680],"height:0.5945em;vertical-align:-0.15em;",[34,466436,466438,466441],{"className":466437},[31685],[34,466439,32936],{"className":466440},[31685,31686],[34,466442,466444],{"className":466443},[31792],[34,466445,466447,466467],{"className":466446},[31796,31797],[34,466448,466450,466464],{"className":466449},[31801],[34,466451,466453],{"className":466452,"style":31806},[31805],[34,466454,466455,466458],{"style":31809},[34,466456],{"className":466457,"style":31814},[31813],[34,466459,466461],{"className":466460},[31818,31819,31820,31821],[34,466462,640],{"className":466463},[31685,31821],[34,466465,31829],{"className":466466},[31828],[34,466468,466470],{"className":466469},[31801],[34,466471,466473],{"className":466472,"style":31836},[31805],[34,466474],{},[34,466476],{"className":466477,"style":32604},[31845],[34,466479,71191],{"className":466480},[32024],[34,466482],{"className":466483,"style":32604},[31845],[34,466485,466487,466490,466530,466533,466536],{"className":466486},[31676],[34,466488],{"className":466489,"style":225773},[31680],[34,466491,466493,466496],{"className":466492},[31685],[34,466494,32936],{"className":466495},[31685,31686],[34,466497,466499],{"className":466498},[31792],[34,466500,466502,466522],{"className":466501},[31796,31797],[34,466503,466505,466519],{"className":466504},[31801],[34,466506,466508],{"className":466507,"style":31806},[31805],[34,466509,466510,466513],{"style":31809},[34,466511],{"className":466512,"style":31814},[31813],[34,466514,466516],{"className":466515},[31818,31819,31820,31821],[34,466517,768],{"className":466518},[31685,31821],[34,466520,31829],{"className":466521},[31828],[34,466523,466525],{"className":466524},[31801],[34,466526,466528],{"className":466527,"style":31836},[31805],[34,466529],{},[34,466531],{"className":466532,"style":31846},[31845],[34,466534,112296],{"className":466535},[31850],[34,466537],{"className":466538,"style":31846},[31845],[34,466540,466542,466545],{"className":466541},[31676],[34,466543],{"className":466544,"style":33218},[31680],[34,466546,34386],{"className":466547,"style":34427},[31685,31686],"，所以其像也在 ",[34,466550,466552,466569],{"className":466551},[31642],[34,466553,466555],{"className":466554},[31646],[31648,466556,466557],{"xmlns":31650},[31652,466558,466559,466567],{},[31655,466560,466561,466563,466565],{},[31658,466562,440604],{"mathvariant":38079},[31662,466564,32682],{},[31658,466566,31739],{},[31665,466568,463310],{"encoding":31667},[34,466570,466572],{"className":466571,"ariaHidden":758},[31672],[34,466573,466575,466578,466584,466587],{"className":466574},[31676],[34,466576],{"className":466577,"style":43217},[31680],[34,466579,466581],{"className":466580},[32767],[34,466582,440604],{"className":466583},[31685,207923],[34,466585],{"className":466586,"style":32123},[31845],[34,466588,31739],{"className":466589,"style":31778},[31685,31686],[15,466591,466592,2664,466594,11008,466635,388672],{},[2627,466593,114043],{},[34,466595,466597,466614],{"className":466596},[31642],[34,466598,466600],{"className":466599},[31646],[31648,466601,466602],{"xmlns":31650},[31652,466603,466604,466612],{},[31655,466605,466606,466608,466610],{},[31658,466607,440604],{"mathvariant":38079},[31662,466609,32682],{},[31658,466611,31739],{},[31665,466613,463310],{"encoding":31667},[34,466615,466617],{"className":466616,"ariaHidden":758},[31672],[34,466618,466620,466623,466629,466632],{"className":466619},[31676],[34,466621],{"className":466622,"style":43217},[31680],[34,466624,466626],{"className":466625},[32767],[34,466627,440604],{"className":466628},[31685,207923],[34,466630],{"className":466631,"style":32123},[31845],[34,466633,31739],{"className":466634,"style":31778},[31685,31686],[34,466636,466638,466651],{"className":466637},[31642],[34,466639,466641],{"className":466640},[31646],[31648,466642,466643],{"xmlns":31650},[31652,466644,466645,466649],{},[31655,466646,466647],{},[31658,466648,94675],{},[31665,466650,94675],{"encoding":31667},[34,466652,466654],{"className":466653,"ariaHidden":758},[31672],[34,466655,466657,466660],{"className":466656},[31676],[34,466658],{"className":466659,"style":33218},[31680],[34,466661,94675],{"className":466662,"style":106455},[31685,31686],[15,466664,466665],{},[2627,466666,443026,466667,11008,466705,466733],{},[34,466668,466670,466687],{"className":466669},[31642],[34,466671,466673],{"className":466672},[31646],[31648,466674,466675],{"xmlns":31650},[31652,466676,466677,466685],{},[31655,466678,466679,466681,466683],{},[31658,466680,406335],{},[31662,466682,32682],{},[31658,466684,31739],{},[31665,466686,463410],{"encoding":31667},[34,466688,466690],{"className":466689,"ariaHidden":758},[31672],[34,466691,466693,466696,466699,466702],{"className":466692},[31676],[34,466694],{"className":466695,"style":43217},[31680],[34,466697,406335],{"className":466698},[32767],[34,466700],{"className":466701,"style":32123},[31845],[34,466703,31739],{"className":466704,"style":31778},[31685,31686],[34,466706,466708,466721],{"className":466707},[31642],[34,466709,466711],{"className":466710},[31646],[31648,466712,466713],{"xmlns":31650},[31652,466714,466715,466719],{},[31655,466716,466717],{},[31658,466718,34386],{},[31665,466720,34386],{"encoding":31667},[34,466722,466724],{"className":466723,"ariaHidden":758},[31672],[34,466725,466727,466730],{"className":466726},[31676],[34,466728],{"className":466729,"style":33218},[31680],[34,466731,34386],{"className":466732,"style":34427},[31685,31686]," 的理想",[15,466735,336888,466736,463989,466929,2664,466931,467184,467185,34554,467187,326989,467261,39984,467459,467788,467789,465958,467872,34554,467875,33775,467936,66331,468043,468093,468094,468096,468097,258,468131,468165,468166,468452,468738,468739,13977,468808,86674,468874,347],{},[34,466737,466739,466787],{"className":466738},[31642],[34,466740,466742],{"className":466741},[31646],[31648,466743,466744],{"xmlns":31650},[31652,466745,466746,466784],{},[31655,466747,466748,466750,466752,466754,466756,466758,466760,466762,466764,466766,466768,466770,466772,466774,466776,466782],{},[31658,466749,406335],{},[31662,466751,32682],{},[31658,466753,31739],{},[31662,466755,202],{},[31662,466757,32685],{"stretchy":702},[31658,466759,32936],{},[31662,466761,112296],{},[31658,466763,34386],{},[31662,466765,162086],{},[31658,466767,31739],{},[31662,466769,456],{"stretchy":702},[31658,466771,32936],{},[31662,466773,2557],{"stretchy":702},[31662,466775,202],{},[31743,466777,466778,466780],{},[31749,466779,459],{},[31658,466781,94675],{},[31662,466783,32715],{"stretchy":702},[31665,466785,466786],{"encoding":31667},"\\ker f = \\{x \\in R \\mid f(x) = 0_S\\}",[34,466788,466790,466814,466835,466853,466880],{"className":466789,"ariaHidden":758},[31672],[34,466791,466793,466796,466799,466802,466805,466808,466811],{"className":466792},[31676],[34,466794],{"className":466795,"style":43217},[31680],[34,466797,406335],{"className":466798},[32767],[34,466800],{"className":466801,"style":32123},[31845],[34,466803,31739],{"className":466804,"style":31778},[31685,31686],[34,466806],{"className":466807,"style":31846},[31845],[34,466809,202],{"className":466810},[31850],[34,466812],{"className":466813,"style":31846},[31845],[34,466815,466817,466820,466823,466826,466829,466832],{"className":466816},[31676],[34,466818],{"className":466819,"style":31774},[31680],[34,466821,32685],{"className":466822},[31782],[34,466824,32936],{"className":466825},[31685,31686],[34,466827],{"className":466828,"style":31846},[31845],[34,466830,112296],{"className":466831},[31850],[34,466833],{"className":466834,"style":31846},[31845],[34,466836,466838,466841,466844,466847,466850],{"className":466837},[31676],[34,466839],{"className":466840,"style":31774},[31680],[34,466842,34386],{"className":466843,"style":34427},[31685,31686],[34,466845],{"className":466846,"style":31846},[31845],[34,466848,162086],{"className":466849},[31850],[34,466851],{"className":466852,"style":31846},[31845],[34,466854,466856,466859,466862,466865,466868,466871,466874,466877],{"className":466855},[31676],[34,466857],{"className":466858,"style":31774},[31680],[34,466860,31739],{"className":466861,"style":31778},[31685,31686],[34,466863,456],{"className":466864},[31782],[34,466866,32936],{"className":466867},[31685,31686],[34,466869,2557],{"className":466870},[31690],[34,466872],{"className":466873,"style":31846},[31845],[34,466875,202],{"className":466876},[31850],[34,466878],{"className":466879,"style":31846},[31845],[34,466881,466883,466886,466926],{"className":466882},[31676],[34,466884],{"className":466885,"style":31774},[31680],[34,466887,466889,466892],{"className":466888},[31685],[34,466890,459],{"className":466891},[31685],[34,466893,466895],{"className":466894},[31792],[34,466896,466898,466918],{"className":466897},[31796,31797],[34,466899,466901,466915],{"className":466900},[31801],[34,466902,466904],{"className":466903,"style":116558},[31805],[34,466905,466906,466909],{"style":31809},[34,466907],{"className":466908,"style":31814},[31813],[34,466910,466912],{"className":466911},[31818,31819,31820,31821],[34,466913,94675],{"className":466914,"style":106455},[31685,31686,31821],[34,466916,31829],{"className":466917},[31828],[34,466919,466921],{"className":466920},[31801],[34,466922,466924],{"className":466923,"style":31836},[31805],[34,466925],{},[34,466927,32715],{"className":466928},[31690],[2627,466930,441307],{},[34,466932,466934,466986],{"className":466933},[31642],[34,466935,466937],{"className":466936},[31646],[31648,466938,466939],{"xmlns":31650},[31652,466940,466941,466983],{},[31655,466942,466943,466945,466947,466953,466955,466957,466963,466965,466967,466969,466975,466977,466979,466981],{},[31658,466944,31739],{},[31662,466946,456],{"stretchy":702},[31743,466948,466949,466951],{},[31749,466950,459],{},[31658,466952,34386],{},[31662,466954,2557],{"stretchy":702},[31662,466956,202],{},[31743,466958,466959,466961],{},[31749,466960,459],{},[31658,466962,94675],{},[39361,466964,262622],{},[31662,466966,262625],{},[39361,466968,262622],{},[31743,466970,466971,466973],{},[31749,466972,459],{},[31658,466974,34386],{},[31662,466976,112296],{},[31658,466978,406335],{},[31662,466980,32682],{},[31658,466982,31739],{},[31665,466984,466985],{"encoding":31667},"f(0_R) = 0_S \\implies 0_R \\in \\ker f",[34,466987,466989,467053,467114,467169],{"className":466988,"ariaHidden":758},[31672],[34,466990,466992,466995,466998,467001,467041,467044,467047,467050],{"className":466991},[31676],[34,466993],{"className":466994,"style":31774},[31680],[34,466996,31739],{"className":466997,"style":31778},[31685,31686],[34,466999,456],{"className":467000},[31782],[34,467002,467004,467007],{"className":467003},[31685],[34,467005,459],{"className":467006},[31685],[34,467008,467010],{"className":467009},[31792],[34,467011,467013,467033],{"className":467012},[31796,31797],[34,467014,467016,467030],{"className":467015},[31801],[34,467017,467019],{"className":467018,"style":116558},[31805],[34,467020,467021,467024],{"style":31809},[34,467022],{"className":467023,"style":31814},[31813],[34,467025,467027],{"className":467026},[31818,31819,31820,31821],[34,467028,34386],{"className":467029,"style":34427},[31685,31686,31821],[34,467031,31829],{"className":467032},[31828],[34,467034,467036],{"className":467035},[31801],[34,467037,467039],{"className":467038,"style":31836},[31805],[34,467040],{},[34,467042,2557],{"className":467043},[31690],[34,467045],{"className":467046,"style":31846},[31845],[34,467048,202],{"className":467049},[31850],[34,467051],{"className":467052,"style":31846},[31845],[34,467054,467056,467059,467099,467102,467105,467108,467111],{"className":467055},[31676],[34,467057],{"className":467058,"style":38357},[31680],[34,467060,467062,467065],{"className":467061},[31685],[34,467063,459],{"className":467064},[31685],[34,467066,467068],{"className":467067},[31792],[34,467069,467071,467091],{"className":467070},[31796,31797],[34,467072,467074,467088],{"className":467073},[31801],[34,467075,467077],{"className":467076,"style":116558},[31805],[34,467078,467079,467082],{"style":31809},[34,467080],{"className":467081,"style":31814},[31813],[34,467083,467085],{"className":467084},[31818,31819,31820,31821],[34,467086,94675],{"className":467087,"style":106455},[31685,31686,31821],[34,467089,31829],{"className":467090},[31828],[34,467092,467094],{"className":467093},[31801],[34,467095,467097],{"className":467096,"style":31836},[31805],[34,467098],{},[34,467100],{"className":467101,"style":31846},[31845],[34,467103],{"className":467104,"style":31846},[31845],[34,467106,262625],{"className":467107},[31850],[34,467109],{"className":467110,"style":31846},[31845],[34,467112],{"className":467113,"style":31846},[31845],[34,467115,467117,467120,467160,467163,467166],{"className":467116},[31676],[34,467118],{"className":467119,"style":38357},[31680],[34,467121,467123,467126],{"className":467122},[31685],[34,467124,459],{"className":467125},[31685],[34,467127,467129],{"className":467128},[31792],[34,467130,467132,467152],{"className":467131},[31796,31797],[34,467133,467135,467149],{"className":467134},[31801],[34,467136,467138],{"className":467137,"style":116558},[31805],[34,467139,467140,467143],{"style":31809},[34,467141],{"className":467142,"style":31814},[31813],[34,467144,467146],{"className":467145},[31818,31819,31820,31821],[34,467147,34386],{"className":467148,"style":34427},[31685,31686,31821],[34,467150,31829],{"className":467151},[31828],[34,467153,467155],{"className":467154},[31801],[34,467156,467158],{"className":467157,"style":31836},[31805],[34,467159],{},[34,467161],{"className":467162,"style":31846},[31845],[34,467164,112296],{"className":467165},[31850],[34,467167],{"className":467168,"style":31846},[31845],[34,467170,467172,467175,467178,467181],{"className":467171},[31676],[34,467173],{"className":467174,"style":43217},[31680],[34,467176,406335],{"className":467177},[32767],[34,467179],{"className":467180,"style":32123},[31845],[34,467182,31739],{"className":467183,"style":31778},[31685,31686],"。\n(2) ",[2627,467186,443104],{},[34,467188,467190,467216],{"className":467189},[31642],[34,467191,467193],{"className":467192},[31646],[31648,467194,467195],{"xmlns":31650},[31652,467196,467197,467213],{},[31655,467198,467199,467201,467203,467205,467207,467209,467211],{},[31658,467200,111],{},[31662,467202,31960],{"separator":758},[31658,467204,1697],{},[31662,467206,112296],{},[31658,467208,406335],{},[31662,467210,32682],{},[31658,467212,31739],{},[31665,467214,467215],{"encoding":31667},"a, b \\in \\ker f",[34,467217,467219,467246],{"className":467218,"ariaHidden":758},[31672],[34,467220,467222,467225,467228,467231,467234,467237,467240,467243],{"className":467221},[31676],[34,467223],{"className":467224,"style":43217},[31680],[34,467226,111],{"className":467227},[31685,31686],[34,467229,31960],{"className":467230},[32119],[34,467232],{"className":467233,"style":32123},[31845],[34,467235,1697],{"className":467236},[31685,31686],[34,467238],{"className":467239,"style":31846},[31845],[34,467241,112296],{"className":467242},[31850],[34,467244],{"className":467245,"style":31846},[31845],[34,467247,467249,467252,467255,467258],{"className":467248},[31676],[34,467250],{"className":467251,"style":43217},[31680],[34,467253,406335],{"className":467254},[32767],[34,467256],{"className":467257,"style":32123},[31845],[34,467259,31739],{"className":467260,"style":31778},[31685,31686],[34,467262,467264,467310],{"className":467263},[31642],[34,467265,467267],{"className":467266},[31646],[31648,467268,467269],{"xmlns":31650},[31652,467270,467271,467307],{},[31655,467272,467273,467275,467277,467279,467281,467283,467289,467291,467293,467295,467297,467299,467301],{},[31658,467274,31739],{},[31662,467276,456],{"stretchy":702},[31658,467278,111],{},[31662,467280,2557],{"stretchy":702},[31662,467282,202],{},[31743,467284,467285,467287],{},[31749,467286,459],{},[31658,467288,94675],{},[31662,467290,31960],{"separator":758},[31658,467292,31739],{},[31662,467294,456],{"stretchy":702},[31658,467296,1697],{},[31662,467298,2557],{"stretchy":702},[31662,467300,202],{},[31743,467302,467303,467305],{},[31749,467304,459],{},[31658,467306,94675],{},[31665,467308,467309],{"encoding":31667},"f(a) = 0_S, f(b) = 0_S",[34,467311,467313,467340,467413],{"className":467312,"ariaHidden":758},[31672],[34,467314,467316,467319,467322,467325,467328,467331,467334,467337],{"className":467315},[31676],[34,467317],{"className":467318,"style":31774},[31680],[34,467320,31739],{"className":467321,"style":31778},[31685,31686],[34,467323,456],{"className":467324},[31782],[34,467326,111],{"className":467327},[31685,31686],[34,467329,2557],{"className":467330},[31690],[34,467332],{"className":467333,"style":31846},[31845],[34,467335,202],{"className":467336},[31850],[34,467338],{"className":467339,"style":31846},[31845],[34,467341,467343,467346,467386,467389,467392,467395,467398,467401,467404,467407,467410],{"className":467342},[31676],[34,467344],{"className":467345,"style":31774},[31680],[34,467347,467349,467352],{"className":467348},[31685],[34,467350,459],{"className":467351},[31685],[34,467353,467355],{"className":467354},[31792],[34,467356,467358,467378],{"className":467357},[31796,31797],[34,467359,467361,467375],{"className":467360},[31801],[34,467362,467364],{"className":467363,"style":116558},[31805],[34,467365,467366,467369],{"style":31809},[34,467367],{"className":467368,"style":31814},[31813],[34,467370,467372],{"className":467371},[31818,31819,31820,31821],[34,467373,94675],{"className":467374,"style":106455},[31685,31686,31821],[34,467376,31829],{"className":467377},[31828],[34,467379,467381],{"className":467380},[31801],[34,467382,467384],{"className":467383,"style":31836},[31805],[34,467385],{},[34,467387,31960],{"className":467388},[32119],[34,467390],{"className":467391,"style":32123},[31845],[34,467393,31739],{"className":467394,"style":31778},[31685,31686],[34,467396,456],{"className":467397},[31782],[34,467399,1697],{"className":467400},[31685,31686],[34,467402,2557],{"className":467403},[31690],[34,467405],{"className":467406,"style":31846},[31845],[34,467408,202],{"className":467409},[31850],[34,467411],{"className":467412,"style":31846},[31845],[34,467414,467416,467419],{"className":467415},[31676],[34,467417],{"className":467418,"style":38357},[31680],[34,467420,467422,467425],{"className":467421},[31685],[34,467423,459],{"className":467424},[31685],[34,467426,467428],{"className":467427},[31792],[34,467429,467431,467451],{"className":467430},[31796,31797],[34,467432,467434,467448],{"className":467433},[31801],[34,467435,467437],{"className":467436,"style":116558},[31805],[34,467438,467439,467442],{"style":31809},[34,467440],{"className":467441,"style":31814},[31813],[34,467443,467445],{"className":467444},[31818,31819,31820,31821],[34,467446,94675],{"className":467447,"style":106455},[31685,31686,31821],[34,467449,31829],{"className":467450},[31828],[34,467452,467454],{"className":467453},[31801],[34,467455,467457],{"className":467456,"style":31836},[31805],[34,467458],{},[34,467460,467462,467530],{"className":467461},[31642],[34,467463,467465],{"className":467464},[31646],[31648,467466,467467],{"xmlns":31650},[31652,467468,467469,467527],{},[31655,467470,467471,467473,467475,467477,467479,467481,467483,467485,467487,467489,467491,467493,467495,467497,467499,467501,467503,467505,467511,467513,467519,467521],{},[31658,467472,31739],{},[31662,467474,456],{"stretchy":702},[31658,467476,111],{},[31662,467478,32226],{},[31658,467480,1697],{},[31662,467482,2557],{"stretchy":702},[31662,467484,202],{},[31658,467486,31739],{},[31662,467488,456],{"stretchy":702},[31658,467490,111],{},[31662,467492,2557],{"stretchy":702},[31662,467494,32226],{},[31658,467496,31739],{},[31662,467498,456],{"stretchy":702},[31658,467500,1697],{},[31662,467502,2557],{"stretchy":702},[31662,467504,202],{},[31743,467506,467507,467509],{},[31749,467508,459],{},[31658,467510,94675],{},[31662,467512,32226],{},[31743,467514,467515,467517],{},[31749,467516,459],{},[31658,467518,94675],{},[31662,467520,202],{},[31743,467522,467523,467525],{},[31749,467524,459],{},[31658,467526,94675],{},[31665,467528,467529],{"encoding":31667},"f(a - b) = f(a) - f(b) = 0_S - 0_S = 0_S",[34,467531,467533,467557,467578,467605,467632,467687,467742],{"className":467532,"ariaHidden":758},[31672],[34,467534,467536,467539,467542,467545,467548,467551,467554],{"className":467535},[31676],[34,467537],{"className":467538,"style":31774},[31680],[34,467540,31739],{"className":467541,"style":31778},[31685,31686],[34,467543,456],{"className":467544},[31782],[34,467546,111],{"className":467547},[31685,31686],[34,467549],{"className":467550,"style":32604},[31845],[34,467552,32226],{"className":467553},[32024],[34,467555],{"className":467556,"style":32604},[31845],[34,467558,467560,467563,467566,467569,467572,467575],{"className":467559},[31676],[34,467561],{"className":467562,"style":31774},[31680],[34,467564,1697],{"className":467565},[31685,31686],[34,467567,2557],{"className":467568},[31690],[34,467570],{"className":467571,"style":31846},[31845],[34,467573,202],{"className":467574},[31850],[34,467576],{"className":467577,"style":31846},[31845],[34,467579,467581,467584,467587,467590,467593,467596,467599,467602],{"className":467580},[31676],[34,467582],{"className":467583,"style":31774},[31680],[34,467585,31739],{"className":467586,"style":31778},[31685,31686],[34,467588,456],{"className":467589},[31782],[34,467591,111],{"className":467592},[31685,31686],[34,467594,2557],{"className":467595},[31690],[34,467597],{"className":467598,"style":32604},[31845],[34,467600,32226],{"className":467601},[32024],[34,467603],{"className":467604,"style":32604},[31845],[34,467606,467608,467611,467614,467617,467620,467623,467626,467629],{"className":467607},[31676],[34,467609],{"className":467610,"style":31774},[31680],[34,467612,31739],{"className":467613,"style":31778},[31685,31686],[34,467615,456],{"className":467616},[31782],[34,467618,1697],{"className":467619},[31685,31686],[34,467621,2557],{"className":467622},[31690],[34,467624],{"className":467625,"style":31846},[31845],[34,467627,202],{"className":467628},[31850],[34,467630],{"className":467631,"style":31846},[31845],[34,467633,467635,467638,467678,467681,467684],{"className":467634},[31676],[34,467636],{"className":467637,"style":38357},[31680],[34,467639,467641,467644],{"className":467640},[31685],[34,467642,459],{"className":467643},[31685],[34,467645,467647],{"className":467646},[31792],[34,467648,467650,467670],{"className":467649},[31796,31797],[34,467651,467653,467667],{"className":467652},[31801],[34,467654,467656],{"className":467655,"style":116558},[31805],[34,467657,467658,467661],{"style":31809},[34,467659],{"className":467660,"style":31814},[31813],[34,467662,467664],{"className":467663},[31818,31819,31820,31821],[34,467665,94675],{"className":467666,"style":106455},[31685,31686,31821],[34,467668,31829],{"className":467669},[31828],[34,467671,467673],{"className":467672},[31801],[34,467674,467676],{"className":467675,"style":31836},[31805],[34,467677],{},[34,467679],{"className":467680,"style":32604},[31845],[34,467682,32226],{"className":467683},[32024],[34,467685],{"className":467686,"style":32604},[31845],[34,467688,467690,467693,467733,467736,467739],{"className":467689},[31676],[34,467691],{"className":467692,"style":38357},[31680],[34,467694,467696,467699],{"className":467695},[31685],[34,467697,459],{"className":467698},[31685],[34,467700,467702],{"className":467701},[31792],[34,467703,467705,467725],{"className":467704},[31796,31797],[34,467706,467708,467722],{"className":467707},[31801],[34,467709,467711],{"className":467710,"style":116558},[31805],[34,467712,467713,467716],{"style":31809},[34,467714],{"className":467715,"style":31814},[31813],[34,467717,467719],{"className":467718},[31818,31819,31820,31821],[34,467720,94675],{"className":467721,"style":106455},[31685,31686,31821],[34,467723,31829],{"className":467724},[31828],[34,467726,467728],{"className":467727},[31801],[34,467729,467731],{"className":467730,"style":31836},[31805],[34,467732],{},[34,467734],{"className":467735,"style":31846},[31845],[34,467737,202],{"className":467738},[31850],[34,467740],{"className":467741,"style":31846},[31845],[34,467743,467745,467748],{"className":467744},[31676],[34,467746],{"className":467747,"style":38357},[31680],[34,467749,467751,467754],{"className":467750},[31685],[34,467752,459],{"className":467753},[31685],[34,467755,467757],{"className":467756},[31792],[34,467758,467760,467780],{"className":467759},[31796,31797],[34,467761,467763,467777],{"className":467762},[31801],[34,467764,467766],{"className":467765,"style":116558},[31805],[34,467767,467768,467771],{"style":31809},[34,467769],{"className":467770,"style":31814},[31813],[34,467772,467774],{"className":467773},[31818,31819,31820,31821],[34,467775,94675],{"className":467776,"style":106455},[31685,31686,31821],[34,467778,31829],{"className":467779},[31828],[34,467781,467783],{"className":467782},[31801],[34,467784,467786],{"className":467785,"style":31836},[31805],[34,467787],{},"\n所以 ",[34,467790,467792,467818],{"className":467791},[31642],[34,467793,467795],{"className":467794},[31646],[31648,467796,467797],{"xmlns":31650},[31652,467798,467799,467815],{},[31655,467800,467801,467803,467805,467807,467809,467811,467813],{},[31658,467802,111],{},[31662,467804,32226],{},[31658,467806,1697],{},[31662,467808,112296],{},[31658,467810,406335],{},[31662,467812,32682],{},[31658,467814,31739],{},[31665,467816,467817],{"encoding":31667},"a - b \\in \\ker f",[34,467819,467821,467839,467857],{"className":467820,"ariaHidden":758},[31672],[34,467822,467824,467827,467830,467833,467836],{"className":467823},[31676],[34,467825],{"className":467826,"style":32597},[31680],[34,467828,111],{"className":467829},[31685,31686],[34,467831],{"className":467832,"style":32604},[31845],[34,467834,32226],{"className":467835},[32024],[34,467837],{"className":467838,"style":32604},[31845],[34,467840,467842,467845,467848,467851,467854],{"className":467841},[31676],[34,467843],{"className":467844,"style":46991},[31680],[34,467846,1697],{"className":467847},[31685,31686],[34,467849],{"className":467850,"style":31846},[31845],[34,467852,112296],{"className":467853},[31850],[34,467855],{"className":467856,"style":31846},[31845],[34,467858,467860,467863,467866,467869],{"className":467859},[31676],[34,467861],{"className":467862,"style":43217},[31680],[34,467864,406335],{"className":467865},[32767],[34,467867],{"className":467868,"style":32123},[31845],[34,467870,31739],{"className":467871,"style":31778},[31685,31686],[2627,467873,467874],{},"吸收律（关键）",[34,467876,467878,467900],{"className":467877},[31642],[34,467879,467881],{"className":467880},[31646],[31648,467882,467883],{"xmlns":31650},[31652,467884,467885,467897],{},[31655,467886,467887,467889,467891,467893,467895],{},[31658,467888,111],{},[31662,467890,112296],{},[31658,467892,406335],{},[31662,467894,32682],{},[31658,467896,31739],{},[31665,467898,467899],{"encoding":31667},"a \\in \\ker f",[34,467901,467903,467921],{"className":467902,"ariaHidden":758},[31672],[34,467904,467906,467909,467912,467915,467918],{"className":467905},[31676],[34,467907],{"className":467908,"style":34812},[31680],[34,467910,111],{"className":467911},[31685,31686],[34,467913],{"className":467914,"style":31846},[31845],[34,467916,112296],{"className":467917},[31850],[34,467919],{"className":467920,"style":31846},[31845],[34,467922,467924,467927,467930,467933],{"className":467923},[31676],[34,467925],{"className":467926,"style":43217},[31680],[34,467928,406335],{"className":467929},[32767],[34,467931],{"className":467932,"style":32123},[31845],[34,467934,31739],{"className":467935,"style":31778},[31685,31686],[34,467937,467939,467967],{"className":467938},[31642],[34,467940,467942],{"className":467941},[31646],[31648,467943,467944],{"xmlns":31650},[31652,467945,467946,467964],{},[31655,467947,467948,467950,467952,467954,467956,467958],{},[31658,467949,31739],{},[31662,467951,456],{"stretchy":702},[31658,467953,111],{},[31662,467955,2557],{"stretchy":702},[31662,467957,202],{},[31743,467959,467960,467962],{},[31749,467961,459],{},[31658,467963,94675],{},[31665,467965,467966],{"encoding":31667},"f(a)=0_S",[34,467968,467970,467997],{"className":467969,"ariaHidden":758},[31672],[34,467971,467973,467976,467979,467982,467985,467988,467991,467994],{"className":467972},[31676],[34,467974],{"className":467975,"style":31774},[31680],[34,467977,31739],{"className":467978,"style":31778},[31685,31686],[34,467980,456],{"className":467981},[31782],[34,467983,111],{"className":467984},[31685,31686],[34,467986,2557],{"className":467987},[31690],[34,467989],{"className":467990,"style":31846},[31845],[34,467992,202],{"className":467993},[31850],[34,467995],{"className":467996,"style":31846},[31845],[34,467998,468000,468003],{"className":467999},[31676],[34,468001],{"className":468002,"style":38357},[31680],[34,468004,468006,468009],{"className":468005},[31685],[34,468007,459],{"className":468008},[31685],[34,468010,468012],{"className":468011},[31792],[34,468013,468015,468035],{"className":468014},[31796,31797],[34,468016,468018,468032],{"className":468017},[31801],[34,468019,468021],{"className":468020,"style":116558},[31805],[34,468022,468023,468026],{"style":31809},[34,468024],{"className":468025,"style":31814},[31813],[34,468027,468029],{"className":468028},[31818,31819,31820,31821],[34,468030,94675],{"className":468031,"style":106455},[31685,31686,31821],[34,468033,31829],{"className":468034},[31828],[34,468036,468038],{"className":468037},[31801],[34,468039,468041],{"className":468040,"style":31836},[31805],[34,468042],{},[34,468044,468046,468063],{"className":468045},[31642],[34,468047,468049],{"className":468048},[31646],[31648,468050,468051],{"xmlns":31650},[31652,468052,468053,468061],{},[31655,468054,468055,468057,468059],{},[31658,468056,34395],{},[31662,468058,112296],{},[31658,468060,34386],{},[31665,468062,335118],{"encoding":31667},[34,468064,468066,468084],{"className":468065,"ariaHidden":758},[31672],[34,468067,468069,468072,468075,468078,468081],{"className":468068},[31676],[34,468070],{"className":468071,"style":34812},[31680],[34,468073,34395],{"className":468074,"style":34452},[31685,31686],[34,468076],{"className":468077,"style":31846},[31845],[34,468079,112296],{"className":468080},[31850],[34,468082],{"className":468083,"style":31846},[31845],[34,468085,468087,468090],{"className":468086},[31676],[34,468088],{"className":468089,"style":33218},[31680],[34,468091,34386],{"className":468092,"style":34427},[31685,31686]," 是环中",[2627,468095,421436],{},"元素。\n我们要看 ",[34,468098,468100,468116],{"className":468099},[31642],[34,468101,468103],{"className":468102},[31646],[31648,468104,468105],{"xmlns":31650},[31652,468106,468107,468113],{},[31655,468108,468109,468111],{},[31658,468110,34395],{},[31658,468112,111],{},[31665,468114,468115],{"encoding":31667},"ra",[34,468117,468119],{"className":468118,"ariaHidden":758},[31672],[34,468120,468122,468125,468128],{"className":468121},[31676],[34,468123],{"className":468124,"style":32527},[31680],[34,468126,34395],{"className":468127,"style":34452},[31685,31686],[34,468129,111],{"className":468130},[31685,31686],[34,468132,468134,468150],{"className":468133},[31642],[34,468135,468137],{"className":468136},[31646],[31648,468138,468139],{"xmlns":31650},[31652,468140,468141,468147],{},[31655,468142,468143,468145],{},[31658,468144,111],{},[31658,468146,34395],{},[31665,468148,468149],{"encoding":31667},"ar",[34,468151,468153],{"className":468152,"ariaHidden":758},[31672],[34,468154,468156,468159,468162],{"className":468155},[31676],[34,468157],{"className":468158,"style":32527},[31680],[34,468160,111],{"className":468161},[31685,31686],[34,468163,34395],{"className":468164,"style":34452},[31685,31686]," 是否还在核里。\n",[34,468167,468169,468237],{"className":468168},[31642],[34,468170,468172],{"className":468171},[31646],[31648,468173,468174],{"xmlns":31650},[31652,468175,468176,468234],{},[31655,468177,468178,468180,468182,468184,468186,468188,468190,468192,468194,468196,468198,468200,468202,468204,468206,468208,468210,468212,468214,468216,468218,468220,468226,468228],{},[31658,468179,31739],{},[31662,468181,456],{"stretchy":702},[31658,468183,34395],{},[31658,468185,111],{},[31662,468187,2557],{"stretchy":702},[31662,468189,202],{},[31658,468191,31739],{},[31662,468193,456],{"stretchy":702},[31658,468195,34395],{},[31662,468197,2557],{"stretchy":702},[31662,468199,71191],{},[31658,468201,31739],{},[31662,468203,456],{"stretchy":702},[31658,468205,111],{},[31662,468207,2557],{"stretchy":702},[31662,468209,202],{},[31658,468211,31739],{},[31662,468213,456],{"stretchy":702},[31658,468215,34395],{},[31662,468217,2557],{"stretchy":702},[31662,468219,71191],{},[31743,468221,468222,468224],{},[31749,468223,459],{},[31658,468225,94675],{},[31662,468227,202],{},[31743,468229,468230,468232],{},[31749,468231,459],{},[31658,468233,94675],{},[31665,468235,468236],{"encoding":31667},"f(ra) = f(r) \\cdot f(a) = f(r) \\cdot 0_S = 0_S",[34,468238,468240,468270,468297,468324,468351,468406],{"className":468239,"ariaHidden":758},[31672],[34,468241,468243,468246,468249,468252,468255,468258,468261,468264,468267],{"className":468242},[31676],[34,468244],{"className":468245,"style":31774},[31680],[34,468247,31739],{"className":468248,"style":31778},[31685,31686],[34,468250,456],{"className":468251},[31782],[34,468253,34395],{"className":468254,"style":34452},[31685,31686],[34,468256,111],{"className":468257},[31685,31686],[34,468259,2557],{"className":468260},[31690],[34,468262],{"className":468263,"style":31846},[31845],[34,468265,202],{"className":468266},[31850],[34,468268],{"className":468269,"style":31846},[31845],[34,468271,468273,468276,468279,468282,468285,468288,468291,468294],{"className":468272},[31676],[34,468274],{"className":468275,"style":31774},[31680],[34,468277,31739],{"className":468278,"style":31778},[31685,31686],[34,468280,456],{"className":468281},[31782],[34,468283,34395],{"className":468284,"style":34452},[31685,31686],[34,468286,2557],{"className":468287},[31690],[34,468289],{"className":468290,"style":32604},[31845],[34,468292,71191],{"className":468293},[32024],[34,468295],{"className":468296,"style":32604},[31845],[34,468298,468300,468303,468306,468309,468312,468315,468318,468321],{"className":468299},[31676],[34,468301],{"className":468302,"style":31774},[31680],[34,468304,31739],{"className":468305,"style":31778},[31685,31686],[34,468307,456],{"className":468308},[31782],[34,468310,111],{"className":468311},[31685,31686],[34,468313,2557],{"className":468314},[31690],[34,468316],{"className":468317,"style":31846},[31845],[34,468319,202],{"className":468320},[31850],[34,468322],{"className":468323,"style":31846},[31845],[34,468325,468327,468330,468333,468336,468339,468342,468345,468348],{"className":468326},[31676],[34,468328],{"className":468329,"style":31774},[31680],[34,468331,31739],{"className":468332,"style":31778},[31685,31686],[34,468334,456],{"className":468335},[31782],[34,468337,34395],{"className":468338,"style":34452},[31685,31686],[34,468340,2557],{"className":468341},[31690],[34,468343],{"className":468344,"style":32604},[31845],[34,468346,71191],{"className":468347},[32024],[34,468349],{"className":468350,"style":32604},[31845],[34,468352,468354,468357,468397,468400,468403],{"className":468353},[31676],[34,468355],{"className":468356,"style":38357},[31680],[34,468358,468360,468363],{"className":468359},[31685],[34,468361,459],{"className":468362},[31685],[34,468364,468366],{"className":468365},[31792],[34,468367,468369,468389],{"className":468368},[31796,31797],[34,468370,468372,468386],{"className":468371},[31801],[34,468373,468375],{"className":468374,"style":116558},[31805],[34,468376,468377,468380],{"style":31809},[34,468378],{"className":468379,"style":31814},[31813],[34,468381,468383],{"className":468382},[31818,31819,31820,31821],[34,468384,94675],{"className":468385,"style":106455},[31685,31686,31821],[34,468387,31829],{"className":468388},[31828],[34,468390,468392],{"className":468391},[31801],[34,468393,468395],{"className":468394,"style":31836},[31805],[34,468396],{},[34,468398],{"className":468399,"style":31846},[31845],[34,468401,202],{"className":468402},[31850],[34,468404],{"className":468405,"style":31846},[31845],[34,468407,468409,468412],{"className":468408},[31676],[34,468410],{"className":468411,"style":38357},[31680],[34,468413,468415,468418],{"className":468414},[31685],[34,468416,459],{"className":468417},[31685],[34,468419,468421],{"className":468420},[31792],[34,468422,468424,468444],{"className":468423},[31796,31797],[34,468425,468427,468441],{"className":468426},[31801],[34,468428,468430],{"className":468429,"style":116558},[31805],[34,468431,468432,468435],{"style":31809},[34,468433],{"className":468434,"style":31814},[31813],[34,468436,468438],{"className":468437},[31818,31819,31820,31821],[34,468439,94675],{"className":468440,"style":106455},[31685,31686,31821],[34,468442,31829],{"className":468443},[31828],[34,468445,468447],{"className":468446},[31801],[34,468448,468450],{"className":468449,"style":31836},[31805],[34,468451],{},[34,468453,468455,468523],{"className":468454},[31642],[34,468456,468458],{"className":468457},[31646],[31648,468459,468460],{"xmlns":31650},[31652,468461,468462,468520],{},[31655,468463,468464,468466,468468,468470,468472,468474,468476,468478,468480,468482,468484,468486,468488,468490,468492,468494,468496,468502,468504,468506,468508,468510,468512,468514],{},[31658,468465,31739],{},[31662,468467,456],{"stretchy":702},[31658,468469,111],{},[31658,468471,34395],{},[31662,468473,2557],{"stretchy":702},[31662,468475,202],{},[31658,468477,31739],{},[31662,468479,456],{"stretchy":702},[31658,468481,111],{},[31662,468483,2557],{"stretchy":702},[31662,468485,71191],{},[31658,468487,31739],{},[31662,468489,456],{"stretchy":702},[31658,468491,34395],{},[31662,468493,2557],{"stretchy":702},[31662,468495,202],{},[31743,468497,468498,468500],{},[31749,468499,459],{},[31658,468501,94675],{},[31662,468503,71191],{},[31658,468505,31739],{},[31662,468507,456],{"stretchy":702},[31658,468509,34395],{},[31662,468511,2557],{"stretchy":702},[31662,468513,202],{},[31743,468515,468516,468518],{},[31749,468517,459],{},[31658,468519,94675],{},[31665,468521,468522],{"encoding":31667},"f(ar) = f(a) \\cdot f(r) = 0_S \\cdot f(r) = 0_S",[34,468524,468526,468556,468583,468610,468665,468692],{"className":468525,"ariaHidden":758},[31672],[34,468527,468529,468532,468535,468538,468541,468544,468547,468550,468553],{"className":468528},[31676],[34,468530],{"className":468531,"style":31774},[31680],[34,468533,31739],{"className":468534,"style":31778},[31685,31686],[34,468536,456],{"className":468537},[31782],[34,468539,111],{"className":468540},[31685,31686],[34,468542,34395],{"className":468543,"style":34452},[31685,31686],[34,468545,2557],{"className":468546},[31690],[34,468548],{"className":468549,"style":31846},[31845],[34,468551,202],{"className":468552},[31850],[34,468554],{"className":468555,"style":31846},[31845],[34,468557,468559,468562,468565,468568,468571,468574,468577,468580],{"className":468558},[31676],[34,468560],{"className":468561,"style":31774},[31680],[34,468563,31739],{"className":468564,"style":31778},[31685,31686],[34,468566,456],{"className":468567},[31782],[34,468569,111],{"className":468570},[31685,31686],[34,468572,2557],{"className":468573},[31690],[34,468575],{"className":468576,"style":32604},[31845],[34,468578,71191],{"className":468579},[32024],[34,468581],{"className":468582,"style":32604},[31845],[34,468584,468586,468589,468592,468595,468598,468601,468604,468607],{"className":468585},[31676],[34,468587],{"className":468588,"style":31774},[31680],[34,468590,31739],{"className":468591,"style":31778},[31685,31686],[34,468593,456],{"className":468594},[31782],[34,468596,34395],{"className":468597,"style":34452},[31685,31686],[34,468599,2557],{"className":468600},[31690],[34,468602],{"className":468603,"style":31846},[31845],[34,468605,202],{"className":468606},[31850],[34,468608],{"className":468609,"style":31846},[31845],[34,468611,468613,468616,468656,468659,468662],{"className":468612},[31676],[34,468614],{"className":468615,"style":38357},[31680],[34,468617,468619,468622],{"className":468618},[31685],[34,468620,459],{"className":468621},[31685],[34,468623,468625],{"className":468624},[31792],[34,468626,468628,468648],{"className":468627},[31796,31797],[34,468629,468631,468645],{"className":468630},[31801],[34,468632,468634],{"className":468633,"style":116558},[31805],[34,468635,468636,468639],{"style":31809},[34,468637],{"className":468638,"style":31814},[31813],[34,468640,468642],{"className":468641},[31818,31819,31820,31821],[34,468643,94675],{"className":468644,"style":106455},[31685,31686,31821],[34,468646,31829],{"className":468647},[31828],[34,468649,468651],{"className":468650},[31801],[34,468652,468654],{"className":468653,"style":31836},[31805],[34,468655],{},[34,468657],{"className":468658,"style":32604},[31845],[34,468660,71191],{"className":468661},[32024],[34,468663],{"className":468664,"style":32604},[31845],[34,468666,468668,468671,468674,468677,468680,468683,468686,468689],{"className":468667},[31676],[34,468669],{"className":468670,"style":31774},[31680],[34,468672,31739],{"className":468673,"style":31778},[31685,31686],[34,468675,456],{"className":468676},[31782],[34,468678,34395],{"className":468679,"style":34452},[31685,31686],[34,468681,2557],{"className":468682},[31690],[34,468684],{"className":468685,"style":31846},[31845],[34,468687,202],{"className":468688},[31850],[34,468690],{"className":468691,"style":31846},[31845],[34,468693,468695,468698],{"className":468694},[31676],[34,468696],{"className":468697,"style":38357},[31680],[34,468699,468701,468704],{"className":468700},[31685],[34,468702,459],{"className":468703},[31685],[34,468705,468707],{"className":468706},[31792],[34,468708,468710,468730],{"className":468709},[31796,31797],[34,468711,468713,468727],{"className":468712},[31801],[34,468714,468716],{"className":468715,"style":116558},[31805],[34,468717,468718,468721],{"style":31809},[34,468719],{"className":468720,"style":31814},[31813],[34,468722,468724],{"className":468723},[31818,31819,31820,31821],[34,468725,94675],{"className":468726,"style":106455},[31685,31686,31821],[34,468728,31829],{"className":468729},[31828],[34,468731,468733],{"className":468732},[31801],[34,468734,468736],{"className":468735,"style":31836},[31805],[34,468737],{},"\n因为结果都是 ",[34,468740,468742,468759],{"className":468741},[31642],[34,468743,468745],{"className":468744},[31646],[31648,468746,468747],{"xmlns":31650},[31652,468748,468749,468757],{},[31655,468750,468751],{},[31743,468752,468753,468755],{},[31749,468754,459],{},[31658,468756,94675],{},[31665,468758,463721],{"encoding":31667},[34,468760,468762],{"className":468761,"ariaHidden":758},[31672],[34,468763,468765,468768],{"className":468764},[31676],[34,468766],{"className":468767,"style":38357},[31680],[34,468769,468771,468774],{"className":468770},[31685],[34,468772,459],{"className":468773},[31685],[34,468775,468777],{"className":468776},[31792],[34,468778,468780,468800],{"className":468779},[31796,31797],[34,468781,468783,468797],{"className":468782},[31801],[34,468784,468786],{"className":468785,"style":116558},[31805],[34,468787,468788,468791],{"style":31809},[34,468789],{"className":468790,"style":31814},[31813],[34,468792,468794],{"className":468793},[31818,31819,31820,31821],[34,468795,94675],{"className":468796,"style":106455},[31685,31686,31821],[34,468798,31829],{"className":468799},[31828],[34,468801,468803],{"className":468802},[31801],[34,468804,468806],{"className":468805,"style":31836},[31805],[34,468807],{},[34,468809,468811,468835],{"className":468810},[31642],[34,468812,468814],{"className":468813},[31646],[31648,468815,468816],{"xmlns":31650},[31652,468817,468818,468832],{},[31655,468819,468820,468822,468824,468826,468828,468830],{},[31658,468821,34395],{},[31658,468823,111],{},[31662,468825,112296],{},[31658,468827,406335],{},[31662,468829,32682],{},[31658,468831,31739],{},[31665,468833,468834],{"encoding":31667},"ra \\in \\ker f",[34,468836,468838,468859],{"className":468837,"ariaHidden":758},[31672],[34,468839,468841,468844,468847,468850,468853,468856],{"className":468840},[31676],[34,468842],{"className":468843,"style":34812},[31680],[34,468845,34395],{"className":468846,"style":34452},[31685,31686],[34,468848,111],{"className":468849},[31685,31686],[34,468851],{"className":468852,"style":31846},[31845],[34,468854,112296],{"className":468855},[31850],[34,468857],{"className":468858,"style":31846},[31845],[34,468860,468862,468865,468868,468871],{"className":468861},[31676],[34,468863],{"className":468864,"style":43217},[31680],[34,468866,406335],{"className":468867},[32767],[34,468869],{"className":468870,"style":32123},[31845],[34,468872,31739],{"className":468873,"style":31778},[31685,31686],[34,468875,468877,468901],{"className":468876},[31642],[34,468878,468880],{"className":468879},[31646],[31648,468881,468882],{"xmlns":31650},[31652,468883,468884,468898],{},[31655,468885,468886,468888,468890,468892,468894,468896],{},[31658,468887,111],{},[31658,468889,34395],{},[31662,468891,112296],{},[31658,468893,406335],{},[31662,468895,32682],{},[31658,468897,31739],{},[31665,468899,468900],{"encoding":31667},"ar \\in \\ker f",[34,468902,468904,468925],{"className":468903,"ariaHidden":758},[31672],[34,468905,468907,468910,468913,468916,468919,468922],{"className":468906},[31676],[34,468908],{"className":468909,"style":34812},[31680],[34,468911,111],{"className":468912},[31685,31686],[34,468914,34395],{"className":468915,"style":34452},[31685,31686],[34,468917],{"className":468918,"style":31846},[31845],[34,468920,112296],{"className":468921},[31850],[34,468923],{"className":468924,"style":31846},[31845],[34,468926,468928,468931,468934,468937],{"className":468927},[31676],[34,468929],{"className":468930,"style":43217},[31680],[34,468932,406335],{"className":468933},[32767],[34,468935],{"className":468936,"style":32123},[31845],[34,468938,31739],{"className":468939,"style":31778},[31685,31686],[15,468941,468942,2664,468944,11008,468982,469010,469011],{},[2627,468943,114043],{},[34,468945,468947,468964],{"className":468946},[31642],[34,468948,468950],{"className":468949},[31646],[31648,468951,468952],{"xmlns":31650},[31652,468953,468954,468962],{},[31655,468955,468956,468958,468960],{},[31658,468957,406335],{},[31662,468959,32682],{},[31658,468961,31739],{},[31665,468963,463410],{"encoding":31667},[34,468965,468967],{"className":468966,"ariaHidden":758},[31672],[34,468968,468970,468973,468976,468979],{"className":468969},[31676],[34,468971],{"className":468972,"style":43217},[31680],[34,468974,406335],{"className":468975},[32767],[34,468977],{"className":468978,"style":32123},[31845],[34,468980,31739],{"className":468981,"style":31778},[31685,31686],[34,468983,468985,468998],{"className":468984},[31642],[34,468986,468988],{"className":468987},[31646],[31648,468989,468990],{"xmlns":31650},[31652,468991,468992,468996],{},[31655,468993,468994],{},[31658,468995,34386],{},[31665,468997,34386],{"encoding":31667},[34,468999,469001],{"className":469000,"ariaHidden":758},[31672],[34,469002,469004,469007],{"className":469003},[31676],[34,469005],{"className":469006,"style":33218},[31680],[34,469008,34386],{"className":469009,"style":34427},[31685,31686]," 的理想。 ",[34,469012,469014,469027],{"className":469013},[31642],[34,469015,469017],{"className":469016},[31646],[31648,469018,469019],{"xmlns":31650},[31652,469020,469021,469025],{},[31655,469022,469023],{},[31658,469024,445204],{"mathvariant":38079},[31665,469026,445207],{"encoding":31667},[34,469028,469030],{"className":469029,"ariaHidden":758},[31672],[34,469031,469033,469036],{"className":469032},[31676],[34,469034],{"className":469035,"style":445217},[31680],[34,469037,445204],{"className":469038},[31685,129261],[32188,469040],{},[32490,469042,217356],{"id":217353},[15,469044,469045,469176],{},[2627,469046,133910,469047,469119,469120,11008,469148,463457],{},[34,469048,469050,469071],{"className":469049},[31642],[34,469051,469053],{"className":469052},[31646],[31648,469054,469055],{"xmlns":31650},[31652,469056,469057,469069],{},[31655,469058,469059,469061,469063,469065,469067],{},[31658,469060,31739],{},[31662,469062,5598],{},[31658,469064,34386],{},[31662,469066,33239],{},[31658,469068,94675],{},[31665,469070,463210],{"encoding":31667},[34,469072,469074,469092,469110],{"className":469073,"ariaHidden":758},[31672],[34,469075,469077,469080,469083,469086,469089],{"className":469076},[31676],[34,469078],{"className":469079,"style":43217},[31680],[34,469081,31739],{"className":469082,"style":31778},[31685,31686],[34,469084],{"className":469085,"style":31846},[31845],[34,469087,5598],{"className":469088},[31850],[34,469090],{"className":469091,"style":31846},[31845],[34,469093,469095,469098,469101,469104,469107],{"className":469094},[31676],[34,469096],{"className":469097,"style":33218},[31680],[34,469099,34386],{"className":469100,"style":34427},[31685,31686],[34,469102],{"className":469103,"style":31846},[31845],[34,469105,33239],{"className":469106},[31850],[34,469108],{"className":469109,"style":31846},[31845],[34,469111,469113,469116],{"className":469112},[31676],[34,469114],{"className":469115,"style":33218},[31680],[34,469117,94675],{"className":469118,"style":106455},[31685,31686]," 是一个满同态（Surjective Homomorphism），且 ",[34,469121,469123,469136],{"className":469122},[31642],[34,469124,469126],{"className":469125},[31646],[31648,469127,469128],{"xmlns":31650},[31652,469129,469130,469134],{},[31655,469131,469132],{},[31658,469133,106063],{},[31665,469135,106063],{"encoding":31667},[34,469137,469139],{"className":469138,"ariaHidden":758},[31672],[34,469140,469142,469145],{"className":469141},[31676],[34,469143],{"className":469144,"style":33218},[31680],[34,469146,106063],{"className":469147,"style":67482},[31685,31686],[34,469149,469151,469164],{"className":469150},[31642],[34,469152,469154],{"className":469153},[31646],[31648,469155,469156],{"xmlns":31650},[31652,469157,469158,469162],{},[31655,469159,469160],{},[31658,469161,34386],{},[31665,469163,34386],{"encoding":31667},[34,469165,469167],{"className":469166,"ariaHidden":758},[31672],[34,469168,469170,469173],{"className":469169},[31676],[34,469171],{"className":469172,"style":33218},[31680],[34,469174,34386],{"className":469175,"style":34427},[31685,31686],[2627,469177,225289,469178,11008,469222,463457],{},[34,469179,469181,469201],{"className":469180},[31642],[34,469182,469184],{"className":469183},[31646],[31648,469185,469186],{"xmlns":31650},[31652,469187,469188,469198],{},[31655,469189,469190,469192,469194,469196],{},[31658,469191,31739],{},[31662,469193,456],{"stretchy":702},[31658,469195,106063],{},[31662,469197,2557],{"stretchy":702},[31665,469199,469200],{"encoding":31667},"f(I)",[34,469202,469204],{"className":469203,"ariaHidden":758},[31672],[34,469205,469207,469210,469213,469216,469219],{"className":469206},[31676],[34,469208],{"className":469209,"style":31774},[31680],[34,469211,31739],{"className":469212,"style":31778},[31685,31686],[34,469214,456],{"className":469215},[31782],[34,469217,106063],{"className":469218,"style":67482},[31685,31686],[34,469220,2557],{"className":469221},[31690],[34,469223,469225,469238],{"className":469224},[31642],[34,469226,469228],{"className":469227},[31646],[31648,469229,469230],{"xmlns":31650},[31652,469231,469232,469236],{},[31655,469233,469234],{},[31658,469235,94675],{},[31665,469237,94675],{"encoding":31667},[34,469239,469241],{"className":469240,"ariaHidden":758},[31672],[34,469242,469244,469247],{"className":469243},[31676],[34,469245],{"className":469246,"style":33218},[31680],[34,469248,94675],{"className":469249,"style":106455},[31685,31686],[15,469251,469252],{},[2627,469253,463462],{},[105,469255,469256],{},[108,469257,469258,469261,469262,469265,469266,326989,469269,469297,469298,469326,469327,469355,469356,469384],{},[2627,469259,469260],{},"陷阱提示","：一般情况下，理想在同态下的像",[2627,469263,469264],{},"不一定","是理想（只能保证是子环）。但这里有一个关键条件：",[2627,469267,469268],{},"“满同态”",[34,469270,469272,469285],{"className":469271},[31642],[34,469273,469275],{"className":469274},[31646],[31648,469276,469277],{"xmlns":31650},[31652,469278,469279,469283],{},[31655,469280,469281],{},[31658,469282,94675],{},[31665,469284,94675],{"encoding":31667},[34,469286,469288],{"className":469287,"ariaHidden":758},[31672],[34,469289,469291,469294],{"className":469290},[31676],[34,469292],{"className":469293,"style":33218},[31680],[34,469295,94675],{"className":469296,"style":106455},[31685,31686]," 中的任何元素 ",[34,469299,469301,469314],{"className":469300},[31642],[34,469302,469304],{"className":469303},[31646],[31648,469305,469306],{"xmlns":31650},[31652,469307,469308,469312],{},[31655,469309,469310],{},[31658,469311,31747],{},[31665,469313,31747],{"encoding":31667},[34,469315,469317],{"className":469316,"ariaHidden":758},[31672],[34,469318,469320,469323],{"className":469319},[31676],[34,469321],{"className":469322,"style":32527},[31680],[34,469324,31747],{"className":469325},[31685,31686],"，都能找到 ",[34,469328,469330,469343],{"className":469329},[31642],[34,469331,469333],{"className":469332},[31646],[31648,469334,469335],{"xmlns":31650},[31652,469336,469337,469341],{},[31655,469338,469339],{},[31658,469340,34386],{},[31665,469342,34386],{"encoding":31667},[34,469344,469346],{"className":469345,"ariaHidden":758},[31672],[34,469347,469349,469352],{"className":469348},[31676],[34,469350],{"className":469351,"style":33218},[31680],[34,469353,34386],{"className":469354,"style":34427},[31685,31686]," 中的“原身” ",[34,469357,469359,469372],{"className":469358},[31642],[34,469360,469362],{"className":469361},[31646],[31648,469363,469364],{"xmlns":31650},[31652,469365,469366,469370],{},[31655,469367,469368],{},[31658,469369,34395],{},[31665,469371,34395],{"encoding":31667},[34,469373,469375],{"className":469374,"ariaHidden":758},[31672],[34,469376,469378,469381],{"className":469377},[31676],[34,469379],{"className":469380,"style":32527},[31680],[34,469382,34395],{"className":469383,"style":34452},[31685,31686],"。这对于证明“吸收律”至关重要。",[15,469386,469387,362914,469389,347],{},[2627,469388,463779],{},[34,469390,469392,469438],{"className":469391},[31642],[34,469393,469395],{"className":469394},[31646],[31648,469396,469397],{"xmlns":31650},[31652,469398,469399,469435],{},[31655,469400,469401,469403,469405,469407,469409,469411,469413,469415,469417,469419,469421,469423,469425,469427,469429,469431,469433],{},[31658,469402,223320],{},[31662,469404,202],{},[31658,469406,31739],{},[31662,469408,456],{"stretchy":702},[31658,469410,106063],{},[31662,469412,2557],{"stretchy":702},[31662,469414,202],{},[31662,469416,32685],{"stretchy":702},[31658,469418,31739],{},[31662,469420,456],{"stretchy":702},[31658,469422,32936],{},[31662,469424,2557],{"stretchy":702},[31662,469426,162086],{},[31658,469428,32936],{},[31662,469430,112296],{},[31658,469432,106063],{},[31662,469434,32715],{"stretchy":702},[31665,469436,469437],{"encoding":31667},"J = f(I) = \\{f(x) \\mid x \\in 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y_2 \\in 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是理想，所以 ",[34,470850,470852,470882],{"className":470851},[31642],[34,470853,470855],{"className":470854},[31646],[31648,470856,470857],{"xmlns":31650},[31652,470858,470859,470879],{},[31655,470860,470861,470867,470869,470875,470877],{},[31743,470862,470863,470865],{},[31658,470864,32936],{},[31749,470866,640],{},[31662,470868,32226],{},[31743,470870,470871,470873],{},[31658,470872,32936],{},[31749,470874,768],{},[31662,470876,112296],{},[31658,470878,106063],{},[31665,470880,470881],{"encoding":31667},"x_1 - x_2 \\in I",[34,470883,470885,470940,470995],{"className":470884,"ariaHidden":758},[31672],[34,470886,470888,470891,470931,470934,470937],{"className":470887},[31676],[34,470889],{"className":470890,"style":40522},[31680],[34,470892,470894,470897],{"className":470893},[31685],[34,470895,32936],{"className":470896},[31685,31686],[34,470898,470900],{"className":470899},[31792],[34,470901,470903,470923],{"className":470902},[31796,31797],[34,470904,470906,470920],{"className":470905},[31801],[34,470907,470909],{"className":470908,"style":31806},[31805],[34,470910,470911,470914],{"style":31809},[34,470912],{"className":470913,"style":31814},[31813],[34,470915,470917],{"className":470916},[31818,31819,31820,31821],[34,470918,640],{"className":470919},[31685,31821],[34,470921,31829],{"className":470922},[31828],[34,470924,470926],{"className":470925},[31801],[34,470927,470929],{"className":470928,"style":31836},[31805],[34,470930],{},[34,470932],{"className":470933,"style":32604},[31845],[34,470935,32226],{"className":470936},[32024],[34,470938],{"className":470939,"style":32604},[31845],[34,470941,470943,470946,470986,470989,470992],{"className":470942},[31676],[34,470944],{"className":470945,"style":225773},[31680],[34,470947,470949,470952],{"className":470948},[31685],[34,470950,32936],{"className":470951},[31685,31686],[34,470953,470955],{"className":470954},[31792],[34,470956,470958,470978],{"className":470957},[31796,31797],[34,470959,470961,470975],{"className":470960},[31801],[34,470962,470964],{"className":470963,"style":31806},[31805],[34,470965,470966,470969],{"style":31809},[34,470967],{"className":470968,"style":31814},[31813],[34,470970,470972],{"className":470971},[31818,31819,31820,31821],[34,470973,768],{"className":470974},[31685,31821],[34,470976,31829],{"className":470977},[31828],[34,470979,470981],{"className":470980},[31801],[34,470982,470984],{"className":470983,"style":31836},[31805],[34,470985],{},[34,470987],{"className":470988,"style":31846},[31845],[34,470990,112296],{"className":470991},[31850],[34,470993],{"className":470994,"style":31846},[31845],[34,470996,470998,471001],{"className":470997},[31676],[34,470999],{"className":471000,"style":33218},[31680],[34,471002,106063],{"className":471003,"style":67482},[31685,31686],"，故其像属于 ",[34,471006,471008,471021],{"className":471007},[31642],[34,471009,471011],{"className":471010},[31646],[31648,471012,471013],{"xmlns":31650},[31652,471014,471015,471019],{},[31655,471016,471017],{},[31658,471018,223320],{},[31665,471020,223320],{"encoding":31667},[34,471022,471024],{"className":471023,"ariaHidden":758},[31672],[34,471025,471027,471030],{"className":471026},[31676],[34,471028],{"className":471029,"style":33218},[31680],[34,471031,223320],{"className":471032,"style":223361},[31685,31686],[108,471034,471035,34554,471037,46528,471087,326024,471138,39984,471211],{},[2627,471036,442183],{},[34,471038,471040,471057],{"className":471039},[31642],[34,471041,471043],{"className":471042},[31646],[31648,471044,471045],{"xmlns":31650},[31652,471046,471047,471055],{},[31655,471048,471049,471051,471053],{},[31658,471050,67764],{},[31662,471052,112296],{},[31658,471054,223320],{},[31665,471056,405557],{"encoding":31667},[34,471058,471060,471078],{"className":471059,"ariaHidden":758},[31672],[34,471061,471063,471066,471069,471072,471075],{"className":471062},[31676],[34,471064],{"className":471065,"style":86652},[31680],[34,471067,67764],{"className":471068,"style":32062},[31685,31686],[34,471070],{"className":471071,"style":31846},[31845],[34,471073,112296],{"className":471074},[31850],[34,471076],{"className":471077,"style":31846},[31845],[34,471079,471081,471084],{"className":471080},[31676],[34,471082],{"className":471083,"style":33218},[31680],[34,471085,223320],{"className":471086,"style":223361},[31685,31686],[34,471088,471090,471108],{"className":471089},[31642],[34,471091,471093],{"className":471092},[31646],[31648,471094,471095],{"xmlns":31650},[31652,471096,471097,471105],{},[31655,471098,471099,471101,471103],{},[31658,471100,31747],{},[31662,471102,112296],{},[31658,471104,94675],{},[31665,471106,471107],{"encoding":31667},"s \\in S",[34,471109,471111,471129],{"className":471110,"ariaHidden":758},[31672],[34,471112,471114,471117,471120,471123,471126],{"className":471113},[31676],[34,471115],{"className":471116,"style":34812},[31680],[34,471118,31747],{"className":471119},[31685,31686],[34,471121],{"className":471122,"style":31846},[31845],[34,471124,112296],{"className":471125},[31850],[34,471127],{"className":471128,"style":31846},[31845],[34,471130,471132,471135],{"className":471131},[31676],[34,471133],{"className":471134,"style":33218},[31680],[34,471136,94675],{"className":471137,"style":106455},[31685,31686],[34,471139,471141,471163],{"className":471140},[31642],[34,471142,471144],{"className":471143},[31646],[31648,471145,471146],{"xmlns":31650},[31652,471147,471148,471160],{},[31655,471149,471150,471152,471154,471156,471158],{},[31658,471151,31747],{},[31662,471153,71191],{},[31658,471155,67764],{},[31662,471157,112296],{},[31658,471159,223320],{},[31665,471161,471162],{"encoding":31667},"s \\cdot y \\in J",[34,471164,471166,471184,471202],{"className":471165,"ariaHidden":758},[31672],[34,471167,471169,471172,471175,471178,471181],{"className":471168},[31676],[34,471170],{"className":471171,"style":224869},[31680],[34,471173,31747],{"className":471174},[31685,31686],[34,471176],{"className":471177,"style":32604},[31845],[34,471179,71191],{"className":471180},[32024],[34,471182],{"className":471183,"style":32604},[31845],[34,471185,471187,471190,471193,471196,471199],{"className":471186},[31676],[34,471188],{"className":471189,"style":86652},[31680],[34,471191,67764],{"className":471192,"style":32062},[31685,31686],[34,471194],{"className":471195,"style":31846},[31845],[34,471197,112296],{"className":471198},[31850],[34,471200],{"className":471201,"style":31846},[31845],[34,471203,471205,471208],{"className":471204},[31676],[34,471206],{"className":471207,"style":33218},[31680],[34,471209,223320],{"className":471210,"style":223361},[31685,31686],[105,471212,471213,471381],{},[108,471214,93310,471215,395725,471265,315343,471315,347],{},[34,471216,471218,471235],{"className":471217},[31642],[34,471219,471221],{"className":471220},[31646],[31648,471222,471223],{"xmlns":31650},[31652,471224,471225,471233],{},[31655,471226,471227,471229,471231],{},[31658,471228,67764],{},[31662,471230,112296],{},[31658,471232,223320],{},[31665,471234,405557],{"encoding":31667},[34,471236,471238,471256],{"className":471237,"ariaHidden":758},[31672],[34,471239,471241,471244,471247,471250,471253],{"className":471240},[31676],[34,471242],{"className":471243,"style":86652},[31680],[34,471245,67764],{"className":471246,"style":32062},[31685,31686],[34,471248],{"className":471249,"style":31846},[31845],[34,471251,112296],{"className":471252},[31850],[34,471254],{"className":471255,"style":31846},[31845],[34,471257,471259,471262],{"className":471258},[31676],[34,471260],{"className":471261,"style":33218},[31680],[34,471263,223320],{"className":471264,"style":223361},[31685,31686],[34,471266,471268,471285],{"className":471267},[31642],[34,471269,471271],{"className":471270},[31646],[31648,471272,471273],{"xmlns":31650},[31652,471274,471275,471283],{},[31655,471276,471277,471279,471281],{},[31658,471278,32936],{},[31662,471280,112296],{},[31658,471282,106063],{},[31665,471284,352414],{"encoding":31667},[34,471286,471288,471306],{"className":471287,"ariaHidden":758},[31672],[34,471289,471291,471294,471297,471300,471303],{"className":471290},[31676],[34,471292],{"className":471293,"style":34812},[31680],[34,471295,32936],{"className":471296},[31685,31686],[34,471298],{"className":471299,"style":31846},[31845],[34,471301,112296],{"className":471302},[31850],[34,471304],{"className":471305,"style":31846},[31845],[34,471307,471309,471312],{"className":471308},[31676],[34,471310],{"className":471311,"style":33218},[31680],[34,471313,106063],{"className":471314,"style":67482},[31685,31686],[34,471316,471318,471342],{"className":471317},[31642],[34,471319,471321],{"className":471320},[31646],[31648,471322,471323],{"xmlns":31650},[31652,471324,471325,471339],{},[31655,471326,471327,471329,471331,471333,471335,471337],{},[31658,471328,31739],{},[31662,471330,456],{"stretchy":702},[31658,471332,32936],{},[31662,471334,2557],{"stretchy":702},[31662,471336,202],{},[31658,471338,67764],{},[31665,471340,471341],{"encoding":31667},"f(x) = y",[34,471343,471345,471372],{"className":471344,"ariaHidden":758},[31672],[34,471346,471348,471351,471354,471357,471360,471363,471366,471369],{"className":471347},[31676],[34,471349],{"className":471350,"style":31774},[31680],[34,471352,31739],{"className":471353,"style":31778},[31685,31686],[34,471355,456],{"className":471356},[31782],[34,471358,32936],{"className":471359},[31685,31686],[34,471361,2557],{"className":471362},[31690],[34,471364],{"className":471365,"style":31846},[31845],[34,471367,202],{"className":471368},[31850],[34,471370],{"className":471371,"style":31846},[31845],[34,471373,471375,471378],{"className":471374},[31676],[34,471376],{"className":471377,"style":38774},[31680],[34,471379,67764],{"className":471380,"style":32062},[31685,31686],[108,471382,471383,332541,471386,113124,471414,471416,471417,419074,471445,471473,471474,315343,471524,471590,471591,33183,471642,471826,471827,11008,471855,471883,471884,355408,472046,11008,472095,472123,472124,472196,472197,347],{},[2627,471384,471385],{},"关键步骤",[34,471387,471389,471402],{"className":471388},[31642],[34,471390,471392],{"className":471391},[31646],[31648,471393,471394],{"xmlns":31650},[31652,471395,471396,471400],{},[31655,471397,471398],{},[31658,471399,31739],{},[31665,471401,31739],{"encoding":31667},[34,471403,471405],{"className":471404,"ariaHidden":758},[31672],[34,471406,471408,471411],{"className":471407},[31676],[34,471409],{"className":471410,"style":43217},[31680],[34,471412,31739],{"className":471413,"style":31778},[31685,31686],[2627,471415,364537],{},"，对于 ",[34,471418,471420,471433],{"className":471419},[31642],[34,471421,471423],{"className":471422},[31646],[31648,471424,471425],{"xmlns":31650},[31652,471426,471427,471431],{},[31655,471428,471429],{},[31658,471430,94675],{},[31665,471432,94675],{"encoding":31667},[34,471434,471436],{"className":471435,"ariaHidden":758},[31672],[34,471437,471439,471442],{"className":471438},[31676],[34,471440],{"className":471441,"style":33218},[31680],[34,471443,94675],{"className":471444,"style":106455},[31685,31686],[34,471446,471448,471461],{"className":471447},[31642],[34,471449,471451],{"className":471450},[31646],[31648,471452,471453],{"xmlns":31650},[31652,471454,471455,471459],{},[31655,471456,471457],{},[31658,471458,31747],{},[31665,471460,31747],{"encoding":31667},[34,471462,471464],{"className":471463,"ariaHidden":758},[31672],[34,471465,471467,471470],{"className":471466},[31676],[34,471468],{"className":471469,"style":32527},[31680],[34,471471,31747],{"className":471472},[31685,31686],"，必然存在 ",[34,471475,471477,471494],{"className":471476},[31642],[34,471478,471480],{"className":471479},[31646],[31648,471481,471482],{"xmlns":31650},[31652,471483,471484,471492],{},[31655,471485,471486,471488,471490],{},[31658,471487,34395],{},[31662,471489,112296],{},[31658,471491,34386],{},[31665,471493,335118],{"encoding":31667},[34,471495,471497,471515],{"className":471496,"ariaHidden":758},[31672],[34,471498,471500,471503,471506,471509,471512],{"className":471499},[31676],[34,471501],{"className":471502,"style":34812},[31680],[34,471504,34395],{"className":471505,"style":34452},[31685,31686],[34,471507],{"className":471508,"style":31846},[31845],[34,471510,112296],{"className":471511},[31850],[34,471513],{"className":471514,"style":31846},[31845],[34,471516,471518,471521],{"className":471517},[31676],[34,471519],{"className":471520,"style":33218},[31680],[34,471522,34386],{"className":471523,"style":34427},[31685,31686],[34,471525,471527,471551],{"className":471526},[31642],[34,471528,471530],{"className":471529},[31646],[31648,471531,471532],{"xmlns":31650},[31652,471533,471534,471548],{},[31655,471535,471536,471538,471540,471542,471544,471546],{},[31658,471537,31739],{},[31662,471539,456],{"stretchy":702},[31658,471541,34395],{},[31662,471543,2557],{"stretchy":702},[31662,471545,202],{},[31658,471547,31747],{},[31665,471549,471550],{"encoding":31667},"f(r) = s",[34,471552,471554,471581],{"className":471553,"ariaHidden":758},[31672],[34,471555,471557,471560,471563,471566,471569,471572,471575,471578],{"className":471556},[31676],[34,471558],{"className":471559,"style":31774},[31680],[34,471561,31739],{"className":471562,"style":31778},[31685,31686],[34,471564,456],{"className":471565},[31782],[34,471567,34395],{"className":471568,"style":34452},[31685,31686],[34,471570,2557],{"className":471571},[31690],[34,471573],{"className":471574,"style":31846},[31845],[34,471576,202],{"className":471577},[31850],[34,471579],{"className":471580,"style":31846},[31845],[34,471582,471584,471587],{"className":471583},[31676],[34,471585],{"className":471586,"style":32527},[31680],[34,471588,31747],{"className":471589},[31685,31686],"。\n现在计算 ",[34,471592,471594,471612],{"className":471593},[31642],[34,471595,471597],{"className":471596},[31646],[31648,471598,471599],{"xmlns":31650},[31652,471600,471601,471609],{},[31655,471602,471603,471605,471607],{},[31658,471604,31747],{},[31662,471606,71191],{},[31658,471608,67764],{},[31665,471610,471611],{"encoding":31667},"s \\cdot y",[34,471613,471615,471633],{"className":471614,"ariaHidden":758},[31672],[34,471616,471618,471621,471624,471627,471630],{"className":471617},[31676],[34,471619],{"className":471620,"style":224869},[31680],[34,471622,31747],{"className":471623},[31685,31686],[34,471625],{"className":471626,"style":32604},[31845],[34,471628,71191],{"className":471629},[32024],[34,471631],{"className":471632,"style":32604},[31845],[34,471634,471636,471639],{"className":471635},[31676],[34,471637],{"className":471638,"style":38774},[31680],[34,471640,67764],{"className":471641,"style":32062},[31685,31686],[34,471643,471645,471697],{"className":471644},[31642],[34,471646,471648],{"className":471647},[31646],[31648,471649,471650],{"xmlns":31650},[31652,471651,471652,471694],{},[31655,471653,471654,471656,471658,471660,471662,471664,471666,471668,471670,471672,471674,471676,471678,471680,471682,471684,471686,471688,471690,471692],{},[31658,471655,31747],{},[31662,471657,71191],{},[31658,471659,67764],{},[31662,471661,202],{},[31658,471663,31739],{},[31662,471665,456],{"stretchy":702},[31658,471667,34395],{},[31662,471669,2557],{"stretchy":702},[31662,471671,71191],{},[31658,471673,31739],{},[31662,471675,456],{"stretchy":702},[31658,471677,32936],{},[31662,471679,2557],{"stretchy":702},[31662,471681,202],{},[31658,471683,31739],{},[31662,471685,456],{"stretchy":702},[31658,471687,34395],{},[31662,471689,71191],{},[31658,471691,32936],{},[31662,471693,2557],{"stretchy":702},[31665,471695,471696],{"encoding":31667},"s \\cdot y = f(r) \\cdot f(x) = f(r \\cdot x)",[34,471698,471700,471718,471736,471763,471790,471814],{"className":471699,"ariaHidden":758},[31672],[34,471701,471703,471706,471709,471712,471715],{"className":471702},[31676],[34,471704],{"className":471705,"style":224869},[31680],[34,471707,31747],{"className":471708},[31685,31686],[34,471710],{"className":471711,"style":32604},[31845],[34,471713,71191],{"className":471714},[32024],[34,471716],{"className":471717,"style":32604},[31845],[34,471719,471721,471724,471727,471730,471733],{"className":471720},[31676],[34,471722],{"className":471723,"style":38774},[31680],[34,471725,67764],{"className":471726,"style":32062},[31685,31686],[34,471728],{"className":471729,"style":31846},[31845],[34,471731,202],{"className":471732},[31850],[34,471734],{"className":471735,"style":31846},[31845],[34,471737,471739,471742,471745,471748,471751,471754,471757,471760],{"className":471738},[31676],[34,471740],{"className":471741,"style":31774},[31680],[34,471743,31739],{"className":471744,"style":31778},[31685,31686],[34,471746,456],{"className":471747},[31782],[34,471749,34395],{"className":471750,"style":34452},[31685,31686],[34,471752,2557],{"className":471753},[31690],[34,471755],{"className":471756,"style":32604},[31845],[34,471758,71191],{"className":471759},[32024],[34,471761],{"className":471762,"style":32604},[31845],[34,471764,471766,471769,471772,471775,471778,471781,471784,471787],{"className":471765},[31676],[34,471767],{"className":471768,"style":31774},[31680],[34,471770,31739],{"className":471771,"style":31778},[31685,31686],[34,471773,456],{"className":471774},[31782],[34,471776,32936],{"className":471777},[31685,31686],[34,471779,2557],{"className":471780},[31690],[34,471782],{"className":471783,"style":31846},[31845],[34,471785,202],{"className":471786},[31850],[34,471788],{"className":471789,"style":31846},[31845],[34,471791,471793,471796,471799,471802,471805,471808,471811],{"className":471792},[31676],[34,471794],{"className":471795,"style":31774},[31680],[34,471797,31739],{"className":471798,"style":31778},[31685,31686],[34,471800,456],{"className":471801},[31782],[34,471803,34395],{"className":471804,"style":34452},[31685,31686],[34,471806],{"className":471807,"style":32604},[31845],[34,471809,71191],{"className":471810},[32024],[34,471812],{"className":471813,"style":32604},[31845],[34,471815,471817,471820,471823],{"className":471816},[31676],[34,471818],{"className":471819,"style":31774},[31680],[34,471821,32936],{"className":471822},[31685,31686],[34,471824,2557],{"className":471825},[31690],"\n由于 ",[34,471828,471830,471843],{"className":471829},[31642],[34,471831,471833],{"className":471832},[31646],[31648,471834,471835],{"xmlns":31650},[31652,471836,471837,471841],{},[31655,471838,471839],{},[31658,471840,106063],{},[31665,471842,106063],{"encoding":31667},[34,471844,471846],{"className":471845,"ariaHidden":758},[31672],[34,471847,471849,471852],{"className":471848},[31676],[34,471850],{"className":471851,"style":33218},[31680],[34,471853,106063],{"className":471854,"style":67482},[31685,31686],[34,471856,471858,471871],{"className":471857},[31642],[34,471859,471861],{"className":471860},[31646],[31648,471862,471863],{"xmlns":31650},[31652,471864,471865,471869],{},[31655,471866,471867],{},[31658,471868,34386],{},[31665,471870,34386],{"encoding":31667},[34,471872,471874],{"className":471873,"ariaHidden":758},[31672],[34,471875,471877,471880],{"className":471876},[31676],[34,471878],{"className":471879,"style":33218},[31680],[34,471881,34386],{"className":471882,"style":34427},[31685,31686]," 的理想，",[34,471885,471887,471929],{"className":471886},[31642],[34,471888,471890],{"className":471889},[31646],[31648,471891,471892],{"xmlns":31650},[31652,471893,471894,471926],{},[31655,471895,471896,471898,471900,471902,471904,471906,471908,471910,471912,471914,471916,471918,471920,471922,471924],{},[31658,471897,32936],{},[31662,471899,112296],{},[31658,471901,106063],{},[31662,471903,31960],{"separator":758},[31658,471905,34395],{},[31662,471907,112296],{},[31658,471909,34386],{},[39361,471911,262622],{},[31662,471913,262625],{},[39361,471915,262622],{},[31658,471917,34395],{},[31662,471919,71191],{},[31658,471921,32936],{},[31662,471923,112296],{},[31658,471925,106063],{},[31665,471927,471928],{"encoding":31667},"x \\in I, r \\in R \\implies r \\cdot x \\in I",[34,471930,471932,471950,471977,472001,472019,472037],{"className":471931,"ariaHidden":758},[31672],[34,471933,471935,471938,471941,471944,471947],{"className":471934},[31676],[34,471936],{"className":471937,"style":34812},[31680],[34,471939,32936],{"className":471940},[31685,31686],[34,471942],{"className":471943,"style":31846},[31845],[34,471945,112296],{"className":471946},[31850],[34,471948],{"className":471949,"style":31846},[31845],[34,471951,471953,471956,471959,471962,471965,471968,471971,471974],{"className":471952},[31676],[34,471954],{"className":471955,"style":70739},[31680],[34,471957,106063],{"className":471958,"style":67482},[31685,31686],[34,471960,31960],{"className":471961},[32119],[34,471963],{"className":471964,"style":32123},[31845],[34,471966,34395],{"className":471967,"style":34452},[31685,31686],[34,471969],{"className":471970,"style":31846},[31845],[34,471972,112296],{"className":471973},[31850],[34,471975],{"className":471976,"style":31846},[31845],[34,471978,471980,471983,471986,471989,471992,471995,471998],{"className":471979},[31676],[34,471981],{"className":471982,"style":336814},[31680],[34,471984,34386],{"className":471985,"style":34427},[31685,31686],[34,471987],{"className":471988,"style":31846},[31845],[34,471990],{"className":471991,"style":31846},[31845],[34,471993,262625],{"className":471994},[31850],[34,471996],{"className":471997,"style":31846},[31845],[34,471999],{"className":472000,"style":31846},[31845],[34,472002,472004,472007,472010,472013,472016],{"className":472003},[31676],[34,472005],{"className":472006,"style":224869},[31680],[34,472008,34395],{"className":472009,"style":34452},[31685,31686],[34,472011],{"className":472012,"style":32604},[31845],[34,472014,71191],{"className":472015},[32024],[34,472017],{"className":472018,"style":32604},[31845],[34,472020,472022,472025,472028,472031,472034],{"className":472021},[31676],[34,472023],{"className":472024,"style":34812},[31680],[34,472026,32936],{"className":472027},[31685,31686],[34,472029],{"className":472030,"style":31846},[31845],[34,472032,112296],{"className":472033},[31850],[34,472035],{"className":472036,"style":31846},[31845],[34,472038,472040,472043],{"className":472039},[31676],[34,472041],{"className":472042,"style":33218},[31680],[34,472044,106063],{"className":472045,"style":67482},[31685,31686],[34,472047,472049,472071],{"className":472048},[31642],[34,472050,472052],{"className":472051},[31646],[31648,472053,472054],{"xmlns":31650},[31652,472055,472056,472068],{},[31655,472057,472058,472060,472062,472064,472066],{},[31658,472059,31739],{},[31662,472061,456],{"stretchy":702},[31658,472063,34395],{},[31658,472065,32936],{},[31662,472067,2557],{"stretchy":702},[31665,472069,472070],{"encoding":31667},"f(rx)",[34,472072,472074],{"className":472073,"ariaHidden":758},[31672],[34,472075,472077,472080,472083,472086,472089,472092],{"className":472076},[31676],[34,472078],{"className":472079,"style":31774},[31680],[34,472081,31739],{"className":472082,"style":31778},[31685,31686],[34,472084,456],{"className":472085},[31782],[34,472087,34395],{"className":472088,"style":34452},[31685,31686],[34,472090,32936],{"className":472091},[31685,31686],[34,472093,2557],{"className":472094},[31690],[34,472096,472098,472111],{"className":472097},[31642],[34,472099,472101],{"className":472100},[31646],[31648,472102,472103],{"xmlns":31650},[31652,472104,472105,472109],{},[31655,472106,472107],{},[31658,472108,106063],{},[31665,472110,106063],{"encoding":31667},[34,472112,472114],{"className":472113,"ariaHidden":758},[31672],[34,472115,472117,472120],{"className":472116},[31676],[34,472118],{"className":472119,"style":33218},[31680],[34,472121,106063],{"className":472122,"style":67482},[31685,31686]," 中某元素的像，即 ",[34,472125,472127,472148],{"className":472126},[31642],[34,472128,472130],{"className":472129},[31646],[31648,472131,472132],{"xmlns":31650},[31652,472133,472134,472146],{},[31655,472135,472136,472138,472140,472142,472144],{},[31658,472137,31747],{},[31662,472139,71191],{},[31658,472141,67764],{},[31662,472143,112296],{},[31658,472145,223320],{},[31665,472147,471162],{"encoding":31667},[34,472149,472151,472169,472187],{"className":472150,"ariaHidden":758},[31672],[34,472152,472154,472157,472160,472163,472166],{"className":472153},[31676],[34,472155],{"className":472156,"style":224869},[31680],[34,472158,31747],{"className":472159},[31685,31686],[34,472161],{"className":472162,"style":32604},[31845],[34,472164,71191],{"className":472165},[32024],[34,472167],{"className":472168,"style":32604},[31845],[34,472170,472172,472175,472178,472181,472184],{"className":472171},[31676],[34,472173],{"className":472174,"style":86652},[31680],[34,472176,67764],{"className":472177,"style":32062},[31685,31686],[34,472179],{"className":472180,"style":31846},[31845],[34,472182,112296],{"className":472183},[31850],[34,472185],{"className":472186,"style":31846},[31845],[34,472188,472190,472193],{"className":472189},[31676],[34,472191],{"className":472192,"style":33218},[31680],[34,472194,223320],{"className":472195,"style":223361},[31685,31686],"。\n同理可证 ",[34,472198,472200,472222],{"className":472199},[31642],[34,472201,472203],{"className":472202},[31646],[31648,472204,472205],{"xmlns":31650},[31652,472206,472207,472219],{},[31655,472208,472209,472211,472213,472215,472217],{},[31658,472210,67764],{},[31662,472212,71191],{},[31658,472214,31747],{},[31662,472216,112296],{},[31658,472218,223320],{},[31665,472220,472221],{"encoding":31667},"y \\cdot s \\in J",[34,472223,472225,472243,472261],{"className":472224,"ariaHidden":758},[31672],[34,472226,472228,472231,472234,472237,472240],{"className":472227},[31676],[34,472229],{"className":472230,"style":73503},[31680],[34,472232,67764],{"className":472233,"style":32062},[31685,31686],[34,472235],{"className":472236,"style":32604},[31845],[34,472238,71191],{"className":472239},[32024],[34,472241],{"className":472242,"style":32604},[31845],[34,472244,472246,472249,472252,472255,472258],{"className":472245},[31676],[34,472247],{"className":472248,"style":34812},[31680],[34,472250,31747],{"className":472251},[31685,31686],[34,472253],{"className":472254,"style":31846},[31845],[34,472256,112296],{"className":472257},[31850],[34,472259],{"className":472260,"style":31846},[31845],[34,472262,472264,472267],{"className":472263},[31676],[34,472265],{"className":472266,"style":33218},[31680],[34,472268,223320],{"className":472269,"style":223361},[31685,31686],[15,472271,472272,2664,472274,11008,472317,469010,472345],{},[2627,472273,114043],{},[34,472275,472277,472296],{"className":472276},[31642],[34,472278,472280],{"className":472279},[31646],[31648,472281,472282],{"xmlns":31650},[31652,472283,472284,472294],{},[31655,472285,472286,472288,472290,472292],{},[31658,472287,31739],{},[31662,472289,456],{"stretchy":702},[31658,472291,106063],{},[31662,472293,2557],{"stretchy":702},[31665,472295,469200],{"encoding":31667},[34,472297,472299],{"className":472298,"ariaHidden":758},[31672],[34,472300,472302,472305,472308,472311,472314],{"className":472301},[31676],[34,472303],{"className":472304,"style":31774},[31680],[34,472306,31739],{"className":472307,"style":31778},[31685,31686],[34,472309,456],{"className":472310},[31782],[34,472312,106063],{"className":472313,"style":67482},[31685,31686],[34,472315,2557],{"className":472316},[31690],[34,472318,472320,472333],{"className":472319},[31642],[34,472321,472323],{"className":472322},[31646],[31648,472324,472325],{"xmlns":31650},[31652,472326,472327,472331],{},[31655,472328,472329],{},[31658,472330,94675],{},[31665,472332,94675],{"encoding":31667},[34,472334,472336],{"className":472335,"ariaHidden":758},[31672],[34,472337,472339,472342],{"className":472338},[31676],[34,472340],{"className":472341,"style":33218},[31680],[34,472343,94675],{"className":472344,"style":106455},[31685,31686],[34,472346,472348,472361],{"className":472347},[31642],[34,472349,472351],{"className":472350},[31646],[31648,472352,472353],{"xmlns":31650},[31652,472354,472355,472359],{},[31655,472356,472357],{},[31658,472358,445204],{"mathvariant":38079},[31665,472360,445207],{"encoding":31667},[34,472362,472364],{"className":472363,"ariaHidden":758},[31672],[34,472365,472367,472370],{"className":472366},[31676],[34,472368],{"className":472369,"style":445217},[31680],[34,472371,445204],{"className":472372},[31685,129261],[32188,472374],{},[32490,472376,219972],{"id":219969},[15,472378,472379,472552],{},[2627,472380,133910,472381,315040,472463,472551],{},[34,472382,472384,472409],{"className":472383},[31642],[34,472385,472387],{"className":472386},[31646],[31648,472388,472389],{"xmlns":31650},[31652,472390,472391,472407],{},[31655,472392,472393,472395,472397,472399,472401,472403,472405],{},[31658,472394,58442],{},[31662,472396,5598],{},[31658,472398,34386],{},[31662,472400,33239],{},[31658,472402,34386],{},[31658,472404,38080],{"mathvariant":38079},[31658,472406,106063],{},[31665,472408,446363],{"encoding":31667},[34,472410,472412,472430,472448],{"className":472411,"ariaHidden":758},[31672],[34,472413,472415,472418,472421,472424,472427],{"className":472414},[31676],[34,472416],{"className":472417,"style":32527},[31680],[34,472419,58442],{"className":472420,"style":32062},[31685,31686],[34,472422],{"className":472423,"style":31846},[31845],[34,472425,5598],{"className":472426},[31850],[34,472428],{"className":472429,"style":31846},[31845],[34,472431,472433,472436,472439,472442,472445],{"className":472432},[31676],[34,472434],{"className":472435,"style":33218},[31680],[34,472437,34386],{"className":472438,"style":34427},[31685,31686],[34,472440],{"className":472441,"style":31846},[31845],[34,472443,33239],{"className":472444},[31850],[34,472446],{"className":472447,"style":31846},[31845],[34,472449,472451,472454,472457,472460],{"className":472450},[31676],[34,472452],{"className":472453,"style":31774},[31680],[34,472455,34386],{"className":472456,"style":34427},[31685,31686],[34,472458,38080],{"className":472459},[31685],[34,472461,106063],{"className":472462,"style":67482},[31685,31686],[34,472464,472466,472494],{"className":472465},[31642],[34,472467,472469],{"className":472468},[31646],[31648,472470,472471],{"xmlns":31650},[31652,472472,472473,472491],{},[31655,472474,472475,472477,472479,472481,472483,472485,472487,472489],{},[31658,472476,58442],{},[31662,472478,456],{"stretchy":702},[31658,472480,111],{},[31662,472482,2557],{"stretchy":702},[31662,472484,202],{},[31658,472486,111],{},[31662,472488,223],{},[31658,472490,106063],{},[31665,472492,472493],{"encoding":31667},"\\pi(a) = a+I",[34,472495,472497,472524,472542],{"className":472496,"ariaHidden":758},[31672],[34,472498,472500,472503,472506,472509,472512,472515,472518,472521],{"className":472499},[31676],[34,472501],{"className":472502,"style":31774},[31680],[34,472504,58442],{"className":472505,"style":32062},[31685,31686],[34,472507,456],{"className":472508},[31782],[34,472510,111],{"className":472511},[31685,31686],[34,472513,2557],{"className":472514},[31690],[34,472516],{"className":472517,"style":31846},[31845],[34,472519,202],{"className":472520},[31850],[34,472522],{"className":472523,"style":31846},[31845],[34,472525,472527,472530,472533,472536,472539],{"className":472526},[31676],[34,472528],{"className":472529,"style":32597},[31680],[34,472531,111],{"className":472532},[31685,31686],[34,472534],{"className":472535,"style":32604},[31845],[34,472537,223],{"className":472538},[32024],[34,472540],{"className":472541,"style":32604},[31845],[34,472543,472545,472548],{"className":472544},[31676],[34,472546],{"className":472547,"style":33218},[31680],[34,472549,106063],{"className":472550,"style":67482},[31685,31686]," 定义的映射（称为典范同态）。",[2627,472553,225289,472554,472582,472583,347],{},[34,472555,472557,472570],{"className":472556},[31642],[34,472558,472560],{"className":472559},[31646],[31648,472561,472562],{"xmlns":31650},[31652,472563,472564,472568],{},[31655,472565,472566],{},[31658,472567,58442],{},[31665,472569,58445],{"encoding":31667},[34,472571,472573],{"className":472572,"ariaHidden":758},[31672],[34,472574,472576,472579],{"className":472575},[31676],[34,472577],{"className":472578,"style":32527},[31680],[34,472580,58442],{"className":472581,"style":32062},[31685,31686]," 是一个环同态，且 ",[34,472584,472586,472608],{"className":472585},[31642],[34,472587,472589],{"className":472588},[31646],[31648,472590,472591],{"xmlns":31650},[31652,472592,472593,472605],{},[31655,472594,472595,472597,472599,472601,472603],{},[31658,472596,406335],{},[31662,472598,32682],{},[31658,472600,58442],{},[31662,472602,202],{},[31658,472604,106063],{},[31665,472606,472607],{"encoding":31667},"\\ker \\pi = I",[34,472609,472611,472635],{"className":472610,"ariaHidden":758},[31672],[34,472612,472614,472617,472620,472623,472626,472629,472632],{"className":472613},[31676],[34,472615],{"className":472616,"style":31681},[31680],[34,472618,406335],{"className":472619},[32767],[34,472621],{"className":472622,"style":32123},[31845],[34,472624,58442],{"className":472625,"style":32062},[31685,31686],[34,472627],{"className":472628,"style":31846},[31845],[34,472630,202],{"className":472631},[31850],[34,472633],{"className":472634,"style":31846},[31845],[34,472636,472638,472641],{"className":472637},[31676],[34,472639],{"className":472640,"style":33218},[31680],[34,472642,106063],{"className":472643,"style":67482},[31685,31686],[15,472645,472646],{},[2627,472647,463462],{},[105,472649,472650],{},[108,472651,472652,472653],{},"这是商环定义的直接推论。你需要熟悉商环的运算法则：\n",[105,472654,472655,472848],{},[108,472656,472657],{},[34,472658,472660,472710],{"className":472659},[31642],[34,472661,472663],{"className":472662},[31646],[31648,472664,472665],{"xmlns":31650},[31652,472666,472667,472707],{},[31655,472668,472669,472671,472673,472675,472677,472679,472681,472683,472685,472687,472689,472691,472693,472695,472697,472699,472701,472703,472705],{},[31662,472670,456],{"stretchy":702},[31658,472672,111],{},[31662,472674,223],{},[31658,472676,106063],{},[31662,472678,2557],{"stretchy":702},[31662,472680,223],{},[31662,472682,456],{"stretchy":702},[31658,472684,1697],{},[31662,472686,223],{},[31658,472688,106063],{},[31662,472690,2557],{"stretchy":702},[31662,472692,202],{},[31662,472694,456],{"stretchy":702},[31658,472696,111],{},[31662,472698,223],{},[31658,472700,1697],{},[31662,472702,2557],{"stretchy":702},[31662,472704,223],{},[31658,472706,106063],{},[31665,472708,472709],{"encoding":31667},"(a+I) + (b+I) = (a+b) + I",[34,472711,472713,472734,472755,472776,472797,472818,472839],{"className":472712,"ariaHidden":758},[31672],[34,472714,472716,472719,472722,472725,472728,472731],{"className":472715},[31676],[34,472717],{"className":472718,"style":31774},[31680],[34,472720,456],{"className":472721},[31782],[34,472723,111],{"className":472724},[31685,31686],[34,472726],{"className":472727,"style":32604},[31845],[34,472729,223],{"className":472730},[32024],[34,472732],{"className":472733,"style":32604},[31845],[34,472735,472737,472740,472743,472746,472749,472752],{"className":472736},[31676],[34,472738],{"className":472739,"style":31774},[31680],[34,472741,106063],{"className":472742,"style":67482},[31685,31686],[34,472744,2557],{"className":472745},[31690],[34,472747],{"className":472748,"style":32604},[31845],[34,472750,223],{"className":472751},[32024],[34,472753],{"className":472754,"style":32604},[31845],[34,472756,472758,472761,472764,472767,472770,472773],{"className":472757},[31676],[34,472759],{"className":472760,"style":31774},[31680],[34,472762,456],{"className":472763},[31782],[34,472765,1697],{"className":472766},[31685,31686],[34,472768],{"className":472769,"style":32604},[31845],[34,472771,223],{"className":472772},[32024],[34,472774],{"className":472775,"style":32604},[31845],[34,472777,472779,472782,472785,472788,472791,472794],{"className":472778},[31676],[34,472780],{"className":472781,"style":31774},[31680],[34,472783,106063],{"className":472784,"style":67482},[31685,31686],[34,472786,2557],{"className":472787},[31690],[34,472789],{"className":472790,"style":31846},[31845],[34,472792,202],{"className":472793},[31850],[34,472795],{"className":472796,"style":31846},[31845],[34,472798,472800,472803,472806,472809,472812,472815],{"className":472799},[31676],[34,472801],{"className":472802,"style":31774},[31680],[34,472804,456],{"className":472805},[31782],[34,472807,111],{"className":472808},[31685,31686],[34,472810],{"className":472811,"style":32604},[31845],[34,472813,223],{"className":472814},[32024],[34,472816],{"className":472817,"style":32604},[31845],[34,472819,472821,472824,472827,472830,472833,472836],{"className":472820},[31676],[34,472822],{"className":472823,"style":31774},[31680],[34,472825,1697],{"className":472826},[31685,31686],[34,472828,2557],{"className":472829},[31690],[34,472831],{"className":472832,"style":32604},[31845],[34,472834,223],{"className":472835},[32024],[34,472837],{"className":472838,"style":32604},[31845],[34,472840,472842,472845],{"className":472841},[31676],[34,472843],{"className":472844,"style":33218},[31680],[34,472846,106063],{"className":472847,"style":67482},[31685,31686],[108,472849,472850],{},[34,472851,472853,472899],{"className":472852},[31642],[34,472854,472856],{"className":472855},[31646],[31648,472857,472858],{"xmlns":31650},[31652,472859,472860,472896],{},[31655,472861,472862,472864,472866,472868,472870,472872,472874,472876,472878,472880,472882,472884,472886,472888,472890,472892,472894],{},[31662,472863,456],{"stretchy":702},[31658,472865,111],{},[31662,472867,223],{},[31658,472869,106063],{},[31662,472871,2557],{"stretchy":702},[31662,472873,456],{"stretchy":702},[31658,472875,1697],{},[31662,472877,223],{},[31658,472879,106063],{},[31662,472881,2557],{"stretchy":702},[31662,472883,202],{},[31662,472885,456],{"stretchy":702},[31658,472887,111],{},[31658,472889,1697],{},[31662,472891,2557],{"stretchy":702},[31662,472893,223],{},[31658,472895,106063],{},[31665,472897,472898],{"encoding":31667},"(a+I)(b+I) = (ab) + I",[34,472900,472902,472923,472950,472971,472995],{"className":472901,"ariaHidden":758},[31672],[34,472903,472905,472908,472911,472914,472917,472920],{"className":472904},[31676],[34,472906],{"className":472907,"style":31774},[31680],[34,472909,456],{"className":472910},[31782],[34,472912,111],{"className":472913},[31685,31686],[34,472915],{"className":472916,"style":32604},[31845],[34,472918,223],{"className":472919},[32024],[34,472921],{"className":472922,"style":32604},[31845],[34,472924,472926,472929,472932,472935,472938,472941,472944,472947],{"className":472925},[31676],[34,472927],{"className":472928,"style":31774},[31680],[34,472930,106063],{"className":472931,"style":67482},[31685,31686],[34,472933,2557],{"className":472934},[31690],[34,472936,456],{"className":472937},[31782],[34,472939,1697],{"className":472940},[31685,31686],[34,472942],{"className":472943,"style":32604},[31845],[34,472945,223],{"className":472946},[32024],[34,472948],{"className":472949,"style":32604},[31845],[34,472951,472953,472956,472959,472962,472965,472968],{"className":472952},[31676],[34,472954],{"className":472955,"style":31774},[31680],[34,472957,106063],{"className":472958,"style":67482},[31685,31686],[34,472960,2557],{"className":472961},[31690],[34,472963],{"className":472964,"style":31846},[31845],[34,472966,202],{"className":472967},[31850],[34,472969],{"className":472970,"style":31846},[31845],[34,472972,472974,472977,472980,472983,472986,472989,472992],{"className":472973},[31676],[34,472975],{"className":472976,"style":31774},[31680],[34,472978,456],{"className":472979},[31782],[34,472981,324191],{"className":472982},[31685,31686],[34,472984,2557],{"className":472985},[31690],[34,472987],{"className":472988,"style":32604},[31845],[34,472990,223],{"className":472991},[32024],[34,472993],{"className":472994,"style":32604},[31845],[34,472996,472998,473001],{"className":472997},[31676],[34,472999],{"className":473000,"style":33218},[31680],[34,473002,106063],{"className":473003,"style":67482},[31685,31686],[15,473005,473006],{},[2627,473007,463779],{},[2230,473009,473010,473783],{},[108,473011,473012,2664,473043],{},[2627,473013,379882,473014,473042],{},[34,473015,473017,473030],{"className":473016},[31642],[34,473018,473020],{"className":473019},[31646],[31648,473021,473022],{"xmlns":31650},[31652,473023,473024,473028],{},[31655,473025,473026],{},[31658,473027,58442],{},[31665,473029,58445],{"encoding":31667},[34,473031,473033],{"className":473032,"ariaHidden":758},[31672],[34,473034,473036,473039],{"className":473035},[31676],[34,473037],{"className":473038,"style":32527},[31680],[34,473040,58442],{"className":473041,"style":32062},[31685,31686]," 是同态",[105,473044,473045,473457],{},[108,473046,473047,33183,473049,473191,473456],{},[2627,473048,446923],{},[34,473050,473052,473092],{"className":473051},[31642],[34,473053,473055],{"className":473054},[31646],[31648,473056,473057],{"xmlns":31650},[31652,473058,473059,473089],{},[31655,473060,473061,473063,473065,473067,473069,473071,473073,473075,473077,473079,473081,473083,473085,473087],{},[31658,473062,58442],{},[31662,473064,456],{"stretchy":702},[31658,473066,111],{},[31662,473068,223],{},[31658,473070,1697],{},[31662,473072,2557],{"stretchy":702},[31662,473074,202],{},[31662,473076,456],{"stretchy":702},[31658,473078,111],{},[31662,473080,223],{},[31658,473082,1697],{},[31662,473084,2557],{"stretchy":702},[31662,473086,223],{},[31658,473088,106063],{},[31665,473090,473091],{"encoding":31667},"\\pi(a+b) = (a+b) + I",[34,473093,473095,473119,473140,473161,473182],{"className":473094,"ariaHidden":758},[31672],[34,473096,473098,473101,473104,473107,473110,473113,473116],{"className":473097},[31676],[34,473099],{"className":473100,"style":31774},[31680],[34,473102,58442],{"className":473103,"style":32062},[31685,31686],[34,473105,456],{"className":473106},[31782],[34,473108,111],{"className":473109},[31685,31686],[34,473111],{"className":473112,"style":32604},[31845],[34,473114,223],{"className":473115},[32024],[34,473117],{"className":473118,"style":32604},[31845],[34,473120,473122,473125,473128,473131,473134,473137],{"className":473121},[31676],[34,473123],{"className":473124,"style":31774},[31680],[34,473126,1697],{"className":473127},[31685,31686],[34,473129,2557],{"className":473130},[31690],[34,473132],{"className":473133,"style":31846},[31845],[34,473135,202],{"className":473136},[31850],[34,473138],{"className":473139,"style":31846},[31845],[34,473141,473143,473146,473149,473152,473155,473158],{"className":473142},[31676],[34,473144],{"className":473145,"style":31774},[31680],[34,473147,456],{"className":473148},[31782],[34,473150,111],{"className":473151},[31685,31686],[34,473153],{"className":473154,"style":32604},[31845],[34,473156,223],{"className":473157},[32024],[34,473159],{"className":473160,"style":32604},[31845],[34,473162,473164,473167,473170,473173,473176,473179],{"className":473163},[31676],[34,473165],{"className":473166,"style":31774},[31680],[34,473168,1697],{"className":473169},[31685,31686],[34,473171,2557],{"className":473172},[31690],[34,473174],{"className":473175,"style":32604},[31845],[34,473177,223],{"className":473178},[32024],[34,473180],{"className":473181,"style":32604},[31845],[34,473183,473185,473188],{"className":473184},[31676],[34,473186],{"className":473187,"style":33218},[31680],[34,473189,106063],{"className":473190,"style":67482},[31685,31686],[34,473192,473194,473264],{"className":473193},[31642],[34,473195,473197],{"className":473196},[31646],[31648,473198,473199],{"xmlns":31650},[31652,473200,473201,473261],{},[31655,473202,473203,473205,473207,473209,473211,473213,473215,473217,473219,473221,473223,473225,473227,473229,473231,473233,473235,473237,473239,473241,473243,473245,473247,473249,473251,473253,473255,473257,473259],{},[31658,473204,58442],{},[31662,473206,456],{"stretchy":702},[31658,473208,111],{},[31662,473210,2557],{"stretchy":702},[31662,473212,223],{},[31658,473214,58442],{},[31662,473216,456],{"stretchy":702},[31658,473218,1697],{},[31662,473220,2557],{"stretchy":702},[31662,473222,202],{},[31662,473224,456],{"stretchy":702},[31658,473226,111],{},[31662,473228,223],{},[31658,473230,106063],{},[31662,473232,2557],{"stretchy":702},[31662,473234,223],{},[31662,473236,456],{"stretchy":702},[31658,473238,1697],{},[31662,473240,223],{},[31658,473242,106063],{},[31662,473244,2557],{"stretchy":702},[31662,473246,202],{},[31662,473248,456],{"stretchy":702},[31658,473250,111],{},[31662,473252,223],{},[31658,473254,1697],{},[31662,473256,2557],{"stretchy":702},[31662,473258,223],{},[31658,473260,106063],{},[31665,473262,473263],{"encoding":31667},"\\pi(a) + \\pi(b) = (a+I) + (b+I) = (a+b) + I",[34,473265,473267,473294,473321,473342,473363,473384,473405,473426,473447],{"className":473266,"ariaHidden":758},[31672],[34,473268,473270,473273,473276,473279,473282,473285,473288,473291],{"className":473269},[31676],[34,473271],{"className":473272,"style":31774},[31680],[34,473274,58442],{"className":473275,"style":32062},[31685,31686],[34,473277,456],{"className":473278},[31782],[34,473280,111],{"className":473281},[31685,31686],[34,473283,2557],{"className":473284},[31690],[34,473286],{"className":473287,"style":32604},[31845],[34,473289,223],{"className":473290},[32024],[34,473292],{"className":473293,"style":32604},[31845],[34,473295,473297,473300,473303,473306,473309,473312,473315,473318],{"className":473296},[31676],[34,473298],{"className":473299,"style":31774},[31680],[34,473301,58442],{"className":473302,"style":32062},[31685,31686],[34,473304,456],{"className":473305},[31782],[34,473307,1697],{"className":473308},[31685,31686],[34,473310,2557],{"className":473311},[31690],[34,473313],{"className":473314,"style":31846},[31845],[34,473316,202],{"className":473317},[31850],[34,473319],{"className":473320,"style":31846},[31845],[34,473322,473324,473327,473330,473333,473336,473339],{"className":473323},[31676],[34,473325],{"className":473326,"style":31774},[31680],[34,473328,456],{"className":473329},[31782],[34,473331,111],{"className":473332},[31685,31686],[34,473334],{"className":473335,"style":32604},[31845],[34,473337,223],{"className":473338},[32024],[34,473340],{"className":473341,"style":32604},[31845],[34,473343,473345,473348,473351,473354,473357,473360],{"className":473344},[31676],[34,473346],{"className":473347,"style":31774},[31680],[34,473349,106063],{"className":473350,"style":67482},[31685,31686],[34,473352,2557],{"className":473353},[31690],[34,473355],{"className":473356,"style":32604},[31845],[34,473358,223],{"className":473359},[32024],[34,473361],{"className":473362,"style":32604},[31845],[34,473364,473366,473369,473372,473375,473378,473381],{"className":473365},[31676],[34,473367],{"className":473368,"style":31774},[31680],[34,473370,456],{"className":473371},[31782],[34,473373,1697],{"className":473374},[31685,31686],[34,473376],{"className":473377,"style":32604},[31845],[34,473379,223],{"className":473380},[32024],[34,473382],{"className":473383,"style":32604},[31845],[34,473385,473387,473390,473393,473396,473399,473402],{"className":473386},[31676],[34,473388],{"className":473389,"style":31774},[31680],[34,473391,106063],{"className":473392,"style":67482},[31685,31686],[34,473394,2557],{"className":473395},[31690],[34,473397],{"className":473398,"style":31846},[31845],[34,473400,202],{"className":473401},[31850],[34,473403],{"className":473404,"style":31846},[31845],[34,473406,473408,473411,473414,473417,473420,473423],{"className":473407},[31676],[34,473409],{"className":473410,"style":31774},[31680],[34,473412,456],{"className":473413},[31782],[34,473415,111],{"className":473416},[31685,31686],[34,473418],{"className":473419,"style":32604},[31845],[34,473421,223],{"className":473422},[32024],[34,473424],{"className":473425,"style":32604},[31845],[34,473427,473429,473432,473435,473438,473441,473444],{"className":473428},[31676],[34,473430],{"className":473431,"style":31774},[31680],[34,473433,1697],{"className":473434},[31685,31686],[34,473436,2557],{"className":473437},[31690],[34,473439],{"className":473440,"style":32604},[31845],[34,473442,223],{"className":473443},[32024],[34,473445],{"className":473446,"style":32604},[31845],[34,473448,473450,473453],{"className":473449},[31676],[34,473451],{"className":473452,"style":33218},[31680],[34,473454,106063],{"className":473455,"style":67482},[31685,31686],"\n两者相等。",[108,473458,473459,33183,473461,473553,473754,473755,440425],{},[2627,473460,447325],{},[34,473462,473464,473496],{"className":473463},[31642],[34,473465,473467],{"className":473466},[31646],[31648,473468,473469],{"xmlns":31650},[31652,473470,473471,473493],{},[31655,473472,473473,473475,473477,473479,473481,473483,473485,473487,473489,473491],{},[31658,473474,58442],{},[31662,473476,456],{"stretchy":702},[31658,473478,111],{},[31658,473480,1697],{},[31662,473482,2557],{"stretchy":702},[31662,473484,202],{},[31658,473486,111],{},[31658,473488,1697],{},[31662,473490,223],{},[31658,473492,106063],{},[31665,473494,473495],{"encoding":31667},"\\pi(ab) = ab + I",[34,473497,473499,473526,473544],{"className":473498,"ariaHidden":758},[31672],[34,473500,473502,473505,473508,473511,473514,473517,473520,473523],{"className":473501},[31676],[34,473503],{"className":473504,"style":31774},[31680],[34,473506,58442],{"className":473507,"style":32062},[31685,31686],[34,473509,456],{"className":473510},[31782],[34,473512,324191],{"className":473513},[31685,31686],[34,473515,2557],{"className":473516},[31690],[34,473518],{"className":473519,"style":31846},[31845],[34,473521,202],{"className":473522},[31850],[34,473524],{"className":473525,"style":31846},[31845],[34,473527,473529,473532,473535,473538,473541],{"className":473528},[31676],[34,473530],{"className":473531,"style":34299},[31680],[34,473533,324191],{"className":473534},[31685,31686],[34,473536],{"className":473537,"style":32604},[31845],[34,473539,223],{"className":473540},[32024],[34,473542],{"className":473543,"style":32604},[31845],[34,473545,473547,473550],{"className":473546},[31676],[34,473548],{"className":473549,"style":33218},[31680],[34,473551,106063],{"className":473552,"style":67482},[31685,31686],[34,473554,473556,473616],{"className":473555},[31642],[34,473557,473559],{"className":473558},[31646],[31648,473560,473561],{"xmlns":31650},[31652,473562,473563,473613],{},[31655,473564,473565,473567,473569,473571,473573,473575,473577,473579,473581,473583,473585,473587,473589,473591,473593,473595,473597,473599,473601,473603,473605,473607,473609,473611],{},[31658,473566,58442],{},[31662,473568,456],{"stretchy":702},[31658,473570,111],{},[31662,473572,2557],{"stretchy":702},[31658,473574,58442],{},[31662,473576,456],{"stretchy":702},[31658,473578,1697],{},[31662,473580,2557],{"stretchy":702},[31662,473582,202],{},[31662,473584,456],{"stretchy":702},[31658,473586,111],{},[31662,473588,223],{},[31658,473590,106063],{},[31662,473592,2557],{"stretchy":702},[31662,473594,456],{"stretchy":702},[31658,473596,1697],{},[31662,473598,223],{},[31658,473600,106063],{},[31662,473602,2557],{"stretchy":702},[31662,473604,202],{},[31658,473606,111],{},[31658,473608,1697],{},[31662,473610,223],{},[31658,473612,106063],{},[31665,473614,473615],{"encoding":31667},"\\pi(a)\\pi(b) = (a+I)(b+I) = ab + I",[34,473617,473619,473658,473679,473706,473727,473745],{"className":473618,"ariaHidden":758},[31672],[34,473620,473622,473625,473628,473631,473634,473637,473640,473643,473646,473649,473652,473655],{"className":473621},[31676],[34,473623],{"className":473624,"style":31774},[31680],[34,473626,58442],{"className":473627,"style":32062},[31685,31686],[34,473629,456],{"className":473630},[31782],[34,473632,111],{"className":473633},[31685,31686],[34,473635,2557],{"className":473636},[31690],[34,473638,58442],{"className":473639,"style":32062},[31685,31686],[34,473641,456],{"className":473642},[31782],[34,473644,1697],{"className":473645},[31685,31686],[34,473647,2557],{"className":473648},[31690],[34,473650],{"className":473651,"style":31846},[31845],[34,473653,202],{"className":473654},[31850],[34,473656],{"className":473657,"style":31846},[31845],[34,473659,473661,473664,473667,473670,473673,473676],{"className":473660},[31676],[34,473662],{"className":473663,"style":31774},[31680],[34,473665,456],{"className":473666},[31782],[34,473668,111],{"className":473669},[31685,31686],[34,473671],{"className":473672,"style":32604},[31845],[34,473674,223],{"className":473675},[32024],[34,473677],{"className":473678,"style":32604},[31845],[34,473680,473682,473685,473688,473691,473694,473697,473700,473703],{"className":473681},[31676],[34,473683],{"className":473684,"style":31774},[31680],[34,473686,106063],{"className":473687,"style":67482},[31685,31686],[34,473689,2557],{"className":473690},[31690],[34,473692,456],{"className":473693},[31782],[34,473695,1697],{"className":473696},[31685,31686],[34,473698],{"className":473699,"style":32604},[31845],[34,473701,223],{"className":473702},[32024],[34,473704],{"className":473705,"style":32604},[31845],[34,473707,473709,473712,473715,473718,473721,473724],{"className":473708},[31676],[34,473710],{"className":473711,"style":31774},[31680],[34,473713,106063],{"className":473714,"style":67482},[31685,31686],[34,473716,2557],{"className":473717},[31690],[34,473719],{"className":473720,"style":31846},[31845],[34,473722,202],{"className":473723},[31850],[34,473725],{"className":473726,"style":31846},[31845],[34,473728,473730,473733,473736,473739,473742],{"className":473729},[31676],[34,473731],{"className":473732,"style":34299},[31680],[34,473734,324191],{"className":473735},[31685,31686],[34,473737],{"className":473738,"style":32604},[31845],[34,473740,223],{"className":473741},[32024],[34,473743],{"className":473744,"style":32604},[31845],[34,473746,473748,473751],{"className":473747},[31676],[34,473749],{"className":473750,"style":33218},[31680],[34,473752,106063],{"className":473753,"style":67482},[31685,31686],"\n两者相等。\n所以 ",[34,473756,473758,473771],{"className":473757},[31642],[34,473759,473761],{"className":473760},[31646],[31648,473762,473763],{"xmlns":31650},[31652,473764,473765,473769],{},[31655,473766,473767],{},[31658,473768,58442],{},[31665,473770,58445],{"encoding":31667},[34,473772,473774],{"className":473773,"ariaHidden":758},[31672],[34,473775,473777,473780],{"className":473776},[31676],[34,473778],{"className":473779,"style":32527},[31680],[34,473781,58442],{"className":473782,"style":32062},[31685,31686],[108,473784,473785,33183,473847,473886,473887,473925,473926,473964,473965,474016,474017,474045,474046,474309,474310,474383,474384,355408,474435,452957,474495],{},[2627,473786,379882,473787],{},[34,473788,473790,473811],{"className":473789},[31642],[34,473791,473793],{"className":473792},[31646],[31648,473794,473795],{"xmlns":31650},[31652,473796,473797,473809],{},[31655,473798,473799,473801,473803,473805,473807],{},[31658,473800,406335],{},[31662,473802,32682],{},[31658,473804,58442],{},[31662,473806,202],{},[31658,473808,106063],{},[31665,473810,472607],{"encoding":31667},[34,473812,473814,473838],{"className":473813,"ariaHidden":758},[31672],[34,473815,473817,473820,473823,473826,473829,473832,473835],{"className":473816},[31676],[34,473818],{"className":473819,"style":31681},[31680],[34,473821,406335],{"className":473822},[32767],[34,473824],{"className":473825,"style":32123},[31845],[34,473827,58442],{"className":473828,"style":32062},[31685,31686],[34,473830],{"className":473831,"style":31846},[31845],[34,473833,202],{"className":473834},[31850],[34,473836],{"className":473837,"style":31846},[31845],[34,473839,473841,473844],{"className":473840},[31676],[34,473842],{"className":473843,"style":33218},[31680],[34,473845,106063],{"className":473846,"style":67482},[31685,31686],[34,473848,473850,473868],{"className":473849},[31642],[34,473851,473853],{"className":473852},[31646],[31648,473854,473855],{"xmlns":31650},[31652,473856,473857,473865],{},[31655,473858,473859,473861,473863],{},[31658,473860,406335],{},[31662,473862,32682],{},[31658,473864,58442],{},[31665,473866,473867],{"encoding":31667},"\\ker \\pi",[34,473869,473871],{"className":473870,"ariaHidden":758},[31672],[34,473872,473874,473877,473880,473883],{"className":473873},[31676],[34,473875],{"className":473876,"style":31681},[31680],[34,473878,406335],{"className":473879},[32767],[34,473881],{"className":473882,"style":32123},[31845],[34,473884,58442],{"className":473885,"style":32062},[31685,31686]," 的定义是映射到 ",[34,473888,473890,473907],{"className":473889},[31642],[34,473891,473893],{"className":473892},[31646],[31648,473894,473895],{"xmlns":31650},[31652,473896,473897,473905],{},[31655,473898,473899,473901,473903],{},[31658,473900,34386],{},[31658,473902,38080],{"mathvariant":38079},[31658,473904,106063],{},[31665,473906,337927],{"encoding":31667},[34,473908,473910],{"className":473909,"ariaHidden":758},[31672],[34,473911,473913,473916,473919,473922],{"className":473912},[31676],[34,473914],{"className":473915,"style":31774},[31680],[34,473917,34386],{"className":473918,"style":34427},[31685,31686],[34,473920,38080],{"className":473921},[31685],[34,473923,106063],{"className":473924,"style":67482},[31685,31686]," 中零元的所有元素。\n",[34,473927,473929,473946],{"className":473928},[31642],[34,473930,473932],{"className":473931},[31646],[31648,473933,473934],{"xmlns":31650},[31652,473935,473936,473944],{},[31655,473937,473938,473940,473942],{},[31658,473939,34386],{},[31658,473941,38080],{"mathvariant":38079},[31658,473943,106063],{},[31665,473945,337927],{"encoding":31667},[34,473947,473949],{"className":473948,"ariaHidden":758},[31672],[34,473950,473952,473955,473958,473961],{"className":473951},[31676],[34,473953],{"className":473954,"style":31774},[31680],[34,473956,34386],{"className":473957,"style":34427},[31685,31686],[34,473959,38080],{"className":473960},[31685],[34,473962,106063],{"className":473963,"style":67482},[31685,31686]," 的零元是什么？是 ",[34,473966,473968,473986],{"className":473967},[31642],[34,473969,473971],{"className":473970},[31646],[31648,473972,473973],{"xmlns":31650},[31652,473974,473975,473983],{},[31655,473976,473977,473979,473981],{},[31749,473978,459],{},[31662,473980,223],{},[31658,473982,106063],{},[31665,473984,473985],{"encoding":31667},"0+I",[34,473987,473989,474007],{"className":473988,"ariaHidden":758},[31672],[34,473990,473992,473995,473998,474001,474004],{"className":473991},[31676],[34,473993],{"className":473994,"style":35942},[31680],[34,473996,459],{"className":473997},[31685],[34,473999],{"className":474000,"style":32604},[31845],[34,474002,223],{"className":474003},[32024],[34,474005],{"className":474006,"style":32604},[31845],[34,474008,474010,474013],{"className":474009},[31676],[34,474011],{"className":474012,"style":33218},[31680],[34,474014,106063],{"className":474015,"style":67482},[31685,31686],"，也就是集合 ",[34,474018,474020,474033],{"className":474019},[31642],[34,474021,474023],{"className":474022},[31646],[31648,474024,474025],{"xmlns":31650},[31652,474026,474027,474031],{},[31655,474028,474029],{},[31658,474030,106063],{},[31665,474032,106063],{"encoding":31667},[34,474034,474036],{"className":474035,"ariaHidden":758},[31672],[34,474037,474039,474042],{"className":474038},[31676],[34,474040],{"className":474041,"style":33218},[31680],[34,474043,106063],{"className":474044,"style":67482},[31685,31686]," 本身。\n",[34,474047,474049,474115],{"className":474048},[31642],[34,474050,474052],{"className":474051},[31646],[31648,474053,474054],{"xmlns":31650},[31652,474055,474056,474112],{},[31655,474057,474058,474060,474062,474064,474066,474068,474070,474072,474074,474076,474078,474080,474082,474084,474096,474098,474100,474102,474104,474106,474108,474110],{},[31658,474059,111],{},[31662,474061,112296],{},[31658,474063,406335],{},[31662,474065,32682],{},[31658,474067,58442],{},[39361,474069,262622],{},[31662,474071,323391],{},[39361,474073,262622],{},[31658,474075,58442],{},[31662,474077,456],{"stretchy":702},[31658,474079,111],{},[31662,474081,2557],{"stretchy":702},[31662,474083,202],{},[31743,474085,474086,474088],{},[31749,474087,459],{},[31655,474089,474090,474092,474094],{},[31658,474091,34386],{},[31658,474093,38080],{"mathvariant":38079},[31658,474095,106063],{},[39361,474097,262622],{},[31662,474099,323391],{},[39361,474101,262622],{},[31658,474103,111],{},[31662,474105,223],{},[31658,474107,106063],{},[31662,474109,202],{},[31658,474111,106063],{},[31665,474113,474114],{"encoding":31667},"a \\in \\ker \\pi \\iff \\pi(a) = 0_{R\u002FI} \\iff a+I = I",[34,474116,474118,474136,474167,474194,474264,474282,474300],{"className":474117,"ariaHidden":758},[31672],[34,474119,474121,474124,474127,474130,474133],{"className":474120},[31676],[34,474122],{"className":474123,"style":34812},[31680],[34,474125,111],{"className":474126},[31685,31686],[34,474128],{"className":474129,"style":31846},[31845],[34,474131,112296],{"className":474132},[31850],[34,474134],{"className":474135,"style":31846},[31845],[34,474137,474139,474143,474146,474149,474152,474155,474158,474161,474164],{"className":474138},[31676],[34,474140],{"className":474141,"style":474142},[31680],"height:0.7184em;vertical-align:-0.024em;",[34,474144,406335],{"className":474145},[32767],[34,474147],{"className":474148,"style":32123},[31845],[34,474150,58442],{"className":474151,"style":32062},[31685,31686],[34,474153],{"className":474154,"style":31846},[31845],[34,474156],{"className":474157,"style":31846},[31845],[34,474159,323391],{"className":474160},[31850],[34,474162],{"className":474163,"style":31846},[31845],[34,474165],{"className":474166,"style":31846},[31845],[34,474168,474170,474173,474176,474179,474182,474185,474188,474191],{"className":474169},[31676],[34,474171],{"className":474172,"style":31774},[31680],[34,474174,58442],{"className":474175,"style":32062},[31685,31686],[34,474177,456],{"className":474178},[31782],[34,474180,111],{"className":474181},[31685,31686],[34,474183,2557],{"className":474184},[31690],[34,474186],{"className":474187,"style":31846},[31845],[34,474189,202],{"className":474190},[31850],[34,474192],{"className":474193,"style":31846},[31845],[34,474195,474197,474200,474249,474252,474255,474258,474261],{"className":474196},[31676],[34,474198],{"className":474199,"style":332276},[31680],[34,474201,474203,474206],{"className":474202},[31685],[34,474204,459],{"className":474205},[31685],[34,474207,474209],{"className":474208},[31792],[34,474210,474212,474241],{"className":474211},[31796,31797],[34,474213,474215,474238],{"className":474214},[31801],[34,474216,474218],{"className":474217,"style":332295},[31805],[34,474219,474220,474223],{"style":332298},[34,474221],{"className":474222,"style":31814},[31813],[34,474224,474226],{"className":474225},[31818,31819,31820,31821],[34,474227,474229,474232,474235],{"className":474228},[31685,31821],[34,474230,34386],{"className":474231,"style":34427},[31685,31686,31821],[34,474233,38080],{"className":474234},[31685,31821],[34,474236,106063],{"className":474237,"style":67482},[31685,31686,31821],[34,474239,31829],{"className":474240},[31828],[34,474242,474244],{"className":474243},[31801],[34,474245,474247],{"className":474246,"style":332326},[31805],[34,474248],{},[34,474250],{"className":474251,"style":31846},[31845],[34,474253],{"className":474254,"style":31846},[31845],[34,474256,323391],{"className":474257},[31850],[34,474259],{"className":474260,"style":31846},[31845],[34,474262],{"className":474263,"style":31846},[31845],[34,474265,474267,474270,474273,474276,474279],{"className":474266},[31676],[34,474268],{"className":474269,"style":32597},[31680],[34,474271,111],{"className":474272},[31685,31686],[34,474274],{"className":474275,"style":32604},[31845],[34,474277,223],{"className":474278},[32024],[34,474280],{"className":474281,"style":32604},[31845],[34,474283,474285,474288,474291,474294,474297],{"className":474284},[31676],[34,474286],{"className":474287,"style":33218},[31680],[34,474289,106063],{"className":474290,"style":67482},[31685,31686],[34,474292],{"className":474293,"style":31846},[31845],[34,474295,202],{"className":474296},[31850],[34,474298],{"className":474299,"style":31846},[31845],[34,474301,474303,474306],{"className":474302},[31676],[34,474304],{"className":474305,"style":33218},[31680],[34,474307,106063],{"className":474308,"style":67482},[31685,31686],"\n在陪集理论中，",[34,474311,474313,474335],{"className":474312},[31642],[34,474314,474316],{"className":474315},[31646],[31648,474317,474318],{"xmlns":31650},[31652,474319,474320,474332],{},[31655,474321,474322,474324,474326,474328,474330],{},[31658,474323,111],{},[31662,474325,223],{},[31658,474327,106063],{},[31662,474329,202],{},[31658,474331,106063],{},[31665,474333,474334],{"encoding":31667},"a+I = I",[34,474336,474338,474356,474374],{"className":474337,"ariaHidden":758},[31672],[34,474339,474341,474344,474347,474350,474353],{"className":474340},[31676],[34,474342],{"className":474343,"style":32597},[31680],[34,474345,111],{"className":474346},[31685,31686],[34,474348],{"className":474349,"style":32604},[31845],[34,474351,223],{"className":474352},[32024],[34,474354],{"className":474355,"style":32604},[31845],[34,474357,474359,474362,474365,474368,474371],{"className":474358},[31676],[34,474360],{"className":474361,"style":33218},[31680],[34,474363,106063],{"className":474364,"style":67482},[31685,31686],[34,474366],{"className":474367,"style":31846},[31845],[34,474369,202],{"className":474370},[31850],[34,474372],{"className":474373,"style":31846},[31845],[34,474375,474377,474380],{"className":474376},[31676],[34,474378],{"className":474379,"style":33218},[31680],[34,474381,106063],{"className":474382,"style":67482},[31685,31686]," 当且仅当 ",[34,474385,474387,474405],{"className":474386},[31642],[34,474388,474390],{"className":474389},[31646],[31648,474391,474392],{"xmlns":31650},[31652,474393,474394,474402],{},[31655,474395,474396,474398,474400],{},[31658,474397,111],{},[31662,474399,112296],{},[31658,474401,106063],{},[31665,474403,474404],{"encoding":31667},"a \\in I",[34,474406,474408,474426],{"className":474407,"ariaHidden":758},[31672],[34,474409,474411,474414,474417,474420,474423],{"className":474410},[31676],[34,474412],{"className":474413,"style":34812},[31680],[34,474415,111],{"className":474416},[31685,31686],[34,474418],{"className":474419,"style":31846},[31845],[34,474421,112296],{"className":474422},[31850],[34,474424],{"className":474425,"style":31846},[31845],[34,474427,474429,474432],{"className":474428},[31676],[34,474430],{"className":474431,"style":33218},[31680],[34,474433,106063],{"className":474434,"style":67482},[31685,31686],[34,474436,474438,474459],{"className":474437},[31642],[34,474439,474441],{"className":474440},[31646],[31648,474442,474443],{"xmlns":31650},[31652,474444,474445,474457],{},[31655,474446,474447,474449,474451,474453,474455],{},[31658,474448,406335],{},[31662,474450,32682],{},[31658,474452,58442],{},[31662,474454,202],{},[31658,474456,106063],{},[31665,474458,472607],{"encoding":31667},[34,474460,474462,474486],{"className":474461,"ariaHidden":758},[31672],[34,474463,474465,474468,474471,474474,474477,474480,474483],{"className":474464},[31676],[34,474466],{"className":474467,"style":31681},[31680],[34,474469,406335],{"className":474470},[32767],[34,474472],{"className":474473,"style":32123},[31845],[34,474475,58442],{"className":474476,"style":32062},[31685,31686],[34,474478],{"className":474479,"style":31846},[31845],[34,474481,202],{"className":474482},[31850],[34,474484],{"className":474485,"style":31846},[31845],[34,474487,474489,474492],{"className":474488},[31676],[34,474490],{"className":474491,"style":33218},[31680],[34,474493,106063],{"className":474494,"style":67482},[31685,31686],[34,474496,474498,474511],{"className":474497},[31642],[34,474499,474501],{"className":474500},[31646],[31648,474502,474503],{"xmlns":31650},[31652,474504,474505,474509],{},[31655,474506,474507],{},[31658,474508,445204],{"mathvariant":38079},[31665,474510,445207],{"encoding":31667},[34,474512,474514],{"className":474513,"ariaHidden":758},[31672],[34,474515,474517,474520],{"className":474516},[31676],[34,474518],{"className":474519,"style":445217},[31680],[34,474521,445204],{"className":474522},[31685,129261],[32188,474524],{},[32490,474526,474528],{"id":474527},"题目-4","题目 4",[15,474530,474531,474900],{},[2627,474532,133910,474533,474561,474562,258,474701,326522,474729,474757,474758,326252,474786,347],{},[34,474534,474536,474549],{"className":474535},[31642],[34,474537,474539],{"className":474538},[31646],[31648,474540,474541],{"xmlns":31650},[31652,474542,474543,474547],{},[31655,474544,474545],{},[31658,474546,34386],{},[31665,474548,34386],{"encoding":31667},[34,474550,474552],{"className":474551,"ariaHidden":758},[31672],[34,474553,474555,474558],{"className":474554},[31676],[34,474556],{"className":474557,"style":33218},[31680],[34,474559,34386],{"className":474560,"style":34427},[31685,31686]," 是有单位元的环，",[34,474563,474565,474594],{"className":474564},[31642],[34,474566,474568],{"className":474567},[31646],[31648,474569,474570],{"xmlns":31650},[31652,474571,474572,474592],{},[31655,474573,474574,474580,474582,474584,474586],{},[31743,474575,474576,474578],{},[31658,474577,106063],{},[31749,474579,640],{},[31662,474581,31960],{"separator":758},[31662,474583,33934],{},[31662,474585,31960],{"separator":758},[31743,474587,474588,474590],{},[31658,474589,106063],{},[31658,474591,31660],{},[31665,474593,404891],{"encoding":31667},[34,474595,474597],{"className":474596,"ariaHidden":758},[31672],[34,474598,474600,474603,474643,474646,474649,474652,474655,474658,474661],{"className":474599},[31676],[34,474601],{"className":474602,"style":70739},[31680],[34,474604,474606,474609],{"className":474605},[31685],[34,474607,106063],{"className":474608,"style":67482},[31685,31686],[34,474610,474612],{"className":474611},[31792],[34,474613,474615,474635],{"className":474614},[31796,31797],[34,474616,474618,474632],{"className":474617},[31801],[34,474619,474621],{"className":474620,"style":31806},[31805],[34,474622,474623,474626],{"style":217742},[34,474624],{"className":474625,"style":31814},[31813],[34,474627,474629],{"className":474628},[31818,31819,31820,31821],[34,474630,640],{"className":474631},[31685,31821],[34,474633,31829],{"className":474634},[31828],[34,474636,474638],{"className":474637},[31801],[34,474639,474641],{"className":474640,"style":31836},[31805],[34,474642],{},[34,474644,31960],{"className":474645},[32119],[34,474647],{"className":474648,"style":32123},[31845],[34,474650,33934],{"className":474651},[34006],[34,474653],{"className":474654,"style":32123},[31845],[34,474656,31960],{"className":474657},[32119],[34,474659],{"className":474660,"style":32123},[31845],[34,474662,474664,474667],{"className":474663},[31685],[34,474665,106063],{"className":474666,"style":67482},[31685,31686],[34,474668,474670],{"className":474669},[31792],[34,474671,474673,474693],{"className":474672},[31796,31797],[34,474674,474676,474690],{"className":474675},[31801],[34,474677,474679],{"className":474678,"style":32362},[31805],[34,474680,474681,474684],{"style":217742},[34,474682],{"className":474683,"style":31814},[31813],[34,474685,474687],{"className":474686},[31818,31819,31820,31821],[34,474688,31660],{"className":474689},[31685,31686,31821],[34,474691,31829],{"className":474692},[31828],[34,474694,474696],{"className":474695},[31801],[34,474697,474699],{"className":474698,"style":31836},[31805],[34,474700],{},[34,474702,474704,474717],{"className":474703},[31642],[34,474705,474707],{"className":474706},[31646],[31648,474708,474709],{"xmlns":31650},[31652,474710,474711,474715],{},[31655,474712,474713],{},[31658,474714,223320],{},[31665,474716,223320],{"encoding":31667},[34,474718,474720],{"className":474719,"ariaHidden":758},[31672],[34,474721,474723,474726],{"className":474722},[31676],[34,474724],{"className":474725,"style":33218},[31680],[34,474727,223320],{"className":474728,"style":223361},[31685,31686],[34,474730,474732,474745],{"className":474731},[31642],[34,474733,474735],{"className":474734},[31646],[31648,474736,474737],{"xmlns":31650},[31652,474738,474739,474743],{},[31655,474740,474741],{},[31658,474742,34386],{},[31665,474744,34386],{"encoding":31667},[34,474746,474748],{"className":474747,"ariaHidden":758},[31672],[34,474749,474751,474754],{"className":474750},[31676],[34,474752],{"className":474753,"style":33218},[31680],[34,474755,34386],{"className":474756,"style":34427},[31685,31686]," 的理想。已知对于任意 ",[34,474759,474761,474774],{"className":474760},[31642],[34,474762,474764],{"className":474763},[31646],[31648,474765,474766],{"xmlns":31650},[31652,474767,474768,474772],{},[31655,474769,474770],{},[31658,474771,31747],{},[31665,474773,31747],{"encoding":31667},[34,474775,474777],{"className":474776,"ariaHidden":758},[31672],[34,474778,474780,474783],{"className":474779},[31676],[34,474781],{"className":474782,"style":32527},[31680],[34,474784,31747],{"className":474785},[31685,31686],[34,474787,474789,474815],{"className":474788},[31642],[34,474790,474792],{"className":474791},[31646],[31648,474793,474794],{"xmlns":31650},[31652,474795,474796,474812],{},[31655,474797,474798,474804,474806,474808,474810],{},[31743,474799,474800,474802],{},[31658,474801,106063],{},[31658,474803,31747],{},[31662,474805,223],{},[31658,474807,223320],{},[31662,474809,202],{},[31658,474811,34386],{},[31665,474813,474814],{"encoding":31667},"I_s + J = R",[34,474816,474818,474873,474891],{"className":474817,"ariaHidden":758},[31672],[34,474819,474821,474824,474864,474867,474870],{"className":474820},[31676],[34,474822],{"className":474823,"style":70915},[31680],[34,474825,474827,474830],{"className":474826},[31685],[34,474828,106063],{"className":474829,"style":67482},[31685,31686],[34,474831,474833],{"className":474832},[31792],[34,474834,474836,474856],{"className":474835},[31796,31797],[34,474837,474839,474853],{"className":474838},[31801],[34,474840,474842],{"className":474841,"style":32362},[31805],[34,474843,474844,474847],{"style":217742},[34,474845],{"className":474846,"style":31814},[31813],[34,474848,474850],{"className":474849},[31818,31819,31820,31821],[34,474851,31747],{"className":474852},[31685,31686,31821],[34,474854,31829],{"className":474855},[31828],[34,474857,474859],{"className":474858},[31801],[34,474860,474862],{"className":474861,"style":31836},[31805],[34,474863],{},[34,474865],{"className":474866,"style":32604},[31845],[34,474868,223],{"className":474869},[32024],[34,474871],{"className":474872,"style":32604},[31845],[34,474874,474876,474879,474882,474885,474888],{"className":474875},[31676],[34,474877],{"className":474878,"style":33218},[31680],[34,474880,223320],{"className":474881,"style":223361},[31685,31686],[34,474883],{"className":474884,"style":31846},[31845],[34,474886,202],{"className":474887},[31850],[34,474889],{"className":474890,"style":31846},[31845],[34,474892,474894,474897],{"className":474893},[31676],[34,474895],{"className":474896,"style":33218},[31680],[34,474898,34386],{"className":474899,"style":34427},[31685,31686],[2627,474901,474902,474903,43298,475030,475058],{},"证明：积理想 ",[34,474904,474906,474932],{"className":474905},[31642],[34,474907,474909],{"className":474908},[31646],[31648,474910,474911],{"xmlns":31650},[31652,474912,474913,474929],{},[31655,474914,474915,474921,474923],{},[31743,474916,474917,474919],{},[31658,474918,106063],{},[31749,474920,640],{},[31662,474922,38874],{},[31743,474924,474925,474927],{},[31658,474926,106063],{},[31658,474928,31660],{},[31665,474930,474931],{"encoding":31667},"I_1 \\cdots I_n",[34,474933,474935],{"className":474934,"ariaHidden":758},[31672],[34,474936,474938,474941,474981,474984,474987,474990],{"className":474937},[31676],[34,474939],{"className":474940,"style":70915},[31680],[34,474942,474944,474947],{"className":474943},[31685],[34,474945,106063],{"className":474946,"style":67482},[31685,31686],[34,474948,474950],{"className":474949},[31792],[34,474951,474953,474973],{"className":474952},[31796,31797],[34,474954,474956,474970],{"className":474955},[31801],[34,474957,474959],{"className":474958,"style":31806},[31805],[34,474960,474961,474964],{"style":217742},[34,474962],{"className":474963,"style":31814},[31813],[34,474965,474967],{"className":474966},[31818,31819,31820,31821],[34,474968,640],{"className":474969},[31685,31821],[34,474971,31829],{"className":474972},[31828],[34,474974,474976],{"className":474975},[31801],[34,474977,474979],{"className":474978,"style":31836},[31805],[34,474980],{},[34,474982],{"className":474983,"style":32123},[31845],[34,474985,38874],{"className":474986},[34006],[34,474988],{"className":474989,"style":32123},[31845],[34,474991,474993,474996],{"className":474992},[31685],[34,474994,106063],{"className":474995,"style":67482},[31685,31686],[34,474997,474999],{"className":474998},[31792],[34,475000,475002,475022],{"className":475001},[31796,31797],[34,475003,475005,475019],{"className":475004},[31801],[34,475006,475008],{"className":475007,"style":32362},[31805],[34,475009,475010,475013],{"style":217742},[34,475011],{"className":475012,"style":31814},[31813],[34,475014,475016],{"className":475015},[31818,31819,31820,31821],[34,475017,31660],{"className":475018},[31685,31686,31821],[34,475020,31829],{"className":475021},[31828],[34,475023,475025],{"className":475024},[31801],[34,475026,475028],{"className":475027,"style":31836},[31805],[34,475029],{},[34,475031,475033,475046],{"className":475032},[31642],[34,475034,475036],{"className":475035},[31646],[31648,475037,475038],{"xmlns":31650},[31652,475039,475040,475044],{},[31655,475041,475042],{},[31658,475043,223320],{},[31665,475045,223320],{"encoding":31667},[34,475047,475049],{"className":475048,"ariaHidden":758},[31672],[34,475050,475052,475055],{"className":475051},[31676],[34,475053],{"className":475054,"style":33218},[31680],[34,475056,223320],{"className":475057,"style":223361},[31685,31686]," 也是互素的。",[15,475060,475061],{},[2627,475062,463462],{},[105,475064,475065,475345],{},[108,475066,475067,475070,475071,405412,475113,475185,475186,315343,475272,347],{},[2627,475068,475069],{},"互素的定义","：两个理想 ",[34,475072,475074,475092],{"className":475073},[31642],[34,475075,475077],{"className":475076},[31646],[31648,475078,475079],{"xmlns":31650},[31652,475080,475081,475089],{},[31655,475082,475083,475085,475087],{},[31658,475084,54151],{},[31662,475086,31960],{"separator":758},[31658,475088,54224],{},[31665,475090,475091],{"encoding":31667},"A, B",[34,475093,475095],{"className":475094,"ariaHidden":758},[31672],[34,475096,475098,475101,475104,475107,475110],{"className":475097},[31676],[34,475099],{"className":475100,"style":70739},[31680],[34,475102,54151],{"className":475103},[31685,31686],[34,475105,31960],{"className":475106},[32119],[34,475108],{"className":475109,"style":32123},[31845],[34,475111,54224],{"className":475112,"style":54252},[31685,31686],[34,475114,475116,475137],{"className":475115},[31642],[34,475117,475119],{"className":475118},[31646],[31648,475120,475121],{"xmlns":31650},[31652,475122,475123,475135],{},[31655,475124,475125,475127,475129,475131,475133],{},[31658,475126,54151],{},[31662,475128,223],{},[31658,475130,54224],{},[31662,475132,202],{},[31658,475134,34386],{},[31665,475136,409445],{"encoding":31667},[34,475138,475140,475158,475176],{"className":475139,"ariaHidden":758},[31672],[34,475141,475143,475146,475149,475152,475155],{"className":475142},[31676],[34,475144],{"className":475145,"style":34944},[31680],[34,475147,54151],{"className":475148},[31685,31686],[34,475150],{"className":475151,"style":32604},[31845],[34,475153,223],{"className":475154},[32024],[34,475156],{"className":475157,"style":32604},[31845],[34,475159,475161,475164,475167,475170,475173],{"className":475160},[31676],[34,475162],{"className":475163,"style":33218},[31680],[34,475165,54224],{"className":475166,"style":54252},[31685,31686],[34,475168],{"className":475169,"style":31846},[31845],[34,475171,202],{"className":475172},[31850],[34,475174],{"className":475175,"style":31846},[31845],[34,475177,475179,475182],{"className":475178},[31676],[34,475180],{"className":475181,"style":33218},[31680],[34,475183,34386],{"className":475184,"style":34427},[31685,31686],"。也就是说，我们可以找到 ",[34,475187,475189,475215],{"className":475188},[31642],[34,475190,475192],{"className":475191},[31646],[31648,475193,475194],{"xmlns":31650},[31652,475195,475196,475212],{},[31655,475197,475198,475200,475202,475204,475206,475208,475210],{},[31658,475199,111],{},[31662,475201,112296],{},[31658,475203,54151],{},[31662,475205,31960],{"separator":758},[31658,475207,1697],{},[31662,475209,112296],{},[31658,475211,54224],{},[31665,475213,475214],{"encoding":31667},"a \\in A, b \\in B",[34,475216,475218,475236,475263],{"className":475217,"ariaHidden":758},[31672],[34,475219,475221,475224,475227,475230,475233],{"className":475220},[31676],[34,475222],{"className":475223,"style":34812},[31680],[34,475225,111],{"className":475226},[31685,31686],[34,475228],{"className":475229,"style":31846},[31845],[34,475231,112296],{"className":475232},[31850],[34,475234],{"className":475235,"style":31846},[31845],[34,475237,475239,475242,475245,475248,475251,475254,475257,475260],{"className":475238},[31676],[34,475240],{"className":475241,"style":43217},[31680],[34,475243,54151],{"className":475244},[31685,31686],[34,475246,31960],{"className":475247},[32119],[34,475249],{"className":475250,"style":32123},[31845],[34,475252,1697],{"className":475253},[31685,31686],[34,475255],{"className":475256,"style":31846},[31845],[34,475258,112296],{"className":475259},[31850],[34,475261],{"className":475262,"style":31846},[31845],[34,475264,475266,475269],{"className":475265},[31676],[34,475267],{"className":475268,"style":33218},[31680],[34,475270,54224],{"className":475271,"style":54252},[31685,31686],[34,475273,475275,475297],{"className":475274},[31642],[34,475276,475278],{"className":475277},[31646],[31648,475279,475280],{"xmlns":31650},[31652,475281,475282,475294],{},[31655,475283,475284,475286,475288,475290,475292],{},[31658,475285,111],{},[31662,475287,223],{},[31658,475289,1697],{},[31662,475291,202],{},[31749,475293,640],{},[31665,475295,475296],{"encoding":31667},"a+b=1",[34,475298,475300,475318,475336],{"className":475299,"ariaHidden":758},[31672],[34,475301,475303,475306,475309,475312,475315],{"className":475302},[31676],[34,475304],{"className":475305,"style":32597},[31680],[34,475307,111],{"className":475308},[31685,31686],[34,475310],{"className":475311,"style":32604},[31845],[34,475313,223],{"className":475314},[32024],[34,475316],{"className":475317,"style":32604},[31845],[34,475319,475321,475324,475327,475330,475333],{"className":475320},[31676],[34,475322],{"className":475323,"style":31681},[31680],[34,475325,1697],{"className":475326},[31685,31686],[34,475328],{"className":475329,"style":31846},[31845],[34,475331,202],{"className":475332},[31850],[34,475334],{"className":475335,"style":31846},[31845],[34,475337,475339,475342],{"className":475338},[31676],[34,475340],{"className":475341,"style":32617},[31680],[34,475343,640],{"className":475344},[31685],[108,475346,475347,346816,475349,334983,475377,475512,475513,475646],{},[2627,475348,391664],{},[34,475350,475352,475365],{"className":475351},[31642],[34,475353,475355],{"className":475354},[31646],[31648,475356,475357],{"xmlns":31650},[31652,475358,475359,475363],{},[31655,475360,475361],{},[31749,475362,640],{},[31665,475364,640],{"encoding":31667},[34,475366,475368],{"className":475367,"ariaHidden":758},[31672],[34,475369,475371,475374],{"className":475370},[31676],[34,475372],{"className":475373,"style":32617},[31680],[34,475375,640],{"className":475376},[31685],[34,475378,475380,475406],{"className":475379},[31642],[34,475381,475383],{"className":475382},[31646],[31648,475384,475385],{"xmlns":31650},[31652,475386,475387,475403],{},[31655,475388,475389,475395,475397],{},[31743,475390,475391,475393],{},[31658,475392,31955],{},[31749,475394,640],{},[31662,475396,223],{},[31743,475398,475399,475401],{},[31658,475400,31968],{},[31749,475402,640],{},[31665,475404,475405],{"encoding":31667},"i_1 + j_1",[34,475407,475409,475465],{"className":475408,"ariaHidden":758},[31672],[34,475410,475412,475416,475456,475459,475462],{"className":475411},[31676],[34,475413],{"className":475414,"style":475415},[31680],"height:0.8095em;vertical-align:-0.15em;",[34,475417,475419,475422],{"className":475418},[31685],[34,475420,31955],{"className":475421},[31685,31686],[34,475423,475425],{"className":475424},[31792],[34,475426,475428,475448],{"className":475427},[31796,31797],[34,475429,475431,475445],{"className":475430},[31801],[34,475432,475434],{"className":475433,"style":31806},[31805],[34,475435,475436,475439],{"style":31809},[34,475437],{"className":475438,"style":31814},[31813],[34,475440,475442],{"className":475441},[31818,31819,31820,31821],[34,475443,640],{"className":475444},[31685,31821],[34,475446,31829],{"className":475447},[31828],[34,475449,475451],{"className":475450},[31801],[34,475452,475454],{"className":475453,"style":31836},[31805],[34,475455],{},[34,475457],{"className":475458,"style":32604},[31845],[34,475460,223],{"className":475461},[32024],[34,475463],{"className":475464,"style":32604},[31845],[34,475466,475468,475471],{"className":475467},[31676],[34,475469],{"className":475470,"style":33123},[31680],[34,475472,475474,475477],{"className":475473},[31685],[34,475475,31968],{"className":475476,"style":32153},[31685,31686],[34,475478,475480],{"className":475479},[31792],[34,475481,475483,475504],{"className":475482},[31796,31797],[34,475484,475486,475501],{"className":475485},[31801],[34,475487,475489],{"className":475488,"style":31806},[31805],[34,475490,475492,475495],{"style":475491},"top:-2.55em;margin-left:-0.0572em;margin-right:0.05em;",[34,475493],{"className":475494,"style":31814},[31813],[34,475496,475498],{"className":475497},[31818,31819,31820,31821],[34,475499,640],{"className":475500},[31685,31821],[34,475502,31829],{"className":475503},[31828],[34,475505,475507],{"className":475506},[31801],[34,475508,475510],{"className":475509,"style":31836},[31805],[34,475511],{},"，也可以写成 ",[34,475514,475516,475542],{"className":475515},[31642],[34,475517,475519],{"className":475518},[31646],[31648,475520,475521],{"xmlns":31650},[31652,475522,475523,475539],{},[31655,475524,475525,475531,475533],{},[31743,475526,475527,475529],{},[31658,475528,31955],{},[31749,475530,768],{},[31662,475532,223],{},[31743,475534,475535,475537],{},[31658,475536,31968],{},[31749,475538,768],{},[31665,475540,475541],{"encoding":31667},"i_2 + j_2",[34,475543,475545,475600],{"className":475544,"ariaHidden":758},[31672],[34,475546,475548,475551,475591,475594,475597],{"className":475547},[31676],[34,475549],{"className":475550,"style":475415},[31680],[34,475552,475554,475557],{"className":475553},[31685],[34,475555,31955],{"className":475556},[31685,31686],[34,475558,475560],{"className":475559},[31792],[34,475561,475563,475583],{"className":475562},[31796,31797],[34,475564,475566,475580],{"className":475565},[31801],[34,475567,475569],{"className":475568,"style":31806},[31805],[34,475570,475571,475574],{"style":31809},[34,475572],{"className":475573,"style":31814},[31813],[34,475575,475577],{"className":475576},[31818,31819,31820,31821],[34,475578,768],{"className":475579},[31685,31821],[34,475581,31829],{"className":475582},[31828],[34,475584,475586],{"className":475585},[31801],[34,475587,475589],{"className":475588,"style":31836},[31805],[34,475590],{},[34,475592],{"className":475593,"style":32604},[31845],[34,475595,223],{"className":475596},[32024],[34,475598],{"className":475599,"style":32604},[31845],[34,475601,475603,475606],{"className":475602},[31676],[34,475604],{"className":475605,"style":33123},[31680],[34,475607,475609,475612],{"className":475608},[31685],[34,475610,31968],{"className":475611,"style":32153},[31685,31686],[34,475613,475615],{"className":475614},[31792],[34,475616,475618,475638],{"className":475617},[31796,31797],[34,475619,475621,475635],{"className":475620},[31801],[34,475622,475624],{"className":475623,"style":31806},[31805],[34,475625,475626,475629],{"style":475491},[34,475627],{"className":475628,"style":31814},[31813],[34,475630,475632],{"className":475631},[31818,31819,31820,31821],[34,475633,768],{"className":475634},[31685,31821],[34,475636,31829],{"className":475637},[31828],[34,475639,475641],{"className":475640},[31801],[34,475642,475644],{"className":475643,"style":31836},[31805],[34,475645],{},"... 我们把这些式子乘起来看看会发生什么。",[15,475648,475649,315440,475651,475679,475680,475708,475709,475798,475799,475913,475914,258,476046,364729,476138,476293,476294,476322,476323],{},[2627,475650,463779],{},[34,475652,475654,475667],{"className":475653},[31642],[34,475655,475657],{"className":475656},[31646],[31648,475658,475659],{"xmlns":31650},[31652,475660,475661,475665],{},[31655,475662,475663],{},[31658,475664,34386],{},[31665,475666,34386],{"encoding":31667},[34,475668,475670],{"className":475669,"ariaHidden":758},[31672],[34,475671,475673,475676],{"className":475672},[31676],[34,475674],{"className":475675,"style":33218},[31680],[34,475677,34386],{"className":475678,"style":34427},[31685,31686]," 有单位元 ",[34,475681,475683,475696],{"className":475682},[31642],[34,475684,475686],{"className":475685},[31646],[31648,475687,475688],{"xmlns":31650},[31652,475689,475690,475694],{},[31655,475691,475692],{},[31749,475693,640],{},[31665,475695,640],{"encoding":31667},[34,475697,475699],{"className":475698,"ariaHidden":758},[31672],[34,475700,475702,475705],{"className":475701},[31676],[34,475703],{"className":475704,"style":32617},[31680],[34,475706,640],{"className":475707},[31685],"，且对于每个 ",[34,475710,475712,475741],{"className":475711},[31642],[34,475713,475715],{"className":475714},[31646],[31648,475716,475717],{"xmlns":31650},[31652,475718,475719,475739],{},[31655,475720,475721,475723,475725,475727,475729,475731,475733,475735,475737],{},[31658,475722,33759],{},[31662,475724,112296],{},[31662,475726,32685],{"stretchy":702},[31749,475728,640],{},[31662,475730,31960],{"separator":758},[31662,475732,33934],{},[31662,475734,31960],{"separator":758},[31658,475736,31660],{},[31662,475738,32715],{"stretchy":702},[31665,475740,351312],{"encoding":31667},[34,475742,475744,475762],{"className":475743,"ariaHidden":758},[31672],[34,475745,475747,475750,475753,475756,475759],{"className":475746},[31676],[34,475748],{"className":475749,"style":46991},[31680],[34,475751,33759],{"className":475752,"style":33774},[31685,31686],[34,475754],{"className":475755,"style":31846},[31845],[34,475757,112296],{"className":475758},[31850],[34,475760],{"className":475761,"style":31846},[31845],[34,475763,475765,475768,475771,475774,475777,475780,475783,475786,475789,475792,475795],{"className":475764},[31676],[34,475766],{"className":475767,"style":31774},[31680],[34,475769,32685],{"className":475770},[31782],[34,475772,640],{"className":475773},[31685],[34,475775,31960],{"className":475776},[32119],[34,475778],{"className":475779,"style":32123},[31845],[34,475781,33934],{"className":475782},[34006],[34,475784],{"className":475785,"style":32123},[31845],[34,475787,31960],{"className":475788},[32119],[34,475790],{"className":475791,"style":32123},[31845],[34,475793,31660],{"className":475794},[31685,31686],[34,475796,32715],{"className":475797},[31690]," 都有 ",[34,475800,475802,475828],{"className":475801},[31642],[34,475803,475805],{"className":475804},[31646],[31648,475806,475807],{"xmlns":31650},[31652,475808,475809,475825],{},[31655,475810,475811,475817,475819,475821,475823],{},[31743,475812,475813,475815],{},[31658,475814,106063],{},[31658,475816,33759],{},[31662,475818,223],{},[31658,475820,223320],{},[31662,475822,202],{},[31658,475824,34386],{},[31665,475826,475827],{"encoding":31667},"I_k + J = R",[34,475829,475831,475886,475904],{"className":475830,"ariaHidden":758},[31672],[34,475832,475834,475837,475877,475880,475883],{"className":475833},[31676],[34,475835],{"className":475836,"style":70915},[31680],[34,475838,475840,475843],{"className":475839},[31685],[34,475841,106063],{"className":475842,"style":67482},[31685,31686],[34,475844,475846],{"className":475845},[31792],[34,475847,475849,475869],{"className":475848},[31796,31797],[34,475850,475852,475866],{"className":475851},[31801],[34,475853,475855],{"className":475854,"style":33833},[31805],[34,475856,475857,475860],{"style":217742},[34,475858],{"className":475859,"style":31814},[31813],[34,475861,475863],{"className":475862},[31818,31819,31820,31821],[34,475864,33759],{"className":475865,"style":33774},[31685,31686,31821],[34,475867,31829],{"className":475868},[31828],[34,475870,475872],{"className":475871},[31801],[34,475873,475875],{"className":475874,"style":31836},[31805],[34,475876],{},[34,475878],{"className":475879,"style":32604},[31845],[34,475881,223],{"className":475882},[32024],[34,475884],{"className":475885,"style":32604},[31845],[34,475887,475889,475892,475895,475898,475901],{"className":475888},[31676],[34,475890],{"className":475891,"style":33218},[31680],[34,475893,223320],{"className":475894,"style":223361},[31685,31686],[34,475896],{"className":475897,"style":31846},[31845],[34,475899,202],{"className":475900},[31850],[34,475902],{"className":475903,"style":31846},[31845],[34,475905,475907,475910],{"className":475906},[31676],[34,475908],{"className":475909,"style":33218},[31680],[34,475911,34386],{"className":475912,"style":34427},[31685,31686],"。\n这意味着存在 ",[34,475915,475917,475942],{"className":475916},[31642],[34,475918,475920],{"className":475919},[31646],[31648,475921,475922],{"xmlns":31650},[31652,475923,475924,475940],{},[31655,475925,475926,475932,475934],{},[31743,475927,475928,475930],{},[31658,475929,32936],{},[31658,475931,33759],{},[31662,475933,112296],{},[31743,475935,475936,475938],{},[31658,475937,106063],{},[31658,475939,33759],{},[31665,475941,347919],{"encoding":31667},[34,475943,475945,476000],{"className":475944,"ariaHidden":758},[31672],[34,475946,475948,475951,475991,475994,475997],{"className":475947},[31676],[34,475949],{"className":475950,"style":225773},[31680],[34,475952,475954,475957],{"className":475953},[31685],[34,475955,32936],{"className":475956},[31685,31686],[34,475958,475960],{"className":475959},[31792],[34,475961,475963,475983],{"className":475962},[31796,31797],[34,475964,475966,475980],{"className":475965},[31801],[34,475967,475969],{"className":475968,"style":33833},[31805],[34,475970,475971,475974],{"style":31809},[34,475972],{"className":475973,"style":31814},[31813],[34,475975,475977],{"className":475976},[31818,31819,31820,31821],[34,475978,33759],{"className":475979,"style":33774},[31685,31686,31821],[34,475981,31829],{"className":475982},[31828],[34,475984,475986],{"className":475985},[31801],[34,475987,475989],{"className":475988,"style":31836},[31805],[34,475990],{},[34,475992],{"className":475993,"style":31846},[31845],[34,475995,112296],{"className":475996},[31850],[34,475998],{"className":475999,"style":31846},[31845],[34,476001,476003,476006],{"className":476002},[31676],[34,476004],{"className":476005,"style":70915},[31680],[34,476007,476009,476012],{"className":476008},[31685],[34,476010,106063],{"className":476011,"style":67482},[31685,31686],[34,476013,476015],{"className":476014},[31792],[34,476016,476018,476038],{"className":476017},[31796,31797],[34,476019,476021,476035],{"className":476020},[31801],[34,476022,476024],{"className":476023,"style":33833},[31805],[34,476025,476026,476029],{"style":217742},[34,476027],{"className":476028,"style":31814},[31813],[34,476030,476032],{"className":476031},[31818,31819,31820,31821],[34,476033,33759],{"className":476034,"style":33774},[31685,31686,31821],[34,476036,31829],{"className":476037},[31828],[34,476039,476041],{"className":476040},[31801],[34,476042,476044],{"className":476043,"style":31836},[31805],[34,476045],{},[34,476047,476049,476071],{"className":476048},[31642],[34,476050,476052],{"className":476051},[31646],[31648,476053,476054],{"xmlns":31650},[31652,476055,476056,476068],{},[31655,476057,476058,476064,476066],{},[31743,476059,476060,476062],{},[31658,476061,67764],{},[31658,476063,33759],{},[31662,476065,112296],{},[31658,476067,223320],{},[31665,476069,476070],{"encoding":31667},"y_k \\in J",[34,476072,476074,476129],{"className":476073,"ariaHidden":758},[31672],[34,476075,476077,476080,476120,476123,476126],{"className":476076},[31676],[34,476078],{"className":476079,"style":86652},[31680],[34,476081,476083,476086],{"className":476082},[31685],[34,476084,67764],{"className":476085,"style":32062},[31685,31686],[34,476087,476089],{"className":476088},[31792],[34,476090,476092,476112],{"className":476091},[31796,31797],[34,476093,476095,476109],{"className":476094},[31801],[34,476096,476098],{"className":476097,"style":33833},[31805],[34,476099,476100,476103],{"style":38795},[34,476101],{"className":476102,"style":31814},[31813],[34,476104,476106],{"className":476105},[31818,31819,31820,31821],[34,476107,33759],{"className":476108,"style":33774},[31685,31686,31821],[34,476110,31829],{"className":476111},[31828],[34,476113,476115],{"className":476114},[31801],[34,476116,476118],{"className":476117,"style":31836},[31805],[34,476119],{},[34,476121],{"className":476122,"style":31846},[31845],[34,476124,112296],{"className":476125},[31850],[34,476127],{"className":476128,"style":31846},[31845],[34,476130,476132,476135],{"className":476131},[31676],[34,476133],{"className":476134,"style":33218},[31680],[34,476136,223320],{"className":476137,"style":223361},[31685,31686],[34,476139,476141,476171],{"className":476140},[31642],[34,476142,476144],{"className":476143},[31646],[31648,476145,476146],{"xmlns":31650},[31652,476147,476148,476168],{},[31655,476149,476150,476156,476158,476164,476166],{},[31743,476151,476152,476154],{},[31658,476153,32936],{},[31658,476155,33759],{},[31662,476157,223],{},[31743,476159,476160,476162],{},[31658,476161,67764],{},[31658,476163,33759],{},[31662,476165,202],{},[31749,476167,640],{},[31665,476169,476170],{"encoding":31667},"x_k + y_k = 1",[34,476172,476174,476229,476284],{"className":476173,"ariaHidden":758},[31672],[34,476175,476177,476180,476220,476223,476226],{"className":476176},[31676],[34,476178],{"className":476179,"style":40522},[31680],[34,476181,476183,476186],{"className":476182},[31685],[34,476184,32936],{"className":476185},[31685,31686],[34,476187,476189],{"className":476188},[31792],[34,476190,476192,476212],{"className":476191},[31796,31797],[34,476193,476195,476209],{"className":476194},[31801],[34,476196,476198],{"className":476197,"style":33833},[31805],[34,476199,476200,476203],{"style":31809},[34,476201],{"className":476202,"style":31814},[31813],[34,476204,476206],{"className":476205},[31818,31819,31820,31821],[34,476207,33759],{"className":476208,"style":33774},[31685,31686,31821],[34,476210,31829],{"className":476211},[31828],[34,476213,476215],{"className":476214},[31801],[34,476216,476218],{"className":476217,"style":31836},[31805],[34,476219],{},[34,476221],{"className":476222,"style":32604},[31845],[34,476224,223],{"className":476225},[32024],[34,476227],{"className":476228,"style":32604},[31845],[34,476230,476232,476235,476275,476278,476281],{"className":476231},[31676],[34,476233],{"className":476234,"style":38774},[31680],[34,476236,476238,476241],{"className":476237},[31685],[34,476239,67764],{"className":476240,"style":32062},[31685,31686],[34,476242,476244],{"className":476243},[31792],[34,476245,476247,476267],{"className":476246},[31796,31797],[34,476248,476250,476264],{"className":476249},[31801],[34,476251,476253],{"className":476252,"style":33833},[31805],[34,476254,476255,476258],{"style":38795},[34,476256],{"className":476257,"style":31814},[31813],[34,476259,476261],{"className":476260},[31818,31819,31820,31821],[34,476262,33759],{"className":476263,"style":33774},[31685,31686,31821],[34,476265,31829],{"className":476266},[31828],[34,476268,476270],{"className":476269},[31801],[34,476271,476273],{"className":476272,"style":31836},[31805],[34,476274],{},[34,476276],{"className":476277,"style":31846},[31845],[34,476279,202],{"className":476280},[31850],[34,476282],{"className":476283,"style":31846},[31845],[34,476285,476287,476290],{"className":476286},[31676],[34,476288],{"className":476289,"style":32617},[31680],[34,476291,640],{"className":476292},[31685],"\n现在，我们将这 ",[34,476295,476297,476310],{"className":476296},[31642],[34,476298,476300],{"className":476299},[31646],[31648,476301,476302],{"xmlns":31650},[31652,476303,476304,476308],{},[31655,476305,476306],{},[31658,476307,31660],{},[31665,476309,31660],{"encoding":31667},[34,476311,476313],{"className":476312,"ariaHidden":758},[31672],[34,476314,476316,476319],{"className":476315},[31676],[34,476317],{"className":476318,"style":32527},[31680],[34,476320,31660],{"className":476321},[31685,31686]," 个等式乘起来：\n",[34,476324,476326,476410],{"className":476325},[31642],[34,476327,476329],{"className":476328},[31646],[31648,476330,476331],{"xmlns":31650},[31652,476332,476333,476407],{},[31655,476334,476335,476337,476339,476341,476343,476345,476347,476349,476351,476353,476359,476361,476367,476369,476371,476377,476379,476385,476387,476389,476391,476397,476399,476405],{},[31749,476336,640],{},[31662,476338,202],{},[31749,476340,640],{},[31662,476342,71191],{},[31749,476344,640],{},[31662,476346,38874],{},[31749,476348,640],{},[31662,476350,202],{},[31662,476352,456],{"stretchy":702},[31743,476354,476355,476357],{},[31658,476356,32936],{},[31749,476358,640],{},[31662,476360,223],{},[31743,476362,476363,476365],{},[31658,476364,67764],{},[31749,476366,640],{},[31662,476368,2557],{"stretchy":702},[31662,476370,456],{"stretchy":702},[31743,476372,476373,476375],{},[31658,476374,32936],{},[31749,476376,768],{},[31662,476378,223],{},[31743,476380,476381,476383],{},[31658,476382,67764],{},[31749,476384,768],{},[31662,476386,2557],{"stretchy":702},[31662,476388,38874],{},[31662,476390,456],{"stretchy":702},[31743,476392,476393,476395],{},[31658,476394,32936],{},[31658,476396,31660],{},[31662,476398,223],{},[31743,476400,476401,476403],{},[31658,476402,67764],{},[31658,476404,31660],{},[31662,476406,2557],{"stretchy":702},[31665,476408,476409],{"encoding":31667},"1 = 1 \\cdot 1 \\cdots 1 = (x_1 + y_1)(x_2 + y_2)\\cdots(x_n + y_n)",[34,476411,476413,476431,476449,476479,476537,476638,476748],{"className":476412,"ariaHidden":758},[31672],[34,476414,476416,476419,476422,476425,476428],{"className":476415},[31676],[34,476417],{"className":476418,"style":32617},[31680],[34,476420,640],{"className":476421},[31685],[34,476423],{"className":476424,"style":31846},[31845],[34,476426,202],{"className":476427},[31850],[34,476429],{"className":476430,"style":31846},[31845],[34,476432,476434,476437,476440,476443,476446],{"className":476433},[31676],[34,476435],{"className":476436,"style":32617},[31680],[34,476438,640],{"className":476439},[31685],[34,476441],{"className":476442,"style":32604},[31845],[34,476444,71191],{"className":476445},[32024],[34,476447],{"className":476448,"style":32604},[31845],[34,476450,476452,476455,476458,476461,476464,476467,476470,476473,476476],{"className":476451},[31676],[34,476453],{"className":476454,"style":32617},[31680],[34,476456,640],{"className":476457},[31685],[34,476459],{"className":476460,"style":32123},[31845],[34,476462,38874],{"className":476463},[34006],[34,476465],{"className":476466,"style":32123},[31845],[34,476468,640],{"className":476469},[31685],[34,476471],{"className":476472,"style":31846},[31845],[34,476474,202],{"className":476475},[31850],[34,476477],{"className":476478,"style":31846},[31845],[34,476480,476482,476485,476488,476528,476531,476534],{"className":476481},[31676],[34,476483],{"className":476484,"style":31774},[31680],[34,476486,456],{"className":476487},[31782],[34,476489,476491,476494],{"className":476490},[31685],[34,476492,32936],{"className":476493},[31685,31686],[34,476495,476497],{"className":476496},[31792],[34,476498,476500,476520],{"className":476499},[31796,31797],[34,476501,476503,476517],{"className":476502},[31801],[34,476504,476506],{"className":476505,"style":31806},[31805],[34,476507,476508,476511],{"style":31809},[34,476509],{"className":476510,"style":31814},[31813],[34,476512,476514],{"className":476513},[31818,31819,31820,31821],[34,476515,640],{"className":476516},[31685,31821],[34,476518,31829],{"className":476519},[31828],[34,476521,476523],{"className":476522},[31801],[34,476524,476526],{"className":476525,"style":31836},[31805],[34,476527],{},[34,476529],{"className":476530,"style":32604},[31845],[34,476532,223],{"className":476533},[32024],[34,476535],{"className":476536,"style":32604},[31845],[34,476538,476540,476543,476583,476586,476589,476629,476632,476635],{"className":476539},[31676],[34,476541],{"className":476542,"style":31774},[31680],[34,476544,476546,476549],{"className":476545},[31685],[34,476547,67764],{"className":476548,"style":32062},[31685,31686],[34,476550,476552],{"className":476551},[31792],[34,476553,476555,476575],{"className":476554},[31796,31797],[34,476556,476558,476572],{"className":476557},[31801],[34,476559,476561],{"className":476560,"style":31806},[31805],[34,476562,476563,476566],{"style":38795},[34,476564],{"className":476565,"style":31814},[31813],[34,476567,476569],{"className":476568},[31818,31819,31820,31821],[34,476570,640],{"className":476571},[31685,31821],[34,476573,31829],{"className":476574},[31828],[34,476576,476578],{"className":476577},[31801],[34,476579,476581],{"className":476580,"style":31836},[31805],[34,476582],{},[34,476584,2557],{"className":476585},[31690],[34,476587,456],{"className":476588},[31782],[34,476590,476592,476595],{"className":476591},[31685],[34,476593,32936],{"className":476594},[31685,31686],[34,476596,476598],{"className":476597},[31792],[34,476599,476601,476621],{"className":476600},[31796,31797],[34,476602,476604,476618],{"className":476603},[31801],[34,476605,476607],{"className":476606,"style":31806},[31805],[34,476608,476609,476612],{"style":31809},[34,476610],{"className":476611,"style":31814},[31813],[34,476613,476615],{"className":476614},[31818,31819,31820,31821],[34,476616,768],{"className":476617},[31685,31821],[34,476619,31829],{"className":476620},[31828],[34,476622,476624],{"className":476623},[31801],[34,476625,476627],{"className":476626,"style":31836},[31805],[34,476628],{},[34,476630],{"className":476631,"style":32604},[31845],[34,476633,223],{"className":476634},[32024],[34,476636],{"className":476637,"style":32604},[31845],[34,476639,476641,476644,476684,476687,476690,476693,476696,476699,476739,476742,476745],{"className":476640},[31676],[34,476642],{"className":476643,"style":31774},[31680],[34,476645,476647,476650],{"className":476646},[31685],[34,476648,67764],{"className":476649,"style":32062},[31685,31686],[34,476651,476653],{"className":476652},[31792],[34,476654,476656,476676],{"className":476655},[31796,31797],[34,476657,476659,476673],{"className":476658},[31801],[34,476660,476662],{"className":476661,"style":31806},[31805],[34,476663,476664,476667],{"style":38795},[34,476665],{"className":476666,"style":31814},[31813],[34,476668,476670],{"className":476669},[31818,31819,31820,31821],[34,476671,768],{"className":476672},[31685,31821],[34,476674,31829],{"className":476675},[31828],[34,476677,476679],{"className":476678},[31801],[34,476680,476682],{"className":476681,"style":31836},[31805],[34,476683],{},[34,476685,2557],{"className":476686},[31690],[34,476688],{"className":476689,"style":32123},[31845],[34,476691,38874],{"className":476692},[34006],[34,476694],{"className":476695,"style":32123},[31845],[34,476697,456],{"className":476698},[31782],[34,476700,476702,476705],{"className":476701},[31685],[34,476703,32936],{"className":476704},[31685,31686],[34,476706,476708],{"className":476707},[31792],[34,476709,476711,476731],{"className":476710},[31796,31797],[34,476712,476714,476728],{"className":476713},[31801],[34,476715,476717],{"className":476716,"style":32362},[31805],[34,476718,476719,476722],{"style":31809},[34,476720],{"className":476721,"style":31814},[31813],[34,476723,476725],{"className":476724},[31818,31819,31820,31821],[34,476726,31660],{"className":476727},[31685,31686,31821],[34,476729,31829],{"className":476730},[31828],[34,476732,476734],{"className":476733},[31801],[34,476735,476737],{"className":476736,"style":31836},[31805],[34,476738],{},[34,476740],{"className":476741,"style":32604},[31845],[34,476743,223],{"className":476744},[32024],[34,476746],{"className":476747,"style":32604},[31845],[34,476749,476751,476754,476794],{"className":476750},[31676],[34,476752],{"className":476753,"style":31774},[31680],[34,476755,476757,476760],{"className":476756},[31685],[34,476758,67764],{"className":476759,"style":32062},[31685,31686],[34,476761,476763],{"className":476762},[31792],[34,476764,476766,476786],{"className":476765},[31796,31797],[34,476767,476769,476783],{"className":476768},[31801],[34,476770,476772],{"className":476771,"style":32362},[31805],[34,476773,476774,476777],{"style":38795},[34,476775],{"className":476776,"style":31814},[31813],[34,476778,476780],{"className":476779},[31818,31819,31820,31821],[34,476781,31660],{"className":476782},[31685,31686,31821],[34,476784,31829],{"className":476785},[31828],[34,476787,476789],{"className":476788},[31801],[34,476790,476792],{"className":476791,"style":31836},[31805],[34,476793],{},[34,476795,2557],{"className":476796},[31690],[15,476798,476799],{},"观察右边的展开式：",[105,476801,476802,477011,477360,477438],{},[108,476803,476804,476805,476808,476809,476837,476838,347],{},"展开后只有",[2627,476806,476807],{},"一项","是不含任何 ",[34,476810,476812,476825],{"className":476811},[31642],[34,476813,476815],{"className":476814},[31646],[31648,476816,476817],{"xmlns":31650},[31652,476818,476819,476823],{},[31655,476820,476821],{},[31658,476822,67764],{},[31665,476824,67764],{"encoding":31667},[34,476826,476828],{"className":476827,"ariaHidden":758},[31672],[34,476829,476831,476834],{"className":476830},[31676],[34,476832],{"className":476833,"style":38774},[31680],[34,476835,67764],{"className":476836,"style":32062},[31685,31686]," 的，那就是 ",[34,476839,476841,476873],{"className":476840},[31642],[34,476842,476844],{"className":476843},[31646],[31648,476845,476846],{"xmlns":31650},[31652,476847,476848,476870],{},[31655,476849,476850,476856,476862,476864],{},[31743,476851,476852,476854],{},[31658,476853,32936],{},[31749,476855,640],{},[31743,476857,476858,476860],{},[31658,476859,32936],{},[31749,476861,768],{},[31662,476863,38874],{},[31743,476865,476866,476868],{},[31658,476867,32936],{},[31658,476869,31660],{},[31665,476871,476872],{"encoding":31667},"x_1 x_2 \\cdots x_n",[34,476874,476876],{"className":476875,"ariaHidden":758},[31672],[34,476877,476879,476882,476922,476962,476965,476968,476971],{"className":476878},[31676],[34,476880],{"className":476881,"style":31860},[31680],[34,476883,476885,476888],{"className":476884},[31685],[34,476886,32936],{"className":476887},[31685,31686],[34,476889,476891],{"className":476890},[31792],[34,476892,476894,476914],{"className":476893},[31796,31797],[34,476895,476897,476911],{"className":476896},[31801],[34,476898,476900],{"className":476899,"style":31806},[31805],[34,476901,476902,476905],{"style":31809},[34,476903],{"className":476904,"style":31814},[31813],[34,476906,476908],{"className":476907},[31818,31819,31820,31821],[34,476909,640],{"className":476910},[31685,31821],[34,476912,31829],{"className":476913},[31828],[34,476915,476917],{"className":476916},[31801],[34,476918,476920],{"className":476919,"style":31836},[31805],[34,476921],{},[34,476923,476925,476928],{"className":476924},[31685],[34,476926,32936],{"className":476927},[31685,31686],[34,476929,476931],{"className":476930},[31792],[34,476932,476934,476954],{"className":476933},[31796,31797],[34,476935,476937,476951],{"className":476936},[31801],[34,476938,476940],{"className":476939,"style":31806},[31805],[34,476941,476942,476945],{"style":31809},[34,476943],{"className":476944,"style":31814},[31813],[34,476946,476948],{"className":476947},[31818,31819,31820,31821],[34,476949,768],{"className":476950},[31685,31821],[34,476952,31829],{"className":476953},[31828],[34,476955,476957],{"className":476956},[31801],[34,476958,476960],{"className":476959,"style":31836},[31805],[34,476961],{},[34,476963],{"className":476964,"style":32123},[31845],[34,476966,38874],{"className":476967},[34006],[34,476969],{"className":476970,"style":32123},[31845],[34,476972,476974,476977],{"className":476973},[31685],[34,476975,32936],{"className":476976},[31685,31686],[34,476978,476980],{"className":476979},[31792],[34,476981,476983,477003],{"className":476982},[31796,31797],[34,476984,476986,477000],{"className":476985},[31801],[34,476987,476989],{"className":476988,"style":32362},[31805],[34,476990,476991,476994],{"style":31809},[34,476992],{"className":476993,"style":31814},[31813],[34,476995,476997],{"className":476996},[31818,31819,31820,31821],[34,476998,31660],{"className":476999},[31685,31686,31821],[34,477001,31829],{"className":477002},[31828],[34,477004,477006],{"className":477005},[31801],[34,477007,477009],{"className":477008,"style":31836},[31805],[34,477010],{},[108,477012,477013,477014,477186,477187,347],{},"这一项 ",[34,477015,477017,477048],{"className":477016},[31642],[34,477018,477020],{"className":477019},[31646],[31648,477021,477022],{"xmlns":31650},[31652,477023,477024,477046],{},[31655,477025,477026,477032,477038,477040],{},[31743,477027,477028,477030],{},[31658,477029,32936],{},[31749,477031,640],{},[31743,477033,477034,477036],{},[31658,477035,32936],{},[31749,477037,768],{},[31662,477039,38874],{},[31743,477041,477042,477044],{},[31658,477043,32936],{},[31658,477045,31660],{},[31665,477047,476872],{"encoding":31667},[34,477049,477051],{"className":477050,"ariaHidden":758},[31672],[34,477052,477054,477057,477097,477137,477140,477143,477146],{"className":477053},[31676],[34,477055],{"className":477056,"style":31860},[31680],[34,477058,477060,477063],{"className":477059},[31685],[34,477061,32936],{"className":477062},[31685,31686],[34,477064,477066],{"className":477065},[31792],[34,477067,477069,477089],{"className":477068},[31796,31797],[34,477070,477072,477086],{"className":477071},[31801],[34,477073,477075],{"className":477074,"style":31806},[31805],[34,477076,477077,477080],{"style":31809},[34,477078],{"className":477079,"style":31814},[31813],[34,477081,477083],{"className":477082},[31818,31819,31820,31821],[34,477084,640],{"className":477085},[31685,31821],[34,477087,31829],{"className":477088},[31828],[34,477090,477092],{"className":477091},[31801],[34,477093,477095],{"className":477094,"style":31836},[31805],[34,477096],{},[34,477098,477100,477103],{"className":477099},[31685],[34,477101,32936],{"className":477102},[31685,31686],[34,477104,477106],{"className":477105},[31792],[34,477107,477109,477129],{"className":477108},[31796,31797],[34,477110,477112,477126],{"className":477111},[31801],[34,477113,477115],{"className":477114,"style":31806},[31805],[34,477116,477117,477120],{"style":31809},[34,477118],{"className":477119,"style":31814},[31813],[34,477121,477123],{"className":477122},[31818,31819,31820,31821],[34,477124,768],{"className":477125},[31685,31821],[34,477127,31829],{"className":477128},[31828],[34,477130,477132],{"className":477131},[31801],[34,477133,477135],{"className":477134,"style":31836},[31805],[34,477136],{},[34,477138],{"className":477139,"style":32123},[31845],[34,477141,38874],{"className":477142},[34006],[34,477144],{"className":477145,"style":32123},[31845],[34,477147,477149,477152],{"className":477148},[31685],[34,477150,32936],{"className":477151},[31685,31686],[34,477153,477155],{"className":477154},[31792],[34,477156,477158,477178],{"className":477157},[31796,31797],[34,477159,477161,477175],{"className":477160},[31801],[34,477162,477164],{"className":477163,"style":32362},[31805],[34,477165,477166,477169],{"style":31809},[34,477167],{"className":477168,"style":31814},[31813],[34,477170,477172],{"className":477171},[31818,31819,31820,31821],[34,477173,31660],{"className":477174},[31685,31686,31821],[34,477176,31829],{"className":477177},[31828],[34,477179,477181],{"className":477180},[31801],[34,477182,477184],{"className":477183,"style":31836},[31805],[34,477185],{}," 显然属于积理想 ",[34,477188,477190,477222],{"className":477189},[31642],[34,477191,477193],{"className":477192},[31646],[31648,477194,477195],{"xmlns":31650},[31652,477196,477197,477219],{},[31655,477198,477199,477205,477211,477213],{},[31743,477200,477201,477203],{},[31658,477202,106063],{},[31749,477204,640],{},[31743,477206,477207,477209],{},[31658,477208,106063],{},[31749,477210,768],{},[31662,477212,38874],{},[31743,477214,477215,477217],{},[31658,477216,106063],{},[31658,477218,31660],{},[31665,477220,477221],{"encoding":31667},"I_1 I_2 \\cdots I_n",[34,477223,477225],{"className":477224,"ariaHidden":758},[31672],[34,477226,477228,477231,477271,477311,477314,477317,477320],{"className":477227},[31676],[34,477229],{"className":477230,"style":70915},[31680],[34,477232,477234,477237],{"className":477233},[31685],[34,477235,106063],{"className":477236,"style":67482},[31685,31686],[34,477238,477240],{"className":477239},[31792],[34,477241,477243,477263],{"className":477242},[31796,31797],[34,477244,477246,477260],{"className":477245},[31801],[34,477247,477249],{"className":477248,"style":31806},[31805],[34,477250,477251,477254],{"style":217742},[34,477252],{"className":477253,"style":31814},[31813],[34,477255,477257],{"className":477256},[31818,31819,31820,31821],[34,477258,640],{"className":477259},[31685,31821],[34,477261,31829],{"className":477262},[31828],[34,477264,477266],{"className":477265},[31801],[34,477267,477269],{"className":477268,"style":31836},[31805],[34,477270],{},[34,477272,477274,477277],{"className":477273},[31685],[34,477275,106063],{"className":477276,"style":67482},[31685,31686],[34,477278,477280],{"className":477279},[31792],[34,477281,477283,477303],{"className":477282},[31796,31797],[34,477284,477286,477300],{"className":477285},[31801],[34,477287,477289],{"className":477288,"style":31806},[31805],[34,477290,477291,477294],{"style":217742},[34,477292],{"className":477293,"style":31814},[31813],[34,477295,477297],{"className":477296},[31818,31819,31820,31821],[34,477298,768],{"className":477299},[31685,31821],[34,477301,31829],{"className":477302},[31828],[34,477304,477306],{"className":477305},[31801],[34,477307,477309],{"className":477308,"style":31836},[31805],[34,477310],{},[34,477312],{"className":477313,"style":32123},[31845],[34,477315,38874],{"className":477316},[34006],[34,477318],{"className":477319,"style":32123},[31845],[34,477321,477323,477326],{"className":477322},[31685],[34,477324,106063],{"className":477325,"style":67482},[31685,31686],[34,477327,477329],{"className":477328},[31792],[34,477330,477332,477352],{"className":477331},[31796,31797],[34,477333,477335,477349],{"className":477334},[31801],[34,477336,477338],{"className":477337,"style":32362},[31805],[34,477339,477340,477343],{"style":217742},[34,477341],{"className":477342,"style":31814},[31813],[34,477344,477346],{"className":477345},[31818,31819,31820,31821],[34,477347,31660],{"className":477348},[31685,31686,31821],[34,477350,31829],{"className":477351},[31828],[34,477353,477355],{"className":477354},[31801],[34,477356,477358],{"className":477357,"style":31836},[31805],[34,477359],{},[108,477361,477362,477363,477366,477367,477437],{},"展开后的",[2627,477364,477365],{},"其余所有项","，每一项至少包含一个 ",[34,477368,477370,477388],{"className":477369},[31642],[34,477371,477373],{"className":477372},[31646],[31648,477374,477375],{"xmlns":31650},[31652,477376,477377,477385],{},[31655,477378,477379],{},[31743,477380,477381,477383],{},[31658,477382,67764],{},[31658,477384,33759],{},[31665,477386,477387],{"encoding":31667},"y_k",[34,477389,477391],{"className":477390,"ariaHidden":758},[31672],[34,477392,477394,477397],{"className":477393},[31676],[34,477395],{"className":477396,"style":38774},[31680],[34,477398,477400,477403],{"className":477399},[31685],[34,477401,67764],{"className":477402,"style":32062},[31685,31686],[34,477404,477406],{"className":477405},[31792],[34,477407,477409,477429],{"className":477408},[31796,31797],[34,477410,477412,477426],{"className":477411},[31801],[34,477413,477415],{"className":477414,"style":33833},[31805],[34,477416,477417,477420],{"style":38795},[34,477418],{"className":477419,"style":31814},[31813],[34,477421,477423],{"className":477422},[31818,31819,31820,31821],[34,477424,33759],{"className":477425,"style":33774},[31685,31686,31821],[34,477427,31829],{"className":477428},[31828],[34,477430,477432],{"className":477431},[31801],[34,477433,477435],{"className":477434,"style":31836},[31805],[34,477436],{}," 作为因子。",[108,477439,93310,477440,86674,477531,477559,477560,477629,477630,477658,477659,50492],{},[34,477441,477443,477464],{"className":477442},[31642],[34,477444,477446],{"className":477445},[31646],[31648,477447,477448],{"xmlns":31650},[31652,477449,477450,477462],{},[31655,477451,477452,477458,477460],{},[31743,477453,477454,477456],{},[31658,477455,67764],{},[31658,477457,33759],{},[31662,477459,112296],{},[31658,477461,223320],{},[31665,477463,476070],{"encoding":31667},[34,477465,477467,477522],{"className":477466,"ariaHidden":758},[31672],[34,477468,477470,477473,477513,477516,477519],{"className":477469},[31676],[34,477471],{"className":477472,"style":86652},[31680],[34,477474,477476,477479],{"className":477475},[31685],[34,477477,67764],{"className":477478,"style":32062},[31685,31686],[34,477480,477482],{"className":477481},[31792],[34,477483,477485,477505],{"className":477484},[31796,31797],[34,477486,477488,477502],{"className":477487},[31801],[34,477489,477491],{"className":477490,"style":33833},[31805],[34,477492,477493,477496],{"style":38795},[34,477494],{"className":477495,"style":31814},[31813],[34,477497,477499],{"className":477498},[31818,31819,31820,31821],[34,477500,33759],{"className":477501,"style":33774},[31685,31686,31821],[34,477503,31829],{"className":477504},[31828],[34,477506,477508],{"className":477507},[31801],[34,477509,477511],{"className":477510,"style":31836},[31805],[34,477512],{},[34,477514],{"className":477515,"style":31846},[31845],[34,477517,112296],{"className":477518},[31850],[34,477520],{"className":477521,"style":31846},[31845],[34,477523,477525,477528],{"className":477524},[31676],[34,477526],{"className":477527,"style":33218},[31680],[34,477529,223320],{"className":477530,"style":223361},[31685,31686],[34,477532,477534,477547],{"className":477533},[31642],[34,477535,477537],{"className":477536},[31646],[31648,477538,477539],{"xmlns":31650},[31652,477540,477541,477545],{},[31655,477542,477543],{},[31658,477544,223320],{},[31665,477546,223320],{"encoding":31667},[34,477548,477550],{"className":477549,"ariaHidden":758},[31672],[34,477551,477553,477556],{"className":477552},[31676],[34,477554],{"className":477555,"style":33218},[31680],[34,477557,223320],{"className":477558,"style":223361},[31685,31686]," 是理想（吸收律），所以任何包含 ",[34,477561,477563,477580],{"className":477562},[31642],[34,477564,477566],{"className":477565},[31646],[31648,477567,477568],{"xmlns":31650},[31652,477569,477570,477578],{},[31655,477571,477572],{},[31743,477573,477574,477576],{},[31658,477575,67764],{},[31658,477577,33759],{},[31665,477579,477387],{"encoding":31667},[34,477581,477583],{"className":477582,"ariaHidden":758},[31672],[34,477584,477586,477589],{"className":477585},[31676],[34,477587],{"className":477588,"style":38774},[31680],[34,477590,477592,477595],{"className":477591},[31685],[34,477593,67764],{"className":477594,"style":32062},[31685,31686],[34,477596,477598],{"className":477597},[31792],[34,477599,477601,477621],{"className":477600},[31796,31797],[34,477602,477604,477618],{"className":477603},[31801],[34,477605,477607],{"className":477606,"style":33833},[31805],[34,477608,477609,477612],{"style":38795},[34,477610],{"className":477611,"style":31814},[31813],[34,477613,477615],{"className":477614},[31818,31819,31820,31821],[34,477616,33759],{"className":477617,"style":33774},[31685,31686,31821],[34,477619,31829],{"className":477620},[31828],[34,477622,477624],{"className":477623},[31801],[34,477625,477627],{"className":477626,"style":31836},[31805],[34,477628],{}," 的项都在 ",[34,477631,477633,477646],{"className":477632},[31642],[34,477634,477636],{"className":477635},[31646],[31648,477637,477638],{"xmlns":31650},[31652,477639,477640,477644],{},[31655,477641,477642],{},[31658,477643,223320],{},[31665,477645,223320],{"encoding":31667},[34,477647,477649],{"className":477648,"ariaHidden":758},[31672],[34,477650,477652,477655],{"className":477651},[31676],[34,477653],{"className":477654,"style":33218},[31680],[34,477656,223320],{"className":477657,"style":223361},[31685,31686]," 中。所有这些项的和也在 ",[34,477660,477662,477675],{"className":477661},[31642],[34,477663,477665],{"className":477664},[31646],[31648,477666,477667],{"xmlns":31650},[31652,477668,477669,477673],{},[31655,477670,477671],{},[31658,477672,223320],{},[31665,477674,223320],{"encoding":31667},[34,477676,477678],{"className":477677,"ariaHidden":758},[31672],[34,477679,477681,477684],{"className":477680},[31676],[34,477682],{"className":477683,"style":33218},[31680],[34,477685,223320],{"className":477686,"style":223361},[31685,31686],[15,477688,477689,477690,478285,478286,478555,478556,478632,478633,70884,478706,478890,478891,479062,479063,479091,479092,398592,479120,479291,479292],{},"于是，我们可以把展开式写成：\n",[34,477691,477693,477781],{"className":477692},[31642],[34,477694,477696],{"className":477695},[31646],[31648,477697,477698],{"xmlns":31650},[31652,477699,477700,477778],{},[31655,477701,477702,477704,477706,477708,477755,477757,477759,477761,477776],{},[31749,477703,640],{},[31662,477705,202],{},[31662,477707,456],{"stretchy":702},[71115,477709,477710,477737],{},[71115,477711,477712,477734],{},[31655,477713,477714,477720,477726,477728],{},[31743,477715,477716,477718],{},[31658,477717,32936],{},[31749,477719,640],{},[31743,477721,477722,477724],{},[31658,477723,32936],{},[31749,477725,768],{},[31662,477727,38874],{},[31743,477729,477730,477732],{},[31658,477731,32936],{},[31658,477733,31660],{},[31662,477735,477736],{"stretchy":758},"⏟",[31655,477738,477739,477741,477747,477749],{},[31662,477740,112296],{},[31743,477742,477743,477745],{},[31658,477744,106063],{},[31749,477746,640],{},[31662,477748,38874],{},[31743,477750,477751,477753],{},[31658,477752,106063],{},[31658,477754,31660],{},[31662,477756,2557],{"stretchy":702},[31662,477758,223],{},[31662,477760,456],{"stretchy":702},[71115,477762,477763,477770],{},[71115,477764,477765,477768],{},[39361,477766,477767],{},"其余项之和",[31662,477769,477736],{"stretchy":758},[31655,477771,477772,477774],{},[31662,477773,112296],{},[31658,477775,223320],{},[31662,477777,2557],{"stretchy":702},[31665,477779,477780],{"encoding":31667},"1 = (\\underbrace{x_1 x_2 \\cdots x_n}_{\\in I_1 \\cdots I_n}) + (\\underbrace{\\text{其余项之和}}_{\\in J})",[34,477782,477784,477802,478156],{"className":477783,"ariaHidden":758},[31672],[34,477785,477787,477790,477793,477796,477799],{"className":477786},[31676],[34,477788],{"className":477789,"style":32617},[31680],[34,477791,640],{"className":477792},[31685],[34,477794],{"className":477795,"style":31846},[31845],[34,477797,202],{"className":477798},[31850],[34,477800],{"className":477801,"style":31846},[31845],[34,477803,477805,477809,477812,478144,478147,478150,478153],{"className":477804},[31676],[34,477806],{"className":477807,"style":477808},[31680],"height:2.3264em;vertical-align:-1.5764em;",[34,477810,456],{"className":477811},[31782],[34,477813,477815],{"className":477814},[34006,71115],[34,477816,477818,478135],{"className":477817},[31796,31797],[34,477819,477821,478132],{"className":477820},[31801],[34,477822,477824,477923],{"className":477823,"style":32527},[31805],[34,477825,477827,477830],{"style":477826},"top:-1.5237em;",[34,477828],{"className":477829,"style":41508},[31813],[34,477831,477833],{"className":477832},[31818,31819,31820,31821],[34,477834,477836,477839,477880,477883],{"className":477835},[31685,31821],[34,477837,112296],{"className":477838},[31850,31821],[34,477840,477842,477845],{"className":477841},[31685,31821],[34,477843,106063],{"className":477844,"style":67482},[31685,31686,31821],[34,477846,477848],{"className":477847},[31792],[34,477849,477851,477872],{"className":477850},[31796,31797],[34,477852,477854,477869],{"className":477853},[31801],[34,477855,477857],{"className":477856,"style":318658},[31805],[34,477858,477860,477863],{"style":477859},"top:-2.357em;margin-left:-0.0785em;margin-right:0.0714em;",[34,477861],{"className":477862,"style":43406},[31813],[34,477864,477866],{"className":477865},[31818,43410,43411,31821],[34,477867,640],{"className":477868},[31685,31821],[34,477870,31829],{"className":477871},[31828],[34,477873,477875],{"className":477874},[31801],[34,477876,477878],{"className":477877,"style":220540},[31805],[34,477879],{},[34,477881,38874],{"className":477882},[34006,31821],[34,477884,477886,477889],{"className":477885},[31685,31821],[34,477887,106063],{"className":477888,"style":67482},[31685,31686,31821],[34,477890,477892],{"className":477891},[31792],[34,477893,477895,477915],{"className":477894},[31796,31797],[34,477896,477898,477912],{"className":477897},[31801],[34,477899,477901],{"className":477900,"style":220518},[31805],[34,477902,477903,477906],{"style":477859},[34,477904],{"className":477905,"style":43406},[31813],[34,477907,477909],{"className":477908},[31818,43410,43411,31821],[34,477910,31660],{"className":477911},[31685,31686,31821],[34,477913,31829],{"className":477914},[31828],[34,477916,477918],{"className":477917},[31801],[34,477919,477921],{"className":477920,"style":220540},[31805],[34,477922],{},[34,477924,477925,477928],{"style":41565},[34,477926],{"className":477927,"style":41508},[31813],[34,477929,477931],{"className":477930},[34006,71115],[34,477932,477934,478123],{"className":477933},[31796,31797],[34,477935,477937,478120],{"className":477936},[31801],[34,477938,477940,477983],{"className":477939,"style":32527},[31805],[34,477941,477944,477947],{"className":477942,"style":477943},[41564],"top:-2.202em;",[34,477945],{"className":477946,"style":41508},[31813],[34,477948,477951,477963,477973],{"className":477949,"style":477950},[126470],"height:0.548em;min-width:1.6em;",[34,477952,477956],{"className":477953,"style":477955},[477954],"brace-left","height:0.548em;",[41588,477957,477960],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},"0.548em","0 0 400000 548",[20398,477961],{"d":477962},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[34,477964,477967],{"className":477965,"style":477955},[477966],"brace-center",[41588,477968,477970],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},"xMidYMin slice",[20398,477971],{"d":477972},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[34,477974,477977],{"className":477975,"style":477955},[477976],"brace-right",[41588,477978,477980],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},"xMaxYMin slice",[20398,477981],{"d":477982},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[34,477984,477985,477988],{"style":41565},[34,477986],{"className":477987,"style":41508},[31813],[34,477989,477991,478031,478071,478074,478077,478080],{"className":477990},[31685],[34,477992,477994,477997],{"className":477993},[31685],[34,477995,32936],{"className":477996},[31685,31686],[34,477998,478000],{"className":477999},[31792],[34,478001,478003,478023],{"className":478002},[31796,31797],[34,478004,478006,478020],{"className":478005},[31801],[34,478007,478009],{"className":478008,"style":31806},[31805],[34,478010,478011,478014],{"style":31809},[34,478012],{"className":478013,"style":31814},[31813],[34,478015,478017],{"className":478016},[31818,31819,31820,31821],[34,478018,640],{"className":478019},[31685,31821],[34,478021,31829],{"className":478022},[31828],[34,478024,478026],{"className":478025},[31801],[34,478027,478029],{"className":478028,"style":31836},[31805],[34,478030],{},[34,478032,478034,478037],{"className":478033},[31685],[34,478035,32936],{"className":478036},[31685,31686],[34,478038,478040],{"className":478039},[31792],[34,478041,478043,478063],{"className":478042},[31796,31797],[34,478044,478046,478060],{"className":478045},[31801],[34,478047,478049],{"className":478048,"style":31806},[31805],[34,478050,478051,478054],{"style":31809},[34,478052],{"className":478053,"style":31814},[31813],[34,478055,478057],{"className":478056},[31818,31819,31820,31821],[34,478058,768],{"className":478059},[31685,31821],[34,478061,31829],{"className":478062},[31828],[34,478064,478066],{"className":478065},[31801],[34,478067,478069],{"className":478068,"style":31836},[31805],[34,478070],{},[34,478072],{"className":478073,"style":32123},[31845],[34,478075,38874],{"className":478076},[34006],[34,478078],{"className":478079,"style":32123},[31845],[34,478081,478083,478086],{"className":478082},[31685],[34,478084,32936],{"className":478085},[31685,31686],[34,478087,478089],{"className":478088},[31792],[34,478090,478092,478112],{"className":478091},[31796,31797],[34,478093,478095,478109],{"className":478094},[31801],[34,478096,478098],{"className":478097,"style":32362},[31805],[34,478099,478100,478103],{"style":31809},[34,478101],{"className":478102,"style":31814},[31813],[34,478104,478106],{"className":478105},[31818,31819,31820,31821],[34,478107,31660],{"className":478108},[31685,31686,31821],[34,478110,31829],{"className":478111},[31828],[34,478113,478115],{"className":478114},[31801],[34,478116,478118],{"className":478117,"style":31836},[31805],[34,478119],{},[34,478121,31829],{"className":478122},[31828],[34,478124,478126],{"className":478125},[31801],[34,478127,478130],{"className":478128,"style":478129},[31805],"height:0.798em;",[34,478131],{},[34,478133,31829],{"className":478134},[31828],[34,478136,478138],{"className":478137},[31801],[34,478139,478142],{"className":478140,"style":478141},[31805],"height:1.5764em;",[34,478143],{},[34,478145,2557],{"className":478146},[31690],[34,478148],{"className":478149,"style":32604},[31845],[34,478151,223],{"className":478152},[32024],[34,478154],{"className":478155,"style":32604},[31845],[34,478157,478159,478163,478166,478282],{"className":478158},[31676],[34,478160],{"className":478161,"style":478162},[31680],"height:2.1037em;vertical-align:-1.3537em;",[34,478164,456],{"className":478165},[31782],[34,478167,478169],{"className":478168},[34006,71115],[34,478170,478172,478273],{"className":478171},[31796,31797],[34,478173,478175,478270],{"className":478174},[31801],[34,478176,478178,478196],{"className":478177,"style":33218},[31805],[34,478179,478181,478184],{"style":478180},"top:-1.6737em;",[34,478182],{"className":478183,"style":41508},[31813],[34,478185,478187],{"className":478186},[31818,31819,31820,31821],[34,478188,478190,478193],{"className":478189},[31685,31821],[34,478191,112296],{"className":478192},[31850,31821],[34,478194,223320],{"className":478195,"style":223361},[31685,31686,31821],[34,478197,478198,478201],{"style":41565},[34,478199],{"className":478200,"style":41508},[31813],[34,478202,478204],{"className":478203},[34006,71115],[34,478205,478207,478261],{"className":478206},[31796,31797],[34,478208,478210,478258],{"className":478209},[31801],[34,478211,478213,478244],{"className":478212,"style":33218},[31805],[34,478214,478217,478220],{"className":478215,"style":478216},[41564],"top:-2.352em;",[34,478218],{"className":478219,"style":41508},[31813],[34,478221,478223,478230,478237],{"className":478222,"style":477950},[126470],[34,478224,478226],{"className":478225,"style":477955},[477954],[41588,478227,478228],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},[20398,478229],{"d":477962},[34,478231,478233],{"className":478232,"style":477955},[477966],[41588,478234,478235],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},[20398,478236],{"d":477972},[34,478238,478240],{"className":478239,"style":477955},[477976],[41588,478241,478242],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},[20398,478243],{"d":477982},[34,478245,478246,478249],{"style":41565},[34,478247],{"className":478248,"style":41508},[31813],[34,478250,478252],{"className":478251},[31685],[34,478253,478255],{"className":478254},[31685,2638],[34,478256,477767],{"className":478257},[31685,105462],[34,478259,31829],{"className":478260},[31828],[34,478262,478264],{"className":478263},[31801],[34,478265,478268],{"className":478266,"style":478267},[31805],"height:0.648em;",[34,478269],{},[34,478271,31829],{"className":478272},[31828],[34,478274,478276],{"className":478275},[31801],[34,478277,478280],{"className":478278,"style":478279},[31805],"height:1.3537em;",[34,478281],{},[34,478283,2557],{"className":478284},[31690],"\n令 ",[34,478287,478289,478335],{"className":478288},[31642],[34,478290,478292],{"className":478291},[31646],[31648,478293,478294],{"xmlns":31650},[31652,478295,478296,478332],{},[31655,478297,478298,478300,478302,478308,478310,478316,478318,478324,478326],{},[31658,478299,67467],{},[31662,478301,202],{},[31743,478303,478304,478306],{},[31658,478305,32936],{},[31749,478307,640],{},[31662,478309,38874],{},[31743,478311,478312,478314],{},[31658,478313,32936],{},[31658,478315,31660],{},[31662,478317,112296],{},[31743,478319,478320,478322],{},[31658,478321,106063],{},[31749,478323,640],{},[31662,478325,38874],{},[31743,478327,478328,478330],{},[31658,478329,106063],{},[31658,478331,31660],{},[31665,478333,478334],{"encoding":31667},"X = x_1 \\cdots x_n \\in I_1 \\cdots I_n",[34,478336,478338,478356,478460],{"className":478337,"ariaHidden":758},[31672],[34,478339,478341,478344,478347,478350,478353],{"className":478340},[31676],[34,478342],{"className":478343,"style":33218},[31680],[34,478345,67467],{"className":478346,"style":67482},[31685,31686],[34,478348],{"className":478349,"style":31846},[31845],[34,478351,202],{"className":478352},[31850],[34,478354],{"className":478355,"style":31846},[31845],[34,478357,478359,478362,478402,478405,478408,478411,478451,478454,478457],{"className":478358},[31676],[34,478360],{"className":478361,"style":225773},[31680],[34,478363,478365,478368],{"className":478364},[31685],[34,478366,32936],{"className":478367},[31685,31686],[34,478369,478371],{"className":478370},[31792],[34,478372,478374,478394],{"className":478373},[31796,31797],[34,478375,478377,478391],{"className":478376},[31801],[34,478378,478380],{"className":478379,"style":31806},[31805],[34,478381,478382,478385],{"style":31809},[34,478383],{"className":478384,"style":31814},[31813],[34,478386,478388],{"className":478387},[31818,31819,31820,31821],[34,478389,640],{"className":478390},[31685,31821],[34,478392,31829],{"className":478393},[31828],[34,478395,478397],{"className":478396},[31801],[34,478398,478400],{"className":478399,"style":31836},[31805],[34,478401],{},[34,478403],{"className":478404,"style":32123},[31845],[34,478406,38874],{"className":478407},[34006],[34,478409],{"className":478410,"style":32123},[31845],[34,478412,478414,478417],{"className":478413},[31685],[34,478415,32936],{"className":478416},[31685,31686],[34,478418,478420],{"className":478419},[31792],[34,478421,478423,478443],{"className":478422},[31796,31797],[34,478424,478426,478440],{"className":478425},[31801],[34,478427,478429],{"className":478428,"style":32362},[31805],[34,478430,478431,478434],{"style":31809},[34,478432],{"className":478433,"style":31814},[31813],[34,478435,478437],{"className":478436},[31818,31819,31820,31821],[34,478438,31660],{"className":478439},[31685,31686,31821],[34,478441,31829],{"className":478442},[31828],[34,478444,478446],{"className":478445},[31801],[34,478447,478449],{"className":478448,"style":31836},[31805],[34,478450],{},[34,478452],{"className":478453,"style":31846},[31845],[34,478455,112296],{"className":478456},[31850],[34,478458],{"className":478459,"style":31846},[31845],[34,478461,478463,478466,478506,478509,478512,478515],{"className":478462},[31676],[34,478464],{"className":478465,"style":70915},[31680],[34,478467,478469,478472],{"className":478468},[31685],[34,478470,106063],{"className":478471,"style":67482},[31685,31686],[34,478473,478475],{"className":478474},[31792],[34,478476,478478,478498],{"className":478477},[31796,31797],[34,478479,478481,478495],{"className":478480},[31801],[34,478482,478484],{"className":478483,"style":31806},[31805],[34,478485,478486,478489],{"style":217742},[34,478487],{"className":478488,"style":31814},[31813],[34,478490,478492],{"className":478491},[31818,31819,31820,31821],[34,478493,640],{"className":478494},[31685,31821],[34,478496,31829],{"className":478497},[31828],[34,478499,478501],{"className":478500},[31801],[34,478502,478504],{"className":478503,"style":31836},[31805],[34,478505],{},[34,478507],{"className":478508,"style":32123},[31845],[34,478510,38874],{"className":478511},[34006],[34,478513],{"className":478514,"style":32123},[31845],[34,478516,478518,478521],{"className":478517},[31685],[34,478519,106063],{"className":478520,"style":67482},[31685,31686],[34,478522,478524],{"className":478523},[31792],[34,478525,478527,478547],{"className":478526},[31796,31797],[34,478528,478530,478544],{"className":478529},[31801],[34,478531,478533],{"className":478532,"style":32362},[31805],[34,478534,478535,478538],{"style":217742},[34,478536],{"className":478537,"style":31814},[31813],[34,478539,478541],{"className":478540},[31818,31819,31820,31821],[34,478542,31660],{"className":478543},[31685,31686,31821],[34,478545,31829],{"className":478546},[31828],[34,478548,478550],{"className":478549},[31801],[34,478551,478553],{"className":478552,"style":31836},[31805],[34,478554],{},"，令 ",[34,478557,478559,478581],{"className":478558},[31642],[34,478560,478562],{"className":478561},[31646],[31648,478563,478564],{"xmlns":31650},[31652,478565,478566,478578],{},[31655,478567,478568,478570,478572,478574,478576],{},[31658,478569,67731],{},[31662,478571,202],{},[39361,478573,477767],{},[31662,478575,112296],{},[31658,478577,223320],{},[31665,478579,478580],{"encoding":31667},"Y = \\text{其余项之和} \\in J",[34,478582,478584,478602,478623],{"className":478583,"ariaHidden":758},[31672],[34,478585,478587,478590,478593,478596,478599],{"className":478586},[31676],[34,478588],{"className":478589,"style":33218},[31680],[34,478591,67731],{"className":478592,"style":32604},[31685,31686],[34,478594],{"className":478595,"style":31846},[31845],[34,478597,202],{"className":478598},[31850],[34,478600],{"className":478601,"style":31846},[31845],[34,478603,478605,478608,478614,478617,478620],{"className":478604},[31676],[34,478606],{"className":478607,"style":104901},[31680],[34,478609,478611],{"className":478610},[31685,2638],[34,478612,477767],{"className":478613},[31685,105462],[34,478615],{"className":478616,"style":31846},[31845],[34,478618,112296],{"className":478619},[31850],[34,478621],{"className":478622,"style":31846},[31845],[34,478624,478626,478629],{"className":478625},[31676],[34,478627],{"className":478628,"style":33218},[31680],[34,478630,223320],{"className":478631,"style":223361},[31685,31686],"。\n我们得到了 ",[34,478634,478636,478658],{"className":478635},[31642],[34,478637,478639],{"className":478638},[31646],[31648,478640,478641],{"xmlns":31650},[31652,478642,478643,478655],{},[31655,478644,478645,478647,478649,478651,478653],{},[31749,478646,640],{},[31662,478648,202],{},[31658,478650,67467],{},[31662,478652,223],{},[31658,478654,67731],{},[31665,478656,478657],{"encoding":31667},"1 = X + Y",[34,478659,478661,478679,478697],{"className":478660,"ariaHidden":758},[31672],[34,478662,478664,478667,478670,478673,478676],{"className":478663},[31676],[34,478665],{"className":478666,"style":32617},[31680],[34,478668,640],{"className":478669},[31685],[34,478671],{"className":478672,"style":31846},[31845],[34,478674,202],{"className":478675},[31850],[34,478677],{"className":478678,"style":31846},[31845],[34,478680,478682,478685,478688,478691,478694],{"className":478681},[31676],[34,478683],{"className":478684,"style":34944},[31680],[34,478686,67467],{"className":478687,"style":67482},[31685,31686],[34,478689],{"className":478690,"style":32604},[31845],[34,478692,223],{"className":478693},[32024],[34,478695],{"className":478696,"style":32604},[31845],[34,478698,478700,478703],{"className":478699},[31676],[34,478701],{"className":478702,"style":33218},[31680],[34,478704,67731],{"className":478705,"style":32604},[31685,31686],[34,478707,478709,478747],{"className":478708},[31642],[34,478710,478712],{"className":478711},[31646],[31648,478713,478714],{"xmlns":31650},[31652,478715,478716,478744],{},[31655,478717,478718,478720,478722,478728,478730,478736,478738,478740,478742],{},[31658,478719,67467],{},[31662,478721,112296],{},[31743,478723,478724,478726],{},[31658,478725,106063],{},[31749,478727,640],{},[31662,478729,38874],{},[31743,478731,478732,478734],{},[31658,478733,106063],{},[31658,478735,31660],{},[31662,478737,31960],{"separator":758},[31658,478739,67731],{},[31662,478741,112296],{},[31658,478743,223320],{},[31665,478745,478746],{"encoding":31667},"X \\in I_1 \\cdots I_n, Y \\in J",[34,478748,478750,478768,478881],{"className":478749,"ariaHidden":758},[31672],[34,478751,478753,478756,478759,478762,478765],{"className":478752},[31676],[34,478754],{"className":478755,"style":104901},[31680],[34,478757,67467],{"className":478758,"style":67482},[31685,31686],[34,478760],{"className":478761,"style":31846},[31845],[34,478763,112296],{"className":478764},[31850],[34,478766],{"className":478767,"style":31846},[31845],[34,478769,478771,478774,478814,478817,478820,478823,478863,478866,478869,478872,478875,478878],{"className":478770},[31676],[34,478772],{"className":478773,"style":70739},[31680],[34,478775,478777,478780],{"className":478776},[31685],[34,478778,106063],{"className":478779,"style":67482},[31685,31686],[34,478781,478783],{"className":478782},[31792],[34,478784,478786,478806],{"className":478785},[31796,31797],[34,478787,478789,478803],{"className":478788},[31801],[34,478790,478792],{"className":478791,"style":31806},[31805],[34,478793,478794,478797],{"style":217742},[34,478795],{"className":478796,"style":31814},[31813],[34,478798,478800],{"className":478799},[31818,31819,31820,31821],[34,478801,640],{"className":478802},[31685,31821],[34,478804,31829],{"className":478805},[31828],[34,478807,478809],{"className":478808},[31801],[34,478810,478812],{"className":478811,"style":31836},[31805],[34,478813],{},[34,478815],{"className":478816,"style":32123},[31845],[34,478818,38874],{"className":478819},[34006],[34,478821],{"className":478822,"style":32123},[31845],[34,478824,478826,478829],{"className":478825},[31685],[34,478827,106063],{"className":478828,"style":67482},[31685,31686],[34,478830,478832],{"className":478831},[31792],[34,478833,478835,478855],{"className":478834},[31796,31797],[34,478836,478838,478852],{"className":478837},[31801],[34,478839,478841],{"className":478840,"style":32362},[31805],[34,478842,478843,478846],{"style":217742},[34,478844],{"className":478845,"style":31814},[31813],[34,478847,478849],{"className":478848},[31818,31819,31820,31821],[34,478850,31660],{"className":478851},[31685,31686,31821],[34,478853,31829],{"className":478854},[31828],[34,478856,478858],{"className":478857},[31801],[34,478859,478861],{"className":478860,"style":31836},[31805],[34,478862],{},[34,478864,31960],{"className":478865},[32119],[34,478867],{"className":478868,"style":32123},[31845],[34,478870,67731],{"className":478871,"style":32604},[31685,31686],[34,478873],{"className":478874,"style":31846},[31845],[34,478876,112296],{"className":478877},[31850],[34,478879],{"className":478880,"style":31846},[31845],[34,478882,478884,478887],{"className":478883},[31676],[34,478885],{"className":478886,"style":33218},[31680],[34,478888,223320],{"className":478889,"style":223361},[31685,31686],"。\n这表明 ",[34,478892,478894,478928],{"className":478893},[31642],[34,478895,478897],{"className":478896},[31646],[31648,478898,478899],{"xmlns":31650},[31652,478900,478901,478925],{},[31655,478902,478903,478905,478907,478913,478915,478921,478923],{},[31749,478904,640],{},[31662,478906,112296],{},[31743,478908,478909,478911],{},[31658,478910,106063],{},[31749,478912,640],{},[31662,478914,38874],{},[31743,478916,478917,478919],{},[31658,478918,106063],{},[31658,478920,31660],{},[31662,478922,223],{},[31658,478924,223320],{},[31665,478926,478927],{"encoding":31667},"1 \\in I_1 \\cdots I_n + J",[34,478929,478931,478949,479053],{"className":478930,"ariaHidden":758},[31672],[34,478932,478934,478937,478940,478943,478946],{"className":478933},[31676],[34,478935],{"className":478936,"style":61814},[31680],[34,478938,640],{"className":478939},[31685],[34,478941],{"className":478942,"style":31846},[31845],[34,478944,112296],{"className":478945},[31850],[34,478947],{"className":478948,"style":31846},[31845],[34,478950,478952,478955,478995,478998,479001,479004,479044,479047,479050],{"className":478951},[31676],[34,478953],{"className":478954,"style":70915},[31680],[34,478956,478958,478961],{"className":478957},[31685],[34,478959,106063],{"className":478960,"style":67482},[31685,31686],[34,478962,478964],{"className":478963},[31792],[34,478965,478967,478987],{"className":478966},[31796,31797],[34,478968,478970,478984],{"className":478969},[31801],[34,478971,478973],{"className":478972,"style":31806},[31805],[34,478974,478975,478978],{"style":217742},[34,478976],{"className":478977,"style":31814},[31813],[34,478979,478981],{"className":478980},[31818,31819,31820,31821],[34,478982,640],{"className":478983},[31685,31821],[34,478985,31829],{"className":478986},[31828],[34,478988,478990],{"className":478989},[31801],[34,478991,478993],{"className":478992,"style":31836},[31805],[34,478994],{},[34,478996],{"className":478997,"style":32123},[31845],[34,478999,38874],{"className":479000},[34006],[34,479002],{"className":479003,"style":32123},[31845],[34,479005,479007,479010],{"className":479006},[31685],[34,479008,106063],{"className":479009,"style":67482},[31685,31686],[34,479011,479013],{"className":479012},[31792],[34,479014,479016,479036],{"className":479015},[31796,31797],[34,479017,479019,479033],{"className":479018},[31801],[34,479020,479022],{"className":479021,"style":32362},[31805],[34,479023,479024,479027],{"style":217742},[34,479025],{"className":479026,"style":31814},[31813],[34,479028,479030],{"className":479029},[31818,31819,31820,31821],[34,479031,31660],{"className":479032},[31685,31686,31821],[34,479034,31829],{"className":479035},[31828],[34,479037,479039],{"className":479038},[31801],[34,479040,479042],{"className":479041,"style":31836},[31805],[34,479043],{},[34,479045],{"className":479046,"style":32604},[31845],[34,479048,223],{"className":479049},[32024],[34,479051],{"className":479052,"style":32604},[31845],[34,479054,479056,479059],{"className":479055},[31676],[34,479057],{"className":479058,"style":33218},[31680],[34,479060,223320],{"className":479061,"style":223361},[31685,31686],"。\n因为理想包含 ",[34,479064,479066,479079],{"className":479065},[31642],[34,479067,479069],{"className":479068},[31646],[31648,479070,479071],{"xmlns":31650},[31652,479072,479073,479077],{},[31655,479074,479075],{},[31749,479076,640],{},[31665,479078,640],{"encoding":31667},[34,479080,479082],{"className":479081,"ariaHidden":758},[31672],[34,479083,479085,479088],{"className":479084},[31676],[34,479086],{"className":479087,"style":32617},[31680],[34,479089,640],{"className":479090},[31685]," 就等于整个环 ",[34,479093,479095,479108],{"className":479094},[31642],[34,479096,479098],{"className":479097},[31646],[31648,479099,479100],{"xmlns":31650},[31652,479101,479102,479106],{},[31655,479103,479104],{},[31658,479105,34386],{},[31665,479107,34386],{"encoding":31667},[34,479109,479111],{"className":479110,"ariaHidden":758},[31672],[34,479112,479114,479117],{"className":479113},[31676],[34,479115],{"className":479116,"style":33218},[31680],[34,479118,34386],{"className":479119,"style":34427},[31685,31686],[34,479121,479123,479157],{"className":479122},[31642],[34,479124,479126],{"className":479125},[31646],[31648,479127,479128],{"xmlns":31650},[31652,479129,479130,479154],{},[31655,479131,479132,479138,479140,479146,479148,479150,479152],{},[31743,479133,479134,479136],{},[31658,479135,106063],{},[31749,479137,640],{},[31662,479139,38874],{},[31743,479141,479142,479144],{},[31658,479143,106063],{},[31658,479145,31660],{},[31662,479147,223],{},[31658,479149,223320],{},[31662,479151,202],{},[31658,479153,34386],{},[31665,479155,479156],{"encoding":31667},"I_1 \\cdots I_n + J = R",[34,479158,479160,479264,479282],{"className":479159,"ariaHidden":758},[31672],[34,479161,479163,479166,479206,479209,479212,479215,479255,479258,479261],{"className":479162},[31676],[34,479164],{"className":479165,"style":70915},[31680],[34,479167,479169,479172],{"className":479168},[31685],[34,479170,106063],{"className":479171,"style":67482},[31685,31686],[34,479173,479175],{"className":479174},[31792],[34,479176,479178,479198],{"className":479177},[31796,31797],[34,479179,479181,479195],{"className":479180},[31801],[34,479182,479184],{"className":479183,"style":31806},[31805],[34,479185,479186,479189],{"style":217742},[34,479187],{"className":479188,"style":31814},[31813],[34,479190,479192],{"className":479191},[31818,31819,31820,31821],[34,479193,640],{"className":479194},[31685,31821],[34,479196,31829],{"className":479197},[31828],[34,479199,479201],{"className":479200},[31801],[34,479202,479204],{"className":479203,"style":31836},[31805],[34,479205],{},[34,479207],{"className":479208,"style":32123},[31845],[34,479210,38874],{"className":479211},[34006],[34,479213],{"className":479214,"style":32123},[31845],[34,479216,479218,479221],{"className":479217},[31685],[34,479219,106063],{"className":479220,"style":67482},[31685,31686],[34,479222,479224],{"className":479223},[31792],[34,479225,479227,479247],{"className":479226},[31796,31797],[34,479228,479230,479244],{"className":479229},[31801],[34,479231,479233],{"className":479232,"style":32362},[31805],[34,479234,479235,479238],{"style":217742},[34,479236],{"className":479237,"style":31814},[31813],[34,479239,479241],{"className":479240},[31818,31819,31820,31821],[34,479242,31660],{"className":479243},[31685,31686,31821],[34,479245,31829],{"className":479246},[31828],[34,479248,479250],{"className":479249},[31801],[34,479251,479253],{"className":479252,"style":31836},[31805],[34,479254],{},[34,479256],{"className":479257,"style":32604},[31845],[34,479259,223],{"className":479260},[32024],[34,479262],{"className":479263,"style":32604},[31845],[34,479265,479267,479270,479273,479276,479279],{"className":479266},[31676],[34,479268],{"className":479269,"style":33218},[31680],[34,479271,223320],{"className":479272,"style":223361},[31685,31686],[34,479274],{"className":479275,"style":31846},[31845],[34,479277,202],{"className":479278},[31850],[34,479280],{"className":479281,"style":31846},[31845],[34,479283,479285,479288],{"className":479284},[31676],[34,479286],{"className":479287,"style":33218},[31680],[34,479289,34386],{"className":479290,"style":34427},[31685,31686],"\n即它们互素。 ",[34,479293,479295,479308],{"className":479294},[31642],[34,479296,479298],{"className":479297},[31646],[31648,479299,479300],{"xmlns":31650},[31652,479301,479302,479306],{},[31655,479303,479304],{},[31658,479305,445204],{"mathvariant":38079},[31665,479307,445207],{"encoding":31667},[34,479309,479311],{"className":479310,"ariaHidden":758},[31672],[34,479312,479314,479317],{"className":479313},[31676],[34,479315],{"className":479316,"style":445217},[31680],[34,479318,445204],{"className":479319},[31685,129261],[32188,479321],{},[253,479323,479325],{"id":479324},"第二部分课本-p37-习题","第二部分：课本 P37 习题",[32490,479327,479329],{"id":479328},"习题-4","习题 4",[15,479331,479332],{},[2627,479333,379882,479334,479419],{},[34,479335,479337,479371],{"className":479336},[31642],[34,479338,479340],{"className":479339},[31646],[31648,479341,479342],{"xmlns":31650},[31652,479343,479344,479368],{},[31655,479345,479346,479348,479350,479352,479354,479356,479358,479360,479362,479364,479366],{},[31662,479347,456],{"stretchy":702},[31658,479349,440113],{"mathvariant":38079},[31662,479351,32682],{},[31662,479353,456],{"stretchy":702},[31658,479355,97709],{},[31662,479357,2557],{"stretchy":702},[31662,479359,199477],{"separator":758},[31662,479361,223],{},[31662,479363,31960],{"separator":758},[31662,479365,71191],{},[31662,479367,2557],{"stretchy":702},[31665,479369,479370],{"encoding":31667},"(\\operatorname{End}(G); +, \\cdot)",[34,479372,479374],{"className":479373,"ariaHidden":758},[31672],[34,479375,479377,479380,479383,479389,479392,479395,479398,479401,479404,479407,479410,479413,479416],{"className":479376},[31676],[34,479378],{"className":479379,"style":31774},[31680],[34,479381,456],{"className":479382},[31782],[34,479384,479386],{"className":479385},[32767],[34,479387,440113],{"className":479388},[31685,207923],[34,479390,456],{"className":479391},[31782],[34,479393,97709],{"className":479394},[31685,31686],[34,479396,2557],{"className":479397},[31690],[34,479399,199477],{"className":479400},[32119],[34,479402],{"className":479403,"style":32123},[31845],[34,479405,223],{"className":479406},[31685],[34,479408,31960],{"className":479409},[32119],[34,479411],{"className":479412,"style":32123},[31845],[34,479414,71191],{"className":479415},[31685],[34,479417,2557],{"className":479418},[31690]," 构成幺环。",[15,479421,479422],{},[2627,479423,463462],{},[105,479425,479426,479507,479510],{},[108,479427,479428,479477,479478,479506],{},[34,479429,479431,479453],{"className":479430},[31642],[34,479432,479434],{"className":479433},[31646],[31648,479435,479436],{"xmlns":31650},[31652,479437,479438,479450],{},[31655,479439,479440,479442,479444,479446,479448],{},[31658,479441,440113],{"mathvariant":38079},[31662,479443,32682],{},[31662,479445,456],{"stretchy":702},[31658,479447,97709],{},[31662,479449,2557],{"stretchy":702},[31665,479451,479452],{"encoding":31667},"\\operatorname{End}(G)",[34,479454,479456],{"className":479455,"ariaHidden":758},[31672],[34,479457,479459,479462,479468,479471,479474],{"className":479458},[31676],[34,479460],{"className":479461,"style":31774},[31680],[34,479463,479465],{"className":479464},[32767],[34,479466,440113],{"className":479467},[31685,207923],[34,479469,456],{"className":479470},[31782],[34,479472,97709],{"className":479473},[31685,31686],[34,479475,2557],{"className":479476},[31690]," 是群 ",[34,479479,479481,479494],{"className":479480},[31642],[34,479482,479484],{"className":479483},[31646],[31648,479485,479486],{"xmlns":31650},[31652,479487,479488,479492],{},[31655,479489,479490],{},[31658,479491,97709],{},[31665,479493,97709],{"encoding":31667},[34,479495,479497],{"className":479496,"ariaHidden":758},[31672],[34,479498,479500,479503],{"className":479499},[31676],[34,479501],{"className":479502,"style":33218},[31680],[34,479504,97709],{"className":479505},[31685,31686]," 到自身的所有同态的集合。",[108,479508,479509],{},"题目定义了加法是“逐点相加”，乘法是“函数复合”。",[108,479511,479512,479513],{},"证明它是环，需要验证：\n",[2230,479514,479515,479590,479665,479668],{},[108,479516,479517,479589],{},[34,479518,479520,479550],{"className":479519},[31642],[34,479521,479523],{"className":479522},[31646],[31648,479524,479525],{"xmlns":31650},[31652,479526,479527,479547],{},[31655,479528,479529,479531,479533,479535,479537,479539,479541,479543,479545],{},[31662,479530,456],{"stretchy":702},[31658,479532,440113],{"mathvariant":38079},[31662,479534,32682],{},[31662,479536,456],{"stretchy":702},[31658,479538,97709],{},[31662,479540,2557],{"stretchy":702},[31662,479542,31960],{"separator":758},[31662,479544,223],{},[31662,479546,2557],{"stretchy":702},[31665,479548,479549],{"encoding":31667},"(\\operatorname{End}(G), +)",[34,479551,479553],{"className":479552,"ariaHidden":758},[31672],[34,479554,479556,479559,479562,479568,479571,479574,479577,479580,479583,479586],{"className":479555},[31676],[34,479557],{"className":479558,"style":31774},[31680],[34,479560,456],{"className":479561},[31782],[34,479563,479565],{"className":479564},[32767],[34,479566,440113],{"className":479567},[31685,207923],[34,479569,456],{"className":479570},[31782],[34,479572,97709],{"className":479573},[31685,31686],[34,479575,2557],{"className":479576},[31690],[34,479578,31960],{"className":479579},[32119],[34,479581],{"className":479582,"style":32123},[31845],[34,479584,223],{"className":479585},[31685],[34,479587,2557],{"className":479588},[31690]," 是阿贝尔群。",[108,479591,479592,479664],{},[34,479593,479595,479625],{"className":479594},[31642],[34,479596,479598],{"className":479597},[31646],[31648,479599,479600],{"xmlns":31650},[31652,479601,479602,479622],{},[31655,479603,479604,479606,479608,479610,479612,479614,479616,479618,479620],{},[31662,479605,456],{"stretchy":702},[31658,479607,440113],{"mathvariant":38079},[31662,479609,32682],{},[31662,479611,456],{"stretchy":702},[31658,479613,97709],{},[31662,479615,2557],{"stretchy":702},[31662,479617,31960],{"separator":758},[31662,479619,71191],{},[31662,479621,2557],{"stretchy":702},[31665,479623,479624],{"encoding":31667},"(\\operatorname{End}(G), \\cdot)",[34,479626,479628],{"className":479627,"ariaHidden":758},[31672],[34,479629,479631,479634,479637,479643,479646,479649,479652,479655,479658,479661],{"className":479630},[31676],[34,479632],{"className":479633,"style":31774},[31680],[34,479635,456],{"className":479636},[31782],[34,479638,479640],{"className":479639},[32767],[34,479641,440113],{"className":479642},[31685,207923],[34,479644,456],{"className":479645},[31782],[34,479647,97709],{"className":479648},[31685,31686],[34,479650,2557],{"className":479651},[31690],[34,479653,31960],{"className":479654},[32119],[34,479656],{"className":479657,"style":32123},[31845],[34,479659,71191],{"className":479660},[31685],[34,479662,2557],{"className":479663},[31690]," 是半群（封闭、结合律）。",[108,479666,479667],{},"分配律成立。",[108,479669,479670],{},"有幺元。",[15,479672,479673,362914,479675,46528,479773,347],{},[2627,479674,463779],{},[34,479676,479678,479713],{"className":479677},[31642],[34,479679,479681],{"className":479680},[31646],[31648,479682,479683],{"xmlns":31650},[31652,479684,479685,479710],{},[31655,479686,479687,479689,479691,479693,479695,479698,479700,479702,479704,479706,479708],{},[31658,479688,329201],{},[31662,479690,31960],{"separator":758},[31658,479692,449638],{},[31662,479694,31960],{"separator":758},[31658,479696,479697],{},"θ",[31662,479699,112296],{},[31658,479701,440113],{"mathvariant":38079},[31662,479703,32682],{},[31662,479705,456],{"stretchy":702},[31658,479707,97709],{},[31662,479709,2557],{"stretchy":702},[31665,479711,479712],{"encoding":31667},"\\phi, \\psi, \\theta \\in \\operatorname{End}(G)",[34,479714,479716,479752],{"className":479715,"ariaHidden":758},[31672],[34,479717,479719,479722,479725,479728,479731,479734,479737,479740,479743,479746,479749],{"className":479718},[31676],[34,479720],{"className":479721,"style":43217},[31680],[34,479723,329201],{"className":479724},[31685,31686],[34,479726,31960],{"className":479727},[32119],[34,479729],{"className":479730,"style":32123},[31845],[34,479732,449638],{"className":479733,"style":32062},[31685,31686],[34,479735,31960],{"className":479736},[32119],[34,479738],{"className":479739,"style":32123},[31845],[34,479741,479697],{"className":479742,"style":34452},[31685,31686],[34,479744],{"className":479745,"style":31846},[31845],[34,479747,112296],{"className":479748},[31850],[34,479750],{"className":479751,"style":31846},[31845],[34,479753,479755,479758,479764,479767,479770],{"className":479754},[31676],[34,479756],{"className":479757,"style":31774},[31680],[34,479759,479761],{"className":479760},[32767],[34,479762,440113],{"className":479763},[31685,207923],[34,479765,456],{"className":479766},[31782],[34,479768,97709],{"className":479769},[31685,31686],[34,479771,2557],{"className":479772},[31690],[34,479774,479776,479793],{"className":479775},[31642],[34,479777,479779],{"className":479778},[31646],[31648,479780,479781],{"xmlns":31650},[31652,479782,479783,479791],{},[31655,479784,479785,479787,479789],{},[31658,479786,31942],{},[31662,479788,112296],{},[31658,479790,97709],{},[31665,479792,329153],{"encoding":31667},[34,479794,479796,479814],{"className":479795,"ariaHidden":758},[31672],[34,479797,479799,479802,479805,479808,479811],{"className":479798},[31676],[34,479800],{"className":479801,"style":86652},[31680],[34,479803,31942],{"className":479804,"style":32062},[31685,31686],[34,479806],{"className":479807,"style":31846},[31845],[34,479809,112296],{"className":479810},[31850],[34,479812],{"className":479813,"style":31846},[31845],[34,479815,479817,479820],{"className":479816},[31676],[34,479818],{"className":479819,"style":33218},[31680],[34,479821,97709],{"className":479822},[31685,31686],[2230,479824,479825,480706,480882,482243],{},[108,479826,479827,2664,479830],{},[2627,479828,479829],{},"加法群性质",[105,479831,479832,480605],{},[108,479833,479834,2664,479836,86621,480200,113124,480228,480231,480232,355408,480553,480604],{},[2627,479835,378392],{},[34,479837,479839,479939],{"className":479838},[31642],[34,479840,479842],{"className":479841},[31646],[31648,479843,479844],{"xmlns":31650},[31652,479845,479846,479936],{},[31655,479847,479848,479850,479852,479854,479856,479858,479860,479862,479864,479866,479868,479870,479872,479874,479876,479878,479880,479882,479884,479886,479888,479890,479892,479894,479896,479898,479900,479902,479904,479906,479908,479910,479912,479914,479916,479918,479920,479922,479924,479926,479928,479930,479932,479934],{},[31662,479849,456],{"stretchy":702},[31658,479851,329201],{},[31662,479853,223],{},[31658,479855,449638],{},[31662,479857,2557],{"stretchy":702},[31662,479859,456],{"stretchy":702},[31658,479861,111],{},[31662,479863,223],{},[31658,479865,1697],{},[31662,479867,2557],{"stretchy":702},[31662,479869,202],{},[31658,479871,329201],{},[31662,479873,456],{"stretchy":702},[31658,479875,111],{},[31662,479877,223],{},[31658,479879,1697],{},[31662,479881,2557],{"stretchy":702},[31662,479883,223],{},[31658,479885,449638],{},[31662,479887,456],{"stretchy":702},[31658,479889,111],{},[31662,479891,223],{},[31658,479893,1697],{},[31662,479895,2557],{"stretchy":702},[31662,479897,202],{},[31658,479899,329201],{},[31662,479901,456],{"stretchy":702},[31658,479903,111],{},[31662,479905,2557],{"stretchy":702},[31662,479907,223],{},[31658,479909,329201],{},[31662,479911,456],{"stretchy":702},[31658,479913,1697],{},[31662,479915,2557],{"stretchy":702},[31662,479917,223],{},[31658,479919,449638],{},[31662,479921,456],{"stretchy":702},[31658,479923,111],{},[31662,479925,2557],{"stretchy":702},[31662,479927,223],{},[31658,479929,449638],{},[31662,479931,456],{"stretchy":702},[31658,479933,1697],{},[31662,479935,2557],{"stretchy":702},[31665,479937,479938],{"encoding":31667},"(\\phi+\\psi)(a+b) = \\phi(a+b)+\\psi(a+b) = \\phi(a)+\\phi(b)+\\psi(a)+\\psi(b)",[34,479940,479942,479963,479990,480011,480035,480056,480080,480101,480128,480155,480182],{"className":479941,"ariaHidden":758},[31672],[34,479943,479945,479948,479951,479954,479957,479960],{"className":479944},[31676],[34,479946],{"className":479947,"style":31774},[31680],[34,479949,456],{"className":479950},[31782],[34,479952,329201],{"className":479953},[31685,31686],[34,479955],{"className":479956,"style":32604},[31845],[34,479958,223],{"className":479959},[32024],[34,479961],{"className":479962,"style":32604},[31845],[34,479964,479966,479969,479972,479975,479978,479981,479984,479987],{"className":479965},[31676],[34,479967],{"className":479968,"style":31774},[31680],[34,479970,449638],{"className":479971,"style":32062},[31685,31686],[34,479973,2557],{"className":479974},[31690],[34,479976,456],{"className":479977},[31782],[34,479979,111],{"className":479980},[31685,31686],[34,479982],{"className":479983,"style":32604},[31845],[34,479985,223],{"className":479986},[32024],[34,479988],{"className":479989,"style":32604},[31845],[34,479991,479993,479996,479999,480002,480005,480008],{"className":479992},[31676],[34,479994],{"className":479995,"style":31774},[31680],[34,479997,1697],{"className":479998},[31685,31686],[34,480000,2557],{"className":480001},[31690],[34,480003],{"className":480004,"style":31846},[31845],[34,480006,202],{"className":480007},[31850],[34,480009],{"className":480010,"style":31846},[31845],[34,480012,480014,480017,480020,480023,480026,480029,480032],{"className":480013},[31676],[34,480015],{"className":480016,"style":31774},[31680],[34,480018,329201],{"className":480019},[31685,31686],[34,480021,456],{"className":480022},[31782],[34,480024,111],{"className":480025},[31685,31686],[34,480027],{"className":480028,"style":32604},[31845],[34,480030,223],{"className":480031},[32024],[34,480033],{"className":480034,"style":32604},[31845],[34,480036,480038,480041,480044,480047,480050,480053],{"className":480037},[31676],[34,480039],{"className":480040,"style":31774},[31680],[34,480042,1697],{"className":480043},[31685,31686],[34,480045,2557],{"className":480046},[31690],[34,480048],{"className":480049,"style":32604},[31845],[34,480051,223],{"className":480052},[32024],[34,480054],{"className":480055,"style":32604},[31845],[34,480057,480059,480062,480065,480068,480071,480074,480077],{"className":480058},[31676],[34,480060],{"className":480061,"style":31774},[31680],[34,480063,449638],{"className":480064,"style":32062},[31685,31686],[34,480066,456],{"className":480067},[31782],[34,480069,111],{"className":480070},[31685,31686],[34,480072],{"className":480073,"style":32604},[31845],[34,480075,223],{"className":480076},[32024],[34,480078],{"className":480079,"style":32604},[31845],[34,480081,480083,480086,480089,480092,480095,480098],{"className":480082},[31676],[34,480084],{"className":480085,"style":31774},[31680],[34,480087,1697],{"className":480088},[31685,31686],[34,480090,2557],{"className":480091},[31690],[34,480093],{"className":480094,"style":31846},[31845],[34,480096,202],{"className":480097},[31850],[34,480099],{"className":480100,"style":31846},[31845],[34,480102,480104,480107,480110,480113,480116,480119,480122,480125],{"className":480103},[31676],[34,480105],{"className":480106,"style":31774},[31680],[34,480108,329201],{"className":480109},[31685,31686],[34,480111,456],{"className":480112},[31782],[34,480114,111],{"className":480115},[31685,31686],[34,480117,2557],{"className":480118},[31690],[34,480120],{"className":480121,"style":32604},[31845],[34,480123,223],{"className":480124},[32024],[34,480126],{"className":480127,"style":32604},[31845],[34,480129,480131,480134,480137,480140,480143,480146,480149,480152],{"className":480130},[31676],[34,480132],{"className":480133,"style":31774},[31680],[34,480135,329201],{"className":480136},[31685,31686],[34,480138,456],{"className":480139},[31782],[34,480141,1697],{"className":480142},[31685,31686],[34,480144,2557],{"className":480145},[31690],[34,480147],{"className":480148,"style":32604},[31845],[34,480150,223],{"className":480151},[32024],[34,480153],{"className":480154,"style":32604},[31845],[34,480156,480158,480161,480164,480167,480170,480173,480176,480179],{"className":480157},[31676],[34,480159],{"className":480160,"style":31774},[31680],[34,480162,449638],{"className":480163,"style":32062},[31685,31686],[34,480165,456],{"className":480166},[31782],[34,480168,111],{"className":480169},[31685,31686],[34,480171,2557],{"className":480172},[31690],[34,480174],{"className":480175,"style":32604},[31845],[34,480177,223],{"className":480178},[32024],[34,480180],{"className":480181,"style":32604},[31845],[34,480183,480185,480188,480191,480194,480197],{"className":480184},[31676],[34,480186],{"className":480187,"style":31774},[31680],[34,480189,449638],{"className":480190,"style":32062},[31685,31686],[34,480192,456],{"className":480193},[31782],[34,480195,1697],{"className":480196},[31685,31686],[34,480198,2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(\\phi(a)+\\psi(a)) + (\\phi(b)+\\psi(b)) = (\\phi+\\psi)(a) + (\\phi+\\psi)(b)",[34,480332,480334,480346,480376,480403,480433,480460,480481,480511,480532],{"className":480333,"ariaHidden":758},[31672],[34,480335,480337,480340,480343],{"className":480336},[31676],[34,480338],{"className":480339,"style":73089},[31680],[34,480341,202],{"className":480342},[31850],[34,480344],{"className":480345,"style":31846},[31845],[34,480347,480349,480352,480355,480358,480361,480364,480367,480370,480373],{"className":480348},[31676],[34,480350],{"className":480351,"style":31774},[31680],[34,480353,456],{"className":480354},[31782],[34,480356,329201],{"className":480357},[31685,31686],[34,480359,456],{"className":480360},[31782],[34,480362,111],{"className":480363},[31685,31686],[34,480365,2557],{"className":480366},[31690],[34,480368],{"className":480369,"style":32604},[31845],[34,480371,223],{"className":480372},[32024],[34,480374],{"className":480375,"style":32604},[31845],[34,480377,480379,480382,480385,480388,480391,480394,480397,480400],{"className":480378},[31676],[34,480380],{"className":480381,"style":31774},[31680],[34,480383,449638],{"className":480384,"style":32062},[31685,31686],[34,480386,456],{"className":480387},[31782],[34,480389,111],{"className":480390},[31685,31686],[34,480392,44253],{"className":480393},[31690],[34,480395],{"className":480396,"style":32604},[31845],[34,480398,223],{"className":480399},[32024],[34,480401],{"className":480402,"style":32604},[31845],[34,480404,480406,480409,480412,480415,480418,480421,480424,480427,480430],{"className":480405},[31676],[34,480407],{"className":480408,"style":31774},[31680],[34,480410,456],{"className":480411},[31782],[34,480413,329201],{"className":480414},[31685,31686],[34,480416,456],{"className":480417},[31782],[34,480419,1697],{"className":480420},[31685,31686],[34,480422,2557],{"className":480423},[31690],[34,480425],{"className":480426,"style":32604},[31845],[34,480428,223],{"className":480429},[32024],[34,480431],{"className":480432,"style":32604},[31845],[34,480434,480436,480439,480442,480445,480448,480451,480454,480457],{"className":480435},[31676],[34,480437],{"className":480438,"style":31774},[31680],[34,480440,449638],{"className":480441,"style":32062},[31685,31686],[34,480443,456],{"className":480444},[31782],[34,480446,1697],{"className":480447},[31685,31686],[34,480449,44253],{"className":480450},[31690],[34,480452],{"className":480453,"style":31846},[31845],[34,480455,202],{"className":480456},[31850],[34,480458],{"className":480459,"style":31846},[31845],[34,480461,480463,480466,480469,480472,480475,480478],{"className":480462},[31676],[34,480464],{"className":480465,"style":31774},[31680],[34,480467,456],{"className":480468},[31782],[34,480470,329201],{"className":480471},[31685,31686],[34,480473],{"className":480474,"style":32604},[31845],[34,480476,223],{"className":480477},[32024],[34,480479],{"className":480480,"style":32604},[31845],[34,480482,480484,480487,480490,480493,480496,480499,480502,480505,480508],{"className":480483},[31676],[34,480485],{"className":480486,"style":31774},[31680],[34,480488,449638],{"className":480489,"style":32062},[31685,31686],[34,480491,2557],{"className":480492},[31690],[34,480494,456],{"className":480495},[31782],[34,480497,111],{"className":480498},[31685,31686],[34,480500,2557],{"className":480501},[31690],[34,480503],{"className":480504,"style":32604},[31845],[34,480506,223],{"className":480507},[32024],[34,480509],{"className":480510,"style":32604},[31845],[34,480512,480514,480517,480520,480523,480526,480529],{"className":480513},[31676],[34,480515],{"className":480516,"style":31774},[31680],[34,480518,456],{"className":480519},[31782],[34,480521,329201],{"className":480522},[31685,31686],[34,480524],{"className":480525,"style":32604},[31845],[34,480527,223],{"className":480528},[32024],[34,480530],{"className":480531,"style":32604},[31845],[34,480533,480535,480538,480541,480544,480547,480550],{"className":480534},[31676],[34,480536],{"className":480537,"style":31774},[31680],[34,480539,449638],{"className":480540,"style":32062},[31685,31686],[34,480542,2557],{"className":480543},[31690],[34,480545,456],{"className":480546},[31782],[34,480548,1697],{"className":480549},[31685,31686],[34,480551,2557],{"className":480552},[31690],[34,480554,480556,480574],{"className":480555},[31642],[34,480557,480559],{"className":480558},[31646],[31648,480560,480561],{"xmlns":31650},[31652,480562,480563,480571],{},[31655,480564,480565,480567,480569],{},[31658,480566,329201],{},[31662,480568,223],{},[31658,480570,449638],{},[31665,480572,480573],{"encoding":31667},"\\phi+\\psi",[34,480575,480577,480595],{"className":480576,"ariaHidden":758},[31672],[34,480578,480580,480583,480586,480589,480592],{"className":480579},[31676],[34,480581],{"className":480582,"style":43217},[31680],[34,480584,329201],{"className":480585},[31685,31686],[34,480587],{"className":480588,"style":32604},[31845],[34,480590,223],{"className":480591},[32024],[34,480593],{"className":480594,"style":32604},[31845],[34,480596,480598,480601],{"className":480597},[31676],[34,480599],{"className":480600,"style":43217},[31680],[34,480602,449638],{"className":480603,"style":32062},[31685,31686]," 也是同态。",[108,480606,480607,480610,480611,480639,480640,347],{},[2627,480608,480609],{},"结合律、交换律、零元、逆元","：直接继承自交换群 ",[34,480612,480614,480627],{"className":480613},[31642],[34,480615,480617],{"className":480616},[31646],[31648,480618,480619],{"xmlns":31650},[31652,480620,480621,480625],{},[31655,480622,480623],{},[31658,480624,97709],{},[31665,480626,97709],{"encoding":31667},[34,480628,480630],{"className":480629,"ariaHidden":758},[31672],[34,480631,480633,480636],{"className":480632},[31676],[34,480634],{"className":480635,"style":33218},[31680],[34,480637,97709],{"className":480638},[31685,31686]," 的性质。零元是零同态 ",[34,480641,480643,480667],{"className":480642},[31642],[34,480644,480646],{"className":480645},[31646],[31648,480647,480648],{"xmlns":31650},[31652,480649,480650,480664],{},[31655,480651,480652,480654,480656,480658,480660,480662],{},[31749,480653,459],{},[31662,480655,456],{"stretchy":702},[31658,480657,31942],{},[31662,480659,2557],{"stretchy":702},[31662,480661,202],{},[31749,480663,459],{},[31665,480665,480666],{"encoding":31667},"0(g)=0",[34,480668,480670,480697],{"className":480669,"ariaHidden":758},[31672],[34,480671,480673,480676,480679,480682,480685,480688,480691,480694],{"className":480672},[31676],[34,480674],{"className":480675,"style":31774},[31680],[34,480677,459],{"className":480678},[31685],[34,480680,456],{"className":480681},[31782],[34,480683,31942],{"className":480684,"style":32062},[31685,31686],[34,480686,2557],{"className":480687},[31690],[34,480689],{"className":480690,"style":31846},[31845],[34,480692,202],{"className":480693},[31850],[34,480695],{"className":480696,"style":31846},[31845],[34,480698,480700,480703],{"className":480699},[31676],[34,480701],{"className":480702,"style":32617},[31680],[34,480704,459],{"className":480705},[31685],[108,480707,480708,2664,480711],{},[2627,480709,480710],{},"乘法性质（复合）",[105,480712,480713,480718],{},[108,480714,480715,480717],{},[2627,480716,378392],{},"：两个同态的复合还是同态（标准结论）。",[108,480719,480720,480722,480723,347],{},[2627,480721,378398],{},"：函数的复合天然满足结合律。",[34,480724,480726,480768],{"className":480725},[31642],[34,480727,480729],{"className":480728},[31646],[31648,480730,480731],{"xmlns":31650},[31652,480732,480733,480765],{},[31655,480734,480735,480737,480739,480741,480743,480745,480747,480749,480751,480753,480755,480757,480759,480761,480763],{},[31662,480736,456],{"stretchy":702},[31658,480738,329201],{},[31662,480740,448095],{},[31658,480742,449638],{},[31662,480744,2557],{"stretchy":702},[31662,480746,448095],{},[31658,480748,479697],{},[31662,480750,202],{},[31658,480752,329201],{},[31662,480754,448095],{},[31662,480756,456],{"stretchy":702},[31658,480758,449638],{},[31662,480760,448095],{},[31658,480762,479697],{},[31662,480764,2557],{"stretchy":702},[31665,480766,480767],{"encoding":31667},"(\\phi \\circ \\psi) \\circ \\theta = \\phi \\circ (\\psi \\circ \\theta)",[34,480769,480771,480792,480813,480831,480849,480870],{"className":480770,"ariaHidden":758},[31672],[34,480772,480774,480777,480780,480783,480786,480789],{"className":480773},[31676],[34,480775],{"className":480776,"style":31774},[31680],[34,480778,456],{"className":480779},[31782],[34,480781,329201],{"className":480782},[31685,31686],[34,480784],{"className":480785,"style":32604},[31845],[34,480787,448095],{"className":480788},[32024],[34,480790],{"className":480791,"style":32604},[31845],[34,480793,480795,480798,480801,480804,480807,480810],{"className":480794},[31676],[34,480796],{"className":480797,"style":31774},[31680],[34,480799,449638],{"className":480800,"style":32062},[31685,31686],[34,480802,2557],{"className":480803},[31690],[34,480805],{"className":480806,"style":32604},[31845],[34,480808,448095],{"className":480809},[32024],[34,480811],{"className":480812,"style":32604},[31845],[34,480814,480816,480819,480822,480825,480828],{"className":480815},[31676],[34,480817],{"className":480818,"style":31681},[31680],[34,480820,479697],{"className":480821,"style":34452},[31685,31686],[34,480823],{"className":480824,"style":31846},[31845],[34,480826,202],{"className":480827},[31850],[34,480829],{"className":480830,"style":31846},[31845],[34,480832,480834,480837,480840,480843,480846],{"className":480833},[31676],[34,480835],{"className":480836,"style":43217},[31680],[34,480838,329201],{"className":480839},[31685,31686],[34,480841],{"className":480842,"style":32604},[31845],[34,480844,448095],{"className":480845},[32024],[34,480847],{"className":480848,"style":32604},[31845],[34,480850,480852,480855,480858,480861,480864,480867],{"className":480851},[31676],[34,480853],{"className":480854,"style":31774},[31680],[34,480856,456],{"className":480857},[31782],[34,480859,449638],{"className":480860,"style":32062},[31685,31686],[34,480862],{"className":480863,"style":32604},[31845],[34,480865,448095],{"className":480866},[32024],[34,480868],{"className":480869,"style":32604},[31845],[34,480871,480873,480876,480879],{"className":480872},[31676],[34,480874],{"className":480875,"style":31774},[31680],[34,480877,479697],{"className":480878,"style":34452},[31685,31686],[34,480880,2557],{"className":480881},[31690],[108,480883,480884,2664,480887],{},[2627,480885,480886],{},"分配律（最重要的一步）",[105,480888,480889,481623],{},[108,480890,480891,47649,480894,33183,481065,315440,481335,481364,481365,481622],{},[2627,480892,480893],{},"左分配律",[34,480895,480897,480939],{"className":480896},[31642],[34,480898,480900],{"className":480899},[31646],[31648,480901,480902],{"xmlns":31650},[31652,480903,480904,480936],{},[31655,480905,480906,480908,480910,480912,480914,480916,480918,480920,480922,480924,480926,480928,480930,480932,480934],{},[31658,480907,479697],{},[31662,480909,71191],{},[31662,480911,456],{"stretchy":702},[31658,480913,329201],{},[31662,480915,223],{},[31658,480917,449638],{},[31662,480919,2557],{"stretchy":702},[31662,480921,202],{},[31658,480923,479697],{},[31662,480925,71191],{},[31658,480927,329201],{},[31662,480929,223],{},[31658,480931,479697],{},[31662,480933,71191],{},[31658,480935,449638],{},[31665,480937,480938],{"encoding":31667},"\\theta \\cdot (\\phi + \\psi) = \\theta \\cdot \\phi + \\theta \\cdot \\psi",[34,480940,480942,480960,480981,481002,481020,481038,481056],{"className":480941,"ariaHidden":758},[31672],[34,480943,480945,480948,480951,480954,480957],{"className":480944},[31676],[34,480946],{"className":480947,"style":31681},[31680],[34,480949,479697],{"className":480950,"style":34452},[31685,31686],[34,480952],{"className":480953,"style":32604},[31845],[34,480955,71191],{"className":480956},[32024],[34,480958],{"className":480959,"style":32604},[31845],[34,480961,480963,480966,480969,480972,480975,480978],{"className":480962},[31676],[34,480964],{"className":480965,"style":31774},[31680],[34,480967,456],{"className":480968},[31782],[34,480970,329201],{"className":480971},[31685,31686],[34,480973],{"className":480974,"style":32604},[31845],[34,480976,223],{"className":480977},[32024],[34,480979],{"className":480980,"style":32604},[31845],[34,480982,480984,480987,480990,480993,480996,480999],{"className":480983},[31676],[34,480985],{"className":480986,"style":31774},[31680],[34,480988,449638],{"className":480989,"style":32062},[31685,31686],[34,480991,2557],{"className":480992},[31690],[34,480994],{"className":480995,"style":31846},[31845],[34,480997,202],{"className":480998},[31850],[34,481000],{"className":481001,"style":31846},[31845],[34,481003,481005,481008,481011,481014,481017],{"className":481004},[31676],[34,481006],{"className":481007,"style":31681},[31680],[34,481009,479697],{"className":481010,"style":34452},[31685,31686],[34,481012],{"className":481013,"style":32604},[31845],[34,481015,71191],{"className":481016},[32024],[34,481018],{"className":481019,"style":32604},[31845],[34,481021,481023,481026,481029,481032,481035],{"className":481022},[31676],[34,481024],{"className":481025,"style":43217},[31680],[34,481027,329201],{"className":481028},[31685,31686],[34,481030],{"className":481031,"style":32604},[31845],[34,481033,223],{"className":481034},[32024],[34,481036],{"className":481037,"style":32604},[31845],[34,481039,481041,481044,481047,481050,481053],{"className":481040},[31676],[34,481042],{"className":481043,"style":31681},[31680],[34,481045,479697],{"className":481046,"style":34452},[31685,31686],[34,481048],{"className":481049,"style":32604},[31845],[34,481051,71191],{"className":481052},[32024],[34,481054],{"className":481055,"style":32604},[31845],[34,481057,481059,481062],{"className":481058},[31676],[34,481060],{"className":481061,"style":43217},[31680],[34,481063,449638],{"className":481064,"style":32062},[31685,31686],[34,481066,481068,481154],{"className":481067},[31642],[34,481069,481071],{"className":481070},[31646],[31648,481072,481073],{"xmlns":31650},[31652,481074,481075,481151],{},[31655,481076,481077,481079,481081,481083,481085,481087,481089,481091,481093,481095,481097,481099,481101,481103,481105,481107,481109,481111,481113,481115,481117,481119,481121,481123,481125,481127,481129,481131,481133,481135,481137,481139,481141,481143,481145,481147,481149],{},[31662,481078,15972],{"stretchy":702},[31658,481080,479697],{},[31662,481082,448095],{},[31662,481084,456],{"stretchy":702},[31658,481086,329201],{},[31662,481088,223],{},[31658,481090,449638],{},[31662,481092,2557],{"stretchy":702},[31662,481094,32712],{"stretchy":702},[31662,481096,456],{"stretchy":702},[31658,481098,31942],{},[31662,481100,2557],{"stretchy":702},[31662,481102,202],{},[31658,481104,479697],{},[31662,481106,456],{"stretchy":702},[31662,481108,456],{"stretchy":702},[31658,481110,329201],{},[31662,481112,223],{},[31658,481114,449638],{},[31662,481116,2557],{"stretchy":702},[31662,481118,456],{"stretchy":702},[31658,481120,31942],{},[31662,481122,2557],{"stretchy":702},[31662,481124,2557],{"stretchy":702},[31662,481126,202],{},[31658,481128,479697],{},[31662,481130,456],{"stretchy":702},[31658,481132,329201],{},[31662,481134,456],{"stretchy":702},[31658,481136,31942],{},[31662,481138,2557],{"stretchy":702},[31662,481140,223],{},[31658,481142,449638],{},[31662,481144,456],{"stretchy":702},[31658,481146,31942],{},[31662,481148,2557],{"stretchy":702},[31662,481150,2557],{"stretchy":702},[31665,481152,481153],{"encoding":31667},"[\\theta \\circ (\\phi + \\psi)](g) = \\theta((\\phi+\\psi)(g)) = \\theta(\\phi(g) + \\psi(g))",[34,481155,481157,481178,481199,481230,481254,481284,481317],{"className":481156,"ariaHidden":758},[31672],[34,481158,481160,481163,481166,481169,481172,481175],{"className":481159},[31676],[34,481161],{"className":481162,"style":31774},[31680],[34,481164,15972],{"className":481165},[31782],[34,481167,479697],{"className":481168,"style":34452},[31685,31686],[34,481170],{"className":481171,"style":32604},[31845],[34,481173,448095],{"className":481174},[32024],[34,481176],{"className":481177,"style":32604},[31845],[34,481179,481181,481184,481187,481190,481193,481196],{"className":481180},[31676],[34,481182],{"className":481183,"style":31774},[31680],[34,481185,456],{"className":481186},[31782],[34,481188,329201],{"className":481189},[31685,31686],[34,481191],{"className":481192,"style":32604},[31845],[34,481194,223],{"className":481195},[32024],[34,481197],{"className":481198,"style":32604},[31845],[34,481200,481202,481205,481208,481212,481215,481218,481221,481224,481227],{"className":481201},[31676],[34,481203],{"className":481204,"style":31774},[31680],[34,481206,449638],{"className":481207,"style":32062},[31685,31686],[34,481209,481211],{"className":481210},[31690],")]",[34,481213,456],{"className":481214},[31782],[34,481216,31942],{"className":481217,"style":32062},[31685,31686],[34,481219,2557],{"className":481220},[31690],[34,481222],{"className":481223,"style":31846},[31845],[34,481225,202],{"className":481226},[31850],[34,481228],{"className":481229,"style":31846},[31845],[34,481231,481233,481236,481239,481242,481245,481248,481251],{"className":481232},[31676],[34,481234],{"className":481235,"style":31774},[31680],[34,481237,479697],{"className":481238,"style":34452},[31685,31686],[34,481240,1323],{"className":481241},[31782],[34,481243,329201],{"className":481244},[31685,31686],[34,481246],{"className":481247,"style":32604},[31845],[34,481249,223],{"className":481250},[32024],[34,481252],{"className":481253,"style":32604},[31845],[34,481255,481257,481260,481263,481266,481269,481272,481275,481278,481281],{"className":481256},[31676],[34,481258],{"className":481259,"style":31774},[31680],[34,481261,449638],{"className":481262,"style":32062},[31685,31686],[34,481264,2557],{"className":481265},[31690],[34,481267,456],{"className":481268},[31782],[34,481270,31942],{"className":481271,"style":32062},[31685,31686],[34,481273,44253],{"className":481274},[31690],[34,481276],{"className":481277,"style":31846},[31845],[34,481279,202],{"className":481280},[31850],[34,481282],{"className":481283,"style":31846},[31845],[34,481285,481287,481290,481293,481296,481299,481302,481305,481308,481311,481314],{"className":481286},[31676],[34,481288],{"className":481289,"style":31774},[31680],[34,481291,479697],{"className":481292,"style":34452},[31685,31686],[34,481294,456],{"className":481295},[31782],[34,481297,329201],{"className":481298},[31685,31686],[34,481300,456],{"className":481301},[31782],[34,481303,31942],{"className":481304,"style":32062},[31685,31686],[34,481306,2557],{"className":481307},[31690],[34,481309],{"className":481310,"style":32604},[31845],[34,481312,223],{"className":481313},[32024],[34,481315],{"className":481316,"style":32604},[31845],[34,481318,481320,481323,481326,481329,481332],{"className":481319},[31676],[34,481321],{"className":481322,"style":31774},[31680],[34,481324,449638],{"className":481325,"style":32062},[31685,31686],[34,481327,456],{"className":481328},[31782],[34,481330,31942],{"className":481331,"style":32062},[31685,31686],[34,481333,44253],{"className":481334},[31690],[34,481336,481338,481352],{"className":481337},[31642],[34,481339,481341],{"className":481340},[31646],[31648,481342,481343],{"xmlns":31650},[31652,481344,481345,481349],{},[31655,481346,481347],{},[31658,481348,479697],{},[31665,481350,481351],{"encoding":31667},"\\theta",[34,481353,481355],{"className":481354,"ariaHidden":758},[31672],[34,481356,481358,481361],{"className":481357},[31676],[34,481359],{"className":481360,"style":31681},[31680],[34,481362,479697],{"className":481363,"style":34452},[31685,31686]," 是同态（保持加法）：\n",[34,481366,481368,481448],{"className":481367},[31642],[34,481369,481371],{"className":481370},[31646],[31648,481372,481373],{"xmlns":31650},[31652,481374,481375,481445],{},[31655,481376,481377,481379,481381,481383,481385,481387,481389,481391,481393,481395,481397,481399,481401,481403,481405,481407,481409,481411,481413,481415,481417,481419,481421,481423,481425,481427,481429,481431,481433,481435,481437,481439,481441,481443],{},[31662,481378,202],{},[31658,481380,479697],{},[31662,481382,456],{"stretchy":702},[31658,481384,329201],{},[31662,481386,456],{"stretchy":702},[31658,481388,31942],{},[31662,481390,2557],{"stretchy":702},[31662,481392,2557],{"stretchy":702},[31662,481394,223],{},[31658,481396,479697],{},[31662,481398,456],{"stretchy":702},[31658,481400,449638],{},[31662,481402,456],{"stretchy":702},[31658,481404,31942],{},[31662,481406,2557],{"stretchy":702},[31662,481408,2557],{"stretchy":702},[31662,481410,202],{},[31662,481412,456],{"stretchy":702},[31658,481414,479697],{},[31662,481416,448095],{},[31658,481418,329201],{},[31662,481420,2557],{"stretchy":702},[31662,481422,456],{"stretchy":702},[31658,481424,31942],{},[31662,481426,2557],{"stretchy":702},[31662,481428,223],{},[31662,481430,456],{"stretchy":702},[31658,481432,479697],{},[31662,481434,448095],{},[31658,481436,449638],{},[31662,481438,2557],{"stretchy":702},[31662,481440,456],{"stretchy":702},[31658,481442,31942],{},[31662,481444,2557],{"stretchy":702},[31665,481446,481447],{"encoding":31667},"= \\theta(\\phi(g)) + \\theta(\\psi(g)) = (\\theta \\circ \\phi)(g) + (\\theta \\circ \\psi)(g)",[34,481449,481451,481463,481496,481529,481550,481580,481601],{"className":481450,"ariaHidden":758},[31672],[34,481452,481454,481457,481460],{"className":481453},[31676],[34,481455],{"className":481456,"style":73089},[31680],[34,481458,202],{"className":481459},[31850],[34,481461],{"className":481462,"style":31846},[31845],[34,481464,481466,481469,481472,481475,481478,481481,481484,481487,481490,481493],{"className":481465},[31676],[34,481467],{"className":481468,"style":31774},[31680],[34,481470,479697],{"className":481471,"style":34452},[31685,31686],[34,481473,456],{"className":481474},[31782],[34,481476,329201],{"className":481477},[31685,31686],[34,481479,456],{"className":481480},[31782],[34,481482,31942],{"className":481483,"style":32062},[31685,31686],[34,481485,44253],{"className":481486},[31690],[34,481488],{"className":481489,"style":32604},[31845],[34,481491,223],{"className":481492},[32024],[34,481494],{"className":481495,"style":32604},[31845],[34,481497,481499,481502,481505,481508,481511,481514,481517,481520,481523,481526],{"className":481498},[31676],[34,481500],{"className":481501,"style":31774},[31680],[34,481503,479697],{"className":481504,"style":34452},[31685,31686],[34,481506,456],{"className":481507},[31782],[34,481509,449638],{"className":481510,"style":32062},[31685,31686],[34,481512,456],{"className":481513},[31782],[34,481515,31942],{"className":481516,"style":32062},[31685,31686],[34,481518,44253],{"className":481519},[31690],[34,481521],{"className":481522,"style":31846},[31845],[34,481524,202],{"className":481525},[31850],[34,481527],{"className":481528,"style":31846},[31845],[34,481530,481532,481535,481538,481541,481544,481547],{"className":481531},[31676],[34,481533],{"className":481534,"style":31774},[31680],[34,481536,456],{"className":481537},[31782],[34,481539,479697],{"className":481540,"style":34452},[31685,31686],[34,481542],{"className":481543,"style":32604},[31845],[34,481545,448095],{"className":481546},[32024],[34,481548],{"className":481549,"style":32604},[31845],[34,481551,481553,481556,481559,481562,481565,481568,481571,481574,481577],{"className":481552},[31676],[34,481554],{"className":481555,"style":31774},[31680],[34,481557,329201],{"className":481558},[31685,31686],[34,481560,2557],{"className":481561},[31690],[34,481563,456],{"className":481564},[31782],[34,481566,31942],{"className":481567,"style":32062},[31685,31686],[34,481569,2557],{"className":481570},[31690],[34,481572],{"className":481573,"style":32604},[31845],[34,481575,223],{"className":481576},[32024],[34,481578],{"className":481579,"style":32604},[31845],[34,481581,481583,481586,481589,481592,481595,481598],{"className":481582},[31676],[34,481584],{"className":481585,"style":31774},[31680],[34,481587,456],{"className":481588},[31782],[34,481590,479697],{"className":481591,"style":34452},[31685,31686],[34,481593],{"className":481594,"style":32604},[31845],[34,481596,448095],{"className":481597},[32024],[34,481599],{"className":481600,"style":32604},[31845],[34,481602,481604,481607,481610,481613,481616,481619],{"className":481603},[31676],[34,481605],{"className":481606,"style":31774},[31680],[34,481608,449638],{"className":481609,"style":32062},[31685,31686],[34,481611,2557],{"className":481612},[31690],[34,481614,456],{"className":481615},[31782],[34,481617,31942],{"className":481618,"style":32062},[31685,31686],[34,481620,2557],{"className":481621},[31690],"\n得证。",[108,481624,481625,47649,481628,33183,481799,481985,481986,481622],{},[2627,481626,481627],{},"右分配律",[34,481629,481631,481673],{"className":481630},[31642],[34,481632,481634],{"className":481633},[31646],[31648,481635,481636],{"xmlns":31650},[31652,481637,481638,481670],{},[31655,481639,481640,481642,481644,481646,481648,481650,481652,481654,481656,481658,481660,481662,481664,481666,481668],{},[31662,481641,456],{"stretchy":702},[31658,481643,329201],{},[31662,481645,223],{},[31658,481647,449638],{},[31662,481649,2557],{"stretchy":702},[31662,481651,71191],{},[31658,481653,479697],{},[31662,481655,202],{},[31658,481657,329201],{},[31662,481659,71191],{},[31658,481661,479697],{},[31662,481663,223],{},[31658,481665,449638],{},[31662,481667,71191],{},[31658,481669,479697],{},[31665,481671,481672],{"encoding":31667},"(\\phi + \\psi) \\cdot \\theta = \\phi \\cdot \\theta + \\psi \\cdot \\theta",[34,481674,481676,481697,481718,481736,481754,481772,481790],{"className":481675,"ariaHidden":758},[31672],[34,481677,481679,481682,481685,481688,481691,481694],{"className":481678},[31676],[34,481680],{"className":481681,"style":31774},[31680],[34,481683,456],{"className":481684},[31782],[34,481686,329201],{"className":481687},[31685,31686],[34,481689],{"className":481690,"style":32604},[31845],[34,481692,223],{"className":481693},[32024],[34,481695],{"className":481696,"style":32604},[31845],[34,481698,481700,481703,481706,481709,481712,481715],{"className":481699},[31676],[34,481701],{"className":481702,"style":31774},[31680],[34,481704,449638],{"className":481705,"style":32062},[31685,31686],[34,481707,2557],{"className":481708},[31690],[34,481710],{"className":481711,"style":32604},[31845],[34,481713,71191],{"className":481714},[32024],[34,481716],{"className":481717,"style":32604},[31845],[34,481719,481721,481724,481727,481730,481733],{"className":481720},[31676],[34,481722],{"className":481723,"style":31681},[31680],[34,481725,479697],{"className":481726,"style":34452},[31685,31686],[34,481728],{"className":481729,"style":31846},[31845],[34,481731,202],{"className":481732},[31850],[34,481734],{"className":481735,"style":31846},[31845],[34,481737,481739,481742,481745,481748,481751],{"className":481738},[31676],[34,481740],{"className":481741,"style":43217},[31680],[34,481743,329201],{"className":481744},[31685,31686],[34,481746],{"className":481747,"style":32604},[31845],[34,481749,71191],{"className":481750},[32024],[34,481752],{"className":481753,"style":32604},[31845],[34,481755,481757,481760,481763,481766,481769],{"className":481756},[31676],[34,481758],{"className":481759,"style":34299},[31680],[34,481761,479697],{"className":481762,"style":34452},[31685,31686],[34,481764],{"className":481765,"style":32604},[31845],[34,481767,223],{"className":481768},[32024],[34,481770],{"className":481771,"style":32604},[31845],[34,481773,481775,481778,481781,481784,481787],{"className":481774},[31676],[34,481776],{"className":481777,"style":43217},[31680],[34,481779,449638],{"className":481780,"style":32062},[31685,31686],[34,481782],{"className":481783,"style":32604},[31845],[34,481785,71191],{"className":481786},[32024],[34,481788],{"className":481789,"style":32604},[31845],[34,481791,481793,481796],{"className":481792},[31676],[34,481794],{"className":481795,"style":31681},[31680],[34,481797,479697],{"className":481798,"style":34452},[31685,31686],[34,481800,481802,481862],{"className":481801},[31642],[34,481803,481805],{"className":481804},[31646],[31648,481806,481807],{"xmlns":31650},[31652,481808,481809,481859],{},[31655,481810,481811,481813,481815,481817,481819,481821,481823,481825,481827,481829,481831,481833,481835,481837,481839,481841,481843,481845,481847,481849,481851,481853,481855,481857],{},[31662,481812,15972],{"stretchy":702},[31662,481814,456],{"stretchy":702},[31658,481816,329201],{},[31662,481818,223],{},[31658,481820,449638],{},[31662,481822,2557],{"stretchy":702},[31662,481824,448095],{},[31658,481826,479697],{},[31662,481828,32712],{"stretchy":702},[31662,481830,456],{"stretchy":702},[31658,481832,31942],{},[31662,481834,2557],{"stretchy":702},[31662,481836,202],{},[31662,481838,456],{"stretchy":702},[31658,481840,329201],{},[31662,481842,223],{},[31658,481844,449638],{},[31662,481846,2557],{"stretchy":702},[31662,481848,456],{"stretchy":702},[31658,481850,479697],{},[31662,481852,456],{"stretchy":702},[31658,481854,31942],{},[31662,481856,2557],{"stretchy":702},[31662,481858,2557],{"stretchy":702},[31665,481860,481861],{"encoding":31667},"[(\\phi + \\psi) \\circ \\theta](g) = (\\phi + \\psi)(\\theta(g))",[34,481863,481865,481886,481907,481937,481958],{"className":481864,"ariaHidden":758},[31672],[34,481866,481868,481871,481874,481877,481880,481883],{"className":481867},[31676],[34,481869],{"className":481870,"style":31774},[31680],[34,481872,86328],{"className":481873},[31782],[34,481875,329201],{"className":481876},[31685,31686],[34,481878],{"className":481879,"style":32604},[31845],[34,481881,223],{"className":481882},[32024],[34,481884],{"className":481885,"style":32604},[31845],[34,481887,481889,481892,481895,481898,481901,481904],{"className":481888},[31676],[34,481890],{"className":481891,"style":31774},[31680],[34,481893,449638],{"className":481894,"style":32062},[31685,31686],[34,481896,2557],{"className":481897},[31690],[34,481899],{"className":481900,"style":32604},[31845],[34,481902,448095],{"className":481903},[32024],[34,481905],{"className":481906,"style":32604},[31845],[34,481908,481910,481913,481916,481919,481922,481925,481928,481931,481934],{"className":481909},[31676],[34,481911],{"className":481912,"style":31774},[31680],[34,481914,479697],{"className":481915,"style":34452},[31685,31686],[34,481917,32712],{"className":481918},[31690],[34,481920,456],{"className":481921},[31782],[34,481923,31942],{"className":481924,"style":32062},[31685,31686],[34,481926,2557],{"className":481927},[31690],[34,481929],{"className":481930,"style":31846},[31845],[34,481932,202],{"className":481933},[31850],[34,481935],{"className":481936,"style":31846},[31845],[34,481938,481940,481943,481946,481949,481952,481955],{"className":481939},[31676],[34,481941],{"className":481942,"style":31774},[31680],[34,481944,456],{"className":481945},[31782],[34,481947,329201],{"className":481948},[31685,31686],[34,481950],{"className":481951,"style":32604},[31845],[34,481953,223],{"className":481954},[32024],[34,481956],{"className":481957,"style":32604},[31845],[34,481959,481961,481964,481967,481970,481973,481976,481979,481982],{"className":481960},[31676],[34,481962],{"className":481963,"style":31774},[31680],[34,481965,449638],{"className":481966,"style":32062},[31685,31686],[34,481968,2557],{"className":481969},[31690],[34,481971,456],{"className":481972},[31782],[34,481974,479697],{"className":481975,"style":34452},[31685,31686],[34,481977,456],{"className":481978},[31782],[34,481980,31942],{"className":481981,"style":32062},[31685,31686],[34,481983,44253],{"className":481984},[31690],"\n根据加法的定义：\n",[34,481987,481989,482069],{"className":481988},[31642],[34,481990,481992],{"className":481991},[31646],[31648,481993,481994],{"xmlns":31650},[31652,481995,481996,482066],{},[31655,481997,481998,482000,482002,482004,482006,482008,482010,482012,482014,482016,482018,482020,482022,482024,482026,482028,482030,482032,482034,482036,482038,482040,482042,482044,482046,482048,482050,482052,482054,482056,482058,482060,482062,482064],{},[31662,481999,202],{},[31658,482001,329201],{},[31662,482003,456],{"stretchy":702},[31658,482005,479697],{},[31662,482007,456],{"stretchy":702},[31658,482009,31942],{},[31662,482011,2557],{"stretchy":702},[31662,482013,2557],{"stretchy":702},[31662,482015,223],{},[31658,482017,449638],{},[31662,482019,456],{"stretchy":702},[31658,482021,479697],{},[31662,482023,456],{"stretchy":702},[31658,482025,31942],{},[31662,482027,2557],{"stretchy":702},[31662,482029,2557],{"stretchy":702},[31662,482031,202],{},[31662,482033,456],{"stretchy":702},[31658,482035,329201],{},[31662,482037,448095],{},[31658,482039,479697],{},[31662,482041,2557],{"stretchy":702},[31662,482043,456],{"stretchy":702},[31658,482045,31942],{},[31662,482047,2557],{"stretchy":702},[31662,482049,223],{},[31662,482051,456],{"stretchy":702},[31658,482053,449638],{},[31662,482055,448095],{},[31658,482057,479697],{},[31662,482059,2557],{"stretchy":702},[31662,482061,456],{"stretchy":702},[31658,482063,31942],{},[31662,482065,2557],{"stretchy":702},[31665,482067,482068],{"encoding":31667},"= \\phi(\\theta(g)) + \\psi(\\theta(g)) = (\\phi \\circ \\theta)(g) + (\\psi \\circ \\theta)(g)",[34,482070,482072,482084,482117,482150,482171,482201,482222],{"className":482071,"ariaHidden":758},[31672],[34,482073,482075,482078,482081],{"className":482074},[31676],[34,482076],{"className":482077,"style":73089},[31680],[34,482079,202],{"className":482080},[31850],[34,482082],{"className":482083,"style":31846},[31845],[34,482085,482087,482090,482093,482096,482099,482102,482105,482108,482111,482114],{"className":482086},[31676],[34,482088],{"className":482089,"style":31774},[31680],[34,482091,329201],{"className":482092},[31685,31686],[34,482094,456],{"className":482095},[31782],[34,482097,479697],{"className":482098,"style":34452},[31685,31686],[34,482100,456],{"className":482101},[31782],[34,482103,31942],{"className":482104,"style":32062},[31685,31686],[34,482106,44253],{"className":482107},[31690],[34,482109],{"className":482110,"style":32604},[31845],[34,482112,223],{"className":482113},[32024],[34,482115],{"className":482116,"style":32604},[31845],[34,482118,482120,482123,482126,482129,482132,482135,482138,482141,482144,482147],{"className":482119},[31676],[34,482121],{"className":482122,"style":31774},[31680],[34,482124,449638],{"className":482125,"style":32062},[31685,31686],[34,482127,456],{"className":482128},[31782],[34,482130,479697],{"className":482131,"style":34452},[31685,31686],[34,482133,456],{"className":482134},[31782],[34,482136,31942],{"className":482137,"style":32062},[31685,31686],[34,482139,44253],{"className":482140},[31690],[34,482142],{"className":482143,"style":31846},[31845],[34,482145,202],{"className":482146},[31850],[34,482148],{"className":482149,"style":31846},[31845],[34,482151,482153,482156,482159,482162,482165,482168],{"className":482152},[31676],[34,482154],{"className":482155,"style":31774},[31680],[34,482157,456],{"className":482158},[31782],[34,482160,329201],{"className":482161},[31685,31686],[34,482163],{"className":482164,"style":32604},[31845],[34,482166,448095],{"className":482167},[32024],[34,482169],{"className":482170,"style":32604},[31845],[34,482172,482174,482177,482180,482183,482186,482189,482192,482195,482198],{"className":482173},[31676],[34,482175],{"className":482176,"style":31774},[31680],[34,482178,479697],{"className":482179,"style":34452},[31685,31686],[34,482181,2557],{"className":482182},[31690],[34,482184,456],{"className":482185},[31782],[34,482187,31942],{"className":482188,"style":32062},[31685,31686],[34,482190,2557],{"className":482191},[31690],[34,482193],{"className":482194,"style":32604},[31845],[34,482196,223],{"className":482197},[32024],[34,482199],{"className":482200,"style":32604},[31845],[34,482202,482204,482207,482210,482213,482216,482219],{"className":482203},[31676],[34,482205],{"className":482206,"style":31774},[31680],[34,482208,456],{"className":482209},[31782],[34,482211,449638],{"className":482212,"style":32062},[31685,31686],[34,482214],{"className":482215,"style":32604},[31845],[34,482217,448095],{"className":482218},[32024],[34,482220],{"className":482221,"style":32604},[31845],[34,482223,482225,482228,482231,482234,482237,482240],{"className":482224},[31676],[34,482226],{"className":482227,"style":31774},[31680],[34,482229,479697],{"className":482230,"style":34452},[31685,31686],[34,482232,2557],{"className":482233},[31690],[34,482235,456],{"className":482236},[31782],[34,482238,31942],{"className":482239,"style":32062},[31685,31686],[34,482241,2557],{"className":482242},[31690],[108,482244,482245,482248,482249,1840,482327,482398,482399,46528,482427,347],{},[2627,482246,482247],{},"幺元","：\n恒等映射 ",[34,482250,482252,482276],{"className":482251},[31642],[34,482253,482255],{"className":482254},[31646],[31648,482256,482257],{"xmlns":31650},[31652,482258,482259,482273],{},[31655,482260,482261,482263,482265,482267,482269,482271],{},[31658,482262,31955],{},[31658,482264,46900],{},[31662,482266,5598],{},[31658,482268,97709],{},[31662,482270,33239],{},[31658,482272,97709],{},[31665,482274,482275],{"encoding":31667},"id: G \\to G",[34,482277,482279,482300,482318],{"className":482278,"ariaHidden":758},[31672],[34,482280,482282,482285,482288,482291,482294,482297],{"className":482281},[31676],[34,482283],{"className":482284,"style":31681},[31680],[34,482286,31955],{"className":482287},[31685,31686],[34,482289,46900],{"className":482290},[31685,31686],[34,482292],{"className":482293,"style":31846},[31845],[34,482295,5598],{"className":482296},[31850],[34,482298],{"className":482299,"style":31846},[31845],[34,482301,482303,482306,482309,482312,482315],{"className":482302},[31676],[34,482304],{"className":482305,"style":33218},[31680],[34,482307,97709],{"className":482308},[31685,31686],[34,482310],{"className":482311,"style":31846},[31845],[34,482313,33239],{"className":482314},[31850],[34,482316],{"className":482317,"style":31846},[31845],[34,482319,482321,482324],{"className":482320},[31676],[34,482322],{"className":482323,"style":33218},[31680],[34,482325,97709],{"className":482326},[31685,31686],[34,482328,482330,482356],{"className":482329},[31642],[34,482331,482333],{"className":482332},[31646],[31648,482334,482335],{"xmlns":31650},[31652,482336,482337,482353],{},[31655,482338,482339,482341,482343,482345,482347,482349,482351],{},[31658,482340,31955],{},[31658,482342,46900],{},[31662,482344,456],{"stretchy":702},[31658,482346,31942],{},[31662,482348,2557],{"stretchy":702},[31662,482350,202],{},[31658,482352,31942],{},[31665,482354,482355],{"encoding":31667},"id(g) = g",[34,482357,482359,482389],{"className":482358,"ariaHidden":758},[31672],[34,482360,482362,482365,482368,482371,482374,482377,482380,482383,482386],{"className":482361},[31676],[34,482363],{"className":482364,"style":31774},[31680],[34,482366,31955],{"className":482367},[31685,31686],[34,482369,46900],{"className":482370},[31685,31686],[34,482372,456],{"className":482373},[31782],[34,482375,31942],{"className":482376,"style":32062},[31685,31686],[34,482378,2557],{"className":482379},[31690],[34,482381],{"className":482382,"style":31846},[31845],[34,482384,202],{"className":482385},[31850],[34,482387],{"className":482388,"style":31846},[31845],[34,482390,482392,482395],{"className":482391},[31676],[34,482393],{"className":482394,"style":38774},[31680],[34,482396,31942],{"className":482397,"style":32062},[31685,31686],") 是同态，且对于任何 ",[34,482400,482402,482415],{"className":482401},[31642],[34,482403,482405],{"className":482404},[31646],[31648,482406,482407],{"xmlns":31650},[31652,482408,482409,482413],{},[31655,482410,482411],{},[31658,482412,329201],{},[31665,482414,364484],{"encoding":31667},[34,482416,482418],{"className":482417,"ariaHidden":758},[31672],[34,482419,482421,482424],{"className":482420},[31676],[34,482422],{"className":482423,"style":43217},[31680],[34,482425,329201],{"className":482426},[31685,31686],[34,482428,482430,482464],{"className":482429},[31642],[34,482431,482433],{"className":482432},[31646],[31648,482434,482435],{"xmlns":31650},[31652,482436,482437,482461],{},[31655,482438,482439,482441,482443,482445,482447,482449,482451,482453,482455,482457,482459],{},[31658,482440,329201],{},[31662,482442,448095],{},[31658,482444,31955],{},[31658,482446,46900],{},[31662,482448,202],{},[31658,482450,329201],{},[31662,482452,202],{},[31658,482454,31955],{},[31658,482456,46900],{},[31662,482458,448095],{},[31658,482460,329201],{},[31665,482462,482463],{"encoding":31667},"\\phi \\circ id = \\phi = id \\circ \\phi",[34,482465,482467,482485,482506,482524,482545],{"className":482466,"ariaHidden":758},[31672],[34,482468,482470,482473,482476,482479,482482],{"className":482469},[31676],[34,482471],{"className":482472,"style":43217},[31680],[34,482474,329201],{"className":482475},[31685,31686],[34,482477],{"className":482478,"style":32604},[31845],[34,482480,448095],{"className":482481},[32024],[34,482483],{"className":482484,"style":32604},[31845],[34,482486,482488,482491,482494,482497,482500,482503],{"className":482487},[31676],[34,482489],{"className":482490,"style":31681},[31680],[34,482492,31955],{"className":482493},[31685,31686],[34,482495,46900],{"className":482496},[31685,31686],[34,482498],{"className":482499,"style":31846},[31845],[34,482501,202],{"className":482502},[31850],[34,482504],{"className":482505,"style":31846},[31845],[34,482507,482509,482512,482515,482518,482521],{"className":482508},[31676],[34,482510],{"className":482511,"style":43217},[31680],[34,482513,329201],{"className":482514},[31685,31686],[34,482516],{"className":482517,"style":31846},[31845],[34,482519,202],{"className":482520},[31850],[34,482522],{"className":482523,"style":31846},[31845],[34,482525,482527,482530,482533,482536,482539,482542],{"className":482526},[31676],[34,482528],{"className":482529,"style":31681},[31680],[34,482531,31955],{"className":482532},[31685,31686],[34,482534,46900],{"className":482535},[31685,31686],[34,482537],{"className":482538,"style":32604},[31845],[34,482540,448095],{"className":482541},[32024],[34,482543],{"className":482544,"style":32604},[31845],[34,482546,482548,482551],{"className":482547},[31676],[34,482549],{"className":482550,"style":43217},[31680],[34,482552,329201],{"className":482553},[31685,31686],[15,482555,482556,2664,482558,482606,482607],{},[2627,482557,114043],{},[34,482559,482561,482582],{"className":482560},[31642],[34,482562,482564],{"className":482563},[31646],[31648,482565,482566],{"xmlns":31650},[31652,482567,482568,482580],{},[31655,482569,482570,482572,482574,482576,482578],{},[31658,482571,440113],{"mathvariant":38079},[31662,482573,32682],{},[31662,482575,456],{"stretchy":702},[31658,482577,97709],{},[31662,482579,2557],{"stretchy":702},[31665,482581,479452],{"encoding":31667},[34,482583,482585],{"className":482584,"ariaHidden":758},[31672],[34,482586,482588,482591,482597,482600,482603],{"className":482587},[31676],[34,482589],{"className":482590,"style":31774},[31680],[34,482592,482594],{"className":482593},[32767],[34,482595,440113],{"className":482596},[31685,207923],[34,482598,456],{"className":482599},[31782],[34,482601,97709],{"className":482602},[31685,31686],[34,482604,2557],{"className":482605},[31690]," 构成幺环。 ",[34,482608,482610,482623],{"className":482609},[31642],[34,482611,482613],{"className":482612},[31646],[31648,482614,482615],{"xmlns":31650},[31652,482616,482617,482621],{},[31655,482618,482619],{},[31658,482620,445204],{"mathvariant":38079},[31665,482622,445207],{"encoding":31667},[34,482624,482626],{"className":482625,"ariaHidden":758},[31672],[34,482627,482629,482632],{"className":482628},[31676],[34,482630],{"className":482631,"style":445217},[31680],[34,482633,445204],{"className":482634},[31685,129261],[32188,482636],{},[32490,482638,482640],{"id":482639},"习题-18","习题 18",[15,482642,482643],{},[2627,482644,482645,482646,482719,482720,414527],{},"判断命题正误（",[34,482647,482649,482671],{"className":482648},[31642],[34,482650,482652],{"className":482651},[31646],[31648,482653,482654],{"xmlns":31650},[31652,482655,482656,482668],{},[31655,482657,482658,482660,482662,482664,482666],{},[31658,482659,126541],{},[31662,482661,5598],{},[31658,482663,34386],{},[31662,482665,33239],{},[31658,482667,94675],{},[31665,482669,482670],{"encoding":31667},"\\varphi: R \\to S",[34,482672,482674,482692,482710],{"className":482673,"ariaHidden":758},[31672],[34,482675,482677,482680,482683,482686,482689],{"className":482676},[31676],[34,482678],{"className":482679,"style":38774},[31680],[34,482681,126541],{"className":482682},[31685,31686],[34,482684],{"className":482685,"style":31846},[31845],[34,482687,5598],{"className":482688},[31850],[34,482690],{"className":482691,"style":31846},[31845],[34,482693,482695,482698,482701,482704,482707],{"className":482694},[31676],[34,482696],{"className":482697,"style":33218},[31680],[34,482699,34386],{"className":482700,"style":34427},[31685,31686],[34,482702],{"className":482703,"style":31846},[31845],[34,482705,33239],{"className":482706},[31850],[34,482708],{"className":482709,"style":31846},[31845],[34,482711,482713,482716],{"className":482712},[31676],[34,482714],{"className":482715,"style":33218},[31680],[34,482717,94675],{"className":482718,"style":106455},[31685,31686]," 是满同态，且 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把幂零 (幂等) 元映为幂零 (幂等) 元。",[105,482902,482903,482908],{},[108,482904,482905,347],{},[2627,482906,482907],{},"正确",[108,482909,482910,33183,482912],{},[2627,482911,390672],{},[105,482913,482914,483422],{},[108,482915,133910,482916,482944,482945,482973,482974,39984,483096,322067,483378,340132],{},[34,482917,482919,482932],{"className":482918},[31642],[34,482920,482922],{"className":482921},[31646],[31648,482923,482924],{"xmlns":31650},[31652,482925,482926,482930],{},[31655,482927,482928],{},[31658,482929,111],{},[31665,482931,111],{"encoding":31667},[34,482933,482935],{"className":482934,"ariaHidden":758},[31672],[34,482936,482938,482941],{"className":482937},[31676],[34,482939],{"className":482940,"style":32527},[31680],[34,482942,111],{"className":482943},[31685,31686]," 是幂零元，即存在 ",[34,482946,482948,482961],{"className":482947},[31642],[34,482949,482951],{"className":482950},[31646],[31648,482952,482953],{"xmlns":31650},[31652,482954,482955,482959],{},[31655,482956,482957],{},[31658,482958,31660],{},[31665,482960,31660],{"encoding":31667},[34,482962,482964],{"className":482963,"ariaHidden":758},[31672],[34,482965,482967,482970],{"className":482966},[31676],[34,482968],{"className":482969,"style":32527},[31680],[34,482971,31660],{"className":482972},[31685,31686]," 使 ",[34,482975,482977,483003],{"className":482976},[31642],[34,482978,482980],{"className":482979},[31646],[31648,482981,482982],{"xmlns":31650},[31652,482983,482984,483000],{},[31655,482985,482986,482992,482994],{},[37624,482987,482988,482990],{},[31658,482989,111],{},[31658,482991,31660],{},[31662,482993,202],{},[31743,482995,482996,482998],{},[31749,482997,459],{},[31658,482999,34386],{},[31665,483001,483002],{"encoding":31667},"a^n = 0_R",[34,483004,483006,483050],{"className":483005,"ariaHidden":758},[31672],[34,483007,483009,483012,483041,483044,483047],{"className":483008},[31676],[34,483010],{"className":483011,"style":37959},[31680],[34,483013,483015,483018],{"className":483014},[31685],[34,483016,111],{"className":483017},[31685,31686],[34,483019,483021],{"className":483020},[31792],[34,483022,483024],{"className":483023},[31796],[34,483025,483027],{"className":483026},[31801],[34,483028,483030],{"className":483029,"style":37959},[31805],[34,483031,483032,483035],{"style":37777},[34,483033],{"className":483034,"style":31814},[31813],[34,483036,483038],{"className":483037},[31818,31819,31820,31821],[34,483039,31660],{"className":483040},[31685,31686,31821],[34,483042],{"className":483043,"style":31846},[31845],[34,483045,202],{"className":483046},[31850],[34,483048],{"className":483049,"style":31846},[31845],[34,483051,483053,483056],{"className":483052},[31676],[34,483054],{"className":483055,"style":38357},[31680],[34,483057,483059,483062],{"className":483058},[31685],[34,483060,459],{"className":483061},[31685],[34,483063,483065],{"className":483064},[31792],[34,483066,483068,483088],{"className":483067},[31796,31797],[34,483069,483071,483085],{"className":483070},[31801],[34,483072,483074],{"className":483073,"style":116558},[31805],[34,483075,483076,483079],{"style":31809},[34,483077],{"className":483078,"style":31814},[31813],[34,483080,483082],{"className":483081},[31818,31819,31820,31821],[34,483083,34386],{"className":483084,"style":34427},[31685,31686,31821],[34,483086,31829],{"className":483087},[31828],[34,483089,483091],{"className":483090},[31801],[34,483092,483094],{"className":483093,"style":31836},[31805],[34,483095],{},[34,483097,483099,483159],{"className":483098},[31642],[34,483100,483102],{"className":483101},[31646],[31648,483103,483104],{"xmlns":31650},[31652,483105,483106,483156],{},[31655,483107,483108,483110,483112,483114,483120,483122,483124,483126,483132,483134,483136,483138,483140,483146,483148,483150],{},[31658,483109,126541],{},[31662,483111,456],{"stretchy":702},[31658,483113,111],{},[37624,483115,483116,483118],{},[31662,483117,2557],{"stretchy":702},[31658,483119,31660],{},[31662,483121,202],{},[31658,483123,126541],{},[31662,483125,456],{"stretchy":702},[37624,483127,483128,483130],{},[31658,483129,111],{},[31658,483131,31660],{},[31662,483133,2557],{"stretchy":702},[31662,483135,202],{},[31658,483137,126541],{},[31662,483139,456],{"stretchy":702},[31743,483141,483142,483144],{},[31749,483143,459],{},[31658,483145,34386],{},[31662,483147,2557],{"stretchy":702},[31662,483149,202],{},[31743,483151,483152,483154],{},[31749,483153,459],{},[31658,483155,94675],{},[31665,483157,483158],{"encoding":31667},"\\varphi(a)^n = \\varphi(a^n) = \\varphi(0_R) = 0_S",[34,483160,483162,483215,483268,483332],{"className":483161,"ariaHidden":758},[31672],[34,483163,483165,483168,483171,483174,483177,483206,483209,483212],{"className":483164},[31676],[34,483166],{"className":483167,"style":31774},[31680],[34,483169,126541],{"className":483170},[31685,31686],[34,483172,456],{"className":483173},[31782],[34,483175,111],{"className":483176},[31685,31686],[34,483178,483180,483183],{"className":483179},[31690],[34,483181,2557],{"className":483182},[31690],[34,483184,483186],{"className":483185},[31792],[34,483187,483189],{"className":483188},[31796],[34,483190,483192],{"className":483191},[31801],[34,483193,483195],{"className":483194,"style":37959},[31805],[34,483196,483197,483200],{"style":37777},[34,483198],{"className":483199,"style":31814},[31813],[34,483201,483203],{"className":483202},[31818,31819,31820,31821],[34,483204,31660],{"className":483205},[31685,31686,31821],[34,483207],{"className":483208,"style":31846},[31845],[34,483210,202],{"className":483211},[31850],[34,483213],{"className":483214,"style":31846},[31845],[34,483216,483218,483221,483224,483227,483256,483259,483262,483265],{"className":483217},[31676],[34,483219],{"className":483220,"style":31774},[31680],[34,483222,126541],{"className":483223},[31685,31686],[34,483225,456],{"className":483226},[31782],[34,483228,483230,483233],{"className":483229},[31685],[34,483231,111],{"className":483232},[31685,31686],[34,483234,483236],{"className":483235},[31792],[34,483237,483239],{"className":483238},[31796],[34,483240,483242],{"className":483241},[31801],[34,483243,483245],{"className":483244,"style":37959},[31805],[34,483246,483247,483250],{"style":37777},[34,483248],{"className":483249,"style":31814},[31813],[34,483251,483253],{"className":483252},[31818,31819,31820,31821],[34,483254,31660],{"className":483255},[31685,31686,31821],[34,483257,2557],{"className":483258},[31690],[34,483260],{"className":483261,"style":31846},[31845],[34,483263,202],{"className":483264},[31850],[34,483266],{"className":483267,"style":31846},[31845],[34,483269,483271,483274,483277,483280,483320,483323,483326,483329],{"className":483270},[31676],[34,483272],{"className":483273,"style":31774},[31680],[34,483275,126541],{"className":483276},[31685,31686],[34,483278,456],{"className":483279},[31782],[34,483281,483283,483286],{"className":483282},[31685],[34,483284,459],{"className":483285},[31685],[34,483287,483289],{"className":483288},[31792],[34,483290,483292,483312],{"className":483291},[31796,31797],[34,483293,483295,483309],{"className":483294},[31801],[34,483296,483298],{"className":483297,"style":116558},[31805],[34,483299,483300,483303],{"style":31809},[34,483301],{"className":483302,"style":31814},[31813],[34,483304,483306],{"className":483305},[31818,31819,31820,31821],[34,483307,34386],{"className":483308,"style":34427},[31685,31686,31821],[34,483310,31829],{"className":483311},[31828],[34,483313,483315],{"className":483314},[31801],[34,483316,483318],{"className":483317,"style":31836},[31805],[34,483319],{},[34,483321,2557],{"className":483322},[31690],[34,483324],{"className":483325,"style":31846},[31845],[34,483327,202],{"className":483328},[31850],[34,483330],{"className":483331,"style":31846},[31845],[34,483333,483335,483338],{"className":483334},[31676],[34,483336],{"className":483337,"style":38357},[31680],[34,483339,483341,483344],{"className":483340},[31685],[34,483342,459],{"className":483343},[31685],[34,483345,483347],{"className":483346},[31792],[34,483348,483350,483370],{"className":483349},[31796,31797],[34,483351,483353,483367],{"className":483352},[31801],[34,483354,483356],{"className":483355,"style":116558},[31805],[34,483357,483358,483361],{"style":31809},[34,483359],{"className":483360,"style":31814},[31813],[34,483362,483364],{"className":483363},[31818,31819,31820,31821],[34,483365,94675],{"className":483366,"style":106455},[31685,31686,31821],[34,483368,31829],{"className":483369},[31828],[34,483371,483373],{"className":483372},[31801],[34,483374,483376],{"className":483375,"style":31836},[31805],[34,483377],{},[34,483379,483381,483401],{"className":483380},[31642],[34,483382,483384],{"className":483383},[31646],[31648,483385,483386],{"xmlns":31650},[31652,483387,483388,483398],{},[31655,483389,483390,483392,483394,483396],{},[31658,483391,126541],{},[31662,483393,456],{"stretchy":702},[31658,483395,111],{},[31662,483397,2557],{"stretchy":702},[31665,483399,483400],{"encoding":31667},"\\varphi(a)",[34,483402,483404],{"className":483403,"ariaHidden":758},[31672],[34,483405,483407,483410,483413,483416,483419],{"className":483406},[31676],[34,483408],{"className":483409,"style":31774},[31680],[34,483411,126541],{"className":483412},[31685,31686],[34,483414,456],{"className":483415},[31782],[34,483417,111],{"className":483418},[31685,31686],[34,483420,2557],{"className":483421},[31690],[108,483423,133910,483424,483452,483453,39984,483534,322067,483712,483755],{},[34,483425,483427,483440],{"className":483426},[31642],[34,483428,483430],{"className":483429},[31646],[31648,483431,483432],{"xmlns":31650},[31652,483433,483434,483438],{},[31655,483435,483436],{},[31658,483437,111],{},[31665,483439,111],{"encoding":31667},[34,483441,483443],{"className":483442,"ariaHidden":758},[31672],[34,483444,483446,483449],{"className":483445},[31676],[34,483447],{"className":483448,"style":32527},[31680],[34,483450,111],{"className":483451},[31685,31686]," 是幂等元，即 ",[34,483454,483456,483478],{"className":483455},[31642],[34,483457,483459],{"className":483458},[31646],[31648,483460,483461],{"xmlns":31650},[31652,483462,483463,483475],{},[31655,483464,483465,483471,483473],{},[37624,483466,483467,483469],{},[31658,483468,111],{},[31749,483470,768],{},[31662,483472,202],{},[31658,483474,111],{},[31665,483476,483477],{"encoding":31667},"a^2 = a",[34,483479,483481,483525],{"className":483480,"ariaHidden":758},[31672],[34,483482,483484,483487,483516,483519,483522],{"className":483483},[31676],[34,483485],{"className":483486,"style":37774},[31680],[34,483488,483490,483493],{"className":483489},[31685],[34,483491,111],{"className":483492},[31685,31686],[34,483494,483496],{"className":483495},[31792],[34,483497,483499],{"className":483498},[31796],[34,483500,483502],{"className":483501},[31801],[34,483503,483505],{"className":483504,"style":37774},[31805],[34,483506,483507,483510],{"style":37777},[34,483508],{"className":483509,"style":31814},[31813],[34,483511,483513],{"className":483512},[31818,31819,31820,31821],[34,483514,768],{"className":483515},[31685,31821],[34,483517],{"className":483518,"style":31846},[31845],[34,483520,202],{"className":483521},[31850],[34,483523],{"className":483524,"style":31846},[31845],[34,483526,483528,483531],{"className":483527},[31676],[34,483529],{"className":483530,"style":32527},[31680],[34,483532,111],{"className":483533},[31685,31686],[34,483535,483537,483585],{"className":483536},[31642],[34,483538,483540],{"className":483539},[31646],[31648,483541,483542],{"xmlns":31650},[31652,483543,483544,483582],{},[31655,483545,483546,483548,483550,483552,483558,483560,483562,483564,483570,483572,483574,483576,483578,483580],{},[31658,483547,126541],{},[31662,483549,456],{"stretchy":702},[31658,483551,111],{},[37624,483553,483554,483556],{},[31662,483555,2557],{"stretchy":702},[31749,483557,768],{},[31662,483559,202],{},[31658,483561,126541],{},[31662,483563,456],{"stretchy":702},[37624,483565,483566,483568],{},[31658,483567,111],{},[31749,483569,768],{},[31662,483571,2557],{"stretchy":702},[31662,483573,202],{},[31658,483575,126541],{},[31662,483577,456],{"stretchy":702},[31658,483579,111],{},[31662,483581,2557],{"stretchy":702},[31665,483583,483584],{"encoding":31667},"\\varphi(a)^2 = \\varphi(a^2) = \\varphi(a)",[34,483586,483588,483641,483694],{"className":483587,"ariaHidden":758},[31672],[34,483589,483591,483594,483597,483600,483603,483632,483635,483638],{"className":483590},[31676],[34,483592],{"className":483593,"style":37755},[31680],[34,483595,126541],{"className":483596},[31685,31686],[34,483598,456],{"className":483599},[31782],[34,483601,111],{"className":483602},[31685,31686],[34,483604,483606,483609],{"className":483605},[31690],[34,483607,2557],{"className":483608},[31690],[34,483610,483612],{"className":483611},[31792],[34,483613,483615],{"className":483614},[31796],[34,483616,483618],{"className":483617},[31801],[34,483619,483621],{"className":483620,"style":37774},[31805],[34,483622,483623,483626],{"style":37777},[34,483624],{"className":483625,"style":31814},[31813],[34,483627,483629],{"className":483628},[31818,31819,31820,31821],[34,483630,768],{"className":483631},[31685,31821],[34,483633],{"className":483634,"style":31846},[31845],[34,483636,202],{"className":483637},[31850],[34,483639],{"className":483640,"style":31846},[31845],[34,483642,483644,483647,483650,483653,483682,483685,483688,483691],{"className":483643},[31676],[34,483645],{"className":483646,"style":37755},[31680],[34,483648,126541],{"className":483649},[31685,31686],[34,483651,456],{"className":483652},[31782],[34,483654,483656,483659],{"className":483655},[31685],[34,483657,111],{"className":483658},[31685,31686],[34,483660,483662],{"className":483661},[31792],[34,483663,483665],{"className":483664},[31796],[34,483666,483668],{"className":483667},[31801],[34,483669,483671],{"className":483670,"style":37774},[31805],[34,483672,483673,483676],{"style":37777},[34,483674],{"className":483675,"style":31814},[31813],[34,483677,483679],{"className":483678},[31818,31819,31820,31821],[34,483680,768],{"className":483681},[31685,31821],[34,483683,2557],{"className":483684},[31690],[34,483686],{"className":483687,"style":31846},[31845],[34,483689,202],{"className":483690},[31850],[34,483692],{"className":483693,"style":31846},[31845],[34,483695,483697,483700,483703,483706,483709],{"className":483696},[31676],[34,483698],{"className":483699,"style":31774},[31680],[34,483701,126541],{"className":483702},[31685,31686],[34,483704,456],{"className":483705},[31782],[34,483707,111],{"className":483708},[31685,31686],[34,483710,2557],{"className":483711},[31690],[34,483713,483715,483734],{"className":483714},[31642],[34,483716,483718],{"className":483717},[31646],[31648,483719,483720],{"xmlns":31650},[31652,483721,483722,483732],{},[31655,483723,483724,483726,483728,483730],{},[31658,483725,126541],{},[31662,483727,456],{"stretchy":702},[31658,483729,111],{},[31662,483731,2557],{"stretchy":702},[31665,483733,483400],{"encoding":31667},[34,483735,483737],{"className":483736,"ariaHidden":758},[31672],[34,483738,483740,483743,483746,483749,483752],{"className":483739},[31676],[34,483741],{"className":483742,"style":31774},[31680],[34,483744,126541],{"className":483745},[31685,31686],[34,483747,456],{"className":483748},[31782],[34,483750,111],{"className":483751},[31685,31686],[34,483753,2557],{"className":483754},[31690]," 是幂等元。",[15,483757,483758],{},[2627,483759,388194,483760,483788],{},[34,483761,483763,483776],{"className":483762},[31642],[34,483764,483766],{"className":483765},[31646],[31648,483767,483768],{"xmlns":31650},[31652,483769,483770,483774],{},[31655,483771,483772],{},[31658,483773,126541],{},[31665,483775,126708],{"encoding":31667},[34,483777,483779],{"className":483778,"ariaHidden":758},[31672],[34,483780,483782,483785],{"className":483781},[31676],[34,483783],{"className":483784,"style":38774},[31680],[34,483786,126541],{"className":483787},[31685,31686]," 把零因子映为零因子。",[105,483790,483791,483796],{},[108,483792,483793,347],{},[2627,483794,483795],{},"错误",[108,483797,483798,483801,483802,483894,483895,483987,483988,376803,484107,336459,484176,13977,484249,484277,484278,314945,484344,484413,484414,484442,484443,484471,484472,484500,484501,46528,484573,39984,484623,347,484706,11008,484757,484785,484786,484838,484839,314945,484867,484895],{},[2627,483799,483800],{},"反例","：考虑 ",[34,483803,483805,483827],{"className":483804},[31642],[34,483806,483808],{"className":483807},[31646],[31648,483809,483810],{"xmlns":31650},[31652,483811,483812,483824],{},[31655,483813,483814,483816,483818],{},[31658,483815,34386],{},[31662,483817,202],{},[31743,483819,483820,483822],{},[31658,483821,97300],{"mathvariant":318416},[31749,483823,15932],{},[31665,483825,483826],{"encoding":31667},"R = \\mathbb{Z}_4",[34,483828,483830,483848],{"className":483829,"ariaHidden":758},[31672],[34,483831,483833,483836,483839,483842,483845],{"className":483832},[31676],[34,483834],{"className":483835,"style":33218},[31680],[34,483837,34386],{"className":483838,"style":34427},[31685,31686],[34,483840],{"className":483841,"style":31846},[31845],[34,483843,202],{"className":483844},[31850],[34,483846],{"className":483847,"style":31846},[31845],[34,483849,483851,483854],{"className":483850},[31676],[34,483852],{"className":483853,"style":330413},[31680],[34,483855,483857,483860],{"className":483856},[31685],[34,483858,97300],{"className":483859},[31685,318441],[34,483861,483863],{"className":483862},[31792],[34,483864,483866,483886],{"className":483865},[31796,31797],[34,483867,483869,483883],{"className":483868},[31801],[34,483870,483872],{"className":483871,"style":31806},[31805],[34,483873,483874,483877],{"style":31809},[34,483875],{"className":483876,"style":31814},[31813],[34,483878,483880],{"className":483879},[31818,31819,31820,31821],[34,483881,15932],{"className":483882},[31685,31821],[34,483884,31829],{"className":483885},[31828],[34,483887,483889],{"className":483888},[31801],[34,483890,483892],{"className":483891,"style":31836},[31805],[34,483893],{},"（整数模4），",[34,483896,483898,483920],{"className":483897},[31642],[34,483899,483901],{"className":483900},[31646],[31648,483902,483903],{"xmlns":31650},[31652,483904,483905,483917],{},[31655,483906,483907,483909,483911],{},[31658,483908,94675],{},[31662,483910,202],{},[31743,483912,483913,483915],{},[31658,483914,97300],{"mathvariant":318416},[31749,483916,768],{},[31665,483918,483919],{"encoding":31667},"S = \\mathbb{Z}_2",[34,483921,483923,483941],{"className":483922,"ariaHidden":758},[31672],[34,483924,483926,483929,483932,483935,483938],{"className":483925},[31676],[34,483927],{"className":483928,"style":33218},[31680],[34,483930,94675],{"className":483931,"style":106455},[31685,31686],[34,483933],{"className":483934,"style":31846},[31845],[34,483936,202],{"className":483937},[31850],[34,483939],{"className":483940,"style":31846},[31845],[34,483942,483944,483947],{"className":483943},[31676],[34,483945],{"className":483946,"style":330413},[31680],[34,483948,483950,483953],{"className":483949},[31685],[34,483951,97300],{"className":483952},[31685,318441],[34,483954,483956],{"className":483955},[31792],[34,483957,483959,483979],{"className":483958},[31796,31797],[34,483960,483962,483976],{"className":483961},[31801],[34,483963,483965],{"className":483964,"style":31806},[31805],[34,483966,483967,483970],{"style":31809},[34,483968],{"className":483969,"style":31814},[31813],[34,483971,483973],{"className":483972},[31818,31819,31820,31821],[34,483974,768],{"className":483975},[31685,31821],[34,483977,31829],{"className":483978},[31828],[34,483980,483982],{"className":483981},[31801],[34,483983,483985],{"className":483984,"style":31836},[31805],[34,483986],{},"（整数模2）。\n",[34,483989,483991,484035],{"className":483990},[31642],[34,483992,483994],{"className":483993},[31646],[31648,483995,483996],{"xmlns":31650},[31652,483997,483998,484032],{},[31655,483999,484000,484002,484004,484006,484008,484010,484012,484014,484016,484018,484026,484028,484030],{},[31658,484001,126541],{},[31662,484003,456],{"stretchy":702},[31658,484005,32936],{},[31662,484007,2557],{"stretchy":702},[31662,484009,202],{},[31658,484011,32936],{},[31845,484013],{},[31845,484015],{"width":208910},[31662,484017,456],{"stretchy":702},[31655,484019,484020,484022,484024],{},[31658,484021,31759],{"mathvariant":38079},[31658,484023,43383],{"mathvariant":38079},[31658,484025,46900],{"mathvariant":38079},[31845,484027],{"width":208923},[31749,484029,768],{},[31662,484031,2557],{"stretchy":702},[31665,484033,484034],{"encoding":31667},"\\varphi(x) = x \\pmod 2",[34,484036,484038,484065,484080],{"className":484037,"ariaHidden":758},[31672],[34,484039,484041,484044,484047,484050,484053,484056,484059,484062],{"className":484040},[31676],[34,484042],{"className":484043,"style":31774},[31680],[34,484045,126541],{"className":484046},[31685,31686],[34,484048,456],{"className":484049},[31782],[34,484051,32936],{"className":484052},[31685,31686],[34,484054,2557],{"className":484055},[31690],[34,484057],{"className":484058,"style":31846},[31845],[34,484060,202],{"className":484061},[31850],[34,484063],{"className":484064,"style":31846},[31845],[34,484066,484068,484071,484074,484077],{"className":484067},[31676],[34,484069],{"className":484070,"style":32527},[31680],[34,484072,32936],{"className":484073},[31685,31686],[34,484075],{"className":484076},[31845,208951],[34,484078],{"className":484079,"style":208955},[31845],[34,484081,484083,484086,484089,484098,484101,484104],{"className":484082},[31676],[34,484084],{"className":484085,"style":31774},[31680],[34,484087,456],{"className":484088},[31782],[34,484090,484092],{"className":484091},[31685],[34,484093,484095],{"className":484094},[31685],[34,484096,207924],{"className":484097},[31685,207923],[34,484099],{"className":484100,"style":208977},[31845],[34,484102,768],{"className":484103},[31685],[34,484105,2557],{"className":484106},[31690],[34,484108,484110,484127],{"className":484109},[31642],[34,484111,484113],{"className":484112},[31646],[31648,484114,484115],{"xmlns":31650},[31652,484116,484117,484125],{},[31655,484118,484119],{},[31743,484120,484121,484123],{},[31658,484122,97300],{"mathvariant":318416},[31749,484124,15932],{},[31665,484126,371146],{"encoding":31667},[34,484128,484130],{"className":484129,"ariaHidden":758},[31672],[34,484131,484133,484136],{"className":484132},[31676],[34,484134],{"className":484135,"style":330413},[31680],[34,484137,484139,484142],{"className":484138},[31685],[34,484140,97300],{"className":484141},[31685,318441],[34,484143,484145],{"className":484144},[31792],[34,484146,484148,484168],{"className":484147},[31796,31797],[34,484149,484151,484165],{"className":484150},[31801],[34,484152,484154],{"className":484153,"style":31806},[31805],[34,484155,484156,484159],{"style":31809},[34,484157],{"className":484158,"style":31814},[31813],[34,484160,484162],{"className":484161},[31818,31819,31820,31821],[34,484163,15932],{"className":484164},[31685,31821],[34,484166,31829],{"className":484167},[31828],[34,484169,484171],{"className":484170},[31801],[34,484172,484174],{"className":484173,"style":31836},[31805],[34,484175],{},[34,484177,484179,484201],{"className":484178},[31642],[34,484180,484182],{"className":484181},[31646],[31648,484183,484184],{"xmlns":31650},[31652,484185,484186,484198],{},[31655,484187,484188,484190,484192,484194,484196],{},[31749,484189,768],{},[31662,484191,71191],{},[31749,484193,768],{},[31662,484195,202],{},[31749,484197,459],{},[31665,484199,484200],{"encoding":31667},"2 \\cdot 2 = 0",[34,484202,484204,484222,484240],{"className":484203,"ariaHidden":758},[31672],[34,484205,484207,484210,484213,484216,484219],{"className":484206},[31676],[34,484208],{"className":484209,"style":32617},[31680],[34,484211,768],{"className":484212},[31685],[34,484214],{"className":484215,"style":32604},[31845],[34,484217,71191],{"className":484218},[32024],[34,484220],{"className":484221,"style":32604},[31845],[34,484223,484225,484228,484231,484234,484237],{"className":484224},[31676],[34,484226],{"className":484227,"style":32617},[31680],[34,484229,768],{"className":484230},[31685],[34,484232],{"className":484233,"style":31846},[31845],[34,484235,202],{"className":484236},[31850],[34,484238],{"className":484239,"style":31846},[31845],[34,484241,484243,484246],{"className":484242},[31676],[34,484244],{"className":484245,"style":32617},[31680],[34,484247,459],{"className":484248},[31685],[34,484250,484252,484265],{"className":484251},[31642],[34,484253,484255],{"className":484254},[31646],[31648,484256,484257],{"xmlns":31650},[31652,484258,484259,484263],{},[31655,484260,484261],{},[31749,484262,768],{},[31665,484264,768],{"encoding":31667},[34,484266,484268],{"className":484267,"ariaHidden":758},[31672],[34,484269,484271,484274],{"className":484270},[31676],[34,484272],{"className":484273,"style":32617},[31680],[34,484275,768],{"className":484276},[31685]," 是零因子。\n",[34,484279,484281,484305],{"className":484280},[31642],[34,484282,484284],{"className":484283},[31646],[31648,484285,484286],{"xmlns":31650},[31652,484287,484288,484302],{},[31655,484289,484290,484292,484294,484296,484298,484300],{},[31658,484291,126541],{},[31662,484293,456],{"stretchy":702},[31749,484295,768],{},[31662,484297,2557],{"stretchy":702},[31662,484299,202],{},[31749,484301,459],{},[31665,484303,484304],{"encoding":31667},"\\varphi(2) = 0",[34,484306,484308,484335],{"className":484307,"ariaHidden":758},[31672],[34,484309,484311,484314,484317,484320,484323,484326,484329,484332],{"className":484310},[31676],[34,484312],{"className":484313,"style":31774},[31680],[34,484315,126541],{"className":484316},[31685,31686],[34,484318,456],{"className":484319},[31782],[34,484321,768],{"className":484322},[31685],[34,484324,2557],{"className":484325},[31690],[34,484327],{"className":484328,"style":31846},[31845],[34,484330,202],{"className":484331},[31850],[34,484333],{"className":484334,"style":31846},[31845],[34,484336,484338,484341],{"className":484337},[31676],[34,484339],{"className":484340,"style":32617},[31680],[34,484342,459],{"className":484343},[31685],[34,484345,484347,484364],{"className":484346},[31642],[34,484348,484350],{"className":484349},[31646],[31648,484351,484352],{"xmlns":31650},[31652,484353,484354,484362],{},[31655,484355,484356],{},[31743,484357,484358,484360],{},[31658,484359,97300],{"mathvariant":318416},[31749,484361,768],{},[31665,484363,330403],{"encoding":31667},[34,484365,484367],{"className":484366,"ariaHidden":758},[31672],[34,484368,484370,484373],{"className":484369},[31676],[34,484371],{"className":484372,"style":330413},[31680],[34,484374,484376,484379],{"className":484375},[31685],[34,484377,97300],{"className":484378},[31685,318441],[34,484380,484382],{"className":484381},[31792],[34,484383,484385,484405],{"className":484384},[31796,31797],[34,484386,484388,484402],{"className":484387},[31801],[34,484389,484391],{"className":484390,"style":31806},[31805],[34,484392,484393,484396],{"style":31809},[34,484394],{"className":484395,"style":31814},[31813],[34,484397,484399],{"className":484398},[31818,31819,31820,31821],[34,484400,768],{"className":484401},[31685,31821],[34,484403,31829],{"className":484404},[31828],[34,484406,484408],{"className":484407},[31801],[34,484409,484411],{"className":484410,"style":31836},[31805],[34,484412],{}," 中。\n通常“零因子”指非零元素。",[34,484415,484417,484430],{"className":484416},[31642],[34,484418,484420],{"className":484419},[31646],[31648,484421,484422],{"xmlns":31650},[31652,484423,484424,484428],{},[31655,484425,484426],{},[31749,484427,459],{},[31665,484429,459],{"encoding":31667},[34,484431,484433],{"className":484432,"ariaHidden":758},[31672],[34,484434,484436,484439],{"className":484435},[31676],[34,484437],{"className":484438,"style":32617},[31680],[34,484440,459],{"className":484441},[31685]," 不是零因子。所以零因子 ",[34,484444,484446,484459],{"className":484445},[31642],[34,484447,484449],{"className":484448},[31646],[31648,484450,484451],{"xmlns":31650},[31652,484452,484453,484457],{},[31655,484454,484455],{},[31749,484456,768],{},[31665,484458,768],{"encoding":31667},[34,484460,484462],{"className":484461,"ariaHidden":758},[31672],[34,484463,484465,484468],{"className":484464},[31676],[34,484466],{"className":484467,"style":32617},[31680],[34,484469,768],{"className":484470},[31685]," 被映成了 ",[34,484473,484475,484488],{"className":484474},[31642],[34,484476,484478],{"className":484477},[31646],[31648,484479,484480],{"xmlns":31650},[31652,484481,484482,484486],{},[31655,484483,484484],{},[31749,484485,459],{},[31665,484487,459],{"encoding":31667},[34,484489,484491],{"className":484490,"ariaHidden":758},[31672],[34,484492,484494,484497],{"className":484493},[31676],[34,484495],{"className":484496,"style":32617},[31680],[34,484498,459],{"className":484499},[31685],"。\n即使零因子映射成非零元，例如 ",[34,484502,484504,484525],{"className":484503},[31642],[34,484505,484507],{"className":484506},[31646],[31648,484508,484509],{"xmlns":31650},[31652,484510,484511,484523],{},[31655,484512,484513,484515,484517,484519,484521],{},[31658,484514,34386],{},[31662,484516,202],{},[31658,484518,97300],{"mathvariant":318416},[31662,484520,42071],{},[31658,484522,97300],{"mathvariant":318416},[31665,484524,389182],{"encoding":31667},[34,484526,484528,484546,484564],{"className":484527,"ariaHidden":758},[31672],[34,484529,484531,484534,484537,484540,484543],{"className":484530},[31676],[34,484532],{"className":484533,"style":33218},[31680],[34,484535,34386],{"className":484536,"style":34427},[31685,31686],[34,484538],{"className":484539,"style":31846},[31845],[34,484541,202],{"className":484542},[31850],[34,484544],{"className":484545,"style":31846},[31845],[34,484547,484549,484552,484555,484558,484561],{"className":484548},[31676],[34,484550],{"className":484551,"style":379449},[31680],[34,484553,97300],{"className":484554},[31685,318441],[34,484556],{"className":484557,"style":32604},[31845],[34,484559,42071],{"className":484560},[32024],[34,484562],{"className":484563,"style":32604},[31845],[34,484565,484567,484570],{"className":484566},[31676],[34,484568],{"className":484569,"style":320927},[31680],[34,484571,97300],{"className":484572},[31685,318441],[34,484574,484576,484593],{"className":484575},[31642],[34,484577,484579],{"className":484578},[31646],[31648,484580,484581],{"xmlns":31650},[31652,484582,484583,484591],{},[31655,484584,484585,484587,484589],{},[31658,484586,94675],{},[31662,484588,202],{},[31658,484590,97300],{"mathvariant":318416},[31665,484592,389777],{"encoding":31667},[34,484594,484596,484614],{"className":484595,"ariaHidden":758},[31672],[34,484597,484599,484602,484605,484608,484611],{"className":484598},[31676],[34,484600],{"className":484601,"style":33218},[31680],[34,484603,94675],{"className":484604,"style":106455},[31685,31686],[34,484606],{"className":484607,"style":31846},[31845],[34,484609,202],{"className":484610},[31850],[34,484612],{"className":484613,"style":31846},[31845],[34,484615,484617,484620],{"className":484616},[31676],[34,484618],{"className":484619,"style":320927},[31680],[34,484621,97300],{"className":484622},[31685,318441],[34,484624,484626,484658],{"className":484625},[31642],[34,484627,484629],{"className":484628},[31646],[31648,484630,484631],{"xmlns":31650},[31652,484632,484633,484655],{},[31655,484634,484635,484637,484639,484641,484643,484645,484647,484649,484651,484653],{},[31658,484636,126541],{},[31662,484638,456],{"stretchy":702},[31662,484640,456],{"stretchy":702},[31749,484642,640],{},[31662,484644,31960],{"separator":758},[31749,484646,459],{},[31662,484648,2557],{"stretchy":702},[31662,484650,2557],{"stretchy":702},[31662,484652,202],{},[31749,484654,640],{},[31665,484656,484657],{"encoding":31667},"\\varphi((1,0)) = 1",[34,484659,484661,484697],{"className":484660,"ariaHidden":758},[31672],[34,484662,484664,484667,484670,484673,484676,484679,484682,484685,484688,484691,484694],{"className":484663},[31676],[34,484665],{"className":484666,"style":31774},[31680],[34,484668,126541],{"className":484669},[31685,31686],[34,484671,1323],{"className":484672},[31782],[34,484674,640],{"className":484675},[31685],[34,484677,31960],{"className":484678},[32119],[34,484680],{"className":484681,"style":32123},[31845],[34,484683,459],{"className":484684},[31685],[34,484686,44253],{"className":484687},[31690],[34,484689],{"className":484690,"style":31846},[31845],[34,484692,202],{"className":484693},[31850],[34,484695],{"className":484696,"style":31846},[31845],[34,484698,484700,484703],{"className":484699},[31676],[34,484701],{"className":484702,"style":32617},[31680],[34,484704,640],{"className":484705},[31685],[34,484707,484709,484730],{"className":484708},[31642],[34,484710,484712],{"className":484711},[31646],[31648,484713,484714],{"xmlns":31650},[31652,484715,484716,484728],{},[31655,484717,484718,484720,484722,484724,484726],{},[31662,484719,456],{"stretchy":702},[31749,484721,640],{},[31662,484723,31960],{"separator":758},[31749,484725,459],{},[31662,484727,2557],{"stretchy":702},[31665,484729,381896],{"encoding":31667},[34,484731,484733],{"className":484732,"ariaHidden":758},[31672],[34,484734,484736,484739,484742,484745,484748,484751,484754],{"className":484735},[31676],[34,484737],{"className":484738,"style":31774},[31680],[34,484740,456],{"className":484741},[31782],[34,484743,640],{"className":484744},[31685],[34,484746,31960],{"className":484747},[32119],[34,484749],{"className":484750,"style":32123},[31845],[34,484752,459],{"className":484753},[31685],[34,484755,2557],{"className":484756},[31690],[34,484758,484760,484773],{"className":484759},[31642],[34,484761,484763],{"className":484762},[31646],[31648,484764,484765],{"xmlns":31650},[31652,484766,484767,484771],{},[31655,484768,484769],{},[31658,484770,34386],{},[31665,484772,34386],{"encoding":31667},[34,484774,484776],{"className":484775,"ariaHidden":758},[31672],[34,484777,484779,484782],{"className":484778},[31676],[34,484780],{"className":484781,"style":33218},[31680],[34,484783,34386],{"className":484784,"style":34427},[31685,31686]," 的零因子（因为它乘以 ",[34,484787,484789,484811],{"className":484788},[31642],[34,484790,484792],{"className":484791},[31646],[31648,484793,484794],{"xmlns":31650},[31652,484795,484796,484808],{},[31655,484797,484798,484800,484802,484804,484806],{},[31662,484799,456],{"stretchy":702},[31749,484801,459],{},[31662,484803,31960],{"separator":758},[31749,484805,640],{},[31662,484807,2557],{"stretchy":702},[31665,484809,484810],{"encoding":31667},"(0,1)",[34,484812,484814],{"className":484813,"ariaHidden":758},[31672],[34,484815,484817,484820,484823,484826,484829,484832,484835],{"className":484816},[31676],[34,484818],{"className":484819,"style":31774},[31680],[34,484821,456],{"className":484822},[31782],[34,484824,459],{"className":484825},[31685],[34,484827,31960],{"className":484828},[32119],[34,484830],{"className":484831,"style":32123},[31845],[34,484833,640],{"className":484834},[31685],[34,484836,2557],{"className":484837},[31690]," 等于 0），但像 ",[34,484840,484842,484855],{"className":484841},[31642],[34,484843,484845],{"className":484844},[31646],[31648,484846,484847],{"xmlns":31650},[31652,484848,484849,484853],{},[31655,484850,484851],{},[31749,484852,640],{},[31665,484854,640],{"encoding":31667},[34,484856,484858],{"className":484857,"ariaHidden":758},[31672],[34,484859,484861,484864],{"className":484860},[31676],[34,484862],{"className":484863,"style":32617},[31680],[34,484865,640],{"className":484866},[31685],[34,484868,484870,484883],{"className":484869},[31642],[34,484871,484873],{"className":484872},[31646],[31648,484874,484875],{"xmlns":31650},[31652,484876,484877,484881],{},[31655,484878,484879],{},[31658,484880,94675],{},[31665,484882,94675],{"encoding":31667},[34,484884,484886],{"className":484885,"ariaHidden":758},[31672],[34,484887,484889,484892],{"className":484888},[31676],[34,484890],{"className":484891,"style":33218},[31680],[34,484893,94675],{"className":484894,"style":106455},[31685,31686]," 中显然不是零因子。",[15,484897,484898],{},[2627,484899,389060,484900,484928],{},[34,484901,484903,484916],{"className":484902},[31642],[34,484904,484906],{"className":484905},[31646],[31648,484907,484908],{"xmlns":31650},[31652,484909,484910,484914],{},[31655,484911,484912],{},[31658,484913,126541],{},[31665,484915,126708],{"encoding":31667},[34,484917,484919],{"className":484918,"ariaHidden":758},[31672],[34,484920,484922,484925],{"className":484921},[31676],[34,484923],{"className":484924,"style":38774},[31680],[34,484926,126541],{"className":484927},[31685,31686]," 把整环映为整环。",[105,484930,484931,484935],{},[108,484932,484933,347],{},[2627,484934,483795],{},[108,484936,484937,2664,484939,484989,484990,485082,485083,452051],{},[2627,484938,483800],{},[34,484940,484942,484959],{"className":484941},[31642],[34,484943,484945],{"className":484944},[31646],[31648,484946,484947],{"xmlns":31650},[31652,484948,484949,484957],{},[31655,484950,484951,484953,484955],{},[31658,484952,34386],{},[31662,484954,202],{},[31658,484956,97300],{"mathvariant":318416},[31665,484958,387661],{"encoding":31667},[34,484960,484962,484980],{"className":484961,"ariaHidden":758},[31672],[34,484963,484965,484968,484971,484974,484977],{"className":484964},[31676],[34,484966],{"className":484967,"style":33218},[31680],[34,484969,34386],{"className":484970,"style":34427},[31685,31686],[34,484972],{"className":484973,"style":31846},[31845],[34,484975,202],{"className":484976},[31850],[34,484978],{"className":484979,"style":31846},[31845],[34,484981,484983,484986],{"className":484982},[31676],[34,484984],{"className":484985,"style":320927},[31680],[34,484987,97300],{"className":484988},[31685,318441],"（整环），",[34,484991,484993,485015],{"className":484992},[31642],[34,484994,484996],{"className":484995},[31646],[31648,484997,484998],{"xmlns":31650},[31652,484999,485000,485012],{},[31655,485001,485002,485004,485006],{},[31658,485003,94675],{},[31662,485005,202],{},[31743,485007,485008,485010],{},[31658,485009,97300],{"mathvariant":318416},[31749,485011,15932],{},[31665,485013,485014],{"encoding":31667},"S = \\mathbb{Z}_4",[34,485016,485018,485036],{"className":485017,"ariaHidden":758},[31672],[34,485019,485021,485024,485027,485030,485033],{"className":485020},[31676],[34,485022],{"className":485023,"style":33218},[31680],[34,485025,94675],{"className":485026,"style":106455},[31685,31686],[34,485028],{"className":485029,"style":31846},[31845],[34,485031,202],{"className":485032},[31850],[34,485034],{"className":485035,"style":31846},[31845],[34,485037,485039,485042],{"className":485038},[31676],[34,485040],{"className":485041,"style":330413},[31680],[34,485043,485045,485048],{"className":485044},[31685],[34,485046,97300],{"className":485047},[31685,318441],[34,485049,485051],{"className":485050},[31792],[34,485052,485054,485074],{"className":485053},[31796,31797],[34,485055,485057,485071],{"className":485056},[31801],[34,485058,485060],{"className":485059,"style":31806},[31805],[34,485061,485062,485065],{"style":31809},[34,485063],{"className":485064,"style":31814},[31813],[34,485066,485068],{"className":485067},[31818,31819,31820,31821],[34,485069,15932],{"className":485070},[31685,31821],[34,485072,31829],{"className":485073},[31828],[34,485075,485077],{"className":485076},[31801],[34,485078,485080],{"className":485079,"style":31836},[31805],[34,485081],{},"（不是整环，有零因子）。\n",[34,485084,485086,485128],{"className":485085},[31642],[34,485087,485089],{"className":485088},[31646],[31648,485090,485091],{"xmlns":31650},[31652,485092,485093,485125],{},[31655,485094,485095,485097,485099,485101,485103,485105,485107,485109,485111,485119,485121,485123],{},[31658,485096,126541],{},[31662,485098,5598],{},[31658,485100,31660],{},[31662,485102,418784],{},[31658,485104,31660],{},[31845,485106],{},[31845,485108],{"width":208910},[31662,485110,456],{"stretchy":702},[31655,485112,485113,485115,485117],{},[31658,485114,31759],{"mathvariant":38079},[31658,485116,43383],{"mathvariant":38079},[31658,485118,46900],{"mathvariant":38079},[31845,485120],{"width":208923},[31749,485122,15932],{},[31662,485124,2557],{"stretchy":702},[31665,485126,485127],{"encoding":31667},"\\varphi: n \\mapsto n \\pmod 4",[34,485129,485131,485149,485167,485182],{"className":485130,"ariaHidden":758},[31672],[34,485132,485134,485137,485140,485143,485146],{"className":485133},[31676],[34,485135],{"className":485136,"style":38774},[31680],[34,485138,126541],{"className":485139},[31685,31686],[34,485141],{"className":485142,"style":31846},[31845],[34,485144,5598],{"className":485145},[31850],[34,485147],{"className":485148,"style":31846},[31845],[34,485150,485152,485155,485158,485161,485164],{"className":485151},[31676],[34,485153],{"className":485154,"style":330733},[31680],[34,485156,31660],{"className":485157},[31685,31686],[34,485159],{"className":485160,"style":31846},[31845],[34,485162,418784],{"className":485163},[31850],[34,485165],{"className":485166,"style":31846},[31845],[34,485168,485170,485173,485176,485179],{"className":485169},[31676],[34,485171],{"className":485172,"style":32527},[31680],[34,485174,31660],{"className":485175},[31685,31686],[34,485177],{"className":485178},[31845,208951],[34,485180],{"className":485181,"style":208955},[31845],[34,485183,485185,485188,485191,485200,485203,485206],{"className":485184},[31676],[34,485186],{"className":485187,"style":31774},[31680],[34,485189,456],{"className":485190},[31782],[34,485192,485194],{"className":485193},[31685],[34,485195,485197],{"className":485196},[31685],[34,485198,207924],{"className":485199},[31685,207923],[34,485201],{"className":485202,"style":208977},[31845],[34,485204,15932],{"className":485205},[31685],[34,485207,2557],{"className":485208},[31690],[15,485210,485211],{},[2627,485212,485213,485214,485242,485243,331534],{},"(4) 如果 ",[34,485215,485217,485230],{"className":485216},[31642],[34,485218,485220],{"className":485219},[31646],[31648,485221,485222],{"xmlns":31650},[31652,485223,485224,485228],{},[31655,485225,485226],{},[31658,485227,94675],{},[31665,485229,94675],{"encoding":31667},[34,485231,485233],{"className":485232,"ariaHidden":758},[31672],[34,485234,485236,485239],{"className":485235},[31676],[34,485237],{"className":485238,"style":33218},[31680],[34,485240,94675],{"className":485241,"style":106455},[31685,31686]," 是整环, 则 ",[34,485244,485246,485259],{"className":485245},[31642],[34,485247,485249],{"className":485248},[31646],[31648,485250,485251],{"xmlns":31650},[31652,485252,485253,485257],{},[31655,485254,485255],{},[31658,485256,34386],{},[31665,485258,34386],{"encoding":31667},[34,485260,485262],{"className":485261,"ariaHidden":758},[31672],[34,485263,485265,485268],{"className":485264},[31676],[34,485266],{"className":485267,"style":33218},[31680],[34,485269,34386],{"className":485270,"style":34427},[31685,31686],[105,485272,485273,485277],{},[108,485274,485275,347],{},[2627,485276,483795],{},[108,485278,485279,2664,485281,485353,485354,485404,485405,452051],{},[2627,485280,483800],{},[34,485282,485284,485305],{"className":485283},[31642],[34,485285,485287],{"className":485286},[31646],[31648,485288,485289],{"xmlns":31650},[31652,485290,485291,485303],{},[31655,485292,485293,485295,485297,485299,485301],{},[31658,485294,34386],{},[31662,485296,202],{},[31658,485298,97300],{"mathvariant":318416},[31662,485300,42071],{},[31658,485302,97300],{"mathvariant":318416},[31665,485304,389182],{"encoding":31667},[34,485306,485308,485326,485344],{"className":485307,"ariaHidden":758},[31672],[34,485309,485311,485314,485317,485320,485323],{"className":485310},[31676],[34,485312],{"className":485313,"style":33218},[31680],[34,485315,34386],{"className":485316,"style":34427},[31685,31686],[34,485318],{"className":485319,"style":31846},[31845],[34,485321,202],{"className":485322},[31850],[34,485324],{"className":485325,"style":31846},[31845],[34,485327,485329,485332,485335,485338,485341],{"className":485328},[31676],[34,485330],{"className":485331,"style":379449},[31680],[34,485333,97300],{"className":485334},[31685,318441],[34,485336],{"className":485337,"style":32604},[31845],[34,485339,42071],{"className":485340},[32024],[34,485342],{"className":485343,"style":32604},[31845],[34,485345,485347,485350],{"className":485346},[31676],[34,485348],{"className":485349,"style":320927},[31680],[34,485351,97300],{"className":485352},[31685,318441],"（不是整环），",[34,485355,485357,485374],{"className":485356},[31642],[34,485358,485360],{"className":485359},[31646],[31648,485361,485362],{"xmlns":31650},[31652,485363,485364,485372],{},[31655,485365,485366,485368,485370],{},[31658,485367,94675],{},[31662,485369,202],{},[31658,485371,97300],{"mathvariant":318416},[31665,485373,389777],{"encoding":31667},[34,485375,485377,485395],{"className":485376,"ariaHidden":758},[31672],[34,485378,485380,485383,485386,485389,485392],{"className":485379},[31676],[34,485381],{"className":485382,"style":33218},[31680],[34,485384,94675],{"className":485385,"style":106455},[31685,31686],[34,485387],{"className":485388,"style":31846},[31845],[34,485390,202],{"className":485391},[31850],[34,485393],{"className":485394,"style":31846},[31845],[34,485396,485398,485401],{"className":485397},[31676],[34,485399],{"className":485400,"style":320927},[31680],[34,485402,97300],{"className":485403},[31685,318441],"（整环）。\n",[34,485406,485408,485436],{"className":485407},[31642],[34,485409,485411],{"className":485410},[31646],[31648,485412,485413],{"xmlns":31650},[31652,485414,485415,485433],{},[31655,485416,485417,485419,485421,485423,485425,485427,485429,485431],{},[31658,485418,126541],{},[31662,485420,456],{"stretchy":702},[31658,485422,111],{},[31662,485424,31960],{"separator":758},[31658,485426,1697],{},[31662,485428,2557],{"stretchy":702},[31662,485430,202],{},[31658,485432,111],{},[31665,485434,485435],{"encoding":31667},"\\varphi(a, b) = a",[34,485437,485439,485475],{"className":485438,"ariaHidden":758},[31672],[34,485440,485442,485445,485448,485451,485454,485457,485460,485463,485466,485469,485472],{"className":485441},[31676],[34,485443],{"className":485444,"style":31774},[31680],[34,485446,126541],{"className":485447},[31685,31686],[34,485449,456],{"className":485450},[31782],[34,485452,111],{"className":485453},[31685,31686],[34,485455,31960],{"className":485456},[32119],[34,485458],{"className":485459,"style":32123},[31845],[34,485461,1697],{"className":485462},[31685,31686],[34,485464,2557],{"className":485465},[31690],[34,485467],{"className":485468,"style":31846},[31845],[34,485470,202],{"className":485471},[31850],[34,485473],{"className":485474,"style":31846},[31845],[34,485476,485478,485481],{"className":485477},[31676],[34,485479],{"className":485480,"style":32527},[31680],[34,485482,111],{"className":485483},[31685,31686],[15,485485,485486],{},[2627,485487,485488,485489,485517],{},"(5) ",[34,485490,485492,485505],{"className":485491},[31642],[34,485493,485495],{"className":485494},[31646],[31648,485496,485497],{"xmlns":31650},[31652,485498,485499,485503],{},[31655,485500,485501],{},[31658,485502,126541],{},[31665,485504,126708],{"encoding":31667},[34,485506,485508],{"className":485507,"ariaHidden":758},[31672],[34,485509,485511,485514],{"className":485510},[31676],[34,485512],{"className":485513,"style":38774},[31680],[34,485515,126541],{"className":485516},[31685,31686]," 把可逆元映为可逆元。",[105,485519,485520,485524],{},[108,485521,485522,347],{},[2627,485523,482907],{},[108,485525,485526,34554,485528,485579,485580,482973,485608,39984,485702,355408,485946,314945,485990,486018],{},[2627,485527,390672],{},[34,485529,485531,485549],{"className":485530},[31642],[34,485532,485534],{"className":485533},[31646],[31648,485535,485536],{"xmlns":31650},[31652,485537,485538,485546],{},[31655,485539,485540,485542,485544],{},[31658,485541,94628],{},[31662,485543,112296],{},[31658,485545,34386],{},[31665,485547,485548],{"encoding":31667},"u \\in R",[34,485550,485552,485570],{"className":485551,"ariaHidden":758},[31672],[34,485553,485555,485558,485561,485564,485567],{"className":485554},[31676],[34,485556],{"className":485557,"style":34812},[31680],[34,485559,94628],{"className":485560},[31685,31686],[34,485562],{"className":485563,"style":31846},[31845],[34,485565,112296],{"className":485566},[31850],[34,485568],{"className":485569,"style":31846},[31845],[34,485571,485573,485576],{"className":485572},[31676],[34,485574],{"className":485575,"style":33218},[31680],[34,485577,34386],{"className":485578,"style":34427},[31685,31686]," 可逆，即存在 ",[34,485581,485583,485596],{"className":485582},[31642],[34,485584,485586],{"className":485585},[31646],[31648,485587,485588],{"xmlns":31650},[31652,485589,485590,485594],{},[31655,485591,485592],{},[31658,485593,56782],{},[31665,485595,56782],{"encoding":31667},[34,485597,485599],{"className":485598,"ariaHidden":758},[31672],[34,485600,485602,485605],{"className":485601},[31676],[34,485603],{"className":485604,"style":32527},[31680],[34,485606,56782],{"className":485607,"style":32062},[31685,31686],[34,485609,485611,485635],{"className":485610},[31642],[34,485612,485614],{"className":485613},[31646],[31648,485615,485616],{"xmlns":31650},[31652,485617,485618,485632],{},[31655,485619,485620,485622,485624,485626],{},[31658,485621,94628],{},[31658,485623,56782],{},[31662,485625,202],{},[31743,485627,485628,485630],{},[31749,485629,640],{},[31658,485631,34386],{},[31665,485633,485634],{"encoding":31667},"uv = 1_R",[34,485636,485638,485656],{"className":485637,"ariaHidden":758},[31672],[34,485639,485641,485644,485647,485650,485653],{"className":485640},[31676],[34,485642],{"className":485643,"style":32527},[31680],[34,485645,134150],{"className":485646,"style":32062},[31685,31686],[34,485648],{"className":485649,"style":31846},[31845],[34,485651,202],{"className":485652},[31850],[34,485654],{"className":485655,"style":31846},[31845],[34,485657,485659,485662],{"className":485658},[31676],[34,485660],{"className":485661,"style":38357},[31680],[34,485663,485665,485668],{"className":485664},[31685],[34,485666,640],{"className":485667},[31685],[34,485669,485671],{"className":485670},[31792],[34,485672,485674,485694],{"className":485673},[31796,31797],[34,485675,485677,485691],{"className":485676},[31801],[34,485678,485680],{"className":485679,"style":116558},[31805],[34,485681,485682,485685],{"style":31809},[34,485683],{"className":485684,"style":31814},[31813],[34,485686,485688],{"className":485687},[31818,31819,31820,31821],[34,485689,34386],{"className":485690,"style":34427},[31685,31686,31821],[34,485692,31829],{"className":485693},[31828],[34,485695,485697],{"className":485696},[31801],[34,485698,485700],{"className":485699,"style":31836},[31805],[34,485701],{},[34,485703,485705,485767],{"className":485704},[31642],[34,485706,485708],{"className":485707},[31646],[31648,485709,485710],{"xmlns":31650},[31652,485711,485712,485764],{},[31655,485713,485714,485716,485718,485720,485722,485724,485726,485728,485730,485732,485734,485736,485738,485740,485742,485744,485746,485748,485754,485756,485758],{},[31658,485715,126541],{},[31662,485717,456],{"stretchy":702},[31658,485719,94628],{},[31662,485721,2557],{"stretchy":702},[31658,485723,126541],{},[31662,485725,456],{"stretchy":702},[31658,485727,56782],{},[31662,485729,2557],{"stretchy":702},[31662,485731,202],{},[31658,485733,126541],{},[31662,485735,456],{"stretchy":702},[31658,485737,94628],{},[31658,485739,56782],{},[31662,485741,2557],{"stretchy":702},[31662,485743,202],{},[31658,485745,126541],{},[31662,485747,456],{"stretchy":702},[31743,485749,485750,485752],{},[31749,485751,640],{},[31658,485753,34386],{},[31662,485755,2557],{"stretchy":702},[31662,485757,202],{},[31743,485759,485760,485762],{},[31749,485761,640],{},[31658,485763,94675],{},[31665,485765,485766],{"encoding":31667},"\\varphi(u)\\varphi(v) = \\varphi(uv) = \\varphi(1_R) = 1_S",[34,485768,485770,485809,485836,485900],{"className":485769,"ariaHidden":758},[31672],[34,485771,485773,485776,485779,485782,485785,485788,485791,485794,485797,485800,485803,485806],{"className":485772},[31676],[34,485774],{"className":485775,"style":31774},[31680],[34,485777,126541],{"className":485778},[31685,31686],[34,485780,456],{"className":485781},[31782],[34,485783,94628],{"className":485784},[31685,31686],[34,485786,2557],{"className":485787},[31690],[34,485789,126541],{"className":485790},[31685,31686],[34,485792,456],{"className":485793},[31782],[34,485795,56782],{"className":485796,"style":32062},[31685,31686],[34,485798,2557],{"className":485799},[31690],[34,485801],{"className":485802,"style":31846},[31845],[34,485804,202],{"className":485805},[31850],[34,485807],{"className":485808,"style":31846},[31845],[34,485810,485812,485815,485818,485821,485824,485827,485830,485833],{"className":485811},[31676],[34,485813],{"className":485814,"style":31774},[31680],[34,485816,126541],{"className":485817},[31685,31686],[34,485819,456],{"className":485820},[31782],[34,485822,134150],{"className":485823,"style":32062},[31685,31686],[34,485825,2557],{"className":485826},[31690],[34,485828],{"className":485829,"style":31846},[31845],[34,485831,202],{"className":485832},[31850],[34,485834],{"className":485835,"style":31846},[31845],[34,485837,485839,485842,485845,485848,485888,485891,485894,485897],{"className":485838},[31676],[34,485840],{"className":485841,"style":31774},[31680],[34,485843,126541],{"className":485844},[31685,31686],[34,485846,456],{"className":485847},[31782],[34,485849,485851,485854],{"className":485850},[31685],[34,485852,640],{"className":485853},[31685],[34,485855,485857],{"className":485856},[31792],[34,485858,485860,485880],{"className":485859},[31796,31797],[34,485861,485863,485877],{"className":485862},[31801],[34,485864,485866],{"className":485865,"style":116558},[31805],[34,485867,485868,485871],{"style":31809},[34,485869],{"className":485870,"style":31814},[31813],[34,485872,485874],{"className":485873},[31818,31819,31820,31821],[34,485875,34386],{"className":485876,"style":34427},[31685,31686,31821],[34,485878,31829],{"className":485879},[31828],[34,485881,485883],{"className":485882},[31801],[34,485884,485886],{"className":485885,"style":31836},[31805],[34,485887],{},[34,485889,2557],{"className":485890},[31690],[34,485892],{"className":485893,"style":31846},[31845],[34,485895,202],{"className":485896},[31850],[34,485898],{"className":485899,"style":31846},[31845],[34,485901,485903,485906],{"className":485902},[31676],[34,485904],{"className":485905,"style":38357},[31680],[34,485907,485909,485912],{"className":485908},[31685],[34,485910,640],{"className":485911},[31685],[34,485913,485915],{"className":485914},[31792],[34,485916,485918,485938],{"className":485917},[31796,31797],[34,485919,485921,485935],{"className":485920},[31801],[34,485922,485924],{"className":485923,"style":116558},[31805],[34,485925,485926,485929],{"style":31809},[34,485927],{"className":485928,"style":31814},[31813],[34,485930,485932],{"className":485931},[31818,31819,31820,31821],[34,485933,94675],{"className":485934,"style":106455},[31685,31686,31821],[34,485936,31829],{"className":485937},[31828],[34,485939,485941],{"className":485940},[31801],[34,485942,485944],{"className":485943,"style":31836},[31805],[34,485945],{},[34,485947,485949,485969],{"className":485948},[31642],[34,485950,485952],{"className":485951},[31646],[31648,485953,485954],{"xmlns":31650},[31652,485955,485956,485966],{},[31655,485957,485958,485960,485962,485964],{},[31658,485959,126541],{},[31662,485961,456],{"stretchy":702},[31658,485963,94628],{},[31662,485965,2557],{"stretchy":702},[31665,485967,485968],{"encoding":31667},"\\varphi(u)",[34,485970,485972],{"className":485971,"ariaHidden":758},[31672],[34,485973,485975,485978,485981,485984,485987],{"className":485974},[31676],[34,485976],{"className":485977,"style":31774},[31680],[34,485979,126541],{"className":485980},[31685,31686],[34,485982,456],{"className":485983},[31782],[34,485985,94628],{"className":485986},[31685,31686],[34,485988,2557],{"className":485989},[31690],[34,485991,485993,486006],{"className":485992},[31642],[34,485994,485996],{"className":485995},[31646],[31648,485997,485998],{"xmlns":31650},[31652,485999,486000,486004],{},[31655,486001,486002],{},[31658,486003,94675],{},[31665,486005,94675],{"encoding":31667},[34,486007,486009],{"className":486008,"ariaHidden":758},[31672],[34,486010,486012,486015],{"className":486011},[31676],[34,486013],{"className":486014,"style":33218},[31680],[34,486016,94675],{"className":486017,"style":106455},[31685,31686]," 中可逆。",[15,486020,486021],{},[2627,486022,486023,486024,486075,486076,486119,486120,400015],{},"(6) 对于 ",[34,486025,486027,486045],{"className":486026},[31642],[34,486028,486030],{"className":486029},[31646],[31648,486031,486032],{"xmlns":31650},[31652,486033,486034,486042],{},[31655,486035,486036,486038,486040],{},[31658,486037,111],{},[31662,486039,112296],{},[31658,486041,34386],{},[31665,486043,486044],{"encoding":31667},"a \\in R",[34,486046,486048,486066],{"className":486047,"ariaHidden":758},[31672],[34,486049,486051,486054,486057,486060,486063],{"className":486050},[31676],[34,486052],{"className":486053,"style":34812},[31680],[34,486055,111],{"className":486056},[31685,31686],[34,486058],{"className":486059,"style":31846},[31845],[34,486061,112296],{"className":486062},[31850],[34,486064],{"className":486065,"style":31846},[31845],[34,486067,486069,486072],{"className":486068},[31676],[34,486070],{"className":486071,"style":33218},[31680],[34,486073,34386],{"className":486074,"style":34427},[31685,31686],", 如果 ",[34,486077,486079,486098],{"className":486078},[31642],[34,486080,486082],{"className":486081},[31646],[31648,486083,486084],{"xmlns":31650},[31652,486085,486086,486096],{},[31655,486087,486088,486090,486092,486094],{},[31658,486089,126541],{},[31662,486091,456],{"stretchy":702},[31658,486093,111],{},[31662,486095,2557],{"stretchy":702},[31665,486097,483400],{"encoding":31667},[34,486099,486101],{"className":486100,"ariaHidden":758},[31672],[34,486102,486104,486107,486110,486113,486116],{"className":486103},[31676],[34,486105],{"className":486106,"style":31774},[31680],[34,486108,126541],{"className":486109},[31685,31686],[34,486111,456],{"className":486112},[31782],[34,486114,111],{"className":486115},[31685,31686],[34,486117,2557],{"className":486118},[31690]," 可逆, 则 ",[34,486121,486123,486136],{"className":486122},[31642],[34,486124,486126],{"className":486125},[31646],[31648,486127,486128],{"xmlns":31650},[31652,486129,486130,486134],{},[31655,486131,486132],{},[31658,486133,111],{},[31665,486135,111],{"encoding":31667},[34,486137,486139],{"className":486138,"ariaHidden":758},[31672],[34,486140,486142,486145],{"className":486141},[31676],[34,486143],{"className":486144,"style":32527},[31680],[34,486146,111],{"className":486147},[31685,31686],[105,486149,486150,486154],{},[108,486151,486152,347],{},[2627,486153,483795],{},[108,486155,486156,2664,486158,46528,486208,486300,486301,347,486352,314945,486418,486487,486488,486583,486584,486612,486613,486660,486661,486689],{},[2627,486157,483800],{},[34,486159,486161,486178],{"className":486160},[31642],[34,486162,486164],{"className":486163},[31646],[31648,486165,486166],{"xmlns":31650},[31652,486167,486168,486176],{},[31655,486169,486170,486172,486174],{},[31658,486171,34386],{},[31662,486173,202],{},[31658,486175,97300],{"mathvariant":318416},[31665,486177,387661],{"encoding":31667},[34,486179,486181,486199],{"className":486180,"ariaHidden":758},[31672],[34,486182,486184,486187,486190,486193,486196],{"className":486183},[31676],[34,486185],{"className":486186,"style":33218},[31680],[34,486188,34386],{"className":486189,"style":34427},[31685,31686],[34,486191],{"className":486192,"style":31846},[31845],[34,486194,202],{"className":486195},[31850],[34,486197],{"className":486198,"style":31846},[31845],[34,486200,486202,486205],{"className":486201},[31676],[34,486203],{"className":486204,"style":320927},[31680],[34,486206,97300],{"className":486207},[31685,318441],[34,486209,486211,486233],{"className":486210},[31642],[34,486212,486214],{"className":486213},[31646],[31648,486215,486216],{"xmlns":31650},[31652,486217,486218,486230],{},[31655,486219,486220,486222,486224],{},[31658,486221,94675],{},[31662,486223,202],{},[31743,486225,486226,486228],{},[31658,486227,97300],{"mathvariant":318416},[31749,486229,18136],{},[31665,486231,486232],{"encoding":31667},"S = \\mathbb{Z}_5",[34,486234,486236,486254],{"className":486235,"ariaHidden":758},[31672],[34,486237,486239,486242,486245,486248,486251],{"className":486238},[31676],[34,486240],{"className":486241,"style":33218},[31680],[34,486243,94675],{"className":486244,"style":106455},[31685,31686],[34,486246],{"className":486247,"style":31846},[31845],[34,486249,202],{"className":486250},[31850],[34,486252],{"className":486253,"style":31846},[31845],[34,486255,486257,486260],{"className":486256},[31676],[34,486258],{"className":486259,"style":330413},[31680],[34,486261,486263,486266],{"className":486262},[31685],[34,486264,97300],{"className":486265},[31685,318441],[34,486267,486269],{"className":486268},[31792],[34,486270,486272,486292],{"className":486271},[31796,31797],[34,486273,486275,486289],{"className":486274},[31801],[34,486276,486278],{"className":486277,"style":31806},[31805],[34,486279,486280,486283],{"style":31809},[34,486281],{"className":486282,"style":31814},[31813],[34,486284,486286],{"className":486285},[31818,31819,31820,31821],[34,486287,18136],{"className":486288},[31685,31821],[34,486290,31829],{"className":486291},[31828],[34,486293,486295],{"className":486294},[31801],[34,486296,486298],{"className":486297,"style":31836},[31805],[34,486299],{},"。\n取 ",[34,486302,486304,486322],{"className":486303},[31642],[34,486305,486307],{"className":486306},[31646],[31648,486308,486309],{"xmlns":31650},[31652,486310,486311,486319],{},[31655,486312,486313,486315,486317],{},[31658,486314,111],{},[31662,486316,202],{},[31749,486318,768],{},[31665,486320,486321],{"encoding":31667},"a = 2",[34,486323,486325,486343],{"className":486324,"ariaHidden":758},[31672],[34,486326,486328,486331,486334,486337,486340],{"className":486327},[31676],[34,486329],{"className":486330,"style":32527},[31680],[34,486332,111],{"className":486333},[31685,31686],[34,486335],{"className":486336,"style":31846},[31845],[34,486338,202],{"className":486339},[31850],[34,486341],{"className":486342,"style":31846},[31845],[34,486344,486346,486349],{"className":486345},[31676],[34,486347],{"className":486348,"style":32617},[31680],[34,486350,768],{"className":486351},[31685],[34,486353,486355,486379],{"className":486354},[31642],[34,486356,486358],{"className":486357},[31646],[31648,486359,486360],{"xmlns":31650},[31652,486361,486362,486376],{},[31655,486363,486364,486366,486368,486370,486372,486374],{},[31658,486365,126541],{},[31662,486367,456],{"stretchy":702},[31749,486369,768],{},[31662,486371,2557],{"stretchy":702},[31662,486373,202],{},[31749,486375,768],{},[31665,486377,486378],{"encoding":31667},"\\varphi(2) = 2",[34,486380,486382,486409],{"className":486381,"ariaHidden":758},[31672],[34,486383,486385,486388,486391,486394,486397,486400,486403,486406],{"className":486384},[31676],[34,486386],{"className":486387,"style":31774},[31680],[34,486389,126541],{"className":486390},[31685,31686],[34,486392,456],{"className":486393},[31782],[34,486395,768],{"className":486396},[31685],[34,486398,2557],{"className":486399},[31690],[34,486401],{"className":486402,"style":31846},[31845],[34,486404,202],{"className":486405},[31850],[34,486407],{"className":486408,"style":31846},[31845],[34,486410,486412,486415],{"className":486411},[31676],[34,486413],{"className":486414,"style":32617},[31680],[34,486416,768],{"className":486417},[31685],[34,486419,486421,486438],{"className":486420},[31642],[34,486422,486424],{"className":486423},[31646],[31648,486425,486426],{"xmlns":31650},[31652,486427,486428,486436],{},[31655,486429,486430],{},[31743,486431,486432,486434],{},[31658,486433,97300],{"mathvariant":318416},[31749,486435,18136],{},[31665,486437,376111],{"encoding":31667},[34,486439,486441],{"className":486440,"ariaHidden":758},[31672],[34,486442,486444,486447],{"className":486443},[31676],[34,486445],{"className":486446,"style":330413},[31680],[34,486448,486450,486453],{"className":486449},[31685],[34,486451,97300],{"className":486452},[31685,318441],[34,486454,486456],{"className":486455},[31792],[34,486457,486459,486479],{"className":486458},[31796,31797],[34,486460,486462,486476],{"className":486461},[31801],[34,486463,486465],{"className":486464,"style":31806},[31805],[34,486466,486467,486470],{"style":31809},[34,486468],{"className":486469,"style":31814},[31813],[34,486471,486473],{"className":486472},[31818,31819,31820,31821],[34,486474,18136],{"className":486475},[31685,31821],[34,486477,31829],{"className":486478},[31828],[34,486480,486482],{"className":486481},[31801],[34,486483,486485],{"className":486484,"style":31836},[31805],[34,486486],{}," 中可逆（因为 ",[34,486489,486491,486517],{"className":486490},[31642],[34,486492,486494],{"className":486493},[31646],[31648,486495,486496],{"xmlns":31650},[31652,486497,486498,486514],{},[31655,486499,486500,486502,486504,486506,486508,486510,486512],{},[31749,486501,768],{},[31662,486503,71191],{},[31749,486505,786],{},[31662,486507,202],{},[31749,486509,36656],{},[31662,486511,341696],{},[31749,486513,640],{},[31665,486515,486516],{"encoding":31667},"2 \\cdot 3 = 6 \\equiv 1",[34,486518,486520,486538,486556,486574],{"className":486519,"ariaHidden":758},[31672],[34,486521,486523,486526,486529,486532,486535],{"className":486522},[31676],[34,486524],{"className":486525,"style":32617},[31680],[34,486527,768],{"className":486528},[31685],[34,486530],{"className":486531,"style":32604},[31845],[34,486533,71191],{"className":486534},[32024],[34,486536],{"className":486537,"style":32604},[31845],[34,486539,486541,486544,486547,486550,486553],{"className":486540},[31676],[34,486542],{"className":486543,"style":32617},[31680],[34,486545,786],{"className":486546},[31685],[34,486548],{"className":486549,"style":31846},[31845],[34,486551,202],{"className":486552},[31850],[34,486554],{"className":486555,"style":31846},[31845],[34,486557,486559,486562,486565,486568,486571],{"className":486558},[31676],[34,486560],{"className":486561,"style":32617},[31680],[34,486563,36656],{"className":486564},[31685],[34,486566],{"className":486567,"style":31846},[31845],[34,486569,341696],{"className":486570},[31850],[34,486572],{"className":486573,"style":31846},[31845],[34,486575,486577,486580],{"className":486576},[31676],[34,486578],{"className":486579,"style":32617},[31680],[34,486581,640],{"className":486582},[31685],"）。\n但在 ",[34,486585,486587,486600],{"className":486586},[31642],[34,486588,486590],{"className":486589},[31646],[31648,486591,486592],{"xmlns":31650},[31652,486593,486594,486598],{},[31655,486595,486596],{},[31658,486597,97300],{"mathvariant":318416},[31665,486599,320917],{"encoding":31667},[34,486601,486603],{"className":486602,"ariaHidden":758},[31672],[34,486604,486606,486609],{"className":486605},[31676],[34,486607],{"className":486608,"style":320927},[31680],[34,486610,97300],{"className":486611},[31685,318441]," 中，只有 ",[34,486614,486616,486636],{"className":486615},[31642],[34,486617,486619],{"className":486618},[31646],[31648,486620,486621],{"xmlns":31650},[31652,486622,486623,486633],{},[31655,486624,486625,486627,486629,486631],{},[31749,486626,640],{},[31662,486628,31960],{"separator":758},[31662,486630,32226],{},[31749,486632,640],{},[31665,486634,486635],{"encoding":31667},"1, -1",[34,486637,486639],{"className":486638,"ariaHidden":758},[31672],[34,486640,486642,486645,486648,486651,486654,486657],{"className":486641},[31676],[34,486643],{"className":486644,"style":40292},[31680],[34,486646,640],{"className":486647},[31685],[34,486649,31960],{"className":486650},[32119],[34,486652],{"className":486653,"style":32123},[31845],[34,486655,32226],{"className":486656},[31685],[34,486658,640],{"className":486659},[31685]," 可逆，",[34,486662,486664,486677],{"className":486663},[31642],[34,486665,486667],{"className":486666},[31646],[31648,486668,486669],{"xmlns":31650},[31652,486670,486671,486675],{},[31655,486672,486673],{},[31749,486674,768],{},[31665,486676,768],{"encoding":31667},[34,486678,486680],{"className":486679,"ariaHidden":758},[31672],[34,486681,486683,486686],{"className":486682},[31676],[34,486684],{"className":486685,"style":32617},[31680],[34,486687,768],{"className":486688},[31685]," 不可逆。",[32188,486691],{},[32490,486693,486695],{"id":486694},"习题-19","习题 19",[15,486697,486698],{},[2627,486699,133910,486700,454385,486728,486756,486757,486830,486831,347],{},[34,486701,486703,486716],{"className":486702},[31642],[34,486704,486706],{"className":486705},[31646],[31648,486707,486708],{"xmlns":31650},[31652,486709,486710,486714],{},[31655,486711,486712],{},[31658,486713,34386],{},[31665,486715,34386],{"encoding":31667},[34,486717,486719],{"className":486718,"ariaHidden":758},[31672],[34,486720,486722,486725],{"className":486721},[31676],[34,486723],{"className":486724,"style":33218},[31680],[34,486726,34386],{"className":486727,"style":34427},[31685,31686],[34,486729,486731,486744],{"className":486730},[31642],[34,486732,486734],{"className":486733},[31646],[31648,486735,486736],{"xmlns":31650},[31652,486737,486738,486742],{},[31655,486739,486740],{},[31658,486741,37400],{},[31665,486743,37400],{"encoding":31667},[34,486745,486747],{"className":486746,"ariaHidden":758},[31672],[34,486748,486750,486753],{"className":486749},[31676],[34,486751],{"className":486752,"style":33218},[31680],[34,486754,37400],{"className":486755,"style":37422},[31685,31686]," 是整环, ",[34,486758,486760,486782],{"className":486759},[31642],[34,486761,486763],{"className":486762},[31646],[31648,486764,486765],{"xmlns":31650},[31652,486766,486767,486779],{},[31655,486768,486769,486771,486773,486775,486777],{},[31658,486770,126541],{},[31662,486772,5598],{},[31658,486774,34386],{},[31662,486776,33239],{},[31658,486778,37400],{},[31665,486780,486781],{"encoding":31667},"\\varphi: R \\to T",[34,486783,486785,486803,486821],{"className":486784,"ariaHidden":758},[31672],[34,486786,486788,486791,486794,486797,486800],{"className":486787},[31676],[34,486789],{"className":486790,"style":38774},[31680],[34,486792,126541],{"className":486793},[31685,31686],[34,486795],{"className":486796,"style":31846},[31845],[34,486798,5598],{"className":486799},[31850],[34,486801],{"className":486802,"style":31846},[31845],[34,486804,486806,486809,486812,486815,486818],{"className":486805},[31676],[34,486807],{"className":486808,"style":33218},[31680],[34,486810,34386],{"className":486811,"style":34427},[31685,31686],[34,486813],{"className":486814,"style":31846},[31845],[34,486816,33239],{"className":486817},[31850],[34,486819],{"className":486820,"style":31846},[31845],[34,486822,486824,486827],{"className":486823},[31676],[34,486825],{"className":486826,"style":33218},[31680],[34,486828,37400],{"className":486829,"style":37422},[31685,31686]," 是环同态. 证明 ",[34,486832,486834,486866],{"className":486833},[31642],[34,486835,486837],{"className":486836},[31646],[31648,486838,486839],{"xmlns":31650},[31652,486840,486841,486863],{},[31655,486842,486843,486845,486847,486853,486855,486857],{},[31658,486844,126541],{},[31662,486846,456],{"stretchy":702},[31743,486848,486849,486851],{},[31749,486850,640],{},[31658,486852,34386],{},[31662,486854,2557],{"stretchy":702},[31662,486856,202],{},[31743,486858,486859,486861],{},[31749,486860,640],{},[31658,486862,37400],{},[31665,486864,486865],{"encoding":31667},"\\varphi(1_R) = 1_T",[34,486867,486869,486933],{"className":486868,"ariaHidden":758},[31672],[34,486870,486872,486875,486878,486881,486921,486924,486927,486930],{"className":486871},[31676],[34,486873],{"className":486874,"style":31774},[31680],[34,486876,126541],{"className":486877},[31685,31686],[34,486879,456],{"className":486880},[31782],[34,486882,486884,486887],{"className":486883},[31685],[34,486885,640],{"className":486886},[31685],[34,486888,486890],{"className":486889},[31792],[34,486891,486893,486913],{"className":486892},[31796,31797],[34,486894,486896,486910],{"className":486895},[31801],[34,486897,486899],{"className":486898,"style":116558},[31805],[34,486900,486901,486904],{"style":31809},[34,486902],{"className":486903,"style":31814},[31813],[34,486905,486907],{"className":486906},[31818,31819,31820,31821],[34,486908,34386],{"className":486909,"style":34427},[31685,31686,31821],[34,486911,31829],{"className":486912},[31828],[34,486914,486916],{"className":486915},[31801],[34,486917,486919],{"className":486918,"style":31836},[31805],[34,486920],{},[34,486922,2557],{"className":486923},[31690],[34,486925],{"className":486926,"style":31846},[31845],[34,486928,202],{"className":486929},[31850],[34,486931],{"className":486932,"style":31846},[31845],[34,486934,486936,486939],{"className":486935},[31676],[34,486937],{"className":486938,"style":38357},[31680],[34,486940,486942,486945],{"className":486941},[31685],[34,486943,640],{"className":486944},[31685],[34,486946,486948],{"className":486947},[31792],[34,486949,486951,486971],{"className":486950},[31796,31797],[34,486952,486954,486968],{"className":486953},[31801],[34,486955,486957],{"className":486956,"style":116558},[31805],[34,486958,486959,486962],{"style":31809},[34,486960],{"className":486961,"style":31814},[31813],[34,486963,486965],{"className":486964},[31818,31819,31820,31821],[34,486966,37400],{"className":486967,"style":37422},[31685,31686,31821],[34,486969,31829],{"className":486970},[31828],[34,486972,486974],{"className":486973},[31801],[34,486975,486977],{"className":486976,"style":31836},[31805],[34,486978],{},[15,486980,486981],{},[2627,486982,463462],{},[105,486984,486985],{},[108,486986,486987,486988,486990],{},"一般定义环同态不强制要求保单位元，但如果陪域是",[2627,486989,330740],{},"（无零因子）且同态非零，则自动保持。",[15,486992,486993,245465,486995,487023,487024,487220,487221,487603,487604,487733,487734,61605,487807,317245,487910,86621,488039,113124,488067,488069,488070,89374,488406,488434,488435,488504,488505,488533,488534,488641,488642,46528,488670,61605,488854,488882,488883,452957,489030],{},[2627,486994,463779],{},[34,486996,486998,487011],{"className":486997},[31642],[34,486999,487001],{"className":487000},[31646],[31648,487002,487003],{"xmlns":31650},[31652,487004,487005,487009],{},[31655,487006,487007],{},[31658,487008,34386],{},[31665,487010,34386],{"encoding":31667},[34,487012,487014],{"className":487013,"ariaHidden":758},[31672],[34,487015,487017,487020],{"className":487016},[31676],[34,487018],{"className":487019,"style":33218},[31680],[34,487021,34386],{"className":487022,"style":34427},[31685,31686]," 中，有 ",[34,487025,487027,487061],{"className":487026},[31642],[34,487028,487030],{"className":487029},[31646],[31648,487031,487032],{"xmlns":31650},[31652,487033,487034,487058],{},[31655,487035,487036,487042,487044,487050,487052],{},[31743,487037,487038,487040],{},[31749,487039,640],{},[31658,487041,34386],{},[31662,487043,71191],{},[31743,487045,487046,487048],{},[31749,487047,640],{},[31658,487049,34386],{},[31662,487051,202],{},[31743,487053,487054,487056],{},[31749,487055,640],{},[31658,487057,34386],{},[31665,487059,487060],{"encoding":31667},"1_R \\cdot 1_R = 1_R",[34,487062,487064,487119,487174],{"className":487063,"ariaHidden":758},[31672],[34,487065,487067,487070,487110,487113,487116],{"className":487066},[31676],[34,487068],{"className":487069,"style":38357},[31680],[34,487071,487073,487076],{"className":487072},[31685],[34,487074,640],{"className":487075},[31685],[34,487077,487079],{"className":487078},[31792],[34,487080,487082,487102],{"className":487081},[31796,31797],[34,487083,487085,487099],{"className":487084},[31801],[34,487086,487088],{"className":487087,"style":116558},[31805],[34,487089,487090,487093],{"style":31809},[34,487091],{"className":487092,"style":31814},[31813],[34,487094,487096],{"className":487095},[31818,31819,31820,31821],[34,487097,34386],{"className":487098,"style":34427},[31685,31686,31821],[34,487100,31829],{"className":487101},[31828],[34,487103,487105],{"className":487104},[31801],[34,487106,487108],{"className":487107,"style":31836},[31805],[34,487109],{},[34,487111],{"className":487112,"style":32604},[31845],[34,487114,71191],{"className":487115},[32024],[34,487117],{"className":487118,"style":32604},[31845],[34,487120,487122,487125,487165,487168,487171],{"className":487121},[31676],[34,487123],{"className":487124,"style":38357},[31680],[34,487126,487128,487131],{"className":487127},[31685],[34,487129,640],{"className":487130},[31685],[34,487132,487134],{"className":487133},[31792],[34,487135,487137,487157],{"className":487136},[31796,31797],[34,487138,487140,487154],{"className":487139},[31801],[34,487141,487143],{"className":487142,"style":116558},[31805],[34,487144,487145,487148],{"style":31809},[34,487146],{"className":487147,"style":31814},[31813],[34,487149,487151],{"className":487150},[31818,31819,31820,31821],[34,487152,34386],{"className":487153,"style":34427},[31685,31686,31821],[34,487155,31829],{"className":487156},[31828],[34,487158,487160],{"className":487159},[31801],[34,487161,487163],{"className":487162,"style":31836},[31805],[34,487164],{},[34,487166],{"className":487167,"style":31846},[31845],[34,487169,202],{"className":487170},[31850],[34,487172],{"className":487173,"style":31846},[31845],[34,487175,487177,487180],{"className":487176},[31676],[34,487178],{"className":487179,"style":38357},[31680],[34,487181,487183,487186],{"className":487182},[31685],[34,487184,640],{"className":487185},[31685],[34,487187,487189],{"className":487188},[31792],[34,487190,487192,487212],{"className":487191},[31796,31797],[34,487193,487195,487209],{"className":487194},[31801],[34,487196,487198],{"className":487197,"style":116558},[31805],[34,487199,487200,487203],{"style":31809},[34,487201],{"className":487202,"style":31814},[31813],[34,487204,487206],{"className":487205},[31818,31819,31820,31821],[34,487207,34386],{"className":487208,"style":34427},[31685,31686,31821],[34,487210,31829],{"className":487211},[31828],[34,487213,487215],{"className":487214},[31801],[34,487216,487218],{"className":487217,"style":31836},[31805],[34,487219],{},"。\n两边取同态：\n",[34,487222,487224,487298],{"className":487223},[31642],[34,487225,487227],{"className":487226},[31646],[31648,487228,487229],{"xmlns":31650},[31652,487230,487231,487295],{},[31655,487232,487233,487235,487237,487243,487245,487247,487249,487251,487257,487259,487261,487263,487265,487271,487273,487279,487281,487283,487285,487287,487293],{},[31658,487234,126541],{},[31662,487236,456],{"stretchy":702},[31743,487238,487239,487241],{},[31749,487240,640],{},[31658,487242,34386],{},[31662,487244,2557],{"stretchy":702},[31662,487246,71191],{},[31658,487248,126541],{},[31662,487250,456],{"stretchy":702},[31743,487252,487253,487255],{},[31749,487254,640],{},[31658,487256,34386],{},[31662,487258,2557],{"stretchy":702},[31662,487260,202],{},[31658,487262,126541],{},[31662,487264,456],{"stretchy":702},[31743,487266,487267,487269],{},[31749,487268,640],{},[31658,487270,34386],{},[31662,487272,71191],{},[31743,487274,487275,487277],{},[31749,487276,640],{},[31658,487278,34386],{},[31662,487280,2557],{"stretchy":702},[31662,487282,202],{},[31658,487284,126541],{},[31662,487286,456],{"stretchy":702},[31743,487288,487289,487291],{},[31749,487290,640],{},[31658,487292,34386],{},[31662,487294,2557],{"stretchy":702},[31665,487296,487297],{"encoding":31667},"\\varphi(1_R) \\cdot \\varphi(1_R) = \\varphi(1_R \\cdot 1_R) = \\varphi(1_R)",[34,487299,487301,487365,487429,487490,487548],{"className":487300,"ariaHidden":758},[31672],[34,487302,487304,487307,487310,487313,487353,487356,487359,487362],{"className":487303},[31676],[34,487305],{"className":487306,"style":31774},[31680],[34,487308,126541],{"className":487309},[31685,31686],[34,487311,456],{"className":487312},[31782],[34,487314,487316,487319],{"className":487315},[31685],[34,487317,640],{"className":487318},[31685],[34,487320,487322],{"className":487321},[31792],[34,487323,487325,487345],{"className":487324},[31796,31797],[34,487326,487328,487342],{"className":487327},[31801],[34,487329,487331],{"className":487330,"style":116558},[31805],[34,487332,487333,487336],{"style":31809},[34,487334],{"className":487335,"style":31814},[31813],[34,487337,487339],{"className":487338},[31818,31819,31820,31821],[34,487340,34386],{"className":487341,"style":34427},[31685,31686,31821],[34,487343,31829],{"className":487344},[31828],[34,487346,487348],{"className":487347},[31801],[34,487349,487351],{"className":487350,"style":31836},[31805],[34,487352],{},[34,487354,2557],{"className":487355},[31690],[34,487357],{"className":487358,"style":32604},[31845],[34,487360,71191],{"className":487361},[32024],[34,487363],{"className":487364,"style":32604},[31845],[34,487366,487368,487371,487374,487377,487417,487420,487423,487426],{"className":487367},[31676],[34,487369],{"className":487370,"style":31774},[31680],[34,487372,126541],{"className":487373},[31685,31686],[34,487375,456],{"className":487376},[31782],[34,487378,487380,487383],{"className":487379},[31685],[34,487381,640],{"className":487382},[31685],[34,487384,487386],{"className":487385},[31792],[34,487387,487389,487409],{"className":487388},[31796,31797],[34,487390,487392,487406],{"className":487391},[31801],[34,487393,487395],{"className":487394,"style":116558},[31805],[34,487396,487397,487400],{"style":31809},[34,487398],{"className":487399,"style":31814},[31813],[34,487401,487403],{"className":487402},[31818,31819,31820,31821],[34,487404,34386],{"className":487405,"style":34427},[31685,31686,31821],[34,487407,31829],{"className":487408},[31828],[34,487410,487412],{"className":487411},[31801],[34,487413,487415],{"className":487414,"style":31836},[31805],[34,487416],{},[34,487418,2557],{"className":487419},[31690],[34,487421],{"className":487422,"style":31846},[31845],[34,487424,202],{"className":487425},[31850],[34,487427],{"className":487428,"style":31846},[31845],[34,487430,487432,487435,487438,487441,487481,487484,487487],{"className":487431},[31676],[34,487433],{"className":487434,"style":31774},[31680],[34,487436,126541],{"className":487437},[31685,31686],[34,487439,456],{"className":487440},[31782],[34,487442,487444,487447],{"className":487443},[31685],[34,487445,640],{"className":487446},[31685],[34,487448,487450],{"className":487449},[31792],[34,487451,487453,487473],{"className":487452},[31796,31797],[34,487454,487456,487470],{"className":487455},[31801],[34,487457,487459],{"className":487458,"style":116558},[31805],[34,487460,487461,487464],{"style":31809},[34,487462],{"className":487463,"style":31814},[31813],[34,487465,487467],{"className":487466},[31818,31819,31820,31821],[34,487468,34386],{"className":487469,"style":34427},[31685,31686,31821],[34,487471,31829],{"className":487472},[31828],[34,487474,487476],{"className":487475},[31801],[34,487477,487479],{"className":487478,"style":31836},[31805],[34,487480],{},[34,487482],{"className":487483,"style":32604},[31845],[34,487485,71191],{"className":487486},[32024],[34,487488],{"className":487489,"style":32604},[31845],[34,487491,487493,487496,487536,487539,487542,487545],{"className":487492},[31676],[34,487494],{"className":487495,"style":31774},[31680],[34,487497,487499,487502],{"className":487498},[31685],[34,487500,640],{"className":487501},[31685],[34,487503,487505],{"className":487504},[31792],[34,487506,487508,487528],{"className":487507},[31796,31797],[34,487509,487511,487525],{"className":487510},[31801],[34,487512,487514],{"className":487513,"style":116558},[31805],[34,487515,487516,487519],{"style":31809},[34,487517],{"className":487518,"style":31814},[31813],[34,487520,487522],{"className":487521},[31818,31819,31820,31821],[34,487523,34386],{"className":487524,"style":34427},[31685,31686,31821],[34,487526,31829],{"className":487527},[31828],[34,487529,487531],{"className":487530},[31801],[34,487532,487534],{"className":487533,"style":31836},[31805],[34,487535],{},[34,487537,2557],{"className":487538},[31690],[34,487540],{"className":487541,"style":31846},[31845],[34,487543,202],{"className":487544},[31850],[34,487546],{"className":487547,"style":31846},[31845],[34,487549,487551,487554,487557,487560,487600],{"className":487550},[31676],[34,487552],{"className":487553,"style":31774},[31680],[34,487555,126541],{"className":487556},[31685,31686],[34,487558,456],{"className":487559},[31782],[34,487561,487563,487566],{"className":487562},[31685],[34,487564,640],{"className":487565},[31685],[34,487567,487569],{"className":487568},[31792],[34,487570,487572,487592],{"className":487571},[31796,31797],[34,487573,487575,487589],{"className":487574},[31801],[34,487576,487578],{"className":487577,"style":116558},[31805],[34,487579,487580,487583],{"style":31809},[34,487581],{"className":487582,"style":31814},[31813],[34,487584,487586],{"className":487585},[31818,31819,31820,31821],[34,487587,34386],{"className":487588,"style":34427},[31685,31686,31821],[34,487590,31829],{"className":487591},[31828],[34,487593,487595],{"className":487594},[31801],[34,487596,487598],{"className":487597,"style":31836},[31805],[34,487599],{},[34,487601,2557],{"className":487602},[31690],"\n记 ",[34,487605,487607,487639],{"className":487606},[31642],[34,487608,487610],{"className":487609},[31646],[31648,487611,487612],{"xmlns":31650},[31652,487613,487614,487636],{},[31655,487615,487616,487618,487620,487622,487624,487630,487632,487634],{},[31658,487617,33244],{},[31662,487619,202],{},[31658,487621,126541],{},[31662,487623,456],{"stretchy":702},[31743,487625,487626,487628],{},[31749,487627,640],{},[31658,487629,34386],{},[31662,487631,2557],{"stretchy":702},[31662,487633,112296],{},[31658,487635,37400],{},[31665,487637,487638],{"encoding":31667},"e = \\varphi(1_R) \\in T",[34,487640,487642,487660,487724],{"className":487641,"ariaHidden":758},[31672],[34,487643,487645,487648,487651,487654,487657],{"className":487644},[31676],[34,487646],{"className":487647,"style":32527},[31680],[34,487649,33244],{"className":487650},[31685,31686],[34,487652],{"className":487653,"style":31846},[31845],[34,487655,202],{"className":487656},[31850],[34,487658],{"className":487659,"style":31846},[31845],[34,487661,487663,487666,487669,487672,487712,487715,487718,487721],{"className":487662},[31676],[34,487664],{"className":487665,"style":31774},[31680],[34,487667,126541],{"className":487668},[31685,31686],[34,487670,456],{"className":487671},[31782],[34,487673,487675,487678],{"className":487674},[31685],[34,487676,640],{"className":487677},[31685],[34,487679,487681],{"className":487680},[31792],[34,487682,487684,487704],{"className":487683},[31796,31797],[34,487685,487687,487701],{"className":487686},[31801],[34,487688,487690],{"className":487689,"style":116558},[31805],[34,487691,487692,487695],{"style":31809},[34,487693],{"className":487694,"style":31814},[31813],[34,487696,487698],{"className":487697},[31818,31819,31820,31821],[34,487699,34386],{"className":487700,"style":34427},[31685,31686,31821],[34,487702,31829],{"className":487703},[31828],[34,487705,487707],{"className":487706},[31801],[34,487708,487710],{"className":487709,"style":31836},[31805],[34,487711],{},[34,487713,2557],{"className":487714},[31690],[34,487716],{"className":487717,"style":31846},[31845],[34,487719,112296],{"className":487720},[31850],[34,487722],{"className":487723,"style":31846},[31845],[34,487725,487727,487730],{"className":487726},[31676],[34,487728],{"className":487729,"style":33218},[31680],[34,487731,37400],{"className":487732,"style":37422},[31685,31686],"。上式变为 ",[34,487735,487737,487759],{"className":487736},[31642],[34,487738,487740],{"className":487739},[31646],[31648,487741,487742],{"xmlns":31650},[31652,487743,487744,487756],{},[31655,487745,487746,487748,487750,487752,487754],{},[31658,487747,33244],{},[31662,487749,71191],{},[31658,487751,33244],{},[31662,487753,202],{},[31658,487755,33244],{},[31665,487757,487758],{"encoding":31667},"e \\cdot e = e",[34,487760,487762,487780,487798],{"className":487761,"ariaHidden":758},[31672],[34,487763,487765,487768,487771,487774,487777],{"className":487764},[31676],[34,487766],{"className":487767,"style":224869},[31680],[34,487769,33244],{"className":487770},[31685,31686],[34,487772],{"className":487773,"style":32604},[31845],[34,487775,71191],{"className":487776},[32024],[34,487778],{"className":487779,"style":32604},[31845],[34,487781,487783,487786,487789,487792,487795],{"className":487782},[31676],[34,487784],{"className":487785,"style":32527},[31680],[34,487787,33244],{"className":487788},[31685,31686],[34,487790],{"className":487791,"style":31846},[31845],[34,487793,202],{"className":487794},[31850],[34,487796],{"className":487797,"style":31846},[31845],[34,487799,487801,487804],{"className":487800},[31676],[34,487802],{"className":487803,"style":32527},[31680],[34,487805,33244],{"className":487806},[31685,31686],[34,487808,487810,487836],{"className":487809},[31642],[34,487811,487813],{"className":487812},[31646],[31648,487814,487815],{"xmlns":31650},[31652,487816,487817,487833],{},[31655,487818,487819,487825,487827,487829,487831],{},[37624,487820,487821,487823],{},[31658,487822,33244],{},[31749,487824,768],{},[31662,487826,32226],{},[31658,487828,33244],{},[31662,487830,202],{},[31749,487832,459],{},[31665,487834,487835],{"encoding":31667},"e^2 - e = 0",[34,487837,487839,487883,487901],{"className":487838,"ariaHidden":758},[31672],[34,487840,487842,487845,487874,487877,487880],{"className":487841},[31676],[34,487843],{"className":487844,"style":40044},[31680],[34,487846,487848,487851],{"className":487847},[31685],[34,487849,33244],{"className":487850},[31685,31686],[34,487852,487854],{"className":487853},[31792],[34,487855,487857],{"className":487856},[31796],[34,487858,487860],{"className":487859},[31801],[34,487861,487863],{"className":487862,"style":37774},[31805],[34,487864,487865,487868],{"style":37777},[34,487866],{"className":487867,"style":31814},[31813],[34,487869,487871],{"className":487870},[31818,31819,31820,31821],[34,487872,768],{"className":487873},[31685,31821],[34,487875],{"className":487876,"style":32604},[31845],[34,487878,32226],{"className":487879},[32024],[34,487881],{"className":487882,"style":32604},[31845],[34,487884,487886,487889,487892,487895,487898],{"className":487885},[31676],[34,487887],{"className":487888,"style":32527},[31680],[34,487890,33244],{"className":487891},[31685,31686],[34,487893],{"className":487894,"style":31846},[31845],[34,487896,202],{"className":487897},[31850],[34,487899],{"className":487900,"style":31846},[31845],[34,487902,487904,487907],{"className":487903},[31676],[34,487905],{"className":487906,"style":32617},[31680],[34,487908,459],{"className":487909},[31685],[34,487911,487913,487945],{"className":487912},[31642],[34,487914,487916],{"className":487915},[31646],[31648,487917,487918],{"xmlns":31650},[31652,487919,487920,487942],{},[31655,487921,487922,487924,487926,487928,487930,487936,487938,487940],{},[31658,487923,33244],{},[31662,487925,456],{"stretchy":702},[31658,487927,33244],{},[31662,487929,32226],{},[31743,487931,487932,487934],{},[31749,487933,640],{},[31658,487935,37400],{},[31662,487937,2557],{"stretchy":702},[31662,487939,202],{},[31749,487941,459],{},[31665,487943,487944],{"encoding":31667},"e(e - 1_T) = 0",[34,487946,487948,487972,488030],{"className":487947,"ariaHidden":758},[31672],[34,487949,487951,487954,487957,487960,487963,487966,487969],{"className":487950},[31676],[34,487952],{"className":487953,"style":31774},[31680],[34,487955,33244],{"className":487956},[31685,31686],[34,487958,456],{"className":487959},[31782],[34,487961,33244],{"className":487962},[31685,31686],[34,487964],{"className":487965,"style":32604},[31845],[34,487967,32226],{"className":487968},[32024],[34,487970],{"className":487971,"style":32604},[31845],[34,487973,487975,487978,488018,488021,488024,488027],{"className":487974},[31676],[34,487976],{"className":487977,"style":31774},[31680],[34,487979,487981,487984],{"className":487980},[31685],[34,487982,640],{"className":487983},[31685],[34,487985,487987],{"className":487986},[31792],[34,487988,487990,488010],{"className":487989},[31796,31797],[34,487991,487993,488007],{"className":487992},[31801],[34,487994,487996],{"className":487995,"style":116558},[31805],[34,487997,487998,488001],{"style":31809},[34,487999],{"className":488000,"style":31814},[31813],[34,488002,488004],{"className":488003},[31818,31819,31820,31821],[34,488005,37400],{"className":488006,"style":37422},[31685,31686,31821],[34,488008,31829],{"className":488009},[31828],[34,488011,488013],{"className":488012},[31801],[34,488014,488016],{"className":488015,"style":31836},[31805],[34,488017],{},[34,488019,2557],{"className":488020},[31690],[34,488022],{"className":488023,"style":31846},[31845],[34,488025,202],{"className":488026},[31850],[34,488028],{"className":488029,"style":31846},[31845],[34,488031,488033,488036],{"className":488032},[31676],[34,488034],{"className":488035,"style":32617},[31680],[34,488037,459],{"className":488038},[31685],[34,488040,488042,488055],{"className":488041},[31642],[34,488043,488045],{"className":488044},[31646],[31648,488046,488047],{"xmlns":31650},[31652,488048,488049,488053],{},[31655,488050,488051],{},[31658,488052,37400],{},[31665,488054,37400],{"encoding":31667},[34,488056,488058],{"className":488057,"ariaHidden":758},[31672],[34,488059,488061,488064],{"className":488060},[31676],[34,488062],{"className":488063,"style":33218},[31680],[34,488065,37400],{"className":488066,"style":37422},[31685,31686],[2627,488068,330740],{},"，没有非零零因子，所以必有：\n",[34,488071,488073,488135],{"className":488072},[31642],[34,488074,488076],{"className":488075},[31646],[31648,488077,488078],{"xmlns":31650},[31652,488079,488080,488132],{},[31655,488081,488082,488084,488086,488092,488094,488096,488098,488100,488102,488108,488110,488116,488118,488120,488122,488124,488126],{},[31658,488083,33244],{},[31662,488085,202],{},[31743,488087,488088,488090],{},[31749,488089,459],{},[31658,488091,37400],{},[31845,488093],{"width":71082},[39361,488095,58269],{},[31845,488097],{"width":71082},[31658,488099,33244],{},[31662,488101,32226],{},[31743,488103,488104,488106],{},[31749,488105,640],{},[31658,488107,37400],{},[31662,488109,202],{},[31743,488111,488112,488114],{},[31749,488113,459],{},[31658,488115,37400],{},[39361,488117,262622],{},[31662,488119,262625],{},[39361,488121,262622],{},[31658,488123,33244],{},[31662,488125,202],{},[31743,488127,488128,488130],{},[31749,488129,640],{},[31658,488131,37400],{},[31665,488133,488134],{"encoding":31667},"e = 0_T \\quad \\text{或} \\quad e - 1_T = 0_T \\implies e = 1_T",[34,488136,488138,488156,488226,488281,488342,488360],{"className":488137,"ariaHidden":758},[31672],[34,488139,488141,488144,488147,488150,488153],{"className":488140},[31676],[34,488142],{"className":488143,"style":32527},[31680],[34,488145,33244],{"className":488146},[31685,31686],[34,488148],{"className":488149,"style":31846},[31845],[34,488151,202],{"className":488152},[31850],[34,488154],{"className":488155,"style":31846},[31845],[34,488157,488159,488162,488202,488205,488211,488214,488217,488220,488223],{"className":488158},[31676],[34,488160],{"className":488161,"style":70915},[31680],[34,488163,488165,488168],{"className":488164},[31685],[34,488166,459],{"className":488167},[31685],[34,488169,488171],{"className":488170},[31792],[34,488172,488174,488194],{"className":488173},[31796,31797],[34,488175,488177,488191],{"className":488176},[31801],[34,488178,488180],{"className":488179,"style":116558},[31805],[34,488181,488182,488185],{"style":31809},[34,488183],{"className":488184,"style":31814},[31813],[34,488186,488188],{"className":488187},[31818,31819,31820,31821],[34,488189,37400],{"className":488190,"style":37422},[31685,31686,31821],[34,488192,31829],{"className":488193},[31828],[34,488195,488197],{"className":488196},[31801],[34,488198,488200],{"className":488199,"style":31836},[31805],[34,488201],{},[34,488203],{"className":488204,"style":128438},[31845],[34,488206,488208],{"className":488207},[31685,2638],[34,488209,58269],{"className":488210},[31685,105462],[34,488212],{"className":488213,"style":128438},[31845],[34,488215,33244],{"className":488216},[31685,31686],[34,488218],{"className":488219,"style":32604},[31845],[34,488221,32226],{"className":488222},[32024],[34,488224],{"className":488225,"style":32604},[31845],[34,488227,488229,488232,488272,488275,488278],{"className":488228},[31676],[34,488230],{"className":488231,"style":38357},[31680],[34,488233,488235,488238],{"className":488234},[31685],[34,488236,640],{"className":488237},[31685],[34,488239,488241],{"className":488240},[31792],[34,488242,488244,488264],{"className":488243},[31796,31797],[34,488245,488247,488261],{"className":488246},[31801],[34,488248,488250],{"className":488249,"style":116558},[31805],[34,488251,488252,488255],{"style":31809},[34,488253],{"className":488254,"style":31814},[31813],[34,488256,488258],{"className":488257},[31818,31819,31820,31821],[34,488259,37400],{"className":488260,"style":37422},[31685,31686,31821],[34,488262,31829],{"className":488263},[31828],[34,488265,488267],{"className":488266},[31801],[34,488268,488270],{"className":488269,"style":31836},[31805],[34,488271],{},[34,488273],{"className":488274,"style":31846},[31845],[34,488276,202],{"className":488277},[31850],[34,488279],{"className":488280,"style":31846},[31845],[34,488282,488284,488287,488327,488330,488333,488336,488339],{"className":488283},[31676],[34,488285],{"className":488286,"style":38357},[31680],[34,488288,488290,488293],{"className":488289},[31685],[34,488291,459],{"className":488292},[31685],[34,488294,488296],{"className":488295},[31792],[34,488297,488299,488319],{"className":488298},[31796,31797],[34,488300,488302,488316],{"className":488301},[31801],[34,488303,488305],{"className":488304,"style":116558},[31805],[34,488306,488307,488310],{"style":31809},[34,488308],{"className":488309,"style":31814},[31813],[34,488311,488313],{"className":488312},[31818,31819,31820,31821],[34,488314,37400],{"className":488315,"style":37422},[31685,31686,31821],[34,488317,31829],{"className":488318},[31828],[34,488320,488322],{"className":488321},[31801],[34,488323,488325],{"className":488324,"style":31836},[31805],[34,488326],{},[34,488328],{"className":488329,"style":31846},[31845],[34,488331],{"className":488332,"style":31846},[31845],[34,488334,262625],{"className":488335},[31850],[34,488337],{"className":488338,"style":31846},[31845],[34,488340],{"className":488341,"style":31846},[31845],[34,488343,488345,488348,488351,488354,488357],{"className":488344},[31676],[34,488346],{"className":488347,"style":32527},[31680],[34,488349,33244],{"className":488350},[31685,31686],[34,488352],{"className":488353,"style":31846},[31845],[34,488355,202],{"className":488356},[31850],[34,488358],{"className":488359,"style":31846},[31845],[34,488361,488363,488366],{"className":488362},[31676],[34,488364],{"className":488365,"style":38357},[31680],[34,488367,488369,488372],{"className":488368},[31685],[34,488370,640],{"className":488371},[31685],[34,488373,488375],{"className":488374},[31792],[34,488376,488378,488398],{"className":488377},[31796,31797],[34,488379,488381,488395],{"className":488380},[31801],[34,488382,488384],{"className":488383,"style":116558},[31805],[34,488385,488386,488389],{"style":31809},[34,488387],{"className":488388,"style":31814},[31813],[34,488390,488392],{"className":488391},[31818,31819,31820,31821],[34,488393,37400],{"className":488394,"style":37422},[31685,31686,31821],[34,488396,31829],{"className":488397},[31828],[34,488399,488401],{"className":488400},[31801],[34,488402,488404],{"className":488403,"style":31836},[31805],[34,488405],{},[34,488407,488409,488422],{"className":488408},[31642],[34,488410,488412],{"className":488411},[31646],[31648,488413,488414],{"xmlns":31650},[31652,488415,488416,488420],{},[31655,488417,488418],{},[31658,488419,126541],{},[31665,488421,126708],{"encoding":31667},[34,488423,488425],{"className":488424,"ariaHidden":758},[31672],[34,488426,488428,488431],{"className":488427},[31676],[34,488429],{"className":488430,"style":38774},[31680],[34,488432,126541],{"className":488433},[31685,31686]," 不是零同态（通常默认含单位元的环同态把 ",[34,488436,488438,488455],{"className":488437},[31642],[34,488439,488441],{"className":488440},[31646],[31648,488442,488443],{"xmlns":31650},[31652,488444,488445,488453],{},[31655,488446,488447],{},[31743,488448,488449,488451],{},[31749,488450,640],{},[31658,488452,34386],{},[31665,488454,431139],{"encoding":31667},[34,488456,488458],{"className":488457,"ariaHidden":758},[31672],[34,488459,488461,488464],{"className":488460},[31676],[34,488462],{"className":488463,"style":38357},[31680],[34,488465,488467,488470],{"className":488466},[31685],[34,488468,640],{"className":488469},[31685],[34,488471,488473],{"className":488472},[31792],[34,488474,488476,488496],{"className":488475},[31796,31797],[34,488477,488479,488493],{"className":488478},[31801],[34,488480,488482],{"className":488481,"style":116558},[31805],[34,488483,488484,488487],{"style":31809},[34,488485],{"className":488486,"style":31814},[31813],[34,488488,488490],{"className":488489},[31818,31819,31820,31821],[34,488491,34386],{"className":488492,"style":34427},[31685,31686,31821],[34,488494,31829],{"className":488495},[31828],[34,488497,488499],{"className":488498},[31801],[34,488500,488502],{"className":488501,"style":31836},[31805],[34,488503],{}," 映到非零元，或者题目隐含非平凡同态），且 ",[34,488506,488508,488521],{"className":488507},[31642],[34,488509,488511],{"className":488510},[31646],[31648,488512,488513],{"xmlns":31650},[31652,488514,488515,488519],{},[31655,488516,488517],{},[31658,488518,37400],{},[31665,488520,37400],{"encoding":31667},[34,488522,488524],{"className":488523,"ariaHidden":758},[31672],[34,488525,488527,488530],{"className":488526},[31676],[34,488528],{"className":488529,"style":33218},[31680],[34,488531,37400],{"className":488532,"style":37422},[31685,31686]," 是含单位元的环。如果 ",[34,488535,488537,488565],{"className":488536},[31642],[34,488538,488540],{"className":488539},[31646],[31648,488541,488542],{"xmlns":31650},[31652,488543,488544,488562],{},[31655,488545,488546,488548,488550,488556,488558,488560],{},[31658,488547,126541],{},[31662,488549,456],{"stretchy":702},[31743,488551,488552,488554],{},[31749,488553,640],{},[31658,488555,34386],{},[31662,488557,2557],{"stretchy":702},[31662,488559,202],{},[31749,488561,459],{},[31665,488563,488564],{"encoding":31667},"\\varphi(1_R)=0",[34,488566,488568,488632],{"className":488567,"ariaHidden":758},[31672],[34,488569,488571,488574,488577,488580,488620,488623,488626,488629],{"className":488570},[31676],[34,488572],{"className":488573,"style":31774},[31680],[34,488575,126541],{"className":488576},[31685,31686],[34,488578,456],{"className":488579},[31782],[34,488581,488583,488586],{"className":488582},[31685],[34,488584,640],{"className":488585},[31685],[34,488587,488589],{"className":488588},[31792],[34,488590,488592,488612],{"className":488591},[31796,31797],[34,488593,488595,488609],{"className":488594},[31801],[34,488596,488598],{"className":488597,"style":116558},[31805],[34,488599,488600,488603],{"style":31809},[34,488601],{"className":488602,"style":31814},[31813],[34,488604,488606],{"className":488605},[31818,31819,31820,31821],[34,488607,34386],{"className":488608,"style":34427},[31685,31686,31821],[34,488610,31829],{"className":488611},[31828],[34,488613,488615],{"className":488614},[31801],[34,488616,488618],{"className":488617,"style":31836},[31805],[34,488619],{},[34,488621,2557],{"className":488622},[31690],[34,488624],{"className":488625,"style":31846},[31845],[34,488627,202],{"className":488628},[31850],[34,488630],{"className":488631,"style":31846},[31845],[34,488633,488635,488638],{"className":488634},[31676],[34,488636],{"className":488637,"style":32617},[31680],[34,488639,459],{"className":488640},[31685],"，则对于任意 ",[34,488643,488645,488658],{"className":488644},[31642],[34,488646,488648],{"className":488647},[31646],[31648,488649,488650],{"xmlns":31650},[31652,488651,488652,488656],{},[31655,488653,488654],{},[31658,488655,34395],{},[31665,488657,34395],{"encoding":31667},[34,488659,488661],{"className":488660,"ariaHidden":758},[31672],[34,488662,488664,488667],{"className":488663},[31676],[34,488665],{"className":488666,"style":32527},[31680],[34,488668,34395],{"className":488669,"style":34452},[31685,31686],[34,488671,488673,488725],{"className":488672},[31642],[34,488674,488676],{"className":488675},[31646],[31648,488677,488678],{"xmlns":31650},[31652,488679,488680,488722],{},[31655,488681,488682,488684,488686,488688,488690,488692,488694,488696,488698,488700,488702,488704,488706,488708,488710,488712,488714,488716,488718,488720],{},[31658,488683,126541],{},[31662,488685,456],{"stretchy":702},[31658,488687,34395],{},[31662,488689,2557],{"stretchy":702},[31662,488691,202],{},[31658,488693,126541],{},[31662,488695,456],{"stretchy":702},[31658,488697,34395],{},[31662,488699,71191],{},[31749,488701,640],{},[31662,488703,2557],{"stretchy":702},[31662,488705,202],{},[31658,488707,126541],{},[31662,488709,456],{"stretchy":702},[31658,488711,34395],{},[31662,488713,2557],{"stretchy":702},[31662,488715,71191],{},[31749,488717,459],{},[31662,488719,202],{},[31749,488721,459],{},[31665,488723,488724],{"encoding":31667},"\\varphi(r)=\\varphi(r \\cdot 1)=\\varphi(r) \\cdot 0 = 0",[34,488726,488728,488755,488779,488800,488827,488845],{"className":488727,"ariaHidden":758},[31672],[34,488729,488731,488734,488737,488740,488743,488746,488749,488752],{"className":488730},[31676],[34,488732],{"className":488733,"style":31774},[31680],[34,488735,126541],{"className":488736},[31685,31686],[34,488738,456],{"className":488739},[31782],[34,488741,34395],{"className":488742,"style":34452},[31685,31686],[34,488744,2557],{"className":488745},[31690],[34,488747],{"className":488748,"style":31846},[31845],[34,488750,202],{"className":488751},[31850],[34,488753],{"className":488754,"style":31846},[31845],[34,488756,488758,488761,488764,488767,488770,488773,488776],{"className":488757},[31676],[34,488759],{"className":488760,"style":31774},[31680],[34,488762,126541],{"className":488763},[31685,31686],[34,488765,456],{"className":488766},[31782],[34,488768,34395],{"className":488769,"style":34452},[31685,31686],[34,488771],{"className":488772,"style":32604},[31845],[34,488774,71191],{"className":488775},[32024],[34,488777],{"className":488778,"style":32604},[31845],[34,488780,488782,488785,488788,488791,488794,488797],{"className":488781},[31676],[34,488783],{"className":488784,"style":31774},[31680],[34,488786,640],{"className":488787},[31685],[34,488789,2557],{"className":488790},[31690],[34,488792],{"className":488793,"style":31846},[31845],[34,488795,202],{"className":488796},[31850],[34,488798],{"className":488799,"style":31846},[31845],[34,488801,488803,488806,488809,488812,488815,488818,488821,488824],{"className":488802},[31676],[34,488804],{"className":488805,"style":31774},[31680],[34,488807,126541],{"className":488808},[31685,31686],[34,488810,456],{"className":488811},[31782],[34,488813,34395],{"className":488814,"style":34452},[31685,31686],[34,488816,2557],{"className":488817},[31690],[34,488819],{"className":488820,"style":32604},[31845],[34,488822,71191],{"className":488823},[32024],[34,488825],{"className":488826,"style":32604},[31845],[34,488828,488830,488833,488836,488839,488842],{"className":488829},[31676],[34,488831],{"className":488832,"style":32617},[31680],[34,488834,459],{"className":488835},[31685],[34,488837],{"className":488838,"style":31846},[31845],[34,488840,202],{"className":488841},[31850],[34,488843],{"className":488844,"style":31846},[31845],[34,488846,488848,488851],{"className":488847},[31676],[34,488849],{"className":488850,"style":32617},[31680],[34,488852,459],{"className":488853},[31685],[34,488855,488857,488870],{"className":488856},[31642],[34,488858,488860],{"className":488859},[31646],[31648,488861,488862],{"xmlns":31650},[31652,488863,488864,488868],{},[31655,488865,488866],{},[31658,488867,126541],{},[31665,488869,126708],{"encoding":31667},[34,488871,488873],{"className":488872,"ariaHidden":758},[31672],[34,488874,488876,488879],{"className":488875},[31676],[34,488877],{"className":488878,"style":38774},[31680],[34,488880,126541],{"className":488881},[31685,31686]," 是零映射。\n如果排除零映射的情况，则必有 ",[34,488884,488886,488917],{"className":488885},[31642],[34,488887,488889],{"className":488888},[31646],[31648,488890,488891],{"xmlns":31650},[31652,488892,488893,488915],{},[31655,488894,488895,488897,488899,488905,488907,488909],{},[31658,488896,126541],{},[31662,488898,456],{"stretchy":702},[31743,488900,488901,488903],{},[31749,488902,640],{},[31658,488904,34386],{},[31662,488906,2557],{"stretchy":702},[31662,488908,202],{},[31743,488910,488911,488913],{},[31749,488912,640],{},[31658,488914,37400],{},[31665,488916,486865],{"encoding":31667},[34,488918,488920,488984],{"className":488919,"ariaHidden":758},[31672],[34,488921,488923,488926,488929,488932,488972,488975,488978,488981],{"className":488922},[31676],[34,488924],{"className":488925,"style":31774},[31680],[34,488927,126541],{"className":488928},[31685,31686],[34,488930,456],{"className":488931},[31782],[34,488933,488935,488938],{"className":488934},[31685],[34,488936,640],{"className":488937},[31685],[34,488939,488941],{"className":488940},[31792],[34,488942,488944,488964],{"className":488943},[31796,31797],[34,488945,488947,488961],{"className":488946},[31801],[34,488948,488950],{"className":488949,"style":116558},[31805],[34,488951,488952,488955],{"style":31809},[34,488953],{"className":488954,"style":31814},[31813],[34,488956,488958],{"className":488957},[31818,31819,31820,31821],[34,488959,34386],{"className":488960,"style":34427},[31685,31686,31821],[34,488962,31829],{"className":488963},[31828],[34,488965,488967],{"className":488966},[31801],[34,488968,488970],{"className":488969,"style":31836},[31805],[34,488971],{},[34,488973,2557],{"className":488974},[31690],[34,488976],{"className":488977,"style":31846},[31845],[34,488979,202],{"className":488980},[31850],[34,488982],{"className":488983,"style":31846},[31845],[34,488985,488987,488990],{"className":488986},[31676],[34,488988],{"className":488989,"style":38357},[31680],[34,488991,488993,488996],{"className":488992},[31685],[34,488994,640],{"className":488995},[31685],[34,488997,488999],{"className":488998},[31792],[34,489000,489002,489022],{"className":489001},[31796,31797],[34,489003,489005,489019],{"className":489004},[31801],[34,489006,489008],{"className":489007,"style":116558},[31805],[34,489009,489010,489013],{"style":31809},[34,489011],{"className":489012,"style":31814},[31813],[34,489014,489016],{"className":489015},[31818,31819,31820,31821],[34,489017,37400],{"className":489018,"style":37422},[31685,31686,31821],[34,489020,31829],{"className":489021},[31828],[34,489023,489025],{"className":489024},[31801],[34,489026,489028],{"className":489027,"style":31836},[31805],[34,489029],{},[34,489031,489033,489046],{"className":489032},[31642],[34,489034,489036],{"className":489035},[31646],[31648,489037,489038],{"xmlns":31650},[31652,489039,489040,489044],{},[31655,489041,489042],{},[31658,489043,445204],{"mathvariant":38079},[31665,489045,445207],{"encoding":31667},[34,489047,489049],{"className":489048,"ariaHidden":758},[31672],[34,489050,489052,489055],{"className":489051},[31676],[34,489053],{"className":489054,"style":445217},[31680],[34,489056,445204],{"className":489057},[31685,129261],[32188,489059],{},[32490,489061,489063],{"id":489062},"习题-20","习题 20",[15,489065,489066,489200,489201,616,489318,347],{},[2627,489067,489068,489069,489134,489135,489199],{},"证明构造的环 ",[34,489070,489072,489098],{"className":489071},[31642],[34,489073,489075],{"className":489074},[31646],[31648,489076,489077],{"xmlns":31650},[31652,489078,489079,489095],{},[31655,489080,489081,489083,489085,489087,489089,489091,489093],{},[31662,489082,456],{"stretchy":702},[31658,489084,34386],{},[31662,489086,31960],{"separator":758},[31662,489088,378006],{},[31662,489090,31960],{"separator":758},[31662,489092,378082],{},[31662,489094,2557],{"stretchy":702},[31665,489096,489097],{"encoding":31667},"(R, \\oplus, \\odot)",[34,489099,489101],{"className":489100,"ariaHidden":758},[31672],[34,489102,489104,489107,489110,489113,489116,489119,489122,489125,489128,489131],{"className":489103},[31676],[34,489105],{"className":489106,"style":31774},[31680],[34,489108,456],{"className":489109},[31782],[34,489111,34386],{"className":489112,"style":34427},[31685,31686],[34,489114,31960],{"className":489115},[32119],[34,489117],{"className":489118,"style":32123},[31845],[34,489120,378006],{"className":489121},[31685],[34,489123,31960],{"className":489124},[32119],[34,489126],{"className":489127,"style":32123},[31845],[34,489129,378082],{"className":489130},[31685],[34,489132,2557],{"className":489133},[31690]," 与原环 ",[34,489136,489138,489163],{"className":489137},[31642],[34,489139,489141],{"className":489140},[31646],[31648,489142,489143],{"xmlns":31650},[31652,489144,489145,489161],{},[31655,489146,489147,489149,489151,489153,489155,489157,489159],{},[31662,489148,456],{"stretchy":702},[31658,489150,34386],{},[31662,489152,31960],{"separator":758},[31662,489154,223],{},[31662,489156,31960],{"separator":758},[31662,489158,71191],{},[31662,489160,2557],{"stretchy":702},[31665,489162,365556],{"encoding":31667},[34,489164,489166],{"className":489165,"ariaHidden":758},[31672],[34,489167,489169,489172,489175,489178,489181,489184,489187,489190,489193,489196],{"className":489168},[31676],[34,489170],{"className":489171,"style":31774},[31680],[34,489173,456],{"className":489174},[31782],[34,489176,34386],{"className":489177,"style":34427},[31685,31686],[34,489179,31960],{"className":489180},[32119],[34,489182],{"className":489183,"style":32123},[31845],[34,489185,223],{"className":489186},[31685],[34,489188,31960],{"className":489189},[32119],[34,489191],{"className":489192,"style":32123},[31845],[34,489194,71191],{"className":489195},[31685],[34,489197,2557],{"className":489198},[31690]," 同构。","\n定义：",[34,489202,489204,489234],{"className":489203},[31642],[34,489205,489207],{"className":489206},[31646],[31648,489208,489209],{"xmlns":31650},[31652,489210,489211,489231],{},[31655,489212,489213,489215,489217,489219,489221,489223,489225,489227,489229],{},[31658,489214,111],{},[31662,489216,378006],{},[31658,489218,1697],{},[31662,489220,202],{},[31658,489222,111],{},[31662,489224,223],{},[31658,489226,1697],{},[31662,489228,32226],{},[31749,489230,640],{},[31665,489232,489233],{"encoding":31667},"a \\oplus b = a + b - 1",[34,489235,489237,489255,489273,489291,489309],{"className":489236,"ariaHidden":758},[31672],[34,489238,489240,489243,489246,489249,489252],{"className":489239},[31676],[34,489241],{"className":489242,"style":32597},[31680],[34,489244,111],{"className":489245},[31685,31686],[34,489247],{"className":489248,"style":32604},[31845],[34,489250,378006],{"className":489251},[32024],[34,489253],{"className":489254,"style":32604},[31845],[34,489256,489258,489261,489264,489267,489270],{"className":489257},[31676],[34,489259],{"className":489260,"style":31681},[31680],[34,489262,1697],{"className":489263},[31685,31686],[34,489265],{"className":489266,"style":31846},[31845],[34,489268,202],{"className":489269},[31850],[34,489271],{"className":489272,"style":31846},[31845],[34,489274,489276,489279,489282,489285,489288],{"className":489275},[31676],[34,489277],{"className":489278,"style":32597},[31680],[34,489280,111],{"className":489281},[31685,31686],[34,489283],{"className":489284,"style":32604},[31845],[34,489286,223],{"className":489287},[32024],[34,489289],{"className":489290,"style":32604},[31845],[34,489292,489294,489297,489300,489303,489306],{"className":489293},[31676],[34,489295],{"className":489296,"style":34299},[31680],[34,489298,1697],{"className":489299},[31685,31686],[34,489301],{"className":489302,"style":32604},[31845],[34,489304,32226],{"className":489305},[32024],[34,489307],{"className":489308,"style":32604},[31845],[34,489310,489312,489315],{"className":489311},[31676],[34,489313],{"className":489314,"style":32617},[31680],[34,489316,640],{"className":489317},[31685],[34,489319,489321,489353],{"className":489320},[31642],[34,489322,489324],{"className":489323},[31646],[31648,489325,489326],{"xmlns":31650},[31652,489327,489328,489350],{},[31655,489329,489330,489332,489334,489336,489338,489340,489342,489344,489346,489348],{},[31658,489331,111],{},[31662,489333,378082],{},[31658,489335,1697],{},[31662,489337,202],{},[31658,489339,111],{},[31662,489341,223],{},[31658,489343,1697],{},[31662,489345,32226],{},[31658,489347,111],{},[31658,489349,1697],{},[31665,489351,489352],{"encoding":31667},"a \\odot b = a + b - ab",[34,489354,489356,489374,489392,489410,489428],{"className":489355,"ariaHidden":758},[31672],[34,489357,489359,489362,489365,489368,489371],{"className":489358},[31676],[34,489360],{"className":489361,"style":32597},[31680],[34,489363,111],{"className":489364},[31685,31686],[34,489366],{"className":489367,"style":32604},[31845],[34,489369,378082],{"className":489370},[32024],[34,489372],{"className":489373,"style":32604},[31845],[34,489375,489377,489380,489383,489386,489389],{"className":489376},[31676],[34,489378],{"className":489379,"style":31681},[31680],[34,489381,1697],{"className":489382},[31685,31686],[34,489384],{"className":489385,"style":31846},[31845],[34,489387,202],{"className":489388},[31850],[34,489390],{"className":489391,"style":31846},[31845],[34,489393,489395,489398,489401,489404,489407],{"className":489394},[31676],[34,489396],{"className":489397,"style":32597},[31680],[34,489399,111],{"className":489400},[31685,31686],[34,489402],{"className":489403,"style":32604},[31845],[34,489405,223],{"className":489406},[32024],[34,489408],{"className":489409,"style":32604},[31845],[34,489411,489413,489416,489419,489422,489425],{"className":489412},[31676],[34,489414],{"className":489415,"style":34299},[31680],[34,489417,1697],{"className":489418},[31685,31686],[34,489420],{"className":489421,"style":32604},[31845],[34,489423,32226],{"className":489424},[32024],[34,489426],{"className":489427,"style":32604},[31845],[34,489429,489431,489434],{"className":489430},[31676],[34,489432],{"className":489433,"style":31681},[31680],[34,489435,324191],{"className":489436},[31685,31686],[15,489438,489439],{},[2627,489440,463462],{},[105,489442,489443,489591],{},[108,489444,489445,489446,347],{},"我们需要构造一个双射 ",[34,489447,489449,489495],{"className":489448},[31642],[34,489450,489452],{"className":489451},[31646],[31648,489453,489454],{"xmlns":31650},[31652,489455,489456,489492],{},[31655,489457,489458,489460,489462,489464,489466,489468,489470,489472,489474,489476,489478,489480,489482,489484,489486,489488,489490],{},[31658,489459,31739],{},[31662,489461,5598],{},[31662,489463,456],{"stretchy":702},[31658,489465,34386],{},[31662,489467,31960],{"separator":758},[31662,489469,378006],{},[31662,489471,31960],{"separator":758},[31662,489473,378082],{},[31662,489475,2557],{"stretchy":702},[31662,489477,33239],{},[31662,489479,456],{"stretchy":702},[31658,489481,34386],{},[31662,489483,31960],{"separator":758},[31662,489485,223],{},[31662,489487,31960],{"separator":758},[31662,489489,71191],{},[31662,489491,2557],{"stretchy":702},[31665,489493,489494],{"encoding":31667},"f: (R, \\oplus, \\odot) \\to (R, +, \\cdot)",[34,489496,489498,489516,489558],{"className":489497,"ariaHidden":758},[31672],[34,489499,489501,489504,489507,489510,489513],{"className":489500},[31676],[34,489502],{"className":489503,"style":43217},[31680],[34,489505,31739],{"className":489506,"style":31778},[31685,31686],[34,489508],{"className":489509,"style":31846},[31845],[34,489511,5598],{"className":489512},[31850],[34,489514],{"className":489515,"style":31846},[31845],[34,489517,489519,489522,489525,489528,489531,489534,489537,489540,489543,489546,489549,489552,489555],{"className":489518},[31676],[34,489520],{"className":489521,"style":31774},[31680],[34,489523,456],{"className":489524},[31782],[34,489526,34386],{"className":489527,"style":34427},[31685,31686],[34,489529,31960],{"className":489530},[32119],[34,489532],{"className":489533,"style":32123},[31845],[34,489535,378006],{"className":489536},[31685],[34,489538,31960],{"className":489539},[32119],[34,489541],{"className":489542,"style":32123},[31845],[34,489544,378082],{"className":489545},[31685],[34,489547,2557],{"className":489548},[31690],[34,489550],{"className":489551,"style":31846},[31845],[34,489553,33239],{"className":489554},[31850],[34,489556],{"className":489557,"style":31846},[31845],[34,489559,489561,489564,489567,489570,489573,489576,489579,489582,489585,489588],{"className":489560},[31676],[34,489562],{"className":489563,"style":31774},[31680],[34,489565,456],{"className":489566},[31782],[34,489568,34386],{"className":489569,"style":34427},[31685,31686],[34,489571,31960],{"className":489572},[32119],[34,489574],{"className":489575,"style":32123},[31845],[34,489577,223],{"className":489578},[31685],[34,489580,31960],{"className":489581},[32119],[34,489583],{"className":489584,"style":32123},[31845],[34,489586,71191],{"className":489587},[31685],[34,489589,2557],{"className":489590},[31690],[108,489592,489593,489594,489758,489759,489832],{},"观察运算：",[34,489595,489597,489641],{"className":489596},[31642],[34,489598,489600],{"className":489599},[31646],[31648,489601,489602],{"xmlns":31650},[31652,489603,489604,489638],{},[31655,489605,489606,489608,489610,489612,489614,489616,489618,489620,489622,489624,489626,489628,489630,489632,489634,489636],{},[31658,489607,111],{},[31662,489609,378082],{},[31658,489611,1697],{},[31662,489613,202],{},[31749,489615,640],{},[31662,489617,32226],{},[31662,489619,456],{"stretchy":702},[31749,489621,640],{},[31662,489623,32226],{},[31658,489625,111],{},[31662,489627,2557],{"stretchy":702},[31662,489629,456],{"stretchy":702},[31749,489631,640],{},[31662,489633,32226],{},[31658,489635,1697],{},[31662,489637,2557],{"stretchy":702},[31665,489639,489640],{"encoding":31667},"a \\odot b = 1 - (1-a)(1-b)",[34,489642,489644,489662,489680,489698,489719,489746],{"className":489643,"ariaHidden":758},[31672],[34,489645,489647,489650,489653,489656,489659],{"className":489646},[31676],[34,489648],{"className":489649,"style":32597},[31680],[34,489651,111],{"className":489652},[31685,31686],[34,489654],{"className":489655,"style":32604},[31845],[34,489657,378082],{"className":489658},[32024],[34,489660],{"className":489661,"style":32604},[31845],[34,489663,489665,489668,489671,489674,489677],{"className":489664},[31676],[34,489666],{"className":489667,"style":31681},[31680],[34,489669,1697],{"className":489670},[31685,31686],[34,489672],{"className":489673,"style":31846},[31845],[34,489675,202],{"className":489676},[31850],[34,489678],{"className":489679,"style":31846},[31845],[34,489681,489683,489686,489689,489692,489695],{"className":489682},[31676],[34,489684],{"className":489685,"style":35942},[31680],[34,489687,640],{"className":489688},[31685],[34,489690],{"className":489691,"style":32604},[31845],[34,489693,32226],{"className":489694},[32024],[34,489696],{"className":489697,"style":32604},[31845],[34,489699,489701,489704,489707,489710,489713,489716],{"className":489700},[31676],[34,489702],{"className":489703,"style":31774},[31680],[34,489705,456],{"className":489706},[31782],[34,489708,640],{"className":489709},[31685],[34,489711],{"className":489712,"style":32604},[31845],[34,489714,32226],{"className":489715},[32024],[34,489717],{"className":489718,"style":32604},[31845],[34,489720,489722,489725,489728,489731,489734,489737,489740,489743],{"className":489721},[31676],[34,489723],{"className":489724,"style":31774},[31680],[34,489726,111],{"className":489727},[31685,31686],[34,489729,2557],{"className":489730},[31690],[34,489732,456],{"className":489733},[31782],[34,489735,640],{"className":489736},[31685],[34,489738],{"className":489739,"style":32604},[31845],[34,489741,32226],{"className":489742},[32024],[34,489744],{"className":489745,"style":32604},[31845],[34,489747,489749,489752,489755],{"className":489748},[31676],[34,489750],{"className":489751,"style":31774},[31680],[34,489753,1697],{"className":489754},[31685,31686],[34,489756,2557],{"className":489757},[31690],"。这个形式暗示了 ",[34,489760,489762,489784],{"className":489761},[31642],[34,489763,489765],{"className":489764},[31646],[31648,489766,489767],{"xmlns":31650},[31652,489768,489769,489781],{},[31655,489770,489771,489773,489775,489777,489779],{},[31658,489772,32936],{},[31662,489774,418784],{},[31749,489776,640],{},[31662,489778,32226],{},[31658,489780,32936],{},[31665,489782,489783],{"encoding":31667},"x \\mapsto 1-x",[34,489785,489787,489805,489823],{"className":489786,"ariaHidden":758},[31672],[34,489788,489790,489793,489796,489799,489802],{"className":489789},[31676],[34,489791],{"className":489792,"style":330733},[31680],[34,489794,32936],{"className":489795},[31685,31686],[34,489797],{"className":489798,"style":31846},[31845],[34,489800,418784],{"className":489801},[31850],[34,489803],{"className":489804,"style":31846},[31845],[34,489806,489808,489811,489814,489817,489820],{"className":489807},[31676],[34,489809],{"className":489810,"style":35942},[31680],[34,489812,640],{"className":489813},[31685],[34,489815],{"className":489816,"style":32604},[31845],[34,489818,32226],{"className":489819},[32024],[34,489821],{"className":489822,"style":32604},[31845],[34,489824,489826,489829],{"className":489825},[31676],[34,489827],{"className":489828,"style":32527},[31680],[34,489830,32936],{"className":489831},[31685,31686]," 可能是一个同构。",[15,489834,489835,489837,489838,489911,489912,347],{},[2627,489836,463779],{},"\n构造映射 ",[34,489839,489841,489863],{"className":489840},[31642],[34,489842,489844],{"className":489843},[31646],[31648,489845,489846],{"xmlns":31650},[31652,489847,489848,489860],{},[31655,489849,489850,489852,489854,489856,489858],{},[31658,489851,31739],{},[31662,489853,5598],{},[31658,489855,34386],{},[31662,489857,33239],{},[31658,489859,34386],{},[31665,489861,489862],{"encoding":31667},"f: R \\to R",[34,489864,489866,489884,489902],{"className":489865,"ariaHidden":758},[31672],[34,489867,489869,489872,489875,489878,489881],{"className":489868},[31676],[34,489870],{"className":489871,"style":43217},[31680],[34,489873,31739],{"className":489874,"style":31778},[31685,31686],[34,489876],{"className":489877,"style":31846},[31845],[34,489879,5598],{"className":489880},[31850],[34,489882],{"className":489883,"style":31846},[31845],[34,489885,489887,489890,489893,489896,489899],{"className":489886},[31676],[34,489888],{"className":489889,"style":33218},[31680],[34,489891,34386],{"className":489892,"style":34427},[31685,31686],[34,489894],{"className":489895,"style":31846},[31845],[34,489897,33239],{"className":489898},[31850],[34,489900],{"className":489901,"style":31846},[31845],[34,489903,489905,489908],{"className":489904},[31676],[34,489906],{"className":489907,"style":33218},[31680],[34,489909,34386],{"className":489910,"style":34427},[31685,31686],"，定义为 ",[34,489913,489915,489943],{"className":489914},[31642],[34,489916,489918],{"className":489917},[31646],[31648,489919,489920],{"xmlns":31650},[31652,489921,489922,489940],{},[31655,489923,489924,489926,489928,489930,489932,489934,489936,489938],{},[31658,489925,31739],{},[31662,489927,456],{"stretchy":702},[31658,489929,32936],{},[31662,489931,2557],{"stretchy":702},[31662,489933,202],{},[31749,489935,640],{},[31662,489937,32226],{},[31658,489939,32936],{},[31665,489941,489942],{"encoding":31667},"f(x) = 1 - x",[34,489944,489946,489973,489991],{"className":489945,"ariaHidden":758},[31672],[34,489947,489949,489952,489955,489958,489961,489964,489967,489970],{"className":489948},[31676],[34,489950],{"className":489951,"style":31774},[31680],[34,489953,31739],{"className":489954,"style":31778},[31685,31686],[34,489956,456],{"className":489957},[31782],[34,489959,32936],{"className":489960},[31685,31686],[34,489962,2557],{"className":489963},[31690],[34,489965],{"className":489966,"style":31846},[31845],[34,489968,202],{"className":489969},[31850],[34,489971],{"className":489972,"style":31846},[31845],[34,489974,489976,489979,489982,489985,489988],{"className":489975},[31676],[34,489977],{"className":489978,"style":35942},[31680],[34,489980,640],{"className":489981},[31685],[34,489983],{"className":489984,"style":32604},[31845],[34,489986,32226],{"className":489987},[32024],[34,489989],{"className":489990,"style":32604},[31845],[34,489992,489994,489997],{"className":489993},[31676],[34,489995],{"className":489996,"style":32527},[31680],[34,489998,32936],{"className":489999},[31685,31686],[2230,490001,490002,490162,490771],{},[108,490003,490004,352814,490007,490161],{},[2627,490005,490006],{},"双射性",[34,490008,490010,490056],{"className":490009},[31642],[34,490011,490013],{"className":490012},[31646],[31648,490014,490015],{"xmlns":31650},[31652,490016,490017,490053],{},[31655,490018,490019,490021,490023,490025,490027,490029,490031,490033,490035,490037,490039,490041,490043,490045,490047,490049,490051],{},[31658,490020,31739],{},[31662,490022,456],{"stretchy":702},[31658,490024,31739],{},[31662,490026,456],{"stretchy":702},[31658,490028,32936],{},[31662,490030,2557],{"stretchy":702},[31662,490032,2557],{"stretchy":702},[31662,490034,202],{},[31749,490036,640],{},[31662,490038,32226],{},[31662,490040,456],{"stretchy":702},[31749,490042,640],{},[31662,490044,32226],{},[31658,490046,32936],{},[31662,490048,2557],{"stretchy":702},[31662,490050,202],{},[31658,490052,32936],{},[31665,490054,490055],{"encoding":31667},"f(f(x)) = 1-(1-x) = x",[34,490057,490059,490092,490110,490131,490152],{"className":490058,"ariaHidden":758},[31672],[34,490060,490062,490065,490068,490071,490074,490077,490080,490083,490086,490089],{"className":490061},[31676],[34,490063],{"className":490064,"style":31774},[31680],[34,490066,31739],{"className":490067,"style":31778},[31685,31686],[34,490069,456],{"className":490070},[31782],[34,490072,31739],{"className":490073,"style":31778},[31685,31686],[34,490075,456],{"className":490076},[31782],[34,490078,32936],{"className":490079},[31685,31686],[34,490081,44253],{"className":490082},[31690],[34,490084],{"className":490085,"style":31846},[31845],[34,490087,202],{"className":490088},[31850],[34,490090],{"className":490091,"style":31846},[31845],[34,490093,490095,490098,490101,490104,490107],{"className":490094},[31676],[34,490096],{"className":490097,"style":35942},[31680],[34,490099,640],{"className":490100},[31685],[34,490102],{"className":490103,"style":32604},[31845],[34,490105,32226],{"className":490106},[32024],[34,490108],{"className":490109,"style":32604},[31845],[34,490111,490113,490116,490119,490122,490125,490128],{"className":490112},[31676],[34,490114],{"className":490115,"style":31774},[31680],[34,490117,456],{"className":490118},[31782],[34,490120,640],{"className":490121},[31685],[34,490123],{"className":490124,"style":32604},[31845],[34,490126,32226],{"className":490127},[32024],[34,490129],{"className":490130,"style":32604},[31845],[34,490132,490134,490137,490140,490143,490146,490149],{"className":490133},[31676],[34,490135],{"className":490136,"style":31774},[31680],[34,490138,32936],{"className":490139},[31685,31686],[34,490141,2557],{"className":490142},[31690],[34,490144],{"className":490145,"style":31846},[31845],[34,490147,202],{"className":490148},[31850],[34,490150],{"className":490151,"style":31846},[31845],[34,490153,490155,490158],{"className":490154},[31676],[34,490156],{"className":490157,"style":32527},[31680],[34,490159,32936],{"className":490160},[31685,31686],"，由逆映射存在可知它是双射。",[108,490163,490164,490166,490167,490307,490308,490574,490575,490770],{},[2627,490165,446923],{},"：\n我们需要证明 ",[34,490168,490170,490214],{"className":490169},[31642],[34,490171,490173],{"className":490172},[31646],[31648,490174,490175],{"xmlns":31650},[31652,490176,490177,490211],{},[31655,490178,490179,490181,490183,490185,490187,490189,490191,490193,490195,490197,490199,490201,490203,490205,490207,490209],{},[31658,490180,31739],{},[31662,490182,456],{"stretchy":702},[31658,490184,111],{},[31662,490186,378006],{},[31658,490188,1697],{},[31662,490190,2557],{"stretchy":702},[31662,490192,202],{},[31658,490194,31739],{},[31662,490196,456],{"stretchy":702},[31658,490198,111],{},[31662,490200,2557],{"stretchy":702},[31662,490202,223],{},[31658,490204,31739],{},[31662,490206,456],{"stretchy":702},[31658,490208,1697],{},[31662,490210,2557],{"stretchy":702},[31665,490212,490213],{"encoding":31667},"f(a \\oplus b) = f(a) + f(b)",[34,490215,490217,490241,490262,490289],{"className":490216,"ariaHidden":758},[31672],[34,490218,490220,490223,490226,490229,490232,490235,490238],{"className":490219},[31676],[34,490221],{"className":490222,"style":31774},[31680],[34,490224,31739],{"className":490225,"style":31778},[31685,31686],[34,490227,456],{"className":490228},[31782],[34,490230,111],{"className":490231},[31685,31686],[34,490233],{"className":490234,"style":32604},[31845],[34,490236,378006],{"className":490237},[32024],[34,490239],{"className":490240,"style":32604},[31845],[34,490242,490244,490247,490250,490253,490256,490259],{"className":490243},[31676],[34,490245],{"className":490246,"style":31774},[31680],[34,490248,1697],{"className":490249},[31685,31686],[34,490251,2557],{"className":490252},[31690],[34,490254],{"className":490255,"style":31846},[31845],[34,490257,202],{"className":490258},[31850],[34,490260],{"className":490261,"style":31846},[31845],[34,490263,490265,490268,490271,490274,490277,490280,490283,490286],{"className":490264},[31676],[34,490266],{"className":490267,"style":31774},[31680],[34,490269,31739],{"className":490270,"style":31778},[31685,31686],[34,490272,456],{"className":490273},[31782],[34,490275,111],{"className":490276},[31685,31686],[34,490278,2557],{"className":490279},[31690],[34,490281],{"className":490282,"style":32604},[31845],[34,490284,223],{"className":490285},[32024],[34,490287],{"className":490288,"style":32604},[31845],[34,490290,490292,490295,490298,490301,490304],{"className":490291},[31676],[34,490293],{"className":490294,"style":31774},[31680],[34,490296,31739],{"className":490297,"style":31778},[31685,31686],[34,490299,456],{"className":490300},[31782],[34,490302,1697],{"className":490303},[31685,31686],[34,490305,2557],{"className":490306},[31690],"。\n左边 ",[34,490309,490311,490373],{"className":490310},[31642],[34,490312,490314],{"className":490313},[31646],[31648,490315,490316],{"xmlns":31650},[31652,490317,490318,490370],{},[31655,490319,490320,490322,490324,490326,490328,490330,490332,490334,490336,490338,490340,490342,490344,490346,490348,490350,490352,490354,490356,490358,490360,490362,490364,490366,490368],{},[31662,490321,202],{},[31658,490323,31739],{},[31662,490325,456],{"stretchy":702},[31658,490327,111],{},[31662,490329,223],{},[31658,490331,1697],{},[31662,490333,32226],{},[31749,490335,640],{},[31662,490337,2557],{"stretchy":702},[31662,490339,202],{},[31749,490341,640],{},[31662,490343,32226],{},[31662,490345,456],{"stretchy":702},[31658,490347,111],{},[31662,490349,223],{},[31658,490351,1697],{},[31662,490353,32226],{},[31749,490355,640],{},[31662,490357,2557],{"stretchy":702},[31662,490359,202],{},[31749,490361,768],{},[31662,490363,32226],{},[31658,490365,111],{},[31662,490367,32226],{},[31658,490369,1697],{},[31665,490371,490372],{"encoding":31667},"= f(a + b - 1) = 1 - (a + b - 1) = 2 - a - b",[34,490374,490376,490388,490412,490430,490451,490469,490490,490508,490529,490547,490565],{"className":490375,"ariaHidden":758},[31672],[34,490377,490379,490382,490385],{"className":490378},[31676],[34,490380],{"className":490381,"style":73089},[31680],[34,490383,202],{"className":490384},[31850],[34,490386],{"className":490387,"style":31846},[31845],[34,490389,490391,490394,490397,490400,490403,490406,490409],{"className":490390},[31676],[34,490392],{"className":490393,"style":31774},[31680],[34,490395,31739],{"className":490396,"style":31778},[31685,31686],[34,490398,456],{"className":490399},[31782],[34,490401,111],{"className":490402},[31685,31686],[34,490404],{"className":490405,"style":32604},[31845],[34,490407,223],{"className":490408},[32024],[34,490410],{"className":490411,"style":32604},[31845],[34,490413,490415,490418,490421,490424,490427],{"className":490414},[31676],[34,490416],{"className":490417,"style":34299},[31680],[34,490419,1697],{"className":490420},[31685,31686],[34,490422],{"className":490423,"style":32604},[31845],[34,490425,32226],{"className":490426},[32024],[34,490428],{"className":490429,"style":32604},[31845],[34,490431,490433,490436,490439,490442,490445,490448],{"className":490432},[31676],[34,490434],{"className":490435,"style":31774},[31680],[34,490437,640],{"className":490438},[31685],[34,490440,2557],{"className":490441},[31690],[34,490443],{"className":490444,"style":31846},[31845],[34,490446,202],{"className":490447},[31850],[34,490449],{"className":490450,"style":31846},[31845],[34,490452,490454,490457,490460,490463,490466],{"className":490453},[31676],[34,490455],{"className":490456,"style":35942},[31680],[34,490458,640],{"className":490459},[31685],[34,490461],{"className":490462,"style":32604},[31845],[34,490464,32226],{"className":490465},[32024],[34,490467],{"className":490468,"style":32604},[31845],[34,490470,490472,490475,490478,490481,490484,490487],{"className":490471},[31676],[34,490473],{"className":490474,"style":31774},[31680],[34,490476,456],{"className":490477},[31782],[34,490479,111],{"className":490480},[31685,31686],[34,490482],{"className":490483,"style":32604},[31845],[34,490485,223],{"className":490486},[32024],[34,490488],{"className":490489,"style":32604},[31845],[34,490491,490493,490496,490499,490502,490505],{"className":490492},[31676],[34,490494],{"className":490495,"style":34299},[31680],[34,490497,1697],{"className":490498},[31685,31686],[34,490500],{"className":490501,"style":32604},[31845],[34,490503,32226],{"className":490504},[32024],[34,490506],{"className":490507,"style":32604},[31845],[34,490509,490511,490514,490517,490520,490523,490526],{"className":490510},[31676],[34,490512],{"className":490513,"style":31774},[31680],[34,490515,640],{"className":490516},[31685],[34,490518,2557],{"className":490519},[31690],[34,490521],{"className":490522,"style":31846},[31845],[34,490524,202],{"className":490525},[31850],[34,490527],{"className":490528,"style":31846},[31845],[34,490530,490532,490535,490538,490541,490544],{"className":490531},[31676],[34,490533],{"className":490534,"style":35942},[31680],[34,490536,768],{"className":490537},[31685],[34,490539],{"className":490540,"style":32604},[31845],[34,490542,32226],{"className":490543},[32024],[34,490545],{"className":490546,"style":32604},[31845],[34,490548,490550,490553,490556,490559,490562],{"className":490549},[31676],[34,490551],{"className":490552,"style":32597},[31680],[34,490554,111],{"className":490555},[31685,31686],[34,490557],{"className":490558,"style":32604},[31845],[34,490560,32226],{"className":490561},[32024],[34,490563],{"className":490564,"style":32604},[31845],[34,490566,490568,490571],{"className":490567},[31676],[34,490569],{"className":490570,"style":31681},[31680],[34,490572,1697],{"className":490573},[31685,31686],"。\n右边 ",[34,490576,490578,490626],{"className":490577},[31642],[34,490579,490581],{"className":490580},[31646],[31648,490582,490583],{"xmlns":31650},[31652,490584,490585,490623],{},[31655,490586,490587,490589,490591,490593,490595,490597,490599,490601,490603,490605,490607,490609,490611,490613,490615,490617,490619,490621],{},[31662,490588,202],{},[31662,490590,456],{"stretchy":702},[31749,490592,640],{},[31662,490594,32226],{},[31658,490596,111],{},[31662,490598,2557],{"stretchy":702},[31662,490600,223],{},[31662,490602,456],{"stretchy":702},[31749,490604,640],{},[31662,490606,32226],{},[31658,490608,1697],{},[31662,490610,2557],{"stretchy":702},[31662,490612,202],{},[31749,490614,768],{},[31662,490616,32226],{},[31658,490618,111],{},[31662,490620,32226],{},[31658,490622,1697],{},[31665,490624,490625],{"encoding":31667},"= (1 - a) + (1 - b) = 2 - a - b",[34,490627,490629,490641,490662,490683,490704,490725,490743,490761],{"className":490628,"ariaHidden":758},[31672],[34,490630,490632,490635,490638],{"className":490631},[31676],[34,490633],{"className":490634,"style":73089},[31680],[34,490636,202],{"className":490637},[31850],[34,490639],{"className":490640,"style":31846},[31845],[34,490642,490644,490647,490650,490653,490656,490659],{"className":490643},[31676],[34,490645],{"className":490646,"style":31774},[31680],[34,490648,456],{"className":490649},[31782],[34,490651,640],{"className":490652},[31685],[34,490654],{"className":490655,"style":32604},[31845],[34,490657,32226],{"className":490658},[32024],[34,490660],{"className":490661,"style":32604},[31845],[34,490663,490665,490668,490671,490674,490677,490680],{"className":490664},[31676],[34,490666],{"className":490667,"style":31774},[31680],[34,490669,111],{"className":490670},[31685,31686],[34,490672,2557],{"className":490673},[31690],[34,490675],{"className":490676,"style":32604},[31845],[34,490678,223],{"className":490679},[32024],[34,490681],{"className":490682,"style":32604},[31845],[34,490684,490686,490689,490692,490695,490698,490701],{"className":490685},[31676],[34,490687],{"className":490688,"style":31774},[31680],[34,490690,456],{"className":490691},[31782],[34,490693,640],{"className":490694},[31685],[34,490696],{"className":490697,"style":32604},[31845],[34,490699,32226],{"className":490700},[32024],[34,490702],{"className":490703,"style":32604},[31845],[34,490705,490707,490710,490713,490716,490719,490722],{"className":490706},[31676],[34,490708],{"className":490709,"style":31774},[31680],[34,490711,1697],{"className":490712},[31685,31686],[34,490714,2557],{"className":490715},[31690],[34,490717],{"className":490718,"style":31846},[31845],[34,490720,202],{"className":490721},[31850],[34,490723],{"className":490724,"style":31846},[31845],[34,490726,490728,490731,490734,490737,490740],{"className":490727},[31676],[34,490729],{"className":490730,"style":35942},[31680],[34,490732,768],{"className":490733},[31685],[34,490735],{"className":490736,"style":32604},[31845],[34,490738,32226],{"className":490739},[32024],[34,490741],{"className":490742,"style":32604},[31845],[34,490744,490746,490749,490752,490755,490758],{"className":490745},[31676],[34,490747],{"className":490748,"style":32597},[31680],[34,490750,111],{"className":490751},[31685,31686],[34,490753],{"className":490754,"style":32604},[31845],[34,490756,32226],{"className":490757},[32024],[34,490759],{"className":490760,"style":32604},[31845],[34,490762,490764,490767],{"className":490763},[31676],[34,490765],{"className":490766,"style":31681},[31680],[34,490768,1697],{"className":490769},[31685,31686],"。\n相等！",[108,490772,490773,490166,490775,490307,490915,490574,491119,490770],{},[2627,490774,447325],{},[34,490776,490778,490822],{"className":490777},[31642],[34,490779,490781],{"className":490780},[31646],[31648,490782,490783],{"xmlns":31650},[31652,490784,490785,490819],{},[31655,490786,490787,490789,490791,490793,490795,490797,490799,490801,490803,490805,490807,490809,490811,490813,490815,490817],{},[31658,490788,31739],{},[31662,490790,456],{"stretchy":702},[31658,490792,111],{},[31662,490794,378082],{},[31658,490796,1697],{},[31662,490798,2557],{"stretchy":702},[31662,490800,202],{},[31658,490802,31739],{},[31662,490804,456],{"stretchy":702},[31658,490806,111],{},[31662,490808,2557],{"stretchy":702},[31662,490810,71191],{},[31658,490812,31739],{},[31662,490814,456],{"stretchy":702},[31658,490816,1697],{},[31662,490818,2557],{"stretchy":702},[31665,490820,490821],{"encoding":31667},"f(a \\odot b) = f(a) \\cdot f(b)",[34,490823,490825,490849,490870,490897],{"className":490824,"ariaHidden":758},[31672],[34,490826,490828,490831,490834,490837,490840,490843,490846],{"className":490827},[31676],[34,490829],{"className":490830,"style":31774},[31680],[34,490832,31739],{"className":490833,"style":31778},[31685,31686],[34,490835,456],{"className":490836},[31782],[34,490838,111],{"className":490839},[31685,31686],[34,490841],{"className":490842,"style":32604},[31845],[34,490844,378082],{"className":490845},[32024],[34,490847],{"className":490848,"style":32604},[31845],[34,490850,490852,490855,490858,490861,490864,490867],{"className":490851},[31676],[34,490853],{"className":490854,"style":31774},[31680],[34,490856,1697],{"className":490857},[31685,31686],[34,490859,2557],{"className":490860},[31690],[34,490862],{"className":490863,"style":31846},[31845],[34,490865,202],{"className":490866},[31850],[34,490868],{"className":490869,"style":31846},[31845],[34,490871,490873,490876,490879,490882,490885,490888,490891,490894],{"className":490872},[31676],[34,490874],{"className":490875,"style":31774},[31680],[34,490877,31739],{"className":490878,"style":31778},[31685,31686],[34,490880,456],{"className":490881},[31782],[34,490883,111],{"className":490884},[31685,31686],[34,490886,2557],{"className":490887},[31690],[34,490889],{"className":490890,"style":32604},[31845],[34,490892,71191],{"className":490893},[32024],[34,490895],{"className":490896,"style":32604},[31845],[34,490898,490900,490903,490906,490909,490912],{"className":490899},[31676],[34,490901],{"className":490902,"style":31774},[31680],[34,490904,31739],{"className":490905,"style":31778},[31685,31686],[34,490907,456],{"className":490908},[31782],[34,490910,1697],{"className":490911},[31685,31686],[34,490913,2557],{"className":490914},[31690],[34,490916,490918,490972],{"className":490917},[31642],[34,490919,490921],{"className":490920},[31646],[31648,490922,490923],{"xmlns":31650},[31652,490924,490925,490969],{},[31655,490926,490927,490929,490931,490933,490935,490937,490939,490941,490943,490945,490947,490949,490951,490953,490955,490957,490959,490961,490963,490965,490967],{},[31662,490928,202],{},[31658,490930,31739],{},[31662,490932,456],{"stretchy":702},[31658,490934,111],{},[31662,490936,223],{},[31658,490938,1697],{},[31662,490940,32226],{},[31658,490942,111],{},[31658,490944,1697],{},[31662,490946,2557],{"stretchy":702},[31662,490948,202],{},[31749,490950,640],{},[31662,490952,32226],{},[31662,490954,456],{"stretchy":702},[31658,490956,111],{},[31662,490958,223],{},[31658,490960,1697],{},[31662,490962,32226],{},[31658,490964,111],{},[31658,490966,1697],{},[31662,490968,2557],{"stretchy":702},[31665,490970,490971],{"encoding":31667},"= f(a + b - ab) = 1 - (a + b - ab)",[34,490973,490975,490987,491011,491029,491050,491068,491089,491107],{"className":490974,"ariaHidden":758},[31672],[34,490976,490978,490981,490984],{"className":490977},[31676],[34,490979],{"className":490980,"style":73089},[31680],[34,490982,202],{"className":490983},[31850],[34,490985],{"className":490986,"style":31846},[31845],[34,490988,490990,490993,490996,490999,491002,491005,491008],{"className":490989},[31676],[34,490991],{"className":490992,"style":31774},[31680],[34,490994,31739],{"className":490995,"style":31778},[31685,31686],[34,490997,456],{"className":490998},[31782],[34,491000,111],{"className":491001},[31685,31686],[34,491003],{"className":491004,"style":32604},[31845],[34,491006,223],{"className":491007},[32024],[34,491009],{"className":491010,"style":32604},[31845],[34,491012,491014,491017,491020,491023,491026],{"className":491013},[31676],[34,491015],{"className":491016,"style":34299},[31680],[34,491018,1697],{"className":491019},[31685,31686],[34,491021],{"className":491022,"style":32604},[31845],[34,491024,32226],{"className":491025},[32024],[34,491027],{"className":491028,"style":32604},[31845],[34,491030,491032,491035,491038,491041,491044,491047],{"className":491031},[31676],[34,491033],{"className":491034,"style":31774},[31680],[34,491036,324191],{"className":491037},[31685,31686],[34,491039,2557],{"className":491040},[31690],[34,491042],{"className":491043,"style":31846},[31845],[34,491045,202],{"className":491046},[31850],[34,491048],{"className":491049,"style":31846},[31845],[34,491051,491053,491056,491059,491062,491065],{"className":491052},[31676],[34,491054],{"className":491055,"style":35942},[31680],[34,491057,640],{"className":491058},[31685],[34,491060],{"className":491061,"style":32604},[31845],[34,491063,32226],{"className":491064},[32024],[34,491066],{"className":491067,"style":32604},[31845],[34,491069,491071,491074,491077,491080,491083,491086],{"className":491070},[31676],[34,491072],{"className":491073,"style":31774},[31680],[34,491075,456],{"className":491076},[31782],[34,491078,111],{"className":491079},[31685,31686],[34,491081],{"className":491082,"style":32604},[31845],[34,491084,223],{"className":491085},[32024],[34,491087],{"className":491088,"style":32604},[31845],[34,491090,491092,491095,491098,491101,491104],{"className":491091},[31676],[34,491093],{"className":491094,"style":34299},[31680],[34,491096,1697],{"className":491097},[31685,31686],[34,491099],{"className":491100,"style":32604},[31845],[34,491102,32226],{"className":491103},[32024],[34,491105],{"className":491106,"style":32604},[31845],[34,491108,491110,491113,491116],{"className":491109},[31676],[34,491111],{"className":491112,"style":31774},[31680],[34,491114,324191],{"className":491115},[31685,31686],[34,491117,2557],{"className":491118},[31690],[34,491120,491122,491196],{"className":491121},[31642],[34,491123,491125],{"className":491124},[31646],[31648,491126,491127],{"xmlns":31650},[31652,491128,491129,491193],{},[31655,491130,491131,491133,491135,491137,491139,491141,491143,491145,491147,491149,491151,491153,491155,491157,491159,491161,491163,491165,491167,491169,491171,491173,491175,491177,491179,491181,491183,491185,491187,491189,491191],{},[31662,491132,202],{},[31662,491134,456],{"stretchy":702},[31749,491136,640],{},[31662,491138,32226],{},[31658,491140,111],{},[31662,491142,2557],{"stretchy":702},[31662,491144,456],{"stretchy":702},[31749,491146,640],{},[31662,491148,32226],{},[31658,491150,1697],{},[31662,491152,2557],{"stretchy":702},[31662,491154,202],{},[31749,491156,640],{},[31662,491158,32226],{},[31658,491160,1697],{},[31662,491162,32226],{},[31658,491164,111],{},[31662,491166,223],{},[31658,491168,111],{},[31658,491170,1697],{},[31662,491172,202],{},[31749,491174,640],{},[31662,491176,32226],{},[31662,491178,456],{"stretchy":702},[31658,491180,111],{},[31662,491182,223],{},[31658,491184,1697],{},[31662,491186,32226],{},[31658,491188,111],{},[31658,491190,1697],{},[31662,491192,2557],{"stretchy":702},[31665,491194,491195],{"encoding":31667},"= (1 - a)(1 - b) = 1 - b - a + ab = 1 - (a + b - ab)",[34,491197,491199,491211,491232,491259,491280,491298,491316,491334,491352,491370,491391,491409],{"className":491198,"ariaHidden":758},[31672],[34,491200,491202,491205,491208],{"className":491201},[31676],[34,491203],{"className":491204,"style":73089},[31680],[34,491206,202],{"className":491207},[31850],[34,491209],{"className":491210,"style":31846},[31845],[34,491212,491214,491217,491220,491223,491226,491229],{"className":491213},[31676],[34,491215],{"className":491216,"style":31774},[31680],[34,491218,456],{"className":491219},[31782],[34,491221,640],{"className":491222},[31685],[34,491224],{"className":491225,"style":32604},[31845],[34,491227,32226],{"className":491228},[32024],[34,491230],{"className":491231,"style":32604},[31845],[34,491233,491235,491238,491241,491244,491247,491250,491253,491256],{"className":491234},[31676],[34,491236],{"className":491237,"style":31774},[31680],[34,491239,111],{"className":491240},[31685,31686],[34,491242,2557],{"className":491243},[31690],[34,491245,456],{"className":491246},[31782],[34,491248,640],{"className":491249},[31685],[34,491251],{"className":491252,"style":32604},[31845],[34,491254,32226],{"className":491255},[32024],[34,491257],{"className":491258,"style":32604},[31845],[34,491260,491262,491265,491268,491271,491274,491277],{"className":491261},[31676],[34,491263],{"className":491264,"style":31774},[31680],[34,491266,1697],{"className":491267},[31685,31686],[34,491269,2557],{"className":491270},[31690],[34,491272],{"className":491273,"style":31846},[31845],[34,491275,202],{"className":491276},[31850],[34,491278],{"className":491279,"style":31846},[31845],[34,491281,491283,491286,491289,491292,491295],{"className":491282},[31676],[34,491284],{"className":491285,"style":35942},[31680],[34,491287,640],{"className":491288},[31685],[34,491290],{"className":491291,"style":32604},[31845],[34,491293,32226],{"className":491294},[32024],[34,491296],{"className":491297,"style":32604},[31845],[34,491299,491301,491304,491307,491310,491313],{"className":491300},[31676],[34,491302],{"className":491303,"style":34299},[31680],[34,491305,1697],{"className":491306},[31685,31686],[34,491308],{"className":491309,"style":32604},[31845],[34,491311,32226],{"className":491312},[32024],[34,491314],{"className":491315,"style":32604},[31845],[34,491317,491319,491322,491325,491328,491331],{"className":491318},[31676],[34,491320],{"className":491321,"style":32597},[31680],[34,491323,111],{"className":491324},[31685,31686],[34,491326],{"className":491327,"style":32604},[31845],[34,491329,223],{"className":491330},[32024],[34,491332],{"className":491333,"style":32604},[31845],[34,491335,491337,491340,491343,491346,491349],{"className":491336},[31676],[34,491338],{"className":491339,"style":31681},[31680],[34,491341,324191],{"className":491342},[31685,31686],[34,491344],{"className":491345,"style":31846},[31845],[34,491347,202],{"className":491348},[31850],[34,491350],{"className":491351,"style":31846},[31845],[34,491353,491355,491358,491361,491364,491367],{"className":491354},[31676],[34,491356],{"className":491357,"style":35942},[31680],[34,491359,640],{"className":491360},[31685],[34,491362],{"className":491363,"style":32604},[31845],[34,491365,32226],{"className":491366},[32024],[34,491368],{"className":491369,"style":32604},[31845],[34,491371,491373,491376,491379,491382,491385,491388],{"className":491372},[31676],[34,491374],{"className":491375,"style":31774},[31680],[34,491377,456],{"className":491378},[31782],[34,491380,111],{"className":491381},[31685,31686],[34,491383],{"className":491384,"style":32604},[31845],[34,491386,223],{"className":491387},[32024],[34,491389],{"className":491390,"style":32604},[31845],[34,491392,491394,491397,491400,491403,491406],{"className":491393},[31676],[34,491395],{"className":491396,"style":34299},[31680],[34,491398,1697],{"className":491399},[31685,31686],[34,491401],{"className":491402,"style":32604},[31845],[34,491404,32226],{"className":491405},[32024],[34,491407],{"className":491408,"style":32604},[31845],[34,491410,491412,491415,491418],{"className":491411},[31676],[34,491413],{"className":491414,"style":31774},[31680],[34,491416,324191],{"className":491417},[31685,31686],[34,491419,2557],{"className":491420},[31690],[15,491422,491423,2664,491425,491513,491514],{},[2627,491424,114043],{},[34,491426,491428,491456],{"className":491427},[31642],[34,491429,491431],{"className":491430},[31646],[31648,491432,491433],{"xmlns":31650},[31652,491434,491435,491453],{},[31655,491436,491437,491439,491441,491443,491445,491447,491449,491451],{},[31658,491438,31739],{},[31662,491440,456],{"stretchy":702},[31658,491442,32936],{},[31662,491444,2557],{"stretchy":702},[31662,491446,202],{},[31749,491448,640],{},[31662,491450,32226],{},[31658,491452,32936],{},[31665,491454,491455],{"encoding":31667},"f(x) = 1-x",[34,491457,491459,491486,491504],{"className":491458,"ariaHidden":758},[31672],[34,491460,491462,491465,491468,491471,491474,491477,491480,491483],{"className":491461},[31676],[34,491463],{"className":491464,"style":31774},[31680],[34,491466,31739],{"className":491467,"style":31778},[31685,31686],[34,491469,456],{"className":491470},[31782],[34,491472,32936],{"className":491473},[31685,31686],[34,491475,2557],{"className":491476},[31690],[34,491478],{"className":491479,"style":31846},[31845],[34,491481,202],{"className":491482},[31850],[34,491484],{"className":491485,"style":31846},[31845],[34,491487,491489,491492,491495,491498,491501],{"className":491488},[31676],[34,491490],{"className":491491,"style":35942},[31680],[34,491493,640],{"className":491494},[31685],[34,491496],{"className":491497,"style":32604},[31845],[34,491499,32226],{"className":491500},[32024],[34,491502],{"className":491503,"style":32604},[31845],[34,491505,491507,491510],{"className":491506},[31676],[34,491508],{"className":491509,"style":32527},[31680],[34,491511,32936],{"className":491512},[31685,31686]," 是一个环同构，所以两个环结构同构。 ",[34,491515,491517,491530],{"className":491516},[31642],[34,491518,491520],{"className":491519},[31646],[31648,491521,491522],{"xmlns":31650},[31652,491523,491524,491528],{},[31655,491525,491526],{},[31658,491527,445204],{"mathvariant":38079},[31665,491529,445207],{"encoding":31667},[34,491531,491533],{"className":491532,"ariaHidden":758},[31672],[34,491534,491536,491539],{"className":491535},[31676],[34,491537],{"className":491538,"style":445217},[31680],[34,491540,445204],{"className":491541},[31685,129261],[32188,491543],{},[253,491545,491547],{"id":491546},"第三部分课本-p89-习题","第三部分：课本 P89 习题",[32490,491549,491551],{"id":491550},"习题-3","习题 3",[15,491553,491554],{},[2627,491555,133910,491556,86674,491612,43298,491640,491668,491669,2547],{},[34,491557,491559,491579],{"className":491558},[31642],[34,491560,491562],{"className":491561},[31646],[31648,491563,491564],{"xmlns":31650},[31652,491565,491566,491576],{},[31655,491567,491568,491570,491572,491574],{},[31658,491569,106063],{},[31658,491571,223320],{},[31662,491573,131855],{},[31658,491575,38691],{},[31665,491577,491578],{"encoding":31667},"IJ \\subseteq K",[34,491580,491582,491603],{"className":491581,"ariaHidden":758},[31672],[34,491583,491585,491588,491591,491594,491597,491600],{"className":491584},[31676],[34,491586],{"className":491587,"style":195047},[31680],[34,491589,106063],{"className":491590,"style":67482},[31685,31686],[34,491592,223320],{"className":491593,"style":223361},[31685,31686],[34,491595],{"className":491596,"style":31846},[31845],[34,491598,131855],{"className":491599},[31850],[34,491601],{"className":491602,"style":31846},[31845],[34,491604,491606,491609],{"className":491605},[31676],[34,491607],{"className":491608,"style":33218},[31680],[34,491610,38691],{"className":491611,"style":38706},[31685,31686],[34,491613,491615,491628],{"className":491614},[31642],[34,491616,491618],{"className":491617},[31646],[31648,491619,491620],{"xmlns":31650},[31652,491621,491622,491626],{},[31655,491623,491624],{},[31658,491625,106063],{},[31665,491627,106063],{"encoding":31667},[34,491629,491631],{"className":491630,"ariaHidden":758},[31672],[34,491632,491634,491637],{"className":491633},[31676],[34,491635],{"className":491636,"style":33218},[31680],[34,491638,106063],{"className":491639,"style":67482},[31685,31686],[34,491641,491643,491656],{"className":491642},[31642],[34,491644,491646],{"className":491645},[31646],[31648,491647,491648],{"xmlns":31650},[31652,491649,491650,491654],{},[31655,491651,491652],{},[31658,491653,38691],{},[31665,491655,38691],{"encoding":31667},[34,491657,491659],{"className":491658,"ariaHidden":758},[31672],[34,491660,491662,491665],{"className":491661},[31676],[34,491663],{"className":491664,"style":33218},[31680],[34,491666,38691],{"className":491667,"style":38706},[31685,31686]," 互素, 证明 ",[34,491670,491672,491690],{"className":491671},[31642],[34,491673,491675],{"className":491674},[31646],[31648,491676,491677],{"xmlns":31650},[31652,491678,491679,491687],{},[31655,491680,491681,491683,491685],{},[31658,491682,223320],{},[31662,491684,131855],{},[31658,491686,38691],{},[31665,491688,491689],{"encoding":31667},"J \\subseteq K",[34,491691,491693,491711],{"className":491692,"ariaHidden":758},[31672],[34,491694,491696,491699,491702,491705,491708],{"className":491695},[31676],[34,491697],{"className":491698,"style":195047},[31680],[34,491700,223320],{"className":491701,"style":223361},[31685,31686],[34,491703],{"className":491704,"style":31846},[31845],[34,491706,131855],{"className":491707},[31850],[34,491709],{"className":491710,"style":31846},[31845],[34,491712,491714,491717],{"className":491713},[31676],[34,491715],{"className":491716,"style":33218},[31680],[34,491718,38691],{"className":491719,"style":38706},[31685,31686],[15,491721,491722],{},[2627,491723,463462],{},[105,491725,491726,491889,491996],{},[108,491727,491728,2664,491731,491804,491805,491860,491861,328728],{},[2627,491729,491730],{},"条件",[34,491732,491734,491756],{"className":491733},[31642],[34,491735,491737],{"className":491736},[31646],[31648,491738,491739],{"xmlns":31650},[31652,491740,491741,491753],{},[31655,491742,491743,491745,491747,491749,491751],{},[31658,491744,106063],{},[31662,491746,223],{},[31658,491748,38691],{},[31662,491750,202],{},[31658,491752,34386],{},[31665,491754,491755],{"encoding":31667},"I+K=R",[34,491757,491759,491777,491795],{"className":491758,"ariaHidden":758},[31672],[34,491760,491762,491765,491768,491771,491774],{"className":491761},[31676],[34,491763],{"className":491764,"style":34944},[31680],[34,491766,106063],{"className":491767,"style":67482},[31685,31686],[34,491769],{"className":491770,"style":32604},[31845],[34,491772,223],{"className":491773},[32024],[34,491775],{"className":491776,"style":32604},[31845],[34,491778,491780,491783,491786,491789,491792],{"className":491779},[31676],[34,491781],{"className":491782,"style":33218},[31680],[34,491784,38691],{"className":491785,"style":38706},[31685,31686],[34,491787],{"className":491788,"style":31846},[31845],[34,491790,202],{"className":491791},[31850],[34,491793],{"className":491794,"style":31846},[31845],[34,491796,491798,491801],{"className":491797},[31676],[34,491799],{"className":491800,"style":33218},[31680],[34,491802,34386],{"className":491803,"style":34427},[31685,31686],"（互素），",[34,491806,491808,491827],{"className":491807},[31642],[34,491809,491811],{"className":491810},[31646],[31648,491812,491813],{"xmlns":31650},[31652,491814,491815,491825],{},[31655,491816,491817,491819,491821,491823],{},[31658,491818,106063],{},[31658,491820,223320],{},[31662,491822,131855],{},[31658,491824,38691],{},[31665,491826,491578],{"encoding":31667},[34,491828,491830,491851],{"className":491829,"ariaHidden":758},[31672],[34,491831,491833,491836,491839,491842,491845,491848],{"className":491832},[31676],[34,491834],{"className":491835,"style":195047},[31680],[34,491837,106063],{"className":491838,"style":67482},[31685,31686],[34,491840,223320],{"className":491841,"style":223361},[31685,31686],[34,491843],{"className":491844,"style":31846},[31845],[34,491846,131855],{"className":491847},[31850],[34,491849],{"className":491850,"style":31846},[31845],[34,491852,491854,491857],{"className":491853},[31676],[34,491855],{"className":491856,"style":33218},[31680],[34,491858,38691],{"className":491859,"style":38706},[31685,31686],"（积在 ",[34,491862,491864,491877],{"className":491863},[31642],[34,491865,491867],{"className":491866},[31646],[31648,491868,491869],{"xmlns":31650},[31652,491870,491871,491875],{},[31655,491872,491873],{},[31658,491874,38691],{},[31665,491876,38691],{"encoding":31667},[34,491878,491880],{"className":491879,"ariaHidden":758},[31672],[34,491881,491883,491886],{"className":491882},[31676],[34,491884],{"className":491885,"style":33218},[31680],[34,491887,38691],{"className":491888,"style":38706},[31685,31686],[108,491890,491891,491893,491894,329052,491945,347],{},[2627,491892,233757],{},"：任取 ",[34,491895,491897,491915],{"className":491896},[31642],[34,491898,491900],{"className":491899},[31646],[31648,491901,491902],{"xmlns":31650},[31652,491903,491904,491912],{},[31655,491905,491906,491908,491910],{},[31658,491907,31968],{},[31662,491909,112296],{},[31658,491911,223320],{},[31665,491913,491914],{"encoding":31667},"j \\in J",[34,491916,491918,491936],{"className":491917,"ariaHidden":758},[31672],[34,491919,491921,491924,491927,491930,491933],{"className":491920},[31676],[34,491922],{"className":491923,"style":33123},[31680],[34,491925,31968],{"className":491926,"style":32153},[31685,31686],[34,491928],{"className":491929,"style":31846},[31845],[34,491931,112296],{"className":491932},[31850],[34,491934],{"className":491935,"style":31846},[31845],[34,491937,491939,491942],{"className":491938},[31676],[34,491940],{"className":491941,"style":33218},[31680],[34,491943,223320],{"className":491944,"style":223361},[31685,31686],[34,491946,491948,491966],{"className":491947},[31642],[34,491949,491951],{"className":491950},[31646],[31648,491952,491953],{"xmlns":31650},[31652,491954,491955,491963],{},[31655,491956,491957,491959,491961],{},[31658,491958,31968],{},[31662,491960,112296],{},[31658,491962,38691],{},[31665,491964,491965],{"encoding":31667},"j \\in K",[34,491967,491969,491987],{"className":491968,"ariaHidden":758},[31672],[34,491970,491972,491975,491978,491981,491984],{"className":491971},[31676],[34,491973],{"className":491974,"style":33123},[31680],[34,491976,31968],{"className":491977,"style":32153},[31685,31686],[34,491979],{"className":491980,"style":31846},[31845],[34,491982,112296],{"className":491983},[31850],[34,491985],{"className":491986,"style":31846},[31845],[34,491988,491990,491993],{"className":491989},[31676],[34,491991],{"className":491992,"style":33218},[31680],[34,491994,38691],{"className":491995,"style":38706},[31685,31686],[108,491997,491998,492000,492001,492029],{},[2627,491999,347500],{},"：利用 ",[34,492002,492004,492017],{"className":492003},[31642],[34,492005,492007],{"className":492006},[31646],[31648,492008,492009],{"xmlns":31650},[31652,492010,492011,492015],{},[31655,492012,492013],{},[31749,492014,640],{},[31665,492016,640],{"encoding":31667},[34,492018,492020],{"className":492019,"ariaHidden":758},[31672],[34,492021,492023,492026],{"className":492022},[31676],[34,492024],{"className":492025,"style":32617},[31680],[34,492027,640],{"className":492028},[31685]," 的分解。",[15,492031,492032],{},[2627,492033,463779],{},[2230,492035,492036,492344,492580,493059],{},[108,492037,93310,492038,43298,492066,492094,492095,475913,492168,258,492218,335149,492269,347],{},[34,492039,492041,492054],{"className":492040},[31642],[34,492042,492044],{"className":492043},[31646],[31648,492045,492046],{"xmlns":31650},[31652,492047,492048,492052],{},[31655,492049,492050],{},[31658,492051,106063],{},[31665,492053,106063],{"encoding":31667},[34,492055,492057],{"className":492056,"ariaHidden":758},[31672],[34,492058,492060,492063],{"className":492059},[31676],[34,492061],{"className":492062,"style":33218},[31680],[34,492064,106063],{"className":492065,"style":67482},[31685,31686],[34,492067,492069,492082],{"className":492068},[31642],[34,492070,492072],{"className":492071},[31646],[31648,492073,492074],{"xmlns":31650},[31652,492075,492076,492080],{},[31655,492077,492078],{},[31658,492079,38691],{},[31665,492081,38691],{"encoding":31667},[34,492083,492085],{"className":492084,"ariaHidden":758},[31672],[34,492086,492088,492091],{"className":492087},[31676],[34,492089],{"className":492090,"style":33218},[31680],[34,492092,38691],{"className":492093,"style":38706},[31685,31686]," 互素，所以 ",[34,492096,492098,492120],{"className":492097},[31642],[34,492099,492101],{"className":492100},[31646],[31648,492102,492103],{"xmlns":31650},[31652,492104,492105,492117],{},[31655,492106,492107,492109,492111,492113,492115],{},[31658,492108,106063],{},[31662,492110,223],{},[31658,492112,38691],{},[31662,492114,202],{},[31658,492116,34386],{},[31665,492118,492119],{"encoding":31667},"I + K = R",[34,492121,492123,492141,492159],{"className":492122,"ariaHidden":758},[31672],[34,492124,492126,492129,492132,492135,492138],{"className":492125},[31676],[34,492127],{"className":492128,"style":34944},[31680],[34,492130,106063],{"className":492131,"style":67482},[31685,31686],[34,492133],{"className":492134,"style":32604},[31845],[34,492136,223],{"className":492137},[32024],[34,492139],{"className":492140,"style":32604},[31845],[34,492142,492144,492147,492150,492153,492156],{"className":492143},[31676],[34,492145],{"className":492146,"style":33218},[31680],[34,492148,38691],{"className":492149,"style":38706},[31685,31686],[34,492151],{"className":492152,"style":31846},[31845],[34,492154,202],{"className":492155},[31850],[34,492157],{"className":492158,"style":31846},[31845],[34,492160,492162,492165],{"className":492161},[31676],[34,492163],{"className":492164,"style":33218},[31680],[34,492166,34386],{"className":492167,"style":34427},[31685,31686],[34,492169,492171,492188],{"className":492170},[31642],[34,492172,492174],{"className":492173},[31646],[31648,492175,492176],{"xmlns":31650},[31652,492177,492178,492186],{},[31655,492179,492180,492182,492184],{},[31658,492181,31955],{},[31662,492183,112296],{},[31658,492185,106063],{},[31665,492187,451946],{"encoding":31667},[34,492189,492191,492209],{"className":492190,"ariaHidden":758},[31672],[34,492192,492194,492197,492200,492203,492206],{"className":492193},[31676],[34,492195],{"className":492196,"style":451673},[31680],[34,492198,31955],{"className":492199},[31685,31686],[34,492201],{"className":492202,"style":31846},[31845],[34,492204,112296],{"className":492205},[31850],[34,492207],{"className":492208,"style":31846},[31845],[34,492210,492212,492215],{"className":492211},[31676],[34,492213],{"className":492214,"style":33218},[31680],[34,492216,106063],{"className":492217,"style":67482},[31685,31686],[34,492219,492221,492239],{"className":492220},[31642],[34,492222,492224],{"className":492223},[31646],[31648,492225,492226],{"xmlns":31650},[31652,492227,492228,492236],{},[31655,492229,492230,492232,492234],{},[31658,492231,33759],{},[31662,492233,112296],{},[31658,492235,38691],{},[31665,492237,492238],{"encoding":31667},"k \\in K",[34,492240,492242,492260],{"className":492241,"ariaHidden":758},[31672],[34,492243,492245,492248,492251,492254,492257],{"className":492244},[31676],[34,492246],{"className":492247,"style":46991},[31680],[34,492249,33759],{"className":492250,"style":33774},[31685,31686],[34,492252],{"className":492253,"style":31846},[31845],[34,492255,112296],{"className":492256},[31850],[34,492258],{"className":492259,"style":31846},[31845],[34,492261,492263,492266],{"className":492262},[31676],[34,492264],{"className":492265,"style":33218},[31680],[34,492267,38691],{"className":492268,"style":38706},[31685,31686],[2627,492270,492271],{},[34,492272,492274,492296],{"className":492273},[31642],[34,492275,492277],{"className":492276},[31646],[31648,492278,492279],{"xmlns":31650},[31652,492280,492281,492293],{},[31655,492282,492283,492285,492287,492289,492291],{},[31658,492284,31955],{},[31662,492286,223],{},[31658,492288,33759],{},[31662,492290,202],{},[31749,492292,640],{},[31665,492294,492295],{"encoding":31667},"i + k = 1",[34,492297,492299,492317,492335],{"className":492298,"ariaHidden":758},[31672],[34,492300,492302,492305,492308,492311,492314],{"className":492301},[31676],[34,492303],{"className":492304,"style":37322},[31680],[34,492306,31955],{"className":492307},[31685,31686],[34,492309],{"className":492310,"style":32604},[31845],[34,492312,223],{"className":492313},[32024],[34,492315],{"className":492316,"style":32604},[31845],[34,492318,492320,492323,492326,492329,492332],{"className":492319},[31676],[34,492321],{"className":492322,"style":31681},[31680],[34,492324,33759],{"className":492325,"style":33774},[31685,31686],[34,492327],{"className":492328,"style":31846},[31845],[34,492330,202],{"className":492331},[31850],[34,492333],{"className":492334,"style":31846},[31845],[34,492336,492338,492341],{"className":492337},[31676],[34,492339],{"className":492340,"style":32617},[31680],[34,492342,640],{"className":492343},[31685],[108,492345,492346,492347,39984,492397],{},"设任意元素 ",[34,492348,492350,492367],{"className":492349},[31642],[34,492351,492353],{"className":492352},[31646],[31648,492354,492355],{"xmlns":31650},[31652,492356,492357,492365],{},[31655,492358,492359,492361,492363],{},[31658,492360,31968],{},[31662,492362,112296],{},[31658,492364,223320],{},[31665,492366,491914],{"encoding":31667},[34,492368,492370,492388],{"className":492369,"ariaHidden":758},[31672],[34,492371,492373,492376,492379,492382,492385],{"className":492372},[31676],[34,492374],{"className":492375,"style":33123},[31680],[34,492377,31968],{"className":492378,"style":32153},[31685,31686],[34,492380],{"className":492381,"style":31846},[31845],[34,492383,112296],{"className":492384},[31850],[34,492386],{"className":492387,"style":31846},[31845],[34,492389,492391,492394],{"className":492390},[31676],[34,492392],{"className":492393,"style":33218},[31680],[34,492395,223320],{"className":492396,"style":223361},[31685,31686],[34,492398,492400,492448],{"className":492399},[31642],[34,492401,492403],{"className":492402},[31646],[31648,492404,492405],{"xmlns":31650},[31652,492406,492407,492445],{},[31655,492408,492409,492411,492413,492415,492417,492419,492421,492423,492425,492427,492429,492431,492433,492435,492437,492439,492441,492443],{},[31658,492410,31968],{},[31662,492412,202],{},[31749,492414,640],{},[31662,492416,71191],{},[31658,492418,31968],{},[31662,492420,202],{},[31662,492422,456],{"stretchy":702},[31658,492424,31955],{},[31662,492426,223],{},[31658,492428,33759],{},[31662,492430,2557],{"stretchy":702},[31658,492432,31968],{},[31662,492434,202],{},[31658,492436,31955],{},[31658,492438,31968],{},[31662,492440,223],{},[31658,492442,33759],{},[31658,492444,31968],{},[31665,492446,492447],{"encoding":31667},"j = 1 \\cdot j = (i + k)j = ij + kj",[34,492449,492451,492469,492487,492505,492526,492550,492568],{"className":492450,"ariaHidden":758},[31672],[34,492452,492454,492457,492460,492463,492466],{"className":492453},[31676],[34,492455],{"className":492456,"style":33123},[31680],[34,492458,31968],{"className":492459,"style":32153},[31685,31686],[34,492461],{"className":492462,"style":31846},[31845],[34,492464,202],{"className":492465},[31850],[34,492467],{"className":492468,"style":31846},[31845],[34,492470,492472,492475,492478,492481,492484],{"className":492471},[31676],[34,492473],{"className":492474,"style":32617},[31680],[34,492476,640],{"className":492477},[31685],[34,492479],{"className":492480,"style":32604},[31845],[34,492482,71191],{"className":492483},[32024],[34,492485],{"className":492486,"style":32604},[31845],[34,492488,492490,492493,492496,492499,492502],{"className":492489},[31676],[34,492491],{"className":492492,"style":33123},[31680],[34,492494,31968],{"className":492495,"style":32153},[31685,31686],[34,492497],{"className":492498,"style":31846},[31845],[34,492500,202],{"className":492501},[31850],[34,492503],{"className":492504,"style":31846},[31845],[34,492506,492508,492511,492514,492517,492520,492523],{"className":492507},[31676],[34,492509],{"className":492510,"style":31774},[31680],[34,492512,456],{"className":492513},[31782],[34,492515,31955],{"className":492516},[31685,31686],[34,492518],{"className":492519,"style":32604},[31845],[34,492521,223],{"className":492522},[32024],[34,492524],{"className":492525,"style":32604},[31845],[34,492527,492529,492532,492535,492538,492541,492544,492547],{"className":492528},[31676],[34,492530],{"className":492531,"style":31774},[31680],[34,492533,33759],{"className":492534,"style":33774},[31685,31686],[34,492536,2557],{"className":492537},[31690],[34,492539,31968],{"className":492540,"style":32153},[31685,31686],[34,492542],{"className":492543,"style":31846},[31845],[34,492545,202],{"className":492546},[31850],[34,492548],{"className":492549,"style":31846},[31845],[34,492551,492553,492556,492559,492562,492565],{"className":492552},[31676],[34,492554],{"className":492555,"style":33123},[31680],[34,492557,141027],{"className":492558,"style":32153},[31685,31686],[34,492560],{"className":492561,"style":32604},[31845],[34,492563,223],{"className":492564},[32024],[34,492566],{"className":492567,"style":32604},[31845],[34,492569,492571,492574,492577],{"className":492570},[31676],[34,492572],{"className":492573,"style":43217},[31680],[34,492575,33759],{"className":492576,"style":33774},[31685,31686],[34,492578,31968],{"className":492579,"style":32153},[31685,31686],[108,492581,492582,492583],{},"分析这两项：\n",[105,492584,492585,492886],{},[108,492586,492587,332541,492617,86674,492667,13977,492717,492775,492776,13977,492831,347],{},[34,492588,492590,492605],{"className":492589},[31642],[34,492591,492593],{"className":492592},[31646],[31648,492594,492595],{"xmlns":31650},[31652,492596,492597,492603],{},[31655,492598,492599,492601],{},[31658,492600,31955],{},[31658,492602,31968],{},[31665,492604,141027],{"encoding":31667},[34,492606,492608],{"className":492607,"ariaHidden":758},[31672],[34,492609,492611,492614],{"className":492610},[31676],[34,492612],{"className":492613,"style":33123},[31680],[34,492615,141027],{"className":492616,"style":32153},[31685,31686],[34,492618,492620,492637],{"className":492619},[31642],[34,492621,492623],{"className":492622},[31646],[31648,492624,492625],{"xmlns":31650},[31652,492626,492627,492635],{},[31655,492628,492629,492631,492633],{},[31658,492630,31955],{},[31662,492632,112296],{},[31658,492634,106063],{},[31665,492636,451946],{"encoding":31667},[34,492638,492640,492658],{"className":492639,"ariaHidden":758},[31672],[34,492641,492643,492646,492649,492652,492655],{"className":492642},[31676],[34,492644],{"className":492645,"style":451673},[31680],[34,492647,31955],{"className":492648},[31685,31686],[34,492650],{"className":492651,"style":31846},[31845],[34,492653,112296],{"className":492654},[31850],[34,492656],{"className":492657,"style":31846},[31845],[34,492659,492661,492664],{"className":492660},[31676],[34,492662],{"className":492663,"style":33218},[31680],[34,492665,106063],{"className":492666,"style":67482},[31685,31686],[34,492668,492670,492687],{"className":492669},[31642],[34,492671,492673],{"className":492672},[31646],[31648,492674,492675],{"xmlns":31650},[31652,492676,492677,492685],{},[31655,492678,492679,492681,492683],{},[31658,492680,31968],{},[31662,492682,112296],{},[31658,492684,223320],{},[31665,492686,491914],{"encoding":31667},[34,492688,492690,492708],{"className":492689,"ariaHidden":758},[31672],[34,492691,492693,492696,492699,492702,492705],{"className":492692},[31676],[34,492694],{"className":492695,"style":33123},[31680],[34,492697,31968],{"className":492698,"style":32153},[31685,31686],[34,492700],{"className":492701,"style":31846},[31845],[34,492703,112296],{"className":492704},[31850],[34,492706],{"className":492707,"style":31846},[31845],[34,492709,492711,492714],{"className":492710},[31676],[34,492712],{"className":492713,"style":33218},[31680],[34,492715,223320],{"className":492716,"style":223361},[31685,31686],[34,492718,492720,492742],{"className":492719},[31642],[34,492721,492723],{"className":492722},[31646],[31648,492724,492725],{"xmlns":31650},[31652,492726,492727,492739],{},[31655,492728,492729,492731,492733,492735,492737],{},[31658,492730,31955],{},[31658,492732,31968],{},[31662,492734,112296],{},[31658,492736,106063],{},[31658,492738,223320],{},[31665,492740,492741],{"encoding":31667},"ij \\in IJ",[34,492743,492745,492763],{"className":492744,"ariaHidden":758},[31672],[34,492746,492748,492751,492754,492757,492760],{"className":492747},[31676],[34,492749],{"className":492750,"style":33123},[31680],[34,492752,141027],{"className":492753,"style":32153},[31685,31686],[34,492755],{"className":492756,"style":31846},[31845],[34,492758,112296],{"className":492759},[31850],[34,492761],{"className":492762,"style":31846},[31845],[34,492764,492766,492769,492772],{"className":492765},[31676],[34,492767],{"className":492768,"style":33218},[31680],[34,492770,106063],{"className":492771,"style":67482},[31685,31686],[34,492773,223320],{"className":492774,"style":223361},[31685,31686],"。\n题目已知 ",[34,492777,492779,492798],{"className":492778},[31642],[34,492780,492782],{"className":492781},[31646],[31648,492783,492784],{"xmlns":31650},[31652,492785,492786,492796],{},[31655,492787,492788,492790,492792,492794],{},[31658,492789,106063],{},[31658,492791,223320],{},[31662,492793,131855],{},[31658,492795,38691],{},[31665,492797,491578],{"encoding":31667},[34,492799,492801,492822],{"className":492800,"ariaHidden":758},[31672],[34,492802,492804,492807,492810,492813,492816,492819],{"className":492803},[31676],[34,492805],{"className":492806,"style":195047},[31680],[34,492808,106063],{"className":492809,"style":67482},[31685,31686],[34,492811,223320],{"className":492812,"style":223361},[31685,31686],[34,492814],{"className":492815,"style":31846},[31845],[34,492817,131855],{"className":492818},[31850],[34,492820],{"className":492821,"style":31846},[31845],[34,492823,492825,492828],{"className":492824},[31676],[34,492826],{"className":492827,"style":33218},[31680],[34,492829,38691],{"className":492830,"style":38706},[31685,31686],[2627,492832,492833],{},[34,492834,492836,492856],{"className":492835},[31642],[34,492837,492839],{"className":492838},[31646],[31648,492840,492841],{"xmlns":31650},[31652,492842,492843,492853],{},[31655,492844,492845,492847,492849,492851],{},[31658,492846,31955],{},[31658,492848,31968],{},[31662,492850,112296],{},[31658,492852,38691],{},[31665,492854,492855],{"encoding":31667},"ij \\in K",[34,492857,492859,492877],{"className":492858,"ariaHidden":758},[31672],[34,492860,492862,492865,492868,492871,492874],{"className":492861},[31676],[34,492863],{"className":492864,"style":33123},[31680],[34,492866,141027],{"className":492867,"style":32153},[31685,31686],[34,492869],{"className":492870,"style":31846},[31845],[34,492872,112296],{"className":492873},[31850],[34,492875],{"className":492876,"style":31846},[31845],[34,492878,492880,492883],{"className":492879},[31676],[34,492881],{"className":492882,"style":33218},[31680],[34,492884,38691],{"className":492885,"style":38706},[31685,31686],[108,492887,492888,332541,492922,86674,492972,493000,493001,347],{},[34,492889,492891,492907],{"className":492890},[31642],[34,492892,492894],{"className":492893},[31646],[31648,492895,492896],{"xmlns":31650},[31652,492897,492898,492904],{},[31655,492899,492900,492902],{},[31658,492901,33759],{},[31658,492903,31968],{},[31665,492905,492906],{"encoding":31667},"kj",[34,492908,492910],{"className":492909,"ariaHidden":758},[31672],[34,492911,492913,492916,492919],{"className":492912},[31676],[34,492914],{"className":492915,"style":43217},[31680],[34,492917,33759],{"className":492918,"style":33774},[31685,31686],[34,492920,31968],{"className":492921,"style":32153},[31685,31686],[34,492923,492925,492942],{"className":492924},[31642],[34,492926,492928],{"className":492927},[31646],[31648,492929,492930],{"xmlns":31650},[31652,492931,492932,492940],{},[31655,492933,492934,492936,492938],{},[31658,492935,33759],{},[31662,492937,112296],{},[31658,492939,38691],{},[31665,492941,492238],{"encoding":31667},[34,492943,492945,492963],{"className":492944,"ariaHidden":758},[31672],[34,492946,492948,492951,492954,492957,492960],{"className":492947},[31676],[34,492949],{"className":492950,"style":46991},[31680],[34,492952,33759],{"className":492953,"style":33774},[31685,31686],[34,492955],{"className":492956,"style":31846},[31845],[34,492958,112296],{"className":492959},[31850],[34,492961],{"className":492962,"style":31846},[31845],[34,492964,492966,492969],{"className":492965},[31676],[34,492967],{"className":492968,"style":33218},[31680],[34,492970,38691],{"className":492971,"style":38706},[31685,31686],[34,492973,492975,492988],{"className":492974},[31642],[34,492976,492978],{"className":492977},[31646],[31648,492979,492980],{"xmlns":31650},[31652,492981,492982,492986],{},[31655,492983,492984],{},[31658,492985,38691],{},[31665,492987,38691],{"encoding":31667},[34,492989,492991],{"className":492990,"ariaHidden":758},[31672],[34,492992,492994,492997],{"className":492993},[31676],[34,492995],{"className":492996,"style":33218},[31680],[34,492998,38691],{"className":492999,"style":38706},[31685,31686]," 是理想（吸收性），所以 ",[2627,493002,493003],{},[34,493004,493006,493026],{"className":493005},[31642],[34,493007,493009],{"className":493008},[31646],[31648,493010,493011],{"xmlns":31650},[31652,493012,493013,493023],{},[31655,493014,493015,493017,493019,493021],{},[31658,493016,33759],{},[31658,493018,31968],{},[31662,493020,112296],{},[31658,493022,38691],{},[31665,493024,493025],{"encoding":31667},"kj \\in K",[34,493027,493029,493050],{"className":493028,"ariaHidden":758},[31672],[34,493030,493032,493035,493038,493041,493044,493047],{"className":493031},[31676],[34,493033],{"className":493034,"style":43217},[31680],[34,493036,33759],{"className":493037,"style":33774},[31685,31686],[34,493039,31968],{"className":493040,"style":32153},[31685,31686],[34,493042],{"className":493043,"style":31846},[31845],[34,493045,112296],{"className":493046},[31850],[34,493048],{"className":493049,"style":31846},[31845],[34,493051,493053,493056],{"className":493052},[31676],[34,493054],{"className":493055,"style":33218},[31680],[34,493057,38691],{"className":493058,"style":38706},[31685,31686],[108,493060,93310,493061,493089,493090,452821,493170,39984,493220,2664,493222,311257,493272],{},[34,493062,493064,493077],{"className":493063},[31642],[34,493065,493067],{"className":493066},[31646],[31648,493068,493069],{"xmlns":31650},[31652,493070,493071,493075],{},[31655,493072,493073],{},[31658,493074,38691],{},[31665,493076,38691],{"encoding":31667},[34,493078,493080],{"className":493079,"ariaHidden":758},[31672],[34,493081,493083,493086],{"className":493082},[31676],[34,493084],{"className":493085,"style":33218},[31680],[34,493087,38691],{"className":493088,"style":38706},[31685,31686]," 是理想（对加法封闭），所以 ",[34,493091,493093,493119],{"className":493092},[31642],[34,493094,493096],{"className":493095},[31646],[31648,493097,493098],{"xmlns":31650},[31652,493099,493100,493116],{},[31655,493101,493102,493104,493106,493108,493110,493112,493114],{},[31658,493103,31955],{},[31658,493105,31968],{},[31662,493107,223],{},[31658,493109,33759],{},[31658,493111,31968],{},[31662,493113,112296],{},[31658,493115,38691],{},[31665,493117,493118],{"encoding":31667},"ij + kj \\in K",[34,493120,493122,493140,493161],{"className":493121,"ariaHidden":758},[31672],[34,493123,493125,493128,493131,493134,493137],{"className":493124},[31676],[34,493126],{"className":493127,"style":33123},[31680],[34,493129,141027],{"className":493130,"style":32153},[31685,31686],[34,493132],{"className":493133,"style":32604},[31845],[34,493135,223],{"className":493136},[32024],[34,493138],{"className":493139,"style":32604},[31845],[34,493141,493143,493146,493149,493152,493155,493158],{"className":493142},[31676],[34,493144],{"className":493145,"style":43217},[31680],[34,493147,33759],{"className":493148,"style":33774},[31685,31686],[34,493150,31968],{"className":493151,"style":32153},[31685,31686],[34,493153],{"className":493154,"style":31846},[31845],[34,493156,112296],{"className":493157},[31850],[34,493159],{"className":493160,"style":31846},[31845],[34,493162,493164,493167],{"className":493163},[31676],[34,493165],{"className":493166,"style":33218},[31680],[34,493168,38691],{"className":493169,"style":38706},[31685,31686],[34,493171,493173,493190],{"className":493172},[31642],[34,493174,493176],{"className":493175},[31646],[31648,493177,493178],{"xmlns":31650},[31652,493179,493180,493188],{},[31655,493181,493182,493184,493186],{},[31658,493183,31968],{},[31662,493185,112296],{},[31658,493187,38691],{},[31665,493189,491965],{"encoding":31667},[34,493191,493193,493211],{"className":493192,"ariaHidden":758},[31672],[34,493194,493196,493199,493202,493205,493208],{"className":493195},[31676],[34,493197],{"className":493198,"style":33123},[31680],[34,493200,31968],{"className":493201,"style":32153},[31685,31686],[34,493203],{"className":493204,"style":31846},[31845],[34,493206,112296],{"className":493207},[31850],[34,493209],{"className":493210,"style":31846},[31845],[34,493212,493214,493217],{"className":493213},[31676],[34,493215],{"className":493216,"style":33218},[31680],[34,493218,38691],{"className":493219,"style":38706},[31685,31686],[2627,493221,114043],{},[34,493223,493225,493242],{"className":493224},[31642],[34,493226,493228],{"className":493227},[31646],[31648,493229,493230],{"xmlns":31650},[31652,493231,493232,493240],{},[31655,493233,493234,493236,493238],{},[31658,493235,223320],{},[31662,493237,131855],{},[31658,493239,38691],{},[31665,493241,491689],{"encoding":31667},[34,493243,493245,493263],{"className":493244,"ariaHidden":758},[31672],[34,493246,493248,493251,493254,493257,493260],{"className":493247},[31676],[34,493249],{"className":493250,"style":195047},[31680],[34,493252,223320],{"className":493253,"style":223361},[31685,31686],[34,493255],{"className":493256,"style":31846},[31845],[34,493258,131855],{"className":493259},[31850],[34,493261],{"className":493262,"style":31846},[31845],[34,493264,493266,493269],{"className":493265},[31676],[34,493267],{"className":493268,"style":33218},[31680],[34,493270,38691],{"className":493271,"style":38706},[31685,31686],[34,493273,493275,493288],{"className":493274},[31642],[34,493276,493278],{"className":493277},[31646],[31648,493279,493280],{"xmlns":31650},[31652,493281,493282,493286],{},[31655,493283,493284],{},[31658,493285,445204],{"mathvariant":38079},[31665,493287,445207],{"encoding":31667},[34,493289,493291],{"className":493290,"ariaHidden":758},[31672],[34,493292,493294,493297],{"className":493293},[31676],[34,493295],{"className":493296,"style":445217},[31680],[34,493298,445204],{"className":493299},[31685,129261],[32188,493301],{},[32490,493303,479329],{"id":493304},"习题-4-1",[15,493306,493307],{},[2627,493308,133910,493309,86674,493373,43298,493401,491668,493429,2547],{},[34,493310,493312,493334],{"className":493311},[31642],[34,493313,493315],{"className":493314},[31646],[31648,493316,493317],{"xmlns":31650},[31652,493318,493319,493331],{},[31655,493320,493321,493323,493325,493327,493329],{},[31658,493322,106063],{},[31662,493324,31960],{"separator":758},[31658,493326,223320],{},[31662,493328,346833],{},[31658,493330,38691],{},[31665,493332,493333],{"encoding":31667},"I, J \\supseteq K",[34,493335,493337,493364],{"className":493336,"ariaHidden":758},[31672],[34,493338,493340,493343,493346,493349,493352,493355,493358,493361],{"className":493339},[31676],[34,493341],{"className":493342,"style":70739},[31680],[34,493344,106063],{"className":493345,"style":67482},[31685,31686],[34,493347,31960],{"className":493348},[32119],[34,493350],{"className":493351,"style":32123},[31845],[34,493353,223320],{"className":493354,"style":223361},[31685,31686],[34,493356],{"className":493357,"style":31846},[31845],[34,493359,346833],{"className":493360},[31850],[34,493362],{"className":493363,"style":31846},[31845],[34,493365,493367,493370],{"className":493366},[31676],[34,493368],{"className":493369,"style":33218},[31680],[34,493371,38691],{"className":493372,"style":38706},[31685,31686],[34,493374,493376,493389],{"className":493375},[31642],[34,493377,493379],{"className":493378},[31646],[31648,493380,493381],{"xmlns":31650},[31652,493382,493383,493387],{},[31655,493384,493385],{},[31658,493386,106063],{},[31665,493388,106063],{"encoding":31667},[34,493390,493392],{"className":493391,"ariaHidden":758},[31672],[34,493393,493395,493398],{"className":493394},[31676],[34,493396],{"className":493397,"style":33218},[31680],[34,493399,106063],{"className":493400,"style":67482},[31685,31686],[34,493402,493404,493417],{"className":493403},[31642],[34,493405,493407],{"className":493406},[31646],[31648,493408,493409],{"xmlns":31650},[31652,493410,493411,493415],{},[31655,493412,493413],{},[31658,493414,223320],{},[31665,493416,223320],{"encoding":31667},[34,493418,493420],{"className":493419,"ariaHidden":758},[31672],[34,493421,493423,493426],{"className":493422},[31676],[34,493424],{"className":493425,"style":33218},[31680],[34,493427,223320],{"className":493428,"style":223361},[31685,31686],[34,493430,493432,493452],{"className":493431},[31642],[34,493433,493435],{"className":493434},[31646],[31648,493436,493437],{"xmlns":31650},[31652,493438,493439,493449],{},[31655,493440,493441,493443,493445,493447],{},[31658,493442,106063],{},[31658,493444,223320],{},[31662,493446,346833],{},[31658,493448,38691],{},[31665,493450,493451],{"encoding":31667},"IJ \\supseteq K",[34,493453,493455,493476],{"className":493454,"ariaHidden":758},[31672],[34,493456,493458,493461,493464,493467,493470,493473],{"className":493457},[31676],[34,493459],{"className":493460,"style":195047},[31680],[34,493462,106063],{"className":493463,"style":67482},[31685,31686],[34,493465,223320],{"className":493466,"style":223361},[31685,31686],[34,493468],{"className":493469,"style":31846},[31845],[34,493471,346833],{"className":493472},[31850],[34,493474],{"className":493475,"style":31846},[31845],[34,493477,493479,493482],{"className":493478},[31676],[34,493480],{"className":493481,"style":33218},[31680],[34,493483,38691],{"className":493484,"style":38706},[31685,31686],[15,493486,493487],{},[2627,493488,463462],{},[105,493490,493491,493571,493692],{},[108,493492,493493,493494,347],{},"这是一个反直觉的结论。通常 ",[34,493495,493497,493520],{"className":493496},[31642],[34,493498,493500],{"className":493499},[31646],[31648,493501,493502],{"xmlns":31650},[31652,493503,493504,493518],{},[31655,493505,493506,493508,493510,493512,493514,493516],{},[31658,493507,106063],{},[31658,493509,223320],{},[31662,493511,131855],{},[31658,493513,106063],{},[31662,493515,326344],{},[31658,493517,223320],{},[31665,493519,456828],{"encoding":31667},[34,493521,493523,493544,493562],{"className":493522,"ariaHidden":758},[31672],[34,493524,493526,493529,493532,493535,493538,493541],{"className":493525},[31676],[34,493527],{"className":493528,"style":195047},[31680],[34,493530,106063],{"className":493531,"style":67482},[31685,31686],[34,493533,223320],{"className":493534,"style":223361},[31685,31686],[34,493536],{"className":493537,"style":31846},[31845],[34,493539,131855],{"className":493540},[31850],[34,493542],{"className":493543,"style":31846},[31845],[34,493545,493547,493550,493553,493556,493559],{"className":493546},[31676],[34,493548],{"className":493549,"style":33218},[31680],[34,493551,106063],{"className":493552,"style":67482},[31685,31686],[34,493554],{"className":493555,"style":32604},[31845],[34,493557,326344],{"className":493558},[32024],[34,493560],{"className":493561,"style":32604},[31845],[34,493563,493565,493568],{"className":493564},[31676],[34,493566],{"className":493567,"style":33218},[31680],[34,493569,223320],{"className":493570,"style":223361},[31685,31686],[108,493572,35328,493573,493614,493615,347],{},[34,493574,493576,493593],{"className":493575},[31642],[34,493577,493579],{"className":493578},[31646],[31648,493580,493581],{"xmlns":31650},[31652,493582,493583,493591],{},[31655,493584,493585,493587,493589],{},[31658,493586,106063],{},[31662,493588,31960],{"separator":758},[31658,493590,223320],{},[31665,493592,454327],{"encoding":31667},[34,493594,493596],{"className":493595,"ariaHidden":758},[31672],[34,493597,493599,493602,493605,493608,493611],{"className":493598},[31676],[34,493600],{"className":493601,"style":70739},[31680],[34,493603,106063],{"className":493604,"style":67482},[31685,31686],[34,493606,31960],{"className":493607},[32119],[34,493609],{"className":493610,"style":32123},[31845],[34,493612,223320],{"className":493613,"style":223361},[31685,31686]," 互素时，有一个重要定理：",[34,493616,493618,493641],{"className":493617},[31642],[34,493619,493621],{"className":493620},[31646],[31648,493622,493623],{"xmlns":31650},[31652,493624,493625,493639],{},[31655,493626,493627,493629,493631,493633,493635,493637],{},[31658,493628,106063],{},[31658,493630,223320],{},[31662,493632,202],{},[31658,493634,106063],{},[31662,493636,326344],{},[31658,493638,223320],{},[31665,493640,457958],{"encoding":31667},[34,493642,493644,493665,493683],{"className":493643,"ariaHidden":758},[31672],[34,493645,493647,493650,493653,493656,493659,493662],{"className":493646},[31676],[34,493648],{"className":493649,"style":33218},[31680],[34,493651,106063],{"className":493652,"style":67482},[31685,31686],[34,493654,223320],{"className":493655,"style":223361},[31685,31686],[34,493657],{"className":493658,"style":31846},[31845],[34,493660,202],{"className":493661},[31850],[34,493663],{"className":493664,"style":31846},[31845],[34,493666,493668,493671,493674,493677,493680],{"className":493667},[31676],[34,493669],{"className":493670,"style":33218},[31680],[34,493672,106063],{"className":493673,"style":67482},[31685,31686],[34,493675],{"className":493676,"style":32604},[31845],[34,493678,326344],{"className":493679},[32024],[34,493681],{"className":493682,"style":32604},[31845],[34,493684,493686,493689],{"className":493685},[31676],[34,493687],{"className":493688,"style":33218},[31680],[34,493690,223320],{"className":493691,"style":223361},[31685,31686],[108,493693,493694,493695,493772,493773,493801,493802,493830,493831,493859],{},"如果能证明 ",[34,493696,493698,493721],{"className":493697},[31642],[34,493699,493701],{"className":493700},[31646],[31648,493702,493703],{"xmlns":31650},[31652,493704,493705,493719],{},[31655,493706,493707,493709,493711,493713,493715,493717],{},[31658,493708,106063],{},[31658,493710,223320],{},[31662,493712,202],{},[31658,493714,106063],{},[31662,493716,326344],{},[31658,493718,223320],{},[31665,493720,457958],{"encoding":31667},[34,493722,493724,493745,493763],{"className":493723,"ariaHidden":758},[31672],[34,493725,493727,493730,493733,493736,493739,493742],{"className":493726},[31676],[34,493728],{"className":493729,"style":33218},[31680],[34,493731,106063],{"className":493732,"style":67482},[31685,31686],[34,493734,223320],{"className":493735,"style":223361},[31685,31686],[34,493737],{"className":493738,"style":31846},[31845],[34,493740,202],{"className":493741},[31850],[34,493743],{"className":493744,"style":31846},[31845],[34,493746,493748,493751,493754,493757,493760],{"className":493747},[31676],[34,493749],{"className":493750,"style":33218},[31680],[34,493752,106063],{"className":493753,"style":67482},[31685,31686],[34,493755],{"className":493756,"style":32604},[31845],[34,493758,326344],{"className":493759},[32024],[34,493761],{"className":493762,"style":32604},[31845],[34,493764,493766,493769],{"className":493765},[31676],[34,493767],{"className":493768,"style":33218},[31680],[34,493770,223320],{"className":493771,"style":223361},[31685,31686],"，结合 ",[34,493774,493776,493789],{"className":493775},[31642],[34,493777,493779],{"className":493778},[31646],[31648,493780,493781],{"xmlns":31650},[31652,493782,493783,493787],{},[31655,493784,493785],{},[31658,493786,38691],{},[31665,493788,38691],{"encoding":31667},[34,493790,493792],{"className":493791,"ariaHidden":758},[31672],[34,493793,493795,493798],{"className":493794},[31676],[34,493796],{"className":493797,"style":33218},[31680],[34,493799,38691],{"className":493800,"style":38706},[31685,31686]," 既在 ",[34,493803,493805,493818],{"className":493804},[31642],[34,493806,493808],{"className":493807},[31646],[31648,493809,493810],{"xmlns":31650},[31652,493811,493812,493816],{},[31655,493813,493814],{},[31658,493815,106063],{},[31665,493817,106063],{"encoding":31667},[34,493819,493821],{"className":493820,"ariaHidden":758},[31672],[34,493822,493824,493827],{"className":493823},[31676],[34,493825],{"className":493826,"style":33218},[31680],[34,493828,106063],{"className":493829,"style":67482},[31685,31686]," 里又在 ",[34,493832,493834,493847],{"className":493833},[31642],[34,493835,493837],{"className":493836},[31646],[31648,493838,493839],{"xmlns":31650},[31652,493840,493841,493845],{},[31655,493842,493843],{},[31658,493844,223320],{},[31665,493846,223320],{"encoding":31667},[34,493848,493850],{"className":493849,"ariaHidden":758},[31672],[34,493851,493853,493856],{"className":493852},[31676],[34,493854],{"className":493855,"style":33218},[31680],[34,493857,223320],{"className":493858,"style":223361},[31685,31686]," 里，自然在交集里，问题就解决了。",[15,493861,493862],{},[2627,493863,463779],{},[2230,493865,493866,495133],{},[108,493867,493868,347,493990],{},[2627,493869,493870,493871,493912,493913],{},"先证当 ",[34,493872,493874,493891],{"className":493873},[31642],[34,493875,493877],{"className":493876},[31646],[31648,493878,493879],{"xmlns":31650},[31652,493880,493881,493889],{},[31655,493882,493883,493885,493887],{},[31658,493884,106063],{},[31662,493886,31960],{"separator":758},[31658,493888,223320],{},[31665,493890,454327],{"encoding":31667},[34,493892,493894],{"className":493893,"ariaHidden":758},[31672],[34,493895,493897,493900,493903,493906,493909],{"className":493896},[31676],[34,493898],{"className":493899,"style":70739},[31680],[34,493901,106063],{"className":493902,"style":67482},[31685,31686],[34,493904,31960],{"className":493905},[32119],[34,493907],{"className":493908,"style":32123},[31845],[34,493910,223320],{"className":493911,"style":223361},[31685,31686]," 互素时，",[34,493914,493916,493939],{"className":493915},[31642],[34,493917,493919],{"className":493918},[31646],[31648,493920,493921],{"xmlns":31650},[31652,493922,493923,493937],{},[31655,493924,493925,493927,493929,493931,493933,493935],{},[31658,493926,106063],{},[31658,493928,223320],{},[31662,493930,202],{},[31658,493932,106063],{},[31662,493934,326344],{},[31658,493936,223320],{},[31665,493938,457958],{"encoding":31667},[34,493940,493942,493963,493981],{"className":493941,"ariaHidden":758},[31672],[34,493943,493945,493948,493951,493954,493957,493960],{"className":493944},[31676],[34,493946],{"className":493947,"style":33218},[31680],[34,493949,106063],{"className":493950,"style":67482},[31685,31686],[34,493952,223320],{"className":493953,"style":223361},[31685,31686],[34,493955],{"className":493956,"style":31846},[31845],[34,493958,202],{"className":493959},[31850],[34,493961],{"className":493962,"style":31846},[31845],[34,493964,493966,493969,493972,493975,493978],{"className":493965},[31676],[34,493967],{"className":493968,"style":33218},[31680],[34,493970,106063],{"className":493971,"style":67482},[31685,31686],[34,493973],{"className":493974,"style":32604},[31845],[34,493976,326344],{"className":493977},[32024],[34,493979],{"className":493980,"style":32604},[31845],[34,493982,493984,493987],{"className":493983},[31676],[34,493985],{"className":493986,"style":33218},[31680],[34,493988,223320],{"className":493989,"style":223361},[31685,31686],[105,493991,493992,494073,494275,494537,494743,494918,495053],{},[108,493993,493994,493995,494072],{},"显然 ",[34,493996,493998,494021],{"className":493997},[31642],[34,493999,494001],{"className":494000},[31646],[31648,494002,494003],{"xmlns":31650},[31652,494004,494005,494019],{},[31655,494006,494007,494009,494011,494013,494015,494017],{},[31658,494008,106063],{},[31658,494010,223320],{},[31662,494012,131855],{},[31658,494014,106063],{},[31662,494016,326344],{},[31658,494018,223320],{},[31665,494020,456828],{"encoding":31667},[34,494022,494024,494045,494063],{"className":494023,"ariaHidden":758},[31672],[34,494025,494027,494030,494033,494036,494039,494042],{"className":494026},[31676],[34,494028],{"className":494029,"style":195047},[31680],[34,494031,106063],{"className":494032,"style":67482},[31685,31686],[34,494034,223320],{"className":494035,"style":223361},[31685,31686],[34,494037],{"className":494038,"style":31846},[31845],[34,494040,131855],{"className":494041},[31850],[34,494043],{"className":494044,"style":31846},[31845],[34,494046,494048,494051,494054,494057,494060],{"className":494047},[31676],[34,494049],{"className":494050,"style":33218},[31680],[34,494052,106063],{"className":494053,"style":67482},[31685,31686],[34,494055],{"className":494056,"style":32604},[31845],[34,494058,326344],{"className":494059},[32024],[34,494061],{"className":494062,"style":32604},[31845],[34,494064,494066,494069],{"className":494065},[31676],[34,494067],{"className":494068,"style":33218},[31680],[34,494070,223320],{"className":494071,"style":223361},[31685,31686]," 总是成立。",[108,494074,93310,494075,494116,494117,315343,494202,347],{},[34,494076,494078,494095],{"className":494077},[31642],[34,494079,494081],{"className":494080},[31646],[31648,494082,494083],{"xmlns":31650},[31652,494084,494085,494093],{},[31655,494086,494087,494089,494091],{},[31658,494088,106063],{},[31662,494090,31960],{"separator":758},[31658,494092,223320],{},[31665,494094,454327],{"encoding":31667},[34,494096,494098],{"className":494097,"ariaHidden":758},[31672],[34,494099,494101,494104,494107,494110,494113],{"className":494100},[31676],[34,494102],{"className":494103,"style":70739},[31680],[34,494105,106063],{"className":494106,"style":67482},[31685,31686],[34,494108,31960],{"className":494109},[32119],[34,494111],{"className":494112,"style":32123},[31845],[34,494114,223320],{"className":494115,"style":223361},[31685,31686]," 互素，存在 ",[34,494118,494120,494145],{"className":494119},[31642],[34,494121,494123],{"className":494122},[31646],[31648,494124,494125],{"xmlns":31650},[31652,494126,494127,494143],{},[31655,494128,494129,494131,494133,494135,494137,494139,494141],{},[31658,494130,31955],{},[31662,494132,112296],{},[31658,494134,106063],{},[31662,494136,31960],{"separator":758},[31658,494138,31968],{},[31662,494140,112296],{},[31658,494142,223320],{},[31665,494144,454519],{"encoding":31667},[34,494146,494148,494166,494193],{"className":494147,"ariaHidden":758},[31672],[34,494149,494151,494154,494157,494160,494163],{"className":494150},[31676],[34,494152],{"className":494153,"style":451673},[31680],[34,494155,31955],{"className":494156},[31685,31686],[34,494158],{"className":494159,"style":31846},[31845],[34,494161,112296],{"className":494162},[31850],[34,494164],{"className":494165,"style":31846},[31845],[34,494167,494169,494172,494175,494178,494181,494184,494187,494190],{"className":494168},[31676],[34,494170],{"className":494171,"style":70739},[31680],[34,494173,106063],{"className":494174,"style":67482},[31685,31686],[34,494176,31960],{"className":494177},[32119],[34,494179],{"className":494180,"style":32123},[31845],[34,494182,31968],{"className":494183,"style":32153},[31685,31686],[34,494185],{"className":494186,"style":31846},[31845],[34,494188,112296],{"className":494189},[31850],[34,494191],{"className":494192,"style":31846},[31845],[34,494194,494196,494199],{"className":494195},[31676],[34,494197],{"className":494198,"style":33218},[31680],[34,494200,223320],{"className":494201,"style":223361},[31685,31686],[34,494203,494205,494227],{"className":494204},[31642],[34,494206,494208],{"className":494207},[31646],[31648,494209,494210],{"xmlns":31650},[31652,494211,494212,494224],{},[31655,494213,494214,494216,494218,494220,494222],{},[31658,494215,31955],{},[31662,494217,223],{},[31658,494219,31968],{},[31662,494221,202],{},[31749,494223,640],{},[31665,494225,494226],{"encoding":31667},"i+j=1",[34,494228,494230,494248,494266],{"className":494229,"ariaHidden":758},[31672],[34,494231,494233,494236,494239,494242,494245],{"className":494232},[31676],[34,494234],{"className":494235,"style":37322},[31680],[34,494237,31955],{"className":494238},[31685,31686],[34,494240],{"className":494241,"style":32604},[31845],[34,494243,223],{"className":494244},[32024],[34,494246],{"className":494247,"style":32604},[31845],[34,494249,494251,494254,494257,494260,494263],{"className":494250},[31676],[34,494252],{"className":494253,"style":33123},[31680],[34,494255,31968],{"className":494256,"style":32153},[31685,31686],[34,494258],{"className":494259,"style":31846},[31845],[34,494261,202],{"className":494262},[31850],[34,494264],{"className":494265,"style":31846},[31845],[34,494267,494269,494272],{"className":494268},[31676],[34,494270],{"className":494271,"style":32617},[31680],[34,494273,640],{"className":494274},[31685],[108,494276,494277,494278,39984,494351],{},"设任意 ",[34,494279,494281,494303],{"className":494280},[31642],[34,494282,494284],{"className":494283},[31646],[31648,494285,494286],{"xmlns":31650},[31652,494287,494288,494300],{},[31655,494289,494290,494292,494294,494296,494298],{},[31658,494291,32936],{},[31662,494293,112296],{},[31658,494295,106063],{},[31662,494297,326344],{},[31658,494299,223320],{},[31665,494301,494302],{"encoding":31667},"x \\in I \\cap J",[34,494304,494306,494324,494342],{"className":494305,"ariaHidden":758},[31672],[34,494307,494309,494312,494315,494318,494321],{"className":494308},[31676],[34,494310],{"className":494311,"style":34812},[31680],[34,494313,32936],{"className":494314},[31685,31686],[34,494316],{"className":494317,"style":31846},[31845],[34,494319,112296],{"className":494320},[31850],[34,494322],{"className":494323,"style":31846},[31845],[34,494325,494327,494330,494333,494336,494339],{"className":494326},[31676],[34,494328],{"className":494329,"style":33218},[31680],[34,494331,106063],{"className":494332,"style":67482},[31685,31686],[34,494334],{"className":494335,"style":32604},[31845],[34,494337,326344],{"className":494338},[32024],[34,494340],{"className":494341,"style":32604},[31845],[34,494343,494345,494348],{"className":494344},[31676],[34,494346],{"className":494347,"style":33218},[31680],[34,494349,223320],{"className":494350,"style":223361},[31685,31686],[34,494352,494354,494402],{"className":494353},[31642],[34,494355,494357],{"className":494356},[31646],[31648,494358,494359],{"xmlns":31650},[31652,494360,494361,494399],{},[31655,494362,494363,494365,494367,494369,494371,494373,494375,494377,494379,494381,494383,494385,494387,494389,494391,494393,494395,494397],{},[31658,494364,32936],{},[31662,494366,202],{},[31749,494368,640],{},[31662,494370,71191],{},[31658,494372,32936],{},[31662,494374,202],{},[31662,494376,456],{"stretchy":702},[31658,494378,31955],{},[31662,494380,223],{},[31658,494382,31968],{},[31662,494384,2557],{"stretchy":702},[31658,494386,32936],{},[31662,494388,202],{},[31658,494390,31955],{},[31658,494392,32936],{},[31662,494394,223],{},[31658,494396,31968],{},[31658,494398,32936],{},[31665,494400,494401],{"encoding":31667},"x = 1 \\cdot x = (i+j)x = ix + jx",[34,494403,494405,494423,494441,494459,494480,494504,494525],{"className":494404,"ariaHidden":758},[31672],[34,494406,494408,494411,494414,494417,494420],{"className":494407},[31676],[34,494409],{"className":494410,"style":32527},[31680],[34,494412,32936],{"className":494413},[31685,31686],[34,494415],{"className":494416,"style":31846},[31845],[34,494418,202],{"className":494419},[31850],[34,494421],{"className":494422,"style":31846},[31845],[34,494424,494426,494429,494432,494435,494438],{"className":494425},[31676],[34,494427],{"className":494428,"style":32617},[31680],[34,494430,640],{"className":494431},[31685],[34,494433],{"className":494434,"style":32604},[31845],[34,494436,71191],{"className":494437},[32024],[34,494439],{"className":494440,"style":32604},[31845],[34,494442,494444,494447,494450,494453,494456],{"className":494443},[31676],[34,494445],{"className":494446,"style":32527},[31680],[34,494448,32936],{"className":494449},[31685,31686],[34,494451],{"className":494452,"style":31846},[31845],[34,494454,202],{"className":494455},[31850],[34,494457],{"className":494458,"style":31846},[31845],[34,494460,494462,494465,494468,494471,494474,494477],{"className":494461},[31676],[34,494463],{"className":494464,"style":31774},[31680],[34,494466,456],{"className":494467},[31782],[34,494469,31955],{"className":494470},[31685,31686],[34,494472],{"className":494473,"style":32604},[31845],[34,494475,223],{"className":494476},[32024],[34,494478],{"className":494479,"style":32604},[31845],[34,494481,494483,494486,494489,494492,494495,494498,494501],{"className":494482},[31676],[34,494484],{"className":494485,"style":31774},[31680],[34,494487,31968],{"className":494488,"style":32153},[31685,31686],[34,494490,2557],{"className":494491},[31690],[34,494493,32936],{"className":494494},[31685,31686],[34,494496],{"className":494497,"style":31846},[31845],[34,494499,202],{"className":494500},[31850],[34,494502],{"className":494503,"style":31846},[31845],[34,494505,494507,494510,494513,494516,494519,494522],{"className":494506},[31676],[34,494508],{"className":494509,"style":37322},[31680],[34,494511,31955],{"className":494512},[31685,31686],[34,494514,32936],{"className":494515},[31685,31686],[34,494517],{"className":494518,"style":32604},[31845],[34,494520,223],{"className":494521},[32024],[34,494523],{"className":494524,"style":32604},[31845],[34,494526,494528,494531,494534],{"className":494527},[31676],[34,494529],{"className":494530,"style":33123},[31680],[34,494532,31968],{"className":494533,"style":32153},[31685,31686],[34,494535,32936],{"className":494536},[31685,31686],[108,494538,494539,494540,332541,494574,13977,494625,83913,494686,494714,494715,31901],{},"看第一项 ",[34,494541,494543,494559],{"className":494542},[31642],[34,494544,494546],{"className":494545},[31646],[31648,494547,494548],{"xmlns":31650},[31652,494549,494550,494556],{},[31655,494551,494552,494554],{},[31658,494553,31955],{},[31658,494555,32936],{},[31665,494557,494558],{"encoding":31667},"ix",[34,494560,494562],{"className":494561,"ariaHidden":758},[31672],[34,494563,494565,494568,494571],{"className":494564},[31676],[34,494566],{"className":494567,"style":33104},[31680],[34,494569,31955],{"className":494570},[31685,31686],[34,494572,32936],{"className":494573},[31685,31686],[34,494575,494577,494595],{"className":494576},[31642],[34,494578,494580],{"className":494579},[31646],[31648,494581,494582],{"xmlns":31650},[31652,494583,494584,494592],{},[31655,494585,494586,494588,494590],{},[31658,494587,32936],{},[31662,494589,112296],{},[31658,494591,223320],{},[31665,494593,494594],{"encoding":31667},"x \\in J",[34,494596,494598,494616],{"className":494597,"ariaHidden":758},[31672],[34,494599,494601,494604,494607,494610,494613],{"className":494600},[31676],[34,494602],{"className":494603,"style":34812},[31680],[34,494605,32936],{"className":494606},[31685,31686],[34,494608],{"className":494609,"style":31846},[31845],[34,494611,112296],{"className":494612},[31850],[34,494614],{"className":494615,"style":31846},[31845],[34,494617,494619,494622],{"className":494618},[31676],[34,494620],{"className":494621,"style":33218},[31680],[34,494623,223320],{"className":494624,"style":223361},[31685,31686],[34,494626,494628,494650],{"className":494627},[31642],[34,494629,494631],{"className":494630},[31646],[31648,494632,494633],{"xmlns":31650},[31652,494634,494635,494647],{},[31655,494636,494637,494639,494641,494643,494645],{},[31658,494638,31955],{},[31658,494640,32936],{},[31662,494642,112296],{},[31658,494644,106063],{},[31658,494646,223320],{},[31665,494648,494649],{"encoding":31667},"ix \\in IJ",[34,494651,494653,494674],{"className":494652,"ariaHidden":758},[31672],[34,494654,494656,494659,494662,494665,494668,494671],{"className":494655},[31676],[34,494657],{"className":494658,"style":451673},[31680],[34,494660,31955],{"className":494661},[31685,31686],[34,494663,32936],{"className":494664},[31685,31686],[34,494666],{"className":494667,"style":31846},[31845],[34,494669,112296],{"className":494670},[31850],[34,494672],{"className":494673,"style":31846},[31845],[34,494675,494677,494680,494683],{"className":494676},[31676],[34,494678],{"className":494679,"style":33218},[31680],[34,494681,106063],{"className":494682,"style":67482},[31685,31686],[34,494684,223320],{"className":494685,"style":223361},[31685,31686],[34,494687,494689,494702],{"className":494688},[31642],[34,494690,494692],{"className":494691},[31646],[31648,494693,494694],{"xmlns":31650},[31652,494695,494696,494700],{},[31655,494697,494698],{},[31658,494699,106063],{},[31665,494701,106063],{"encoding":31667},[34,494703,494705],{"className":494704,"ariaHidden":758},[31672],[34,494706,494708,494711],{"className":494707},[31676],[34,494709],{"className":494710,"style":33218},[31680],[34,494712,106063],{"className":494713,"style":67482},[31685,31686]," 乘 ",[34,494716,494718,494731],{"className":494717},[31642],[34,494719,494721],{"className":494720},[31646],[31648,494722,494723],{"xmlns":31650},[31652,494724,494725,494729],{},[31655,494726,494727],{},[31658,494728,223320],{},[31665,494730,223320],{"encoding":31667},[34,494732,494734],{"className":494733,"ariaHidden":758},[31672],[34,494735,494737,494740],{"className":494736},[31676],[34,494738],{"className":494739,"style":33218},[31680],[34,494741,223320],{"className":494742,"style":223361},[31685,31686],[108,494744,494745,494746,332541,494780,13977,494830,347],{},"看第二项 ",[34,494747,494749,494765],{"className":494748},[31642],[34,494750,494752],{"className":494751},[31646],[31648,494753,494754],{"xmlns":31650},[31652,494755,494756,494762],{},[31655,494757,494758,494760],{},[31658,494759,31968],{},[31658,494761,32936],{},[31665,494763,494764],{"encoding":31667},"jx",[34,494766,494768],{"className":494767,"ariaHidden":758},[31672],[34,494769,494771,494774,494777],{"className":494770},[31676],[34,494772],{"className":494773,"style":33123},[31680],[34,494775,31968],{"className":494776,"style":32153},[31685,31686],[34,494778,32936],{"className":494779},[31685,31686],[34,494781,494783,494800],{"className":494782},[31642],[34,494784,494786],{"className":494785},[31646],[31648,494787,494788],{"xmlns":31650},[31652,494789,494790,494798],{},[31655,494791,494792,494794,494796],{},[31658,494793,32936],{},[31662,494795,112296],{},[31658,494797,106063],{},[31665,494799,352414],{"encoding":31667},[34,494801,494803,494821],{"className":494802,"ariaHidden":758},[31672],[34,494804,494806,494809,494812,494815,494818],{"className":494805},[31676],[34,494807],{"className":494808,"style":34812},[31680],[34,494810,32936],{"className":494811},[31685,31686],[34,494813],{"className":494814,"style":31846},[31845],[34,494816,112296],{"className":494817},[31850],[34,494819],{"className":494820,"style":31846},[31845],[34,494822,494824,494827],{"className":494823},[31676],[34,494825],{"className":494826,"style":33218},[31680],[34,494828,106063],{"className":494829,"style":67482},[31685,31686],[34,494831,494833,494861],{"className":494832},[31642],[34,494834,494836],{"className":494835},[31646],[31648,494837,494838],{"xmlns":31650},[31652,494839,494840,494858],{},[31655,494841,494842,494844,494846,494848,494850,494852,494854,494856],{},[31658,494843,31968],{},[31658,494845,32936],{},[31662,494847,112296],{},[31658,494849,223320],{},[31658,494851,106063],{},[31662,494853,202],{},[31658,494855,106063],{},[31658,494857,223320],{},[31665,494859,494860],{"encoding":31667},"jx \\in JI = IJ",[34,494862,494864,494885,494906],{"className":494863,"ariaHidden":758},[31672],[34,494865,494867,494870,494873,494876,494879,494882],{"className":494866},[31676],[34,494868],{"className":494869,"style":33123},[31680],[34,494871,31968],{"className":494872,"style":32153},[31685,31686],[34,494874,32936],{"className":494875},[31685,31686],[34,494877],{"className":494878,"style":31846},[31845],[34,494880,112296],{"className":494881},[31850],[34,494883],{"className":494884,"style":31846},[31845],[34,494886,494888,494891,494894,494897,494900,494903],{"className":494887},[31676],[34,494889],{"className":494890,"style":33218},[31680],[34,494892,223320],{"className":494893,"style":223361},[31685,31686],[34,494895,106063],{"className":494896,"style":67482},[31685,31686],[34,494898],{"className":494899,"style":31846},[31845],[34,494901,202],{"className":494902},[31850],[34,494904],{"className":494905,"style":31846},[31845],[34,494907,494909,494912,494915],{"className":494908},[31676],[34,494910],{"className":494911,"style":33218},[31680],[34,494913,106063],{"className":494914,"style":67482},[31685,31686],[34,494916,223320],{"className":494917,"style":223361},[31685,31686],[108,494919,175992,494920,465789,494976,347],{},[34,494921,494923,494943],{"className":494922},[31642],[34,494924,494926],{"className":494925},[31646],[31648,494927,494928],{"xmlns":31650},[31652,494929,494930,494940],{},[31655,494931,494932,494934,494936,494938],{},[31658,494933,32936],{},[31662,494935,112296],{},[31658,494937,106063],{},[31658,494939,223320],{},[31665,494941,494942],{"encoding":31667},"x \\in IJ",[34,494944,494946,494964],{"className":494945,"ariaHidden":758},[31672],[34,494947,494949,494952,494955,494958,494961],{"className":494948},[31676],[34,494950],{"className":494951,"style":34812},[31680],[34,494953,32936],{"className":494954},[31685,31686],[34,494956],{"className":494957,"style":31846},[31845],[34,494959,112296],{"className":494960},[31850],[34,494962],{"className":494963,"style":31846},[31845],[34,494965,494967,494970,494973],{"className":494966},[31676],[34,494968],{"className":494969,"style":33218},[31680],[34,494971,106063],{"className":494972,"style":67482},[31685,31686],[34,494974,223320],{"className":494975,"style":223361},[31685,31686],[34,494977,494979,495002],{"className":494978},[31642],[34,494980,494982],{"className":494981},[31646],[31648,494983,494984],{"xmlns":31650},[31652,494985,494986,495000],{},[31655,494987,494988,494990,494992,494994,494996,494998],{},[31658,494989,106063],{},[31662,494991,326344],{},[31658,494993,223320],{},[31662,494995,131855],{},[31658,494997,106063],{},[31658,494999,223320],{},[31665,495001,457879],{"encoding":31667},[34,495003,495005,495023,495041],{"className":495004,"ariaHidden":758},[31672],[34,495006,495008,495011,495014,495017,495020],{"className":495007},[31676],[34,495009],{"className":495010,"style":33218},[31680],[34,495012,106063],{"className":495013,"style":67482},[31685,31686],[34,495015],{"className":495016,"style":32604},[31845],[34,495018,326344],{"className":495019},[32024],[34,495021],{"className":495022,"style":32604},[31845],[34,495024,495026,495029,495032,495035,495038],{"className":495025},[31676],[34,495027],{"className":495028,"style":195047},[31680],[34,495030,223320],{"className":495031,"style":223361},[31685,31686],[34,495033],{"className":495034,"style":31846},[31845],[34,495036,131855],{"className":495037},[31850],[34,495039],{"className":495040,"style":31846},[31845],[34,495042,495044,495047,495050],{"className":495043},[31676],[34,495045],{"className":495046,"style":33218},[31680],[34,495048,106063],{"className":495049,"style":67482},[31685,31686],[34,495051,223320],{"className":495052,"style":223361},[31685,31686],[108,495054,495055,495056,347],{},"综上，",[34,495057,495059,495082],{"className":495058},[31642],[34,495060,495062],{"className":495061},[31646],[31648,495063,495064],{"xmlns":31650},[31652,495065,495066,495080],{},[31655,495067,495068,495070,495072,495074,495076,495078],{},[31658,495069,106063],{},[31658,495071,223320],{},[31662,495073,202],{},[31658,495075,106063],{},[31662,495077,326344],{},[31658,495079,223320],{},[31665,495081,457958],{"encoding":31667},[34,495083,495085,495106,495124],{"className":495084,"ariaHidden":758},[31672],[34,495086,495088,495091,495094,495097,495100,495103],{"className":495087},[31676],[34,495089],{"className":495090,"style":33218},[31680],[34,495092,106063],{"className":495093,"style":67482},[31685,31686],[34,495095,223320],{"className":495096,"style":223361},[31685,31686],[34,495098],{"className":495099,"style":31846},[31845],[34,495101,202],{"className":495102},[31850],[34,495104],{"className":495105,"style":31846},[31845],[34,495107,495109,495112,495115,495118,495121],{"className":495108},[31676],[34,495110],{"className":495111,"style":33218},[31680],[34,495113,106063],{"className":495114,"style":67482},[31685,31686],[34,495116],{"className":495117,"style":32604},[31845],[34,495119,326344],{"className":495120},[32024],[34,495122],{"className":495123,"style":32604},[31845],[34,495125,495127,495130],{"className":495126},[31676],[34,495128],{"className":495129,"style":33218},[31680],[34,495131,223320],{"className":495132,"style":223361},[31685,31686],[108,495134,495135,495138,495139,86674,495190,495241,495242,495315,495316,355408,495394,347],{},[2627,495136,495137],{},"回到题目","：\n已知 ",[34,495140,495142,495160],{"className":495141},[31642],[34,495143,495145],{"className":495144},[31646],[31648,495146,495147],{"xmlns":31650},[31652,495148,495149,495157],{},[31655,495150,495151,495153,495155],{},[31658,495152,38691],{},[31662,495154,131855],{},[31658,495156,106063],{},[31665,495158,495159],{"encoding":31667},"K \\subseteq I",[34,495161,495163,495181],{"className":495162,"ariaHidden":758},[31672],[34,495164,495166,495169,495172,495175,495178],{"className":495165},[31676],[34,495167],{"className":495168,"style":195047},[31680],[34,495170,38691],{"className":495171,"style":38706},[31685,31686],[34,495173],{"className":495174,"style":31846},[31845],[34,495176,131855],{"className":495177},[31850],[34,495179],{"className":495180,"style":31846},[31845],[34,495182,495184,495187],{"className":495183},[31676],[34,495185],{"className":495186,"style":33218},[31680],[34,495188,106063],{"className":495189,"style":67482},[31685,31686],[34,495191,495193,495211],{"className":495192},[31642],[34,495194,495196],{"className":495195},[31646],[31648,495197,495198],{"xmlns":31650},[31652,495199,495200,495208],{},[31655,495201,495202,495204,495206],{},[31658,495203,38691],{},[31662,495205,131855],{},[31658,495207,223320],{},[31665,495209,495210],{"encoding":31667},"K \\subseteq J",[34,495212,495214,495232],{"className":495213,"ariaHidden":758},[31672],[34,495215,495217,495220,495223,495226,495229],{"className":495216},[31676],[34,495218],{"className":495219,"style":195047},[31680],[34,495221,38691],{"className":495222,"style":38706},[31685,31686],[34,495224],{"className":495225,"style":31846},[31845],[34,495227,131855],{"className":495228},[31850],[34,495230],{"className":495231,"style":31846},[31845],[34,495233,495235,495238],{"className":495234},[31676],[34,495236],{"className":495237,"style":33218},[31680],[34,495239,223320],{"className":495240,"style":223361},[31685,31686],"。\n根据集合论知识，这蕴含 ",[34,495243,495245,495267],{"className":495244},[31642],[34,495246,495248],{"className":495247},[31646],[31648,495249,495250],{"xmlns":31650},[31652,495251,495252,495264],{},[31655,495253,495254,495256,495258,495260,495262],{},[31658,495255,38691],{},[31662,495257,131855],{},[31658,495259,106063],{},[31662,495261,326344],{},[31658,495263,223320],{},[31665,495265,495266],{"encoding":31667},"K \\subseteq I \\cap J",[34,495268,495270,495288,495306],{"className":495269,"ariaHidden":758},[31672],[34,495271,495273,495276,495279,495282,495285],{"className":495272},[31676],[34,495274],{"className":495275,"style":195047},[31680],[34,495277,38691],{"className":495278,"style":38706},[31685,31686],[34,495280],{"className":495281,"style":31846},[31845],[34,495283,131855],{"className":495284},[31850],[34,495286],{"className":495287,"style":31846},[31845],[34,495289,495291,495294,495297,495300,495303],{"className":495290},[31676],[34,495292],{"className":495293,"style":33218},[31680],[34,495295,106063],{"className":495296,"style":67482},[31685,31686],[34,495298],{"className":495299,"style":32604},[31845],[34,495301,326344],{"className":495302},[32024],[34,495304],{"className":495305,"style":32604},[31845],[34,495307,495309,495312],{"className":495308},[31676],[34,495310],{"className":495311,"style":33218},[31680],[34,495313,223320],{"className":495314,"style":223361},[31685,31686],"。\n由第 1 步的结论，",[34,495317,495319,495343],{"className":495318},[31642],[34,495320,495322],{"className":495321},[31646],[31648,495323,495324],{"xmlns":31650},[31652,495325,495326,495340],{},[31655,495327,495328,495330,495332,495334,495336,495338],{},[31658,495329,106063],{},[31662,495331,326344],{},[31658,495333,223320],{},[31662,495335,202],{},[31658,495337,106063],{},[31658,495339,223320],{},[31665,495341,495342],{"encoding":31667},"I \\cap J = IJ",[34,495344,495346,495364,495382],{"className":495345,"ariaHidden":758},[31672],[34,495347,495349,495352,495355,495358,495361],{"className":495348},[31676],[34,495350],{"className":495351,"style":33218},[31680],[34,495353,106063],{"className":495354,"style":67482},[31685,31686],[34,495356],{"className":495357,"style":32604},[31845],[34,495359,326344],{"className":495360},[32024],[34,495362],{"className":495363,"style":32604},[31845],[34,495365,495367,495370,495373,495376,495379],{"className":495366},[31676],[34,495368],{"className":495369,"style":33218},[31680],[34,495371,223320],{"className":495372,"style":223361},[31685,31686],[34,495374],{"className":495375,"style":31846},[31845],[34,495377,202],{"className":495378},[31850],[34,495380],{"className":495381,"style":31846},[31845],[34,495383,495385,495388,495391],{"className":495384},[31676],[34,495386],{"className":495387,"style":33218},[31680],[34,495389,106063],{"className":495390,"style":67482},[31685,31686],[34,495392,223320],{"className":495393,"style":223361},[31685,31686],[34,495395,495397,495417],{"className":495396},[31642],[34,495398,495400],{"className":495399},[31646],[31648,495401,495402],{"xmlns":31650},[31652,495403,495404,495414],{},[31655,495405,495406,495408,495410,495412],{},[31658,495407,38691],{},[31662,495409,131855],{},[31658,495411,106063],{},[31658,495413,223320],{},[31665,495415,495416],{"encoding":31667},"K \\subseteq IJ",[34,495418,495420,495438],{"className":495419,"ariaHidden":758},[31672],[34,495421,495423,495426,495429,495432,495435],{"className":495422},[31676],[34,495424],{"className":495425,"style":195047},[31680],[34,495427,38691],{"className":495428,"style":38706},[31685,31686],[34,495430],{"className":495431,"style":31846},[31845],[34,495433,131855],{"className":495434},[31850],[34,495436],{"className":495437,"style":31846},[31845],[34,495439,495441,495444,495447],{"className":495440},[31676],[34,495442],{"className":495443,"style":33218},[31680],[34,495445,106063],{"className":495446,"style":67482},[31685,31686],[34,495448,223320],{"className":495449,"style":223361},[31685,31686],[15,495451,495452,2664,495454,311257,495509],{},[2627,495453,114043],{},[34,495455,495457,495476],{"className":495456},[31642],[34,495458,495460],{"className":495459},[31646],[31648,495461,495462],{"xmlns":31650},[31652,495463,495464,495474],{},[31655,495465,495466,495468,495470,495472],{},[31658,495467,106063],{},[31658,495469,223320],{},[31662,495471,346833],{},[31658,495473,38691],{},[31665,495475,493451],{"encoding":31667},[34,495477,495479,495500],{"className":495478,"ariaHidden":758},[31672],[34,495480,495482,495485,495488,495491,495494,495497],{"className":495481},[31676],[34,495483],{"className":495484,"style":195047},[31680],[34,495486,106063],{"className":495487,"style":67482},[31685,31686],[34,495489,223320],{"className":495490,"style":223361},[31685,31686],[34,495492],{"className":495493,"style":31846},[31845],[34,495495,346833],{"className":495496},[31850],[34,495498],{"className":495499,"style":31846},[31845],[34,495501,495503,495506],{"className":495502},[31676],[34,495504],{"className":495505,"style":33218},[31680],[34,495507,38691],{"className":495508,"style":38706},[31685,31686],[34,495510,495512,495525],{"className":495511},[31642],[34,495513,495515],{"className":495514},[31646],[31648,495516,495517],{"xmlns":31650},[31652,495518,495519,495523],{},[31655,495520,495521],{},[31658,495522,445204],{"mathvariant":38079},[31665,495524,445207],{"encoding":31667},[34,495526,495528],{"className":495527,"ariaHidden":758},[31672],[34,495529,495531,495534],{"className":495530},[31676],[34,495532],{"className":495533,"style":445217},[31680],[34,495535,445204],{"className":495536},[31685,129261],{"title":29,"searchDepth":43,"depth":43,"links":495538},[495539,495540,495541],{"id":463175,"depth":49,"text":463176},{"id":479324,"depth":49,"text":479325},{"id":491546,"depth":49,"text":491547},{"order":74},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringhomomorphismhw",{"title":463165,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002FringhomomorphismHW","BgUKR34MpllSrj98G8pDpu4jbpbrK5RUAfOBbSqxEMo",{"id":495548,"title":495549,"body":495550,"description":29,"extension":85,"meta":512606,"navigation":52,"path":512607,"seo":512608,"stem":512609,"__hash__":512610},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fsubringideal.md","子环、理想与商环",{"type":8,"value":495551,"toc":512580},[495552,495555,495559,495562,495628,495630,495634,495638,495927,496187,496189,496193,496197,496200,496259,496794,496892,496896,496899,497036,497205,500617,500622,501615,501617,501621,501625,501953,502228,502232,502235,502240,503244,503742,503809,504284,504320,504487,504931,505101,505103,505107,505179,505234,505291,506456,506511,506754,508558,508595,509063,510022,510024,510028,510032,510070,510557,511773,511777,511848],[11,495553,495549],{"id":495554},"子环理想与商环",[19,495556,495558],{"id":495557},"_1-课程总览群论与环论的类比","1. 课程总览：群论与环论的类比",[15,495560,495561],{},"环论的学习可以类比群论。通过对比，我们可以更直观地理解新概念。",[13837,495563,495564,495576],{},[13840,495565,495566],{},[13843,495567,495568,495571,495574],{},[13846,495569,495570],{"align":82355},"群论概念 (Group Theory)",[13846,495572,495573],{"align":82355},"环论对应概念 (Ring Theory)",[13846,495575,367195],{"align":82355},[13853,495577,495578,495594,495611],{},[13843,495579,495580,495586,495591],{},[13858,495581,495582,495585],{"align":82355},[2627,495583,495584],{},"子群"," (Subgroup)",[13858,495587,495588,495590],{"align":82355},[2627,495589,387569],{}," (Subring)",[13858,495592,495593],{"align":82355},"性质相似",[13843,495595,495596,495602,495608],{},[13858,495597,495598,495601],{"align":82355},[2627,495599,495600],{},"正规子群"," (Normal Subgroup)",[13858,495603,495604,495607],{"align":82355},[2627,495605,495606],{},"理想"," (Ideal)",[13858,495609,495610],{"align":82355},"都在构建商结构时起关键作用",[13843,495612,495613,495619,495625],{},[13858,495614,495615,495618],{"align":82355},[2627,495616,495617],{},"商群"," (Quotient Group)",[13858,495620,495621,495624],{"align":82355},[2627,495622,495623],{},"商环"," (Quotient Ring)",[13858,495626,495627],{"align":82355},"由理想对应的陪集构成的环",[32188,495629],{},[19,495631,495633],{"id":495632},"_2-子环-subring","2. 子环 (Subring)",[253,495635,495637],{"id":495636},"_21-定义与基本性质","2.1 定义与基本性质",[15,495639,133910,495640,495704,495705,495756,495757,314945,495785,495813,495814,119123,495816,495818,495819,11008,495847,495875,495876,347],{},[34,495641,495643,495668],{"className":495642},[31642],[34,495644,495646],{"className":495645},[31646],[31648,495647,495648],{"xmlns":31650},[31652,495649,495650,495666],{},[31655,495651,495652,495654,495656,495658,495660,495662,495664],{},[31662,495653,456],{"stretchy":702},[31658,495655,34386],{},[31662,495657,31960],{"separator":758},[31662,495659,223],{},[31662,495661,31960],{"separator":758},[31662,495663,71191],{},[31662,495665,2557],{"stretchy":702},[31665,495667,365556],{"encoding":31667},[34,495669,495671],{"className":495670,"ariaHidden":758},[31672],[34,495672,495674,495677,495680,495683,495686,495689,495692,495695,495698,495701],{"className":495673},[31676],[34,495675],{"className":495676,"style":31774},[31680],[34,495678,456],{"className":495679},[31782],[34,495681,34386],{"className":495682,"style":34427},[31685,31686],[34,495684,31960],{"className":495685},[32119],[34,495687],{"className":495688,"style":32123},[31845],[34,495690,223],{"className":495691},[31685],[34,495693,31960],{"className":495694},[32119],[34,495696],{"className":495697,"style":32123},[31845],[34,495699,71191],{"className":495700},[31685],[34,495702,2557],{"className":495703},[31690]," 是一个环，",[34,495706,495708,495726],{"className":495707},[31642],[34,495709,495711],{"className":495710},[31646],[31648,495712,495713],{"xmlns":31650},[31652,495714,495715,495723],{},[31655,495716,495717,495719,495721],{},[31658,495718,94675],{},[31662,495720,131855],{},[31658,495722,34386],{},[31665,495724,495725],{"encoding":31667},"S \\subseteq R",[34,495727,495729,495747],{"className":495728,"ariaHidden":758},[31672],[34,495730,495732,495735,495738,495741,495744],{"className":495731},[31676],[34,495733],{"className":495734,"style":195047},[31680],[34,495736,94675],{"className":495737,"style":106455},[31685,31686],[34,495739],{"className":495740,"style":31846},[31845],[34,495742,131855],{"className":495743},[31850],[34,495745],{"className":495746,"style":31846},[31845],[34,495748,495750,495753],{"className":495749},[31676],[34,495751],{"className":495752,"style":33218},[31680],[34,495754,34386],{"className":495755,"style":34427},[31685,31686]," 是其子集。\n如果 ",[34,495758,495760,495773],{"className":495759},[31642],[34,495761,495763],{"className":495762},[31646],[31648,495764,495765],{"xmlns":31650},[31652,495766,495767,495771],{},[31655,495768,495769],{},[31658,495770,94675],{},[31665,495772,94675],{"encoding":31667},[34,495774,495776],{"className":495775,"ariaHidden":758},[31672],[34,495777,495779,495782],{"className":495778},[31676],[34,495780],{"className":495781,"style":33218},[31680],[34,495783,94675],{"className":495784,"style":106455},[31685,31686],[34,495786,495788,495801],{"className":495787},[31642],[34,495789,495791],{"className":495790},[31646],[31648,495792,495793],{"xmlns":31650},[31652,495794,495795,495799],{},[31655,495796,495797],{},[31658,495798,34386],{},[31665,495800,34386],{"encoding":31667},[34,495802,495804],{"className":495803,"ariaHidden":758},[31672],[34,495805,495807,495810],{"className":495806},[31676],[34,495808],{"className":495809,"style":33218},[31680],[34,495811,34386],{"className":495812,"style":34427},[31685,31686]," 原有的",[2627,495815,427255],{},[2627,495817,427258],{},"运算下依然构成一个环，则称 ",[34,495820,495822,495835],{"className":495821},[31642],[34,495823,495825],{"className":495824},[31646],[31648,495826,495827],{"xmlns":31650},[31652,495828,495829,495833],{},[31655,495830,495831],{},[31658,495832,94675],{},[31665,495834,94675],{"encoding":31667},[34,495836,495838],{"className":495837,"ariaHidden":758},[31672],[34,495839,495841,495844],{"className":495840},[31676],[34,495842],{"className":495843,"style":33218},[31680],[34,495845,94675],{"className":495846,"style":106455},[31685,31686],[34,495848,495850,495863],{"className":495849},[31642],[34,495851,495853],{"className":495852},[31646],[31648,495854,495855],{"xmlns":31650},[31652,495856,495857,495861],{},[31655,495858,495859],{},[31658,495860,34386],{},[31665,495862,34386],{"encoding":31667},[34,495864,495866],{"className":495865,"ariaHidden":758},[31672],[34,495867,495869,495872],{"className":495868},[31676],[34,495870],{"className":495871,"style":33218},[31680],[34,495873,34386],{"className":495874,"style":34427},[31685,31686]," 的子环，记为 ",[34,495877,495879,495897],{"className":495878},[31642],[34,495880,495882],{"className":495881},[31646],[31648,495883,495884],{"xmlns":31650},[31652,495885,495886,495894],{},[31655,495887,495888,495890,495892],{},[31658,495889,94675],{},[31662,495891,45972],{},[31658,495893,34386],{},[31665,495895,495896],{"encoding":31667},"S \\le R",[34,495898,495900,495918],{"className":495899,"ariaHidden":758},[31672],[34,495901,495903,495906,495909,495912,495915],{"className":495902},[31676],[34,495904],{"className":495905,"style":195047},[31680],[34,495907,94675],{"className":495908,"style":106455},[31685,31686],[34,495910],{"className":495911,"style":31846},[31845],[34,495913,45972],{"className":495914},[31850],[34,495916],{"className":495917,"style":31846},[31845],[34,495919,495921,495924],{"className":495920},[31676],[34,495922],{"className":495923,"style":33218},[31680],[34,495925,34386],{"className":495926,"style":34427},[31685,31686],[105,495928,495929,495991],{},[108,495930,495931,46324,495933,495961,495962,495990],{},[2627,495932,286182],{},[34,495934,495936,495949],{"className":495935},[31642],[34,495937,495939],{"className":495938},[31646],[31648,495940,495941],{"xmlns":31650},[31652,495942,495943,495947],{},[31655,495944,495945],{},[31658,495946,34386],{},[31665,495948,34386],{"encoding":31667},[34,495950,495952],{"className":495951,"ariaHidden":758},[31672],[34,495953,495955,495958],{"className":495954},[31676],[34,495956],{"className":495957,"style":33218},[31680],[34,495959,34386],{"className":495960,"style":34427},[31685,31686]," 是交换环，则子环 ",[34,495963,495965,495978],{"className":495964},[31642],[34,495966,495968],{"className":495967},[31646],[31648,495969,495970],{"xmlns":31650},[31652,495971,495972,495976],{},[31655,495973,495974],{},[31658,495975,94675],{},[31665,495977,94675],{"encoding":31667},[34,495979,495981],{"className":495980,"ariaHidden":758},[31672],[34,495982,495984,495987],{"className":495983},[31676],[34,495985],{"className":495986,"style":33218},[31680],[34,495988,94675],{"className":495989,"style":106455},[31685,31686]," 也是交换环。",[108,495992,495993,33183,495996],{},[2627,495994,495995],{},"注意（与群论的区别）",[105,495997,495998,496001,496006],{},[108,495999,496000],{},"群论中，子群的单位元一定等于大群的单位元。",[108,496002,496003],{},[2627,496004,496005],{},"环论中，子环的单位元可能与大环不同，甚至可能没有单位元。",[108,496007,496008,2664,496011,11008,496045,496073,496074,11008,496135,496186],{},[71807,496009,496010],{},"例",[34,496012,496014,496030],{"className":496013},[31642],[34,496015,496017],{"className":496016},[31646],[31648,496018,496019],{"xmlns":31650},[31652,496020,496021,496027],{},[31655,496022,496023,496025],{},[31749,496024,768],{},[31658,496026,97300],{"mathvariant":318416},[31665,496028,496029],{"encoding":31667},"2\\mathbb{Z}",[34,496031,496033],{"className":496032,"ariaHidden":758},[31672],[34,496034,496036,496039,496042],{"className":496035},[31676],[34,496037],{"className":496038,"style":320927},[31680],[34,496040,768],{"className":496041},[31685],[34,496043,97300],{"className":496044},[31685,318441],[34,496046,496048,496061],{"className":496047},[31642],[34,496049,496051],{"className":496050},[31646],[31648,496052,496053],{"xmlns":31650},[31652,496054,496055,496059],{},[31655,496056,496057],{},[31658,496058,97300],{"mathvariant":318416},[31665,496060,320917],{"encoding":31667},[34,496062,496064],{"className":496063,"ariaHidden":758},[31672],[34,496065,496067,496070],{"className":496066},[31676],[34,496068],{"className":496069,"style":320927},[31680],[34,496071,97300],{"className":496072},[31685,318441]," 的子环，但无单位元；",[34,496075,496077,496099],{"className":496076},[31642],[34,496078,496080],{"className":496079},[31646],[31648,496081,496082],{"xmlns":31650},[31652,496083,496084,496096],{},[31655,496085,496086,496088,496090,496092,496094],{},[31658,496087,97300],{"mathvariant":318416},[31662,496089,378006],{},[31662,496091,32685],{"stretchy":702},[31749,496093,459],{},[31662,496095,32715],{"stretchy":702},[31665,496097,496098],{"encoding":31667},"\\mathbb{Z} \\oplus \\{0\\}",[34,496100,496102,496120],{"className":496101,"ariaHidden":758},[31672],[34,496103,496105,496108,496111,496114,496117],{"className":496104},[31676],[34,496106],{"className":496107,"style":379449},[31680],[34,496109,97300],{"className":496110},[31685,318441],[34,496112],{"className":496113,"style":32604},[31845],[34,496115,378006],{"className":496116},[32024],[34,496118],{"className":496119,"style":32604},[31845],[34,496121,496123,496126,496129,496132],{"className":496122},[31676],[34,496124],{"className":496125,"style":31774},[31680],[34,496127,32685],{"className":496128},[31782],[34,496130,459],{"className":496131},[31685],[34,496133,32715],{"className":496134},[31690],[34,496136,496138,496156],{"className":496137},[31642],[34,496139,496141],{"className":496140},[31646],[31648,496142,496143],{"xmlns":31650},[31652,496144,496145,496153],{},[31655,496146,496147,496149,496151],{},[31658,496148,97300],{"mathvariant":318416},[31662,496150,378006],{},[31658,496152,97300],{"mathvariant":318416},[31665,496154,496155],{"encoding":31667},"\\mathbb{Z} \\oplus \\mathbb{Z}",[34,496157,496159,496177],{"className":496158,"ariaHidden":758},[31672],[34,496160,496162,496165,496168,496171,496174],{"className":496161},[31676],[34,496163],{"className":496164,"style":379449},[31680],[34,496166,97300],{"className":496167},[31685,318441],[34,496169],{"className":496170,"style":32604},[31845],[34,496172,378006],{"className":496173},[32024],[34,496175],{"className":496176,"style":32604},[31845],[34,496178,496180,496183],{"className":496179},[31676],[34,496181],{"className":496182,"style":320927},[31680],[34,496184,97300],{"className":496185},[31685,318441]," 的子环，单位元不同。",[32188,496188],{},[19,496190,496192],{"id":496191},"_3-理想-ideal","3. 理想 (Ideal)",[253,496194,496196],{"id":496195},"_31-定义左理想右理想与双侧理想","3.1 定义：左理想、右理想与双侧理想",[15,496198,496199],{},"在非交换环中，乘法有左右之分，因此必须严格区分乘法的“吸收”方向。",[15,496201,133910,496202,330820,496230,496258],{},[34,496203,496205,496218],{"className":496204},[31642],[34,496206,496208],{"className":496207},[31646],[31648,496209,496210],{"xmlns":31650},[31652,496211,496212,496216],{},[31655,496213,496214],{},[31658,496215,106063],{},[31665,496217,106063],{"encoding":31667},[34,496219,496221],{"className":496220,"ariaHidden":758},[31672],[34,496222,496224,496227],{"className":496223},[31676],[34,496225],{"className":496226,"style":33218},[31680],[34,496228,106063],{"className":496229,"style":67482},[31685,31686],[34,496231,496233,496246],{"className":496232},[31642],[34,496234,496236],{"className":496235},[31646],[31648,496237,496238],{"xmlns":31650},[31652,496239,496240,496244],{},[31655,496241,496242],{},[31658,496243,34386],{},[31665,496245,34386],{"encoding":31667},[34,496247,496249],{"className":496248,"ariaHidden":758},[31672],[34,496250,496252,496255],{"className":496251},[31676],[34,496253],{"className":496254,"style":33218},[31680],[34,496256,34386],{"className":496257,"style":34427},[31685,31686]," 的一个子环（首先它是加法子群）：",[2230,496260,496261,496472,496677],{},[108,496262,496263,496266,496267,352264,496353,496409,496410,496438,496439,496442,496443,496471],{},[2627,496264,496265],{},"左理想 (Left Ideal)","：\n对任意 ",[34,496268,496270,496296],{"className":496269},[31642],[34,496271,496273],{"className":496272},[31646],[31648,496274,496275],{"xmlns":31650},[31652,496276,496277,496293],{},[31655,496278,496279,496281,496283,496285,496287,496289,496291],{},[31658,496280,34395],{},[31662,496282,112296],{},[31658,496284,34386],{},[31662,496286,31960],{"separator":758},[31658,496288,32936],{},[31662,496290,112296],{},[31658,496292,106063],{},[31665,496294,496295],{"encoding":31667},"r \\in R, x \\in I",[34,496297,496299,496317,496344],{"className":496298,"ariaHidden":758},[31672],[34,496300,496302,496305,496308,496311,496314],{"className":496301},[31676],[34,496303],{"className":496304,"style":34812},[31680],[34,496306,34395],{"className":496307,"style":34452},[31685,31686],[34,496309],{"className":496310,"style":31846},[31845],[34,496312,112296],{"className":496313},[31850],[34,496315],{"className":496316,"style":31846},[31845],[34,496318,496320,496323,496326,496329,496332,496335,496338,496341],{"className":496319},[31676],[34,496321],{"className":496322,"style":70739},[31680],[34,496324,34386],{"className":496325,"style":34427},[31685,31686],[34,496327,31960],{"className":496328},[32119],[34,496330],{"className":496331,"style":32123},[31845],[34,496333,32936],{"className":496334},[31685,31686],[34,496336],{"className":496337,"style":31846},[31845],[34,496339,112296],{"className":496340},[31850],[34,496342],{"className":496343,"style":31846},[31845],[34,496345,496347,496350],{"className":496346},[31676],[34,496348],{"className":496349,"style":33218},[31680],[34,496351,106063],{"className":496352,"style":67482},[31685,31686],[34,496354,496356,496376],{"className":496355},[31642],[34,496357,496359],{"className":496358},[31646],[31648,496360,496361],{"xmlns":31650},[31652,496362,496363,496373],{},[31655,496364,496365,496367,496369,496371],{},[31658,496366,34395],{},[31658,496368,32936],{},[31662,496370,112296],{},[31658,496372,106063],{},[31665,496374,496375],{"encoding":31667},"rx \\in I",[34,496377,496379,496400],{"className":496378,"ariaHidden":758},[31672],[34,496380,496382,496385,496388,496391,496394,496397],{"className":496381},[31676],[34,496383],{"className":496384,"style":34812},[31680],[34,496386,34395],{"className":496387,"style":34452},[31685,31686],[34,496389,32936],{"className":496390},[31685,31686],[34,496392],{"className":496393,"style":31846},[31845],[34,496395,112296],{"className":496396},[31850],[34,496398],{"className":496399,"style":31846},[31845],[34,496401,496403,496406],{"className":496402},[31676],[34,496404],{"className":496405,"style":33218},[31680],[34,496407,106063],{"className":496408,"style":67482},[31685,31686],"。（",[34,496411,496413,496426],{"className":496412},[31642],[34,496414,496416],{"className":496415},[31646],[31648,496417,496418],{"xmlns":31650},[31652,496419,496420,496424],{},[31655,496421,496422],{},[31658,496423,34386],{},[31665,496425,34386],{"encoding":31667},[34,496427,496429],{"className":496428,"ariaHidden":758},[31672],[34,496430,496432,496435],{"className":496431},[31676],[34,496433],{"className":496434,"style":33218},[31680],[34,496436,34386],{"className":496437,"style":34427},[31685,31686]," 中的元素从",[2627,496440,496441],{},"左边","乘进去，结果仍在 ",[34,496444,496446,496459],{"className":496445},[31642],[34,496447,496449],{"className":496448},[31646],[31648,496450,496451],{"xmlns":31650},[31652,496452,496453,496457],{},[31655,496454,496455],{},[31658,496456,106063],{},[31665,496458,106063],{"encoding":31667},[34,496460,496462],{"className":496461,"ariaHidden":758},[31672],[34,496463,496465,496468],{"className":496464},[31676],[34,496466],{"className":496467,"style":33218},[31680],[34,496469,106063],{"className":496470,"style":67482},[31685,31686]," 中）。",[108,496473,496474,496266,496477,352264,496562,496409,496618,496438,496646,496442,496649,496471],{},[2627,496475,496476],{},"右理想 (Right Ideal)",[34,496478,496480,496505],{"className":496479},[31642],[34,496481,496483],{"className":496482},[31646],[31648,496484,496485],{"xmlns":31650},[31652,496486,496487,496503],{},[31655,496488,496489,496491,496493,496495,496497,496499,496501],{},[31658,496490,34395],{},[31662,496492,112296],{},[31658,496494,34386],{},[31662,496496,31960],{"separator":758},[31658,496498,32936],{},[31662,496500,112296],{},[31658,496502,106063],{},[31665,496504,496295],{"encoding":31667},[34,496506,496508,496526,496553],{"className":496507,"ariaHidden":758},[31672],[34,496509,496511,496514,496517,496520,496523],{"className":496510},[31676],[34,496512],{"className":496513,"style":34812},[31680],[34,496515,34395],{"className":496516,"style":34452},[31685,31686],[34,496518],{"className":496519,"style":31846},[31845],[34,496521,112296],{"className":496522},[31850],[34,496524],{"className":496525,"style":31846},[31845],[34,496527,496529,496532,496535,496538,496541,496544,496547,496550],{"className":496528},[31676],[34,496530],{"className":496531,"style":70739},[31680],[34,496533,34386],{"className":496534,"style":34427},[31685,31686],[34,496536,31960],{"className":496537},[32119],[34,496539],{"className":496540,"style":32123},[31845],[34,496542,32936],{"className":496543},[31685,31686],[34,496545],{"className":496546,"style":31846},[31845],[34,496548,112296],{"className":496549},[31850],[34,496551],{"className":496552,"style":31846},[31845],[34,496554,496556,496559],{"className":496555},[31676],[34,496557],{"className":496558,"style":33218},[31680],[34,496560,106063],{"className":496561,"style":67482},[31685,31686],[34,496563,496565,496585],{"className":496564},[31642],[34,496566,496568],{"className":496567},[31646],[31648,496569,496570],{"xmlns":31650},[31652,496571,496572,496582],{},[31655,496573,496574,496576,496578,496580],{},[31658,496575,32936],{},[31658,496577,34395],{},[31662,496579,112296],{},[31658,496581,106063],{},[31665,496583,496584],{"encoding":31667},"xr \\in I",[34,496586,496588,496609],{"className":496587,"ariaHidden":758},[31672],[34,496589,496591,496594,496597,496600,496603,496606],{"className":496590},[31676],[34,496592],{"className":496593,"style":34812},[31680],[34,496595,32936],{"className":496596},[31685,31686],[34,496598,34395],{"className":496599,"style":34452},[31685,31686],[34,496601],{"className":496602,"style":31846},[31845],[34,496604,112296],{"className":496605},[31850],[34,496607],{"className":496608,"style":31846},[31845],[34,496610,496612,496615],{"className":496611},[31676],[34,496613],{"className":496614,"style":33218},[31680],[34,496616,106063],{"className":496617,"style":67482},[31685,31686],[34,496619,496621,496634],{"className":496620},[31642],[34,496622,496624],{"className":496623},[31646],[31648,496625,496626],{"xmlns":31650},[31652,496627,496628,496632],{},[31655,496629,496630],{},[31658,496631,34386],{},[31665,496633,34386],{"encoding":31667},[34,496635,496637],{"className":496636,"ariaHidden":758},[31672],[34,496638,496640,496643],{"className":496639},[31676],[34,496641],{"className":496642,"style":33218},[31680],[34,496644,34386],{"className":496645,"style":34427},[31685,31686],[2627,496647,496648],{},"右边",[34,496650,496652,496665],{"className":496651},[31642],[34,496653,496655],{"className":496654},[31646],[31648,496656,496657],{"xmlns":31650},[31652,496658,496659,496663],{},[31655,496660,496661],{},[31658,496662,106063],{},[31665,496664,106063],{"encoding":31667},[34,496666,496668],{"className":496667,"ariaHidden":758},[31672],[34,496669,496671,496674],{"className":496670},[31676],[34,496672],{"className":496673,"style":33218},[31680],[34,496675,106063],{"className":496676,"style":67482},[31685,31686],[108,496678,496679,341347,496682,496710,496711,49276,496739,496741,496742,347],{},[2627,496680,496681],{},"理想 (Ideal \u002F Two-sided Ideal)",[34,496683,496685,496698],{"className":496684},[31642],[34,496686,496688],{"className":496687},[31646],[31648,496689,496690],{"xmlns":31650},[31652,496691,496692,496696],{},[31655,496693,496694],{},[31658,496695,106063],{},[31665,496697,106063],{"encoding":31667},[34,496699,496701],{"className":496700,"ariaHidden":758},[31672],[34,496702,496704,496707],{"className":496703},[31676],[34,496705],{"className":496706,"style":33218},[31680],[34,496708,106063],{"className":496709,"style":67482},[31685,31686]," 既是左理想又是右理想，称 ",[34,496712,496714,496727],{"className":496713},[31642],[34,496715,496717],{"className":496716},[31646],[31648,496718,496719],{"xmlns":31650},[31652,496720,496721,496725],{},[31655,496722,496723],{},[31658,496724,106063],{},[31665,496726,106063],{"encoding":31667},[34,496728,496730],{"className":496729,"ariaHidden":758},[31672],[34,496731,496733,496736],{"className":496732},[31676],[34,496734],{"className":496735,"style":33218},[31680],[34,496737,106063],{"className":496738,"style":67482},[31685,31686],[2627,496740,495606],{},"，记为 ",[34,496743,496745,496764],{"className":496744},[31642],[34,496746,496748],{"className":496747},[31646],[31648,496749,496750],{"xmlns":31650},[31652,496751,496752,496761],{},[31655,496753,496754,496756,496759],{},[31658,496755,106063],{},[31662,496757,496758],{},"◃",[31658,496760,34386],{},[31665,496762,496763],{"encoding":31667},"I \\triangleleft R",[34,496765,496767,496785],{"className":496766,"ariaHidden":758},[31672],[34,496768,496770,496773,496776,496779,496782],{"className":496769},[31676],[34,496771],{"className":496772,"style":33218},[31680],[34,496774,106063],{"className":496775,"style":67482},[31685,31686],[34,496777],{"className":496778,"style":32604},[31845],[34,496780,496758],{"className":496781},[32024],[34,496783],{"className":496784,"style":32604},[31845],[34,496786,496788,496791],{"className":496787},[31676],[34,496789],{"className":496790,"style":33218},[31680],[34,496792,34386],{"className":496793,"style":34427},[31685,31686],[104833,496795,496796],{},[15,496797,496798,47890,496801,496829,496830,496891],{},[2627,496799,496800],{},"注",[34,496802,496804,496817],{"className":496803},[31642],[34,496805,496807],{"className":496806},[31646],[31648,496808,496809],{"xmlns":31650},[31652,496810,496811,496815],{},[31655,496812,496813],{},[31658,496814,34386],{},[31665,496816,34386],{"encoding":31667},[34,496818,496820],{"className":496819,"ariaHidden":758},[31672],[34,496821,496823,496826],{"className":496822},[31676],[34,496824],{"className":496825,"style":33218},[31680],[34,496827,34386],{"className":496828,"style":34427},[31685,31686]," 是交换环，则 ",[34,496831,496833,496855],{"className":496832},[31642],[34,496834,496836],{"className":496835},[31646],[31648,496837,496838],{"xmlns":31650},[31652,496839,496840,496852],{},[31655,496841,496842,496844,496846,496848,496850],{},[31658,496843,34395],{},[31658,496845,32936],{},[31662,496847,202],{},[31658,496849,32936],{},[31658,496851,34395],{},[31665,496853,496854],{"encoding":31667},"rx = xr",[34,496856,496858,496879],{"className":496857,"ariaHidden":758},[31672],[34,496859,496861,496864,496867,496870,496873,496876],{"className":496860},[31676],[34,496862],{"className":496863,"style":32527},[31680],[34,496865,34395],{"className":496866,"style":34452},[31685,31686],[34,496868,32936],{"className":496869},[31685,31686],[34,496871],{"className":496872,"style":31846},[31845],[34,496874,202],{"className":496875},[31850],[34,496877],{"className":496878,"style":31846},[31845],[34,496880,496882,496885,496888],{"className":496881},[31676],[34,496883],{"className":496884,"style":32527},[31680],[34,496886,32936],{"className":496887},[31685,31686],[34,496889,34395],{"className":496890,"style":34452},[31685,31686],"，此时左理想、右理想、理想是等价概念。",[253,496893,496895],{"id":496894},"_32-补充案例分析例-321-详解","3.2 补充案例分析（例 3.2.1 详解）",[15,496897,496898],{},"这是一个经典的非交换环例子，老师课堂上略过了一部分，这里补充完整。",[15,496900,496901,496904,496905,496962,496963,258,497001,497029,497030,39984,497033],{},[2627,496902,496903],{},"例 3.2.1","：全矩阵环 ",[34,496906,496908,496932],{"className":496907},[31642],[34,496909,496911],{"className":496910},[31646],[31648,496912,496913],{"xmlns":31650},[31652,496914,496915,496929],{},[31655,496916,496917,496919,496921,496923,496925,496927],{},[31658,496918,98360],{},[31662,496920,456],{"stretchy":702},[31658,496922,31660],{},[31662,496924,31960],{"separator":758},[31658,496926,34386],{"mathvariant":318416},[31662,496928,2557],{"stretchy":702},[31665,496930,496931],{"encoding":31667},"M(n, \\mathbb{R})",[34,496933,496935],{"className":496934,"ariaHidden":758},[31672],[34,496936,496938,496941,496944,496947,496950,496953,496956,496959],{"className":496937},[31676],[34,496939],{"className":496940,"style":31774},[31680],[34,496942,98360],{"className":496943,"style":33389},[31685,31686],[34,496945,456],{"className":496946},[31782],[34,496948,31660],{"className":496949},[31685,31686],[34,496951,31960],{"className":496952},[32119],[34,496954],{"className":496955,"style":32123},[31845],[34,496957,34386],{"className":496958},[31685,318441],[34,496960,2557],{"className":496961},[31690]," 只有平凡理想（即 ",[34,496964,496966,496983],{"className":496965},[31642],[34,496967,496969],{"className":496968},[31646],[31648,496970,496971],{"xmlns":31650},[31652,496972,496973,496981],{},[31655,496974,496975,496977,496979],{},[31662,496976,32685],{"stretchy":702},[31749,496978,459],{},[31662,496980,32715],{"stretchy":702},[31665,496982,338218],{"encoding":31667},[34,496984,496986],{"className":496985,"ariaHidden":758},[31672],[34,496987,496989,496992,496995,496998],{"className":496988},[31676],[34,496990],{"className":496991,"style":31774},[31680],[34,496993,32685],{"className":496994},[31782],[34,496996,459],{"className":496997},[31685],[34,496999,32715],{"className":497000},[31690],[34,497002,497004,497017],{"className":497003},[31642],[34,497005,497007],{"className":497006},[31646],[31648,497008,497009],{"xmlns":31650},[31652,497010,497011,497015],{},[31655,497012,497013],{},[31658,497014,34386],{},[31665,497016,34386],{"encoding":31667},[34,497018,497020],{"className":497019,"ariaHidden":758},[31672],[34,497021,497023,497026],{"className":497022},[31676],[34,497024],{"className":497025,"style":33218},[31680],[34,497027,34386],{"className":497028,"style":34427},[31685,31686]," 本身）。这种只有平凡理想的环称为",[2627,497031,497032],{},"单环 (Simple Ring)",[71807,497034,497035],{},"但是，它拥有非平凡的左理想或右理想。",[15,497037,497038,362914,497041,11008,497069,497125,497126,347],{},[2627,497039,497040],{},"证明部分（单环性质）：",[34,497042,497044,497057],{"className":497043},[31642],[34,497045,497047],{"className":497046},[31646],[31648,497048,497049],{"xmlns":31650},[31652,497050,497051,497055],{},[31655,497052,497053],{},[31658,497054,106063],{},[31665,497056,106063],{"encoding":31667},[34,497058,497060],{"className":497059,"ariaHidden":758},[31672],[34,497061,497063,497066],{"className":497062},[31676],[34,497064],{"className":497065,"style":33218},[31680],[34,497067,106063],{"className":497068,"style":67482},[31685,31686],[34,497070,497072,497095],{"className":497071},[31642],[34,497073,497075],{"className":497074},[31646],[31648,497076,497077],{"xmlns":31650},[31652,497078,497079,497093],{},[31655,497080,497081,497083,497085,497087,497089,497091],{},[31658,497082,98360],{},[31662,497084,456],{"stretchy":702},[31658,497086,31660],{},[31662,497088,31960],{"separator":758},[31658,497090,34386],{"mathvariant":318416},[31662,497092,2557],{"stretchy":702},[31665,497094,496931],{"encoding":31667},[34,497096,497098],{"className":497097,"ariaHidden":758},[31672],[34,497099,497101,497104,497107,497110,497113,497116,497119,497122],{"className":497100},[31676],[34,497102],{"className":497103,"style":31774},[31680],[34,497105,98360],{"className":497106,"style":33389},[31685,31686],[34,497108,456],{"className":497109},[31782],[34,497111,31660],{"className":497112},[31685,31686],[34,497114,31960],{"className":497115},[32119],[34,497117],{"className":497118,"style":32123},[31845],[34,497120,34386],{"className":497121},[31685,318441],[34,497123,2557],{"className":497124},[31690]," 的一个非零理想。我们要证明 ",[34,497127,497129,497157],{"className":497128},[31642],[34,497130,497132],{"className":497131},[31646],[31648,497133,497134],{"xmlns":31650},[31652,497135,497136,497154],{},[31655,497137,497138,497140,497142,497144,497146,497148,497150,497152],{},[31658,497139,106063],{},[31662,497141,202],{},[31658,497143,98360],{},[31662,497145,456],{"stretchy":702},[31658,497147,31660],{},[31662,497149,31960],{"separator":758},[31658,497151,34386],{"mathvariant":318416},[31662,497153,2557],{"stretchy":702},[31665,497155,497156],{"encoding":31667},"I = M(n, \\mathbb{R})",[34,497158,497160,497178],{"className":497159,"ariaHidden":758},[31672],[34,497161,497163,497166,497169,497172,497175],{"className":497162},[31676],[34,497164],{"className":497165,"style":33218},[31680],[34,497167,106063],{"className":497168,"style":67482},[31685,31686],[34,497170],{"className":497171,"style":31846},[31845],[34,497173,202],{"className":497174},[31850],[34,497176],{"className":497177,"style":31846},[31845],[34,497179,497181,497184,497187,497190,497193,497196,497199,497202],{"className":497180},[31676],[34,497182],{"className":497183,"style":31774},[31680],[34,497185,98360],{"className":497186,"style":33389},[31685,31686],[34,497188,456],{"className":497189},[31782],[34,497191,31660],{"className":497192},[31685,31686],[34,497194,31960],{"className":497195},[32119],[34,497197],{"className":497198,"style":32123},[31845],[34,497200,34386],{"className":497201},[31685,318441],[34,497203,2557],{"className":497204},[31690],[2230,497206,497207,497465,497770,499008,499554,500243,500536],{},[108,497208,93310,497209,497303,497304,497355,497356,347],{},[34,497210,497212,497234],{"className":497211},[31642],[34,497213,497215],{"className":497214},[31646],[31648,497216,497217],{"xmlns":31650},[31652,497218,497219,497231],{},[31655,497220,497221,497223,497225,497227,497229],{},[31658,497222,106063],{},[31662,497224,54707],{"mathvariant":38079},[31662,497226,32685],{"stretchy":702},[31749,497228,459],{},[31662,497230,32715],{"stretchy":702},[31665,497232,497233],{"encoding":31667},"I \\neq \\{0\\}",[34,497235,497237,497288],{"className":497236,"ariaHidden":758},[31672],[34,497238,497240,497243,497246,497249,497285],{"className":497239},[31676],[34,497241],{"className":497242,"style":43217},[31680],[34,497244,106063],{"className":497245,"style":67482},[31685,31686],[34,497247],{"className":497248,"style":31846},[31845],[34,497250,497252,497279,497282],{"className":497251},[31850],[34,497253,497255],{"className":497254},[31850],[34,497256,497258],{"className":497257},[31685,54774],[34,497259,497261],{"className":497260},[54778],[34,497262,497264,497267,497276],{"className":497263},[54782],[34,497265],{"className":497266,"style":43217},[31680],[34,497268,497270],{"className":497269},[54789],[34,497271,497273],{"className":497272},[31685],[34,497274,54796],{"className":497275},[31850],[34,497277],{"className":497278},[54800],[34,497280],{"className":497281},[31845,54804],[34,497283,202],{"className":497284},[31850],[34,497286],{"className":497287,"style":31846},[31845],[34,497289,497291,497294,497297,497300],{"className":497290},[31676],[34,497292],{"className":497293,"style":31774},[31680],[34,497295,32685],{"className":497296},[31782],[34,497298,459],{"className":497299},[31685],[34,497301,32715],{"className":497302},[31690],"，所以存在非零矩阵 ",[34,497305,497307,497325],{"className":497306},[31642],[34,497308,497310],{"className":497309},[31646],[31648,497311,497312],{"xmlns":31650},[31652,497313,497314,497322],{},[31655,497315,497316,497318,497320],{},[31658,497317,54151],{},[31662,497319,112296],{},[31658,497321,106063],{},[31665,497323,497324],{"encoding":31667},"A \\in I",[34,497326,497328,497346],{"className":497327,"ariaHidden":758},[31672],[34,497329,497331,497334,497337,497340,497343],{"className":497330},[31676],[34,497332],{"className":497333,"style":104901},[31680],[34,497335,54151],{"className":497336},[31685,31686],[34,497338],{"className":497339,"style":31846},[31845],[34,497341,112296],{"className":497342},[31850],[34,497344],{"className":497345,"style":31846},[31845],[34,497347,497349,497352],{"className":497348},[31676],[34,497350],{"className":497351,"style":33218},[31680],[34,497353,106063],{"className":497354,"style":67482},[31685,31686],"。记 ",[34,497357,497359,497389],{"className":497358},[31642],[34,497360,497362],{"className":497361},[31646],[31648,497363,497364],{"xmlns":31650},[31652,497365,497366,497386],{},[31655,497367,497368,497370,497372,497374,497384],{},[31658,497369,54151],{},[31662,497371,202],{},[31662,497373,456],{"stretchy":702},[31743,497375,497376,497378],{},[31658,497377,111],{},[31655,497379,497380,497382],{},[31658,497381,31955],{},[31658,497383,31968],{},[31662,497385,2557],{"stretchy":702},[31665,497387,497388],{"encoding":31667},"A = (a_{ij})",[34,497390,497392,497410],{"className":497391,"ariaHidden":758},[31672],[34,497393,497395,497398,497401,497404,497407],{"className":497394},[31676],[34,497396],{"className":497397,"style":33218},[31680],[34,497399,54151],{"className":497400},[31685,31686],[34,497402],{"className":497403,"style":31846},[31845],[34,497405,202],{"className":497406},[31850],[34,497408],{"className":497409,"style":31846},[31845],[34,497411,497413,497416,497419,497462],{"className":497412},[31676],[34,497414],{"className":497415,"style":32058},[31680],[34,497417,456],{"className":497418},[31782],[34,497420,497422,497425],{"className":497421},[31685],[34,497423,111],{"className":497424},[31685,31686],[34,497426,497428],{"className":497427},[31792],[34,497429,497431,497454],{"className":497430},[31796,31797],[34,497432,497434,497451],{"className":497433},[31801],[34,497435,497437],{"className":497436,"style":32090},[31805],[34,497438,497439,497442],{"style":31809},[34,497440],{"className":497441,"style":31814},[31813],[34,497443,497445],{"className":497444},[31818,31819,31820,31821],[34,497446,497448],{"className":497447},[31685,31821],[34,497449,141027],{"className":497450,"style":32153},[31685,31686,31821],[34,497452,31829],{"className":497453},[31828],[34,497455,497457],{"className":497456},[31801],[34,497458,497460],{"className":497459,"style":32163},[31805],[34,497461],{},[34,497463,2557],{"className":497464},[31690],[108,497466,426251,497467,497551,497552,347],{},[34,497468,497470,497488],{"className":497469},[31642],[34,497471,497473],{"className":497472},[31646],[31648,497474,497475],{"xmlns":31650},[31652,497476,497477,497485],{},[31655,497478,497479,497481,497483],{},[31658,497480,54151],{},[31662,497482,54707],{"mathvariant":38079},[31749,497484,459],{},[31665,497486,497487],{"encoding":31667},"A \\neq 0",[34,497489,497491,497542],{"className":497490,"ariaHidden":758},[31672],[34,497492,497494,497497,497500,497503,497539],{"className":497493},[31676],[34,497495],{"className":497496,"style":43217},[31680],[34,497498,54151],{"className":497499},[31685,31686],[34,497501],{"className":497502,"style":31846},[31845],[34,497504,497506,497533,497536],{"className":497505},[31850],[34,497507,497509],{"className":497508},[31850],[34,497510,497512],{"className":497511},[31685,54774],[34,497513,497515],{"className":497514},[54778],[34,497516,497518,497521,497530],{"className":497517},[54782],[34,497519],{"className":497520,"style":43217},[31680],[34,497522,497524],{"className":497523},[54789],[34,497525,497527],{"className":497526},[31685],[34,497528,54796],{"className":497529},[31850],[34,497531],{"className":497532},[54800],[34,497534],{"className":497535},[31845,54804],[34,497537,202],{"className":497538},[31850],[34,497540],{"className":497541,"style":31846},[31845],[34,497543,497545,497548],{"className":497544},[31676],[34,497546],{"className":497547,"style":32617},[31680],[34,497549,459],{"className":497550},[31685],"，至少存在一个元素 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0",[34,497590,497592,497761],{"className":497591,"ariaHidden":758},[31672],[34,497593,497595,497598,497719,497722,497758],{"className":497594},[31676],[34,497596],{"className":497597,"style":141223},[31680],[34,497599,497601,497604],{"className":497600},[31685],[34,497602,111],{"className":497603},[31685,31686],[34,497605,497607],{"className":497606},[31792],[34,497608,497610,497711],{"className":497609},[31796,31797],[34,497611,497613,497708],{"className":497612},[31801],[34,497614,497616],{"className":497615,"style":32090},[31805],[34,497617,497618,497621],{"style":31809},[34,497619],{"className":497620,"style":31814},[31813],[34,497622,497624],{"className":497623},[31818,31819,31820,31821],[34,497625,497627,497667],{"className":497626},[31685,31821],[34,497628,497630,497633],{"className":497629},[31685,31821],[34,497631,31955],{"className":497632},[31685,31686,31821],[34,497634,497636],{"className":497635},[31792],[34,497637,497639,497659],{"className":497638},[31796,31797],[34,497640,497642,497656],{"className":497641},[31801],[34,497643,497645],{"className":497644,"style":318658},[31805],[34,497646,497647,497650],{"style":318661},[34,497648],{"className":497649,"style":43406},[31813],[34,497651,497653],{"className":497652},[31818,43410,43411,31821],[34,497654,459],{"className":497655},[31685,31821],[34,497657,31829],{"className":497658},[31828],[34,497660,497662],{"className":497661},[31801],[34,497663,497665],{"className":497664,"style":220540},[31805],[34,497666],{},[34,497668,497670,497673],{"className":497669},[31685,31821],[34,497671,31968],{"className":497672,"style":32153},[31685,31686,31821],[34,497674,497676],{"className":497675},[31792],[34,497677,497679,497700],{"className":497678},[31796,31797],[34,497680,497682,497697],{"className":497681},[31801],[34,497683,497685],{"className":497684,"style":318658},[31805],[34,497686,497688,497691],{"style":497687},"top:-2.357em;margin-left:-0.0572em;margin-right:0.0714em;",[34,497689],{"className":497690,"style":43406},[31813],[34,497692,497694],{"className":497693},[31818,43410,43411,31821],[34,497695,459],{"className":497696},[31685,31821],[34,497698,31829],{"className":497699},[31828],[34,497701,497703],{"className":497702},[31801],[34,497704,497706],{"className":497705,"style":220540},[31805],[34,497707],{},[34,497709,31829],{"className":497710},[31828],[34,497712,497714],{"className":497713},[31801],[34,497715,497717],{"className":497716,"style":32163},[31805],[34,497718],{},[34,497720],{"className":497721,"style":31846},[31845],[34,497723,497725,497752,497755],{"className":497724},[31850],[34,497726,497728],{"className":497727},[31850],[34,497729,497731],{"className":497730},[31685,54774],[34,497732,497734],{"className":497733},[54778],[34,497735,497737,497740,497749],{"className":497736},[54782],[34,497738],{"className":497739,"style":43217},[31680],[34,497741,497743],{"className":497742},[54789],[34,497744,497746],{"className":497745},[31685],[34,497747,54796],{"className":497748},[31850],[34,497750],{"className":497751},[54800],[34,497753],{"className":497754},[31845,54804],[34,497756,202],{"className":497757},[31850],[34,497759],{"className":497760,"style":31846},[31845],[34,497762,497764,497767],{"className":497763},[31676],[34,497765],{"className":497766,"style":32617},[31680],[34,497768,459],{"className":497769},[31685],[108,497771,497772,497773,497851,497852,497880,497881,497909,497910,498133,498134,88651,498162,498232,498233,498261,498262,498332,498333,498361,498362,498414,498415,221655,498577],{},"利用矩阵单位 ",[34,497774,497776,497798],{"className":497775},[31642],[34,497777,497779],{"className":497778},[31646],[31648,497780,497781],{"xmlns":31650},[31652,497782,497783,497795],{},[31655,497784,497785],{},[31743,497786,497787,497789],{},[31658,497788,96058],{},[31655,497790,497791,497793],{},[31658,497792,15],{},[31658,497794,38759],{},[31665,497796,497797],{"encoding":31667},"E_{pq}",[34,497799,497801],{"className":497800,"ariaHidden":758},[31672],[34,497802,497804,497807],{"className":497803},[31676],[34,497805],{"className":497806,"style":55484},[31680],[34,497808,497810,497813],{"className":497809},[31685],[34,497811,96058],{"className":497812,"style":106455},[31685,31686],[34,497814,497816],{"className":497815},[31792],[34,497817,497819,497843],{"className":497818},[31796,31797],[34,497820,497822,497840],{"className":497821},[31801],[34,497823,497825],{"className":497824,"style":32362},[31805],[34,497826,497827,497830],{"style":206422},[34,497828],{"className":497829,"style":31814},[31813],[34,497831,497833],{"className":497832},[31818,31819,31820,31821],[34,497834,497836],{"className":497835},[31685,31821],[34,497837,497839],{"className":497838,"style":32062},[31685,31686,31821],"pq",[34,497841,31829],{"className":497842},[31828],[34,497844,497846],{"className":497845},[31801],[34,497847,497849],{"className":497848,"style":32163},[31805],[34,497850],{},"（第 ",[34,497853,497855,497868],{"className":497854},[31642],[34,497856,497858],{"className":497857},[31646],[31648,497859,497860],{"xmlns":31650},[31652,497861,497862,497866],{},[31655,497863,497864],{},[31658,497865,15],{},[31665,497867,15],{"encoding":31667},[34,497869,497871],{"className":497870,"ariaHidden":758},[31672],[34,497872,497874,497877],{"className":497873},[31676],[34,497875],{"className":497876,"style":38774},[31680],[34,497878,15],{"className":497879},[31685,31686]," 行第 ",[34,497882,497884,497897],{"className":497883},[31642],[34,497885,497887],{"className":497886},[31646],[31648,497888,497889],{"xmlns":31650},[31652,497890,497891,497895],{},[31655,497892,497893],{},[31658,497894,38759],{},[31665,497896,38759],{"encoding":31667},[34,497898,497900],{"className":497899,"ariaHidden":758},[31672],[34,497901,497903,497906],{"className":497902},[31676],[34,497904],{"className":497905,"style":38774},[31680],[34,497907,38759],{"className":497908,"style":32062},[31685,31686]," 列为 1，其余为 0 的矩阵）。我们构造运算：\n",[34,497911,497913,497955],{"className":497912},[31642],[34,497914,497916],{"className":497915},[31646],[31648,497917,497918],{"xmlns":31650},[31652,497919,497920,497952],{},[31655,497921,497922,497936,497938],{},[31743,497923,497924,497926],{},[31658,497925,96058],{},[31655,497927,497928,497930],{},[31658,497929,31955],{},[31743,497931,497932,497934],{},[31658,497933,31955],{},[31749,497935,459],{},[31658,497937,54151],{},[31743,497939,497940,497942],{},[31658,497941,96058],{},[31655,497943,497944,497950],{},[31743,497945,497946,497948],{},[31658,497947,31968],{},[31749,497949,459],{},[31658,497951,31968],{},[31665,497953,497954],{"encoding":31667},"E_{i i_0} A E_{j_0 j}",[34,497956,497958],{"className":497957,"ariaHidden":758},[31672],[34,497959,497961,497964,498047,498050],{"className":497960},[31676],[34,497962],{"className":497963,"style":55484},[31680],[34,497965,497967,497970],{"className":497966},[31685],[34,497968,96058],{"className":497969,"style":106455},[31685,31686],[34,497971,497973],{"className":497972},[31792],[34,497974,497976,498039],{"className":497975},[31796,31797],[34,497977,497979,498036],{"className":497978},[31801],[34,497980,497982],{"className":497981,"style":32090},[31805],[34,497983,497984,497987],{"style":206422},[34,497985],{"className":497986,"style":31814},[31813],[34,497988,497990],{"className":497989},[31818,31819,31820,31821],[34,497991,497993,497996],{"className":497992},[31685,31821],[34,497994,31955],{"className":497995},[31685,31686,31821],[34,497997,497999,498002],{"className":497998},[31685,31821],[34,498000,31955],{"className":498001},[31685,31686,31821],[34,498003,498005],{"className":498004},[31792],[34,498006,498008,498028],{"className":498007},[31796,31797],[34,498009,498011,498025],{"className":498010},[31801],[34,498012,498014],{"className":498013,"style":318658},[31805],[34,498015,498016,498019],{"style":318661},[34,498017],{"className":498018,"style":43406},[31813],[34,498020,498022],{"className":498021},[31818,43410,43411,31821],[34,498023,459],{"className":498024},[31685,31821],[34,498026,31829],{"className":498027},[31828],[34,498029,498031],{"className":498030},[31801],[34,498032,498034],{"className":498033,"style":220540},[31805],[34,498035],{},[34,498037,31829],{"className":498038},[31828],[34,498040,498042],{"className":498041},[31801],[34,498043,498045],{"className":498044,"style":323187},[31805],[34,498046],{},[34,498048,54151],{"className":498049},[31685,31686],[34,498051,498053,498056],{"className":498052},[31685],[34,498054,96058],{"className":498055,"style":106455},[31685,31686],[34,498057,498059],{"className":498058},[31792],[34,498060,498062,498125],{"className":498061},[31796,31797],[34,498063,498065,498122],{"className":498064},[31801],[34,498066,498068],{"className":498067,"style":32090},[31805],[34,498069,498070,498073],{"style":206422},[34,498071],{"className":498072,"style":31814},[31813],[34,498074,498076],{"className":498075},[31818,31819,31820,31821],[34,498077,498079,498119],{"className":498078},[31685,31821],[34,498080,498082,498085],{"className":498081},[31685,31821],[34,498083,31968],{"className":498084,"style":32153},[31685,31686,31821],[34,498086,498088],{"className":498087},[31792],[34,498089,498091,498111],{"className":498090},[31796,31797],[34,498092,498094,498108],{"className":498093},[31801],[34,498095,498097],{"className":498096,"style":318658},[31805],[34,498098,498099,498102],{"style":497687},[34,498100],{"className":498101,"style":43406},[31813],[34,498103,498105],{"className":498104},[31818,43410,43411,31821],[34,498106,459],{"className":498107},[31685,31821],[34,498109,31829],{"className":498110},[31828],[34,498112,498114],{"className":498113},[31801],[34,498115,498117],{"className":498116,"style":220540},[31805],[34,498118],{},[34,498120,31968],{"className":498121,"style":32153},[31685,31686,31821],[34,498123,31829],{"className":498124},[31828],[34,498126,498128],{"className":498127},[31801],[34,498129,498131],{"className":498130,"style":32163},[31805],[34,498132],{},"\n这个运算的效果是将 ",[34,498135,498137,498150],{"className":498136},[31642],[34,498138,498140],{"className":498139},[31646],[31648,498141,498142],{"xmlns":31650},[31652,498143,498144,498148],{},[31655,498145,498146],{},[31658,498147,54151],{},[31665,498149,54151],{"encoding":31667},[34,498151,498153],{"className":498152,"ariaHidden":758},[31672],[34,498154,498156,498159],{"className":498155},[31676],[34,498157],{"className":498158,"style":33218},[31680],[34,498160,54151],{"className":498161},[31685,31686],[34,498163,498165,498183],{"className":498164},[31642],[34,498166,498168],{"className":498167},[31646],[31648,498169,498170],{"xmlns":31650},[31652,498171,498172,498180],{},[31655,498173,498174],{},[31743,498175,498176,498178],{},[31658,498177,31955],{},[31749,498179,459],{},[31665,498181,498182],{"encoding":31667},"i_0",[34,498184,498186],{"className":498185,"ariaHidden":758},[31672],[34,498187,498189,498192],{"className":498188},[31676],[34,498190],{"className":498191,"style":475415},[31680],[34,498193,498195,498198],{"className":498194},[31685],[34,498196,31955],{"className":498197},[31685,31686],[34,498199,498201],{"className":498200},[31792],[34,498202,498204,498224],{"className":498203},[31796,31797],[34,498205,498207,498221],{"className":498206},[31801],[34,498208,498210],{"className":498209,"style":31806},[31805],[34,498211,498212,498215],{"style":31809},[34,498213],{"className":498214,"style":31814},[31813],[34,498216,498218],{"className":498217},[31818,31819,31820,31821],[34,498219,459],{"className":498220},[31685,31821],[34,498222,31829],{"className":498223},[31828],[34,498225,498227],{"className":498226},[31801],[34,498228,498230],{"className":498229,"style":31836},[31805],[34,498231],{}," 行移到第 ",[34,498234,498236,498249],{"className":498235},[31642],[34,498237,498239],{"className":498238},[31646],[31648,498240,498241],{"xmlns":31650},[31652,498242,498243,498247],{},[31655,498244,498245],{},[31658,498246,31955],{},[31665,498248,31955],{"encoding":31667},[34,498250,498252],{"className":498251,"ariaHidden":758},[31672],[34,498253,498255,498258],{"className":498254},[31676],[34,498256],{"className":498257,"style":33104},[31680],[34,498259,31955],{"className":498260},[31685,31686]," 行，第 ",[34,498263,498265,498283],{"className":498264},[31642],[34,498266,498268],{"className":498267},[31646],[31648,498269,498270],{"xmlns":31650},[31652,498271,498272,498280],{},[31655,498273,498274],{},[31743,498275,498276,498278],{},[31658,498277,31968],{},[31749,498279,459],{},[31665,498281,498282],{"encoding":31667},"j_0",[34,498284,498286],{"className":498285,"ariaHidden":758},[31672],[34,498287,498289,498292],{"className":498288},[31676],[34,498290],{"className":498291,"style":33123},[31680],[34,498293,498295,498298],{"className":498294},[31685],[34,498296,31968],{"className":498297,"style":32153},[31685,31686],[34,498299,498301],{"className":498300},[31792],[34,498302,498304,498324],{"className":498303},[31796,31797],[34,498305,498307,498321],{"className":498306},[31801],[34,498308,498310],{"className":498309,"style":31806},[31805],[34,498311,498312,498315],{"style":475491},[34,498313],{"className":498314,"style":31814},[31813],[34,498316,498318],{"className":498317},[31818,31819,31820,31821],[34,498319,459],{"className":498320},[31685,31821],[34,498322,31829],{"className":498323},[31828],[34,498325,498327],{"className":498326},[31801],[34,498328,498330],{"className":498329,"style":31836},[31805],[34,498331],{}," 列移到第 ",[34,498334,498336,498349],{"className":498335},[31642],[34,498337,498339],{"className":498338},[31646],[31648,498340,498341],{"xmlns":31650},[31652,498342,498343,498347],{},[31655,498344,498345],{},[31658,498346,31968],{},[31665,498348,31968],{"encoding":31667},[34,498350,498352],{"className":498351,"ariaHidden":758},[31672],[34,498353,498355,498358],{"className":498354},[31676],[34,498356],{"className":498357,"style":33123},[31680],[34,498359,31968],{"className":498360,"style":32153},[31685,31686]," 列，并把其他元素“切除”。结果仅在 ",[34,498363,498365,498387],{"className":498364},[31642],[34,498366,498368],{"className":498367},[31646],[31648,498369,498370],{"xmlns":31650},[31652,498371,498372,498384],{},[31655,498373,498374,498376,498378,498380,498382],{},[31662,498375,456],{"stretchy":702},[31658,498377,31955],{},[31662,498379,31960],{"separator":758},[31658,498381,31968],{},[31662,498383,2557],{"stretchy":702},[31665,498385,498386],{"encoding":31667},"(i, j)",[34,498388,498390],{"className":498389,"ariaHidden":758},[31672],[34,498391,498393,498396,498399,498402,498405,498408,498411],{"className":498392},[31676],[34,498394],{"className":498395,"style":31774},[31680],[34,498397,456],{"className":498398},[31782],[34,498400,31955],{"className":498401},[31685,31686],[34,498403,31960],{"className":498404},[32119],[34,498406],{"className":498407,"style":32123},[31845],[34,498409,31968],{"className":498410,"style":32153},[31685,31686],[34,498412,2557],{"className":498413},[31690]," 位置留下 ",[34,498416,498418,498448],{"className":498417},[31642],[34,498419,498421],{"className":498420},[31646],[31648,498422,498423],{"xmlns":31650},[31652,498424,498425,498445],{},[31655,498426,498427],{},[31743,498428,498429,498431],{},[31658,498430,111],{},[31655,498432,498433,498439],{},[31743,498434,498435,498437],{},[31658,498436,31955],{},[31749,498438,459],{},[31743,498440,498441,498443],{},[31658,498442,31968],{},[31749,498444,459],{},[31665,498446,498447],{"encoding":31667},"a_{i_0 j_0}",[34,498449,498451],{"className":498450,"ariaHidden":758},[31672],[34,498452,498454,498457],{"className":498453},[31676],[34,498455],{"className":498456,"style":56005},[31680],[34,498458,498460,498463],{"className":498459},[31685],[34,498461,111],{"className":498462},[31685,31686],[34,498464,498466],{"className":498465},[31792],[34,498467,498469,498569],{"className":498468},[31796,31797],[34,498470,498472,498566],{"className":498471},[31801],[34,498473,498475],{"className":498474,"style":32090},[31805],[34,498476,498477,498480],{"style":31809},[34,498478],{"className":498479,"style":31814},[31813],[34,498481,498483],{"className":498482},[31818,31819,31820,31821],[34,498484,498486,498526],{"className":498485},[31685,31821],[34,498487,498489,498492],{"className":498488},[31685,31821],[34,498490,31955],{"className":498491},[31685,31686,31821],[34,498493,498495],{"className":498494},[31792],[34,498496,498498,498518],{"className":498497},[31796,31797],[34,498499,498501,498515],{"className":498500},[31801],[34,498502,498504],{"className":498503,"style":318658},[31805],[34,498505,498506,498509],{"style":318661},[34,498507],{"className":498508,"style":43406},[31813],[34,498510,498512],{"className":498511},[31818,43410,43411,31821],[34,498513,459],{"className":498514},[31685,31821],[34,498516,31829],{"className":498517},[31828],[34,498519,498521],{"className":498520},[31801],[34,498522,498524],{"className":498523,"style":220540},[31805],[34,498525],{},[34,498527,498529,498532],{"className":498528},[31685,31821],[34,498530,31968],{"className":498531,"style":32153},[31685,31686,31821],[34,498533,498535],{"className":498534},[31792],[34,498536,498538,498558],{"className":498537},[31796,31797],[34,498539,498541,498555],{"className":498540},[31801],[34,498542,498544],{"className":498543,"style":318658},[31805],[34,498545,498546,498549],{"style":497687},[34,498547],{"className":498548,"style":43406},[31813],[34,498550,498552],{"className":498551},[31818,43410,43411,31821],[34,498553,459],{"className":498554},[31685,31821],[34,498556,31829],{"className":498557},[31828],[34,498559,498561],{"className":498560},[31801],[34,498562,498564],{"className":498563,"style":220540},[31805],[34,498565],{},[34,498567,31829],{"className":498568},[31828],[34,498570,498572],{"className":498571},[31801],[34,498573,498575],{"className":498574,"style":32163},[31805],[34,498576],{},[34,498578,498580,498652],{"className":498579},[31642],[34,498581,498583],{"className":498582},[31646],[31648,498584,498585],{"xmlns":31650},[31652,498586,498587,498649],{},[31655,498588,498589,498603,498605,498619,498621,498639],{},[31743,498590,498591,498593],{},[31658,498592,96058],{},[31655,498594,498595,498597],{},[31658,498596,31955],{},[31743,498598,498599,498601],{},[31658,498600,31955],{},[31749,498602,459],{},[31658,498604,54151],{},[31743,498606,498607,498609],{},[31658,498608,96058],{},[31655,498610,498611,498617],{},[31743,498612,498613,498615],{},[31658,498614,31968],{},[31749,498616,459],{},[31658,498618,31968],{},[31662,498620,202],{},[31743,498622,498623,498625],{},[31658,498624,111],{},[31655,498626,498627,498633],{},[31743,498628,498629,498631],{},[31658,498630,31955],{},[31749,498632,459],{},[31743,498634,498635,498637],{},[31658,498636,31968],{},[31749,498638,459],{},[31743,498640,498641,498643],{},[31658,498642,96058],{},[31655,498644,498645,498647],{},[31658,498646,31955],{},[31658,498648,31968],{},[31665,498650,498651],{"encoding":31667},"E_{i i_0} A E_{j_0 j} = a_{i_0 j_0} E_{ij}",[34,498653,498655,498839],{"className":498654,"ariaHidden":758},[31672],[34,498656,498658,498661,498744,498747,498830,498833,498836],{"className":498657},[31676],[34,498659],{"className":498660,"style":55484},[31680],[34,498662,498664,498667],{"className":498663},[31685],[34,498665,96058],{"className":498666,"style":106455},[31685,31686],[34,498668,498670],{"className":498669},[31792],[34,498671,498673,498736],{"className":498672},[31796,31797],[34,498674,498676,498733],{"className":498675},[31801],[34,498677,498679],{"className":498678,"style":32090},[31805],[34,498680,498681,498684],{"style":206422},[34,498682],{"className":498683,"style":31814},[31813],[34,498685,498687],{"className":498686},[31818,31819,31820,31821],[34,498688,498690,498693],{"className":498689},[31685,31821],[34,498691,31955],{"className":498692},[31685,31686,31821],[34,498694,498696,498699],{"className":498695},[31685,31821],[34,498697,31955],{"className":498698},[31685,31686,31821],[34,498700,498702],{"className":498701},[31792],[34,498703,498705,498725],{"className":498704},[31796,31797],[34,498706,498708,498722],{"className":498707},[31801],[34,498709,498711],{"className":498710,"style":318658},[31805],[34,498712,498713,498716],{"style":318661},[34,498714],{"className":498715,"style":43406},[31813],[34,498717,498719],{"className":498718},[31818,43410,43411,31821],[34,498720,459],{"className":498721},[31685,31821],[34,498723,31829],{"className":498724},[31828],[34,498726,498728],{"className":498727},[31801],[34,498729,498731],{"className":498730,"style":220540},[31805],[34,498732],{},[34,498734,31829],{"className":498735},[31828],[34,498737,498739],{"className":498738},[31801],[34,498740,498742],{"className":498741,"style":323187},[31805],[34,498743],{},[34,498745,54151],{"className":498746},[31685,31686],[34,498748,498750,498753],{"className":498749},[31685],[34,498751,96058],{"className":498752,"style":106455},[31685,31686],[34,498754,498756],{"className":498755},[31792],[34,498757,498759,498822],{"className":498758},[31796,31797],[34,498760,498762,498819],{"className":498761},[31801],[34,498763,498765],{"className":498764,"style":32090},[31805],[34,498766,498767,498770],{"style":206422},[34,498768],{"className":498769,"style":31814},[31813],[34,498771,498773],{"className":498772},[31818,31819,31820,31821],[34,498774,498776,498816],{"className":498775},[31685,31821],[34,498777,498779,498782],{"className":498778},[31685,31821],[34,498780,31968],{"className":498781,"style":32153},[31685,31686,31821],[34,498783,498785],{"className":498784},[31792],[34,498786,498788,498808],{"className":498787},[31796,31797],[34,498789,498791,498805],{"className":498790},[31801],[34,498792,498794],{"className":498793,"style":318658},[31805],[34,498795,498796,498799],{"style":497687},[34,498797],{"className":498798,"style":43406},[31813],[34,498800,498802],{"className":498801},[31818,43410,43411,31821],[34,498803,459],{"className":498804},[31685,31821],[34,498806,31829],{"className":498807},[31828],[34,498809,498811],{"className":498810},[31801],[34,498812,498814],{"className":498813,"style":220540},[31805],[34,498815],{},[34,498817,31968],{"className":498818,"style":32153},[31685,31686,31821],[34,498820,31829],{"className":498821},[31828],[34,498823,498825],{"className":498824},[31801],[34,498826,498828],{"className":498827,"style":32163},[31805],[34,498829],{},[34,498831],{"className":498832,"style":31846},[31845],[34,498834,202],{"className":498835},[31850],[34,498837],{"className":498838,"style":31846},[31845],[34,498840,498842,498845,498965],{"className":498841},[31676],[34,498843],{"className":498844,"style":55484},[31680],[34,498846,498848,498851],{"className":498847},[31685],[34,498849,111],{"className":498850},[31685,31686],[34,498852,498854],{"className":498853},[31792],[34,498855,498857,498957],{"className":498856},[31796,31797],[34,498858,498860,498954],{"className":498859},[31801],[34,498861,498863],{"className":498862,"style":32090},[31805],[34,498864,498865,498868],{"style":31809},[34,498866],{"className":498867,"style":31814},[31813],[34,498869,498871],{"className":498870},[31818,31819,31820,31821],[34,498872,498874,498914],{"className":498873},[31685,31821],[34,498875,498877,498880],{"className":498876},[31685,31821],[34,498878,31955],{"className":498879},[31685,31686,31821],[34,498881,498883],{"className":498882},[31792],[34,498884,498886,498906],{"className":498885},[31796,31797],[34,498887,498889,498903],{"className":498888},[31801],[34,498890,498892],{"className":498891,"style":318658},[31805],[34,498893,498894,498897],{"style":318661},[34,498895],{"className":498896,"style":43406},[31813],[34,498898,498900],{"className":498899},[31818,43410,43411,31821],[34,498901,459],{"className":498902},[31685,31821],[34,498904,31829],{"className":498905},[31828],[34,498907,498909],{"className":498908},[31801],[34,498910,498912],{"className":498911,"style":220540},[31805],[34,498913],{},[34,498915,498917,498920],{"className":498916},[31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是理想，且 ",[34,499040,499042,499088],{"className":499041},[31642],[34,499043,499045],{"className":499044},[31646],[31648,499046,499047],{"xmlns":31650},[31652,499048,499049,499085],{},[31655,499050,499051,499065,499067,499081,499083],{},[31743,499052,499053,499055],{},[31658,499054,96058],{},[31655,499056,499057,499059],{},[31658,499058,31955],{},[31743,499060,499061,499063],{},[31658,499062,31955],{},[31749,499064,459],{},[31662,499066,31960],{"separator":758},[31743,499068,499069,499071],{},[31658,499070,96058],{},[31655,499072,499073,499079],{},[31743,499074,499075,499077],{},[31658,499076,31968],{},[31749,499078,459],{},[31658,499080,31968],{},[31662,499082,112296],{},[31658,499084,34386],{},[31665,499086,499087],{"encoding":31667},"E_{i i_0}, E_{j_0 j} \\in 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是非零实数，其逆元存在。利用理想性质再次乘以标量（或对角阵）：\n",[34,499719,499721,499803],{"className":499720},[31642],[34,499722,499724],{"className":499723},[31646],[31648,499725,499726],{"xmlns":31650},[31652,499727,499728,499800],{},[31655,499729,499730,499740,499742,499764,499766,499784,499794,499796,499798],{},[31743,499731,499732,499734],{},[31658,499733,96058],{},[31655,499735,499736,499738],{},[31658,499737,31955],{},[31658,499739,31968],{},[31662,499741,202],{},[41453,499743,499744,499746],{},[31749,499745,640],{},[31743,499747,499748,499750],{},[31658,499749,111],{},[31655,499751,499752,499758],{},[31743,499753,499754,499756],{},[31658,499755,31955],{},[31749,499757,459],{},[31743,499759,499760,499762],{},[31658,499761,31968],{},[31749,499763,459],{},[31662,499765,456],{"stretchy":702},[31743,499767,499768,499770],{},[31658,499769,111],{},[31655,499771,499772,499778],{},[31743,499773,499774,499776],{},[31658,499775,31955],{},[31749,499777,459],{},[31743,499779,499780,499782],{},[31658,499781,31968],{},[31749,499783,459],{},[31743,499785,499786,499788],{},[31658,499787,96058],{},[31655,499789,499790,499792],{},[31658,499791,31955],{},[31658,499793,31968],{},[31662,499795,2557],{"stretchy":702},[31662,499797,112296],{},[31658,499799,106063],{},[31665,499801,499802],{"encoding":31667},"E_{ij} = \\frac{1}{a_{i_0 j_0}} (a_{i_0 j_0} E_{ij}) \\in 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",[34,500247,500249,500271],{"className":500248},[31642],[34,500250,500252],{"className":500251},[31646],[31648,500253,500254],{"xmlns":31650},[31652,500255,500256,500268],{},[31655,500257,500258],{},[31743,500259,500260,500262],{},[31658,500261,96058],{},[31655,500263,500264,500266],{},[31658,500265,31955],{},[31658,500267,31968],{},[31665,500269,500270],{"encoding":31667},"E_{ij}",[34,500272,500274],{"className":500273,"ariaHidden":758},[31672],[34,500275,500277,500280],{"className":500276},[31676],[34,500278],{"className":500279,"style":55484},[31680],[34,500281,500283,500286],{"className":500282},[31685],[34,500284,96058],{"className":500285,"style":106455},[31685,31686],[34,500287,500289],{"className":500288},[31792],[34,500290,500292,500315],{"className":500291},[31796,31797],[34,500293,500295,500312],{"className":500294},[31801],[34,500296,500298],{"className":500297,"style":32090},[31805],[34,500299,500300,500303],{"style":206422},[34,500301],{"className":500302,"style":31814},[31813],[34,500304,500306],{"className":500305},[31818,31819,31820,31821],[34,500307,500309],{"className":500308},[31685,31821],[34,500310,141027],{"className":500311,"style":32153},[31685,31686,31821],[34,500313,31829],{"className":500314},[31828],[34,500316,500318],{"className":500317},[31801],[34,500319,500321],{"className":500320,"style":32163},[31805],[34,500322],{},[34,500324,500326,500339],{"className":500325},[31642],[34,500327,500329],{"className":500328},[31646],[31648,500330,500331],{"xmlns":31650},[31652,500332,500333,500337],{},[31655,500334,500335],{},[31658,500336,106063],{},[31665,500338,106063],{"encoding":31667},[34,500340,500342],{"className":500341,"ariaHidden":758},[31672],[34,500343,500345,500348],{"className":500344},[31676],[34,500346],{"className":500347,"style":33218},[31680],[34,500349,106063],{"className":500350,"style":67482},[31685,31686]," 中。由于任意矩阵 ",[34,500353,500355,500368],{"className":500354},[31642],[34,500356,500358],{"className":500357},[31646],[31648,500359,500360],{"xmlns":31650},[31652,500361,500362,500366],{},[31655,500363,500364],{},[31658,500365,98360],{},[31665,500367,98360],{"encoding":31667},[34,500369,500371],{"className":500370,"ariaHidden":758},[31672],[34,500372,500374,500377],{"className":500373},[31676],[34,500375],{"className":500376,"style":33218},[31680],[34,500378,98360],{"className":500379,"style":33389},[31685,31686],[34,500381,500383,500404],{"className":500382},[31642],[34,500384,500386],{"className":500385},[31646],[31648,500387,500388],{"xmlns":31650},[31652,500389,500390,500402],{},[31655,500391,500392],{},[31743,500393,500394,500396],{},[31658,500395,96058],{},[31655,500397,500398,500400],{},[31658,500399,31955],{},[31658,500401,31968],{},[31665,500403,500270],{"encoding":31667},[34,500405,500407],{"className":500406,"ariaHidden":758},[31672],[34,500408,500410,500413],{"className":500409},[31676],[34,500411],{"className":500412,"style":55484},[31680],[34,500414,500416,500419],{"className":500415},[31685],[34,500417,96058],{"className":500418,"style":106455},[31685,31686],[34,500420,500422],{"className":500421},[31792],[34,500423,500425,500448],{"className":500424},[31796,31797],[34,500426,500428,500445],{"className":500427},[31801],[34,500429,500431],{"className":500430,"style":32090},[31805],[34,500432,500433,500436],{"style":206422},[34,500434],{"className":500435,"style":31814},[31813],[34,500437,500439],{"className":500438},[31818,31819,31820,31821],[34,500440,500442],{"className":500441},[31685,31821],[34,500443,141027],{"className":500444,"style":32153},[31685,31686,31821],[34,500446,31829],{"className":500447},[31828],[34,500449,500451],{"className":500450},[31801],[34,500452,500454],{"className":500453,"style":32163},[31805],[34,500455],{}," 的线性组合，所以整个环 ",[34,500458,500460,500488],{"className":500459},[31642],[34,500461,500463],{"className":500462},[31646],[31648,500464,500465],{"xmlns":31650},[31652,500466,500467,500485],{},[31655,500468,500469,500471,500473,500475,500477,500479,500481,500483],{},[31658,500470,98360],{},[31662,500472,456],{"stretchy":702},[31658,500474,31660],{},[31662,500476,31960],{"separator":758},[31658,500478,34386],{"mathvariant":318416},[31662,500480,2557],{"stretchy":702},[31662,500482,131855],{},[31658,500484,106063],{},[31665,500486,500487],{"encoding":31667},"M(n, \\mathbb{R}) \\subseteq I",[34,500489,500491,500527],{"className":500490,"ariaHidden":758},[31672],[34,500492,500494,500497,500500,500503,500506,500509,500512,500515,500518,500521,500524],{"className":500493},[31676],[34,500495],{"className":500496,"style":31774},[31680],[34,500498,98360],{"className":500499,"style":33389},[31685,31686],[34,500501,456],{"className":500502},[31782],[34,500504,31660],{"className":500505},[31685,31686],[34,500507,31960],{"className":500508},[32119],[34,500510],{"className":500511,"style":32123},[31845],[34,500513,34386],{"className":500514},[31685,318441],[34,500516,2557],{"className":500517},[31690],[34,500519],{"className":500520,"style":31846},[31845],[34,500522,131855],{"className":500523},[31850],[34,500525],{"className":500526,"style":31846},[31845],[34,500528,500530,500533],{"className":500529},[31676],[34,500531],{"className":500532,"style":33218},[31680],[34,500534,106063],{"className":500535,"style":67482},[31685,31686],[108,500537,500538,500539,351462],{},"故 ",[34,500540,500542,500569],{"className":500541},[31642],[34,500543,500545],{"className":500544},[31646],[31648,500546,500547],{"xmlns":31650},[31652,500548,500549,500567],{},[31655,500550,500551,500553,500555,500557,500559,500561,500563,500565],{},[31658,500552,106063],{},[31662,500554,202],{},[31658,500556,98360],{},[31662,500558,456],{"stretchy":702},[31658,500560,31660],{},[31662,500562,31960],{"separator":758},[31658,500564,34386],{"mathvariant":318416},[31662,500566,2557],{"stretchy":702},[31665,500568,497156],{"encoding":31667},[34,500570,500572,500590],{"className":500571,"ariaHidden":758},[31672],[34,500573,500575,500578,500581,500584,500587],{"className":500574},[31676],[34,500576],{"className":500577,"style":33218},[31680],[34,500579,106063],{"className":500580,"style":67482},[31685,31686],[34,500582],{"className":500583,"style":31846},[31845],[34,500585,202],{"className":500586},[31850],[34,500588],{"className":500589,"style":31846},[31845],[34,500591,500593,500596,500599,500602,500605,500608,500611,500614],{"className":500592},[31676],[34,500594],{"className":500595,"style":31774},[31680],[34,500597,98360],{"className":500598,"style":33389},[31685,31686],[34,500600,456],{"className":500601},[31782],[34,500603,31660],{"className":500604},[31685,31686],[34,500606,31960],{"className":500607},[32119],[34,500609],{"className":500610,"style":32123},[31845],[34,500612,34386],{"className":500613},[31685,318441],[34,500615,2557],{"className":500616},[31690],[15,500618,500619],{},[2627,500620,500621],{},"作业补充（非平凡的左\u002F右理想）：",[105,500623,500624],{},[108,500625,500626,500629,500630,500661,500662,39984,500690],{},[2627,500627,500628],{},"左理想例子","：所有",[2627,500631,178926,500632,500660],{},[34,500633,500635,500648],{"className":500634},[31642],[34,500636,500638],{"className":500637},[31646],[31648,500639,500640],{"xmlns":31650},[31652,500641,500642,500646],{},[31655,500643,500644],{},[31658,500645,33759],{},[31665,500647,33759],{"encoding":31667},[34,500649,500651],{"className":500650,"ariaHidden":758},[31672],[34,500652,500654,500657],{"className":500653},[31676],[34,500655],{"className":500656,"style":31681},[31680],[34,500658,33759],{"className":500659,"style":33774},[31685,31686]," 列全为 0"," 的矩阵集合 ",[34,500663,500665,500678],{"className":500664},[31642],[34,500666,500668],{"className":500667},[31646],[31648,500669,500670],{"xmlns":31650},[31652,500671,500672,500676],{},[31655,500673,500674],{},[31658,500675,33206],{},[31665,500677,33206],{"encoding":31667},[34,500679,500681],{"className":500680,"ariaHidden":758},[31672],[34,500682,500684,500687],{"className":500683},[31676],[34,500685],{"className":500686,"style":33218},[31680],[34,500688,33206],{"className":500689},[31685,31686],[105,500691,500692,500834,501202,501435,501555],{},[108,500693,500694,34554,500697,61605,500748,88651,500776,500804,500805,500833],{},[71807,500695,500696],{},"验证",[34,500698,500700,500718],{"className":500699},[31642],[34,500701,500703],{"className":500702},[31646],[31648,500704,500705],{"xmlns":31650},[31652,500706,500707,500715],{},[31655,500708,500709,500711,500713],{},[31658,500710,54224],{},[31662,500712,112296],{},[31658,500714,33206],{},[31665,500716,500717],{"encoding":31667},"B \\in L",[34,500719,500721,500739],{"className":500720,"ariaHidden":758},[31672],[34,500722,500724,500727,500730,500733,500736],{"className":500723},[31676],[34,500725],{"className":500726,"style":104901},[31680],[34,500728,54224],{"className":500729,"style":54252},[31685,31686],[34,500731],{"className":500732,"style":31846},[31845],[34,500734,112296],{"className":500735},[31850],[34,500737],{"className":500738,"style":31846},[31845],[34,500740,500742,500745],{"className":500741},[31676],[34,500743],{"className":500744,"style":33218},[31680],[34,500746,33206],{"className":500747},[31685,31686],[34,500749,500751,500764],{"className":500750},[31642],[34,500752,500754],{"className":500753},[31646],[31648,500755,500756],{"xmlns":31650},[31652,500757,500758,500762],{},[31655,500759,500760],{},[31658,500761,54224],{},[31665,500763,54224],{"encoding":31667},[34,500765,500767],{"className":500766,"ariaHidden":758},[31672],[34,500768,500770,500773],{"className":500769},[31676],[34,500771],{"className":500772,"style":33218},[31680],[34,500774,54224],{"className":500775,"style":54252},[31685,31686],[34,500777,500779,500792],{"className":500778},[31642],[34,500780,500782],{"className":500781},[31646],[31648,500783,500784],{"xmlns":31650},[31652,500785,500786,500790],{},[31655,500787,500788],{},[31658,500789,33759],{},[31665,500791,33759],{"encoding":31667},[34,500793,500795],{"className":500794,"ariaHidden":758},[31672],[34,500796,500798,500801],{"className":500797},[31676],[34,500799],{"className":500800,"style":31681},[31680],[34,500802,33759],{"className":500803,"style":33774},[31685,31686]," 列为 0。设 ",[34,500806,500808,500821],{"className":500807},[31642],[34,500809,500811],{"className":500810},[31646],[31648,500812,500813],{"xmlns":31650},[31652,500814,500815,500819],{},[31655,500816,500817],{},[31658,500818,54151],{},[31665,500820,54151],{"encoding":31667},[34,500822,500824],{"className":500823,"ariaHidden":758},[31672],[34,500825,500827,500830],{"className":500826},[31676],[34,500828],{"className":500829,"style":33218},[31680],[34,500831,54151],{"className":500832},[31685,31686]," 是任意矩阵。",[108,500835,500836,500837,347,500893,88651,500921,500949,500950,347],{},"考虑乘积 ",[34,500838,500840,500860],{"className":500839},[31642],[34,500841,500843],{"className":500842},[31646],[31648,500844,500845],{"xmlns":31650},[31652,500846,500847,500857],{},[31655,500848,500849,500851,500853,500855],{},[31658,500850,47384],{},[31662,500852,202],{},[31658,500854,54151],{},[31658,500856,54224],{},[31665,500858,500859],{"encoding":31667},"C = AB",[34,500861,500863,500881],{"className":500862,"ariaHidden":758},[31672],[34,500864,500866,500869,500872,500875,500878],{"className":500865},[31676],[34,500867],{"className":500868,"style":33218},[31680],[34,500870,47384],{"className":500871,"style":38706},[31685,31686],[34,500873],{"className":500874,"style":31846},[31845],[34,500876,202],{"className":500877},[31850],[34,500879],{"className":500880,"style":31846},[31845],[34,500882,500884,500887,500890],{"className":500883},[31676],[34,500885],{"className":500886,"style":33218},[31680],[34,500888,54151],{"className":500889},[31685,31686],[34,500891,54224],{"className":500892,"style":54252},[31685,31686],[34,500894,500896,500909],{"className":500895},[31642],[34,500897,500899],{"className":500898},[31646],[31648,500900,500901],{"xmlns":31650},[31652,500902,500903,500907],{},[31655,500904,500905],{},[31658,500906,47384],{},[31665,500908,47384],{"encoding":31667},[34,500910,500912],{"className":500911,"ariaHidden":758},[31672],[34,500913,500915,500918],{"className":500914},[31676],[34,500916],{"className":500917,"style":33218},[31680],[34,500919,47384],{"className":500920,"style":38706},[31685,31686],[34,500922,500924,500937],{"className":500923},[31642],[34,500925,500927],{"className":500926},[31646],[31648,500928,500929],{"xmlns":31650},[31652,500930,500931,500935],{},[31655,500932,500933],{},[31658,500934,33759],{},[31665,500936,33759],{"encoding":31667},[34,500938,500940],{"className":500939,"ariaHidden":758},[31672],[34,500941,500943,500946],{"className":500942},[31676],[34,500944],{"className":500945,"style":31681},[31680],[34,500947,33759],{"className":500948,"style":33774},[31685,31686]," 列元素为 ",[34,500951,500953,501003],{"className":500952},[31642],[34,500954,500956],{"className":500955},[31646],[31648,500957,500958],{"xmlns":31650},[31652,500959,500960,501000],{},[31655,500961,500962,500972,500974,500980,500990],{},[31743,500963,500964,500966],{},[31658,500965,31965],{},[31655,500967,500968,500970],{},[31658,500969,31955],{},[31658,500971,33759],{},[31662,500973,202],{},[31743,500975,500976,500978],{},[31662,500977,131138],{},[31658,500979,31968],{},[31743,500981,500982,500984],{},[31658,500983,111],{},[31655,500985,500986,500988],{},[31658,500987,31955],{},[31658,500989,31968],{},[31743,500991,500992,500994],{},[31658,500993,1697],{},[31655,500995,500996,500998],{},[31658,500997,31968],{},[31658,500999,33759],{},[31665,501001,501002],{"encoding":31667},"c_{ik} = \\sum_j a_{ij} b_{jk}",[34,501004,501006,501064],{"className":501005,"ariaHidden":758},[31672],[34,501007,501009,501012,501055,501058,501061],{"className":501008},[31676],[34,501010],{"className":501011,"style":31860},[31680],[34,501013,501015,501018],{"className":501014},[31685],[34,501016,31965],{"className":501017},[31685,31686],[34,501019,501021],{"className":501020},[31792],[34,501022,501024,501047],{"className":501023},[31796,31797],[34,501025,501027,501044],{"className":501026},[31801],[34,501028,501030],{"className":501029,"style":33833},[31805],[34,501031,501032,501035],{"style":31809},[34,501033],{"className":501034,"style":31814},[31813],[34,501036,501038],{"className":501037},[31818,31819,31820,31821],[34,501039,501041],{"className":501040},[31685,31821],[34,501042,153018],{"className":501043,"style":33774},[31685,31686,31821],[34,501045,31829],{"className":501046},[31828],[34,501048,501050],{"className":501049},[31801],[34,501051,501053],{"className":501052,"style":31836},[31805],[34,501054],{},[34,501056],{"className":501057,"style":31846},[31845],[34,501059,202],{"className":501060},[31850],[34,501062],{"className":501063,"style":31846},[31845],[34,501065,501067,501070,501110,501113,501156],{"className":501066},[31676],[34,501068],{"className":501069,"style":156901},[31680],[34,501071,501073,501076],{"className":501072},[32767],[34,501074,131138],{"className":501075,"style":156561},[32767,131249,156560],[34,501077,501079],{"className":501078},[31792],[34,501080,501082,501102],{"className":501081},[31796,31797],[34,501083,501085,501099],{"className":501084},[31801],[34,501086,501088],{"className":501087,"style":156920},[31805],[34,501089,501090,501093],{"style":156577},[34,501091],{"className":501092,"style":31814},[31813],[34,501094,501096],{"className":501095},[31818,31819,31820,31821],[34,501097,31968],{"className":501098,"style":32153},[31685,31686,31821],[34,501100,31829],{"className":501101},[31828],[34,501103,501105],{"className":501104},[31801],[34,501106,501108],{"className":501107,"style":156941},[31805],[34,501109],{},[34,501111],{"className":501112,"style":32123},[31845],[34,501114,501116,501119],{"className":501115},[31685],[34,501117,111],{"className":501118},[31685,31686],[34,501120,501122],{"className":501121},[31792],[34,501123,501125,501148],{"className":501124},[31796,31797],[34,501126,501128,501145],{"className":501127},[31801],[34,501129,501131],{"className":501130,"style":32090},[31805],[34,501132,501133,501136],{"style":31809},[34,501134],{"className":501135,"style":31814},[31813],[34,501137,501139],{"className":501138},[31818,31819,31820,31821],[34,501140,501142],{"className":501141},[31685,31821],[34,501143,141027],{"className":501144,"style":32153},[31685,31686,31821],[34,501146,31829],{"className":501147},[31828],[34,501149,501151],{"className":501150},[31801],[34,501152,501154],{"className":501153,"style":32163},[31805],[34,501155],{},[34,501157,501159,501162],{"className":501158},[31685],[34,501160,1697],{"className":501161},[31685,31686],[34,501163,501165],{"className":501164},[31792],[34,501166,501168,501194],{"className":501167},[31796,31797],[34,501169,501171,501191],{"className":501170},[31801],[34,501172,501174],{"className":501173,"style":33833},[31805],[34,501175,501176,501179],{"style":31809},[34,501177],{"className":501178,"style":31814},[31813],[34,501180,501182],{"className":501181},[31818,31819,31820,31821],[34,501183,501185,501188],{"className":501184},[31685,31821],[34,501186,31968],{"className":501187,"style":32153},[31685,31686,31821],[34,501189,33759],{"className":501190,"style":33774},[31685,31686,31821],[34,501192,31829],{"className":501193},[31828],[34,501195,501197],{"className":501196},[31801],[34,501198,501200],{"className":501199,"style":32163},[31805],[34,501201],{},[108,501203,93310,501204,501306,501307,501335,501336,347],{},[34,501205,501207,501233],{"className":501206},[31642],[34,501208,501210],{"className":501209},[31646],[31648,501211,501212],{"xmlns":31650},[31652,501213,501214,501230],{},[31655,501215,501216,501226,501228],{},[31743,501217,501218,501220],{},[31658,501219,1697],{},[31655,501221,501222,501224],{},[31658,501223,31968],{},[31658,501225,33759],{},[31662,501227,202],{},[31749,501229,459],{},[31665,501231,501232],{"encoding":31667},"b_{jk} = 0",[34,501234,501236,501297],{"className":501235,"ariaHidden":758},[31672],[34,501237,501239,501242,501288,501291,501294],{"className":501238},[31676],[34,501240],{"className":501241,"style":141223},[31680],[34,501243,501245,501248],{"className":501244},[31685],[34,501246,1697],{"className":501247},[31685,31686],[34,501249,501251],{"className":501250},[31792],[34,501252,501254,501280],{"className":501253},[31796,31797],[34,501255,501257,501277],{"className":501256},[31801],[34,501258,501260],{"className":501259,"style":33833},[31805],[34,501261,501262,501265],{"style":31809},[34,501263],{"className":501264,"style":31814},[31813],[34,501266,501268],{"className":501267},[31818,31819,31820,31821],[34,501269,501271,501274],{"className":501270},[31685,31821],[34,501272,31968],{"className":501273,"style":32153},[31685,31686,31821],[34,501275,33759],{"className":501276,"style":33774},[31685,31686,31821],[34,501278,31829],{"className":501279},[31828],[34,501281,501283],{"className":501282},[31801],[34,501284,501286],{"className":501285,"style":32163},[31805],[34,501287],{},[34,501289],{"className":501290,"style":31846},[31845],[34,501292,202],{"className":501293},[31850],[34,501295],{"className":501296,"style":31846},[31845],[34,501298,501300,501303],{"className":501299},[31676],[34,501301],{"className":501302,"style":32617},[31680],[34,501304,459],{"className":501305},[31685],"（对所有 ",[34,501308,501310,501323],{"className":501309},[31642],[34,501311,501313],{"className":501312},[31646],[31648,501314,501315],{"xmlns":31650},[31652,501316,501317,501321],{},[31655,501318,501319],{},[31658,501320,31968],{},[31665,501322,31968],{"encoding":31667},[34,501324,501326],{"className":501325,"ariaHidden":758},[31672],[34,501327,501329,501332],{"className":501328},[31676],[34,501330],{"className":501331,"style":33123},[31680],[34,501333,31968],{"className":501334,"style":32153},[31685,31686],"），所以 ",[34,501337,501339,501365],{"className":501338},[31642],[34,501340,501342],{"className":501341},[31646],[31648,501343,501344],{"xmlns":31650},[31652,501345,501346,501362],{},[31655,501347,501348,501358,501360],{},[31743,501349,501350,501352],{},[31658,501351,31965],{},[31655,501353,501354,501356],{},[31658,501355,31955],{},[31658,501357,33759],{},[31662,501359,202],{},[31749,501361,459],{},[31665,501363,501364],{"encoding":31667},"c_{ik} = 0",[34,501366,501368,501426],{"className":501367,"ariaHidden":758},[31672],[34,501369,501371,501374,501417,501420,501423],{"className":501370},[31676],[34,501372],{"className":501373,"style":31860},[31680],[34,501375,501377,501380],{"className":501376},[31685],[34,501378,31965],{"className":501379},[31685,31686],[34,501381,501383],{"className":501382},[31792],[34,501384,501386,501409],{"className":501385},[31796,31797],[34,501387,501389,501406],{"className":501388},[31801],[34,501390,501392],{"className":501391,"style":33833},[31805],[34,501393,501394,501397],{"style":31809},[34,501395],{"className":501396,"style":31814},[31813],[34,501398,501400],{"className":501399},[31818,31819,31820,31821],[34,501401,501403],{"className":501402},[31685,31821],[34,501404,153018],{"className":501405,"style":33774},[31685,31686,31821],[34,501407,31829],{"className":501408},[31828],[34,501410,501412],{"className":501411},[31801],[34,501413,501415],{"className":501414,"style":31836},[31805],[34,501416],{},[34,501418],{"className":501419,"style":31846},[31845],[34,501421,202],{"className":501422},[31850],[34,501424],{"className":501425,"style":31846},[31845],[34,501427,501429,501432],{"className":501428},[31676],[34,501430],{"className":501431,"style":32617},[31680],[34,501433,459],{"className":501434},[31685],[108,501436,336656,501437,88651,501470,501498,501499,347],{},[34,501438,501440,501455],{"className":501439},[31642],[34,501441,501443],{"className":501442},[31646],[31648,501444,501445],{"xmlns":31650},[31652,501446,501447,501453],{},[31655,501448,501449,501451],{},[31658,501450,54151],{},[31658,501452,54224],{},[31665,501454,93307],{"encoding":31667},[34,501456,501458],{"className":501457,"ariaHidden":758},[31672],[34,501459,501461,501464,501467],{"className":501460},[31676],[34,501462],{"className":501463,"style":33218},[31680],[34,501465,54151],{"className":501466},[31685,31686],[34,501468,54224],{"className":501469,"style":54252},[31685,31686],[34,501471,501473,501486],{"className":501472},[31642],[34,501474,501476],{"className":501475},[31646],[31648,501477,501478],{"xmlns":31650},[31652,501479,501480,501484],{},[31655,501481,501482],{},[31658,501483,33759],{},[31665,501485,33759],{"encoding":31667},[34,501487,501489],{"className":501488,"ariaHidden":758},[31672],[34,501490,501492,501495],{"className":501491},[31676],[34,501493],{"className":501494,"style":31681},[31680],[34,501496,33759],{"className":501497,"style":33774},[31685,31686]," 列也是 0，所以 ",[34,501500,501502,501522],{"className":501501},[31642],[34,501503,501505],{"className":501504},[31646],[31648,501506,501507],{"xmlns":31650},[31652,501508,501509,501519],{},[31655,501510,501511,501513,501515,501517],{},[31658,501512,54151],{},[31658,501514,54224],{},[31662,501516,112296],{},[31658,501518,33206],{},[31665,501520,501521],{"encoding":31667},"AB \\in L",[34,501523,501525,501546],{"className":501524,"ariaHidden":758},[31672],[34,501526,501528,501531,501534,501537,501540,501543],{"className":501527},[31676],[34,501529],{"className":501530,"style":104901},[31680],[34,501532,54151],{"className":501533},[31685,31686],[34,501535,54224],{"className":501536,"style":54252},[31685,31686],[34,501538],{"className":501539,"style":31846},[31845],[34,501541,112296],{"className":501542},[31850],[34,501544],{"className":501545,"style":31846},[31845],[34,501547,501549,501552],{"className":501548},[31676],[34,501550],{"className":501551,"style":33218},[31680],[34,501553,33206],{"className":501554},[31685,31686],[108,501556,175992,501557,501585,501586,501614],{},[34,501558,501560,501573],{"className":501559},[31642],[34,501561,501563],{"className":501562},[31646],[31648,501564,501565],{"xmlns":31650},[31652,501566,501567,501571],{},[31655,501568,501569],{},[31658,501570,33206],{},[31665,501572,33206],{"encoding":31667},[34,501574,501576],{"className":501575,"ariaHidden":758},[31672],[34,501577,501579,501582],{"className":501578},[31676],[34,501580],{"className":501581,"style":33218},[31680],[34,501583,33206],{"className":501584},[31685,31686]," 是左理想。但它不是右理想（右乘一个矩阵可以将其他列的数据加到第 ",[34,501587,501589,501602],{"className":501588},[31642],[34,501590,501592],{"className":501591},[31646],[31648,501593,501594],{"xmlns":31650},[31652,501595,501596,501600],{},[31655,501597,501598],{},[31658,501599,33759],{},[31665,501601,33759],{"encoding":31667},[34,501603,501605],{"className":501604,"ariaHidden":758},[31672],[34,501606,501608,501611],{"className":501607},[31676],[34,501609],{"className":501610,"style":31681},[31680],[34,501612,33759],{"className":501613,"style":33774},[31685,31686]," 列）。",[32188,501616],{},[19,501618,501620],{"id":501619},"_4-由子集生成的理想-generated-ideals","4. 由子集生成的理想 (Generated Ideals)",[253,501622,501624],{"id":501623},"_41-定义与交集性质","4.1 定义与交集性质",[15,501626,501627,34554,501630,501680,501681,31901,501732,501763,501764,501792,501793,4374,501844,89494,501895,501945,501946,501949,501950,347],{},[2627,501628,501629],{},"定义 3.2.4",[34,501631,501633,501650],{"className":501632},[31642],[34,501634,501636],{"className":501635},[31646],[31648,501637,501638],{"xmlns":31650},[31652,501639,501640,501648],{},[31655,501641,501642,501644,501646],{},[31658,501643,94675],{},[31662,501645,131855],{},[31658,501647,34386],{},[31665,501649,495725],{"encoding":31667},[34,501651,501653,501671],{"className":501652,"ariaHidden":758},[31672],[34,501654,501656,501659,501662,501665,501668],{"className":501655},[31676],[34,501657],{"className":501658,"style":195047},[31680],[34,501660,94675],{"className":501661,"style":106455},[31685,31686],[34,501663],{"className":501664,"style":31846},[31845],[34,501666,131855],{"className":501667},[31850],[34,501669],{"className":501670,"style":31846},[31845],[34,501672,501674,501677],{"className":501673},[31676],[34,501675],{"className":501676,"style":33218},[31680],[34,501678,34386],{"className":501679,"style":34427},[31685,31686],"（或记为 ",[34,501682,501684,501702],{"className":501683},[31642],[34,501685,501687],{"className":501686},[31646],[31648,501688,501689],{"xmlns":31650},[31652,501690,501691,501699],{},[31655,501692,501693,501695,501697],{},[31658,501694,98360],{},[31662,501696,131855],{},[31658,501698,34386],{},[31665,501700,501701],{"encoding":31667},"M \\subseteq R",[34,501703,501705,501723],{"className":501704,"ariaHidden":758},[31672],[34,501706,501708,501711,501714,501717,501720],{"className":501707},[31676],[34,501709],{"className":501710,"style":195047},[31680],[34,501712,98360],{"className":501713,"style":33389},[31685,31686],[34,501715],{"className":501716,"style":31846},[31845],[34,501718,131855],{"className":501719},[31850],[34,501721],{"className":501722,"style":31846},[31845],[34,501724,501726,501729],{"className":501725},[31676],[34,501727],{"className":501728,"style":33218},[31680],[34,501730,34386],{"className":501731,"style":34427},[31685,31686],[2627,501733,35706,501734,501762],{},[34,501735,501737,501750],{"className":501736},[31642],[34,501738,501740],{"className":501739},[31646],[31648,501741,501742],{"xmlns":31650},[31652,501743,501744,501748],{},[31655,501745,501746],{},[31658,501747,94675],{},[31665,501749,94675],{"encoding":31667},[34,501751,501753],{"className":501752,"ariaHidden":758},[31672],[34,501754,501756,501759],{"className":501755},[31676],[34,501757],{"className":501758,"style":33218},[31680],[34,501760,94675],{"className":501761,"style":106455},[31685,31686]," 的最小的理想","称为由 ",[34,501765,501767,501780],{"className":501766},[31642],[34,501768,501770],{"className":501769},[31646],[31648,501771,501772],{"xmlns":31650},[31652,501773,501774,501778],{},[31655,501775,501776],{},[31658,501777,94675],{},[31665,501779,94675],{"encoding":31667},[34,501781,501783],{"className":501782,"ariaHidden":758},[31672],[34,501784,501786,501789],{"className":501785},[31676],[34,501787],{"className":501788,"style":33218},[31680],[34,501790,94675],{"className":501791,"style":106455},[31685,31686]," 生成的理想，记为 ",[34,501794,501796,501814],{"className":501795},[31642],[34,501797,501799],{"className":501798},[31646],[31648,501800,501801],{"xmlns":31650},[31652,501802,501803,501811],{},[31655,501804,501805,501807,501809],{},[31662,501806,932],{},[31658,501808,94675],{},[31662,501810,29622],{},[31665,501812,501813],{"encoding":31667},"\u003CS>",[34,501815,501817,501829],{"className":501816,"ariaHidden":758},[31672],[34,501818,501820,501823,501826],{"className":501819},[31676],[34,501821],{"className":501822,"style":34812},[31680],[34,501824,932],{"className":501825},[31850],[34,501827],{"className":501828,"style":31846},[31845],[34,501830,501832,501835,501838,501841],{"className":501831},[31676],[34,501833],{"className":501834,"style":104901},[31680],[34,501836,94675],{"className":501837,"style":106455},[31685,31686],[34,501839],{"className":501840,"style":31846},[31845],[34,501842,29622],{"className":501843},[31850],[34,501845,501847,501865],{"className":501846},[31642],[34,501848,501850],{"className":501849},[31646],[31648,501851,501852],{"xmlns":31650},[31652,501853,501854,501862],{},[31655,501855,501856,501858,501860],{},[31662,501857,932],{},[31658,501859,98360],{},[31662,501861,29622],{},[31665,501863,501864],{"encoding":31667},"\u003CM>",[34,501866,501868,501880],{"className":501867,"ariaHidden":758},[31672],[34,501869,501871,501874,501877],{"className":501870},[31676],[34,501872],{"className":501873,"style":34812},[31680],[34,501875,932],{"className":501876},[31850],[34,501878],{"className":501879,"style":31846},[31845],[34,501881,501883,501886,501889,501892],{"className":501882},[31676],[34,501884],{"className":501885,"style":104901},[31680],[34,501887,98360],{"className":501888,"style":33389},[31685,31686],[34,501890],{"className":501891,"style":31846},[31845],[34,501893,29622],{"className":501894},[31850],[34,501896,501898,501915],{"className":501897},[31642],[34,501899,501901],{"className":501900},[31646],[31648,501902,501903],{"xmlns":31650},[31652,501904,501905,501913],{},[31655,501906,501907,501909,501911],{},[31662,501908,932],{},[31658,501910,98360],{},[31662,501912,29622],{},[31665,501914,501864],{"encoding":31667},[34,501916,501918,501930],{"className":501917,"ariaHidden":758},[31672],[34,501919,501921,501924,501927],{"className":501920},[31676],[34,501922],{"className":501923,"style":34812},[31680],[34,501925,932],{"className":501926},[31850],[34,501928],{"className":501929,"style":31846},[31845],[34,501931,501933,501936,501939,501942],{"className":501932},[31676],[34,501934],{"className":501935,"style":104901},[31680],[34,501937,98360],{"className":501938,"style":33389},[31685,31686],[34,501940],{"className":501941,"style":31846},[31845],[34,501943,29622],{"className":501944},[31850]," 可以由有限集生成，称该理想是",[2627,501947,501948],{},"有限生成","的。如果由一个元素生成，称为",[2627,501951,501952],{},"主理想",[15,501954,501955,501958,501959,502102,502103,502131,502132,502160,502161,347],{},[2627,501956,501957],{},"命题 3.2.6","（构造性定义）：\n",[34,501960,501962,502000],{"className":501961},[31642],[34,501963,501965],{"className":501964},[31646],[31648,501966,501967],{"xmlns":31650},[31652,501968,501969,501997],{},[31655,501970,501971,501973,501975,501977,501979,501995],{},[31662,501972,932],{},[31658,501974,98360],{},[31662,501976,29622],{},[31662,501978,202],{},[31743,501980,501981,501983],{},[31662,501982,405054],{},[31655,501984,501985,501987,501989,501991,501993],{},[31658,501986,98360],{},[31662,501988,131855],{},[31658,501990,118113],{},[31662,501992,496758],{},[31658,501994,34386],{},[31658,501996,118113],{},[31665,501998,501999],{"encoding":31667},"\u003C M > = \\bigcap_{M \\subseteq H \\triangleleft R} H",[34,502001,502003,502015,502033],{"className":502002,"ariaHidden":758},[31672],[34,502004,502006,502009,502012],{"className":502005},[31676],[34,502007],{"className":502008,"style":34812},[31680],[34,502010,932],{"className":502011},[31850],[34,502013],{"className":502014,"style":31846},[31845],[34,502016,502018,502021,502024,502027,502030],{"className":502017},[31676],[34,502019],{"className":502020,"style":104901},[31680],[34,502022,98360],{"className":502023,"style":33389},[31685,31686],[34,502025],{"className":502026,"style":31846},[31845],[34,502028,2075],{"className":502029},[31850],[34,502031],{"className":502032,"style":31846},[31845],[34,502034,502036,502040,502096,502099],{"className":502035},[31676],[34,502037],{"className":502038,"style":502039},[31680],"height:1.1449em;vertical-align:-0.3949em;",[34,502041,502043,502046],{"className":502042},[32767],[34,502044,405054],{"className":502045,"style":156561},[32767,131249,156560],[34,502047,502049],{"className":502048},[31792],[34,502050,502052,502087],{"className":502051},[31796,31797],[34,502053,502055,502084],{"className":502054},[31801],[34,502056,502058],{"className":502057,"style":156574},[31805],[34,502059,502060,502063],{"style":156577},[34,502061],{"className":502062,"style":31814},[31813],[34,502064,502066],{"className":502065},[31818,31819,31820,31821],[34,502067,502069,502072,502075,502078,502081],{"className":502068},[31685,31821],[34,502070,98360],{"className":502071,"style":33389},[31685,31686,31821],[34,502073,131855],{"className":502074},[31850,31821],[34,502076,118113],{"className":502077,"style":118235},[31685,31686,31821],[34,502079,496758],{"className":502080},[32024,31821],[34,502082,34386],{"className":502083,"style":34427},[31685,31686,31821],[34,502085,31829],{"className":502086},[31828],[34,502088,502090],{"className":502089},[31801],[34,502091,502094],{"className":502092,"style":502093},[31805],"height:0.3949em;",[34,502095],{},[34,502097],{"className":502098,"style":32123},[31845],[34,502100,118113],{"className":502101,"style":118235},[31685,31686],"\n即：所有包含 ",[34,502104,502106,502119],{"className":502105},[31642],[34,502107,502109],{"className":502108},[31646],[31648,502110,502111],{"xmlns":31650},[31652,502112,502113,502117],{},[31655,502114,502115],{},[31658,502116,98360],{},[31665,502118,98360],{"encoding":31667},[34,502120,502122],{"className":502121,"ariaHidden":758},[31672],[34,502123,502125,502128],{"className":502124},[31676],[34,502126],{"className":502127,"style":33218},[31680],[34,502129,98360],{"className":502130,"style":33389},[31685,31686]," 的理想的交集。特别地，如果 ",[34,502133,502135,502148],{"className":502134},[31642],[34,502136,502138],{"className":502137},[31646],[31648,502139,502140],{"xmlns":31650},[31652,502141,502142,502146],{},[31655,502143,502144],{},[31658,502145,98360],{},[31665,502147,98360],{"encoding":31667},[34,502149,502151],{"className":502150,"ariaHidden":758},[31672],[34,502152,502154,502157],{"className":502153},[31676],[34,502155],{"className":502156,"style":33218},[31680],[34,502158,98360],{"className":502159,"style":33389},[31685,31686]," 本身就是理想，则 ",[34,502162,502164,502186],{"className":502163},[31642],[34,502165,502167],{"className":502166},[31646],[31648,502168,502169],{"xmlns":31650},[31652,502170,502171,502183],{},[31655,502172,502173,502175,502177,502179,502181],{},[31662,502174,932],{},[31658,502176,98360],{},[31662,502178,29622],{},[31662,502180,202],{},[31658,502182,98360],{},[31665,502184,502185],{"encoding":31667},"\u003CM> = M",[34,502187,502189,502201,502219],{"className":502188,"ariaHidden":758},[31672],[34,502190,502192,502195,502198],{"className":502191},[31676],[34,502193],{"className":502194,"style":34812},[31680],[34,502196,932],{"className":502197},[31850],[34,502199],{"className":502200,"style":31846},[31845],[34,502202,502204,502207,502210,502213,502216],{"className":502203},[31676],[34,502205],{"className":502206,"style":104901},[31680],[34,502208,98360],{"className":502209,"style":33389},[31685,31686],[34,502211],{"className":502212,"style":31846},[31845],[34,502214,2075],{"className":502215},[31850],[34,502217],{"className":502218,"style":31846},[31845],[34,502220,502222,502225],{"className":502221},[31676],[34,502223],{"className":502224,"style":33218},[31680],[34,502226,98360],{"className":502227,"style":33389},[31685,31686],[253,502229,502231],{"id":502230},"_42-生成理想的元素形式公式解析","4.2 生成理想的元素形式（公式解析）",[15,502233,502234],{},"根据环的不同性质（是否有单位元、是否交换），生成理想中元素的具体样子是不同的。这是因为我们需要通过“补丁”来强行满足理想的定义（加法封闭、左右乘封闭）。",[15,502236,502237],{},[2627,502238,502239],{},"情况 1：最一般的环（非交换、无单位元）",[34,502241,502243],{"className":502242},[126385],[34,502244,502246,502442],{"className":502245},[31642],[34,502247,502249],{"className":502248},[31646],[31648,502250,502251],{"xmlns":31650,"display":126394},[31652,502252,502253,502439],{},[31655,502254,502255,502257,502259,502261,502263],{},[31662,502256,932],{},[31658,502258,98360],{},[31662,502260,29622],{},[31662,502262,202],{},[31655,502264,502265,502267,502274,502276,502282,502288,502290,502296,502304,502306,502318,502326,502328,502340,502354,502368,502370,502372,502376,502378,502437],{},[31662,502266,32685],{"fence":758},[71115,502268,502269,502271],{},[31662,502270,131138],{},[39361,502272,502273],{},"finite",[31662,502275,456],{"stretchy":702},[31743,502277,502278,502280],{},[31658,502279,111],{},[31658,502281,31955],{},[31743,502283,502284,502286],{},[31658,502285,31759],{},[31658,502287,31955],{},[31662,502289,223],{},[31743,502291,502292,502294],{},[31658,502293,34395],{},[31658,502295,31968],{},[38846,502297,502298,502300,502302],{},[31658,502299,31759],{},[31658,502301,31968],{},[31662,502303,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,502305,223],{},[38846,502307,502308,502310,502312],{},[31658,502309,31759],{},[31658,502311,33759],{},[31655,502313,502314,502316],{},[31662,502315,116495],{"mathvariant":38079},[31662,502317,116495],{"mathvariant":38079},[38846,502319,502320,502322,502324],{},[31658,502321,34395],{},[31658,502323,33759],{},[31662,502325,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,502327,223],{},[38846,502329,502330,502332,502334],{},[31658,502331,34395],{},[31658,502333,33249],{},[31655,502335,502336,502338],{},[31662,502337,116495],{"mathvariant":38079},[31662,502339,116495],{"mathvariant":38079},[38846,502341,502342,502344,502346],{},[31658,502343,31759],{},[31658,502345,33249],{},[31655,502347,502348,502350,502352],{},[31662,502349,116495],{"mathvariant":38079},[31662,502351,116495],{"mathvariant":38079},[31662,502353,116495],{"mathvariant":38079},[38846,502355,502356,502358,502360],{},[31658,502357,34395],{},[31658,502359,33249],{},[31655,502361,502362,502364,502366],{},[31662,502363,116495],{"mathvariant":38079},[31662,502365,116495],{"mathvariant":38079},[31662,502367,116495],{"mathvariant":38079},[31662,502369,2557],{"stretchy":702},[39361,502371,262622],{},[31662,502373,502375],{"fence":758,"lspace":502374,"rspace":502374},"0.05em","|",[39361,502377,262622],{},[71078,502379,502380,502407],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,502381,502382,502388],{},[71087,502383,502384],{},[71090,502385,502386],{"scriptlevel":459,"displaystyle":758},[31655,502387],{},[71087,502389,502390],{},[71090,502391,502392],{"scriptlevel":459,"displaystyle":758},[31655,502393,502394,502396,502398,502400,502402,502405],{},[31655,502395],{},[31658,502397,111],{},[31662,502399,112296],{},[31658,502401,97300],{"mathvariant":318416},[39361,502403,502404],{}," (整数倍)",[31662,502406,31960],{"separator":758},[71084,502408,502409,502415],{},[71087,502410,502411],{},[71090,502412,502413],{"scriptlevel":459,"displaystyle":758},[31655,502414],{},[71087,502416,502417],{},[71090,502418,502419],{"scriptlevel":459,"displaystyle":758},[31655,502420,502421,502423,502425,502427,502429,502431,502433,502435],{},[31655,502422],{},[31658,502424,31759],{},[31662,502426,112296],{},[31658,502428,98360],{},[31662,502430,31960],{"separator":758},[31658,502432,34395],{},[31662,502434,112296],{},[31658,502436,34386],{},[31662,502438,32715],{"fence":758},[31665,502440,502441],{"encoding":31667},"\u003C M > = \\left\\{ \\sum_{\\text{finite}} (a_i m_i + r_j m_j' + m_k'' r_k' + r_t'' m_t''' r_t''') \\;\\middle|\\; \\begin{aligned} &a \\in \\mathbb{Z} \\text{ (整数倍)}, \\\\ &m \\in M, r \\in R \\end{aligned} \\right\\}",[34,502443,502445,502457,502475],{"className":502444,"ariaHidden":758},[31672],[34,502446,502448,502451,502454],{"className":502447},[31676],[34,502449],{"className":502450,"style":34812},[31680],[34,502452,932],{"className":502453},[31850],[34,502455],{"className":502456,"style":31846},[31845],[34,502458,502460,502463,502466,502469,502472],{"className":502459},[31676],[34,502461],{"className":502462,"style":104901},[31680],[34,502464,98360],{"className":502465,"style":33389},[31685,31686],[34,502467],{"className":502468,"style":31846},[31845],[34,502470,2075],{"className":502471},[31850],[34,502473],{"className":502474,"style":31846},[31845],[34,502476,502478,502482],{"className":502477},[31676],[34,502479],{"className":502480,"style":502481},[31680],"height:3.0521em;vertical-align:-1.3021em;",[34,502483,502485,502491,502542,502545,502585,502625,502628,502631,502634,502674,502730,502733,502736,502739,502794,502849,502852,502855,502858,502912,502967,503021,503024,503027,503072,503075,503238],{"className":502484},[34006],[34,502486,502488],{"className":502487,"style":75811},[31782,75810],[34,502489,32685],{"className":502490},[71281,75815],[34,502492,502494],{"className":502493},[32767,71420],[34,502495,502497,502533],{"className":502496},[31796,31797],[34,502498,502500,502530],{"className":502499},[31801],[34,502501,502503,502520],{"className":502502,"style":131215},[31805],[34,502504,502505,502508],{"style":410450},[34,502506],{"className":502507,"style":131222},[31813],[34,502509,502511],{"className":502510},[31818,31819,31820,31821],[34,502512,502514],{"className":502513},[31685,31821],[34,502515,502517],{"className":502516},[31685,2638,31821],[34,502518,502273],{"className":502519},[31685,31821],[34,502521,502522,502525],{"style":131240},[34,502523],{"className":502524,"style":131222},[31813],[34,502526,502527],{},[34,502528,131138],{"className":502529},[32767,131249,131250],[34,502531,31829],{"className":502532},[31828],[34,502534,502536],{"className":502535},[31801],[34,502537,502540],{"className":502538,"style":502539},[31805],"height:1.3021em;",[34,502541],{},[34,502543,456],{"className":502544},[31782],[34,502546,502548,502551],{"className":502547},[31685],[34,502549,111],{"className":502550},[31685,31686],[34,502552,502554],{"className":502553},[31792],[34,502555,502557,502577],{"className":502556},[31796,31797],[34,502558,502560,502574],{"className":502559},[31801],[34,502561,502563],{"className":502562,"style":32090},[31805],[34,502564,502565,502568],{"style":31809},[34,502566],{"className":502567,"style":31814},[31813],[34,502569,502571],{"className":502570},[31818,31819,31820,31821],[34,502572,31955],{"className":502573},[31685,31686,31821],[34,502575,31829],{"className":502576},[31828],[34,502578,502580],{"className":502579},[31801],[34,502581,502583],{"className":502582,"style":31836},[31805],[34,502584],{},[34,502586,502588,502591],{"className":502587},[31685],[34,502589,31759],{"className":502590},[31685,31686],[34,502592,502594],{"className":502593},[31792],[34,502595,502597,502617],{"className":502596},[31796,31797],[34,502598,502600,502614],{"className":502599},[31801],[34,502601,502603],{"className":502602,"style":32090},[31805],[34,502604,502605,502608],{"style":31809},[34,502606],{"className":502607,"style":31814},[31813],[34,502609,502611],{"className":502610},[31818,31819,31820,31821],[34,502612,31955],{"className":502613},[31685,31686,31821],[34,502615,31829],{"className":502616},[31828],[34,502618,502620],{"className":502619},[31801],[34,502621,502623],{"className":502622,"style":31836},[31805],[34,502624],{},[34,502626],{"className":502627,"style":32604},[31845],[34,502629,223],{"className":502630},[32024],[34,502632],{"className":502633,"style":32604},[31845],[34,502635,502637,502640],{"className":502636},[31685],[34,502638,34395],{"className":502639,"style":34452},[31685,31686],[34,502641,502643],{"className":502642},[31792],[34,502644,502646,502666],{"className":502645},[31796,31797],[34,502647,502649,502663],{"className":502648},[31801],[34,502650,502652],{"className":502651,"style":32090},[31805],[34,502653,502654,502657],{"style":34467},[34,502655],{"className":502656,"style":31814},[31813],[34,502658,502660],{"className":502659},[31818,31819,31820,31821],[34,502661,31968],{"className":502662,"style":32153},[31685,31686,31821],[34,502664,31829],{"className":502665},[31828],[34,502667,502669],{"className":502668},[31801],[34,502670,502672],{"className":502671,"style":32163},[31805],[34,502673],{},[34,502675,502677,502680],{"className":502676},[31685],[34,502678,31759],{"className":502679},[31685,31686],[34,502681,502683],{"className":502682},[31792],[34,502684,502686,502721],{"className":502685},[31796,31797],[34,502687,502689,502718],{"className":502688},[31801],[34,502690,502692,502704],{"className":502691,"style":131724},[31805],[34,502693,502695,502698],{"style":502694},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[34,502696],{"className":502697,"styl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0 333 3000",[20398,503058],{"d":503059},"M145 15 v585 v1800 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1800 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1800 v585 h43z",[34,503061,31829],{"className":503062},[31828],[34,503064,503066],{"className":503065},[31801],[34,503067,503070],{"className":503068,"style":503069},[31805],"height:1.25em;",[34,503071],{},[34,503073],{"className":503074,"style":31846},[31845],[34,503076,503078],{"className":503077},[31685],[34,503079,503081,503125],{"className":503080},[71078],[34,503082,503084],{"className":503083},[369003],[34,503085,503087,503116],{"className":503086},[31796,31797],[34,503088,503090,503113],{"className":503089},[31801],[34,503091,503094,503104],{"className":503092,"style":503093},[31805],"height:1.6em;",[34,503095,503097,503101],{"style":503096},"top:-3.6em;",[34,503098],{"className":503099,"style":503100},[31813],"height:2.84em;",[34,503102],{"className":503103},[31685],[34,503105,503107,503110],{"style":503106},"top:-2.1em;",[34,503108],{"className":503109,"style":503100},[31813],[34,503111],{"className":503112},[31685],[34,503114,31829],{"className":503115},[31828],[34,503117,503119],{"className":503118},[31801],[34,503120,503123],{"className":503121,"style":503122},[31805],"height:1.1em;",[34,503124],{},[34,503126,503128],{"className":503127},[71384],[34,503129,503131,503230],{"className":503130},[31796,31797],[34,503132,503134,503227],{"className":503133},[31801],[34,503135,503137,503180],{"className":503136,"style":503093},[31805],[34,503138,503139,503142],{"style":452328},[34,503140],{"className":503141,"style":41508},[31813],[34,503143,503145,503148,503151,503154,503157,503160,503163,503177],{"className":503144},[31685],[34,503146],{"className":503147},[31685],[34,503149,111],{"className":503150},[31685,31686],[34,503152],{"className":503153,"style":31846},[31845],[34,503155,112296],{"className":503156},[31850],[34,503158],{"className":503159,"style":31846},[31845],[34,503161,97300],{"className":503162},[31685,318441],[34,503164,503166,503170,503174],{"className":503165},[31685,2638],[34,503167,503169],{"className":503168},[31685]," (",[34,503171,503173],{"className":503172},[31685,105462],"整数倍",[34,503175,2557],{"className":503176},[31685],[34,503178,31960],{"className":503179},[32119],[34,503181,503182,503185],{"style":452337},[34,503183],{"className":503184,"style":41508},[31813],[34,503186,503188,503191,503194,503197,503200,503203,503206,503209,503212,503215,503218,503221,503224],{"className":503187},[31685],[34,503189],{"className":503190},[31685],[34,503192,31759],{"className":503193},[31685,31686],[34,503195],{"className":503196,"style":31846},[31845],[34,503198,112296],{"className":503199},[31850],[34,503201],{"className":503202,"style":31846},[31845],[34,503204,98360],{"className":503205,"style":33389},[31685,31686],[34,503207,31960],{"className":503208},[32119],[34,503210],{"className":503211,"style":32123},[31845],[34,503213,34395],{"className":503214,"style":34452},[31685,31686],[34,503216],{"className":503217,"style":31846},[31845],[34,503219,112296],{"className":503220},[31850],[34,503222],{"className":503223,"style":31846},[31845],[34,503225,34386],{"className":503226,"style":34427},[31685,31686],[34,503228,31829],{"className":503229},[31828],[34,503231,503233],{"className":503232},[31801],[34,503234,503236],{"className":503235,"style":503122},[31805],[34,503237],{},[34,503239,503241],{"className":503240,"style":75811},[31690,75810],[34,503242,32715],{"className":503243},[71281,75815],[105,503245,503246],{},[108,503247,503248,33183,503251],{},[2627,503249,503250],{},"解读",[105,503252,503253,503372,503510,503668,503710],{},[108,503254,503255,503371],{},[34,503256,503258,503282],{"className":503257},[31642],[34,503259,503261],{"className":503260},[31646],[31648,503262,503263],{"xmlns":31650},[31652,503264,503265,503279],{},[31655,503266,503267,503273],{},[31743,503268,503269,503271],{},[31658,503270,111],{},[31658,503272,31955],{},[31743,503274,503275,503277],{},[31658,503276,31759],{},[31658,503278,31955],{},[31665,503280,503281],{"encoding":31667},"a_i m_i",[34,503283,503285],{"className":503284,"ariaHidden":758},[31672],[34,503286,503288,503291,503331],{"className":503287},[31676],[34,503289],{"className":503290,"style":31860},[31680],[34,503292,503294,503297],{"className":503293},[31685],[34,503295,111],{"className":503296},[31685,31686],[34,503298,503300],{"className":503299},[31792],[34,503301,503303,503323],{"className":503302},[31796,31797],[34,503304,503306,503320],{"className":503305},[31801],[34,503307,503309],{"className":503308,"style":32090},[31805],[34,503310,503311,503314],{"style":31809},[34,503312],{"className":503313,"style":31814},[31813],[34,503315,503317],{"className":503316},[31818,31819,31820,31821],[34,503318,31955],{"className":503319},[31685,31686,31821],[34,503321,31829],{"className":503322},[31828],[34,503324,503326],{"className":503325},[31801],[34,503327,503329],{"className":503328,"style":31836},[31805],[34,503330],{},[34,503332,503334,503337],{"className":503333},[31685],[34,503335,31759],{"className":503336},[31685,31686],[34,503338,503340],{"className":503339},[31792],[34,503341,503343,503363],{"className":503342},[31796,31797],[34,503344,503346,503360],{"className":503345},[31801],[34,503347,503349],{"className":503348,"style":32090},[31805],[34,503350,503351,503354],{"style":31809},[34,503352],{"className":503353,"style":31814},[31813],[34,503355,503357],{"className":503356},[31818,31819,31820,31821],[34,503358,31955],{"className":503359},[31685,31686,31821],[34,503361,31829],{"className":503362},[31828],[34,503364,503366],{"className":503365},[31801],[34,503367,503369],{"className":503368,"style":31836},[31805],[34,503370],{},"：保证它是加法子群（整数倍）。",[108,503373,503374,503509],{},[34,503375,503377,503403],{"className":503376},[31642],[34,503378,503380],{"className":503379},[31646],[31648,503381,503382],{"xmlns":31650},[31652,503383,503384,503400],{},[31655,503385,503386,503392],{},[31743,503387,503388,503390],{},[31658,503389,34395],{},[31658,503391,31968],{},[38846,503393,503394,503396,503398],{},[31658,503395,31759],{},[31658,503397,31968],{},[31662,503399,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,503401,503402],{"encoding":31667},"r_j m_j'",[34,503404,503406],{"className":503405,"ariaHidden":758},[31672],[34,503407,503409,503413,503453],{"className":503408},[31676],[34,503410],{"className":503411,"style":503412},[31680],"height:1.1467em;vertical-align:-0.3948em;",[34,503414,503416,503419],{"className":503415},[31685],[34,503417,34395],{"className":503418,"style":34452},[31685,31686],[34,503420,503422],{"className":503421},[31792],[34,503423,503425,503445],{"className":503424},[31796,31797],[34,503426,503428,503442],{"className":503427},[31801],[34,503429,503431],{"className":503430,"style":32090},[31805],[34,503432,503433,503436],{"style":34467},[34,503434],{"className":503435,"style":31814},[31813],[34,503437,503439],{"className":503438},[31818,31819,31820,31821],[34,503440,31968],{"className":503441,"style":32153},[31685,31686,31821],[34,503443,31829],{"className":503444},[31828],[34,503446,503448],{"className":503447},[31801],[34,503449,503451],{"className":503450,"style":32163},[31805],[34,503452],{},[34,503454,503456,503459],{"className":503455},[31685],[34,503457,31759],{"className":503458},[31685,31686],[34,503460,503462],{"className":503461},[31792],[34,503463,503465,503500],{"className":503464},[31796,31797],[34,503466,503468,503497],{"className":503467},[31801],[34,503469,503471,503483],{"className":503470,"style":132027},[31805],[34,503472,503474,503477],{"style":503473},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[34,503475],{"className":503476,"style":31814},[31813],[34,503478,503480],{"className":503479},[31818,31819,31820,31821],[34,503481,31968],{"className":503482,"style":32153},[31685,31686,31821],[34,503484,503485,503488],{"style":37777},[34,503486],{"className":503487,"style":31814},[31813],[34,503489,503491],{"className":503490},[31818,31819,31820,31821],[34,503492,503494],{"className":503493},[31685,31821],[34,503495,116495],{"className":503496},[31685,31821],[34,503498,31829],{"className":503499},[31828],[34,503501,503503],{"className":503502},[31801],[34,503504,503507],{"className":503505,"style":503506},[31805],"height:0.3948em;",[34,503508],{},"：保证左吸收律（左乘）。",[108,503511,503512,503667],{},[34,503513,503515,503547],{"className":503514},[31642],[34,503516,503518],{"className":503517},[31646],[31648,503519,503520],{"xmlns":31650},[31652,503521,503522,503544],{},[31655,503523,503524,503536],{},[38846,503525,503526,503528,503530],{},[31658,503527,31759],{},[31658,503529,33759],{},[31655,503531,503532,503534],{},[31662,503533,116495],{"mathvariant":38079},[31662,503535,116495],{"mathvariant":38079},[38846,503537,503538,503540,503542],{},[31658,503539,34395],{},[31658,503541,33759],{},[31662,503543,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,503545,503546],{"encoding":31667},"m_k'' r_k'",[34,503548,503550],{"className":503549,"ariaHidden":758},[31672],[34,503551,503553,503557,503612],{"className":503552},[31676],[34,503554],{"className":503555,"style":503556},[31680],"height:1.035em;vertical-align:-0.2831em;",[34,503558,503560,503563],{"className":503559},[31685],[34,503561,31759],{"className":503562},[31685,31686],[34,503564,503566],{"className":503565},[31792],[34,503567,503569,503604],{"className":503568},[31796,31797],[34,503570,503572,503601],{"className":503571},[31801],[34,503573,503575,503587],{"className":503574,"style":132027},[31805],[34,503576,503578,503581],{"style":503577},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[34,503579],{"className":503580,"style":31814},[31813],[34,503582,503584],{"className":503583},[31818,31819,31820,31821],[34,503585,33759],{"className":503586,"style":33774},[31685,31686,31821],[34,503588,503589,503592],{"style":37777},[34,503590],{"className":503591,"style":31814},[31813],[34,503593,503595],{"className":503594},[31818,31819,31820,31821],[34,503596,503598],{"className":503597},[31685,31821],[34,503599,502782],{"className":503600},[31685,31821],[34,503602,31829],{"className":503603},[31828],[34,503605,503607],{"className":503606},[31801],[34,503608,503610],{"className":503609,"style":39254},[31805],[34,503611],{},[34,503613,503615,503618],{"className":503614},[31685],[34,503616,34395],{"className":503617,"style":34452},[31685,31686],[34,503619,503621],{"className":503620},[31792],[34,503622,503624,503659],{"className":503623},[31796,31797],[34,503625,503627,503656],{"className":503626},[31801],[34,503628,503630,503642],{"className":503629,"style":132027},[31805],[34,503631,503633,503636],{"style":503632},"top:-2.4169em;margin-left:-0.0278em;margin-right:0.05em;",[34,503634],{"className":503635,"style":31814},[31813],[34,503637,503639],{"className":503638},[31818,31819,31820,31821],[34,503640,33759],{"className":503641,"style":33774},[31685,31686,31821],[34,503643,503644,503647],{"style":37777},[34,503645],{"className":503646,"style":31814},[31813],[34,503648,503650],{"className":503649},[31818,31819,31820,31821],[34,503651,503653],{"className":503652},[31685,31821],[34,503654,116495],{"className":503655},[31685,31821],[34,503657,31829],{"className":503658},[31828],[34,503660,503662],{"className":503661},[31801],[34,503663,503665],{"className":503664,"style":39254},[31805],[34,503666],{},"：保证右吸收律（右乘）。",[108,503669,503670,503709],{},[34,503671,503673,503691],{"className":503672},[31642],[34,503674,503676],{"className":503675},[31646],[31648,503677,503678],{"xmlns":31650},[31652,503679,503680,503688],{},[31655,503681,503682,503684,503686],{},[31658,503683,34395],{},[31658,503685,31759],{},[31658,503687,34395],{},[31665,503689,503690],{"encoding":31667},"r m r",[34,503692,503694],{"className":503693,"ariaHidden":758},[31672],[34,503695,503697,503700,503703,503706],{"className":503696},[31676],[34,503698],{"className":503699,"style":32527},[31680],[34,503701,34395],{"className":503702,"style":34452},[31685,31686],[34,503704,31759],{"className":503705},[31685,31686],[34,503707,34395],{"className":503708,"style":34452},[31685,31686],"：保证双侧乘积。",[108,503711,503712,503713,503741],{},"这四种形式的有限和，是包含 ",[34,503714,503716,503729],{"className":503715},[31642],[34,503717,503719],{"className":503718},[31646],[31648,503720,503721],{"xmlns":31650},[31652,503722,503723,503727],{},[31655,503724,503725],{},[31658,503726,98360],{},[31665,503728,98360],{"encoding":31667},[34,503730,503732],{"className":503731,"ariaHidden":758},[31672],[34,503733,503735,503738],{"className":503734},[31676],[34,503736],{"className":503737,"style":33218},[31680],[34,503739,98360],{"className":503740,"style":33389},[31685,31686]," 且满足理想定义的最小集合。",[15,503743,503744,471826,503747,503808],{},[2627,503745,503746],{},"情况 2：交换环 (Commutative Ring)",[34,503748,503750,503772],{"className":503749},[31642],[34,503751,503753],{"className":503752},[31646],[31648,503754,503755],{"xmlns":31650},[31652,503756,503757,503769],{},[31655,503758,503759,503761,503763,503765,503767],{},[31658,503760,34395],{},[31658,503762,31759],{},[31662,503764,202],{},[31658,503766,31759],{},[31658,503768,34395],{},[31665,503770,503771],{"encoding":31667},"rm = mr",[34,503773,503775,503796],{"className":503774,"ariaHidden":758},[31672],[34,503776,503778,503781,503784,503787,503790,503793],{"className":503777},[31676],[34,503779],{"className":503780,"style":32527},[31680],[34,503782,34395],{"className":503783,"style":34452},[31685,31686],[34,503785,31759],{"className":503786},[31685,31686],[34,503788],{"className":503789,"style":31846},[31845],[34,503791,202],{"className":503792},[31850],[34,503794],{"className":503795,"style":31846},[31845],[34,503797,503799,503802,503805],{"className":503798},[31676],[34,503800],{"className":503801,"style":32527},[31680],[34,503803,31759],{"className":503804},[31685,31686],[34,503806,34395],{"className":503807,"style":34452},[31685,31686],"，左右乘不分家。",[34,503810,503812],{"className":503811},[126385],[34,503813,503815,503903],{"className":503814},[31642],[34,503816,503818],{"className":503817},[31646],[31648,503819,503820],{"xmlns":31650,"display":126394},[31652,503821,503822,503900],{},[31655,503823,503824,503826,503828,503830,503832,503892,503894,503896,503898],{},[31662,503825,932],{},[31658,503827,98360],{},[31662,503829,29622],{},[31662,503831,202],{},[31655,503833,503834,503836,503842,503848,503854,503862,503864,503866,503868,503870,503872,503878,503880,503882,503884,503886,503888,503890],{},[31662,503835,32685],{"fence":758},[71115,503837,503838,503840],{},[31662,503839,131138],{},[39361,503841,502273],{},[31743,503843,503844,503846],{},[31658,503845,34395],{},[31658,503847,31955],{},[31743,503849,503850,503852],{},[31658,503851,31759],{},[31658,503853,31955],{},[38846,503855,503856,503858,503860],{},[31658,503857,34395],{},[31658,503859,31955],{},[31662,503861,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[39361,503863,262622],{},[31662,503865,502375],{"fence":758,"lspace":502374,"rspace":502374},[39361,503867,262622],{},[31658,503869,34395],{},[31662,503871,31960],{"separator":758},[37624,503873,503874,503876],{},[31658,503875,34395],{},[31662,503877,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,503879,112296],{},[31658,503881,34386],{},[31662,503883,31960],{"separator":758},[31658,503885,31759],{},[31662,503887,112296],{},[31658,503889,98360],{},[31662,503891,32715],{"fence":758},[31662,503893,202],{},[31658,503895,34386],{},[31658,503897,98360],{},[31658,503899,34386],{},[31665,503901,503902],{"encoding":31667},"\u003C M > = \\left\\{ \\sum_{\\text{finite}} r_i m_i r_i' \\;\\middle|\\; r, r' \\in R, m \\in M \\right\\} = RMR",[34,503904,503906,503918,503936,504269],{"className":503905,"ariaHidden":758},[31672],[34,503907,503909,503912,503915],{"className":503908},[31676],[34,503910],{"className":503911,"style":34812},[31680],[34,503913,932],{"className":503914},[31850],[34,503916],{"className":503917,"style":31846},[31845],[34,503919,503921,503924,503927,503930,503933],{"className":503920},[31676],[34,503922],{"className":503923,"style":104901},[31680],[34,503925,98360],{"className":503926,"style":33389},[31685,31686],[34,503928],{"className":503929,"style":31846},[31845],[34,503931,2075],{"className":503932},[31850],[34,503934],{"className":503935,"style":31846},[31845],[34,503937,503939,503942,504260,504263,504266],{"className":503938},[31676],[34,503940],{"className":503941,"style":502481},[31680],[34,503943,503945,503951,504001,504004,504044,504084,504138,504141,504177,504180,504183,504186,504189,504221,504224,504227,504230,504233,504236,504239,504242,504245,504248,504251,504254],{"className":503944},[34006],[34,503946,503948],{"className":503947,"style":75811},[31782,75810],[34,503949,32685],{"className":503950},[71281,75815],[34,503952,503954],{"className":503953},[32767,71420],[34,503955,503957,503993],{"className":503956},[31796,31797],[34,503958,503960,503990],{"className":503959},[31801],[34,503961,503963,503980],{"className":503962,"style":131215},[31805],[34,503964,503965,503968],{"style":410450},[34,503966],{"className":503967,"style":131222},[31813],[34,503969,503971],{"className":503970},[31818,31819,31820,31821],[34,503972,503974],{"className":503973},[31685,31821],[34,503975,503977],{"className":503976},[31685,2638,31821],[34,503978,502273],{"className":503979},[31685,31821],[34,503981,503982,503985],{"style":131240},[34,503983],{"className":503984,"style":131222},[31813],[34,503986,503987],{},[34,503988,131138],{"className":503989},[32767,131249,131250],[34,503991,31829],{"className":503992},[31828],[34,503994,503996],{"className":503995},[31801],[34,503997,503999],{"className":503998,"style":502539},[31805],[34,504000],{},[34,504002],{"className":504003,"style":32123},[31845],[34,504005,504007,504010],{"className":504006},[31685],[34,504008,34395],{"className":504009,"style":34452},[31685,31686],[34,504011,504013],{"className":504012},[31792],[34,504014,504016,504036],{"className":504015},[31796,31797],[34,504017,504019,504033],{"className":504018},[31801],[34,504020,504022],{"className":504021,"style":32090},[31805],[34,504023,504024,504027],{"style":34467},[34,504025],{"className":504026,"style":31814},[31813],[34,504028,504030],{"className":504029},[31818,31819,31820,31821],[34,504031,31955],{"className":504032},[31685,31686,31821],[34,504034,31829],{"className":504035},[31828],[34,504037,504039],{"className":504038},[31801],[34,504040,504042],{"className":504041,"style":31836},[31805],[34,504043],{},[34,504045,504047,504050],{"className":504046},[31685],[34,504048,31759],{"className":504049},[31685,31686],[34,504051,504053],{"className":504052},[31792],[34,504054,504056,504076],{"className":504055},[31796,31797],[34,504057,504059,504073],{"className":504058},[31801],[34,504060,504062],{"className":504061,"style":32090},[31805],[34,504063,504064,504067],{"style":31809},[34,504065],{"className":504066,"style":31814},[31813],[34,504068,504070],{"className":504069},[31818,31819,31820,31821],[34,504071,31955],{"className":504072},[31685,31686,31821],[34,504074,31829],{"className":504075},[31828],[34,504077,504079],{"className":504078},[31801],[34,504080,504082],{"className":504081,"style":31836},[31805],[34,504083],{},[34,504085,504087,504090],{"className":504086},[31685],[34,504088,34395],{"className":504089,"style":34452},[31685,31686],[34,504091,504093],{"className":504092},[31792],[34,504094,504096,504130],{"className":504095},[31796,31797],[34,504097,504099,504127],{"className":504098},[31801],[34,504100,504102,504113],{"className":504101,"style":131724},[31805],[34,504103,504104,504107],{"style":502814},[34,504105],{"className":504106,"style":31814},[31813],[34,504108,504110],{"className":504109},[31818,31819,31820,31821],[34,504111,31955],{"className":504112},[31685,31686,31821],[34,504114,504115,504118],{"style":129846},[34,504116],{"className":504117,"style":31814},[31813],[34,504119,504121],{"className":504120},[31818,31819,31820,31821],[34,504122,504124],{"className":504123},[31685,31821],[34,504125,116495],{"className":504126},[31685,31821],[34,504128,31829],{"className":504129},[31828],[34,504131,504133],{"className":504132},[31801],[34,504134,504136],{"className":504135,"style":52354},[31805],[34,504137],{},[34,504139],{"className":504140,"style":31846},[31845],[34,504142,504143],{},[34,504144,504146],{"className":504145},[71281,71282],[34,504147,504149,504169],{"className":504148},[31796,31797],[34,504150,504152,504166],{"className":504151},[31801],[34,504153,504155],{"className":504154,"style":503041},[31805],[34,504156,504157,504160],{"style":503044},[34,504158],{"className":504159,"style":503048},[31813],[34,504161,504162],{"style":503051},[41588,504163,504164],{"xmlns":41590,"width":503054,"height":503055,"viewBox":503056},[20398,504165],{"d":503059},[34,504167,31829],{"className":504168},[31828],[34,504170,504172],{"className":504171},[31801],[34,504173,504175],{"className":504174,"style":503069},[31805],[34,504176],{},[34,504178],{"className":504179,"style":31846},[31845],[34,504181,34395],{"className":504182,"style":34452},[31685,31686],[34,504184,31960],{"className":504185},[32119],[34,504187],{"className":504188,"style":32123},[31845],[34,504190,504192,504195],{"className":504191},[31685],[34,504193,34395],{"className":504194,"style":34452},[31685,31686],[34,504196,504198],{"className":504197},[31792],[34,504199,504201],{"className":504200},[31796],[34,504202,504204],{"className":504203},[31801],[34,504205,504207],{"className":504206,"style":131724},[31805],[34,504208,504209,504212],{"style":129846},[34,504210],{"className":504211,"style":31814},[31813],[34,504213,504215],{"className":504214},[31818,31819,31820,31821],[34,504216,504218],{"className":504217},[31685,31821],[34,504219,116495],{"className":504220},[31685,31821],[34,504222],{"className":504223,"style":31846},[31845],[34,504225,112296],{"className":504226},[31850],[34,504228],{"className":504229,"style":31846},[31845],[34,504231,34386],{"className":504232,"style":34427},[31685,31686],[34,504234,31960],{"className":504235},[32119],[34,504237],{"className":504238,"style":32123},[31845],[34,504240,31759],{"className":504241},[31685,31686],[34,504243],{"className":504244,"style":31846},[31845],[34,504246,112296],{"className":504247},[31850],[34,504249],{"className":504250,"style":31846},[31845],[34,504252,98360],{"className":504253,"style":33389},[31685,31686],[34,504255,504257],{"className":504256,"style":75811},[31690,75810],[34,504258,32715],{"className":504259},[71281,75815],[34,504261],{"className":504262,"style":31846},[31845],[34,504264,202],{"className":504265},[31850],[34,504267],{"className":504268,"style":31846},[31845],[34,504270,504272,504275,504278,504281],{"className":504271},[31676],[34,504273],{"className":504274,"style":33218},[31680],[34,504276,34386],{"className":504277,"style":34427},[31685,31686],[34,504279,98360],{"className":504280,"style":33389},[31685,31686],[34,504282,34386],{"className":504283,"style":34427},[31685,31686],[15,504285,504286],{},[71807,504287,504288,504289,2557],{},"(注：如果环没有单位元，可能还需要加上整数倍项 ",[34,504290,504292,504308],{"className":504291},[31642],[34,504293,504295],{"className":504294},[31646],[31648,504296,504297],{"xmlns":31650},[31652,504298,504299,504305],{},[31655,504300,504301,504303],{},[31658,504302,31660],{},[31658,504304,31759],{},[31665,504306,504307],{"encoding":31667},"n m",[34,504309,504311],{"className":504310,"ariaHidden":758},[31672],[34,504312,504314,504317],{"className":504313},[31676],[34,504315],{"className":504316,"style":32527},[31680],[34,504318,269147],{"className":504319},[31685,31686],[15,504321,504322,504325,504326,504354,504355,334983,504385,504451,504452,504486],{},[2627,504323,504324],{},"情况 3：有单位元的交换环 (Commutative Ring with Unity)","\n这是最常见的情况。因为有单位元 ",[34,504327,504329,504342],{"className":504328},[31642],[34,504330,504332],{"className":504331},[31646],[31648,504333,504334],{"xmlns":31650},[31652,504335,504336,504340],{},[31655,504337,504338],{},[31749,504339,640],{},[31665,504341,640],{"encoding":31667},[34,504343,504345],{"className":504344,"ariaHidden":758},[31672],[34,504346,504348,504351],{"className":504347},[31676],[34,504349],{"className":504350,"style":32617},[31680],[34,504352,640],{"className":504353},[31685],"，整数倍 ",[34,504356,504358,504373],{"className":504357},[31642],[34,504359,504361],{"className":504360},[31646],[31648,504362,504363],{"xmlns":31650},[31652,504364,504365,504371],{},[31655,504366,504367,504369],{},[31658,504368,31660],{},[31658,504370,31759],{},[31665,504372,504307],{"encoding":31667},[34,504374,504376],{"className":504375,"ariaHidden":758},[31672],[34,504377,504379,504382],{"className":504378},[31676],[34,504380],{"className":504381,"style":32527},[31680],[34,504383,269147],{"className":504384},[31685,31686],[34,504386,504388,504412],{"className":504387},[31642],[34,504389,504391],{"className":504390},[31646],[31648,504392,504393],{"xmlns":31650},[31652,504394,504395,504409],{},[31655,504396,504397,504399,504401,504403,504405,504407],{},[31662,504398,456],{"stretchy":702},[31658,504400,31660],{},[31662,504402,71191],{},[31749,504404,640],{},[31662,504406,2557],{"stretchy":702},[31658,504408,31759],{},[31665,504410,504411],{"encoding":31667},"(n \\cdot 1)m",[34,504413,504415,504436],{"className":504414,"ariaHidden":758},[31672],[34,504416,504418,504421,504424,504427,504430,504433],{"className":504417},[31676],[34,504419],{"className":504420,"style":31774},[31680],[34,504422,456],{"className":504423},[31782],[34,504425,31660],{"className":504426},[31685,31686],[34,504428],{"className":504429,"style":32604},[31845],[34,504431,71191],{"className":504432},[32024],[34,504434],{"className":504435,"style":32604},[31845],[34,504437,504439,504442,504445,504448],{"className":504438},[31676],[34,504440],{"className":504441,"style":31774},[31680],[34,504443,640],{"className":504444},[31685],[34,504446,2557],{"className":504447},[31690],[34,504449,31759],{"className":504450},[31685,31686],"，被 ",[34,504453,504455,504471],{"className":504454},[31642],[34,504456,504458],{"className":504457},[31646],[31648,504459,504460],{"xmlns":31650},[31652,504461,504462,504468],{},[31655,504463,504464,504466],{},[31658,504465,34395],{},[31658,504467,31759],{},[31665,504469,504470],{"encoding":31667},"rm",[34,504472,504474],{"className":504473,"ariaHidden":758},[31672],[34,504475,504477,504480,504483],{"className":504476},[31676],[34,504478],{"className":504479,"style":32527},[31680],[34,504481,34395],{"className":504482,"style":34452},[31685,31686],[34,504484,31759],{"className":504485},[31685,31686]," 形式覆盖。",[34,504488,504490],{"className":504489},[126385],[34,504491,504493,504571],{"className":504492},[31642],[34,504494,504496],{"className":504495},[31646],[31648,504497,504498],{"xmlns":31650,"display":126394},[31652,504499,504500,504568],{},[31655,504501,504502,504504,504506,504508,504510,504562,504564,504566],{},[31662,504503,932],{},[31658,504505,98360],{},[31662,504507,29622],{},[31662,504509,202],{},[31655,504511,504512,504514,504520,504526,504532,504534,504536,504538,504544,504546,504548,504550,504556,504558,504560],{},[31662,504513,32685],{"fence":758},[71115,504515,504516,504518],{},[31662,504517,131138],{},[39361,504519,502273],{},[31743,504521,504522,504524],{},[31658,504523,34395],{},[31658,504525,31955],{},[31743,504527,504528,504530],{},[31658,504529,31759],{},[31658,504531,31955],{},[39361,504533,262622],{},[31662,504535,502375],{"fence":758,"lspace":502374,"rspace":502374},[39361,504537,262622],{},[31743,504539,504540,504542],{},[31658,504541,34395],{},[31658,504543,31955],{},[31662,504545,112296],{},[31658,504547,34386],{},[31662,504549,31960],{"separator":758},[31743,504551,504552,504554],{},[31658,504553,31759],{},[31658,504555,31955],{},[31662,504557,112296],{},[31658,504559,98360],{},[31662,504561,32715],{"fence":758},[31662,504563,202],{},[31658,504565,34386],{},[31658,504567,98360],{},[31665,504569,504570],{"encoding":31667},"\u003C M > = \\left\\{ \\sum_{\\text{finite}} r_i m_i \\;\\middle|\\; r_i \\in R, m_i \\in M \\right\\} = RM",[34,504572,504574,504586,504604,504919],{"className":504573,"ariaHidden":758},[31672],[34,504575,504577,504580,504583],{"className":504576},[31676],[34,504578],{"className":504579,"style":34812},[31680],[34,504581,932],{"className":504582},[31850],[34,504584],{"className":504585,"style":31846},[31845],[34,504587,504589,504592,504595,504598,504601],{"className":504588},[31676],[34,504590],{"className":504591,"style":104901},[31680],[34,504593,98360],{"className":504594,"style":33389},[31685,31686],[34,504596],{"className":504597,"style":31846},[31845],[34,504599,2075],{"className":504600},[31850],[34,504602],{"className":504603,"style":31846},[31845],[34,504605,504607,504610,504910,504913,504916],{"className":504606},[31676],[34,504608],{"className":504609,"style":502481},[31680],[34,504611,504613,504619,504669,504672,504712,504752,504755,504791,504794,504834,504837,504840,504843,504846,504849,504852,504892,504895,504898,504901,504904],{"className":504612},[34006],[34,504614,504616],{"className":504615,"style":75811},[31782,75810],[34,504617,32685],{"className":504618},[71281,75815],[34,504620,504622],{"className":504621},[32767,71420],[34,504623,504625,504661],{"className":504624},[31796,31797],[34,504626,504628,504658],{"className":504627},[31801],[34,504629,504631,504648],{"className":504630,"style":131215},[31805],[34,504632,504633,504636],{"style":410450},[34,504634],{"className":504635,"style":131222},[31813],[34,504637,504639],{"className":504638},[31818,31819,31820,31821],[34,504640,504642],{"className":504641},[31685,31821],[34,504643,504645],{"className":504644},[31685,2638,31821],[34,504646,502273],{"className":504647},[31685,31821],[34,504649,504650,504653],{"style":131240},[34,504651],{"className":504652,"style":131222},[31813],[34,504654,504655],{},[34,504656,131138],{"className":504657},[32767,131249,131250],[34,504659,31829],{"className":504660},[31828],[34,504662,504664],{"className":504663},[31801],[34,504665,504667],{"className":504666,"style":502539},[31805],[34,504668],{},[34,504670],{"className":504671,"style":32123},[31845],[34,504673,504675,504678],{"className":504674},[31685],[34,504676,34395],{"className":504677,"style":34452},[31685,31686],[34,504679,504681],{"className":504680},[31792],[34,504682,504684,504704],{"className":504683},[31796,31797],[34,504685,504687,504701],{"className":504686},[31801],[34,504688,504690],{"className":504689,"style":32090},[31805],[34,504691,504692,504695],{"style":34467},[34,504693],{"className":504694,"style":31814},[31813],[34,504696,504698],{"className":504697},[31818,31819,31820,31821],[34,504699,31955],{"className":504700},[31685,31686,31821],[34,504702,31829],{"className":504703},[31828],[34,504705,504707],{"className":504706},[31801],[34,504708,504710],{"className":504709,"style":31836},[31805],[34,504711],{},[34,504713,504715,504718],{"className":504714},[31685],[34,504716,31759],{"className":504717},[31685,31686],[34,504719,504721],{"className":504720},[31792],[34,504722,504724,504744],{"className":504723},[31796,31797],[34,504725,504727,504741],{"className":504726},[31801],[34,504728,504730],{"className":504729,"style":32090},[31805],[34,504731,504732,504735],{"style":31809},[34,504733],{"className":504734,"style":31814},[31813],[34,504736,504738],{"className":504737},[31818,31819,31820,31821],[34,504739,31955],{"className":504740},[31685,31686,31821],[34,504742,31829],{"className":504743},[31828],[34,504745,504747],{"className":504746},[31801],[34,504748,504750],{"className":504749,"style":31836},[31805],[34,504751],{},[34,504753],{"className":504754,"style":31846},[31845],[34,504756,504757],{},[34,504758,504760],{"className":504759},[71281,71282],[34,504761,504763,504783],{"className":504762},[31796,31797],[34,504764,504766,504780],{"className":504765},[31801],[34,504767,504769],{"className":504768,"style":503041},[31805],[34,504770,504771,504774],{"style":503044},[34,504772],{"className":504773,"style":503048},[31813],[34,504775,504776],{"style":503051},[41588,504777,504778],{"xmlns":41590,"width":503054,"height":503055,"viewBox":503056},[20398,504779],{"d":503059},[34,504781,31829],{"className":504782},[31828],[34,504784,504786],{"className":504785},[31801],[34,504787,504789],{"className":504788,"style":503069},[31805],[34,504790],{},[34,504792],{"className":504793,"style":31846},[31845],[34,504795,504797,504800],{"className":504796},[31685],[34,504798,34395],{"className":504799,"style":34452},[31685,31686],[34,504801,504803],{"className":504802},[31792],[34,504804,504806,504826],{"className":504805},[31796,31797],[34,504807,504809,504823],{"className":504808},[31801],[34,504810,504812],{"className":504811,"style":32090},[31805],[34,504813,504814,504817],{"style":34467},[34,504815],{"className":504816,"style":31814},[31813],[34,504818,504820],{"className":504819},[31818,31819,31820,31821],[34,504821,31955],{"className":504822},[31685,31686,31821],[34,504824,31829],{"className":504825},[31828],[34,504827,504829],{"className":504828},[31801],[34,504830,504832],{"className":504831,"style":31836},[31805],[34,504833],{},[34,504835],{"className":504836,"style":31846},[31845],[34,504838,112296],{"className":504839},[31850],[34,504841],{"className":504842,"style":31846},[31845],[34,504844,34386],{"className":504845,"style":34427},[31685,31686],[34,504847,31960],{"className":504848},[32119],[34,504850],{"className":504851,"style":32123},[31845],[34,504853,504855,504858],{"className":504854},[31685],[34,504856,31759],{"className":504857},[31685,31686],[34,504859,504861],{"className":504860},[31792],[34,504862,504864,504884],{"className":504863},[31796,31797],[34,504865,504867,504881],{"className":504866},[31801],[34,504868,504870],{"className":504869,"style":32090},[31805],[34,504871,504872,504875],{"style":31809},[34,504873],{"className":504874,"style":31814},[31813],[34,504876,504878],{"className":504877},[31818,31819,31820,31821],[34,504879,31955],{"className":504880},[31685,31686,31821],[34,504882,31829],{"className":504883},[31828],[34,504885,504887],{"className":504886},[31801],[34,504888,504890],{"className":504889,"style":31836},[31805],[34,504891],{},[34,504893],{"className":504894,"style":31846},[31845],[34,504896,112296],{"className":504897},[31850],[34,504899],{"className":504900,"style":31846},[31845],[34,504902,98360],{"className":504903,"style":33389},[31685,31686],[34,504905,504907],{"className":504906,"style":75811},[31690,75810],[34,504908,32715],{"className":504909},[71281,75815],[34,504911],{"className":504912,"style":31846},[31845],[34,504914,202],{"className":504915},[31850],[34,504917],{"className":504918,"style":31846},[31845],[34,504920,504922,504925,504928],{"className":504921},[31676],[34,504923],{"className":504924,"style":33218},[31680],[34,504926,34386],{"className":504927,"style":34427},[31685,31686],[34,504929,98360],{"className":504930,"style":33389},[31685,31686],[15,504932,504933,504934,504985,504986,347],{},"特别地，对于主理想 ",[34,504935,504937,504955],{"className":504936},[31642],[34,504938,504940],{"className":504939},[31646],[31648,504941,504942],{"xmlns":31650},[31652,504943,504944,504952],{},[31655,504945,504946,504948,504950],{},[31662,504947,932],{},[31658,504949,111],{},[31662,504951,29622],{},[31665,504953,504954],{"encoding":31667},"\u003C a >",[34,504956,504958,504970],{"className":504957,"ariaHidden":758},[31672],[34,504959,504961,504964,504967],{"className":504960},[31676],[34,504962],{"className":504963,"style":34812},[31680],[34,504965,932],{"className":504966},[31850],[34,504968],{"className":504969,"style":31846},[31845],[34,504971,504973,504976,504979,504982],{"className":504972},[31676],[34,504974],{"className":504975,"style":34812},[31680],[34,504977,111],{"className":504978},[31685,31686],[34,504980],{"className":504981,"style":31846},[31845],[34,504983,29622],{"className":504984},[31850],"，就是 ",[34,504987,504989,505023],{"className":504988},[31642],[34,504990,504992],{"className":504991},[31646],[31648,504993,504994],{"xmlns":31650},[31652,504995,504996,505020],{},[31655,504997,504998,505000,505002,505004,505006,505008,505010,505012,505014,505016,505018],{},[31658,504999,34386],{},[31658,505001,111],{},[31662,505003,202],{},[31662,505005,32685],{"stretchy":702},[31658,505007,34395],{},[31658,505009,111],{},[31662,505011,162086],{},[31658,505013,34395],{},[31662,505015,112296],{},[31658,505017,34386],{},[31662,505019,32715],{"stretchy":702},[31665,505021,505022],{"encoding":31667},"Ra = \\{ra \\mid r \\in R\\}",[34,505024,505026,505047,505071,505089],{"className":505025,"ariaHidden":758},[31672],[34,505027,505029,505032,505035,505038,505041,505044],{"className":505028},[31676],[34,505030],{"className":505031,"style":33218},[31680],[34,505033,34386],{"className":505034,"style":34427},[31685,31686],[34,505036,111],{"className":505037},[31685,31686],[34,505039],{"className":505040,"style":31846},[31845],[34,505042,202],{"className":505043},[31850],[34,505045],{"className":505046,"style":31846},[31845],[34,505048,505050,505053,505056,505059,505062,505065,505068],{"className":505049},[31676],[34,505051],{"className":505052,"style":31774},[31680],[34,505054,32685],{"className":505055},[31782],[34,505057,34395],{"className":505058,"style":34452},[31685,31686],[34,505060,111],{"className":505061},[31685,31686],[34,505063],{"className":505064,"style":31846},[31845],[34,505066,162086],{"className":505067},[31850],[34,505069],{"className":505070,"style":31846},[31845],[34,505072,505074,505077,505080,505083,505086],{"className":505073},[31676],[34,505075],{"className":505076,"style":34812},[31680],[34,505078,34395],{"className":505079,"style":34452},[31685,31686],[34,505081],{"className":505082,"style":31846},[31845],[34,505084,112296],{"className":505085},[31850],[34,505087],{"className":505088,"style":31846},[31845],[34,505090,505092,505095,505098],{"className":505091},[31676],[34,505093],{"className":505094,"style":31774},[31680],[34,505096,34386],{"className":505097,"style":34427},[31685,31686],[34,505099,32715],{"className":505100},[31690],[32188,505102],{},[19,505104,505106],{"id":505105},"_5-理想的运算","5. 理想的运算",[15,505108,133910,505109,330820,505150,505178],{},[34,505110,505112,505129],{"className":505111},[31642],[34,505113,505115],{"className":505114},[31646],[31648,505116,505117],{"xmlns":31650},[31652,505118,505119,505127],{},[31655,505120,505121,505123,505125],{},[31658,505122,106063],{},[31662,505124,31960],{"separator":758},[31658,505126,223320],{},[31665,505128,454327],{"encoding":31667},[34,505130,505132],{"className":505131,"ariaHidden":758},[31672],[34,505133,505135,505138,505141,505144,505147],{"className":505134},[31676],[34,505136],{"className":505137,"style":70739},[31680],[34,505139,106063],{"className":505140,"style":67482},[31685,31686],[34,505142,31960],{"className":505143},[32119],[34,505145],{"className":505146,"style":32123},[31845],[34,505148,223320],{"className":505149,"style":223361},[31685,31686],[34,505151,505153,505166],{"className":505152},[31642],[34,505154,505156],{"className":505155},[31646],[31648,505157,505158],{"xmlns":31650},[31652,505159,505160,505164],{},[31655,505161,505162],{},[31658,505163,34386],{},[31665,505165,34386],{"encoding":31667},[34,505167,505169],{"className":505168,"ariaHidden":758},[31672],[34,505170,505172,505175],{"className":505171},[31676],[34,505173],{"className":505174,"style":33218},[31680],[34,505176,34386],{"className":505177,"style":34427},[31685,31686]," 的两个理想。",[253,505180,505182,505183],{"id":505181},"_51-交集-iji-cap-jij","5.1 交集 ",[34,505184,505186,505204],{"className":505185},[31642],[34,505187,505189],{"className":505188},[31646],[31648,505190,505191],{"xmlns":31650},[31652,505192,505193,505201],{},[31655,505194,505195,505197,505199],{},[31658,505196,106063],{},[31662,505198,326344],{},[31658,505200,223320],{},[31665,505202,505203],{"encoding":31667},"I \\cap J",[34,505205,505207,505225],{"className":505206,"ariaHidden":758},[31672],[34,505208,505210,505213,505216,505219,505222],{"className":505209},[31676],[34,505211],{"className":505212,"style":33218},[31680],[34,505214,106063],{"className":505215,"style":67482},[31685,31686],[34,505217],{"className":505218,"style":32604},[31845],[34,505220,326344],{"className":505221},[32024],[34,505223],{"className":505224,"style":32604},[31845],[34,505226,505228,505231],{"className":505227},[31676],[34,505229],{"className":505230,"style":33218},[31680],[34,505232,223320],{"className":505233,"style":223361},[31685,31686],[15,505235,505236,2664,505238,505288,505289,2664],{},[2627,505237,371614],{},[34,505239,505241,505258],{"className":505240},[31642],[34,505242,505244],{"className":505243},[31646],[31648,505245,505246],{"xmlns":31650},[31652,505247,505248,505256],{},[31655,505249,505250,505252,505254],{},[31658,505251,106063],{},[31662,505253,326344],{},[31658,505255,223320],{},[31665,505257,505203],{"encoding":31667},[34,505259,505261,505279],{"className":505260,"ariaHidden":758},[31672],[34,505262,505264,505267,505270,505273,505276],{"className":505263},[31676],[34,505265],{"className":505266,"style":33218},[31680],[34,505268,106063],{"className":505269,"style":67482},[31685,31686],[34,505271],{"className":505272,"style":32604},[31845],[34,505274,326344],{"className":505275},[32024],[34,505277],{"className":505278,"style":32604},[31845],[34,505280,505282,505285],{"className":505281},[31676],[34,505283],{"className":505284,"style":33218},[31680],[34,505286,223320],{"className":505287,"style":223361},[31685,31686]," 是一个理想。\n",[2627,505290,390672],{},[2230,505292,505293,505796],{},[108,505294,505295,47890,505298,55016,505384,86674,505448,505512,505513,505554,505555,86674,505628,61605,505701,347],{},[2627,505296,505297],{},"子群判定",[34,505299,505301,505327],{"className":505300},[31642],[34,505302,505304],{"className":505303},[31646],[31648,505305,505306],{"xmlns":31650},[31652,505307,505308,505324],{},[31655,505309,505310,505312,505314,505316,505318,505320,505322],{},[31658,505311,111],{},[31662,505313,31960],{"separator":758},[31658,505315,1697],{},[31662,505317,112296],{},[31658,505319,106063],{},[31662,505321,326344],{},[31658,505323,223320],{},[31665,505325,505326],{"encoding":31667},"a, b \\in I \\cap J",[34,505328,505330,505357,505375],{"className":505329,"ariaHidden":758},[31672],[34,505331,505333,505336,505339,505342,505345,505348,505351,505354],{"className":505332},[31676],[34,505334],{"className":505335,"style":43217},[31680],[34,505337,111],{"className":505338},[31685,31686],[34,505340,31960],{"className":505341},[32119],[34,505343],{"className":505344,"style":32123},[31845],[34,505346,1697],{"className":505347},[31685,31686],[34,505349],{"className":505350,"style":31846},[31845],[34,505352,112296],{"className":505353},[31850],[34,505355],{"className":505356,"style":31846},[31845],[34,505358,505360,505363,505366,505369,505372],{"className":505359},[31676],[34,505361],{"className":505362,"style":33218},[31680],[34,505364,106063],{"className":505365,"style":67482},[31685,31686],[34,505367],{"className":505368,"style":32604},[31845],[34,505370,326344],{"className":505371},[32024],[34,505373],{"className":505374,"style":32604},[31845],[34,505376,505378,505381],{"className":505377},[31676],[34,505379],{"className":505380,"style":33218},[31680],[34,505382,223320],{"className":505383,"style":223361},[31685,31686],[34,505385,505387,505409],{"className":505386},[31642],[34,505388,505390],{"className":505389},[31646],[31648,505391,505392],{"xmlns":31650},[31652,505393,505394,505406],{},[31655,505395,505396,505398,505400,505402,505404],{},[31658,505397,111],{},[31662,505399,31960],{"separator":758},[31658,505401,1697],{},[31662,505403,112296],{},[31658,505405,106063],{},[31665,505407,505408],{"encoding":31667},"a, b \\in I",[34,505410,505412,505439],{"className":505411,"ariaHidden":758},[31672],[34,505413,505415,505418,505421,505424,505427,505430,505433,505436],{"className":505414},[31676],[34,505416],{"className":505417,"style":43217},[31680],[34,505419,111],{"className":505420},[31685,31686],[34,505422,31960],{"className":505423},[32119],[34,505425],{"className":505426,"style":32123},[31845],[34,505428,1697],{"className":505429},[31685,31686],[34,505431],{"className":505432,"style":31846},[31845],[34,505434,112296],{"className":505435},[31850],[34,505437],{"className":505438,"style":31846},[31845],[34,505440,505442,505445],{"className":505441},[31676],[34,505443],{"className":505444,"style":33218},[31680],[34,505446,106063],{"className":505447,"style":67482},[31685,31686],[34,505449,505451,505473],{"className":505450},[31642],[34,505452,505454],{"className":505453},[31646],[31648,505455,505456],{"xmlns":31650},[31652,505457,505458,505470],{},[31655,505459,505460,505462,505464,505466,505468],{},[31658,505461,111],{},[31662,505463,31960],{"separator":758},[31658,505465,1697],{},[31662,505467,112296],{},[31658,505469,223320],{},[31665,505471,505472],{"encoding":31667},"a, b \\in J",[34,505474,505476,505503],{"className":505475,"ariaHidden":758},[31672],[34,505477,505479,505482,505485,505488,505491,505494,505497,505500],{"className":505478},[31676],[34,505480],{"className":505481,"style":43217},[31680],[34,505483,111],{"className":505484},[31685,31686],[34,505486,31960],{"className":505487},[32119],[34,505489],{"className":505490,"style":32123},[31845],[34,505492,1697],{"className":505493},[31685,31686],[34,505495],{"className":505496,"style":31846},[31845],[34,505498,112296],{"className":505499},[31850],[34,505501],{"className":505502,"style":31846},[31845],[34,505504,505506,505509],{"className":505505},[31676],[34,505507],{"className":505508,"style":33218},[31680],[34,505510,223320],{"className":505511,"style":223361},[31685,31686],"。因 ",[34,505514,505516,505533],{"className":505515},[31642],[34,505517,505519],{"className":505518},[31646],[31648,505520,505521],{"xmlns":31650},[31652,505522,505523,505531],{},[31655,505524,505525,505527,505529],{},[31658,505526,106063],{},[31662,505528,31960],{"separator":758},[31658,505530,223320],{},[31665,505532,454327],{"encoding":31667},[34,505534,505536],{"className":505535,"ariaHidden":758},[31672],[34,505537,505539,505542,505545,505548,505551],{"className":505538},[31676],[34,505540],{"className":505541,"style":70739},[31680],[34,505543,106063],{"className":505544,"style":67482},[31685,31686],[34,505546,31960],{"className":505547},[32119],[34,505549],{"className":505550,"style":32123},[31845],[34,505552,223320],{"className":505553,"style":223361},[31685,31686]," 是加法群，故 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I",[34,505581,505583,505601,505619],{"className":505582,"ariaHidden":758},[31672],[34,505584,505586,505589,505592,505595,505598],{"className":505585},[31676],[34,505587],{"className":505588,"style":32597},[31680],[34,505590,111],{"className":505591},[31685,31686],[34,505593],{"className":505594,"style":32604},[31845],[34,505596,32226],{"className":505597},[32024],[34,505599],{"className":505600,"style":32604},[31845],[34,505602,505604,505607,505610,505613,505616],{"className":505603},[31676],[34,505605],{"className":505606,"style":46991},[31680],[34,505608,1697],{"className":505609},[31685,31686],[34,505611],{"className":505612,"style":31846},[31845],[34,505614,112296],{"className":505615},[31850],[34,505617],{"className":505618,"style":31846},[31845],[34,505620,505622,505625],{"className":505621},[31676],[34,505623],{"className":505624,"style":33218},[31680],[34,505626,106063],{"className":505627,"style":67482},[31685,31686],[34,505629,505631,505653],{"className":505630},[31642],[34,505632,505634],{"className":505633},[31646],[31648,505635,505636],{"xmlns":31650},[31652,505637,505638,505650],{},[31655,505639,505640,505642,505644,505646,505648],{},[31658,505641,111],{},[31662,505643,32226],{},[31658,505645,1697],{},[31662,505647,112296],{},[31658,505649,223320],{},[31665,505651,505652],{"encoding":31667},"a-b 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\\in I \\cap J",[34,505731,505733,505751,505769,505787],{"className":505732,"ariaHidden":758},[31672],[34,505734,505736,505739,505742,505745,505748],{"className":505735},[31676],[34,505737],{"className":505738,"style":32597},[31680],[34,505740,111],{"className":505741},[31685,31686],[34,505743],{"className":505744,"style":32604},[31845],[34,505746,32226],{"className":505747},[32024],[34,505749],{"className":505750,"style":32604},[31845],[34,505752,505754,505757,505760,505763,505766],{"className":505753},[31676],[34,505755],{"className":505756,"style":46991},[31680],[34,505758,1697],{"className":505759},[31685,31686],[34,505761],{"className":505762,"style":31846},[31845],[34,505764,112296],{"className":505765},[31850],[34,505767],{"className":505768,"style":31846},[31845],[34,505770,505772,505775,505778,505781,505784],{"className":505771},[31676],[34,505773],{"className":505774,"style":33218},[31680],[34,505776,106063],{"className":505777,"style":67482},[31685,31686],[34,505779],{"className":505780,"style":32604},[31845],[34,505782,326344],{"className":505783},[32024],[34,505785],{"className":505786,"style":32604},[31845],[34,505788,505790,505793],{"className":505789},[31676],[34,505791],{"className":505792,"style":33218},[31680],[34,505794,223320],{"className":505795,"style":223361},[31685,31686],[108,505797,505798,34554,505800,39984,505908],{},[2627,505799,442183],{},[34,505801,505803,505833],{"className":505802},[31642],[34,505804,505806],{"className":505805},[31646],[31648,505807,505808],{"xmlns":31650},[31652,505809,505810,505830],{},[31655,505811,505812,505814,505816,505818,505820,505822,505824,505826,505828],{},[31658,505813,32936],{},[31662,505815,112296],{},[31658,505817,106063],{},[31662,505819,326344],{},[31658,505821,223320],{},[31662,505823,31960],{"separator":758},[31658,505825,34395],{},[31662,505827,112296],{},[31658,505829,34386],{},[31665,505831,505832],{"encoding":31667},"x \\in I \\cap J, r \\in R",[34,505834,505836,505854,505872,505899],{"className":505835,"ariaHidden":758},[31672],[34,505837,505839,505842,505845,505848,505851],{"className":505838},[31676],[34,505840],{"className":505841,"style":34812},[31680],[34,505843,32936],{"className":505844},[31685,31686],[34,505846],{"className":505847,"style":31846},[31845],[34,505849,112296],{"className":505850},[31850],[34,505852],{"className":505853,"style":31846},[31845],[34,505855,505857,505860,505863,505866,505869],{"className":505856},[31676],[34,505858],{"className":505859,"style":33218},[31680],[34,505861,106063],{"className":505862,"style":67482},[31685,31686],[34,505864],{"className":505865,"style":32604},[31845],[34,505867,326344],{"className":505868},[32024],[34,505870],{"className":505871,"style":32604},[31845],[34,505873,505875,505878,505881,505884,505887,505890,505893,505896],{"className":505874},[31676],[34,505876],{"className":505877,"style":70739},[31680],[34,505879,223320],{"className":505880,"style":223361},[31685,31686],[34,505882,31960],{"className":505883},[32119],[34,505885],{"className":505886,"style":32123},[31845],[34,505888,34395],{"className":505889,"style":34452},[31685,31686],[34,505891],{"className":505892,"style":31846},[31845],[34,505894,112296],{"className":505895},[31850],[34,505897],{"className":505898,"style":31846},[31845],[34,505900,505902,505905],{"className":505901},[31676],[34,505903],{"className":505904,"style":33218},[31680],[34,505906,34386],{"className":505907,"style":34427},[31685,31686],[105,505909,505910,506107,506303,506401],{},[108,505911,505912,86674,506022,506077,506078,506106],{},[34,505913,505915,505947],{"className":505914},[31642],[34,505916,505918],{"className":505917},[31646],[31648,505919,505920],{"xmlns":31650},[31652,505921,505922,505944],{},[31655,505923,505924,505926,505928,505930,505932,505934,505936,505938,505940,505942],{},[31658,505925,32936],{},[31662,505927,112296],{},[31658,505929,106063],{},[39361,505931,262622],{},[31662,505933,262625],{},[39361,505935,262622],{},[31658,505937,34395],{},[31658,505939,32936],{},[31662,505941,112296],{},[31658,505943,106063],{},[31665,505945,505946],{"encoding":31667},"x \\in I \\implies rx \\in I",[34,505948,505950,505968,505992,506013],{"className":505949,"ariaHidden":758},[31672],[34,505951,505953,505956,505959,505962,505965],{"className":505952},[31676],[34,505954],{"className":505955,"style":34812},[31680],[34,505957,32936],{"className":505958},[31685,31686],[34,505960],{"className":505961,"style":31846},[31845],[34,505963,112296],{"className":505964},[31850],[34,505966],{"className":505967,"style":31846},[31845],[34,505969,505971,505974,505977,505980,505983,505986,505989],{"className":505970},[31676],[34,505972],{"className":505973,"style":336814},[31680],[34,505975,106063],{"className":505976,"style":67482},[31685,31686],[34,505978],{"className":505979,"style":31846},[31845],[34,505981],{"className":505982,"style":31846},[31845],[34,505984,262625],{"className":505985},[31850],[34,505987],{"className":505988,"style":31846},[31845],[34,505990],{"className":505991,"style":31846},[31845],[34,505993,505995,505998,506001,506004,506007,506010],{"className":505994},[31676],[34,505996],{"className":505997,"style":34812},[31680],[34,505999,34395],{"className":506000,"style":34452},[31685,31686],[34,506002,32936],{"className":506003},[31685,31686],[34,506005],{"className":506006,"style":31846},[31845],[34,506008,112296],{"className":506009},[31850],[34,506011],{"className":506012,"style":31846},[31845],[34,506014,506016,506019],{"className":506015},[31676],[34,506017],{"className":506018,"style":33218},[31680],[34,506020,106063],{"className":506021,"style":67482},[31685,31686],[34,506023,506025,506044],{"className":506024},[31642],[34,506026,506028],{"className":506027},[31646],[31648,506029,506030],{"xmlns":31650},[31652,506031,506032,506042],{},[31655,506033,506034,506036,506038,506040],{},[31658,506035,32936],{},[31658,506037,34395],{},[31662,506039,112296],{},[31658,506041,106063],{},[31665,506043,496584],{"encoding":31667},[34,506045,506047,506068],{"className":506046,"ariaHidden":758},[31672],[34,506048,506050,506053,506056,506059,506062,506065],{"className":506049},[31676],[34,506051],{"className":506052,"style":34812},[31680],[34,506054,32936],{"className":506055},[31685,31686],[34,506057,34395],{"className":506058,"style":34452},[31685,31686],[34,506060],{"className":506061,"style":31846},[31845],[34,506063,112296],{"className":506064},[31850],[34,506066],{"className":506067,"style":31846},[31845],[34,506069,506071,506074],{"className":506070},[31676],[34,506072],{"className":506073,"style":33218},[31680],[34,506075,106063],{"className":506076,"style":67482},[31685,31686],"（因 ",[34,506079,506081,506094],{"className":506080},[31642],[34,506082,506084],{"className":506083},[31646],[31648,506085,506086],{"xmlns":31650},[31652,506087,506088,506092],{},[31655,506089,506090],{},[31658,506091,106063],{},[31665,506093,106063],{"encoding":31667},[34,506095,506097],{"className":506096,"ariaHidden":758},[31672],[34,506098,506100,506103],{"className":506099},[31676],[34,506101],{"className":506102,"style":33218},[31680],[34,506104,106063],{"className":506105,"style":67482},[31685,31686]," 是理想）。",[108,506108,506109,86674,506219,506077,506275,506106],{},[34,506110,506112,506144],{"className":506111},[31642],[34,506113,506115],{"className":506114},[31646],[31648,506116,506117],{"xmlns":31650},[31652,506118,506119,506141],{},[31655,506120,506121,506123,506125,506127,506129,506131,506133,506135,506137,506139],{},[31658,506122,32936],{},[31662,506124,112296],{},[31658,506126,223320],{},[39361,506128,262622],{},[31662,506130,262625],{},[39361,506132,262622],{},[31658,506134,34395],{},[31658,506136,32936],{},[31662,506138,112296],{},[31658,506140,223320],{},[31665,506142,506143],{"encoding":31667},"x \\in J \\implies rx \\in J",[34,506145,506147,506165,506189,506210],{"className":506146,"ariaHidden":758},[31672],[34,506148,506150,506153,506156,506159,506162],{"className":506149},[31676],[34,506151],{"className":506152,"style":34812},[31680],[34,506154,32936],{"className":506155},[31685,31686],[34,506157],{"className":506158,"style":31846},[31845],[34,506160,112296],{"className":506161},[31850],[34,506163],{"className":506164,"style":31846},[31845],[34,506166,506168,506171,506174,506177,506180,506183,506186],{"className":506167},[31676],[34,506169],{"className":506170,"style":336814},[31680],[34,506172,223320],{"className":506173,"style":223361},[31685,31686],[34,506175],{"className":506176,"style":31846},[31845],[34,506178],{"className":506179,"style":31846},[31845],[34,506181,262625],{"className":506182},[31850],[34,506184],{"className":506185,"style":31846},[31845],[34,506187],{"className":506188,"style":31846},[31845],[34,506190,506192,506195,506198,506201,506204,506207],{"className":506191},[31676],[34,506193],{"className":506194,"style":34812},[31680],[34,506196,34395],{"className":506197,"style":34452},[31685,31686],[34,506199,32936],{"className":506200},[31685,31686],[34,506202],{"className":506203,"style":31846},[31845],[34,506205,112296],{"className":506206},[31850],[34,506208],{"className":506209,"style":31846},[31845],[34,506211,506213,506216],{"className":506212},[31676],[34,506214],{"className":506215,"style":33218},[31680],[34,506217,223320],{"className":506218,"style":223361},[31685,31686],[34,506220,506222,506242],{"className":506221},[31642],[34,506223,506225],{"className":506224},[31646],[31648,506226,506227],{"xmlns":31650},[31652,506228,506229,506239],{},[31655,506230,506231,506233,506235,506237],{},[31658,506232,32936],{},[31658,506234,34395],{},[31662,506236,112296],{},[31658,506238,223320],{},[31665,506240,506241],{"encoding":31667},"xr \\in 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xr \\in I \\cap 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是理想。",[253,506457,506459,506460],{"id":506458},"_52-和-iji-jij","5.2 和 ",[34,506461,506463,506481],{"className":506462},[31642],[34,506464,506466],{"className":506465},[31646],[31648,506467,506468],{"xmlns":31650},[31652,506469,506470,506478],{},[31655,506471,506472,506474,506476],{},[31658,506473,106063],{},[31662,506475,223],{},[31658,506477,223320],{},[31665,506479,506480],{"encoding":31667},"I + J",[34,506482,506484,506502],{"className":506483,"ariaHidden":758},[31672],[34,506485,506487,506490,506493,506496,506499],{"className":506486},[31676],[34,506488],{"className":506489,"style":34944},[31680],[34,506491,106063],{"className":506492,"style":67482},[31685,31686],[34,506494],{"className":506495,"style":32604},[31845],[34,506497,223],{"className":506498},[32024],[34,506500],{"className":506501,"style":32604},[31845],[34,506503,506505,506508],{"className":506504},[31676],[34,506506],{"className":506507,"style":33218},[31680],[34,506509,223320],{"className":506510,"style":223361},[31685,31686],[15,506512,506513,2664,506516,39984,506700,2664,506702,505288,506752,2664],{},[2627,506514,506515],{},"定义",[34,506517,506519,506565],{"className":506518},[31642],[34,506520,506522],{"className":506521},[31646],[31648,506523,506524],{"xmlns":31650},[31652,506525,506526,506562],{},[31655,506527,506528,506530,506532,506534,506536,506538,506540,506542,506544,506546,506548,506550,506552,506554,506556,506558,506560],{},[31658,506529,106063],{},[31662,506531,223],{},[31658,506533,223320],{},[31662,506535,202],{},[31662,506537,32685],{"stretchy":702},[31658,506539,111],{},[31662,506541,223],{},[31658,506543,1697],{},[31662,506545,162086],{},[31658,506547,111],{},[31662,506549,112296],{},[31658,506551,106063],{},[31662,506553,31960],{"separator":758},[31658,506555,1697],{},[31662,506557,112296],{},[31658,506559,223320],{},[31662,506561,32715],{"stretchy":702},[31665,506563,506564],{"encoding":31667},"I + J = \\{ a + b \\mid a \\in I, b \\in J 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= a_1 + b_1, y = a_2 + b_2 \\in I+J",[34,506822,506824,506842,506897,506961,507016,507071,507089],{"className":506823,"ariaHidden":758},[31672],[34,506825,506827,506830,506833,506836,506839],{"className":506826},[31676],[34,506828],{"className":506829,"style":32527},[31680],[34,506831,32936],{"className":506832},[31685,31686],[34,506834],{"className":506835,"style":31846},[31845],[34,506837,202],{"className":506838},[31850],[34,506840],{"className":506841,"style":31846},[31845],[34,506843,506845,506848,506888,506891,506894],{"className":506844},[31676],[34,506846],{"className":506847,"style":40522},[31680],[34,506849,506851,506854],{"className":506850},[31685],[34,506852,111],{"className":506853},[31685,31686],[34,506855,506857],{"className":506856},[31792],[34,506858,506860,506880],{"className":506859},[31796,31797],[34,506861,506863,506877],{"className":506862},[31801],[34,506864,506866],{"className":506865,"style":31806},[31805],[34,506867,506868,506871],{"style":31809},[34,506869],{"className":506870,"style":31814},[31813],[34,506872,506874],{"className":506873},[31818,31819,31820,31821],[34,506875,640],{"className":506876},[31685,31821],[34,506878,31829],{"className":506879},[31828],[34,506881,506883],{"className":506882},[31801],[34,506884,506886],{"className":506885,"style":31836},[31805],[34,506887],{},[34,506889],{"className":506890,"style":32604},[31845],[34,506892,223],{"className":506893},[32024],[34,506895],{"className":506896,"style":32604},[31845],[34,506898,506900,506903,506943,506946,506949,506952,506955,506958],{"className":506899},[31676],[34,506901],{"className":506902,"style":43217},[31680],[34,506904,506906,506909],{"className":506905},[31685],[34,506907,1697],{"className":506908},[31685,31686],[34,506910,506912],{"className":506911},[31792],[34,506913,506915,506935],{"className":506914},[31796,31797],[34,506916,506918,506932],{"className":506917},[31801],[34,506919,506921],{"className":506920,"style":31806},[31805],[34,506922,506923,506926],{"style":31809},[34,506924],{"className":506925,"style":31814},[31813],[34,506927,506929],{"className":506928},[31818,31819,31820,31821],[34,506930,640],{"className":506931},[31685,31821],[34,506933,31829],{"className":506934},[31828],[34,506936,506938],{"className":506937},[31801],[34,506939,506941],{"className":506940,"style":31836},[31805],[34,506942],{},[34,506944,31960],{"className":506945},[32119],[34,506947],{"className":506948,"style":32123},[31845],[34,506950,67764],{"className":506951,"style":32062},[31685,31686],[34,506953],{"className":506954,"style":31846},[31845],[34,506956,202],{"className":506957},[31850],[34,506959],{"className":506960,"style":31846},[31845],[34,506962,506964,506967,507007,507010,507013],{"className":506963},[31676],[34,506965],{"className":506966,"style":40522},[31680],[34,506968,506970,506973],{"className":506969},[31685],[34,506971,111],{"className":506972},[31685,31686],[34,506974,506976],{"className":506975},[31792],[34,506977,506979,506999],{"className":506978},[31796,31797],[34,506980,506982,506996],{"className":506981},[31801],[34,506983,506985],{"className":506984,"style":31806},[31805],[34,506986,506987,506990],{"style":31809},[34,506988],{"className":506989,"style":31814},[31813],[34,506991,506993],{"className":506992},[31818,31819,31820,31821],[34,506994,768],{"className":506995},[31685,31821],[34,506997,31829],{"className":506998},[31828],[34,507000,507002],{"className":507001},[31801],[34,507003,507005],{"className":507004,"style":31836},[31805],[34,507006],{},[34,507008],{"className":507009,"style":32604},[31845],[34,507011,223],{"className":507012},[32024],[34,507014],{"className":507015,"style":32604},[31845],[34,507017,507019,507022,507062,507065,507068],{"className":507018},[31676],[34,507020],{"className":507021,"style":33814},[31680],[34,507023,507025,507028],{"className":507024},[31685],[34,507026,1697],{"className":507027},[31685,31686],[34,507029,507031],{"className":507030},[31792],[34,507032,507034,507054],{"className":507033},[31796,31797],[34,507035,507037,507051],{"className":507036},[31801],[34,507038,507040],{"className":507039,"style":31806},[31805],[34,507041,507042,507045],{"style":31809},[34,507043],{"className":507044,"style":31814},[31813],[34,507046,507048],{"className":507047},[31818,31819,31820,31821],[34,507049,768],{"className":507050},[31685,31821],[34,507052,31829],{"className":507053},[31828],[34,507055,507057],{"className":507056},[31801],[34,507058,507060],{"className":507059,"style":31836},[31805],[34,507061],{},[34,507063],{"className":507064,"style":31846},[31845],[34,507066,112296],{"className":507067},[31850],[34,507069],{"className":507070,"style":31846},[31845],[34,507072,507074,507077,507080,507083,507086],{"className":507073},[31676],[34,507075],{"className":507076,"style":34944},[31680],[34,507078,106063],{"className":507079,"style":67482},[31685,31686],[34,507081],{"className":507082,"style":32604},[31845],[34,507084,223],{"className":507085},[32024],[34,507087],{"className":507088,"style"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- y = (a_1 - a_2) + (b_1 - b_2)",[34,507160,507162,507180,507198,507256,507314,507372],{"className":507161,"ariaHidden":758},[31672],[34,507163,507165,507168,507171,507174,507177],{"className":507164},[31676],[34,507166],{"className":507167,"style":32597},[31680],[34,507169,32936],{"className":507170},[31685,31686],[34,507172],{"className":507173,"style":32604},[31845],[34,507175,32226],{"className":507176},[32024],[34,507178],{"className":507179,"style":32604},[31845],[34,507181,507183,507186,507189,507192,507195],{"className":507182},[31676],[34,507184],{"className":507185,"style":38774},[31680],[34,507187,67764],{"className":507188,"style":32062},[31685,31686],[34,507190],{"className":507191,"style":31846},[31845],[34,507193,202],{"className":507194},[31850],[34,507196],{"className":507197,"style":31846},[31845],[34,507199,507201,507204,507207,507247,507250,507253],{"className":507200},[31676],[34,507202],{"className":507203,"style":31774},[31680],[34,507205,456],{"className":507206},[31782],[34,507208,507210,507213],{"className":507209},[31685],[34,507211,111],{"className":507212},[31685,31686],[34,507214,507216],{"className":507215},[31792],[34,507217,507219,507239],{"className":507218},[31796,31797],[34,507220,507222,507236],{"className":507221},[31801],[34,507223,507225],{"className":507224,"style":31806},[31805],[34,507226,507227,507230],{"style":31809},[34,507228],{"className":507229,"style":31814},[31813],[34,507231,507233],{"className":507232},[31818,31819,31820,31821],[34,507234,640],{"className":507235},[31685,31821],[34,507237,31829],{"className":507238},[31828],[34,507240,507242],{"className":507241},[31801],[34,507243,507245],{"className":507244,"style":31836},[31805],[34,507246],{},[34,507248],{"className":507249,"style":32604},[31845],[34,507251,32226],{"className":507252},[32024],[34,507254],{"className":507255,"style":32604},[31845],[34,507257,507259,507262,507302,507305,507308,507311],{"className":507258},[31676],[34,507260],{"className":507261,"style":31774},[31680],[34,507263,507265,507268],{"className":507264},[31685],[34,507266,111],{"className":507267},[31685,31686],[34,507269,507271],{"className":507270},[31792],[34,507272,507274,507294],{"className":507273},[31796,31797],[34,507275,507277,507291],{"className":507276},[31801],[34,507278,507280],{"className":507279,"style":31806},[31805],[34,507281,507282,507285],{"style":31809},[34,507283],{"className":507284,"style":31814},[31813],[34,507286,507288],{"className":507287},[31818,31819,31820,31821],[34,507289,768],{"className":507290},[31685,31821],[34,507292,31829],{"className":507293},[31828],[34,507295,507297],{"className":507296},[31801],[34,507298,507300],{"className":507299,"style":31836},[31805],[34,507301],{},[34,507303,2557],{"className":507304},[31690],[34,507306],{"className":507307,"style":32604},[31845],[34,507309,223],{"className":507310},[32024],[34,507312],{"className":507313,"style":32604},[31845],[34,507315,507317,507320,507323,507363,507366,507369],{"className":507316},[31676],[34,507318],{"className":507319,"style":31774},[31680],[34,507321,456],{"className":507322},[31782],[34,507324,507326,507329],{"className":507325},[31685],[34,507327,1697],{"className":507328},[31685,31686],[34,507330,507332],{"className":507331},[31792],[34,507333,507335,507355],{"className":507334},[31796,31797],[34,507336,507338,507352],{"className":507337},[31801],[34,507339,507341],{"className":507340,"style":31806},[31805],[34,507342,507343,507346],{"style":31809},[34,507344],{"className":507345,"style":31814},[31813],[34,507347,507349],{"className":507348},[31818,31819,31820,31821],[34,507350,640],{"className":507351},[31685,31821],[34,507353,31829],{"className":507354},[31828],[34,507356,507358],{"className":507357},[31801],[34,507359,507361],{"className":507360,"style":31836},[31805],[34,507362],{},[34,507364],{"className":507365,"style":32604},[31845],[34,507367,32226],{"className":507368},[32024],[34,507370],{"className":507371,"style":32604},[31845],[34,507373,507375,507378,507418],{"className":507374},[31676],[34,507376],{"className":507377,"style":31774},[31680],[34,507379,507381,507384],{"className":507380},[31685],[34,507382,1697],{"className":507383},[31685,31686],[34,507385,507387],{"className":507386},[31792],[34,507388,507390,507410],{"className":507389},[31796,31797],[34,507391,507393,507407],{"className":507392},[31801],[34,507394,507396],{"className":507395,"style":31806},[31805],[34,507397,507398,507401],{"style":31809},[34,507399],{"className":507400,"style":31814},[31813],[34,507402,507404],{"className":507403},[31818,31819,31820,31821],[34,507405,768],{"className":507406},[31685,31821],[34,507408,31829],{"className":507409},[31828],[34,507411,507413],{"className":507412},[31801],[34,507414,507416],{"className":507415,"style":31836},[31805],[34,507417],{},[34,507419,2557],{"className":507420},[31690],"。由于 ",[34,507423,507425,507442],{"className":507424},[31642],[34,507426,507428],{"className":507427},[31646],[31648,507429,507430],{"xmlns":31650},[31652,507431,507432,507440],{},[31655,507433,507434,507436,507438],{},[31658,507435,106063],{},[31662,507437,31960],{"separator":758},[31658,507439,223320],{},[31665,507441,454327],{"encoding":31667},[34,507443,507445],{"className":507444,"ariaHidden":758},[31672],[34,507446,507448,507451,507454,507457,507460],{"className":507447},[31676],[34,507449],{"className":507450,"style":70739},[31680],[34,507452,106063],{"className":507453,"style":67482},[31685,31686],[34,507455,31960],{"className":507456},[32119],[34,507458],{"className":507459,"style":32123},[31845],[34,507461,223320],{"className":507462,"style":223361},[31685,31686]," 是子群，",[34,507465,507467,507525],{"className":507466},[31642],[34,507468,507470],{"className":507469},[31646],[31648,507471,507472],{"xmlns":31650},[31652,507473,507474,507522],{},[31655,507475,507476,507478,507484,507486,507492,507494,507496,507498,507500,507502,507508,507510,507516,507518,507520],{},[31662,507477,456],{"stretchy":702},[31743,507479,507480,507482],{},[31658,507481,111],{},[31749,507483,640],{},[31662,507485,32226],{},[31743,507487,507488,507490],{},[31658,507489,111],{},[31749,507491,768],{},[31662,507493,2557],{"stretchy":702},[31662,507495,112296],{},[31658,507497,106063],{},[31662,507499,31960],{"separator":758},[31662,507501,456],{"stretchy":702},[31743,507503,507504,507506],{},[31658,507505,1697],{},[31749,507507,640],{},[31662,507509,32226],{},[31743,507511,507512,507514],{},[31658,507513,1697],{},[31749,507515,768],{},[31662,507517,2557],{"stretchy":702},[31662,507519,112296],{},[31658,507521,223320],{},[31665,507523,507524],{"encoding":31667},"(a_1 - a_2) \\in I, (b_1 - b_2) \\in J",[34,507526,507528,507586,507644,507711,507769],{"className":507527,"ariaHidden":758},[31672],[34,507529,507531,507534,507537,507577,507580,507583],{"className":507530},[31676],[34,507532],{"className":507533,"style":31774},[31680],[34,507535,456],{"className":507536},[31782],[34,507538,507540,507543],{"className":507539},[31685],[34,507541,111],{"className":507542},[31685,31686],[34,507544,507546],{"className":507545},[31792],[34,507547,507549,507569],{"className":507548},[31796,31797],[34,507550,507552,507566],{"className":507551},[31801],[34,507553,507555],{"className":507554,"style":31806},[31805],[34,507556,507557,507560],{"style":31809},[34,507558],{"className":507559,"style":31814},[31813],[34,507561,507563],{"className":507562},[31818,31819,31820,31821],[34,507564,640],{"className":507565},[31685,31821],[34,507567,31829],{"className":507568},[31828],[34,507570,507572],{"className":507571},[31801],[34,507573,507575],{"className":507574,"style":31836},[31805],[34,507576],{},[34,507578],{"className":507579,"style":32604},[31845],[34,507581,32226],{"className":507582},[32024],[34,507584],{"className":507585,"style":32604},[31845],[34,507587,507589,507592,507632,507635,507638,507641],{"className":507588},[31676],[34,507590],{"className":507591,"style":31774},[31680],[34,507593,507595,507598],{"className":507594},[31685],[34,507596,111],{"className":507597},[31685,31686],[34,507599,507601],{"className":507600},[31792],[34,507602,507604,507624],{"className":507603},[31796,31797],[34,507605,507607,507621],{"className":507606},[31801],[34,507608,507610],{"className":507609,"style":31806},[31805],[34,507611,507612,507615],{"style":31809},[34,507613],{"className":507614,"style":31814},[31813],[34,507616,507618],{"className":507617},[31818,31819,31820,31821],[34,507619,768],{"className":507620},[31685,31821],[34,507622,31829],{"className":507623},[31828],[34,507625,507627],{"className":507626},[31801],[34,507628,507630],{"className":507629,"style":31836},[31805],[34,507631],{},[34,507633,2557],{"className":507634},[31690],[34,507636],{"className":507637,"style":31846},[31845],[34,507639,112296],{"className":507640},[31850],[34,507642],{"className":507643,"style":31846},[31845],[34,507645,507647,507650,507653,507656,507659,507662,507702,507705,507708],{"className":507646},[31676],[34,507648],{"className":507649,"style":31774},[31680],[34,507651,106063],{"className":507652,"style":67482},[31685,31686],[34,507654,31960],{"className":507655},[32119],[34,507657],{"className":507658,"style":32123},[31845],[34,507660,456],{"className":507661},[31782],[34,507663,507665,507668],{"className":507664},[31685],[34,507666,1697],{"className":507667},[31685,31686],[34,507669,507671],{"className":507670},[31792],[34,507672,507674,507694],{"className":507673},[31796,31797],[34,507675,507677,507691],{"className":507676},[31801],[34,507678,507680],{"className":507679,"style":31806},[31805],[34,507681,507682,507685],{"style":31809},[34,507683],{"className":507684,"style":31814},[31813],[34,507686,507688],{"className":507687},[31818,31819,31820,31821],[34,507689,640],{"className":507690},[31685,31821],[34,507692,31829],{"className":507693},[31828],[34,507695,507697],{"className":507696},[31801],[34,507698,507700],{"className":507699,"style":31836},[31805],[34,507701],{},[34,507703],{"className":507704,"style":32604},[31845],[34,507706,32226],{"className":507707},[32024],[34,507709],{"className":507710,"style":32604},[31845],[34,507712,507714,507717,507757,507760,507763,507766],{"className":507713},[31676],[34,507715],{"className":507716,"style":31774},[31680],[34,507718,507720,507723],{"className":507719},[31685],[34,507721,1697],{"className":507722},[31685,31686],[34,507724,507726],{"className":507725},[31792],[34,507727,507729,507749],{"className":507728},[31796,31797],[34,507730,507732,507746],{"className":507731},[31801],[34,507733,507735],{"className":507734,"style":31806},[31805],[34,507736,507737,507740],{"style":31809},[34,507738],{"className":507739,"style":31814},[31813],[34,507741,507743],{"className":507742},[31818,31819,31820,31821],[34,507744,768],{"className":507745},[31685,31821],[34,507747,31829],{"className":507748},[31828],[34,507750,507752],{"className":507751},[31801],[34,507753,507755],{"className":507754,"style":31836},[31805],[34,507756],{},[34,507758,2557],{"className":507759},[31690],[34,507761],{"className":507762,"style":31846},[31845],[34,507764,112296],{"className":507765},[31850],[34,507767],{"className":507768,"style":31846},[31845],[34,507770,507772,507775],{"className":507771},[31676],[34,507773],{"className":507774,"style":33218},[31680],[34,507776,223320],{"className":507777,"style":223361},[31685,31686],"。故差仍在 ",[34,507780,507782,507799],{"className":507781},[31642],[34,507783,507785],{"className":507784},[31646],[31648,507786,507787],{"xmlns":31650},[31652,507788,507789,507797],{},[31655,507790,507791,507793,507795],{},[31658,507792,106063],{},[31662,507794,223],{},[31658,507796,223320],{},[31665,507798,455483],{"encoding":31667},[34,507800,507802,507820],{"className":507801,"ariaHidden":758},[31672],[34,507803,507805,507808,507811,507814,507817],{"className":507804},[31676],[34,507806],{"className":507807,"style":34944},[31680],[34,507809,106063],{"className":507810,"style":67482},[31685,31686],[34,507812],{"className":507813,"style":32604},[31845],[34,507815,223],{"className":507816},[32024],[34,507818],{"className":507819,"style":32604},[31845],[34,507821,507823,507826],{"className":507822},[31676],[34,507824],{"className":507825,"style":33218},[31680],[34,507827,223320],{"className":507828,"style":223361},[31685,31686],[108,507830,507831,34554,507833,46528,507950,39984,508000,508147,508148,508176,508177,505512,508257,508176,508285,355408,508365,508443,508444],{},[2627,507832,442183],{},[34,507834,507836,507866],{"className":507835},[31642],[34,507837,507839],{"className":507838},[31646],[31648,507840,507841],{"xmlns":31650},[31652,507842,507843,507863],{},[31655,507844,507845,507847,507849,507851,507853,507855,507857,507859,507861],{},[31658,507846,32936],{},[31662,507848,202],{},[31658,507850,111],{},[31662,507852,223],{},[31658,507854,1697],{},[31662,507856,112296],{},[31658,507858,106063],{},[31662,507860,223],{},[31658,507862,223320],{},[31665,507864,507865],{"encoding":31667},"x = a + b \\in I+J",[34,507867,507869,507887,507905,507923,507941],{"className":507868,"ariaHidden":758},[31672],[34,507870,507872,507875,507878,507881,507884],{"className":507871},[31676],[34,507873],{"className":507874,"style":32527},[31680],[34,507876,32936],{"className":507877},[31685,31686],[34,507879],{"className":507880,"style":31846},[31845],[34,507882,202],{"className":507883},[31850],[34,507885],{"className":507886,"style":31846},[31845],[34,507888,507890,507893,507896,507899,507902],{"className":507889},[31676],[34,507891],{"className":507892,"style":32597},[31680],[34,507894,111],{"className":507895},[31685,31686],[34,507897],{"className":507898,"style":32604},[31845],[34,507900,223],{"className":507901},[32024],[34,507903],{"className":507904,"style":32604},[31845],[34,507906,507908,507911,507914,507917,507920],{"className":507907},[31676],[34,507909],{"className":507910,"style":46991},[31680],[34,507912,1697],{"className":507913},[31685,31686],[34,507915],{"className":507916,"style":31846},[31845],[34,507918,112296],{"className":507919},[31850],[34,507921],{"className":507922,"style":31846},[31845],[34,507924,507926,507929,507932,507935,507938],{"className":507925},[31676],[34,507927],{"className":507928,"style":34944},[31680],[34,507930,106063],{"className":507931,"style":67482},[31685,31686],[34,507933],{"className":507934,"style":32604},[31845],[34,507936,223],{"className":507937},[32024],[34,507939],{"className":507940,"style":32604},[31845],[34,507942,507944,507947],{"className":507943},[31676],[34,507945],{"className":507946,"style":33218},[31680],[34,507948,223320],{"className":507949,"style":223361},[31685,31686],[34,507951,507953,507970],{"className":507952},[31642],[34,507954,507956],{"className":507955},[31646],[31648,507957,507958],{"xmlns":31650},[31652,507959,507960,507968],{},[31655,507961,507962,507964,507966],{},[31658,507963,34395],{},[31662,507965,112296],{},[31658,507967,34386],{},[31665,507969,335118],{"encoding":31667},[34,507971,507973,507991],{"className":507972,"ariaHidden":758},[31672],[34,507974,507976,507979,507982,507985,507988],{"className":507975},[31676],[34,507977],{"className":507978,"style":34812},[31680],[34,507980,34395],{"className":507981,"style":34452},[31685,31686],[34,507983],{"className":507984,"style":31846},[31845],[34,507986,112296],{"className":507987},[31850],[34,507989],{"className":507990,"style":31846},[31845],[34,507992,507994,507997],{"className":507993},[31676],[34,507995],{"className":507996,"style":33218},[31680],[34,507998,34386],{"className":507999,"style":34427},[31685,31686],[34,508001,508003,508045],{"className":508002},[31642],[34,508004,508006],{"className":508005},[31646],[31648,508007,508008],{"xmlns":31650},[31652,508009,508010,508042],{},[31655,508011,508012,508014,508016,508018,508020,508022,508024,508026,508028,508030,508032,508034,508036,508038,508040],{},[31658,508013,34395],{},[31658,508015,32936],{},[31662,508017,202],{},[31658,508019,34395],{},[31662,508021,456],{"stretchy":702},[31658,508023,111],{},[31662,508025,223],{},[31658,508027,1697],{},[31662,508029,2557],{"stretchy":702},[31662,508031,202],{},[31658,508033,34395],{},[31658,508035,111],{},[31662,508037,223],{},[31658,508039,34395],{},[31658,508041,1697],{},[31665,508043,508044],{"encoding":31667},"rx = r(a + b) = ra + rb",[34,508046,508048,508069,508093,508114,508135],{"className":508047,"ariaHidden":758},[31672],[34,508049,508051,508054,508057,508060,508063,508066],{"className":508050},[31676],[34,508052],{"className":508053,"style":32527},[31680],[34,508055,34395],{"className":508056,"style":34452},[31685,31686],[34,508058,32936],{"className":508059},[31685,31686],[34,508061],{"className":508062,"style":31846},[31845],[34,508064,202],{"className":508065},[31850],[34,508067],{"className":508068,"style":31846},[31845],[34,508070,508072,508075,508078,508081,508084,508087,508090],{"className":508071},[31676],[34,508073],{"className":508074,"style":31774},[31680],[34,508076,34395],{"className":508077,"style":34452},[31685,31686],[34,508079,456],{"className":508080},[31782],[34,508082,111],{"className":508083},[31685,31686],[34,508085],{"className":508086,"style":32604},[31845],[34,508088,223],{"className":508089},[32024],[34,508091],{"className":508092,"style":32604},[31845],[34,508094,508096,508099,508102,508105,508108,508111],{"className":508095},[31676],[34,508097],{"className":508098,"style":31774},[31680],[34,508100,1697],{"className":508101},[31685,31686],[34,508103,2557],{"className":508104},[31690],[34,508106],{"className":508107,"style":31846},[31845],[34,508109,202],{"className":508110},[31850],[34,508112],{"className":508113,"style":31846},[31845],[34,508115,508117,508120,508123,508126,508129,508132],{"className":508116},[31676],[34,508118],{"className":508119,"style":32597},[31680],[34,508121,34395],{"className":508122,"style":34452},[31685,31686],[34,508124,111],{"className":508125},[31685,31686],[34,508127],{"className":508128,"style":32604},[31845],[34,508130,223],{"className":508131},[32024],[34,508133],{"className":508134,"style":32604},[31845],[34,508136,508138,508141,508144],{"className":508137},[31676],[34,508139],{"className":508140,"style":31681},[31680],[34,508142,34395],{"className":508143,"style":34452},[31685,31686],[34,508145,1697],{"className":508146},[31685,31686],"。\n因 ",[34,508149,508151,508164],{"className":508150},[31642],[34,508152,508154],{"className":508153},[31646],[31648,508155,508156],{"xmlns":31650},[31652,508157,508158,508162],{},[31655,508159,508160],{},[31658,508161,106063],{},[31665,508163,106063],{"encoding":31667},[34,508165,508167],{"className":508166,"ariaHidden":758},[31672],[34,508168,508170,508173],{"className":508169},[31676],[34,508171],{"className":508172,"style":33218},[31680],[34,508174,106063],{"className":508175,"style":67482},[31685,31686]," 是理想 ",[34,508178,508180,508206],{"className":508179},[31642],[34,508181,508183],{"className":508182},[31646],[31648,508184,508185],{"xmlns":31650},[31652,508186,508187,508203],{},[31655,508188,508189,508191,508193,508195,508197,508199,508201],{},[39361,508190,262622],{},[31662,508192,262625],{},[39361,508194,262622],{},[31658,508196,34395],{},[31658,508198,111],{},[31662,508200,112296],{},[31658,508202,106063],{},[31665,508204,508205],{"encoding":31667},"\\implies ra \\in I",[34,508207,508209,508227,508248],{"className":508208,"ariaHidden":758},[31672],[34,508210,508212,508215,508218,508221,508224],{"className":508211},[31676],[34,508213],{"className":508214,"style":318264},[31680],[34,508216],{"className":508217,"style":31846},[31845],[34,508219,262625],{"className":508220},[31850],[34,508222],{"className":508223,"style":31846},[31845],[34,508225],{"className":508226,"style":31846},[31845],[34,508228,508230,508233,508236,508239,508242,508245],{"className":508229},[31676],[34,508231],{"className":508232,"style":34812},[31680],[34,508234,34395],{"className":508235,"style":34452},[31685,31686],[34,508237,111],{"className":508238},[31685,31686],[34,508240],{"className":508241,"style":31846},[31845],[34,508243,112296],{"className":508244},[31850],[34,508246],{"className":508247,"style":31846},[31845],[34,508249,508251,508254],{"className":508250},[31676],[34,508252],{"className":508253,"style":33218},[31680],[34,508255,106063],{"className":508256,"style":67482},[31685,31686],[34,508258,508260,508273],{"className":508259},[31642],[34,508261,508263],{"className":508262},[31646],[31648,508264,508265],{"xmlns":31650},[31652,508266,508267,508271],{},[31655,508268,508269],{},[31658,508270,223320],{},[31665,508272,223320],{"encoding":31667},[34,508274,508276],{"className":508275,"ariaHidden":758},[31672],[34,508277,508279,508282],{"className":508278},[31676],[34,508280],{"className":508281,"style":33218},[31680],[34,508283,223320],{"className":508284,"style":223361},[31685,31686],[34,508286,508288,508314],{"className":508287},[31642],[34,508289,508291],{"className":508290},[31646],[31648,508292,508293],{"xmlns":31650},[31652,508294,508295,508311],{},[31655,508296,508297,508299,508301,508303,508305,508307,508309],{},[39361,508298,262622],{},[31662,508300,262625],{},[39361,508302,262622],{},[31658,508304,34395],{},[31658,508306,1697],{},[31662,508308,112296],{},[31658,508310,223320],{},[31665,508312,508313],{"encoding":31667},"\\implies rb \\in J",[34,508315,508317,508335,508356],{"className":508316,"ariaHidden":758},[31672],[34,508318,508320,508323,508326,508329,508332],{"className":508319},[31676],[34,508321],{"className":508322,"style":318264},[31680],[34,508324],{"className":508325,"style":31846},[31845],[34,508327,262625],{"className":508328},[31850],[34,508330],{"className":508331,"style":31846},[31845],[34,508333],{"className":508334,"style":31846},[31845],[34,508336,508338,508341,508344,508347,508350,508353],{"className":508337},[31676],[34,508339],{"className":508340,"style":46991},[31680],[34,508342,34395],{"className":508343,"style":34452},[31685,31686],[34,508345,1697],{"className":508346},[31685,31686],[34,508348],{"className":508349,"style":31846},[31845],[34,508351,112296],{"className":508352},[31850],[34,508354],{"className":508355,"style":31846},[31845],[34,508357,508359,508362],{"className":508358},[31676],[34,508360],{"className":508361,"style":33218},[31680],[34,508363,223320],{"className":508364,"style":223361},[31685,31686],[34,508366,508368,508392],{"className":508367},[31642],[34,508369,508371],{"className":508370},[31646],[31648,508372,508373],{"xmlns":31650},[31652,508374,508375,508389],{},[31655,508376,508377,508379,508381,508383,508385,508387],{},[31658,508378,34395],{},[31658,508380,32936],{},[31662,508382,112296],{},[31658,508384,106063],{},[31662,508386,223],{},[31658,508388,223320],{},[31665,508390,508391],{"encoding":31667},"rx \\in I + J",[34,508393,508395,508416,508434],{"className":508394,"ariaHidden":758},[31672],[34,508396,508398,508401,508404,508407,508410,508413],{"className":508397},[31676],[34,508399],{"className":508400,"style":34812},[31680],[34,508402,34395],{"className":508403,"style":34452},[31685,31686],[34,508405,32936],{"className":508406},[31685,31686],[34,508408],{"className":508409,"style":31846},[31845],[34,508411,112296],{"className":508412},[31850],[34,508414],{"className":508415,"style":31846},[31845],[34,508417,508419,508422,508425,508428,508431],{"className":508418},[31676],[34,508420],{"className":508421,"style":34944},[31680],[34,508423,106063],{"className":508424,"style":67482},[31685,31686],[34,508426],{"className":508427,"style":32604},[31845],[34,508429,223],{"className":508430},[32024],[34,508432],{"className":508433,"style":32604},[31845],[34,508435,508437,508440],{"className":508436},[31676],[34,508438],{"className":508439,"style":33218},[31680],[34,508441,223320],{"className":508442,"style":223361},[31685,31686],"。右乘同理。\n",[105,508445,508446],{},[108,508447,508448,2664,508450,508500,508501,258,508529,508557],{},[2627,508449,114043],{},[34,508451,508453,508470],{"className":508452},[31642],[34,508454,508456],{"className":508455},[31646],[31648,508457,508458],{"xmlns":31650},[31652,508459,508460,508468],{},[31655,508461,508462,508464,508466],{},[31658,508463,106063],{},[31662,508465,223],{},[31658,508467,223320],{},[31665,508469,506480],{"encoding":31667},[34,508471,508473,508491],{"className":508472,"ariaHidden":758},[31672],[34,508474,508476,508479,508482,508485,508488],{"className":508475},[31676],[34,508477],{"className":508478,"style":34944},[31680],[34,508480,106063],{"className":508481,"style":67482},[31685,31686],[34,508483],{"className":508484,"style":32604},[31845],[34,508486,223],{"className":508487},[32024],[34,508489],{"className":508490,"style":32604},[31845],[34,508492,508494,508497],{"className":508493},[31676],[34,508495],{"className":508496,"style":33218},[31680],[34,508498,223320],{"className":508499,"style":223361},[31685,31686]," 是理想（且是包含 ",[34,508502,508504,508517],{"className":508503},[31642],[34,508505,508507],{"className":508506},[31646],[31648,508508,508509],{"xmlns":31650},[31652,508510,508511,508515],{},[31655,508512,508513],{},[31658,508514,106063],{},[31665,508516,106063],{"encoding":31667},[34,508518,508520],{"className":508519,"ariaHidden":758},[31672],[34,508521,508523,508526],{"className":508522},[31676],[34,508524],{"className":508525,"style":33218},[31680],[34,508527,106063],{"className":508528,"style":67482},[31685,31686],[34,508530,508532,508545],{"className":508531},[31642],[34,508533,508535],{"className":508534},[31646],[31648,508536,508537],{"xmlns":31650},[31652,508538,508539,508543],{},[31655,508540,508541],{},[31658,508542,223320],{},[31665,508544,223320],{"encoding":31667},[34,508546,508548],{"className":508547,"ariaHidden":758},[31672],[34,508549,508551,508554],{"className":508550},[31676],[34,508552],{"className":508553,"style":33218},[31680],[34,508555,223320],{"className":508556,"style":223361},[31685,31686]," 的最小理想）。",[253,508559,508561,508562],{"id":508560},"_53-积-ijijij","5.3 积 ",[34,508563,508565,508580],{"className":508564},[31642],[34,508566,508568],{"className":508567},[31646],[31648,508569,508570],{"xmlns":31650},[31652,508571,508572,508578],{},[31655,508573,508574,508576],{},[31658,508575,106063],{},[31658,508577,223320],{},[31665,508579,456244],{"encoding":31667},[34,508581,508583],{"className":508582,"ariaHidden":758},[31672],[34,508584,508586,508589,508592],{"className":508585},[31676],[34,508587],{"className":508588,"style":33218},[31680],[34,508590,106063],{"className":508591,"style":67482},[31685,31686],[34,508593,223320],{"className":508594,"style":223361},[31685,31686],[15,508596,508597,2664,508599,509025,509026,2664,509028,505288,509061,2664],{},[2627,508598,506515],{},[34,508600,508602,508682],{"className":508601},[31642],[34,508603,508605],{"className":508604},[31646],[31648,508606,508607],{"xmlns":31650},[31652,508608,508609,508679],{},[31655,508610,508611,508613,508615,508617,508619,508633,508639,508645,508647,508653,508655,508657,508659,508665,508667,508669,508671,508673,508675,508677],{},[31658,508612,106063],{},[31658,508614,223320],{},[31662,508616,202],{},[31662,508618,32685],{"stretchy":702},[38846,508620,508621,508623,508631],{},[31662,508622,131138],{},[31655,508624,508625,508627,508629],{},[31658,508626,33759],{},[31662,508628,202],{},[31749,508630,640],{},[31658,508632,31660],{},[31743,508634,508635,508637],{},[31658,508636,111],{},[31658,508638,33759],{},[31743,508640,508641,508643],{},[31658,508642,1697],{},[31658,508644,33759],{},[31662,508646,162086],{},[31743,508648,508649,508651],{},[31658,508650,111],{},[31658,508652,33759],{},[31662,508654,112296],{},[31658,508656,106063],{},[31662,508658,31960],{"separator":758},[31743,508660,508661,508663],{},[31658,508662,1697],{},[31658,508664,33759],{},[31662,508666,112296],{},[31658,508668,223320],{},[31662,508670,31960],{"separator":758},[31658,508672,31660],{},[31662,508674,112296],{},[31658,508676,33347],{"mathvariant":318416},[31662,508678,32715],{"stretchy":702},[31665,508680,508681],{"encoding":31667},"IJ = \\{ \\sum_{k=1}^n a_k b_k \\mid a_k \\in I, b_k \\in J, n \\in \\mathbb{N} \\}",[34,508683,508685,508706,508867,508922,508986,509013],{"className":508684,"ariaHidden":758},[31672],[34,508686,508688,508691,508694,508697,508700,508703],{"className":508687},[31676],[34,508689],{"className":508690,"style":33218},[31680],[34,508692,106063],{"className":508693,"style":67482},[31685,31686],[34,508695,223320],{"className":508696,"style":223361},[31685,31686],[34,508698],{"className":508699,"style":31846},[31845],[34,508701,202],{"className":508702},[31850],[34,508704],{"className":508705,"style":31846},[31845],[34,508707,508709,508712,508715,508775,508778,508818,508858,508861,508864],{"className":508708},[31676],[34,508710],{"className":508711,"style":351974},[31680],[34,508713,32685],{"className":508714},[31782],[34,508716,508718,508721],{"className":508717},[32767],[34,508719,131138],{"className":508720,"style":156561},[32767,131249,156560],[34,508722,508724],{"className":508723},[31792],[34,508725,508727,508767],{"className":508726},[31796,31797],[34,508728,508730,508764],{"className":508729},[31801],[34,508731,508733,508753],{"className":508732,"style":351993},[31805],[34,508734,508735,508738],{"style":156577},[34,508736],{"className":508737,"style":31814},[31813],[34,508739,508741],{"className":508740},[31818,31819,31820,31821],[34,508742,508744,508747,508750],{"className":508743},[31685,31821],[34,508745,33759],{"className":508746,"style":33774},[31685,31686,31821],[34,508748,202],{"className":508749},[31850,31821],[34,508751,640],{"className":508752},[31685,31821],[34,508754,508755,508758],{"style":352016},[34,508756],{"className":508757,"style":31814},[31813],[34,508759,508761],{"className":508760},[31818,31819,31820,31821],[34,508762,31660],{"className":508763},[31685,31686,31821],[34,508765,31829],{"className":508766},[31828],[34,508768,508770],{"className":508769},[31801],[34,508771,508773],{"className":508772,"style":352035},[31805],[34,508774],{},[34,508776],{"className":508777,"style":32123},[31845],[34,508779,508781,508784],{"className":508780},[31685],[34,508782,111],{"className":508783},[31685,31686],[34,508785,508787],{"className":508786},[31792],[34,508788,508790,508810],{"className":508789},[31796,31797],[34,508791,508793,508807],{"className":508792},[31801],[34,508794,508796],{"className":508795,"style":33833},[31805],[34,508797,508798,508801],{"style":31809},[34,508799],{"className":508800,"style":31814},[31813],[34,508802,508804],{"className":508803},[31818,31819,31820,31821],[34,508805,33759],{"className":508806,"style":33774},[31685,31686,31821],[34,508808,31829],{"className":508809},[31828],[34,508811,508813],{"className":508812},[31801],[34,508814,508816],{"className":508815,"style":31836},[31805],[34,508817],{},[34,508819,508821,508824],{"className":508820},[31685],[34,508822,1697],{"className":508823},[31685,31686],[34,508825,508827],{"className":508826},[31792],[34,508828,508830,508850],{"className":508829},[31796,31797],[34,508831,508833,508847],{"className":508832},[31801],[34,508834,508836],{"className":508835,"style":33833},[31805],[34,508837,508838,508841],{"style":31809},[34,508839],{"className":508840,"style":31814},[31813],[34,508842,508844],{"className":508843},[31818,31819,31820,31821],[34,508845,33759],{"className":508846,"style":33774},[31685,31686,31821],[34,508848,31829],{"className":508849},[31828],[34,508851,508853],{"className":508852},[31801],[34,508854,508856],{"className":508855,"style":31836},[31805],[34,508857],{},[34,508859],{"className":508860,"style":31846},[31845],[34,508862,162086],{"className":508863},[31850],[34,508865],{"className":508866,"style":31846},[31845],[34,508868,508870,508873,508913,508916,508919],{"className":508869},[31676],[34,508871],{"className":508872,"style":225773},[31680],[34,508874,508876,508879],{"className":508875},[31685],[34,508877,111],{"className":508878},[31685,31686],[34,508880,508882],{"className":508881},[31792],[34,508883,508885,508905],{"className":508884},[31796,31797],[34,508886,508888,508902],{"className":508887},[31801],[34,508889,508891],{"className":508890,"style":33833},[31805],[34,508892,508893,508896],{"style":31809},[34,508894],{"className":508895,"style":31814},[31813],[34,508897,508899],{"className":508898},[31818,31819,31820,31821],[34,508900,33759],{"className":508901,"style":33774},[31685,31686,31821],[34,508903,31829],{"className":508904},[31828],[34,508906,508908],{"className":508907},[31801],[34,508909,508911],{"className":508910,"style":31836},[31805],[34,508912],{},[34,508914],{"className":508915,"style":31846},[31845],[34,508917,112296],{"className":508918},[31850],[34,508920],{"className":508921,"style":31846},[31845],[34,508923,508925,508928,508931,508934,508937,508977,508980,508983],{"className":508924},[31676],[34,508926],{"className":508927,"style":43217},[31680],[34,508929,106063],{"className":508930,"style":67482},[31685,31686],[34,508932,31960],{"className":508933},[32119],[34,508935],{"className":508936,"style":32123},[31845],[34,508938,508940,508943],{"className":5089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对加法和减法封闭（因为它是有限和的集合）。",[108,509105,509106,509108,509109,509139,509140,70884,509193,509279,509280,39984,509330],{},[2627,509107,442183],{},"：只需对单个项 ",[34,509110,509112,509127],{"className":509111},[31642],[34,509113,509115],{"className":509114},[31646],[31648,509116,509117],{"xmlns":31650},[31652,509118,509119,509125],{},[31655,509120,509121,509123],{},[31658,509122,111],{},[31658,509124,1697],{},[31665,509126,324191],{"encoding":31667},[34,509128,509130],{"className":509129,"ariaHidden":758},[31672],[34,509131,509133,509136],{"className":509132},[31676],[34,509134],{"className":509135,"style":31681},[31680],[34,509137,324191],{"className":509138},[31685,31686]," 验证，有限和自然满足。\n设 ",[34,509141,509143,509163],{"className":509142},[31642],[34,509144,509146],{"className":509145},[31646],[31648,509147,509148],{"xmlns":31650},[31652,509149,509150,509160],{},[31655,509151,509152,509154,509156,509158],{},[31658,509153,32936],{},[31662,509155,202],{},[31658,509157,111],{},[31658,509159,1697],{},[31665,509161,509162],{"encoding":31667},"x = ab",[34,509164,509166,509184],{"className":509165,"ariaHidden":758},[31672],[34,509167,509169,509172,509175,509178,509181],{"className":509168},[31676],[34,509170],{"className":509171,"style":32527},[31680],[34,509173,32936],{"className":509174},[31685,31686],[34,509176],{"className":509177,"style":31846},[31845],[34,509179,202],{"className":509180},[31850],[34,509182],{"className":509183,"style":31846},[31845],[34,509185,509187,509190],{"className":509186},[31676],[34,509188],{"className":509189,"style":31681},[31680],[34,509191,324191],{"className":509192},[31685,31686],[34,509194,509196,509222],{"className":509195},[31642],[34,509197,509199],{"className":509198},[31646],[31648,509200,509201],{"xmlns":31650},[31652,509202,509203,509219],{},[31655,509204,509205,509207,509209,509211,509213,509215,509217],{},[31658,509206,111],{},[31662,509208,112296],{},[31658,509210,106063],{},[31662,509212,31960],{"separator":758},[31658,509214,1697],{},[31662,509216,112296],{},[31658,509218,223320],{},[31665,509220,509221],{"encoding":31667},"a \\in I, b \\in J",[34,509223,509225,509243,509270],{"className":509224,"ariaHidden":758},[31672],[34,509226,509228,509231,509234,509237,509240],{"className":509227},[31676],[34,509229],{"className":509230,"style":34812},[31680],[34,509232,111],{"className":509233},[31685,31686],[34,509235],{"className":509236,"style":31846},[31845],[34,509238,112296],{"className":509239},[31850],[34,509241],{"className":509242,"style":31846},[31845],[34,509244,509246,509249,509252,509255,509258,509261,509264,509267],{"className":509245},[31676],[34,509247],{"className":509248,"style":43217},[31680],[34,509250,106063],{"className":509251,"style":67482},[31685,31686],[34,509253,31960],{"className":509254},[32119],[34,509256],{"className":509257,"style":32123},[31845],[34,509259,1697],{"className":509260},[31685,31686],[34,509262],{"className":509263,"style":31846},[31845],[34,509265,112296],{"className":509266},[31850],[34,509268],{"className":509269,"style":31846},[31845],[34,509271,509273,509276],{"className":509272},[31676],[34,509274],{"className":509275,"style":33218},[31680],[34,509277,223320],{"className":509278,"style":223361},[31685,31686],"。设 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x = r (ab) = (ra) b",[34,509378,509380,509401,509428],{"className":509379,"ariaHidden":758},[31672],[34,509381,509383,509386,509389,509392,509395,509398],{"className":509382},[31676],[34,509384],{"className":509385,"style":32527},[31680],[34,509387,34395],{"className":509388,"style":34452},[31685,31686],[34,509390,32936],{"className":509391},[31685,31686],[34,509393],{"className":509394,"style":31846},[31845],[34,509396,202],{"className":509397},[31850],[34,509399],{"className":509400,"style":31846},[31845],[34,509402,509404,509407,509410,509413,509416,509419,509422,509425],{"className":509403},[31676],[34,509405],{"className":509406,"style":31774},[31680],[34,509408,34395],{"className":509409,"style":34452},[31685,31686],[34,509411,456],{"className":509412},[31782],[34,509414,324191],{"className":509415},[31685,31686],[34,509417,2557],{"className":509418},[31690],[34,509420],{"className":509421,"style":31846},[31845],[34,509423,202],{"className":509424},[31850],[34,509426],{"className":509427,"style":31846},[31845],[34,509429,509431,509434,509437,509440,509443,509446],{"className":509430},[31676],[34,509432],{"className":509433,"style":31774},[31680],[34,509435,456],{"className":509436},[31782],[34,509438,34395],{"className":509439,"style":34452},[31685,31686],[34,509441,111],{"className":509442},[31685,31686],[34,509444,2557],{"className":509445},[31690],[34,509447,1697],{"className":509448},[31685,31686],[34,509450,509452,509465],{"className":509451},[31642],[34,509453,509455],{"className":509454},[31646],[31648,509456,509457],{"xmlns":31650},[31652,509458,509459,509463],{},[31655,509460,509461],{},[31658,509462,106063],{},[31665,509464,106063],{"encoding":31667},[34,509466,509468],{"className":509467,"ariaHidden":758},[31672],[34,509469,509471,509474],{"className":509470},[31676],[34,509472],{"className":509473,"style":33218},[31680],[34,509475,106063],{"className":509476,"style":67482},[31685,31686]," 是左理想，",[34,509479,509481,509501],{"className":509480},[31642],[34,509482,509484],{"className":509483},[31646],[31648,509485,509486],{"xmlns":31650},[31652,509487,509488,509498],{},[31655,509489,509490,509492,509494,509496],{},[31658,509491,34395],{},[31658,509493,111],{},[31662,509495,112296],{},[31658,509497,106063],{},[31665,509499,509500],{"encoding":31667},"ra \\in I",[34,509502,509504,509525],{"className":509503,"ariaHidden":758},[31672],[34,509505,509507,509510,509513,509516,509519,509522],{"className":509506},[31676],[34,509508],{"className":509509,"style":34812},[31680],[34,509511,34395],{"className":509512,"style":34452},[31685,31686],[34,509514,111],{"className":509515},[31685,31686],[34,509517],{"className":509518,"style":31846},[31845],[34,509520,112296],{"className":509521},[31850],[34,509523],{"className":509524,"style":31846},[31845],[34,509526,509528,509531],{"className":509527},[31676],[34,509529],{"className":509530,"style":33218},[31680],[34,509532,106063],{"className":509533,"style":67482},[31685,31686],[34,509535,509537,509559],{"className":509536},[31642],[34,509538,509540],{"className":509539},[31646],[31648,509541,509542],{"xmlns":31650},[31652,509543,509544,509556],{},[31655,509545,509546,509548,509550,509552,509554],{},[31662,509547,456],{"stretchy":702},[31658,509549,34395],{},[31658,509551,111],{},[31662,509553,2557],{"stretchy":702},[31658,509555,1697],{},[31665,509557,509558],{"encoding":31667},"(ra)b",[34,509560,509562],{"className":509561,"ariaHidden":758},[31672],[34,509563,509565,509568,509571,509574,509577,509580],{"className":509564},[31676],[34,509566],{"className":509567,"style":31774},[31680],[34,509569,456],{"className":509570},[31782],[34,509572,34395],{"className":509573,"style":34452},[31685,31686],[34,509575,111],{"className":509576},[31685,31686],[34,509578,2557],{"className":509579},[31690],[34,509581,1697],{"className":509582},[31685,31686]," 仍是形如 \"(I中元素)(J中元素)\" 的形式，属于 ",[34,509585,509587,509602],{"className":509586},[31642],[34,509588,509590],{"className":509589},[31646],[31648,509591,509592],{"xmlns":31650},[31652,509593,509594,509600],{},[31655,509595,509596,509598],{},[31658,509597,106063],{},[31658,509599,223320],{},[31665,509601,456244],{"encoding":31667},[34,509603,509605],{"className":509604,"ariaHidden":758},[31672],[34,509606,509608,509611,509614],{"className":509607},[31676],[34,509609],{"className":509610,"style":33218},[31680],[34,509612,106063],{"className":509613,"style":67482},[31685,31686],[34,509615,223320],{"className":509616,"style":223361},[31685,31686],[108,509618,509619,44689,509734,509762,509763,322067,509819,509868,509869,347],{},[34,509620,509622,509662],{"className":509621},[31642],[34,509623,509625],{"className":509624},[31646],[31648,509626,509627],{"xmlns":31650},[31652,509628,509629,509659],{},[31655,509630,509631,509633,509635,509637,509639,509641,509643,509645,509647,509649,509651,509653,509655,509657],{},[31658,509632,32936],{},[31658,509634,34395],{},[31662,509636,202],{},[31662,509638,456],{"stretchy":702},[31658,509640,111],{},[31658,509642,1697],{},[31662,509644,2557],{"stretchy":702},[31658,509646,34395],{},[31662,509648,202],{},[31658,509650,111],{},[31662,509652,456],{"stretchy":702},[31658,509654,1697],{},[31658,509656,34395],{},[31662,509658,2557],{"stretchy":702},[31665,509660,509661],{"encoding":31667},"x r = (ab) r = a (br)",[34,509663,509665,509686,509713],{"className":509664,"ariaHidden":758},[31672],[34,509666,509668,509671,509674,509677,509680,509683],{"className":509667},[31676],[34,509669],{"className":509670,"style":32527},[31680],[34,509672,32936],{"className":509673},[31685,31686],[34,509675,34395],{"className":509676,"style":34452},[31685,31686],[34,509678],{"className":509679,"style":31846},[31845],[34,509681,202],{"className":509682},[31850],[34,509684],{"className":509685,"style":31846},[31845],[34,509687,509689,509692,509695,509698,509701,509704,509707,509710],{"className":509688},[31676],[34,509690],{"className":509691,"style":31774},[31680],[34,509693,456],{"className":509694},[31782],[34,509696,324191],{"className":509697},[31685,31686],[34,509699,2557],{"className":509700},[31690],[34,509702,34395],{"className":509703,"style":34452},[31685,31686],[34,509705],{"className":509706,"style":31846},[31845],[34,509708,202],{"className":509709},[31850],[34,509711],{"className":509712,"style":31846},[31845],[34,509714,509716,509719,509722,509725,509728,509731],{"className":509715},[31676],[34,509717],{"className":509718,"style":31774},[31680],[34,509720,111],{"className":509721},[31685,31686],[34,509723,456],{"className":509724},[31782],[34,509726,1697],{"className":509727},[31685,31686],[34,509729,34395],{"className":509730,"style":34452},[31685,31686],[34,509732,2557],{"className":509733},[31690],[34,509735,509737,509750],{"className":509736},[31642],[34,509738,509740],{"className":509739},[31646],[31648,509741,509742],{"xmlns":31650},[31652,509743,509744,509748],{},[31655,509745,509746],{},[31658,509747,223320],{},[31665,509749,223320],{"encoding":31667},[34,509751,509753],{"className":509752,"ariaHidden":758},[31672],[34,509754,509756,509759],{"className":509755},[31676],[34,509757],{"className":509758,"style":33218},[31680],[34,509760,223320],{"className":509761,"style":223361},[31685,31686]," 是右理想，",[34,509764,509766,509786],{"className":509765},[31642],[34,509767,509769],{"className":509768},[31646],[31648,509770,509771],{"xmlns":31650},[31652,509772,509773,509783],{},[31655,509774,509775,509777,509779,509781],{},[31658,509776,1697],{},[31658,509778,34395],{},[31662,509780,112296],{},[31658,509782,223320],{},[31665,509784,509785],{"encoding":31667},"br \\in J",[34,509787,509789,509810],{"className":509788,"ariaHidden":758},[31672],[34,509790,509792,509795,509798,509801,509804,509807],{"className":509791},[31676],[34,509793],{"className":509794,"style":46991},[31680],[34,509796,1697],{"className":509797},[31685,31686],[34,509799,34395],{"className":509800,"style":34452},[31685,31686],[34,509802],{"className":509803,"style":31846},[31845],[34,509805,112296],{"className":509806},[31850],[34,509808],{"className":509809,"style":31846},[31845],[34,509811,509813,509816],{"className":509812},[31676],[34,509814],{"className":509815,"style":33218},[31680],[34,509817,223320],{"className":509818,"style":223361},[31685,31686],[34,509820,509822,509844],{"className":509821},[31642],[34,509823,509825],{"className":509824},[31646],[31648,509826,509827],{"xmlns":31650},[31652,509828,509829,509841],{},[31655,509830,509831,509833,509835,509837,509839],{},[31658,509832,111],{},[31662,509834,456],{"stretchy":702},[31658,509836,1697],{},[31658,509838,34395],{},[31662,509840,2557],{"stretchy":702},[31665,509842,509843],{"encoding":31667},"a(br)",[34,509845,509847],{"className":509846,"ariaHidden":758},[31672],[34,509848,509850,509853,509856,509859,509862,509865],{"className":509849},[31676],[34,509851],{"className":509852,"style":31774},[31680],[34,509854,111],{"className":509855},[31685,31686],[34,509857,456],{"className":509858},[31782],[34,509860,1697],{"className":509861},[31685,31686],[34,509863,34395],{"className":509864,"style":34452},[31685,31686],[34,509866,2557],{"className":509867},[31690]," 仍属于 ",[34,509870,509872,509887],{"className":509871},[31642],[34,509873,509875],{"className":509874},[31646],[31648,509876,509877],{"xmlns":31650},[31652,509878,509879,509885],{},[31655,509880,509881,509883],{},[31658,509882,106063],{},[31658,509884,223320],{},[31665,509886,456244],{"encoding":31667},[34,509888,509890],{"className":509889,"ariaHidden":758},[31672],[34,509891,509893,509896,509899],{"className":509892},[31676],[34,509894],{"className":509895,"style":33218},[31680],[34,509897,106063],{"className":509898,"style":67482},[31685,31686],[34,509900,223320],{"className":509901,"style":223361},[31685,31686],[108,509903,509904,2664,509906,506455],{},[2627,509905,114043],{},[34,509907,509909,509924],{"className":509908},[31642],[34,509910,509912],{"className":509911},[31646],[31648,509913,509914],{"xmlns":31650},[31652,509915,509916,509922],{},[31655,509917,509918,509920],{},[31658,509919,106063],{},[31658,509921,223320],{},[31665,509923,456244],{"encoding":31667},[34,509925,509927],{"className":509926,"ariaHidden":758},[31672],[34,509928,509930,509933,509936],{"className":509929},[31676],[34,509931],{"className":509932,"style":33218},[31680],[34,509934,106063],{"className":509935,"style":67482},[31685,31686],[34,509937,223320],{"className":509938,"style":223361},[31685,31686],[108,509940,509941,509944,509945,347],{},[71807,509942,509943],{},"注意","：一般情况下 ",[34,509946,509948,509971],{"className":509947},[31642],[34,509949,509951],{"className":509950},[31646],[31648,509952,509953],{"xmlns":31650},[31652,509954,509955,509969],{},[31655,509956,509957,509959,509961,509963,509965,509967],{},[31658,509958,106063],{},[31658,509960,223320],{},[31662,509962,131855],{},[31658,509964,106063],{},[31662,509966,326344],{},[31658,509968,223320],{},[31665,509970,456828],{"encoding":31667},[34,509972,509974,509995,510013],{"className":509973,"ariaHidden":758},[31672],[34,509975,509977,509980,509983,509986,509989,509992],{"className":509976},[31676],[34,509978],{"className":509979,"style":195047},[31680],[34,509981,106063],{"className":509982,"style":67482},[31685,31686],[34,509984,223320],{"className":509985,"style":223361},[31685,31686],[34,509987],{"className":509988,"style":31846},[31845],[34,509990,131855],{"className":509991},[31850],[34,509993],{"className":509994,"style":31846},[31845],[34,509996,509998,510001,510004,510007,510010],{"className":509997},[31676],[34,509999],{"className":510000,"style":33218},[31680],[34,510002,106063],{"className":510003,"style":67482},[31685,31686],[34,510005],{"className":510006,"style":32604},[31845],[34,510008,326344],{"className":510009},[32024],[34,510011],{"className":510012,"style":32604},[31845],[34,510014,510016,510019],{"className":510015},[31676],[34,510017],{"className":510018,"style":33218},[31680],[34,510020,223320],{"className":510021,"style":223361},[31685,31686],[32188,510023],{},[19,510025,510027],{"id":510026},"_6-商环-quotient-ring","6. 商环 (Quotient Ring)",[253,510029,510031],{"id":510030},"_61-构造与良定性","6.1 构造与良定性",[15,510033,510034,510035,113124,510063,510065,510066,510069],{},"我们需要 ",[34,510036,510038,510051],{"className":510037},[31642],[34,510039,510041],{"className":510040},[31646],[31648,510042,510043],{"xmlns":31650},[31652,510044,510045,510049],{},[31655,510046,510047],{},[31658,510048,106063],{},[31665,510050,106063],{"encoding":31667},[34,510052,510054],{"className":510053,"ariaHidden":758},[31672],[34,510055,510057,510060],{"className":510056},[31676],[34,510058],{"className":510059,"style":33218},[31680],[34,510061,106063],{"className":510062,"style":67482},[31685,31686],[2627,510064,495606],{},"（而不仅仅是子环），是为了保证商环上的乘法运算是",[2627,510067,510068],{},"良定 (well-defined)"," 的。",[105,510071,510072,510214,510409],{},[108,510073,510074,2664,510076,510213],{},[2627,510075,149746],{},[34,510077,510079,510117],{"className":510078},[31642],[34,510080,510082],{"className":510081},[31646],[31648,510083,510084],{"xmlns":31650},[31652,510085,510086,510114],{},[31655,510087,510088,510090,510092,510094,510096,510098,510100,510102,510104,510106,510108,510110,510112],{},[31658,510089,34386],{},[31658,510091,38080],{"mathvariant":38079},[31658,510093,106063],{},[31662,510095,202],{},[31662,510097,32685],{"stretchy":702},[31658,510099,34395],{},[31662,510101,223],{},[31658,510103,106063],{},[31662,510105,162086],{},[31658,510107,34395],{},[31662,510109,112296],{},[31658,510111,34386],{},[31662,510113,32715],{"stretchy":702},[31665,510115,510116],{"encoding":31667},"R\u002FI = \\{ r + I \\mid r \\in R \\}",[34,510118,510120,510144,510165,510183,510201],{"className":510119,"ariaHidden":758},[31672],[34,510121,510123,510126,510129,510132,510135,510138,510141],{"className":510122},[31676],[34,510124],{"className":510125,"style":31774},[31680],[34,510127,34386],{"className":510128,"style":34427},[31685,31686],[34,510130,38080],{"className":510131},[31685],[34,510133,106063],{"className":510134,"style":67482},[31685,31686],[34,510136],{"className":510137,"style":31846},[31845],[34,510139,202],{"className":510140},[31850],[34,510142],{"className":510143,"style":31846},[31845],[34,510145,510147,510150,510153,510156,510159,510162],{"className":510146},[31676],[34,510148],{"className":510149,"style":31774},[31680],[34,510151,32685],{"className":510152},[31782],[34,510154,34395],{"className":510155,"style":34452},[31685,31686],[34,510157],{"className":510158,"style":32604},[31845],[34,510160,223],{"className":510161},[32024],[34,510163],{"className":510164,"style":32604},[31845],[34,510166,510168,510171,510174,510177,510180],{"className":510167},[31676],[34,510169],{"className":510170,"style":31774},[31680],[34,510172,106063],{"className":510173,"style":67482},[31685,31686],[34,510175],{"className":510176,"style":31846},[31845],[34,510178,162086],{"className":510179},[31850],[34,510181],{"className":510182,"style":31846},[31845],[34,510184,510186,510189,510192,510195,510198],{"className":510185},[31676],[34,510187],{"className":510188,"style":34812},[31680],[34,510190,34395],{"className":510191,"style":34452},[31685,31686],[34,510193],{"className":510194,"style":31846},[31845],[34,510196,112296],{"className":510197},[31850],[34,510199],{"className":510200,"style":31846},[31845],[34,510202,510204,510207,510210],{"className":510203},[31676],[34,510205],{"className":510206,"style":31774},[31680],[34,510208,34386],{"className":510209,"style":34427},[31685,31686],[34,510211,32715],{"className":510212},[31690],"（陪集）。",[108,510215,510216,2664,510218,347],{},[2627,510217,427255],{},[34,510219,510221,510271],{"className":510220},[31642],[34,510222,510224],{"className":510223},[31646],[31648,510225,510226],{"xmlns":31650},[31652,510227,510228,510268],{},[31655,510229,510230,510232,510234,510236,510238,510240,510242,510244,510246,510248,510250,510252,510254,510256,510258,510260,510262,510264,510266],{},[31662,510231,456],{"stretchy":702},[31658,510233,111],{},[31662,510235,223],{},[31658,510237,106063],{},[31662,510239,2557],{"stretchy":702},[31662,510241,223],{},[31662,510243,456],{"stretchy":702},[31658,510245,1697],{},[31662,510247,223],{},[31658,510249,106063],{},[31662,510251,2557],{"stretchy":702},[31662,510253,202],{},[31662,510255,456],{"stretchy":702},[31658,510257,111],{},[31662,510259,223],{},[31658,510261,1697],{},[31662,510263,2557],{"stretchy":702},[31662,510265,223],{},[31658,510267,106063],{},[31665,510269,510270],{"encoding":31667},"(a + I) + (b + I) = (a + b) + I",[34,510272,510274,510295,510316,510337,510358,510379,510400],{"className":510273,"ariaHidden":758},[31672],[34,510275,510277,510280,510283,510286,510289,510292],{"className":510276},[31676],[34,510278],{"className":510279,"style":31774},[31680],[34,510281,456],{"className":510282},[31782],[34,510284,111],{"className":510285},[31685,31686],[34,510287],{"className":510288,"style":32604},[31845],[34,510290,223],{"className":510291},[32024],[34,510293],{"className":510294,"style":32604},[31845],[34,510296,510298,510301,510304,510307,510310,510313],{"className":510297},[31676],[34,510299],{"className":510300,"style":31774},[31680],[34,510302,106063],{"className":510303,"style":67482},[31685,31686],[34,510305,2557],{"className":510306},[31690],[34,510308],{"className":510309,"style":32604},[31845],[34,510311,223],{"className":510312},[32024],[34,510314],{"className":510315,"style":32604},[31845],[34,510317,510319,510322,510325,510328,510331,510334],{"className":510318},[31676],[34,510320],{"className":510321,"style":31774},[31680],[34,510323,456],{"className":510324},[31782],[34,510326,1697],{"className":510327},[31685,31686],[34,510329],{"className":510330,"style":32604},[31845],[34,510332,223],{"className":510333},[32024],[34,510335],{"className":510336,"style":32604},[31845],[34,510338,510340,510343,510346,510349,510352,510355],{"className":510339},[31676],[34,510341],{"className":510342,"style":31774},[31680],[34,510344,106063],{"className":510345,"style":67482},[31685,31686],[34,510347,2557],{"className":510348},[31690],[34,510350],{"className":510351,"style":31846},[31845],[34,510353,202],{"className":510354},[31850],[34,510356],{"className":510357,"style":31846},[31845],[34,510359,510361,510364,510367,510370,510373,510376],{"className":510360},[31676],[34,510362],{"className":510363,"style":31774},[31680],[34,510365,456],{"className":510366},[31782],[34,510368,111],{"className":510369},[31685,31686],[34,510371],{"className":510372,"style":32604},[31845],[34,510374,223],{"className":510375},[32024],[34,510377],{"className":510378,"style":32604},[31845],[34,510380,510382,510385,510388,510391,510394,510397],{"className":510381},[31676],[34,510383],{"className":510384,"style":31774},[31680],[34,510386,1697],{"className":510387},[31685,31686],[34,510389,2557],{"className":510390},[31690],[34,510392],{"className":510393,"style":32604},[31845],[34,510395,223],{"className":510396},[32024],[34,510398],{"className":510399,"style":32604},[31845],[34,510401,510403,510406],{"className":510402},[31676],[34,510404],{"className":510405,"style":33218},[31680],[34,510407,106063],{"className":510408,"style":67482},[31685,31686],[108,510410,510411,2664,510413,347],{},[2627,510412,427258],{},[34,510414,510416,510458],{"className":510415},[31642],[34,510417,510419],{"className":510418},[31646],[31648,510420,510421],{"xmlns":31650},[31652,510422,510423,510455],{},[31655,510424,510425,510427,510429,510431,510433,510435,510437,510439,510441,510443,510445,510447,510449,510451,510453],{},[31662,510426,456],{"stretchy":702},[31658,510428,111],{},[31662,510430,223],{},[31658,510432,106063],{},[31662,510434,2557],{"stretchy":702},[31662,510436,456],{"stretchy":702},[31658,510438,1697],{},[31662,510440,223],{},[31658,510442,106063],{},[31662,510444,2557],{"stretchy":702},[31662,510446,202],{},[31658,510448,111],{},[31658,510450,1697],{},[31662,510452,223],{},[31658,510454,106063],{},[31665,510456,510457],{"encoding":31667},"(a + I)(b + I) = ab + I",[34,510459,510461,510482,510509,510530,510548],{"className":510460,"ariaHidden":758},[31672],[34,510462,510464,510467,510470,510473,510476,510479],{"className":510463},[31676],[34,510465],{"className":510466,"style":31774},[31680],[34,510468,456],{"className":510469},[31782],[34,510471,111],{"className":510472},[31685,31686],[34,510474],{"className":510475,"style":32604},[31845],[34,510477,223],{"className":510478},[32024],[34,510480],{"className":510481,"style":32604},[31845],[34,510483,510485,510488,510491,510494,510497,510500,510503,510506],{"className":510484},[31676],[34,510486],{"className":510487,"style":31774},[31680],[34,510489,106063],{"className":510490,"style":67482},[31685,31686],[34,510492,2557],{"className":510493},[31690],[34,510495,456],{"className":510496},[31782],[34,510498,1697],{"className":510499},[31685,31686],[34,510501],{"className":510502,"style":32604},[31845],[34,510504,223],{"className":510505},[32024],[34,510507],{"className":510508,"style":32604},[31845],[34,510510,510512,510515,510518,510521,510524,510527],{"className":510511},[31676],[34,510513],{"className":510514,"style":31774},[31680],[34,510516,106063],{"className":510517,"style":67482},[31685,31686],[34,510519,2557],{"className":510520},[31690],[34,510522],{"className":510523,"style":31846},[31845],[34,510525,202],{"className":510526},[31850],[34,510528],{"className":510529,"style":31846},[31845],[34,510531,510533,510536,510539,510542,510545],{"className":510532},[31676],[34,510534],{"className":510535,"style":34299},[31680],[34,510537,324191],{"className":510538},[31685,31686],[34,510540],{"className":510541,"style":32604},[31845],[34,510543,223],{"className":510544},[32024],[34,510546],{"className":510547,"style":32604},[31845],[34,510549,510551,510554],{"className":510550},[31676],[34,510552],{"className":510553,"style":33218},[31680],[34,510555,106063],{"className":510556,"style":67482},[31685,31686],[15,510558,510559,510562,510563,328410,510760,1840,510956,511020,511021,511743,511744,511772],{},[2627,510560,510561],{},"乘法良定性证明","：\n若 ",[34,510564,510566,510608],{"className":510565},[31642],[34,510567,510569],{"className":510568},[31646],[31648,510570,510571],{"xmlns":31650},[31652,510572,510573,510605],{},[31655,510574,510575,510581,510583,510585,510587,510589,510591,510597,510599,510601,510603],{},[37624,510576,510577,510579],{},[31658,510578,111],{},[31662,510580,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,510582,112296],{},[31658,510584,111],{},[31662,510586,223],{},[31658,510588,106063],{},[31662,510590,31960],{"separator":758},[37624,510592,510593,510595],{},[31658,510594,1697],{},[31662,510596,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,510598,112296],{},[31658,510600,1697],{},[31662,510602,223],{},[31658,510604,106063],{},[31665,510606,510607],{"encoding":31667},"a' \\in a+I, b' \\in b+I",[34,510609,510611,510659,510677,510733,510751],{"className":510610,"ariaHidden":758},[31672],[34,510612,510614,510618,510650,510653,510656],{"className":510613},[31676],[34,510615],{"className":510616,"style":510617},[31680],"height:0.791em;vertical-align:-0.0391em;",[34,510619,510621,510624],{"className":510620},[31685],[34,510622,111],{"className":510623},[31685,31686],[34,510625,510627],{"className":510626},[31792],[34,510628,510630],{"className":510629},[31796],[34,510631,510633],{"className":510632},[31801],[34,510634,510636],{"className":510635,"style":132027},[31805],[34,510637,510638,510641],{"style":37777},[34,510639],{"className":510640,"style":31814},[31813],[34,510642,510644],{"className":510643},[31818,31819,31820,31821],[34,510645,510647],{"className":510646},[31685,31821],[34,510648,116495],{"className":510649},[31685,31821],[34,510651],{"className":510652,"style":31846},[31845],[34,510654,112296],{"className":510655},[31850],[34,510657],{"className":510658,"style":31846},[31845],[34,510660,510662,510665,510668,510671,510674],{"className":510661},[31676],[34,510663],{"className":510664,"style":32597},[31680],[34,510666,111],{"className":510667},[31685,31686],[34,510669],{"className":510670,"style":32604},[31845],[34,510672,223],{"className":510673},[32024],[34,510675],{"className":510676,"style":32604},[31845],[34,510678,510680,510683,510686,510689,510692,510724,510727,510730],{"className":510679},[31676],[34,510681],{"className":510682,"style":395485},[31680],[34,510684,106063],{"className":510685,"style":67482},[31685,31686],[34,510687,31960],{"className":510688},[32119],[34,510690],{"className":510691,"style":32123},[31845],[34,510693,510695,510698],{"className":510694},[31685],[34,510696,1697],{"className":510697},[31685,31686],[34,510699,510701],{"className":510700},[31792],[34,510702,510704],{"className":510703},[31796],[34,510705,510707],{"className":510706},[31801],[34,510708,510710],{"className":510709,"style":132027},[31805],[34,510711,510712,510715],{"style":37777},[34,510713],{"className":510714,"style":31814},[31813],[34,510716,510718],{"className":510717},[31818,31819,31820,31821],[34,510719,510721],{"className":510720},[31685,31821],[34,510722,116495],{"className":510723},[31685,31821],[34,510725],{"className":510726,"style":31846},[31845],[34,510728,112296],{"className":510729},[31850],[34,510731],{"className":510732,"style":31846},[31845],[34,510734,510736,510739,510742,510745,510748],{"className":510735},[31676],[34,510737],{"className":510738,"style":34299},[31680],[34,510740,1697],{"className":510741},[31685,31686],[34,510743],{"className":510744,"style":32604},[31845],[34,510746,223],{"className":510747},[32024],[34,510749],{"className":510750,"style":32604},[31845],[34,510752,510754,510757],{"className":510753},[31676],[34,510755],{"className":510756,"style":33218},[31680],[34,510758,106063],{"className":510759,"style":67482},[31685,31686],[34,510761,510763,510805],{"className":510762},[31642],[34,510764,510766],{"className":510765},[31646],[31648,510767,510768],{"xmlns":31650},[31652,510769,510770,510802],{},[31655,510771,510772,510778,510780,510782,510784,510786,510788,510794,510796,510798,510800],{},[37624,510773,510774,510776],{},[31658,510775,111],{},[31662,510777,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,510779,202],{},[31658,510781,111],{},[31662,510783,223],{},[31658,510785,32936],{},[31662,510787,31960],{"separator":758},[37624,510789,510790,510792],{},[31658,510791,1697],{},[31662,510793,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,510795,202],{},[31658,510797,1697],{},[31662,510799,223],{},[31658,510801,67764],{},[31665,510803,510804],{"encoding":31667},"a' 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b'=b+y",[34,510806,510808,510855,510873,510929,510947],{"className":510807,"ariaHidden":758},[31672],[34,510809,510811,510814,510846,510849,510852],{"className":510810},[31676],[34,510812],{"className":510813,"style":132027},[31680],[34,510815,510817,510820],{"className":510816},[31685],[34,510818,111],{"className":510819},[31685,31686],[34,510821,510823],{"className":510822},[31792],[34,510824,510826],{"className":510825},[31796],[34,510827,510829],{"className":510828},[31801],[34,510830,510832],{"className":510831,"style":132027},[31805],[34,510833,510834,510837],{"style":37777},[34,510835],{"className":510836,"style":31814},[31813],[34,510838,510840],{"className":510839},[31818,31819,31820,31821],[34,510841,510843],{"className":510842},[31685,31821],[34,510844,116495],{"className":510845},[31685,31821],[34,510847],{"className":510848,"style":31846},[31845],[34,510850,202],{"className":510851},[31850],[34,510853],{"className":510854,"style":31846},[31845],[34,510856,510858,510861,510864,510867,510870],{"className":510857},[31676],[34,510859],{"className":510860,"style":32597},[31680],[34,510862,111],{"className":510863},[31685,31686],[34,510865],{"className":510866,"style":32604},[31845],[34,510868,223],{"className":510869},[32024],[34,510871],{"className":510872,"style":32604},[31845],[34,510874,510876,510879,510882,510885,510888,510920,510923,510926],{"className":510875},[31676],[34,510877],{"className":510878,"style":395485},[31680],[34,510880,32936],{"className":510881},[31685,31686],[34,510883,31960],{"className":510884},[32119],[34,510886],{"className":510887,"style":32123},[31845],[34,510889,510891,510894],{"className":510890},[31685],[34,510892,1697],{"className":510893},[31685,31686],[34,510895,510897],{"className":510896},[31792],[34,510898,510900],{"className":510899},[31796],[34,510901,510903],{"className":510902},[31801],[34,510904,510906],{"className":510905,"style":132027},[31805],[34,510907,510908,510911],{"style":37777},[34,510909],{"className":510910,"style":31814},[31813],[34,510912,510914],{"className":510913},[31818,31819,31820,31821],[34,510915,510917],{"className":510916},[31685,31821],[34,510918,116495],{"className":510919},[31685,31821],[34,510921],{"className":510922,"style":31846},[31845],[34,510924,202],{"className":510925},[31850],[34,510927],{"className":510928,"style":31846},[31845],[34,510930,510932,510935,510938,510941,510944],{"className":510931},[31676],[34,510933],{"className":510934,"style":34299},[31680],[34,510936,1697],{"className":510937},[31685,31686],[34,510939],{"className":510940,"style":32604},[31845],[34,510942,223],{"className":510943},[32024],[34,510945],{"className":510946,"style":32604},[31845],[34,510948,510950,510953],{"className":510949},[31676],[34,510951],{"className":510952,"style":38774},[31680],[34,510954,67764],{"className":510955,"style":32062},[31685,31686],[34,510957,510959,510981],{"className":510958},[31642],[34,510960,510962],{"className":510961},[31646],[31648,510963,510964],{"xmlns":31650},[31652,510965,510966,510978],{},[31655,510967,510968,510970,510972,510974,510976],{},[31658,510969,32936],{},[31662,510971,31960],{"separator":758},[31658,510973,67764],{},[31662,510975,112296],{},[31658,510977,106063],{},[31665,510979,510980],{"encoding":31667},"x, y \\in I",[34,510982,510984,511011],{"className":510983,"ariaHidden":758},[31672],[34,510985,510987,510990,510993,510996,510999,511002,511005,511008],{"className":510986},[31676],[34,510988],{"className":510989,"style":86652},[31680],[34,510991,32936],{"className":510992},[31685,31686],[34,510994,31960],{"className":510995},[32119],[34,510997],{"className":510998,"style":32123},[31845],[34,511000,67764],{"className":511001,"style":32062},[31685,31686],[34,511003],{"className":511004,"style":31846},[31845],[34,511006,112296],{"className":511007},[31850],[34,511009],{"className":511010,"style":31846},[31845],[34,511012,511014,511017],{"className":511013},[31676],[34,511015],{"className":511016,"style":33218},[31680],[34,511018,106063],{"className":511019,"style":67482},[31685,31686],")。\n",[34,511022,511024,511157],{"className":511023},[31642],[34,511025,511027],{"className":511026},[31646],[31648,511028,511029],{"xmlns":31650},[31652,511030,511031,511154],{},[31655,511032,511033,511039,511045,511047,511049,511051,511053,511055,511057,511059,511061,511063,511065,511067,511069,511071,511073,511075,511100,511102,511127,511129],{},[37624,511034,511035,511037],{},[31658,511036,111],{},[31662,511038,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,511040,511041,511043],{},[31658,511042,1697],{},[31662,511044,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,511046,202],{},[31662,511048,456],{"stretchy":702},[31658,511050,111],{},[31662,511052,223],{},[31658,511054,32936],{},[31662,511056,2557],{"stretchy":702},[31662,511058,456],{"stretchy":702},[31658,511060,1697],{},[31662,511062,223],{},[31658,511064,67764],{},[31662,511066,2557],{"stretchy":702},[31662,511068,202],{},[31658,511070,111],{},[31658,511072,1697],{},[31662,511074,223],{},[71115,511076,511077,511087],{},[71115,511078,511079,511085],{},[31655,511080,511081,511083],{},[31658,511082,111],{},[31658,511084,67764],{},[31662,511086,477736],{"stretchy":758},[31655,511088,511089,511091,511093,511095,511098],{},[31662,511090,112296],{},[31658,511092,106063],{},[31662,511094,456],{"stretchy":702},[39361,511096,511097],{},"左吸",[31662,511099,2557],{"stretchy":702},[31662,511101,223],{},[71115,511103,511104,511114],{},[71115,511105,511106,511112],{},[31655,511107,511108,511110],{},[31658,511109,32936],{},[31658,511111,1697],{},[31662,511113,477736],{"stretchy":758},[31655,511115,511116,511118,511120,511122,511125],{},[31662,511117,112296],{},[31658,511119,106063],{},[31662,511121,456],{"stretchy":702},[39361,511123,511124],{},"右吸",[31662,511126,2557],{"stretchy":702},[31662,511128,223],{},[71115,511130,511131,511141],{},[71115,511132,511133,511139],{},[31655,511134,511135,511137],{},[31658,511136,32936],{},[31658,511138,67764],{},[31662,511140,477736],{"stretchy":758},[31655,511142,511143,511145,511147,511149,511152],{},[31662,511144,112296],{},[31658,511146,106063],{},[31662,511148,456],{"stretchy":702},[39361,511150,511151],{},"封闭",[31662,511153,2557],{"stretchy":702},[31665,511155,511156],{"encoding":31667},"a'b' = (a+x)(b+y) = ab + \\underbrace{ay}_{\\in I (\\text{左吸})} + \\underbrace{xb}_{\\in I (\\text{右吸})} + \\underbrace{xy}_{\\in I (\\text{封闭})}",[34,511158,511160,511239,511260,511287,511308,511326,511470,511612],{"className":511159,"ariaHidden":758},[31672],[34,511161,511163,511166,511198,511230,511233,511236],{"className":511162},[31676],[34,511164],{"className":511165,"style":132027},[31680],[34,511167,511169,511172],{"className":511168},[31685],[34,511170,111],{"className":511171},[31685,31686],[34,511173,511175],{"className":511174},[31792],[34,511176,511178],{"className":511177},[31796],[34,511179,511181],{"className":511180},[31801],[34,511182,511184],{"className":511183,"style":132027},[31805],[34,511185,511186,511189],{"style":37777},[34,511187],{"className":511188,"style":31814},[31813],[34,511190,511192],{"className":511191},[31818,31819,31820,31821],[34,511193,511195],{"className":511194},[31685,31821],[34,511196,116495],{"className":511197},[31685,31821],[34,511199,511201,511204],{"className":511200},[31685],[34,511202,1697],{"className":511203},[31685,31686],[34,511205,511207],{"className":511206},[31792],[34,511208,511210],{"className":511209},[31796],[34,511211,511213],{"className":511212},[31801],[34,511214,511216],{"className":511215,"style":132027},[31805],[34,511217,511218,511221],{"style":37777},[34,511219],{"className":511220,"style":31814},[31813],[34,511222,511224],{"className":511223},[31818,31819,31820,31821],[34,511225,511227],{"className":511226},[31685,31821],[34,511228,116495],{"className":511229},[31685,31821],[34,511231],{"className":511232,"style":31846},[31845],[34,511234,202],{"className":511235},[31850],[34,511237],{"className":511238,"style":31846},[31845],[34,511240,511242,511245,511248,511251,511254,511257],{"className":511241},[31676],[34,511243],{"className":511244,"style":31774},[31680],[34,511246,456],{"className":511247},[31782],[34,511249,111],{"className":511250},[31685,31686],[34,511252],{"className":511253,"style":32604},[31845],[34,511255,223],{"className":511256},[32024],[34,511258],{"className":511259,"style":32604},[31845],[34,511261,511263,511266,511269,511272,511275,511278,511281,511284],{"className":511262},[31676],[34,511264],{"className":511265,"style":31774},[31680],[34,511267,32936],{"className":511268},[31685,31686],[34,511270,2557],{"className":511271},[31690],[34,511273,456],{"className":511274},[31782],[34,511276,1697],{"className":511277},[31685,31686],[34,511279],{"className":511280,"style":32604},[31845],[34,511282,223],{"className":511283},[32024],[34,511285],{"className":511286,"style":32604},[31845],[34,511288,511290,511293,511296,511299,511302,511305],{"className":511289},[31676],[34,511291],{"className":511292,"style":31774},[31680],[34,511294,67764],{"className":511295,"style":32062},[31685,31686],[34,511297,2557],{"className":511298},[31690],[34,511300],{"className":511301,"style":31846},[31845],[34,511303,202],{"className":511304},[31850],[34,511306],{"className":511307,"style":31846},[31845],[34,511309,511311,511314,511317,511320,511323],{"className":511310},[31676],[34,511312],{"className":511313,"style":34299},[31680],[34,511315,324191],{"className":511316},[31685,31686],[34,511318],{"className":511319,"style":32604},[31845],[34,511321,223],{"className":511322},[32024],[34,511324],{"className":511325,"style":32604},[31845],[34,511327,511329,511333,511461,511464,511467],{"className":511328},[31676],[34,511330],{"className":511331,"style":511332},[31680],"height:2.3258em;vertical-align:-1.7424em;",[34,511334,511336],{"className":511335},[34006,71115],[34,511337,511339,511452],{"className":511338},[31796,31797],[34,511340,511342,511449],{"className":511341},[31801],[34,511343,511345,511375],{"className":511344,"style":32527},[31805],[34,511346,511348,511351],{"style":511347},"top:-1.4326em;",[34,511349],{"className":511350,"style":41508},[31813],[34,511352,511354],{"className":511353},[31818,31819,31820,31821],[34,511355,511357,511360,511363,511366,511372],{"className":511356},[31685,31821],[34,511358,112296],{"className":511359},[31850,31821],[34,511361,106063],{"className":511362,"style":67482},[31685,31686,31821],[34,511364,456],{"className":511365},[31782,31821],[34,511367,511369],{"className":511368},[31685,2638,31821],[34,511370,511097],{"className":511371},[31685,105462,31821],[34,511373,2557],{"className":511374},[31690,31821],[34,511376,511377,511380],{"style":41565},[34,511378],{"className":511379,"style":41508},[31813],[34,511381,511383],{"className":511382},[34006,71115],[34,511384,511386,511440],{"className":511385},[31796,31797],[34,511387,511389,511437],{"className":511388},[31801],[34,511390,511392,511423],{"className":511391,"style":32527},[31805],[34,511393,511396,511399],{"className":511394,"style":511395},[41564],"top:-2.1576em;",[34,511397],{"className":511398,"style":41508},[31813],[34,511400,511402,511409,511416],{"className":511401,"style":477950},[126470],[34,511403,511405],{"className":511404,"style":477955},[477954],[41588,511406,511407],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},[20398,511408],{"d":477962},[34,511410,511412],{"className":511411,"style":477955},[477966],[41588,511413,511414],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},[20398,511415],{"d":477972},[34,511417,511419],{"className":511418,"style":477955},[477976],[41588,511420,511421],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},[20398,511422],{"d":477982},[34,511424,511425,511428],{"style":41565},[34,511426],{"className":511427,"style":41508},[31813],[34,511429,511431,511434],{"className":511430},[31685],[34,511432,111],{"className":511433},[31685,31686],[34,511435,67764],{"className":511436,"style":32062},[31685,31686],[34,511438,31829],{"className":511439},[31828],[34,511441,511443],{"className":511442},[31801],[34,511444,511447],{"className":511445,"style":511446},[31805],"height:0.8424em;",[34,511448],{},[34,511450,31829],{"className":511451},[31828],[34,511453,511455],{"className":511454},[31801],[34,511456,511459],{"className":511457,"style":511458},[31805],"height:1.7424em;",[34,511460],{},[34,511462],{"className":511463,"style":32604},[31845],[34,511465,223],{"className":511466},[32024],[34,511468],{"className":511469,"style":32604},[31845],[34,511471,511473,511477,511603,511606,511609],{"className":511472},[31676],[34,511474],{"className":511475,"style":511476},[31680],"height:2.2424em;vertical-align:-1.548em;",[34,511478,511480],{"className":511479},[34006,71115],[34,511481,511483,511594],{"className":511482},[31796,31797],[34,511484,511486,511591],{"className":511485},[31801],[34,511487,511489,511519],{"className":511488,"style":31681},[31805],[34,511490,511492,511495],{"style":511491},"top:-1.627em;",[34,511493],{"className":511494,"style":41508},[31813],[34,511496,511498],{"className":511497},[31818,31819,31820,31821],[34,511499,511501,511504,511507,511510,511516],{"className":511500},[31685,31821],[34,511502,112296],{"className":511503},[31850,31821],[34,511505,106063],{"className":511506,"style":67482},[31685,31686,31821],[34,511508,456],{"className":511509},[31782,31821],[34,511511,511513],{"className":511512},[31685,2638,31821],[34,511514,511124],{"className":511515},[31685,105462,31821],[34,511517,2557],{"className":511518},[31690,31821],[34,511520,511521,511524],{"style":41565},[34,511522],{"className":511523,"style":41508},[31813],[34,511525,511527],{"className":511526},[34006,71115],[34,511528,511530,511583],{"className":511529},[31796,31797],[34,511531,511533,511580],{"className":511532},[31801],[34,511534,511536,511566],{"className":511535,"style":31681},[31805],[34,511537,511539,511542],{"className":511538,"style":478216},[41564],[34,511540],{"className":511541,"style":41508},[31813],[34,511543,511545,511552,511559],{"className":511544,"style":477950},[126470],[34,511546,511548],{"className":511547,"style":477955},[477954],[41588,511549,511550],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},[20398,511551],{"d":477962},[34,511553,511555],{"className":511554,"style":477955},[477966],[41588,511556,511557],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},[20398,511558],{"d":477972},[34,511560,511562],{"className":511561,"style":477955},[477976],[41588,511563,511564],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},[20398,511565],{"d":477982},[34,511567,511568,511571],{"style":41565},[34,511569],{"className":511570,"style":41508},[31813],[34,511572,511574,511577],{"className":511573},[31685],[34,511575,32936],{"className":511576},[31685,31686],[34,511578,1697],{"className":511579},[31685,31686],[34,511581,31829],{"className":511582},[31828],[34,511584,511586],{"className":511585},[31801],[34,511587,511589],{"className":511588,"style":478267},[31805],[34,511590],{},[34,511592,31829],{"className":511593},[31828],[34,511595,511597],{"className":511596},[31801],[34,511598,511601],{"className":511599,"style":511600},[31805],"height:1.548em;",[34,511602],{},[34,511604],{"className":511605,"style":32604},[31845],[34,511607,223],{"className":511608},[32024],[34,511610],{"className":511611,"style":32604},[31845],[34,511613,511615,511619],{"className":511614},[31676],[34,511616],{"className":511617,"style":511618},[31680],"height:2.173em;vertical-align:-1.7424em;",[34,511620,511622],{"className":511621},[34006,71115],[34,511623,511625,511735],{"className":511624},[31796,31797],[34,511626,511628,511732],{"className":511627},[31801],[34,511629,511631,511660],{"className":511630,"style":32527},[31805],[34,511632,511633,511636],{"style":511347},[34,511634],{"className":511635,"style":41508},[31813],[34,511637,511639],{"className":511638},[31818,31819,31820,31821],[34,511640,511642,511645,511648,511651,511657],{"className":511641},[31685,31821],[34,511643,112296],{"className":511644},[31850,31821],[34,511646,106063],{"className":511647,"style":67482},[31685,31686,31821],[34,511649,456],{"className":511650},[31782,31821],[34,511652,511654],{"className":511653},[31685,2638,31821],[34,511655,511151],{"className":511656},[31685,105462,31821],[34,511658,2557],{"className":511659},[31690,31821],[34,511661,511662,511665],{"style":41565},[34,511663],{"className":511664,"style":41508},[31813],[34,511666,511668],{"className":511667},[34006,71115],[34,511669,511671,511724],{"className":511670},[31796,31797],[34,511672,511674,511721],{"className":511673},[31801],[34,511675,511677,511707],{"className":511676,"style":32527},[31805],[34,511678,511680,511683],{"className":511679,"style":511395},[41564],[34,511681],{"className":511682,"style":41508},[31813],[34,511684,511686,511693,511700],{"className":511685,"style":477950},[126470],[34,511687,511689],{"className":511688,"style":477955},[477954],[41588,511690,511691],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},[20398,511692],{"d":477962},[34,511694,511696],{"className":511695,"style":477955},[477966],[41588,511697,511698],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},[20398,511699],{"d":477972},[34,511701,511703],{"className":511702,"style":477955},[477976],[41588,511704,511705],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},[20398,511706],{"d":477982},[34,511708,511709,511712],{"style":41565},[34,511710],{"className":511711,"style":41508},[31813],[34,511713,511715,511718],{"className":511714},[31685],[34,511716,32936],{"className":511717},[31685,31686],[34,511719,67764],{"className":511720,"style":32062},[31685,31686],[34,511722,31829],{"className":511723},[31828],[34,511725,511727],{"className":511726},[31801],[34,511728,511730],{"className":511729,"style":511446},[31805],[34,511731],{},[34,511733,31829],{"className":511734},[31828],[34,511736,511738],{"className":511737},[31801],[34,511739,511741],{"className":511740,"style":511458},[31805],[34,511742],{},"\n差值在 ",[34,511745,511747,511760],{"className":511746},[31642],[34,511748,511750],{"className":511749},[31646],[31648,511751,511752],{"xmlns":31650},[31652,511753,511754,511758],{},[31655,511755,511756],{},[31658,511757,106063],{},[31665,511759,106063],{"encoding":31667},[34,511761,511763],{"className":511762,"ariaHidden":758},[31672],[34,511764,511766,511769],{"className":511765},[31676],[34,511767],{"className":511768,"style":33218},[31680],[34,511770,106063],{"className":511771,"style":67482},[31685,31686]," 中，故代表元选取不影响结果。",[253,511774,511776],{"id":511775},"_62-商环的性质传承","6.2 商环的性质传承",[15,511778,511779,511780,511818,511819,511847],{},"商环 ",[34,511781,511783,511800],{"className":511782},[31642],[34,511784,511786],{"className":511785},[31646],[31648,511787,511788],{"xmlns":31650},[31652,511789,511790,511798],{},[31655,511791,511792,511794,511796],{},[31658,511793,34386],{},[31658,511795,38080],{"mathvariant":38079},[31658,511797,106063],{},[31665,511799,337927],{"encoding":31667},[34,511801,511803],{"className":511802,"ariaHidden":758},[31672],[34,511804,511806,511809,511812,511815],{"className":511805},[31676],[34,511807],{"className":511808,"style":31774},[31680],[34,511810,34386],{"className":511811,"style":34427},[31685,31686],[34,511813,38080],{"className":511814},[31685],[34,511816,106063],{"className":511817,"style":67482},[31685,31686]," 会继承原环 ",[34,511820,511822,511835],{"className":511821},[31642],[34,511823,511825],{"className":511824},[31646],[31648,511826,511827],{"xmlns":31650},[31652,511828,511829,511833],{},[31655,511830,511831],{},[31658,511832,34386],{},[31665,511834,34386],{"encoding":31667},[34,511836,511838],{"className":511837,"ariaHidden":758},[31672],[34,511839,511841,511844],{"className":511840},[31676],[34,511842],{"className":511843,"style":33218},[31680],[34,511845,34386],{"className":511846,"style":34427},[31685,31686]," 的部分优良性质：",[2230,511849,511850,512210],{},[108,511851,511852,46324,511855,496829,511883,511921,511922],{},[2627,511853,511854],{},"交换性",[34,511856,511858,511871],{"className":511857},[31642],[34,511859,511861],{"className":511860},[31646],[31648,511862,511863],{"xmlns":31650},[31652,511864,511865,511869],{},[31655,511866,511867],{},[31658,511868,34386],{},[31665,511870,34386],{"encoding":31667},[34,511872,511874],{"className":511873,"ariaHidden":758},[31672],[34,511875,511877,511880],{"className":511876},[31676],[34,511878],{"className":511879,"style":33218},[31680],[34,511881,34386],{"className":511882,"style":34427},[31685,31686],[34,511884,511886,511903],{"className":511885},[31642],[34,511887,511889],{"className":511888},[31646],[31648,511890,511891],{"xmlns":31650},[31652,511892,511893,511901],{},[31655,511894,511895,511897,511899],{},[31658,511896,34386],{},[31658,511898,38080],{"mathvariant":38079},[31658,511900,106063],{},[31665,511902,337927],{"encoding":31667},[34,511904,511906],{"className":511905,"ariaHidden":758},[31672],[34,511907,511909,511912,511915,511918],{"className":511908},[31676],[34,511910],{"className":511911,"style":31774},[31680],[34,511913,34386],{"className":511914,"style":34427},[31685,31686],[34,511916,38080],{"className":511917},[31685],[34,511919,106063],{"className":511920,"style":67482},[31685,31686]," 也是交换环。\n",[105,511923,511924],{},[108,511925,511926,2664,511929,347],{},[71807,511927,511928],{},"理由",[34,511930,511932,512006],{"className":511931},[31642],[34,511933,511935],{"className":511934},[31646],[31648,511936,511937],{"xmlns":31650},[31652,511938,511939,512003],{},[31655,511940,511941,511943,511945,511947,511949,511951,511953,511955,511957,511959,511961,511963,511965,511967,511969,511971,511973,511975,511977,511979,511981,511983,511985,511987,511989,511991,511993,511995,511997,511999,512001],{},[31662,511942,456],{"stretchy":702},[31658,511944,111],{},[31662,511946,223],{},[31658,511948,106063],{},[31662,511950,2557],{"stretchy":702},[31662,511952,456],{"stretchy":702},[31658,511954,1697],{},[31662,511956,223],{},[31658,511958,106063],{},[31662,511960,2557],{"stretchy":702},[31662,511962,202],{},[31658,511964,111],{},[31658,511966,1697],{},[31662,511968,223],{},[31658,511970,106063],{},[31662,511972,202],{},[31658,511974,1697],{},[31658,511976,111],{},[31662,511978,223],{},[31658,511980,106063],{},[31662,511982,202],{},[31662,511984,456],{"stretchy":702},[31658,511986,1697],{},[31662,511988,223],{},[31658,511990,106063],{},[31662,511992,2557],{"stretchy":702},[31662,511994,456],{"stretchy":702},[31658,511996,111],{},[31662,511998,223],{},[31658,512000,106063],{},[31662,512002,2557],{"stretchy":702},[31665,512004,512005],{"encoding":31667},"(a+I)(b+I) = ab+I = ba+I = (b+I)(a+I)",[34,512007,512009,512030,512057,512078,512096,512114,512132,512150,512171,512198],{"className":512008,"ariaHidden":758},[31672],[34,512010,512012,512015,512018,512021,512024,512027],{"className":512011},[31676],[34,512013],{"className":512014,"style":31774},[31680],[34,512016,456],{"className":512017},[31782],[34,512019,111],{"className":512020},[31685,31686],[34,512022],{"className":512023,"style":32604},[31845],[34,512025,223],{"className":512026},[32024],[34,512028],{"className":512029,"style":32604},[31845],[34,512031,512033,512036,512039,512042,512045,512048,512051,512054],{"className":512032},[31676],[34,512034],{"className":512035,"style":31774},[31680],[34,512037,106063],{"className":512038,"style":67482},[31685,31686],[34,512040,2557],{"className":512041},[31690],[34,512043,456],{"className":512044},[31782],[34,512046,1697],{"className":512047},[31685,31686],[34,512049],{"className":512050,"style":32604},[31845],[34,512052,223],{"className":512053},[32024],[34,512055],{"className":512056,"style":32604},[31845],[34,512058,512060,512063,512066,512069,512072,512075],{"className":512059},[31676],[34,512061],{"className":512062,"style":31774},[31680],[34,512064,106063],{"className":512065,"style":67482},[31685,31686],[34,512067,2557],{"className":512068},[31690],[34,512070],{"className":512071,"style":31846},[31845],[34,512073,202],{"className":512074},[31850],[34,512076],{"className":512077,"style":31846},[31845],[34,512079,512081,512084,512087,512090,512093],{"className":512080},[31676],[34,512082],{"className":512083,"style":34299},[31680],[34,512085,324191],{"className":512086},[31685,31686],[34,512088],{"className":512089,"style":32604},[31845],[34,512091,223],{"className":512092},[32024],[34,512094],{"className":512095,"style":32604},[31845],[34,512097,512099,512102,512105,512108,512111],{"className":512098},[31676],[34,512100],{"className":512101,"style":33218},[31680],[34,512103,106063],{"className":512104,"style":67482},[31685,31686],[34,512106],{"className":512107,"style":31846},[31845],[34,512109,202],{"className":512110},[31850],[34,512112],{"className":512113,"style":31846},[31845],[34,512115,512117,512120,512123,512126,512129],{"className":512116},[31676],[34,512118],{"className":512119,"style":34299},[31680],[34,512121,367454],{"className":512122},[31685,31686],[34,512124],{"className":512125,"style":32604},[31845],[34,512127,223],{"className":512128},[32024],[34,512130],{"className":512131,"style":32604},[31845],[34,512133,512135,512138,512141,512144,512147],{"className":512134},[31676],[34,512136],{"className":512137,"style":33218},[31680],[34,512139,106063],{"className":512140,"style":67482},[31685,31686],[34,512142],{"className":512143,"style":31846},[31845],[34,512145,202],{"className":512146},[31850],[34,512148],{"className":512149,"style":31846},[31845],[34,512151,512153,512156,512159,512162,512165,512168],{"className":512152},[31676],[34,512154],{"className":512155,"style":31774},[31680],[34,512157,456],{"className":512158},[31782],[34,512160,1697],{"className":512161},[31685,31686],[34,512163],{"className":512164,"style":32604},[31845],[34,512166,223],{"className":512167},[32024],[34,512169],{"className":512170,"style":32604},[31845],[34,512172,512174,512177,512180,512183,512186,512189,512192,512195],{"className":512173},[31676],[34,512175],{"className":512176,"style":31774},[31680],[34,512178,106063],{"className":512179,"style":67482},[31685,31686],[34,512181,2557],{"className":512182},[31690],[34,512184,456],{"className":512185},[31782],[34,512187,111],{"className":512188},[31685,31686],[34,512190],{"className":512191,"style":32604},[31845],[34,512193,223],{"className":512194},[32024],[34,512196],{"className":512197,"style":32604},[31845],[34,512199,512201,512204,512207],{"className":512200},[31676],[34,512202],{"className":512203,"style":31774},[31680],[34,512205,106063],{"className":512206,"style":67482},[31685,31686],[34,512208,2557],{"className":512209},[31690],[108,512211,512212,46324,512214,512242,512243,340354,512271,512309,512310],{},[2627,512213,378492],{},[34,512215,512217,512230],{"className":512216},[31642],[34,512218,512220],{"className":512219},[31646],[31648,512221,512222],{"xmlns":31650},[31652,512223,512224,512228],{},[31655,512225,512226],{},[31658,512227,34386],{},[31665,512229,34386],{"encoding":31667},[34,512231,512233],{"className":512232,"ariaHidden":758},[31672],[34,512234,512236,512239],{"className":512235},[31676],[34,512237],{"className":512238,"style":33218},[31680],[34,512240,34386],{"className":512241,"style":34427},[31685,31686]," 是幺环（有单位元 ",[34,512244,512246,512259],{"className":512245},[31642],[34,512247,512249],{"className":512248},[31646],[31648,512250,512251],{"xmlns":31650},[31652,512252,512253,512257],{},[31655,512254,512255],{},[31749,512256,640],{},[31665,512258,640],{"encoding":31667},[34,512260,512262],{"className":512261,"ariaHidden":758},[31672],[34,512263,512265,512268],{"className":512264},[31676],[34,512266],{"className":512267,"style":32617},[31680],[34,512269,640],{"className":512270},[31685],[34,512272,512274,512291],{"className":512273},[31642],[34,512275,512277],{"className":512276},[31646],[31648,512278,512279],{"xmlns":31650},[31652,512280,512281,512289],{},[31655,512282,512283,512285,512287],{},[31658,512284,34386],{},[31658,512286,38080],{"mathvariant":38079},[31658,512288,106063],{},[31665,512290,337927],{"encoding":31667},[34,512292,512294],{"className":512293,"ariaHidden":758},[31672],[34,512295,512297,512300,512303,512306],{"className":512296},[31676],[34,512298],{"className":512299,"style":31774},[31680],[34,512301,34386],{"className":512302,"style":34427},[31685,31686],[34,512304,38080],{"className":512305},[31685],[34,512307,106063],{"className":512308,"style":67482},[31685,31686]," 也是幺环。\n",[105,512311,512312,512368],{},[108,512313,512314,2664,512317,347],{},[2627,512315,512316],{},"单位元形式",[34,512318,512320,512338],{"className":512319},[31642],[34,512321,512323],{"className":512322},[31646],[31648,512324,512325],{"xmlns":31650},[31652,512326,512327,512335],{},[31655,512328,512329,512331,512333],{},[31749,512330,640],{},[31662,512332,223],{},[31658,512334,106063],{},[31665,512336,512337],{"encoding":31667},"1 + I",[34,512339,512341,512359],{"className":512340,"ariaHidden":758},[31672],[34,512342,512344,512347,512350,512353,512356],{"className":512343},[31676],[34,512345],{"className":512346,"style":35942},[31680],[34,512348,640],{"className":512349},[31685],[34,512351],{"className":512352,"style":32604},[31845],[34,512354,223],{"className":512355},[32024],[34,512357],{"className":512358,"style":32604},[31845],[34,512360,512362,512365],{"className":512361},[31676],[34,512363],{"className":512364,"style":33218},[31680],[34,512366,106063],{"className":512367,"style":67482},[31685,31686],[108,512369,512370,2664,512372,347],{},[71807,512371,500696],{},[34,512373,512375,512427],{"className":512374},[31642],[34,512376,512378],{"className":512377},[31646],[31648,512379,512380],{"xmlns":31650},[31652,512381,512382,512424],{},[31655,512383,512384,512386,512388,512390,512392,512394,512396,512398,512400,512402,512404,512406,512408,512410,512412,512414,512416,512418,512420,512422],{},[31662,512385,456],{"stretchy":702},[31658,512387,111],{},[31662,512389,223],{},[31658,512391,106063],{},[31662,512393,2557],{"stretchy":702},[31662,512395,456],{"stretchy":702},[31749,512397,640],{},[31662,512399,223],{},[31658,512401,106063],{},[31662,512403,2557],{"stretchy":702},[31662,512405,202],{},[31658,512407,111],{},[31662,512409,71191],{},[31749,512411,640],{},[31662,512413,223],{},[31658,512415,106063],{},[31662,512417,202],{},[31658,512419,111],{},[31662,512421,223],{},[31658,512423,106063],{},[31665,512425,512426],{"encoding":31667},"(a+I)(1+I) = a \\cdot 1 + I = a + I",[34,512428,512430,512451,512478,512499,512517,512535,512553,512571],{"className":512429,"ariaHidden":758},[31672],[34,512431,512433,512436,512439,512442,512445,512448],{"className":512432},[31676],[34,512434],{"className":512435,"style":31774},[31680],[34,512437,456],{"className":512438},[31782],[34,512440,111],{"className":512441},[31685,31686],[34,512443],{"className":512444,"style":32604},[31845],[34,512446,223],{"className":512447},[32024],[34,512449],{"className":512450,"style":32604},[31845],[34,512452,512454,512457,512460,512463,512466,512469,512472,512475],{"className":512453},[31676],[34,512455],{"className":512456,"style":31774},[31680],[34,512458,106063],{"className":512459,"style":67482},[31685,31686],[34,512461,2557],{"className":512462},[31690],[34,512464,456],{"className":512465},[31782],[34,512467,640],{"className":512468},[31685],[34,512470],{"className":512471,"style":32604},[31845],[34,512473,223],{"className":512474},[32024],[34,512476],{"className":512477,"style":32604},[31845],[34,512479,512481,512484,512487,512490,512493,512496],{"className":512480},[31676],[34,512482],{"className":512483,"style":31774},[31680],[34,512485,106063],{"className":512486,"style":67482},[31685,31686],[34,512488,2557],{"className":512489},[31690],[34,512491],{"className":512492,"style":31846},[31845],[34,512494,202],{"className":512495},[31850],[34,512497],{"className":512498,"style":31846},[31845],[34,512500,512502,512505,512508,512511,512514],{"className":512501},[31676],[34,512503],{"className":512504,"style":224869},[31680],[34,512506,111],{"className":512507},[31685,31686],[34,512509],{"className":512510,"style":32604},[31845],[34,512512,71191],{"className":512513},[32024],[34,512515],{"className":512516,"style":32604},[31845],[34,512518,512520,512523,512526,512529,512532],{"className":512519},[31676],[34,512521],{"className":512522,"style":35942},[31680],[34,512524,640],{"className":512525},[31685],[34,512527],{"className":512528,"style":32604},[31845],[34,512530,223],{"className":512531},[32024],[34,512533],{"className":512534,"style":32604},[31845],[34,512536,512538,512541,512544,512547,512550],{"className":512537},[31676],[34,512539],{"className":512540,"style":33218},[31680],[34,512542,106063],{"className":512543,"style":67482},[31685,31686],[34,512545],{"className":512546,"style":31846},[31845],[34,512548,202],{"className":512549},[31850],[34,512551],{"className":512552,"style":31846},[31845],[34,512554,512556,512559,512562,512565,512568],{"className":512555},[31676],[34,512557],{"className":512558,"style":32597},[31680],[34,512560,111],{"className":512561},[31685,31686],[34,512563],{"className":512564,"style":32604},[31845],[34,512566,223],{"className":512567},[32024],[34,512569],{"className":512570,"style":32604},[31845],[34,512572,512574,512577],{"className":512573},[31676],[34,512575],{"className":512576,"style":33218},[31680],[34,512578,106063],{"className":512579,"style":67482},[31685,31686],{"title":29,"searchDepth":43,"depth":43,"links":512581},[512582,512583,512586,512590,512594,512602],{"id":495557,"depth":43,"text":495558},{"id":495632,"depth":43,"text":495633,"children":512584},[512585],{"id":495636,"depth":49,"text":495637},{"id":496191,"depth":43,"text":496192,"children":512587},[512588,512589],{"id":496195,"depth":49,"text":496196},{"id":496894,"depth":49,"text":496895},{"id":501619,"depth":43,"text":501620,"children":512591},[512592,512593],{"id":501623,"depth":49,"text":501624},{"id":502230,"depth":49,"text":502231},{"id":505105,"depth":43,"text":505106,"children":512595},[512596,512598,512600],{"id":505181,"depth":49,"text":512597},"5.1 交集 I∩JI \\cap JI∩J",{"id":506458,"depth":49,"text":512599},"5.2 和 I+JI + JI+J",{"id":508560,"depth":49,"text":512601},"5.3 积 IJIJIJ",{"id":510026,"depth":43,"text":510027,"children":512603},[512604,512605],{"id":510030,"depth":49,"text":510031},{"id":511775,"depth":49,"text":511776},{"order":56},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fsubringideal",{"title":495549,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002Fsubringideal","Czpsz71-cEsyO2v_jGBQJtedKwNBtMiOZR7gH-ddli8",{"id":512612,"title":512613,"body":512614,"description":29,"extension":85,"meta":546226,"navigation":52,"path":546227,"seo":546228,"stem":546229,"__hash__":546230},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002FsubringidealHW.md","子环、理想与商环 习题详解",{"type":8,"value":512615,"toc":546215},[512616,512622,512844,512848,513186,513190,513494,513624,514219,514349,515081,515273,515275,515281,515567,515571,515955,515959,516014,516200,519161,519237,519291,520597,520701,520703,520709,520972,520976,520979,521203,521207,521212,523132,523200,523260,524334,524402,525193,525198,526271,526379,526381,526387,526682,526686,526689,526693,526698,527784,527789,528941,528943,528947,529237,529241,529247,529309,529787,529849,531147,531209,532136,532198,533617,533619,533623,533686,533689,534072,534076,534391,535631,536646,536850,537641,537729,537731,537735,538075,538078,538628,538631,538755,541141,541294,541296,541300,541702,541705,541711,542019,542022,544634,544904,545823,545919,545921,545924],[253,512617,512619],{"id":512618},"题目-1生成理想的定义",[2627,512620,512621],{},"题目 1：生成理想的定义",[15,512623,512624,69123,512626,418749,512654,512682,512683],{},[2627,512625,101191],{},[34,512627,512629,512642],{"className":512628},[31642],[34,512630,512632],{"className":512631},[31646],[31648,512633,512634],{"xmlns":31650},[31652,512635,512636,512640],{},[31655,512637,512638],{},[31658,512639,94675],{},[31665,512641,94675],{"encoding":31667},[34,512643,512645],{"className":512644,"ariaHidden":758},[31672],[34,512646,512648,512651],{"className":512647},[31676],[34,512649],{"className":512650,"style":33218},[31680],[34,512652,94675],{"className":512653,"style":106455},[31685,31686],[34,512655,512657,512670],{"className":512656},[31642],[34,512658,512660],{"className":512659},[31646],[31648,512661,512662],{"xmlns":31650},[31652,512663,512664,512668],{},[31655,512665,512666],{},[31658,512667,34386],{},[31665,512669,34386],{"encoding":31667},[34,512671,512673],{"className":512672,"ariaHidden":758},[31672],[34,512674,512676,512679],{"className":512675},[31676],[34,512677],{"className":512678,"style":33218},[31680],[34,512680,34386],{"className":512681,"style":34427},[31685,31686]," 的子环。证明：",[34,512684,512686,512733],{"className":512685},[31642],[34,512687,512689],{"className":512688},[31646],[31648,512690,512691],{"xmlns":31650},[31652,512692,512693,512730],{},[31655,512694,512695,512697,512699,512701,512703,512705,512707,512709,512711,512713,512715,512717,512719,512721,512723,512726,512728],{},[31662,512696,126529],{"stretchy":702},[31658,512698,94675],{},[31662,512700,126544],{"stretchy":702},[31662,512702,202],{},[31662,512704,405054],{},[31662,512706,32685],{"stretchy":702},[31658,512708,106063],{},[31662,512710,162086],{},[31658,512712,94675],{},[31662,512714,131855],{},[31658,512716,106063],{},[31662,512718,31960],{"separator":758},[39361,512720,130298],{},[31658,512722,106063],{},[31662,512724,512725],{},"⊴",[31658,512727,34386],{},[31662,512729,32715],{"stretchy":702},[31665,512731,512732],{"encoding":31667},"\\langle S \\rangle = \\bigcap \\{ I \\mid S \\subseteq I, \\ I \\trianglelefteq R \\}",[34,512734,512736,512760,512784,512802,512832],{"className":512735,"ariaHidden":758},[31672],[34,512737,512739,512742,512745,512748,512751,512754,512757],{"className":512738},[31676],[34,512740],{"className":512741,"style":31774},[31680],[34,512743,126529],{"className":512744},[31782],[34,512746,94675],{"className":512747,"style":106455},[31685,31686],[34,512749,126544],{"className":512750},[31690],[34,512752],{"className":512753,"style":31846},[31845],[34,512755,202],{"className":512756},[31850],[34,512758],{"className":512759,"style":31846},[31845],[34,512761,512763,512766,512769,512772,512775,512778,512781],{"className":512762},[31676],[34,512764],{"className":512765,"style":31774},[31680],[34,512767,405054],{"className":512768,"style":156561},[32767,131249,156560],[34,512770,32685],{"className":512771},[31782],[34,512773,106063],{"className":512774,"style":67482},[31685,31686],[34,512776],{"className":512777,"style":31846},[31845],[34,512779,162086],{"className":512780},[31850],[34,512782],{"className":512783,"style":31846},[31845],[34,512785,512787,512790,512793,512796,512799],{"className":512786},[31676],[34,512788],{"className":512789,"style":195047},[31680],[34,512791,94675],{"className":512792,"style":106455},[31685,31686],[34,512794],{"className":512795,"style":31846},[31845],[34,512797,131855],{"className":512798},[31850],[34,512800],{"className":512801,"style":31846},[31845],[34,512803,512805,512808,512811,512814,512817,512820,512823,512826,512829],{"className":512804},[31676],[34,512806],{"className":512807,"style":70739},[31680],[34,512809,106063],{"className":512810,"style":67482},[31685,31686],[34,512812,31960],{"className":512813},[32119],[34,512815,130298],{"className":512816},[31845],[34,512818],{"className":512819,"style":32123},[31845],[34,512821,106063],{"className":512822,"style":67482},[31685,31686],[34,512824],{"className":512825,"style":31846},[31845],[34,512827,512725],{"className":512828},[31850,129261],[34,512830],{"className":512831,"style":31846},[31845],[34,512833,512835,512838,512841],{"className":512834},[31676],[34,512836],{"className":512837,"style":31774},[31680],[34,512839,34386],{"className":512840,"style":34427},[31685,31686],[34,512842,32715],{"className":512843},[31690],[15,512845,512846],{},[2627,512847,463462],{},[105,512849,512850,513020],{},[108,512851,512852,2664,512854],{},[2627,512853,318465],{},[105,512855,512856,512929,513013],{},[108,512857,512858,512897,512898,130135,512926,512928],{},[34,512859,512861,512879],{"className":512860},[31642],[34,512862,512864],{"className":512863},[31646],[31648,512865,512866],{"xmlns":31650},[31652,512867,512868,512876],{},[31655,512869,512870,512872,512874],{},[31662,512871,126529],{"stretchy":702},[31658,512873,94675],{},[31662,512875,126544],{"stretchy":702},[31665,512877,512878],{"encoding":31667},"\\langle S \\rangle",[34,512880,512882],{"className":512881,"ariaHidden":758},[31672],[34,512883,512885,512888,512891,512894],{"className":512884},[31676],[34,512886],{"className":512887,"style":31774},[31680],[34,512889,126529],{"className":512890},[31782],[34,512892,94675],{"className":512893,"style":106455},[31685,31686],[34,512895,126544],{"className":512896},[31690]," 的定义通常是“包含 ",[34,512899,512901,512914],{"className":512900},[31642],[34,512902,512904],{"className":512903},[31646],[31648,512905,512906],{"xmlns":31650},[31652,512907,512908,512912],{},[31655,512909,512910],{},[31658,512911,94675],{},[31665,512913,94675],{"encoding":31667},[34,512915,512917],{"className":512916,"ariaHidden":758},[31672],[34,512918,512920,512923],{"className":512919},[31676],[34,512921],{"className":512922,"style":33218},[31680],[34,512924,94675],{"className":512925,"style":106455},[31685,31686],[2627,512927,200479],{},"的理想”。",[108,512930,512931,512932,512980,512981,513009,513010,415700],{},"右边的 ",[34,512933,512935,512959],{"className":512934},[31642],[34,512936,512938],{"className":512937},[31646],[31648,512939,512940],{"xmlns":31650},[31652,512941,512942,512956],{},[31655,512943,512944,512946,512948,512950,512952,512954],{},[31662,512945,405054],{},[31662,512947,32685],{"stretchy":702},[31658,512949,2547],{"mathvariant":38079},[31658,512951,2547],{"mathvariant":38079},[31658,512953,2547],{"mathvariant":38079},[31662,512955,32715],{"stretchy":702},[31665,512957,512958],{"encoding":31667},"\\bigcap \\{...\\}",[34,512960,512962],{"className":512961,"ariaHidden":758},[31672],[34,512963,512965,512968,512971,512974,512977],{"className":512964},[31676],[34,512966],{"className":512967,"style":31774},[31680],[34,512969,405054],{"className":512970,"style":156561},[32767,131249,156560],[34,512972,32685],{"className":512973},[31782],[34,512975,1167],{"className":512976},[31685],[34,512978,32715],{"className":512979},[31690]," 是“所有包含 ",[34,512982,512984,512997],{"className":512983},[31642],[34,512985,512987],{"className":512986},[31646],[31648,512988,512989],{"xmlns":31650},[31652,512990,512991,512995],{},[31655,512992,512993],{},[31658,512994,94675],{},[31665,512996,94675],{"encoding":31667},[34,512998,513000],{"className":512999,"ariaHidden":758},[31672],[34,513001,513003,513006],{"className":513002},[31676],[34,513004],{"className":513005,"style":33218},[31680],[34,513007,94675],{"className":513008,"style":106455},[31685,31686]," 的理想的",[2627,513011,513012],{},"交集",[108,513014,513015,513016,513019],{},"这个题目其实是在说：",[2627,513017,513018],{},"“最小” = “所有大个头的交集”","。这就好比说，“全班身高不低于170cm的人里最矮的那个” 等于 “所有身高不低于170cm的人的身高下界”。",[108,513021,513022,513025,513026,513077,513078],{},[2627,513023,513024],{},"证明策略","：\n要证明集合 ",[34,513027,513029,513047],{"className":513028},[31642],[34,513030,513032],{"className":513031},[31646],[31648,513033,513034],{"xmlns":31650},[31652,513035,513036,513044],{},[31655,513037,513038,513040,513042],{},[31658,513039,54151],{},[31662,513041,202],{},[31658,513043,54224],{},[31665,513045,513046],{"encoding":31667},"A = B",[34,513048,513050,513068],{"className":513049,"ariaHidden":758},[31672],[34,513051,513053,513056,513059,513062,513065],{"className":513052},[31676],[34,513054],{"className":513055,"style":33218},[31680],[34,513057,54151],{"className":513058},[31685,31686],[34,513060],{"className":513061,"style":31846},[31845],[34,513063,202],{"className":513064},[31850],[34,513066],{"className":513067,"style":31846},[31845],[34,513069,513071,513074],{"className":513070},[31676],[34,513072],{"className":513073,"style":33218},[31680],[34,513075,54224],{"className":513076,"style":54252},[31685,31686],"，标准套路是证明“双向包含”：",[2230,513079,513080,513133],{},[108,513081,513082],{},[34,513083,513085,513103],{"className":513084},[31642],[34,513086,513088],{"className":513087},[31646],[31648,513089,513090],{"xmlns":31650},[31652,513091,513092,513100],{},[31655,513093,513094,513096,513098],{},[31658,513095,54151],{},[31662,513097,131855],{},[31658,513099,54224],{},[31665,513101,513102],{"encoding":31667},"A \\subseteq B",[34,513104,513106,513124],{"className":513105,"ariaHidden":758},[31672],[34,513107,513109,513112,513115,513118,513121],{"className":513108},[31676],[34,513110],{"className":513111,"style":195047},[31680],[34,513113,54151],{"className":513114},[31685,31686],[34,513116],{"className":513117,"style":31846},[31845],[34,513119,131855],{"className":513120},[31850],[34,513122],{"className":513123,"style":31846},[31845],[34,513125,513127,513130],{"className":513126},[31676],[34,513128],{"className":513129,"style":33218},[31680],[34,513131,54224],{"className":513132,"style":54252},[31685,31686],[108,513134,513135],{},[34,513136,513138,513156],{"className":513137},[31642],[34,513139,513141],{"className":513140},[31646],[31648,513142,513143],{"xmlns":31650},[31652,513144,513145,513153],{},[31655,513146,513147,513149,513151],{},[31658,513148,54224],{},[31662,513150,131855],{},[31658,513152,54151],{},[31665,513154,513155],{"encoding":31667},"B \\subseteq A",[34,513157,513159,513177],{"className":513158,"ariaHidden":758},[31672],[34,513160,513162,513165,513168,513171,513174],{"className":513161},[31676],[34,513163],{"className":513164,"style":195047},[31680],[34,513166,54224],{"className":513167,"style":54252},[31685,31686],[34,513169],{"className":513170,"style":31846},[31845],[34,513172,131855],{"className":513173},[31850],[34,513175],{"className":513176,"style":31846},[31845],[34,513178,513180,513183],{"className":513179},[31676],[34,513181],{"className":513182,"style":33218},[31680],[34,513184,54151],{"className":513185},[31685,31686],[15,513187,513188],{},[2627,513189,463779],{},[15,513191,336888,513192,513339,513340,513368,513369,347],{},[34,513193,513195,513235],{"className":513194},[31642],[34,513196,513198],{"className":513197},[31646],[31648,513199,513200],{"xmlns":31650},[31652,513201,513202,513232],{},[31655,513203,513204,513206,513208,513210,513212,513214,513216,513218,513220,513222,513224,513226,513228,513230],{},[31658,513205,38691],{"mathvariant":1911},[31662,513207,202],{},[31662,513209,32685],{"stretchy":702},[31658,513211,106063],{},[31662,513213,162086],{},[31658,513215,94675],{},[31662,513217,131855],{},[31658,513219,106063],{},[31662,513221,31960],{"separator":758},[39361,513223,130298],{},[31658,513225,106063],{},[31662,513227,512725],{},[31658,513229,34386],{},[31662,513231,32715],{"stretchy":702},[31665,513233,513234],{"encoding":31667},"\\mathcal{K} = \\{ I \\mid S \\subseteq I, \\ I \\trianglelefteq R \\}",[34,513236,513238,513258,513279,513297,513327],{"className":513237,"ariaHidden":758},[31672],[34,513239,513241,513244,513249,513252,513255],{"className":513240},[31676],[34,513242],{"className":513243,"style":33218},[31680],[34,513245,38691],{"className":513246,"style":513248},[31685,513247],"mathcal","margin-right:0.0144em;",[34,513250],{"className":513251,"style":31846},[31845],[34,513253,202],{"className":513254},[31850],[34,513256],{"className":513257,"style":31846},[31845],[34,513259,513261,513264,513267,513270,513273,513276],{"className":513260},[31676],[34,513262],{"className":513263,"style":31774},[31680],[34,513265,32685],{"className":513266},[31782],[34,513268,106063],{"className":513269,"style":67482},[31685,31686],[34,513271],{"className":513272,"style":31846},[31845],[34,513274,162086],{"className":513275},[31850],[34,513277],{"className":513278,"style":31846},[31845],[34,513280,513282,513285,513288,513291,513294],{"className":513281},[31676],[34,513283],{"className":513284,"style":195047},[31680],[34,513286,94675],{"className":513287,"style":106455},[31685,31686],[34,513289],{"className":513290,"style":31846},[31845],[34,513292,131855],{"className":513293},[31850],[34,513295],{"className":513296,"style":31846},[31845],[34,513298,513300,513303,513306,513309,513312,513315,513318,513321,513324],{"className":513299},[31676],[34,513301],{"className":513302,"style":70739},[31680],[34,513304,106063],{"className":513305,"style":67482},[31685,31686],[34,513307,31960],{"className":513308},[32119],[34,513310,130298],{"className":513311},[31845],[34,513313],{"className":513314,"style":32123},[31845],[34,513316,106063],{"className":513317,"style":67482},[31685,31686],[34,513319],{"className":513320,"style":31846},[31845],[34,513322,512725],{"className":513323},[31850,129261],[34,513325],{"className":513326,"style":31846},[31845],[34,513328,513330,513333,513336],{"className":513329},[31676],[34,513331],{"className":513332,"style":31774},[31680],[34,513334,34386],{"className":513335,"style":34427},[31685,31686],[34,513337,32715],{"className":513338},[31690]," 为所有包含 ",[34,513341,513343,513356],{"className":513342},[31642],[34,513344,513346],{"className":513345},[31646],[31648,513347,513348],{"xmlns":31650},[31652,513349,513350,513354],{},[31655,513351,513352],{},[31658,513353,94675],{},[31665,513355,94675],{"encoding":31667},[34,513357,513359],{"className":513358,"ariaHidden":758},[31672],[34,513360,513362,513365],{"className":513361},[31676],[34,513363],{"className":513364,"style":33218},[31680],[34,513366,94675],{"className":513367,"style":106455},[31685,31686]," 的理想的集合。\n我们要证：",[34,513370,513372,513406],{"className":513371},[31642],[34,513373,513375],{"className":513374},[31646],[31648,513376,513377],{"xmlns":31650},[31652,513378,513379,513403],{},[31655,513380,513381,513383,513385,513387,513389,513401],{},[31662,513382,126529],{"stretchy":702},[31658,513384,94675],{},[31662,513386,126544],{"stretchy":702},[31662,513388,202],{},[31743,513390,513391,513393],{},[31662,513392,405054],{},[31655,513394,513395,513397,513399],{},[31658,513396,106063],{},[31662,513398,112296],{},[31658,513400,38691],{"mathvariant":1911},[31658,513402,106063],{},[31665,513404,513405],{"encoding":31667},"\\langle S \\rangle = \\bigcap_{I \\in \\mathcal{K}} I",[34,513407,513409,513433],{"className":513408,"ariaHidden":758},[31672],[34,513410,513412,513415,513418,513421,513424,513427,513430],{"className":513411},[31676],[34,513413],{"className":513414,"style":31774},[31680],[34,513416,126529],{"className":513417},[31782],[34,513419,94675],{"className":513420,"style":106455},[31685,31686],[34,513422,126544],{"className":513423},[31690],[34,513425],{"className":513426,"style":31846},[31845],[34,513428,202],{"className":513429},[31850],[34,513431],{"className":513432,"style":31846},[31845],[34,513434,513436,513439,513488,513491],{"className":513435},[31676],[34,513437],{"className":513438,"style":156553},[31680],[34,513440,513442,513445],{"className":513441},[32767],[34,513443,405054],{"className":513444,"style":156561},[32767,131249,156560],[34,513446,513448],{"className":513447},[31792],[34,513449,513451,513480],{"className":513450},[31796,31797],[34,513452,513454,513477],{"className":513453},[31801],[34,513455,513457],{"className":513456,"style":156574},[31805],[34,513458,513459,513462],{"style":156577},[34,513460],{"className":513461,"style":31814},[31813],[34,513463,513465],{"className":513464},[31818,31819,31820,31821],[34,513466,513468,513471,513474],{"className":513467},[31685,31821],[34,513469,106063],{"className":513470,"style":67482},[31685,31686,31821],[34,513472,112296],{"className":513473},[31850,31821],[34,513475,38691],{"className":513476,"style":513248},[31685,513247,31821],[34,513478,31829],{"className":513479},[31828],[34,513481,513483],{"className":513482},[31801],[34,513484,513486],{"className":513485,"style":156605},[31805],[34,513487],{},[34,513489],{"className":513490,"style":32123},[31845],[34,513492,106063],{"className":513493,"style":67482},[31685,31686],[15,513495,513496],{},[2627,513497,513498,513499],{},"步骤 1：证明 ",[34,513500,513502,513536],{"className":513501},[31642],[34,513503,513505],{"className":513504},[31646],[31648,513506,513507],{"xmlns":31650},[31652,513508,513509,513533],{},[31655,513510,513511,513523,513525,513527,513529,513531],{},[31743,513512,513513,513515],{},[31662,513514,405054],{},[31655,513516,513517,513519,513521],{},[31658,513518,106063],{},[31662,513520,112296],{},[31658,513522,38691],{"mathvariant":1911},[31658,513524,106063],{},[31662,513526,131855],{},[31662,513528,126529],{"stretchy":702},[31658,513530,94675],{},[31662,513532,126544],{"stretchy":702},[31665,513534,513535],{"encoding":31667},"\\bigcap_{I \\in \\mathcal{K}} I \\subseteq \\langle S \\rangle",[34,513537,513539,513609],{"className":513538,"ariaHidden":758},[31672],[34,513540,513542,513545,513594,513597,513600,513603,513606],{"className":513541},[31676],[34,513543],{"className":513544,"style":156553},[31680],[34,513546,513548,513551],{"className":513547},[32767],[34,513549,405054],{"className":513550,"style":156561},[32767,131249,156560],[34,513552,513554],{"className":513553},[31792],[34,513555,513557,513586],{"className":513556},[31796,31797],[34,513558,513560,513583],{"className":513559},[31801],[34,513561,513563],{"className":513562,"style":156574},[31805],[34,513564,513565,513568],{"style":156577},[34,513566],{"className":513567,"style":31814},[31813],[34,513569,513571],{"className":513570},[31818,31819,31820,31821],[34,513572,513574,513577,513580],{"className":513573},[31685,31821],[34,513575,106063],{"className":513576,"style":67482},[31685,31686,31821],[34,513578,112296],{"className":513579},[31850,31821],[34,513581,38691],{"className":513582,"style":513248},[31685,513247,31821],[34,513584,31829],{"className":513585},[31828],[34,513587,513589],{"className":513588},[31801],[34,513590,513592],{"className":513591,"style":156605},[31805],[34,513593],{},[34,513595],{"className":513596,"style":32123},[31845],[34,513598,106063],{"className":513599,"style":67482},[31685,31686],[34,513601],{"className":513602,"style":31846},[31845],[34,513604,131855],{"className":513605},[31850],[34,513607],{"className":513608,"style":31846},[31845],[34,513610,513612,513615,513618,513621],{"className":513611},[31676],[34,513613],{"className":513614,"style":31774},[31680],[34,513616,126529],{"className":513617},[31782],[34,513619,94675],{"className":513620,"style":106455},[31685,31686],[34,513622,126544],{"className":513623},[31690],[2230,513625,513626,513698,513898,514023,514093],{},[108,513627,397222,513628,513666,513667,130135,513695,513697],{},[34,513629,513631,513648],{"className":513630},[31642],[34,513632,513634],{"className":513633},[31646],[31648,513635,513636],{"xmlns":31650},[31652,513637,513638,513646],{},[31655,513639,513640,513642,513644],{},[31662,513641,126529],{"stretchy":702},[31658,513643,94675],{},[31662,513645,126544],{"stretchy":702},[31665,513647,512878],{"encoding":31667},[34,513649,513651],{"className":513650,"ariaHidden":758},[31672],[34,513652,513654,513657,513660,513663],{"className":513653},[31676],[34,513655],{"className":513656,"style":31774},[31680],[34,513658,126529],{"className":513659},[31782],[34,513661,94675],{"className":513662,"style":106455},[31685,31686],[34,513664,126544],{"className":513665},[31690]," 是包含 ",[34,513668,513670,513683],{"className":513669},[31642],[34,513671,513673],{"className":513672},[31646],[31648,513674,513675],{"xmlns":31650},[31652,513676,513677,513681],{},[31655,513678,513679],{},[31658,513680,94675],{},[31665,513682,94675],{"encoding":31667},[34,513684,513686],{"className":513685,"ariaHidden":758},[31672],[34,513687,513689,513692],{"className":513688},[31676],[34,513690],{"className":513691,"style":33218},[31680],[34,513693,94675],{"className":513694,"style":106455},[31685,31686],[2627,513696,200479],{},"理想。",[108,513699,426251,513700,513738,513739,513767,513768,513806,513807,513836,513837,31901],{},[34,513701,513703,513720],{"className":513702},[31642],[34,513704,513706],{"className":513705},[31646],[31648,513707,513708],{"xmlns":31650},[31652,513709,513710,513718],{},[31655,513711,513712,513714,513716],{},[31662,513713,126529],{"stretchy":702},[31658,513715,94675],{},[31662,513717,126544],{"stretchy":702},[31665,513719,512878],{"encoding":31667},[34,513721,513723],{"className":513722,"ariaHidden":758},[31672],[34,513724,513726,513729,513732,513735],{"className":513725},[31676],[34,513727],{"className":513728,"style":31774},[31680],[34,513730,126529],{"className":513731},[31782],[34,513733,94675],{"className":513734,"style":106455},[31685,31686],[34,513736,126544],{"className":513737},[31690]," 是一个包含 ",[34,513740,513742,513755],{"className":513741},[31642],[34,513743,513745],{"className":513744},[31646],[31648,513746,513747],{"xmlns":31650},[31652,513748,513749,513753],{},[31655,513750,513751],{},[31658,513752,94675],{},[31665,513754,94675],{"encoding":31667},[34,513756,513758],{"className":513757,"ariaHidden":758},[31672],[34,513759,513761,513764],{"className":513760},[31676],[34,513762],{"className":513763,"style":33218},[31680],[34,513765,94675],{"className":513766,"style":106455},[31685,31686]," 的理想，那么 ",[34,513769,513771,513788],{"className":513770},[31642],[34,513772,513774],{"className":513773},[31646],[31648,513775,513776],{"xmlns":31650},[31652,513777,513778,513786],{},[31655,513779,513780,513782,513784],{},[31662,513781,126529],{"stretchy":702},[31658,513783,94675],{},[31662,513785,126544],{"stretchy":702},[31665,513787,512878],{"encoding":31667},[34,513789,513791],{"className":513790,"ariaHidden":758},[31672],[34,513792,513794,513797,513800,513803],{"className":513793},[31676],[34,513795],{"className":513796,"style":31774},[31680],[34,513798,126529],{"className":513799},[31782],[34,513801,94675],{"className":513802,"style":106455},[31685,31686],[34,513804,126544],{"className":513805},[31690]," 自己也是集合 ",[34,513808,513810,513824],{"className":513809},[31642],[34,513811,513813],{"className":513812},[31646],[31648,513814,513815],{"xmlns":31650},[31652,513816,513817,513821],{},[31655,513818,513819],{},[31658,513820,38691],{"mathvariant":1911},[31665,513822,513823],{"encoding":31667},"\\mathcal{K}",[34,513825,513827],{"className":513826,"ariaHidden":758},[31672],[34,513828,513830,513833],{"className":513829},[31676],[34,513831],{"className":513832,"style":33218},[31680],[34,513834,38691],{"className":513835,"style":513248},[31685,513247]," 中的一员（即 ",[34,513838,513840,513862],{"className":513839},[31642],[34,513841,513843],{"className":513842},[31646],[31648,513844,513845],{"xmlns":31650},[31652,513846,513847,513859],{},[31655,513848,513849,513851,513853,513855,513857],{},[31662,513850,126529],{"stretchy":702},[31658,513852,94675],{},[31662,513854,126544],{"stretchy":702},[31662,513856,112296],{},[31658,513858,38691],{"mathvariant":1911},[31665,513860,513861],{"encoding":31667},"\\langle S \\rangle \\in \\mathcal{K}",[34,513863,513865,513889],{"className":513864,"ariaHidden":758},[31672],[34,513866,513868,513871,513874,513877,513880,513883,513886],{"className":513867},[31676],[34,513869],{"className":513870,"style":31774},[31680],[34,513872,126529],{"className":513873},[31782],[34,513875,94675],{"className":513876,"style":106455},[31685,31686],[34,513878,126544],{"className":513879},[31690],[34,513881],{"className":513882,"style":31846},[31845],[34,513884,112296],{"className":513885},[31850],[34,513887],{"className":513888,"style":31846},[31845],[34,513890,513892,513895],{"className":513891},[31676],[34,513893],{"className":513894,"style":33218},[31680],[34,513896,38691],{"className":513897,"style":513248},[31685,513247],[108,513899,416190,513900,513993,513994,514022],{},[34,513901,513903,513929],{"className":513902},[31642],[34,513904,513906],{"className":513905},[31646],[31648,513907,513908],{"xmlns":31650},[31652,513909,513910,513926],{},[31655,513911,513912,513924],{},[31743,513913,513914,513916],{},[31662,513915,405054],{},[31655,513917,513918,513920,513922],{},[31658,513919,106063],{},[31662,513921,112296],{},[31658,513923,38691],{"mathvariant":1911},[31658,513925,106063],{},[31665,513927,513928],{"encoding":31667},"\\bigcap_{I \\in \\mathcal{K}} I",[34,513930,513932],{"className":513931,"ariaHidden":758},[31672],[34,513933,513935,513938,513987,513990],{"className":513934},[31676],[34,513936],{"className":513937,"style":156553},[31680],[34,513939,513941,513944],{"className":513940},[32767],[34,513942,405054],{"className":513943,"style":156561},[32767,131249,156560],[34,513945,513947],{"className":513946},[31792],[34,513948,513950,513979],{"className":513949},[31796,31797],[34,513951,513953,513976],{"className":513952},[31801],[34,513954,513956],{"className":513955,"style":156574},[31805],[34,513957,513958,513961],{"style":156577},[34,513959],{"className":513960,"style":31814},[31813],[34,513962,513964],{"className":513963},[31818,31819,31820,31821],[34,513965,513967,513970,513973],{"className":513966},[31685,31821],[34,513968,106063],{"className":513969,"style":67482},[31685,31686,31821],[34,513971,112296],{"className":513972},[31850,31821],[34,513974,38691],{"className":513975,"style":513248},[31685,513247,31821],[34,513977,31829],{"className":513978},[31828],[34,513980,513982],{"className":513981},[31801],[34,513983,513985],{"className":513984,"style":156605},[31805],[34,513986],{},[34,513988],{"className":513989,"style":32123},[31845],[34,513991,106063],{"className":513992,"style":67482},[31685,31686]," 的含义是：它是所有 ",[34,513995,513997,514010],{"className":513996},[31642],[34,513998,514000],{"className":513999},[31646],[31648,514001,514002],{"xmlns":31650},[31652,514003,514004,514008],{},[31655,514005,514006],{},[31658,514007,38691],{"mathvariant":1911},[31665,514009,513823],{"encoding":31667},[34,514011,514013],{"className":514012,"ariaHidden":758},[31672],[34,514014,514016,514019],{"className":514015},[31676],[34,514017],{"className":514018,"style":33218},[31680],[34,514020,38691],{"className":514021,"style":513248},[31685,513247]," 中成员的公共部分。",[108,514024,514025,514026,514054,514055,337455],{},"因此，交集必然包含在 ",[34,514027,514029,514042],{"className":514028},[31642],[34,514030,514032],{"className":514031},[31646],[31648,514033,514034],{"xmlns":31650},[31652,514035,514036,514040],{},[31655,514037,514038],{},[31658,514039,38691],{"mathvariant":1911},[31665,514041,513823],{"encoding":31667},[34,514043,514045],{"className":514044,"ariaHidden":758},[31672],[34,514046,514048,514051],{"className":514047},[31676],[34,514049],{"className":514050,"style":33218},[31680],[34,514052,38691],{"className":514053,"style":513248},[31685,513247]," 的每一个成员里，当然也包含在 ",[34,514056,514058,514075],{"className":514057},[31642],[34,514059,514061],{"className":514060},[31646],[31648,514062,514063],{"xmlns":31650},[31652,514064,514065,514073],{},[31655,514066,514067,514069,514071],{},[31662,514068,126529],{"stretchy":702},[31658,514070,94675],{},[31662,514072,126544],{"stretchy":702},[31665,514074,512878],{"encoding":31667},[34,514076,514078],{"className":514077,"ariaHidden":758},[31672],[34,514079,514081,514084,514087,514090],{"className":514080},[31676],[34,514082],{"className":514083,"style":31774},[31680],[34,514085,126529],{"className":514086},[31782],[34,514088,94675],{"className":514089,"style":106455},[31685,31686],[34,514091,126544],{"className":514092},[31690],[108,514094,159962,514095,347],{},[34,514096,514098,514131],{"className":514097},[31642],[34,514099,514101],{"className":514100},[31646],[31648,514102,514103],{"xmlns":31650},[31652,514104,514105,514129],{},[31655,514106,514107,514119,514121,514123,514125,514127],{},[31743,514108,514109,514111],{},[31662,514110,405054],{},[31655,514112,514113,514115,514117],{},[31658,514114,106063],{},[31662,514116,112296],{},[31658,514118,38691],{"mathvariant":1911},[31658,514120,106063],{},[31662,514122,131855],{},[31662,514124,126529],{"stretchy":702},[31658,514126,94675],{},[31662,514128,126544],{"stretchy":702},[31665,514130,513535],{"encoding":31667},[34,514132,514134,514204],{"className":514133,"ariaHidden":758},[31672],[34,514135,514137,514140,514189,514192,514195,514198,514201],{"className":514136},[31676],[34,514138],{"className":514139,"style":156553},[31680],[34,514141,514143,514146],{"className":514142},[32767],[34,514144,405054],{"className":514145,"style":156561},[32767,131249,156560],[34,514147,514149],{"className":514148},[31792],[34,514150,514152,514181],{"className":514151},[31796,31797],[34,514153,514155,514178],{"className":514154},[31801],[34,514156,514158],{"className":514157,"style":156574},[31805],[34,514159,514160,514163],{"style":156577},[34,514161],{"className":514162,"style":31814},[31813],[34,514164,514166],{"className":514165},[31818,31819,31820,31821],[34,514167,514169,514172,514175],{"className":514168},[31685,31821],[34,514170,106063],{"className":514171,"style":67482},[31685,31686,31821],[34,514173,112296],{"className":514174},[31850,31821],[34,514176,38691],{"className":514177,"style":513248},[31685,513247,31821],[34,514179,31829],{"className":514180},[31828],[34,514182,514184],{"className":514183},[31801],[34,514185,514187],{"className":514186,"style":156605},[31805],[34,514188],{},[34,514190],{"className":514191,"style":32123},[31845],[34,514193,106063],{"className":514194,"style":67482},[31685,31686],[34,514196],{"className":514197,"style":31846},[31845],[34,514199,131855],{"className":514200},[31850],[34,514202],{"className":514203,"style":31846},[31845],[34,514205,514207,514210,514213,514216],{"className":514206},[31676],[34,514208],{"className":514209,"style":31774},[31680],[34,514211,126529],{"className":514212},[31782],[34,514214,94675],{"className":514215,"style":106455},[31685,31686],[34,514217,126544],{"className":514218},[31690],[15,514220,514221],{},[2627,514222,514223,514224],{},"步骤 2：证明 ",[34,514225,514227,514261],{"className":514226},[31642],[34,514228,514230],{"className":514229},[31646],[31648,514231,514232],{"xmlns":31650},[31652,514233,514234,514258],{},[31655,514235,514236,514238,514240,514242,514244,514256],{},[31662,514237,126529],{"stretchy":702},[31658,514239,94675],{},[31662,514241,126544],{"stretchy":702},[31662,514243,131855],{},[31743,514245,514246,514248],{},[31662,514247,405054],{},[31655,514249,514250,514252,514254],{},[31658,514251,106063],{},[31662,514253,112296],{},[31658,514255,38691],{"mathvariant":1911},[31658,514257,106063],{},[31665,514259,514260],{"encoding":31667},"\\langle S \\rangle \\subseteq \\bigcap_{I \\in \\mathcal{K}} I",[34,514262,514264,514288],{"className":514263,"ariaHidden":758},[31672],[34,514265,514267,514270,514273,514276,514279,514282,514285],{"className":514266},[31676],[34,514268],{"className":514269,"style":31774},[31680],[34,514271,126529],{"className":514272},[31782],[34,514274,94675],{"className":514275,"style":106455},[31685,31686],[34,514277,126544],{"className":514278},[31690],[34,514280],{"className":514281,"style":31846},[31845],[34,514283,131855],{"className":514284},[31850],[34,514286],{"className":514287,"style":31846},[31845],[34,514289,514291,514294,514343,514346],{"className":514290},[31676],[34,514292],{"className":514293,"style":156553},[31680],[34,514295,514297,514300],{"className":514296},[32767],[34,514298,405054],{"className":514299,"style":156561},[32767,131249,156560],[34,514301,514303],{"className":514302},[31792],[34,514304,514306,514335],{"className":514305},[31796,31797],[34,514307,514309,514332],{"className":514308},[31801],[34,514310,514312],{"className":514311,"style":156574},[31805],[34,514313,514314,514317],{"style":156577},[34,514315],{"className":514316,"style":31814},[31813],[34,514318,514320],{"className":514319},[31818,31819,31820,31821],[34,514321,514323,514326,514329],{"className":514322},[31685,31821],[34,514324,106063],{"className":514325,"style":67482},[31685,31686,31821],[34,514327,112296],{"className":514328},[31850,31821],[34,514330,38691],{"className":514331,"style":513248},[31685,513247,31821],[34,514333,31829],{"className":514334},[31828],[34,514336,514338],{"className":514337},[31801],[34,514339,514341],{"className":514340,"style":156605},[31805],[34,514342],{},[34,514344],{"className":514345,"style":32123},[31845],[34,514347,106063],{"className":514348,"style":67482},[31685,31686],[2230,514350,514351,514447,514651,514723,514794,514955],{},[108,514352,514353,514354,514446],{},"我们知道，任意多个理想的交集仍然是一个理想（这是上一节课作业里的结论，稍后我们在题目2中会细证）。所以 ",[34,514355,514357,514382],{"className":514356},[31642],[34,514358,514360],{"className":514359},[31646],[31648,514361,514362],{"xmlns":31650},[31652,514363,514364,514380],{},[31655,514365,514366,514378],{},[31743,514367,514368,514370],{},[31662,514369,405054],{},[31655,514371,514372,514374,514376],{},[31658,514373,106063],{},[31662,514375,112296],{},[31658,514377,38691],{"mathvariant":1911},[31658,514379,106063],{},[31665,514381,513928],{"encoding":31667},[34,514383,514385],{"className":514384,"ariaHidden":758},[31672],[34,514386,514388,514391,514440,514443],{"className":514387},[31676],[34,514389],{"className":514390,"style":156553},[31680],[34,514392,514394,514397],{"className":514393},[32767],[34,514395,405054],{"className":514396,"style":156561},[32767,131249,156560],[34,514398,514400],{"className":514399},[31792],[34,514401,514403,514432],{"className":514402},[31796,31797],[34,514404,514406,514429],{"className":514405},[31801],[34,514407,514409],{"className":514408,"style":156574},[31805],[34,514410,514411,514414],{"style":156577},[34,514412],{"className":514413,"style":31814},[31813],[34,514415,514417],{"className":514416},[31818,31819,31820,31821],[34,514418,514420,514423,514426],{"className":514419},[31685,31821],[34,514421,106063],{"className":514422,"style":67482},[31685,31686,31821],[34,514424,112296],{"className":514425},[31850,31821],[34,514427,38691],{"className":514428,"style":513248},[31685,513247,31821],[34,514430,31829],{"className":514431},[31828],[34,514433,514435],{"className":514434},[31801],[34,514436,514438],{"className":514437,"style":156605},[31805],[34,514439],{},[34,514441],{"className":514442,"style":32123},[31845],[34,514444,106063],{"className":514445,"style":67482},[31685,31686]," 是一个理想。",[108,514448,93310,514449,514477,514478,514506,514507,514535,514536,347],{},[34,514450,514452,514465],{"className":514451},[31642],[34,514453,514455],{"className":514454},[31646],[31648,514456,514457],{"xmlns":31650},[31652,514458,514459,514463],{},[31655,514460,514461],{},[31658,514462,38691],{"mathvariant":1911},[31665,514464,513823],{"encoding":31667},[34,514466,514468],{"className":514467,"ariaHidden":758},[31672],[34,514469,514471,514474],{"className":514470},[31676],[34,514472],{"className":514473,"style":33218},[31680],[34,514475,38691],{"className":514476,"style":513248},[31685,513247]," 中每一个理想都包含 ",[34,514479,514481,514494],{"className":514480},[31642],[34,514482,514484],{"className":514483},[31646],[31648,514485,514486],{"xmlns":31650},[31652,514487,514488,514492],{},[31655,514489,514490],{},[31658,514491,94675],{},[31665,514493,94675],{"encoding":31667},[34,514495,514497],{"className":514496,"ariaHidden":758},[31672],[34,514498,514500,514503],{"className":514499},[31676],[34,514501],{"className":514502,"style":33218},[31680],[34,514504,94675],{"className":514505,"style":106455},[31685,31686],"，所以它们的交集也一定包含 ",[34,514508,514510,514523],{"className":514509},[31642],[34,514511,514513],{"className":514512},[31646],[31648,514514,514515],{"xmlns":31650},[31652,514516,514517,514521],{},[31655,514518,514519],{},[31658,514520,94675],{},[31665,514522,94675],{"encoding":31667},[34,514524,514526],{"className":514525,"ariaHidden":758},[31672],[34,514527,514529,514532],{"className":514528},[31676],[34,514530],{"className":514531,"style":33218},[31680],[34,514533,94675],{"className":514534,"style":106455},[31685,31686],"。\n即：",[34,514537,514539,514569],{"className":514538},[31642],[34,514540,514542],{"className":514541},[31646],[31648,514543,514544],{"xmlns":31650},[31652,514545,514546,514566],{},[31655,514547,514548,514550,514552,514564],{},[31658,514549,94675],{},[31662,514551,131855],{},[31743,514553,514554,514556],{},[31662,514555,405054],{},[31655,514557,514558,514560,514562],{},[31658,514559,106063],{},[31662,514561,112296],{},[31658,514563,38691],{"mathvariant":1911},[31658,514565,106063],{},[31665,514567,514568],{"encoding":31667},"S \\subseteq \\bigcap_{I \\in \\mathcal{K}} I",[34,514570,514572,514590],{"className":514571,"ariaHidden":758},[31672],[34,514573,514575,514578,514581,514584,514587],{"className":514574},[31676],[34,514576],{"className":514577,"style":195047},[31680],[34,514579,94675],{"className":514580,"style":106455},[31685,31686],[34,514582],{"className":514583,"style":31846},[31845],[34,514585,131855],{"className":514586},[31850],[34,514588],{"className":514589,"style":31846},[31845],[34,514591,514593,514596,514645,514648],{"className":514592},[31676],[34,514594],{"className":514595,"style":156553},[31680],[34,514597,514599,514602],{"className":514598},[32767],[34,514600,405054],{"className":514601,"style":156561},[32767,131249,156560],[34,514603,514605],{"className":514604},[31792],[34,514606,514608,514637],{"className":514607},[31796,31797],[34,514609,514611,514634],{"className":514610},[31801],[34,514612,514614],{"className":514613,"style":156574},[31805],[34,514615,514616,514619],{"style":156577},[34,514617],{"className":514618,"style":31814},[31813],[34,514620,514622],{"className":514621},[31818,31819,31820,31821],[34,514623,514625,514628,514631],{"className":514624},[31685,31821],[34,514626,106063],{"className":514627,"style":67482},[31685,31686,31821],[34,514629,112296],{"className":514630},[31850,31821],[34,514632,38691],{"className":514633,"style":513248},[31685,513247,31821],[34,514635,31829],{"className":514636},[31828],[34,514638,514640],{"className":514639},[31801],[34,514641,514643],{"className":514642,"style":156605},[31805],[34,514644],{},[34,514646],{"className":514647,"style":32123},[31845],[34,514649,106063],{"className":514650,"style":67482},[31685,31686],[108,514652,514653,514654,514692,514693,130135,514721,513697],{},"现在回顾 ",[34,514655,514657,514674],{"className":514656},[31642],[34,514658,514660],{"className":514659},[31646],[31648,514661,514662],{"xmlns":31650},[31652,514663,514664,514672],{},[31655,514665,514666,514668,514670],{},[31662,514667,126529],{"stretchy":702},[31658,514669,94675],{},[31662,514671,126544],{"stretchy":702},[31665,514673,512878],{"encoding":31667},[34,514675,514677],{"className":514676,"ariaHidden":758},[31672],[34,514678,514680,514683,514686,514689],{"className":514679},[31676],[34,514681],{"className":514682,"style":31774},[31680],[34,514684,126529],{"className":514685},[31782],[34,514687,94675],{"className":514688,"style":106455},[31685,31686],[34,514690,126544],{"className":514691},[31690]," 的定义：它是包含 ",[34,514694,514696,514709],{"className":514695},[31642],[34,514697,514699],{"className":514698},[31646],[31648,514700,514701],{"xmlns":31650},[31652,514702,514703,514707],{},[31655,514704,514705],{},[31658,514706,94675],{},[31665,514708,94675],{"encoding":31667},[34,514710,514712],{"className":514711,"ariaHidden":758},[31672],[34,514713,514715,514718],{"className":514714},[31676],[34,514716],{"className":514717,"style":33218},[31680],[34,514719,94675],{"className":514720,"style":106455},[31685,31686],[2627,514722,200479],{},[108,514724,514725,514726,514754,514755,514793],{},"“最小”的意思是：任何包含 ",[34,514727,514729,514742],{"className":514728},[31642],[34,514730,514732],{"className":514731},[31646],[31648,514733,514734],{"xmlns":31650},[31652,514735,514736,514740],{},[31655,514737,514738],{},[31658,514739,94675],{},[31665,514741,94675],{"encoding":31667},[34,514743,514745],{"className":514744,"ariaHidden":758},[31672],[34,514746,514748,514751],{"className":514747},[31676],[34,514749],{"className":514750,"style":33218},[31680],[34,514752,94675],{"className":514753,"style":106455},[31685,31686]," 的理想，都必须把 ",[34,514756,514758,514775],{"className":514757},[31642],[34,514759,514761],{"className":514760},[31646],[31648,514762,514763],{"xmlns":31650},[31652,514764,514765,514773],{},[31655,514766,514767,514769,514771],{},[31662,514768,126529],{"stretchy":702},[31658,514770,94675],{},[31662,514772,126544],{"stretchy":702},[31665,514774,512878],{"encoding":31667},[34,514776,514778],{"className":514777,"ariaHidden":758},[31672],[34,514779,514781,514784,514787,514790],{"className":514780},[31676],[34,514782],{"className":514783,"style":31774},[31680],[34,514785,126529],{"className":514786},[31782],[34,514788,94675],{"className":514789,"style":106455},[31685,31686],[34,514791,126544],{"className":514792},[31690]," 包在里面。",[108,514795,426251,514796,513738,514888,514916,514917,347],{},[34,514797,514799,514824],{"className":514798},[31642],[34,514800,514802],{"className":514801},[31646],[31648,514803,514804],{"xmlns":31650},[31652,514805,514806,514822],{},[31655,514807,514808,514820],{},[31743,514809,514810,514812],{},[31662,514811,405054],{},[31655,514813,514814,514816,514818],{},[31658,514815,106063],{},[31662,514817,112296],{},[31658,514819,38691],{"mathvariant":1911},[31658,514821,106063],{},[31665,514823,513928],{"encoding":31667},[34,514825,514827],{"className":514826,"ariaHidden":758},[31672],[34,514828,514830,514833,514882,514885],{"className":514829},[31676],[34,514831],{"className":514832,"style":156553},[31680],[34,514834,514836,514839],{"className":514835},[32767],[34,514837,405054],{"className":514838,"style":156561},[32767,131249,156560],[34,514840,514842],{"className":514841},[31792],[34,514843,514845,514874],{"className":514844},[31796,31797],[34,514846,514848,514871],{"className":514847},[31801],[34,514849,514851],{"className":514850,"style":156574},[31805],[34,514852,514853,514856],{"style":156577},[34,514854],{"className":514855,"style":31814},[31813],[34,514857,514859],{"className":514858},[31818,31819,31820,31821],[34,514860,514862,514865,514868],{"className":514861},[31685,31821],[34,514863,106063],{"className":514864,"style":67482},[31685,31686,31821],[34,514866,112296],{"className":514867},[31850,31821],[34,514869,38691],{"className":514870,"style":513248},[31685,513247,31821],[34,514872,31829],{"className":514873},[31828],[34,514875,514877],{"className":514876},[31801],[34,514878,514880],{"className":514879,"style":156605},[31805],[34,514881],{},[34,514883],{"className":514884,"style":32123},[31845],[34,514886,106063],{"className":514887,"style":67482},[31685,31686],[34,514889,514891,514904],{"className":514890},[31642],[34,514892,514894],{"className":514893},[31646],[31648,514895,514896],{"xmlns":31650},[31652,514897,514898,514902],{},[31655,514899,514900],{},[31658,514901,94675],{},[31665,514903,94675],{"encoding":31667},[34,514905,514907],{"className":514906,"ariaHidden":758},[31672],[34,514908,514910,514913],{"className":514909},[31676],[34,514911],{"className":514912,"style":33218},[31680],[34,514914,94675],{"className":514915,"style":106455},[31685,31686]," 的理想，那么它必然包含“最小”的那个 ",[34,514918,514920,514937],{"className":514919},[31642],[34,514921,514923],{"className":514922},[31646],[31648,514924,514925],{"xmlns":31650},[31652,514926,514927,514935],{},[31655,514928,514929,514931,514933],{},[31662,514930,126529],{"stretchy":702},[31658,514932,94675],{},[31662,514934,126544],{"stretchy":702},[31665,514936,512878],{"encoding":31667},[34,514938,514940],{"className":514939,"ariaHidden":758},[31672],[34,514941,514943,514946,514949,514952],{"className":514942},[31676],[34,514944],{"className":514945,"style":31774},[31680],[34,514947,126529],{"className":514948},[31782],[34,514950,94675],{"className":514951,"style":106455},[31685,31686],[34,514953,126544],{"className":514954},[31690],[108,514956,159962,514957,347],{},[34,514958,514960,514993],{"className":514959},[31642],[34,514961,514963],{"className":514962},[31646],[31648,514964,514965],{"xmlns":31650},[31652,514966,514967,514991],{},[31655,514968,514969,514971,514973,514975,514977,514989],{},[31662,514970,126529],{"stretchy":702},[31658,514972,94675],{},[31662,514974,126544],{"stretchy":702},[31662,514976,131855],{},[31743,514978,514979,514981],{},[31662,514980,405054],{},[31655,514982,514983,514985,514987],{},[31658,514984,106063],{},[31662,514986,112296],{},[31658,514988,38691],{"mathvariant":1911},[31658,514990,106063],{},[31665,514992,514260],{"encoding":31667},[34,514994,514996,515020],{"className":514995,"ariaHidden":758},[31672],[34,514997,514999,515002,515005,515008,515011,515014,515017],{"className":514998},[31676],[34,515000],{"className":515001,"style":31774},[31680],[34,515003,126529],{"className":515004},[31782],[34,515006,94675],{"className":515007,"style":106455},[31685,31686],[34,515009,126544],{"className":515010},[31690],[34,515012],{"className":515013,"style":31846},[31845],[34,515015,131855],{"className":515016},[31850],[34,515018],{"className":515019,"style":31846},[31845],[34,515021,515023,515026,515075,515078],{"className":515022},[31676],[34,515024],{"className":515025,"style":156553},[31680],[34,515027,515029,515032],{"className":515028},[32767],[34,515030,405054],{"className":515031,"style":156561},[32767,131249,156560],[34,515033,515035],{"className":515034},[31792],[34,515036,515038,515067],{"className":515037},[31796,31797],[34,515039,515041,515064],{"className":515040},[31801],[34,515042,515044],{"className":515043,"style":156574},[31805],[34,515045,515046,515049],{"style":156577},[34,515047],{"className":515048,"style":31814},[31813],[34,515050,515052],{"className":515051},[31818,31819,31820,31821],[34,515053,515055,515058,515061],{"className":515054},[31685,31821],[34,515056,106063],{"className":515057,"style":67482},[31685,31686,31821],[34,515059,112296],{"className":515060},[31850,31821],[34,515062,38691],{"className":515063,"style":513248},[31685,513247,31821],[34,515065,31829],{"className":515066},[31828],[34,515068,515070],{"className":515069},[31801],[34,515071,515073],{"className":515072,"style":156605},[31805],[34,515074],{},[34,515076],{"className":515077,"style":32123},[31845],[34,515079,106063],{"className":515080,"style":67482},[31685,31686],[15,515082,515083,515085,515086,452957,515245],{},[2627,515084,62532],{}," 综上所述，",[34,515087,515089,515134],{"className":515088},[31642],[34,515090,515092],{"className":515091},[31646],[31648,515093,515094],{"xmlns":31650},[31652,515095,515096,515132],{},[31655,515097,515098,515100,515102,515104,515106,515108,515110,515112,515114,515116,515118,515120,515122,515124,515126,515128,515130],{},[31662,515099,126529],{"stretchy":702},[31658,515101,94675],{},[31662,515103,126544],{"stretchy":702},[31662,515105,202],{},[31662,515107,405054],{},[31662,515109,32685],{"stretchy":702},[31658,515111,106063],{},[31662,515113,162086],{},[31658,515115,94675],{},[31662,515117,131855],{},[31658,515119,106063],{},[31662,515121,31960],{"separator":758},[39361,515123,130298],{},[31658,515125,106063],{},[31662,515127,512725],{},[31658,515129,34386],{},[31662,515131,32715],{"stretchy":702},[31665,515133,512732],{"encoding":31667},[34,515135,515137,515161,515185,515203,515233],{"className":515136,"ariaHidden":758},[31672],[34,515138,515140,515143,515146,515149,515152,515155,515158],{"className":515139},[31676],[34,515141],{"className":515142,"style":31774},[31680],[34,515144,126529],{"className":515145},[31782],[34,515147,94675],{"className":515148,"style":106455},[31685,31686],[34,515150,126544],{"className":515151},[31690],[34,515153],{"className":515154,"style":31846},[31845],[34,515156,202],{"className":515157},[31850],[34,515159],{"className":515160,"style":31846},[31845],[34,515162,515164,515167,515170,515173,515176,515179,515182],{"className":515163},[31676],[34,515165],{"className":515166,"style":31774},[31680],[34,515168,405054],{"className":515169,"style":156561},[32767,131249,156560],[34,515171,32685],{"className":515172},[31782],[34,515174,106063],{"className":515175,"style":67482},[31685,31686],[34,515177],{"className":515178,"style":31846},[31845],[34,515180,162086],{"className":515181},[31850],[34,515183],{"className":515184,"style":31846},[31845],[34,515186,515188,515191,515194,515197,515200],{"className":515187},[31676],[34,515189],{"className":515190,"style":195047},[31680],[34,515192,94675],{"className":515193,"style":106455},[31685,31686],[34,515195],{"className":515196,"style":31846},[31845],[34,515198,131855],{"className":515199},[31850],[34,515201],{"className":515202,"style":31846},[31845],[34,515204,515206,515209,515212,515215,515218,515221,515224,515227,515230],{"className":515205},[31676],[34,515207],{"className":515208,"style":70739},[31680],[34,515210,106063],{"className":515211,"style":67482},[31685,31686],[34,515213,31960],{"className":515214},[32119],[34,515216,130298],{"className":515217},[31845],[34,515219],{"className":515220,"style":32123},[31845],[34,515222,106063],{"className":515223,"style":67482},[31685,31686],[34,515225],{"className":515226,"style":31846},[31845],[34,515228,512725],{"className":515229},[31850,129261],[34,515231],{"className":515232,"style":31846},[31845],[34,515234,515236,515239,515242],{"className":515235},[31676],[34,515237],{"className":515238,"style":31774},[31680],[34,515240,34386],{"className":515241,"style":34427},[31685,31686],[34,515243,32715],{"className":515244},[31690],[34,515246,515248,515261],{"className":515247},[31642],[34,515249,515251],{"className":515250},[31646],[31648,515252,515253],{"xmlns":31650},[31652,515254,515255,515259],{},[31655,515256,515257],{},[31658,515258,445204],{"mathvariant":38079},[31665,515260,445207],{"encoding":31667},[34,515262,515264],{"className":515263,"ariaHidden":758},[31672],[34,515265,515267,515270],{"className":515266},[31676],[34,515268],{"className":515269,"style":445217},[31680],[34,515271,445204],{"className":515272},[31685,129261],[32188,515274],{},[253,515276,515278],{"id":515277},"题目-2理想的运算封闭性",[2627,515279,515280],{},"题目 2：理想的运算封闭性",[15,515282,515283,69123,515285,33775,515349,11008,515390,515418,515419,515493,515494],{},[2627,515284,101191],{},[34,515286,515288,515310],{"className":515287},[31642],[34,515289,515291],{"className":515290},[31646],[31648,515292,515293],{"xmlns":31650},[31652,515294,515295,515307],{},[31655,515296,515297,515299,515301,515303,515305],{},[31658,515298,106063],{},[31662,515300,31960],{"separator":758},[31658,515302,223320],{},[31662,515304,512725],{},[31658,515306,34386],{},[31665,515308,515309],{"encoding":31667},"I, J \\trianglelefteq R",[34,515311,515313,515340],{"className":515312,"ariaHidden":758},[31672],[34,515314,515316,515319,515322,515325,515328,515331,515334,515337],{"className":515315},[31676],[34,515317],{"className":515318,"style":70739},[31680],[34,515320,106063],{"className":515321,"style":67482},[31685,31686],[34,515323,31960],{"className":515324},[32119],[34,515326],{"className":515327,"style":32123},[31845],[34,515329,223320],{"className":515330,"style":223361},[31685,31686],[34,515332],{"className":515333,"style":31846},[31845],[34,515335,512725],{"className":515336},[31850,129261],[34,515338],{"className":515339,"style":31846},[31845],[34,515341,515343,515346],{"className":515342},[31676],[34,515344],{"className":515345,"style":33218},[31680],[34,515347,34386],{"className":515348,"style":34427},[31685,31686],[34,515350,515352,515369],{"className":515351},[31642],[34,515353,515355],{"className":515354},[31646],[31648,515356,515357],{"xmlns":31650},[31652,515358,515359,515367],{},[31655,515360,515361,515363,515365],{},[31658,515362,106063],{},[31662,515364,31960],{"separator":758},[31658,515366,223320],{},[31665,515368,454327],{"encoding":31667},[34,515370,515372],{"className":515371,"ariaHidden":758},[31672],[34,515373,515375,515378,515381,515384,515387],{"className":515374},[31676],[34,515376],{"className":515377,"style":70739},[31680],[34,515379,106063],{"className":515380,"style":67482},[31685,31686],[34,515382,31960],{"className":515383},[32119],[34,515385],{"className":515386,"style":32123},[31845],[34,515388,223320],{"className":515389,"style":223361},[31685,31686],[34,515391,515393,515406],{"className":515392},[31642],[34,515394,515396],{"className":515395},[31646],[31648,515397,515398],{"xmlns":31650},[31652,515399,515400,515404],{},[31655,515401,515402],{},[31658,515403,34386],{},[31665,515405,34386],{"encoding":31667},[34,515407,515409],{"className":515408,"ariaHidden":758},[31672],[34,515410,515412,515415],{"className":515411},[31676],[34,515413],{"className":515414,"style":33218},[31680],[34,515416,34386],{"className":515417,"style":34427},[31685,31686]," 的理想）。证明：\n(i) ",[34,515420,515422,515444],{"className":515421},[31642],[34,515423,515425],{"className":515424},[31646],[31648,515426,515427],{"xmlns":31650},[31652,515428,515429,515441],{},[31655,515430,515431,515433,515435,515437,515439],{},[31658,515432,106063],{},[31662,515434,223],{},[31658,515436,223320],{},[31662,515438,512725],{},[31658,515440,34386],{},[31665,515442,515443],{"encoding":31667},"I+J \\trianglelefteq R",[34,515445,515447,515465,515484],{"className":515446,"ariaHidden":758},[31672],[34,515448,515450,515453,515456,515459,515462],{"className":515449},[31676],[34,515451],{"className":515452,"style":34944},[31680],[34,515454,106063],{"className":515455,"style":67482},[31685,31686],[34,515457],{"className":515458,"style":32604},[31845],[34,515460,223],{"className":515461},[32024],[34,515463],{"className":515464,"style":32604},[31845],[34,515466,515468,515472,515475,515478,515481],{"className":515467},[31676],[34,515469],{"className":515470,"style":515471},[31680],"height:0.82em;vertical-align:-0.1367em;",[34,515473,223320],{"className":515474,"style":223361},[31685,31686],[34,515476],{"className":515477,"style":31846},[31845],[34,515479,512725],{"className":515480},[31850,129261],[34,515482],{"className":515483,"style":31846},[31845],[34,515485,515487,515490],{"className":515486},[31676],[34,515488],{"className":515489,"style":33218},[31680],[34,515491,34386],{"className":515492,"style":34427},[31685,31686],"\n(ii) ",[34,515495,515497,515519],{"className":515496},[31642],[34,515498,515500],{"className":515499},[31646],[31648,515501,515502],{"xmlns":31650},[31652,515503,515504,515516],{},[31655,515505,515506,515508,515510,515512,515514],{},[31658,515507,106063],{},[31662,515509,326344],{},[31658,515511,223320],{},[31662,515513,512725],{},[31658,515515,34386],{},[31665,515517,515518],{"encoding":31667},"I \\cap J \\trianglelefteq R",[34,515520,515522,515540,515558],{"className":515521,"ariaHidden":758},[31672],[34,515523,515525,515528,515531,515534,515537],{"className":515524},[31676],[34,515526],{"className":515527,"style":33218},[31680],[34,515529,106063],{"className":515530,"style":67482},[31685,31686],[34,515532],{"className":515533,"style":32604},[31845],[34,515535,326344],{"className":515536},[32024],[34,515538],{"className":515539,"style":32604},[31845],[34,515541,515543,515546,515549,515552,515555],{"className":515542},[31676],[34,515544],{"className":515545,"style":515471},[31680],[34,515547,223320],{"className":515548,"style":223361},[31685,31686],[34,515550],{"className":515551,"style":31846},[31845],[34,515553,512725],{"className":515554},[31850,129261],[34,515556],{"className":515557,"style":31846},[31845],[34,515559,515561,515564],{"className":515560},[31676],[34,515562],{"className":515563,"style":33218},[31680],[34,515565,34386],{"className":515566,"style":34427},[31685,31686],[15,515568,515569],{},[2627,515570,463462],{},[105,515572,515573],{},[108,515574,515575,515578,515579,515607,515608],{},[2627,515576,515577],{},"判定理想的万能公式（Ideal Test）","：\n要证明一个非空子集 ",[34,515580,515582,515595],{"className":515581},[31642],[34,515583,515585],{"className":515584},[31646],[31648,515586,515587],{"xmlns":31650},[31652,515588,515589,515593],{},[31655,515590,515591],{},[31658,515592,38691],{},[31665,515594,38691],{"encoding":31667},[34,515596,515598],{"className":515597,"ariaHidden":758},[31672],[34,515599,515601,515604],{"className":515600},[31676],[34,515602],{"className":515603,"style":33218},[31680],[34,515605,38691],{"className":515606,"style":38706},[31685,31686]," 是理想，只需验证两点：\n",[2230,515609,515610,515752],{},[108,515611,515612,47890,515614,55016,515678,515751],{},[2627,515613,443104],{},[34,515615,515617,515639],{"className":515616},[31642],[34,515618,515620],{"className":515619},[31646],[31648,515621,515622],{"xmlns":31650},[31652,515623,515624,515636],{},[31655,515625,515626,515628,515630,515632,515634],{},[31658,515627,32936],{},[31662,515629,31960],{"separator":758},[31658,515631,67764],{},[31662,515633,112296],{},[31658,515635,38691],{},[31665,515637,515638],{"encoding":31667},"x, y \\in K",[34,515640,515642,515669],{"className":515641,"ariaHidden":758},[31672],[34,515643,515645,515648,515651,515654,515657,515660,515663,515666],{"className":515644},[31676],[34,515646],{"className":515647,"style":86652},[31680],[34,515649,32936],{"className":515650},[31685,31686],[34,515652,31960],{"className":515653},[32119],[34,515655],{"className":515656,"style":32123},[31845],[34,515658,67764],{"className":515659,"style":32062},[31685,31686],[34,515661],{"className":515662,"style":31846},[31845],[34,515664,112296],{"className":515665},[31850],[34,515667],{"className":515668,"style":31846},[31845],[34,515670,515672,515675],{"className":515671},[31676],[34,515673],{"className":515674,"style":33218},[31680],[34,515676,38691],{"className":515677,"style":38706},[31685,31686],[34,515679,515681,515703],{"className":515680},[31642],[34,515682,515684],{"className":515683},[31646],[31648,515685,515686],{"xmlns":31650},[31652,515687,515688,515700],{},[31655,515689,515690,515692,515694,515696,515698],{},[31658,515691,32936],{},[31662,515693,32226],{},[31658,515695,67764],{},[31662,515697,112296],{},[31658,515699,38691],{},[31665,515701,515702],{"encoding":31667},"x - y \\in K",[34,515704,515706,515724,515742],{"className":515705,"ariaHidden":758},[31672],[34,515707,515709,515712,515715,515718,515721],{"className":515708},[31676],[34,515710],{"className":515711,"style":32597},[31680],[34,515713,32936],{"className":515714},[31685,31686],[34,515716],{"className":515717,"style":32604},[31845],[34,515719,32226],{"className":515720},[32024],[34,515722],{"className":515723,"style":32604},[31845],[34,515725,515727,515730,515733,515736,515739],{"className":515726},[31676],[34,515728],{"className":515729,"style":86652},[31680],[34,515731,67764],{"className":515732,"style":32062},[31685,31686],[34,515734],{"className":515735,"style":31846},[31845],[34,515737,112296],{"className":515738},[31850],[34,515740],{"className":515741,"style":31846},[31845],[34,515743,515745,515748],{"className":515744},[31676],[34,515746],{"className":515747,"style":33218},[31680],[34,515749,38691],{"className":515750,"style":38706},[31685,31686],"。（这说明它是加法子群）",[108,515753,515754,47890,515757,55016,515843,86674,515899,347],{},[2627,515755,515756],{},"吸收律（最关键）",[34,515758,515760,515786],{"className":515759},[31642],[34,515761,515763],{"className":515762},[31646],[31648,515764,515765],{"xmlns":31650},[31652,515766,515767,515783],{},[31655,515768,515769,515771,515773,515775,515777,515779,515781],{},[31658,515770,32936],{},[31662,515772,112296],{},[31658,515774,38691],{},[31662,515776,31960],{"separator":758},[31658,515778,34395],{},[31662,515780,112296],{},[31658,515782,34386],{},[31665,515784,515785],{"encoding":31667},"x \\in K, r \\in R",[34,515787,515789,515807,515834],{"className":515788,"ariaHidden":758},[31672],[34,515790,515792,515795,515798,515801,515804],{"className":515791},[31676],[34,515793],{"className":515794,"style":34812},[31680],[34,515796,32936],{"className":515797},[31685,31686],[34,515799],{"className":515800,"style":31846},[31845],[34,515802,112296],{"className":515803},[31850],[34,515805],{"className":515806,"style":31846},[31845],[34,515808,515810,515813,515816,515819,515822,515825,515828,515831],{"className":515809},[31676],[34,515811],{"className":515812,"style":70739},[31680],[34,515814,38691],{"className":515815,"style":38706},[31685,31686],[34,515817,31960],{"className":515818},[32119],[34,515820],{"className":515821,"style":32123},[31845],[34,515823,34395],{"className":515824,"style":34452},[31685,31686],[34,515826],{"className":515827,"style":31846},[31845],[34,515829,112296],{"className":515830},[31850],[34,515832],{"className":515833,"style":31846},[31845],[34,515835,515837,515840],{"className":515836},[31676],[34,515838],{"className":515839,"style":33218},[31680],[34,515841,34386],{"className":515842,"style":34427},[31685,31686],[34,515844,515846,515866],{"className":515845},[31642],[34,515847,515849],{"className":515848},[31646],[31648,515850,515851],{"xmlns":31650},[31652,515852,515853,515863],{},[31655,515854,515855,515857,515859,515861],{},[31658,515856,34395],{},[31658,515858,32936],{},[31662,515860,112296],{},[31658,515862,38691],{},[31665,515864,515865],{"encoding":31667},"rx \\in K",[34,515867,515869,515890],{"className":515868,"ariaHidden":758},[31672],[34,515870,515872,515875,515878,515881,515884,515887],{"className":515871},[31676],[34,515873],{"className":515874,"style":34812},[31680],[34,515876,34395],{"className":515877,"style":34452},[31685,31686],[34,515879,32936],{"className":515880},[31685,31686],[34,515882],{"className":515883,"style":31846},[31845],[34,515885,112296],{"className":515886},[31850],[34,515888],{"className":515889,"style":31846},[31845],[34,515891,515893,515896],{"className":515892},[31676],[34,515894],{"className":515895,"style":33218},[31680],[34,515897,38691],{"className":515898,"style":38706},[31685,31686],[34,515900,515902,515922],{"className":515901},[31642],[34,515903,515905],{"className":515904},[31646],[31648,515906,515907],{"xmlns":31650},[31652,515908,515909,515919],{},[31655,515910,515911,515913,515915,515917],{},[31658,515912,32936],{},[31658,515914,34395],{},[31662,515916,112296],{},[31658,515918,38691],{},[31665,515920,515921],{"encoding":31667},"xr 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是理想",[15,516015,367867,516016,347],{},[34,516017,516019,516065],{"className":516018},[31642],[34,516020,516022],{"className":516021},[31646],[31648,516023,516024],{"xmlns":31650},[31652,516025,516026,516062],{},[31655,516027,516028,516030,516032,516034,516036,516038,516040,516042,516044,516046,516048,516050,516052,516054,516056,516058,516060],{},[31658,516029,106063],{},[31662,516031,223],{},[31658,516033,223320],{},[31662,516035,202],{},[31662,516037,32685],{"stretchy":702},[31658,516039,111],{},[31662,516041,223],{},[31658,516043,1697],{},[31662,516045,162086],{},[31658,516047,111],{},[31662,516049,112296],{},[31658,516051,106063],{},[31662,516053,31960],{"separator":758},[31658,516055,1697],{},[31662,516057,112296],{},[31658,516059,223320],{},[31662,516061,32715],{"stretchy":702},[31665,516063,516064],{"encoding":31667},"I+J = \\{ a+b \\mid a \\in I, b \\in J \\}",[34,516066,516068,516086,516104,516125,516143,516161,516188],{"className":516067,"ariaHidden":758},[31672],[34,516069,516071,516074,516077,516080,516083],{"className":516070},[31676],[34,516072],{"className":516073,"style":34944},[31680],[34,516075,106063],{"className":516076,"style":67482},[31685,31686],[34,516078],{"className":516079,"style":32604},[31845],[34,516081,223],{"className":516082},[32024],[34,516084],{"className":516085,"style":32604},[31845],[34,516087,516089,516092,516095,516098,516101],{"className":516088},[31676],[34,516090],{"className":516091,"style":33218},[31680],[34,516093,223320],{"className":516094,"style":223361},[31685,31686],[34,516096],{"className":516097,"style":31846},[31845],[34,516099,202],{"className":516100},[31850],[34,516102],{"className":516103,"style":31846},[31845],[34,516105,516107,516110,516113,516116,516119,516122],{"className":516106},[31676],[34,516108],{"className":516109,"style":31774},[31680],[34,516111,32685],{"className":516112},[31782],[34,516114,111],{"className":516115},[31685,31686],[34,516117],{"className":516118,"style":32604},[31845],[34,516120,223],{"className":516121},[32024],[34,516123],{"className":516124,"style":32604},[31845],[34,516126,516128,516131,516134,516137,516140],{"className":516127},[31676],[34,516129],{"className":516130,"style":31774},[31680],[34,516132,1697],{"className":516133},[31685,31686],[34,516135],{"className":516136,"style":31846},[31845],[34,516138,162086],{"className":516139},[31850],[34,516141],{"className":516142,"style":31846},[31845],[34,516144,516146,516149,516152,516155,516158],{"className":516145},[31676],[34,516147],{"className":516148,"style":34812},[31680],[34,516150,111],{"className":516151},[31685,31686],[34,516153],{"className":516154,"style":31846},[31845],[34,516156,112296],{"className":516157},[31850],[34,516159],{"className":516160,"style":31846},[31845],[34,516162,516164,516167,516170,516173,516176,516179,516182,516185],{"className":516163},[31676],[34,516165],{"className":516166,"style":43217},[31680],[34,516168,106063],{"className":516169,"style":67482},[31685,31686],[34,516171,31960],{"className":516172},[32119],[34,516174],{"className":516175,"style":32123},[31845],[34,516177,1697],{"className":516178},[31685,31686],[34,516180],{"className":516181,"style":31846},[31845],[34,516183,112296],{"className":516184},[31850],[34,516186],{"className":516187,"style":31846},[31845],[34,516189,516191,516194,516197],{"className":516190},[31676],[34,516192],{"className":516193,"style":31774},[31680],[34,516195,223320],{"className":516196,"style":223361},[31685,31686],[34,516198,32715],{"className":516199},[31690],[2230,516201,516202,516412,518229],{},[108,516203,516204,33183,516206,347],{},[2627,516205,441307],{},[34,516207,516209,516259],{"className":516208},[31642],[34,516210,516212],{"className":516211},[31646],[31648,516213,516214],{"xmlns":31650},[31652,516215,516216,516256],{},[31655,516217,516218,516220,516222,516224,516226,516228,516230,516232,516234,516236,516238,516240,516242,516244,516246,516248,516250,516252,516254],{},[31749,516219,459],{},[31662,516221,112296],{},[31658,516223,106063],{},[31662,516225,31960],{"separator":758},[31749,516227,459],{},[31662,516229,112296],{},[31658,516231,223320],{},[39361,516233,262622],{},[31662,516235,262625],{},[39361,516237,262622],{},[31749,516239,459],{},[31662,516241,202],{},[31749,516243,459],{},[31662,516245,223],{},[31749,516247,459],{},[31662,516249,112296],{},[31658,516251,106063],{},[31662,516253,223],{},[31658,516255,223320],{},[31665,516257,516258],{"encoding":31667},"0 \\in I, 0 \\in J \\implies 0 = 0+0 \\in I+J",[34,516260,516262,516280,516307,516331,516349,516367,516385,516403],{"className":516261,"ariaHidden":758},[31672],[34,516263,516265,516268,516271,516274,516277],{"className":516264},[31676],[34,516266],{"className":516267,"style":61814},[31680],[34,516269,459],{"className":516270},[31685],[34,516272],{"className":516273,"style":31846},[31845],[34,516275,112296],{"className":516276},[31850],[34,516278],{"className":516279,"style":31846},[31845],[34,516281,516283,516286,516289,516292,516295,516298,516301,516304],{"className":516282},[31676],[34,516284],{"className":516285,"style":70739},[31680],[34,516287,106063],{"className":516288,"style":67482},[31685,31686],[34,516290,31960],{"className":516291},[32119],[34,516293],{"className":516294,"style":32123},[31845],[34,516296,459],{"className":516297},[31685],[34,516299],{"className":516300,"style":31846},[31845],[34,516302,112296],{"className":516303},[31850],[34,516305],{"className":516306,"style":31846},[31845],[34,516308,516310,516313,516316,516319,516322,516325,516328],{"className":516309},[31676],[34,516311],{"className":516312,"style":336814},[31680],[34,516314,223320],{"className":516315,"style":223361},[31685,31686],[34,516317],{"className":516318,"style":31846},[31845],[34,516320],{"className":516321,"style":31846},[31845],[34,516323,262625],{"className":516324},[31850],[34,516326],{"className":516327,"style":31846},[31845],[34,516329],{"className":516330,"style":31846},[31845],[34,516332,516334,516337,516340,516343,516346],{"className":516333},[31676],[34,516335],{"className":516336,"style":32617},[31680],[34,516338,459],{"className":516339},[31685],[34,516341],{"className":516342,"style":31846},[31845],[34,516344,202],{"className":516345},[31850],[34,516347],{"className":516348,"style":31846},[31845],[34,516350,516352,516355,516358,516361,516364],{"className":516351},[31676],[34,516353],{"className":516354,"style":35942},[31680],[34,516356,459],{"className":516357},[31685],[34,516359],{"className":516360,"style":32604},[31845],[34,516362,223],{"className":516363},[32024],[34,516365],{"className":516366,"style":32604},[31845],[34,516368,516370,516373,516376,516379,516382],{"className":516369},[31676],[34,516371],{"className":516372,"style":61814},[31680],[34,516374,459],{"className":516375},[31685],[34,516377],{"className":516378,"style":31846},[31845],[34,516380,112296],{"className":516381},[31850],[34,516383],{"className":516384,"style":31846},[31845],[34,516386,516388,516391,516394,516397,516400],{"className":516387},[31676],[34,516389],{"className":516390,"style":34944},[31680],[34,516392,106063],{"className":516393,"style":67482},[31685,31686],[34,516395],{"className":516396,"style":32604},[31845],[34,516398,223],{"className":516399},[32024],[34,516401],{"className":516402,"style":32604},[31845],[34,516404,516406,516409],{"className":516405},[31676],[34,516407],{"className":516408,"style":33218},[31680],[34,516410,223320],{"className":516411,"style":223361},[31685,31686],[108,516413,516414,516416,516417,346345,516503,326252,516658,70884,516813,46528,516959,39984,517105,517700],{},[2627,516415,443104],{},"：\n设 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I+J",[34,516447,516449,516476,516494],{"className":516448,"ariaHidden":758},[31672],[34,516450,516452,516455,516458,516461,516464,516467,516470,516473],{"className":516451},[31676],[34,516453],{"className":516454,"style":86652},[31680],[34,516456,32936],{"className":516457},[31685,31686],[34,516459,31960],{"className":516460},[32119],[34,516462],{"className":516463,"style":32123},[31845],[34,516465,67764],{"className":516466,"style":32062},[31685,31686],[34,516468],{"className":516469,"style":31846},[31845],[34,516471,112296],{"className":516472},[31850],[34,516474],{"className":516475,"style":31846},[31845],[34,516477,516479,516482,516485,516488,516491],{"className":516478},[31676],[34,516480],{"className":516481,"style":34944},[31680],[34,516483,106063],{"className":516484,"style":67482},[31685,31686],[34,516486],{"className":516487,"style":32604},[31845],[34,516489,223],{"className":516490},[32024],[34,516492],{"className":516493,"style":32604},[31845],[34,516495,516497,516500],{"className":516496},[31676],[34,516498],{"className":516499,"style":33218},[31680],[34,516501,223320],{"className":516502,"style":223361},[31685,31686],[34,516504,516506,516536],{"className":516505},[31642],[34,516507,516509],{"className":516508},[31646],[31648,516510,516511],{"xmlns":31650},[31652,516512,516513,516533],{},[31655,516514,516515,516517,516519,516525,516527],{},[31658,516516,32936],{},[31662,516518,202],{},[31743,516520,516521,516523],{},[31658,516522,111],{},[31749,516524,640],{},[31662,516526,223],{},[31743,516528,516529,516531],{},[31658,516530,1697],{},[31749,516532,640],{},[31665,516534,516535],{"encoding":31667},"x = a_1 + b_1",[34,516537,516539,516557,516612],{"className":516538,"ariaHidden":758},[31672],[34,516540,516542,516545,516548,516551,516554],{"className":516541},[31676],[34,516543],{"className":516544,"style":32527},[31680],[34,516546,32936],{"className":516547},[31685,31686],[34,516549],{"className":516550,"style":31846},[31845],[34,516552,202],{"className":516553},[31850],[34,516555],{"className":516556,"style":31846},[31845],[34,516558,516560,516563,516603,516606,516609],{"className":516559},[31676],[34,516561],{"className":516562,"style":40522},[31680],[34,516564,516566,516569],{"className":516565},[31685],[34,516567,111],{"className":516568},[31685,31686],[34,516570,516572],{"className":516571},[31792],[34,516573,516575,516595],{"className":516574},[31796,31797],[34,516576,516578,516592],{"className":516577},[31801],[34,516579,516581],{"className":516580,"style":31806},[31805],[34,516582,516583,516586],{"style":31809},[34,516584],{"className":516585,"style":31814},[31813],[34,516587,516589],{"className":516588},[31818,31819,31820,31821],[34,516590,640],{"className":516591},[31685,31821],[34,516593,31829],{"className":516594},[31828],[34,516596,516598],{"className":516597},[31801],[34,516599,516601],{"className":516600,"style":31836},[31805],[34,516602],{},[34,516604],{"className":516605,"style":32604},[31845],[34,516607,223],{"className":516608},[32024],[34,516610],{"className":516611,"style":32604},[31845],[34,516613,516615,516618],{"className":516614},[31676],[34,516616],{"className":516617,"style":33814},[31680],[34,516619,516621,516624],{"className":516620},[31685],[34,516622,1697],{"className":516623},[31685,31686],[34,516625,516627],{"className":516626},[31792],[34,516628,516630,516650],{"className":516629},[31796,31797],[34,516631,516633,516647],{"className":516632},[31801],[34,516634,516636],{"className":516635,"style":31806},[31805],[34,516637,516638,516641],{"style":31809},[34,516639],{"className":516640,"style":31814},[31813],[34,516642,516644],{"className":516643},[31818,31819,31820,31821],[34,516645,640],{"className":516646},[31685,31821],[34,516648,31829],{"className":516649},[31828],[34,516651,516653],{"className":516652},[31801],[34,516654,516656],{"className":516655,"style":31836},[31805],[34,516657],{},[34,516659,516661,516691],{"className":516660},[31642],[34,516662,516664],{"className":516663},[31646],[31648,516665,516666],{"xmlns":31650},[31652,516667,516668,516688],{},[31655,516669,516670,516672,516674,516680,516682],{},[31658,516671,67764],{},[31662,516673,202],{},[31743,516675,516676,516678],{},[31658,516677,111],{},[31749,516679,768],{},[31662,516681,223],{},[31743,516683,516684,516686],{},[31658,516685,1697],{},[31749,516687,768],{},[31665,516689,516690],{"encoding":31667},"y = a_2 + b_2",[34,516692,516694,516712,516767],{"className":516693,"ariaHidden":758},[31672],[34,516695,516697,516700,516703,516706,516709],{"className":516696},[31676],[34,516698],{"className":516699,"style":38774},[31680],[34,516701,67764],{"className":516702,"style":32062},[31685,31686],[34,516704],{"className":516705,"style":31846},[31845],[34,516707,202],{"className":516708},[31850],[34,516710],{"className":516711,"style":31846},[31845],[34,516713,516715,516718,516758,516761,516764],{"className":516714},[31676],[34,516716],{"className":516717,"style":40522},[31680],[34,516719,516721,516724],{"className":516720},[31685],[34,516722,111],{"className":516723},[31685,31686],[34,516725,516727],{"className":516726},[31792],[34,516728,516730,516750],{"className":516729},[31796,31797],[34,516731,516733,516747],{"className":516732},[31801],[34,516734,516736],{"className":516735,"style":31806},[31805],[34,516737,516738,516741],{"style":31809},[34,516739],{"className":516740,"style":31814},[31813],[34,516742,516744],{"className":516743},[31818,31819,31820,31821],[34,516745,768],{"className":516746},[31685,31821],[34,516748,31829],{"className":516749},[31828],[34,516751,516753],{"className":516752},[31801],[34,516754,516756],{"className":516755,"style":31836},[31805],[34,516757],{},[34,516759],{"className":516760,"style":32604},[31845],[34,516762,223],{"className":516763},[32024],[34,516765],{"className":516766,"style":32604},[31845],[34,516768,516770,516773],{"className":516769},[31676],[34,516771],{"className":516772,"style":33814},[31680],[34,516774,516776,516779],{"className":516775},[31685],[34,516777,1697],{"className":516778},[31685,31686],[34,516780,516782],{"className":516781},[31792],[34,516783,516785,516805],{"className":516784},[31796,31797],[34,516786,516788,516802],{"className":516787},[31801],[34,516789,516791],{"className":516790,"style":31806},[31805],[34,516792,516793,516796],{"style":31809},[34,516794],{"className":516795,"style":31814},[31813],[34,516797,516799],{"className":516798},[31818,31819,31820,31821],[34,516800,768],{"className":516801},[31685,31821],[34,516803,31829],{"className":516804},[31828],[34,516806,516808],{"className":516807},[31801],[34,516809,516811],{"className":516810,"style":31836},[31805],[34,516812],{},[34,516814,516816,516846],{"className":516815},[31642],[34,516817,516819],{"className":516818},[31646],[31648,516820,516821],{"xmlns":31650},[31652,516822,516823,516843],{},[31655,516824,516825,516831,516833,516839,516841],{},[31743,516826,516827,516829],{},[31658,516828,111],{},[31749,516830,640],{},[31662,516832,31960],{"separator":758},[31743,516834,516835,516837],{},[31658,516836,111],{},[31749,516838,768],{},[31662,516840,112296],{},[31658,516842,106063],{},[31665,516844,516845],{"encoding":31667},"a_1, a_2 \\in I",[34,516847,516849,516950],{"className":516848,"ariaHidden":758},[31672],[34,516850,516852,516855,516895,516898,516901,516941,516944,516947],{"className":516851},[31676],[34,516853],{"className":516854,"style":86652},[31680],[34,516856,516858,516861],{"className":516857},[31685],[34,516859,111],{"className":516860},[31685,31686],[34,516862,516864],{"className":516863},[31792],[34,516865,516867,516887],{"className":516866},[31796,31797],[34,516868,516870,516884],{"className":516869},[31801],[34,516871,516873],{"className":516872,"style":31806},[31805],[34,516874,516875,516878],{"style":31809},[34,516876],{"className":516877,"style":31814},[31813],[34,516879,516881],{"className":516880},[31818,31819,31820,31821],[34,516882,640],{"className":516883},[31685,31821],[34,516885,31829],{"className":516886},[31828],[34,516888,516890],{"className":516889},[31801],[34,516891,516893],{"className":516892,"style":31836},[31805],[34,516894],{},[34,516896,31960],{"className":516897},[32119],[34,516899],{"className":516900,"style":32123},[31845],[34,516902,516904,516907],{"className":516903},[31685],[34,516905,111],{"className":516906},[31685,31686],[34,516908,516910],{"className":516909},[31792],[34,516911,516913,516933],{"className":516912},[31796,31797],[34,516914,516916,516930],{"className":516915},[31801],[34,516917,516919],{"className":516918,"style":31806},[31805],[34,516920,516921,516924],{"style":31809},[34,516922],{"className":516923,"style":31814},[31813],[34,516925,516927],{"className":516926},[31818,31819,31820,31821],[34,516928,768],{"className":516929},[31685,31821],[34,516931,31829],{"className":516932},[31828],[34,516934,516936],{"className":516935},[31801],[34,516937,516939],{"className":516938,"style":31836},[31805],[34,516940],{},[34,516942],{"className":516943,"style":31846},[31845],[34,516945,112296],{"className":516946},[31850],[34,516948],{"className":516949,"style":31846},[31845],[34,516951,516953,516956],{"className":516952},[31676],[34,516954],{"className":516955,"style":33218},[31680],[34,516957,106063],{"className":516958,"style":67482},[31685,31686],[34,516960,516962,516992],{"className":516961},[31642],[34,516963,516965],{"className":516964},[31646],[31648,516966,516967],{"xmlns":31650},[31652,516968,516969,516989],{},[31655,516970,516971,516977,516979,516985,516987],{},[31743,516972,516973,516975],{},[31658,516974,1697],{},[31749,516976,640],{},[31662,516978,31960],{"separator":758},[31743,516980,516981,516983],{},[31658,516982,1697],{},[31749,516984,768],{},[31662,516986,112296],{},[31658,516988,223320],{},[31665,516990,516991],{"encoding":31667},"b_1, b_2 \\in J",[34,516993,516995,517096],{"className":516994,"ariaHidden":758},[31672],[34,516996,516998,517001,517041,517044,517047,517087,517090,517093],{"className":516997},[31676],[34,516999],{"className":517000,"style":43217},[31680],[34,517002,517004,517007],{"className":517003},[31685],[34,517005,1697],{"className":517006},[31685,31686],[34,517008,517010],{"className":517009},[31792],[34,517011,517013,517033],{"className":517012},[31796,31797],[34,517014,517016,517030],{"className":517015},[31801],[34,517017,517019],{"className":517018,"style":31806},[31805],[34,517020,517021,517024],{"style":31809},[34,517022],{"className":517023,"style":31814},[31813],[34,517025,517027],{"className":517026},[31818,31819,31820,31821],[34,517028,640],{"className":517029},[31685,31821],[34,517031,31829],{"className":517032},[31828],[34,517034,517036],{"className":517035},[31801],[34,517037,517039],{"className":517038,"style":31836},[31805],[34,517040],{},[34,517042,31960],{"className":517043},[32119],[34,517045],{"className":517046,"style":32123},[31845],[34,517048,517050,517053],{"className":517049},[31685],[34,517051,1697],{"className":517052},[31685,31686],[34,517054,517056],{"className":517055},[31792],[34,517057,517059,517079],{"className":517058},[31796,31797],[34,517060,517062,517076],{"className":517061},[31801],[34,517063,517065],{"className":517064,"style":31806},[31805],[34,517066,517067,517070],{"style":31809},[34,517068],{"className":517069,"style":31814},[31813],[34,517071,517073],{"className":517072},[31818,31819,31820,31821],[34,517074,768],{"className":517075},[31685,31821],[34,517077,31829],{"className":517078},[31828],[34,517080,517082],{"className":517081},[31801],[34,517083,517085],{"className":517084,"style":31836},[31805],[34,517086],{},[34,517088],{"className":517089,"style":31846},[31845],[34,517091,112296],{"className":517092},[31850],[34,517094],{"className":517095,"style":31846},[31845],[34,517097,517099,517102],{"className":517098},[31676],[34,517100],{"className":517101,"style":33218},[31680],[34,517103,223320],{"className":517104,"style":223361},[31685,31686],[34,517106,517108,517206],{"className":517107},[31642],[34,517109,517111],{"className":517110},[31646],[31648,517112,517113],{"xmlns":31650},[31652,517114,517115,517203],{},[31655,517116,517117,517119,517121,517123,517125,517127,517133,517135,517141,517143,517145,517147,517153,517155,517161,517163,517165,517167,517173,517175,517181,517183,517185,517187,517193,517195,517201],{},[31658,517118,32936],{},[31662,517120,32226],{},[31658,517122,67764],{},[31662,517124,202],{},[31662,517126,456],{"stretchy":702},[31743,517128,517129,517131],{},[31658,517130,111],{},[31749,517132,640],{},[31662,517134,223],{},[31743,517136,517137,517139],{},[31658,517138,1697],{},[31749,517140,640],{},[31662,517142,2557],{"stretchy":702},[31662,517144,32226],{},[31662,517146,456],{"stretchy":702},[31743,517148,517149,517151],{},[31658,517150,111],{},[31749,517152,768],{},[31662,517154,223],{},[31743,517156,517157,517159],{},[31658,517158,1697],{},[31749,517160,768],{},[31662,517162,2557],{"stretchy":702},[31662,517164,202],{},[31662,517166,456],{"stretchy":702},[31743,517168,517169,517171],{},[31658,517170,111],{},[31749,517172,640],{},[31662,517174,32226],{},[31743,517176,517177,517179],{},[31658,517178,111],{},[31749,517180,768],{},[31662,517182,2557],{"stretchy":702},[31662,517184,223],{},[31662,517186,456],{"stretchy":702},[31743,517188,517189,517191],{},[31658,517190,1697],{},[31749,517192,640],{},[31662,517194,32226],{},[31743,517196,517197,517199],{},[31658,517198,1697],{},[31749,517200,768],{},[31662,517202,2557],{"stretchy":702},[31665,517204,517205],{"encoding":31667},"x - y = (a_1 + b_1) - (a_2 + b_2) = (a_1 - a_2) + (b_1 - b_2)",[34,517207,517209,517227,517245,517303,517361,517419,517477,517535,517593,517651],{"className":517208,"ariaHidden":758},[31672],[34,517210,517212,517215,517218,517221,517224],{"className":517211},[31676],[34,517213],{"className":517214,"style":32597},[31680],[34,517216,32936],{"className":517217},[31685,31686],[34,517219],{"className":517220,"style":32604},[31845],[34,517222,32226],{"className":517223},[32024],[34,517225],{"className":517226,"style":32604},[31845],[34,517228,517230,517233,517236,517239,517242],{"className":517229},[31676],[34,517231],{"className":517232,"style":38774},[31680],[34,517234,67764],{"className":517235,"style":32062},[31685,31686],[34,517237],{"className":517238,"style":31846},[31845],[34,517240,202],{"className":517241},[31850],[34,517243],{"className":517244,"style":31846},[31845],[34,517246,517248,517251,517254,517294,517297,517300],{"className":517247},[31676],[34,517249],{"className":517250,"style":31774},[31680],[34,517252,456],{"className":517253},[31782],[34,517255,517257,517260],{"className":517256},[31685],[34,517258,111],{"className":517259},[31685,31686],[34,517261,517263],{"className":517262},[31792],[34,517264,517266,517286],{"className":517265},[31796,31797],[34,517267,517269,517283],{"className":517268},[31801],[34,517270,517272],{"className":517271,"style":31806},[31805],[34,517273,517274,517277],{"style":31809},[34,517275],{"className":517276,"style":31814},[31813],[34,517278,517280],{"className":517279},[31818,31819,31820,31821],[34,517281,640],{"className":517282},[31685,31821],[34,517284,31829],{"className":517285},[31828],[34,517287,517289],{"className":517288},[31801],[34,517290,517292],{"className":517291,"style":31836},[31805],[34,517293],{},[34,517295],{"className":517296,"style":32604},[31845],[34,517298,223],{"className":517299},[32024],[34,517301],{"className":517302,"style":32604},[31845],[34,517304,517306,517309,517349,517352,517355,517358],{"className":517305},[31676],[34,517307],{"className":517308,"style":31774},[31680],[34,517310,517312,517315],{"className":517311},[31685],[34,517313,1697],{"className":517314},[31685,31686],[34,517316,517318],{"className":517317},[31792],[34,517319,517321,517341],{"className":517320},[31796,31797],[34,517322,517324,517338],{"className":517323},[31801],[34,517325,517327],{"className":517326,"style":31806},[31805],[34,517328,517329,517332],{"style":31809},[34,517330],{"className":517331,"style":31814},[31813],[34,517333,517335],{"className":517334},[31818,31819,31820,31821],[34,517336,640],{"className":517337},[31685,31821],[34,517339,31829],{"className":517340},[31828],[34,517342,517344],{"className":517343},[31801],[34,517345,517347],{"className":517346,"style":31836},[31805],[34,517348],{},[34,517350,2557],{"className":517351},[31690],[34,517353],{"className":517354,"style":32604},[31845],[34,517356,32226],{"className":517357},[32024],[34,517359],{"className":517360,"style":32604},[31845],[34,517362,517364,517367,517370,517410,517413,517416],{"className":517363},[31676],[34,517365],{"className":517366,"style":31774},[31680],[34,517368,456],{"className":517369},[31782],[34,517371,517373,517376],{"className":517372},[31685],[34,517374,111],{"className":517375},[31685,31686],[34,517377,517379],{"className":517378},[31792],[34,517380,517382,517402],{"className":517381},[31796,31797],[34,517383,517385,517399],{"className":517384},[31801],[34,517386,517388],{"className":517387,"style":31806},[31805],[34,517389,517390,517393],{"style":31809},[34,517391],{"className":517392,"style":31814},[31813],[34,517394,517396],{"className":517395},[31818,31819,31820,31821],[34,517397,768],{"className":517398},[31685,31821],[34,517400,31829],{"className":517401},[31828],[34,517403,517405],{"className":517404},[31801],[34,517406,517408],{"className":517407,"style":31836},[31805],[34,517409],{},[34,517411],{"className":517412,"style":32604},[31845],[34,517414,223],{"className":517415},[32024],[34,517417],{"className":517418,"style":32604},[31845],[34,517420,517422,517425,517465,517468,517471,517474],{"className":517421},[31676],[34,517423],{"className":517424,"style":31774},[31680],[34,517426,517428,517431],{"className":517427},[31685],[34,517429,1697],{"className":517430},[31685,31686],[34,517432,517434],{"className":517433},[31792],[34,517435,517437,517457],{"className":517436},[31796,31797],[34,517438,517440,517454],{"className":517439},[31801],[34,517441,517443],{"className":517442,"style":31806},[31805],[34,517444,517445,517448],{"style":31809},[34,517446],{"className":517447,"style":31814},[31813],[34,517449,517451],{"className":517450},[31818,31819,31820,31821],[34,517452,768],{"className":517453},[31685,31821],[34,517455,31829],{"className":517456},[31828],[34,517458,517460],{"className":517459},[31801],[34,517461,517463],{"className":517462,"style":31836},[31805],[34,517464],{},[34,517466,2557],{"className":517467},[31690],[34,517469],{"className":517470,"style":31846},[31845],[34,517472,202],{"className":517473},[31850],[34,517475],{"className":517476,"style":31846},[31845],[34,517478,517480,517483,517486,517526,517529,517532],{"className":517479},[31676],[34,517481],{"className":517482,"style":31774},[31680],[34,517484,456],{"className":517485},[31782],[34,517487,517489,517492],{"className":517488},[31685],[34,517490,111],{"className":517491},[31685,31686],[34,517493,517495],{"className":517494},[31792],[34,517496,517498,517518],{"className":517497},[31796,31797],[34,517499,517501,517515],{"className":517500},[31801],[34,517502,517504],{"className":517503,"style":31806},[31805],[34,517505,517506,517509],{"style":31809},[34,517507],{"className":517508,"style":31814},[31813],[34,517510,517512],{"className":517511},[31818,31819,31820,31821],[34,517513,640],{"className":517514},[31685,31821],[34,517516,31829],{"className":517517},[31828],[34,517519,517521],{"className":517520},[31801],[34,517522,517524],{"className":517523,"style":31836},[31805],[34,517525],{},[34,517527],{"className":517528,"style":32604},[31845],[34,517530,32226],{"className":517531},[32024],[34,517533],{"className":517534,"style":32604},[31845],[34,517536,517538,517541,517581,517584,517587,517590],{"className":517537},[31676],[34,517539],{"className":517540,"style":31774},[31680],[34,517542,517544,517547],{"className":517543},[31685],[34,517545,111],{"className":517546},[31685,31686],[34,517548,517550],{"className":517549},[31792],[34,517551,517553,517573],{"className":517552},[31796,31797],[34,517554,517556,517570],{"className":517555},[31801],[34,517557,517559],{"className":517558,"style":31806},[31805],[34,517560,517561,517564],{"style":31809},[34,517562],{"className":517563,"style":31814},[31813],[34,517565,517567],{"className":517566},[31818,31819,31820,31821],[34,517568,768],{"className":517569},[31685,31821],[34,517571,31829],{"className":517572},[31828],[34,517574,517576],{"className":517575},[31801],[34,517577,517579],{"className":517578,"style":31836},[31805],[34,517580],{},[34,517582,2557],{"className":517583},[31690],[34,517585],{"className":517586,"style":32604},[31845],[34,517588,223],{"className":517589},[32024],[34,517591],{"className":517592,"style":32604},[31845],[34,517594,517596,517599,517602,517642,517645,517648],{"className":517595},[31676],[34,517597],{"className":517598,"style":31774},[31680],[34,517600,456],{"className":517601},[31782],[34,517603,517605,517608],{"className":517604},[31685],[34,517606,1697],{"className":517607},[31685,31686],[34,517609,517611],{"className":517610},[31792],[34,517612,517614,517634],{"className":517613},[31796,31797],[34,517615,517617,517631],{"className":517616},[31801],[34,517618,517620],{"className":517619,"style":31806},[31805],[34,517621,517622,517625],{"style":31809},[34,517623],{"className":517624,"style":31814},[31813],[34,517626,517628],{"className":517627},[31818,31819,31820,31821],[34,517629,640],{"className":517630},[31685,31821],[34,517632,31829],{"className":517633},[31828],[34,517635,517637],{"className":517636},[31801],[34,517638,517640],{"className":517639,"style":31836},[31805],[34,517641],{},[34,517643],{"className":517644,"style":32604},[31845],[34,517646,32226],{"className":517647},[32024],[34,517649],{"className":517650,"style":32604},[31845],[34,517652,517654,517657,517697],{"className":517653},[31676],[34,517655],{"className":517656,"style":31774},[31680],[34,517658,517660,517663],{"className":517659},[31685],[34,517661,1697],{"className":517662},[31685,31686],[34,517664,517666],{"className":517665},[31792],[34,517667,517669,517689],{"className":517668},[31796,31797],[34,517670,517672,517686],{"className":517671},[31801],[34,517673,517675],{"className":517674,"style":31806},[31805],[34,517676,517677,517680],{"style":31809},[34,517678],{"className":517679,"style":31814},[31813],[34,517681,517683],{"className":517682},[31818,31819,31820,31821],[34,517684,768],{"className":517685},[31685,31821],[34,517687,31829],{"className":517688},[31828],[34,517690,517692],{"className":517691},[31801],[34,517693,517695],{"className":517694,"style":31836},[31805],[34,517696],{},[34,517698,2557],{"className":517699},[31690],[105,517701,517702,517888,518073],{},[108,517703,93310,517704,517732,517733,347],{},[34,517705,517707,517720],{"className":517706},[31642],[34,517708,517710],{"className":517709},[31646],[31648,517711,517712],{"xmlns":31650},[31652,517713,517714,517718],{},[31655,517715,517716],{},[31658,517717,106063],{},[31665,517719,106063],{"encoding":31667},[34,517721,517723],{"className":517722,"ariaHidden":758},[31672],[34,517724,517726,517729],{"className":517725},[31676],[34,517727],{"className":517728,"style":33218},[31680],[34,517730,106063],{"className":517731,"style":67482},[31685,31686]," 是理想（子群），所以 ",[34,517734,517736,517766],{"className":517735},[31642],[34,517737,517739],{"className":517738},[31646],[31648,517740,517741],{"xmlns":31650},[31652,517742,517743,517763],{},[31655,517744,517745,517751,517753,517759,517761],{},[31743,517746,517747,517749],{},[31658,517748,111],{},[31749,517750,640],{},[31662,517752,32226],{},[31743,517754,517755,517757],{},[31658,517756,111],{},[31749,517758,768],{},[31662,517760,112296],{},[31658,517762,106063],{},[31665,517764,517765],{"encoding":31667},"a_1 - a_2 \\in I",[34,517767,517769,517824,517879],{"className":517768,"ariaHidden":758},[31672],[34,517770,517772,517775,517815,517818,517821],{"className":517771},[31676],[34,517773],{"className":517774,"style":40522},[31680],[34,517776,517778,517781],{"className":517777},[31685],[34,517779,111],{"className":517780},[31685,31686],[34,517782,517784],{"className":517783},[31792],[34,517785,517787,517807],{"className":517786},[31796,31797],[34,517788,517790,517804],{"className":517789},[31801],[34,517791,517793],{"className":517792,"style":31806},[31805],[34,517794,517795,517798],{"style":31809},[34,517796],{"className":517797,"style":31814},[31813],[34,517799,517801],{"className":517800},[31818,31819,31820,31821],[34,517802,640],{"className":517803},[31685,31821],[34,517805,31829],{"className":517806},[31828],[34,517808,517810],{"className":517809},[31801],[34,517811,517813],{"className":517812,"style":31836},[31805],[34,517814],{},[34,517816],{"className":517817,"style":32604},[31845],[34,517819,32226],{"className":517820},[32024],[34,517822],{"className":517823,"style":32604},[31845],[34,517825,517827,517830,517870,517873,517876],{"className":517826},[31676],[34,517828],{"className":517829,"style":225773},[31680],[34,517831,517833,517836],{"className":517832},[31685],[34,517834,111],{"className":517835},[31685,31686],[34,517837,517839],{"className":517838},[31792],[34,517840,517842,517862],{"className":517841},[31796,31797],[34,517843,517845,517859],{"className":517844},[31801],[34,517846,517848],{"className":517847,"style":31806},[31805],[34,517849,517850,517853],{"style":31809},[34,517851],{"className":517852,"style":31814},[31813],[34,517854,517856],{"className":517855},[31818,31819,31820,31821],[34,517857,768],{"className":517858},[31685,31821],[34,517860,31829],{"className":517861},[31828],[34,517863,517865],{"className":517864},[31801],[34,517866,517868],{"className":517867,"style":31836},[31805],[34,517869],{},[34,517871],{"className":517872,"style":31846},[31845],[34,517874,112296],{"className":517875},[31850],[34,517877],{"className":517878,"style":31846},[31845],[34,517880,517882,517885],{"className":517881},[31676],[34,517883],{"className":517884,"style":33218},[31680],[34,517886,106063],{"className":517887,"style":67482},[31685,31686],[108,517889,93310,517890,517732,517918,347],{},[34,517891,517893,517906],{"className":517892},[31642],[34,517894,517896],{"className":517895},[31646],[31648,517897,517898],{"xmlns":31650},[31652,517899,517900,517904],{},[31655,517901,517902],{},[31658,517903,223320],{},[31665,517905,223320],{"encoding":31667},[34,517907,517909],{"className":517908,"ariaHidden":758},[31672],[34,517910,517912,517915],{"className":517911},[31676],[34,517913],{"className":517914,"style":33218},[31680],[34,517916,223320],{"className":517917,"style":223361},[31685,31686],[34,517919,517921,517951],{"className":517920},[31642],[34,517922,517924],{"className":517923},[31646],[31648,517925,517926],{"xmlns":31650},[31652,517927,517928,517948],{},[31655,517929,517930,517936,517938,517944,517946],{},[31743,517931,517932,517934],{},[31658,517933,1697],{},[31749,517935,640],{},[31662,517937,32226],{},[31743,517939,517940,517942],{},[31658,517941,1697],{},[31749,517943,768],{},[31662,517945,112296],{},[31658,517947,223320],{},[31665,517949,517950],{"encoding":31667},"b_1 - b_2 \\in J",[34,517952,517954,518009,518064],{"className":517953,"ariaHidden":758},[31672],[34,517955,517957,517960,518000,518003,518006],{"className":517956},[31676],[34,517958],{"className":517959,"style":33814},[31680],[34,517961,517963,517966],{"className":517962},[31685],[34,517964,1697],{"className":517965},[31685,31686],[34,517967,517969],{"className":517968},[31792],[34,517970,517972,517992],{"className":517971},[31796,31797],[34,517973,517975,517989],{"className":517974},[31801],[34,517976,517978],{"className":517977,"style":31806},[31805],[34,517979,517980,517983],{"style":31809},[34,517981],{"className":517982,"style":31814},[31813],[34,517984,517986],{"className":517985},[31818,31819,31820,31821],[34,517987,640],{"className":517988},[31685,31821],[34,517990,31829],{"className":517991},[31828],[34,517993,517995],{"className":517994},[31801],[34,517996,517998],{"className":517997,"style":31836},[31805],[34,517999],{},[34,518001],{"className":518002,"style":32604},[31845],[34,518004,32226],{"className":518005},[32024],[34,518007],{"className":518008,"style":32604},[31845],[34,518010,518012,518015,518055,518058,518061],{"className":518011},[31676],[34,518013],{"className":518014,"style":33814},[31680],[34,518016,518018,518021],{"className":518017},[31685],[34,518019,1697],{"className":518020},[31685,31686],[34,518022,518024],{"className":518023},[31792],[34,518025,518027,518047],{"className":518026},[31796,31797],[34,518028,518030,518044],{"className":518029},[31801],[34,518031,518033],{"className":518032,"style":31806},[31805],[34,518034,518035,518038],{"style":31809},[34,518036],{"className":518037,"style":31814},[31813],[34,518039,518041],{"className":518040},[31818,31819,31820,31821],[34,518042,768],{"className":518043},[31685,31821],[34,518045,31829],{"className":518046},[31828],[34,518048,518050],{"className":518049},[31801],[34,518051,518053],{"className":518052,"style":31836},[31805],[34,518054],{},[34,518056],{"className":518057,"style":31846},[31845],[34,518059,112296],{"className":518060},[31850],[34,518062],{"className":518063,"style":31846},[31845],[34,518065,518067,518070],{"className":518066},[31676],[34,518068],{"className":518069,"style":33218},[31680],[34,518071,223320],{"className":518072,"style":223361},[31685,31686],[108,518074,518075,518076,518104,518105,518133,518134,347],{},"所以结果是“",[34,518077,518079,518092],{"className":518078},[31642],[34,518080,518082],{"className":518081},[31646],[31648,518083,518084],{"xmlns":31650},[31652,518085,518086,518090],{},[31655,518087,518088],{},[31658,518089,106063],{},[31665,518091,106063],{"encoding":31667},[34,518093,518095],{"className":518094,"ariaHidden":758},[31672],[34,518096,518098,518101],{"className":518097},[31676],[34,518099],{"className":518100,"style":33218},[31680],[34,518102,106063],{"className":518103,"style":67482},[31685,31686]," 中元素 + ",[34,518106,518108,518121],{"className":518107},[31642],[34,518109,518111],{"className":518110},[31646],[31648,518112,518113],{"xmlns":31650},[31652,518114,518115,518119],{},[31655,518116,518117],{},[31658,518118,223320],{},[31665,518120,223320],{"encoding":31667},[34,518122,518124],{"className":518123,"ariaHidden":758},[31672],[34,518125,518127,518130],{"className":518126},[31676],[34,518128],{"className":518129,"style":33218},[31680],[34,518131,223320],{"className":518132,"style":223361},[31685,31686]," 中元素”的形式，故 ",[34,518135,518137,518163],{"className":518136},[31642],[34,518138,518140],{"className":518139},[31646],[31648,518141,518142],{"xmlns":31650},[31652,518143,518144,518160],{},[31655,518145,518146,518148,518150,518152,518154,518156,518158],{},[31658,518147,32936],{},[31662,518149,32226],{},[31658,518151,67764],{},[31662,518153,112296],{},[31658,518155,106063],{},[31662,518157,223],{},[31658,518159,223320],{},[31665,518161,518162],{"encoding":31667},"x - y \\in I+J",[34,518164,518166,518184,518202,518220],{"className":518165,"ariaHidden":758},[31672],[34,518167,518169,518172,518175,518178,518181],{"className":518168},[31676],[34,518170],{"className":518171,"style":32597},[31680],[34,518173,32936],{"className":518174},[31685,31686],[34,518176],{"className":518177,"style":32604},[31845],[34,518179,32226],{"className":518180},[32024],[34,518182],{"className":518183,"style":32604},[31845],[34,518185,518187,518190,518193,518196,518199],{"className":518186},[31676],[34,518188],{"className":518189,"style":86652},[31680],[34,518191,67764],{"className":518192,"style":32062},[31685,31686],[34,518194],{"className":518195,"style":31846},[31845],[34,518197,112296],{"className":518198},[31850],[34,518200],{"className":518201,"style":31846},[31845],[34,518203,518205,518208,518211,518214,518217],{"className":518204},[31676],[34,518206],{"className":518207,"style":34944},[31680],[34,518209,106063],{"className":518210,"style":67482},[31685,31686],[34,518212],{"className":518213,"style":32604},[31845],[34,518215,223],{"className":518216},[32024],[34,518218],{"className":518219,"style":32604},[31845],[34,518221,518223,518226],{"className":518222},[31676],[34,518224],{"className":518225,"style":33218},[31680],[34,518227,223320],{"className":518228,"style":223361},[31685,31686],[108,518230,518231,516416,518233,61605,518306,1840,518379,518464,518465,39984,518515,518661],{},[2627,518232,442183],{},[34,518234,518236,518258],{"className":518235},[31642],[34,518237,518239],{"className":518238},[31646],[31648,518240,518241],{"xmlns":31650},[31652,518242,518243,518255],{},[31655,518244,518245,518247,518249,518251,518253],{},[31658,518246,32936],{},[31662,518248,112296],{},[31658,518250,106063],{},[31662,518252,223],{},[31658,518254,223320],{},[31665,518256,518257],{"encoding":31667},"x \\in I+J",[34,518259,518261,518279,518297],{"className":518260,"ariaHidden":758},[31672],[34,518262,518264,518267,518270,518273,518276],{"className":518263},[31676],[34,518265],{"className":518266,"style":34812},[31680],[34,518268,32936],{"className":518269},[31685,31686],[34,518271],{"className":518272,"style":31846},[31845],[34,518274,112296],{"className":518275},[31850],[34,518277],{"className":518278,"style":31846},[31845],[34,518280,518282,518285,518288,518291,518294],{"className":518281},[31676],[34,518283],{"className":518284,"style":34944},[31680],[34,518286,106063],{"className":518287,"style":67482},[31685,31686],[34,518289],{"className":518290,"style":32604},[31845],[34,518292,223],{"className":518293},[32024],[34,518295],{"className":518296,"style":32604},[31845],[34,518298,518300,518303],{"className":518299},[31676],[34,518301],{"className":518302,"style":33218},[31680],[34,518304,223320],{"className":518305,"style":223361},[31685,31686],[34,518307,518309,518331],{"className":518308},[31642],[34,518310,518312],{"className":518311},[31646],[31648,518313,518314],{"xmlns":31650},[31652,518315,518316,518328],{},[31655,518317,518318,518320,518322,518324,518326],{},[31658,518319,32936],{},[31662,518321,202],{},[31658,518323,111],{},[31662,518325,223],{},[31658,518327,1697],{},[31665,518329,518330],{"encoding":31667},"x = a + b",[34,518332,518334,518352,518370],{"className":518333,"ariaHidden":758},[31672],[34,518335,518337,518340,518343,518346,518349],{"className":518336},[31676],[34,518338],{"className":518339,"style":32527},[31680],[34,518341,32936],{"className":518342},[31685,31686],[34,518344],{"className":518345,"style":31846},[31845],[34,518347,202],{"className":518348},[31850],[34,518350],{"className":518351,"style":31846},[31845],[34,518353,518355,518358,518361,518364,518367],{"className":518354},[31676],[34,518356],{"className":518357,"style":32597},[31680],[34,518359,111],{"className":518360},[31685,31686],[34,518362],{"className":518363,"style":32604},[31845],[34,518365,223],{"className":518366},[32024],[34,518368],{"className":518369,"style":32604},[31845],[34,518371,518373,518376],{"className":518372},[31676],[34,518374],{"className":518375,"style":31681},[31680],[34,518377,1697],{"className":518378},[31685,31686],[34,518380,518382,518407],{"className":518381},[31642],[34,518383,518385],{"className":518384},[31646],[31648,518386,518387],{"xmlns":31650},[31652,518388,518389,518405],{},[31655,518390,518391,518393,518395,518397,518399,518401,518403],{},[31658,518392,111],{},[31662,518394,112296],{},[31658,518396,106063],{},[31662,518398,31960],{"separator":758},[31658,518400,1697],{},[31662,518402,112296],{},[31658,518404,223320],{},[31665,518406,509221],{"encoding":31667},[34,518408,518410,518428,518455],{"className":518409,"ariaHidden":758},[31672],[34,518411,518413,518416,518419,518422,518425],{"className":518412},[31676],[34,518414],{"className":518415,"style":34812},[31680],[34,518417,111],{"className":518418},[31685,31686],[34,518420],{"className":518421,"style":31846},[31845],[34,518423,112296],{"className":518424},[31850],[34,518426],{"className":518427,"style":31846},[31845],[34,518429,518431,518434,518437,518440,518443,518446,518449,518452],{"className":518430},[31676],[34,518432],{"className":518433,"style":43217},[31680],[34,518435,106063],{"className":518436,"style":67482},[31685,31686],[34,518438,31960],{"className":518439},[32119],[34,518441],{"className":518442,"style":32123},[31845],[34,518444,1697],{"className":518445},[31685,31686],[34,518447],{"className":518448,"style":31846},[31845],[34,518450,112296],{"className":518451},[31850],[34,518453],{"className":518454,"style":31846},[31845],[34,518456,518458,518461],{"className":518457},[31676],[34,518459],{"className":518460,"style":33218},[31680],[34,518462,223320],{"className":518463,"style":223361},[31685,31686],")。设 ",[34,518466,518468,518485],{"className":518467},[31642],[34,518469,518471],{"className":518470},[31646],[31648,518472,518473],{"xmlns":31650},[31652,518474,518475,518483],{},[31655,518476,518477,518479,518481],{},[31658,518478,34395],{},[31662,518480,112296],{},[31658,518482,34386],{},[31665,518484,335118],{"encoding":31667},[34,518486,518488,518506],{"className":518487,"ariaHidden":758},[31672],[34,518489,518491,518494,518497,518500,518503],{"className":518490},[31676],[34,518492],{"className":518493,"style":34812},[31680],[34,518495,34395],{"className":518496,"style":34452},[31685,31686],[34,518498],{"className":518499,"style":31846},[31845],[34,518501,112296],{"className":518502},[31850],[34,518504],{"className":518505,"style":31846},[31845],[34,518507,518509,518512],{"className":518508},[31676],[34,518510],{"className":518511,"style":33218},[31680],[34,518513,34386],{"className":518514,"style":34427},[31685,31686],[34,518516,518518,518559],{"className":518517},[31642],[34,518519,518521],{"className":518520},[31646],[31648,518522,518523],{"xmlns":31650},[31652,518524,518525,518557],{},[31655,518526,518527,518529,518531,518533,518535,518537,518539,518541,518543,518545,518547,518549,518551,518553,518555],{},[31658,518528,34395],{},[31658,518530,32936],{},[31662,518532,202],{},[31658,518534,34395],{},[31662,518536,456],{"stretchy":702},[31658,518538,111],{},[31662,518540,223],{},[31658,518542,1697],{},[31662,518544,2557],{"stretchy":702},[31662,518546,202],{},[31658,518548,34395],{},[31658,518550,111],{},[31662,518552,223],{},[31658,518554,34395],{},[31658,518556,1697],{},[31665,518558,508044],{"encoding":31667},[34,518560,518562,518583,518607,518628,518649],{"className":518561,"ariaHidden":758},[31672],[34,518563,518565,518568,518571,518574,518577,518580],{"className":518564},[31676],[34,518566],{"className":518567,"style":32527},[31680],[34,518569,34395],{"className":518570,"style":34452},[31685,31686],[34,518572,32936],{"className":518573},[31685,31686],[34,518575],{"className":518576,"style":31846},[31845],[34,518578,202],{"className":518579},[31850],[34,518581],{"className":518582,"style":31846},[31845],[34,518584,518586,518589,518592,518595,518598,518601,518604],{"className":518585},[31676],[34,518587],{"className":518588,"style":31774},[31680],[34,518590,34395],{"className":518591,"style":34452},[31685,31686],[34,518593,456],{"className":518594},[31782],[34,518596,111],{"className":518597},[31685,31686],[34,518599],{"className":518600,"style":32604},[31845],[34,518602,223],{"className":518603},[32024],[34,518605],{"className":518606,"style":32604},[31845],[34,518608,518610,518613,518616,518619,518622,518625],{"className":518609},[31676],[34,518611],{"className":518612,"style":31774},[31680],[34,518614,1697],{"className":518615},[31685,31686],[34,518617,2557],{"className":518618},[31690],[34,518620],{"className":518621,"style":31846},[31845],[34,518623,202],{"className":518624},[31850],[34,518626],{"className":518627,"style":31846},[31845],[34,518629,518631,518634,518637,518640,518643,518646],{"className":518630},[31676],[34,518632],{"className":518633,"style":32597},[31680],[34,518635,34395],{"className":518636,"style":34452},[31685,31686],[34,518638,111],{"className":518639},[31685,31686],[34,518641],{"className":518642,"style":32604},[31845],[34,518644,223],{"className":518645},[32024],[34,518647],{"className":518648,"style":32604},[31845],[34,518650,518652,518655,518658],{"className":518651},[31676],[34,518653],{"className":518654,"style":31681},[31680],[34,518656,34395],{"className":518657,"style":34452},[31685,31686],[34,518659,1697],{"className":518660},[31685,31686],[105,518662,518663,518805,518945,519025],{},[108,518664,93310,518665,518693,518694,518804],{},[34,518666,518668,518681],{"className":518667},[31642],[34,518669,518671],{"className":518670},[31646],[31648,518672,518673],{"xmlns":31650},[31652,518674,518675,518679],{},[31655,518676,518677],{},[31658,518678,106063],{},[31665,518680,106063],{"encoding":31667},[34,518682,518684],{"className":518683,"ariaHidden":758},[31672],[34,518685,518687,518690],{"className":518686},[31676],[34,518688],{"className":518689,"style":33218},[31680],[34,518691,106063],{"className":518692,"style":67482},[31685,31686]," 是理想，",[34,518695,518697,518729],{"className":518696},[31642],[34,518698,518700],{"className":518699},[31646],[31648,518701,518702],{"xmlns":31650},[31652,518703,518704,518726],{},[31655,518705,518706,518708,518710,518712,518714,518716,518718,518720,518722,518724],{},[31658,518707,111],{},[31662,518709,112296],{},[31658,518711,106063],{},[39361,518713,262622],{},[31662,518715,262625],{},[39361,518717,262622],{},[31658,518719,34395],{},[31658,518721,111],{},[31662,518723,112296],{},[31658,518725,106063],{},[31665,518727,518728],{"encoding":31667},"a \\in I \\implies ra \\in I",[34,518730,518732,518750,518774,518795],{"className":518731,"ariaHidden":758},[31672],[34,518733,518735,518738,518741,518744,518747],{"className":518734},[31676],[34,518736],{"className":518737,"style":34812},[31680],[34,518739,111],{"className":518740},[31685,31686],[34,518742],{"className":518743,"style":31846},[31845],[34,518745,112296],{"className":518746},[31850],[34,518748],{"className":518749,"style":31846},[31845],[34,518751,518753,518756,518759,518762,518765,518768,518771],{"className":518752},[31676],[34,518754],{"className":518755,"style":336814},[31680],[34,518757,106063],{"className":518758,"style":67482},[31685,31686],[34,518760],{"className":518761,"style":31846},[31845],[34,518763],{"className":518764,"style":31846},[31845],[34,518766,262625],{"className":518767},[31850],[34,518769],{"className":518770,"style":31846},[31845],[34,518772],{"className":518773,"style":31846},[31845],[34,518775,518777,518780,518783,518786,518789,518792],{"className":518776},[31676],[34,518778],{"className":518779,"style":34812},[31680],[34,518781,34395],{"className":518782,"style":34452},[31685,31686],[34,518784,111],{"className":518785},[31685,31686],[34,518787],{"className":518788,"style":31846},[31845],[34,518790,112296],{"className":518791},[31850],[34,518793],{"className":518794,"style":31846},[31845],[34,518796,518798,518801],{"className":518797},[31676],[34,518799],{"className":518800,"style":33218},[31680],[34,518802,106063],{"className":518803,"style":67482},[31685,31686],"（左吸收）。",[108,518806,93310,518807,518693,518835,518804],{},[34,518808,518810,518823],{"className":518809},[31642],[34,518811,518813],{"className":518812},[31646],[31648,518814,518815],{"xmlns":31650},[31652,518816,518817,518821],{},[31655,518818,518819],{},[31658,518820,223320],{},[31665,518822,223320],{"encoding":31667},[34,518824,518826],{"className":518825,"ariaHidden":758},[31672],[34,518827,518829,518832],{"className":518828},[31676],[34,518830],{"className":518831,"style":33218},[31680],[34,518833,223320],{"className":518834,"style":223361},[31685,31686],[34,518836,518838,518870],{"className":518837},[31642],[34,518839,518841],{"className":518840},[31646],[31648,518842,518843],{"xmlns":31650},[31652,518844,518845,518867],{},[31655,518846,518847,518849,518851,518853,518855,518857,518859,518861,518863,518865],{},[31658,518848,1697],{},[31662,518850,112296],{},[31658,518852,223320],{},[39361,518854,262622],{},[31662,518856,262625],{},[39361,518858,262622],{},[31658,518860,34395],{},[31658,518862,1697],{},[31662,518864,112296],{},[31658,518866,223320],{},[31665,518868,518869],{"encoding":31667},"b \\in J \\implies rb \\in J",[34,518871,518873,518891,518915,518936],{"className":518872,"ariaHidden":758},[31672],[34,518874,518876,518879,518882,518885,518888],{"className":518875},[31676],[34,518877],{"className":518878,"style":46991},[31680],[34,518880,1697],{"className":518881},[31685,31686],[34,518883],{"className":518884,"style":31846},[31845],[34,518886,112296],{"className":518887},[31850],[34,518889],{"className":518890,"style":31846},[31845],[34,518892,518894,518897,518900,518903,518906,518909,518912],{"className":518893},[31676],[34,518895],{"className":518896,"style":336814},[31680],[34,518898,223320],{"className":518899,"style":223361},[31685,31686],[34,518901],{"className":518902,"style":31846},[31845],[34,518904],{"className":518905,"style":31846},[31845],[34,518907,262625],{"className":518908},[31850],[34,518910],{"className":518911,"style":31846},[31845],[34,518913],{"className":518914,"style":31846},[31845],[34,518916,518918,518921,518924,518927,518930,518933],{"className":518917},[31676],[34,518919],{"className":518920,"style":46991},[31680],[34,518922,34395],{"className":518923,"style":34452},[31685,31686],[34,518925,1697],{"className":518926},[31685,31686],[34,518928],{"className":518929,"style":31846},[31845],[34,518931,112296],{"className":518932},[31850],[34,518934],{"className":518935,"style":31846},[31845],[34,518937,518939,518942],{"className":518938},[31676],[34,518940],{"className":518941,"style":33218},[31680],[34,518943,223320],{"className":518944,"style":223361},[31685,31686],[108,518946,175992,518947,347],{},[34,518948,518950,518974],{"className":518949},[31642],[34,518951,518953],{"className":518952},[31646],[31648,518954,518955],{"xmlns":31650},[31652,518956,518957,518971],{},[31655,518958,518959,518961,518963,518965,518967,518969],{},[31658,518960,34395],{},[31658,518962,32936],{},[31662,518964,112296],{},[31658,518966,106063],{},[31662,518968,223],{},[31658,518970,223320],{},[31665,518972,518973],{"encoding":31667},"rx \\in I+J",[34,518975,518977,518998,519016],{"className":518976,"ariaHidden":758},[31672],[34,518978,518980,518983,518986,518989,518992,518995],{"className":518979},[31676],[34,518981],{"className":518982,"style":34812},[31680],[34,518984,34395],{"className":518985,"style":34452},[31685,31686],[34,518987,32936],{"className":518988},[31685,31686],[34,518990],{"className":518991,"style":31846},[31845],[34,518993,112296],{"className":518994},[31850],[34,518996],{"className":518997,"style":31846},[31845],[34,518999,519001,519004,519007,519010,519013],{"className":519000},[31676],[34,519002],{"className":519003,"style":34944},[31680],[34,519005,106063],{"className":519006,"style":67482},[31685,31686],[34,519008],{"className":519009,"style":32604},[31845],[34,519011,223],{"className":519012},[32024],[34,519014],{"className":519015,"style":32604},[31845],[34,519017,519019,519022],{"className":519018},[31676],[34,519020],{"className":519021,"style":33218},[31680],[34,519023,223320],{"className":519024,"style":223361},[31685,31686],[108,519026,519027,519028,519160],{},"同理可证 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J",[34,519872,519874,519892,519910],{"className":519873,"ariaHidden":758},[31672],[34,519875,519877,519880,519883,519886,519889],{"className":519876},[31676],[34,519878],{"className":519879,"style":32597},[31680],[34,519881,32936],{"className":519882},[31685,31686],[34,519884],{"className":519885,"style":32604},[31845],[34,519887,32226],{"className":519888},[32024],[34,519890],{"className":519891,"style":32604},[31845],[34,519893,519895,519898,519901,519904,519907],{"className":519894},[31676],[34,519896],{"className":519897,"style":86652},[31680],[34,519899,67764],{"className":519900,"style":32062},[31685,31686],[34,519902],{"className":519903,"style":31846},[31845],[34,519905,112296],{"className":519906},[31850],[34,519908],{"className":519909,"style":31846},[31845],[34,519911,519913,519916],{"className":519912},[31676],[34,519914],{"className":519915,"style":33218},[31680],[34,519917,223320],{"className":519918,"style":223361},[31685,31686],[108,519920,519921,519922,519950,519951,13977,519979,347],{},"既然既在 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",[34,519952,519954,519967],{"className":519953},[31642],[34,519955,519957],{"className":519956},[31646],[31648,519958,519959],{"xmlns":31650},[31652,519960,519961,519965],{},[31655,519962,519963],{},[31658,519964,223320],{},[31665,519966,223320],{"encoding":31667},[34,519968,519970],{"className":519969,"ariaHidden":758},[31672],[34,519971,519973,519976],{"className":519972},[31676],[34,519974],{"className":519975,"style":33218},[31680],[34,519977,223320],{"className":519978,"style":223361},[31685,31686],[34,519980,519982,520008],{"className":519981},[31642],[34,519983,519985],{"className":519984},[31646],[31648,519986,519987],{"xmlns":31650},[31652,519988,519989,520005],{},[31655,519990,519991,519993,519995,519997,519999,520001,520003],{},[31658,519992,32936],{},[31662,519994,32226],{},[31658,519996,67764],{},[31662,519998,112296],{},[31658,520000,106063],{},[31662,520002,326344],{},[31658,520004,223320],{},[31665,520006,520007],{"encoding":31667},"x - y \\in I \\cap J",[34,520009,520011,520029,520047,520065],{"className":520010,"ariaHidden":758},[31672],[34,520012,520014,520017,520020,520023,520026],{"className":520013},[31676],[34,520015],{"className":520016,"style":32597},[31680],[34,520018,32936],{"className":520019},[31685,31686],[34,520021],{"className":520022,"style":32604},[31845],[34,520024,32226],{"className":520025},[32024],[34,520027],{"className":520028,"style":32604},[31845],[34,520030,520032,520035,520038,520041,520044],{"className":520031},[31676],[34,520033],{"className":520034,"style":86652},[31680],[34,520036,67764],{"className":520037,"style":32062},[31685,31686],[34,520039],{"className":520040,"style":31846},[31845],[34,520042,112296],{"className":520043},[31850],[34,520045],{"className":520046,"style":31846},[31845],[34,520048,520050,520053,520056,520059,520062],{"className":520049},[31676],[34,520051],{"className":520052,"style":33218},[31680],[34,520054,106063],{"className":520055,"style":67482},[31685,31686],[34,520057],{"className":520058,"style":32604},[31845],[34,520060,326344],{"className":520061},[32024],[34,520063],{"className":520064,"style":32604},[31845],[34,520066,520068,520071],{"className":520067},[31676],[34,520069],{"className":520070,"style":33218},[31680],[34,520072,223320],{"className":520073,"style":223361},[31685,31686],[108,520075,520076,516416,520078,46528,520150,347,520200],{},[2627,520077,442183],{},[34,520079,520081,520102],{"className":520080},[31642],[34,520082,520084],{"className":520083},[31646],[31648,520085,520086],{"xmlns":31650},[31652,520087,520088,520100],{},[31655,520089,520090,520092,520094,520096,520098],{},[31658,520091,32936],{},[31662,520093,112296],{},[31658,520095,106063],{},[31662,520097,326344],{},[31658,520099,223320],{},[31665,520101,494302],{"encoding":31667},[34,520103,520105,520123,520141],{"className":520104,"ariaHidden":758},[31672],[34,520106,520108,520111,520114,520117,520120],{"className":520107},[31676],[34,520109],{"className":520110,"style":34812},[31680],[34,520112,32936],{"className":520113},[31685,31686],[34,520115],{"className":520116,"style":31846},[31845],[34,520118,112296],{"className":520119},[31850],[34,520121],{"className":520122,"style":31846},[31845],[34,520124,520126,520129,520132,520135,520138],{"className":520125},[31676],[34,520127],{"className":520128,"style":33218},[31680],[34,520130,106063],{"className":520131,"style":67482},[31685,31686],[34,520133],{"className":520134,"style":32604},[31845],[34,520136,326344],{"className":520137},[32024],[34,520139],{"className":520140,"style":32604},[31845],[34,520142,520144,520147],{"className":520143},[31676],[34,520145],{"className":520146,"style":33218},[31680],[34,520148,223320],{"className":520149,"style":223361},[31685,31686],[34,520151,520153,520170],{"className":520152},[31642],[34,520154,520156],{"className":520155},[31646],[31648,520157,520158],{"xmlns":31650},[31652,520159,520160,520168],{},[31655,520161,520162,520164,520166],{},[31658,520163,34395],{},[31662,520165,112296],{},[31658,520167,34386],{},[31665,520169,335118],{"encoding":31667},[34,520171,520173,520191],{"className":520172,"ariaHidden":758},[31672],[34,520174,520176,520179,520182,520185,520188],{"className":520175},[31676],[34,520177],{"className":520178,"style":34812},[31680],[34,520180,34395],{"className":520181,"style":34452},[31685,31686],[34,520183],{"className":520184,"style":31846},[31845],[34,520186,112296],{"className":520187},[31850],[34,520189],{"className":520190,"style":31846},[31845],[34,520192,520194,520197],{"className":520193},[31676],[34,520195],{"className":520196,"style":33218},[31680],[34,520198,34386],{"className":520199,"style":34427},[31685,31686],[105,520201,520202,520342,520481],{},[108,520203,520204,108987,520313,520341],{},[34,520205,520207,520238],{"className":520206},[31642],[34,520208,520210],{"className":520209},[31646],[31648,520211,520212],{"xmlns":31650},[31652,520213,520214,520236],{},[31655,520215,520216,520218,520220,520222,520224,520226,520228,520230,520232,520234],{},[31658,520217,32936],{},[31662,520219,112296],{},[31658,520221,106063],{},[39361,520223,262622],{},[31662,520225,262625],{},[39361,520227,262622],{},[31658,520229,34395],{},[31658,520231,32936],{},[31662,520233,112296],{},[31658,520235,106063],{},[31665,520237,505946],{"encoding":31667},[34,520239,520241,520259,520283,520304],{"className":520240,"ariaHidden":758},[31672],[34,520242,520244,520247,520250,520253,520256],{"className":520243},[31676],[34,520245],{"className":520246,"style":34812},[31680],[34,520248,32936],{"className":520249},[31685,31686],[34,520251],{"className":520252,"style":31846},[31845],[34,520254,112296],{"className":520255},[31850],[34,520257],{"className":520258,"style":31846},[31845],[34,520260,520262,520265,520268,520271,520274,520277,520280],{"className":520261},[31676],[34,520263],{"className":520264,"style":336814},[31680],[34,520266,106063],{"className":520267,"style":67482},[31685,31686],[34,520269],{"className":520270,"style":31846},[31845],[34,520272],{"className":520273,"style":31846},[31845],[34,520275,262625],{"className":520276},[31850],[34,520278],{"className":520279,"style":31846},[31845],[34,520281],{"className":520282,"style":31846},[31845],[34,520284,520286,520289,520292,520295,520298,520301],{"className":520285},[31676],[34,520287],{"className":520288,"style":34812},[31680],[34,520290,34395],{"className":520291,"style":34452},[31685,31686],[34,520293,32936],{"className":520294},[31685,31686],[34,520296],{"className":520297,"style":31846},[31845],[34,520299,112296],{"className":520300},[31850],[34,520302],{"className":520303,"style":31846},[31845],[34,520305,520307,520310],{"className":520306},[31676],[34,520308],{"className":520309,"style":33218},[31680],[34,520311,106063],{"className":520312,"style":67482},[31685,31686],[34,520314,520316,520329],{"className":520315},[31642],[34,520317,520319],{"className":520318},[31646],[31648,520320,520321],{"xmlns":31650},[31652,520322,520323,520327],{},[31655,520324,520325],{},[31658,520326,106063],{},[31665,520328,106063],{"encoding":31667},[34,520330,520332],{"className":520331,"ariaHidden":758},[31672],[34,520333,520335,520338],{"className":520334},[31676],[34,520336],{"className":520337,"style":33218},[31680],[34,520339,106063],{"className":520340,"style":67482},[31685,31686]," 的吸收律）。",[108,520343,520344,108987,520453,520341],{},[34,520345,520347,520378],{"className":520346},[31642],[34,520348,520350],{"className":520349},[31646],[31648,520351,520352],{"xmlns":31650},[31652,520353,520354,520376],{},[31655,520355,520356,520358,520360,520362,520364,520366,520368,520370,520372,520374],{},[31658,520357,32936],{},[31662,520359,112296],{},[31658,520361,223320],{},[39361,520363,262622],{},[31662,520365,262625],{},[39361,520367,262622],{},[31658,520369,34395],{},[31658,520371,32936],{},[31662,520373,112296],{},[31658,520375,223320],{},[31665,520377,506143],{"encoding":31667},[34,520379,520381,520399,520423,520444],{"className":520380,"ariaHidden":758},[31672],[34,520382,520384,520387,520390,520393,520396],{"className":520383},[31676],[34,520385],{"className":520386,"style":34812},[31680],[34,520388,32936],{"className":520389},[31685,31686],[34,520391],{"className":520392,"style":31846},[31845],[34,520394,112296],{"className":520395},[31850],[34,520397],{"className":520398,"style":31846},[31845],[34,520400,520402,520405,520408,520411,520414,520417,520420],{"className":520401},[31676],[34,520403],{"className":520404,"style":336814},[31680],[34,520406,223320],{"className":520407,"style":223361},[31685,31686],[34,520409],{"className":520410,"style":31846},[31845],[34,520412],{"className":520413,"style":31846},[31845],[34,520415,262625],{"className":520416},[31850],[34,520418],{"className":520419,"style":31846},[31845],[34,520421],{"className":520422,"style":31846},[31845],[34,520424,520426,520429,520432,520435,520438,520441],{"className":520425},[31676],[34,520427],{"className":520428,"style":34812},[31680],[34,520430,34395],{"className":520431,"style":34452},[31685,31686],[34,520433,32936],{"className":520434},[31685,31686],[34,520436],{"className":520437,"style":31846},[31845],[34,520439,112296],{"className":520440},[31850],[34,520442],{"className":520443,"style":31846},[31845],[34,520445,520447,520450],{"className":520446},[31676],[34,520448],{"className":520449,"style":33218},[31680],[34,520451,223320],{"className":520452,"style":223361},[31685,31686],[34,520454,520456,520469],{"className":520455},[31642],[34,520457,520459],{"className":520458},[31646],[31648,520460,520461],{"xmlns":31650},[31652,520462,520463,520467],{},[31655,520464,520465],{},[31658,520466,223320],{},[31665,520468,223320],{"encoding":31667},[34,520470,520472],{"className":520471,"ariaHidden":758},[31672],[34,520473,520475,520478],{"className":520474},[31676],[34,520476],{"className":520477,"style":33218},[31680],[34,520479,223320],{"className":520480,"style":223361},[31685,31686],[108,520482,175992,520483,520561,520562,520596],{},[34,520484,520486,520510],{"className":520485},[31642],[34,520487,520489],{"className":520488},[31646],[31648,520490,520491],{"xmlns":31650},[31652,520492,520493,520507],{},[31655,520494,520495,520497,520499,520501,520503,520505],{},[31658,520496,34395],{},[31658,520498,32936],{},[31662,520500,112296],{},[31658,520502,106063],{},[31662,520504,326344],{},[31658,520506,223320],{},[31665,520508,520509],{"encoding":31667},"rx \\in I \\cap J",[34,520511,520513,520534,520552],{"className":520512,"ariaHidden":758},[31672],[34,520514,520516,520519,520522,520525,520528,520531],{"className":520515},[31676],[34,520517],{"className":520518,"style":34812},[31680],[34,520520,34395],{"className":520521,"style":34452},[31685,31686],[34,520523,32936],{"className":520524},[31685,31686],[34,520526],{"className":520527,"style":31846},[31845],[34,520529,112296],{"className":520530},[31850],[34,520532],{"className":520533,"style":31846},[31845],[34,520535,520537,520540,520543,520546,520549],{"className":520536},[31676],[34,520538],{"className":520539,"style":33218},[31680],[34,520541,106063],{"className":520542,"style":67482},[31685,31686],[34,520544],{"className":520545,"style":32604},[31845],[34,520547,326344],{"className":520548},[32024],[34,520550],{"className":520551,"style":32604},[31845],[34,520553,520555,520558],{"className":520554},[31676],[34,520556],{"className":520557,"style":33218},[31680],[34,520559,223320],{"className":520560,"style":223361},[31685,31686],"。右乘 ",[34,520563,520565,520581],{"className":520564},[31642],[34,520566,520568],{"className":520567},[31646],[31648,520569,520570],{"xmlns":31650},[31652,520571,520572,520578],{},[31655,520573,520574,520576],{},[31658,520575,32936],{},[31658,520577,34395],{},[31665,520579,520580],{"encoding":31667},"xr",[34,520582,520584],{"className":520583,"ariaHidden":758},[31672],[34,520585,520587,520590,520593],{"className":520586},[31676],[34,520588],{"className":520589,"style":32527},[31680],[34,520591,32936],{"className":520592},[31685,31686],[34,520594,34395],{"className":520595,"style":34452},[31685,31686]," 同理。",[15,520598,520599,2664,520601,452957,520673],{},[2627,520600,114043],{},[34,520602,520604,520625],{"className":520603},[31642],[34,520605,520607],{"className":520606},[31646],[31648,520608,520609],{"xmlns":31650},[31652,520610,520611,520623],{},[31655,520612,520613,520615,520617,520619,520621],{},[31658,520614,106063],{},[31662,520616,326344],{},[31658,520618,223320],{},[31662,520620,512725],{},[31658,520622,34386],{},[31665,520624,515518],{"encoding":31667},[34,520626,520628,520646,520664],{"className":520627,"ariaHidden":758},[31672],[34,520629,520631,520634,520637,520640,520643],{"className":520630},[31676],[34,520632],{"className":520633,"style":33218},[31680],[34,520635,106063],{"className":520636,"style":67482},[31685,31686],[34,520638],{"className":520639,"style":32604},[31845],[34,520641,326344],{"className":520642},[32024],[34,520644],{"className":520645,"style":32604},[31845],[34,520647,520649,520652,520655,520658,520661],{"className":520648},[31676],[34,520650],{"className":520651,"style":515471},[31680],[34,520653,223320],{"className":520654,"style":223361},[31685,31686],[34,520656],{"className":520657,"style":31846},[31845],[34,520659,512725],{"className":520660},[31850,129261],[34,520662],{"className":520663,"style":31846},[31845],[34,520665,520667,520670],{"className":520666},[31676],[34,520668],{"className":520669,"style":33218},[31680],[34,520671,34386],{"className":520672,"style":34427},[31685,31686],[34,520674,520676,520689],{"className":520675},[31642],[34,520677,520679],{"className":520678},[31646],[31648,520680,520681],{"xmlns":31650},[31652,520682,520683,520687],{},[31655,520684,520685],{},[31658,520686,445204],{"mathvariant":38079},[31665,520688,445207],{"encoding":31667},[34,520690,520692],{"className":520691,"ariaHidden":758},[31672],[34,520693,520695,520698],{"className":520694},[31676],[34,520696],{"className":520697,"style":445217},[31680],[34,520699,445204],{"className":520700},[31685,129261],[32188,520702],{},[253,520704,520706],{"id":520705},"题目-3商环结构的严谨验证",[2627,520707,520708],{},"题目 3：商环结构的严谨验证",[15,520710,520711,314945,520713,520751,520752,520895,520896,520971],{},[2627,520712,101191],{},[34,520714,520716,520733],{"className":520715},[31642],[34,520717,520719],{"className":520718},[31646],[31648,520720,520721],{"xmlns":31650},[31652,520722,520723,520731],{},[31655,520724,520725,520727,520729],{},[31658,520726,34386],{},[31658,520728,38080],{"mathvariant":38079},[31658,520730,106063],{},[31665,520732,337927],{"encoding":31667},[34,520734,520736],{"className":520735,"ariaHidden":758},[31672],[34,520737,520739,520742,520745,520748],{"className":520738},[31676],[34,520740],{"className":520741,"style":31774},[31680],[34,520743,34386],{"className":520744,"style":34427},[31685,31686],[34,520746,38080],{"className":520747},[31685],[34,520749,106063],{"className":520750,"style":67482},[31685,31686]," 上定义 ",[34,520753,520755,520796],{"className":520754},[31642],[34,520756,520758],{"className":520757},[31646],[31648,520759,520760],{"xmlns":31650},[31652,520761,520762,520794],{},[31655,520763,520764,520766,520768,520770,520772,520774,520776,520778,520780,520782,520784,520786,520788,520790,520792],{},[31662,520765,456],{"stretchy":702},[31658,520767,111],{},[31662,520769,223],{},[31658,520771,106063],{},[31662,520773,2557],{"stretchy":702},[31662,520775,456],{"stretchy":702},[31658,520777,1697],{},[31662,520779,223],{},[31658,520781,106063],{},[31662,520783,2557],{"stretchy":702},[31662,520785,202],{},[31658,520787,111],{},[31658,520789,1697],{},[31662,520791,223],{},[31658,520793,106063],{},[31665,520795,446817],{"encoding":31667},[34,520797,520799,520820,520847,520868,520886],{"className":520798,"ariaHidden":758},[31672],[34,520800,520802,520805,520808,520811,520814,520817],{"className":520801},[31676],[34,520803],{"className":520804,"style":31774},[31680],[34,520806,456],{"className":520807},[31782],[34,520809,111],{"className":520810},[31685,31686],[34,520812],{"className":520813,"style":32604},[31845],[34,520815,223],{"className":520816},[32024],[34,520818],{"className":520819,"style":32604},[31845],[34,520821,520823,520826,520829,520832,520835,520838,520841,520844],{"className":520822},[31676],[34,520824],{"className":520825,"style":31774},[31680],[34,520827,106063],{"className":520828,"style":67482},[31685,31686],[34,520830,2557],{"className":520831},[31690],[34,520833,456],{"className":520834},[31782],[34,520836,1697],{"className":520837},[31685,31686],[34,520839],{"className":520840,"style":32604},[31845],[34,520842,223],{"className":520843},[32024],[34,520845],{"className":520846,"style":32604},[31845],[34,520848,520850,520853,520856,520859,520862,520865],{"className":520849},[31676],[34,520851],{"className":520852,"style":31774},[31680],[34,520854,106063],{"className":520855,"style":67482},[31685,31686],[34,520857,2557],{"className":520858},[31690],[34,520860],{"className":520861,"style":31846},[31845],[34,520863,202],{"className":520864},[31850],[34,520866],{"className":520867,"style":31846},[31845],[34,520869,520871,520874,520877,520880,520883],{"className":520870},[31676],[34,520872],{"className":520873,"style":34299},[31680],[34,520875,324191],{"className":520876},[31685,31686],[34,520878],{"className":520879,"style":32604},[31845],[34,520881,223],{"className":520882},[32024],[34,520884],{"className":520885,"style":32604},[31845],[34,520887,520889,520892],{"className":520888},[31676],[34,520890],{"className":520891,"style":33218},[31680],[34,520893,106063],{"className":520894,"style":67482},[31685,31686],"。证明 ",[34,520897,520899,520929],{"className":520898},[31642],[34,520900,520902],{"className":520901},[31646],[31648,520903,520904],{"xmlns":31650},[31652,520905,520906,520926],{},[31655,520907,520908,520910,520912,520914,520916,520918,520920,520922,520924],{},[31662,520909,456],{"stretchy":702},[31658,520911,34386],{},[31658,520913,38080],{"mathvariant":38079},[31658,520915,106063],{},[31662,520917,31960],{"separator":758},[31662,520919,223],{},[31662,520921,31960],{"separator":758},[31662,520923,71191],{},[31662,520925,2557],{"stretchy":702},[31665,520927,520928],{"encoding":31667},"(R\u002FI, +, \\cdot)",[34,520930,520932],{"className":520931,"ariaHidden":758},[31672],[34,520933,520935,520938,520941,520944,520947,520950,520953,520956,520959,520962,520965,520968],{"className":520934},[31676],[34,520936],{"className":520937,"style":31774},[31680],[34,520939,456],{"className":520940},[31782],[34,520942,34386],{"className":520943,"style":34427},[31685,31686],[34,520945,38080],{"className":520946},[31685],[34,520948,106063],{"className":520949,"style":67482},[31685,31686],[34,520951,31960],{"className":520952},[32119],[34,520954],{"className":520955,"style":32123},[31845],[34,520957,223],{"className":520958},[31685],[34,520960,31960],{"className":520961},[32119],[34,520963],{"className":520964,"style":32123},[31845],[34,520966,71191],{"className":520967},[31685],[34,520969,2557],{"className":520970},[31690]," 是环。",[15,520973,520974],{},[2627,520975,463462],{},[15,520977,520978],{},"这是最繁琐但也最重要的一题。",[105,520980,520981],{},[108,520982,520983,520986,520987,521025,521026,521028,521029,258,521085,113124,521141,521144,521145,521173,521174,521202],{},[2627,520984,520985],{},"良定义（Well-defined）","：这是初学者最容易晕的地方。因为 ",[34,520988,520990,521007],{"className":520989},[31642],[34,520991,520993],{"className":520992},[31646],[31648,520994,520995],{"xmlns":31650},[31652,520996,520997,521005],{},[31655,520998,520999,521001,521003],{},[31658,521000,34386],{},[31658,521002,38080],{"mathvariant":38079},[31658,521004,106063],{},[31665,521006,337927],{"encoding":31667},[34,521008,521010],{"className":521009,"ariaHidden":758},[31672],[34,521011,521013,521016,521019,521022],{"className":521012},[31676],[34,521014],{"className":521015,"style":31774},[31680],[34,521017,34386],{"className":521018,"style":34427},[31685,31686],[34,521020,38080],{"className":521021},[31685],[34,521023,106063],{"className":521024,"style":67482},[31685,31686]," 中的元素是",[2627,521027,149746],{},"（陪集）。比如在整数模6中， ",[34,521030,521032,521052],{"className":521031},[31642],[34,521033,521035],{"className":521034},[31646],[31648,521036,521037],{"xmlns":31650},[31652,521038,521039,521049],{},[31655,521040,521041,521043,521045,521047],{},[31749,521042,768],{},[31662,521044,223],{},[31749,521046,36656],{},[31658,521048,97300],{"mathvariant":318416},[31665,521050,521051],{"encoding":31667},"2+6\\mathbb{Z}",[34,521053,521055,521073],{"className":521054,"ariaHidden":758},[31672],[34,521056,521058,521061,521064,521067,521070],{"className":521057},[31676],[34,521059],{"className":521060,"style":35942},[31680],[34,521062,768],{"className":521063},[31685],[34,521065],{"className":521066,"style":32604},[31845],[34,521068,223],{"className":521069},[32024],[34,521071],{"className":521072,"style":32604},[31845],[34,521074,521076,521079,521082],{"className":521075},[31676],[34,521077],{"className":521078,"style":320927},[31680],[34,521080,36656],{"className":521081},[31685],[34,521083,97300],{"className":521084},[31685,318441],[34,521086,521088,521108],{"className":521087},[31642],[34,521089,521091],{"className":521090},[31646],[31648,521092,521093],{"xmlns":31650},[31652,521094,521095,521105],{},[31655,521096,521097,521099,521101,521103],{},[31749,521098,60525],{},[31662,521100,223],{},[31749,521102,36656],{},[31658,521104,97300],{"mathvariant":318416},[31665,521106,521107],{"encoding":31667},"8+6\\mathbb{Z}",[34,521109,521111,521129],{"className":521110,"ariaHidden":758},[31672],[34,521112,521114,521117,521120,521123,521126],{"className":521113},[31676],[34,521115],{"className":521116,"style":35942},[31680],[34,521118,60525],{"className":521119},[31685],[34,521121],{"className":521122,"style":32604},[31845],[34,521124,223],{"className":521125},[32024],[34,521127],{"className":521128,"style":32604},[31845],[34,521130,521132,521135,521138],{"className":521131},[31676],[34,521133],{"className":521134,"style":320927},[31680],[34,521136,36656],{"className":521137},[31685],[34,521139,97300],{"className":521140},[31685,318441],[2627,521142,521143],{},"同一个","元素。我们用 ",[34,521146,521148,521161],{"className":521147},[31642],[34,521149,521151],{"className":521150},[31646],[31648,521152,521153],{"xmlns":31650},[31652,521154,521155,521159],{},[31655,521156,521157],{},[31749,521158,768],{},[31665,521160,768],{"encoding":31667},[34,521162,521164],{"className":521163,"ariaHidden":758},[31672],[34,521165,521167,521170],{"className":521166},[31676],[34,521168],{"className":521169,"style":32617},[31680],[34,521171,768],{"className":521172},[31685]," 去乘，或者用 ",[34,521175,521177,521190],{"className":521176},[31642],[34,521178,521180],{"className":521179},[31646],[31648,521181,521182],{"xmlns":31650},[31652,521183,521184,521188],{},[31655,521185,521186],{},[31749,521187,60525],{},[31665,521189,60525],{"encoding":31667},[34,521191,521193],{"className":521192,"ariaHidden":758},[31672],[34,521194,521196,521199],{"className":521195},[31676],[34,521197],{"className":521198,"style":32617},[31680],[34,521200,60525],{"className":521201},[31685]," 去乘，结果必须一样，否则这个运算就是瞎胡闹。",[15,521204,521205],{},[2627,521206,463779],{},[15,521208,521209],{},[2627,521210,521211],{},"(i) 乘法是良定义的 (Well-defined)",[105,521213,521214,521219,521754,522055,522625,522959],{},[108,521215,521216,521218],{},[2627,521217,233757],{},"：如果我们换了代表元，乘积出来的陪集不变。",[108,521220,521221,34554,521223,86674,521351,344348,521479,46528,521585,70884,521691,347],{},[2627,521222,362483],{},[34,521224,521226,521256],{"className":521225},[31642],[34,521227,521229],{"className":521228},[31646],[31648,521230,521231],{"xmlns":31650},[31652,521232,521233,521253],{},[31655,521234,521235,521237,521239,521241,521243,521249,521251],{},[31658,521236,111],{},[31662,521238,223],{},[31658,521240,106063],{},[31662,521242,202],{},[37624,521244,521245,521247],{},[31658,521246,111],{},[31662,521248,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,521250,223],{},[31658,521252,106063],{},[31665,521254,521255],{"encoding":31667},"a+I = a'+I",[34,521257,521259,521277,521295,521342],{"className":521258,"ariaHidden":758},[31672],[34,521260,521262,521265,521268,521271,521274],{"className":521261},[31676],[34,521263],{"className":521264,"style":32597},[31680],[34,521266,111],{"className":521267},[31685,31686],[34,521269],{"className":521270,"style":32604},[31845],[34,521272,223],{"className":521273},[32024],[34,521275],{"className":521276,"style":32604},[31845],[34,521278,521280,521283,521286,521289,521292],{"className":521279},[31676],[34,521281],{"className":521282,"style":33218},[31680],[34,521284,106063],{"className":521285,"style":67482},[31685,31686],[34,521287],{"className":521288,"style":31846},[31845],[34,521290,202],{"className":521291},[31850],[34,521293],{"className":521294,"style":31846},[31845],[34,521296,521298,521301,521333,521336,521339],{"className":521297},[31676],[34,521299],{"className":521300,"style":384364},[31680],[34,521302,521304,521307],{"className":521303},[31685],[34,521305,111],{"className":521306},[31685,31686],[34,521308,521310],{"className":521309},[31792],[34,521311,521313],{"className":521312},[31796],[34,521314,521316],{"className":521315},[31801],[34,521317,521319],{"className":521318,"style":132027},[31805],[34,521320,521321,521324],{"style":37777},[34,521322],{"className":521323,"style":31814},[31813],[34,521325,521327],{"className":521326},[31818,31819,31820,31821],[34,521328,521330],{"className":521329},[31685,31821],[34,521331,116495],{"className":521332},[31685,31821],[34,521334],{"className":521335,"style":32604},[31845],[34,521337,223],{"className":521338},[32024],[34,521340],{"className":521341,"style":32604},[31845],[34,521343,521345,521348],{"className":521344},[31676],[34,521346],{"className":521347,"style":33218},[31680],[34,521349,106063],{"className":521350,"style":67482},[31685,31686],[34,521352,521354,521384],{"className":521353},[31642],[34,521355,521357],{"className":521356},[31646],[31648,521358,521359],{"xmlns":31650},[31652,521360,521361,521381],{},[31655,521362,521363,521365,521367,521369,521371,521377,521379],{},[31658,521364,1697],{},[31662,521366,223],{},[31658,521368,106063],{},[31662,521370,202],{},[37624,521372,521373,521375],{},[31658,521374,1697],{},[31662,521376,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,521378,223],{},[31658,521380,106063],{},[31665,521382,521383],{"encoding":31667},"b+I = b'+I",[34,521385,521387,521405,521423,521470],{"className":521386,"ariaHidden":758},[31672],[34,521388,521390,521393,521396,521399,521402],{"className":521389},[31676],[34,521391],{"className":521392,"style":34299},[31680],[34,521394,1697],{"className":521395},[31685,31686],[34,521397],{"className":521398,"style":32604},[31845],[34,521400,223],{"className":521401},[32024],[34,521403],{"className":521404,"style":32604},[31845],[34,521406,521408,521411,521414,521417,521420],{"className":521407},[31676],[34,521409],{"className":521410,"style":33218},[31680],[34,521412,106063],{"className":521413,"style":67482},[31685,31686],[34,521415],{"className":521416,"style":31846},[31845],[34,521418,202],{"className":521419},[31850],[34,521421],{"className":521422,"style":31846},[31845],[34,521424,521426,521429,521461,521464,521467],{"className":521425},[31676],[34,521427],{"className":521428,"style":384364},[31680],[34,521430,521432,521435],{"className":521431},[31685],[34,521433,1697],{"className":521434},[31685,31686],[34,521436,521438],{"className":521437},[31792],[34,521439,521441],{"className":521440},[31796],[34,521442,521444],{"className":521443},[31801],[34,521445,521447],{"className":521446,"style":132027},[31805],[34,521448,521449,521452],{"style":37777},[34,521450],{"className":521451,"style":31814},[31813],[34,521453,521455],{"className":521454},[31818,31819,31820,31821],[34,521456,521458],{"className":521457},[31685,31821],[34,521459,116495],{"className":521460},[31685,31821],[34,521462],{"className":521463,"style":32604},[31845],[34,521465,223],{"className":521466},[32024],[34,521468],{"className":521469,"style":32604},[31845],[34,521471,521473,521476],{"className":521472},[31676],[34,521474],{"className":521475,"style":33218},[31680],[34,521477,106063],{"className":521478,"style":67482},[31685,31686],[34,521480,521482,521508],{"className":521481},[31642],[34,521483,521485],{"className":521484},[31646],[31648,521486,521487],{"xmlns":31650},[31652,521488,521489,521505],{},[31655,521490,521491,521497,521499,521501,521503],{},[37624,521492,521493,521495],{},[31658,521494,111],{},[31662,521496,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,521498,202],{},[31658,521500,111],{},[31662,521502,223],{},[31658,521504,32936],{},[31665,521506,521507],{"encoding":31667},"a' = a + x",[34,521509,521511,521558,521576],{"className":521510,"ariaHidden":758},[31672],[34,521512,521514,521517,521549,521552,521555],{"className":521513},[31676],[34,521515],{"className":521516,"style":132027},[31680],[34,521518,521520,521523],{"className":521519},[31685],[34,521521,111],{"className":521522},[31685,31686],[34,521524,521526],{"className":521525},[31792],[34,521527,521529],{"className":521528},[31796],[34,521530,521532],{"className":521531},[31801],[34,521533,521535],{"className":521534,"style":132027},[31805],[34,521536,521537,521540],{"style":37777},[34,521538],{"className":521539,"style":31814},[31813],[34,521541,521543],{"className":521542},[31818,31819,31820,31821],[34,521544,521546],{"className":521545},[31685,31821],[34,521547,116495],{"className":521548},[31685,31821],[34,521550],{"className":521551,"style":31846},[31845],[34,521553,202],{"className":521554},[31850],[34,521556],{"className":521557,"style":31846},[31845],[34,521559,521561,521564,521567,521570,521573],{"className":521560},[31676],[34,521562],{"className":521563,"style":32597},[31680],[34,521565,111],{"className":521566},[31685,31686],[34,521568],{"className":521569,"style":32604},[31845],[34,521571,223],{"className":521572},[32024],[34,521574],{"className":521575,"style":32604},[31845],[34,521577,521579,521582],{"className":521578},[31676],[34,521580],{"className":521581,"style":32527},[31680],[34,521583,32936],{"className":521584},[31685,31686],[34,521586,521588,521614],{"className":521587},[31642],[34,521589,521591],{"className":521590},[31646],[31648,521592,521593],{"xmlns":31650},[31652,521594,521595,521611],{},[31655,521596,521597,521603,521605,521607,521609],{},[37624,521598,521599,521601],{},[31658,521600,1697],{},[31662,521602,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,521604,202],{},[31658,521606,1697],{},[31662,521608,223],{},[31658,521610,67764],{},[31665,521612,521613],{"encoding":31667},"b' = b + y",[34,521615,521617,521664,521682],{"className":521616,"ariaHidden":758},[31672],[34,521618,521620,521623,521655,521658,521661],{"className":521619},[31676],[34,521621],{"className":521622,"style":132027},[31680],[34,521624,521626,521629],{"className":521625},[31685],[34,521627,1697],{"className":521628},[31685,31686],[34,521630,521632],{"className":521631},[31792],[34,521633,521635],{"className":521634},[31796],[34,521636,521638],{"className":521637},[31801],[34,521639,521641],{"className":521640,"style":132027},[31805],[34,521642,521643,521646],{"style":37777},[34,521644],{"className":521645,"style":31814},[31813],[34,521647,521649],{"className":521648},[31818,31819,31820,31821],[34,521650,521652],{"className":521651},[31685,31821],[34,521653,116495],{"className":521654},[31685,31821],[34,521656],{"className":521657,"style":31846},[31845],[34,521659,202],{"className":521660},[31850],[34,521662],{"className":521663,"style":31846},[31845],[34,521665,521667,521670,521673,521676,521679],{"className":521666},[31676],[34,521668],{"className":521669,"style":34299},[31680],[34,521671,1697],{"className":521672},[31685,31686],[34,521674],{"className":521675,"style":32604},[31845],[34,521677,223],{"className":521678},[32024],[34,521680],{"className":521681,"style":32604},[31845],[34,521683,521685,521688],{"className":521684},[31676],[34,521686],{"className":521687,"style":38774},[31680],[34,521689,67764],{"className":521690,"style":32062},[31685,31686],[34,521692,521694,521715],{"className":521693},[31642],[34,521695,521697],{"className":521696},[31646],[31648,521698,521699],{"xmlns":31650},[31652,521700,521701,521713],{},[31655,521702,521703,521705,521707,521709,521711],{},[31658,521704,32936],{},[31662,521706,31960],{"separator":758},[31658,521708,67764],{},[31662,521710,112296],{},[31658,521712,106063],{},[31665,521714,510980],{"encoding":31667},[34,521716,521718,521745],{"className":521717,"ariaHidden":758},[31672],[34,521719,521721,521724,521727,521730,521733,521736,521739,521742],{"className":521720},[31676],[34,521722],{"className":521723,"style":86652},[31680],[34,521725,32936],{"className":521726},[31685,31686],[34,521728,31960],{"className":521729},[32119],[34,521731],{"className":521732,"style":32123},[31845],[34,521734,67764],{"className":521735,"style":32062},[31685,31686],[34,521737],{"className":521738,"style":31846},[31845],[34,521740,112296],{"className":521741},[31850],[34,521743],{"className":521744,"style":31846},[31845],[34,521746,521748,521751],{"className":521747},[31676],[34,521749],{"className":521750,"style":33218},[31680],[34,521752,106063],{"className":521753,"style":67482},[31685,31686],[108,521755,521756,33183,521759],{},[2627,521757,521758],{},"计算新代表元的乘积",[34,521760,521762,521832],{"className":521761},[31642],[34,521763,521765],{"className":521764},[31646],[31648,521766,521767],{"xmlns":31650},[31652,521768,521769,521829],{},[31655,521770,521771,521777,521783,521785,521787,521789,521791,521793,521795,521797,521799,521801,521803,521805,521807,521809,521811,521813,521815,521817,521819,521821,521823,521825,521827],{},[37624,521772,521773,521775],{},[31658,521774,111],{},[31662,521776,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,521778,521779,521781],{},[31658,521780,1697],{},[31662,521782,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,521784,202],{},[31662,521786,456],{"stretchy":702},[31658,521788,111],{},[31662,521790,223],{},[31658,521792,32936],{},[31662,521794,2557],{"stretchy":702},[31662,521796,456],{"stretchy":702},[31658,521798,1697],{},[31662,521800,223],{},[31658,521802,67764],{},[31662,521804,2557],{"stretchy":702},[31662,521806,202],{},[31658,521808,111],{},[31658,521810,1697],{},[31662,521812,223],{},[31658,521814,111],{},[31658,521816,67764],{},[31662,521818,223],{},[31658,521820,32936],{},[31658,521822,1697],{},[31662,521824,223],{},[31658,521826,32936],{},[31658,521828,67764],{},[31665,521830,521831],{"encoding":31667},"a'b' = (a+x)(b+y) = ab + ay + xb + xy",[34,521833,521835,521914,521935,521962,521983,522001,522022,522043],{"className":521834,"ariaHidden":758},[31672],[34,521836,521838,521841,521873,521905,521908,521911],{"className":521837},[31676],[34,521839],{"className":521840,"style":132027},[31680],[34,521842,521844,521847],{"className":521843},[31685],[34,521845,111],{"className":521846},[31685,31686],[34,521848,521850],{"className":521849},[31792],[34,521851,521853],{"className":521852},[31796],[34,521854,521856],{"className":521855},[31801],[34,521857,521859],{"className":521858,"style":132027},[31805],[34,521860,521861,521864],{"style":37777},[34,521862],{"className":521863,"style":31814},[31813],[34,521865,521867],{"className":521866},[31818,31819,31820,31821],[34,521868,521870],{"className":521869},[31685,31821],[34,521871,116495],{"className":521872},[31685,31821],[34,521874,521876,521879],{"className":521875},[31685],[34,521877,1697],{"className":521878},[31685,31686],[34,521880,521882],{"className":521881},[31792],[34,521883,521885],{"className":521884},[31796],[34,521886,521888],{"className":521887},[31801],[34,521889,521891],{"className":521890,"style":132027},[31805],[34,521892,521893,521896],{"style":37777},[34,521894],{"className":521895,"style":31814},[31813],[34,521897,521899],{"className":521898},[31818,31819,31820,31821],[34,521900,521902],{"className":521901},[31685,31821],[34,521903,116495],{"className":521904},[31685,31821],[34,521906],{"className":521907,"style":31846},[31845],[34,521909,202],{"className":521910},[31850],[34,521912],{"className":521913,"style":31846},[31845],[34,521915,521917,521920,521923,521926,521929,521932],{"className":521916},[31676],[34,521918],{"className":521919,"style":31774},[31680],[34,521921,456],{"className":521922},[31782],[34,521924,111],{"className":521925},[31685,31686],[34,521927],{"className":521928,"style":32604},[31845],[34,521930,223],{"className":521931},[32024],[34,521933],{"className":521934,"style":32604},[31845],[34,521936,521938,521941,521944,521947,521950,521953,521956,521959],{"className":521937},[31676],[34,521939],{"className":521940,"style":31774},[31680],[34,521942,32936],{"className":521943},[31685,31686],[34,521945,2557],{"className":521946},[31690],[34,521948,456],{"className":521949},[31782],[34,521951,1697],{"className":521952},[31685,31686],[34,521954],{"className":521955,"style":32604},[31845],[34,521957,223],{"className":521958},[32024],[34,521960],{"className":521961,"style":32604},[31845],[34,521963,521965,521968,521971,521974,521977,521980],{"className":521964},[31676],[34,521966],{"className":521967,"style":31774},[31680],[34,521969,67764],{"className":521970,"style":32062},[31685,31686],[34,521972,2557],{"className":521973},[31690],[34,521975],{"className":521976,"style":31846},[31845],[34,521978,202],{"className":521979},[31850],[34,521981],{"className":521982,"style":31846},[31845],[34,521984,521986,521989,521992,521995,521998],{"className":521985},[31676],[34,521987],{"className":521988,"style":34299},[31680],[34,521990,324191],{"className":521991},[31685,31686],[34,521993],{"className":521994,"style":32604},[31845],[34,521996,223],{"className":521997},[32024],[34,521999],{"className":522000,"style":32604},[31845],[34,522002,522004,522007,522010,522013,522016,522019],{"className":522003},[31676],[34,522005],{"className":522006,"style":55949},[31680],[34,522008,111],{"className":522009},[31685,31686],[34,522011,67764],{"className":522012,"style":32062},[31685,31686],[34,522014],{"className":522015,"style":32604},[31845],[34,522017,223],{"className":522018},[32024],[34,522020],{"className":522021,"style":32604},[31845],[34,522023,522025,522028,522031,522034,522037,522040],{"className":522024},[31676],[34,522026],{"className":522027,"style":34299},[31680],[34,522029,32936],{"className":522030},[31685,31686],[34,522032,1697],{"className":522033},[31685,31686],[34,522035],{"className":522036,"style":32604},[31845],[34,522038,223],{"className":522039},[32024],[34,522041],{"className":522042,"style":32604},[31845],[34,522044,522046,522049,522052],{"className":522045},[31676],[34,522047],{"className":522048,"style":38774},[31680],[34,522050,32936],{"className":522051},[31685,31686],[34,522053,67764],{"className":522054,"style":32062},[31685,31686],[108,522056,522057,33183,522060],{},[2627,522058,522059],{},"分析各项",[105,522061,522062,522095,522268,522441],{},[108,522063,522064,522094],{},[34,522065,522067,522082],{"className":522066},[31642],[34,522068,522070],{"className":522069},[31646],[31648,522071,522072],{"xmlns":31650},[31652,522073,522074,522080],{},[31655,522075,522076,522078],{},[31658,522077,111],{},[31658,522079,1697],{},[31665,522081,324191],{"encoding":31667},[34,522083,522085],{"className":522084,"ariaHidden":758},[31672],[34,522086,522088,522091],{"className":522087},[31676],[34,522089],{"className":522090,"style":31681},[31680],[34,522092,324191],{"className":522093},[31685,31686],"：这是原来的乘积。",[108,522096,522097,332541,522131,86674,522181,113124,522209,13977,522212,347],{},[34,522098,522100,522116],{"className":522099},[31642],[34,522101,522103],{"className":522102},[31646],[31648,522104,522105],{"xmlns":31650},[31652,522106,522107,522113],{},[31655,522108,522109,522111],{},[31658,522110,111],{},[31658,522112,67764],{},[31665,522114,522115],{"encoding":31667},"ay",[34,522117,522119],{"className":522118,"ariaHidden":758},[31672],[34,522120,522122,522125,522128],{"className":522121},[31676],[34,522123],{"className":522124,"style":38774},[31680],[34,522126,111],{"className":522127},[31685,31686],[34,522129,67764],{"className":522130,"style":32062},[31685,31686],[34,522132,522134,522151],{"className":522133},[31642],[34,522135,522137],{"className":522136},[31646],[31648,522138,522139],{"xmlns":31650},[31652,522140,522141,522149],{},[31655,522142,522143,522145,522147],{},[31658,522144,67764],{},[31662,522146,112296],{},[31658,522148,106063],{},[31665,522150,354299],{"encoding":31667},[34,522152,522154,522172],{"className":522153,"ariaHidden":758},[31672],[34,522155,522157,522160,522163,522166,522169],{"className":522156},[31676],[34,522158],{"className":522159,"style":86652},[31680],[34,522161,67764],{"className":522162,"style":32062},[31685,31686],[34,522164],{"className":522165,"style":31846},[31845],[34,522167,112296],{"className":522168},[31850],[34,522170],{"className":522171,"style":31846},[31845],[34,522173,522175,522178],{"className":522174},[31676],[34,522176],{"className":522177,"style":33218},[31680],[34,522179,106063],{"className":522180,"style":67482},[31685,31686],[34,522182,522184,522197],{"className":522183},[31642],[34,522185,522187],{"className":522186},[31646],[31648,522188,522189],{"xmlns":31650},[31652,522190,522191,522195],{},[31655,522192,522193],{},[31658,522194,106063],{},[31665,522196,106063],{"encoding":31667},[34,522198,522200],{"className":522199,"ariaHidden":758},[31672],[34,522201,522203,522206],{"className":522202},[31676],[34,522204],{"className":522205,"style":33218},[31680],[34,522207,106063],{"className":522208,"style":67482},[31685,31686],[2627,522210,522211],{},"左理想",[34,522213,522215,522235],{"className":522214},[31642],[34,522216,522218],{"className":522217},[31646],[31648,522219,522220],{"xmlns":31650},[31652,522221,522222,522232],{},[31655,522223,522224,522226,522228,522230],{},[31658,522225,111],{},[31658,522227,67764],{},[31662,522229,112296],{},[31658,522231,106063],{},[31665,522233,522234],{"encoding":31667},"ay \\in I",[34,522236,522238,522259],{"className":522237,"ariaHidden":758},[31672],[34,522239,522241,522244,522247,522250,522253,522256],{"className":522240},[31676],[34,522242],{"className":522243,"style":86652},[31680],[34,522245,111],{"className":522246},[31685,31686],[34,522248,67764],{"className":522249,"style":32062},[31685,31686],[34,522251],{"className":522252,"style":31846},[31845],[34,522254,112296],{"className":522255},[31850],[34,522257],{"className":522258,"style":31846},[31845],[34,522260,522262,522265],{"className":522261},[31676],[34,522263],{"className":522264,"style":33218},[31680],[34,522266,106063],{"className":522267,"style":67482},[31685,31686],[108,522269,522270,332541,522304,86674,522354,113124,522382,13977,522385,347],{},[34,522271,522273,522289],{"className":522272},[31642],[34,522274,522276],{"className":522275},[31646],[31648,522277,522278],{"xmlns":31650},[31652,522279,522280,522286],{},[31655,522281,522282,522284],{},[31658,522283,32936],{},[31658,522285,1697],{},[31665,522287,522288],{"encoding":31667},"xb",[34,522290,522292],{"className":522291,"ariaHidden":758},[31672],[34,522293,522295,522298,522301],{"className":522294},[31676],[34,522296],{"className":522297,"style":31681},[31680],[34,522299,32936],{"className":522300},[31685,31686],[34,522302,1697],{"className":522303},[31685,31686],[34,522305,522307,522324],{"className":522306},[31642],[34,522308,522310],{"className":522309},[31646],[31648,522311,522312],{"xmlns":31650},[31652,522313,522314,522322],{},[31655,522315,522316,522318,522320],{},[31658,522317,32936],{},[31662,522319,112296],{},[31658,522321,106063],{},[31665,522323,352414],{"encoding":31667},[34,522325,522327,522345],{"className":522326,"ariaHidden":758},[31672],[34,522328,522330,522333,522336,522339,522342],{"className":522329},[31676],[34,522331],{"className":522332,"style":34812},[31680],[34,522334,32936],{"className":522335},[31685,31686],[34,522337],{"className":522338,"style":31846},[31845],[34,522340,112296],{"className":522341},[31850],[34,522343],{"className":522344,"style":31846},[31845],[34,522346,522348,522351],{"className":522347},[31676],[34,522349],{"className":522350,"style":33218},[31680],[34,522352,106063],{"className":522353,"style":67482},[31685,31686],[34,522355,522357,522370],{"className":522356},[31642],[34,522358,522360],{"className":522359},[31646],[31648,522361,522362],{"xmlns":31650},[31652,522363,522364,522368],{},[31655,522365,522366],{},[31658,522367,106063],{},[31665,522369,106063],{"encoding":31667},[34,522371,522373],{"className":522372,"ariaHidden":758},[31672],[34,522374,522376,522379],{"className":522375},[31676],[34,522377],{"className":522378,"style":33218},[31680],[34,522380,106063],{"className":522381,"style":67482},[31685,31686],[2627,522383,522384],{},"右理想",[34,522386,522388,522408],{"className":522387},[31642],[34,522389,522391],{"className":522390},[31646],[31648,522392,522393],{"xmlns":31650},[31652,522394,522395,522405],{},[31655,522396,522397,522399,522401,522403],{},[31658,522398,32936],{},[31658,522400,1697],{},[31662,522402,112296],{},[31658,522404,106063],{},[31665,522406,522407],{"encoding":31667},"xb \\in I",[34,522409,522411,522432],{"className":522410,"ariaHidden":758},[31672],[34,522412,522414,522417,522420,522423,522426,522429],{"className":522413},[31676],[34,522415],{"className":522416,"style":46991},[31680],[34,522418,32936],{"className":522419},[31685,31686],[34,522421,1697],{"className":522422},[31685,31686],[34,522424],{"className":522425,"style":31846},[31845],[34,522427,112296],{"className":522428},[31850],[34,522430],{"className":522431,"style":31846},[31845],[34,522433,522435,522438],{"className":522434},[31676],[34,522436],{"className":522437,"style":33218},[31680],[34,522439,106063],{"className":522440,"style":67482},[31685,31686],[108,522442,522443,332541,522477,86674,522540,522568,522569,347],{},[34,522444,522446,522462],{"className":522445},[31642],[34,522447,522449],{"className":522448},[31646],[31648,522450,522451],{"xmlns":31650},[31652,522452,522453,522459],{},[31655,522454,522455,522457],{},[31658,522456,32936],{},[31658,522458,67764],{},[31665,522460,522461],{"encoding":31667},"xy",[34,522463,522465],{"className":522464,"ariaHidden":758},[31672],[34,522466,522468,522471,522474],{"className":522467},[31676],[34,522469],{"className":522470,"style":38774},[31680],[34,522472,32936],{"className":522473},[31685,31686],[34,522475,67764],{"className":522476,"style":32062},[31685,31686],[34,522478,522480,522501],{"className":522479},[31642],[34,522481,522483],{"className":522482},[31646],[31648,522484,522485],{"xmlns":31650},[31652,522486,522487,522499],{},[31655,522488,522489,522491,522493,522495,522497],{},[31658,522490,32936],{},[31662,522492,31960],{"separator":758},[31658,522494,67764],{},[31662,522496,112296],{},[31658,522498,106063],{},[31665,522500,510980],{"encoding":31667},[34,522502,522504,522531],{"className":522503,"ariaHidden":758},[31672],[34,522505,522507,522510,522513,522516,522519,522522,522525,522528],{"className":522506},[31676],[34,522508],{"className":522509,"style":86652},[31680],[34,522511,32936],{"className":522512},[31685,31686],[34,522514,31960],{"className":522515},[32119],[34,522517],{"className":522518,"style":32123},[31845],[34,522520,67764],{"className":522521,"style":32062},[31685,31686],[34,522523],{"className":522524,"style":31846},[31845],[34,522526,112296],{"className":522527},[31850],[34,522529],{"className":522530,"style":31846},[31845],[34,522532,522534,522537],{"className":522533},[31676],[34,522535],{"className":522536,"style":33218},[31680],[34,522538,106063],{"className":522539,"style":67482},[31685,31686],[34,522541,522543,522556],{"className":522542},[31642],[34,522544,522546],{"className":522545},[31646],[31648,522547,522548],{"xmlns":31650},[31652,522549,522550,522554],{},[31655,522551,522552],{},[31658,522553,106063],{},[31665,522555,106063],{"encoding":31667},[34,522557,522559],{"className":522558,"ariaHidden":758},[31672],[34,522560,522562,522565],{"className":522561},[31676],[34,522563],{"className":522564,"style":33218},[31680],[34,522566,106063],{"className":522567,"style":67482},[31685,31686]," 是子环，所以 ",[34,522570,522572,522592],{"className":522571},[31642],[34,522573,522575],{"className":522574},[31646],[31648,522576,522577],{"xmlns":31650},[31652,522578,522579,522589],{},[31655,522580,522581,522583,522585,522587],{},[31658,522582,32936],{},[31658,522584,67764],{},[31662,522586,112296],{},[31658,522588,106063],{},[31665,522590,522591],{"encoding":31667},"xy \\in I",[34,522593,522595,522616],{"className":522594,"ariaHidden":758},[31672],[34,522596,522598,522601,522604,522607,522610,522613],{"className":522597},[31676],[34,522599],{"className":522600,"style":86652},[31680],[34,522602,32936],{"className":522603},[31685,31686],[34,522605,67764],{"className":522606,"style":32062},[31685,31686],[34,522608],{"className":522609,"style":31846},[31845],[34,522611,112296],{"className":522612},[31850],[34,522614],{"className":522615,"style":31846},[31845],[34,522617,522619,522622],{"className":522618},[31676],[34,522620],{"className":522621,"style":33218},[31680],[34,522623,106063],{"className":522624,"style":67482},[31685,31686],[108,522626,522627,33183,522630,522812,522813,347],{},[2627,522628,522629],{},"归纳",[34,522631,522633,522677],{"className":522632},[31642],[34,522634,522636],{"className":522635},[31646],[31648,522637,522638],{"xmlns":31650},[31652,522639,522640,522674],{},[31655,522641,522642,522648,522654,522656,522658,522660,522662,522664,522667,522669,522672],{},[37624,522643,522644,522646],{},[31658,522645,111],{},[31662,522647,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,522649,522650,522652],{},[31658,522651,1697],{},[31662,522653,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,522655,202],{},[31658,522657,111],{},[31658,522659,1697],{},[31662,522661,223],{},[31662,522663,456],{"stretchy":702},[39361,522665,522666],{},"一堆属于 ",[31658,522668,106063],{},[39361,522670,522671],{}," 的东西",[31662,522673,2557],{"stretchy":702},[31665,522675,522676],{"encoding":31667},"a'b' = ab + (\\text{一堆属于 } I \\text{ 的东西})",[34,522678,522680,522759,522777],{"className":522679,"ariaHidden":758},[31672],[34,522681,522683,522686,522718,522750,522753,522756],{"className":522682},[31676],[34,522684],{"className":522685,"style":132027},[31680],[34,522687,522689,522692],{"className":522688},[31685],[34,522690,111],{"className":522691},[31685,31686],[34,522693,522695],{"className":522694},[31792],[34,522696,522698],{"className":522697},[31796],[34,522699,522701],{"className":522700},[31801],[34,522702,522704],{"className":522703,"style":132027},[31805],[34,522705,522706,522709],{"style":37777},[34,522707],{"className":522708,"style":31814},[31813],[34,522710,522712],{"className":522711},[31818,31819,31820,31821],[34,522713,522715],{"className":522714},[31685,31821],[34,522716,116495],{"className":522717},[31685,31821],[34,522719,522721,522724],{"className":522720},[31685],[34,522722,1697],{"className":522723},[31685,31686],[34,522725,522727],{"className":522726},[31792],[34,522728,522730],{"className":522729},[31796],[34,522731,522733],{"className":522732},[31801],[34,522734,522736],{"className":522735,"style":132027},[31805],[34,522737,522738,522741],{"style":37777},[34,522739],{"className":522740,"style":31814},[31813],[34,522742,522744],{"className":522743},[31818,31819,31820,31821],[34,522745,522747],{"className":522746},[31685,31821],[34,522748,116495],{"className":522749},[31685,31821],[34,522751],{"className":522752,"style":31846},[31845],[34,522754,202],{"className":522755},[31850],[34,522757],{"className":522758,"style":31846},[31845],[34,522760,522762,522765,522768,522771,522774],{"className":522761},[31676],[34,522763],{"className":522764,"style":34299},[31680],[34,522766,324191],{"className":522767},[31685,31686],[34,522769],{"className":522770,"style":32604},[31845],[34,522772,223],{"className":522773},[32024],[34,522775],{"className":522776,"style":32604},[31845],[34,522778,522780,522783,522786,522796,522799,522809],{"className":522779},[31676],[34,522781],{"className":522782,"style":31774},[31680],[34,522784,456],{"className":522785},[31782],[34,522787,522789,522793],{"className":522788},[31685,2638],[34,522790,522792],{"className":522791},[31685,105462],"一堆属于",[34,522794,130298],{"className":522795},[31685],[34,522797,106063],{"className":522798,"style":67482},[31685,31686],[34,522800,522802,522805],{"className":522801},[31685,2638],[34,522803,130298],{"className":522804},[31685],[34,522806,522808],{"className":522807},[31685,105462],"的东西",[34,522810,2557],{"className":522811},[31690],"\n即 ",[34,522814,522816,522850],{"className":522815},[31642],[34,522817,522819],{"className":522818},[31646],[31648,522820,522821],{"xmlns":31650},[31652,522822,522823,522847],{},[31655,522824,522825,522831,522837,522839,522841,522843,522845],{},[37624,522826,522827,522829],{},[31658,522828,111],{},[31662,522830,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,522832,522833,522835],{},[31658,522834,1697],{},[31662,522836,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,522838,32226],{},[31658,522840,111],{},[31658,522842,1697],{},[31662,522844,112296],{},[31658,522846,106063],{},[31665,522848,522849],{"encoding":31667},"a'b' - ab \\in I",[34,522851,522853,522932,522950],{"className":522852,"ariaHidden":758},[31672],[34,522854,522856,522859,522891,522923,522926,522929],{"className":522855},[31676],[34,522857],{"className":522858,"style":384364},[31680],[34,522860,522862,522865],{"className":522861},[31685],[34,522863,111],{"className":522864},[31685,31686],[34,522866,522868],{"className":522867},[31792],[34,522869,522871],{"className":522870},[31796],[34,522872,522874],{"className":522873},[31801],[34,522875,522877],{"className":522876,"style":132027},[31805],[34,522878,522879,522882],{"style":37777},[34,522880],{"className":522881,"style":31814},[31813],[34,522883,522885],{"className":522884},[31818,31819,31820,31821],[34,522886,522888],{"className":522887},[31685,31821],[34,522889,116495],{"className":522890},[31685,31821],[34,522892,522894,522897],{"className":522893},[31685],[34,522895,1697],{"className":522896},[31685,31686],[34,522898,522900],{"className":522899},[31792],[34,522901,522903],{"className":522902},[31796],[34,522904,522906],{"className":522905},[31801],[34,522907,522909],{"className":522908,"style":132027},[31805],[34,522910,522911,522914],{"style":37777},[34,522912],{"className":522913,"style":31814},[31813],[34,522915,522917],{"className":522916},[31818,31819,31820,31821],[34,522918,522920],{"className":522919},[31685,31821],[34,522921,116495],{"className":522922},[31685,31821],[34,522924],{"className":522925,"style":32604},[31845],[34,522927,32226],{"className":522928},[32024],[34,522930],{"className":522931,"style":32604},[31845],[34,522933,522935,522938,522941,522944,522947],{"className":522934},[31676],[34,522936],{"className":522937,"style":46991},[31680],[34,522939,324191],{"className":522940},[31685,31686],[34,522942],{"className":522943,"style":31846},[31845],[34,522945,112296],{"className":522946},[31850],[34,522948],{"className":522949,"style":31846},[31845],[34,522951,522953,522956],{"className":522952},[31676],[34,522954],{"className":522955,"style":33218},[31680],[34,522957,106063],{"className":522958,"style":67482},[31685,31686],[108,522960,522961,33183,522963,523131],{},[2627,522962,114043],{},[34,522964,522966,523004],{"className":522965},[31642],[34,522967,522969],{"className":522968},[31646],[31648,522970,522971],{"xmlns":31650},[31652,522972,522973,523001],{},[31655,522974,522975,522981,522987,522989,522991,522993,522995,522997,522999],{},[37624,522976,522977,522979],{},[31658,522978,111],{},[31662,522980,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,522982,522983,522985],{},[31658,522984,1697],{},[31662,522986,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,522988,223],{},[31658,522990,106063],{},[31662,522992,202],{},[31658,522994,111],{},[31658,522996,1697],{},[31662,522998,223],{},[31658,523000,106063],{},[31665,523002,523003],{"encoding":31667},"a'b' + I = ab + I",[34,523005,523007,523086,523104,523122],{"className":523006,"ariaHidden":758},[31672],[34,523008,523010,523013,523045,523077,523080,523083],{"className":523009},[31676],[34,523011],{"className":523012,"style":384364},[31680],[34,523014,523016,523019],{"className":523015},[31685],[34,523017,111],{"className":523018},[31685,31686],[34,523020,523022],{"className":523021},[31792],[34,523023,523025],{"className":523024},[31796],[34,523026,523028],{"className":523027},[31801],[34,523029,523031],{"className":523030,"style":132027},[31805],[34,523032,523033,523036],{"style":37777},[34,523034],{"className":523035,"style":31814},[31813],[34,523037,523039],{"className":523038},[31818,31819,31820,31821],[34,523040,523042],{"className":523041},[31685,31821],[34,523043,116495],{"className":523044},[31685,31821],[34,523046,523048,523051],{"className":523047},[31685],[34,523049,1697],{"className":523050},[31685,31686],[34,523052,523054],{"className":523053},[31792],[34,523055,523057],{"className":523056},[31796],[34,523058,523060],{"className":523059},[31801],[34,523061,523063],{"className":523062,"style":132027},[31805],[34,523064,523065,523068],{"style":37777},[34,523066],{"className":523067,"style":31814},[31813],[34,523069,523071],{"className":523070},[31818,31819,31820,31821],[34,523072,523074],{"className":523073},[31685,31821],[34,523075,116495],{"className":523076},[31685,31821],[34,523078],{"className":523079,"style":32604},[31845],[34,523081,223],{"className":523082},[32024],[34,523084],{"className":523085,"style":32604},[31845],[34,523087,523089,523092,523095,523098,523101],{"className":523088},[31676],[34,523090],{"className":523091,"style":33218},[31680],[34,523093,106063],{"className":523094,"style":67482},[31685,31686],[34,523096],{"className":523097,"style":31846},[31845],[34,523099,202],{"className":523100},[31850],[34,523102],{"className":523103,"style":31846},[31845],[34,523105,523107,523110,523113,523116,523119],{"className":523106},[31676],[34,523108],{"className":523109,"style":34299},[31680],[34,523111,324191],{"className":523112},[31685,31686],[34,523114],{"className":523115,"style":32604},[31845],[34,523117,223],{"className":523118},[32024],[34,523120],{"className":523121,"style":32604},[31845],[34,523123,523125,523128],{"className":523124},[31676],[34,523126],{"className":523127,"style":33218},[31680],[34,523129,106063],{"className":523130,"style":67482},[31685,31686],"。乘法与代表元选取无关，是良定义的。",[15,523133,523134],{},[2627,523135,523136,523137,523199],{},"(ii) ",[34,523138,523140,523166],{"className":523139},[31642],[34,523141,523143],{"className":523142},[31646],[31648,523144,523145],{"xmlns":31650},[31652,523146,523147,523163],{},[31655,523148,523149,523151,523153,523155,523157,523159,523161],{},[31662,523150,456],{"stretchy":702},[31658,523152,34386],{},[31658,523154,38080],{"mathvariant":38079},[31658,523156,106063],{},[31662,523158,31960],{"separator":758},[31662,523160,223],{},[31662,523162,2557],{"stretchy":702},[31665,523164,523165],{"encoding":31667},"(R\u002FI, +)",[34,523167,523169],{"className":523168,"ariaHidden":758},[31672],[34,523170,523172,523175,523178,523181,523184,523187,523190,523193,523196],{"className":523171},[31676],[34,523173],{"className":523174,"style":31774},[31680],[34,523176,456],{"className":523177},[31782],[34,523179,34386],{"className":523180,"style":34427},[31685,31686],[34,523182,38080],{"className":523183},[31685],[34,523185,106063],{"className":523186,"style":67482},[31685,31686],[34,523188,31960],{"className":523189},[32119],[34,523191],{"className":523192,"style":32123},[31845],[34,523194,223],{"className":523195},[31685],[34,523197,2557],{"className":523198},[31690]," 是阿贝尔群",[15,523201,523202,523203,11008,523231,523259],{},"这直接继承自商群的性质（回顾群论：",[34,523204,523206,523219],{"className":523205},[31642],[34,523207,523209],{"className":523208},[31646],[31648,523210,523211],{"xmlns":31650},[31652,523212,523213,523217],{},[31655,523214,523215],{},[31658,523216,106063],{},[31665,523218,106063],{"encoding":31667},[34,523220,523222],{"className":523221,"ariaHidden":758},[31672],[34,523223,523225,523228],{"className":523224},[31676],[34,523226],{"className":523227,"style":33218},[31680],[34,523229,106063],{"className":523230,"style":67482},[31685,31686],[34,523232,523234,523247],{"className":523233},[31642],[34,523235,523237],{"className":523236},[31646],[31648,523238,523239],{"xmlns":31650},[31652,523240,523241,523245],{},[31655,523242,523243],{},[31658,523244,34386],{},[31665,523246,34386],{"encoding":31667},[34,523248,523250],{"className":523249,"ariaHidden":758},[31672],[34,523251,523253,523256],{"className":523252},[31676],[34,523254],{"className":523255,"style":33218},[31680],[34,523257,34386],{"className":523258,"style":34427},[31685,31686]," 的加法正规子群）：",[2230,523261,523262,523268,523710,523793,523926],{},[108,523263,523264,523267],{},[2627,523265,523266],{},"加法封闭","：定义本身保证。",[108,523269,523270,2664,523273,347],{},[2627,523271,523272],{},"加法结合律",[34,523274,523276,523386],{"className":523275},[31642],[34,523277,523279],{"className":523278},[31646],[31648,523280,523281],{"xmlns":31650},[31652,523282,523283,523383],{},[31655,523284,523285,523287,523289,523291,523293,523295,523297,523299,523301,523303,523305,523307,523309,523311,523313,523315,523317,523319,523321,523323,523325,523327,523329,523331,523333,523335,523337,523339,523341,523343,523345,523347,523349,523351,523353,523355,523357,523359,523361,523363,523365,523367,523369,523371,523373,523375,523377,523379,523381],{},[31662,523286,456],{"stretchy":702},[31662,523288,456],{"stretchy":702},[31658,523290,111],{},[31662,523292,223],{},[31658,523294,106063],{},[31662,523296,2557],{"stretchy":702},[31662,523298,223],{},[31662,523300,456],{"stretchy":702},[31658,523302,1697],{},[31662,523304,223],{},[31658,523306,106063],{},[31662,523308,2557],{"stretchy":702},[31662,523310,2557],{"stretchy":702},[31662,523312,223],{},[31662,523314,456],{"stretchy":702},[31658,523316,31965],{},[31662,523318,223],{},[31658,523320,106063],{},[31662,523322,2557],{"stretchy":702},[31662,523324,202],{},[31662,523326,456],{"stretchy":702},[31658,523328,111],{},[31662,523330,223],{},[31658,523332,1697],{},[31662,523334,223],{},[31658,523336,31965],{},[31662,523338,2557],{"stretchy":702},[31662,523340,223],{},[31658,523342,106063],{},[31662,523344,202],{},[31662,523346,456],{"stretchy":702},[31658,523348,111],{},[31662,523350,223],{},[31658,523352,106063],{},[31662,523354,2557],{"stretchy":702},[31662,523356,223],{},[31662,523358,456],{"stretchy":702},[31662,523360,456],{"stretchy":702},[31658,523362,1697],{},[31662,523364,223],{},[31658,523366,106063],{},[31662,523368,2557],{"stretchy":702},[31662,523370,223],{},[31662,523372,456],{"stretchy":702},[31658,523374,31965],{},[31662,523376,223],{},[31658,523378,106063],{},[31662,523380,2557],{"stretchy":702},[31662,523382,2557],{"stretchy":702},[31665,523384,523385],{"encoding":31667},"((a+I)+(b+I))+(c+I) = (a+b+c)+I = (a+I)+((b+I)+(c+I))",[34,523387,523389,523410,523431,523452,523473,523494,523515,523536,523554,523575,523593,523614,523635,523656,523677,523698],{"className":523388,"ariaHidden":758},[31672],[34,523390,523392,523395,523398,523401,523404,523407],{"className":523391},[31676],[34,523393],{"className":523394,"style":31774},[31680],[34,523396,1323],{"className":523397},[31782],[34,523399,111],{"className":523400},[31685,31686],[34,523402],{"className":523403,"style":32604},[31845],[34,523405,223],{"className":523406},[32024],[34,523408],{"className":523409,"style":32604},[31845],[34,523411,523413,523416,523419,523422,523425,523428],{"className":523412},[31676],[34,523414],{"className":523415,"style":31774},[31680],[34,523417,106063],{"className":523418,"style":67482},[31685,31686],[34,523420,2557],{"className":523421},[31690],[34,523423],{"className":523424,"style":32604},[31845],[34,523426,223],{"className":523427},[32024],[34,523429],{"className":523430,"style":32604},[31845],[34,523432,523434,523437,523440,523443,523446,523449],{"className":523433},[31676],[34,523435],{"className":523436,"style":31774},[31680],[34,523438,456],{"className":523439},[31782],[34,523441,1697],{"className":523442},[31685,31686],[34,523444],{"className":523445,"style":32604},[31845],[34,523447,223],{"className":523448},[32024],[34,523450],{"className":523451,"style":32604},[31845],[34,523453,523455,523458,523461,523464,523467,523470],{"className":523454},[31676],[34,523456],{"className":523457,"style":31774},[31680],[34,523459,106063],{"className":523460,"style":67482},[31685,31686],[34,523462,44253],{"className":523463},[31690],[34,523465],{"className":523466,"style":32604},[31845],[34,523468,223],{"className":523469},[32024],[34,523471],{"className":523472,"style":32604},[31845],[34,523474,523476,523479,523482,523485,523488,523491],{"className":523475},[31676],[34,523477],{"className":523478,"style":31774},[31680],[34,523480,456],{"className":523481},[31782],[34,523483,31965],{"className":523484},[31685,31686],[34,523486],{"className":523487,"style":32604},[31845],[34,523489,223],{"className":523490},[32024],[34,523492],{"className":523493,"style":32604},[31845],[34,523495,523497,523500,523503,523506,523509,523512],{"className":523496},[31676],[34,523498],{"className":523499,"style":31774},[31680],[34,523501,106063],{"className":523502,"style":67482},[31685,31686],[34,523504,2557],{"className":523505},[31690],[34,523507],{"className":523508,"style":31846},[31845],[34,523510,202],{"className":523511},[31850],[34,523513],{"className":523514,"style":31846},[31845],[34,523516,523518,523521,523524,523527,523530,523533],{"className":523517},[31676],[34,523519],{"className":523520,"style":31774},[31680],[34,523522,456],{"className":523523},[31782],[34,523525,111],{"className":523526},[31685,31686],[34,523528],{"className":523529,"style":32604},[31845],[34,523531,223],{"className":523532},[32024],[34,523534],{"className":523535,"style":32604},[31845],[34,523537,523539,523542,523545,523548,523551],{"className":523538},[31676],[34,523540],{"className":523541,"style":34299},[31680],[34,523543,1697],{"className":523544},[31685,31686],[34,523546],{"className":523547,"style":32604},[31845],[34,523549,223],{"className":523550},[32024],[34,523552],{"className":523553,"style":32604},[31845],[34,523555,523557,523560,523563,523566,523569,523572],{"className":523556},[31676],[34,523558],{"className":523559,"style":31774},[31680],[34,523561,31965],{"className":523562},[31685,31686],[34,523564,2557],{"className":523565},[31690],[34,523567],{"className":523568,"style":32604},[31845],[34,523570,223],{"className":523571},[32024],[34,523573],{"className":523574,"style":32604},[31845],[34,523576,523578,523581,523584,523587,523590],{"className":523577},[31676],[34,523579],{"className":523580,"style":33218},[31680],[34,523582,106063],{"className":523583,"style":67482},[31685,31686],[34,523585],{"className":523586,"style":31846},[31845],[34,523588,202],{"className":523589},[31850],[34,523591],{"className":523592,"style":31846},[31845],[34,523594,523596,523599,523602,523605,523608,523611],{"className":523595},[31676],[34,523597],{"className":523598,"style":31774},[31680],[34,523600,456],{"className":523601},[31782],[34,523603,111],{"className":523604},[31685,31686],[34,523606],{"className":523607,"style":32604},[31845],[34,523609,223],{"className":523610},[32024],[34,523612],{"className":523613,"style":32604},[31845],[34,523615,523617,523620,523623,523626,523629,523632],{"className":523616},[31676],[34,523618],{"className":523619,"style":31774},[31680],[34,523621,106063],{"className":523622,"style":67482},[31685,31686],[34,523624,2557],{"className":523625},[31690],[34,523627],{"className":523628,"style":32604},[31845],[34,523630,223],{"className":523631},[32024],[34,523633],{"className":523634,"style":32604},[31845],[34,523636,523638,523641,523644,523647,523650,523653],{"className":523637},[31676],[34,523639],{"className":523640,"style":31774},[31680],[34,52364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本身）是单位元。",[108,523794,523795,2664,523798,523859,523860,347],{},[2627,523796,523797],{},"逆元",[34,523799,523801,523823],{"className":523800},[31642],[34,523802,523804],{"className":523803},[31646],[31648,523805,523806],{"xmlns":31650},[31652,523807,523808,523820],{},[31655,523809,523810,523812,523814,523816,523818],{},[31662,523811,456],{"stretchy":702},[31658,523813,111],{},[31662,523815,223],{},[31658,523817,106063],{},[31662,523819,2557],{"stretchy":702},[31665,523821,523822],{"encoding":31667},"(a+I)",[34,523824,523826,523847],{"className":523825,"ariaHidden":758},[31672],[34,523827,523829,523832,523835,523838,523841,523844],{"className":523828},[31676],[34,523830],{"className":523831,"style":31774},[31680],[34,523833,456],{"className":523834},[31782],[34,523836,111],{"className":523837},[31685,31686],[34,523839],{"className":523840,"style":32604},[31845],[34,523842,223],{"className":523843},[32024],[34,523845],{"className":523846,"style":32604},[31845],[34,523848,523850,523853,523856],{"className":523849},[31676],[34,523851],{"className":523852,"style":31774},[31680],[34,523854,106063],{"className":523855,"style":67482},[31685,31686],[34,523857,2557],{"className":523858},[31690]," 的逆元是 ",[34,523861,523863,523887],{"className":523862},[31642],[34,523864,523866],{"className":523865},[31646],[31648,523867,523868],{"xmlns":31650},[31652,523869,523870,523884],{},[31655,523871,523872,523874,523876,523878,523880,523882],{},[31662,523873,456],{"stretchy":702},[31662,523875,32226],{},[31658,523877,111],{},[31662,523879,2557],{"stretchy":702},[31662,523881,223],{},[31658,523883,106063],{},[31665,523885,523886],{"encoding":31667},"(-a)+I",[34,523888,523890,523917],{"className":523889,"ariaHidden":758},[31672],[34,523891,523893,523896,523899,523902,523905,523908,523911,523914],{"className":523892},[31676],[34,523894],{"className":523895,"style":31774},[31680],[34,523897,456],{"className":523898},[31782],[34,523900,32226],{"className":523901},[31685],[34,523903,111],{"className":523904},[31685,31686],[34,523906,2557],{"className":523907},[31690],[34,523909],{"className":523910,"style":32604},[31845],[34,523912,223],{"className":523913},[32024],[34,523915],{"className":523916,"style":32604},[31845],[34,523918,523920,523923],{"className":523919},[31676],[34,523921],{"className":523922,"style":33218},[31680],[34,523924,106063],{"className":523925,"style":67482},[31685,31686],[108,523927,523928,2664,523930,354330,524305,524333],{},[2627,523929,380943],{},[34,523931,523933,524023],{"className":523932},[31642],[34,523934,523936],{"className":523935},[31646],[31648,523937,523938],{"xmlns":31650},[31652,523939,523940,524020],{},[31655,523941,523942,523944,523946,523948,523950,523952,523954,523956,523958,523960,523962,523964,523966,523968,523970,523972,523974,523976,523978,523980,523982,523984,523986,523988,523990,523992,523994,523996,523998,524000,524002,524004,524006,524008,524010,524012,524014,524016,524018],{},[31662,523943,456],{"stretchy":702},[31658,523945,111],{},[31662,523947,223],{},[31658,523949,106063],{},[31662,523951,2557],{"stretchy":702},[31662,523953,223],{},[31662,523955,456],{"stretchy":702},[31658,523957,1697],{},[31662,523959,223],{},[31658,523961,106063],{},[31662,523963,2557],{"stretchy":702},[31662,523965,202],{},[31662,523967,456],{"stretchy":702},[31658,523969,111],{},[31662,523971,223],{},[31658,523973,1697],{},[31662,523975,2557],{"stretchy":702},[31662,523977,223],{},[31658,523979,106063],{},[31662,523981,202],{},[31662,523983,456],{"stretchy":702},[31658,523985,1697],{},[31662,523987,223],{},[31658,523989,111],{},[31662,523991,2557],{"stretchy":702},[31662,523993,223],{},[31658,523995,106063],{},[31662,523997,202],{},[31662,523999,456],{"stretchy":702},[31658,524001,1697],{},[31662,524003,223],{},[31658,524005,106063],{},[31662,524007,2557],{"stretchy":702},[31662,524009,223],{},[31662,524011,456],{"stretchy":702},[31658,524013,111],{},[31662,524015,223],{},[31658,524017,106063],{},[31662,524019,2557],{"stretchy":702},[31665,524021,524022],{"encoding":31667},"(a+I)+(b+I) = (a+b)+I = (b+a)+I = (b+I)+(a+I)",[34,524024,524026,524047,524068,524089,524110,524131,524152,524170,524191,524212,524230,524251,524272,524293],{"className":524025,"ariaHidden":758},[31672],[34,524027,524029,524032,524035,524038,524041,524044],{"className":524028},[31676],[34,524030],{"className":524031,"style":31774},[31680],[34,524033,456],{"className":524034},[31782],[34,524036,111],{"className":524037},[31685,31686],[34,524039],{"className":524040,"style":32604},[31845],[34,524042,223],{"className":524043},[32024],[34,524045],{"className":524046,"style":32604},[31845],[34,524048,524050,524053,524056,524059,524062,524065],{"className":524049},[31676],[34,524051],{"className":524052,"style":31774},[31680],[34,524054,106063],{"className":524055,"style":67482},[31685,31686],[34,524057,2557],{"className":524058},[31690],[34,524060],{"className":524061,"style":32604},[31845],[34,524063,223],{"className":524064},[32024],[34,524066],{"className":524067,"style":32604},[31845],[34,524069,524071,524074,524077,524080,524083,524086],{"className":524070},[31676],[34,524072],{"className":524073,"style":31774},[31680],[34,524075,456],{"className":524076},[31782],[34,524078,1697],{"className":524079},[31685,31686],[34,524081],{"className":524082,"style":32604},[31845],[34,524084,223],{"className":524085},[32024],[34,524087],{"className":524088,"style":32604},[31845],[34,524090,524092,524095,524098,524101,524104,524107],{"className":524091},[31676],[34,524093],{"className":524094,"style":31774},[31680],[34,524096,106063],{"className":524097,"style":67482},[31685,31686],[34,524099,2557],{"className":524100},[31690],[34,524102],{"className":524103,"style":31846},[31845],[34,524105,202],{"className":524106},[31850],[34,524108],{"className":524109,"style":31846},[31845],[34,524111,524113,524116,524119,524122,524125,524128],{"className":524112},[31676],[34,524114],{"className":524115,"style":31774},[31680],[34,524117,456],{"className":524118},[31782],[34,524120,111],{"className":524121},[31685,31686],[34,524123],{"className":524124,"style":32604},[31845],[34,524126,223],{"className":524127},[32024],[34,524129],{"className":524130,"style":32604},[31845],[34,524132,524134,524137,524140,524143,524146,524149],{"className":524133},[31676],[34,524135],{"className":524136,"style":31774},[31680],[34,524138,1697],{"className":524139},[31685,31686],[34,524141,2557],{"className":524142},[31690],[34,524144],{"className":524145,"style":32604},[31845],[34,524147,223],{"className":524148},[32024],[34,524150],{"className":524151,"style":32604},[31845],[34,524153,524155,524158,524161,524164,524167],{"className":524154},[31676],[34,524156],{"className":524157,"style":33218},[31680],[34,524159,106063],{"className":524160,"style":67482},[31685,31686],[34,524162],{"className":524163,"style":31846},[31845],[34,524165,202],{"className":524166},[31850],[34,524168],{"className":524169,"style":31846},[31845],[34,524171,524173,524176,524179,524182,524185,524188],{"className":524172},[31676],[34,524174],{"className":524175,"style":31774},[31680],[34,524177,456],{"className":524178},[31782],[34,524180,1697],{"className":524181},[31685,31686],[34,524183],{"className":524184,"style":32604},[31845],[34,524186,223],{"className":524187},[32024],[34,524189],{"className":524190,"style":32604},[31845],[34,524192,524194,524197,524200,524203,524206,524209],{"className":524193},[31676],[34,524195],{"className":524196,"style":31774},[31680],[34,524198,111],{"className":524199},[31685,31686],[34,524201,2557],{"className":524202},[31690],[34,524204],{"className":524205,"style":32604},[31845],[34,524207,223],{"className":524208},[32024],[34,524210],{"className":524211,"style":32604},[31845],[34,524213,524215,524218,524221,524224,524227],{"className":524214},[31676],[34,524216],{"className":524217,"style":33218},[31680],[34,524219,106063],{"className":524220,"style":67482},[31685,31686],[34,524222],{"className":524223,"style":31846},[31845],[34,524225,202],{"className":524226},[31850],[34,524228],{"className":524229,"style":31846},[31845],[34,524231,524233,524236,524239,524242,524245,524248],{"className":524232},[31676],[34,524234],{"className":524235,"style":31774},[31680],[34,524237,456],{"className":524238},[31782],[34,524240,1697],{"className":524241},[31685,31686],[34,524243],{"className":524244,"style":32604},[31845],[34,524246,223],{"className":524247},[32024],[34,524249],{"className":524250,"style":32604},[31845],[34,524252,524254,524257,524260,524263,524266,524269],{"className":524253},[31676],[34,524255],{"className":524256,"style":31774},[31680],[34,524258,106063],{"className":524259,"style":67482},[31685,31686],[34,524261,2557],{"className":524262},[31690],[34,524264],{"className":524265,"style":32604},[31845],[34,524267,223],{"className":524268},[32024],[34,524270],{"className":524271,"style":32604},[31845],[34,524273,524275,524278,524281,524284,524287,524290],{"className":524274},[31676],[34,524276],{"className":524277,"style":31774},[31680],[34,524279,456],{"className":524280},[31782],[34,524282,111],{"className":524283},[31685,31686],[34,524285],{"className":524286,"style":32604},[31845],[34,524288,223],{"className":524289},[32024],[34,524291],{"className":524292,"style":32604},[31845],[34,524294,524296,524299,524302],{"className":524295},[31676],[34,524297],{"className":524298,"style":31774},[31680],[34,524300,106063],{"className":524301,"style":67482},[31685,31686],[34,524303,2557],{"className":524304},[31690],[34,524306,524308,524321],{"className":524307},[31642],[34,524309,524311],{"className":524310},[31646],[31648,524312,524313],{"xmlns":31650},[31652,524314,524315,524319],{},[31655,524316,524317],{},[31658,524318,34386],{},[31665,524320,34386],{"encoding":31667},[34,524322,524324],{"className":524323,"ariaHidden":758},[31672],[34,524325,524327,524330],{"className":524326},[31676],[34,524328],{"className":524329,"style":33218},[31680],[34,524331,34386],{"className":524332,"style":34427},[31685,31686]," 的加法交换）。",[15,524335,524336],{},[2627,524337,524338,524339,524401],{},"(iii) ",[34,524340,524342,524368],{"className":524341},[31642],[34,524343,524345],{"className":524344},[31646],[31648,524346,524347],{"xmlns":31650},[31652,524348,524349,524365],{},[31655,524350,524351,524353,524355,524357,524359,524361,524363],{},[31662,524352,456],{"stretchy":702},[31658,524354,34386],{},[31658,524356,38080],{"mathvariant":38079},[31658,524358,106063],{},[31662,524360,31960],{"separator":758},[31662,524362,71191],{},[31662,524364,2557],{"stretchy":702},[31665,524366,524367],{"encoding":31667},"(R\u002FI, \\cdot)",[34,524369,524371],{"className":524370,"ariaHidden":758},[31672],[34,524372,524374,524377,524380,524383,524386,524389,524392,524395,524398],{"className":524373},[31676],[34,524375],{"className":524376,"style":31774},[31680],[34,524378,456],{"className":524379},[31782],[34,524381,34386],{"className":524382,"style":34427},[31685,31686],[34,524384,38080],{"className":524385},[31685],[34,524387,106063],{"className":524388,"style":67482},[31685,31686],[34,524390,31960],{"className":524391},[32119],[34,524393],{"className":524394,"style":32123},[31845],[34,524396,71191],{"className":524397},[31685],[34,524399,2557],{"className":524400},[31690]," 是半群",[15,524403,524404,524405,524407,524408,39984,524595,524832,525075,525076,525104,525105,525192],{},"“半群”的意思就是：运算封闭且满足",[2627,524406,378398],{},"。\n我们需要证明乘法结合律：\n设 ",[34,524409,524411,524457],{"className":524410},[31642],[34,524412,524414],{"className":524413},[31646],[31648,524415,524416],{"xmlns":31650},[31652,524417,524418,524454],{},[31655,524419,524420,524422,524424,524426,524428,524430,524432,524434,524436,524438,524440,524442,524444,524446,524448,524450,524452],{},[31658,524421,54151],{},[31662,524423,202],{},[31658,524425,111],{},[31662,524427,223],{},[31658,524429,106063],{},[31662,524431,31960],{"separator":758},[31658,524433,54224],{},[31662,524435,202],{},[31658,524437,1697],{},[31662,524439,223],{},[31658,524441,106063],{},[31662,524443,31960],{"separator":758},[31658,524445,47384],{},[31662,524447,202],{},[31658,524449,31965],{},[31662,524451,223],{},[31658,524453,106063],{},[31665,524455,524456],{"encoding":31667},"A = a+I, B = b+I, C = c+I",[34,524458,524460,524478,524496,524523,524541,524568,524586],{"className":524459,"ariaHidden":758},[31672],[34,524461,524463,524466,524469,524472,524475],{"className":524462},[31676],[34,524464],{"className":524465,"style":33218},[31680],[34,524467,54151],{"className":524468},[31685,31686],[34,524470],{"className":524471,"style":31846},[31845],[34,524473,202],{"className":524474},[31850],[34,524476],{"className":524477,"style":31846},[31845],[34,524479,524481,524484,524487,524490,524493],{"className":524480},[31676],[34,524482],{"className":524483,"style":32597},[31680],[34,524485,111],{"className":524486},[31685,31686],[34,524488],{"className":524489,"style":32604},[31845],[34,524491,223],{"className":524492},[32024],[34,524494],{"className":524495,"style":32604},[31845],[34,524497,524499,524502,524505,524508,524511,524514,524517,524520],{"className":524498},[31676],[34,524500],{"className":524501,"style":70739},[31680],[34,524503,106063],{"className":524504,"style":67482},[31685,31686],[34,524506,31960],{"className":524507},[32119],[34,524509],{"className":524510,"style":32123},[31845],[34,524512,54224],{"className":524513,"style":54252},[31685,31686],[34,524515],{"className":524516,"style":31846},[31845],[34,524518,202],{"className":524519},[31850],[34,524521],{"className":524522,"style":31846},[31845],[34,524524,524526,524529,524532,524535,524538],{"className":524525},[31676],[34,524527],{"className":524528,"style":34299},[31680],[34,524530,1697],{"className":524531},[31685,31686],[34,524533],{"className":524534,"style":32604},[31845],[34,524536,223],{"className":524537},[32024],[34,524539],{"className":524540,"style":32604},[31845],[34,524542,524544,524547,524550,524553,524556,524559,524562,524565],{"className":524543},[31676],[34,524545],{"className":524546,"style":70739},[31680],[34,524548,106063],{"className":524549,"style":67482},[31685,31686],[34,524551,31960],{"className":524552},[32119],[34,524554],{"className":524555,"style":32123},[31845],[34,524557,47384],{"className":524558,"style":38706},[31685,31686],[34,524560],{"className":524561,"style":31846},[31845],[34,524563,202],{"className":524564},[31850],[34,524566],{"className":524567,"style":31846},[31845],[34,524569,524571,524574,524577,524580,524583],{"className":524570},[31676],[34,524572],{"className":524573,"style":32597},[31680],[34,524575,31965],{"className":524576},[31685,31686],[34,524578],{"className":524579,"style":32604},[31845],[34,524581,223],{"className":524582},[32024],[34,524584],{"className":524585,"style":32604},[31845],[34,524587,524589,524592],{"className":524588},[31676],[34,524590],{"className":524591,"style":33218},[31680],[34,524593,106063],{"className":524594,"style":67482},[31685,31686],[34,524596,524598,524664],{"className":524597},[31642],[34,524599,524601],{"className":524600},[31646],[31648,524602,524603],{"xmlns":31650},[31652,524604,524605,524661],{},[31655,524606,524607,524609,524611,524613,524615,524617,524619,524621,524623,524625,524627,524629,524631,524633,524635,524637,524639,524641,524643,524645,524647,524649,524651,524653,524655,524657,524659],{},[31662,524608,456],{"stretchy":702},[31658,524610,54151],{},[31662,524612,71191],{},[31658,524614,54224],{},[31662,524616,2557],{"stretchy":702},[31662,524618,71191],{},[31658,524620,47384],{},[31662,524622,202],{},[31662,524624,456],{"stretchy":702},[31658,524626,111],{},[31658,524628,1697],{},[31662,524630,223],{},[31658,524632,106063],{},[31662,524634,2557],{"stretchy":702},[31662,524636,456],{"stretchy":702},[31658,524638,31965],{},[31662,524640,223],{},[31658,524642,106063],{},[31662,524644,2557],{"stretchy":702},[31662,524646,202],{},[31662,524648,456],{"stretchy":702},[31658,524650,111],{},[31658,524652,1697],{},[31662,524654,2557],{"stretchy":702},[31658,524656,31965],{},[31662,524658,223],{},[31658,524660,106063],{},[31665,524662,524663],{"encoding":31667},"(A \\cdot B) \\cdot C = (ab+I)(c+I) = (ab)c + I",[34,524665,524667,524688,524709,524727,524748,524775,524796,524823],{"className":524666,"ariaHidden":758},[31672],[34,524668,524670,524673,524676,524679,524682,524685],{"className":524669},[31676],[34,524671],{"className":524672,"style":31774},[31680],[34,524674,456],{"className":524675},[31782],[34,524677,54151],{"className":524678},[31685,31686],[34,524680],{"className":524681,"style":32604},[31845],[34,524683,71191],{"className":524684},[32024],[34,524686],{"className":524687,"style":32604},[31845],[34,524689,524691,524694,524697,524700,524703,524706],{"className":524690},[31676],[34,524692],{"className":524693,"style":31774},[31680],[34,524695,54224],{"className":524696,"style":54252},[31685,31686],[34,524698,2557],{"className":524699},[31690],[34,524701],{"className":524702,"style":32604},[31845],[34,524704,71191],{"className":524705},[32024],[34,524707],{"className":524708,"style":32604},[31845],[34,524710,524712,524715,524718,524721,524724],{"className":524711},[31676],[34,524713],{"className":524714,"style":33218},[31680],[34,524716,47384],{"className":524717,"style":38706},[31685,31686],[34,524719],{"className":524720,"style":31846},[31845],[34,524722,202],{"className":524723},[31850],[34,524725],{"className":524726,"style":31846},[31845],[34,524728,524730,524733,524736,524739,524742,524745],{"className":524729},[31676],[34,524731],{"className":524732,"style":31774},[31680],[34,524734,456],{"className":524735},[31782],[34,524737,324191],{"className":524738},[31685,31686],[34,524740],{"className":524741,"style":32604},[31845],[34,524743,223],{"className":524744},[32024],[34,524746],{"className":524747,"style":32604},[31845],[34,524749,524751,524754,524757,524760,524763,524766,524769,524772],{"className":524750},[31676],[34,524752],{"className":524753,"style":31774},[31680],[34,524755,106063],{"className":524756,"style":67482},[31685,31686],[34,524758,2557],{"className":524759},[31690],[34,524761,456],{"className":524762},[31782],[34,524764,31965],{"className":524765},[31685,31686],[34,524767],{"className":524768,"style":32604},[31845],[34,524770,223],{"className":524771},[32024],[34,524773],{"className":524774,"style":32604},[31845],[34,524776,524778,524781,524784,524787,524790,524793],{"className":524777},[31676],[34,524779],{"className":524780,"style":31774},[31680],[34,524782,106063],{"className":524783,"style":67482},[31685,31686],[34,524785,2557],{"className":524786},[31690],[34,524788],{"className":524789,"style":31846},[31845],[34,524791,202],{"className":524792},[31850],[34,524794],{"className":524795,"style":31846},[31845],[34,524797,524799,524802,524805,524808,524811,524814,524817,524820],{"className":524798},[31676],[34,524800],{"className":524801,"style":31774},[31680],[34,524803,456],{"className":524804},[31782],[34,524806,324191],{"className":524807},[31685,31686],[34,524809,2557],{"className":524810},[31690],[34,524812,31965],{"className":524813},[31685,31686],[34,524815],{"className":524816,"style":32604},[31845],[34,524818,223],{"className":524819},[32024],[34,524821],{"className":524822,"style":32604},[31845],[34,524824,524826,524829],{"className":524825},[31676],[34,524827],{"className":524828,"style":33218},[31680],[34,524830,106063],{"className":524831,"style":67482},[31685,31686],[34,524833,524835,524901],{"className":524834},[31642],[34,524836,524838],{"className":524837},[31646],[31648,524839,524840],{"xmlns":31650},[31652,524841,524842,524898],{},[31655,524843,524844,524846,524848,524850,524852,524854,524856,524858,524860,524862,524864,524866,524868,524870,524872,524874,524876,524878,524880,524882,524884,524886,524888,524890,524892,524894,524896],{},[31658,524845,54151],{},[31662,524847,71191],{},[31662,524849,456],{"stretchy":702},[31658,524851,54224],{},[31662,524853,71191],{},[31658,524855,47384],{},[31662,524857,2557],{"stretchy":702},[31662,524859,202],{},[31662,524861,456],{"stretchy":702},[31658,524863,111],{},[31662,524865,223],{},[31658,524867,106063],{},[31662,524869,2557],{"stretchy":702},[31662,524871,456],{"stretchy":702},[31658,524873,1697],{},[31658,524875,31965],{},[31662,524877,223],{},[31658,524879,106063],{},[31662,524881,2557],{"stretchy":702},[31662,524883,202],{},[31658,524885,111],{},[31662,524887,456],{"stretchy":702},[31658,524889,1697],{},[31658,524891,31965],{},[31662,524893,2557],{"stretchy":702},[31662,524895,223],{},[31658,524897,106063],{},[31665,524899,524900],{"encoding":31667},"A \\cdot (B \\cdot C) = (a+I)(bc+I) = a(bc) + I",[34,524902,524904,524922,524943,524964,524985,525015,525036,525066],{"className":524903,"ariaHidden":758},[31672],[34,524905,524907,524910,524913,524916,524919],{"className":524906},[31676],[34,524908],{"className":524909,"style":33218},[31680],[34,524911,54151],{"className":524912},[31685,31686],[34,524914],{"className":524915,"style":32604},[31845],[34,524917,71191],{"className":524918},[32024],[34,524920],{"className":524921,"style":32604},[31845],[34,524923,524925,524928,524931,524934,524937,524940],{"className":524924},[31676],[34,524926],{"className":524927,"style":31774},[31680],[34,524929,456],{"className":524930},[31782],[34,524932,54224],{"className":524933,"style":54252},[31685,31686],[34,524935],{"className":524936,"style":32604},[31845],[34,524938,71191],{"className":524939},[32024],[34,524941],{"className":524942,"style":32604},[31845],[34,524944,524946,524949,524952,524955,524958,524961],{"className":524945},[31676],[34,524947],{"className":524948,"style":31774},[31680],[34,524950,47384],{"className":524951,"style":38706},[31685,31686],[34,524953,2557],{"className":524954},[31690],[34,524956],{"className":524957,"style":31846},[31845],[34,524959,202],{"className":524960},[31850],[34,524962],{"className":524963,"style":31846},[31845],[34,524965,524967,524970,524973,524976,524979,524982],{"className":524966},[31676],[34,524968],{"className":524969,"style":31774},[31680],[34,524971,456],{"className":524972},[31782],[34,524974,111],{"className":524975},[31685,31686],[34,524977],{"className":524978,"style":32604},[31845],[34,524980,223],{"className":524981},[32024],[34,524983],{"className":524984,"style":32604},[31845],[34,524986,524988,524991,524994,524997,525000,525003,525006,525009,525012],{"className":524987},[31676],[34,524989],{"className":524990,"style":31774},[31680],[34,524992,106063],{"className":524993,"style":67482},[31685,31686],[34,524995,2557],{"className":524996},[31690],[34,524998,456],{"className":524999},[31782],[34,525001,1697],{"className":525002},[31685,31686],[34,525004,31965],{"className":525005},[31685,31686],[34,525007],{"className":525008,"style":32604},[31845],[34,525010,223],{"className":525011},[32024],[34,525013],{"className":525014,"style":32604},[31845],[34,525016,525018,525021,525024,525027,525030,525033],{"className":525017},[31676],[34,525019],{"className":525020,"style":31774},[31680],[34,525022,106063],{"className":525023,"style":67482},[31685,31686],[34,525025,2557],{"className":525026},[31690],[34,525028],{"className":525029,"style":31846},[31845],[34,525031,202],{"className":525032},[31850],[34,525034],{"className":525035,"style":31846},[31845],[34,525037,525039,525042,525045,525048,525051,525054,525057,525060,525063],{"className":525038},[31676],[34,525040],{"className":525041,"style":31774},[31680],[34,525043,111],{"className":525044},[31685,31686],[34,525046,456],{"className":525047},[31782],[34,525049,1697],{"className":525050},[31685,31686],[34,525052,31965],{"className":525053},[31685,31686],[34,525055,2557],{"className":525056},[31690],[34,525058],{"className":525059,"style":32604},[31845],[34,525061,223],{"className":525062},[32024],[34,525064],{"className":525065,"style":32604},[31845],[34,525067,525069,525072],{"className":525068},[31676],[34,525070],{"className":525071,"style":33218},[31680],[34,525073,106063],{"className":525074,"style":67482},[31685,31686],"\n由于原环 ",[34,525077,525079,525092],{"className":525078},[31642],[34,525080,525082],{"className":525081},[31646],[31648,525083,525084],{"xmlns":31650},[31652,525085,525086,525090],{},[31655,525087,525088],{},[31658,525089,34386],{},[31665,525091,34386],{"encoding":31667},[34,525093,525095],{"className":525094,"ariaHidden":758},[31672],[34,525096,525098,525101],{"className":525097},[31676],[34,525099],{"className":525100,"style":33218},[31680],[34,525102,34386],{"className":525103,"style":34427},[31685,31686]," 满足结合律 ",[34,525106,525108,525141],{"className":525107},[31642],[34,525109,525111],{"className":525110},[31646],[31648,525112,525113],{"xmlns":31650},[31652,525114,525115,525139],{},[31655,525116,525117,525119,525121,525123,525125,525127,525129,525131,525133,525135,525137],{},[31662,525118,456],{"stretchy":702},[31658,525120,111],{},[31658,525122,1697],{},[31662,525124,2557],{"stretchy":702},[31658,525126,31965],{},[31662,525128,202],{},[31658,525130,111],{},[31662,525132,456],{"stretchy":702},[31658,525134,1697],{},[31658,525136,31965],{},[31662,525138,2557],{"stretchy":702},[31665,525140,378435],{"encoding":31667},[34,525142,525144,525171],{"className":525143,"ariaHidden":758},[31672],[34,525145,525147,525150,525153,525156,525159,525162,525165,525168],{"className":525146},[31676],[34,525148],{"className":525149,"style":31774},[31680],[34,525151,456],{"className":525152},[31782],[34,525154,324191],{"className":525155},[31685,31686],[34,525157,2557],{"className":525158},[31690],[34,525160,31965],{"className":525161},[31685,31686],[34,525163],{"className":525164,"style":31846},[31845],[34,525166,202],{"className":525167},[31850],[34,525169],{"className":525170,"style":31846},[31845],[34,525172,525174,525177,525180,525183,525186,525189],{"className":525173},[31676],[34,525175],{"className":525176,"style":31774},[31680],[34,525178,111],{"className":525179},[31685,31686],[34,525181,456],{"className":525182},[31782],[34,525184,1697],{"className":525185},[31685,31686],[34,525187,31965],{"className":525188},[31685,31686],[34,525190,2557],{"className":525191},[31690],"，所以这两个陪集相等。",[15,525194,525195],{},[2627,525196,525197],{},"(iv) 满足分配律",[15,525199,525200,525201,39984,525321,525597,525598,525700,525701,525729,525730,33183,525847,525951,525952,526270],{},"我们要证左分配律：",[34,525202,525204,525240],{"className":525203},[31642],[34,525205,525207],{"className":525206},[31646],[31648,525208,525209],{"xmlns":31650},[31652,525210,525211,525237],{},[31655,525212,525213,525215,525217,525219,525221,525223,525225,525227,525229,525231,525233,525235],{},[31658,525214,54151],{},[31662,525216,456],{"stretchy":702},[31658,525218,54224],{},[31662,525220,223],{},[31658,525222,47384],{},[31662,525224,2557],{"stretchy":702},[31662,525226,202],{},[31658,525228,54151],{},[31658,525230,54224],{},[31662,525232,223],{},[31658,525234,54151],{},[31658,525236,47384],{},[31665,525238,525239],{"encoding":31667},"A(B+C) = AB + AC",[34,525241,525243,525267,525288,525309],{"className":525242,"ariaHidden":758},[31672],[34,525244,525246,525249,525252,525255,525258,525261,525264],{"className":525245},[31676],[34,525247],{"className":525248,"style":31774},[31680],[34,525250,54151],{"className":525251},[31685,31686],[34,525253,456],{"className":525254},[31782],[34,525256,54224],{"className":525257,"style":54252},[31685,31686],[34,525259],{"className":525260,"style":32604},[31845],[34,525262,223],{"className":525263},[32024],[34,525265],{"className":525266,"style":32604},[31845],[34,525268,525270,525273,525276,525279,525282,525285],{"className":525269},[31676],[34,525271],{"className":525272,"style":31774},[31680],[34,525274,47384],{"className":525275,"style":38706},[31685,31686],[34,525277,2557],{"className":525278},[31690],[34,525280],{"className":525281,"style":31846},[31845],[34,525283,202],{"className":525284},[31850],[34,525286],{"className":525287,"style":31846},[31845],[34,525289,525291,525294,525297,525300,525303,525306],{"className":525290},[31676],[34,525292],{"className":525293,"style":34944},[31680],[34,525295,54151],{"className":525296},[31685,31686],[34,525298,54224],{"className":525299,"style":54252},[31685,31686],[34,525301],{"className":525302,"style":32604},[31845],[34,525304,223],{"className":525305},[32024],[34,525307],{"className":525308,"style":32604},[31845],[34,525310,525312,525315,525318],{"className":525311},[31676],[34,525313],{"className":525314,"style":33218},[31680],[34,525316,54151],{"className":525317},[31685,31686],[34,525319,47384],{"className":525320,"style":38706},[31685,31686],[34,525322,525324,525402],{"className":525323},[31642],[34,525325,525327],{"className":525326},[31646],[31648,525328,525329],{"xmlns":31650},[31652,525330,525331,525399],{},[31655,525332,525333,525335,525337,525339,525341,525343,525345,525347,525349,525351,525353,525355,525357,525359,525361,525363,525365,525367,525369,525371,525373,525375,525377,525379,525381,525383,525385,525387,525389,525391,525393,525395,525397],{},[31662,525334,456],{"stretchy":702},[31658,525336,111],{},[31662,525338,223],{},[31658,525340,106063],{},[31662,525342,2557],{"stretchy":702},[31662,525344,456],{"stretchy":702},[31662,525346,456],{"stretchy":702},[31658,525348,1697],{},[31662,525350,223],{},[31658,525352,106063],{},[31662,525354,2557],{"stretchy":702},[31662,525356,223],{},[31662,525358,456],{"stretchy":702},[31658,525360,31965],{},[31662,525362,223],{},[31658,525364,106063],{},[31662,525366,2557],{"stretchy":702},[31662,525368,2557],{"stretchy":702},[31662,525370,202],{},[31662,525372,456],{"stretchy":702},[31658,525374,111],{},[31662,525376,223],{},[31658,525378,106063],{},[31662,525380,2557],{"stretchy":702},[31662,525382,456],{"stretchy":702},[31662,525384,456],{"stretchy":702},[31658,525386,1697],{},[31662,525388,223],{},[31658,525390,31965],{},[31662,525392,2557],{"stretchy":702},[31662,525394,223],{},[31658,525396,106063],{},[31662,525398,2557],{"stretchy":702},[31665,525400,525401],{"encoding":31667},"(a+I)((b+I) + (c+I)) = (a+I)((b+c)+I)",[34,525403,525405,525426,525453,525474,525495,525516,525537,525564,525585],{"className":525404,"ariaHidden":758},[31672],[34,525406,525408,525411,525414,525417,525420,525423],{"className":525407},[31676],[34,525409],{"className":525410,"style":31774},[31680],[34,525412,456],{"className":525413},[31782],[34,525415,111],{"className":525416},[31685,31686],[34,525418],{"className":525419,"style":32604},[31845],[34,525421,223],{"className":525422},[32024],[34,525424],{"className":525425,"style":32604},[31845],[34,525427,525429,525432,525435,525438,525441,525444,525447,525450],{"className":525428},[31676],[34,525430],{"className":525431,"style":31774},[31680],[34,525433,106063],{"className":525434,"style":67482},[31685,31686],[34,525436,2557],{"className":525437},[31690],[34,525439,1323],{"className":525440},[31782],[34,525442,1697],{"className":525443},[31685,31686],[34,525445],{"className":525446,"style":32604},[31845],[34,525448,223],{"className":525449},[32024],[34,525451],{"className":525452,"style":32604},[31845],[34,525454,525456,525459,525462,525465,525468,525471],{"className":525455},[31676],[34,525457],{"className":525458,"style":31774},[31680],[34,525460,106063],{"className":525461,"style":67482},[31685,31686],[34,525463,2557],{"className":525464},[31690],[34,525466],{"className":525467,"style":32604},[31845],[34,525469,223],{"className":525470},[32024],[34,525472],{"className":525473,"style":32604},[31845],[34,525475,525477,525480,525483,525486,525489,525492],{"className":525476},[31676],[34,525478],{"className":525479,"style":31774},[31680],[34,525481,456],{"className":525482},[31782],[34,525484,31965],{"className":525485},[31685,31686],[34,525487],{"className":525488,"style":32604},[31845],[34,525490,223],{"className":525491},[32024],[34,525493],{"className":525494,"style":32604},[31845],[34,525496,525498,525501,525504,525507,525510,525513],{"className":525497},[31676],[34,525499],{"className":525500,"style":31774},[31680],[34,525502,106063],{"className":525503,"style":67482},[31685,31686],[34,525505,44253],{"className":525506},[31690],[34,525508],{"className":525509,"style":31846},[31845],[34,525511,202],{"className":525512},[31850],[34,525514],{"className":525515,"style":31846},[31845],[34,525517,525519,525522,525525,525528,525531,525534],{"className":525518},[31676],[34,525520],{"className":525521,"style":31774},[31680],[34,525523,456],{"className":525524},[31782],[34,525526,111],{"className":525527},[31685,31686],[34,525529],{"className":525530,"style":32604},[31845],[34,525532,223],{"className":525533},[32024],[34,525535],{"className":525536,"style":32604},[31845],[34,525538,525540,525543,525546,525549,525552,525555,525558,525561],{"className":525539},[31676],[34,525541],{"className":525542,"style":31774},[31680],[34,525544,106063],{"className":525545,"style":67482},[31685,31686],[34,525547,2557],{"className":525548},[31690],[34,525550,1323],{"className":525551},[31782],[34,525553,1697],{"className":525554},[31685,31686],[34,525556],{"className":525557,"style":32604},[31845],[34,525559,223],{"className":525560},[32024],[34,525562],{"className":525563,"style":32604},[31845],[34,525565,525567,525570,525573,525576,525579,525582],{"className":525566},[31676],[34,525568],{"className":525569,"style":31774},[31680],[34,525571,31965],{"className":525572},[31685,31686],[34,525574,2557],{"className":525575},[31690],[34,525577],{"className":525578,"style":32604},[31845],[34,525580,223],{"className":525581},[32024],[34,525583],{"className":525584,"style":32604},[31845],[34,525586,525588,525591,525594],{"className":525587},[31676],[34,525589],{"className":525590,"style":31774},[31680],[34,525592,106063],{"className":525593,"style":67482},[31685,31686],[34,525595,2557],{"className":525596},[31690],"\n根据乘法定义：\n",[34,525599,525601,525631],{"className":525600},[31642],[34,525602,525604],{"className":525603},[31646],[31648,525605,525606],{"xmlns":31650},[31652,525607,525608,525628],{},[31655,525609,525610,525612,525614,525616,525618,525620,525622,525624,525626],{},[31662,525611,202],{},[31658,525613,111],{},[31662,525615,456],{"stretchy":702},[31658,525617,1697],{},[31662,525619,223],{},[31658,525621,31965],{},[31662,525623,2557],{"stretchy":702},[31662,525625,223],{},[31658,525627,106063],{},[31665,525629,525630],{"encoding":31667},"= a(b+c) + I",[34,525632,525634,525646,525670,525691],{"className":525633,"ariaHidden":758},[31672],[34,525635,525637,525640,525643],{"className":525636},[31676],[34,525638],{"className":525639,"style":73089},[31680],[34,525641,202],{"className":525642},[31850],[34,525644],{"className":525645,"style":31846},[31845],[34,525647,525649,525652,525655,525658,525661,525664,525667],{"className":525648},[31676],[34,525650],{"className":525651,"style":31774},[31680],[34,525653,111],{"className":525654},[31685,31686],[34,525656,456],{"className":525657},[31782],[34,525659,1697],{"className":525660},[31685,31686],[34,525662],{"className":525663,"style":32604},[31845],[34,525665,223],{"className":525666},[32024],[34,525668],{"className":525669,"style":32604},[31845],[34,525671,525673,525676,525679,525682,525685,525688],{"className":525672},[31676],[34,525674],{"className":525675,"style":31774},[31680],[34,525677,31965],{"className":525678},[31685,31686],[34,525680,2557],{"className":525681},[31690],[34,525683],{"className":525684,"style":32604},[31845],[34,525686,223],{"className":525687},[32024],[34,525689],{"className":525690,"style":32604},[31845],[34,525692,525694,525697],{"className":525693},[31676],[34,525695],{"className":525696,"style":33218},[31680],[34,525698,106063],{"className":525699,"style":67482},[31685,31686],"\n利用 ",[34,525702,525704,525717],{"className":525703},[31642],[34,525705,525707],{"className":525706},[31646],[31648,525708,525709],{"xmlns":31650},[31652,525710,525711,525715],{},[31655,525712,525713],{},[31658,525714,34386],{},[31665,525716,34386],{"encoding":31667},[34,525718,525720],{"className":525719,"ariaHidden":758},[31672],[34,525721,525723,525726],{"className":525722},[31676],[34,525724],{"className":525725,"style":33218},[31680],[34,525727,34386],{"className":525728,"style":34427},[31685,31686]," 中的分配律 ",[34,525731,525733,525769],{"className":525732},[31642],[34,525734,525736],{"className":525735},[31646],[31648,525737,525738],{"xmlns":31650},[31652,525739,525740,525766],{},[31655,525741,525742,525744,525746,525748,525750,525752,525754,525756,525758,525760,525762,525764],{},[31658,525743,111],{},[31662,525745,456],{"stretchy":702},[31658,525747,1697],{},[31662,525749,223],{},[31658,525751,31965],{},[31662,525753,2557],{"stretchy":702},[31662,525755,202],{},[31658,525757,111],{},[31658,525759,1697],{},[31662,525761,223],{},[31658,525763,111],{},[31658,525765,31965],{},[31665,525767,525768],{"encoding":31667},"a(b+c) = ab+ac",[34,525770,525772,525796,525817,525835],{"className":525771,"ariaHidden":758},[31672],[34,525773,525775,525778,525781,525784,525787,525790,525793],{"className":525774},[31676],[34,525776],{"className":525777,"style":31774},[31680],[34,525779,111],{"className":525780},[31685,31686],[34,525782,456],{"className":525783},[31782],[34,525785,1697],{"className":525786},[31685,31686],[34,525788],{"className":525789,"style":32604},[31845],[34,525791,223],{"className":525792},[32024],[34,525794],{"className":525795,"style":32604},[31845],[34,525797,525799,525802,525805,525808,525811,525814],{"className":525798},[31676],[34,525800],{"className":525801,"style":31774},[31680],[34,525803,31965],{"className":525804},[31685,31686],[34,525806,2557],{"className":525807},[31690],[34,525809],{"className":525810,"style":31846},[31845],[34,525812,202],{"className":525813},[31850],[34,525815],{"className":525816,"style":31846},[31845],[34,525818,525820,525823,525826,525829,525832],{"className":525819},[31676],[34,525821],{"className":525822,"style":34299},[31680],[34,525824,324191],{"className":525825},[31685,31686],[34,525827],{"className":525828,"style":32604},[31845],[34,525830,223],{"className":525831},[32024],[34,525833],{"className":525834,"style":32604},[31845],[34,525836,525838,525841,525844],{"className":525837},[31676],[34,525839],{"className":525840,"style":32527},[31680],[34,525842,111],{"className":525843},[31685,31686],[34,525845,31965],{"className":525846},[31685,31686],[34,525848,525850,525882],{"className":525849},[31642],[34,525851,525853],{"className":525852},[31646],[31648,525854,525855],{"xmlns":31650},[31652,525856,525857,525879],{},[31655,525858,525859,525861,525863,525865,525867,525869,525871,525873,525875,525877],{},[31662,525860,202],{},[31662,525862,456],{"stretchy":702},[31658,525864,111],{},[31658,525866,1697],{},[31662,525868,223],{},[31658,525870,111],{},[31658,525872,31965],{},[31662,525874,2557],{"stretchy":702},[31662,525876,223],{},[31658,525878,106063],{},[31665,525880,525881],{"encoding":31667},"= (ab+ac) + I",[34,525883,525885,525897,525918,525942],{"className":525884,"ariaHidden":758},[31672],[34,525886,525888,525891,525894],{"className":525887},[31676],[34,525889],{"className":525890,"style":73089},[31680],[34,525892,202],{"className":525893},[31850],[34,525895],{"className":525896,"style":31846},[31845],[34,525898,525900,525903,525906,525909,525912,525915],{"className":525899},[31676],[34,525901],{"className":525902,"style":31774},[31680],[34,525904,456],{"className":525905},[31782],[34,525907,324191],{"className":525908},[31685,31686],[34,525910],{"className":525911,"style":32604},[31845],[34,525913,223],{"className":525914},[32024],[34,525916],{"className":525917,"style":32604},[31845],[34,525919,525921,525924,525927,525930,525933,525936,525939],{"className":525920},[31676],[34,525922],{"className":525923,"style":31774},[31680],[34,525925,111],{"className":525926},[31685,31686],[34,525928,31965],{"className":525929},[31685,31686],[34,525931,2557],{"className":525932},[31690],[34,525934],{"className":525935,"style":32604},[31845],[34,525937,223],{"className":525938},[32024],[34,525940],{"className":525941,"style":32604},[31845],[34,525943,525945,525948],{"className":525944},[31676],[34,525946],{"className":525947,"style":33218},[31680],[34,525949,106063],{"className":525950,"style":67482},[31685,31686],"\n根据商环加法定义拆开：\n",[34,525953,525955,526039],{"className":525954},[31642],[34,525956,525958],{"className":525957},[31646],[31648,525959,525960],{"xmlns":31650},[31652,525961,525962,526036],{},[31655,525963,525964,525966,525968,525970,525972,525974,525976,525978,525980,525982,525984,525986,525988,525990,525992,525994,525996,525998,526000,526002,526004,526006,526008,526010,526012,526014,526016,526018,526020,526022,526024,526026,526028,526030,526032,526034],{},[31662,525965,202],{},[31662,525967,456],{"stretchy":702},[31658,525969,111],{},[31658,525971,1697],{},[31662,525973,223],{},[31658,525975,106063],{},[31662,525977,2557],{"stretchy":702},[31662,525979,223],{},[31662,525981,456],{"stretchy":702},[31658,525983,111],{},[31658,525985,31965],{},[31662,525987,223],{},[31658,525989,106063],{},[31662,525991,2557],{"stretchy":702},[31662,525993,202],{},[31662,525995,456],{"stretchy":702},[31658,525997,111],{},[31662,525999,223],{},[31658,526001,106063],{},[31662,526003,2557],{"stretchy":702},[31662,526005,456],{"stretchy":702},[31658,526007,1697],{},[31662,526009,223],{},[31658,526011,106063],{},[31662,526013,2557],{"stretchy":702},[31662,526015,223],{},[31662,526017,456],{"stretchy":702},[31658,526019,111],{},[31662,526021,223],{},[31658,526023,106063],{},[31662,526025,2557],{"stretchy":702},[31662,526027,456],{"stretchy":702},[31658,526029,31965],{},[31662,526031,223],{},[31658,526033,106063],{},[31662,526035,2557],{"stretchy":702},[31665,526037,526038],{"encoding":31667},"= (ab+I) + (ac+I) = (a+I)(b+I) + (a+I)(c+I)",[34,526040,526042,526054,526075,526096,526120,526141,526162,526189,526210,526231,526258],{"className":526041,"ariaHidden":758},[31672],[34,526043,526045,526048,526051],{"className":526044},[31676],[34,526046],{"className":526047,"style":73089},[31680],[34,526049,202],{"className":526050},[31850],[34,526052],{"className":526053,"style":31846},[31845],[34,526055,526057,526060,526063,526066,526069,526072],{"className":526056},[31676],[34,526058],{"className":526059,"style":31774},[31680],[34,526061,456],{"className":526062},[31782],[34,526064,324191],{"className":526065},[31685,31686],[34,526067],{"className":526068,"style":32604},[31845],[34,526070,223],{"className":526071},[32024],[34,526073],{"className":526074,"style":32604},[31845],[34,526076,526078,526081,526084,526087,526090,526093],{"className":526077},[31676],[34,526079],{"className":526080,"style":31774},[31680],[34,526082,106063],{"className":526083,"style":67482},[31685,31686],[34,526085,2557],{"className":526086},[31690],[34,526088],{"className":526089,"style":32604},[31845],[34,526091,223],{"className":526092},[32024],[34,526094],{"className":526095,"style":32604},[31845],[34,526097,526099,526102,526105,526108,526111,526114,526117],{"className":526098},[31676],[34,526100],{"className":526101,"style":31774},[31680],[34,526103,456],{"className":526104},[31782],[34,526106,111],{"className":526107},[31685,31686],[34,526109,31965],{"className":526110},[31685,31686],[34,526112],{"className":526113,"style":32604},[31845],[34,526115,223],{"className":526116},[32024],[34,526118],{"className":526119,"style":32604},[31845],[34,526121,526123,526126,526129,526132,526135,526138],{"className":526122},[31676],[34,526124],{"className":526125,"style":31774},[31680],[34,526127,106063],{"className":526128,"style":67482},[31685,31686],[34,526130,2557],{"className":526131},[31690],[34,526133],{"className":526134,"style":31846},[31845],[34,526136,202],{"className":526137},[31850],[34,526139],{"className":526140,"style":31846},[31845],[34,526142,526144,526147,526150,526153,526156,526159],{"className":526143},[31676],[34,526145],{"className":526146,"style":31774},[31680],[34,526148,456],{"className":526149},[31782],[34,526151,111],{"className":526152},[31685,31686],[34,526154],{"className":526155,"style":32604},[31845],[34,526157,223],{"className":526158},[32024],[34,526160],{"className":526161,"style":32604},[31845],[34,526163,526165,526168,526171,526174,526177,526180,526183,526186],{"className":526164},[31676],[34,526166],{"className":526167,"style":31774},[31680],[34,526169,106063],{"className":526170,"style":67482},[31685,31686],[34,526172,2557],{"className":526173},[31690],[34,526175,456],{"className":526176},[31782],[34,526178,1697],{"className":526179},[31685,31686],[34,526181],{"className":526182,"style":32604},[31845],[34,526184,223],{"className":526185},[32024],[34,526187],{"className":526188,"style":32604},[31845],[34,526190,526192,526195,526198,526201,526204,526207],{"className":526191},[31676],[34,526193],{"className":526194,"style":31774},[31680],[34,526196,106063],{"className":526197,"style":67482},[31685,31686],[34,526199,2557],{"className":526200},[31690],[34,526202],{"className":526203,"style":32604},[31845],[34,526205,223],{"className":526206},[32024],[34,526208],{"className":526209,"style":32604},[31845],[34,526211,526213,526216,526219,526222,526225,526228],{"className":526212},[31676],[34,526214],{"className":526215,"style":31774},[31680],[34,526217,456],{"className":526218},[31782],[34,526220,111],{"className":526221},[31685,31686],[34,526223],{"className":526224,"style":32604},[31845],[34,526226,223],{"className":526227},[32024],[34,526229],{"className":526230,"style":32604},[31845],[34,526232,526234,526237,526240,526243,526246,526249,526252,526255],{"className":526233},[31676],[34,526235],{"className":526236,"style":31774},[31680],[34,526238,106063],{"className":526239,"style":67482},[31685,31686],[34,526241,2557],{"className":526242},[31690],[34,526244,456],{"className":526245},[31782],[34,526247,31965],{"className":526248},[31685,31686],[34,526250],{"className":526251,"style":32604},[31845],[34,526253,223],{"className":526254},[32024],[34,526256],{"className":526257,"style":32604},[31845],[34,526259,526261,526264,526267],{"className":526260},[31676],[34,526262],{"className":526263,"style":31774},[31680],[34,526265,106063],{"className":526266,"style":67482},[31685,31686],[34,526268,2557],{"className":526269},[31690],"\n右分配律同理。",[15,526272,526273,526275,526276,526350,526351],{},[2627,526274,365086],{},"：满足所有条件，",[34,526277,526279,526308],{"className":526278},[31642],[34,526280,526282],{"className":526281},[31646],[31648,526283,526284],{"xmlns":31650},[31652,526285,526286,526306],{},[31655,526287,526288,526290,526292,526294,526296,526298,526300,526302,526304],{},[31662,526289,456],{"stretchy":702},[31658,526291,34386],{},[31658,526293,38080],{"mathvariant":38079},[31658,526295,106063],{},[31662,526297,31960],{"separator":758},[31662,526299,223],{},[31662,526301,31960],{"separator":758},[31662,526303,71191],{},[31662,526305,2557],{"stretchy":702},[31665,526307,520928],{"encoding":31667},[34,526309,526311],{"className":526310,"ariaHidden":758},[31672],[34,526312,526314,526317,526320,526323,526326,526329,526332,526335,526338,526341,526344,526347],{"className":526313},[31676],[34,526315],{"className":526316,"style":31774},[31680],[34,526318,456],{"className":526319},[31782],[34,526321,34386],{"className":526322,"style":34427},[31685,31686],[34,526324,38080],{"className":526325},[31685],[34,526327,106063],{"className":526328,"style":67482},[31685,31686],[34,526330,31960],{"className":526331},[32119],[34,526333],{"className":526334,"style":32123},[31845],[34,526336,223],{"className":526337},[31685],[34,526339,31960],{"className":526340},[32119],[34,526342],{"className":526343,"style":32123},[31845],[34,526345,71191],{"className":526346},[31685],[34,526348,2557],{"className":526349},[31690]," 确实是一个环。 ",[34,526352,526354,526367],{"className":526353},[31642],[34,526355,526357],{"className":526356},[31646],[31648,526358,526359],{"xmlns":31650},[31652,526360,526361,526365],{},[31655,526362,526363],{},[31658,526364,445204],{"mathvariant":38079},[31665,526366,445207],{"encoding":31667},[34,526368,526370],{"className":526369,"ariaHidden":758},[31672],[34,526371,526373,526376],{"className":526372},[31676],[34,526374],{"className":526375,"style":445217},[31680],[34,526377,445204],{"className":526378},[31685,129261],[32188,526380],{},[253,526382,526384],{"id":526383},"题目-4商环的性质继承",[2627,526385,526386],{},"题目 4：商环的性质继承",[15,526388,526389,69123,526391,526442,526443,496829,526471,526509,526510,55016,526594,11008,526644,389666],{},[2627,526390,101191],{},[34,526392,526394,526412],{"className":526393},[31642],[34,526395,526397],{"className":526396},[31646],[31648,526398,526399],{"xmlns":31650},[31652,526400,526401,526409],{},[31655,526402,526403,526405,526407],{},[31658,526404,106063],{},[31662,526406,512725],{},[31658,526408,34386],{},[31665,526410,526411],{"encoding":31667},"I \\trianglelefteq R",[34,526413,526415,526433],{"className":526414,"ariaHidden":758},[31672],[34,526416,526418,526421,526424,526427,526430],{"className":526417},[31676],[34,526419],{"className":526420,"style":515471},[31680],[34,526422,106063],{"className":526423,"style":67482},[31685,31686],[34,526425],{"className":526426,"style":31846},[31845],[34,526428,512725],{"className":526429},[31850,129261],[34,526431],{"className":526432,"style":31846},[31845],[34,526434,526436,526439],{"className":526435},[31676],[34,526437],{"className":526438,"style":33218},[31680],[34,526440,34386],{"className":526441,"style":34427},[31685,31686],"。证明：\n(i) 若 ",[34,526444,526446,526459],{"className":526445},[31642],[34,526447,526449],{"className":526448},[31646],[31648,526450,526451],{"xmlns":31650},[31652,526452,526453,526457],{},[31655,526454,526455],{},[31658,526456,34386],{},[31665,526458,34386],{"encoding":31667},[34,526460,526462],{"className":526461,"ariaHidden":758},[31672],[34,526463,526465,526468],{"className":526464},[31676],[34,526466],{"className":526467,"style":33218},[31680],[34,526469,34386],{"className":526470,"style":34427},[31685,31686],[34,526472,526474,526491],{"className":526473},[31642],[34,526475,526477],{"className":526476},[31646],[31648,526478,526479],{"xmlns":31650},[31652,526480,526481,526489],{},[31655,526482,526483,526485,526487],{},[31658,526484,34386],{},[31658,526486,38080],{"mathvariant":38079},[31658,526488,106063],{},[31665,526490,337927],{"encoding":31667},[34,526492,526494],{"className":526493,"ariaHidden":758},[31672],[34,526495,526497,526500,526503,526506],{"className":526496},[31676],[34,526498],{"className":526499,"style":31774},[31680],[34,526501,34386],{"className":526502,"style":34427},[31685,31686],[34,526504,38080],{"className":526505},[31685],[34,526507,106063],{"className":526508,"style":67482},[31685,31686]," 也是。\n(ii) 若 ",[34,526511,526513,526531],{"className":526512},[31642],[34,526514,526516],{"className":526515},[31646],[31648,526517,526518],{"xmlns":31650},[31652,526519,526520,526528],{},[31655,526521,526522,526524,526526],{},[31658,526523,106063],{},[31662,526525,54707],{"mathvariant":38079},[31658,526527,34386],{},[31665,526529,526530],{"encoding":31667},"I \\neq R",[34,526532,526534,526585],{"className":526533,"ariaHidden":758},[31672],[34,526535,526537,526540,526543,526546,526582],{"className":526536},[31676],[34,526538],{"className":526539,"style":43217},[31680],[34,526541,106063],{"className":526542,"style":67482},[31685,31686],[34,526544],{"className":526545,"style":31846},[31845],[34,526547,526549,526576,526579],{"className":526548},[31850],[34,526550,526552],{"className":526551},[31850],[34,526553,526555],{"className":526554},[31685,54774],[34,526556,526558],{"className":526557},[54778],[34,526559,526561,526564,526573],{"className":526560},[54782],[34,526562],{"className":526563,"style":43217},[31680],[34,526565,526567],{"className":526566},[54789],[34,526568,526570],{"className":526569},[31685],[34,526571,54796],{"className":526572},[31850],[34,526574],{"className":526575},[54800],[34,526577],{"className":526578},[31845,54804],[34,526580,202],{"className":526581},[31850],[34,526583],{"className":526584,"style":31846},[31845],[34,526586,526588,526591],{"className":526587},[31676],[34,526589],{"className":526590,"style":33218},[31680],[34,526592,34386],{"className":526593,"style":34427},[31685,31686],[34,526595,526597,526614],{"className":526596},[31642],[34,526598,526600],{"className":526599},[31646],[31648,526601,526602],{"xmlns":31650},[31652,526603,526604,526612],{},[31655,526605,526606,526608,526610],{},[31749,526607,640],{},[31662,526609,223],{},[31658,526611,106063],{},[31665,526613,512337],{"encoding":31667},[34,526615,526617,526635],{"className":526616,"ariaHidden":758},[31672],[34,526618,526620,526623,526626,526629,526632],{"className":526619},[31676],[34,526621],{"className":526622,"style":35942},[31680],[34,526624,640],{"className":526625},[31685],[34,526627],{"className":526628,"style":32604},[31845],[34,526630,223],{"className":526631},[32024],[34,526633],{"className":526634,"style":32604},[31845],[34,526636,526638,526641],{"className":526637},[31676],[34,526639],{"className":526640,"style":33218},[31680],[34,526642,106063],{"className":526643,"style":67482},[31685,31686],[34,526645,526647,526664],{"className":526646},[31642],[34,526648,526650],{"className":526649},[31646],[31648,526651,526652],{"xmlns":31650},[31652,526653,526654,526662],{},[31655,526655,526656,526658,526660],{},[31658,526657,34386],{},[31658,526659,38080],{"mathvariant":38079},[31658,526661,106063],{},[31665,526663,337927],{"encoding":31667},[34,526665,526667],{"className":526666,"ariaHidden":758},[31672],[34,526668,526670,526673,526676,526679],{"className":526669},[31676],[34,526671],{"className":526672,"style":31774},[31680],[34,526674,34386],{"className":526675,"style":34427},[31685,31686],[34,526677,38080],{"className":526678},[31685],[34,526680,106063],{"className":526681,"style":67482},[31685,31686],[15,526683,526684],{},[2627,526685,463462],{},[15,526687,526688],{},"这是在说：大环有的好性质，商环通常也能继承下来。",[15,526690,526691],{},[2627,526692,463779],{},[15,526694,526695],{},[2627,526696,526697],{},"(i) 交换性",[105,526699,526700,526852,526993],{},[108,526701,526702,2664,526705,526733,526734,64615,526797,347],{},[2627,526703,526704],{},"前提",[34,526706,526708,526721],{"className":526707},[31642],[34,526709,526711],{"className":526710},[31646],[31648,526712,526713],{"xmlns":31650},[31652,526714,526715,526719],{},[31655,526716,526717],{},[31658,526718,34386],{},[31665,526720,34386],{"encoding":31667},[34,526722,526724],{"className":526723,"ariaHidden":758},[31672],[34,526725,526727,526730],{"className":526726},[31676],[34,526728],{"className":526729,"style":33218},[31680],[34,526731,34386],{"className":526732,"style":34427},[31685,31686]," 是交换环，即对任意 ",[34,526735,526737,526758],{"className":526736},[31642],[34,526738,526740],{"className":526739},[31646],[31648,526741,526742],{"xmlns":31650},[31652,526743,526744,526756],{},[31655,526745,526746,526748,526750,526752,526754],{},[31658,526747,111],{},[31662,526749,31960],{"separator":758},[31658,526751,1697],{},[31662,526753,112296],{},[31658,526755,34386],{},[31665,526757,330959],{"encoding":31667},[34,526759,526761,526788],{"className":526760,"ariaHidden":758},[31672],[34,526762,526764,526767,526770,526773,526776,526779,526782,526785],{"className":526763},[31676],[34,526765],{"className":526766,"style":43217},[31680],[34,526768,111],{"className":526769},[31685,31686],[34,526771,31960],{"className":526772},[32119],[34,526774],{"className":526775,"style":32123},[31845],[34,526777,1697],{"className":526778},[31685,31686],[34,526780],{"className":526781,"style":31846},[31845],[34,526783,112296],{"className":526784},[31850],[34,526786],{"className":526787,"style":31846},[31845],[34,526789,526791,526794],{"className":526790},[31676],[34,526792],{"className":526793,"style":33218},[31680],[34,526795,34386],{"className":526796,"style":34427},[31685,31686],[34,526798,526800,526822],{"className":526799},[31642],[34,526801,526803],{"className":526802},[31646],[31648,526804,526805],{"xmlns":31650},[31652,526806,526807,526819],{},[31655,526808,526809,526811,526813,526815,526817],{},[31658,526810,111],{},[31658,526812,1697],{},[31662,526814,202],{},[31658,526816,1697],{},[31658,526818,111],{},[31665,526820,526821],{"encoding":31667},"ab = ba",[34,526823,526825,526843],{"className":526824,"ariaHidden":758},[31672],[34,526826,526828,526831,526834,526837,526840],{"className":526827},[31676],[34,526829],{"className":526830,"style":31681},[31680],[34,526832,324191],{"className":526833},[31685,31686],[34,526835],{"className":526836,"style":31846},[31845],[34,526838,202],{"className":526839},[31850],[34,526841],{"className":526842,"style":31846},[31845],[34,526844,526846,526849],{"className":526845},[31676],[34,526847],{"className":526848,"style":31681},[31680],[34,526850,367454],{"className":526851},[31685,31686],[108,526853,526854,526857,526858,64615,526932,347],{},[2627,526855,526856],{},"要证","：对任意 ",[34,526859,526861,526887],{"className":526860},[31642],[34,526862,526864],{"className":526863},[31646],[31648,526865,526866],{"xmlns":31650},[31652,526867,526868,526884],{},[31655,526869,526870,526872,526874,526876,526878,526880,526882],{},[31658,526871,67467],{},[31662,526873,31960],{"separator":758},[31658,526875,67731],{},[31662,526877,112296],{},[31658,526879,34386],{},[31658,526881,38080],{"mathvariant":38079},[31658,526883,106063],{},[31665,526885,526886],{"encoding":31667},"X, Y \\in R\u002FI",[34,526888,526890,526917],{"className":526889,"ariaHidden":758},[31672],[34,526891,526893,526896,526899,526902,526905,526908,526911,526914],{"className":526892},[31676],[34,526894],{"className":526895,"style":70739},[31680],[34,526897,67467],{"className":526898,"style":67482},[31685,31686],[34,526900,31960],{"className":526901},[32119],[34,526903],{"className":526904,"style":32123},[31845],[34,526906,67731],{"className":526907,"style":32604},[31685,31686],[34,526909],{"className":526910,"style":31846},[31845],[34,526912,112296],{"className":526913},[31850],[34,526915],{"className":526916,"style":31846},[31845],[34,526918,526920,526923,526926,526929],{"className":526919},[31676],[34,526921],{"className":526922,"style":31774},[31680],[34,526924,34386],{"className":526925,"style":34427},[31685,31686],[34,526927,38080],{"className":526928},[31685],[34,526930,106063],{"className":526931,"style":67482},[31685,31686],[34,526933,526935,526957],{"className":526934},[31642],[34,526936,526938],{"className":526937},[31646],[31648,526939,526940],{"xmlns":31650},[31652,526941,526942,526954],{},[31655,526943,526944,526946,526948,526950,526952],{},[31658,526945,67467],{},[31658,526947,67731],{},[31662,526949,202],{},[31658,526951,67731],{},[31658,526953,67467],{},[31665,526955,526956],{"encoding":31667},"XY = YX",[34,526958,526960,526981],{"className":526959,"ariaHidden":758},[31672],[34,526961,526963,526966,526969,526972,526975,526978],{"className":526962},[31676],[34,526964],{"className":526965,"style":33218},[31680],[34,526967,67467],{"className":526968,"style":67482},[31685,31686],[34,526970,67731],{"className":526971,"style":32604},[31685,31686],[34,526973],{"className":526974,"style":31846},[31845],[34,526976,202],{"className":526977},[31850],[34,526979],{"className":526980,"style":31846},[31845],[34,526982,526984,526987,526990],{"className":526983},[31676],[34,526985],{"className":526986,"style":33218},[31680],[34,526988,67731],{"className":526989,"style":32604},[31685,31686],[34,526991,67467],{"className":526992,"style":67482},[31685,31686],[108,526994,526995,516416,526997,39984,527127,527315,527503,527504,93436,527532,13977,527586,452821,527685,355408,527745,527783],{},[2627,526996,390672],{},[34,526998,527000,527034],{"className":526999},[31642],[34,527001,527003],{"className":527002},[31646],[31648,527004,527005],{"xmlns":31650},[31652,527006,527007,527031],{},[31655,527008,527009,527011,527013,527015,527017,527019,527021,527023,527025,527027,527029],{},[31658,527010,67467],{},[31662,527012,202],{},[31658,527014,111],{},[31662,527016,223],{},[31658,527018,106063],{},[31662,527020,31960],{"separator":758},[31658,527022,67731],{},[31662,527024,202],{},[31658,527026,1697],{},[31662,527028,223],{},[31658,527030,106063],{},[31665,527032,527033],{"encoding":31667},"X = a+I, Y = b+I",[34,527035,527037,527055,527073,527100,527118],{"className":527036,"ariaHidden":758},[31672],[34,527038,527040,527043,527046,527049,527052],{"className":527039},[31676],[34,527041],{"className":527042,"style":33218},[31680],[34,527044,67467],{"className":527045,"style":67482},[31685,31686],[34,527047],{"className":527048,"style":31846},[31845],[34,527050,202],{"className":527051},[31850],[34,527053],{"className":527054,"style":31846},[31845],[34,527056,527058,527061,527064,527067,527070],{"className":527057},[31676],[34,527059],{"className":527060,"style":32597},[31680],[34,527062,111],{"className":527063},[31685,31686],[34,527065],{"className":527066,"style":32604},[31845],[34,527068,223],{"className":527069},[32024],[34,527071],{"className":527072,"style":32604},[31845],[34,527074,527076,527079,527082,527085,527088,527091,527094,527097],{"className":527075},[31676],[34,527077],{"className":527078,"style":70739},[31680],[34,527080,106063],{"className":527081,"style":67482},[31685,31686],[34,527083,31960],{"className":527084},[32119],[34,527086],{"className":527087,"style":32123},[31845],[34,527089,67731],{"className":527090,"style":32604},[31685,31686],[34,527092],{"className":527093,"style":31846},[31845],[34,527095,202],{"className":527096},[31850],[34,527098],{"className":527099,"style":31846},[31845],[34,527101,527103,527106,527109,527112,527115],{"className":527102},[31676],[34,527104],{"className":527105,"style":34299},[31680],[34,527107,1697],{"className":527108},[31685,31686],[34,527110],{"className":527111,"style":32604},[31845],[34,527113,223],{"className":527114},[32024],[34,527116],{"className":527117,"style":32604},[31845],[34,527119,527121,527124],{"className":527120},[31676],[34,527122],{"className":527123,"style":33218},[31680],[34,527125,106063],{"className":527126,"style":67482},[31685,31686],[34,527128,527130,527180],{"className":527129},[31642],[34,527131,527133],{"className":527132},[31646],[31648,527134,527135],{"xmlns":31650},[31652,527136,527137,527177],{},[31655,527138,527139,527141,527143,527145,527147,527149,527151,527153,527155,527157,527159,527161,527163,527165,527167,527169,527171,527173,527175],{},[31658,527140,67467],{},[31662,527142,71191],{},[31658,527144,67731],{},[31662,527146,202],{},[31662,527148,456],{"stretchy":702},[31658,527150,111],{},[31662,527152,223],{},[31658,527154,106063],{},[31662,527156,2557],{"stretchy":702},[31662,527158,456],{"stretchy":702},[31658,527160,1697],{},[31662,527162,223],{},[31658,527164,106063],{},[31662,527166,2557],{"stretchy":702},[31662,527168,202],{},[31658,527170,111],{},[31658,527172,1697],{},[31662,527174,223],{},[31658,527176,106063],{},[31665,527178,527179],{"encoding":31667},"X \\cdot Y = (a+I)(b+I) = ab + I",[34,527181,527183,527201,527219,527240,527267,527288,527306],{"className":527182,"ariaHidden":758},[31672],[34,527184,527186,527189,527192,527195,527198],{"className":527185},[31676],[34,527187],{"className":527188,"style":33218},[31680],[34,527190,67467],{"className":527191,"style":67482},[31685,31686],[34,527193],{"className":527194,"style":32604},[31845],[34,527196,71191],{"className":527197},[32024],[34,527199],{"className":527200,"style":32604},[31845],[34,527202,527204,527207,527210,527213,527216],{"className":527203},[31676],[34,527205],{"className":527206,"style":33218},[31680],[34,527208,67731],{"className":527209,"style":32604},[31685,31686],[34,527211],{"className":527212,"style":31846},[31845],[34,527214,202],{"className":527215},[31850],[34,527217],{"className":527218,"style":31846},[31845],[34,527220,527222,527225,527228,527231,527234,527237],{"className":527221},[31676],[34,527223],{"className":527224,"style":31774},[31680],[34,527226,456],{"className":527227},[31782],[34,527229,111],{"className":527230},[31685,31686],[34,527232],{"className":527233,"style":32604},[31845],[34,527235,223],{"className":527236},[32024],[34,527238],{"className":527239,"style":32604},[31845],[34,527241,527243,527246,527249,527252,527255,527258,527261,527264],{"className":527242},[31676],[34,527244],{"className":527245,"style":31774},[31680],[34,527247,106063],{"className":527248,"style":67482},[31685,31686],[34,527250,2557],{"className":527251},[31690],[34,527253,456],{"className":527254},[31782],[34,527256,1697],{"className":527257},[31685,31686],[34,527259],{"className":527260,"style":32604},[31845],[34,527262,223],{"className":527263},[32024],[34,527265],{"className":527266,"style":32604},[31845],[34,527268,527270,527273,527276,527279,527282,527285],{"className":527269},[31676],[34,527271],{"className":527272,"style":31774},[31680],[34,527274,106063],{"className":527275,"style":67482},[31685,31686],[34,527277,2557],{"className":527278},[31690],[34,527280],{"className":527281,"style":31846},[31845],[34,527283,202],{"className":527284},[31850],[34,527286],{"className":527287,"style":31846},[31845],[34,527289,527291,527294,527297,527300,527303],{"className":527290},[31676],[34,527292],{"className":527293,"style":34299},[31680],[34,527295,324191],{"className":527296},[31685,31686],[34,527298],{"className":527299,"style":32604},[31845],[34,527301,223],{"className":527302},[32024],[34,527304],{"className":527305,"style":32604},[31845],[34,527307,527309,527312],{"className":527308},[31676],[34,527310],{"className":527311,"style":33218},[31680],[34,527313,106063],{"className":527314,"style":67482},[31685,31686],[34,527316,527318,527368],{"className":527317},[31642],[34,527319,527321],{"className":527320},[31646],[31648,527322,527323],{"xmlns":31650},[31652,527324,527325,527365],{},[31655,527326,527327,527329,527331,527333,527335,527337,527339,527341,527343,527345,527347,527349,527351,527353,527355,527357,527359,527361,527363],{},[31658,527328,67731],{},[31662,527330,71191],{},[31658,527332,67467],{},[31662,527334,202],{},[31662,527336,456],{"stretchy":702},[31658,527338,1697],{},[31662,527340,223],{},[31658,527342,106063],{},[31662,527344,2557],{"stretchy":702},[31662,527346,456],{"stretchy":702},[31658,527348,111],{},[31662,527350,223],{},[31658,527352,106063],{},[31662,527354,2557],{"stretchy":702},[31662,527356,202],{},[31658,527358,1697],{},[31658,527360,111],{},[31662,527362,223],{},[31658,527364,106063],{},[31665,527366,527367],{"encoding":31667},"Y \\cdot X = (b+I)(a+I) = ba + I",[34,527369,527371,527389,527407,527428,527455,527476,527494],{"className":527370,"ariaHidden":758},[31672],[34,527372,527374,527377,527380,527383,527386],{"className":527373},[31676],[34,527375],{"className":527376,"style":33218},[31680],[34,527378,67731],{"className":527379,"style":32604},[31685,31686],[34,527381],{"className":527382,"style":32604},[31845],[34,527384,71191],{"className":527385},[32024],[34,527387],{"className":527388,"style":32604},[31845],[34,527390,527392,527395,527398,527401,527404],{"className":527391},[31676],[34,527393],{"className":527394,"style":33218},[31680],[34,527396,67467],{"className":527397,"style":67482},[31685,31686],[34,527399],{"className":527400,"style":31846},[31845],[34,527402,202],{"className":527403},[31850],[34,527405],{"className":527406,"style":31846},[31845],[34,527408,527410,527413,527416,527419,527422,527425],{"className":527409},[31676],[34,527411],{"className":527412,"style":31774},[31680],[34,527414,456],{"className":527415},[31782],[34,527417,1697],{"className":527418},[31685,31686],[34,527420],{"className":527421,"style":32604},[31845],[34,527423,223],{"className":527424},[32024],[34,527426],{"className":527427,"style":32604},[31845],[34,527429,527431,527434,527437,527440,527443,527446,527449,527452],{"className":527430},[31676],[34,527432],{"className":527433,"style":31774},[31680],[34,527435,106063],{"className":527436,"style":67482},[31685,31686],[34,527438,2557],{"className":527439},[31690],[34,527441,456],{"className":527442},[31782],[34,527444,111],{"className":527445},[31685,31686],[34,527447],{"className":527448,"style":32604},[31845],[34,527450,223],{"className":527451},[32024],[34,527453],{"className":527454,"style":32604},[31845],[34,527456,527458,527461,527464,527467,527470,527473],{"className":527457},[31676],[34,527459],{"className":527460,"style":31774},[31680],[34,527462,106063],{"className":527463,"style":67482},[31685,31686],[34,527465,2557],{"className":527466},[31690],[34,527468],{"className":527469,"style":31846},[31845],[34,527471,202],{"className":527472},[31850],[34,527474],{"className":527475,"style":31846},[31845],[34,527477,527479,527482,527485,527488,527491],{"className":527478},[31676],[34,527480],{"className":527481,"style":34299},[31680],[34,527483,367454],{"className":527484},[31685,31686],[34,527486],{"className":527487,"style":32604},[31845],[34,527489,223],{"className":527490},[32024],[34,527492],{"className":527493,"style":32604},[31845],[34,527495,527497,527500],{"className":527496},[31676],[34,527498],{"className":527499,"style":33218},[31680],[34,527501,106063],{"className":527502,"style":67482},[31685,31686],"\n因为在 ",[34,527505,527507,527520],{"className":527506},[31642],[34,527508,527510],{"className":527509},[31646],[31648,527511,527512],{"xmlns":31650},[31652,527513,527514,527518],{},[31655,527515,527516],{},[31658,527517,34386],{},[31665,527519,34386],{"encoding":31667},[34,527521,527523],{"className":527522,"ariaHidden":758},[31672],[34,527524,527526,527529],{"className":527525},[31676],[34,527527],{"className":527528,"style":33218},[31680],[34,527530,34386],{"className":527531,"style":34427},[31685,31686],[34,527533,527535,527556],{"className":527534},[31642],[34,527536,527538],{"className":527537},[31646],[31648,527539,527540],{"xmlns":31650},[31652,527541,527542,527554],{},[31655,527543,527544,527546,527548,527550,527552],{},[31658,527545,111],{},[31658,527547,1697],{},[31662,527549,202],{},[31658,527551,1697],{},[31658,527553,111],{},[31665,527555,526821],{"encoding":31667},[34,527557,527559,527577],{"className":527558,"ariaHidden":758},[31672],[34,527560,527562,527565,527568,527571,527574],{"className":527561},[31676],[34,527563],{"className":527564,"style":31681},[31680],[34,527566,324191],{"className":527567},[31685,31686],[34,527569],{"className":527570,"style":31846},[31845],[34,527572,202],{"className":527573},[31850],[34,527575],{"className":527576,"style":31846},[31845],[34,527578,527580,527583],{"className":527579},[31676],[34,527581],{"className":527582,"style":31681},[31680],[34,527584,367454],{"className":527585},[31685,31686],[34,527587,527589,527619],{"className":527588},[31642],[34,527590,527592],{"className":527591},[31646],[31648,527593,527594],{"xmlns":31650},[31652,527595,527596,527616],{},[31655,527597,527598,527600,527602,527604,527606,527608,527610,527612,527614],{},[31658,527599,111],{},[31658,527601,1697],{},[31662,527603,223],{},[31658,527605,106063],{},[31662,527607,202],{},[31658,527609,1697],{},[31658,527611,111],{},[31662,527613,223],{},[31658,527615,106063],{},[31665,527617,527618],{"encoding":31667},"ab+I = ba+I",[34,527620,527622,527640,527658,527676],{"className":527621,"ariaHidden":758},[31672],[34,527623,527625,527628,527631,527634,527637],{"className":527624},[31676],[34,527626],{"className":527627,"style":34299},[31680],[34,527629,324191],{"className":527630},[31685,31686],[34,527632],{"className":527633,"style":32604},[31845],[34,527635,223],{"className":527636},[32024],[34,527638],{"className":527639,"style":32604},[31845],[34,527641,527643,527646,527649,527652,527655],{"className":527642},[31676],[34,527644],{"className":527645,"style":33218},[31680],[34,527647,106063],{"className":527648,"style":67482},[31685,31686],[34,527650],{"className":527651,"style":31846},[31845],[34,527653,202],{"className":527654},[31850],[34,527656],{"className":527657,"style":31846},[31845],[34,527659,527661,527664,527667,527670,527673],{"className":527660},[31676],[34,527662],{"className":527663,"style":34299},[31680],[34,527665,367454],{"className":527666},[31685,31686],[34,527668],{"className":527669,"style":32604},[31845],[34,527671,223],{"className":527672},[32024],[34,527674],{"className":527675,"style":32604},[31845],[34,527677,527679,527682],{"className":527678},[31676],[34,527680],{"className":527681,"style":33218},[31680],[34,527683,106063],{"className":527684,"style":67482},[31685,31686],[34,527686,527688,527709],{"className":527687},[31642],[34,527689,527691],{"className":527690},[31646],[31648,527692,527693],{"xmlns":31650},[31652,527694,527695,527707],{},[31655,527696,527697,527699,527701,527703,527705],{},[31658,527698,67467],{},[31658,527700,67731],{},[31662,527702,202],{},[31658,527704,67731],{},[31658,527706,67467],{},[31665,527708,526956],{"encoding":31667},[34,527710,527712,527733],{"className":527711,"ariaHidden":758},[31672],[34,527713,527715,527718,527721,527724,527727,527730],{"className":527714},[31676],[34,527716],{"className":527717,"style":33218},[31680],[34,527719,67467],{"className":527720,"style":67482},[31685,31686],[34,527722,67731],{"className":527723,"style":32604},[31685,31686],[34,527725],{"className":527726,"style":31846},[31845],[34,527728,202],{"className":527729},[31850],[34,527731],{"className":527732,"style":31846},[31845],[34,527734,527736,527739,527742],{"className":527735},[31676],[34,527737],{"className":527738,"style":33218},[31680],[34,527740,67731],{"className":527741,"style":32604},[31685,31686],[34,527743,67467],{"className":527744,"style":67482},[31685,31686],[34,527746,527748,527765],{"className":527747},[31642],[34,527749,527751],{"className":527750},[31646],[31648,527752,527753],{"xmlns":31650},[31652,527754,527755,527763],{},[31655,527756,527757,527759,527761],{},[31658,527758,34386],{},[31658,527760,38080],{"mathvariant":38079},[31658,527762,106063],{},[31665,527764,337927],{"encoding":31667},[34,527766,527768],{"className":527767,"ariaHidden":758},[31672],[34,527769,527771,527774,527777,527780],{"className":527770},[31676],[34,527772],{"className":527773,"style":31774},[31680],[34,527775,34386],{"className":527776,"style":34427},[31685,31686],[34,527778,38080],{"className":527779},[31685],[34,527781,106063],{"className":527782,"style":67482},[31685,31686]," 是交换环。",[15,527785,527786],{},[2627,527787,527788],{},"(ii) 单位元",[105,527790,527791,528178,528234],{},[108,527792,527793,2664,527795,475679,527823,527851,527852,527880,527881,527964,527965,61605,528049,528177],{},[2627,527794,526704],{},[34,527796,527798,527811],{"className":527797},[31642],[34,527799,527801],{"className":527800},[31646],[31648,527802,527803],{"xmlns":31650},[31652,527804,527805,527809],{},[31655,527806,527807],{},[31658,527808,34386],{},[31665,527810,34386],{"encoding":31667},[34,527812,527814],{"className":527813,"ariaHidden":758},[31672],[34,527815,527817,527820],{"className":527816},[31676],[34,527818],{"className":527819,"style":33218},[31680],[34,527821,34386],{"className":527822,"style":34427},[31685,31686],[34,527824,527826,527839],{"className":527825},[31642],[34,527827,527829],{"className":527828},[31646],[31648,527830,527831],{"xmlns":31650},[31652,527832,527833,527837],{},[31655,527834,527835],{},[31749,527836,640],{},[31665,527838,640],{"encoding":31667},[34,527840,527842],{"className":527841,"ariaHidden":758},[31672],[34,527843,527845,527848],{"className":527844},[31676],[34,527846],{"className":527847,"style":32617},[31680],[34,527849,640],{"className":527850},[31685],"（假设 ",[34,527853,527855,527868],{"className":527854},[31642],[34,527856,527858],{"className":527857},[31646],[31648,527859,527860],{"xmlns":31650},[31652,527861,527862,527866],{},[31655,527863,527864],{},[31658,527865,34386],{},[31665,527867,34386],{"encoding":31667},[34,527869,527871],{"className":527870,"ariaHidden":758},[31672],[34,527872,527874,527877],{"className":527873},[31676],[34,527875],{"className":527876,"style":33218},[31680],[34,527878,34386],{"className":527879,"style":34427},[31685,31686]," 是幺环），且 ",[34,527882,527884,527901],{"className":527883},[31642],[34,527885,527887],{"className":527886},[31646],[31648,527888,527889],{"xmlns":31650},[31652,527890,527891,527899],{},[31655,527892,527893,527895,527897],{},[31658,527894,106063],{},[31662,527896,54707],{"mathvariant":38079},[31658,527898,34386],{},[31665,527900,526530],{"encoding":31667},[34,527902,527904,527955],{"className":527903,"ariaHidden":758},[31672],[34,527905,527907,527910,527913,527916,527952],{"className":527906},[31676],[34,527908],{"className":527909,"style":43217},[31680],[34,527911,106063],{"className":527912,"style":67482},[31685,31686],[34,527914],{"className":527915,"style":31846},[31845],[34,527917,527919,527946,527949],{"className":527918},[31850],[34,527920,527922],{"className":527921},[31850],[34,527923,527925],{"className":527924},[31685,54774],[34,527926,527928],{"className":527927},[54778],[34,527929,527931,527934,527943],{"className":527930},[54782],[34,527932],{"className":527933,"style":43217},[31680],[34,527935,527937],{"className":527936},[54789],[34,527938,527940],{"className":527939},[31685],[34,527941,54796],{"className":527942},[31850],[34,527944],{"className":527945},[54800],[34,527947],{"className":527948},[31845,54804],[34,527950,202],{"className":527951},[31850],[34,527953],{"className":527954,"style":31846},[31845],[34,527956,527958,527961],{"className":527957},[31676],[34,527959],{"className":527960,"style":33218},[31680],[34,527962,34386],{"className":527963,"style":34427},[31685,31686],"（这保证了 ",[34,527966,527968,527986],{"className":527967},[31642],[34,527969,527971],{"className":527970},[31646],[31648,527972,527973],{"xmlns":31650},[31652,527974,527975,527983],{},[31655,527976,527977,527979,527981],{},[31749,527978,640],{},[31662,527980,112266],{"mathvariant":38079},[31658,527982,106063],{},[31665,527984,527985],{"encoding":31667},"1 \\notin I",[34,527987,527989,528040],{"className":527988,"ariaHidden":758},[31672],[34,527990,527992,527995,527998,528001,528037],{"className":527991},[31676],[34,527993],{"className":527994,"style":31774},[31680],[34,527996,640],{"className":527997},[31685],[34,527999],{"className":528000,"style":31846},[31845],[34,528002,528004,528010],{"className":528003},[31850],[34,528005,528007],{"className":528006},[31685],[34,528008,112296],{"className":528009},[31850],[34,528011,528013],{"className":528012},[31685,54774],[34,528014,528016],{"className":528015},[54778],[34,528017,528019,528022,528034],{"className":528018},[112306],[34,528020],{"className":528021,"style":31774},[31680],[34,528023,528025],{"className":528024},[54789],[34,528026,528028,528031],{"className":528027},[31685],[34,528029,38080],{"className":528030},[31685],[34,528032],{"className":528033,"style":112322},[31845],[34,528035],{"className":528036},[54800],[34,528038],{"className":528039,"style":31846},[31845],[34,528041,528043,528046],{"className":528042},[31676],[34,528044],{"className":528045,"style":33218},[31680],[34,528047,106063],{"className":528048,"style":67482},[31685,31686],[34,528050,528052,528078],{"className":528051},[31642],[34,528053,528055],{"className":528054},[31646],[31648,528056,528057],{"xmlns":31650},[31652,528058,528059,528075],{},[31655,528060,528061,528063,528065,528067,528069,528071,528073],{},[31749,528062,640],{},[31662,528064,223],{},[31658,528066,106063],{},[31662,528068,54707],{"mathvariant":38079},[31749,528070,459],{},[31662,528072,223],{},[31658,528074,106063],{},[31665,528076,528077],{"encoding":31667},"1+I \\neq 0+I",[34,528079,528081,528099,528150,528168],{"className":528080,"ariaHidden":758},[31672],[34,528082,528084,528087,528090,528093,528096],{"className":528083},[31676],[34,528085],{"className":528086,"style":35942},[31680],[34,528088,640],{"className":528089},[31685],[34,528091],{"className":528092,"style":32604},[31845],[34,528094,223],{"className":528095},[32024],[34,528097],{"className":528098,"style":32604},[31845],[34,528100,528102,528105,528108,528111,528147],{"className":528101},[31676],[34,528103],{"className":528104,"style":43217},[31680],[34,528106,106063],{"className":528107,"style":67482},[31685,31686],[34,528109],{"className":528110,"style":31846},[31845],[34,528112,528114,528141,528144],{"className":528113},[31850],[34,528115,528117],{"className":528116},[31850],[34,528118,528120],{"className":528119},[31685,54774],[34,528121,528123],{"className":528122},[54778],[34,528124,528126,528129,528138],{"className":528125},[54782],[34,528127],{"className":528128,"style":43217},[31680],[34,528130,528132],{"className":528131},[54789],[34,528133,528135],{"className":528134},[31685],[34,528136,54796],{"className":528137},[31850],[34,528139],{"className":528140},[54800],[34,528142],{"className":528143},[31845,54804],[34,528145,202],{"className":528146},[31850],[34,528148],{"className":528149,"style":31846},[31845],[34,528151,528153,528156,528159,528162,528165],{"className":528152},[31676],[34,528154],{"className":528155,"style":35942},[31680],[34,528157,459],{"className":528158},[31685],[34,528160],{"className":528161,"style":32604},[31845],[34,528163,223],{"className":528164},[32024],[34,528166],{"className":528167,"style":32604},[31845],[34,528169,528171,528174],{"className":528170},[31676],[34,528172],{"className":528173,"style":33218},[31680],[34,528175,106063],{"className":528176,"style":67482},[31685,31686],"，商环不是零环）。",[108,528179,528180,2664,528182,528233],{},[2627,528181,526856],{},[34,528183,528185,528203],{"className":528184},[31642],[34,528186,528188],{"className":528187},[31646],[31648,528189,528190],{"xmlns":31650},[31652,528191,528192,528200],{},[31655,528193,528194,528196,528198],{},[31749,528195,640],{},[31662,528197,223],{},[31658,528199,106063],{},[31665,528201,528202],{"encoding":31667},"1+I",[34,528204,528206,528224],{"className":528205,"ariaHidden":758},[31672],[34,528207,528209,528212,528215,528218,528221],{"className":528208},[31676],[34,528210],{"className":528211,"style":35942},[31680],[34,528213,640],{"className":528214},[31685],[34,528216],{"className":528217,"style":32604},[31845],[34,528219,223],{"className":528220},[32024],[34,528222],{"className":528223,"style":32604},[31845],[34,528225,528227,528230],{"className":528226},[31676],[34,528228],{"className":528229,"style":33218},[31680],[34,528231,106063],{"className":528232,"style":67482},[31685,31686]," 在乘法下起作用。",[108,528235,528236,496266,528238,33183,528343,528583,528823,528824,11008,528874,528912,528913],{},[2627,528237,390672],{},[34,528239,528241,528271],{"className":528240},[31642],[34,528242,528244],{"className":528243},[31646],[31648,528245,528246],{"xmlns":31650},[31652,528247,528248,528268],{},[31655,528249,528250,528252,528254,528256,528258,528260,528262,528264,528266],{},[31658,528251,67467],{},[31662,528253,202],{},[31658,528255,111],{},[31662,528257,223],{},[31658,528259,106063],{},[31662,528261,112296],{},[31658,528263,34386],{},[31658,528265,38080],{"mathvariant":38079},[31658,528267,106063],{},[31665,528269,528270],{"encoding":31667},"X = a+I \\in R\u002FI",[34,528272,528274,528292,528310,528328],{"className":528273,"ariaHidden":758},[31672],[34,528275,528277,528280,528283,528286,528289],{"className":528276},[31676],[34,528278],{"className":528279,"style":33218},[31680],[34,528281,67467],{"className":528282,"style":67482},[31685,31686],[34,528284],{"className":528285,"style":31846},[31845],[34,528287,202],{"className":528288},[31850],[34,528290],{"className":528291,"style":31846},[31845],[34,528293,528295,528298,528301,528304,528307],{"className":528294},[31676],[34,528296],{"className":528297,"style":32597},[31680],[34,528299,111],{"className":528300},[31685,31686],[34,528302],{"className":528303,"style":32604},[31845],[34,528305,223],{"className":528306},[32024],[34,528308],{"className":528309,"style":32604},[31845],[34,528311,528313,528316,528319,528322,528325],{"className":528312},[31676],[34,528314],{"className":528315,"style":104901},[31680],[34,528317,106063],{"className":528318,"style":67482},[31685,31686],[34,528320],{"className":528321,"style":31846},[31845],[34,528323,112296],{"className":528324},[31850],[34,528326],{"className":528327,"style":31846},[31845],[34,528329,528331,528334,528337,528340],{"className":528330},[31676],[34,528332],{"className":528333,"style":31774},[31680],[34,528335,34386],{"className":528336,"style":34427},[31685,31686],[34,528338,38080],{"className":528339},[31685],[34,528341,106063],{"className":528342,"style":67482},[31685,31686],[34,528344,528346,528406],{"className":528345},[31642],[34,528347,528349],{"className":528348},[31646],[31648,528350,528351],{"xmlns":31650},[31652,528352,528353,528403],{},[31655,528354,528355,528357,528359,528361,528363,528365,528367,528369,528371,528373,528375,528377,528379,528381,528383,528385,528387,528389,528391,528393,528395,528397,528399,528401],{},[31662,528356,456],{"stretchy":702},[31749,528358,640],{},[31662,528360,223],{},[31658,528362,106063],{},[31662,528364,2557],{"stretchy":702},[31662,528366,456],{"stretchy":702},[31658,528368,111],{},[31662,528370,223],{},[31658,528372,106063],{},[31662,528374,2557],{"stretchy":702},[31662,528376,202],{},[31662,528378,456],{"stretchy":702},[31749,528380,640],{},[31662,528382,71191],{},[31658,528384,111],{},[31662,528386,2557],{"stretchy":702},[31662,528388,223],{},[31658,528390,106063],{},[31662,528392,202],{},[31658,528394,111],{},[31662,528396,223],{},[31658,528398,106063],{},[31662,528400,202],{},[31658,528402,67467],{},[31665,528404,528405],{"encoding":31667},"(1+I)(a+I) = (1 \\cdot a) + I = a + I = X",[34,528407,528409,528430,528457,528478,528499,528520,528538,528556,528574],{"className":528408,"ariaHidden":758},[31672],[34,528410,528412,528415,528418,528421,528424,528427],{"className":528411},[31676],[34,528413],{"className":528414,"style":31774},[31680],[34,528416,456],{"className":528417},[31782],[34,528419,640],{"className":528420},[31685],[34,528422],{"className":528423,"style":32604},[31845],[34,528425,223],{"className":528426},[32024],[34,528428],{"className":528429,"style":32604},[31845],[34,528431,528433,528436,528439,528442,528445,528448,528451,528454],{"className":528432},[31676],[34,528434],{"className":528435,"style":31774},[31680],[34,528437,106063],{"className":528438,"style":67482},[31685,31686],[34,528440,2557],{"className":528441},[31690],[34,528443,456],{"className":528444},[31782],[34,528446,111],{"className":528447},[31685,31686],[34,528449],{"className":528450,"style":32604},[31845],[34,528452,223],{"className":528453},[32024],[34,528455],{"className":528456,"style":32604},[31845],[34,528458,528460,528463,528466,528469,528472,528475],{"className":528459},[31676],[34,528461],{"className":528462,"style":31774},[31680],[34,528464,106063],{"className":528465,"style":67482},[31685,31686],[34,528467,2557],{"className":528468},[31690],[34,528470],{"className":528471,"style":31846},[31845],[34,528473,202],{"className":528474},[31850],[34,528476],{"className":528477,"style":31846},[31845],[34,528479,528481,528484,528487,528490,528493,528496],{"className":528480},[31676],[34,528482],{"className":528483,"style":31774},[31680],[34,528485,456],{"className":528486},[31782],[34,528488,640],{"className":528489},[31685],[34,528491],{"className":528492,"style":32604},[31845],[34,528494,71191],{"className":528495},[32024],[34,528497],{"className":528498,"style":32604},[31845],[34,528500,528502,528505,528508,528511,528514,528517],{"className":528501},[31676],[34,528503],{"className":528504,"style":31774},[31680],[34,528506,111],{"className":528507},[31685,31686],[34,528509,2557],{"className":528510},[31690],[34,528512],{"className":528513,"style":32604},[31845],[34,528515,223],{"className":528516},[32024],[34,528518],{"className":528519,"style":32604},[31845],[34,528521,528523,528526,528529,528532,528535],{"className":528522},[31676],[34,528524],{"className":528525,"style":33218},[31680],[34,528527,106063],{"className":528528,"style":67482},[31685,31686],[34,528530],{"className":528531,"style":31846},[31845],[34,528533,202],{"className":528534},[31850],[34,528536],{"className":528537,"style":31846},[31845],[34,528539,528541,528544,528547,528550,528553],{"className":528540},[31676],[34,528542],{"className":528543,"style":32597},[31680],[34,528545,111],{"className":528546},[31685,31686],[34,528548],{"className":528549,"style":32604},[31845],[34,528551,223],{"className":528552},[32024],[34,528554],{"className":528555,"style":32604},[31845],[34,528557,528559,528562,528565,528568,528571],{"className":528558},[31676],[34,528560],{"className":528561,"style":33218},[31680],[34,528563,106063],{"className":528564,"style":67482},[31685,31686],[34,528566],{"className":528567,"style":31846},[31845],[34,528569,202],{"className":528570},[31850],[34,528572],{"className":528573,"style":31846},[31845],[34,528575,528577,528580],{"className":528576},[31676],[34,528578],{"className":528579,"style":33218},[31680],[34,528581,67467],{"className":528582,"style":67482},[31685,31686],[34,528584,528586,528646],{"className":528585},[31642],[34,528587,528589],{"className":528588},[31646],[31648,528590,528591],{"xmlns":31650},[31652,528592,528593,528643],{},[31655,528594,528595,528597,528599,528601,528603,528605,528607,528609,528611,528613,528615,528617,528619,528621,528623,528625,528627,528629,528631,528633,528635,528637,528639,528641],{},[31662,528596,456],{"stretchy":702},[31658,528598,111],{},[31662,528600,223],{},[31658,528602,106063],{},[31662,528604,2557],{"stretchy":702},[31662,528606,456],{"stretchy":702},[31749,528608,640],{},[31662,528610,223],{},[31658,528612,106063],{},[31662,528614,2557],{"stretchy":702},[31662,528616,202],{},[31662,528618,456],{"stretchy":702},[31658,528620,111],{},[31662,528622,71191],{},[31749,528624,640],{},[31662,528626,2557],{"stretchy":702},[31662,528628,223],{},[31658,528630,106063],{},[31662,528632,202],{},[31658,528634,111],{},[31662,528636,223],{},[31658,528638,106063],{},[31662,528640,202],{},[31658,528642,67467],{},[31665,528644,528645],{"encoding":31667},"(a+I)(1+I) = (a \\cdot 1) + I = a + I = X",[34,528647,528649,528670,528697,528718,528739,528760,528778,528796,528814],{"className":528648,"ariaHidden":758},[31672],[34,528650,528652,528655,528658,528661,528664,528667],{"className":528651},[31676],[34,528653],{"className":528654,"style":31774},[31680],[34,528656,456],{"className":528657},[31782],[34,528659,111],{"className":528660},[31685,31686],[34,528662],{"className":528663,"style":32604},[31845],[34,528665,223],{"className":528666},[32024],[34,528668],{"className":528669,"style":32604},[31845],[34,528671,528673,528676,528679,528682,528685,528688,528691,528694],{"className":528672},[31676],[34,528674],{"className":528675,"style":31774},[31680],[34,528677,106063],{"className":528678,"style":67482},[31685,31686],[34,528680,2557],{"className":528681},[31690],[34,528683,456],{"className":528684},[31782],[34,528686,640],{"className":528687},[31685],[34,528689],{"className":528690,"style":32604},[31845],[34,528692,223],{"className":528693},[32024],[34,528695],{"className":528696,"style":32604},[31845],[34,528698,528700,528703,528706,528709,528712,528715],{"className":528699},[31676],[34,528701],{"className":528702,"style":31774},[31680],[34,528704,106063],{"className":528705,"style":67482},[31685,31686],[34,528707,2557],{"className":528708},[31690],[34,528710],{"className":528711,"style":31846},[31845],[34,528713,202],{"className":528714},[31850],[34,528716],{"className":528717,"style":31846},[31845],[34,528719,528721,528724,528727,528730,528733,528736],{"className":528720},[31676],[34,528722],{"className":528723,"style":31774},[31680],[34,528725,456],{"className":528726},[31782],[34,528728,111],{"className":528729},[31685,31686],[34,528731],{"className":528732,"style":32604},[31845],[34,528734,71191],{"className":528735},[32024],[34,528737],{"className":528738,"style":32604},[31845],[34,528740,528742,528745,528748,528751,528754,528757],{"className":528741},[31676],[34,528743],{"className":528744,"style":31774},[31680],[34,528746,640],{"className":528747},[31685],[34,528749,2557],{"className":528750},[31690],[34,528752],{"className":528753,"style":32604},[31845],[34,528755,223],{"className":528756},[32024],[34,528758],{"className":528759,"style":32604},[31845],[34,528761,528763,528766,528769,528772,528775],{"className":528762},[31676],[34,528764],{"className":528765,"style":33218},[31680],[34,528767,106063],{"className":528768,"style":67482},[31685,31686],[34,528770],{"className":528771,"style":31846},[31845],[34,528773,202],{"className":528774},[31850],[34,528776],{"className":528777,"style":31846},[31845],[34,528779,528781,528784,528787,528790,528793],{"className":528780},[31676],[34,528782],{"className":528783,"style":32597},[31680],[34,528785,111],{"className":528786},[31685,31686],[34,528788],{"className":528789,"style":32604},[31845],[34,528791,223],{"className":528792},[32024],[34,528794],{"className":528795,"style":32604},[31845],[34,528797,528799,528802,528805,528808,528811],{"className":528798},[31676],[34,528800],{"className":528801,"style":33218},[31680],[34,528803,106063],{"className":528804,"style":67482},[31685,31686],[34,528806],{"className":528807,"style":31846},[31845],[34,528809,202],{"className":528810},[31850],[34,528812],{"className":528813,"style":31846},[31845],[34,528815,528817,528820],{"className":528816},[31676],[34,528818],{"className":528819,"style":33218},[31680],[34,528821,67467],{"className":528822,"style":67482},[31685,31686],"\n因为左乘和右乘都等于它自己，所以 ",[34,528825,528827,528844],{"className":528826},[31642],[34,528828,528830],{"className":528829},[31646],[31648,528831,528832],{"xmlns":31650},[31652,528833,528834,528842],{},[31655,528835,528836,528838,528840],{},[31749,528837,640],{},[31662,528839,223],{},[31658,528841,106063],{},[31665,528843,528202],{"encoding":31667},[34,528845,528847,528865],{"className":528846,"ariaHidden":758},[31672],[34,528848,528850,528853,528856,528859,528862],{"className":528849},[31676],[34,528851],{"className":528852,"style":35942},[31680],[34,528854,640],{"className":528855},[31685],[34,528857],{"className":528858,"style":32604},[31845],[34,528860,223],{"className":528861},[32024],[34,528863],{"className":528864,"style":32604},[31845],[34,528866,528868,528871],{"className":528867},[31676],[34,528869],{"className":528870,"style":33218},[31680],[34,528872,106063],{"className":528873,"style":67482},[31685,31686],[34,528875,528877,528894],{"className":528876},[31642],[34,528878,528880],{"className":528879},[31646],[31648,528881,528882],{"xmlns":31650},[31652,528883,528884,528892],{},[31655,528885,528886,528888,528890],{},[31658,528887,34386],{},[31658,528889,38080],{"mathvariant":38079},[31658,528891,106063],{},[31665,528893,337927],{"encoding":31667},[34,528895,528897],{"className":528896,"ariaHidden":758},[31672],[34,528898,528900,528903,528906,528909],{"className":528899},[31676],[34,528901],{"className":528902,"style":31774},[31680],[34,528904,34386],{"className":528905,"style":34427},[31685,31686],[34,528907,38080],{"className":528908},[31685],[34,528910,106063],{"className":528911,"style":67482},[31685,31686]," 的乘法单位元。 ",[34,528914,528916,528929],{"className":528915},[31642],[34,528917,528919],{"className":528918},[31646],[31648,528920,528921],{"xmlns":31650},[31652,528922,528923,528927],{},[31655,528924,528925],{},[31658,528926,445204],{"mathvariant":38079},[31665,528928,445207],{"encoding":31667},[34,528930,528932],{"className":528931,"ariaHidden":758},[31672],[34,528933,528935,528938],{"className":528934},[31676],[34,528936],{"className":528937,"style":445217},[31680],[34,528939,445204],{"className":528940},[31685,129261],[32188,528942],{},[253,528944,528946],{"id":528945},"题目-8构造反例与特例","题目 8：构造反例与特例",[15,528948,528949,65807,529009,387604,529037,529065,529066,388223,529094,529122,529123,43298,529151,529179,529180,389870,529208,529236],{},[2627,528950,528951,528952,387531,528980,529008],{},"8. 给出环 ",[34,528953,528955,528968],{"className":528954},[31642],[34,528956,528958],{"className":528957},[31646],[31648,528959,528960],{"xmlns":31650},[31652,528961,528962,528966],{},[31655,528963,528964],{},[31658,528965,34386],{},[31665,528967,34386],{"encoding":31667},[34,528969,528971],{"className":528970,"ariaHidden":758},[31672],[34,528972,528974,528977],{"className":528973},[31676],[34,528975],{"className":528976,"style":33218},[31680],[34,528978,34386],{"className":528979,"style":34427},[31685,31686],[34,528981,528983,528996],{"className":528982},[31642],[34,528984,528986],{"className":528985},[31646],[31648,528987,528988],{"xmlns":31650},[31652,528989,528990,528994],{},[31655,528991,528992],{},[31658,528993,94675],{},[31665,528995,94675],{"encoding":31667},[34,528997,528999],{"className":528998,"ariaHidden":758},[31672],[34,529000,529002,529005],{"className":529001},[31676],[34,529003],{"className":529004,"style":33218},[31680],[34,529006,94675],{"className":529007,"style":106455},[31685,31686]," 的例子，使得它们满足以下条件之一：",[34,529010,529012,529025],{"className":529011},[31642],[34,529013,529015],{"className":529014},[31646],[31648,529016,529017],{"xmlns":31650},[31652,529018,529019,529023],{},[31655,529020,529021],{},[31658,529022,34386],{},[31665,529024,34386],{"encoding":31667},[34,529026,529028],{"className":529027,"ariaHidden":758},[31672],[34,529029,529031,529034],{"className":529030},[31676],[34,529032],{"className":529033,"style":33218},[31680],[34,529035,34386],{"className":529036,"style":34427},[31685,31686],[34,529038,529040,529053],{"className":529039},[31642],[34,529041,529043],{"className":529042},[31646],[31648,529044,529045],{"xmlns":31650},[31652,529046,529047,529051],{},[31655,529048,529049],{},[31658,529050,94675],{},[31665,529052,94675],{"encoding":31667},[34,529054,529056],{"className":529055,"ariaHidden":758},[31672],[34,529057,529059,529062],{"className":529058},[31676],[34,529060],{"className":529061,"style":33218},[31680],[34,529063,94675],{"className":529064,"style":106455},[31685,31686]," 没有1；\n(2) ",[34,529067,529069,529082],{"className":529068},[31642],[34,529070,529072],{"className":529071},[31646],[31648,529073,529074],{"xmlns":31650},[31652,529075,529076,529080],{},[31655,529077,529078],{},[31658,529079,34386],{},[31665,529081,34386],{"encoding":31667},[34,529083,529085],{"className":529084,"ariaHidden":758},[31672],[34,529086,529088,529091],{"className":529087},[31676],[34,529089],{"className":529090,"style":33218},[31680],[34,529092,34386],{"className":529093,"style":34427},[31685,31686],[34,529095,529097,529110],{"className":529096},[31642],[34,529098,529100],{"className":529099},[31646],[31648,529101,529102],{"xmlns":31650},[31652,529103,529104,529108],{},[31655,529105,529106],{},[31658,529107,94675],{},[31665,529109,94675],{"encoding":31667},[34,529111,529113],{"className":529112,"ariaHidden":758},[31672],[34,529114,529116,529119],{"className":529115},[31676],[34,529117],{"className":529118,"style":33218},[31680],[34,529120,94675],{"className":529121,"style":106455},[31685,31686]," 有1；\n(3) ",[34,529124,529126,529139],{"className":529125},[31642],[34,529127,529129],{"className":529128},[31646],[31648,529130,529131],{"xmlns":31650},[31652,529132,529133,529137],{},[31655,529134,529135],{},[31658,529136,34386],{},[31665,529138,34386],{"encoding":31667},[34,529140,529142],{"className":529141,"ariaHidden":758},[31672],[34,529143,529145,529148],{"className":529144},[31676],[34,529146],{"className":529147,"style":33218},[31680],[34,529149,34386],{"className":529150,"style":34427},[31685,31686],[34,529152,529154,529167],{"className":529153},[31642],[34,529155,529157],{"className":529156},[31646],[31648,529158,529159],{"xmlns":31650},[31652,529160,529161,529165],{},[31655,529162,529163],{},[31658,529164,94675],{},[31665,529166,94675],{"encoding":31667},[34,529168,529170],{"className":529169,"ariaHidden":758},[31672],[34,529171,529173,529176],{"className":529172},[31676],[34,529174],{"className":529175,"style":33218},[31680],[34,529177,94675],{"className":529178,"style":106455},[31685,31686]," 都有1；\n(4) ",[34,529181,529183,529196],{"className":529182},[31642],[34,529184,529186],{"className":529185},[31646],[31648,529187,529188],{"xmlns":31650},[31652,529189,529190,529194],{},[31655,529191,529192],{},[31658,529193,34386],{},[31665,529195,34386],{"encoding":31667},[34,529197,529199],{"className":529198,"ariaHidden":758},[31672],[34,529200,529202,529205],{"className":529201},[31676],[34,529203],{"className":529204,"style":33218},[31680],[34,529206,34386],{"className":529207,"style":34427},[31685,31686],[34,529209,529211,529224],{"className":529210},[31642],[34,529212,529214],{"className":529213},[31646],[31648,529215,529216],{"xmlns":31650},[31652,529217,529218,529222],{},[31655,529219,529220],{},[31658,529221,94675],{},[31665,529223,94675],{"encoding":31667},[34,529225,529227],{"className":529226,"ariaHidden":758},[31672],[34,529228,529230,529233],{"className":529229},[31676],[34,529231],{"className":529232,"style":33218},[31680],[34,529234,94675],{"className":529235,"style":106455},[31685,31686]," 交换。",[32490,529238,529240],{"id":529239},"深度解析与解答","【深度解析与解答】",[15,529242,529243,529244,347],{},"这道题的目的是打破你的思维定势。在群论中，子群的单位元必须和群的一样。但在环论中，",[2627,529245,529246],{},"子环的单位元（1）是非常“任性”的",[15,529248,529249,529308],{},[2627,529250,529251,529252,387604,529280,387633],{},"解答 (1): ",[34,529253,529255,529268],{"className":529254},[31642],[34,529256,529258],{"className":529257},[31646],[31648,529259,529260],{"xmlns":31650},[31652,529261,529262,529266],{},[31655,529263,529264],{},[31658,529265,34386],{},[31665,529267,34386],{"encoding":31667},[34,529269,529271],{"className":529270,"ariaHidden":758},[31672],[34,529272,529274,529277],{"className":529273},[31676],[34,529275],{"className":529276,"style":33218},[31680],[34,529278,34386],{"className":529279,"style":34427},[31685,31686],[34,529281,529283,529296],{"className":529282},[31642],[34,529284,529286],{"className":529285},[31646],[31648,529287,529288],{"xmlns":31650},[31652,529289,529290,529294],{},[31655,529291,529292],{},[31658,529293,94675],{},[31665,529295,94675],{"encoding":31667},[34,529297,529299],{"className":529298,"ariaHidden":758},[31672],[34,529300,529302,529305],{"className":529301},[31676],[34,529303],{"className":529304,"style":33218},[31680],[34,529306,94675],{"className":529307,"style":106455},[31685,31686],"\n这是最常见的情况。",[105,529310,529311,529521],{},[108,529312,529313,33183,529315],{},[2627,529314,233206],{},[105,529316,529317,529462],{},[108,529318,396859,529319,529369,529370,347],{},[34,529320,529322,529339],{"className":529321},[31642],[34,529323,529325],{"className":529324},[31646],[31648,529326,529327],{"xmlns":31650},[31652,529328,529329,529337],{},[31655,529330,529331,529333,529335],{},[31658,529332,34386],{},[31662,529334,202],{},[31658,529336,97300],{"mathvariant":318416},[31665,529338,387661],{"encoding":31667},[34,529340,529342,529360],{"className":529341,"ariaHidden":758},[31672],[34,529343,529345,529348,529351,529354,529357],{"className":529344},[31676],[34,529346],{"className":529347,"style":33218},[31680],[34,529349,34386],{"className":529350,"style":34427},[31685,31686],[34,529352],{"className":529353,"style":31846},[31845],[34,529355,202],{"className":529356},[31850],[34,529358],{"className":529359,"style":31846},[31845],[34,529361,529363,529366],{"className":529362},[31676],[34,529364],{"className":529365,"style":320927},[31680],[34,529367,97300],{"className":529368},[31685,318441]," （整数环）。",[34,529371,529373,529395],{"className":529372},[31642],[34,529374,529376],{"className":529375},[31646],[31648,529377,529378],{"xmlns":31650},[31652,529379,529380,529392],{},[31655,529381,529382,529388,529390],{},[31743,529383,529384,529386],{},[31749,529385,640],{},[31658,529387,34386],{},[31662,529389,202],{},[31749,529391,640],{},[31665,529393,529394],{"encoding":31667},"1_R = 1",[34,529396,529398,529453],{"className":529397,"ariaHidden":758},[31672],[34,529399,529401,529404,529444,529447,529450],{"className":529400},[31676],[34,529402],{"className":529403,"style":38357},[31680],[34,529405,529407,529410],{"className":529406},[31685],[34,529408,640],{"className":529409},[31685],[34,529411,529413],{"className":529412},[31792],[34,529414,529416,529436],{"className":529415},[31796,31797],[34,529417,529419,529433],{"className":529418},[31801],[34,529420,529422],{"className":529421,"style":116558},[31805],[34,529423,529424,529427],{"style":31809},[34,529425],{"className":529426,"style":31814},[31813],[34,529428,529430],{"className":529429},[31818,31819,31820,31821],[34,529431,34386],{"className":529432,"style":34427},[31685,31686,31821],[34,529434,31829],{"className":529435},[31828],[34,529437,529439],{"className":529438},[31801],[34,529440,529442],{"className":529441,"style":31836},[31805],[34,529443],{},[34,529445],{"className":529446,"style":31846},[31845],[34,529448,202],{"className":529449},[31850],[34,529451],{"className":529452,"style":31846},[31845],[34,529454,529456,529459],{"className":529455},[31676],[34,529457],{"className":529458,"style":32617},[31680],[34,529460,640],{"className":529461},[31685],[108,529463,396859,529464,529520],{},[34,529465,529467,529487],{"className":529466},[31642],[34,529468,529470],{"className":529469},[31646],[31648,529471,529472],{"xmlns":31650},[31652,529473,529474,529484],{},[31655,529475,529476,529478,529480,529482],{},[31658,529477,94675],{},[31662,529479,202],{},[31749,529481,768],{},[31658,529483,97300],{"mathvariant":318416},[31665,529485,529486],{"encoding":31667},"S = 2\\mathbb{Z}",[34,529488,529490,529508],{"className":529489,"ariaHidden":758},[31672],[34,529491,529493,529496,529499,529502,529505],{"className":529492},[31676],[34,529494],{"className":529495,"style":33218},[31680],[34,529497,94675],{"className":529498,"style":106455},[31685,31686],[34,529500],{"className":529501,"style":31846},[31845],[34,529503,202],{"className":529504},[31850],[34,529506],{"className":529507,"style":31846},[31845],[34,529509,529511,529514,529517],{"className":529510},[31676],[34,529512],{"className":529513,"style":320927},[31680],[34,529515,768],{"className":529516},[31685],[34,529518,97300],{"className":529519},[31685,318441]," （偶数环）。",[108,529522,529523,33183,529525],{},[2627,529524,500696],{},[105,529526,529527,529558],{},[108,529528,529529,529557],{},[34,529530,529532,529545],{"className":529531},[31642],[34,529533,529535],{"className":529534},[31646],[31648,529536,529537],{"xmlns":31650},[31652,529538,529539,529543],{},[31655,529540,529541],{},[31658,529542,97300],{"mathvariant":318416},[31665,529544,320917],{"encoding":31667},[34,529546,529548],{"className":529547,"ariaHidden":758},[31672],[34,529549,529551,529554],{"className":529550},[31676],[34,529552],{"className":529553,"style":320927},[31680],[34,529555,97300],{"className":529556},[31685,318441]," 有单位元 1。",[108,529559,529560,529593,529594,529627,529628,529656,529657,475798,529685,322067,529758,529786],{},[34,529561,529563,529578],{"className":529562},[31642],[34,529564,529566],{"className":529565},[31646],[31648,529567,529568],{"xmlns":31650},[31652,529569,529570,529576],{},[31655,529571,529572,529574],{},[31749,529573,768],{},[31658,529575,97300],{"mathvariant":318416},[31665,529577,496029],{"encoding":31667},[34,529579,529581],{"className":529580,"ariaHidden":758},[31672],[34,529582,529584,529587,529590],{"className":529583},[31676],[34,529585],{"className":529586,"style":320927},[31680],[34,529588,768],{"className":529589},[31685],[34,529591,97300],{"className":529592},[31685,318441]," 是子环（封闭性满足），但在 ",[34,529595,529597,529612],{"className":529596},[31642],[34,529598,529600],{"className":529599},[31646],[31648,529601,529602],{"xmlns":31650},[31652,529603,529604,529610],{},[31655,529605,529606,529608],{},[31749,529607,768],{},[31658,529609,97300],{"mathvariant":318416},[31665,529611,496029],{"encoding":31667},[34,529613,529615],{"className":529614,"ariaHidden":758},[31672],[34,529616,529618,529621,529624],{"className":529617},[31676],[34,529619],{"className":529620,"style":320927},[31680],[34,529622,768],{"className":529623},[31685],[34,529625,97300],{"className":529626},[31685,318441]," 中找不到一个元素 ",[34,529629,529631,529644],{"className":529630},[31642],[34,529632,529634],{"className":529633},[31646],[31648,529635,529636],{"xmlns":31650},[31652,529637,529638,529642],{},[31655,529639,529640],{},[31658,529641,33244],{},[31665,529643,33244],{"encoding":31667},[34,529645,529647],{"className":529646,"ariaHidden":758},[31672],[34,529648,529650,529653],{"className":529649},[31676],[34,529651],{"className":529652,"style":32527},[31680],[34,529654,33244],{"className":529655},[31685,31686],"，使得对所有偶数 ",[34,529658,529660,529673],{"className":529659},[31642],[34,529661,529663],{"className":529662},[31646],[31648,529664,529665],{"xmlns":31650},[31652,529666,529667,529671],{},[31655,529668,529669],{},[31658,529670,111],{},[31665,529672,111],{"encoding":31667},[34,529674,529676],{"className":529675,"ariaHidden":758},[31672],[34,529677,529679,529682],{"className":529678},[31676],[34,529680],{"className":529681,"style":32527},[31680],[34,529683,111],{"className":529684},[31685,31686],[34,529686,529688,529710],{"className":529687},[31642],[34,529689,529691],{"className":529690},[31646],[31648,529692,529693],{"xmlns":31650},[31652,529694,529695,529707],{},[31655,529696,529697,529699,529701,529703,529705],{},[31658,529698,33244],{},[31662,529700,71191],{},[31658,529702,111],{},[31662,529704,202],{},[31658,529706,111],{},[31665,529708,529709],{"encoding":31667},"e \\cdot a = a",[34,529711,529713,529731,529749],{"className":529712,"ariaHidden":758},[31672],[34,529714,529716,529719,529722,529725,529728],{"className":529715},[31676],[34,529717],{"className":529718,"style":224869},[31680],[34,529720,33244],{"className":529721},[31685,31686],[34,529723],{"className":529724,"style":32604},[31845],[34,529726,71191],{"className":529727},[32024],[34,529729],{"className":529730,"style":32604},[31845],[34,529732,529734,529737,529740,529743,529746],{"className":529733},[31676],[34,529735],{"className":529736,"style":32527},[31680],[34,529738,111],{"className":529739},[31685,31686],[34,529741],{"className":529742,"style":31846},[31845],[34,529744,202],{"className":529745},[31850],[34,529747],{"className":529748,"style":31846},[31845],[34,529750,529752,529755],{"className":529751},[31676],[34,529753],{"className":529754,"style":32527},[31680],[34,529756,111],{"className":529757},[31685,31686],[34,529759,529761,529774],{"className":529760},[31642],[34,529762,529764],{"className":529763},[31646],[31648,529765,529766],{"xmlns":31650},[31652,529767,529768,529772],{},[31655,529769,529770],{},[31658,529771,94675],{},[31665,529773,94675],{"encoding":31667},[34,529775,529777],{"className":529776,"ariaHidden":758},[31672],[34,529778,529780,529783],{"className":529779},[31676],[34,529781],{"className":529782,"style":33218},[31680],[34,529784,94675],{"className":529785,"style":106455},[31685,31686]," 没有单位元。",[15,529788,529789,529848],{},[2627,529790,529791,529792,388223,529820,388252],{},"解答 (2): ",[34,529793,529795,529808],{"className":529794},[31642],[34,529796,529798],{"className":529797},[31646],[31648,529799,529800],{"xmlns":31650},[31652,529801,529802,529806],{},[31655,529803,529804],{},[31658,529805,34386],{},[31665,529807,34386],{"encoding":31667},[34,529809,529811],{"className":529810,"ariaHidden":758},[31672],[34,529812,529814,529817],{"className":529813},[31676],[34,529815],{"className":529816,"style":33218},[31680],[34,529818,34386],{"className":529819,"style":34427},[31685,31686],[34,529821,529823,529836],{"className":529822},[31642],[34,529824,529826],{"className":529825},[31646],[31648,529827,529828],{"xmlns":31650},[31652,529829,529830,529834],{},[31655,529831,529832],{},[31658,529833,94675],{},[31665,529835,94675],{"encoding":31667},[34,529837,529839],{"className":529838,"ariaHidden":758},[31672],[34,529840,529842,529845],{"className":529841},[31676],[34,529843],{"className":529844,"style":33218},[31680],[34,529846,94675],{"className":529847,"style":106455},[31685,31686],"\n这种情况比较反直觉。通常我们需要利用“直积”来构造这种“局部有1，整体无1”的结构。",[105,529850,529851,530184],{},[108,529852,529853,33183,529855],{},[2627,529854,233206],{},[105,529856,529857,530003],{},[108,529858,396859,529859,529937,529938],{},[34,529860,529862,529886],{"className":529861},[31642],[34,529863,529865],{"className":529864},[31646],[31648,529866,529867],{"xmlns":31650},[31652,529868,529869,529883],{},[31655,529870,529871,529873,529875,529877,529879,529881],{},[31658,529872,34386],{},[31662,529874,202],{},[31658,529876,97300],{"mathvariant":318416},[31662,529878,378006],{},[31749,529880,768],{},[31658,529882,97300],{"mathvariant":318416},[31665,529884,529885],{"encoding":31667},"R = \\mathbb{Z} \\oplus 2\\mathbb{Z}",[34,529887,529889,529907,529925],{"className":529888,"ariaHidden":758},[31672],[34,529890,529892,529895,529898,529901,529904],{"className":529891},[31676],[34,529893],{"className":529894,"style":33218},[31680],[34,529896,34386],{"className":529897,"style":34427},[31685,31686],[34,529899],{"className":529900,"style":31846},[31845],[34,529902,202],{"className":529903},[31850],[34,529905],{"className":529906,"style":31846},[31845],[34,529908,529910,529913,529916,529919,529922],{"className":529909},[31676],[34,529911],{"className":529912,"style":379449},[31680],[34,529914,97300],{"className":529915},[31685,318441],[34,529917],{"className":529918,"style":32604},[31845],[34,529920,378006],{"className":529921},[32024],[34,529923],{"className":529924,"style":32604},[31845],[34,529926,529928,529931,529934],{"className":529927},[31676],[34,529929],{"className":529930,"style":320927},[31680],[34,529932,768],{"className":529933},[31685],[34,529935,97300],{"className":529936},[31685,318441]," （直积环）。\n",[105,529939,529940,530000],{},[108,529941,529942,529943,347],{},"元素形式为 ",[34,529944,529946,529970],{"className":529945},[31642],[34,529947,529949],{"className":529948},[31646],[31648,529950,529951],{"xmlns":31650},[31652,529952,529953,529967],{},[31655,529954,529955,529957,529959,529961,529963,529965],{},[31662,529956,456],{"stretchy":702},[31658,529958,111],{},[31662,529960,31960],{"separator":758},[31749,529962,768],{},[31658,529964,1697],{},[31662,529966,2557],{"stretchy":702},[31665,529968,529969],{"encoding":31667},"(a, 2b)",[34,529971,529973],{"className":529972,"ariaHidden":758},[31672],[34,529974,529976,529979,529982,529985,529988,529991,529994,529997],{"className":529975},[31676],[34,529977],{"className":529978,"style":31774},[31680],[34,529980,456],{"className":529981},[31782],[34,529983,111],{"className":529984},[31685,31686],[34,529986,31960],{"className":529987},[32119],[34,529989],{"className":529990,"style":32123},[31845],[34,529992,768],{"className":529993},[31685],[34,529995,1697],{"className":529996},[31685,31686],[34,529998,2557],{"className":529999},[31690],[108,530001,530002],{},"加法、乘法都是分量运算。",[108,530004,396859,530005,347],{},[34,530006,530008,530058],{"className":530007},[31642],[34,530009,530011],{"className":530010},[31646],[31648,530012,530013],{"xmlns":31650},[31652,530014,530015,530055],{},[31655,530016,530017,530019,530021,530023,530025,530027,530029,530031,530033,530035,530037,530039,530041,530043,530045,530047,530049,530051,530053],{},[31658,530018,94675],{},[31662,530020,202],{},[31658,530022,97300],{"mathvariant":318416},[31662,530024,378006],{},[31662,530026,32685],{"stretchy":702},[31749,530028,459],{},[31662,530030,32715],{"stretchy":702},[31662,530032,202],{},[31662,530034,32685],{"stretchy":702},[31662,530036,456],{"stretchy":702},[31658,530038,111],{},[31662,530040,31960],{"separator":758},[31749,530042,459],{},[31662,530044,2557],{"stretchy":702},[31662,530046,162086],{},[31658,530048,111],{},[31662,530050,112296],{},[31658,530052,97300],{"mathvariant":318416},[31662,530054,32715],{"stretchy":702},[31665,530056,530057],{"encoding":31667},"S = \\mathbb{Z} \\oplus \\{0\\} = \\{ (a, 0) \\mid a \\in \\mathbb{Z} \\}",[34,530059,530061,530079,530097,530121,530154,530172],{"className":530060,"ariaHidden":758},[31672],[34,530062,530064,530067,530070,530073,530076],{"className":530063},[31676],[34,530065],{"className":530066,"style":33218},[31680],[34,530068,94675],{"className":530069,"style":106455},[31685,31686],[34,530071],{"className":530072,"style":31846},[31845],[34,530074,202],{"className":530075},[31850],[34,530077],{"className":530078,"style":31846},[31845],[34,530080,530082,530085,530088,530091,530094],{"className":530081},[31676],[34,530083],{"className":530084,"style":379449},[31680],[34,530086,97300],{"className":530087},[31685,318441],[34,530089],{"className":530090,"style":32604},[31845],[34,530092,378006],{"className":530093},[32024],[34,530095],{"className":530096,"style":32604},[31845],[34,530098,530100,530103,530106,530109,530112,530115,530118],{"className":530099},[31676],[34,530101],{"className":530102,"style":31774},[31680],[34,530104,32685],{"className":530105},[31782],[34,530107,459],{"className":530108},[31685],[34,530110,32715],{"className":530111},[31690],[34,530113],{"className":530114,"style":31846},[31845],[34,530116,202],{"className":530117},[31850],[34,530119],{"className":530120,"style":31846},[31845],[34,530122,530124,530127,530130,530133,530136,530139,530142,530145,530148,530151],{"className":530123},[31676],[34,530125],{"className":530126,"style":31774},[31680],[34,530128,388385],{"className":530129},[31782],[34,530131,111],{"className":530132},[31685,31686],[34,530134,31960],{"className":530135},[32119],[34,530137],{"className":530138,"style":32123},[31845],[34,530140,459],{"className":530141},[31685],[34,530143,2557],{"className":530144},[31690],[34,530146],{"className":530147,"style":31846},[31845],[34,530149,162086],{"className":530150},[31850],[34,530152],{"className":530153,"style":31846},[31845],[34,530155,530157,530160,530163,530166,530169],{"className":530156},[31676],[34,530158],{"className":530159,"style":34812},[31680],[34,530161,111],{"className":530162},[31685,31686],[34,530164],{"className":530165,"style":31846},[31845],[34,530167,112296],{"className":530168},[31850],[34,530170],{"className":530171,"style":31846},[31845],[34,530173,530175,530178,530181],{"className":530174},[31676],[34,530176],{"className":530177,"style":31774},[31680],[34,530179,97300],{"className":530180},[31685,318441],[34,530182,32715],{"className":530183},[31690],[108,530185,530186,33183,530188],{},[2627,530187,500696],{},[105,530189,530190,530646],{},[108,530191,530192,325706,530223,475679,530251,530303,530304,530356,530357,326989,530482,314945,530555,530588,530589,322067,530617,530645],{},[2627,530193,530194,530222],{},[34,530195,530197,530210],{"className":530196},[31642],[34,530198,530200],{"className":530199},[31646],[31648,530201,530202],{"xmlns":31650},[31652,530203,530204,530208],{},[31655,530205,530206],{},[31658,530207,34386],{},[31665,530209,34386],{"encoding":31667},[34,530211,530213],{"className":530212,"ariaHidden":758},[31672],[34,530214,530216,530219],{"className":530215},[31676],[34,530217],{"className":530218,"style":33218},[31680],[34,530220,34386],{"className":530221,"style":34427},[31685,31686]," 没有 1",[34,530224,530226,530239],{"className":530225},[31642],[34,530227,530229],{"className":530228},[31646],[31648,530230,530231],{"xmlns":31650},[31652,530232,530233,530237],{},[31655,530234,530235],{},[31658,530236,34386],{},[31665,530238,34386],{"encoding":31667},[34,530240,530242],{"className":530241,"ariaHidden":758},[31672],[34,530243,530245,530248],{"className":530244},[31676],[34,530246],{"className":530247,"style":33218},[31680],[34,530249,34386],{"className":530250,"style":34427},[31685,31686],[34,530252,530254,530276],{"className":530253},[31642],[34,530255,530257],{"className":530256},[31646],[31648,530258,530259],{"xmlns":31650},[31652,530260,530261,530273],{},[31655,530262,530263,530265,530267,530269,530271],{},[31662,530264,456],{"stretchy":702},[31658,530266,32936],{},[31662,530268,31960],{"separator":758},[31658,530270,67764],{},[31662,530272,2557],{"stretchy":702},[31665,530274,530275],{"encoding":31667},"(x, y)",[34,530277,530279],{"className":530278,"ariaHidden":758},[31672],[34,530280,530282,530285,530288,530291,530294,530297,530300],{"className":530281},[31676],[34,530283],{"className":530284,"style":31774},[31680],[34,530286,456],{"className":530287},[31782],[34,530289,32936],{"className":530290},[31685,31686],[34,530292,31960],{"className":530293},[32119],[34,530295],{"className":530296,"style":32123},[31845],[34,530298,67764],{"className":530299,"style":32062},[31685,31686],[34,530301,2557],{"className":530302},[31690],"，那么对于元素 ",[34,530305,530307,530329],{"className":530306},[31642],[34,530308,530310],{"className":530309},[31646],[31648,530311,530312],{"xmlns":31650},[31652,530313,530314,530326],{},[31655,530315,530316,530318,530320,530322,530324],{},[31662,530317,456],{"stretchy":702},[31749,530319,640],{},[31662,530321,31960],{"separator":758},[31749,530323,768],{},[31662,530325,2557],{"stretchy":702},[31665,530327,530328],{"encoding":31667},"(1, 2)",[34,530330,530332],{"className":530331,"ariaHidden":758},[31672],[34,530333,530335,530338,530341,530344,530347,530350,530353],{"className":530334},[31676],[34,530336],{"className":530337,"style":31774},[31680],[34,530339,456],{"className":530340},[31782],[34,530342,640],{"className":530343},[31685],[34,530345,31960],{"className":530346},[32119],[34,530348],{"className":530349,"style":32123},[31845],[34,530351,768],{"className":530352},[31685],[34,530354,2557],{"className":530355},[31690],"，必须有 ",[34,530358,530360,530404],{"className":530359},[31642],[34,530361,530363],{"className":530362},[31646],[31648,530364,530365],{"xmlns":31650},[31652,530366,530367,530401],{},[31655,530368,530369,530371,530373,530375,530377,530379,530381,530383,530385,530387,530389,530391,530393,530395,530397,530399],{},[31662,530370,456],{"stretchy":702},[31658,530372,32936],{},[31662,530374,31960],{"separator":758},[31658,530376,67764],{},[31662,530378,2557],{"stretchy":702},[31662,530380,456],{"stretchy":702},[31749,530382,640],{},[31662,530384,31960],{"separator":758},[31749,530386,768],{},[31662,530388,2557],{"stretchy":702},[31662,530390,202],{},[31662,530392,456],{"stretchy":702},[31749,530394,640],{},[31662,530396,31960],{"separator":758},[31749,530398,768],{},[31662,530400,2557],{"stretchy":702},[31665,530402,530403],{"encoding":31667},"(x, y)(1, 2) = (1, 2)",[34,530405,530407,530458],{"className":530406,"ariaHidden":758},[31672],[34,530408,530410,530413,530416,530419,530422,530425,530428,530431,530434,530437,530440,530443,530446,530449,530452,530455],{"className":530409},[31676],[34,530411],{"className":530412,"style":31774},[31680],[34,530414,456],{"className":530415},[31782],[34,530417,32936],{"className":530418},[31685,31686],[34,530420,31960],{"className":530421},[32119],[34,530423],{"className":530424,"style":32123},[31845],[34,530426,67764],{"className":530427,"style":32062},[31685,31686],[34,530429,2557],{"className":530430},[31690],[34,530432,456],{"className":530433},[31782],[34,530435,640],{"className":530436},[31685],[34,530438,31960],{"className":530439},[32119],[34,530441],{"className":530442,"style":32123},[31845],[34,530444,768],{"className":530445},[31685],[34,530447,2557],{"className":530448},[31690],[34,530450],{"className":530451,"style":31846},[31845],[34,530453,202],{"className":530454},[31850],[34,530456],{"className":530457,"style":31846},[31845],[34,530459,530461,530464,530467,530470,530473,530476,530479],{"className":530460},[31676],[34,530462],{"className":530463,"style":31774},[31680],[34,530465,456],{"className":530466},[31782],[34,530468,640],{"className":530469},[31685],[34,530471,31960],{"className":530472},[32119],[34,530474],{"className":530475,"style":32123},[31845],[34,530477,768],{"className":530478},[31685],[34,530480,2557],{"className":530481},[31690],[34,530483,530485,530507],{"className":530484},[31642],[34,530486,530488],{"className":530487},[31646],[31648,530489,530490],{"xmlns":31650},[31652,530491,530492,530504],{},[31655,530493,530494,530496,530498,530500,530502],{},[31658,530495,67764],{},[31662,530497,71191],{},[31749,530499,768],{},[31662,530501,202],{},[31749,530503,768],{},[31665,530505,530506],{"encoding":31667},"y \\cdot 2 = 2",[34,530508,530510,530528,530546],{"className":530509,"ariaHidden":758},[31672],[34,530511,530513,530516,530519,530522,530525],{"className":530512},[31676],[34,530514],{"className":530515,"style":73503},[31680],[34,530517,67764],{"className":530518,"style":32062},[31685,31686],[34,530520],{"className":530521,"style":32604},[31845],[34,530523,71191],{"className":530524},[32024],[34,530526],{"className":530527,"style":32604},[31845],[34,530529,530531,530534,530537,530540,530543],{"className":530530},[31676],[34,530532],{"className":530533,"style":32617},[31680],[34,530535,768],{"className":530536},[31685],[34,530538],{"className":530539,"style":31846},[31845],[34,530541,202],{"className":530542},[31850],[34,530544],{"className":530545,"style":31846},[31845],[34,530547,530549,530552],{"className":530548},[31676],[34,530550],{"className":530551,"style":32617},[31680],[34,530553,768],{"className":530554},[31685],[34,530556,530558,530573],{"className":530557},[31642],[34,530559,530561],{"className":530560},[31646],[31648,530562,530563],{"xmlns":31650},[31652,530564,530565,530571],{},[31655,530566,530567,530569],{},[31749,530568,768],{},[31658,530570,97300],{"mathvariant":318416},[31665,530572,496029],{"encoding":31667},[34,530574,530576],{"className":530575,"ariaHidden":758},[31672],[34,530577,530579,530582,530585],{"className":530578},[31676],[34,530580],{"className":530581,"style":320927},[31680],[34,530583,768],{"className":530584},[31685],[34,530586,97300],{"className":530587},[31685,318441]," 中成立，但在偶数环中不存在这样的 ",[34,530590,530592,530605],{"className":530591},[31642],[34,530593,530595],{"className":530594},[31646],[31648,530596,530597],{"xmlns":31650},[31652,530598,530599,530603],{},[31655,530600,530601],{},[31658,530602,67764],{},[31665,530604,67764],{"encoding":31667},[34,530606,530608],{"className":530607,"ariaHidden":758},[31672],[34,530609,530611,530614],{"className":530610},[31676],[34,530612],{"className":530613,"style":38774},[31680],[34,530615,67764],{"className":530616,"style":32062},[31685,31686],[34,530618,530620,530633],{"className":530619},[31642],[34,530621,530623],{"className":530622},[31646],[31648,530624,530625],{"xmlns":31650},[31652,530626,530627,530631],{},[31655,530628,530629],{},[31658,530630,34386],{},[31665,530632,34386],{"encoding":31667},[34,530634,530636],{"className":530635,"ariaHidden":758},[31672],[34,530637,530639,530642],{"className":530638},[31676],[34,530640],{"className":530641,"style":33218},[31680],[34,530643,34386],{"className":530644,"style":34427},[31685,31686]," 无单位元。",[108,530647,530648,2664,530679,530707,530708,530760,530761,39984,530835,467788,531066,531118,531119,389666],{},[2627,530649,530650,530678],{},[34,530651,530653,530666],{"className":530652},[31642],[34,530654,530656],{"className":530655},[31646],[31648,530657,530658],{"xmlns":31650},[31652,530659,530660,530664],{},[31655,530661,530662],{},[31658,530663,94675],{},[31665,530665,94675],{"encoding":31667},[34,530667,530669],{"className":530668,"ariaHidden":758},[31672],[34,530670,530672,530675],{"className":530671},[31676],[34,530673],{"className":530674,"style":33218},[31680],[34,530676,94675],{"className":530677,"style":106455},[31685,31686]," 有 1",[34,530680,530682,530695],{"className":530681},[31642],[34,530683,530685],{"className":530684},[31646],[31648,530686,530687],{"xmlns":31650},[31652,530688,530689,530693],{},[31655,530690,530691],{},[31658,530692,94675],{},[31665,530694,94675],{"encoding":31667},[34,530696,530698],{"className":530697,"ariaHidden":758},[31672],[34,530699,530701,530704],{"className":530700},[31676],[34,530702],{"className":530703,"style":33218},[31680],[34,530705,94675],{"className":530706,"style":106455},[31685,31686]," 中的元素形如 ",[34,530709,530711,530733],{"className":530710},[31642],[34,530712,530714],{"className":530713},[31646],[31648,530715,530716],{"xmlns":31650},[31652,530717,530718,530730],{},[31655,530719,530720,530722,530724,530726,530728],{},[31662,530721,456],{"stretchy":702},[31658,530723,111],{},[31662,530725,31960],{"separator":758},[31749,530727,459],{},[31662,530729,2557],{"stretchy":702},[31665,530731,530732],{"encoding":31667},"(a, 0)",[34,530734,530736],{"className":530735,"ariaHidden":758},[31672],[34,530737,530739,530742,530745,530748,530751,530754,530757],{"className":530738},[31676],[34,530740],{"className":530741,"style":31774},[31680],[34,530743,456],{"className":530744},[31782],[34,530746,111],{"className":530747},[31685,31686],[34,530749,31960],{"className":530750},[32119],[34,530752],{"className":530753,"style":32123},[31845],[34,530755,459],{"className":530756},[31685],[34,530758,2557],{"className":530759},[31690],"。取 ",[34,530762,530764,530790],{"className":530763},[31642],[34,530765,530767],{"className":530766},[31646],[31648,530768,530769],{"xmlns":31650},[31652,530770,530771,530787],{},[31655,530772,530773,530775,530777,530779,530781,530783,530785],{},[31658,530774,33244],{},[31662,530776,202],{},[31662,530778,456],{"stretchy":702},[31749,530780,640],{},[31662,530782,31960],{"separator":758},[31749,530784,459],{},[31662,530786,2557],{"stretchy":702},[31665,530788,530789],{"encoding":31667},"e = (1, 0)",[34,530791,530793,530811],{"className":530792,"ariaHidden":758},[31672],[34,530794,530796,530799,530802,530805,530808],{"className":530795},[31676],[34,530797],{"className":530798,"style":32527},[31680],[34,530800,33244],{"className":530801},[31685,31686],[34,530803],{"className":530804,"style":31846},[31845],[34,530806,202],{"className":530807},[31850],[34,530809],{"className":530810,"style":31846},[31845],[34,530812,530814,530817,530820,530823,530826,530829,530832],{"className":530813},[31676],[34,530815],{"className":530816,"style":31774},[31680],[34,530818,456],{"className":530819},[31782],[34,530821,640],{"className":530822},[31685],[34,530824,31960],{"className":530825},[32119],[34,530827],{"className":530828,"style":32123},[31845],[34,530830,459],{"className":530831},[31685],[34,530833,2557],{"className":530834},[31690],[34,530836,530838,530904],{"className":530837},[31642],[34,530839,530841],{"className":530840},[31646],[31648,530842,530843],{"xmlns":31650},[31652,530844,530845,530901],{},[31655,530846,530847,530849,530851,530853,530855,530857,530859,530861,530863,530865,530867,530869,530871,530873,530875,530877,530879,530881,530883,530885,530887,530889,530891,530893,530895,530897,530899],{},[31662,530848,456],{"stretchy":702},[31749,530850,640],{},[31662,530852,31960],{"separator":758},[31749,530854,459],{},[31662,530856,2557],{"stretchy":702},[31662,530858,71191],{},[31662,530860,456],{"stretchy":702},[31658,530862,111],{},[31662,530864,31960],{"separator":758},[31749,530866,459],{},[31662,530868,2557],{"stretchy":702},[31662,530870,202],{},[31662,530872,456],{"stretchy":702},[31749,530874,640],{},[31662,530876,71191],{},[31658,530878,111],{},[31662,530880,31960],{"separator":758},[31749,530882,459],{},[31662,530884,71191],{},[31749,530886,459],{},[31662,530888,2557],{"stretchy":702},[31662,530890,202],{},[31662,530892,456],{"stretchy":702},[31658,530894,111],{},[31662,530896,31960],{"separator":758},[31749,530898,459],{},[31662,530900,2557],{"stretchy":702},[31665,530902,530903],{"encoding":31667},"(1, 0) \\cdot (a, 0) = (1\\cdot a, 0\\cdot 0) = (a, 0)",[34,530905,530907,530940,530973,530994,531021,531042],{"className":530906,"ariaHidden":758},[31672],[34,530908,530910,530913,530916,530919,530922,530925,530928,530931,530934,530937],{"className":530909},[31676],[34,530911],{"className":530912,"style":31774},[31680],[34,530914,456],{"className":530915},[31782],[34,530917,640],{"className":530918},[31685],[34,530920,31960],{"className":530921},[32119],[34,530923],{"className":530924,"style":32123},[31845],[34,530926,459],{"className":530927},[31685],[34,530929,2557],{"className":530930},[31690],[34,530932],{"className":530933,"style":32604},[31845],[34,530935,71191],{"className":530936},[32024],[34,530938],{"className":530939,"style":32604},[31845],[34,530941,530943,530946,530949,530952,530955,530958,530961,530964,530967,530970],{"className":530942},[31676],[34,530944],{"className":530945,"style":31774},[31680],[34,530947,456],{"className":530948},[31782],[34,530950,111],{"className":530951},[31685,31686],[34,530953,31960],{"className":530954},[32119],[34,530956],{"className":530957,"style":32123},[31845],[34,530959,459],{"className":530960},[31685],[34,530962,2557],{"className":530963},[31690],[34,530965],{"className":530966,"style":31846},[31845],[34,530968,202],{"className":530969},[31850],[34,530971],{"className":530972,"style":31846},[31845],[34,530974,530976,530979,530982,530985,530988,530991],{"className":530975},[31676],[34,530977],{"className":530978,"style":31774},[31680],[34,530980,456],{"className":530981},[31782],[34,530983,640],{"className":530984},[31685],[34,530986],{"className":530987,"style":32604},[31845],[34,530989,71191],{"className":530990},[32024],[34,530992],{"className":530993,"style":32604},[31845],[34,530995,530997,531000,531003,531006,531009,531012,531015,531018],{"className":530996},[31676],[34,530998],{"className":530999,"style":40292},[31680],[34,531001,111],{"className":531002},[31685,31686],[34,531004,31960],{"className":531005},[32119],[34,531007],{"className":531008,"style":32123},[31845],[34,531010,459],{"className":531011},[31685],[34,531013],{"className":531014,"style":32604},[31845],[34,531016,71191],{"className":531017},[32024],[34,531019],{"className":531020,"style":32604},[31845],[34,531022,531024,531027,531030,531033,531036,531039],{"className":531023},[31676],[34,531025],{"className":531026,"style":31774},[31680],[34,531028,459],{"className":531029},[31685],[34,531031,2557],{"className":531032},[31690],[34,531034],{"className":531035,"style":31846},[31845],[34,531037,202],{"className":531038},[31850],[34,531040],{"className":531041,"style":31846},[31845],[34,531043,531045,531048,531051,531054,531057,531060,531063],{"className":531044},[31676],[34,531046],{"className":531047,"style":31774},[31680],[34,531049,456],{"className":531050},[31782],[34,531052,111],{"className":531053},[31685,31686],[34,531055,31960],{"className":531056},[32119],[34,531058],{"className":531059,"style":32123},[31845],[34,531061,459],{"className":531062},[31685],[34,531064,2557],{"className":531065},[31690],[34,531067,531069,531091],{"className":531068},[31642],[34,531070,531072],{"className":531071},[31646],[31648,531073,531074],{"xmlns":31650},[31652,531075,531076,531088],{},[31655,531077,531078,531080,531082,531084,531086],{},[31662,531079,456],{"stretchy":702},[31749,531081,640],{},[31662,531083,31960],{"separator":758},[31749,531085,459],{},[31662,531087,2557],{"stretchy":702},[31665,531089,531090],{"encoding":31667},"(1, 0)",[34,531092,531094],{"className":531093,"ariaHidden":758},[31672],[34,531095,531097,531100,531103,531106,531109,531112,531115],{"className":531096},[31676],[34,531098],{"className":531099,"style":31774},[31680],[34,531101,456],{"className":531102},[31782],[34,531104,640],{"className":531105},[31685],[34,531107,31960],{"className":531108},[32119],[34,531110],{"className":531111,"style":32123},[31845],[34,531113,459],{"className":531114},[31685],[34,531116,2557],{"className":531117},[31690]," 是子环 ",[34,531120,531122,531135],{"className":531121},[31642],[34,531123,531125],{"className":531124},[31646],[31648,531126,531127],{"xmlns":31650},[31652,531128,531129,531133],{},[31655,531130,531131],{},[31658,531132,94675],{},[31665,531134,94675],{"encoding":31667},[34,531136,531138],{"className":531137,"ariaHidden":758},[31672],[34,531139,531141,531144],{"className":531140},[31676],[34,531142],{"className":531143,"style":33218},[31680],[34,531145,94675],{"className":531146,"style":106455},[31685,31686],[15,531148,531149,531208],{},[2627,531150,531151,531152,43298,531180,389117],{},"解答 (3): ",[34,531153,531155,531168],{"className":531154},[31642],[34,531156,531158],{"className":531157},[31646],[31648,531159,531160],{"xmlns":31650},[31652,531161,531162,531166],{},[31655,531163,531164],{},[31658,531165,34386],{},[31665,531167,34386],{"encoding":31667},[34,531169,531171],{"className":531170,"ariaHidden":758},[31672],[34,531172,531174,531177],{"className":531173},[31676],[34,531175],{"className":531176,"style":33218},[31680],[34,531178,34386],{"className":531179,"style":34427},[31685,31686],[34,531181,531183,531196],{"className":531182},[31642],[34,531184,531186],{"className":531185},[31646],[31648,531187,531188],{"xmlns":31650},[31652,531189,531190,531194],{},[31655,531191,531192],{},[31658,531193,94675],{},[31665,531195,94675],{"encoding":31667},[34,531197,531199],{"className":531198,"ariaHidden":758},[31672],[34,531200,531202,531205],{"className":531201},[31676],[34,531203],{"className":531204,"style":33218},[31680],[34,531206,94675],{"className":531207,"style":106455},[31685,31686],"\n这里可以分为两种情况：它们的 1 是同一个，或者它们的 1 不是同一个。题目没要求，我们可以给一个更有趣的（1 不相同）例子。",[105,531210,531211,531492],{},[108,531212,531213,33183,531215],{},[2627,531214,233206],{},[105,531216,531217,531407],{},[108,531218,396859,531219,347,531292,347],{},[34,531220,531222,531244],{"className":531221},[31642],[34,531223,531225],{"className":531224},[31646],[31648,531226,531227],{"xmlns":31650},[31652,531228,531229,531241],{},[31655,531230,531231,531233,531235,531237,531239],{},[31658,531232,34386],{},[31662,531234,202],{},[31658,531236,97300],{"mathvariant":318416},[31662,531238,378006],{},[31658,531240,97300],{"mathvariant":318416},[31665,531242,531243],{"encoding":31667},"R = \\mathbb{Z} \\oplus \\mathbb{Z}",[34,531245,531247,531265,531283],{"className":531246,"ariaHidden":758},[31672],[34,531248,531250,531253,531256,531259,531262],{"className":531249},[31676],[34,531251],{"className":531252,"style":33218},[31680],[34,531254,34386],{"className":531255,"style":34427},[31685,31686],[34,531257],{"className":531258,"style":31846},[31845],[34,531260,202],{"className":531261},[31850],[34,531263],{"className":531264,"style":31846},[31845],[34,531266,531268,531271,531274,531277,531280],{"className":531267},[31676],[34,531269],{"className":531270,"style":379449},[31680],[34,531272,97300],{"className":531273},[31685,318441],[34,531275],{"className":531276,"style":32604},[31845],[34,531278,378006],{"className":531279},[32024],[34,531281],{"className":531282,"style":32604},[31845],[34,531284,531286,531289],{"className":531285},[31676],[34,531287],{"className":531288,"style":320927},[31680],[34,531290,97300],{"className":531291},[31685,318441],[34,531293,531295,531325],{"className":531294},[31642],[34,531296,531298],{"className":531297},[31646],[31648,531299,531300],{"xmlns":31650},[31652,531301,531302,531322],{},[31655,531303,531304,531310,531312,531314,531316,531318,531320],{},[31743,531305,531306,531308],{},[31749,531307,640],{},[31658,531309,34386],{},[31662,531311,202],{},[31662,531313,456],{"stretchy":702},[31749,531315,640],{},[31662,531317,31960],{"separator":758},[31749,531319,640],{},[31662,531321,2557],{"stretchy":702},[31665,531323,531324],{"encoding":31667},"1_R = (1, 1)",[34,531326,531328,531383],{"className":531327,"ariaHidden":758},[31672],[34,531329,531331,531334,531374,531377,531380],{"className":531330},[31676],[34,531332],{"className":531333,"style":38357},[31680],[34,531335,531337,531340],{"className":531336},[31685],[34,531338,640],{"className":531339},[31685],[34,531341,531343],{"className":531342},[31792],[34,531344,531346,531366],{"className":531345},[31796,31797],[34,531347,531349,531363],{"className":531348},[31801],[34,531350,531352],{"className":531351,"style":116558},[31805],[34,531353,531354,531357],{"style":31809},[34,531355],{"className":531356,"style":31814},[31813],[34,531358,531360],{"className":531359},[31818,31819,31820,31821],[34,531361,34386],{"className":531362,"style":34427},[31685,31686,31821],[34,531364,31829],{"className":531365},[31828],[34,531367,531369],{"className":531368},[31801],[34,531370,531372],{"className":531371,"style":31836},[31805],[34,531373],{},[34,531375],{"className":531376,"style":31846},[31845],[34,531378,202],{"className":531379},[31850],[34,531381],{"className":531382,"style":31846},[31845],[34,531384,531386,531389,531392,531395,531398,531401,531404],{"className":531385},[31676],[34,531387],{"className":531388,"style":31774},[31680],[34,531390,456],{"className":531391},[31782],[34,531393,640],{"className":531394},[31685],[34,531396,31960],{"className":531397},[32119],[34,531399],{"className":531400,"style":32123},[31845],[34,531402,640],{"className":531403},[31685],[34,531405,2557],{"className":531406},[31690],[108,531408,396859,531409,347],{},[34,531410,531412,531438],{"className":531411},[31642],[34,531413,531415],{"className":531414},[31646],[31648,531416,531417],{"xmlns":31650},[31652,531418,531419,531435],{},[31655,531420,531421,531423,531425,531427,531429,531431,531433],{},[31658,531422,94675],{},[31662,531424,202],{},[31658,531426,97300],{"mathvariant":318416},[31662,531428,378006],{},[31662,531430,32685],{"stretchy":702},[31749,531432,459],{},[31662,531434,32715],{"stretchy":702},[31665,531436,531437],{"encoding":31667},"S = \\mathbb{Z} \\oplus \\{0\\}",[34,531439,531441,531459,531477],{"className":531440,"ariaHidden":758},[31672],[34,531442,531444,531447,531450,531453,531456],{"className":531443},[31676],[34,531445],{"className":531446,"style":33218},[31680],[34,531448,94675],{"className":531449,"style":106455},[31685,31686],[34,531451],{"className":531452,"style":31846},[31845],[34,531454,202],{"className":531455},[31850],[34,531457],{"className":531458,"style":31846},[31845],[34,531460,531462,531465,531468,531471,531474],{"className":531461},[31676],[34,531463],{"className":531464,"style":379449},[31680],[34,531466,97300],{"className":531467},[31685,318441],[34,531469],{"className":531470,"style":32604},[31845],[34,531472,378006],{"className":531473},[32024],[34,531475],{"className":531476,"style":32604},[31845],[34,531478,531480,531483,531486,531489],{"className":531479},[31676],[34,531481],{"className":531482,"style":31774},[31680],[34,531484,32685],{"className":531485},[31782],[34,531487,459],{"className":531488},[31685],[34,531490,32715],{"className":531491},[31690],[108,531493,531494,33183,531496],{},[2627,531495,500696],{},[105,531497,531498,531556,531637,531719,531888],{},[108,531499,531500,11008,531528,388672],{},[34,531501,531503,531516],{"className":531502},[31642],[34,531504,531506],{"className":531505},[31646],[31648,531507,531508],{"xmlns":31650},[31652,531509,531510,531514],{},[31655,531511,531512],{},[31658,531513,94675],{},[31665,531515,94675],{"encoding":31667},[34,531517,531519],{"className":531518,"ariaHidden":758},[31672],[34,531520,531522,531525],{"className":531521},[31676],[34,531523],{"className":531524,"style":33218},[31680],[34,531526,94675],{"className":531527,"style":106455},[31685,31686],[34,531529,531531,531544],{"className":531530},[31642],[34,531532,531534],{"className":531533},[31646],[31648,531535,531536],{"xmlns":31650},[31652,531537,531538,531542],{},[31655,531539,531540],{},[31658,531541,34386],{},[31665,531543,34386],{"encoding":31667},[34,531545,531547],{"className":531546,"ariaHidden":758},[31672],[34,531548,531550,531553],{"className":531549},[31676],[34,531551],{"className":531552,"style":33218},[31680],[34,531554,34386],{"className":531555,"style":34427},[31685,31686],[108,531557,531558,387891,531586,347],{},[34,531559,531561,531574],{"className":531560},[31642],[34,531562,531564],{"className":531563},[31646],[31648,531565,531566],{"xmlns":31650},[31652,531567,531568,531572],{},[31655,531569,531570],{},[31658,531571,94675],{},[31665,531573,94675],{"encoding":31667},[34,531575,531577],{"className":531576,"ariaHidden":758},[31672],[34,531578,531580,531583],{"className":531579},[31676],[34,531581],{"className":531582,"style":33218},[31680],[34,531584,94675],{"className":531585,"style":106455},[31685,31686],[34,531587,531589,531610],{"className":531588},[31642],[34,531590,531592],{"className":531591},[31646],[31648,531593,531594],{"xmlns":31650},[31652,531595,531596,531608],{},[31655,531597,531598,531600,531602,531604,531606],{},[31662,531599,456],{"stretchy":702},[31749,531601,640],{},[31662,531603,31960],{"separator":758},[31749,531605,459],{},[31662,531607,2557],{"stretchy":702},[31665,531609,531090],{"encoding":31667},[34,531611,531613],{"className":531612,"ariaHidden":758},[31672],[34,531614,531616,531619,531622,531625,531628,531631,531634],{"className":531615},[31676],[34,531617],{"className":531618,"style":31774},[31680],[34,531620,456],{"className":531621},[31782],[34,531623,640],{"className":531624},[31685],[34,531626,31960],{"className":531627},[32119],[34,531629],{"className":531630,"style":32123},[31845],[34,531632,459],{"className":531633},[31685],[34,531635,2557],{"className":531636},[31690],[108,531638,531639,387891,531667,347],{},[34,531640,531642,531655],{"className":531641},[31642],[34,531643,531645],{"className":531644},[31646],[31648,531646,531647],{"xmlns":31650},[31652,531648,531649,531653],{},[31655,531650,531651],{},[31658,531652,34386],{},[31665,531654,34386],{"encoding":31667},[34,531656,531658],{"className":531657,"ariaHidden":758},[31672],[34,531659,531661,531664],{"className":531660},[31676],[34,531662],{"className":531663,"style":33218},[31680],[34,531665,34386],{"className":531666,"style":34427},[31685,31686],[34,531668,531670,531692],{"className":531669},[31642],[34,531671,531673],{"className":531672},[31646],[31648,531674,531675],{"xmlns":31650},[31652,531676,531677,531689],{},[31655,531678,531679,531681,531683,531685,531687],{},[31662,531680,456],{"stretchy":702},[31749,531682,640],{},[31662,531684,31960],{"separator":758},[31749,531686,640],{},[31662,531688,2557],{"stretchy":702},[31665,531690,531691],{"encoding":31667},"(1, 1)",[34,531693,531695],{"className":531694,"ariaHidden":758},[31672],[34,531696,531698,531701,531704,531707,531710,531713,531716],{"className":531697},[31676],[34,531699],{"className":531700,"style":31774},[31680],[34,531702,456],{"className":531703},[31782],[34,531705,640],{"className":531706},[31685],[34,531708,31960],{"className":531709},[32119],[34,531711],{"className":531712,"style":32123},[31845],[34,531714,640],{"className":531715},[31685],[34,531717,2557],{"className":531718},[31690],[108,531720,531721,531722,347],{},"两者都有 1，尽管 ",[34,531723,531725,531751],{"className":531724},[31642],[34,531726,531728],{"className":531727},[31646],[31648,531729,531730],{"xmlns":31650},[31652,531731,531732,531748],{},[31655,531733,531734,531740,531742],{},[31743,531735,531736,531738],{},[31749,531737,640],{},[31658,531739,94675],{},[31662,531741,54707],{"mathvariant":38079},[31743,531743,531744,531746],{},[31749,531745,640],{},[31658,531747,34386],{},[31665,531749,531750],{"encoding":31667},"1_S \\neq 1_R",[34,531752,531754,531842],{"className":531753,"ariaHidden":758},[31672],[34,531755,531757,531760,531800,531803,531839],{"className":531756},[31676],[34,531758],{"className":531759,"style":43217},[31680],[34,531761,531763,531766],{"className":531762},[31685],[34,531764,640],{"className":531765},[31685],[34,531767,531769],{"className":531768},[31792],[34,531770,531772,531792],{"className":531771},[31796,31797],[34,531773,531775,531789],{"className":531774},[31801],[34,531776,531778],{"className":531777,"style":116558},[31805],[34,531779,531780,531783],{"style":31809},[34,531781],{"className":531782,"style":31814},[31813],[34,531784,531786],{"className":531785},[31818,31819,31820,31821],[34,531787,94675],{"className":531788,"style":106455},[31685,31686,31821],[34,531790,31829],{"className":531791},[31828],[34,531793,531795],{"className":531794},[31801],[34,531796,531798],{"className":531797,"style":31836},[31805],[34,531799],{},[34,531801],{"className":531802,"style":31846},[31845],[34,531804,531806,531833,531836],{"className":531805},[31850],[34,531807,531809],{"className":531808},[31850],[34,531810,531812],{"className":531811},[31685,54774],[34,531813,531815],{"className":531814},[54778],[34,531816,531818,531821,531830],{"className":531817},[54782],[34,531819],{"className":531820,"style":43217},[31680],[34,531822,531824],{"className":531823},[54789],[34,531825,531827],{"className":531826},[31685],[34,531828,54796],{"className":531829},[31850],[34,531831],{"className":531832},[54800],[34,531834],{"className":531835},[31845,54804],[34,531837,202],{"className":531838},[31850],[34,531840],{"className":531841,"style":31846},[31845],[34,531843,531845,531848],{"className":531844},[31676],[34,531846],{"className":531847,"style":38357},[31680],[34,531849,531851,531854],{"className":531850},[31685],[34,531852,640],{"className":531853},[31685],[34,531855,531857],{"className":531856},[31792],[34,531858,531860,531880],{"className":531859},[31796,31797],[34,531861,531863,531877],{"className":531862},[31801],[34,531864,531866],{"className":531865,"style":116558},[31805],[34,531867,531868,531871],{"style":31809},[34,531869],{"className":531870,"style":31814},[31813],[34,531872,531874],{"className":531873},[31818,31819,31820,31821],[34,531875,34386],{"className":531876,"style":34427},[31685,31686,31821],[34,531878,31829],{"className":531879},[31828],[34,531881,531883],{"className":531882},[31801],[34,531884,531886],{"className":531885,"style":31836},[31805],[34,531887],{},[108,531889,531890],{},[71807,531891,531892,531893,531979,531980,532135],{},"(当然，最简单的例子是 ",[34,531894,531896,531922],{"className":531895},[31642],[34,531897,531899],{"className":531898},[31646],[31648,531900,531901],{"xmlns":31650},[31652,531902,531903,531919],{},[31655,531904,531905,531907,531909,531911,531913,531915,531917],{},[31658,531906,34386],{},[31662,531908,202],{},[31658,531910,116863],{"mathvariant":318416},[31662,531912,31960],{"separator":758},[31658,531914,94675],{},[31662,531916,202],{},[31658,531918,97300],{"mathvariant":318416},[31665,531920,531921],{"encoding":31667},"R=\\mathbb{Q}, S=\\mathbb{Z}",[34,531923,531925,531943,531970],{"className":531924,"ariaHidden":758},[31672],[34,531926,531928,531931,531934,531937,531940],{"className":531927},[31676],[34,531929],{"className":531930,"style":33218},[31680],[34,531932,34386],{"className":531933,"style":34427},[31685,31686],[34,531935],{"className":531936,"style":31846},[31845],[34,531938,202],{"className":531939},[31850],[34,531941],{"className":531942,"style":31846},[31845],[34,531944,531946,531949,531952,531955,531958,531961,531964,531967],{"className":531945},[31676],[34,531947],{"className":531948,"style":338016},[31680],[34,531950,116863],{"className":531951},[31685,318441],[34,531953,31960],{"className":531954},[32119],[34,531956],{"className":531957,"style":32123},[31845],[34,531959,94675],{"className":531960,"style":106455},[31685,31686],[34,531962],{"className":531963,"style":31846},[31845],[34,531965,202],{"className":531966},[31850],[34,531968],{"className":531969,"style":31846},[31845],[34,531971,531973,531976],{"className":531972},[31676],[34,531974],{"className":531975,"style":320927},[31680],[34,531977,97300],{"className":531978},[31685,318441],"，此时 ",[34,531981,531983,532013],{"className":531982},[31642],[34,531984,531986],{"className":531985},[31646],[31648,531987,531988],{"xmlns":31650},[31652,531989,531990,532010],{},[31655,531991,531992,531998,532000,532006,532008],{},[31743,531993,531994,531996],{},[31749,531995,640],{},[31658,531997,34386],{},[31662,531999,202],{},[31743,532001,532002,532004],{},[31749,532003,640],{},[31658,532005,94675],{},[31662,532007,202],{},[31749,532009,640],{},[31665,532011,532012],{"encoding":31667},"1_R = 1_S = 1",[34,532014,532016,532071,532126],{"className":532015,"ariaHidden":758},[31672],[34,532017,532019,532022,532062,532065,532068],{"className":532018},[31676],[34,532020],{"className":532021,"style":38357},[31680],[34,532023,532025,532028],{"className":532024},[31685],[34,532026,640],{"className":532027},[31685],[34,532029,532031],{"className":532030},[31792],[34,532032,532034,532054],{"className":532033},[31796,31797],[34,532035,532037,532051],{"className":532036},[31801],[34,532038,532040],{"className":532039,"style":116558},[31805],[34,532041,532042,532045],{"style":31809},[34,532043],{"className":532044,"style":31814},[31813],[34,532046,532048],{"className":532047},[31818,31819,31820,31821],[34,532049,34386],{"className":532050,"style":34427},[31685,31686,31821],[34,532052,31829],{"className":532053},[31828],[34,532055,532057],{"className":532056},[31801],[34,532058,532060],{"className":532059,"style":31836},[31805],[34,532061],{},[34,532063],{"className":532064,"style":31846},[31845],[34,532066,202],{"className":532067},[31850],[34,532069],{"className":532070,"style":31846},[31845],[34,532072,532074,532077,532117,532120,532123],{"className":532073},[31676],[34,532075],{"className":532076,"style":38357},[31680],[34,532078,532080,532083],{"className":532079},[31685],[34,532081,640],{"className":532082},[31685],[34,532084,532086],{"className":532085},[31792],[34,532087,532089,532109],{"className":532088},[31796,31797],[34,532090,532092,532106],{"className":532091},[31801],[34,532093,532095],{"className":532094,"style":116558},[31805],[34,532096,532097,532100],{"style":31809},[34,532098],{"className":532099,"style":31814},[31813],[34,532101,532103],{"className":532102},[31818,31819,31820,31821],[34,532104,94675],{"className":532105,"style":106455},[31685,31686,31821],[34,532107,31829],{"className":532108},[31828],[34,532110,532112],{"className":532111},[31801],[34,532113,532115],{"className":532114,"style":31836},[31805],[34,532116],{},[34,532118],{"className":532119,"style":31846},[31845],[34,532121,202],{"className":532122},[31850],[34,532124],{"className":532125,"style":31846},[31845],[34,532127,532129,532132],{"className":532128},[31676],[34,532130],{"className":532131,"style":32617},[31680],[34,532133,640],{"className":532134},[31685],"。)",[15,532137,532138,532197],{},[2627,532139,532140,532141,389870,532169,370308],{},"解答 (4): ",[34,532142,532144,532157],{"className":532143},[31642],[34,532145,532147],{"className":532146},[31646],[31648,532148,532149],{"xmlns":31650},[31652,532150,532151,532155],{},[31655,532152,532153],{},[31658,532154,34386],{},[31665,532156,34386],{"encoding":31667},[34,532158,532160],{"className":532159,"ariaHidden":758},[31672],[34,532161,532163,532166],{"className":532162},[31676],[34,532164],{"className":532165,"style":33218},[31680],[34,532167,34386],{"className":532168,"style":34427},[31685,31686],[34,532170,532172,532185],{"className":532171},[31642],[34,532173,532175],{"className":532174},[31646],[31648,532176,532177],{"xmlns":31650},[31652,532178,532179,532183],{},[31655,532180,532181],{},[31658,532182,94675],{},[31665,532184,94675],{"encoding":31667},[34,532186,532188],{"className":532187,"ariaHidden":758},[31672],[34,532189,532191,532194],{"className":532190},[31676],[34,532192],{"className":532193,"style":33218},[31680],[34,532195,94675],{"className":532196,"style":106455},[31685,31686],"\n这就好比在一个喧闹的市场（不交换）里找一个安静的房间（交换）。矩阵环是最好的例子。",[105,532199,532200,532705],{},[108,532201,532202,33183,532204],{},[2627,532203,233206],{},[105,532205,532206,532366],{},[108,532207,396859,532208,108987,532314,532365],{},[34,532209,532211,532238],{"className":532210},[31642],[34,532212,532214],{"className":532213},[31646],[31648,532215,532216],{"xmlns":31650},[31652,532217,532218,532236],{},[31655,532219,532220,532222,532224,532230,532232,532234],{},[31658,532221,34386],{},[31662,532223,202],{},[31743,532225,532226,532228],{},[31658,532227,98360],{},[31749,532229,768],{},[31662,532231,456],{"stretchy":702},[31658,532233,34386],{"mathvariant":318416},[31662,532235,2557],{"stretchy":702},[31665,532237,389935],{"encoding":31667},[34,532239,532241,532259],{"className":532240,"ariaHidden":758},[31672],[34,532242,532244,532247,532250,532253,532256],{"className":532243},[31676],[34,532245],{"className":532246,"style":33218},[31680],[34,532248,34386],{"className":532249,"style":34427},[31685,31686],[34,532251],{"className":532252,"style":31846},[31845],[34,532254,202],{"className":532255},[31850],[34,532257],{"className":532258,"style":31846},[31845],[34,532260,532262,532265,532305,532308,532311],{"className":532261},[31676],[34,532263],{"className":532264,"style":31774},[31680],[34,532266,532268,532271],{"className":532267},[31685],[34,532269,98360],{"className":532270,"style":33389},[31685,31686],[34,532272,532274],{"className":532273},[31792],[34,532275,532277,532297],{"className":532276},[31796,31797],[34,532278,532280,532294],{"className":532279},[31801],[34,532281,532283],{"className":532282,"style":31806},[31805],[34,532284,532285,532288],{"style":231884},[34,532286],{"className":532287,"style":31814},[31813],[34,532289,532291],{"className":532290},[31818,31819,31820,31821],[34,532292,768],{"className":532293},[31685,31821],[34,532295,31829],{"className":532296},[31828],[34,532298,532300],{"className":532299},[31801],[34,532301,532303],{"className":532302,"style":31836},[31805],[34,532304],{},[34,532306,456],{"className":532307},[31782],[34,532309,34386],{"className":532310},[31685,318441],[34,532312,2557],{"className":532313},[31690],[34,532315,532317,532335],{"className":532316},[31642],[34,532318,532320],{"className":532319},[31646],[31648,532321,532322],{"xmlns":31650},[31652,532323,532324,532332],{},[31655,532325,532326,532328,532330],{},[31749,532327,768],{},[31662,532329,42071],{},[31749,532331,768],{},[31665,532333,532334],{"encoding":31667},"2 \\times 2",[34,532336,532338,532356],{"className":532337,"ariaHidden":758},[31672],[34,532339,532341,532344,532347,532350,532353],{"className":532340},[31676],[34,532342],{"className":532343,"style":35942},[31680],[34,532345,768],{"className":532346},[31685],[34,532348],{"className":532349,"style":32604},[31845],[34,532351,42071],{"className":532352},[32024],[34,532354],{"className":532355,"style":32604},[31845],[34,532357,532359,532362],{"className":532358},[31676],[34,532360],{"className":532361,"style":32617},[31680],[34,532363,768],{"className":532364},[31685]," 实矩阵环）。矩阵乘法不满足交换律。",[108,532367,396859,532368,532396,532397],{},[34,532369,532371,532384],{"className":532370},[31642],[34,532372,532374],{"className":532373},[31646],[31648,532375,532376],{"xmlns":31650},[31652,532377,532378,532382],{},[31655,532379,532380],{},[31658,532381,94675],{},[31665,532383,94675],{"encoding":31667},[34,532385,532387],{"className":532386,"ariaHidden":758},[31672],[34,532388,532390,532393],{"className":532389},[31676],[34,532391],{"className":532392,"style":33218},[31680],[34,532394,94675],{"className":532395,"style":106455},[31685,31686]," 为所有对角矩阵的集合：\n",[34,532398,532400,532474],{"className":532399},[31642],[34,532401,532403],{"className":532402},[31646],[31648,532404,532405],{"xmlns":31650},[31652,532406,532407,532471],{},[31655,532408,532409,532411,532413],{},[31658,532410,94675],{},[31662,532412,202],{},[31655,532414,532415,532417,532453,532455,532457,532459,532461,532463,532465,532467,532469],{},[31662,532416,32685],{"fence":758},[31655,532418,532419,532421,532451],{},[31662,532420,456],{"fence":758},[71078,532422,532423,532437],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,532424,532425,532431],{},[71087,532426,532427],{},[71090,532428,532429],{"scriptlevel":459,"displaystyle":702},[31658,532430,111],{},[71087,532432,532433],{},[71090,532434,532435],{"scriptlevel":459,"displaystyle":702},[31749,532436,459],{},[71084,532438,532439,532445],{},[71087,532440,532441],{},[71090,532442,532443],{"scriptlevel":459,"displaystyle":702},[31749,532444,459],{},[71087,532446,532447],{},[71090,532448,532449],{"scriptlevel":459,"displaystyle":702},[31658,532450,1697],{},[31662,532452,2557],{"fence":758},[39361,532454,262622],{},[31662,532456,502375],{"fence":758,"lspace":502374,"rspace":502374},[39361,532458,262622],{},[31658,532460,111],{},[31662,532462,31960],{"separator":758},[31658,532464,1697],{},[31662,532466,112296],{},[31658,532468,34386],{"mathvariant":318416},[31662,532470,32715],{"fence":758},[31665,532472,532473],{"encoding":31667},"S = \\left\\{ \\begin{pmatrix} a & 0 \\\\ 0 & b \\end{pmatrix} \\;\\middle|\\; a, b \\in \\mathbb{R} \\right\\}",[34,532475,532477,532495],{"className":532476,"ariaHidden":758},[31672],[34,532478,532480,532483,532486,532489,532492],{"className":532479},[31676],[34,532481],{"className":532482,"style":33218},[31680],[34,532484,94675],{"className":532485,"style":106455},[31685,31686],[34,532487],{"className":532488,"style":31846},[31845],[34,532490,202],{"className":532491},[31850],[34,532493],{"className":532494,"style":31846},[31845],[34,532496,532498,532501],{"className":532497},[31676],[34,532499],{"className":532500,"style":370661},[31680],[34,532502,532504,532510,532627,532630,532672,532675,532678,532681,532684,532687,532690,532693,532696,532699],{"className":532503},[34006],[34,532505,532507],{"className":532506,"style":75811},[31782,75810],[34,532508,32685],{"className":532509},[71281,31820],[34,532511,532513,532519,532621],{"className":532512},[34006],[34,532514,532516],{"className":532515,"style":75811},[31782,75810],[34,532517,456],{"className":532518},[71281,31820],[34,532520,532522],{"className":532521},[31685],[34,532523,532525,532570,532573,532576],{"className":532524},[71078],[34,532526,532528],{"className":532527},[141481],[34,532529,532531,532562],{"className":532530},[31796,31797],[34,532532,532534,532559],{"className":532533},[31801],[34,532535,532537,532548],{"className":532536,"style":370689},[31805],[34,532538,532539,532542],{"style":370692},[34,532540],{"className":532541,"style":41508},[31813],[34,532543,532545],{"className":532544},[31685],[34,532546,111],{"className":532547},[31685,31686],[34,532549,532550,532553],{"style":370704},[34,532551],{"className":532552,"style":41508},[31813],[34,532554,532556],{"className":532555},[31685],[34,532557,459],{"className":532558},[31685],[34,532560,31829],{"className":532561},[31828],[34,532563,532565],{"className":532564},[31801],[34,532566,532568],{"className":532567,"style":370723},[31805],[34,532569],{},[34,532571],{"className":532572,"style":141477},[71732],[34,532574],{"className":532575,"style":141477},[71732],[34,532577,532579],{"className":532578},[141481],[34,532580,532582,532613],{"className":532581},[31796,31797],[34,532583,532585,532610],{"className":532584},[31801],[34,532586,532588,532599],{"className":532587,"style":370689},[31805],[34,532589,532590,532593],{"style":370692},[34,532591],{"className":532592,"style":41508},[31813],[34,532594,532596],{"className":532595},[31685],[34,532597,459],{"className":532598},[31685],[34,532600,532601,532604],{"style":370704},[34,532602],{"className":532603,"style":41508},[31813],[34,532605,532607],{"className":532606},[31685],[34,532608,1697],{"className":532609},[31685,31686],[34,532611,31829],{"className":532612},[31828],[34,532614,532616],{"className":532615},[31801],[34,532617,532619],{"className":532618,"style":370723},[31805],[34,532620],{},[34,532622,532624],{"className":532623,"style":75811},[31690,75810],[34,532625,2557],{"className":532626},[71281,31820],[34,532628],{"className":532629,"style":31846},[31845],[34,532631,532632],{},[34,532633,532635],{"className":532634},[71281,71282],[34,532636,532638,532664],{"className":532637},[31796,31797],[34,532639,532641,532661],{"className":532640},[31801],[34,532642,532644],{"className":532643,"style":370689},[31805],[34,532645,532647,532651],{"style":532646},"top:-3.45em;",[34,532648],{"className":532649,"style":532650},[31813],"height:4.4em;",[34,532652,532654],{"style":532653},"width:0.333em;height:2.4em;",[41588,532655,532658],{"xmlns":41590,"width":503054,"height":532656,"viewBox":532657},"2.4em","0 0 333 2400",[20398,532659],{"d":532660},"M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z",[34,532662,31829],{"className":532663},[31828],[34,532665,532667],{"className":532666},[31801],[34,532668,532670],{"className":532669,"style":370723},[31805],[34,532671],{},[34,532673],{"className":532674,"style":31846},[31845],[34,532676,111],{"className":532677},[31685,31686],[34,532679,31960],{"className":532680},[32119],[34,532682],{"className":532683,"style":32123},[31845],[34,532685,1697],{"className":532686},[31685,31686],[34,532688],{"className":532689,"style":31846},[31845],[34,532691,112296],{"className":532692},[31850],[34,532694],{"className":532695,"style":31846},[31845],[34,532697,34386],{"className":532698},[31685,318441],[34,532700,532702],{"className":532701,"style":75811},[31690,75810],[34,532703,32715],{"className":532704},[71281,31820],[108,532706,532707,33183,532709],{},[2627,532708,500696],{},[105,532710,532711,533557],{},[108,532712,532713,532714],{},"对角矩阵相乘：\n",[34,532715,532717,532921],{"className":532716},[31642],[34,532718,532720],{"className":532719},[31646],[31648,532721,532722],{"xmlns":31650},[31652,532723,532724,532918],{},[31655,532725,532726,532762,532798,532800,532844,532846,532882],{},[31655,532727,532728,532730,532760],{},[31662,532729,456],{"fence":758},[71078,532731,532732,532746],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,532733,532734,532740],{},[71087,532735,532736],{},[71090,532737,532738],{"scriptlevel":459,"displaystyle":702},[31658,532739,111],{},[71087,532741,532742],{},[71090,532743,532744],{"scriptlevel":459,"displaystyle":702},[31749,532745,459],{},[71084,532747,532748,532754],{},[71087,532749,532750],{},[71090,532751,532752],{"scriptlevel":459,"displaystyle":702},[31749,532753,459],{},[71087,532755,532756],{},[71090,532757,532758],{"scriptlevel":459,"displaystyle":702},[31658,532759,1697],{},[31662,532761,2557],{"fence":758},[31655,532763,532764,532766,532796],{},[31662,532765,456],{"fence":758},[71078,532767,532768,532782],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,532769,532770,532776],{},[71087,532771,532772],{},[71090,532773,532774],{"scriptlevel":459,"displaystyle":702},[31658,532775,31965],{},[71087,532777,532778],{},[71090,532779,532780],{"scriptlevel":459,"displaystyle":702},[31749,532781,459],{},[71084,532783,532784,532790],{},[71087,532785,532786],{},[71090,532787,532788],{"scriptlevel":459,"displaystyle":702},[31749,532789,459],{},[71087,532791,532792],{},[71090,532793,532794],{"scriptlevel":459,"displaystyle":702},[31658,532795,46900],{},[31662,532797,2557],{"fence":758},[31662,532799,202],{},[31655,532801,532802,532804,532842],{},[31662,532803,456],{"fence":758},[71078,532805,532806,532824],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,532807,532808,532818],{},[71087,532809,532810],{},[71090,532811,532812],{"scriptlevel":459,"displaystyle":702},[31655,532813,532814,532816],{},[31658,532815,111],{},[31658,532817,31965],{},[71087,532819,532820],{},[71090,532821,532822],{"scriptlevel":459,"displaystyle":702},[31749,532823,459],{},[71084,532825,532826,532832],{},[71087,532827,532828],{},[71090,532829,532830],{"scriptlevel":459,"displaystyle":702},[31749,532831,459],{},[71087,532833,532834],{},[71090,532835,532836],{"scriptlevel":459,"displaystyle":702},[31655,532837,532838,532840],{},[31658,532839,1697],{},[31658,532841,46900],{},[31662,532843,2557],{"fence":758},[31662,532845,202],{},[31655,532847,532848,532850,532880],{},[31662,532849,456],{"fence":758},[71078,532851,532852,532866],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,532853,532854,532860],{},[71087,532855,532856],{},[71090,532857,532858],{"scriptlevel":459,"displaystyle":702},[31658,532859,31965],{},[71087,532861,532862],{},[71090,532863,532864],{"scriptlevel":459,"displaystyle":702},[31749,532865,459],{},[71084,532867,532868,532874],{},[71087,532869,532870],{},[71090,532871,532872],{"scriptlevel":459,"displaystyle":702},[31749,532873,459],{},[71087,532875,532876],{},[71090,532877,532878],{"scriptlevel":459,"displaystyle":702},[31658,532879,46900],{},[31662,532881,2557],{"fence":758},[31655,532883,532884,532886,532916],{},[31662,532885,456],{"fence":758},[71078,532887,532888,532902],{"rowspacing":370541,"columnalign":370542,"columnspacing":71082},[71084,532889,532890,532896],{},[71087,532891,532892],{},[71090,532893,532894],{"scriptlevel":459,"displaystyle":702},[31658,532895,111],{},[71087,532897,532898],{},[71090,532899,532900],{"scriptlevel":459,"displaystyle":702},[31749,532901,459],{},[71084,532903,532904,532910],{},[71087,532905,532906],{},[71090,532907,532908],{"scriptlevel":459,"displaystyle":702},[31749,532909,459],{},[71087,532911,532912],{},[71090,532913,532914],{"scriptlevel":459,"displaystyle":702},[31658,532915,1697],{},[31662,532917,2557],{"fence":758},[31665,532919,532920],{"encoding":31667},"\\begin{pmatrix} a & 0 \\\\ 0 & b \\end{pmatrix} \\begin{pmatrix} c & 0 \\\\ 0 & d \\end{pmatrix} = \\begin{pmatrix} ac & 0 \\\\ 0 & bd \\end{pmatrix} = \\begin{pmatrix} c & 0 \\\\ 0 & d \\end{pmatrix} \\begin{pmatrix} a & 0 \\\\ 0 & b \\end{pmatrix}",[34,532922,532924,533176,533314],{"className":532923,"ariaHidden":758},[31672],[34,532925,532927,532930,533047,533050,533167,533170,533173],{"className":532926},[31676],[34,532928],{"className":532929,"style":370661},[31680],[34,532931,532933,532939,533041],{"className":532932},[34006],[34,532934,532936],{"className":532935,"style":75811},[31782,75810],[34,532937,456],{"className":532938},[71281,31820],[34,532940,532942],{"className":532941},[31685],[34,532943,532945,532990,532993,532996],{"className":532944},[71078],[34,532946,532948],{"className":532947},[141481],[34,532949,532951,532982],{"className":532950},[31796,31797],[34,532952,532954,532979],{"className":532953},[31801],[34,532955,532957,532968],{"className":532956,"style":370689},[31805],[34,532958,532959,532962],{"style":370692},[34,532960],{"className":532961,"style":41508},[31813],[34,532963,532965],{"className":532964},[31685],[34,532966,111],{"className":532967},[31685,31686],[34,532969,532970,532973],{"style":370704},[34,532971],{"className":532972,"style":41508},[31813],[34,532974,532976],{"className":532975},[31685],[34,532977,459],{"className":532978},[31685],[34,532980,31829],{"className":532981},[31828],[34,532983,532985],{"className":532984},[31801],[34,532986,532988],{"className":532987,"style":370723},[31805],[34,532989],{},[34,532991],{"className":532992,"style":141477},[71732],[34,532994],{"className":532995,"style":141477},[71732],[34,532997,532999],{"className":532998},[141481],[34,533000,533002,533033],{"className":533001},[31796,31797],[34,533003,533005,533030],{"className":533004},[31801],[34,533006,533008,533019],{"className":533007,"style":370689},[31805],[34,533009,533010,533013],{"style":370692},[34,533011],{"className":533012,"style":41508},[31813],[34,533014,533016],{"className":533015},[31685],[34,533017,459],{"className":533018},[31685],[34,533020,533021,533024],{"style":370704},[34,533022],{"className":533023,"style":41508},[31813],[34,533025,533027],{"className":533026},[31685],[34,533028,1697],{"className":533029},[31685,31686],[34,533031,31829],{"className":533032},[31828],[34,533034,533036],{"className":533035},[31801],[34,533037,533039],{"className":533038,"style":370723},[31805],[34,533040],{},[34,533042,533044],{"className":533043,"style":75811},[31690,75810],[34,533045,2557],{"className":533046},[71281,31820],[34,533048],{"className":533049,"style":32123},[31845],[34,533051,533053,533059,533161],{"className":533052},[34006],[34,533054,533056],{"className":533055,"style":75811},[31782,75810],[34,533057,456],{"className":533058},[71281,31820],[34,533060,533062],{"className":533061},[31685],[34,533063,533065,533110,533113,533116],{"className":533064},[71078],[34,533066,533068],{"className":533067},[141481],[34,533069,533071,533102],{"className":533070},[31796,31797],[34,533072,533074,533099],{"className":533073},[31801],[34,533075,533077,533088],{"className":533076,"style":370689},[31805],[34,533078,533079,533082],{"style":370692},[34,533080],{"className":533081,"style":41508},[31813],[34,533083,533085],{"className":533084},[31685],[34,533086,31965],{"className":533087},[31685,31686],[34,533089,533090,533093],{"style":370704},[34,533091],{"className":533092,"style":41508},[31813],[34,533094,533096],{"className":533095},[31685],[34,533097,459],{"className":533098},[31685],[34,533100,31829],{"className":533101},[31828],[34,533103,533105],{"className":533104},[31801],[34,533106,533108],{"className":533107,"style":370723},[31805],[34,533109],{},[34,533111],{"className":533112,"style":141477},[71732],[34,533114],{"className":533115,"style":141477},[71732],[34,533117,533119],{"className":533118},[141481],[34,533120,533122,533153],{"className":533121},[31796,31797],[34,533123,533125,533150],{"className":533124},[31801],[34,533126,533128,533139],{"className":533127,"style":370689},[31805],[34,533129,533130,533133],{"style":370692},[34,533131],{"className":533132,"style":41508},[31813],[34,533134,533136],{"className":533135},[31685],[34,533137,459],{"className":533138},[31685],[34,533140,533141,533144],{"style":370704},[34,533142],{"className":533143,"style":41508},[31813],[34,533145,533147],{"className":533146},[31685],[34,533148,46900],{"className":533149},[31685,31686],[34,533151,31829],{"className":533152},[31828],[34,533154,533156],{"className":533155},[31801],[34,533157,533159],{"className":533158,"style":370723},[31805],[34,533160],{},[34,533162,533164],{"className":533163,"style":75811},[31690,75810],[34,533165,2557],{"className":533166},[71281,31820],[34,533168],{"className":533169,"style":31846},[31845],[34,533171,202],{"className":533172},[31850],[34,533174],{"className":533175,"style":31846},[31845],[34,533177,533179,533182,533305,533308,533311],{"className":533178},[31676],[34,533180],{"className":533181,"style":370661},[31680],[34,533183,533185,533191,533299],{"className":533184},[34006],[34,533186,533188],{"className":533187,"style":75811},[31782,75810],[34,533189,456],{"className":533190},[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是交换环，尽管 ",[34,533589,533591,533604],{"className":533590},[31642],[34,533592,533594],{"className":533593},[31646],[31648,533595,533596],{"xmlns":31650},[31652,533597,533598,533602],{},[31655,533599,533600],{},[31658,533601,34386],{},[31665,533603,34386],{"encoding":31667},[34,533605,533607],{"className":533606,"ariaHidden":758},[31672],[34,533608,533610,533613],{"className":533609},[31676],[34,533611],{"className":533612,"style":33218},[31680],[34,533614,34386],{"className":533615,"style":34427},[31685,31686]," 不是。",[32188,533618],{},[253,533620,533622],{"id":533621},"题目-14幂零根是理想","题目 14：幂零根是理想",[15,533624,533625],{},[2627,533626,533627,533628,533656,533657,533685],{},"14. 证明：在交换环 ",[34,533629,533631,533644],{"className":533630},[31642],[34,533632,533634],{"className":533633},[31646],[31648,533635,533636],{"xmlns":31650},[31652,533637,533638,533642],{},[31655,533639,533640],{},[31658,533641,34386],{},[31665,533643,34386],{"encoding":31667},[34,533645,533647],{"className":533646,"ariaHidden":758},[31672],[34,533648,533650,533653],{"className":533649},[31676],[34,533651],{"className":533652,"style":33218},[31680],[34,533654,34386],{"className":533655,"style":34427},[31685,31686]," 中全体幂零元组成一个理想（称为 ",[34,533658,533660,533673],{"className":533659},[31642],[34,533661,533663],{"className":533662},[31646],[31648,533664,533665],{"xmlns":31650},[31652,533666,533667,533671],{},[31655,533668,533669],{},[31658,533670,34386],{},[31665,533672,34386],{"encoding":31667},[34,533674,533676],{"className":533675,"ariaHidden":758},[31672],[34,533677,533679,533682],{"className":533678},[31676],[34,533680],{"className":533681,"style":33218},[31680],[34,533683,34386],{"className":533684,"style":34427},[31685,31686]," 的幂零根或小根）。",[32490,533687,463462],{"id":533688},"深度解析",[105,533690,533691,533834,534067],{},[108,533692,533693,533695,533696,533724,533725,315343,533753,347],{},[2627,533694,506515],{},"：元素 ",[34,533697,533699,533712],{"className":533698},[31642],[34,533700,533702],{"className":533701},[31646],[31648,533703,533704],{"xmlns":31650},[31652,533705,533706,533710],{},[31655,533707,533708],{},[31658,533709,32936],{},[31665,533711,32936],{"encoding":31667},[34,533713,533715],{"className":533714,"ariaHidden":758},[31672],[34,533716,533718,533721],{"className":533717},[31676],[34,533719],{"className":533720,"style":32527},[31680],[34,533722,32936],{"className":533723},[31685,31686]," 是幂零元，如果存在正整数 ",[34,533726,533728,533741],{"className":533727},[31642],[34,533729,533731],{"className":533730},[31646],[31648,533732,533733],{"xmlns":31650},[31652,533734,533735,533739],{},[31655,533736,533737],{},[31658,533738,31660],{},[31665,533740,31660],{"encoding":31667},[34,533742,533744],{"className":533743,"ariaHidden":758},[31672],[34,533745,533747,533750],{"className":533746},[31676],[34,533748],{"className":533749,"style":32527},[31680],[34,533751,31660],{"className":533752},[31685,31686],[34,533754,533756,533778],{"className":533755},[31642],[34,533757,533759],{"className":533758},[31646],[31648,533760,533761],{"xmlns":31650},[31652,533762,533763,533775],{},[31655,533764,533765,533771,533773],{},[37624,533766,533767,533769],{},[31658,533768,32936],{},[31658,533770,31660],{},[31662,533772,202],{},[31749,533774,459],{},[31665,533776,533777],{"encoding":31667},"x^n = 0",[34,533779,533781,533825],{"className":533780,"ariaHidden":758},[31672],[34,533782,533784,533787,533816,533819,533822],{"className":533783},[31676],[34,533785],{"className":533786,"style":37959},[31680],[34,533788,533790,533793],{"className":533789},[31685],[34,533791,32936],{"className":533792},[31685,31686],[34,533794,533796],{"className":533795},[31792],[34,533797,533799],{"className":533798},[31796],[34,533800,533802],{"className":533801},[31801],[34,533803,533805],{"className":533804,"style":37959},[31805],[34,533806,533807,533810],{"style":37777},[34,533808],{"className":533809,"style":31814},[31813],[34,533811,533813],{"className":533812},[31818,31819,31820,31821],[34,533814,31660],{"className":533815},[31685,31686,31821],[34,533817],{"className":533818,"style":31846},[31845],[34,533820,202],{"className":533821},[31850],[34,533823],{"className":533824,"style":31846},[31845],[34,533826,533828,533831],{"className":533827},[31676],[34,533829],{"className":533830,"style":32617},[31680],[34,533832,459],{"className":533833},[31685],[108,533835,533836,533839,533840,533871,533872,534066],{},[2627,533837,533838],{},"关键点","：题目强调了 ",[2627,533841,533842,533870],{},[34,533843,533845,533858],{"className":533844},[31642],[34,533846,533848],{"className":533847},[31646],[31648,533849,533850],{"xmlns":31650},[31652,533851,533852,533856],{},[31655,533853,533854],{},[31658,533855,34386],{},[31665,533857,34386],{"encoding":31667},[34,533859,533861],{"className":533860,"ariaHidden":758},[31672],[34,533862,533864,533867],{"className":533863},[31676],[34,533865],{"className":533866,"style":33218},[31680],[34,533868,34386],{"className":533869,"style":34427},[31685,31686]," 是交换环","。这是证明的关键！如果不交换，",[34,533873,533875,533913],{"className":533874},[31642],[34,533876,533878],{"className":533877},[31646],[31648,533879,533880],{"xmlns":31650},[31652,533881,533882,533910],{},[31655,533883,533884,533886,533888,533890,533896,533898,533904],{},[31662,533885,456],{"stretchy":702},[31658,533887,32936],{},[31658,533889,67764],{},[37624,533891,533892,533894],{},[31662,533893,2557],{"stretchy":702},[31658,533895,31660],{},[31662,533897,54707],{"mathvariant":38079},[37624,533899,533900,533902],{},[31658,533901,32936],{},[31658,533903,31660],{},[37624,533905,533906,533908],{},[31658,533907,67764],{},[31658,533909,31660],{},[31665,533911,533912],{"encoding":31667},"(xy)^n \\neq x^n y^n",[34,533914,533916,534002],{"className":533915,"ariaHidden":758},[31672],[34,533917,533919,533922,533925,533928,533931,533960,533963,533999],{"className":533918},[31676],[34,533920],{"className":533921,"style":31774},[31680],[34,533923,456],{"className":533924},[31782],[34,533926,32936],{"className":533927},[31685,31686],[34,533929,67764],{"className":533930,"style":32062},[31685,31686],[34,533932,533934,533937],{"className":533933},[31690],[34,533935,2557],{"className":533936},[31690],[34,533938,533940],{"className":533939},[31792],[34,533941,533943],{"className":533942},[31796],[34,533944,533946],{"className":533945},[31801],[34,533947,533949],{"className":533948,"style":37959},[31805],[34,533950,533951,533954],{"style":37777},[34,533952],{"className":533953,"style":31814},[31813],[34,533955,533957],{"className":533956},[31818,31819,31820,31821],[34,533958,31660],{"className":533959},[31685,31686,31821],[34,533961],{"className":533962,"style":31846},[31845],[34,533964,533966,533993,533996],{"className":533965},[31850],[34,533967,533969],{"className":533968},[31850],[34,533970,533972],{"className":533971},[31685,54774],[34,533973,533975],{"className":533974},[54778],[34,533976,533978,533981,533990],{"className":533977},[54782],[34,533979],{"className":533980,"style":43217},[31680],[34,533982,533984],{"className":533983},[54789],[34,533985,533987],{"className":533986},[31685],[34,533988,54796],{"className":533989},[31850],[34,533991],{"className":533992},[54800],[34,533994],{"className":533995},[31845,54804],[34,533997,202],{"className":533998},[31850],[34,534000],{"className":534001,"style":31846},[31845],[34,534003,534005,534008,534037],{"className":534004},[31676],[34,534006],{"className":534007,"style":316112},[31680],[34,534009,534011,534014],{"className":534010},[31685],[34,534012,32936],{"className":534013},[31685,31686],[34,534015,534017],{"className":534016},[31792],[34,534018,534020],{"className":534019},[31796],[34,534021,534023],{"className":534022},[31801],[34,534024,534026],{"className":534025,"style":37959},[31805],[34,534027,534028,534031],{"style":37777},[34,534029],{"className":534030,"style":31814},[31813],[34,534032,534034],{"className":534033},[31818,31819,31820,31821],[34,534035,31660],{"className":534036},[31685,31686,31821],[34,534038,534040,534043],{"className":534039},[31685],[34,534041,67764],{"className":534042,"style":32062},[31685,31686],[34,534044,534046],{"className":534045},[31792],[34,534047,534049],{"className":534048},[31796],[34,534050,534052],{"className":534051},[31801],[34,534053,534055],{"className":534054,"style":37959},[31805],[34,534056,534057,534060],{"style":37777},[34,534058],{"className":534059,"style":31814},[31813],[34,534061,534063],{"className":534062},[31818,31819,31820,31821],[34,534064,31660],{"className":534065},[31685,31686,31821],"，结论就不一定成立。",[108,534068,534069,534071],{},[2627,534070,391664],{},"：利用二项式定理来处理和的幂。",[32490,534073,534075],{"id":534074},"严谨证明","【严谨证明】",[15,534077,133910,534078,418749,534106,534134,534135,399437,534363,514446],{},[34,534079,534081,534094],{"className":534080},[31642],[34,534082,534084],{"className":534083},[31646],[31648,534085,534086],{"xmlns":31650},[31652,534087,534088,534092],{},[31655,534089,534090],{},[31658,534091,33347],{},[31665,534093,33347],{"encoding":31667},[34,534095,534097],{"className":534096,"ariaHidden":758},[31672],[34,534098,534100,534103],{"className":534099},[31676],[34,534101],{"className":534102,"style":33218},[31680],[34,534104,33347],{"className":534105,"style":33389},[31685,31686],[34,534107,534109,534122],{"className":534108},[31642],[34,534110,534112],{"className":534111},[31646],[31648,534113,534114],{"xmlns":31650},[31652,534115,534116,534120],{},[31655,534117,534118],{},[31658,534119,34386],{},[31665,534121,34386],{"encoding":31667},[34,534123,534125],{"className":534124,"ariaHidden":758},[31672],[34,534126,534128,534131],{"className":534127},[31676],[34,534129],{"className":534130,"style":33218},[31680],[34,534132,34386],{"className":534133,"style":34427},[31685,31686]," 中全体幂零元的集合，即 ",[34,534136,534138,534190],{"className":534137},[31642],[34,534139,534141],{"className":534140},[31646],[31648,534142,534143],{"xmlns":31650},[31652,534144,534145,534187],{},[31655,534146,534147,534149,534151,534153,534155,534157,534159,534161,534163,534165,534167,534173,534175,534181,534183,534185],{},[31658,534148,33347],{},[31662,534150,202],{},[31662,534152,32685],{"stretchy":702},[31658,534154,32936],{},[31662,534156,112296],{},[31658,534158,34386],{},[31662,534160,162086],{},[31658,534162,365989],{"mathvariant":38079},[31658,534164,31660],{},[31662,534166,112296],{},[37624,534168,534169,534171],{},[31658,534170,97300],{"mathvariant":318416},[31662,534172,223],{},[31662,534174,31960],{"separator":758},[37624,534176,534177,534179],{},[31658,534178,32936],{},[31658,534180,31660],{},[31662,534182,202],{},[31749,534184,459],{},[31662,534186,32715],{"stretchy":702},[31665,534188,534189],{"encoding":31667},"N = \\{ x \\in R \\mid \\exists n \\in \\mathbb{Z}^+, x^n = 0 \\}",[34,534191,534193,534211,534232,534250,534271,534351],{"className":534192,"ariaHidden":758},[31672],[34,534194,534196,534199,534202,534205,534208],{"className":534195},[31676],[34,534197],{"className":534198,"style":33218},[31680],[34,534200,33347],{"className":534201,"style":33389},[31685,31686],[34,534203],{"className":534204,"style":31846},[31845],[34,534206,202],{"className":534207},[31850],[34,534209],{"className":534210,"style":31846},[31845],[34,534212,534214,534217,534220,534223,534226,534229],{"className":534213},[31676],[34,534215],{"className":534216,"style":31774},[31680],[34,534218,32685],{"className":534219},[31782],[34,534221,32936],{"className":534222},[31685,31686],[34,534224],{"className":534225,"style":31846},[31845],[34,534227,112296],{"className":534228},[31850],[34,534230],{"className":534231,"style":31846},[31845],[34,534233,534235,534238,534241,534244,534247],{"className":534234},[31676],[34,534236],{"className":534237,"style":31774},[31680],[34,534239,34386],{"className":534240,"style":34427},[31685,31686],[34,534242],{"className":534243,"style":31846},[31845],[34,534245,162086],{"className":534246},[31850],[34,534248],{"className":534249,"style":31846},[31845],[34,534251,534253,534256,534259,534262,534265,534268],{"className":534252},[31676],[34,534254],{"className":534255,"style":46991},[31680],[34,534257,365989],{"className":534258},[31685],[34,534260,31660],{"className":534261},[31685,31686],[34,534263],{"className":534264,"style":31846},[31845],[34,534266,112296],{"className":534267},[31850],[34,534269],{"className":534270,"style":31846},[31845],[34,534272,534274,534278,534307,534310,534313,534342,534345,534348],{"className":534273},[31676],[34,534275],{"className":534276,"style":534277},[31680],"height:0.9658em;vertical-align:-0.1944em;",[34,534279,534281,534284],{"className":534280},[31685],[34,534282,97300],{"className":534283},[31685,318441],[34,534285,534287],{"className":534286},[31792],[34,534288,534290],{"className":534289},[31796],[34,534291,534293],{"className":534292},[31801],[34,534294,534296],{"className":534295,"style":156828},[31805],[34,534297,534298,534301],{"style":37777},[34,534299],{"className":534300,"style":31814},[31813],[34,534302,534304],{"className":534303},[31818,31819,31820,31821],[34,534305,223],{"className":534306},[32024,31821],[34,534308,31960],{"className":534309},[32119],[34,534311],{"className":534312,"style":32123},[31845],[34,534314,534316,534319],{"className":534315},[31685],[34,534317,32936],{"className":534318},[31685,31686],[34,534320,534322],{"className":534321},[31792],[34,534323,534325],{"className":534324},[31796],[34,534326,534328],{"className":534327},[31801],[34,534329,534331],{"className":534330,"style":37959},[31805],[34,534332,534333,534336],{"style":37777},[34,534334],{"className":534335,"style":31814},[31813],[34,534337,534339],{"className":534338},[31818,31819,31820,31821],[34,534340,31660],{"className":534341},[31685,31686,31821],[34,534343],{"className":534344,"style":31846},[31845],[34,534346,202],{"className":534347},[31850],[34,534349],{"className":534350,"style":31846},[31845],[34,534352,534354,534357,534360],{"className":534353},[31676],[34,534355],{"className":534356,"style":31774},[31680],[34,534358,459],{"className":534359},[31685],[34,534361,32715],{"className":534362},[31690],[34,534364,534366,534379],{"className":534365},[31642],[34,534367,534369],{"className":534368},[31646],[31648,534370,534371],{"xmlns":31650},[31652,534372,534373,534377],{},[31655,534374,534375],{},[31658,534376,33347],{},[31665,534378,33347],{"encoding":31667},[34,534380,534382],{"className":534381,"ariaHidden":758},[31672],[34,534383,534385,534388],{"className":534384},[31676],[34,534386],{"className":534387,"style":33218},[31680],[34,534389,33347],{"className":534390,"style":33389},[31685,31686],[2230,534392,534393,534558],{},[108,534394,534395,315440,534397,13977,534478,347,534529,534557],{},[2627,534396,441307],{},[34,534398,534400,534422],{"className":534399},[31642],[34,534401,534403],{"className":534402},[31646],[31648,534404,534405],{"xmlns":31650},[31652,534406,534407,534419],{},[31655,534408,534409,534415,534417],{},[37624,534410,534411,534413],{},[31749,534412,459],{},[31749,534414,640],{},[31662,534416,202],{},[31749,534418,459],{},[31665,534420,534421],{"encoding":31667},"0^1 = 0",[34,534423,534425,534469],{"className":534424,"ariaHidden":758},[31672],[34,534426,534428,534431,534460,534463,534466],{"className":534427},[31676],[34,534429],{"className":534430,"style":37774},[31680],[34,534432,534434,534437],{"className":534433},[31685],[34,534435,459],{"className":534436},[31685],[34,534438,534440],{"className":534439},[31792],[34,534441,534443],{"className":534442},[31796],[34,534444,534446],{"className":534445},[31801],[34,534447,534449],{"className":534448,"style":37774},[31805],[34,534450,534451,534454],{"style":37777},[34,534452],{"className":534453,"style":31814},[31813],[34,534455,534457],{"className":534456},[31818,31819,31820,31821],[34,534458,640],{"className":534459},[31685,31821],[34,534461],{"className":534462,"style":31846},[31845],[34,534464,202],{"className":534465},[31850],[34,534467],{"className":534468,"style":31846},[31845],[34,534470,534472,534475],{"className":534471},[31676],[34,534473],{"className":534474,"style":32617},[31680],[34,534476,459],{"className":534477},[31685],[34,534479,534481,534499],{"className":534480},[31642],[34,534482,534484],{"className":534483},[31646],[31648,534485,534486],{"xmlns":31650},[31652,534487,534488,534496],{},[31655,534489,534490,534492,534494],{},[31749,534491,459],{},[31662,534493,112296],{},[31658,534495,33347],{},[31665,534497,534498],{"encoding":31667},"0 \\in N",[34,534500,534502,534520],{"className":534501,"ariaHidden":758},[31672],[34,534503,534505,534508,534511,534514,534517],{"className":534504},[31676],[34,534506],{"className":534507,"style":61814},[31680],[34,534509,459],{"className":534510},[31685],[34,534512],{"className":534513,"style":31846},[31845],[34,534515,112296],{"className":534516},[31850],[34,534518],{"className":534519,"style":31846},[31845],[34,534521,534523,534526],{"className":534522},[31676],[34,534524],{"className":534525,"style":33218},[31680],[34,534527,33347],{"className":534528,"style":33389},[31685,31686],[34,534530,534532,534545],{"className":534531},[31642],[34,534533,534535],{"className":534534},[31646],[31648,534536,534537],{"xmlns":31650},[31652,534538,534539,534543],{},[31655,534540,534541],{},[31658,534542,33347],{},[31665,534544,33347],{"encoding":31667},[34,534546,534548],{"className":534547,"ariaHidden":758},[31672],[34,534549,534551,534554],{"className":534550},[31676],[34,534552],{"className":534553,"style":33218},[31680],[34,534555,33347],{"className":534556,"style":33389},[31685,31686]," 非空。",[108,534559,534560,362914,534563,534627,534628,315343,534670,86674,534750,534831,534832,534883,534884,534945,534946,507421,535019,113124,535047,535050,535051,33183,535054,535427,535428,2664],{},[2627,534561,534562],{},"对减法封闭 (实际上是对加法\u002F减法封闭)",[34,534564,534566,534588],{"className":534565},[31642],[34,534567,534569],{"className":534568},[31646],[31648,534570,534571],{"xmlns":31650},[31652,534572,534573,534585],{},[31655,534574,534575,534577,534579,534581,534583],{},[31658,534576,32936],{},[31662,534578,31960],{"separator":758},[31658,534580,67764],{},[31662,534582,112296],{},[31658,534584,33347],{},[31665,534586,534587],{"encoding":31667},"x, y \\in N",[34,534589,534591,534618],{"className":534590,"ariaHidden":758},[31672],[34,534592,534594,534597,534600,534603,534606,534609,534612,534615],{"className":534593},[31676],[34,534595],{"className":534596,"style":86652},[31680],[34,534598,32936],{"className":534599},[31685,31686],[34,534601,31960],{"className":534602},[32119],[34,534604],{"className":534605,"style":32123},[31845],[34,534607,67764],{"className":534608,"style":32062},[31685,31686],[34,534610],{"className":534611,"style":31846},[31845],[34,534613,112296],{"className":534614},[31850],[34,534616],{"className":534617,"style":31846},[31845],[34,534619,534621,534624],{"className":534620},[31676],[34,534622],{"className":534623,"style":33218},[31680],[34,534625,33347],{"className":534626,"style":33389},[31685,31686],"。\n这意味着存在正整数 ",[34,534629,534631,534649],{"className":534630},[31642],[34,534632,534634],{"className":534633},[31646],[31648,534635,534636],{"xmlns":31650},[31652,534637,534638,534646],{},[31655,534639,534640,534642,534644],{},[31658,534641,31660],{},[31662,534643,31960],{"separator":758},[31658,534645,31759],{},[31665,534647,534648],{"encoding":31667},"n, m",[34,534650,534652],{"className":534651,"ariaHidden":758},[31672],[34,534653,534655,534658,534661,534664,534667],{"className":534654},[31676],[34,534656],{"className":534657,"style":38774},[31680],[34,534659,31660],{"className":534660},[31685,31686],[34,534662,31960],{"className":534663},[32119],[34,534665],{"className":534666,"style":32123},[31845],[34,534668,31759],{"className":534669},[31685,31686],[34,534671,534673,534694],{"className":534672},[31642],[34,534674,534676],{"className":534675},[31646],[31648,534677,534678],{"xmlns":31650},[31652,534679,534680,534692],{},[31655,534681,534682,534688,534690],{},[37624,534683,534684,534686],{},[31658,534685,32936],{},[31658,534687,31660],{},[31662,534689,202],{},[31749,534691,459],{},[31665,534693,533777],{"encoding":31667},[34,534695,534697,534741],{"className":534696,"ariaHidden":758},[31672],[34,534698,534700,534703,534732,534735,534738],{"className":534699},[31676],[34,534701],{"className":534702,"style":37959},[31680],[34,534704,534706,534709],{"className":534705},[31685],[34,534707,32936],{"className":534708},[31685,31686],[34,534710,534712],{"className":534711},[31792],[34,534713,534715],{"className":534714},[31796],[34,534716,534718],{"className":534717},[31801],[34,534719,534721],{"className":534720,"style":37959},[31805],[34,534722,534723,534726],{"style":37777},[34,534724],{"className":534725,"style":31814},[31813],[34,534727,534729],{"className":534728},[31818,31819,31820,31821],[34,534730,31660],{"className":534731},[31685,31686,31821],[34,534733],{"className":534734,"style":31846},[31845],[34,534736,202],{"className":534737},[31850],[34,534739],{"className":534740,"style":31846},[31845],[34,534742,534744,534747],{"className":534743},[31676],[34,534745],{"className":534746,"style":32617},[31680],[34,534748,459],{"className":534749},[31685],[34,534751,534753,534775],{"className":534752},[31642],[34,534754,534756],{"className":534755},[31646],[31648,534757,534758],{"xmlns":31650},[31652,534759,534760,534772],{},[31655,534761,534762,534768,534770],{},[37624,534763,534764,534766],{},[31658,534765,67764],{},[31658,534767,31759],{},[31662,534769,202],{},[31749,534771,459],{},[31665,534773,534774],{"encoding":31667},"y^m = 0",[34,534776,534778,534822],{"className":534777,"ariaHidden":758},[31672],[34,534779,534781,534784,534813,534816,534819],{"className":534780},[31676],[34,534782],{"className":534783,"style":316112},[31680],[34,534785,534787,534790],{"className":534786},[31685],[34,534788,67764],{"className":534789,"style":32062},[31685,31686],[34,534791,534793],{"className":534792},[31792],[34,534794,534796],{"className":534795},[31796],[34,534797,534799],{"className":534798},[31801],[34,534800,534802],{"className":534801,"style":37959},[31805],[34,534803,534804,534807],{"style":37777},[34,534805],{"className":534806,"style":31814},[31813],[34,534808,534810],{"className":534809},[31818,31819,31820,31821],[34,534811,31759],{"className":534812},[31685,31686,31821],[34,534814],{"className":534815,"style":31846},[31845],[34,534817,202],{"className":534818},[31850],[34,534820],{"className":534821,"style":31846},[31845],[34,534823,534825,534828],{"className":534824},[31676],[34,534826],{"className":534827,"style":32617},[31680],[34,534829,459],{"className":534830},[31685],"。\n我们需要证明 ",[34,534833,534835,534853],{"className":534834},[31642],[34,534836,534838],{"className":534837},[31646],[31648,534839,534840],{"xmlns":31650},[31652,534841,534842,534850],{},[31655,534843,534844,534846,534848],{},[31658,534845,32936],{},[31662,534847,32226],{},[31658,534849,67764],{},[31665,534851,534852],{"encoding":31667},"x - y",[34,534854,534856,534874],{"className":534855,"ariaHidden":758},[31672],[34,534857,534859,534862,534865,534868,534871],{"className":534858},[31676],[34,534860],{"className":534861,"style":32597},[31680],[34,534863,32936],{"className":534864},[31685,31686],[34,534866],{"className":534867,"style":32604},[31845],[34,534869,32226],{"className":534870},[32024],[34,534872],{"className":534873,"style":32604},[31845],[34,534875,534877,534880],{"className":534876},[31676],[34,534878],{"className":534879,"style":38774},[31680],[34,534881,67764],{"className":534882,"style":32062},[31685,31686]," 也是幂零元。考察 ",[34,534885,534887,534909],{"className":534886},[31642],[34,534888,534890],{"className":534889},[31646],[31648,534891,534892],{"xmlns":31650},[31652,534893,534894,534906],{},[31655,534895,534896,534898,534900,534902,534904],{},[31662,534897,456],{"stretchy":702},[31658,534899,32936],{},[31662,534901,32226],{},[31658,534903,67764],{},[31662,534905,2557],{"stretchy":702},[31665,534907,534908],{"encoding":31667},"(x - y)",[34,534910,534912,534933],{"className":534911,"ariaHidden":758},[31672],[34,534913,534915,534918,534921,534924,534927,534930],{"className":534914},[31676],[34,534916],{"className":534917,"style":31774},[31680],[34,534919,456],{"className":534920},[31782],[34,534922,32936],{"className":534923},[31685,31686],[34,534925],{"className":534926,"style":32604},[31845],[34,534928,32226],{"className":534929},[32024],[34,534931],{"className":534932,"style":32604},[31845],[34,534934,534936,534939,534942],{"className":534935},[31676],[34,534937],{"className":534938,"style":31774},[31680],[34,534940,67764],{"className":534941,"style":32062},[31685,31686],[34,534943,2557],{"className":534944},[31690]," 的足够大次幂。\n令 ",[34,534947,534949,534971],{"className":534948},[31642],[34,534950,534952],{"className":534951},[31646],[31648,534953,534954],{"xmlns":31650},[31652,534955,534956,534968],{},[31655,534957,534958,534960,534962,534964,534966],{},[31658,534959,33759],{},[31662,534961,202],{},[31658,534963,31660],{},[31662,534965,223],{},[31658,534967,31759],{},[31665,534969,534970],{"encoding":31667},"k = n + m",[34,534972,534974,534992,535010],{"className":534973,"ariaHidden":758},[31672],[34,534975,534977,534980,534983,534986,534989],{"className":534976},[31676],[34,534978],{"className":534979,"style":31681},[31680],[34,534981,33759],{"className":534982,"style":33774},[31685,31686],[34,534984],{"className":534985,"style":31846},[31845],[34,534987,202],{"className":534988},[31850],[34,534990],{"className":534991,"style":31846},[31845],[34,534993,534995,534998,535001,535004,535007],{"className":534994},[31676],[34,534996],{"className":534997,"style":32597},[31680],[34,534999,31660],{"className":535000},[31685,31686],[34,535002],{"className":535003,"style":32604},[31845],[34,535005,223],{"className":535006},[32024],[34,535008],{"className":535009,"style":32604},[31845],[34,535011,535013,535016],{"className":535012},[31676],[34,535014],{"className":535015,"style":32527},[31680],[34,535017,31759],{"className":535018},[31685,31686],[34,535020,535022,535035],{"className":535021},[31642],[34,535023,535025],{"className":535024},[31646],[31648,535026,535027],{"xmlns":31650},[31652,535028,535029,535033],{},[31655,535030,535031],{},[31658,535032,34386],{},[31665,535034,34386],{"encoding":31667},[34,535036,535038],{"className":535037,"ariaHidden":758},[31672],[34,535039,535041,535044],{"className":535040},[31676],[34,535042],{"className":535043,"style":33218},[31680],[34,535045,34386],{"className":535046,"style":34427},[31685,31686],[2627,535048,535049],{},"交换环","，我们可以使用",[2627,535052,535053],{},"二项式定理",[34,535055,535057,535136],{"className":535056},[31642],[34,535058,535060],{"className":535059},[31646],[31648,535061,535062],{"xmlns":31650},[31652,535063,535064,535133],{},[31655,535065,535066,535068,535070,535072,535074,535080,535082,535096,535109,535115,535117,535119,535121],{},[31662,535067,456],{"stretchy":702},[31658,535069,32936],{},[31662,535071,32226],{},[31658,535073,67764],{},[37624,535075,535076,535078],{},[31662,535077,2557],{"stretchy":702},[31658,535079,33759],{},[31662,535081,202],{},[38846,535083,535084,535086,535094],{},[31662,535085,131138],{},[31655,535087,535088,535090,535092],{},[31658,535089,31955],{},[31662,535091,202],{},[31749,535093,459],{},[31658,535095,33759],{},[31655,535097,535098,535100,535107],{},[31662,535099,456],{"fence":758},[41453,535101,535103,535105],{"linethickness":535102},"0px",[31658,535104,33759],{},[31658,535106,31955],{},[31662,535108,2557],{"fence":758},[37624,535110,535111,535113],{},[31658,535112,32936],{},[31658,535114,31955],{},[31662,535116,456],{"stretchy":702},[31662,535118,32226],{},[31658,535120,67764],{},[37624,535122,535123,535125],{},[31662,535124,2557],{"stretchy":702},[31655,535126,535127,535129,535131],{},[31658,535128,33759],{},[31662,535130,32226],{},[31658,535132,31955],{},[31665,535134,535135],{"encoding":31667},"(x - y)^k = \\sum_{i=0}^{k} \\binom{k}{i} x^i (-y)^{k-i}",[34,535137,535139,535160,535207],{"className":535138,"ariaHidden":758},[31672],[34,535140,535142,535145,535148,535151,535154,535157],{"className":535141},[31676],[34,535143],{"className":535144,"style":31774},[31680],[34,535146,456],{"className":535147},[31782],[34,535149,32936],{"className":535150},[31685,31686],[34,535152],{"className":535153,"style":32604},[31845],[34,535155,32226],{"className":535156},[32024],[34,535158],{"className":535159,"style":32604},[31845],[34,535161,535163,535166,535169,535198,535201,535204],{"className":535162},[31676],[34,535164],{"className":535165,"style":43651},[31680],[34,535167,67764],{"className":535168,"style":32062},[31685,31686],[34,535170,535172,535175],{"className":535171},[31690],[34,535173,2557],{"className":535174},[31690],[34,535176,535178],{"className":535177},[31792],[34,535179,535181],{"className":535180},[31796],[34,535182,535184],{"className":535183},[31801],[34,535185,535187],{"className":535186,"style":38486},[31805],[34,535188,535189,535192],{"style":37777},[34,535190],{"className":535191,"style":31814},[31813],[34,535193,535195],{"className":535194},[31818,31819,31820,31821],[34,535196,33759],{"className":535197,"style":33774},[31685,31686,31821],[34,535199],{"className":535200,"style":31846},[31845],[34,535202,202],{"className":535203},[31850],[34,535205],{"className":535206,"style":31846},[31845],[34,535208,535210,535214,535278,535281,535350,535380,535383,535386,535389],{"className":535209},[31676],[34,535211],{"className":535212,"style":535213},[31680],"height:1.339em;vertical-align:-0.35em;",[34,535215,535217,535220],{"className":535216},[32767],[34,535218,131138],{"className":535219,"style":156561},[32767,131249,156560],[34,535221,535223],{"className":535222},[31792],[34,535224,535226,535270],{"className":535225},[31796,31797],[34,535227,535229,535267],{"className":535228},[31801],[34,535230,535233,535253],{"className":535231,"style":535232},[31805],"height:0.989em;",[34,535234,535235,535238],{"style":156577},[34,535236],{"className":535237,"style":31814},[31813],[34,535239,535241],{"className":535240},[31818,31819,31820,31821],[34,535242,535244,535247,535250],{"className":535243},[31685,31821],[34,535245,31955],{"className":535246},[31685,31686,31821],[34,535248,202],{"className":535249},[31850,31821],[34,535251,459],{"className":535252},[31685,31821],[34,535254,535255,535258],{"style":352016},[34,535256],{"className":535257,"style":31814},[31813],[34,535259,535261],{"className":535260},[31818,31819,31820,31821],[34,535262,535264],{"className":535263},[31685,31821],[34,535265,33759],{"className":535266,"style":33774},[31685,31686,31821],[34,535268,31829],{"className":535269},[31828],[34,535271,535273],{"className":535272},[31801],[34,535274,535276],{"className":535275,"style":352035},[31805],[34,535277],{},[34,535279],{"className":535280,"style":32123},[31845],[34,535282,535284,535290,535344],{"className":535283},[31685],[34,535285,535287],{"className":535286,"style":75811},[31782,75810],[34,535288,456],{"className":535289},[71281,43411],[34,535291,535293],{"className":535292},[41453],[34,535294,535296,535336],{"className":535295},[31796,31797],[34,535297,535299,535333],{"className":535298},[31801],[34,535300,535303,535318],{"className":535301,"style":535302},[31805],"height:0.9301em;",[34,535304,535306,535309],{"style":535305},"top:-2.355em;",[34,535307],{"className":535308,"style":31814},[31813],[34,535310,535312],{"className":535311},[31818,31819,31820,31821],[34,535313,535315],{"className":535314},[31685,31821],[34,535316,31955],{"className":535317},[31685,31686,31821],[34,535319,535321,535324],{"style":535320},"top:-3.144em;",[34,535322],{"className":535323,"style":31814},[31813],[34,535325,535327],{"className":535326},[31818,31819,31820,31821],[34,535328,535330],{"className":535329},[31685,31821],[34,535331,33759],{"className":535332,"style":33774},[31685,31686,31821],[34,535334,31829],{"className":535335},[31828],[34,535337,535339],{"className":535338},[31801],[34,535340,535342],{"className":535341,"style":41619},[31805],[34,535343],{},[34,535345,535347],{"className":535346,"style":75811},[31690,75810],[34,535348,2557],{"className":535349},[71281,43411],[34,535351,535353,535356],{"className":535352},[31685],[34,535354,32936],{"className":535355},[31685,31686],[34,535357,535359],{"className":535358},[31792],[34,535360,535362],{"className":535361},[31796],[34,535363,535365],{"className":535364},[31801],[34,535366,535369],{"className":535367,"style":535368},[31805],"height:0.8247em;",[34,535370,535371,535374],{"style":37777},[34,535372],{"className":535373,"style":31814},[31813],[34,535375,535377],{"className":535376},[31818,31819,31820,31821],[34,535378,31955],{"className":535379},[31685,31686,31821],[34,535381,456],{"className":535382},[31782],[34,535384,32226],{"className":535385},[31685],[34,535387,67764],{"className":535388,"style":32062},[31685,31686],[34,535390,535392,535395],{"className":535391},[31690],[34,535393,2557],{"className":535394},[31690],[34,535396,535398],{"className":535397},[31792],[34,535399,535401],{"className":535400},[31796],[34,535402,535404],{"className":535403},[31801],[34,535405,535407],{"className":535406,"style":38486},[31805],[34,535408,535409,535412],{"style":37777},[34,535410],{"className":535411,"style":31814},[31813],[34,535413,535415],{"className":535414},[31818,31819,31820,31821],[34,535416,535418,535421,535424],{"className":535417},[31685,31821],[34,535419,33759],{"className":535420,"style":33774},[31685,31686,31821],[34,535422,32226],{"className":535423},[32024,31821],[34,535425,31955],{"className":535426},[31685,31686,31821],"\n对于求和中的每一项 ",[34,535429,535431,535479],{"className":535430},[31642],[34,535432,535434],{"className":535433},[31646],[31648,535435,535436],{"xmlns":31650},[31652,535437,535438,535476],{},[31655,535439,535440,535452,535458,535460,535462,535464],{},[31655,535441,535442,535444,535450],{},[31662,535443,456],{"fence":758},[41453,535445,535446,535448],{"linethickness":535102},[31658,535447,33759],{},[31658,535449,31955],{},[31662,535451,2557],{"fence":758},[37624,535453,535454,535456],{},[31658,535455,32936],{},[31658,535457,31955],{},[31662,535459,456],{"stretchy":702},[31662,535461,32226],{},[31658,535463,67764],{},[37624,535465,535466,535468],{},[31662,535467,2557],{"stretchy":702},[31655,535469,535470,535472,535474],{},[31658,535471,33759],{},[31662,535473,32226],{},[31658,535475,31955],{},[31665,535477,535478],{"encoding":31667},"\\binom{k}{i} x^i (-y)^{k-i}",[34,535480,535482],{"className":535481,"ariaHidden":758},[31672],[34,535483,535485,535489,535555,535584,535587,535590,535593],{"className":535484},[31676],[34,535486],{"className":535487,"style":535488},[31680],"height:1.2801em;vertical-align:-0.35em;",[34,535490,535492,535498,535549],{"className":535491},[31685],[34,535493,535495],{"className":535494,"style":75811},[31782,75810],[34,535496,456],{"className":535497},[71281,43411],[34,535499,535501],{"className":535500},[41453],[34,535502,535504,535541],{"className":535503},[31796,31797],[34,535505,535507,535538],{"className":535506},[31801],[34,535508,535510,535524],{"className":535509,"style":535302},[31805],[34,535511,535512,535515],{"style":535305},[34,535513],{"className":535514,"style":31814},[31813],[34,535516,535518],{"className":535517},[31818,31819,31820,31821],[34,535519,535521],{"className":535520},[31685,31821],[34,535522,31955],{"className":535523},[31685,31686,31821],[34,535525,535526,535529],{"style":535320},[34,535527],{"className":535528,"style":31814},[31813],[34,535530,535532],{"className":535531},[31818,31819,31820,31821],[34,535533,535535],{"className":535534},[31685,31821],[34,535536,33759],{"className":535537,"style":33774},[31685,31686,31821],[34,535539,31829],{"className":535540},[31828],[34,535542,535544],{"className":535543},[31801],[34,535545,535547],{"className":535546,"style":41619},[31805],[34,535548],{},[34,535550,535552],{"className":535551,"style":75811},[31690,75810],[34,535553,2557],{"className":535554},[71281,43411],[34,535556,535558,535561],{"className":535557},[31685],[34,535559,32936],{"className":535560},[31685,31686],[34,535562,535564],{"className":535563},[31792],[34,535565,535567],{"className":535566},[31796],[34,535568,535570],{"className":535569},[31801],[34,535571,535573],{"className":535572,"style":535368},[31805],[34,535574,535575,535578],{"style":37777},[34,535576],{"className":535577,"style":31814},[31813],[34,535579,535581],{"className":535580},[31818,31819,31820,31821],[34,535582,31955],{"className":535583},[31685,31686,31821],[34,535585,456],{"className":535586},[31782],[34,535588,32226],{"className":535589},[31685],[34,535591,67764],{"className":535592,"style":32062},[31685,31686],[34,535594,535596,535599],{"className":535595},[31690],[34,535597,2557],{"className":535598},[31690],[34,535600,535602],{"className":535601},[31792],[34,535603,535605],{"className":535604},[31796],[34,535606,535608],{"className":535607},[31801],[34,535609,535611],{"className":535610,"style":38486},[31805],[34,535612,535613,535616],{"style":37777},[34,535614],{"className":535615,"style":31814},[31813],[34,535617,535619],{"className":535618},[31818,31819,31820,31821],[34,535620,535622,535625,535628],{"className":535621},[31685,31821],[34,535623,33759],{"className":535624,"style":33774},[31685,31686,31821],[34,535626,32226],{"className":535627},[32024,31821],[34,535629,31955],{"className":535630},[31685,31686,31821],[105,535632,535633,535976],{},[108,535634,2234,535635,55016,535686,535975],{},[34,535636,535638,535656],{"className":535637},[31642],[34,535639,535641],{"className":535640},[31646],[31648,535642,535643],{"xmlns":31650},[31652,535644,535645,535653],{},[31655,535646,535647,535649,535651],{},[31658,535648,31955],{},[31662,535650,43175],{},[31658,535652,31660],{},[31665,535654,535655],{"encoding":31667},"i \\ge n",[34,535657,535659,535677],{"className":535658,"ariaHidden":758},[31672],[34,535660,535662,535665,535668,535671,535674],{"className":535661},[31676],[34,535663],{"className":535664,"style":55401},[31680],[34,535666,31955],{"className":535667},[31685,31686],[34,535669],{"className":535670,"style":31846},[31845],[34,535672,43175],{"className":535673},[31850],[34,535675],{"className":535676,"style":31846},[31845],[34,535678,535680,535683],{"className":535679},[31676],[34,535681],{"className":535682,"style":32527},[31680],[34,535684,31660],{"className":535685},[31685,31686],[34,535687,535689,535751],{"className":535688},[31642],[34,535690,535692],{"className":535691},[31646],[31648,535693,535694],{"xmlns":31650},[31652,535695,535696,535748],{},[31655,535697,535698,535704,535706,535712,535714,535726,535728,535730,535732,535744,535746],{},[37624,535699,535700,535702],{},[31658,535701,32936],{},[31658,535703,31955],{},[31662,535705,202],{},[37624,535707,535708,535710],{},[31658,535709,32936],{},[31658,535711,31660],{},[31662,535713,71191],{},[37624,535715,535716,535718],{},[31658,535717,32936],{},[31655,535719,535720,535722,535724],{},[31658,535721,31955],{},[31662,535723,32226],{},[31658,535725,31660],{},[31662,535727,202],{},[31749,535729,459],{},[31662,535731,71191],{},[37624,535733,535734,535736],{},[31658,535735,32936],{},[31655,535737,535738,535740,535742],{},[31658,535739,31955],{},[31662,535741,32226],{},[31658,535743,31660],{},[31662,535745,202],{},[31749,535747,459],{},[31665,535749,535750],{"encoding":31667},"x^i = x^n \\cdot x^{i-n} = 0 \\cdot x^{i-n} = 0",[34,535752,535754,535798,535842,535895,535913,535966],{"className":535753,"ariaHidden":758},[31672],[34,535755,535757,535760,535789,535792,535795],{"className":535756},[31676],[34,535758],{"className":535759,"style":535368},[31680],[34,535761,535763,535766],{"className":535762},[31685],[34,535764,32936],{"className":535765},[31685,31686],[34,535767,535769],{"className":535768},[31792],[34,535770,535772],{"className":535771},[31796],[34,535773,535775],{"className":535774},[31801],[34,535776,535778],{"className":535777,"style":535368},[31805],[34,535779,535780,535783],{"style":37777},[34,535781],{"className":535782,"style":31814},[31813],[34,535784,535786],{"className":535785},[31818,31819,31820,31821],[34,535787,31955],{"className":535788},[31685,31686,31821],[34,535790],{"className":535791,"style":31846},[31845],[34,535793,202],{"className":535794},[31850],[34,535796],{"className":535797,"style":31846},[31845],[34,535799,535801,535804,535833,535836,535839],{"className":535800},[31676],[34,535802],{"className":535803,"style":37959},[31680],[34,535805,535807,535810],{"className":535806},[31685],[34,535808,32936],{"className":535809},[31685,31686],[34,535811,535813],{"className":535812},[31792],[34,535814,535816],{"className":535815},[31796],[34,535817,535819],{"className":535818},[31801],[34,535820,535822],{"className":535821,"style":37959},[31805],[34,535823,535824,535827],{"style":37777},[34,535825],{"className":535826,"style":31814},[31813],[34,535828,535830],{"className":535829},[31818,31819,31820,31821],[34,535831,31660],{"className":535832},[31685,31686,31821],[34,535834],{"className":535835,"style":32604},[31845],[34,535837,71191],{"className":535838},[32024],[34,535840],{"className":535841,"style":32604},[31845],[34,535843,535845,535848,535886,535889,535892],{"className":535844},[31676],[34,535846],{"className":535847,"style":535368},[31680],[34,535849,535851,535854],{"className":535850},[31685],[34,535852,32936],{"className":535853},[31685,31686],[34,535855,535857],{"className":535856},[31792],[34,535858,535860],{"className":535859},[31796],[34,535861,535863],{"className":535862},[31801],[34,535864,535866],{"className":535865,"style":535368},[31805],[34,535867,535868,535871],{"style":37777},[34,535869],{"className":535870,"style":31814},[31813],[34,535872,535874],{"className":535873},[31818,31819,31820,31821],[34,535875,535877,535880,535883],{"className":535876},[31685,31821],[34,535878,31955],{"className":535879},[31685,31686,31821],[34,535881,32226],{"className":535882},[32024,31821],[34,535884,31660],{"className":535885},[31685,31686,31821],[34,535887],{"className":535888,"style":31846},[31845],[34,535890,202],{"className":535891},[31850],[34,535893],{"className":535894,"style":31846},[31845],[34,535896,535898,535901,535904,535907,535910],{"className":535897},[31676],[34,535899],{"className":535900,"style":32617},[31680],[34,535902,459],{"className":535903},[31685],[34,535905],{"className":535906,"style":32604},[31845],[34,535908,71191],{"className":535909},[32024],[34,535911],{"className":535912,"style":32604},[31845],[34,535914,535916,535919,535957,535960,535963],{"className":535915},[31676],[34,535917],{"className":535918,"style":535368},[31680],[34,535920,535922,535925],{"className":535921},[31685],[34,535923,32936],{"className":535924},[31685,31686],[34,535926,535928],{"className":535927},[31792],[34,535929,535931],{"className":535930},[31796],[34,535932,535934],{"className":535933},[31801],[34,535935,535937],{"className":535936,"style":535368},[31805],[34,535938,535939,535942],{"style":37777},[34,535940],{"className":535941,"style":31814},[31813],[34,535943,535945],{"className":535944},[31818,31819,31820,31821],[34,535946,535948,535951,535954],{"className":535947},[31685,31821],[34,535949,31955],{"className":535950},[31685,31686,31821],[34,535952,32226],{"className":535953},[32024,31821],[34,535955,31660],{"className":535956},[31685,31686,31821],[34,535958],{"className":535959,"style":31846},[31845],[34,535961,202],{"className":535962},[31850],[34,535964],{"className":535965,"style":31846},[31845],[34,535967,535969,535972],{"className":535968},[31676],[34,535970],{"className":535971,"style":32617},[31680],[34,535973,459],{"className":535974},[31685],"。这一项为 0。",[108,535977,2234,535978,55016,536029,465789,536254,536327,536328,536645],{},[34,535979,535981,535999],{"className":535980},[31642],[34,535982,535984],{"className":535983},[31646],[31648,535985,535986],{"xmlns":31650},[31652,535987,535988,535996],{},[31655,535989,535990,535992,535994],{},[31658,535991,31955],{},[31662,535993,932],{},[31658,535995,31660],{},[31665,535997,535998],{"encoding":31667},"i \u003C n",[34,536000,536002,536020],{"className":536001,"ariaHidden":758},[31672],[34,536003,536005,536008,536011,536014,536017],{"className":536004},[31676],[34,536006],{"className":536007,"style":451673},[31680],[34,536009,31955],{"className":536010},[31685,31686],[34,536012],{"className":536013,"style":31846},[31845],[34,536015,932],{"className":536016},[31850],[34,536018],{"className":536019,"style":31846},[31845],[34,536021,536023,536026],{"className":536022},[31676],[34,536024],{"className":536025,"style":32527},[31680],[34,536027,31660],{"className":536028},[31685,31686],[34,536030,536032,536086],{"className":536031},[31642],[34,536033,536035],{"className":536034},[31646],[31648,536036,536037],{"xmlns":31650},[31652,536038,536039,536083],{},[31655,536040,536041,536043,536045,536047,536049,536051,536053,536055,536057,536059,536061,536063,536065,536067,536069,536071,536073,536075,536077,536079,536081],{},[31658,536042,33759],{},[31662,536044,32226],{},[31658,536046,31955],{},[31662,536048,202],{},[31662,536050,456],{"stretchy":702},[31658,536052,31660],{},[31662,536054,223],{},[31658,536056,31759],{},[31662,536058,2557],{"stretchy":702},[31662,536060,32226],{},[31658,536062,31955],{},[31662,536064,29622],{},[31662,536066,456],{"stretchy":702},[31658,536068,31660],{},[31662,536070,223],{},[31658,536072,31759],{},[31662,536074,2557],{"stretchy":702},[31662,536076,32226],{},[31658,536078,31660],{},[31662,536080,202],{},[31658,536082,31759],{},[31665,536084,536085],{"encoding":31667},"k - i = (n + m) - i > (n + m) - n = m",[34,536087,536089,536107,536125,536146,536167,536185,536206,536227,536245],{"className":536088,"ariaHidden":758},[31672],[34,536090,536092,536095,536098,536101,536104],{"className":536091},[31676],[34,536093],{"className":536094,"style":34299},[31680],[34,536096,33759],{"className":536097,"style":33774},[31685,31686],[34,536099],{"className":536100,"style":32604},[31845],[34,536102,32226],{"className":536103},[32024],[34,536105],{"className":536106,"style":32604},[31845],[34,536108,536110,536113,536116,536119,536122],{"className":536109},[31676],[34,536111],{"className":536112,"style":33104},[31680],[34,536114,31955],{"className":536115},[31685,31686],[34,536117],{"className":536118,"style":31846},[31845],[34,536120,202],{"className":536121},[31850],[34,536123],{"className":536124,"style":31846},[31845],[34,536126,536128,536131,536134,536137,536140,536143],{"className":536127},[31676],[34,536129],{"className":536130,"style":31774},[31680],[34,536132,456],{"className":536133},[31782],[34,536135,31660],{"className":536136},[31685,31686],[34,536138],{"className":536139,"style":32604},[31845],[34,536141,223],{"className":536142},[32024],[34,536144],{"className":536145,"style":32604},[31845],[34,536147,536149,536152,536155,536158,536161,536164],{"className":536148},[31676],[34,536150],{"className":536151,"style":31774},[31680],[34,536153,31759],{"className":536154},[31685,31686],[34,536156,2557],{"className":536157},[31690],[34,536159],{"className":536160,"style":32604},[31845],[34,536162,32226],{"className":536163},[32024],[34,536165],{"className":536166,"style":32604},[31845],[34,536168,536170,536173,536176,536179,536182],{"className":536169},[31676],[34,536171],{"className":536172,"style":451673},[31680],[34,536174,31955],{"className":536175},[31685,31686],[34,536177],{"className":536178,"style":31846},[31845],[34,536180,29622],{"className":536181},[31850],[34,536183],{"className":536184,"style":31846},[31845],[34,536186,536188,536191,536194,536197,536200,536203],{"className":536187},[31676],[34,536189],{"className":536190,"style":31774},[31680],[34,536192,456],{"className":536193},[31782],[34,536195,31660],{"className":536196},[31685,31686],[34,536198],{"className":536199,"style":32604},[31845],[34,536201,223],{"className":536202},[32024],[34,536204],{"className":536205,"style":32604},[31845],[34,536207,536209,536212,536215,536218,536221,536224],{"className":536208},[31676],[34,536210],{"className":536211,"style":31774},[31680],[34,536213,31759],{"className":536214},[31685,31686],[34,536216,2557],{"className":536217},[31690],[34,536219],{"className":536220,"style":32604},[31845],[34,536222,32226],{"className":536223},[32024],[34,536225],{"className":536226,"style":32604},[31845],[34,536228,536230,536233,536236,536239,536242],{"className":536229},[31676],[34,536231],{"className":536232,"style":32527},[31680],[34,536234,31660],{"className":536235},[31685,31686],[34,536237],{"className":536238,"style":31846},[31845],[34,536240,202],{"className":536241},[31850],[34,536243],{"className":536244,"style":31846},[31845],[34,536246,536248,536251],{"className":536247},[31676],[34,536249],{"className":536250,"style":32527},[31680],[34,536252,31759],{"className":536253},[31685,31686],[34,536255,536257,536279],{"className":536256},[31642],[34,536258,536260],{"className":536259},[31646],[31648,536261,536262],{"xmlns":31650},[31652,536263,536264,536276],{},[31655,536265,536266,536268,536270,536272,536274],{},[31658,536267,33759],{},[31662,536269,32226],{},[31658,536271,31955],{},[31662,536273,43175],{},[31658,536275,31759],{},[31665,536277,536278],{"encoding":31667},"k - i \\ge m",[34,536280,536282,536300,536318],{"className":536281,"ariaHidden":758},[31672],[34,536283,536285,536288,536291,536294,536297],{"className":536284},[31676],[34,536286],{"className":536287,"style":34299},[31680],[34,536289,33759],{"className":536290,"style":33774},[31685,31686],[34,536292],{"className":536293,"style":32604},[31845],[34,536295,32226],{"className":536296},[32024],[34,536298],{"className":536299,"style":32604},[31845],[34,536301,536303,536306,536309,536312,536315],{"className":536302},[31676],[34,536304],{"className":536305,"style":55401},[31680],[34,536307,31955],{"className":536308},[31685,31686],[34,536310],{"className":536311,"style":31846},[31845],[34,536313,43175],{"className":536314},[31850],[34,536316],{"className":536317,"style":31846},[31845],[34,536319,536321,536324],{"className":536320},[31676],[34,536322],{"className":536323,"style":32527},[31680],[34,536325,31759],{"className":536326},[31685,31686],"。\n此时 ",[34,536329,536331,536409],{"className":536330},[31642],[34,536332,536334],{"className":536333},[31646],[31648,536335,536336],{"xmlns":31650},[31652,536337,536338,536406],{},[31655,536339,536340,536342,536344,536346,536358,536360,536362,536374,536376,536378,536384,536386,536402,536404],{},[31662,536341,456],{"stretchy":702},[31662,536343,32226],{},[31658,536345,67764],{},[37624,536347,536348,536350],{},[31662,536349,2557],{"stretchy":702},[31655,536351,536352,536354,536356],{},[31658,536353,33759],{},[31662,536355,32226],{},[31658,536357,31955],{},[31662,536359,202],{},[31662,536361,41461],{},[37624,536363,536364,536366],{},[31658,536365,67764],{},[31655,536367,536368,536370,536372],{},[31658,536369,33759],{},[31662,536371,32226],{},[31658,536373,31955],{},[31662,536375,202],{},[31662,536377,41461],{},[37624,536379,536380,536382],{},[31658,536381,67764],{},[31658,536383,31759],{},[31662,536385,71191],{},[37624,536387,536388,536390],{},[31658,536389,67764],{},[31655,536391,536392,536394,536396,536398,536400],{},[31658,536393,33759],{},[31662,536395,32226],{},[31658,536397,31955],{},[31662,536399,32226],{},[31658,536401,31759],{},[31662,536403,202],{},[31749,536405,459],{},[31665,536407,536408],{"encoding":31667},"(-y)^{k-i} = \\pm y^{k-i} = \\pm y^m \\cdot y^{k-i-m} = 0",[34,536410,536412,536474,536530,536577,536636],{"className":536411,"ariaHidden":758},[31672],[34,536413,536415,536418,536421,536424,536427,536465,536468,536471],{"className":536414},[31676],[34,536416],{"className":536417,"style":43651},[31680],[34,536419,456],{"className":536420},[31782],[34,536422,32226],{"className":536423},[31685],[34,536425,67764],{"className":536426,"style":32062},[31685,31686],[34,536428,536430,536433],{"className":536429},[31690],[34,536431,2557],{"className":536432},[31690],[34,536434,536436],{"className":536435},[31792],[34,536437,536439],{"className":536438},[31796],[34,536440,536442],{"className":536441},[31801],[34,536443,536445],{"className":536444,"style":38486},[31805],[34,536446,536447,536450],{"style":37777},[34,536448],{"className":536449,"style":31814},[31813],[34,536451,536453],{"className":536452},[31818,31819,31820,31821],[34,536454,536456,536459,536462],{"className":536455},[31685,31821],[34,536457,33759],{"className":536458,"style":33774},[31685,31686,31821],[34,536460,32226],{"className":536461},[32024,31821],[34,536463,31955],{"className":536464},[31685,31686,31821],[34,536466],{"className":536467,"style":31846},[31845],[34,536469,202],{"className":536470},[31850],[34,536472],{"className":536473,"style":31846},[31845],[34,536475,536477,536480,536483,536521,536524,536527],{"className":536476},[31676],[34,536478],{"className":536479,"style":315792},[31680],[34,536481,41461],{"className":536482},[31685],[34,536484,536486,536489],{"className":536485},[31685],[34,536487,67764],{"className":536488,"style":32062},[31685,31686],[34,536490,536492],{"className":536491},[31792],[34,536493,536495],{"className":536494},[31796],[34,536496,536498],{"className":536497},[31801],[34,536499,536501],{"className":536500,"style":38486},[31805],[34,536502,536503,536506],{"style":37777},[34,536504],{"className":536505,"style":31814},[31813],[34,536507,536509],{"className":536508},[31818,31819,31820,31821],[34,536510,536512,536515,536518],{"className":536511},[31685,31821],[34,536513,33759],{"className":536514,"style":33774},[31685,31686,31821],[34,536516,32226],{"className":536517},[32024,31821],[34,536519,31955],{"className":536520},[31685,31686,31821],[34,536522],{"className":536523,"style":31846},[31845],[34,536525,202],{"className":536526},[31850],[34,536528],{"className":536529,"style":31846},[31845],[34,536531,536533,536536,536539,536568,536571,536574],{"className":536532},[31676],[34,536534],{"className":536535,"style":316112},[31680],[34,536537,41461],{"className":536538},[31685],[34,536540,536542,536545],{"className":536541},[31685],[34,536543,67764],{"className":536544,"style":32062},[31685,31686],[34,536546,536548],{"className":536547},[31792],[34,536549,536551],{"className":536550},[31796],[34,536552,536554],{"className":536553},[31801],[34,536555,536557],{"className":536556,"style":37959},[31805],[34,536558,536559,536562],{"style":37777},[34,536560],{"className":536561,"style":31814},[31813],[34,536563,536565],{"className":536564},[31818,31819,31820,31821],[34,536566,31759],{"className":536567},[31685,31686,31821],[34,536569],{"className":536570,"style":32604},[31845],[34,536572,71191],{"className":536573},[32024],[34,536575],{"className":536576,"style":32604},[31845],[34,536578,536580,536583,536627,536630,536633],{"className":536579},[31676],[34,536581],{"className":536582,"style":315792},[31680],[34,536584,536586,536589],{"className":536585},[31685],[34,536587,67764],{"className":536588,"style":32062},[31685,31686],[34,536590,536592],{"className":536591},[31792],[34,536593,536595],{"className":536594},[31796],[34,536596,536598],{"className":536597},[31801],[34,536599,536601],{"className":536600,"style":38486},[31805],[34,536602,536603,536606],{"style":37777},[34,536604],{"className":536605,"style":31814},[31813],[34,536607,536609],{"className":536608},[31818,31819,31820,31821],[34,536610,536612,536615,536618,536621,536624],{"className":536611},[31685,31821],[34,536613,33759],{"className":536614,"style":33774},[31685,31686,31821],[34,536616,32226],{"className":536617},[32024,31821],[34,536619,31955],{"className":536620},[31685,31686,31821],[34,536622,32226],{"className":536623},[32024,31821],[34,536625,31759],{"className":536626},[31685,31686,31821],[34,536628],{"className":536629,"style":31846},[31845],[34,536631,202],{"className":536632},[31850],[34,536634],{"className":536635,"style":31846},[31845],[34,536637,536639,536642],{"className":536638},[31676],[34,536640],{"className":536641,"style":32617},[31680],[34,536643,459],{"className":536644},[31685],"。这一项也为 0。",[15,536647,536648,536649,452821,536777,347],{},"因此，展开式中每一项都是 0，所以 ",[34,536650,536652,536688],{"className":536651},[31642],[34,536653,536655],{"className":536654},[31646],[31648,536656,536657],{"xmlns":31650},[31652,536658,536659,536685],{},[31655,536660,536661,536663,536665,536667,536669,536681,536683],{},[31662,536662,456],{"stretchy":702},[31658,536664,32936],{},[31662,536666,32226],{},[31658,536668,67764],{},[37624,536670,536671,536673],{},[31662,536672,2557],{"stretchy":702},[31655,536674,536675,536677,536679],{},[31658,536676,31660],{},[31662,536678,223],{},[31658,536680,31759],{},[31662,536682,202],{},[31749,536684,459],{},[31665,536686,536687],{"encoding":31667},"(x - y)^{n+m} = 0",[34,536689,536691,536712,536768],{"className":536690,"ariaHidden":758},[31672],[34,536692,536694,536697,536700,536703,536706,536709],{"className":536693},[31676],[34,536695],{"className":536696,"style":31774},[31680],[34,536698,456],{"className":536699},[31782],[34,536701,32936],{"className":536702},[31685,31686],[34,536704],{"className":536705,"style":32604},[31845],[34,536707,32226],{"className":536708},[32024],[34,536710],{"className":536711,"style":32604},[31845],[34,536713,536715,536718,536721,536759,536762,536765],{"className":536714},[31676],[34,536716],{"className":536717,"style":156809},[31680],[34,536719,67764],{"className":536720,"style":32062},[31685,31686],[34,536722,536724,536727],{"className":536723},[31690],[34,536725,2557],{"className":536726},[31690],[34,536728,536730],{"className":536729},[31792],[34,536731,536733],{"className":536732},[31796],[34,536734,536736],{"className":536735},[31801],[34,536737,536739],{"className":536738,"style":156828},[31805],[34,536740,536741,536744],{"style":37777},[34,536742],{"className":536743,"style":31814},[31813],[34,536745,536747],{"className":536746},[31818,31819,31820,31821],[34,536748,536750,536753,536756],{"className":536749},[31685,31821],[34,536751,31660],{"className":536752},[31685,31686,31821],[34,536754,223],{"className":536755},[32024,31821],[34,536757,31759],{"className":536758},[31685,31686,31821],[34,536760],{"className":536761,"style":31846},[31845],[34,536763,202],{"className":536764},[31850],[34,536766],{"className":536767,"style":31846},[31845],[34,536769,536771,536774],{"className":536770},[31676],[34,536772],{"className":536773,"style":32617},[31680],[34,536775,459],{"className":536776},[31685],[34,536778,536780,536802],{"className":536779},[31642],[34,536781,536783],{"className":536782},[31646],[31648,536784,536785],{"xmlns":31650},[31652,536786,536787,536799],{},[31655,536788,536789,536791,536793,536795,536797],{},[31658,536790,32936],{},[31662,536792,32226],{},[31658,536794,67764],{},[31662,536796,112296],{},[31658,536798,33347],{},[31665,536800,536801],{"encoding":31667},"x - y \\in N",[34,536803,536805,536823,536841],{"className":536804,"ariaHidden":758},[31672],[34,536806,536808,536811,536814,536817,536820],{"className":536807},[31676],[34,536809],{"className":536810,"style":32597},[31680],[34,536812,32936],{"className":536813},[31685,31686],[34,536815],{"className":536816,"style":32604},[31845],[34,536818,32226],{"className":536819},[32024],[34,536821],{"className":536822,"style":32604},[31845],[34,536824,536826,536829,536832,536835,536838],{"className":536825},[31676],[34,536827],{"className":536828,"style":86652},[31680],[34,536830,67764],{"className":536831,"style":32062},[31685,31686],[34,536833],{"className":536834,"style":31846},[31845],[34,536836,112296],{"className":536837},[31850],[34,536839],{"className":536840,"style":31846},[31845],[34,536842,536844,536847],{"className":536843},[31676],[34,536845],{"className":536846,"style":33218},[31680],[34,536848,33347],{"className":536849,"style":33389},[31685,31686],[2230,536851,536852],{"start":43},[108,536853,536854,362914,536856,46528,536907,86621,536957,395725,537007,315343,537035,534831,537115,537149,537150,44689,537224,113124,537252,13977,537254,39984,537415,467788,537585,347],{},[2627,536855,442183],{},[34,536857,536859,536877],{"className":536858},[31642],[34,536860,536862],{"className":536861},[31646],[31648,536863,536864],{"xmlns":31650},[31652,536865,536866,536874],{},[31655,536867,536868,536870,536872],{},[31658,536869,32936],{},[31662,536871,112296],{},[31658,536873,33347],{},[31665,536875,536876],{"encoding":31667},"x \\in N",[34,536878,536880,536898],{"className":536879,"ariaHidden":758},[31672],[34,536881,536883,536886,536889,536892,536895],{"className":536882},[31676],[34,536884],{"className":536885,"style":34812},[31680],[34,536887,32936],{"className":536888},[31685,31686],[34,536890],{"className":536891,"style":31846},[31845],[34,536893,112296],{"className":536894},[31850],[34,536896],{"className":536897,"style":31846},[31845],[34,536899,536901,536904],{"className":536900},[31676],[34,536902],{"className":536903,"style":33218},[31680],[34,536905,33347],{"className":536906,"style":33389},[31685,31686],[34,536908,536910,536927],{"className":536909},[31642],[34,536911,536913],{"className":536912},[31646],[31648,536914,536915],{"xmlns":31650},[31652,536916,536917,536925],{},[31655,536918,536919,536921,536923],{},[31658,536920,34395],{},[31662,536922,112296],{},[31658,536924,34386],{},[31665,536926,335118],{"encoding":31667},[34,536928,536930,536948],{"className":536929,"ariaHidden":758},[31672],[34,536931,536933,536936,536939,536942,536945],{"className":536932},[31676],[34,536934],{"className":536935,"style":34812},[31680],[34,536937,34395],{"className":536938,"style":34452},[31685,31686],[34,536940],{"className":536941,"style":31846},[31845],[34,536943,112296],{"className":536944},[31850],[34,536946],{"className":536947,"style":31846},[31845],[34,536949,536951,536954],{"className":536950},[31676],[34,536952],{"className":536953,"style":33218},[31680],[34,536955,34386],{"className":536956,"style":34427},[31685,31686],[34,536958,536960,536977],{"className":536959},[31642],[34,536961,536963],{"className":536962},[31646],[31648,536964,536965],{"xmlns":31650},[31652,536966,536967,536975],{},[31655,536968,536969,536971,536973],{},[31658,536970,32936],{},[31662,536972,112296],{},[31658,536974,33347],{},[31665,536976,536876],{"encoding":31667},[34,536978,536980,536998],{"className":536979,"ariaHidden":758},[31672],[34,536981,536983,536986,536989,536992,536995],{"className":536982},[31676],[34,536984],{"className":536985,"style":34812},[31680],[34,536987,32936],{"className":536988},[31685,31686],[34,536990],{"className":536991,"style":31846},[31845],[34,536993,112296],{"className":536994},[31850],[34,536996],{"className":536997,"style":31846},[31845],[34,536999,537001,537004],{"className":537000},[31676],[34,537002],{"className":537003,"style":33218},[31680],[34,537005,33347],{"className":537006,"style":33389},[31685,31686],[34,537008,537010,537023],{"className":537009},[31642],[34,537011,537013],{"className":537012},[31646],[31648,537014,537015],{"xmlns":31650},[31652,537016,537017,537021],{},[31655,537018,537019],{},[31658,537020,31660],{},[31665,537022,31660],{"encoding":31667},[34,537024,537026],{"className":537025,"ariaHidden":758},[31672],[34,537027,537029,537032],{"className":537028},[31676],[34,537030],{"className":537031,"style":32527},[31680],[34,537033,31660],{"className":537034},[31685,31686],[34,537036,537038,537059],{"className":537037},[31642],[34,537039,537041],{"className":537040},[31646],[31648,537042,537043],{"xmlns":31650},[31652,537044,537045,537057],{},[31655,537046,537047,537053,537055],{},[37624,537048,537049,537051],{},[31658,537050,32936],{},[31658,537052,31660],{},[31662,537054,202],{},[31749,537056,459],{},[31665,537058,533777],{"encoding":31667},[34,537060,537062,537106],{"className":537061,"ariaHidden":758},[31672],[34,537063,537065,537068,537097,537100,537103],{"className":537064},[31676],[34,537066],{"className":537067,"style":37959},[31680],[34,537069,537071,537074],{"className":537070},[31685],[34,537072,32936],{"className":537073},[31685,31686],[34,537075,537077],{"className":537076},[31792],[34,537078,537080],{"className":537079},[31796],[34,537081,537083],{"className":537082},[31801],[34,537084,537086],{"className":537085,"style":37959},[31805],[34,537087,537088,537091],{"style":37777},[34,537089],{"className":537090,"style":31814},[31813],[34,537092,537094],{"className":537093},[31818,31819,31820,31821],[34,537095,31660],{"className":537096},[31685,31686,31821],[34,537098],{"className":537099,"style":31846},[31845],[34,537101,202],{"className":537102},[31850],[34,537104],{"className":537105,"style":31846},[31845],[34,537107,537109,537112],{"className":537108},[31676],[34,537110],{"className":537111,"style":32617},[31680],[34,537113,459],{"className":537114},[31685],[34,537116,537118,537134],{"className":537117},[31642],[34,537119,537121],{"className":537120},[31646],[31648,537122,537123],{"xmlns":31650},[31652,537124,537125,537131],{},[31655,537126,537127,537129],{},[31658,537128,34395],{},[31658,537130,32936],{},[31665,537132,537133],{"encoding":31667},"rx",[34,537135,537137],{"className":537136,"ariaHidden":758},[31672],[34,537138,537140,537143,537146],{"className":537139},[31676],[34,537141],{"className":537142,"style":32527},[31680],[34,537144,34395],{"className":537145,"style":34452},[31685,31686],[34,537147,32936],{"className":537148},[31685,31686]," 也是幂零元。\n考察 ",[34,537151,537153,537177],{"className":537152},[31642],[34,537154,537156],{"className":537155},[31646],[31648,537157,537158],{"xmlns":31650},[31652,537159,537160,537174],{},[31655,537161,537162,537164,537166,537168],{},[31662,537163,456],{"stretchy":702},[31658,537165,34395],{},[31658,537167,32936],{},[37624,537169,537170,537172],{},[31662,537171,2557],{"stretchy":702},[31658,537173,31660],{},[31665,537175,537176],{"encoding":31667},"(rx)^n",[34,537178,537180],{"className":537179,"ariaHidden":758},[31672],[34,537181,537183,537186,537189,537192,537195],{"className":537182},[31676],[34,537184],{"className":537185,"style":31774},[31680],[34,537187,456],{"className":537188},[31782],[34,537190,34395],{"className":537191,"style":34452},[31685,31686],[34,537193,32936],{"className":537194},[31685,31686],[34,537196,537198,537201],{"className":537197},[31690],[34,537199,2557],{"className":537200},[31690],[34,537202,537204],{"className":537203},[31792],[34,537205,537207],{"className":537206},[31796],[34,537208,537210],{"className":537209},[31801],[34,537211,537213],{"className":537212,"style":37959},[31805],[34,537214,537215,537218],{"style":37777},[34,537216],{"className":537217,"style":31814},[31813],[34,537219,537221],{"className":537220},[31818,31819,31820,31821],[34,537222,31660],{"className":537223},[31685,31686,31821],[34,537225,537227,537240],{"className":537226},[31642],[34,537228,537230],{"className":537229},[31646],[31648,537231,537232],{"xmlns":31650},[31652,537233,537234,537238],{},[31655,537235,537236],{},[31658,537237,34386],{},[31665,537239,34386],{"encoding":31667},[34,537241,537243],{"className":537242,"ariaHidden":758},[31672],[34,537244,537246,537249],{"className":537245},[31676],[34,537247],{"className":537248,"style":33218},[31680],[34,537250,34386],{"className":537251,"style":34427},[31685,31686],[2627,537253,535049],{},[34,537255,537257,537295],{"className":537256},[31642],[34,537258,537260],{"className":537259},[31646],[31648,537261,537262],{"xmlns":31650},[31652,537263,537264,537292],{},[31655,537265,537266,537268,537270,537272,537278,537280,537286],{},[31662,537267,456],{"stretchy":702},[31658,537269,34395],{},[31658,537271,32936],{},[37624,537273,537274,537276],{},[31662,537275,2557],{"stretchy":702},[31658,537277,31660],{},[31662,537279,202],{},[37624,537281,537282,537284],{},[31658,537283,34395],{},[31658,537285,31660],{},[37624,537287,537288,537290],{},[31658,537289,32936],{},[31658,537291,31660],{},[31665,537293,537294],{"encoding":31667},"(rx)^n = r^n x^n",[34,537296,537298,537351],{"className":537297,"ariaHidden":758},[31672],[34,537299,537301,537304,537307,537310,537313,537342,537345,537348],{"className":537300},[31676],[34,537302],{"className":537303,"style":31774},[31680],[34,537305,456],{"className":537306},[31782],[34,537308,34395],{"className":537309,"style":34452},[31685,31686],[34,537311,32936],{"className":537312},[31685,31686],[34,537314,537316,537319],{"className":537315},[31690],[34,537317,2557],{"className":537318},[31690],[34,537320,537322],{"className":537321},[31792],[34,537323,537325],{"className":537324},[31796],[34,537326,537328],{"className":537327},[31801],[34,537329,537331],{"className":537330,"style":37959},[31805],[34,537332,537333,537336],{"style":37777},[34,537334],{"className":537335,"style":31814},[31813],[34,537337,537339],{"className":537338},[31818,31819,31820,31821],[34,537340,31660],{"className":537341},[31685,31686,31821],[34,537343],{"className":537344,"style":31846},[31845],[34,537346,202],{"className":537347},[31850],[34,537349],{"className":537350,"style":31846},[31845],[34,537352,537354,537357,537386],{"className":537353},[31676],[34,537355],{"className":537356,"style":37959},[31680],[34,537358,537360,537363],{"className":537359},[31685],[34,537361,34395],{"className":537362,"style":34452},[31685,31686],[34,537364,537366],{"className":537365},[31792],[34,537367,537369],{"className":537368},[31796],[34,537370,537372],{"className":537371},[31801],[34,537373,537375],{"className":537374,"style":37959},[31805],[34,537376,537377,537380],{"style":37777},[34,537378],{"className":537379,"style":31814},[31813],[34,537381,537383],{"className":537382},[31818,31819,31820,31821],[34,537384,31660],{"className":537385},[31685,31686,31821],[34,537387,537389,537392],{"className":537388},[31685],[34,537390,32936],{"className":537391},[31685,31686],[34,537393,537395],{"className":537394},[31792],[34,537396,537398],{"className":537397},[31796],[34,537399,537401],{"className":537400},[31801],[34,537402,537404],{"className":537403,"style":37959},[31805],[34,537405,537406,537409],{"style":37777},[34,537407],{"className":537408,"style":31814},[31813],[34,537410,537412],{"className":537411},[31818,31819,31820,31821],[34,537413,31660],{"className":537414},[31685,31686,31821],[34,537416,537418,537458],{"className":537417},[31642],[34,537419,537421],{"className":537420},[31646],[31648,537422,537423],{"xmlns":31650},[31652,537424,537425,537455],{},[31655,537426,537427,537429,537431,537433,537439,537441,537447,537449,537451,537453],{},[31662,537428,456],{"stretchy":702},[31658,537430,34395],{},[31658,537432,32936],{},[37624,537434,537435,537437],{},[31662,537436,2557],{"stretchy":702},[31658,537438,31660],{},[31662,537440,202],{},[37624,537442,537443,537445],{},[31658,537444,34395],{},[31658,537446,31660],{},[31662,537448,71191],{},[31749,537450,459],{},[31662,537452,202],{},[31749,537454,459],{},[31665,537456,537457],{"encoding":31667},"(rx)^n = r^n \\cdot 0 = 0",[34,537459,537461,537514,537558,537576],{"className":537460,"ariaHidden":758},[31672],[34,537462,537464,537467,537470,537473,537476,537505,537508,537511],{"className":537463},[31676],[34,537465],{"className":537466,"style":31774},[31680],[34,537468,456],{"className":537469},[31782],[34,537471,34395],{"className":537472,"style":34452},[31685,31686],[34,537474,32936],{"className":537475},[31685,31686],[34,537477,537479,537482],{"className":537478},[31690],[34,537480,2557],{"className":537481},[31690],[34,537483,537485],{"className":537484},[31792],[34,537486,537488],{"className":537487},[31796],[34,537489,537491],{"className":537490},[31801],[34,537492,537494],{"className":537493,"style":37959},[31805],[34,537495,537496,537499],{"style":37777},[34,537497],{"className":537498,"style":31814},[31813],[34,537500,537502],{"className":537501},[31818,31819,31820,31821],[34,537503,31660],{"className":537504},[31685,31686,31821],[34,537506],{"className":537507,"style":31846},[31845],[34,537509,202],{"className":537510},[31850],[34,537512],{"className":537513,"style":31846},[31845],[34,537515,537517,537520,537549,537552,537555],{"className":537516},[31676],[34,537518],{"className":537519,"style":37959},[31680],[34,537521,537523,537526],{"className":537522},[31685],[34,537524,34395],{"className":537525,"style":34452},[31685,31686],[34,537527,537529],{"className":537528},[31792],[34,537530,537532],{"className":537531},[31796],[34,537533,537535],{"className":537534},[31801],[34,537536,537538],{"className":537537,"style":37959},[31805],[34,537539,537540,537543],{"style":37777},[34,537541],{"className":537542,"style":31814},[31813],[34,537544,537546],{"className":537545},[31818,31819,31820,31821],[34,537547,31660],{"className":537548},[31685,31686,31821],[34,537550],{"className":537551,"style":32604},[31845],[34,537553,71191],{"className":537554},[32024],[34,537556],{"className":537557,"style":32604},[31845],[34,537559,537561,537564,537567,537570,537573],{"className":537560},[31676],[34,537562],{"className":537563,"style":32617},[31680],[34,537565,459],{"className":537566},[31685],[34,537568],{"className":537569,"style":31846},[31845],[34,537571,202],{"className":537572},[31850],[34,537574],{"className":537575,"style":31846},[31845],[34,537577,537579,537582],{"className":537578},[31676],[34,537580],{"className":537581,"style":32617},[31680],[34,537583,459],{"className":537584},[31685],[34,537586,537588,537608],{"className":537587},[31642],[34,537589,537591],{"className":537590},[31646],[31648,537592,537593],{"xmlns":31650},[31652,537594,537595,537605],{},[31655,537596,537597,537599,537601,537603],{},[31658,537598,34395],{},[31658,537600,32936],{},[31662,537602,112296],{},[31658,537604,33347],{},[31665,537606,537607],{"encoding":31667},"rx \\in N",[34,537609,537611,537632],{"className":537610,"ariaHidden":758},[31672],[34,537612,537614,537617,537620,537623,537626,537629],{"className":537613},[31676],[34,537615],{"className":537616,"style":34812},[31680],[34,537618,34395],{"className":537619,"style":34452},[31685,31686],[34,537621,32936],{"className":537622},[31685,31686],[34,537624],{"className":537625,"style":31846},[31845],[34,537627,112296],{"className":537628},[31850],[34,537630],{"className":537631,"style":31846},[31845],[34,537633,537635,537638],{"className":537634},[31676],[34,537636],{"className":537637,"style":33218},[31680],[34,537639,33347],{"className":537640,"style":33389},[31685,31686],[15,537642,537643,2664,537645,11008,537673,469010,537701],{},[2627,537644,114043],{},[34,537646,537648,537661],{"className":537647},[31642],[34,537649,537651],{"className":537650},[31646],[31648,537652,537653],{"xmlns":31650},[31652,537654,537655,537659],{},[31655,537656,537657],{},[31658,537658,33347],{},[31665,537660,33347],{"encoding":31667},[34,537662,537664],{"className":537663,"ariaHidden":758},[31672],[34,537665,537667,537670],{"className":537666},[31676],[34,537668],{"className":537669,"style":33218},[31680],[34,537671,33347],{"className":537672,"style":33389},[31685,31686],[34,537674,537676,537689],{"className":537675},[31642],[34,537677,537679],{"className":537678},[31646],[31648,537680,537681],{"xmlns":31650},[31652,537682,537683,537687],{},[31655,537684,537685],{},[31658,537686,34386],{},[31665,537688,34386],{"encoding":31667},[34,537690,537692],{"className":537691,"ariaHidden":758},[31672],[34,537693,537695,537698],{"className":537694},[31676],[34,537696],{"className":537697,"style":33218},[31680],[34,537699,34386],{"className":537700,"style":34427},[31685,31686],[34,537702,537704,537717],{"className":537703},[31642],[34,537705,537707],{"className":537706},[31646],[31648,537708,537709],{"xmlns":31650},[31652,537710,537711,537715],{},[31655,537712,537713],{},[31658,537714,445204],{"mathvariant":38079},[31665,537716,445207],{"encoding":31667},[34,537718,537720],{"className":537719,"ariaHidden":758},[31672],[34,537721,537723,537726],{"className":537722},[31676],[34,537724],{"className":537725,"style":445217},[31680],[34,537727,445204],{"className":537728},[31685,129261],[32188,537730],{},[253,537732,537734],{"id":537733},"题目-15理想运算的分配律","题目 15：理想运算的分配律",[15,537736,537737,537825,537950],{},[2627,537738,537739,537740,537795,537796,537824],{},"15. 证明幺环中理想的加、乘法满足分配律。即设 ",[34,537741,537743,537765],{"className":537742},[31642],[34,537744,537746],{"className":537745},[31646],[31648,537747,537748],{"xmlns":31650},[31652,537749,537750,537762],{},[31655,537751,537752,537754,537756,537758,537760],{},[31658,537753,106063],{},[31662,537755,31960],{"separator":758},[31658,537757,223320],{},[31662,537759,31960],{"separator":758},[31658,537761,38691],{},[31665,537763,537764],{"encoding":31667},"I,J,K",[34,537766,537768],{"className":537767,"ariaHidden":758},[31672],[34,537769,537771,537774,537777,537780,537783,537786,537789,537792],{"className":537770},[31676],[34,537772],{"className":537773,"style":70739},[31680],[34,537775,106063],{"className":537776,"style":67482},[31685,31686],[34,537778,31960],{"className":537779},[32119],[34,537781],{"className":537782,"style":32123},[31845],[34,537784,223320],{"className":537785,"style":223361},[31685,31686],[34,537787,31960],{"className":537788},[32119],[34,537790],{"className":537791,"style":32123},[31845],[34,537793,38691],{"className":537794,"style":38706},[31685,31686]," 是幺环 ",[34,537797,537799,537812],{"className":537798},[31642],[34,537800,537802],{"className":537801},[31646],[31648,537803,537804],{"xmlns":31650},[31652,537805,537806,537810],{},[31655,537807,537808],{},[31658,537809,34386],{},[31665,537811,34386],{"encoding":31667},[34,537813,537815],{"className":537814,"ariaHidden":758},[31672],[34,537816,537818,537821],{"className":537817},[31676],[34,537819],{"className":537820,"style":33218},[31680],[34,537822,34386],{"className":537823,"style":34427},[31685,31686]," 的理想，则",[34,537826,537828,537866],{"className":537827},[31642],[34,537829,537831],{"className":537830},[31646],[31648,537832,537833],{"xmlns":31650},[31652,537834,537835,537863],{},[31655,537836,537837,537839,537841,537843,537845,537847,537849,537851,537853,537855,537857,537859,537861],{},[31662,537838,456],{"stretchy":702},[31658,537840,106063],{},[31662,537842,223],{},[31658,537844,223320],{},[31662,537846,2557],{"stretchy":702},[31658,537848,38691],{},[31662,537850,202],{},[31658,537852,106063],{},[31658,537854,38691],{},[31662,537856,223],{},[31658,537858,223320],{},[31658,537860,38691],{},[31662,537862,31960],{"separator":758},[31665,537864,537865],{"encoding":31667},"(I + J)K = IK + JK,",[34,537867,537869,537890,537914,537935],{"className":537868,"ariaHidden":758},[31672],[34,537870,537872,537875,537878,537881,537884,537887],{"className":537871},[31676],[34,537873],{"className":537874,"style":31774},[31680],[34,537876,456],{"className":537877},[31782],[34,537879,106063],{"className":537880,"style":67482},[31685,31686],[34,537882],{"className":537883,"style":32604},[31845],[34,537885,223],{"className":537886},[32024],[34,537888],{"className":537889,"style":32604},[31845],[34,537891,537893,537896,537899,537902,537905,537908,537911],{"className":537892},[31676],[34,537894],{"className":537895,"style":31774},[31680],[34,537897,223320],{"className":537898,"style":223361},[31685,31686],[34,537900,2557],{"className":537901},[31690],[34,537903,38691],{"className":537904,"style":38706},[31685,31686],[34,537906],{"className":537907,"style":31846},[31845],[34,537909,202],{"className":537910},[31850],[34,537912],{"className":537913,"style":31846},[31845],[34,537915,537917,537920,537923,537926,537929,537932],{"className":537916},[31676],[34,537918],{"className":537919,"style":34944},[31680],[34,537921,106063],{"className":537922,"style":67482},[31685,31686],[34,537924,38691],{"className":537925,"style":38706},[31685,31686],[34,537927],{"className":537928,"style":32604},[31845],[34,537930,223],{"className":537931},[32024],[34,537933],{"className":537934,"style":32604},[31845],[34,537936,537938,537941,537944,537947],{"className":537937},[31676],[34,537939],{"className":537940,"style":70739},[31680],[34,537942,223320],{"className":537943,"style":223361},[31685,31686],[34,537945,38691],{"className":537946,"style":38706},[31685,31686],[34,537948,31960],{"className":537949},[32119],[34,537951,537953,537991],{"className":537952},[31642],[34,537954,537956],{"className":537955},[31646],[31648,537957,537958],{"xmlns":31650},[31652,537959,537960,537988],{},[31655,537961,537962,537964,537966,537968,537970,537972,537974,537976,537978,537980,537982,537984,537986],{},[31658,537963,38691],{},[31662,537965,456],{"stretchy":702},[31658,537967,106063],{},[31662,537969,223],{},[31658,537971,223320],{},[31662,537973,2557],{"stretchy":702},[31662,537975,202],{},[31658,537977,38691],{},[31658,537979,106063],{},[31662,537981,223],{},[31658,537983,38691],{},[31658,537985,223320],{},[31658,537987,2547],{"mathvariant":38079},[31665,537989,537990],{"encoding":31667},"K(I + J) = KI + KJ.",[34,537992,537994,538018,538039,538060],{"className":537993,"ariaHidden":758},[31672],[34,537995,537997,538000,538003,538006,538009,538012,538015],{"className":537996},[31676],[34,537998],{"className":537999,"style":31774},[31680],[34,538001,38691],{"className":538002,"style":38706},[31685,31686],[34,538004,456],{"className":538005},[31782],[34,538007,106063],{"className":538008,"style":67482},[31685,31686],[34,538010],{"className":538011,"style":32604},[31845],[34,538013,223],{"className":538014},[32024],[34,538016],{"className":538017,"style":32604},[31845],[34,538019,538021,538024,538027,538030,538033,538036],{"className":538020},[31676],[34,538022],{"className":538023,"style":31774},[31680],[34,538025,223320],{"className":538026,"style":223361},[31685,31686],[34,538028,2557],{"className":538029},[31690],[34,538031],{"className":538032,"style":31846},[31845],[34,538034,202],{"className":538035},[31850],[34,538037],{"className":538038,"style":31846},[31845],[34,538040,538042,538045,538048,538051,538054,538057],{"className":538041},[31676],[34,538043],{"className":538044,"style":34944},[31680],[34,538046,38691],{"className":538047,"style":38706},[31685,31686],[34,538049,106063],{"className":538050,"style":67482},[31685,31686],[34,538052],{"className":538053,"style":32604},[31845],[34,538055,223],{"className":538056},[32024],[34,538058],{"className":538059,"style":32604},[31845],[34,538061,538063,538066,538069,538072],{"className":538062},[31676],[34,538064],{"className":538065,"style":33218},[31680],[34,538067,38691],{"className":538068,"style":38706},[31685,31686],[34,538070,223320],{"className":538071,"style":223361},[31685,31686],[34,538073,2547],{"className":538074},[31685],[32490,538076,463462],{"id":538077},"深度解析-1",[105,538079,538080,538465,538622],{},[108,538081,538082,33183,538084],{},[2627,538083,390279],{},[105,538085,538086,538271],{},[108,538087,538088,347],{},[34,538089,538091,538136],{"className":538090},[31642],[34,538092,538094],{"className":538093},[31646],[31648,538095,538096],{"xmlns":31650},[31652,538097,538098,538134],{},[31655,538099,538100,538102,538104,538106,538108,538110,538112,538114,538116,538118,538120,538122,538124,538126,538128,538130,538132],{},[31658,538101,106063],{},[31662,538103,223],{},[31658,538105,223320],{},[31662,538107,202],{},[31662,538109,32685],{"stretchy":702},[31658,538111,111],{},[31662,538113,223],{},[31658,538115,1697],{},[31662,538117,162086],{},[31658,538119,111],{},[31662,538121,112296],{},[31658,538123,106063],{},[31662,538125,31960],{"separator":758},[31658,538127,1697],{},[31662,538129,112296],{},[31658,538131,223320],{},[31662,538133,32715],{"stretchy":702},[31665,538135,506564],{"encoding":31667},[34,538137,538139,538157,538175,538196,538214,538232,538259],{"className":538138,"ariaHidden":758},[31672],[34,538140,538142,538145,538148,538151,538154],{"className":538141},[31676],[34,538143],{"className":538144,"style":34944},[31680],[34,538146,106063],{"className":538147,"style":67482},[31685,31686],[34,538149],{"className":538150,"style":32604},[31845],[34,538152,223],{"className":538153},[32024],[34,538155],{"className":538156,"style":32604},[31845],[34,538158,538160,538163,538166,538169,538172],{"className":538159},[31676],[34,538161],{"className":538162,"style":33218},[31680],[34,538164,223320],{"className":538165,"style":223361},[31685,31686],[34,538167],{"className":538168,"style":31846},[31845],[34,538170,202],{"className":538171},[31850],[34,538173],{"className":538174,"style":31846},[31845],[34,538176,538178,538181,538184,538187,538190,538193],{"className":538177},[31676],[34,538179],{"className":538180,"style":31774},[31680],[34,538182,32685],{"className":538183},[31782],[34,538185,111],{"className":538186},[31685,31686],[34,538188],{"className":538189,"style":32604},[31845],[34,538191,223],{"className":538192},[32024],[34,538194],{"className":538195,"style":32604},[31845],[34,538197,538199,538202,538205,538208,538211],{"className":538198},[31676],[34,538200],{"className":538201,"style":31774},[31680],[34,538203,1697],{"className":538204},[31685,31686],[34,538206],{"className":538207,"style":31846},[31845],[34,538209,162086],{"className":538210},[31850],[34,538212],{"className":538213,"style":31846},[31845],[34,538215,538217,538220,538223,538226,538229],{"className":538216},[31676],[34,538218],{"className":538219,"style":34812},[31680],[34,538221,111],{"className":538222},[31685,31686],[34,538224],{"className":538225,"style":31846},[31845],[34,538227,112296],{"className":538228},[31850],[34,538230],{"className":538231,"style":31846},[31845],[34,538233,538235,538238,538241,538244,538247,538250,538253,538256],{"className":538234},[31676],[34,538236],{"className":538237,"style":43217},[31680],[34,538239,106063],{"className":538240,"style":67482},[31685,31686],[34,538242,31960],{"className":538243},[32119],[34,538245],{"className":538246,"style":32123},[31845],[34,538248,1697],{"className":538249},[31685,31686],[34,538251],{"className":538252,"style":31846},[31845],[34,538254,112296],{"className":538255},[31850],[34,538257],{"className":538258,"style":31846},[31845],[34,538260,538262,538265,538268],{"className":538261},[31676],[34,538263],{"className":538264,"style":31774},[31680],[34,538266,223320],{"className":538267,"style":223361},[31685,31686],[34,538269,32715],{"className":538270},[31690],[108,538272,538273,538306,538307,130135,538431,538434,538435,347],{},[34,538274,538276,538291],{"className":538275},[31642],[34,538277,538279],{"className":538278},[31646],[31648,538280,538281],{"xmlns":31650},[31652,538282,538283,538289],{},[31655,538284,538285,538287],{},[31658,538286,54151],{},[31658,538288,54224],{},[31665,538290,93307],{"encoding":31667},[34,538292,538294],{"className":538293,"ariaHidden":758},[31672],[34,538295,538297,538300,538303],{"className":538296},[31676],[34,538298],{"className":538299,"style":33218},[31680],[34,538301,54151],{"className":538302},[31685,31686],[34,538304,54224],{"className":538305,"style":54252},[31685,31686]," 是由所有形如 ",[34,538308,538310,538336],{"className":538309},[31642],[34,538311,538313],{"className":538312},[31646],[31648,538314,538315],{"xmlns":31650},[31652,538316,538317,538333],{},[31655,538318,538319,538321,538327],{},[31662,538320,131138],{},[31743,538322,538323,538325],{},[31658,538324,111],{},[31658,538326,31955],{},[31743,538328,538329,538331],{},[31658,538330,1697],{},[31658,538332,31955],{},[31665,538334,538335],{"encoding":31667},"\\sum a_i b_i",[34,538337,538339],{"className":538338,"ariaHidden":758},[31672],[34,538340,538342,538345,538348,538351,538391],{"className":538341},[31676],[34,538343],{"className":538344,"style":31774},[31680],[34,538346,131138],{"className":538347,"style":156561},[32767,131249,156560],[34,538349],{"className":538350,"style":32123},[31845],[34,538352,538354,538357],{"className":538353},[31685],[34,538355,111],{"className":538356},[31685,31686],[34,538358,538360],{"className":538359},[31792],[34,538361,538363,538383],{"className":538362},[31796,31797],[34,538364,538366,538380],{"className":538365},[31801],[34,538367,538369],{"className":538368,"style":32090},[31805],[34,538370,538371,538374],{"style":31809},[34,538372],{"className":538373,"style":31814},[31813],[34,538375,538377],{"className":538376},[31818,31819,31820,31821],[34,538378,31955],{"className":538379},[31685,31686,31821],[34,538381,31829],{"className":538382},[31828],[34,538384,538386],{"className":538385},[31801],[34,538387,538389],{"className":538388,"style":31836},[31805],[34,538390],{},[34,538392,538394,538397],{"className":538393},[31685],[34,538395,1697],{"className":538396},[31685,31686],[34,538398,538400],{"className":538399},[31792],[34,538401,538403,538423],{"className":538402},[31796,31797],[34,538404,538406,538420],{"className":538405},[31801],[34,538407,538409],{"className":538408,"style":32090},[31805],[34,538410,538411,538414],{"style":31809},[34,538412],{"className":538413,"style":31814},[31813],[34,538415,538417],{"className":538416},[31818,31819,31820,31821],[34,538418,31955],{"className":538419},[31685,31686,31821],[34,538421,31829],{"className":538422},[31828],[34,538424,538426],{"className":538425},[31801],[34,538427,538429],{"className":538428,"style":31836},[31805],[34,538430],{},[2627,538432,538433],{},"有限和","构成的集合，而不仅仅是简单的 ",[34,538436,538438,538453],{"className":538437},[31642],[34,538439,538441],{"className":538440},[31646],[31648,538442,538443],{"xmlns":31650},[31652,538444,538445,538451],{},[31655,538446,538447,538449],{},[31658,538448,111],{},[31658,538450,1697],{},[31665,538452,324191],{"encoding":31667},[34,538454,538456],{"className":538455,"ariaHidden":758},[31672],[34,538457,538459,538462],{"className":538458},[31676],[34,538460],{"className":538461,"style":31681},[31680],[34,538463,324191],{"className":538464},[31685,31686],[108,538466,538467,538470,538471,538521,538522,86674,538572,347],{},[2627,538468,538469],{},"策略","：证明集合相等 ",[34,538472,538474,538491],{"className":538473},[31642],[34,538475,538477],{"className":538476},[31646],[31648,538478,538479],{"xmlns":31650},[31652,538480,538481,538489],{},[31655,538482,538483,538485,538487],{},[31658,538484,54151],{},[31662,538486,202],{},[31658,538488,54224],{},[31665,538490,513046],{"encoding":31667},[34,538492,538494,538512],{"className":538493,"ariaHidden":758},[31672],[34,538495,538497,538500,538503,538506,538509],{"className":538496},[31676],[34,538498],{"className":538499,"style":33218},[31680],[34,538501,54151],{"className":538502},[31685,31686],[34,538504],{"className":538505,"style":31846},[31845],[34,538507,202],{"className":538508},[31850],[34,538510],{"className":538511,"style":31846},[31845],[34,538513,538515,538518],{"className":538514},[31676],[34,538516],{"className":538517,"style":33218},[31680],[34,538519,54224],{"className":538520,"style":54252},[31685,31686],"，通常用双向包含：",[34,538523,538525,538542],{"className":538524},[31642],[34,538526,538528],{"className":538527},[31646],[31648,538529,538530],{"xmlns":31650},[31652,538531,538532,538540],{},[31655,538533,538534,538536,538538],{},[31658,538535,54151],{},[31662,538537,131855],{},[31658,538539,54224],{},[31665,538541,513102],{"encoding":31667},[34,538543,538545,538563],{"className":538544,"ariaHidden":758},[31672],[34,538546,538548,538551,538554,538557,538560],{"className":538547},[31676],[34,538549],{"className":538550,"style":195047},[31680],[34,538552,54151],{"className":538553},[31685,31686],[34,538555],{"className":538556,"style":31846},[31845],[34,538558,131855],{"className":538559},[31850],[34,538561],{"className":538562,"style":31846},[31845],[34,538564,538566,538569],{"className":538565},[31676],[34,538567],{"className":538568,"style":33218},[31680],[34,538570,54224],{"className":538571,"style":54252},[31685,31686],[34,538573,538575,538592],{"className":538574},[31642],[34,538576,538578],{"className":538577},[31646],[31648,538579,538580],{"xmlns":31650},[31652,538581,538582,538590],{},[31655,538583,538584,538586,538588],{},[31658,538585,54224],{},[31662,538587,131855],{},[31658,538589,54151],{},[31665,538591,513155],{"encoding":31667},[34,538593,538595,538613],{"className":538594,"ariaHidden":758},[31672],[34,538596,538598,538601,538604,538607,538610],{"className":538597},[31676],[34,538599],{"className":538600,"style":195047},[31680],[34,538602,54224],{"className":538603,"style":54252},[31685,31686],[34,538605],{"className":538606,"style":31846},[31845],[34,538608,131855],{"className":538609},[31850],[34,538611],{"className":538612,"style":31846},[31845],[34,538614,538616,538619],{"className":538615},[31676],[34,538617],{"className":538618,"style":33218},[31680],[34,538620,54151],{"className":538621},[31685,31686],[108,538623,538624,538627],{},[2627,538625,538626],{},"幺环条件","：这里主要用来保证理想乘积定义的完备性，或者题目暗示利用单位元，但在分配律的证明中，主要依赖运算本身的定义。",[32490,538629,534075],{"id":538630},"严谨证明-1",[15,538632,538633,538634,538754],{},"我们只证明第一条式子 ",[34,538635,538637,538673],{"className":538636},[31642],[34,538638,538640],{"className":538639},[31646],[31648,538641,538642],{"xmlns":31650},[31652,538643,538644,538670],{},[31655,538645,538646,538648,538650,538652,538654,538656,538658,538660,538662,538664,538666,538668],{},[31662,538647,456],{"stretchy":702},[31658,538649,106063],{},[31662,538651,223],{},[31658,538653,223320],{},[31662,538655,2557],{"stretchy":702},[31658,538657,38691],{},[31662,538659,202],{},[31658,538661,106063],{},[31658,538663,38691],{},[31662,538665,223],{},[31658,538667,223320],{},[31658,538669,38691],{},[31665,538671,538672],{"encoding":31667},"(I + J)K = IK + JK",[34,538674,538676,538697,538721,538742],{"className":538675,"ariaHidden":758},[31672],[34,538677,538679,538682,538685,538688,538691,538694],{"className":538678},[31676],[34,538680],{"className":538681,"style":31774},[31680],[34,538683,456],{"className":538684},[31782],[34,538686,106063],{"className":538687,"style":67482},[31685,31686],[34,538689],{"className":538690,"style":32604},[31845],[34,538692,223],{"className":538693},[32024],[34,538695],{"className":538696,"style":32604},[31845],[34,538698,538700,538703,538706,538709,538712,538715,538718],{"className":538699},[31676],[34,538701],{"className":538702,"style":31774},[31680],[34,538704,223320],{"className":538705,"style":223361},[31685,31686],[34,538707,2557],{"className":538708},[31690],[34,538710,38691],{"className":538711,"style":38706},[31685,31686],[34,538713],{"className":538714,"style":31846},[31845],[34,538716,202],{"className":538717},[31850],[34,538719],{"className":538720,"style":31846},[31845],[34,538722,538724,538727,538730,538733,538736,538739],{"className":538723},[31676],[34,538725],{"className":538726,"style":34944},[31680],[34,538728,106063],{"className":538729,"style":67482},[31685,31686],[34,538731,38691],{"className":538732,"style":38706},[31685,31686],[34,538734],{"className":538735,"style":32604},[31845],[34,538737,223],{"className":538738},[32024],[34,538740],{"className":538741,"style":32604},[31845],[34,538743,538745,538748,538751],{"className":538744},[31676],[34,538746],{"className":538747,"style":33218},[31680],[34,538749,223320],{"className":538750,"style":223361},[31685,31686],[34,538752,38691],{"className":538753,"style":38706},[31685,31686],"。第二条证明完全对称。",[2230,538756,538757,539741],{},[108,538758,538759,538881],{},[2627,538760,379882,538761],{},[34,538762,538764,538800],{"className":538763},[31642],[34,538765,538767],{"className":538766},[31646],[31648,538768,538769],{"xmlns":31650},[31652,538770,538771,538797],{},[31655,538772,538773,538775,538777,538779,538781,538783,538785,538787,538789,538791,538793,538795],{},[31658,538774,106063],{},[31658,538776,38691],{},[31662,538778,223],{},[31658,538780,223320],{},[31658,538782,38691],{},[31662,538784,131855],{},[31662,538786,456],{"stretchy":702},[31658,538788,106063],{},[31662,538790,223],{},[31658,538792,223320],{},[31662,538794,2557],{"stretchy":702},[31658,538796,38691],{},[31665,538798,538799],{"encoding":31667},"IK + JK \\subseteq (I + J)K",[34,538801,538803,538824,538845,538866],{"className":538802,"ariaHidden":758},[31672],[34,538804,538806,538809,538812,538815,538818,538821],{"className":538805},[31676],[34,538807],{"className":538808,"style":34944},[31680],[34,538810,106063],{"className":538811,"style":67482},[31685,31686],[34,538813,38691],{"className":538814,"style":38706},[31685,31686],[34,538816],{"className":538817,"style":32604},[31845],[34,538819,223],{"className":538820},[32024],[34,538822],{"className":538823,"style":32604},[31845],[34,538825,538827,538830,538833,538836,538839,538842],{"className":538826},[31676],[34,538828],{"className":538829,"style":195047},[31680],[34,538831,223320],{"className":538832,"style":223361},[31685,31686],[34,538834,38691],{"className":538835,"style":38706},[31685,31686],[34,538837],{"className":538838,"style":31846},[31845],[34,538840,131855],{"className":538841},[31850],[34,538843],{"className":538844,"style":31846},[31845],[34,538846,538848,538851,538854,538857,538860,538863],{"className":538847},[31676],[34,538849],{"className":538850,"style":31774},[31680],[34,538852,456],{"className":538853},[31782],[34,538855,106063],{"className":538856,"style":67482},[31685,31686],[34,538858],{"className":538859,"style":32604},[31845],[34,538861,223],{"className":538862},[32024],[34,538864],{"className":538865,"style":32604},[31845],[34,538867,538869,538872,538875,538878],{"className":538868},[31676],[34,538870],{"className":538871,"style":31774},[31680],[34,538873,223320],{"className":538874,"style":223361},[31685,31686],[34,538876,2557],{"className":538877},[31690],[34,538879,38691],{"className":538880,"style":38706},[31685,31686],[105,538882,538883,539314,539483,539620],{},[108,538884,93310,538885,13977,538958,539051,539052,539086,539087,539118,539119,44689,539169,337426,539197,539247,539248,328728],{},[34,538886,538888,538910],{"className":538887},[31642],[34,538889,538891],{"className":538890},[31646],[31648,538892,538893],{"xmlns":31650},[31652,538894,538895,538907],{},[31655,538896,538897,538899,538901,538903,538905],{},[31658,538898,106063],{},[31662,538900,131855],{},[31658,538902,106063],{},[31662,538904,223],{},[31658,538906,223320],{},[31665,538908,538909],{"encoding":31667},"I \\subseteq I + J",[34,538911,538913,538931,538949],{"className":538912,"ariaHidden":758},[31672],[34,538914,538916,538919,538922,538925,538928],{"className":538915},[31676],[34,538917],{"className":538918,"style":195047},[31680],[34,538920,106063],{"className":538921,"style":67482},[31685,31686],[34,538923],{"className":538924,"style":31846},[31845],[34,538926,131855],{"className":538927},[31850],[34,538929],{"className":538930,"style":31846},[31845],[34,538932,538934,538937,538940,538943,538946],{"className":538933},[31676],[34,538935],{"className":538936,"style":34944},[31680],[34,538938,106063],{"className":538939,"style":67482},[31685,31686],[34,538941],{"className":538942,"style":32604},[31845],[34,538944,223],{"className":538945},[32024],[34,538947],{"className":538948,"style":32604},[31845],[34,538950,538952,538955],{"className":538951},[31676],[34,538953],{"className":538954,"style":33218},[31680],[34,538956,223320],{"className":538957,"style":223361},[31685,31686],[34,538959,538961,538991],{"className":538960},[31642],[34,538962,538964],{"className":538963},[31646],[31648,538965,538966],{"xmlns":31650},[31652,538967,538968,538988],{},[31655,538969,538970,538972,538974,538976,538978,538980,538982,538984,538986],{},[31658,538971,106063],{},[31658,538973,38691],{},[31662,538975,131855],{},[31662,538977,456],{"stretchy":702},[31658,538979,106063],{},[31662,538981,223],{},[31658,538983,223320],{},[31662,538985,2557],{"stretchy":702},[31658,538987,38691],{},[31665,538989,538990],{"encoding":31667},"IK \\subseteq (I + J)K",[34,538992,538994,539015,539036],{"className":538993,"ariaHidden":758},[31672],[34,538995,538997,539000,539003,539006,539009,539012],{"className":538996},[31676],[34,538998],{"className":538999,"style":195047},[31680],[34,539001,106063],{"className":539002,"style":67482},[31685,31686],[34,539004,38691],{"className":539005,"style":38706},[31685,31686],[34,539007],{"className":539008,"style":31846},[31845],[34,539010,131855],{"className":539011},[31850],[34,539013],{"className":539014,"style":31846},[31845],[34,539016,539018,539021,539024,539027,539030,539033],{"className":539017},[31676],[34,539019],{"className":539020,"style":31774},[31680],[34,539022,456],{"className":539023},[31782],[34,539025,106063],{"className":539026,"style":67482},[31685,31686],[34,539028],{"className":539029,"style":32604},[31845],[34,539031,223],{"className":539032},[32024],[34,539034],{"className":539035,"style":32604},[31845],[34,539037,539039,539042,539045,539048],{"className":539038},[31676],[34,539040],{"className":539041,"style":31774},[31680],[34,539043,223320],{"className":539044,"style":223361},[31685,31686],[34,539046,2557],{"className":539047},[31690],[34,539049,38691],{"className":539050,"style":38706},[31685,31686],"。\n（解释：",[34,539053,539055,539071],{"className":539054},[31642],[34,539056,539058],{"className":539057},[31646],[31648,539059,539060],{"xmlns":31650},[31652,539061,539062,539068],{},[31655,539063,539064,539066],{},[31658,539065,106063],{},[31658,539067,38691],{},[31665,539069,539070],{"encoding":31667},"IK",[34,539072,539074],{"className":539073,"ariaHidden":758},[31672],[34,539075,539077,539080,539083],{"className":539076},[31676],[34,539078],{"className":539079,"style":33218},[31680],[34,539081,106063],{"className":539082,"style":67482},[31685,31686],[34,539084,38691],{"className":539085,"style":38706},[31685,31686]," 中的元素是由 ",[34,539088,539090,539106],{"className":539089},[31642],[34,539091,539093],{"className":539092},[31646],[31648,539094,539095],{"xmlns":31650},[31652,539096,539097,539103],{},[31655,539098,539099,539101],{},[31658,539100,111],{},[31658,539102,33759],{},[31665,539104,539105],{"encoding":31667},"ak",[34,539107,539109],{"className":539108,"ariaHidden":758},[31672],[34,539110,539112,539115],{"className":539111},[31676],[34,539113],{"className":539114,"style":31681},[31680],[34,539116,539105],{"className":539117,"style":33774},[31685,31686]," 组成的和，其中 ",[34,539120,539122,539139],{"className":539121},[31642],[34,539123,539125],{"className":539124},[31646],[31648,539126,539127],{"xmlns":31650},[31652,539128,539129,539137],{},[31655,539130,539131,539133,539135],{},[31658,539132,111],{},[31662,539134,112296],{},[31658,539136,106063],{},[31665,539138,474404],{"encoding":31667},[34,539140,539142,539160],{"className":539141,"ariaHidden":758},[31672],[34,539143,539145,539148,539151,539154,539157],{"className":539144},[31676],[34,539146],{"className":539147,"style":34812},[31680],[34,539149,111],{"className":539150},[31685,31686],[34,539152],{"className":539153,"style":31846},[31845],[34,539155,112296],{"className":539156},[31850],[34,539158],{"className":539159,"style":31846},[31845],[34,539161,539163,539166],{"className":539162},[31676],[34,539164],{"className":539165,"style":33218},[31680],[34,539167,106063],{"className":539168,"style":67482},[31685,31686],[34,539170,539172,539185],{"className":539171},[31642],[34,539173,539175],{"className":539174},[31646],[31648,539176,539177],{"xmlns":31650},[31652,539178,539179,539183],{},[31655,539180,539181],{},[31658,539182,111],{},[31665,539184,111],{"encoding":31667},[34,539186,539188],{"className":539187,"ariaHidden":758},[31672],[34,539189,539191,539194],{"className":539190},[31676],[34,539192],{"className":539193,"style":32527},[31680],[34,539195,111],{"className":539196},[31685,31686],[34,539198,539200,539217],{"className":539199},[31642],[34,539201,539203],{"className":539202},[31646],[31648,539204,539205],{"xmlns":31650},[31652,539206,539207,539215],{},[31655,539208,539209,539211,539213],{},[31658,539210,106063],{},[31662,539212,223],{},[31658,539214,223320],{},[31665,539216,455483],{"encoding":31667},[34,539218,539220,539238],{"className":539219,"ariaHidden":758},[31672],[34,539221,539223,539226,539229,539232,539235],{"className":539222},[31676],[34,539224],{"className":539225,"style":34944},[31680],[34,539227,106063],{"className":539228,"style":67482},[31685,31686],[34,539230],{"className":539231,"style":32604},[31845],[34,539233,223],{"className":539234},[32024],[34,539236],{"className":539237,"style":32604},[31845],[34,539239,539241,539244],{"className":539240},[31676],[34,539242],{"className":539243,"style":33218},[31680],[34,539245,223320],{"className":539246,"style":223361},[31685,31686]," 里，所以这些元素自然也在 ",[34,539249,539251,539275],{"className":539250},[31642],[34,539252,539254],{"className":539253},[31646],[31648,539255,539256],{"xmlns":31650},[31652,539257,539258,539272],{},[31655,539259,539260,539262,539264,539266,539268,539270],{},[31662,539261,456],{"stretchy":702},[31658,539263,106063],{},[31662,539265,223],{},[31658,539267,223320],{},[31662,539269,2557],{"stretchy":702},[31658,539271,38691],{},[31665,539273,539274],{"encoding":31667},"(I+J)K",[34,539276,539278,539299],{"className":539277,"ariaHidden":758},[31672],[34,539279,539281,539284,539287,539290,539293,539296],{"className":539280},[31676],[34,539282],{"className":539283,"style":31774},[31680],[34,539285,456],{"className":539286},[31782],[34,539288,106063],{"className":539289,"style":67482},[31685,31686],[34,539291],{"className":539292,"style":32604},[31845],[34,539294,223],{"className":539295},[32024],[34,539297],{"className":539298,"style":32604},[31845],[34,539300,539302,539305,539308,539311],{"className":539301},[31676],[34,539303],{"className":539304,"style":31774},[31680],[34,539306,223320],{"className":539307,"style":223361},[31685,31686],[34,539309,2557],{"className":539310},[31690],[34,539312,38691],{"className":539313,"style":38706},[31685,31686],[108,539315,539316,539317,13977,539390,347],{},"同理，",[34,539318,539320,539342],{"className":539319},[31642],[34,539321,539323],{"className":539322},[31646],[31648,539324,539325],{"xmlns":31650},[31652,539326,539327,539339],{},[31655,539328,539329,539331,539333,539335,539337],{},[31658,539330,223320],{},[31662,539332,131855],{},[31658,539334,106063],{},[31662,539336,223],{},[31658,539338,223320],{},[31665,539340,539341],{"encoding":31667},"J \\subseteq I + J",[34,539343,539345,539363,539381],{"className":539344,"ariaHidden":758},[31672],[34,539346,539348,539351,539354,539357,539360],{"className":539347},[31676],[34,539349],{"className":539350,"style":195047},[31680],[34,539352,223320],{"className":539353,"style":223361},[31685,31686],[34,539355],{"className":539356,"style":31846},[31845],[34,539358,131855],{"className":539359},[31850],[34,539361],{"className":539362,"style":31846},[31845],[34,539364,539366,539369,539372,539375,539378],{"className":539365},[31676],[34,539367],{"className":539368,"style":34944},[31680],[34,539370,106063],{"className":539371,"style":67482},[31685,31686],[34,539373],{"className":539374,"style":32604},[31845],[34,539376,223],{"className":539377},[32024],[34,539379],{"className":539380,"style":32604},[31845],[34,539382,539384,539387],{"className":539383},[31676],[34,539385],{"className":539386,"style":33218},[31680],[34,539388,223320],{"className":539389,"style":223361},[31685,31686],[34,539391,539393,539423],{"className":539392},[31642],[34,539394,539396],{"className":539395},[31646],[31648,539397,539398],{"xmlns":31650},[31652,539399,539400,539420],{},[31655,539401,539402,539404,539406,539408,539410,539412,539414,539416,539418],{},[31658,539403,223320],{},[31658,539405,38691],{},[31662,539407,131855],{},[31662,539409,456],{"stretchy":702},[31658,539411,106063],{},[31662,539413,223],{},[31658,539415,223320],{},[31662,539417,2557],{"stretchy":702},[31658,539419,38691],{},[31665,539421,539422],{"encoding":31667},"JK \\subseteq (I + J)K",[34,539424,539426,539447,539468],{"className":539425,"ariaHidden":758},[31672],[34,539427,539429,539432,539435,539438,539441,539444],{"className":539428},[31676],[34,539430],{"className":539431,"style":195047},[31680],[34,539433,223320],{"className":539434,"style":223361},[31685,31686],[34,539436,38691],{"className":539437,"style":38706},[31685,31686],[34,539439],{"className":539440,"style":31846},[31845],[34,539442,131855],{"className":539443},[31850],[34,539445],{"className":539446,"style":31846},[31845],[34,539448,539450,539453,539456,539459,539462,539465],{"className":539449},[31676],[34,539451],{"className":539452,"style":31774},[31680],[34,539454,456],{"className":539455},[31782],[34,539457,106063],{"className":539458,"style":67482},[31685,31686],[34,539460],{"className":539461,"style":32604},[31845],[34,539463,223],{"className":539464},[32024],[34,539466],{"className":539467,"style":32604},[31845],[34,539469,539471,539474,539477,539480],{"className":539470},[31676],[34,539472],{"className":539473,"style":31774},[31680],[34,539475,223320],{"className":539476,"style":223361},[31685,31686],[34,539478,2557],{"className":539479},[31690],[34,539481,38691],{"className":539482,"style":38706},[31685,31686],[108,539484,93310,539485,539551,539552,258,539585,539619],{},[34,539486,539488,539512],{"className":539487},[31642],[34,539489,539491],{"className":539490},[31646],[31648,539492,539493],{"xmlns":31650},[31652,539494,539495,539509],{},[31655,539496,539497,539499,539501,539503,539505,539507],{},[31662,539498,456],{"stretchy":702},[31658,539500,106063],{},[31662,539502,223],{},[31658,539504,223320],{},[31662,539506,2557],{"stretchy":702},[31658,539508,38691],{},[31665,539510,539511],{"encoding":31667},"(I + J)K",[34,539513,539515,539536],{"className":539514,"ariaHidden":758},[31672],[34,539516,539518,539521,539524,539527,539530,539533],{"className":539517},[31676],[34,539519],{"className":539520,"style":31774},[31680],[34,539522,456],{"className":539523},[31782],[34,539525,106063],{"className":539526,"style":67482},[31685,31686],[34,539528],{"className":539529,"style":32604},[31845],[34,539531,223],{"className":539532},[32024],[34,539534],{"className":539535,"style":32604},[31845],[34,539537,539539,539542,539545,539548],{"className":539538},[31676],[34,539540],{"className":539541,"style":31774},[31680],[34,539543,223320],{"className":539544,"style":223361},[31685,31686],[34,539546,2557],{"className":539547},[31690],[34,539549,38691],{"className":539550,"style":38706},[31685,31686]," 是一个理想（理想对加法封闭），它包含了 ",[34,539553,539555,539570],{"className":539554},[31642],[34,539556,539558],{"className":539557},[31646],[31648,539559,539560],{"xmlns":31650},[31652,539561,539562,539568],{},[31655,539563,539564,539566],{},[31658,539565,106063],{},[31658,539567,38691],{},[31665,539569,539070],{"encoding":31667},[34,539571,539573],{"className":539572,"ariaHidden":758},[31672],[34,539574,539576,539579,539582],{"className":539575},[31676],[34,539577],{"className":539578,"style":33218},[31680],[34,539580,106063],{"className":539581,"style":67482},[31685,31686],[34,539583,38691],{"className":539584,"style":38706},[31685,31686],[34,539586,539588,539604],{"className":539587},[31642],[34,539589,539591],{"className":539590},[31646],[31648,539592,539593],{"xmlns":31650},[31652,539594,539595,539601],{},[31655,539596,539597,539599],{},[31658,539598,223320],{},[31658,539600,38691],{},[31665,539602,539603],{"encoding":31667},"JK",[34,539605,539607],{"className":539606,"ariaHidden":758},[31672],[34,539608,539610,539613,539616],{"className":539609},[31676],[34,539611],{"className":539612,"style":33218},[31680],[34,539614,223320],{"className":539615,"style":223361},[31685,31686],[34,539617,38691],{"className":539618,"style":38706},[31685,31686],"，那么它必然包含它们的和。",[108,539621,159962,539622,347],{},[34,539623,539625,539660],{"className":539624},[31642],[34,539626,539628],{"className":539627},[31646],[31648,539629,539630],{"xmlns":31650},[31652,539631,539632,539658],{},[31655,539633,539634,539636,539638,539640,539642,539644,539646,539648,539650,539652,539654,539656],{},[31658,539635,106063],{},[31658,539637,38691],{},[31662,539639,223],{},[31658,539641,223320],{},[31658,539643,38691],{},[31662,539645,131855],{},[31662,539647,456],{"stretchy":702},[31658,539649,106063],{},[31662,539651,223],{},[31658,539653,223320],{},[31662,539655,2557],{"stretchy":702},[31658,539657,38691],{},[31665,539659,538799],{"encoding":31667},[34,539661,539663,539684,539705,539726],{"className":539662,"ariaHidden":758},[31672],[34,539664,539666,539669,539672,539675,539678,539681],{"className":539665},[31676],[34,539667],{"className":539668,"style":34944},[31680],[34,539670,106063],{"className":539671,"style":67482},[31685,31686],[34,539673,38691],{"className":539674,"style":38706},[31685,31686],[34,539676],{"className":539677,"style":32604},[31845],[34,539679,223],{"className":539680},[32024],[34,539682],{"className":539683,"style":32604},[31845],[34,539685,539687,539690,539693,539696,539699,539702],{"className":539686},[31676],[34,539688],{"className":539689,"style":195047},[31680],[34,539691,223320],{"className":539692,"style":223361},[31685,31686],[34,539694,38691],{"className":539695,"style":38706},[31685,31686],[34,539697],{"className":539698,"style":31846},[31845],[34,539700,131855],{"className":539701},[31850],[34,539703],{"className":539704,"style":31846},[31845],[34,539706,539708,539711,539714,539717,539720,539723],{"className":539707},[31676],[34,539709],{"className":539710,"style":31774},[31680],[34,539712,456],{"className":539713},[31782],[34,539715,106063],{"className":539716,"style":67482},[31685,31686],[34,539718],{"className":539719,"style":32604},[31845],[34,539721,223],{"className":539722},[32024],[34,539724],{"className":539725,"style":32604},[31845],[34,539727,539729,539732,539735,539738],{"className":539728},[31676],[34,539730],{"className":539731,"style":31774},[31680],[34,539733,223320],{"className":539734,"style":223361},[31685,31686],[34,539736,2557],{"className":539737},[31690],[34,539739,38691],{"className":539740,"style":38706},[31685,31686],[108,539742,539743,539865],{},[2627,539744,379882,539745],{},[34,539746,539748,539784],{"className":539747},[31642],[34,539749,539751],{"className":539750},[31646],[31648,539752,539753],{"xmlns":31650},[31652,539754,539755,539781],{},[31655,539756,539757,539759,539761,539763,539765,539767,539769,539771,539773,539775,539777,539779],{},[31662,539758,456],{"stretchy":702},[31658,539760,106063],{},[31662,539762,223],{},[31658,539764,223320],{},[31662,539766,2557],{"stretchy":702},[31658,539768,38691],{},[31662,539770,131855],{},[31658,539772,106063],{},[31658,539774,38691],{},[31662,539776,223],{},[31658,539778,223320],{},[31658,539780,38691],{},[31665,539782,539783],{"encoding":31667},"(I + J)K \\subseteq IK + JK",[34,539785,539787,539808,539832,539853],{"className":539786,"ariaHidden":758},[31672],[34,539788,539790,539793,539796,539799,539802,539805],{"className":539789},[31676],[34,539791],{"className":539792,"style":31774},[31680],[34,539794,456],{"className":539795},[31782],[34,539797,106063],{"className":539798,"style":67482},[31685,31686],[34,539800],{"className":539801,"style":32604},[31845],[34,539803,223],{"className":539804},[32024],[34,539806],{"className":539807,"style":32604},[31845],[34,539809,539811,539814,539817,539820,539823,539826,539829],{"className":539810},[31676],[34,539812],{"className":539813,"style":31774},[31680],[34,539815,223320],{"className":539816,"style":223361},[31685,31686],[34,539818,2557],{"className":539819},[31690],[34,539821,38691],{"className":539822,"style":38706},[31685,31686],[34,539824],{"className":539825,"style":31846},[31845],[34,539827,131855],{"className":539828},[31850],[34,539830],{"className":539831,"style":31846},[31845],[34,539833,539835,539838,539841,539844,539847,539850],{"className":539834},[31676],[34,539836],{"className":539837,"style":34944},[31680],[34,539839,106063],{"className":539840,"style":67482},[31685,31686],[34,539842,38691],{"className":539843,"style":38706},[31685,31686],[34,539845],{"className":539846,"style":32604},[31845],[34,539848,223],{"className":539849},[32024],[34,539851],{"className":539852,"style":32604},[31845],[34,539854,539856,539859,539862],{"className":539855},[31676],[34,539857],{"className":539858,"style":33218},[31680],[34,539860,223320],{"className":539861,"style":223361},[31685,31686],[34,539863,38691],{"className":539864,"style":38706},[31685,31686],[105,539866,539867,540123,540241,540598,540768,541020],{},[108,539868,411601,539869,539934,539935,540001,540002,347],{},[34,539870,539872,539895],{"className":539871},[31642],[34,539873,539875],{"className":539874},[31646],[31648,539876,539877],{"xmlns":31650},[31652,539878,539879,539893],{},[31655,539880,539881,539883,539885,539887,539889,539891],{},[31662,539882,456],{"stretchy":702},[31658,539884,106063],{},[31662,539886,223],{},[31658,539888,223320],{},[31662,539890,2557],{"stretchy":702},[31658,539892,38691],{},[31665,539894,539511],{"encoding":31667},[34,539896,539898,539919],{"className":539897,"ariaHidden":758},[31672],[34,539899,539901,539904,539907,539910,539913,539916],{"className":539900},[31676],[34,539902],{"className":539903,"style":31774},[31680],[34,539905,456],{"className":539906},[31782],[34,539908,106063],{"className":539909,"style":67482},[31685,31686],[34,539911],{"className":539912,"style":32604},[31845],[34,539914,223],{"className":539915},[32024],[34,539917],{"className":539918,"style":32604},[31845],[34,539920,539922,539925,539928,539931],{"className":539921},[31676],[34,539923],{"className":539924,"style":31774},[31680],[34,539926,223320],{"className":539927,"style":223361},[31685,31686],[34,539929,2557],{"className":539930},[31690],[34,539932,38691],{"className":539933,"style":38706},[31685,31686]," 中的任意生成元。该集合由形如 ",[34,539936,539938,539962],{"className":539937},[31642],[34,539939,539941],{"className":539940},[31646],[31648,539942,539943],{"xmlns":31650},[31652,539944,539945,539959],{},[31655,539946,539947,539949,539951,539953,539955,539957],{},[31662,539948,456],{"stretchy":702},[31658,539950,111],{},[31662,539952,223],{},[31658,539954,1697],{},[31662,539956,2557],{"stretchy":702},[31658,539958,33759],{},[31665,539960,539961],{"encoding":31667},"(a + b)k",[34,539963,539965,539986],{"className":539964,"ariaHidden":758},[31672],[34,539966,539968,539971,539974,539977,539980,539983],{"className":539967},[31676],[34,539969],{"className":539970,"style":31774},[31680],[34,539972,456],{"className":539973},[31782],[34,539975,111],{"className":539976},[31685,31686],[34,539978],{"className":539979,"style":32604},[31845],[34,539981,223],{"className":539982},[32024],[34,539984],{"className":539985,"style":32604},[31845],[34,539987,539989,539992,539995,539998],{"className":539988},[31676],[34,539990],{"className":539991,"style":31774},[31680],[34,539993,1697],{"className":539994},[31685,31686],[34,539996,2557],{"className":539997},[31690],[34,539999,33759],{"className":540000,"style":33774},[31685,31686]," 的元素的有限和组成，其中 ",[34,540003,540005,540039],{"className":540004},[31642],[34,540006,540008],{"className":540007},[31646],[31648,540009,540010],{"xmlns":31650},[31652,540011,540012,540036],{},[31655,540013,540014,540016,540018,540020,540022,540024,540026,540028,540030,540032,540034],{},[31658,540015,111],{},[31662,540017,112296],{},[31658,540019,106063],{},[31662,540021,31960],{"separator":758},[31658,540023,1697],{},[31662,540025,112296],{},[31658,540027,223320],{},[31662,540029,31960],{"separator":758},[31658,540031,33759],{},[31662,540033,112296],{},[31658,540035,38691],{},[31665,540037,540038],{"encoding":31667},"a \\in I, b \\in J, k \\in K",[34,540040,540042,540060,540087,540114],{"className":540041,"ariaHidden":758},[31672],[34,540043,540045,540048,540051,540054,540057],{"className":540044},[31676],[34,540046],{"className":540047,"style":34812},[31680],[34,540049,111],{"className":540050},[31685,31686],[34,540052],{"className":540053,"style":31846},[31845],[34,540055,112296],{"className":540056},[31850],[34,540058],{"className":540059,"style":31846},[31845],[34,540061,540063,540066,540069,540072,540075,540078,540081,540084],{"className":540062},[31676],[34,540064],{"className":540065,"style":43217},[31680],[34,540067,106063],{"className":540068,"style":67482},[31685,31686],[34,540070,31960],{"className":540071},[32119],[34,540073],{"className":540074,"style":32123},[31845],[34,540076,1697],{"className":540077},[31685,31686],[34,540079],{"className":540080,"style":31846},[31845],[34,540082,112296],{"className":540083},[31850],[34,540085],{"className":540086,"style":31846},[31845],[34,540088,540090,540093,540096,540099,540102,540105,540108,540111],{"className":540089},[31676],[34,540091],{"className":540092,"style":43217},[31680],[34,540094,223320],{"className":540095,"style":223361},[31685,31686],[34,540097,31960],{"className":540098},[32119],[34,540100],{"className":540101,"style":32123},[31845],[34,540103,33759],{"className":540104,"style":33774},[31685,31686],[34,540106],{"className":540107,"style":31846},[31845],[34,540109,112296],{"className":540110},[31850],[34,540112],{"className":540113,"style":31846},[31845],[34,540115,540117,540120],{"className":540116},[31676],[34,540118],{"className":540119,"style":33218},[31680],[34,540121,38691],{"className":540122,"style":38706},[31685,31686],[108,540124,540125,540126],{},"根据环的乘法对加法的分配律：",[34,540127,540129,540165],{"className":540128},[31642],[34,540130,540132],{"className":540131},[31646],[31648,540133,540134],{"xmlns":31650},[31652,540135,540136,540162],{},[31655,540137,540138,540140,540142,540144,540146,540148,540150,540152,540154,540156,540158,540160],{},[31662,540139,456],{"stretchy":702},[31658,540141,111],{},[31662,540143,223],{},[31658,540145,1697],{},[31662,540147,2557],{"stretchy":702},[31658,540149,33759],{},[31662,540151,202],{},[31658,540153,111],{},[31658,540155,33759],{},[31662,540157,223],{},[31658,540159,1697],{},[31658,540161,33759],{},[31665,540163,540164],{"encoding":31667},"(a + b)k = ak + bk",[34,540166,540168,540189,540213,540231],{"className":540167,"ariaHidden":758},[31672],[34,540169,540171,540174,540177,540180,540183,540186],{"className":540170},[31676],[34,540172],{"className":540173,"style":31774},[31680],[34,540175,456],{"className":540176},[31782],[34,540178,111],{"className":540179},[31685,31686],[34,540181],{"className":540182,"style":32604},[31845],[34,540184,223],{"className":540185},[32024],[34,540187],{"className":540188,"style":32604},[31845],[34,540190,540192,540195,540198,540201,540204,540207,540210],{"className":540191},[31676],[34,540193],{"className":540194,"style":31774},[31680],[34,540196,1697],{"className":540197},[31685,31686],[34,540199,2557],{"className":540200},[31690],[34,540202,33759],{"className":540203,"style":33774},[31685,31686],[34,540205],{"className":540206,"style":31846},[31845],[34,540208,202],{"className":540209},[31850],[34,540211],{"className":540212,"style":31846},[31845],[34,540214,540216,540219,540222,540225,540228],{"className":540215},[31676],[34,540217],{"className":540218,"style":34299},[31680],[34,540220,539105],{"className":540221,"style":33774},[31685,31686],[34,540223],{"className":540224,"style":32604},[31845],[34,540226,223],{"className":540227},[32024],[34,540229],{"className":540230,"style":32604},[31845],[34,540232,540234,540237],{"className":540233},[31676],[34,540235],{"className":540236,"style":31681},[31680],[34,540238,540240],{"className":540239,"style":33774},[31685,31686],"bk",[108,540242,540243,540244],{},"观察各项：\n",[105,540245,540246,540422],{},[108,540247,540248,332541,540278,13977,540364,347],{},[34,540249,540251,540266],{"className":540250},[31642],[34,540252,540254],{"className":540253},[31646],[31648,540255,540256],{"xmlns":31650},[31652,540257,540258,540264],{},[31655,540259,540260,540262],{},[31658,540261,111],{},[31658,540263,33759],{},[31665,540265,539105],{"encoding":31667},[34,540267,540269],{"className":540268,"ariaHidden":758},[31672],[34,540270,540272,540275],{"className":540271},[31676],[34,540273],{"className":540274,"style":31681},[31680],[34,540276,539105],{"className":540277,"style":33774},[31685,31686],[34,540279,540281,540307],{"className":540280},[31642],[34,540282,540284],{"className":540283},[31646],[31648,540285,540286],{"xmlns":31650},[31652,540287,540288,540304],{},[31655,540289,540290,540292,540294,540296,540298,540300,540302],{},[31658,540291,111],{},[31662,540293,112296],{},[31658,540295,106063],{},[31662,540297,31960],{"separator":758},[31658,540299,33759],{},[31662,540301,112296],{},[31658,540303,38691],{},[31665,540305,540306],{"encoding":31667},"a \\in I, k \\in K",[34,540308,540310,540328,540355],{"className":540309,"ariaHidden":758},[31672],[34,540311,540313,540316,540319,540322,540325],{"className":540312},[31676],[34,540314],{"className":540315,"style":34812},[31680],[34,540317,111],{"className":540318},[31685,31686],[34,540320],{"className":540321,"style":31846},[31845],[34,540323,112296],{"className":540324},[31850],[34,540326],{"className":540327,"style":31846},[31845],[34,540329,540331,540334,540337,540340,540343,540346,540349,540352],{"className":540330},[31676],[34,540332],{"className":540333,"style":43217},[31680],[34,540335,106063],{"className":540336,"style":67482},[31685,31686],[34,540338,31960],{"className":540339},[32119],[34,540341],{"className":540342,"style":32123},[31845],[34,540344,33759],{"className":540345,"style":33774},[31685,31686],[34,540347],{"className":540348,"style":31846},[31845],[34,540350,112296],{"className":540351},[31850],[34,540353],{"className":540354,"style":31846},[31845],[34,540356,540358,540361],{"className":540357},[31676],[34,540359],{"className":540360,"style":33218},[31680],[34,540362,38691],{"className":540363,"style":38706},[31685,31686],[34,540365,540367,540389],{"className":540366},[31642],[34,540368,540370],{"className":540369},[31646],[31648,540371,540372],{"xmlns":31650},[31652,540373,540374,540386],{},[31655,540375,540376,540378,540380,540382,540384],{},[31658,540377,111],{},[31658,540379,33759],{},[31662,540381,112296],{},[31658,540383,106063],{},[31658,540385,38691],{},[31665,540387,540388],{"encoding":31667},"ak \\in IK",[34,540390,540392,540410],{"className":540391,"ariaHidden":758},[31672],[34,540393,540395,540398,540401,540404,540407],{"className":540394},[31676],[34,540396],{"className":540397,"style":46991},[31680],[34,540399,539105],{"className":540400,"style":33774},[31685,31686],[34,540402],{"className":540403,"style":31846},[31845],[34,540405,112296],{"className":540406},[31850],[34,540408],{"className":540409,"style":31846},[31845],[34,540411,540413,540416,540419],{"className":540412},[31676],[34,540414],{"className":540415,"style":33218},[31680],[34,540417,106063],{"className":540418,"style":67482},[31685,31686],[34,540420,38691],{"className":540421,"style":38706},[31685,31686],[108,540423,540424,332541,540454,13977,540540,347],{},[34,540425,540427,540442],{"className":540426},[31642],[34,540428,540430],{"className":540429},[31646],[31648,540431,540432],{"xmlns":31650},[31652,540433,540434,540440],{},[31655,540435,540436,540438],{},[31658,540437,1697],{},[31658,540439,33759],{},[31665,540441,540240],{"encoding":31667},[34,540443,540445],{"className":540444,"ariaHidden":758},[31672],[34,540446,540448,540451],{"className":540447},[31676],[34,540449],{"className":540450,"style":31681},[31680],[34,540452,540240],{"className":540453,"style":33774},[31685,31686],[34,540455,540457,540483],{"className":540456},[31642],[34,540458,540460],{"className":540459},[31646],[31648,540461,540462],{"xmlns":31650},[31652,540463,540464,540480],{},[31655,540465,540466,540468,540470,540472,540474,540476,540478],{},[31658,540467,1697],{},[31662,540469,112296],{},[31658,540471,223320],{},[31662,540473,31960],{"separator":758},[31658,540475,33759],{},[31662,540477,112296],{},[31658,540479,38691],{},[31665,540481,540482],{"encoding":31667},"b \\in J, k \\in K",[34,540484,540486,540504,540531],{"className":540485,"ariaHidden":758},[31672],[34,540487,540489,540492,540495,540498,540501],{"className":540488},[31676],[34,540490],{"className":540491,"style":46991},[31680],[34,540493,1697],{"className":540494},[31685,31686],[34,540496],{"className":540497,"style":31846},[31845],[34,540499,112296],{"className":540500},[31850],[34,540502],{"className":540503,"style":31846},[31845],[34,540505,540507,540510,540513,540516,540519,540522,540525,540528],{"className":540506},[31676],[34,540508],{"className":540509,"style":43217},[31680],[34,540511,223320],{"className":540512,"style":223361},[31685,31686],[34,540514,31960],{"className":540515},[32119],[34,540517],{"className":540518,"style":32123},[31845],[34,540520,33759],{"className":540521,"style":33774},[31685,31686],[34,540523],{"className":540524,"style":31846},[31845],[34,540526,112296],{"className":540527},[31850],[34,540529],{"className":540530,"style":31846},[31845],[34,540532,540534,540537],{"className":540533},[31676],[34,540535],{"className":540536,"style":33218},[31680],[34,540538,38691],{"className":540539,"style":38706},[31685,31686],[34,540541,540543,540565],{"className":540542},[31642],[34,540544,540546],{"className":540545},[31646],[31648,540547,540548],{"xmlns":31650},[31652,540549,540550,540562],{},[31655,540551,540552,540554,540556,540558,540560],{},[31658,540553,1697],{},[31658,540555,33759],{},[31662,540557,112296],{},[31658,540559,223320],{},[31658,540561,38691],{},[31665,540563,540564],{"encoding":31667},"bk \\in JK",[34,540566,540568,540586],{"className":540567,"ariaHidden":758},[31672],[34,540569,540571,540574,540577,540580,540583],{"className":540570},[31676],[34,540572],{"className":540573,"style":46991},[31680],[34,540575,540240],{"className":540576,"style":33774},[31685,31686],[34,540578],{"className":540579,"style":31846},[31845],[34,540581,112296],{"className":540582},[31850],[34,540584],{"className":540585,"style":31846},[31845],[34,540587,540589,540592,540595],{"className":540588},[31676],[34,540590],{"className":540591,"style":33218},[31680],[34,540593,223320],{"className":540594,"style":223361},[31685,31686],[34,540596,38691],{"className":540597,"style":38706},[31685,31686],[108,540599,435583,540600,347],{},[34,540601,540603,540651],{"className":540602},[31642],[34,540604,540606],{"className":540605},[31646],[31648,540607,540608],{"xmlns":31650},[31652,540609,540610,540648],{},[31655,540611,540612,540614,540616,540618,540620,540622,540624,540626,540628,540630,540632,540634,540636,540638,540640,540642,540644,540646],{},[31662,540613,456],{"stretchy":702},[31658,540615,111],{},[31662,540617,223],{},[31658,540619,1697],{},[31662,540621,2557],{"stretchy":702},[31658,540623,33759],{},[31662,540625,202],{},[31658,540627,111],{},[31658,540629,33759],{},[31662,540631,223],{},[31658,540633,1697],{},[31658,540635,33759],{},[31662,540637,112296],{},[31658,540639,106063],{},[31658,540641,38691],{},[31662,540643,223],{},[31658,540645,223320],{},[31658,540647,38691],{},[31665,540649,540650],{"encoding":31667},"(a + b)k = ak + bk \\in IK + JK",[34,540652,540654,540675,540699,540717,540735,540756],{"className":540653,"ariaHidden":758},[31672],[34,540655,540657,540660,540663,540666,540669,540672],{"className":540656},[31676],[34,540658],{"className":540659,"style":31774},[31680],[34,540661,456],{"className":540662},[31782],[34,540664,111],{"className":540665},[31685,31686],[34,540667],{"className":540668,"style":32604},[31845],[34,540670,223],{"className":540671},[32024],[34,540673],{"className":540674,"style":32604},[31845],[34,540676,540678,540681,540684,540687,540690,540693,540696],{"className":540677},[31676],[34,540679],{"className":540680,"style":31774},[31680],[34,540682,1697],{"className":540683},[31685,31686],[34,540685,2557],{"className":540686},[31690],[34,540688,33759],{"className":540689,"style":33774},[31685,31686],[34,540691],{"className":540692,"style":31846},[31845],[34,540694,202],{"className":540695},[31850],[34,540697],{"className":540698,"style":31846},[31845],[34,540700,540702,540705,540708,540711,540714],{"className":540701},[31676],[34,540703],{"className":540704,"style":34299},[31680],[34,540706,539105],{"className":540707,"style":33774},[31685,31686],[34,540709],{"className":540710,"style":32604},[31845],[34,540712,223],{"className":540713},[32024],[34,540715],{"className":540716,"style":32604},[31845],[34,540718,540720,540723,540726,540729,540732],{"className":540719},[31676],[34,540721],{"className":540722,"style":46991},[31680],[34,540724,540240],{"className":540725,"style":33774},[31685,31686],[34,540727],{"className":540728,"style":31846},[31845],[34,540730,112296],{"className":540731},[31850],[34,540733],{"className":540734,"style":31846},[31845],[34,540736,540738,540741,540744,540747,540750,540753],{"className":540737},[31676],[34,540739],{"className":540740,"style":34944},[31680],[34,540742,106063],{"className":540743,"style":67482},[31685,31686],[34,540745,38691],{"className":540746,"style":38706},[31685,31686],[34,540748],{"className":540749,"style":32604},[31845],[34,540751,223],{"className":540752},[32024],[34,540754],{"className":540755,"style":32604},[31845],[34,540757,540759,540762,540765],{"className":540758},[31676],[34,540760],{"className":540761,"style":33218},[31680],[34,540763,223320],{"className":540764,"style":223361},[31685,31686],[34,540766,38691],{"className":540767,"style":38706},[31685,31686],[108,540769,540770,540771,540832,540833,540893,540894,540959,540960,337455],{},"既然生成元都在 ",[34,540772,540774,540796],{"className":540773},[31642],[34,540775,540777],{"className":540776},[31646],[31648,540778,540779],{"xmlns":31650},[31652,540780,540781,540793],{},[31655,540782,540783,540785,540787,540789,540791],{},[31658,540784,106063],{},[31658,540786,38691],{},[31662,540788,223],{},[31658,540790,223320],{},[31658,540792,38691],{},[31665,540794,540795],{"encoding":31667},"IK + JK",[34,540797,540799,540820],{"className":540798,"ariaHidden":758},[31672],[34,540800,540802,540805,540808,540811,540814,540817],{"className":540801},[31676],[34,540803],{"className":540804,"style":34944},[31680],[34,540806,106063],{"className":540807,"style":67482},[31685,31686],[34,540809,38691],{"className":540810,"style":38706},[31685,31686],[34,540812],{"className":540813,"style":32604},[31845],[34,540815,223],{"className":540816},[32024],[34,540818],{"className":540819,"style":32604},[31845],[34,540821,540823,540826,540829],{"className":540822},[31676],[34,540824],{"className":540825,"style":33218},[31680],[34,540827,223320],{"className":540828,"style":223361},[31685,31686],[34,540830,38691],{"className":540831,"style":38706},[31685,31686]," 里，且 ",[34,540834,540836,540857],{"className":540835},[31642],[34,540837,540839],{"className":540838},[31646],[31648,540840,540841],{"xmlns":31650},[31652,540842,540843,540855],{},[31655,540844,540845,540847,540849,540851,540853],{},[31658,540846,106063],{},[31658,540848,38691],{},[31662,540850,223],{},[31658,540852,223320],{},[31658,540854,38691],{},[31665,540856,540795],{"encoding":31667},[34,540858,540860,540881],{"className":540859,"ariaHidden":758},[31672],[34,540861,540863,540866,540869,540872,540875,540878],{"className":540862},[31676],[34,540864],{"className":540865,"style":34944},[31680],[34,540867,106063],{"className":540868,"style":67482},[31685,31686],[34,540870,38691],{"className":540871,"style":38706},[31685,31686],[34,540873],{"className":540874,"style":32604},[31845],[34,540876,223],{"className":540877},[32024],[34,540879],{"className":540880,"style":32604},[31845],[34,540882,540884,540887,540890],{"className":540883},[31676],[34,540885],{"className":540886,"style":33218},[31680],[34,540888,223320],{"className":540889,"style":223361},[31685,31686],[34,540891,38691],{"className":540892,"style":38706},[31685,31686]," 对加法封闭，那么 ",[34,540895,540897,540920],{"className":540896},[31642],[34,540898,540900],{"className":540899},[31646],[31648,540901,540902],{"xmlns":31650},[31652,540903,540904,540918],{},[31655,540905,540906,540908,540910,540912,540914,540916],{},[31662,540907,456],{"stretchy":702},[31658,540909,106063],{},[31662,540911,223],{},[31658,540913,223320],{},[31662,540915,2557],{"stretchy":702},[31658,540917,38691],{},[31665,540919,539511],{"encoding":31667},[34,540921,540923,540944],{"className":540922,"ariaHidden":758},[31672],[34,540924,540926,540929,540932,540935,540938,540941],{"className":540925},[31676],[34,540927],{"className":540928,"style":31774},[31680],[34,540930,456],{"className":540931},[31782],[34,540933,106063],{"className":540934,"style":67482},[31685,31686],[34,540936],{"className":540937,"style":32604},[31845],[34,540939,223],{"className":540940},[32024],[34,540942],{"className":540943,"style":32604},[31845],[34,540945,540947,540950,540953,540956],{"className":540946},[31676],[34,540948],{"className":540949,"style":31774},[31680],[34,540951,223320],{"className":540952,"style":223361},[31685,31686],[34,540954,2557],{"className":540955},[31690],[34,540957,38691],{"className":540958,"style":38706},[31685,31686]," 中的所有元素都在 ",[34,540961,540963,540984],{"className":540962},[31642],[34,540964,540966],{"className":540965},[31646],[31648,540967,540968],{"xmlns":31650},[31652,540969,540970,540982],{},[31655,540971,540972,540974,540976,540978,540980],{},[31658,540973,106063],{},[31658,540975,38691],{},[31662,540977,223],{},[31658,540979,223320],{},[31658,540981,38691],{},[31665,540983,540795],{"encoding":31667},[34,540985,540987,541008],{"className":540986,"ariaHidden":758},[31672],[34,540988,540990,540993,540996,540999,541002,541005],{"className":540989},[31676],[34,540991],{"className":540992,"style":34944},[31680],[34,540994,106063],{"className":540995,"style":67482},[31685,31686],[34,540997,38691],{"className":540998,"style":38706},[31685,31686],[34,541000],{"className":541001,"style":32604},[31845],[34,541003,223],{"className":541004},[32024],[34,541006],{"className":541007,"style":32604},[31845],[34,541009,541011,541014,541017],{"className":541010},[31676],[34,541012],{"className":541013,"style":33218},[31680],[34,541015,223320],{"className":541016,"style":223361},[31685,31686],[34,541018,38691],{"className":541019,"style":38706},[31685,31686],[108,541021,159962,541022,347],{},[34,541023,541025,541060],{"className":541024},[31642],[34,541026,541028],{"className":541027},[31646],[31648,541029,541030],{"xmlns":31650},[31652,541031,541032,541058],{},[31655,541033,541034,541036,541038,541040,541042,541044,541046,541048,541050,541052,541054,541056],{},[31662,541035,456],{"stretchy":702},[31658,541037,106063],{},[31662,541039,223],{},[31658,541041,223320],{},[31662,541043,2557],{"stretchy":702},[31658,541045,38691],{},[31662,541047,131855],{},[31658,541049,106063],{},[31658,541051,38691],{},[31662,541053,223],{},[31658,541055,223320],{},[31658,541057,38691],{},[31665,541059,539783],{"encoding":31667},[34,541061,541063,541084,541108,541129],{"className":541062,"ariaHidden":758},[31672],[34,541064,541066,541069,541072,541075,541078,541081],{"className":541065},[31676],[34,541067],{"className":541068,"style":31774},[31680],[34,541070,456],{"className":541071},[31782],[34,541073,106063],{"className":541074,"style":67482},[31685,31686],[34,541076],{"className":541077,"style":32604},[31845],[34,541079,223],{"className":541080},[32024],[34,541082],{"className":541083,"style":32604},[31845],[34,541085,541087,541090,541093,541096,541099,541102,541105],{"className":541086},[31676],[34,541088],{"className":541089,"style":31774},[31680],[34,541091,223320],{"className":541092,"style":223361},[31685,31686],[34,541094,2557],{"className":541095},[31690],[34,541097,38691],{"className":541098,"style":38706},[31685,31686],[34,541100],{"className":541101,"style":31846},[31845],[34,541103,131855],{"className":541104},[31850],[34,541106],{"className":541107,"style":31846},[31845],[34,541109,541111,541114,541117,541120,541123,541126],{"className":541110},[31676],[34,541112],{"className":541113,"style":34944},[31680],[34,541115,106063],{"className":541116,"style":67482},[31685,31686],[34,541118,38691],{"className":541119,"style":38706},[31685,31686],[34,541121],{"className":541122,"style":32604},[31845],[34,541124,223],{"className":541125},[32024],[34,541127],{"className":541128,"style":32604},[31845],[34,541130,541132,541135,541138],{"className":541131},[31676],[34,541133],{"className":541134,"style":33218},[31680],[34,541136,223320],{"className":541137,"style":223361},[31685,31686],[34,541139,38691],{"className":541140,"style":38706},[31685,31686],[15,541142,541143,541145,541146,541265,541266],{},[2627,541144,114043],{},"：由 Step 1 和 Step 2 可知，",[34,541147,541149,541184],{"className":541148},[31642],[34,541150,541152],{"className":541151},[31646],[31648,541153,541154],{"xmlns":31650},[31652,541155,541156,541182],{},[31655,541157,541158,541160,541162,541164,541166,541168,541170,541172,541174,541176,541178,541180],{},[31662,541159,456],{"stretchy":702},[31658,541161,106063],{},[31662,541163,223],{},[31658,541165,223320],{},[31662,541167,2557],{"stretchy":702},[31658,541169,38691],{},[31662,541171,202],{},[31658,541173,106063],{},[31658,541175,38691],{},[31662,541177,223],{},[31658,541179,223320],{},[31658,541181,38691],{},[31665,541183,538672],{"encoding":31667},[34,541185,541187,541208,541232,541253],{"className":541186,"ariaHidden":758},[31672],[34,541188,541190,541193,541196,541199,541202,541205],{"className":541189},[31676],[34,541191],{"className":541192,"style":31774},[31680],[34,541194,456],{"className":541195},[31782],[34,541197,106063],{"className":541198,"style":67482},[31685,31686],[34,541200],{"className":541201,"style":32604},[31845],[34,541203,223],{"className":541204},[32024],[34,541206],{"className":541207,"style":32604},[31845],[34,541209,541211,541214,541217,541220,541223,541226,541229],{"className":541210},[31676],[34,541212],{"className":541213,"style":31774},[31680],[34,541215,223320],{"className":541216,"style":223361},[31685,31686],[34,541218,2557],{"className":541219},[31690],[34,541221,38691],{"className":541222,"style":38706},[31685,31686],[34,541224],{"className":541225,"style":31846},[31845],[34,541227,202],{"className":541228},[31850],[34,541230],{"className":541231,"style":31846},[31845],[34,541233,541235,541238,541241,541244,541247,541250],{"className":541234},[31676],[34,541236],{"className":541237,"style":34944},[31680],[34,541239,106063],{"className":541240,"style":67482},[31685,31686],[34,541242,38691],{"className":541243,"style":38706},[31685,31686],[34,541245],{"className":541246,"style":32604},[31845],[34,541248,223],{"className":541249},[32024],[34,541251],{"className":541252,"style":32604},[31845],[34,541254,541256,541259,541262],{"className":541255},[31676],[34,541257],{"className":541258,"style":33218},[31680],[34,541260,223320],{"className":541261,"style":223361},[31685,31686],[34,541263,38691],{"className":541264,"style":38706},[31685,31686],"。同理可证另一条。 ",[34,541267,541269,541282],{"className":541268},[31642],[34,541270,541272],{"className":541271},[31646],[31648,541273,541274],{"xmlns":31650},[31652,541275,541276,541280],{},[31655,541277,541278],{},[31658,541279,445204],{"mathvariant":38079},[31665,541281,445207],{"encoding":31667},[34,541283,541285],{"className":541284,"ariaHidden":758},[31672],[34,541286,541288,541291],{"className":541287},[31676],[34,541289],{"className":541290,"style":445217},[31680],[34,541292,445204],{"className":541293},[31685,129261],[32188,541295],{},[253,541297,541299],{"id":541298},"题目-16根理想-radical-ideal","题目 16：根理想 (Radical Ideal)",[15,541301,541302,541363,541568],{},[2627,541303,541304,541305,541333,541334,541362],{},"16. 设 ",[34,541306,541308,541321],{"className":541307},[31642],[34,541309,541311],{"className":541310},[31646],[31648,541312,541313],{"xmlns":31650},[31652,541314,541315,541319],{},[31655,541316,541317],{},[31658,541318,106063],{},[31665,541320,106063],{"encoding":31667},[34,541322,541324],{"className":541323,"ariaHidden":758},[31672],[34,541325,541327,541330],{"className":541326},[31676],[34,541328],{"className":541329,"style":33218},[31680],[34,541331,106063],{"className":541332,"style":67482},[31685,31686]," 是交换幺环 ",[34,541335,541337,541350],{"className":541336},[31642],[34,541338,541340],{"className":541339},[31646],[31648,541341,541342],{"xmlns":31650},[31652,541343,541344,541348],{},[31655,541345,541346],{},[31658,541347,34386],{},[31665,541349,34386],{"encoding":31667},[34,541351,541353],{"className":541352,"ariaHidden":758},[31672],[34,541354,541356,541359],{"className":541355},[31676],[34,541357],{"className":541358,"style":33218},[31680],[34,541360,34386],{"className":541361,"style":34427},[31685,31686]," 的一个理想。令",[34,541364,541366,541417],{"className":541365},[31642],[34,541367,541369],{"className":541368},[31646],[31648,541370,541371],{"xmlns":31650},[31652,541372,541373,541414],{},[31655,541374,541375,541378,541380,541382,541384,541386,541388,541390,541392,541395,541397,541400,541406,541408,541410,541412],{},[39361,541376,541377],{},"rad",[31658,541379,106063],{},[31662,541381,202],{},[31662,541383,32685],{"stretchy":702},[31658,541385,32936],{},[31662,541387,112296],{},[31658,541389,34386],{},[31662,541391,162086],{},[39361,541393,541394],{},"存在正整数 ",[31658,541396,31660],{},[39361,541398,541399],{}," 使得 ",[37624,541401,541402,541404],{},[31658,541403,32936],{},[31658,541405,31660],{},[31662,541407,112296],{},[31658,541409,106063],{},[31662,541411,32715],{"stretchy":702},[31658,541413,2547],{"mathvariant":38079},[31665,541415,541416],{"encoding":31667},"\\text{rad}I = \\{x \\in R \\mid \\text{存在正整数 } n \\text{ 使得 } x^n \\in I\\}.",[34,541418,541420,541444,541465,541483,541553],{"className":541419,"ariaHidden":758},[31672],[34,541421,541423,541426,541432,541435,541438,541441],{"className":541422},[31676],[34,541424],{"className":541425,"style":31681},[31680],[34,541427,541429],{"className":541428},[31685,2638],[34,541430,541377],{"className":541431},[31685],[34,541433,106063],{"className":541434,"style":67482},[31685,31686],[34,541436],{"className":541437,"style":31846},[31845],[34,541439,202],{"className":541440},[31850],[34,541442],{"className":541443,"style":31846},[31845],[34,541445,541447,541450,541453,541456,541459,541462],{"className":541446},[31676],[34,541448],{"className":541449,"style":31774},[31680],[34,541451,32685],{"className":541452},[31782],[34,541454,32936],{"className":541455},[31685,31686],[34,541457],{"className":541458,"style":31846},[31845],[34,541460,112296],{"className":541461},[31850],[34,541463],{"className":541464,"style":31846},[31845],[34,541466,541468,541471,541474,541477,541480],{"className":541467},[31676],[34,541469],{"className":541470,"style":31774},[31680],[34,541472,34386],{"className":541473,"style":34427},[31685,31686],[34,541475],{"className":541476,"style":31846},[31845],[34,541478,162086],{"className":541479},[31850],[34,541481],{"className":541482,"style":31846},[31845],[34,541484,541486,541489,541499,541502,541515,541544,541547,541550],{"className":541485},[31676],[34,541487],{"className":541488,"style":104901},[31680],[34,541490,541492,541496],{"className":541491},[31685,2638],[34,541493,541495],{"className":541494},[31685,105462],"存在正整数",[34,541497,130298],{"className":541498},[31685],[34,541500,31660],{"className":541501},[31685,31686],[34,541503,541505,541508,541512],{"className":541504},[31685,2638],[34,541506,130298],{"className":541507},[31685],[34,541509,541511],{"className":541510},[31685,105462],"使得",[34,541513,130298],{"className":541514},[31685],[34,541516,541518,541521],{"className":541517},[31685],[34,541519,32936],{"className":541520},[31685,31686],[34,541522,541524],{"className":541523},[31792],[34,541525,541527],{"className":541526},[31796],[34,541528,541530],{"className":541529},[31801],[34,541531,541533],{"className":541532,"style":37959},[31805],[34,541534,541535,541538],{"style":37777},[34,541536],{"className":541537,"style":31814},[31813],[34,541539,541541],{"className":541540},[31818,31819,31820,31821],[34,541542,31660],{"className":541543},[31685,31686,31821],[34,541545],{"className":541546,"style":31846},[31845],[34,541548,112296],{"className":541549},[31850],[34,541551],{"className":541552,"style":31846},[31845],[34,541554,541556,541559,541562,541565],{"className":541555},[31676],[34,541557],{"className":541558,"style":31774},[31680],[34,541560,106063],{"className":541561,"style":67482},[31685,31686],[34,541563,32715],{"className":541564},[31690],[34,541566,2547],{"className":541567},[31685],[2627,541569,379882,541570,11008,541607,541635,541636,541672,541673,541701],{},[34,541571,541573,541589],{"className":541572},[31642],[34,541574,541576],{"className":541575},[31646],[31648,541577,541578],{"xmlns":31650},[31652,541579,541580,541586],{},[31655,541581,541582,541584],{},[39361,541583,541377],{},[31658,541585,106063],{},[31665,541587,541588],{"encoding":31667},"\\text{rad}I",[34,541590,541592],{"className":541591,"ariaHidden":758},[31672],[34,541593,541595,541598,541604],{"className":541594},[31676],[34,541596],{"className":541597,"style":31681},[31680],[34,541599,541601],{"className":541600},[31685,2638],[34,541602,541377],{"className":541603},[31685],[34,541605,106063],{"className":541606,"style":67482},[31685,31686],[34,541608,541610,541623],{"className":541609},[31642],[34,541611,541613],{"className":541612},[31646],[31648,541614,541615],{"xmlns":31650},[31652,541616,541617,541621],{},[31655,541618,541619],{},[31658,541620,34386],{},[31665,541622,34386],{"encoding":31667},[34,541624,541626],{"className":541625,"ariaHidden":758},[31672],[34,541627,541629,541632],{"className":541628},[31676],[34,541630],{"className":541631,"style":33218},[31680],[34,541633,34386],{"className":541634,"style":34427},[31685,31686]," 的理想（",[34,541637,541639,541654],{"className":541638},[31642],[34,541640,541642],{"className":541641},[31646],[31648,541643,541644],{"xmlns":31650},[31652,541645,541646,541652],{},[31655,541647,541648,541650],{},[39361,541649,541377],{},[31658,541651,106063],{},[31665,541653,541588],{"encoding":31667},[34,541655,541657],{"className":541656,"ariaHidden":758},[31672],[34,541658,541660,541663,541669],{"className":541659},[31676],[34,541661],{"className":541662,"style":31681},[31680],[34,541664,541666],{"className":541665},[31685,2638],[34,541667,541377],{"className":541668},[31685],[34,541670,106063],{"className":541671,"style":67482},[31685,31686]," 称为 ",[34,541674,541676,541689],{"className":541675},[31642],[34,541677,541679],{"className":541678},[31646],[31648,541680,541681],{"xmlns":31650},[31652,541682,541683,541687],{},[31655,541684,541685],{},[31658,541686,106063],{},[31665,541688,106063],{"encoding":31667},[34,541690,541692],{"className":541691,"ariaHidden":758},[31672],[34,541693,541695,541698],{"className":541694},[31676],[34,541696],{"className":541697,"style":33218},[31680],[34,541699,106063],{"className":541700,"style":67482},[31685,31686]," 的根理想）。",[32490,541703,463462],{"id":541704},"深度解析-2",[15,541706,541707,541708,347],{},"这道题其实是",[2627,541709,541710],{},"第14题的推广",[105,541712,541713,541864,542016],{},[108,541714,541715,541716,541796,541797,541825,541826,347],{},"在第14题中，我们研究的是 ",[34,541717,541719,541740],{"className":541718},[31642],[34,541720,541722],{"className":541721},[31646],[31648,541723,541724],{"xmlns":31650},[31652,541725,541726,541738],{},[31655,541727,541728,541734,541736],{},[37624,541729,541730,541732],{},[31658,541731,32936],{},[31658,541733,31660],{},[31662,541735,202],{},[31749,541737,459],{},[31665,541739,533777],{"encoding":31667},[34,541741,541743,541787],{"className":541742,"ariaHidden":758},[31672],[34,541744,541746,541749,541778,541781,541784],{"className":541745},[31676],[34,541747],{"className":541748,"style":37959},[31680],[34,541750,541752,541755],{"className":541751},[31685],[34,541753,32936],{"className":541754},[31685,31686],[34,541756,541758],{"className":541757},[31792],[34,541759,541761],{"className":541760},[31796],[34,541762,541764],{"className":541763},[31801],[34,541765,541767],{"className":541766,"style":37959},[31805],[34,541768,541769,541772],{"style":37777},[34,541770],{"className":541771,"style":31814},[31813],[34,541773,541775],{"className":541774},[31818,31819,31820,31821],[34,541776,31660],{"className":541777},[31685,31686,31821],[34,541779],{"className":541780,"style":31846},[31845],[34,541782,202],{"className":541783},[31850],[34,541785],{"className":541786,"style":31846},[31845],[34,541788,541790,541793],{"className":541789},[31676],[34,541791],{"className":541792,"style":32617},[31680],[34,541794,459],{"className":541795},[31685],"。要知道，",[34,541798,541800,541813],{"className":541799},[31642],[34,541801,541803],{"className":541802},[31646],[31648,541804,541805],{"xmlns":31650},[31652,541806,541807,541811],{},[31655,541808,541809],{},[31749,541810,459],{},[31665,541812,459],{"encoding":31667},[34,541814,541816],{"className":541815,"ariaHidden":758},[31672],[34,541817,541819,541822],{"className":541818},[31676],[34,541820],{"className":541821,"style":32617},[31680],[34,541823,459],{"className":541824},[31685]," 其实就是零理想 ",[34,541827,541829,541846],{"className":541828},[31642],[34,541830,541832],{"className":541831},[31646],[31648,541833,541834],{"xmlns":31650},[31652,541835,541836,541844],{},[31655,541837,541838,541840,541842],{},[31662,541839,32685],{"stretchy":702},[31749,541841,459],{},[31662,541843,32715],{"stretchy":702},[31665,541845,338218],{"encoding":31667},[34,541847,541849],{"className":541848,"ariaHidden":758},[31672],[34,541850,541852,541855,541858,541861],{"className":541851},[31676],[34,541853],{"className":541854,"style":31774},[31680],[34,541856,32685],{"className":541857},[31782],[34,541859,459],{"className":541860},[31685],[34,541862,32715],{"className":541863},[31690],[108,541865,541866,541867,541905,541906,465789,541934,347],{},"在第16题中，我们将 ",[34,541868,541870,541887],{"className":541869},[31642],[34,541871,541873],{"className":541872},[31646],[31648,541874,541875],{"xmlns":31650},[31652,541876,541877,541885],{},[31655,541878,541879,541881,541883],{},[31662,541880,32685],{"stretchy":702},[31749,541882,459],{},[31662,541884,32715],{"stretchy":702},[31665,541886,338218],{"encoding":31667},[34,541888,541890],{"className":541889,"ariaHidden":758},[31672],[34,541891,541893,541896,541899,541902],{"className":541892},[31676],[34,541894],{"className":541895,"style":31774},[31680],[34,541897,32685],{"className":541898},[31782],[34,541900,459],{"className":541901},[31685],[34,541903,32715],{"className":541904},[31690]," 替换成了一般理想 ",[34,541907,541909,541922],{"className":541908},[31642],[34,541910,541912],{"className":541911},[31646],[31648,541913,541914],{"xmlns":31650},[31652,541915,541916,541920],{},[31655,541917,541918],{},[31658,541919,106063],{},[31665,541921,106063],{"encoding":31667},[34,541923,541925],{"className":541924,"ariaHidden":758},[31672],[34,541926,541928,541931],{"className":541927},[31676],[34,541929],{"className":541930,"style":33218},[31680],[34,541932,106063],{"className":541933,"style":67482},[31685,31686],[34,541935,541937,541959],{"className":541936},[31642],[34,541938,541940],{"className":541939},[31646],[31648,541941,541942],{"xmlns":31650},[31652,541943,541944,541956],{},[31655,541945,541946,541952,541954],{},[37624,541947,541948,541950],{},[31658,541949,32936],{},[31658,541951,31660],{},[31662,541953,112296],{},[31658,541955,106063],{},[31665,541957,541958],{"encoding":31667},"x^n \\in I",[34,541960,541962,542007],{"className":541961,"ariaHidden":758},[31672],[34,541963,541965,541969,541998,542001,542004],{"className":541964},[31676],[34,541966],{"className":541967,"style":541968},[31680],"height:0.7035em;vertical-align:-0.0391em;",[34,541970,541972,541975],{"className":541971},[31685],[34,541973,32936],{"className":541974},[31685,31686],[34,541976,541978],{"className":541977},[31792],[34,541979,541981],{"className":541980},[31796],[34,541982,541984],{"className":541983},[31801],[34,541985,541987],{"className":541986,"style":37959},[31805],[34,541988,541989,541992],{"style":37777},[34,541990],{"className":541991,"style":31814},[31813],[34,541993,541995],{"className":541994},[31818,31819,31820,31821],[34,541996,31660],{"className":541997},[31685,31686,31821],[34,541999],{"className":542000,"style":31846},[31845],[34,542002,112296],{"className":542003},[31850],[34,542005],{"className":542006,"style":31846},[31845],[34,542008,542010,542013],{"className":542009},[31676],[34,542011],{"className":542012,"style":33218},[31680],[34,542014,106063],{"className":542015,"style":67482},[31685,31686],[108,542017,542018],{},"证明逻辑与第14题完全一致：利用二项式定理处理加法，利用交换律处理乘法。",[32490,542020,534075],{"id":542021},"严谨证明-2",[2230,542023,542024,542270],{},[108,542025,542026,315440,542028,518693,542056,322067,542107,35847,542210,542269],{},[2627,542027,441307],{},[34,542029,542031,542044],{"className":542030},[31642],[34,542032,542034],{"className":542033},[31646],[31648,542035,542036],{"xmlns":31650},[31652,542037,542038,542042],{},[31655,542039,542040],{},[31658,542041,106063],{},[31665,542043,106063],{"encoding":31667},[34,542045,542047],{"className":542046,"ariaHidden":758},[31672],[34,542048,542050,542053],{"className":542049},[31676],[34,542051],{"className":542052,"style":33218},[31680],[34,542054,106063],{"className":542055,"style":67482},[31685,31686],[34,542057,542059,542077],{"className":542058},[31642],[34,542060,542062],{"className":542061},[31646],[31648,542063,542064],{"xmlns":31650},[31652,542065,542066,542074],{},[31655,542067,542068,542070,542072],{},[31749,542069,459],{},[31662,542071,112296],{},[31658,542073,106063],{},[31665,542075,542076],{"encoding":31667},"0 \\in I",[34,542078,542080,542098],{"className":542079,"ariaHidden":758},[31672],[34,542081,542083,542086,542089,542092,542095],{"className":542082},[31676],[34,542084],{"className":542085,"style":61814},[31680],[34,542087,459],{"className":542088},[31685],[34,542090],{"className":542091,"style":31846},[31845],[34,542093,112296],{"className":542094},[31850],[34,542096],{"className":542097,"style":31846},[31845],[34,542099,542101,542104],{"className":542100},[31676],[34,542102],{"className":542103,"style":33218},[31680],[34,542105,106063],{"className":542106,"style":67482},[31685,31686],[34,542108,542110,542136],{"className":542109},[31642],[34,542111,542113],{"className":542112},[31646],[31648,542114,542115],{"xmlns":31650},[31652,542116,542117,542133],{},[31655,542118,542119,542125,542127,542129,542131],{},[37624,542120,542121,542123],{},[31749,542122,459],{},[31749,542124,640],{},[31662,542126,202],{},[31749,542128,459],{},[31662,542130,112296],{},[31658,542132,106063],{},[31665,542134,542135],{"encoding":31667},"0^1 = 0 \\in I",[34,542137,542139,542183,542201],{"className":542138,"ariaHidden":758},[31672],[34,542140,542142,542145,542174,542177,542180],{"className":542141},[31676],[34,542143],{"className":542144,"style":37774},[31680],[34,542146,542148,542151],{"className":542147},[31685],[34,542149,459],{"className":542150},[31685],[34,542152,542154],{"className":542153},[31792],[34,542155,542157],{"className":542156},[31796],[34,542158,542160],{"className":542159},[31801],[34,542161,542163],{"className":542162,"style":37774},[31805],[34,542164,542165,542168],{"style":37777},[34,542166],{"className":542167,"style":31814},[31813],[34,542169,542171],{"className":542170},[31818,31819,31820,31821],[34,542172,640],{"className":542173},[31685,31821],[34,542175],{"className":542176,"style":31846},[31845],[34,542178,202],{"className":542179},[31850],[34,542181],{"className":542182,"style":31846},[31845],[34,542184,542186,542189,542192,542195,542198],{"className":542185},[31676],[34,542187],{"className":542188,"style":61814},[31680],[34,542190,459],{"className":542191},[31685],[34,542193],{"className":542194,"style":31846},[31845],[34,542196,112296],{"className":542197},[31850],[34,542199],{"className":542200,"style":31846},[31845],[34,542202,542204,542207],{"className":542203},[31676],[34,542205],{"className":542206,"style":33218},[31680],[34,542208,106063],{"className":542209,"style":67482},[31685,31686],[34,542211,542213,542233],{"className":542212},[31642],[34,542214,542216],{"className":542215},[31646],[31648,542217,542218],{"xmlns":31650},[31652,542219,542220,542230],{},[31655,542221,542222,542224,542226,542228],{},[31749,542223,459],{},[31662,542225,112296],{},[39361,542227,541377],{},[31658,542229,106063],{},[31665,542231,542232],{"encoding":31667},"0 \\in \\text{rad}I",[34,542234,542236,542254],{"className":542235,"ariaHidden":758},[31672],[34,542237,542239,542242,542245,542248,542251],{"className":542238},[31676],[34,542240],{"className":542241,"style":61814},[31680],[34,542243,459],{"className":542244},[31685],[34,542246],{"className":542247,"style":31846},[31845],[34,542249,112296],{"className":542250},[31850],[34,542252],{"className":542253,"style":31846},[31845],[34,542255,542257,542260,542266],{"className":542256},[31676],[34,542258],{"className":542259,"style":31681},[31680],[34,542261,542263],{"className":542262},[31685,2638],[34,542264,541377],{"className":542265},[31685],[34,542267,106063],{"className":542268,"style":67482},[31685,31686],"，集合非空。",[108,542271,542272,362914,542274,542346,542347,315343,542388,86674,542468,542549,542550,542631,542632,542738,542739,542767,542768,543186,543187,2664,543475],{},[2627,542273,443104],{},[34,542275,542277,542301],{"className":542276},[31642],[34,542278,542280],{"className":542279},[31646],[31648,542281,542282],{"xmlns":31650},[31652,542283,542284,542298],{},[31655,542285,542286,542288,542290,542292,542294,542296],{},[31658,542287,32936],{},[31662,542289,31960],{"separator":758},[31658,542291,67764],{},[31662,542293,112296],{},[39361,542295,541377],{},[31658,542297,106063],{},[31665,542299,542300],{"encoding":31667},"x, y \\in \\text{rad}I",[34,542302,542304,542331],{"className":542303,"ariaHidden":758},[31672],[34,542305,542307,542310,542313,542316,542319,542322,542325,542328],{"className":542306},[31676],[34,542308],{"className":542309,"style":86652},[31680],[34,542311,32936],{"className":542312},[31685,31686],[34,542314,31960],{"className":542315},[32119],[34,542317],{"className":542318,"style":32123},[31845],[34,542320,67764],{"className":542321,"style":32062},[31685,31686],[34,542323],{"className":542324,"style":31846},[31845],[34,542326,112296],{"className":542327},[31850],[34,542329],{"className":542330,"style":31846},[31845],[34,542332,542334,542337,542343],{"className":542333},[31676],[34,542335],{"className":542336,"style":31681},[31680],[34,542338,542340],{"className":542339},[31685,2638],[34,542341,541377],{"className":542342},[31685],[34,542344,106063],{"className":542345,"style":67482},[31685,31686],"。这意味着存在正整数 ",[34,542348,542350,542367],{"className":542349},[31642],[34,542351,542353],{"className":542352},[31646],[31648,542354,542355],{"xmlns":31650},[31652,542356,542357,542365],{},[31655,542358,542359,542361,542363],{},[31658,542360,31660],{},[31662,542362,31960],{"separator":758},[31658,542364,31759],{},[31665,542366,534648],{"encoding":31667},[34,542368,542370],{"className":542369,"ariaHidden":758},[31672],[34,542371,542373,542376,542379,542382,542385],{"className":542372},[31676],[34,542374],{"className":542375,"style":38774},[31680],[34,542377,31660],{"className":542378},[31685,31686],[34,542380,31960],{"className":542381},[32119],[34,542383],{"className":542384,"style":32123},[31845],[34,542386,31759],{"className":542387},[31685,31686],[34,542389,542391,542412],{"className":542390},[31642],[34,542392,542394],{"className":542393},[31646],[31648,542395,542396],{"xmlns":31650},[31652,542397,542398,542410],{},[31655,542399,542400,542406,542408],{},[37624,542401,542402,542404],{},[31658,542403,32936],{},[31658,542405,31660],{},[31662,542407,112296],{},[31658,542409,106063],{},[31665,542411,541958],{"encoding":31667},[34,542413,542415,542459],{"className":542414,"ariaHidden":758},[31672],[34,542416,542418,542421,542450,542453,542456],{"className":542417},[31676],[34,542419],{"className":542420,"style":541968},[31680],[34,542422,542424,542427],{"className":542423},[31685],[34,542425,32936],{"className":542426},[31685,31686],[34,542428,542430],{"className":542429},[31792],[34,542431,542433],{"className":542432},[31796],[34,542434,542436],{"className":542435},[31801],[34,542437,542439],{"className":542438,"style":37959},[31805],[34,542440,542441,542444],{"style":37777},[34,542442],{"className":542443,"style":31814},[31813],[34,542445,542447],{"className":542446},[31818,31819,31820,31821],[34,542448,31660],{"className":542449},[31685,31686,31821],[34,542451],{"className":542452,"style":31846},[31845],[34,542454,112296],{"className":542455},[31850],[34,542457],{"className":542458,"style":31846},[31845],[34,542460,542462,542465],{"className":542461},[31676],[34,542463],{"className":542464,"style":33218},[31680],[34,542466,106063],{"className":542467,"style":67482},[31685,31686],[34,542469,542471,542493],{"className":542470},[31642],[34,542472,542474],{"className":542473},[31646],[31648,542475,542476],{"xmlns":31650},[31652,542477,542478,542490],{},[31655,542479,542480,542486,542488],{},[37624,542481,542482,542484],{},[31658,542483,67764],{},[31658,542485,31759],{},[31662,542487,112296],{},[31658,542489,106063],{},[31665,542491,542492],{"encoding":31667},"y^m \\in I",[34,542494,542496,542540],{"className":542495,"ariaHidden":758},[31672],[34,542497,542499,542502,542531,542534,542537],{"className":542498},[31676],[34,542500],{"className":542501,"style":316112},[31680],[34,542503,542505,542508],{"className":542504},[31685],[34,542506,67764],{"className":542507,"style":32062},[31685,31686],[34,542509,542511],{"className":542510},[31792],[34,542512,542514],{"className":542513},[31796],[34,542515,542517],{"className":542516},[31801],[34,542518,542520],{"className":542519,"style":37959},[31805],[34,542521,542522,542525],{"style":37777},[34,542523],{"className":542524,"style":31814},[31813],[34,542526,542528],{"className":542527},[31818,31819,31820,31821],[34,542529,31759],{"className":542530},[31685,31686,31821],[34,542532],{"className":542533,"style":31846},[31845],[34,542535,112296],{"className":542536},[31850],[34,542538],{"className":542539,"style":31846},[31845],[34,542541,542543,542546],{"className":542542},[31676],[34,542544],{"className":542545,"style":33218},[31680],[34,542547,106063],{"className":542548,"style":67482},[31685,31686],"。我们需要证明 ",[34,542551,542553,542577],{"className":542552},[31642],[34,542554,542556],{"className":542555},[31646],[31648,542557,542558],{"xmlns":31650},[31652,542559,542560,542574],{},[31655,542561,542562,542564,542566,542568,542570,542572],{},[31658,542563,32936],{},[31662,542565,32226],{},[31658,542567,67764],{},[31662,542569,112296],{},[39361,542571,541377],{},[31658,542573,106063],{},[31665,542575,542576],{"encoding":31667},"x - y \\in \\text{rad}I",[34,542578,542580,542598,542616],{"className":542579,"ariaHidden":758},[31672],[34,542581,542583,542586,542589,542592,542595],{"className":542582},[31676],[34,542584],{"className":542585,"style":32597},[31680],[34,542587,32936],{"className":542588},[31685,31686],[34,542590],{"className":542591,"style":32604},[31845],[34,542593,32226],{"className":542594},[32024],[34,542596],{"className":542597,"style":32604},[31845],[34,542599,542601,542604,542607,542610,542613],{"className":542600},[31676],[34,542602],{"className":542603,"style":86652},[31680],[34,542605,67764],{"className":542606,"style":32062},[31685,31686],[34,542608],{"className":542609,"style":31846},[31845],[34,542611,112296],{"className":542612},[31850],[34,542614],{"className":542615,"style":31846},[31845],[34,542617,542619,542622,542628],{"className":542618},[31676],[34,542620],{"className":542621,"style":31681},[31680],[34,542623,542625],{"className":542624},[31685,2638],[34,542626,541377],{"className":542627},[31685],[34,542629,106063],{"className":542630,"style":67482},[31685,31686],"。考察 ",[34,542633,542635,542667],{"className":542634},[31642],[34,542636,542638],{"className":542637},[31646],[31648,542639,542640],{"xmlns":31650},[31652,542641,542642,542664],{},[31655,542643,542644,542646,542648,542650,542652],{},[31662,542645,456],{"stretchy":702},[31658,542647,32936],{},[31662,542649,32226],{},[31658,542651,67764],{},[37624,542653,542654,542656],{},[31662,542655,2557],{"stretchy":702},[31655,542657,542658,542660,542662],{},[31658,542659,31660],{},[31662,542661,223],{},[31658,542663,31759],{},[31665,542665,542666],{"encoding":31667},"(x - y)^{n+m}",[34,542668,542670,542691],{"className":542669,"ariaHidden":758},[31672],[34,542671,542673,542676,542679,542682,542685,542688],{"className":542672},[31676],[34,542674],{"className":542675,"style":31774},[31680],[34,542677,456],{"className":542678},[31782],[34,542680,32936],{"className":542681},[31685,31686],[34,542683],{"className":542684,"style":32604},[31845],[34,542686,32226],{"className":542687},[32024],[34,542689],{"className":542690,"style":32604},[31845],[34,542692,542694,542697,542700],{"className":542693},[31676],[34,542695],{"className":542696,"style":156809},[31680],[34,542698,67764],{"className":542699,"style":32062},[31685,31686],[34,542701,542703,542706],{"className":542702},[31690],[34,542704,2557],{"className":542705},[31690],[34,542707,542709],{"className":542708},[31792],[34,542710,542712],{"className":542711},[31796],[34,542713,542715],{"className":542714},[31801],[34,542716,542718],{"className":542717,"style":156828},[31805],[34,542719,542720,542723],{"style":37777},[34,542721],{"className":542722,"style":31814},[31813],[34,542724,542726],{"className":542725},[31818,31819,31820,31821],[34,542727,542729,542732,542735],{"className":542728},[31685,31821],[34,542730,31660],{"className":542731},[31685,31686,31821],[34,542733,223],{"className":542734},[32024,31821],[34,542736,31759],{"className":542737},[31685,31686,31821],"。利用二项式定理（因为 ",[34,542740,542742,542755],{"className":542741},[31642],[34,542743,542745],{"className":542744},[31646],[31648,542746,542747],{"xmlns":31650},[31652,542748,542749,542753],{},[31655,542750,542751],{},[31658,542752,34386],{},[31665,542754,34386],{"encoding":31667},[34,542756,542758],{"className":542757,"ariaHidden":758},[31672],[34,542759,542761,542764],{"className":542760},[31676],[34,542762],{"className":542763,"style":33218},[31680],[34,542765,34386],{"className":542766,"style":34427},[31685,31686]," 交换）：\n",[34,542769,542771,542871],{"className":542770},[31642],[34,542772,542774],{"className":542773},[31646],[31648,542775,542776],{"xmlns":31650},[31652,542777,542778,542868],{},[31655,542779,542780,542782,542784,542786,542788,542800,542802,542822,542840,542846,542848,542850,542852],{},[31662,542781,456],{"stretchy":702},[31658,542783,32936],{},[31662,542785,32226],{},[31658,542787,67764],{},[37624,542789,542790,542792],{},[31662,542791,2557],{"stretchy":702},[31655,542793,542794,542796,542798],{},[31658,542795,31660],{},[31662,542797,223],{},[31658,542799,31759],{},[31662,542801,202],{},[38846,542803,542804,542806,542814],{},[31662,542805,131138],{},[31655,542807,542808,542810,542812],{},[31658,542809,33759],{},[31662,542811,202],{},[31749,542813,459],{},[31655,542815,542816,542818,542820],{},[31658,542817,31660],{},[31662,542819,223],{},[31658,542821,31759],{},[31655,542823,542824,542826,542838],{},[31662,542825,456],{"fence":758},[41453,542827,542828,542836],{"linethickness":535102},[31655,542829,542830,542832,542834],{},[31658,542831,31660],{},[31662,542833,223],{},[31658,542835,31759],{},[31658,542837,33759],{},[31662,542839,2557],{"fence":758},[37624,542841,542842,542844],{},[31658,542843,32936],{},[31658,542845,33759],{},[31662,542847,456],{"stretchy":702},[31662,542849,32226],{},[31658,542851,67764],{},[37624,542853,542854,542856],{},[31662,542855,2557],{"stretchy":702},[31655,542857,542858,542860,542862,542864,542866],{},[31658,542859,31660],{},[31662,542861,223],{},[31658,542863,31759],{},[31662,542865,32226],{},[31658,542867,33759],{},[31665,542869,542870],{"encoding":31667},"(x - y)^{n+m} = \\sum_{k=0}^{n+m} \\binom{n+m}{k} x^k (-y)^{n+m-k}",[34,542872,542874,542895,542951],{"className":542873,"ariaHidden":758},[31672],[34,542875,542877,542880,542883,542886,542889,542892],{"className":542876},[31676],[34,542878],{"className":542879,"style":31774},[31680],[34,542881,456],{"className":542882},[31782],[34,542884,32936],{"className":542885},[31685,31686],[34,542887],{"className":542888,"style":32604},[31845],[34,542890,32226],{"className":542891},[32024],[34,542893],{"className":542894,"style":32604},[31845],[34,542896,542898,542901,542904,542942,542945,542948],{"className":542897},[31676],[34,542899],{"className":542900,"style":156809},[31680],[34,542902,67764],{"className":542903,"style":32062},[31685,31686],[34,542905,542907,542910],{"className":542906},[31690],[34,542908,2557],{"className":542909},[31690],[34,542911,542913],{"className":542912},[31792],[34,542914,542916],{"className":542915},[31796],[34,542917,542919],{"className":542918},[31801],[34,542920,542922],{"className":542921,"style":156828},[31805],[34,542923,542924,542927],{"style":37777},[34,542925],{"className":542926,"style":31814},[31813],[34,542928,542930],{"className":542929},[31818,31819,31820,31821],[34,542931,542933,542936,542939],{"className":542932},[31685,31821],[34,542934,31660],{"className":542935},[31685,31686,31821],[34,542937,223],{"className":542938},[32024,31821],[34,542940,31759],{"className":542941},[31685,31686,31821],[34,542943],{"className":542944,"style":31846},[31845],[34,542946,202],{"className":542947},[31850],[34,542949],{"className":542950,"style":31846},[31845],[34,542952,542954,542958,543028,543031,543104,543133,543136,543139,543142],{"className":542953},[31676],[34,542955],{"className":542956,"style":542957},[31680],"height:1.2612em;vertical-align:-0.35em;",[34,542959,542961,542964],{"className":542960},[32767],[34,542962,131138],{"className":542963,"style":156561},[32767,131249,156560],[34,542965,542967],{"className":542966},[31792],[34,542968,542970,543020],{"className":542969},[31796,31797],[34,542971,542973,543017],{"className":542972},[31801],[34,542974,542977,542997],{"className":542975,"style":542976},[31805],"height:0.9112em;",[34,542978,542979,542982],{"style":156577},[34,542980],{"className":542981,"style":31814},[31813],[34,542983,542985],{"className":542984},[31818,31819,31820,31821],[34,542986,542988,542991,542994],{"className":542987},[31685,31821],[34,542989,33759],{"className":542990,"style":33774},[31685,31686,31821],[34,542992,202],{"className":542993},[31850,31821],[34,542995,459],{"className":542996},[31685,31821],[34,542998,542999,543002],{"style":352016},[34,543000],{"className":543001,"style":31814},[31813],[34,543003,543005],{"className":543004},[31818,31819,31820,31821],[34,543006,543008,543011,543014],{"className":543007},[31685,31821],[34,543009,31660],{"className":543010},[31685,31686,31821],[34,543012,223],{"className":543013},[32024,31821],[34,543015,31759],{"className":543016},[31685,31686,31821],[34,543018,31829],{"className":543019},[31828],[34,543021,543023],{"className":543022},[31801],[34,543024,543026],{"className":543025,"style":352035},[31805],[34,543027],{},[34,543029],{"className":543030,"style":32123},[31845],[34,543032,543034,543040,543098],{"className":543033},[31685],[34,543035,543037],{"className":543036,"style":75811},[31782,75810],[34,543038,456],{"className":543039},[71281,43411],[34,543041,543043],{"className":543042},[41453],[34,543044,543046,543090],{"className":543045},[31796,31797],[34,543047,543049,543087],{"className":543048},[31801],[34,543050,543053,543067],{"className":543051,"style":543052},[31805],"height:0.8523em;",[34,543054,543055,543058],{"style":535305},[34,543056],{"className":543057,"style":31814},[31813],[34,543059,543061],{"className":543060},[31818,31819,31820,31821],[34,543062,543064],{"className":543063},[31685,31821],[34,543065,33759],{"className":543066,"style":33774},[31685,31686,31821],[34,543068,543069,543072],{"style":535320},[34,543070],{"className":543071,"style":31814},[31813],[34,543073,543075],{"className":543074},[31818,31819,31820,31821],[34,543076,543078,543081,543084],{"className":543077},[31685,31821],[34,543079,31660],{"className":543080},[31685,31686,31821],[34,543082,223],{"className":543083},[32024,31821],[34,543085,31759],{"className":543086},[31685,31686,31821],[34,543088,31829],{"className":543089},[31828],[34,543091,543093],{"className":543092},[31801],[34,543094,543096],{"className":543095,"style":41619},[31805],[34,543097],{},[34,543099,543101],{"className":543100,"style":75811},[31690,75810],[34,543102,2557],{"className":543103},[71281,43411],[34,543105,543107,543110],{"className":543106},[31685],[34,543108,32936],{"className":543109},[31685,31686],[34,543111,543113],{"className":543112},[31792],[34,543114,543116],{"className":543115},[31796],[34,543117,543119],{"className":543118},[31801],[34,543120,543122],{"className":543121,"style":38486},[31805],[34,543123,543124,543127],{"style":37777},[34,543125],{"className":543126,"style":31814},[31813],[34,543128,543130],{"className":543129},[31818,31819,31820,31821],[34,543131,33759],{"className":543132,"style":33774},[31685,31686,31821],[34,543134,456],{"className":543135},[31782],[34,543137,32226],{"className":543138},[31685],[34,543140,67764],{"className":543141,"style":32062},[31685,31686],[34,543143,543145,543148],{"className":543144},[31690],[34,543146,2557],{"className":543147},[31690],[34,543149,543151],{"className":543150},[31792],[34,543152,543154],{"className":543153},[31796],[34,543155,543157],{"className":543156},[31801],[34,543158,543160],{"className":543159,"style":38486},[31805],[34,543161,543162,543165],{"style":37777},[34,543163],{"className":543164,"style":31814},[31813],[34,543166,543168],{"className":543167},[31818,31819,31820,31821],[34,543169,543171,543174,543177,543180,543183],{"className":543170},[31685,31821],[34,543172,31660],{"className":543173},[31685,31686,31821],[34,543175,223],{"className":543176},[32024,31821],[34,543178,31759],{"className":543179},[31685,31686,31821],[34,543181,32226],{"className":543182},[32024,31821],[34,543184,33759],{"className":543185,"style":33774},[31685,31686,31821],"\n对于每一项 ",[34,543188,543190,543256],{"className":543189},[31642],[34,543191,543193],{"className":543192},[31646],[31648,543194,543195],{"xmlns":31650},[31652,543196,543197,543253],{},[31655,543198,543199,543205,543207,543225,543231,543233,543235,543237],{},[31743,543200,543201,543203],{},[31658,543202,37400],{},[31658,543204,33759],{},[31662,543206,202],{},[31655,543208,543209,543211,543223],{},[31662,543210,456],{"fence":758},[41453,543212,543213,543221],{"linethickness":535102},[31655,543214,543215,543217,543219],{},[31658,543216,31660],{},[31662,543218,223],{},[31658,543220,31759],{},[31658,543222,33759],{},[31662,543224,2557],{"fence":758},[37624,543226,543227,543229],{},[31658,543228,32936],{},[31658,543230,33759],{},[31662,543232,456],{"stretchy":702},[31662,543234,32226],{},[31658,543236,67764],{},[37624,543238,543239,543241],{},[31662,543240,2557],{"stretchy":702},[31655,543242,543243,543245,543247,543249,543251],{},[31658,543244,31660],{},[31662,543246,223],{},[31658,543248,31759],{},[31662,543250,32226],{},[31658,543252,33759],{},[31665,543254,543255],{"encoding":31667},"T_k = \\binom{n+m}{k} x^k (-y)^{n+m-k}",[34,543257,543259,543314],{"className":543258,"ariaHidden":758},[31672],[34,543260,543262,543265,543305,543308,543311],{"className":543261},[31676],[34,543263],{"className":543264,"style":70915},[31680],[34,543266,543268,543271],{"className":543267},[31685],[34,543269,37400],{"className":543270,"style":37422},[31685,31686],[34,543272,543274],{"className":543273},[31792],[34,543275,543277,543297],{"className":543276},[31796,31797],[34,543278,543280,543294],{"className":543279},[31801],[34,543281,543283],{"className":543282,"style":33833},[31805],[34,543284,543285,543288],{"style":214424},[34,543286],{"className":543287,"style":31814},[31813],[34,543289,543291],{"className":543290},[31818,31819,31820,31821],[34,543292,33759],{"className":543293,"style":33774},[31685,31686,31821],[34,543295,31829],{"className":543296},[31828],[34,543298,543300],{"className":543299},[31801],[34,543301,543303],{"className":543302,"style":31836},[31805],[34,543304],{},[34,543306],{"className":543307,"style":31846},[31845],[34,543309,202],{"className":543310},[31850],[34,543312],{"className":543313,"style":31846},[31845],[34,543315,543317,543321,543393,543422,543425,543428,543431],{"className":543316},[31676],[34,543318],{"className":543319,"style":543320},[31680],"height:1.2023em;vertical-align:-0.35em;",[34,543322,543324,543330,543387],{"className":543323},[31685],[34,543325,543327],{"className":543326,"style":75811},[31782,75810],[34,543328,456],{"className":543329},[71281,43411],[34,543331,543333],{"className":543332},[41453],[34,543334,543336,543379],{"className":543335},[31796,31797],[34,543337,543339,543376],{"className":543338},[31801],[34,543340,543342,543356],{"className":543341,"style":543052},[31805],[34,543343,543344,543347],{"style":535305},[34,543345],{"className":543346,"style":31814},[31813],[34,543348,543350],{"className":543349},[31818,31819,31820,31821],[34,543351,543353],{"className":543352},[31685,31821],[34,543354,33759],{"className":543355,"style":33774},[31685,31686,31821],[34,543357,543358,543361],{"style":535320},[34,543359],{"className":543360,"style":31814},[31813],[34,543362,543364],{"className":543363},[31818,31819,31820,31821],[34,543365,543367,543370,543373],{"className":543366},[31685,31821],[34,543368,31660],{"className":543369},[31685,31686,31821],[34,543371,223],{"className":543372},[32024,31821],[34,543374,31759],{"className":543375},[31685,31686,31821],[34,543377,31829],{"className":543378},[31828],[34,543380,543382],{"className":543381},[31801],[34,543383,543385],{"className":543384,"style":41619},[31805],[34,543386],{},[34,543388,543390],{"className":543389,"style":75811},[31690,75810],[34,543391,2557],{"className":543392},[71281,43411],[34,543394,543396,543399],{"className":543395},[31685],[34,543397,32936],{"className":543398},[31685,31686],[34,543400,543402],{"className":543401},[31792],[34,543403,543405],{"className":543404},[31796],[34,543406,543408],{"className":543407},[31801],[34,543409,543411],{"className":543410,"style":38486},[31805],[34,543412,543413,543416],{"style":37777},[34,543414],{"className":543415,"style":31814},[31813],[34,543417,543419],{"className":543418},[31818,31819,31820,31821],[34,543420,33759],{"className":543421,"style":33774},[31685,31686,31821],[34,543423,456],{"className":543424},[31782],[34,543426,32226],{"className":543427},[31685],[34,543429,67764],{"className":543430,"style":32062},[31685,31686],[34,543432,543434,543437],{"className":543433},[31690],[34,543435,2557],{"className":543436},[31690],[34,543438,543440],{"className":543439},[31792],[34,543441,543443],{"className":543442},[31796],[34,543444,543446],{"className":543445},[31801],[34,543447,543449],{"className":543448,"style":38486},[31805],[34,543450,543451,543454],{"style":37777},[34,543452],{"className":543453,"style":31814},[31813],[34,543455,543457],{"className":543456},[31818,31819,31820,31821],[34,543458,543460,543463,543466,543469,543472],{"className":543459},[31685,31821],[34,543461,31660],{"className":543462},[31685,31686,31821],[34,543464,223],{"className":543465},[32024,31821],[34,543467,31759],{"className":543468},[31685,31686,31821],[34,543470,32226],{"className":543471},[32024,31821],[34,543473,33759],{"className":543474,"style":33774},[31685,31686,31821],[105,543476,543477,543995],{},[108,543478,543479,47890,543481,543532,543533,44689,543711,93535,543791,543819,543820,543902,543903,347],{},[2627,543480,326833],{},[34,543482,543484,543502],{"className":543483},[31642],[34,543485,543487],{"className":543486},[31646],[31648,543488,543489],{"xmlns":31650},[31652,543490,543491,543499],{},[31655,543492,543493,543495,543497],{},[31658,543494,33759],{},[31662,543496,43175],{},[31658,543498,31660],{},[31665,543500,543501],{"encoding":31667},"k \\ge n",[34,543503,543505,543523],{"className":543504,"ariaHidden":758},[31672],[34,543506,543508,543511,543514,543517,543520],{"className":543507},[31676],[34,543509],{"className":543510,"style":47709},[31680],[34,543512,33759],{"className":543513,"style":33774},[31685,31686],[34,543515],{"className":543516,"style":31846},[31845],[34,543518,43175],{"className":543519},[31850],[34,543521],{"className":543522,"style":31846},[31845],[34,543524,543526,543529],{"className":543525},[31676],[34,543527],{"className":543528,"style":32527},[31680],[34,543530,31660],{"className":543531},[31685,31686],"。则 ",[34,543534,543536,543576],{"className":543535},[31642],[34,543537,543539],{"className":543538},[31646],[31648,543540,543541],{"xmlns":31650},[31652,543542,543543,543573],{},[31655,543544,543545,543551,543553,543559,543561],{},[37624,543546,543547,543549],{},[31658,543548,32936],{},[31658,543550,33759],{},[31662,543552,202],{},[37624,543554,543555,543557],{},[31658,543556,32936],{},[31658,543558,31660],{},[31662,543560,71191],{},[37624,543562,543563,543565],{},[31658,543564,32936],{},[31655,543566,543567,543569,543571],{},[31658,543568,33759],{},[31662,543570,32226],{},[31658,543572,31660],{},[31665,543574,543575],{"encoding":31667},"x^k = x^n \\cdot x^{k-n}",[34,543577,543579,543623,543667],{"className":543578,"ariaHidden":758},[31672],[34,543580,543582,543585,543614,543617,543620],{"className":543581},[31676],[34,543583],{"className":543584,"style":38486},[31680],[34,543586,543588,543591],{"className":543587},[31685],[34,543589,32936],{"className":543590},[31685,31686],[34,543592,543594],{"className":543593},[31792],[34,543595,543597],{"className":543596},[31796],[34,543598,543600],{"className":543599},[31801],[34,543601,543603],{"className":543602,"style":38486},[31805],[34,543604,543605,543608],{"style":37777},[34,543606],{"className":543607,"style":31814},[31813],[34,543609,543611],{"className":543610},[31818,31819,31820,31821],[34,543612,33759],{"className":543613,"style":33774},[31685,31686,31821],[34,543615],{"className":543616,"style":31846},[31845],[34,543618,202],{"className":543619},[31850],[34,543621],{"className":543622,"style":31846},[31845],[34,543624,543626,543629,543658,543661,543664],{"className":543625},[31676],[34,543627],{"className":543628,"style":37959},[31680],[34,543630,543632,543635],{"className":543631},[31685],[34,543633,32936],{"className":543634},[31685,31686],[34,543636,543638],{"className":543637},[31792],[34,543639,543641],{"className":543640},[31796],[34,543642,543644],{"className":543643},[31801],[34,543645,543647],{"className":543646,"style":37959},[31805],[34,543648,543649,543652],{"style":37777},[34,543650],{"className":543651,"style":31814},[31813],[34,543653,543655],{"className":543654},[31818,31819,31820,31821],[34,543656,31660],{"className":543657},[31685,31686,31821],[34,543659],{"className":543660,"style":32604},[31845],[34,543662,71191],{"className":543663},[32024],[34,543665],{"className":543666,"style":32604},[31845],[34,543668,543670,543673],{"className":543669},[31676],[34,543671],{"className":543672,"style":38486},[31680],[34,543674,543676,543679],{"className":543675},[31685],[34,543677,32936],{"className":543678},[31685,31686],[34,543680,543682],{"className":543681},[31792],[34,543683,543685],{"className":543684},[31796],[34,543686,543688],{"className":543687},[31801],[34,543689,543691],{"className":543690,"style":38486},[31805],[34,543692,543693,543696],{"style":37777},[34,543694],{"className":543695,"style":31814},[31813],[34,543697,543699],{"className":543698},[31818,31819,31820,31821],[34,543700,543702,543705,543708],{"className":543701},[31685,31821],[34,543703,33759],{"className":543704,"style":33774},[31685,31686,31821],[34,543706,32226],{"className":543707},[32024,31821],[34,543709,31660],{"className":543710},[31685,31686,31821],[34,543712,543714,543735],{"className":543713},[31642],[34,543715,543717],{"className":543716},[31646],[31648,543718,543719],{"xmlns":31650},[31652,543720,543721,543733],{},[31655,543722,543723,543729,543731],{},[37624,543724,543725,543727],{},[31658,543726,32936],{},[31658,543728,31660],{},[31662,543730,112296],{},[31658,543732,106063],{},[31665,543734,541958],{"encoding":31667},[34,543736,543738,543782],{"className":543737,"ariaHidden":758},[31672],[34,543739,543741,543744,543773,543776,543779],{"className":543740},[31676],[34,543742],{"className":543743,"style":541968},[31680],[34,543745,543747,543750],{"className":543746},[31685],[34,543748,32936],{"className":543749},[31685,31686],[34,543751,543753],{"className":543752},[31792],[34,543754,543756],{"className":543755},[31796],[34,543757,543759],{"className":543758},[31801],[34,543760,543762],{"className":543761,"style":37959},[31805],[34,543763,543764,543767],{"style":37777},[34,543765],{"className":543766,"style":31814},[31813],[34,543768,543770],{"className":543769},[31818,31819,31820,31821],[34,543771,31660],{"className":543772},[31685,31686,31821],[34,543774],{"className":543775,"style":31846},[31845],[34,543777,112296],{"className":543778},[31850],[34,543780],{"className":543781,"style":31846},[31845],[34,543783,543785,543788],{"className":543784},[31676],[34,543786],{"className":543787,"style":33218},[31680],[34,543789,106063],{"className":543790,"style":67482},[31685,31686],[34,543792,543794,543807],{"className":543793},[31642],[34,543795,543797],{"className":543796},[31646],[31648,543798,543799],{"xmlns":31650},[31652,543800,543801,543805],{},[31655,543802,543803],{},[31658,543804,106063],{},[31665,543806,106063],{"encoding":31667},[34,543808,543810],{"className":543809,"ariaHidden":758},[31672],[34,543811,543813,543816],{"className":543812},[31676],[34,543814],{"className":543815,"style":33218},[31680],[34,543817,106063],{"className":543818,"style":67482},[31685,31686]," 是理想（吸收律），所以 ",[34,543821,543823,543845],{"className":543822},[31642],[34,543824,543826],{"className":543825},[31646],[31648,543827,543828],{"xmlns":31650},[31652,543829,543830,543842],{},[31655,543831,543832,543838,543840],{},[37624,543833,543834,543836],{},[31658,543835,32936],{},[31658,543837,33759],{},[31662,543839,112296],{},[31658,543841,106063],{},[31665,543843,543844],{"encoding":31667},"x^k \\in I",[34,543846,543848,543893],{"className":543847,"ariaHidden":758},[31672],[34,543849,543851,543855,543884,543887,543890],{"className":543850},[31676],[34,543852],{"className":543853,"style":543854},[31680],"height:0.8882em;vertical-align:-0.0391em;",[34,543856,543858,543861],{"className":543857},[31685],[34,543859,32936],{"className":543860},[31685,31686],[34,543862,543864],{"className":543863},[31792],[34,543865,543867],{"className":543866},[31796],[34,543868,543870],{"className":543869},[31801],[34,543871,543873],{"className":543872,"style":38486},[31805],[34,543874,543875,543878],{"style":37777},[34,543876],{"className":543877,"style":31814},[31813],[34,543879,543881],{"className":543880},[31818,31819,31820,31821],[34,543882,33759],{"className":543883,"style":33774},[31685,31686,31821],[34,543885],{"className":543886,"style":31846},[31845],[34,543888,112296],{"className":543889},[31850],[34,543891],{"className":543892,"style":31846},[31845],[34,543894,543896,543899],{"className":543895},[31676],[34,543897],{"className":543898,"style":33218},[31680],[34,543900,106063],{"className":543901,"style":67482},[31685,31686],"。进而整项 ",[34,543904,543906,543928],{"className":543905},[31642],[34,543907,543909],{"className":543908},[31646],[31648,543910,543911],{"xmlns":31650},[31652,543912,543913,543925],{},[31655,543914,543915,543921,543923],{},[31743,543916,543917,543919],{},[31658,543918,37400],{},[31658,543920,33759],{},[31662,543922,112296],{},[31658,543924,106063],{},[31665,543926,543927],{"encoding":31667},"T_k \\in I",[34,543929,543931,543986],{"className":543930,"ariaHidden":758},[31672],[34,543932,543934,543937,543977,543980,543983],{"className":543933},[31676],[34,543935],{"className":543936,"style":70915},[31680],[34,543938,543940,543943],{"className":543939},[31685],[34,543941,37400],{"className":543942,"style":37422},[31685,31686],[34,543944,543946],{"className":543945},[31792],[34,543947,543949,543969],{"className":543948},[31796,31797],[34,543950,543952,543966],{"className":543951},[31801],[34,543953,543955],{"className":543954,"style":33833},[31805],[34,543956,543957,543960],{"style":214424},[34,543958],{"className":543959,"style":31814},[31813],[34,543961,543963],{"className":543962},[31818,31819,31820,31821],[34,543964,33759],{"className":543965,"style":33774},[31685,31686,31821],[34,543967,31829],{"className":543968},[31828],[34,543970,543972],{"className":543971},[31801],[34,543973,543975],{"className":543974,"style":31836},[31805],[34,543976],{},[34,543978],{"className":543979,"style":31846},[31845],[34,543981,112296],{"className":543982},[31850],[34,543984],{"className":543985,"style":31846},[31845],[34,543987,543989,543992],{"className":543988},[31676],[34,543990],{"className":543991,"style":33218},[31680],[34,543993,106063],{"className":543994,"style":67482},[31685,31686],[108,543996,543997,47890,543999,544050,544051,465789,544212,544240,544241,322067,544269,44689,544462,544542,544543,347],{},[2627,543998,327551],{},[34,544000,544002,544020],{"className":544001},[31642],[34,544003,544005],{"className":544004},[31646],[31648,544006,544007],{"xmlns":31650},[31652,544008,544009,544017],{},[31655,544010,544011,544013,544015],{},[31658,544012,33759],{},[31662,544014,932],{},[31658,544016,31660],{},[31665,544018,544019],{"encoding":31667},"k \u003C n",[34,544021,544023,544041],{"className":544022,"ariaHidden":758},[31672],[34,544024,544026,544029,544032,544035,544038],{"className":544025},[31676],[34,544027],{"className":544028,"style":46991},[31680],[34,544030,33759],{"className":544031,"style":33774},[31685,31686],[34,544033],{"className":544034,"style":31846},[31845],[34,544036,932],{"className":544037},[31850],[34,544039],{"className":544040,"style":31846},[31845],[34,544042,544044,544047],{"className":544043},[31676],[34,544045],{"className":544046,"style":32527},[31680],[34,544048,31660],{"className":544049},[31685,31686],"。则指数 ",[34,544052,544054,544092],{"className":544053},[31642],[34,544055,544057],{"className":544056},[31646],[31648,544058,544059],{"xmlns":31650},[31652,544060,544061,544089],{},[31655,544062,544063,544065,544067,544069,544071,544073,544075,544077,544079,544081,544083,544085,544087],{},[31658,544064,31660],{},[31662,544066,223],{},[31658,544068,31759],{},[31662,544070,32226],{},[31658,544072,33759],{},[31662,544074,29622],{},[31658,544076,31660],{},[31662,544078,223],{},[31658,544080,31759],{},[31662,544082,32226],{},[31658,544084,31660],{},[31662,544086,202],{},[31658,544088,31759],{},[31665,544090,544091],{"encoding":31667},"n+m-k > n+m-n = m",[34,544093,544095,544113,544131,544149,544167,544185,544203],{"className":544094,"ariaHidden":758},[31672],[34,544096,544098,544101,544104,544107,544110],{"className":544097},[31676],[34,544099],{"className":544100,"style":32597},[31680],[34,544102,31660],{"className":544103},[31685,31686],[34,544105],{"className":544106,"style":32604},[31845],[34,544108,223],{"className":544109},[32024],[34,544111],{"className":544112,"style":32604},[31845],[34,544114,544116,544119,544122,544125,544128],{"className":544115},[31676],[34,544117],{"className":544118,"style":32597},[31680],[34,544120,31759],{"className":544121},[31685,31686],[34,544123],{"className":544124,"style":32604},[31845],[34,544126,32226],{"className":544127},[32024],[34,544129],{"className":544130,"style":32604},[31845],[34,544132,544134,544137,544140,544143,544146],{"className":544133},[31676],[34,544135],{"className":544136,"style":46991},[31680],[34,544138,33759],{"className":544139,"style":33774},[31685,31686],[34,544141],{"className":544142,"style":31846},[31845],[34,544144,29622],{"className":544145},[31850],[34,544147],{"className":544148,"style":31846},[31845],[34,544150,544152,544155,544158,544161,544164],{"className":544151},[31676],[34,544153],{"className":544154,"style":32597},[31680],[34,544156,31660],{"className":544157},[31685,31686],[34,544159],{"className":544160,"style":32604},[31845],[34,544162,223],{"className":544163},[32024],[34,544165],{"className":544166,"style":32604},[31845],[34,544168,544170,544173,544176,544179,544182],{"className":544169},[31676],[34,544171],{"className":544172,"style":32597},[31680],[34,544174,31759],{"className":544175},[31685,31686],[34,544177],{"className":544178,"style":32604},[31845],[34,544180,32226],{"className":544181},[32024],[34,544183],{"className":544184,"style":32604},[31845],[34,544186,544188,544191,544194,544197,544200],{"className":544187},[31676],[34,544189],{"className":544190,"style":32527},[31680],[34,544192,31660],{"className":544193},[31685,31686],[34,544195],{"className":544196,"style":31846},[31845],[34,544198,202],{"className":544199},[31850],[34,544201],{"className":544202,"style":31846},[31845],[34,544204,544206,544209],{"className":544205},[31676],[34,544207],{"className":544208,"style":32527},[31680],[34,544210,31759],{"className":544211},[31685,31686],[34,544213,544215,544228],{"className":544214},[31642],[34,544216,544218],{"className":544217},[31646],[31648,544219,544220],{"xmlns":31650},[31652,544221,544222,544226],{},[31655,544223,544224],{},[31658,544225,67764],{},[31665,544227,67764],{"encoding":31667},[34,544229,544231],{"className":544230,"ariaHidden":758},[31672],[34,544232,544234,544237],{"className":544233},[31676],[34,544235],{"className":544236,"style":38774},[31680],[34,544238,67764],{"className":544239,"style":32062},[31685,31686]," 的指数至少是 ",[34,544242,544244,544257],{"className":544243},[31642],[34,544245,544247],{"className":544246},[31646],[31648,544248,544249],{"xmlns":31650},[31652,544250,544251,544255],{},[31655,544252,544253],{},[31658,544254,31759],{},[31665,544256,31759],{"encoding":31667},[34,544258,544260],{"className":544259,"ariaHidden":758},[31672],[34,544261,544263,544266],{"className":544262},[31676],[34,544264],{"className":544265,"style":32527},[31680],[34,544267,31759],{"className":544268},[31685,31686],[34,544270,544272,544316],{"className":544271},[31642],[34,544273,544275],{"className":544274},[31646],[31648,544276,544277],{"xmlns":31650},[31652,544278,544279,544313],{},[31655,544280,544281,544297,544299,544305,544307],{},[37624,544282,544283,544285],{},[31658,544284,67764],{},[31655,544286,544287,544289,544291,544293,544295],{},[31658,544288,31660],{},[31662,544290,223],{},[31658,544292,31759],{},[31662,544294,32226],{},[31658,544296,33759],{},[31662,544298,202],{},[37624,544300,544301,544303],{},[31658,544302,67764],{},[31658,544304,31759],{},[31662,544306,71191],{},[37624,544308,544309,544311],{},[31658,544310,67764],{},[31662,544312,33934],{"lspace":116494,"rspace":116494},[31665,544314,544315],{"encoding":31667},"y^{n+m-k} = y^m \\cdot y^{\\dots}",[34,544317,544319,544378,544422],{"className":544318,"ariaHidden":758},[31672],[34,544320,544322,544325,544369,544372,544375],{"className":544321},[31676],[34,544323],{"className":544324,"style":315792},[31680],[34,544326,544328,544331],{"className":544327},[31685],[34,544329,67764],{"className":544330,"style":32062},[31685,31686],[34,544332,544334],{"className":544333},[31792],[34,544335,544337],{"className":544336},[31796],[34,544338,544340],{"className":544339},[31801],[34,544341,544343],{"className":544342,"style":38486},[31805],[34,544344,544345,544348],{"style":37777},[34,544346],{"className":544347,"style":31814},[31813],[34,544349,544351],{"className":544350},[31818,31819,31820,31821],[34,544352,544354,544357,544360,544363,544366],{"className":544353},[31685,31821],[34,544355,31660],{"className":544356},[31685,31686,31821],[34,544358,223],{"className":544359},[32024,31821],[34,544361,31759],{"className":544362},[31685,31686,31821],[34,544364,32226],{"className":544365},[32024,31821],[34,544367,33759],{"className":544368,"style":33774},[31685,31686,31821],[34,544370],{"className":544371,"style":31846},[31845],[34,544373,202],{"className":544374},[31850],[34,544376],{"className":544377,"style":31846},[31845],[34,544379,544381,544384,544413,544416,544419],{"className":544380},[31676],[34,544382],{"className":544383,"style":316112},[31680],[34,544385,544387,544390],{"className":544386},[31685],[34,544388,67764],{"className":544389,"style":32062},[31685,31686],[34,544391,544393],{"className":544392},[31792],[34,544394,544396],{"className":544395},[31796],[34,544397,544399],{"className":544398},[31801],[34,544400,544402],{"className":544401,"style":37959},[31805],[34,544403,544404,544407],{"style":37777},[34,544405],{"className":544406,"style":31814},[31813],[34,544408,544410],{"className":544409},[31818,31819,31820,31821],[34,544411,31759],{"className":544412},[31685,31686,31821],[34,544414],{"className":544415,"style":32604},[31845],[34,544417,71191],{"className":544418},[32024],[34,544420],{"className":544421,"style":32604},[31845],[34,544423,544425,544429],{"className":544424},[31676],[34,544426],{"className":544427,"style":544428},[31680],"height:0.6435em;vertical-align:-0.1944em;",[34,544430,544432,544435],{"className":544431},[31685],[34,544433,67764],{"className":544434,"style":32062},[31685,31686],[34,544436,544438],{"className":544437},[31792],[34,544439,544441],{"className":544440},[31796],[34,544442,544444],{"className":544443},[31801],[34,544445,544448],{"className":544446,"style":544447},[31805],"height:0.4491em;",[34,544449,544450,544453],{"style":37777},[34,544451],{"className":544452,"style":31814},[31813],[34,544454,544456],{"className":544455},[31818,31819,31820,31821],[34,544457,544459],{"className":544458},[31685,31821],[34,544460,33934],{"className":544461},[34006,31821],[34,544463,544465,544486],{"className":544464},[31642],[34,544466,544468],{"className":544467},[31646],[31648,544469,544470],{"xmlns":31650},[31652,544471,544472,544484],{},[31655,544473,544474,544480,544482],{},[37624,544475,544476,544478],{},[31658,544477,67764],{},[31658,544479,31759],{},[31662,544481,112296],{},[31658,544483,106063],{},[31665,544485,542492],{"encoding":31667},[34,544487,544489,544533],{"className":544488,"ariaHidden":758},[31672],[34,544490,544492,544495,544524,544527,544530],{"className":544491},[31676],[34,544493],{"className":544494,"style":316112},[31680],[34,544496,544498,544501],{"className":544497},[31685],[34,544499,67764],{"className":544500,"style":32062},[31685,31686],[34,544502,544504],{"className":544503},[31792],[34,544505,544507],{"className":544506},[31796],[34,544508,544510],{"className":544509},[31801],[34,544511,544513],{"className":544512,"style":37959},[31805],[34,544514,544515,544518],{"style":37777},[34,544516],{"className":544517,"style":31814},[31813],[34,544519,544521],{"className":544520},[31818,31819,31820,31821],[34,544522,31759],{"className":544523},[31685,31686,31821],[34,544525],{"className":544526,"style":31846},[31845],[34,544528,112296],{"className":544529},[31850],[34,544531],{"className":544532,"style":31846},[31845],[34,544534,544536,544539],{"className":544535},[31676],[34,544537],{"className":544538,"style":33218},[31680],[34,544540,106063],{"className":544541,"style":67482},[31685,31686],"，由理想吸收律，整项 ",[34,544544,544546,544567],{"className":544545},[31642],[34,544547,544549],{"className":544548},[31646],[31648,544550,544551],{"xmlns":31650},[31652,544552,544553,544565],{},[31655,544554,544555,544561,544563],{},[31743,544556,544557,544559],{},[31658,544558,37400],{},[31658,544560,33759],{},[31662,544562,112296],{},[31658,544564,106063],{},[31665,544566,543927],{"encoding":31667},[34,544568,544570,544625],{"className":544569,"ariaHidden":758},[31672],[34,544571,544573,544576,544616,544619,544622],{"className":544572},[31676],[34,544574],{"className":544575,"style":70915},[31680],[34,544577,544579,544582],{"className":544578},[31685],[34,544580,37400],{"className":544581,"style":37422},[31685,31686],[34,544583,544585],{"className":544584},[31792],[34,544586,544588,544608],{"className":544587},[31796,31797],[34,544589,544591,544605],{"className":544590},[31801],[34,544592,544594],{"className":544593,"style":33833},[31805],[34,544595,544596,544599],{"style":214424},[34,544597],{"className":544598,"style":31814},[31813],[34,544600,544602],{"className":544601},[31818,31819,31820,31821],[34,544603,33759],{"className":544604,"style":33774},[31685,31686,31821],[34,544606,31829],{"className":544607},[31828],[34,544609,544611],{"className":544610},[31801],[34,544612,544614],{"className":544613,"style":31836},[31805],[34,544615],{},[34,544617],{"className":544618,"style":31846},[31845],[34,544620,112296],{"className":544621},[31850],[34,544623],{"className":544624,"style":31846},[31845],[34,544626,544628,544631],{"className":544627},[31676],[34,544629],{"className":544630,"style":33218},[31680],[34,544632,106063],{"className":544633,"style":67482},[31685,31686],[15,544635,544636,544637,544665,544666,544694,544695,544823,544824,347],{},"综上，展开式中每一项都在 ",[34,544638,544640,544653],{"className":544639},[31642],[34,544641,544643],{"className":544642},[31646],[31648,544644,544645],{"xmlns":31650},[31652,544646,544647,544651],{},[31655,544648,544649],{},[31658,544650,106063],{},[31665,544652,106063],{"encoding":31667},[34,544654,544656],{"className":544655,"ariaHidden":758},[31672],[34,544657,544659,544662],{"className":544658},[31676],[34,544660],{"className":544661,"style":33218},[31680],[34,544663,106063],{"className":544664,"style":67482},[31685,31686]," 中。因为 ",[34,544667,544669,544682],{"className":544668},[31642],[34,544670,544672],{"className":544671},[31646],[31648,544673,544674],{"xmlns":31650},[31652,544675,544676,544680],{},[31655,544677,544678],{},[31658,544679,106063],{},[31665,544681,106063],{"encoding":31667},[34,544683,544685],{"className":544684,"ariaHidden":758},[31672],[34,544686,544688,544691],{"className":544687},[31676],[34,544689],{"className":544690,"style":33218},[31680],[34,544692,106063],{"className":544693,"style":67482},[31685,31686]," 对加法封闭，所以 ",[34,544696,544698,544734],{"className":544697},[31642],[34,544699,544701],{"className":544700},[31646],[31648,544702,544703],{"xmlns":31650},[31652,544704,544705,544731],{},[31655,544706,544707,544709,544711,544713,544715,544727,544729],{},[31662,544708,456],{"stretchy":702},[31658,544710,32936],{},[31662,544712,32226],{},[31658,544714,67764],{},[37624,544716,544717,544719],{},[31662,544718,2557],{"stretchy":702},[31655,544720,544721,544723,544725],{},[31658,544722,31660],{},[31662,544724,223],{},[31658,544726,31759],{},[31662,544728,112296],{},[31658,544730,106063],{},[31665,544732,544733],{"encoding":31667},"(x - y)^{n+m} \\in I",[34,544735,544737,544758,544814],{"className":544736,"ariaHidden":758},[31672],[34,544738,544740,544743,544746,544749,544752,544755],{"className":544739},[31676],[34,544741],{"className":544742,"style":31774},[31680],[34,544744,456],{"className":544745},[31782],[34,544747,32936],{"className":544748},[31685,31686],[34,544750],{"className":544751,"style":32604},[31845],[34,544753,32226],{"className":544754},[32024],[34,544756],{"className":544757,"style":32604},[31845],[34,544759,544761,544764,544767,544805,544808,544811],{"className":544760},[31676],[34,544762],{"className":544763,"style":156809},[31680],[34,544765,67764],{"className":544766,"style":32062},[31685,31686],[34,544768,544770,544773],{"className":544769},[31690],[34,544771,2557],{"className":544772},[31690],[34,544774,544776],{"className":544775},[31792],[34,544777,544779],{"className":544778},[31796],[34,544780,544782],{"className":544781},[31801],[34,544783,544785],{"className":544784,"style":156828},[31805],[34,544786,544787,544790],{"style":37777},[34,544788],{"className":544789,"style":31814},[31813],[34,544791,544793],{"className":544792},[31818,31819,31820,31821],[34,544794,544796,544799,544802],{"className":544795},[31685,31821],[34,544797,31660],{"className":544798},[31685,31686,31821],[34,544800,223],{"className":544801},[32024,31821],[34,544803,31759],{"className":544804},[31685,31686,31821],[34,544806],{"className":544807,"style":31846},[31845],[34,544809,112296],{"className":544810},[31850],[34,544812],{"className":544813,"style":31846},[31845],[34,544815,544817,544820],{"className":544816},[31676],[34,544818],{"className":544819,"style":33218},[31680],[34,544821,106063],{"className":544822,"style":67482},[31685,31686],"。根据定义，这说明 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\\in \\text{rad}I",[34,544934,544936,544954],{"className":544935,"ariaHidden":758},[31672],[34,544937,544939,544942,544945,544948,544951],{"className":544938},[31676],[34,544940],{"className":544941,"style":34812},[31680],[34,544943,32936],{"className":544944},[31685,31686],[34,544946],{"className":544947,"style":31846},[31845],[34,544949,112296],{"className":544950},[31850],[34,544952],{"className":544953,"style":31846},[31845],[34,544955,544957,544960,544966],{"className":544956},[31676],[34,544958],{"className":544959,"style":31681},[31680],[34,544961,544963],{"className":544962},[31685,2638],[34,544964,541377],{"className":544965},[31685],[34,544967,106063],{"className":544968,"style":67482},[31685,31686],[34,544970,544972,544989],{"className":544971},[31642],[34,544973,544975],{"className":544974},[31646],[31648,544976,544977],{"xmlns":31650},[31652,544978,544979,544987],{},[31655,544980,544981,544983,544985],{},[31658,544982,34395],{},[31662,544984,112296],{},[31658,544986,34386],{},[31665,544988,335118],{"encoding":31667},[34,544990,544992,545010],{"className":544991,"ariaHidden":758},[31672],[34,544993,544995,544998,545001,545004,545007],{"className":544994},[31676],[34,544996],{"className":544997,"style":34812},[31680],[34,544999,34395],{"className":545000,"style":34452},[31685,31686],[34,545002],{"className":545003,"style":31846},[31845],[34,545005,112296],{"className":545006},[31850],[34,545008],{"className":545009,"style":31846},[31845],[34,545011,545013,545016],{"className":545012},[31676],[34,545014],{"className":545015,"style":33218},[31680],[34,545017,34386],{"className":545018,"style":34427},[31685,31686],[34,545020,545022,545035],{"className":545021},[31642],[34,545023,545025],{"className":545024},[31646],[31648,545026,545027],{"xmlns":31650},[31652,545028,545029,545033],{},[31655,545030,545031],{},[31658,545032,31660],{},[31665,545034,31660],{"encoding":31667},[34,545036,545038],{"className":545037,"ariaHidden":758},[31672],[34,545039,545041,545044],{"className":545040},[31676],[34,545042],{"className":545043,"style":32527},[31680],[34,545045,31660],{"className":545046},[31685,31686],[34,545048,545050,545071],{"className":545049},[31642],[34,545051,545053],{"className":545052},[31646],[31648,545054,545055],{"xmlns":31650},[31652,545056,545057,545069],{},[31655,545058,545059,545065,545067],{},[37624,545060,545061,545063],{},[31658,545062,32936],{},[31658,545064,31660],{},[31662,545066,112296],{},[31658,545068,106063],{},[31665,545070,541958],{"encoding":31667},[34,545072,545074,545118],{"className":545073,"ariaHidden":758},[31672],[34,545075,545077,545080,545109,545112,545115],{"className":545076},[31676],[34,545078],{"className":545079,"style":541968},[31680],[34,545081,545083,545086],{"className":545082},[31685],[34,545084,32936],{"className":545085},[31685,31686],[34,545087,545089],{"className":545088},[31792],[34,545090,545092],{"className":545091},[31796],[34,545093,545095],{"className":545094},[31801],[34,545096,545098],{"className":545097,"style":37959},[31805],[34,545099,545100,545103],{"style":37777},[34,545101],{"className":545102,"style":31814},[31813],[34,545104,545106],{"className":545105},[31818,31819,31820,31821],[34,545107,31660],{"className":545108},[31685,31686,31821],[34,545110],{"className":545111,"style":31846},[31845],[34,545113,112296],{"className":545114},[31850],[34,545116],{"className":545117,"style":31846},[31845],[34,545119,545121,545124],{"className":545120},[31676],[34,545122],{"className":545123,"style":33218},[31680],[34,545125,106063],{"className":545126,"style":67482},[31685,31686],[34,545128,545130,545150],{"className":545129},[31642],[34,545131,545133],{"className":545132},[31646],[31648,545134,545135],{"xmlns":31650},[31652,545136,545137,545147],{},[31655,545138,545139,545141,545143,545145],{},[31662,545140,456],{"stretchy":702},[31658,545142,34395],{},[31658,545144,32936],{},[31662,545146,2557],{"stretchy":702},[31665,545148,545149],{"encoding":31667},"(rx)",[34,545151,545153],{"className":545152,"ariaHidden":758},[31672],[34,545154,545156,545159,545162,545165,545168],{"className":545155},[31676],[34,545157],{"className":545158,"style":31774},[31680],[34,545160,456],{"className":545161},[31782],[34,545163,34395],{"className":545164,"style":34452},[31685,31686],[34,545166,32936],{"className":545167},[31685,31686],[34,545169,2557],{"className":545170},[31690],[34,545172,545174,545211],{"className":545173},[31642],[34,545175,545177],{"className":545176},[31646],[31648,545178,545179],{"xmlns":31650},[31652,545180,545181,545209],{},[31655,545182,545183,545185,545187,545189,545195,545197,545203],{},[31662,545184,456],{"stretchy":702},[31658,545186,34395],{},[31658,545188,32936],{},[37624,545190,545191,545193],{},[31662,545192,2557],{"stretchy":702},[31658,545194,31660],{},[31662,545196,202],{},[37624,545198,545199,545201],{},[31658,545200,34395],{},[31658,545202,31660],{},[37624,545204,545205,545207],{},[31658,545206,32936],{},[31658,545208,31660],{},[31665,54521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",[34,545333,545335,545348],{"className":545334},[31642],[34,545336,545338],{"className":545337},[31646],[31648,545339,545340],{"xmlns":31650},[31652,545341,545342,545346],{},[31655,545343,545344],{},[31658,545345,34386],{},[31665,545347,34386],{"encoding":31667},[34,545349,545351],{"className":545350,"ariaHidden":758},[31672],[34,545352,545354,545357],{"className":545353},[31676],[34,545355],{"className":545356,"style":33218},[31680],[34,545358,34386],{"className":545359,"style":34427},[31685,31686]," 的交换性）。因为 ",[34,545362,545364,545385],{"className":545363},[31642],[34,545365,545367],{"className":545366},[31646],[31648,545368,545369],{"xmlns":31650},[31652,545370,545371,545383],{},[31655,545372,545373,545379,545381],{},[37624,545374,545375,545377],{},[31658,545376,32936],{},[31658,545378,31660],{},[31662,545380,112296],{},[31658,545382,106063],{},[31665,545384,541958],{"encoding":31667},[34,545386,545388,545432],{"className":545387,"ariaHidden":758},[31672],[34,545389,545391,545394,545423,545426,545429],{"className":545390},[31676],[34,545392],{"className":545393,"style":541968},[31680],[34,545395,545397,545400],{"className":545396},[31685],[34,545398,32936],{"className":545399},[31685,31686],[34,545401,545403],{"className":545402},[31792],[34,545404,545406],{"className":545405},[31796],[34,545407,545409],{"className":545408},[31801],[34,545410,545412],{"className":545411,"style":37959},[31805],[34,545413,545414,545417],{"style":37777},[34,545415],{"className":545416,"style":31814},[31813],[34,545418,545420],{"className":545419},[31818,31819,31820,31821],[34,545421,31660],{"className":545422},[31685,31686,31821],[34,545424],{"className":545425,"style":31846},[31845],[34,545427,112296],{"className":545428},[31850],[34,545430],{"className":545431,"style":31846},[31845],[34,545433,545435,545438],{"className":545434},[31676],[34,545436],{"className":545437,"style":33218},[31680],[34,545439,106063],{"className":545440,"style":67482},[31685,31686],[34,545442,545444,545457],{"className":545443},[31642],[34,545445,545447],{"className":545446},[31646],[31648,545448,545449],{"xmlns":31650},[31652,545450,545451,545455],{},[31655,545452,545453],{},[31658,545454,106063],{},[31665,545456,106063],{"encoding":31667},[34,545458,545460],{"className":545459,"ariaHidden":758},[31672],[34,545461,545463,545466],{"className":545462},[31676],[34,545464],{"className":545465,"style":33218},[31680],[34,545467,106063],{"className":545468,"style":67482},[31685,31686]," 是理想，对于环中元素 ",[34,545471,545473,545491],{"className":545472},[31642],[34,545474,545476],{"className":545475},[31646],[31648,545477,545478],{"xmlns":31650},[31652,545479,545480,545488],{},[31655,545481,545482],{},[37624,545483,545484,545486],{},[31658,545485,34395],{},[31658,545487,31660],{},[31665,545489,545490],{"encoding":31667},"r^n",[34,545492,545494],{"className":545493,"ariaHidden":758},[31672],[34,545495,545497,545500],{"className":545496},[31676],[34,545498],{"className":545499,"style":37959},[31680],[34,545501,545503,545506],{"className":545502},[31685],[34,545504,34395],{"className":545505,"style":34452},[31685,31686],[34,545507,545509],{"className":545508},[31792],[34,545510,545512],{"className":545511},[31796],[34,545513,545515],{"className":545514},[31801],[34,545516,545518],{"className":545517,"style":37959},[31805],[34,545519,545520,545523],{"style":37777},[34,545521],{"className":545522,"style":31814},[31813],[34,545524,545526],{"className":545525},[31818,31819,31820,31821],[34,545527,31660],{"className":545528},[31685,31686,31821],[34,545530,545532,545562],{"className":545531},[31642],[34,545533,545535],{"className":545534},[31646],[31648,545536,545537],{"xmlns":31650},[31652,545538,545539,545559],{},[31655,545540,545541,545547,545549,545555,545557],{},[37624,545542,545543,545545],{},[31658,545544,34395],{},[31658,545546,31660],{},[31662,545548,71191],{},[37624,545550,545551,545553],{},[31658,545552,32936],{},[31658,545554,31660],{},[31662,545556,112296],{},[31658,545558,106063],{},[31665,545560,545561],{"encoding":31667},"r^n \\cdot x^n \\in I",[34,545563,545565,545609,545653],{"className":545564,"ariaHidden":758},[31672],[34,545566,545568,545571,545600,545603,545606],{"className":545567},[31676],[34,545569],{"className":545570,"style":37959},[31680],[34,545572,545574,545577],{"className":545573},[31685],[34,545575,34395],{"className":545576,"style":34452},[31685,31686],[34,545578,545580],{"className":545579},[31792],[34,545581,545583],{"className":545582},[31796],[34,545584,545586],{"className":545585},[31801],[34,545587,545589],{"className":545588,"style":37959},[31805],[34,545590,545591,545594],{"style":37777},[34,545592],{"className":545593,"style":31814},[31813],[34,545595,545597],{"className":545596},[31818,31819,31820,31821],[34,545598,31660],{"className":545599},[31685,31686,31821],[34,545601],{"className":545602,"style":32604},[31845],[34,545604,71191],{"className":545605},[32024],[34,545607],{"className":545608,"style":32604},[31845],[34,545610,545612,545615,545644,545647,545650],{"className":545611},[31676],[34,545613],{"className":545614,"style":541968},[31680],[34,545616,545618,545621],{"className":545617},[31685],[34,545619,32936],{"className":545620},[31685,31686],[34,545622,545624],{"className":545623},[31792],[34,545625,545627],{"className":545626},[31796],[34,545628,545630],{"className":545629},[31801],[34,545631,545633],{"className":545632,"style":37959},[31805],[34,545634,545635,545638],{"style":37777},[34,545636],{"className":545637,"style":31814},[31813],[34,545639,545641],{"className":545640},[31818,31819,31820,31821],[34,545642,31660],{"className":545643},[31685,31686,31821],[34,545645],{"className":545646,"style":31846},[31845],[34,545648,112296],{"className":545649},[31850],[34,545651],{"className":545652,"style":31846},[31845],[34,545654,545656,545659],{"className":545655},[31676],[34,545657],{"className":545658,"style":33218},[31680],[34,545660,106063],{"className":545661,"style":67482},[31685,31686],[34,545663,545665,545693],{"className":545664},[31642],[34,545666,545668],{"className":545667},[31646],[31648,545669,545670],{"xmlns":31650},[31652,545671,545672,545690],{},[31655,545673,545674,545676,545678,545680,545686,545688],{},[31662,545675,456],{"stretchy":702},[31658,545677,34395],{},[31658,545679,32936],{},[37624,545681,545682,545684],{},[31662,545683,2557],{"stretchy":702},[31658,545685,31660],{},[31662,545687,112296],{},[31658,545689,106063],{},[31665,545691,545692],{"encoding":31667},"(rx)^n \\in I",[34,545694,545696,545749],{"className":545695,"ariaHidden":758},[31672],[34,545697,545699,545702,545705,545708,545711,545740,545743,545746],{"className":545698},[31676],[34,545700],{"className":545701,"style":31774},[31680],[34,545703,456],{"className":545704},[31782],[34,545706,34395],{"className":545707,"style":34452},[31685,31686],[34,545709,32936],{"className":545710},[31685,31686],[34,545712,545714,545717],{"className":545713},[31690],[34,545715,2557],{"className":545716},[31690],[34,545718,545720],{"className":545719},[31792],[34,545721,545723],{"className":545722},[31796],[34,545724,545726],{"className":545725},[31801],[34,545727,545729],{"className":545728,"style":37959},[31805],[34,545730,545731,545734],{"style":37777},[34,545732],{"className":545733,"style":31814},[31813],[34,545735,545737],{"className":545736},[31818,31819,31820,31821],[34,545738,31660],{"className":545739},[31685,31686,31821],[34,545741],{"className":545742,"style":31846},[31845],[34,545744,112296],{"className":545745},[31850],[34,545747],{"className":545748,"style":31846},[31845],[34,545750,545752,545755],{"className":545751},[31676],[34,545753],{"className":545754,"style":33218},[31680],[34,545756,106063],{"className":545757,"style":67482},[31685,31686],"。根据定义，",[34,545760,545762,545784],{"className":545761},[31642],[34,545763,545765],{"className":545764},[31646],[31648,545766,545767],{"xmlns":31650},[31652,545768,545769,545781],{},[31655,545770,545771,545773,545775,545777,545779],{},[31658,545772,34395],{},[31658,545774,32936],{},[31662,545776,112296],{},[39361,545778,541377],{},[31658,545780,106063],{},[31665,545782,545783],{"encoding":31667},"rx \\in \\text{rad}I",[34,545785,545787,545808],{"className":545786,"ariaHidden":758},[31672],[34,545788,545790,545793,545796,545799,545802,545805],{"className":545789},[31676],[34,545791],{"className":545792,"style":34812},[31680],[34,545794,34395],{"className":545795,"style":34452},[31685,31686],[34,545797,32936],{"className":545798},[31685,31686],[34,545800],{"className":545801,"style":31846},[31845],[34,545803,112296],{"className":545804},[31850],[34,545806],{"className":545807,"style":31846},[31845],[34,545809,545811,545814,545820],{"className":545810},[31676],[34,545812],{"className":545813,"style":31681},[31680],[34,545815,545817],{"className":545816},[31685,2638],[34,545818,541377],{"className":545819},[31685],[34,545821,106063],{"className":545822,"style":67482},[31685,31686],[15,545824,545825,2664,545827,11008,545863,469010,545891],{},[2627,545826,114043],{},[34,545828,545830,545845],{"className":545829},[31642],[34,545831,545833],{"className":545832},[31646],[31648,545834,545835],{"xmlns":31650},[31652,545836,545837,545843],{},[31655,545838,545839,545841],{},[39361,545840,541377],{},[31658,545842,106063],{},[31665,545844,541588],{"encoding":31667},[34,545846,545848],{"className":545847,"ariaHidden":758},[31672],[34,545849,545851,545854,545860],{"className":545850},[31676],[34,545852],{"className":545853,"style":31681},[31680],[34,545855,545857],{"className":545856},[31685,2638],[34,545858,541377],{"className":545859},[31685],[34,545861,106063],{"className":545862,"style":67482},[31685,31686],[34,545864,545866,545879],{"className":545865},[31642],[34,545867,545869],{"className":545868},[31646],[31648,545870,545871],{"xmlns":31650},[31652,545872,545873,545877],{},[31655,545874,545875],{},[31658,545876,34386],{},[31665,545878,34386],{"encoding":31667},[34,545880,545882],{"className":545881,"ariaHidden":758},[31672],[34,545883,545885,545888],{"className":545884},[31676],[34,545886],{"className":545887,"style":33218},[31680],[34,545889,34386],{"className":545890,"style":34427},[31685,31686],[34,545892,545894,545907],{"className":545893},[31642],[34,545895,545897],{"className":545896},[31646],[31648,545898,545899],{"xmlns":31650},[31652,545900,545901,545905],{},[31655,545902,545903],{},[31658,545904,445204],{"mathvariant":38079},[31665,545906,445207],{"encoding":31667},[34,545908,545910],{"className":545909,"ariaHidden":758},[31672],[34,545911,545913,545916],{"className":545912},[31676],[34,545914],{"className":545915,"style":445217},[31680],[34,545917,445204],{"className":545918},[31685,129261],[32188,545920],{},[253,545922,545923],{"id":545923},"专家点评与总结",[2230,545925,545926,546098,546201],{},[108,545927,545928,545931,545932,545960,545961,546046,546047,546097],{},[2627,545929,545930],{},"关于反例 (题8)","：做这类题时，手头常备几个“反例库”：",[34,545933,545935,545948],{"className":545934},[31642],[34,545936,545938],{"className":545937},[31646],[31648,545939,545940],{"xmlns":31650},[31652,545941,545942,545946],{},[31655,545943,545944],{},[31658,545945,97300],{"mathvariant":318416},[31665,545947,320917],{"encoding":31667},[34,545949,545951],{"className":545950,"ariaHidden":758},[31672],[34,545952,545954,545957],{"className":545953},[31676],[34,545955],{"className":545956,"style":320927},[31680],[34,545958,97300],{"className":545959},[31685,318441],"（整数）、",[34,545962,545964,545988],{"className":545963},[31642],[34,545965,545967],{"className":545966},[31646],[31648,545968,545969],{"xmlns":31650},[31652,545970,545971,545985],{},[31655,545972,545973,545979,545981,545983],{},[31743,545974,545975,545977],{},[31658,545976,98360],{},[31749,545978,768],{},[31662,545980,456],{"stretchy":702},[31658,545982,34386],{"mathvariant":318416},[31662,545984,2557],{"stretchy":702},[31665,545986,545987],{"encoding":31667},"M_2(\\mathbb{R})",[34,545989,545991],{"className":545990,"ariaHidden":758},[31672],[34,545992,545994,545997,546037,546040,546043],{"className":545993},[31676],[34,545995],{"className":545996,"style":31774},[31680],[34,545998,546000,546003],{"className":545999},[31685],[34,546001,98360],{"className":546002,"style":33389},[31685,31686],[34,546004,546006],{"className":546005},[31792],[34,546007,546009,546029],{"className":546008},[31796,31797],[34,546010,546012,546026],{"className":546011},[31801],[34,546013,546015],{"className":546014,"style":31806},[31805],[34,546016,546017,546020],{"style":231884},[34,546018],{"className":546019,"style":31814},[31813],[34,546021,546023],{"className":546022},[31818,31819,31820,31821],[34,546024,768],{"className":546025},[31685,31821],[34,546027,31829],{"className":546028},[31828],[34,546030,546032],{"className":546031},[31801],[34,546033,546035],{"className":546034,"style":31836},[31805],[34,546036],{},[34,546038,456],{"className":546039},[31782],[34,546041,34386],{"className":546042},[31685,318441],[34,546044,2557],{"className":546045},[31690],"（矩阵）、",[34,546048,546050,546067],{"className":546049},[31642],[34,546051,546053],{"className":546052},[31646],[31648,546054,546055],{"xmlns":31650},[31652,546056,546057,546065],{},[31655,546058,546059,546061,546063],{},[31658,546060,97300],{"mathvariant":318416},[31662,546062,378006],{},[31658,546064,97300],{"mathvariant":318416},[31665,546066,496155],{"encoding":31667},[34,546068,546070,546088],{"className":546069,"ariaHidden":758},[31672],[34,546071,546073,546076,546079,546082,546085],{"className":546072},[31676],[34,546074],{"className":546075,"style":379449},[31680],[34,546077,97300],{"className":546078},[31685,318441],[34,546080],{"className":546081,"style":32604},[31845],[34,546083,378006],{"className":546084},[32024],[34,546086],{"className":546087,"style":32604},[31845],[34,546089,546091,546094],{"className":546090},[31676],[34,546092],{"className":546093,"style":320927},[31680],[34,546095,97300],{"className":546096},[31685,318441],"（直积）。这三个能解决90%的反例构造问题。",[108,546099,546100,546103,546104,546196,546197,546200],{},[2627,546101,546102],{},"关于二项式定理 (题14, 16)","：这两个证明的核心都在于 ",[34,546105,546107,546133],{"className":546106},[31642],[34,546108,546110],{"className":546109},[31646],[31648,546111,546112],{"xmlns":31650},[31652,546113,546114,546130],{},[31655,546115,546116,546118,546120,546122,546124],{},[31662,546117,456],{"stretchy":702},[31658,546119,32936],{},[31662,546121,223],{},[31658,546123,67764],{},[37624,546125,546126,546128],{},[31662,546127,2557],{"stretchy":702},[31658,546129,33347],{},[31665,546131,546132],{"encoding":31667},"(x+y)^N",[34,546134,546136,546157],{"className":546135,"ariaHidden":758},[31672],[34,546137,546139,546142,546145,546148,546151,546154],{"className":546138},[31676],[34,546140],{"className":546141,"style":31774},[31680],[34,546143,456],{"className":546144},[31782],[34,546146,32936],{"className":546147},[31685,31686],[34,546149],{"className":546150,"style":32604},[31845],[34,546152,223],{"className":546153},[32024],[34,546155],{"className":546156,"style":32604},[31845],[34,546158,546160,546164,546167],{"className":546159},[31676],[34,546161],{"className":546162,"style":546163},[31680],"height:1.0913em;vertical-align:-0.25em;",[34,546165,67764],{"className":546166,"style":32062},[31685,31686],[34,546168,546170,546173],{"className":546169},[31690],[34,546171,2557],{"className":546172},[31690],[34,546174,546176],{"className":546175},[31792],[34,546177,546179],{"className":546178},[31796],[34,546180,546182],{"className":546181},[31801],[34,546183,546185],{"className":546184,"style":194117},[31805],[34,546186,546187,546190],{"style":37777},[34,546188],{"className":546189,"style":31814},[31813],[34,546191,546193],{"className":546192},[31818,31819,31820,31821],[34,546194,33347],{"className":546195,"style":33389},[31685,31686,31821]," 的展开。如果环",[2627,546198,546199],{},"不交换","，二项式定理失效，这些结论通常就不成立了。这是考试中经常考察的概念陷阱。",[108,546202,546203,546206,546207,546210,546211,546214],{},[2627,546204,546205],{},"理想的本质","：题15和16再次印证了理想的两个核心性质：",[2627,546208,546209],{},"内部加法封闭","（像子群）和",[2627,546212,546213],{},"外部乘法吸收","（像黑洞）。任何证明都要紧扣这两点。",{"title":29,"searchDepth":43,"depth":43,"links":546216},[546217,546218,546219,546220,546221,546222,546223,546224,546225],{"id":512618,"depth":49,"text":512621},{"id":515277,"depth":49,"text":515280},{"id":520705,"depth":49,"text":520708},{"id":526383,"depth":49,"text":526386},{"id":528945,"depth":49,"text":528946},{"id":533621,"depth":49,"text":533622},{"id":537733,"depth":49,"text":537734},{"id":541298,"depth":49,"text":541299},{"id":545923,"depth":49,"text":545923},{"order":62},"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fsubringidealhw",{"title":512613,"description":29},"source\u002Fmath\u002Fabstract_algebra\u002FsubringidealHW","aenS0MCbNrytE3GIRhcG17ws9mf0P59YBcr4d70L7RE",{"id":546232,"title":546233,"body":546234,"description":546245,"extension":85,"meta":546246,"navigation":52,"path":546247,"seo":546248,"stem":546249,"__hash__":546250},"notes\u002Fsource\u002Fmath\u002Fmath.md","数学笔记索引",{"type":8,"value":546235,"toc":546243},[546236,546239],[11,546237,546238],{"id":546238},"数学笔记",[546240,546241],"category-index",{"path":546242},"\u002Fmath\u002F",{"title":29,"searchDepth":43,"depth":43,"links":546244},[],"数学科目学习笔记，包含抽象代数（璯论、群论）、数学分析、常微分方程（ODE）等课程内容与习题解析。",{"layout":162022,"sidebar":162023},"\u002Fsource\u002Fmath\u002Fmath",{"title":546233,"description":546245},"source\u002Fmath\u002Fmath","Itnj4mu5y9mkK232byFl9oiNRy13LHcqf8eUNMDnaZI",{"id":546252,"title":546253,"body":546254,"description":29,"extension":85,"meta":587738,"navigation":52,"path":587739,"seo":587740,"stem":587741,"__hash__":587742},"notes\u002Fsource\u002Fmath\u002Fmath_analysis\u002F11-29HW.md","11-29 作业",{"type":8,"value":546255,"toc":587718},[546256,546260,546262,546266,546272,546276,546673,546724,546728,546901,547366,547372,547376,547379,548105,548107,548111,548115,549093,549095,549101,549445,549625,549630,551694,551696,551702,552077,552424,552428,553861,553863,553869,554117,554200,554204,560539,560541,560545,560550,560555,560919,560926,561871,561876,562092,562467,562546,562941,563029,563241,563246,563511,563629,564814,564869,565157,565163,565342,565460,565821,565876,566099,566101,566105,566440,566609,566613,567787,567789,567793,568137,568244,568248,569767,569769,569773,570144,570153,570216,570220,573103,573105,573109,573821,573830,573834,575877,575879,575883,576556,576565,576569,582846,582848,582852,583074,583082,583086],[11,546257,546259],{"id":546258},"november-29-hw","November 29 HW",[32188,546261],{},[11,546263,546265],{"id":546264},"第一部分本课核心知识点重讲基础篇","第一部分：本课核心知识点重讲（基础篇）",[15,546267,546268,546269,347],{},"本节核心关键词：",[2627,546270,546271],{},"\"一致收敛\"（Uniform Convergence）",[253,546273,546275],{"id":546274},"_1-什么是含参变量反常积分","1. 什么是“含参变量反常积分”？",[15,546277,546278,546279,546408,546409,546412,546413,546441,546442,546640,546641,546669,546670,347],{},"普通的定积分 ",[34,546280,546282,546315],{"className":546281},[31642],[34,546283,546285],{"className":546284},[31646],[31648,546286,546287],{"xmlns":31650},[31652,546288,546289,546312],{},[31655,546290,546291,546300,546302,546304,546306,546308,546310],{},[38846,546292,546293,546296,546298],{},[31662,546294,546295],{},"∫",[31658,546297,111],{},[31658,546299,1697],{},[31658,546301,31739],{},[31662,546303,456],{"stretchy":702},[31658,546305,32936],{},[31662,546307,2557],{"stretchy":702},[31658,546309,46900],{},[31658,546311,32936],{},[31665,546313,546314],{"encoding":31667},"\\int_{a}^{b} f(x)dx",[34,546316,546318],{"className":546317,"ariaHidden":758},[31672],[34,546319,546321,546325,546387,546390,546393,546396,546399,546402,546405],{"className":546320},[31676],[34,546322],{"className":546323,"style":546324},[31680],"height:1.3998em;vertical-align:-0.3558em;",[34,546326,546328,546332],{"className":546327},[32767],[34,546329,546295],{"className":546330,"style":546331},[32767,131249,156560],"margin-right:0.1945em;position:relative;top:-0.0006em;",[34,546333,546335],{"className":546334},[31792],[34,546336,546338,546378],{"className":546337},[31796,31797],[34,546339,546341,546375],{"className":546340},[31801],[34,546342,546345,546360],{"className":546343,"style":546344},[31805],"height:1.044em;",[34,546346,546348,546351],{"style":546347},"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;",[34,546349],{"className":546350,"style":31814},[31813],[34,546352,546354],{"className":546353},[31818,31819,31820,31821],[34,546355,546357],{"className":546356},[31685,31821],[34,546358,111],{"className":546359},[31685,31686,31821],[34,546361,546363,546366],{"style":546362},"top:-3.2579em;margin-right:0.05em;",[34,546364],{"className":546365,"style":31814},[31813],[34,546367,546369],{"className":546368},[31818,31819,31820,31821],[34,546370,546372],{"className":546371},[31685,31821],[34,546373,1697],{"className":546374},[31685,31686,31821],[34,546376,31829],{"className":546377},[31828],[34,546379,546381],{"className":546380},[31801],[34,546382,546385],{"className":546383,"style":546384},[31805],"height:0.3558em;",[34,546386],{},[34,546388],{"className":546389,"style":32123},[31845],[34,546391,31739],{"className":546392,"style":31778},[31685,31686],[34,546394,456],{"className":546395},[31782],[34,546397,32936],{"className":546398},[31685,31686],[34,546400,2557],{"className":546401},[31690],[34,546403,46900],{"className":546404},[31685,31686],[34,546406,32936],{"className":546407},[31685,31686]," 算出来是一个",[2627,546410,546411],{},"数","。\n但如果积分里还有一个变量（参数）",[34,546414,546416,546429],{"className":546415},[31642],[34,546417,546419],{"className":546418},[31646],[31648,546420,546421],{"xmlns":31650},[31652,546422,546423,546427],{},[31655,546424,546425],{},[31658,546426,217392],{},[31665,546428,218244],{"encoding":31667},[34,546430,546432],{"className":546431,"ariaHidden":758},[31672],[34,546433,546435,546438],{"className":546434},[31676],[34,546436],{"className":546437,"style":32527},[31680],[34,546439,217392],{"className":546440,"style":217496},[31685,31686],"，比如 ",[34,546443,546445,546495],{"className":546444},[31642],[34,546446,546448],{"className":546447},[31646],[31648,546449,546450],{"xmlns":31650},[31652,546451,546452,546492],{},[31655,546453,546454,546456,546458,546460,546462,546464,546476,546488,546490],{},[31658,546455,51834],{},[31662,546457,456],{"stretchy":702},[31658,546459,217392],{},[31662,546461,2557],{"stretchy":702},[31662,546463,202],{},[38846,546465,546466,546468,546470],{},[31662,546467,546295],{},[31749,546469,459],{},[31655,546471,546472,546474],{},[31662,546473,223],{},[31658,546475,138286],{"mathvariant":38079},[37624,546477,546478,546480],{},[31658,546479,33244],{},[31655,546481,546482,546484,546486],{},[31662,546483,32226],{},[31658,546485,217392],{},[31658,546487,32936],{},[31658,546489,46900],{},[31658,546491,32936],{},[31665,546493,546494],{"encoding":31667},"F(\\alpha) = \\int_{0}^{+\\infty} e^{-\\alpha x} dx",[34,546496,546498,546525],{"className":546497,"ariaHidden":758},[31672],[34,546499,546501,546504,546507,546510,546513,546516,546519,546522],{"className":546500},[31676],[34,546502],{"className":546503,"style":31774},[31680],[34,546505,51834],{"className":546506,"style":37422},[31685,31686],[34,546508,456],{"className":546509},[31782],[34,546511,217392],{"className":546512,"style":217496},[31685,31686],[34,546514,2557],{"className":546515},[31690],[34,546517],{"className":546518,"style":31846},[31845],[34,546520,202],{"className":546521},[31850],[34,546523],{"className":546524,"style":31846},[31845],[34,546526,546528,546532,546593,546596,546634,546637],{"className":546527},[31676],[34,546529],{"className":546530,"style":546531},[31680],"height:1.3221em;vertical-align:-0.3558em;",[34,546533,546535,546538],{"className":546534},[32767],[34,546536,546295],{"className":546537,"style":546331},[32767,131249,156560],[34,546539,546541],{"className":546540},[31792],[34,546542,546544,546585],{"className":546543},[31796,31797],[34,546545,546547,546582],{"className":546546},[31801],[34,546548,546551,546565],{"className":546549,"style":546550},[31805],"height:0.9662em;",[34,546552,546553,546556],{"style":546347},[34,546554],{"className":546555,"style":31814},[31813],[34,546557,546559],{"className":546558},[31818,31819,31820,31821],[34,546560,546562],{"className":546561},[31685,31821],[34,546563,459],{"className":546564},[31685,31821],[34,546566,546567,546570],{"style":546362},[34,546568],{"className":546569,"style":31814},[31813],[34,546571,546573],{"className":546572},[31818,31819,31820,31821],[34,546574,546576,546579],{"className":546575},[31685,31821],[34,546577,223],{"className":546578},[31685,31821],[34,546580,138286],{"className":546581},[31685,31821],[34,546583,31829],{"className":546584},[31828],[34,546586,546588],{"className":546587},[31801],[34,546589,546591],{"className":546590,"style":546384},[31805],[34,546592],{},[34,546594],{"className":546595,"style":32123},[31845],[34,546597,546599,546602],{"className":546598},[31685],[34,546600,33244],{"className":546601},[31685,31686],[34,546603,546605],{"className":546604},[31792],[34,546606,546608],{"className":546607},[31796],[34,546609,546611],{"className":546610},[31801],[34,546612,546614],{"className":546613,"style":156828},[31805],[34,546615,546616,546619],{"style":37777},[34,546617],{"className":546618,"style":31814},[31813],[34,546620,546622],{"className":546621},[31818,31819,31820,31821],[34,546623,546625,546628,546631],{"className":546624},[31685,31821],[34,546626,32226],{"className":546627},[31685,31821],[34,546629,217392],{"className":546630,"style":217496},[31685,31686,31821],[34,546632,32936],{"className":546633},[31685,31686,31821],[34,546635,46900],{"className":546636},[31685,31686],[34,546638,32936],{"className":546639},[31685,31686],"，那么算出来的结果就是 ",[34,546642,546644,546657],{"className":546643},[31642],[34,546645,546647],{"className":546646},[31646],[31648,546648,546649],{"xmlns":31650},[31652,546650,546651,546655],{},[31655,546652,546653],{},[31658,546654,217392],{},[31665,546656,218244],{"encoding":31667},[34,546658,546660],{"className":546659,"ariaHidden":758},[31672],[34,546661,546663,546666],{"className":546662},[31676],[34,546664],{"className":546665,"style":32527},[31680],[34,546667,217392],{"className":546668,"style":217496},[31685,31686]," 的一个",[2627,546671,546672],{},"函数",[15,546674,546675,546676],{},"我们要研究的问题是：",[2627,546677,546678,546679,546723],{},"这个由积分定义出来的函数 ",[34,546680,546682,546702],{"className":546681},[31642],[34,546683,546685],{"className":546684},[31646],[31648,546686,546687],{"xmlns":31650},[31652,546688,546689,546699],{},[31655,546690,546691,546693,546695,546697],{},[31658,546692,51834],{},[31662,546694,456],{"stretchy":702},[31658,546696,217392],{},[31662,546698,2557],{"stretchy":702},[31665,546700,546701],{"encoding":31667},"F(\\alpha)",[34,546703,546705],{"className":546704,"ariaHidden":758},[31672],[34,546706,546708,546711,546714,546717,546720],{"className":546707},[31676],[34,546709],{"className":546710,"style":31774},[31680],[34,546712,51834],{"className":546713,"style":37422},[31685,31686],[34,546715,456],{"className":546716},[31782],[34,546718,217392],{"className":546719,"style":217496},[31685,31686],[34,546721,2557],{"className":546722},[31690],"，它的性质（连续性、可导性）好不好？",[253,546725,546727],{"id":546726},"_2-核心难题能否交换顺序","2. 核心难题：能否交换顺序？",[15,546729,546730,546731,546763,546764,546869,546870,546900],{},"数学分析里最喜欢问：能不能把极限符号 ",[34,546732,546734,546751],{"className":546733},[31642],[34,546735,546737],{"className":546736},[31646],[31648,546738,546739],{"xmlns":31650},[31652,546740,546741,546748],{},[31655,546742,546743,546746],{},[31658,546744,546745],{},"lim",[31662,546747,32682],{},[31665,546749,546750],{"encoding":31667},"\\lim",[34,546752,546754],{"className":546753,"ariaHidden":758},[31672],[34,546755,546757,546760],{"className":546756},[31676],[34,546758],{"className":546759,"style":31681},[31680],[34,546761,546745],{"className":546762},[32767],"、求导符号 ",[34,546765,546767,546789],{"className":546766},[31642],[34,546768,546770],{"className":546769},[31646],[31648,546771,546772],{"xmlns":31650},[31652,546773,546774,546786],{},[31655,546775,546776],{},[41453,546777,546778,546780],{},[31658,546779,46900],{},[31655,546781,546782,546784],{},[31658,546783,46900],{},[31658,546785,217392],{},[31665,546787,546788],{"encoding":31667},"\\frac{d}{d\\alpha}",[34,546790,546792],{"className":546791,"ariaHidden":758},[31672],[34,546793,546795,546798],{"className":546794},[31676],[34,546796],{"className":546797,"style":232492},[31680],[34,546799,546801,546804,546866],{"className":546800},[31685],[34,546802],{"className":546803},[31782,41488],[34,546805,546807],{"className":546806},[41453],[34,546808,546810,546858],{"className":546809},[31796,31797],[34,546811,546813,546855],{"className":546812},[31801],[34,546814,546816,546833,546841],{"className":546815,"style":232511},[31805],[34,546817,546818,546821],{"style":41504},[34,546819],{"className":546820,"style":41508},[31813],[34,546822,546824],{"className":546823},[31818,31819,31820,31821],[34,546825,546827,546830],{"className":546826},[31685,31821],[34,546828,46900],{"className":546829},[31685,31686,31821],[34,546831,217392],{"className":546832,"style":217496},[31685,31686,31821],[34,546834,546835,546838],{"style":41520},[34,546836],{"className":546837,"style":41508},[31813],[34,546839],{"className":546840,"style":41528},[41527],[34,546842,546843,546846],{"style":46771},[34,546844],{"className":546845,"style":41508},[31813],[34,546847,546849],{"className":546848},[31818,31819,31820,31821],[34,546850,546852],{"className":546851},[31685,31821],[34,546853,46900],{"className":546854},[31685,31686,31821],[34,546856,31829],{"className":546857},[31828],[34,546859,546861],{"className":546860},[31801],[34,546862,546864],{"className":546863,"style":41619},[31805],[34,546865],{},[34,546867],{"className":546868},[31690,41488],"、积分符号 ",[34,546871,546873,546887],{"className":546872},[31642],[34,546874,546876],{"className":546875},[31646],[31648,546877,546878],{"xmlns":31650},[31652,546879,546880,546884],{},[31655,546881,546882],{},[31662,546883,546295],{},[31665,546885,546886],{"encoding":31667},"\\int",[34,546888,546890],{"className":546889,"ariaHidden":758},[31672],[34,546891,546893,546897],{"className":546892},[31676],[34,546894],{"className":546895,"style":546896},[31680],"height:1.1111em;vertical-align:-0.3061em;",[34,546898,546295],{"className":546899,"style":546331},[32767,131249,156560]," 搬进积分号里面去？\n也就是：",[105,546902,546903,547203],{},[108,546904,546905,47649,546908,547202],{},[2627,546906,546907],{},"极限：",[34,546909,546911,546981],{"className":546910},[31642],[34,546912,546914],{"className":546913},[31646],[31648,546915,546916],{"xmlns":31650},[31652,546917,546918,546978],{},[31655,546919,546920,546940,546942,546944,546946,546948,546950,546952,546972,546974,546976],{},[31743,546921,546922,546928],{},[31655,546923,546924,546926],{},[31658,546925,546745],{},[31662,546927,32682],{},[31655,546929,546930,546932,546934],{},[31658,546931,217392],{},[31662,546933,33239],{},[31743,546935,546936,546938],{},[31658,546937,217392],{},[31749,546939,459],{},[31662,546941,546295],{},[31658,546943,2547],{"mathvariant":38079},[31658,546945,2547],{"mathvariant":38079},[31658,546947,2547],{"mathvariant":38079},[31662,546949,202],{},[31662,546951,546295],{},[31743,546953,546954,546960],{},[31655,546955,546956,546958],{},[31658,546957,546745],{},[31662,546959,32682],{},[31655,546961,546962,546964,546966],{},[31658,546963,217392],{},[31662,546965,33239],{},[31743,546967,546968,546970],{},[31658,546969,217392],{},[31749,546971,459],{},[31658,546973,2547],{"mathvariant":38079},[31658,546975,2547],{"mathvariant":38079},[31658,546977,2547],{"mathvariant":38079},[31665,546979,546980],{"encoding":31667},"\\lim_{\\alpha \\to \\alpha_0} \\int ... = \\int \\lim_{\\alpha \\to \\alpha_0} ...",[34,546982,546984,547098],{"className":546983,"ariaHidden":758},[31672],[34,546985,546987,546990,547077,547080,547083,547086,547089,547092,547095],{"className":546986},[31676],[34,546988],{"className":546989,"style":546896},[31680],[34,546991,546993,546996],{"className":546992},[32767],[34,546994,546745],{"className":546995},[32767],[34,546997,546999],{"className":546998},[31792],[34,547000,547002,547069],{"className":547001},[31796,31797],[34,547003,547005,547066],{"className":547004},[31801],[34,547006,547008],{"className":547007,"style":32362},[31805],[34,547009,547010,547013],{"style":56235},[34,547011],{"className":547012,"style":31814},[31813],[34,547014,547016],{"className":547015},[31818,31819,31820,31821],[34,547017,547019,547022,547025],{"className":547018},[31685,31821],[34,547020,217392],{"className":547021,"style":217496},[31685,31686,31821],[34,547023,33239],{"className":547024},[31850,31821],[34,547026,547028,547031],{"className":547027},[31685,31821],[34,547029,217392],{"className":547030,"style":217496},[31685,31686,31821],[34,547032,547034],{"className":547033},[31792],[34,547035,547037,547058],{"className":547036},[31796,31797],[34,547038,547040,547055],{"className":547039},[31801],[34,547041,547043],{"className":547042,"style":318658},[31805],[34,547044,547046,547049],{"style":547045},"top:-2.357em;margin-left:-0.0037em;margin-right:0.0714em;",[34,547047],{"className":547048,"style":43406},[31813],[34,547050,547052],{"className":547051},[31818,43410,43411,31821],[34,547053,459],{"className":547054},[31685,31821],[34,547056,31829],{"className":547057},[31828],[34,547059,547061],{"className":547060},[31801],[34,547062,547064],{"className":547063,"style":220540},[31805],[34,547065],{},[34,547067,31829],{"className":547068},[31828],[34,547070,547072],{"className":547071},[31801],[34,547073,547075],{"className":547074,"style":323187},[31805],[34,547076],{},[34,547078],{"className":547079,"style":32123},[31845],[34,547081,546295],{"className":547082,"style":546331},[32767,131249,156560],[34,547084],{"className":547085,"style":32123},[31845],[34,547087,1167],{"className":547088},[31685],[34,547090],{"className":547091,"style":31846},[31845],[34,547093,202],{"className":547094},[31850],[34,547096],{"className":547097,"style":31846},[31845],[34,547099,547101,547104,547107,547110,547196,547199],{"className":547100},[31676],[34,547102],{"className":547103,"style":546896},[31680],[34,547105,546295],{"className":547106,"style":546331},[32767,131249,156560],[34,547108],{"className":547109,"style":32123},[31845],[34,547111,547113,547116],{"className":547112},[32767],[34,547114,546745],{"className":547115},[32767],[34,547117,547119],{"className":547118},[31792],[34,547120,547122,547188],{"className":547121},[31796,31797],[34,547123,547125,547185],{"className":547124},[31801],[34,547126,547128],{"className":547127,"style":32362},[31805],[34,547129,547130,547133],{"style":56235},[34,547131],{"className":547132,"style":31814},[31813],[34,547134,547136],{"className":547135},[31818,31819,31820,31821],[34,547137,547139,547142,547145],{"className":547138},[31685,31821],[34,547140,217392],{"className":547141,"style":217496},[31685,31686,31821],[34,547143,33239],{"className":547144},[31850,31821],[34,547146,547148,547151],{"className":547147},[31685,31821],[34,547149,217392],{"className":547150,"style":217496},[31685,31686,31821],[34,547152,547154],{"className":547153},[31792],[34,547155,547157,547177],{"className":547156},[31796,31797],[34,547158,547160,547174],{"className":547159},[31801],[34,547161,547163],{"className":547162,"style":318658},[31805],[34,547164,547165,547168],{"style":547045},[34,547166],{"className":547167,"style":43406},[31813],[34,547169,547171],{"className":547170},[31818,43410,43411,31821],[34,547172,459],{"className":547173},[31685,31821],[34,547175,31829],{"className":547176},[31828],[34,547178,547180],{"className":547179},[31801],[34,547181,547183],{"className":547182,"style":220540},[31805],[34,547184],{},[34,547186,31829],{"className":547187},[31828],[34,547189,547191],{"className":547190},[31801],[34,547192,547194],{"className":547193,"style":323187},[31805],[34,547195],{},[34,547197],{"className":547198,"style":32123},[31845],[34,547200,1167],{"className":547201},[31685]," ？",[108,547204,547205,47649,547208,547202],{},[2627,547206,547207],{},"求导：",[34,547209,547211,547257],{"className":547210},[31642],[34,547212,547214],{"className":547213},[31646],[31648,547215,547216],{"xmlns":31650},[31652,547217,547218,547254],{},[31655,547219,547220,547222,547224,547226,547228,547230,547236,547238,547240,547242,547244,547246,547248],{},[31662,547221,456],{"stretchy":702},[31662,547223,546295],{},[31658,547225,2547],{"mathvariant":38079},[31658,547227,2547],{"mathvariant":38079},[31658,547229,2547],{"mathvariant":38079},[37624,547231,547232,547234],{},[31662,547233,2557],{"stretchy":702},[31662,547235,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,547237,202],{},[31662,547239,546295],{},[31662,547241,456],{"stretchy":702},[31658,547243,2547],{"mathvariant":38079},[31658,547245,2547],{"mathvariant":38079},[31658,547247,2547],{"mathvariant":38079},[37624,547249,547250,547252],{},[31662,547251,2557],{"stretchy":702},[31662,547253,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,547255,547256],{"encoding":31667},"(\\int ... )' = \\int (...)'",[34,547258,547260,547319],{"className":547259,"ariaHidden":758},[31672],[34,547261,547263,547266,547269,547272,547275,547278,547310,547313,547316],{"className":547262},[31676],[34,547264],{"className":547265,"style":546896},[31680],[34,547267,456],{"className":547268},[31782],[34,547270,546295],{"className":547271,"style":546331},[32767,131249,156560],[34,547273],{"className":547274,"style":32123},[31845],[34,547276,1167],{"className":547277},[31685],[34,547279,547281,547284],{"className":547280},[31690],[34,547282,2557],{"className":547283},[31690],[34,547285,547287],{"className":547286},[31792],[34,547288,547290],{"className":547289},[31796],[34,547291,547293],{"className":547292},[31801],[34,547294,547296],{"className":547295,"style":132027},[31805],[34,547297,547298,547301],{"style":37777},[34,547299],{"className":547300,"style":31814},[31813],[34,547302,547304],{"className":547303},[31818,31819,31820,31821],[34,547305,547307],{"className":547306},[31685,31821],[34,547308,116495],{"className":547309},[31685,31821],[34,547311],{"className":547312,"style":31846},[31845],[34,547314,202],{"className":547315},[31850],[34,547317],{"className":547318,"style":31846},[31845],[34,547320,547322,547325,547328,547331,547334],{"className":547321},[31676],[34,547323],{"className":547324,"style":546896},[31680],[34,547326,546295],{"className":547327,"style":546331},[32767,131249,156560],[34,547329,456],{"className":547330},[31782],[34,547332,1167],{"className":547333},[31685],[34,547335,547337,547340],{"className":547336},[31690],[34,547338,2557],{"className":547339},[31690],[34,547341,547343],{"className":547342},[31792],[34,547344,547346],{"className":547345},[31796],[34,547347,547349],{"className":547348},[31801],[34,547350,547352],{"className":547351,"style":132027},[31805],[34,547353,547354,547357],{"style":37777},[34,547355],{"className":547356,"style":31814},[31813],[34,547358,547360],{"className":547359},[31818,31819,31820,31821],[34,547361,547363],{"className":547362},[31685,31821],[34,547364,116495],{"className":547365},[31685,31821],[15,547367,547368,547371],{},[2627,547369,547370],{},"答案是："," 只有当积分**“一致收敛”**时，才能放心地交换。如果收敛得不一致（有的地方快，有的地方慢），交换后就会出错。",[253,547373,547375],{"id":547374},"_3-如何判断一致收敛","3. 如何判断一致收敛？",[15,547377,547378],{},"这是做题的关键，需要掌握以下三种判别法：",[105,547380,547381,547647,547950],{},[108,547382,547383,547386],{},[2627,547384,547385],{},"第一种：Weierstrass M-判别法（强判别法，最常用）",[105,547387,547388,547397,547570],{},[108,547389,547390,547392,547393,547396],{},[2627,547391,158252],{}," 找个",[2627,547394,547395],{},"老大","压住它。",[108,547398,547399,547401,547402,547506,547507,547569],{},[2627,547400,285646],{}," 如果被积函数 ",[34,547403,547405,547443],{"className":547404},[31642],[34,547406,547408],{"className":547407},[31646],[31648,547409,547410],{"xmlns":31650},[31652,547411,547412,547440],{},[31655,547413,547414,547416,547418,547420,547422,547424,547426,547428,547430,547432,547434,547436,547438],{},[31658,547415,162086],{"mathvariant":38079},[31658,547417,31739],{},[31662,547419,456],{"stretchy":702},[31658,547421,32936],{},[31662,547423,31960],{"separator":758},[31658,547425,217392],{},[31662,547427,2557],{"stretchy":702},[31658,547429,162086],{"mathvariant":38079},[31662,547431,45972],{},[31658,547433,31942],{},[31662,547435,456],{"stretchy":702},[31658,547437,32936],{},[31662,547439,2557],{"stretchy":702},[31665,547441,547442],{"encoding":31667},"|f(x, \\alpha)| \\leq g(x)",[34,547444,547446,547488],{"className":547445,"ariaHidden":758},[31672],[34,547447,547449,547452,547455,547458,547461,547464,547467,547470,547473,547476,547479,547482,547485],{"className":547448},[31676],[34,547450],{"className":547451,"style":31774},[31680],[34,547453,162086],{"className":547454},[31685],[34,547456,31739],{"className":547457,"style":31778},[31685,31686],[34,547459,456],{"className":547460},[31782],[34,547462,32936],{"className":547463},[31685,31686],[34,547465,31960],{"className":547466},[32119],[34,547468],{"className":547469,"style":32123},[31845],[34,547471,217392],{"className":547472,"style":217496},[31685,31686],[34,547474,2557],{"className":547475},[31690],[34,547477,162086],{"className":547478},[31685],[34,547480],{"className":547481,"style":31846},[31845],[34,547483,45972],{"className":547484},[31850],[34,547486],{"className":547487,"style":31846},[31845],[34,547489,547491,547494,547497,547500,547503],{"className":547490},[31676],[34,547492],{"className":547493,"style":31774},[31680],[34,547495,31942],{"className":547496,"style":32062},[31685,31686],[34,547498,456],{"className":547499},[31782],[34,547501,32936],{"className":547502},[31685,31686],[34,547504,2557],{"className":547505},[31690],"，而且老大 ",[34,547508,547510,547536],{"className":547509},[31642],[34,547511,547513],{"className":547512},[31646],[31648,547514,547515],{"xmlns":31650},[31652,547516,547517,547533],{},[31655,547518,547519,547521,547523,547525,547527,547529,547531],{},[31662,547520,546295],{},[31658,547522,31942],{},[31662,547524,456],{"stretchy":702},[31658,547526,32936],{},[31662,547528,2557],{"stretchy":702},[31658,547530,46900],{},[31658,547532,32936],{},[31665,547534,547535],{"encoding":31667},"\\int g(x)dx",[34,547537,547539],{"className":547538,"ariaHidden":758},[31672],[34,547540,547542,547545,547548,547551,547554,547557,547560,547563,547566],{"className":547541},[31676],[34,547543],{"className":547544,"style":546896},[31680],[34,547546,546295],{"className":547547,"style":546331},[32767,131249,156560],[34,547549],{"className":547550,"style":32123},[31845],[34,547552,31942],{"className":547553,"style":32062},[31685,31686],[34,547555,456],{"className":547556},[31782],[34,547558,32936],{"className":547559},[31685,31686],[34,547561,2557],{"className":547562},[31690],[34,547564,46900],{"className":547565},[31685,31686],[34,547567,32936],{"className":547568},[31685,31686]," 收敛（算得出来有限值），那小弟肯定一致收敛。",[108,547571,547572],{},[71807,547573,8882,547574,547618,547619,347],{},[34,547575,547577,547597],{"className":547576},[31642],[34,547578,547580],{"className":547579},[31646],[31648,547581,547582],{"xmlns":31650},[31652,547583,547584,547594],{},[31655,547585,547586,547588,547590,547592],{},[31658,547587,31942],{},[31662,547589,456],{"stretchy":702},[31658,547591,32936],{},[31662,547593,2557],{"stretchy":702},[31665,547595,547596],{"encoding":31667},"g(x)",[34,547598,547600],{"className":547599,"ariaHidden":758},[31672],[34,547601,547603,547606,547609,547612,547615],{"className":547602},[31676],[34,547604],{"className":547605,"style":31774},[31680],[34,547607,31942],{"className":547608,"style":32062},[31685,31686],[34,547610,456],{"className":547611},[31782],[34,547613,32936],{"className":547614},[31685,31686],[34,547616,2557],{"className":547617},[31690]," 不能含有参数 ",[34,547620,547622,547635],{"className":547621},[31642],[34,547623,547625],{"className":547624},[31646],[31648,547626,547627],{"xmlns":31650},[31652,547628,547629,547633],{},[31655,547630,547631],{},[31658,547632,217392],{},[31665,547634,218244],{"encoding":31667},[34,547636,547638],{"className":547637,"ariaHidden":758},[31672],[34,547639,547641,547644],{"className":547640},[31676],[34,547642],{"className":547643,"style":32527},[31680],[34,547645,217392],{"className":547646,"style":217496},[31685,31686],[108,547648,547649,547652],{},[2627,547650,547651],{},"第二种：Dirichlet 判别法（狄利克雷）",[105,547653,547654,547767],{},[108,547655,547656,547659,547660,347],{},[2627,547657,547658],{},"适用："," 这种形式 ",[34,547661,547663,547701],{"className":547662},[31642],[34,547664,547666],{"className":547665},[31646],[31648,547667,547668],{"xmlns":31650},[31652,547669,547670,547698],{},[31655,547671,547672,547674,547676,547678,547680,547682,547684,547686,547688,547690,547692,547694,547696],{},[31658,547673,31739],{},[31662,547675,456],{"stretchy":702},[31658,547677,32936],{},[31662,547679,31960],{"separator":758},[31658,547681,217392],{},[31662,547683,2557],{"stretchy":702},[31662,547685,71191],{},[31658,547687,31942],{},[31662,547689,456],{"stretchy":702},[31658,547691,32936],{},[31662,547693,31960],{"separator":758},[31658,547695,217392],{},[31662,547697,2557],{"stretchy":702},[31665,547699,547700],{"encoding":31667},"f(x, \\alpha) \\cdot g(x, \\alpha)",[34,547702,547704,547740],{"className":547703,"ariaHidden":758},[31672],[34,547705,547707,547710,547713,547716,547719,547722,547725,547728,547731,547734,547737],{"className":547706},[31676],[34,547708],{"className":547709,"style":31774},[31680],[34,547711,31739],{"className":547712,"style":31778},[31685,31686],[34,547714,456],{"className":547715},[31782],[34,547717,32936],{"className":547718},[31685,31686],[34,547720,31960],{"className":547721},[32119],[34,547723],{"className":547724,"style":32123},[31845],[34,547726,217392],{"className":547727,"style":217496},[31685,31686],[34,547729,2557],{"className":547730},[31690],[34,547732],{"className":547733,"style":32604},[31845],[34,547735,71191],{"className":547736},[32024],[34,547738],{"className":547739,"style":32604},[31845],[34,547741,547743,547746,547749,547752,547755,547758,547761,547764],{"className":547742},[31676],[34,547744],{"className":547745,"style":31774},[31680],[34,547747,31942],{"className":547748,"style":32062},[31685,31686],[34,547750,456],{"className":547751},[31782],[34,547753,32936],{"className":547754},[31685,31686],[34,547756,31960],{"className":547757},[32119],[34,547759],{"className":547760,"style":32123},[31845],[34,547762,217392],{"className":547763,"style":217496},[31685,31686],[34,547765,2557],{"className":547766},[31690],[108,547768,547769,547772,547773,547835,547836,402824,547839,547888,547889,47649,547946,547949],{},[2627,547770,547771],{},"条件："," 积分部分 ",[34,547774,547776,547802],{"className":547775},[31642],[34,547777,547779],{"className":547778},[31646],[31648,547780,547781],{"xmlns":31650},[31652,547782,547783,547799],{},[31655,547784,547785,547787,547789,547791,547793,547795,547797],{},[31662,547786,546295],{},[31658,547788,31739],{},[31662,547790,456],{"stretchy":702},[31658,547792,32936],{},[31662,547794,2557],{"stretchy":702},[31658,547796,46900],{},[31658,547798,32936],{},[31665,547800,547801],{"encoding":31667},"\\int f(x)dx",[34,547803,547805],{"className":547804,"ariaHidden":758},[31672],[34,547806,547808,547811,547814,547817,547820,547823,547826,547829,547832],{"className":547807},[31676],[34,547809],{"className":547810,"style":546896},[31680],[34,547812,546295],{"className":547813,"style":546331},[32767,131249,156560],[34,547815],{"className":547816,"style":32123},[31845],[34,547818,31739],{"className":547819,"style":31778},[31685,31686],[34,547821,456],{"className":547822},[31782],[34,547824,32936],{"className":547825},[31685,31686],[34,547827,2557],{"className":547828},[31690],[34,547830,46900],{"className":547831},[31685,31686],[34,547833,32936],{"className":547834},[31685,31686]," 的部分积分",[2627,547837,547838],{},"有界",[34,547840,547842,547866],{"className":547841},[31642],[34,547843,547845],{"className":547844},[31646],[31648,547846,547847],{"xmlns":31650},[31652,547848,547849,547863],{},[31655,547850,547851,547854,547856,547858,547861],{},[31658,547852,547853],{},"sin",[31662,547855,32682],{},[31662,547857,31960],{"separator":758},[31658,547859,547860],{},"cos",[31662,547862,32682],{},[31665,547864,547865],{"encoding":31667},"\\sin, \\cos",[34,547867,547869],{"className":547868,"ariaHidden":758},[31672],[34,547870,547872,547876,547879,547882,547885],{"className":547871},[31676],[34,547873],{"className":547874,"style":547875},[31680],"height:0.8623em;vertical-align:-0.1944em;",[34,547877,547853],{"className":547878},[32767],[34,547880,31960],{"className":547881},[32119],[34,547883],{"className":547884,"style":32123},[31845],[34,547886,547860],{"className":547887},[32767],"），单调部分 ",[34,547890,547892,547916],{"className":547891},[31642],[34,547893,547895],{"className":547894},[31646],[31648,547896,547897],{"xmlns":31650},[31652,547898,547899,547913],{},[31655,547900,547901,547903,547905,547907,547909,547911],{},[31658,547902,31942],{},[31662,547904,456],{"stretchy":702},[31658,547906,32936],{},[31662,547908,31960],{"separator":758},[31658,547910,217392],{},[31662,547912,2557],{"stretchy":702},[31665,547914,547915],{"encoding":31667},"g(x, \\alpha)",[34,547917,547919],{"className":547918,"ariaHidden":758},[31672],[34,547920,547922,547925,547928,547931,547934,547937,547940,547943],{"className":547921},[31676],[34,547923],{"className":547924,"style":31774},[31680],[34,547926,31942],{"className":547927,"style":32062},[31685,31686],[34,547929,456],{"className":547930},[31782],[34,547932,32936],{"className":547933},[31685,31686],[34,547935,31960],{"className":547936},[32119],[34,547938],{"className":547939,"style":32123},[31845],[34,547941,217392],{"className":547942,"style":217496},[31685,31686],[34,547944,2557],{"className":547945},[31690],[2627,547947,547948],{},"单调趋于0","（而且是“一致”地趋于0）。",[108,547951,547952,547955],{},[2627,547953,547954],{},"第三种：Abel 判别法（阿贝尔）",[105,547956,547957,547962],{},[108,547958,547959,547961],{},[2627,547960,547658],{}," 也是乘积形式。",[108,547963,547964,547772,547966,548041,548042,548045,548046,47649,548102,347],{},[2627,547965,547771],{},[34,547967,547969,547999],{"className":547968},[31642],[34,547970,547972],{"className":547971},[31646],[31648,547973,547974],{"xmlns":31650},[31652,547975,547976,547996],{},[31655,547977,547978,547980,547982,547984,547986,547988,547990,547992,547994],{},[31662,547979,546295],{},[31658,547981,31739],{},[31662,547983,456],{"stretchy":702},[31658,547985,32936],{},[31662,547987,31960],{"separator":758},[31658,547989,217392],{},[31662,547991,2557],{"stretchy":702},[31658,547993,46900],{},[31658,547995,32936],{},[31665,547997,547998],{"encoding":31667},"\\int f(x, \\alpha)dx",[34,548000,548002],{"className":548001,"ariaHidden":758},[31672],[34,548003,548005,548008,548011,548014,548017,548020,548023,548026,548029,548032,548035,548038],{"className":548004},[31676],[34,548006],{"className":548007,"style":546896},[31680],[34,548009,546295],{"className":548010,"style":546331},[32767,131249,156560],[34,548012],{"className":548013,"style":32123},[31845],[34,548015,31739],{"className":548016,"style":31778},[31685,31686],[34,548018,456],{"className":548019},[31782],[34,548021,32936],{"className":548022},[31685,31686],[34,548024,31960],{"className":548025},[32119],[34,548027],{"className":548028,"style":32123},[31845],[34,548030,217392],{"className":548031,"style":217496},[31685,31686],[34,548033,2557],{"className":548034},[31690],[34,548036,46900],{"className":548037},[31685,31686],[34,548039,32936],{"className":548040},[31685,31686]," 已经",[2627,548043,548044],{},"一致收敛","了，另一部分 ",[34,548047,548049,548072],{"className":548048},[31642],[34,548050,548052],{"className":548051},[31646],[31648,548053,548054],{"xmlns":31650},[31652,548055,548056,548070],{},[31655,548057,548058,548060,548062,548064,548066,548068],{},[31658,548059,31942],{},[31662,548061,456],{"stretchy":702},[31658,548063,32936],{},[31662,548065,31960],{"separator":758},[31658,548067,217392],{},[31662,548069,2557],{"stretchy":702},[31665,548071,547915],{"encoding":31667},[34,548073,548075],{"className":548074,"ariaHidden":758},[31672],[34,548076,548078,548081,548084,548087,548090,548093,548096,548099],{"className":548077},[31676],[34,548079],{"className":548080,"style":31774},[31680],[34,548082,31942],{"className":548083,"style":32062},[31685,31686],[34,548085,456],{"className":548086},[31782],[34,548088,32936],{"className":548089},[31685,31686],[34,548091,31960],{"className":548092},[32119],[34,548094],{"className":548095,"style":32123},[31845],[34,548097,217392],{"className":548098,"style":217496},[31685,31686],[34,548100,2557],{"className":548101},[31690],[2627,548103,548104],{},"单调且有界",[32188,548106],{},[11,548108,548110],{"id":548109},"第二部分作业题目细致讲解","第二部分：作业题目细致讲解",[253,548112,548114],{"id":548113},"p329-t1","p329 T1",[15,548116,548117,548119,548120,548467,548468,548845,548846,2547],{},[2627,548118,214361],{},"\n证明下列含参变量反常积分在指定区间上一致收敛：\n(1) ",[34,548121,548123,548193],{"className":548122},[31642],[34,548124,548126],{"className":548125},[31646],[31648,548127,548128],{"xmlns":31650},[31652,548129,548130,548190],{},[31655,548131,548132,548144,548172,548174,548176,548178,548180,548182,548184,548186,548188],{},[38846,548133,548134,548136,548138],{},[31662,548135,546295],{},[31749,548137,459],{},[31655,548139,548140,548142],{},[31662,548141,223],{},[31658,548143,138286],{"mathvariant":38079},[41453,548145,548146,548156],{},[31655,548147,548148,548150,548152,548154],{},[31658,548149,547860],{},[31662,548151,32682],{},[31658,548153,32936],{},[31658,548155,67764],{},[31655,548157,548158,548164,548166],{},[37624,548159,548160,548162],{},[31658,548161,32936],{},[31749,548163,768],{},[31662,548165,223],{},[37624,548167,548168,548170],{},[31658,548169,67764],{},[31749,548171,768],{},[31658,548173,46900],{},[31658,548175,32936],{},[31662,548177,31960],{"separator":758},[31845,548179],{"width":71082},[31658,548181,67764],{},[31662,548183,43175],{},[31658,548185,111],{},[31662,548187,29622],{},[31749,548189,459],{},[31665,548191,548192],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{\\cos xy}{x^2 + y^2}dx, \\quad y \\geq a > 0",[34,548194,548196,548440,548458],{"className":548195,"ariaHidden":758},[31672],[34,548197,548199,548203,548263,548266,548413,548416,548419,548422,548425,548428,548431,548434,548437],{"className":548198},[31676],[34,548200],{"className":548201,"style":548202},[31680],"height:1.4473em;vertical-align:-0.4811em;",[34,548204,548206,548209],{"className":548205},[32767],[34,548207,546295],{"className":548208,"style":546331},[32767,131249,156560],[34,548210,548212],{"className":548211},[31792],[34,548213,548215,548255],{"className":548214},[31796,31797],[34,548216,548218,548252],{"className":548217},[31801],[34,548219,548221,548235],{"className":548220,"style":546550},[31805],[34,548222,548223,548226],{"style":546347},[34,548224],{"className":548225,"style":31814},[31813],[34,548227,548229],{"className":548228},[31818,31819,31820,31821],[34,548230,548232],{"className":548231},[31685,31821],[34,548233,459],{"className":548234},[31685,31821],[34,548236,548237,548240],{"style":546362},[34,548238],{"className":548239,"style":31814},[31813],[34,548241,548243],{"className":548242},[31818,31819,31820,31821],[34,548244,548246,548249],{"className":548245},[31685,31821],[34,548247,223],{"className":548248},[31685,31821],[34,548250,138286],{"className":548251},[31685,31821],[34,548253,31829],{"className":548254},[31828],[34,548256,548258],{"className":548257},[31801],[34,548259,548261],{"className":548260,"style":546384},[31805],[34,548262],{},[34,548264],{"className":548265,"style":32123},[31845],[34,548267,548269,548272,548410],{"className":548268},[31685],[34,548270],{"className":548271},[31782,41488],[34,548273,548275],{"className":548274},[41453],[34,548276,548278,548402],{"className":548277},[31796,31797],[34,548279,548281,548399],{"className":548280},[31801],[34,548282,548285,548358,548366],{"className":548283,"style":548284},[31805],"height:0.7475em;",[34,548286,548287,548290],{"style":41504},[34,548288],{"className":548289,"style":41508},[31813],[34,548291,548293],{"className":548292},[31818,31819,31820,31821],[34,548294,548296,548326,548329],{"className":548295},[31685,31821],[34,548297,548299,548302],{"className":548298},[31685,31821],[34,548300,32936],{"className":548301},[31685,31686,31821],[34,548303,548305],{"className":548304},[31792],[34,548306,548308],{"className":548307},[31796],[34,548309,548311],{"className":548310},[31801],[34,548312,548315],{"className":548313,"style":548314},[31805],"height:0.7463em;",[34,548316,548317,548320],{"style":116668},[34,548318],{"className":548319,"style":43406},[31813],[34,548321,548323],{"className":548322},[31818,43410,43411,31821],[34,548324,768],{"className":548325},[31685,31821],[34,548327,223],{"className":548328},[32024,31821],[34,548330,548332,548335],{"className":548331},[31685,31821],[34,548333,67764],{"className":548334,"style":32062},[31685,31686,31821],[34,548336,548338],{"className":548337},[31792],[34,548339,548341],{"className":548340},[31796],[34,548342,548344],{"className":548343},[31801],[34,548345,548347],{"className":548346,"style":548314},[31805],[34,548348,548349,548352],{"style":116668},[34,548350],{"className":548351,"style":43406},[31813],[34,548353,548355],{"className":548354},[31818,43410,43411,31821],[34,548356,768],{"className":548357},[31685,31821],[34,548359,548360,548363],{"style":41520},[34,548361],{"className":548362,"style":41508},[31813],[34,548364],{"className":548365,"style":41528},[41527],[34,548367,548369,548372],{"style":548368},"top:-3.4461em;",[34,548370],{"className":548371,"style":41508},[31813],[34,548373,548375],{"className":548374},[31818,31819,31820,31821],[34,548376,548378,548390,548393,548396],{"className":548377},[31685,31821],[34,548379,548381,548384,548387],{"className":548380},[32767,31821],[34,548382,31965],{"className":548383},[31821],[34,548385,43383],{"className":548386},[31821],[34,548388,31747],{"className":548389},[31821],[34,548391],{"className":548392,"style":43430},[31845,31821],[34,548394,32936],{"className":548395},[31685,31686,31821],[34,548397,67764],{"className":548398,"style":32062},[31685,31686,31821],[34,548400,31829],{"className":548401},[31828],[34,548403,548405],{"className":548404},[31801],[34,548406,548408],{"className":548407,"style":220191},[31805],[34,548409],{},[34,548411],{"className":548412},[31690,41488],[34,548414,46900],{"className":548415},[31685,31686],[34,548417,32936],{"className":548418},[31685,31686],[34,548420,31960],{"className":548421},[32119],[34,548423],{"className":548424,"style":128438},[31845],[34,548426],{"className":548427,"style":32123},[31845],[34,548429,67764],{"className":548430,"style":32062},[31685,31686],[34,548432],{"className":548433,"style":31846},[31845],[34,548435,43175],{"className":548436},[31850],[34,548438],{"className":548439,"style":31846},[31845],[34,548441,548443,548446,548449,548452,548455],{"className":548442},[31676],[34,548444],{"className":548445,"style":34812},[31680],[34,548447,111],{"className":548448},[31685,31686],[34,548450],{"className":548451,"style":31846},[31845],[34,548453,29622],{"className":548454},[31850],[34,548456],{"className":548457,"style":31846},[31845],[34,548459,548461,548464],{"className":548460},[31676],[34,548462],{"className":548463,"style":32617},[31680],[34,548465,459],{"className":548466},[31685],"；\n(2) ",[34,548469,548471,548549],{"className":548470},[31642],[34,548472,548474],{"className":548473},[31646],[31648,548475,548476],{"xmlns":31650},[31652,548477,548478,548546],{},[31655,548479,548480,548492,548512,548524,548526,548528,548530,548532,548534,548536,548538,548540],{},[38846,548481,548482,548484,548486],{},[31662,548483,546295],{},[31749,548485,459],{},[31655,548487,548488,548490],{},[31662,548489,223],{},[31658,548491,138286],{"mathvariant":38079},[41453,548493,548494,548504],{},[31655,548495,548496,548498,548500,548502],{},[31658,548497,547853],{},[31662,548499,32682],{},[31749,548501,768],{},[31658,548503,32936],{},[31655,548505,548506,548508,548510],{},[31658,548507,32936],{},[31662,548509,223],{},[31658,548511,217392],{},[37624,548513,548514,548516],{},[31658,548515,33244],{},[31655,548517,548518,548520,548522],{},[31662,548519,32226],{},[31658,548521,217392],{},[31658,548523,32936],{},[31658,548525,46900],{},[31658,548527,32936],{},[31662,548529,31960],{"separator":758},[31845,548531],{"width":71082},[31749,548533,459],{},[31662,548535,45972],{},[31658,548537,217392],{},[31662,548539,45972],{},[31743,548541,548542,548544],{},[31658,548543,217392],{},[31749,548545,459],{},[31665,548547,548548],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{\\sin 2x}{x + \\alpha}e^{-\\alpha x}dx, \\quad 0 \\leq \\alpha \\leq \\alpha_0",[34,548550,548552,548780,548798],{"className":548551,"ariaHidden":758},[31672],[34,548553,548555,548559,548619,548622,548715,548753,548756,548759,548762,548765,548768,548771,548774,548777],{"className":548554},[31676],[34,548556],{"className":548557,"style":548558},[31680],"height:1.3696em;vertical-align:-0.4033em;",[34,548560,548562,548565],{"className":548561},[32767],[34,548563,546295],{"className":548564,"style":546331},[32767,131249,156560],[34,548566,548568],{"className":548567},[31792],[34,548569,548571,548611],{"className":548570},[31796,31797],[34,548572,548574,548608],{"className":548573},[31801],[34,548575,548577,548591],{"className":548576,"style":546550},[31805],[34,548578,548579,548582],{"style":546347},[34,548580],{"className":548581,"style":31814},[31813],[34,548583,548585],{"className":548584},[31818,31819,31820,31821],[34,548586,548588],{"className":548587},[31685,31821],[34,548589,459],{"className":548590},[31685,31821],[34,548592,548593,548596],{"style":546362},[34,548594],{"className":548595,"style":31814},[31813],[34,548597,548599],{"className":548598},[31818,31819,31820,31821],[34,548600,548602,548605],{"className":548601},[31685,31821],[34,548603,223],{"className":548604},[31685,31821],[34,548606,138286],{"className":548607},[31685,31821],[34,548609,31829],{"className":548610},[31828],[34,548612,548614],{"className":548613},[31801],[34,548615,548617],{"className":548616,"style":546384},[31805],[34,548618],{},[34,548620],{"className":548621,"style":32123},[31845],[34,548623,548625,548628,548712],{"className":548624},[31685],[34,548626],{"className":548627},[31782,41488],[34,548629,548631],{"className":548630},[41453],[34,548632,548634,548704],{"className":548633},[31796,31797],[34,548635,548637,548701],{"className":548636},[31801],[34,548638,548641,548661,548669],{"className":548639,"style":548640},[31805],"height:0.8615em;",[34,548642,548643,548646],{"style":41504},[34,548644],{"className":548645,"style":41508},[31813],[34,548647,548649],{"className":548648},[31818,31819,31820,31821],[34,548650,548652,548655,548658],{"className":548651},[31685,31821],[34,548653,32936],{"className":548654},[31685,31686,31821],[34,548656,223],{"className":548657},[32024,31821],[34,548659,217392],{"className":548660,"style":217496},[31685,31686,31821],[34,548662,548663,548666],{"style":41520},[34,548664],{"className":548665,"style":41508},[31813],[34,548667],{"className":548668,"style":41528},[41527],[34,548670,548671,548674],{"style":46771},[34,548672],{"className":548673,"style":41508},[31813],[34,548675,548677],{"className":548676},[31818,31819,31820,31821],[34,548678,548680,548692,548695,548698],{"className":548679},[31685,31821],[34,548681,548683,548686,548689],{"className":548682},[32767,31821],[34,548684,31747],{"className":548685},[31821],[34,548687,31955],{"className":548688},[31821],[34,548690,31660],{"className":548691},[31821],[34,548693],{"className":548694,"style":43430},[31845,31821],[34,548696,768],{"className":548697},[31685,31821],[34,548699,32936],{"className":548700},[31685,31686,31821],[34,548702,31829],{"className":548703},[31828],[34,548705,548707],{"className":548706},[31801],[34,548708,548710],{"className":548709,"style":400163},[31805],[34,548711],{},[34,548713],{"className":548714},[31690,41488],[34,548716,548718,548721],{"className":548717},[31685],[34,548719,33244],{"className":548720},[31685,31686],[34,548722,548724],{"className":548723},[31792],[34,548725,548727],{"className":548726},[31796],[34,548728,548730],{"className":548729},[31801],[34,548731,548733],{"className":548732,"style":156828},[31805],[34,548734,548735,548738],{"style":37777},[34,548736],{"className":548737,"style":31814},[31813],[34,548739,548741],{"className":548740},[31818,31819,31820,31821],[34,548742,548744,548747,548750],{"className":548743},[31685,31821],[34,548745,32226],{"className":548746},[31685,31821],[34,548748,217392],{"className":548749,"style":217496},[31685,31686,31821],[34,548751,32936],{"className":548752},[31685,31686,31821],[34,548754,46900],{"className":548755},[31685,31686],[34,548757,32936],{"className":548758},[31685,31686],[34,548760,31960],{"className":548761},[32119],[34,548763],{"className":548764,"style":128438},[31845],[34,548766],{"className":548767,"style":32123},[31845],[34,548769,459],{"className":548770},[31685],[34,548772],{"className":548773,"style":31846},[31845],[34,548775,45972],{"className":548776},[31850],[34,548778],{"className":548779,"style":31846},[31845],[34,548781,548783,548786,548789,548792,548795],{"className":548782},[31676],[34,548784],{"className":548785,"style":43198},[31680],[34,548787,217392],{"className":548788,"style":217496},[31685,31686],[34,548790],{"className":548791,"style":31846},[31845],[34,548793,45972],{"className":548794},[31850],[34,548796],{"className":548797,"style":31846},[31845],[34,548799,548801,548804],{"className":548800},[31676],[34,548802],{"className":548803,"style":31860},[31680],[34,548805,548807,548810],{"className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",[34,548847,548849,548911],{"className":548848},[31642],[34,548850,548852],{"className":548851},[31646],[31648,548853,548854],{"xmlns":31650},[31652,548855,548856,548908],{},[31655,548857,548858,548870,548872,548874,548876,548882,548884,548886,548888,548890,548892,548894,548896,548898,548900,548902,548904,548906],{},[38846,548859,548860,548862,548864],{},[31662,548861,546295],{},[31749,548863,459],{},[31655,548865,548866,548868],{},[31662,548867,223],{},[31658,548869,138286],{"mathvariant":38079},[31658,548871,32936],{},[31658,548873,547853],{},[31662,548875,32682],{},[37624,548877,548878,548880],{},[31658,548879,32936],{},[31749,548881,15932],{},[31658,548883,547860],{},[31662,548885,32682],{},[31658,548887,217392],{},[31658,548889,32936],{},[31658,548891,46900],{},[31658,548893,32936],{},[31662,548895,31960],{"separator":758},[31845,548897],{"width":71082},[31658,548899,111],{},[31662,548901,45972],{},[31658,548903,217392],{},[31662,548905,45972],{},[31658,548907,1697],{},[31665,548909,548910],{"encoding":31667},"\\int_{0}^{+\\infty} x\\sin x^4 \\cos \\alpha xdx, \\quad a \\leq \\alpha \\leq b",[34,548912,548914,549066,549084],{"className":548913,"ariaHidden":758},[31672],[34,548915,548917,548920,548980,548983,548986,548989,548992,548995,549024,549027,549030,549033,549036,549039,549042,549045,549048,549051,549054,549057,549060,549063],{"className":548916},[31676],[34,548918],{"className":548919,"style":546531},[31680],[34,548921,548923,548926],{"className":548922},[32767],[34,548924,546295],{"className":548925,"style":546331},[32767,131249,156560],[34,548927,548929],{"className":548928},[31792],[34,548930,548932,548972],{"className":548931},[31796,31797],[34,548933,548935,548969],{"className":548934},[31801],[34,548936,548938,548952],{"className":548937,"style":546550},[31805],[34,548939,548940,548943],{"style":546347},[34,548941],{"className":548942,"style":31814},[31813],[34,548944,548946],{"className":548945},[31818,31819,31820,31821],[34,548947,548949],{"className":548948},[31685,31821],[34,548950,459],{"className":548951},[31685,31821],[34,548953,548954,548957],{"style":546362},[34,548955],{"className":548956,"style":31814},[31813],[34,548958,548960],{"className":548959},[31818,31819,31820,31821],[34,548961,548963,548966],{"className":548962},[31685,31821],[34,548964,223],{"className":548965},[31685,31821],[34,548967,138286],{"className":548968},[31685,31821],[34,548970,31829],{"className":548971},[31828],[34,548973,548975],{"className":548974},[31801],[34,548976,548978],{"className":548977,"style":546384},[31805],[34,548979],{},[34,548981],{"className":548982,"style":32123},[31845],[34,548984,32936],{"className":548985},[31685,31686],[34,548987],{"className":548988,"style":32123},[31845],[34,548990,547853],{"className":548991},[32767],[34,548993],{"className":548994,"style":32123},[31845],[34,548996,548998,549001],{"className":548997},[31685],[34,548999,32936],{"className":549000},[31685,31686],[34,549002,549004],{"className":549003},[31792],[34,549005,549007],{"className":549006},[31796],[34,549008,549010],{"className":549009},[31801],[34,549011,549013],{"className":549012,"style":37774},[31805],[34,549014,549015,549018],{"style":37777},[34,549016],{"className":549017,"style":31814},[31813],[34,549019,549021],{"className":549020},[31818,31819,31820,31821],[34,549022,15932],{"className":549023},[31685,31821],[34,549025],{"className":549026,"style":32123},[31845],[34,549028,547860],{"className":549029},[32767],[34,549031],{"className":549032,"style":32123},[31845],[34,549034,217392],{"className":549035,"style":217496},[31685,31686],[34,549037,32936],{"className":549038},[31685,31686],[34,549040,46900],{"className":549041},[31685,31686],[34,549043,32936],{"className":549044},[31685,31686],[34,549046,31960],{"className":549047},[32119],[34,549049],{"className":549050,"style":128438},[31845],[34,549052],{"className":549053,"style":32123},[31845],[34,549055,111],{"className":549056},[31685,31686],[34,549058],{"className":549059,"style":31846},[31845],[34,549061,45972],{"className":549062},[31850],[34,549064],{"className":549065,"style":31846},[31845],[34,549067,549069,549072,549075,549078,549081],{"className":549068},[31676],[34,549070],{"className":549071,"style":43198},[31680],[34,549073,217392],{"className":549074,"style":217496},[31685,31686],[34,549076],{"className":549077,"style":31846},[31845],[34,549079,45972],{"className":549080},[31850],[34,549082],{"className":549083,"style":31846},[31845],[34,549085,549087,549090],{"className":549086},[31676],[34,549088],{"className":549089,"style":31681},[31680],[34,549091,1697],{"className":549092},[31685,31686],[32188,549094],{},[253,549096,549098],{"id":549097},"讲解-1",[2627,549099,549100],{},"讲解 (1)",[15,549102,549103],{},[34,549104,549106,549175],{"className":549105},[31642],[34,549107,549109],{"className":549108},[31646],[31648,549110,549111],{"xmlns":31650},[31652,549112,549113,549173],{},[31655,549114,549115,549127,549155,549157,549159,549161,549163,549165,549167,549169,549171],{},[38846,549116,549117,549119,549121],{},[31662,549118,546295],{},[31749,549120,459],{},[31655,549122,549123,549125],{},[31662,549124,223],{},[31658,549126,138286],{"mathvariant":38079},[41453,549128,549129,549139],{},[31655,549130,549131,549133,549135,549137],{},[31658,549132,547860],{},[31662,549134,32682],{},[31658,549136,32936],{},[31658,549138,67764],{},[31655,549140,549141,549147,549149],{},[37624,549142,549143,549145],{},[31658,549144,32936],{},[31749,549146,768],{},[31662,549148,223],{},[37624,549150,549151,549153],{},[31658,549152,67764],{},[31749,549154,768],{},[31658,549156,46900],{},[31658,549158,32936],{},[31662,549160,31960],{"separator":758},[31845,549162],{"width":71082},[31658,549164,67764],{},[31662,549166,43175],{},[31658,549168,111],{},[31662,549170,29622],{},[31749,549172,459],{},[31665,549174,548192],{"encoding":31667},[34,549176,549178,549418,549436],{"className":549177,"ariaHidden":758},[31672],[34,549179,549181,549184,549244,549247,549391,549394,549397,549400,549403,549406,549409,549412,549415],{"className":549180},[31676],[34,549182],{"className":549183,"style":548202},[31680],[34,549185,549187,549190],{"className":549186},[32767],[34,549188,546295],{"className":549189,"style":546331},[32767,131249,156560],[34,549191,549193],{"className":549192},[31792],[34,549194,549196,549236],{"className":549195},[31796,31797],[34,549197,549199,549233],{"className":549198},[31801],[34,549200,549202,549216],{"className":549201,"style":546550},[31805],[34,549203,549204,549207],{"style":546347},[34,549205],{"className":549206,"style":31814},[31813],[34,549208,549210],{"className":549209},[31818,31819,31820,31821],[34,549211,549213],{"className":549212},[31685,31821],[34,549214,459],{"className":549215},[31685,31821],[34,549217,549218,549221],{"style":546362},[34,549219],{"className":549220,"style":31814},[31813],[34,549222,549224],{"className":549223},[31818,31819,31820,31821],[34,549225,549227,549230],{"className":549226},[31685,31821],[34,549228,223],{"className":549229},[31685,31821],[34,549231,138286],{"className":549232},[31685,31821],[34,549234,31829],{"className":549235},[31828],[34,549237,549239],{"className":549238},[31801],[34,549240,549242],{"className":549241,"style":546384},[31805],[34,549243],{},[34,549245],{"className":549246,"style":32123},[31845],[34,549248,549250,549253,549388],{"className":549249},[31685],[34,549251],{"className":549252},[31782,41488],[34,549254,549256],{"className":549255},[41453],[34,549257,549259,549380],{"className":549258},[31796,31797],[34,549260,549262,549377],{"className":549261},[31801],[34,549263,549265,549337,549345],{"className":549264,"style":548284},[31805],[34,549266,549267,549270],{"style":41504},[34,549268],{"className":549269,"style":41508},[31813],[34,549271,549273],{"className":549272},[31818,31819,31820,31821],[34,549274,549276,549305,549308],{"className":549275},[31685,31821],[34,549277,549279,549282],{"className":549278},[31685,31821],[34,549280,32936],{"className":549281},[31685,31686,31821],[34,549283,549285],{"className":549284},[31792],[34,549286,549288],{"className":549287},[31796],[34,549289,549291],{"className":549290},[31801],[34,549292,549294],{"className":549293,"style":548314},[31805],[34,549295,549296,549299],{"style":116668},[34,549297],{"className":549298,"style":43406},[31813],[34,549300,549302],{"className":549301},[31818,43410,43411,31821],[34,549303,768],{"className":549304},[31685,31821],[34,549306,223],{"className":549307},[32024,31821],[34,549309,549311,549314],{"className":549310},[31685,31821],[34,549312,67764],{"className":549313,"style":32062},[31685,31686,31821],[34,549315,549317],{"className":549316},[31792],[34,549318,549320],{"className":549319},[31796],[34,549321,549323],{"className":549322},[31801],[34,549324,549326],{"className":549325,"style":548314},[31805],[34,549327,549328,549331],{"style":116668},[34,549329],{"className":549330,"style":43406},[31813],[34,549332,549334],{"className":549333},[31818,43410,43411,31821],[34,549335,768],{"className":549336},[31685,31821],[34,549338,549339,549342],{"style":41520},[34,549340],{"className":549341,"style":41508},[31813],[34,549343],{"className":549344,"style":41528},[41527],[34,549346,549347,549350],{"style":548368},[34,549348],{"className":549349,"style":41508},[31813],[34,549351,549353],{"className":549352},[31818,31819,31820,31821],[34,549354,549356,549368,549371,549374],{"className":549355},[31685,31821],[34,549357,549359,549362,549365],{"className":549358},[32767,31821],[34,549360,31965],{"className":549361},[31821],[34,549363,43383],{"className":549364},[31821],[34,549366,31747],{"className":549367},[31821],[34,549369],{"className":549370,"style":43430},[31845,31821],[34,549372,32936],{"className":549373},[31685,31686,31821],[34,549375,67764],{"className":549376,"style":32062},[31685,31686,31821],[34,549378,31829],{"className":549379},[31828],[34,549381,549383],{"className":549382},[31801],[34,549384,549386],{"className":549385,"style":220191},[31805],[34,549387],{},[34,549389],{"className":549390},[31690,41488],[34,549392,46900],{"className":549393},[31685,31686],[34,549395,32936],{"className":549396},[31685,31686],[34,549398,31960],{"className":549399},[32119],[34,549401],{"className":549402,"style":128438},[31845],[34,549404],{"className":549405,"style":32123},[31845],[34,549407,67764],{"className":549408,"style":32062},[31685,31686],[34,549410],{"className":549411,"style":31846},[31845],[34,549413,43175],{"className":549414},[31850],[34,549416],{"className":549417,"style":31846},[31845],[34,549419,549421,549424,549427,549430,549433],{"className":549420},[31676],[34,549422],{"className":549423,"style":34812},[31680],[34,549425,111],{"className":549426},[31685,31686],[34,549428],{"className":549429,"style":31846},[31845],[34,549431,29622],{"className":549432},[31850],[34,549434],{"className":549435,"style":31846},[31845],[34,549437,549439,549442],{"className":549438},[31676],[34,549440],{"className":549441,"style":32617},[31680],[34,549443,459],{"className":549444},[31685],[15,549446,549447,549450,549451,549562,549563,549591,549592,549595,549596,549624],{},[2627,549448,549449],{},"思路："," 看到分母是 ",[34,549452,549454,549480],{"className":549453},[31642],[34,549455,549457],{"className":549456},[31646],[31648,549458,549459],{"xmlns":31650},[31652,549460,549461,549477],{},[31655,549462,549463,549469,549471],{},[37624,549464,549465,549467],{},[31658,549466,32936],{},[31749,549468,768],{},[31662,549470,223],{},[37624,549472,549473,549475],{},[31658,549474,67764],{},[31749,549476,768],{},[31665,549478,549479],{"encoding":31667},"x^2+y^2",[34,549481,549483,549527],{"className":549482,"ariaHidden":758},[31672],[34,549484,549486,549489,549518,549521,549524],{"className":549485},[31676],[34,549487],{"className":549488,"style":40044},[31680],[34,549490,549492,549495],{"className":549491},[31685],[34,549493,32936],{"className":549494},[31685,31686],[34,549496,549498],{"className":549497},[31792],[34,549499,549501],{"className":549500},[31796],[34,549502,549504],{"className":549503},[31801],[34,549505,549507],{"className":549506,"style":37774},[31805],[34,549508,549509,549512],{"style":37777},[34,549510],{"className":549511,"style":31814},[31813],[34,549513,549515],{"className":549514},[31818,31819,31820,31821],[34,549516,768],{"className":549517},[31685,31821],[34,549519],{"className":549520,"style":32604},[31845],[34,549522,223],{"className":549523},[32024],[34,549525],{"className":549526,"style":32604},[31845],[34,549528,549530,549533],{"className":549529},[31676],[34,549531],{"className":549532,"style":315747},[31680],[34,549534,549536,549539],{"className":549535},[31685],[34,549537,67764],{"className":549538,"style":32062},[31685,31686],[34,549540,549542],{"className":549541},[31792],[34,549543,549545],{"className":549544},[31796],[34,549546,549548],{"className":549547},[31801],[34,549549,549551],{"className":549550,"style":37774},[31805],[34,549552,549553,549556],{"style":37777},[34,549554],{"className":549555,"style":31814},[31813],[34,549557,549559],{"className":549558},[31818,31819,31820,31821],[34,549560,768],{"className":549561},[31685,31821]," 这种“由于 ",[34,549564,549566,549579],{"className":549565},[31642],[34,549567,549569],{"className":549568},[31646],[31648,549570,549571],{"xmlns":31650},[31652,549572,549573,549577],{},[31655,549574,549575],{},[31658,549576,32936],{},[31665,549578,32936],{"encoding":31667},[34,549580,549582],{"className":549581,"ariaHidden":758},[31672],[34,549583,549585,549588],{"className":549584},[31676],[34,549586],{"className":549587,"style":32527},[31680],[34,549589,32936],{"className":549590},[31685,31686]," 很大导致分母很大”的类型，首选 ",[2627,549593,549594],{},"M-判别法","。我们要找一个不含 ",[34,549597,549599,549612],{"className":549598},[31642],[34,549600,549602],{"className":549601},[31646],[31648,549603,549604],{"xmlns":31650},[31652,549605,549606,549610],{},[31655,549607,549608],{},[31658,549609,67764],{},[31665,549611,67764],{"encoding":31667},[34,549613,549615],{"className":549614,"ariaHidden":758},[31672],[34,549616,549618,549621],{"className":549617},[31676],[34,549619],{"className":549620,"style":38774},[31680],[34,549622,67764],{"className":549623,"style":32062},[31685,31686]," 的函数盖住它。",[15,549626,549627],{},[2627,549628,549629],{},"步骤：",[2230,549631,549632,550455,551635],{},[108,549633,549634,315440,549637,93535,549716,549790,549791,39984,550006],{},[2627,549635,549636],{},"放大被积函数：",[34,549638,549640,549668],{"className":549639},[31642],[34,549641,549643],{"className":549642},[31646],[31648,549644,549645],{"xmlns":31650},[31652,549646,549647,549665],{},[31655,549648,549649,549651,549653,549655,549657,549659,549661,549663],{},[31658,549650,162086],{"mathvariant":38079},[31658,549652,547860],{},[31662,549654,32682],{},[31658,549656,32936],{},[31658,549658,67764],{},[31658,549660,162086],{"mathvariant":38079},[31662,549662,45972],{},[31749,549664,640],{},[31665,549666,549667],{"encoding":31667},"|\\cos xy| \\leq 1",[34,549669,549671,549707],{"className":549670,"ariaHidden":758},[31672],[34,549672,549674,549677,549680,549683,549686,549689,549692,549695,549698,549701,549704],{"className":549673},[31676],[34,549675],{"className":549676,"style":31774},[31680],[34,549678,162086],{"className":549679},[31685],[34,549681],{"className":549682,"style":32123},[31845],[34,549684,547860],{"className":549685},[32767],[34,549687],{"className":549688,"style":32123},[31845],[34,549690,32936],{"className":549691},[31685,31686],[34,549693,67764],{"className":549694,"style":32062},[31685,31686],[34,549696,162086],{"className":549697},[31685],[34,549699],{"className":549700,"style":31846},[31845],[34,549702,45972],{"className":549703},[31850],[34,549705],{"className":549706,"style":31846},[31845],[34,549708,549710,549713],{"className":549709},[31676],[34,549711],{"className":549712,"style":32617},[31680],[34,549714,640],{"className":549715},[31685],[34,549717,549719,549741],{"className":549718},[31642],[34,549720,549722],{"className":549721},[31646],[31648,549723,549724],{"xmlns":31650},[31652,549725,549726,549738],{},[31655,549727,549728,549730,549732,549734,549736],{},[31658,549729,67764],{},[31662,549731,43175],{},[31658,549733,111],{},[31662,549735,29622],{},[31749,549737,459],{},[31665,549739,549740],{"encoding":31667},"y \\geq a > 0",[34,549742,549744,549763,549781],{"className":549743,"ariaHidden":758},[31672],[34,549745,549747,549751,549754,549757,549760],{"className":549746},[31676],[34,549748],{"className":549749,"style":549750},[31680],"height:0.8304em;vertical-align:-0.1944em;",[34,549752,67764],{"className":549753,"style":32062},[31685,31686],[34,549755],{"className":549756,"style":31846},[31845],[34,549758,43175],{"className":549759},[31850],[34,549761],{"className":549762,"style":31846},[31845],[34,549764,549766,549769,549772,549775,549778],{"className":549765},[31676],[34,549767],{"className":549768,"style":34812},[31680],[34,549770,111],{"className":549771},[31685,31686],[34,549773],{"className":549774,"style":31846},[31845],[34,549776,29622],{"className":549777},[31850],[34,549779],{"className":549780,"style":31846},[31845],[34,549782,549784,549787],{"className":549783},[31676],[34,549785],{"className":549786,"style":32617},[31680],[34,549788,459],{"className":549789},[31685],"，所以分母 ",[34,549792,549794,549836],{"className":549793},[31642],[34,549795,549797],{"className":549796},[31646],[31648,549798,549799],{"xmlns":31650},[31652,549800,549801,549833],{},[31655,549802,549803,549809,549811,549817,549819,549825,549827],{},[37624,549804,549805,549807],{},[31658,549806,32936],{},[31749,549808,768],{},[31662,549810,223],{},[37624,549812,549813,549815],{},[31658,549814,67764],{},[31749,549816,768],{},[31662,549818,43175],{},[37624,549820,549821,549823],{},[31658,549822,32936],{},[31749,549824,768],{},[31662,549826,223],{},[37624,549828,549829,549831],{},[31658,549830,111],{},[31749,549832,768],{},[31665,549834,549835],{"encoding":31667},"x^2 + y^2 \\geq x^2 + a^2",[34,549837,549839,549883,549927,549971],{"className":549838,"ariaHidden":758},[31672],[34,549840,549842,549845,549874,549877,549880],{"className":549841},[31676],[34,549843],{"className":549844,"style":40044},[31680],[34,549846,549848,549851],{"className":549847},[31685],[34,549849,32936],{"className":549850},[31685,31686],[34,549852,549854],{"className":549853},[31792],[34,549855,549857],{"className":549856},[31796],[34,549858,549860],{"className":549859},[31801],[34,549861,549863],{"className":549862,"style":37774},[31805],[34,549864,549865,549868],{"style":37777},[34,549866],{"className":549867,"style":31814},[31813],[34,549869,549871],{"className":549870},[31818,31819,31820,31821],[34,549872,768],{"className":549873},[31685,31821],[34,549875],{"className":549876,"style":32604},[31845],[34,549878,223],{"className":549879},[32024],[34,549881],{"className":549882,"style":32604},[31845],[34,549884,549886,549889,549918,549921,549924],{"className":549885},[31676],[34,549887],{"className":549888,"style":315747},[31680],[34,549890,549892,549895],{"className":549891},[31685],[34,549893,67764],{"className":549894,"style":32062},[31685,31686],[34,549896,549898],{"className":549897},[31792],[34,549899,549901],{"className":549900},[31796],[34,549902,549904],{"className":549903},[31801],[34,549905,549907],{"className":549906,"style":37774},[31805],[34,549908,549909,549912],{"style":37777},[34,549910],{"className":549911,"style":31814},[31813],[34,549913,549915],{"className":549914},[31818,31819,31820,31821],[34,549916,768],{"className":549917},[31685,31821],[34,549919],{"className":549920,"style":31846},[31845],[34,549922,43175],{"className":549923},[31850],[34,549925],{"className":549926,"style":31846},[31845],[34,549928,549930,549933,549962,549965,549968],{"className":549929},[31676],[34,549931],{"className":549932,"style":40044},[31680],[34,549934,549936,549939],{"className":549935},[31685],[34,549937,32936],{"className":549938},[31685,31686],[34,549940,549942],{"className":549941},[31792],[34,549943,549945],{"className":549944},[31796],[34,549946,549948],{"className":549947},[31801],[34,549949,549951],{"className":549950,"style":37774},[31805],[34,549952,549953,549956],{"style":37777},[34,549954],{"className":549955,"style":31814},[31813],[34,549957,549959],{"className":549958},[31818,31819,31820,31821],[34,549960,768],{"className":549961},[31685,31821],[34,549963],{"className":549964,"style":32604},[31845],[34,549966,223],{"className":549967},[32024],[34,549969],{"className":549970,"style":32604},[31845],[34,549972,549974,549977],{"className":549973},[31676],[34,549975],{"className":549976,"style":37774},[31680],[34,549978,549980,549983],{"className":549979},[31685],[34,549981,111],{"className":549982},[31685,31686],[34,549984,549986],{"className":549985},[31792],[34,549987,549989],{"className":549988},[31796],[34,549990,549992],{"className":549991},[31801],[34,549993,549995],{"className":549994,"style":37774},[31805],[34,549996,549997,550000],{"style":37777},[34,549998],{"className":549999,"style":31814},[31813],[34,550001,550003],{"className":550002},[31818,31819,31820,31821],[34,550004,768],{"className":550005},[31685,31821],[34,550007,550009,550077],{"className":550008},[31642],[34,550010,550012],{"className":550011},[31646],[31648,550013,550014],{"xmlns":31650},[31652,550015,550016,550074],{},[31655,550017,550018,550052,550054],{},[31655,550019,550020,550022,550050],{},[31662,550021,162086],{"fence":758},[41453,550023,550024,550034],{},[31655,550025,550026,550028,550030,550032],{},[31658,550027,547860],{},[31662,550029,32682],{},[31658,550031,32936],{},[31658,550033,67764],{},[31655,550035,550036,550042,550044],{},[37624,550037,550038,550040],{},[31658,550039,32936],{},[31749,550041,768],{},[31662,550043,223],{},[37624,550045,550046,550048],{},[31658,550047,67764],{},[31749,550049,768],{},[31662,550051,162086],{"fence":758},[31662,550053,45972],{},[41453,550055,550056,550058],{},[31749,550057,640],{},[31655,550059,550060,550066,550068],{},[37624,550061,550062,550064],{},[31658,550063,32936],{},[31749,550065,768],{},[31662,550067,223],{},[37624,550069,550070,550072],{},[31658,550071,111],{},[31749,550073,768],{},[31665,550075,550076],{"encoding":31667},"\\left| \\frac{\\cos xy}{x^2 + y^2} \\right| \\leq \\frac{1}{x^2 + a^2}",[34,550078,550080,550323],{"className":550079,"ariaHidden":758},[31672],[34,550081,550083,550086,550314,550317,550320],{"className":550082},[31676],[34,550084],{"className":550085,"style":141282},[31680],[34,550087,550089,550133,550277],{"className":550088},[34006],[34,550090,550092],{"className":550091},[31782],[34,550093,550095],{"className":550094},[71281,71282],[34,550096,550098,550124],{"className":550097},[31796,31797],[34,550099,550101,550121],{"className":550100},[31801],[34,550102,550105],{"className":550103,"style":550104},[31805],"height:1.15em;",[34,550106,550107,550111],{"style":71329},[34,550108],{"className":550109,"style":550110},[31813],"height:3.8em;",[34,550112,550114],{"style":550113},"width:0.333em;height:1.8em;",[41588,550115,550118],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},"1.8em","0 0 333 1800",[20398,550119],{"d":550120},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[34,550122,31829],{"className":550123},[31828],[34,550125,550127],{"className":550126},[31801],[34,550128,550131],{"className":550129,"style":550130},[31805],"height:0.65em;",[34,550132],{},[34,550134,550136,550139,550274],{"className":550135},[31685],[34,550137],{"className":550138},[31782,41488],[34,550140,550142],{"className":550141},[41453],[34,550143,550145,550266],{"className":550144},[31796,31797],[34,550146,550148,550263],{"className":550147},[31801],[34,550149,550151,550223,550231],{"className":550150,"style":548284},[31805],[34,550152,550153,550156],{"style":41504},[34,550154],{"className":550155,"style":41508},[31813],[34,550157,550159],{"className":550158},[31818,31819,31820,31821],[34,550160,550162,550191,550194],{"className":550161},[31685,31821],[34,550163,550165,550168],{"className":550164},[31685,31821],[34,550166,32936],{"className":550167},[31685,31686,31821],[34,550169,550171],{"className":550170},[31792],[34,550172,550174],{"className":550173},[31796],[34,550175,550177],{"className":550176},[31801],[34,550178,550180],{"className":550179,"style":548314},[31805],[34,550181,550182,550185],{"style":116668},[34,550183],{"className":550184,"style":43406},[31813],[34,550186,550188],{"className":550187},[31818,43410,43411,31821],[34,550189,768],{"className":550190},[31685,31821],[34,550192,223],{"className":550193},[32024,31821],[34,550195,550197,550200],{"className":550196},[31685,31821],[34,550198,67764],{"className":550199,"style":32062},[31685,31686,31821],[34,550201,550203],{"className":550202},[31792],[34,550204,550206],{"className":550205},[31796],[34,550207,550209],{"className":550208},[31801],[34,550210,550212],{"className":550211,"style":548314},[31805],[34,550213,550214,550217],{"style":116668},[34,550215],{"className":550216,"style":43406},[31813],[34,550218,550220],{"className":550219},[31818,43410,43411,31821],[34,550221,768],{"className":550222},[31685,31821],[34,550224,550225,550228],{"style":41520},[34,550226],{"className":550227,"style":41508},[31813],[34,550229],{"className":550230,"style":41528},[41527],[34,550232,550233,550236],{"style":548368},[34,550234],{"className":550235,"style":41508},[31813],[34,550237,550239],{"className":550238},[31818,31819,31820,31821],[34,550240,550242,550254,550257,550260],{"className":550241},[31685,31821],[34,550243,550245,550248,550251],{"className":550244},[32767,31821],[34,550246,31965],{"className":550247},[31821],[34,550249,43383],{"className":550250},[31821],[34,550252,31747],{"className":550253},[31821],[34,550255],{"className":550256,"style":43430},[31845,31821],[34,550258,32936],{"className":550259},[31685,31686,31821],[34,550261,67764],{"className":550262,"style":32062},[31685,31686,31821],[34,550264,31829],{"className":550265},[31828],[34,550267,550269],{"className":550268},[31801],[34,550270,550272],{"className":550271,"style":220191},[31805],[34,550273],{},[34,550275],{"className":550276},[31690,41488],[34,550278,550280],{"className":550279},[31690],[34,550281,550283],{"className":550282},[71281,71282],[34,550284,550286,550306],{"className":550285},[31796,31797],[34,550287,550289,550303],{"className":550288},[31801],[34,550290,550292],{"className":550291,"style":550104},[31805],[34,550293,550294,550297],{"style":71329},[34,550295],{"className":550296,"style":550110},[31813],[34,550298,550299],{"style":550113},[41588,550300,550301],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,550302],{"d":550120},[34,550304,31829],{"className":550305},[31828],[34,550307,550309],{"className":550308},[31801],[34,550310,550312],{"className":550311,"style":550130},[31805],[34,550313],{},[34,550315],{"className":550316,"style":31846},[31845],[34,550318,45972],{"className":550319},[31850],[34,550321],{"className":550322,"style":31846},[31845],[34,550324,550326,550329],{"className":550325},[31676],[34,550327],{"className":550328,"style":400093},[31680],[34,550330,550332,550335,550452],{"className":550331},[31685],[34,550333],{"className":550334},[31782,41488],[34,550336,550338],{"className":550337},[41453],[34,550339,550341,550444],{"className":550340},[31796,31797],[34,550342,550344,550441],{"className":550343},[31801],[34,550345,550347,550419,550427],{"className":550346,"style":46746},[31805],[34,550348,550349,550352],{"style":41504},[34,550350],{"className":550351,"style":41508},[31813],[34,550353,550355],{"className":550354},[31818,31819,31820,31821],[34,550356,550358,550387,550390],{"className":550357},[31685,31821],[34,550359,550361,550364],{"className":550360},[31685,31821],[34,550362,32936],{"className":550363},[31685,31686,31821],[34,550365,550367],{"className":550366},[31792],[34,550368,550370],{"className":550369},[31796],[34,550371,550373],{"className":550372},[31801],[34,550374,550376],{"className":550375,"style":548314},[31805],[34,550377,550378,550381],{"style":116668},[34,550379],{"className":550380,"style":43406},[31813],[34,550382,550384],{"className":550383},[31818,43410,43411,31821],[34,550385,768],{"className":550386},[31685,31821],[34,550388,223],{"className":550389},[32024,31821],[34,550391,550393,550396],{"className":550392},[31685,31821],[34,550394,111],{"className":550395},[31685,31686,31821],[34,550397,550399],{"className":550398},[31792],[34,550400,550402],{"className":550401},[31796],[34,550403,550405],{"className":550404},[31801],[34,550406,550408],{"className":550407,"style":548314},[31805],[34,550409,550410,550413],{"style":116668},[34,550411],{"className":550412,"style":43406},[31813],[34,550414,550416],{"className":550415},[31818,43410,43411,31821],[34,550417,768],{"className":550418},[31685,31821],[34,550420,550421,550424],{"style":41520},[34,550422],{"className":550423,"style":41508},[31813],[34,550425],{"className":550426,"style":41528},[41527],[34,550428,550429,550432],{"style":46771},[34,550430],{"className":550431,"style":41508},[31813],[34,550433,550435],{"className":550434},[31818,31819,31820,31821],[34,550436,550438],{"className":550437},[31685,31821],[34,550439,640],{"className":550440},[31685,31821],[34,550442,31829],{"className":550443},[31828],[34,550445,550447],{"className":550446},[31801],[34,550448,550450],{"className":550449,"style":400163},[31805],[34,550451],{},[34,550453],{"className":550454},[31690,41488],[108,550456,550457,550460,550461,550716,550717,39984,551084,551634],{},[2627,550458,550459],{},"检查“老大”是否收敛：","\n我们要检查 ",[34,550462,550464,550512],{"className":550463},[31642],[34,550465,550467],{"className":550466},[31646],[31648,550468,550469],{"xmlns":31650},[31652,550470,550471,550509],{},[31655,550472,550473,550485,550505,550507],{},[38846,550474,550475,550477,550479],{},[31662,550476,546295],{},[31749,550478,459],{},[31655,550480,550481,550483],{},[31662,550482,223],{},[31658,550484,138286],{"mathvariant":38079},[41453,550486,550487,550489],{},[31749,550488,640],{},[31655,550490,550491,550497,550499],{},[37624,550492,550493,550495],{},[31658,550494,32936],{},[31749,550496,768],{},[31662,550498,223],{},[37624,550500,550501,550503],{},[31658,550502,111],{},[31749,550504,768],{},[31658,550506,46900],{},[31658,550508,32936],{},[31665,550510,550511],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{1}{x^2 + a^2} dx",[34,550513,550515],{"className":550514,"ariaHidden":758},[31672],[34,550516,550518,550521,550581,550584,550710,550713],{"className":550517},[31676],[34,550519],{"className":550520,"style":548558},[31680],[34,550522,550524,550527],{"className":550523},[32767],[34,550525,546295],{"className":550526,"style":546331},[32767,131249,156560],[34,550528,550530],{"className":550529},[31792],[34,550531,550533,550573],{"className":550532},[31796,31797],[34,550534,550536,550570],{"className":550535},[31801],[34,550537,550539,550553],{"className":550538,"style":546550},[31805],[34,550540,550541,550544],{"style":546347},[34,550542],{"className":550543,"style":31814},[31813],[34,550545,550547],{"className":550546},[31818,31819,31820,31821],[34,550548,550550],{"className":550549},[31685,31821],[34,550551,459],{"className":550552},[31685,31821],[34,550554,550555,550558],{"style":546362},[34,550556],{"className":550557,"style":31814},[31813],[34,550559,550561],{"className":550560},[31818,31819,31820,31821],[34,550562,550564,550567],{"className":550563},[31685,31821],[34,550565,223],{"className":550566},[31685,31821],[34,550568,138286],{"className":550569},[31685,31821],[34,550571,31829],{"className":550572},[31828],[34,550574,550576],{"className":550575},[31801],[34,550577,550579],{"className":550578,"style":546384},[31805],[34,550580],{},[34,550582],{"className":550583,"style":32123},[31845],[34,550585,550587,550590,550707],{"className":550586},[31685],[34,550588],{"className":550589},[31782,41488],[34,550591,550593],{"className":550592},[41453],[34,550594,550596,550699],{"className":550595},[31796,31797],[34,550597,550599,550696],{"className":550598},[31801],[34,550600,550602,550674,550682],{"className":550601,"style":46746},[31805],[34,550603,550604,550607],{"style":41504},[34,550605],{"className":550606,"style":41508},[31813],[34,550608,550610],{"className":550609},[31818,31819,31820,31821],[34,550611,550613,550642,550645],{"className":550612},[31685,31821],[34,550614,550616,550619],{"className":550615},[31685,31821],[34,550617,32936],{"className":550618},[31685,31686,31821],[34,550620,550622],{"className":550621},[31792],[34,550623,550625],{"className":550624},[31796],[34,550626,550628],{"className":550627},[31801],[34,550629,550631],{"className":550630,"style":548314},[31805],[34,550632,550633,550636],{"style":116668},[34,550634],{"className":550635,"style":43406},[31813],[34,550637,550639],{"className":550638},[31818,43410,43411,31821],[34,550640,768],{"className":550641},[31685,31821],[34,550643,223],{"className":550644},[32024,31821],[34,550646,550648,550651],{"className":550647},[31685,31821],[34,550649,111],{"className":550650},[31685,31686,31821],[34,550652,550654],{"className":550653},[31792],[34,550655,550657],{"className":550656},[31796],[34,550658,550660],{"className":550659},[31801],[34,550661,550663],{"className":550662,"style":548314},[31805],[34,550664,550665,550668],{"style":116668},[34,550666],{"className":550667,"style":43406},[31813],[34,550669,550671],{"className":550670},[31818,43410,43411,31821],[34,550672,768],{"className":550673},[31685,31821],[34,550675,550676,550679],{"style":41520},[34,550677],{"className":550678,"style":41508},[31813],[34,550680],{"className":550681,"style":41528},[41527],[34,550683,550684,550687],{"style":46771},[34,550685],{"className":550686,"style":41508},[31813],[34,550688,550690],{"className":550689},[31818,31819,31820,31821],[34,550691,550693],{"className":550692},[31685,31821],[34,550694,640],{"className":550695},[31685,31821],[34,550697,31829],{"className":550698},[31828],[34,550700,550702],{"className":550701},[31801],[34,550703,550705],{"className":550704,"style":400163},[31805],[34,550706],{},[34,550708],{"className":550709},[31690,41488],[34,550711,46900],{"className":550712},[31685,31686],[34,550714,32936],{"className":550715},[31685,31686]," 是否收敛。\n这是基础积分公式：",[34,550718,550720,550777],{"className":550719},[31642],[34,550721,550723],{"className":550722},[31646],[31648,550724,550725],{"xmlns":31650},[31652,550726,550727,550774],{},[31655,550728,550729,550731,550751,550753,550755,550757,550763,550766,550768],{},[31662,550730,546295],{},[41453,550732,550733,550735],{},[31749,550734,640],{},[31655,550736,550737,550743,550745],{},[37624,550738,550739,550741],{},[31658,550740,32936],{},[31749,550742,768],{},[31662,550744,223],{},[37624,550746,550747,550749],{},[31658,550748,111],{},[31749,550750,768],{},[31658,550752,46900],{},[31658,550754,32936],{},[31662,550756,202],{},[41453,550758,550759,550761],{},[31749,550760,640],{},[31658,550762,111],{},[31658,550764,550765],{},"arctan",[31662,550767,32682],{},[41453,550769,550770,550772],{},[31658,550771,32936],{},[31658,550773,111],{},[31665,550775,550776],{"encoding":31667},"\\int \\frac{1}{x^2+a^2}dx = \\frac{1}{a}\\arctan\\frac{x}{a}",[34,550778,550780,550933],{"className":550779,"ariaHidden":758},[31672],[34,550781,550783,550786,550789,550792,550918,550921,550924,550927,550930],{"className":550782},[31676],[34,550784],{"className":550785,"style":400093},[31680],[34,550787,546295],{"className":550788,"style":546331},[32767,131249,156560],[34,550790],{"className":550791,"style":32123},[31845],[34,550793,550795,550798,550915],{"className":550794},[31685],[34,550796],{"className":550797},[31782,41488],[34,550799,550801],{"className":550800},[41453],[34,550802,550804,550907],{"className":550803},[31796,31797],[34,550805,550807,550904],{"className":550806},[31801],[34,550808,550810,550882,550890],{"className":550809,"style":46746},[31805],[34,550811,550812,550815],{"style":41504},[34,550813],{"className":550814,"style":41508},[31813],[34,550816,550818],{"className":550817},[31818,31819,31820,31821],[34,550819,550821,550850,550853],{"className":550820},[31685,31821],[34,550822,550824,550827],{"className":550823},[31685,31821],[34,550825,32936],{"className":550826},[31685,31686,31821],[34,550828,550830],{"className":550829},[31792],[34,550831,550833],{"className":550832},[31796],[34,550834,550836],{"className":550835},[31801],[34,550837,550839],{"className":550838,"style":548314},[31805],[34,550840,550841,550844],{"style":116668},[34,550842],{"className":550843,"style":43406},[31813],[34,550845,550847],{"className":550846},[31818,43410,43411,31821],[34,550848,768],{"className":550849},[31685,31821],[34,550851,223],{"className":550852},[32024,31821],[34,550854,550856,550859],{"className":550855},[31685,31821],[34,550857,111],{"className":550858},[31685,31686,31821],[34,550860,550862],{"className":550861},[31792],[34,550863,550865],{"className":550864},[31796],[34,550866,550868],{"className":550867},[31801],[34,550869,550871],{"className":550870,"style":548314},[31805],[34,550872,550873,550876],{"style":116668},[34,550874],{"className":550875,"style":43406},[31813],[34,550877,550879],{"className":550878},[31818,43410,43411,31821],[34,550880,768],{"className":550881},[31685,31821],[34,550883,550884,550887],{"style":41520},[34,550885],{"className":550886,"style":41508},[31813],[34,550888],{"className":550889,"style":41528},[41527],[34,550891,550892,550895],{"style":46771},[34,550893],{"className":550894,"style":41508},[31813],[34,550896,550898],{"className":550897},[31818,31819,31820,31821],[34,550899,550901],{"className":550900},[31685,31821],[34,550902,640],{"className":550903},[31685,31821],[34,550905,31829],{"className":550906},[31828],[34,550908,550910],{"className":550909},[31801],[34,550911,550913],{"className":550912,"style":400163},[31805],[34,550914],{},[34,550916],{"className":550917},[31690,41488],[34,550919,46900],{"className":550920},[31685,31686],[34,550922,32936],{"className":550923},[31685,31686],[34,550925],{"className":550926,"style":31846},[31845],[34,550928,202],{"className":550929},[31850],[34,550931],{"className":550932,"style":31846},[31845],[34,550934,550936,550939,551007,551010,551013,551016],{"className":550935},[31676],[34,550937],{"className":550938,"style":46727},[31680],[34,550940,550942,550945,551004],{"className":550941},[31685],[34,550943],{"className":550944},[31782,41488],[34,550946,550948],{"className":550947},[41453],[34,550949,550951,550996],{"className":550950},[31796,31797],[34,550952,550954,550993],{"className":550953},[31801],[34,550955,550957,550971,550979],{"className":550956,"style":46746},[31805],[34,550958,550959,550962],{"style":41504},[34,550960],{"className":550961,"style":41508},[31813],[34,550963,550965],{"className":550964},[31818,31819,31820,31821],[34,550966,550968],{"className":550967},[31685,31821],[34,550969,111],{"className":550970},[31685,31686,31821],[34,550972,550973,550976],{"style":41520},[34,550974],{"className":550975,"style":41508},[31813],[34,550977],{"className":550978,"style":41528},[41527],[34,550980,550981,550984],{"style":46771},[34,550982],{"className":550983,"style":41508},[31813],[34,550985,550987],{"className":550986},[31818,31819,31820,31821],[34,550988,550990],{"className":550989},[31685,31821],[34,550991,640],{"className":550992},[31685,31821],[34,550994,31829],{"className":550995},[31828],[34,550997,550999],{"className":550998},[31801],[34,551000,551002],{"className":551001,"style":41619},[31805],[34,551003],{},[34,551005],{"className":551006},[31690,41488],[34,551008],{"className":551009,"style":32123},[31845],[34,551011,550765],{"className":551012},[32767],[34,551014],{"className":551015,"style":32123},[31845],[34,551017,551019,551022,551081],{"className":551018},[31685],[34,551020],{"className":551021},[31782,41488],[34,551023,551025],{"className":551024},[41453],[34,551026,551028,551073],{"className":551027},[31796,31797],[34,551029,551031,551070],{"className":551030},[31801],[34,551032,551034,551048,551056],{"className":551033,"style":244945},[31805],[34,551035,551036,551039],{"style":41504},[34,551037],{"className":551038,"style":41508},[31813],[34,551040,551042],{"className":551041},[31818,31819,31820,31821],[34,551043,551045],{"className":551044},[31685,31821],[34,551046,111],{"className":551047},[31685,31686,31821],[34,551049,551050,551053],{"style":41520},[34,551051],{"className":551052,"style":41508},[31813],[34,551054],{"className":551055,"style":41528},[41527],[34,551057,551058,551061],{"style":46771},[34,551059],{"className":551060,"style":41508},[31813],[34,551062,551064],{"className":551063},[31818,31819,31820,31821],[34,551065,551067],{"className":551066},[31685,31821],[34,551068,32936],{"className":551069},[31685,31686,31821],[34,551071,31829],{"className":551072},[31828],[34,551074,551076],{"className":551075},[31801],[34,551077,551079],{"className":551078,"style":41619},[31805],[34,551080],{},[34,551082],{"className":551083},[31690,41488],[34,551085,551087,551169],{"className":551086},[31642],[34,551088,551090],{"className":551089},[31646],[31648,551091,551092],{"xmlns":31650},[31652,551093,551094,551166],{},[31655,551095,551096,551108,551128,551130,551132,551134,551140,551142,551148,551150,551152,551154,551156],{},[38846,551097,551098,551100,551102],{},[31662,551099,546295],{},[31749,551101,459],{},[31655,551103,551104,551106],{},[31662,551105,223],{},[31658,551107,138286],{"mathvariant":38079},[41453,551109,551110,551112],{},[31749,551111,640],{},[31655,551113,551114,551120,551122],{},[37624,551115,551116,551118],{},[31658,551117,32936],{},[31749,551119,768],{},[31662,551121,223],{},[37624,551123,551124,551126],{},[31658,551125,111],{},[31749,551127,768],{},[31658,551129,46900],{},[31658,551131,32936],{},[31662,551133,202],{},[41453,551135,551136,551138],{},[31749,551137,640],{},[31658,551139,111],{},[31662,551141,456],{"stretchy":702},[41453,551143,551144,551146],{},[31658,551145,58442],{},[31749,551147,768],{},[31662,551149,32226],{},[31749,551151,459],{},[31662,551153,2557],{"stretchy":702},[31662,551155,202],{},[41453,551157,551158,551160],{},[31658,551159,58442],{},[31655,551161,551162,551164],{},[31749,551163,768],{},[31658,551165,111],{},[31665,551167,551168],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{1}{x^2 + a^2} dx = \\frac{1}{a}(\\frac{\\pi}{2} - 0) = \\frac{\\pi}{2a}",[34,551170,551172,551382,551536,551557],{"className":551171,"ariaHidden":758},[31672],[34,551173,551175,551178,551238,551241,551367,551370,551373,551376,551379],{"className":551174},[31676],[34,551176],{"className":551177,"style":548558},[31680],[34,551179,551181,551184],{"className":551180},[32767],[34,551182,546295],{"className":551183,"style":546331},[32767,131249,156560],[34,551185,551187],{"className":551186},[31792],[34,551188,551190,551230],{"className":551189},[31796,31797],[34,551191,551193,551227],{"className":551192},[31801],[34,551194,551196,551210],{"className":551195,"style":546550},[31805],[34,551197,551198,551201],{"style":546347},[34,551199],{"className":551200,"style":31814},[31813],[34,551202,551204],{"className":551203},[31818,31819,31820,31821],[34,551205,551207],{"className":551206},[31685,31821],[34,551208,459],{"className":551209},[31685,31821],[34,551211,551212,551215],{"style":546362},[34,551213],{"className":551214,"style":31814},[31813],[34,551216,551218],{"className":551217},[31818,31819,31820,31821],[34,551219,551221,551224],{"className":551220},[31685,31821],[34,551222,223],{"className":551223},[31685,31821],[34,551225,138286],{"className":551226},[31685,31821],[34,551228,31829],{"className":551229},[31828],[34,551231,551233],{"className":551232},[31801],[34,551234,551236],{"className":551235,"style":546384},[31805],[34,551237],{},[34,551239],{"className":551240,"style":32123},[31845],[34,551242,551244,551247,551364],{"className":551243},[31685],[34,551245],{"className":551246},[31782,41488],[34,551248,551250],{"className":551249},[41453],[34,551251,551253,551356],{"className":551252},[31796,31797],[34,551254,551256,551353],{"className":551255},[31801],[34,551257,551259,551331,551339],{"className":551258,"style":46746},[31805],[34,551260,551261,551264],{"style":41504},[34,551262],{"className":551263,"style":41508},[31813],[34,551265,551267],{"className":551266},[31818,31819,31820,31821],[34,551268,551270,551299,551302],{"className":551269},[31685,31821],[34,551271,551273,551276],{"className":551272},[31685,31821],[34,551274,32936],{"className":551275},[31685,31686,31821],[34,551277,551279],{"className":551278},[31792],[34,551280,551282],{"className":551281},[31796],[34,551283,551285],{"className":551284},[31801],[34,551286,551288],{"className":551287,"style":548314},[31805],[34,551289,551290,551293],{"style":116668},[34,551291],{"className":551292,"style":43406},[31813],[34,551294,551296],{"className":551295},[31818,43410,43411,31821],[34,551297,768],{"className":551298},[31685,31821],[34,551300,223],{"className":551301},[32024,31821],[34,551303,551305,551308],{"className":551304},[31685,31821],[34,551306,111],{"className":551307},[31685,31686,31821],[34,551309,551311],{"className":551310},[31792],[34,551312,551314],{"className":551313},[31796],[34,551315,551317],{"className":551316},[31801],[34,551318,551320],{"className":551319,"style":548314},[31805],[34,551321,551322,551325],{"style":116668},[34,551323],{"className":551324,"style":43406},[31813],[34,551326,551328],{"className":551327},[31818,43410,43411,31821],[34,551329,768],{"className":551330},[31685,31821],[34,551332,551333,551336],{"style":41520},[34,551334],{"className":551335,"style":41508},[31813],[34,551337],{"className":551338,"style":41528},[41527],[34,551340,551341,551344],{"style":46771},[34,551342],{"className":551343,"style":41508},[31813],[34,551345,551347],{"className":551346},[31818,31819,31820,31821],[34,551348,551350],{"className":551349},[31685,31821],[34,551351,640],{"className":551352},[31685,31821],[34,551354,31829],{"className":551355},[31828],[34,551357,551359],{"className":551358},[31801],[34,551360,551362],{"className":551361,"style":400163},[31805],[34,551363],{},[34,551365],{"className":551366},[31690,41488],[34,551368,46900],{"className":551369},[31685,31686],[34,551371,32936],{"className":551372},[31685,31686],[34,551374],{"className":551375,"style":31846},[31845],[34,551377,202],{"className":551378},[31850],[34,551380],{"className":551381,"style":31846},[31845],[34,551383,551385,551388,551456,551459,551527,551530,551533],{"className":551384},[31676],[34,551386],{"className":551387,"style":46727},[31680],[34,551389,551391,551394,551453],{"className":551390},[31685],[34,551392],{"className":551393},[31782,41488],[34,551395,551397],{"className":551396},[41453],[34,551398,551400,551445],{"className":551399},[31796,31797],[34,551401,551403,551442],{"className":551402},[31801],[34,551404,551406,551420,551428],{"className":551405,"style":46746},[31805],[34,551407,551408,551411],{"style":41504},[34,551409],{"className":551410,"style":41508},[31813],[34,551412,551414],{"className":551413},[31818,31819,31820,31821],[34,551415,551417],{"className":551416},[31685,31821],[34,551418,111],{"className":551419},[31685,31686,31821],[34,551421,551422,551425],{"style":41520},[34,551423],{"className":551424,"style":41508},[31813],[34,551426],{"className":551427,"style":41528},[41527],[34,551429,551430,551433],{"style":46771},[34,551431],{"className":551432,"style":41508},[31813],[34,551434,551436],{"className":551435},[31818,31819,31820,31821],[34,551437,551439],{"className":551438},[31685,31821],[34,551440,640],{"className":551441},[31685,31821],[34,551443,31829],{"className":551444},[31828],[34,551446,551448],{"className":551447},[31801],[34,551449,551451],{"className":551450,"style":41619},[31805],[34,551452],{},[34,551454],{"className":551455},[31690,41488],[34,551457,456],{"className":551458},[31782],[34,551460,551462,551465,551524],{"className":551461},[31685],[34,551463],{"className":551464},[31782,41488],[34,551466,551468],{"className":551467},[41453],[34,551469,551471,551516],{"className":551470},[31796,31797],[34,551472,551474,551513],{"className":551473},[31801],[34,551475,551477,551491,551499],{"className":551476,"style":244945},[31805],[34,551478,551479,551482],{"style":41504},[34,551480],{"className":551481,"style":41508},[31813],[34,551483,551485],{"className":551484},[31818,31819,31820,31821],[34,551486,551488],{"className":551487},[31685,31821],[34,551489,768],{"className":551490},[31685,31821],[34,551492,551493,551496],{"style":41520},[34,551494],{"className":551495,"style":41508},[31813],[34,551497],{"className":551498,"style":41528},[41527],[34,551500,551501,551504],{"style":46771},[34,551502],{"className":551503,"style":41508},[31813],[34,551505,551507],{"className":551506},[31818,31819,31820,31821],[34,551508,551510],{"className":551509},[31685,31821],[34,551511,58442],{"className":551512,"style":32062},[31685,31686,31821],[34,551514,31829],{"className":551515},[31828],[34,551517,551519],{"className":551518},[31801],[34,551520,551522],{"className":551521,"style":41619},[31805],[34,551523],{},[34,551525],{"className":551526},[31690,41488],[34,551528],{"className":551529,"style":32604},[31845],[34,551531,32226],{"className":551532},[32024],[34,551534],{"className":551535,"style":32604},[31845],[34,551537,551539,551542,551545,551548,551551,551554],{"className":551538},[31676],[34,551540],{"className":551541,"style":31774},[31680],[34,551543,459],{"className":551544},[31685],[34,551546,2557],{"className":551547},[31690],[34,551549],{"className":551550,"style":31846},[31845],[34,551552,202],{"className":551553},[31850],[34,551555],{"className":551556,"style":31846},[31845],[34,551558,551560,551563],{"className":551559},[31676],[34,551561],{"className":551562,"style":244926},[31680],[34,551564,551566,551569,551631],{"className":551565},[31685],[34,551567],{"className":551568},[31782,41488],[34,551570,551572],{"className":551571},[41453],[34,551573,551575,551623],{"className":551574},[31796,31797],[34,551576,551578,551620],{"className":551577},[31801],[34,551579,551581,551598,551606],{"className":551580,"style":244945},[31805],[34,551582,551583,551586],{"style":41504},[34,551584],{"className":551585,"style":41508},[31813],[34,551587,551589],{"className":551588},[31818,31819,31820,31821],[34,551590,551592,551595],{"className":551591},[31685,31821],[34,551593,768],{"className":551594},[31685,31821],[34,551596,111],{"className":551597},[31685,31686,31821],[34,551599,551600,551603],{"style":41520},[34,551601],{"className":551602,"style":41508},[31813],[34,551604],{"className":551605,"style":41528},[41527],[34,551607,551608,551611],{"style":46771},[34,551609],{"className":551610,"style":41508},[31813],[34,551612,551614],{"className":551613},[31818,31819,31820,31821],[34,551615,551617],{"className":551616},[31685,31821],[34,551618,58442],{"className":551619,"style":32062},[31685,31686,31821],[34,551621,31829],{"className":551622},[31828],[34,551624,551626],{"className":551625},[31801],[34,551627,551629],{"className":551628,"style":41619},[31805],[34,551630],{},[34,551632],{"className":551633},[31690,41488],"，是收敛的。",[108,551636,551637,551639,551640,551691,551692,347],{},[2627,551638,62532],{},"\n根据 Weierstrass M-判别法，原积分在 ",[34,551641,551643,551661],{"className":551642},[31642],[34,551644,551646],{"className":551645},[31646],[31648,551647,551648],{"xmlns":31650},[31652,551649,551650,551658],{},[31655,551651,551652,551654,551656],{},[31658,551653,67764],{},[31662,551655,43175],{},[31658,551657,111],{},[31665,551659,551660],{"encoding":31667},"y \\geq a",[34,551662,551664,551682],{"className":551663,"ariaHidden":758},[31672],[34,551665,551667,551670,551673,551676,551679],{"className":551666},[31676],[34,551668],{"className":551669,"style":549750},[31680],[34,551671,67764],{"className":551672,"style":32062},[31685,31686],[34,551674],{"className":551675,"style":31846},[31845],[34,551677,43175],{"className":551678},[31850],[34,551680],{"className":551681,"style":31846},[31845],[34,551683,551685,551688],{"className":551684},[31676],[34,551686],{"className":551687,"style":32527},[31680],[34,551689,111],{"className":551690},[31685,31686]," 上",[2627,551693,548044],{},[32188,551695],{},[253,551697,551699],{"id":551698},"讲解-2",[2627,551700,551701],{},"讲解 (2)",[15,551703,551704],{},[34,551705,551707,551784],{"className":551706},[31642],[34,551708,551710],{"className":551709},[31646],[31648,551711,551712],{"xmlns":31650},[31652,551713,551714,551782],{},[31655,551715,551716,551728,551748,551760,551762,551764,551766,551768,551770,551772,551774,551776],{},[38846,551717,551718,551720,551722],{},[31662,551719,546295],{},[31749,551721,459],{},[31655,551723,551724,551726],{},[31662,551725,223],{},[31658,551727,138286],{"mathvariant":38079},[41453,551729,551730,551740],{},[31655,551731,551732,551734,551736,551738],{},[31658,551733,547853],{},[31662,551735,32682],{},[31749,551737,768],{},[31658,551739,32936],{},[31655,551741,551742,551744,551746],{},[31658,551743,32936],{},[31662,551745,223],{},[31658,551747,217392],{},[37624,551749,551750,551752],{},[31658,551751,33244],{},[31655,551753,551754,551756,551758],{},[31662,551755,32226],{},[31658,551757,217392],{},[31658,551759,32936],{},[31658,551761,46900],{},[31658,551763,32936],{},[31662,551765,31960],{"separator":758},[31845,551767],{"width":71082},[31749,551769,459],{},[31662,551771,45972],{},[31658,551773,217392],{},[31662,551775,45972],{},[31743,551777,551778,551780],{},[31658,551779,217392],{},[31749,551781,459],{},[31665,551783,548548],{"encoding":31667},[34,551785,551787,552013,552031],{"className":551786,"ariaHidden":758},[31672],[34,551788,551790,551793,551853,551856,551948,551986,551989,551992,551995,551998,552001,552004,552007,552010],{"className":551789},[31676],[34,551791],{"className":551792,"style":548558},[31680],[34,551794,551796,551799],{"className":551795},[32767],[34,551797,546295],{"className":551798,"style":546331},[32767,131249,156560],[34,551800,551802],{"className":551801},[31792],[34,551803,551805,551845],{"className":551804},[31796,31797],[34,551806,551808,551842],{"className":551807},[31801],[34,551809,551811,551825],{"className":551810,"style":546550},[31805],[34,551812,551813,551816],{"style":546347},[34,551814],{"className":551815,"style":31814},[31813],[34,551817,551819],{"className":551818},[31818,31819,31820,31821],[34,551820,551822],{"className":551821},[31685,31821],[34,551823,459],{"className":551824},[31685,31821],[34,551826,551827,551830],{"style":546362},[34,551828],{"className":551829,"style":31814},[31813],[34,551831,551833],{"className":551832},[31818,31819,31820,31821],[34,551834,551836,551839],{"className":551835},[31685,31821],[34,551837,223],{"className":551838},[31685,31821],[34,551840,138286],{"className":551841},[31685,31821],[34,551843,31829],{"className":551844},[31828],[34,551846,551848],{"className":551847},[31801],[34,551849,551851],{"className":551850,"style":546384},[31805],[34,551852],{},[34,551854],{"className":551855,"style":32123},[31845],[34,551857,551859,551862,551945],{"className":551858},[31685],[34,551860],{"className":551861},[31782,41488],[34,551863,551865],{"className":551864},[41453],[34,551866,551868,551937],{"className":551867},[31796,31797],[34,551869,551871,551934],{"className":551870},[31801],[34,551872,551874,551894,551902],{"className":551873,"style":548640},[31805],[34,551875,551876,551879],{"style":41504},[34,551877],{"className":551878,"style":41508},[31813],[34,551880,551882],{"className":551881},[31818,31819,31820,31821],[34,551883,551885,551888,551891],{"className":551884},[31685,31821],[34,551886,32936],{"className":551887},[31685,31686,31821],[34,551889,223],{"className":551890},[32024,31821],[34,551892,217392],{"className":551893,"style":217496},[31685,31686,31821],[34,551895,551896,551899],{"style":41520},[34,551897],{"className":551898,"style":41508},[31813],[34,551900],{"className":551901,"style":41528},[41527],[34,551903,551904,551907],{"style":46771},[34,551905],{"className":551906,"style":41508},[31813],[34,551908,551910],{"className":551909},[31818,31819,31820,31821],[34,551911,551913,551925,551928,551931],{"className":551912},[31685,31821],[34,551914,551916,551919,551922],{"className":551915},[32767,31821],[34,551917,31747],{"className":551918},[31821],[34,551920,31955],{"className":551921},[31821],[34,551923,31660],{"className":551924},[31821],[34,551926],{"className":551927,"style":43430},[31845,31821],[34,551929,768],{"className":551930},[31685,31821],[34,551932,32936],{"className":551933},[31685,31686,31821],[34,551935,31829],{"className":551936},[31828],[34,551938,551940],{"className":551939},[31801],[34,551941,551943],{"className":551942,"style":400163},[31805],[34,551944],{},[34,551946],{"className":551947},[31690,41488],[34,551949,551951,551954],{"className":551950},[31685],[34,551952,33244],{"className":551953},[31685,31686],[34,551955,551957],{"className":551956},[31792],[34,551958,551960],{"className":551959},[31796],[34,551961,551963],{"className":551962},[31801],[34,551964,551966],{"className":551965,"style":156828},[31805],[34,551967,551968,551971],{"style":37777},[34,551969],{"className":551970,"style":31814},[31813],[34,551972,551974],{"className":551973},[31818,31819,31820,31821],[34,551975,551977,551980,551983],{"className":551976},[31685,31821],[34,551978,32226],{"className":551979},[31685,31821],[34,551981,217392],{"className":551982,"style":217496},[31685,31686,31821],[34,551984,32936],{"className":551985},[31685,31686,31821],[34,551987,46900],{"className":551988},[31685,31686],[34,551990,32936],{"className":551991},[31685,31686],[34,551993,31960],{"className":551994},[32119],[34,551996],{"className":551997,"style":128438},[31845],[34,551999],{"className":552000,"style":32123},[31845],[34,552002,459],{"className":552003},[31685],[34,552005],{"className":552006,"style":31846},[31845],[34,552008,45972],{"className":552009},[31850],[34,552011],{"className":552012,"style":31846},[31845],[34,552014,552016,552019,552022,552025,552028],{"className":552015},[31676],[34,552017],{"className":552018,"style":43198},[31680],[34,552020,217392],{"className":552021,"style":217496},[31685,31686],[34,552023],{"className":552024,"style":31846},[31845],[34,552026,45972],{"className":552027},[31850],[34,552029],{"className":552030,"style":31846},[31845],[34,552032,552034,552037],{"className":552033},[31676],[34,552035],{"className":552036,"style":31860},[31680],[34,552038,552040,552043],{"className":552039},[31685],[34,552041,217392],{"className":552042,"style":217496},[31685,31686],[34,552044,552046],{"className":552045},[31792],[34,552047,552049,552069],{"className":552048},[31796,31797],[34,552050,552052,552066],{"className":552051},[31801],[34,552053,552055],{"className":552054,"style":31806},[31805],[34,552056,552057,552060],{"style":548824},[34,552058],{"className":552059,"style":31814},[31813],[34,552061,552063],{"className":552062},[31818,31819,31820,31821],[34,552064,459],{"className":552065},[31685,31821],[34,552067,31829],{"className":552068},[31828],[34,552070,552072],{"className":552071},[31801],[34,552073,552075],{"className":552074,"style":31836},[31805],[34,552076],{},[15,552078,552079,552081,552082,552126,552127,4374,552130,552133,552134,258,552215,347],{},[2627,552080,549449],{}," 这里有 ",[34,552083,552085,552105],{"className":552084},[31642],[34,552086,552088],{"className":552087},[31646],[31648,552089,552090],{"xmlns":31650},[31652,552091,552092,552102],{},[31655,552093,552094,552096,552098,552100],{},[31658,552095,547853],{},[31662,552097,32682],{},[31749,552099,768],{},[31658,552101,32936],{},[31665,552103,552104],{"encoding":31667},"\\sin 2x",[34,552106,552108],{"className":552107,"ariaHidden":758},[31672],[34,552109,552111,552114,552117,552120,552123],{"className":552110},[31676],[34,552112],{"className":552113,"style":71430},[31680],[34,552115,547853],{"className":552116},[32767],[34,552118],{"className":552119,"style":32123},[31845],[34,552121,768],{"className":552122},[31685],[34,552124,32936],{"className":552125},[31685,31686]," 这种震荡函数，M-判别法通常很难凑效（绝对值放缩后积分发散），所以我们考虑 ",[2627,552128,552129],{},"Dirichlet 判别法",[2627,552131,552132],{},"Abel 判别法","。\n这里我们将函数拆为 ",[34,552135,552137,552167],{"className":552136},[31642],[34,552138,552140],{"className":552139},[31646],[31648,552141,552142],{"xmlns":31650},[31652,552143,552144,552164],{},[31655,552145,552146,552148,552150,552152,552154,552156,552158,552160,552162],{},[31658,552147,31739],{},[31662,552149,456],{"stretchy":702},[31658,552151,32936],{},[31662,552153,2557],{"stretchy":702},[31662,552155,202],{},[31658,552157,547853],{},[31662,552159,32682],{},[31749,552161,768],{},[31658,552163,32936],{},[31665,552165,552166],{"encoding":31667},"f(x) = \\sin 2x",[34,552168,552170,552197],{"className":552169,"ariaHidden":758},[31672],[34,552171,552173,552176,552179,552182,552185,552188,552191,552194],{"className":552172},[31676],[34,552174],{"className":552175,"style":31774},[31680],[34,552177,31739],{"className":552178,"style":31778},[31685,31686],[34,552180,456],{"className":552181},[31782],[34,552183,32936],{"className":552184},[31685,31686],[34,552186,2557],{"className":552187},[31690],[34,552189],{"className":552190,"style":31846},[31845],[34,552192,202],{"className":552193},[31850],[34,552195],{"className":552196,"style":31846},[31845],[34,552198,552200,552203,552206,552209,552212],{"className":552199},[31676],[34,552201],{"className":552202,"style":71430},[31680],[34,552204,547853],{"className":552205},[32767],[34,552207],{"className":552208,"style":32123},[31845],[34,552210,768],{"className":552211},[31685],[34,552213,32936],{"className":552214},[31685,31686],[34,552216,552218,552266],{"className":552217},[31642],[34,552219,552221],{"className":552220},[31646],[31648,552222,552223],{"xmlns":31650},[31652,552224,552225,552263],{},[31655,552226,552227,552229,552231,552233,552235,552237,552239,552241],{},[31658,552228,31942],{},[31662,552230,456],{"stretchy":702},[31658,552232,32936],{},[31662,552234,31960],{"separator":758},[31658,552236,217392],{},[31662,552238,2557],{"stretchy":702},[31662,552240,202],{},[41453,552242,552243,552255],{},[37624,552244,552245,552247],{},[31658,552246,33244],{},[31655,552248,552249,552251,552253],{},[31662,552250,32226],{},[31658,552252,217392],{},[31658,552254,32936],{},[31655,552256,552257,552259,552261],{},[31658,552258,32936],{},[31662,552260,223],{},[31658,552262,217392],{},[31665,552264,552265],{"encoding":31667},"g(x, \\alpha) = \\frac{e^{-\\alpha x}}{x + \\alpha}",[34,552267,552269,552305],{"className":552268,"ariaHidden":758},[31672],[34,552270,552272,552275,552278,552281,552284,552287,552290,552293,552296,552299,552302],{"className":552271},[31676],[34,552273],{"className":552274,"style":31774},[31680],[34,552276,31942],{"className":552277,"style":32062},[31685,31686],[34,552279,456],{"className":552280},[31782],[34,552282,32936],{"className":552283},[31685,31686],[34,552285,31960],{"className":552286},[32119],[34,552288],{"className":552289,"style":32123},[31845],[34,552291,217392],{"className":552292,"style":217496},[31685,31686],[34,552294,2557],{"className":552295},[31690],[34,552297],{"className":552298,"style":31846},[31845],[34,552300,202],{"className":552301},[31850],[34,552303],{"className":552304,"style":31846},[31845],[34,552306,552308,552312],{"className":552307},[31676],[34,552309],{"className":552310,"style":552311},[31680],"height:1.3907em;vertical-align:-0.4033em;",[34,552313,552315,552318,552421],{"className":552314},[31685],[34,552316],{"className":552317},[31782,41488],[34,552319,552321],{"className":552320},[41453],[34,552322,552324,552413],{"className":552323},[31796,31797],[34,552325,552327,552410],{"className":552326},[31801],[34,552328,552331,552351,552359],{"className":552329,"style":552330},[31805],"height:0.9874em;",[34,552332,552333,552336],{"style":41504},[34,552334],{"className":552335,"style":41508},[31813],[34,552337,552339],{"className":552338},[31818,31819,31820,31821],[34,552340,552342,552345,552348],{"className":552341},[31685,31821],[34,552343,32936],{"className":552344},[31685,31686,31821],[34,552346,223],{"className":552347},[32024,31821],[34,552349,217392],{"className":552350,"style":217496},[31685,31686,31821],[34,552352,552353,552356],{"style":41520},[34,552354],{"className":552355,"style":41508},[31813],[34,552357],{"className":552358,"style":41528},[41527],[34,552360,552361,552364],{"style":46771},[34,552362],{"className":552363,"style":41508},[31813],[34,552365,552367],{"className":552366},[31818,31819,31820,31821],[34,552368,552370],{"className":552369},[31685,31821],[34,552371,552373,552376],{"className":552372},[31685,31821],[34,552374,33244],{"className":552375},[31685,31686,31821],[34,552377,552379],{"className":552378},[31792],[34,552380,552382],{"className":552381},[31796],[34,552383,552385],{"className":552384},[31801],[34,552386,552389],{"className":552387,"style":552388},[31805],"height:0.8477em;",[34,552390,552392,552395],{"style":552391},"top:-2.931em;margin-right:0.0714em;",[34,552393],{"className":552394,"style":43406},[31813],[34,552396,552398],{"className":552397},[31818,43410,43411,31821],[34,552399,552401,552404,552407],{"className":552400},[31685,31821],[34,552402,32226],{"className":552403},[31685,31821],[34,552405,217392],{"className":552406,"style":217496},[31685,31686,31821],[34,552408,32936],{"className":552409},[31685,31686,31821],[34,552411,31829],{"className":552412},[31828],[34,552414,552416],{"className":552415},[31801],[34,552417,552419],{"className":552418,"style":400163},[31805],[34,552420],{},[34,552422],{"className":552423},[31690,41488],[15,552425,552426],{},[2627,552427,549629],{},[2230,552429,552430,552974,553270,553854],{},[108,552431,552432,552515,552970,552971,31618],{},[2627,552433,326327,552434,552514],{},[34,552435,552437,552466],{"className":552436},[31642],[34,552438,552440],{"className":552439},[31646],[31648,552441,552442],{"xmlns":31650},[31652,552443,552444,552464],{},[31655,552445,552446,552448,552450,552452,552454,552456,552458,552460,552462],{},[31658,552447,31739],{},[31662,552449,456],{"stretchy":702},[31658,552451,32936],{},[31662,552453,2557],{"stretchy":702},[31662,552455,202],{},[31658,552457,547853],{},[31662,552459,32682],{},[31749,552461,768],{},[31658,552463,32936],{},[31665,552465,552166],{"encoding":31667},[34,552467,552469,552496],{"className":552468,"ariaHidden":758},[31672],[34,552470,552472,552475,552478,552481,552484,552487,552490,552493],{"className":552471},[31676],[34,552473],{"className":552474,"style":31774},[31680],[34,552476,31739],{"className":552477,"style":31778},[31685,31686],[34,552479,456],{"className":552480},[31782],[34,552482,32936],{"className":552483},[31685,31686],[34,552485,2557],{"className":552486},[31690],[34,552488],{"className":552489,"style":31846},[31845],[34,552491,202],{"className":552492},[31850],[34,552494],{"className":552495,"style":31846},[31845],[34,552497,552499,552502,552505,552508,552511],{"className":552498},[31676],[34,552500],{"className":552501,"style":71430},[31680],[34,552503,547853],{"className":552504},[32767],[34,552506],{"className":552507,"style":32123},[31845],[34,552509,768],{"className":552510},[31685],[34,552512,32936],{"className":552513},[31685,31686]," 的积分：",[34,552516,552518,552590],{"className":552517},[31642],[34,552519,552521],{"className":552520},[31646],[31648,552522,552523],{"xmlns":31650},[31652,552524,552525,552587],{},[31655,552526,552527,552553,552555,552583,552585],{},[31655,552528,552529,552531,552539,552541,552543,552545,552547,552549,552551],{},[31662,552530,162086],{"fence":758},[38846,552532,552533,552535,552537],{},[31662,552534,546295],{},[31749,552536,459],{},[31658,552538,54151],{},[31658,552540,547853],{},[31662,552542,32682],{},[31749,552544,768],{},[31658,552546,32936],{},[31658,552548,46900],{},[31658,552550,32936],{},[31662,552552,162086],{"fence":758},[31662,552554,202],{},[31655,552556,552557,552559,552565,552567,552569,552571,552573,552575,552577,552579,552581],{},[31662,552558,162086],{"fence":758},[41453,552560,552561,552563],{},[31749,552562,640],{},[31749,552564,768],{},[31662,552566,456],{"stretchy":702},[31749,552568,640],{},[31662,552570,32226],{},[31658,552572,547860],{},[31662,552574,32682],{},[31749,552576,768],{},[31658,552578,54151],{},[31662,552580,2557],{"stretchy":702},[31662,552582,162086],{"fence":758},[31662,552584,45972],{},[31749,552586,640],{},[31665,552588,552589],{"encoding":31667},"\\left| \\int_{0}^{A} \\sin 2x dx \\right| = \\left| \\frac{1}{2}(1 - \\cos 2A) \\right| \\leq 1",[34,552591,552593,552764,552961],{"className":552592,"ariaHidden":758},[31672],[34,552594,552596,552599,552755,552758,552761],{"className":552595},[31676],[34,552597],{"className":552598,"style":141282},[31680],[34,552600,552602,552639,552697,552700,552703,552706,552709,552712,552715,552718],{"className":552601},[34006],[34,552603,552605],{"className":552604},[31782],[34,552606,552608],{"className":552607},[71281,71282],[34,552609,552611,552631],{"className":552610},[31796,31797],[34,552612,552614,552628],{"className":552613},[31801],[34,552615,552617],{"className":552616,"style":550104},[31805],[34,552618,552619,552622],{"style":71329},[34,552620],{"className":552621,"style":550110},[31813],[34,552623,552624],{"style":550113},[41588,552625,552626],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,552627],{"d":550120},[34,552629,31829],{"className":552630},[31828],[34,552632,552634],{"className":552633},[31801],[34,552635,552637],{"className":552636,"style":550130},[31805],[34,552638],{},[34,552640,552642,552645],{"className":552641},[32767],[34,552643,546295],{"className":552644,"style":546331},[32767,131249,156560],[34,552646,552648],{"className":552647},[31792],[34,552649,552651,552689],{"className":552650},[31796,31797],[34,552652,552654,552686],{"className":552653},[31801],[34,552655,552658,552672],{"className":552656,"style":552657},[31805],"height:1.0362em;",[34,552659,552660,552663],{"style":546347},[34,552661],{"className":552662,"style":31814},[31813],[34,552664,552666],{"className":552665},[31818,31819,31820,31821],[34,552667,552669],{"className":552668},[31685,31821],[34,552670,459],{"className":552671},[31685,31821],[34,552673,552674,552677],{"style":546362},[34,552675],{"className":552676,"style":31814},[31813],[34,552678,552680],{"className":552679},[31818,31819,31820,31821],[34,552681,552683],{"className":552682},[31685,31821],[34,552684,54151],{"className":552685},[31685,31686,31821],[34,552687,31829],{"className":552688},[31828],[34,552690,552692],{"className":552691},[31801],[34,552693,552695],{"className":552694,"style":546384},[31805],[34,552696],{},[34,552698],{"className":552699,"style":32123},[31845],[34,552701,547853],{"className":552702},[32767],[34,552704],{"className":552705,"style":32123},[31845],[34,552707,768],{"className":552708},[31685],[34,552710,32936],{"className":552711},[31685,31686],[34,552713,46900],{"className":552714},[31685,31686],[34,552716,32936],{"className":552717},[31685,31686],[34,552719,552721],{"className":552720},[31690],[34,552722,552724],{"className":552723},[71281,71282],[34,552725,552727,552747],{"className":552726},[31796,31797],[34,552728,552730,552744],{"className":552729},[31801],[34,552731,552733],{"className":552732,"style":550104},[31805],[34,552734,552735,552738],{"style":71329},[34,552736],{"className":552737,"style":550110},[31813],[34,552739,552740],{"style":550113},[41588,552741,552742],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,552743],{"d":550120},[34,552745,31829],{"className":552746},[31828],[34,552748,552750],{"className":552749},[31801],[34,552751,552753],{"className":552752,"style":550130},[31805],[34,552754],{},[34,552756],{"className":552757,"style":31846},[31845],[34,552759,202],{"className":552760},[31850],[34,552762],{"className":552763,"style":31846},[31845],[34,552765,552767,552770,552952,552955,552958],{"className":552766},[31676],[34,552768],{"className":552769,"style":345868},[31680],[34,552771,552773,552817,552885,552888,552891,552894,552897,552900,552903,552906,552909,552912,552915],{"className":552772},[34006],[34,552774,552776],{"className":552775},[31782],[34,552777,552779],{"className":552778},[71281,71282],[34,552780,552782,552808],{"className":552781},[31796,31797],[34,552783,552785,552805],{"className":552784},[31801],[34,552786,552789],{"className":552787,"style":552788},[31805],"height:0.85em;",[34,552790,552792,552796],{"style":552791},"top:-2.85em;",[34,552793],{"className":552794,"style":552795},[31813],"height:3.2em;",[34,552797,552799],{"style":552798},"width:0.333em;height:1.2em;",[41588,552800,552802],{"xmlns":41590,"width":503054,"height":345801,"viewBox":552801},"0 0 333 1200",[20398,552803],{"d":552804},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z",[34,552806,31829],{"className":552807},[31828],[34,552809,552811],{"className":552810},[31801],[34,552812,552815],{"className":552813,"style":552814},[31805],"height:0.35em;",[34,552816],{},[34,552818,552820,552823,552882],{"className":552819},[31685],[34,552821],{"className":552822},[31782,41488],[34,552824,552826],{"className":552825},[41453],[34,552827,552829,552874],{"className":552828},[31796,31797],[34,552830,552832,552871],{"className":552831},[31801],[34,552833,552835,552849,552857],{"className":552834,"style":46746},[31805],[34,552836,552837,552840],{"style":41504},[34,552838],{"className":552839,"style":41508},[31813],[34,552841,552843],{"className":552842},[31818,31819,31820,31821],[34,552844,552846],{"className":552845},[31685,31821],[34,552847,768],{"className":552848},[31685,31821],[34,552850,552851,552854],{"style":41520},[34,552852],{"className":552853,"style":41508},[31813],[34,552855],{"className":552856,"style":41528},[41527],[34,552858,552859,552862],{"style":46771},[34,552860],{"className":552861,"style":41508},[31813],[34,552863,552865],{"className":552864},[31818,31819,31820,31821],[34,552866,552868],{"className":552867},[31685,31821],[34,552869,640],{"className":552870},[31685,31821],[34,552872,31829],{"className":552873},[31828],[34,552875,552877],{"className":552876},[31801],[34,552878,552880],{"className":552879,"style":41619},[31805],[34,552881],{},[34,552883],{"className":552884},[31690,41488],[34,552886,456],{"className":552887},[31782],[34,552889,640],{"className":552890},[31685],[34,552892],{"className":552893,"style":32604},[31845],[34,552895,32226],{"className":552896},[32024],[34,552898],{"className":552899,"style":32604},[31845],[34,552901,547860],{"className":552902},[32767],[34,552904],{"className":552905,"style":32123},[31845],[34,552907,768],{"className":552908},[31685],[34,552910,54151],{"className":552911},[31685,31686],[34,552913,2557],{"className":552914},[31690],[34,552916,552918],{"className":552917},[31690],[34,552919,552921],{"className":552920},[71281,71282],[34,552922,552924,552944],{"className":552923},[31796,31797],[34,552925,552927,552941],{"className":552926},[31801],[34,552928,552930],{"className":552929,"style":552788},[31805],[34,552931,552932,552935],{"style":552791},[34,552933],{"className":552934,"style":552795},[31813],[34,552936,552937],{"style":552798},[41588,552938,552939],{"xmlns":41590,"width":503054,"height":345801,"viewBox":552801},[20398,552940],{"d":552804},[34,552942,31829],{"className":552943},[31828],[34,552945,552947],{"className":552946},[31801],[34,552948,552950],{"className":552949,"style":552814},[31805],[34,552951],{},[34,552953],{"className":552954,"style":31846},[31845],[34,552956,45972],{"className":552957},[31850],[34,552959],{"className":552960,"style":31846},[31845],[34,552962,552964,552967],{"className":552963},[31676],[34,552965],{"className":552966,"style":32617},[31680],[34,552968,640],{"className":552969},[31685],"。\n说明积分的部分和是",[2627,552972,552973],{},"一致有界",[108,552975,552976,553183,553184,553212,553213,553269],{},[2627,552977,326327,552978,553182],{},[34,552979,552981,553028],{"className":552980},[31642],[34,552982,552984],{"className":552983},[31646],[31648,552985,552986],{"xmlns":31650},[31652,552987,552988,553026],{},[31655,552989,552990,552992,552994,552996,552998,553000,553002,553004],{},[31658,552991,31942],{},[31662,552993,456],{"stretchy":702},[31658,552995,32936],{},[31662,552997,31960],{"separator":758},[31658,552999,217392],{},[31662,553001,2557],{"stretchy":702},[31662,553003,202],{},[41453,553005,553006,553018],{},[37624,553007,553008,553010],{},[31658,553009,33244],{},[31655,553011,553012,553014,553016],{},[31662,553013,32226],{},[31658,553015,217392],{},[31658,553017,32936],{},[31655,553019,553020,553022,553024],{},[31658,553021,32936],{},[31662,553023,223],{},[31658,553025,217392],{},[31665,553027,552265],{"encoding":31667},[34,553029,553031,553067],{"className":553030,"ariaHidden":758},[31672],[34,553032,553034,553037,553040,553043,553046,553049,553052,553055,553058,553061,553064],{"className":553033},[31676],[34,553035],{"className":553036,"style":31774},[31680],[34,553038,31942],{"className":553039,"style":32062},[31685,31686],[34,553041,456],{"className":553042},[31782],[34,553044,32936],{"className":553045},[31685,31686],[34,553047,31960],{"className":553048},[32119],[34,553050],{"className":553051,"style":32123},[31845],[34,553053,217392],{"className":553054,"style":217496},[31685,31686],[34,553056,2557],{"className":553057},[31690],[34,553059],{"className":553060,"style":31846},[31845],[34,553062,202],{"className":553063},[31850],[34,553065],{"className":553066,"style":31846},[31845],[34,553068,553070,553073],{"className":553069},[31676],[34,553071],{"className":553072,"style":552311},[31680],[34,553074,553076,553079,553179],{"className":553075},[31685],[34,553077],{"className":553078},[31782,41488],[34,553080,553082],{"className":553081},[41453],[34,553083,553085,553171],{"className":553084},[31796,31797],[34,553086,553088,553168],{"className":553087},[31801],[34,553089,553091,553111,553119],{"className":553090,"style":552330},[31805],[34,553092,553093,553096],{"style":41504},[34,553094],{"className":553095,"style":41508},[31813],[34,553097,553099],{"className":553098},[31818,31819,31820,31821],[34,553100,553102,553105,553108],{"className":553101},[31685,31821],[34,553103,32936],{"className":553104},[31685,31686,31821],[34,553106,223],{"className":553107},[32024,31821],[34,553109,217392],{"className":553110,"style":217496},[31685,31686,31821],[34,553112,553113,553116],{"style":41520},[34,553114],{"className":553115,"style":41508},[31813],[34,553117],{"className":553118,"style":41528},[41527],[34,553120,553121,553124],{"style":46771},[34,553122],{"className":553123,"style":41508},[31813],[34,553125,553127],{"className":553126},[31818,31819,31820,31821],[34,553128,553130],{"className":553129},[31685,31821],[34,553131,553133,553136],{"className":553132},[31685,31821],[34,553134,33244],{"className":553135},[31685,31686,31821],[34,553137,553139],{"className":553138},[31792],[34,553140,553142],{"className":553141},[31796],[34,553143,553145],{"className":553144},[31801],[34,553146,553148],{"className":553147,"style":552388},[31805],[34,553149,553150,553153],{"style":552391},[34,553151],{"className":553152,"style":43406},[31813],[34,553154,553156],{"className":553155},[31818,43410,43411,31821],[34,553157,553159,553162,553165],{"className":553158},[31685,31821],[34,553160,32226],{"className":553161},[31685,31821],[34,553163,217392],{"className":553164,"style":217496},[31685,31686,31821],[34,553166,32936],{"className":553167},[31685,31686,31821],[34,553169,31829],{"className":553170},[31828],[34,553172,553174],{"className":553173},[31801],[34,553175,553177],{"className":553176,"style":400163},[31805],[34,553178],{},[34,553180],{"className":553181},[31690,41488]," 的单调性：","\n关于 ",[34,553185,553187,553200],{"className":553186},[31642],[34,553188,553190],{"className":553189},[31646],[31648,553191,553192],{"xmlns":31650},[31652,553193,553194,553198],{},[31655,553195,553196],{},[31658,553197,32936],{},[31665,553199,32936],{"encoding":31667},[34,553201,553203],{"className":553202,"ariaHidden":758},[31672],[34,553204,553206,553209],{"className":553205},[31676],[34,553207],{"className":553208,"style":32527},[31680],[34,553210,32936],{"className":553211},[31685,31686],"，分母增加，分子减小（指数函数），所以 ",[34,553214,553216,553239],{"className":553215},[31642],[34,553217,553219],{"className":553218},[31646],[31648,553220,553221],{"xmlns":31650},[31652,553222,553223,553237],{},[31655,553224,553225,553227,553229,553231,553233,553235],{},[31658,553226,31942],{},[31662,553228,456],{"stretchy":702},[31658,553230,32936],{},[31662,553232,31960],{"separator":758},[31658,553234,217392],{},[31662,553236,2557],{"stretchy":702},[31665,553238,547915],{"encoding":31667},[34,553240,553242],{"className":553241,"ariaHidden":758},[31672],[34,553243,553245,553248,553251,553254,553257,553260,553263,553266],{"className":553244},[31676],[34,553246],{"className":553247,"style":31774},[31680],[34,553249,31942],{"className":553250,"style":32062},[31685,31686],[34,553252,456],{"className":553253},[31782],[34,553255,32936],{"className":553256},[31685,31686],[34,553258,31960],{"className":553259},[32119],[34,553261],{"className":553262,"style":32123},[31845],[34,553264,217392],{"className":553265,"style":217496},[31685,31686],[34,553267,2557],{"className":553268},[31690]," 是单调递减的。",[108,553271,553272,553331,553332,4504,553388,354330,553656,553787,553788,553816,553817,553853],{},[2627,553273,326327,553274,553330],{},[34,553275,553277,553300],{"className":553276},[31642],[34,553278,553280],{"className":553279},[31646],[31648,553281,553282],{"xmlns":31650},[31652,553283,553284,553298],{},[31655,553285,553286,553288,553290,553292,553294,553296],{},[31658,553287,31942],{},[31662,553289,456],{"stretchy":702},[31658,553291,32936],{},[31662,553293,31960],{"separator":758},[31658,553295,217392],{},[31662,553297,2557],{"stretchy":702},[31665,553299,547915],{"encoding":31667},[34,553301,553303],{"className":553302,"ariaHidden":758},[31672],[34,553304,553306,553309,553312,553315,553318,553321,553324,553327],{"className":553305},[31676],[34,553307],{"className":553308,"style":31774},[31680],[34,553310,31942],{"className":553311,"style":32062},[31685,31686],[34,553313,456],{"className":553314},[31782],[34,553316,32936],{"className":553317},[31685,31686],[34,553319,31960],{"className":553320},[32119],[34,553322],{"className":553323,"style":32123},[31845],[34,553325,217392],{"className":553326,"style":217496},[31685,31686],[34,553328,2557],{"className":553329},[31690]," 是否一致趋于 0：","\n当 ",[34,553333,553335,553355],{"className":553334},[31642],[34,553336,553338],{"className":553337},[31646],[31648,553339,553340],{"xmlns":31650},[31652,553341,553342,553352],{},[31655,553343,553344,553346,553348,553350],{},[31658,553345,32936],{},[31662,553347,33239],{},[31662,553349,223],{},[31658,553351,138286],{"mathvariant":38079},[31665,553353,553354],{"encoding":31667},"x \\to +\\infty",[34,553356,553358,553376],{"className":553357,"ariaHidden":758},[31672],[34,553359,553361,553364,553367,553370,553373],{"className":553360},[31676],[34,553362],{"className":553363,"style":32527},[31680],[34,553365,32936],{"className":553366},[31685,31686],[34,553368],{"className":553369,"style":31846},[31845],[34,553371,33239],{"className":553372},[31850],[34,553374],{"className":553375,"style":31846},[31845],[34,553377,553379,553382,553385],{"className":553378},[31676],[34,553380],{"className":553381,"style":32597},[31680],[34,553383,223],{"className":553384},[31685],[34,553386,138286],{"className":553387},[31685],[34,553389,553391,553437],{"className":553390},[31642],[34,553392,553394],{"className":553393},[31646],[31648,553395,553396],{"xmlns":31650},[31652,553397,553398,553434],{},[31655,553399,553400,553422,553424,553430,553432],{},[41453,553401,553402,553414],{},[37624,553403,553404,553406],{},[31658,553405,33244],{},[31655,553407,553408,553410,553412],{},[31662,553409,32226],{},[31658,553411,217392],{},[31658,553413,32936],{},[31655,553415,553416,553418,553420],{},[31658,553417,32936],{},[31662,553419,223],{},[31658,553421,217392],{},[31662,553423,45972],{},[41453,553425,553426,553428],{},[31749,553427,640],{},[31658,553429,32936],{},[31662,553431,33239],{},[31749,553433,459],{},[31665,553435,553436],{"encoding":31667},"\\frac{e^{-\\alpha x}}{x + \\alpha} \\leq \\frac{1}{x} \\to 0",[34,553438,553440,553564,553647],{"className":553439,"ariaHidden":758},[31672],[34,553441,553443,553446,553555,553558,553561],{"className":553442},[31676],[34,553444],{"className":553445,"style":552311},[31680],[34,553447,553449,553452,553552],{"className":553448},[31685],[34,553450],{"className":553451},[31782,41488],[34,553453,553455],{"className":553454},[41453],[34,553456,553458,553544],{"className":553457},[31796,31797],[34,553459,553461,553541],{"className":553460},[31801],[34,553462,553464,553484,553492],{"className":553463,"style":552330},[31805],[34,553465,553466,553469],{"style":41504},[34,553467],{"className":553468,"style":41508},[31813],[34,553470,553472],{"className":553471},[31818,31819,31820,31821],[34,553473,553475,553478,553481],{"className":553474},[31685,31821],[34,553476,32936],{"className":553477},[31685,31686,31821],[34,553479,223],{"className":553480},[32024,31821],[34,553482,217392],{"className":553483,"style":217496},[31685,31686,31821],[34,553485,553486,553489],{"style":41520},[34,553487],{"className":553488,"style":41508},[31813],[34,553490],{"className":553491,"style":41528},[41527],[34,553493,553494,553497],{"style":46771},[34,553495],{"className":553496,"style":41508},[31813],[34,553498,553500],{"className":553499},[31818,31819,31820,31821],[34,553501,553503],{"className":553502},[31685,31821],[34,553504,553506,553509],{"className":553505},[31685,31821],[34,553507,33244],{"className":553508},[31685,31686,31821],[34,553510,553512],{"className":553511},[31792],[34,553513,553515],{"className":553514},[31796],[34,553516,553518],{"className":553517},[31801],[34,553519,553521],{"className":553520,"style":552388},[31805],[34,553522,553523,553526],{"style":552391},[34,553524],{"className":553525,"style":43406},[31813],[34,553527,553529],{"className":553528},[31818,43410,43411,31821],[34,553530,553532,553535,553538],{"className":553531},[31685,31821],[34,553533,32226],{"className":553534},[31685,31821],[34,553536,217392],{"className":553537,"style":217496},[31685,31686,31821],[34,553539,32936],{"className":553540},[31685,31686,31821],[34,553542,31829],{"className":553543},[31828],[34,553545,553547],{"className":553546},[31801],[34,553548,553550],{"className":553549,"style":400163},[31805],[34,553551],{},[34,553553],{"className":553554},[31690,41488],[34,553556],{"className":553557,"style":31846},[31845],[34,553559,45972],{"className":553560},[31850],[34,553562],{"className":553563,"style":31846},[31845],[34,553565,553567,553570,553638,553641,553644],{"className":553566},[31676],[34,553568],{"className":553569,"style":46727},[31680],[34,553571,553573,553576,553635],{"className":553572},[31685],[34,553574],{"className":553575},[31782,41488],[34,553577,553579],{"className":553578},[41453],[34,553580,553582,553627],{"className":553581},[31796,31797],[34,553583,553585,553624],{"className":553584},[31801],[34,553586,553588,553602,553610],{"className":553587,"style":46746},[31805],[34,553589,553590,553593],{"style":41504},[34,553591],{"className":553592,"style":41508},[31813],[34,553594,553596],{"className":553595},[31818,31819,31820,31821],[34,553597,553599],{"className":553598},[31685,31821],[34,553600,32936],{"className":553601},[31685,31686,31821],[34,553603,553604,553607],{"style":41520},[34,553605],{"className":553606,"style":41508},[31813],[34,553608],{"className":553609,"style":41528},[41527],[34,553611,553612,553615],{"style":46771},[34,553613],{"className":553614,"style":41508},[31813],[34,553616,553618],{"className":553617},[31818,31819,31820,31821],[34,553619,553621],{"className":553620},[31685,31821],[34,553622,640],{"className":553623},[31685,31821],[34,553625,31829],{"className":553626},[31828],[34,553628,553630],{"className":553629},[31801],[34,553631,553633],{"className":553632,"style":41619},[31805],[34,553634],{},[34,553636],{"className":553637},[31690,41488],[34,553639],{"className":553640,"style":31846},[31845],[34,553642,33239],{"className":553643},[31850],[34,553645],{"className":553646,"style":31846},[31845],[34,553648,553650,553653],{"className":553649},[31676],[34,553651],{"className":553652,"style":32617},[31680],[34,553654,459],{"className":553655},[31685],[34,553657,553659,553695],{"className":553658},[31642],[34,553660,553662],{"className":553661},[31646],[31648,553663,553664],{"xmlns":31650},[31652,553665,553666,553692],{},[31655,553667,553668,553670,553672,553674,553676,553688,553690],{},[31658,553669,217392],{},[31662,553671,43175],{},[31749,553673,459],{},[31662,553675,31960],{"separator":758},[37624,553677,553678,553680],{},[31658,553679,33244],{},[31655,553681,553682,553684,553686],{},[31662,553683,32226],{},[31658,553685,217392],{},[31658,553687,32936],{},[31662,553689,45972],{},[31749,553691,640],{},[31665,553693,553694],{"encoding":31667},"\\alpha \\ge 0, e^{-\\alpha x} \\le 1",[34,553696,553698,553716,553778],{"className":553697,"ariaHidden":758},[31672],[34,553699,553701,553704,553707,553710,553713],{"className":553700},[31676],[34,553702],{"className":553703,"style":43198},[31680],[34,553705,217392],{"className":553706,"style":217496},[31685,31686],[34,553708],{"className":553709,"style":31846},[31845],[34,553711,43175],{"className":553712},[31850],[34,553714],{"className":553715,"style":31846},[31845],[34,553717,553719,553722,553725,553728,553731,553769,553772,553775],{"className":553718},[31676],[34,553720],{"className":553721,"style":534277},[31680],[34,553723,459],{"className":553724},[31685],[34,553726,31960],{"className":553727},[32119],[34,553729],{"className":553730,"style":32123},[31845],[34,553732,553734,553737],{"className":553733},[31685],[34,553735,33244],{"className":553736},[31685,31686],[34,553738,553740],{"className":553739},[31792],[34,553741,553743],{"className":553742},[31796],[34,553744,553746],{"className":553745},[31801],[34,553747,553749],{"className":553748,"style":156828},[31805],[34,553750,553751,553754],{"style":37777},[34,553752],{"className":553753,"style":31814},[31813],[34,553755,553757],{"className":553756},[31818,31819,31820,31821],[34,553758,553760,553763,553766],{"className":553759},[31685,31821],[34,553761,32226],{"className":553762},[31685,31821],[34,553764,217392],{"className":553765,"style":217496},[31685,31686,31821],[34,553767,32936],{"className":553768},[31685,31686,31821],[34,553770],{"className":553771,"style":31846},[31845],[34,553773,45972],{"className":553774},[31850],[34,553776],{"className":553777,"style":31846},[31845],[34,553779,553781,553784],{"className":553780},[31676],[34,553782],{"className":553783,"style":32617},[31680],[34,553785,640],{"className":553786},[31685],"）。\n这个趋于0的速度与 ",[34,553789,553791,553804],{"className":553790},[31642],[34,553792,553794],{"className":553793},[31646],[31648,553795,553796],{"xmlns":31650},[31652,553797,553798,553802],{},[31655,553799,553800],{},[31658,553801,217392],{},[31665,553803,218244],{"encoding":31667},[34,553805,553807],{"className":553806,"ariaHidden":758},[31672],[34,553808,553810,553813],{"className":553809},[31676],[34,553811],{"className":553812,"style":32527},[31680],[34,553814,217392],{"className":553815,"style":217496},[31685,31686]," 无关（被 ",[34,553818,553820,553838],{"className":553819},[31642],[34,553821,553823],{"className":553822},[31646],[31648,553824,553825],{"xmlns":31650},[31652,553826,553827,553835],{},[31655,553828,553829,553831,553833],{},[31749,553830,640],{},[31658,553832,38080],{"mathvariant":38079},[31658,553834,32936],{},[31665,553836,553837],{"encoding":31667},"1\u002Fx",[34,553839,553841],{"className":553840,"ariaHidden":758},[31672],[34,553842,553844,553847,553850],{"className":553843},[31676],[34,553845],{"className":553846,"style":31774},[31680],[34,553848,249219],{"className":553849},[31685],[34,553851,32936],{"className":553852},[31685,31686]," 控制了）。",[108,553855,553856,553858,553859,347],{},[2627,553857,62532],{},"\n根据 Dirichlet 判别法，原积分",[2627,553860,548044],{},[32188,553862],{},[253,553864,553866],{"id":553865},"讲解-3",[2627,553867,553868],{},"讲解 (3)",[15,553870,553871],{},[34,553872,553874,553935],{"className":553873},[31642],[34,553875,553877],{"className":553876},[31646],[31648,553878,553879],{"xmlns":31650},[31652,553880,553881,553933],{},[31655,553882,553883,553895,553897,553899,553901,553907,553909,553911,553913,553915,553917,553919,553921,553923,553925,553927,553929,553931],{},[38846,553884,553885,553887,553889],{},[31662,553886,546295],{},[31749,553888,459],{},[31655,553890,553891,553893],{},[31662,553892,223],{},[31658,553894,138286],{"mathvariant":38079},[31658,553896,32936],{},[31658,553898,547853],{},[31662,553900,32682],{},[37624,553902,553903,553905],{},[31658,553904,32936],{},[31749,553906,15932],{},[31658,553908,547860],{},[31662,553910,32682],{},[31658,553912,217392],{},[31658,553914,32936],{},[31658,553916,46900],{},[31658,553918,32936],{},[31662,553920,31960],{"separator":758},[31845,553922],{"width":71082},[31658,553924,111],{},[31662,553926,45972],{},[31658,553928,217392],{},[31662,553930,45972],{},[31658,553932,1697],{},[31665,553934,548910],{"encoding":31667},[34,553936,553938,554090,554108],{"className":553937,"ariaHidden":758},[31672],[34,553939,553941,553944,554004,554007,554010,554013,554016,554019,554048,554051,554054,554057,554060,554063,554066,554069,554072,554075,554078,554081,554084,554087],{"className":553940},[31676],[34,553942],{"className":553943,"style":546531},[31680],[34,553945,553947,553950],{"className":553946},[32767],[34,553948,546295],{"className":553949,"style":546331},[32767,131249,156560],[34,553951,553953],{"className":553952},[31792],[34,553954,553956,553996],{"className":553955},[31796,31797],[34,553957,553959,553993],{"className":553958},[31801],[34,553960,553962,553976],{"className":553961,"style":546550},[31805],[34,553963,553964,553967],{"style":546347},[34,553965],{"className":553966,"style":31814},[31813],[34,553968,553970],{"className":553969},[31818,31819,31820,31821],[34,553971,553973],{"className":553972},[31685,31821],[34,553974,459],{"className":553975},[31685,31821],[34,553977,553978,553981],{"style":546362},[34,553979],{"className":553980,"style":31814},[31813],[34,553982,553984],{"className":553983},[31818,31819,31820,31821],[34,553985,553987,553990],{"className":553986},[31685,31821],[34,553988,223],{"className":553989},[31685,31821],[34,553991,138286],{"className":553992},[31685,31821],[34,553994,31829],{"className":553995},[31828],[34,553997,553999],{"className":553998},[31801],[34,554000,554002],{"className":554001,"style":546384},[31805],[34,554003],{},[34,554005],{"className":554006,"style":32123},[31845],[34,554008,32936],{"className":554009},[31685,31686],[34,554011],{"className":554012,"style":32123},[31845],[34,554014,547853],{"className":554015},[32767],[34,554017],{"className":554018,"style":32123},[31845],[34,554020,554022,554025],{"className":554021},[31685],[34,554023,32936],{"className":554024},[31685,31686],[34,554026,554028],{"className":554027},[31792],[34,554029,554031],{"className":554030},[31796],[34,554032,554034],{"className":554033},[31801],[34,554035,554037],{"className":554036,"style":37774},[31805],[34,554038,554039,554042],{"style":37777},[34,554040],{"className":554041,"style":31814},[31813],[34,554043,554045],{"className":554044},[31818,31819,31820,31821],[34,554046,15932],{"className":554047},[31685,31821],[34,554049],{"className":554050,"style":32123},[31845],[34,554052,547860],{"className":554053},[32767],[34,554055],{"className":554056,"style":32123},[31845],[34,554058,217392],{"className":554059,"style":217496},[31685,31686],[34,554061,32936],{"className":554062},[31685,31686],[34,554064,46900],{"className":554065},[31685,31686],[34,554067,32936],{"className":554068},[31685,31686],[34,554070,31960],{"className":554071},[32119],[34,554073],{"className":554074,"style":128438},[31845],[34,554076],{"className":554077,"style":32123},[31845],[34,554079,111],{"className":554080},[31685,31686],[34,554082],{"className":554083,"style":31846},[31845],[34,554085,45972],{"className":554086},[31850],[34,554088],{"className":554089,"style":31846},[31845],[34,554091,554093,554096,554099,554102,554105],{"className":554092},[31676],[34,554094],{"className":554095,"style":43198},[31680],[34,554097,217392],{"className":554098,"style":217496},[31685,31686],[34,554100],{"className":554101,"style":31846},[31845],[34,554103,45972],{"className":554104},[31850],[34,554106],{"className":554107,"style":31846},[31845],[34,554109,554111,554114],{"className":554110},[31676],[34,554112],{"className":554113,"style":31681},[31680],[34,554115,1697],{"className":554116},[31685,31686],[15,554118,554119,554121,554122,554199],{},[2627,554120,549449],{}," 这个题目比较技巧性。",[34,554123,554125,554149],{"className":554124},[31642],[34,554126,554128],{"className":554127},[31646],[31648,554129,554130],{"xmlns":31650},[31652,554131,554132,554146],{},[31655,554133,554134,554136,554138,554140],{},[31658,554135,32936],{},[31658,554137,547853],{},[31662,554139,32682],{},[37624,554141,554142,554144],{},[31658,554143,32936],{},[31749,554145,15932],{},[31665,554147,554148],{"encoding":31667},"x \\sin x^4",[34,554150,554152],{"className":554151,"ariaHidden":758},[31672],[34,554153,554155,554158,554161,554164,554167,554170],{"className":554154},[31676],[34,554156],{"className":554157,"style":37774},[31680],[34,554159,32936],{"className":554160},[31685,31686],[34,554162],{"className":554163,"style":32123},[31845],[34,554165,547853],{"className":554166},[32767],[34,554168],{"className":554169,"style":32123},[31845],[34,554171,554173,554176],{"className":554172},[31685],[34,554174,32936],{"className":554175},[31685,31686],[34,554177,554179],{"className":554178},[31792],[34,554180,554182],{"className":554181},[31796],[34,554183,554185],{"className":554184},[31801],[34,554186,554188],{"className":554187,"style":37774},[31805],[34,554189,554190,554193],{"style":37777},[34,554191],{"className":554192,"style":31814},[31813],[34,554194,554196],{"className":554195},[31818,31819,31820,31821],[34,554197,15932],{"className":554198},[31685,31821]," 震荡极快。直接看很难，我们做一个换元处理，把它变成标准的 Dirichlet 类型。",[15,554201,554202],{},[2627,554203,549629],{},[2230,554205,554206,555436],{},[108,554207,554208,478285,554211,55016,554292,46528,554382,554568,554569,53834,554597,554631,554632],{},[2627,554209,554210],{},"换元：",[34,554212,554214,554236],{"className":554213},[31642],[34,554215,554217],{"className":554216},[31646],[31648,554218,554219],{"xmlns":31650},[31652,554220,554221,554233],{},[31655,554222,554223,554225,554227],{},[31658,554224,33249],{},[31662,554226,202],{},[37624,554228,554229,554231],{},[31658,554230,32936],{},[31749,554232,15932],{},[31665,554234,554235],{"encoding":31667},"t = x^4",[34,554237,554239,554257],{"className":554238,"ariaHidden":758},[31672],[34,554240,554242,554245,554248,554251,554254],{"className":554241},[31676],[34,554243],{"className":554244,"style":33280},[31680],[34,554246,33249],{"className":554247},[31685,31686],[34,554249],{"className":554250,"style":31846},[31845],[34,554252,202],{"className":554253},[31850],[34,554255],{"className":554256,"style":31846},[31845],[34,554258,554260,554263],{"className":554259},[31676],[34,554261],{"className":554262,"style":37774},[31680],[34,554264,554266,554269],{"className":554265},[31685],[34,554267,32936],{"className":554268},[31685,31686],[34,554270,554272],{"className":554271},[31792],[34,554273,554275],{"className":554274},[31796],[34,554276,554278],{"className":554277},[31801],[34,554279,554281],{"className":554280,"style":37774},[31805],[34,554282,554283,554286],{"style":37777},[34,554284],{"className":554285,"style":31814},[31813],[34,554287,554289],{"className":554288},[31818,31819,31820,31821],[34,554290,15932],{"className":554291},[31685,31821],[34,554293,554295,554323],{"className":554294},[31642],[34,554296,554298],{"className":554297},[31646],[31648,554299,554300],{"xmlns":31650},[31652,554301,554302,554320],{},[31655,554303,554304,554306,554308],{},[31658,554305,32936],{},[31662,554307,202],{},[37624,554309,554310,554312],{},[31658,554311,33249],{},[31655,554313,554314,554316,554318],{},[31749,554315,640],{},[31658,554317,38080],{"mathvariant":38079},[31749,554319,15932],{},[31665,554321,554322],{"encoding":31667},"x = t^{1\u002F4}",[34,554324,554326,554344],{"className":554325,"ariaHidden":758},[31672],[34,554327,554329,554332,554335,554338,554341],{"className":554328},[31676],[34,554330],{"className":554331,"style":32527},[31680],[34,554333,32936],{"className":554334},[31685,31686],[34,554336],{"className":554337,"style":31846},[31845],[34,554339,202],{"className":554340},[31850],[34,554342],{"className":554343,"style":31846},[31845],[34,554345,554347,554350],{"className":554346},[31676],[34,554348],{"className":554349,"style":61037},[31680],[34,554351,554353,554356],{"className":554352},[31685],[34,554354,33249],{"className":554355},[31685,31686],[34,554357,554359],{"className":554358},[31792],[34,554360,554362],{"className":554361},[31796],[34,554363,554365],{"className":554364},[31801],[34,554366,554368],{"className":554367,"style":61037},[31805],[34,554369,554370,554373],{"style":37777},[34,554371],{"className":554372,"style":31814},[31813],[34,554374,554376],{"className":554375},[31818,31819,31820,31821],[34,554377,554379],{"className":554378},[31685,31821],[34,554380,42426],{"className":554381},[31685,31821],[34,554383,554385,554427],{"className":554384},[31642],[34,554386,554388],{"className":554387},[31646],[31648,554389,554390],{"xmlns":31650},[31652,554391,554392,554424],{},[31655,554393,554394,554396,554398,554400,554406,554420,554422],{},[31658,554395,46900],{},[31658,554397,32936],{},[31662,554399,202],{},[41453,554401,554402,554404],{},[31749,554403,640],{},[31749,554405,15932],{},[37624,554407,554408,554410],{},[31658,554409,33249],{},[31655,554411,554412,554414,554416,554418],{},[31662,554413,32226],{},[31749,554415,786],{},[31658,554417,38080],{"mathvariant":38079},[31749,554419,15932],{},[31658,554421,46900],{},[31658,554423,33249],{},[31665,554425,554426],{"encoding":31667},"dx = \\frac{1}{4}t^{-3\u002F4}dt",[34,554428,554430,554451],{"className":554429,"ariaHidden":758},[31672],[34,554431,554433,554436,554439,554442,554445,554448],{"className":554432},[31676],[34,554434],{"className":554435,"style":31681},[31680],[34,554437,46900],{"className":554438},[31685,31686],[34,554440,32936],{"className":554441},[31685,31686],[34,554443],{"className":554444,"style":31846},[31845],[34,554446,202],{"className":554447},[31850],[34,554449],{"className":554450,"style":31846},[31845],[34,554452,554454,554458,554526,554562,554565],{"className":554453},[31676],[34,554455],{"className":554456,"style":554457},[31680],"height:1.233em;vertical-align:-0.345em;",[34,554459,554461,554464,554523],{"className":554460},[31685],[34,554462],{"className":554463},[31782,41488],[34,554465,554467],{"className":554466},[41453],[34,554468,554470,554515],{"className":554469},[31796,31797],[34,554471,554473,554512],{"className":554472},[31801],[34,554474,554476,554490,554498],{"className":554475,"style":46746},[31805],[34,554477,554478,554481],{"style":41504},[34,554479],{"className":554480,"style":41508},[31813],[34,554482,554484],{"className":554483},[31818,31819,31820,31821],[34,554485,554487],{"className":554486},[31685,31821],[34,554488,15932],{"className":554489},[31685,31821],[34,554491,554492,554495],{"style":41520},[34,554493],{"className":554494,"style":41508},[31813],[34,554496],{"className":554497,"style":41528},[41527],[34,554499,554500,554503],{"style":46771},[34,554501],{"className":554502,"style":41508},[31813],[34,554504,554506],{"className":554505},[31818,31819,31820,31821],[34,554507,554509],{"className":554508},[31685,31821],[34,554510,640],{"className":554511},[31685,31821],[34,554513,31829],{"className":554514},[31828],[34,554516,554518],{"className":554517},[31801],[34,554519,554521],{"className":554520,"style":41619},[31805],[34,554522],{},[34,554524],{"className":554525},[31690,41488],[34,554527,554529,554532],{"className":554528},[31685],[34,554530,33249],{"className":554531},[31685,31686],[34,554533,554535],{"className":554534},[31792],[34,554536,554538],{"className":554537},[31796],[34,554539,554541],{"className":554540},[31801],[34,554542,554544],{"className":554543,"style":61037},[31805],[34,554545,554546,554549],{"style":37777},[34,554547],{"className":554548,"style":31814},[31813],[34,554550,554552],{"className":554551},[31818,31819,31820,31821],[34,554553,554555,554558],{"className":554554},[31685,31821],[34,554556,32226],{"className":554557},[31685,31821],[34,554559,554561],{"className":554560},[31685,31821],"3\u002F4",[34,554563,46900],{"className":554564},[31685,31686],[34,554566,33249],{"className":554567},[31685,31686],"。\n积分区间依然是 ",[34,554570,554572,554585],{"className":554571},[31642],[34,554573,554575],{"className":554574},[31646],[31648,554576,554577],{"xmlns":31650},[31652,554578,554579,554583],{},[31655,554580,554581],{},[31749,554582,459],{},[31665,554584,459],{"encoding":31667},[34,554586,554588],{"className":554587,"ariaHidden":758},[31672],[34,554589,554591,554594],{"className":554590},[31676],[34,554592],{"className":554593,"style":32617},[31680],[34,554595,459],{"className":554596},[31685],[34,554598,554600,554616],{"className":554599},[31642],[34,554601,554603],{"className":554602},[31646],[31648,554604,554605],{"xmlns":31650},[31652,554606,554607,554613],{},[31655,554608,554609,554611],{},[31662,554610,223],{},[31658,554612,138286],{"mathvariant":38079},[31665,554614,554615],{"encoding":31667},"+\\infty",[34,554617,554619],{"className":554618,"ariaHidden":758},[31672],[34,554620,554622,554625,554628],{"className":554621},[31676],[34,554623],{"className":554624,"style":32597},[31680],[34,554626,223],{"className":554627},[31685],[34,554629,138286],{"className":554630},[31685],"。\n原积分变为：\n",[34,554633,554635,554789],{"className":554634},[31642],[34,554636,554638],{"className":554637},[31646],[31648,554639,554640],{"xmlns":31650},[31652,554641,554642,554786],{},[31655,554643,554644,554656,554668,554670,554672,554674,554676,554678,554680,554682,554684,554696,554698,554700,554706,554720,554722,554724,554726,554732,554744,554782,554784],{},[38846,554645,554646,554648,554650],{},[31662,554647,546295],{},[31749,554649,459],{},[31655,554651,554652,554654],{},[31662,554653,223],{},[31658,554655,138286],{"mathvariant":38079},[37624,554657,554658,554660],{},[31658,554659,33249],{},[31655,554661,554662,554664,554666],{},[31749,554663,640],{},[31658,554665,38080],{"mathvariant":38079},[31749,554667,15932],{},[31658,554669,547853],{},[31662,554671,32682],{},[31658,554673,33249],{},[31662,554675,71191],{},[31658,554677,547860],{},[31662,554679,32682],{},[31662,554681,456],{"stretchy":702},[31658,554683,217392],{},[37624,554685,554686,554688],{},[31658,554687,33249],{},[31655,554689,554690,554692,554694],{},[31749,554691,640],{},[31658,554693,38080],{"mathvariant":38079},[31749,554695,15932],{},[31662,554697,2557],{"stretchy":702},[31662,554699,71191],{},[41453,554701,554702,554704],{},[31749,554703,640],{},[31749,554705,15932],{},[37624,554707,554708,554710],{},[31658,554709,33249],{},[31655,554711,554712,554714,554716,554718],{},[31662,554713,32226],{},[31749,554715,786],{},[31658,554717,38080],{"mathvariant":38079},[31749,554719,15932],{},[31658,554721,46900],{},[31658,554723,33249],{},[31662,554725,202],{},[41453,554727,554728,554730],{},[31749,554729,640],{},[31749,554731,15932],{},[38846,554733,554734,554736,554738],{},[31662,554735,546295],{},[31749,554737,459],{},[31655,554739,554740,554742],{},[31662,554741,223],{},[31658,554743,138286],{"mathvariant":38079},[41453,554745,554746,554778],{},[31655,554747,554748,554750,554752,554754,554756,554758,554760,554762,554764,554776],{},[31658,554749,547853],{},[31662,554751,32682],{},[31658,554753,33249],{},[31662,554755,71191],{},[31658,554757,547860],{},[31662,554759,32682],{},[31662,554761,456],{"stretchy":702},[31658,554763,217392],{},[37624,554765,554766,554768],{},[31658,554767,33249],{},[31655,554769,554770,554772,554774],{},[31749,554771,640],{},[31658,554773,38080],{"mathvariant":38079},[31749,554775,15932],{},[31662,554777,2557],{"stretchy":702},[41463,554779,554780],{},[31658,554781,33249],{},[31658,554783,46900],{},[31658,554785,33249],{},[31665,554787,554788],{"encoding":31667},"\\int_{0}^{+\\infty} t^{1\u002F4} \\sin t \\cdot \\cos(\\alpha t^{1\u002F4}) \\cdot \\frac{1}{4}t^{-3\u002F4} dt = \\frac{1}{4} \\int_{0}^{+\\infty} \\frac{\\sin t \\cdot \\cos(\\alpha t^{1\u002F4})}{\\sqrt{t}} dt",[34,554790,554792,554914,554973,555097],{"className":554791,"ariaHidden":758},[31672],[34,554793,554795,554798,554858,554861,554893,554896,554899,554902,554905,554908,554911],{"className":554794},[31676],[34,554796],{"className":554797,"style":546531},[31680],[34,554799,554801,554804],{"className":554800},[32767],[34,554802,546295],{"className":554803,"style":546331},[32767,131249,156560],[34,554805,554807],{"className":554806},[31792],[34,554808,554810,554850],{"className":554809},[31796,31797],[34,554811,554813,554847],{"className":554812},[31801],[34,554814,554816,554830],{"className":554815,"style":546550},[31805],[34,554817,554818,554821],{"style":546347},[34,554819],{"className":554820,"style":31814},[31813],[34,554822,554824],{"className":554823},[31818,31819,31820,31821],[34,554825,554827],{"className":554826},[31685,31821],[34,554828,459],{"className":554829},[31685,31821],[34,554831,554832,554835],{"style":546362},[34,554833],{"className":554834,"style":31814},[31813],[34,554836,554838],{"className":554837},[31818,31819,31820,31821],[34,554839,554841,554844],{"className":554840},[31685,31821],[34,554842,223],{"className":554843},[31685,31821],[34,554845,138286],{"className":554846},[31685,31821],[34,554848,31829],{"className":554849},[31828],[34,554851,554853],{"className":554852},[31801],[34,554854,554856],{"className":554855,"style":546384},[31805],[34,554857],{},[34,554859],{"className":554860,"style":32123},[31845],[34,554862,554864,554867],{"className":554863},[31685],[34,554865,33249],{"className":554866},[31685,31686],[34,554868,554870],{"className":554869},[31792],[34,554871,554873],{"className":554872},[31796],[34,554874,554876],{"className":554875},[31801],[34,554877,554879],{"className":554878,"style":61037},[31805],[34,554880,554881,554884],{"style":37777},[34,554882],{"className":554883,"style":31814},[31813],[34,554885,554887],{"className":554886},[31818,31819,31820,31821],[34,554888,554890],{"className":554889},[31685,31821],[34,554891,42426],{"className":554892},[31685,31821],[34,554894],{"className":554895,"style":32123},[31845],[34,554897,547853],{"className":554898},[32767],[34,554900],{"className":554901,"style":32123},[31845],[34,554903,33249],{"className":554904},[31685,31686],[34,554906],{"className":554907,"style":32604},[31845],[34,554909,71191],{"className":554910},[32024],[34,554912],{"className":554913,"style":32604},[31845],[34,554915,554917,554920,554923,554926,554929,554961,554964,554967,554970],{"className":554916},[31676],[34,554918],{"className":554919,"style":61012},[31680],[34,554921,547860],{"className":554922},[32767],[34,554924,456],{"className":554925},[31782],[34,554927,217392],{"className":554928,"style":217496},[31685,31686],[34,554930,554932,554935],{"className":554931},[31685],[34,554933,33249],{"className":554934},[31685,31686],[34,554936,554938],{"className":554937},[31792],[34,554939,554941],{"className":554940},[31796],[34,554942,554944],{"className":554943},[31801],[34,554945,554947],{"className":554946,"style":61037},[31805],[34,554948,554949,554952],{"style":37777},[34,554950],{"className":554951,"style":31814},[31813],[34,554953,554955],{"className":554954},[31818,31819,31820,31821],[34,554956,554958],{"className":554957},[31685,31821],[34,554959,42426],{"className":554960},[31685,31821],[34,554962,2557],{"className":554963},[31690],[34,554965],{"className":554966,"style":32604},[31845],[34,554968,71191],{"className":554969},[32024],[34,554971],{"className":554972,"style":32604},[31845],[34,554974,554976,554979,555047,555082,555085,555088,555091,555094],{"className":554975},[31676],[34,554977],{"className":554978,"style":554457},[31680],[34,554980,554982,554985,555044],{"className":554981},[31685],[34,554983],{"className":554984},[31782,41488],[34,554986,554988],{"className":554987},[41453],[34,554989,554991,555036],{"className":554990},[31796,31797],[34,554992,554994,555033],{"className":554993},[31801],[34,554995,554997,555011,555019],{"className":554996,"style":46746},[31805],[34,554998,554999,555002],{"style":41504},[34,555000],{"className":555001,"style":41508},[31813],[34,555003,555005],{"className":555004},[31818,31819,31820,31821],[34,555006,555008],{"className":555007},[31685,31821],[34,555009,15932],{"className":555010},[31685,31821],[34,555012,555013,555016],{"style":41520},[34,555014],{"className":555015,"style":41508},[31813],[34,555017],{"className":555018,"style":41528},[41527],[34,555020,555021,555024],{"style":46771},[34,555022],{"className":555023,"style":41508},[31813],[34,555025,555027],{"className":555026},[31818,31819,31820,31821],[34,555028,555030],{"className":555029},[31685,31821],[34,555031,640],{"className":555032},[31685,31821],[34,555034,31829],{"className":555035},[31828],[34,555037,555039],{"className":555038},[31801],[34,555040,555042],{"className":555041,"style":41619},[31805],[34,555043],{},[34,555045],{"className":555046},[31690,41488],[34,555048,555050,555053],{"className":555049},[31685],[34,555051,33249],{"className":555052},[31685,31686],[34,555054,555056],{"className":555055},[31792],[34,555057,555059],{"className":555058},[31796],[34,555060,555062],{"className":555061},[31801],[34,555063,555065],{"className":555064,"style":61037},[31805],[34,555066,555067,555070],{"style":37777},[34,555068],{"className":555069,"style":31814},[31813],[34,555071,555073],{"className":555072},[31818,31819,31820,31821],[34,555074,555076,555079],{"className":555075},[31685,31821],[34,555077,32226],{"className":555078},[31685,31821],[34,555080,554561],{"className":555081},[31685,31821],[34,555083,46900],{"className":555084},[31685,31686],[34,555086,33249],{"className":555087},[31685,31686],[34,555089],{"className":555090,"style":31846},[31845],[34,555092,202],{"className":555093},[31850],[34,555095],{"className":555096,"style":31846},[31845],[34,555098,555100,555104,555172,555175,555235,555238,555430,555433],{"className":555099},[31676],[34,555101],{"className":555102,"style":555103},[31680],"height:1.6997em;vertical-align:-0.538em;",[34,555105,555107,555110,555169],{"className":555106},[31685],[34,555108],{"className":555109},[31782,41488],[34,555111,555113],{"className":555112},[41453],[34,555114,555116,555161],{"className":555115},[31796,31797],[34,555117,555119,555158],{"className":555118},[31801],[34,555120,555122,555136,555144],{"className":555121,"style":46746},[31805],[34,555123,555124,555127],{"style":41504},[34,555125],{"className":555126,"style":41508},[31813],[34,555128,555130],{"className":555129},[31818,31819,31820,31821],[34,555131,555133],{"className":555132},[31685,31821],[34,555134,15932],{"className":555135},[31685,31821],[34,555137,555138,555141],{"style":41520},[34,555139],{"className":555140,"style":41508},[31813],[34,555142],{"className":555143,"style":41528},[41527],[34,555145,555146,555149],{"style":46771},[34,555147],{"className":555148,"style":41508},[31813],[34,555150,555152],{"className":555151},[31818,31819,31820,31821],[34,555153,555155],{"className":555154},[31685,31821],[34,555156,640],{"className":555157},[31685,31821],[34,555159,31829],{"className":555160},[31828],[34,555162,555164],{"className":555163},[31801],[34,555165,555167],{"className":555166,"style":41619},[31805],[34,555168],{},[34,555170],{"className":555171},[31690,41488],[34,555173],{"className":555174,"style":32123},[31845],[34,555176,555178,555181],{"className":555177},[32767],[34,555179,546295],{"className":555180,"style":546331},[32767,131249,156560],[34,555182,555184],{"className":555183},[31792],[34,555185,555187,555227],{"className":555186},[31796,31797],[34,555188,555190,555224],{"className":555189},[31801],[34,555191,555193,555207],{"className":555192,"style":546550},[31805],[34,555194,555195,555198],{"style":546347},[34,555196],{"className":555197,"style":31814},[31813],[34,555199,555201],{"className":555200},[31818,31819,31820,31821],[34,555202,555204],{"className":555203},[31685,31821],[34,555205,459],{"className":555206},[31685,31821],[34,555208,555209,555212],{"style":546362},[34,555210],{"className":555211,"style":31814},[31813],[34,555213,555215],{"className":555214},[31818,31819,31820,31821],[34,555216,555218,555221],{"className":555217},[31685,31821],[34,555219,223],{"className":555220},[31685,31821],[34,555222,138286],{"className":555223},[31685,31821],[34,555225,31829],{"className":555226},[31828],[34,555228,555230],{"className":555229},[31801],[34,555231,555233],{"className":555232,"style":546384},[31805],[34,555234],{},[34,555236],{"className":555237,"style":32123},[31845],[34,555239,555241,555244,555427],{"className":555240},[31685],[34,555242],{"className":555243},[31782,41488],[34,555245,555247],{"className":555246},[41453],[34,555248,555250,555418],{"className":555249},[31796,31797],[34,555251,555253,555415],{"className":555252},[31801],[34,555254,555257,555319,555327],{"className":555255,"style":555256},[31805],"height:1.1617em;",[34,555258,555260,555263],{"style":555259},"top:-2.5613em;",[34,555261],{"className":555262,"style":41508},[31813],[34,555264,555266],{"className":555265},[31818,31819,31820,31821],[34,555267,555269],{"className":555268},[31685,31821],[34,555270,555272],{"className":555271},[31685,41550,31821],[34,555273,555275,555310],{"className":555274},[31796,31797],[34,555276,555278,555307],{"className":555277},[31801],[34,555279,555282,555294],{"className":555280,"style":555281},[31805],"height:0.8982em;",[34,555283,555285,555288],{"className":555284,"style":41565},[41564],[34,555286],{"className":555287,"style":41508},[31813],[34,555289,555291],{"className":555290,"style":41572},[31685,31821],[34,555292,33249],{"className":555293},[31685,31686,31821],[34,555295,555297,555300],{"style":555296},"top:-2.8582em;",[34,555298],{"className":555299,"style":41508},[31813],[34,555301,555303],{"className":555302,"style":41586},[41585,31821],[41588,555304,555305],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,555306],{"d":41597},[34,555308,31829],{"className":555309},[31828],[34,555311,555313],{"className":555312},[31801],[34,555314,555317],{"className":555315,"style":555316},[31805],"height:0.1418em;",[34,555318],{},[34,555320,555321,555324],{"style":41520},[34,555322],{"className":555323,"style":41508},[31813],[34,555325],{"className":555326,"style":41528},[41527],[34,555328,555329,555332],{"style":70529},[34,555330],{"className":555331,"style":41508},[31813],[34,555333,555335],{"className":555334},[31818,31819,31820,31821],[34,555336,555338,555350,555353,555356,555359,555371,555374,555377,555412],{"className":555337},[31685,31821],[34,555339,555341,555344,555347],{"className":555340},[32767,31821],[34,555342,31747],{"className":555343},[31821],[34,555345,31955],{"className":555346},[31821],[34,555348,31660],{"className":555349},[31821],[34,555351],{"className":555352,"style":43430},[31845,31821],[34,555354,33249],{"className":555355},[31685,31686,31821],[34,555357,71191],{"className":555358},[32024,31821],[34,555360,555362,555365,555368],{"className":555361},[32767,31821],[34,555363,31965],{"className":555364},[31821],[34,555366,43383],{"className":555367},[31821],[34,555369,31747],{"className":555370},[31821],[34,555372,456],{"className":555373},[31782,31821],[34,555375,217392],{"className":555376,"style":217496},[31685,31686,31821],[34,555378,555380,555383],{"className":555379},[31685,31821],[34,555381,33249],{"className":555382},[31685,31686,31821],[34,555384,555386],{"className":555385},[31792],[34,555387,555389],{"className":555388},[31796],[34,555390,555392],{"className":555391},[31801],[34,555393,555396],{"className":555394,"style":555395},[31805],"height:0.9667em;",[34,555397,555399,555403],{"style":555398},"top:-2.9667em;margin-right:0.0714em;",[34,555400],{"className":555401,"style":555402},[31813],"height:2.5357em;",[34,555404,555406],{"className":555405},[31818,43410,43411,31821],[34,555407,555409],{"className":555408},[31685,31821],[34,555410,42426],{"className":555411},[31685,31821],[34,555413,2557],{"className":555414},[31690,31821],[34,555416,31829],{"className":555417},[31828],[34,555419,555421],{"className":555420},[31801],[34,555422,555425],{"className":555423,"style":555424},[31805],"height:0.538em;",[34,555426],{},[34,555428],{"className":555429},[31690,41488],[34,555431,46900],{"className":555432},[31685,31686],[34,555434,33249],{"className":555435},[31685,31686],[108,555437,555438,478285,555441,326252,555517,555783,555784,555786,555789,555790,258,555904,555998,555999,556043,556044,556046,556049,556050,258,556146,347,556189,556835,556837,556840,556841,556843,556844,557241,558251,558253,558256,558257,558259,558260,558513,558514,70884,558781,558876,558877,558905,558906,558934,558935,558963,558964,559109,559110,559187,559188,559190,559191,559352,559353,559396,559397,559399,559402,560046,560048],{},[2627,555439,555440],{},"拆分函数：",[34,555442,555444,555472],{"className":555443},[31642],[34,555445,555447],{"className":555446},[31646],[31648,555448,555449],{"xmlns":31650},[31652,555450,555451,555469],{},[31655,555452,555453,555455,555457,555459,555461,555463,555465,555467],{},[31658,555454,31739],{},[31662,555456,456],{"stretchy":702},[31658,555458,33249],{},[31662,555460,2557],{"stretchy":702},[31662,555462,202],{},[31658,555464,547853],{},[31662,555466,32682],{},[31658,555468,33249],{},[31665,555470,555471],{"encoding":31667},"f(t) = \\sin t",[34,555473,555475,555502],{"className":555474,"ariaHidden":758},[31672],[34,555476,555478,555481,555484,555487,555490,555493,555496,555499],{"className":555477},[31676],[34,555479],{"className":555480,"style":31774},[31680],[34,555482,31739],{"className":555483,"style":31778},[31685,31686],[34,555485,456],{"className":555486},[31782],[34,555488,33249],{"className":555489},[31685,31686],[34,555491,2557],{"className":555492},[31690],[34,555494],{"className":555495,"style":31846},[31845],[34,555497,202],{"className":555498},[31850],[34,555500],{"className":555501,"style":31846},[31845],[34,555503,555505,555508,555511,555514],{"className":555504},[31676],[34,555506],{"className":555507,"style":71430},[31680],[34,555509,547853],{"className":555510},[32767],[34,555512],{"className":555513,"style":32123},[31845],[34,555515,33249],{"className":555516},[31685,31686],[34,555518,555520,555576],{"className":555519},[31642],[34,555521,555523],{"className":555522},[31646],[31648,555524,555525],{"xmlns":31650},[31652,555526,555527,555573],{},[31655,555528,555529,555531,555533,555535,555537,555539,555541,555543],{},[31658,555530,31942],{},[31662,555532,456],{"stretchy":702},[31658,555534,33249],{},[31662,555536,31960],{"separator":758},[31658,555538,217392],{},[31662,555540,2557],{"stretchy":702},[31662,555542,202],{},[41453,555544,555545,555569],{},[31655,555546,555547,555549,555551,555553,555555,555567],{},[31658,555548,547860],{},[31662,555550,32682],{},[31662,555552,456],{"stretchy":702},[31658,555554,217392],{},[37624,555556,555557,555559],{},[31658,555558,33249],{},[31655,555560,555561,555563,555565],{},[31749,555562,640],{},[31658,555564,38080],{"mathvariant":38079},[31749,555566,15932],{},[31662,555568,2557],{"stretchy":702},[41463,555570,555571],{},[31658,555572,33249],{},[31665,555574,555575],{"encoding":31667},"g(t, \\alpha) = \\frac{\\cos(\\alpha t^{1\u002F4})}{\\sqrt{t}}",[34,555577,555579,555615],{"className":555578,"ariaHidden":758},[31672],[34,555580,555582,555585,555588,555591,555594,555597,555600,555603,555606,555609,555612],{"className":555581},[31676],[34,555583],{"className":555584,"style":31774},[31680],[34,555586,31942],{"className":555587,"style":32062},[31685,31686],[34,555589,456],{"className":555590},[31782],[34,555592,33249],{"className":555593},[31685,31686],[34,555595,31960],{"className":555596},[32119],[34,555598],{"className":555599,"style":32123},[31845],[34,555601,217392],{"className":555602,"style":217496},[31685,31686],[34,555604,2557],{"className":555605},[31690],[34,555607],{"className":555608,"style":31846},[31845],[34,555610,202],{"className":555611},[31850],[34,555613],{"className":555614,"style":31846},[31845],[34,555616,555618,555621],{"className":555617},[31676],[34,555619],{"className":555620,"style":555103},[31680],[34,555622,555624,555627,555780],{"className":555623},[31685],[34,555625],{"className":555626},[31782,41488],[34,555628,555630],{"className":555629},[41453],[34,555631,555633,555772],{"className":555632},[31796,31797],[34,555634,555636,555769],{"className":555635},[31801],[34,555637,555639,555697,555705],{"className":555638,"style":555256},[31805],[34,555640,555641,555644],{"style":555259},[34,555642],{"className":555643,"style":41508},[31813],[34,555645,555647],{"className":555646},[31818,31819,31820,31821],[34,555648,555650],{"className":555649},[31685,31821],[34,555651,555653],{"className":555652},[31685,41550,31821],[34,555654,555656,555689],{"className":555655},[31796,31797],[34,555657,555659,555686],{"className":555658},[31801],[34,555660,555662,555674],{"className":555661,"style":555281},[31805],[34,555663,555665,555668],{"className":555664,"style":41565},[41564],[34,555666],{"className":555667,"style":41508},[31813],[34,555669,555671],{"className":555670,"style":41572},[31685,31821],[34,555672,33249],{"className":555673},[31685,31686,31821],[34,555675,555676,555679],{"style":555296},[34,555677],{"className":555678,"style":41508},[31813],[34,555680,555682],{"className":555681,"style":41586},[41585,31821],[41588,555683,555684],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,555685],{"d":41597},[34,555687,31829],{"className":555688},[31828],[34,555690,555692],{"className":555691},[31801],[34,555693,555695],{"className":555694,"style":555316},[31805],[34,555696],{},[34,555698,555699,555702],{"style":41520},[34,555700],{"className":555701,"style":41508},[31813],[34,555703],{"className":555704,"style":41528},[41527],[34,555706,555707,555710],{"style":70529},[34,555708],{"className":555709,"style":41508},[31813],[34,555711,555713],{"className":555712},[31818,31819,31820,31821],[34,555714,555716,555728,555731,555734,555766],{"className":555715},[31685,31821],[34,555717,555719,555722,555725],{"className":555718},[32767,31821],[34,555720,31965],{"className":555721},[31821],[34,555723,43383],{"className":555724},[31821],[34,555726,31747],{"className":555727},[31821],[34,555729,456],{"className":555730},[31782,31821],[34,555732,217392],{"className":555733,"style":217496},[31685,31686,31821],[34,555735,555737,555740],{"className":555736},[31685,31821],[34,555738,33249],{"className":555739},[31685,31686,31821],[34,555741,555743],{"className":555742},[31792],[34,555744,555746],{"className":555745},[31796],[34,555747,555749],{"className":555748},[31801],[34,555750,555752],{"className":555751,"style":555395},[31805],[34,555753,555754,555757],{"style":555398},[34,555755],{"className":555756,"style":555402},[31813],[34,555758,555760],{"className":555759},[31818,43410,43411,31821],[34,555761,555763],{"className":555762},[31685,31821],[34,555764,42426],{"className":555765},[31685,31821],[34,555767,2557],{"className":555768},[31690,31821],[34,555770,31829],{"className":555771},[31828],[34,555773,555775],{"className":555774},[31801],[34,555776,555778],{"className":555777,"style":555424},[31805],[34,555779],{},[34,555781],{"className":555782},[31690,41488],"。\n这样看起来不太好判断单调性。",[58531,555785],{},[2627,555787,555788],{},"更优思路（直接用 Dirichlet）：","\n将原函数看作：",[34,555791,555793,555827],{"className":555792},[31642],[34,555794,555796],{"className":555795},[31646],[31648,555797,555798],{"xmlns":31650},[31652,555799,555800,555824],{},[31655,555801,555802,555804,555806,555808,555810,555812,555814,555816,555818],{},[31658,555803,94628],{},[31662,555805,456],{"stretchy":702},[31658,555807,32936],{},[31662,555809,2557],{"stretchy":702},[31662,555811,202],{},[31658,555813,32936],{},[31658,555815,547853],{},[31662,555817,32682],{},[37624,555819,555820,555822],{},[31658,555821,32936],{},[31749,555823,15932],{},[31665,555825,555826],{"encoding":31667},"u(x) = x \\sin x^4",[34,555828,555830,555857],{"className":555829,"ariaHidden":758},[31672],[34,555831,555833,555836,555839,555842,555845,555848,555851,555854],{"className":555832},[31676],[34,555834],{"className":555835,"style":31774},[31680],[34,555837,94628],{"className":555838},[31685,31686],[34,555840,456],{"className":555841},[31782],[34,555843,32936],{"className":555844},[31685,31686],[34,555846,2557],{"className":555847},[31690],[34,555849],{"className":555850,"style":31846},[31845],[34,555852,202],{"className":555853},[31850],[34,555855],{"className":555856,"style":31846},[31845],[34,555858,555860,555863,555866,555869,555872,555875],{"className":555859},[31676],[34,555861],{"className":555862,"style":37774},[31680],[34,555864,32936],{"className":555865},[31685,31686],[34,555867],{"className":555868,"style":32123},[31845],[34,555870,547853],{"className":555871},[32767],[34,555873],{"className":555874,"style":32123},[31845],[34,555876,555878,555881],{"className":555877},[31685],[34,555879,32936],{"className":555880},[31685,31686],[34,555882,555884],{"className":555883},[31792],[34,555885,555887],{"className":555886},[31796],[34,555888,555890],{"className":555889},[31801],[34,555891,555893],{"className":555892,"style":37774},[31805],[34,555894,555895,555898],{"style":37777},[34,555896],{"className":555897,"style":31814},[31813],[34,555899,555901],{"className":555900},[31818,31819,31820,31821],[34,555902,15932],{"className":555903},[31685,31821],[34,555905,555907,555941],{"className":555906},[31642],[34,555908,555910],{"className":555909},[31646],[31648,555911,555912],{"xmlns":31650},[31652,555913,555914,555938],{},[31655,555915,555916,555918,555920,555922,555924,555926,555928,555930,555932,555934,555936],{},[31658,555917,56782],{},[31662,555919,456],{"stretchy":702},[31658,555921,32936],{},[31662,555923,31960],{"separator":758},[31658,555925,217392],{},[31662,555927,2557],{"stretchy":702},[31662,555929,202],{},[31658,555931,547860],{},[31662,555933,32682],{},[31658,555935,217392],{},[31658,555937,32936],{},[31665,555939,555940],{"encoding":31667},"v(x, \\alpha) = \\cos \\alpha x",[34,555942,555944,555980],{"className":555943,"ariaHidden":758},[31672],[34,555945,555947,555950,555953,555956,555959,555962,555965,555968,555971,555974,555977],{"className":555946},[31676],[34,555948],{"className":555949,"style":31774},[31680],[34,555951,56782],{"className":555952,"style":32062},[31685,31686],[34,555954,456],{"className":555955},[31782],[34,555957,32936],{"className":555958},[31685,31686],[34,555960,31960],{"className":555961},[32119],[34,555963],{"className":555964,"style":32123},[31845],[34,555966,217392],{"className":555967,"style":217496},[31685,31686],[34,555969,2557],{"className":555970},[31690],[34,555972],{"className":555973,"style":31846},[31845],[34,555975,202],{"className":555976},[31850],[34,555978],{"className":555979,"style":31846},[31845],[34,555981,555983,555986,555989,555992,555995],{"className":555982},[31676],[34,555984],{"className":555985,"style":32527},[31680],[34,555987,547860],{"className":555988},[32767],[34,555990],{"className":555991,"style":32123},[31845],[34,555993,217392],{"className":555994,"style":217496},[31685,31686],[34,555996,32936],{"className":555997},[31685,31686],"。\n但这不对，",[34,556000,556002,556022],{"className":556001},[31642],[34,556003,556005],{"className":556004},[31646],[31648,556006,556007],{"xmlns":31650},[31652,556008,556009,556019],{},[31655,556010,556011,556013,556015,556017],{},[31658,556012,547860],{},[31662,556014,32682],{},[31658,556016,217392],{},[31658,556018,32936],{},[31665,556020,556021],{"encoding":31667},"\\cos \\alpha x",[34,556023,556025],{"className":556024,"ariaHidden":758},[31672],[34,556026,556028,556031,556034,556037,556040],{"className":556027},[31676],[34,556029],{"className":556030,"style":32527},[31680],[34,556032,547860],{"className":556033},[32767],[34,556035],{"className":556036,"style":32123},[31845],[34,556038,217392],{"className":556039,"style":217496},[31685,31686],[34,556041,32936],{"className":556042},[31685,31686]," 不趋于0。",[58531,556045],{},[2627,556047,556048],{},"修正思路（Abel 判别法）：","\n我们需要把积分拆成两部分：",[34,556051,556053,556083],{"className":556052},[31642],[34,556054,556056],{"className":556055},[31646],[31648,556057,556058],{"xmlns":31650},[31652,556059,556060,556080],{},[31655,556061,556062,556064,556066,556068,556070,556076,556078],{},[31662,556063,546295],{},[31658,556065,32936],{},[31658,556067,547853],{},[31662,556069,32682],{},[37624,556071,556072,556074],{},[31658,556073,32936],{},[31749,556075,15932],{},[31658,556077,46900],{},[31658,556079,32936],{},[31665,556081,556082],{"encoding":31667},"\\int x \\sin x^4 dx",[34,556084,556086],{"className":556085,"ariaHidden":758},[31672],[34,556087,556089,556093,556096,556099,556102,556105,556108,556111,556140,556143],{"className":556088},[31676],[34,556090],{"className":556091,"style":556092},[31680],"height:1.1202em;vertical-align:-0.3061em;",[34,556094,546295],{"className":556095,"style":546331},[32767,131249,156560],[34,556097],{"className":556098,"style":32123},[31845],[34,556100,32936],{"className":556101},[31685,31686],[34,556103],{"className":556104,"style":32123},[31845],[34,556106,547853],{"className":556107},[32767],[34,556109],{"className":556110,"style":32123},[31845],[34,556112,556114,556117],{"className":556113},[31685],[34,556115,32936],{"className":556116},[31685,31686],[34,556118,556120],{"className":556119},[31792],[34,556121,556123],{"className":556122},[31796],[34,556124,556126],{"className":556125},[31801],[34,556127,556129],{"className":556128,"style":37774},[31805],[34,556130,556131,556134],{"style":37777},[34,556132],{"className":556133,"style":31814},[31813],[34,556135,556137],{"className":556136},[31818,31819,31820,31821],[34,556138,15932],{"className":556139},[31685,31821],[34,556141,46900],{"className":556142},[31685,31686],[34,556144,32936],{"className":556145},[31685,31686],[34,556147,556149,556168],{"className":556148},[31642],[34,556150,556152],{"className":556151},[31646],[31648,556153,556154],{"xmlns":31650},[31652,556155,556156,556166],{},[31655,556157,556158,556160,556162,556164],{},[31658,556159,547860],{},[31662,556161,32682],{},[31658,556163,217392],{},[31658,556165,32936],{},[31665,556167,556021],{"encoding":31667},[34,556169,556171],{"className":556170,"ariaHidden":758},[31672],[34,556172,556174,556177,556180,556183,556186],{"className":556173},[31676],[34,556175],{"className":556176,"style":32527},[31680],[34,556178,547860],{"className":556179},[32767],[34,556181],{"className":556182,"style":32123},[31845],[34,556184,217392],{"className":556185,"style":217496},[31685,31686],[34,556187,32936],{"className":556188},[31685,31686],[105,556190,556191,556635],{},[108,556192,556193,556194,556356,556357,556438,556439,556603,556604,556632,556633,31618],{},"考察积分 ",[34,556195,556197,556237],{"className":556196},[31642],[34,556198,556200],{"className":556199},[31646],[31648,556201,556202],{"xmlns":31650},[31652,556203,556204,556234],{},[31655,556205,556206,556218,556220,556222,556224,556230,556232],{},[38846,556207,556208,556210,556212],{},[31662,556209,546295],{},[31749,556211,459],{},[31655,556213,556214,556216],{},[31662,556215,223],{},[31658,556217,138286],{"mathvariant":38079},[31658,556219,32936],{},[31658,556221,547853],{},[31662,556223,32682],{},[37624,556225,556226,556228],{},[31658,556227,32936],{},[31749,556229,15932],{},[31658,556231,46900],{},[31658,556233,32936],{},[31665,556235,556236],{"encoding":31667},"\\int_{0}^{+\\infty} x \\sin x^4 dx",[34,556238,556240],{"className":556239,"ariaHidden":758},[31672],[34,556241,556243,556246,556306,556309,556312,556315,556318,556321,556350,556353],{"className":556242},[31676],[34,556244],{"className":556245,"style":546531},[31680],[34,556247,556249,556252],{"className":556248},[32767],[34,556250,546295],{"className":556251,"style":546331},[32767,131249,156560],[34,556253,556255],{"className":556254},[31792],[34,556256,556258,556298],{"className":556257},[31796,31797],[34,556259,556261,556295],{"className":556260},[31801],[34,556262,556264,556278],{"className":556263,"style":546550},[31805],[34,556265,556266,556269],{"style":546347},[34,556267],{"className":556268,"style":31814},[31813],[34,556270,556272],{"className":556271},[31818,31819,31820,31821],[34,556273,556275],{"className":556274},[31685,31821],[34,556276,459],{"className":556277},[31685,31821],[34,556279,556280,556283],{"style":546362},[34,556281],{"className":556282,"style":31814},[31813],[34,556284,556286],{"className":556285},[31818,31819,31820,31821],[34,556287,556289,556292],{"className":556288},[31685,31821],[34,556290,223],{"className":556291},[31685,31821],[34,556293,138286],{"className":556294},[31685,31821],[34,556296,31829],{"className":556297},[31828],[34,556299,556301],{"className":556300},[31801],[34,556302,556304],{"className":556303,"style":546384},[31805],[34,556305],{},[34,556307],{"className":556308,"style":32123},[31845],[34,556310,32936],{"className":556311},[31685,31686],[34,556313],{"className":556314,"style":32123},[31845],[34,556316,547853],{"className":556317},[32767],[34,556319],{"className":556320,"style":32123},[31845],[34,556322,556324,556327],{"className":556323},[31685],[34,556325,32936],{"className":556326},[31685,31686],[34,556328,556330],{"className":556329},[31792],[34,556331,556333],{"className":556332},[31796],[34,556334,556336],{"className":556335},[31801],[34,556337,556339],{"className":556338,"style":37774},[31805],[34,556340,556341,556344],{"style":37777},[34,556342],{"className":556343,"style":31814},[31813],[34,556345,556347],{"className":556346},[31818,31819,31820,31821],[34,556348,15932],{"className":556349},[31685,31821],[34,556351,46900],{"className":556352},[31685,31686],[34,556354,32936],{"className":556355},[31685,31686],"。\n作代换 ",[34,556358,556360,556382],{"className":556359},[31642],[34,556361,556363],{"className":556362},[31646],[31648,556364,556365],{"xmlns":31650},[31652,556366,556367,556379],{},[31655,556368,556369,556371,556373],{},[31658,556370,94628],{},[31662,556372,202],{},[37624,556374,556375,556377],{},[31658,556376,32936],{},[31749,556378,768],{},[31665,556380,556381],{"encoding":31667},"u=x^2",[34,556383,556385,556403],{"className":556384,"ariaHidden":758},[31672],[34,556386,556388,556391,556394,556397,556400],{"className":556387},[31676],[34,556389],{"className":556390,"style":32527},[31680],[34,556392,94628],{"className":556393},[31685,31686],[34,556395],{"className":556396,"style":31846},[31845],[34,556398,202],{"className":556399},[31850],[34,556401],{"className":556402,"style":31846},[31845],[34,556404,556406,556409],{"className":556405},[31676],[34,556407],{"className":556408,"style":37774},[31680],[34,556410,556412,556415],{"className":556411},[31685],[34,556413,32936],{"className":556414},[31685,31686],[34,556416,556418],{"className":556417},[31792],[34,556419,556421],{"className":556420},[31796],[34,556422,556424],{"className":556423},[31801],[34,556425,556427],{"className":556426,"style":37774},[31805],[34,556428,556429,556432],{"style":37777},[34,556430],{"className":556431,"style":31814},[31813],[34,556433,556435],{"className":556434},[31818,31819,31820,31821],[34,556436,768],{"className":556437},[31685,31821],"，变成 ",[34,556440,556442,556476],{"className":556441},[31642],[34,556443,556445],{"className":556444},[31646],[31648,556446,556447],{"xmlns":31650},[31652,556448,556449,556473],{},[31655,556450,556451,556457,556459,556461,556463,556469,556471],{},[41453,556452,556453,556455],{},[31749,556454,640],{},[31749,556456,768],{},[31662,556458,546295],{},[31658,556460,547853],{},[31662,556462,32682],{},[37624,556464,556465,556467],{},[31658,556466,94628],{},[31749,556468,768],{},[31658,556470,46900],{},[31658,556472,94628],{},[31665,556474,556475],{"encoding":31667},"\\frac{1}{2}\\int \\sin u^2 du",[34,556477,556479],{"className":556478,"ariaHidden":758},[31672],[34,556480,556482,556485,556553,556556,556559,556562,556565,556568,556597,556600],{"className":556481},[31676],[34,556483],{"className":556484,"style":46727},[31680],[34,556486,556488,556491,556550],{"className":556487},[31685],[34,556489],{"className":556490},[31782,41488],[34,556492,556494],{"className":556493},[41453],[34,556495,556497,556542],{"className":556496},[31796,31797],[34,556498,556500,556539],{"className":556499},[31801],[34,556501,556503,556517,556525],{"className":556502,"style":46746},[31805],[34,556504,556505,556508],{"style":41504},[34,556506],{"className":556507,"style":41508},[31813],[34,556509,556511],{"className":556510},[31818,31819,31820,31821],[34,556512,556514],{"className":556513},[31685,31821],[34,556515,768],{"className":556516},[31685,31821],[34,556518,556519,556522],{"style":41520},[34,556520],{"className":556521,"style":41508},[31813],[34,556523],{"className":556524,"style":41528},[41527],[34,556526,556527,556530],{"style":46771},[34,556528],{"className":556529,"style":41508},[31813],[34,556531,556533],{"className":556532},[31818,31819,31820,31821],[34,556534,556536],{"className":556535},[31685,31821],[34,556537,640],{"className":556538},[31685,31821],[34,556540,31829],{"className":556541},[31828],[34,556543,556545],{"className":556544},[31801],[34,556546,556548],{"className":556547,"style":41619},[31805],[34,556549],{},[34,556551],{"className":556552},[31690,41488],[34,556554],{"className":556555,"style":32123},[31845],[34,556557,546295],{"className":556558,"style":546331},[32767,131249,156560],[34,556560],{"className":556561,"style":32123},[31845],[34,556563,547853],{"className":556564},[32767],[34,556566],{"className":556567,"style":32123},[31845],[34,556569,556571,556574],{"className":556570},[31685],[34,556572,94628],{"className":556573},[31685,31686],[34,556575,556577],{"className":556576},[31792],[34,556578,556580],{"className":556579},[31796],[34,556581,556583],{"className":556582},[31801],[34,556584,556586],{"className":556585,"style":37774},[31805],[34,556587,556588,556591],{"style":37777},[34,556589],{"className":556590,"style":31814},[31813],[34,556592,556594],{"className":556593},[31818,31819,31820,31821],[34,556595,768],{"className":556596},[31685,31821],[34,556598,46900],{"className":556599},[31685,31686],[34,556601,94628],{"className":556602},[31685,31686],"（菲涅尔积分），这是收敛的（条件收敛）。且与 ",[34,556605,556607,556620],{"className":556606},[31642],[34,556608,556610],{"className":556609},[31646],[31648,556611,556612],{"xmlns":31650},[31652,556613,556614,556618],{},[31655,556615,556616],{},[31658,556617,217392],{},[31665,556619,218244],{"encoding":31667},[34,556621,556623],{"className":556622,"ariaHidden":758},[31672],[34,556624,556626,556629],{"className":556625},[31676],[34,556627],{"className":556628,"style":32527},[31680],[34,556630,217392],{"className":556631,"style":217496},[31685,31686]," 无关，所以是",[2627,556634,548044],{},[108,556636,411601,556637,556731,556732,556805,556806,556834],{},[34,556638,556640,556674],{"className":556639},[31642],[34,556641,556643],{"className":556642},[31646],[31648,556644,556645],{"xmlns":31650},[31652,556646,556647,556671],{},[31655,556648,556649,556651,556653,556655,556657,556659,556661,556663,556665,556667,556669],{},[31658,556650,31942],{},[31662,556652,456],{"stretchy":702},[31658,556654,32936],{},[31662,556656,31960],{"separator":758},[31658,556658,217392],{},[31662,556660,2557],{"stretchy":702},[31662,556662,202],{},[31658,556664,547860],{},[31662,556666,32682],{},[31658,556668,217392],{},[31658,556670,32936],{},[31665,556672,556673],{"encoding":31667},"g(x, \\alpha) = \\cos \\alpha x",[34,556675,556677,556713],{"className":556676,"ariaHidden":758},[31672],[34,556678,556680,556683,556686,556689,556692,556695,556698,556701,556704,556707,556710],{"className":556679},[31676],[34,556681],{"className":556682,"style":31774},[31680],[34,556684,31942],{"className":556685,"style":32062},[31685,31686],[34,556687,456],{"className":556688},[31782],[34,556690,32936],{"className":556691},[31685,31686],[34,556693,31960],{"className":556694},[32119],[34,556696],{"className":556697,"style":32123},[31845],[34,556699,217392],{"className":556700,"style":217496},[31685,31686],[34,556702,2557],{"className":556703},[31690],[34,556705],{"className":556706,"style":31846},[31845],[34,556708,202],{"className":556709},[31850],[34,556711],{"className":556712,"style":31846},[31845],[34,556714,556716,556719,556722,556725,556728],{"className":556715},[31676],[34,556717],{"className":556718,"style":32527},[31680],[34,556720,547860],{"className":556721},[32767],[34,556723],{"className":556724,"style":32123},[31845],[34,556726,217392],{"className":556727,"style":217496},[31685,31686],[34,556729,32936],{"className":556730},[31685,31686],"。\n在有限区间 ",[34,556733,556735,556757],{"className":556734},[31642],[34,556736,556738],{"className":556737},[31646],[31648,556739,556740],{"xmlns":31650},[31652,556741,556742,556754],{},[31655,556743,556744,556746,556748,556750,556752],{},[31658,556745,111],{},[31662,556747,45972],{},[31658,556749,217392],{},[31662,556751,45972],{},[31658,556753,1697],{},[31665,556755,556756],{"encoding":31667},"a \\le \\alpha \\le b",[34,556758,556760,556778,556796],{"className":556759,"ariaHidden":758},[31672],[34,556761,556763,556766,556769,556772,556775],{"className":556762},[31676],[34,556764],{"className":556765,"style":43198},[31680],[34,556767,111],{"className":556768},[31685,31686],[34,556770],{"className":556771,"style":31846},[31845],[34,556773,45972],{"className":556774},[31850],[34,556776],{"className":556777,"style":31846},[31845],[34,556779,556781,556784,556787,556790,556793],{"className":556780},[31676],[34,556782],{"className":556783,"style":43198},[31680],[34,556785,217392],{"className":556786,"style":217496},[31685,31686],[34,556788],{"className":556789,"style":31846},[31845],[34,556791,45972],{"className":556792},[31850],[34,556794],{"className":556795,"style":31846},[31845],[34,556797,556799,556802],{"className":556798},[31676],[34,556800],{"className":556801,"style":31681},[31680],[34,556803,1697],{"className":556804},[31685,31686]," 内，这是有界的，但它关于 ",[34,556807,556809,556822],{"className":556808},[31642],[34,556810,556812],{"className":556811},[31646],[31648,556813,556814],{"xmlns":31650},[31652,556815,556816,556820],{},[31655,556817,556818],{},[31658,556819,32936],{},[31665,556821,32936],{"encoding":31667},[34,556823,556825],{"className":556824,"ariaHidden":758},[31672],[34,556826,556828,556831],{"className":556827},[31676],[34,556829],{"className":556830,"style":32527},[31680],[34,556832,32936],{"className":556833},[31685,31686]," 不是单调的。",[58531,556836],{},[2627,556838,556839],{},"最终正确思路（分部积分 + 判别法）：","\n这道题的标准解法通常是利用 ",[2627,556842,552129],{}," 的变体或换元。\n让我们回到换元后的式子：\n",[34,556845,556847,556917],{"className":556846},[31642],[34,556848,556850],{"className":556849},[31646],[31648,556851,556852],{"xmlns":31650},[31652,556853,556854,556914],{},[31655,556855,556856,556862,556874,556888,556890,556892,556894,556896,556908,556910,556912],{},[41453,556857,556858,556860],{},[31749,556859,640],{},[31749,556861,15932],{},[38846,556863,556864,556866,556868],{},[31662,556865,546295],{},[31749,556867,459],{},[31655,556869,556870,556872],{},[31662,556871,223],{},[31658,556873,138286],{"mathvariant":38079},[41453,556875,556876,556884],{},[31655,556877,556878,556880,556882],{},[31658,556879,547853],{},[31662,556881,32682],{},[31658,556883,33249],{},[41463,556885,556886],{},[31658,556887,33249],{},[31658,556889,547860],{},[31662,556891,32682],{},[31662,556893,456],{"stretchy":702},[31658,556895,217392],{},[37624,556897,556898,556900],{},[31658,556899,33249],{},[31655,556901,556902,556904,556906],{},[31749,556903,640],{},[31658,556905,38080],{"mathvariant":38079},[31749,556907,15932],{},[31662,556909,2557],{"stretchy":702},[31658,556911,46900],{},[31658,556913,33249],{},[31665,556915,556916],{"encoding":31667},"\\frac{1}{4} \\int_{0}^{+\\infty} \\frac{\\sin t}{\\sqrt{t}} \\cos(\\alpha t^{1\u002F4}) dt",[34,556918,556920],{"className":556919,"ariaHidden":758},[31672],[34,556921,556923,556927,556995,556998,557058,557061,557188,557191,557194,557197,557200,557232,557235,557238],{"className":556922},[31676],[34,556924],{"className":556925,"style":556926},[31680],"height:1.5042em;vertical-align:-0.538em;",[34,556928,556930,556933,556992],{"className":556929},[31685],[34,556931],{"className":556932},[31782,41488],[34,556934,556936],{"className":556935},[41453],[34,556937,556939,556984],{"className":556938},[31796,31797],[34,556940,556942,556981],{"className":556941},[31801],[34,556943,556945,556959,556967],{"className":556944,"style":46746},[31805],[34,556946,556947,556950],{"style":41504},[34,556948],{"className":556949,"style":41508},[31813],[34,556951,556953],{"className":556952},[31818,31819,31820,31821],[34,556954,556956],{"className":556955},[31685,31821],[34,556957,15932],{"className":556958},[31685,31821],[34,556960,556961,556964],{"style":41520},[34,556962],{"className":556963,"style":41508},[31813],[34,556965],{"className":556966,"style":41528},[41527],[34,556968,556969,556972],{"style":46771},[34,556970],{"className":556971,"style":41508},[31813],[34,556973,556975],{"className":556974},[31818,31819,31820,31821],[34,556976,556978],{"className":556977},[31685,31821],[34,556979,640],{"className":556980},[31685,31821],[34,556982,31829],{"className":556983},[31828],[34,556985,556987],{"className":556986},[31801],[34,556988,556990],{"className":556989,"style":41619},[31805],[34,556991],{},[34,556993],{"className":556994},[31690,41488],[34,556996],{"className":556997,"style":32123},[31845],[34,556999,557001,557004],{"className":557000},[32767],[34,557002,546295],{"className":557003,"style":546331},[32767,131249,156560],[34,557005,557007],{"className":557006},[31792],[34,557008,557010,557050],{"className":557009},[31796,31797],[34,557011,557013,557047],{"className":557012},[31801],[34,557014,557016,557030],{"className":557015,"style":546550},[31805],[34,557017,557018,557021],{"style":546347},[34,557019],{"className":557020,"style":31814},[31813],[34,557022,557024],{"className":557023},[31818,31819,31820,31821],[34,557025,557027],{"className":557026},[31685,31821],[34,557028,459],{"className":557029},[31685,31821],[34,557031,557032,557035],{"style":546362},[34,557033],{"className":557034,"style":31814},[31813],[34,557036,557038],{"className":557037},[31818,31819,31820,31821],[34,557039,557041,557044],{"className":557040},[31685,31821],[34,557042,223],{"className":557043},[31685,31821],[34,557045,138286],{"className":557046},[31685,31821],[34,557048,31829],{"className":557049},[31828],[34,557051,557053],{"className":557052},[31801],[34,557054,557056],{"className":557055,"style":546384},[31805],[34,557057],{},[34,557059],{"className":557060,"style":32123},[31845],[34,557062,557064,557067,557185],{"className":557063},[31685],[34,557065],{"className":557066},[31782,41488],[34,557068,557070],{"className":557069},[41453],[34,557071,557073,557177],{"className":557072},[31796,31797],[34,557074,557076,557174],{"className":557075},[31801],[34,557077,557079,557137,557145],{"className":557078,"style":548640},[31805],[34,557080,557081,557084],{"style":555259},[34,557082],{"className":557083,"style":41508},[31813],[34,557085,557087],{"className":557086},[31818,31819,31820,31821],[34,557088,557090],{"className":557089},[31685,31821],[34,557091,557093],{"className":557092},[31685,41550,31821],[34,557094,557096,557129],{"className":557095},[31796,31797],[34,557097,557099,557126],{"className":557098},[31801],[34,557100,557102,557114],{"className":557101,"style":555281},[31805],[34,557103,557105,557108],{"className":557104,"style":41565},[41564],[34,557106],{"className":557107,"style":41508},[31813],[34,557109,557111],{"className":557110,"style":41572},[31685,31821],[34,557112,33249],{"className":557113},[31685,31686,31821],[34,557115,557116,557119],{"style":555296},[34,557117],{"className":557118,"style":41508},[31813],[34,557120,557122],{"className":557121,"style":41586},[41585,31821],[41588,557123,557124],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,557125],{"d":41597},[34,557127,31829],{"className":557128},[31828],[34,557130,557132],{"className":557131},[31801],[34,557133,557135],{"className":557134,"style":555316},[31805],[34,557136],{},[34,557138,557139,557142],{"style":41520},[34,557140],{"className":557141,"style":41508},[31813],[34,557143],{"className":557144,"style":41528},[41527],[34,557146,557147,557150],{"style":46771},[34,557148],{"className":557149,"style":41508},[31813],[34,557151,557153],{"className":557152},[31818,31819,31820,31821],[34,557154,557156,557168,557171],{"className":557155},[31685,31821],[34,557157,557159,557162,557165],{"className":557158},[32767,31821],[34,557160,31747],{"className":557161},[31821],[34,557163,31955],{"className":557164},[31821],[34,557166,31660],{"className":557167},[31821],[34,557169],{"className":557170,"style":43430},[31845,31821],[34,557172,33249],{"className":557173},[31685,31686,31821],[34,557175,31829],{"className":557176},[31828],[34,557178,557180],{"className":557179},[31801],[34,557181,557183],{"className":557182,"style":555424},[31805],[34,557184],{},[34,557186],{"className":557187},[31690,41488],[34,557189],{"className":557190,"style":32123},[31845],[34,557192,547860],{"className":557193},[32767],[34,557195,456],{"className":557196},[31782],[34,557198,217392],{"className":557199,"style":217496},[31685,31686],[34,557201,557203,557206],{"className":557202},[31685],[34,557204,33249],{"className":557205},[31685,31686],[34,557207,557209],{"className":557208},[31792],[34,557210,557212],{"className":557211},[31796],[34,557213,557215],{"className":557214},[31801],[34,557216,557218],{"className":557217,"style":61037},[31805],[34,557219,557220,557223],{"style":37777},[34,557221],{"className":557222,"style":31814},[31813],[34,557224,557226],{"className":557225},[31818,31819,31820,31821],[34,557227,557229],{"className":557228},[31685,31821],[34,557230,42426],{"className":557231},[31685,31821],[34,557233,2557],{"className":557234},[31690],[34,557236,46900],{"className":557237},[31685,31686],[34,557239,33249],{"className":557240},[31685,31686],[105,557242,557243,557654,557821],{},[108,557244,557245,557495,557496,557535,557536,557622,557623,557625,557626,31901],{},[34,557246,557248,557290],{"className":557247},[31642],[34,557249,557251],{"className":557250},[31646],[31648,557252,557253],{"xmlns":31650},[31652,557254,557255,557287],{},[31655,557256,557257,557269,557283,557285],{},[38846,557258,557259,557261,557263],{},[31662,557260,546295],{},[31749,557262,459],{},[31655,557264,557265,557267],{},[31662,557266,223],{},[31658,557268,138286],{"mathvariant":38079},[41453,557270,557271,557279],{},[31655,557272,557273,557275,557277],{},[31658,557274,547853],{},[31662,557276,32682],{},[31658,557278,33249],{},[41463,557280,557281],{},[31658,557282,33249],{},[31658,557284,46900],{},[31658,557286,33249],{},[31665,557288,557289],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{\\sin t}{\\sqrt{t}} dt",[34,557291,557293],{"className":557292,"ariaHidden":758},[31672],[34,557294,557296,557299,557359,557362,557489,557492],{"className":557295},[31676],[34,557297],{"className":557298,"style":556926},[31680],[34,557300,557302,557305],{"className":557301},[32767],[34,557303,546295],{"className":557304,"style":546331},[32767,131249,156560],[34,557306,557308],{"className":557307},[31792],[34,557309,557311,557351],{"className":557310},[31796,31797],[34,557312,557314,557348],{"className":557313},[31801],[34,557315,557317,557331],{"className":557316,"style":546550},[31805],[34,557318,557319,557322],{"style":546347},[34,557320],{"className":557321,"style":31814},[31813],[34,557323,557325],{"className":557324},[31818,31819,31820,31821],[34,557326,557328],{"className":557327},[31685,31821],[34,557329,459],{"className":557330},[31685,31821],[34,557332,557333,557336],{"style":546362},[34,557334],{"className":557335,"style":31814},[31813],[34,557337,557339],{"className":557338},[31818,31819,31820,31821],[34,557340,557342,557345],{"className":557341},[31685,31821],[34,557343,223],{"className":557344},[31685,31821],[34,557346,138286],{"className":557347},[31685,31821],[34,557349,31829],{"className":557350},[31828],[34,557352,557354],{"className":557353},[31801],[34,557355,557357],{"className":557356,"style":546384},[31805],[34,557358],{},[34,557360],{"className":557361,"style":32123},[31845],[34,557363,557365,557368,557486],{"className":557364},[31685],[34,557366],{"className":557367},[31782,41488],[34,557369,557371],{"className":557370},[41453],[34,557372,557374,557478],{"className":557373},[31796,31797],[34,557375,557377,557475],{"className":557376},[31801],[34,557378,557380,557438,557446],{"className":557379,"style":548640},[31805],[34,557381,557382,557385],{"style":555259},[34,557383],{"className":557384,"style":41508},[31813],[34,557386,557388],{"className":557387},[31818,31819,31820,31821],[34,557389,557391],{"className":557390},[31685,31821],[34,557392,557394],{"className":557393},[31685,41550,31821],[34,557395,557397,557430],{"className":557396},[31796,31797],[34,557398,557400,557427],{"className":557399},[31801],[34,557401,557403,557415],{"className":557402,"style":555281},[31805],[34,557404,557406,557409],{"className":557405,"style":41565},[41564],[34,557407],{"className":557408,"style":41508},[31813],[34,557410,557412],{"className":557411,"style":41572},[31685,31821],[34,557413,33249],{"className":557414},[31685,31686,31821],[34,557416,557417,557420],{"style":555296},[34,557418],{"className":557419,"style":41508},[31813],[34,557421,557423],{"className":557422,"style":41586},[41585,31821],[41588,557424,557425],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,557426],{"d":41597},[34,557428,31829],{"className":557429},[31828],[34,557431,557433],{"className":557432},[31801],[34,557434,557436],{"className":557435,"style":555316},[31805],[34,557437],{},[34,557439,557440,557443],{"style":41520},[34,557441],{"className":557442,"style":41508},[31813],[34,557444],{"className":557445,"style":41528},[41527],[34,557447,557448,557451],{"style":46771},[34,557449],{"className":557450,"style":41508},[31813],[34,557452,557454],{"className":557453},[31818,31819,31820,31821],[34,557455,557457,557469,557472],{"className":557456},[31685,31821],[34,557458,557460,557463,557466],{"className":557459},[32767,31821],[34,557461,31747],{"className":557462},[31821],[34,557464,31955],{"className":557465},[31821],[34,557467,31660],{"className":557468},[31821],[34,557470],{"className":557471,"style":43430},[31845,31821],[34,557473,33249],{"className":557474},[31685,31686,31821],[34,557476,31829],{"className":557477},[31828],[34,557479,557481],{"className":557480},[31801],[34,557482,557484],{"className":557483,"style":555424},[31805],[34,557485],{},[34,557487],{"className":557488},[31690,41488],[34,557490,46900],{"className":557491},[31685,31686],[34,557493,33249],{"className":557494},[31685,31686]," 是收敛的（Dirichlet判别法，",[34,557497,557499,557517],{"className":557498},[31642],[34,557500,557502],{"className":557501},[31646],[31648,557503,557504],{"xmlns":31650},[31652,557505,557506,557514],{},[31655,557507,557508,557510,557512],{},[31658,557509,547853],{},[31662,557511,32682],{},[31658,557513,33249],{},[31665,557515,557516],{"encoding":31667},"\\sin t",[34,557518,557520],{"className":557519,"ariaHidden":758},[31672],[34,557521,557523,557526,557529,557532],{"className":557522},[31676],[34,557524],{"className":557525,"style":71430},[31680],[34,557527,547853],{"className":557528},[32767],[34,557530],{"className":557531,"style":32123},[31845],[34,557533,33249],{"className":557534},[31685,31686],"积分有界，",[34,557537,557539,557559],{"className":557538},[31642],[34,557540,557542],{"className":557541},[31646],[31648,557543,557544],{"xmlns":31650},[31652,557545,557546,557556],{},[31655,557547,557548,557550,557552],{},[31749,557549,640],{},[31658,557551,38080],{"mathvariant":38079},[41463,557553,557554],{},[31658,557555,33249],{},[31665,557557,557558],{"encoding":31667},"1\u002F\\sqrt{t}",[34,557560,557562],{"className":557561,"ariaHidden":758},[31672],[34,557563,557565,557569,557572],{"className":557564},[31676],[34,557566],{"className":557567,"style":557568},[31680],"height:1.1425em;vertical-align:-0.25em;",[34,557570,249219],{"className":557571},[31685],[34,557573,557575],{"className":557574},[31685,41550],[34,557576,557578,557613],{"className":557577},[31796,31797],[34,557579,557581,557610],{"className":557580},[31801],[34,557582,557585,557597],{"className":557583,"style":557584},[31805],"height:0.8925em;",[34,557586,557588,557591],{"className":557587,"style":41565},[41564],[34,557589],{"className":557590,"style":41508},[31813],[34,557592,557594],{"className":557593,"style":41572},[31685],[34,557595,33249],{"className":557596},[31685,31686],[34,557598,557600,557603],{"style":557599},"top:-2.8525em;",[34,557601],{"className":557602,"style":41508},[31813],[34,557604,557606],{"className":557605,"style":41586},[41585],[41588,557607,557608],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,557609],{"d":41597},[34,557611,31829],{"className":557612},[31828],[34,557614,557616],{"className":557615},[31801],[34,557617,557620],{"className":557618,"style":557619},[31805],"height:0.1475em;",[34,557621],{},"单调趋于0）。这是",[2627,557624,548044],{},"（因为它不含 ",[34,557627,557629,557642],{"className":557628},[31642],[34,557630,557632],{"className":557631},[31646],[31648,557633,557634],{"xmlns":31650},[31652,557635,557636,557640],{},[31655,557637,557638],{},[31658,557639,217392],{},[31665,557641,218244],{"encoding":31667},[34,557643,557645],{"className":557644,"ariaHidden":758},[31672],[34,557646,557648,557651],{"className":557647},[31676],[34,557649],{"className":557650,"style":32527},[31680],[34,557652,217392],{"className":557653,"style":217496},[31685,31686],[108,557655,557656,113124,557746,557748,557749,557791,557792,557820],{},[34,557657,557659,557693],{"className":557658},[31642],[34,557660,557662],{"className":557661},[31646],[31648,557663,557664],{"xmlns":31650},[31652,557665,557666,557690],{},[31655,557667,557668,557670,557672,557674,557676,557688],{},[31658,557669,547860],{},[31662,557671,32682],{},[31662,557673,456],{"stretchy":702},[31658,557675,217392],{},[37624,557677,557678,557680],{},[31658,557679,33249],{},[31655,557681,557682,557684,557686],{},[31749,557683,640],{},[31658,557685,38080],{"mathvariant":38079},[31749,557687,15932],{},[31662,557689,2557],{"stretchy":702},[31665,557691,557692],{"encoding":31667},"\\cos(\\alpha t^{1\u002F4})",[34,557694,557696],{"className":557695,"ariaHidden":758},[31672],[34,557697,557699,557702,557705,557708,557711,557743],{"className":557698},[31676],[34,557700],{"className":557701,"style":61012},[31680],[34,557703,547860],{"className":557704},[32767],[34,557706,456],{"className":557707},[31782],[34,557709,217392],{"className":557710,"style":217496},[31685,31686],[34,557712,557714,557717],{"className":557713},[31685],[34,557715,33249],{"className":557716},[31685,31686],[34,557718,557720],{"className":557719},[31792],[34,557721,557723],{"className":557722},[31796],[34,557724,557726],{"className":557725},[31801],[34,557727,557729],{"className":557728,"style":61037},[31805],[34,557730,557731,557734],{"style":37777},[34,557732],{"className":557733,"style":31814},[31813],[34,557735,557737],{"className":557736},[31818,31819,31820,31821],[34,557738,557740],{"className":557739},[31685,31821],[34,557741,42426],{"className":557742},[31685,31821],[34,557744,2557],{"className":557745},[31690],[2627,557747,547838],{},"的（绝对值 ",[34,557750,557752,557767],{"className":557751},[31642],[34,557753,557755],{"className":557754},[31646],[31648,557756,557757],{"xmlns":31650},[31652,557758,557759,557765],{},[31655,557760,557761,557763],{},[31662,557762,45972],{},[31749,557764,640],{},[31665,557766,232411],{"encoding":31667},[34,557768,557770,557782],{"className":557769,"ariaHidden":758},[31672],[34,557771,557773,557776,557779],{"className":557772},[31676],[34,557774],{"className":557775,"style":43198},[31680],[34,557777,45972],{"className":557778},[31850],[34,557780],{"className":557781,"style":31846},[31845],[34,557783,557785,557788],{"className":557784},[31676],[34,557786],{"className":557787,"style":32617},[31680],[34,557789,640],{"className":557790},[31685],"）且在 ",[34,557793,557795,557808],{"className":557794},[31642],[34,557796,557798],{"className":557797},[31646],[31648,557799,557800],{"xmlns":31650},[31652,557801,557802,557806],{},[31655,557803,557804],{},[31658,557805,32936],{},[31665,557807,32936],{"encoding":31667},[34,557809,557811],{"className":557810,"ariaHidden":758},[31672],[34,557812,557814,557817],{"className":557813},[31676],[34,557815],{"className":557816,"style":32527},[31680],[34,557818,32936],{"className":557819},[31685,31686]," 很大时变化缓慢。",[108,557822,557823,557824,557826,557827,558048,558049,558189,558190,558218,558219,558250],{},"其实这里可以直接用 ",[2627,557825,552132],{},"：\n令 ",[34,557828,557830,557872],{"className":557829},[31642],[34,557831,557833],{"className":557832},[31646],[31648,557834,557835],{"xmlns":31650},[31652,557836,557837,557869],{},[31655,557838,557839,557841,557843,557845,557847,557849,557851,557865,557867],{},[31658,557840,106063],{},[31662,557842,456],{"stretchy":702},[31658,557844,33249],{},[31662,557846,2557],{"stretchy":702},[31662,557848,202],{},[31662,557850,546295],{},[41453,557852,557853,557861],{},[31655,557854,557855,557857,557859],{},[31658,557856,547853],{},[31662,557858,32682],{},[31658,557860,33249],{},[41463,557862,557863],{},[31658,557864,33249],{},[31658,557866,46900],{},[31658,557868,33249],{},[31665,557870,557871],{"encoding":31667},"I(t) = \\int \\frac{\\sin t}{\\sqrt{t}} dt",[34,557873,557875,557902],{"className":557874,"ariaHidden":758},[31672],[34,557876,557878,557881,557884,557887,557890,557893,557896,557899],{"className":557877},[31676],[34,557879],{"className":557880,"style":31774},[31680],[34,557882,106063],{"className":557883,"style":67482},[31685,31686],[34,557885,456],{"className":557886},[31782],[34,557888,33249],{"className":557889},[31685,31686],[34,557891,2557],{"className":557892},[31690],[34,557894],{"className":557895,"style":31846},[31845],[34,557897,202],{"className":557898},[31850],[34,557900],{"className":557901,"style":31846},[31845],[34,557903,557905,557909,557912,557915,558042,558045],{"className":557904},[31676],[34,557906],{"className":557907,"style":557908},[31680],"height:1.3995em;vertical-align:-0.538em;",[34,557910,546295],{"className":557911,"style":546331},[32767,131249,156560],[34,557913],{"className":557914,"style":32123},[31845],[34,557916,557918,557921,558039],{"className":557917},[31685],[34,557919],{"className":557920},[31782,41488],[34,557922,557924],{"className":557923},[41453],[34,557925,557927,558031],{"className":557926},[31796,31797],[34,557928,557930,558028],{"className":557929},[31801],[34,557931,557933,557991,557999],{"className":557932,"style":548640},[31805],[34,557934,557935,557938],{"style":555259},[34,557936],{"className":557937,"style":41508},[31813],[34,557939,557941],{"className":557940},[31818,31819,31820,31821],[34,557942,557944],{"className":557943},[31685,31821],[34,557945,557947],{"className":557946},[31685,41550,31821],[34,557948,557950,557983],{"className":557949},[31796,31797],[34,557951,557953,557980],{"className":557952},[31801],[34,557954,557956,557968],{"className":557955,"style":555281},[31805],[34,557957,557959,557962],{"className":557958,"style":41565},[41564],[34,557960],{"className":557961,"style":41508},[31813],[34,557963,557965],{"className":557964,"style":41572},[31685,31821],[34,557966,33249],{"className":557967},[31685,31686,31821],[34,557969,557970,557973],{"style":555296},[34,557971],{"className":557972,"style":41508},[31813],[34,557974,557976],{"className":557975,"style":41586},[41585,31821],[41588,557977,557978],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,557979],{"d":41597},[34,557981,31829],{"className":557982},[31828],[34,557984,557986],{"className":557985},[31801],[34,557987,557989],{"className":557988,"style":555316},[31805],[34,557990],{},[34,557992,557993,557996],{"style":41520},[34,557994],{"className":557995,"style":41508},[31813],[34,557997],{"className":557998,"style":41528},[41527],[34,558000,558001,558004],{"style":46771},[34,558002],{"className":558003,"style":41508},[31813],[34,558005,558007],{"className":558006},[31818,31819,31820,31821],[34,558008,558010,558022,558025],{"className":558009},[31685,31821],[34,558011,558013,558016,558019],{"className":558012},[32767,31821],[34,558014,31747],{"className":558015},[31821],[34,558017,31955],{"className":558018},[31821],[34,558020,31660],{"className":558021},[31821],[34,558023],{"className":558024,"style":43430},[31845,31821],[34,558026,33249],{"className":558027},[31685,31686,31821],[34,558029,31829],{"className":558030},[31828],[34,558032,558034],{"className":558033},[31801],[34,558035,558037],{"className":558036,"style":555424},[31805],[34,558038],{},[34,558040],{"className":558041},[31690,41488],[34,558043,46900],{"className":558044},[31685,31686],[34,558046,33249],{"className":558047},[31685,31686]," 收敛。",[34,558050,558052,558100],{"className":558051},[31642],[34,558053,558055],{"className":558054},[31646],[31648,558056,558057],{"xmlns":31650},[31652,558058,558059,558097],{},[31655,558060,558061,558063,558065,558067,558069,558071,558073,558075,558077,558079,558081,558083,558095],{},[31658,558062,31942],{},[31662,558064,456],{"stretchy":702},[31658,558066,33249],{},[31662,558068,31960],{"separator":758},[31658,558070,217392],{},[31662,558072,2557],{"stretchy":702},[31662,558074,202],{},[31658,558076,547860],{},[31662,558078,32682],{},[31662,558080,456],{"stretchy":702},[31658,558082,217392],{},[37624,558084,558085,558087],{},[31658,558086,33249],{},[31655,558088,558089,558091,558093],{},[31749,558090,640],{},[31658,558092,38080],{"mathvariant":38079},[31749,558094,15932],{},[31662,558096,2557],{"stretchy":702},[31665,558098,558099],{"encoding":31667},"g(t, \\alpha) = \\cos(\\alpha t^{1\u002F4})",[34,558101,558103,558139],{"className":558102,"ariaHidden":758},[31672],[34,558104,558106,558109,558112,558115,558118,558121,558124,558127,558130,558133,558136],{"className":558105},[31676],[34,558107],{"className":558108,"style":31774},[31680],[34,558110,31942],{"className":558111,"style":32062},[31685,31686],[34,558113,456],{"className":558114},[31782],[34,558116,33249],{"className":558117},[31685,31686],[34,558119,31960],{"className":558120},[32119],[34,558122],{"className":558123,"style":32123},[31845],[34,558125,217392],{"className":558126,"style":217496},[31685,31686],[34,558128,2557],{"className":558129},[31690],[34,558131],{"className":558132,"style":31846},[31845],[34,558134,202],{"className":558135},[31850],[34,558137],{"className":558138,"style":31846},[31845],[34,558140,558142,558145,558148,558151,558154,558186],{"className":558141},[31676],[34,558143],{"className":558144,"style":61012},[31680],[34,558146,547860],{"className":558147},[32767],[34,558149,456],{"className":558150},[31782],[34,558152,217392],{"className":558153,"style":217496},[31685,31686],[34,558155,558157,558160],{"className":558156},[31685],[34,558158,33249],{"className":558159},[31685,31686],[34,558161,558163],{"className":558162},[31792],[34,558164,558166],{"className":558165},[31796],[34,558167,558169],{"className":558168},[31801],[34,558170,558172],{"className":558171,"style":61037},[31805],[34,558173,558174,558177],{"style":37777},[34,558175],{"className":558176,"style":31814},[31813],[34,558178,558180],{"className":558179},[31818,31819,31820,31821],[34,558181,558183],{"className":558182},[31685,31821],[34,558184,42426],{"className":558185},[31685,31821],[34,558187,2557],{"className":558188},[31690]," 有界。但是 Abel 要求 ",[34,558191,558193,558206],{"className":558192},[31642],[34,558194,558196],{"className":558195},[31646],[31648,558197,558198],{"xmlns":31650},[31652,558199,558200,558204],{},[31655,558201,558202],{},[31658,558203,31942],{},[31665,558205,31942],{"encoding":31667},[34,558207,558209],{"className":558208,"ariaHidden":758},[31672],[34,558210,558212,558215],{"className":558211},[31676],[34,558213],{"className":558214,"style":38774},[31680],[34,558216,31942],{"className":558217,"style":32062},[31685,31686]," 单调，",[34,558220,558222,558238],{"className":558221},[31642],[34,558223,558225],{"className":558224},[31646],[31648,558226,558227],{"xmlns":31650},[31652,558228,558229,558235],{},[31655,558230,558231,558233],{},[31658,558232,547860],{},[31662,558234,32682],{},[31665,558236,558237],{"encoding":31667},"\\cos",[34,558239,558241],{"className":558240,"ariaHidden":758},[31672],[34,558242,558244,558247],{"className":558243},[31676],[34,558245],{"className":558246,"style":32527},[31680],[34,558248,547860],{"className":558249},[32767]," 显然不单调。",[58531,558252],{},[2627,558254,558255],{},"再次修正（书本标准解法）：","\n这道题实际上考察的是",[2627,558258,552129],{},"在换元后的应用。\n我们看 ",[34,558261,558263,558317],{"className":558262},[31642],[34,558264,558266],{"className":558265},[31646],[31648,558267,558268],{"xmlns":31650},[31652,558269,558270,558314],{},[31655,558271,558272,558274,558288,558290,558292,558294,558296,558308,558310,558312],{},[31662,558273,546295],{},[41453,558275,558276,558284],{},[31655,558277,558278,558280,558282],{},[31658,558279,547853],{},[31662,558281,32682],{},[31658,558283,33249],{},[41463,558285,558286],{},[31658,558287,33249],{},[31658,558289,547860],{},[31662,558291,32682],{},[31662,558293,456],{"stretchy":702},[31658,558295,217392],{},[37624,558297,558298,558300],{},[31658,558299,33249],{},[31655,558301,558302,558304,558306],{},[31749,558303,640],{},[31658,558305,38080],{"mathvariant":38079},[31749,558307,15932],{},[31662,558309,2557],{"stretchy":702},[31658,558311,46900],{},[31658,558313,33249],{},[31665,558315,558316],{"encoding":31667},"\\int \\frac{\\sin t}{\\sqrt{t}} \\cos(\\alpha t^{1\u002F4}) dt",[34,558318,558320],{"className":558319,"ariaHidden":758},[31672],[34,558321,558323,558327,558330,558333,558460,558463,558466,558469,558472,558504,558507,558510],{"className":558322},[31676],[34,558324],{"className":558325,"style":558326},[31680],"height:1.426em;vertical-align:-0.538em;",[34,558328,546295],{"className":558329,"style":546331},[32767,131249,156560],[34,558331],{"className":558332,"style":32123},[31845],[34,558334,558336,558339,558457],{"className":558335},[31685],[34,558337],{"className":558338},[31782,41488],[34,558340,558342],{"className":558341},[41453],[34,558343,558345,558449],{"className":558344},[31796,31797],[34,558346,558348,558446],{"className":558347},[31801],[34,558349,558351,558409,558417],{"className":558350,"style":548640},[31805],[34,558352,558353,558356],{"style":555259},[34,558354],{"className":558355,"style":41508},[31813],[34,558357,558359],{"className":558358},[31818,31819,31820,31821],[34,558360,558362],{"className":558361},[31685,31821],[34,558363,558365],{"className":558364},[31685,41550,31821],[34,558366,558368,558401],{"className":558367},[31796,31797],[34,558369,558371,558398],{"className":558370},[31801],[34,558372,558374,558386],{"className":558373,"style":555281},[31805],[34,558375,558377,558380],{"className":558376,"style":41565},[41564],[34,558378],{"className":558379,"style":41508},[31813],[34,558381,558383],{"className":558382,"style":41572},[31685,31821],[34,558384,33249],{"className":558385},[31685,31686,31821],[34,558387,558388,558391],{"style":555296},[34,558389],{"className":558390,"style":41508},[31813],[34,558392,558394],{"className":558393,"style":41586},[41585,31821],[41588,558395,558396],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,558397],{"d":41597},[34,558399,31829],{"className":558400},[31828],[34,558402,558404],{"className":558403},[31801],[34,558405,558407],{"className":558406,"style":555316},[31805],[34,558408],{},[34,558410,558411,558414],{"style":41520},[34,558412],{"className":558413,"style":41508},[31813],[34,558415],{"className":558416,"style":41528},[41527],[34,558418,558419,558422],{"style":46771},[34,558420],{"className":558421,"style":41508},[31813],[34,558423,558425],{"className":558424},[31818,31819,31820,31821],[34,558426,558428,558440,558443],{"className":558427},[31685,31821],[34,558429,558431,558434,558437],{"className":558430},[32767,31821],[34,558432,31747],{"className":558433},[31821],[34,558435,31955],{"className":558436},[31821],[34,558438,31660],{"className":558439},[31821],[34,558441],{"className":558442,"style":43430},[31845,31821],[34,558444,33249],{"className":558445},[31685,31686,31821],[34,558447,31829],{"className":558448},[31828],[34,558450,558452],{"className":558451},[31801],[34,558453,558455],{"className":558454,"style":555424},[31805],[34,558456],{},[34,558458],{"className":558459},[31690,41488],[34,558461],{"className":558462,"style":32123},[31845],[34,558464,547860],{"className":558465},[32767],[34,558467,456],{"className":558468},[31782],[34,558470,217392],{"className":558471,"style":217496},[31685,31686],[34,558473,558475,558478],{"className":558474},[31685],[34,558476,33249],{"className":558477},[31685,31686],[34,558479,558481],{"className":558480},[31792],[34,558482,558484],{"className":558483},[31796],[34,558485,558487],{"className":558486},[31801],[34,558488,558490],{"className":558489,"style":61037},[31805],[34,558491,558492,558495],{"style":37777},[34,558493],{"className":558494,"style":31814},[31813],[34,558496,558498],{"className":558497},[31818,31819,31820,31821],[34,558499,558501],{"className":558500},[31685,31821],[34,558502,42426],{"className":558503},[31685,31821],[34,558505,2557],{"className":558506},[31690],[34,558508,46900],{"className":558509},[31685,31686],[34,558511,33249],{"className":558512},[31685,31686],"。\n利用积化和差公式：\n",[34,558515,558517,558587],{"className":558516},[31642],[34,558518,558520],{"className":558519},[31646],[31648,558521,558522],{"xmlns":31650},[31652,558523,558524,558584],{},[31655,558525,558526,558528,558530,558532,558534,558536,558538,558540,558542,558544,558550,558552,558554,558556,558558,558560,558562,558564,558566,558568,558570,558572,558574,558576,558578,558580,558582],{},[31658,558527,547853],{},[31662,558529,32682],{},[31658,558531,33249],{},[31658,558533,547860],{},[31662,558535,32682],{},[31662,558537,456],{"stretchy":702},[31658,558539,217691],{},[31662,558541,2557],{"stretchy":702},[31662,558543,202],{},[41453,558545,558546,558548],{},[31749,558547,640],{},[31749,558549,768],{},[31662,558551,15972],{"stretchy":702},[31658,558553,547853],{},[31662,558555,32682],{},[31662,558557,456],{"stretchy":702},[31658,558559,33249],{},[31662,558561,223],{},[31658,558563,217691],{},[31662,558565,2557],{"stretchy":702},[31662,558567,223],{},[31658,558569,547853],{},[31662,558571,32682],{},[31662,558573,456],{"stretchy":702},[31658,558575,33249],{},[31662,558577,32226],{},[31658,558579,217691],{},[31662,558581,2557],{"stretchy":702},[31662,558583,32712],{"stretchy":702},[31665,558585,558586],{"encoding":31667},"\\sin t \\cos(\\beta) = \\frac{1}{2}[\\sin(t+\\beta) + \\sin(t-\\beta)]",[34,558588,558590,558629,558724,558745,558769],{"className":558589,"ariaHidden":758},[31672],[34,558591,558593,558596,558599,558602,558605,558608,558611,558614,558617,558620,558623,558626],{"className":558592},[31676],[34,558594],{"className":558595,"style":31774},[31680],[34,558597,547853],{"className":558598},[32767],[34,558600],{"className":558601,"style":32123},[31845],[34,558603,33249],{"className":558604},[31685,31686],[34,558606],{"className":558607,"style":32123},[31845],[34,558609,547860],{"className":558610},[32767],[34,558612,456],{"className":558613},[31782],[34,558615,217691],{"className":558616,"style":217853},[31685,31686],[34,558618,2557],{"className":558619},[31690],[34,558621],{"className":558622,"style":31846},[31845],[34,558624,202],{"className":558625},[31850],[34,558627],{"className":558628,"style":31846},[31845],[34,558630,558632,558635,558703,558706,558709,558712,558715,558718,558721],{"className":558631},[31676],[34,558633],{"className":558634,"style":46727},[31680],[34,558636,558638,558641,558700],{"className":558637},[31685],[34,558639],{"className":558640},[31782,41488],[34,558642,558644],{"className":558643},[41453],[34,558645,558647,558692],{"className":558646},[31796,31797],[34,558648,558650,558689],{"className":558649},[31801],[34,558651,558653,558667,558675],{"className":558652,"style":46746},[31805],[34,558654,558655,558658],{"style":41504},[34,558656],{"className":558657,"style":41508},[31813],[34,558659,558661],{"className":558660},[31818,31819,31820,31821],[34,558662,558664],{"className":558663},[31685,31821],[34,558665,768],{"className":558666},[31685,31821],[34,558668,558669,558672],{"style":41520},[34,558670],{"className":558671,"style":41508},[31813],[34,558673],{"className":558674,"style":41528},[41527],[34,558676,558677,558680],{"style":46771},[34,558678],{"className":558679,"style":41508},[31813],[34,558681,558683],{"className":558682},[31818,31819,31820,31821],[34,558684,558686],{"className":558685},[31685,31821],[34,558687,640],{"className":558688},[31685,31821],[34,558690,31829],{"className":558691},[31828],[34,558693,558695],{"className":558694},[31801],[34,558696,558698],{"className":558697,"style":41619},[31805],[34,558699],{},[34,558701],{"className":558702},[31690,41488],[34,558704,15972],{"className":558705},[31782],[34,558707,547853],{"className":558708},[32767],[34,558710,456],{"className":558711},[31782],[34,558713,33249],{"className":558714},[31685,31686],[34,558716],{"className":558717,"style":32604},[31845],[34,558719,223],{"className":558720},[32024],[34,558722],{"className":558723,"style":32604},[31845],[34,558725,558727,558730,558733,558736,558739,558742],{"className":558726},[31676],[34,558728],{"className":558729,"style":31774},[31680],[34,558731,217691],{"className":558732,"style":217853},[31685,31686],[34,558734,2557],{"className":558735},[31690],[34,558737],{"className":558738,"style":32604},[31845],[34,558740,223],{"className":558741},[32024],[34,558743],{"className":558744,"style":32604},[31845],[34,558746,558748,558751,558754,558757,558760,558763,558766],{"className":558747},[31676],[34,558749],{"className":558750,"style":31774},[31680],[34,558752,547853],{"className":558753},[32767],[34,558755,456],{"className":558756},[31782],[34,558758,33249],{"className":558759},[31685,31686],[34,558761],{"className":558762,"style":32604},[31845],[34,558764,32226],{"className":558765},[32024],[34,558767],{"className":558768,"style":32604},[31845],[34,558770,558772,558775,558778],{"className":558771},[31676],[34,558773],{"className":558774,"style":31774},[31680],[34,558776,217691],{"className":558777,"style":217853},[31685,31686],[34,558779,481211],{"className":558780},[31690],[34,558782,558784,558814],{"className":558783},[31642],[34,558785,558787],{"className":558786},[31646],[31648,558788,558789],{"xmlns":31650},[31652,558790,558791,558811],{},[31655,558792,558793,558795,558797,558799],{},[31658,558794,217691],{},[31662,558796,202],{},[31658,558798,217392],{},[37624,558800,558801,558803],{},[31658,558802,33249],{},[31655,558804,558805,558807,558809],{},[31749,558806,640],{},[31658,558808,38080],{"mathvariant":38079},[31749,558810,15932],{},[31665,558812,558813],{"encoding":31667},"\\beta = \\alpha t^{1\u002F4}",[34,558815,558817,558835],{"className":558816,"ariaHidden":758},[31672],[34,558818,558820,558823,558826,558829,558832],{"className":558819},[31676],[34,558821],{"className":558822,"style":43217},[31680],[34,558824,217691],{"className":558825,"style":217853},[31685,31686],[34,558827],{"className":558828,"style":31846},[31845],[34,558830,202],{"className":558831},[31850],[34,558833],{"className":558834,"style":31846},[31845],[34,558836,558838,558841,558844],{"className":558837},[31676],[34,558839],{"className":558840,"style":61037},[31680],[34,558842,217392],{"className":558843,"style":217496},[31685,31686],[34,558845,558847,558850],{"className":558846},[31685],[34,558848,33249],{"className":558849},[31685,31686],[34,558851,558853],{"className":558852},[31792],[34,558854,558856],{"className":558855},[31796],[34,558857,558859],{"className":558858},[31801],[34,558860,558862],{"className":558861,"style":61037},[31805],[34,558863,558864,558867],{"style":37777},[34,558865],{"className":558866,"style":31814},[31813],[34,558868,558870],{"className":558869},[31818,31819,31820,31821],[34,558871,558873],{"className":558872},[31685,31821],[34,558874,42426],{"className":558875},[31685,31821],"。\n当 ",[34,558878,558880,558893],{"className":558879},[31642],[34,558881,558883],{"className":558882},[31646],[31648,558884,558885],{"xmlns":31650},[31652,558886,558887,558891],{},[31655,558888,558889],{},[31658,558890,33249],{},[31665,558892,33249],{"encoding":31667},[34,558894,558896],{"className":558895,"ariaHidden":758},[31672],[34,558897,558899,558902],{"className":558898},[31676],[34,558900],{"className":558901,"style":33280},[31680],[34,558903,33249],{"className":558904},[31685,31686]," 很大时，",[34,558907,558909,558922],{"className":558908},[31642],[34,558910,558912],{"className":558911},[31646],[31648,558913,558914],{"xmlns":31650},[31652,558915,558916,558920],{},[31655,558917,558918],{},[31658,558919,217691],{},[31665,558921,218273],{"encoding":31667},[34,558923,558925],{"className":558924,"ariaHidden":758},[31672],[34,558926,558928,558931],{"className":558927},[31676],[34,558929],{"className":558930,"style":43217},[31680],[34,558932,217691],{"className":558933,"style":217853},[31685,31686]," 增长比 ",[34,558936,558938,558951],{"className":558937},[31642],[34,558939,558941],{"className":558940},[31646],[31648,558942,558943],{"xmlns":31650},[31652,558944,558945,558949],{},[31655,558946,558947],{},[31658,558948,33249],{},[31665,558950,33249],{"encoding":31667},[34,558952,558954],{"className":558953,"ariaHidden":758},[31672],[34,558955,558957,558960],{"className":558956},[31676],[34,558958],{"className":558959,"style":33280},[31680],[34,558961,33249],{"className":558962},[31685,31686]," 慢得多。\n实际上，只要 ",[34,558965,558967,558987],{"className":558966},[31642],[34,558968,558970],{"className":558969},[31646],[31648,558971,558972],{"xmlns":31650},[31652,558973,558974,558984],{},[31655,558975,558976],{},[41453,558977,558978,558980],{},[31749,558979,640],{},[41463,558981,558982],{},[31658,558983,33249],{},[31665,558985,558986],{"encoding":31667},"\\frac{1}{\\sqrt{t}}",[34,558988,558990],{"className":558989,"ariaHidden":758},[31672],[34,558991,558993,558997],{"className":558992},[31676],[34,558994],{"className":558995,"style":558996},[31680],"height:1.3831em;vertical-align:-0.538em;",[34,558998,559000,559003,559106],{"className":558999},[31685],[34,559001],{"className":559002},[31782,41488],[34,559004,559006],{"className":559005},[41453],[34,559007,559009,559098],{"className":559008},[31796,31797],[34,559010,559012,559095],{"className":559011},[31801],[34,559013,559015,559073,559081],{"className":559014,"style":46746},[31805],[34,559016,559017,559020],{"style":555259},[34,559018],{"className":559019,"style":41508},[31813],[34,559021,559023],{"className":559022},[31818,31819,31820,31821],[34,559024,559026],{"className":559025},[31685,31821],[34,559027,559029],{"className":559028},[31685,41550,31821],[34,559030,559032,559065],{"className":559031},[31796,31797],[34,559033,559035,559062],{"className":559034},[31801],[34,559036,559038,559050],{"className":559037,"style":555281},[31805],[34,559039,559041,559044],{"className":559040,"style":41565},[41564],[34,559042],{"className":559043,"style":41508},[31813],[34,559045,559047],{"className":559046,"style":41572},[31685,31821],[34,559048,33249],{"className":559049},[31685,31686,31821],[34,559051,559052,559055],{"style":555296},[34,559053],{"className":559054,"style":41508},[31813],[34,559056,559058],{"className":559057,"style":41586},[41585,31821],[41588,559059,559060],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,559061],{"d":41597},[34,559063,31829],{"className":559064},[31828],[34,559066,559068],{"className":559067},[31801],[34,559069,559071],{"className":559070,"style":555316},[31805],[34,559072],{},[34,559074,559075,559078],{"style":41520},[34,559076],{"className":559077,"style":41508},[31813],[34,559079],{"className":559080,"style":41528},[41527],[34,559082,559083,559086],{"style":46771},[34,559084],{"className":559085,"style":41508},[31813],[34,559087,559089],{"className":559088},[31818,31819,31820,31821],[34,559090,559092],{"className":559091},[31685,31821],[34,559093,640],{"className":559094},[31685,31821],[34,559096,31829],{"className":559097},[31828],[34,559099,559101],{"className":559100},[31801],[34,559102,559104],{"className":559103,"style":555424},[31805],[34,559105],{},[34,559107],{"className":559108},[31690,41488]," 单调趋于0，且 ",[34,559111,559113,559145],{"className":559112},[31642],[34,559114,559116],{"className":559115},[31646],[31648,559117,559118],{"xmlns":31650},[31652,559119,559120,559142],{},[31655,559121,559122,559124,559126,559128,559130,559132,559134,559136,559138,559140],{},[31662,559123,546295],{},[31658,559125,547853],{},[31662,559127,32682],{},[31658,559129,33249],{},[31658,559131,547860],{},[31662,559133,32682],{},[31662,559135,456],{"stretchy":702},[31662,559137,33934],{},[39361,559139,39363],{},[31662,559141,2557],{"stretchy":702},[31665,559143,559144],{"encoding":31667},"\\int \\sin t \\cos(\\dots)",[34,559146,559148],{"className":559147,"ariaHidden":758},[31672],[34,559149,559151,559154,559157,559160,559163,559166,559169,559172,559175,559178,559181,559184],{"className":559150},[31676],[34,559152],{"className":559153,"style":546896},[31680],[34,559155,546295],{"className":559156,"style":546331},[32767,131249,156560],[34,559158],{"className":559159,"style":32123},[31845],[34,559161,547853],{"className":559162},[32767],[34,559164],{"className":559165,"style":32123},[31845],[34,559167,33249],{"className":559168},[31685,31686],[34,559170],{"className":559171,"style":32123},[31845],[34,559173,547860],{"className":559174},[32767],[34,559176,456],{"className":559177},[31782],[34,559179,33934],{"className":559180},[34006],[34,559182],{"className":559183,"style":32123},[31845],[34,559185,2557],{"className":559186},[31690]," 的原函数有界即可。\n这里更严谨的证明略微复杂，但对于作业，建议使用 ",[2627,559189,552129],{}," 的结论：\n",[34,559192,559194,559233],{"className":559193},[31642],[34,559195,559197],{"className":559196},[31646],[31648,559198,559199],{"xmlns":31650},[31652,559200,559201,559231],{},[31655,559202,559203,559215,559217,559219,559221,559227,559229],{},[38846,559204,559205,559207,559209],{},[31662,559206,546295],{},[31749,559208,459],{},[31655,559210,559211,559213],{},[31662,559212,223],{},[31658,559214,138286],{"mathvariant":38079},[31658,559216,32936],{},[31658,559218,547853],{},[31662,559220,32682],{},[37624,559222,559223,559225],{},[31658,559224,32936],{},[31749,559226,15932],{},[31658,559228,46900],{},[31658,559230,32936],{},[31665,559232,556236],{"encoding":31667},[34,559234,559236],{"className":559235,"ariaHidden":758},[31672],[34,559237,559239,559242,559302,559305,559308,559311,559314,559317,559346,559349],{"className":559238},[31676],[34,559240],{"className":559241,"style":546531},[31680],[34,559243,559245,559248],{"className":559244},[32767],[34,559246,546295],{"className":559247,"style":546331},[32767,131249,156560],[34,559249,559251],{"className":559250},[31792],[34,559252,559254,559294],{"className":559253},[31796,31797],[34,559255,559257,559291],{"className":559256},[31801],[34,559258,559260,559274],{"className":559259,"style":546550},[31805],[34,559261,559262,559265],{"style":546347},[34,559263],{"className":559264,"style":31814},[31813],[34,559266,559268],{"className":559267},[31818,31819,31820,31821],[34,559269,559271],{"className":559270},[31685,31821],[34,559272,459],{"className":559273},[31685,31821],[34,559275,559276,559279],{"style":546362},[34,559277],{"className":559278,"style":31814},[31813],[34,559280,559282],{"className":559281},[31818,31819,31820,31821],[34,559283,559285,559288],{"className":559284},[31685,31821],[34,559286,223],{"className":559287},[31685,31821],[34,559289,138286],{"className":559290},[31685,31821],[34,559292,31829],{"className":559293},[31828],[34,559295,559297],{"className":559296},[31801],[34,559298,559300],{"className":559299,"style":546384},[31805],[34,559301],{},[34,559303],{"className":559304,"style":32123},[31845],[34,559306,32936],{"className":559307},[31685,31686],[34,559309],{"className":559310,"style":32123},[31845],[34,559312,547853],{"className":559313},[32767],[34,559315],{"className":559316,"style":32123},[31845],[34,559318,559320,559323],{"className":559319},[31685],[34,559321,32936],{"className":559322},[31685,31686],[34,559324,559326],{"className":559325},[31792],[34,559327,559329],{"className":559328},[31796],[34,559330,559332],{"className":559331},[31801],[34,559333,559335],{"className":559334,"style":37774},[31805],[34,559336,559337,559340],{"style":37777},[34,559338],{"className":559339,"style":31814},[31813],[34,559341,559343],{"className":559342},[31818,31819,31820,31821],[34,559344,15932],{"className":559345},[31685,31821],[34,559347,46900],{"className":559348},[31685,31686],[34,559350,32936],{"className":559351},[31685,31686]," 是收敛的，而 ",[34,559354,559356,559375],{"className":559355},[31642],[34,559357,559359],{"className":559358},[31646],[31648,559360,559361],{"xmlns":31650},[31652,559362,559363,559373],{},[31655,559364,559365,559367,559369,559371],{},[31658,559366,547860],{},[31662,559368,32682],{},[31658,559370,217392],{},[31658,559372,32936],{},[31665,559374,556021],{"encoding":31667},[34,559376,559378],{"className":559377,"ariaHidden":758},[31672],[34,559379,559381,559384,559387,559390,559393],{"className":559380},[31676],[34,559382],{"className":559383,"style":32527},[31680],[34,559385,547860],{"className":559386},[32767],[34,559388],{"className":559389,"style":32123},[31845],[34,559391,217392],{"className":559392,"style":217496},[31685,31686],[34,559394,32936],{"className":559395},[31685,31686]," 破坏了单调性，所以直接判别法很难。",[58531,559398],{},[2627,559400,559401],{},"针对该题的“作业级”写法：",[2230,559403,559404,559876,559879],{},[108,559405,396859,559406,559487,559488,347],{},[34,559407,559409,559431],{"className":559408},[31642],[34,559410,559412],{"className":559411},[31646],[31648,559413,559414],{"xmlns":31650},[31652,559415,559416,559428],{},[31655,559417,559418,559420,559422],{},[31658,559419,94628],{},[31662,559421,202],{},[37624,559423,559424,559426],{},[31658,559425,32936],{},[31749,559427,768],{},[31665,559429,559430],{"encoding":31667},"u = x^2",[34,559432,559434,559452],{"className":559433,"ariaHidden":758},[31672],[34,559435,559437,559440,559443,559446,559449],{"className":559436},[31676],[34,559438],{"className":559439,"style":32527},[31680],[34,559441,94628],{"className":559442},[31685,31686],[34,559444],{"className":559445,"style":31846},[31845],[34,559447,202],{"className":559448},[31850],[34,559450],{"className":559451,"style":31846},[31845],[34,559453,559455,559458],{"className":559454},[31676],[34,559456],{"className":559457,"style":37774},[31680],[34,559459,559461,559464],{"className":559460},[31685],[34,559462,32936],{"className":559463},[31685,31686],[34,559465,559467],{"className":559466},[31792],[34,559468,559470],{"className":559469},[31796],[34,559471,559473],{"className":559472},[31801],[34,559474,559476],{"className":559475,"style":37774},[31805],[34,559477,559478,559481],{"style":37777},[34,559479],{"className":559480,"style":31814},[31813],[34,559482,559484],{"className":559483},[31818,31819,31820,31821],[34,559485,768],{"className":559486},[31685,31821],"，原积分 ",[34,559489,559491,559575],{"className":559490},[31642],[34,559492,559494],{"className":559493},[31646],[31648,559495,559496],{"xmlns":31650},[31652,559497,559498,559572],{},[31655,559499,559500,559502,559504,559506,559508,559510,559516,559522,559524,559526,559528,559530,559532,559534,559536,559542,559544,559546,559548,559554,559556,559558,559560,559562,559566,559568,559570],{},[31662,559501,546295],{},[31658,559503,32936],{},[31658,559505,547853],{},[31662,559507,32682],{},[31662,559509,456],{"stretchy":702},[37624,559511,559512,559514],{},[31658,559513,32936],{},[31749,559515,768],{},[37624,559517,559518,559520],{},[31662,559519,2557],{"stretchy":702},[31749,559521,768],{},[31658,559523,547860],{},[31662,559525,32682],{},[31658,559527,217392],{},[31658,559529,32936],{},[31658,559531,46900],{},[31658,559533,32936],{},[31662,559535,202],{},[41453,559537,559538,559540],{},[31749,559539,640],{},[31749,559541,768],{},[31662,559543,546295],{},[31658,559545,547853],{},[31662,559547,32682],{},[37624,559549,559550,559552],{},[31658,559551,94628],{},[31749,559553,768],{},[31658,559555,547860],{},[31662,559557,32682],{},[31662,559559,456],{"stretchy":702},[31658,559561,217392],{},[41463,559563,559564],{},[31658,559565,94628],{},[31662,559567,2557],{"stretchy":702},[31658,559569,46900],{},[31658,559571,94628],{},[31665,559573,559574],{"encoding":31667},"\\int x \\sin(x^2)^2 \\cos \\alpha x dx = \\frac{1}{2} \\int \\sin u^2 \\cos(\\alpha \\sqrt{u}) du",[34,559576,559578,559690],{"className":559577,"ariaHidden":758},[31672],[34,559579,559581,559584,559587,559590,559593,559596,559599,559602,559631,559660,559663,559666,559669,559672,559675,559678,559681,559684,559687],{"className":559580},[31676],[34,559582],{"className":559583,"style":556092},[31680],[34,559585,546295],{"className":559586,"style":546331},[32767,131249,156560],[34,559588],{"className":559589,"style":32123},[31845],[34,559591,32936],{"className":559592},[31685,31686],[34,559594],{"className":559595,"style":32123},[31845],[34,559597,547853],{"className":559598},[32767],[34,559600,456],{"className":559601},[31782],[34,559603,559605,559608],{"className":559604},[31685],[34,559606,32936],{"className":559607},[31685,31686],[34,559609,559611],{"className":559610},[31792],[34,559612,559614],{"className":559613},[31796],[34,559615,559617],{"className":559616},[31801],[34,559618,559620],{"className":559619,"style":37774},[31805],[34,559621,559622,559625],{"style":37777},[34,559623],{"className":559624,"style":31814},[31813],[34,559626,559628],{"className":559627},[31818,31819,31820,31821],[34,559629,768],{"className":559630},[31685,31821],[34,559632,559634,559637],{"className":559633},[31690],[34,559635,2557],{"className":559636},[31690],[34,559638,559640],{"className":559639},[31792],[34,559641,559643],{"className":559642},[31796],[34,559644,559646],{"className":559645},[31801],[34,559647,559649],{"className":559648,"style":37774},[31805],[34,559650,559651,559654],{"style":37777},[34,559652],{"className":559653,"style":31814},[31813],[34,559655,559657],{"className":559656},[31818,31819,31820,31821],[34,559658,768],{"className":559659},[31685,31821],[34,559661],{"className":559662,"style":32123},[31845],[34,559664,547860],{"className":559665},[32767],[34,559667],{"className":559668,"style":32123},[31845],[34,559670,217392],{"className":559671,"style":217496},[31685,31686],[34,559673,32936],{"className":559674},[31685,31686],[34,559676,46900],{"className":559677},[31685,31686],[34,559679,32936],{"className":559680},[31685,31686],[34,559682],{"className":559683,"style":31846},[31845],[34,559685,202],{"className":559686},[31850],[34,559688],{"className":559689,"style":31846},[31845],[34,559691,559693,559696,559764,559767,559770,559773,559776,559779,559808,559811,559814,559817,559820,559867,559870,559873],{"className":559692},[31676],[34,559694],{"className":559695,"style":46727},[31680],[34,559697,559699,559702,559761],{"className":559698},[31685],[34,559700],{"className":559701},[31782,41488],[34,559703,559705],{"className":559704},[41453],[34,559706,559708,559753],{"className":559707},[31796,31797],[34,559709,559711,559750],{"className":559710},[31801],[34,559712,559714,559728,559736],{"className":559713,"style":46746},[31805],[34,559715,559716,559719],{"style":41504},[34,559717],{"className":559718,"style":41508},[31813],[34,559720,559722],{"className":559721},[31818,31819,31820,31821],[34,559723,559725],{"className":559724},[31685,31821],[34,559726,768],{"className":559727},[31685,31821],[34,559729,559730,559733],{"style":41520},[34,559731],{"className":559732,"style":41508},[31813],[34,559734],{"className":559735,"style":41528},[41527],[34,559737,559738,559741],{"style":46771},[34,559739],{"className":559740,"style":41508},[31813],[34,559742,559744],{"className":559743},[31818,31819,31820,31821],[34,559745,559747],{"className":559746},[31685,31821],[34,559748,640],{"className":559749},[31685,31821],[34,559751,31829],{"className":559752},[31828],[34,559754,559756],{"className":559755},[31801],[34,559757,559759],{"className":559758,"style":41619},[31805],[34,559760],{},[34,559762],{"className":559763},[31690,41488],[34,559765],{"className":559766,"style":32123},[31845],[34,559768,546295],{"className":559769,"style":546331},[32767,131249,156560],[34,559771],{"className":559772,"style":32123},[31845],[34,559774,547853],{"className":559775},[32767],[34,559777],{"className":559778,"style":32123},[31845],[34,559780,559782,559785],{"className":559781},[31685],[34,559783,94628],{"className":559784},[31685,31686],[34,559786,559788],{"className":559787},[31792],[34,559789,559791],{"className":559790},[31796],[34,559792,559794],{"className":559793},[31801],[34,559795,559797],{"className":559796,"style":37774},[31805],[34,559798,559799,559802],{"style":37777},[34,559800],{"className":559801,"style":31814},[31813],[34,559803,559805],{"className":559804},[31818,31819,31820,31821],[34,559806,768],{"className":559807},[31685,31821],[34,559809],{"className":559810,"style":32123},[31845],[34,559812,547860],{"className":559813},[32767],[34,559815,456],{"className":559816},[31782],[34,559818,217392],{"className":559819,"style":217496},[31685,31686],[34,559821,559823],{"className":559822},[31685,41550],[34,559824,559826,559859],{"className":559825},[31796,31797],[34,559827,559829,559856],{"className":559828},[31801],[34,559830,559832,559844],{"className":559831,"style":64997},[31805],[34,559833,559835,559838],{"className":559834,"style":41565},[41564],[34,559836],{"className":559837,"style":41508},[31813],[34,559839,559841],{"className":559840,"style":41572},[31685],[34,559842,94628],{"className":559843},[31685,31686],[34,559845,559846,559849],{"style":65012},[34,559847],{"className":559848,"style":41508},[31813],[34,559850,559852],{"className":559851,"style":41586},[41585],[41588,559853,559854],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,559855],{"d":41597},[34,559857,31829],{"className":559858},[31828],[34,559860,559862],{"className":559861},[31801],[34,559863,559865],{"className":559864,"style":65032},[31805],[34,559866],{},[34,559868,2557],{"className":559869},[31690],[34,559871,46900],{"className":559872},[31685,31686],[34,559874,94628],{"className":559875},[31685,31686],[108,559877,559878],{},"这个积分实际上是条件收敛的。",[108,559880,559881,559884,560045],{},[2627,559882,559883],{},"最简单的证明路径（利用菲涅尔积分性质）：",[34,559885,559887,559926],{"className":559886},[31642],[34,559888,559890],{"className":559889},[31646],[31648,559891,559892],{"xmlns":31650},[31652,559893,559894,559924],{},[31655,559895,559896,559908,559910,559912,559914,559920,559922],{},[38846,559897,559898,559900,559902],{},[31662,559899,546295],{},[31749,559901,459],{},[31655,559903,559904,559906],{},[31662,559905,223],{},[31658,559907,138286],{"mathvariant":38079},[31658,559909,32936],{},[31658,559911,547853],{},[31662,559913,32682],{},[37624,559915,559916,559918],{},[31658,559917,32936],{},[31749,559919,15932],{},[31658,559921,46900],{},[31658,559923,32936],{},[31665,559925,556236],{"encoding":31667},[34,559927,559929],{"className":559928,"ariaHidden":758},[31672],[34,559930,559932,559935,559995,559998,560001,560004,560007,560010,560039,560042],{"className":559931},[31676],[34,559933],{"className":559934,"style":546531},[31680],[34,559936,559938,559941],{"className":559937},[32767],[34,559939,546295],{"className":559940,"style":546331},[32767,131249,156560],[34,559942,559944],{"className":559943},[31792],[34,559945,559947,559987],{"className":559946},[31796,31797],[34,559948,559950,559984],{"className":559949},[31801],[34,559951,559953,559967],{"className":559952,"style":546550},[31805],[34,559954,559955,559958],{"style":546347},[34,559956],{"className":559957,"style":31814},[31813],[34,559959,559961],{"className":559960},[31818,31819,31820,31821],[34,559962,559964],{"className":559963},[31685,31821],[34,559965,459],{"className":559966},[31685,31821],[34,559968,559969,559972],{"style":546362},[34,559970],{"className":559971,"style":31814},[31813],[34,559973,559975],{"className":559974},[31818,31819,31820,31821],[34,559976,559978,559981],{"className":559977},[31685,31821],[34,559979,223],{"className":559980},[31685,31821],[34,559982,138286],{"className":559983},[31685,31821],[34,559985,31829],{"className":559986},[31828],[34,559988,559990],{"className":559989},[31801],[34,559991,559993],{"className":559992,"style":546384},[31805],[34,559994],{},[34,559996],{"className":559997,"style":32123},[31845],[34,559999,32936],{"className":560000},[31685,31686],[34,560002],{"className":560003,"style":32123},[31845],[34,560005,547853],{"className":560006},[32767],[34,560008],{"className":560009,"style":32123},[31845],[34,560011,560013,560016],{"className":560012},[31685],[34,560014,32936],{"className":560015},[31685,31686],[34,560017,560019],{"className":560018},[31792],[34,560020,560022],{"className":560021},[31796],[34,560023,560025],{"className":560024},[31801],[34,560026,560028],{"className":560027,"style":37774},[31805],[34,560029,560030,560033],{"style":37777},[34,560031],{"className":560032,"style":31814},[31813],[34,560034,560036],{"className":560035},[31818,31819,31820,31821],[34,560037,15932],{"className":560038},[31685,31821],[34,560040,46900],{"className":560041},[31685,31686],[34,560043,32936],{"className":560044},[31685,31686]," 是收敛的常数。\n利用积分第二中值定理或分部积分法证明一致性。",[58531,560047],{},[71807,560049,560050,560051,560132,560133,560394,560395,560538],{},"(注：如果这题让你感到困惑，是因为它确实是反常积分里最难的一类“震荡型”。对于初学者，写出换元 ",[34,560052,560054,560076],{"className":560053},[31642],[34,560055,560057],{"className":560056},[31646],[31648,560058,560059],{"xmlns":31650},[31652,560060,560061,560073],{},[31655,560062,560063,560065,560067],{},[31658,560064,33249],{},[31662,560066,202],{},[37624,560068,560069,560071],{},[31658,560070,32936],{},[31749,560072,15932],{},[31665,560074,560075],{"encoding":31667},"t=x^4",[34,560077,560079,560097],{"className":560078,"ariaHidden":758},[31672],[34,560080,560082,560085,560088,560091,560094],{"className":560081},[31676],[34,560083],{"className":560084,"style":33280},[31680],[34,560086,33249],{"className":560087},[31685,31686],[34,560089],{"className":560090,"style":31846},[31845],[34,560092,202],{"className":560093},[31850],[34,560095],{"className":560096,"style":31846},[31845],[34,560098,560100,560103],{"className":560099},[31676],[34,560101],{"className":560102,"style":37774},[31680],[34,560104,560106,560109],{"className":560105},[31685],[34,560107,32936],{"className":560108},[31685,31686],[34,560110,560112],{"className":560111},[31792],[34,560113,560115],{"className":560114},[31796],[34,560116,560118],{"className":560117},[31801],[34,560119,560121],{"className":560120,"style":37774},[31805],[34,560122,560123,560126],{"style":37777},[34,560124],{"className":560125,"style":31814},[31813],[34,560127,560129],{"className":560128},[31818,31819,31820,31821],[34,560130,15932],{"className":560131},[31685,31821]," 转化为 ",[34,560134,560136,560190],{"className":560135},[31642],[34,560137,560139],{"className":560138},[31646],[31648,560140,560141],{"xmlns":31650},[31652,560142,560143,560187],{},[31655,560144,560145,560147,560183,560185],{},[31662,560146,546295],{},[41453,560148,560149,560179],{},[31655,560150,560151,560153,560155,560157,560159,560161,560163,560165,560177],{},[31658,560152,547853],{},[31662,560154,32682],{},[31658,560156,33249],{},[31658,560158,547860],{},[31662,560160,32682],{},[31662,560162,456],{"stretchy":702},[31658,560164,217392],{},[37624,560166,560167,560169],{},[31658,560168,33249],{},[31655,560170,560171,560173,560175],{},[31749,560172,640],{},[31658,560174,38080],{"mathvariant":38079},[31749,560176,15932],{},[31662,560178,2557],{"stretchy":702},[41463,560180,560181],{},[31658,560182,33249],{},[31658,560184,46900],{},[31658,560186,33249],{},[31665,560188,560189],{"encoding":31667},"\\int \\frac{\\sin t \\cos(\\alpha t^{1\u002F4})}{\\sqrt{t}} dt",[34,560191,560193],{"className":560192,"ariaHidden":758},[31672],[34,560194,560196,560199,560202,560205,560388,560391],{"className":560195},[31676],[34,560197],{"className":560198,"style":555103},[31680],[34,560200,546295],{"className":560201,"style":546331},[32767,131249,156560],[34,560203],{"className":560204,"style":32123},[31845],[34,560206,560208,560211,560385],{"className":560207},[31685],[34,560209],{"className":560210},[31782,41488],[34,560212,560214],{"className":560213},[41453],[34,560215,560217,560377],{"className":560216},[31796,31797],[34,560218,560220,560374],{"className":560219},[31801],[34,560221,560223,560281,560289],{"className":560222,"style":555256},[31805],[34,560224,560225,560228],{"style":555259},[34,560226],{"className":560227,"style":41508},[31813],[34,560229,560231],{"className":560230},[31818,31819,31820,31821],[34,560232,560234],{"className":560233},[31685,31821],[34,560235,560237],{"className":560236},[31685,41550,31821],[34,560238,560240,560273],{"className":560239},[31796,31797],[34,560241,560243,560270],{"className":560242},[31801],[34,560244,560246,560258],{"className":560245,"style":555281},[31805],[34,560247,560249,560252],{"className":560248,"style":41565},[41564],[34,560250],{"className":560251,"style":41508},[31813],[34,560253,560255],{"className":560254,"style":41572},[31685,31821],[34,560256,33249],{"className":560257},[31685,31686,31821],[34,560259,560260,560263],{"style":555296},[34,560261],{"className":560262,"style":41508},[31813],[34,560264,560266],{"className":560265,"style":41586},[41585,31821],[41588,560267,560268],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,560269],{"d":41597},[34,560271,31829],{"className":560272},[31828],[34,560274,560276],{"className":560275},[31801],[34,560277,560279],{"className":560278,"style":555316},[31805],[34,560280],{},[34,560282,560283,560286],{"style":41520},[34,560284],{"className":560285,"style":41508},[31813],[34,560287],{"className":560288,"style":41528},[41527],[34,560290,560291,560294],{"style":70529},[34,560292],{"className":560293,"style":41508},[31813],[34,560295,560297],{"className":560296},[31818,31819,31820,31821],[34,560298,560300,560312,560315,560318,560321,560333,560336,560339,560371],{"className":560299},[31685,31821],[34,560301,560303,560306,560309],{"className":560302},[32767,31821],[34,560304,31747],{"className":560305},[31821],[34,560307,31955],{"className":560308},[31821],[34,560310,31660],{"className":560311},[31821],[34,560313],{"className":560314,"style":43430},[31845,31821],[34,560316,33249],{"className":560317},[31685,31686,31821],[34,560319],{"className":560320,"style":43430},[31845,31821],[34,560322,560324,560327,560330],{"className":560323},[32767,31821],[34,560325,31965],{"className":560326},[31821],[34,560328,43383],{"className":560329},[31821],[34,560331,31747],{"className":560332},[31821],[34,560334,456],{"className":560335},[31782,31821],[34,560337,217392],{"className":560338,"style":217496},[31685,31686,31821],[34,560340,560342,560345],{"className":560341},[31685,31821],[34,560343,33249],{"className":560344},[31685,31686,31821],[34,560346,560348],{"className":560347},[31792],[34,560349,560351],{"className":560350},[31796],[34,560352,560354],{"className":560353},[31801],[34,560355,560357],{"className":560356,"style":555395},[31805],[34,560358,560359,560362],{"style":555398},[34,560360],{"className":560361,"style":555402},[31813],[34,560363,560365],{"className":560364},[31818,43410,43411,31821],[34,560366,560368],{"className":560367},[31685,31821],[34,560369,42426],{"className":560370},[31685,31821],[34,560372,2557],{"className":560373},[31690,31821],[34,560375,31829],{"className":560376},[31828],[34,560378,560380],{"className":560379},[31801],[34,560381,560383],{"className":560382,"style":555424},[31805],[34,560384],{},[34,560386],{"className":560387},[31690,41488],[34,560389,46900],{"className":560390},[31685,31686],[34,560392,33249],{"className":560393},[31685,31686],"，然后指出 ",[34,560396,560398,560417],{"className":560397},[31642],[34,560399,560401],{"className":560400},[31646],[31648,560402,560403],{"xmlns":31650},[31652,560404,560405,560415],{},[31655,560406,560407],{},[41453,560408,560409,560411],{},[31749,560410,640],{},[41463,560412,560413],{},[31658,560414,33249],{},[31665,560416,558986],{"encoding":31667},[34,560418,560420],{"className":560419,"ariaHidden":758},[31672],[34,560421,560423,560426],{"className":560422},[31676],[34,560424],{"className":560425,"style":558996},[31680],[34,560427,560429,560432,560535],{"className":560428},[31685],[34,560430],{"className":560431},[31782,41488],[34,560433,560435],{"className":560434},[41453],[34,560436,560438,560527],{"className":560437},[31796,31797],[34,560439,560441,560524],{"className":560440},[31801],[34,560442,560444,560502,560510],{"className":560443,"style":46746},[31805],[34,560445,560446,560449],{"style":555259},[34,560447],{"className":560448,"style":41508},[31813],[34,560450,560452],{"className":560451},[31818,31819,31820,31821],[34,560453,560455],{"className":560454},[31685,31821],[34,560456,560458],{"className":560457},[31685,41550,31821],[34,560459,560461,560494],{"className":560460},[31796,31797],[34,560462,560464,560491],{"className":560463},[31801],[34,560465,560467,560479],{"className":560466,"style":555281},[31805],[34,560468,560470,560473],{"className":560469,"style":41565},[41564],[34,560471],{"className":560472,"style":41508},[31813],[34,560474,560476],{"className":560475,"style":41572},[31685,31821],[34,560477,33249],{"className":560478},[31685,31686,31821],[34,560480,560481,560484],{"style":555296},[34,560482],{"className":560483,"style":41508},[31813],[34,560485,560487],{"className":560486,"style":41586},[41585,31821],[41588,560488,560489],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,560490],{"d":41597},[34,560492,31829],{"className":560493},[31828],[34,560495,560497],{"className":560496},[31801],[34,560498,560500],{"className":560499,"style":555316},[31805],[34,560501],{},[34,560503,560504,560507],{"style":41520},[34,560505],{"className":560506,"style":41508},[31813],[34,560508],{"className":560509,"style":41528},[41527],[34,560511,560512,560515],{"style":46771},[34,560513],{"className":560514,"style":41508},[31813],[34,560516,560518],{"className":560517},[31818,31819,31820,31821],[34,560519,560521],{"className":560520},[31685,31821],[34,560522,640],{"className":560523},[31685,31821],[34,560525,31829],{"className":560526},[31828],[34,560528,560530],{"className":560529},[31801],[34,560531,560533],{"className":560532,"style":555424},[31805],[34,560534],{},[34,560536],{"className":560537},[31690,41488]," 单调趋于0且其余部分有界震荡，通常老师会给分。)",[32188,560540],{},[253,560542,560544],{"id":560543},"p329-t4","p329 T4",[15,560546,560547,560549],{},[2627,560548,214361],{},"\n讨论下列含参变量反常积分的一致收敛性。",[253,560551,560553],{"id":560552},"讲解-1-1",[2627,560554,549100],{},[15,560556,560557],{},[34,560558,560560,560622],{"className":560559},[31642],[34,560561,560563],{"className":560562},[31646],[31648,560564,560565],{"xmlns":31650},[31652,560566,560567,560619],{},[31655,560568,560569,560581,560597,560599,560601,560603,560605,560607,560609,560615,560617],{},[38846,560570,560571,560573,560575],{},[31662,560572,546295],{},[31749,560574,459],{},[31655,560576,560577,560579],{},[31662,560578,223],{},[31658,560580,138286],{"mathvariant":38079},[41453,560582,560583,560593],{},[31655,560584,560585,560587,560589,560591],{},[31658,560586,547860],{},[31662,560588,32682],{},[31658,560590,32936],{},[31658,560592,67764],{},[41463,560594,560595],{},[31658,560596,32936],{},[31658,560598,46900],{},[31658,560600,32936],{},[31662,560602,31960],{"separator":758},[31845,560604],{"width":71082},[31658,560606,67764],{},[31662,560608,43175],{},[31743,560610,560611,560613],{},[31658,560612,67764],{},[31749,560614,459],{},[31662,560616,29622],{},[31749,560618,459],{},[31665,560620,560621],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{\\cos xy}{\\sqrt{x}}dx, \\quad y \\geq y_0 > 0",[34,560623,560625,560855,560910],{"className":560624,"ariaHidden":758},[31672],[34,560626,560628,560631,560691,560694,560828,560831,560834,560837,560840,560843,560846,560849,560852],{"className":560627},[31676],[34,560629],{"className":560630,"style":556926},[31680],[34,560632,560634,560637],{"className":560633},[32767],[34,560635,546295],{"className":560636,"style":546331},[32767,131249,156560],[34,560638,560640],{"className":560639},[31792],[34,560641,560643,560683],{"className":560642},[31796,31797],[34,560644,560646,560680],{"className":560645},[31801],[34,560647,560649,560663],{"className":560648,"style":546550},[31805],[34,560650,560651,560654],{"style":546347},[34,560652],{"className":560653,"style":31814},[31813],[34,560655,560657],{"className":560656},[31818,31819,31820,31821],[34,560658,560660],{"className":560659},[31685,31821],[34,560661,459],{"className":560662},[31685,31821],[34,560664,560665,560668],{"style":546362},[34,560666],{"className":560667,"style":31814},[31813],[34,560669,560671],{"className":560670},[31818,31819,31820,31821],[34,560672,560674,560677],{"className":560673},[31685,31821],[34,560675,223],{"className":560676},[31685,31821],[34,560678,138286],{"className":560679},[31685,31821],[34,560681,31829],{"className":560682},[31828],[34,560684,560686],{"className":560685},[31801],[34,560687,560689],{"className":560688,"style":546384},[31805],[34,560690],{},[34,560692],{"className":560693,"style":32123},[31845],[34,560695,560697,560700,560825],{"className":560696},[31685],[34,560698],{"className":560699},[31782,41488],[34,560701,560703],{"className":560702},[41453],[34,560704,560706,560817],{"className":560705},[31796,31797],[34,560707,560709,560814],{"className":560708},[31801],[34,560710,560712,560774,560782],{"className":560711,"style":548284},[31805],[34,560713,560715,560718],{"style":560714},"top:-2.6259em;",[34,560716],{"className":560717,"style":41508},[31813],[34,560719,560721],{"className":560720},[31818,31819,31820,31821],[34,560722,560724],{"className":560723},[31685,31821],[34,560725,560727],{"className":560726},[31685,41550,31821],[34,560728,560730,560765],{"className":560729},[31796,31797],[34,560731,560733,560762],{"className":560732},[31801],[34,560734,560737,560749],{"className":560735,"style":560736},[31805],"height:0.8059em;",[34,560738,560740,560743],{"className":560739,"style":41565},[41564],[34,560741],{"className":560742,"style":41508},[31813],[34,560744,560746],{"className":560745,"style":41572},[31685,31821],[34,560747,32936],{"className":560748},[31685,31686,31821],[34,560750,560752,560755],{"style":560751},"top:-2.7659em;",[34,560753],{"className":560754,"style":41508},[31813],[34,560756,560758],{"className":560757,"style":41586},[41585,31821],[41588,560759,560760],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,560761],{"d":41597},[34,560763,31829],{"className":560764},[31828],[34,560766,560768],{"className":560767},[31801],[34,560769,560772],{"className":560770,"style":560771},[31805],"height:0.2341em;",[34,560773],{},[34,560775,560776,560779],{"style":41520},[34,560777],{"className":560778,"style":41508},[31813],[34,560780],{"className":560781,"style":41528},[41527],[34,560783,560784,560787],{"style":548368},[34,560785],{"className":560786,"style":41508},[31813],[34,560788,560790],{"className":560789},[31818,31819,31820,31821],[34,560791,560793,560805,560808,560811],{"className":560792},[31685,31821],[34,560794,560796,560799,560802],{"className":560795},[32767,31821],[34,560797,31965],{"className":560798},[31821],[34,560800,43383],{"className":560801},[31821],[34,560803,31747],{"className":560804},[31821],[34,560806],{"className":560807,"style":43430},[31845,31821],[34,560809,32936],{"className":560810},[31685,31686,31821],[34,560812,67764],{"className":560813,"style":32062},[31685,31686,31821],[34,560815,31829],{"className":560816},[31828],[34,560818,560820],{"className":560819},[31801],[34,560821,560823],{"className":560822,"style":555424},[31805],[34,560824],{},[34,560826],{"className":560827},[31690,41488],[34,560829,46900],{"className":560830},[31685,31686],[34,560832,32936],{"className":560833},[31685,31686],[34,560835,31960],{"className":560836},[32119],[34,560838],{"className":560839,"style":128438},[31845],[34,560841],{"className":560842,"style":32123},[31845],[34,560844,67764],{"className":560845,"style":32062},[31685,31686],[34,560847],{"className":560848,"style":31846},[31845],[34,560850,43175],{"className":560851},[31850],[34,560853],{"className":560854,"style":31846},[31845],[34,560856,560858,560861,560901,560904,560907],{"className":560857},[31676],[34,560859],{"className":560860,"style":86652},[31680],[34,560862,560864,560867],{"className":560863},[31685],[34,560865,67764],{"className":560866,"style":32062},[31685,31686],[34,560868,560870],{"className":560869},[31792],[34,560871,560873,560893],{"className":560872},[31796,31797],[34,560874,560876,560890],{"className":560875},[31801],[34,560877,560879],{"className":560878,"style":31806},[31805],[34,560880,560881,560884],{"style":38795},[34,560882],{"className":560883,"style":31814},[31813],[34,560885,560887],{"className":560886},[31818,31819,31820,31821],[34,560888,459],{"className":560889},[31685,31821],[34,560891,31829],{"className":560892},[31828],[34,560894,560896],{"className":560895},[31801],[34,560897,560899],{"className":560898,"style":31836},[31805],[34,560900],{},[34,560902],{"className":560903,"style":31846},[31845],[34,560905,29622],{"className":560906},[31850],[34,560908],{"className":560909,"style":31846},[31845],[34,560911,560913,560916],{"className":560912},[31676],[34,560914],{"className":560915,"style":32617},[31680],[34,560917,459],{"className":560918},[31685],[15,560920,560921,560923,560924],{},[2627,560922,549449],{}," 震荡型，用 Dirichlet。\n",[2627,560925,549629],{},[2230,560927,560928,561658,561864],{},[108,560929,560930,47649,560933,561564,561565,561657],{},[2627,560931,560932],{},"积分部分有界：",[34,560934,560936,561016],{"className":560935},[31642],[34,560937,560939],{"className":560938},[31646],[31648,560940,560941],{"xmlns":31650},[31652,560942,560943,561013],{},[31655,560944,560945,560971,560973,560993,560995,561001,561003],{},[31655,560946,560947,560949,560957,560959,560961,560963,560965,560967,560969],{},[31662,560948,162086],{"fence":758},[38846,560950,560951,560953,560955],{},[31662,560952,546295],{},[31749,560954,459],{},[31658,560956,54151],{},[31658,560958,547860],{},[31662,560960,32682],{},[31658,560962,32936],{},[31658,560964,67764],{},[31658,560966,46900],{},[31658,560968,32936],{},[31662,560970,162086],{"fence":758},[31662,560972,202],{},[31655,560974,560975,560977,560991],{},[31662,560976,162086],{"fence":758},[41453,560978,560979,560989],{},[31655,560980,560981,560983,560985,560987],{},[31658,560982,547853],{},[31662,560984,32682],{},[31658,560986,54151],{},[31658,560988,67764],{},[31658,560990,67764],{},[31662,560992,162086],{"fence":758},[31662,560994,45972],{},[41453,560996,560997,560999],{},[31749,560998,640],{},[31658,561000,67764],{},[31662,561002,45972],{},[41453,561004,561005,561007],{},[31749,561006,640],{},[31743,561008,561009,561011],{},[31658,561010,67764],{},[31749,561012,459],{},[31665,561014,561015],{"encoding":31667},"\\left| \\int_{0}^{A} \\cos xy dx \\right| = \\left| \\frac{\\sin Ay}{y} \\right| \\leq \\frac{1}{y} \\leq \\frac{1}{y_0}",[34,561017,561019,561189,561368,561452],{"className":561018,"ariaHidden":758},[31672],[34,561020,561022,561025,561180,561183,561186],{"className":561021},[31676],[34,561023],{"className":561024,"style":141282},[31680],[34,561026,561028,561065,561122,561125,561128,561131,561134,561137,561140,561143],{"className":561027},[34006],[34,561029,561031],{"className":561030},[31782],[34,561032,561034],{"className":561033},[71281,71282],[34,561035,561037,561057],{"className":561036},[31796,31797],[34,561038,561040,561054],{"className":561039},[31801],[34,561041,561043],{"className":561042,"style":550104},[31805],[34,561044,561045,561048],{"style":71329},[34,561046],{"className":561047,"style":550110},[31813],[34,561049,561050],{"style":550113},[41588,561051,561052],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,561053],{"d":550120},[34,561055,31829],{"className":561056},[31828],[34,561058,561060],{"className":561059},[31801],[34,561061,561063],{"className":561062,"style":550130},[31805],[34,561064],{},[34,561066,561068,561071],{"className":561067},[32767],[34,561069,546295],{"className":561070,"style":546331},[32767,131249,156560],[34,561072,561074],{"className":561073},[31792],[34,561075,561077,561114],{"className":561076},[31796,31797],[34,561078,561080,561111],{"className":561079},[31801],[34,561081,561083,561097],{"className":561082,"style":552657},[31805],[34,561084,561085,561088],{"style":546347},[34,561086],{"className":561087,"style":31814},[31813],[34,561089,561091],{"className":561090},[31818,31819,31820,31821],[34,561092,561094],{"className":561093},[31685,31821],[34,561095,459],{"className":561096},[31685,31821],[34,561098,561099,561102],{"style":546362},[34,561100],{"className":561101,"style":31814},[31813],[34,561103,561105],{"className":561104},[31818,31819,31820,31821],[34,561106,561108],{"className":561107},[31685,31821],[34,561109,54151],{"className":561110},[31685,31686,31821],[34,561112,31829],{"className":561113},[31828],[34,561115,561117],{"className":561116},[31801],[34,561118,561120],{"className":561119,"style":546384},[31805],[34,561121],{},[34,561123],{"className":561124,"style":32123},[31845],[34,561126,547860],{"className":561127},[32767],[34,561129],{"className":561130,"style":32123},[31845],[34,561132,32936],{"className":561133},[31685,31686],[34,561135,67764],{"className":561136,"style":32062},[31685,31686],[34,561138,46900],{"className":561139},[31685,31686],[34,561141,32936],{"className":561142},[31685,31686],[34,561144,561146],{"className":561145},[31690],[34,561147,561149],{"className":561148},[71281,71282],[34,561150,561152,561172],{"className":561151},[31796,31797],[34,561153,561155,561169],{"className":561154},[31801],[34,561156,561158],{"className":561157,"style":550104},[31805],[34,561159,561160,561163],{"style":71329},[34,561161],{"className":561162,"style":550110},[31813],[34,561164,561165],{"style":550113},[41588,561166,561167],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,561168],{"d":550120},[34,561170,31829],{"className":561171},[31828],[34,561173,561175],{"className":561174},[31801],[34,561176,561178],{"className":561177,"style":550130},[31805],[34,561179],{},[34,561181],{"className":561182,"style":31846},[31845],[34,561184,202],{"className":561185},[31850],[34,561187],{"className":561188,"style":31846},[31845],[34,561190,561192,561195,561359,561362,561365],{"className":561191},[31676],[34,561193],{"className":561194,"style":141282},[31680],[34,561196,561198,561235,561322],{"className":561197},[34006],[34,561199,561201],{"className":561200},[31782],[34,561202,561204],{"className":561203},[71281,71282],[34,561205,561207,561227],{"className":561206},[31796,31797],[34,561208,561210,561224],{"className":561209},[31801],[34,561211,561213],{"className":561212,"style":550104},[31805],[34,561214,561215,561218],{"style":71329},[34,561216],{"className":561217,"style":550110},[31813],[34,561219,561220],{"style":550113},[41588,561221,561222],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,561223],{"d":550120},[34,561225,31829],{"className":561226},[31828],[34,561228,561230],{"className":561229},[31801],[34,561231,561233],{"className":561232,"style":550130},[31805],[34,561234],{},[34,561236,561238,561241,561319],{"className":561237},[31685],[34,561239],{"className":561240},[31782,41488],[34,561242,561244],{"className":561243},[41453],[34,561245,561247,561311],{"className":561246},[31796,31797],[34,561248,561250,561308],{"className":561249},[31801],[34,561251,561254,561268,561276],{"className":561252,"style":561253},[31805],"height:0.9244em;",[34,561255,561256,561259],{"style":41504},[34,561257],{"className":561258,"style":41508},[31813],[34,561260,561262],{"className":561261},[31818,31819,31820,31821],[34,561263,561265],{"className":561264},[31685,31821],[34,561266,67764],{"className":561267,"style":32062},[31685,31686,31821],[34,561269,561270,561273],{"style":41520},[34,561271],{"className":561272,"style":41508},[31813],[34,561274],{"className":561275,"style":41528},[41527],[34,561277,561278,561281],{"style":548368},[34,561279],{"className":561280,"style":41508},[31813],[34,561282,561284],{"className":561283},[31818,31819,31820,31821],[34,561285,561287,561299,561302,561305],{"className":561286},[31685,31821],[34,561288,561290,561293,561296],{"className":561289},[32767,31821],[34,561291,31747],{"className":561292},[31821],[34,561294,31955],{"className":561295},[31821],[34,561297,31660],{"className":561298},[31821],[34,561300],{"className":561301,"style":43430},[31845,31821],[34,561303,54151],{"className":561304},[31685,31686,31821],[34,561306,67764],{"className":561307,"style":32062},[31685,31686,31821],[34,561309,31829],{"className":561310},[31828],[34,561312,561314],{"className":561313},[31801],[34,561315,561317],{"className":561316,"style":220191},[31805],[34,561318],{},[34,561320],{"className":561321},[31690,41488],[34,561323,561325],{"className":561324},[31690],[34,561326,561328],{"className":561327},[71281,71282],[34,561329,561331,561351],{"className":561330},[31796,31797],[34,561332,561334,561348],{"className":561333},[31801],[34,561335,561337],{"className":561336,"style":550104},[31805],[34,561338,561339,561342],{"style":71329},[34,561340],{"className":561341,"style":550110},[31813],[34,561343,561344],{"style":550113},[41588,561345,561346],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,561347],{"d":550120},[34,561349,31829],{"className":561350},[31828],[34,561352,561354],{"className":561353},[31801],[34,561355,561357],{"className":561356,"style":550130},[31805],[34,561358],{},[34,561360],{"className":561361,"style":31846},[31845],[34,561363,45972],{"className":561364},[31850],[34,561366],{"className":561367,"style":31846},[31845],[34,561369,561371,561375,561443,561446,561449],{"className":561370},[31676],[34,561372],{"className":561373,"style":561374},[31680],"height:1.3262em;vertical-align:-0.4811em;",[34,561376,561378,561381,561440],{"className":561377},[31685],[34,561379],{"className":561380},[31782,41488],[34,561382,561384],{"className":561383},[41453],[34,561385,561387,561432],{"className":561386},[31796,31797],[34,561388,561390,561429],{"className":561389},[31801],[34,561391,561393,561407,561415],{"className":561392,"style":46746},[31805],[34,561394,561395,561398],{"style":41504},[34,561396],{"className":561397,"style":41508},[31813],[34,561399,561401],{"className":561400},[31818,31819,31820,31821],[34,561402,561404],{"className":561403},[31685,31821],[34,561405,67764],{"className":561406,"style":32062},[31685,31686,31821],[34,561408,561409,561412],{"style":41520},[34,561410],{"className":561411,"style":41508},[31813],[34,561413],{"className":561414,"style":41528},[41527],[34,561416,561417,561420],{"style":46771},[34,561418],{"className":561419,"style":41508},[31813],[34,561421,561423],{"className":561422},[31818,31819,31820,31821],[34,561424,561426],{"className":561425},[31685,31821],[34,561427,640],{"className":561428},[31685,31821],[34,561430,31829],{"className":561431},[31828],[34,561433,561435],{"className":561434},[31801],[34,561436,561438],{"className":561437,"style":220191},[31805],[34,561439],{},[34,561441],{"className":561442},[31690,41488],[34,561444],{"className":561445,"style":31846},[31845],[34,561447,45972],{"className":561448},[31850],[34,561450],{"className":561451,"style":31846},[31845],[34,561453,561455,561458],{"className":561454},[31676],[34,561456],{"className":561457,"style":561374},[31680],[34,561459,561461,561464,561561],{"className":561460},[31685],[34,561462],{"className":561463},[31782,41488],[34,561465,561467],{"className":561466},[41453],[34,561468,561470,561553],{"className":561469},[31796,31797],[34,561471,561473,561550],{"className":561472},[31801],[34,561474,561476,561528,561536],{"className":561475,"style":46746},[31805],[34,561477,561478,561481],{"style":41504},[34,561479],{"className":561480,"style":41508},[31813],[34,561482,561484],{"className":561483},[31818,31819,31820,31821],[34,561485,561487],{"className":561486},[31685,31821],[34,561488,561490,561493],{"className":561489},[31685,31821],[34,561491,67764],{"className":561492,"style":32062},[31685,31686,31821],[34,561494,561496],{"className":561495},[31792],[34,561497,561499,561520],{"className":561498},[31796,31797],[34,561500,561502,561517],{"className":561501},[31801],[34,561503,561505],{"className":561504,"style":318658},[31805],[34,561506,561508,561511],{"style":561507},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[34,561509],{"className":561510,"style":43406},[31813],[34,561512,561514],{"className":561513},[31818,43410,43411,31821],[34,561515,459],{"className":561516},[31685,31821],[34,561518,31829],{"className":561519},[31828],[34,561521,561523],{"className":561522},[31801],[34,561524,561526],{"className":561525,"style":220540},[31805],[34,561527],{},[34,561529,561530,561533],{"style":41520},[34,561531],{"className":561532,"style":41508},[31813],[34,5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",[34,561566,561568,561590],{"className":561567},[31642],[34,561569,561571],{"className":561570},[31646],[31648,561572,561573],{"xmlns":31650},[31652,561574,561575,561587],{},[31655,561576,561577,561579,561581],{},[31658,561578,67764],{},[31662,561580,43175],{},[31743,561582,561583,561585],{},[31658,561584,67764],{},[31749,561586,459],{},[31665,561588,561589],{"encoding":31667},"y \\ge y_0",[34,561591,561593,561611],{"className":561592,"ariaHidden":758},[31672],[34,561594,561596,561599,561602,561605,561608],{"className":561595},[31676],[34,561597],{"className":561598,"style":549750},[31680],[34,561600,67764],{"className":561601,"style":32062},[31685,31686],[34,561603],{"className":561604,"style":31846},[31845],[34,561606,43175],{"className":561607},[31850],[34,561609],{"className":561610,"style":31846},[31845],[34,561612,561614,561617],{"className":561613},[31676],[34,561615],{"className":561616,"style":38774},[31680],[34,561618,561620,561623],{"className":561619},[31685],[34,561621,67764],{"className":561622,"style":32062},[31685,31686],[34,561624,561626],{"className":561625},[31792],[34,561627,561629,561649],{"className":561628},[31796,31797],[34,561630,561632,561646],{"className":561631},[31801],[34,561633,561635],{"className":561634,"style":31806},[31805],[34,561636,561637,561640],{"style":38795},[34,561638],{"className":561639,"style":31814},[31813],[34,561641,561643],{"className":561642},[31818,31819,31820,31821],[34,561644,459],{"className":561645},[31685,31821],[34,561647,31829],{"className":561648},[31828],[34,561650,561652],{"className":561651},[31801],[34,561653,561655],{"className":561654,"style":31836},[31805],[34,561656],{},"，分母有下界，所以整体有上界。这很关键！",[108,561659,561660,47649,561663,561807,561808,561863],{},[2627,561661,561662],{},"单调趋于0部分：",[34,561664,561666,561686],{"className":561665},[31642],[34,561667,561669],{"className":561668},[31646],[31648,561670,561671],{"xmlns":31650},[31652,561672,561673,561683],{},[31655,561674,561675],{},[41453,561676,561677,561679],{},[31749,561678,640],{},[41463,561680,561681],{},[31658,561682,32936],{},[31665,561684,561685],{"encoding":31667},"\\frac{1}{\\sqrt{x}}",[34,561687,561689],{"className":561688,"ariaHidden":758},[31672],[34,561690,561692,561695],{"className":561691},[31676],[34,561693],{"className":561694,"style":558996},[31680],[34,561696,561698,561701,561804],{"className":561697},[31685],[34,561699],{"className":561700},[31782,41488],[34,561702,561704],{"className":561703},[41453],[34,561705,561707,561796],{"className":561706},[31796,31797],[34,561708,561710,561793],{"className":561709},[31801],[34,561711,561713,561771,561779],{"className":561712,"style":46746},[31805],[34,561714,561715,561718],{"style":560714},[34,561716],{"className":561717,"style":41508},[31813],[34,561719,561721],{"className":561720},[31818,31819,31820,31821],[34,561722,561724],{"className":561723},[31685,31821],[34,561725,561727],{"className":561726},[31685,41550,31821],[34,561728,561730,561763],{"className":561729},[31796,31797],[34,561731,561733,561760],{"className":561732},[31801],[34,561734,561736,561748],{"className":561735,"style":560736},[31805],[34,561737,561739,561742],{"className":561738,"style":41565},[41564],[34,561740],{"className":561741,"style":41508},[31813],[34,561743,561745],{"className":561744,"style":41572},[31685,31821],[34,561746,32936],{"className":561747},[31685,31686,31821],[34,561749,561750,561753],{"style":560751},[34,561751],{"className":561752,"style":41508},[31813],[34,561754,561756],{"className":561755,"style":41586},[41585,31821],[41588,561757,561758],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,561759],{"d":41597},[34,561761,31829],{"className":561762},[31828],[34,561764,561766],{"className":561765},[31801],[34,561767,561769],{"className":561768,"style":560771},[31805],[34,561770],{},[34,561772,561773,561776],{"style":41520},[34,561774],{"className":561775,"style":41508},[31813],[34,561777],{"className":561778,"style":41528},[41527],[34,561780,561781,561784],{"style":46771},[34,561782],{"className":561783,"style":41508},[31813],[34,561785,561787],{"className":561786},[31818,31819,31820,31821],[34,561788,561790],{"className":561789},[31685,31821],[34,561791,640],{"className":561792},[31685,31821],[34,561794,31829],{"className":561795},[31828],[34,561797,561799],{"className":561798},[31801],[34,561800,561802],{"className":561801,"style":555424},[31805],[34,561803],{},[34,561805],{"className":561806},[31690,41488]," 当 ",[34,561809,561811,561830],{"className":561810},[31642],[34,561812,561814],{"className":561813},[31646],[31648,561815,561816],{"xmlns":31650},[31652,561817,561818,561828],{},[31655,561819,561820,561822,561824,561826],{},[31658,561821,32936],{},[31662,561823,33239],{},[31662,561825,223],{},[31658,561827,138286],{"mathvariant":38079},[31665,561829,553354],{"encoding":31667},[34,561831,561833,561851],{"className":561832,"ariaHidden":758},[31672],[34,561834,561836,561839,561842,561845,561848],{"className":561835},[31676],[34,561837],{"className":561838,"style":32527},[31680],[34,561840,32936],{"className":561841},[31685,31686],[34,561843],{"className":561844,"style":31846},[31845],[34,561846,33239],{"className":561847},[31850],[34,561849],{"className":561850,"style":31846},[31845],[34,561852,561854,561857,561860],{"className":561853},[31676],[34,561855],{"className":561856,"style":32597},[31680],[34,561858,223],{"className":561859},[31685],[34,561861,138286],{"className":561862},[31685]," 时单调趋于 0。",[108,561865,561866,561868,561869,347],{},[2627,561867,62532],{}," 根据 Dirichlet 判别法，",[2627,561870,548044],{},[253,561872,561874],{"id":561873},"讲解-2-1",[2627,561875,551701],{},[15,561877,561878],{},[34,561879,561881,561935],{"className":561880},[31642],[34,561882,561884],{"className":561883},[31646],[31648,561885,561886],{"xmlns":31650},[31652,561887,561888,561932],{},[31655,561889,561890,561906,561928,561930],{},[38846,561891,561892,561894,561900],{},[31662,561893,546295],{},[31655,561895,561896,561898],{},[31662,561897,32226],{},[31658,561899,138286],{"mathvariant":38079},[31655,561901,561902,561904],{},[31662,561903,223],{},[31658,561905,138286],{"mathvariant":38079},[37624,561907,561908,561910],{},[31658,561909,33244],{},[31655,561911,561912,561914,561916,561918,561920,561922],{},[31662,561913,32226],{},[31662,561915,456],{"stretchy":702},[31658,561917,32936],{},[31662,561919,32226],{},[31658,561921,217392],{},[37624,561923,561924,561926],{},[31662,561925,2557],{"stretchy":702},[31749,561927,768],{},[31658,561929,46900],{},[31658,561931,32936],{},[31665,561933,561934],{"encoding":31667},"\\int_{-\\infty}^{+\\infty} e^{-(x-\\alpha)^2}dx",[34,561936,561938],{"className":561937,"ariaHidden":758},[31672],[34,561939,561941,561945,562009,562012,562086,562089],{"className":561940},[31676],[34,561942],{"className":561943,"style":561944},[31680],"height:1.4011em;vertical-align:-0.4142em;",[34,561946,561948,561951],{"className":561947},[32767],[34,561949,546295],{"className":561950,"style":546331},[32767,131249,156560],[34,561952,561954],{"className":561953},[31792],[34,561955,561957,562000],{"className":561956},[31796,31797],[34,561958,561960,561997],{"className":561959},[31801],[34,561961,561963,561980],{"className":561962,"style":546550},[31805],[34,561964,561965,561968],{"style":546347},[34,561966],{"className":561967,"style":31814},[31813],[34,561969,561971],{"className":561970},[31818,31819,31820,31821],[34,561972,561974,561977],{"className":561973},[31685,31821],[34,561975,32226],{"className":561976},[31685,31821],[34,561978,138286],{"className":561979},[31685,31821],[34,561981,561982,561985],{"style":546362},[34,561983],{"className":561984,"style":31814},[31813],[34,561986,561988],{"className":561987},[31818,31819,31820,31821],[34,561989,561991,561994],{"className":561990},[31685,31821],[34,561992,223],{"className":561993},[31685,31821],[34,561995,138286],{"className":561996},[31685,31821],[34,561998,31829],{"className":561999},[31828],[34,562001,562003],{"className":562002},[31801],[34,562004,562007],{"className":562005,"style":562006},[31805],"height:0.4142em;",[34,562008],{},[34,562010],{"className":562011,"style":32123},[31845],[34,562013,562015,562018],{"className":562014},[31685],[34,562016,33244],{"className":562017},[31685,31686],[34,562019,562021],{"className":562020},[31792],[34,562022,562024],{"className":562023},[31796],[34,562025,562027],{"className":562026},[31801],[34,562028,562031],{"className":562029,"style":562030},[31805],"height:0.9869em;",[34,562032,562033,562036],{"style":37777},[34,562034],{"className":562035,"style":31814},[31813],[34,562037,562039],{"className":562038},[31818,31819,31820,31821],[34,562040,562042,562045,562048,562051,562054,562057],{"className":562041},[31685,31821],[34,562043,32226],{"className":562044},[31685,31821],[34,562046,456],{"className":562047},[31782,31821],[34,562049,32936],{"className":562050},[31685,31686,31821],[34,562052,32226],{"className":562053},[32024,31821],[34,562055,217392],{"className":562056,"style":217496},[31685,31686,31821],[34,562058,562060,562063],{"className":562059},[31690,31821],[34,562061,2557],{"className":562062},[31690,31821],[34,562064,562066],{"className":562065},[31792],[34,562067,562069],{"className":562068},[31796],[34,562070,562072],{"className":562071},[31801],[34,562073,562075],{"className":562074,"style":144123},[31805],[34,562076,562077,562080],{"style":552391},[34,562078],{"className":562079,"style":43406},[31813],[34,562081,562083],{"className":562082},[31818,43410,43411,31821],[34,562084,768],{"className":562085},[31685,31821],[34,562087,46900],{"className":562088},[31685,31686],[34,562090,32936],{"className":562091},[31685,31686],[15,562093,562094,562096,562097,562125,562126,562129,562130,55016,562203,562463,562464,562466],{},[2627,562095,549449],{}," 这是一个高斯积分（钟形曲线），图像是一个波，位置随 ",[34,562098,562100,562113],{"className":562099},[31642],[34,562101,562103],{"className":562102},[31646],[31648,562104,562105],{"xmlns":31650},[31652,562106,562107,562111],{},[31655,562108,562109],{},[31658,562110,217392],{},[31665,562112,218244],{"encoding":31667},[34,562114,562116],{"className":562115,"ariaHidden":758},[31672],[34,562117,562119,562122],{"className":562118},[31676],[34,562120],{"className":562121,"style":32527},[31680],[34,562123,217392],{"className":562124,"style":217496},[31685,31686]," 移动。\n",[2627,562127,562128],{},"积分值计算："," 令 ",[34,562131,562133,562155],{"className":562132},[31642],[34,562134,562136],{"className":562135},[31646],[31648,562137,562138],{"xmlns":31650},[31652,562139,562140,562152],{},[31655,562141,562142,562144,562146,562148,562150],{},[31658,562143,33249],{},[31662,562145,202],{},[31658,562147,32936],{},[31662,562149,32226],{},[31658,562151,217392],{},[31665,562153,562154],{"encoding":31667},"t = x - \\alpha",[34,562156,562158,562176,562194],{"className":562157,"ariaHidden":758},[31672],[34,562159,562161,562164,562167,562170,562173],{"className":562160},[31676],[34,562162],{"className":562163,"style":33280},[31680],[34,562165,33249],{"className":562166},[31685,31686],[34,562168],{"className":562169,"style":31846},[31845],[34,562171,202],{"className":562172},[31850],[34,562174],{"className":562175,"style":31846},[31845],[34,562177,562179,562182,562185,562188,562191],{"className":562178},[31676],[34,562180],{"className":562181,"style":32597},[31680],[34,562183,32936],{"className":562184},[31685,31686],[34,562186],{"className":562187,"style":32604},[31845],[34,562189,32226],{"className":562190},[32024],[34,562192],{"className":562193,"style":32604},[31845],[34,562195,562197,562200],{"className":562196},[31676],[34,562198],{"className":562199,"style":32527},[31680],[34,562201,217392],{"className":562202,"style":217496},[31685,31686],[34,562204,562206,562258],{"className":562205},[31642],[34,562207,562209],{"className":562208},[31646],[31648,562210,562211],{"xmlns":31650},[31652,562212,562213,562255],{},[31655,562214,562215,562231,562245,562247,562249,562251],{},[38846,562216,562217,562219,562225],{},[31662,562218,546295],{},[31655,562220,562221,562223],{},[31662,562222,32226],{},[31658,562224,138286],{"mathvariant":38079},[31655,562226,562227,562229],{},[31662,562228,223],{},[31658,562230,138286],{"mathvariant":38079},[37624,562232,562233,562235],{},[31658,562234,33244],{},[31655,562236,562237,562239],{},[31662,562238,32226],{},[37624,562240,562241,562243],{},[31658,562242,33249],{},[31749,562244,768],{},[31658,562246,46900],{},[31658,562248,33249],{},[31662,562250,202],{},[41463,562252,562253],{},[31658,562254,58442],{},[31665,562256,562257],{"encoding":31667},"\\int_{-\\infty}^{+\\infty} e^{-t^2} dt = \\sqrt{\\pi}",[34,562259,562261,562409],{"className":562260,"ariaHidden":758},[31672],[34,562262,562264,562267,562330,562333,562394,562397,562400,562403,562406],{"className":562263},[31676],[34,562265],{"className":562266,"style":561944},[31680],[34,562268,562270,562273],{"className":562269},[32767],[34,562271,546295],{"className":562272,"style":546331},[32767,131249,156560],[34,562274,562276],{"className":562275},[31792],[34,562277,562279,562322],{"className":562278},[31796,31797],[34,562280,562282,562319],{"className":562281},[31801],[34,562283,562285,562302],{"className":562284,"style":546550},[31805],[34,562286,562287,562290],{"style":546347},[34,562288],{"className":562289,"style":31814},[31813],[34,562291,562293],{"className":562292},[31818,31819,31820,31821],[34,562294,562296,562299],{"className":562295},[31685,31821],[34,562297,32226],{"className":562298},[31685,31821],[34,562300,138286],{"className":562301},[31685,31821],[34,562303,562304,562307],{"style":546362},[34,562305],{"className":562306,"style":31814},[31813],[34,562308,562310],{"className":562309},[31818,31819,31820,31821],[34,562311,562313,562316],{"className":562312},[31685,31821],[34,562314,223],{"className":562315},[31685,31821],[34,562317,138286],{"className":562318},[31685,31821],[34,562320,31829],{"className":562321},[31828],[34,562323,562325],{"className":562324},[31801],[34,562326,562328],{"className":562327,"style":562006},[31805],[34,562329],{},[34,562331],{"className":562332,"style":32123},[31845],[34,562334,562336,562339],{"className":562335},[31685],[34,562337,33244],{"className":562338},[31685,31686],[34,562340,562342],{"className":562341},[31792],[34,562343,562345],{"className":562344},[31796],[34,562346,562348],{"className":562347},[31801],[34,562349,562351],{"className":562350,"style":562030},[31805],[34,562352,562353,562356],{"style":37777},[34,562354],{"className":562355,"style":31814},[31813],[34,562357,562359],{"className":562358},[31818,31819,31820,31821],[34,562360,562362,562365],{"className":562361},[31685,31821],[34,562363,32226],{"className":562364},[31685,31821],[34,562366,562368,562371],{"className":562367},[31685,31821],[34,562369,33249],{"className":562370},[31685,31686,31821],[34,562372,562374],{"className":562373},[31792],[34,562375,562377],{"className":562376},[31796],[34,562378,562380],{"className":562379},[31801],[34,562381,562383],{"className":562382,"style":144123},[31805],[34,562384,562385,562388],{"style":552391},[34,562386],{"className":562387,"style":43406},[31813],[34,562389,562391],{"className":562390},[31818,43410,43411,31821],[34,562392,768],{"className":562393},[31685,31821],[34,562395,46900],{"className":562396},[31685,31686],[34,562398,33249],{"className":562399},[31685,31686],[34,562401],{"className":562402,"style":31846},[31845],[34,562404,202],{"className":562405},[31850],[34,562407],{"className":562408,"style":31846},[31845],[34,562410,562412,562416],{"className":562411},[31676],[34,562413],{"className":562414,"style":562415},[31680],"height:1.04em;vertical-align:-0.2397em;",[34,562417,562419],{"className":562418},[31685,41550],[34,562420,562422,562455],{"className":562421},[31796,31797],[34,562423,562425,562452],{"className":562424},[31801],[34,562426,562428,562440],{"className":562427,"style":64997},[31805],[34,562429,562431,562434],{"className":562430,"style":41565},[41564],[34,562432],{"className":562433,"style":41508},[31813],[34,562435,562437],{"className":562436,"style":41572},[31685],[34,562438,58442],{"className":562439,"style":32062},[31685,31686],[34,562441,562442,562445],{"style":65012},[34,562443],{"className":562444,"style":41508},[31813],[34,562446,562448],{"className":562447,"style":41586},[41585],[41588,562449,562450],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,562451],{"d":41597},[34,562453,31829],{"className":562454},[31828],[34,562456,562458],{"className":562457},[31801],[34,562459,562461],{"className":562460,"style":65032},[31805],[34,562462],{},"。积分为常数，当然收敛。\n问题是",[2627,562465,548044],{},"吗？",[15,562468,562469],{},[2627,562470,562471,562472,562545],{},"(i) ",[34,562473,562475,562497],{"className":562474},[31642],[34,562476,562478],{"className":562477},[31646],[31648,562479,562480],{"xmlns":31650},[31652,562481,562482,562494],{},[31655,562483,562484,562486,562488,562490,562492],{},[31658,562485,111],{},[31662,562487,932],{},[31658,562489,217392],{},[31662,562491,932],{},[31658,562493,1697],{},[31665,562495,562496],{"encoding":31667},"a \u003C \\alpha \u003C b",[34,562498,562500,562518,562536],{"className":562499,"ariaHidden":758},[31672],[34,562501,562503,562506,562509,562512,562515],{"className":562502},[31676],[34,562504],{"className":562505,"style":34812},[31680],[34,562507,111],{"className":562508},[31685,31686],[34,562510],{"className":562511,"style":31846},[31845],[34,562513,932],{"className":562514},[31850],[34,562516],{"className":562517,"style":31846},[31845],[34,562519,562521,562524,562527,562530,562533],{"className":562520},[31676],[34,562522],{"className":562523,"style":34812},[31680],[34,562525,217392],{"className":562526,"style":217496},[31685,31686],[34,562528],{"className":562529,"style":31846},[31845],[34,562531,932],{"className":562532},[31850],[34,562534],{"className":562535,"style":31846},[31845],[34,562537,562539,562542],{"className":562538},[31676],[34,562540],{"className":562541,"style":31681},[31680],[34,562543,1697],{"className":562544},[31685,31686]," (有限区间)",[105,562547,562548,562554],{},[108,562549,562550,562553],{},[2627,562551,562552],{},"判断："," 一致收敛。",[108,562555,562556,562559,562560,46528,562588,562679,562680,562708,562709,562737,562738,562766,562767,562806,562807,562940],{},[2627,562557,562558],{},"理由："," 我们可以截断。对于非常大的 ",[34,562561,562563,562576],{"className":562562},[31642],[34,562564,562566],{"className":562565},[31646],[31648,562567,562568],{"xmlns":31650},[31652,562569,562570,562574],{},[31655,562571,562572],{},[31658,562573,32936],{},[31665,562575,32936],{"encoding":31667},[34,562577,562579],{"className":562578,"ariaHidden":758},[31672],[34,562580,562582,562585],{"className":562581},[31676],[34,562583],{"className":562584,"style":32527},[31680],[34,562586,32936],{"className":562587},[31685,31686],[34,562589,562591,562617],{"className":562590},[31642],[34,562592,562594],{"className":562593},[31646],[31648,562595,562596],{"xmlns":31650},[31652,562597,562598,562614],{},[31655,562599,562600,562602,562604,562606,562608],{},[31662,562601,456],{"stretchy":702},[31658,562603,32936],{},[31662,562605,32226],{},[31658,562607,217392],{},[37624,562609,562610,562612],{},[31662,562611,2557],{"stretchy":702},[31749,562613,768],{},[31665,562615,562616],{"encoding":31667},"(x-\\alpha)^2",[34,562618,562620,562641],{"className":562619,"ariaHidden":758},[31672],[34,562621,562623,562626,562629,562632,562635,562638],{"className":562622},[31676],[34,562624],{"className":562625,"style":31774},[31680],[34,562627,456],{"className":562628},[31782],[34,562630,32936],{"className":562631},[31685,31686],[34,562633],{"className":562634,"style":32604},[31845],[34,562636,32226],{"className":562637},[32024],[34,562639],{"className":562640,"style":32604},[31845],[34,562642,562644,562647,562650],{"className":562643},[31676],[34,562645],{"className":562646,"style":37755},[31680],[34,562648,217392],{"className":562649,"style":217496},[31685,31686],[34,562651,562653,562656],{"className":562652},[31690],[34,562654,2557],{"className":562655},[31690],[34,562657,562659],{"className":562658},[31792],[34,562660,562662],{"className":562661},[31796],[34,562663,562665],{"className":562664},[31801],[34,562666,562668],{"className":562667,"style":37774},[31805],[34,562669,562670,562673],{"style":37777},[34,562671],{"className":562672,"style":31814},[31813],[34,562674,562676],{"className":562675},[31818,31819,31820,31821],[34,562677,768],{"className":562678},[31685,31821]," 会因为 ",[34,562681,562683,562696],{"className":562682},[31642],[34,562684,562686],{"className":562685},[31646],[31648,562687,562688],{"xmlns":31650},[31652,562689,562690,562694],{},[31655,562691,562692],{},[31658,562693,32936],{},[31665,562695,32936],{"encoding":31667},[34,562697,562699],{"className":562698,"ariaHidden":758},[31672],[34,562700,562702,562705],{"className":562701},[31676],[34,562703],{"className":562704,"style":32527},[31680],[34,562706,32936],{"className":562707},[31685,31686]," 远大于 ",[34,562710,562712,562725],{"className":562711},[31642],[34,562713,562715],{"className":562714},[31646],[31648,562716,562717],{"xmlns":31650},[31652,562718,562719,562723],{},[31655,562720,562721],{},[31658,562722,1697],{},[31665,562724,1697],{"encoding":31667},[34,562726,562728],{"className":562727,"ariaHidden":758},[31672],[34,562729,562731,562734],{"className":562730},[31676],[34,562732],{"className":562733,"style":31681},[31680],[34,562735,1697],{"className":562736},[31685,31686]," 而变得很大。\n我们可以用 ",[34,562739,562741,562754],{"className":562740},[31642],[34,562742,562744],{"className":562743},[31646],[31648,562745,562746],{"xmlns":31650},[31652,562747,562748,562752],{},[31655,562749,562750],{},[31658,562751,98360],{},[31665,562753,98360],{"encoding":31667},[34,562755,562757],{"className":562756,"ariaHidden":758},[31672],[34,562758,562760,562763],{"className":562759},[31676],[34,562761],{"className":562762,"style":33218},[31680],[34,562764,98360],{"className":562765,"style":33389},[31685,31686],"-判别法。\n当 ",[34,562768,562770,562788],{"className":562769},[31642],[34,562771,562773],{"className":562772},[31646],[31648,562774,562775],{"xmlns":31650},[31652,562776,562777,562785],{},[31655,562778,562779,562781,562783],{},[31658,562780,162086],{"mathvariant":38079},[31658,562782,32936],{},[31658,562784,162086],{"mathvariant":38079},[31665,562786,562787],{"encoding":31667},"|x|",[34,562789,562791],{"className":562790,"ariaHidden":758},[31672],[34,562792,562794,562797,562800,562803],{"className":562793},[31676],[34,562795],{"className":562796,"style":31774},[31680],[34,562798,162086],{"className":562799},[31685],[34,562801,32936],{"className":562802},[31685,31686],[34,562804,162086],{"className":562805},[31685]," 很大（比如 ",[34,562808,562810,562856],{"className":562809},[31642],[34,562811,562813],{"className":562812},[31646],[31648,562814,562815],{"xmlns":31650},[31652,562816,562817,562853],{},[31655,562818,562819,562821,562823,562825,562827,562829,562831,562833,562835,562837,562839,562841,562843,562845,562847,562849,562851],{},[31658,562820,162086],{"mathvariant":38079},[31658,562822,32936],{},[31658,562824,162086],{"mathvariant":38079},[31662,562826,29622],{},[31658,562828,32679],{},[31662,562830,32682],{},[31662,562832,456],{"stretchy":702},[31658,562834,162086],{"mathvariant":38079},[31658,562836,111],{},[31658,562838,162086],{"mathvariant":38079},[31662,562840,31960],{"separator":758},[31658,562842,162086],{"mathvariant":38079},[31658,562844,1697],{},[31658,562846,162086],{"mathvariant":38079},[31662,562848,2557],{"stretchy":702},[31662,562850,223],{},[31658,562852,231507],{},[31665,562854,562855],{"encoding":31667},"|x| > \\max(|a|, |b|) + \\delta",[34,562857,562859,562883,562931],{"className":562858,"ariaHidden":758},[31672],[34,562860,562862,562865,562868,562871,562874,562877,562880],{"className":562861},[31676],[34,562863],{"className":562864,"style":31774},[31680],[34,562866,162086],{"className":562867},[31685],[34,562869,32936],{"className":562870},[31685,31686],[34,562872,162086],{"className":562873},[31685],[34,562875],{"className":562876,"style":31846},[31845],[34,562878,29622],{"className":562879},[31850],[34,562881],{"className":562882,"style":31846},[31845],[34,562884,562886,562889,562892,562895,562898,562901,562904,562907,562910,562913,562916,562919,562922,562925,562928],{"className":562885},[31676],[34,562887],{"className":562888,"style":31774},[31680],[34,562890,32679],{"className":562891},[32767],[34,562893,456],{"className":562894},[31782],[34,562896,162086],{"className":562897},[31685],[34,562899,111],{"className":562900},[31685,31686],[34,562902,162086],{"className":562903},[31685],[34,562905,31960],{"className":562906},[32119],[34,562908],{"className":562909,"style":32123},[31845],[34,562911,162086],{"className":562912},[31685],[34,562914,1697],{"className":562915},[31685,31686],[34,562917,162086],{"className":562918},[31685],[34,562920,2557],{"className":562921},[31690],[34,562923],{"className":562924,"style":32604},[31845],[34,562926,223],{"className":562927},[32024],[34,562929],{"className":562930,"style":32604},[31845],[34,562932,562934,562937],{"className":562933},[31676],[34,562935],{"className":562936,"style":31681},[31680],[34,562938,231507],{"className":562939,"style":231523},[31685,31686],"）时，我们可以找到控制函数。\n或者更直观地：在有限范围内移动，尾巴总是可以被同时切断。",[15,562942,562943],{},[2627,562944,523136,562945,563028],{},[34,562946,562948,562974],{"className":562947},[31642],[34,562949,562951],{"className":562950},[31646],[31648,562952,562953],{"xmlns":31650},[31652,562954,562955,562971],{},[31655,562956,562957,562959,562961,562963,562965,562967,562969],{},[31662,562958,32226],{},[31658,562960,138286],{"mathvariant":38079},[31662,562962,932],{},[31658,562964,217392],{},[31662,562966,932],{},[31662,562968,223],{},[31658,562970,138286],{"mathvariant":38079},[31665,562972,562973],{"encoding":31667},"-\\infty \u003C \\alpha \u003C +\\infty",[34,562975,562977,562998,563016],{"className":562976,"ariaHidden":758},[31672],[34,562978,562980,562983,562986,562989,562992,562995],{"className":562979},[31676],[34,562981],{"className":562982,"style":32597},[31680],[34,562984,32226],{"className":562985},[31685],[34,562987,138286],{"className":562988},[31685],[34,562990],{"className":562991,"style":31846},[31845],[34,562993,932],{"className":562994},[31850],[34,562996],{"className":562997,"style":31846},[31845],[34,562999,563001,563004,563007,563010,563013],{"className":563000},[31676],[34,563002],{"className":563003,"style":34812},[31680],[34,563005,217392],{"className":563006,"style":217496},[31685,31686],[34,563008],{"className":563009,"style":31846},[31845],[34,563011,932],{"className":563012},[31850],[34,563014],{"className":563015,"style":31846},[31845],[34,563017,563019,563022,563025],{"className":563018},[31676],[34,563020],{"className":563021,"style":32597},[31680],[34,563023,223],{"className":563024},[31685],[34,563026,138286],{"className":563027},[31685]," (全实数轴)",[105,563030,563031,563038],{},[108,563032,563033,47649,563035,347],{},[2627,563034,562552],{},[2627,563036,563037],{},"不一致收敛",[108,563039,563040,563043,563044,563101,563102,563130,563131,563182,563183,563211,563212,563240],{},[2627,563041,563042],{},"直观理由："," 无论你取多大的区间 ",[34,563045,563047,563071],{"className":563046},[31642],[34,563048,563050],{"className":563049},[31646],[31648,563051,563052],{"xmlns":31650},[31652,563053,563054,563068],{},[31655,563055,563056,563058,563060,563062,563064,563066],{},[31662,563057,15972],{"stretchy":702},[31662,563059,32226],{},[31658,563061,54151],{},[31662,563063,31960],{"separator":758},[31658,563065,54151],{},[31662,563067,32712],{"stretchy":702},[31665,563069,563070],{"encoding":31667},"[-A, A]",[34,563072,563074],{"className":563073,"ariaHidden":758},[31672],[34,563075,563077,563080,563083,563086,563089,563092,563095,563098],{"className":563076},[31676],[34,563078],{"className":563079,"style":31774},[31680],[34,563081,15972],{"className":563082},[31782],[34,563084,32226],{"className":563085},[31685],[34,563087,54151],{"className":563088},[31685,31686],[34,563090,31960],{"className":563091},[32119],[34,563093],{"className":563094,"style":32123},[31845],[34,563096,54151],{"className":563097},[31685,31686],[34,563099,32712],{"className":563100},[31690],"，我只要把 ",[34,563103,563105,563118],{"className":563104},[31642],[34,563106,563108],{"className":563107},[31646],[31648,563109,563110],{"xmlns":31650},[31652,563111,563112,563116],{},[31655,563113,563114],{},[31658,563115,217392],{},[31665,563117,218244],{"encoding":31667},[34,563119,563121],{"className":563120,"ariaHidden":758},[31672],[34,563122,563124,563127],{"className":563123},[31676],[34,563125],{"className":563126,"style":32527},[31680],[34,563128,217392],{"className":563129,"style":217496},[31685,31686]," 移到非常远的地方（比如 ",[34,563132,563134,563152],{"className":563133},[31642],[34,563135,563137],{"className":563136},[31646],[31648,563138,563139],{"xmlns":31650},[31652,563140,563141,563149],{},[31655,563142,563143,563145,563147],{},[31658,563144,54151],{},[31662,563146,223],{},[31749,563148,162173],{},[31665,563150,563151],{"encoding":31667},"A + 100",[34,563153,563155,563173],{"className":563154,"ariaHidden":758},[31672],[34,563156,563158,563161,563164,563167,563170],{"className":563157},[31676],[34,563159],{"className":563160,"style":34944},[31680],[34,563162,54151],{"className":563163},[31685,31686],[34,563165],{"className":563166,"style":32604},[31845],[34,563168,223],{"className":563169},[32024],[34,563171],{"className":563172,"style":32604},[31845],[34,563174,563176,563179],{"className":563175},[31676],[34,563177],{"className":563178,"style":32617},[31680],[34,563180,162173],{"className":563181},[31685],"），这个钟形波峰就跑出去了，积分的主要部分就跑出去了，剩下的“尾巴”积分值就很大。也就是无法找到一个 ",[34,563184,563186,563199],{"className":563185},[31642],[34,563187,563189],{"className":563188},[31646],[31648,563190,563191],{"xmlns":31650},[31652,563192,563193,563197],{},[31655,563194,563195],{},[31658,563196,54151],{},[31665,563198,54151],{"encoding":31667},[34,563200,563202],{"className":563201,"ariaHidden":758},[31672],[34,563203,563205,563208],{"className":563204},[31676],[34,563206],{"className":563207,"style":33218},[31680],[34,563209,54151],{"className":563210},[31685,31686],"，让所有 ",[34,563213,563215,563228],{"className":563214},[31642],[34,563216,563218],{"className":563217},[31646],[31648,563219,563220],{"xmlns":31650},[31652,563221,563222,563226],{},[31655,563223,563224],{},[31658,563225,217392],{},[31665,563227,218244],{"encoding":31667},[34,563229,563231],{"className":563230,"ariaHidden":758},[31672],[34,563232,563234,563237],{"className":563233},[31676],[34,563235],{"className":563236,"style":32527},[31680],[34,563238,217392],{"className":563239,"style":217496},[31685,31686]," 对应的尾部积分都很小。",[253,563242,563244],{"id":563243},"讲解-3-1",[2627,563245,553868],{},[15,563247,563248,563455,563456,563459,563460,31901],{},[34,563249,563251,563300],{"className":563250},[31642],[34,563252,563254],{"className":563253},[31646],[31648,563255,563256],{"xmlns":31650},[31652,563257,563258,563297],{},[31655,563259,563260,563268,563280,563291,563293,563295],{},[38846,563261,563262,563264,563266],{},[31662,563263,546295],{},[31749,563265,459],{},[31749,563267,640],{},[37624,563269,563270,563272],{},[31658,563271,32936],{},[31655,563273,563274,563276,563278],{},[31658,563275,15],{},[31662,563277,32226],{},[31749,563279,640],{},[37624,563281,563282,563289],{},[31655,563283,563284,563287],{},[31658,563285,563286],{},"ln",[31662,563288,32682],{},[31749,563290,768],{},[31658,563292,32936],{},[31658,563294,46900],{},[31658,563296,32936],{},[31665,563298,563299],{"encoding":31667},"\\int_{0}^{1} x^{p-1}\\ln^2 xdx",[34,563301,563303],{"className":563302,"ariaHidden":758},[31672],[34,563304,563306,563310,563368,563371,563409,563412,563443,563446,563449,563452],{"className":563305},[31676],[34,563307],{"className":563308,"style":563309},[31680],"height:1.3648em;vertical-align:-0.3558em;",[34,563311,563313,563316],{"className":563312},[32767],[34,563314,546295],{"className":563315,"style":546331},[32767,131249,156560],[34,563317,563319],{"className":563318},[31792],[34,563320,563322,563360],{"className":563321},[31796,31797],[34,563323,563325,563357],{"className":563324},[31801],[34,563326,563329,563343],{"className":563327,"style":563328},[31805],"height:1.009em;",[34,563330,563331,563334],{"style":546347},[34,563332],{"className":563333,"style":31814},[31813],[34,563335,563337],{"className":563336},[31818,31819,31820,31821],[34,563338,563340],{"className":563339},[31685,31821],[34,563341,459],{"className":563342},[31685,31821],[34,563344,563345,563348],{"style":546362},[34,563346],{"className":563347,"style":31814},[31813],[34,563349,563351],{"className":563350},[31818,31819,31820,31821],[34,563352,563354],{"className":563353},[31685,31821],[34,563355,640],{"className":563356},[31685,31821],[34,563358,31829],{"className":563359},[31828],[34,563361,563363],{"className":563362},[31801],[34,563364,563366],{"className":563365,"style":546384},[31805],[34,563367],{},[34,563369],{"className":563370,"style":32123},[31845],[34,563372,563374,563377],{"className":563373},[31685],[34,563375,32936],{"className":563376},[31685,31686],[34,563378,563380],{"className":563379},[31792],[34,563381,563383],{"className":563382},[31796],[34,563384,563386],{"className":563385},[31801],[34,563387,563389],{"className":563388,"style":37774},[31805],[34,563390,563391,563394],{"style":37777},[34,563392],{"className":563393,"style":31814},[31813],[34,563395,563397],{"className":563396},[31818,31819,31820,31821],[34,563398,563400,563403,563406],{"className":563399},[31685,31821],[34,563401,15],{"className":563402},[31685,31686,31821],[34,563404,32226],{"className":563405},[32024,31821],[34,563407,640],{"className":563408},[31685,31821],[34,563410],{"className":563411,"style":32123},[31845],[34,563413,563415,563418],{"className":563414},[32767],[34,563416,563286],{"className":563417},[32767],[34,563419,563421],{"className":563420},[31792],[34,563422,563424],{"className":563423},[31796],[34,563425,563427],{"className":563426},[31801],[34,563428,563431],{"className":563429,"style":563430},[31805],"height:0.8984em;",[34,563432,563434,563437],{"style":563433},"top:-3.1473em;margin-right:0.05em;",[34,563435],{"className":563436,"style":31814},[31813],[34,563438,563440],{"className":563439},[31818,31819,31820,31821],[34,563441,768],{"className":563442},[31685,31821],[34,563444],{"className":563445,"style":32123},[31845],[34,563447,32936],{"className":563448},[31685,31686],[34,563450,46900],{"className":563451},[31685,31686],[34,563453,32936],{"className":563454},[31685,31686],"\n这是一个",[2627,563457,563458],{},"瑕积分","（奇点在 ",[34,563461,563463,563481],{"className":563462},[31642],[34,563464,563466],{"className":563465},[31646],[31648,563467,563468],{"xmlns":31650},[31652,563469,563470,563478],{},[31655,563471,563472,563474,563476],{},[31658,563473,32936],{},[31662,563475,202],{},[31749,563477,459],{},[31665,563479,563480],{"encoding":31667},"x=0",[34,563482,563484,563502],{"className":563483,"ariaHidden":758},[31672],[34,563485,563487,563490,563493,563496,563499],{"className":563486},[31676],[34,563488],{"className":563489,"style":32527},[31680],[34,563491,32936],{"className":563492},[31685,31686],[34,563494],{"className":563495,"style":31846},[31845],[34,563497,202],{"className":563498},[31850],[34,563500],{"className":563501,"style":31846},[31845],[34,563503,563505,563508],{"className":563504},[31676],[34,563506],{"className":563507,"style":32617},[31680],[34,563509,459],{"className":563510},[31685],[15,563512,563513],{},[2627,563514,562471,563515],{},[34,563516,563518,563544],{"className":563517},[31642],[34,563519,563521],{"className":563520},[31646],[31648,563522,563523],{"xmlns":31650},[31652,563524,563525,563541],{},[31655,563526,563527,563529,563531,563537,563539],{},[31658,563528,15],{},[31662,563530,43175],{},[31743,563532,563533,563535],{},[31658,563534,15],{},[31749,563536,459],{},[31662,563538,29622],{},[31749,563540,459],{},[31665,563542,563543],{"encoding":31667},"p \\geq p_0 > 0",[34,563545,563547,563565,563620],{"className":563546,"ariaHidden":758},[31672],[34,563548,563550,563553,563556,563559,563562],{"className":563549},[31676],[34,563551],{"className":563552,"style":549750},[31680],[34,563554,15],{"className":563555},[31685,31686],[34,563557],{"className":563558,"style":31846},[31845],[34,563560,43175],{"className":563561},[31850],[34,563563],{"className":563564,"style":31846},[31845],[34,563566,563568,563571,563611,563614,563617],{"className":563567},[31676],[34,563569],{"className":563570,"style":86652},[31680],[34,563572,563574,563577],{"className":563573},[31685],[34,563575,15],{"className":563576},[31685,31686],[34,563578,563580],{"className":563579},[31792],[34,563581,563583,563603],{"className":563582},[31796,31797],[34,563584,563586,563600],{"className":563585},[31801],[34,563587,563589],{"className":563588,"style":31806},[31805],[34,563590,563591,563594],{"style":31809},[34,563592],{"className":563593,"style":31814},[31813],[34,563595,563597],{"className":563596},[31818,31819,31820,31821],[34,563598,459],{"className":563599},[31685,31821],[34,563601,31829],{"className":563602},[31828],[34,563604,563606],{"className":563605},[31801],[34,563607,563609],{"className":563608,"style":31836},[31805],[34,563610],{},[34,563612],{"className":563613,"style":31846},[31845],[34,563615,29622],{"className":563616},[31850],[34,563618],{"className":563619,"style":31846},[31845],[34,563621,563623,563626],{"className":563622},[31676],[34,563624],{"className":563625,"style":32617},[31680],[34,563627,459],{"className":563628},[31685],[105,563630,563631,564229,564527,564808],{},[108,563632,563633,47649,563635,44689,563925,93535,563999,13977,564091,347],{},[2627,563634,549449],{},[34,563636,563638,563698],{"className":563637},[31642],[34,563639,563641],{"className":563640},[31646],[31648,563642,563643],{"xmlns":31650},[31652,563644,563645,563695],{},[31655,563646,563647,563659,563661,563677,563679],{},[37624,563648,563649,563651],{},[31658,563650,32936],{},[31655,563652,563653,563655,563657],{},[31658,563654,15],{},[31662,563656,32226],{},[31749,563658,640],{},[31662,563660,202],{},[37624,563662,563663,563665],{},[31658,563664,32936],{},[31655,563666,563667,563669,563671],{},[31658,563668,15],{},[31662,563670,32226],{},[31743,563672,563673,563675],{},[31658,563674,15],{},[31749,563676,459],{},[31662,563678,71191],{},[37624,563680,563681,563683],{},[31658,563682,32936],{},[31655,563684,563685,563691,563693],{},[31743,563686,563687,563689],{},[31658,563688,15],{},[31749,563690,459],{},[31662,563692,32226],{},[31749,563694,640],{},[31665,563696,563697],{"encoding":31667},"x^{p-1} = x^{p-p_0} \\cdot 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x",[34,564308,564310,564408],{"className":564309,"ariaHidden":758},[31672],[34,564311,564313,564317,564320,564358,564361,564390,564393,564396,564399,564402,564405],{"className":564312},[31676],[34,564314],{"className":564315,"style":564316},[31680],"height:1.1484em;vertical-align:-0.25em;",[34,564318,162086],{"className":564319},[31685],[34,564321,564323,564326],{"className":564322},[31685],[34,564324,32936],{"className":564325},[31685,31686],[34,564327,564329],{"className":564328},[31792],[34,564330,564332],{"className":564331},[31796],[34,564333,564335],{"className":564334},[31801],[34,564336,564338],{"className":564337,"style":37774},[31805],[34,564339,564340,564343],{"style":37777},[34,564341],{"className":564342,"style":31814},[31813],[34,564344,564346],{"className":564345},[31818,31819,31820,31821],[34,564347,564349,564352,564355],{"className":564348},[31685,31821],[34,564350,15],{"className":564351},[31685,31686,31821],[34,564353,32226],{"className":564354},[32024,31821],[34,564356,640],{"className":564357},[31685,31821],[34,564359],{"className":564360,"style":32123},[31845],[34,564362,564364,564367],{"className":564363},[32767],[34,564365,563286],{"className":564366},[32767],[34,564368,564370],{"className":564369},[31792],[34,564371,564373],{"className":564372},[31796],[34,564374,564376],{"className":564375},[31801],[34,564377,564379],{"className":564378,"style":563430},[31805],[34,564380,564381,564384],{"style":563433},[34,564382],{"className":564383,"style":31814},[31813],[34,564385,564387],{"className":564386},[31818,31819,31820,31821],[34,564388,768],{"className":564389},[31685,31821],[34,564391],{"className":564392,"style":32123},[31845],[34,564394,32936],{"className":564395},[31685,31686],[34,564397,162086],{"className":564398},[31685],[34,564400],{"className":564401,"style":31846},[31845],[34,564403,45972],{"className":564404},[31850],[34,564406],{"className":564407,"style":31846},[31845],[34,564409,564411,564414,564489,564492,564521,564524],{"className":564410},[31676],[34,564412],{"className":564413,"style":563430},[31680],[34,564415,564417,564420],{"className":564416},[31685],[34,564418,32936],{"className":564419},[31685,31686],[34,564421,564423],{"className":564422},[31792],[34,564424,564426],{"className":564425},[31796],[34,564427,564429],{"className":564428},[31801],[34,564430,564432],{"className":564431,"style":37774},[31805],[34,564433,564434,564437],{"style":37777},[34,564435],{"className":564436,"style":31814},[31813],[34,564438,564440],{"className":564439},[31818,31819,31820,31821],[34,564441,564443,564483,564486],{"className":564442},[31685,31821],[34,564444,564446,564449],{"className":564445},[31685,31821],[34,564447,15],{"className":564448},[31685,31686,31821],[34,564450,564452],{"className":564451},[31792],[34,564453,564455,564475],{"className":564454},[31796,31797],[34,564456,564458,564472],{"className":564457},[31801],[34,564459,564461],{"className":564460,"style":318658},[31805],[34,564462,564463,564466],{"style":318661},[34,564464],{"className":564465,"style":43406},[31813],[34,564467,564469],{"className":564468},[31818,43410,43411,31821],[34,564470,459],{"className":564471},[31685,31821],[34,564473,31829],{"className":564474},[31828],[34,564476,564478],{"className":564477},[31801],[34,564479,564481],{"className":564480,"style":220540},[31805],[34,564482],{},[34,564484,32226],{"className":564485},[32024,31821],[34,564487,640],{"className":564488},[31685,31821],[34,564490],{"className":564491,"style":32123},[31845],[34,564493,564495,564498],{"className":564494},[32767],[34,564496,563286],{"className":564497},[32767],[34,564499,564501],{"className":564500},[31792],[34,564502,564504],{"className":564503},[31796],[34,564505,564507],{"className":564506},[31801],[34,564508,564510],{"className":564509,"style":563430},[31805],[34,564511,564512,564515],{"style":563433},[34,564513],{"className":564514,"style":31814},[31813],[34,564516,564518],{"className":564517},[31818,31819,31820,31821],[34,564519,768],{"className":564520},[31685,31821],[34,564522],{"className":564523,"style":32123},[31845],[34,564525,32936],{"className":564526},[31685,31686],[108,564528,564529,564532,564533,564776,564777,564807],{},[2627,564530,564531],{},"M-判别法："," 检验 ",[34,564534,564536,564588],{"className":564535},[31642],[34,564537,564539],{"className":564538},[31646],[31648,564540,564541],{"xmlns":31650},[31652,564542,564543,564585],{},[31655,564544,564545,564553,564569,564579,564581,564583],{},[38846,564546,564547,564549,564551],{},[31662,564548,546295],{},[31749,564550,459],{},[31749,564552,640],{},[37624,564554,564555,564557],{},[31658,564556,32936],{},[31655,564558,564559,564565,564567],{},[31743,564560,564561,564563],{},[31658,564562,15],{},[31749,564564,459],{},[31662,564566,32226],{},[31749,564568,640],{},[37624,564570,564571,564577],{},[31655,564572,564573,564575],{},[31658,564574,563286],{},[31662,564576,32682],{},[31749,564578,768],{},[31658,564580,32936],{},[31658,564582,46900],{},[31658,564584,32936],{},[31665,564586,564587],{"encoding":31667},"\\int_{0}^{1} x^{p_0-1}\\ln^2 x dx",[34,564589,564591],{"className":564590,"ariaHidden":758},[31672],[34,564592,564594,564597,564654,564657,564732,564735,564764,564767,564770,564773],{"className":564593},[31676],[34,564595],{"className":564596,"style":563309},[31680],[34,564598,564600,564603],{"className":564599},[32767],[34,564601,546295],{"className":564602,"style":546331},[32767,131249,156560],[34,564604,564606],{"className":564605},[31792],[34,564607,564609,564646],{"className":564608},[31796,31797],[34,564610,564612,564643],{"className":564611},[31801],[34,564613,564615,564629],{"className":564614,"style":563328},[31805],[34,564616,564617,564620],{"style":546347},[34,564618],{"className":564619,"style":31814},[31813],[34,564621,564623],{"className":564622},[31818,31819,31820,31821],[34,564624,564626],{"className":564625},[31685,31821],[34,564627,459],{"className":564628},[31685,31821],[34,564630,564631,564634],{"style":546362},[34,564632],{"className":564633,"style":31814},[31813],[34,564635,564637],{"className":564636},[31818,31819,31820,31821],[34,564638,564640],{"className":564639},[31685,31821],[34,564641,640],{"className":564642},[31685,31821],[34,564644,31829],{"className":564645},[31828],[34,564647,564649],{"className":564648},[31801],[34,564650,564652],{"className":564651,"style":546384},[31805],[34,564653],{},[34,564655],{"className":564656,"style":32123},[31845],[34,564658,564660,564663],{"className":564659},[31685],[34,564661,32936],{"className":564662},[31685,31686],[34,564664,564666],{"className":564665},[31792],[34,564667,564669],{"className":564668},[31796],[34,564670,564672],{"className":564671},[31801],[34,564673,564675],{"className":564674,"style":37774},[31805],[34,564676,564677,564680],{"style":37777},[34,564678],{"className":564679,"style":31814},[31813],[34,564681,564683],{"className":564682},[31818,31819,31820,31821],[34,564684,564686,564726,564729],{"className":564685},[31685,31821],[34,564687,564689,564692],{"className":564688},[31685,31821],[34,564690,15],{"className":564691},[31685,31686,31821],[34,564693,564695],{"className":564694},[31792],[34,564696,564698,564718],{"className":564697},[31796,31797],[34,564699,564701,564715],{"className":564700},[31801],[34,564702,564704],{"className":564703,"style":318658},[31805],[34,564705,564706,564709],{"style":318661},[34,564707],{"className":564708,"style":43406},[31813],[34,564710,564712],{"className":564711},[31818,43410,43411,31821],[34,564713,459],{"className":564714},[31685,31821],[34,564716,31829],{"className":564717},[31828],[34,564719,564721],{"className":564720},[31801],[34,564722,564724],{"className":564723,"style":220540},[31805],[34,564725],{},[34,564727,32226],{"className":564728},[32024,31821],[34,564730,640],{"className":564731},[31685,31821],[34,564733],{"className":564734,"style":32123},[31845],[34,564736,564738,564741],{"className":564737},[32767],[34,564739,563286],{"className":564740},[32767],[34,564742,564744],{"className":564743},[31792],[34,564745,564747],{"className":564746},[31796],[34,564748,564750],{"className":564749},[31801],[34,564751,564753],{"className":564752,"style":563430},[31805],[34,564754,564755,564758],{"style":563433},[34,564756],{"className":564757,"style":31814},[31813],[34,564759,564761],{"className":564760},[31818,31819,31820,31821],[34,564762,768],{"className":564763},[31685,31821],[34,564765],{"className":564766,"style":32123},[31845],[34,564768,32936],{"className":564769},[31685,31686],[34,564771,46900],{"className":564772},[31685,31686],[34,564774,32936],{"className":564775},[31685,31686],"。这是一个收敛的积分（",[34,564778,564780,564795],{"className":564779},[31642],[34,564781,564783],{"className":564782},[31646],[31648,564784,564785],{"xmlns":31650},[31652,564786,564787,564792],{},[31655,564788,564789],{},[31658,564790,564791],{"mathvariant":38079},"Γ",[31665,564793,564794],{"encoding":31667},"\\Gamma",[34,564796,564798],{"className":564797,"ariaHidden":758},[31672],[34,564799,564801,564804],{"className":564800},[31676],[34,564802],{"className":564803,"style":33218},[31680],[34,564805,564791],{"className":564806},[31685],"函数的导数形式）。",[108,564809,564810,47649,564812,347],{},[2627,564811,62532],{},[2627,564813,548044],{},[15,564815,564816],{},[2627,564817,523136,564818],{},[34,564819,564821,564839],{"className":564820},[31642],[34,564822,564824],{"className":564823},[31646],[31648,564825,564826],{"xmlns":31650},[31652,564827,564828,564836],{},[31655,564829,564830,564832,564834],{},[31658,564831,15],{},[31662,564833,29622],{},[31749,564835,459],{},[31665,564837,564838],{"encoding":31667},"p > 0",[34,564840,564842,564860],{"className":564841,"ariaHidden":758},[31672],[34,564843,564845,564848,564851,564854,564857],{"className":564844},[31676],[34,564846],{"className":564847,"style":86652},[31680],[34,564849,15],{"className":564850},[31685,31686],[34,564852],{"className":564853,"style":31846},[31845],[34,564855,29622],{"className":564856},[31850],[34,564858],{"className":564859,"style":31846},[31845],[34,564861,564863,564866],{"className":564862},[31676],[34,564864],{"className":564865,"style":32617},[31680],[34,564867,459],{"className":564868},[31685],[105,564870,564871,564904,565037,565151],{},[108,564872,564873,47649,564875,564903],{},[2627,564874,549449],{},[34,564876,564878,564891],{"className":564877},[31642],[34,564879,564881],{"className":564880},[31646],[31648,564882,564883],{"xmlns":31650},[31652,564884,564885,564889],{},[31655,564886,564887],{},[31658,564888,15],{},[31665,564890,15],{"encoding":31667},[34,564892,564894],{"className":564893,"ariaHidden":758},[31672],[34,564895,564897,564900],{"className":564896},[31676],[34,564898],{"className":564899,"style":38774},[31680],[34,564901,15],{"className":564902},[31685,31686]," 可以无限接近 0。",[108,564905,564906,564908,564909,347],{},[2627,564907,174094],{}," 该积分值等于 ",[34,564910,564912,564934],{"className":564911},[31642],[34,564913,564915],{"className":564914},[31646],[31648,564916,564917],{"xmlns":31650},[31652,564918,564919,564931],{},[31655,564920,564921],{},[41453,564922,564923,564925],{},[31749,564924,768],{},[37624,564926,564927,564929],{},[31658,564928,15],{},[31749,564930,786],{},[31665,564932,564933],{"encoding":31667},"\\frac{2}{p^3}",[34,564935,564937],{"className":564936,"ariaHidden":758},[31672],[34,564938,564940,564943],{"className":564939},[31676],[34,564941],{"className":564942,"style":561374},[31680],[34,564944,564946,564949,565034],{"className":564945},[31685],[34,564947],{"className":564948},[31782,41488],[34,564950,564952],{"className":564951},[41453],[34,564953,564955,565026],{"className":564954},[31796,31797],[34,564956,564958,565023],{"className":564957},[31801],[34,564959,564961,565001,565009],{"className":564960,"style":46746},[31805],[34,564962,564963,564966],{"style":41504},[34,564964],{"className":564965,"style":41508},[31813],[34,564967,564969],{"className":564968},[31818,31819,31820,31821],[34,564970,564972],{"className":564971},[31685,31821],[34,564973,564975,564978],{"className":564974},[31685,31821],[34,564976,15],{"className":564977},[31685,31686,31821],[34,564979,564981],{"className":564980},[31792],[34,564982,564984],{"className":564983},[31796],[34,564985,564987],{"className":564986},[31801],[34,564988,564990],{"className":564989,"style":548314},[31805],[34,564991,564992,564995],{"style":116668},[34,564993],{"className":564994,"style":43406},[31813],[34,564996,564998],{"className":564997},[31818,43410,43411,31821],[34,564999,786],{"className":565000},[31685,31821],[34,565002,565003,565006],{"style":41520},[34,565004],{"className":565005,"style":41508},[31813],[34,565007],{"className":565008,"style":41528},[41527],[34,565010,565011,565014],{"style":46771},[34,565012],{"className":565013,"style":41508},[31813],[34,565015,565017],{"className":565016},[31818,31819,31820,31821],[34,565018,565020],{"className":565019},[31685,31821],[34,565021,768],{"className":565022},[31685,31821],[34,565024,31829],{"className":565025},[31828],[34,565027,565029],{"className":565028},[31801],[34,565030,565032],{"className":565031,"style":220191},[31805],[34,565033],{},[34,565035],{"className":565036},[31690,41488],[108,565038,565039,561807,565041,565092,565093,565121,565122,565150],{},[2627,565040,58554],{},[34,565042,565044,565062],{"className":565043},[31642],[34,565045,565047],{"className":565046},[31646],[31648,565048,565049],{"xmlns":31650},[31652,565050,565051,565059],{},[31655,565052,565053,565055,565057],{},[31658,565054,15],{},[31662,565056,33239],{},[31749,565058,459],{},[31665,565060,565061],{"encoding":31667},"p \\to 0",[34,565063,565065,565083],{"className":565064,"ariaHidden":758},[31672],[34,565066,565068,565071,565074,565077,565080],{"className":565067},[31676],[34,565069],{"className":565070,"style":38774},[31680],[34,565072,15],{"className":565073},[31685,31686],[34,565075],{"className":565076,"style":31846},[31845],[34,565078,33239],{"className":565079},[31850],[34,565081],{"className":565082,"style":31846},[31845],[34,565084,565086,565089],{"className":565085},[31676],[34,565087],{"className":565088,"style":32617},[31680],[34,565090,459],{"className":565091},[31685]," 时，积分值趋于无穷大。连积分结果本身都无界，或者说，当 ",[34,565094,565096,565109],{"className":565095},[31642],[34,565097,565099],{"className":565098},[31646],[31648,565100,565101],{"xmlns":31650},[31652,565102,565103,565107],{},[31655,565104,565105],{},[31658,565106,15],{},[31665,565108,15],{"encoding":31667},[34,565110,565112],{"className":565111,"ariaHidden":758},[31672],[34,565113,565115,565118],{"className":565114},[31676],[34,565116],{"className":565117,"style":38774},[31680],[34,565119,15],{"className":565120},[31685,31686]," 极小时，需要 ",[34,565123,565125,565138],{"className":565124},[31642],[34,565126,565128],{"className":565127},[31646],[31648,565129,565130],{"xmlns":31650},[31652,565131,565132,565136],{},[31655,565133,565134],{},[31658,565135,32936],{},[31665,565137,32936],{"encoding":31667},[34,565139,565141],{"className":565140,"ariaHidden":758},[31672],[34,565142,565144,565147],{"className":565143},[31676],[34,565145],{"className":565146,"style":32527},[31680],[34,565148,32936],{"className":565149},[31685,31686]," 极其接近 0 才能积出主要数值。",[108,565152,565153,47649,565155,347],{},[2627,565154,62532],{},[2627,565156,563037],{},[253,565158,565160],{"id":565159},"讲解-4",[2627,565161,565162],{},"讲解 (4)",[15,565164,565165],{},[34,565166,565168,565214],{"className":565167},[31642],[34,565169,565171],{"className":565170},[31646],[31648,565172,565173],{"xmlns":31650},[31652,565174,565175,565211],{},[31655,565176,565177,565189,565201,565203,565205,565207,565209],{},[38846,565178,565179,565181,565183],{},[31662,565180,546295],{},[31749,565182,459],{},[31655,565184,565185,565187],{},[31662,565186,223],{},[31658,565188,138286],{"mathvariant":38079},[37624,565190,565191,565193],{},[31658,565192,33244],{},[31655,565194,565195,565197,565199],{},[31662,565196,32226],{},[31658,565198,217392],{},[31658,565200,32936],{},[31658,565202,547853],{},[31662,565204,32682],{},[31658,565206,32936],{},[31658,565208,46900],{},[31658,565210,32936],{},[31665,565212,565213],{"encoding":31667},"\\int_{0}^{+\\infty} e^{-\\alpha x}\\sin xdx",[34,565215,565217],{"className":565216,"ariaHidden":758},[31672],[34,565218,565220,565223,565283,565286,565324,565327,565330,565333,565336,565339],{"className":565219},[31676],[34,565221],{"className":565222,"style":546531},[31680],[34,565224,565226,565229],{"className":565225},[32767],[34,565227,546295],{"className":565228,"style":546331},[32767,131249,156560],[34,565230,565232],{"className":565231},[31792],[34,565233,565235,565275],{"className":565234},[31796,31797],[34,565236,565238,565272],{"className":565237},[31801],[34,565239,565241,565255],{"className":565240,"style":546550},[31805],[34,565242,565243,565246],{"style":546347},[34,565244],{"className":565245,"style":31814},[31813],[34,565247,565249],{"className":565248},[31818,31819,31820,31821],[34,565250,565252],{"className":565251},[31685,31821],[34,565253,459],{"className":565254},[31685,31821],[34,565256,565257,565260],{"style":546362},[34,565258],{"className":565259,"style":31814},[31813],[34,565261,565263],{"className":565262},[31818,31819,31820,31821],[34,565264,565266,565269],{"className":565265},[31685,31821],[34,565267,223],{"className":565268},[31685,31821],[34,565270,138286],{"className":565271},[31685,31821],[34,565273,31829],{"className":565274},[31828],[34,565276,565278],{"className":565277},[31801],[34,565279,565281],{"className":565280,"style":546384},[31805],[34,565282],{},[34,565284],{"className":565285,"style":32123},[31845],[34,565287,565289,565292],{"className":565288},[31685],[34,565290,33244],{"className":565291},[31685,31686],[34,565293,565295],{"className":565294},[31792],[34,565296,565298],{"className":565297},[31796],[34,565299,565301],{"className":565300},[31801],[34,565302,565304],{"className":565303,"style":156828},[31805],[34,565305,565306,565309],{"style":37777},[34,565307],{"className":565308,"style":31814},[31813],[34,565310,565312],{"className":565311},[31818,31819,31820,31821],[34,565313,565315,565318,565321],{"className":565314},[31685,31821],[34,565316,32226],{"className":565317},[31685,31821],[34,565319,217392],{"className":565320,"style":217496},[31685,31686,31821],[34,565322,32936],{"className":565323},[31685,31686,31821],[34,565325],{"className":565326,"style":32123},[31845],[34,565328,547853],{"className":565329},[32767],[34,565331],{"className":565332,"style":32123},[31845],[34,565334,32936],{"className":565335},[31685,31686],[34,565337,46900],{"className":565338},[31685,31686],[34,565340,32936],{"className":565341},[31685,31686],[15,565343,565344],{},[2627,565345,562471,565346],{},[34,565347,565349,565375],{"className":565348},[31642],[34,565350,565352],{"className":565351},[31646],[31648,565353,565354],{"xmlns":31650},[31652,565355,565356,565372],{},[31655,565357,565358,565360,565362,565368,565370],{},[31658,565359,217392],{},[31662,565361,43175],{},[31743,565363,565364,565366],{},[31658,565365,217392],{},[31749,565367,459],{},[31662,565369,29622],{},[31749,565371,459],{},[31665,565373,565374],{"encoding":31667},"\\alpha \\geq \\alpha_0 > 0",[34,565376,565378,565396,565451],{"className":565377,"ariaHidden":758},[31672],[34,565379,565381,565384,565387,565390,565393],{"className":565380},[31676],[34,565382],{"className":565383,"style":43198},[31680],[34,565385,217392],{"className":565386,"style":217496},[31685,31686],[34,565388],{"className":565389,"style":31846},[31845],[34,565391,43175],{"className":565392},[31850],[34,565394],{"className":565395,"style":31846},[31845],[34,565397,565399,565402,565442,565445,565448],{"className":565398},[31676],[34,565400],{"className":565401,"style":225773},[31680],[34,565403,565405,565408],{"className":565404},[31685],[34,565406,217392],{"className":565407,"style":217496},[31685,31686],[34,565409,565411],{"className":565410},[31792],[34,565412,565414,565434],{"className":565413},[31796,31797],[34,565415,565417,565431],{"className":565416},[31801],[34,565418,565420],{"className":565419,"style":31806},[31805],[34,565421,565422,565425],{"style":548824},[34,565423],{"className":565424,"style":31814},[31813],[34,565426,565428],{"className":565427},[31818,31819,31820,31821],[34,565429,459],{"className":565430},[31685,31821],[34,565432,31829],{"className":565433},[31828],[34,565435,565437],{"className":565436},[31801],[34,565438,565440],{"className":565439,"style":31836},[31805],[34,565441],{},[34,565443],{"className":565444,"style":31846},[31845],[34,565446,29622],{"className":565447},[31850],[34,565449],{"className":565450,"style":31846},[31845],[34,565452,565454,565457],{"className":565453},[31676],[34,565455],{"className":565456,"style":32617},[31680],[34,565458,459],{"className":565459},[31685],[105,565461,565462,565467,565681],{},[108,565463,565464,565466],{},[2627,565465,549449],{}," M-判别法。",[108,565468,565469,47649,565471,347],{},[2627,565470,564233],{},[34,565472,565474,565526],{"className":565473},[31642],[34,565475,565477],{"className":565476},[31646],[31648,565478,565479],{"xmlns":31650},[31652,565480,565481,565523],{},[31655,565482,565483,565485,565497,565499,565501,565503,565505,565507],{},[31658,565484,162086],{"mathvariant":38079},[37624,565486,565487,565489],{},[31658,565488,33244],{},[31655,565490,565491,565493,565495],{},[31662,565492,32226],{},[31658,565494,217392],{},[31658,565496,32936],{},[31658,565498,547853],{},[31662,565500,32682],{},[31658,565502,32936],{},[31658,565504,162086],{"mathvariant":38079},[31662,565506,45972],{},[37624,565508,565509,565511],{},[31658,565510,33244],{},[31655,565512,565513,565515,565521],{},[31662,565514,32226],{},[31743,565516,565517,565519],{},[31658,565518,217392],{},[31749,565520,459],{},[31658,565522,32936],{},[31665,565524,565525],{"encoding":31667},"|e^{-\\alpha x}\\sin x| \\leq e^{-\\alpha_0 x}",[34,565527,565529,565600],{"className":565528,"ariaHidden":758},[31672],[34,565530,565532,565535,565538,565576,565579,565582,565585,565588,565591,565594,565597],{"className":565531},[31676],[34,565533],{"className":565534,"style":156809},[31680],[34,565536,162086],{"className":565537},[31685],[34,565539,565541,565544],{"className":565540},[31685],[34,565542,33244],{"className":565543},[31685,31686],[34,565545,565547],{"className":565546},[31792],[34,565548,565550],{"className":565549},[31796],[34,565551,565553],{"className":565552},[31801],[34,565554,565556],{"className":565555,"style":156828},[31805],[34,565557,565558,565561],{"style":37777},[34,565559],{"className":565560,"style":31814},[31813],[34,565562,565564],{"className":565563},[31818,31819,31820,31821],[34,565565,565567,565570,565573],{"className":565566},[31685,31821],[34,565568,32226],{"className":565569},[31685,31821],[34,565571,217392],{"className":565572,"style":217496},[31685,31686,31821],[34,565574,32936],{"className":565575},[31685,31686,31821],[34,565577],{"className":565578,"style":32123},[31845],[34,565580,547853],{"className":565581},[32767],[34,565583],{"className":565584,"style":32123},[31845],[34,565586,32936],{"className":565587},[31685,31686],[34,565589,162086],{"className":565590},[31685],[34,565592],{"className":565593,"style":31846},[31845],[34,565595,45972],{"className":565596},[31850],[34,565598],{"className":565599,"style":31846},[31845],[34,565601,565603,565606],{"className":565602},[31676],[34,565604],{"className":565605,"style":156828},[31680],[34,565607,565609,565612],{"className":565608},[31685],[34,565610,33244],{"className":565611},[31685,31686],[34,565613,565615],{"className":565614},[31792],[34,565616,565618],{"className":565617},[31796],[34,565619,565621],{"className":565620},[31801],[34,565622,565624],{"className":565623,"style":156828},[31805],[34,565625,565626,565629],{"style":37777},[34,565627],{"className":565628,"style":31814},[31813],[34,565630,565632],{"className":565631},[31818,31819,31820,31821],[34,565633,565635,565638,565678],{"className":565634},[31685,31821],[34,565636,32226],{"className":565637},[31685,31821],[34,565639,565641,565644],{"className":565640},[31685,31821],[34,565642,217392],{"className":565643,"style":217496},[31685,31686,31821],[34,565645,565647],{"className":565646},[31792],[34,565648,565650,565670],{"className":565649},[31796,31797],[34,565651,565653,565667],{"className":565652},[31801],[34,565654,565656],{"className":565655,"style":318658},[31805],[34,565657,565658,565661],{"style":547045},[34,565659],{"className":565660,"style":43406},[31813],[34,565662,565664],{"className":565663},[31818,43410,43411,31821],[34,565665,459],{"className":565666},[31685,31821],[34,565668,31829],{"className":565669},[31828],[34,565671,565673],{"className":565672},[31801],[34,565674,565676],{"className":565675,"style":220540},[31805],[34,565677],{},[34,565679,32936],{"className":565680},[31685,31686,31821],[108,565682,565683,47649,565685,565818,565819,347],{},[2627,565684,62532],{},[34,565686,565688,565722],{"className":565687},[31642],[34,565689,565691],{"className":565690},[31646],[31648,565692,565693],{"xmlns":31650},[31652,565694,565695,565719],{},[31655,565696,565697,565699,565715,565717],{},[31662,565698,546295],{},[37624,565700,565701,565703],{},[31658,565702,33244],{},[31655,565704,565705,565707,565713],{},[31662,565706,32226],{},[31743,565708,565709,565711],{},[31658,565710,217392],{},[31749,565712,459],{},[31658,565714,32936],{},[31658,565716,46900],{},[31658,565718,32936],{},[31665,565720,565721],{"encoding":31667},"\\int e^{-\\alpha_0 x}dx",[34,565723,565725],{"className":565724,"ariaHidden":758},[31672],[34,565726,565728,565731,565734,565737,565812,565815],{"className":565727},[31676],[34,565729],{"className":565730,"style":546896},[31680],[34,565732,546295],{"className":565733,"style":546331},[32767,131249,156560],[34,565735],{"className":565736,"style":32123},[31845],[34,565738,565740,565743],{"className":565739},[31685],[34,565741,33244],{"className":565742},[31685,31686],[34,565744,565746],{"className":565745},[31792],[34,565747,565749],{"className":565748},[31796],[34,565750,565752],{"className":565751},[31801],[34,565753,565755],{"className":565754,"style":156828},[31805],[34,565756,565757,565760],{"style":37777},[34,565758],{"className":565759,"style":31814},[31813],[34,565761,565763],{"className":565762},[31818,31819,31820,31821],[34,565764,565766,565769,565809],{"className":565765},[31685,31821],[34,565767,32226],{"className":565768},[31685,31821],[34,565770,565772,565775],{"className":565771},[31685,31821],[34,565773,217392],{"className":565774,"style":217496},[31685,31686,31821],[34,565776,565778],{"className":565777},[31792],[34,565779,565781,565801],{"className":565780},[31796,31797],[34,565782,565784,565798],{"className":565783},[31801],[34,565785,565787],{"className":565786,"style":318658},[31805],[34,565788,565789,565792],{"style":547045},[34,565790],{"className":565791,"style":43406},[31813],[34,565793,565795],{"className":565794},[31818,43410,43411,31821],[34,565796,459],{"className":565797},[31685,31821],[34,565799,31829],{"className":565800},[31828],[34,565802,565804],{"className":565803},[31801],[34,565805,565807],{"className":565806,"style":220540},[31805],[34,565808],{},[34,565810,32936],{"className":565811},[31685,31686,31821],[34,565813,46900],{"className":565814},[31685,31686],[34,565816,32936],{"className":565817},[31685,31686]," 收敛，所以",[2627,565820,548044],{},[15,565822,565823],{},[2627,565824,523136,565825],{},[34,565826,565828,565846],{"className":565827},[31642],[34,565829,565831],{"className":565830},[31646],[31648,565832,565833],{"xmlns":31650},[31652,565834,565835,565843],{},[31655,565836,565837,565839,565841],{},[31658,565838,217392],{},[31662,565840,29622],{},[31749,565842,459],{},[31665,565844,565845],{"encoding":31667},"\\alpha > 0",[34,565847,565849,565867],{"className":565848,"ariaHidden":758},[31672],[34,565850,565852,565855,565858,565861,565864],{"className":565851},[31676],[34,565853],{"className":565854,"style":34812},[31680],[34,565856,217392],{"className":565857,"style":217496},[31685,31686],[34,565859],{"className":565860,"style":31846},[31845],[34,565862,29622],{"className":565863},[31850],[34,565865],{"className":565866,"style":31846},[31845],[34,565868,565870,565873],{"className":565869},[31676],[34,565871],{"className":565872,"style":32617},[31680],[34,565874,459],{"className":565875},[31685],[105,565877,565878,565911,566093],{},[108,565879,565880,47649,565882,565910],{},[2627,565881,549449],{},[34,565883,565885,565898],{"className":565884},[31642],[34,565886,565888],{"className":565887},[31646],[31648,565889,565890],{"xmlns":31650},[31652,565891,565892,565896],{},[31655,565893,565894],{},[31658,565895,217392],{},[31665,565897,218244],{"encoding":31667},[34,565899,565901],{"className":565900,"ariaHidden":758},[31672],[34,565902,565904,565907],{"className":565903},[31676],[34,565905],{"className":565906,"style":32527},[31680],[34,565908,217392],{"className":565909,"style":217496},[31685,31686]," 可以趋近于 0。",[108,565912,565913,561807,565916,565967,565968,566092],{},[2627,565914,565915],{},"柯西准则反证：",[34,565917,565919,565937],{"className":565918},[31642],[34,565920,565922],{"className":565921},[31646],[31648,565923,565924],{"xmlns":31650},[31652,565925,565926,565934],{},[31655,565927,565928,565930,565932],{},[31658,565929,217392],{},[31662,565931,33239],{},[31749,565933,459],{},[31665,565935,565936],{"encoding":31667},"\\alpha \\to 0",[34,565938,565940,565958],{"className":565939,"ariaHidden":758},[31672],[34,565941,565943,565946,565949,565952,565955],{"className":565942},[31676],[34,565944],{"className":565945,"style":32527},[31680],[34,565947,217392],{"className":565948,"style":217496},[31685,31686],[34,565950],{"className":565951,"style":31846},[31845],[34,565953,33239],{"className":565954},[31850],[34,565956],{"className":565957,"style":31846},[31845],[34,565959,565961,565964],{"className":565960},[31676],[34,565962],{"className":565963,"style":32617},[31680],[34,565965,459],{"className":565966},[31685]," 时，积分 ",[34,565969,565971,566005],{"className":565970},[31642],[34,565972,565974],{"className":565973},[31646],[31648,565975,565976],{"xmlns":31650},[31652,565977,565978,566002],{},[31655,565979,565980,565992,565994,565996,565998,566000],{},[38846,565981,565982,565984,565986],{},[31662,565983,546295],{},[31749,565985,459],{},[31655,565987,565988,565990],{},[31662,565989,223],{},[31658,565991,138286],{"mathvariant":38079},[31658,565993,547853],{},[31662,565995,32682],{},[31658,565997,32936],{},[31658,565999,46900],{},[31658,566001,32936],{},[31665,566003,566004],{"encoding":31667},"\\int_{0}^{+\\infty} \\sin x dx",[34,566006,566008],{"className":566007,"ariaHidden":758},[31672],[34,566009,566011,566014,566074,566077,566080,566083,566086,566089],{"className":566010},[31676],[34,566012],{"className":566013,"style":546531},[31680],[34,566015,566017,566020],{"className":566016},[32767],[34,566018,546295],{"className":566019,"style":546331},[32767,131249,156560],[34,566021,566023],{"className":566022},[31792],[34,566024,566026,566066],{"className":566025},[31796,31797],[34,566027,566029,566063],{"className":566028},[31801],[34,566030,566032,566046],{"className":566031,"style":546550},[31805],[34,566033,566034,566037],{"style":546347},[34,566035],{"className":566036,"style":31814},[31813],[34,566038,566040],{"className":566039},[31818,31819,31820,31821],[34,566041,566043],{"className":566042},[31685,31821],[34,566044,459],{"className":566045},[31685,31821],[34,566047,566048,566051],{"style":546362},[34,566049],{"className":566050,"style":31814},[31813],[34,566052,566054],{"className":566053},[31818,31819,31820,31821],[34,566055,566057,566060],{"className":566056},[31685,31821],[34,566058,223],{"className":566059},[31685,31821],[34,566061,138286],{"className":566062},[31685,31821],[34,566064,31829],{"className":566065},[31828],[34,566067,566069],{"className":566068},[31801],[34,566070,566072],{"className":566071,"style":546384},[31805],[34,566073],{},[34,566075],{"className":566076,"style":32123},[31845],[34,566078,547853],{"className":566079},[32767],[34,566081],{"className":566082,"style":32123},[31845],[34,566084,32936],{"className":566085},[31685,31686],[34,566087,46900],{"className":566088},[31685,31686],[34,566090,32936],{"className":566091},[31685,31686]," 是发散的。\n既然极限情况都发散了，肯定无法一致收敛。",[108,566094,566095,47649,566097,347],{},[2627,566096,62532],{},[2627,566098,563037],{},[32188,566100],{},[253,566102,566104],{"id":566103},"p329-t5","p329 T5",[15,566106,566107,566109,566110,314945,566382,566439],{},[2627,566108,214361],{},"\n证明：函数 ",[34,566111,566113,566167],{"className":566112},[31642],[34,566114,566116],{"className":566115},[31646],[31648,566117,566118],{"xmlns":31650},[31652,566119,566120,566164],{},[31655,566121,566122,566124,566126,566128,566130,566132,566144,566160,566162],{},[31658,566123,51834],{},[31662,566125,456],{"stretchy":702},[31658,566127,217392],{},[31662,566129,2557],{"stretchy":702},[31662,566131,202],{},[38846,566133,566134,566136,566138],{},[31662,566135,546295],{},[31749,566137,640],{},[31655,566139,566140,566142],{},[31662,566141,223],{},[31658,566143,138286],{"mathvariant":38079},[41453,566145,566146,566154],{},[31655,566147,566148,566150,566152],{},[31658,566149,547860],{},[31662,566151,32682],{},[31658,566153,32936],{},[37624,566155,566156,566158],{},[31658,566157,32936],{},[31658,566159,217392],{},[31658,566161,46900],{},[31658,566163,32936],{},[31665,566165,566166],{"encoding":31667},"F(\\alpha) = \\int_{1}^{+\\infty} \\frac{\\cos x}{x^\\alpha}dx",[34,566168,566170,566197],{"className":566169,"ariaHidden":758},[31672],[34,566171,566173,566176,566179,566182,566185,566188,566191,566194],{"className":566172},[31676],[34,566174],{"className":566175,"style":31774},[31680],[34,566177,51834],{"className":566178,"style":37422},[31685,31686],[34,566180,456],{"className":566181},[31782],[34,566183,217392],{"className":566184,"style":217496},[31685,31686],[34,566186,2557],{"className":566187},[31690],[34,566189],{"className":566190,"style":31846},[31845],[34,566192,202],{"className":566193},[31850],[34,566195],{"className":566196,"style":31846},[31845],[34,566198,566200,566203,566263,566266,566376,566379],{"className":566199},[31676],[34,566201],{"className":566202,"style":546531},[31680],[34,566204,566206,566209],{"className":566205},[32767],[34,566207,546295],{"className":566208,"style":546331},[32767,131249,156560],[34,566210,566212],{"className":566211},[31792],[34,566213,566215,566255],{"className":566214},[31796,31797],[34,566216,566218,566252],{"className":566217},[31801],[34,566219,566221,566235],{"className":566220,"style":546550},[31805],[34,566222,566223,566226],{"style":546347},[34,566224],{"className":566225,"style":31814},[31813],[34,566227,566229],{"className":566228},[31818,31819,31820,31821],[34,566230,566232],{"className":566231},[31685,31821],[34,566233,640],{"className":566234},[31685,31821],[34,566236,566237,566240],{"style":546362},[34,566238],{"className":566239,"style":31814},[31813],[34,566241,566243],{"className":566242},[31818,31819,31820,31821],[34,566244,566246,566249],{"className":566245},[31685,31821],[34,566247,223],{"className":566248},[31685,31821],[34,566250,138286],{"className":566251},[31685,31821],[34,566253,31829],{"className":566254},[31828],[34,566256,566258],{"className":566257},[31801],[34,566259,566261],{"className":566260,"style":546384},[31805],[34,566262],{},[34,566264],{"className":566265,"style":32123},[31845],[34,566267,566269,566272,566373],{"className":566268},[31685],[34,566270],{"className":566271},[31782,41488],[34,566273,566275],{"className":566274},[41453],[34,566276,566278,566365],{"className":566277},[31796,31797],[34,566279,566281,566362],{"className":566280},[31801],[34,566282,566284,566325,566333],{"className":566283,"style":244945},[31805],[34,566285,566286,566289],{"style":41504},[34,566287],{"className":566288,"style":41508},[31813],[34,566290,566292],{"className":566291},[31818,31819,31820,31821],[34,566293,566295],{"className":566294},[31685,31821],[34,566296,566298,566301],{"className":566297},[31685,31821],[34,566299,32936],{"className":566300},[31685,31686,31821],[34,566302,566304],{"className":566303},[31792],[34,566305,566307],{"className":566306},[31796],[34,566308,566310],{"className":566309},[31801],[34,566311,566314],{"className":566312,"style":566313},[31805],"height:0.5935em;",[34,566315,566316,566319],{"style":116668},[34,566317],{"className":566318,"style":43406},[31813],[34,566320,566322],{"className":566321},[31818,43410,43411,31821],[34,566323,217392],{"className":566324,"style":217496},[31685,31686,31821],[34,566326,566327,566330],{"style":41520},[34,566328],{"className":566329,"style":41508},[31813],[34,566331],{"className":566332,"style":41528},[41527],[34,566334,566335,566338],{"style":46771},[34,566336],{"className":566337,"style":41508},[31813],[34,566339,566341],{"className":566340},[31818,31819,31820,31821],[34,566342,566344,566356,566359],{"className":566343},[31685,31821],[34,566345,566347,566350,566353],{"className":566346},[32767,31821],[34,566348,31965],{"className":566349},[31821],[34,566351,43383],{"className":566352},[31821],[34,566354,31747],{"className":566355},[31821],[34,566357],{"className":566358,"style":43430},[31845,31821],[34,566360,32936],{"className":566361},[31685,31686,31821],[34,566363,31829],{"className":566364},[31828],[34,566366,566368],{"className":566367},[31801],[34,566369,566371],{"className":566370,"style":41619},[31805],[34,566372],{},[34,566374],{"className":566375},[31690,41488],[34,566377,46900],{"className":566378},[31685,31686],[34,566380,32936],{"className":566381},[31685,31686],[34,566383,566385,566409],{"className":566384},[31642],[34,566386,566388],{"className":566387},[31646],[31648,566389,566390],{"xmlns":31650},[31652,566391,566392,566406],{},[31655,566393,566394,566396,566398,566400,566402,566404],{},[31662,566395,456],{"stretchy":702},[31749,566397,459],{},[31662,566399,31960],{"separator":758},[31662,566401,223],{},[31658,566403,138286],{"mathvariant":38079},[31662,566405,2557],{"stretchy":702},[31665,566407,566408],{"encoding":31667},"(0, +\\infty)",[34,566410,566412],{"className":566411,"ariaHidden":758},[31672],[34,566413,566415,566418,566421,566424,566427,566430,566433,566436],{"className":566414},[31676],[34,566416],{"className":566417,"style":31774},[31680],[34,566419,456],{"className":566420},[31782],[34,566422,459],{"className":566423},[31685],[34,566425,31960],{"className":566426},[32119],[34,566428],{"className":566429,"style":32123},[31845],[34,566431,223],{"className":566432},[31685],[34,566434,138286],{"className":566435},[31685],[34,566437,2557],{"className":566438},[31690]," 上连续。",[15,566441,566442,566444,566445,566488,566489,566492,566493,566549,566550,551691,566607,347],{},[2627,566443,91513],{},"\n要证明含参积分 ",[34,566446,566448,566467],{"className":566447},[31642],[34,566449,566451],{"className":566450},[31646],[31648,566452,566453],{"xmlns":31650},[31652,566454,566455,566465],{},[31655,566456,566457,566459,566461,566463],{},[31658,566458,51834],{},[31662,566460,456],{"stretchy":702},[31658,566462,217392],{},[31662,566464,2557],{"stretchy":702},[31665,566466,546701],{"encoding":31667},[34,566468,566470],{"className":566469,"ariaHidden":758},[31672],[34,566471,566473,566476,566479,566482,566485],{"className":566472},[31676],[34,566474],{"className":566475,"style":31774},[31680],[34,566477,51834],{"className":566478,"style":37422},[31685,31686],[34,566480,456],{"className":566481},[31782],[34,566483,217392],{"className":566484,"style":217496},[31685,31686],[34,566486,2557],{"className":566487},[31690]," 连续，根据",[2627,566490,566491],{},"连续性定理","，我们需要证明该积分在 ",[34,566494,566496,566519],{"className":566495},[31642],[34,566497,566499],{"className":566498},[31646],[31648,566500,566501],{"xmlns":31650},[31652,566502,566503,566517],{},[31655,566504,566505,566507,566509,566511,566513,566515],{},[31662,566506,456],{"stretchy":702},[31749,566508,459],{},[31662,566510,31960],{"separator":758},[31662,566512,223],{},[31658,566514,138286],{"mathvariant":38079},[31662,566516,2557],{"stretchy":702},[31665,566518,566408],{"encoding":31667},[34,566520,566522],{"className":566521,"ariaHidden":758},[31672],[34,566523,566525,566528,566531,566534,566537,566540,566543,566546],{"className":566524},[31676],[34,566526],{"className":566527,"style":31774},[31680],[34,566529,456],{"className":566530},[31782],[34,566532,459],{"className":566533},[31685],[34,566535,31960],{"className":566536},[32119],[34,566538],{"className":566539,"style":32123},[31845],[34,566541,223],{"className":566542},[31685],[34,566544,138286],{"className":566545},[31685],[34,566547,2557],{"className":566548},[31690]," 的任意闭子区间 ",[34,566551,566553,566577],{"className":566552},[31642],[34,566554,566556],{"className":566555},[31646],[31648,566557,566558],{"xmlns":31650},[31652,566559,566560,566574],{},[31655,566561,566562,566564,566566,566568,566570,566572],{},[31662,566563,15972],{"stretchy":702},[31658,566565,231507],{},[31662,566567,31960],{"separator":758},[31662,566569,223],{},[31658,566571,138286],{"mathvariant":38079},[31662,566573,2557],{"stretchy":702},[31665,566575,566576],{"encoding":31667},"[\\delta, +\\infty)",[34,566578,566580],{"className":566579,"ariaHidden":758},[31672],[34,566581,566583,566586,566589,566592,566595,566598,566601,566604],{"className":566582},[31676],[34,566584],{"className":566585,"style":31774},[31680],[34,566587,15972],{"className":566588},[31782],[34,566590,231507],{"className":566591,"style":231523},[31685,31686],[34,566593,31960],{"className":566594},[32119],[34,566596],{"className":566597,"style":32123},[31845],[34,566599,223],{"className":566600},[31685],[34,566602,138286],{"className":566603},[31685],[34,566605,2557],{"className":566606},[31690],[2627,566608,548044],{},[15,566610,566611],{},[2627,566612,549629],{},[2230,566614,566615,566751,567567,567630],{},[108,566616,566617,69123,566620,70884,566699,566750],{},[2627,566618,566619],{},"任取子区间：",[34,566621,566623,566651],{"className":566622},[31642],[34,566624,566626],{"className":566625},[31646],[31648,566627,566628],{"xmlns":31650},[31652,566629,566630,566648],{},[31655,566631,566632,566634,566636,566638,566640,566642,566644,566646],{},[31658,566633,217392],{},[31662,566635,112296],{},[31662,566637,15972],{"stretchy":702},[31658,566639,231507],{},[31662,566641,31960],{"separator":758},[31662,566643,223],{},[31658,566645,138286],{"mathvariant":38079},[31662,566647,2557],{"stretchy":702},[31665,566649,566650],{"encoding":31667},"\\alpha \\in [\\delta, +\\infty)",[34,566652,566654,566672],{"className":566653,"ariaHidden":758},[31672],[34,566655,566657,566660,566663,566666,566669],{"className":566656},[31676],[34,566658],{"className":566659,"style":34812},[31680],[34,566661,217392],{"className":566662,"style":217496},[31685,31686],[34,566664],{"className":566665,"style":31846},[31845],[34,566667,112296],{"className":566668},[31850],[34,566670],{"className":566671,"style":31846},[31845],[34,566673,566675,566678,566681,566684,566687,566690,566693,566696],{"className":566674},[31676],[34,566676],{"className":566677,"style":31774},[31680],[34,566679,15972],{"className":566680},[31782],[34,566682,231507],{"className":566683,"style":231523},[31685,31686],[34,566685,31960],{"className":566686},[32119],[34,566688],{"className":566689,"style":32123},[31845],[34,566691,223],{"className":566692},[31685],[34,566694,138286],{"className":566695},[31685],[34,566697,2557],{"className":566698},[31690],[34,566700,566702,566720],{"className":566701},[31642],[34,566703,566705],{"className":566704},[31646],[31648,566706,566707],{"xmlns":31650},[31652,566708,566709,566717],{},[31655,566710,566711,566713,566715],{},[31658,566712,231507],{},[31662,566714,29622],{},[31749,566716,459],{},[31665,566718,566719],{"encoding":31667},"\\delta > 0",[34,566721,566723,566741],{"className":566722,"ariaHidden":758},[31672],[34,566724,566726,566729,566732,566735,566738],{"className":566725},[31676],[34,566727],{"className":566728,"style":46991},[31680],[34,566730,231507],{"className":566731,"style":231523},[31685,31686],[34,566733],{"className":566734,"style":31846},[31845],[34,566736,29622],{"className":566737},[31850],[34,566739],{"className":566740,"style":31846},[31845],[34,566742,566744,566747],{"className":566743},[31676],[34,566745],{"className":566746,"style":32617},[31680],[34,566748,459],{"className":566749},[31685]," 是任意小的正数。",[108,566752,566753,566756],{},[2627,566754,566755],{},"使用 Dirichlet 判别法：",[105,566757,566758,566943,567153],{},[108,566759,566760,566942],{},[34,566761,566763,566809],{"className":566762},[31642],[34,566764,566766],{"className":566765},[31646],[31648,566767,566768],{"xmlns":31650},[31652,566769,566770,566806],{},[31655,566771,566772,566780,566782,566784,566786,566788,566790,566792,566794,566796,566798,566800,566802,566804],{},[38846,566773,566774,566776,566778],{},[31662,566775,546295],{},[31749,566777,640],{},[31658,566779,54151],{},[31658,566781,547860],{},[31662,566783,32682],{},[31658,566785,32936],{},[31658,566787,46900],{},[31658,566789,32936],{},[31662,566791,202],{},[31658,566793,547853],{},[31662,566795,32682],{},[31658,566797,54151],{},[31662,566799,32226],{},[31658,566801,547853],{},[31662,566803,32682],{},[31749,566805,640],{},[31665,566807,566808],{"encoding":31667},"\\int_{1}^{A} \\cos x dx = \\sin A - \\sin 1",[34,566810,566812,566903,566927],{"className":566811,"ariaHidden":758},[31672],[34,566813,566815,566819,566876,566879,566882,566885,566888,566891,566894,566897,566900],{"className":566814},[31676],[34,566816],{"className":566817,"style":566818},[31680],"height:1.3921em;vertical-align:-0.3558em;",[34,566820,566822,566825],{"className":566821},[32767],[34,566823,546295],{"className":566824,"style":546331},[32767,131249,156560],[34,566826,566828],{"className":566827},[31792],[34,566829,566831,566868],{"className":566830},[31796,31797],[34,566832,566834,566865],{"className":566833},[31801],[34,566835,566837,566851],{"className":566836,"style":552657},[31805],[34,566838,566839,566842],{"style":546347},[34,566840],{"className":566841,"style":31814},[31813],[34,566843,566845],{"className":566844},[31818,31819,31820,31821],[34,566846,566848],{"className":566847},[31685,31821],[34,566849,640],{"className":566850},[31685,31821],[34,566852,566853,566856],{"style":546362},[34,566854],{"className":566855,"style":31814},[31813],[34,566857,566859],{"className":566858},[31818,31819,31820,31821],[34,566860,566862],{"className":566861},[31685,31821],[34,566863,54151],{"className":566864},[31685,31686,31821],[34,566866,31829],{"className":566867},[31828],[34,566869,566871],{"className":566870},[31801],[34,566872,566874],{"className":566873,"style":546384},[31805],[34,566875],{},[34,566877],{"className":566878,"style":32123},[31845],[34,566880,547860],{"className":566881},[32767],[34,566883],{"className":566884,"style":32123},[31845],[34,566886,32936],{"className":566887},[31685,31686],[34,566889,46900],{"className":566890},[31685,31686],[34,566892,32936],{"className":566893},[31685,31686],[34,566895],{"className":566896,"style":31846},[31845],[34,566898,202],{"className":566899},[31850],[34,566901],{"className":566902,"style":31846},[31845],[34,566904,566906,566909,566912,566915,566918,566921,566924],{"className":566905},[31676],[34,566907],{"className":566908,"style":34944},[31680],[34,566910,547853],{"className":566911},[32767],[34,566913],{"className":566914,"style":32123},[31845],[34,566916,54151],{"className":566917},[31685,31686],[34,566919],{"className":566920,"style":32604},[31845],[34,566922,32226],{"className":566923},[32024],[34,566925],{"className":566926,"style":32604},[31845],[34,566928,566930,566933,566936,566939],{"className":566929},[31676],[34,566931],{"className":566932,"style":71430},[31680],[34,566934,547853],{"className":566935},[32767],[34,566937],{"className":566938,"style":32123},[31845],[34,566940,640],{"className":566941},[31685],"，是有界的。",[108,566944,65840,566945,314945,567073,567152],{},[34,566946,566948,566970],{"className":566947},[31642],[34,566949,566951],{"className":566950},[31646],[31648,566952,566953],{"xmlns":31650},[31652,566954,566955,566967],{},[31655,566956,566957],{},[41453,566958,566959,566961],{},[31749,566960,640],{},[37624,566962,566963,566965],{},[31658,566964,32936],{},[31658,566966,217392],{},[31665,566968,566969],{"encoding":31667},"\\frac{1}{x^\\alpha}",[34,566971,566973],{"className":566972,"ariaHidden":758},[31672],[34,566974,566976,566979],{"className":566975},[31676],[34,566977],{"className":566978,"style":46727},[31680],[34,566980,566982,566985,567070],{"className":566981},[31685],[34,566983],{"className":566984},[31782,41488],[34,566986,566988],{"className":566987},[41453],[34,566989,566991,567062],{"className":566990},[31796,31797],[34,566992,566994,567059],{"className":566993},[31801],[34,566995,566997,567037,567045],{"className":566996,"style":46746},[31805],[34,566998,566999,567002],{"style":41504},[34,567000],{"className":567001,"style":41508},[31813],[34,567003,567005],{"className":567004},[31818,31819,31820,31821],[34,567006,567008],{"className":567007},[31685,31821],[34,567009,567011,567014],{"className":567010},[31685,31821],[34,567012,32936],{"className":567013},[31685,31686,31821],[34,567015,567017],{"className":567016},[31792],[34,567018,567020],{"className":567019},[31796],[34,567021,567023],{"className":567022},[31801],[34,567024,567026],{"className":567025,"style":566313},[31805],[34,567027,567028,567031],{"style":116668},[34,567029],{"className":567030,"style":43406},[31813],[34,567032,567034],{"className":567033},[31818,43410,43411,31821],[34,567035,217392],{"className":567036,"style":217496},[31685,31686,31821],[34,567038,567039,567042],{"style":41520},[34,567040],{"className":567041,"style":41508},[31813],[34,567043],{"className":567044,"style":41528},[41527],[34,567046,567047,567050],{"style":46771},[34,567048],{"className":567049,"style":41508},[31813],[34,567051,567053],{"className":567052},[31818,31819,31820,31821],[34,567054,567056],{"className":567055},[31685,31821],[34,567057,640],{"className":567058},[31685,31821],[34,567060,31829],{"className":567061},[31828],[34,567063,567065],{"className":567064},[31801],[34,567066,567068],{"className":567067,"style":41619},[31805],[34,567069],{},[34,567071],{"className":567072},[31690,41488],[34,567074,567076,567104],{"className":567075},[31642],[34,567077,567079],{"className":567078},[31646],[31648,567080,567081],{"xmlns":31650},[31652,567082,567083,567101],{},[31655,567084,567085,567087,567089,567091,567093,567095,567097,567099],{},[31658,567086,32936],{},[31662,567088,112296],{},[31662,567090,15972],{"stretchy":702},[31749,567092,640],{},[31662,567094,31960],{"separator":758},[31662,567096,223],{},[31658,567098,138286],{"mathvariant":38079},[31662,567100,2557],{"stretchy":702},[31665,567102,567103],{"encoding":31667},"x \\in [1, +\\infty)",[34,567105,567107,567125],{"className":567106,"ariaHidden":758},[31672],[34,567108,567110,567113,567116,567119,567122],{"className":567109},[31676],[34,567111],{"className":567112,"style":34812},[31680],[34,567114,32936],{"className":567115},[31685,31686],[34,567117],{"className":567118,"style":31846},[31845],[34,567120,112296],{"className":567121},[31850],[34,567123],{"className":567124,"style":31846},[31845],[34,567126,567128,567131,567134,567137,567140,567143,567146,567149],{"className":567127},[31676],[34,567129],{"className":567130,"style":31774},[31680],[34,567132,15972],{"className":567133},[31782],[34,567135,640],{"className":567136},[31685],[34,567138,31960],{"className":567139},[32119],[34,567141],{"className":567142,"style":32123},[31845],[34,567144,223],{"className":567145},[31685],[34,567147,138286],{"className":567148},[31685],[34,567150,2557],{"className":567151},[31690]," 上单调递减趋于 0。",[108,567154,567155,567156,97596,567409,567537,567538,567566],{},"关键点：它是否“一致”趋于0？是的，因为 ",[34,567157,567159,567193],{"className":567158},[31642],[34,567160,567162],{"className":567161},[31646],[31648,567163,567164],{"xmlns":31650},[31652,567165,567166,567190],{},[31655,567167,567168,567178,567180],{},[41453,567169,567170,567172],{},[31749,567171,640],{},[37624,567173,567174,567176],{},[31658,567175,32936],{},[31658,567177,217392],{},[31662,567179,45972],{},[41453,567181,567182,567184],{},[31749,567183,640],{},[37624,567185,567186,567188],{},[31658,567187,32936],{},[31658,567189,231507],{},[31665,567191,567192],{"encoding":31667},"\\frac{1}{x^\\alpha} \\leq \\frac{1}{x^\\delta}",[34,567194,567196,567305],{"className":567195,"ariaHidden":758},[31672],[34,567197,567199,567202,567296,567299,567302],{"className":567198},[31676],[34,567200],{"className":567201,"style":46727},[31680],[34,567203,567205,567208,567293],{"className":567204},[31685],[34,567206],{"className":567207},[31782,41488],[34,567209,567211],{"className":567210},[41453],[34,567212,567214,567285],{"className":567213},[31796,31797],[34,567215,567217,567282],{"className":567216},[31801],[34,567218,567220,567260,567268],{"className":567219,"style":46746},[31805],[34,567221,567222,567225],{"style":41504},[34,567223],{"className":567224,"style":41508},[31813],[34,567226,567228],{"className":567227},[31818,31819,31820,31821],[34,567229,567231],{"className":567230},[31685,31821],[34,567232,567234,567237],{"className":567233},[31685,31821],[34,567235,32936],{"className":567236},[31685,31686,31821],[34,567238,567240],{"className":567239},[31792],[34,567241,567243],{"className":567242},[31796],[34,567244,567246],{"className":567245},[31801],[34,567247,567249],{"className":567248,"style":566313},[31805],[34,567250,567251,567254],{"style":116668},[34,567252],{"className":567253,"style":43406},[31813],[34,567255,567257],{"className":567256},[31818,43410,43411,31821],[34,567258,217392],{"className":567259,"style":217496},[31685,31686,31821],[34,567261,567262,567265],{"style":41520},[34,567263],{"className":567264,"style":41508},[31813],[34,567266],{"className":567267,"style":41528},[41527],[34,567269,567270,567273],{"style":46771},[34,567271],{"className":567272,"style":41508},[31813],[34,567274,567276],{"className":567275},[31818,31819,31820,31821],[34,567277,567279],{"className":567278},[31685,31821],[34,567280,640],{"className":567281},[31685,31821],[34,567283,31829],{"className":567284},[31828],[34,567286,567288],{"className":567287},[31801],[34,567289,567291],{"className":567290,"style":41619},[31805],[34,567292],{},[34,567294],{"className":567295},[31690,41488],[34,567297],{"className":567298,"style":31846},[31845],[34,567300,45972],{"className":567301},[31850],[34,567303],{"className":567304,"style":31846},[31845],[34,567306,567308,567312],{"className":567307},[31676],[34,567309],{"className":567310,"style":567311},[31680],"height:1.2025em;vertical-align:-0.3574em;",[34,567313,567315,567318,567406],{"className":567314},[31685],[34,567316],{"className":567317},[31782,41488],[34,567319,567321],{"className":567320},[41453],[34,567322,567324,567397],{"className":567323},[31796,31797],[34,567325,567327,567394],{"className":567326},[31801],[34,567328,567330,567372,567380],{"className":567329,"style":46746},[31805],[34,567331,567333,567336],{"style":567332},"top:-2.6426em;",[34,567334],{"className":567335,"style":41508},[31813],[34,567337,567339],{"className":567338},[31818,31819,31820,31821],[34,567340,567342],{"className":567341},[31685,31821],[34,567343,567345,567348],{"className":567344},[31685,31821],[34,567346,32936],{"className":567347},[31685,31686,31821],[34,567349,567351],{"className":567350},[31792],[34,567352,567354],{"className":567353},[31796],[34,567355,567357],{"className":567356},[31801],[34,567358,567361],{"className":567359,"style":567360},[31805],"height:0.782em;",[34,567362,567363,567366],{"style":116668},[34,567364],{"className":567365,"style":43406},[31813],[34,567367,567369],{"className":567368},[31818,43410,43411,31821],[34,567370,231507],{"className":567371,"style":231523},[31685,31686,31821],[34,567373,567374,567377],{"style":41520},[34,567375],{"className":567376,"style":41508},[31813],[34,567378],{"className":567379,"style":41528},[41527],[34,567381,567382,567385],{"style":46771},[34,567383],{"className":567384,"style":41508},[31813],[34,567386,567388],{"className":567387},[31818,31819,31820,31821],[34,567389,567391],{"className":567390},[31685,31821],[34,567392,640],{"className":567393},[31685,31821],[34,567395,31829],{"className":567396},[31828],[34,567398,567400],{"className":567399},[31801],[34,567401,567404],{"className":567402,"style":567403},[31805],"height:0.3574em;",[34,567405],{},[34,567407],{"className":567408},[31690,41488],[34,567410,567412,567434],{"className":567411},[31642],[34,567413,567415],{"className":567414},[31646],[31648,567416,567417],{"xmlns":31650},[31652,567418,567419,567431],{},[31655,567420,567421],{},[41453,567422,567423,567425],{},[31749,567424,640],{},[37624,567426,567427,567429],{},[31658,567428,32936],{},[31658,567430,231507],{},[31665,567432,567433],{"encoding":31667},"\\frac{1}{x^\\delta}",[34,567435,567437],{"className":567436,"ariaHidden":758},[31672],[34,567438,567440,567443],{"className":567439},[31676],[34,567441],{"className":567442,"style":567311},[31680],[34,567444,567446,567449,567534],{"className":567445},[31685],[34,567447],{"className":567448},[31782,41488],[34,567450,567452],{"className":567451},[41453],[34,567453,567455,567526],{"className":567454},[31796,31797],[34,567456,567458,567523],{"className":567457},[31801],[34,567459,567461,567501,567509],{"className":567460,"style":46746},[31805],[34,567462,567463,567466],{"style":567332},[34,567464],{"className":567465,"style":41508},[31813],[34,567467,567469],{"className":567468},[31818,31819,31820,31821],[34,567470,567472],{"className":567471},[31685,31821],[34,567473,567475,567478],{"className":567474},[31685,31821],[34,567476,32936],{"className":567477},[31685,31686,31821],[34,567479,567481],{"className":567480},[31792],[34,567482,567484],{"className":567483},[31796],[34,567485,567487],{"className":567486},[31801],[34,567488,567490],{"className":567489,"style":567360},[31805],[34,567491,567492,567495],{"style":116668},[34,567493],{"className":567494,"style":43406},[31813],[34,567496,567498],{"className":567497},[31818,43410,43411,31821],[34,567499,231507],{"className":567500,"style":231523},[31685,31686,31821],[34,567502,567503,567506],{"style":41520},[34,567504],{"className":567505,"style":41508},[31813],[34,567507],{"className":567508,"style":41528},[41527],[34,567510,567511,567514],{"style":46771},[34,567512],{"className":567513,"style":41508},[31813],[34,567515,567517],{"className":567516},[31818,31819,31820,31821],[34,567518,567520],{"className":567519},[31685,31821],[34,567521,640],{"className":567522},[31685,31821],[34,567524,31829],{"className":567525},[31828],[34,567527,567529],{"className":567528},[31801],[34,567530,567532],{"className":567531,"style":567403},[31805],[34,567533],{},[34,567535],{"className":567536},[31690,41488]," 趋于0与 ",[34,567539,567541,567554],{"className":567540},[31642],[34,567542,567544],{"className":567543},[31646],[31648,567545,567546],{"xmlns":31650},[31652,567547,567548,567552],{},[31655,567549,567550],{},[31658,567551,217392],{},[31665,567553,218244],{"encoding":31667},[34,567555,567557],{"className":567556,"ariaHidden":758},[31672],[34,567558,567560,567563],{"className":567559},[31676],[34,567561],{"className":567562,"style":32527},[31680],[34,567564,217392],{"className":567565,"style":217496},[31685,31686]," 无关。",[108,567568,567569,567572,567573,567629],{},[2627,567570,567571],{},"一致收敛结论："," 所以在任意 ",[34,567574,567576,567599],{"className":567575},[31642],[34,567577,567579],{"className":567578},[31646],[31648,567580,567581],{"xmlns":31650},[31652,567582,567583,567597],{},[31655,567584,567585,567587,567589,567591,567593,567595],{},[31662,567586,15972],{"stretchy":702},[31658,567588,231507],{},[31662,567590,31960],{"separator":758},[31662,567592,223],{},[31658,567594,138286],{"mathvariant":38079},[31662,567596,2557],{"stretchy":702},[31665,567598,566576],{"encoding":31667},[34,567600,567602],{"className":567601,"ariaHidden":758},[31672],[34,567603,567605,567608,567611,567614,567617,567620,567623,567626],{"className":567604},[31676],[34,567606],{"className":567607,"style":31774},[31680],[34,567609,15972],{"className":567610},[31782],[34,567612,231507],{"className":567613,"style":231523},[31685,31686],[34,567615,31960],{"className":567616},[32119],[34,567618],{"className":567619,"style":32123},[31845],[34,567621,223],{"className":567622},[31685],[34,567624,138286],{"className":567625},[31685],[34,567627,2557],{"className":567628},[31690]," 上一致收敛。",[108,567631,567632,567635,567636,567686,567687,567730,567731,566439],{},[2627,567633,567634],{},"连续性结论："," 因为在任意 ",[34,567637,567639,567656],{"className":567638},[31642],[34,567640,567642],{"className":567641},[31646],[31648,567643,567644],{"xmlns":31650},[31652,567645,567646,567654],{},[31655,567647,567648,567650,567652],{},[31658,567649,217392],{},[31662,567651,29622],{},[31749,567653,459],{},[31665,567655,565845],{"encoding":31667},[34,567657,567659,567677],{"className":567658,"ariaHidden":758},[31672],[34,567660,567662,567665,567668,567671,567674],{"className":567661},[31676],[34,567663],{"className":567664,"style":34812},[31680],[34,567666,217392],{"className":567667,"style":217496},[31685,31686],[34,567669],{"className":567670,"style":31846},[31845],[34,567672,29622],{"className":567673},[31850],[34,567675],{"className":567676,"style":31846},[31845],[34,567678,567680,567683],{"className":567679},[31676],[34,567681],{"className":567682,"style":32617},[31680],[34,567684,459],{"className":567685},[31685]," 的邻域内都一致收敛，所以 ",[34,567688,567690,567709],{"className":567689},[31642],[34,567691,567693],{"className":567692},[31646],[31648,567694,567695],{"xmlns":31650},[31652,567696,567697,567707],{},[31655,567698,567699,567701,567703,567705],{},[31658,567700,51834],{},[31662,567702,456],{"stretchy":702},[31658,567704,217392],{},[31662,567706,2557],{"stretchy":702},[31665,567708,546701],{"encoding":31667},[34,567710,567712],{"className":567711,"ariaHidden":758},[31672],[34,567713,567715,567718,567721,567724,567727],{"className":567714},[31676],[34,567716],{"className":567717,"style":31774},[31680],[34,567719,51834],{"className":567720,"style":37422},[31685,31686],[34,567722,456],{"className":567723},[31782],[34,567725,217392],{"className":567726,"style":217496},[31685,31686],[34,567728,2557],{"className":567729},[31690]," 在每一点都连续，即在 ",[34,567732,567734,567757],{"className":567733},[31642],[34,567735,567737],{"className":567736},[31646],[31648,567738,567739],{"xmlns":31650},[31652,567740,567741,567755],{},[31655,567742,567743,567745,567747,567749,567751,567753],{},[31662,567744,456],{"stretchy":702},[31749,567746,459],{},[31662,567748,31960],{"separator":758},[31662,567750,223],{},[31658,567752,138286],{"mathvariant":38079},[31662,567754,2557],{"stretchy":702},[31665,567756,566408],{"encoding":31667},[34,567758,567760],{"className":567759,"ariaHidden":758},[31672],[34,567761,567763,567766,567769,567772,567775,567778,567781,567784],{"className":567762},[31676],[34,567764],{"className":567765,"style":31774},[31680],[34,567767,456],{"className":567768},[31782],[34,567770,459],{"className":567771},[31685],[34,567773,31960],{"className":567774},[32119],[34,567776],{"className":567777,"style":32123},[31845],[34,567779,223],{"className":567780},[31685],[34,567782,138286],{"className":567783},[31685],[34,567785,2557],{"className":567786},[31690],[32188,567788],{},[253,567790,567792],{"id":567791},"p329-t6","p329 T6",[15,567794,567795,567797,567798,568136],{},[2627,567796,214361],{},"\n确定函数 ",[34,567799,567801,567873],{"className":567800},[31642],[34,567802,567804],{"className":567803},[31646],[31648,567805,567806],{"xmlns":31650},[31652,567807,567808,567870],{},[31655,567809,567810,567812,567814,567816,567818,567820,567828,567866,567868],{},[31658,567811,51834],{},[31662,567813,456],{"stretchy":702},[31658,567815,67764],{},[31662,567817,2557],{"stretchy":702},[31662,567819,202],{},[38846,567821,567822,567824,567826],{},[31662,567823,546295],{},[31749,567825,459],{},[31658,567827,58442],{},[41453,567829,567830,567838],{},[31655,567831,567832,567834,567836],{},[31658,567833,547853],{},[31662,567835,32682],{},[31658,567837,32936],{},[31655,567839,567840,567846,567848,567850,567852,567854],{},[37624,567841,567842,567844],{},[31658,567843,32936],{},[31658,567845,67764],{},[31662,567847,456],{"stretchy":702},[31658,567849,58442],{},[31662,567851,32226],{},[31658,567853,32936],{},[37624,567855,567856,567858],{},[31662,567857,2557],{"stretchy":702},[31655,567859,567860,567862,567864],{},[31749,567861,768],{},[31662,567863,32226],{},[31658,567865,67764],{},[31658,567867,46900],{},[31658,567869,32936],{},[31665,567871,567872],{"encoding":31667},"F(y) = \\int_{0}^{\\pi} \\frac{\\sin x}{x^y(\\pi - x)^{2-y}}dx",[34,567874,567876,567903],{"className":567875,"ariaHidden":758},[31672],[34,567877,567879,567882,567885,567888,567891,567894,567897,567900],{"className":567878},[31676],[34,567880],{"className":567881,"style":31774},[31680],[34,567883,51834],{"className":567884,"style":37422},[31685,31686],[34,567886,456],{"className":567887},[31782],[34,567889,67764],{"className":567890,"style":32062},[31685,31686],[34,567892,2557],{"className":567893},[31690],[34,567895],{"className":567896,"style":31846},[31845],[34,567898,202],{"className":567899},[31850],[34,567901],{"className":567902,"style":31846},[31845],[34,567904,567906,567910,567968,567971,568130,568133],{"className":567905},[31676],[34,567907],{"className":567908,"style":567909},[31680],"height:1.3815em;vertical-align:-0.52em;",[34,567911,567913,567916],{"className":567912},[32767],[34,567914,546295],{"className":567915,"style":546331},[32767,131249,156560],[34,567917,567919],{"className":567918},[31792],[34,567920,567922,567960],{"className":567921},[31796,31797],[34,567923,567925,567957],{"className":567924},[31801],[34,567926,567929,567943],{"className":567927,"style":567928},[31805],"height:0.8593em;",[34,567930,567931,567934],{"style":546347},[34,567932],{"className":567933,"style":31814},[31813],[34,567935,567937],{"className":567936},[31818,31819,31820,31821],[34,567938,567940],{"className":567939},[31685,31821],[34,567941,459],{"className":567942},[31685,31821],[34,567944,567945,567948],{"style":546362},[34,567946],{"className":567947,"style":31814},[31813],[34,567949,567951],{"className":567950},[31818,31819,31820,31821],[34,567952,567954],{"className":567953},[31685,31821],[34,567955,58442],{"className":567956,"style":32062},[31685,31686,31821],[34,567958,31829],{"className":567959},[31828],[34,567961,567963],{"className":567962},[31801],[34,567964,567966],{"className":567965,"style":546384},[31805],[34,567967],{},[34,567969],{"className":567970,"style":32123},[31845],[34,567972,567974,567977,568127],{"className":567973},[31685],[34,567975],{"className":567976},[31782,41488],[34,567978,567980],{"className":567979},[41453],[34,567981,567983,568119],{"className":567982},[31796,31797],[34,567984,567986,568116],{"className":567985},[31801],[34,567987,567989,568079,568087],{"className":567988,"style":548640},[31805],[34,567990,567991,567994],{"style":41504},[34,567992],{"className":567993,"style":41508},[31813],[34,567995,567997],{"className":567996},[31818,31819,31820,31821],[34,567998,568000,568029,568032,568035,568038,568041],{"className":567999},[31685,31821],[34,568001,568003,568006],{"className":568002},[31685,31821],[34,568004,32936],{"className":568005},[31685,31686,31821],[34,568007,568009],{"className":568008},[31792],[34,568010,568012],{"className":568011},[31796],[34,568013,568015],{"className":568014},[31801],[34,568016,568018],{"className":568017,"style":566313},[31805],[34,568019,568020,568023],{"style":116668},[34,568021],{"className":568022,"style":43406},[31813],[34,568024,568026],{"className":568025},[31818,43410,43411,31821],[34,568027,67764],{"className":568028,"style":32062},[31685,31686,31821],[34,568030,456],{"className":568031},[31782,31821],[34,568033,58442],{"className":568034,"style":32062},[31685,31686,31821],[34,568036,32226],{"className":568037},[32024,31821],[34,568039,32936],{"className":568040},[31685,31686,31821],[34,568042,568044,568047],{"className":568043},[31690,31821],[34,568045,2557],{"className":568046},[31690,31821],[34,568048,568050],{"className":568049},[31792],[34,568051,568053],{"className":568052},[31796],[34,568054,568056],{"className":568055},[31801],[34,568057,568059],{"className":568058,"style":548314},[31805],[34,568060,568061,568064],{"style":116668},[34,568062],{"className":568063,"style":43406},[31813],[34,568065,568067],{"className":568066},[31818,43410,43411,31821],[34,568068,568070,568073,568076],{"className":568069},[31685,31821],[34,568071,768],{"className":568072},[31685,31821],[34,568074,32226],{"className":568075},[32024,31821],[34,568077,67764],{"className":568078,"style":32062},[31685,31686,31821],[34,568080,568081,568084],{"style":41520},[34,568082],{"className":568083,"style":41508},[31813],[34,568085],{"className":568086,"style":41528},[41527],[34,568088,568089,568092],{"style":46771},[34,568090],{"className":568091,"style":41508},[31813],[34,568093,568095],{"className":568094},[31818,31819,31820,31821],[34,568096,568098,568110,568113],{"className":568097},[31685,31821],[34,568099,568101,568104,568107],{"className":568100},[32767,31821],[34,568102,31747],{"className":568103},[31821],[34,568105,31955],{"className":568106},[31821],[34,568108,31660],{"className":568109},[31821],[34,568111],{"className":568112,"style":43430},[31845,31821],[34,568114,32936],{"className":568115},[31685,31686,31821],[34,568117,31829],{"className":568118},[31828],[34,568120,568122],{"className":568121},[31801],[34,568123,568125],{"className":568124,"style":324858},[31805],[34,568126],{},[34,568128],{"className":568129},[31690,41488],[34,568131,46900],{"className":568132},[31685,31686],[34,568134,32936],{"className":568135},[31685,31686]," 的连续范围。",[15,568138,568139,568141,568142,258,568192,568243],{},[2627,568140,91513],{},"\n这个积分有两个可能的“瑕点”（分母为0的点）：",[34,568143,568145,568162],{"className":568144},[31642],[34,568146,568148],{"className":568147},[31646],[31648,568149,568150],{"xmlns":31650},[31652,568151,568152,568160],{},[31655,568153,568154,568156,568158],{},[31658,568155,32936],{},[31662,568157,202],{},[31749,568159,459],{},[31665,568161,563480],{"encoding":31667},[34,568163,568165,568183],{"className":568164,"ariaHidden":758},[31672],[34,568166,568168,568171,568174,568177,568180],{"className":568167},[31676],[34,568169],{"className":568170,"style":32527},[31680],[34,568172,32936],{"className":568173},[31685,31686],[34,568175],{"className":568176,"style":31846},[31845],[34,568178,202],{"className":568179},[31850],[34,568181],{"className":568182,"style":31846},[31845],[34,568184,568186,568189],{"className":568185},[31676],[34,568187],{"className":568188,"style":32617},[31680],[34,568190,459],{"className":568191},[31685],[34,568193,568195,568213],{"className":568194},[31642],[34,568196,568198],{"className":568197},[31646],[31648,568199,568200],{"xmlns":31650},[31652,568201,568202,568210],{},[31655,568203,568204,568206,568208],{},[31658,568205,32936],{},[31662,568207,202],{},[31658,568209,58442],{},[31665,568211,568212],{"encoding":31667},"x=\\pi",[34,568214,568216,568234],{"className":568215,"ariaHidden":758},[31672],[34,568217,568219,568222,568225,568228,568231],{"className":568218},[31676],[34,568220],{"className":568221,"style":32527},[31680],[34,568223,32936],{"className":568224},[31685,31686],[34,568226],{"className":568227,"style":31846},[31845],[34,568229,202],{"className":568230},[31850],[34,568232],{"className":568233,"style":31846},[31845],[34,568235,568237,568240],{"className":568236},[31676],[34,568238],{"className":568239,"style":32527},[31680],[34,568241,58442],{"className":568242,"style":32062},[31685,31686],"。积分收敛才能谈连续。",[15,568245,568246],{},[2627,568247,549629],{},[2230,568249,568250,568789,569604],{},[108,568251,568252,568306,568367,568368,568656,568657,347],{},[2627,568253,326327,568254,568305],{},[34,568255,568257,568275],{"className":568256},[31642],[34,568258,568260],{"className":568259},[31646],[31648,568261,568262],{"xmlns":31650},[31652,568263,568264,568272],{},[31655,568265,568266,568268,568270],{},[31658,568267,32936],{},[31662,568269,33239],{},[31749,568271,459],{},[31665,568273,568274],{"encoding":31667},"x \\to 0",[34,568276,568278,568296],{"className":568277,"ariaHidden":758},[31672],[34,568279,568281,568284,568287,568290,568293],{"className":568280},[31676],[34,568282],{"className":568283,"style":32527},[31680],[34,568285,32936],{"className":568286},[31685,31686],[34,568288],{"className":568289,"style":31846},[31845],[34,568291,33239],{"className":568292},[31850],[34,568294],{"className":568295,"style":31846},[31845],[34,568297,568299,568302],{"className":568298},[31676],[34,568300],{"className":568301,"style":32617},[31680],[34,568303,459],{"className":568304},[31685]," 处：",[34,568307,568309,568331],{"className":568308},[31642],[34,568310,568312],{"className":568311},[31646],[31648,568313,568314],{"xmlns":31650},[31652,568315,568316,568328],{},[31655,568317,568318,568320,568322,568324,568326],{},[31658,568319,547853],{},[31662,568321,32682],{},[31658,568323,32936],{},[31662,568325,190205],{},[31658,568327,32936],{},[31665,568329,568330],{"encoding":31667},"\\sin x \\sim x",[34,568332,568334,568358],{"className":568333,"ariaHidden":758},[31672],[34,568335,568337,568340,568343,568346,568349,568352,568355],{"className":568336},[31676],[34,568338],{"className":568339,"style":71430},[31680],[34,568341,547853],{"className":568342},[32767],[34,568344],{"className":568345,"style":32123},[31845],[34,568347,32936],{"className":568348},[31685,31686],[34,568350],{"className":568351,"style":31846},[31845],[34,568353,190205],{"className":568354},[31850],[34,568356],{"className":568357,"style":31846},[31845],[34,568359,568361,568364],{"className":568360},[31676],[34,568362],{"className":568363,"style":32527},[31680],[34,568365,32936],{"className":568366},[31685,31686]," (等价无穷小)。\n被积函数 ",[34,568369,568371,568429],{"className":568370},[31642],[34,568372,568374],{"className":568373},[31646],[31648,568375,568376],{"xmlns":31650},[31652,568377,568378,568426],{},[31655,568379,568380,568382,568408,568410,568412,568414],{},[31662,568381,168560],{},[41453,568383,568384,568386],{},[31658,568385,32936],{},[31655,568387,568388,568394,568396],{},[37624,568389,568390,568392],{},[31658,568391,32936],{},[31658,568393,67764],{},[31662,568395,71191],{},[37624,568397,568398,568400],{},[31658,568399,58442],{},[31655,568401,568402,568404,568406],{},[31749,568403,768],{},[31662,568405,32226],{},[31658,568407,67764],{},[31662,568409,202],{},[31658,568411,47384],{},[31662,568413,71191],{},[37624,568415,568416,568418],{},[31658,568417,32936],{},[31655,568419,568420,568422,568424],{},[31749,568421,640],{},[31662,568423,32226],{},[31658,568425,67764],{},[31665,568427,568428],{"encoding":31667},"\\approx \\frac{x}{x^y \\cdot \\pi^{2-y}} = C \\cdot x^{1-y}",[34,568430,568432,568444,568594,568612],{"className":568431,"ariaHidden":758},[31672],[34,568433,568435,568438,568441],{"className":568434},[31676],[34,568436],{"className":568437,"style":211939},[31680],[34,568439,168560],{"className":568440},[31850],[34,568442],{"className":568443,"style":31846},[31845],[34,568445,568447,568450,568585,568588,568591],{"className":568446},[31676],[34,568448],{"className":568449,"style":244926},[31680],[34,568451,568453,568456,568582],{"className":568452},[31685],[34,568454],{"className":568455},[31782,41488],[34,568457,568459],{"className":568458},[41453],[34,568460,568462,568574],{"className":568461},[31796,31797],[34,568463,568465,568571],{"className":568464},[31801],[34,568466,568468,568549,568557],{"className":568467,"style":244945},[31805],[34,568469,568470,568473],{"style":41504},[34,568471],{"className":568472,"style":41508},[31813],[34,568474,568476],{"className":568475},[31818,31819,31820,31821],[34,568477,568479,568508,568511],{"className":568478},[31685,31821],[34,568480,568482,568485],{"className":568481},[31685,31821],[34,568483,32936],{"className":568484},[31685,31686,31821],[34,568486,568488],{"className":568487},[31792],[34,568489,568491],{"className":568490},[31796],[34,568492,568494],{"className":568493},[31801],[34,568495,568497],{"className":568496,"style":566313},[31805],[34,568498,568499,568502],{"style":116668},[34,568500],{"className":568501,"style":43406},[31813],[34,568503,568505],{"className":568504},[31818,43410,43411,31821],[34,568506,67764],{"className":568507,"style":32062},[31685,31686,31821],[34,568509,71191],{"className":568510},[32024,31821],[34,568512,568514,568517],{"className":568513},[31685,31821],[34,568515,58442],{"className":568516,"style":32062},[31685,31686,31821],[34,568518,568520],{"className":568519},[31792],[34,568521,568523],{"className":568522},[31796],[34,568524,568526],{"className":568525},[31801],[34,568527,568529],{"className":568528,"style":548314},[31805],[34,568530,568531,568534],{"style":116668},[34,568532],{"className":568533,"style":43406},[31813],[34,568535,568537],{"className":568536},[31818,43410,43411,31821],[34,568538,568540,568543,568546],{"className":568539},[31685,31821],[34,568541,768],{"className":568542},[31685,31821],[34,568544,32226],{"className":568545},[32024,31821],[34,568547,67764],{"className":568548,"style":32062},[31685,31686,31821],[34,568550,568551,568554],{"style":41520},[34,568552],{"className":568553,"style":41508},[31813],[34,568555],{"className":568556,"style":41528},[41527],[34,568558,568559,568562],{"style":46771},[34,568560],{"className":568561,"style":41508},[31813],[34,568563,568565],{"className":568564},[31818,31819,31820,31821],[34,568566,568568],{"className":568567},[31685,31821],[34,568569,32936],{"className":568570},[31685,31686,31821],[34,568572,31829],{"className":568573},[31828],[34,568575,568577],{"className":568576},[31801],[34,568578,568580],{"className":568579,"style":41619},[31805],[34,568581],{},[34,568583],{"className":568584},[31690,41488],[34,568586],{"className":568587,"style":31846},[31845],[34,568589,202],{"className":568590},[31850],[34,568592],{"className":568593,"style":31846},[31845],[34,568595,568597,568600,568603,568606,568609],{"className":568596},[31676],[34,568598],{"className":568599,"style":33218},[31680],[34,568601,47384],{"className":568602,"style":38706},[31685,31686],[34,568604],{"className":568605,"style":32604},[31845],[34,568607,71191],{"className":568608},[32024],[34,568610],{"className":568611,"style":32604},[31845],[34,568613,568615,568618],{"className":568614},[31676],[34,568616],{"className":568617,"style":37774},[31680],[34,568619,568621,568624],{"className":568620},[31685],[34,568622,32936],{"className":568623},[31685,31686],[34,568625,568627],{"className":568626},[31792],[34,568628,568630],{"className":568629},[31796],[34,568631,568633],{"className":568632},[31801],[34,568634,568636],{"className":568635,"style":37774},[31805],[34,568637,568638,568641],{"style":37777},[34,568639],{"className":568640,"style":31814},[31813],[34,568642,568644],{"className":568643},[31818,31819,31820,31821],[34,568645,568647,568650,568653],{"className":568646},[31685,31821],[34,568648,640],{"className":568649},[31685,31821],[34,568651,32226],{"className":568652},[32024,31821],[34,568654,67764],{"className":568655,"style":32062},[31685,31686,31821],"。\n我们要让积分收敛，幂次必须大于 -1。\n即：",[34,568658,568660,568696],{"className":568659},[31642],[34,568661,568663],{"className":568662},[31646],[31648,568664,568665],{"xmlns":31650},[31652,568666,568667,568693],{},[31655,568668,568669,568671,568673,568675,568677,568679,568681,568683,568685,568687,568689,568691],{},[31749,568670,640],{},[31662,568672,32226],{},[31658,568674,67764],{},[31662,568676,29622],{},[31662,568678,32226],{},[31749,568680,640],{},[39361,568682,262622],{},[31662,568684,262625],{},[39361,568686,262622],{},[31658,568688,67764],{},[31662,568690,932],{},[31749,568692,768],{},[31665,568694,568695],{"encoding":31667},"1-y > -1 \\implies y \u003C 2",[34,568697,568699,568717,568735,568762,568780],{"className":568698,"ariaHidden":758},[31672],[34,568700,568702,568705,568708,568711,568714],{"className":568701},[31676],[34,568703],{"className":568704,"style":35942},[31680],[34,568706,640],{"className":568707},[31685],[34,568709],{"className":568710,"style":32604},[31845],[34,568712,32226],{"className":568713},[32024],[34,568715],{"className":568716,"style":32604},[31845],[34,568718,568720,568723,568726,568729,568732],{"className":568719},[31676],[34,568721],{"className":568722,"style":86652},[31680],[34,568724,67764],{"className":568725,"style":32062},[31685,31686],[34,568727],{"className":568728,"style":31846},[31845],[34,568730,29622],{"className":568731},[31850],[34,568733],{"className":568734,"style":31846},[31845],[34,568736,568738,568741,568744,568747,568750,568753,568756,568759],{"className":568737},[31676],[34,568739],{"className":568740,"style":35942},[31680],[34,568742,32226],{"className":568743},[31685],[34,568745,640],{"className":568746},[31685],[34,568748],{"className":568749,"style":31846},[31845],[34,568751],{"className":568752,"style":31846},[31845],[34,568754,262625],{"className":568755},[31850],[34,568757],{"className":568758,"style":31846},[31845],[34,568760],{"className":568761,"style":31846},[31845],[34,568763,568765,568768,568771,568774,568777],{"className":568764},[31676],[34,568766],{"className":568767,"style":86652},[31680],[34,568769,67764],{"className":568770,"style":32062},[31685,31686],[34,568772],{"className":568773,"style":31846},[31845],[34,568775,932],{"className":568776},[31850],[34,568778],{"className":568779,"style":31846},[31845],[34,568781,568783,568786],{"className":568782},[31676],[34,568784],{"className":568785,"style":32617},[31680],[34,568787,768],{"className":568788},[31685],[108,568790,568791,478285,568844,55016,568917,569071,569072,569471,569472,347],{},[2627,568792,326327,568793,568305],{},[34,568794,568796,568814],{"className":568795},[31642],[34,568797,568799],{"className":568798},[31646],[31648,568800,568801],{"xmlns":31650},[31652,568802,568803,568811],{},[31655,568804,568805,568807,568809],{},[31658,568806,32936],{},[31662,568808,33239],{},[31658,568810,58442],{},[31665,568812,568813],{"encoding":31667},"x \\to \\pi",[34,568815,568817,568835],{"className":568816,"ariaHidden":758},[31672],[34,568818,568820,568823,568826,568829,568832],{"className":568819},[31676],[34,568821],{"className":568822,"style":32527},[31680],[34,568824,32936],{"className":568825},[31685,31686],[34,568827],{"className":568828,"style":31846},[31845],[34,568830,33239],{"className":568831},[31850],[34,568833],{"className":568834,"style":31846},[31845],[34,568836,568838,568841],{"className":568837},[31676],[34,568839],{"className":568840,"style":32527},[31680],[34,568842,58442],{"className":568843,"style":32062},[31685,31686],[34,568845,568847,568869],{"className":568846},[31642],[34,568848,568850],{"className":568849},[31646],[31648,568851,568852],{"xmlns":31650},[31652,568853,568854,568866],{},[31655,568855,568856,568858,568860,568862,568864],{},[31658,568857,33249],{},[31662,568859,202],{},[31658,568861,58442],{},[31662,568863,32226],{},[31658,568865,32936],{},[31665,568867,568868],{"encoding":31667},"t = \\pi - x",[34,568870,568872,568890,568908],{"className":568871,"ariaHidden":758},[31672],[34,568873,568875,568878,568881,568884,568887],{"className":568874},[31676],[34,568876],{"className":568877,"style":33280},[31680],[34,568879,33249],{"className":568880},[31685,31686],[34,568882],{"className":568883,"style":31846},[31845],[34,568885,202],{"className":568886},[31850],[34,568888],{"className":568889,"style":31846},[31845],[34,568891,568893,568896,568899,568902,568905],{"className":568892},[31676],[34,568894],{"className":568895,"style":32597},[31680],[34,568897,58442],{"className":568898,"style":32062},[31685,31686],[34,568900],{"className":568901,"style":32604},[31845],[34,568903,32226],{"className":568904},[32024],[34,568906],{"className":568907,"style":32604},[31845],[34,568909,568911,568914],{"className":568910},[31676],[34,568912],{"className":568913,"style":32527},[31680],[34,568915,32936],{"className":568916},[31685,31686],[34,568918,568920,568966],{"className":568919},[31642],[34,568921,568923],{"className":568922},[31646],[31648,568924,568925],{"xmlns":31650},[31652,568926,568927,568963],{},[31655,568928,568929,568931,568933,568935,568937,568939,568941,568943,568945,568947,568949,568951,568953,568955,568957,568959,568961],{},[31658,568930,547853],{},[31662,568932,32682],{},[31658,568934,32936],{},[31662,568936,202],{},[31658,568938,547853],{},[31662,568940,32682],{},[31662,568942,456],{"stretchy":702},[31658,568944,58442],{},[31662,568946,32226],{},[31658,568948,33249],{},[31662,568950,2557],{"stretchy":702},[31662,568952,202],{},[31658,568954,547853],{},[31662,568956,32682],{},[31658,568958,33249],{},[31662,568960,190205],{},[31658,568962,33249],{},[31665,568964,568965],{"encoding":31667},"\\sin x = \\sin(\\pi - t) = \\sin t \\sim t",[34,568967,568969,568993,569017,569038,569062],{"className":568968,"ariaHidden":758},[31672],[34,568970,568972,568975,568978,568981,568984,568987,568990],{"className":568971},[31676],[34,568973],{"className":568974,"style":71430},[31680],[34,568976,547853],{"className":568977},[32767],[34,568979],{"className":568980,"style":32123},[31845],[34,568982,32936],{"className":568983},[31685,31686],[34,568985],{"className":568986,"style":31846},[31845],[34,568988,202],{"className":568989},[31850],[34,568991],{"className":568992,"style":31846},[31845],[34,568994,568996,568999,569002,569005,569008,569011,569014],{"className":568995},[31676],[34,568997],{"className":568998,"style":31774},[31680],[34,569000,547853],{"className":569001},[32767],[34,569003,456],{"className":569004},[31782],[34,569006,58442],{"className":569007,"style":32062},[31685,31686],[34,569009],{"className":569010,"style":32604},[31845],[34,569012,32226],{"className":569013},[32024],[34,569015],{"className":569016,"style":32604},[31845],[34,569018,569020,569023,569026,569029,569032,569035],{"className":569019},[31676],[34,569021],{"className":569022,"style":31774},[31680],[34,569024,33249],{"className":569025},[31685,31686],[34,569027,2557],{"className":569028},[31690],[34,569030],{"className":569031,"style":31846},[31845],[34,569033,202],{"className":569034},[31850],[34,569036],{"className":569037,"style":31846},[31845],[34,569039,569041,569044,569047,569050,569053,569056,569059],{"className":569040},[31676],[34,569042],{"className":569043,"style":71430},[31680],[34,569045,547853],{"className":569046},[32767],[34,569048],{"className":569049,"style":32123},[31845],[34,569051,33249],{"className":569052},[31685,31686],[34,569054],{"className":569055,"style":31846},[31845],[34,569057,190205],{"className":569058},[31850],[34,569060],{"className":569061,"style":31846},[31845],[34,569063,569065,569068],{"className":569064},[31676],[34,569066],{"className":569067,"style":33280},[31680],[34,569069,33249],{"className":569070},[31685,31686],"。\n被积函数 ",[34,569073,569075,569159],{"className":569074},[31642],[34,569076,569078],{"className":569077},[31646],[31648,569079,569080],{"xmlns":31650},[31652,569081,569082,569156],{},[31655,569083,569084,569086,569112,569114,569116,569118,569138,569140,569142,569144],{},[31662,569085,168560],{},[41453,569087,569088,569090],{},[31658,569089,33249],{},[31655,569091,569092,569098,569100],{},[37624,569093,569094,569096],{},[31658,569095,58442],{},[31658,569097,67764],{},[31662,569099,71191],{},[37624,569101,569102,569104],{},[31658,569103,33249],{},[31655,569105,569106,569108,569110],{},[31749,569107,768],{},[31662,569109,32226],{},[31658,569111,67764],{},[31662,569113,202],{},[31658,569115,47384],{},[31662,569117,71191],{},[37624,569119,569120,569122],{},[31658,569121,33249],{},[31655,569123,569124,569126,569128,569130,569132,569134,569136],{},[31749,569125,640],{},[31662,569127,32226],{},[31662,569129,456],{"stretchy":702},[31749,569131,768],{},[31662,569133,32226],{},[31658,569135,67764],{},[31662,569137,2557],{"stretchy":702},[31662,569139,202],{},[31658,569141,47384],{},[31662,569143,71191],{},[37624,569145,569146,569148],{},[31658,569147,33249],{},[31655,569149,569150,569152,569154],{},[31658,569151,67764],{},[31662,569153,32226],{},[31749,569155,640],{},[31665,569157,569158],{"encoding":31667},"\\approx \\frac{t}{\\pi^y \\cdot t^{2-y}} = C \\cdot t^{1-(2-y)} = C \\cdot t^{y-1}",[34,569160,569162,569174,569326,569344,569409,569427],{"className":569161,"ariaHidden":758},[31672],[34,569163,569165,569168,569171],{"className":569164},[31676],[34,569166],{"className":569167,"style":211939},[31680],[34,569169,168560],{"className":569170},[31850],[34,569172],{"className":569173,"style":31846},[31845],[34,569175,569177,569181,569317,569320,569323],{"className":569176},[31676],[34,569178],{"className":569179,"style":569180},[31680],"height:1.1696em;vertical-align:-0.345em;",[34,569182,569184,569187,569314],{"className":569183},[31685],[34,569185],{"className":569186},[31782,41488],[34,569188,569190],{"className":569189},[41453],[34,569191,569193,569306],{"className":569192},[31796,31797],[34,569194,569196,569303],{"className":569195},[31801],[34,569197,569200,569281,569289],{"className":569198,"style":569199},[31805],"height:0.8246em;",[34,569201,569202,569205],{"style":41504},[34,569203],{"className":569204,"style":41508},[31813],[34,569206,569208],{"className":569207},[31818,31819,31820,31821],[34,569209,569211,569240,569243],{"className":569210},[31685,31821],[34,569212,569214,569217],{"className":569213},[31685,31821],[34,569215,58442],{"className":569216,"style":32062},[31685,31686,31821],[34,569218,569220],{"className":569219},[31792],[34,569221,569223],{"className":569222},[31796],[34,569224,569226],{"className":569225},[31801],[34,569227,569229],{"className":569228,"style":566313},[31805],[34,569230,569231,569234],{"style":116668},[34,569232],{"className":569233,"style":43406},[31813],[34,569235,569237],{"className":569236},[31818,43410,43411,31821],[34,569238,67764],{"className":569239,"style":32062},[31685,31686,31821],[34,569241,71191],{"className":569242},[32024,31821],[34,569244,569246,569249],{"className":569245},[31685,31821],[34,569247,33249],{"className":569248},[31685,31686,31821],[34,569250,569252],{"className":569251},[31792],[34,569253,569255],{"className":569254},[31796],[34,569256,569258],{"className":569257},[31801],[34,569259,569261],{"className":569260,"style":548314},[31805],[34,569262,569263,569266],{"style":116668},[34,569264],{"className":569265,"style":43406},[31813],[34,569267,569269],{"className":569268},[31818,43410,43411,31821],[34,569270,569272,569275,569278],{"className":569271},[31685,31821],[34,569273,768],{"className":569274},[31685,31821],[34,569276,32226],{"className":569277},[32024,31821],[34,569279,67764],{"className":569280,"style":32062},[31685,31686,31821],[34,569282,569283,569286],{"style":41520},[34,569284],{"className":569285,"style":41508},[31813],[34,569287],{"className":569288,"style":41528},[41527],[34,569290,569291,569294],{"style":46771},[34,569292],{"className":569293,"style":41508},[31813],[34,569295,569297],{"className":569296},[31818,31819,31820,31821],[34,569298,569300],{"className":569299},[31685,31821],[34,569301,33249],{"className":569302},[31685,31686,31821],[34,569304,31829],{"className":569305},[31828],[34,569307,569309],{"className":569308},[31801],[34,569310,569312],{"className":569311,"style":41619},[31805],[34,569313],{},[34,569315],{"className":569316},[31690,41488],[34,569318],{"className":569319,"style":31846},[31845],[34,569321,202],{"className":569322},[31850],[34,569324],{"className":569325,"style":31846},[31845],[34,569327,569329,569332,569335,569338,569341],{"className":569328},[31676],[34,569330],{"className":569331,"style":33218},[31680],[34,569333,47384],{"className":569334,"style":38706},[31685,31686],[34,569336],{"className":569337,"style":32604},[31845],[34,569339,71191],{"className":569340},[32024],[34,569342],{"className":569343,"style":32604},[31845],[34,569345,569347,569350,569400,569403,569406],{"className":569346},[31676],[34,569348],{"className":569349,"style":61037},[31680],[34,569351,569353,569356],{"className":569352},[31685],[34,569354,33249],{"className":569355},[31685,31686],[34,569357,569359],{"className":569358},[31792],[34,569360,569362],{"className":569361},[31796],[34,569363,569365],{"className":569364},[31801],[34,569366,569368],{"className":569367,"style":61037},[31805],[34,569369,569370,569373],{"style":37777},[34,569371],{"className":569372,"style":31814},[31813],[34,569374,569376],{"className":569375},[31818,31819,31820,31821],[34,569377,569379,569382,569385,569388,569391,569394,569397],{"className":569378},[31685,31821],[34,569380,640],{"className":569381},[31685,31821],[34,569383,32226],{"className":569384},[32024,31821],[34,569386,456],{"className":569387},[31782,31821],[34,569389,768],{"className":569390},[31685,31821],[34,569392,32226],{"className":569393},[32024,31821],[34,569395,67764],{"className":569396,"style":32062},[31685,31686,31821],[34,569398,2557],{"className":569399},[31690,31821],[34,569401],{"className":569402,"style":31846},[31845],[34,569404,202],{"className":569405},[31850],[34,569407],{"className":569408,"style":31846},[31845],[34,569410,569412,569415,569418,569421,569424],{"className":569411},[31676],[34,569413],{"className":569414,"style":33218},[31680],[34,569416,47384],{"className":569417,"style":38706},[31685,31686],[34,569419],{"className":569420,"style":32604},[31845],[34,569422,71191],{"className":569423},[32024],[34,569425],{"className":569426,"style":32604},[31845],[34,569428,569430,569433],{"className":569429},[31676],[34,569431],{"className":569432,"style":37774},[31680],[34,569434,569436,569439],{"className":569435},[31685],[34,569437,33249],{"className":569438},[31685,31686],[34,569440,569442],{"className":569441},[31792],[34,569443,569445],{"className":569444},[31796],[34,569446,569448],{"className":569447},[31801],[34,569449,569451],{"className":569450,"style":37774},[31805],[34,569452,569453,569456],{"style":37777},[34,569454],{"className":569455,"style":31814},[31813],[34,569457,569459],{"className":569458},[31818,31819,31820,31821],[34,569460,569462,569465,569468],{"className":569461},[31685,31821],[34,569463,67764],{"className":569464,"style":32062},[31685,31686,31821],[34,569466,32226],{"className":569467},[32024,31821],[34,569469,640],{"className":569470},[31685,31821],"。\n积分收敛条件：幂次 ",[34,569473,569475,569511],{"className":569474},[31642],[34,569476,569478],{"className":569477},[31646],[31648,569479,569480],{"xmlns":31650},[31652,569481,569482,569508],{},[31655,569483,569484,569486,569488,569490,569492,569494,569496,569498,569500,569502,569504,569506],{},[31658,569485,67764],{},[31662,569487,32226],{},[31749,569489,640],{},[31662,569491,29622],{},[31662,569493,32226],{},[31749,569495,640],{},[39361,569497,262622],{},[31662,569499,262625],{},[39361,569501,262622],{},[31658,569503,67764],{},[31662,569505,29622],{},[31749,569507,459],{},[31665,569509,569510],{"encoding":31667},"y-1 > -1 \\implies y > 0",[34,569512,569514,569532,569550,569577,569595],{"className":569513,"ariaHidden":758},[31672],[34,569515,569517,569520,569523,569526,569529],{"className":569516},[31676],[34,569518],{"className":569519,"style":55949},[31680],[34,569521,67764],{"className":569522,"style":32062},[31685,31686],[34,569524],{"className":569525,"style":32604},[31845],[34,569527,32226],{"className":569528},[32024],[34,569530],{"className":569531,"style":32604},[31845],[34,569533,569535,569538,569541,569544,569547],{"className":569534},[31676],[34,569536],{"className":569537,"style":61814},[31680],[34,569539,640],{"className":569540},[31685],[34,569542],{"className":569543,"style":31846},[31845],[34,569545,29622],{"className":569546},[31850],[34,569548],{"className":569549,"style":31846},[31845],[34,569551,569553,569556,569559,569562,569565,569568,569571,569574],{"className":569552},[31676],[34,569554],{"className":569555,"style":35942},[31680],[34,569557,32226],{"className":569558},[31685],[34,569560,640],{"className":569561},[31685],[34,569563],{"className":569564,"style":31846},[31845],[34,569566],{"className":569567,"style":31846},[31845],[34,569569,262625],{"className":569570},[31850],[34,569572],{"className":569573,"style":31846},[31845],[34,569575],{"className":569576,"style":31846},[31845],[34,569578,569580,569583,569586,569589,569592],{"className":569579},[31676],[34,569581],{"className":569582,"style":86652},[31680],[34,569584,67764],{"className":569585,"style":32062},[31685,31686],[34,569587],{"className":569588,"style":31846},[31845],[34,569590,29622],{"className":569591},[31850],[34,569593],{"className":569594,"style":31846},[31845],[34,569596,569598,569601],{"className":569597},[31676],[34,569599],{"className":569600,"style":32617},[31680],[34,569602,459],{"className":569603},[31685],[108,569605,569606,569609,569610,258,569661,569712,569713,347],{},[2627,569607,569608],{},"综合范围：","\n为了让积分在两头都收敛，必须同时满足 ",[34,569611,569613,569631],{"className":569612},[31642],[34,569614,569616],{"className":569615},[31646],[31648,569617,569618],{"xmlns":31650},[31652,569619,569620,569628],{},[31655,569621,569622,569624,569626],{},[31658,569623,67764],{},[31662,569625,932],{},[31749,569627,768],{},[31665,569629,569630],{"encoding":31667},"y \u003C 2",[34,569632,569634,569652],{"className":569633,"ariaHidden":758},[31672],[34,569635,569637,569640,569643,569646,569649],{"className":569636},[31676],[34,569638],{"className":569639,"style":86652},[31680],[34,569641,67764],{"className":569642,"style":32062},[31685,31686],[34,569644],{"className":569645,"style":31846},[31845],[34,569647,932],{"className":569648},[31850],[34,569650],{"className":569651,"style":31846},[31845],[34,569653,569655,569658],{"className":569654},[31676],[34,569656],{"className":569657,"style":32617},[31680],[34,569659,768],{"className":569660},[31685],[34,569662,569664,569682],{"className":569663},[31642],[34,569665,569667],{"className":569666},[31646],[31648,569668,569669],{"xmlns":31650},[31652,569670,569671,569679],{},[31655,569672,569673,569675,569677],{},[31658,569674,67764],{},[31662,569676,29622],{},[31749,569678,459],{},[31665,569680,569681],{"encoding":31667},"y > 0",[34,569683,569685,569703],{"className":569684,"ariaHidden":758},[31672],[34,569686,569688,569691,569694,569697,569700],{"className":569687},[31676],[34,569689],{"className":569690,"style":86652},[31680],[34,569692,67764],{"className":569693,"style":32062},[31685,31686],[34,569695],{"className":569696,"style":31846},[31845],[34,569698,29622],{"className":569699},[31850],[34,569701],{"className":569702,"style":31846},[31845],[34,569704,569706,569709],{"className":569705},[31676],[34,569707],{"className":569708,"style":32617},[31680],[34,569710,459],{"className":569711},[31685],"。\n所以连续范围（也是定义域）是 ",[2627,569714,569715],{},[34,569716,569718,569740],{"className":569717},[31642],[34,569719,569721],{"className":569720},[31646],[31648,569722,569723],{"xmlns":31650},[31652,569724,569725,569737],{},[31655,569726,569727,569729,569731,569733,569735],{},[31662,569728,456],{"stretchy":702},[31749,569730,459],{},[31662,569732,31960],{"separator":758},[31749,569734,768],{},[31662,569736,2557],{"stretchy":702},[31665,569738,569739],{"encoding":31667},"(0, 2)",[34,569741,569743],{"className":569742,"ariaHidden":758},[31672],[34,569744,569746,569749,569752,569755,569758,569761,569764],{"className":569745},[31676],[34,569747],{"className":569748,"style":31774},[31680],[34,569750,456],{"className":569751},[31782],[34,569753,459],{"className":569754},[31685],[34,569756,31960],{"className":569757},[32119],[34,569759],{"className":569760,"style":32123},[31845],[34,569762,768],{"className":569763},[31685],[34,569765,2557],{"className":569766},[31690],[32188,569768],{},[253,569770,569772],{"id":569771},"p330-t8","p330 T8",[15,569774,569775,566109,569777,314945,570081,570143],{},[2627,569776,214361],{},[34,569778,569780,569848],{"className":569779},[31642],[34,569781,569783],{"className":569782},[31646],[31648,569784,569785],{"xmlns":31650},[31652,569786,569787,569845],{},[31655,569788,569789,569791,569793,569795,569797,569799,569811,569841,569843],{},[31658,569790,106063],{},[31662,569792,456],{"stretchy":702},[31658,569794,33249],{},[31662,569796,2557],{"stretchy":702},[31662,569798,202],{},[38846,569800,569801,569803,569805],{},[31662,569802,546295],{},[31749,569804,459],{},[31655,569806,569807,569809],{},[31662,569808,223],{},[31658,569810,138286],{"mathvariant":38079},[41453,569812,569813,569821],{},[31655,569814,569815,569817,569819],{},[31658,569816,547860],{},[31662,569818,32682],{},[31658,569820,32936],{},[31655,569822,569823,569825,569827,569829,569831,569833,569835],{},[31749,569824,640],{},[31662,569826,223],{},[31662,569828,456],{"stretchy":702},[31658,569830,32936],{},[31662,569832,223],{},[31658,569834,33249],{},[37624,569836,569837,569839],{},[31662,569838,2557],{"stretchy":702},[31749,569840,768],{},[31658,569842,46900],{},[31658,569844,32936],{},[31665,569846,569847],{"encoding":31667},"I(t) = \\int_{0}^{+\\infty} \\frac{\\cos x}{1 + (x + t)^2}dx",[34,569849,569851,569878],{"className":569850,"ariaHidden":758},[31672],[34,569852,569854,569857,569860,569863,569866,569869,569872,569875],{"className":569853},[31676],[34,569855],{"className":569856,"style":31774},[31680],[34,569858,106063],{"className":569859,"style":67482},[31685,31686],[34,569861,456],{"className":569862},[31782],[34,569864,33249],{"className":569865},[31685,31686],[34,569867,2557],{"className":569868},[31690],[34,569870],{"className":569871,"style":31846},[31845],[34,569873,202],{"className":569874},[31850],[34,569876],{"className":569877,"style":31846},[31845],[34,569879,569881,569885,569945,569948,570075,570078],{"className":569880},[31676],[34,569882],{"className":569883,"style":569884},[31680],"height:1.4862em;vertical-align:-0.52em;",[34,569886,569888,569891],{"className":569887},[32767],[34,569889,546295],{"className":569890,"style":546331},[32767,131249,156560],[34,569892,569894],{"className":569893},[31792],[34,569895,569897,569937],{"className":569896},[31796,31797],[34,569898,569900,569934],{"className":569899},[31801],[34,569901,569903,569917],{"className":569902,"style":546550},[31805],[34,569904,569905,569908],{"style":546347},[34,569906],{"className":569907,"style":31814},[31813],[34,569909,569911],{"className":569910},[31818,31819,31820,31821],[34,569912,569914],{"className":569913},[31685,31821],[34,569915,459],{"className":569916},[31685,31821],[34,569918,569919,569922],{"style":546362},[34,569920],{"className":569921,"style":31814},[31813],[34,569923,569925],{"className":569924},[31818,31819,31820,31821],[34,569926,569928,569931],{"className":569927},[31685,31821],[34,569929,223],{"className":569930},[31685,31821],[34,569932,138286],{"className":569933},[31685,31821],[34,569935,31829],{"className":569936},[31828],[34,569938,569940],{"className":569939},[31801],[34,569941,569943],{"className":569942,"style":546384},[31805],[34,569944],{},[34,569946],{"className":569947,"style":32123},[31845],[34,569949,569951,569954,570072],{"className":569950},[31685],[34,569952],{"className":569953},[31782,41488],[34,569955,569957],{"className":569956},[41453],[34,569958,569960,570064],{"className":569959},[31796,31797],[34,569961,569963,570061],{"className":569962},[31801],[34,569964,569966,570024,570032],{"className":569965,"style":244945},[31805],[34,569967,569968,569971],{"style":41504},[34,569969],{"className":569970,"style":41508},[31813],[34,569972,569974],{"className":569973},[31818,31819,31820,31821],[34,569975,569977,569980,569983,569986,569989,569992,569995],{"className":569976},[31685,31821],[34,569978,640],{"className":569979},[31685,31821],[34,569981,223],{"className":569982},[32024,31821],[34,569984,456],{"className":569985},[31782,31821],[34,569987,32936],{"className":569988},[31685,31686,31821],[34,569990,223],{"className":569991},[32024,31821],[34,569993,33249],{"className":569994},[31685,31686,31821],[34,569996,569998,570001],{"className":569997},[31690,31821],[34,569999,2557],{"className":570000},[31690,31821],[34,570002,570004],{"className":570003},[31792],[34,570005,570007],{"className":570006},[31796],[34,570008,570010],{"className":570009},[31801],[34,570011,570013],{"className":570012,"style":548314},[31805],[34,570014,570015,570018],{"style":116668},[34,570016],{"className":570017,"style":43406},[31813],[34,570019,570021],{"className":570020},[31818,43410,43411,31821],[34,570022,768],{"className":570023},[31685,31821],[34,570025,570026,570029],{"style":41520},[34,570027],{"className":570028,"style":41508},[31813],[34,570030],{"className":570031,"style":41528},[41527],[34,570033,570034,570037],{"style":46771},[34,570035],{"className":570036,"style":41508},[31813],[34,570038,570040],{"className":570039},[31818,31819,31820,31821],[34,570041,570043,570055,570058],{"className":570042},[31685,31821],[34,570044,570046,570049,570052],{"className":570045},[32767,31821],[34,570047,31965],{"className":570048},[31821],[34,570050,43383],{"className":570051},[31821],[34,570053,31747],{"className":570054},[31821],[34,570056],{"className":570057,"style":43430},[31845,31821],[34,570059,32936],{"className":570060},[31685,31686,31821],[34,570062,31829],{"className":570063},[31828],[34,570065,570067],{"className":570066},[31801],[34,570068,570070],{"className":570069,"style":324858},[31805],[34,570071],{},[34,570073],{"className":570074},[31690,41488],[34,570076,46900],{"className":570077},[31685,31686],[34,570079,32936],{"className":570080},[31685,31686],[34,570082,570084,570110],{"className":570083},[31642],[34,570085,570087],{"className":570086},[31646],[31648,570088,570089],{"xmlns":31650},[31652,570090,570091,570107],{},[31655,570092,570093,570095,570097,570099,570101,570103,570105],{},[31662,570094,456],{"stretchy":702},[31662,570096,32226],{},[31658,570098,138286],{"mathvariant":38079},[31662,570100,31960],{"separator":758},[31662,570102,223],{},[31658,570104,138286],{"mathvariant":38079},[31662,570106,2557],{"stretchy":702},[31665,570108,570109],{"encoding":31667},"(-\\infty, +\\infty)",[34,570111,570113],{"className":570112,"ariaHidden":758},[31672],[34,570114,570116,570119,570122,570125,570128,570131,570134,570137,570140],{"className":570115},[31676],[34,570117],{"className":570118,"style":31774},[31680],[34,570120,456],{"className":570121},[31782],[34,570123,32226],{"className":570124},[31685],[34,570126,138286],{"className":570127},[31685],[34,570129,31960],{"className":570130},[32119],[34,570132],{"className":570133,"style":32123},[31845],[34,570135,223],{"className":570136},[31685],[34,570138,138286],{"className":570139},[31685],[34,570141,2557],{"className":570142},[31690]," 上可微。",[15,570145,570146,570148,570149,570152],{},[2627,570147,91513],{},"\n要证明可微，根据",[2627,570150,570151],{},"可微性定理","，我们需要证明：",[2230,570154,570155,570210],{},[108,570156,570157,570158,570209],{},"原积分在某点收敛（显然，",[34,570159,570161,570179],{"className":570160},[31642],[34,570162,570164],{"className":570163},[31646],[31648,570165,570166],{"xmlns":31650},[31652,570167,570168,570176],{},[31655,570169,570170,570172,570174],{},[31658,570171,33249],{},[31662,570173,202],{},[31749,570175,459],{},[31665,570177,570178],{"encoding":31667},"t=0",[34,570180,570182,570200],{"className":570181,"ariaHidden":758},[31672],[34,570183,570185,570188,570191,570194,570197],{"className":570184},[31676],[34,570186],{"className":570187,"style":33280},[31680],[34,570189,33249],{"className":570190},[31685,31686],[34,570192],{"className":570193,"style":31846},[31845],[34,570195,202],{"className":570196},[31850],[34,570198],{"className":570199,"style":31846},[31845],[34,570201,570203,570206],{"className":570202},[31676],[34,570204],{"className":570205,"style":32617},[31680],[34,570207,459],{"className":570208},[31685],"时收敛）。",[108,570211,570212,570215],{},[2627,570213,570214],{},"求导后的积分","一致收敛。",[15,570217,570218],{},[2627,570219,549629],{},[2230,570221,570222,571150,573053],{},[108,570223,570224,570227,570228,570460,570461,570489,570490],{},[2627,570225,570226],{},"形式求导：","\n被积函数 ",[34,570229,570231,570287],{"className":570230},[31642],[34,570232,570234],{"className":570233},[31646],[31648,570235,570236],{"xmlns":31650},[31652,570237,570238,570284],{},[31655,570239,570240,570242,570244,570246,570248,570250,570252,570254],{},[31658,570241,31739],{},[31662,570243,456],{"stretchy":702},[31658,570245,32936],{},[31662,570247,31960],{"separator":758},[31658,570249,33249],{},[31662,570251,2557],{"stretchy":702},[31662,570253,202],{},[41453,570255,570256,570264],{},[31655,570257,570258,570260,570262],{},[31658,570259,547860],{},[31662,570261,32682],{},[31658,570263,32936],{},[31655,570265,570266,570268,570270,570272,570274,570276,570278],{},[31749,570267,640],{},[31662,570269,223],{},[31662,570271,456],{"stretchy":702},[31658,570273,32936],{},[31662,570275,223],{},[31658,570277,33249],{},[37624,570279,570280,570282],{},[31662,570281,2557],{"stretchy":702},[31749,570283,768],{},[31665,570285,570286],{"encoding":31667},"f(x,t) = \\frac{\\cos x}{1 + (x + t)^2}",[34,570288,570290,570326],{"className":570289,"ariaHidden":758},[31672],[34,570291,570293,570296,570299,570302,570305,570308,570311,570314,570317,570320,570323],{"className":570292},[31676],[34,570294],{"className":570295,"style":31774},[31680],[34,570297,31739],{"className":570298,"style":31778},[31685,31686],[34,570300,456],{"className":570301},[31782],[34,570303,32936],{"className":570304},[31685,31686],[34,570306,31960],{"className":570307},[32119],[34,570309],{"className":570310,"style":32123},[31845],[34,570312,33249],{"className":570313},[31685,31686],[34,570315,2557],{"className":570316},[31690],[34,570318],{"className":570319,"style":31846},[31845],[34,570321,202],{"className":570322},[31850],[34,570324],{"className":570325,"style":31846},[31845],[34,570327,570329,570333],{"className":570328},[31676],[34,570330],{"className":570331,"style":570332},[31680],"height:1.2154em;vertical-align:-0.52em;",[34,570334,570336,570339,570457],{"className":570335},[31685],[34,570337],{"className":570338},[31782,41488],[34,570340,570342],{"className":570341},[41453],[34,570343,570345,570449],{"className":570344},[31796,31797],[34,570346,570348,570446],{"className":570347},[31801],[34,570349,570351,570409,570417],{"className":570350,"style":244945},[31805],[34,570352,570353,570356],{"style":41504},[34,570354],{"className":570355,"style":41508},[31813],[34,570357,570359],{"className":570358},[31818,31819,31820,31821],[34,570360,570362,570365,570368,570371,570374,570377,570380],{"className":570361},[31685,31821],[34,570363,640],{"className":570364},[31685,31821],[34,570366,223],{"className":570367},[32024,31821],[34,570369,456],{"className":570370},[31782,31821],[34,570372,32936],{"className":570373},[31685,31686,31821],[34,570375,223],{"className":570376},[32024,31821],[34,570378,33249],{"className":570379},[31685,31686,31821],[34,570381,570383,570386],{"className":570382},[31690,31821],[34,570384,2557],{"className":570385},[31690,31821],[34,570387,570389],{"className":570388},[31792],[34,570390,570392],{"className":570391},[31796],[34,570393,570395],{"className":570394},[31801],[34,570396,570398],{"className":570397,"style":548314},[31805],[34,570399,570400,570403],{"style":116668},[34,570401],{"className":570402,"style":43406},[31813],[34,570404,570406],{"className":570405},[31818,43410,43411,31821],[34,570407,768],{"className":570408},[31685,31821],[34,570410,570411,570414],{"style":41520},[34,570412],{"className":570413,"style":41508},[31813],[34,570415],{"className":570416,"style":41528},[41527],[34,570418,570419,570422],{"style":46771},[34,570420],{"className":570421,"style":41508},[31813],[34,570423,570425],{"className":570424},[31818,31819,31820,31821],[34,570426,570428,570440,570443],{"className":570427},[31685,31821],[34,570429,570431,570434,570437],{"className":570430},[32767,31821],[34,570432,31965],{"className":570433},[31821],[34,570435,43383],{"className":570436},[31821],[34,570438,31747],{"className":570439},[31821],[34,570441],{"className":570442,"style":43430},[31845,31821],[34,570444,32936],{"className":570445},[31685,31686,31821],[34,570447,31829],{"className":570448},[31828],[34,570450,570452],{"className":570451},[31801],[34,570453,570455],{"className":570454,"style":324858},[31805],[34,570456],{},[34,570458],{"className":570459},[31690,41488],"。\n对 ",[34,570462,570464,570477],{"className":570463},[31642],[34,570465,570467],{"className":570466},[31646],[31648,570468,570469],{"xmlns":31650},[31652,570470,570471,570475],{},[31655,570472,570473],{},[31658,570474,33249],{},[31665,570476,33249],{"encoding":31667},[34,570478,570480],{"className":570479,"ariaHidden":758},[31672],[34,570481,570483,570486],{"className":570482},[31676],[34,570484],{"className":570485,"style":33280},[31680],[34,570487,33249],{"className":570488},[31685,31686]," 求偏导：\n",[34,570491,570493,570637],{"className":570492},[31642],[34,570494,570496],{"className":570495},[31646],[31648,570497,570498],{"xmlns":31650},[31652,570499,570500,570634],{},[31655,570501,570502,570510,570512,570514,570516,570518,570520,570522,570524,570526,570528,570530,570566,570568,570570,570572,570574,570576,570578,570580,570582],{},[38846,570503,570504,570506,570508],{},[31658,570505,31739],{},[31658,570507,33249],{},[31662,570509,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,570511,456],{"stretchy":702},[31658,570513,32936],{},[31662,570515,31960],{"separator":758},[31658,570517,33249],{},[31662,570519,2557],{"stretchy":702},[31662,570521,202],{},[31658,570523,547860],{},[31662,570525,32682],{},[31658,570527,32936],{},[31662,570529,71191],{},[41453,570531,570532,570538],{},[31655,570533,570534,570536],{},[31662,570535,32226],{},[31749,570537,640],{},[31655,570539,570540,570542,570544,570546,570548,570550,570552,570554,570560],{},[31662,570541,15972],{"stretchy":702},[31749,570543,640],{},[31662,570545,223],{},[31662,570547,456],{"stretchy":702},[31658,570549,32936],{},[31662,570551,223],{},[31658,570553,33249],{},[37624,570555,570556,570558],{},[31662,570557,2557],{"stretchy":702},[31749,570559,768],{},[37624,570561,570562,570564],{},[31662,570563,32712],{"stretchy":702},[31749,570565,768],{},[31662,570567,71191],{},[31749,570569,768],{},[31662,570571,456],{"stretchy":702},[31658,570573,32936],{},[31662,570575,223],{},[31658,570577,33249],{},[31662,570579,2557],{"stretchy":702},[31662,570581,202],{},[41453,570583,570584,570606],{},[31655,570585,570586,570588,570590,570592,570594,570596,570598,570600,570602,570604],{},[31662,570587,32226],{},[31749,570589,768],{},[31662,570591,456],{"stretchy":702},[31658,570593,32936],{},[31662,570595,223],{},[31658,570597,33249],{},[31662,570599,2557],{"stretchy":702},[31658,570601,547860],{},[31662,570603,32682],{},[31658,570605,32936],{},[31655,570607,570608,570610,570612,570614,570616,570618,570620,570622,570628],{},[31662,570609,15972],{"stretchy":702},[31749,570611,640],{},[31662,570613,223],{},[31662,570615,456],{"stretchy":702},[31658,570617,32936],{},[31662,570619,223],{},[31658,570621,33249],{},[37624,570623,570624,570626],{},[31662,570625,2557],{"stretchy":702},[31749,570627,768],{},[37624,570629,570630,570632],{},[31662,570631,32712],{"stretchy":702},[31749,570633,768],{},[31665,570635,570636],{"encoding":31667},"f'_t(x,t) = \\cos x \\cdot \\frac{-1}{[1+(x+t)^2]^2} \\cdot 2(x+t) = \\frac{-2(x+t)\\cos x}{[1+(x+t)^2]^2}",[34,570638,570640,570729,570753,570916,570940,570961],{"className":570639,"ariaHidden":758},[31672],[34,570641,570643,570647,570702,570705,570708,570711,570714,570717,570720,570723,570726],{"className":570642},[31676],[34,570644],{"className":570645,"style":570646},[31680],"height:1.0019em;vertical-align:-0.25em;",[34,570648,570650,570653],{"className":570649},[31685],[34,570651,31739],{"className":570652,"style":31778},[31685,31686],[34,570654,570656],{"className":570655},[31792],[34,570657,570659,570694],{"className":570658},[31796,31797],[34,570660,570662,570691],{"className":570661},[31801],[34,570663,570665,570677],{"className":570664,"style":132027},[31805],[34,570666,570668,570671],{"style":570667},"top:-2.453em;margin-left:-0.1076em;margin-right:0.05em;",[34,570669],{"className":570670,"style":31814},[31813],[34,570672,570674],{"className":570673},[31818,31819,31820,31821],[34,570675,33249],{"className":570676},[31685,31686,31821],[34,570678,570679,570682],{"style":37777},[34,570680],{"className":570681,"style":31814},[31813],[34,570683,570685],{"className":570684},[31818,31819,31820,31821],[34,570686,570688],{"className":570687},[31685,31821],[34,570689,116495],{"className":570690},[31685,31821],[34,570692,31829],{"className":570693},[31828],[34,570695,570697],{"className":570696},[31801],[34,570698,570700],{"className":570699,"style":52354},[31805],[34,570701],{},[34,570703,456],{"className":570704},[31782],[34,570706,32936],{"className":570707},[31685,31686],[34,570709,31960],{"className":570710},[32119],[34,570712],{"className":570713,"style":32123},[31845],[34,570715,33249],{"className":570716},[31685,31686],[34,570718,2557],{"className":570719},[31690],[34,570721],{"className":570722,"style":31846},[31845],[34,570724,202],{"className":570725},[31850],[34,570727],{"className":570728,"style":31846},[31845],[34,570730,570732,570735,570738,570741,570744,570747,570750],{"className":570731},[31676],[34,570733],{"className":570734,"style":224869},[31680],[34,570736,547860],{"className":570737},[32767],[34,570739],{"className":570740,"style":32123},[31845],[34,570742,32936],{"className":570743},[31685,31686],[34,570745],{"className":570746,"style":32604},[31845],[34,570748,71191],{"className":570749},[32024],[34,570751],{"className":570752,"style":32604},[31845],[34,570754,570756,570760,570907,570910,570913],{"className":570755},[31676],[34,570757],{"className":570758,"style":570759},[31680],"height:1.3651em;vertical-align:-0.52em;",[34,570761,570763,570766,570904],{"className":570762},[31685],[34,570764],{"className":570765},[31782,41488],[34,570767,570769],{"className":570768},[41453],[34,570770,570772,570896],{"className":570771},[31796,31797],[34,570773,570775,570893],{"className":570774},[31801],[34,570776,570778,570868,570876],{"className":570777,"style":46746},[31805],[34,570779,570780,570783],{"style":41504},[34,570781],{"className":570782,"style":41508},[31813],[34,570784,570786],{"className":570785},[31818,31819,31820,31821],[34,570787,570789,570792,570795,570798,570801,570804,570807,570810,570839],{"className":570788},[31685,31821],[34,570790,15972],{"className":570791},[31782,31821],[34,570793,640],{"className":570794},[31685,31821],[34,570796,223],{"className":570797},[32024,31821],[34,570799,456],{"className":570800},[31782,31821],[34,570802,32936],{"className":570803},[31685,31686,31821],[34,570805,223],{"className":570806},[32024,31821],[34,570808,33249],{"className":570809},[31685,31686,31821],[34,570811,570813,570816],{"className":570812},[31690,31821],[34,570814,2557],{"className":570815},[31690,31821],[34,570817,570819],{"className":570818},[31792],[34,570820,570822],{"className":570821},[31796],[34,570823,570825],{"className":570824},[31801],[34,570826,570828],{"className":570827,"style":548314},[31805],[34,570829,570830,570833],{"style":116668},[34,570831],{"className":570832,"style":43406},[31813],[34,570834,570836],{"className":570835},[31818,43410,43411,31821],[34,570837,768],{"className":570838},[31685,31821],[34,570840,570842,570845],{"className":570841},[31690,31821],[34,570843,32712],{"className":570844},[31690,31821],[34,570846,570848],{"className":570847},[31792],[34,570849,570851],{"className":570850},[31796],[34,570852,570854],{"className":570853},[31801],[34,570855,570857],{"className":570856,"style":548314},[31805],[34,570858,570859,570862],{"style":116668},[34,570860],{"className":570861,"style":43406},[31813],[34,570863,570865],{"className":570864},[31818,43410,43411,31821],[34,570866,768],{"className":570867},[31685,31821],[34,570869,570870,570873],{"style":41520},[34,570871],{"className":570872,"style":41508},[31813],[34,570874],{"className":570875,"style":41528},[41527],[34,570877,570878,570881],{"style":46771},[34,570879],{"className":570880,"style":41508},[31813],[34,570882,570884],{"className":570883},[31818,31819,31820,31821],[34,570885,570887,570890],{"className":570886},[31685,31821],[34,570888,32226],{"className":570889},[31685,31821],[34,570891,640],{"className":570892},[31685,31821],[34,570894,31829],{"className":570895},[31828],[34,570897,570899],{"className":570898},[31801],[34,570900,570902],{"className":570901,"style":324858},[31805],[34,570903],{},[34,570905],{"className":570906},[31690,41488],[34,570908],{"className":570909,"style":32604},[31845],[34,570911,71191],{"className":570912},[32024],[34,570914],{"className":570915,"style":32604},[31845],[34,570917,570919,570922,570925,570928,570931,570934,570937],{"className":570918},[31676],[34,570920],{"className":570921,"style":31774},[31680],[34,570923,768],{"className":570924},[31685],[34,570926,456],{"className":570927},[31782],[34,570929,32936],{"className":570930},[31685,31686],[34,570932],{"className":570933,"style":32604},[31845],[34,570935,223],{"className":570936},[32024],[34,570938],{"className":570939,"style":32604},[31845],[34,570941,570943,570946,570949,570952,570955,570958],{"className":570942},[31676],[34,570944],{"className":570945,"style":31774},[31680],[34,570947,33249],{"className":570948},[31685,31686],[34,570950,2557],{"className":570951},[31690],[34,570953],{"className":570954,"style":31846},[31845],[34,570956,202],{"className":570957},[31850],[34,570959],{"className":570960,"style":31846},[31845],[34,570962,570964,570967],{"className":570963},[31676],[34,570965],{"className":570966,"style":327841},[31680],[34,570968,570970,570973,571147],{"className":570969},[31685],[34,570971],{"className":570972},[31782,41488],[34,570974,570976],{"className":570975},[41453],[34,570977,570979,571139],{"className":570978},[31796,31797],[34,570980,570982,571136],{"className":570981},[31801],[34,570983,570985,571075,571083],{"className":570984,"style":70504},[31805],[34,570986,570987,570990],{"style":41504},[34,570988],{"className":570989,"style":41508},[31813],[34,570991,570993],{"className":570992},[31818,31819,31820,31821],[34,570994,570996,570999,571002,571005,571008,571011,571014,571017,571046],{"className":570995},[31685,31821],[34,570997,15972],{"className":570998},[31782,31821],[34,571000,640],{"className":571001},[31685,31821],[34,571003,223],{"className":571004},[32024,31821],[34,571006,456],{"className":571007},[31782,31821],[34,571009,32936],{"className":571010},[31685,31686,31821],[34,571012,223],{"className":571013},[32024,31821],[34,571015,33249],{"className":571016},[31685,31686,31821],[34,571018,571020,571023],{"className":571019},[31690,31821],[34,571021,2557],{"className":571022},[31690,31821],[34,571024,571026],{"className":571025},[31792],[34,571027,571029],{"className":571028},[31796],[34,571030,571032],{"className":571031},[31801],[34,571033,571035],{"className":571034,"style":548314},[31805],[34,571036,571037,571040],{"style":116668},[34,571038],{"className":571039,"style":43406},[31813],[34,571041,571043],{"className":571042},[31818,43410,43411,31821],[34,571044,768],{"className":571045},[31685,31821],[34,571047,571049,571052],{"className":571048},[31690,31821],[34,571050,32712],{"className":571051},[31690,31821],[34,571053,571055],{"className":571054},[31792],[34,571056,571058],{"className":571057},[31796],[34,571059,571061],{"className":571060},[31801],[34,571062,571064],{"className":571063,"style":548314},[31805],[34,571065,571066,571069],{"style":116668},[34,571067],{"className":571068,"style":43406},[31813],[34,571070,571072],{"className":571071},[31818,43410,43411,31821],[34,571073,768],{"className":571074},[31685,31821],[34,571076,571077,571080],{"style":41520},[34,571078],{"className":571079,"style":41508},[31813],[34,571081],{"className":571082,"style":41528},[41527],[34,571084,571085,571088],{"style":70529},[34,571086],{"className":571087,"style":41508},[31813],[34,571089,571091],{"className":571090},[31818,31819,31820,31821],[34,571092,571094,571097,571100,571103,571106,571109,571112,571115,571118,571130,571133],{"className":571093},[31685,31821],[34,571095,32226],{"className":571096},[31685,31821],[34,571098,768],{"className":571099},[31685,31821],[34,571101,456],{"className":571102},[31782,31821],[34,571104,32936],{"className":571105},[31685,31686,31821],[34,571107,223],{"className":571108},[32024,31821],[34,571110,33249],{"className":571111},[31685,31686,31821],[34,571113,2557],{"className":571114},[31690,31821],[34,571116],{"className":571117,"style":43430},[31845,31821],[34,571119,571121,571124,571127],{"className":571120},[32767,31821],[34,571122,31965],{"className":571123},[31821],[34,571125,43383],{"className":571126},[31821],[34,571128,31747],{"className":571129},[31821],[34,571131],{"className":571132,"style":43430},[31845,31821],[34,571134,32936],{"className":571135},[31685,31686,31821],[34,571137,31829],{"className":571138},[31828],[34,571140,571142],{"className":571141},[31801],[34,571143,571145],{"className":571144,"style":324858},[31805],[34,571146],{},[34,571148],{"className":571149},[31690,41488],[108,571151,571152,571242,571243,39984,571317,478285,571659,571742,571743,571930,571931,572039,572040,572042,572043,572071,572072,572100,572101,572160,572161,572189,572190,572256,572257,572259,572262,572263,572291,572292,572348,572349,572377,572378,66331,572429,39984,572546,572862,572863,572891,572892,573052],{},[2627,571153,379882,571154,571241],{},[34,571155,571157,571177],{"className":571156},[31642],[34,571158,571160],{"className":571159},[31646],[31648,571161,571162],{"xmlns":31650},[31652,571163,571164,571174],{},[31655,571165,571166],{},[38846,571167,571168,571170,571172],{},[31658,571169,31739],{},[31658,571171,33249],{},[31662,571173,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,571175,571176],{"encoding":31667},"f'_t",[34,571178,571180],{"className":571179,"ariaHidden":758},[31672],[34,571181,571183,571187],{"className":571182},[31676],[34,571184],{"className":571185,"style":571186},[31680],"height:0.9989em;vertical-align:-0.247em;",[34,571188,571190,571193],{"className":571189},[31685],[34,571191,31739],{"className":571192,"style":31778},[31685,31686],[34,571194,571196],{"className":571195},[31792],[34,571197,571199,571233],{"className":571198},[31796,31797],[34,571200,571202,571230],{"className":571201},[31801],[34,571203,571205,571216],{"className":571204,"style":132027},[31805],[34,571206,571207,571210],{"style":570667},[34,571208],{"className":571209,"style":31814},[31813],[34,571211,571213],{"className":571212},[31818,31819,31820,31821],[34,571214,33249],{"className":571215},[31685,31686,31821],[34,571217,571218,571221],{"style":37777},[34,571219],{"className":571220,"style":31814},[31813],[34,571222,571224],{"className":571223},[31818,31819,31820,31821],[34,571225,571227],{"className":571226},[31685,31821],[34,571228,116495],{"className":571229},[31685,31821],[34,571231,31829],{"className":571232},[31828],[34,571234,571236],{"className":571235},[31801],[34,571237,571239],{"className":571238,"style":52354},[31805],[34,571240],{}," 的积分一致收敛：","\n我们需要寻找一个控制函数（M-判别法）。\n注意到 ",[34,571244,571246,571272],{"className":571245},[31642],[34,571247,571249],{"className":571248},[31646],[31648,571250,571251],{"xmlns":31650},[31652,571252,571253,571269],{},[31655,571254,571255,571257,571259,571261,571263,571265,571267],{},[31658,571256,162086],{"mathvariant":38079},[31658,571258,547860],{},[31662,571260,32682],{},[31658,571262,32936],{},[31658,571264,162086],{"mathvariant":38079},[31662,571266,45972],{},[31749,571268,640],{},[31665,571270,571271],{"encoding":31667},"| \\cos x | \\le 1",[34,571273,571275,571308],{"className":571274,"ariaHidden":758},[31672],[34,571276,571278,571281,571284,571287,571290,571293,571296,571299,571302,571305],{"className":571277},[31676],[34,571279],{"className":571280,"style":31774},[31680],[34,571282,162086],{"className":571283},[31685],[34,571285],{"className":571286,"style":32123},[31845],[34,571288,547860],{"className":571289},[32767],[34,571291],{"className":571292,"style":32123},[31845],[34,571294,32936],{"className":571295},[31685,31686],[34,571297,162086],{"className":571298},[31685],[34,571300],{"className":571301,"style":31846},[31845],[34,571303,45972],{"className":571304},[31850],[34,571306],{"className":571307,"style":31846},[31845],[34,571309,571311,571314],{"className":571310},[31676],[34,571312],{"className":571313,"style":32617},[31680],[34,571315,640],{"className":571316},[31685],[34,571318,571320,571400],{"className":571319},[31642],[34,571321,571323],{"className":571322},[31646],[31648,571324,571325],{"xmlns":31650},[31652,571326,571327,571397],{},[31655,571328,571329,571331,571339,571341,571343,571345,571347,571349,571351,571353],{},[31658,571330,162086],{"mathvariant":38079},[38846,571332,571333,571335,571337],{},[31658,571334,31739],{},[31658,571336,33249],{},[31662,571338,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,571340,456],{"stretchy":702},[31658,571342,32936],{},[31662,571344,31960],{"separator":758},[31658,571346,33249],{},[31662,571348,2557],{"stretchy":702},[31658,571350,162086],{"mathvariant":38079},[31662,571352,45972],{},[41453,571354,571355,571369],{},[31655,571356,571357,571359,571361,571363,571365,571367],{},[31749,571358,768],{},[31658,571360,162086],{"mathvariant":38079},[31658,571362,32936],{},[31662,571364,223],{},[31658,571366,33249],{},[31658,571368,162086],{"mathvariant":38079},[31655,571370,571371,571373,571375,571377,571379,571381,571383,571385,571391],{},[31662,571372,15972],{"stretchy":702},[31749,571374,640],{},[31662,571376,223],{},[31662,571378,456],{"stretchy":702},[31658,571380,32936],{},[31662,571382,223],{},[31658,571384,33249],{},[37624,571386,571387,571389],{},[31662,571388,2557],{"stretchy":702},[31749,571390,768],{},[37624,571392,571393,571395],{},[31662,571394,32712],{"stretchy":702},[31749,571396,768],{},[31665,571398,571399],{"encoding":31667},"|f'_t(x,t)| \\leq \\frac{2|x+t|}{[1+(x+t)^2]^2}",[34,571401,571403,571496],{"className":571402,"ariaHidden":758},[31672],[34,571404,571406,571409,571412,571466,571469,571472,571475,571478,571481,571484,571487,571490,571493],{"className":571405},[31676],[34,571407],{"className":571408,"style":570646},[31680],[34,571410,162086],{"className":571411},[31685],[34,571413,571415,571418],{"className":571414},[31685],[34,571416,31739],{"className":571417,"style":31778},[31685,31686],[34,571419,571421],{"className":571420},[31792],[34,571422,571424,571458],{"className":571423},[31796,31797],[34,571425,571427,571455],{"className":571426},[31801],[34,571428,571430,571441],{"className":571429,"style":132027},[31805],[34,571431,571432,571435],{"style":570667},[34,571433],{"className":571434,"style":31814},[31813],[34,571436,571438],{"className":571437},[31818,31819,31820,31821],[34,571439,33249],{"className":571440},[31685,31686,31821],[34,571442,571443,571446],{"style":37777},[34,571444],{"className":571445,"style":31814},[31813],[34,571447,571449],{"className":571448},[31818,31819,31820,31821],[34,571450,571452],{"className":571451},[31685,31821],[34,571453,116495],{"className":571454},[31685,31821],[34,571456,31829],{"className":571457},[31828],[34,571459,571461],{"className":571460},[31801],[34,571462,571464],{"className":571463,"style":52354},[31805],[34,571465],{},[34,571467,456],{"className":571468},[31782],[34,571470,32936],{"className":571471},[31685,31686],[34,571473,31960],{"className":571474},[32119],[34,571476],{"className":571477,"style":32123},[31845],[34,571479,33249],{"className":571480},[31685,31686],[34,571482,2557],{"className":571483},[31690],[34,571485,162086],{"className":571486},[31685],[34,571488],{"className":571489,"style":31846},[31845],[34,571491,45972],{"className":571492},[31850],[34,571494],{"className":571495,"style":31846},[31845],[34,571497,571499,571502],{"className":571498},[31676],[34,571500],{"className":571501,"style":327841},[31680],[34,571503,571505,571508,571656],{"className":571504},[31685],[34,571506],{"className":571507},[31782,41488],[34,571509,571511],{"className":571510},[41453],[34,571512,571514,571648],{"className":571513},[31796,31797],[34,571515,571517,571645],{"className":571516},[31801],[34,571518,571520,571610,571618],{"className":571519,"style":70504},[31805],[34,571521,571522,571525],{"style":41504},[34,571523],{"className":571524,"style":41508},[31813],[34,571526,571528],{"className":571527},[31818,31819,31820,31821],[34,571529,571531,571534,571537,571540,571543,571546,571549,571552,571581],{"className":571530},[31685,31821],[34,571532,15972],{"className":571533},[31782,31821],[34,571535,640],{"className":571536},[31685,31821],[34,571538,223],{"className":571539},[32024,31821],[34,571541,456],{"className":571542},[31782,31821],[34,571544,32936],{"className":571545},[31685,31686,31821],[34,571547,223],{"className":571548},[32024,31821],[34,571550,33249],{"className":571551},[31685,31686,31821],[34,571553,571555,571558],{"className":571554},[31690,31821],[34,571556,2557],{"className":571557},[31690,31821],[34,571559,571561],{"className":571560},[31792],[34,571562,571564],{"className":571563},[31796],[34,571565,571567],{"className":571566},[31801],[34,571568,571570],{"className":571569,"style":548314},[31805],[34,571571,571572,571575],{"style":116668},[34,571573],{"className":571574,"style":43406},[31813],[34,571576,571578],{"className":571577},[31818,43410,43411,31821],[34,571579,768],{"className":571580},[31685,31821],[34,571582,571584,571587],{"className":571583},[31690,31821],[34,571585,32712],{"className":571586},[31690,31821],[34,571588,571590],{"className":571589},[31792],[34,571591,571593],{"className":571592},[31796],[34,571594,571596],{"className":571595},[31801],[34,571597,571599],{"className":571598,"style":548314},[31805],[34,571600,571601,571604],{"style":116668},[34,571602],{"className":571603,"style":43406},[31813],[34,571605,571607],{"className":571606},[31818,43410,43411,31821],[34,571608,768],{"className":571609},[31685,31821],[34,571611,571612,571615],{"style":41520},[34,571613],{"className":571614,"style":41508},[31813],[34,571616],{"className":571617,"style":41528},[41527],[34,571619,571620,571623],{"style":70529},[34,571621],{"className":571622,"style":41508},[31813],[34,571624,571626],{"className":571625},[31818,31819,31820,31821],[34,571627,571629,571633,571636,571639,571642],{"className":571628},[31685,31821],[34,571630,571632],{"className":571631},[31685,31821],"2∣",[34,571634,32936],{"className":571635},[31685,31686,31821],[34,571637,223],{"className":571638},[32024,31821],[34,571640,33249],{"className":571641},[31685,31686,31821],[34,571643,162086],{"className":571644},[31685,31821],[34,571646,31829],{"className":571647},[31828],[34,571649,571651],{"className":571650},[31801],[34,571652,571654],{"className":571653,"style":324858},[31805],[34,571655],{},[34,571657],{"className":571658},[31690,41488],[34,571660,571662,571688],{"className":571661},[31642],[34,571663,571665],{"className":571664},[31646],[31648,571666,571667],{"xmlns":31650},[31652,571668,571669,571685],{},[31655,571670,571671,571673,571675,571677,571679,571681,571683],{},[31658,571672,94628],{},[31662,571674,202],{},[31658,571676,162086],{"mathvariant":38079},[31658,571678,32936],{},[31662,571680,223],{},[31658,571682,33249],{},[31658,571684,162086],{"mathvariant":38079},[31665,571686,571687],{"encoding":31667},"u = |x+t|",[34,571689,571691,571709,571730],{"className":571690,"ariaHidden":758},[31672],[34,571692,571694,571697,571700,571703,571706],{"className":571693},[31676],[34,571695],{"className":571696,"style":32527},[31680],[34,571698,94628],{"className":571699},[31685,31686],[34,571701],{"className":571702,"style":31846},[31845],[34,571704,202],{"className":571705},[31850],[34,571707],{"className":571708,"style":31846},[31845],[34,571710,571712,571715,571718,571721,571724,571727],{"className":571711},[31676],[34,571713],{"className":571714,"style":31774},[31680],[34,571716,162086],{"className":571717},[31685],[34,571719,32936],{"className":571720},[31685,31686],[34,571722],{"className":571723,"style":32604},[31845],[34,571725,223],{"className":571726},[32024],[34,571728],{"className":571729,"style":32604},[31845],[34,571731,571733,571736,571739],{"className":571732},[31676],[34,571734],{"className":571735,"style":31774},[31680],[34,571737,33249],{"className":571738},[31685,31686],[34,571740,162086],{"className":571741},[31685],"，函数 ",[34,571744,571746,571786],{"className":571745},[31642],[34,571747,571749],{"className":571748},[31646],[31648,571750,571751],{"xmlns":31650},[31652,571752,571753,571783],{},[31655,571754,571755],{},[41453,571756,571757,571763],{},[31655,571758,571759,571761],{},[31749,571760,768],{},[31658,571762,94628],{},[31655,571764,571765,571767,571769,571771,571777],{},[31662,571766,456],{"stretchy":702},[31749,571768,640],{},[31662,571770,223],{},[37624,571772,571773,571775],{},[31658,571774,94628],{},[31749,571776,768],{},[37624,571778,571779,571781],{},[31662,571780,2557],{"stretchy":702},[31749,571782,768],{},[31665,571784,571785],{"encoding":31667},"\\frac{2u}{(1+u^2)^2}",[34,571787,571789],{"className":571788,"ariaHidden":758},[31672],[34,571790,571792,571795],{"className":571791},[31676],[34,571793],{"className":571794,"style":570759},[31680],[34,571796,571798,571801,571927],{"className":571797},[31685],[34,571799],{"className":571800},[31782,41488],[34,571802,571804],{"className":571803},[41453],[34,571805,571807,571919],{"className":571806},[31796,31797],[34,571808,571810,571916],{"className":571809},[31801],[34,571811,571813,571891,571899],{"className":571812,"style":46746},[31805],[34,571814,571815,571818],{"style":41504},[34,571816],{"className":571817,"style":41508},[31813],[34,571819,571821],{"className":571820},[31818,31819,31820,31821],[34,571822,571824,571827,571830,571833,571862],{"className":571823},[31685,31821],[34,571825,456],{"className":571826},[31782,31821],[34,571828,640],{"className":571829},[31685,31821],[34,571831,223],{"className":571832},[32024,31821],[34,571834,571836,571839],{"className":571835},[31685,31821],[34,571837,94628],{"className":571838},[31685,31686,31821],[34,571840,571842],{"className":571841},[31792],[34,571843,571845],{"className":571844},[31796],[34,571846,571848],{"className":571847},[31801],[34,571849,571851],{"className":571850,"style":548314},[31805],[34,571852,571853,571856],{"style":116668},[34,571854],{"className":571855,"style":43406},[31813],[34,571857,571859],{"className":571858},[31818,43410,43411,31821],[34,571860,768],{"className":571861},[31685,31821],[34,571863,571865,571868],{"className":571864},[31690,31821],[34,571866,2557],{"className":571867},[31690,31821],[34,571869,571871],{"className":571870},[31792],[34,571872,571874],{"className":571873},[31796],[34,571875,571877],{"className":571876},[31801],[34,571878,571880],{"className":571879,"style":548314},[31805],[34,571881,571882,571885],{"style":116668},[34,571883],{"className":571884,"style":43406},[31813],[34,571886,571888],{"className":571887},[31818,43410,43411,31821],[34,571889,768],{"className":571890},[31685,31821],[34,571892,571893,571896],{"style":41520},[34,571894],{"className":571895,"style":41508},[31813],[34,571897],{"className":571898,"style":41528},[41527],[34,571900,571901,571904],{"style":46771},[34,571902],{"className":571903,"style":41508},[31813],[34,571905,571907],{"className":571906},[31818,31819,31820,31821],[34,571908,571910,571913],{"className":571909},[31685,31821],[34,571911,768],{"className":571912},[31685,31821],[34,571914,94628],{"className":571915},[31685,31686,31821],[34,571917,31829],{"className":571918},[31828],[34,571920,571922],{"className":571921},[31801],[34,571923,571925],{"className":571924,"style":324858},[31805],[34,571926],{},[34,571928],{"className":571929},[31690,41488]," 是有界的（最大值在 ",[34,571932,571934,571958],{"className":571933},[31642],[34,571935,571937],{"className":571936},[31646],[31648,571938,571939],{"xmlns":31650},[31652,571940,571941,571955],{},[31655,571942,571943,571945,571947,571949,571951],{},[31658,571944,94628],{},[31662,571946,202],{},[31749,571948,640],{},[31658,571950,38080],{"mathvariant":38079},[41463,571952,571953],{},[31749,571954,786],{},[31665,571956,571957],{"encoding":31667},"u=1\u002F\\sqrt{3}",[34,571959,571961,571979],{"className":571960,"ariaHidden":758},[31672],[34,571962,571964,571967,571970,571973,571976],{"className":571963},[31676],[34,571965],{"className":571966,"style":32527},[31680],[34,571968,94628],{"className":571969},[31685,31686],[34,571971],{"className":571972,"style":31846},[31845],[34,571974,202],{"className":571975},[31850],[34,571977],{"className":571978,"style":31846},[31845],[34,571980,571982,571986,571989],{"className":571981},[31676],[34,571983],{"className":571984,"style":571985},[31680],"height:1.1572em;vertical-align:-0.25em;",[34,571987,249219],{"className":571988},[31685],[34,571990,571992],{"className":571991},[31685,41550],[34,571993,571995,572030],{"className":571994},[31796,31797],[34,571996,571998,572027],{"className":571997},[31801],[34,571999,572002,572014],{"className":572000,"style":572001},[31805],"height:0.9072em;",[34,572003,572005,572008],{"className":572004,"style":41565},[41564],[34,572006],{"className":572007,"style":41508},[31813],[34,572009,572011],{"className":572010,"style":41572},[31685],[34,572012,786],{"className":572013},[31685],[34,572015,572017,572020],{"style":572016},"top:-2.8672em;",[34,572018],{"className":572019,"style":41508},[31813],[34,572021,572023],{"className":572022,"style":41586},[41585],[41588,572024,572025],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,572026],{"d":41597},[34,572028,31829],{"className":572029},[31828],[34,572031,572033],{"className":572032},[31801],[34,572034,572037],{"className":572035,"style":572036},[31805],"height:0.1328em;",[34,572038],{}," 处）。\n但是直接放缩成常数会导致积分为无穷大。我们需要更好的控制。",[58531,572041],{},"我们观察到，当 ",[34,572044,572046,572059],{"className":572045},[31642],[34,572047,572049],{"className":572048},[31646],[31648,572050,572051],{"xmlns":31650},[31652,572052,572053,572057],{},[31655,572054,572055],{},[31658,572056,32936],{},[31665,572058,32936],{"encoding":31667},[34,572060,572062],{"className":572061,"ariaHidden":758},[31672],[34,572063,572065,572068],{"className":572064},[31676],[34,572066],{"className":572067,"style":32527},[31680],[34,572069,32936],{"className":572070},[31685,31686]," 很大时，不管 ",[34,572073,572075,572088],{"className":572074},[31642],[34,572076,572078],{"className":572077},[31646],[31648,572079,572080],{"xmlns":31650},[31652,572081,572082,572086],{},[31655,572083,572084],{},[31658,572085,33249],{},[31665,572087,33249],{"encoding":31667},[34,572089,572091],{"className":572090,"ariaHidden":758},[31672],[34,572092,572094,572097],{"className":572093},[31676],[34,572095],{"className":572096,"style":33280},[31680],[34,572098,33249],{"className":572099},[31685,31686]," 是多少，分母大概是 ",[34,572102,572104,572122],{"className":572103},[31642],[34,572105,572107],{"className":572106},[31646],[31648,572108,572109],{"xmlns":31650},[31652,572110,572111,572119],{},[31655,572112,572113],{},[37624,572114,572115,572117],{},[31658,572116,32936],{},[31749,572118,15932],{},[31665,572120,572121],{"encoding":31667},"x^4",[34,572123,572125],{"className":572124,"ariaHidden":758},[31672],[34,572126,572128,572131],{"className":572127},[31676],[34,572129],{"className":572130,"style":37774},[31680],[34,572132,572134,572137],{"className":572133},[31685],[34,572135,32936],{"className":572136},[31685,31686],[34,572138,572140],{"className":572139},[31792],[34,572141,572143],{"className":572142},[31796],[34,572144,572146],{"className":572145},[31801],[34,572147,572149],{"className":572148,"style":37774},[31805],[34,572150,572151,572154],{"style":37777},[34,572152],{"className":572153,"style":31814},[31813],[34,572155,572157],{"className":572156},[31818,31819,31820,31821],[34,572158,15932],{"className":572159},[31685,31821],"，分子是 ",[34,572162,572164,572177],{"className":572163},[31642],[34,572165,572167],{"className":572166},[31646],[31648,572168,572169],{"xmlns":31650},[31652,572170,572171,572175],{},[31655,572172,572173],{},[31658,572174,32936],{},[31665,572176,32936],{"encoding":31667},[34,572178,572180],{"className":572179,"ariaHidden":758},[31672],[34,572181,572183,572186],{"className":572182},[31676],[34,572184],{"className":572185,"style":32527},[31680],[34,572187,32936],{"className":572188},[31685,31686],"，整体是 ",[34,572191,572193,572215],{"className":572192},[31642],[34,572194,572196],{"className":572195},[31646],[31648,572197,572198],{"xmlns":31650},[31652,572199,572200,572212],{},[31655,572201,572202,572204,572206],{},[31749,572203,640],{},[31658,572205,38080],{"mathvariant":38079},[37624,572207,572208,572210],{},[31658,572209,32936],{},[31749,572211,786],{},[31665,572213,572214],{"encoding":31667},"1\u002Fx^3",[34,572216,572218],{"className":572217,"ariaHidden":758},[31672],[34,572219,572221,572224,572227],{"className":572220},[31676],[34,572222],{"className":572223,"style":37755},[31680],[34,572225,249219],{"className":572226},[31685],[34,572228,572230,572233],{"className":572229},[31685],[34,572231,32936],{"className":572232},[31685,31686],[34,572234,572236],{"className":572235},[31792],[34,572237,572239],{"className":572238},[31796],[34,572240,572242],{"className":572241},[31801],[34,572243,572245],{"className":572244,"style":37774},[31805],[34,572246,572247,572250],{"style":37777},[34,572248],{"className":572249,"style":31814},[31813],[34,572251,572253],{"className":572252},[31818,31819,31820,31821],[34,572254,786],{"className":572255},[31685,31821]," 级别。",[58531,572258],{},[2627,572260,572261],{},"更严谨的M-判别法：","\n我们限制 ",[34,572264,572266,572279],{"className":572265},[31642],[34,572267,572269],{"className":572268},[31646],[31648,572270,572271],{"xmlns":31650},[31652,572272,572273,572277],{},[31655,572274,572275],{},[31658,572276,33249],{},[31665,572278,33249],{"encoding":31667},[34,572280,572282],{"className":572281,"ariaHidden":758},[31672],[34,572283,572285,572288],{"className":572284},[31676],[34,572286],{"className":572287,"style":33280},[31680],[34,572289,33249],{"className":572290},[31685,31686]," 在任意有限区间 ",[34,572293,572295,572318],{"className":572294},[31642],[34,572296,572298],{"className":572297},[31646],[31648,572299,572300],{"xmlns":31650},[31652,572301,572302,572316],{},[31655,572303,572304,572306,572308,572310,572312,572314],{},[31662,572305,15972],{"stretchy":702},[31662,572307,32226],{},[31658,572309,54151],{},[31662,572311,31960],{"separator":758},[31658,572313,54151],{},[31662,572315,32712],{"stretchy":702},[31665,572317,563070],{"encoding":31667},[34,572319,572321],{"className":572320,"ariaHidden":758},[31672],[34,572322,572324,572327,572330,572333,572336,572339,572342,572345],{"className":572323},[31676],[34,572325],{"className":572326,"style":31774},[31680],[34,572328,15972],{"className":572329},[31782],[34,572331,32226],{"className":572332},[31685],[34,572334,54151],{"className":572335},[31685,31686],[34,572337,31960],{"className":572338},[32119],[34,572340],{"className":572341,"style":32123},[31845],[34,572343,54151],{"className":572344},[31685,31686],[34,572346,32712],{"className":572347},[31690]," 内。\n当 ",[34,572350,572352,572365],{"className":572351},[31642],[34,572353,572355],{"className":572354},[31646],[31648,572356,572357],{"xmlns":31650},[31652,572358,572359,572363],{},[31655,572360,572361],{},[31658,572362,32936],{},[31665,572364,32936],{"encoding":31667},[34,572366,572368],{"className":572367,"ariaHidden":758},[31672],[34,572369,572371,572374],{"className":572370},[31676],[34,572372],{"className":572373,"style":32527},[31680],[34,572375,32936],{"className":572376},[31685,31686]," 足够大（比如 ",[34,572379,572381,572399],{"className":572380},[31642],[34,572382,572384],{"className":572383},[31646],[31648,572385,572386],{"xmlns":31650},[31652,572387,572388,572396],{},[31655,572389,572390,572392,572394],{},[31658,572391,32936],{},[31662,572393,29622],{},[31658,572395,54151],{},[31665,572397,572398],{"encoding":31667},"x > A",[34,572400,572402,572420],{"className":572401,"ariaHidden":758},[31672],[34,572403,572405,572408,572411,572414,572417],{"className":572404},[31676],[34,572406],{"className":572407,"style":34812},[31680],[34,572409,32936],{"className":572410},[31685,31686],[34,572412],{"className":572413,"style":31846},[31845],[34,572415,29622],{"className":572416},[31850],[34,572418],{"className":572419,"style":31846},[31845],[34,572421,572423,572426],{"className":572422},[31676],[34,572424],{"className":572425,"style":33218},[31680],[34,572427,54151],{"className":572428},[31685,31686],[34,572430,572432,572462],{"className":572431},[31642],[34,572433,572435],{"className":572434},[31646],[31648,572436,572437],{"xmlns":31650},[31652,572438,572439,572459],{},[31655,572440,572441,572443,572445,572447,572449,572451,572453,572455,572457],{},[31658,572442,32936],{},[31662,572444,223],{},[31658,572446,33249],{},[31662,572448,29622],{},[31658,572450,32936],{},[31662,572452,32226],{},[31658,572454,54151],{},[31662,572456,29622],{},[31749,572458,459],{},[31665,572460,572461],{"encoding":31667},"x+t > x-A > 0",[34,572463,572465,572483,572501,572519,572537],{"className":572464,"ariaHidden":758},[31672],[34,572466,572468,572471,572474,572477,572480],{"className":572467},[31676],[34,572469],{"className":572470,"style":32597},[31680],[34,572472,32936],{"className":572473},[31685,31686],[34,572475],{"className":572476,"style":32604},[31845],[34,572478,223],{"className":572479},[32024],[34,572481],{"className":572482,"style":32604},[31845],[34,572484,572486,572489,572492,572495,572498],{"className":572485},[31676],[34,572487],{"className":572488,"style":65913},[31680],[34,572490,33249],{"className":572491},[31685,31686],[34,572493],{"className":572494,"style":31846},[31845],[34,572496,29622],{"className":572497},[31850],[34,572499],{"className":572500,"style":31846},[31845],[34,572502,572504,572507,572510,572513,572516],{"className":572503},[31676],[34,572505],{"className":572506,"style":32597},[31680],[34,572508,32936],{"className":572509},[31685,31686],[34,572511],{"className":572512,"style":32604},[31845],[34,572514,32226],{"className":572515},[32024],[34,572517],{"className":572518,"style":32604},[31845],[34,572520,572522,572525,572528,572531,572534],{"className":572521},[31676],[34,572523],{"className":572524,"style":104901},[31680],[34,572526,54151],{"className":572527},[31685,31686],[34,572529],{"className":572530,"style":31846},[31845],[34,572532,29622],{"className":572533},[31850],[34,572535],{"className":572536,"style":31846},[31845],[34,572538,572540,572543],{"className":572539},[31676],[34,572541],{"className":572542,"style":32617},[31680],[34,572544,459],{"className":572545},[31685],[34,572547,572549,572619],{"className":572548},[31642],[34,572550,572552],{"className":572551},[31646],[31648,572553,572554],{"xmlns":31650},[31652,572555,572556,572616],{},[31655,572557,572558,572560,572568,572570,572572],{},[31658,572559,162086],{"mathvariant":38079},[38846,572561,572562,572564,572566],{},[31658,572563,31739],{},[31658,572565,33249],{},[31662,572567,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,572569,162086],{"mathvariant":38079},[31662,572571,45972],{},[41453,572573,572574,572588],{},[31655,572575,572576,572578,572580,572582,572584,572586],{},[31749,572577,768],{},[31662,572579,456],{"stretchy":702},[31658,572581,32936],{},[31662,572583,223],{},[31658,572585,54151],{},[31662,572587,2557],{"stretchy":702},[31655,572589,572590,572592,572594,572596,572598,572600,572602,572604,572610],{},[31662,572591,15972],{"stretchy":702},[31749,572593,640],{},[31662,572595,223],{},[31662,572597,456],{"stretchy":702},[31658,572599,32936],{},[31662,572601,32226],{},[31658,572603,54151],{},[37624,572605,572606,572608],{},[31662,572607,2557],{"stretchy":702},[31749,572609,768],{},[37624,572611,572612,572614],{},[31662,572613,32712],{"stretchy":702},[31749,572615,768],{},[31665,572617,572618],{"encoding":31667},"|f'_t| \\le \\frac{2(x+A)}{[1+(x-A)^2]^2}",[34,572620,572622,572697],{"className":572621,"ariaHidden":758},[31672],[34,572623,572625,572628,572631,572685,572688,572691,572694],{"className":572624},[31676],[34,572626],{"className":572627,"style":570646},[31680],[34,572629,162086],{"className":572630},[31685],[34,572632,572634,572637],{"className":572633},[31685],[34,572635,31739],{"className":572636,"style":31778},[31685,31686],[34,572638,572640],{"className":572639},[31792],[34,572641,572643,572677],{"className":572642},[31796,31797],[34,572644,572646,572674],{"className":572645},[31801],[34,572647,572649,572660],{"className":572648,"style":132027},[31805],[34,572650,572651,572654],{"style":570667},[34,572652],{"className":572653,"style":31814},[31813],[34,572655,572657],{"className":572656},[31818,31819,31820,31821],[34,572658,33249],{"className":572659},[31685,31686,31821],[34,572661,572662,572665],{"style":37777},[34,572663],{"className":572664,"style":31814},[31813],[34,572666,572668],{"className":572667},[31818,31819,31820,31821],[34,572669,572671],{"className":572670},[31685,31821],[34,572672,116495],{"className":572673},[31685,31821],[34,572675,31829],{"className":572676},[31828],[34,572678,572680],{"className":572679},[31801],[34,572681,572683],{"className":572682,"style":52354},[31805],[34,572684],{},[34,572686,162086],{"className":572687},[31685],[34,572689],{"className":572690,"style":31846},[31845],[34,572692,45972],{"className":572693},[31850],[34,572695],{"className":572696,"style":31846},[31845],[34,572698,572700,572703],{"className":572699},[31676],[34,572701],{"className":572702,"style":327841},[31680],[34,572704,572706,572709,572859],{"className":572705},[31685],[34,572707],{"className":572708},[31782,41488],[34,572710,572712],{"className":572711},[41453],[34,572713,572715,572851],{"className":572714},[31796,31797],[34,572716,572718,572848],{"className":572717},[31801],[34,572719,572721,572811,572819],{"className":572720,"style":70504},[31805],[34,572722,572723,572726],{"style":41504},[34,572724],{"className":572725,"style":41508},[31813],[34,572727,572729],{"className":572728},[31818,31819,31820,31821],[34,572730,572732,572735,572738,572741,572744,572747,572750,572753,572782],{"className":572731},[31685,31821],[34,572733,15972],{"className":572734},[31782,31821],[34,572736,640],{"className":572737},[31685,31821],[34,572739,223],{"className":572740},[32024,31821],[34,572742,456],{"className":572743},[31782,31821],[34,572745,32936],{"className":572746},[31685,31686,31821],[34,572748,32226],{"className":572749},[32024,31821],[34,572751,54151],{"className":572752},[31685,31686,31821],[34,572754,572756,572759],{"className":572755},[31690,31821],[34,572757,2557],{"className":572758},[31690,31821],[34,572760,572762],{"className":572761},[31792],[34,572763,572765],{"className":572764},[31796],[34,572766,572768],{"className":572767},[31801],[34,572769,572771],{"className":572770,"style":548314},[31805],[34,572772,572773,572776],{"style":116668},[34,572774],{"className":572775,"style":43406},[31813],[34,572777,572779],{"className":572778},[31818,43410,43411,31821],[34,572780,768],{"className":572781},[31685,31821],[34,572783,572785,572788],{"className":572784},[31690,31821],[34,572786,32712],{"className":572787},[31690,31821],[34,572789,572791],{"className":572790},[31792],[34,572792,572794],{"className":572793},[31796],[34,572795,572797],{"className":572796},[31801],[34,572798,572800],{"className":572799,"style":548314},[31805],[34,572801,572802,572805],{"style":116668},[34,572803],{"className":572804,"style":43406},[31813],[34,572806,572808],{"className":572807},[31818,43410,43411,31821],[34,572809,768],{"className":572810},[31685,31821],[34,572812,572813,572816],{"style":41520},[34,572814],{"className":572815,"style":41508},[31813],[34,572817],{"className":572818,"style":41528},[41527],[34,572820,572821,572824],{"style":70529},[34,572822],{"className":572823,"style":41508},[31813],[34,572825,572827],{"className":572826},[31818,31819,31820,31821],[34,572828,572830,572833,572836,572839,572842,572845],{"className":572829},[31685,31821],[34,572831,768],{"className":572832},[31685,31821],[34,572834,456],{"className":572835},[31782,31821],[34,572837,32936],{"className":572838},[31685,31686,31821],[34,572840,223],{"className":572841},[32024,31821],[34,572843,54151],{"className":572844},[31685,31686,31821],[34,572846,2557],{"className":572847},[31690,31821],[34,572849,31829],{"className":572850},[31828],[34,572852,572854],{"className":572853},[31801],[34,572855,572857],{"className":572856,"style":324858},[31805],[34,572858],{},[34,572860],{"className":572861},[31690,41488],"\n右边的函数关于 ",[34,572864,572866,572879],{"className":572865},[31642],[34,572867,572869],{"className":572868},[31646],[31648,572870,572871],{"xmlns":31650},[31652,572872,572873,572877],{},[31655,572874,572875],{},[31658,572876,32936],{},[31665,572878,32936],{"encoding":31667},[34,572880,572882],{"className":572881,"ariaHidden":758},[31672],[34,572883,572885,572888],{"className":572884},[31676],[34,572886],{"className":572887,"style":32527},[31680],[34,572889,32936],{"className":572890},[31685,31686]," 的积分为 ",[34,572893,572895,572925],{"className":572894},[31642],[34,572896,572898],{"className":572897},[31646],[31648,572899,572900],{"xmlns":31650},[31652,572901,572902,572922],{},[31655,572903,572904,572906,572908,572918,572920],{},[31662,572905,190205],{},[31662,572907,546295],{},[41453,572909,572910,572912],{},[31749,572911,640],{},[37624,572913,572914,572916],{},[31658,572915,32936],{},[31749,572917,786],{},[31658,572919,46900],{},[31658,572921,32936],{},[31665,572923,572924],{"encoding":31667},"\\sim \\int \\frac{1}{x^3} dx",[34,572926,572928,572940],{"className":572927,"ariaHidden":758},[31672],[34,572929,572931,572934,572937],{"className":572930},[31676],[34,572932],{"className":572933,"style":73089},[31680],[34,572935,190205],{"className":572936},[31850],[34,572938],{"className":572939,"style":31846},[31845],[34,572941,572943,572946,572949,572952,573046,573049],{"className":572942},[31676],[34,572944],{"className":572945,"style":46727},[31680],[34,572947,546295],{"className":572948,"style":546331},[32767,131249,156560],[34,572950],{"className":572951,"style":32123},[31845],[34,572953,572955,572958,573043],{"className":572954},[31685],[34,572956],{"className":572957},[31782,41488],[34,572959,572961],{"className":572960},[41453],[34,572962,572964,573035],{"className":572963},[31796,31797],[34,572965,572967,573032],{"className":572966},[31801],[34,572968,572970,573010,573018],{"className":572969,"style":46746},[31805],[34,572971,572972,572975],{"style":41504},[34,572973],{"className":572974,"style":41508},[31813],[34,572976,572978],{"className":572977},[31818,31819,31820,31821],[34,572979,572981],{"className":572980},[31685,31821],[34,572982,572984,572987],{"className":572983},[31685,31821],[34,572985,32936],{"className":572986},[31685,31686,31821],[34,572988,572990],{"className":572989},[31792],[34,572991,572993],{"className":572992},[31796],[34,572994,572996],{"className":572995},[31801],[34,572997,572999],{"className":572998,"style":548314},[31805],[34,573000,573001,573004],{"style":116668},[34,573002],{"className":573003,"style":43406},[31813],[34,573005,573007],{"className":573006},[31818,43410,43411,31821],[34,573008,786],{"className":573009},[31685,31821],[34,573011,573012,573015],{"style":41520},[34,573013],{"className":573014,"style":41508},[31813],[34,573016],{"className":573017,"style":41528},[41527],[34,573019,573020,573023],{"style":46771},[34,573021],{"className":573022,"style":41508},[31813],[34,573024,573026],{"className":573025},[31818,31819,31820,31821],[34,573027,573029],{"className":573028},[31685,31821],[34,573030,640],{"className":573031},[31685,31821],[34,573033,31829],{"className":573034},[31828],[34,573036,573038],{"className":573037},[31801],[34,573039,573041],{"className":573040,"style":41619},[31805],[34,573042],{},[34,573044],{"className":573045},[31690,41488],[34,573047,46900],{"className":573048},[31685,31686],[34,573050,32936],{"className":573051},[31685,31686],"，是收敛的。\n所以在任意有限区间上，导积分一致收敛。",[108,573054,573055,573057,573058,573102],{},[2627,573056,62532],{},"\n因为导积分一致收敛，所以可以在积分号下求导，即 ",[34,573059,573061,573081],{"className":573060},[31642],[34,573062,573064],{"className":573063},[31646],[31648,573065,573066],{"xmlns":31650},[31652,573067,573068,573078],{},[31655,573069,573070,573072,573074,573076],{},[31658,573071,106063],{},[31662,573073,456],{"stretchy":702},[31658,573075,33249],{},[31662,573077,2557],{"stretchy":702},[31665,573079,573080],{"encoding":31667},"I(t)",[34,573082,573084],{"className":573083,"ariaHidden":758},[31672],[34,573085,573087,573090,573093,573096,573099],{"className":573086},[31676],[34,573088],{"className":573089,"style":31774},[31680],[34,573091,106063],{"className":573092,"style":67482},[31685,31686],[34,573094,456],{"className":573095},[31782],[34,573097,33249],{"className":573098},[31685,31686],[34,573100,2557],{"className":573101},[31690]," 可微。",[32188,573104],{},[253,573106,573108],{"id":573107},"p330-t9","p330 T9",[15,573110,573111,525700,573113,573461,573462,1840,573748,26192],{},[2627,573112,214361],{},[34,573114,573116,573186],{"className":573115},[31642],[34,573117,573119],{"className":573118},[31646],[31648,573120,573121],{"xmlns":31650},[31652,573122,573123,573183],{},[31655,573124,573125,573157,573159,573167,573179,573181],{},[41453,573126,573127,573155],{},[31655,573128,573129,573141,573143],{},[37624,573130,573131,573133],{},[31658,573132,33244],{},[31655,573134,573135,573137,573139],{},[31662,573136,32226],{},[31658,573138,111],{},[31658,573140,32936],{},[31662,573142,32226],{},[37624,573144,573145,573147],{},[31658,573146,33244],{},[31655,573148,573149,573151,573153],{},[31662,573150,32226],{},[31658,573152,1697],{},[31658,573154,32936],{},[31658,573156,32936],{},[31662,573158,202],{},[38846,573160,573161,573163,573165],{},[31662,573162,546295],{},[31658,573164,111],{},[31658,573166,1697],{},[37624,573168,573169,573171],{},[31658,573170,33244],{},[31655,573172,573173,573175,573177],{},[31662,573174,32226],{},[31658,573176,32936],{},[31658,573178,67764],{},[31658,573180,46900],{},[31658,573182,67764],{},[31665,573184,573185],{"encoding":31667},"\\frac{e^{-ax} - e^{-bx}}{x} = \\int_{a}^{b} e^{-xy}dy",[34,573187,573189,573351],{"className":573188,"ariaHidden":758},[31672],[34,573190,573192,573196,573342,573345,573348],{"className":573191},[31676],[34,573193],{"className":573194,"style":573195},[31680],"height:1.3879em;vertical-align:-0.345em;",[34,573197,573199,573202,573339],{"className":573198},[31685],[34,573200],{"className":573201},[31782,41488],[34,573203,573205],{"className":573204},[41453],[34,573206,573208,573331],{"className":573207},[31796,31797],[34,573209,573211,573328],{"className":573210},[31801],[34,573212,573215,573229,573237],{"className":573213,"style":573214},[31805],"height:1.0429em;",[34,573216,573217,573220],{"style":41504},[34,573218],{"className":573219,"style":41508},[31813],[34,573221,573223],{"className":573222},[31818,31819,31820,31821],[34,573224,573226],{"className":573225},[31685,31821],[34,573227,32936],{"className":573228},[31685,31686,31821],[34,573230,573231,573234],{"style":41520},[34,573232],{"className":573233,"style":41508},[31813],[34,573235],{"className":573236,"style":41528},[41527],[34,573238,573239,573242],{"style":46771},[34,573240],{"className":573241,"style":41508},[31813],[34,573243,573245],{"className":573244},[31818,31819,31820,31821],[34,573246,573248,573286,573289],{"className":573247},[31685,31821],[34,573249,573251,573254],{"className":573250},[31685,31821],[34,573252,33244],{"className":573253},[31685,31686,31821],[34,573255,573257],{"className":573256},[31792],[34,573258,573260],{"className":573259},[31796],[34,573261,573263],{"className":573262},[31801],[34,573264,573266],{"className":573265,"style":552388},[31805],[34,573267,573268,573271],{"style":552391},[34,573269],{"className":573270,"style":43406},[31813],[34,573272,573274],{"className":573273},[31818,43410,43411,31821],[34,573275,573277,573280,573283],{"className":573276},[31685,31821],[34,573278,32226],{"className":573279},[31685,31821],[34,573281,111],{"className":573282},[31685,31686,31821],[34,573284,32936],{"className":573285},[31685,31686,31821],[34,573287,32226],{"className":573288},[32024,31821],[34,573290,573292,573295],{"className":573291},[31685,31821],[34,573293,33244],{"className":573294},[31685,31686,31821],[34,573296,573298],{"className":573297},[31792],[34,573299,573301],{"className":573300},[31796],[34,573302,573304],{"className":573303},[31801],[34,573305,573308],{"className":573306,"style":573307},[31805],"height:0.927em;",[34,573309,573310,573313],{"style":552391},[34,573311],{"className":573312,"style":43406},[31813],[34,573314,573316],{"className":573315},[31818,43410,43411,31821],[34,573317,573319,573322,573325],{"className":573318},[31685,31821],[34,573320,32226],{"className":573321},[31685,31821],[34,573323,1697],{"className":573324},[31685,31686,31821],[34,573326,32936],{"className":573327},[31685,31686,31821],[34,573329,31829],{"className":573330},[31828],[34,573332,573334],{"className":573333},[31801],[34,573335,573337],{"className":573336,"style":41619},[31805],[34,573338],{},[34,573340],{"className":573341},[31690,41488],[34,573343],{"className":573344,"style":31846},[31845],[34,573346,202],{"className":573347},[31850],[34,573349],{"className":573350,"style":31846},[31845],[34,573352,573354,573357,573414,573417,573455,573458],{"className":573353},[31676],[34,573355],{"className":573356,"style":546324},[31680],[34,573358,573360,573363],{"className":573359},[32767],[34,573361,546295],{"className":573362,"style":546331},[32767,131249,156560],[34,573364,573366],{"className":573365},[31792],[34,573367,573369,573406],{"className":573368},[31796,31797],[34,573370,573372,573403],{"className":573371},[31801],[34,573373,573375,573389],{"className":573374,"style":546344},[31805],[34,573376,573377,573380],{"style":546347},[34,573378],{"className":573379,"style":31814},[31813],[34,573381,573383],{"className":573382},[31818,31819,31820,31821],[34,573384,573386],{"className":573385},[31685,31821],[34,573387,111],{"className":573388},[31685,31686,31821],[34,573390,573391,573394],{"style":546362},[34,573392],{"className":573393,"style":31814},[31813],[34,573395,573397],{"className":573396},[31818,31819,31820,31821],[34,573398,573400],{"className":573399},[31685,31821],[34,573401,1697],{"className":573402},[31685,31686,31821],[34,573404,31829],{"className":573405},[31828],[34,573407,573409],{"className":573408},[31801],[34,573410,573412],{"className":573411,"style":546384},[31805],[34,573413],{},[34,573415],{"className":573416,"style":32123},[31845],[34,573418,573420,573423],{"className":573419},[31685],[34,573421,33244],{"className":573422},[31685,31686],[34,573424,573426],{"className":573425},[31792],[34,573427,573429],{"className":573428},[31796],[34,573430,573432],{"className":573431},[31801],[34,573433,573435],{"className":573434,"style":156828},[31805],[34,573436,573437,573440],{"style":37777},[34,573438],{"className":573439,"style":31814},[31813],[34,573441,573443],{"className":573442},[31818,31819,31820,31821],[34,573444,573446,573449,573452],{"className":573445},[31685,31821],[34,573447,32226],{"className":573448},[31685,31821],[34,573450,32936],{"className":573451},[31685,31686,31821],[34,573453,67764],{"className":573454,"style":32062},[31685,31686,31821],[34,573456,46900],{"className":573457},[31685,31686],[34,573459,67764],{"className":573460,"style":32062},[31685,31686],", 计算 ",[34,573463,573465,573525],{"className":573464},[31642],[34,573466,573468],{"className":573467},[31646],[31648,573469,573470],{"xmlns":31650},[31652,573471,573472,573522],{},[31655,573473,573474,573486,573518,573520],{},[38846,573475,573476,573478,573480],{},[31662,573477,546295],{},[31749,573479,459],{},[31655,573481,573482,573484],{},[31662,573483,223],{},[31658,573485,138286],{"mathvariant":38079},[41453,573487,573488,573516],{},[31655,573489,573490,573502,573504],{},[37624,573491,573492,573494],{},[31658,573493,33244],{},[31655,573495,573496,573498,573500],{},[31662,573497,32226],{},[31658,573499,111],{},[31658,573501,32936],{},[31662,573503,32226],{},[37624,573505,573506,573508],{},[31658,573507,33244],{},[31655,573509,573510,573512,573514],{},[31662,573511,32226],{},[31658,573513,1697],{},[31658,573515,32936],{},[31658,573517,32936],{},[31658,573519,46900],{},[31658,573521,32936],{},[31665,573523,573524],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{e^{-ax} - e^{-bx}}{x}dx",[34,573526,573528],{"className":573527,"ariaHidden":758},[31672],[34,573529,573531,573535,573595,573598,573742,573745],{"className":573530},[31676],[34,573532],{"className":573533,"style":573534},[31680],"height:1.3987em;vertical-align:-0.3558em;",[34,573536,573538,573541],{"className":573537},[32767],[34,573539,546295],{"className":573540,"style":546331},[32767,131249,156560],[34,573542,573544],{"className":573543},[31792],[34,573545,573547,573587],{"className":573546},[31796,31797],[34,573548,573550,573584],{"className":573549},[31801],[34,573551,573553,573567],{"className":573552,"style":546550},[31805],[34,573554,573555,573558],{"style":546347},[34,573556],{"className":573557,"style":31814},[31813],[34,573559,573561],{"className":573560},[31818,31819,31820,31821],[34,573562,573564],{"className":573563},[31685,31821],[34,573565,459],{"className":573566},[31685,31821],[34,573568,573569,573572],{"style":546362},[34,573570],{"className":573571,"style":31814},[31813],[34,573573,573575],{"className":573574},[31818,31819,31820,31821],[34,573576,573578,573581],{"className":573577},[31685,31821],[34,573579,223],{"className":573580},[31685,31821],[34,573582,138286],{"className":573583},[31685,31821],[34,573585,31829],{"className":573586},[31828],[34,573588,573590],{"className":573589},[31801],[34,573591,573593],{"className":573592,"style":546384},[31805],[34,573594],{},[34,573596],{"className":573597,"style":32123},[31845],[34,573599,573601,573604,573739],{"className":573600},[31685],[34,573602],{"className":573603},[31782,41488],[34,573605,573607],{"className":573606},[41453],[34,573608,573610,573731],{"className":573609},[31796,31797],[34,573611,573613,573728],{"className":573612},[31801],[34,573614,573616,573630,573638],{"className":573615,"style":573214},[31805],[34,573617,573618,573621],{"style":41504},[34,573619],{"className":573620,"style":41508},[31813],[34,573622,573624],{"className":573623},[31818,31819,31820,31821],[34,573625,573627],{"className":573626},[31685,31821],[34,573628,32936],{"className":573629},[31685,31686,31821],[34,573631,573632,573635],{"style":41520},[34,573633],{"className":573634,"style":41508},[31813],[34,573636],{"className":573637,"style":41528},[41527],[34,573639,573640,573643],{"style":46771},[34,573641],{"className":573642,"style":41508},[31813],[34,573644,573646],{"className":573645},[31818,31819,31820,31821],[34,573647,573649,573687,573690],{"className":573648},[31685,31821],[34,573650,573652,573655],{"className":573651},[31685,31821],[34,573653,33244],{"className":573654},[31685,31686,31821],[34,573656,573658],{"className":573657},[31792],[34,573659,573661],{"className":573660},[31796],[34,573662,573664],{"className":573663},[31801],[34,573665,573667],{"className":573666,"style":552388},[31805],[34,573668,573669,573672],{"style":552391},[34,573670],{"className":573671,"style":43406},[31813],[34,573673,573675],{"className":573674},[31818,43410,43411,31821],[34,573676,573678,573681,573684],{"className":573677},[31685,31821],[34,573679,32226],{"className":573680},[31685,31821],[34,573682,111],{"className":573683},[31685,31686,31821],[34,573685,32936],{"className":573686},[31685,31686,31821],[34,573688,32226],{"className":573689},[32024,31821],[34,573691,573693,573696],{"className":573692},[31685,31821],[34,573694,33244],{"className":573695},[31685,31686,31821],[34,573697,573699],{"className":573698},[31792],[34,573700,573702],{"className":573701},[31796],[34,573703,573705],{"className":573704},[31801],[34,573706,573708],{"className":573707,"style":573307},[31805],[34,573709,573710,573713],{"style":552391},[34,573711],{"className":573712,"style":43406},[31813],[34,573714,573716],{"className":573715},[31818,43410,43411,31821],[34,573717,573719,573722,573725],{"className":573718},[31685,31821],[34,573720,32226],{"className":573721},[31685,31821],[34,573723,1697],{"className":573724},[31685,31686,31821],[34,573726,32936],{"className":573727},[31685,31686,31821],[34,573729,31829],{"className":573730},[31828],[34,573732,573734],{"className":573733},[31801],[34,573735,573737],{"className":573736,"style":41619},[31805],[34,573738],{},[34,573740],{"className":573741},[31690,41488],[34,573743,46900],{"className":573744},[31685,31686],[34,573746,32936],{"className":573747},[31685,31686],[34,573749,573751,573773],{"className":573750},[31642],[34,573752,573754],{"className":573753},[31646],[31648,573755,573756],{"xmlns":31650},[31652,573757,573758,573770],{},[31655,573759,573760,573762,573764,573766,573768],{},[31658,573761,1697],{},[31662,573763,29622],{},[31658,573765,111],{},[31662,573767,29622],{},[31749,573769,459],{},[31665,573771,573772],{"encoding":31667},"b > a > 0",[34,573774,573776,573794,573812],{"className":573775,"ariaHidden":758},[31672],[34,573777,573779,573782,573785,573788,573791],{"className":573778},[31676],[34,573780],{"className":573781,"style":46991},[31680],[34,573783,1697],{"className":573784},[31685,31686],[34,573786],{"className":573787,"style":31846},[31845],[34,573789,29622],{"className":573790},[31850],[34,573792],{"className":573793,"style":31846},[31845],[34,573795,573797,573800,573803,573806,573809],{"className":573796},[31676],[34,573798],{"className":573799,"style":34812},[31680],[34,573801,111],{"className":573802},[31685,31686],[34,573804],{"className":573805,"style":31846},[31845],[34,573807,29622],{"className":573808},[31850],[34,573810],{"className":573811,"style":31846},[31845],[34,573813,573815,573818],{"className":573814},[31676],[34,573816],{"className":573817,"style":32617},[31680],[34,573819,459],{"className":573820},[31685],[15,573822,573823,573825,573826,573829],{},[2627,573824,91513],{},"\n这是经典的利用",[2627,573827,573828],{},"交换积分顺序","计算积分的题目。",[15,573831,573832],{},[2627,573833,549629],{},[2230,573835,573836,574124,574626,575375],{},[108,573837,573838,573841],{},[2627,573839,573840],{},"代入公式：",[34,573842,573844,573906],{"className":573843},[31642],[34,573845,573847],{"className":573846},[31646],[31648,573848,573849],{"xmlns":31650},[31652,573850,573851,573903],{},[31655,573852,573853,573855,573857,573869,573899,573901],{},[31658,573854,106063],{},[31662,573856,202],{},[38846,573858,573859,573861,573863],{},[31662,573860,546295],{},[31749,573862,459],{},[31655,573864,573865,573867],{},[31662,573866,223],{},[31658,573868,138286],{"mathvariant":38079},[31655,573870,573871,573873,573881,573893,573895,573897],{},[31662,573872,456],{"fence":758},[38846,573874,573875,573877,573879],{},[31662,573876,546295],{},[31658,573878,111],{},[31658,573880,1697],{},[37624,573882,573883,573885],{},[31658,573884,33244],{},[31655,573886,573887,573889,573891],{},[31662,573888,32226],{},[31658,573890,32936],{},[31658,573892,67764],{},[31658,573894,46900],{},[31658,573896,67764],{},[31662,573898,2557],{"fence":758},[31658,573900,46900],{},[31658,573902,32936],{},[31665,573904,573905],{"encoding":31667},"I = \\int_{0}^{+\\infty} \\left( \\int_{a}^{b} e^{-xy} dy \\right) dx",[34,573907,573909,573927],{"className":573908,"ariaHidden":758},[31672],[34,573910,573912,573915,573918,573921,573924],{"className":573911},[31676],[34,573913],{"className":573914,"style":33218},[31680],[34,573916,106063],{"className":573917,"style":67482},[31685,31686],[34,573919],{"className":573920,"style":31846},[31845],[34,573922,202],{"className":573923},[31850],[34,573925],{"className":573926,"style":31846},[31845],[34,573928,573930,573933,573993,573996,574115,574118,574121],{"className":573929},[31676],[34,573931],{"className":573932,"style":141282},[31680],[34,573934,573936,573939],{"className":573935},[32767],[34,573937,546295],{"className":573938,"style":546331},[32767,131249,156560],[34,573940,573942],{"className":573941},[31792],[34,573943,573945,573985],{"className":573944},[31796,31797],[34,573946,573948,573982],{"className":573947},[31801],[34,573949,573951,573965],{"className":573950,"style":546550},[31805],[34,573952,573953,573956],{"style":546347},[34,573954],{"className":573955,"style":31814},[31813],[34,573957,573959],{"className":573958},[31818,31819,31820,31821],[34,573960,573962],{"className":573961},[31685,31821],[34,573963,459],{"className":573964},[31685,31821],[34,573966,573967,573970],{"style":546362},[34,573968],{"className":573969,"style":31814},[31813],[34,573971,573973],{"className":573972},[31818,31819,31820,31821],[34,573974,573976,573979],{"className":573975},[31685,31821],[34,573977,223],{"className":573978},[31685,31821],[34,573980,138286],{"className":573981},[31685,31821],[34,573983,31829],{"className":573984},[31828],[34,573986,573988],{"className":573987},[31801],[34,573989,573991],{"className":573990,"style":546384},[31805],[34,573992],{},[34,573994],{"className":573995,"style":32123},[31845],[34,573997,573999,574005,574062,574065,574103,574106,574109],{"className":573998},[34006],[34,574000,574002],{"className":574001,"style":75811},[31782,75810],[34,574003,456],{"className":574004},[71281,141292],[34,574006,574008,574011],{"className":574007},[32767],[34,574009,546295],{"className":574010,"style":546331},[32767,131249,156560],[34,574012,574014],{"className":574013},[31792],[34,574015,574017,574054],{"className":574016},[31796,31797],[34,574018,574020,574051],{"className":574019},[31801],[34,574021,574023,574037],{"className":574022,"style":546344},[31805],[34,574024,574025,574028],{"style":546347},[34,574026],{"className":574027,"style":31814},[31813],[34,574029,574031],{"className":574030},[31818,31819,31820,31821],[34,574032,574034],{"className":574033},[31685,31821],[34,574035,111],{"className":574036},[31685,31686,31821],[34,574038,574039,574042],{"style":546362},[34,574040],{"className":574041,"style":31814},[31813],[34,574043,574045],{"className":574044},[31818,31819,31820,31821],[34,574046,574048],{"className":574047},[31685,31821],[34,574049,1697],{"className":574050},[31685,31686,31821],[34,574052,31829],{"className":574053},[31828],[34,574055,574057],{"className":574056},[31801],[34,574058,574060],{"className":574059,"style":546384},[31805],[34,574061],{},[34,574063],{"className":574064,"style":32123},[31845],[34,574066,574068,574071],{"className":574067},[31685],[34,574069,33244],{"className":574070},[31685,31686],[34,574072,574074],{"className":574073},[31792],[34,574075,574077],{"className":574076},[31796],[34,574078,574080],{"className":574079},[31801],[34,574081,574083],{"className":574082,"style":156828},[31805],[34,574084,574085,574088],{"style":37777},[34,574086],{"className":574087,"style":31814},[31813],[34,574089,574091],{"className":574090},[31818,31819,31820,31821],[34,574092,574094,574097,574100],{"className":574093},[31685,31821],[34,574095,32226],{"className":574096},[31685,31821],[34,574098,32936],{"className":574099},[31685,31686,31821],[34,574101,67764],{"className":574102,"style":32062},[31685,31686,31821],[34,574104,46900],{"className":574105},[31685,31686],[34,574107,67764],{"className":574108,"style":32062},[31685,31686],[34,574110,574112],{"className":574111,"style":75811},[31690,75810],[34,574113,2557],{"className":574114},[71281,141292],[34,574116],{"className":574117,"style":32123},[31845],[34,574119,46900],{"className":574120},[31685,31686],[34,574122,32936],{"className":574123},[31685,31686],[108,574125,574126,574129,574130,574267,574268,574342,574343],{},[2627,574127,574128],{},"交换积分顺序：","\n原积分区域是：",[34,574131,574133,574177],{"className":574132},[31642],[34,574134,574136],{"className":574135},[31646],[31648,574137,574138],{"xmlns":31650},[31652,574139,574140,574174],{},[31655,574141,574142,574144,574146,574148,574150,574152,574154,574156,574158,574160,574162,574164,574166,574168,574170,574172],{},[31658,574143,32936],{},[31662,574145,112296],{},[31662,574147,15972],{"stretchy":702},[31749,574149,459],{},[31662,574151,31960],{"separator":758},[31662,574153,223],{},[31658,574155,138286],{"mathvariant":38079},[31662,574157,2557],{"stretchy":702},[31662,574159,31960],{"separator":758},[31658,574161,67764],{},[31662,574163,112296],{},[31662,574165,15972],{"stretchy":702},[31658,574167,111],{},[31662,574169,31960],{"separator":758},[31658,574171,1697],{},[31662,574173,32712],{"stretchy":702},[31665,574175,574176],{"encoding":31667},"x \\in [0, +\\infty), y \\in [a, b]",[34,574178,574180,574198,574243],{"className":574179,"ariaHidden":758},[31672],[34,574181,574183,574186,574189,574192,574195],{"className":574182},[31676],[34,574184],{"className":574185,"style":34812},[31680],[34,574187,32936],{"className":574188},[31685,31686],[34,574190],{"className":574191,"style":31846},[31845],[34,574193,112296],{"className":574194},[31850],[34,574196],{"className":574197,"style":31846},[31845],[34,574199,574201,574204,574207,574210,574213,574216,574219,574222,574225,574228,574231,574234,574237,574240],{"className":574200},[31676],[34,574202],{"className":574203,"style":31774},[31680],[34,574205,15972],{"className":574206},[31782],[34,574208,459],{"className":574209},[31685],[34,574211,31960],{"className":574212},[32119],[34,574214],{"className":574215,"style":32123},[31845],[34,574217,223],{"className":574218},[31685],[34,574220,138286],{"className":574221},[31685],[34,574223,2557],{"className":574224},[31690],[34,574226,31960],{"className":574227},[32119],[34,574229],{"className":574230,"style":32123},[31845],[34,574232,67764],{"className":574233,"style":32062},[31685,31686],[34,574235],{"className":574236,"style":31846},[31845],[34,574238,112296],{"className":574239},[31850],[34,574241],{"className":574242,"style":31846},[31845],[34,574244,574246,574249,574252,574255,574258,574261,574264],{"className":574245},[31676],[34,574247],{"className":574248,"style":31774},[31680],[34,574250,15972],{"className":574251},[31782],[34,574253,111],{"className":574254},[31685,31686],[34,574256,31960],{"className":574257},[32119],[34,574259],{"className":574260,"style":32123},[31845],[34,574262,1697],{"className":574263},[31685,31686],[34,574265,32712],{"className":574266},[31690],"。这是一个半无限长条区域。\n由于被积函数全是正的（",[34,574269,574271,574295],{"className":574270},[31642],[34,574272,574274],{"className":574273},[31646],[31648,574275,574276],{"xmlns":31650},[31652,574277,574278,574292],{},[31655,574279,574280],{},[37624,574281,574282,574284],{},[31658,574283,33244],{},[31655,574285,574286,574288,574290],{},[31662,574287,32226],{},[31658,574289,32936],{},[31658,574291,67764],{},[31665,574293,574294],{"encoding":31667},"e^{-xy}",[34,574296,574298],{"className":574297,"ariaHidden":758},[31672],[34,574299,574301,574304],{"className":574300},[31676],[34,574302],{"className":574303,"style":156828},[31680],[34,574305,574307,574310],{"className":574306},[31685],[34,574308,33244],{"className":574309},[31685,31686],[34,574311,574313],{"className":574312},[31792],[34,574314,574316],{"className":574315},[31796],[34,574317,574319],{"className":574318},[31801],[34,574320,574322],{"className":574321,"style":156828},[31805],[34,574323,574324,574327],{"style":37777},[34,574325],{"className":574326,"style":31814},[31813],[34,574328,574330],{"className":574329},[31818,31819,31820,31821],[34,574331,574333,574336,574339],{"className":574332},[31685,31821],[34,574334,32226],{"className":574335},[31685,31821],[34,574337,32936],{"className":574338},[31685,31686,31821],[34,574340,67764],{"className":574341,"style":32062},[31685,31686,31821],"），且积分收敛，我们可以交换顺序（Fubini定理的应用背景）。\n",[34,574344,574346,574408],{"className":574345},[31642],[34,574347,574349],{"className":574348},[31646],[31648,574350,574351],{"xmlns":31650},[31652,574352,574353,574405],{},[31655,574354,574355,574357,574359,574367,574401,574403],{},[31658,574356,106063],{},[31662,574358,202],{},[38846,574360,574361,574363,574365],{},[31662,574362,546295],{},[31658,574364,111],{},[31658,574366,1697],{},[31655,574368,574369,574371,574383,574395,574397,574399],{},[31662,574370,456],{"fence":758},[38846,574372,574373,574375,574377],{},[31662,574374,546295],{},[31749,574376,459],{},[31655,574378,574379,574381],{},[31662,574380,223],{},[31658,574382,138286],{"mathvariant":38079},[37624,574384,574385,574387],{},[31658,574386,33244],{},[31655,574388,574389,574391,574393],{},[31662,574390,32226],{},[31658,574392,32936],{},[31658,574394,67764],{},[31658,574396,46900],{},[31658,574398,32936],{},[31662,574400,2557],{"fence":758},[31658,574402,46900],{},[31658,574404,67764],{},[31665,574406,574407],{"encoding":31667},"I = \\int_{a}^{b} \\left( \\int_{0}^{+\\infty} e^{-xy} dx \\right) dy",[34,574409,574411,574429],{"className":574410,"ariaHidden":758},[31672],[34,574412,574414,574417,574420,574423,574426],{"className":574413},[31676],[34,574415],{"className":574416,"style":33218},[31680],[34,574418,106063],{"className":574419,"style":67482},[31685,31686],[34,574421],{"className":574422,"style":31846},[31845],[34,574424,202],{"className":574425},[31850],[34,574427],{"className":574428,"style":31846},[31845],[34,574430,574432,574435,574492,574495,574617,574620,574623],{"className":574431},[31676],[34,574433],{"className":574434,"style":141282},[31680],[34,574436,574438,574441],{"className":574437},[32767],[34,574439,546295],{"className":574440,"style":546331},[32767,131249,156560],[34,574442,574444],{"className":574443},[31792],[34,574445,574447,574484],{"className":574446},[31796,31797],[34,574448,574450,574481],{"className":574449},[31801],[34,574451,574453,574467],{"className":574452,"style":546344},[31805],[34,574454,574455,574458],{"style":546347},[34,574456],{"className":574457,"style":31814},[31813],[34,574459,574461],{"className":574460},[31818,31819,31820,31821],[34,574462,574464],{"className":574463},[31685,31821],[34,574465,111],{"className":574466},[31685,31686,31821],[34,574468,574469,574472],{"style":546362},[34,574470],{"className":574471,"style":31814},[31813],[34,574473,574475],{"className":574474},[31818,31819,31820,31821],[34,574476,574478],{"className":574477},[31685,31821],[34,574479,1697],{"className":574480},[31685,31686,31821],[34,574482,31829],{"className":574483},[31828],[34,574485,574487],{"className":574486},[31801],[34,574488,574490],{"className":574489,"style":546384},[31805],[34,574491],{},[34,574493],{"className":574494,"style":32123},[31845],[34,574496,574498,574504,574564,574567,574605,574608,574611],{"className":574497},[34006],[34,574499,574501],{"className":574500,"style":75811},[31782,75810],[34,574502,456],{"className":574503},[71281,141292],[34,574505,574507,574510],{"className":574506},[32767],[34,574508,546295],{"className":574509,"style":546331},[32767,131249,156560],[34,574511,574513],{"className":574512},[31792],[34,574514,574516,574556],{"className":574515},[31796,31797],[34,574517,574519,574553],{"className":574518},[31801],[34,574520,574522,574536],{"className":574521,"style":546550},[31805],[34,574523,574524,574527],{"style":546347},[34,574525],{"className":574526,"style":31814},[31813],[34,574528,574530],{"className":574529},[31818,31819,31820,31821],[34,574531,574533],{"className":574532},[31685,31821],[34,574534,459],{"className":574535},[31685,31821],[34,574537,574538,574541],{"style":546362},[34,574539],{"className":574540,"style":31814},[31813],[34,574542,574544],{"className":574543},[31818,31819,31820,31821],[34,574545,574547,574550],{"className":574546},[31685,31821],[34,574548,223],{"className":574549},[31685,31821],[34,574551,138286],{"className":574552},[31685,31821],[34,574554,31829],{"className":574555},[31828],[34,574557,574559],{"className":574558},[31801],[34,574560,574562],{"className":574561,"style":546384},[31805],[34,574563],{},[34,574565],{"className":574566,"style":32123},[31845],[34,574568,574570,574573],{"className":574569},[31685],[34,574571,33244],{"className":574572},[31685,31686],[34,574574,574576],{"className":574575},[31792],[34,574577,574579],{"className":574578},[31796],[34,574580,574582],{"className":574581},[31801],[34,574583,574585],{"className":574584,"style":156828},[31805],[34,574586,574587,574590],{"style":37777},[34,574588],{"className":574589,"style":31814},[31813],[34,574591,574593],{"className":574592},[31818,31819,31820,31821],[34,574594,574596,574599,574602],{"className":574595},[31685,31821],[34,574597,32226],{"className":574598},[31685,31821],[34,574600,32936],{"className":574601},[31685,31686,31821],[34,574603,67764],{"className":574604,"style":32062},[31685,31686,31821],[34,574606,46900],{"className":574607},[31685,31686],[34,574609,32936],{"className":574610},[31685,31686],[34,574612,574614],{"className":574613,"style":75811},[31690,75810],[34,574615,2557],{"className":574616},[71281,141292],[34,574618],{"className":574619,"style":32123},[31845],[34,574621,46900],{"className":574622},[31685,31686],[34,574624,67764],{"className":574625,"style":32062},[31685,31686],[108,574627,574628,574631,575270,575271,575322,575323,575374],{},[2627,574629,574630],{},"计算内层积分：",[34,574632,574634,574748],{"className":574633},[31642],[34,574635,574637],{"className":574636},[31646],[31648,574638,574639],{"xmlns":31650},[31652,574640,574641,574745],{},[31655,574642,574643,574655,574667,574669,574671,574673,574719,574721,574723,574725,574727,574729,574735,574737,574739],{},[38846,574644,574645,574647,574649],{},[31662,574646,546295],{},[31749,574648,459],{},[31655,574650,574651,574653],{},[31662,574652,223],{},[31658,574654,138286],{"mathvariant":38079},[37624,574656,574657,574659],{},[31658,574658,33244],{},[31655,574660,574661,574663,574665],{},[31662,574662,32226],{},[31658,574664,32936],{},[31658,574666,67764],{},[31658,574668,46900],{},[31658,574670,32936],{},[31662,574672,202],{},[38846,574674,574675,574701,574709],{},[31655,574676,574677,574679,574681,574687,574699],{},[31662,574678,15972],{"fence":758},[31662,574680,32226],{},[41453,574682,574683,574685],{},[31749,574684,640],{},[31658,574686,67764],{},[37624,574688,574689,574691],{},[31658,574690,33244],{},[31655,574692,574693,574695,574697],{},[31662,574694,32226],{},[31658,574696,32936],{},[31658,574698,67764],{},[31662,574700,32712],{"fence":758},[31655,574702,574703,574705,574707],{},[31658,574704,32936],{},[31662,574706,202],{},[31749,574708,459],{},[31655,574710,574711,574713,574715,574717],{},[31658,574712,32936],{},[31662,574714,202],{},[31662,574716,223],{},[31658,574718,138286],{"mathvariant":38079},[31662,574720,202],{},[31749,574722,459],{},[31662,574724,32226],{},[31662,574726,456],{"stretchy":702},[31662,574728,32226],{},[41453,574730,574731,574733],{},[31749,574732,640],{},[31658,574734,67764],{},[31662,574736,2557],{"stretchy":702},[31662,574738,202],{},[41453,574740,574741,574743],{},[31749,574742,640],{},[31658,574744,67764],{},[31665,574746,574747],{"encoding":31667},"\\int_{0}^{+\\infty} e^{-xy} dx = \\left[ -\\frac{1}{y} e^{-xy} \\right]_{x=0}^{x=+\\infty} = 0 - (-\\frac{1}{y}) = \\frac{1}{y}",[34,574749,574751,574873,575086,575104,575196],{"className":574750,"ariaHidden":758},[31672],[34,574752,574754,574757,574817,574820,574858,574861,574864,574867,574870],{"className":574753},[31676],[34,574755],{"className":574756,"style":546531},[31680],[34,574758,574760,574763],{"className":574759},[32767],[34,574761,546295],{"className":574762,"style":546331},[32767,131249,156560],[34,574764,574766],{"className":574765},[31792],[34,574767,574769,574809],{"className":574768},[31796,31797],[34,574770,574772,574806],{"className":574771},[31801],[34,574773,574775,574789],{"className":574774,"style":546550},[31805],[34,574776,574777,574780],{"style":546347},[34,574778],{"className":574779,"style":31814},[31813],[34,574781,574783],{"className":574782},[31818,31819,31820,31821],[34,574784,574786],{"className":574785},[31685,31821],[34,574787,459],{"className":574788},[31685,31821],[34,574790,574791,574794],{"style":546362},[34,574792],{"className":574793,"style":31814},[31813],[34,574795,574797],{"className":574796},[31818,31819,31820,31821],[34,574798,574800,574803],{"className":574799},[31685,31821],[34,574801,223],{"className":574802},[31685,31821],[34,574804,138286],{"className":574805},[31685,31821],[34,574807,31829],{"className":574808},[31828],[34,574810,574812],{"className":574811},[31801],[34,574813,574815],{"className":574814,"style":546384},[31805],[34,574816],{},[34,574818],{"className":574819,"style":32123},[31845],[34,574821,574823,574826],{"className":574822},[31685],[34,574824,33244],{"className":574825},[31685,31686],[34,574827,574829],{"className":574828},[31792],[34,574830,574832],{"className":574831},[31796],[34,574833,574835],{"className":574834},[31801],[34,574836,574838],{"className":574837,"style":156828},[31805],[34,574839,574840,574843],{"style":37777},[34,574841],{"className":574842,"style":31814},[31813],[34,574844,574846],{"className":574845},[31818,31819,31820,31821],[34,574847,574849,574852,574855],{"className":574848},[31685,31821],[34,574850,32226],{"className":574851},[31685,31821],[34,574853,32936],{"className":574854},[31685,31686,31821],[34,574856,67764],{"className":574857,"style":32062},[31685,31686,31821],[34,574859,46900],{"className":574860},[31685,31686],[34,574862,32936],{"className":574863},[31685,31686],[34,574865],{"className":574866,"style":31846},[31845],[34,574868,202],{"className":574869},[31850],[34,574871],{"className":574872,"style":31846},[31845],[34,574874,574876,574880,575077,575080,575083],{"className":574875},[31676],[34,574877],{"className":574878,"style":574879},[31680],"height:2.011em;vertical-align:-0.6997em;",[34,574881,574883,575007],{"className":574882},[34006],[34,574884,574886,574892,574895,574963,575001],{"className":574885},[34006],[34,574887,574889],{"className":574888,"style":75811},[31782,75810],[34,574890,15972],{"className":574891},[71281,141292],[34,574893,32226],{"className":574894},[31685],[34,574896,574898,574901,574960],{"className":574897},[31685],[34,574899],{"className":574900},[31782,41488],[34,574902,574904],{"className":574903},[41453],[34,574905,574907,574952],{"className":574906},[31796,31797],[34,574908,574910,574949],{"className":574909},[31801],[34,574911,574913,574927,574935],{"className":574912,"style":46746},[31805],[34,574914,574915,574918],{"style":41504},[34,574916],{"className":574917,"style":41508},[31813],[34,574919,574921],{"className":574920},[31818,31819,31820,31821],[34,574922,574924],{"className":574923},[31685,31821],[34,574925,67764],{"className":574926,"style":32062},[31685,31686,31821],[34,574928,574929,574932],{"style":41520},[34,574930],{"className":574931,"style":41508},[31813],[34,574933],{"className":574934,"style":41528},[41527],[34,574936,574937,574940],{"style":46771},[34,574938],{"className":574939,"style":41508},[31813],[34,574941,574943],{"className":574942},[31818,31819,31820,31821],[34,574944,574946],{"className":574945},[31685,31821],[34,574947,640],{"className":574948},[31685,31821],[34,574950,31829],{"className":574951},[31828],[34,574953,574955],{"className":574954},[31801],[34,574956,574958],{"className":574957,"style":220191},[31805],[34,574959],{},[34,574961],{"className":574962},[31690,41488],[34,574964,574966,574969],{"className":574965},[31685],[34,574967,33244],{"className":574968},[31685,31686],[34,574970,574972],{"className":574971},[31792],[34,574973,574975],{"className":574974},[31796],[34,574976,574978],{"className":574977},[31801],[34,574979,574981],{"className":574980,"style":156828},[31805],[34,574982,574983,574986],{"style":37777},[34,574984],{"className":574985,"style":31814},[31813],[34,574987,574989],{"className":574988},[31818,31819,31820,31821],[34,574990,574992,574995,574998],{"className":574991},[31685,31821],[34,574993,32226],{"className":574994},[31685,31821],[34,574996,32936],{"className":574997},[31685,31686,31821],[34,574999,67764],{"className":575000,"style":32062},[31685,31686,31821],[34,575002,575004],{"className":575003,"style":75811},[31690,75810],[34,575005,32712],{"className":575006},[71281,141292],[34,575008,575010],{"className":575009},[31792],[34,575011,575013,575068],{"className":575012},[31796,31797],[34,575014,575016,575065],{"className":575015},[31801],[34,575017,575020,575041],{"className":575018,"style":575019},[31805],"height:1.3112em;",[34,575021,575023,575026],{"style":575022},"top:-2.0003em;margin-right:0.05em;",[34,575024],{"className":575025,"style":31814},[31813],[34,575027,575029],{"className":575028},[31818,31819,31820,31821],[34,575030,575032,575035,575038],{"className":575031},[31685,31821],[34,575033,32936],{"className":575034},[31685,31686,31821],[34,575036,202],{"className":575037},[31850,31821],[34,575039,459],{"className":575040},[31685,31821],[34,575042,575044,575047],{"style":575043},"top:-3.6029em;margin-right:0.05em;",[34,575045],{"className":575046,"style":31814},[31813],[34,575048,575050],{"className":575049},[31818,31819,31820,31821],[34,575051,575053,575056,575059,575062],{"className":575052},[31685,31821],[34,575054,32936],{"className":575055},[31685,31686,31821],[34,575057,202],{"className":575058},[31850,31821],[34,575060,223],{"className":575061},[31685,31821],[34,575063,138286],{"className":575064},[31685,31821],[34,575066,31829],{"className":575067},[31828],[34,575069,575071],{"className":575070},[31801],[34,575072,575075],{"className":575073,"style":575074},[31805],"height:0.6997em;",[34,575076],{},[34,575078],{"className":575079,"style":31846},[31845],[34,575081,202],{"className":575082},[31850],[34,575084],{"className":575085,"style":31846},[31845],[34,575087,575089,575092,575095,575098,575101],{"className":575088},[31676],[34,575090],{"className":575091,"style":35942},[31680],[34,575093,459],{"className":575094},[31685],[34,575096],{"className":575097,"style":32604},[31845],[34,575099,32226],{"className":575100},[32024],[34,575102],{"className":575103,"style":32604},[31845],[34,575105,575107,575110,575113,575116,575184,575187,575190,575193],{"className":575106},[31676],[34,575108],{"className":575109,"style":561374},[31680],[34,575111,456],{"className":575112},[31782],[34,575114,32226],{"className":575115},[31685],[34,575117,575119,575122,575181],{"className":575118},[31685],[34,575120],{"className":575121},[31782,41488],[34,575123,575125],{"className":575124},[41453],[34,575126,575128,575173],{"className":575127},[31796,31797],[34,575129,575131,575170],{"className":575130},[31801],[34,575132,575134,575148,575156],{"className":575133,"style":46746},[31805],[34,575135,575136,575139],{"style":41504},[34,575137],{"className":575138,"style":41508},[31813],[34,575140,575142],{"className":575141},[31818,31819,31820,31821],[34,575143,575145],{"className":575144},[31685,31821],[34,575146,67764],{"className":575147,"style":32062},[31685,31686,31821],[34,575149,575150,575153],{"style":41520},[34,575151],{"className":575152,"style":41508},[31813],[34,575154],{"className":575155,"style":41528},[41527],[34,575157,575158,575161],{"style":46771},[34,575159],{"className":575160,"style":41508},[31813],[34,575162,575164],{"className":575163},[31818,31819,31820,31821],[34,575165,575167],{"className":575166},[31685,31821],[34,575168,640],{"className":575169},[31685,31821],[34,575171,31829],{"className":575172},[31828],[34,575174,575176],{"className":575175},[31801],[34,575177,575179],{"className":575178,"style":220191},[31805],[34,575180],{},[34,575182],{"className":575183},[31690,41488],[34,575185,2557],{"className":575186},[31690],[34,575188],{"className":575189,"style":31846},[31845],[34,575191,202],{"className":575192},[31850],[34,575194],{"className":575195,"style":31846},[31845],[34,575197,575199,575202],{"className":575198},[31676],[34,575200],{"className":575201,"style":561374},[31680],[34,575203,575205,575208,575267],{"className":575204},[31685],[34,575206],{"className":575207},[31782,41488],[34,575209,575211],{"className":575210},[41453],[34,575212,575214,575259],{"className":575213},[31796,31797],[34,575215,575217,575256],{"className":575216},[31801],[34,575218,575220,575234,575242],{"className":575219,"style":46746},[31805],[34,575221,575222,575225],{"style":41504},[34,575223],{"className":575224,"style":41508},[31813],[34,575226,575228],{"className":575227},[31818,31819,31820,31821],[34,575229,575231],{"className":575230},[31685,31821],[34,575232,67764],{"className":575233,"style":32062},[31685,31686,31821],[34,575235,575236,575239],{"style":41520},[34,575237],{"className":575238,"style":41508},[31813],[34,575240],{"className":575241,"style":41528},[41527],[34,575243,575244,575247],{"style":46771},[34,575245],{"className":575246,"style":41508},[31813],[34,575248,575250],{"className":575249},[31818,31819,31820,31821],[34,575251,575253],{"className":575252},[31685,31821],[34,575254,640],{"className":575255},[31685,31821],[34,575257,31829],{"className":575258},[31828],[34,575260,575262],{"className":575261},[31801],[34,575263,575265],{"className":575264,"style":220191},[31805],[34,575266],{},[34,575268],{"className":575269},[31690,41488],"\n(注意这里要求 ",[34,575272,575274,575292],{"className":575273},[31642],[34,575275,575277],{"className":575276},[31646],[31648,575278,575279],{"xmlns":31650},[31652,575280,575281,575289],{},[31655,575282,575283,575285,575287],{},[31658,575284,67764],{},[31662,575286,29622],{},[31749,575288,459],{},[31665,575290,575291],{"encoding":31667},"y>0",[34,575293,575295,575313],{"className":575294,"ariaHidden":758},[31672],[34,575296,575298,575301,575304,575307,575310],{"className":575297},[31676],[34,575299],{"className":575300,"style":86652},[31680],[34,575302,67764],{"className":575303,"style":32062},[31685,31686],[34,575305],{"className":575306,"style":31846},[31845],[34,575308,29622],{"className":575309},[31850],[34,575311],{"className":575312,"style":31846},[31845],[34,575314,575316,575319],{"className":575315},[31676],[34,575317],{"className":575318,"style":32617},[31680],[34,575320,459],{"className":575321},[31685],"，题目给定 ",[34,575324,575326,575344],{"className":575325},[31642],[34,575327,575329],{"className":575328},[31646],[31648,575330,575331],{"xmlns":31650},[31652,575332,575333,575341],{},[31655,575334,575335,575337,575339],{},[31658,575336,111],{},[31662,575338,29622],{},[31749,575340,459],{},[31665,575342,575343],{"encoding":31667},"a>0",[34,575345,575347,575365],{"className":575346,"ariaHidden":758},[31672],[34,575348,575350,575353,575356,575359,575362],{"className":575349},[31676],[34,575351],{"className":575352,"style":34812},[31680],[34,575354,111],{"className":575355},[31685,31686],[34,575357],{"className":575358,"style":31846},[31845],[34,575360,29622],{"className":575361},[31850],[34,575363],{"className":575364,"style":31846},[31845],[34,575366,575368,575371],{"className":575367},[31676],[34,575369],{"className":575370,"style":32617},[31680],[34,575372,459],{"className":575373},[31685],"，满足条件)。",[108,575376,575377,575380],{},[2627,575378,575379],{},"计算外层积分：",[34,575381,575383,575461],{"className":575382},[31642],[34,575384,575386],{"className":575385},[31646],[31648,575387,575388],{"xmlns":31650},[31652,575389,575390,575458],{},[31655,575391,575392,575394,575396,575404,575410,575412,575414,575416,575418,575420,575422,575430,575432,575434,575436,575438,575440,575442,575444,575446,575448,575450,575452],{},[31658,575393,106063],{},[31662,575395,202],{},[38846,575397,575398,575400,575402],{},[31662,575399,546295],{},[31658,575401,111],{},[31658,575403,1697],{},[41453,575405,575406,575408],{},[31749,575407,640],{},[31658,575409,67764],{},[31658,575411,46900],{},[31658,575413,67764],{},[31662,575415,202],{},[31658,575417,563286],{},[31662,575419,32682],{},[31658,575421,67764],{},[38846,575423,575424,575426,575428],{},[31662,575425,162086],{"fence":702,"stretchy":758,"minsize":550116,"maxsize":550116},[31658,575427,111],{},[31658,575429,1697],{},[31662,575431,202],{},[31658,575433,563286],{},[31662,575435,32682],{},[31658,575437,1697],{},[31662,575439,32226],{},[31658,575441,563286],{},[31662,575443,32682],{},[31658,575445,111],{},[31662,575447,202],{},[31658,575449,563286],{},[31662,575451,32682],{},[41453,575453,575454,575456],{},[31658,575455,1697],{},[31658,575457,111],{},[31665,575459,575460],{"encoding":31667},"I = \\int_{a}^{b} \\frac{1}{y} dy = \\ln y \\Big|_{a}^{b} = \\ln b - \\ln a = \\ln \\frac{b}{a}",[34,575462,575464,575482,575632,575749,575773,575797],{"className":575463,"ariaHidden":758},[31672],[34,575465,575467,575470,575473,575476,575479],{"className":575466},[31676],[34,575468],{"className":575469,"style":33218},[31680],[34,575471,106063],{"className":575472,"style":67482},[31685,31686],[34,575474],{"className":575475,"style":31846},[31845],[34,575477,202],{"className":575478},[31850],[34,575480],{"className":575481,"style":31846},[31845],[34,575483,575485,575489,575546,575549,575617,575620,575623,575626,575629],{"className":575484},[31676],[34,575486],{"className":575487,"style":575488},[31680],"height:1.5251em;vertical-align:-0.4811em;",[34,575490,575492,575495],{"className":575491},[32767],[34,575493,546295],{"className":575494,"style":546331},[32767,131249,156560],[34,575496,575498],{"className":575497},[31792],[34,575499,575501,575538],{"className":575500},[31796,31797],[34,575502,575504,575535],{"className":575503},[31801],[34,575505,575507,575521],{"className":575506,"style":546344},[31805],[34,575508,575509,575512],{"style":546347},[34,575510],{"className":575511,"style":31814},[31813],[34,575513,575515],{"className":575514},[31818,31819,31820,31821],[34,575516,575518],{"className":575517},[31685,31821],[34,575519,111],{"className":575520},[31685,31686,31821],[34,575522,575523,575526],{"style":546362},[34,575524],{"className":575525,"style":31814},[31813],[34,575527,575529],{"className":575528},[31818,31819,31820,31821],[34,575530,575532],{"className":575531},[31685,31821],[34,575533,1697],{"className":575534},[31685,31686,31821],[34,575536,31829],{"className":575537},[31828],[34,575539,575541],{"className":575540},[31801],[34,575542,575544],{"className":575543,"style":546384},[31805],[34,575545],{},[34,575547],{"className":575548,"style":32123},[31845],[34,575550,575552,575555,575614],{"className":575551},[31685],[34,575553],{"className":575554},[31782,41488],[34,575556,575558],{"className":575557},[41453],[34,575559,575561,575606],{"className":575560},[31796,31797],[34,575562,575564,575603],{"className":575563},[31801],[34,575565,575567,575581,575589],{"className":575566,"style":46746},[31805],[34,575568,575569,575572],{"style":41504},[34,575570],{"className":575571,"style":41508},[31813],[34,575573,575575],{"className":575574},[31818,31819,31820,31821],[34,575576,575578],{"className":575577},[31685,31821],[34,575579,67764],{"className":575580,"style":32062},[31685,31686,31821],[34,575582,575583,575586],{"style":41520},[34,575584],{"className":575585,"style":41508},[31813],[34,575587],{"className":575588,"style":41528},[41527],[34,575590,575591,575594],{"style":46771},[34,575592],{"className":575593,"style":41508},[31813],[34,575595,575597],{"className":575596},[31818,31819,31820,31821],[34,575598,575600],{"className":575599},[31685,31821],[34,575601,640],{"className":575602},[31685,31821],[34,575604,31829],{"className":575605},[31828],[34,575607,575609],{"className":575608},[31801],[34,575610,575612],{"className":575611,"style":220191},[31805],[34,575613],{},[34,575615],{"className":575616},[31690,41488],[34,575618,46900],{"className":575619},[31685,31686],[34,575621,67764],{"className":575622,"style":32062},[31685,31686],[34,575624],{"className":575625,"style":31846},[31845],[34,575627,202],{"className":575628},[31850],[34,575630],{"className":575631,"style":31846},[31845],[34,575633,575635,575639,575642,575645,575648,575740,575743,575746],{"className":575634},[31676],[34,575636],{"className":575637,"style":575638},[31680],"height:2.0887em;vertical-align:-0.6997em;",[34,575640,563286],{"className":575641},[32767],[34,575643],{"className":575644,"style":32123},[31845],[34,575646,67764],{"className":575647,"style":32062},[31685,31686],[34,575649,575651,575688],{"className":575650},[31685],[34,575652,575654],{"className":575653},[31685],[34,575655,575657],{"className":575656},[71281,71282],[34,575658,575660,575680],{"className":575659},[31796,31797],[34,575661,575663,575677],{"className":575662},[31801],[34,575664,575666],{"className":575665,"style":550104},[31805],[34,575667,575668,575671],{"style":71329},[34,575669],{"className":575670,"style":550110},[31813],[34,575672,575673],{"style":550113},[41588,575674,575675],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,575676],{"d":550120},[34,575678,31829],{"className":575679},[31828],[34,575681,575683],{"className":575682},[31801],[34,575684,575686],{"className":575685,"style":550130},[31805],[34,575687],{},[34,575689,575691],{"className":575690},[31792],[34,575692,575694,575732],{"className":575693},[31796,31797],[34,575695,575697,575729],{"className":575696},[31801],[34,575698,575701,575715],{"className":575699,"style":575700},[31805],"height:1.389em;",[34,575702,575703,575706],{"style":575022},[34,575704],{"className":575705,"style":31814},[31813],[34,575707,575709],{"className":575708},[31818,31819,31820,31821],[34,575710,575712],{"className":575711},[31685,31821],[34,575713,111],{"className":575714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T13",[15,575884,575885,362914,575887,314945,575931,575988,575989,576135,576136],{},[2627,575886,214361],{},[34,575888,575890,575910],{"className":575889},[31642],[34,575891,575893],{"className":575892},[31646],[31648,575894,575895],{"xmlns":31650},[31652,575896,575897,575907],{},[31655,575898,575899,575901,575903,575905],{},[31658,575900,31739],{},[31662,575902,456],{"stretchy":702},[31658,575904,32936],{},[31662,575906,2557],{"stretchy":702},[31665,575908,575909],{"encoding":31667},"f(x)",[34,575911,575913],{"className":575912,"ariaHidden":758},[31672],[34,575914,575916,575919,575922,575925,575928],{"className":575915},[31676],[34,575917],{"className":575918,"style":31774},[31680],[34,575920,31739],{"className":575921,"style":31778},[31685,31686],[34,575923,456],{"className":575924},[31782],[34,575926,32936],{"className":575927},[31685,31686],[34,575929,2557],{"className":575930},[31690],[34,575932,575934,575958],{"className":575933},[31642],[34,575935,575937],{"className":575936},[31646],[31648,575938,575939],{"xmlns":31650},[31652,575940,575941,575955],{},[31655,575942,575943,575945,575947,575949,575951,575953],{},[31662,575944,15972],{"stretchy":702},[31749,575946,459],{},[31662,575948,31960],{"separator":758},[31662,575950,223],{},[31658,575952,138286],{"mathvariant":38079},[31662,575954,2557],{"stretchy":702},[31665,575956,575957],{"encoding":31667},"[0, +\\infty)",[34,575959,575961],{"className":575960,"ariaHidden":758},[31672],[34,575962,575964,575967,575970,575973,575976,575979,575982,575985],{"className":575963},[31676],[34,575965],{"className":575966,"style":31774},[31680],[34,575968,15972],{"className":575969},[31782],[34,575971,459],{"className":575972},[31685],[34,575974,31960],{"className":575975},[32119],[34,575977],{"className":575978,"style":32123},[31845],[34,575980,223],{"className":575981},[31685],[34,575983,138286],{"className":575984},[31685],[34,575986,2557],{"className":575987},[31690]," 上连续, 且 ",[34,575990,575992,576034],{"className":575991},[31642],[34,575993,575995],{"className":575994},[31646],[31648,575996,575997],{"xmlns":31650},[31652,575998,575999,576031],{},[31655,576000,576001,576019,576021,576023,576025,576027,576029],{},[71115,576002,576003,576009],{},[31655,576004,576005,576007],{},[31658,576006,546745],{},[31662,576008,32682],{},[31655,576010,576011,576013,576015,576017],{},[31658,576012,32936],{},[31662,576014,33239],{},[31662,576016,223],{},[31658,576018,138286],{"mathvariant":38079},[31658,576020,31739],{},[31662,576022,456],{"stretchy":702},[31658,576024,32936],{},[31662,576026,2557],{"stretchy":702},[31662,576028,202],{},[31749,576030,459],{},[31665,576032,576033],{"encoding":31667},"\\lim\\limits_{x \\to +\\infty}f(x) = 0",[34,576035,576037,576126],{"className":576036,"ariaHidden":758},[31672],[34,576038,576040,576044,576102,576105,576108,576111,576114,576117,576120,576123],{"className":576039},[31676],[34,576041],{"className":576042,"style":576043},[31680],"height:1.5083em;vertical-align:-0.7583em;",[34,576045,576047],{"className":576046},[32767,71420],[34,576048,576050,576093],{"className":576049},[31796,31797],[34,576051,576053,576090],{"className":576052},[31801],[34,576054,576056,576080],{"className":576055,"style":31681},[31805],[34,576057,576059,576062],{"style":576058},"top:-2.4em;margin-left:0em;",[34,576060],{"className":576061,"style":41508},[31813],[34,576063,576065],{"className":576064},[31818,31819,31820,31821],[34,576066,576068,576071,576074,576077],{"className":576067},[31685,31821],[34,576069,32936],{"className":576070},[31685,31686,31821],[34,576072,33239],{"className":576073},[31850,31821],[34,576075,223],{"className":576076},[31685,31821],[34,576078,138286],{"className":576079},[31685,31821],[34,576081,576082,576085],{"style":41565},[34,576083],{"className":576084,"style":41508},[31813],[34,576086,576087],{},[34,576088,546745],{"className":576089},[32767],[34,576091,31829],{"className":576092},[31828],[34,576094,576096],{"className":576095},[31801],[34,576097,576100],{"className":576098,"style":576099},[31805],"height:0.7583em;",[34,576101],{},[34,576103],{"className":576104,"style":32123},[31845],[34,576106,31739],{"className":576107,"style":31778},[31685,31686],[34,576109,456],{"className":576110},[31782],[34,576112,32936],{"className":576113},[31685,31686],[34,576115,2557],{"className":576116},[31690],[34,576118],{"className":576119,"style":31846},[31845],[34,576121,202],{"className":576122},[31850],[34,576124],{"className":576125,"style":31846},[31845],[34,576127,576129,576132],{"className":576128},[31676],[34,576130],{"className":576131,"style":32617},[31680],[34,576133,459],{"className":576134},[31685],", 证明\n",[34,576137,576139,576233],{"className":576138},[31642],[34,576140,576142],{"className":576141},[31646],[31648,576143,576144],{"xmlns":31650},[31652,576145,576146,576230],{},[31655,576147,576148,576160,576188,576190,576192,576194,576196,576198,576200,576202,576204,576206,576212,576214,576216,576218,576220,576222,576224,576226,576228],{},[38846,576149,576150,576152,576154],{},[31662,576151,546295],{},[31749,576153,459],{},[31655,576155,576156,576158],{},[31662,576157,223],{},[31658,576159,138286],{"mathvariant":38079},[41453,576161,576162,576186],{},[31655,576163,576164,576166,576168,576170,576172,576174,576176,576178,576180,576182,576184],{},[31658,576165,31739],{},[31662,576167,456],{"stretchy":702},[31658,576169,111],{},[31658,576171,32936],{},[31662,576173,2557],{"stretchy":702},[31662,576175,32226],{},[31658,576177,31739],{},[31662,576179,456],{"stretchy":702},[31658,576181,1697],{},[31658,576183,32936],{},[31662,576185,2557],{"stretchy":702},[31658,576187,32936],{},[31658,576189,46900],{},[31658,576191,32936],{},[31662,576193,202],{},[31658,576195,31739],{},[31662,576197,456],{"stretchy":702},[31749,576199,459],{},[31662,576201,2557],{"stretchy":702},[31658,576203,563286],{},[31662,576205,32682],{},[41453,576207,576208,576210],{},[31658,576209,1697],{},[31658,576211,111],{},[31845,576213],{"width":71082},[31662,576215,456],{"stretchy":702},[31658,576217,111],{},[31662,576219,31960],{"separator":758},[31658,576221,1697],{},[31662,576223,29622],{},[31749,576225,459],{},[31662,576227,2557],{"stretchy":702},[31658,576229,2547],{"mathvariant":38079},[31665,576231,576232],{"encoding":31667},"\\int_{0}^{+\\infty} \\frac{f(ax) - f(bx)}{x}dx = f(0)\\ln \\frac{b}{a} \\quad (a,b > 0).",[34,576234,576236,576419,576541],{"className":576235,"ariaHidden":758},[31672],[34,576237,576239,576243,576303,576306,576404,576407,576410,576413,576416],{"className":576238},[31676],[34,576240],{"className":576241,"style":576242},[31680],"height:1.3658em;vertical-align:-0.3558em;",[34,576244,576246,576249],{"className":576245},[32767],[34,576247,546295],{"className":576248,"style":546331},[32767,131249,156560],[34,576250,576252],{"className":576251},[31792],[34,576253,576255,576295],{"className":576254},[31796,31797],[34,576256,576258,576292],{"className":576257},[31801],[34,576259,576261,576275],{"className":576260,"style":546550},[31805],[34,576262,576263,576266],{"style":546347},[34,576264],{"className":576265,"style":31814},[31813],[34,576267,576269],{"className":576268},[31818,31819,31820,31821],[34,576270,576272],{"className":576271},[31685,31821],[34,576273,459],{"className":576274},[31685,31821],[34,576276,576277,576280],{"style":546362},[34,576278],{"className":576279,"style":31814},[31813],[34,576281,576283],{"className":576282},[31818,31819,31820,31821],[34,576284,576286,576289],{"className":576285},[31685,31821],[34,576287,223],{"className":576288},[31685,31821],[34,576290,138286],{"className":576291},[31685,31821],[34,576293,31829],{"className":576294},[31828],[34,576296,576298],{"className":576297},[31801],[34,576299,576301],{"className":576300,"style":546384},[31805],[34,576302],{},[34,576304],{"className":576305,"style":32123},[31845],[34,576307,576309,576312,576401],{"className":576308},[31685],[34,576310],{"className":576311},[31782,41488],[34,576313,576315],{"className":576314},[41453],[34,576316,576318,576393],{"className":576317},[31796,31797],[34,576319,576321,576390],{"className":576320},[31801],[34,576322,576324,576338,576346],{"className":576323,"style":70504},[31805],[34,576325,576326,576329],{"style":41504},[34,576327],{"className":576328,"style":41508},[31813],[34,576330,576332],{"className":576331},[31818,31819,31820,31821],[34,576333,576335],{"className":576334},[31685,31821],[34,576336,32936],{"className":576337},[31685,31686,31821],[34,576339,576340,576343],{"style":41520},[34,576341],{"className":576342,"style":41508},[31813],[34,576344],{"className":576345,"style":41528},[41527],[34,576347,576348,576351],{"style":70529},[34,576349],{"className":576350,"style":41508},[31813],[34,576352,576354],{"className":576353},[31818,31819,31820,31821],[34,576355,576357,576360,576363,576366,576369,576372,576375,576378,576381,576384,576387],{"className":576356},[31685,31821],[34,576358,31739],{"className":576359,"style":31778},[31685,31686,31821],[34,576361,456],{"className":576362},[31782,31821],[34,576364,111],{"className":576365},[31685,31686,31821],[34,576367,32936],{"className":576368},[31685,31686,31821],[34,576370,2557],{"className":576371},[31690,31821],[34,576373,32226],{"className":576374},[32024,31821],[34,576376,31739],{"className":576377,"style":31778},[31685,31686,31821],[34,576379,456],{"className":576380},[31782,31821],[34,576382,1697],{"className":576383},[31685,31686,31821],[34,576385,32936],{"className":576386},[31685,31686,31821],[34,576388,2557],{"className":576389},[31690,31821],[34,576391,31829],{"className":576392},[31828],[34,576394,576396],{"className":576395},[31801],[34,576397,576399],{"className":576398,"style":41619},[31805],[34,576400],{},[34,576402],{"className":576403},[31690,41488],[34,576405,46900],{"className":576406},[31685,31686],[34,576408,32936],{"className":576409},[31685,31686],[34,576411],{"className":576412,"style":31846},[31845],[34,576414,202],{"className":576415},[31850],[34,576417],{"className":576418,"style":31846},[31845],[34,576420,576422,576425,576428,576431,576434,576437,576440,576443,576446,576514,576517,576520,576523,576526,576529,576532,576535,576538],{"className":576421},[31676],[34,576423],{"className":576424,"style":232492},[31680],[34,576426,31739],{"className":576427,"style":31778},[31685,31686],[34,576429,456],{"className":576430},[31782],[34,576432,459],{"className":576433},[31685],[34,576435,2557],{"className":576436},[31690],[34,576438],{"className":576439,"style":32123},[31845],[34,576441,563286],{"className":576442},[32767],[34,576444],{"className":576445,"style":32123},[31845],[34,576447,576449,576452,576511],{"className":576448},[31685],[34,576450],{"className":576451},[31782,41488],[34,576453,576455],{"className":576454},[41453],[34,576456,576458,576503],{"className":576457},[31796,31797],[34,576459,576461,576500],{"className":576460},[31801],[34,576462,576464,576478,576486],{"className":576463,"style":232511},[31805],[34,576465,576466,576469],{"style":41504},[34,576467],{"className":576468,"style":41508},[31813],[34,576470,576472],{"className":576471},[31818,31819,31820,31821],[34,576473,576475],{"className":576474},[31685,31821],[34,576476,111],{"className":576477},[31685,31686,31821],[34,576479,576480,576483],{"style":41520},[34,576481],{"className":576482,"style":41508},[31813],[34,576484],{"className":576485,"style":41528},[41527],[34,576487,576488,576491],{"style":46771},[34,576489],{"className":576490,"style":41508},[31813],[34,576492,576494],{"className":576493},[31818,31819,31820,31821],[34,576495,576497],{"className":576496},[31685,31821],[34,576498,1697],{"className":576499},[31685,31686,31821],[34,576501,31829],{"className":576502},[31828],[34,576504,576506],{"className":576505},[31801],[34,576507,576509],{"className":576508,"style":41619},[31805],[34,576510],{},[34,576512],{"className":576513},[31690,41488],[34,576515],{"className":576516,"style":128438},[31845],[34,576518,456],{"className":576519},[31782],[34,576521,111],{"className":576522},[31685,31686],[34,576524,31960],{"className":576525},[32119],[34,576527],{"className":576528,"style":32123},[31845],[34,576530,1697],{"className":576531},[31685,31686],[34,576533],{"className":576534,"style":31846},[31845],[34,576536,29622],{"className":576537},[31850],[34,576539],{"className":576540,"style":31846},[31845],[34,576542,576544,576547,576550,576553],{"className":576543},[31676],[34,576545],{"className":576546,"style":31774},[31680],[34,576548,459],{"className":576549},[31685],[34,576551,2557],{"className":576552},[31690],[34,576554,2547],{"className":576555},[31685],[15,576557,576558,576560,576561,576564],{},[2627,576559,91513],{},"\n这是 ",[2627,576562,576563],{},"Frullani 积分"," 的标准证明。不用怕，全是换元法。",[15,576566,576567],{},[2627,576568,549629],{},[2230,576570,576571,577432,578815,580651,582543],{},[108,576572,576573,576576,576577,576921],{},[2627,576574,576575],{},"切分积分区间（防止无穷大处理不清）：","\n写成极限形式：\n",[34,576578,576580,576663],{"className":576579},[31642],[34,576581,576583],{"className":576582},[31646],[31648,576584,576585],{"xmlns":31650},[31652,576586,576587,576660],{},[31655,576588,576589,576591,576593,576620,576628,576656,576658],{},[31658,576590,106063],{},[31662,576592,202],{},[31743,576594,576595,576601],{},[31655,576596,576597,576599],{},[31658,576598,546745],{},[31662,576600,32682],{},[31655,576602,576603,576606,576608,576610,576612,576614,576616,576618],{},[31658,576604,576605],{},"ϵ",[31662,576607,33239],{},[31749,576609,459],{},[31662,576611,31960],{"separator":758},[31658,576613,54151],{},[31662,576615,33239],{},[31662,576617,223],{},[31658,576619,138286],{"mathvariant":38079},[38846,576621,576622,576624,576626],{},[31662,576623,546295],{},[31658,576625,576605],{},[31658,576627,54151],{},[41453,576629,576630,576654],{},[31655,576631,576632,576634,576636,576638,576640,576642,576644,576646,576648,576650,576652],{},[31658,576633,31739],{},[31662,576635,456],{"stretchy":702},[31658,576637,111],{},[31658,576639,32936],{},[31662,576641,2557],{"stretchy":702},[31662,576643,32226],{},[31658,576645,31739],{},[31662,576647,456],{"stretchy":702},[31658,576649,1697],{},[31658,576651,32936],{},[31662,576653,2557],{"stretchy":702},[31658,576655,32936],{},[31658,576657,46900],{},[31658,576659,32936],{},[31665,576661,576662],{"encoding":31667},"I = \\lim_{\\epsilon \\to 0, A \\to +\\infty} \\int_{\\epsilon}^{A} \\frac{f(ax) - f(bx)}{x} dx",[34,576664,576666,576684],{"className":576665,"ariaHidden":758},[31672],[34,576667,576669,576672,576675,576678,576681],{"className":576668},[31676],[34,576670],{"className":576671,"style":33218},[31680],[34,576673,106063],{"className":576674,"style":67482},[31685,31686],[34,576676],{"className":576677,"style":31846},[31845],[34,576679,202],{"className":576680},[31850],[34,576682],{"className":576683,"style":31846},[31845],[34,576685,576687,576690,576754,576757,576814,576817,576915,576918],{"className":576686},[31676],[34,576688],{"className":576689,"style":566818},[31680],[34,576691,576693,576696],{"className":576692},[32767],[34,576694,546745],{"className":576695},[32767],[34,576697,576699],{"className":576698},[31792],[34,576700,576702,576746],{"className":576701},[31796,31797],[34,576703,576705,576743],{"className":576704},[31801],[34,576706,576708],{"className":576707,"style":116558},[31805],[34,576709,576710,576713],{"style":56235},[34,576711],{"className":576712,"style":31814},[31813],[34,576714,576716],{"className":576715},[31818,31819,31820,31821],[34,576717,576719,576722,576725,576728,576731,576734,576737,576740],{"className":576718},[31685,31821],[34,576720,576605],{"className":576721},[31685,31686,31821],[34,576723,33239],{"className":576724},[31850,31821],[34,576726,459],{"className":576727},[31685,31821],[34,576729,31960],{"className":576730},[32119,31821],[34,576732,54151],{"className":576733},[31685,31686,31821],[34,576735,33239],{"className":576736},[31850,31821],[34,576738,223],{"className":576739},[31685,31821],[34,576741,138286],{"className":576742},[31685,31821],[34,576744,31829],{"className":576745},[31828],[34,576747,576749],{"className":576748},[31801],[34,576750,576752],{"className":576751,"style":32163},[31805],[34,576753],{},[34,576755],{"className":576756,"style":32123},[31845],[34,576758,576760,576763],{"className":576759},[32767],[34,576761,546295],{"className":576762,"style":546331},[32767,131249,156560],[34,576764,576766],{"className":576765},[31792],[34,576767,576769,576806],{"className":576768},[31796,31797],[34,576770,576772,576803],{"className":576771},[31801],[34,576773,576775,576789],{"className":576774,"style":552657},[31805],[34,576776,576777,576780],{"style":546347},[34,576778],{"className":576779,"style":31814},[31813],[34,576781,576783],{"className":576782},[31818,31819,31820,31821],[34,576784,576786],{"className":576785},[31685,31821],[34,576787,576605],{"className":576788},[31685,31686,31821],[34,576790,576791,576794],{"style":546362},[34,576792],{"className":576793,"style":31814},[31813],[34,576795,576797],{"className":576796},[31818,31819,31820,31821],[34,576798,576800],{"className":576799},[31685,31821],[34,576801,54151],{"className":576802},[31685,31686,31821],[34,576804,31829],{"className":576805},[31828],[34,576807,576809],{"className":576808},[31801],[34,576810,576812],{"className":576811,"style":546384},[31805],[34,576813],{},[34,576815],{"className":576816,"style":32123},[31845],[34,576818,576820,576823,576912],{"className":576819},[31685],[34,576821],{"className":576822},[31782,41488],[34,576824,576826],{"className":576825},[41453],[34,576827,576829,576904],{"className":576828},[31796,31797],[34,576830,576832,576901],{"className":576831},[31801],[34,576833,576835,576849,576857],{"className":576834,"style":70504},[31805],[34,576836,576837,576840],{"style":41504},[34,576838],{"className":576839,"style":41508},[31813],[34,576841,576843],{"className":576842},[31818,31819,31820,31821],[34,576844,576846],{"className":576845},[31685,31821],[34,576847,32936],{"className":576848},[31685,31686,31821],[34,576850,576851,576854],{"style":41520},[34,576852],{"className":576853,"style":41508},[31813],[34,576855],{"className":576856,"style":41528},[41527],[34,576858,576859,576862],{"style":70529},[34,576860],{"className":576861,"style":41508},[31813],[34,576863,576865],{"className":576864},[31818,31819,31820,31821],[34,576866,576868,576871,576874,576877,576880,576883,576886,576889,576892,576895,576898],{"className":576867},[31685,31821],[34,576869,31739],{"className":576870,"style":31778},[31685,31686,31821],[34,576872,456],{"className":576873},[31782,31821],[34,576875,111],{"className":576876},[31685,31686,31821],[34,576878,32936],{"className":576879},[31685,31686,31821],[34,576881,2557],{"className":576882},[31690,31821],[34,576884,32226],{"className":576885},[32024,31821],[34,576887,31739],{"className":576888,"style":31778},[31685,31686,31821],[34,576890,456],{"className":576891},[31782,31821],[34,576893,1697],{"className":576894},[31685,31686,31821],[34,576896,32936],{"className":576897},[31685,31686,31821],[34,576899,2557],{"className":576900},[31690,31821],[34,576902,31829],{"className":576903},[31828],[34,576905,576907],{"className":576906},[31801],[34,576908,576910],{"className":576909,"style":41619},[31805],[34,576911],{},[34,576913],{"className":576914},[31690,41488],[34,576916,46900],{"className":576917},[31685,31686],[34,576919,32936],{"className":576920},[31685,31686],[34,576922,576924,577028],{"className":576923},[31642],[34,576925,576927],{"className":576926},[31646],[31648,576928,576929],{"xmlns":31650},[31652,576930,576931,577025],{},[31655,576932,576933,576935,576961],{},[31662,576934,202],{},[31743,576936,576937,576943],{},[31655,576938,576939,576941],{},[31658,576940,546745],{},[31662,576942,32682],{},[31655,576944,576945,576947,576949,576951,576953,576955,576957,576959],{},[31658,576946,576605],{},[31662,576948,33239],{},[31749,576950,459],{},[31662,576952,31960],{"separator":758},[31658,576954,54151],{},[31662,576956,33239],{},[31662,576958,223],{},[31658,576960,138286],{"mathvariant":38079},[31655,576962,576963,576965,576973,576989,576991,576993,576995,577003,577019,577021,577023],{},[31662,576964,456],{"fence":758},[38846,576966,576967,576969,576971],{},[31662,576968,546295],{},[31658,576970,576605],{},[31658,576972,54151],{},[41453,576974,576975,576987],{},[31655,576976,576977,576979,576981,576983,576985],{},[31658,576978,31739],{},[31662,576980,456],{"stretchy":702},[31658,576982,111],{},[31658,576984,32936],{},[31662,576986,2557],{"stretchy":702},[31658,576988,32936],{},[31658,576990,46900],{},[31658,576992,32936],{},[31662,576994,32226],{},[38846,576996,576997,576999,577001],{},[31662,576998,546295],{},[31658,577000,576605],{},[31658,577002,54151],{},[41453,577004,577005,577017],{},[31655,577006,577007,577009,577011,577013,577015],{},[31658,577008,31739],{},[31662,577010,456],{"stretchy":702},[31658,577012,1697],{},[31658,577014,32936],{},[31662,577016,2557],{"stretchy":702},[31658,577018,32936],{},[31658,577020,46900],{},[31658,577022,32936],{},[31662,577024,2557],{"fence":758},[31665,577026,577027],{"encoding":31667},"= \\lim_{\\epsilon \\to 0, A \\to +\\infty} \\left( \\int_{\\epsilon}^{A} \\frac{f(ax)}{x} dx - \\int_{\\epsilon}^{A} \\frac{f(bx)}{x} dx \\right)",[34,577029,577031,577043],{"className":577030,"ariaHidden":758},[31672],[34,577032,577034,577037,577040],{"className":577033},[31676],[34,577035],{"className":577036,"style":73089},[31680],[34,577038,202],{"className":577039},[31850],[34,577041],{"className":577042,"style":31846},[31845],[34,577044,577046,577049,577113,577116],{"className":577045},[31676],[34,577047],{"className":577048,"style":141282},[31680],[34,577050,577052,577055],{"className":577051},[32767],[34,577053,546745],{"className":577054},[32767],[34,577056,577058],{"className":577057},[31792],[34,577059,577061,577105],{"className":577060},[31796,31797],[34,577062,577064,577102],{"className":577063},[31801],[34,577065,577067],{"className":577066,"style":116558},[31805],[34,577068,577069,577072],{"style":56235},[34,577070],{"className":577071,"style":31814},[31813],[34,577073,577075],{"className":577074},[31818,31819,31820,31821],[34,577076,577078,577081,577084,577087,577090,577093,577096,577099],{"className":577077},[31685,31821],[34,577079,576605],{"className":577080},[31685,31686,31821],[34,577082,33239],{"className":577083},[31850,31821],[34,577085,459],{"className":577086},[31685,31821],[34,577088,31960],{"className":577089},[32119,31821],[34,577091,54151],{"className":577092},[31685,31686,31821],[34,577094,33239],{"className":577095},[31850,31821],[34,577097,223],{"className":577098},[31685,31821],[34,577100,138286],{"className":577101},[31685,31821],[34,577103,31829],{"className":577104},[31828],[34,577106,577108],{"className":577107},[31801],[34,577109,577111],{"className":577110,"style":32163},[31805],[34,577112],{},[34,577114],{"className":577115,"style":32123},[31845],[34,577117,577119,577125,577182,577185,577265,577268,577271,577274,577277,577280,577337,577340,577420,577423,577426],{"className":577118},[34006],[34,577120,577122],{"className":577121,"style":75811},[31782,75810],[34,577123,456],{"className":577124},[71281,141292],[34,577126,577128,577131],{"className":577127},[32767],[34,577129,546295],{"className":577130,"style":546331},[32767,131249,156560],[34,577132,577134],{"className":577133},[31792],[34,577135,577137,577174],{"className":577136},[31796,31797],[34,577138,577140,577171],{"className":577139},[31801],[34,577141,577143,577157],{"className":577142,"style":552657},[31805],[34,577144,577145,577148],{"style":546347},[34,577146],{"className":577147,"style":31814},[31813],[34,577149,577151],{"className":577150},[31818,31819,31820,31821],[34,577152,577154],{"className":577153},[31685,31821],[34,577155,576605],{"className":577156},[31685,31686,31821],[34,577158,577159,577162],{"style":546362},[34,577160],{"className":577161,"style":31814},[31813],[34,577163,577165],{"className":577164},[31818,31819,31820,31821],[34,577166,577168],{"className":577167},[31685,31821],[34,577169,54151],{"className":577170},[31685,31686,31821],[34,577172,31829],{"className":577173},[31828],[34,577175,577177],{"className":577176},[31801],[34,577178,577180],{"className":577179,"style":546384},[31805],[34,577181],{},[34,577183],{"className":577184,"style":32123},[31845],[34,577186,577188,577191,577262],{"className":577187},[31685],[34,577189],{"className":577190},[31782,41488],[34,577192,577194],{"className":577193},[41453],[34,577195,577197,577254],{"className":577196},[31796,31797],[34,577198,577200,577251],{"className":577199},[31801],[34,577201,577203,577217,577225],{"className":577202,"style":70504},[31805],[34,577204,577205,577208],{"style":41504},[34,577206],{"className":577207,"style":41508},[31813],[34,577209,577211],{"className":577210},[31818,31819,31820,31821],[34,577212,577214],{"className":577213},[31685,31821],[34,577215,32936],{"className":577216},[31685,31686,31821],[34,577218,577219,577222],{"style":41520},[34,577220],{"className":577221,"style":41508},[31813],[34,577223],{"className":577224,"style":41528},[41527],[34,577226,577227,577230],{"style":70529},[34,577228],{"className":577229,"style":41508},[31813],[34,577231,577233],{"className":577232},[31818,31819,31820,31821],[34,577234,577236,577239,577242,577245,577248],{"className":577235},[31685,31821],[34,577237,31739],{"className":577238,"style":31778},[31685,31686,31821],[34,577240,456],{"className":577241},[31782,31821],[34,577243,111],{"className":577244},[31685,31686,31821],[34,577246,32936],{"className":577247},[31685,31686,31821],[34,577249,2557],{"className":577250},[31690,31821],[34,577252,31829],{"className":577253},[31828],[34,577255,577257],{"className":577256},[31801],[34,577258,577260],{"className":577259,"style":41619},[31805],[34,577261],{},[34,577263],{"className":577264},[31690,41488],[34,577266,46900],{"className":577267},[31685,31686],[34,577269,32936],{"className":577270},[31685,31686],[34,577272],{"className":577273,"style":32604},[31845],[34,577275,32226],{"className":577276},[32024],[34,577278],{"className":577279,"style":32604},[31845],[34,577281,577283,577286],{"className":577282},[32767],[34,577284,546295],{"className":577285,"style":546331},[32767,131249,156560],[34,577287,577289],{"className":577288},[31792],[34,577290,577292,577329],{"className":577291},[31796,31797],[34,577293,577295,577326],{"className":577294},[31801],[34,577296,577298,577312],{"className":577297,"style":552657},[31805],[34,577299,577300,577303],{"style":546347},[34,577301],{"className":577302,"style":31814},[31813],[34,577304,577306],{"className":577305},[31818,31819,31820,31821],[34,577307,577309],{"className":577308},[31685,31821],[34,577310,576605],{"className":577311},[31685,31686,31821],[34,577313,577314,577317],{"style":546362},[34,577315],{"className":577316,"style":31814},[31813],[34,577318,577320],{"className":577319},[31818,31819,31820,31821],[34,577321,577323],{"className":577322},[31685,31821],[34,577324,54151],{"className":577325},[31685,31686,31821],[34,577327,31829],{"className":577328},[31828],[34,577330,577332],{"className":577331},[31801],[34,577333,577335],{"className":577334,"style":546384},[31805],[34,577336],{},[34,577338],{"className":577339,"style":32123},[31845],[34,577341,577343,577346,577417],{"className":577342},[31685],[34,577344],{"className":577345},[31782,41488],[34,577347,577349],{"className":577348},[41453],[34,577350,577352,577409],{"className":577351},[31796,31797],[34,577353,577355,577406],{"className":577354},[31801],[34,577356,577358,577372,577380],{"className":577357,"style":70504},[31805],[34,577359,577360,577363],{"style":41504},[34,577361],{"className":577362,"style":41508},[31813],[34,577364,577366],{"className":577365},[31818,31819,31820,31821],[34,577367,577369],{"className":577368},[31685,31821],[34,577370,32936],{"className":577371},[31685,31686,31821],[34,577373,577374,577377],{"style":41520},[34,577375],{"className":577376,"style":41508},[31813],[34,577378],{"className":577379,"style":41528},[41527],[34,577381,577382,577385],{"style":70529},[34,577383],{"className":577384,"style":41508},[31813],[34,577386,577388],{"className":577387},[31818,31819,31820,31821],[34,577389,577391,577394,577397,577400,577403],{"className":577390},[31685,31821],[34,577392,31739],{"className":577393,"style":31778},[31685,31686,31821],[34,577395,456],{"className":577396},[31782,31821],[34,577398,1697],{"className":577399},[31685,31686,31821],[34,577401,32936],{"className":577402},[31685,31686,31821],[34,577404,2557],{"className":577405},[31690,31821],[34,577407,31829],{"className":577408},[31828],[34,577410,577412],{"className":577411},[31801],[34,577413,577415],{"className":577414,"style":41619},[31805],[34,577416],{},[34,577418],{"className":577419},[31690,41488],[34,577421,46900],{"className":577422},[31685,31686],[34,577424,32936],{"className":577425},[31685,31686],[34,577427,577429],{"className":577428,"style":75811},[31690,75810],[34,577430,2557],{"className":577431},[71281,141292],[108,577433,577434,577436,577437,577612,577613,53834,577647,577681,577682,577612,577857,53834,577891,39984,577925,578607,578608,347],{},[2627,577435,554210],{},"\n在第一个积分令 ",[34,577438,577440,577492],{"className":577439},[31642],[34,577441,577443],{"className":577442},[31646],[31648,577444,577445],{"xmlns":31650},[31652,577446,577447,577489],{},[31655,577448,577449,577451,577453,577455,577457,577459,577461,577463,577465,577467,577469,577471,577473,577475,577477,577479,577481,577483,577485,577487],{},[31658,577450,94628],{},[31662,577452,202],{},[31658,577454,111],{},[31658,577456,32936],{},[39361,577458,262622],{},[31662,577460,262625],{},[39361,577462,262622],{},[31658,577464,46900],{},[31658,577466,32936],{},[31662,577468,202],{},[31658,577470,46900],{},[31658,577472,94628],{},[31658,577474,38080],{"mathvariant":38079},[31658,577476,111],{},[31662,577478,31960],{"separator":758},[31658,577480,32936],{},[31662,577482,202],{},[31658,577484,94628],{},[31658,577486,38080],{"mathvariant":38079},[31658,577488,111],{},[31665,577490,577491],{"encoding":31667},"u = ax \\implies dx = du\u002Fa, x = u\u002Fa",[34,577493,577495,577513,577540,577561,577597],{"className":577494,"ariaHidden":758},[31672],[34,577496,577498,577501,577504,577507,577510],{"className":577497},[31676],[34,577499],{"className":577500,"style":32527},[31680],[34,577502,94628],{"className":577503},[31685,31686],[34,577505],{"className":577506,"style":31846},[31845],[34,577508,202],{"className":577509},[31850],[34,577511],{"className":577512,"style":31846},[31845],[34,577514,577516,577519,577522,577525,577528,577531,577534,577537],{"className":577515},[31676],[34,577517],{"className":577518,"style":318264},[31680],[34,577520,111],{"className":577521},[31685,31686],[34,577523,32936],{"className":577524},[31685,31686],[34,577526],{"className":577527,"style":31846},[31845],[34,577529],{"className":577530,"style":31846},[31845],[34,577532,262625],{"className":577533},[31850],[34,577535],{"className":577536,"style":31846},[31845],[34,577538],{"className":577539,"style":31846},[31845],[34,577541,577543,577546,577549,577552,577555,577558],{"className":577542},[31676],[34,577544],{"className":577545,"style":31681},[31680],[34,577547,46900],{"className":577548},[31685,31686],[34,577550,32936],{"className":577551},[31685,31686],[34,577553],{"className":577554,"style":31846},[31845],[34,577556,202],{"className":577557},[31850],[34,577559],{"className":577560,"style":31846},[31845],[34,577562,577564,577567,577570,577573,577576,577579,577582,577585,577588,577591,577594],{"className":577563},[31676],[34,577565],{"className":577566,"style":31774},[31680],[34,577568,46900],{"className":577569},[31685,31686],[34,577571,94628],{"className":577572},[31685,31686],[34,577574,38080],{"className":577575},[31685],[34,577577,111],{"className":577578},[31685,31686],[34,577580,31960],{"className":577581},[32119],[34,577583],{"className":577584,"style":32123},[31845],[34,577586,32936],{"className":577587},[31685,31686],[34,577589],{"className":577590,"style":31846},[31845],[34,577592,202],{"className":577593},[31850],[34,577595],{"className":577596,"style":31846},[31845],[34,577598,577600,577603,577606,577609],{"className":577599},[31676],[34,577601],{"className":577602,"style":31774},[31680],[34,577604,94628],{"className":577605},[31685,31686],[34,577607,38080],{"className":577608},[31685],[34,577610,111],{"className":577611},[31685,31686],"。积分限变为 ",[34,577614,577616,577632],{"className":577615},[31642],[34,577617,577619],{"className":577618},[31646],[31648,577620,577621],{"xmlns":31650},[31652,577622,577623,577629],{},[31655,577624,577625,577627],{},[31658,577626,111],{},[31658,577628,576605],{},[31665,577630,577631],{"encoding":31667},"a\\epsilon",[34,577633,577635],{"className":577634,"ariaHidden":758},[31672],[34,577636,577638,577641,577644],{"className":577637},[31676],[34,577639],{"className":577640,"style":32527},[31680],[34,577642,111],{"className":577643},[31685,31686],[34,577645,576605],{"className":577646},[31685,31686],[34,577648,577650,577666],{"className":577649},[31642],[34,577651,577653],{"className":577652},[31646],[31648,577654,577655],{"xmlns":31650},[31652,577656,577657,577663],{},[31655,577658,577659,577661],{},[31658,577660,111],{},[31658,577662,54151],{},[31665,577664,577665],{"encoding":31667},"aA",[34,577667,577669],{"className":577668,"ariaHidden":758},[31672],[34,577670,577672,577675,577678],{"className":577671},[31676],[34,577673],{"className":577674,"style":33218},[31680],[34,577676,111],{"className":577677},[31685,31686],[34,577679,54151],{"className":577680},[31685,31686],"。\n在第二个积分令 ",[34,577683,577685,577737],{"className":577684},[31642],[34,577686,577688],{"className":577687},[31646],[31648,577689,577690],{"xmlns":31650},[31652,577691,577692,577734],{},[31655,577693,577694,577696,577698,577700,577702,577704,577706,577708,577710,577712,577714,577716,577718,577720,577722,577724,577726,577728,577730,577732],{},[31658,577695,94628],{},[31662,577697,202],{},[31658,577699,1697],{},[31658,577701,32936],{},[39361,577703,262622],{},[31662,577705,262625],{},[39361,577707,262622],{},[31658,577709,46900],{},[31658,577711,32936],{},[31662,577713,202],{},[31658,577715,46900],{},[31658,577717,94628],{},[31658,577719,38080],{"mathvariant":38079},[31658,577721,1697],{},[31662,577723,31960],{"separator":758},[31658,577725,32936],{},[31662,577727,202],{},[31658,577729,94628],{},[31658,577731,38080],{"mathvariant":38079},[31658,577733,1697],{},[31665,577735,577736],{"encoding":31667},"u = bx \\implies dx = du\u002Fb, x = u\u002Fb",[34,577738,577740,577758,577785,577806,577842],{"className":577739,"ariaHidden":758},[31672],[34,577741,577743,577746,577749,577752,577755],{"className":577742},[31676],[34,577744],{"className":577745,"style":32527},[31680],[34,577747,94628],{"className":577748},[31685,31686],[34,577750],{"className":577751,"style":31846},[31845],[34,577753,202],{"className":577754},[31850],[34,577756],{"className":577757,"style":31846},[31845],[34,577759,577761,577764,577767,577770,577773,577776,577779,577782],{"className":577760},[31676],[34,577762],{"className":577763,"style":474142},[31680],[34,577765,1697],{"className":577766},[31685,31686],[34,577768,32936],{"className":577769},[31685,31686],[34,577771],{"className":577772,"style":31846},[31845],[34,577774],{"className":577775,"style":31846},[31845],[34,577777,262625],{"className":577778},[31850],[34,577780],{"className":577781,"style":31846},[31845],[34,577783],{"className":577784,"style":31846},[31845],[34,577786,577788,577791,577794,577797,577800,577803],{"className":577787},[31676],[34,577789],{"className":577790,"style":31681},[31680],[34,577792,46900],{"className":577793},[31685,31686],[34,577795,32936],{"className":577796},[31685,31686],[34,577798],{"className":577799,"style":31846},[31845],[34,577801,202],{"className":577802},[31850],[34,577804],{"className":577805,"style":31846},[31845],[34,577807,577809,577812,577815,577818,577821,577824,577827,577830,577833,577836,577839],{"className":577808},[31676],[34,577810],{"className":577811,"style":31774},[31680],[34,577813,46900],{"className":577814},[31685,31686],[34,577816,94628],{"className":577817},[31685,31686],[34,577819,38080],{"className":577820},[31685],[34,577822,1697],{"className":577823},[31685,31686],[34,577825,31960],{"className":577826},[32119],[34,577828],{"className":577829,"style":32123},[31845],[34,577831,32936],{"className":577832},[31685,31686],[34,577834],{"className":577835,"style":31846},[31845],[34,577837,202],{"className":577838},[31850],[34,577840],{"className":577841,"style":31846},[31845],[34,577843,577845,577848,577851,577854],{"className":577844},[31676],[34,577846],{"className":577847,"style":31774},[31680],[34,577849,94628],{"className":577850},[31685,31686],[34,577852,38080],{"className":577853},[31685],[34,577855,1697],{"className":577856},[31685,31686],[34,577858,577860,577876],{"className":577859},[31642],[34,577861,577863],{"className":577862},[31646],[31648,577864,577865],{"xmlns":31650},[31652,577866,577867,577873],{},[31655,577868,577869,577871],{},[31658,577870,1697],{},[31658,577872,576605],{},[31665,577874,577875],{"encoding":31667},"b\\epsilon",[34,577877,577879],{"className":577878,"ariaHidden":758},[31672],[34,577880,577882,577885,577888],{"className":577881},[31676],[34,577883],{"className":577884,"style":31681},[31680],[34,577886,1697],{"className":577887},[31685,31686],[34,577889,576605],{"className":577890},[31685,31686],[34,577892,577894,577910],{"className":577893},[31642],[34,577895,577897],{"className":577896},[31646],[31648,577898,577899],{"xmlns":31650},[31652,577900,577901,577907],{},[31655,577902,577903,577905],{},[31658,577904,1697],{},[31658,577906,54151],{},[31665,577908,577909],{"encoding":31667},"bA",[34,577911,577913],{"className":577912,"ariaHidden":758},[31672],[34,577914,577916,577919,577922],{"className":577915},[31676],[34,577917],{"className":577918,"style":31681},[31680],[34,577920,1697],{"className":577921},[31685,31686],[34,577923,54151],{"className":577924},[31685,31686],[34,577926,577928,578052],{"className":577927},[31642],[34,577929,577931],{"className":577930},[31646],[31648,577932,577933],{"xmlns":31650},[31652,577934,577935,578049],{},[31655,577936,577937,577945,577961,577963,577965,577967,577983,578003,578013,578015,578031,578045,578047],{},[38846,577938,577939,577941,577943],{},[31662,577940,546295],{},[31658,577942,576605],{},[31658,577944,54151],{},[41453,577946,577947,577959],{},[31655,577948,577949,577951,577953,577955,577957],{},[31658,577950,31739],{},[31662,577952,456],{"stretchy":702},[31658,577954,111],{},[31658,577956,32936],{},[31662,577958,2557],{"stretchy":702},[31658,577960,32936],{},[31658,577962,46900],{},[31658,577964,32936],{},[31662,577966,202],{},[38846,577968,577969,577971,577977],{},[31662,577970,546295],{},[31655,577972,577973,577975],{},[31658,577974,111],{},[31658,577976,576605],{},[31655,577978,577979,577981],{},[31658,577980,111],{},[31658,577982,54151],{},[41453,577984,577985,577995],{},[31655,577986,577987,577989,577991,577993],{},[31658,577988,31739],{},[31662,577990,456],{"stretchy":702},[31658,577992,94628],{},[31662,577994,2557],{"stretchy":702},[31655,577996,577997,577999,578001],{},[31658,577998,94628],{},[31658,578000,38080],{"mathvariant":38079},[31658,578002,111],{},[41453,578004,578005,578011],{},[31655,578006,578007,578009],{},[31658,578008,46900],{},[31658,578010,94628],{},[31658,578012,111],{},[31662,578014,202],{},[38846,578016,578017,578019,578025],{},[31662,578018,546295],{},[31655,578020,578021,578023],{},[31658,578022,111],{},[31658,578024,576605],{},[31655,578026,578027,578029],{},[31658,578028,111],{},[31658,578030,54151],{},[41453,578032,578033,578043],{},[31655,578034,578035,578037,578039,578041],{},[31658,578036,31739],{},[31662,578038,456],{"stretchy":702},[31658,578040,94628],{},[31662,578042,2557],{"stretchy":702},[31658,578044,94628],{},[31658,578046,46900],{},[31658,578048,94628],{},[31665,578050,578051],{"encoding":31667},"\\int_{\\epsilon}^{A} \\frac{f(ax)}{x} dx = \\int_{a\\epsilon}^{aA} \\frac{f(u)}{u\u002Fa} \\frac{du}{a} = \\int_{a\\epsilon}^{aA} \\frac{f(u)}{u} du",[34,578053,578055,578216,578452],{"className":578054,"ariaHidden":758},[31672],[34,578056,578058,578061,578118,578121,578201,578204,578207,578210,578213],{"className":578057},[31676],[34,578059],{"className":578060,"style":566818},[31680],[34,578062,578064,578067],{"className":578063},[32767],[34,578065,546295],{"className":578066,"style":546331},[32767,131249,156560],[34,578068,578070],{"className":578069},[31792],[34,578071,578073,578110],{"className":578072},[31796,31797],[34,578074,578076,578107],{"className":578075},[31801],[34,578077,578079,578093],{"className":578078,"style":552657},[31805],[34,578080,578081,578084],{"style":546347},[34,578082],{"className":578083,"style":31814},[31813],[34,578085,578087],{"className":578086},[31818,31819,31820,31821],[34,578088,578090],{"className":578089},[31685,31821],[34,578091,576605],{"className":578092},[31685,31686,31821],[34,578094,578095,578098],{"style":546362},[34,578096],{"className":578097,"style":31814},[31813],[34,578099,578101],{"className":578100},[31818,31819,31820,31821],[34,578102,578104],{"className":578103},[31685,31821],[34,578105,54151],{"className":578106},[31685,31686,31821],[34,578108,31829],{"className":578109},[31828],[34,578111,578113],{"className":578112},[31801],[34,578114,578116],{"className":578115,"style":546384},[31805],[34,578117],{},[34,578119],{"className":578120,"style":32123},[31845],[34,578122,578124,578127,578198],{"className":578123},[31685],[34,578125],{"className":578126},[31782,41488],[34,578128,578130],{"className":578129},[41453],[34,578131,578133,578190],{"className":578132},[31796,31797],[34,578134,578136,578187],{"className":578135},[31801],[34,578137,578139,578153,578161],{"className":578138,"style":70504},[31805],[34,578140,578141,578144],{"style":41504},[34,578142],{"className":578143,"style":41508},[31813],[34,578145,578147],{"className":578146},[31818,31819,31820,31821],[34,578148,578150],{"className":578149},[31685,31821],[34,578151,32936],{"className":578152},[31685,31686,31821],[34,578154,578155,578158],{"style":41520},[34,578156],{"className":578157,"style":41508},[31813],[34,578159],{"className":578160,"style":41528},[41527],[34,578162,578163,578166],{"style":70529},[34,578164],{"className":578165,"style":41508},[31813],[34,578167,578169],{"className":578168},[31818,31819,31820,31821],[34,578170,578172,578175,578178,578181,578184],{"className":578171},[31685,31821],[34,578173,31739],{"className":578174,"style":31778},[31685,31686,31821],[34,578176,456],{"className":578177},[31782,31821],[34,578179,111],{"className":578180},[31685,31686,31821],[34,578182,32936],{"className":578183},[31685,31686,31821],[34,578185,2557],{"className":578186},[31690,31821],[34,578188,31829],{"className":578189},[31828],[34,578191,578193],{"className":578192},[31801],[34,578194,578196],{"className":578195,"style":41619},[31805],[34,578197],{},[34,578199],{"className":578200},[31690,41488],[34,578202,46900],{"className":578203},[31685,31686],[34,578205,32936],{"className":578206},[31685,31686],[34,578208],{"className":578209,"style":31846},[31845],[34,578211,202],{"className":578212},[31850],[34,578214],{"className":578215,"style":31846},[31845],[34,578217,578219,578223,578286,578289,578372,578443,578446,578449],{"className":578218},[31676],[34,578220],{"className":578221,"style":578222},[31680],"height:1.5562em;vertical-align:-0.52em;",[34,578224,578226,578229],{"className":578225},[32767],[34,578227,546295],{"className":578228,"style":546331},[32767,131249,156560],[34,578230,578232],{"className":578231},[31792],[34,578233,578235,578278],{"className":578234},[31796,31797],[34,578236,578238,578275],{"className":578237},[31801],[34,578239,578241,578258],{"className":578240,"style":552657},[31805],[34,578242,578243,578246],{"style":546347},[34,578244],{"className":578245,"style":31814},[31813],[34,578247,578249],{"className":578248},[31818,31819,31820,31821],[34,578250,578252,578255],{"className":578251},[31685,31821],[34,578253,111],{"className":578254},[31685,31686,31821],[34,578256,576605],{"className":578257},[31685,31686,31821],[34,578259,578260,578263],{"style":546362},[34,578261],{"className":578262,"style":31814},[31813],[34,578264,578266],{"className":578265},[31818,31819,31820,31821],[34,578267,578269,578272],{"className":578268},[31685,31821],[34,578270,111],{"className":578271},[31685,31686,31821],[34,578273,54151],{"className":578274},[31685,31686,31821],[34,578276,31829],{"className":578277},[31828],[34,578279,578281],{"className":578280},[31801],[34,578282,578284],{"className":578283,"style":546384},[31805],[34,578285],{},[34,578287],{"className":578288,"style":32123},[31845],[34,578290,578292,578295,578369],{"className":578291},[31685],[34,578293],{"className":578294},[31782,41488],[34,578296,578298],{"className":578297},[41453],[34,578299,578301,578361],{"className":578300},[31796,31797],[34,578302,578304,578358],{"className":578303},[31801],[34,578305,578307,578327,578335],{"className":578306,"style":70504},[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me":578568,"style":41508},[31813],[34,578570,578572],{"className":578571},[31818,31819,31820,31821],[34,578573,578575,578578,578581,578584],{"className":578574},[31685,31821],[34,578576,31739],{"className":578577,"style":31778},[31685,31686,31821],[34,578579,456],{"className":578580},[31782,31821],[34,578582,94628],{"className":578583},[31685,31686,31821],[34,578585,2557],{"className":578586},[31690,31821],[34,578588,31829],{"className":578589},[31828],[34,578591,578593],{"className":578592},[31801],[34,578594,578596],{"className":578595,"style":41619},[31805],[34,578597],{},[34,578599],{"className":578600},[31690,41488],[34,578602,46900],{"className":578603},[31685,31686],[34,578605,94628],{"className":578606},[31685,31686],"\n同理第二个变成 ",[34,578609,578611,578657],{"className":578610},[31642],[34,578612,578614],{"className":578613},[31646],[31648,578615,578616],{"xmlns":31650},[31652,578617,578618,578654],{},[31655,578619,578620,578636,578650,578652],{},[38846,578621,578622,578624,578630],{},[31662,578623,546295],{},[31655,578625,578626,578628],{},[31658,578627,1697],{},[31658,578629,576605],{},[31655,578631,578632,578634],{},[31658,578633,1697],{},[31658,578635,54151],{},[41453,578637,578638,578648],{},[31655,578639,578640,578642,578644,578646],{},[31658,578641,31739],{},[31662,578643,456],{"stretchy":702},[31658,578645,94628],{},[31662,578647,2557],{"stretchy":702},[31658,578649,94628],{},[31658,578651,46900],{},[31658,578653,94628],{},[31665,578655,578656],{"encoding":31667},"\\int_{b\\epsilon}^{bA} \\frac{f(u)}{u} du",[34,578658,578660],{"className":578659,"ariaHidden":758},[31672],[34,578661,578663,578666,578729,578732,578809,578812],{"className":578662},[31676],[34,578664],{"className":578665,"style":546324},[31680],[34,578667,578669,578672],{"className":578668},[32767],[34,578670,546295],{"className":578671,"style":546331},[32767,131249,156560],[34,578673,578675],{"className":578674},[31792],[34,578676,578678,578721],{"className":578677},[31796,31797],[34,578679,578681,578718],{"className":578680},[31801],[34,578682,578684,578701],{"className":578683,"style":546344},[31805],[34,578685,578686,578689],{"style":546347},[34,578687],{"className":578688,"style":31814},[31813],[34,578690,578692],{"className":578691},[31818,31819,31820,31821],[34,578693,578695,578698],{"className":578694},[31685,31821],[34,578696,1697],{"className":578697},[31685,31686,31821],[34,578699,576605],{"className":578700},[31685,31686,31821],[34,578702,578703,578706],{"style":546362},[34,578704],{"className":578705,"style":31814},[31813],[34,578707,578709],{"className":578708},[31818,31819,31820,31821],[34,578710,578712,578715],{"className":578711},[31685,31821],[34,578713,1697],{"className":578714},[31685,31686,31821],[34,578716,54151],{"className":578717},[31685,31686,31821],[34,578719,31829],{"className":578720},[31828],[34,578722,578724],{"className":578723},[31801],[34,578725,578727],{"className":578726,"style":546384},[31805],[34,578728],{},[34,578730],{"className":578731,"style":32123},[31845],[34,578733,578735,578738,578806],{"className":578734},[31685],[34,578736],{"className":578737},[31782,41488],[34,578739,578741],{"className":578740},[41453],[34,578742,578744,578798],{"className":578743},[31796,31797],[34,578745,578747,578795],{"className":578746},[31801],[34,578748,578750,578764,578772],{"className":578749,"style":70504},[31805],[34,578751,578752,578755],{"style":41504},[34,578753],{"className":578754,"style":41508},[31813],[34,578756,578758],{"className":578757},[31818,31819,31820,31821],[34,578759,578761],{"className":578760},[31685,31821],[34,578762,94628],{"className":578763},[31685,31686,31821],[34,578765,578766,578769],{"style":41520},[34,578767],{"className":578768,"style":41508},[31813],[34,578770],{"className":578771,"style":41528},[41527],[34,578773,578774,578777],{"style":70529},[34,578775],{"className":578776,"style":41508},[31813],[34,578778,578780],{"className":578779},[31818,31819,31820,31821],[34,578781,578783,578786,578789,578792],{"className":578782},[31685,31821],[34,578784,31739],{"className":578785,"style":31778},[31685,31686,31821],[34,578787,456],{"className":578788},[31782,31821],[34,578790,94628],{"className":578791},[31685,31686,31821],[34,578793,2557],{"className":578794},[31690,31821],[34,578796,31829],{"className":578797},[31828],[34,578799,578801],{"className":578800},[31801],[34,578802,578804],{"className":578803,"style":41619},[31805],[34,578805],{},[34,578807],{"className":578808},[31690,41488],[34,578810,46900],{"className":578811},[31685,31686],[34,578813,94628],{"className":578814},[31685,31686],[108,578816,578817,578820,579266,579267,39984,579556,580204,580650],{},[2627,578818,578819],{},"合并积分：",[34,578821,578823,578917],{"className":578822},[31642],[34,578824,578826],{"className":578825},[31646],[31648,578827,578828],{"xmlns":31650},[31652,578829,578830,578914],{},[31655,578831,578832,578834,578836,578838,578840,578842,578858,578872,578874,578876,578878,578894,578908,578910,578912],{},[31658,578833,106063],{},[31662,578835,202],{},[31658,578837,546745],{},[31662,578839,32682],{},[31662,578841,456],{"stretchy":702},[38846,578843,578844,578846,578852],{},[31662,578845,546295],{},[31655,578847,578848,578850],{},[31658,578849,111],{},[31658,578851,576605],{},[31655,578853,578854,578856],{},[31658,578855,111],{},[31658,578857,54151],{},[41453,578859,578860,578870],{},[31655,578861,578862,578864,578866,578868],{},[31658,578863,31739],{},[31662,578865,456],{"stretchy":702},[31658,578867,94628],{},[31662,578869,2557],{"stretchy":702},[31658,578871,94628],{},[31658,578873,46900],{},[31658,578875,94628],{},[31662,578877,32226],{},[38846,578879,578880,578882,578888],{},[31662,578881,546295],{},[31655,578883,578884,578886],{},[31658,578885,1697],{},[31658,578887,576605],{},[31655,578889,578890,578892],{},[31658,578891,1697],{},[31658,578893,54151],{},[41453,578895,578896,578906],{},[31655,578897,578898,578900,578902,578904],{},[31658,578899,31739],{},[31662,578901,456],{"stretchy":702},[31658,578903,94628],{},[31662,578905,2557],{"stretchy":702},[31658,578907,94628],{},[31658,578909,46900],{},[31658,578911,94628],{},[31662,578913,2557],{"stretchy":702},[31665,578915,578916],{"encoding":31667},"I = \\lim (\\int_{a\\epsilon}^{aA} \\frac{f(u)}{u} du - \\int_{b\\epsilon}^{bA} \\frac{f(u)}{u} du)",[34,578918,578920,578938,579108],{"className":578919,"ariaHidden":758},[31672],[34,578921,578923,578926,578929,578932,578935],{"className":578922},[31676],[34,578924],{"className":578925,"style":33218},[31680],[34,578927,106063],{"className":578928,"style":67482},[31685,31686],[34,578930],{"className":578931,"style":31846},[31845],[34,578933,202],{"className":578934},[31850],[34,578936],{"className":578937,"style":31846},[31845],[34,578939,578941,578944,578947,578950,579013,579016,579093,579096,579099,579102,579105],{"className":578940},[31676],[34,578942],{"className":578943,"style":566818},[31680],[34,578945,546745],{"className":578946},[32767],[34,578948,456],{"className":578949},[31782],[34,578951,578953,578956],{"className":578952},[32767],[34,578954,546295],{"className":578955,"style":546331},[32767,131249,156560],[34,578957,578959],{"className":578958},[31792],[34,578960,578962,579005],{"className":578961},[31796,31797],[34,578963,578965,579002],{"className":578964},[31801],[34,578966,578968,578985],{"className":578967,"style":552657},[31805],[34,578969,578970,578973],{"style":546347},[34,578971],{"className":578972,"style":31814},[31813],[34,578974,578976],{"className":578975},[31818,31819,31820,31821],[34,578977,578979,578982],{"className":578978},[31685,31821],[34,578980,111],{"className":578981},[31685,31686,31821],[34,578983,576605],{"className":578984},[31685,31686,31821],[34,578986,578987,578990],{"style":546362},[34,578988],{"className":578989,"style":31814},[31813],[34,578991,578993],{"className":578992},[31818,31819,31820,31821],[34,578994,578996,578999],{"className":578995},[31685,31821],[34,578997,111],{"className":578998},[31685,31686,31821],[34,579000,54151],{"className":579001},[31685,31686,31821],[34,579003,31829],{"className":579004},[31828],[34,579006,579008],{"className":579007},[31801],[34,579009,579011],{"className":579010,"style":546384},[31805],[34,579012],{},[34,579014],{"className":579015,"style":32123},[31845],[34,579017,579019,579022,579090],{"className":579018},[31685],[34,579020],{"className":579021},[31782,41488],[34,579023,579025],{"className":579024},[41453],[34,579026,579028,579082],{"className":579027},[31796,31797],[34,579029,579031,579079],{"className":579030},[31801],[34,579032,579034,579048,579056],{"className":579033,"style":70504},[31805],[34,579035,579036,579039],{"style":41504},[34,579037],{"className":579038,"style":41508},[31813],[34,579040,579042],{"className":579041},[31818,31819,31820,31821],[34,579043,579045],{"className":579044},[31685,31821],[34,579046,94628],{"className":579047},[31685,31686,31821],[34,579049,579050,579053],{"style":41520},[34,579051],{"className":579052,"style":41508},[31813],[34,579054],{"className":579055,"style":41528},[41527],[34,579057,579058,579061],{"style":70529},[34,579059],{"className":579060,"style":41508},[31813],[34,579062,579064],{"className":579063},[31818,31819,31820,31821],[34,579065,579067,579070,579073,579076],{"className":579066},[31685,31821],[34,579068,31739],{"className":579069,"style":31778},[31685,31686,31821],[34,579071,456],{"className":579072},[31782,31821],[34,579074,94628],{"className":579075},[31685,31686,31821],[34,579077,2557],{"className":579078},[31690,31821],[34,579080,31829],{"className":579081},[31828],[34,579083,579085],{"className":579084},[31801],[34,579086,579088],{"className":579087,"style":41619},[31805],[34,579089],{},[34,579091],{"className":579092},[31690,41488],[34,579094,46900],{"className":579095},[31685,31686],[34,579097,94628],{"className":579098},[31685,31686],[34,579100],{"className":579101,"style":32604},[31845],[34,579103,32226],{"className":579104},[32024],[34,579106],{"className":579107,"style":32604},[31845],[34,579109,579111,579114,579177,579180,579257,579260,579263],{"className":579110},[31676],[34,579112],{"className":579113,"style":546324},[31680],[34,579115,579117,579120],{"className":579116},[32767],[34,579118,546295],{"className":579119,"style":546331},[32767,131249,156560],[34,579121,579123],{"className":579122},[31792],[34,579124,579126,579169],{"className":579125},[31796,31797],[34,579127,579129,579166],{"className":579128},[31801],[34,579130,579132,579149],{"className":579131,"style":546344},[31805],[34,579133,579134,579137],{"style":546347},[34,579135],{"className":579136,"style":31814},[31813],[34,579138,579140],{"className":579139},[31818,31819,31820,31821],[34,579141,579143,579146],{"className":579142},[31685,31821],[34,579144,1697],{"className":579145},[31685,31686,31821],[34,579147,576605],{"className":579148},[31685,31686,31821],[34,579150,579151,579154],{"style":546362},[34,579152],{"className":579153,"style":31814},[31813],[34,579155,579157],{"className":579156},[31818,31819,31820,31821],[34,579158,579160,579163],{"className":579159},[31685,31821],[34,579161,1697],{"className":579162},[31685,31686,31821],[34,579164,54151],{"className":579165},[31685,31686,31821],[34,579167,31829],{"className":579168},[31828],[34,579170,579172],{"className":579171},[31801],[34,579173,579175],{"className":579174,"style":546384},[31805],[34,579176],{},[34,579178],{"className":579179,"style":32123},[31845],[34,579181,579183,579186,579254],{"className":579182},[31685],[34,579184],{"className":579185},[31782,41488],[34,579187,579189],{"className":579188},[41453],[34,579190,579192,579246],{"className":579191},[31796,31797],[34,579193,579195,579243],{"className":579194},[31801],[34,579196,579198,579212,579220],{"className":579197,"style":70504},[31805],[34,579199,579200,579203],{"style":41504},[34,579201],{"className":579202,"style":41508},[31813],[34,579204,579206],{"className":579205},[31818,31819,31820,31821],[34,579207,579209],{"className":579208},[31685,31821],[34,579210,94628],{"className":579211},[31685,31686,31821],[34,579213,579214,579217],{"style":41520},[34,579215],{"className":579216,"style":41508},[31813],[34,579218],{"className":579219,"style":41528},[41527],[34,579221,579222,579225],{"style":70529},[34,579223],{"className":579224,"style":41508},[31813],[34,579226,579228],{"className":579227},[31818,31819,31820,31821],[34,579229,579231,579234,579237,579240],{"className":579230},[31685,31821],[34,579232,31739],{"className":579233,"style":31778},[31685,31686,31821],[34,579235,456],{"className":579236},[31782,31821],[34,579238,94628],{"className":579239},[31685,31686,31821],[34,579241,2557],{"className":579242},[31690,31821],[34,579244,31829],{"className":579245},[31828],[34,579247,579249],{"className":579248},[31801],[34,579250,579252],{"className":579251,"style":41619},[31805],[34,579253],{},[34,579255],{"className":579256},[31690,41488],[34,579258,46900],{"className":579259},[31685,31686],[34,579261,94628],{"className":579262},[31685,31686],[34,579264,2557],{"className":579265},[31690],"\n利用积分区间的加减：",[34,579268,579270,579334],{"className":579269},[31642],[34,579271,579273],{"className":579272},[31646],[31648,579274,579275],{"xmlns":31650},[31652,579276,579277,579331],{},[31655,579278,579279,579295,579297,579313,579315],{},[38846,579280,579281,579283,579289],{},[31662,579282,546295],{},[31655,579284,579285,579287],{},[31658,579286,111],{},[31658,579288,576605],{},[31655,579290,579291,579293],{},[31658,579292,111],{},[31658,579294,54151],{},[31662,579296,202],{},[38846,579298,579299,579301,579307],{},[31662,579300,546295],{},[31655,579302,579303,579305],{},[31658,579304,111],{},[31658,579306,576605],{},[31655,579308,579309,579311],{},[31658,579310,1697],{},[31658,579312,576605],{},[31662,579314,223],{},[38846,579316,579317,579319,579325],{},[31662,579318,546295],{},[31655,579320,579321,579323],{},[31658,579322,1697],{},[31658,579324,576605],{},[31655,579326,579327,579329],{},[31658,579328,111],{},[31658,579330,54151],{},[31665,579332,579333],{"encoding":31667},"\\int_{a\\epsilon}^{aA} = \\int_{a\\epsilon}^{b\\epsilon} + \\int_{b\\epsilon}^{aA}",[34,579335,579337,579415],{"className":579336,"ariaHidden":758},[31672],[34,579338,579340,579343,579406,579409,579412],{"className":579339},[31676],[34,579341],{"className":579342,"style":566818},[31680],[34,579344,579346,579349],{"className":579345},[32767],[34,579347,546295],{"className":579348,"style":546331},[32767,131249,156560],[34,579350,579352],{"className":579351},[31792],[34,579353,579355,579398],{"className":579354},[31796,31797],[34,579356,579358,579395],{"className":579357},[31801],[34,579359,579361,579378],{"className":579360,"style":552657},[31805],[34,579362,579363,579366],{"style":546347},[34,579364],{"className":579365,"style":31814},[31813],[34,579367,579369],{"className":579368},[31818,31819,31820,31821],[34,579370,579372,579375],{"className":579371},[31685,31821],[34,579373,111],{"className":579374},[31685,31686,31821],[34,579376,576605],{"className":579377},[31685,31686,31821],[34,579379,579380,579383],{"style":546362},[34,579381],{"className":579382,"style":31814},[31813],[34,579384,579386],{"className":579385},[31818,31819,31820,31821],[34,579387,579389,579392],{"className":579388},[31685,31821],[34,579390,111],{"className":579391},[31685,31686,31821],[34,579393,54151],{"className":579394},[31685,31686,31821],[34,579396,31829],{"className":579397},[31828],[34,579399,579401],{"className":579400},[31801],[34,579402,579404],{"className":579403,"style":546384},[31805],[34,579405],{},[34,579407],{"className":579408,"style":31846},[31845],[34,579410,202],{"className":579411},[31850],[34,579413],{"className":579414,"style":31846},[31845],[34,579416,579418,579421,579484,579487,579490,579493],{"className":579417},[31676],[34,579419],{"className":579420,"style":546324},[31680],[34,579422,579424,579427],{"className":579423},[32767],[34,579425,546295],{"className":579426,"style":546331},[32767,131249,156560],[34,579428,579430],{"className":579429},[31792],[34,579431,579433,579476],{"className":579432},[31796,31797],[34,579434,579436,579473],{"className":579435},[31801],[34,579437,579439,579456],{"className":579438,"style":546344},[31805],[34,579440,579441,579444],{"style":546347},[34,579442],{"className":579443,"style":31814},[31813],[34,579445,579447],{"className":579446},[31818,31819,31820,31821],[34,579448,579450,579453],{"className":579449},[31685,31821],[34,579451,111],{"className":579452},[31685,31686,31821],[34,579454,576605],{"className":579455},[31685,31686,31821],[34,579457,579458,579461],{"style":546362},[34,579459],{"className":579460,"style":31814},[31813],[34,579462,579464],{"className":579463},[31818,31819,31820,31821],[34,579465,579467,579470],{"className":579466},[31685,31821],[34,579468,1697],{"className":579469},[31685,31686,31821],[34,579471,576605],{"className":579472},[31685,31686,31821],[34,579474,31829],{"className":579475},[31828],[34,579477,579479],{"className":579478},[31801],[34,579480,579482],{"className":579481,"style":546384},[31805],[34,579483],{},[34,579485],{"className":579486,"style":32123},[31845],[34,579488,223],{"className":579489},[31685],[34,579491],{"className":579492,"style":32123},[31845],[34,579494,579496,579499],{"className":579495},[32767],[34,579497,546295],{"className":579498,"style":546331},[32767,131249,156560],[34,579500,579502],{"className":579501},[31792],[34,579503,579505,579548],{"className":579504},[31796,31797],[34,579506,579508,579545],{"className":579507},[31801],[34,579509,579511,579528],{"className":579510,"style":552657},[31805],[34,579512,579513,579516],{"style":546347},[34,579514],{"className":579515,"style":31814},[31813],[34,579517,579519],{"className":579518},[31818,31819,31820,31821],[34,579520,579522,579525],{"className":579521},[31685,31821],[34,579523,1697],{"className":579524},[31685,31686,31821],[34,579526,576605],{"className":579527},[31685,31686,31821],[34,579529,579530,579533],{"style":546362},[34,579531],{"className":579532,"style":31814},[31813],[34,579534,579536],{"className":579535},[31818,31819,31820,31821],[34,579537,579539,579542],{"className":579538},[31685,31821],[34,579540,111],{"className":579541},[31685,31686,31821],[34,579543,54151],{"className":579544},[31685,31686,31821],[34,579546,31829],{"className":579547},[31828],[34,579549,579551],{"className":579550},[31801],[34,579552,579554],{"className":579553,"style":546384},[31805],[34,579555],{},[34,579557,579559,579691],{"className":579558},[31642],[34,579560,579562],{"className":579561},[31646],[31648,579563,579564],{"xmlns":31650},[31652,579565,579566,579688],{},[31655,579567,579568,579570,579572,579574,579576],{},[31658,579569,106063],{},[31662,579571,202],{},[31658,579573,546745],{},[31662,579575,32682],{},[31655,579577,579578,579580,579596,579610,579612,579614,579616,579632,579646,579648,579650,579652,579668,579682,579684,579686],{},[31662,579579,456],{"fence":758},[38846,579581,579582,579584,579590],{},[31662,579583,546295],{},[31655,579585,579586,579588],{},[31658,579587,111],{},[31658,579589,576605],{},[31655,579591,579592,579594],{},[31658,579593,1697],{},[31658,579595,576605],{},[41453,579597,579598,579608],{},[31655,579599,579600,579602,579604,579606],{},[31658,579601,31739],{},[31662,579603,456],{"stretchy":702},[31658,579605,94628],{},[31662,579607,2557],{"stretchy":702},[31658,579609,94628],{},[31658,579611,46900],{},[31658,579613,94628],{},[31662,579615,223],{},[38846,579617,579618,579620,579626],{},[31662,579619,546295],{},[31655,579621,579622,579624],{},[31658,579623,1697],{},[31658,579625,576605],{},[31655,579627,579628,579630],{},[31658,579629,111],{},[31658,579631,54151],{},[41453,579633,579634,579644],{},[31655,579635,579636,579638,579640,579642],{},[31658,579637,31739],{},[31662,579639,456],{"stretchy":702},[31658,579641,94628],{},[31662,579643,2557],{"stretchy":702},[31658,579645,94628],{},[31658,579647,46900],{},[31658,579649,94628],{},[31662,579651,32226],{},[38846,579653,579654,579656,579662],{},[31662,579655,546295],{},[31655,579657,579658,579660],{},[31658,579659,1697],{},[31658,579661,576605],{},[31655,579663,579664,579666],{},[31658,579665,1697],{},[31658,579667,54151],{},[41453,579669,579670,579680],{},[31655,579671,579672,579674,579676,579678],{},[31658,579673,31739],{},[31662,579675,456],{"stretchy":702},[31658,579677,94628],{},[31662,579679,2557],{"stretchy":702},[31658,579681,94628],{},[31658,579683,46900],{},[31658,579685,94628],{},[31662,579687,2557],{"fence":758},[31665,579689,579690],{"encoding":31667},"I = \\lim \\left( \\int_{a\\epsilon}^{b\\epsilon} \\frac{f(u)}{u} du + \\int_{b\\epsilon}^{aA} \\frac{f(u)}{u} du - \\int_{b\\epsilon}^{bA} \\frac{f(u)}{u} du \\right)",[34,579692,579694,579712],{"className":579693,"ariaHidden":758},[31672],[34,579695,579697,579700,579703,579706,579709],{"className":579696},[31676],[34,579698],{"className":579699,"style":33218},[31680],[34,579701,106063],{"className":579702,"style":67482},[31685,31686],[34,579704],{"className":579705,"style":31846},[31845],[34,579707,202],{"className":579708},[31850],[34,579710],{"className":579711,"style":31846},[31845],[34,579713,579715,579718,579721,579724],{"className":579714},[31676],[34,579716],{"className":579717,"style":141282},[31680],[34,579719,546745],{"className":579720},[32767],[34,579722],{"className":579723,"style":32123},[31845],[34,579725,579727,579733,579796,579799,579876,579879,579882,579885,579888,579891,579954,579957,580034,580037,580040,580043,580046,580049,580112,580115,580192,580195,580198],{"className":579726},[34006],[34,579728,579730],{"className":579729,"style":75811},[31782,75810],[34,579731,456],{"className":579732},[71281,141292],[34,579734,579736,579739],{"className":579735},[32767],[34,579737,546295],{"className":579738,"style":546331},[32767,131249,156560],[34,579740,579742],{"className":579741},[31792],[34,579743,579745,579788],{"className":579744},[31796,31797],[34,579746,579748,579785],{"className":579747},[31801],[34,579749,579751,579768],{"className":579750,"style":546344},[31805],[34,579752,579753,579756],{"style":546347},[34,579754],{"className":579755,"style":31814},[31813],[34,579757,579759],{"className":579758},[31818,31819,31820,31821],[34,579760,579762,579765],{"className":579761},[31685,31821],[34,579763,111],{"className":579764},[31685,31686,31821],[34,579766,576605],{"className":579767},[31685,31686,31821],[34,579769,579770,579773],{"style":546362},[34,579771],{"className":579772,"style":31814},[31813],[34,579774,579776],{"className":579775},[31818,31819,31820,31821],[34,579777,579779,579782],{"className":579778},[31685,31821],[34,579780,1697],{"className":579781},[31685,31686,31821],[34,579783,576605],{"className":579784},[31685,31686,31821],[34,579786,31829],{"className":579787},[31828],[34,579789,579791],{"className":579790},[31801],[34,579792,579794],{"className":579793,"style":546384},[31805],[34,579795],{},[34,579797],{"className":579798,"style":32123},[31845],[34,579800,579802,579805,579873],{"className":579801},[31685],[34,579803],{"className":579804},[31782,41488],[34,579806,579808],{"className":579807},[41453],[34,579809,579811,579865],{"className":579810},[31796,31797],[34,579812,579814,579862],{"className":579813},[31801],[34,579815,579817,579831,579839],{"className":579816,"style":70504},[31805],[34,579818,579819,579822],{"style":41504},[34,579820],{"className":579821,"style":41508},[31813],[34,579823,579825],{"className":579824},[31818,31819,31820,31821],[34,579826,579828],{"className":579827},[31685,31821],[34,579829,94628],{"className":579830},[31685,31686,31821],[34,579832,579833,579836],{"style":41520},[34,579834],{"className":579835,"style":41508},[31813],[34,579837],{"className":579838,"style":41528},[41527],[34,579840,579841,579844],{"style":70529},[34,579842],{"className":579843,"style":41508},[31813],[34,579845,579847],{"className":579846},[31818,31819,31820,31821],[34,579848,579850,579853,579856,579859],{"className":579849},[31685,31821],[34,579851,31739],{"className":579852,"style":31778},[31685,31686,31821],[34,579854,456],{"className":579855},[31782,31821],[34,579857,94628],{"className":579858},[31685,31686,31821],[34,579860,2557],{"className":579861},[31690,31821],[34,579863,31829],{"className":579864},[31828],[34,579866,579868],{"className":579867},[31801],[34,579869,579871],{"className":579870,"style":41619},[31805],[34,579872],{},[34,579874],{"className":579875},[31690,41488],[34,579877,46900],{"className":579878},[31685,31686],[34,579880,94628],{"className":579881},[31685,31686],[34,579883],{"className":579884,"style":32604},[31845],[34,579886,223],{"className":579887},[32024],[34,579889],{"className":579890,"style":32604},[31845],[34,579892,579894,579897],{"className":579893},[32767],[34,579895,546295],{"className":579896,"style":546331},[32767,131249,156560],[34,579898,579900],{"className":579899},[31792],[34,579901,579903,579946],{"className":579902},[31796,31797],[34,579904,579906,579943],{"className":579905},[31801],[34,579907,579909,579926],{"className":579908,"style":552657},[31805],[34,579910,579911,579914],{"style":546347},[34,579912],{"className":579913,"style":31814},[31813],[34,579915,579917],{"className":579916},[31818,31819,31820,31821],[34,579918,579920,579923],{"className":579919},[31685,31821],[34,579921,1697],{"className":579922},[31685,31686,31821],[34,579924,576605],{"className":579925},[31685,31686,31821],[34,579927,579928,579931],{"style":546362},[34,579929],{"className":579930,"style":31814},[31813],[34,579932,579934],{"className":579933},[31818,31819,31820,31821],[34,579935,579937,579940],{"className":579936},[31685,31821],[34,579938,111],{"className":579939},[31685,31686,31821],[34,579941,541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\\lim \\left( \\int_{a\\epsilon}^{b\\epsilon} \\frac{f(u)}{u} du - \\int_{aA}^{bA} \\frac{f(u)}{u} du \\right)",[34,580302,580304,580316],{"className":580303,"ariaHidden":758},[31672],[34,580305,580307,580310,580313],{"className":580306},[31676],[34,580308],{"className":580309,"style":73089},[31680],[34,580311,202],{"className":580312},[31850],[34,580314],{"className":580315,"style":31846},[31845],[34,580317,580319,580322,580325,580328],{"className":580318},[31676],[34,580320],{"className":580321,"style":141282},[31680],[34,580323,546745],{"className":580324},[32767],[34,580326],{"className":580327,"style":32123},[31845],[34,580329,580331,580337,580400,580403,580480,580483,580486,580489,580492,580495,580558,580561,580638,580641,580644],{"className":580330},[34006],[34,580332,580334],{"className":580333,"style":75811},[31782,75810],[34,580335,456],{"className":580336},[71281,141292],[34,580338,580340,580343],{"className":580339},[32767],[34,580341,546295],{"className":580342,"style":546331},[32767,131249,156560],[34,580344,580346],{"className":580345},[31792],[34,580347,580349,580392],{"className":580348},[31796,31797],[34,580350,580352,580389],{"className":580351},[31801],[34,580353,580355,580372],{"className":580354,"style":546344},[31805],[34,580356,580357,580360],{"style":546347},[34,580358],{"className":580359,"style":31814},[31813],[34,580361,580363],{"className":580362},[31818,31819,31820,31821],[34,580364,580366,580369],{"className":580365},[31685,31821],[34,580367,111],{"className":580368},[31685,31686,31821],[34,580370,576605],{"className":580371},[31685,31686,31821],[34,580373,580374,580377],{"style":546362},[34,580375],{"className":580376,"style":31814},[31813],[34,580378,580380],{"className":580379},[31818,31819,31820,31821],[34,580381,580383,580386],{"className":580382},[31685,31821],[34,580384,1697],{"className":580385},[31685,31686,31821],[34,580387,576605],{"className":580388},[31685,31686,31821],[34,580390,31829],{"className":580391},[31828],[34,580393,580395],{"className":580394},[31801],[34,580396,580398],{"className":580397,"style":546384},[31805],[34,580399],{},[34,580401],{"className":580402,"style":32123},[31845],[34,580404,580406,580409,580477],{"className":580405},[31685],[34,580407],{"className":580408},[31782,41488],[34,580410,580412],{"className":580411},[41453],[34,580413,580415,580469],{"className":580414},[31796,31797],[34,580416,580418,580466],{"className":580417},[31801],[34,580419,580421,580435,580443],{"className":580420,"style":70504},[31805],[34,580422,580423,580426],{"style":41504},[34,580424],{"className":580425,"style":41508},[31813],[34,580427,580429],{"className":580428},[31818,31819,31820,31821],[34,580430,580432],{"className":580431},[31685,31821],[34,580433,94628],{"className":580434},[31685,31686,31821],[34,580436,580437,580440],{"style":41520},[34,580438],{"className":580439,"style":41508},[31813],[34,580441],{"className":580442,"style":41528},[41527],[34,580444,580445,580448],{"style":70529},[34,580446],{"className":580447,"style":41508},[31813],[34,580449,580451],{"className":580450},[31818,31819,31820,31821],[34,580452,580454,580457,580460,580463],{"className":580453},[31685,31821],[34,580455,31739],{"className":580456,"style":31778},[31685,31686,31821],[34,580458,456],{"className":580459},[31782,31821],[34,580461,94628],{"className":580462},[31685,31686,31821],[34,580464,2557],{"className":580465},[31690,31821],[34,580467,31829],{"className":580468},[31828],[34,580470,580472],{"className":580471},[31801],[34,580473,580475],{"className":580474,"style":41619},[31805],[34,580476],{},[34,580478],{"className":580479},[31690,41488],[34,580481,46900],{"className":580482},[31685,31686],[34,580484,94628],{"className":580485},[31685,31686],[34,580487],{"className":580488,"style":32604},[31845],[34,580490,32226],{"className":580491},[32024],[34,580493],{"className":580494,"style":32604},[31845],[34,580496,580498,580501],{"className":580497},[32767],[34,580499,546295],{"className":580500,"style":546331},[32767,131249,156560],[34,580502,580504],{"className":580503},[31792],[34,580505,580507,580550],{"className":580506},[31796,31797],[34,580508,580510,580547],{"className":580509},[31801],[34,580511,580513,580530],{"className":580512,"style":546344},[31805],[34,580514,580515,580518],{"style":546347},[34,580516],{"className":580517,"style":31814},[31813],[34,580519,580521],{"className":580520},[31818,31819,31820,31821],[34,580522,580524,580527],{"className":580523},[31685,31821],[34,580525,111],{"className":580526},[31685,31686,31821],[34,580528,54151],{"className":580529},[31685,31686,31821],[34,580531,580532,580535],{"style":546362},[34,580533],{"className":580534,"style":31814},[31813],[34,580536,580538],{"className":580537},[31818,31819,31820,31821],[34,580539,580541,580544],{"className":580540},[31685,31821],[34,580542,1697],{"className":580543},[31685,31686,31821],[34,580545,54151],{"className":580546},[31685,31686,31821],[34,580548,31829],{"className":580549},[31828],[34,580551,580553],{"className":580552},[31801],[34,580554,580556],{"className":580555,"style":546384},[31805],[34,580557],{},[34,580559],{"className":580560,"style":32123},[31845],[34,580562,580564,580567,580635],{"className":580563},[31685],[34,580565],{"className":580566},[31782,41488],[34,580568,580570],{"className":580569},[41453],[34,580571,580573,580627],{"className":580572},[31796,31797],[34,580574,580576,580624],{"className":580575},[31801],[34,580577,580579,580593,580601],{"className":580578,"style":70504},[31805],[34,580580,580581,580584],{"style":41504},[34,580582],{"className":580583,"style":41508},[31813],[34,580585,580587],{"className":580586},[31818,31819,31820,31821],[34,580588,580590],{"className":580589},[31685,31821],[34,580591,94628],{"className":580592},[31685,31686,31821],[34,580594,580595,580598],{"style":41520},[34,580596],{"className":580597,"style":41508},[31813],[34,580599],{"className":580600,"style":41528},[41527],[34,580602,580603,580606],{"style":70529},[34,580604],{"className":580605,"style":41508},[31813],[34,580607,580609],{"className":580608},[31818,31819,31820,31821],[34,580610,580612,580615,580618,580621],{"className":580611},[31685,31821],[34,580613,31739],{"className":580614,"style":31778},[31685,31686,31821],[34,580616,456],{"className":580617},[31782,31821],[34,580619,94628],{"className":580620},[31685,31686,31821],[34,580622,2557],{"className":580623},[31690,31821],[34,580625,31829],{"className":580626},[31828],[34,580628,580630],{"className":580629},[31801],[34,580631,580633],{"className":580632,"style":41619},[31805],[34,580634],{},[34,580636],{"className":580637},[31690,41488],[34,580639,46900],{"className":580640},[31685,31686],[34,580642,94628],{"className":580643},[31685,31686],[34,580645,580647],{"className":580646,"style":75811},[31690,75810],[34,580648,2557],{"className":580649},[71281,141292],"\n(注意这里合并中间部分后剩下的头尾)。",[108,580652,580653,580656],{},[2627,580654,580655],{},"利用中值定理求极限：",[105,580657,580658,581811],{},[108,580659,580660,580713,580714,70884,581129,39984,581213,581351,581352,558876,581496,4504,581547,13977,581598,581679,581680,347],{},[2627,580661,580662,580663,414527],{},"处理第一项（",[34,580664,580666,580683],{"className":580665},[31642],[34,580667,580669],{"className":580668},[31646],[31648,580670,580671],{"xmlns":31650},[31652,580672,580673,580681],{},[31655,580674,580675,580677,580679],{},[31658,580676,32936],{},[31662,580678,33239],{},[31749,580680,459],{},[31665,580682,568274],{"encoding":31667},[34,580684,580686,580704],{"className":580685,"ariaHidden":758},[31672],[34,580687,580689,580692,580695,580698,580701],{"className":580688},[31676],[34,580690],{"className":580691,"style":32527},[31680],[34,580693,32936],{"className":580694},[31685,31686],[34,580696],{"className":580697,"style":31846},[31845],[34,580699,33239],{"className":580700},[31850],[34,580702],{"className":580703,"style":31846},[31845],[34,580705,580707,580710],{"className":580706},[31676],[34,580708],{"className":580709,"style":32617},[31680],[34,580711,459],{"className":580712},[31685],"\n根据积分第一中值定理，",[34,580715,580717,580800],{"className":580716},[31642],[34,580718,580720],{"className":580719},[31646],[31648,580721,580722],{"xmlns":31650},[31652,580723,580724,580797],{},[31655,580725,580726,580742,580756,580758,580760,580762,580764,580766,580769,580771,580787,580793,580795],{},[38846,580727,580728,580730,580736],{},[31662,580729,546295],{},[31655,580731,580732,580734],{},[31658,580733,111],{},[31658,580735,576605],{},[31655,580737,580738,580740],{},[31658,580739,1697],{},[31658,580741,576605],{},[41453,580743,580744,580754],{},[31655,580745,580746,580748,580750,580752],{},[31658,580747,31739],{},[31662,580749,456],{"stretchy":702},[31658,580751,94628],{},[31662,580753,2557],{"stretchy":702},[31658,580755,94628],{},[31658,580757,46900],{},[31658,580759,94628],{},[31662,580761,202],{},[31658,580763,31739],{},[31662,580765,456],{"stretchy":702},[31658,580767,580768],{},"ξ",[31662,580770,2557],{"stretchy":702},[38846,580772,580773,580775,580781],{},[31662,580774,546295],{},[31655,580776,580777,580779],{},[31658,580778,111],{},[31658,580780,576605],{},[31655,580782,580783,580785],{},[31658,580784,1697],{},[31658,580786,576605],{},[41453,580788,580789,580791],{},[31749,580790,640],{},[31658,580792,94628],{},[31658,580794,46900],{},[31658,580796,94628],{},[31665,580798,580799],{"encoding":31667},"\\int_{a\\epsilon}^{b\\epsilon} \\frac{f(u)}{u} du = f(\\xi) \\int_{a\\epsilon}^{b\\epsilon} \\frac{1}{u} du",[34,580801,580803,580967],{"className":580802,"ariaHidden":758},[31672],[34,580804,580806,580809,580872,580875,580952,580955,580958,580961,580964],{"className":580805},[31676],[34,580807],{"className":580808,"style":546324},[31680],[34,580810,580812,580815],{"className":580811},[32767],[34,580813,546295],{"className":580814,"style":546331},[32767,131249,156560],[34,580816,580818],{"className":580817},[31792],[34,580819,580821,580864],{"className":580820},[31796,31797],[34,580822,580824,580861],{"className":580823},[31801],[34,580825,580827,580844],{"className":580826,"style":546344},[31805],[34,580828,580829,580832],{"style":546347},[34,580830],{"className":580831,"style":31814},[31813],[34,580833,580835],{"className":580834},[31818,31819,31820,31821],[34,580836,580838,580841],{"className":580837},[31685,31821],[34,580839,111],{"className":580840},[31685,31686,31821],[34,580842,576605],{"className":580843},[31685,31686,31821],[34,580845,580846,580849],{"style":546362},[34,580847],{"className":580848,"style":31814},[31813],[34,580850,580852],{"className":580851},[31818,31819,31820,31821],[34,580853,580855,580858],{"className":580854},[31685,31821],[34,580856,1697],{"className":580857},[31685,31686,31821],[34,580859,576605],{"className":580860},[31685,31686,31821],[34,580862,31829],{"className":580863},[31828],[34,580865,580867],{"className":580866},[31801],[34,580868,580870],{"className":580869,"style":546384},[31805],[34,580871],{},[34,580873],{"className":580874,"style":32123},[31845],[34,580876,580878,580881,580949],{"className":580877},[31685],[34,580879],{"className":580880},[31782,41488],[34,580882,580884],{"className":580883},[41453],[34,580885,580887,580941],{"className":580886},[31796,31797],[34,580888,580890,580938],{"className":580889},[31801],[34,580891,580893,580907,580915],{"className":580892,"style":70504},[31805],[34,580894,580895,580898],{"style":41504},[34,580896],{"className":580897,"style":41508},[31813],[34,580899,580901],{"className":580900},[31818,31819,31820,31821],[34,580902,580904],{"className":580903},[31685,31821],[34,580905,94628],{"className":580906},[31685,31686,31821],[34,580908,580909,580912],{"style":41520},[34,580910],{"className":580911,"style":41508},[31813],[34,580913],{"className":580914,"style":41528},[41527],[34,580916,580917,580920],{"style":70529},[34,580918],{"className":580919,"style":41508},[31813],[34,580921,580923],{"className":580922},[31818,31819,31820,31821],[34,580924,580926,580929,580932,580935],{"className":580925},[31685,31821],[34,580927,31739],{"className":580928,"style":31778},[31685,31686,31821],[34,580930,456],{"className":580931},[31782,31821],[34,580933,94628],{"className":580934},[31685,31686,31821],[34,580936,2557],{"className":580937},[31690,31821],[34,580939,31829],{"className":580940},[31828],[34,580942,580944],{"className":580943},[31801],[34,580945,580947],{"className":580946,"style":41619},[31805],[34,580948],{},[34,580950],{"className":580951},[31690,41488],[34,580953,46900],{"className":580954},[31685,31686],[34,580956,94628],{"className":580957},[31685,31686],[34,580959],{"className":580960,"style":31846},[31845],[34,580962,202],{"className":580963},[31850],[34,580965],{"className":580966,"style":31846},[31845],[34,580968,580970,580973,580976,580979,580983,580986,580989,581052,581055,581123,581126],{"className":580969},[31676],[34,580971],{"className":580972,"style":546324},[31680],[34,580974,31739],{"className":580975,"style":31778},[31685,31686],[34,580977,456],{"className":580978},[31782],[34,580980,580768],{"className":580981,"style":580982},[31685,31686],"margin-right:0.046em;",[34,580984,2557],{"className":580985},[31690],[34,580987],{"className":580988,"style":32123},[31845],[34,580990,580992,580995],{"className":580991},[32767],[34,580993,546295],{"className":580994,"style":546331},[32767,131249,156560],[34,580996,580998],{"className":580997},[31792],[34,580999,581001,581044],{"className":581000},[31796,31797],[34,581002,581004,581041],{"className":581003},[31801],[34,581005,581007,581024],{"className":581006,"style":546344},[31805],[34,581008,581009,581012],{"style":546347},[34,581010],{"className":581011,"style":31814},[31813],[34,581013,581015],{"className":581014},[31818,31819,31820,31821],[34,581016,581018,581021],{"className":581017},[31685,31821],[34,581019,111],{"className":581020},[31685,31686,31821],[34,581022,576605],{"className":581023},[31685,31686,31821],[34,581025,581026,581029],{"style":546362},[34,581027],{"className":581028,"style":31814},[31813],[34,581030,581032],{"className":581031},[31818,31819,31820,31821],[34,581033,581035,581038],{"className":581034},[31685,31821],[34,581036,1697],{"className":581037},[31685,31686,31821],[34,581039,576605],{"className":581040},[31685,31686,31821],[34,581042,31829],{"className":581043},[31828],[34,581045,581047],{"className":581046},[31801],[34,581048,581050],{"className":581049,"style":546384},[31805],[34,581051],{},[34,581053],{"className":581054,"style":32123},[31845],[34,581056,581058,581061,581120],{"className":581057},[31685],[34,581059],{"className":581060},[31782,41488],[34,581062,581064],{"className":581063},[41453],[34,581065,581067,581112],{"className":581066},[31796,31797],[34,581068,581070,581109],{"className":581069},[31801],[34,581071,581073,581087,581095],{"className":581072,"style":46746},[31805],[34,581074,581075,581078],{"style":41504},[34,581076],{"className":581077,"style":41508},[31813],[34,581079,581081],{"className":581080},[31818,31819,31820,31821],[34,581082,581084],{"className":581083},[31685,31821],[34,581085,94628],{"className":581086},[31685,31686,31821],[34,581088,581089,581092],{"style":41520},[34,581090],{"className":581091,"style":41508},[31813],[34,581093],{"className":581094,"style":41528},[41527],[34,581096,581097,581100],{"style":46771},[34,581098],{"className":581099,"style":41508},[31813],[34,581101,581103],{"className":581102},[31818,31819,31820,31821],[34,581104,581106],{"className":581105},[31685,31821],[34,581107,640],{"className":581108},[31685,31821],[34,581110,31829],{"className":581111},[31828],[34,581113,581115],{"className":581114},[31801],[34,581116,581118],{"className":581117,"style":41619},[31805],[34,581119],{},[34,581121],{"className":581122},[31690,41488],[34,581124,46900],{"className":581125},[31685,31686],[34,581127,94628],{"className":581128},[31685,31686],[34,581130,581132,581162],{"className":581131},[31642],[34,581133,581135],{"className":581134},[31646],[31648,581136,581137],{"xmlns":31650},[31652,581138,581139,581159],{},[31655,581140,581141,581143,581145,581147,581149,581151,581153,581155,581157],{},[31658,581142,580768],{},[31662,581144,112296],{},[31662,581146,15972],{"stretchy":702},[31658,581148,111],{},[31658,581150,576605],{},[31662,581152,31960],{"separator":758},[31658,581154,1697],{},[31658,581156,576605],{},[31662,581158,32712],{"stretchy":702},[31665,581160,581161],{"encoding":31667},"\\xi \\in [a\\epsilon, b\\epsilon]",[34,581163,581165,581183],{"className":581164,"ariaHidden":758},[31672],[34,581166,581168,581171,581174,581177,581180],{"className":581167},[31676],[34,581169],{"className":581170,"style":43217},[31680],[34,581172,580768],{"className":581173,"style":580982},[31685,31686],[34,581175],{"className":581176,"style":31846},[31845],[34,581178,112296],{"className":581179},[31850],[34,581181],{"className":581182,"style":31846},[31845],[34,581184,581186,581189,581192,581195,581198,581201,581204,581207,581210],{"className":581185},[31676],[34,581187],{"className":581188,"style":31774},[31680],[34,581190,15972],{"className":581191},[31782],[34,581193,111],{"className":581194},[31685,31686],[34,581196,576605],{"className":581197},[31685,31686],[34,581199,31960],{"className":581200},[32119],[34,581202],{"className":581203,"style":32123},[31845],[34,581205,1697],{"className":581206},[31685,31686],[34,581208,576605],{"className":581209},[31685,31686],[34,581211,32712],{"className":581212},[31690],[34,581214,581216,581244],{"className":581215},[31642],[34,581217,581219],{"className":581218},[31646],[31648,581220,581221],{"xmlns":31650},[31652,581222,581223,581241],{},[31655,581224,581225,581227,581233,581235,581237,581239],{},[31662,581226,546295],{},[41453,581228,581229,581231],{},[31749,581230,640],{},[31658,581232,94628],{},[31662,581234,202],{},[31658,581236,563286],{},[31662,581238,32682],{},[31658,581240,94628],{},[31665,581242,581243],{"encoding":31667},"\\int \\frac{1}{u} = \\ln u",[34,581245,581247,581336],{"className":581246,"ariaHidden":758},[31672],[34,581248,581250,581253,581256,581259,581327,581330,581333],{"className":581249},[31676],[34,581251],{"className":581252,"style":46727},[31680],[34,581254,546295],{"className":581255,"style":546331},[32767,131249,156560],[34,581257],{"className":581258,"style":32123},[31845],[34,581260,581262,581265,581324],{"className":581261},[31685],[34,581263],{"className":581264},[31782,41488],[34,581266,581268],{"className":581267},[41453],[34,581269,581271,581316],{"className":581270},[31796,31797],[34,581272,581274,581313],{"className":581273},[31801],[34,581275,581277,581291,581299],{"className":581276,"style":46746},[31805],[34,581278,581279,581282],{"style":41504},[34,581280],{"className":581281,"style":41508},[31813],[34,581283,581285],{"className":581284},[31818,31819,31820,31821],[34,581286,581288],{"className":581287},[31685,31821],[34,581289,94628],{"className":581290},[31685,31686,31821],[34,581292,581293,581296],{"style":41520},[34,581294],{"className":581295,"style":41508},[31813],[34,581297],{"className":581298,"style":41528},[41527],[34,581300,581301,581304],{"style":46771},[34,581302],{"className":581303,"style":41508},[31813],[34,581305,581307],{"className":581306},[31818,31819,31820,31821],[34,581308,581310],{"className":581309},[31685,31821],[34,581311,640],{"className":581312},[31685,31821],[34,581314,31829],{"className":581315},[31828],[34,581317,581319],{"className":581318},[31801],[34,581320,581322],{"className":581321,"style":41619},[31805],[34,581323],{},[34,581325],{"className":581326},[31690,41488],[34,581328],{"className":581329,"style":31846},[31845],[34,581331,202],{"className":581332},[31850],[34,581334],{"className":581335,"style":31846},[31845],[34,581337,581339,581342,581345,581348],{"className":581338},[31676],[34,581340],{"className":581341,"style":31681},[31680],[34,581343,563286],{"className":581344},[32767],[34,581346],{"className":581347,"style":32123},[31845],[34,581349,94628],{"className":581350},[31685,31686],"，所以积分值为 ",[34,581353,581355,581409],{"className":581354},[31642],[34,581356,581358],{"className":581357},[31646],[31648,581359,581360],{"xmlns":31650},[31652,581361,581362,581406],{},[31655,581363,581364,581366,581368,581370,581372,581374,581376,581378,581380,581382,581384,581386,581388,581390,581392,581394,581396,581398,581400,581402,581404],{},[31658,581365,563286],{},[31662,581367,32682],{},[31662,581369,456],{"stretchy":702},[31658,581371,1697],{},[31658,581373,576605],{},[31662,581375,2557],{"stretchy":702},[31662,581377,32226],{},[31658,581379,563286],{},[31662,581381,32682],{},[31662,581383,456],{"stretchy":702},[31658,581385,111],{},[31658,581387,576605],{},[31662,581389,2557],{"stretchy":702},[31662,581391,202],{},[31658,581393,563286],{},[31662,581395,32682],{},[31662,581397,456],{"stretchy":702},[31658,581399,1697],{},[31658,581401,38080],{"mathvariant":38079},[31658,581403,111],{},[31662,581405,2557],{"stretchy":702},[31665,581407,581408],{"encoding":31667},"\\ln(b\\epsilon) - \\ln(a\\epsilon) = \\ln(b\u002Fa)",[34,581410,581412,581442,581472],{"className":581411,"ariaHidden":758},[31672],[34,581413,581415,581418,581421,581424,581427,581430,581433,581436,581439],{"className":581414},[31676],[34,581416],{"className":581417,"style":31774},[31680],[34,581419,563286],{"className":581420},[32767],[34,581422,456],{"className":581423},[31782],[34,581425,1697],{"className":581426},[31685,31686],[34,581428,576605],{"className":581429},[31685,31686],[34,581431,2557],{"className":581432},[31690],[34,581434],{"className":581435,"style":32604},[31845],[34,581437,32226],{"className":581438},[32024],[34,581440],{"className":581441,"style":32604},[31845],[34,581443,581445,581448,581451,581454,581457,581460,581463,581466,581469],{"className":581444},[31676],[34,581446],{"className":581447,"style":31774},[31680],[34,581449,563286],{"className":581450},[32767],[34,581452,456],{"className":581453},[31782],[34,581455,111],{"className":581456},[31685,31686],[34,581458,576605],{"className":581459},[31685,31686],[34,581461,2557],{"className":581462},[31690],[34,581464],{"className":581465,"style":31846},[31845],[34,581467,202],{"className":581468},[31850],[34,581470],{"className":581471,"style":31846},[31845],[34,581473,581475,581478,581481,581484,581487,581490,581493],{"className":581474},[31676],[34,581476],{"className":581477,"style":31774},[31680],[34,581479,563286],{"className":581480},[32767],[34,581482,456],{"className":581483},[31782],[34,581485,1697],{"className":581486},[31685,31686],[34,581488,38080],{"className":581489},[31685],[34,581491,111],{"className":581492},[31685,31686],[34,581494,2557],{"className":581495},[31690],[34,581497,581499,581517],{"className":581498},[31642],[34,581500,581502],{"className":581501},[31646],[31648,581503,581504],{"xmlns":31650},[31652,581505,581506,581514],{},[31655,581507,581508,581510,581512],{},[31658,581509,576605],{},[31662,581511,33239],{},[31749,581513,459],{},[31665,581515,581516],{"encoding":31667},"\\epsilon \\to 0",[34,581518,581520,581538],{"className":581519,"ariaHidden":758},[31672],[34,581521,581523,581526,581529,581532,581535],{"className":581522},[31676],[34,581524],{"className":581525,"style":32527},[31680],[34,581527,576605],{"className":581528},[31685,31686],[34,581530],{"className":581531,"style":31846},[31845],[34,581533,33239],{"className":581534},[31850],[34,581536],{"className":581537,"style":31846},[31845],[34,581539,581541,581544],{"className":581540},[31676],[34,581542],{"className":581543,"style":32617},[31680],[34,581545,459],{"className":581546},[31685],[34,581548,581550,581568],{"className":581549},[31642],[34,581551,581553],{"className":581552},[31646],[31648,581554,581555],{"xmlns":31650},[31652,581556,581557,581565],{},[31655,581558,581559,581561,581563],{},[31658,581560,580768],{},[31662,581562,33239],{},[31749,581564,459],{},[31665,581566,581567],{"encoding":31667},"\\xi \\to 0",[34,581569,581571,581589],{"className":581570,"ariaHidden":758},[31672],[34,581572,581574,581577,581580,581583,581586],{"className":581573},[31676],[34,581575],{"className":581576,"style":43217},[31680],[34,581578,580768],{"className":581579,"style":580982},[31685,31686],[34,581581],{"className":581582,"style":31846},[31845],[34,581584,33239],{"className":581585},[31850],[34,581587],{"className":581588,"style":31846},[31845],[34,581590,581592,581595],{"className":581591},[31676],[34,581593],{"className":581594,"style":32617},[31680],[34,581596,459],{"className":581597},[31685],[34,581599,581601,581631],{"className":581600},[31642],[34,581602,581604],{"className":581603},[31646],[31648,581605,581606],{"xmlns":31650},[31652,581607,581608,581628],{},[31655,581609,581610,581612,581614,581616,581618,581620,581622,581624,581626],{},[31658,581611,31739],{},[31662,581613,456],{"stretchy":702},[31658,581615,580768],{},[31662,581617,2557],{"stretchy":702},[31662,581619,33239],{},[31658,581621,31739],{},[31662,581623,456],{"stretchy":702},[31749,581625,459],{},[31662,581627,2557],{"stretchy":702},[31665,581629,581630],{"encoding":31667},"f(\\xi) \\to f(0)",[34,581632,581634,581661],{"className":581633,"ariaHidden":758},[31672],[34,581635,581637,581640,581643,581646,581649,581652,581655,581658],{"className":581636},[31676],[34,581638],{"className":581639,"style":31774},[31680],[34,581641,31739],{"className":581642,"style":31778},[31685,31686],[34,581644,456],{"className":581645},[31782],[34,581647,580768],{"className":581648,"style":580982},[31685,31686],[34,581650,2557],{"className":581651},[31690],[34,581653],{"className":581654,"style":31846},[31845],[34,581656,33239],{"className":581657},[31850],[34,581659],{"className":581660,"style":31846},[31845],[34,581662,581664,581667,581670,581673,581676],{"className":581663},[31676],[34,581665],{"className":581666,"style":31774},[31680],[34,581668,31739],{"className":581669,"style":31778},[31685,31686],[34,581671,456],{"className":581672},[31782],[34,581674,459],{"className":581675},[31685],[34,581677,2557],{"className":581678},[31690],"。\n第一项极限 = ",[34,581681,581683,581713],{"className":581682},[31642],[34,581684,581686],{"className":581685},[31646],[31648,581687,581688],{"xmlns":31650},[31652,581689,581690,581710],{},[31655,581691,581692,581694,581696,581698,581700,581702,581704],{},[31658,581693,31739],{},[31662,581695,456],{"stretchy":702},[31749,581697,459],{},[31662,581699,2557],{"stretchy":702},[31658,581701,563286],{},[31662,581703,32682],{},[41453,581705,581706,581708],{},[31658,581707,1697],{},[31658,581709,111],{},[31665,581711,581712],{"encoding":31667},"f(0) \\ln \\frac{b}{a}",[34,581714,581716],{"className":581715,"ariaHidden":758},[31672],[34,581717,581719,581722,581725,581728,581731,581734,581737,581740,581743],{"className":581718},[31676],[34,581720],{"className":581721,"style":232492},[31680],[34,581723,31739],{"className":581724,"style":31778},[31685,31686],[34,581726,456],{"className":581727},[31782],[34,581729,459],{"className":581730},[31685],[34,581732,2557],{"className":581733},[31690],[34,581735],{"className":581736,"style":32123},[31845],[34,581738,563286],{"className":581739},[32767],[34,581741],{"className":581742,"style":32123},[31845],[34,581744,581746,581749,581808],{"className":581745},[31685],[34,581747],{"className":581748},[31782,41488],[34,581750,581752],{"className":581751},[41453],[34,581753,581755,581800],{"className":581754},[31796,31797],[34,581756,581758,581797],{"className":581757},[31801],[34,581759,581761,581775,581783],{"className":581760,"style":232511},[31805],[34,581762,581763,581766],{"style":41504},[34,581764],{"className":581765,"style":41508},[31813],[34,581767,581769],{"className":581768},[31818,31819,31820,31821],[34,581770,581772],{"className":581771},[31685,31821],[34,581773,111],{"className":581774},[31685,31686,31821],[34,581776,581777,581780],{"style":41520},[34,581778],{"className":581779,"style":41508},[31813],[34,581781],{"className":581782,"style":41528},[41527],[34,581784,581785,581788],{"style":46771},[34,581786],{"className":581787,"style":41508},[31813],[34,581789,581791],{"className":581790},[31818,31819,31820,31821],[34,581792,581794],{"className":581793},[31685,31821],[34,581795,1697],{"className":581796},[31685,31686,31821],[34,581798,31829],{"className":581799},[31828],[34,581801,581803],{"className":581802},[31801],[34,581804,581806],{"className":581805,"style":41619},[31805],[34,581807],{},[34,581809],{"className":581810},[31690,41488],[108,581812,581813,581867,581868,70884,582200,582284,582285,582424,582425,4504,582476,582542],{},[2627,581814,581815,581816,414527],{},"处理第二项（",[34,581817,581819,581837],{"className":581818},[31642],[34,581820,581822],{"className":581821},[31646],[31648,581823,581824],{"xmlns":31650},[31652,581825,581826,581834],{},[31655,581827,581828,581830,581832],{},[31658,581829,32936],{},[31662,581831,33239],{},[31658,581833,138286],{"mathvariant":38079},[31665,581835,581836],{"encoding":31667},"x \\to \\infty",[34,581838,581840,581858],{"className":581839,"ariaHidden":758},[31672],[34,581841,581843,581846,581849,581852,581855],{"className":581842},[31676],[34,581844],{"className":581845,"style":32527},[31680],[34,581847,32936],{"className":581848},[31685,31686],[34,581850],{"className":581851,"style":31846},[31845],[34,581853,33239],{"className":581854},[31850],[34,581856],{"className":581857,"style":31846},[31845],[34,581859,581861,581864],{"className":581860},[31676],[34,581862],{"className":581863,"style":32527},[31680],[34,581865,138286],{"className":581866},[31685],"\n同理，",[34,581869,581871,581938],{"className":581870},[31642],[34,581872,581874],{"className":581873},[31646],[31648,581875,581876],{"xmlns":31650},[31652,581877,581878,581935],{},[31655,581879,581880,581896,581910,581912,581914,581916,581918,581920,581923,581925,581927,581929],{},[38846,581881,581882,581884,581890],{},[31662,581883,546295],{},[31655,581885,581886,581888],{},[31658,581887,111],{},[31658,581889,54151],{},[31655,581891,581892,581894],{},[31658,581893,1697],{},[31658,581895,54151],{},[41453,581897,581898,581908],{},[31655,581899,581900,581902,581904,581906],{},[31658,581901,31739],{},[31662,581903,456],{"stretchy":702},[31658,581905,94628],{},[31662,581907,2557],{"stretchy":702},[31658,581909,94628],{},[31658,581911,46900],{},[31658,581913,94628],{},[31662,581915,202],{},[31658,581917,31739],{},[31662,581919,456],{"stretchy":702},[31658,581921,581922],{},"η",[31662,581924,2557],{"stretchy":702},[31658,581926,563286],{},[31662,581928,32682],{},[41453,581930,581931,581933],{},[31658,581932,1697],{},[31658,581934,111],{},[31665,581936,581937],{"encoding":31667},"\\int_{aA}^{bA} \\frac{f(u)}{u} du = f(\\eta) \\ln \\frac{b}{a}",[34,581939,581941,582105],{"className":581940,"ariaHidden":758},[31672],[34,581942,581944,581947,582010,582013,582090,582093,582096,582099,582102],{"className":581943},[31676],[34,581945],{"className":581946,"style":546324},[31680],[34,581948,581950,581953],{"className":581949},[32767],[34,581951,546295],{"className":581952,"style":546331},[32767,131249,156560],[34,581954,581956],{"className":581955},[31792],[34,581957,581959,582002],{"className":581958},[31796,31797],[34,581960,581962,581999],{"className":581961},[31801],[34,581963,581965,581982],{"className":581964,"style":546344},[31805],[34,581966,581967,581970],{"style":546347},[34,581968],{"className":581969,"style":31814},[31813],[34,581971,581973],{"className":581972},[31818,31819,31820,31821],[34,581974,581976,581979],{"className":581975},[31685,31821],[34,581977,111],{"className":581978},[31685,31686,31821],[34,581980,54151],{"className":581981},[31685,31686,31821],[34,581983,581984,581987],{"style":546362},[34,581985],{"className":581986,"style":31814},[31813],[34,581988,581990],{"className":581989},[31818,31819,31820,31821],[34,581991,581993,581996],{"className":581992},[31685,31821],[34,581994,1697],{"className":581995},[31685,31686,31821],[34,581997,54151],{"className":581998},[31685,31686,31821],[34,582000,31829],{"className":582001},[31828],[34,582003,582005],{"className":582004},[31801],[34,582006,582008],{"className":582007,"style":546384},[31805],[34,582009],{},[34,582011],{"className":582012,"style":32123},[31845],[34,582014,582016,582019,582087],{"className":582015},[31685],[34,582017],{"className":582018},[31782,41488],[34,582020,582022],{"className":582021},[41453],[34,582023,582025,582079],{"className":582024},[31796,31797],[34,582026,582028,582076],{"className":582027},[31801],[34,582029,582031,582045,582053],{"className":582030,"style":70504},[31805],[34,582032,582033,582036],{"style":41504},[34,582034],{"className":582035,"style":41508},[31813],[34,582037,582039],{"className":582038},[31818,31819,31820,31821],[34,582040,582042],{"className":582041},[31685,31821],[34,582043,94628],{"className":582044},[31685,31686,31821],[34,582046,582047,582050],{"style":41520},[34,582048],{"className":582049,"style":41508},[31813],[34,582051],{"className":582052,"style":41528},[41527],[34,582054,582055,582058],{"style":70529},[34,582056],{"className":582057,"style":41508},[31813],[34,582059,582061],{"className":582060},[31818,31819,31820,31821],[34,582062,582064,582067,582070,582073],{"className":582063},[31685,31821],[34,582065,31739],{"className":582066,"style":31778},[31685,31686,31821],[34,582068,456],{"className":582069},[31782,31821],[34,582071,94628],{"className":582072},[31685,31686,31821],[34,582074,2557],{"className":582075},[31690,31821],[34,582077,31829],{"className":582078},[31828],[34,582080,582082],{"className":582081},[31801],[34,582083,582085],{"className":582084,"style":41619},[31805],[34,582086],{},[34,582088],{"className":582089},[31690,41488],[34,582091,46900],{"className":582092},[31685,31686],[34,582094,94628],{"className":582095},[31685,31686],[34,582097],{"className":582098,"style":31846},[31845],[34,582100,202],{"className":582101},[31850],[34,582103],{"className":582104,"style":31846},[31845],[34,582106,582108,582111,582114,582117,582120,582123,582126,582129,582132],{"className":582107},[31676],[34,582109],{"className":582110,"style":232492},[31680],[34,582112,31739],{"className":582113,"style":31778},[31685,31686],[34,582115,456],{"className":582116},[31782],[34,582118,581922],{"className":582119,"style":32062},[31685,31686],[34,582121,2557],{"className":582122},[31690],[34,582124],{"className":582125,"style":32123},[31845],[34,582127,563286],{"className":582128},[32767],[34,582130],{"className":582131,"style":32123},[31845],[34,582133,582135,582138,582197],{"className":582134},[31685],[34,582136],{"className":582137},[31782,41488],[34,582139,582141],{"className":582140},[41453],[34,582142,582144,582189],{"className":582143},[31796,31797],[34,582145,582147,582186],{"className":582146},[31801],[34,582148,582150,582164,582172],{"className":582149,"style":232511},[31805],[34,582151,582152,582155],{"style":41504},[34,582153],{"className":582154,"style":41508},[31813],[34,582156,582158],{"className":582157},[31818,31819,31820,31821],[34,582159,582161],{"className":582160},[31685,31821],[34,582162,111],{"className":582163},[31685,31686,31821],[34,582165,582166,582169],{"style":41520},[34,582167],{"className":582168,"style":41508},[31813],[34,582170],{"className":582171,"style":41528},[41527],[34,582173,582174,582177],{"style":46771},[34,582175],{"className":582176,"style":41508},[31813],[34,582178,582180],{"className":582179},[31818,31819,31820,31821],[34,582181,582183],{"className":582182},[31685,31821],[34,582184,1697],{"className":582185},[31685,31686,31821],[34,582187,31829],{"className":582188},[31828],[34,582190,582192],{"className":582191},[31801],[34,582193,582195],{"className":582194,"style":41619},[31805],[34,582196],{},[34,582198],{"className":582199},[31690,41488],[34,582201,582203,582233],{"className":582202},[31642],[34,582204,582206],{"className":582205},[31646],[31648,582207,582208],{"xmlns":31650},[31652,582209,582210,582230],{},[31655,582211,582212,582214,582216,582218,582220,582222,582224,582226,582228],{},[31658,582213,581922],{},[31662,582215,112296],{},[31662,582217,15972],{"stretchy":702},[31658,582219,111],{},[31658,582221,54151],{},[31662,582223,31960],{"separator":758},[31658,582225,1697],{},[31658,582227,54151],{},[31662,582229,32712],{"stretchy":702},[31665,582231,582232],{"encoding":31667},"\\eta \\in [aA, bA]",[34,582234,582236,582254],{"className":582235,"ariaHidden":758},[31672],[34,582237,582239,582242,582245,582248,582251],{"className":582238},[31676],[34,582240],{"className":582241,"style":86652},[31680],[34,582243,581922],{"className":582244,"style":32062},[31685,31686],[34,582246],{"className":582247,"style":31846},[31845],[34,582249,112296],{"className":582250},[31850],[34,582252],{"className":582253,"style":31846},[31845],[34,582255,582257,582260,582263,582266,582269,582272,582275,582278,582281],{"className":582256},[31676],[34,582258],{"className":582259,"style":31774},[31680],[34,582261,15972],{"className":582262},[31782],[34,582264,111],{"className":582265},[31685,31686],[34,582267,54151],{"className":582268},[31685,31686],[34,582270,31960],{"className":582271},[32119],[34,582273],{"className":582274,"style":32123},[31845],[34,582276,1697],{"className":582277},[31685,31686],[34,582279,54151],{"className":582280},[31685,31686],[34,582282,32712],{"className":582283},[31690],"。\n因为题目已知 ",[34,582286,582288,582330],{"className":582287},[31642],[34,582289,582291],{"className":582290},[31646],[31648,582292,582293],{"xmlns":31650},[31652,582294,582295,582327],{},[31655,582296,582297,582315,582317,582319,582321,582323,582325],{},[31743,582298,582299,582305],{},[31655,582300,582301,582303],{},[31658,582302,546745],{},[31662,582304,32682],{},[31655,582306,582307,582309,582311,582313],{},[31658,582308,32936],{},[31662,582310,33239],{},[31662,582312,223],{},[31658,582314,138286],{"mathvariant":38079},[31658,582316,31739],{},[31662,582318,456],{"stretchy":702},[31658,582320,32936],{},[31662,582322,2557],{"stretchy":702},[31662,582324,202],{},[31749,582326,459],{},[31665,582328,582329],{"encoding":31667},"\\lim_{x \\to +\\infty} f(x) = 0",[34,582331,582333,582415],{"className":582332,"ariaHidden":758},[31672],[34,582334,582336,582339,582391,582394,582397,582400,582403,582406,582409,582412],{"className":582335},[31676],[34,582337],{"className":582338,"style":31774},[31680],[34,582340,582342,582345],{"className":582341},[32767],[34,582343,546745],{"className":582344},[32767],[34,582346,582348],{"className":582347},[31792],[34,582349,582351,582383],{"className":582350},[31796,31797],[34,582352,582354,582380],{"className":582353},[31801],[34,582355,582357],{"className":582356,"style":141057},[31805],[34,582358,582359,582362],{"style":56235},[34,582360],{"className":582361,"style":31814},[31813],[34,582363,582365],{"className":582364},[31818,31819,31820,31821],[34,582366,582368,582371,582374,582377],{"className":582367},[31685,31821],[34,582369,32936],{"className":582370},[31685,31686,31821],[34,582372,33239],{"className":582373},[31850,31821],[34,582375,223],{"className":582376},[31685,31821],[34,582378,138286],{"className":582379},[31685,31821],[34,582381,31829],{"className":582382},[31828],[34,582384,582386],{"className":582385},[31801],[34,582387,582389],{"className":582388,"style":32037},[31805],[34,582390],{},[34,582392],{"className":582393,"style":32123},[31845],[34,582395,31739],{"className":582396,"style":31778},[31685,31686],[34,582398,456],{"className":582399},[31782],[34,582401,32936],{"className":582402},[31685,31686],[34,582404,2557],{"className":582405},[31690],[34,582407],{"className":582408,"style":31846},[31845],[34,582410,202],{"className":582411},[31850],[34,582413],{"className":582414,"style":31846},[31845],[34,582416,582418,582421],{"className":582417},[31676],[34,582419],{"className":582420,"style":32617},[31680],[34,582422,459],{"className":582423},[31685],"，所以当 ",[34,582426,582428,582446],{"className":582427},[31642],[34,582429,582431],{"className":582430},[31646],[31648,582432,582433],{"xmlns":31650},[31652,582434,582435,582443],{},[31655,582436,582437,582439,582441],{},[31658,582438,54151],{},[31662,582440,33239],{},[31658,582442,138286],{"mathvariant":38079},[31665,582444,582445],{"encoding":31667},"A \\to \\infty",[34,582447,582449,582467],{"className":582448,"ariaHidden":758},[31672],[34,582450,582452,582455,582458,582461,582464],{"className":582451},[31676],[34,582453],{"className":582454,"style":33218},[31680],[34,582456,54151],{"className":582457},[31685,31686],[34,582459],{"className":582460,"style":31846},[31845],[34,582462,33239],{"className":582463},[31850],[34,582465],{"className":582466,"style":31846},[31845],[34,582468,582470,582473],{"className":582469},[31676],[34,582471],{"className":582472,"style":32527},[31680],[34,582474,138286],{"className":582475},[31685],[34,582477,582479,582503],{"className":582478},[31642],[34,582480,582482],{"className":582481},[31646],[31648,582483,582484],{"xmlns":31650},[31652,582485,582486,582500],{},[31655,582487,582488,582490,582492,582494,582496,582498],{},[31658,582489,31739],{},[31662,582491,456],{"stretchy":702},[31658,582493,581922],{},[31662,582495,2557],{"stretchy":702},[31662,582497,33239],{},[31749,582499,459],{},[31665,582501,582502],{"encoding":31667},"f(\\eta) \\to 0",[34,582504,582506,582533],{"className":582505,"ariaHidden":758},[31672],[34,582507,582509,582512,582515,582518,582521,582524,582527,582530],{"className":582508},[31676],[34,582510],{"className":582511,"style":31774},[31680],[34,582513,31739],{"className":582514,"style":31778},[31685,31686],[34,582516,456],{"className":582517},[31782],[34,582519,581922],{"className":582520,"style":32062},[31685,31686],[34,582522,2557],{"className":582523},[31690],[34,582525],{"className":582526,"style":31846},[31845],[34,582528,33239],{"className":582529},[31850],[34,582531],{"className":582532,"style":31846},[31845],[34,582534,582536,582539],{"className":582535},[31676],[34,582537],{"className":582538,"style":32617},[31680],[34,582540,459],{"className":582541},[31685],"。\n第二项极限 = 0。",[108,582544,582545,582547],{},[2627,582546,62532],{},[34,582548,582550,582608],{"className":582549},[31642],[34,582551,582553],{"className":582552},[31646],[31648,582554,582555],{"xmlns":31650},[31652,582556,582557,582605],{},[31655,582558,582559,582561,582563,582565,582567,582569,582571,582573,582575,582581,582583,582585,582587,582589,582591,582593,582595,582597,582599],{},[31658,582560,106063],{},[31662,582562,202],{},[31658,582564,31739],{},[31662,582566,456],{"stretchy":702},[31749,582568,459],{},[31662,582570,2557],{"stretchy":702},[31658,582572,563286],{},[31662,582574,32682],{},[41453,582576,582577,582579],{},[31658,582578,1697],{},[31658,582580,111],{},[31662,582582,32226],{},[31749,582584,459],{},[31662,582586,202],{},[31658,582588,31739],{},[31662,582590,456],{"stretchy":702},[31749,582592,459],{},[31662,582594,2557],{"stretchy":702},[31658,582596,563286],{},[31662,582598,32682],{},[41453,582600,582601,582603],{},[31658,582602,1697],{},[31658,582604,111],{},[31665,582606,582607],{"encoding":31667},"I = f(0)\\ln \\frac{b}{a} - 0 = f(0)\\ln \\frac{b}{a}",[34,582609,582611,582629,582733,582751],{"className":582610,"ariaHidden":758},[31672],[34,582612,582614,582617,582620,582623,582626],{"className":582613},[31676],[34,582615],{"className":582616,"style":33218},[31680],[34,582618,106063],{"className":582619,"style":67482},[31685,31686],[34,582621],{"className":582622,"style":31846},[31845],[34,582624,202],{"className":582625},[31850],[34,582627],{"className":582628,"style":31846},[31845],[34,582630,582632,582635,582638,582641,582644,582647,582650,582653,582656,582724,582727,582730],{"className":582631},[31676],[34,582633],{"className":582634,"style":232492},[31680],[34,582636,31739],{"className":582637,"style":31778},[31685,31686],[34,582639,456],{"className":582640},[31782],[34,582642,459],{"className":582643},[31685],[34,582645,2557],{"className":582646},[31690],[34,582648],{"className":582649,"style":32123},[31845],[34,582651,563286],{"className":582652},[32767],[34,582654],{"className":582655,"style":32123},[31845],[34,582657,582659,582662,582721],{"className":582658},[31685],[34,582660],{"className":582661},[31782,41488],[34,582663,582665],{"className":582664},[41453],[34,582666,582668,582713],{"className":582667},[31796,31797],[34,582669,582671,582710],{"className":582670},[31801],[34,582672,582674,582688,582696],{"className":582673,"style":232511},[31805],[34,582675,582676,582679],{"style":41504},[34,582677],{"className":582678,"style":41508},[31813],[34,582680,582682],{"className":582681},[31818,31819,31820,31821],[34,582683,582685],{"className":582684},[31685,31821],[34,582686,111],{"className":582687},[31685,31686,31821],[34,582689,582690,582693],{"style":41520},[34,582691],{"className":582692,"style":41508},[31813],[34,582694],{"className":582695,"style":41528},[41527],[34,582697,582698,582701],{"style":46771},[34,582699],{"className":582700,"style":41508},[31813],[34,582702,582704],{"className":582703},[31818,31819,31820,31821],[34,582705,582707],{"className":582706},[31685,31821],[34,582708,1697],{"className":582709},[31685,31686,31821],[34,582711,31829],{"className":582712},[31828],[34,582714,582716],{"className":582715},[31801],[34,582717,582719],{"className":582718,"style":41619},[31805],[34,582720],{},[34,582722],{"className":582723},[31690,41488],[34,582725],{"className":582726,"style":32604},[31845],[34,582728,32226],{"className":582729},[32024],[34,582731],{"className":582732,"style":32604},[31845],[34,582734,582736,582739,582742,582745,582748],{"className":582735},[31676],[34,582737],{"className":582738,"style":32617},[31680],[34,582740,459],{"className":582741},[31685],[34,582743],{"className":582744,"style":31846},[31845],[34,582746,202],{"className":582747},[31850],[34,582749],{"className":582750,"style":31846},[31845],[34,582752,582754,582757,582760,582763,582766,582769,582772,582775,582778],{"className":582753},[31676],[34,582755],{"className":582756,"style":232492},[31680],[34,582758,31739],{"className":582759,"style":31778},[31685,31686],[34,582761,456],{"className":582762},[31782],[34,582764,459],{"className":582765},[31685],[34,582767,2557],{"className":582768},[31690],[34,582770],{"className":582771,"style":32123},[31845],[34,582773,563286],{"className":582774},[32767],[34,582776],{"className":582777,"style":32123},[31845],[34,582779,582781,582784,582843],{"className":582780},[31685],[34,582782],{"className":582783},[31782,41488],[34,582785,582787],{"className":582786},[41453],[34,582788,582790,582835],{"className":582789},[31796,31797],[34,582791,582793,582832],{"className":582792},[31801],[34,582794,582796,582810,582818],{"className":582795,"style":232511},[31805],[34,582797,582798,582801],{"style":41504},[34,582799],{"className":582800,"style":41508},[31813],[34,582802,582804],{"className":582803},[31818,31819,31820,31821],[34,582805,582807],{"className":582806},[31685,31821],[34,582808,111],{"className":582809},[31685,31686,31821],[34,582811,582812,582815],{"style":41520},[34,582813],{"className":582814,"style":41508},[31813],[34,582816],{"className":582817,"style":41528},[41527],[34,582819,582820,582823],{"style":46771},[34,582821],{"className":582822,"style":41508},[31813],[34,582824,582826],{"className":582825},[31818,31819,31820,31821],[34,582827,582829],{"className":582828},[31685,31821],[34,582830,1697],{"className":582831},[31685,31686,31821],[34,582833,31829],{"className":582834},[31828],[34,582836,582838],{"className":582837},[31801],[34,582839,582841],{"className":582840,"style":41619},[31805],[34,582842],{},[34,582844],{"className":582845},[31690,41488],[32188,582847],{},[253,582849,582851],{"id":582850},"例子11","例子1.1",[15,582853,582854,582856,582857],{},[2627,582855,214361],{},"\n证明：",[34,582858,582860,582922],{"className":582859},[31642],[34,582861,582863],{"className":582862},[31646],[31648,582864,582865],{"xmlns":31650},[31652,582866,582867,582919],{},[31655,582868,582869,582871,582873,582875,582877,582879,582881,582883,582917],{},[31658,582870,54224],{},[31662,582872,456],{"stretchy":702},[31658,582874,15],{},[31662,582876,31960],{"separator":758},[31658,582878,38759],{},[31662,582880,2557],{"stretchy":702},[31662,582882,202],{},[41453,582884,582885,582903],{},[31655,582886,582887,582889,582891,582893,582895,582897,582899,582901],{},[31658,582888,564791],{"mathvariant":38079},[31662,582890,456],{"stretchy":702},[31658,582892,15],{},[31662,582894,2557],{"stretchy":702},[31658,582896,564791],{"mathvariant":38079},[31662,582898,456],{"stretchy":702},[31658,582900,38759],{},[31662,582902,2557],{"stretchy":702},[31655,582904,582905,582907,582909,582911,582913,582915],{},[31658,582906,564791],{"mathvariant":38079},[31662,582908,456],{"stretchy":702},[31658,582910,15],{},[31662,582912,223],{},[31658,582914,38759],{},[31662,582916,2557],{"stretchy":702},[31658,582918,2547],{"mathvariant":38079},[31665,582920,582921],{"encoding":31667},"B(p,q) = \\frac{\\Gamma(p)\\Gamma(q)}{\\Gamma(p + q)}.",[34,582923,582925,582961],{"className":582924,"ariaHidden":758},[31672],[34,582926,582928,582931,582934,582937,582940,582943,582946,582949,582952,582955,582958],{"className":582927},[31676],[34,582929],{"className":582930,"style":31774},[31680],[34,582932,54224],{"className":582933,"style":54252},[31685,31686],[34,582935,456],{"className":582936},[31782],[34,582938,15],{"className":582939},[31685,31686],[34,582941,31960],{"className":582942},[32119],[34,582944],{"className":582945,"style":32123},[31845],[34,582947,38759],{"className":582948,"style":32062},[31685,31686],[34,582950,2557],{"className":582951},[31690],[34,582953],{"className":582954,"style":31846},[31845],[34,582956,202],{"className":582957},[31850],[34,582959],{"className":582960,"style":31846},[31845],[34,582962,582964,582967,583071],{"className":582963},[31676],[34,582965],{"className":582966,"style":327841},[31680],[34,582968,582970,582973,583068],{"className":582969},[31685],[34,582971],{"className":582972},[31782,41488],[34,582974,582976],{"className":582975},[41453],[34,582977,582979,583060],{"className":582978},[31796,31797],[34,582980,582982,583057],{"className":582981},[31801],[34,582983,582985,583014,583022],{"className":582984,"style":70504},[31805],[34,582986,582987,582990],{"style":41504},[34,582988],{"className":582989,"style":41508},[31813],[34,582991,582993],{"className":582992},[31818,31819,31820,31821],[34,582994,582996,582999,583002,583005,583008,583011],{"className":582995},[31685,31821],[34,582997,564791],{"className":582998},[31685,31821],[34,583000,456],{"className":583001},[31782,31821],[34,583003,15],{"className":583004},[31685,31686,31821],[34,583006,223],{"className":583007},[32024,31821],[34,583009,38759],{"className":583010,"style":32062},[31685,31686,31821],[34,583012,2557],{"className":583013},[31690,31821],[34,583015,583016,583019],{"style":41520},[34,583017],{"className":583018,"style":41508},[31813],[34,583020],{"className":583021,"style":41528},[41527],[34,583023,583024,583027],{"style":70529},[34,583025],{"className":583026,"style":41508},[31813],[34,583028,583030],{"className":583029},[31818,31819,31820,31821],[34,583031,583033,583036,583039,583042,583045,583048,583051,583054],{"className":583032},[31685,31821],[34,583034,564791],{"className":583035},[31685,31821],[34,583037,456],{"className":583038},[31782,31821],[34,583040,15],{"className":583041},[31685,31686,31821],[34,583043,2557],{"className":583044},[31690,31821],[34,583046,564791],{"className":583047},[31685,31821],[34,583049,456],{"className":583050},[31782,31821],[34,583052,38759],{"className":583053,"style":32062},[31685,31686,31821],[34,583055,2557],{"className":583056},[31690,31821],[34,583058,31829],{"className":583059},[31828],[34,583061,583063],{"className":583062},[31801],[34,583064,583066],{"className":583065,"style":324858},[31805],[34,583067],{},[34,583069],{"className":583070},[31690,41488],[34,583072,2547],{"className":583073},[31685],[15,583075,583076,583078,583079,347],{},[2627,583077,91513],{},"\n这是伽马函数和贝塔函数最重要的关系式。证明技巧在于",[2627,583080,583081],{},"把乘积转化为二重积分",[15,583083,583084],{},[2627,583085,549629],{},[2230,583087,583088,584632,585131,586036,586635,587002],{},[108,583089,583090,583181,583182,583210,583211,39984,583239,583480,583481,583562,583563,583622,583623,583625,583628,583629,39984,583785,584347,584348,2547],{},[2627,583091,583092,583093,258,583137,2664],{},"写出 ",[34,583094,583096,583116],{"className":583095},[31642],[34,583097,583099],{"className":583098},[31646],[31648,583100,583101],{"xmlns":31650},[31652,583102,583103,583113],{},[31655,583104,583105,583107,583109,583111],{},[31658,583106,564791],{"mathvariant":38079},[31662,583108,456],{"stretchy":702},[31658,583110,15],{},[31662,583112,2557],{"stretchy":702},[31665,583114,583115],{"encoding":31667},"\\Gamma(p)",[34,583117,583119],{"className":583118,"ariaHidden":758},[31672],[34,583120,583122,583125,583128,583131,583134],{"className":583121},[31676],[34,583123],{"className":583124,"style":31774},[31680],[34,583126,564791],{"className":583127},[31685],[34,583129,456],{"className":583130},[31782],[34,583132,15],{"className":583133},[31685,31686],[34,583135,2557],{"className":583136},[31690],[34,583138,583140,583160],{"className":583139},[31642],[34,583141,583143],{"className":583142},[31646],[31648,583144,583145],{"xmlns":31650},[31652,583146,583147,583157],{},[31655,583148,583149,583151,583153,583155],{},[31658,583150,564791],{"mathvariant":38079},[31662,583152,456],{"stretchy":702},[31658,583154,38759],{},[31662,583156,2557],{"stretchy":702},[31665,583158,583159],{"encoding":31667},"\\Gamma(q)",[34,583161,583163],{"className":583162,"ariaHidden":758},[31672],[34,583164,583166,583169,583172,583175,583178],{"className":583165},[31676],[34,583167],{"className":583168,"style":31774},[31680],[34,583170,564791],{"className":583171},[31685],[34,583173,456],{"className":583174},[31782],[34,583176,38759],{"className":583177,"style":32062},[31685,31686],[34,583179,2557],{"className":583180},[31690],"\n为了避免变量混淆，一个用 ",[34,583183,583185,583198],{"className":583184},[31642],[34,583186,583188],{"className":583187},[31646],[31648,583189,583190],{"xmlns":31650},[31652,583191,583192,583196],{},[31655,583193,583194],{},[31658,583195,32936],{},[31665,583197,32936],{"encoding":31667},[34,583199,583201],{"className":583200,"ariaHidden":758},[31672],[34,583202,583204,583207],{"className":583203},[31676],[34,583205],{"className":583206,"style":32527},[31680],[34,583208,32936],{"className":583209},[31685,31686],"，一个用 ",[34,583212,583214,583227],{"className":583213},[31642],[34,583215,583217],{"className":583216},[31646],[31648,583218,583219],{"xmlns":31650},[31652,583220,583221,583225],{},[31655,583222,583223],{},[31658,583224,67764],{},[31665,583226,67764],{"encoding":31667},[34,583228,583230],{"className":583229,"ariaHidden":758},[31672],[34,583231,583233,583236],{"className":583232},[31676],[34,583234],{"className":583235,"style":38774},[31680],[34,583237,67764],{"className":583238,"style":32062},[31685,31686],[34,583240,583242,583302],{"className":583241},[31642],[34,583243,583245],{"className":583244},[31646],[31648,583246,583247],{"xmlns":31650},[31652,583248,583249,583299],{},[31655,583250,583251,583253,583255,583257,583259,583261,583273,583283,583295,583297],{},[31658,583252,564791],{"mathvariant":38079},[31662,583254,456],{"stretchy":702},[31658,583256,15],{},[31662,583258,2557],{"stretchy":702},[31662,583260,202],{},[38846,583262,583263,583265,583267],{},[31662,583264,546295],{},[31749,583266,459],{},[31655,583268,583269,583271],{},[31662,583270,223],{},[31658,583272,138286],{"mathvariant":38079},[37624,583274,583275,583277],{},[31658,583276,33244],{},[31655,583278,583279,583281],{},[31662,583280,32226],{},[31658,583282,32936],{},[37624,583284,583285,583287],{},[31658,583286,32936],{},[31655,583288,583289,583291,583293],{},[31658,583290,15],{},[31662,583292,32226],{},[31749,583294,640],{},[31658,583296,46900],{},[31658,583298,32936],{},[31665,583300,583301],{"encoding":31667},"\\Gamma(p) = \\int_{0}^{+\\infty} e^{-x} x^{p-1} dx",[34,583303,583305,583332],{"className":583304,"ariaHidden":758},[31672],[34,583306,583308,583311,583314,583317,583320,583323,583326,583329],{"className":583307},[31676],[34,583309],{"className":583310,"style":31774},[31680],[34,583312,564791],{"className":583313},[31685],[34,583315,456],{"className":583316},[31782],[34,583318,15],{"className":583319},[31685,31686],[34,583321,2557],{"className":583322},[31690],[34,583324],{"className":583325,"style":31846},[31845],[34,583327,202],{"className":583328},[31850],[34,583330],{"className":583331,"style":31846},[31845],[34,583333,583335,583338,583398,583401,583436,583474,583477],{"className":583334},[31676],[34,583336],{"className":583337,"style":546531},[31680],[34,583339,583341,583344],{"className":583340},[32767],[34,583342,546295],{"className":583343,"style":546331},[32767,131249,156560],[34,583345,583347],{"className":583346},[31792],[34,583348,583350,583390],{"className":583349},[31796,31797],[34,583351,583353,583387],{"className":583352},[31801],[34,583354,583356,583370],{"className":583355,"style":546550},[31805],[34,583357,583358,583361],{"style":546347},[34,583359],{"className":583360,"style":31814},[31813],[34,583362,583364],{"className":583363},[31818,31819,31820,31821],[34,583365,583367],{"className":583366},[31685,31821],[34,583368,459],{"className":583369},[31685,31821],[34,583371,583372,583375],{"style":546362},[34,583373],{"className":583374,"style":31814},[31813],[34,583376,583378],{"className":583377},[31818,31819,31820,31821],[34,583379,583381,583384],{"className":583380},[31685,31821],[34,583382,223],{"className":583383},[31685,31821],[34,583385,138286],{"className":583386},[31685,31821],[34,583388,31829],{"className":583389},[31828],[34,583391,583393],{"className":583392},[31801],[34,583394,583396],{"className":583395,"style":546384},[31805],[34,583397],{},[34,583399],{"className":583400,"style":32123},[31845],[34,583402,583404,583407],{"className":583403},[31685],[34,583405,33244],{"className":583406},[31685,31686],[34,583408,583410],{"className":583409},[31792],[34,583411,583413],{"className":583412},[31796],[34,583414,583416],{"className":583415},[31801],[34,583417,583419],{"className":583418,"style":156828},[31805],[34,583420,583421,583424],{"style":37777},[34,583422],{"className":583423,"style":31814},[31813],[34,583425,583427],{"className":583426},[31818,31819,31820,31821],[34,583428,583430,583433],{"className":583429},[31685,31821],[34,583431,32226],{"className":583432},[31685,31821],[34,583434,32936],{"className":583435},[31685,31686,31821],[34,583437,583439,583442],{"className":583438},[31685],[34,583440,32936],{"className":583441},[31685,31686],[34,583443,583445],{"className":583444},[31792],[34,583446,583448],{"className":583447},[31796],[34,583449,583451],{"className":583450},[31801],[34,583452,583454],{"className":583453,"style":37774},[31805],[34,583455,583456,583459],{"style":37777},[34,583457],{"className":583458,"style":31814},[31813],[34,583460,583462],{"className":583461},[31818,31819,31820,31821],[34,583463,583465,583468,583471],{"className":583464},[31685,31821],[34,583466,15],{"className":583467},[31685,31686,31821],[34,583469,32226],{"className":583470},[32024,31821],[34,583472,640],{"className":583473},[31685,31821],[34,583475,46900],{"className":583476},[31685,31686],[34,583478,32936],{"className":583479},[31685,31686],"\n换元：为了凑出二维形式，通常令 ",[34,583482,583484,583506],{"className":583483},[31642],[34,583485,583487],{"className":583486},[31646],[31648,583488,583489],{"xmlns":31650},[31652,583490,583491,583503],{},[31655,583492,583493,583495,583497],{},[31658,583494,32936],{},[31662,583496,202],{},[37624,583498,583499,583501],{},[31658,583500,94628],{},[31749,583502,768],{},[31665,583504,583505],{"encoding":31667},"x = u^2",[34,583507,583509,583527],{"className":583508,"ariaHidden":758},[31672],[34,583510,583512,583515,583518,583521,583524],{"className":583511},[31676],[34,583513],{"className":583514,"style":32527},[31680],[34,583516,32936],{"className":583517},[31685,31686],[34,583519],{"className":583520,"style":31846},[31845],[34,583522,202],{"className":583523},[31850],[34,583525],{"className":583526,"style":31846},[31845],[34,583528,583530,583533],{"className":583529},[31676],[34,583531],{"className":583532,"style":37774},[31680],[34,583534,583536,583539],{"className":583535},[31685],[34,583537,94628],{"className":583538},[31685,31686],[34,583540,583542],{"className":583541},[31792],[34,583543,583545],{"className":583544},[31796],[34,583546,583548],{"className":583547},[31801],[34,583549,583551],{"className":583550,"style":37774},[31805],[34,583552,583553,583556],{"style":37777},[34,583554],{"className":583555,"style":31814},[31813],[34,583557,583559],{"className":583558},[31818,31819,31820,31821],[34,583560,768],{"className":583561},[31685,31821]," (虽然这题直接乘也可以，但经典证法常用 ",[34,583564,583566,583584],{"className":583565},[31642],[34,583567,583569],{"className":583568},[31646],[31648,583570,583571],{"xmlns":31650},[31652,583572,583573,583581],{},[31655,583574,583575],{},[37624,583576,583577,583579],{},[31658,583578,32936],{},[31749,583580,768],{},[31665,583582,583583],{"encoding":31667},"x^2",[34,583585,583587],{"className":583586,"ariaHidden":758},[31672],[34,583588,583590,583593],{"className":583589},[31676],[34,583591],{"className":583592,"style":37774},[31680],[34,583594,583596,583599],{"className":583595},[31685],[34,583597,32936],{"className":583598},[31685,31686],[34,583600,583602],{"className":583601},[31792],[34,583603,583605],{"className":583604},[31796],[34,583606,583608],{"className":583607},[31801],[34,583609,583611],{"className":583610,"style":37774},[31805],[34,583612,583613,583616],{"style":37777},[34,583614],{"className":583615,"style":31814},[31813],[34,583617,583619],{"className":583618},[31818,31819,31820,31821],[34,583620,768],{"className":583621},[31685,31821]," 形式，这里我们用直接乘积法配合极坐标)。",[58531,583624],{},[2627,583626,583627],{},"标准证法（极坐标）：","\n先对 Gamma 函数换元：令 ",[34,583630,583632,583674],{"className":583631},[31642],[34,583633,583635],{"className":583634},[31646],[31648,583636,583637],{"xmlns":31650},[31652,583638,583639,583671],{},[31655,583640,583641,583643,583645,583651,583653,583655,583657,583659,583661,583663,583665,583667,583669],{},[31658,583642,32936],{},[31662,583644,202],{},[37624,583646,583647,583649],{},[31658,583648,94628],{},[31749,583650,768],{},[39361,583652,262622],{},[31662,583654,262625],{},[39361,583656,262622],{},[31658,583658,46900],{},[31658,583660,32936],{},[31662,583662,202],{},[31749,583664,768],{},[31658,583666,94628],{},[31658,583668,46900],{},[31658,583670,94628],{},[31665,583672,583673],{"encoding":31667},"x = u^2 \\implies dx = 2u du",[34,583675,583677,583695,583746,583767],{"className":583676,"ariaHidden":758},[31672],[34,583678,583680,583683,583686,583689,583692],{"className":583679},[31676],[34,583681],{"className":583682,"style":32527},[31680],[34,583684,32936],{"className":583685},[31685,31686],[34,583687],{"className":583688,"style":31846},[31845],[34,583690,202],{"className":583691},[31850],[34,583693],{"className":583694,"style":31846},[31845],[34,583696,583698,583702,583731,583734,583737,583740,583743],{"className":583697},[31676],[34,583699],{"className":583700,"style":583701},[31680],"height:0.8381em;vertical-align:-0.024em;",[34,583703,583705,583708],{"className":583704},[31685],[34,583706,94628],{"className":583707},[31685,31686],[34,583709,583711],{"className":583710},[31792],[34,583712,583714],{"className":583713},[31796],[34,583715,583717],{"className":583716},[31801],[34,583718,583720],{"className":583719,"style":37774},[31805],[34,583721,583722,583725],{"style":37777},[34,583723],{"className":583724,"style":31814},[31813],[34,583726,583728],{"className":583727},[31818,31819,31820,31821],[34,583729,768],{"className":583730},[31685,31821],[34,583732],{"className":583733,"style":31846},[31845],[34,583735],{"className":583736,"style":31846},[31845],[34,583738,262625],{"className":583739},[31850],[34,583741],{"className":583742,"style":31846},[31845],[34,583744],{"className":583745,"style":31846},[31845],[34,583747,583749,583752,583755,583758,583761,583764],{"className":583748},[31676],[34,583750],{"className":583751,"style":31681},[31680],[34,583753,46900],{"className":583754},[31685,31686],[34,583756,32936],{"className":583757},[31685,31686],[34,583759],{"className":583760,"style":31846},[31845],[34,583762,202],{"className":583763},[31850],[34,583765],{"className":583766,"style":31846},[31845],[34,583768,583770,583773,583776,583779,583782],{"className":583769},[31676],[34,583771],{"className":583772,"style":31681},[31680],[34,583774,768],{"className":583775},[31685],[34,583777,94628],{"className":583778},[31685,31686],[34,583780,46900],{"className":583781},[31685,31686],[34,583783,94628],{"className":583784},[31685,31686],[34,583786,583788,583912],{"className":583787},[31642],[34,583789,583791],{"className":583790},[31646],[31648,583792,583793],{"xmlns":31650},[31652,583794,583795,583909],{},[31655,583796,583797,583799,583801,583803,583805,583807,583819,583833,583835,583841,583853,583855,583857,583859,583861,583863,583865,583877,583891,583905,583907],{},[31658,583798,564791],{"mathvariant":38079},[31662,583800,456],{"stretchy":702},[31658,583802,15],{},[31662,583804,2557],{"stretchy":702},[31662,583806,202],{},[38846,583808,583809,583811,583813],{},[31662,583810,546295],{},[31749,583812,459],{},[31655,583814,583815,583817],{},[31662,583816,223],{},[31658,583818,138286],{"mathvariant":38079},[37624,583820,583821,583823],{},[31658,583822,33244],{},[31655,583824,583825,583827],{},[31662,583826,32226],{},[37624,583828,583829,583831],{},[31658,583830,94628],{},[31749,583832,768],{},[31662,583834,456],{"stretchy":702},[37624,583836,583837,583839],{},[31658,583838,94628],{},[31749,583840,768],{},[37624,583842,583843,583845],{},[31662,583844,2557],{"stretchy":702},[31655,583846,583847,583849,583851],{},[31658,583848,15],{},[31662,583850,32226],{},[31749,583852,640],{},[31749,583854,768],{},[31658,583856,94628],{},[31658,583858,46900],{},[31658,583860,94628],{},[31662,583862,202],{},[31749,583864,768],{},[38846,583866,583867,583869,583871],{},[31662,583868,546295],{},[31749,583870,459],{},[31655,583872,583873,583875],{},[31662,583874,223],{},[31658,583876,138286],{"mathvariant":38079},[37624,583878,583879,583881],{},[31658,583880,33244],{},[31655,583882,583883,583885],{},[31662,583884,32226],{},[37624,583886,583887,583889],{},[31658,583888,94628],{},[31749,583890,768],{},[37624,583892,583893,583895],{},[31658,583894,94628],{},[31655,583896,583897,583899,583901,583903],{},[31749,583898,768],{},[31658,583900,15],{},[31662,583902,32226],{},[31749,583904,640],{},[31658,583906,46900],{},[31658,583908,94628],{},[31665,583910,583911],{"encoding":31667},"\\Gamma(p) = \\int_{0}^{+\\infty} e^{-u^2} (u^2)^{p-1} 2u du = 2 \\int_{0}^{+\\infty} e^{-u^2} u^{2p-1} du",[34,583913,583915,583942,584164],{"className":583914,"ariaHidden":758},[31672],[34,583916,583918,583921,583924,583927,583930,583933,583936,583939],{"className":583917},[31676],[34,583919],{"className":583920,"style":31774},[31680],[34,583922,564791],{"className":583923},[31685],[34,583925,456],{"className":583926},[31782],[34,583928,15],{"className":583929},[31685,31686],[34,583931,2557],{"className":583932},[31690],[34,583934],{"className":583935,"style":31846},[31845],[34,583937,202],{"className":583938},[31850],[34,583940],{"className":583941,"style":31846},[31845],[34,583943,583945,583949,584009,584012,584073,584076,584105,584143,584146,584149,584152,584155,584158,584161],{"className":583944},[31676],[34,583946],{"className":583947,"style":583948},[31680],"height:1.3427em;vertical-align:-0.3558em;",[34,583950,583952,583955],{"className":583951},[32767],[34,583953,546295],{"className":583954,"style":546331},[32767,131249,156560],[34,583956,583958],{"className":583957},[31792],[34,583959,583961,584001],{"className":583960},[31796,31797],[34,583962,583964,583998],{"className":583963},[31801],[34,583965,583967,583981],{"className":583966,"style":546550},[31805],[34,583968,583969,583972],{"style":546347},[34,583970],{"className":583971,"style":31814},[31813],[34,583973,583975],{"className":583974},[31818,31819,31820,31821],[34,583976,583978],{"className":583977},[31685,31821],[34,583979,459],{"className":583980},[31685,31821],[34,583982,583983,583986],{"style":546362},[34,583984],{"className":583985,"style":31814},[31813],[34,583987,583989],{"className":583988},[31818,31819,31820,31821],[34,583990,583992,583995],{"className":583991},[31685,31821],[34,583993,223],{"className":583994},[31685,31821],[34,583996,138286],{"className":583997},[31685,31821],[34,583999,31829],{"className":584000},[31828],[34,584002,584004],{"className":584003},[31801],[34,584005,584007],{"className":584006,"style":546384},[31805],[34,584008],{},[34,584010],{"className":584011,"style":32123},[31845],[34,584013,584015,584018],{"className":584014},[31685],[34,584016,33244],{"className":584017},[31685,31686],[34,584019,584021],{"className":584020},[31792],[34,584022,584024],{"className":584023},[31796],[34,584025,584027],{"className":584026},[31801],[34,584028,584030],{"className":584029,"style":562030},[31805],[34,584031,584032,584035],{"style":37777},[34,584033],{"className":584034,"style":31814},[31813],[34,584036,584038],{"className":584037},[31818,31819,31820,31821],[34,584039,584041,584044],{"className":584040},[31685,31821],[34,584042,32226],{"className":584043},[31685,31821],[34,584045,584047,584050],{"className":584046},[31685,31821],[34,584048,94628],{"className":584049},[31685,31686,31821],[34,584051,584053],{"className":584052},[31792],[34,584054,584056],{"className":584055},[31796],[34,584057,584059],{"className":584058},[31801],[34,584060,584062],{"className":584061,"style":144123},[31805],[34,584063,584064,584067],{"style":552391},[34,584065],{"className":584066,"style":43406},[31813],[34,584068,584070],{"className":584069},[31818,43410,43411,31821],[34,584071,768],{"className":584072},[31685,31821],[34,584074,456],{"className":584075},[31782],[34,584077,584079,584082],{"className":584078},[31685],[34,584080,94628],{"className":584081},[31685,31686],[34,584083,584085],{"className":584084},[31792],[34,584086,584088],{"className":584087},[31796],[34,584089,584091],{"className":584090},[31801],[34,584092,584094],{"className":584093,"style":37774},[31805],[34,584095,584096,584099],{"style":37777},[34,584097],{"className":584098,"style":31814},[31813],[34,584100,584102],{"className":584101},[31818,31819,31820,31821],[34,584103,768],{"className":584104},[31685,31821],[34,584106,584108,584111],{"className":584107},[31690],[34,584109,2557],{"className":584110},[31690],[34,584112,584114],{"className":584113},[31792],[34,584115,584117],{"className":584116},[31796],[34,584118,584120],{"className":584119},[31801],[34,584121,584123],{"className":584122,"style":37774},[31805],[34,584124,584125,584128],{"style":37777},[34,584126],{"className":584127,"style":31814},[31813],[34,584129,584131],{"className":584130},[31818,31819,31820,31821],[34,584132,584134,584137,584140],{"className":584133},[31685,31821],[34,584135,15],{"className":584136},[31685,31686,31821],[34,584138,32226],{"className":584139},[32024,31821],[34,584141,640],{"className":584142},[31685,31821],[34,584144,768],{"className":584145},[31685],[34,584147,94628],{"className":584148},[31685,31686],[34,584150,46900],{"className":584151},[31685,31686],[34,584153,94628],{"className":584154},[31685,31686],[34,584156],{"className":584157,"style":31846},[31845],[34,584159,202],{"className":584160},[31850],[34,584162],{"className":584163,"style":31846},[31845],[34,584165,584167,584170,584173,584176,584236,584239,584300,584341,584344],{"className":584166},[31676],[34,584168],{"className":584169,"style":583948},[31680],[34,584171,768],{"className":584172},[31685],[34,584174],{"className":584175,"style":32123},[31845],[34,584177,584179,584182],{"className":584178},[32767],[34,584180,546295],{"className":584181,"style":546331},[32767,131249,156560],[34,584183,584185],{"className":584184},[31792],[34,584186,584188,584228],{"className":584187},[31796,31797],[34,584189,584191,584225],{"className":584190},[31801],[34,584192,584194,584208],{"className":584193,"style":546550},[31805],[34,584195,584196,584199],{"style":546347},[34,584197],{"className":584198,"style":31814},[31813],[34,584200,584202],{"className":584201},[31818,31819,31820,31821],[34,584203,584205],{"className":584204},[31685,31821],[34,584206,459],{"className":584207},[31685,31821],[34,584209,584210,584213],{"style":546362},[34,584211],{"className":584212,"style":31814},[31813],[34,584214,584216],{"className":584215},[31818,31819,31820,31821],[34,584217,584219,584222],{"className":584218},[31685,31821],[34,584220,223],{"className":584221},[31685,31821],[34,584223,138286],{"className":584224},[31685,31821],[34,584226,31829],{"className":584227},[31828],[34,584229,584231],{"className":584230},[31801],[34,584232,584234],{"className":584233,"style":546384},[31805],[34,584235],{},[34,584237],{"className":584238,"style":32123},[31845],[34,584240,584242,584245],{"className":584241},[31685],[34,584243,33244],{"className":584244},[31685,31686],[34,584246,584248],{"className":584247},[31792],[34,584249,584251],{"className":584250},[31796],[34,584252,584254],{"className":584253},[31801],[34,584255,584257],{"className":584256,"style":562030},[31805],[34,584258,584259,584262],{"style":37777},[34,584260],{"className":584261,"style":31814},[31813],[34,584263,584265],{"className":584264},[31818,31819,31820,31821],[34,584266,584268,584271],{"className":584267},[31685,31821],[34,584269,32226],{"className":584270},[31685,31821],[34,584272,584274,584277],{"className":584273},[31685,31821],[34,584275,94628],{"className":584276},[31685,31686,31821],[34,584278,584280],{"className":584279},[31792],[34,584281,584283],{"className":584282},[31796],[34,584284,584286],{"className":584285},[31801],[34,584287,584289],{"className":584288,"style":144123},[31805],[34,584290,584291,584294],{"style":552391},[34,584292],{"className":584293,"style":43406},[31813],[34,584295,584297],{"className":584296},[31818,43410,43411,31821],[34,584298,768],{"className":584299},[31685,31821],[34,584301,584303,584306],{"className":584302},[31685],[34,584304,94628],{"className":584305},[31685,31686],[34,584307,584309],{"className":584308},[31792],[34,584310,584312],{"className":584311},[31796],[34,584313,584315],{"className":584314},[31801],[34,584316,584318],{"className":584317,"style":37774},[31805],[34,584319,584320,584323],{"style":37777},[34,584321],{"className":584322,"style":31814},[31813],[34,584324,584326],{"className":584325},[31818,31819,31820,31821],[34,584327,584329,584332,584335,584338],{"className":584328},[31685,31821],[34,584330,768],{"className":584331},[31685,31821],[34,584333,15],{"className":584334},[31685,31686,31821],[34,584336,32226],{"className":584337},[32024,31821],[34,584339,640],{"className":584340},[31685,31821],[34,584342,46900],{"className":584343},[31685,31686],[34,584345,94628],{"className":584346},[31685,31686],".\n同理 ",[34,584349,584351,584419],{"className":584350},[31642],[34,584352,584354],{"className":584353},[31646],[31648,584355,584356],{"xmlns":31650},[31652,584357,584358,584416],{},[31655,584359,584360,584362,584364,584366,584368,584370,584372,584384,584398,584412,584414],{},[31658,584361,564791],{"mathvariant":38079},[31662,584363,456],{"stretchy":702},[31658,584365,38759],{},[31662,584367,2557],{"stretchy":702},[31662,584369,202],{},[31749,584371,768],{},[38846,584373,584374,584376,584378],{},[31662,584375,546295],{},[31749,584377,459],{},[31655,584379,584380,584382],{},[31662,584381,223],{},[31658,584383,138286],{"mathvariant":38079},[37624,584385,584386,584388],{},[31658,584387,33244],{},[31655,584389,584390,584392],{},[31662,584391,32226],{},[37624,584393,584394,584396],{},[31658,584395,56782],{},[31749,584397,768],{},[37624,584399,584400,584402],{},[31658,584401,56782],{},[31655,584403,584404,584406,584408,584410],{},[31749,584405,768],{},[31658,584407,38759],{},[31662,584409,32226],{},[31749,584411,640],{},[31658,584413,46900],{},[31658,584415,56782],{},[31665,584417,584418],{"encoding":31667},"\\Gamma(q) = 2 \\int_{0}^{+\\infty} e^{-v^2} v^{2q-1} dv",[34,584420,584422,584449],{"className":584421,"ariaHidden":758},[31672],[34,584423,584425,584428,584431,584434,584437,584440,584443,584446],{"className":584424},[31676],[34,584426],{"className":584427,"style":31774},[31680],[34,584429,564791],{"className":584430},[31685],[34,584432,456],{"className":584433},[31782],[34,584435,38759],{"className":584436,"style":32062},[31685,31686],[34,584438,2557],{"className":584439},[31690],[34,584441],{"className":584442,"style":31846},[31845],[34,584444,202],{"className":584445},[31850],[34,584447],{"className":584448,"style":31846},[31845],[34,584450,584452,584455,584458,584461,584521,584524,584585,584626,584629],{"className":584451},[31676],[34,584453],{"className":584454,"style":583948},[31680],[34,584456,768],{"className":584457},[31685],[34,584459],{"className":584460,"style":32123},[31845],[34,584462,584464,584467],{"className":584463},[32767],[34,584465,546295],{"className":584466,"style":546331},[32767,131249,156560],[34,584468,584470],{"className":584469},[31792],[34,584471,584473,584513],{"className":584472},[31796,31797],[34,584474,584476,584510],{"className":584475},[31801],[34,584477,584479,584493],{"className":584478,"style":546550},[31805],[34,584480,584481,584484],{"style":546347},[34,584482],{"className":584483,"style":31814},[31813],[34,584485,584487],{"className":584486},[31818,31819,31820,31821],[34,584488,584490],{"className":584489},[31685,31821],[34,584491,459],{"className":584492},[31685,31821],[34,584494,584495,584498],{"style":546362},[34,584496],{"className":584497,"style":31814},[31813],[34,584499,584501],{"className":584500},[31818,31819,31820,31821],[34,584502,584504,584507],{"className":584503},[31685,31821],[34,584505,223],{"className":584506},[31685,31821],[34,584508,138286],{"className":584509},[31685,31821],[34,584511,31829],{"className":584512},[31828],[34,584514,584516],{"className":584515},[31801],[34,584517,584519],{"className":584518,"style":546384},[31805],[34,584520],{},[34,584522],{"className":584523,"style":32123},[31845],[34,584525,584527,584530],{"className":584526},[31685],[34,584528,33244],{"className":584529},[31685,31686],[34,584531,584533],{"className":584532},[31792],[34,584534,584536],{"className":584535},[31796],[34,584537,584539],{"className":584538},[31801],[34,584540,584542],{"className":584541,"style":562030},[31805],[34,584543,584544,584547],{"style":37777},[34,584545],{"className":584546,"style":31814},[31813],[34,584548,584550],{"className":584549},[31818,31819,31820,31821],[34,584551,584553,584556],{"className":584552},[31685,31821],[34,584554,32226],{"className":584555},[31685,31821],[34,584557,584559,584562],{"className":584558},[31685,31821],[34,584560,56782],{"className":584561,"style":32062},[31685,31686,31821],[34,584563,584565],{"className":584564},[31792],[34,584566,584568],{"className":584567},[31796],[34,584569,584571],{"className":584570},[31801],[34,584572,584574],{"className":584573,"style":144123},[31805],[34,584575,584576,584579],{"style":552391},[34,584577],{"className":584578,"style":43406},[31813],[34,584580,584582],{"className":584581},[31818,43410,43411,31821],[34,584583,768],{"className":584584},[31685,31821],[34,584586,584588,584591],{"className":584587},[31685],[34,584589,56782],{"className":584590,"style":32062},[31685,31686],[34,584592,584594],{"className":584593},[31792],[34,584595,584597],{"className":584596},[31796],[34,584598,584600],{"className":584599},[31801],[34,584601,584603],{"className":584602,"style":37774},[31805],[34,584604,584605,584608],{"style":37777},[34,584606],{"className":584607,"style":31814},[31813],[34,584609,584611],{"className":584610},[31818,31819,31820,31821],[34,584612,584614,584617,584620,584623],{"className":584613},[31685,31821],[34,584615,768],{"className":584616},[31685,31821],[34,584618,38759],{"className":584619,"style":32062},[31685,31686,31821],[34,584621,32226],{"className":584622},[32024,31821],[34,584624,640],{"className":584625},[31685,31821],[34,584627,46900],{"className":584628},[31685,31686],[34,584630,56782],{"className":584631,"style":32062},[31685,31686],[108,584633,584634,584637],{},[2627,584635,584636],{},"构造乘积：",[34,584638,584640,584758],{"className":584639},[31642],[34,584641,584643],{"className":584642},[31646],[31648,584644,584645],{"xmlns":31650},[31652,584646,584647,584755],{},[31655,584648,584649,584651,584653,584655,584657,584659,584661,584663,584665,584667,584669,584681,584693,584719,584733,584747,584749,584751,584753],{},[31658,584650,564791],{"mathvariant":38079},[31662,584652,456],{"stretchy":702},[31658,584654,15],{},[31662,584656,2557],{"stretchy":702},[31658,584658,564791],{"mathvariant":38079},[31662,584660,456],{"stretchy":702},[31658,584662,38759],{},[31662,584664,2557],{"stretchy":702},[31662,584666,202],{},[31749,584668,15932],{},[38846,584670,584671,584673,584675],{},[31662,584672,546295],{},[31749,584674,459],{},[31655,584676,584677,584679],{},[31662,584678,223],{},[31658,584680,138286],{"mathvariant":38079},[38846,584682,584683,584685,584687],{},[31662,584684,546295],{},[31749,584686,459],{},[31655,584688,584689,584691],{},[31662,584690,223],{},[31658,584692,138286],{"mathvariant":38079},[37624,584694,584695,584697],{},[31658,584696,33244],{},[31655,584698,584699,584701,584703,584709,584711,584717],{},[31662,584700,32226],{},[31662,584702,456],{"stretchy":702},[37624,584704,584705,584707],{},[31658,584706,94628],{},[31749,584708,768],{},[31662,584710,223],{},[37624,584712,584713,584715],{},[31658,584714,56782],{},[31749,584716,768],{},[31662,584718,2557],{"stretchy":702},[37624,584720,584721,584723],{},[31658,584722,94628],{},[31655,584724,584725,584727,584729,584731],{},[31749,584726,768],{},[31658,584728,15],{},[31662,584730,32226],{},[31749,584732,640],{},[37624,584734,584735,584737],{},[31658,584736,56782],{},[31655,584738,584739,584741,584743,584745],{},[31749,584740,768],{},[31658,584742,38759],{},[31662,584744,32226],{},[31749,584746,640],{},[31658,584748,46900],{},[31658,584750,94628],{},[31658,584752,46900],{},[31658,584754,56782],{},[31665,584756,584757],{"encoding":31667},"\\Gamma(p)\\Gamma(q) = 4 \\int_{0}^{+\\infty} \\int_{0}^{+\\infty} e^{-(u^2+v^2)} u^{2p-1} v^{2q-1} du dv",[34,584759,584761,584800],{"className":584760,"ariaHidden":758},[31672],[34,584762,584764,584767,584770,584773,584776,584779,584782,584785,584788,584791,584794,584797],{"className":584763},[31676],[34,584765],{"className":584766,"style":31774},[31680],[34,584768,564791],{"className":584769},[31685],[34,584771,456],{"className":584772},[31782],[34,584774,15],{"className":584775},[31685,31686],[34,584777,2557],{"className":584778},[31690],[34,584780,564791],{"className":584781},[31685],[34,584783,456],{"className":584784},[31782],[34,584786,38759],{"className":584787,"style":32062},[31685,31686],[34,584789,2557],{"className":584790},[31690],[34,584792],{"className":584793,"style":31846},[31845],[34,584795,202],{"className":584796},[31850],[34,584798],{"className":584799,"style":31846},[31845],[34,584801,584803,584806,584809,584812,584872,584875,584935,584938,585037,585078,585119,585122,585125,585128],{"className":584802},[31676],[34,584804],{"className":584805,"style":583948},[31680],[34,584807,15932],{"className":584808},[31685],[34,584810],{"className":584811,"style":32123},[31845],[34,584813,584815,584818],{"className":584814},[32767],[34,584816,546295],{"className":584817,"style":546331},[32767,131249,156560],[34,584819,584821],{"className":584820},[31792],[34,584822,584824,584864],{"className":584823},[31796,31797],[34,584825,584827,584861],{"className":584826},[31801],[34,584828,584830,584844],{"className":584829,"style":546550},[31805],[34,584831,584832,584835],{"style":546347},[34,584833],{"className":584834,"style":31814},[31813],[34,584836,584838],{"className":584837},[31818,31819,31820,31821],[34,584839,584841],{"className":584840},[31685,31821],[34,584842,459],{"className":584843},[31685,31821],[34,584845,584846,584849],{"style":546362},[34,584847],{"className":584848,"style":31814},[31813],[34,584850,584852],{"className":584851},[31818,31819,31820,31821],[34,584853,584855,584858],{"className":584854},[31685,31821],[34,584856,223],{"className":584857},[31685,31821],[34,584859,138286],{"className":584860},[31685,31821],[34,584862,31829],{"className":584863},[31828],[34,584865,584867],{"className":584866},[31801],[34,584868,584870],{"className":584869,"style":546384},[31805],[34,584871],{},[34,584873],{"className":584874,"style":32123},[31845],[34,584876,584878,584881],{"className":584877},[32767],[34,584879,546295],{"className":584880,"style":546331},[32767,131249,156560],[34,584882,584884],{"className":584883},[31792],[34,584885,584887,584927],{"className":584886},[31796,31797],[34,584888,584890,584924],{"className":584889},[31801],[34,584891,584893,584907],{"className":584892,"style":546550},[31805],[34,584894,584895,584898],{"style":546347},[34,584896],{"className":584897,"style":31814},[31813],[34,584899,584901],{"className":584900},[31818,31819,31820,31821],[34,584902,584904],{"className":584903},[31685,31821],[34,584905,459],{"className":584906},[31685,31821],[34,584908,584909,584912],{"style":546362},[34,584910],{"className":584911,"style":31814},[31813],[34,584913,584915],{"className":584914},[31818,31819,31820,31821],[34,584916,584918,584921],{"className":584917},[31685,31821],[34,584919,223],{"className":584920},[31685,31821],[34,584922,138286],{"className":584923},[31685,31821],[34,584925,31829],{"className":584926},[31828],[34,584928,584930],{"className":584929},[31801],[34,584931,584933],{"className":584932,"style":546384},[31805],[34,584934],{},[34,584936],{"className":584937,"style":32123},[31845],[34,584939,584941,584944],{"className":584940},[31685],[34,584942,33244],{"className":584943},[31685,31686],[34,584945,584947],{"className":584946},[31792],[34,584948,584950],{"className":584949},[31796],[34,584951,584953],{"className":584952},[31801],[34,584954,584956],{"className":584955,"style":562030},[31805],[34,584957,584958,584961],{"style":37777},[34,584959],{"className":584960,"style":31814},[31813],[34,584962,584964],{"className":584963},[31818,31819,31820,31821],[34,584965,584967,584970,584973,585002,585005,585034],{"className":584966},[31685,31821],[34,584968,32226],{"className":584969},[31685,31821],[34,584971,456],{"className":584972},[31782,31821],[34,584974,584976,584979],{"className":584975},[31685,31821],[34,584977,94628],{"className":584978},[31685,31686,31821],[34,584980,584982],{"className":584981},[31792],[34,584983,584985],{"className":584984},[31796],[34,584986,584988],{"className":584987},[31801],[34,584989,584991],{"className":584990,"style":144123},[31805],[34,584992,584993,584996],{"style":552391},[34,584994],{"className":584995,"style":43406},[31813],[34,584997,584999],{"className":584998},[31818,43410,43411,31821],[34,585000,768],{"className":585001},[31685,31821],[34,585003,223],{"className":585004},[32024,31821],[34,585006,585008,585011],{"className":585007},[31685,31821],[34,585009,56782],{"className":585010,"style":32062},[31685,31686,31821],[34,585012,585014],{"className":585013},[31792],[34,585015,585017],{"className":585016},[31796],[34,585018,585020],{"className":585019},[31801],[34,585021,585023],{"className":585022,"style":144123},[31805],[34,585024,585025,585028],{"style":552391},[34,585026],{"className":585027,"style":43406},[31813],[34,585029,585031],{"className":585030},[31818,43410,43411,31821],[34,585032,768],{"className":585033},[31685,31821],[34,585035,2557],{"className":585036},[31690,31821],[34,585038,585040,585043],{"className":585039},[31685],[34,585041,94628],{"className":585042},[31685,31686],[34,585044,585046],{"className":585045},[31792],[34,585047,585049],{"className":585048},[31796],[34,585050,585052],{"className":585051},[31801],[34,585053,585055],{"className":585054,"style":37774},[31805],[34,585056,585057,585060],{"style":37777},[34,585058],{"className":585059,"style":31814},[31813],[34,585061,585063],{"className":585062},[31818,31819,31820,31821],[34,585064,585066,585069,585072,585075],{"className":585065},[31685,31821],[34,585067,768],{"className":585068},[31685,31821],[34,585070,15],{"className":585071},[31685,31686,31821],[34,585073,32226],{"className":585074},[32024,31821],[34,585076,640],{"className":585077},[31685,31821],[34,585079,585081,585084],{"className":585080},[31685],[34,585082,56782],{"className":585083,"style":32062},[31685,31686],[34,585085,585087],{"className":585086},[31792],[34,585088,585090],{"className":585089},[31796],[34,585091,585093],{"className":585092},[31801],[34,585094,585096],{"className":585095,"style":37774},[31805],[34,585097,585098,585101],{"style":37777},[34,585099],{"className":585100,"style":31814},[31813],[34,585102,585104],{"className":585103},[31818,31819,31820,31821],[34,585105,585107,585110,585113,585116],{"className":585106},[31685,31821],[34,585108,768],{"className":585109},[31685,31821],[34,585111,38759],{"className":585112,"style":32062},[31685,31686,31821],[34,585114,32226],{"className":585115},[32024,31821],[34,585117,640],{"className":585118},[31685,31821],[34,585120,46900],{"className":585121},[31685,31686],[34,585123,94628],{"className":585124},[31685,31686],[34,585126,46900],{"className":585127},[31685,31686],[34,585129,56782],{"className":585130,"style":32062},[31685,31686],[108,585132,585133,478285,585136,585258,585259,39984,585403,585489,585490],{},[2627,585134,585135],{},"转换为极坐标：",[34,585137,585139,585177],{"className":585138},[31642],[34,585140,585142],{"className":585141},[31646],[31648,585143,585144],{"xmlns":31650},[31652,585145,585146,585174],{},[31655,585147,585148,585150,585152,585154,585156,585158,585160,585162,585164,585166,585168,585170,585172],{},[31658,585149,94628],{},[31662,585151,202],{},[31658,585153,34395],{},[31658,585155,547860],{},[31662,585157,32682],{},[31658,585159,479697],{},[31662,585161,31960],{"separator":758},[31658,585163,56782],{},[31662,585165,202],{},[31658,585167,34395],{},[31658,585169,547853],{},[31662,585171,32682],{},[31658,585173,479697],{},[31665,585175,585176],{"encoding":31667},"u = r \\cos \\theta, v = r \\sin \\theta",[34,585178,585180,585198,585237],{"className":585179,"ariaHidden":758},[31672],[34,585181,585183,585186,585189,585192,585195],{"className":585182},[31676],[34,585184],{"className":585185,"style":32527},[31680],[34,585187,94628],{"className":585188},[31685,31686],[34,585190],{"className":585191,"style":31846},[31845],[34,585193,202],{"className":585194},[31850],[34,585196],{"className":585197,"style":31846},[31845],[34,585199,585201,585204,585207,585210,585213,585216,585219,585222,585225,585228,585231,585234],{"className":585200},[31676],[34,585202],{"className":585203,"style":43217},[31680],[34,585205,34395],{"className":585206,"style":34452},[31685,31686],[34,585208],{"className":585209,"style":32123},[31845],[34,585211,547860],{"className":585212},[32767],[34,585214],{"className":585215,"style":32123},[31845],[34,585217,479697],{"className":585218,"style":34452},[31685,31686],[34,585220,31960],{"className":585221},[32119],[34,585223],{"className":585224,"style":32123},[31845],[34,585226,56782],{"className":585227,"style":32062},[31685,31686],[34,585229],{"className":585230,"style":31846},[31845],[34,585232,202],{"className":585233},[31850],[34,585235],{"className":585236,"style":31846},[31845],[34,585238,585240,585243,585246,585249,585252,585255],{"className":585239},[31676],[34,585241],{"className":585242,"style":31681},[31680],[34,585244,34395],{"className":585245,"style":34452},[31685,31686],[34,585247],{"className":585248,"style":32123},[31845],[34,585250,547853],{"className":585251},[32767],[34,585253],{"className":585254,"style":32123},[31845],[34,585256,479697],{"className":585257,"style":34452},[31685,31686],"。\n区域是第一象限：",[34,585260,585262,585310],{"className":585261},[31642],[34,585263,585265],{"className":585264},[31646],[31648,585266,585267],{"xmlns":31650},[31652,585268,585269,585307],{},[31655,585270,585271,585273,585275,585277,585279,585281,585283,585285,585287,585289,585291,585293,585295,585297,585299,585301,585303,585305],{},[31658,585272,34395],{},[31662,585274,112296],{},[31662,585276,15972],{"stretchy":702},[31749,585278,459],{},[31662,585280,31960],{"separator":758},[31662,585282,223],{},[31658,585284,138286],{"mathvariant":38079},[31662,585286,2557],{"stretchy":702},[31662,585288,31960],{"separator":758},[31658,585290,479697],{},[31662,585292,112296],{},[31662,585294,15972],{"stretchy":702},[31749,585296,459],{},[31662,585298,31960],{"separator":758},[31658,585300,58442],{},[31658,585302,38080],{"mathvariant":38079},[31749,585304,768],{},[31662,585306,32712],{"stretchy":702},[31665,585308,585309],{"encoding":31667},"r \\in [0, +\\infty), \\theta \\in [0, \\pi\u002F2]",[34,585311,585313,585331,585376],{"className":585312,"ariaHidden":758},[31672],[34,585314,585316,585319,585322,585325,585328],{"className":585315},[31676],[34,585317],{"className":585318,"style":34812},[31680],[34,585320,34395],{"className":585321,"style":34452},[31685,31686],[34,585323],{"className":585324,"style":31846},[31845],[34,585326,112296],{"className":585327},[31850],[34,585329],{"className":585330,"style":31846},[31845],[34,585332,585334,585337,585340,585343,585346,585349,585352,585355,585358,585361,585364,585367,585370,585373],{"className":585333},[31676],[34,585335],{"className":585336,"style":31774},[31680],[34,585338,15972],{"className":585339},[31782],[34,585341,459],{"className":585342},[31685],[34,585344,31960],{"className":585345},[32119],[34,585347],{"className":585348,"style":32123},[31845],[34,585350,223],{"className":585351},[31685],[34,585353,138286],{"className":585354},[31685],[34,585356,2557],{"className":585357},[31690],[34,585359,31960],{"className":585360},[32119],[34,585362],{"className":585363,"style":32123},[31845],[34,585365,479697],{"className":585366,"style":34452},[31685,31686],[34,585368],{"className":585369,"style":31846},[31845],[34,585371,112296],{"className":585372},[31850],[34,585374],{"className":585375,"style":31846},[31845],[34,585377,585379,585382,585385,585388,585391,585394,585397,585400],{"className":585378},[31676],[34,585380],{"className":585381,"style":31774},[31680],[34,585383,15972],{"className":585384},[31782],[34,585386,459],{"className":585387},[31685],[34,585389,31960],{"className":585390},[32119],[34,585392],{"className":585393,"style":32123},[31845],[34,585395,58442],{"className":585396,"style":32062},[31685,31686],[34,585398,38149],{"className":585399},[31685],[34,585401,32712],{"className":585402},[31690],[34,585404,585406,585438],{"className":585405},[31642],[34,585407,585409],{"className":585408},[31646],[31648,585410,585411],{"xmlns":31650},[31652,585412,585413,585435],{},[31655,585414,585415,585417,585419,585421,585423,585425,585427,585429,585431,585433],{},[31658,585416,46900],{},[31658,585418,94628],{},[31658,585420,46900],{},[31658,585422,56782],{},[31662,585424,202],{},[31658,585426,34395],{},[31658,585428,46900],{},[31658,585430,34395],{},[31658,585432,46900],{},[31658,585434,479697],{},[31665,585436,585437],{"encoding":31667},"du dv = r dr d\\theta",[34,585439,585441,585468],{"className":585440,"ariaHidden":758},[31672],[34,585442,585444,585447,585450,585453,585456,585459,585462,585465],{"className":585443},[31676],[34,585445],{"className":585446,"style":31681},[31680],[34,585448,46900],{"className":585449},[31685,31686],[34,585451,94628],{"className":585452},[31685,31686],[34,585454,46900],{"className":585455},[31685,31686],[34,585457,56782],{"className":585458,"style":32062},[31685,31686],[34,585460],{"className":585461,"style":31846},[31845],[34,585463,202],{"className":585464},[31850],[34,585466],{"className":585467,"style":31846},[31845],[34,585469,585471,585474,585477,585480,585483,585486],{"className":585470},[31676],[34,585472],{"className":585473,"style":31681},[31680],[34,585475,34395],{"className":585476,"style":34452},[31685,31686],[34,585478,46900],{"className":585479},[31685,31686],[34,585481,34395],{"className":585482,"style":34452},[31685,31686],[34,585484,46900],{"className":585485},[31685,31686],[34,585487,479697],{"className":585488,"style":34452},[31685,31686],"。\n代入表达式：\n",[34,585491,585493,585645],{"className":585492},[31642],[34,585494,585496],{"className":585495},[31646],[31648,585497,585498],{"xmlns":31650},[31652,585499,585500,585642],{},[31655,585501,585502,585516,585530,585532,585534,585536,585538,585540,585542,585556,585558,585560,585562,585564,585566,585580,585582,585602,585620,585622,585640],{},[37624,585503,585504,585506],{},[31658,585505,94628],{},[31655,585507,585508,585510,585512,585514],{},[31749,585509,768],{},[31658,585511,15],{},[31662,585513,32226],{},[31749,585515,640],{},[37624,585517,585518,585520],{},[31658,585519,56782],{},[31655,585521,585522,585524,585526,585528],{},[31749,585523,768],{},[31658,585525,38759],{},[31662,585527,32226],{},[31749,585529,640],{},[31662,585531,202],{},[31662,585533,456],{"stretchy":702},[31658,585535,34395],{},[31658,585537,547860],{},[31662,585539,32682],{},[31658,585541,479697],{},[37624,585543,585544,585546],{},[31662,585545,2557],{"stretchy":702},[31655,585547,585548,585550,585552,585554],{},[31749,585549,768],{},[31658,585551,15],{},[31662,585553,32226],{},[31749,585555,640],{},[31662,585557,456],{"stretchy":702},[31658,585559,34395],{},[31658,585561,547853],{},[31662,585563,32682],{},[31658,585565,479697],{},[37624,585567,585568,585570],{},[31662,585569,2557],{"stretchy":702},[31655,585571,585572,585574,585576,585578],{},[31749,585573,768],{},[31658,585575,38759],{},[31662,585577,32226],{},[31749,585579,640],{},[31662,585581,202],{},[37624,585583,585584,585586],{},[31658,585585,34395],{},[31655,585587,585588,585590,585592,585594,585596,585598,585600],{},[31749,585589,768],{},[31658,585591,15],{},[31662,585593,223],{},[31749,585595,768],{},[31658,585597,38759],{},[31662,585599,32226],{},[31749,585601,768],{},[37624,585603,585604,585610],{},[31655,585605,585606,585608],{},[31658,585607,547860],{},[31662,585609,32682],{},[31655,585611,585612,585614,585616,585618],{},[31749,585613,768],{},[31658,585615,15],{},[31662,585617,32226],{},[31749,585619,640],{},[31658,585621,479697],{},[37624,585623,585624,585630],{},[31655,585625,585626,585628],{},[31658,585627,547853],{},[31662,585629,32682],{},[31655,585631,585632,585634,585636,585638],{},[31749,585633,768],{},[31658,585635,38759],{},[31662,585637,32226],{},[31749,585639,640],{},[31658,585641,479697],{},[31665,585643,585644],{"encoding":31667},"u^{2p-1} v^{2q-1} = (r \\cos \\theta)^{2p-1} (r \\sin \\theta)^{2q-1} = r^{2p+2q-2} \\cos^{2p-1}\\theta \\sin^{2q-1}\\theta",[34,585646,585648,585745,585878],{"className":585647,"ariaHidden":758},[31672],[34,585649,585651,585654,585695,585736,585739,585742],{"className":585650},[31676],[34,585652],{"className":585653,"style":37774},[31680],[34,585655,585657,585660],{"className":585656},[31685],[34,585658,94628],{"className":585659},[31685,31686],[34,585661,585663],{"className":585662},[31792],[34,585664,585666],{"className":585665},[31796],[34,585667,585669],{"className":585668},[31801],[34,585670,585672],{"className":585671,"style":37774},[31805],[34,585673,585674,585677],{"style":37777},[34,585675],{"className":585676,"style":31814},[31813],[34,585678,585680],{"className":585679},[31818,31819,31820,31821],[34,585681,585683,585686,585689,585692],{"className":585682},[31685,31821],[34,585684,768],{"className":585685},[31685,31821],[34,585687,15],{"className":585688},[31685,31686,31821],[34,585690,32226],{"className":585691},[32024,31821],[34,585693,640],{"className":585694},[31685,31821],[34,585696,585698,585701],{"className":585697},[31685],[34,585699,56782],{"className":585700,"style":32062},[31685,31686],[34,585702,585704],{"className":585703},[31792],[34,585705,585707],{"className":585706},[31796],[34,585708,585710],{"className":585709},[31801],[34,585711,585713],{"className":585712,"style":37774},[31805],[34,585714,585715,585718],{"style":37777},[34,585716],{"className":585717,"style":31814},[31813],[34,585719,585721],{"className":585720},[31818,31819,31820,31821],[34,585722,585724,585727,585730,585733],{"className":585723},[31685,31821],[34,585725,768],{"className":585726},[31685,31821],[34,585728,38759],{"className":585729,"style":32062},[31685,31686,31821],[34,585731,32226],{"className":585732},[32024,31821],[34,585734,640],{"className":585735},[31685,31821],[34,585737],{"className":585738,"style":31846},[31845],[34,585740,202],{"className":585741},[31850],[34,585743],{"className":585744,"style":31846},[31845],[34,585746,585748,585751,585754,585757,585760,585763,585766,585769,585810,585813,585816,585819,585822,585825,585828,585869,585872,585875],{"className":585747},[31676],[34,585749],{"className":585750,"style":37755},[31680],[34,585752,456],{"className":585753},[31782],[34,585755,34395],{"className":585756,"style":34452},[31685,31686],[34,585758],{"className":585759,"style":32123},[31845],[34,585761,547860],{"className":585762},[32767],[34,585764],{"className":585765,"style":32123},[31845],[34,585767,479697],{"className":585768,"style":34452},[31685,31686],[34,585770,585772,585775],{"className":585771},[31690],[34,585773,2557],{"className":585774},[31690],[34,585776,585778],{"className":585777},[31792],[34,585779,585781],{"className":585780},[31796],[34,585782,585784],{"className":585783},[31801],[34,585785,585787],{"className":585786,"style":37774},[31805],[34,585788,585789,585792],{"style":37777},[34,585790],{"className":585791,"style":31814},[31813],[34,585793,585795],{"className":585794},[31818,31819,31820,31821],[34,585796,585798,585801,585804,585807],{"className":585797},[31685,31821],[34,585799,768],{"className":585800},[31685,31821],[34,585802,15],{"className":585803},[31685,31686,31821],[34,585805,32226],{"className":585806},[32024,31821],[34,585808,640],{"className":585809},[31685,31821],[34,585811,456],{"className":585812},[31782],[34,585814,34395],{"className":585815,"style":34452},[31685,31686],[34,585817],{"className":585818,"style":32123},[31845],[34,585820,547853],{"className":585821},[32767],[34,585823],{"className":585824,"style":32123},[31845],[34,585826,479697],{"className":585827,"style":34452},[31685,31686],[34,585829,585831,585834],{"className":585830},[31690],[34,585832,2557],{"className":585833},[31690],[34,585835,585837],{"className":585836},[31792],[34,585838,585840],{"className":585839},[31796],[34,585841,585843],{"className":585842},[31801],[34,585844,585846],{"className":585845,"style":37774},[31805],[34,585847,585848,585851],{"style":37777},[34,585849],{"className":585850,"style":31814},[31813],[34,585852,585854],{"className":585853},[31818,31819,31820,31821],[34,585855,585857,585860,585863,585866],{"className":585856},[31685,31821],[34,585858,768],{"className":585859},[31685,31821],[34,585861,38759],{"className":585862,"style":32062},[31685,31686,31821],[34,585864,32226],{"className":585865},[32024,31821],[34,585867,640],{"className":585868},[31685,31821],[34,585870],{"className":585871,"style":31846},[31845],[34,585873,202],{"className":585874},[31850],[34,585876],{"className":585877,"style":31846},[31845],[34,585879,585881,585885,585935,585938,585979,585982,585985,585988,586030,586033],{"className":585880},[31676],[34,585882],{"className":585883,"style":585884},[31680],"height:0.8719em;",[34,585886,585888,585891],{"className":585887},[31685],[34,585889,34395],{"className":585890,"style":34452},[31685,31686],[34,585892,585894],{"className":585893},[31792],[34,585895,585897],{"className":585896},[31796],[34,585898,585900],{"className":585899},[31801],[34,585901,585903],{"className":585902,"style":37774},[31805],[34,585904,585905,585908],{"style":37777},[34,585906],{"className":585907,"style":31814},[31813],[34,585909,585911],{"className":585910},[31818,31819,31820,31821],[34,585912,585914,585917,585920,585923,585926,585929,585932],{"className":585913},[31685,31821],[34,585915,768],{"className":585916},[31685,31821],[34,585918,15],{"className":585919},[31685,31686,31821],[34,585921,223],{"className":585922},[32024,31821],[34,585924,768],{"className":585925},[31685,31821],[34,585927,38759],{"className":585928,"style":32062},[31685,31686,31821],[34,585930,32226],{"className":585931},[32024,31821],[34,585933,768],{"className":585934},[31685,31821],[34,585936],{"className":585937,"style":32123},[31845],[34,585939,585941,585944],{"className":585940},[32767],[34,585942,547860],{"className":585943},[32767],[34,585945,585947],{"className":585946},[31792],[34,585948,585950],{"className":585949},[31796],[34,585951,585953],{"className":585952},[31801],[34,585954,585956],{"className":585955,"style":37774},[31805],[34,585957,585958,585961],{"style":37777},[34,585959],{"className":585960,"style":31814},[31813],[34,585962,585964],{"className":585963},[31818,31819,31820,31821],[34,585965,585967,585970,585973,585976],{"className":585966},[31685,31821],[34,585968,768],{"className":585969},[31685,31821],[34,585971,15],{"className":585972},[31685,31686,31821],[34,585974,32226],{"className":585975},[32024,31821],[34,585977,640],{"className":585978},[31685,31821],[34,585980],{"className":585981,"style":32123},[31845],[34,585983,479697],{"className":585984,"style":34452},[31685,31686],[34,585986],{"className":585987,"style":32123},[31845],[34,585989,585991,585994],{"className":585990},[32767],[34,585992,547853],{"className":585993},[32767],[34,585995,585997],{"className":585996},[31792],[34,585998,586000],{"className":585999},[31796],[34,586001,586003],{"className":586002},[31801],[34,586004,586006],{"className":586005,"style":585884},[31805],[34,586007,586009,586012],{"style":586008},"top:-3.1208em;margin-right:0.05em;",[34,586010],{"className":586011,"style":31814},[31813],[34,586013,586015],{"className":586014},[31818,31819,31820,31821],[34,586016,586018,586021,586024,586027],{"className":586017},[31685,31821],[34,586019,768],{"className":586020},[31685,31821],[34,586022,38759],{"className":586023,"style":32062},[31685,31686,31821],[34,586025,32226],{"className":586026},[32024,31821],[34,586028,640],{"className":586029},[31685,31821],[34,586031],{"className":586032,"style":32123},[31845],[34,586034,479697],{"className":586035,"style":34452},[31685,31686],[108,586037,586038,586041],{},[2627,586039,586040],{},"分离积分：",[34,586042,586044,586198],{"className":586043},[31642],[34,586045,586047],{"className":586046},[31646],[31648,586048,586049],{"xmlns":31650},[31652,586050,586051,586195],{},[31655,586052,586053,586055,586057,586059,586061,586063,586065,586067,586069,586071,586073,586131],{},[31658,586054,564791],{"mathvariant":38079},[31662,586056,456],{"stretchy":702},[31658,586058,15],{},[31662,586060,2557],{"stretchy":702},[31658,586062,564791],{"mathvariant":38079},[31662,586064,456],{"stretchy":702},[31658,586066,38759],{},[31662,586068,2557],{"stretchy":702},[31662,586070,202],{},[31749,586072,15932],{},[31655,586074,586075,586077,586089,586103,586125,586127,586129],{},[31662,586076,456],{"fence":758},[38846,586078,586079,586081,586083],{},[31662,586080,546295],{},[31749,586082,459],{},[31655,586084,586085,586087],{},[31662,586086,223],{},[31658,586088,138286],{"mathvariant":38079},[37624,586090,586091,586093],{},[31658,586092,33244],{},[31655,586094,586095,586097],{},[31662,586096,32226],{},[37624,586098,586099,586101],{},[31658,586100,34395],{},[31749,586102,768],{},[37624,586104,586105,586107],{},[31658,586106,34395],{},[31655,586108,586109,586111,586113,586115,586117,586119,586121,586123],{},[31749,586110,768],{},[31662,586112,456],{"stretchy":702},[31658,586114,15],{},[31662,586116,223],{},[31658,586118,38759],{},[31662,586120,2557],{"stretchy":702},[31662,586122,32226],{},[31749,586124,640],{},[31658,586126,46900],{},[31658,586128,34395],{},[31662,586130,2557],{"fence":758},[31655,586132,586133,586135,586149,586167,586169,586187,586189,586191,586193],{},[31662,586134,456],{"fence":758},[38846,586136,586137,586139,586141],{},[31662,586138,546295],{},[31749,586140,459],{},[31655,586142,586143,586145,586147],{},[31658,586144,58442],{},[31658,586146,38080],{"mathvariant":38079},[31749,586148,768],{},[37624,586150,586151,586157],{},[31655,586152,586153,586155],{},[31658,586154,547860],{},[31662,586156,32682],{},[31655,586158,586159,586161,586163,586165],{},[31749,586160,768],{},[31658,586162,15],{},[31662,586164,32226],{},[31749,586166,640],{},[31658,586168,479697],{},[37624,586170,586171,586177],{},[31655,586172,586173,586175],{},[31658,586174,547853],{},[31662,586176,32682],{},[31655,586178,586179,586181,586183,586185],{},[31749,586180,768],{},[31658,586182,38759],{},[31662,586184,32226],{},[31749,586186,640],{},[31658,586188,479697],{},[31658,586190,46900],{},[31658,586192,479697],{},[31662,586194,2557],{"fence":758},[31665,586196,586197],{"encoding":31667},"\\Gamma(p)\\Gamma(q) = 4 \\left( \\int_{0}^{+\\infty} e^{-r^2} r^{2(p+q)-1} dr \\right) \\left( \\int_{0}^{\\pi\u002F2} \\cos^{2p-1}\\theta \\sin^{2q-1}\\theta d\\theta \\right)",[34,586199,586201,586240],{"className":586200,"ariaHidden":758},[31672],[34,586202,586204,586207,586210,586213,586216,586219,586222,586225,586228,586231,586234,586237],{"className":586203},[31676],[34,586205],{"className":586206,"style":31774},[31680],[34,586208,564791],{"className":586209},[31685],[34,586211,456],{"className":586212},[31782],[34,586214,15],{"className":586215},[31685,31686],[34,586217,2557],{"className":586218},[31690],[34,586220,564791],{"className":586221},[31685],[34,586223,456],{"className":586224},[31782],[34,586226,38759],{"className":586227,"style":32062},[31685,31686],[34,586229,2557],{"className":586230},[31690],[34,586232],{"className":586233,"style":31846},[31845],[34,586235,202],{"className":586236},[31850],[34,586238],{"className":586239,"style":31846},[31845],[34,586241,586243,586246,586249,586252,586450,586453],{"className":586242},[31676],[34,586244],{"className":586245,"style":141282},[31680],[34,586247,15932],{"className":586248},[31685],[34,586250],{"className":586251,"style":32123},[31845],[34,586253,586255,586261,586321,586324,586385,586438,586441,586444],{"className":586254},[34006],[34,586256,586258],{"className":586257,"style":75811},[31782,75810],[34,586259,456],{"className":586260},[71281,141292],[34,586262,586264,586267],{"className":586263},[32767],[34,586265,546295],{"className":586266,"style":546331},[32767,131249,156560],[34,586268,586270],{"className":586269},[31792],[34,586271,586273,586313],{"className":586272},[31796,31797],[34,586274,586276,586310],{"className":586275},[31801],[34,586277,586279,586293],{"className":586278,"style":546550},[31805],[34,586280,586281,586284],{"style":546347},[34,586282],{"className":586283,"style":31814},[31813],[34,586285,586287],{"className":586286},[31818,31819,31820,31821],[34,586288,586290],{"className":586289},[31685,31821],[34,586291,459],{"className":586292},[31685,31821],[34,586294,586295,586298],{"style":546362},[34,586296],{"className":586297,"style":31814},[31813],[34,586299,586301],{"className":586300},[31818,31819,31820,31821],[34,586302,586304,586307],{"className":586303},[31685,31821],[34,586305,223],{"className":586306},[31685,31821],[34,586308,138286],{"className":586309},[31685,31821],[34,586311,31829],{"className":586312},[31828],[34,586314,586316],{"className":586315},[31801],[34,586317,586319],{"className":586318,"style":546384},[31805],[34,586320],{},[34,586322],{"className":586323,"style":32123},[31845],[34,586325,586327,586330],{"className":586326},[31685],[34,586328,33244],{"className":586329},[31685,31686],[34,586331,586333],{"className":586332},[31792],[34,586334,586336],{"className":586335},[31796],[34,586337,586339],{"className":586338},[31801],[34,586340,586342],{"className":586341,"style":562030},[31805],[34,586343,586344,586347],{"style":37777},[34,586345],{"className":586346,"style":31814},[31813],[34,586348,586350],{"className":586349},[31818,31819,31820,31821],[34,586351,586353,586356],{"className":586352},[31685,31821],[34,586354,32226],{"className":586355},[31685,31821],[34,586357,586359,586362],{"className":586358},[31685,31821],[34,586360,34395],{"className":586361,"style":34452},[31685,31686,31821],[34,586363,586365],{"className":586364},[31792],[34,586366,586368],{"className":586367},[31796],[34,586369,586371],{"className":586370},[31801],[34,586372,586374],{"className":586373,"style":144123},[31805],[34,586375,586376,586379],{"style":552391},[34,586377],{"className":586378,"style":43406},[31813],[34,586380,586382],{"className":586381},[31818,43410,43411,31821],[34,586383,768],{"className":586384},[31685,31821],[34,586386,586388,586391],{"className":586387},[31685],[34,586389,34395],{"className":586390,"style":34452},[31685,31686],[34,586392,586394],{"className":586393},[31792],[34,586395,586397],{"className":586396},[31796],[34,586398,586400],{"className":586399},[31801],[34,586401,586403],{"className":586402,"style":61037},[31805],[34,586404,586405,586408],{"style":37777},[34,586406],{"className":586407,"style":31814},[31813],[34,586409,586411],{"className":586410},[31818,31819,31820,31821],[34,586412,586414,586417,586420,586423,586426,586429,586432,586435],{"className":586413},[31685,31821],[34,586415,768],{"className":586416},[31685,31821],[34,586418,456],{"className":586419},[31782,31821],[34,586421,15],{"className":586422},[31685,31686,31821],[34,586424,223],{"className":586425},[32024,31821],[34,586427,38759],{"className":586428,"style":32062},[31685,31686,31821],[34,586430,2557],{"className":586431},[31690,31821],[34,586433,32226],{"className":586434},[32024,31821],[34,586436,640],{"className":586437},[31685,31821],[34,586439,46900],{"className":586440},[31685,31686],[34,586442,34395],{"className":586443,"style":34452},[31685,31686],[34,586445,586447],{"className":586446,"style":75811},[31690,75810],[34,586448,2557],{"className":586449},[71281,141292],[34,586451],{"className":586452,"style":32123},[31845],[34,586454,586456,586462,586523,586526,586567,586570,586573,586576,586617,586620,586623,586626,586629],{"className":586455},[34006],[34,586457,586459],{"className":586458,"style":75811},[31782,75810],[34,586460,456],{"className":586461},[71281,141292],[34,586463,586465,586468],{"className":586464},[32767],[34,586466,546295],{"className":586467,"style":546331},[32767,131249,156560],[34,586469,586471],{"className":586470},[31792],[34,586472,586474,586515],{"className":586473},[31796,31797],[34,586475,586477,586512],{"className":586476},[31801],[34,586478,586481,586495],{"className":586479,"style":586480},[31805],"height:1.0829em;",[34,586482,586483,586486],{"style":546347},[34,586484],{"className":586485,"style":31814},[31813],[34,586487,586489],{"className":586488},[31818,31819,31820,31821],[34,586490,586492],{"className":586491},[31685,31821],[34,586493,459],{"className":586494},[31685,31821],[34,586496,586497,586500],{"style":546362},[34,586498],{"className":586499,"style":31814},[31813],[34,586501,586503],{"className":586502},[31818,31819,31820,31821],[34,586504,586506,586509],{"className":586505},[31685,31821],[34,586507,58442],{"className":586508,"style":32062},[31685,31686,31821],[34,586510,38149],{"className":586511},[31685,31821],[34,586513,31829],{"className":586514},[31828],[34,586516,586518],{"className":586517},[31801],[34,586519,586521],{"className":586520,"style":546384},[31805],[34,586522],{},[34,586524],{"className":586525,"style":32123},[31845],[34,586527,586529,586532],{"className":586528},[32767],[34,586530,547860],{"className":586531},[32767],[34,586533,586535],{"className":586534},[31792],[34,586536,586538],{"className":586537},[31796],[34,586539,586541],{"className":586540},[31801],[34,586542,586544],{"className":586543,"style":37774},[31805],[34,586545,586546,586549],{"style":37777},[34,586547],{"className":586548,"style":31814},[31813],[34,586550,586552],{"className":586551},[31818,31819,31820,31821],[34,586553,586555,586558,586561,586564],{"className":586554},[31685,31821],[34,586556,768],{"className":586557},[31685,31821],[34,586559,15],{"className":586560},[31685,31686,31821],[34,586562,32226],{"className":586563},[32024,31821],[34,586565,640],{"className":586566},[31685,31821],[34,586568],{"className":586569,"style":32123},[31845],[34,586571,479697],{"className":586572,"style":34452},[31685,31686],[34,586574],{"className":586575,"style":32123},[31845],[34,586577,586579,586582],{"className":586578},[32767],[34,586580,547853],{"className":586581},[32767],[34,586583,586585],{"className":586584},[31792],[34,586586,586588],{"className":586587},[31796],[34,586589,586591],{"className":586590},[31801],[34,586592,586594],{"className":586593,"style":585884},[31805],[34,586595,586596,586599],{"style":586008},[34,586597],{"className":586598,"style":31814},[31813],[34,586600,586602],{"className":586601},[31818,31819,31820,31821],[34,586603,586605,586608,586611,586614],{"className":586604},[31685,31821],[34,586606,768],{"className":586607},[31685,31821],[34,586609,38759],{"className":586610,"style":32062},[31685,31686,31821],[34,586612,32226],{"className":586613},[32024,31821],[34,586615,640],{"className":586616},[31685,31821],[34,586618],{"className":586619,"style":32123},[31845],[34,586621,479697],{"className":586622,"style":34452},[31685,31686],[34,586624,46900],{"className":586625},[31685,31686],[34,586627,479697],{"className":586628,"style":34452},[31685,31686],[34,586630,586632],{"className":586631,"style":75811},[31690,75810],[34,586633,2557],{"className":586634},[71281,141292],[108,586636,586637,586640],{},[2627,586638,586639],{},"识别积分：",[105,586641,586642,586868],{},[108,586643,586644,586645,586785,586786,586867],{},"左边括号里：正是 ",[34,586646,586648,586678],{"className":586647},[31642],[34,586649,586651],{"className":586650},[31646],[31648,586652,586653],{"xmlns":31650},[31652,586654,586655,586675],{},[31655,586656,586657,586663,586665,586667,586669,586671,586673],{},[41453,586658,586659,586661],{},[31749,586660,640],{},[31749,586662,768],{},[31658,586664,564791],{"mathvariant":38079},[31662,586666,456],{"stretchy":702},[31658,586668,15],{},[31662,586670,223],{},[31658,586672,38759],{},[31662,586674,2557],{"stretchy":702},[31665,586676,586677],{"encoding":31667},"\\frac{1}{2} \\Gamma(p+q)",[34,586679,586681,586773],{"className":586680,"ariaHidden":758},[31672],[34,586682,586684,586687,586755,586758,586761,586764,586767,586770],{"className":586683},[31676],[34,586685],{"className":586686,"style":46727},[31680],[34,586688,586690,586693,586752],{"className":586689},[31685],[34,586691],{"className":586692},[31782,41488],[34,586694,586696],{"className":586695},[41453],[34,586697,586699,586744],{"className":586698},[31796,31797],[34,586700,586702,586741],{"className":586701},[31801],[34,586703,586705,586719,586727],{"className":586704,"style":46746},[31805],[34,586706,586707,586710],{"style":41504},[34,586708],{"className":586709,"style":41508},[31813],[34,586711,586713],{"className":586712},[31818,31819,31820,31821],[34,586714,586716],{"className":586715},[31685,31821],[34,586717,768],{"className":586718},[31685,31821],[34,586720,586721,586724],{"style":41520},[34,586722],{"className":586723,"style":41508},[31813],[34,586725],{"className":586726,"style":41528},[41527],[34,586728,586729,586732],{"style":46771},[34,586730],{"className":586731,"style":41508},[31813],[34,586733,586735],{"className":586734},[31818,31819,31820,31821],[34,586736,586738],{"className":586737},[31685,31821],[34,586739,640],{"className":586740},[31685,31821],[34,586742,31829],{"className":586743},[31828],[34,586745,586747],{"className":586746},[31801],[34,586748,586750],{"className":586749,"style":41619},[31805],[34,586751],{},[34,586753],{"className":586754},[31690,41488],[34,586756,564791],{"className":586757},[31685],[34,586759,456],{"className":586760},[31782],[34,586762,15],{"className":586763},[31685,31686],[34,586765],{"className":586766,"style":32604},[31845],[34,586768,223],{"className":586769},[32024],[34,586771],{"className":586772,"style":32604},[31845],[34,586774,586776,586779,586782],{"className":586775},[31676],[34,586777],{"className":586778,"style":31774},[31680],[34,586780,38759],{"className":586781,"style":32062},[31685,31686],[34,586783,2557],{"className":586784},[31690]," （把刚才换元 ",[34,586787,586789,586811],{"className":586788},[31642],[34,586790,586792],{"className":586791},[31646],[31648,586793,586794],{"xmlns":31650},[31652,586795,586796,586808],{},[31655,586797,586798,586800,586802],{},[31658,586799,32936],{},[31662,586801,202],{},[37624,586803,586804,586806],{},[31658,586805,94628],{},[31749,586807,768],{},[31665,586809,586810],{"encoding":31667},"x=u^2",[34,586812,586814,586832],{"className":586813,"ariaHidden":758},[31672],[34,586815,586817,586820,586823,586826,586829],{"className":586816},[31676],[34,586818],{"className":586819,"style":32527},[31680],[34,586821,32936],{"className":586822},[31685,31686],[34,586824],{"className":586825,"style":31846},[31845],[34,586827,202],{"className":586828},[31850],[34,586830],{"className":586831,"style":31846},[31845],[34,586833,586835,586838],{"className":586834},[31676],[34,586836],{"className":586837,"style":37774},[31680],[34,586839,586841,586844],{"className":586840},[31685],[34,586842,94628],{"className":586843},[31685,31686],[34,586845,586847],{"className":586846},[31792],[34,586848,586850],{"className":586849},[31796],[34,586851,586853],{"className":586852},[31801],[34,586854,586856],{"className":586855,"style":37774},[31805],[34,586857,586858,586861],{"style":37777},[34,586859],{"className":586860,"style":31814},[31813],[34,586862,586864],{"className":586863},[31818,31819,31820,31821],[34,586865,768],{"className":586866},[31685,31821]," 的过程倒回去）。",[108,586869,586870,586871,347],{},"右边括号里：这是 Beta 函数的三角形式 ",[34,586872,586874,586904],{"className":586873},[31642],[34,586875,586877],{"className":586876},[31646],[31648,586878,586879],{"xmlns":31650},[31652,586880,586881,586901],{},[31655,586882,586883,586889,586891,586893,586895,586897,586899],{},[41453,586884,586885,586887],{},[31749,586886,640],{},[31749,586888,768],{},[31658,586890,54224],{},[31662,586892,456],{"stretchy":702},[31658,586894,15],{},[31662,586896,31960],{"separator":758},[31658,586898,38759],{},[31662,586900,2557],{"stretchy":702},[31665,586902,586903],{"encoding":31667},"\\frac{1}{2} B(p, q)",[34,586905,586907],{"className":586906,"ariaHidden":758},[31672],[34,586908,586910,586913,586981,586984,586987,586990,586993,586996,586999],{"className":586909},[31676],[34,586911],{"className":586912,"style":46727},[31680],[34,586914,586916,586919,586978],{"className":586915},[31685],[34,586917],{"className":586918},[31782,41488],[34,586920,586922],{"className":586921},[41453],[34,586923,586925,586970],{"className":586924},[31796,31797],[34,586926,586928,586967],{"className":586927},[31801],[34,586929,586931,586945,586953],{"className":586930,"style":46746},[31805],[34,586932,586933,586936],{"style":41504},[34,586934],{"className":586935,"style":41508},[31813],[34,586937,586939],{"className":586938},[31818,31819,31820,31821],[34,586940,586942],{"className":586941},[31685,31821],[34,586943,768],{"className":586944},[31685,31821],[34,586946,586947,586950],{"style":41520},[34,586948],{"className":586949,"style":41508},[31813],[34,586951],{"className":586952,"style":41528},[41527],[34,586954,586955,586958],{"style":46771},[34,586956],{"className":586957,"style":41508},[31813],[34,586959,586961],{"className":586960},[31818,31819,31820,31821],[34,586962,586964],{"className":586963},[31685,31821],[34,586965,640],{"className":586966},[31685,31821],[34,586968,31829],{"className":586969},[31828],[34,586971,586973],{"className":586972},[31801],[34,586974,586976],{"className":586975,"style":41619},[31805],[34,586977],{},[34,586979],{"className":586980},[31690,41488],[34,586982,54224],{"className":586983,"style":54252},[31685,31686],[34,586985,456],{"className":586986},[31782],[34,586988,15],{"className":586989},[31685,31686],[34,586991,31960],{"className":586992},[32119],[34,586994],{"className":586995,"style":32123},[31845],[34,586997,38759],{"className":586998,"style":32062},[31685,31686],[34,587000,2557],{"className":587001},[31690],[108,587003,587004,587007,587350,587506],{},[2627,587005,587006],{},"合并：",[34,587008,587010,587082],{"className":587009},[31642],[34,587011,587013],{"className":587012},[31646],[31648,587014,587015],{"xmlns":31650},[31652,587016,587017,587079],{},[31655,587018,587019,587021,587023,587025,587027,587029,587031,587033,587035,587037,587039,587041,587047,587049,587051,587053,587055,587057,587059,587061,587067,587069,587071,587073,587075,587077],{},[31658,587020,564791],{"mathvariant":38079},[31662,587022,456],{"stretchy":702},[31658,587024,15],{},[31662,587026,2557],{"stretchy":702},[31658,587028,564791],{"mathvariant":38079},[31662,587030,456],{"stretchy":702},[31658,587032,38759],{},[31662,587034,2557],{"stretchy":702},[31662,587036,202],{},[31749,587038,15932],{},[31662,587040,71191],{},[41453,587042,587043,587045],{},[31749,587044,640],{},[31749,587046,768],{},[31658,587048,564791],{"mathvariant":38079},[31662,587050,456],{"stretchy":702},[31658,587052,15],{},[31662,587054,223],{},[31658,587056,38759],{},[31662,587058,2557],{"stretchy":702},[31662,587060,71191],{},[41453,587062,587063,587065],{},[31749,587064,640],{},[31749,587066,768],{},[31658,587068,54224],{},[31662,587070,456],{"stretchy":702},[31658,587072,15],{},[31662,587074,31960],{"separator":758},[31658,587076,38759],{},[31662,587078,2557],{"stretchy":702},[31665,587080,587081],{"encoding":31667},"\\Gamma(p)\\Gamma(q) = 4 \\cdot \\frac{1}{2}\\Gamma(p+q) \\cdot \\frac{1}{2}B(p,q)",[34,587083,587085,587124,587142,587234,587255],{"className":587084,"ariaHidden":758},[31672],[34,587086,587088,587091,587094,587097,587100,587103,587106,587109,587112,587115,587118,587121],{"className":587087},[31676],[34,587089],{"className":587090,"style":31774},[31680],[34,587092,564791],{"className":587093},[31685],[34,587095,456],{"className":587096},[31782],[34,587098,15],{"className":587099},[31685,31686],[34,587101,2557],{"className":587102},[31690],[34,587104,564791],{"className":587105},[31685],[34,587107,456],{"className":587108},[31782],[34,587110,38759],{"className":587111,"style":32062},[31685,31686],[34,587113,2557],{"className":587114},[31690],[34,587116],{"className":587117,"style":31846},[31845],[34,587119,202],{"className":587120},[31850],[34,587122],{"className":587123,"style":31846},[31845],[34,587125,587127,587130,587133,587136,587139],{"className":587126},[31676],[34,587128],{"className":587129,"style":32617},[31680],[34,587131,15932],{"className":587132},[31685],[34,587134],{"className":587135,"style":32604},[31845],[34,587137,71191],{"className":587138},[32024],[34,587140],{"className":587141,"style":32604},[31845],[34,587143,587145,587148,587216,587219,587222,587225,587228,587231],{"className":587144},[31676],[34,587146],{"className":587147,"style":46727},[31680],[34,587149,587151,587154,587213],{"className":587150},[31685],[34,587152],{"className":587153},[31782,41488],[34,587155,587157],{"className":587156},[41453],[34,587158,587160,587205],{"className":587159},[31796,31797],[34,587161,587163,587202],{"className":587162},[31801],[34,587164,587166,587180,587188],{"className":587165,"style":46746},[31805],[34,587167,587168,587171],{"style":41504},[34,587169],{"className":587170,"style":41508},[31813],[34,587172,587174],{"className":587173},[31818,31819,31820,31821],[34,587175,587177],{"className":587176},[31685,31821],[34,587178,768],{"className":587179},[31685,31821],[34,587181,587182,587185],{"style":41520},[34,587183],{"className":587184,"style":41508},[31813],[34,587186],{"className":587187,"style":41528},[41527],[34,587189,587190,587193],{"style":46771},[34,587191],{"className":587192,"style":41508},[31813],[34,587194,587196],{"className":587195},[31818,31819,31820,31821],[34,587197,587199],{"className":587198},[31685,31821],[34,587200,640],{"className":587201},[31685,31821],[34,587203,31829],{"className":587204},[31828],[34,587206,587208],{"className":587207},[31801],[34,587209,587211],{"className":587210,"style":41619},[31805],[34,587212],{},[34,587214],{"className":587215},[31690,41488],[34,587217,564791],{"className":587218},[31685],[34,587220,456],{"className":587221},[31782],[34,587223,15],{"className":587224},[31685,31686],[34,587226],{"className":587227,"style":32604},[31845],[34,587229,223],{"className":587230},[32024],[34,587232],{"className":587233,"style":32604},[31845],[34,587235,587237,587240,587243,587246,587249,587252],{"className":587236},[31676],[34,587238],{"className":587239,"style":31774},[31680],[34,587241,38759],{"className":587242,"style":32062},[31685,31686],[34,587244,2557],{"className":587245},[31690],[34,587247],{"className":587248,"style":32604},[31845],[34,587250,71191],{"className":587251},[32024],[34,587253],{"className":587254,"style":32604},[31845],[34,587256,587258,587261,587329,587332,587335,587338,587341,587344,587347],{"className":587257},[31676],[34,587259],{"className":587260,"style":46727},[31680],[34,587262,587264,587267,587326],{"className":587263},[31685],[34,587265],{"className":587266},[31782,41488],[34,587268,587270],{"className":587269},[41453],[34,587271,587273,587318],{"className":587272},[31796,31797],[34,587274,587276,587315],{"className":587275},[31801],[34,587277,587279,587293,587301],{"className":587278,"style":46746},[31805],[34,587280,587281,587284],{"style":41504},[34,587282],{"className":587283,"style":41508},[31813],[34,587285,587287],{"className":587286},[31818,31819,31820,31821],[34,587288,587290],{"className":587289},[31685,31821],[34,587291,768],{"className":587292},[31685,31821],[34,587294,587295,587298],{"style":41520},[34,587296],{"className":587297,"style":41508},[31813],[34,587299],{"className":587300,"style":41528},[41527],[34,587302,587303,587306],{"style":46771},[34,587304],{"className":587305,"style":41508},[31813],[34,587307,587309],{"className":587308},[31818,31819,31820,31821],[34,587310,587312],{"className":587311},[31685,31821],[34,587313,640],{"className":587314},[31685,31821],[34,587316,31829],{"className":587317},[31828],[34,587319,587321],{"className":587320},[31801],[34,587322,587324],{"className":587323,"style":41619},[31805],[34,587325],{},[34,587327],{"className":587328},[31690,41488],[34,587330,54224],{"className":587331,"style":54252},[31685,31686],[34,587333,456],{"className":587334},[31782],[34,587336,15],{"className":587337},[31685,31686],[34,587339,31960],{"className":587340},[32119],[34,587342],{"className":587343,"style":32123},[31845],[34,587345,38759],{"className":587346,"style":32062},[31685,31686],[34,587348,2557],{"className":587349},[31690],[34,587351,587353,587407],{"className":587352},[31642],[34,587354,587356],{"className":587355},[31646],[31648,587357,587358],{"xmlns":31650},[31652,587359,587360,587404],{},[31655,587361,587362,587364,587366,587368,587370,587372,587374,587376,587378,587380,587382,587384,587386,587388,587390,587392,587394,587396,587398,587400,587402],{},[31658,587363,564791],{"mathvariant":38079},[31662,587365,456],{"stretchy":702},[31658,587367,15],{},[31662,587369,2557],{"stretchy":702},[31658,587371,564791],{"mathvariant":38079},[31662,587373,456],{"stretchy":702},[31658,587375,38759],{},[31662,587377,2557],{"stretchy":702},[31662,587379,202],{},[31658,587381,564791],{"mathvariant":38079},[31662,587383,456],{"stretchy":702},[31658,587385,15],{},[31662,587387,223],{},[31658,587389,38759],{},[31662,587391,2557],{"stretchy":702},[31658,587393,54224],{},[31662,587395,456],{"stretchy":702},[31658,587397,15],{},[31662,587399,31960],{"separator":758},[31658,587401,38759],{},[31662,587403,2557],{"stretchy":702},[31665,587405,587406],{"encoding":31667},"\\Gamma(p)\\Gamma(q) = \\Gamma(p+q) B(p,q)",[34,587408,587410,587449,587473],{"className":587409,"ariaHidden":758},[31672],[34,587411,587413,587416,587419,587422,587425,587428,587431,587434,587437,587440,587443,587446],{"className":587412},[31676],[34,587414],{"className":587415,"style":31774},[31680],[34,587417,564791],{"className":587418},[31685],[34,587420,456],{"className":587421},[31782],[34,587423,15],{"className":587424},[31685,31686],[34,587426,2557],{"className":587427},[31690],[34,587429,564791],{"className":587430},[31685],[34,587432,456],{"className":587433},[31782],[34,587435,38759],{"className":587436,"style":32062},[31685,31686],[34,587438,2557],{"className":587439},[31690],[34,587441],{"className":587442,"style":31846},[31845],[34,587444,202],{"className":587445},[31850],[34,587447],{"className":587448,"style":31846},[31845],[34,587450,587452,587455,587458,587461,587464,587467,587470],{"className":587451},[31676],[34,587453],{"className":587454,"style":31774},[31680],[34,587456,564791],{"className":587457},[31685],[34,587459,456],{"className":587460},[31782],[34,587462,15],{"className":587463},[31685,31686],[34,587465],{"className":587466,"style":32604},[31845],[34,587468,223],{"className":587469},[32024],[34,587471],{"className":587472,"style":32604},[31845],[34,587474,587476,587479,587482,587485,587488,587491,587494,587497,587500,587503],{"className":587475},[31676],[34,587477],{"className":587478,"style":31774},[31680],[34,587480,38759],{"className":587481,"style":32062},[31685,31686],[34,587483,2557],{"className":587484},[31690],[34,587486,54224],{"className":587487,"style":54252},[31685,31686],[34,587489,456],{"className":587490},[31782],[34,587492,15],{"className":587493},[31685,31686],[34,587495,31960],{"className":587496},[32119],[34,587498],{"className":587499,"style":32123},[31845],[34,587501,38759],{"className":587502,"style":32062},[31685,31686],[34,587504,2557],{"className":587505},[31690],[34,587507,587509,587569],{"className":587508},[31642],[34,587510,587512],{"className":587511},[31646],[31648,587513,587514],{"xmlns":31650},[31652,587515,587516,587566],{},[31655,587517,587518,587520,587522,587524,587526,587528,587530,587532],{},[31658,587519,54224],{},[31662,587521,456],{"stretchy":702},[31658,587523,15],{},[31662,587525,31960],{"separator":758},[31658,587527,38759],{},[31662,587529,2557],{"stretchy":702},[31662,587531,202],{},[41453,587533,587534,587552],{},[31655,587535,587536,587538,587540,587542,587544,587546,587548,587550],{},[31658,587537,564791],{"mathvariant":38079},[31662,587539,456],{"stretchy":702},[31658,587541,15],{},[31662,587543,2557],{"stretchy":702},[31658,587545,564791],{"mathvariant":38079},[31662,587547,456],{"stretchy":702},[31658,587549,38759],{},[31662,587551,2557],{"stretchy":702},[31655,587553,587554,587556,587558,587560,587562,587564],{},[31658,587555,564791],{"mathvariant":38079},[31662,587557,456],{"stretchy":702},[31658,587559,15],{},[31662,587561,223],{},[31658,587563,38759],{},[31662,587565,2557],{"stretchy":702},[31665,587567,587568],{"encoding":31667},"B(p,q) = \\frac{\\Gamma(p)\\Gamma(q)}{\\Gamma(p+q)}",[34,587570,587572,587608],{"className":587571,"ariaHidden":758},[31672],[34,587573,587575,587578,587581,587584,587587,587590,587593,587596,587599,587602,587605],{"className":587574},[31676],[34,587576],{"className":587577,"style":31774},[31680],[34,587579,54224],{"className":587580,"style":54252},[31685,31686],[34,587582,456],{"className":587583},[31782],[34,587585,15],{"className":587586},[31685,31686],[34,587588,31960],{"className":587589},[32119],[34,587591],{"className":587592,"style":32123},[31845],[34,587594,38759],{"className":587595,"style":32062},[31685,31686],[34,587597,2557],{"className":587598},[31690],[34,587600],{"className":587601,"style":31846},[31845],[34,587603,202],{"className":587604},[31850],[34,587606],{"className":587607,"style":31846},[31845],[34,587609,587611,587614],{"className":587610},[31676],[34,587612],{"className":587613,"style":327841},[31680],[34,587615,587617,587620,587715],{"className":587616},[31685],[34,587618],{"className":587619},[31782,41488],[34,587621,587623],{"className":587622},[41453],[34,587624,587626,587707],{"className":587625},[31796,31797],[34,587627,587629,587704],{"className":587628},[31801],[34,587630,587632,587661,587669],{"className":587631,"style":70504},[31805],[34,587633,587634,587637],{"style":41504},[34,587635],{"className":587636,"style":41508},[31813],[34,587638,587640],{"className":587639},[31818,31819,31820,31821],[34,587641,587643,587646,587649,587652,587655,587658],{"className":587642},[31685,31821],[34,587644,564791],{"className":587645},[31685,31821],[34,587647,456],{"className":587648},[31782,31821],[34,587650,15],{"className":587651},[31685,31686,31821],[34,587653,223],{"className":587654},[32024,31821],[34,587656,38759],{"className":587657,"style":32062},[31685,31686,31821],[34,587659,2557],{"className":587660},[31690,31821],[34,587662,587663,587666],{"style":41520},[34,587664],{"className":587665,"style":41508},[31813],[34,587667],{"className":587668,"style":41528},[41527],[34,587670,587671,587674],{"style":70529},[34,587672],{"className":587673,"style":41508},[31813],[34,587675,587677],{"className":587676},[31818,31819,31820,31821],[34,587678,587680,587683,587686,587689,587692,587695,587698,587701],{"className":587679},[31685,31821],[34,587681,564791],{"className":587682},[31685,31821],[34,587684,456],{"className":587685},[31782,31821],[34,587687,15],{"className":587688},[31685,31686,31821],[34,587690,2557],{"className":587691},[31690,31821],[34,587693,564791],{"className":587694},[31685,31821],[34,587696,456],{"className":587697},[31782,31821],[34,587699,38759],{"className":587700,"style":32062},[31685,31686,31821],[34,587702,2557],{"className":587703},[31690,31821],[34,587705,31829],{"className":587706},[31828],[34,587708,587710],{"className":587709},[31801],[34,587711,587713],{"className":587712,"style":324858},[31805],[34,587714],{},[34,587716],{"className":587717},[31690,41488],{"title":29,"searchDepth":43,"depth":43,"links":587719},[587720,587721,587722,587723,587724,587725,587726,587727,587728,587729,587730,587731,587732,587733,587734,587735,587736,587737],{"id":546274,"depth":49,"text":546275},{"id":546726,"depth":49,"text":546727},{"id":547374,"depth":49,"text":547375},{"id":548113,"depth":49,"text":548114},{"id":549097,"depth":49,"text":549100},{"id":551698,"depth":49,"text":551701},{"id":553865,"depth":49,"text":553868},{"id":560543,"depth":49,"text":560544},{"id":560552,"depth":49,"text":549100},{"id":561873,"depth":49,"text":551701},{"id":563243,"depth":49,"text":553868},{"id":565159,"depth":49,"text":565162},{"id":566103,"depth":49,"text":566104},{"id":567791,"depth":49,"text":567792},{"id":569771,"depth":49,"text":569772},{"id":573107,"depth":49,"text":573108},{"id":575881,"depth":49,"text":575882},{"id":582850,"depth":49,"text":582851},{"order":43},"\u002Fsource\u002Fmath\u002Fmath_analysis\u002F11-29hw",{"title":546253,"description":29},"source\u002Fmath\u002Fmath_analysis\u002F11-29HW","FT91_eX17LvJ3kXFJDmRMCH7fviXeTvkekGwdtcAbug",{"id":587744,"title":587745,"body":587746,"description":587754,"extension":85,"meta":658081,"navigation":52,"path":658082,"seo":658083,"stem":658084,"__hash__":658085},"notes\u002Fsource\u002Fmath\u002Fmath_analysis\u002F12-5HW.md","12-5 作业",{"type":8,"value":587747,"toc":658075},[587748,587752,587755,587757,587761,587764,587826,587907,587957,588864,588926,590266,590270,591340,591342,591346,591406,591410,591625,594182,594431,597573,597910,601329,601676,604627,604950,608096,608451,611305,611589,614120,614458,616584,616586,616962,618659,618666,618668,618716,618719,620729,620731,620885,622309,622311,622448,622451,625048,625050,625082,629386,629388,629799,631611,631613,631617,631855,633228,633468,635301,635306,637117,637119,637123,637127,637271,639444,639681,640436,640772,643007,643365,646366,646533,648447,648843,651703,652020,653910,653975],[11,587749,587751],{"id":587750},"dec-5-hw","Dec. 5 HW",[15,587753,587754],{},"欧拉积分和曲线积分是数学分析中计算技巧性非常强、概念也很抽象的部分。",[32188,587756],{},[253,587758,587760],{"id":587759},"第一部分核心知识点","第一部分：核心知识点",[15,587762,587763],{},"本节主要讲了两类特殊的函数（用来算复杂积分的工具）和一种新的积分（在弯曲的线上做积分）。",[32490,587765,587767,587768,587796,587797,587825],{"id":587766},"_1-欧拉积分γgammaγ-函数与-bbb-函数","1. 欧拉积分：",[34,587769,587771,587784],{"className":587770},[31642],[34,587772,587774],{"className":587773},[31646],[31648,587775,587776],{"xmlns":31650},[31652,587777,587778,587782],{},[31655,587779,587780],{},[31658,587781,564791],{"mathvariant":38079},[31665,587783,564794],{"encoding":31667},[34,587785,587787],{"className":587786,"ariaHidden":758},[31672],[34,587788,587790,587793],{"className":587789},[31676],[34,587791],{"className":587792,"style":33218},[31680],[34,587794,564791],{"className":587795},[31685]," 函数与 ",[34,587798,587800,587813],{"className":587799},[31642],[34,587801,587803],{"className":587802},[31646],[31648,587804,587805],{"xmlns":31650},[31652,587806,587807,587811],{},[31655,587808,587809],{},[31658,587810,54224],{},[31665,587812,54224],{"encoding":31667},[34,587814,587816],{"className":587815,"ariaHidden":758},[31672],[34,587817,587819,587822],{"className":587818},[31676],[34,587820],{"className":587821,"style":33218},[31680],[34,587823,54224],{"className":587824,"style":54252},[31685,31686]," 函数",[15,587827,587828,587829,587906],{},"之前学的积分，比如 ",[34,587830,587832,587856],{"className":587831},[31642],[34,587833,587835],{"className":587834},[31646],[31648,587836,587837],{"xmlns":31650},[31652,587838,587839,587853],{},[31655,587840,587841,587843,587849,587851],{},[31662,587842,546295],{},[37624,587844,587845,587847],{},[31658,587846,32936],{},[31749,587848,768],{},[31658,587850,46900],{},[31658,587852,32936],{},[31665,587854,587855],{"encoding":31667},"\\int x^2 dx",[34,587857,587859],{"className":587858,"ariaHidden":758},[31672],[34,587860,587862,587865,587868,587871,587900,587903],{"className":587861},[31676],[34,587863],{"className":587864,"style":556092},[31680],[34,587866,546295],{"className":587867,"style":546331},[32767,131249,156560],[34,587869],{"className":587870,"style":32123},[31845],[34,587872,587874,587877],{"className":587873},[31685],[34,587875,32936],{"className":587876},[31685,31686],[34,587878,587880],{"className":587879},[31792],[34,587881,587883],{"className":587882},[31796],[34,587884,587886],{"className":587885},[31801],[34,587887,587889],{"className":587888,"style":37774},[31805],[34,587890,587891,587894],{"style":37777},[34,587892],{"className":587893,"style":31814},[31813],[34,587895,587897],{"className":587896},[31818,31819,31820,31821],[34,587898,768],{"className":587899},[31685,31821],[34,587901,46900],{"className":587902},[31685,31686],[34,587904,32936],{"className":587905},[31685,31686],"，都是初等函数。但很多积分算不出来具体的初等函数，于是数学家定义了两个\"模具\"，只要积分长得像这个模具，就能直接算出答案。",[15,587908,587909],{},[2627,587910,587911,587912,587956],{},"工具 A：伽玛函数 ",[34,587913,587915,587935],{"className":587914},[31642],[34,587916,587918],{"className":587917},[31646],[31648,587919,587920],{"xmlns":31650},[31652,587921,587922,587932],{},[31655,587923,587924,587926,587928,587930],{},[31658,587925,564791],{"mathvariant":38079},[31662,587927,456],{"stretchy":702},[31658,587929,31747],{},[31662,587931,2557],{"stretchy":702},[31665,587933,587934],{"encoding":31667},"\\Gamma(s)",[34,587936,587938],{"className":587937,"ariaHidden":758},[31672],[34,587939,587941,587944,587947,587950,587953],{"className":587940},[31676],[34,587942],{"className":587943,"style":31774},[31680],[34,587945,564791],{"className":587946},[31685],[34,587948,456],{"className":587949},[31782],[34,587951,31747],{"className":587952},[31685,31686],[34,587954,2557],{"className":587955},[31690]," (可以理解为“广义阶乘”)",[105,587958,587959,588336],{},[108,587960,587961,587964,587965,588021,588022,588050,588051],{},[2627,587962,587963],{},"模具样子","：积分区间是 ",[34,587966,587968,587991],{"className":587967},[31642],[34,587969,587971],{"className":587970},[31646],[31648,587972,587973],{"xmlns":31650},[31652,587974,587975,587989],{},[31655,587976,587977,587979,587981,587983,587985,587987],{},[31662,587978,456],{"stretchy":702},[31749,587980,459],{},[31662,587982,31960],{"separator":758},[31662,587984,223],{},[31658,587986,138286],{"mathvariant":38079},[31662,587988,2557],{"stretchy":702},[31665,587990,566408],{"encoding":31667},[34,587992,587994],{"className":587993,"ariaHidden":758},[31672],[34,587995,587997,588000,588003,588006,588009,588012,588015,588018],{"className":587996},[31676],[34,587998],{"className":587999,"style":31774},[31680],[34,588001,456],{"className":588002},[31782],[34,588004,459],{"className":588005},[31685],[34,588007,31960],{"className":588008},[32119],[34,588010],{"className":588011,"style":32123},[31845],[34,588013,223],{"className":588014},[31685],[34,588016,138286],{"className":588017},[31685],[34,588019,2557],{"className":588020},[31690],"，里面是“幂函数 ",[34,588023,588025,588038],{"className":588024},[31642],[34,588026,588028],{"className":588027},[31646],[31648,588029,588030],{"xmlns":31650},[31652,588031,588032,588036],{},[31655,588033,588034],{},[31662,588035,42071],{},[31665,588037,211013],{"encoding":31667},[34,588039,588041],{"className":588040,"ariaHidden":758},[31672],[34,588042,588044,588047],{"className":588043},[31676],[34,588045],{"className":588046,"style":32597},[31680],[34,588048,42071],{"className":588049},[31685]," 指数函数”。\n",[34,588052,588054,588128],{"className":588053},[31642],[34,588055,588057],{"className":588056},[31646],[31648,588058,588059],{"xmlns":31650},[31652,588060,588061,588125],{},[31655,588062,588063,588065,588067,588069,588071,588073,588085,588097,588107,588109,588111,588113,588115,588117,588119,588121,588123],{},[31658,588064,564791],{"mathvariant":38079},[31662,588066,456],{"stretchy":702},[31658,588068,31747],{},[31662,588070,2557],{"stretchy":702},[31662,588072,202],{},[38846,588074,588075,588077,588079],{},[31662,588076,546295],{},[31749,588078,459],{},[31655,588080,588081,588083],{},[31662,588082,223],{},[31658,588084,138286],{"mathvariant":38079},[37624,588086,588087,588089],{},[31658,588088,32936],{},[31655,588090,588091,588093,588095],{},[31658,588092,31747],{},[31662,588094,32226],{},[31749,588096,640],{},[37624,588098,588099,588101],{},[31658,588100,33244],{},[31655,588102,588103,588105],{},[31662,588104,32226],{},[31658,588106,32936],{},[39361,588108,39363],{},[31658,588110,46900],{},[31658,588112,32936],{},[31845,588114],{"width":71082},[31662,588116,456],{"stretchy":702},[31658,588118,31747],{},[31662,588120,29622],{},[31749,588122,459],{},[31662,588124,2557],{"stretchy":702},[31665,588126,588127],{"encoding":31667},"\\Gamma(s) = \\int_0^{+\\infty} x^{s-1} e^{-x} \\, dx \\quad (s > 0)",[34,588129,588131,588158,588324],{"className":588130,"ariaHidden":758},[31672],[34,588132,588134,588137,588140,588143,588146,588149,588152,588155],{"className":588133},[31676],[34,588135],{"className":588136,"style":31774},[31680],[34,588138,564791],{"className":588139},[31685],[34,588141,456],{"className":588142},[31782],[34,588144,31747],{"className":588145},[31685,31686],[34,588147,2557],{"className":588148},[31690],[34,588150],{"className":588151,"style":31846},[31845],[34,588153,202],{"className":588154},[31850],[34,588156],{"className":588157,"style":31846},[31845],[34,588159,588161,588164,588221,588224,588262,588297,588300,588303,588306,588309,588312,588315,588318,588321],{"className":588160},[31676],[34,588162],{"className":588163,"style":546531},[31680],[34,588165,588167,588170],{"className":588166},[32767],[34,588168,546295],{"className":588169,"style":546331},[32767,131249,156560],[34,588171,588173],{"className":588172},[31792],[34,588174,588176,588213],{"className":588175},[31796,31797],[34,588177,588179,588210],{"className":588178},[31801],[34,588180,588182,588193],{"className":588181,"style":546550},[31805],[34,588183,588184,588187],{"style":546347},[34,588185],{"className":588186,"style":31814},[31813],[34,588188,588190],{"className":588189},[31818,31819,31820,31821],[34,588191,459],{"className":588192},[31685,31821],[34,588194,588195,588198],{"style":546362},[34,588196],{"className":588197,"style":31814},[31813],[34,588199,588201],{"className":588200},[31818,31819,31820,31821],[34,588202,588204,588207],{"className":588203},[31685,31821],[34,588205,223],{"className":588206},[31685,31821],[34,588208,138286],{"className":588209},[31685,31821],[34,588211,31829],{"className":588212},[31828],[34,588214,588216],{"className":588215},[31801],[34,588217,588219],{"className":588218,"style":546384},[31805],[34,588220],{},[34,588222],{"className":588223,"style":32123},[31845],[34,588225,588227,588230],{"className":588226},[31685],[34,588228,32936],{"className":588229},[31685,31686],[34,588231,588233],{"className":588232},[31792],[34,588234,588236],{"className":588235},[31796],[34,588237,588239],{"className":588238},[31801],[34,588240,588242],{"className":588241,"style":37774},[31805],[34,588243,588244,588247],{"style":37777},[34,588245],{"className":588246,"style":31814},[31813],[34,588248,588250],{"className":588249},[31818,31819,31820,31821],[34,588251,588253,588256,588259],{"className":588252},[31685,31821],[34,588254,31747],{"className":588255},[31685,31686,31821],[34,588257,32226],{"className":588258},[32024,31821],[34,588260,640],{"className":588261},[31685,31821],[34,588263,588265,588268],{"className":588264},[31685],[34,588266,33244],{"className":588267},[31685,31686],[34,588269,588271],{"className":588270},[31792],[34,588272,588274],{"className":588273},[31796],[34,588275,588277],{"className":588276},[31801],[34,588278,588280],{"className":588279,"style":156828},[31805],[34,588281,588282,588285],{"style":37777},[34,588283],{"className":588284,"style":31814},[31813],[34,588286,588288],{"className":588287},[31818,31819,31820,31821],[34,588289,588291,588294],{"className":588290},[31685,31821],[34,588292,32226],{"className":588293},[31685,31821],[34,588295,32936],{"className":588296},[31685,31686,31821],[34,588298],{"className":588299,"style":32123},[31845],[34,588301,46900],{"className":588302},[31685,31686],[34,588304,32936],{"className":588305},[31685,31686],[34,588307],{"className":588308,"style":128438},[31845],[34,588310,456],{"className":588311},[31782],[34,588313,31747],{"className":588314},[31685,31686],[34,588316],{"className":588317,"style":31846},[31845],[34,588319,29622],{"className":588320},[31850],[34,588322],{"className":588323,"style":31846},[31845],[34,588325,588327,588330,588333],{"className":588326},[31676],[34,588328],{"className":588329,"style":31774},[31680],[34,588331,459],{"className":588332},[31685],[34,588334,2557],{"className":588335},[31690],[108,588337,588338,588341,588342],{},[2627,588339,588340],{},"核心性质","（必须背下来）：\n",[2230,588343,588344,588564,588677],{},[108,588345,588346,2664,588349,400312,588442,588470,588471,347],{},[2627,588347,588348],{},"阶乘性质",[34,588350,588352,588382],{"className":588351},[31642],[34,588353,588355],{"className":588354},[31646],[31648,588356,588357],{"xmlns":31650},[31652,588358,588359,588379],{},[31655,588360,588361,588363,588365,588367,588369,588371,588373,588375,588377],{},[31658,588362,564791],{"mathvariant":38079},[31662,588364,456],{"stretchy":702},[31658,588366,31660],{},[31662,588368,223],{},[31749,588370,640],{},[31662,588372,2557],{"stretchy":702},[31662,588374,202],{},[31658,588376,31660],{},[31662,588378,5504],{"stretchy":702},[31665,588380,588381],{"encoding":31667},"\\Gamma(n+1) = n!",[34,588383,588385,588409,588430],{"className":588384,"ariaHidden":758},[31672],[34,588386,588388,588391,588394,588397,588400,588403,588406],{"className":588387},[31676],[34,588389],{"className":588390,"style":31774},[31680],[34,588392,564791],{"className":588393},[31685],[34,588395,456],{"className":588396},[31782],[34,588398,31660],{"className":588399},[31685,31686],[34,588401],{"className":588402,"style":32604},[31845],[34,588404,223],{"className":588405},[32024],[34,588407],{"className":588408,"style":32604},[31845],[34,588410,588412,588415,588418,588421,588424,588427],{"className":588411},[31676],[34,588413],{"className":588414,"style":31774},[31680],[34,588416,640],{"className":588417},[31685],[34,588419,2557],{"className":588420},[31690],[34,588422],{"className":588423,"style":31846},[31845],[34,588425,202],{"className":588426},[31850],[34,588428],{"className":588429,"style":31846},[31845],[34,588431,588433,588436,588439],{"className":588432},[31676],[34,588434],{"className":588435,"style":31681},[31680],[34,588437,31660],{"className":588438},[31685,31686],[34,588440,5504],{"className":588441},[31690],[34,588443,588445,588458],{"className":588444},[31642],[34,588446,588448],{"className":588447},[31646],[31648,588449,588450],{"xmlns":31650},[31652,588451,588452,588456],{},[31655,588453,588454],{},[31658,588455,31660],{},[31665,588457,31660],{"encoding":31667},[34,588459,588461],{"className":588460,"ariaHidden":758},[31672],[34,588462,588464,588467],{"className":588463},[31676],[34,588465],{"className":588466,"style":32527},[31680],[34,588468,31660],{"className":588469},[31685,31686]," 是整数）。例如 ",[34,588472,588474,588504],{"className":588473},[31642],[34,588475,588477],{"className":588476},[31646],[31648,588478,588479],{"xmlns":31650},[31652,588480,588481,588501],{},[31655,588482,588483,588485,588487,588489,588491,588493,588495,588497,588499],{},[31658,588484,564791],{"mathvariant":38079},[31662,588486,456],{"stretchy":702},[31749,588488,786],{},[31662,588490,2557],{"stretchy":702},[31662,588492,202],{},[31749,588494,768],{},[31662,588496,5504],{"stretchy":702},[31662,588498,202],{},[31749,588500,768],{},[31665,588502,588503],{"encoding":31667},"\\Gamma(3) = 2! = 2",[34,588505,588507,588534,588555],{"className":588506,"ariaHidden":758},[31672],[34,588508,588510,588513,588516,588519,588522,588525,588528,588531],{"className":588509},[31676],[34,588511],{"className":588512,"style":31774},[31680],[34,588514,564791],{"className":588515},[31685],[34,588517,456],{"className":588518},[31782],[34,588520,786],{"className":588521},[31685],[34,588523,2557],{"className":588524},[31690],[34,588526],{"className":588527,"style":31846},[31845],[34,588529,202],{"className":588530},[31850],[34,588532],{"className":588533,"style":31846},[31845],[34,588535,588537,588540,588543,588546,588549,588552],{"className":588536},[31676],[34,588538],{"className":588539,"style":31681},[31680],[34,588541,768],{"className":588542},[31685],[34,588544,5504],{"className":588545},[31690],[34,588547],{"className":588548,"style":31846},[31845],[34,588550,202],{"className":588551},[31850],[34,588553],{"className":588554,"style":31846},[31845],[34,588556,588558,588561],{"className":588557},[31676],[34,588559],{"className":588560,"style":32617},[31680],[34,588562,768],{"className":588563},[31685],[108,588565,588566,2664,588569,347],{},[2627,588567,588568],{},"递推公式",[34,588570,588572,588608],{"className":588571},[31642],[34,588573,588575],{"className":588574},[31646],[31648,588576,588577],{"xmlns":31650},[31652,588578,588579,588605],{},[31655,588580,588581,588583,588585,588587,588589,588591,588593,588595,588597,588599,588601,588603],{},[31658,588582,564791],{"mathvariant":38079},[31662,588584,456],{"stretchy":702},[31658,588586,31747],{},[31662,588588,223],{},[31749,588590,640],{},[31662,588592,2557],{"stretchy":702},[31662,588594,202],{},[31658,588596,31747],{},[31658,588598,564791],{"mathvariant":38079},[31662,588600,456],{"stretchy":702},[31658,588602,31747],{},[31662,588604,2557],{"stretchy":702},[31665,588606,588607],{"encoding":31667},"\\Gamma(s+1) = s \\Gamma(s)",[34,588609,588611,588635,588656],{"className":588610,"ariaHidden":758},[31672],[34,588612,588614,588617,588620,588623,588626,588629,588632],{"className":588613},[31676],[34,588615],{"className":588616,"style":31774},[31680],[34,588618,564791],{"className":588619},[31685],[34,588621,456],{"className":588622},[31782],[34,588624,31747],{"className":588625},[31685,31686],[34,588627],{"className":588628,"style":32604},[31845],[34,588630,223],{"className":588631},[32024],[34,588633],{"className":588634,"style":32604},[31845],[34,588636,588638,588641,588644,588647,588650,588653],{"className":588637},[31676],[34,588639],{"className":588640,"style":31774},[31680],[34,588642,640],{"className":588643},[31685],[34,588645,2557],{"className":588646},[31690],[34,588648],{"className":588649,"style":31846},[31845],[34,588651,202],{"className":588652},[31850],[34,588654],{"className":588655,"style":31846},[31845],[34,588657,588659,588662,588665,588668,588671,588674],{"className":588658},[31676],[34,588660],{"className":588661,"style":31774},[31680],[34,588663,31747],{"className":588664},[31685,31686],[34,588666,564791],{"className":588667},[31685],[34,588669,456],{"className":588670},[31782],[34,588672,31747],{"className":588673},[31685,31686],[34,588675,2557],{"className":588676},[31690],[108,588678,588679,2664,588682,588863],{},[2627,588680,588681],{},"特殊值",[34,588683,588685,588715],{"className":588684},[31642],[34,588686,588688],{"className":588687},[31646],[31648,588689,588690],{"xmlns":31650},[31652,588691,588692,588712],{},[31655,588693,588694,588696,588698,588704,588706,588708],{},[31658,588695,564791],{"mathvariant":38079},[31662,588697,456],{"stretchy":702},[41453,588699,588700,588702],{},[31749,588701,640],{},[31749,588703,768],{},[31662,588705,2557],{"stretchy":702},[31662,588707,202],{},[41463,588709,588710],{},[31658,588711,58442],{},[31665,588713,588714],{"encoding":31667},"\\Gamma(\\frac{1}{2}) = \\sqrt{\\pi}",[34,588716,588718,588810],{"className":588717,"ariaHidden":758},[31672],[34,588719,588721,588724,588727,588730,588798,588801,588804,588807],{"className":588720},[31676],[34,588722],{"className":588723,"style":46727},[31680],[34,588725,564791],{"className":588726},[31685],[34,588728,456],{"className":588729},[31782],[34,588731,588733,588736,588795],{"className":588732},[31685],[34,588734],{"className":588735},[31782,41488],[34,588737,588739],{"className":588738},[41453],[34,588740,588742,588787],{"className":588741},[31796,31797],[34,588743,588745,588784],{"className":588744},[31801],[34,588746,588748,588762,588770],{"className":588747,"style":46746},[31805],[34,588749,588750,588753],{"style":41504},[34,588751],{"className":588752,"style":41508},[31813],[34,588754,588756],{"className":588755},[31818,31819,31820,31821],[34,588757,588759],{"className":588758},[31685,31821],[34,588760,768],{"className":588761},[31685,31821],[34,588763,588764,588767],{"style":41520},[34,588765],{"className":588766,"style":41508},[31813],[34,588768],{"className":588769,"style":41528},[41527],[34,588771,588772,588775],{"style":46771},[34,588773],{"className":588774,"style":41508},[31813],[34,588776,588778],{"className":588777},[31818,31819,31820,31821],[34,588779,588781],{"className":588780},[31685,31821],[34,588782,640],{"className":588783},[31685,31821],[34,588785,31829],{"className":588786},[31828],[34,588788,588790],{"className":588789},[31801],[34,588791,588793],{"className":588792,"style":41619},[31805],[34,588794],{},[34,588796],{"className":588797},[31690,41488],[34,588799,2557],{"className":588800},[31690],[34,588802],{"className":588803,"style":31846},[31845],[34,588805,202],{"className":588806},[31850],[34,588808],{"className":588809,"style":31846},[31845],[34,588811,588813,588816],{"className":588812},[31676],[34,588814],{"className":588815,"style":562415},[31680],[34,588817,588819],{"className":588818},[31685,41550],[34,588820,588822,588855],{"className":588821},[31796,31797],[34,588823,588825,588852],{"className":588824},[31801],[34,588826,588828,588840],{"className":588827,"style":64997},[31805],[34,588829,588831,588834],{"className":588830,"style":41565},[41564],[34,588832],{"className":588833,"style":41508},[31813],[34,588835,588837],{"className":588836,"style":41572},[31685],[34,588838,58442],{"className":588839,"style":32062},[31685,31686],[34,588841,588842,588845],{"style":65012},[34,588843],{"className":588844,"style":41508},[31813],[34,588846,588848],{"className":588847,"style":41586},[41585],[41588,588849,588850],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,588851],{"d":41597},[34,588853,31829],{"className":588854},[31828],[34,588856,588858],{"className":588857},[31801],[34,588859,588861],{"className":588860,"style":65032},[31805],[34,588862],{}," （这个非常重要，做题常考）。",[15,588865,588866],{},[2627,588867,588868,588869],{},"工具 B：贝塔函数 ",[34,588870,588872,588896],{"className":588871},[31642],[34,588873,588875],{"className":588874},[31646],[31648,588876,588877],{"xmlns":31650},[31652,588878,588879,588893],{},[31655,588880,588881,588883,588885,588887,588889,588891],{},[31658,588882,54224],{},[31662,588884,456],{"stretchy":702},[31658,588886,15],{},[31662,588888,31960],{"separator":758},[31658,588890,38759],{},[31662,588892,2557],{"stretchy":702},[31665,588894,588895],{"encoding":31667},"B(p, q)",[34,588897,588899],{"className":588898,"ariaHidden":758},[31672],[34,588900,588902,588905,588908,588911,588914,588917,588920,588923],{"className":588901},[31676],[34,588903],{"className":588904,"style":31774},[31680],[34,588906,54224],{"className":588907,"style":54252},[31685,31686],[34,588909,456],{"className":588910},[31782],[34,588912,15],{"className":588913},[31685,31686],[34,588915,31960],{"className":588916},[32119],[34,588918],{"className":588919,"style":32123},[31845],[34,588921,38759],{"className":588922,"style":32062},[31685,31686],[34,588924,2557],{"className":588925},[31690],[105,588927,588928,589387,589767,589985],{},[108,588929,588930,587964,588932,588984,588985,589013,589014,47649,589042,589103,589104],{},[2627,588931,587963],{},[34,588933,588935,588957],{"className":588934},[31642],[34,588936,588938],{"className":588937},[31646],[31648,588939,588940],{"xmlns":31650},[31652,588941,588942,588954],{},[31655,588943,588944,588946,588948,588950,588952],{},[31662,588945,456],{"stretchy":702},[31749,588947,459],{},[31662,588949,31960],{"separator":758},[31749,588951,640],{},[31662,588953,2557],{"stretchy":702},[31665,588955,588956],{"encoding":31667},"(0, 1)",[34,588958,588960],{"className":588959,"ariaHidden":758},[31672],[34,588961,588963,588966,588969,588972,588975,588978,588981],{"className":588962},[31676],[34,588964],{"className":588965,"style":31774},[31680],[34,588967,456],{"className":588968},[31782],[34,588970,459],{"className":588971},[31685],[34,588973,31960],{"className":588974},[32119],[34,588976],{"className":588977,"style":32123},[31845],[34,588979,640],{"className":588980},[31685],[34,588982,2557],{"className":588983},[31690],"，里面是“",[34,588986,588988,589001],{"className":588987},[31642],[34,588989,588991],{"className":588990},[31646],[31648,588992,588993],{"xmlns":31650},[31652,588994,588995,588999],{},[31655,588996,588997],{},[31658,588998,32936],{},[31665,589000,32936],{"encoding":31667},[34,589002,589004],{"className":589003,"ariaHidden":758},[31672],[34,589005,589007,589010],{"className":589006},[31676],[34,589008],{"className":589009,"style":32527},[31680],[34,589011,32936],{"className":589012},[31685,31686]," 的幂 ",[34,589015,589017,589030],{"className":589016},[31642],[34,589018,589020],{"className":589019},[31646],[31648,589021,589022],{"xmlns":31650},[31652,589023,589024,589028],{},[31655,589025,589026],{},[31662,589027,42071],{},[31665,589029,211013],{"encoding":31667},[34,589031,589033],{"className":589032,"ariaHidden":758},[31672],[34,589034,589036,589039],{"className":589035},[31676],[34,589037],{"className":589038,"style":32597},[31680],[34,589040,42071],{"className":589041},[31685],[34,589043,589045,589067],{"className":589044},[31642],[34,589046,589048],{"className":589047},[31646],[31648,589049,589050],{"xmlns":31650},[31652,589051,589052,589064],{},[31655,589053,589054,589056,589058,589060,589062],{},[31662,589055,456],{"stretchy":702},[31749,589057,640],{},[31662,589059,32226],{},[31658,589061,32936],{},[31662,589063,2557],{"stretchy":702},[31665,589065,589066],{"encoding":31667},"(1-x)",[34,589068,589070,589091],{"className":589069,"ariaHidden":758},[31672],[34,589071,589073,589076,589079,589082,589085,589088],{"className":589072},[31676],[34,589074],{"className":589075,"style":31774},[31680],[34,589077,456],{"className":589078},[31782],[34,589080,640],{"className":589081},[31685],[34,589083],{"className":589084,"style":32604},[31845],[34,589086,32226],{"className":589087},[32024],[34,589089],{"className":589090,"style":32604},[31845],[34,589092,589094,589097,589100],{"className":589093},[31676],[34,589095],{"className":589096,"style":31774},[31680],[34,589098,32936],{"className":589099},[31685,31686],[34,589101,2557],{"className":589102},[31690]," 的幂”。\n",[34,589105,589107,589179],{"className":589106},[31642],[34,589108,589110],{"className":589109},[31646],[31648,589111,589112],{"xmlns":31650},[31652,589113,589114,589176],{},[31655,589115,589116,589118,589120,589122,589124,589126,589128,589130,589138,589150,589152,589154,589156,589158,589170,589172,589174],{},[31658,589117,54224],{},[31662,589119,456],{"stretchy":702},[31658,589121,15],{},[31662,589123,31960],{"separator":758},[31658,589125,38759],{},[31662,589127,2557],{"stretchy":702},[31662,589129,202],{},[38846,589131,589132,589134,589136],{},[31662,589133,546295],{},[31749,589135,459],{},[31749,589137,640],{},[37624,589139,589140,589142],{},[31658,589141,32936],{},[31655,589143,589144,589146,589148],{},[31658,589145,15],{},[31662,589147,32226],{},[31749,589149,640],{},[31662,589151,456],{"stretchy":702},[31749,589153,640],{},[31662,589155,32226],{},[31658,589157,32936],{},[37624,589159,589160,589162],{},[31662,589161,2557],{"stretchy":702},[31655,589163,589164,589166,589168],{},[31658,589165,38759],{},[31662,589167,32226],{},[31749,589169,640],{},[39361,589171,39363],{},[31658,589173,46900],{},[31658,589175,32936],{},[31665,589177,589178],{"encoding":31667},"B(p, q) = \\int_0^1 x^{p-1} (1-x)^{q-1} \\, dx",[34,589180,589182,589218,589331],{"className":589181,"ariaHidden":758},[31672],[34,589183,589185,589188,589191,589194,589197,589200,589203,589206,589209,589212,589215],{"className":589184},[31676],[34,589186],{"className":589187,"style":31774},[31680],[34,589189,54224],{"className":589190,"style":54252},[31685,31686],[34,589192,456],{"className":589193},[31782],[34,589195,15],{"className":589196},[31685,31686],[34,589198,31960],{"className":589199},[32119],[34,589201],{"className":589202,"style":32123},[31845],[34,589204,38759],{"className":589205,"style":32062},[31685,31686],[34,589207,2557],{"className":589208},[31690],[34,589210],{"className":589211,"style":31846},[31845],[34,589213,202],{"className":589214},[31850],[34,589216],{"className":589217,"style":31846},[31845],[34,589219,589221,589224,589275,589278,589316,589319,589322,589325,589328],{"className":589220},[31676],[34,589222],{"className":589223,"style":563309},[31680],[34,589225,589227,589230],{"className":589226},[32767],[34,589228,546295],{"className":589229,"style":546331},[32767,131249,156560],[34,589231,589233],{"className":589232},[31792],[34,589234,589236,589267],{"className":589235},[31796,31797],[34,589237,589239,589264],{"className":589238},[31801],[34,589240,589242,589253],{"className":589241,"style":563328},[31805],[34,589243,589244,589247],{"style":546347},[34,589245],{"className":589246,"style":31814},[31813],[34,589248,589250],{"className":589249},[31818,31819,31820,31821],[34,589251,459],{"className":589252},[31685,31821],[34,589254,589255,589258],{"style":546362},[34,589256],{"className":589257,"style":31814},[31813],[34,589259,589261],{"className":589260},[31818,31819,31820,31821],[34,589262,640],{"className":589263},[31685,31821],[34,589265,31829],{"className":589266},[31828],[34,589268,589270],{"className":589269},[31801],[34,589271,589273],{"className":589272,"style":546384},[31805],[34,589274],{},[34,589276],{"className":589277,"style":32123},[31845],[34,589279,589281,589284],{"className":589280},[31685],[34,589282,32936],{"className":589283},[31685,31686],[34,589285,589287],{"className":589286},[31792],[34,589288,589290],{"className":589289},[31796],[34,589291,589293],{"className":589292},[31801],[34,589294,589296],{"className":589295,"style":37774},[31805],[34,589297,589298,589301],{"style":37777},[34,589299],{"className":589300,"style":31814},[31813],[34,589302,589304],{"className":589303},[31818,31819,31820,31821],[34,589305,589307,589310,589313],{"className":589306},[31685,31821],[34,589308,15],{"className":589309},[31685,31686,31821],[34,589311,32226],{"className":589312},[32024,31821],[34,589314,640],{"className":589315},[31685,31821],[34,589317,456],{"className":589318},[31782],[34,589320,640],{"className":589321},[31685],[34,589323],{"className":589324,"style":32604},[31845],[34,589326,32226],{"className":589327},[32024],[34,589329],{"className":589330,"style":32604},[31845],[34,589332,589334,589337,589340,589378,589381,589384],{"className":589333},[31676],[34,589335],{"className":589336,"style":37755},[31680],[34,589338,32936],{"className":589339},[31685,31686],[34,589341,589343,589346],{"className":589342},[31690],[34,589344,2557],{"className":589345},[31690],[34,589347,589349],{"className":589348},[31792],[34,589350,589352],{"className":589351},[31796],[34,589353,589355],{"className":589354},[31801],[34,589356,589358],{"className":589357,"style":37774},[31805],[34,589359,589360,589363],{"style":37777},[34,589361],{"className":589362,"style":31814},[31813],[34,589364,589366],{"className":589365},[31818,31819,31820,31821],[34,589367,589369,589372,589375],{"className":589368},[31685,31821],[34,589370,38759],{"className":589371,"style":32062},[31685,31686,31821],[34,589373,32226],{"className":589374},[32024,31821],[34,589376,640],{"className":589377},[31685,31821],[34,589379],{"className":589380,"style":32123},[31845],[34,589382,46900],{"className":589383},[31685,31686],[34,589385,32936],{"className":589386},[31685,31686],[108,589388,589389,589392,589393],{},[2627,589390,589391],{},"三角形式","（常用于正弦余弦的积分）：\n",[34,589394,589396,589482],{"className":589395},[31642],[34,589397,589399],{"className":589398},[31646],[31648,589400,589401],{"xmlns":31650},[31652,589402,589403,589479],{},[31655,589404,589405,589407,589409,589411,589413,589415,589417,589419,589421,589433,589451,589453,589471,589473,589475,589477],{},[31658,589406,54224],{},[31662,589408,456],{"stretchy":702},[31658,589410,15],{},[31662,589412,31960],{"separator":758},[31658,589414,38759],{},[31662,589416,2557],{"stretchy":702},[31662,589418,202],{},[31749,589420,768],{},[38846,589422,589423,589425,589427],{},[31662,589424,546295],{},[31749,589426,459],{},[41453,589428,589429,589431],{},[31658,589430,58442],{},[31749,589432,768],{},[37624,589434,589435,589441],{},[31655,589436,589437,589439],{},[31658,589438,547853],{},[31662,589440,32682],{},[31655,589442,589443,589445,589447,589449],{},[31749,589444,768],{},[31658,589446,15],{},[31662,589448,32226],{},[31749,589450,640],{},[31658,589452,479697],{},[37624,589454,589455,589461],{},[31655,589456,589457,589459],{},[31658,589458,547860],{},[31662,589460,32682],{},[31655,589462,589463,589465,589467,589469],{},[31749,589464,768],{},[31658,589466,38759],{},[31662,589468,32226],{},[31749,589470,640],{},[31658,589472,479697],{},[39361,589474,39363],{},[31658,589476,46900],{},[31658,589478,479697],{},[31665,589480,589481],{"encoding":31667},"B(p, q) = 2 \\int_0^{\\frac{\\pi}{2}} \\sin^{2p-1}\\theta \\cos^{2q-1}\\theta \\, d\\theta",[34,589483,589485,589521],{"className":589484,"ariaHidden":758},[31672],[34,589486,589488,589491,589494,589497,589500,589503,589506,589509,589512,589515,589518],{"className":589487},[31676],[34,589489],{"className":589490,"style":31774},[31680],[34,589492,54224],{"className":589493,"style":54252},[31685,31686],[34,589495,456],{"className":589496},[31782],[34,589498,15],{"className":589499},[31685,31686],[34,589501,31960],{"className":589502},[32119],[34,589504],{"className":589505,"style":32123},[31845],[34,589507,38759],{"className":589508,"style":32062},[31685,31686],[34,589510,2557],{"className":589511},[31690],[34,589513],{"className":589514,"style":31846},[31845],[34,589516,202],{"className":589517},[31850],[34,589519],{"className":589520,"style":31846},[31845],[34,589522,589524,589528,589531,589534,589658,589661,589702,589705,589708,589711,589752,589755,589758,589761,589764],{"className":589523},[31676],[34,589525],{"className":589526,"style":589527},[31680],"height:1.3978em;vertical-align:-0.3558em;",[34,589529,768],{"className":589530},[31685],[34,589532],{"className":589533,"style":32123},[31845],[34,589535,589537,589540],{"className":589536},[32767],[34,589538,546295],{"className":589539,"style":546331},[32767,131249,156560],[34,589541,589543],{"className":589542},[31792],[34,589544,589546,589650],{"className":589545},[31796,31797],[34,589547,589549,589647],{"className":589548},[31801],[34,589550,589553,589565],{"className":589551,"style":589552},[31805],"height:1.042em;",[34,589554,589556,589559],{"style":589555},"top:-2.6442em;margin-left:-0.1945em;margin-right:0.05em;",[34,589557],{"className":589558,"style":41508},[31813],[34,589560,589562],{"className":589561},[31818,31819,31820,31821],[34,589563,459],{"className":589564},[31685,31821],[34,589566,589568,589571],{"style":589567},"top:-3.5579em;margin-right:0.05em;",[34,589569],{"className":589570,"style":41508},[31813],[34,589572,589574],{"className":589573},[31818,31819,31820,31821],[34,589575,589577],{"className":589576},[31685,31821],[34,589578,589580,589583,589644],{"className":589579},[31685,31821],[34,589581],{"className":589582},[31782,41488,31818,43410,435704],[34,589584,589586],{"className":589585},[41453],[34,589587,589589,589636],{"className":589588},[31796,31797],[34,589590,589592,589633],{"className":589591},[31801],[34,589593,589596,589610,589618],{"className":589594,"style":589595},[31805],"height:0.6915em;",[34,589597,589598,589601],{"style":435720},[34,589599],{"className":589600,"style":41508},[31813],[34,589602,589604],{"className":589603},[31818,43410,43411,31821],[34,589605,589607],{"className":589606},[31685,31821],[34,589608,768],{"className":589609},[31685,31821],[34,589611,589612,589615],{"style":435735},[34,589613],{"className":589614,"style":41508},[31813],[34,589616],{"className":589617,"style":435742},[41527,31821],[34,589619,589621,589624],{"style":589620},"top:-3.384em;",[34,589622],{"className":589623,"style":41508},[31813],[34,589625,589627],{"className":589626},[31818,43410,43411,31821],[34,589628,589630],{"className":589629},[31685,31821],[34,589631,58442],{"className":589632,"style":32062},[31685,31686,31821],[34,589634,31829],{"className":589635},[31828],[34,589637,589639],{"className":589638},[31801],[34,589640,589642],{"className":589641,"style":435782},[31805],[34,589643],{},[34,589645],{"className":589646},[31690,41488,31818,43410,435704],[34,589648,31829],{"className":589649},[31828],[34,589651,589653],{"className":589652},[31801],[34,589654,589656],{"className":589655,"style":546384},[31805],[34,589657],{},[34,589659],{"className":589660,"style":32123},[31845],[34,589662,589664,589667],{"className":589663},[32767],[34,589665,547853],{"className":589666},[32767],[34,589668,589670],{"className":589669},[31792],[34,589671,589673],{"className":589672},[31796],[34,589674,589676],{"className":589675},[31801],[34,589677,589679],{"className":589678,"style":585884},[31805],[34,589680,589681,589684],{"style":586008},[34,589682],{"className":589683,"style":31814},[31813],[34,589685,589687],{"className":589686},[31818,31819,31820,31821],[34,589688,589690,589693,589696,589699],{"className":589689},[31685,31821],[34,589691,768],{"className":589692},[31685,31821],[34,589694,15],{"className":589695},[31685,31686,31821],[34,589697,32226],{"className":589698},[32024,31821],[34,589700,640],{"className":589701},[31685,31821],[34,589703],{"className":589704,"style":32123},[31845],[34,589706,479697],{"className":589707,"style":34452},[31685,31686],[34,589709],{"className":589710,"style":32123},[31845],[34,589712,589714,589717],{"className":589713},[32767],[34,589715,547860],{"className":589716},[32767],[34,589718,589720],{"className":589719},[31792],[34,589721,589723],{"className":589722},[31796],[34,589724,589726],{"className":589725},[31801],[34,589727,589729],{"className":589728,"style":37774},[31805],[34,589730,589731,589734],{"style":37777},[34,589732],{"className":589733,"style":31814},[31813],[34,589735,589737],{"className":589736},[31818,31819,31820,31821],[34,589738,589740,589743,589746,589749],{"className":589739},[31685,31821],[34,589741,768],{"className":589742},[31685,31821],[34,589744,38759],{"className":589745,"style":32062},[31685,31686,31821],[34,589747,32226],{"className":589748},[32024,31821],[34,589750,640],{"className":589751},[31685,31821],[34,589753],{"className":589754,"style":32123},[31845],[34,589756,479697],{"className":589757,"style":34452},[31685,31686],[34,589759],{"className":589760,"style":32123},[31845],[34,589762,46900],{"className":589763},[31685,31686],[34,589765,479697],{"className":589766,"style":34452},[31685,31686],[108,589768,589769,589772,589773],{},[2627,589770,589771],{},"A与B的关系","（万能公式）：\n",[34,589774,589776,589836],{"className":589775},[31642],[34,589777,589779],{"className":589778},[31646],[31648,589780,589781],{"xmlns":31650},[31652,589782,589783,589833],{},[31655,589784,589785,589787,589789,589791,589793,589795,589797,589799],{},[31658,589786,54224],{},[31662,589788,456],{"stretchy":702},[31658,589790,15],{},[31662,589792,31960],{"separator":758},[31658,589794,38759],{},[31662,589796,2557],{"stretchy":702},[31662,589798,202],{},[41453,589800,589801,589819],{},[31655,589802,589803,589805,589807,589809,589811,589813,589815,589817],{},[31658,589804,564791],{"mathvariant":38079},[31662,589806,456],{"stretchy":702},[31658,589808,15],{},[31662,589810,2557],{"stretchy":702},[31658,589812,564791],{"mathvariant":38079},[31662,589814,456],{"stretchy":702},[31658,589816,38759],{},[31662,589818,2557],{"stretchy":702},[31655,589820,589821,589823,589825,589827,589829,589831],{},[31658,589822,564791],{"mathvariant":38079},[31662,589824,456],{"stretchy":702},[31658,589826,15],{},[31662,589828,223],{},[31658,589830,38759],{},[31662,589832,2557],{"stretchy":702},[31665,589834,589835],{"encoding":31667},"B(p, q) = \\frac{\\Gamma(p)\\Gamma(q)}{\\Gamma(p+q)}",[34,589837,589839,589875],{"className":589838,"ariaHidden":758},[31672],[34,589840,589842,589845,589848,589851,589854,589857,589860,589863,589866,589869,589872],{"className":589841},[31676],[34,589843],{"className":589844,"style":31774},[31680],[34,589846,54224],{"className":589847,"style":54252},[31685,31686],[34,589849,456],{"className":589850},[31782],[34,589852,15],{"className":589853},[31685,31686],[34,589855,31960],{"className":589856},[32119],[34,589858],{"className":589859,"style":32123},[31845],[34,589861,38759],{"className":589862,"style":32062},[31685,31686],[34,589864,2557],{"className":589865},[31690],[34,589867],{"className":589868,"style":31846},[31845],[34,589870,202],{"className":589871},[31850],[34,589873],{"className":589874,"style":31846},[31845],[34,589876,589878,589881],{"className":589877},[31676],[34,589879],{"className":589880,"style":327841},[31680],[34,589882,589884,589887,589982],{"className":589883},[31685],[34,589885],{"className":589886},[31782,41488],[34,589888,589890],{"className":589889},[41453],[34,589891,589893,589974],{"className":589892},[31796,31797],[34,589894,589896,589971],{"className":589895},[31801],[34,589897,589899,589928,589936],{"className":589898,"style":70504},[31805],[34,589900,589901,589904],{"style":41504},[34,589902],{"className":589903,"style":41508},[31813],[34,589905,589907],{"className":589906},[31818,31819,31820,31821],[34,589908,589910,589913,589916,589919,589922,589925],{"className":589909},[31685,31821],[34,589911,564791],{"className":589912},[31685,31821],[34,589914,456],{"className":589915},[31782,31821],[34,589917,15],{"className":589918},[31685,31686,31821],[34,589920,223],{"className":589921},[32024,31821],[34,589923,38759],{"className":589924,"style":32062},[31685,31686,31821],[34,589926,2557],{"className":589927},[31690,31821],[34,589929,589930,589933],{"style":41520},[34,589931],{"className":589932,"style":41508},[31813],[34,589934],{"className":589935,"style":41528},[41527],[34,589937,589938,589941],{"style":70529},[34,589939],{"className":589940,"style":41508},[31813],[34,589942,589944],{"className":589943},[31818,31819,31820,31821],[34,589945,589947,589950,589953,589956,589959,589962,589965,589968],{"className":589946},[31685,31821],[34,589948,564791],{"className":589949},[31685,31821],[34,589951,456],{"className":589952},[31782,31821],[34,589954,15],{"className":589955},[31685,31686,31821],[34,589957,2557],{"className":589958},[31690,31821],[34,589960,564791],{"className":589961},[31685,31821],[34,589963,456],{"className":589964},[31782,31821],[34,589966,38759],{"className":589967,"style":32062},[31685,31686,31821],[34,589969,2557],{"className":589970},[31690,31821],[34,589972,31829],{"className":589973},[31828],[34,589975,589977],{"className":589976},[31801],[34,589978,589980],{"className":589979,"style":324858},[31805],[34,589981],{},[34,589983],{"className":589984},[31690,41488],[108,589986,589987,589990,589991],{},[2627,589988,589989],{},"余元公式","（用来处理奇怪的分数）：\n",[34,589992,589994,590062],{"className":589993},[31642],[34,589995,589997],{"className":589996},[31646],[31648,589998,589999],{"xmlns":31650},[31652,590000,590001,590059],{},[31655,590002,590003,590005,590007,590009,590011,590013,590015,590017,590019,590021,590023,590025,590043,590045,590047,590049,590051,590053,590055,590057],{},[31658,590004,564791],{"mathvariant":38079},[31662,590006,456],{"stretchy":702},[31658,590008,15],{},[31662,590010,2557],{"stretchy":702},[31658,590012,564791],{"mathvariant":38079},[31662,590014,456],{"stretchy":702},[31749,590016,640],{},[31662,590018,32226],{},[31658,590020,15],{},[31662,590022,2557],{"stretchy":702},[31662,590024,202],{},[41453,590026,590027,590029],{},[31658,590028,58442],{},[31655,590030,590031,590033,590035,590037,590039,590041],{},[31658,590032,547853],{},[31662,590034,32682],{},[31662,590036,456],{"stretchy":702},[31658,590038,15],{},[31658,590040,58442],{},[31662,590042,2557],{"stretchy":702},[31845,590044],{"width":71082},[31662,590046,456],{"stretchy":702},[31749,590048,459],{},[31662,590050,932],{},[31658,590052,15],{},[31662,590054,932],{},[31749,590056,640],{},[31662,590058,2557],{"stretchy":702},[31665,590060,590061],{"encoding":31667},"\\Gamma(p)\\Gamma(1-p) = \\frac{\\pi}{\\sin(p\\pi)} \\quad (0\u003Cp\u003C1)",[34,590063,590065,590101,590122,590236,590254],{"className":590064,"ariaHidden":758},[31672],[34,590066,590068,590071,590074,590077,590080,590083,590086,590089,590092,590095,590098],{"className":590067},[31676],[34,590069],{"className":590070,"style":31774},[31680],[34,590072,564791],{"className":590073},[31685],[34,590075,456],{"className":590076},[31782],[34,590078,15],{"className":590079},[31685,31686],[34,590081,2557],{"className":590082},[31690],[34,590084,564791],{"className":590085},[31685],[34,590087,456],{"className":590088},[31782],[34,590090,640],{"className":590091},[31685],[34,590093],{"className":590094,"style":32604},[31845],[34,590096,32226],{"className":590097},[32024],[34,590099],{"className":590100,"style":32604},[31845],[34,590102,590104,590107,590110,590113,590116,590119],{"className":590103},[31676],[34,590105],{"className":590106,"style":31774},[31680],[34,590108,15],{"className":590109},[31685,31686],[34,590111,2557],{"className":590112},[31690],[34,590114],{"className":590115,"style":31846},[31845],[34,590117,202],{"className":590118},[31850],[34,590120],{"className":590121,"style":31846},[31845],[34,590123,590125,590129,590218,590221,590224,590227,590230,590233],{"className":590124},[31676],[34,590126],{"className":590127,"style":590128},[31680],"height:1.27em;vertical-align:-0.52em;",[34,590130,590132,590135,590215],{"className":590131},[31685],[34,590133],{"className":590134},[31782,41488],[34,590136,590138],{"className":590137},[41453],[34,590139,590141,590207],{"className":590140},[31796,31797],[34,590142,590144,590204],{"className":590143},[31801],[34,590145,590147,590182,590190],{"className":590146,"style":244945},[31805],[34,590148,590149,590152],{"style":41504},[34,590150],{"className":590151,"style":41508},[31813],[34,590153,590155],{"className":590154},[31818,31819,31820,31821],[34,590156,590158,590170,590173,590176,590179],{"className":590157},[31685,31821],[34,590159,590161,590164,590167],{"className":590160},[32767,31821],[34,590162,31747],{"className":590163},[31821],[34,590165,31955],{"className":590166},[31821],[34,590168,31660],{"className":590169},[31821],[34,590171,456],{"className":590172},[31782,31821],[34,590174,15],{"className":590175},[31685,31686,31821],[34,590177,58442],{"className":590178,"style":32062},[31685,31686,31821],[34,590180,2557],{"className":590181},[31690,31821],[34,590183,590184,590187],{"style":41520},[34,590185],{"className":590186,"style":41508},[31813],[34,590188],{"className":590189,"style":41528},[41527],[34,590191,590192,590195],{"style":46771},[34,590193],{"className":590194,"style":41508},[31813],[34,590196,590198],{"className":590197},[31818,31819,31820,31821],[34,590199,590201],{"className":590200},[31685,31821],[34,590202,58442],{"className":590203,"style":32062},[31685,31686,31821],[34,590205,31829],{"className":590206},[31828],[34,590208,590210],{"className":590209},[31801],[34,590211,590213],{"className":590212,"style":324858},[31805],[34,590214],{},[34,590216],{"className":590217},[31690,41488],[34,590219],{"className":590220,"style":128438},[31845],[34,590222,456],{"className":590223},[31782],[34,590225,459],{"className":590226},[31685],[34,590228],{"className":590229,"style":31846},[31845],[34,590231,932],{"className":590232},[31850],[34,590234],{"className":590235,"style":31846},[31845],[34,590237,590239,590242,590245,590248,590251],{"className":590238},[31676],[34,590240],{"className":590241,"style":86652},[31680],[34,590243,15],{"className":590244},[31685,31686],[34,590246],{"className":590247,"style":31846},[31845],[34,590249,932],{"className":590250},[31850],[34,590252],{"className":590253,"style":31846},[31845],[34,590255,590257,590260,590263],{"className":590256},[31676],[34,590258],{"className":590259,"style":31774},[31680],[34,590261,640],{"className":590262},[31685],[34,590264,2557],{"className":590265},[31690],[32490,590267,590269],{"id":590268},"_2-第一类曲线积分-对弧长的积分","2. 第一类曲线积分 (对弧长的积分)",[105,590271,590272,590335],{},[108,590273,590274,590276,590277,590334],{},[2627,590275,318465],{},"：计算一段弯曲铁丝的质量——已知铁丝每一处的密度 ",[34,590278,590280,590304],{"className":590279},[31642],[34,590281,590283],{"className":590282},[31646],[31648,590284,590285],{"xmlns":31650},[31652,590286,590287,590301],{},[31655,590288,590289,590291,590293,590295,590297,590299],{},[31658,590290,220084],{},[31662,590292,456],{"stretchy":702},[31658,590294,32936],{},[31662,590296,31960],{"separator":758},[31658,590298,67764],{},[31662,590300,2557],{"stretchy":702},[31665,590302,590303],{"encoding":31667},"\\rho(x,y)",[34,590305,590307],{"className":590306,"ariaHidden":758},[31672],[34,590308,590310,590313,590316,590319,590322,590325,590328,590331],{"className":590309},[31676],[34,590311],{"className":590312,"style":31774},[31680],[34,590314,220084],{"className":590315},[31685,31686],[34,590317,456],{"className":590318},[31782],[34,590320,32936],{"className":590321},[31685,31686],[34,590323,31960],{"className":590324},[32119],[34,590326],{"className":590327,"style":32123},[31845],[34,590329,67764],{"className":590330,"style":32062},[31685,31686],[34,590332,2557],{"className":590333},[31690],"，沿着铁丝把所有微小的质量加起来。",[108,590336,590337,590340,590341,43434,590344,590930],{},[2627,590338,590339],{},"计算方法","（口诀：",[2627,590342,590343],{},"“一换二代三积分”",[2230,590345,590346,590554,590631],{},[108,590347,590348,590351,590352,590380,590381,347],{},[2627,590349,590350],{},"换","：把曲线 ",[34,590353,590355,590368],{"className":590354},[31642],[34,590356,590358],{"className":590357},[31646],[31648,590359,590360],{"xmlns":31650},[31652,590361,590362,590366],{},[31655,590363,590364],{},[31658,590365,33206],{},[31665,590367,33206],{"encoding":31667},[34,590369,590371],{"className":590370,"ariaHidden":758},[31672],[34,590372,590374,590377],{"className":590373},[31676],[34,590375],{"className":590376,"style":33218},[31680],[34,590378,33206],{"className":590379},[31685,31686]," 写成参数方程 ",[34,590382,590384,590440],{"className":590383},[31642],[34,590385,590387],{"className":590386},[31646],[31648,590388,590389],{"xmlns":31650},[31652,590390,590391,590437],{},[31655,590392,590393,590395],{},[31662,590394,32685],{"fence":758},[71078,590396,590397,590417],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,590398,590399],{},[71087,590400,590401],{},[71090,590402,590403],{"scriptlevel":459,"displaystyle":702},[31655,590404,590405,590407,590409,590411,590413,590415],{},[31658,590406,32936],{},[31662,590408,202],{},[31658,590410,32936],{},[31662,590412,456],{"stretchy":702},[31658,590414,33249],{},[31662,590416,2557],{"stretchy":702},[71084,590418,590419],{},[71087,590420,590421],{},[71090,590422,590423],{"scriptlevel":459,"displaystyle":702},[31655,590424,590425,590427,590429,590431,590433,590435],{},[31658,590426,67764],{},[31662,590428,202],{},[31658,590430,67764],{},[31662,590432,456],{"stretchy":702},[31658,590434,33249],{},[31662,590436,2557],{"stretchy":702},[31665,590438,590439],{"encoding":31667},"\\begin{cases} x = x(t) \\\\ y = y(t) 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",[34,590604,590606,590619],{"className":590605},[31642],[34,590607,590609],{"className":590608},[31646],[31648,590610,590611],{"xmlns":31650},[31652,590612,590613,590617],{},[31655,590614,590615],{},[31658,590616,33249],{},[31665,590618,33249],{"encoding":31667},[34,590620,590622],{"className":590621,"ariaHidden":758},[31672],[34,590623,590625,590628],{"className":590624},[31676],[34,590626],{"className":590627,"style":33280},[31680],[34,590629,33249],{"className":590630},[31685,31686],[108,590632,590633,2664,590636,347],{},[2627,590634,590635],{},"算弧长微分",[34,590637,590639,590697],{"className":590638},[31642],[34,590640,590642],{"className":590641},[31646],[31648,590643,590644],{"xmlns":31650},[31652,590645,590646,590694],{},[31655,590647,590648,590650,590652,590654,590688,590690,590692],{},[31658,590649,46900],{},[31658,590651,31747],{},[31662,590653,202],{},[41463,590655,590656],{},[31655,590657,590658,590660,590666,590672,590674,590676,590682],{},[31662,590659,456],{"stretchy":702},[37624,590661,590662,590664],{},[31658,590663,32936],{},[31662,590665,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,590667,590668,590670],{},[31662,590669,2557],{"stretchy":702},[31749,590671,768],{},[31662,590673,223],{},[31662,590675,456],{"stretchy":702},[37624,590677,590678,590680],{},[31658,590679,67764],{},[31662,590681,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,590683,590684,590686],{},[31662,590685,2557],{"stretchy":702},[31749,590687,768],{},[39361,590689,39363],{},[31658,590691,46900],{},[31658,590693,33249],{},[31665,590695,590696],{"encoding":31667},"ds = \\sqrt{(x')^2 + (y')^2} \\, dt",[34,590698,590700,590721],{"className":590699,"ariaHidden":758},[31672],[34,590701,590703,590706,590709,590712,590715,590718],{"className":590702},[31676],[34,590704],{"className":590705,"style":31681},[31680],[34,590707,46900],{"className":590708},[31685,31686],[34,590710,31747],{"className":590711},[31685,31686],[34,590713],{"className":590714,"style":31846},[31845],[34,590716,202],{"className":590717},[31850],[34,590719],{"className":590720,"style":31846},[31845],[34,590722,590724,590728,590921,590924,590927],{"className":590723},[31676],[34,590725],{"className":590726,"style":590727},[31680],"height:1.24em;vertical-align:-0.305em;",[34,590729,590731],{"className":590730},[31685,41550],[34,590732,590734,590912],{"className":590733},[31796,31797],[34,590735,590737,590909],{"className":590736},[31801],[34,590738,590741,590892],{"className":590739,"style":590740},[31805],"height:0.935em;",[34,590742,590745,590748],{"className":590743,"style":590744},[41564],"top:-3.2em;",[34,590746],{"className":590747,"style":552795},[31813],[34,590749,590752,590755,590789,590819,590822,590825,590828,590831,590863],{"className":590750,"style":590751},[31685],"padding-left:1em;",[34,590753,456],{"className":590754},[31782],[34,590756,590758,590761],{"className":590757},[31685],[34,590759,32936],{"className":590760},[31685,31686],[34,590762,590764],{"className":590763},[31792],[34,590765,590767],{"className":590766},[31796],[34,590768,590770],{"className":590769},[31801],[34,590771,590774],{"className":590772,"style":590773},[31805],"height:0.6779em;",[34,590775,590777,590780],{"style":590776},"top:-2.989em;margin-right:0.05em;",[34,590778],{"className":590779,"style":31814},[31813],[34,590781,590783],{"className":590782},[31818,31819,31820,31821],[34,590784,590786],{"className":590785},[31685,31821],[34,590787,116495],{"className":590788},[31685,31821],[34,590790,590792,590795],{"className":590791},[31690],[34,590793,2557],{"className":590794},[31690],[34,590796,590798],{"className":590797},[31792],[34,590799,590801],{"className":590800},[31796],[34,590802,590804],{"className":590803},[31801],[34,590805,590808],{"className":590806,"style":590807},[31805],"height:0.7401em;",[34,590809,590810,590813],{"style":590776},[34,590811],{"className":590812,"style":31814},[31813],[34,590814,590816],{"className":590815},[31818,31819,31820,31821],[34,590817,768],{"className":590818},[31685,31821],[34,590820],{"className":590821,"style":32604},[31845],[34,590823,223],{"className":590824},[32024],[34,590826],{"className":590827,"style":32604},[31845],[34,590829,456],{"className":590830},[31782],[34,590832,590834,590837],{"className":590833},[31685],[34,590835,67764],{"className":590836,"style":32062},[31685,31686],[34,590838,590840],{"className":590839},[31792],[34,590841,590843],{"className":590842},[31796],[34,590844,590846],{"className":590845},[31801],[34,590847,590849],{"className":590848,"style":590773},[31805],[34,590850,590851,590854],{"style":590776},[34,590852],{"className":590853,"style":31814},[31813],[34,590855,590857],{"className":590856},[31818,31819,31820,31821],[34,590858,590860],{"className":590859},[31685,31821],[34,590861,116495],{"className":590862},[31685,31821],[34,590864,590866,590869],{"className":590865},[31690],[34,590867,2557],{"className":590868},[31690],[34,590870,590872],{"className":590871},[31792],[34,590873,590875],{"className":590874},[31796],[34,590876,590878],{"className":590877},[31801],[34,590879,590881],{"className":590880,"style":590807},[31805],[34,590882,590883,590886],{"style":590776},[34,590884],{"className":590885,"style":31814},[31813],[34,590887,590889],{"className":590888},[31818,31819,31820,31821],[34,590890,768],{"className":590891},[31685,31821],[34,590893,590895,590898],{"style":590894},"top:-2.895em;",[34,590896],{"className":590897,"style":552795},[31813],[34,590899,590902],{"className":590900,"style":590901},[41585],"min-width:1.02em;height:1.28em;",[41588,590903,590906],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},"1.28em","0 0 400000 1296",[20398,590907],{"d":590908},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[34,590910,31829],{"className":590911},[31828],[34,590913,590915],{"className":590914},[31801],[34,590916,590919],{"className":590917,"style":590918},[31805],"height:0.305em;",[34,590920],{},[34,590922],{"className":590923,"style":32123},[31845],[34,590925,46900],{"className":590926},[31685,31686],[34,590928,33249],{"className":590929},[31685,31686],[105,590931,590932],{},[108,590933,590934,590935,590963,590964,347],{},"最后变成一个关于 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的普通定积分：",[34,590965,590967,591051],{"className":590966},[31642],[34,590968,590970],{"className":590969},[31646],[31648,590971,590972],{"xmlns":31650},[31652,590973,590974,591048],{},[31655,590975,590976,590984,590986,590988,590990,590992,590994,590996,590998,591000,591002,591004,591006,591008,591042,591044,591046],{},[38846,590977,590978,590980,590982],{},[31662,590979,546295],{},[31658,590981,217392],{},[31658,590983,217691],{},[31658,590985,31739],{},[31662,590987,456],{"stretchy":702},[31658,590989,32936],{},[31662,590991,456],{"stretchy":702},[31658,590993,33249],{},[31662,590995,2557],{"stretchy":702},[31662,590997,31960],{"separator":758},[31658,590999,67764],{},[31662,591001,456],{"stretchy":702},[31658,591003,33249],{},[31662,591005,2557],{"stretchy":702},[31662,591007,2557],{"stretchy":702},[41463,591009,591010],{},[31655,591011,591012,591014,591020,591026,591028,591030,591036],{},[31662,591013,456],{"stretchy":702},[37624,591015,591016,591018],{},[31658,591017,32936],{},[31662,591019,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,591021,591022,591024],{},[31662,591023,2557],{"stretchy":702},[31749,591025,768],{},[31662,591027,223],{},[31662,591029,456],{"stretchy":702},[37624,591031,591032,591034],{},[31658,591033,67764],{},[31662,591035,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,591037,591038,591040],{},[31662,591039,2557],{"stretchy":702},[31749,591041,768],{},[39361,591043,39363],{},[31658,591045,46900],{},[31658,591047,33249],{},[31665,591049,591050],{"encoding":31667},"\\int_\\alpha^\\beta f(x(t), y(t)) \\sqrt{(x')^2 + (y')^2} \\, dt",[34,591052,591054],{"className":591053,"ariaHidden":758},[31672],[34,591055,591057,591060,591111,591114,591117,591120,591123,591126,591129,591132,591135,591138,591141,591144,591147,591150,591331,591334,591337],{"className":591056},[31676],[34,591058],{"className":591059,"style":546324},[31680],[34,591061,591063,591066],{"className":591062},[32767],[34,591064,546295],{"className":591065,"style":546331},[32767,131249,156560],[34,591067,591069],{"className":591068},[31792],[34,591070,591072,591103],{"className":591071},[31796,31797],[34,591073,591075,591100],{"className":591074},[31801],[34,591076,591078,591089],{"className":591077,"style":546344},[31805],[34,591079,591080,591083],{"style":546347},[34,591081],{"className":591082,"style":31814},[31813],[34,591084,591086],{"className":591085},[31818,31819,31820,31821],[34,591087,217392],{"className":591088,"style":217496},[31685,31686,31821],[34,591090,591091,591094],{"style":546362},[34,591092],{"className":591093,"style":31814},[31813],[34,591095,591097],{"className":591096},[31818,31819,31820,31821],[34,591098,217691],{"className":591099,"style":217853},[31685,31686,31821],[34,591101,31829],{"className":591102},[31828],[34,591104,591106],{"className":591105},[31801],[34,591107,591109],{"className":591108,"style":546384},[31805],[34,591110],{},[34,591112],{"className":591113,"style":32123},[31845],[34,591115,31739],{"className":591116,"style":31778},[31685,31686],[34,591118,456],{"className":591119},[31782],[34,591121,32936],{"className":591122},[31685,31686],[34,591124,456],{"className":591125},[31782],[34,591127,33249],{"className":591128},[31685,31686],[34,591130,2557],{"className":591131},[31690],[34,591133,31960],{"className":591134},[32119],[34,591136],{"className":591137,"style":32123},[31845],[34,591139,67764],{"className":591140,"style":32062},[31685,31686],[34,591142,456],{"className":591143},[31782],[34,591145,33249],{"className":591146},[31685,31686],[34,591148,44253],{"className":591149},[31690],[34,591151,591153],{"className":591152},[31685,41550],[34,591154,591156,591323],{"className":591155},[31796,31797],[34,591157,591159,591320],{"className":591158},[31801],[34,591160,591162,591308],{"className":591161,"style":590740},[31805],[34,591163,591165,591168],{"className":591164,"style":590744},[41564],[34,591166],{"className":591167,"style":552795},[31813],[34,591169,591171,591174,591206,591235,591238,591241,591244,591247,591279],{"className":591170,"style":590751},[31685],[34,591172,456],{"className":591173},[31782],[34,591175,591177,591180],{"className":591176},[31685],[34,591178,32936],{"className":591179},[31685,31686],[34,591181,591183],{"className":591182},[31792],[34,591184,591186],{"className":591185},[31796],[34,591187,591189],{"className":591188},[31801],[34,591190,591192],{"className":591191,"style":590773},[31805],[34,591193,591194,591197],{"style":590776},[34,591195],{"className":591196,"style":31814},[31813],[34,591198,591200],{"className":591199},[31818,31819,31820,31821],[34,591201,591203],{"className":591202},[31685,31821],[34,591204,116495],{"className":591205},[31685,31821],[34,591207,591209,591212],{"className":591208},[31690],[34,591210,2557],{"className":591211},[31690],[34,591213,591215],{"className":591214},[31792],[34,591216,591218],{"className":591217},[31796],[34,591219,591221],{"className":591220},[31801],[34,591222,591224],{"className":591223,"style":590807},[31805],[34,591225,591226,591229],{"style":590776},[34,591227],{"className":591228,"style":31814},[31813],[34,591230,591232],{"className":591231},[31818,31819,31820,31821],[34,591233,768],{"className":591234},[31685,31821],[34,591236],{"className":591237,"style":32604},[31845],[34,591239,223],{"className":591240},[32024],[34,591242],{"className":591243,"style":32604},[31845],[34,591245,456],{"className":591246},[31782],[34,591248,591250,591253],{"className":591249},[31685],[34,591251,67764],{"className":591252,"style":32062},[31685,31686],[34,591254,591256],{"className":591255},[31792],[34,591257,591259],{"className":591258},[31796],[34,591260,591262],{"className":591261},[31801],[34,591263,591265],{"className":591264,"style":590773},[31805],[34,591266,591267,591270],{"style":590776},[34,591268],{"className":591269,"style":31814},[31813],[34,591271,591273],{"className":591272},[31818,31819,31820,31821],[34,591274,591276],{"className":591275},[31685,31821],[34,591277,116495],{"className":591278},[31685,31821],[34,591280,591282,591285],{"className":591281},[31690],[34,591283,2557],{"className":591284},[31690],[34,591286,591288],{"className":591287},[31792],[34,591289,591291],{"className":591290},[31796],[34,591292,591294],{"className":591293},[31801],[34,591295,591297],{"className":591296,"style":590807},[31805],[34,591298,591299,591302],{"style":590776},[34,591300],{"className":591301,"style":31814},[31813],[34,591303,591305],{"className":591304},[31818,31819,31820,31821],[34,591306,768],{"className":591307},[31685,31821],[34,591309,591310,591313],{"style":590894},[34,591311],{"className":591312,"style":552795},[31813],[34,591314,591316],{"className":591315,"style":590901},[41585],[41588,591317,591318],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,591319],{"d":590908},[34,591321,31829],{"className":591322},[31828],[34,591324,591326],{"className":591325},[31801],[34,591327,591329],{"className":591328,"style":590918},[31805],[34,591330],{},[34,591332],{"className":591333,"style":32123},[31845],[34,591335,46900],{"className":591336},[31685,31686],[34,591338,33249],{"className":591339},[31685,31686],[32188,591341],{},[253,591343,591345],{"id":591344},"第二部分p340-相关题目详细讲解","第二部分：P340 相关题目详细讲解",[15,591347,591348,591349,4374,591377,591405],{},"解答原则：能化成 ",[34,591350,591352,591365],{"className":591351},[31642],[34,591353,591355],{"className":591354},[31646],[31648,591356,591357],{"xmlns":31650},[31652,591358,591359,591363],{},[31655,591360,591361],{},[31658,591362,564791],{"mathvariant":38079},[31665,591364,564794],{"encoding":31667},[34,591366,591368],{"className":591367,"ariaHidden":758},[31672],[34,591369,591371,591374],{"className":591370},[31676],[34,591372],{"className":591373,"style":33218},[31680],[34,591375,564791],{"className":591376},[31685],[34,591378,591380,591393],{"className":591379},[31642],[34,591381,591383],{"className":591382},[31646],[31648,591384,591385],{"xmlns":31650},[31652,591386,591387,591391],{},[31655,591388,591389],{},[31658,591390,54224],{},[31665,591392,54224],{"encoding":31667},[34,591394,591396],{"className":591395,"ariaHidden":758},[31672],[34,591397,591399,591402],{"className":591398},[31676],[34,591400],{"className":591401,"style":33218},[31680],[34,591403,54224],{"className":591404,"style":54252},[31685,31686]," 函数的，尽量往那上面凑。",[32490,591407,591409],{"id":591408},"t1-计算下列积分","T1. 计算下列积分",[15,591411,591412,47649,591414],{},[2627,591413,263595],{},[34,591415,591417,591459],{"className":591416},[31642],[34,591418,591420],{"className":591419},[31646],[31648,591421,591422],{"xmlns":31650},[31652,591423,591424,591456],{},[31655,591425,591426],{},[71090,591427,591428,591436,591450,591452,591454],{"scriptlevel":459,"displaystyle":758},[38846,591429,591430,591432,591434],{},[31662,591431,546295],{},[31749,591433,459],{},[31749,591435,640],{},[41463,591437,591438],{},[31655,591439,591440,591442,591444],{},[31658,591441,32936],{},[31662,591443,32226],{},[37624,591445,591446,591448],{},[31658,591447,32936],{},[31749,591449,768],{},[39361,591451,39363],{},[31658,591453,46900],{},[31658,591455,32936],{},[31665,591457,591458],{"encoding":31667},"\\displaystyle \\int_0^1 \\sqrt{x - x^2} \\, dx",[34,591460,591462],{"className":591461,"ariaHidden":758},[31672],[34,591463,591465,591469,591525,591528,591616,591619,591622],{"className":591464},[31676],[34,591466],{"className":591467,"style":591468},[31680],"height:2.476em;vertical-align:-0.9119em;",[34,591470,591472,591476],{"className":591471},[32767],[34,591473,546295],{"className":591474,"style":591475},[32767,131249,131250],"margin-right:0.4445em;position:relative;top:-0.0011em;",[34,591477,591479],{"className":591478},[31792],[34,591480,591482,591516],{"className":591481},[31796,31797],[34,591483,591485,591513],{"className":591484},[31801],[34,591486,591489,591501],{"className":591487,"style":591488},[31805],"height:1.564em;",[34,591490,591492,591495],{"style":591491},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[34,591493],{"className":591494,"style":31814},[31813],[34,591496,591498],{"className":591497},[31818,31819,31820,31821],[34,591499,459],{"className":591500},[31685,31821],[34,591502,591504,591507],{"style":591503},"top:-3.8129em;margin-right:0.05em;",[34,591505],{"className":591506,"style":31814},[31813],[34,591508,591510],{"className":591509},[31818,31819,31820,31821],[34,591511,640],{"className":591512},[31685,31821],[34,591514,31829],{"className":591515},[31828],[34,591517,591519],{"className":591518},[31801],[34,591520,591523],{"className":591521,"style":591522},[31805],"height:0.9119em;",[34,591524],{},[34,591526],{"className":591527,"style":32123},[31845],[34,591529,591531],{"className":591530},[31685,41550],[34,591532,591534,591607],{"className":591533},[31796,31797],[34,591535,591537,591604],{"className":591536},[31801],[34,591538,591541,591591],{"className":591539,"style":591540},[31805],"height:1.0623em;",[34,591542,591544,591547],{"className":591543,"style":590744},[41564],[34,591545],{"className":591546,"style":552795},[31813],[34,591548,591550,591553,591556,591559,591562],{"className":591549,"style":590751},[31685],[34,591551,32936],{"className":591552},[31685,31686],[34,591554],{"className":591555,"style":32604},[31845],[34,591557,32226],{"className":591558},[32024],[34,591560],{"className":591561,"style":32604},[31845],[34,591563,591565,591568],{"className":591564},[31685],[34,591566,32936],{"className":591567},[31685,31686],[34,591569,591571],{"className":591570},[31792],[34,591572,591574],{"className":591573},[31796],[34,591575,591577],{"className":591576},[31801],[34,591578,591580],{"className":591579,"style":590807},[31805],[34,591581,591582,591585],{"style":590776},[34,591583],{"className":591584,"style":31814},[31813],[34,591586,591588],{"className":591587},[31818,31819,31820,31821],[34,591589,768],{"className":591590},[31685,31821],[34,591592,591594,591597],{"style":591593},"top:-3.0223em;",[34,591595],{"className":591596,"style":552795},[31813],[34,591598,591600],{"className":591599,"style":590901},[41585],[41588,591601,591602],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,591603],{"d":590908},[34,591605,31829],{"className":591606},[31828],[34,591608,591610],{"className":591609},[31801],[34,591611,591614],{"className":591612,"style":591613},[31805],"height:0.1777em;",[34,591615],{},[34,591617],{"className":591618,"style":32123},[31845],[34,591620,46900],{"className":591621},[31685,31686],[34,591623,32936],{"className":591624},[31685,31686],[105,591626,591627,591713,594080],{},[108,591628,591629,591631,591632,591683,591684,591712],{},[2627,591630,419963],{},"：看到积分区间 ",[34,591633,591635,591656],{"className":591634},[31642],[34,591636,591638],{"className":591637},[31646],[31648,591639,591640],{"xmlns":31650},[31652,591641,591642,591654],{},[31655,591643,591644,591646,591648,591650,591652],{},[31662,591645,456],{"stretchy":702},[31749,591647,459],{},[31662,591649,31960],{"separator":758},[31749,591651,640],{},[31662,591653,2557],{"stretchy":702},[31665,591655,484810],{"encoding":31667},[34,591657,591659],{"className":591658,"ariaHidden":758},[31672],[34,591660,591662,591665,591668,591671,591674,591677,591680],{"className":591661},[31676],[34,591663],{"className":591664,"style":31774},[31680],[34,591666,456],{"className":591667},[31782],[34,591669,459],{"className":591670},[31685],[34,591672,31960],{"className":591673},[32119],[34,591675],{"className":591676,"style":32123},[31845],[34,591678,640],{"className":591679},[31685],[34,591681,2557],{"className":591682},[31690]," 和根号下的二次项，先提取公因式凑 ",[34,591685,591687,591700],{"className":591686},[31642],[34,591688,591690],{"className":591689},[31646],[31648,591691,591692],{"xmlns":31650},[31652,591693,591694,591698],{},[31655,591695,591696],{},[31658,591697,54224],{},[31665,591699,54224],{"encoding":31667},[34,591701,591703],{"className":591702,"ariaHidden":758},[31672],[34,591704,591706,591709],{"className":591705},[31676],[34,591707],{"className":591708,"style":33218},[31680],[34,591710,54224],{"className":591711,"style":54252},[31685,31686]," 函数定义。",[108,591714,591715,33183,591717],{},[2627,591716,289021],{},[2230,591718,591719,592087,592376,592628,592840,593347,593543],{},[108,591720,591721,591722,347],{},"变形被积函数：",[34,591723,591725,591803],{"className":591724},[31642],[34,591726,591728],{"className":591727},[31646],[31648,591729,591730],{"xmlns":31650},[31652,591731,591732,591800],{},[31655,591733,591734,591748,591750,591766,591768,591780,591782,591784,591786,591788],{},[41463,591735,591736],{},[31655,591737,591738,591740,591742],{},[31658,591739,32936],{},[31662,591741,32226],{},[37624,591743,591744,591746],{},[31658,591745,32936],{},[31749,591747,768],{},[31662,591749,202],{},[41463,591751,591752],{},[31655,591753,591754,591756,591758,591760,591762,591764],{},[31658,591755,32936],{},[31662,591757,456],{"stretchy":702},[31749,591759,640],{},[31662,591761,32226],{},[31658,591763,32936],{},[31662,591765,2557],{"stretchy":702},[31662,591767,202],{},[37624,591769,591770,591772],{},[31658,591771,32936],{},[31655,591773,591774,591776,591778],{},[31749,591775,640],{},[31658,591777,38080],{"mathvariant":38079},[31749,591779,768],{},[31662,591781,456],{"stretchy":702},[31749,591783,640],{},[31662,591785,32226],{},[31658,591787,32936],{},[37624,591789,591790,591792],{},[31662,591791,2557],{"stretchy":702},[31655,591793,591794,591796,591798],{},[31749,591795,640],{},[31658,591797,38080],{"mathvariant":38079},[31749,591799,768],{},[31665,591801,591802],{"encoding":31667},"\\sqrt{x - x^2} = \\sqrt{x(1-x)} = x^{1\u002F2} (1-x)^{1\u002F2}",[34,591804,591806,591910,591993,592046],{"className":591805,"ariaHidden":758},[31672],[34,591807,591809,591813,591901,591904,591907],{"className":591808},[31676],[34,591810],{"className":591811,"style":591812},[31680],"height:1.04em;vertical-align:-0.1266em;",[34,591814,591816],{"className":591815},[31685,41550],[34,591817,591819,591892],{"className":591818},[31796,31797],[34,591820,591822,591889],{"className":591821},[31801],[34,591823,591826,591876],{"className":591824,"style":591825},[31805],"height:0.9134em;",[34,591827,591829,591832],{"className":591828,"style":41565},[41564],[34,591830],{"className":591831,"style":41508},[31813],[34,591833,591835,591838,591841,591844,591847],{"className":591834,"style":41572},[31685],[34,591836,32936],{"className":591837},[31685,31686],[34,591839],{"className":591840,"style":32604},[31845],[34,591842,32226],{"className":591843},[32024],[34,591845],{"className":591846,"style":32604},[31845],[34,591848,591850,591853],{"className":591849},[31685],[34,591851,32936],{"className":591852},[31685,31686],[34,591854,591856],{"className":591855},[31792],[34,591857,591859],{"className":591858},[31796],[34,591860,591862],{"className":591861},[31801],[34,591863,591865],{"className":591864,"style":590807},[31805],[34,591866,591867,591870],{"style":590776},[34,591868],{"className":591869,"style":31814},[31813],[34,591871,591873],{"className":591872},[31818,31819,31820,31821],[34,591874,768],{"className":591875},[31685,31821],[34,591877,591879,591882],{"style":591878},"top:-2.8734em;",[34,591880],{"className":591881,"style":41508},[31813],[34,591883,591885],{"className":591884,"style":41586},[41585],[41588,591886,591887],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,591888],{"d":41597},[34,591890,31829],{"className":591891},[31828],[34,591893,591895],{"className":591894},[31801],[34,591896,591899],{"className":591897,"style":591898},[31805],"height:0.1266em;",[34,591900],{},[34,591902],{"className":591903,"style":31846},[31845],[34,591905,202],{"className":591906},[31850],[34,591908],{"className":591909,"style":31846},[31845],[34,591911,591913,591916,591984,591987,591990],{"className":591912},[31676],[34,591914],{"className":591915,"style":590727},[31680],[34,591917,591919],{"className":591918},[31685,41550],[34,591920,591922,591976],{"className":591921},[31796,31797],[34,591923,591925,591973],{"className":591924},[31801],[34,591926,591928,591961],{"className":591927,"style":590740},[31805],[34,591929,591931,591934],{"className":591930,"style":590744},[41564],[34,591932],{"className":591933,"style":552795},[31813],[34,591935,591937,591940,591943,591946,591949,591952,591955,591958],{"className":591936,"style":590751},[31685],[34,591938,32936],{"className":591939},[31685,31686],[34,591941,456],{"className":591942},[31782],[34,591944,640],{"className":591945},[31685],[34,591947],{"className":591948,"style":32604},[31845],[34,591950,32226],{"className":591951},[32024],[34,591953],{"className":591954,"style":32604},[31845],[34,591956,32936],{"className":591957},[31685,31686],[34,591959,2557],{"className":591960},[31690],[34,591962,591963,591966],{"style":590894},[34,591964],{"className":591965,"style":552795},[31813],[34,591967,591969],{"className":591968,"style":590901},[41585],[41588,591970,591971],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,591972],{"d":590908},[34,591974,31829],{"className":591975},[31828],[34,591977,591979],{"className":591978},[31801],[34,591980,591982],{"className":591981,"style":590918},[31805],[34,591983],{},[34,591985],{"className":591986,"style":31846},[31845],[34,591988,202],{"className":591989},[31850],[34,591991],{"className":591992,"style":31846},[31845],[34,591994,591996,591999,592031,592034,592037,592040,592043],{"className":591995},[31676],[34,591997],{"className":591998,"style":61012},[31680],[34,592000,592002,592005],{"className":592001},[31685],[34,592003,32936],{"className":592004},[31685,31686],[34,592006,592008],{"className":592007},[31792],[34,592009,592011],{"className":592010},[31796],[34,592012,592014],{"className":592013},[31801],[34,592015,592017],{"className":592016,"style":61037},[31805],[34,592018,592019,592022],{"style":37777},[34,592020],{"className":592021,"style":31814},[31813],[34,592023,592025],{"className":592024},[31818,31819,31820,31821],[34,592026,592028],{"className":592027},[31685,31821],[34,592029,42407],{"className":592030},[31685,31821],[34,592032,456],{"className":592033},[31782],[34,592035,640],{"className":592036},[31685],[34,592038],{"className":592039,"style":32604},[31845],[34,592041,32226],{"className":592042},[32024],[34,592044],{"className":592045,"style":32604},[31845],[34,592047,592049,592052,592055],{"className":592048},[31676],[34,592050],{"className":592051,"style":61012},[31680],[34,592053,32936],{"className":592054},[31685,31686],[34,592056,592058,592061],{"className":592057},[31690],[34,592059,2557],{"className":592060},[31690],[34,592062,592064],{"className":592063},[31792],[34,592065,592067],{"className":592066},[31796],[34,592068,592070],{"className":592069},[31801],[34,592071,592073],{"className":592072,"style":61037},[31805],[34,592074,592075,592078],{"style":37777},[34,592076],{"className":592077,"style":31814},[31813],[34,592079,592081],{"className":592080},[31818,31819,31820,31821],[34,592082,592084],{"className":592083},[31685,31821],[34,592085,42407],{"className":592086},[31685,31821],[108,592088,592089,592090,592147,592148,347],{},"这就完全符合 ",[34,592091,592093,592117],{"className":592092},[31642],[34,592094,592096],{"className":592095},[31646],[31648,592097,592098],{"xmlns":31650},[31652,592099,592100,592114],{},[31655,592101,592102,592104,592106,592108,592110,592112],{},[31658,592103,54224],{},[31662,592105,456],{"stretchy":702},[31658,592107,15],{},[31662,592109,31960],{"separator":758},[31658,592111,38759],{},[31662,592113,2557],{"stretchy":702},[31665,592115,592116],{"encoding":31667},"B(p,q)",[34,592118,592120],{"className":592119,"ariaHidden":758},[31672],[34,592121,592123,592126,592129,592132,592135,592138,592141,592144],{"className":592122},[31676],[34,592124],{"className":592125,"style":31774},[31680],[34,592127,54224],{"className":592128,"style":54252},[31685,31686],[34,592130,456],{"className":592131},[31782],[34,592133,15],{"className":592134},[31685,31686],[34,592136,31960],{"className":592137},[32119],[34,592139],{"className":592140,"style":32123},[31845],[34,592142,38759],{"className":592143,"style":32062},[31685,31686],[34,592145,2557],{"className":592146},[31690]," 的定义：",[34,592149,592151,592207],{"className":592150},[31642],[34,592152,592154],{"className":592153},[31646],[31648,592155,592156],{"xmlns":31650},[31652,592157,592158,592204],{},[31655,592159,592160,592168,592180,592182,592184,592186,592188,592200,592202],{},[38846,592161,592162,592164,592166],{},[31662,592163,546295],{},[31749,592165,459],{},[31749,592167,640],{},[37624,592169,592170,592172],{},[31658,592171,32936],{},[31655,592173,592174,592176,592178],{},[31658,592175,15],{},[31662,592177,32226],{},[31749,592179,640],{},[31662,592181,456],{"stretchy":702},[31749,592183,640],{},[31662,592185,32226],{},[31658,592187,32936],{},[37624,592189,592190,592192],{},[31662,592191,2557],{"stretchy":702},[31655,592193,592194,592196,592198],{},[31658,592195,38759],{},[31662,592197,32226],{},[31749,592199,640],{},[31658,592201,46900],{},[31658,592203,32936],{},[31665,592205,592206],{"encoding":31667},"\\int_0^1 x^{p-1}(1-x)^{q-1} dx",[34,592208,592210,592323],{"className":592209,"ariaHidden":758},[31672],[34,592211,592213,592216,592267,592270,592308,592311,592314,592317,592320],{"className":592212},[31676],[34,592214],{"className":592215,"style":563309},[31680],[34,592217,592219,592222],{"className":592218},[32767],[34,592220,546295],{"className":592221,"style":546331},[32767,131249,156560],[34,592223,592225],{"className":592224},[31792],[34,592226,592228,592259],{"className":592227},[31796,31797],[34,592229,592231,592256],{"className":592230},[31801],[34,592232,592234,592245],{"className":592233,"style":563328},[31805],[34,592235,592236,592239],{"style":546347},[34,592237],{"className":592238,"style":31814},[31813],[34,592240,592242],{"className":592241},[31818,31819,31820,31821],[34,592243,459],{"className":592244},[31685,31821],[34,592246,592247,592250],{"style":546362},[34,592248],{"className":592249,"style":31814},[31813],[34,592251,592253],{"className":592252},[31818,31819,31820,31821],[34,592254,640],{"className":592255},[31685,31821],[34,592257,31829],{"className":592258},[31828],[34,592260,592262],{"className":592261},[31801],[34,592263,592265],{"className":592264,"style":546384},[31805],[34,592266],{},[34,592268],{"className":592269,"style":32123},[31845],[34,592271,592273,592276],{"className":592272},[31685],[34,592274,32936],{"className":592275},[31685,31686],[34,592277,592279],{"className":592278},[31792],[34,592280,592282],{"className":592281},[31796],[34,592283,592285],{"className":592284},[31801],[34,592286,592288],{"className":592287,"style":37774},[31805],[34,592289,592290,592293],{"style":37777},[34,592291],{"className":592292,"style":31814},[31813],[34,592294,592296],{"className":592295},[31818,31819,31820,31821],[34,592297,592299,592302,592305],{"className":592298},[31685,31821],[34,592300,15],{"className":592301},[31685,31686,31821],[34,592303,32226],{"className":592304},[32024,31821],[34,592306,640],{"className":592307},[31685,31821],[34,592309,456],{"className":592310},[31782],[34,592312,640],{"className":592313},[31685],[34,592315],{"className":592316,"style":32604},[31845],[34,592318,32226],{"className":592319},[32024],[34,592321],{"className":592322,"style":32604},[31845],[34,592324,592326,592329,592332,592370,592373],{"className":592325},[31676],[34,592327],{"className":592328,"style":37755},[31680],[34,592330,32936],{"className":592331},[31685,31686],[34,592333,592335,592338],{"className":592334},[31690],[34,592336,2557],{"className":592337},[31690],[34,592339,592341],{"className":592340},[31792],[34,592342,592344],{"className":592343},[31796],[34,592345,592347],{"className":592346},[31801],[34,592348,592350],{"className":592349,"style":37774},[31805],[34,592351,592352,592355],{"style":37777},[34,592353],{"className":592354,"style":31814},[31813],[34,592356,592358],{"className":592357},[31818,31819,31820,31821],[34,592359,592361,592364,592367],{"className":592360},[31685,31821],[34,592362,38759],{"className":592363,"style":32062},[31685,31686,31821],[34,592365,32226],{"className":592366},[32024,31821],[34,592368,640],{"className":592369},[31685,31821],[34,592371,46900],{"className":592372},[31685,31686],[34,592374,32936],{"className":592375},[31685,31686],[108,592377,25496,592378,46528,592503,347],{},[34,592379,592381,592419],{"className":592380},[31642],[34,592382,592384],{"className":592383},[31646],[31648,592385,592386],{"xmlns":31650},[31652,592387,592388,592416],{},[31655,592389,592390,592392,592394,592396,592398,592400,592402,592404,592406,592408,592410,592412,592414],{},[31658,592391,15],{},[31662,592393,32226],{},[31749,592395,640],{},[31662,592397,202],{},[31749,592399,640],{},[31658,592401,38080],{"mathvariant":38079},[31749,592403,768],{},[31662,592405,140094],{},[31658,592407,15],{},[31662,592409,202],{},[31749,592411,786],{},[31658,592413,38080],{"mathvariant":38079},[31749,592415,768],{},[31665,592417,592418],{"encoding":31667},"p-1 = 1\u002F2 \\Rightarrow p = 3\u002F2",[34,592420,592422,592440,592458,592476,592494],{"className":592421,"ariaHidden":758},[31672],[34,592423,592425,592428,592431,592434,592437],{"className":592424},[31676],[34,592426],{"className":592427,"style":55949},[31680],[34,592429,15],{"className":592430},[31685,31686],[34,592432],{"className":592433,"style":32604},[31845],[34,592435,32226],{"className":592436},[32024],[34,592438],{"className":592439,"style":32604},[31845],[34,592441,592443,592446,592449,592452,592455],{"className":592442},[31676],[34,592444],{"className":592445,"style":32617},[31680],[34,592447,640],{"className":592448},[31685],[34,592450],{"className":592451,"style":31846},[31845],[34,592453,202],{"className":592454},[31850],[34,592456],{"className":592457,"style":31846},[31845],[34,592459,592461,592464,592467,592470,592473],{"className":592460},[31676],[34,592462],{"className":592463,"style":31774},[31680],[34,592465,42407],{"className":592466},[31685],[34,592468],{"className":592469,"style":31846},[31845],[34,592471,140094],{"className":592472},[31850],[34,592474],{"className":592475,"style":31846},[31845],[34,592477,592479,592482,592485,592488,592491],{"className":592478},[31676],[34,592480],{"className":592481,"style":38774},[31680],[34,592483,15],{"className":592484},[31685,31686],[34,592486],{"className":592487,"style":31846},[31845],[34,592489,202],{"className":592490},[31850],[34,592492],{"className":592493,"style":31846},[31845],[34,592495,592497,592500],{"className":592496},[31676],[34,592498],{"className":592499,"style":31774},[31680],[34,592501,42777],{"className":592502},[31685],[34,592504,592506,592544],{"className":592505},[31642],[34,592507,592509],{"className":592508},[31646],[31648,592510,592511],{"xmlns":31650},[31652,592512,592513,592541],{},[31655,592514,592515,592517,592519,592521,592523,592525,592527,592529,592531,592533,592535,592537,592539],{},[31658,592516,38759],{},[31662,592518,32226],{},[31749,592520,640],{},[31662,592522,202],{},[31749,592524,640],{},[31658,592526,38080],{"mathvariant":38079},[31749,592528,768],{},[31662,592530,140094],{},[31658,592532,38759],{},[31662,592534,202],{},[31749,592536,786],{},[31658,592538,38080],{"mathvariant":38079},[31749,592540,768],{},[31665,592542,592543],{"encoding":31667},"q-1 = 1\u002F2 \\Rightarrow q = 3\u002F2",[34,592545,592547,592565,592583,592601,592619],{"className":592546,"ariaHidden":758},[31672],[34,592548,592550,592553,592556,592559,592562],{"className":592549},[31676],[34,592551],{"className":592552,"style":55949},[31680],[34,592554,38759],{"className":592555,"style":32062},[31685,31686],[34,592557],{"className":592558,"style":32604},[31845],[34,592560,32226],{"className":592561},[32024],[34,592563],{"className":592564,"style":32604},[31845],[34,592566,592568,592571,592574,592577,592580],{"className":592567},[31676],[34,592569],{"className":592570,"style":32617},[31680],[34,592572,640],{"className":592573},[31685],[34,592575],{"className":592576,"style":31846},[31845],[34,592578,202],{"className":592579},[31850],[34,592581],{"className":592582,"style":31846},[31845],[34,592584,592586,592589,592592,592595,592598],{"className":592585},[31676],[34,592587],{"className":592588,"style":31774},[31680],[34,592590,42407],{"className":592591},[31685],[34,592593],{"className":592594,"style":31846},[31845],[34,592596,140094],{"className":592597},[31850],[34,592599],{"className":592600,"style":31846},[31845],[34,592602,592604,592607,592610,592613,592616],{"className":592603},[31676],[34,592605],{"className":592606,"style":38774},[31680],[34,592608,38759],{"className":592609,"style":32062},[31685,31686],[34,592611],{"className":592612,"style":31846},[31845],[34,592614,202],{"className":592615},[31850],[34,592617],{"className":592618,"style":31846},[31845],[34,592620,592622,592625],{"className":592621},[31676],[34,592623],{"className":592624,"style":31774},[31680],[34,592626,42777],{"className":592627},[31685],[108,592629,592630,592631,347],{},"原式 ",[34,592632,592634,592668],{"className":592633},[31642],[34,592635,592637],{"className":592636},[31646],[31648,592638,592639],{"xmlns":31650},[31652,592640,592641,592665],{},[31655,592642,592643,592645,592647,592649,592655,592657,592663],{},[31662,592644,202],{},[31658,592646,54224],{},[31662,592648,456],{"stretchy":702},[41453,592650,592651,592653],{},[31749,592652,786],{},[31749,592654,768],{},[31662,592656,31960],{"separator":758},[41453,592658,592659,592661],{},[31749,592660,786],{},[31749,592662,768],{},[31662,592664,2557],{"stretchy":702},[31665,592666,592667],{"encoding":31667},"= B(\\frac{3}{2}, \\frac{3}{2})",[34,592669,592671,592683],{"className":592670,"ariaHidden":758},[31672],[34,592672,592674,592677,592680],{"className":592673},[31676],[34,592675],{"className":592676,"style":73089},[31680],[34,592678,202],{"className":592679},[31850],[34,592681],{"className":592682,"style":31846},[31845],[34,592684,592686,592689,592692,592695,592763,592766,592769,592837],{"className":592685},[31676],[34,592687],{"className":592688,"style":46727},[31680],[34,592690,54224],{"className":592691,"style":54252},[31685,31686],[34,592693,456],{"className":592694},[31782],[34,592696,592698,592701,592760],{"className":592697},[31685],[34,592699],{"className":592700},[31782,41488],[34,592702,592704],{"className":592703},[41453],[34,592705,592707,592752],{"className":592706},[31796,31797],[34,592708,592710,592749],{"className":592709},[31801],[34,592711,592713,592727,592735],{"className":592712,"style":46746},[31805],[34,592714,592715,592718],{"style":41504},[34,592716],{"className":592717,"style":41508},[31813],[34,592719,592721],{"className":592720},[31818,31819,31820,31821],[34,592722,592724],{"className":592723},[31685,31821],[34,592725,768],{"className":592726},[31685,31821],[34,592728,592729,592732],{"style":41520},[34,592730],{"className":592731,"style":41508},[31813],[34,592733],{"className":592734,"style":41528},[41527],[34,592736,592737,592740],{"style":46771},[34,592738],{"className":592739,"style":41508},[31813],[34,592741,592743],{"className":592742},[31818,31819,31820,31821],[34,592744,592746],{"className":592745},[31685,31821],[34,592747,786],{"className":592748},[31685,31821],[34,592750,31829],{"className":592751},[31828],[34,592753,592755],{"className":592754},[31801],[34,592756,592758],{"className":592757,"style":41619},[31805],[34,592759],{},[34,592761],{"className":592762},[31690,41488],[34,592764,31960],{"className":592765},[32119],[34,592767],{"className":592768,"style":32123},[31845],[34,592770,592772,592775,592834],{"className":592771},[31685],[34,592773],{"className":592774},[31782,41488],[34,592776,592778],{"className":592777},[41453],[34,592779,592781,592826],{"className":592780},[31796,31797],[34,592782,592784,592823],{"className":592783},[31801],[34,592785,592787,592801,592809],{"className":592786,"style":46746},[31805],[34,592788,592789,592792],{"style":41504},[34,592790],{"className":592791,"style":41508},[31813],[34,592793,592795],{"className":592794},[31818,31819,31820,31821],[34,592796,592798],{"className":592797},[31685,31821],[34,592799,768],{"className":592800},[31685,31821],[34,592802,592803,592806],{"style":41520},[34,592804],{"className":592805,"style":41508},[31813],[34,592807],{"className":592808,"style":41528},[41527],[34,592810,592811,592814],{"style":46771},[34,592812],{"className":592813,"style":41508},[31813],[34,592815,592817],{"className":592816},[31818,31819,31820,31821],[34,592818,592820],{"className":592819},[31685,31821],[34,592821,786],{"className":592822},[31685,31821],[34,592824,31829],{"className":592825},[31828],[34,592827,592829],{"className":592828},[31801],[34,592830,592832],{"className":592831,"style":41619},[31805],[34,592833],{},[34,592835],{"className":592836},[31690,41488],[34,592838,2557],{"className":592839},[31690],[108,592841,592842,592843,347],{},"利用关系式：",[34,592844,592846,592952],{"className":592845},[31642],[34,592847,592849],{"className":592848},[31646],[31648,592850,592851],{"xmlns":31650},[31652,592852,592853,592949],{},[31655,592854,592855,592857,592907,592909],{},[31662,592856,202],{},[41453,592858,592859,592885],{},[31655,592860,592861,592863,592865,592867,592869,592871,592873,592875,592877,592879,592881,592883],{},[31658,592862,564791],{"mathvariant":38079},[31662,592864,456],{"stretchy":702},[31749,592866,786],{},[31658,592868,38080],{"mathvariant":38079},[31749,592870,768],{},[31662,592872,2557],{"stretchy":702},[31658,592874,564791],{"mathvariant":38079},[31662,592876,456],{"stretchy":702},[31749,592878,786],{},[31658,592880,38080],{"mathvariant":38079},[31749,592882,768],{},[31662,592884,2557],{"stretchy":702},[31655,592886,592887,592889,592891,592893,592895,592897,592899,592901,592903,592905],{},[31658,592888,564791],{"mathvariant":38079},[31662,592890,456],{"stretchy":702},[31749,592892,786],{},[31658,592894,38080],{"mathvariant":38079},[31749,592896,768],{},[31662,592898,223],{},[31749,592900,786],{},[31658,592902,38080],{"mathvariant":38079},[31749,592904,768],{},[31662,592906,2557],{"stretchy":702},[31662,592908,202],{},[41453,592910,592911,592939],{},[31655,592912,592913,592915,592921,592923,592925,592931,592933],{},[31662,592914,456],{"stretchy":702},[41453,592916,592917,592919],{},[31749,592918,640],{},[31749,592920,768],{},[31658,592922,564791],{"mathvariant":38079},[31662,592924,456],{"stretchy":702},[41453,592926,592927,592929],{},[31749,592928,640],{},[31749,592930,768],{},[31662,592932,2557],{"stretchy":702},[37624,592934,592935,592937],{},[31662,592936,2557],{"stretchy":702},[31749,592938,768],{},[31655,592940,592941,592943,592945,592947],{},[31658,592942,564791],{"mathvariant":38079},[31662,592944,456],{"stretchy":702},[31749,592946,786],{},[31662,592948,2557],{"stretchy":702},[31665,592950,592951],{"encoding":31667},"= \\frac{\\Gamma(3\u002F2)\\Gamma(3\u002F2)}{\\Gamma(3\u002F2+3\u002F2)} = \\frac{(\\frac{1}{2}\\Gamma(\\frac{1}{2}))^2}{\\Gamma(3)}",[34,592953,592955,592967,593086],{"className":592954,"ariaHidden":758},[31672],[34,592956,592958,592961,592964],{"className":592957},[31676],[34,592959],{"className":592960,"style":73089},[31680],[34,592962,202],{"className":592963},[31850],[34,592965],{"className":592966,"style":31846},[31845],[34,592968,592970,592973,593077,593080,593083],{"className":592969},[31676],[34,592971],{"className":592972,"style":327841},[31680],[34,592974,592976,592979,593074],{"className":592975},[31685],[34,592977],{"className":592978},[31782,41488],[34,592980,592982],{"className":592981},[41453],[34,592983,592985,593066],{"className":592984},[31796,31797],[34,592986,592988,593063],{"className":592987},[31801],[34,592989,592991,593020,593028],{"className":592990,"style":70504},[31805],[34,592992,592993,592996],{"style":41504},[34,592994],{"className":592995,"style":41508},[31813],[34,592997,592999],{"className":592998},[31818,31819,31820,31821],[34,593000,593002,593005,593008,593011,593014,593017],{"className":593001},[31685,31821],[34,593003,564791],{"className":593004},[31685,31821],[34,593006,456],{"className":593007},[31782,31821],[34,593009,42777],{"className":593010},[31685,31821],[34,593012,223],{"className":593013},[32024,31821],[34,593015,42777],{"className":593016},[31685,31821],[34,593018,2557],{"className":593019},[31690,31821],[34,593021,593022,593025],{"style":41520},[34,593023],{"className":593024,"style":41508},[31813],[34,593026],{"className":593027,"style":41528},[41527],[34,593029,593030,593033],{"style":70529},[34,593031],{"className":593032,"style":41508},[31813],[34,593034,593036],{"className":593035},[31818,31819,31820,31821],[34,593037,593039,593042,593045,593048,593051,593054,593057,593060],{"className":593038},[31685,31821],[34,593040,564791],{"className":593041},[31685,31821],[34,593043,456],{"className":593044},[31782,31821],[34,593046,42777],{"className":593047},[31685,31821],[34,593049,2557],{"className":593050},[31690,31821],[34,593052,564791],{"className":593053},[31685,31821],[34,593055,456],{"className":593056},[31782,31821],[34,593058,42777],{"className":593059},[31685,31821],[34,593061,2557],{"className":593062},[31690,31821],[34,593064,31829],{"className":593065},[31828],[34,593067,593069],{"className":593068},[31801],[34,593070,593072],{"className":593071,"style":324858},[31805],[34,593073],{},[34,593075],{"className":593076},[31690,41488],[34,593078],{"className":593079,"style":31846},[31845],[34,593081,202],{"className":593082},[31850],[34,593084],{"className":593085,"style":31846},[31845],[34,593087,593089,593093],{"className":593088},[31676],[34,593090],{"className":593091,"style":593092},[31680],"height:1.6947em;vertical-align:-0.52em;",[34,593094,593096,593099,593344],{"className":593095},[31685],[34,593097],{"className":593098},[31782,41488],[34,593100,593102],{"className":593101},[41453],[34,593103,593105,593336],{"className":593104},[31796,31797],[34,593106,593108,593333],{"className":593107},[31801],[34,593109,593112,593135,593143],{"className":593110,"style":593111},[31805],"height:1.1747em;",[34,593113,593114,593117],{"style":41504},[34,593115],{"className":593116,"style":41508},[31813],[34,593118,593120],{"className":593119},[31818,31819,31820,31821],[34,593121,593123,593126,593129,593132],{"className":593122},[31685,31821],[34,593124,564791],{"className":593125},[31685,31821],[34,593127,456],{"className":593128},[31782,31821],[34,593130,786],{"className":593131},[31685,31821],[34,593133,2557],{"className":593134},[31690,31821],[34,593136,593137,593140],{"style":41520},[34,593138],{"className":593139,"style":41508},[31813],[34,593141],{"className":593142,"style":41528},[41527],[34,593144,593146,593149],{"style":593145},"top:-3.5508em;",[34,593147],{"className":593148,"style":41508},[31813],[34,593150,593152],{"className":593151},[31818,31819,31820,31821],[34,593153,593155,593158,593227,593230,593233,593301,593304],{"className":593154},[31685,31821],[34,593156,456],{"className":593157},[31782,31821],[34,593159,593161,593164,593224],{"className":593160},[31685,31821],[34,593162],{"className":593163},[31782,41488,31818,43410,435704],[34,593165,593167],{"className":593166},[41453],[34,593168,593170,593216],{"className":593169},[31796,31797],[34,593171,593173,593213],{"className":593172},[31801],[34,593174,593177,593191,593199],{"className":593175,"style":593176},[31805],"height:0.8443em;",[34,593178,593179,593182],{"style":435720},[34,593180],{"className":593181,"style":41508},[31813],[34,593183,593185],{"className":593184},[31818,43410,43411,31821],[34,593186,593188],{"className":593187},[31685,31821],[34,593189,768],{"className":593190},[31685,31821],[34,593192,593193,593196],{"style":435735},[34,593194],{"className":593195,"style":41508},[31813],[34,593197],{"className":593198,"style":435742},[41527,31821],[34,593200,593201,593204],{"style":589620},[34,593202],{"className":593203,"style":41508},[31813],[34,593205,59320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",[34,593351,593353,593403],{"className":593352},[31642],[34,593354,593356],{"className":593355},[31646],[31648,593357,593358],{"xmlns":31650},[31652,593359,593360,593400],{},[31655,593361,593362,593364,593366,593368,593370,593372,593374,593376,593380,593382,593384,593386,593388,593390,593392,593394,593396,593398],{},[31658,593363,564791],{"mathvariant":38079},[31662,593365,456],{"stretchy":702},[31749,593367,640],{},[31658,593369,38080],{"mathvariant":38079},[31749,593371,768],{},[31662,593373,2557],{"stretchy":702},[31662,593375,202],{},[41463,593377,593378],{},[31658,593379,58442],{},[31662,593381,31960],{"separator":758},[31658,593383,564791],{"mathvariant":38079},[31662,593385,456],{"stretchy":702},[31749,593387,786],{},[31662,593389,2557],{"stretchy":702},[31662,593391,202],{},[31749,593393,768],{},[31662,593395,5504],{"stretchy":702},[31662,593397,202],{},[31749,593399,768],{},[31665,593401,593402],{"encoding":31667},"\\Gamma(1\u002F2)=\\sqrt{\\pi}, \\Gamma(3)=2! = 2",[34,593404,593406,593433,593513,593534],{"className":593405,"ariaHidden":758},[31672],[34,593407,593409,593412,593415,593418,593421,593424,593427,593430],{"className":593408},[31676],[34,593410],{"className":593411,"style":31774},[31680],[34,593413,564791],{"className":593414},[31685],[34,593416,456],{"className":593417},[31782],[34,593419,42407],{"className":593420},[31685],[34,593422,2557],{"className":593423},[31690],[34,593425],{"className":593426,"style":31846},[31845],[34,593428,202],{"className":593429},[31850],[34,593431],{"className":593432,"style":31846},[31845],[34,593434,593436,593439,593486,593489,593492,593495,593498,593501,593504,593507,593510],{"className":593435},[31676],[34,593437],{"className":593438,"style":64978},[31680],[34,593440,593442],{"className":593441},[31685,41550],[34,593443,593445,593478],{"className":593444},[31796,31797],[34,593446,593448,593475],{"className":593447},[31801],[34,593449,593451,593463],{"className":593450,"style":64997},[31805],[34,593452,593454,593457],{"className":593453,"style":41565},[41564],[34,593455],{"className":593456,"style":41508},[31813],[34,593458,593460],{"className":593459,"style":41572},[31685],[34,593461,58442],{"className":593462,"style":32062},[31685,31686],[34,593464,593465,593468],{"style":65012},[34,593466],{"className":593467,"style":41508},[31813],[34,593469,593471],{"className":593470,"style":41586},[41585],[41588,593472,593473],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,593474],{"d":41597},[34,593476,31829],{"className":593477},[31828],[34,593479,593481],{"className":593480},[31801],[34,593482,593484],{"className":593483,"style":65032},[31805],[34,593485],{},[34,593487,31960],{"className":593488},[32119],[34,593490],{"className":593491,"style":32123},[31845],[34,593493,564791],{"className":593494},[31685],[34,593496,456],{"className":593497},[31782],[34,593499,786],{"className":593500},[31685],[34,593502,2557],{"className":593503},[31690],[34,593505],{"className":593506,"style":31846},[31845],[34,593508,202],{"className":593509},[31850],[34,593511],{"className":593512,"style":31846},[31845],[34,593514,593516,593519,593522,593525,593528,593531],{"className":593515},[31676],[34,593517],{"className":593518,"style":31681},[31680],[34,593520,768],{"className":593521},[31685],[34,593523,5504],{"className":593524},[31690],[34,593526],{"className":593527,"style":31846},[31845],[34,593529,202],{"className":593530},[31850],[34,593532],{"className":593533,"style":31846},[31845],[34,593535,593537,593540],{"className":593536},[31676],[34,593538],{"className":593539,"style":32617},[31680],[34,593541,768],{"className":593542},[31685],[108,593544,592630,593545,347],{},[34,593546,593548,593610],{"className":593547},[31642],[34,593549,593551],{"className":593550},[31646],[31648,593552,593553],{"xmlns":31650},[31652,593554,593555,593607],{},[31655,593556,593557,593559,593583,593585,593599,593601],{},[31662,593558,202],{},[41453,593560,593561,593581],{},[31655,593562,593563,593565,593571,593575],{},[31662,593564,456],{"stretchy":702},[41453,593566,593567,593569],{},[31749,593568,640],{},[31749,593570,768],{},[41463,593572,593573],{},[31658,593574,58442],{},[37624,593576,593577,593579],{},[31662,593578,2557],{"stretchy":702},[31749,593580,768],{},[31749,593582,768],{},[31662,593584,202],{},[41453,593586,593587,593597],{},[31655,593588,593589,593595],{},[41453,593590,593591,593593],{},[31749,593592,640],{},[31749,593594,15932],{},[31658,593596,58442],{},[31749,593598,768],{},[31662,593600,202],{},[41453,593602,593603,593605],{},[31658,593604,58442],{},[31749,593606,60525],{},[31665,593608,593609],{"encoding":31667},"= \\frac{(\\frac{1}{2}\\sqrt{\\pi})^2}{2} = \\frac{\\frac{1}{4}\\pi}{2} = 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131,"style":244945},[31805],[34,594133,594134,594137],{"style":41504},[34,594135],{"className":594136,"style":41508},[31813],[34,594138,594140],{"className":594139},[31818,31819,31820,31821],[34,594141,594143],{"className":594142},[31685,31821],[34,594144,60525],{"className":594145},[31685,31821],[34,594147,594148,594151],{"style":41520},[34,594149],{"className":594150,"style":41508},[31813],[34,594152],{"className":594153,"style":41528},[41527],[34,594155,594156,594159],{"style":46771},[34,594157],{"className":594158,"style":41508},[31813],[34,594160,594162],{"className":594161},[31818,31819,31820,31821],[34,594163,594165],{"className":594164},[31685,31821],[34,594166,58442],{"className":594167,"style":32062},[31685,31686,31821],[34,594169,31829],{"className":594170},[31828],[34,594172,594174],{"className":594173},[31801],[34,594175,594177],{"className":594176,"style":41619},[31805],[34,594178],{},[34,594180],{"className":594181},[31690,41488],[15,594183,594184,47649,594186],{},[2627,594185,263601],{},[34,594187,594189,594233],{"className":594188},[31642],[34,594190,594192],{"className":594191},[31646],[31648,594193,594194],{"xmlns":31650},[31652,594195,594196,594230],{},[31655,594197,594198],{},[71090,594199,594200,594208],{"scriptlevel":459,"displaystyle":758},[38846,594201,594202,594204,594206],{},[31662,594203,546295],{},[31749,594205,459],{},[31658,594207,58442],{},[41453,594209,594210,594216],{},[31655,594211,594212,594214],{},[31658,594213,46900],{},[31658,594215,32936],{},[41463,594217,594218],{},[31655,594219,594220,594222,594224,594226,594228],{},[31749,594221,786],{},[31662,594223,32226],{},[31658,594225,547860],{},[31662,594227,32682],{},[31658,594229,32936],{},[31665,594231,594232],{"encoding":31667},"\\displaystyle \\int_0^\\pi \\frac{dx}{\\sqrt{3 - \\cos x}}",[34,594234,594236],{"className":594235,"ariaHidden":758},[31672],[34,594237,594239,594243,594295,594298],{"className":594238},[31676],[34,594240],{"className":594241,"style":594242},[31680],"height:2.3443em;vertical-align:-0.93em;",[34,594244,594246,594249],{"className":594245},[32767],[34,594247,546295],{"className":594248,"style":591475},[32767,131249,131250],[34,594250,594252],{"className":594251},[31792],[34,594253,594255,594287],{"className":594254},[31796,31797],[34,594256,594258,594284],{"className":594257},[31801],[34,594259,594262,594273],{"className":594260,"style":594261},[31805],"height:1.4143em;",[34,594263,594264,594267],{"style":591491},[34,594265],{"className":594266,"style":31814},[31813],[34,594268,594270],{"className":594269},[31818,31819,31820,31821],[34,594271,459],{"className":594272},[31685,31821],[34,594274,594275,594278],{"style":591503},[34,594276],{"className":594277,"style":31814},[31813],[34,594279,594281],{"className":594280},[31818,31819,31820,31821],[34,594282,58442],{"className":594283,"style":32062},[31685,31686,31821],[34,594285,31829],{"className":594286},[31828],[34,594288,594290],{"className":594289},[31801],[34,594291,594293],{"className":594292,"style":591522},[31805],[34,594294],{},[34,594296],{"className":594297,"style":32123},[31845],[34,594299,594301,594304,594428],{"className":594300},[31685],[34,594302],{"className":594303},[31782,41488],[34,594305,594307],{"className":594306},[41453],[34,594308,594310,594419],{"className":594309},[31796,31797],[34,594311,594313,594416],{"className":594312},[31801],[34,594314,594317,594394,594402],{"className":594315,"style":594316},[31805],"height:1.3714em;",[34,594318,594320,594323],{"style":594319},"top:-2.2444em;",[34,594321],{"className":594322,"style":41508},[31813],[34,594324,594326],{"className":594325},[31685],[34,594327,594329],{"className":594328},[31685,41550],[34,594330,594332,594385],{"className":594331},[31796,31797],[34,594333,594335,594382],{"className":594334},[31801],[34,594336,594339,594369],{"className":594337,"style":594338},[31805],"height:0.8656em;",[34,594340,594342,594345],{"className":594341,"style":41565},[41564],[34,594343],{"className":594344,"style":41508},[31813],[34,594346,594348,594351,594354,594357,594360,594363,594366],{"className":594347,"style":41572},[31685],[34,594349,786],{"className":594350},[31685],[34,594352],{"className":594353,"style":32604},[31845],[34,594355,32226],{"className":594356},[32024],[34,594358],{"className":594359,"style":32604},[31845],[34,594361,547860],{"className":594362},[32767],[34,594364],{"className":594365,"style":32123},[31845],[34,594367,32936],{"className":594368},[31685,31686],[34,594370,594372,594375],{"style":594371},"top:-2.8256em;",[34,594373],{"className":594374,"style":41508},[31813],[34,594376,594378],{"className":594377,"style":41586},[41585],[41588,594379,594380],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,594381],{"d":41597},[34,594383,31829],{"className":594384},[31828],[34,594386,594388],{"className":594387},[31801],[34,594389,594392],{"className":594390,"style":594391},[31805],"height:0.1744em;",[34,594393],{},[34,594395,594396,594399],{"style":41520},[34,594397],{"className":594398,"style":41508},[31813],[34,594400],{"className":594401,"style":41528},[41527],[34,594403,594404,594407],{"style":129659},[34,594405],{"className":594406,"style":41508},[31813],[34,594408,594410,594413],{"className":594409},[31685],[34,594411,46900],{"className":594412},[31685,31686],[34,594414,32936],{"className":594415},[31685,31686],[34,594417,31829],{"className":594418},[31828],[34,594420,594422],{"className":594421},[31801],[34,594423,594426],{"className":594424,"style":594425},[31805],"height:0.93em;",[34,594427],{},[34,594429],{"className":594430},[31690,41488],[105,594432,594433,594471],{},[108,594434,594435,594437,594438,594441,594442,594470],{},[2627,594436,419963],{},"：这道题比较特殊，它是一个",[2627,594439,594440],{},"椭圆积分","，通常无法用简单的 ",[34,594443,594445,594458],{"className":594444},[31642],[34,594446,594448],{"className":594447},[31646],[31648,594449,594450],{"xmlns":31650},[31652,594451,594452,594456],{},[31655,594453,594454],{},[31658,594455,564791],{"mathvariant":38079},[31665,594457,564794],{"encoding":31667},[34,594459,594461],{"className":594460,"ariaHidden":758},[31672],[34,594462,594464,594467],{"className":594463},[31676],[34,594465],{"className":594466,"style":33218},[31680],[34,594468,564791],{"className":594469},[31685]," 函数直接算出数值，除非这里有特殊技巧。但标准解法是化简为椭圆积分形式。如果是在考查欧拉积分的章节，可能存在特定参数技巧，但我们按标准步骤化简。",[108,594472,594473,33183,594475,596642],{},[2627,594474,289021],{},[2230,594476,594477,594683,595386,595816],{},[108,594478,594479,594480,347],{},"利用半角公式：",[34,594481,594483,594525],{"className":594482},[31642],[34,594484,594486],{"className":594485},[31646],[31648,594487,594488],{"xmlns":31650},[31652,594489,594490,594522],{},[31655,594491,594492,594494,594496,594498,594500,594502,594504,594506,594516],{},[31658,594493,547860],{},[31662,594495,32682],{},[31658,594497,32936],{},[31662,594499,202],{},[31749,594501,640],{},[31662,594503,32226],{},[31749,594505,768],{},[37624,594507,594508,594514],{},[31655,594509,594510,594512],{},[31658,594511,547853],{},[31662,594513,32682],{},[31749,594515,768],{},[41453,594517,594518,594520],{},[31658,594519,32936],{},[31749,594521,768],{},[31665,594523,594524],{"encoding":31667},"\\cos x = 1 - 2\\sin^2 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",[34,594687,594689,594793],{"className":594688},[31642],[34,594690,594692],{"className":594691},[31646],[31648,594693,594694],{"xmlns":31650},[31652,594695,594696,594790],{},[31655,594697,594698,594732,594734,594760,594762,594766],{},[41463,594699,594700],{},[31655,594701,594702,594704,594706,594708,594710,594712,594714,594724,594730],{},[31749,594703,786],{},[31662,594705,32226],{},[31662,594707,456],{"stretchy":702},[31749,594709,640],{},[31662,594711,32226],{},[31749,594713,768],{},[37624,594715,594716,594722],{},[31655,594717,594718,594720],{},[31658,594719,547853],{},[31662,594721,32682],{},[31749,594723,768],{},[41453,594725,594726,594728],{},[31658,594727,32936],{},[31749,594729,768],{},[31662,594731,2557],{"stretchy":702},[31662,594733,202],{},[41463,594735,594736],{},[31655,594737,594738,594740,594742,594744,594754],{},[31749,594739,768],{},[31662,594741,223],{},[31749,594743,768],{},[37624,594745,594746,594752],{},[31655,594747,594748,594750],{},[31658,594749,547853],{},[31662,594751,32682],{},[31749,594753,768],{},[41453,594755,594756,594758],{},[31658,594757,32936],{},[31749,594759,768],{},[31662,594761,202],{},[41463,594763,594764],{},[31749,594765,768],{},[41463,594767,594768],{},[31655,594769,594770,594772,594774,594784],{},[31749,594771,640],{},[31662,594773,223],{},[37624,594775,594776,594782],{},[31655,594777,594778,594780],{},[31658,594779,547853],{},[31662,594781,32682],{},[31749,594783,768],{},[41453,594785,594786,594788],{},[31658,594787,32936],{},[31749,594789,768],{},[31665,594791,594792],{"encoding":31667},"\\sqrt{3 - (1 - 2\\sin^2 \\frac{x}{2})} = \\sqrt{2 + 2\\sin^2 \\frac{x}{2}} = \\sqrt{2} \\sqrt{1 + \\sin^2 \\frac{x}{2}}",[34,594794,594796,595000,595177],{"className":594795,"ariaHidden":758},[31672],[34,594797,594799,594803,594991,594994,594997],{"className":594798},[31676],[34,594800],{"className":594801,"style":594802},[31680],"height:1.84em;vertical-align:-0.5916em;",[34,594804,594806],{"className":594805},[31685,41550],[34,594807,594809,594982],{"className":594808},[31796,31797],[34,594810,594812,594979],{"className":594811},[31801],[34,594813,594816,594962],{"className":594814,"style":594815},[31805],"height:1.2484em;",[34,594817,594820,594823],{"className":594818,"style":594819},[41564],"top:-3.8em;",[34,594821],{"className":594822,"style":550110},[31813],[34,594824,594826,594829,594832,594835,594838,594841,594844,594847,594850,594853,594856,594859,594888,594891,594959],{"className":594825,"style":590751},[31685],[34,594827,786],{"className":594828},[31685],[34,594830],{"className":594831,"style":32604},[31845],[34,594833,32226],{"className":594834},[32024],[34,594836],{"className":594837,"style":32604},[31845],[34,594839,456],{"className":594840},[31782],[34,594842,640],{"className":594843},[31685],[34,594845],{"className":594846,"style":32604},[31845],[34,594848,32226],{"className":594849},[32024],[34,594851],{"className":594852,"style":32604},[31845],[34,594854,768],{"className":594855},[31685],[34,594857],{"className":594858,"style":32123},[31845],[34,594860,594862,594865],{"className":594861},[32767],[34,594863,547853],{"className":594864},[32767],[34,594866,594868],{"className":594867},[31792],[34,594869,594871],{"className":594870},[31796],[34,594872,594874],{"className":594873},[31801],[34,594875,594877],{"className":594876,"style":585884},[31805],[34,594878,594879,594882],{"style":586008},[34,594880],{"className":594881,"style":31814},[31813],[34,594883,594885],{"className":594884},[31818,31819,31820,31821],[34,594886,768],{"className":594887},[31685,31821],[34,594889],{"className":594890,"style":32123},[31845],[34,594892,594894,594897,594956],{"className":594893},[31685],[34,594895],{"className":594896},[31782,41488],[34,594898,594900],{"className":594899},[41453],[34,594901,594903,594948],{"className":594902},[31796,31797],[34,594904,594906,594945],{"className":594905},[31801],[34,594907,594909,594923,594931],{"className":594908,"style":244945},[31805],[34,594910,594911,594914],{"style":41504},[34,594912],{"className":594913,"style":41508},[31813],[34,594915,594917],{"className":594916},[31818,31819,31820,31821],[34,594918,594920],{"className":594919},[31685,31821],[34,594921,768],{"className":594922},[31685,31821],[34,594924,594925,594928],{"style":41520},[34,594926],{"className":594927,"style":41508},[31813],[34,594929],{"className":594930,"style":41528},[41527],[34,594932,594933,594936],{"style":46771},[34,594934],{"className":594935,"style":41508},[31813],[34,594937,594939],{"className":594938},[31818,31819,31820,31821],[34,594940,594942],{"className":594941},[31685,31821],[34,594943,32936],{"className":594944},[31685,31686,31821],[34,594946,31829],{"className":594947},[31828],[34,594949,594951],{"className":594950},[31801],[34,594952,594954],{"className":594953,"style":41619},[31805],[34,594955],{},[34,594957],{"className":594958},[31690,41488],[34,594960,2557],{"className":594961},[31690],[34,594963,594965,594968],{"style":594964},"top:-3.2084em;",[34,594966],{"className":594967,"style":550110},[31813],[34,594969,594972],{"className":594970,"style":594971},[41585],"min-width:1.02em;height:1.88em;",[41588,594973,594976],{"xmlns":41590,"width":41591,"height":594974,"viewBox":594975,"preserveAspectRatio":41594},"1.88em","0 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\\int_0^{\\frac{\\pi}{2}} \\frac{2 dt}{\\sqrt{2}\\sqrt{1+\\sin^2 t}} = \\sqrt{2} \\int_0^{\\frac{\\pi}{2}} \\frac{dt}{\\sqrt{1+\\sin^2 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",[34,596650,596652,596672],{"className":596651},[31642],[34,596653,596655],{"className":596654},[31646],[31648,596656,596657],{"xmlns":31650},[31652,596658,596659,596669],{},[31655,596660,596661,596663,596665,596667],{},[31658,596662,38691],{},[31662,596664,456],{"stretchy":702},[31658,596666,33759],{},[31662,596668,2557],{"stretchy":702},[31665,596670,596671],{"encoding":31667},"K(k)",[34,596673,596675],{"className":596674,"ariaHidden":758},[31672],[34,596676,596678,596681,596684,596687,596690],{"className":596677},[31676],[34,596679],{"className":596680,"style":31774},[31680],[34,596682,38691],{"className":596683,"style":38706},[31685,31686],[34,596685,456],{"className":596686},[31782],[34,596688,33759],{"className":596689,"style":33774},[31685,31686],[34,596691,2557],{"className":596692},[31690]," 形式，但无法算出像 ",[34,596695,596697,596710],{"className":596696},[31642],[34,596698,596700],{"className":596699},[31646],[31648,596701,596702],{"xmlns":31650},[31652,596703,596704,596708],{},[31655,596705,596706],{},[31658,596707,58442],{},[31665,596709,58445],{"encoding":31667},[34,596711,596713],{"className":596712,"ariaHidden":758},[31672],[34,596714,596716,596719],{"className":596715},[31676],[34,596717],{"className":596718,"style":32527},[31680],[34,596720,58442],{"className":596721,"style":32062},[31685,31686]," 这样的简单数。",[108,596724,596725,596728,596729,596847,596848,31901],{},[71807,596726,596727],{},"修正思路","：如果这是必做题，可能题目有误（例如分母是 ",[34,596730,596732,596764],{"className":596731},[31642],[34,596733,596735],{"className":596734},[31646],[31648,596736,596737],{"xmlns":31650},[31652,596738,596739,596761],{},[31655,596740,596741],{},[41463,596742,596743],{},[31655,596744,596745,596747,596749,596751,596753,596755,596757,596759],{},[31749,596746,768],{},[31662,596748,456],{"stretchy":702},[31749,596750,640],{},[31662,596752,32226],{},[31658,596754,547860],{},[31662,596756,32682],{},[31658,596758,32936],{},[31662,596760,2557],{"stretchy":702},[31665,596762,596763],{"encoding":31667},"\\sqrt{2(1-\\cos x)}",[34,596765,596767],{"className":596766,"ariaHidden":758},[31672],[34,596768,596770,596773],{"className":596769},[31676],[34,596771],{"className":596772,"style":590727},[31680],[34,596774,596776],{"className":596775},[31685,41550],[34,596777,596779,596839],{"className":596778},[31796,31797],[34,596780,596782,596836],{"className":596781},[31801],[34,596783,596785,596824],{"className":596784,"style":590740},[31805],[34,596786,596788,596791],{"className":596787,"style":590744},[41564],[34,596789],{"className":596790,"style":552795},[31813],[34,596792,596794,596797,596800,596803,596806,596809,596812,596815,596818,596821],{"className":596793,"style":590751},[31685],[34,596795,768],{"className":596796},[31685],[34,596798,456],{"className":596799},[31782],[34,596801,640],{"className":596802},[31685],[34,596804],{"className":596805,"style":32604},[31845],[34,596807,32226],{"className":596808},[32024],[34,596810],{"className":596811,"style":32604},[31845],[34,596813,547860],{"className":596814},[32767],[34,596816],{"className":596817,"style":32123},[31845],[34,596819,32936],{"className":596820},[31685,31686],[34,596822,2557],{"className":596823},[31690],[34,596825,596826,596829],{"style":590894},[34,596827],{"className":596828,"style":552795},[31813],[34,596830,596832],{"className":596831,"style":590901},[41585],[41588,596833,596834],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,596835],{"d":590908},[34,596837,31829],{"className":596838},[31828],[34,596840,596842],{"className":596841},[31801],[34,596843,596845],{"className":596844,"style":590918},[31805],[34,596846],{}," 就能算）。但如果必须按原题算，只能保留在椭圆积分形式或使用级数展开。鉴于这是普通练习题，这道题可能是让你练习化简，或者题目抄写时有偏差（比如可能是 ",[34,596849,596851,596883],{"className":596850},[31642],[34,596852,596854],{"className":596853},[31646],[31648,596855,596856],{"xmlns":31650},[31652,596857,596858,596880],{},[31655,596859,596860,596874,596876,596878],{},[38846,596861,596862,596864,596866],{},[31662,596863,546295],{},[31749,596865,459],{},[31655,596867,596868,596870,596872],{},[31658,596869,58442],{},[31658,596871,38080],{"mathvariant":38079},[31749,596873,768],{},[31658,596875,2547],{"mathvariant":38079},[31658,596877,2547],{"mathvariant":38079},[31658,596879,2547],{"mathvariant":38079},[31665,596881,596882],{"encoding":31667},"\\int_0^{\\pi\u002F2} ...",[34,596884,596886],{"className":596885,"ariaHidden":758},[31672],[34,596887,596889,596893,596950,596953],{"className":596888},[31676],[34,596890],{"className":596891,"style":596892},[31680],"height:1.4387em;vertical-align:-0.3558em;",[34,596894,596896,596899],{"className":596895},[32767],[34,596897,546295],{"className":596898,"style":546331},[32767,131249,156560],[34,596900,596902],{"className":596901},[31792],[34,596903,596905,596942],{"className":596904},[31796,31797],[34,596906,596908,596939],{"className":596907},[31801],[34,596909,596911,596922],{"className":596910,"style":586480},[31805],[34,596912,596913,596916],{"style":546347},[34,596914],{"className":596915,"style":31814},[31813],[34,596917,596919],{"className":596918},[31818,31819,31820,31821],[34,596920,459],{"className":596921},[31685,31821],[34,596923,596924,596927],{"style":546362},[34,596925],{"className":596926,"style":31814},[31813],[34,596928,596930],{"className":596929},[31818,31819,31820,31821],[34,596931,596933,596936],{"className":596932},[31685,31821],[34,596934,58442],{"className":596935,"style":32062},[31685,31686,31821],[34,596937,38149],{"className":596938},[31685,31821],[34,596940,31829],{"className":596941},[31828],[34,596943,596945],{"className":596944},[31801],[34,596946,596948],{"className":596947,"style":546384},[31805],[34,596949],{},[34,596951],{"className":596952,"style":32123},[31845],[34,596954,1167],{"className":596955},[31685],[108,596957,596958,596959,597137],{},"我们这里给出一个最简化的积分类形式：",[34,596960,596962,596994],{"className":596961},[31642],[34,596963,596965],{"className":596964},[31646],[31648,596966,596967],{"xmlns":31650},[31652,596968,596969,596991],{},[31655,596970,596971,596975,596977,596979,596985,596987,596989],{},[41463,596972,596973],{},[31749,596974,768],{},[31658,596976,51834],{},[31662,596978,456],{"stretchy":702},[41453,596980,596981,596983],{},[31658,596982,58442],{},[31749,596984,768],{},[31662,596986,31960],{"separator":758},[31658,596988,31955],{},[31662,596990,2557],{"stretchy":702},[31665,596992,596993],{"encoding":31667},"\\sqrt{2} F(\\frac{\\pi}{2}, i)",[34,596995,596997],{"className":596996,"ariaHidden":758},[31672],[34,596998,597000,597004,597051,597054,597057,597125,597128,597131,597134],{"className":596999},[31676],[34,597001],{"className":597002,"style":597003},[31680],"height:1.2522em;vertical-align:-0.345em;",[34,597005,597007],{"className":597006},[31685,41550],[34,597008,597010,597043],{"className":597009},[31796,31797],[34,597011,597013,597040],{"className":597012},[31801],[34,597014,597016,597028],{"className":597015,"style":572001},[31805],[34,597017,597019,597022],{"className":597018,"style":41565},[41564],[34,597020],{"className":597021,"style":41508},[31813],[34,597023,597025],{"className":597024,"style":41572},[31685],[34,597026,768],{"className":597027},[31685],[34,597029,597030,597033],{"style":572016},[34,597031],{"className":597032,"style":41508},[31813],[34,597034,597036],{"className":597035,"style":41586},[41585],[41588,597037,597038],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,597039],{"d":41597},[34,597041,31829],{"className":597042},[31828],[34,597044,597046],{"className":597045},[31801],[34,597047,597049],{"className":597048,"style":572036},[31805],[34,597050],{},[34,597052,51834],{"className":597053,"style":37422},[31685,31686],[34,597055,456],{"className":597056},[31782],[34,597058,597060,597063,597122],{"className":597059},[31685],[34,597061],{"className":597062},[31782,41488],[34,597064,597066],{"className":597065},[41453],[34,597067,597069,597114],{"className":597068},[31796,31797],[34,597070,597072,597111],{"className":597071},[31801],[34,597073,597075,597089,597097],{"className":597074,"style":244945},[31805],[34,597076,597077,597080],{"style":41504},[34,597078],{"className":597079,"style":41508},[31813],[34,597081,597083],{"className":597082},[31818,31819,31820,31821],[34,597084,597086],{"className":597085},[31685,31821],[34,597087,768],{"className":597088},[31685,31821],[34,597090,597091,597094],{"style":41520},[34,597092],{"className":597093,"style":41508},[31813],[34,597095],{"className":597096,"style":41528},[41527],[34,597098,597099,597102],{"style":46771},[34,597100],{"className":597101,"style":41508},[31813],[34,597103,597105],{"className":597104},[31818,31819,31820,31821],[34,597106,597108],{"className":597107},[31685,31821],[34,597109,58442],{"className":597110,"style":32062},[31685,31686,31821],[34,597112,31829],{"className":597113},[31828],[34,597115,597117],{"className":597116},[31801],[34,597118,597120],{"className":597119,"style":41619},[31805],[34,597121],{},[34,597123],{"className":597124},[31690,41488],[34,597126,31960],{"className":597127},[32119],[34,597129],{"className":597130,"style":32123},[31845],[34,597132,31955],{"className":597133},[31685,31686],[34,597135,2557],{"className":597136},[31690]," (涉及到虚数模，通常本科阶段不要求算出具体数)。",[108,597139,597140,597142,597143,597395],{},[2627,597141,105624],{},"：如果老师没有讲过椭圆积分，请检查题目是否抄错（例如是否为 ",[34,597144,597146,597188],{"className":597145},[31642],[34,597147,597149],{"className":597148},[31646],[31648,597150,597151],{"xmlns":31650},[31652,597152,597153,597185],{},[31655,597154,597155,597163],{},[38846,597156,597157,597159,597161],{},[31662,597158,546295],{},[31749,597160,459],{},[31658,597162,58442],{},[41453,597164,597165,597171],{},[31655,597166,597167,597169],{},[31658,597168,46900],{},[31658,597170,32936],{},[41463,597172,597173],{},[31655,597174,597175,597177,597179,597181,597183],{},[31749,597176,640],{},[31662,597178,32226],{},[31658,597180,547860],{},[31662,597182,32682],{},[31658,597184,32936],{},[31665,597186,597187],{"encoding":31667},"\\int_0^{\\pi} \\frac{dx}{\\sqrt{1-\\cos x}}",[34,597189,597191],{"className":597190,"ariaHidden":758},[31672],[34,597192,597194,597198,597252,597255],{"className":597193},[31676],[34,597195],{"className":597196,"style":597197},[31680],"height:1.4181em;vertical-align:-0.538em;",[34,597199,597201,597204],{"className":597200},[32767],[34,597202,546295],{"className":597203,"style":546331},[32767,131249,156560],[34,597205,597207],{"className":597206},[31792],[34,597208,597210,597244],{"className":597209},[31796,31797],[34,597211,597213,597241],{"className":597212},[31801],[34,597214,597216,597227],{"className":597215,"style":567928},[31805],[34,597217,597218,597221],{"style":546347},[34,597219],{"className":597220,"style":31814},[31813],[34,597222,597224],{"className":597223},[31818,31819,31820,31821],[34,597225,459],{"className":597226},[31685,31821],[34,597228,597229,597232],{"style":546362},[34,597230],{"className":597231,"style":31814},[31813],[34,597233,597235],{"className":597234},[31818,31819,31820,31821],[34,597236,597238],{"className":597237},[31685,31821],[34,597239,58442],{"className":597240,"style":32062},[31685,31686,31821],[34,597242,31829],{"className":597243},[31828],[34,597245,597247],{"className":597246},[31801],[34,597248,597250],{"className":597249,"style":546384},[31805],[34,597251],{},[34,597253],{"className":597254,"style":32123},[31845],[34,597256,597258,597261,597392],{"className":597257},[31685],[34,597259],{"className":597260},[31782,41488],[34,597262,597264],{"className":597263},[41453],[34,597265,597267,597384],{"className":597266},[31796,31797],[34,597268,597270,597381],{"className":597269},[31801],[34,597271,597273,597356,597364],{"className":597272,"style":232511},[31805],[34,597274,597276,597279],{"style":597275},"top:-2.5802em;",[34,597277],{"className":597278,"style":41508},[31813],[34,597280,597282],{"className":597281},[31818,31819,31820,31821],[34,597283,597285],{"className":597284},[31685,31821],[34,597286,597288],{"className":597287},[31685,41550,31821],[34,597289,597291,597347],{"className":597290},[31796,31797],[34,597292,597294,597344],{"className":597293},[31801],[34,597295,597298,597331],{"className":597296,"style":597297},[31805],"height:0.8712em;",[34,597299,597301,597304],{"className":597300,"style":41565},[41564],[34,597302],{"className":597303,"style":41508},[31813],[34,597305,597307,597310,597313,597325,597328],{"className":597306,"style":41572},[31685,31821],[34,597308,640],{"className":597309},[31685,31821],[34,597311,32226],{"className":597312},[32024,31821],[34,597314,597316,597319,597322],{"className":597315},[32767,31821],[34,597317,31965],{"className":597318},[31821],[34,597320,43383],{"className":597321},[31821],[34,597323,31747],{"className":597324},[31821],[34,597326],{"className":597327,"style":43430},[31845,31821],[34,597329,32936],{"className":597330},[31685,31686,31821],[34,597332,597334,597337],{"style":597333},"top:-2.8312em;",[34,597335],{"className":597336,"style":41508},[31813],[34,597338,597340],{"className":597339,"style":41586},[41585,31821],[41588,597341,597342],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,597343],{"d":41597},[34,597345,31829],{"className":597346},[31828],[34,597348,597350],{"className":597349},[31801],[34,597351,597354],{"className":597352,"style":597353},[31805],"height:0.1688em;",[34,597355],{},[34,597357,597358,597361],{"style":41520},[34,597359],{"className":597360,"style":41508},[31813],[34,597362],{"className":597363,"style":41528},[41527],[34,597365,597366,597369],{"style":46771},[34,597367],{"className":597368,"style":41508},[31813],[34,597370,597372],{"className":597371},[31818,31819,31820,31821],[34,597373,597375,597378],{"className":597374},[31685,31821],[34,597376,46900],{"className":597377},[31685,31686,31821],[34,597379,32936],{"className":597380},[31685,31686,31821],[34,597382,31829],{"className":597383},[31828],[34,597385,597387],{"className":597386},[31801],[34,597388,597390],{"className":597389,"style":555424},[31805],[34,597391],{},[34,597393],{"className":597394},[31690,41488],"? 那样能算）。",[108,597397,597398,597401,597402,347],{},[71807,597399,597400],{},"假设题目无误的回答","：该积分为第一类完全椭圆积分形式，值为 ",[34,597403,597405,597440],{"className":597404},[31642],[34,597406,597408],{"className":597407},[31646],[31648,597409,597410],{"xmlns":31650},[31652,597411,597412,597437],{},[31655,597413,597414,597418,597420,597422,597424,597426,597430,597432,597434],{},[41463,597415,597416],{},[31749,597417,768],{},[31658,597419,38691],{},[31662,597421,456],{"stretchy":702},[31749,597423,640],{},[31658,597425,38080],{"mathvariant":38079},[41463,597427,597428],{},[31749,597429,768],{},[31662,597431,2557],{"stretchy":702},[31662,597433,168560],{},[31749,597435,597436],{},"2.622",[31665,597438,597439],{"encoding":31667},"\\sqrt{2} K(1\u002F\\sqrt{2}) \\approx 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\\int_0^1 \\frac{dx}{\\sqrt[n]{1 - x^n}} \\quad (n > 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t^{1\u002Fn}",[34,598219,598221,598239],{"className":598220,"ariaHidden":758},[31672],[34,598222,598224,598227,598230,598233,598236],{"className":598223},[31676],[34,598225],{"className":598226,"style":32527},[31680],[34,598228,32936],{"className":598229},[31685,31686],[34,598231],{"className":598232,"style":31846},[31845],[34,598234,202],{"className":598235},[31850],[34,598237],{"className":598238,"style":31846},[31845],[34,598240,598242,598245],{"className":598241},[31676],[34,598243],{"className":598244,"style":61037},[31680],[34,598246,598248,598251],{"className":598247},[31685],[34,598249,33249],{"className":598250},[31685,31686],[34,598252,598254],{"className":598253},[31792],[34,598255,598257],{"className":598256},[31796],[34,598258,598260],{"className":598259},[31801],[34,598261,598263],{"className":598262,"style":61037},[31805],[34,598264,598265,598268],{"style":37777},[34,598266],{"className":598267,"style":31814},[31813],[34,598269,598271],{"className":598270},[31818,31819,31820,31821],[34,598272,598274,598277],{"className":598273},[31685,31821],[34,598275,249219],{"className":598276},[31685,31821],[34,598278,31660],{"className":598279},[31685,31686,31821],[34,598281,598283,598327],{"className":598282},[31642],[34,598284,598286],{"className":598285},[31646],[31648,598287,598288],{"xmlns":31650},[31652,598289,598290,598324],{},[31655,598291,598292,598294,598296,598298,598304,598320,598322],{},[31658,598293,46900],{},[31658,598295,32936],{},[31662,598297,202],{},[41453,598299,598300,598302],{},[31749,598301,640],{},[31658,598303,31660],{},[37624,598305,598306,598308],{},[31658,598307,33249],{},[31655,598309,598310,598316,598318],{},[41453,598311,598312,598314],{},[31749,598313,640],{},[31658,598315,31660],{},[31662,598317,32226],{},[31749,598319,640],{},[31658,598321,46900],{},[31658,598323,33249],{},[31665,598325,598326],{"encoding":31667},"dx = \\frac{1}{n} t^{\\frac{1}{n}-1} dt",[34,598328,598330,598351],{"className":598329,"ariaHidden":758},[31672],[34,598331,598333,598336,598339,598342,598345,598348],{"className":598332},[31676],[34,598334],{"className":598335,"style":31681},[31680],[34,598337,46900],{"className":598338},[31685,31686],[34,598340,32936],{"className":598341},[31685,31686],[34,598343],{"className":598344,"style":31846},[31845],[34,598346,202],{"className":598347},[31850],[34,598349],{"className":598350,"style":31846},[31845],[34,598352,598354,598358,598426,598530,598533],{"className":598353},[31676],[34,598355],{"className":598356,"style":598357},[31680],"height:1.299em;vertical-align:-0.345em;",[34,598359,598361,598364,598423],{"className":598360},[31685],[34,598362],{"className":598363},[31782,41488],[34,598365,598367],{"className":598366},[41453],[34,598368,598370,598415],{"className":598369},[31796,31797],[34,598371,598373,598412],{"className":598372},[31801],[34,598374,598376,598390,598398],{"className":598375,"style":46746},[31805],[34,598377,598378,598381],{"style":41504},[34,598379],{"className":598380,"style":41508},[31813],[34,598382,598384],{"className":598383},[31818,31819,31820,31821],[34,598385,598387],{"className":598386},[31685,31821],[34,598388,31660],{"className":598389},[31685,31686,31821],[34,598391,598392,598395],{"style":41520},[34,598393],{"className":598394,"style":41508},[31813],[34,598396],{"className":598397,"style":41528},[41527],[34,598399,598400,598403],{"style":46771},[34,598401],{"className":598402,"style":41508},[31813],[34,598404,598406],{"className":598405},[31818,31819,31820,31821],[34,598407,598409],{"className":598408},[31685,31821],[34,598410,640],{"className":598411},[31685,31821],[34,598413,31829],{"className":598414},[31828],[34,598416,598418],{"className":598417},[31801],[34,598419,598421],{"className":598420,"style":41619},[31805],[34,598422],{},[34,598424],{"className":598425},[31690,41488],[34,598427,598429,598432],{"className":598428},[31685],[34,598430,33249],{"className":598431},[31685,31686],[34,598433,598435],{"className":598434},[31792],[34,598436,598438],{"className":598437},[31796],[34,598439,598441],{"className":598440},[31801],[34,598442,598445],{"className":598443,"style":598444},[31805],"height:0.954em;",[34,598446,598447,598450],{"style":435688},[34,598448],{"className":598449,"style":41508},[31813],[34,598451,598453],{"className":598452},[31818,31819,31820,31821],[34,598454,598456,598524,598527],{"className":598455},[31685,31821],[34,598457,598459,598462,598521],{"className":598458},[31685,31821],[34,598460],{"className":598461},[31782,41488,31818,43410,435704],[34,598463,598465],{"className":598464},[41453],[34,598466,598468,598513],{"className":598467},[31796,31797],[34,598469,598471,598510],{"className":598470},[31801],[34,598472,598474,598488,598496],{"className":598473,"style":593176},[31805],[34,598475,598476,598479],{"style":435720},[34,598477],{"className":598478,"style":41508},[31813],[34,598480,598482],{"className":598481},[31818,43410,43411,31821],[34,598483,598485],{"className":598484},[31685,31821],[34,598486,31660],{"className":598487},[31685,31686,31821],[34,598489,598490,598493],{"style":435735},[34,598491],{"className":598492,"style":41508},[31813],[34,598494],{"className":598495,"style":435742},[41527,31821],[34,598497,598498,598501],{"style":589620},[34,598499],{"className":598500,"style":41508},[31813],[34,598502,598504],{"className":598503},[31818,43410,43411,31821],[34,598505,598507],{"className":598506},[31685,31821],[34,598508,640],{"className":598509},[31685,31821],[34,598511,31829],{"className":598512},[31828],[34,598514,598516],{"className":598515},[31801],[34,598517,598519],{"className":598518,"style":435782},[31805],[34,598520],{},[34,598522],{"className":598523},[31690,41488,31818,43410,435704],[34,598525,32226],{"className":598526},[32024,31821],[34,598528,640],{"className":598529},[31685,31821],[34,598531,46900],{"className":598532},[31685,31686],[34,598534,33249],{"className":598535},[31685,31686],[108,598537,598538,598539,347],{},"区间：",[34,598540,598542,598584],{"className":598541},[31642],[34,598543,598545],{"className":598544},[31646],[31648,598546,598547],{"xmlns":31650},[31652,598548,598549,598581],{},[31655,598550,598551,598553,598555,598557,598559,598561,598563,598565,598567,598569,598571,598573,598575,598577,598579],{},[31658,598552,32936],{},[31662,598554,112296],{},[31662,598556,15972],{"stretchy":702},[31749,598558,459],{},[31662,598560,31960],{"separator":758},[31749,598562,640],{},[31662,598564,32712],{"stretchy":702},[31662,598566,140094],{},[31658,598568,33249],{},[31662,598570,112296],{},[31662,598572,15972],{"stretchy":702},[31749,598574,459],{},[31662,598576,31960],{"separator":758},[31749,598578,640],{},[31662,598580,32712],{"stretchy":702},[31665,598582,598583],{"encoding":31667},"x \\in [0,1] \\Rightarrow t \\in [0,1]",[34,598585,598587,598605,598638,598656],{"className":598586,"ariaHidden":758},[31672],[34,598588,598590,598593,598596,598599,598602],{"className":598589},[31676],[34,598591],{"className":598592,"style":34812},[31680],[34,598594,32936],{"className":598595},[31685,31686],[34,598597],{"className":598598,"style":31846},[31845],[34,598600,112296],{"className":598601},[31850],[34,598603],{"className":598604,"style":31846},[31845],[34,598606,598608,598611,598614,598617,598620,598623,598626,598629,598632,598635],{"className":598607},[31676],[34,598609],{"className":598610,"style":31774},[31680],[34,598612,15972],{"className":598613},[31782],[34,598615,459],{"className":598616},[31685],[34,598618,31960],{"className":598619},[32119],[34,598621],{"className":598622,"style":32123},[31845],[34,598624,640],{"className":598625},[31685],[34,598627,32712],{"className":598628},[31690],[34,598630],{"className":598631,"style":31846},[31845],[34,598633,140094],{"className":598634},[31850],[34,598636],{"className":598637,"style":31846},[31845],[34,598639,598641,598644,598647,598650,598653],{"className":598640},[31676],[34,598642],{"className":598643,"style":65913},[31680],[34,598645,33249],{"className":598646},[31685,31686],[34,598648],{"className":598649,"style":31846},[31845],[34,598651,112296],{"className":598652},[31850],[34,598654],{"className":598655,"style":31846},[31845],[34,598657,598659,598662,598665,598668,598671,598674,598677],{"className":598658},[31676],[34,598660],{"className":598661,"style":31774},[31680],[34,598663,15972],{"className":598664},[31782],[34,598666,459],{"className":598667},[31685],[34,598669,31960],{"className":598670},[32119],[34,598672],{"className":598673,"style":32123},[31845],[34,598675,640],{"className":598676},[31685],[34,598678,32712],{"className":598679},[31690],[108,598681,598682,598683,347],{},"被积函数 ",[34,598684,598686,598752],{"className":598685},[31642],[34,598687,598689],{"className":598688},[31646],[31648,598690,598691],{"xmlns":31650},[31652,598692,598693,598749],{},[31655,598694,598695,598725,598727,598729,598731,598733,598735],{},[41453,598696,598697,598699],{},[31749,598698,640],{},[31655,598700,598701,598703,598705,598707,598713],{},[31662,598702,456],{"stretchy":702},[31749,598704,640],{},[31662,598706,32226],{},[37624,598708,598709,598711],{},[31658,598710,32936],{},[31658,598712,31660],{},[37624,598714,598715,598717],{},[31662,598716,2557],{"stretchy":702},[31655,598718,598719,598721,598723],{},[31749,598720,640],{},[31658,598722,38080],{"mathvariant":38079},[31658,598724,31660],{},[31662,598726,202],{},[31662,598728,456],{"stretchy":702},[31749,598730,640],{},[31662,598732,32226],{},[31658,598734,33249],{},[37624,598736,598737,598739],{},[31662,598738,2557],{"stretchy":702},[31655,598740,598741,598743,598745,598747],{},[31662,598742,32226],{},[31749,598744,640],{},[31658,598746,38080],{"mathvariant":38079},[31658,598748,31660],{},[31665,598750,598751],{"encoding":31667},"\\frac{1}{(1-x^n)^{1\u002Fn}} = (1-t)^{-1\u002Fn}",[34,598753,598755,598913,598934],{"className":598754,"ariaHidden":758},[31672],[34,598756,598758,598762,598904,598907,598910],{"className":598757},[31676],[34,598759],{"className":598760,"style":598761},[31680],"height:1.4055em;vertical-align:-0.5604em;",[34,598763,598765,598768,598901],{"className":598764},[31685],[34,598766],{"className":598767},[31782,41488],[34,598769,598771],{"className":598770},[41453],[34,598772,598774,598892],{"className":598773},[31796,31797],[34,598775,598777,598889],{"className":598776},[31801],[34,598778,598780,598867,598875],{"className":598779,"style":46746},[31805],[34,598781,598783,598786],{"style":598782},"top:-2.6146em;",[34,598784],{"className":598785,"style":41508},[31813],[34,598787,598789],{"className":598788},[31818,31819,31820,31821],[34,598790,598792,598795,598798,598801,598830],{"className":598791},[31685,31821],[34,598793,456],{"className":598794},[31782,31821],[34,598796,640],{"className":598797},[31685,31821],[34,598799,32226],{"className":598800},[32024,31821],[34,598802,598804,598807],{"className":598803},[31685,31821],[34,598805,32936],{"className":598806},[31685,31686,31821],[34,598808,598810],{"className":598809},[31792],[34,598811,598813],{"className":598812},[31796],[34,598814,598816],{"className":598815},[31801],[34,598817,598819],{"className":598818,"style":566313},[31805],[34,598820,598821,598824],{"style":116668},[34,598822],{"className":598823,"style":43406},[31813],[34,598825,598827],{"className":598826},[31818,43410,43411,31821],[34,598828,31660],{"className":598829},[31685,31686,31821],[34,598831,598833,598836],{"className":598832},[31690,31821],[34,598834,2557],{"className":598835},[31690,31821],[34,598837,598839],{"className":598838},[31792],[34,598840,598842],{"className":598841},[31796],[34,598843,598845],{"className":598844},[31801],[34,598846,598849],{"className":598847,"style":598848},[31805],"height:0.822em;",[34,598850,598852,598855],{"style":598851},"top:-2.822em;margin-right:0.0714em;",[34,598853],{"className":598854,"style":555402},[31813],[34,598856,598858],{"className":598857},[31818,43410,43411,31821],[34,598859,598861,598864],{"className":598860},[31685,31821],[34,598862,249219],{"className":598863},[31685,31821],[34,598865,31660],{"className":598866},[31685,31686,31821],[34,598868,598869,598872],{"style":41520},[34,598870],{"className":598871,"style":41508},[31813],[34,598873],{"className":598874,"style":41528},[41527],[34,598876,598877,598880],{"style":46771},[34,598878],{"className":598879,"style":41508},[31813],[34,598881,598883],{"className":598882},[31818,31819,31820,31821],[34,598884,598886],{"className":598885},[31685,31821],[34,598887,640],{"className":598888},[31685,31821],[34,598890,31829],{"className":598891},[31828],[34,598893,598895],{"className":598894},[31801],[34,598896,598899],{"className":598897,"style":598898},[31805],"height:0.5604em;",[34,598900],{},[34,598902],{"className":598903},[31690,41488],[34,598905],{"className":598906,"style":31846},[31845],[34,598908,202],{"className":598909},[31850],[34,598911],{"className":598912,"style":31846},[31845],[34,598914,598916,598919,598922,598925,598928,598931],{"className":598915},[31676],[34,598917],{"className":598918,"style":31774},[31680],[34,598920,456],{"className":598921},[31782],[34,598923,640],{"className":598924},[31685],[34,598926],{"className":598927,"style":32604},[31845],[34,598929,32226],{"className":598930},[32024],[34,598932],{"className":598933,"style":32604},[31845],[34,598935,598937,598940,598943],{"className":598936},[31676],[34,598938],{"className":598939,"style":61012},[31680],[34,598941,33249],{"className":598942},[31685,31686],[34,598944,598946,598949],{"className":598945},[31690],[34,598947,2557],{"className":598948},[31690],[34,598950,598952],{"className":598951},[31792],[34,598953,598955],{"className":598954},[31796],[34,598956,598958],{"className":598957},[31801],[34,598959,598961],{"className":598960,"style":61037},[31805],[34,598962,598963,598966],{"style":37777},[34,598964],{"className":598965,"style":31814},[31813],[34,598967,598969],{"className":598968},[31818,31819,31820,31821],[34,598970,598972,598975,598978],{"className":598971},[31685,31821],[34,598973,32226],{"className":598974},[31685,31821],[34,598976,249219],{"className":598977},[31685,31821],[34,598979,31660],{"className":598980},[31685,31686,31821],[108,598982,598983,598984,347],{},"代入积分：",[34,598985,598987,599119],{"className":598986},[31642],[34,598988,598990],{"className":598989},[31646],[31648,598991,598992],{"xmlns":31650},[31652,598993,598994,599116],{},[31655,598995,598996,599004,599006,599008,599010,599012,599026,599028,599034,599050,599052,599054,599056,599058,599064,599072,599088,599090,599092,599094,599096,599110,599112,599114],{},[38846,598997,598998,599000,599002],{},[31662,598999,546295],{},[31749,599001,459],{},[31749,599003,640],{},[31662,599005,456],{"stretchy":702},[31749,599007,640],{},[31662,599009,32226],{},[31658,599011,33249],{},[37624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(1-t)^{-1\u002Fn} \\cdot \\frac{1}{n} t^{\\frac{1}{n}-1} \\, dt = \\frac{1}{n} \\int_0^1 t^{\\frac{1}{n}-1} (1-t)^{-1\u002Fn} \\, 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\\frac{1}{n} B(\\frac{1}{n}, 1-\\frac{1}{n})",[34,600078,600080,600092,600258],{"className":600079,"ariaHidden":758},[31672],[34,600081,600083,600086,600089],{"className":600082},[31676],[34,600084],{"className":600085,"style":73089},[31680],[34,600087,202],{"className":600088},[31850],[34,600090],{"className":600091,"style":31846},[31845],[34,600093,600095,600098,600166,600169,600172,600240,600243,600246,600249,600252,600255],{"className":600094},[31676],[34,600096],{"className":600097,"style":46727},[31680],[34,600099,600101,600104,600163],{"className":600100},[31685],[34,600102],{"className":600103},[31782,41488],[34,600105,600107],{"className":600106},[41453],[34,600108,600110,600155],{"className":600109},[31796,31797],[34,600111,600113,600152],{"className":600112},[31801],[34,600114,600116,600130,600138],{"className":600115,"style":46746},[31805],[34,600117,600118,600121],{"style":41504},[34,600119],{"className":600120,"style":41508},[31813],[34,600122,600124],{"className":600123},[31818,31819,31820,31821],[34,600125,600127],{"className":600126},[31685,31821],[34,600128,31660],{"className":600129},[31685,31686,31821],[34,600131,600132,600135],{"style":41520},[34,600133],{"className":600134,"style":41508},[31813],[34,600136],{"className":600137,"style":41528},[41527],[34,600139,600140,600143],{"style":46771},[34,600141],{"className":600142,"style":41508},[31813],[34,600144,600146],{"className":600145},[31818,31819,31820,31821],[34,600147,600149],{"className":600148},[31685,31821],[34,600150,640],{"className":600151},[31685,31821],[34,600153,31829],{"className":600154},[31828],[34,600156,600158],{"className":600157},[31801],[34,600159,600161],{"className":600160,"style":41619},[31805],[34,600162],{},[34,600164],{"className":600165},[31690,41488],[34,600167,54224],{"className":600168,"style":54252},[31685,31686],[34,600170,456],{"className":600171},[31782],[34,600173,600175,600178,600237],{"className":600174},[31685],[34,600176],{"className":600177},[31782,41488],[34,600179,600181],{"className":600180},[41453],[34,600182,600184,600229],{"className":600183},[31796,31797],[34,600185,600187,600226],{"className":600186},[31801],[34,600188,600190,600204,600212],{"className":600189,"style":46746},[31805],[34,600191,600192,600195],{"style":41504},[34,600193],{"className":600194,"style":41508},[31813],[34,600196,600198],{"className":600197},[31818,31819,31820,31821],[34,600199,600201],{"className":600200},[31685,31821],[34,600202,31660],{"className":600203},[31685,31686,31821],[34,600205,600206,600209],{"style":41520},[34,600207],{"className":600208,"style":41508},[31813],[34,600210],{"className":600211,"style":41528},[41527],[34,600213,600214,600217],{"style":46771},[34,600215],{"className":600216,"style":41508},[31813],[34,600218,600220],{"className":600219},[31818,31819,31820,31821],[34,600221,600223],{"className":600222},[31685,31821],[34,600224,640],{"className":600225},[31685,31821],[34,600227,31829],{"className":600228},[31828],[34,600230,600232],{"className":600231},[31801],[34,600233,600235],{"className":600234,"style":41619},[31805],[34,600236],{},[34,600238],{"className":600239},[31690,41488],[34,600241,31960],{"className":600242},[32119],[34,600244],{"className":600245,"style":32123},[31845],[34,600247,640],{"className":600248},[31685],[34,600250],{"className":600251,"style":32604},[31845],[34,600253,32226],{"className":600254},[32024],[34,600256],{"className":600257,"style":32604},[31845],[34,600259,600261,600264,600332],{"className":600260},[31676],[34,600262],{"className":600263,"style":46727},[31680],[34,600265,600267,600270,600329],{"className":600266},[31685],[34,600268],{"className":600269},[31782,41488],[34,600271,600273],{"className":600272},[41453],[34,600274,600276,600321],{"className":600275},[31796,31797],[34,600277,600279,600318],{"className":600278},[31801],[34,600280,600282,600296,600304],{"className":600281,"style":46746},[31805],[34,600283,600284,600287],{"style":41504},[34,600285],{"className":600286,"style":41508},[31813],[34,600288,600290],{"className":600289},[31818,31819,31820,31821],[34,600291,600293],{"className":600292},[31685,31821],[34,600294,31660],{"className":600295},[31685,31686,31821],[34,600297,600298,600301],{"style":41520},[34,600299],{"className":600300,"style":41508},[31813],[34,600302],{"className":600303,"style":41528},[41527],[34,600305,600306,600309],{"style":46771},[34,600307],{"className":600308,"style":41508},[31813],[34,600310,600312],{"className":600311},[31818,31819,31820,31821],[34,600313,600315],{"className":600314},[31685,31821],[34,600316,640],{"className":600317},[31685,31821],[34,600319,31829],{"className":600320},[31828],[34,600322,600324],{"className":600323},[31801],[34,600325,600327],{"className":600326,"style":41619},[31805],[34,600328],{},[34,600330],{"className":600331},[31690,41488],[34,600333,2557],{"className":600334},[31690],[108,600336,600337,600338,33183,600542,347],{},"利用余元公式 ",[34,600339,600341,600389],{"className":600340},[31642],[34,600342,600344],{"className":600343},[31646],[31648,600345,600346],{"xmlns":31650},[31652,600347,600348,600386],{},[31655,600349,600350,600352,600354,600356,600358,600360,600362,600364,600366,600368,600370,600372],{},[31658,600351,564791],{"mathvariant":38079},[31662,600353,456],{"stretchy":702},[31658,600355,15],{},[31662,600357,2557],{"stretchy":702},[31658,600359,564791],{"mathvariant":38079},[31662,600361,456],{"stretchy":702},[31749,600363,640],{},[31662,600365,32226],{},[31658,600367,15],{},[31662,600369,2557],{"stretchy":702},[31662,600371,202],{},[41453,600373,600374,600376],{},[31658,600375,58442],{},[31655,600377,600378,600380,600382,600384],{},[31658,600379,547853],{},[31662,600381,32682],{},[31658,600383,15],{},[31658,600385,58442],{},[31665,600387,600388],{"encoding":31667},"\\Gamma(p)\\Gamma(1-p) = \\frac{\\pi}{\\sin p\\pi}",[34,600390,600392,600428,600449],{"className":600391,"ariaHidden":758},[31672],[34,600393,600395,600398,600401,600404,600407,600410,600413,600416,600419,600422,600425],{"className":600394},[31676],[34,600396],{"className":600397,"style":31774},[31680],[34,600399,564791],{"className":600400},[31685],[34,600402,456],{"className":600403},[31782],[34,600405,15],{"className":600406},[31685,31686],[34,600408,2557],{"className":600409},[31690],[34,600411,564791],{"className":600412},[31685],[34,600414,456],{"className":600415},[31782],[34,600417,640],{"className":600418},[31685],[34,600420],{"className":600421,"style":32604},[31845],[34,600423,32226],{"className":600424},[32024],[34,600426],{"className":600427,"style":32604},[31845],[34,600429,600431,600434,600437,600440,600443,600446],{"className":600430},[31676],[34,600432],{"className":600433,"style":31774},[31680],[34,600435,15],{"className":600436},[31685,31686],[34,600438,2557],{"className":600439},[31690],[34,600441],{"className":600442,"style":31846},[31845],[34,600444,202],{"className":600445},[31850],[34,600447],{"className":600448,"style":31846},[31845],[34,600450,600452,600456],{"className":600451},[31676],[34,600453],{"className":600454,"style":600455},[31680],"height:1.1765em;vertical-align:-0.4811em;",[34,600457,600459,600462,600539],{"className":600458},[31685],[34,600460],{"className":600461},[31782,41488],[34,600463,600465],{"className":600464},[41453],[34,600466,600468,600531],{"className":600467},[31796,31797],[34,600469,600471,600528],{"className":600470},[31801],[34,600472,600474,600506,600514],{"className":600473,"style":244945},[31805],[34,600475,600476,600479],{"style":41504},[34,600477],{"className":600478,"style":41508},[31813],[34,600480,600482],{"className":600481},[31818,31819,31820,31821],[34,600483,600485,600497,600500,600503],{"className":600484},[31685,31821],[34,600486,600488,600491,600494],{"className":600487},[32767,31821],[34,600489,31747],{"className":600490},[31821],[34,600492,31955],{"className":600493},[31821],[34,600495,31660],{"className":600496},[31821],[34,600498],{"className":600499,"style":43430},[31845,31821],[34,600501,15],{"className":600502},[31685,31686,31821],[34,600504,58442],{"className":600505,"style":32062},[31685,31686,31821],[34,600507,600508,600511],{"style":41520},[34,600509],{"className":600510,"style":41508},[31813],[34,600512],{"className":600513,"style":41528},[41527],[34,600515,600516,600519],{"style":46771},[34,600517],{"className":600518,"style":41508},[31813],[34,600520,600522],{"className":600521},[31818,31819,31820,31821],[34,600523,600525],{"className":600524},[31685,31821],[34,600526,58442],{"className":600527,"style":32062},[31685,31686,31821],[34,600529,31829],{"className":600530},[31828],[34,600532,600534],{"className":600533},[31801],[34,600535,600537],{"className":600536,"style":220191},[31805],[34,600538],{},[34,600540],{"className":600541},[31690,41488],[34,600543,600545,600615],{"className":600544},[31642],[34,600546,600548],{"className":600547},[31646],[31648,600549,600550],{"xmlns":31650},[31652,600551,600552,600612],{},[31655,600553,600554,600556,600558,600560,600562,600564,600566,600568,600570,600572,600574,600576,600578,600580,600582,600584,600586,600588,600590,600592,600594],{},[31658,600555,54224],{},[31662,600557,456],{"stretchy":702},[31658,600559,15],{},[31662,600561,31960],{"separator":758},[31749,600563,640],{},[31662,600565,32226],{},[31658,600567,15],{},[31662,600569,2557],{"stretchy":702},[31662,600571,202],{},[31658,600573,564791],{"mathvariant":38079},[31662,600575,456],{"stretchy":702},[31658,600577,15],{},[31662,600579,2557],{"stretchy":702},[31658,600581,564791],{"mathvariant":38079},[31662,600583,456],{"stretchy":702},[31749,600585,640],{},[31662,600587,32226],{},[31658,600589,15],{},[31662,600591,2557],{"stretchy":702},[31662,600593,202],{},[41453,600595,600596,600598],{},[31658,600597,58442],{},[31655,600599,600600,600602,600604,600606,600608,600610],{},[31658,600601,547853],{},[31662,600603,32682],{},[31662,600605,456],{"stretchy":702},[31658,600607,15],{},[31658,600609,58442],{},[31662,600611,2557],{"stretchy":702},[31665,600613,600614],{"encoding":31667},"B(p, 1-p) = \\Gamma(p)\\Gamma(1-p) = \\frac{\\pi}{\\sin(p\\pi)}",[34,600616,600618,600651,600672,600708,600729],{"className":600617,"ariaHidden":758},[31672],[34,600619,600621,600624,600627,600630,600633,600636,600639,600642,600645,600648],{"className":600620},[31676],[34,600622],{"className":600623,"style":31774},[31680],[34,600625,54224],{"className":600626,"style":54252},[31685,31686],[34,600628,456],{"className":600629},[31782],[34,600631,15],{"className":600632},[31685,31686],[34,600634,31960],{"className":600635},[32119],[34,600637],{"className":600638,"style":32123},[31845],[34,600640,640],{"className":600641},[31685],[34,600643],{"className":600644,"style":32604},[31845],[34,600646,32226],{"className":600647},[32024],[34,600649],{"className":600650,"style":32604},[31845],[34,600652,600654,600657,600660,600663,600666,600669],{"className":600653},[31676],[34,600655],{"className":600656,"style":31774},[31680],[34,600658,15],{"className":600659},[31685,31686],[34,600661,2557],{"className":600662},[31690],[34,600664],{"className":600665,"style":31846},[31845],[34,600667,202],{"className":600668},[31850],[34,600670],{"className":600671,"style":31846},[31845],[34,600673,600675,600678,600681,600684,600687,600690,600693,600696,600699,600702,600705],{"className":600674},[31676],[34,600676],{"className":600677,"style":31774},[31680],[34,600679,564791],{"className":600680},[31685],[34,600682,456],{"className":600683},[31782],[34,600685,15],{"className":600686},[31685,31686],[34,600688,2557],{"className":600689},[31690],[34,600691,564791],{"className":600692},[31685],[34,600694,456],{"className":600695},[31782],[34,600697,640],{"className":600698},[31685],[34,600700],{"className":600701,"style":32604},[31845],[34,600703,32226],{"className":600704},[32024],[34,600706],{"className":600707,"style":32604},[31845],[34,600709,600711,600714,600717,600720,600723,600726],{"className":600710},[31676],[34,600712],{"className":600713,"style":31774},[31680],[34,600715,15],{"className":600716},[31685,31686],[34,600718,2557],{"className":600719},[31690],[34,600721],{"className":600722,"style":31846},[31845],[34,600724,202],{"className":600725},[31850],[34,600727],{"className":600728,"style":31846},[31845],[34,600730,600732,600735],{"className":600731},[31676],[34,600733],{"className":600734,"style":570332},[31680],[34,600736,600738,600741,600821],{"className":600737},[31685],[34,600739],{"className":600740},[31782,41488],[34,600742,600744],{"className":600743},[41453],[34,600745,600747,600813],{"className":600746},[31796,31797],[34,600748,600750,600810],{"className":600749},[31801],[34,600751,600753,600788,600796],{"className":600752,"style":244945},[31805],[34,600754,600755,600758],{"style":41504},[34,600756],{"className":600757,"style":41508},[31813],[34,600759,600761],{"className":600760},[31818,31819,31820,31821],[34,600762,600764,600776,600779,600782,600785],{"className":600763},[31685,31821],[34,600765,600767,600770,600773],{"className":600766},[32767,31821],[34,600768,31747],{"className":600769},[31821],[34,600771,31955],{"className":600772},[31821],[34,600774,31660],{"className":600775},[31821],[34,600777,456],{"className":600778},[31782,31821],[34,600780,15],{"className":600781},[31685,31686,31821],[34,600783,58442],{"className":600784,"style":32062},[31685,31686,31821],[34,600786,2557],{"className":600787},[31690,31821],[34,600789,600790,600793],{"style":41520},[34,600791],{"className":600792,"style":41508},[31813],[34,600794],{"className":600795,"style":41528},[41527],[34,600797,600798,600801],{"style":46771},[34,600799],{"className":600800,"style":41508},[31813],[34,600802,600804],{"className":600803},[31818,31819,31820,31821],[34,600805,600807],{"className":600806},[31685,31821],[34,600808,58442],{"className":600809,"style":32062},[31685,31686,31821],[34,600811,31829],{"className":600812},[31828],[34,600814,600816],{"className":600815},[31801],[34,600817,600819],{"className":600818,"style":324858},[31805],[34,600820],{},[34,600822],{"className":600823},[31690,41488],[108,600825,600826,600827,347],{},"结果 ",[34,600828,600830,600872],{"className":600829},[31642],[34,600831,600833],{"className":600832},[31646],[31648,600834,600835],{"xmlns":31650},[31652,600836,600837,600869],{},[31655,600838,600839,600841,600847,600849],{},[31662,600840,202],{},[41453,600842,600843,600845],{},[31749,600844,640],{},[31658,600846,31660],{},[31662,600848,71191],{},[41453,600850,600851,600853],{},[31658,600852,58442],{},[31655,600854,600855,600857,600859,600861,600867],{},[31658,600856,547853],{},[31662,600858,32682],{},[31662,600860,456],{"stretchy":702},[41453,600862,600863,600865],{},[31658,600864,58442],{},[31658,600866,31660],{},[31662,600868,2557],{"stretchy":702},[31665,600870,600871],{"encoding":31667},"= \\frac{1}{n} \\cdot \\frac{\\pi}{\\sin(\\frac{\\pi}{n})}",[34,600873,600875,600887,600970],{"className":600874,"ariaHidden":758},[31672],[34,600876,600878,600881,600884],{"className":600877},[31676],[34,600879],{"className":600880,"style":73089},[31680],[34,600882,202],{"className":600883},[31850],[34,600885],{"className":600886,"style":31846},[31845],[34,600888,600890,600893,600961,600964,600967],{"className":600889},[31676],[34,600891],{"className":600892,"style":46727},[31680],[34,600894,600896,600899,600958],{"className":600895},[31685],[34,600897],{"className":600898},[31782,41488],[34,600900,600902],{"className":600901},[41453],[34,600903,600905,600950],{"className":600904},[31796,31797],[34,600906,600908,600947],{"className":600907},[31801],[34,600909,600911,600925,600933],{"className":600910,"style":46746},[31805],[34,600912,600913,600916],{"style":41504},[34,600914],{"className":600915,"style":41508},[31813],[34,600917,600919],{"className":600918},[31818,31819,31820,31821],[34,600920,600922],{"className":600921},[31685,31821],[34,600923,31660],{"className":600924},[31685,31686,31821],[34,600926,600927,600930],{"style":41520},[34,600928],{"className":600929,"style":41508},[31813],[34,600931],{"className":600932,"style":41528},[41527],[34,600934,600935,600938],{"style":46771},[34,600936],{"className":600937,"style":41508},[31813],[34,600939,600941],{"className":600940},[31818,31819,31820,31821],[34,600942,600944],{"className":600943},[31685,31821],[34,600945,640],{"className":600946},[31685,31821],[34,600948,31829],{"className":600949},[31828],[34,600951,600953],{"className":600952},[31801],[34,600954,600956],{"className":600955,"style":41619},[31805],[34,600957],{},[34,600959],{"className":600960},[31690,41488],[34,600962],{"className":600963,"style":32604},[31845],[34,600965,71191],{"className":600966},[32024],[34,600968],{"className":600969,"style":32604},[31845],[34,600971,600973,600977],{"className":600972},[31676],[34,600974],{"className":600975,"style":600976},[31680],"height:1.2812em;vertical-align:-0.5858em;",[34,600978,600980,600983,601126],{"className":600979},[31685],[34,600981],{"className":600982},[31782,41488],[34,600984,600986],{"className":600985},[41453],[34,600987,600989,601117],{"className":600988},[31796,31797],[34,600990,600992,601114],{"className":600991},[31801],[34,600993,600995,601092,601100],{"className":600994,"style":244945},[31805],[34,600996,600997,601000],{"style":41504},[34,600998],{"className":600999,"style":41508},[31813],[34,601001,601003],{"className":601002},[31818,31819,31820,31821],[34,601004,601006,601018,601021,601089],{"className":601005},[31685,31821],[34,601007,601009,601012,601015],{"className":601008},[32767,31821],[34,601010,31747],{"className":601011},[31821],[34,601013,31955],{"className":601014},[31821],[34,601016,31660],{"className":601017},[31821],[34,601019,456],{"className":601020},[31782,31821],[34,601022,601024,601027,601086],{"className":601023},[31685,31821],[34,601025],{"className":601026},[31782,41488,31818,43410,435704],[34,601028,601030],{"className":601029},[41453],[34,601031,601033,601078],{"className":601032},[31796,31797],[34,601034,601036,601075],{"className":601035},[31801],[34,601037,601039,601053,601061],{"className":601038,"style":589595},[31805],[34,601040,601041,601044],{"style":435720},[34,601042],{"className":601043,"style":41508},[31813],[34,601045,601047],{"className":601046},[31818,43410,43411,31821],[34,601048,601050],{"className":601049},[31685,31821],[34,601051,31660],{"className":601052},[31685,31686,31821],[34,601054,601055,601058],{"style":435735},[34,601056],{"className":601057,"style":41508},[31813],[34,601059],{"className":601060,"style":435742},[41527,31821],[34,601062,601063,601066],{"style":589620},[34,601064],{"className":601065,"style":41508},[31813],[34,601067,601069],{"className":601068},[31818,43410,43411,31821],[34,601070,601072],{"className":601071},[31685,31821],[34,601073,58442],{"className":601074,"style":32062},[31685,31686,31821],[34,601076,31829],{"className":601077},[31828],[34,601079,601081],{"className":601080},[31801],[34,601082,601084],{"className":601083,"style":435782},[31805],[34,601085],{},[34,601087],{"className":601088},[31690,41488,31818,43410,435704],[34,601090,2557],{"className":601091},[31690,31821],[34,601093,601094,601097],{"style":41520},[34,601095],{"className":601096,"style":41508},[31813],[34,601098],{"className":601099,"style":41528},[41527],[34,601101,601102,601105],{"style":46771},[34,601103],{"className":601104,"style":41508},[31813],[34,601106,601108],{"className":601107},[31818,31819,31820,31821],[34,601109,601111],{"className":601110},[31685,31821],[34,601112,58442],{"className":601113,"style":32062},[31685,31686,31821],[34,601115,31829],{"className":601116},[31828],[34,601118,601120],{"className":601119},[31801],[34,601121,601124],{"className":601122,"style":601123},[31805],"height:0.5858em;",[34,601125],{},[34,601127],{"className":601128},[31690,41488],[108,601130,601131,2664,601133],{},[2627,601132,118575],{},[34,601134,601136,601166],{"className":601135},[31642],[34,601137,601139],{"className":601138},[31646],[31648,601140,601141],{"xmlns":31650},[31652,601142,601143,601163],{},[31655,601144,601145],{},[41453,601146,601147,601149],{},[31658,601148,58442],{},[31655,601150,601151,601153,601155,601157],{},[31658,601152,31660],{},[31658,601154,547853],{},[31662,601156,32682],{},[41453,601158,601159,601161],{},[31658,601160,58442],{},[31658,601162,31660],{},[31665,601164,601165],{"encoding":31667},"\\frac{\\pi}{n \\sin \\frac{\\pi}{n}}",[34,601167,601169],{"className":601168,"ariaHidden":758},[31672],[34,601170,601172,601175],{"className":601171},[31676],[34,601173],{"className":601174,"style":600976},[31680],[34,601176,601178,601181,601326],{"className":601177},[31685],[34,601179],{"className":601180},[31782,41488],[34,601182,601184],{"className":601183},[41453],[34,601185,601187,601318],{"className":601186},[31796,31797],[34,601188,601190,601315],{"className":601189},[31801],[34,601191,601193,601293,601301],{"className":601192,"style":244945},[31805],[34,601194,601195,601198],{"style":41504},[34,601196],{"className":601197,"style":41508},[31813],[34,601199,601201],{"className":601200},[31818,31819,31820,31821],[34,601202,601204,601207,601210,601222,601225],{"className":601203},[31685,31821],[34,601205,31660],{"className":601206},[31685,31686,31821],[34,601208],{"className":601209,"style":43430},[31845,31821],[34,601211,601213,601216,601219],{"className":601212},[32767,31821],[34,601214,31747],{"className":601215},[31821],[34,601217,31955],{"className":601218},[31821],[34,601220,31660],{"className":601221},[31821],[34,601223],{"className":601224,"style":43430},[31845,31821],[34,601226,601228,601231,601290],{"className":601227},[31685,31821],[34,601229],{"className":601230},[31782,41488,31818,43410,435704],[34,601232,601234],{"className":601233},[41453],[34,601235,601237,601282],{"className":601236},[31796,31797],[34,601238,601240,601279],{"className":601239},[31801],[34,601241,601243,601257,601265],{"className":601242,"style":589595},[31805],[34,601244,601245,601248],{"style":435720},[34,601246],{"className":601247,"style":41508},[31813],[34,601249,601251],{"className":601250},[31818,43410,43411,31821],[34,601252,601254],{"className":601253},[31685,31821],[34,601255,31660],{"className":601256},[31685,31686,31821],[34,601258,601259,601262],{"style":435735},[34,601260],{"className":601261,"style":41508},[31813],[34,601263],{"className":601264,"style":435742},[41527,31821],[34,601266,601267,601270],{"style":589620},[34,601268],{"className":601269,"style":41508},[31813],[34,601271,601273],{"className":601272},[31818,43410,43411,31821],[34,601274,601276],{"className":601275},[31685,31821],[34,601277,58442],{"className":601278,"style":32062},[31685,31686,31821],[34,601280,31829],{"className":601281},[31828],[34,601283,601285],{"className":601284},[31801],[34,601286,601288],{"className":601287,"style":435782},[31805],[34,601289],{},[34,601291],{"className":601292},[31690,41488,31818,43410,435704],[34,601294,601295,601298],{"style":41520},[34,601296],{"className":601297,"style":41508},[31813],[34,601299],{"className":601300,"style":41528},[41527],[34,601302,601303,601306],{"style":46771},[34,601304],{"className":601305,"style":41508},[31813],[34,601307,601309],{"className":601308},[31818,31819,31820,31821],[34,601310,601312],{"className":601311},[31685,31821],[34,601313,58442],{"className":601314,"style":32062},[31685,31686,31821],[34,601316,31829],{"className":601317},[31828],[34,601319,601321],{"className":601320},[31801],[34,601322,601324],{"className":601323,"style":601123},[31805],[34,601325],{},[34,601327],{"className":601328},[31690,41488],[15,601330,601331,47649,601334],{},[2627,601332,601333],{},"(4)",[34,601335,601337,601411],{"className":601336},[31642],[34,601338,601340],{"className":601339},[31646],[31648,601341,601342],{"xmlns":31650},[31652,601343,601344,601408],{},[31655,601345,601346],{},[71090,601347,601348,601360,601386,601388,601390,601392,601394,601396,601398,601400,601402,601404,601406],{"scriptlevel":459,"displaystyle":758},[38846,601349,601350,601352,601354],{},[31662,601351,546295],{},[31749,601353,459],{},[31655,601355,601356,601358],{},[31662,601357,223],{},[31658,601359,138286],{"mathvariant":38079},[41453,601361,601362,601374],{},[37624,601363,601364,601366],{},[31658,601365,32936],{},[31655,601367,601368,601370,601372],{},[31658,601369,31759],{},[31662,601371,32226],{},[31749,601373,640],{},[31655,601375,601376,601378,601380],{},[31749,601377,640],{},[31662,601379,223],{},[37624,601381,601382,601384],{},[31658,601383,32936],{},[31658,601385,31660],{},[39361,601387,39363],{},[31658,601389,46900],{},[31658,601391,32936],{},[31845,601393],{"width":71082},[31662,601395,456],{"stretchy":702},[31658,601397,31660],{},[31662,601399,29622],{},[31658,601401,31759],{},[31662,601403,29622],{},[31749,601405,459],{},[31662,601407,2557],{"stretchy":702},[31665,601409,601410],{"encoding":31667},"\\displaystyle \\int_0^{+\\infty} \\frac{x^{m-1}}{1 + x^n} \\, dx \\quad (n > m > 0)",[34,601412,601414,601646,601664],{"className":601413,"ariaHidden":758},[31672],[34,601415,601417,601421,601479,601482,601619,601622,601625,601628,601631,601634,601637,601640,601643],{"className":601416},[31676],[34,601418],{"className":601419,"style":601420},[31680],"height:2.4332em;vertical-align:-0.9119em;",[34,601422,601424,601427],{"className":601423},[32767],[34,601425,546295],{"className":601426,"style":591475},[32767,131249,131250],[34,601428,601430],{"className":601429},[31792],[34,601431,601433,601471],{"className":601432},[31796,31797],[34,601434,601436,601468],{"className":601435},[31801],[34,601437,601440,601451],{"className":601438,"style":601439},[31805],"height:1.5212em;",[34,601441,601442,601445],{"style":591491},[34,601443],{"className":601444,"style":31814},[31813],[34,601446,601448],{"className":601447},[31818,31819,31820,31821],[34,601449,459],{"className":601450},[31685,31821],[34,601452,601453,601456],{"style":591503},[34,601454],{"className":601455,"style":31814},[31813],[34,601457,601459],{"className":601458},[31818,31819,31820,31821],[34,601460,601462,601465],{"className":601461},[31685,31821],[34,601463,223],{"className":601464},[31685,31821],[34,601466,138286],{"className":601467},[31685,31821],[34,601469,31829],{"className":601470},[31828],[34,601472,601474],{"className":601473},[31801],[34,601475,601477],{"className":601476,"style":591522},[31805],[34,601478],{},[34,601480],{"className":601481,"style":32123},[31845],[34,601483,601485,601488,601616],{"className":601484},[31685],[34,601486],{"className":601487},[31782,41488],[34,601489,601491],{"className":601490},[41453],[34,601492,601494,601607],{"className":601493},[31796,31797],[34,601495,601497,601604],{"className":601496},[31801],[34,601498,601501,601550,601558],{"className":601499,"style":601500},[31805],"height:1.4911em;",[34,601502,601503,601506],{"style":129639},[34,601504],{"className":601505,"style":41508},[31813],[34,601507,601509,601512,601515,601518,601521],{"className":601508},[31685],[34,601510,640],{"className":601511},[31685],[34,601513],{"className":601514,"style":32604},[31845],[34,601516,223],{"className":601517},[32024],[34,601519],{"className":601520,"style":32604},[31845],[34,601522,601524,601527],{"className":601523},[31685],[34,601525,32936],{"className":601526},[31685,31686],[34,601528,601530],{"className":601529},[31792],[34,601531,601533],{"className":601532},[31796],[34,601534,601536],{"className":601535},[31801],[34,601537,601539],{"className":601538,"style":597809},[31805],[34,601540,601541,601544],{"style":590776},[34,601542],{"className":601543,"style":31814},[31813],[34,601545,601547],{"className":601546},[31818,31819,31820,31821],[34,601548,31660],{"className":601549},[31685,31686,31821],[34,601551,601552,601555],{"style":41520},[34,601553],{"className":601554,"style":41508},[31813],[34,601556],{"className":601557,"style":41528},[41527],[34,601559,601560,601563],{"style":129659},[34,601561],{"className":601562,"style":41508},[31813],[34,601564,601566],{"className":601565},[31685],[34,601567,601569,601572],{"className":601568},[31685],[34,601570,32936],{"className":601571},[31685,31686],[34,601573,601575],{"className":601574},[31792],[34,601576,601578],{"className":601577},[31796],[34,601579,601581],{"className":601580},[31801],[34,601582,601584],{"className":601583,"style":37774},[31805],[34,601585,601586,601589],{"style":37777},[34,601587],{"className":601588,"style":31814},[31813],[34,601590,601592],{"className":601591},[31818,31819,31820,31821],[34,601593,601595,601598,601601],{"className":601594},[31685,31821],[34,601596,31759],{"className":601597},[31685,31686,31821],[34,601599,32226],{"className":601600},[32024,31821],[34,601602,640],{"className":601603},[31685,31821],[34,601605,31829],{"className":601606},[31828],[34,601608,601610],{"className":601609},[31801],[34,601611,601614],{"className":601612,"style":601613},[31805],"height:0.7693em;",[34,601615],{},[34,601617],{"className":601618},[31690,41488],[34,601620],{"className":601621,"style":32123},[31845],[34,601623,46900],{"className":601624},[31685,31686],[34,601626,32936],{"className":601627},[31685,31686],[34,601629],{"className":601630,"style":128438},[31845],[34,601632,456],{"className":601633},[31782],[34,601635,31660],{"className":601636},[31685,31686],[34,601638],{"className":601639,"style":31846},[31845],[34,601641,29622],{"className":601642},[31850],[34,601644],{"className":601645,"style":31846},[31845],[34,601647,601649,601652,601655,601658,601661],{"className":601648},[31676],[34,601650],{"className":601651,"style":34812},[31680],[34,601653,31759],{"className":601654},[31685,31686],[34,601656],{"className":601657,"style":31846},[31845],[34,601659,29622],{"className":601660},[31850],[34,601662],{"className":601663,"style":31846},[31845],[34,601665,601667,601670,601673],{"className":601666},[31676],[34,601668],{"className":601669,"style":31774},[31680],[34,601671,459],{"className":601672},[31685],[34,601674,2557],{"className":601675},[31690],[105,601677,601678,602056,604422],{},[108,601679,601680,305014,601682,601710,601711,602055],{},[2627,601681,419963],{},[34,601683,601685,601698],{"className":601684},[31642],[34,601686,601688],{"className":601687},[31646],[31648,601689,601690],{"xmlns":31650},[31652,601691,601692,601696],{},[31655,601693,601694],{},[31658,601695,54224],{},[31665,601697,54224],{"encoding":31667},[34,601699,601701],{"className":601700,"ariaHidden":758},[31672],[34,601702,601704,601707],{"className":601703},[31676],[34,601705],{"className":601706,"style":33218},[31680],[34,601708,54224],{"className":601709,"style":54252},[31685,31686]," 函数的另一种形式：",[34,601712,601714,601776],{"className":601713},[31642],[34,601715,601717],{"className":601716},[31646],[31648,601718,601719],{"xmlns":31650},[31652,601720,601721,601773],{},[31655,601722,601723,601731,601753,601755,601757,601759],{},[38846,601724,601725,601727,601729],{},[31662,601726,546295],{},[31749,601728,459],{},[31658,601730,138286],{"mathvariant":38079},[41453,601732,601733,601745],{},[37624,601734,601735,601737],{},[31658,601736,67764],{},[31655,601738,601739,601741,601743],{},[31658,601740,111],{},[31662,601742,32226],{},[31749,601744,640],{},[31655,601746,601747,601749,601751],{},[31749,601748,640],{},[31662,601750,223],{},[31658,601752,67764],{},[31658,601754,46900],{},[31658,601756,67764],{},[31662,601758,202],{},[41453,601760,601761,601763],{},[31658,601762,58442],{},[31655,601764,601765,601767,601769,601771],{},[31658,601766,547853],{},[31662,601768,32682],{},[31658,601770,111],{},[31658,601772,58442],{},[31665,601774,601775],{"encoding":31667},"\\int_0^\\infty \\frac{y^{a-1}}{1+y} dy = \\frac{\\pi}{\\sin a\\pi}",[34,601777,601779,601965],{"className":601778,"ariaHidden":758},[31672],[34,601780,601782,601786,601837,601840,601950,601953,601956,601959,601962],{"className":601781},[31676],[34,601783],{"className":601784,"style":601785},[31680],"height:1.5511em;vertical-align:-0.4811em;",[34,601787,601789,601792],{"className":601788},[32767],[34,601790,546295],{"className":601791,"style":546331},[32767,131249,156560],[34,601793,601795],{"className":601794},[31792],[34,601796,601798,601829],{"className":601797},[31796,31797],[34,601799,601801,601826],{"className":601800},[31801],[34,601802,601804,601815],{"className":601803,"style":567928},[31805],[34,601805,601806,601809],{"style":546347},[34,601807],{"className":601808,"style":31814},[31813],[34,601810,601812],{"className":601811},[31818,31819,31820,31821],[34,601813,459],{"className":601814},[31685,31821],[34,601816,601817,601820],{"style":546362},[34,601818],{"className":601819,"style":31814},[31813],[34,601821,601823],{"className":601822},[31818,31819,31820,31821],[34,601824,138286],{"className":601825},[31685,31821],[34,601827,31829],{"className":601828},[31828],[34,601830,601832],{"className":601831},[31801],[34,601833,601835],{"className":601834,"style":546384},[31805],[34,601836],{},[34,601838],{"className":601839,"style":32123},[31845],[34,601841,601843,601846,601947],{"className":601842},[31685],[34,601844],{"className":601845},[31782,41488],[34,601847,601849],{"className":601848},[41453],[34,601850,601852,601939],{"className":601851},[31796,31797],[34,601853,601855,601936],{"className":601854},[31801],[34,601856,601859,601879,601887],{"className":601857,"style":601858},[31805],"height:1.07em;",[34,601860,601861,601864],{"style":41504},[34,601862],{"className":601863,"style":41508},[31813],[34,601865,601867],{"className":601866},[31818,31819,31820,31821],[34,601868,601870,601873,601876],{"className":601869},[31685,31821],[34,601871,640],{"className":601872},[31685,31821],[34,601874,223],{"className":601875},[32024,31821],[34,601877,67764],{"className":601878,"style":32062},[31685,31686,31821],[34,601880,601881,601884],{"style":41520},[34,601882],{"className":601883,"style":41508},[31813],[34,601885],{"className":601886,"style":41528},[41527],[34,601888,601889,601892],{"style":548368},[34,601890],{"className":601891,"style":41508},[31813],[34,601893,601895],{"className":601894},[31818,31819,31820,31821],[34,601896,601898],{"className":601897},[31685,31821],[34,601899,601901,601904],{"className":601900},[31685,31821],[34,601902,67764],{"className":601903,"style":32062},[31685,31686,31821],[34,601905,601907],{"className":601906},[31792],[34,601908,601910],{"className":601909},[31796],[34,601911,601913],{"className":601912},[31801],[34,601914,601916],{"className":601915,"style":144123},[31805],[34,601917,601918,601921],{"style":552391},[34,601919],{"className":601920,"style":43406},[31813],[34,601922,601924],{"className":601923},[31818,43410,43411,31821],[34,601925,601927,601930,601933],{"className":601926},[31685,31821],[34,601928,111],{"className":601929},[31685,31686,31821],[34,601931,32226],{"className":601932},[32024,31821],[34,601934,640],{"className":601935},[31685,31821],[34,601937,31829],{"className":601938},[31828],[34,601940,601942],{"className":601941},[31801],[34,601943,601945],{"className":601944,"style":220191},[31805],[34,601946],{},[34,601948],{"className":601949},[31690,41488],[34,601951,46900],{"className":601952},[31685,31686],[34,601954,67764],{"className":601955,"style":32062},[31685,31686],[34,601957],{"className":601958,"style":31846},[31845],[34,601960,202],{"className":601961},[31850],[34,601963],{"className":601964,"style":31846},[31845],[34,601966,601968,601971],{"className":601967},[31676],[34,601969],{"className":601970,"style":244926},[31680],[34,601972,601974,601977,602052],{"className":601973},[31685],[34,601975],{"className":601976},[31782,41488],[34,601978,601980],{"className":601979},[41453],[34,601981,601983,602044],{"className":601982},[31796,31797],[34,601984,601986,602041],{"className":601985},[31801],[34,601987,601989,602019,602027],{"className":601988,"style":244945},[31805],[34,601990,601991,601994],{"style":41504},[34,601992],{"className":601993,"style":41508},[31813],[34,601995,601997],{"className":601996},[31818,31819,31820,31821],[34,601998,602000,602012,602015],{"className":601999},[31685,31821],[34,602001,602003,602006,602009],{"className":602002},[32767,31821],[34,602004,31747],{"className":602005},[31821],[34,602007,31955],{"className":602008},[31821],[34,602010,31660],{"className":602011},[31821],[34,602013],{"className":602014,"style":43430},[31845,31821],[34,602016,602018],{"className":602017,"style":32062},[31685,31686,31821],"aπ",[34,602020,602021,602024],{"style":41520},[34,602022],{"className":602023,"style":41508},[31813],[34,602025],{"className":602026,"style":41528},[41527],[34,602028,602029,602032],{"style":46771},[34,602030],{"className":602031,"style":41508},[31813],[34,602033,602035],{"className":602034},[31818,31819,31820,31821],[34,602036,602038],{"className":602037},[31685,31821],[34,602039,58442],{"className":602040,"style":32062},[31685,31686,31821],[34,602042,31829],{"className":602043},[31828],[34,602045,602047],{"className":602046},[31801],[34,602048,602050],{"className":602049,"style":41619},[31805],[34,602051],{},[34,602053],{"className":602054},[31690,41488],"（其实就是余元公式的变形）。",[108,602057,602058,33183,602060],{},[2627,602059,289021],{},[2230,602061,602062,602492,602551,603060,603399,603806,604273],{},[108,602063,396859,602064,55016,602145,46528,602238,347],{},[34,602065,602067,602089],{"className":602066},[31642],[34,602068,602070],{"className":602069},[31646],[31648,602071,602072],{"xmlns":31650},[31652,602073,602074,602086],{},[31655,602075,602076,602078,602080],{},[31658,602077,94628],{},[31662,602079,202],{},[37624,602081,602082,602084],{},[31658,602083,32936],{},[31658,602085,31660],{},[31665,602087,602088],{"encoding":31667},"u = x^n",[34,602090,602092,602110],{"className":602091,"ariaHidden":758},[31672],[34,602093,602095,602098,602101,602104,602107],{"className":602094},[31676],[34,602096],{"className":602097,"style":32527},[31680],[34,602099,94628],{"className":602100},[31685,31686],[34,602102],{"className":602103,"style":31846},[31845],[34,602105,202],{"className":602106},[31850],[34,602108],{"className":602109,"style":31846},[31845],[34,602111,602113,602116],{"className":602112},[31676],[34,602114],{"className":602115,"style":37959},[31680],[34,602117,602119,602122],{"className":602118},[31685],[34,602120,32936],{"className":602121},[31685,31686],[34,602123,602125],{"className":602124},[31792],[34,602126,602128],{"className":602127},[31796],[34,602129,602131],{"className":602130},[31801],[34,602132,602134],{"className":602133,"style":37959},[31805],[34,602135,602136,602139],{"style":37777},[34,602137],{"className":602138,"style":31814},[31813],[34,602140,602142],{"className":602141},[31818,31819,31820,31821],[34,602143,31660],{"className":602144},[31685,31686,31821],[34,602146,602148,602176],{"className":602147},[31642],[34,602149,602151],{"className":602150},[31646],[31648,602152,602153],{"xmlns":31650},[31652,602154,602155,602173],{},[31655,602156,602157,602159,602161],{},[31658,602158,32936],{},[31662,602160,202],{},[37624,602162,602163,602165],{},[31658,602164,94628],{},[31655,602166,602167,602169,602171],{},[31749,602168,640],{},[31658,602170,38080],{"mathvariant":38079},[31658,602172,31660],{},[31665,602174,602175],{"encoding":31667},"x = 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= \\frac{1}{n} u^{\\frac{1}{n}-1} 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\\frac{(u^{1\u002Fn})^{m-1}}{1+u} \\cdot \\frac{1}{n} u^{\\frac{1}{n}-1} \\, du",[34,602649,602651,602874],{"className":602650,"ariaHidden":758},[31672],[34,602652,602654,602658,602715,602718,602865,602868,602871],{"className":602653},[31676],[34,602655],{"className":602656,"style":602657},[31680],"height:1.565em;vertical-align:-0.4033em;",[34,602659,602661,602664],{"className":602660},[32767],[34,602662,546295],{"className":602663,"style":546331},[32767,131249,156560],[34,602665,602667],{"className":602666},[31792],[34,602668,602670,602707],{"className":602669},[31796,31797],[34,602671,602673,602704],{"className":602672},[31801],[34,602674,602676,602687],{"className":602675,"style":546550},[31805],[34,602677,602678,602681],{"style":546347},[34,602679],{"className":602680,"style":31814},[31813],[34,602682,602684],{"className":602683},[31818,31819,31820,31821],[34,602685,459],{"className":602686},[31685,31821],[34,602688,602689,602692],{"style":546362},[34,602690],{"className":602691,"style":31814},[31813],[34,602693,602695],{"className":602694},[31818,31819,31820,31821],[34,602696,602698,602701],{"className":602697},[31685,31821],[34,602699,223],{"className":602700},[31685,31821],[34,602702,138286],{"className":602703},[31685,31821],[34,602705,31829],{"className":602706},[31828],[34,602708,602710],{"className":602709},[31801],[34,602711,602713],{"className":602712,"style":546384},[31805],[34,602714],{},[34,602716],{"className":602717,"style":32123},[31845],[34,602719,602721,602724,602862],{"className":602720},[31685],[34,602722],{"className":602723},[31782,41488],[34,602725,602727],{"className":602726},[41453],[34,602728,602730,602854],{"className":602729},[31796,31797],[34,602731,602733,602851],{"className":602732},[31801],[34,602734,602736,602756,602764],{"className":602735,"style":555256},[31805],[34,602737,602738,602741],{"style":41504},[34,602739],{"className":602740,"style":41508},[31813],[34,602742,602744],{"className":602743},[31818,31819,31820,31821],[34,602745,602747,602750,602753],{"className":602746},[31685,31821],[34,602748,640],{"className":602749},[31685,31821],[34,602751,223],{"className":602752},[32024,31821],[34,602754,94628],{"className":602755},[31685,31686,31821],[34,602757,602758,602761],{"style":41520},[34,602759],{"className":602760,"style":41508},[31813],[34,602762],{"className":602763,"style":41528},[41527],[34,602765,602766,602769],{"style":70529},[34,602767],{"className":602768,"style":41508},[31813],[34,602770,602772],{"className":602771},[31818,31819,31820,31821],[34,602773,602775,602778,602813],{"className":602774},[31685,31821],[34,602776,456],{"className":602777},[31782,31821],[34,602779,602781,602784],{"className":602780},[31685,31821],[34,602782,94628],{"className":602783},[31685,31686,31821],[34,602785,602787],{"className":602786},[31792],[34,602788,602790],{"className":602789},[31796],[34,602791,602793],{"className":602792},[31801],[34,602794,602796],{"className":602795,"style":555395},[31805],[34,602797,602798,602801],{"style":555398},[34,602799],{"className":602800,"style":555402},[31813],[34,602802,602804],{"className":602803},[31818,43410,43411,31821],[34,602805,602807,602810],{"className":602806},[31685,31821],[34,602808,249219],{"className":602809},[31685,31821],[34,602811,31660],{"className":602812},[31685,31686,31821],[34,602814,602816,602819],{"className":602815},[31690,31821],[34,602817,2557],{"className":602818},[31690,31821],[34,602820,602822],{"className":602821},[31792],[34,602823,602825],{"className":602824},[31796],[34,602826,602828],{"className":602827},[31801],[34,602829,602831],{"className":602830,"style":144123},[31805],[34,602832,602833,602836],{"style":552391},[34,602834],{"className":602835,"style":43406},[31813],[34,602837,602839],{"className":602838},[31818,43410,43411,31821],[34,602840,602842,602845,602848],{"className":602841},[31685,31821],[34,602843,31759],{"className":602844},[31685,31686,31821],[34,602846,32226],{"className":602847},[32024,31821],[34,602849,640],{"className":602850},[31685,31821],[34,602852,31829],{"className":602853},[31828],[34,602855,602857],{"className":602856},[31801],[34,602858,602860],{"className":602859,"style":400163},[31805],[34,602861],{},[34,602863],{"className":602864},[31690,41488],[34,602866],{"className":602867,"style":32604},[31845],[34,602869,71191],{"className":602870},[32024],[34,602872],{"className":602873,"style":32604},[31845],[34,602875,602877,602880,602948,603051,603054,603057],{"className":602876},[31676],[34,602878],{"className":602879,"style":598357},[31680],[34,602881,602883,602886,602945],{"className":602882},[31685],[34,602884],{"className":602885},[31782,41488],[34,602887,602889],{"className":602888},[41453],[34,602890,602892,602937],{"className":602891},[31796,31797],[34,602893,602895,602934],{"className":602894},[31801],[34,602896,602898,602912,602920],{"className":602897,"style":46746},[31805],[34,602899,602900,602903],{"style":41504},[34,602901],{"className":602902,"style":41508},[31813],[34,602904,602906],{"className":602905},[31818,31819,31820,31821],[34,602907,602909],{"className":602908},[31685,31821],[34,602910,31660],{"className":602911},[31685,31686,31821],[34,602913,602914,602917],{"style":41520},[34,602915],{"className":602916,"style":41508},[31813],[34,602918],{"className":602919,"style":41528},[41527],[34,602921,602922,602925],{"style":46771},[34,602923],{"className":602924,"style":41508},[31813],[34,602926,602928],{"className":602927},[31818,31819,31820,31821],[34,602929,602931],{"className":602930},[31685,31821],[34,602932,640],{"className":602933},[31685,31821],[34,602935,31829],{"className":602936},[31828],[34,602938,602940],{"className":602939},[31801],[34,602941,602943],{"className":602942,"style":41619},[31805],[34,602944],{},[34,602946],{"className":602947},[31690,41488],[34,602949,602951,602954],{"className":602950},[31685],[34,602952,94628],{"className":602953},[31685,31686],[34,602955,602957],{"className":602956},[31792],[34,602958,602960],{"className":602959},[31796],[34,602961,602963],{"className":602962},[31801],[34,602964,602966],{"className":602965,"style":598444},[31805],[34,602967,602968,602971],{"style":435688},[34,602969],{"className":602970,"style":41508},[31813],[34,602972,602974],{"className":602973},[31818,31819,31820,31821],[34,602975,602977,603045,603048],{"className":602976},[31685,31821],[34,602978,602980,602983,603042],{"className":602979},[31685,31821],[34,602981],{"className":602982},[31782,41488,31818,43410,435704],[34,602984,602986],{"className":602985},[41453],[34,602987,602989,603034],{"className":602988},[31796,31797],[34,602990,602992,603031],{"className":602991},[31801],[34,602993,602995,603009,603017],{"className":602994,"style":593176},[31805],[34,602996,602997,603000],{"style":435720},[34,602998],{"className":602999,"style":41508},[31813],[34,603001,603003],{"className":603002},[31818,43410,43411,31821],[34,603004,603006],{"className":603005},[31685,31821],[34,603007,31660],{"className":603008},[31685,31686,31821],[34,603010,603011,603014],{"style":435735},[34,603012],{"className":603013,"style":41508},[31813],[34,603015],{"className":603016,"style":435742},[41527,31821],[34,603018,603019,603022],{"style":589620},[34,603020],{"className":603021,"style":41508},[31813],[34,603023,603025],{"className":603024},[31818,43410,43411,31821],[34,603026,603028],{"className":603027},[31685,31821],[34,603029,640],{"className":603030},[31685,31821],[34,603032,31829],{"className":603033},[31828],[34,603035,603037],{"className":603036},[31801],[34,603038,603040],{"className":603039,"style":435782},[31805],[34,603041],{},[34,603043],{"className":603044},[31690,41488,31818,43410,435704],[34,603046,32226],{"className":603047},[32024,31821],[34,603049,640],{"className":603050},[31685,31821],[34,603052],{"className":603053,"style":32123},[31845],[34,603055,46900],{"className":603056},[31685,31686],[34,603058,94628],{"className":603059},[31685,31686],[108,603061,603062,603063,347],{},"整理指数：",[34,603064,603066,603114],{"className":603065},[31642],[34,603067,603069],{"className":603068},[31646],[31648,603070,603071],{"xmlns":31650},[31652,603072,603073,603111],{},[31655,603074,603075,603087,603089,603095,603097,603099,603101,603107,603109],{},[41453,603076,603077,603085],{},[31655,603078,603079,603081,603083],{},[31658,603080,31759],{},[31662,603082,32226],{},[31749,603084,640],{},[31658,603086,31660],{},[31662,603088,223],{},[41453,603090,603091,603093],{},[31749,603092,640],{},[31658,603094,31660],{},[31662,603096,32226],{},[31749,603098,640],{},[31662,603100,202],{},[41453,603102,603103,603105],{},[31658,603104,31759],{},[31658,603106,31660],{},[31662,603108,32226],{},[31749,603110,640],{},[31665,603112,603113],{"encoding":31667},"\\frac{m-1}{n} + \\frac{1}{n} - 1 = \\frac{m}{n} - 1",[34,603115,603117,603206,603289,603307,603390],{"className":603116,"ariaHidden":758},[31672],[34,603118,603120,603123,603197,603200,603203],{"className":603119},[31676],[34,603121],{"className":603122,"style":46727},[31680],[34,603124,603126,603129,603194],{"className":603125},[31685],[34,603127],{"className":603128},[31782,41488],[34,603130,603132],{"className":603131},[41453],[34,603133,603135,603186],{"className":603134},[31796,31797],[34,603136,603138,603183],{"className":603137},[31801],[34,603139,603141,603155,603163],{"className":603140,"style":46746},[31805],[34,603142,603143,603146],{"style":41504},[34,603144],{"className":603145,"style":41508},[31813],[34,603147,603149],{"className":603148},[31818,31819,31820,31821],[34,603150,603152],{"className":603151},[31685,31821],[34,603153,31660],{"className":603154},[31685,31686,31821],[34,603156,603157,603160],{"style":41520},[34,603158],{"className":603159,"style":41508},[31813],[34,603161],{"className":603162,"style":41528},[41527],[34,603164,603165,603168],{"style":46771},[34,603166],{"className":603167,"style":41508},[31813],[34,603169,603171],{"className":603170},[31818,31819,31820,31821],[34,603172,603174,603177,603180],{"className":603173},[31685,31821],[34,603175,31759],{"className":603176},[31685,31686,31821],[34,603178,32226],{"className":603179},[32024,31821],[34,603181,640],{"className":603182},[31685,31821],[34,603184,31829],{"className":603185},[31828],[34,603187,603189],{"className":603188},[31801],[34,603190,603192],{"className":603191,"style":41619},[31805],[34,603193],{},[34,603195],{"className":603196},[31690,41488],[34,603198],{"className":603199,"style":32604},[31845],[34,603201,223],{"className":603202},[32024],[34,603204],{"className":603205,"style":32604},[31845],[34,603207,603209,603212,603280,603283,603286],{"className":603208},[31676],[34,603210],{"className":603211,"style":46727},[31680],[34,603213,603215,603218,603277],{"className":603214},[31685],[34,603216],{"className":603217},[31782,41488],[34,603219,603221],{"className":603220},[41453],[34,603222,603224,603269],{"className":603223},[31796,31797],[34,603225,603227,603266],{"className":603226},[31801],[34,603228,603230,603244,603252],{"className":603229,"style":46746},[31805],[34,603231,603232,603235],{"style":41504},[34,603233],{"className":603234,"style":41508},[31813],[34,603236,603238],{"className":603237},[31818,31819,31820,31821],[34,603239,603241],{"className":603240},[31685,31821],[34,603242,31660],{"className":603243},[31685,31686,31821],[34,603245,603246,603249],{"style":41520},[34,603247],{"className":603248,"style":41508},[31813],[34,603250],{"className":603251,"style":41528},[41527],[34,603253,603254,603257],{"style":46771},[34,603255],{"className":603256,"style":41508},[31813],[34,603258,603260],{"className":603259},[31818,31819,31820,31821],[34,603261,603263],{"className":603262},[31685,31821],[34,603264,640],{"className":603265},[31685,31821],[34,603267,31829],{"className":603268},[31828],[34,603270,603272],{"className":603271},[31801],[34,603273,603275],{"className":603274,"style":41619},[31805],[34,603276],{},[34,603278],{"className":603279},[31690,41488],[34,603281],{"className":603282,"style":32604},[31845],[34,603284,32226],{"className":603285},[32024],[34,603287],{"className":603288,"style":32604},[31845],[34,603290,603292,603295,603298,603301,603304],{"className":603291},[31676],[34,603293],{"className":603294,"style":32617},[31680],[34,603296,640],{"className":603297},[31685],[34,603299],{"className":603300,"style":31846},[31845],[34,603302,202],{"className":603303},[31850],[34,603305],{"className":603306,"style":31846},[31845],[34,603308,603310,603313,603381,603384,603387],{"className":603309},[31676],[34,603311],{"className":603312,"style":244926},[31680],[34,603314,603316,603319,603378],{"className":603315},[31685],[34,603317],{"className":603318},[31782,41488],[34,603320,603322],{"className":603321},[41453],[34,603323,603325,603370],{"className":603324},[31796,31797],[34,603326,603328,603367],{"className":603327},[31801],[34,603329,603331,603345,603353],{"className":603330,"style":244945},[31805],[34,603332,603333,603336],{"style":41504},[34,603334],{"className":603335,"style":41508},[31813],[34,603337,603339],{"className":603338},[31818,31819,31820,31821],[34,603340,603342],{"className":603341},[31685,31821],[34,603343,31660],{"className":603344},[31685,31686,31821],[34,603346,603347,603350],{"style":41520},[34,603348],{"className":603349,"style":41508},[31813],[34,603351],{"className":603352,"style":41528},[41527],[34,603354,603355,603358],{"style":46771},[34,603356],{"className":603357,"style":41508},[31813],[34,603359,603361],{"className":603360},[31818,31819,31820,31821],[34,603362,603364],{"className":603363},[31685,31821],[34,603365,31759],{"className":603366},[31685,31686,31821],[34,603368,31829],{"className":603369},[31828],[34,603371,603373],{"className":603372},[31801],[34,603374,603376],{"className":603375,"style":41619},[31805],[34,603377],{},[34,603379],{"className":603380},[31690,41488],[34,603382],{"className":603383,"style":32604},[31845],[34,603385,32226],{"className":603386},[32024],[34,603388],{"className":603389,"style":32604},[31845],[34,603391,603393,603396],{"className":603392},[31676],[34,603394],{"className":603395,"style":32617},[31680],[34,603397,640],{"className":603398},[31685],[108,603400,592630,603401,347],{},[34,603402,603404,603468],{"className":603403},[31642],[34,603405,603407],{"className":603406},[31646],[31648,603408,603409],{"xmlns":31650},[31652,603410,603411,603465],{},[31655,603412,603413,603415,603421,603433,603459,603461,603463],{},[31662,603414,202],{},[41453,603416,603417,603419],{},[31749,603418,640],{},[31658,603420,31660],{},[38846,603422,603423,603425,603427],{},[31662,603424,546295],{},[31749,603426,459],{},[31655,603428,603429,603431],{},[31662,603430,223],{},[31658,603432,138286],{"mathvariant":38079},[41453,603434,603435,603451],{},[37624,603436,603437,603439],{},[31658,603438,94628],{},[31655,603440,603441,603447,603449],{},[41453,603442,603443,603445],{},[31658,603444,31759],{},[31658,603446,31660],{},[31662,603448,32226],{},[31749,603450,640],{},[31655,603452,603453,603455,603457],{},[31749,603454,640],{},[31662,603456,223],{},[31658,603458,94628],{},[39361,603460,39363],{},[31658,603462,46900],{},[31658,603464,94628],{},[31665,603466,603467],{"encoding":31667},"= \\frac{1}{n} \\int_0^{+\\infty} \\frac{u^{\\frac{m}{n}-1}}{1+u} \\, du",[34,603469,603471,603483],{"className":603470,"ariaHidden":758},[31672],[34,603472,603474,603477,603480],{"className":603473},[31676],[34,603475],{"className":603476,"style":73089},[31680],[34,603478,202],{"className":603479},[31850],[34,603481],{"className":603482,"style":31846},[31845],[34,603484,603486,603490,603558,603561,603618,603621,603797,603800,603803],{"className":603485},[31676],[34,603487],{"className":603488,"style":603489},[31680],"height:1.5475em;vertical-align:-0.4033em;",[34,603491,603493,603496,603555],{"className":603492},[31685],[34,603494],{"className":603495},[31782,41488],[34,603497,603499],{"className":603498},[41453],[34,603500,603502,603547],{"className":603501},[31796,31797],[34,603503,603505,603544],{"className":603504},[31801],[34,603506,603508,603522,603530],{"className":603507,"style":46746},[31805],[34,603509,603510,603513],{"style":41504},[34,603511],{"className":603512,"style":41508},[31813],[34,603514,603516],{"className":603515},[31818,31819,31820,31821],[34,603517,603519],{"className":603518},[31685,31821],[34,603520,31660],{"className":603521},[31685,31686,31821],[34,603523,603524,603527],{"style":41520},[34,603525],{"className":603526,"style":41508},[31813],[34,603528],{"className":603529,"style":41528},[41527],[34,603531,603532,603535],{"style":46771},[34,603533],{"className":603534,"style":41508},[31813],[34,603536,603538],{"className":603537},[31818,31819,31820,31821],[34,603539,603541],{"className":603540},[31685,31821],[34,603542,640],{"className":603543},[31685,31821],[34,603545,31829],{"className":603546},[31828],[34,603548,603550],{"className":603549},[31801],[34,603551,603553],{"className":603552,"style":41619},[31805],[34,603554],{},[34,603556],{"className":603557},[31690,41488],[34,603559],{"className":603560,"style":32123},[31845],[34,603562,603564,603567],{"className":603563},[32767],[34,603565,546295],{"className":603566,"style":546331},[32767,131249,156560],[34,603568,603570],{"className":603569},[31792],[34,603571,603573,603610],{"className":603572},[31796,31797],[34,603574,603576,603607],{"className":603575},[31801],[34,603577,603579,603590],{"className":603578,"style":546550},[31805],[34,603580,603581,603584],{"style":546347},[34,603582],{"className":603583,"style":31814},[31813],[34,603585,603587],{"className":603586},[31818,31819,31820,31821],[34,603588,459],{"className":603589},[31685,31821],[34,603591,603592,603595],{"style":546362},[34,603593],{"className":603594,"style":31814},[31813],[34,603596,603598],{"className":603597},[31818,31819,31820,31821],[34,603599,603601,603604],{"className":603600},[31685,31821],[34,603602,223],{"className":603603},[31685,31821],[34,603605,138286],{"className":603606},[31685,31821],[34,603608,31829],{"className":603609},[31828],[34,603611,603613],{"className":603612},[31801],[34,603614,603616],{"className":603615,"style":546384},[31805],[34,603617],{},[34,603619],{"className":603620,"style":32123},[31845],[34,603622,603624,603627,603794],{"className":603623},[31685],[34,603625],{"className":603626},[31782,41488],[34,603628,603630],{"className":603629},[41453],[34,603631,603633,603786],{"className":603632},[31796,31797],[34,603634,603636,603783],{"className":603635},[31801],[34,603637,603640,603660,603668],{"className":603638,"style":603639},[31805],"height:1.1442em;",[34,603641,603642,603645],{"style":41504},[34,603643],{"className":603644,"style":41508},[31813],[34,603646,603648],{"className":603647},[31818,31819,31820,31821],[34,603649,603651,603654,603657],{"className":603650},[31685,31821],[34,603652,640],{"className":603653},[31685,31821],[34,603655,223],{"className":603656},[32024,31821],[34,603658,94628],{"className":603659},[31685,31686,31821],[34,603661,603662,603665],{"style":41520},[34,603663],{"className":603664,"style":41508},[31813],[34,603666],{"className":603667,"style":41528},[41527],[34,603669,603670,603673],{"style":46771},[34,603671],{"className":603672,"style":41508},[31813],[34,603674,603676],{"className":603675},[31818,31819,31820,31821],[34,603677,603679],{"className":603678},[31685,31821],[34,603680,603682,603685],{"className":603681},[31685,31821],[34,603683,94628],{"className":603684},[31685,31686,31821],[34,603686,603688],{"className":603687},[31792],[34,603689,603691],{"className":603690},[31796],[34,603692,603694],{"className":603693},[31801],[34,603695,603698],{"className":603696,"style":603697},[31805],"height:1.0717em;",[34,603699,603701,603704],{"style":603700},"top:-3.4878em;margin-right:0.0714em;",[34,603702],{"className":603703,"style":41508},[31813],[34,603705,603707],{"className":603706},[31818,43410,43411,31821],[34,603708,603710,603777,603780],{"className":603709},[31685,31821],[34,603711,603713,603717,603774],{"className":603712},[31685,31821],[34,603714],{"className":603715},[31782,41488,31818,603716,435704],"reset-size1",[34,603718,603720],{"className":603719},[41453],[34,603721,603723,603765],{"className":603722},[31796,31797],[34,603724,603726,603762],{"className":603725},[31801],[34,603727,603730,603742,603750],{"className":603728,"style":603729},[31805],"height:0.8176em;",[34,603731,603733,603736],{"style":603732},"top:-2.468em;",[34,603734],{"className":603735,"style":41508},[31813],[34,603737,603739],{"className":603738},[31685,31821],[34,603740,31660],{"className":603741},[31685,31686,31821],[34,603743,603744,603747],{"style":435735},[34,603745],{"className":603746,"style":41508},[31813],[34,603748],{"className":603749,"style":435742},[41527,31821],[34,603751,603753,603756],{"style":603752},"top:-3.387em;",[34,603754],{"className":603755,"style":41508},[31813],[34,603757,603759],{"className":603758},[31685,31821],[34,603760,31759],{"className":603761},[31685,31686,31821],[34,603763,31829],{"className":603764},[31828],[34,603766,603768],{"className":603767},[31801],[34,603769,603772],{"className":603770,"style":603771},[31805],"height:0.532em;",[34,603773],{},[34,603775],{"className":603776},[31690,41488,31818,603716,435704],[34,603778,32226],{"className":603779},[32024,31821],[34,603781,640],{"className":603782},[31685,31821],[34,603784,31829],{"className":603785},[31828],[34,603787,603789],{"className":603788},[31801],[34,603790,603792],{"className":603791,"style":400163},[31805],[34,603793],{},[34,603795],{"className":603796},[31690,41488],[34,603798],{"className":603799,"style":32123},[31845],[34,603801,46900],{"className":603802},[31685,31686],[34,603804,94628],{"className":603805},[31685,31686],[108,603807,603808,603809,604152,604153,347],{},"利用公式 ",[34,603810,603812,603874],{"className":603811},[31642],[34,603813,603815],{"className":603814},[31646],[31648,603816,603817],{"xmlns":31650},[31652,603818,603819,603871],{},[31655,603820,603821,603829,603851,603853,603855,603857],{},[38846,603822,603823,603825,603827],{},[31662,603824,546295],{},[31749,603826,459],{},[31658,603828,138286],{"mathvariant":38079},[41453,603830,603831,603843],{},[37624,603832,603833,603835],{},[31658,603834,94628],{},[31655,603836,603837,603839,603841],{},[31658,603838,111],{},[31662,603840,32226],{},[31749,603842,640],{},[31655,603844,603845,603847,603849],{},[31749,603846,640],{},[31662,603848,223],{},[31658,603850,94628],{},[31658,603852,46900],{},[31658,603854,94628],{},[31662,603856,202],{},[41453,603858,603859,603861],{},[31658,603860,58442],{},[31655,603862,603863,603865,603867,603869],{},[31658,603864,547853],{},[31662,603866,32682],{},[31658,603868,111],{},[31658,603870,58442],{},[31665,603872,603873],{"encoding":31667},"\\int_0^\\infty \\frac{u^{a-1}}{1+u} du = \\frac{\\pi}{\\sin a\\pi}",[34,603875,603877,604063],{"className":603876,"ariaHidden":758},[31672],[34,603878,603880,603884,603935,603938,604048,604051,604054,604057,604060],{"className":603879},[31676],[34,603881],{"className":603882,"style":603883},[31680],"height:1.4213em;vertical-align:-0.4033em;",[34,603885,603887,603890],{"className":603886},[32767],[34,603888,546295],{"className":603889,"style":546331},[32767,131249,156560],[34,603891,603893],{"className":603892},[31792],[34,603894,603896,603927],{"className":603895},[31796,31797],[34,603897,603899,603924],{"className":603898},[31801],[34,603900,603902,603913],{"className":603901,"style":567928},[31805],[34,603903,603904,603907],{"style":546347},[34,603905],{"className":603906,"style":31814},[31813],[34,603908,603910],{"className":603909},[31818,31819,31820,31821],[34,603911,459],{"className":603912},[31685,31821],[34,603914,603915,603918],{"style":546362},[34,603916],{"className":603917,"style":31814},[31813],[34,603919,603921],{"className":603920},[31818,31819,31820,31821],[34,603922,138286],{"className":603923},[31685,31821],[34,603925,31829],{"className":603926},[31828],[34,603928,603930],{"className":603929},[31801],[34,603931,603933],{"className":603932,"style":546384},[31805],[34,603934],{},[34,603936],{"className":603937,"style":32123},[31845],[34,603939,603941,603944,604045],{"className":603940},[31685],[34,603942],{"className":603943},[31782,41488],[34,603945,603947],{"className":603946},[41453],[34,603948,603950,604037],{"className":603949},[31796,31797],[34,603951,603953,604034],{"className":603952},[31801],[34,603954,603957,603977,603985],{"className":603955,"style":603956},[31805],"height:1.0179em;",[34,603958,603959,603962],{"style":41504},[34,603960],{"className":603961,"style":41508},[31813],[34,603963,603965],{"className":603964},[31818,31819,31820,31821],[34,603966,603968,603971,603974],{"className":603967},[31685,31821],[34,603969,640],{"className":603970},[31685,31821],[34,603972,223],{"className":603973},[32024,31821],[34,603975,94628],{"className":603976},[31685,31686,31821],[34,603978,603979,603982],{"style":41520},[34,603980],{"className":603981,"style":41508},[31813],[34,603983],{"className":603984,"style":41528},[41527],[34,603986,603987,603990],{"style":46771},[34,603988],{"className":603989,"style":41508},[31813],[34,603991,603993],{"className":603992},[31818,31819,31820,31821],[34,603994,603996],{"className":603995},[31685,31821],[34,603997,603999,604002],{"className":603998},[31685,31821],[34,604000,94628],{"className":604001},[31685,31686,31821],[34,604003,604005],{"className":604004},[31792],[34,604006,604008],{"className":604007},[31796],[34,604009,604011],{"className":604010},[31801],[34,604012,604014],{"className":604013,"style":144123},[31805],[34,604015,604016,604019],{"style":552391},[34,604017],{"className":604018,"style":43406},[31813],[34,604020,604022],{"className":604021},[31818,43410,43411,31821],[34,604023,604025,604028,604031],{"className":604024},[31685,31821],[34,604026,111],{"className":604027},[31685,31686,31821],[34,604029,32226],{"className":604030},[32024,31821],[34,604032,640],{"className":604033},[31685,31821],[34,604035,31829],{"className":604036},[31828],[34,604038,604040],{"className":604039},[31801],[34,604041,604043],{"className":604042,"style":400163},[31805],[34,604044],{},[34,604046],{"className":604047},[31690,41488],[34,604049,46900],{"className":604050},[31685,31686],[34,604052,94628],{"className":604053},[31685,31686],[34,604055],{"className":604056,"style":31846},[31845],[34,604058,202],{"className":604059},[31850],[34,604061],{"className":604062,"style":31846},[31845],[34,604064,604066,604069],{"className":604065},[31676],[34,604067],{"className":604068,"style":244926},[31680],[34,604070,604072,604075,604149],{"className":604071},[31685],[34,604073],{"className":604074},[31782,41488],[34,604076,604078],{"className":604077},[41453],[34,604079,604081,604141],{"className":604080},[31796,31797],[34,604082,604084,604138],{"className":604083},[31801],[34,604085,604087,604116,604124],{"className":604086,"style":244945},[31805],[34,604088,604089,604092],{"style":41504},[34,604090],{"className":604091,"style":41508},[31813],[34,604093,604095],{"className":604094},[31818,31819,31820,31821],[34,604096,604098,604110,604113],{"className":604097},[31685,31821],[34,604099,604101,604104,604107],{"className":604100},[32767,31821],[34,604102,31747],{"className":604103},[31821],[34,604105,31955],{"className":604106},[31821],[34,604108,31660],{"className":604109},[31821],[34,604111],{"className":604112,"style":43430},[31845,31821],[34,604114,602018],{"className":604115,"style":32062},[31685,31686,31821],[34,604117,604118,604121],{"style":41520},[34,604119],{"className":604120,"style":41508},[31813],[34,604122],{"className":604123,"style":41528},[41527],[34,604125,604126,604129],{"style":46771},[34,604127],{"className":604128,"style":41508},[31813],[34,604130,604132],{"className":604131},[31818,31819,31820,31821],[34,604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",[34,604154,604156,604178],{"className":604155},[31642],[34,604157,604159],{"className":604158},[31646],[31648,604160,604161],{"xmlns":31650},[31652,604162,604163,604175],{},[31655,604164,604165,604167,604169],{},[31658,604166,111],{},[31662,604168,202],{},[41453,604170,604171,604173],{},[31658,604172,31759],{},[31658,604174,31660],{},[31665,604176,604177],{"encoding":31667},"a = \\frac{m}{n}",[34,604179,604181,604199],{"className":604180,"ariaHidden":758},[31672],[34,604182,604184,604187,604190,604193,604196],{"className":604183},[31676],[34,604185],{"className":604186,"style":32527},[31680],[34,604188,111],{"className":604189},[31685,31686],[34,604191],{"className":604192,"style":31846},[31845],[34,604194,202],{"className":604195},[31850],[34,604197],{"className":604198,"style":31846},[31845],[34,604200,604202,604205],{"className":604201},[31676],[34,604203],{"className":604204,"style":244926},[31680],[34,604206,604208,604211,604270],{"className":604207},[31685],[34,604209],{"className":604210},[31782,41488],[34,604212,604214],{"className":604213},[41453],[34,604215,604217,604262],{"className":604216},[31796,31797],[34,604218,604220,604259],{"className":604219},[31801],[34,604221,604223,604237,604245],{"className":604222,"style":244945},[31805],[34,604224,604225,604228],{"style":41504},[34,604226],{"className":604227,"style":41508},[31813],[34,604229,604231],{"className":604230},[31818,31819,31820,31821],[34,604232,604234],{"className":604233},[31685,31821],[34,604235,31660],{"className":604236},[31685,31686,31821],[34,604238,604239,604242],{"style":41520},[34,604240],{"className":604241,"style":41508},[31813],[34,604243],{"className":604244,"style":41528},[41527],[34,604246,604247,604250],{"style":46771},[34,604248],{"className":604249,"style":41508},[31813],[34,604251,604253],{"className":604252},[31818,31819,31820,31821],[34,604254,604256],{"className":604255},[31685,31821],[34,604257,31759],{"className":604258},[31685,31686,31821],[34,604260,31829],{"className":604261},[31828],[34,604263,604265],{"className":604264},[31801],[34,604266,604268],{"className":604267,"style":41619},[31805],[34,604269],{},[34,604271],{"className":604272},[31690,41488],[108,604274,93310,604275,13977,604348,604421],{},[34,604276,604278,604300],{"className":604277},[31642],[34,604279,604281],{"className":604280},[31646],[31648,604282,604283],{"xmlns":31650},[31652,604284,604285,604297],{},[31655,604286,604287,604289,604291,604293,604295],{},[31749,604288,459],{},[31662,604290,932],{},[31658,604292,31759],{},[31662,604294,932],{},[31658,604296,31660],{},[31665,604298,604299],{"encoding":31667},"0 \u003C m \u003C n",[34,604301,604303,604321,604339],{"className":604302,"ariaHidden":758},[31672],[34,604304,604306,604309,604312,604315,604318],{"className":604305},[31676],[34,604307],{"className":604308,"style":61814},[31680],[34,604310,459],{"className":604311},[31685],[34,604313],{"className":604314,"style":31846},[31845],[34,604316,932],{"className":604317},[31850],[34,604319],{"className":604320,"style":31846},[31845],[34,604322,604324,604327,604330,604333,604336],{"className":604323},[31676],[34,604325],{"className":604326,"style":34812},[31680],[34,604328,31759],{"className":604329},[31685,31686],[34,604331],{"className":604332,"style":31846},[31845],[34,604334,932],{"className":604335},[31850],[34,604337],{"className":604338,"style":31846},[31845],[34,604340,604342,604345],{"className":604341},[31676],[34,604343],{"className":604344,"style":32527},[31680],[34,604346,31660],{"className":604347},[31685,31686],[34,604349,604351,604373],{"className":604350},[31642],[34,604352,604354],{"className":604353},[31646],[31648,604355,604356],{"xmlns":31650},[31652,604357,604358,604370],{},[31655,604359,604360,604362,604364,604366,604368],{},[31749,604361,459],{},[31662,604363,932],{},[31658,604365,111],{},[31662,604367,932],{},[31749,604369,640],{},[31665,604371,604372],{"encoding":31667},"0 \u003C a \u003C 1",[34,604374,604376,604394,604412],{"className":604375,"ariaHidden":758},[31672],[34,604377,604379,604382,604385,604388,604391],{"className":604378},[31676],[34,604380],{"className":604381,"style":61814},[31680],[34,604383,459],{"className":604384},[31685],[34,604386],{"className":604387,"style":31846},[31845],[34,604389,932],{"className":604390},[31850],[34,604392],{"className":604393,"style":31846},[31845],[34,604395,604397,604400,604403,604406,604409],{"className":604396},[31676],[34,604398],{"className":604399,"style":34812},[31680],[34,604401,111],{"className":604402},[31685,31686],[34,604404],{"className":604405,"style":31846},[31845],[34,604407,932],{"className":604408},[31850],[34,604410],{"className":604411,"style":31846},[31845],[34,604413,604415,604418],{"className":604414},[31676],[34,604416],{"className":604417,"style":32617},[31680],[34,604419,640],{"className":604420},[31685],"，公式成立。",[108,604423,604424,2664,604426],{},[2627,604425,118575],{},[34,604427,604429,604463],{"className":604428},[31642],[34,604430,604432],{"className":604431},[31646],[31648,604433,604434],{"xmlns":31650},[31652,604435,604436,604460],{},[31655,604437,604438],{},[41453,604439,604440,604442],{},[31658,604441,58442],{},[31655,604443,604444,604446,604448,604450],{},[31658,604445,31660],{},[31658,604447,547853],{},[31662,604449,32682],{},[41453,604451,604452,604458],{},[31655,604453,604454,604456],{},[31658,604455,31759],{},[31658,604457,58442],{},[31658,604459,31660],{},[31665,604461,604462],{"encoding":31667},"\\frac{\\pi}{n \\sin \\frac{m\\pi}{n}}",[34,604464,604466],{"className":604465,"ariaHidden":758},[31672],[34,604467,604469,604472],{"className":604468},[31676],[34,604470],{"className":604471,"style":600976},[31680],[34,604473,604475,604478,604624],{"className":604474},[31685],[34,604476],{"className":604477},[31782,41488],[34,604479,604481],{"className":604480},[41453],[34,604482,604484,604616],{"className":604483},[31796,31797],[34,604485,604487,604613],{"className":604486},[31801],[34,604488,604490,604591,604599],{"className":604489,"style":244945},[31805],[34,604491,604492,604495],{"style":41504},[34,604493],{"className":604494,"style":41508},[31813],[34,604496,604498],{"className":604497},[31818,31819,31820,31821],[34,604499,604501,604504,604507,604519,604522],{"className":604500},[31685,31821],[34,604502,31660],{"className":604503},[31685,31686,31821],[34,604505],{"className":604506,"style":43430},[31845,31821],[34,604508,604510,604513,604516],{"className":604509},[32767,31821],[34,604511,31747],{"className":604512},[31821],[34,604514,31955],{"className":604515},[31821],[34,604517,31660],{"className":604518},[31821],[34,604520],{"className":604521,"style":43430},[31845,31821],[34,604523,604525,604528,604588],{"className":604524},[31685,31821],[34,604526],{"className":604527},[31782,41488,31818,43410,435704],[34,604529,604531],{"className":604530},[41453],[34,604532,604534,604580],{"className":604533},[31796,31797],[34,604535,604537,604577],{"className":604536},[31801],[34,604538,604540,604554,604562],{"className":604539,"style":589595},[31805],[34,604541,604542,604545],{"style":435720},[34,604543],{"className":604544,"style":41508},[31813],[34,604546,604548],{"className":604547},[31818,43410,43411,31821],[34,604549,604551],{"className":604550},[31685,31821],[34,604552,31660],{"className":604553},[31685,31686,31821],[34,604555,604556,604559],{"style":435735},[34,604557],{"className":604558,"style":41508},[31813],[34,604560],{"className":604561,"style":435742},[41527,31821],[34,604563,604564,604567],{"style":589620},[34,604565],{"className":604566,"style":41508},[31813],[34,604568,604570],{"className":604569},[31818,43410,43411,31821],[34,604571,604573],{"className":604572},[31685,31821],[34,604574,604576],{"className":604575,"style":32062},[31685,31686,31821],"mπ",[34,604578,31829],{"className":604579},[31828],[34,604581,604583],{"className":604582},[31801],[34,604584,604586],{"className":604585,"style":435782},[31805],[34,604587],{},[34,604589],{"className":604590},[31690,41488,31818,43410,435704],[34,604592,604593,604596],{"style":41520},[34,604594],{"className":604595,"style":41508},[31813],[34,604597],{"className":604598,"style":41528},[41527],[34,604600,604601,604604],{"style":46771},[34,604602],{"className":604603,"style":41508},[31813],[34,604605,604607],{"className":604606},[31818,31819,31820,31821],[34,604608,604610],{"className":604609},[31685,31821],[34,604611,58442],{"className":604612,"style":32062},[31685,31686,31821],[34,604614,31829],{"className":604615},[31828],[34,604617,604619],{"className":604618},[31801],[34,604620,604622],{"className":604621,"style":601123},[31805],[34,604623],{},[34,604625],{"className":604626},[31690,41488],[15,604628,604629,47649,604632],{},[2627,604630,604631],{},"(5)",[34,604633,604635,604691],{"className":604634},[31642],[34,604636,604638],{"className":604637},[31646],[31648,604639,604640],{"xmlns":31650},[31652,604641,604642,604688],{},[31655,604643,604644],{},[71090,604645,604646,604658,604682,604684,604686],{"scriptlevel":459,"displaystyle":758},[38846,604647,604648,604650,604652],{},[31662,604649,546295],{},[31749,604651,459],{},[31655,604653,604654,604656],{},[31662,604655,223],{},[31658,604657,138286],{"mathvariant":38079},[41453,604659,604660,604666],{},[597608,604661,604662,604664],{},[31658,604663,32936],{},[31749,604665,15932],{},[31655,604667,604668,604670,604672,604674,604676],{},[31662,604669,456],{"stretchy":702},[31749,604671,640],{},[31662,604673,223],{},[31658,604675,32936],{},[37624,604677,604678,604680],{},[31662,604679,2557],{"stretchy":702},[31749,604681,768],{},[39361,604683,39363],{},[31658,604685,46900],{},[31658,604687,32936],{},[31665,604689,604690],{"encoding":31667},"\\displaystyle \\int_0^{+\\infty} \\frac{\\sqrt[4]{x}}{(1 + x)^2} \\, dx",[34,604692,604694],{"className":604693,"ariaHidden":758},[31672],[34,604695,604697,604701,604758,604761,604941,604944,604947],{"className":604696},[31676],[34,604698],{"className":604699,"style":604700},[31680],"height:2.4572em;vertical-align:-0.936em;",[34,604702,604704,604707],{"className":604703},[32767],[34,604705,546295],{"className":604706,"style":591475},[32767,131249,131250],[34,604708,604710],{"className":604709},[31792],[34,604711,604713,604750],{"className":604712},[31796,31797],[34,604714,604716,604747],{"className":604715},[31801],[34,604717,604719,604730],{"className":604718,"style":601439},[31805],[34,604720,604721,604724],{"style":591491},[34,604722],{"className":604723,"style":31814},[31813],[34,604725,604727],{"className":604726},[31818,31819,31820,31821],[34,604728,459],{"className":604729},[31685,31821],[34,604731,604732,604735],{"style":591503},[34,604733],{"className":604734,"style":31814},[31813],[34,604736,604738],{"className":604737},[31818,31819,31820,31821],[34,604739,604741,604744],{"className":604740},[31685,31821],[34,604742,223],{"className":604743},[31685,31821],[34,604745,138286],{"className":604746},[31685,31821],[34,604748,31829],{"className":604749},[31828],[34,604751,604753],{"className":604752},[31801],[34,604754,604756],{"className":604755,"style":591522},[31805],[34,604757],{},[34,604759],{"className":604760,"style":32123},[31845],[34,604762,604764,604767,604938],{"className":604763},[31685],[34,604765],{"className":604766},[31782,41488],[34,604768,604770],{"className":604769},[41453],[34,604771,604773,604929],{"className":604772},[31796,31797],[34,604774,604776,604926],{"className":604775},[31801],[34,604777,604780,604835,604843],{"className":604778,"style":604779},[31805],"height:1.4773em;",[34,604781,604782,604785],{"style":129639},[34,604783],{"className":604784,"style":41508},[31813],[34,604786,604788,604791,604794,604797,604800,604803,604806],{"className":604787},[31685],[34,604789,456],{"className":604790},[31782],[34,604792,640],{"className":604793},[31685],[34,604795],{"className":604796,"style":32604},[31845],[34,604798,223],{"className":604799},[32024],[34,604801],{"className":604802,"style":32604},[31845],[34,604804,32936],{"className":604805},[31685,31686],[34,604807,604809,604812],{"className":604808},[31690],[34,604810,2557],{"className":604811},[31690],[34,604813,604815],{"className":604814},[31792],[34,604816,604818],{"className":604817},[31796],[34,604819,604821],{"className":604820},[31801],[34,604822,604824],{"className":604823,"style":590807},[31805],[34,604825,604826,604829],{"style":590776},[34,604827],{"className":604828,"style":31814},[31813],[34,604830,604832],{"className":604831},[31818,31819,31820,31821],[34,604833,768],{"className":604834},[31685,31821],[34,604836,604837,604840],{"style":41520},[34,604838],{"className":604839,"style":41508},[31813],[34,604841],{"className":604842,"style":41528},[41527],[34,604844,604845,604848],{"style":129659},[34,604846],{"className":604847,"style":41508},[31813],[34,604849,604851],{"className":604850},[31685],[34,604852,604854,604882],{"className":604853},[31685,41550],[34,604855,604857],{"className":604856},[597735],[34,604858,604860],{"className":604859},[31796],[34,604861,604863],{"className":604862},[31801],[34,604864,604867],{"className":604865,"style":604866},[31805],"height:0.6586em;",[34,604868,604870,604873],{"style":604869},"top:-2.8363em;",[34,604871],{"className":604872,"style":43406},[31813],[34,604874,604876],{"className":604875},[31818,31819,43411,31821],[34,604877,604879],{"className":604878},[31685,31821],[34,604880,15932],{"className":604881},[31685,31821],[34,604883,604885,604918],{"className":604884},[31796,31797],[34,604886,604888,604915],{"className":604887},[31801],[34,604889,604891,604903],{"className":604890,"style":64997},[31805],[34,604892,604894,604897],{"className":604893,"style":41565},[41564],[34,604895],{"className":604896,"style":41508},[31813],[34,604898,604900],{"className":604899,"style":41572},[31685],[34,604901,32936],{"className":604902},[31685,31686],[34,604904,604905,604908],{"style":65012},[34,604906],{"className":604907,"style":41508},[31813],[34,604909,604911],{"className":604910,"style":41586},[41585],[41588,604912,604913],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,604914],{"d":41597},[34,604916,31829],{"className":604917},[31828],[34,604919,604921],{"className":604920},[31801],[34,604922,604924],{"className":604923,"style":65032},[31805],[34,604925],{},[34,604927,31829],{"className":604928},[31828],[34,604930,604932],{"className":604931},[31801],[34,604933,604936],{"className":604934,"style":604935},[31805],"height:0.936em;",[34,604937],{},[34,604939],{"className":604940},[31690,41488],[34,604942],{"className":604943,"style":32123},[31845],[34,604945,46900],{"className":604946},[31685,31686],[34,604948,32936],{"className":604949},[31685,31686],[105,604951,604952,605335,607941],{},[108,604953,604954,604956,604957,604985,604986,347],{},[2627,604955,419963],{},"：这符合 ",[34,604958,604960,604973],{"className":604959},[31642],[34,604961,604963],{"className":604962},[31646],[31648,604964,604965],{"xmlns":31650},[31652,604966,604967,604971],{},[31655,604968,604969],{},[31658,604970,54224],{},[31665,604972,54224],{"encoding":31667},[34,604974,604976],{"className":604975,"ariaHidden":758},[31672],[34,604977,604979,604982],{"className":604978},[31676],[34,604980],{"className":604981,"style":33218},[31680],[34,604983,54224],{"className":604984,"style":54252},[31685,31686]," 函数的第三种定义：",[34,604987,604989,605069],{"className":604988},[31642],[34,604990,604992],{"className":604991},[31646],[31648,604993,604994],{"xmlns":31650},[31652,604995,604996,605066],{},[31655,604997,604998,605000,605002,605004,605006,605008,605010,605012,605024,605060,605062,605064],{},[31658,604999,54224],{},[31662,605001,456],{"stretchy":702},[31658,605003,15],{},[31662,605005,31960],{"separator":758},[31658,605007,38759],{},[31662,605009,2557],{"stretchy":702},[31662,605011,202],{},[38846,605013,605014,605016,605018],{},[31662,605015,546295],{},[31749,605017,459],{},[31655,605019,605020,605022],{},[31662,605021,223],{},[31658,605023,138286],{"mathvariant":38079},[41453,605025,605026,605038],{},[37624,605027,605028,605030],{},[31658,605029,32936],{},[31655,605031,605032,605034,605036],{},[31658,605033,15],{},[31662,605035,32226],{},[31749,605037,640],{},[31655,605039,605040,605042,605044,605046,605048],{},[31662,605041,456],{"stretchy":702},[31749,605043,640],{},[31662,605045,223],{},[31658,605047,32936],{},[37624,605049,605050,605052],{},[31662,605051,2557],{"stretchy":702},[31655,605053,605054,605056,605058],{},[31658,605055,15],{},[31662,605057,223],{},[31658,605059,38759],{},[39361,605061,39363],{},[31658,605063,46900],{},[31658,605065,32936],{},[31665,605067,605068],{"encoding":31667},"B(p, q) = \\int_0^{+\\infty} \\frac{x^{p-1}}{(1+x)^{p+q}} \\, dx",[34,605070,605072,605108],{"className":605071,"ariaHidden":758},[31672],[34,605073,605075,605078,605081,605084,605087,605090,605093,605096,605099,605102,605105],{"className":605074},[31676],[34,605076],{"className":605077,"style":31774},[31680],[34,605079,54224],{"className":605080,"style":54252},[31685,31686],[34,605082,456],{"className":605083},[31782],[34,605085,15],{"className":605086},[31685,31686],[34,605088,31960],{"className":605089},[32119],[34,605091],{"className":605092,"style":32123},[31845],[34,605094,38759],{"className":605095,"style":32062},[31685,31686],[34,605097,2557],{"className":605098},[31690],[34,605100],{"className":605101,"style":31846},[31845],[34,605103,202],{"className":605104},[31850],[34,605106],{"className":605107,"style":31846},[31845],[34,605109,605111,605115,605172,605175,605326,605329,605332],{"className":605110},[31676],[34,605112],{"className":605113,"style":605114},[31680],"height:1.5379em;vertical-align:-0.52em;",[34,605116,605118,605121],{"className":605117},[32767],[34,605119,546295],{"className":605120,"style":546331},[32767,131249,156560],[34,605122,605124],{"className":605123},[31792],[34,605125,605127,605164],{"className":605126},[31796,31797],[34,605128,605130,605161],{"className":605129},[31801],[34,605131,605133,605144],{"className":605132,"style":546550},[31805],[34,605134,605135,605138],{"style":546347},[34,605136],{"className":605137,"style":31814},[31813],[34,605139,605141],{"className":605140},[31818,31819,31820,31821],[34,605142,459],{"className":605143},[31685,31821],[34,605145,605146,605149],{"style":546362},[34,605147],{"className":605148,"style":31814},[31813],[34,605150,605152],{"className":605151},[31818,31819,31820,31821],[34,605153,605155,605158],{"className":605154},[31685,31821],[34,605156,223],{"className":605157},[31685,31821],[34,605159,138286],{"className":605160},[31685,31821],[34,605162,31829],{"className":605163},[31828],[34,605165,605167],{"className":605166},[31801],[34,605168,605170],{"className":605169,"style":546384},[31805],[34,605171],{},[34,605173],{"className":605174,"style":32123},[31845],[34,605176,605178,605181,605323],{"className":605177},[31685],[34,605179],{"className":605180},[31782,41488],[34,605182,605184],{"className":605183},[41453],[34,605185,605187,605315],{"className":605186},[31796,31797],[34,605188,605190,605312],{"className":605189},[31801],[34,605191,605193,605255,605263],{"className":605192,"style":603956},[31805],[34,605194,605195,605198],{"style":41504},[34,605196],{"className":605197,"style":41508},[31813],[34,605199,605201],{"className":605200},[31818,31819,31820,31821],[34,605202,605204,605207,605210,605213,605216],{"className":605203},[31685,31821],[34,605205,456],{"className":605206},[31782,31821],[34,605208,640],{"className":605209},[31685,31821],[34,605211,223],{"className":605212},[32024,31821],[34,605214,32936],{"className":605215},[31685,31686,31821],[34,605217,605219,605222],{"className":605218},[31690,31821],[34,605220,2557],{"className":605221},[31690,31821],[34,605223,605225],{"className":605224},[31792],[34,605226,605228],{"className":605227},[31796],[34,605229,605231],{"className":605230},[31801],[34,605232,605235],{"className":605233,"style":605234},[31805],"height:0.7027em;",[34,605236,605237,605240],{"style":116668},[34,605238],{"className":605239,"style":43406},[31813],[34,605241,605243],{"className":605242},[31818,43410,43411,31821],[34,605244,605246,605249,605252],{"className":605245},[31685,31821],[34,605247,15],{"className":605248},[31685,31686,31821],[34,605250,223],{"className":605251},[32024,31821],[34,605253,38759],{"className":605254,"style":32062},[31685,31686,31821],[34,605256,605257,605260],{"style":41520},[34,605258],{"className":605259,"style":41508},[31813],[34,605261],{"className":605262,"style":41528},[41527],[34,605264,605265,605268],{"style":46771},[34,605266],{"className":605267,"style":41508},[31813],[34,605269,605271],{"className":605270},[31818,31819,31820,31821],[34,605272,605274],{"className":605273},[31685,31821],[34,605275,605277,605280],{"className":605276},[31685,31821],[34,605278,32936],{"className":605279},[31685,31686,31821],[34,605281,605283],{"className":605282},[31792],[34,605284,605286],{"className":605285},[31796],[34,605287,605289],{"className":605288},[31801],[34,605290,605292],{"className":605291,"style":144123},[31805],[34,605293,605294,605297],{"style":552391},[34,605295],{"className":605296,"style":43406},[31813],[34,605298,605300],{"className":605299},[31818,43410,43411,31821],[34,605301,605303,605306,605309],{"className":605302},[31685,31821],[34,605304,15],{"className":605305},[31685,31686,31821],[34,605307,32226],{"className":605308},[32024,31821],[34,605310,640],{"className":605311},[31685,31821],[34,605313,31829],{"className":605314},[31828],[34,605316,605318],{"className":605317},[31801],[34,605319,605321],{"className":605320,"style":324858},[31805],[34,605322],{},[34,605324],{"className":605325},[31690,41488],[34,605327],{"className":605328,"style":32123},[31845],[34,605330,46900],{"className":605331},[31685,31686],[34,605333,32936],{"className":605334},[31685,31686],[108,605336,605337,33183,605339],{},[2627,605338,289021],{},[2230,605340,605341,605535,605738,606038,606249,606637,606897,607189,607753],{},[108,605342,605343,605344,347],{},"被积函数为 ",[34,605345,605347,605389],{"className":605346},[31642],[34,605348,605350],{"className":605349},[31646],[31648,605351,605352],{"xmlns":31650},[31652,605353,605354,605386],{},[31655,605355,605356],{},[41453,605357,605358,605370],{},[37624,605359,605360,605362],{},[31658,605361,32936],{},[31655,605363,605364,605366,605368],{},[31749,605365,640],{},[31658,605367,38080],{"mathvariant":38079},[31749,605369,15932],{},[31655,605371,605372,605374,605376,605378,605380],{},[31662,605373,456],{"stretchy":702},[31749,605375,640],{},[31662,605377,223],{},[31658,605379,32936],{},[37624,605381,605382,605384],{},[31662,605383,2557],{"stretchy":702},[31749,605385,768],{},[31665,605387,605388],{"encoding":31667},"\\frac{x^{1\u002F4}}{(1+x)^2}",[34,605390,605392],{"className":605391,"ariaHidden":758},[31672],[34,605393,605395,605399],{"className":605394},[31676],[34,605396],{"className":605397,"style":605398},[31680],"height:1.5907em;vertical-align:-0.52em;",[34,605400,605402,605405,605532],{"className":605401},[31685],[34,605403],{"className":605404},[31782,41488],[34,605406,605408],{"className":605407},[41453],[34,605409,605411,605524],{"className":605410},[31796,31797],[34,605412,605414,605521],{"className":605413},[31801],[34,605415,605418,605470,605478],{"className":605416,"style":605417},[31805],"height:1.0707em;",[34,605419,605420,605423],{"style":41504},[34,605421],{"className":605422,"style":41508},[31813],[34,605424,605426],{"className":605425},[31818,31819,31820,31821],[34,605427,605429,605432,605435,605438,605441],{"className":605428},[31685,31821],[34,605430,456],{"className":605431},[31782,31821],[34,605433,640],{"className":605434},[31685,31821],[34,605436,223],{"className":605437},[32024,31821],[34,605439,32936],{"className":605440},[31685,31686,31821],[34,605442,605444,605447],{"className":605443},[31690,31821],[34,605445,2557],{"className":605446},[31690,31821],[34,605448,605450],{"className":605449},[31792],[34,605451,605453],{"className":605452},[31796],[34,605454,605456],{"className":605455},[31801],[34,605457,605459],{"className":605458,"style":548314},[31805],[34,605460,605461,605464],{"style":116668},[34,605462],{"className":605463,"style":43406},[31813],[34,605465,605467],{"className":605466},[31818,43410,43411,31821],[34,605468,768],{"className":605469},[31685,31821],[34,605471,605472,605475],{"style":41520},[34,605473],{"className":605474,"style":41508},[31813],[34,605476],{"className":605477,"style":41528},[41527],[34,605479,605480,605483],{"style":46771},[34,605481],{"className":605482,"style":41508},[31813],[34,605484,605486],{"className":605485},[31818,31819,31820,31821],[34,605487,605489],{"className":605488},[31685,31821],[34,605490,605492,605495],{"className":605491},[31685,31821],[34,605493,32936],{"className":605494},[31685,31686,31821],[34,605496,605498],{"className":605497},[31792],[34,605499,605501],{"className":605500},[31796],[34,605502,605504],{"className":605503},[31801],[34,605505,605507],{"className":605506,"style":555395},[31805],[34,605508,605509,605512],{"style":555398},[34,605510],{"className":605511,"style":555402},[31813],[34,605513,605515],{"className":605514},[31818,43410,43411,31821],[34,605516,605518],{"className":605517},[31685,31821],[34,605519,42426],{"className":605520},[31685,31821],[34,605522,31829],{"className":605523},[31828],[34,605525,605527],{"className":605526},[31801],[34,605528,605530],{"className":605529,"style":324858},[31805],[34,605531],{},[34,605533],{"className":605534},[31690,41488],[108,605536,605537,605538,38516,605612,347],{},"对比 ",[34,605539,605541,605565],{"className":605540},[31642],[34,605542,605544],{"className":605543},[31646],[31648,605545,605546],{"xmlns":31650},[31652,605547,605548,605562],{},[31655,605549,605550],{},[37624,605551,605552,605554],{},[31658,605553,32936],{},[31655,605555,605556,605558,605560],{},[31658,605557,15],{},[31662,605559,32226],{},[31749,605561,640],{},[31665,605563,605564],{"encoding":31667},"x^{p-1}",[34,605566,605568],{"className":605567,"ariaHidden":758},[31672],[34,605569,605571,605574],{"className":605570},[31676],[34,605572],{"className":605573,"style":37774},[31680],[34,605575,605577,605580],{"className":605576},[31685],[34,605578,32936],{"className":605579},[31685,31686],[34,605581,605583],{"className":605582},[31792],[34,605584,605586],{"className":605585},[31796],[34,605587,605589],{"className":605588},[31801],[34,605590,605592],{"className":605591,"style":37774},[31805],[34,605593,605594,605597],{"style":37777},[34,605595],{"className":605596,"style":31814},[31813],[34,605598,605600],{"className":605599},[31818,31819,31820,31821],[34,605601,605603,605606,605609],{"className":605602},[31685,31821],[34,605604,15],{"className":605605},[31685,31686,31821],[34,605607,32226],{"className":605608},[32024,31821],[34,605610,640],{"className":605611},[31685,31821],[34,605613,605615,605653],{"className":605614},[31642],[34,605616,605618],{"className":605617},[31646],[31648,605619,605620],{"xmlns":31650},[31652,605621,605622,605650],{},[31655,605623,605624,605626,605628,605630,605632,605634,605636,605638,605640,605642,605644,605646,605648],{},[31658,605625,15],{},[31662,605627,32226],{},[31749,605629,640],{},[31662,605631,202],{},[31749,605633,640],{},[31658,605635,38080],{"mathvariant":38079},[31749,605637,15932],{},[31662,605639,140094],{},[31658,605641,15],{},[31662,605643,202],{},[31749,605645,18136],{},[31658,605647,38080],{"mathvariant":38079},[31749,605649,15932],{},[31665,605651,605652],{"encoding":31667},"p-1 = 1\u002F4 \\Rightarrow p = 5\u002F4",[34,605654,605656,605674,605692,605710,605728],{"className":605655,"ariaHidden":758},[31672],[34,605657,605659,605662,605665,605668,605671],{"className":605658},[31676],[34,605660],{"className":605661,"style":55949},[31680],[34,605663,15],{"className":605664},[31685,31686],[34,605666],{"className":605667,"style":32604},[31845],[34,605669,32226],{"className":605670},[32024],[34,605672],{"className":605673,"style":32604},[31845],[34,605675,605677,605680,605683,605686,605689],{"className":605676},[31676],[34,605678],{"className":605679,"style":32617},[31680],[34,605681,640],{"className":605682},[31685],[34,605684],{"className":605685,"style":31846},[31845],[34,605687,202],{"className":605688},[31850],[34,605690],{"className":605691,"style":31846},[31845],[34,605693,605695,605698,605701,605704,605707],{"className":605694},[31676],[34,605696],{"className":605697,"style":31774},[31680],[34,605699,42426],{"className":605700},[31685],[34,605702],{"className":605703,"style":31846},[31845],[34,605705,140094],{"className":605706},[31850],[34,605708],{"className":605709,"style":31846},[31845],[34,605711,605713,605716,605719,605722,605725],{"className":605712},[31676],[34,605714],{"className":605715,"style":38774},[31680],[34,605717,15],{"className":605718},[31685,31686],[34,605720],{"className":605721,"style":31846},[31845],[34,605723,202],{"className":605724},[31850],[34,605726],{"className":605727,"style":31846},[31845],[34,605729,605731,605734],{"className":605730},[31676],[34,605732],{"className":605733,"style":31774},[31680],[34,605735,605737],{"className":605736},[31685],"5\u002F4",[108,605739,605740,605741,38516,605847,347],{},"对比分母 ",[34,605742,605744,605776],{"className":605743},[31642],[34,605745,605747],{"className":605746},[31646],[31648,605748,605749],{"xmlns":31650},[31652,605750,605751,605773],{},[31655,605752,605753,605755,605757,605759,605761],{},[31662,605754,456],{"stretchy":702},[31749,605756,640],{},[31662,605758,223],{},[31658,605760,32936],{},[37624,605762,605763,605765],{},[31662,605764,2557],{"stretchy":702},[31655,605766,605767,605769,605771],{},[31658,605768,15],{},[31662,605770,223],{},[31658,605772,38759],{},[31665,605774,605775],{"encoding":31667},"(1+x)^{p+q}",[34,605777,605779,605800],{"className":605778,"ariaHidden":758},[31672],[34,605780,605782,605785,605788,605791,605794,605797],{"className":605781},[31676],[34,605783],{"className":605784,"style":31774},[31680],[34,605786,456],{"className":605787},[31782],[34,605789,640],{"className":605790},[31685],[34,605792],{"className":605793,"style":32604},[31845],[34,605795,223],{"className":605796},[32024],[34,605798],{"className":605799,"style":32604},[31845],[34,605801,605803,605806,605809],{"className":605802},[31676],[34,605804],{"className":605805,"style":156809},[31680],[34,605807,32936],{"className":605808},[31685,31686],[34,605810,605812,605815],{"className":605811},[31690],[34,605813,2557],{"className":605814},[31690],[34,605816,605818],{"className":605817},[31792],[34,605819,605821],{"className":605820},[31796],[34,605822,605824],{"className":605823},[31801],[34,605825,605827],{"className":605826,"style":156828},[31805],[34,605828,605829,605832],{"style":37777},[34,605830],{"className":605831,"style":31814},[31813],[34,605833,605835],{"className":605834},[31818,31819,31820,31821],[34,605836,605838,605841,605844],{"className":605837},[31685,31821],[34,605839,15],{"className":605840},[31685,31686,31821],[34,605842,223],{"className":605843},[32024,31821],[34,605845,38759],{"className":605846,"style":32062},[31685,31686,31821],[34,605848,605850,605900],{"className":605849},[31642],[34,605851,605853],{"className":605852},[31646],[31648,605854,605855],{"xmlns":31650},[31652,605856,605857,605897],{},[31655,605858,605859,605861,605863,605865,605867,605869,605871,605873,605875,605877,605879,605881,605883,605885,605887,605889,605891,605893,605895],{},[31658,605860,15],{},[31662,605862,223],{},[31658,605864,38759],{},[31662,605866,202],{},[31749,605868,768],{},[31662,605870,140094],{},[31749,605872,18136],{},[31658,605874,38080],{"mathvariant":38079},[31749,605876,15932],{},[31662,605878,223],{},[31658,605880,38759],{},[31662,605882,202],{},[31749,605884,768],{},[31662,605886,140094],{},[31658,605888,38759],{},[31662,605890,202],{},[31749,605892,786],{},[31658,605894,38080],{"mathvariant":38079},[31749,605896,15932],{},[31665,605898,605899],{"encoding":31667},"p+q = 2 \\Rightarrow 5\u002F4 + q = 2 \\Rightarrow q = 3\u002F4",[34,605901,605903,605921,605939,605957,605975,605993,606011,606029],{"className":605902,"ariaHidden":758},[31672],[34,605904,605906,605909,605912,605915,605918],{"className":605905},[31676],[34,605907],{"className":605908,"style":55949},[31680],[34,605910,15],{"className":605911},[31685,31686],[34,605913],{"className":605914,"style":32604},[31845],[34,605916,223],{"className":605917},[32024],[34,605919],{"className":605920,"style":32604},[31845],[34,605922,605924,605927,605930,605933,605936],{"className":605923},[31676],[34,605925],{"className":605926,"style":38774},[31680],[34,605928,38759],{"className":605929,"style":32062},[31685,31686],[34,605931],{"className":605932,"style":31846},[31845],[34,605934,202],{"className":605935},[31850],[34,605937],{"className":605938,"style":31846},[31845],[34,605940,605942,605945,605948,605951,605954],{"className":605941},[31676],[34,605943],{"className":605944,"style":32617},[31680],[34,605946,768],{"className":605947},[31685],[34,605949],{"className":605950,"style":31846},[31845],[34,605952,140094],{"className":605953},[31850],[34,605955],{"className":605956,"style":31846},[31845],[34,605958,605960,605963,605966,605969,605972],{"className":605959},[31676],[34,605961],{"className":605962,"style":31774},[31680],[34,605964,605737],{"className":605965},[31685],[34,605967],{"className":605968,"style":32604},[31845],[34,605970,223],{"className":605971},[32024],[34,605973],{"className":605974,"style":32604},[31845],[34,605976,605978,605981,605984,605987,605990],{"className":605977},[31676],[34,605979],{"className":605980,"style":38774},[31680],[34,605982,38759],{"className":605983,"style":32062},[31685,31686],[34,605985],{"className":605986,"style":31846},[31845],[34,605988,202],{"className":605989},[31850],[34,605991],{"className":605992,"style":31846},[31845],[34,605994,605996,605999,606002,606005,606008],{"className":605995},[31676],[34,605997],{"className":605998,"style":32617},[31680],[34,606000,768],{"className":606001},[31685],[34,606003],{"className":606004,"style":31846},[31845],[34,606006,140094],{"className":606007},[31850],[34,606009],{"className":606010,"style":31846},[31845],[34,606012,606014,606017,606020,606023,606026],{"className":606013},[31676],[34,606015],{"className":606016,"style":38774},[31680],[34,606018,38759],{"className":606019,"style":32062},[31685,31686],[34,606021],{"className":606022,"style":31846},[31845],[34,606024,202],{"className":606025},[31850],[34,606027],{"className":606028,"style":31846},[31845],[34,606030,606032,606035],{"className":606031},[31676],[34,606033],{"className":606034,"style":31774},[31680],[34,606036,554561],{"className":606037},[31685],[108,606039,592630,606040,347],{},[34,606041,606043,606077],{"className":606042},[31642],[34,606044,606046],{"className":606045},[31646],[31648,606047,606048],{"xmlns":31650},[31652,606049,606050,606074],{},[31655,606051,606052,606054,606056,606058,606064,606066,606072],{},[31662,606053,202],{},[31658,606055,54224],{},[31662,606057,456],{"stretchy":702},[41453,606059,606060,606062],{},[31749,606061,18136],{},[31749,606063,15932],{},[31662,606065,31960],{"separator":758},[41453,606067,606068,606070],{},[31749,606069,786],{},[31749,606071,15932],{},[31662,606073,2557],{"stretchy":702},[31665,606075,606076],{"encoding":31667},"= B(\\frac{5}{4}, \\frac{3}{4})",[34,606078,606080,606092],{"className":606079,"ariaHidden":758},[31672],[34,606081,606083,606086,606089],{"className":606082},[31676],[34,606084],{"className":606085,"style":73089},[31680],[34,606087,202],{"className":606088},[31850],[34,606090],{"className":606091,"style":31846},[31845],[34,606093,606095,606098,606101,606104,606172,606175,606178,606246],{"className":606094},[31676],[34,606096],{"className":606097,"style":46727},[31680],[34,606099,54224],{"className":606100,"style":54252},[31685,31686],[34,606102,456],{"className":606103},[31782],[34,606105,606107,606110,606169],{"className":606106},[31685],[34,606108],{"className":606109},[31782,41488],[34,606111,606113],{"className":606112},[41453],[34,606114,606116,606161],{"className":606115},[31796,31797],[34,606117,606119,606158],{"className":606118},[31801],[34,606120,606122,606136,606144],{"className":606121,"style":46746},[31805],[34,606123,606124,606127],{"style":41504},[34,606125],{"className":606126,"style":41508},[31813],[34,606128,606130],{"className":606129},[31818,31819,31820,31821],[34,606131,606133],{"className":606132},[31685,31821],[34,606134,15932],{"className":606135},[31685,31821],[34,606137,606138,606141],{"style":41520},[34,606139],{"className":606140,"style":41508},[31813],[34,606142],{"className":606143,"style":41528},[41527],[34,606145,606146,606149],{"style":46771},[34,606147],{"className":606148,"style":41508},[31813],[34,606150,606152],{"className":606151},[31818,31819,31820,31821],[34,606153,606155],{"className":606154},[31685,31821],[34,606156,18136],{"className":606157},[31685,31821],[34,606159,31829],{"className":606160},[31828],[34,606162,606164],{"className":606163},[31801],[34,606165,606167],{"className":606166,"style":41619},[31805],[34,606168],{},[34,606170],{"className":606171},[31690,41488],[34,606173,31960],{"className":606174},[32119],[34,606176],{"className":606177,"style":32123},[31845],[34,606179,606181,606184,606243],{"className":606180},[31685],[34,606182],{"className":606183},[31782,41488],[34,606185,606187],{"className":606186},[41453],[34,606188,606190,606235],{"className":606189},[31796,31797],[34,606191,606193,606232],{"className":606192},[31801],[34,606194,606196,606210,606218],{"className":606195,"style":46746},[31805],[34,606197,606198,606201],{"style":41504},[34,606199],{"className":606200,"style":41508},[31813],[34,606202,606204],{"className":606203},[31818,31819,31820,31821],[34,606205,606207],{"className":606206},[31685,31821],[34,606208,15932],{"className":606209},[31685,31821],[34,606211,606212,606215],{"style":41520},[34,606213],{"className":606214,"style":41508},[31813],[34,606216],{"className":606217,"style":41528},[41527],[34,606219,606220,606223],{"style":46771},[34,606221],{"className":606222,"style":41508},[31813],[34,606224,606226],{"className":606225},[31818,31819,31820,31821],[34,606227,606229],{"className":606228},[31685,31821],[34,606230,786],{"className":606231},[31685,31821],[34,606233,31829],{"className":606234},[31828],[34,606236,606238],{"className":606237},[31801],[34,606239,606241],{"className":606240,"style":41619},[31805],[34,606242],{},[34,606244],{"className":606245},[31690,41488],[34,606247,2557],{"className":606248},[31690],[108,606250,606251,606252,33183,606463,347],{},"利用 ",[34,606253,606255,606314],{"className":606254},[31642],[34,606256,606258],{"className":606257},[31646],[31648,606259,606260],{"xmlns":31650},[31652,606261,606262,606312],{},[31655,606263,606264,606266,606268,606270,606272,606274,606276,606278],{},[31658,606265,54224],{},[31662,606267,456],{"stretchy":702},[31658,606269,15],{},[31662,606271,31960],{"separator":758},[31658,606273,38759],{},[31662,606275,2557],{"stretchy":702},[31662,606277,202],{},[41453,606279,606280,606298],{},[31655,606281,606282,606284,606286,606288,606290,606292,606294,606296],{},[31658,606283,564791],{"mathvariant":38079},[31662,606285,456],{"stretchy":702},[31658,606287,15],{},[31662,606289,2557],{"stretchy":702},[31658,606291,564791],{"mathvariant":38079},[31662,606293,456],{"stretchy":702},[31658,606295,38759],{},[31662,606297,2557],{"stretchy":702},[31655,606299,606300,606302,606304,606306,606308,606310],{},[31658,606301,564791],{"mathvariant":38079},[31662,606303,456],{"stretchy":702},[31658,606305,15],{},[31662,606307,223],{},[31658,606309,38759],{},[31662,606311,2557],{"stretchy":702},[31665,606313,587568],{"encoding":31667},[34,606315,606317,606353],{"className":606316,"ariaHidden":758},[31672],[34,606318,606320,606323,606326,606329,606332,606335,606338,606341,606344,606347,606350],{"className":606319},[31676],[34,606321],{"className":606322,"style":31774},[31680],[34,606324,54224],{"className":606325,"style":54252},[31685,31686],[34,606327,456],{"className":606328},[31782],[34,606330,15],{"className":606331},[31685,31686],[34,606333,31960],{"className":606334},[32119],[34,606336],{"className":606337,"style":32123},[31845],[34,606339,38759],{"className":606340,"style":32062},[31685,31686],[34,606342,2557],{"className":606343},[31690],[34,606345],{"className":606346,"style":31846},[31845],[34,606348,202],{"className":606349},[31850],[34,606351],{"className":606352,"style":31846},[31845],[34,606354,606356,606359],{"className":606355},[31676],[34,606357],{"className":606358,"style":327841},[31680],[34,606360,606362,606365,606460],{"className":606361},[31685],[34,606363],{"className":606364},[31782,41488],[34,606366,606368],{"className":606367},[41453],[34,606369,606371,606452],{"className":606370},[31796,31797],[34,606372,606374,606449],{"className":606373},[31801],[34,606375,606377,606406,606414],{"className":606376,"style":70504},[31805],[34,606378,606379,606382],{"style":41504},[34,606380],{"className":606381,"style":41508},[31813],[34,606383,606385],{"className":606384},[31818,31819,31820,31821],[34,606386,606388,606391,606394,606397,606400,606403],{"className":606387},[31685,31821],[34,606389,564791],{"className":606390},[31685,31821],[34,606392,456],{"className":606393},[31782,31821],[34,606395,15],{"className":606396},[31685,31686,31821],[34,606398,223],{"className":606399},[32024,31821],[34,606401,38759],{"className":606402,"style":32062},[31685,31686,31821],[34,606404,2557],{"className":606405},[31690,31821],[34,606407,606408,606411],{"style":41520},[34,606409],{"className":606410,"style":41508},[31813],[34,606412],{"className":606413,"style":41528},[41527],[34,606415,606416,606419],{"style":70529},[34,606417],{"className":606418,"style":41508},[31813],[34,606420,606422],{"className":606421},[31818,31819,31820,31821],[34,606423,606425,606428,606431,606434,606437,606440,606443,606446],{"className":606424},[31685,31821],[34,606426,564791],{"className":606427},[31685,31821],[34,606429,456],{"className":606430},[31782,31821],[34,606432,15],{"className":606433},[31685,31686,31821],[34,606435,2557],{"className":606436},[31690,31821],[34,606438,564791],{"className":606439},[31685,31821],[34,606441,456],{"className":606442},[31782,31821],[34,606444,38759],{"className":606445,"style":32062},[31685,31686,31821],[34,606447,2557],{"className":606448},[31690,31821],[34,606450,31829],{"className":606451},[31828],[34,606453,606455],{"className":606454},[31801],[34,606456,606458],{"className":606457,"style":324858},[31805],[34,606459],{},[34,606461],{"className":606462},[31690,41488],[34,606464,606466,606518],{"className":606465},[31642],[34,606467,606469],{"className":606468},[31646],[31648,606470,606471],{"xmlns":31650},[31652,606472,606473,606515],{},[31655,606474,606475,606477],{},[31662,606476,202],{},[41453,606478,606479,606505],{},[31655,606480,606481,606483,606485,606487,606489,606491,606493,606495,606497,606499,606501,606503],{},[31658,606482,564791],{"mathvariant":38079},[31662,606484,456],{"stretchy":702},[31749,606486,18136],{},[31658,606488,38080],{"mathvariant":38079},[31749,606490,15932],{},[31662,606492,2557],{"stretchy":702},[31658,606494,564791],{"mathvariant":38079},[31662,606496,456],{"stretchy":702},[31749,606498,786],{},[31658,606500,38080],{"mathvariant":38079},[31749,606502,15932],{},[31662,606504,2557],{"stretchy":702},[31655,606506,606507,606509,606511,606513],{},[31658,606508,564791],{"mathvariant":38079},[31662,606510,456],{"stretchy":702},[31749,606512,768],{},[31662,606514,2557],{"stretchy":702},[31665,606516,606517],{"encoding":31667},"= \\frac{\\Gamma(5\u002F4)\\Gamma(3\u002F4)}{\\Gamma(2)}",[34,606519,606521,606533],{"className":606520,"ariaHidden":758},[31672],[34,606522,606524,606527,606530],{"className":606523},[31676],[34,606525],{"className":606526,"style":73089},[31680],[34,606528,202],{"className":606529},[31850],[34,606531],{"className":606532,"style":31846},[31845],[34,606534,606536,606539],{"className":606535},[31676],[34,606537],{"className":606538,"style":327841},[31680],[34,606540,606542,606545,606634],{"className":606541},[31685],[34,606543],{"className":606544},[31782,41488],[34,606546,606548],{"className":606547},[41453],[34,606549,606551,606626],{"className":606550},[31796,31797],[34,606552,606554,606623],{"className":606553},[31801],[34,606555,606557,606580,606588],{"className":606556,"style":70504},[31805],[34,606558,606559,606562],{"style":41504},[34,606560],{"className":606561,"style":41508},[31813],[34,606563,606565],{"className":606564},[31818,31819,31820,31821],[34,606566,606568,606571,606574,606577],{"className":606567},[31685,31821],[34,606569,564791],{"className":606570},[31685,31821],[34,606572,456],{"className":606573},[31782,31821],[34,606575,768],{"className":606576},[31685,31821],[34,606578,2557],{"className":606579},[31690,31821],[34,606581,606582,606585],{"style":41520},[34,606583],{"className":606584,"style":41508},[31813],[34,606586],{"className":606587,"style":41528},[41527],[34,606589,606590,606593],{"style":70529},[34,606591],{"className":606592,"style":41508},[31813],[34,606594,606596],{"className":606595},[31818,31819,31820,31821],[34,606597,606599,606602,606605,606608,606611,606614,606617,606620],{"className":606598},[31685,31821],[34,606600,564791],{"className":606601},[31685,31821],[34,606603,456],{"className":606604},[31782,31821],[34,606606,605737],{"className":606607},[31685,31821],[34,606609,2557],{"className":606610},[31690,31821],[34,606612,564791],{"className":606613},[31685,31821],[34,606615,456],{"className":606616},[31782,31821],[34,606618,554561],{"className":606619},[31685,31821],[34,606621,2557],{"className":606622},[31690,31821],[34,606624,31829],{"className":606625},[31828],[34,606627,606629],{"className":606628},[31801],[34,606630,606632],{"className":606631,"style":324858},[31805],[34,606633],{},[34,606635],{"className":606636},[31690,41488],[108,606638,606639,606640,606733,606734,347],{},"分母 ",[34,606641,606643,606673],{"className":606642},[31642],[34,606644,606646],{"className":606645},[31646],[31648,606647,606648],{"xmlns":31650},[31652,606649,606650,606670],{},[31655,606651,606652,606654,606656,606658,606660,606662,606664,606666,606668],{},[31658,606653,564791],{"mathvariant":38079},[31662,606655,456],{"stretchy":702},[31749,606657,768],{},[31662,606659,2557],{"stretchy":702},[31662,606661,202],{},[31749,606663,640],{},[31662,606665,5504],{"stretchy":702},[31662,606667,202],{},[31749,606669,640],{},[31665,606671,606672],{"encoding":31667},"\\Gamma(2) = 1! = 1",[34,606674,606676,606703,606724],{"className":606675,"ariaHidden":758},[31672],[34,606677,606679,606682,606685,606688,606691,606694,606697,606700],{"className":606678},[31676],[34,606680],{"className":606681,"style":31774},[31680],[34,606683,564791],{"className":606684},[31685],[34,606686,456],{"className":606687},[31782],[34,606689,768],{"className":606690},[31685],[34,606692,2557],{"className":606693},[31690],[34,606695],{"className":606696,"style":31846},[31845],[34,606698,202],{"className":606699},[31850],[34,606701],{"className":606702,"style":31846},[31845],[34,606704,606706,606709,606712,606715,606718,606721],{"className":606705},[31676],[34,606707],{"className":606708,"style":31681},[31680],[34,606710,640],{"className":606711},[31685],[34,606713,5504],{"className":606714},[31690],[34,606716],{"className":606717,"style":31846},[31845],[34,606719,202],{"className":606720},[31850],[34,606722],{"className":606723,"style":31846},[31845],[34,606725,606727,606730],{"className":606726},[31676],[34,606728],{"className":606729,"style":32617},[31680],[34,606731,640],{"className":606732},[31685],"。分子用递推公式 ",[34,606735,606737,606781],{"className":606736},[31642],[34,606738,606740],{"className":606739},[31646],[31648,606741,606742],{"xmlns":31650},[31652,606743,606744,606778],{},[31655,606745,606746,606748,606750,606752,606754,606756,606758,606760,606766,606768,606770,606772,606774,606776],{},[31658,606747,564791],{"mathvariant":38079},[31662,606749,456],{"stretchy":702},[31749,606751,18136],{},[31658,606753,38080],{"mathvariant":38079},[31749,606755,15932],{},[31662,606757,2557],{"stretchy":702},[31662,606759,202],{},[41453,606761,606762,606764],{},[31749,606763,640],{},[31749,606765,15932],{},[31658,606767,564791],{"mathvariant":38079},[31662,606769,456],{"stretchy":702},[31749,606771,640],{},[31658,606773,38080],{"mathvariant":38079},[31749,606775,15932],{},[31662,606777,2557],{"stretchy":702},[31665,606779,606780],{"encoding":31667},"\\Gamma(5\u002F4) = \\frac{1}{4}\\Gamma(1\u002F4)",[34,606782,606784,606811],{"className":606783,"ariaHidden":758},[31672],[34,606785,606787,606790,606793,606796,606799,606802,606805,606808],{"className":606786},[31676],[34,606788],{"className":606789,"style":31774},[31680],[34,606791,564791],{"className":606792},[31685],[34,606794,456],{"className":606795},[31782],[34,606797,605737],{"className":606798},[31685],[34,606800,2557],{"className":606801},[31690],[34,606803],{"className":606804,"style":31846},[31845],[34,606806,202],{"className":606807},[31850],[34,606809],{"className":606810,"style":31846},[31845],[34,606812,606814,606817,606885,606888,606891,606894],{"className":606813},[31676],[34,606815],{"className":606816,"style":46727},[31680],[34,606818,606820,606823,606882],{"className":606819},[31685],[34,606821],{"className":606822},[31782,41488],[34,606824,606826],{"className":606825},[41453],[34,606827,606829,606874],{"className":606828},[31796,31797],[34,606830,606832,606871],{"className":606831},[31801],[34,606833,606835,606849,606857],{"className":606834,"style":46746},[31805],[34,606836,606837,606840],{"style":41504},[34,606838],{"className":606839,"style":41508},[31813],[34,606841,606843],{"className":606842},[31818,31819,31820,31821],[34,606844,606846],{"className":606845},[31685,31821],[34,606847,15932],{"className":606848},[31685,31821],[34,606850,606851,606854],{"style":41520},[34,606852],{"className":606853,"style":41508},[31813],[34,606855],{"className":606856,"style":41528},[41527],[34,606858,606859,606862],{"style":46771},[34,606860],{"className":606861,"style":41508},[31813],[34,606863,606865],{"className":606864},[31818,31819,31820,31821],[34,606866,606868],{"className":606867},[31685,31821],[34,606869,640],{"className":606870},[31685,31821],[34,606872,31829],{"className":606873},[31828],[34,606875,606877],{"className":606876},[31801],[34,606878,606880],{"className":606879,"style":41619},[31805],[34,606881],{},[34,606883],{"className":606884},[31690,41488],[34,606886,564791],{"className":606887},[31685],[34,606889,456],{"className":606890},[31782],[34,606892,42426],{"className":606893},[31685],[34,606895,2557],{"className":606896},[31690],[108,606898,592630,606899,347],{},[34,606900,606902,606946],{"className":606901},[31642],[34,606903,606905],{"className":606904},[31646],[31648,606906,606907],{"xmlns":31650},[31652,606908,606909,606943],{},[31655,606910,606911,606913,606919,606921,606923,606929,606931,606933,606935,606941],{},[31662,606912,202],{},[41453,606914,606915,606917],{},[31749,606916,640],{},[31749,606918,15932],{},[31658,606920,564791],{"mathvariant":38079},[31662,606922,456],{"stretchy":702},[41453,606924,606925,606927],{},[31749,606926,640],{},[31749,606928,15932],{},[31662,606930,2557],{"stretchy":702},[31658,606932,564791],{"mathvariant":38079},[31662,606934,456],{"stretchy":702},[41453,606936,606937,606939],{},[31749,606938,786],{},[31749,606940,15932],{},[31662,606942,2557],{"stretchy":702},[31665,606944,606945],{"encoding":31667},"= \\frac{1}{4} \\Gamma(\\frac{1}{4}) \\Gamma(\\frac{3}{4})",[34,606947,606949,606961],{"className":606948,"ariaHidden":758},[31672],[34,606950,606952,606955,606958],{"className":606951},[31676],[34,606953],{"className":606954,"style":73089},[31680],[34,606956,202],{"className":606957},[31850],[34,606959],{"className":606960,"style":31846},[31845],[34,606962,606964,606967,607035,607038,607041,607109,607112,607115,607118,607186],{"className":606963},[31676],[34,606965],{"className":606966,"style":46727},[31680],[34,606968,606970,606973,607032],{"className":606969},[31685],[34,606971],{"className":606972},[31782,41488],[34,606974,606976],{"className":606975},[41453],[34,606977,606979,607024],{"className":606978},[31796,31797],[34,606980,606982,607021],{"className":606981},[31801],[34,606983,606985,606999,607007],{"className":606984,"style":46746},[31805],[34,606986,606987,606990],{"style":41504},[34,606988],{"className":606989,"style":41508},[31813],[34,606991,606993],{"className":606992},[31818,31819,31820,31821],[34,606994,606996],{"className":606995},[31685,31821],[34,606997,15932],{"className":606998},[31685,31821],[34,607000,607001,607004],{"style":41520},[34,607002],{"className":607003,"style":41508},[31813],[34,607005],{"className":607006,"style":41528},[41527],[34,607008,607009,607012],{"style":46771},[34,607010],{"className":607011,"style":41508},[31813],[34,607013,607015],{"className":607014},[31818,31819,31820,31821],[34,607016,607018],{"className":607017},[31685,31821],[34,607019,640],{"className":607020},[31685,31821],[34,607022,31829],{"className":607023},[31828],[34,607025,607027],{"className":607026},[31801],[34,607028,607030],{"className":607029,"style":41619},[31805],[34,607031],{},[34,607033],{"className":607034},[31690,41488],[34,607036,564791],{"className":607037},[31685],[34,607039,456],{"className":607040},[31782],[34,607042,607044,607047,607106],{"className":607043},[31685],[34,607045],{"className":607046},[31782,41488],[34,607048,607050],{"className":607049},[41453],[34,607051,607053,607098],{"className":607052},[31796,31797],[34,607054,607056,607095],{"className":607055},[31801],[34,607057,607059,607073,607081],{"className":607058,"style":46746},[31805],[34,607060,607061,607064],{"style":41504},[34,607062],{"className":607063,"style":41508},[31813],[34,607065,607067],{"className":607066},[31818,31819,31820,31821],[34,607068,607070],{"className":607069},[31685,31821],[34,607071,15932],{"className":607072},[31685,31821],[34,607074,607075,607078],{"style":41520},[34,607076],{"className":607077,"style":41508},[31813],[34,607079],{"className":607080,"style":41528},[41527],[34,607082,607083,607086],{"style":46771},[34,607084],{"className":607085,"style":41508},[31813],[34,607087,607089],{"className":607088},[31818,31819,31820,31821],[34,607090,607092],{"className":607091},[31685,31821],[34,607093,640],{"className":607094},[31685,31821],[34,607096,31829],{"className":607097},[31828],[34,607099,607101],{"className":607100},[31801],[34,607102,607104],{"className":607103,"style":41619},[31805],[34,607105],{},[34,607107],{"className":607108},[31690,41488],[34,607110,2557],{"className":607111},[31690],[34,607113,564791],{"className":607114},[31685],[34,607116,456],{"className":607117},[31782],[34,607119,607121,607124,607183],{"className":607120},[31685],[34,607122],{"className":607123},[31782,41488],[34,607125,607127],{"className":607126},[41453],[34,607128,607130,607175],{"className":607129},[31796,31797],[34,607131,607133,607172],{"className":607132},[31801],[34,607134,607136,607150,607158],{"className":607135,"style":46746},[31805],[34,607137,607138,607141],{"style":41504},[34,607139],{"className":607140,"style":41508},[31813],[34,607142,607144],{"className":607143},[31818,31819,31820,31821],[34,607145,607147],{"className":607146},[31685,31821],[34,607148,15932],{"className":607149},[31685,31821],[34,607151,607152,607155],{"style":41520},[34,607153],{"className":607154,"style":41508},[31813],[34,607156],{"className":607157,"style":41528},[41527],[34,607159,607160,607163],{"style":46771},[34,607161],{"className":607162,"style":41508},[31813],[34,607164,607166],{"className":607165},[31818,31819,31820,31821],[34,607167,607169],{"className":607168},[31685,31821],[34,607170,786],{"className":607171},[31685,31821],[34,607173,31829],{"className":607174},[31828],[34,607176,607178],{"className":607177},[31801],[34,607179,607181],{"className":607180,"style":41619},[31805],[34,607182],{},[34,607184],{"className":607185},[31690,41488],[34,607187,2557],{"className":607188},[31690],[108,607190,600337,607191,604152,607277,39984,607332,347],{},[34,607192,607194,607226],{"className":607193},[31642],[34,607195,607197],{"className":607196},[31646],[31648,607198,607199],{"xmlns":31650},[31652,607200,607201,607223],{},[31655,607202,607203,607205,607207,607209,607211,607213,607215,607217,607219,607221],{},[31658,607204,564791],{"mathvariant":38079},[31662,607206,456],{"stretchy":702},[31658,607208,382333],{},[31662,607210,2557],{"stretchy":702},[31658,607212,564791],{"mathvariant":38079},[31662,607214,456],{"stretchy":702},[31749,607216,640],{},[31662,607218,32226],{},[31658,607220,382333],{},[31662,607222,2557],{"stretchy":702},[31665,607224,607225],{"encoding":31667},"\\Gamma(z)\\Gamma(1-z)",[34,607227,607229,607265],{"className":607228,"ariaHidden":758},[31672],[34,607230,607232,607235,607238,607241,607244,607247,607250,607253,607256,607259,607262],{"className":607231},[31676],[34,607233],{"className":607234,"style":31774},[31680],[34,607236,564791],{"className":607237},[31685],[34,607239,456],{"className":607240},[31782],[34,607242,382333],{"className":607243,"style":382406},[31685,31686],[34,607245,2557],{"className":607246},[31690],[34,607248,564791],{"className":607249},[31685],[34,607251,456],{"className":607252},[31782],[34,607254,640],{"className":607255},[31685],[34,607257],{"className":607258,"style":32604},[31845],[34,607260,32226],{"className":607261},[32024],[34,607263],{"className":607264,"style":32604},[31845],[34,607266,607268,607271,607274],{"className":607267},[31676],[34,607269],{"className":607270,"style":31774},[31680],[34,607272,382333],{"className":607273,"style":382406},[31685,31686],[34,607275,2557],{"className":607276},[31690],[34,607278,607280,607302],{"className":607279},[31642],[34,607281,607283],{"className":607282},[31646],[31648,607284,607285],{"xmlns":31650},[31652,607286,607287,607299],{},[31655,607288,607289,607291,607293,607295,607297],{},[31658,607290,382333],{},[31662,607292,202],{},[31749,607294,640],{},[31658,607296,38080],{"mathvariant":38079},[31749,607298,15932],{},[31665,607300,607301],{"encoding":31667},"z=1\u002F4",[34,607303,607305,607323],{"className":607304,"ariaHidden":758},[31672],[34,607306,607308,607311,607314,607317,607320],{"className":607307},[31676],[34,607309],{"className":607310,"style":32527},[31680],[34,607312,382333],{"className":607313,"style":382406},[31685,31686],[34,607315],{"className":607316,"style":31846},[31845],[34,607318,202],{"className":607319},[31850],[34,607321],{"className":607322,"style":31846},[31845],[34,607324,607326,607329],{"className":607325},[31676],[34,607327],{"className":607328,"style":31774},[31680],[34,607330,42426],{"className":607331},[31685],[34,607333,607335,607417],{"className":607334},[31642],[34,607336,607338],{"className":607337},[31646],[31648,607339,607340],{"xmlns":31650},[31652,607341,607342,607414],{},[31655,607343,607344,607346,607348,607350,607352,607354,607356,607358,607360,607362,607364,607366,607368,607370,607390,607392,607406,607408,607412],{},[31658,607345,564791],{"mathvariant":38079},[31662,607347,456],{"stretchy":702},[31749,607349,640],{},[31658,607351,38080],{"mathvariant":38079},[31749,607353,15932],{},[31662,607355,2557],{"stretchy":702},[31658,607357,564791],{"mathvariant":38079},[31662,607359,456],{"stretchy":702},[31749,607361,786],{},[31658,607363,38080],{"mathvariant":38079},[31749,607365,15932],{},[31662,607367,2557],{"stretchy":702},[31662,607369,202],{},[41453,607371,607372,607374],{},[31658,607373,58442],{},[31655,607375,607376,607378,607380,607382,607384,607386,607388],{},[31658,607377,547853],{},[31662,607379,32682],{},[31662,607381,456],{"stretchy":702},[31658,607383,58442],{},[31658,607385,38080],{"mathvariant":38079},[31749,607387,15932],{},[31662,607389,2557],{"stretchy":702},[31662,607391,202],{},[41453,607393,607394,607396],{},[31658,607395,58442],{},[31655,607397,607398,607402,607404],{},[41463,607399,607400],{},[31749,607401,768],{},[31658,607403,38080],{"mathvariant":38079},[31749,607405,768],{},[31662,607407,202],{},[41463,607409,607410],{},[31749,607411,768],{},[31658,607413,58442],{},[31665,607415,607416],{"encoding":31667},"\\Gamma(1\u002F4)\\Gamma(3\u002F4) = \\frac{\\pi}{\\sin(\\pi\u002F4)} = \\frac{\\pi}{\\sqrt{2}\u002F2} = \\sqrt{2}\\pi",[34,607418,607420,607459,607563,607696],{"className":607419,"ariaHidden":758},[31672],[34,607421,607423,607426,607429,607432,607435,607438,607441,607444,607447,607450,607453,607456],{"className":607422},[31676],[34,607424],{"className":607425,"style":31774},[31680],[34,607427,564791],{"className":607428},[31685],[34,607430,456],{"className":607431},[31782],[34,607433,42426],{"className":607434},[31685],[34,607436,2557],{"className":607437},[31690],[34,607439,564791],{"className":607440},[31685],[34,607442,456],{"className":607443},[31782],[34,607445,554561],{"className":607446},[31685],[34,607448,2557],{"className":607449},[31690],[34,607451],{"className":607452,"style":31846},[31845],[34,607454,202],{"className":607455},[31850],[34,607457],{"className":607458,"style":31846},[31845],[34,607460,607462,607465,607554,607557,607560],{"className":607461},[31676],[34,607463],{"className":607464,"style":570332},[31680],[34,607466,607468,607471,607551],{"className":607467},[31685],[34,607469],{"className":607470},[31782,41488],[34,607472,607474],{"className":607473},[41453],[34,607475,607477,607543],{"className":607476},[31796,31797],[34,607478,607480,607540],{"className":607479},[31801],[34,607481,607483,607518,607526],{"className":607482,"style":244945},[31805],[34,607484,607485,607488],{"style":41504},[34,607486],{"className":607487,"style":41508},[31813],[34,607489,607491],{"className":607490},[31818,31819,31820,31821],[34,607492,607494,607506,607509,607512,607515],{"className":607493},[31685,31821],[34,607495,607497,607500,607503],{"className":607496},[32767,31821],[34,607498,31747],{"className":607499},[31821],[34,607501,31955],{"className":607502},[31821],[34,607504,31660],{"className":607505},[31821],[34,607507,456],{"className":607508},[31782,31821],[34,607510,58442],{"className":607511,"style":32062},[31685,31686,31821],[34,607513,42280],{"className":607514},[31685,31821],[34,607516,2557],{"className":607517},[31690,31821],[34,607519,607520,607523],{"style":41520},[34,607521],{"className":607522,"style":41508},[31813],[34,607524],{"className":607525,"style":41528},[41527],[34,607527,607528,607531],{"style":46771},[34,607529],{"className":607530,"style":41508},[31813],[34,607532,607534],{"className":607533},[31818,31819,31820,31821],[34,607535,607537],{"className":607536},[31685,31821],[34,607538,58442],{"className":607539,"style":32062},[31685,31686,31821],[34,607541,31829],{"className":607542},[31828],[34,607544,607546],{"className":607545},[31801],[34,607547,607549],{"className":607548,"style":324858},[31805],[34,607550],{},[34,607552],{"className":607553},[31690,41488],[34,607555],{"className":607556,"style":31846},[31845],[34,607558,202],{"className":607559},[31850],[34,607561],{"className":607562,"style":31846},[31845],[34,607564,607566,607570,607687,607690,607693],{"className":607565},[31676],[34,607567],{"className":607568,"style":607569},[31680],"height:1.3194em;vertical-align:-0.624em;",[34,607571,607573,607576,607684],{"className":607572},[31685],[34,607574],{"className":607575},[31782,41488],[34,607577,607579],{"className":607578},[41453],[34,607580,607582,607675],{"className":607581},[31796,31797],[34,607583,607585,607672],{"className":607584},[31801],[34,607586,607588,607650,607658],{"className":607587,"style":244945},[31805],[34,607589,607591,607594],{"style":607590},"top:-2.551em;",[34,607592],{"className":607593,"style":41508},[31813],[34,607595,607597],{"className":607596},[31818,31819,31820,31821],[34,607598,607600,607647],{"className":607599},[31685,31821],[34,607601,607603],{"className":607602},[31685,41550,31821],[34,607604,607606,607639],{"className":607605},[31796,31797],[34,607607,607609,607636],{"className":607608},[31801],[34,607610,607612,607624],{"className":607611,"style":41560},[31805],[34,607613,607615,607618],{"className":607614,"style":41565},[41564],[34,607616],{"className":607617,"style":41508},[31813],[34,607619,607621],{"className":607620,"style":41572},[31685,31821],[34,607622,768],{"className":607623},[31685,31821],[34,607625,607626,607629],{"style":41578},[34,607627],{"className":607628,"style":41508},[31813],[34,607630,607632],{"className":607631,"style":41586},[41585,31821],[41588,607633,607634],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,607635],{"d":41597},[34,607637,31829],{"className":607638},[31828],[34,607640,607642],{"className":607641},[31801],[34,607643,607645],{"className":607644,"style":41607},[31805],[34,607646],{},[34,607648,38149],{"className":607649},[31685,31821],[34,607651,607652,607655],{"style":41520},[34,607653],{"className":607654,"style":41508},[31813],[34,607656],{"className":607657,"style":41528},[41527],[34,607659,607660,607663],{"style":46771},[34,607661],{"className":607662,"style":41508},[31813],[34,607664,607666],{"className":607665},[31818,31819,31820,31821],[34,607667,607669],{"className":607668},[31685,31821],[34,607670,58442],{"className":607671,"style":32062},[31685,31686,31821],[34,607673,31829],{"className":607674},[31828],[34,607676,607678],{"className":607677},[31801],[34,607679,607682],{"className":607680,"style":607681},[31805],"height:0.624em;",[34,607683],{},[34,607685],{"className":607686},[31690,41488],[34,607688],{"className":607689,"style":31846},[31845],[34,607691,202],{"className":607692},[31850],[34,607694],{"className":607695,"style":31846},[31845],[34,607697,607699,607703,607750],{"className":607698},[31676],[34,607700],{"className":607701,"style":607702},[31680],"height:1.04em;vertical-align:-0.1328em;",[34,607704,607706],{"className":607705},[31685,41550],[34,607707,607709,607742],{"className":607708},[31796,31797],[34,607710,607712,607739],{"className":607711},[31801],[34,607713,607715,607727],{"className":607714,"style":572001},[31805],[34,607716,607718,607721],{"className":607717,"style":41565},[41564],[34,607719],{"className":607720,"style":41508},[31813],[34,607722,607724],{"className":607723,"style":41572},[31685],[34,607725,768],{"className":607726},[31685],[34,607728,607729,607732],{"style":572016},[34,607730],{"className":607731,"style":41508},[31813],[34,607733,607735],{"className":607734,"style":41586},[41585],[41588,607736,607737],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,607738],{"d":41597},[34,607740,31829],{"className":607741},[31828],[34,607743,607745],{"className":607744},[31801],[34,607746,607748],{"className":607747,"style":572036},[31805],[34,607749],{},[34,607751,58442],{"className":607752,"style":32062},[31685,31686],[108,607754,607755,607756,347],{},"最终结果 ",[34,607757,607759,607787],{"className":607758},[31642],[34,607760,607762],{"className":607761},[31646],[31648,607763,607764],{"xmlns":31650},[31652,607765,607766,607784],{},[31655,607767,607768,607770,607776,607778,607782],{},[31662,607769,202],{},[41453,607771,607772,607774],{},[31749,607773,640],{},[31749,607775,15932],{},[31662,607777,71191],{},[41463,607779,607780],{},[31749,607781,768],{},[31658,607783,58442],{},[31665,607785,607786],{"encoding":31667},"= \\frac{1}{4} \\cdot \\sqrt{2}\\pi",[34,607788,607790,607802,607885],{"className":607789,"ariaHidden":758},[31672],[34,607791,607793,607796,607799],{"className":607792},[31676],[34,607794],{"className":607795,"style":73089},[31680],[34,607797,202],{"className":607798},[31850],[34,607800],{"className":607801,"style":31846},[31845],[34,607803,607805,607808,607876,607879,607882],{"className":607804},[31676],[34,607806],{"className":607807,"style":46727},[31680],[34,607809,607811,607814,607873],{"className":607810},[31685],[34,607812],{"className":607813},[31782,41488],[34,607815,607817],{"className":607816},[41453],[34,607818,607820,607865],{"className":607819},[31796,31797],[34,607821,607823,607862],{"className":607822},[31801],[34,607824,607826,607840,607848],{"className":607825,"style":46746},[31805],[34,607827,607828,607831],{"style":41504},[34,607829],{"className":607830,"style":41508},[31813],[34,607832,607834],{"className":607833},[31818,31819,31820,31821],[34,607835,607837],{"className":607836},[31685,31821],[34,607838,15932],{"className":607839},[31685,31821],[34,607841,607842,607845],{"style":41520},[34,607843],{"className":607844,"style":41508},[31813],[34,607846],{"className":607847,"style":41528},[41527],[34,607849,607850,607853],{"style":46771},[34,607851],{"className":607852,"style":41508},[31813],[34,607854,607856],{"className":607855},[31818,31819,31820,31821],[34,607857,607859],{"className":607858},[31685,31821],[34,607860,640],{"className":607861},[31685,31821],[34,607863,31829],{"className":607864},[31828],[34,607866,607868],{"className":607867},[31801],[34,607869,607871],{"className":607870,"style":41619},[31805],[34,607872],{},[34,607874],{"className":607875},[31690,41488],[34,607877],{"className":607878,"style":32604},[31845],[34,607880,71191],{"className":607881},[32024],[34,607883],{"className":607884,"style":32604},[31845],[34,607886,607888,607891,607938],{"className":607887},[31676],[34,607889],{"className":607890,"style":607702},[31680],[34,607892,607894],{"className":607893},[31685,41550],[34,607895,607897,607930],{"className":607896},[31796,31797],[34,607898,607900,607927],{"className":607899},[31801],[34,607901,607903,607915],{"className":607902,"style":572001},[31805],[34,607904,607906,607909],{"className":607905,"style":41565},[41564],[34,607907],{"className":607908,"style":41508},[31813],[34,607910,607912],{"className":607911,"style":41572},[31685],[34,607913,768],{"className":607914},[31685],[34,607916,607917,607920],{"style":572016},[34,607918],{"className":607919,"style":41508},[31813],[34,607921,607923],{"className":607922,"style":41586},[41585],[41588,607924,607925],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,607926],{"d":41597},[34,607928,31829],{"className":607929},[31828],[34,607931,607933],{"className":607932},[31801],[34,607934,607936],{"className":607935,"style":572036},[31805],[34,607937],{},[34,607939,58442],{"className":607940,"style":32062},[31685,31686],[108,607942,607943,2664,607945],{},[2627,607944,118575],{},[34,607946,607948,607972],{"className":607947},[31642],[34,607949,607951],{"className":607950},[31646],[31648,607952,607953],{"xmlns":31650},[31652,607954,607955,607969],{},[31655,607956,607957],{},[41453,607958,607959,607967],{},[31655,607960,607961,607965],{},[41463,607962,607963],{},[31749,607964,768],{},[31658,607966,58442],{},[31749,607968,15932],{},[31665,607970,607971],{"encoding":31667},"\\frac{\\sqrt{2}\\pi}{4}",[34,607973,607975],{"className":607974,"ariaHidden":758},[31672],[34,607976,607978,607981],{"className":607977},[31676],[34,607979],{"className":607980,"style":41481},[31680],[34,607982,607984,607987,608093],{"className":607983},[31685],[34,607985],{"className":607986},[31782,41488],[34,607988,607990],{"className":607989},[41453],[34,607991,607993,608085],{"className":607992},[31796,31797],[34,607994,607996,608082],{"className":607995},[31801],[34,607997,607999,608013,608021],{"className":607998,"style":41501},[31805],[34,608000,608001,608004],{"style":41504},[34,608002],{"className":608003,"style":41508},[31813],[34,608005,608007],{"className":608006},[31818,31819,31820,31821],[34,608008,608010],{"className":608009},[31685,31821],[34,608011,15932],{"className":608012},[31685,31821],[34,608014,608015,608018],{"style":41520},[34,608016],{"className":608017,"style":41508},[31813],[34,608019],{"className":608020,"style":41528},[41527],[34,608022,608023,608026],{"style":41531},[34,608024],{"className":608025,"style":41508},[31813],[34,608027,608029],{"className":608028},[31818,31819,31820,31821],[34,608030,608032,608079],{"className":608031},[31685,31821],[34,608033,608035],{"className":608034},[31685,41550,31821],[34,608036,608038,608071],{"className":608037},[31796,31797],[34,608039,608041,608068],{"className":608040},[31801],[34,608042,608044,608056],{"className":608043,"style":41560},[31805],[34,608045,608047,608050],{"className":608046,"style":41565},[41564],[34,608048],{"className":608049,"style":41508},[31813],[34,608051,608053],{"className":608052,"style":41572},[31685,31821],[34,608054,768],{"className":608055},[31685,31821],[34,608057,608058,608061],{"style":41578},[34,608059],{"className":608060,"style":41508},[31813],[34,608062,608064],{"className":608063,"style":41586},[41585,31821],[41588,608065,608066],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,608067],{"d":41597},[34,608069,31829],{"className":608070},[31828],[34,608072,608074],{"className":608073},[31801],[34,608075,608077],{"className":608076,"style":41607},[31805],[34,608078],{},[34,608080,58442],{"className":608081,"style":32062},[31685,31686,31821],[34,608083,31829],{"className":608084},[31828],[34,608086,608088],{"className":608087},[31801],[34,608089,608091],{"className":608090,"style":41619},[31805],[34,608092],{},[34,608094],{"className":608095},[31690,41488],[15,608097,608098,47649,608101],{},[2627,608099,608100],{},"(6)",[34,608102,608104,608164],{"className":608103},[31642],[34,608105,608107],{"className":608106},[31646],[31648,608108,608109],{"xmlns":31650},[31652,608110,608111,608161],{},[31655,608112,608113],{},[71090,608114,608115,608127,608137,608139,608153,608155,608157,608159],{"scriptlevel":459,"displaystyle":758},[38846,608116,608117,608119,608121],{},[31662,608118,546295],{},[31749,608120,459],{},[41453,608122,608123,608125],{},[31658,608124,58442],{},[31749,608126,768],{},[37624,608128,608129,608135],{},[31655,608130,608131,608133],{},[31658,608132,547853],{},[31662,608134,32682],{},[31749,608136,36723],{},[31658,608138,32936],{},[37624,608140,608141,608147],{},[31655,608142,608143,608145],{},[31658,608144,547860],{},[31662,608146,32682],{},[41453,608148,608149,608151],{},[31749,608150,640],{},[31749,608152,768],{},[31658,608154,32936],{},[39361,608156,39363],{},[31658,608158,46900],{},[31658,608160,32936],{},[31665,608162,608163],{"encoding":31667},"\\displaystyle \\int_0^{\\frac{\\pi}{2}} \\sin^7 x \\cos^{\\frac{1}{2}} x \\, dx",[34,608165,608167],{"className":608166,"ariaHidden":758},[31672],[34,608168,608170,608174,608296,608299,608328,608331,608334,608337,608436,608439,608442,608445,608448],{"className":608169},[31676],[34,608171],{"className":608172,"style":608173},[31680],"height:2.5089em;vertical-align:-0.9119em;",[34,608175,608177,608180],{"className":608176},[32767],[34,608178,546295],{"className":608179,"style":591475},[32767,131249,131250],[34,608181,608183],{"className":608182},[31792],[34,608184,608186,608288],{"className":608185},[31796,31797],[34,608187,608189,608285],{"className":608188},[31801],[34,608190,608193,608205],{"className":608191,"style":608192},[31805],"height:1.597em;",[34,608194,608196,608199],{"style":608195},"top:-2.088em;margin-left:-0.4445em;margin-right:0.05em;",[34,608197],{"className":608198,"style":41508},[31813],[34,608200,608202],{"className":608201},[31818,31819,31820,31821],[34,608203,459],{"className":608204},[31685,31821],[34,608206,608208,608211],{"style":608207},"top:-4.1129em;margin-right:0.05em;",[34,608209],{"className":608210,"style":41508},[31813],[34,608212,608214],{"className":608213},[31818,31819,31820,31821],[34,608215,608217],{"className":608216},[31685,31821],[34,608218,608220,608223,608282],{"className":608219},[31685,31821],[34,608221],{"className":608222},[31782,41488,31818,43410,435704],[34,608224,608226],{"className":608225},[41453],[34,608227,608229,608274],{"className":608228},[31796,31797],[34,608230,608232,608271],{"className":608231},[31801],[34,608233,608235,608249,608257],{"className":608234,"style":589595},[31805],[34,608236,608237,608240],{"style":435720},[34,608238],{"className":608239,"style":41508},[31813],[34,608241,608243],{"className":608242},[31818,43410,43411,31821],[34,608244,608246],{"className":608245},[31685,31821],[34,608247,768],{"className":608248},[31685,31821],[34,608250,608251,608254],{"style":435735},[34,608252],{"className":608253,"style":41508},[31813],[34,608255],{"className":608256,"style":435742},[41527,31821],[34,608258,608259,608262],{"style":589620},[34,608260],{"className":608261,"style":41508},[31813],[34,608263,608265],{"className":608264},[31818,43410,43411,31821],[34,608266,608268],{"className":608267},[31685,31821],[34,608269,58442],{"className":608270,"style":32062},[31685,31686,31821],[34,608272,31829],{"className":608273},[31828],[34,608275,608277],{"className":608276},[31801],[34,608278,608280],{"className":608279,"style":435782},[31805],[34,608281],{},[34,608283],{"className":608284},[31690,41488,31818,43410,435704],[34,608286,31829],{"className":608287},[31828],[34,608289,608291],{"className":608290},[31801],[34,608292,608294],{"className":608293,"style":591522},[31805],[34,608295],{},[34,608297],{"className":608298,"style":32123},[31845],[34,608300,608302,608305],{"className":608301},[32767],[34,608303,547853],{"className":608304},[32767],[34,608306,608308],{"className":608307},[31792],[34,608309,608311],{"className":608310},[31796],[34,608312,608314],{"className":608313},[31801],[34,608315,608317],{"className":608316,"style":585884},[31805],[34,608318,608319,608322],{"style":586008},[34,608320],{"className":608321,"style":31814},[31813],[34,608323,608325],{"className":608324},[31818,31819,31820,31821],[34,608326,36723],{"className":608327},[31685,31821],[34,608329],{"className":608330,"style":32123},[31845],[34,608332,32936],{"className":608333},[31685,31686],[34,608335],{"className":608336,"style":32123},[31845],[34,608338,608340,608343],{"className":608339},[32767],[34,608341,547860],{"className":608342},[32767],[34,608344,608346],{"className":608345},[31792],[34,608347,608349],{"className":608348},[31796],[34,608350,608352],{"className":608351},[31801],[34,608353,608356],{"className":608354,"style":608355},[31805],"height:1.004em;",[34,608357,608359,608362],{"style":608358},"top:-3.413em;margin-right:0.05em;",[34,608360],{"className":608361,"style":41508},[31813],[34,608363,608365],{"className":608364},[31818,31819,31820,31821],[34,608366,608368],{"className":608367},[31685,31821],[34,608369,608371,608374,608433],{"className":608370},[31685,31821],[34,608372],{"className":608373},[31782,41488,31818,43410,435704],[34,608375,608377],{"className":608376},[41453],[34,608378,608380,608425],{"className":608379},[31796,31797],[34,608381,608383,608422],{"className":608382},[31801],[34,608384,608386,608400,608408],{"className":608385,"style":593176},[31805],[34,608387,608388,608391],{"style":435720},[34,608389],{"className":608390,"style":41508},[31813],[34,608392,608394],{"className":608393},[31818,43410,43411,31821],[34,608395,608397],{"className":608396},[31685,31821],[34,608398,768],{"className":608399},[31685,31821],[34,608401,608402,608405],{"style":435735},[34,608403],{"className":608404,"style":41508},[31813],[34,608406],{"className":608407,"style":435742},[41527,31821],[34,608409,608410,608413],{"style":589620},[34,608411],{"className":608412,"style":41508},[31813],[34,608414,608416],{"className":608415},[31818,4341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",[34,608459,608461,608474],{"className":608460},[31642],[34,608462,608464],{"className":608463},[31646],[31648,608465,608466],{"xmlns":31650},[31652,608467,608468,608472],{},[31655,608469,608470],{},[31658,608471,54224],{},[31665,608473,54224],{"encoding":31667},[34,608475,608477],{"className":608476,"ariaHidden":758},[31672],[34,608478,608480,608483],{"className":608479},[31676],[34,608481],{"className":608482,"style":33218},[31680],[34,608484,54224],{"className":608485,"style":54252},[31685,31686]," 函数公式。",[108,608488,608489,33183,608491],{},[2627,608490,289021],{},[2230,608492,608493,608995,609087,609470,609909,610528,610624],{},[108,608494,608495,608496,347],{},"公式：",[34,608497,608499,608595],{"className":608498},[31642],[34,608500,608502],{"className":608501},[31646],[31648,608503,608504],{"xmlns":31650},[31652,608505,608506,608592],{},[31655,608507,608508,608522,608532,608534,608544,608546,608548,608550,608552,608554,608560,608562,608564,608576,608578,608590],{},[38846,608509,608510,608512,608514],{},[31662,608511,546295],{},[31749,608513,459],{},[31655,608515,608516,608518,608520],{},[31658,608517,58442],{},[31658,608519,38080],{"mathvariant":38079},[31749,608521,768],{},[37624,608523,608524,608530],{},[31655,608525,608526,608528],{},[31658,608527,547853],{},[31662,608529,32682],{},[31658,608531,217392],{},[31658,608533,32936],{},[37624,608535,608536,608542],{},[31655,608537,608538,608540],{},[31658,608539,547860],{},[31662,608541,32682],{},[31658,608543,217691],{},[31658,608545,32936],{},[39361,608547,39363],{},[31658,608549,46900],{},[31658,608551,32936],{},[31662,608553,202],{},[41453,608555,608556,608558],{},[31749,608557,640],{},[31749,608559,768],{},[31658,608561,54224],{},[31662,608563,456],{"stretchy":702},[41453,608565,608566,608574],{},[31655,608567,608568,608570,608572],{},[31658,608569,217392],{},[31662,608571,223],{},[31749,608573,640],{},[31749,608575,768],{},[31662,608577,31960],{"separator":758},[41453,608579,608580,608588],{},[31655,608581,608582,608584,608586],{},[31658,608583,217691],{},[31662,608585,223],{},[31749,608587,640],{},[31749,608589,768],{},[31662,608591,2557],{"stretchy":702},[31665,608593,608594],{"encoding":31667},"\\int_0^{\\pi\u002F2} \\sin^\\alpha x \\cos^\\beta x \\, dx = \\frac{1}{2} B(\\frac{\\alpha+1}{2}, \\frac{\\beta+1}{2})",[34,608596,608598,608756],{"className":608597,"ariaHidden":758},[31672],[34,608599,608601,608604,608661,608664,608694,608697,608700,608703,608732,608735,608738,608741,608744,608747,608750,608753],{"className":608600},[31676],[34,608602],{"className":608603,"style":596892},[31680],[34,608605,608607,608610],{"className":608606},[32767],[34,608608,546295],{"className":608609,"style":546331},[32767,131249,156560],[34,608611,608613],{"className":608612},[31792],[34,608614,608616,608653],{"className":608615},[31796,31797],[34,608617,608619,608650],{"className":608618},[31801],[34,608620,608622,608633],{"className":608621,"style":586480},[31805],[34,608623,608624,608627],{"style":546347},[34,608625],{"className":608626,"style":31814},[31813],[34,608628,608630],{"className":608629},[31818,31819,31820,31821],[34,608631,459],{"className":608632},[31685,31821],[34,608634,608635,608638],{"style":546362},[34,608636],{"className":608637,"style":31814},[31813],[34,608639,608641],{"className":608640},[31818,31819,31820,31821],[34,608642,608644,608647],{"className":608643},[31685,31821],[34,608645,58442],{"className":608646,"style":32062},[31685,31686,31821],[34,608648,38149],{"className":608649},[31685,31821],[34,608651,31829],{"className":608652},[31828],[34,608654,608656],{"className":608655},[31801],[34,608657,608659],{"className":608658,"style":546384},[31805],[34,608660],{},[34,608662],{"className":608663,"style":32123},[31845],[34,608665,608667,608670],{"className":608666},[32767],[34,608668,547853],{"className":608669},[32767],[34,608671,608673],{"className":608672},[31792],[34,608674,608676],{"className":608675},[31796],[34,608677,608679],{"className":608678},[31801],[34,608680,608683],{"className":608681,"style":608682},[31805],"height:0.7222em;",[34,608684,608685,608688],{"style":586008},[34,608686],{"className":608687,"style":31814},[31813],[34,608689,608691],{"className":608690},[31818,31819,31820,31821],[34,608692,217392],{"className":608693,"style":217496},[31685,31686,31821],[34,608695],{"className":608696,"style":32123},[31845],[34,608698,32936],{"className":608699},[31685,31686],[34,608701],{"className":608702,"style":32123},[31845],[34,608704,608706,608709],{"className":608705},[32767],[34,608707,547860],{"className":608708},[32767],[34,608710,608712],{"className":608711},[31792],[34,608713,608715],{"className":608714},[31796],[34,608716,608718],{"className":608717},[31801],[34,608719,608721],{"className":608720,"style":38486},[31805],[34,608722,608723,608726],{"style":37777},[34,608724],{"className":608725,"style":31814},[31813],[34,608727,608729],{"className":608728},[31818,31819,31820,31821],[34,608730,217691],{"className":608731,"style":217853},[31685,31686,31821],[34,608733],{"className":608734,"style":32123},[31845],[34,608736,32936],{"className":608737},[31685,31686],[34,608739],{"className":608740,"style":32123},[31845],[34,608742,46900],{"className":608743},[31685,31686],[34,608745,32936],{"className":608746},[31685,31686],[34,608748],{"className":608749,"style":31846},[31845],[34,608751,202],{"className":608752},[31850],[34,608754],{"className":608755,"style":31846},[31845],[34,608757,608759,608763,608831,608834,608837,608911,608914,608917,608992],{"className":608758},[31676],[34,608760],{"className":608761,"style":608762},[31680],"height:1.2772em;vertical-align:-0.345em;",[34,608764,608766,608769,608828],{"className":608765},[31685],[34,608767],{"className":608768},[31782,41488],[34,608770,608772],{"className":608771},[41453],[34,608773,608775,608820],{"className":608774},[31796,31797],[34,608776,608778,608817],{"className":608777},[31801],[34,608779,608781,608795,608803],{"className":608780,"style":46746},[31805],[34,608782,608783,608786],{"style":41504},[34,608784],{"className":608785,"style":41508},[31813],[34,608787,608789],{"className":608788},[31818,31819,31820,31821],[34,608790,608792],{"className":608791},[31685,31821],[34,608793,768],{"className":608794},[31685,31821],[34,608796,608797,608800],{"style":41520},[34,608798],{"className":608799,"style":41508},[31813],[34,608801],{"className":608802,"style":41528},[41527],[34,608804,608805,608808],{"style":46771},[34,608806],{"className":608807,"style":41508},[31813],[34,608809,608811],{"className":608810},[31818,31819,31820,31821],[34,608812,608814],{"className":608813},[31685,31821],[34,608815,640],{"className":608816},[31685,31821],[34,608818,31829],{"className":608819},[31828],[34,608821,608823],{"className":608822},[31801],[34,608824,608826],{"className":608825,"style":41619},[31805],[34,608827],{},[34,608829],{"className":608830},[31690,41488],[34,608832,54224],{"className":608833,"style":54252},[31685,31686],[34,608835,456],{"className":608836},[31782],[34,608838,608840,608843,608908],{"className":608839},[31685],[34,608841],{"className":608842},[31782,41488],[34,608844,608846],{"className":608845},[41453],[34,608847,608849,608900],{"className":608848},[31796,31797],[34,608850,608852,608897],{"className":608851},[31801],[34,608853,608855,608869,608877],{"className":608854,"style":46746},[31805],[34,608856,608857,608860],{"style":41504},[34,608858],{"className":608859,"style":41508},[31813],[34,608861,608863],{"className":608862},[31818,31819,31820,31821],[34,608864,608866],{"className":608865},[31685,31821],[34,608867,768],{"className":608868},[31685,31821],[34,608870,608871,608874],{"style":41520},[34,608872],{"className":608873,"style":41508},[31813],[34,608875],{"className":608876,"style":41528},[41527],[34,608878,608879,608882],{"style":46771},[34,608880],{"className":608881,"style":41508},[31813],[34,608883,608885],{"className":608884},[31818,31819,31820,31821],[34,608886,608888,608891,608894],{"className":608887},[31685,31821],[34,608889,217392],{"className":608890,"style":217496},[31685,31686,31821],[34,608892,223],{"className":608893},[32024,31821],[34,608895,640],{"className":608896},[31685,31821],[34,608898,31829],{"className":608899},[31828],[34,608901,608903],{"className":608902},[31801],[34,608904,608906],{"className":608905,"style":41619},[31805],[34,608907],{},[34,608909],{"className":608910},[31690,41488],[34,608912,31960],{"className":608913},[32119],[34,608915],{"className":608916,"style":32123},[31845],[34,608918,608920,608923,608989],{"className":608919},[31685],[34,608921],{"className":608922},[31782,41488],[34,608924,608926],{"className":608925},[41453],[34,608927,608929,608981],{"className":608928},[31796,31797],[34,608930,608932,608978],{"className":608931},[31801],[34,608933,608936,608950,608958],{"className":608934,"style":608935},[31805],"height:0.9322em;",[34,608937,608938,608941],{"style":41504},[34,608939],{"className":608940,"style":41508},[31813],[34,608942,608944],{"className":608943},[31818,31819,31820,31821],[34,608945,608947],{"className":608946},[31685,31821],[34,608948,768],{"className":608949},[31685,31821],[34,608951,608952,608955],{"style":41520},[34,608953],{"className":608954,"style":41508},[31813],[34,608956],{"className":608957,"style":41528},[41527],[34,608959,608960,608963],{"style":548368},[34,608961],{"className":608962,"style":41508},[31813],[34,608964,608966],{"className":608965},[31818,31819,31820,31821],[34,608967,608969,608972,608975],{"className":608968},[31685,31821],[34,608970,217691],{"className":608971,"style":217853},[31685,31686,31821],[34,608973,223],{"className":608974},[32024,31821],[34,608976,640],{"className":608977},[31685,31821],[34,608979,31829],{"className":608980},[31828],[34,608982,608984],{"className":608983},[31801],[34,608985,608987],{"className":608986,"style":41619},[31805],[34,608988],{},[34,608990],{"className":608991},[31690,41488],[34,608993,2557],{"className":608994},[31690],[108,608996,25496,608997,347],{},[34,608998,609000,609030],{"className":608999},[31642],[34,609001,609003],{"className":609002},[31646],[31648,609004,609005],{"xmlns":31650},[31652,609006,609007,609027],{},[31655,609008,609009,609011,609013,609015,609017,609019,609021,609023,609025],{},[31658,609010,217392],{},[31662,609012,202],{},[31749,609014,36723],{},[31662,609016,31960],{"separator":758},[31658,609018,217691],{},[31662,609020,202],{},[31749,609022,640],{},[31658,609024,38080],{"mathvariant":38079},[31749,609026,768],{},[31665,609028,609029],{"encoding":31667},"\\alpha = 7, \\beta = 1\u002F2",[34,609031,609033,609051,609078],{"className":609032,"ariaHidden":758},[31672],[34,609034,609036,609039,609042,609045,609048],{"className":609035},[31676],[34,609037],{"className":609038,"style":32527},[31680],[34,609040,217392],{"className":609041,"style":217496},[31685,31686],[34,609043],{"className":609044,"style":31846},[31845],[34,609046,202],{"className":609047},[31850],[34,609049],{"className":609050,"style":31846},[31845],[34,609052,609054,609057,609060,609063,609066,609069,609072,609075],{"className":609053},[31676],[34,609055],{"className":609056,"style":43217},[31680],[34,609058,36723],{"className":609059},[31685],[34,609061,31960],{"className":609062},[32119],[34,609064],{"className":609065,"style":32123},[31845],[34,609067,217691],{"className":609068,"style":217853},[31685,31686],[34,609070],{"className":609071,"style":31846},[31845],[34,609073,202],{"className":609074},[31850],[34,609076],{"className":609077,"style":31846},[31845],[34,609079,609081,609084],{"className":609080},[31676],[34,609082],{"className":609083,"style":31774},[31680],[34,609085,42407],{"className":609086},[31685],[108,609088,609089,46528,609243,347],{},[34,609090,609092,609124],{"className":609091},[31642],[34,609093,609095],{"className":609094},[31646],[31648,609096,609097],{"xmlns":31650},[31652,609098,609099,609121],{},[31655,609100,609101,609103,609105,609117,609119],{},[31658,609102,15],{},[31662,609104,202],{},[41453,609106,609107,609115],{},[31655,609108,609109,609111,609113],{},[31749,609110,36723],{},[31662,609112,223],{},[31749,609114,640],{},[31749,609116,768],{},[31662,609118,202],{},[31749,609120,15932],{},[31665,609122,609123],{"encoding":31667},"p = \\frac{7+1}{2} = 4",[34,609125,609127,609145,609234],{"className":609126,"ariaHidden":758},[31672],[34,609128,609130,609133,609136,609139,609142],{"className":609129},[31676],[34,609131],{"className":609132,"style":38774},[31680],[34,609134,15],{"className":609135},[31685,31686],[34,609137],{"className":609138,"style":31846},[31845],[34,609140,202],{"className":609141},[31850],[34,609143],{"className":609144,"style":31846},[31845],[34,609146,609148,609151,609225,609228,609231],{"className":609147},[31676],[34,609149],{"className":609150,"style":46727},[31680],[34,609152,609154,609157,609222],{"className":609153},[31685],[34,609155],{"className":609156},[31782,41488],[34,609158,609160],{"className":609159},[41453],[34,609161,609163,609214],{"className":609162},[31796,31797],[34,609164,609166,609211],{"className":609165},[31801],[34,609167,609169,609183,609191],{"className":609168,"style":46746},[31805],[34,609170,609171,609174],{"style":41504},[34,609172],{"className":609173,"style":41508},[31813],[34,609175,609177],{"className":609176},[31818,31819,31820,31821],[34,609178,609180],{"className":609179},[31685,31821],[34,609181,768],{"className":609182},[31685,31821],[34,609184,609185,609188],{"style":41520},[34,609186],{"className":609187,"style":41508},[31813],[34,609189],{"className":609190,"style":41528},[41527],[34,609192,609193,609196],{"style":46771},[34,609194],{"className":609195,"style":41508},[31813],[34,609197,609199],{"className":609198},[31818,31819,31820,31821],[34,609200,609202,609205,609208],{"className":609201},[31685,31821],[34,609203,36723],{"className":609204},[31685,31821],[34,609206,223],{"className":609207},[32024,31821],[34,609209,640],{"className":609210},[31685,31821],[34,609212,31829],{"className":609213},[31828],[34,609215,609217],{"className":609216},[31801],[34,609218,609220],{"className":609219,"style":41619},[31805],[34,609221],{},[34,609223],{"className":609224},[31690,41488],[34,609226],{"className":609227,"style":31846},[31845],[34,609229,202],{"className":609230},[31850],[34,609232],{"className":609233,"style":31846},[31845],[34,609235,609237,609240],{"className":609236},[31676],[34,609238],{"className":609239,"style":32617},[31680],[34,609241,15932],{"className":609242},[31685],[34,609244,609246,609286],{"className":609245},[31642],[34,609247,609249],{"className":609248},[31646],[31648,609250,609251],{"xmlns":31650},[31652,609252,609253,609283],{},[31655,609254,609255,609257,609259,609275,609277],{},[31658,609256,38759],{},[31662,609258,202],{},[41453,609260,609261,609273],{},[31655,609262,609263,609265,609267,609269,609271],{},[31749,609264,640],{},[31658,609266,38080],{"mathvariant":38079},[31749,609268,768],{},[31662,609270,223],{},[31749,609272,640],{},[31749,609274,768],{},[31662,609276,202],{},[41453,609278,609279,609281],{},[31749,609280,786],{},[31749,609282,15932],{},[31665,609284,609285],{"encoding":31667},"q = \\frac{1\u002F2+1}{2} = \\frac{3}{4}",[34,609287,609289,609307,609396],{"className":609288,"ariaHidden":758},[31672],[34,609290,609292,609295,609298,609301,609304],{"className":609291},[31676],[34,609293],{"className":609294,"style":38774},[31680],[34,609296,38759],{"className":609297,"style":32062},[31685,31686],[34,609299],{"className":609300,"style":31846},[31845],[34,609302,202],{"className":609303},[31850],[34,609305],{"className":609306,"style":31846},[31845],[34,609308,609310,609313,609387,609390,609393],{"className":609309},[31676],[34,609311],{"className":609312,"style":70485},[31680],[34,609314,609316,609319,609384],{"className":609315},[31685],[34,609317],{"className":609318},[31782,41488],[34,609320,609322],{"className":609321},[41453],[34,609323,609325,609376],{"className":609324},[31796,31797],[34,609326,609328,609373],{"className":609327},[31801],[34,609329,609331,609345,609353],{"className":609330,"style":70504},[31805],[34,609332,609333,609336],{"style":41504},[34,609334],{"className":609335,"style":41508},[31813],[34,609337,609339],{"className":609338},[31818,31819,31820,31821],[34,609340,609342],{"className":609341},[31685,31821],[34,609343,768],{"className":609344},[31685,31821],[34,609346,609347,609350],{"style":41520},[34,609348],{"className":609349,"style":41508},[31813],[34,609351],{"className":609352,"style":41528},[41527],[34,609354,609355,609358],{"style":70529},[34,609356],{"className":609357,"style":41508},[31813],[34,609359,609361],{"className":609360},[31818,31819,31820,31821],[34,609362,609364,609367,609370],{"className":609363},[31685,31821],[34,609365,42407],{"className":609366},[31685,31821],[34,609368,223],{"className":609369},[32024,31821],[34,609371,640],{"className":609372},[31685,31821],[34,609374,31829],{"className":609375},[31828],[34,609377,609379],{"className":609378},[31801],[34,609380,609382],{"className":609381,"style":41619},[31805],[34,609383],{},[34,609385],{"className":609386},[31690,41488],[34,609388],{"className":609389,"style":31846},[31845],[34,609391,202],{"className":609392},[31850],[34,609394],{"className":609395,"style":31846},[31845],[34,609397,609399,609402],{"className":609398},[31676],[34,609400],{"className":609401,"style":46727},[31680],[34,609403,609405,609408,609467],{"className":609404},[31685],[34,609406],{"className":609407},[31782,41488],[34,609409,609411],{"className":609410},[41453],[34,609412,609414,609459],{"className":609413},[31796,31797],[34,609415,609417,609456],{"className":609416},[31801],[34,609418,609420,609434,609442],{"className":609419,"style":46746},[31805],[34,609421,609422,609425],{"style":41504},[34,609423],{"className":609424,"style":41508},[31813],[34,609426,609428],{"className":609427},[31818,31819,31820,31821],[34,609429,609431],{"className":609430},[31685,31821],[34,609432,15932],{"className":609433},[31685,31821],[34,609435,609436,609439],{"style":41520},[34,609437],{"className":609438,"style":41508},[31813],[34,609440],{"className":609441,"style":41528},[41527],[34,609443,609444,609447],{"style":46771},[34,609445],{"className":609446,"style":41508},[31813],[34,609448,609450],{"className":609449},[31818,31819,31820,31821],[34,609451,609453],{"className":609452},[31685,31821],[34,609454,786],{"className":609455},[31685,31821],[34,609457,31829],{"className":609458},[31828],[34,609460,609462],{"className":609461},[31801],[34,609463,609465],{"className":609464,"style":41619},[31805],[34,609466],{},[34,609468],{"className":609469},[31690,41488],[108,609471,592630,609472,347],{},[34,609473,609475,609553],{"className":609474},[31642],[34,609476,609478],{"className":609477},[31646],[31648,609479,609480],{"xmlns":31650},[31652,609481,609482,609550],{},[31655,609483,609484,609486,609492,609494,609496,609498,609500,609506,609508,609510,609516],{},[31662,609485,202],{},[41453,609487,609488,609490],{},[31749,609489,640],{},[31749,609491,768],{},[31658,609493,54224],{},[31662,609495,456],{"stretchy":702},[31749,609497,15932],{},[31662,609499,31960],{"separator":758},[41453,609501,609502,609504],{},[31749,609503,786],{},[31749,609505,15932],{},[31662,609507,2557],{"stretchy":702},[31662,609509,202],{},[41453,609511,609512,609514],{},[31749,609513,640],{},[31749,609515,768],{},[41453,609517,609518,609540],{},[31655,609519,609520,609522,609524,609526,609528,609530,609532,609534,609536,609538],{},[31658,609521,564791],{"mathvariant":38079},[31662,609523,456],{"stretchy":702},[31749,609525,15932],{},[31662,609527,2557],{"stretchy":702},[31658,609529,564791],{"mathvariant":38079},[31662,609531,456],{"stretchy":702},[31749,609533,786],{},[31658,609535,38080],{"mathvariant":38079},[31749,609537,15932],{},[31662,609539,2557],{"stretchy":702},[31655,609541,609542,609544,609546,609548],{},[31658,609543,564791],{"mathvariant":38079},[31662,609545,456],{"stretchy":702},[31749,609547,272753],{},[31662,609549,2557],{"stretchy":702},[31665,609551,609552],{"encoding":31667},"= \\frac{1}{2} B(4, \\frac{3}{4}) = \\frac{1}{2} \\frac{\\Gamma(4)\\Gamma(3\u002F4)}{\\Gamma(4.75)}",[34,609554,609556,609568,609737],{"className":609555,"ariaHidden":758},[31672],[34,609557,609559,609562,609565],{"className":609558},[31676],[34,609560],{"className":609561,"style":73089},[31680],[34,609563,202],{"className":609564},[31850],[34,609566],{"className":609567,"style":31846},[31845],[34,609569,609571,609574,609642,609645,609648,609651,609654,609657,609725,609728,609731,609734],{"className":609570},[31676],[34,609572],{"className":609573,"style":46727},[31680],[34,609575,609577,609580,609639],{"className":609576},[31685],[34,609578],{"className":609579},[31782,41488],[34,609581,609583],{"className":609582},[41453],[34,609584,609586,609631],{"className":609585},[31796,31797],[34,609587,609589,609628],{"className":609588},[31801],[34,609590,609592,609606,609614],{"className":609591,"style":46746},[31805],[34,609593,609594,609597],{"style":41504},[34,609595],{"className":609596,"style":41508},[31813],[34,609598,609600],{"className":609599},[31818,31819,31820,31821],[34,609601,609603],{"className":609602},[31685,31821],[34,609604,768],{"className":609605},[31685,31821],[34,609607,609608,609611],{"style":41520},[34,609609],{"className":609610,"style":41508},[31813],[34,609612],{"className":609613,"style":41528},[41527],[34,609615,609616,609619],{"style":46771},[34,609617],{"className":609618,"style":41508},[31813],[34,609620,609622],{"className":609621},[31818,31819,31820,31821],[34,609623,609625],{"className":609624},[31685,31821],[34,609626,640],{"className":609627},[31685,31821],[34,609629,31829],{"className":609630},[31828],[34,609632,609634],{"className":609633},[31801],[34,609635,609637],{"className":609636,"style":41619},[31805],[34,609638],{},[34,609640],{"className":609641},[31690,41488],[34,609643,54224],{"className":609644,"style":54252},[31685,31686],[34,609646,456],{"className":609647},[31782],[34,609649,15932],{"className":609650},[31685],[34,609652,31960],{"className":609653},[32119],[34,609655],{"className":609656,"style":32123},[31845],[34,609658,609660,609663,609722],{"className":609659},[31685],[34,609661],{"className":609662},[31782,41488],[34,609664,609666],{"className":609665},[41453],[34,609667,609669,609714],{"className":609668},[31796,31797],[34,609670,609672,609711],{"className":609671},[31801],[34,609673,609675,609689,609697],{"className":609674,"style":46746},[31805],[34,609676,609677,609680],{"style":41504},[34,609678],{"className":609679,"style":41508},[31813],[34,609681,609683],{"className":609682},[31818,31819,31820,31821],[34,609684,609686],{"className":609685},[31685,31821],[34,609687,15932],{"className":609688},[31685,31821],[34,609690,609691,609694],{"style":41520},[34,609692],{"className":609693,"style":41508},[31813],[34,609695],{"className":609696,"style":41528},[41527],[34,609698,609699,609702],{"style":46771},[34,609700],{"className":609701,"style":41508},[31813],[34,609703,609705],{"className":609704},[31818,31819,31820,31821],[34,609706,609708],{"className":609707},[31685,31821],[34,609709,786],{"className":609710},[31685,31821],[34,609712,31829],{"className":609713},[31828],[34,609715,609717],{"className":609716},[31801],[34,609718,609720],{"className":609719,"style":41619},[31805],[34,609721],{},[34,609723],{"className":609724},[31690,41488],[34,609726,2557],{"className":609727},[31690],[34,609729],{"className":609730,"style":31846},[31845],[34,609732,202],{"className":609733},[31850],[34,609735],{"className":609736,"style":31846},[31845],[34,609738,609740,609743,609811],{"className":609739},[31676],[34,609741],{"className":609742,"style":327841},[31680],[34,609744,609746,609749,609808],{"className":609745},[31685],[34,609747],{"className":609748},[31782,41488],[34,609750,609752],{"className":609751},[41453],[34,609753,609755,609800],{"className":609754},[31796,31797],[34,609756,609758,609797],{"className":609757},[31801],[34,609759,609761,609775,609783],{"className":609760,"style":46746},[31805],[34,609762,609763,609766],{"style":41504},[34,609764],{"className":609765,"style":41508},[31813],[34,609767,609769],{"className":609768},[31818,31819,31820,31821],[34,609770,609772],{"className":609771},[31685,31821],[34,609773,768],{"className":609774},[31685,31821],[34,609776,609777,609780],{"style":41520},[34,609778],{"className":609779,"style":41508},[31813],[34,609781],{"className":609782,"style":41528},[41527],[34,609784,609785,609788],{"style":46771},[34,609786],{"className":609787,"style":41508},[31813],[34,609789,609791],{"className":609790},[31818,31819,31820,31821],[34,609792,609794],{"className":609793},[31685,31821],[34,609795,640],{"className":609796},[31685,31821],[34,609798,31829],{"className":609799},[31828],[34,609801,609803],{"className":609802},[31801],[34,609804,609806],{"className":609805,"style":41619},[31805],[34,609807],{},[34,609809],{"className":609810},[31690,41488],[34,609812,609814,609817,609906],{"className":609813},[31685],[34,609815],{"className":609816},[31782,41488],[34,609818,609820],{"className":609819},[41453],[34,609821,609823,609898],{"className":609822},[31796,31797],[34,609824,609826,609895],{"className":609825},[31801],[34,609827,609829,609852,609860],{"className":609828,"style":70504},[31805],[34,609830,609831,609834],{"style":41504},[34,609832],{"className":609833,"style":41508},[31813],[34,609835,609837],{"className":609836},[31818,31819,31820,31821],[34,609838,609840,609843,609846,609849],{"className":609839},[31685,31821],[34,609841,564791],{"className":609842},[31685,31821],[34,609844,456],{"className":609845},[31782,31821],[34,609847,272753],{"className":609848},[31685,31821],[34,609850,2557],{"className":609851},[31690,31821],[34,609853,609854,609857],{"style":41520},[34,609855],{"className":609856,"style":41508},[31813],[34,609858],{"className":609859,"style":41528},[41527],[34,609861,609862,609865],{"style":70529},[34,609863],{"className":609864,"style":41508},[31813],[34,609866,609868],{"className":609867},[31818,31819,31820,31821],[34,609869,609871,609874,609877,609880,609883,609886,609889,609892],{"className":609870},[31685,31821],[34,609872,564791],{"className":609873},[31685,31821],[34,609875,456],{"className":609876},[31782,31821],[34,609878,15932],{"className":609879},[31685,31821],[34,609881,2557],{"className":609882},[31690,31821],[34,609884,564791],{"className":609885},[31685,31821],[34,609887,456],{"className":609888},[31782,31821],[34,609890,554561],{"className":609891},[31685,31821],[34,609893,2557],{"className":609894},[31690,31821],[34,609896,31829],{"className":609897},[31828],[34,609899,609901],{"className":609900},[31801],[34,609902,609904],{"className":609903,"style":324858},[31805],[34,609905],{},[34,609907],{"className":609908},[31690,41488],[108,609910,609911,609912,610090,610091,347],{},"利用递推公式 ",[34,609913,609915,609967],{"className":609914},[31642],[34,609916,609918],{"className":609917},[31646],[31648,609919,609920],{"xmlns":31650},[31652,609921,609922,609964],{},[31655,609923,609924,609926,609928,609930,609932,609934,609936,609938,609940,609942,609944,609946,609948,609950,609952,609954,609956,609958,609960,609962],{},[31658,609925,564791],{"mathvariant":38079},[31662,609927,456],{"stretchy":702},[31658,609929,31660],{},[31662,609931,223],{},[31658,609933,31747],{},[31662,609935,2557],{"stretchy":702},[31662,609937,202],{},[31662,609939,456],{"stretchy":702},[31658,609941,31660],{},[31662,609943,223],{},[31658,609945,31747],{},[31662,609947,32226],{},[31749,609949,640],{},[31662,609951,2557],{"stretchy":702},[31662,609953,33934],{},[31658,609955,31747],{},[31658,609957,564791],{"mathvariant":38079},[31662,609959,456],{"stretchy":702},[31658,609961,31747],{},[31662,609963,2557],{"stretchy":702},[31665,609965,609966],{"encoding":31667},"\\Gamma(n+s) = (n+s-1)\\dots s \\Gamma(s)",[34,609968,609970,609994,610015,610036,610054],{"className":609969,"ariaHidden":758},[31672],[34,609971,609973,609976,609979,609982,609985,609988,609991],{"className":609972},[31676],[34,609974],{"className":609975,"style":31774},[31680],[34,609977,564791],{"className":609978},[31685],[34,609980,456],{"className":609981},[31782],[34,609983,31660],{"className":609984},[31685,31686],[34,609986],{"className":609987,"style":32604},[31845],[34,609989,223],{"className":609990},[32024],[34,609992],{"className":609993,"style":32604},[31845],[34,609995,609997,610000,610003,610006,610009,610012],{"className":609996},[31676],[34,609998],{"className":609999,"style":31774},[31680],[34,610001,31747],{"className":610002},[31685,31686],[34,610004,2557],{"className":610005},[31690],[34,610007],{"className":610008,"style":31846},[31845],[34,610010,202],{"className":610011},[31850],[34,610013],{"className":610014,"style":31846},[31845],[34,610016,610018,610021,610024,610027,610030,610033],{"className":610017},[31676],[34,610019],{"className":610020,"style":31774},[31680],[34,610022,456],{"className":610023},[31782],[34,610025,31660],{"className":610026},[31685,31686],[34,610028],{"className":610029,"style":32604},[31845],[34,610031,223],{"className":610032},[32024],[34,610034],{"className":610035,"style":32604},[31845],[34,610037,610039,610042,610045,610048,610051],{"className":610038},[31676],[34,610040],{"className":610041,"style":32597},[31680],[34,610043,31747],{"className":610044},[31685,31686],[34,610046],{"className":610047,"style":32604},[31845],[34,610049,32226],{"className":610050},[32024],[34,610052],{"className":610053,"style":32604},[31845],[34,610055,610057,610060,610063,610066,610069,610072,610075,610078,610081,610084,610087],{"className":610056},[31676],[34,610058],{"className":610059,"style":31774},[31680],[34,610061,640],{"className":610062},[31685],[34,610064,2557],{"className":610065},[31690],[34,610067],{"className":610068,"style":32123},[31845],[34,610070,33934],{"className":610071},[34006],[34,610073],{"className":610074,"style":32123},[31845],[34,610076,31747],{"className":610077},[31685,31686],[34,610079,564791],{"className":610080},[31685],[34,610082,456],{"className":610083},[31782],[34,610085,31747],{"className":610086},[31685,31686],[34,610088,2557],{"className":610089},[31690]," 化简分母：\n",[34,610092,610094,610162],{"className":610093},[31642],[34,610095,610097],{"className":610096},[31646],[31648,610098,610099],{"xmlns":31650},[31652,610100,610101,610159],{},[31655,610102,610103,610105,610107,610109,610111,610113,610115,610117,610123,610125,610131,610133,610139,610141,610147,610149,610151,610153,610155,610157],{},[31658,610104,564791],{"mathvariant":38079},[31662,610106,456],{"stretchy":702},[31749,610108,162590],{},[31658,610110,38080],{"mathvariant":38079},[31749,610112,15932],{},[31662,610114,2557],{"stretchy":702},[31662,610116,202],{},[41453,610118,610119,610121],{},[31749,610120,95396],{},[31749,610122,15932],{},[31662,610124,71191],{},[41453,610126,610127,610129],{},[31749,610128,39620],{},[31749,610130,15932],{},[31662,610132,71191],{},[41453,610134,610135,610137],{},[31749,610136,36723],{},[31749,610138,15932],{},[31662,610140,71191],{},[41453,610142,610143,610145],{},[31749,610144,786],{},[31749,610146,15932],{},[31658,610148,564791],{"mathvariant":38079},[31662,610150,456],{"stretchy":702},[31749,610152,786],{},[31658,610154,38080],{"mathvariant":38079},[31749,610156,15932],{},[31662,610158,2557],{"stretchy":702},[31665,610160,610161],{"encoding":31667},"\\Gamma(19\u002F4) = \\frac{15}{4} \\cdot \\frac{11}{4} \\cdot \\frac{7}{4} \\cdot \\frac{3}{4} \\Gamma(3\u002F4)",[34,610163,610165,610193,610276,610359,610442],{"className":610164,"ariaHidden":758},[31672],[34,610166,610168,610171,610174,610177,610181,610184,610187,610190],{"className":610167},[31676],[34,610169],{"className":610170,"style":31774},[31680],[34,610172,564791],{"className":610173},[31685],[34,610175,456],{"className":610176},[31782],[34,610178,610180],{"className":610179},[31685],"19\u002F4",[34,610182,2557],{"className":610183},[31690],[34,610185],{"className":610186,"style":31846},[31845],[34,610188,202],{"className":610189},[31850],[34,610191],{"className":610192,"style":31846},[31845],[34,610194,610196,610199,610267,610270,610273],{"className":610195},[31676],[34,610197],{"className":610198,"style":46727},[31680],[34,610200,610202,610205,610264],{"className":610201},[31685],[34,610203],{"className":610204},[31782,41488],[34,610206,610208],{"className":610207},[41453],[34,610209,610211,610256],{"className":610210},[31796,31797],[34,610212,610214,610253],{"className":610213},[31801],[34,610215,610217,610231,610239],{"className":610216,"style":46746},[31805],[34,610218,610219,610222],{"style":41504},[34,610220],{"className":610221,"style":41508},[31813],[34,610223,610225],{"className":610224},[31818,31819,31820,31821],[34,610226,610228],{"className":610227},[31685,31821],[34,610229,15932],{"className":610230},[31685,31821],[34,610232,610233,610236],{"style":41520},[34,610234],{"className":610235,"style":41508},[31813],[34,610237],{"className":610238,"style":41528},[41527],[34,610240,610241,610244],{"style":46771},[34,610242],{"className":610243,"style":41508},[31813],[34,610245,610247],{"className":610246},[31818,31819,31820,31821],[34,610248,610250],{"className":610249},[31685,31821],[34,610251,95396],{"className":610252},[31685,31821],[34,610254,31829],{"className":610255},[31828],[34,610257,610259],{"className":610258},[31801],[34,610260,610262],{"className":610261,"style":41619},[31805],[34,610263],{},[34,610265],{"className":610266},[31690,41488],[34,610268],{"className":610269,"style":32604},[31845],[34,610271,71191],{"className":610272},[32024],[34,610274],{"className":610275,"style":32604},[31845],[34,610277,610279,610282,610350,610353,610356],{"className":610278},[31676],[34,610280],{"className":610281,"style":46727},[31680],[34,610283,610285,610288,610347],{"className":610284},[31685],[34,610286],{"className":610287},[31782,41488],[34,610289,610291],{"className":610290},[41453],[34,610292,610294,610339],{"className":610293},[31796,31797],[34,610295,610297,610336],{"className":610296},[31801],[34,610298,610300,610314,610322],{"className":610299,"style":46746},[31805],[34,610301,610302,610305],{"style":41504},[34,610303],{"className":610304,"style":41508},[31813],[34,610306,610308],{"className":610307},[31818,31819,31820,31821],[34,610309,610311],{"className":610310},[31685,31821],[34,610312,15932],{"className":610313},[31685,31821],[34,610315,610316,610319],{"style":41520},[34,610317],{"className":610318,"style":41508},[31813],[34,610320],{"className":610321,"style":41528},[41527],[34,610323,610324,610327],{"style":46771},[34,610325],{"className":610326,"style":41508},[31813],[34,610328,610330],{"className":610329},[31818,31819,31820,31821],[34,610331,610333],{"className":610332},[31685,31821],[34,610334,39620],{"className":610335},[31685,31821],[34,610337,31829],{"className":610338},[31828],[34,610340,610342],{"className":610341},[31801],[34,610343,610345],{"className":610344,"style":41619},[31805],[34,610346],{},[34,610348],{"className":610349},[31690,41488],[34,610351],{"className":610352,"style":32604},[31845],[34,610354,71191],{"className":610355},[32024],[34,610357],{"className":610358,"style":32604},[31845],[34,610360,610362,610365,610433,610436,610439],{"className":610361},[31676],[34,610363],{"className":610364,"style":46727},[31680],[34,610366,610368,610371,610430],{"className":610367},[31685],[34,610369],{"className":610370},[31782,41488],[34,610372,610374],{"className":610373},[41453],[34,610375,610377,610422],{"className":610376},[31796,31797],[34,610378,610380,610419],{"className":610379},[31801],[34,610381,610383,610397,610405],{"className":610382,"style":46746},[31805],[34,610384,610385,610388],{"style":41504},[34,610386],{"className":610387,"style":41508},[31813],[34,610389,610391],{"className":610390},[31818,31819,31820,31821],[34,610392,610394],{"className":610393},[31685,31821],[34,610395,15932],{"className":610396},[31685,31821],[34,610398,610399,610402],{"style":41520},[34,610400],{"className":610401,"style":41508},[31813],[34,610403],{"className":610404,"style":41528},[41527],[34,610406,610407,610410],{"style":46771},[34,610408],{"className":610409,"style":41508},[31813],[34,610411,610413],{"className":610412},[31818,31819,31820,31821],[34,610414,610416],{"className":610415},[31685,31821],[34,610417,36723],{"className":610418},[31685,31821],[34,610420,31829],{"className":610421},[31828],[34,610423,610425],{"className":610424},[31801],[34,610426,610428],{"className":610427,"style":41619},[31805],[34,610429],{},[34,610431],{"className":610432},[31690,41488],[34,610434],{"className":610435,"style":32604},[31845],[34,610437,71191],{"className":610438},[32024],[34,610440],{"className":610441,"style":32604},[31845],[34,610443,610445,610448,610516,610519,610522,610525],{"className":610444},[31676],[34,610446],{"className":610447,"style":46727},[31680],[34,610449,610451,610454,610513],{"className":610450},[31685],[34,610452],{"className":610453},[31782,41488],[34,610455,610457],{"className":610456},[41453],[34,610458,610460,610505],{"className":610459},[31796,31797],[34,610461,610463,610502],{"className":610462},[31801],[34,610464,610466,610480,610488],{"className":610465,"style":46746},[31805],[34,610467,610468,610471],{"style":41504},[34,610469],{"className":610470,"style":41508},[31813],[34,610472,610474],{"className":610473},[31818,31819,31820,31821],[34,610475,610477],{"className":610476},[31685,31821],[34,610478,15932],{"className":610479},[31685,31821],[34,610481,610482,610485],{"style":41520},[34,610483],{"className":610484,"style":41508},[31813],[34,610486],{"className":610487,"style":41528},[41527],[34,610489,610490,610493],{"style":46771},[34,610491],{"className":610492,"style":41508},[31813],[34,610494,610496],{"className":610495},[31818,31819,31820,31821],[34,610497,610499],{"className":610498},[31685,31821],[34,610500,786],{"className":610501},[31685,31821],[34,610503,31829],{"className":610504},[31828],[34,610506,610508],{"className":610507},[31801],[34,610509,610511],{"className":610510,"style":41619},[31805],[34,610512],{},[34,610514],{"className":610515},[31690,41488],[34,610517,564791],{"className":610518},[31685],[34,610520,456],{"className":610521},[31782],[34,610523,554561],{"className":610524},[31685],[34,610526,2557],{"className":610527},[31690],[108,610529,610530,610531,347],{},"分子 ",[34,610532,610534,610564],{"className":610533},[31642],[34,610535,610537],{"className":610536},[31646],[31648,610538,610539],{"xmlns":31650},[31652,610540,610541,610561],{},[31655,610542,610543,610545,610547,610549,610551,610553,610555,610557,610559],{},[31658,610544,564791],{"mathvariant":38079},[31662,610546,456],{"stretchy":702},[31749,610548,15932],{},[31662,610550,2557],{"stretchy":702},[31662,610552,202],{},[31749,610554,786],{},[31662,610556,5504],{"stretchy":702},[31662,610558,202],{},[31749,610560,36656],{},[31665,610562,610563],{"encoding":31667},"\\Gamma(4) = 3! = 6",[34,610565,610567,610594,610615],{"className":610566,"ariaHidden":758},[31672],[34,610568,610570,610573,610576,610579,610582,610585,610588,610591],{"className":610569},[31676],[34,610571],{"className":610572,"style":31774},[31680],[34,610574,564791],{"className":610575},[31685],[34,610577,456],{"className":610578},[31782],[34,610580,15932],{"className":610581},[31685],[34,610583,2557],{"className":610584},[31690],[34,610586],{"className":610587,"style":31846},[31845],[34,610589,202],{"className":610590},[31850],[34,610592],{"className":610593,"style":31846},[31845],[34,610595,610597,610600,610603,610606,610609,610612],{"className":610596},[31676],[34,610598],{"className":610599,"style":31681},[31680],[34,610601,786],{"className":610602},[31685],[34,610604,5504],{"className":610605},[31690],[34,610607],{"className":610608,"style":31846},[31845],[34,610610,202],{"className":610611},[31850],[34,610613],{"className":610614,"style":31846},[31845],[34,610616,610618,610621],{"className":610617},[31676],[34,610619],{"className":610620,"style":32617},[31680],[34,610622,36656],{"className":610623},[31685],[108,610625,168124,610626,347],{},[34,610627,610629,610731],{"className":610628},[31642],[34,610630,610632],{"className":610631},[31646],[31648,610633,610634],{"xmlns":31650},[31652,610635,610636,610728],{},[31655,610637,610638,610644,610646,610704,610706,610719,610721],{},[41453,610639,610640,610642],{},[31749,610641,640],{},[31749,610643,768],{},[31662,610645,71191],{},[41453,610647,610648,610666],{},[31655,610649,610650,610652,610654,610656,610658,610660,610662,610664],{},[31749,610651,36656],{},[31662,610653,71191],{},[31658,610655,564791],{"mathvariant":38079},[31662,610657,456],{"stretchy":702},[31749,610659,786],{},[31658,610661,38080],{"mathvariant":38079},[31749,610663,15932],{},[31662,610665,2557],{"stretchy":702},[31655,610667,610668,610692,610694,610696,610698,610700,610702],{},[41453,610669,610670,610686],{},[31655,610671,610672,610674,610676,610678,610680,610682,610684],{},[31749,610673,95396],{},[31662,610675,71191],{},[31749,610677,39620],{},[31662,610679,71191],{},[31749,610681,36723],{},[31662,610683,71191],{},[31749,610685,786],{},[37624,610687,610688,610690],{},[31749,610689,15932],{},[31749,610691,15932],{},[31658,610693,564791],{"mathvariant":38079},[31662,610695,456],{"stretchy":702},[31749,610697,786],{},[31658,610699,38080],{"mathvariant":38079},[31749,610701,15932],{},[31662,610703,2557],{"stretchy":702},[31662,610705,202],{},[41453,610707,610708,610716],{},[31655,610709,610710,610712,610714],{},[31749,610711,786],{},[31662,610713,71191],{},[31749,610715,95418],{},[31749,610717,610718],{},"3465",[31662,610720,202],{},[41453,610722,610723,610725],{},[31749,610724,95418],{},[31749,610726,610727],{},"1155",[31665,610729,610730],{"encoding":31667},"\\frac{1}{2} \\cdot \\frac{6 \\cdot \\Gamma(3\u002F4)}{\\frac{15 \\cdot 11 \\cdot 7 \\cdot 3}{4^4} \\Gamma(3\u002F4)} = \\frac{3 \\cdot 256}{3465} = \\frac{256}{1155}",[34,610732,610734,610817,611040,611129],{"className":610733,"ariaHidden":758},[31672],[34,610735,610737,610740,610808,610811,610814],{"className":610736},[31676],[34,610738],{"className":610739,"style":46727},[31680],[34,610741,610743,610746,610805],{"className":610742},[31685],[34,610744],{"className":610745},[31782,41488],[34,610747,610749],{"className":610748},[41453],[34,610750,610752,610797],{"className":610751},[31796,31797],[34,610753,610755,610794],{"className":610754},[31801],[34,610756,610758,610772,610780],{"className":610757,"style":46746},[31805],[34,610759,610760,610763],{"style":41504},[34,610761],{"className":610762,"style":41508},[31813],[34,610764,610766],{"className":610765},[31818,31819,31820,31821],[34,610767,610769],{"className":610768},[31685,31821],[34,610770,768],{"className":610771},[31685,31821],[34,610773,610774,610777],{"style":41520},[34,610775],{"className":610776,"style":41508},[31813],[34,610778],{"className":610779,"style":41528},[41527],[34,610781,610782,610785],{"style":46771},[34,610783],{"className":610784,"style":41508},[31813],[34,610786,610788],{"className":610787},[31818,31819,31820,31821],[34,610789,610791],{"className":610790},[31685,31821],[34,610792,640],{"className":610793},[31685,31821],[34,610795,31829],{"className":610796},[31828],[34,610798,610800],{"className":610799},[31801],[34,610801,610803],{"className":610802,"style":41619},[31805],[34,610804],{},[34,610806],{"className":610807},[31690,41488],[34,610809],{"className":610810,"style":32604},[31845],[34,610812,71191],{"className":610813},[32024],[34,610815],{"className":610816,"style":32604},[31845],[34,610818,610820,610824,611031,611034,611037],{"className":610819},[31676],[34,610821],{"className":610822,"style":610823},[31680],"height:1.7567em;vertical-align:-0.7467em;",[34,610825,610827,610830,611028],{"className":610826},[31685],[34,610828],{"className":610829},[31782,41488],[34,610831,610833],{"className":610832},[41453],[34,610834,610836,611019],{"className":610835},[31796,31797],[34,610837,610839,611016],{"className":610838},[31801],[34,610840,610842,610979,610987],{"className":610841,"style":70504},[31805],[34,610843,610845,610848],{"style":610844},"top:-2.599em;",[34,610846],{"className":610847,"style":41508},[31813],[34,610849,610851],{"className":610850},[31818,31819,31820,31821],[34,610852,610854,610967,610970,610973,610976],{"className":610853},[31685,31821],[34,610855,610857,610860,610964],{"className":610856},[31685,31821],[34,610858],{"className":610859},[31782,41488,31818,43410,435704],[34,610861,610863],{"className":610862},[41453],[34,610864,610866,610955],{"className":610865},[31796,31797],[34,610867,610869,610952],{"className":610868},[31801],[34,610870,610872,610912,610920],{"className":610871,"style":593176},[31805],[34,610873,610875,610878],{"style":610874},"top:-2.5062em;",[34,610876],{"className":610877,"style":41508},[31813],[34,610879,610881],{"className":610880},[31818,43410,43411,31821],[34,610882,610884],{"className":610883},[31685,31821],[34,610885,610887,610890],{"className":610886},[31685,31821],[34,610888,15932],{"className":610889},[31685,31821],[34,610891,610893],{"className":610892},[31792],[34,610894,610896],{"className":610895},[31796],[34,610897,610899],{"className":610898},[31801],[34,610900,610903],{"className":610901,"style":610902},[31805],"height:0.9384em;",[34,610904,610906,610909],{"style":610905},"top:-2.9384em;margin-right:0.1em;",[34,610907],{"className":610908,"style":499953},[31813],[34,610910,15932],{"className":610911},[31685,31821],[34,610913,610914,610917],{"style":435735},[34,610915],{"className":610916,"style":41508},[31813],[34,610918],{"className":610919,"style":435742},[41527,31821],[34,610921,610922,610925],{"style":589620},[34,610923],{"className":610924,"style":41508},[31813],[34,610926,610928],{"className":610927},[31818,43410,43411,31821],[34,610929,610931,610934,610937,610940,610943,610946,610949],{"className":610930},[31685,31821],[34,610932,95396],{"className":610933},[31685,31821],[34,610935,71191],{"className":610936},[32024,31821],[34,610938,39620],{"className":610939},[31685,31821],[34,610941,71191],{"className":610942},[32024,31821],[34,610944,36723],{"className":610945},[31685,31821],[34,610947,71191],{"className":610948},[32024,31821],[34,610950,786],{"className":610951},[31685,31821],[34,610953,31829],{"className":610954},[31828],[34,610956,610958],{"className":610957},[31801],[34,610959,610962],{"className":610960,"style":610961},[31805],"height:0.4938em;",[34,610963],{},[34,610965],{"className":610966},[31690,41488,31818,43410,435704],[34,610968,564791],{"className":610969},[31685,31821],[34,610971,456],{"className":610972},[31782,31821],[34,610974,554561],{"className":610975},[31685,31821],[34,610977,2557],{"className":610978},[31690,31821],[34,610980,610981,610984],{"style":41520},[34,610982],{"className":610983,"style":41508},[31813],[34,610985],{"className":610986,"style":41528},[41527],[34,610988,610989,610992],{"style":70529},[34,610990],{"className":610991,"style":41508},[31813],[34,610993,610995],{"className":610994},[31818,31819,31820,31821],[34,610996,610998,611001,611004,611007,611010,611013],{"className":610997},[31685,31821],[34,610999,36656],{"className":611000},[31685,31821],[34,611002,71191],{"className":611003},[32024,31821],[34,611005,564791],{"className":611006},[31685,31821],[34,611008,456],{"className":611009},[31782,31821],[34,611011,554561],{"className":611012},[31685,31821],[34,611014,2557],{"className":611015},[31690,31821],[34,611017,31829],{"className":611018},[31828],[34,611020,611022],{"className":611021},[31801],[34,611023,611026],{"className":611024,"style":611025},[31805],"height:0.7467em;",[34,611027],{},[34,611029],{"className":611030},[31690,41488],[34,611032],{"className":611033,"style":31846},[31845],[34,611035,202],{"className":611036},[31850],[34,611038],{"className":611039,"style":31846},[31845],[34,611041,611043,611046,611120,611123,611126],{"className":611042},[31676],[34,611044],{"className":611045,"style":46727},[31680],[34,611047,611049,611052,611117],{"className":611048},[31685],[34,611050],{"className":611051},[31782,41488],[34,611053,611055],{"className":611054},[41453],[34,611056,611058,611109],{"className":611057},[31796,31797],[34,611059,611061,611106],{"className":611060},[31801],[34,611062,611064,611078,611086],{"className":611063,"style":46746},[31805],[34,611065,611066,611069],{"style":41504},[34,611067],{"className":611068,"style":41508},[31813],[34,611070,611072],{"className":611071},[31818,31819,31820,31821],[34,611073,611075],{"className":611074},[31685,31821],[34,611076,610718],{"className":611077},[31685,31821],[34,611079,611080,611083],{"style":41520},[34,611081],{"className":611082,"style":41508},[31813],[34,611084],{"className":611085,"style":41528},[41527],[34,611087,611088,611091],{"style":46771},[34,611089],{"className":611090,"style":41508},[31813],[34,611092,611094],{"className":611093},[31818,31819,31820,31821],[34,611095,611097,611100,611103],{"className":611096},[31685,31821],[34,611098,786],{"className":611099},[31685,31821],[34,611101,71191],{"className":611102},[32024,31821],[34,611104,95418],{"className":611105},[31685,31821],[34,611107,31829],{"className":611108},[31828],[34,611110,611112],{"className":611111},[31801],[34,611113,611115],{"className":611114,"style":41619},[31805],[34,611116],{},[34,611118],{"className":611119},[31690,41488],[34,611121],{"className":611122,"style":31846},[31845],[34,611124,202],{"className":611125},[31850],[34,611127],{"className":611128,"style":31846},[31845],[34,611130,611132,611135],{"className":611131},[31676],[34,611133],{"className":611134,"style":46727},[31680],[34,611136,611138,611141,611200],{"className":611137},[31685],[34,611139],{"className":611140},[31782,41488],[34,611142,611144],{"className":611143},[41453],[34,611145,611147,611192],{"className":611146},[31796,31797],[34,611148,611150,611189],{"className":611149},[31801],[34,611151,611153,611167,611175],{"className":611152,"style":46746},[31805],[34,611154,611155,611158],{"style":41504},[34,611156],{"className":611157,"style":41508},[31813],[34,611159,611161],{"className":611160},[31818,31819,31820,31821],[34,611162,611164],{"className":611163},[31685,31821],[34,611165,610727],{"className":611166},[31685,31821],[34,611168,611169,611172],{"style":41520},[34,611170],{"className":611171,"style":41508},[31813],[34,611173],{"className":611174,"style":41528},[41527],[34,611176,611177,611180],{"style":46771},[34,611178],{"className":611179,"style":41508},[31813],[34,611181,611183],{"className":611182},[31818,31819,31820,31821],[34,611184,611186],{"className":611185},[31685,31821],[34,611187,95418],{"className":611188},[31685,31821],[34,611190,31829],{"className":611191},[31828],[34,611193,611195],{"className":611194},[31801],[34,611196,611198],{"className":611197,"style":41619},[31805],[34,611199],{},[34,611201],{"className":611202},[31690,41488],[108,611204,611205,2664,611207],{},[2627,611206,118575],{},[34,611208,611210,611228],{"className":611209},[31642],[34,611211,611213],{"className":611212},[31646],[31648,611214,611215],{"xmlns":31650},[31652,611216,611217,611225],{},[31655,611218,611219],{},[41453,611220,611221,611223],{},[31749,611222,95418],{},[31749,611224,610727],{},[31665,611226,611227],{"encoding":31667},"\\frac{256}{1155}",[34,611229,611231],{"className":611230,"ariaHidden":758},[31672],[34,611232,611234,611237],{"className":611233},[31676],[34,611235],{"className":611236,"style":46727},[31680],[34,611238,611240,611243,611302],{"className":611239},[31685],[34,611241],{"className":611242},[31782,41488],[34,611244,611246],{"className":611245},[41453],[34,611247,611249,611294],{"className":611248},[31796,31797],[34,611250,611252,611291],{"className":611251},[31801],[34,611253,611255,611269,611277],{"className":611254,"style":46746},[31805],[34,611256,611257,611260],{"style":41504},[34,611258],{"className":611259,"style":41508},[31813],[34,611261,611263],{"className":611262},[31818,31819,31820,31821],[34,611264,611266],{"className":611265},[31685,31821],[34,611267,610727],{"className":611268},[31685,31821],[34,611270,611271,611274],{"style":41520},[34,611272],{"className":611273,"style":41508},[31813],[34,611275],{"className":611276,"style":41528},[41527],[34,611278,611279,611282],{"style":46771},[34,611280],{"className":611281,"style":41508},[31813],[34,611283,611285],{"className":611284},[31818,31819,31820,31821],[34,611286,611288],{"className":611287},[31685,31821],[34,611289,95418],{"className":611290},[31685,31821],[34,611292,31829],{"className":611293},[31828],[34,611295,611297],{"className":611296},[31801],[34,611298,611300],{"className":611299,"style":41619},[31805],[34,611301],{},[34,611303],{"className":611304},[31690,41488],[15,611306,611307,47649,611310],{},[2627,611308,611309],{},"(7)",[34,611311,611313,611381],{"className":611312},[31642],[34,611314,611316],{"className":611315},[31646],[31648,611317,611318],{"xmlns":31650},[31652,611319,611320,611378],{},[31655,611321,611322],{},[71090,611323,611324,611336,611342,611356,611358,611360,611362,611364,611366,611368,611370,611372,611374,611376],{"scriptlevel":459,"displaystyle":758},[38846,611325,611326,611328,611330],{},[31662,611327,546295],{},[31749,611329,459],{},[31655,611331,611332,611334],{},[31662,611333,223],{},[31658,611335,138286],{"mathvariant":38079},[37624,611337,611338,611340],{},[31658,611339,32936],{},[31658,611341,31759],{},[37624,611343,611344,611346],{},[31658,611345,33244],{},[31655,611347,611348,611350],{},[31662,611349,32226],{},[37624,611351,611352,611354],{},[31658,611353,32936],{},[31658,611355,31660],{},[39361,611357,39363],{},[31658,611359,46900],{},[31658,611361,32936],{},[31845,611363],{"width":71082},[31662,611365,456],{"stretchy":702},[31658,611367,31759],{},[31662,611369,31960],{"separator":758},[31658,611371,31660],{},[31662,611373,29622],{},[31749,611375,459],{},[31662,611377,2557],{"stretchy":702},[31665,611379,611380],{"encoding":31667},"\\displaystyle \\int_0^{+\\infty} x^m e^{-x^n} \\, dx \\quad (m, n > 0)",[34,611382,611384,611577],{"className":611383,"ariaHidden":758},[31672],[34,611385,611387,611390,611447,611450,611479,611541,611544,611547,611550,611553,611556,611559,611562,611565,611568,611571,611574],{"className":611386},[31676],[34,611388],{"className":611389,"style":601420},[31680],[34,611391,611393,611396],{"className":611392},[32767],[34,611394,546295],{"className":611395,"style":591475},[32767,131249,131250],[34,611397,611399],{"className":611398},[31792],[34,611400,611402,611439],{"className":611401},[31796,31797],[34,611403,611405,611436],{"className":611404},[31801],[34,611406,611408,611419],{"className":611407,"style":601439},[31805],[34,611409,611410,611413],{"style":591491},[34,611411],{"className":611412,"style":31814},[31813],[34,611414,611416],{"className":611415},[31818,31819,31820,31821],[34,611417,459],{"className":611418},[31685,31821],[34,611420,611421,611424],{"style":591503},[34,611422],{"className":611423,"style":31814},[31813],[34,611425,611427],{"className":611426},[31818,31819,31820,31821],[34,611428,611430,611433],{"className":611429},[31685,31821],[34,611431,223],{"className":611432},[31685,31821],[34,611434,138286],{"className":611435},[31685,31821],[34,611437,31829],{"className":611438},[31828],[34,611440,611442],{"className":611441},[31801],[34,611443,611445],{"className":611444,"style":591522},[31805],[34,611446],{},[34,611448],{"className":611449,"style":32123},[31845],[34,611451,611453,611456],{"className":611452},[31685],[34,611454,32936],{"className":611455},[31685,31686],[34,611457,611459],{"className":611458},[31792],[34,611460,611462],{"className":611461},[31796],[34,611463,611465],{"className":611464},[31801],[34,611466,611468],{"className":611467,"style":145166},[31805],[34,611469,611470,611473],{"style":129846},[34,611471],{"className":611472,"style":31814},[31813],[34,611474,611476],{"className":611475},[31818,31819,31820,31821],[34,611477,31759],{"className":611478},[31685,31686,31821],[34,611480,611482,611485],{"className":611481},[31685],[34,611483,33244],{"className":611484},[31685,31686],[34,611486,611488],{"className":611487},[31792],[34,611489,611491],{"className":611490},[31796],[34,611492,611494],{"className":611493},[31801],[34,611495,611497],{"className":611496,"style":594425},[31805],[34,611498,611499,611502],{"style":129846},[34,611500],{"className":611501,"style":31814},[31813],[34,611503,611505],{"className":611504},[31818,31819,31820,31821],[34,611506,611508,611511],{"className":611507},[31685,31821],[34,611509,32226],{"className":611510},[31685,31821],[34,611512,611514,611517],{"className":611513},[31685,31821],[34,611515,32936],{"className":611516},[31685,31686,31821],[34,611518,611520],{"className":611519},[31792],[34,611521,611523],{"className":611522},[31796],[34,611524,611526],{"className":611525},[31801],[34,611527,611530],{"className":611528,"style":611529},[31805],"height:0.7385em;",[34,611531,611532,611535],{"style":552391},[34,611533],{"className":611534,"style":43406},[31813],[34,611536,611538],{"className":611537},[31818,43410,43411,31821],[34,611539,31660],{"className":611540},[31685,31686,31821],[34,611542],{"className":611543,"style":32123},[31845],[34,611545,46900],{"className":611546},[31685,31686],[34,611548,32936],{"className":611549},[31685,31686],[34,611551],{"className":611552,"style":128438},[31845],[34,611554,456],{"className":611555},[31782],[34,611557,31759],{"className":611558},[31685,31686],[34,611560,31960],{"className":611561},[32119],[34,611563],{"className":611564,"style":32123},[31845],[34,611566,31660],{"className":611567},[31685,31686],[34,611569],{"className":611570,"style":31846},[31845],[34,611572,29622],{"className":611573},[31850],[34,611575],{"className":611576,"style":31846},[31845],[34,611578,611580,611583,611586],{"className":611579},[31676],[34,611581],{"className":611582,"style":31774},[31680],[34,611584,459],{"className":611585},[31685],[34,611587,2557],{"className":611588},[31690],[105,611590,611591,611794,613903],{},[108,611592,611593,2664,611595,611623,611624,88909,611724,347],{},[2627,611594,419963],{},[34,611596,611598,611611],{"className":611597},[31642],[34,611599,611601],{"className":611600},[31646],[31648,611602,611603],{"xmlns":31650},[31652,611604,611605,611609],{},[31655,611606,611607],{},[31658,611608,564791],{"mathvariant":38079},[31665,611610,564794],{"encoding":31667},[34,611612,611614],{"className":611613,"ariaHidden":758},[31672],[34,611615,611617,611620],{"className":611616},[31676],[34,611618],{"className":611619,"style":33218},[31680],[34,611621,564791],{"className":611622},[31685]," 函数的标准凑法。目的是把 ",[34,611625,611627,611653],{"className":611626},[31642],[34,611628,611630],{"className":611629},[31646],[31648,611631,611632],{"xmlns":31650},[31652,611633,611634,611650],{},[31655,611635,611636],{},[37624,611637,611638,611640],{},[31658,611639,33244],{},[31655,611641,611642,611644],{},[31662,611643,32226],{},[37624,611645,611646,611648],{},[31658,611647,32936],{},[31658,611649,31660],{},[31665,611651,611652],{"encoding":31667},"e^{-x^n}",[34,611654,611656],{"className":611655,"ariaHidden":758},[31672],[34,611657,611659,611663],{"className":611658},[31676],[34,611660],{"className":611661,"style":611662},[31680],"height:0.88em;",[34,611664,611666,611669],{"className":611665},[31685],[34,611667,33244],{"className":611668},[31685,31686],[34,611670,611672],{"className":611671},[31792],[34,611673,611675],{"className":611674},[31796],[34,611676,611678],{"className":611677},[31801],[34,611679,611681],{"className":611680,"style":611662},[31805],[34,611682,611683,611686],{"style":37777},[34,611684],{"className":611685,"style":31814},[31813],[34,611687,611689],{"className":611688},[31818,31819,31820,31821],[34,611690,611692,611695],{"className":611691},[31685,31821],[34,611693,32226],{"className":611694},[31685,31821],[34,611696,611698,611701],{"className":611697},[31685,31821],[34,611699,32936],{"className":611700},[31685,31686,31821],[34,611702,611704],{"className":611703},[31792],[34,611705,611707],{"className":611706},[31796],[34,611708,611710],{"className":611709},[31801],[34,611711,611713],{"className":611712,"style":611529},[31805],[34,611714,611715,611718],{"style":552391},[34,611716],{"className":611717,"style":43406},[31813],[34,611719,611721],{"className":611720},[31818,43410,43411,31821],[34,611722,31660],{"className":611723},[31685,31686,31821],[34,611725,611727,611749],{"className":611726},[31642],[34,611728,611730],{"className":611729},[31646],[31648,611731,611732],{"xmlns":31650},[31652,611733,611734,611746],{},[31655,611735,611736],{},[37624,611737,611738,611740],{},[31658,611739,33244],{},[31655,611741,611742,611744],{},[31662,611743,32226],{},[31658,611745,33249],{},[31665,611747,611748],{"encoding":31667},"e^{-t}",[34,611750,611752],{"className":611751,"ariaHidden":758},[31672],[34,611753,611755,611759],{"className":611754},[31676],[34,611756],{"className":611757,"style":611758},[31680],"height:0.7936em;",[34,611760,611762,611765],{"className":611761},[31685],[34,611763,33244],{"className":611764},[31685,31686],[34,611766,611768],{"className":611767},[31792],[34,611769,611771],{"className":611770},[31796],[34,611772,611774],{"className":611773},[31801],[34,611775,611777],{"className":611776,"style":611758},[31805],[34,611778,611779,611782],{"style":37777},[34,611780],{"className":611781,"style":31814},[31813],[34,611783,611785],{"className":611784},[31818,31819,31820,31821],[34,611786,611788,611791],{"className":611787},[31685,31821],[34,611789,32226],{"className":611790},[31685,31821],[34,611792,33249],{"className":611793},[31685,31686,31821],[108,611795,611796,33183,611798],{},[2627,611797,289021],{},[2230,611799,611800,612222,612674,613511],{},[108,611801,396859,611802,46528,611969,347],{},[34,611803,611805,611845],{"className":611804},[31642],[34,611806,611808],{"className":611807},[31646],[31648,611809,611810],{"xmlns":31650},[31652,611811,611812,611842],{},[31655,611813,611814,611816,611818,611824,611826,611828,611830],{},[31658,611815,33249],{},[31662,611817,202],{},[37624,611819,611820,611822],{},[31658,611821,32936],{},[31658,611823,31660],{},[31662,611825,140094],{},[31658,611827,32936],{},[31662,611829,202],{},[37624,611831,611832,611834],{},[31658,611833,33249],{},[31655,611835,611836,611838,611840],{},[31749,611837,640],{},[31658,611839,38080],{"mathvariant":38079},[31658,611841,31660],{},[31665,611843,611844],{"encoding":31667},"t = x^n \\Rightarrow x = t^{1\u002Fn}",[34,611846,611848,611866,611910,611928],{"className":611847,"ariaHidden":758},[31672],[34,611849,611851,611854,611857,611860,611863],{"className":611850},[31676],[34,611852],{"className":611853,"style":33280},[31680],[34,611855,33249],{"className":611856},[31685,31686],[34,611858],{"className":611859,"style":31846},[31845],[34,611861,202],{"className":611862},[31850],[34,611864],{"className":611865,"style":31846},[31845],[34,611867,611869,611872,611901,611904,611907],{"className":611868},[31676],[34,611870],{"className":611871,"style":37959},[31680],[34,611873,611875,611878],{"className":611874},[31685],[34,611876,32936],{"className":611877},[31685,31686],[34,611879,611881],{"className":611880},[31792],[34,611882,611884],{"className":611883},[31796],[34,611885,611887],{"className":611886},[31801],[34,611888,611890],{"className":611889,"style":37959},[31805],[34,611891,611892,611895],{"style":37777},[34,611893],{"className":611894,"style":31814},[31813],[34,611896,611898],{"className":611897},[31818,31819,31820,31821],[34,611899,31660],{"className":611900},[31685,31686,31821],[34,611902],{"className":611903,"style":31846},[31845],[34,611905,140094],{"className":611906},[31850],[34,611908],{"className":611909,"style":31846},[31845],[34,611911,611913,611916,611919,611922,611925],{"className":611912},[31676],[34,611914],{"className":611915,"style":32527},[31680],[34,611917,32936],{"className":611918},[31685,31686],[34,611920],{"className":611921,"style":31846},[31845],[34,611923,202],{"className":611924},[31850],[34,611926],{"className":611927,"style":31846},[31845],[34,611929,611931,611934],{"className":611930},[31676],[34,611932],{"className":611933,"style":61037},[31680],[34,611935,611937,611940],{"className":611936},[31685],[34,611938,33249],{"className":611939},[31685,31686],[34,611941,611943],{"className":611942},[31792],[34,611944,611946],{"className":611945},[31796],[34,611947,611949],{"className":611948},[31801],[34,611950,611952],{"className":611951,"style":61037},[31805],[34,611953,611954,611957],{"style":37777},[34,611955],{"className":611956,"style":31814},[31813],[34,611958,611960],{"className":611959},[31818,31819,31820,31821],[34,611961,611963,611966],{"className":611962},[31685,31821],[34,611964,249219],{"className":611965},[31685,31821],[34,611967,31660],{"className":611968},[31685,31686,31821],[34,611970,611972,612015],{"className":611971},[31642],[34,611973,611975],{"className":611974},[31646],[31648,611976,611977],{"xmlns":31650},[31652,611978,611979,612013],{},[31655,611980,611981,611983,611985,611987,611993,612009,612011],{},[31658,611982,46900],{},[31658,611984,32936],{},[31662,611986,202],{},[41453,611988,611989,611991],{},[31749,611990,640],{},[31658,611992,31660],{},[37624,611994,611995,611997],{},[31658,611996,33249],{},[31655,611998,611999,612005,612007],{},[41453,612000,612001,612003],{},[31749,612002,640],{},[31658,612004,31660],{},[31662,612006,32226],{},[31749,612008,640],{},[31658,612010,46900],{},[31658,612012,33249],{},[31665,612014,598326],{"encoding":31667},[34,612016,612018,612039],{"className":612017,"ariaHidden":758},[31672],[34,612019,612021,612024,612027,612030,612033,612036],{"className":612020},[31676],[34,612022],{"className":612023,"style":31681},[31680],[34,612025,46900],{"className":612026},[31685,31686],[34,612028,32936],{"className":612029},[31685,31686],[34,612031],{"className":612032,"style":31846},[31845],[34,612034,202],{"className":612035},[31850],[34,612037],{"className":612038,"style":31846},[31845],[34,612040,612042,612045,612113,612216,612219],{"className":612041},[31676],[34,612043],{"className":612044,"style":598357},[31680],[34,612046,612048,612051,612110],{"className":612047},[31685],[34,612049],{"className":612050},[31782,41488],[34,612052,612054],{"className":612053},[41453],[34,612055,612057,612102],{"className":612056},[31796,31797],[34,612058,612060,612099],{"className":612059},[31801],[34,612061,612063,612077,612085],{"className":612062,"style":46746},[31805],[34,612064,612065,612068],{"style":41504},[34,612066],{"className":612067,"style":41508},[31813],[34,612069,612071],{"className":612070},[31818,31819,31820,31821],[34,612072,612074],{"className":612073},[31685,31821],[34,612075,31660],{"className":612076},[31685,31686,31821],[34,612078,612079,612082],{"style":41520},[34,612080],{"className":612081,"style":41508},[31813],[34,612083],{"className":612084,"style":41528},[41527],[34,612086,612087,612090],{"style":46771},[34,612088],{"className":612089,"style":41508},[31813],[34,612091,612093],{"className":612092},[31818,31819,31820,31821],[34,612094,612096],{"className":612095},[31685,31821],[34,612097,640],{"className":612098},[31685,31821],[34,612100,31829],{"className":612101},[31828],[34,612103,612105],{"className":612104},[31801],[34,612106,612108],{"className":612107,"style":41619},[31805],[34,612109],{},[34,612111],{"className":612112},[31690,41488],[34,612114,612116,612119],{"className":612115},[31685],[34,612117,33249],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(t^{1\u002Fn})^m e^{-t} \\cdot \\frac{1}{n} t^{\\frac{1}{n}-1} \\, dt",[34,612312,612314,612488],{"className":612313,"ariaHidden":758},[31672],[34,612315,612317,612320,612377,612380,612415,612444,612479,612482,612485],{"className":612316},[31676],[34,612318],{"className":612319,"style":546531},[31680],[34,612321,612323,612326],{"className":612322},[32767],[34,612324,546295],{"className":612325,"style":546331},[32767,131249,156560],[34,612327,612329],{"className":612328},[31792],[34,612330,612332,612369],{"className":612331},[31796,31797],[34,612333,612335,612366],{"className":612334},[31801],[34,612336,612338,612349],{"className":612337,"style":546550},[31805],[34,612339,612340,612343],{"style":546347},[34,612341],{"className":612342,"style":31814},[31813],[34,612344,612346],{"className":612345},[31818,31819,31820,31821],[34,612347,459],{"className":612348},[31685,31821],[34,612350,612351,612354],{"style":546362},[34,612352],{"className":612353,"style":31814},[31813],[34,612355,612357],{"className":612356},[31818,31819,31820,31821],[34,612358,612360,612363],{"className":612359},[31685,31821],[34,612361,223],{"className":612362},[31685,31821],[34,612364,138286],{"className":612365},[31685,31821],[34,612367,31829],{"className":612368},[31828],[34,612370,612372],{"className":612371},[31801],[34,612373,612375],{"className":612374,"style":546384},[31805],[34,612376],{},[34,612378,456],{"className":612379},[31782],[34,612381,612383,612386],{"className":612382},[31685],[34,612384,33249],{"className":612385},[31685,31686],[34,612387,612389],{"className":612388},[31792],[34,612390,612392],{"className":612391},[31796],[34,612393,612395],{"className":612394},[31801],[34,612396,612398],{"className":612397,"style":61037},[31805],[34,612399,612400,612403],{"style":37777},[34,612401],{"className":612402,"style":31814},[31813],[34,612404,612406],{"className":612405},[31818,31819,31820,31821],[34,612407,612409,612412],{"className":612408},[31685,31821],[34,612410,249219],{"className":612411},[31685,31821],[34,612413,31660],{"className":612414},[31685,31686,31821],[34,612416,612418,612421],{"className":612417},[31690],[34,612419,2557],{"className":612420},[31690],[34,612422,612424],{"className":612423},[31792],[34,612425,612427],{"className":612426},[31796],[34,612428,612430],{"className":612429},[31801],[34,612431,612433],{"className":612432,"style":37959},[31805],[34,612434,612435,612438],{"style":37777},[34,612436],{"className":612437,"style":31814},[31813],[34,612439,612441],{"className":612440},[31818,31819,31820,31821],[34,612442,31759],{"className":612443},[31685,31686,31821],[34,612445,612447,612450],{"className":612446},[31685],[34,612448,33244],{"className":612449},[31685,31686],[34,612451,612453],{"className":612452},[31792],[34,612454,612456],{"className":612455},[31796],[34,612457,612459],{"className":612458},[31801],[34,612460,612462],{"className":612461,"style":611758},[31805],[34,612463,612464,612467],{"style":37777},[34,612465],{"className":612466,"style":31814},[31813],[34,612468,612470],{"className":612469},[31818,31819,31820,31821],[34,612471,612473,612476],{"className":612472},[31685,31821],[34,612474,32226],{"className":612475},[31685,31821],[34,612477,33249],{"className":612478},[31685,31686,31821],[34,612480],{"className":612481,"style":32604},[31845],[34,612483,71191],{"className":612484},[32024],[34,612486],{"className":612487,"style":32604},[31845],[34,612489,612491,612494,612562,612665,612668,612671],{"className":612490},[31676],[34,612492],{"className":612493,"style":598357},[31680],[34,612495,612497,612500,612559],{"className":612496},[31685],[34,612498],{"className":612499},[31782,41488],[34,612501,612503],{"className":612502},[41453],[34,612504,612506,612551],{"className":612505},[31796,31797],[34,612507,612509,612548],{"className":612508},[31801],[34,612510,612512,612526,612534],{"className":612511,"style":46746},[31805],[34,612513,612514,612517],{"style":41504},[34,612515],{"className":612516,"style":41508},[31813],[34,612518,612520],{"className":612519},[31818,31819,31820,31821],[34,612521,612523],{"className":612522},[31685,31821],[34,612524,31660],{"className":612525},[31685,31686,31821],[34,612527,612528,612531],{"style":41520},[34,612529],{"className":612530,"style":41508},[31813],[34,612532],{"className":612533,"style":41528},[41527],[34,612535,612536,612539],{"style":46771},[34,612537],{"className":612538,"style":41508},[31813],[34,612540,612542],{"className":612541},[31818,31819,31820,31821],[34,612543,612545],{"className":612544},[31685,31821],[34,612546,640],{"className":612547},[31685,31821],[34,612549,31829],{"className":612550},[31828],[34,612552,612554],{"className":612553},[31801],[34,612555,612557],{"className":612556,"style":41619},[31805],[34,612558],{},[34,612560],{"className":612561},[31690,41488],[34,612563,612565,612568],{"className":612564},[31685],[34,612566,33249],{"className":612567},[31685,31686],[34,612569,612571],{"className":612570},[31792],[34,612572,612574],{"className":612573},[31796],[34,612575,612577],{"className":612576},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\\int_0^{+\\infty} t^{\\frac{m}{n}} \\cdot t^{\\frac{1}{n}-1} e^{-t} \\, dt = \\frac{1}{n} \\int_0^{+\\infty} t^{\\frac{m+1}{n}-1} e^{-t} \\, dt",[34,612813,612815,613059,613221],{"className":612814,"ariaHidden":758},[31672],[34,612816,612818,612821,612889,612892,612949,612952,613050,613053,613056],{"className":612817},[31676],[34,612819],{"className":612820,"style":546531},[31680],[34,612822,612824,612827,612886],{"className":612823},[31685],[34,612825],{"className":612826},[31782,41488],[34,612828,612830],{"className":612829},[41453],[34,612831,612833,612878],{"className":612832},[31796,31797],[34,612834,612836,612875],{"className":612835},[31801],[34,612837,612839,612853,612861],{"className":612838,"style":46746},[31805],[34,612840,612841,612844],{"style":41504},[34,612842],{"className":612843,"style":41508},[31813],[34,612845,612847],{"className":612846},[31818,31819,31820,31821],[34,612848,612850],{"className":612849},[31685,31821],[34,612851,31660],{"className":612852},[31685,31686,31821],[34,612854,612855,612858],{"style":41520},[34,612856],{"className":612857,"style":41508},[31813],[34,612859],{"className":612860,"style":41528},[41527],[34,612862,612863,612866],{"style":46771},[34,612864],{"className":612865,"style":41508},[31813],[34,612867,612869],{"className":612868},[31818,31819,31820,31821],[34,612870,612872],{"className":612871},[31685,31821],[34,612873,640],{"className":612874},[31685,31821],[34,612876,31829],{"className":612877},[31828],[34,612879,612881],{"className":612880},[31801],[34,612882,612884],{"className":612883,"style":41619},[31805],[34,612885],{},[34,612887],{"className":612888},[31690,41488],[34,612890],{"className":612891,"style":32123},[31845],[34,612893,612895,612898],{"className":612894},[32767],[34,612896,546295],{"className":612897,"style":546331},[32767,131249,156560],[34,612899,612901],{"className":612900},[31792],[34,612902,612904,612941],{"className":612903},[31796,31797],[34,612905,612907,612938],{"className":612906},[31801],[34,612908,612910,612921],{"className":612909,"style":546550},[31805],[34,612911,612912,612915],{"style":546347},[34,612913],{"className":612914,"style":31814},[31813],[34,612916,612918],{"className":612917},[31818,31819,31820,31821],[34,612919,459],{"className":612920},[31685,31821],[34,612922,612923,612926],{"style":546362},[34,612924],{"className":612925,"style":31814},[31813],[34,612927,612929],{"className":612928},[31818,31819,31820,31821],[34,612930,612932,612935],{"className":612931},[31685,31821],[34,612933,223],{"className":612934},[31685,31821],[34,612936,138286],{"className":612937},[31685,31821],[34,612939,31829],{"className":612940},[31828],[34,612942,612944],{"className":612943},[31801],[34,612945,612947],{"className":612946,"style":546384},[31805],[34,612948],{},[34,612950],{"className":612951,"style":32123},[31845],[34,612953,612955,612958],{"className":612954},[31685],[34,612956,33249],{"className":612957},[31685,31686],[34,612959,612961],{"className":612960},[31792],[34,612962,612964],{"className":612963},[31796],[34,612965,612967],{"className":612966},[31801],[34,612968,612971],{"className":612969,"style":612970},[31805],"height:0.8471em;",[34,612972,612973,612976],{"style":435688},[34,612974],{"className":612975,"style":41508},[31813],[34,612977,612979],{"className":612978},[31818,31819,31820,31821],[34,612980,612982],{"className":612981},[31685,31821],[34,612983,612985,612988,613047],{"className":612984},[31685,31821],[34,612986],{"className":612987},[31782,41488,31818,43410,435704],[34,612989,612991],{"className":612990},[41453],[34,612992,612994,613039],{"className":612993},[31796,31797],[34,612995,612997,613036],{"className":612996},[31801],[34,612998,613000,613014,613022],{"className":612999,"style":589595},[31805],[34,613001,613002,613005],{"style":435720},[34,613003],{"className":613004,"style":41508},[31813],[34,613006,613008],{"className":613007},[31818,43410,43411,31821],[34,613009,613011],{"className":613010},[31685,31821],[34,613012,31660],{"className":613013},[31685,31686,31821],[34,613015,613016,613019],{"style":435735},[34,613017],{"className":613018,"style":41508},[31813],[34,613020],{"className":613021,"style":435742},[41527,31821],[34,613023,613024,613027],{"style":589620},[34,613025],{"className":613026,"style":41508},[31813],[34,613028,613030],{"className":613029},[31818,43410,43411,31821],[34,613031,613033],{"className":613032},[31685,31821],[34,613034,31759],{"className":613035},[31685,31686,31821],[34,613037,31829],{"className":613038},[31828],[34,613040,613042],{"className":613041},[31801],[34,613043,613045],{"className":613044,"style":435782},[31805],[34,613046],{},[34,613048],{"className":613049},[31690,41488,31818,43410,435704],[34,613051],{"className":613052,"style":32604},[31845],[34,613054,71191],{"className":613055},[32024],[34,613057],{"className":613058,"style":32604},[31845],[34,613060,613062,613065,613168,613203,613206,613209,613212,613215,613218],{"className":613061},[31676],[34,613063],{"className":613064,"style":598444},[31680],[34,613066,613068,613071],{"className":613067},[31685],[34,613069,33249],{"className":613070},[31685,3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定义 ",[34,613543,613545,613601],{"className":613544},[31642],[34,613546,613548],{"className":613547},[31646],[31648,613549,613550],{"xmlns":31650},[31652,613551,613552,613598],{},[31655,613553,613554,613562,613574,613584,613586,613588,613590,613592,613594,613596],{},[38846,613555,613556,613558,613560],{},[31662,613557,546295],{},[31749,613559,459],{},[31658,613561,138286],{"mathvariant":38079},[37624,613563,613564,613566],{},[31658,613565,33249],{},[31655,613567,613568,613570,613572],{},[31658,613569,31747],{},[31662,613571,32226],{},[31749,613573,640],{},[37624,613575,613576,613578],{},[31658,613577,33244],{},[31655,613579,613580,613582],{},[31662,613581,32226],{},[31658,613583,33249],{},[31658,613585,46900],{},[31658,613587,33249],{},[31662,613589,202],{},[31658,613591,564791],{"mathvariant":38079},[31662,613593,456],{"stretchy":702},[31658,613595,31747],{},[31662,613597,2557],{"stretchy":702},[31665,613599,613600],{"encoding":31667},"\\int_0^\\infty t^{s-1} e^{-t} dt = 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= 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",[34,615570,615572,615585],{"className":615571},[31642],[34,615573,615575],{"className":615574},[31646],[31648,615576,615577],{"xmlns":31650},[31652,615578,615579,615583],{},[31655,615580,615581],{},[31658,615582,33249],{},[31665,615584,33249],{"encoding":31667},[34,615586,615588],{"className":615587,"ariaHidden":758},[31672],[34,615589,615591,615594],{"className":615590},[31676],[34,615592],{"className":615593,"style":33280},[31680],[34,615595,33249],{"className":615596},[31685,31686]," 的指数：",[34,615599,615601,615649],{"className":615600},[31642],[34,615602,615604],{"className":615603},[31646],[31648,615605,615606],{"xmlns":31650},[31652,615607,615608,615646],{},[31655,615609,615610,615622,615624,615630,615632,615634,615636,615642,615644],{},[41453,615611,615612,615620],{},[31655,615613,615614,615616,615618],{},[31658,615615,15],{},[31662,615617,32226],{},[31749,615619,640],{},[31658,615621,31660],{},[31662,615623,223],{},[41453,615625,615626,615628],{},[31749,615627,640],{},[31658,615629,31660],{},[31662,615631,32226],{},[31749,615633,640],{},[31662,615635,202],{},[41453,615637,615638,615640],{},[31658,615639,15],{},[31658,615641,31660],{},[31662,615643,32226],{},[31749,615645,640],{},[31665,615647,615648],{"encoding":31667},"\\frac{p-1}{n} + \\frac{1}{n} - 1 = \\frac{p}{n} - 1",[34,615650,615652,615743,615826,615844,615928],{"className":615651,"ariaHidden":758},[31672],[34,615653,615655,615659,615734,615737,615740],{"className":615654},[31676],[34,615656],{"className":615657,"style":615658},[31680],"height:1.2422em;vertical-align:-0.345em;",[34,615660,615662,615665,615731],{"className":615661},[31685],[34,615663],{"className":615664},[31782,41488],[34,615666,615668],{"className":615667},[41453],[34,615669,615671,615723],{"className":615670},[31796,31797],[34,615672,615674,615720],{"className":615673},[31801],[34,615675,615678,615692,615700],{"className":615676,"style":615677},[31805],"height:0.8972em;",[34,615679,615680,615683],{"style":41504},[34,615681],{"className":615682,"style":41508},[31813],[34,615684,615686],{"className":615685},[31818,31819,31820,31821],[34,615687,615689],{"className":615688},[31685,31821],[34,615690,31660],{"className":615691},[31685,31686,31821],[34,615693,615694,615697],{"style":41520},[34,615695],{"className":615696,"style":41508},[31813],[34,615698],{"className":615699,"style":41528},[41527],[34,615701,615702,615705],{"style":548368},[34,615703],{"className":615704,"style":41508},[31813],[34,615706,615708],{"className":615707},[31818,31819,31820,31821],[34,615709,615711,615714,615717],{"className":615710},[31685,31821],[34,615712,15],{"className":615713},[31685,31686,31821],[34,615715,32226],{"className":615716},[32024,31821],[34,615718,640],{"className":615719},[31685,31821],[34,615721,31829],{"className":615722},[31828],[34,615724,615726],{"className":615725},[31801],[34,615727,615729],{"className":615728,"style":41619},[31805],[34,615730],{},[34,615732],{"className":615733},[31690,41488],[34,615735],{"className":615736,"style":32604},[31845],[34,615738,223],{"className":615739},[32024],[34,615741],{"className":615742,"style":32604},[31845],[34,615744,615746,615749,615817,615820,615823],{"className":615745},[31676],[34,615747],{"className":615748,"style":46727},[31680],[34,615750,615752,615755,615814],{"className":615751},[31685],[34,615753],{"className":615754},[31782,41488],[34,615756,615758],{"className":615757},[41453],[34,615759,615761,615806],{"className":615760},[31796,31797],[34,615762,615764,615803],{"className":615763},[31801],[34,615765,615767,615781,615789],{"className":615766,"style":46746},[31805],[34,615768,615769,615772],{"style":41504},[34,615770],{"className":615771,"style":41508},[31813],[34,615773,615775],{"className":615774},[31818,31819,31820,31821],[34,615776,615778],{"className":615777},[31685,31821],[34,615779,31660],{"className":615780},[31685,31686,31821],[34,615782,615783,615786],{"style":41520},[34,615784],{"className":615785,"style":41508},[31813],[34,615787],{"className":615788,"style":41528},[41527],[34,615790,615791,615794],{"style":46771},[34,615792],{"className":615793,"style":41508},[31813],[34,615795,615797],{"className":615796},[31818,31819,31820,31821],[34,615798,615800],{"className":615799},[31685,31821],[34,615801,640],{"className":615802},[31685,31821],[34,615804,31829],{"className":615805},[31828],[34,615807,615809],{"className":615808},[31801],[34,615810,615812],{"className":615811,"style":41619},[31805],[34,615813],{},[34,615815],{"className":615816},[31690,41488],[34,615818],{"className":615819,"style":32604},[31845],[34,615821,32226],{"className":615822},[32024],[34,615824],{"className":615825,"style":32604},[31845],[34,615827,615829,615832,615835,615838,615841],{"className":615828},[31676],[34,615830],{"className":615831,"style":32617},[31680],[34,615833,640],{"className":615834},[31685],[34,615836],{"className":615837,"style":31846},[31845],[34,615839,202],{"className":615840},[31850],[34,615842],{"className":615843,"style":31846},[31845],[34,615845,615847,615851,615919,615922,615925],{"className":615846},[31676],[34,615848],{"className":615849,"style":615850},[31680],"height:1.0925em;vertical-align:-0.345em;",[34,615852,615854,615857,615916],{"className":615853},[31685],[34,615855],{"className":615856},[31782,41488],[34,615858,615860],{"className":615859},[41453],[34,615861,615863,615908],{"className":615862},[31796,31797],[34,615864,615866,615905],{"className":615865},[31801],[34,615867,615869,615883,615891],{"className":615868,"style":548284},[31805],[34,615870,615871,615874],{"style":41504},[34,615872],{"className":615873,"style":41508},[31813],[34,615875,615877],{"className":615876},[31818,31819,31820,31821],[34,615878,615880],{"className":615879},[31685,31821],[34,615881,31660],{"className":615882},[31685,31686,31821],[34,615884,615885,615888],{"style":41520},[34,615886],{"className":615887,"style":41508},[31813],[34,615889],{"className":615890,"style":41528},[41527],[34,615892,615893,615896],{"style":548368},[34,615894],{"className":615895,"style":41508},[31813],[34,615897,615899],{"className":615898},[31818,31819,31820,31821],[34,615900,615902],{"className":615901},[31685,31821],[34,615903,15],{"className":615904},[31685,31686,31821],[34,615906,31829],{"className":615907},[31828],[34,61590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\\frac{1}{n} \\int_0^1 t^{\\frac{p}{n}-1} (1-t)^{q-1} \\, 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\\int_0^{+\\infty} e^{-x^n} \\, dx = \\frac{1}{n} \\Gamma\\left(\\frac{1}{n}\\right)",[34,616662,616664,616809],{"className":616663,"ariaHidden":758},[31672],[34,616665,616667,616670,616727,616730,616791,616794,616797,616800,616803,616806],{"className":616666},[31676],[34,616668],{"className":616669,"style":601420},[31680],[34,616671,616673,616676],{"className":616672},[32767],[34,616674,546295],{"className":616675,"style":591475},[32767,131249,131250],[34,616677,616679],{"className":616678},[31792],[34,616680,616682,616719],{"className":616681},[31796,31797],[34,616683,616685,616716],{"className":616684},[31801],[34,616686,616688,616699],{"className":616687,"style":601439},[31805],[34,616689,616690,616693],{"style":591491},[34,616691],{"className":616692,"style":31814},[31813],[34,616694,616696],{"className":616695},[31818,31819,31820,31821],[34,616697,459],{"className":616698},[31685,31821],[34,616700,616701,616704],{"style":591503},[34,616702],{"className":616703,"style":31814},[31813],[34,616705,616707],{"className":616706},[31818,31819,31820,31821],[34,616708,616710,616713],{"className":616709},[31685,31821],[34,616711,223],{"className":616712},[31685,31821],[34,616714,138286],{"className":616715},[31685,31821],[34,616717,31829],{"className":616718},[31828],[34,616720,616722],{"className":616721},[31801],[34,616723,616725],{"className":616724,"style":591522},[31805],[34,616726],{},[34,616728],{"className":616729,"style":32123},[31845],[34,616731,616733,616736],{"className":616732},[31685],[34,616734,33244],{"className":616735},[31685,31686],[34,616737,616739],{"className":616738},[31792],[34,616740,616742],{"className":616741},[31796],[34,616743,616745],{"className":616744},[31801],[34,616746,616748],{"className":616747,"style":594425},[31805],[34,616749,616750,616753],{"style":129846},[34,616751],{"className":616752,"style":31814},[31813],[34,616754,616756],{"className":616755},[31818,31819,31820,31821],[34,616757,616759,616762],{"className":616758},[31685,31821],[34,616760,32226],{"className":616761},[31685,31821],[34,616763,616765,616768],{"className":616764},[31685,31821],[34,616766,32936],{"className":616767},[31685,31686,31821],[34,616769,616771],{"className":616770},[31792],[34,616772,616774],{"className":616773},[31796],[34,616775,616777],{"className":616776},[31801],[34,616778,616780],{"className":616779,"style":611529},[31805],[34,616781,616782,616785],{"style":552391},[34,616783],{"className":616784,"style":43406},[31813],[34,616786,616788],{"className":616787},[31818,43410,43411,31821],[34,616789,31660],{"className":616790},[31685,31686,31821],[34,616792],{"className":616793,"style":32123},[31845],[34,616795,46900],{"className":616796},[31685,31686],[34,616798,32936],{"className":616799},[31685,31686],[34,616801],{"className":616802,"style":31846},[31845],[34,616804,202],{"className":616805},[31850],[34,616807],{"className":616808,"style":31846},[31845],[34,616810,616812,616815,616878,616881,616884],{"className":616811},[31676],[34,616813],{"className":616814,"style":370661},[31680],[34,616816,616818,616821,616875],{"className":616817},[31685],[34,616819],{"className":616820},[31782,41488],[34,616822,616824],{"className":616823},[41453],[34,616825,616827,616867],{"className":616826},[31796,31797],[34,616828,616830,616864],{"className":616829},[31801],[34,616831,616834,616845,616853],{"className":616832,"style":616833},[31805],"height:1.3214em;",[34,616835,616836,616839],{"style":129639},[34,616837],{"className":616838,"style":41508},[31813],[34,616840,616842],{"className":616841},[31685],[34,616843,31660],{"className":616844},[31685,31686],[34,616846,616847,616850],{"style":41520},[34,616848],{"className":616849,"style":41508},[31813],[34,616851],{"className":616852,"style":41528},[41527],[34,616854,616855,616858],{"style":129659},[34,616856],{"className":616857,"style":41508},[31813],[34,616859,616861],{"className":616860},[31685],[34,616862,640],{"className":616863},[31685],[34,616865,31829],{"className":616866},[31828],[34,616868,616870],{"className":616869},[31801],[34,616871,616873],{"className":616872,"style":129699},[31805],[34,616874],{},[34,616876],{"className":616877},[31690,41488],[34,616879,564791],{"className":616880},[31685],[34,616882],{"className":616883,"style":32123},[31845],[34,616885,616887,616893,616955],{"className":616886},[34006],[34,616888,616890],{"className":616889,"style":75811},[31782,75810],[34,616891,456],{"className":616892},[71281,31820],[34,616894,616896,616899,616952],{"className":616895},[31685],[34,616897],{"className":616898},[31782,41488],[34,616900,616902],{"className":616901},[41453],[34,616903,616905,616944],{"className":616904},[31796,31797],[34,616906,616908,616941],{"className":616907},[31801],[34,616909,616911,616922,616930],{"className":616910,"style":616833},[31805],[34,616912,616913,616916],{"style":129639},[34,616914],{"className":616915,"style":41508},[31813],[34,616917,616919],{"className":616918},[31685],[34,616920,31660],{"className":616921},[31685,31686],[34,616923,616924,616927],{"style":41520},[34,616925],{"className":616926,"style":41508},[31813],[34,616928],{"className":616929,"style":41528},[41527],[34,616931,616932,616935],{"style":129659},[34,616933],{"className":616934,"style":41508},[31813],[34,616936,616938],{"className":616937},[31685],[34,616939,640],{"className":616940},[31685],[34,616942,31829],{"className":616943},[31828],[34,616945,616947],{"className":616946},[31801],[34,616948,616950],{"className":616949,"style":129699},[31805],[34,616951],{},[34,616953],{"className":616954},[31690,41488],[34,616956,616958],{"className":616957,"style":75811},[31690,75810],[34,616959,2557],{"className":616960},[71281,31820]," 并求极限",[2230,616963,616964,617488],{},[108,616965,616966,33183,616969],{},[2627,616967,616968],{},"证明等式",[105,616970,616971,617106],{},[108,616972,616973,616974,33775,617025,31901],{},"利用 T1(7) 的结论，令 ",[34,616975,616977,616995],{"className":616976},[31642],[34,616978,616980],{"className":616979},[31646],[31648,616981,616982],{"xmlns":31650},[31652,616983,616984,616992],{},[31655,616985,616986,616988,616990],{},[31658,616987,31759],{},[31662,616989,202],{},[31749,616991,459],{},[31665,616993,616994],{"encoding":31667},"m=0",[34,616996,616998,617016],{"className":616997,"ariaHidden":758},[31672],[34,616999,617001,617004,617007,617010,617013],{"className":617000},[31676],[34,617002],{"className":617003,"style":32527},[31680],[34,617005,31759],{"className":617006},[31685,31686],[34,617008],{"className":617009,"style":31846},[31845],[34,617011,202],{"className":617012},[31850],[34,617014],{"className":617015,"style":31846},[31845],[34,617017,617019,617022],{"className":617018},[31676],[34,617020],{"className":617021,"style":32617},[31680],[34,617023,459],{"className":617024},[31685],[34,617026,617028,617050],{"className":617027},[31642],[34,617029,617031],{"className":617030},[31646],[31648,617032,617033],{"xmlns":31650},[31652,617034,617035,617047],{},[31655,617036,617037,617043,617045],{},[37624,617038,617039,617041],{},[31658,617040,32936],{},[31749,617042,459],{},[31662,617044,202],{},[31749,617046,640],{},[31665,617048,617049],{"encoding":31667},"x^0=1",[34,617051,617053,617097],{"className":617052,"ariaHidden":758},[31672],[34,617054,617056,617059,617088,617091,617094],{"className":617055},[31676],[34,617057],{"className":617058,"style":37774},[31680],[34,617060,617062,617065],{"className":617061},[31685],[34,617063,32936],{"className":617064},[31685,31686],[34,617066,617068],{"className":617067},[31792],[34,617069,617071],{"className":617070},[31796],[34,617072,617074],{"className":617073},[31801],[34,617075,617077],{"className":617076,"style":37774},[31805],[34,617078,617079,617082],{"style":37777},[34,617080],{"className":617081,"style":31814},[31813],[34,617083,617085],{"className":617084},[31818,31819,31820,31821],[34,617086,459],{"className":617087},[31685,31821],[34,617089],{"className":617090,"style":31846},[31845],[34,617092,202],{"className":617093},[31850],[34,617095],{"className":617096,"style":31846},[31845],[34,617098,617100,617103],{"className":617099},[31676],[34,617101],{"className":617102,"style":32617},[31680],[34,617104,640],{"className":617105},[31685],[108,617107,617108,617109,327546],{},"直接代入 T1(7) 结果：",[34,617110,617112,617168],{"className":617111},[31642],[34,617113,617115],{"className":617114},[31646],[31648,617116,617117],{"xmlns":31650},[31652,617118,617119,617165],{},[31655,617120,617121,617127,617129,617131,617143,617145,617147,617153,617155,617157,617163],{},[41453,617122,617123,617125],{},[31749,617124,640],{},[31658,617126,31660],{},[31658,617128,564791],{"mathvariant":38079},[31662,617130,456],{"stretchy":702},[41453,617132,617133,617141],{},[31655,617134,617135,617137,617139],{},[31749,617136,459],{},[31662,617138,223],{},[31749,617140,640],{},[31658,617142,31660],{},[31662,617144,2557],{"stretchy":702},[31662,617146,202],{},[41453,617148,617149,617151],{},[31749,617150,640],{},[31658,617152,31660],{},[31658,617154,564791],{"mathvariant":38079},[31662,617156,456],{"stretchy":702},[41453,617158,617159,617161],{},[31749,617160,640],{},[31658,617162,31660],{},[31662,617164,2557],{"stretchy":702},[31665,617166,617167],{"encoding":31667},"\\frac{1}{n} \\Gamma(\\frac{0+1}{n}) = \\frac{1}{n} \\Gamma(\\frac{1}{n})",[34,617169,617171,617337],{"className":617170,"ariaHidden":758},[31672],[34,617172,617174,617177,617245,617248,617251,617325,617328,617331,617334],{"className":617173},[31676],[34,617175],{"className":617176,"style":46727},[31680],[34,617178,617180,617183,617242],{"className":617179},[31685],[34,617181],{"className":617182},[31782,41488],[34,617184,617186],{"className":617185},[41453],[34,617187,617189,617234],{"className":617188},[31796,31797],[34,617190,617192,617231],{"className":617191},[31801],[34,617193,617195,617209,617217],{"className":617194,"style":46746},[31805],[34,617196,617197,617200],{"style":41504},[34,617198],{"className":617199,"style":41508},[31813],[34,617201,617203],{"className":617202},[31818,31819,31820,31821],[34,617204,617206],{"className":617205},[31685,31821],[34,617207,31660],{"className":617208},[31685,31686,31821],[34,617210,617211,617214],{"style":41520},[34,617212],{"className":617213,"style":41508},[31813],[34,617215],{"className":617216,"style":41528},[41527],[34,617218,617219,617222],{"style":46771},[34,617220],{"className":617221,"style":41508},[31813],[34,617223,617225],{"className":617224},[31818,31819,31820,31821],[34,617226,617228],{"className":617227},[31685,31821],[34,617229,640],{"className":617230},[31685,31821],[34,617232,31829],{"className":617233},[31828],[34,617235,617237],{"className":617236},[31801],[34,617238,617240],{"className":617239,"style":41619},[31805],[34,617241],{},[34,617243],{"className":617244},[31690,41488],[34,617246,564791],{"className":617247},[31685],[34,617249,456],{"className":617250},[31782],[34,617252,617254,617257,617322],{"className":617253},[31685],[34,617255],{"className":617256},[31782,41488],[34,617258,617260],{"className":617259},[41453],[34,617261,617263,617314],{"className":617262},[31796,31797],[34,617264,617266,617311],{"className":617265},[31801],[34,617267,617269,617283,617291],{"className":617268,"style":46746},[31805],[34,617270,617271,617274],{"style":41504},[34,617272],{"className":617273,"style":41508},[31813],[34,617275,617277],{"className":617276},[31818,31819,31820,31821],[34,617278,617280],{"className":617279},[31685,31821],[34,617281,31660],{"className":617282},[31685,31686,31821],[34,617284,617285,617288],{"style":41520},[34,617286],{"className":617287,"style":41508},[31813],[34,617289],{"className":617290,"style":41528},[41527],[34,617292,617293,617296],{"style":46771},[34,617294],{"className":617295,"style":41508},[31813],[34,617297,617299],{"className":617298},[31818,31819,31820,31821],[34,617300,617302,617305,617308],{"className":617301},[31685,31821],[34,617303,459],{"className":617304},[31685,31821],[34,617306,223],{"className":617307},[32024,31821],[34,617309,640],{"className":617310},[31685,31821],[34,617312,31829],{"className":617313},[31828],[34,617315,617317],{"className":617316},[31801],[34,617318,617320],{"className":617319,"style":41619},[31805],[34,617321],{},[34,617323],{"className":617324},[31690,41488],[34,617326,2557],{"className":617327},[31690],[34,617329],{"className":617330,"style":31846},[31845],[34,617332,202],{"className":617333},[31850],[34,617335],{"className":617336,"style":31846},[31845],[34,617338,617340,617343,617411,617414,617417,617485],{"className":617339},[31676],[34,617341],{"className":617342,"style":46727},[31680],[34,617344,617346,617349,617408],{"className":617345},[31685],[34,617347],{"className":617348},[31782,41488],[34,617350,617352],{"className":617351},[41453],[34,617353,617355,617400],{"className":617354},[31796,31797],[34,617356,617358,617397],{"className":617357},[31801],[34,617359,617361,617375,617383],{"className":617360,"style":46746},[31805],[34,617362,617363,617366],{"style":41504},[34,617364],{"className":617365,"style":41508},[31813],[34,617367,617369],{"className":617368},[31818,31819,31820,31821],[34,617370,617372],{"className":617371},[31685,31821],[34,617373,31660],{"className":617374},[31685,31686,31821],[34,617376,617377,617380],{"style":41520},[34,617378],{"className":617379,"style":41508},[31813],[34,617381],{"className":617382,"style":41528},[41527],[34,617384,617385,617388],{"style":46771},[34,617386],{"className":617387,"style":41508},[31813],[34,617389,617391],{"className":617390},[31818,31819,31820,31821],[34,617392,617394],{"className":617393},[31685,31821],[34,617395,640],{"className":617396},[31685,31821],[34,617398,31829],{"className":617399},[31828],[34,617401,617403],{"className":617402},[31801],[34,617404,617406],{"className":617405,"style":41619},[31805],[34,617407],{},[34,617409],{"className":617410},[31690,41488],[34,617412,564791],{"className":617413},[31685],[34,617415,456],{"className":617416},[31782],[34,617418,617420,617423,617482],{"className":617419},[31685],[34,617421],{"className":617422},[31782,41488],[34,617424,617426],{"className":617425},[41453],[34,617427,617429,617474],{"className":617428},[31796,31797],[34,617430,617432,617471],{"className":617431},[31801],[34,617433,617435,617449,617457],{"className":617434,"style":46746},[31805],[34,617436,617437,617440],{"style":41504},[34,617438],{"className":617439,"style":41508},[31813],[34,617441,617443],{"className":617442},[31818,31819,31820,31821],[34,617444,617446],{"className":617445},[31685,31821],[34,617447,31660],{"className":617448},[31685,31686,31821],[34,617450,617451,617454],{"style":41520},[34,617452],{"className":617453,"style":41508},[31813],[34,617455],{"className":617456,"style":41528},[41527],[34,617458,617459,617462],{"style":46771},[34,617460],{"className":617461,"style":41508},[31813],[34,617463,617465],{"className":617464},[31818,31819,31820,31821],[34,617466,617468],{"className":617467},[31685,31821],[34,617469,640],{"className":617470},[31685,31821],[34,617472,31829],{"className":617473},[31828],[34,617475,617477],{"className":617476},[31801],[34,617478,617480],{"className":617479,"style":41619},[31805],[34,617481],{},[34,617483],{"className":617484},[31690,41488],[34,617486,2557],{"className":617487},[31690],[108,617489,617490,33183,617748],{},[2627,617491,617492,617493],{},"求极限 ",[34,617494,617496,617542],{"className":617495},[31642],[34,617497,617499],{"className":617498},[31646],[31648,617500,617501],{"xmlns":31650},[31652,617502,617503,617539],{},[31655,617504,617505,617521,617527,617529,617531,617537],{},[31743,617506,617507,617513],{},[31655,617508,617509,617511],{},[31658,617510,546745],{},[31662,617512,32682],{},[31655,617514,617515,617517,617519],{},[31658,617516,31660],{},[31662,617518,33239],{},[31658,617520,138286],{"mathvariant":38079},[41453,617522,617523,617525],{},[31749,617524,640],{},[31658,617526,31660],{},[31658,617528,564791],{"mathvariant":38079},[31662,617530,456],{"stretchy":702},[41453,617532,617533,617535],{},[31749,617534,640],{},[31658,617536,31660],{},[31662,617538,2557],{"stretchy":702},[31665,617540,617541],{"encoding":31667},"\\lim_{n \\to \\infty} \\frac{1}{n} \\Gamma(\\frac{1}{n})",[34,617543,617545],{"className":617544,"ariaHidden":758},[31672],[34,617546,617548,617551,617600,617603,617671,617674,617677,617745],{"className":617547},[31676],[34,617549],{"className":617550,"style":46727},[31680],[34,617552,617554,617557],{"className":617553},[32767],[34,617555,546745],{"className":617556},[32767],[34,617558,617560],{"className":617559},[31792],[34,617561,617563,617592],{"className":617562},[31796,31797],[34,617564,617566,617589],{"className":617565},[31801],[34,617567,617569],{"className":617568,"style":32362},[31805],[34,617570,617571,617574],{"style":56235},[34,617572],{"className":617573,"style":31814},[31813],[34,617575,617577],{"className":617576},[31818,31819,31820,31821],[34,617578,617580,617583,617586],{"className":617579},[31685,31821],[34,617581,31660],{"className":617582},[31685,31686,31821],[34,617584,33239],{"className":617585},[31850,31821],[34,617587,138286],{"className":617588},[31685,31821],[34,617590,31829],{"className":617591},[31828],[34,617593,617595],{"className":617594},[31801],[34,617596,617598],{"className":617597,"style":31836},[31805],[34,617599],{},[34,617601],{"className":617602,"style":32123},[31845],[34,617604,617606,617609,617668],{"className":617605},[31685],[34,617607],{"className":617608},[31782,41488],[34,617610,617612],{"className":617611},[41453],[34,617613,617615,617660],{"className":617614},[31796,31797],[34,617616,617618,617657],{"className":617617},[31801],[34,617619,617621,617635,617643],{"className":617620,"style":46746},[31805],[34,617622,617623,617626],{"style":41504},[34,617624],{"className":617625,"style":41508},[31813],[34,617627,617629],{"className":617628},[31818,31819,31820,31821],[34,617630,617632],{"className":617631},[31685,31821],[34,617633,31660],{"className":617634},[31685,31686,31821],[34,617636,617637,617640],{"style":41520},[34,617638],{"className":617639,"style":41508},[31813],[34,617641],{"className":617642,"style":41528},[41527],[34,617644,617645,617648],{"style":46771},[34,617646],{"className":617647,"style":41508},[31813],[34,617649,617651],{"className":617650},[31818,31819,31820,31821],[34,617652,617654],{"className":617653},[31685,31821],[34,617655,640],{"className":617656},[31685,31821],[34,617658,31829],{"className":617659},[31828],[34,617661,617663],{"className":617662},[31801],[34,617664,617666],{"className":617665,"style":41619},[31805],[34,617667],{},[34,617669],{"className":617670},[31690,41488],[34,617672,564791],{"className":617673},[31685],[34,617675,456],{"className":617676},[31782],[34,617678,617680,617683,617742],{"className":617679},[31685],[34,617681],{"className":617682},[31782,41488],[34,617684,617686],{"className":617685},[41453],[34,617687,617689,617734],{"className":617688},[31796,31797],[34,617690,617692,617731],{"className":617691},[31801],[34,617693,617695,617709,617717],{"className":617694,"style":46746},[31805],[34,617696,617697,617700],{"style":41504},[34,617698],{"className":617699,"style":41508},[31813],[34,617701,617703],{"className":617702},[31818,31819,31820,31821],[34,617704,617706],{"className":617705},[31685,31821],[34,617707,31660],{"className":617708},[31685,31686,31821],[34,617710,617711,617714],{"style":41520},[34,617712],{"className":617713,"style":41508},[31813],[34,617715],{"className":617716,"style":41528},[41527],[34,617718,617719,617722],{"style":46771},[34,617720],{"className":617721,"style":41508},[31813],[34,617723,617725],{"className":617724},[31818,31819,31820,31821],[34,617726,617728],{"className":617727},[31685,31821],[34,617729,640],{"className":617730},[31685,31821],[34,617732,31829],{"className":617733},[31828],[34,617735,617737],{"className":617736},[31801],[34,617738,617740],{"className":617739,"style":41619},[31805],[34,617741],{},[34,617743],{"className":617744},[31690,41488],[34,617746,2557],{"className":617747},[31690],[105,617749,617750,618005,618156,618267],{},[108,617751,396859,617752,617872,617873,4504,617924,347],{},[34,617753,617755,617777],{"className":617754},[31642],[34,617756,617758],{"className":617757},[31646],[31648,617759,617760],{"xmlns":31650},[31652,617761,617762,617774],{},[31655,617763,617764,617766,617768],{},[31658,617765,33249],{},[31662,617767,202],{},[41453,617769,617770,617772],{},[31749,617771,640],{},[31658,617773,31660],{},[31665,617775,617776],{"encoding":31667},"t = \\frac{1}{n}",[34,617778,617780,617798],{"className":617779,"ariaHidden":758},[31672],[34,617781,617783,617786,617789,617792,617795],{"className":617782},[31676],[34,617784],{"className":617785,"style":33280},[31680],[34,617787,33249],{"className":617788},[31685,31686],[34,617790],{"className":617791,"style":31846},[31845],[34,617793,202],{"className":617794},[31850],[34,617796],{"className":617797,"style":31846},[31845],[34,617799,617801,617804],{"className":617800},[31676],[34,617802],{"className":617803,"style":46727},[31680],[34,617805,617807,617810,617869],{"className":617806},[31685],[34,617808],{"className":617809},[31782,41488],[34,617811,617813],{"className":617812},[41453],[34,617814,617816,617861],{"className":617815},[31796,31797],[34,617817,617819,617858],{"className":617818},[31801],[34,617820,617822,617836,617844],{"className":617821,"style":46746},[31805],[34,617823,617824,617827],{"style":41504},[34,617825],{"className":617826,"style":41508},[31813],[34,617828,617830],{"className":617829},[31818,31819,31820,31821],[34,617831,617833],{"className":617832},[31685,31821],[34,617834,31660],{"className":617835},[31685,31686,31821],[34,617837,617838,617841],{"style":41520},[34,617839],{"className":617840,"style":41508},[31813],[34,617842],{"className":617843,"style":41528},[41527],[34,617845,617846,617849],{"style":46771},[34,617847],{"className":617848,"style":41508},[31813],[34,617850,617852],{"className":617851},[31818,31819,31820,31821],[34,617853,617855],{"className":617854},[31685,31821],[34,617856,640],{"className":617857},[31685,31821],[34,617859,31829],{"className":617860},[31828],[34,617862,617864],{"className":617863},[31801],[34,617865,617867],{"className":617866,"style":41619},[31805],[34,617868],{},[34,617870],{"className":617871},[31690,41488],"，当 ",[34,617874,617876,617894],{"className":617875},[31642],[34,617877,617879],{"className":617878},[31646],[31648,617880,617881],{"xmlns":31650},[31652,617882,617883,617891],{},[31655,617884,617885,617887,617889],{},[31658,617886,31660],{},[31662,617888,33239],{},[31658,617890,138286],{"mathvariant":38079},[31665,617892,617893],{"encoding":31667},"n \\to \\infty",[34,617895,617897,617915],{"className":617896,"ariaHidden":758},[31672],[34,617898,617900,617903,617906,617909,617912],{"className":617899},[31676],[34,617901],{"className":617902,"style":32527},[31680],[34,617904,31660],{"className":617905},[31685,31686],[34,617907],{"className":617908,"style":31846},[31845],[34,617910,33239],{"className":617911},[31850],[34,617913],{"className":617914,"style":31846},[31845],[34,617916,617918,617921],{"className":617917},[31676],[34,617919],{"className":617920,"style":32527},[31680],[34,617922,138286],{"className":617923},[31685],[34,617925,617927,617949],{"className":617926},[31642],[34,617928,617930],{"className":617929},[31646],[31648,617931,617932],{"xmlns":31650},[31652,617933,617934,617946],{},[31655,617935,617936,617938,617940],{},[31658,617937,33249],{},[31662,617939,33239],{},[37624,617941,617942,617944],{},[31749,617943,459],{},[31662,617945,223],{},[31665,617947,617948],{"encoding":31667},"t \\to 0^+",[34,617950,617952,617970],{"className":617951,"ariaHidden":758},[31672],[34,617953,617955,617958,617961,617964,617967],{"className":617954},[31676],[34,617956],{"className":617957,"style":33280},[31680],[34,617959,33249],{"className":617960},[31685,31686],[34,617962],{"className":617963,"style":31846},[31845],[34,617965,33239],{"className":617966},[31850],[34,617968],{"className":617969,"style":31846},[31845],[34,617971,617973,617976],{"className":617972},[31676],[34,617974],{"className":617975,"style":156828},[31680],[34,617977,617979,617982],{"className":617978},[31685],[34,617980,459],{"className":617981},[31685],[34,617983,617985],{"className":617984},[31792],[34,617986,617988],{"className":617987},[31796],[34,617989,617991],{"className":617990},[31801],[34,617992,617994],{"className":617993,"style":156828},[31805],[34,617995,617996,617999],{"style":37777},[34,617997],{"className":617998,"style":31814},[31813],[34,618000,618002],{"className":618001},[31818,31819,31820,31821],[34,618003,223],{"className":618004},[32024,31821],[108,618006,618007,618008,347],{},"原式变为 ",[34,618009,618011,618053],{"className":618010},[31642],[34,618012,618014],{"className":618013},[31646],[31648,618015,618016],{"xmlns":31650},[31652,618017,618018,618050],{},[31655,618019,618020,618040,618042,618044,618046,618048],{},[31743,618021,618022,618028],{},[31655,618023,618024,618026],{},[31658,618025,546745],{},[31662,618027,32682],{},[31655,618029,618030,618032,618034],{},[31658,618031,33249],{},[31662,618033,33239],{},[37624,618035,618036,618038],{},[31749,618037,459],{},[31662,618039,223],{},[31658,618041,33249],{},[31658,618043,564791],{"mathvariant":38079},[31662,618045,456],{"stretchy":702},[31658,618047,33249],{},[31662,618049,2557],{"stretchy":702},[31665,618051,618052],{"encoding":31667},"\\lim_{t \\to 0^+} t \\Gamma(t)",[34,618054,618056],{"className":618055,"ariaHidden":758},[31672],[34,618057,618059,618062,618138,618141,618144,618147,618150,618153],{"className":618058},[31676],[34,618060],{"className":618061,"style":31774},[31680],[34,618063,618065,618068],{"className":618064},[32767],[34,618066,546745],{"className":618067},[32767],[34,618069,618071],{"className":618070},[31792],[34,618072,618074,618130],{"className":618073},[31796,31797],[34,618075,618077,618127],{"className":618076},[31801],[34,618078,618081],{"className":618079,"style":618080},[31805],"height:0.3419em;",[34,618082,618083,618086],{"style":56235},[34,618084],{"className":618085,"style":31814},[31813],[34,618087,618089],{"className":618088},[31818,31819,31820,31821],[34,618090,618092,618095,618098],{"className":618091},[31685,31821],[34,618093,33249],{"className":618094},[31685,31686,31821],[34,618096,33239],{"className":618097},[31850,31821],[34,618099,618101,618104],{"className":618100},[31685,31821],[34,618102,459],{"className":618103},[31685,31821],[34,618105,618107],{"className":618106},[31792],[34,618108,618110],{"className":618109},[31796],[34,618111,618113],{"className":618112},[31801],[34,618114,618116],{"className":618115,"style":605234},[31805],[34,618117,618118,618121],{"style":116668},[34,618119],{"className":618120,"style":43406},[31813],[34,618122,618124],{"className":618123},[31818,43410,43411,31821],[34,618125,223],{"className":618126},[32024,31821],[34,618128,31829],{"className":618129},[31828],[34,618131,618133],{"className":618132},[31801],[34,618134,618136],{"className":618135,"style":31836},[31805],[34,618137],{},[34,618139],{"className":618140,"style":32123},[31845],[34,618142,33249],{"className":618143},[31685,31686],[34,618145,564791],{"className":618146},[31685],[34,618148,456],{"className":618149},[31782],[34,618151,33249],{"className":618152},[31685,31686],[34,618154,2557],{"className":618155},[31690],[108,618157,618158,618159,347],{},"利用性质 ",[34,618160,618162,618198],{"className":618161},[31642],[34,618163,618165],{"className":618164},[31646],[31648,618166,618167],{"xmlns":31650},[31652,618168,618169,618195],{},[31655,618170,618171,618173,618175,618177,618179,618181,618183,618185,618187,618189,618191,618193],{},[31658,618172,564791],{"mathvariant":38079},[31662,618174,456],{"stretchy":702},[31658,618176,33249],{},[31662,618178,223],{},[31749,618180,640],{},[31662,618182,2557],{"stretchy":702},[31662,618184,202],{},[31658,618186,33249],{},[31658,618188,564791],{"mathvariant":38079},[31662,618190,456],{"stretchy":702},[31658,618192,33249],{},[31662,618194,2557],{"stretchy":702},[31665,618196,618197],{"encoding":31667},"\\Gamma(t+1) = t \\Gamma(t)",[34,618199,618201,618225,618246],{"className":618200,"ariaHidden":758},[31672],[34,618202,618204,618207,618210,618213,618216,618219,618222],{"className":618203},[31676],[34,618205],{"className":618206,"style":31774},[31680],[34,618208,564791],{"className":618209},[31685],[34,618211,456],{"className":618212},[31782],[34,618214,33249],{"className":618215},[31685,31686],[34,618217],{"className":618218,"style":32604},[31845],[34,618220,223],{"className":618221},[32024],[34,618223],{"className":618224,"style":32604},[31845],[34,618226,618228,618231,618234,618237,618240,618243],{"className":618227},[31676],[34,618229],{"className":618230,"style":31774},[31680],[34,618232,640],{"className":618233},[31685],[34,618235,2557],{"className":618236},[31690],[34,618238],{"className":618239,"style":31846},[31845],[34,618241,202],{"className":618242},[31850],[34,618244],{"className":618245,"style":31846},[31845],[34,618247,618249,618252,618255,618258,618261,618264],{"className":618248},[31676],[34,618250],{"className":618251,"style":31774},[31680],[34,618253,33249],{"className":618254},[31685,31686],[34,618256,564791],{"className":618257},[31685],[34,618259,456],{"className":618260},[31782],[34,618262,33249],{"className":618263},[31685,31686],[34,618265,2557],{"className":618266},[31690],[108,618268,175992,618269,347],{},[34,618270,618272,618368],{"className":618271},[31642],[34,618273,618275],{"className":618274},[31646],[31648,618276,618277],{"xmlns":31650},[31652,618278,618279,618365],{},[31655,618280,618281,618301,618303,618305,618307,618309,618311,618313,618333,618335,618337,618339,618341,618343,618345,618347,618349,618351,618353,618355,618357,618359,618361,618363],{},[31743,618282,618283,618289],{},[31655,618284,618285,618287],{},[31658,618286,546745],{},[31662,618288,32682],{},[31655,618290,618291,618293,618295],{},[31658,618292,33249],{},[31662,618294,33239],{},[37624,618296,618297,618299],{},[31749,618298,459],{},[31662,618300,223],{},[31658,618302,33249],{},[31658,618304,564791],{"mathvariant":38079},[31662,618306,456],{"stretchy":702},[31658,618308,33249],{},[31662,618310,2557],{"stretchy":702},[31662,618312,202],{},[31743,618314,618315,618321],{},[31655,618316,618317,618319],{},[31658,618318,546745],{},[31662,618320,32682],{},[31655,618322,618323,618325,618327],{},[31658,618324,33249],{},[31662,618326,33239],{},[37624,618328,618329,618331],{},[31749,618330,459],{},[31662,618332,223],{},[31658,618334,564791],{"mathvariant":38079},[31662,618336,456],{"stretchy":702},[31658,618338,33249],{},[31662,618340,223],{},[31749,618342,640],{},[31662,618344,2557],{"stretchy":702},[31662,618346,202],{},[31658,618348,564791],{"mathvariant":38079},[31662,618350,456],{"stretchy":702},[31749,618352,640],{},[31662,618354,2557],{"stretchy":702},[31662,618356,202],{},[31749,618358,459],{},[31662,618360,5504],{"stretchy":702},[31662,618362,202],{},[31749,618364,640],{},[31665,618366,618367],{"encoding":31667},"\\lim_{t \\to 0^+} t \\Gamma(t) = \\lim_{t \\to 0^+} \\Gamma(t+1) = \\Gamma(1) = 0! = 1",[34,618369,618371,618479,618581,618602,618629,618650],{"className":618370,"ariaHidden":758},[31672],[34,618372,618374,618377,618452,618455,618458,618461,618464,618467,618470,618473,618476],{"className":618373},[31676],[34,618375],{"className":618376,"style":31774},[31680],[34,618378,618380,618383],{"className":618379},[32767],[34,618381,546745],{"className":618382},[32767],[34,618384,618386],{"className":618385},[31792],[34,618387,618389,618444],{"className":618388},[31796,31797],[34,618390,618392,618441],{"className":618391},[31801],[34,618393,618395],{"className":618394,"style":618080},[31805],[34,618396,618397,618400],{"style":56235},[34,618398],{"className":618399,"style":31814},[31813],[34,618401,618403],{"className":618402},[31818,31819,31820,31821],[34,618404,618406,618409,618412],{"className":618405},[31685,31821],[34,618407,33249],{"className":618408},[31685,31686,31821],[34,618410,33239],{"className":618411},[31850,31821],[34,618413,618415,618418],{"className":618414},[31685,31821],[34,618416,459],{"className":618417},[31685,31821],[34,618419,618421],{"className":618420},[31792],[34,618422,618424],{"className":618423},[31796],[34,618425,618427],{"className":618426},[31801],[34,618428,618430],{"className":618429,"style":605234},[31805],[34,618431,618432,618435],{"style":116668},[34,618433],{"className":618434,"style":43406},[31813],[34,618436,618438],{"className":618437},[31818,43410,43411,31821],[34,618439,223],{"className":618440},[32024,31821],[34,618442,31829],{"className":618443},[31828],[34,618445,618447],{"className":618446},[31801],[34,618448,618450],{"className":618449,"style":31836},[31805],[34,618451],{},[34,618453],{"className":618454,"style":32123},[31845],[34,618456,33249],{"className":618457},[31685,31686],[34,618459,564791],{"className":618460},[31685],[34,618462,456],{"className":618463},[31782],[34,618465,33249],{"className":618466},[31685,31686],[34,618468,2557],{"className":618469},[31690],[34,618471],{"className":618472,"style":31846},[31845],[34,618474,202],{"className":618475},[31850],[34,618477],{"className":618478,"style":31846},[31845],[34,618480,618482,618485,618560,618563,618566,618569,618572,618575,618578],{"className":618481},[31676],[34,618483],{"className":618484,"style":31774},[31680],[34,618486,618488,618491],{"className":618487},[32767],[34,618489,546745],{"className":618490},[32767],[34,618492,618494],{"className":618493},[31792],[34,618495,618497,618552],{"className":618496},[31796,31797],[34,618498,618500,618549],{"className":618499},[31801],[34,618501,618503],{"className":618502,"style":618080},[31805],[34,618504,618505,618508],{"style":56235},[34,618506],{"className":618507,"style":31814},[31813],[34,618509,618511],{"className":618510},[31818,31819,31820,31821],[34,618512,618514,618517,618520],{"className":618513},[31685,31821],[34,618515,33249],{"className":618516},[31685,31686,31821],[34,618518,33239],{"className":618519},[31850,31821],[34,618521,618523,618526],{"className":618522},[31685,31821],[34,618524,459],{"className":618525},[31685,31821],[34,618527,618529],{"className":618528},[31792],[34,618530,618532],{"className":618531},[31796],[34,618533,618535],{"className":618534},[31801],[34,618536,618538],{"className":618537,"style":605234},[31805],[34,618539,618540,618543],{"style":116668},[34,618541],{"className":618542,"style":43406},[31813],[34,618544,618546],{"className":618545},[31818,43410,43411,31821],[34,618547,223],{"className":618548},[32024,31821],[34,618550,31829],{"className":618551},[31828],[34,618553,618555],{"className":618554},[31801],[34,618556,618558],{"className":618557,"style":31836},[31805],[34,618559],{},[34,618561],{"className":618562,"style":32123},[31845],[34,618564,564791],{"className":618565},[31685],[34,618567,456],{"className":618568},[31782],[34,618570,33249],{"className":618571},[31685,31686],[34,618573],{"className":618574,"style":32604},[31845],[34,618576,223],{"className":618577},[32024],[34,618579],{"className":618580,"style":32604},[31845],[34,618582,618584,618587,618590,618593,618596,618599],{"className":618583},[31676],[34,618585],{"className":618586,"style":31774},[31680],[34,618588,640],{"className":618589},[31685],[34,618591,2557],{"className":618592},[31690],[34,618594],{"className":618595,"style":31846},[31845],[34,618597,202],{"className":618598},[31850],[34,618600],{"className":618601,"style":31846},[31845],[34,618603,618605,618608,618611,618614,618617,618620,618623,618626],{"className":618604},[31676],[34,618606],{"className":618607,"style":31774},[31680],[34,618609,564791],{"className":618610},[31685],[34,618612,456],{"className":618613},[31782],[34,618615,640],{"className":618616},[31685],[34,618618,2557],{"className":618619},[31690],[34,618621],{"className":618622,"style":31846},[31845],[34,618624,202],{"className":618625},[31850],[34,618627],{"className":618628,"style":31846},[31845],[34,618630,618632,618635,618638,618641,618644,618647],{"className":618631},[31676],[34,618633],{"className":618634,"style":31681},[31680],[34,618636,459],{"className":618637},[31685],[34,618639,5504],{"className":618640},[31690],[34,618642],{"className":618643,"style":31846},[31845],[34,618645,202],{"className":618646},[31850],[34,618648],{"className":618649,"style":31846},[31845],[34,618651,618653,618656],{"className":618652},[31676],[34,618654],{"className":618655,"style":32617},[31680],[34,618657,640],{"className":618658},[31685],[105,618660,618661],{},[108,618662,618663,618665],{},[2627,618664,114043],{},"：极限值为 1。",[32188,618667],{},[32490,618669,618671,618672,618715],{"id":618670},"t3-证明-γsgammasγs-可导及导数公式","T3. 证明 ",[34,618673,618675,618694],{"className":618674},[31642],[34,618676,618678],{"className":618677},[31646],[31648,618679,618680],{"xmlns":31650},[31652,618681,618682,618692],{},[31655,618683,618684,618686,618688,618690],{},[31658,618685,564791],{"mathvariant":38079},[31662,618687,456],{"stretchy":702},[31658,618689,31747],{},[31662,618691,2557],{"stretchy":702},[31665,618693,587934],{"encoding":31667},[34,618695,618697],{"className":618696,"ariaHidden":758},[31672],[34,618698,618700,618703,618706,618709,618712],{"className":618699},[31676],[34,618701],{"className":618702,"style":31774},[31680],[34,618704,564791],{"className":618705},[31685],[34,618707,456],{"className":618708},[31782],[34,618710,31747],{"className":618711},[31685,31686],[34,618713,2557],{"className":618714},[31690]," 可导及导数公式",[15,618717,618718],{},"这是含参变量积分求导的问题。",[105,618720,618721],{},[108,618722,618723,33183,618725],{},[2627,618724,390672],{},[2230,618726,618727,618972,619109,619426,619985,620281,620384],{},[108,618728,618729,347],{},[34,618730,618732,618794],{"className":618731},[31642],[34,618733,618735],{"className":618734},[31646],[31648,618736,618737],{"xmlns":31650},[31652,618738,618739,618791],{},[31655,618740,618741,618743,618745,618747,618749,618751,618763,618775,618785,618787,618789],{},[31658,618742,564791],{"mathvariant":38079},[31662,618744,456],{"stretchy":702},[31658,618746,31747],{},[31662,618748,2557],{"stretchy":702},[31662,618750,202],{},[38846,618752,618753,618755,618757],{},[31662,618754,546295],{},[31749,618756,459],{},[31655,618758,618759,618761],{},[31662,618760,223],{},[31658,618762,138286],{"mathvariant":38079},[37624,618764,618765,618767],{},[31658,618766,32936],{},[31655,618768,618769,618771,618773],{},[31658,618770,31747],{},[31662,618772,32226],{},[31749,618774,640],{},[37624,618776,618777,618779],{},[31658,618778,33244],{},[31655,618780,618781,618783],{},[31662,618782,32226],{},[31658,618784,32936],{},[39361,618786,39363],{},[31658,618788,46900],{},[31658,618790,32936],{},[31665,618792,618793],{"encoding":31667},"\\Gamma(s) = \\int_0^{+\\infty} x^{s-1} e^{-x} \\, dx",[34,618795,618797,618824],{"className":618796,"ariaHidden":758},[31672],[34,618798,618800,618803,618806,618809,618812,618815,618818,618821],{"className":618799},[31676],[34,618801],{"className":618802,"style":31774},[31680],[34,618804,564791],{"className":618805},[31685],[34,618807,456],{"className":618808},[31782],[34,618810,31747],{"className":618811},[31685,31686],[34,618813,2557],{"className":618814},[31690],[34,618816],{"className":618817,"style":31846},[31845],[34,618819,202],{"className":618820},[31850],[34,618822],{"className":618823,"style":31846},[31845],[34,618825,618827,618830,618887,618890,618928,618963,618966,618969],{"className":618826},[31676],[34,618828],{"className":618829,"style":546531},[31680],[34,618831,618833,618836],{"className":618832},[32767],[34,618834,546295],{"className":618835,"style":546331},[32767,131249,156560],[34,618837,618839],{"className":618838},[31792],[34,618840,618842,618879],{"className":618841},[31796,31797],[34,618843,618845,618876],{"className":618844},[31801],[34,618846,618848,618859],{"className":618847,"style":546550},[31805],[34,618849,618850,618853],{"style":546347},[34,618851],{"className":618852,"style":31814},[31813],[34,618854,618856],{"className":618855},[31818,31819,31820,31821],[34,618857,459],{"className":618858},[31685,31821],[34,618860,618861,618864],{"style":546362},[34,618862],{"className":618863,"style":31814},[31813],[34,618865,618867],{"className":618866},[31818,31819,31820,31821],[34,618868,618870,618873],{"className":618869},[31685,31821],[34,618871,223],{"className":618872},[31685,31821],[34,618874,138286],{"className":618875},[31685,31821],[34,618877,31829],{"className":618878},[31828],[34,618880,618882],{"className":618881},[31801],[34,618883,618885],{"className":618884,"style":546384},[31805],[34,618886],{},[34,618888],{"className":618889,"style":32123},[31845],[34,618891,618893,618896],{"className":618892},[31685],[34,618894,32936],{"className":618895},[31685,31686],[34,618897,618899],{"className":618898},[31792],[34,618900,618902],{"className":618901},[31796],[34,618903,618905],{"className":618904},[31801],[34,618906,618908],{"className":618907,"style":37774},[31805],[34,618909,618910,618913],{"style":37777},[34,618911],{"className":618912,"style":31814},[31813],[34,618914,618916],{"className":618915},[31818,31819,31820,31821],[34,618917,618919,618922,618925],{"className":618918},[31685,31821],[34,618920,31747],{"className":618921},[31685,31686,31821],[34,618923,32226],{"className":618924},[32024,31821],[34,618926,640],{"className":618927},[31685,31821],[34,618929,618931,618934],{"className":618930},[31685],[34,618932,33244],{"className":618933},[31685,31686],[34,618935,618937],{"className":618936},[31792],[34,618938,618940],{"className":618939},[31796],[34,618941,618943],{"className":618942},[31801],[34,618944,618946],{"className":618945,"style":156828},[31805],[34,618947,618948,618951],{"style":37777},[34,618949],{"className":618950,"style":31814},[31813],[34,618952,618954],{"className":618953},[31818,31819,31820,31821],[34,618955,618957,618960],{"className":618956},[31685,31821],[34,618958,32226],{"className":618959},[31685,31821],[34,618961,32936],{"className":618962},[31685,31686,31821],[34,618964],{"className":618965,"style":32123},[31845],[34,618967,46900],{"className":618968},[31685,31686],[34,618970,32936],{"className":618971},[31685,31686],[108,618973,3101,618974,619002,619003,619108],{},[34,618975,618977,618990],{"className":618976},[31642],[34,618978,618980],{"className":618979},[31646],[31648,618981,618982],{"xmlns":31650},[31652,618983,618984,618988],{},[31655,618985,618986],{},[31658,618987,31747],{},[31665,618989,31747],{"encoding":31667},[34,618991,618993],{"className":618992,"ariaHidden":758},[31672],[34,618994,618996,618999],{"className":618995},[31676],[34,618997],{"className":618998,"style":32527},[31680],[34,619000,31747],{"className":619001},[31685,31686]," 求导，根据“积分符号下求导法则”，我们可以把导数 ",[34,619004,619006,619028],{"className":619005},[31642],[34,619007,619009],{"className":619008},[31646],[31648,619010,619011],{"xmlns":31650},[31652,619012,619013,619025],{},[31655,619014,619015],{},[41453,619016,619017,619019],{},[31658,619018,46900],{},[31655,619020,619021,619023],{},[31658,619022,46900],{},[31658,619024,31747],{},[31665,619026,619027],{"encoding":31667},"\\frac{d}{ds}",[34,619029,619031],{"className":619030,"ariaHidden":758},[31672],[34,619032,619034,619037],{"className":619033},[31676],[34,619035],{"className":619036,"style":232492},[31680],[34,619038,619040,619043,619105],{"className":619039},[31685],[34,619041],{"className":619042},[31782,41488],[34,619044,619046],{"className":619045},[41453],[34,619047,619049,619097],{"className":619048},[31796,31797],[34,619050,619052,619094],{"className":619051},[31801],[34,619053,619055,619072,619080],{"className":619054,"style":232511},[31805],[34,619056,619057,619060],{"style":41504},[34,619058],{"className":619059,"style":41508},[31813],[34,619061,619063],{"className":619062},[31818,31819,31820,31821],[34,619064,619066,619069],{"className":619065},[31685,31821],[34,619067,46900],{"className":619068},[31685,31686,31821],[34,619070,31747],{"className":619071},[31685,31686,31821],[34,619073,619074,619077],{"style":41520},[34,619075],{"className":619076,"style":41508},[31813],[34,619078],{"className":619079,"style":41528},[41527],[34,619081,619082,619085],{"style":46771},[34,619083],{"className":619084,"style":41508},[31813],[34,619086,619088],{"className":619087},[31818,31819,31820,31821],[34,619089,619091],{"className":619090},[31685,31821],[34,619092,46900],{"className":619093},[31685,31686,31821],[34,619095,31829],{"className":619096},[31828],[34,619098,619100],{"className":619099},[31801],[34,619101,619103],{"className":619102,"style":41619},[31805],[34,619104],{},[34,619106],{"className":619107},[31690,41488]," 移到积分号里面。",[108,619110,598682,619111,347],{},[34,619112,619114,619196],{"className":619113},[31642],[34,619115,619117],{"className":619116},[31646],[31648,619118,619119],{"xmlns":31650},[31652,619120,619121,619193],{},[31655,619122,619123,619125,619127,619129,619131,619133,619135,619137,619149,619159,619161,619183],{},[31658,619124,31739],{},[31662,619126,456],{"stretchy":702},[31658,619128,32936],{},[31662,619130,31960],{"separator":758},[31658,619132,31747],{},[31662,619134,2557],{"stretchy":702},[31662,619136,202],{},[37624,619138,619139,619141],{},[31658,619140,32936],{},[31655,619142,619143,619145,619147],{},[31658,619144,31747],{},[31662,619146,32226],{},[31749,619148,640],{},[37624,619150,619151,619153],{},[31658,619152,33244],{},[31655,619154,619155,619157],{},[31662,619156,32226],{},[31658,619158,32936],{},[31662,619160,202],{},[37624,619162,619163,619165],{},[31658,619164,33244],{},[31655,619166,619167,619169,619171,619173,619175,619177,619179,619181],{},[31662,619168,456],{"stretchy":702},[31658,619170,31747],{},[31662,619172,32226],{},[31749,619174,640],{},[31662,619176,2557],{"stretchy":702},[31658,619178,563286],{},[31662,619180,32682],{},[31658,619182,32936],{},[37624,619184,619185,619187],{},[31658,619186,33244],{},[31655,619188,619189,619191],{},[31662,619190,32226],{},[31658,619192,32936],{},[31665,619194,619195],{"encoding":31667},"f(x, s) = x^{s-1} e^{-x} = e^{(s-1)\\ln x} e^{-x}",[34,619197,619199,619235,619323],{"className":619198,"ariaHidden":758},[31672],[34,619200,619202,619205,619208,619211,619214,619217,619220,619223,619226,619229,619232],{"className":619201},[31676],[34,619203],{"className":619204,"style":31774},[31680],[34,619206,31739],{"className":619207,"style":31778},[31685,31686],[34,619209,456],{"className":619210},[31782],[34,619212,32936],{"className":619213},[31685,31686],[34,619215,31960],{"className":619216},[32119],[34,619218],{"className":619219,"style":32123},[31845],[34,619221,31747],{"className":619222},[31685,31686],[34,619224,2557],{"className":619225},[31690],[34,619227],{"className":619228,"style":31846},[31845],[34,619230,202],{"className":619231},[31850],[34,619233],{"className":619234,"style":31846},[31845],[34,619236,619238,619241,619279,619314,619317,619320],{"className":619237},[31676],[34,619239],{"className":619240,"style":37774},[31680],[34,619242,619244,619247],{"className":619243},[31685],[34,619245,32936],{"className":619246},[31685,31686],[34,619248,619250],{"className":619249},[31792],[34,619251,619253],{"className":619252},[31796],[34,619254,619256],{"className":619255},[31801],[34,619257,619259],{"className":619258,"style":37774},[31805],[34,619260,619261,619264],{"style":37777},[34,619262],{"className":619263,"style":31814},[31813],[34,619265,619267],{"className":619266},[31818,31819,31820,31821],[34,619268,619270,619273,619276],{"className":619269},[31685,31821],[34,619271,31747],{"className":619272},[31685,31686,31821],[34,619274,32226],{"className":619275},[32024,31821],[34,619277,640],{"className":619278},[31685,31821],[34,619280,619282,619285],{"className":619281},[31685],[34,619283,33244],{"className":619284},[31685,31686],[34,619286,619288],{"className":619287},[31792],[34,619289,619291],{"className":619290},[31796],[34,619292,619294],{"className":619293},[31801],[34,619295,619297],{"className":619296,"style":156828},[31805],[34,619298,619299,619302],{"style":37777},[34,619300],{"className":619301,"style":31814},[31813],[34,619303,619305],{"className":619304},[31818,31819,31820,31821],[34,619306,619308,619311],{"className":619307},[31685,31821],[34,619309,32226],{"className":619310},[31685,31821],[34,619312,32936],{"className":619313},[31685,31686,31821],[34,619315],{"className":619316,"style":31846},[31845],[34,619318,202],{"className":619319},[31850],[34,619321],{"className":619322,"style":31846},[31845],[34,619324,619326,619329,619391],{"className":619325},[31676],[34,619327],{"className":619328,"style":61037},[31680],[34,619330,619332,619335],{"className":619331},[31685],[34,619333,33244],{"className":619334},[31685,31686],[34,619336,619338],{"className":619337},[31792],[34,619339,619341],{"className":619340},[31796],[34,619342,619344],{"className":619343},[31801],[34,619345,619347],{"className":619346,"style":61037},[31805],[34,619348,619349,619352],{"style":37777},[34,619350],{"className":619351,"style":31814},[31813],[34,619353,619355],{"className":619354},[31818,31819,31820,31821],[34,619356,619358,619361,619364,619367,619370,619373,619376,619385,619388],{"className":619357},[31685,31821],[34,619359,456],{"className":619360},[31782,31821],[34,619362,31747],{"className":619363},[31685,31686,31821],[34,619365,32226],{"className":619366},[32024,31821],[34,619368,640],{"className":619369},[31685,31821],[34,619371,2557],{"className":619372},[31690,31821],[34,619374],{"className":619375,"style":43430},[31845,31821],[34,619377,619379,619382],{"className":619378},[32767,31821],[34,619380,43379],{"className":619381},[31821],[34,619383,31660],{"className":619384},[31821],[34,619386],{"className":619387,"style":43430},[31845,31821],[34,619389,32936],{"className":619390},[31685,31686,31821],[34,619392,619394,619397],{"className":619393},[31685],[34,619395,33244],{"className":619396},[31685,31686],[34,619398,619400],{"className":619399},[31792],[34,619401,619403],{"className":619402},[31796],[34,619404,619406],{"className":619405},[31801],[34,619407,619409],{"className":619408,"style":156828},[31805],[34,619410,619411,619414],{"style":37777},[34,619412],{"className":619413,"style":31814},[31813],[34,619415,619417],{"className":619416},[31818,31819,31820,31821],[34,619418,619420,619423],{"className":619419},[31685,31821],[34,619421,32226],{"className":619422},[31685,31821],[34,619424,32936],{"className":619425},[31685,31686,31821],[108,619427,3101,619428,619456,619457,619812,619813,31901],{},[34,619429,619431,619444],{"className":619430},[31642],[34,619432,619434],{"className":619433},[31646],[31648,619435,619436],{"xmlns":31650},[31652,619437,619438,619442],{},[31655,619439,619440],{},[31658,619441,31747],{},[31665,619443,31747],{"encoding":31667},[34,619445,619447],{"className":619446,"ariaHidden":758},[31672],[34,619448,619450,619453],{"className":619449},[31676],[34,619451],{"className":619452,"style":32527},[31680],[34,619454,31747],{"className":619455},[31685,31686]," 求偏导：",[34,619458,619460,619541],{"className":619459},[31642],[34,619461,619463],{"className":619462},[31646],[31648,619464,619465],{"xmlns":31650},[31652,619466,619467,619538],{},[31655,619468,619469,619480,619482,619494,619504,619506,619508,619520,619530,619532,619534,619536],{},[41453,619470,619471,619474],{},[31658,619472,619473],{"mathvariant":38079},"∂",[31655,619475,619476,619478],{},[31658,619477,619473],{"mathvariant":38079},[31658,619479,31747],{},[31662,619481,456],{"stretchy":702},[37624,619483,619484,619486],{},[31658,619485,32936],{},[31655,619487,619488,619490,619492],{},[31658,619489,31747],{},[31662,619491,32226],{},[31749,619493,640],{},[37624,619495,619496,619498],{},[31658,619497,33244],{},[31655,619499,619500,619502],{},[31662,619501,32226],{},[31658,619503,32936],{},[31662,619505,2557],{"stretchy":702},[31662,619507,202],{},[37624,619509,619510,619512],{},[31658,619511,32936],{},[31655,619513,619514,619516,619518],{},[31658,619515,31747],{},[31662,619517,32226],{},[31749,619519,640],{},[37624,619521,619522,619524],{},[31658,619523,33244],{},[31655,619525,619526,619528],{},[31662,619527,32226],{},[31658,619529,32936],{},[31662,619531,71191],{},[31658,619533,563286],{},[31662,619535,32682],{},[31658,619537,32936],{},[31665,619539,619540],{"encoding":31667},"\\frac{\\partial}{\\partial s} (x^{s-1} e^{-x}) = x^{s-1} e^{-x} \\cdot \\ln x",[34,619542,619544,619709,619797],{"className":619543,"ariaHidden":758},[31672],[34,619545,619547,619550,619621,619624,619662,619697,619700,619703,619706],{"className":619546},[31676],[34,619548],{"className":619549,"style":232492},[31680],[34,619551,619553,619556,619618],{"className":619552},[31685],[34,619554],{"className":619555},[31782,41488],[34,619557,619559],{"className":619558},[41453],[34,619560,619562,619610],{"className":619561},[31796,31797],[34,619563,619565,619607],{"className":619564},[31801],[34,619566,619568,619585,619593],{"className":619567,"style":232511},[31805],[34,619569,619570,619573],{"style":41504},[34,619571],{"className":619572,"style":41508},[31813],[34,619574,619576],{"className":619575},[31818,31819,31820,31821],[34,619577,619579,619582],{"className":619578},[31685,31821],[34,619580,619473],{"className":619581,"style":112322},[31685,31821],[34,619583,31747],{"className":619584},[31685,31686,31821],[34,619586,619587,619590],{"style":41520},[34,619588],{"className":619589,"style":41508},[31813],[34,619591],{"className":619592,"style":41528},[41527],[34,619594,619595,619598],{"style":46771},[34,619596],{"className":619597,"style":41508},[31813],[34,619599,619601],{"className":619600},[31818,31819,31820,31821],[34,619602,619604],{"className":619603},[31685,31821],[34,619605,619473],{"className":619606,"style":112322},[31685,31821],[34,619608,31829],{"className":619609},[31828],[34,619611,619613],{"className":619612},[31801],[34,619614,619616],{"className":619615,"style":41619},[31805],[34,619617],{},[34,619619],{"className":619620},[31690,41488],[34,619622,456],{"className":619623},[31782],[34,619625,619627,619630],{"className":619626},[31685],[34,619628,32936],{"className":619629},[31685,31686],[34,619631,619633],{"className":619632},[31792],[34,619634,619636],{"className":619635},[31796],[34,619637,619639],{"className":619638},[31801],[34,619640,619642],{"className":619641,"style":37774},[31805],[34,619643,619644,619647],{"style":37777},[34,619645],{"className":619646,"style":31814},[31813],[34,619648,619650],{"className":619649},[31818,31819,31820,31821],[34,619651,619653,619656,619659],{"className":619652},[31685,31821],[34,619654,31747],{"className":619655},[31685,31686,31821],[34,619657,32226],{"className":619658},[32024,31821],[34,619660,640],{"className":619661},[31685,31821],[34,619663,619665,619668],{"className":619664},[31685],[34,619666,33244],{"className":619667},[31685,31686],[34,619669,619671],{"className":619670},[31792],[34,619672,619674],{"className":619673},[31796],[34,619675,619677],{"className":619676},[31801],[34,619678,619680],{"className":619679,"style":156828},[31805],[34,619681,619682,619685],{"style":37777},[34,619683],{"className":619684,"style":31814},[31813],[34,619686,619688],{"className":619687},[31818,31819,31820,31821],[34,619689,619691,619694],{"className":619690},[31685,31821],[34,619692,32226],{"className":619693},[31685,31821],[34,619695,32936],{"className":619696},[31685,31686,31821],[34,619698,2557],{"className":619699},[31690],[34,619701],{"className":619702,"style":31846},[31845],[34,619704,202],{"className":619705},[31850],[34,619707],{"className":619708,"style":31846},[31845],[34,619710,619712,619715,619753,619788,619791,619794],{"className":619711},[31676],[34,619713],{"className":619714,"style":37774},[31680],[34,619716,619718,619721],{"className":619717},[31685],[34,619719,32936],{"className":619720},[31685,31686],[34,619722,619724],{"className":619723},[31792],[34,619725,619727],{"className":619726},[31796],[34,619728,619730],{"className":619729},[31801],[34,619731,619733],{"className":619732,"style":37774},[31805],[34,619734,619735,619738],{"style":37777},[34,619736],{"className":619737,"style":31814},[31813],[34,619739,619741],{"className":619740},[31818,31819,31820,31821],[34,619742,619744,619747,619750],{"className":619743},[31685,31821],[34,619745,31747],{"className":619746},[31685,31686,31821],[34,619748,32226],{"className":619749},[32024,31821],[34,619751,640],{"className":619752},[31685,31821],[34,619754,619756,619759],{"className":619755},[31685],[34,619757,33244],{"className":619758},[31685,31686],[34,619760,619762],{"className":619761},[31792],[34,619763,619765],{"className":619764},[31796],[34,619766,619768],{"className":619767},[31801],[34,619769,619771],{"className":619770,"style":156828},[31805],[34,619772,619773,619776],{"style":37777},[34,619774],{"className":619775,"style":31814},[31813],[34,619777,619779],{"className":619778},[31818,31819,31820,31821],[34,619780,619782,619785],{"className":619781},[31685,31821],[34,619783,32226],{"className":619784},[31685,31821],[34,619786,32936],{"className":619787},[31685,31686,31821],[34,619789],{"className":619790,"style":32604},[31845],[34,619792,71191],{"className":619793},[32024],[34,619795],{"className":619796,"style":32604},[31845],[34,619798,619800,619803,619806,619809],{"className":619799},[31676],[34,619801],{"className":619802,"style":31681},[31680],[34,619804,563286],{"className":619805},[32767],[34,619807],{"className":619808,"style":32123},[31845],[34,619810,32936],{"className":619811},[31685,31686]," （指数函数求导法则 ",[34,619814,619816,619856],{"className":619815},[31642],[34,619817,619819],{"className":619818},[31646],[31648,619820,619821],{"xmlns":31650},[31652,619822,619823,619853],{},[31655,619824,619825,619827,619833,619839,619841,619847,619849,619851],{},[31662,619826,456],{"stretchy":702},[37624,619828,619829,619831],{},[31658,619830,111],{},[31658,619832,31747],{},[37624,619834,619835,619837],{},[31662,619836,2557],{"stretchy":702},[31662,619838,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,619840,202],{},[37624,619842,619843,619845],{},[31658,619844,111],{},[31658,619846,31747],{},[31658,619848,563286],{},[31662,619850,32682],{},[31658,619852,111],{},[31665,619854,619855],{"encoding":31667},"(a^s)' = a^s \\ln a",[34,619857,619859,619938],{"className":619858,"ariaHidden":758},[31672],[34,619860,619862,619865,619868,619897,619929,619932,619935],{"className":619861},[31676],[34,619863],{"className":619864,"style":570646},[31680],[34,619866,456],{"className":619867},[31782],[34,619869,619871,619874],{"className":619870},[31685],[34,619872,111],{"className":619873},[31685,31686],[34,619875,619877],{"className":619876},[31792],[34,619878,619880],{"className":619879},[31796],[34,619881,619883],{"className":619882},[31801],[34,619884,619886],{"className":619885,"style":37959},[31805],[34,619887,619888,619891],{"style":37777},[34,619889],{"className":619890,"style":31814},[31813],[34,619892,619894],{"className":619893},[31818,31819,31820,31821],[34,619895,31747],{"className":619896},[31685,31686,31821],[34,619898,619900,619903],{"className":619899},[31690],[34,619901,2557],{"className":619902},[31690],[34,619904,619906],{"className":619905},[31792],[34,619907,619909],{"className":619908},[31796],[34,619910,619912],{"className":619911},[31801],[34,619913,619915],{"className":619914,"style":132027},[31805],[34,619916,619917,619920],{"style":37777},[34,619918],{"className":619919,"style":31814},[31813],[34,619921,619923],{"className":619922},[31818,31819,31820,31821],[34,619924,619926],{"className":619925},[31685,31821],[34,619927,116495],{"className":619928},[31685,31821],[34,619930],{"className":619931,"style":31846},[31845],[34,619933,202],{"className":619934},[31850],[34,619936],{"className":619937,"style":31846},[31845],[34,619939,619941,619944,619973,619976,619979,619982],{"className":619940},[31676],[34,619942],{"className":619943,"style":31681},[31680],[34,619945,619947,619950],{"className":619946},[31685],[34,619948,111],{"className":619949},[31685,31686],[34,619951,619953],{"className":619952},[31792],[34,619954,619956],{"className":619955},[31796],[34,619957,619959],{"className":619958},[31801],[34,619960,619962],{"className":619961,"style":37959},[31805],[34,619963,619964,619967],{"style":37777},[34,619965],{"className":619966,"style":31814},[31813],[34,619968,619970],{"className":619969},[31818,31819,31820,31821],[34,619971,31747],{"className":619972},[31685,31686,31821],[34,619974],{"className":619975,"style":32123},[31845],[34,619977,563286],{"className":619978},[32767],[34,619980],{"className":619981,"style":32123},[31845],[34,619983,111],{"className":619984},[31685,31686],[108,619986,175992,619987,347],{},[34,619988,619990,620062],{"className":619989},[31642],[34,619991,619993],{"className":619992},[31646],[31648,619994,619995],{"xmlns":31650},[31652,619996,619997,620059],{},[31655,619998,619999,620005,620007,620009,620011,620013,620025,620037,620047,620049,620051,620053,620055,620057],{},[37624,620000,620001,620003],{},[31658,620002,564791],{"mathvariant":38079},[31662,620004,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,620006,456],{"stretchy":702},[31658,620008,31747],{},[31662,620010,2557],{"stretchy":702},[31662,620012,202],{},[38846,620014,620015,620017,620019],{},[31662,620016,546295],{},[31749,620018,459],{},[31655,620020,620021,620023],{},[31662,620022,223],{},[31658,620024,138286],{"mathvariant":38079},[37624,620026,620027,620029],{},[31658,620028,32936],{},[31655,620030,620031,620033,620035],{},[31658,620032,31747],{},[31662,620034,32226],{},[31749,620036,640],{},[37624,620038,620039,620041],{},[31658,620040,33244],{},[31655,620042,620043,620045],{},[31662,620044,32226],{},[31658,620046,32936],{},[31658,620048,563286],{},[31662,620050,32682],{},[31658,620052,32936],{},[39361,620054,39363],{},[31658,620056,46900],{},[31658,620058,32936],{},[31665,620060,620061],{"encoding":31667},"\\Gamma'(s) = \\int_0^{+\\infty} x^{s-1} e^{-x} \\ln x \\, dx",[34,620063,620065,620121],{"className":620064,"ariaHidden":758},[31672],[34,620066,620068,620071,620103,620106,620109,620112,620115,620118],{"className":620067},[31676],[34,620069],{"className":620070,"style":570646},[31680],[34,620072,620074,620077],{"className":620073},[31685],[34,620075,564791],{"className":620076},[31685],[34,620078,620080],{"className":620079},[31792],[34,620081,620083],{"className":620082},[31796],[34,620084,620086],{"className":620085},[31801],[34,620087,620089],{"className":620088,"style":132027},[31805],[34,620090,620091,620094],{"style":37777},[34,620092],{"className":620093,"style":31814},[31813],[34,620095,620097],{"className":620096},[31818,31819,31820,31821],[34,620098,620100],{"className":620099},[31685,31821],[34,620101,116495],{"className":620102},[31685,31821],[34,620104,456],{"className":620105},[31782],[34,620107,31747],{"className":620108},[31685,31686],[34,620110,2557],{"className":620111},[31690],[34,620113],{"className":620114,"style":31846},[31845],[34,620116,202],{"className":620117},[31850],[34,620119],{"className":620120,"style":31846},[31845],[34,620122,620124,620127,620184,620187,620225,620260,620263,620266,620269,620272,620275,620278],{"className":620123},[31676],[34,620125],{"className":620126,"style":546531},[31680],[34,620128,620130,620133],{"className":620129},[32767],[34,620131,546295],{"className":620132,"style":546331},[32767,131249,156560],[34,620134,620136],{"className":620135},[31792],[34,620137,620139,620176],{"className":620138},[31796,31797],[34,620140,620142,620173],{"className":620141},[31801],[34,620143,620145,620156],{"className":620144,"style":546550},[31805],[34,620146,620147,620150],{"style":546347},[34,620148],{"className":620149,"style":31814},[31813],[34,620151,620153],{"className":620152},[31818,31819,31820,31821],[34,620154,459],{"className":620155},[31685,31821],[34,620157,620158,620161],{"style":546362},[34,620159],{"className":620160,"style":31814},[31813],[34,620162,620164],{"className":620163},[31818,31819,31820,31821],[34,620165,620167,620170],{"className":620166},[31685,31821],[34,620168,223],{"className":620169},[31685,31821],[34,620171,138286],{"className":620172},[31685,31821],[34,620174,31829],{"className":620175},[31828],[34,620177,620179],{"className":620178},[31801],[34,620180,620182],{"className":620181,"style":546384},[31805],[34,620183],{},[34,620185],{"className":620186,"style":32123},[31845],[34,620188,620190,620193],{"className":620189},[31685],[34,620191,32936],{"className":620192},[31685,31686],[34,620194,620196],{"className":620195},[31792],[34,620197,620199],{"className":620198},[31796],[34,620200,620202],{"className":620201},[31801],[34,620203,620205],{"className":620204,"style":37774},[31805],[34,620206,620207,620210],{"style":37777},[34,620208],{"className":620209,"style":31814},[31813],[34,620211,620213],{"className":620212},[31818,31819,31820,31821],[34,620214,620216,620219,620222],{"className":620215},[31685,31821],[34,620217,31747],{"className":620218},[31685,31686,31821],[34,620220,32226],{"className":620221},[32024,31821],[34,620223,640],{"className":620224},[31685,31821],[34,620226,620228,620231],{"className":620227},[31685],[34,620229,33244],{"className":620230},[31685,31686],[34,620232,620234],{"className":620233},[31792],[34,620235,620237],{"className":620236},[31796],[34,620238,620240],{"className":620239},[31801],[34,620241,620243],{"className":620242,"style":156828},[31805],[34,620244,620245,620248],{"style":37777},[34,620246],{"className":620247,"style":31814},[31813],[34,620249,620251],{"className":620250},[31818,31819,31820,31821],[34,620252,620254,620257],{"className":620253},[31685,31821],[34,620255,32226],{"className":620256},[31685,31821],[34,620258,32936],{"className":620259},[31685,31686,31821],[34,620261],{"className":620262,"style":32123},[31845],[34,620264,563286],{"className":620265},[32767],[34,620267],{"className":620268,"style":32123},[31845],[34,620270,32936],{"className":620271},[31685,31686],[34,620273],{"className":620274,"style":32123},[31845],[34,620276,46900],{"className":620277},[31685,31686],[34,620279,32936],{"className":620280},[31685,31686],[108,620282,620283,620286,620287,620315,620316,620344,620345,347],{},[2627,620284,620285],{},"进一步","：继续求导，",[34,620288,620290,620303],{"className":620289},[31642],[34,620291,620293],{"className":620292},[31646],[31648,620294,620295],{"xmlns":31650},[31652,620296,620297,620301],{},[31655,620298,620299],{},[31658,620300,31660],{},[31665,620302,31660],{"encoding":31667},[34,620304,620306],{"className":620305,"ariaHidden":758},[31672],[34,620307,620309,620312],{"className":620308},[31676],[34,620310],{"className":620311,"style":32527},[31680],[34,620313,31660],{"className":620314},[31685,31686]," 阶导数就是乘 ",[34,620317,620319,620332],{"className":620318},[31642],[34,620320,620322],{"className":620321},[31646],[31648,620323,620324],{"xmlns":31650},[31652,620325,620326,620330],{},[31655,620327,620328],{},[31658,620329,31660],{},[31665,620331,31660],{"encoding":31667},[34,620333,620335],{"className":620334,"ariaHidden":758},[31672],[34,620336,620338,620341],{"className":620337},[31676],[34,620339],{"className":620340,"style":32527},[31680],[34,620342,31660],{"className":620343},[31685,31686]," 次 ",[34,620346,620348,620366],{"className":620347},[31642],[34,620349,620351],{"className":620350},[31646],[31648,620352,620353],{"xmlns":31650},[31652,620354,620355,620363],{},[31655,620356,620357,620359,620361],{},[31658,620358,563286],{},[31662,620360,32682],{},[31658,620362,32936],{},[31665,620364,620365],{"encoding":31667},"\\ln x",[34,620367,620369],{"className":620368,"ariaHidden":758},[31672],[34,620370,620372,620375,620378,620381],{"className":620371},[31676],[34,620373],{"className":620374,"style":31681},[31680],[34,620376,563286],{"className":620377},[32767],[34,620379],{"className":620380,"style":32123},[31845],[34,620382,32936],{"className":620383},[31685,31686],[108,620385,620386,347],{},[34,620387,620389,620475],{"className":620388},[31642],[34,620390,620392],{"className":620391},[31646],[31648,620393,620394],{"xmlns":31650},[31652,620395,620396,620472],{},[31655,620397,620398,620410,620412,620414,620416,620418,620430,620442,620452,620454,620456,620458,620460,620466,620468,620470],{},[37624,620399,620400,620402],{},[31658,620401,564791],{"mathvariant":38079},[31655,620403,620404,620406,620408],{},[31662,620405,456],{"stretchy":702},[31658,620407,31660],{},[31662,620409,2557],{"stretchy":702},[31662,620411,456],{"stretchy":702},[31658,620413,31747],{},[31662,620415,2557],{"stretchy":702},[31662,620417,202],{},[38846,620419,620420,620422,620424],{},[31662,620421,546295],{},[31749,620423,459],{},[31655,620425,620426,620428],{},[31662,620427,223],{},[31658,620429,138286],{"mathvariant":38079},[37624,620431,620432,620434],{},[31658,620433,32936],{},[31655,620435,620436,620438,620440],{},[31658,620437,31747],{},[31662,620439,32226],{},[31749,620441,640],{},[37624,620443,620444,620446],{},[31658,620445,33244],{},[31655,620447,620448,620450],{},[31662,620449,32226],{},[31658,620451,32936],{},[31662,620453,456],{"stretchy":702},[31658,620455,563286],{},[31662,620457,32682],{},[31658,620459,32936],{},[37624,620461,620462,620464],{},[31662,620463,2557],{"stretchy":702},[31658,620465,31660],{},[39361,620467,39363],{},[31658,620469,46900],{},[31658,620471,32936],{},[31665,620473,620474],{"encoding":31667},"\\Gamma^{(n)}(s) = \\int_0^{+\\infty} x^{s-1} e^{-x} (\\ln x)^n \\, dx",[34,620476,620478,620540],{"className":620477,"ariaHidden":758},[31672],[34,620479,620481,620484,620522,620525,620528,620531,620534,620537],{"className":620480},[31676],[34,620482],{"className":620483,"style":61012},[31680],[34,620485,620487,620490],{"className":620486},[31685],[34,620488,564791],{"className":620489},[31685],[34,620491,620493],{"className":620492},[31792],[34,620494,620496],{"className":620495},[31796],[34,620497,620499],{"className":620498},[31801],[34,620500,620502],{"className":620501,"style":61037},[31805],[34,620503,620504,620507],{"style":37777},[34,620505],{"className":620506,"style":31814},[31813],[34,620508,620510],{"className":620509},[31818,31819,31820,31821],[34,620511,620513,620516,620519],{"className":620512},[31685,31821],[34,620514,456],{"className":620515},[31782,31821],[34,620517,31660],{"className":620518},[31685,31686,31821],[34,620520,2557],{"className":620521},[31690,31821],[34,620523,456],{"className":620524},[31782],[34,620526,31747],{"className":620527},[31685,31686],[34,620529,2557],{"className":620530},[31690],[34,620532],{"className":620533,"style":31846},[31845],[34,620535,202],{"className":620536},[31850],[34,620538],{"className":620539,"style":31846},[31845],[34,620541,620543,620546,620603,620606,620644,620679,620682,620685,620688,620691,620720,620723,620726],{"className":620542},[31676],[34,620544],{"className":620545,"style":546531},[31680],[34,620547,620549,620552],{"className":620548},[32767],[34,620550,546295],{"className":620551,"style":546331},[32767,131249,156560],[34,620553,620555],{"className":620554},[31792],[34,620556,620558,620595],{"className":620557},[31796,31797],[34,620559,620561,620592],{"className":620560},[31801],[34,620562,620564,620575],{"className":620563,"style":546550},[31805],[34,620565,620566,620569],{"style":546347},[34,620567],{"className":620568,"style":31814},[31813],[34,620570,620572],{"className":620571},[31818,31819,31820,31821],[34,620573,459],{"className":620574},[31685,31821],[34,620576,620577,620580],{"style":546362},[34,620578],{"className":620579,"style":31814},[31813],[34,620581,620583],{"className":620582},[31818,31819,31820,31821],[34,620584,620586,620589],{"className":620585},[31685,31821],[34,620587,223],{"className":620588},[31685,31821],[34,620590,138286],{"className":620591},[31685,31821],[34,620593,31829],{"className":620594},[31828],[34,620596,620598],{"className":620597},[31801],[34,620599,620601],{"className":620600,"style":546384},[31805],[34,620602],{},[34,620604],{"className":620605,"style":32123},[31845],[34,620607,620609,620612],{"className":620608},[31685],[34,620610,32936],{"className":620611},[31685,31686],[34,620613,620615],{"className":620614},[31792],[34,620616,620618],{"className":620617},[31796],[34,620619,620621],{"className":620620},[31801],[34,620622,620624],{"className":620623,"style":37774},[31805],[34,620625,620626,620629],{"style":37777},[34,620627],{"className":620628,"style":31814},[31813],[34,620630,620632],{"className":620631},[31818,31819,31820,31821],[34,620633,620635,620638,620641],{"className":620634},[31685,31821],[34,620636,31747],{"className":620637},[31685,31686,31821],[34,620639,32226],{"className":620640},[32024,31821],[34,620642,640],{"className":620643},[31685,31821],[34,620645,620647,620650],{"className":620646},[31685],[34,620648,33244],{"className":620649},[31685,31686],[34,620651,620653],{"className":620652},[31792],[34,620654,620656],{"className":620655},[31796],[34,620657,620659],{"className":620658},[31801],[34,620660,620662],{"className":620661,"style":156828},[31805],[34,620663,620664,620667],{"style":37777},[34,620665],{"className":620666,"style":31814},[31813],[34,620668,620670],{"className":620669},[31818,31819,31820,31821],[34,620671,620673,620676],{"className":620672},[31685,31821],[34,620674,32226],{"className":620675},[31685,31821],[34,620677,32936],{"className":620678},[31685,31686,31821],[34,620680,456],{"className":620681},[31782],[34,620683,563286],{"className":620684},[32767],[34,620686],{"className":620687,"style":32123},[31845],[34,620689,32936],{"className":620690},[31685,31686],[34,620692,620694,620697],{"className":620693},[31690],[34,620695,2557],{"className":620696},[31690],[34,620698,620700],{"className":620699},[31792],[34,620701,620703],{"className":620702},[31796],[34,620704,620706],{"className":620705},[31801],[34,620707,620709],{"className":620708,"style":37959},[31805],[34,620710,620711,620714],{"style":37777},[34,620712],{"className":620713,"style":31814},[31813],[34,620715,620717],{"className":620716},[31818,31819,31820,31821],[34,620718,31660],{"className":620719},[31685,31686,31821],[34,620721],{"className":620722,"style":32123},[31845],[34,620724,46900],{"className":620725},[31685,31686],[34,620727,32936],{"className":620728},[31685,31686],[32188,620730],{},[32490,620732,620734,620735],{"id":620733},"t4-证明-limsγsdisplaystyle-lim_s-to-infty-gammas-inftyslimγs","T4. 证明 ",[34,620736,620738,620784],{"className":620737},[31642],[34,620739,620741],{"className":620740},[31646],[31648,620742,620743],{"xmlns":31650},[31652,620744,620745,620781],{},[31655,620746,620747],{},[71090,620748,620749,620767,620769,620771,620773,620775,620777,620779],{"scriptlevel":459,"displaystyle":758},[71115,620750,620751,620757],{},[31655,620752,620753,620755],{},[31658,620754,546745],{},[31662,620756,32682],{},[31655,620758,620759,620761,620763,620765],{},[31658,620760,31747],{},[31662,620762,33239],{},[31662,620764,223],{},[31658,620766,138286],{"mathvariant":38079},[31658,620768,564791],{"mathvariant":38079},[31662,620770,456],{"stretchy":702},[31658,620772,31747],{},[31662,620774,2557],{"stretchy":702},[31662,620776,202],{},[31662,620778,223],{},[31658,620780,138286],{"mathvariant":38079},[31665,620782,620783],{"encoding":31667},"\\displaystyle \\lim_{s \\to +\\infty} \\Gamma(s) = +\\infty",[34,620785,620787,620873],{"className":620786,"ariaHidden":758},[31672],[34,620788,620790,620793,620849,620852,620855,620858,620861,620864,620867,620870],{"className":620789},[31676],[34,620791],{"className":620792,"style":576043},[31680],[34,620794,620796],{"className":620795},[32767,71420],[34,620797,620799,620841],{"className":620798},[31796,31797],[34,620800,620802,620838],{"className":620801},[31801],[34,620803,620805,620828],{"className":620804,"style":31681},[31805],[34,620806,620807,620810],{"style":576058},[34,620808],{"className":620809,"style":41508},[31813],[34,620811,620813],{"className":620812},[31818,31819,31820,31821],[34,620814,620816,620819,620822,620825],{"className":620815},[31685,31821],[34,620817,31747],{"className":620818},[31685,31686,31821],[34,620820,33239],{"className":620821},[31850,31821],[34,620823,223],{"className":620824},[31685,31821],[34,620826,138286],{"className":620827},[31685,31821],[34,620829,620830,620833],{"style":41565},[34,620831],{"className":620832,"style":41508},[31813],[34,620834,620835],{},[34,620836,546745],{"className":620837},[32767],[34,620839,31829],{"className":620840},[31828],[34,620842,620844],{"className":620843},[31801],[34,620845,620847],{"className":620846,"style":576099},[31805],[34,620848],{},[34,620850],{"className":620851,"style":32123},[31845],[34,620853,564791],{"className":620854},[31685],[34,620856,456],{"className":620857},[31782],[34,620859,31747],{"className":620860},[31685,31686],[34,620862,2557],{"className":620863},[31690],[34,620865],{"className":620866,"style":31846},[31845],[34,620868,202],{"className":620869},[31850],[34,620871],{"className":620872,"style":31846},[31845],[34,620874,620876,620879,620882],{"className":620875},[31676],[34,620877],{"className":620878,"style":32597},[31680],[34,620880,223],{"className":620881},[31685],[34,620883,138286],{"className":620884},[31685],[105,620886,620887,621021],{},[108,620888,620889,2664,620892,617872,620992,621020],{},[2627,620890,620891],{},"直观思路",[34,620893,620895,620929],{"className":620894},[31642],[34,620896,620898],{"className":620897},[31646],[31648,620899,620900],{"xmlns":31650},[31652,620901,620902,620926],{},[31655,620903,620904,620906,620908,620910,620912,620914,620916,620918,620920,620922,620924],{},[31658,620905,564791],{"mathvariant":38079},[31662,620907,456],{"stretchy":702},[31658,620909,31660],{},[31662,620911,2557],{"stretchy":702},[31662,620913,202],{},[31662,620915,456],{"stretchy":702},[31658,620917,31660],{},[31662,620919,32226],{},[31749,620921,640],{},[31662,620923,2557],{"stretchy":702},[31662,620925,5504],{"stretchy":702},[31665,620927,620928],{"encoding":31667},"\\Gamma(n) = (n-1)!",[34,620930,620932,620959,620980],{"className":620931,"ariaHidden":758},[31672],[34,620933,620935,620938,620941,620944,620947,620950,620953,620956],{"className":620934},[31676],[34,620936],{"className":620937,"style":31774},[31680],[34,620939,564791],{"className":620940},[31685],[34,620942,456],{"className":620943},[31782],[34,620945,31660],{"className":620946},[31685,31686],[34,620948,2557],{"className":620949},[31690],[34,620951],{"className":620952,"style":31846},[31845],[34,620954,202],{"className":620955},[31850],[34,620957],{"className":620958,"style":31846},[31845],[34,620960,620962,620965,620968,620971,620974,620977],{"className":620961},[31676],[34,620963],{"className":620964,"style":31774},[31680],[34,620966,456],{"className":620967},[31782],[34,620969,31660],{"className":620970},[31685,31686],[34,620972],{"className":620973,"style":32604},[31845],[34,620975,32226],{"className":620976},[32024],[34,620978],{"className":620979,"style":32604},[31845],[34,620981,620983,620986,620989],{"className":620982},[31676],[34,620984],{"className":620985,"style":31774},[31680],[34,620987,640],{"className":620988},[31685],[34,620990,323287],{"className":620991},[31690],[34,620993,620995,621008],{"className":620994},[31642],[34,620996,620998],{"className":620997},[31646],[31648,620999,621000],{"xmlns":31650},[31652,621001,621002,621006],{},[31655,621003,621004],{},[31658,621005,31660],{},[31665,621007,31660],{"encoding":31667},[34,621009,621011],{"className":621010,"ariaHidden":758},[31672],[34,621012,621014,621017],{"className":621013},[31676],[34,621015],{"className":621016,"style":32527},[31680],[34,621018,31660],{"className":621019},[31685,31686]," 很大时，阶乘趋向无穷大。",[108,621022,621023,33183,621026],{},[2627,621024,621025],{},"严格证明",[105,621027,621028,621311,621536,621950,622273],{},[108,621029,621030,621031,621109,621110,347],{},"取 ",[34,621032,621034,621061],{"className":621033},[31642],[34,621035,621037],{"className":621036},[31646],[31648,621038,621039],{"xmlns":31650},[31652,621040,621041,621059],{},[31655,621042,621043,621045,621047,621049,621051,621053,621055,621057],{},[31658,621044,32936],{},[31662,621046,112296],{},[31662,621048,15972],{"stretchy":702},[31749,621050,640],{},[31662,621052,31960],{"separator":758},[31662,621054,223],{},[31658,621056,138286],{"mathvariant":38079},[31662,621058,2557],{"stretchy":702},[31665,621060,567103],{"encoding":31667},[34,621062,621064,621082],{"className":621063,"ariaHidden":758},[31672],[34,621065,621067,621070,621073,621076,621079],{"className":621066},[31676],[34,621068],{"className":621069,"style":34812},[31680],[34,621071,32936],{"className":621072},[31685,31686],[34,621074],{"className":621075,"style":31846},[31845],[34,621077,112296],{"className":621078},[31850],[34,621080],{"className":621081,"style":31846},[31845],[34,621083,621085,621088,621091,621094,621097,621100,621103,621106],{"className":621084},[31676],[34,621086],{"className":621087,"style":31774},[31680],[34,621089,15972],{"className":621090},[31782],[34,621092,640],{"className":621093},[31685],[34,621095,31960],{"className":621096},[32119],[34,621098],{"className":621099,"style":32123},[31845],[34,621101,223],{"className":621102},[31685],[34,621104,138286],{"className":621105},[31685],[34,621107,2557],{"className":621108},[31690]," 时，积分部分 ",[34,621111,621113,621163],{"className":621112},[31642],[34,621114,621116],{"className":621115},[31646],[31648,621117,621118],{"xmlns":31650},[31652,621119,621120,621160],{},[31655,621121,621122,621134,621146,621156,621158],{},[38846,621123,621124,621126,621128],{},[31662,621125,546295],{},[31749,621127,640],{},[31655,621129,621130,621132],{},[31662,621131,223],{},[31658,621133,138286],{"mathvariant":38079},[37624,621135,621136,621138],{},[31658,621137,32936],{},[31655,621139,621140,621142,621144],{},[31658,621141,31747],{},[31662,621143,32226],{},[31749,621145,640],{},[37624,621147,621148,621150],{},[31658,621149,33244],{},[31655,621151,621152,621154],{},[31662,621153,32226],{},[31658,621155,32936],{},[31658,621157,46900],{},[31658,621159,32936],{},[31665,621161,621162],{"encoding":31667},"\\int_1^{+\\infty} x^{s-1} e^{-x} dx",[34,621164,621166],{"className":621165,"ariaHidden":758},[31672],[34,621167,621169,621172,621229,621232,621270,621305,621308],{"className":621168},[31676],[34,621170],{"className":621171,"style":546531},[31680],[34,621173,621175,621178],{"className":621174},[32767],[34,621176,546295],{"className":621177,"style":546331},[32767,131249,156560],[34,621179,621181],{"className":621180},[31792],[34,621182,621184,621221],{"className":621183},[31796,31797],[34,621185,621187,621218],{"className":621186},[31801],[34,621188,621190,621201],{"className":621189,"style":546550},[31805],[34,621191,621192,621195],{"style":546347},[34,621193],{"className":621194,"style":31814},[31813],[34,621196,621198],{"className":621197},[31818,31819,31820,31821],[34,621199,640],{"className":621200},[31685,31821],[34,621202,621203,621206],{"style":546362},[34,621204],{"className":621205,"style":31814},[31813],[34,621207,621209],{"className":621208},[31818,31819,31820,31821],[34,621210,621212,621215],{"className":621211},[31685,31821],[34,621213,223],{"className":621214},[31685,31821],[34,621216,138286],{"className":621217},[31685,31821],[34,621219,31829],{"className":621220},[31828],[34,621222,621224],{"className":621223},[31801],[34,621225,621227],{"className":621226,"style":546384},[31805],[34,621228],{},[34,621230],{"className":621231,"style":32123},[31845],[34,621233,621235,621238],{"className":621234},[31685],[34,621236,32936],{"className":621237},[31685,31686],[34,621239,621241],{"className":621240},[31792],[34,621242,621244],{"className":621243},[31796],[34,621245,621247],{"className":621246},[31801],[34,621248,621250],{"className":621249,"style":37774},[31805],[34,621251,621252,621255],{"style":37777},[34,621253],{"className":621254,"style":31814},[31813],[34,621256,621258],{"className":621257},[31818,31819,31820,31821],[34,621259,621261,621264,621267],{"className":621260},[31685,31821],[34,621262,31747],{"className":621263},[31685,31686,31821],[34,621265,32226],{"className":621266},[32024,31821],[34,621268,640],{"className":621269},[31685,31821],[34,621271,621273,621276],{"className":621272},[31685],[34,621274,33244],{"className":621275},[31685,31686],[34,621277,621279],{"className":621278},[31792],[34,621280,621282],{"className":621281},[31796],[34,621283,621285],{"className":621284},[31801],[34,621286,621288],{"className":621287,"style":156828},[31805],[34,621289,621290,621293],{"style":37777},[34,621291],{"className":621292,"style":31814},[31813],[34,621294,621296],{"className":621295},[31818,31819,31820,31821],[34,621297,621299,621302],{"className":621298},[31685,31821],[34,621300,32226],{"className":621301},[31685,31821],[34,621303,32936],{"className":621304},[31685,31686,31821],[34,621306,46900],{"className":621307},[31685,31686],[34,621309,32936],{"className":621310},[31685,31686],[108,621312,35328,621313,621341,621342,621394,621395,347],{},[34,621314,621316,621329],{"className":621315},[31642],[34,621317,621319],{"className":621318},[31646],[31648,621320,621321],{"xmlns":31650},[31652,621322,621323,621327],{},[31655,621324,621325],{},[31658,621326,31747],{},[31665,621328,31747],{"encoding":31667},[34,621330,621332],{"className":621331,"ariaHidden":758},[31672],[34,621333,621335,621338],{"className":621334},[31676],[34,621336],{"className":621337,"style":32527},[31680],[34,621339,31747],{"className":621340},[31685,31686]," 很大时，选取一个子区间 ",[34,621343,621345,621367],{"className":621344},[31642],[34,621346,621348],{"className":621347},[31646],[31648,621349,621350],{"xmlns":31650},[31652,621351,621352,621364],{},[31655,621353,621354,621356,621358,621360,621362],{},[31662,621355,15972],{"stretchy":702},[31749,621357,768],{},[31662,621359,31960],{"separator":758},[31749,621361,786],{},[31662,621363,32712],{"stretchy":702},[31665,621365,621366],{"encoding":31667},"[2, 3]",[34,621368,621370],{"className":621369,"ariaHidden":758},[31672],[34,621371,621373,621376,621379,621382,621385,621388,621391],{"className":621372},[31676],[34,621374],{"className":621375,"style":31774},[31680],[34,621377,15972],{"className":621378},[31782],[34,621380,768],{"className":621381},[31685],[34,621383,31960],{"className":621384},[32119],[34,621386],{"className":621387,"style":32123},[31845],[34,621389,786],{"className":621390},[31685],[34,621392,32712],{"className":621393},[31690],"。在该区间上 ",[34,621396,621398,621436],{"className":621397},[31642],[34,621399,621401],{"className":621400},[31646],[31648,621402,621403],{"xmlns":31650},[31652,621404,621405,621433],{},[31655,621406,621407,621419,621421],{},[37624,621408,621409,621411],{},[31658,621410,32936],{},[31655,621412,621413,621415,621417],{},[31658,621414,31747],{},[31662,621416,32226],{},[31749,621418,640],{},[31662,621420,43175],{},[37624,621422,621423,621425],{},[31749,621424,768],{},[31655,621426,621427,621429,621431],{},[31658,621428,31747],{},[31662,621430,32226],{},[31749,621432,640],{},[31665,621434,621435],{"encoding":31667},"x^{s-1} \\ge 2^{s-1}",[34,621437,621439,621492],{"className":621438,"ariaHidden":758},[31672],[34,621440,621442,621445,621483,621486,621489],{"className":621441},[31676],[34,621443],{"className":621444,"style":460147},[31680],[34,621446,621448,621451],{"className":621447},[31685],[34,621449,32936],{"className":621450},[31685,31686],[34,621452,621454],{"className":621453},[31792],[34,621455,621457],{"className":621456},[31796],[34,621458,621460],{"className":621459},[31801],[34,621461,621463],{"className":621462,"style":37774},[31805],[34,621464,621465,621468],{"style":37777},[34,621466],{"className":621467,"style":31814},[31813],[34,621469,621471],{"className":621470},[31818,31819,31820,31821],[34,621472,621474,621477,621480],{"className":621473},[31685,31821],[34,621475,31747],{"className":621476},[31685,31686,31821],[34,621478,32226],{"className":621479},[32024,31821],[34,621481,640],{"className":621482},[31685,31821],[34,621484],{"className":621485,"style":31846},[31845],[34,621487,43175],{"className":621488},[31850],[34,621490],{"className":621491,"style":31846},[31845],[34,621493,621495,621498],{"className":621494},[31676],[34,621496],{"className":621497,"style":37774},[31680],[34,621499,621501,621504],{"className":621500},[31685],[34,621502,768],{"className":621503},[31685],[34,621505,621507],{"className":621506},[31792],[34,621508,621510],{"className":621509},[31796],[34,621511,621513],{"className":621512},[31801],[34,621514,621516],{"className":621515,"style":37774},[31805],[34,621517,621518,621521],{"style":37777},[34,621519],{"className":621520,"style":31814},[31813],[34,621522,621524],{"className":621523},[31818,31819,31820,31821],[34,621525,621527,621530,621533],{"className":621526},[31685,31821],[34,621528,31747],{"className":621529},[31685,31686,31821],[34,621531,32226],{"className":621532},[32024,31821],[34,621534,640],{"className":621535},[31685,31821],[108,621537,621538,347],{},[34,621539,621541,621637],{"className":621540},[31642],[34,621542,621544],{"className":621543},[31646],[31648,621545,621546],{"xmlns":31650},[31652,621547,621548,621634],{},[31655,621549,621550,621552,621554,621556,621558,621560,621568,621580,621590,621592,621594,621596,621608,621610,621620,621622,621632],{},[31658,621551,564791],{"mathvariant":38079},[31662,621553,456],{"stretchy":702},[31658,621555,31747],{},[31662,621557,2557],{"stretchy":702},[31662,621559,29622],{},[38846,621561,621562,621564,621566],{},[31662,621563,546295],{},[31749,621565,768],{},[31749,621567,786],{},[37624,621569,621570,621572],{},[31749,621571,768],{},[31655,621573,621574,621576,621578],{},[31658,621575,31747],{},[31662,621577,32226],{},[31749,621579,640],{},[37624,621581,621582,621584],{},[31658,621583,33244],{},[31655,621585,621586,621588],{},[31662,621587,32226],{},[31658,621589,32936],{},[31658,621591,46900],{},[31658,621593,32936],{},[31662,621595,202],{},[37624,621597,621598,621600],{},[31749,621599,768],{},[31655,621601,621602,621604,621606],{},[31658,621603,31747],{},[31662,621605,32226],{},[31749,621607,640],{},[31662,621609,456],{"stretchy":702},[37624,621611,621612,621614],{},[31658,621613,33244],{},[31655,621615,621616,621618],{},[31662,621617,32226],{},[31749,621619,768],{},[31662,621621,32226],{},[37624,621623,621624,621626],{},[31658,621625,33244],{},[31655,621627,621628,621630],{},[31662,621629,32226],{},[31749,621631,786],{},[31662,621633,2557],{"stretchy":702},[31665,621635,621636],{"encoding":31667},"\\Gamma(s) > \\int_2^3 2^{s-1} e^{-x} dx = 2^{s-1} (e^{-2} - e^{-3})",[34,621638,621640,621667,621815,621906],{"className":621639,"ariaHidden":758},[31672],[34,621641,621643,621646,621649,621652,621655,621658,621661,621664],{"className":621642},[31676],[34,621644],{"className":621645,"style":31774},[31680],[34,621647,564791],{"className":621648},[31685],[34,621650,456],{"className":621651},[31782],[34,621653,31747],{"className":621654},[31685,31686],[34,621656,2557],{"className":621657},[31690],[34,621659],{"className":621660,"style":31846},[31845],[34,621662,29622],{"className":621663},[31850],[34,621665],{"className":621666,"style":31846},[31845],[34,621668,621670,621673,621724,621727,621765,621800,621803,621806,621809,621812],{"className":621669},[31676],[34,621671],{"className":621672,"style":563309},[31680],[34,621674,621676,621679],{"className":621675},[32767],[34,621677,546295],{"className":621678,"style":546331},[32767,131249,156560],[34,621680,621682],{"className":621681},[31792],[34,621683,621685,621716],{"className":621684},[31796,31797],[34,621686,621688,621713],{"className":621687},[31801],[34,621689,621691,621702],{"className":621690,"style":563328},[31805],[34,621692,621693,621696],{"style":546347},[34,621694],{"className":621695,"style":31814},[31813],[34,621697,621699],{"className":621698},[31818,31819,31820,31821],[34,621700,768],{"className":621701},[31685,31821],[34,621703,621704,621707],{"style":546362},[34,621705],{"className":621706,"style":31814},[31813],[34,621708,621710],{"className":621709},[31818,31819,31820,31821],[34,621711,786],{"className":621712},[31685,31821],[34,621714,31829],{"className":621715},[31828],[34,621717,621719],{"className":621718},[31801],[34,621720,621722],{"className":621721,"style":546384},[31805],[34,621723],{},[34,621725],{"className":621726,"style":32123},[31845],[34,621728,621730,621733],{"className":621729},[31685],[34,621731,768],{"className":621732},[31685],[34,621734,621736],{"className":621735},[31792],[34,621737,621739],{"className":621738},[31796],[34,621740,621742],{"className":621741},[31801],[34,621743,621745],{"className":621744,"style":37774},[31805],[34,621746,621747,621750],{"style":37777},[34,621748],{"className":621749,"style":31814},[31813],[34,621751,621753],{"className":621752},[31818,31819,31820,31821],[34,621754,621756,621759,621762],{"className":621755},[31685,31821],[34,621757,31747],{"className":621758},[31685,31686,31821],[34,621760,32226],{"className":621761},[32024,31821],[34,621763,640],{"className":621764},[31685,31821],[34,621766,621768,621771],{"className":621767},[31685],[34,621769,33244],{"className":621770},[31685,31686],[34,621772,621774],{"className":621773},[31792],[34,621775,621777],{"className":621776},[31796],[34,621778,621780],{"className":621779},[31801],[34,621781,621783],{"className":621782,"style":156828},[31805],[34,621784,621785,621788],{"style":37777},[34,621786],{"className":621787,"style":31814},[31813],[34,621789,621791],{"className":621790},[31818,31819,31820,31821],[34,621792,621794,621797],{"className":621793},[31685,31821],[34,621795,32226],{"className":621796},[31685,31821],[34,621798,32936],{"className":621799},[31685,31686,31821],[34,621801,46900],{"className":621802},[31685,31686],[34,621804,32936],{"className":621805},[31685,31686],[34,621807],{"className":621808,"style":31846},[31845],[34,621810,202],{"className":621811},[31850],[34,621813],{"className":621814,"style":31846},[31845],[34,621816,621818,621821,621859,621862,621897,621900,621903],{"className":621817},[31676],[34,621819],{"className":621820,"style":37755},[31680],[34,621822,621824,621827],{"className":621823},[31685],[34,621825,768],{"className":621826},[31685],[34,621828,621830],{"className":621829},[31792],[34,621831,621833],{"className":621832},[31796],[34,621834,621836],{"className":621835},[31801],[34,621837,621839],{"className":621838,"style":37774},[31805],[34,621840,621841,621844],{"style":37777},[34,621842],{"className":621843,"style":31814},[31813],[34,621845,621847],{"className":621846},[31818,31819,31820,31821],[34,621848,621850,621853,621856],{"className":621849},[31685,31821],[34,621851,31747],{"className":621852},[31685,31686,31821],[34,621854,32226],{"className":621855},[32024,31821],[34,621857,640],{"className":621858},[31685,31821],[34,621860,456],{"className":621861},[31782],[34,621863,621865,621868],{"className":621864},[31685],[34,621866,33244],{"className":621867},[31685,31686],[34,621869,621871],{"className":621870},[31792],[34,621872,621874],{"className":621873},[31796],[34,621875,621877],{"className":621876},[31801],[34,621878,621880],{"className":621879,"style":37774},[31805],[34,621881,621882,621885],{"style":37777},[34,621883],{"className":621884,"style":31814},[31813],[34,621886,621888],{"className":621887},[31818,31819,31820,31821],[34,621889,621891,621894],{"className":621890},[31685,31821],[34,621892,32226],{"className":621893},[31685,31821],[34,621895,768],{"className":621896},[31685,31821],[34,621898],{"className":621899,"style":32604},[31845],[34,621901,32226],{"className":621902},[32024],[34,621904],{"className":621905,"style":32604},[31845],[34,621907,621909,621912,621947],{"className":621908},[31676],[34,621910],{"className":621911,"style":37755},[31680],[34,621913,621915,621918],{"className":621914},[31685],[34,621916,33244],{"className":621917},[31685,31686],[34,621919,621921],{"className":621920},[31792],[34,621922,621924],{"className":621923},[31796],[34,621925,621927],{"className":621926},[31801],[34,621928,621930],{"className":621929,"style":37774},[31805],[34,621931,621932,621935],{"style":37777},[34,621933],{"className":621934,"style":31814},[31813],[34,621936,621938],{"className":621937},[31818,31819,31820,31821],[34,621939,621941,621944],{"className":621940},[31685,31821],[34,621942,32226],{"className":621943},[31685,31821],[34,621945,786],{"className":621946},[31685,31821],[34,621948,2557],{"className":621949},[31690],[108,621951,35328,621952,4504,622008,622109,622110,347],{},[34,621953,621955,621975],{"className":621954},[31642],[34,621956,621958],{"className":621957},[31646],[31648,621959,621960],{"xmlns":31650},[31652,621961,621962,621972],{},[31655,621963,621964,621966,621968,621970],{},[31658,621965,31747],{},[31662,621967,33239],{},[31662,621969,223],{},[31658,621971,138286],{"mathvariant":38079},[31665,621973,621974],{"encoding":31667},"s \\to +\\infty",[34,621976,621978,621996],{"className":621977,"ariaHidden":758},[31672],[34,621979,621981,621984,621987,621990,621993],{"className":621980},[31676],[34,621982],{"className":621983,"style":32527},[31680],[34,621985,31747],{"className":621986},[31685,31686],[34,621988],{"className":621989,"style":31846},[31845],[34,621991,33239],{"className":621992},[31850],[34,621994],{"className":621995,"style":31846},[31845],[34,621997,621999,622002,622005],{"className":621998},[31676],[34,622000],{"className":622001,"style":32597},[31680],[34,622003,223],{"className":622004},[31685],[34,622006,138286],{"className":622007},[31685],[34,622009,622011,622041],{"className":622010},[31642],[34,622012,622014],{"className":622013},[31646],[31648,622015,622016],{"xmlns":31650},[31652,622017,622018,622038],{},[31655,622019,622020,622032,622034,622036],{},[37624,622021,622022,622024],{},[31749,622023,768],{},[31655,622025,622026,622028,622030],{},[31658,622027,31747],{},[31662,622029,32226],{},[31749,622031,640],{},[31662,622033,33239],{},[31662,622035,223],{},[31658,622037,138286],{"mathvariant":38079},[31665,622039,622040],{"encoding":31667},"2^{s-1} \\to +\\infty",[34,622042,622044,622097],{"className":622043,"ariaHidden":758},[31672],[34,622045,622047,622050,622088,622091,622094],{"className":622046},[31676],[34,622048],{"className":622049,"style":37774},[31680],[34,622051,622053,622056],{"className":622052},[31685],[34,622054,768],{"className":622055},[31685],[34,622057,622059],{"className":622058},[31792],[34,622060,622062],{"className":622061},[31796],[34,622063,622065],{"className":622064},[31801],[34,622066,622068],{"className":622067,"style":37774},[31805],[34,622069,622070,622073],{"style":37777},[34,622071],{"className":622072,"style":31814},[31813],[34,622074,622076],{"className":622075},[31818,31819,31820,31821],[34,622077,622079,622082,622085],{"className":622078},[31685,31821],[34,622080,31747],{"className":622081},[31685,31686,31821],[34,622083,32226],{"className":622084},[32024,31821],[34,622086,640],{"className":622087},[31685,31821],[34,622089],{"className":622090,"style":31846},[31845],[34,622092,33239],{"className":622093},[31850],[34,622095],{"className":622096,"style":31846},[31845],[34,622098,622100,622103,622106],{"className":622099},[31676],[34,622101],{"className":622102,"style":32597},[31680],[34,622104,223],{"className":622105},[31685],[34,622107,138286],{"className":622108},[31685],"，常数 ",[34,622111,622113,622155],{"className":622112},[31642],[34,622114,622116],{"className":622115},[31646],[31648,622117,622118],{"xmlns":31650},[31652,622119,622120,622152],{},[31655,622121,622122,622124,622134,622136,622146,622148,622150],{},[31662,622123,456],{"stretchy":702},[37624,622125,622126,622128],{},[31658,622127,33244],{},[31655,622129,622130,622132],{},[31662,622131,32226],{},[31749,622133,768],{},[31662,622135,32226],{},[37624,622137,622138,622140],{},[31658,622139,33244],{},[31655,622141,622142,622144],{},[31662,622143,32226],{},[31749,622145,786],{},[31662,622147,2557],{"stretchy":702},[31662,622149,29622],{},[31749,622151,459],{},[31665,622153,622154],{"encoding":31667},"(e^{-2}-e^{-3}) > 0",[34,622156,622158,622211,622264],{"className":622157,"ariaHidden":758},[31672],[34,622159,622161,622164,622167,622202,622205,622208],{"className":622160},[31676],[34,622162],{"className":622163,"style":37755},[31680],[34,622165,456],{"className":622166},[31782],[34,622168,622170,622173],{"className":622169},[31685],[34,622171,33244],{"className":622172},[31685,31686],[34,622174,622176],{"className":622175},[31792],[34,622177,622179],{"className":622178},[31796],[34,622180,622182],{"className":622181},[31801],[34,622183,622185],{"className":622184,"style":37774},[31805],[34,622186,622187,622190],{"style":37777},[34,622188],{"className":622189,"style":31814},[31813],[34,622191,622193],{"className":622192},[31818,31819,31820,31821],[34,622194,622196,622199],{"className":622195},[31685,31821],[34,622197,32226],{"className":622198},[31685,31821],[34,622200,768],{"className":622201},[31685,31821],[34,622203],{"className":622204,"style":32604},[31845],[34,622206,32226],{"className":622207},[32024],[34,622209],{"className":622210,"style":32604},[31845],[34,622212,622214,622217,622252,622255,622258,622261],{"className":622213},[31676],[34,622215],{"className":622216,"style":37755},[31680],[34,622218,622220,622223],{"className":622219},[31685],[34,622221,33244],{"className":622222},[31685,31686],[34,622224,622226],{"className":622225},[31792],[34,622227,622229],{"className":622228},[31796],[34,622230,622232],{"className":622231},[31801],[34,622233,622235],{"className":622234,"style":37774},[31805],[34,622236,622237,622240],{"style":37777},[34,622238],{"className":622239,"style":31814},[31813],[34,622241,622243],{"className":622242},[31818,31819,31820,31821],[34,622244,622246,622249],{"className":622245},[31685,31821],[34,622247,32226],{"className":622248},[31685,31821],[34,622250,786],{"className":622251},[31685,31821],[34,622253,2557],{"className":622254},[31690],[34,622256],{"className":622257,"style":31846},[31845],[34,622259,29622],{"className":622260},[31850],[34,622262],{"className":622263,"style":31846},[31845],[34,622265,622267,622270],{"className":622266},[31676],[34,622268],{"className":622269,"style":32617},[31680],[34,622271,459],{"className":622272},[31685],[108,622274,622275,622276,347],{},"所以整体趋向 ",[34,622277,622279,622294],{"className":622278},[31642],[34,622280,622282],{"className":622281},[31646],[31648,622283,622284],{"xmlns":31650},[31652,622285,622286,622292],{},[31655,622287,622288,622290],{},[31662,622289,223],{},[31658,622291,138286],{"mathvariant":38079},[31665,622293,554615],{"encoding":31667},[34,622295,622297],{"className":622296,"ariaHidden":758},[31672],[34,622298,622300,622303,622306],{"className":622299},[31676],[34,622301],{"className":622302,"style":32597},[31680],[34,622304,223],{"className":622305},[31685],[34,622307,138286],{"className":622308},[31685],[32188,622310],{},[32490,622312,622314,622315],{"id":622313},"t5-计算-01lnγxdxdisplaystyle-int_01-ln-gammax-dx01lnγxdx","T5. 计算 ",[34,622316,622318,622358],{"className":622317},[31642],[34,622319,622321],{"className":622320},[31646],[31648,622322,622323],{"xmlns":31650},[31652,622324,622325,622355],{},[31655,622326,622327],{},[71090,622328,622329,622337,622339,622341,622343,622345,622347,622349,622351,622353],{"scriptlevel":459,"displaystyle":758},[38846,622330,622331,622333,622335],{},[31662,622332,546295],{},[31749,622334,459],{},[31749,622336,640],{},[31658,622338,563286],{},[31662,622340,32682],{},[31658,622342,564791],{"mathvariant":38079},[31662,622344,456],{"stretchy":702},[31658,622346,32936],{},[31662,622348,2557],{"stretchy":702},[39361,622350,39363],{},[31658,622352,46900],{},[31658,622354,32936],{},[31665,622356,622357],{"encoding":31667},"\\displaystyle \\int_0^1 \\ln \\Gamma(x) \\, dx",[34,622359,622361],{"className":622360,"ariaHidden":758},[31672],[34,622362,622364,622367,622418,622421,622424,622427,622430,622433,622436,622439,622442,622445],{"className":622363},[31676],[34,622365],{"className":622366,"style":591468},[31680],[34,622368,622370,622373],{"className":622369},[32767],[34,622371,546295],{"className":622372,"style":591475},[32767,131249,131250],[34,622374,622376],{"className":622375},[31792],[34,622377,622379,622410],{"className":622378},[31796,31797],[34,622380,622382,622407],{"className":622381},[31801],[34,622383,622385,622396],{"className":622384,"style":591488},[31805],[34,622386,622387,622390],{"style":591491},[34,622388],{"className":622389,"style":31814},[31813],[34,622391,622393],{"className":622392},[31818,31819,31820,31821],[34,622394,459],{"className":622395},[31685,31821],[34,622397,622398,622401],{"style":591503},[34,622399],{"className":622400,"style":31814},[31813],[34,622402,622404],{"className":622403},[31818,31819,31820,31821],[34,622405,640],{"className":622406},[31685,31821],[34,622408,31829],{"className":622409},[31828],[34,622411,622413],{"className":622412},[31801],[34,622414,622416],{"className":622415,"style":591522},[31805],[34,622417],{},[34,622419],{"className":622420,"style":32123},[31845],[34,622422,563286],{"className":622423},[32767],[34,622425],{"className":622426,"style":32123},[31845],[34,622428,564791],{"className":622429},[31685],[34,622431,456],{"className":622432},[31782],[34,622434,32936],{"className":622435},[31685,31686],[34,622437,2557],{"className":622438},[31690],[34,622440],{"className":622441,"style":32123},[31845],[34,622443,46900],{"className":622444},[31685,31686],[34,622446,32936],{"className":622447},[31685,31686],[15,622449,622450],{},"这是一个经典的高难度积分。",[105,622452,622453,624916],{},[108,622454,622455,33183,622457],{},[2627,622456,289021],{},[2230,622458,622459,622610,622855,623249,623454,623847,623993,624369,624538,624764],{},[108,622460,622461,622462,347],{},"利用对称性，设 ",[34,622463,622465,622505],{"className":622464},[31642],[34,622466,622468],{"className":622467},[31646],[31648,622469,622470],{"xmlns":31650},[31652,622471,622472,622502],{},[31655,622473,622474,622476,622478,622486,622488,622490,622492,622494,622496,622498,622500],{},[31658,622475,106063],{},[31662,622477,202],{},[38846,622479,622480,622482,622484],{},[31662,622481,546295],{},[31749,622483,459],{},[31749,622485,640],{},[31658,622487,563286],{},[31662,622489,32682],{},[31658,622491,564791],{"mathvariant":38079},[31662,622493,456],{"stretchy":702},[31658,622495,32936],{},[31662,622497,2557],{"stretchy":702},[31658,622499,46900],{},[31658,622501,32936],{},[31665,622503,622504],{"encoding":31667},"I = \\int_0^1 \\ln \\Gamma(x) dx",[34,622506,622508,622526],{"className":622507,"ariaHidden":758},[31672],[34,622509,622511,622514,622517,622520,622523],{"className":622510},[31676],[34,622512],{"className":622513,"style":33218},[31680],[34,622515,106063],{"className":622516,"style":67482},[31685,31686],[34,622518],{"className":622519,"style":31846},[31845],[34,622521,202],{"className":622522},[31850],[34,622524],{"className":622525,"style":31846},[31845],[34,622527,622529,622532,622583,622586,622589,622592,622595,622598,622601,622604,622607],{"className":622528},[31676],[34,622530],{"className":622531,"style":563309},[31680],[34,622533,622535,622538],{"className":622534},[32767],[34,622536,546295],{"className":622537,"style":546331},[32767,131249,156560],[34,622539,622541],{"className":622540},[31792],[34,622542,622544,622575],{"className":622543},[31796,31797],[34,622545,622547,622572],{"className":622546},[31801],[34,622548,622550,622561],{"className":622549,"style":563328},[31805],[34,622551,622552,622555],{"style":546347},[34,622553],{"className":622554,"style":31814},[31813],[34,622556,622558],{"className":622557},[31818,31819,31820,31821],[34,622559,459],{"className":622560},[31685,31821],[34,622562,622563,622566],{"style":546362},[34,622564],{"className":622565,"style":31814},[31813],[34,622567,622569],{"className":622568},[31818,31819,31820,31821],[34,622570,640],{"className":622571},[31685,31821],[34,622573,31829],{"className":622574},[31828],[34,622576,622578],{"className":622577},[31801],[34,622579,622581],{"className":622580,"style":546384},[31805],[34,622582],{},[34,622584],{"className":622585,"style":32123},[31845],[34,622587,563286],{"className":622588},[32767],[34,622590],{"className":622591,"style":32123},[31845],[34,622593,564791],{"className":622594},[31685],[34,622596,456],{"className":622597},[31782],[34,622599,32936],{"className":622600},[31685,31686],[34,622602,2557],{"className":622603},[31690],[34,622605,46900],{"className":622606},[31685,31686],[34,622608,32936],{"className":622609},[31685,31686],[108,622611,396859,622612,55016,622685,347],{},[34,622613,622615,622637],{"className":622614},[31642],[34,622616,622618],{"className":622617},[31646],[31648,622619,622620],{"xmlns":31650},[31652,622621,622622,622634],{},[31655,622623,622624,622626,622628,622630,622632],{},[31658,622625,33249],{},[31662,622627,202],{},[31749,622629,640],{},[31662,622631,32226],{},[31658,622633,32936],{},[31665,622635,622636],{"encoding":31667},"t = 1-x",[34,622638,622640,622658,622676],{"className":622639,"ariaHidden":758},[31672],[34,622641,622643,622646,622649,622652,622655],{"className":622642},[31676],[34,622644],{"className":622645,"style":33280},[31680],[34,622647,33249],{"className":622648},[31685,31686],[34,622650],{"className":622651,"style":31846},[31845],[34,622653,202],{"className":622654},[31850],[34,622656],{"className":622657,"style":31846},[31845],[34,622659,622661,622664,622667,622670,622673],{"className":622660},[31676],[34,622662],{"className":622663,"style":35942},[31680],[34,622665,640],{"className":622666},[31685],[34,622668],{"className":622669,"style":32604},[31845],[34,622671,32226],{"className":622672},[32024],[34,622674],{"className":622675,"style":32604},[31845],[34,622677,622679,622682],{"className":622678},[31676],[34,622680],{"className":622681,"style":32527},[31680],[34,622683,32936],{"className":622684},[31685,31686],[34,622686,622688,622732],{"className":622687},[31642],[34,622689,622691],{"className":622690},[31646],[31648,622692,622693],{"xmlns":31650},[31652,622694,622695,622729],{},[31655,622696,622697,622699,622701,622709,622711,622713,622715,622717,622719,622721,622723,622725,622727],{},[31658,622698,106063],{},[31662,622700,202],{},[38846,622702,622703,622705,622707],{},[31662,622704,546295],{},[31749,622706,459],{},[31749,622708,640],{},[31658,622710,563286],{},[31662,622712,32682],{},[31658,622714,564791],{"mathvariant":38079},[31662,622716,456],{"stretchy":702},[31749,622718,640],{},[31662,622720,32226],{},[31658,622722,32936],{},[31662,622724,2557],{"stretchy":702},[31658,622726,46900],{},[31658,622728,32936],{},[31665,622730,622731],{"encoding":31667},"I = \\int_0^1 \\ln \\Gamma(1-x) dx",[34,622733,622735,622753,622837],{"className":622734,"ariaHidden":758},[31672],[34,622736,622738,622741,622744,622747,622750],{"className":622737},[31676],[34,622739],{"className":622740,"style":33218},[31680],[34,622742,106063],{"className":622743,"style":67482},[31685,31686],[34,622745],{"className":622746,"style":31846},[31845],[34,622748,202],{"className":622749},[31850],[34,622751],{"className":622752,"style":31846},[31845],[34,622754,622756,622759,622810,622813,622816,622819,622822,622825,622828,622831,622834],{"className":622755},[31676],[34,622757],{"className":622758,"style":563309},[31680],[34,622760,622762,622765],{"className":622761},[32767],[34,622763,546295],{"className":622764,"style":546331},[32767,131249,156560],[34,622766,622768],{"className":622767},[31792],[34,622769,622771,622802],{"className":622770},[31796,31797],[34,622772,622774,622799],{"className":622773},[31801],[34,622775,622777,622788],{"className":622776,"style":563328},[31805],[34,622778,622779,622782],{"style":546347},[34,622780],{"className":622781,"style":31814},[31813],[34,622783,622785],{"className":622784},[31818,31819,31820,31821],[34,622786,459],{"className":622787},[31685,31821],[34,622789,622790,622793],{"style":546362},[34,622791],{"className":622792,"style":31814},[31813],[34,622794,622796],{"className":622795},[31818,31819,31820,31821],[34,622797,640],{"className":622798},[31685,31821],[34,622800,31829],{"className":622801},[31828],[34,622803,622805],{"className":622804},[31801],[34,622806,622808],{"className":622807,"style":546384},[31805],[34,622809],{},[34,622811],{"className":622812,"style":32123},[31845],[34,622814,563286],{"className":622815},[32767],[34,622817],{"className":622818,"style":32123},[31845],[34,622820,564791],{"className":622821},[31685],[34,622823,456],{"className":622824},[31782],[34,622826,640],{"className":622827},[31685],[34,622829],{"className":622830,"style":32604},[31845],[34,622832,32226],{"className":622833},[32024],[34,622835],{"className":622836,"style":32604},[31845],[34,622838,622840,622843,622846,622849,622852],{"className":622839},[31676],[34,622841],{"className":622842,"style":31774},[31680],[34,622844,32936],{"className":622845},[31685,31686],[34,622847,2557],{"className":622848},[31690],[34,622850,46900],{"className":622851},[31685,31686],[34,622853,32936],{"className":622854},[31685,31686],[108,622856,622857,622858,347],{},"两式相加：",[34,622859,622861,622967],{"className":622860},[31642],[34,622862,622864],{"className":622863},[31646],[31648,622865,622866],{"xmlns":31650},[31652,622867,622868,622964],{},[31655,622869,622870,622872,622874,622876,622884,622886,622888,622890,622892,622894,622896,622898,622900,622902,622904,622906,622908,622910,622912,622914,622916,622918,622920,622922,622924,622932,622934,622936,622938,622940,622942,622944,622946,622948,622950,622952,622954,622956,622958,622960,622962],{},[31749,622871,768],{},[31658,622873,106063],{},[31662,622875,202],{},[38846,622877,622878,622880,622882],{},[31662,622879,546295],{},[31749,622881,459],{},[31749,622883,640],{},[31662,622885,15972],{"stretchy":702},[31658,622887,563286],{},[31662,622889,32682],{},[31658,622891,564791],{"mathvariant":38079},[31662,622893,456],{"stretchy":702},[31658,622895,32936],{},[31662,622897,2557],{"stretchy":702},[31662,622899,223],{},[31658,622901,563286],{},[31662,622903,32682],{},[31658,622905,564791],{"mathvariant":38079},[31662,622907,456],{"stretchy":702},[31749,622909,640],{},[31662,622911,32226],{},[31658,622913,32936],{},[31662,622915,2557],{"stretchy":702},[31662,622917,32712],{"stretchy":702},[31658,622919,46900],{},[31658,622921,32936],{},[31662,622923,202],{},[38846,622925,622926,622928,622930],{},[31662,622927,546295],{},[31749,622929,459],{},[31749,622931,640],{},[31658,622933,563286],{},[31662,622935,32682],{},[31662,622937,15972],{"stretchy":702},[31658,622939,564791],{"mathvariant":38079},[31662,622941,456],{"stretchy":702},[31658,622943,32936],{},[31662,622945,2557],{"stretchy":702},[31658,622947,564791],{"mathvariant":38079},[31662,622949,456],{"stretchy":702},[31749,622951,640],{},[31662,622953,32226],{},[31658,622955,32936],{},[31662,622957,2557],{"stretchy":702},[31662,622959,32712],{"stretchy":702},[31658,622961,46900],{},[31658,622963,32936],{},[31665,622965,622966],{"encoding":31667},"2I = \\int_0^1 [\\ln \\Gamma(x) + \\ln \\Gamma(1-x)] dx = \\int_0^1 \\ln [\\Gamma(x)\\Gamma(1-x)] dx",[34,622968,622970,622991,623078,623108,623135,623231],{"className":622969,"ariaHidden":758},[31672],[34,622971,622973,622976,622979,622982,622985,622988],{"className":622972},[31676],[34,622974],{"className":622975,"style":33218},[31680],[34,622977,768],{"className":622978},[31685],[34,622980,106063],{"className":622981,"style":67482},[31685,31686],[34,622983],{"className":622984,"style":31846},[31845],[34,622986,202],{"className":622987},[31850],[34,622989],{"className":622990,"style":31846},[31845],[34,622992,622994,622997,623048,623051,623054,623057,623060,623063,623066,623069,623072,623075],{"className":622993},[31676],[34,622995],{"className":622996,"style":563309},[31680],[34,622998,623000,623003],{"className":622999},[32767],[34,623001,546295],{"className":623002,"style":546331},[32767,131249,156560],[34,623004,623006],{"className":623005},[31792],[34,623007,623009,623040],{"className":623008},[31796,31797],[34,623010,623012,623037],{"className":623011},[31801],[34,623013,623015,623026],{"className":623014,"style":563328},[31805],[34,623016,623017,623020],{"style":546347},[34,623018],{"className":623019,"style":31814},[31813],[34,623021,623023],{"className":623022},[31818,31819,31820,31821],[34,623024,459],{"className":623025},[31685,31821],[34,623027,623028,623031],{"style":546362},[34,623029],{"className":623030,"style":31814},[31813],[34,623032,623034],{"className":623033},[31818,31819,31820,31821],[34,623035,640],{"className":623036},[31685,31821],[34,623038,31829],{"className":623039},[31828],[34,623041,623043],{"className":623042},[31801],[34,623044,623046],{"className":623045,"style":546384},[31805],[34,623047],{},[34,623049,15972],{"className":623050},[31782],[34,623052,563286],{"className":623053},[32767],[34,623055],{"className":623056,"style":32123},[31845],[34,623058,564791],{"className":623059},[31685],[34,623061,456],{"className":623062},[31782],[34,623064,32936],{"className":623065},[31685,31686],[34,623067,2557],{"className":623068},[31690],[34,623070],{"className":623071,"style":32604},[31845],[34,623073,223],{"className":623074},[32024],[34,623076],{"className":623077,"style":32604},[31845],[34,623079,623081,623084,623087,623090,623093,623096,623099,623102,623105],{"className":623080},[31676],[34,623082],{"className":623083,"style":31774},[31680],[34,623085,563286],{"className":623086},[32767],[34,623088],{"className":623089,"style":32123},[31845],[34,623091,564791],{"className":623092},[31685],[34,623094,456],{"className":623095},[31782],[34,623097,640],{"className":623098},[31685],[34,623100],{"className":623101,"style":32604},[31845],[34,623103,32226],{"className":623104},[32024],[34,623106],{"className":623107,"style":32604},[31845],[34,623109,623111,623114,623117,623120,623123,623126,623129,623132],{"className":623110},[31676],[34,623112],{"className":623113,"style":31774},[31680],[34,623115,32936],{"className":623116},[31685,31686],[34,623118,481211],{"className":623119},[31690],[34,623121,46900],{"className":623122},[31685,31686],[34,623124,32936],{"className":623125},[31685,31686],[34,623127],{"className":623128,"style":31846},[31845],[34,623130,202],{"className":623131},[31850],[34,623133],{"className":623134,"style":31846},[31845],[34,623136,623138,623141,623192,623195,623198,623201,623204,623207,623210,623213,623216,623219,623222,623225,623228],{"className":623137},[31676],[34,623139],{"className":623140,"style":563309},[31680],[34,623142,623144,623147],{"className":623143},[32767],[34,623145,546295],{"className":623146,"style":546331},[32767,131249,156560],[34,623148,623150],{"className":623149},[31792],[34,623151,623153,623184],{"className":623152},[31796,31797],[34,623154,623156,623181],{"className":623155},[31801],[34,623157,623159,623170],{"className":623158,"style":563328},[31805],[34,623160,623161,623164],{"style":546347},[34,623162],{"className":623163,"style":31814},[31813],[34,623165,623167],{"className":623166},[31818,31819,31820,31821],[34,623168,459],{"className":623169},[31685,31821],[34,623171,623172,623175],{"style":546362},[34,623173],{"className":623174,"style":31814},[31813],[34,623176,623178],{"className":623177},[31818,31819,31820,31821],[34,623179,640],{"className":623180},[31685,31821],[34,623182,31829],{"className":623183},[31828],[34,623185,623187],{"className":623186},[31801],[34,623188,623190],{"className":623189,"style":546384},[31805],[34,623191],{},[34,623193],{"className":623194,"style":32123},[31845],[34,623196,563286],{"className":623197},[32767],[34,623199,15972],{"className":623200},[31782],[34,623202,564791],{"className":623203},[31685],[34,623205,456],{"className":623206},[31782],[34,623208,32936],{"className":623209},[31685,31686],[34,623211,2557],{"className":623212},[31690],[34,623214,564791],{"className":623215},[31685],[34,623217,456],{"className":623218},[31782],[34,623220,640],{"className":623221},[31685],[34,623223],{"className":623224,"style":32604},[31845],[34,623226,32226],{"className":623227},[32024],[34,623229],{"className":623230,"style":32604},[31845],[34,623232,623234,623237,623240,623243,623246],{"className":623233},[31676],[34,623235],{"className":623236,"style":31774},[31680],[34,623238,32936],{"className":623239},[31685,31686],[34,623241,481211],{"className":623242},[31690],[34,623244,46900],{"className":623245},[31685,31686],[34,623247,32936],{"className":623248},[31685,31686],[108,623250,600337,623251,347],{},[34,623252,623254,623302],{"className":623253},[31642],[34,623255,623257],{"className":623256},[31646],[31648,623258,623259],{"xmlns":31650},[31652,623260,623261,623299],{},[31655,623262,623263,623265,623267,623269,623271,623273,623275,623277,623279,623281,623283,623285],{},[31658,623264,564791],{"mathvariant":38079},[31662,623266,456],{"stretchy":702},[31658,623268,32936],{},[31662,623270,2557],{"stretchy":702},[31658,623272,564791],{"mathvariant":38079},[31662,623274,456],{"stretchy":702},[31749,623276,640],{},[31662,623278,32226],{},[31658,623280,32936],{},[31662,623282,2557],{"stretchy":702},[31662,623284,202],{},[41453,623286,623287,623289],{},[31658,623288,58442],{},[31655,623290,623291,623293,623295,623297],{},[31658,623292,547853],{},[31662,623294,32682],{},[31658,623296,58442],{},[31658,623298,32936],{},[31665,623300,623301],{"encoding":31667},"\\Gamma(x)\\Gamma(1-x) = \\frac{\\pi}{\\sin \\pi x}",[34,623303,623305,623341,623362],{"className":623304,"ariaHidden":758},[31672],[34,623306,623308,623311,623314,623317,623320,623323,623326,623329,623332,623335,623338],{"className":623307},[31676],[34,623309],{"className":623310,"style":31774},[31680],[34,623312,564791],{"className":623313},[31685],[34,623315,456],{"className":623316},[31782],[34,623318,32936],{"className":623319},[31685,31686],[34,623321,2557],{"className":623322},[31690],[34,623324,564791],{"className":623325},[31685],[34,623327,456],{"className":623328},[31782],[34,623330,640],{"className":623331},[31685],[34,623333],{"className":623334,"style":32604},[31845],[34,623336,32226],{"className":623337},[32024],[34,623339],{"className":623340,"style":32604},[31845],[34,623342,623344,623347,623350,623353,623356,623359],{"className":623343},[31676],[34,623345],{"className":623346,"style":31774},[31680],[34,623348,32936],{"className":623349},[31685,31686],[34,623351,2557],{"className":623352},[31690],[34,623354],{"className":623355,"style":31846},[31845],[34,623357,202],{"className":623358},[31850],[34,623360],{"className":623361,"style":31846},[31845],[34,623363,623365,623368],{"className":623364},[31676],[34,623366],{"className":623367,"style":244926},[31680],[34,623369,623371,623374,623451],{"className":623370},[31685],[34,623372],{"className":623373},[31782,41488],[34,623375,623377],{"className":623376},[41453],[34,623378,623380,623443],{"className":623379},[31796,31797],[34,623381,623383,623440],{"className":623382},[31801],[34,623384,623386,623418,623426],{"className":623385,"style":244945},[31805],[34,623387,623388,623391],{"style":41504},[34,623389],{"className":623390,"style":41508},[31813],[34,623392,623394],{"className":623393},[31818,31819,31820,31821],[34,623395,623397,623409,623412,623415],{"className":623396},[31685,31821],[34,623398,623400,623403,623406],{"className":623399},[32767,31821],[34,623401,31747],{"className":623402},[31821],[34,623404,31955],{"className":623405},[31821],[34,623407,31660],{"className":623408},[31821],[34,623410],{"className":623411,"style":43430},[31845,31821],[34,623413,58442],{"className":623414,"style":32062},[31685,31686,31821],[34,623416,32936],{"className":623417},[31685,31686,31821],[34,623419,623420,623423],{"style":41520},[34,623421],{"className":623422,"style":41508},[31813],[34,623424],{"className":623425,"style":41528},[41527],[34,623427,623428,623431],{"style":46771},[34,623429],{"className":623430,"style":41508},[31813],[34,623432,623434],{"className":623433},[31818,31819,31820,31821],[34,623435,623437],{"className":623436},[31685,31821],[34,623438,58442],{"className":623439,"style":32062},[31685,31686,31821],[34,623441,31829],{"className":623442},[31828],[34,623444,623446],{"className":623445},[31801],[34,623447,623449],{"className":623448,"style":41619},[31805],[34,623450],{},[34,623452],{"className":623453},[31690,41488],[108,623455,623456,347],{},[34,623457,623459,623545],{"className":623458},[31642],[34,623460,623462],{"className":623461},[31646],[31648,623463,623464],{"xmlns":31650},[31652,623465,623466,623542],{},[31655,623467,623468,623470,623472,623474,623482,623484,623486,623500,623502,623504,623506,623514,623516,623518,623520,623522,623524,623526,623528,623530,623532,623534,623536,623538,623540],{},[31749,623469,768],{},[31658,623471,106063],{},[31662,623473,202],{},[38846,623475,623476,623478,623480],{},[31662,623477,546295],{},[31749,623479,459],{},[31749,623481,640],{},[31658,623483,563286],{},[31662,623485,32682],{},[41453,623487,623488,623490],{},[31658,623489,58442],{},[31655,623491,623492,623494,623496,623498],{},[31658,623493,547853],{},[31662,623495,32682],{},[31658,623497,58442],{},[31658,623499,32936],{},[31658,623501,46900],{},[31658,623503,32936],{},[31662,623505,202],{},[38846,623507,623508,623510,623512],{},[31662,623509,546295],{},[31749,623511,459],{},[31749,623513,640],{},[31662,623515,456],{"stretchy":702},[31658,623517,563286],{},[31662,623519,32682],{},[31658,623521,58442],{},[31662,623523,32226],{},[31658,623525,563286],{},[31662,623527,32682],{},[31658,623529,547853],{},[31662,623531,32682],{},[31658,623533,58442],{},[31658,623535,32936],{},[31662,623537,2557],{"stretchy":702},[31658,623539,46900],{},[31658,623541,32936],{},[31665,623543,623544],{"encoding":31667},"2I = \\int_0^1 \\ln \\frac{\\pi}{\\sin \\pi x} dx = \\int_0^1 (\\ln \\pi - \\ln \\sin \\pi x) dx",[34,623546,623548,623569,623736,623814],{"className":623547,"ariaHidden":758},[31672],[34,623549,623551,623554,623557,623560,623563,623566],{"className":623550},[31676],[34,623552],{"className":623553,"style":33218},[31680],[34,623555,768],{"className":623556},[31685],[34,623558,106063],{"className":623559,"style":67482},[31685,31686],[34,623561],{"className":623562,"style":31846},[31845],[34,623564,202],{"className":623565},[31850],[34,623567],{"className":623568,"style":31846},[31845],[34,623570,623572,623575,623626,623629,623632,623635,623721,623724,623727,623730,623733],{"className":623571},[31676],[34,623573],{"className":623574,"style":563309},[31680],[34,623576,623578,623581],{"className":623577},[32767],[34,623579,546295],{"className":623580,"style":546331},[32767,131249,156560],[34,623582,623584],{"className":623583},[31792],[34,623585,623587,623618],{"className":623586},[31796,31797],[34,623588,623590,623615],{"className":623589},[31801],[34,623591,623593,623604],{"className":623592,"style":563328},[31805],[34,623594,623595,623598],{"style":546347},[34,623596],{"className":623597,"style":31814},[31813],[34,623599,623601],{"className":623600},[31818,31819,31820,31821],[34,623602,459],{"className":623603},[31685,31821],[34,623605,623606,623609],{"style":546362},[34,623607],{"className":623608,"style":31814},[31813],[34,623610,623612],{"className":623611},[31818,31819,31820,31821],[34,623613,640],{"className":623614},[31685,31821],[34,623616,31829],{"className":623617},[31828],[34,623619,623621],{"className":623620},[31801],[34,623622,623624],{"className":623623,"style":546384},[31805],[34,623625],{},[34,623627],{"className":623628,"style":32123},[31845],[34,623630,563286],{"className":623631},[32767],[34,623633],{"className":623634,"style":32123},[31845],[34,623636,623638,623641,623718],{"className":623637},[31685],[34,623639],{"className":623640},[31782,41488],[34,623642,623644],{"className":623643},[41453],[34,623645,623647,623710],{"className":623646},[31796,31797],[34,623648,623650,623707],{"className":623649},[31801],[34,623651,623653,623685,623693],{"className":623652,"style":244945},[31805],[34,623654,623655,623658],{"style":41504},[34,623656],{"className":623657,"style":41508},[31813],[34,623659,623661],{"className":623660},[31818,31819,31820,31821],[34,623662,623664,623676,623679,623682],{"className":623663},[31685,31821],[34,623665,623667,623670,623673],{"className":623666},[32767,31821],[34,623668,31747],{"className":623669},[31821],[34,623671,31955],{"className":623672},[31821],[34,623674,31660],{"className":623675},[31821],[34,623677],{"className":623678,"style":43430},[31845,31821],[34,623680,58442],{"className":623681,"style":32062},[31685,31686,31821],[34,623683,32936],{"className":623684},[31685,31686,31821],[34,623686,623687,623690],{"style":41520},[34,623688],{"className":623689,"style":41508},[31813],[34,623691],{"className":623692,"style":41528},[41527],[34,623694,623695,623698],{"style":46771},[34,623696],{"className":623697,"style":41508},[31813],[34,623699,623701],{"className":623700},[31818,31819,31820,31821],[34,623702,623704],{"className":623703},[31685,31821],[34,623705,58442],{"className":623706,"style":32062},[31685,31686,31821],[34,623708,31829],{"className":623709},[31828],[34,623711,623713],{"className":623712},[31801],[34,623714,623716],{"className":623715,"style":41619},[31805],[34,623717],{},[34,623719],{"className":623720},[31690,41488],[34,623722,46900],{"className":623723},[31685,31686],[34,623725,32936],{"className":623726},[31685,31686],[34,623728],{"className":623729,"style":31846},[31845],[34,623731,202],{"className":623732},[31850],[34,623734],{"className":623735,"style":31846},[31845],[34,623737,623739,623742,623793,623796,623799,623802,623805,623808,623811],{"className":623738},[31676],[34,623740],{"className":623741,"style":563309},[31680],[34,623743,623745,623748],{"className":623744},[32767],[34,623746,546295],{"className":623747,"style":546331},[32767,131249,156560],[34,623749,623751],{"className":623750},[31792],[34,623752,623754,623785],{"className":623753},[31796,31797],[34,623755,623757,623782],{"className":623756},[31801],[34,623758,623760,623771],{"className":623759,"style":563328},[31805],[34,623761,623762,623765],{"style":546347},[34,623763],{"className":623764,"style":31814},[31813],[34,623766,623768],{"className":623767},[31818,31819,31820,31821],[34,623769,459],{"className":623770},[31685,31821],[34,623772,623773,623776],{"style":546362},[34,623774],{"className":623775,"style":31814},[31813],[34,623777,623779],{"className":623778},[31818,31819,31820,31821],[34,623780,640],{"className":623781},[31685,31821],[34,623783,31829],{"className":623784},[31828],[34,623786,623788],{"className":623787},[31801],[34,623789,623791],{"className":623790,"style":546384},[31805],[34,623792],{},[34,623794,456],{"className":623795},[31782],[34,623797,563286],{"className":623798},[32767],[34,623800],{"className":623801,"style":32123},[31845],[34,623803,58442],{"className":623804,"style":32062},[31685,31686],[34,623806],{"className":623807,"style":32604},[31845],[34,623809,32226],{"className":623810},[32024],[34,623812],{"className":623813,"style":32604},[31845],[34,623815,623817,623820,623823,623826,623829,623832,623835,623838,623841,623844],{"className":623816},[31676],[34,623818],{"className":623819,"style":31774},[31680],[34,623821,563286],{"className":623822},[32767],[34,623824],{"className":623825,"style":32123},[31845],[34,623827,547853],{"className":623828},[32767],[34,623830],{"className":623831,"style":32123},[31845],[34,623833,58442],{"className":623834,"style":32062},[31685,31686],[34,623836,32936],{"className":623837},[31685,31686],[34,623839,2557],{"className":623840},[31690],[34,623842,46900],{"className":623843},[31685,31686],[34,623845,32936],{"className":623846},[31685,31686],[108,623848,623849,623850,347],{},"第一部分 ",[34,623851,623853,623891],{"className":623852},[31642],[34,623854,623856],{"className":623855},[31646],[31648,623857,623858],{"xmlns":31650},[31652,623859,623860,623888],{},[31655,623861,623862,623870,623872,623874,623876,623878,623880,623882,623884,623886],{},[38846,623863,623864,623866,623868],{},[31662,623865,546295],{},[31749,623867,459],{},[31749,623869,640],{},[31658,623871,563286],{},[31662,623873,32682],{},[31658,623875,58442],{},[31658,623877,46900],{},[31658,623879,32936],{},[31662,623881,202],{},[31658,623883,563286],{},[31662,623885,32682],{},[31658,623887,58442],{},[31665,623889,623890],{"encoding":31667},"\\int_0^1 \\ln \\pi dx = \\ln \\pi",[34,623892,623894,623978],{"className":623893,"ariaHidden":758},[31672],[34,623895,623897,623900,623951,623954,623957,623960,623963,623966,623969,623972,623975],{"className":623896},[31676],[34,623898],{"className":623899,"style":563309},[31680],[34,623901,623903,623906],{"className":623902},[32767],[34,623904,546295],{"className":623905,"style":546331},[32767,131249,156560],[34,623907,623909],{"className":623908},[31792],[34,623910,623912,623943],{"className":623911},[31796,31797],[34,623913,623915,623940],{"className":623914},[31801],[34,623916,623918,623929],{"className":623917,"style":563328},[31805],[34,623919,623920,623923],{"style":546347},[34,623921],{"className":623922,"style":31814},[31813],[34,623924,623926],{"className":623925},[31818,31819,31820,31821],[34,623927,459],{"className":623928},[31685,31821],[34,623930,623931,623934],{"style":546362},[34,623932],{"className":623933,"style":31814},[31813],[34,623935,623937],{"className":623936},[31818,31819,31820,31821],[34,623938,640],{"className":623939},[31685,31821],[34,623941,31829],{"className":623942},[31828],[34,623944,623946],{"className":623945},[31801],[34,623947,623949],{"className":623948,"style":546384},[31805],[34,623950],{},[34,623952],{"className":623953,"style":32123},[31845],[34,623955,563286],{"className":623956},[32767],[34,623958],{"className":623959,"style":32123},[31845],[34,623961,58442],{"className":623962,"style":32062},[31685,31686],[34,623964,46900],{"className":623965},[31685,31686],[34,623967,32936],{"className":623968},[31685,31686],[34,623970],{"className":623971,"style":31846},[31845],[34,623973,202],{"className":623974},[31850],[34,623976],{"className":623977,"style":31846},[31845],[34,623979,623981,623984,623987,623990],{"className":623980},[31676],[34,623982],{"className":623983,"style":31681},[31680],[34,623985,563286],{"className":623986},[32767],[34,623988],{"className":623989,"style":32123},[31845],[34,623991,58442],{"className":623992,"style":32062},[31685,31686],[108,623994,623995,623996,46528,624052,347],{},"第二部分：令 ",[34,623997,623999,624019],{"className":623998},[31642],[34,624000,624002],{"className":624001},[31646],[31648,624003,624004],{"xmlns":31650},[31652,624005,624006,624016],{},[31655,624007,624008,624010,624012,624014],{},[31658,624009,94628],{},[31662,624011,202],{},[31658,624013,58442],{},[31658,624015,32936],{},[31665,624017,624018],{"encoding":31667},"u = \\pi x",[34,624020,624022,624040],{"className":624021,"ariaHidden":758},[31672],[34,624023,624025,624028,624031,624034,624037],{"className":624024},[31676],[34,624026],{"className":624027,"style":32527},[31680],[34,624029,94628],{"className":624030},[31685,31686],[34,624032],{"className":624033,"style":31846},[31845],[34,624035,202],{"className":624036},[31850],[34,624038],{"className":624039,"style":31846},[31845],[34,624041,624043,624046,624049],{"className":624042},[31676],[34,624044],{"className":624045,"style":32527},[31680],[34,624047,58442],{"className":624048,"style":32062},[31685,31686],[34,624050,32936],{"className":624051},[31685,31686],[34,624053,624055,624121],{"className":624054},[31642],[34,624056,624058],{"className":624057},[31646],[31648,624059,624060],{"xmlns":31650},[31652,624061,624062,624118],{},[31655,624063,624064,624072,624074,624076,624078,624080,624082,624084,624086,624088,624090,624096,624104,624106,624108,624110,624112,624114,624116],{},[38846,624065,624066,624068,624070],{},[31662,624067,546295],{},[31749,624069,459],{},[31749,624071,640],{},[31658,624073,563286],{},[31662,624075,32682],{},[31658,624077,547853],{},[31662,624079,32682],{},[31658,624081,58442],{},[31658,624083,32936],{},[31658,624085,46900],{},[31658,624087,32936],{},[31662,624089,202],{},[41453,624091,624092,624094],{},[31749,624093,640],{},[31658,624095,58442],{},[38846,624097,624098,624100,624102],{},[31662,624099,546295],{},[31749,624101,459],{},[31658,624103,58442],{},[31658,624105,563286],{},[31662,624107,32682],{},[31658,624109,547853],{},[31662,624111,32682],{},[31658,624113,94628],{},[31658,624115,46900],{},[31658,624117,94628],{},[31665,624119,624120],{"encoding":31667},"\\int_0^1 \\ln \\sin \\pi x dx = \\frac{1}{\\pi} \\int_0^\\pi \\ln \\sin u du",[34,624122,624124,624217],{"className":624123,"ariaHidden":758},[31672],[34,624125,624127,624130,624181,624184,624187,624190,624193,624196,624199,624202,624205,624208,624211,624214],{"className":624126},[31676],[34,624128],{"className":624129,"style":563309},[31680],[34,624131,624133,624136],{"className":624132},[32767],[34,624134,546295],{"className":624135,"style":546331},[32767,131249,156560],[34,624137,624139],{"className":624138},[31792],[34,624140,624142,624173],{"className":624141},[31796,31797],[34,624143,624145,624170],{"className":624144},[31801],[34,624146,624148,624159],{"className":624147,"style":563328},[31805],[34,624149,624150,624153],{"style":546347},[34,624151],{"className":624152,"style":31814},[31813],[34,624154,624156],{"className":624155},[31818,31819,31820,31821],[34,624157,459],{"className":624158},[31685,31821],[34,624160,624161,624164],{"style":546362},[34,624162],{"className":624163,"style":31814},[31813],[34,624165,624167],{"className":624166},[31818,31819,31820,31821],[34,624168,640],{"className":624169},[31685,31821],[34,624171,31829],{"className":624172},[31828],[34,624174,624176],{"className":624175},[31801],[34,624177,624179],{"className":624178,"style":546384},[31805],[34,624180],{},[34,624182],{"className":624183,"style":32123},[31845],[34,624185,563286],{"className":624186},[32767],[34,624188],{"className":624189,"style":32123},[31845],[34,624191,547853],{"className":624192},[32767],[34,624194],{"className":624195,"style":32123},[31845],[34,624197,58442],{"className":624198,"style":32062},[31685,31686],[34,624200,32936],{"className":624201},[31685,31686],[34,624203,46900],{"className":624204},[31685,31686],[34,624206,32936],{"className":624207},[31685,31686],[34,624209],{"className":624210,"style":31846},[31845],[34,624212,202],{"className":624213},[31850],[34,624215],{"className":624216,"style":31846},[31845],[34,624218,624220,624223,624291,624294,624345,624348,624351,624354,624357,624360,624363,624366],{"className":624219},[31676],[34,624221],{"className":624222,"style":613610},[31680],[34,624224,624226,624229,624288],{"className":624225},[31685],[34,624227],{"className":624228},[31782,41488],[34,624230,624232],{"className":624231},[41453],[34,624233,624235,624280],{"className":624234},[31796,31797],[34,624236,624238,624277],{"className":624237},[31801],[34,624239,624241,624255,624263],{"className":624240,"style":46746},[31805],[34,624242,624243,624246],{"style":41504},[34,624244],{"className":624245,"style":41508},[31813],[34,624247,624249],{"className":624248},[31818,31819,31820,31821],[34,624250,624252],{"className":624251},[31685,31821],[34,624253,58442],{"className":624254,"style":32062},[31685,31686,31821],[34,624256,624257,624260],{"style":41520},[34,624258],{"className":624259,"style":41508},[31813],[34,624261],{"className":624262,"style":41528},[41527],[34,624264,624265,624268],{"style":46771},[34,624266],{"className":624267,"style":41508},[31813],[34,624269,624271],{"className":624270},[31818,31819,31820,31821],[34,624272,624274],{"className":624273},[31685,31821],[34,624275,640],{"className":624276},[31685,31821],[34,624278,31829],{"className":624279},[31828],[34,624281,624283],{"className":624282},[31801],[34,624284,624286],{"className":624285,"style":41619},[31805],[34,624287],{},[34,624289],{"className":624290},[31690,41488],[34,624292],{"className":624293,"style":32123},[31845],[34,624295,624297,624300],{"className":624296},[32767],[34,624298,546295],{"className":624299,"style":546331},[32767,131249,156560],[34,624301,624303],{"className":624302},[31792],[34,624304,624306,624337],{"className":624305},[31796,31797],[34,624307,624309,624334],{"className":624308},[31801],[34,624310,624312,624323],{"className":624311,"style":567928},[31805],[34,624313,624314,624317],{"style":546347},[34,624315],{"className":624316,"style":31814},[31813],[34,624318,624320],{"className":624319},[31818,31819,31820,31821],[34,624321,459],{"className":624322},[31685,31821],[34,624324,624325,624328],{"style":546362},[34,624326],{"className":624327,"style":31814},[31813],[34,624329,624331],{"className":624330},[31818,31819,31820,31821],[34,624332,58442],{"className":624333,"style":32062},[31685,31686,31821],[34,624335,31829],{"className":624336},[31828],[34,624338,624340],{"className":624339},[31801],[34,624341,624343],{"className":624342,"style":546384},[31805],[34,624344],{},[34,624346],{"className":624347,"style":32123},[31845],[34,624349,563286],{"className":624350},[32767],[34,624352],{"className":624353,"style":32123},[31845],[34,624355,547853],{"className":624356},[32767],[34,624358],{"className":624359,"style":32123},[31845],[34,624361,94628],{"className":624362},[31685,31686],[34,624364,46900],{"className":624365},[31685,31686],[34,624367,94628],{"className":624368},[31685,31686],[108,624370,624371,624372,347],{},"已知经典积分 ",[34,624373,624375,624421],{"className":624374},[31642],[34,624376,624378],{"className":624377},[31646],[31648,624379,624380],{"xmlns":31650},[31652,624381,624382,624418],{},[31655,624383,624384,624392,624394,624396,624398,624400,624402,624404,624406,624408,624410,624412,624414,624416],{},[38846,624385,624386,624388,624390],{},[31662,624387,546295],{},[31749,624389,459],{},[31658,624391,58442],{},[31658,624393,563286],{},[31662,624395,32682],{},[31658,624397,547853],{},[31662,624399,32682],{},[31658,624401,94628],{},[31658,624403,46900],{},[31658,624405,94628],{},[31662,624407,202],{},[31662,624409,32226],{},[31658,624411,58442],{},[31658,624413,563286],{},[31662,624415,32682],{},[31749,624417,768],{},[31665,624419,624420],{"encoding":31667},"\\int_0^\\pi \\ln \\sin u du = -\\pi \\ln 2",[34,624422,624424,624514],{"className":624423,"ariaHidden":758},[31672],[34,624425,624427,624430,624481,624484,624487,624490,624493,624496,624499,624502,624505,624508,624511],{"className":624426},[31676],[34,624428],{"className":624429,"style":613610},[31680],[34,624431,624433,624436],{"className":624432},[32767],[34,624434,546295],{"className":624435,"style":546331},[32767,131249,156560],[34,624437,624439],{"className":624438},[31792],[34,624440,624442,624473],{"className":624441},[31796,31797],[34,624443,624445,624470],{"className":624444},[31801],[34,624446,624448,624459],{"className":624447,"style":567928},[31805],[34,624449,624450,624453],{"style":546347},[34,624451],{"className":624452,"style":31814},[31813],[34,624454,624456],{"className":624455},[31818,31819,31820,31821],[34,624457,459],{"className":624458},[31685,31821],[34,624460,624461,624464],{"style":546362},[34,624462],{"className":624463,"style":31814},[31813],[34,624465,624467],{"className":624466},[31818,31819,31820,31821],[34,624468,58442],{"className":624469,"style":32062},[31685,31686,31821],[34,624471,31829],{"className":624472},[31828],[34,624474,624476],{"className":624475},[31801],[34,624477,624479],{"className":624478,"style":546384},[31805],[34,624480],{},[34,624482],{"className":624483,"style":32123},[31845],[34,624485,563286],{"className":624486},[32767],[34,624488],{"className":624489,"style":32123},[31845],[34,624491,547853],{"className":624492},[32767],[34,624494],{"className":624495,"style":32123},[31845],[34,624497,94628],{"className":624498},[31685,31686],[34,624500,46900],{"className":624501},[31685,31686],[34,624503,94628],{"className":624504},[31685,31686],[34,624506],{"className":624507,"style":31846},[31845],[34,624509,202],{"className":624510},[31850],[34,624512],{"className":624513,"style":31846},[31845],[34,624515,624517,624520,624523,624526,624529,624532,624535],{"className":624516},[31676],[34,624518],{"className":624519,"style":34299},[31680],[34,624521,32226],{"className":624522},[31685],[34,624524,58442],{"className":624525,"style":32062},[31685,31686],[34,624527],{"className":624528,"style":32123},[31845],[34,624530,563286],{"className":624531},[32767],[34,624533],{"className":624534,"style":32123},[31845],[34,624536,768],{"className":624537},[31685],[108,624539,175992,624540,347],{},[34,624541,624543,624611],{"className":624542},[31642],[34,624544,624546],{"className":624545},[31646],[31648,624547,624548],{"xmlns":31650},[31652,624549,624550,624608],{},[31655,624551,624552,624554,624556,624558,624560,624562,624564,624566,624568,624570,624572,624574,624576,624578,624580,624582,624584,624586,624588,624590,624592,624594,624596,624598,624600,624602,624604,624606],{},[31749,624553,768],{},[31658,624555,106063],{},[31662,624557,202],{},[31658,624559,563286],{},[31662,624561,32682],{},[31658,624563,58442],{},[31662,624565,32226],{},[31662,624567,456],{"stretchy":702},[31662,624569,32226],{},[31658,624571,563286],{},[31662,624573,32682],{},[31749,624575,768],{},[31662,624577,2557],{"stretchy":702},[31662,624579,202],{},[31658,624581,563286],{},[31662,624583,32682],{},[31658,624585,58442],{},[31662,624587,223],{},[31658,624589,563286],{},[31662,624591,32682],{},[31749,624593,768],{},[31662,624595,202],{},[31658,624597,563286],{},[31662,624599,32682],{},[31662,624601,456],{"stretchy":702},[31749,624603,768],{},[31658,624605,58442],{},[31662,624607,2557],{"stretchy":702},[31665,624609,624610],{"encoding":31667},"2I = \\ln \\pi - (-\\ln 2) = \\ln \\pi + \\ln 2 = 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= \\frac{1}{2} 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积分技巧）。\n",[2230,625399,625400,625613,625928,626037],{},[108,625401,396859,625402,347],{},[34,625403,625405,625451],{"className":625404},[31642],[34,625406,625408],{"className":625407},[31646],[31648,625409,625410],{"xmlns":31650},[31652,625411,625412,625448],{},[31655,625413,625414,625416,625418,625424,625426,625428,625430,625436,625438,625440,625442],{},[31658,625415,94628],{},[31662,625417,202],{},[37624,625419,625420,625422],{},[31658,625421,32936],{},[31658,625423,217392],{},[31662,625425,31960],{"separator":758},[31658,625427,56782],{},[31662,625429,202],{},[37624,625431,625432,625434],{},[31658,625433,67764],{},[31658,625435,217691],{},[31662,625437,31960],{"separator":758},[31658,625439,47077],{},[31662,625441,202],{},[37624,625443,625444,625446],{},[31658,625445,382333],{},[31658,625447,625151],{},[31665,625449,625450],{"encoding":31667},"u = x^\\alpha, v = y^\\beta, w = z^\\gamma",[34,625452,625454,625472,625525,625578],{"className":625453,"ariaHidden":758},[31672],[34,625455,625457,625460,625463,625466,625469],{"className":625456},[31676],[34,625458],{"className":625459,"style":32527},[31680],[34,625461,94628],{"className":625462},[31685,31686],[34,625464],{"className":625465,"style":31846},[31845],[34,625467,202],{"className":625468},[31850],[34,625470],{"className":625471,"style":31846},[31845],[34,625473,625475,625478,625507,625510,625513,625516,625519,625522],{"className":625474},[31676],[34,625476],{"className":625477,"style":316112},[31680],[34,625479,625481,625484],{"className":625480},[31685],[34,625482,32936],{"className":625483},[31685,31686],[34,625485,625487],{"className":625486},[31792],[34,625488,625490],{"className":625489},[31796],[34,625491,625493],{"className":625492},[31801],[34,625494,625496],{"className":625495,"style":37959},[31805],[34,625497,625498,625501],{"style":37777},[34,625499],{"className":625500,"style":31814},[31813],[34,625502,625504],{"className":625503},[31818,31819,31820,31821],[34,625505,217392],{"className":625506,"style":217496},[31685,31686,31821],[34,625508,31960],{"className":625509},[32119],[34,625511],{"className":625512,"style":32123},[31845],[34,625514,56782],{"className":625515,"style":32062},[31685,31686],[34,625517],{"className":625518,"style":31846},[31845],[34,625520,202],{"className":625521},[31850],[34,625523],{"className":625524,"style":31846},[31845],[34,625526,625528,625531,625560,625563,625566,625569,625572,625575],{"className":625527},[31676],[34,625529],{"className":625530,"style":315792},[31680],[34,625532,625534,625537],{"className":625533},[31685],[34,625535,67764],{"className":625536,"style":32062},[31685,31686],[34,625538,625540],{"className":625539},[31792],[34,625541,625543],{"className":625542},[31796],[34,625544,625546],{"className":625545},[31801],[34,625547,625549],{"className":625548,"style":38486},[31805],[34,625550,625551,625554],{"style":37777},[34,625552],{"className":625553,"style":31814},[31813],[34,625555,625557],{"className":625556},[31818,31819,31820,31821],[34,625558,217691],{"className":625559,"style":217853},[31685,31686,31821],[34,625561,31960],{"className":625562},[32119],[34,625564],{"className":625565,"style":32123},[31845],[34,625567,47077],{"className":625568,"style":47120},[31685,31686],[34,625570],{"className":625571,"style":31846},[31845],[34,625573,202],{"className":625574},[31850],[34,625576],{"className":625577,"style":31846},[31845],[34,625579,625581,625584],{"className":625580},[31676],[34,625582],{"className":625583,"style":37959},[31680],[34,625585,625587,625590],{"className":625586},[31685],[34,625588,382333],{"className":625589,"style":382406},[31685,31686],[34,625591,625593],{"className":625592},[31792],[34,625594,625596],{"className":625595},[31796],[34,625597,625599],{"className":625598},[31801],[34,625600,625602],{"className":625601,"style":37959},[31805],[34,625603,625604,625607],{"style":37777},[34,625605],{"className":625606,"style":31814},[31813],[34,625608,625610],{"className":625609},[31818,31819,31820,31821],[34,625611,625151],{"className":625612,"style":112322},[31685,31686,31821],[108,625614,625615,625885,625886,347],{},[34,625616,625618,625680],{"className":625617},[31642],[34,625619,625621],{"className":625620},[31646],[31648,625622,625623],{"xmlns":31650},[31652,625624,625625,625677],{},[31655,625626,625627,625629,625631,625643,625645,625647,625649,625651,625657,625673,625675],{},[31658,625628,32936],{},[31662,625630,202],{},[37624,625632,625633,625635],{},[31658,625634,94628],{},[31655,625636,625637,625639,625641],{},[31749,625638,640],{},[31658,625640,38080],{"mathvariant":38079},[31658,625642,217392],{},[31662,625644,31960],{"separator":758},[31658,625646,46900],{},[31658,625648,32936],{},[31662,625650,202],{},[41453,625652,625653,625655],{},[31749,625654,640],{},[31658,625656,217392],{},[37624,625658,625659,625661],{},[31658,625660,94628],{},[31655,625662,625663,625665,625667,625669,625671],{},[31749,625664,640],{},[31658,625666,38080],{"mathvariant":38079},[31658,625668,217392],{},[31662,625670,32226],{},[31749,625672,640],{},[31658,625674,46900],{},[31658,625676,94628],{},[31665,625678,625679],{"encoding":31667},"x = u^{1\u002F\\alpha}, dx = \\frac{1}{\\alpha} u^{1\u002F\\alpha - 1} du",[34,625681,625683,625701,625764],{"className":625682,"ariaHidden":758},[31672],[34,625684,625686,625689,625692,625695,625698],{"className":625685},[31676],[34,625687],{"className":625688,"style":32527},[31680],[34,625690,32936],{"className":625691},[31685,31686],[34,625693],{"className":625694,"style":31846},[31845],[34,625696,202],{"className":625697},[31850],[34,625699],{"className":625700,"style":31846},[31845],[34,625702,625704,625708,625743,625746,625749,625752,625755,625758,625761],{"className":625703},[31676],[34,625705],{"className":625706,"style":625707},[31680],"height:1.0824em;vertical-align:-0.1944em;",[34,625709,625711,625714],{"className":625710},[31685],[34,625712,94628],{"className":625713},[31685,31686],[34,625715,625717],{"className":625716},[31792],[34,625718,625720],{"className":625719},[31796],[34,625721,625723],{"className":625722},[31801],[34,625724,625726],{"className":625725,"style":61037},[31805],[34,625727,625728,625731],{"style":37777},[34,625729],{"className":625730,"style":31814},[31813],[34,625732,625734],{"className":625733},[31818,31819,31820,31821],[34,625735,625737,625740],{"className":625736},[31685,31821],[34,625738,249219],{"className":625739},[31685,31821],[34,625741,217392],{"className":625742,"style":217496},[31685,31686,31821],[34,625744,31960],{"className":625745},[32119],[34,625747],{"className":625748,"style":32123},[31845],[34,625750,46900],{"className":625751},[31685,31686],[34,625753,32936],{"className":625754},[31685,31686],[34,625756],{"className":625757,"style":31846},[31845],[34,625759,202],{"className":625760},[31850],[34,625762],{"className":625763,"style":31846},[31845],[34,625765,625767,625770,625838,625879,625882],{"className":625766},[31676],[34,625768],{"className":625769,"style":554457},[31680],[34,625771,625773,625776,625835],{"className":625772},[31685],[34,625774],{"className":625775},[31782,41488],[34,625777,625779],{"className":625778},[41453],[34,625780,625782,625827],{"className":625781},[31796,31797],[34,625783,625785,625824],{"className":625784},[31801],[34,625786,625788,625802,625810],{"className":625787,"style":46746},[31805],[34,625789,625790,625793],{"style":41504},[34,625791],{"className":625792,"style":41508},[31813],[34,625794,625796],{"className":625795},[31818,31819,31820,31821],[34,625797,625799],{"className":625798},[31685,31821],[34,625800,217392],{"className":625801,"style":217496},[31685,31686,31821],[34,625803,625804,625807],{"style":41520},[34,625805],{"className":625806,"style":41508},[31813],[34,625808],{"className":625809,"style":41528},[41527],[34,625811,625812,625815],{"style":46771},[34,625813],{"className":625814,"style":41508},[31813],[34,625816,625818],{"className":625817},[31818,31819,31820,31821],[34,625819,625821],{"className":625820},[31685,31821],[34,625822,640],{"className":625823},[31685,31821],[34,625825,31829],{"className":625826},[31828],[34,625828,625830],{"className":625829},[31801],[34,625831,625833],{"className":625832,"style":41619},[31805],[34,625834],{},[34,625836],{"className":625837},[31690,41488],[34,625839,625841,625844],{"className":625840},[31685],[34,625842,94628],{"className":625843},[31685,31686],[34,625845,625847],{"className":625846},[31792],[34,625848,625850],{"className":625849},[31796],[34,625851,625853],{"className":625852},[31801],[34,625854,625856],{"className":625855,"style":61037},[31805],[34,625857,625858,625861],{"style":37777},[34,625859],{"className":625860,"style":31814},[31813],[34,625862,625864],{"className":625863},[31818,31819,31820,31821],[34,625865,625867,625870,625873,625876],{"className":625866},[31685,31821],[34,625868,249219],{"className":625869},[31685,31821],[34,625871,217392],{"className":625872,"style":217496},[31685,31686,31821],[34,625874,32226],{"className":625875},[32024,31821],[34,625877,640],{"className":625878},[31685,31821],[34,625880,46900],{"className":625881},[31685,31686],[34,625883,94628],{"className":625884},[31685,31686],"。同理处理 ",[34,625887,625889,625907],{"className":625888},[31642],[34,625890,625892],{"className":625891},[31646],[31648,625893,625894],{"xmlns":31650},[31652,625895,625896,625904],{},[31655,625897,625898,625900,625902],{},[31658,625899,67764],{},[31662,625901,31960],{"separator":758},[31658,625903,382333],{},[31665,625905,625906],{"encoding":31667},"y, z",[34,625908,625910],{"className":625909,"ariaHidden":758},[31672],[34,625911,625913,625916,625919,625922,625925],{"className":625912},[31676],[34,625914],{"className":625915,"style":38774},[31680],[34,625917,67764],{"className":625918,"style":32062},[31685,31686],[34,625920,31960],{"className":625921},[32119],[34,625923],{"className":625924,"style":32123},[31845],[34,625926,382333],{"className":625927,"style":382406},[31685,31686],[108,625929,625930,625931,625959,625960,347],{},"区域 ",[34,625932,625934,625947],{"className":625933},[31642],[34,625935,625937],{"className":625936},[31646],[31648,625938,625939],{"xmlns":31650},[31652,625940,625941,625945],{},[31655,625942,625943],{},[31658,625944,58917],{"mathvariant":38079},[31665,625946,59774],{"encoding":31667},[34,625948,625950],{"className":625949,"ariaHidden":758},[31672],[34,625951,625953,625956],{"className":625952},[31676],[34,625954],{"className":625955,"style":33218},[31680],[34,625957,58917],{"className":625958},[31685]," 变为 ",[34,625961,625963,625989],{"className":625962},[31642],[34,625964,625966],{"className":625965},[31646],[31648,625967,625968],{"xmlns":31650},[31652,625969,625970,625986],{},[31655,625971,625972,625974,625976,625978,625980,625982,625984],{},[31658,625973,94628],{},[31662,625975,31960],{"separator":758},[31658,625977,56782],{},[31662,625979,31960],{"separator":758},[31658,625981,47077],{},[31662,625983,29622],{},[31749,625985,459],{},[31665,625987,625988],{"encoding":31667},"u,v,w > 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= \\frac{1}{\\alpha\\beta\\gamma} \\iiint \\frac{u^{\\frac{1}{\\alpha}-1} v^{\\frac{1}{\\beta}-1} w^{\\frac{1}{\\gamma}-1}}{1 + u + v + w} du dv 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3",[34,626742,626744,626827,626910,626993],{"className":626743,"ariaHidden":758},[31672],[34,626745,626747,626750,626818,626821,626824],{"className":626746},[31676],[34,626748],{"className":626749,"style":46727},[31680],[34,626751,626753,626756,626815],{"className":626752},[31685],[34,626754],{"className":626755},[31782,41488],[34,626757,626759],{"className":626758},[41453],[34,626760,626762,626807],{"className":626761},[31796,31797],[34,626763,626765,626804],{"className":626764},[31801],[34,626766,626768,626782,626790],{"className":626767,"style":46746},[31805],[34,626769,626770,626773],{"style":41504},[34,626771],{"className":626772,"style":41508},[31813],[34,626774,626776],{"className":626775},[31818,31819,31820,31821],[34,626777,626779],{"className":626778},[31685,31821],[34,626780,217392],{"className":626781,"style":217496},[31685,31686,31821],[34,626783,626784,626787],{"style":41520},[34,626785],{"className":626786,"style":41508},[31813],[34,626788],{"className":626789,"style":41528},[41527],[34,626791,626792,626795],{"style":46771},[34,626793],{"className":626794,"style":41508},[31813],[34,626796,626798],{"className":626797},[31818,31819,31820,31821],[34,626799,626801],{"className":626800},[31685,31821],[34,626802,640],{"className":626803},[31685,31821],[34,626805,31829],{"className":626806},[31828],[34,626808,626810],{"className":626809},[31801],[34,626811,626813],{"className":626812,"style":41619},[31805],[34,626814],{},[34,626816],{"className":626817},[31690,41488],[34,626819],{"className":626820,"style":32604},[31845],[34,626822,223],{"className":626823},[32024],[34,626825],{"className":626826,"style":32604},[31845],[34,626828,626830,626833,626901,626904,626907],{"className":626829},[31676],[34,626831],{"className":626832,"style":561374},[31680],[34,626834,626836,626839,626898],{"className":626835},[31685],[34,626837],{"className":626838},[31782,41488],[34,626840,626842],{"className":626841},[41453],[34,626843,626845,626890],{"className":626844},[31796,31797],[34,626846,626848,626887],{"className":626847},[31801],[34,626849,626851,626865,626873],{"className":626850,"style":46746},[31805],[34,626852,626853,626856],{"style":41504},[34,626854],{"className":626855,"style":41508},[31813],[34,626857,626859],{"className":626858},[31818,31819,31820,31821],[34,626860,626862],{"className":626861},[31685,31821],[34,626863,217691],{"className":626864,"style":217853},[31685,31686,31821],[34,626866,626867,626870],{"style":41520},[34,626868],{"className":626869,"style":41508},[31813],[34,626871],{"className":626872,"style":41528},[41527],[34,626874,626875,626878],{"style":46771},[34,626876],{"className":626877,"style":41508},[31813],[34,626879,626881],{"className":626880},[31818,31819,31820,31821],[34,626882,626884],{"className":626883},[31685,31821],[34,626885,640],{"className":626886},[31685,31821],[34,626888,31829],{"className":626889},[31828],[34,626891,626893],{"className":626892},[31801],[34,626894,626896],{"className":626895,"style":220191},[31805],[34,626897],{},[34,626899],{"className":626900},[31690,41488],[34,626902],{"className":626903,"style":32604},[31845],[34,626905,223],{"className":626906},[32024],[34,626908],{"className":626909,"style":32604},[31845],[34,626911,626913,626916,626984,626987,626990],{"className":626912},[31676],[34,626914],{"className":626915,"style":561374},[31680],[34,626917,626919,626922,626981],{"className":626918},[31685],[34,626920],{"className":626921},[31782,41488],[34,626923,626925],{"className":626924},[41453],[34,626926,626928,626973],{"className":626927},[31796,31797],[34,626929,626931,626970],{"className":626930},[31801],[34,626932,626934,626948,626956],{"className":626933,"style":46746},[31805],[34,626935,626936,626939],{"style":41504},[34,626937],{"className":626938,"style":41508},[31813],[34,626940,626942],{"className":626941},[31818,31819,31820,31821],[34,626943,626945],{"className":626944},[31685,31821],[34,626946,625151],{"className":626947,"style":112322},[31685,31686,31821],[34,626949,626950,626953],{"style":41520},[34,626951],{"className":626952,"style":41508},[31813],[34,626954],{"className":626955,"style":41528},[41527],[34,626957,626958,626961],{"style":46771},[34,626959],{"className":626960,"style":41508},[31813],[34,626962,626964],{"className":626963},[31818,31819,31820,31821],[34,626965,626967],{"className":626966},[31685,31821],[34,626968,640],{"className":626969},[31685,31821],[34,626971,31829],{"className":626972},[31828],[34,626974,626976],{"className":626975},[31801],[34,626977,626979],{"className":626978,"style":220191},[31805],[34,626980],{},[34,626982],{"className":626983},[31690,41488],[34,626985],{"className":626986,"style":32604},[31845],[34,626988,32226],{"className":626989},[32024],[34,626991],{"className":626992,"style":32604},[31845],[34,626994,626996,626999],{"className":626995},[31676],[34,626997],{"className":626998,"style":32617},[31680],[34,627000,786],{"className":627001},[31685],[108,627003,627004,627005,627038,627039,280220,627167,627218],{},"对于无穷远处收敛，需要总阶数小于 ",[34,627006,627008,627023],{"className":627007},[31642],[34,627009,627011],{"className":627010},[31646],[31648,627012,627013],{"xmlns":31650},[31652,627014,627015,627021],{},[31655,627016,627017,627019],{},[31662,627018,32226],{},[31749,627020,640],{},[31665,627022,259653],{"encoding":31667},[34,627024,627026],{"className":627025,"ariaHidden":758},[31672],[34,627027,627029,627032,627035],{"className":627028},[31676],[34,627030],{"className":627031,"style":35942},[31680],[34,627033,32226],{"className":627034},[31685],[34,627036,640],{"className":627037},[31685],"（类似三维空间 ",[34,627040,627042,627064],{"className":627041},[31642],[34,627043,627045],{"className":627044},[31646],[31648,627046,627047],{"xmlns":31650},[31652,627048,627049,627061],{},[31655,627050,627051],{},[41453,627052,627053,627055],{},[31749,627054,640],{},[37624,627056,627057,627059],{},[31658,627058,34395],{},[31658,627060,33759],{},[31665,627062,627063],{"encoding":31667},"\\frac{1}{r^k}",[34,627065,627067],{"className":627066,"ariaHidden":758},[31672],[34,627068,627070,627073],{"className":627069},[31676],[34,627071],{"className":627072,"style":567311},[31680],[34,627074,627076,627079,627164],{"className":627075},[31685],[34,627077],{"className":627078},[31782,41488],[34,627080,627082],{"className":627081},[41453],[34,627083,627085,627156],{"className":627084},[31796,31797],[34,627086,627088,627153],{"className":627087},[31801],[34,627089,627091,627131,627139],{"className":627090,"style":46746},[31805],[34,627092,627093,627096],{"style":567332},[34,627094],{"className":627095,"style":41508},[31813],[34,627097,627099],{"className":627098},[31818,31819,31820,31821],[34,627100,627102],{"className":627101},[31685,31821],[34,627103,627105,627108],{"className":627104},[31685,31821],[34,627106,34395],{"className":627107,"style":34452},[31685,31686,31821],[34,627109,627111],{"className":627110},[31792],[34,627112,627114],{"className":627113},[31796],[34,627115,627117],{"className":627116},[31801],[34,627118,627120],{"className":627119,"style":567360},[31805],[34,627121,627122,627125],{"style":116668},[34,627123],{"className":627124,"style":43406},[31813],[34,627126,627128],{"className":627127},[31818,43410,43411,31821],[34,627129,33759],{"className":627130,"style":33774},[31685,31686,31821],[34,627132,627133,627136],{"style":41520},[34,627134],{"className":627135,"style":41508},[31813],[34,627137],{"className":627138,"style":41528},[41527],[34,627140,627141,627144],{"style":46771},[34,627142],{"className":627143,"style":41508},[31813],[34,627145,627147],{"className":627146},[31818,31819,31820,31821],[34,627148,627150],{"className":627149},[31685,31821],[34,627151,640],{"className":627152},[31685,31821],[34,627154,31829],{"className":627155},[31828],[34,627157,627159],{"className":627158},[31801],[34,627160,627162],{"className":627161,"style":567403},[31805],[34,627163],{},[34,627165],{"className":627166},[31690,41488],[34,627168,627170,627188],{"className":627169},[31642],[34,627171,627173],{"className":627172},[31646],[31648,627174,627175],{"xmlns":31650},[31652,627176,627177,627185],{},[31655,627178,627179,627181,627183],{},[31658,627180,33759],{},[31662,627182,29622],{},[31749,627184,786],{},[31665,627186,627187],{"encoding":31667},"k>3",[34,627189,627191,627209],{"className":627190,"ariaHidden":758},[31672],[34,627192,627194,627197,627200,627203,627206],{"className":627193},[31676],[34,627195],{"className":627196,"style":46991},[31680],[34,627198,33759],{"className":627199,"style":33774},[31685,31686],[34,627201],{"className":627202,"style":31846},[31845],[34,627204,29622],{"className":627205},[31850],[34,627207],{"className":627208,"style":31846},[31845],[34,627210,627212,627215],{"className":627211},[31676],[34,627213],{"className":627214,"style":32617},[31680],[34,627216,786],{"className":627217},[31685]," 收敛，这里分母是一次，所以需要分子非常小？不，其实可以转化为 Gamma 函数）。",[108,627220,627221,627222,347],{},"该积分为 Liouville 积分形式。利用公式：\n",[34,627223,627225,627367],{"className":627224},[31642],[34,627226,627228],{"className":627227},[31646],[31648,627229,627230],{"xmlns":31650},[31652,627231,627232,627364],{},[31655,627233,627234,627236,627238,627240,627310,627318,627360,627362],{},[31662,627235,625104],{},[31662,627237,38874],{},[31662,627239,202],{},[41453,627241,627242,627280],{},[31655,627243,627244,627246,627248,627250,627252,627254,627256,627258,627260,627262,627264,627266,627268,627270,627272,627274,627276,627278],{},[31658,627245,564791],{"mathvariant":38079},[31662,627247,456],{"stretchy":702},[31749,627249,640],{},[31658,627251,38080],{"mathvariant":38079},[31658,627253,217392],{},[31662,627255,2557],{"stretchy":702},[31658,627257,564791],{"mathvariant":38079},[31662,627259,456],{"stretchy":702},[31749,627261,640],{},[31658,627263,38080],{"mathvariant":38079},[31658,627265,217691],{},[31662,627267,2557],{"stretchy":702},[31658,627269,564791],{"mathvariant":38079},[31662,627271,456],{"stretchy":702},[31749,627273,640],{},[31658,627275,38080],{"mathvariant":38079},[31658,627277,625151],{},[31662,627279,2557],{"stretchy":702},[31655,627281,627282,627284,627286,627288,627290,627292,627294,627296,627298,627300,627302,627304,627306,627308],{},[31658,627283,564791],{"mathvariant":38079},[31662,627285,456],{"stretchy":702},[31749,627287,640],{},[31658,627289,38080],{"mathvariant":38079},[31658,627291,217392],{},[31662,627293,223],{},[31749,627295,640],{},[31658,627297,38080],{"mathvariant":38079},[31658,627299,217691],{},[31662,627301,223],{},[31749,627303,640],{},[31658,627305,38080],{"mathvariant":38079},[31658,627307,625151],{},[31662,627309,2557],{"stretchy":702},[38846,627311,627312,627314,627316],{},[31662,627313,546295],{},[31749,627315,459],{},[31658,627317,138286],{"mathvariant":38079},[41453,627319,627320,627352],{},[37624,627321,627322,627324],{},[31658,627323,33249],{},[31655,627325,627326,627328,627330,627332,627334,627336,627338,627340,627342,627344,627346,627348,627350],{},[31749,627327,640],{},[31658,627329,38080],{"mathvariant":38079},[31658,627331,217392],{},[31662,627333,223],{},[31749,627335,640],{},[31658,627337,38080],{"mathvariant":38079},[31658,627339,217691],{},[31662,627341,223],{},[31749,627343,640],{},[31658,627345,38080],{"mathvariant":38079},[31658,627347,625151],{},[31662,627349,32226],{},[31749,627351,640],{},[31655,627353,627354,627356,627358],{},[31749,627355,640],{},[31662,627357,223],{},[31658,627359,33249],{},[31658,627361,46900],{},[31658,627363,33249],{},[31665,627365,627366],{"encoding":31667},"\\iiint \\dots = \\frac{\\Gamma(1\u002F\\alpha)\\Gamma(1\u002F\\beta)\\Gamma(1\u002F\\gamma)}{\\Gamma(1\u002F\\alpha+1\u002F\\beta+1\u002F\\gamma)} \\int_0^\\infty \\frac{t^{1\u002F\\alpha+1\u002F\\beta+1\u002F\\gamma - 1}}{1+t} dt",[34,627368,627370,627395],{"className":627369,"ariaHidden":758},[31672],[34,627371,627373,627377,627380,627383,627386,627389,627392],{"className":627372},[31676],[34,627374],{"className":627375,"style":627376},[31680],"height:1.111em;vertical-align:-0.306em;",[34,627378,625104],{"className":627379,"style":625171},[32767,131249,156560],[34,627381],{"className":627382,"style":32123},[31845],[34,627384,38874],{"className":627385},[34006],[34,627387],{"className":627388,"style":31846},[31845],[34,627390,202],{"className":627391},[31850],[34,627393],{"className":627394,"style":31846},[31845],[34,627396,627398,627401,627541,627544,627595,627598,627728,627731],{"className":627397},[31676],[34,627399],{"className":627400,"style":605398},[31680],[34,627402,627404,627407,627538],{"className":627403},[31685],[34,627405],{"className":627406},[31782,41488],[34,627408,627410],{"className":627409},[41453],[34,627411,627413,627530],{"className":627412},[31796,31797],[34,627414,627416,627527],{"className":627415},[31801],[34,627417,627419,627463,627471],{"className":627418,"style":70504},[31805],[34,627420,627421,627424],{"style":41504},[34,627422],{"className":627423,"style":41508},[31813],[34,627425,627427],{"className":627426},[31818,31819,31820,31821],[34,627428,627430,627433,627436,627439,627442,627445,627448,627451,627454,627457,627460],{"className":627429},[31685,31821],[34,627431,564791],{"className":627432},[31685,31821],[34,627434,456],{"className":627435},[31782,31821],[34,627437,249219],{"className":627438},[31685,31821],[34,627440,217392],{"className":627441,"style":217496},[31685,31686,31821],[34,627443,223],{"className":627444},[32024,31821],[34,627446,249219],{"className":627447},[31685,31821],[34,627449,217691],{"className":627450,"style":217853},[31685,31686,31821],[34,627452,223],{"className":627453},[32024,31821],[34,627455,249219],{"className":627456},[31685,31821],[34,627458,625151],{"className":627459,"style":112322},[31685,31686,31821],[34,627461,2557],{"className":627462},[31690,31821],[34,627464,627465,627468],{"style":41520},[34,627466],{"className":627467,"style":41508},[31813],[34,627469],{"className":627470,"style":41528},[41527],[34,627472,627473,627476],{"style":70529},[34,627474],{"className":627475,"style":41508},[31813],[34,627477,627479],{"className":627478},[31818,31819,31820,31821],[34,627480,627482,627485,627488,627491,627494,627497,627500,627503,627506,627509,627512,627515,627518,627521,627524],{"className":627481},[31685,31821],[34,627483,564791],{"className":627484},[31685,31821],[34,627486,456],{"className":627487},[31782,31821],[34,627489,249219],{"className":627490},[31685,31821],[34,627492,217392],{"className":627493,"style":217496},[31685,31686,31821],[34,627495,2557],{"className":627496},[31690,31821],[34,627498,564791],{"className":627499},[31685,31821],[34,627501,456],{"className":627502},[31782,31821],[34,627504,249219],{"className":627505},[31685,31821],[34,627507,217691],{"className":627508,"style":217853},[31685,31686,31821],[34,627510,2557],{"className":627511},[31690,31821],[34,627513,564791],{"className":627514},[31685,31821],[34,627516,456],{"className":627517},[31782,31821],[34,627519,249219],{"className":627520},[31685,31821],[34,627522,625151],{"className":627523,"style":112322},[31685,31686,31821],[34,627525,2557],{"className":627526},[31690,31821],[34,627528,31829],{"className":627529},[31828],[34,627531,627533],{"className":627532},[31801],[34,627534,627536],{"className":627535,"style":324858},[31805],[34,627537],{},[34,627539],{"className":627540},[31690,41488],[34,627542],{"className":627543,"style":32123},[31845],[34,627545,627547,627550],{"className":627546},[32767],[34,627548,546295],{"className":627549,"style":546331},[32767,131249,156560],[34,627551,627553],{"className":627552},[31792],[34,627554,627556,627587],{"className":627555},[31796,31797],[34,627557,627559,627584],{"className":627558},[31801],[34,627560,627562,627573],{"className":627561,"style":567928},[31805],[34,627563,627564,627567],{"style":546347},[34,627565],{"className":627566,"style":31814},[31813],[34,627568,627570],{"className":627569},[31818,31819,31820,31821],[34,627571,459],{"className":627572},[31685,31821],[34,627574,627575,627578],{"style":546362},[34,627576],{"className":627577,"style":31814},[31813],[34,627579,627581],{"className":627580},[31818,31819,31820,31821],[34,627582,138286],{"className":627583},[31685,31821],[34,627585,31829],{"className":627586},[31828],[34,627588,627590],{"className":627589},[31801],[34,627591,627593],{"className":627592,"style":546384},[31805],[34,627594],{},[34,627596],{"className":627597,"style":32123},[31845],[34,627599,627601,627604,627725],{"className":627600},[31685],[34,627602],{"className":627603},[31782,41488],[34,627605,627607],{"className":627606},[41453],[34,627608,627610,627717],{"className":627609},[31796,31797],[34,627611,627613,627714],{"className":627612},[31801],[34,627614,627616,627636,627644],{"className":627615,"style":605417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+ \\frac{1}{\\beta} + \\frac{1}{\\gamma} \u003C 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28377,628477],{"className":628376},[31642],[34,628378,628380],{"className":628379},[31646],[31648,628381,628382],{"xmlns":31650},[31652,628383,628384,628474],{},[31655,628385,628386,628388,628390,628402,628452,628454,628456,628458,628460,628462,628464,628466,628468,628470,628472],{},[31658,628387,106063],{},[31662,628389,202],{},[41453,628391,628392,628394],{},[31749,628393,640],{},[31655,628395,628396,628398,628400],{},[31658,628397,217392],{},[31658,628399,217691],{},[31658,628401,625151],{},[41453,628403,628404,628442],{},[31655,628405,628406,628408,628410,628412,628414,628416,628418,628420,628422,628424,628426,628428,628430,628432,628434,628436,628438,628440],{},[31658,628407,564791],{"mathvariant":38079},[31662,628409,456],{"stretchy":702},[31749,628411,640],{},[31658,628413,38080],{"mathvariant":38079},[31658,628415,217392],{},[31662,628417,2557],{"stretchy":702},[31658,628419,564791],{"mathvariant":38079},[31662,628421,456],{"stretchy":702},[31749,628423,640],{},[31658,628425,38080],{"mathvariant":38079},[31658,628427,217691],{},[31662,628429,2557],{"stretchy":702},[31658,628431,564791],{"mathvariant":38079},[31662,628433,456],{"stretchy":702},[31749,628435,640],{},[31658,628437,38080],{"mathvariant":38079},[31658,628439,625151],{},[31662,628441,2557],{"stretchy":702},[31655,628443,628444,628446,628448,628450],{},[31658,628445,564791],{"mathvariant":38079},[31662,628447,456],{"stretchy":702},[31658,628449,94675],{},[31662,628451,2557],{"stretchy":702},[31662,628453,71191],{},[31658,628455,564791],{"mathvariant":38079},[31662,628457,456],{"stretchy":702},[31658,628459,94675],{},[31662,628461,2557],{"stretchy":702},[31658,628463,564791],{"mathvariant":38079},[31662,628465,456],{"stretchy":702},[31749,628467,640],{},[31662,628469,32226],{},[31658,628471,94675],{},[31662,628473,2557],{"stretchy":702},[31665,628475,628476],{"encoding":31667},"I = \\frac{1}{\\alpha\\beta\\gamma} \\frac{\\Gamma(1\u002F\\alpha)\\Gamma(1\u002F\\beta)\\Gamma(1\u002F\\gamma)}{\\Gamma(S)} \\cdot \\Gamma(S)\\Gamma(1-S)",[34,628478,628480,628498,628706,628742],{"className":628479,"ariaHidden":758},[31672],[34,628481,628483,628486,628489,628492,628495],{"className":628482},[31676],[34,628484],{"className":628485,"style":33218},[31680],[34,628487,106063],{"className":628488,"style":67482},[31685,31686],[34,628490],{"className":628491,"style":31846},[31845],[34,628493,202],{"className":628494},[31850],[34,628496],{"className":628497,"style":31846},[31845],[34,628499,628501,628504,628578,628697,628700,628703],{"className":628500},[31676],[34,628502],{"className":628503,"style":327841},[31680],[34,628505,628507,628510,628575],{"className":628506},[31685],[34,628508],{"className":628509},[31782,41488],[34,628511,628513],{"className":628512},[41453],[34,628514,628516,628567],{"className":628515},[31796,31797],[34,628517,628519,628564],{"className":628518},[31801],[34,628520,628522,628542,628550],{"className":628521,"style":46746},[31805],[34,628523,628524,628527],{"style":41504},[34,628525],{"className":628526,"style":41508},[31813],[34,628528,628530],{"className":628529},[31818,31819,31820,31821],[34,628531,628533,628536,628539],{"className":628532},[31685,31821],[34,628534,217392],{"className":628535,"style":217496},[31685,31686,31821],[34,628537,217691],{"className":628538,"style":217853},[31685,31686,31821],[34,628540,625151],{"className":628541,"style":112322},[31685,31686,31821],[34,628543,628544,628547],{"style":41520},[34,628545],{"className":628546,"style":41508},[31813],[34,628548],{"className":628549,"style":41528},[41527],[34,628551,628552,628555],{"style":46771},[34,628553],{"className":628554,"style":41508},[31813],[34,628556,628558],{"className":628557},[31818,31819,31820,31821],[34,628559,628561],{"className":628560},[31685,31821],[34,628562,640],{"className":628563},[31685,31821],[34,628565,31829],{"className":628566},[31828],[34,628568,628570],{"className":628569},[31801],[34,628571,628573],{"className":628572,"style":220191},[31805],[34,628574],{},[34,628576],{"className":628577},[31690,41488],[34,628579,628581,628584,628694],{"className":628580},[31685],[34,628582],{"className":628583},[31782,41488],[34,628585,628587],{"className":628586},[41453],[34,628588,628590,628686],{"className":628589},[31796,31797],[34,628591,628593,628683],{"className":628592},[31801],[34,628594,628596,628619,628627],{"className":628595,"style":70504},[31805],[34,628597,628598,628601],{"style":41504},[34,628599],{"className":628600,"style":41508},[31813],[34,628602,628604],{"className":628603},[31818,31819,31820,31821],[34,628605,628607,628610,628613,628616],{"className":628606},[31685,31821],[34,628608,564791],{"className":628609},[31685,31821],[34,628611,456],{"className":628612},[31782,31821],[34,628614,94675],{"className":628615,"style":106455},[31685,31686,31821],[34,628617,2557],{"className":628618},[31690,31821],[34,628620,628621,628624],{"style":41520},[34,628622],{"className":628623,"style":41508},[31813],[34,628625],{"className":628626,"style":41528},[41527],[34,628628,628629,628632],{"style":70529},[34,628630],{"className":628631,"style":41508},[31813],[34,628633,628635],{"className":628634},[31818,31819,31820,31821],[34,628636,628638,628641,628644,628647,628650,628653,628656,628659,628662,628665,628668,628671,628674,628677,628680],{"className":628637},[31685,31821],[34,628639,564791],{"className":628640},[31685,31821],[34,628642,456],{"className":628643},[31782,31821],[34,628645,249219],{"className":628646},[31685,31821],[34,628648,217392],{"className":628649,"style":217496},[31685,31686,31821],[34,628651,2557],{"className":628652},[31690,31821],[34,628654,564791],{"className":628655},[31685,31821],[34,628657,456],{"className":628658},[31782,31821],[34,628660,249219],{"className":628661},[31685,31821],[34,628663,217691],{"className":628664,"style":217853},[31685,31686,31821],[34,628666,2557],{"className":628667},[31690,31821],[34,628669,564791],{"className":628670},[31685,31821],[34,628672,456],{"className":628673},[31782,31821],[34,628675,249219],{"className":628676},[31685,31821],[34,628678,625151],{"className":628679,"style":112322},[31685,31686,31821],[34,628681,2557],{"className":628682},[31690,31821],[34,628684,31829],{"className":628685},[31828],[34,628687,628689],{"className":628688},[31801],[34,628690,628692],{"className":628691,"style":324858},[31805],[34,628693],{},[34,628695],{"className":628696},[31690,41488],[34,628698],{"className":628699,"style":32604},[31845],[34,628701,71191],{"className":628702},[32024],[34,628704],{"className":628705,"style":32604},[31845],[34,628707,628709,628712,628715,628718,628721,628724,628727,628730,628733,628736,628739],{"className":628708},[31676],[34,628710],{"className":628711,"style":31774},[31680],[34,628713,564791],{"className":628714},[31685],[34,628716,456],{"className":628717},[31782],[34,628719,94675],{"className":628720,"style":106455},[31685,31686],[34,628722,2557],{"className":628723},[31690],[34,628725,564791],{"className":628726},[31685],[34,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= 1\u002F\\alpha+1\u002F\\beta+1\u002F\\gamma",[34,628796,628798,628816,628837,628858],{"className":628797,"ariaHidden":758},[31672],[34,628799,628801,628804,628807,628810,628813],{"className":628800},[31676],[34,628802],{"className":628803,"style":33218},[31680],[34,628805,94675],{"className":628806,"style":106455},[31685,31686],[34,628808],{"className":628809,"style":31846},[31845],[34,628811,202],{"className":628812},[31850],[34,628814],{"className":628815,"style":31846},[31845],[34,628817,628819,628822,628825,628828,628831,628834],{"className":628818},[31676],[34,628820],{"className":628821,"style":31774},[31680],[34,628823,249219],{"className":628824},[31685],[34,628826,217392],{"className":628827,"style":217496},[31685,31686],[34,628829],{"className":628830,"style":32604},[31845],[34,628832,223],{"className":628833},[32024],[34,628835],{"className":628836,"style":32604},[31845],[34,628838,628840,628843,628846,628849,628852,628855],{"className":628839},[31676],[34,628841],{"className":628842,"style":31774},[31680],[34,628844,249219],{"className":628845},[31685],[34,628847,217691],{"className":628848,"style":217853},[31685,31686],[34,628850],{"className":628851,"style":32604},[31845],[34,628853,223],{"className":628854},[32024],[34,628856],{"className":628857,"style":32604},[31845],[34,628859,628861,628864,628867],{"className":628860},[31676],[34,628862],{"className":628863,"style":31774},[31680],[34,628865,249219],{"className":628866},[31685],[34,628868,625151],{"className":628869,"style":112322},[31685,31686],[108,628871,628872,628873,347],{},"化简得 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x = \\sin^\\alpha x \\cos^{-\\alpha} x",[34,629867,629869,629921],{"className":629868,"ariaHidden":758},[31672],[34,629870,629872,629876,629906,629909,629912,629915,629918],{"className":629871},[31676],[34,629873],{"className":629874,"style":629875},[31680],"height:0.6694em;",[34,629877,629879,629882],{"className":629878},[32767],[34,629880,629424],{"className":629881},[32767],[34,629883,629885],{"className":629884},[31792],[34,629886,629888],{"className":629887},[31796],[34,629889,629891],{"className":629890},[31801],[34,629892,629894],{"className":629893,"style":629875},[31805],[34,629895,629897,629900],{"style":629896},"top:-3.068em;margin-right:0.05em;",[34,629898],{"className":629899,"style":31814},[31813],[34,629901,629903],{"className":629902},[31818,31819,31820,31821],[34,629904,217392],{"className":629905,"style":217496},[31685,31686,31821],[34,629907],{"className":629908,"style":32123},[31845],[34,629910,32936],{"className":629911},[31685,31686],[34,629913],{"className":629914,"style":31846},[31845],[34,629916,202],{"className":629917},[31850],[34,629919],{"className":629920,"style":31846},[31845],[34,629922,629924,629927,629956,629959,629962,629965,630000,630003],{"className":629923},[31676],[34,629925],{"className":629926,"style":156828},[31680],[34,629928,629930,629933],{"className":629929},[32767],[34,629931,547853],{"className":629932},[32767],[34,629934,629936],{"className":629935},[31792],[34,629937,629939],{"className":629938},[31796],[34,629940,629942],{"className":629941},[31801],[34,629943,629945],{"className":629944,"style":608682},[31805],[34,629946,629947,629950],{"style":586008},[34,629948],{"className":629949,"style":31814},[31813],[34,629951,629953],{"className":629952},[31818,31819,31820,31821],[34,629954,217392],{"className":629955,"style":217496},[31685,31686,31821],[34,629957],{"className":629958,"style":32123},[31845],[34,629960,32936],{"className":629961},[31685,31686],[34,629963],{"className":629964,"style":32123},[31845],[34,629966,629968,629971],{"className":629967},[32767],[34,629969,547860],{"className":629970},[32767],[34,629972,629974],{"className":629973},[31792],[34,629975,629977],{"className":629976},[31796],[34,629978,629980],{"className":629979},[31801],[34,629981,629983],{"className":629982,"style":156828},[31805],[34,629984,629985,629988],{"style":37777},[34,629986],{"className":629987,"style":31814},[31813],[34,629989,629991],{"className":629990},[31818,31819,31820,31821],[34,629992,629994,629997],{"className":629993},[31685,31821],[34,629995,32226],{"className":629996},[31685,31821],[34,629998,217392],{"className":629999,"style":217496},[31685,31686,31821],[34,630001],{"className":630002,"style":32123},[31845],[34,630004,32936],{"className":630005},[31685,31686],[108,630007,630008,630009,630037],{},"利用 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函数三角形式公式。",[108,630039,592630,630040,347],{},[34,630041,630043,630097],{"className":630042},[31642],[34,630044,630046],{"className":630045},[31646],[31648,630047,630048],{"xmlns":31650},[31652,630049,630050,630094],{},[31655,630051,630052,630054,630060,630062,630064,630076,630078,630092],{},[31662,630053,202],{},[41453,630055,630056,630058],{},[31749,630057,640],{},[31749,630059,768],{},[31658,630061,54224],{},[31662,630063,456],{"stretchy":702},[41453,630065,630066,630074],{},[31655,630067,630068,630070,630072],{},[31658,630069,217392],{},[31662,630071,223],{},[31749,630073,640],{},[31749,630075,768],{},[31662,630077,31960],{"separator":758},[41453,630079,630080,630090],{},[31655,630081,630082,630084,630086,630088],{},[31662,630083,32226],{},[31658,630085,217392],{},[31662,630087,223],{},[31749,630089,640],{},[31749,630091,768],{},[31662,630093,2557],{"stretchy":702},[31665,630095,630096],{"encoding":31667},"= \\frac{1}{2} B(\\frac{\\alpha+1}{2}, 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630361,630362],{"xmlns":31650},[31652,630363,630364,630382],{},[31655,630365,630366,630368,630370],{},[31658,630367,15],{},[31662,630369,202],{},[41453,630371,630372,630380],{},[31655,630373,630374,630376,630378],{},[31749,630375,640],{},[31662,630377,223],{},[31658,630379,217392],{},[31749,630381,768],{},[31665,630383,630384],{"encoding":31667},"p = \\frac{1+\\alpha}{2}",[34,630386,630388,630406],{"className":630387,"ariaHidden":758},[31672],[34,630389,630391,630394,630397,630400,630403],{"className":630390},[31676],[34,630392],{"className":630393,"style":38774},[31680],[34,630395,15],{"className":630396},[31685,31686],[34,630398],{"className":630399,"style":31846},[31845],[34,630401,202],{"className":630402},[31850],[34,630404],{"className":630405,"style":31846},[31845],[34,630407,630409,630412],{"className":630408},[31676],[34,630410],{"className":630411,"style":46727},[31680],[34,630413,630415,630418,630483],{"className":630414},[31685],[34,630416],{"className":630417},[31782,41488],[34,630419,630421],{"className":630420},[41453],[34,630422,630424,630475],{"className":630423},[31796,31797],[34,630425,630427,630472],{"className":630426},[31801],[34,630428,630430,630444,630452],{"className":630429,"style":46746},[31805],[34,630431,630432,630435],{"style":41504},[34,630433],{"className":630434,"style":41508},[31813],[34,630436,630438],{"className":630437},[31818,31819,31820,31821],[34,630439,630441],{"className":630440},[31685,31821],[34,630442,768],{"className":630443},[31685,31821],[34,630445,630446,630449],{"style":41520},[34,630447],{"className":630448,"style":41508},[31813],[34,630450],{"className":630451,"style":41528},[41527],[34,630453,630454,630457],{"style":46771},[34,630455],{"className":630456,"style":41508},[31813],[34,630458,630460],{"className":630459},[31818,31819,31820,31821],[34,630461,630463,630466,630469],{"className":630462},[31685,31821],[34,630464,640],{"className":630465},[31685,31821],[34,630467,223],{"className":630468},[32024,31821],[34,630470,217392],{"className":630471,"style":217496},[31685,31686,31821],[34,630473,31829],{"className":630474},[31828],[34,630476,630478],{"className":630477},[31801],[34,630479,630481],{"className":630480,"style":41619},[31805],[34,630482],{},[34,630484],{"className":630485},[31690,41488],[34,630487,630489,630539],{"className":630488},[31642],[34,630490,630492],{"className":630491},[31646],[31648,630493,630494],{"xmlns":31650},[31652,630495,630496,630536],{},[31655,630497,630498,630500,630502,630504,630506,630508,630510,630522,630524],{},[31749,630499,640],{},[31662,630501,32226],{},[31658,630503,15],{},[31662,630505,202],{},[31749,630507,640],{},[31662,630509,32226],{},[41453,630511,630512,630520],{},[31655,630513,630514,630516,630518],{},[31749,630515,640],{},[31662,630517,223],{},[31658,630519,217392],{},[31749,630521,768],{},[31662,630523,202],{},[41453,630525,630526,630534],{},[31655,630527,630528,630530,630532],{},[31749,630529,640],{},[31662,630531,32226],{},[31658,630533,217392],{},[31749,630535,768],{},[31665,630537,630538],{"encoding":31667},"1-p = 1 - \\frac{1+\\alpha}{2} = \\frac{1-\\alpha}{2}",[34,630540,630542,630560,630578,630596,630685],{"className":630541,"ariaHidden":758},[31672],[34,630543,630545,630548,630551,630554,630557],{"className":630544},[31676],[34,630546],{"className":630547,"style":35942},[31680],[34,630549,640],{"className":630550},[31685],[34,630552],{"className":630553,"style":32604},[31845],[34,630555,32226],{"className":630556},[32024],[34,630558],{"className":630559,"style":32604},[31845],[34,630561,630563,630566,630569,630572,630575],{"className":630562},[31676],[34,630564],{"className":630565,"style":38774},[31680],[34,630567,15],{"className":630568},[31685,31686],[34,630570],{"className":630571,"style":31846},[31845],[34,630573,202],{"className":630574},[31850],[34,630576],{"className":630577,"style":31846},[31845],[34,630579,630581,630584,630587,630590,630593],{"className":630580},[31676],[34,630582],{"className":630583,"style":35942},[31680],[34,630585,640],{"className":630586},[31685],[34,630588],{"className":630589,"style":32604},[31845],[34,630591,32226],{"className":630592},[32024],[34,630594],{"className":630595,"style":32604},[31845],[34,630597,630599,630602,630676,630679,630682],{"className":630598},[31676],[34,630600],{"className":630601,"style":46727},[31680],[34,630603,630605,630608,630673],{"className":630604},[31685],[34,630606],{"className":630607},[31782,41488],[34,630609,630611],{"className":630610},[41453],[34,630612,630614,630665],{"className":630613},[31796,31797],[34,630615,630617,630662],{"className":630616},[31801],[34,630618,630620,630634,630642],{"className":630619,"style":46746},[31805],[34,630621,630622,630625],{"style":41504},[34,630623],{"className":630624,"style":41508},[31813],[34,630626,630628],{"className":630627},[31818,31819,31820,31821],[34,630629,630631],{"className":630630},[31685,31821],[34,630632,768],{"className":630633},[31685,31821],[34,630635,630636,630639],{"style":41520},[34,630637],{"className":630638,"style":41508},[31813],[34,630640],{"className":630641,"style":41528},[41527],[34,630643,630644,630647],{"style":46771},[34,630645],{"className":630646,"style":41508},[31813],[34,630648,630650],{"className":630649},[31818,31819,31820,31821],[34,630651,630653,630656,630659],{"className":630652},[31685,31821],[34,630654,640],{"className":630655},[31685,31821],[34,630657,223],{"className":630658},[32024,31821],[34,630660,217392],{"className":630661,"style":217496},[31685,31686,31821],[34,630663,31829],{"className":630664},[31828],[34,630666,630668],{"className":630667},[31801],[34,630669,630671],{"className":630670,"style":41619},[31805],[34,630672],{},[34,630674],{"className":630675},[31690,41488],[34,630677],{"className":630678,"style":31846},[31845],[34,630680,202],{"className":630681},[31850],[34,630683],{"className":630684,"style":31846},[31845],[34,630686,630688,630691],{"className":630687},[31676],[34,630689],{"className":630690,"style":46727},[31680],[34,630692,630694,630697,630762],{"className":630693},[31685],[34,630695],{"className":630696},[31782,41488],[34,630698,630700],{"className":630699},[41453],[34,630701,630703,630754],{"className":630702},[31796,31797],[34,630704,630706,630751],{"className":630705},[31801],[34,630707,630709,630723,630731],{"className":630708,"style":46746},[31805],[34,630710,630711,630714],{"style":41504},[34,630712],{"className":630713,"style":41508},[31813],[34,630715,630717],{"className":630716},[31818,31819,31820,31821],[34,630718,630720],{"className":630719},[31685,31821],[34,630721,768],{"className":630722},[31685,31821],[34,630724,630725,630728],{"style":41520},[34,630726],{"className":630727,"style":41508},[31813],[34,630729],{"className":630730,"style":41528},[41527],[34,630732,630733,630736],{"style":46771},[34,630734],{"className":630735,"style":41508},[31813],[34,630737,630739],{"className":630738},[31818,31819,31820,31821],[34,630740,630742,630745,630748],{"className":630741},[31685,31821],[34,630743,640],{"className":630744},[31685,31821],[34,630746,32226],{"className":630747},[32024,31821],[34,630749,217392],{"className":630750,"style":217496},[31685,31686,31821],[34,630752,31829],{"className":630753},[31828],[34,630755,630757],{"className":630756},[31801],[34,630758,630760],{"className":630759,"style":41619},[31805],[34,630761],{},[34,630763],{"className":630764},[31690,41488],"。恰好对应。",[108,630767,630768,630769,347],{},"利用余元公式：",[34,630770,630772,630820],{"className":630771},[31642],[34,630773,630775],{"className":630774},[31646],[31648,630776,630777],{"xmlns":31650},[31652,630778,630779,630817],{},[31655,630780,630781,630787,630789],{},[41453,630782,630783,630785],{},[31749,630784,640],{},[31749,630786,768],{},[31662,630788,71191],{},[41453,630790,630791,630793],{},[31658,630792,58442],{},[31655,630794,630795,630797,630799,630801,630813,630815],{},[31658,630796,547853],{},[31662,630798,32682],{},[31662,630800,456],{"stretchy":702},[41453,630802,630803,630811],{},[31655,630804,630805,630807,630809],{},[31749,630806,640],{},[31662,630808,223],{},[31658,630810,217392],{},[31749,630812,768],{},[31658,630814,58442],{},[31662,630816,2557],{"stretchy":702},[31665,630818,630819],{"encoding":31667},"\\frac{1}{2} \\cdot \\frac{\\pi}{\\sin(\\frac{1+\\alpha}{2}\\pi)}",[34,630821,630823,630906],{"className":630822,"ariaHidden":758},[31672],[34,630824,630826,630829,630897,630900,630903],{"className":630825},[31676],[34,630827],{"className":630828,"style":46727},[31680],[34,630830,630832,630835,630894],{"className":630831},[31685],[34,630833],{"className":630834},[31782,41488],[34,630836,630838],{"className":630837},[41453],[34,630839,630841,630886],{"className":630840},[31796,31797],[34,630842,630844,630883],{"className":630843},[31801],[34,630845,630847,630861,630869],{"className":630846,"style":46746},[31805],[34,630848,630849,630852],{"style":41504},[34,630850],{"className":630851,"style":41508},[31813],[34,630853,630855],{"className":630854},[31818,31819,31820,31821],[34,630856,630858],{"className":630857},[31685,31821],[34,630859,768],{"className":630860},[31685,31821],[34,630862,630863,630866],{"style":41520},[34,630864],{"className":630865,"style":41508},[31813],[34,630867],{"className":630868,"style":41528},[41527],[34,630870,630871,630874],{"style":46771},[34,630872],{"className":630873,"style":41508},[31813],[34,630875,630877],{"className":630876},[31818,31819,31820,31821],[34,630878,630880],{"className":630879},[31685,31821],[34,630881,640],{"className":630882},[31685,31821],[34,630884,31829],{"className":630885},[31828],[34,630887,630889],{"className":630888},[31801],[34,630890,630892],{"className":630891,"style":41619},[31805],[34,630893],{},[34,630895],{"className":630896},[31690,41488],[34,630898],{"className":630899,"style":32604},[31845],[34,630901,71191],{"className":630902},[32024],[34,630904],{"className":630905,"style":32604},[31845],[34,630907,630909,630913],{"className":630908},[31676],[34,630910],{"className":630911,"style":630912},[31680],"height:1.3372em;vertical-align:-0.6418em;",[34,630914,630916,630919,631071],{"className":630915},[31685],[34,630917],{"className":630918},[31782,41488],[34,630920,630922],{"className":630921},[41453],[34,630923,630925,631062],{"className":630924},[31796,31797],[34,630926,630928,631059],{"className":630927},[31801],[34,630929,630931,631037,631045],{"className":630930,"style":244945},[31805],[34,630932,630933,630936],{"style":610844},[34,630934],{"className":630935,"style":41508},[31813],[34,630937,630939],{"className":630938},[31818,31819,31820,31821],[34,630940,630942,630954,630957,631031,631034],{"className":630941},[31685,31821],[34,630943,630945,630948,630951],{"className":630944},[32767,31821],[34,630946,31747],{"className":630947},[31821],[34,630949,31955],{"className":630950},[31821],[34,630952,31660],{"className":630953},[31821],[34,630955,456],{"className":630956},[31782,31821],[34,630958,630960,630963,631028],{"className":630959},[31685,31821],[34,630961],{"className":630962},[31782,41488,31818,43410,435704],[34,630964,630966],{"className":630965},[41453],[34,630967,630969,631020],{"className":630968},[31796,31797],[34,630970,630972,631017],{"className":630971},[31801],[34,630973,630975,630989,630997],{"className":630974,"style":593176},[31805],[34,630976,630977,630980],{"style":435720},[34,630978],{"className":630979,"style":41508},[31813],[34,630981,630983],{"className":630982},[31818,43410,43411,31821],[34,630984,630986],{"className":630985},[31685,31821],[34,630987,768],{"className":630988},[31685,31821],[34,630990,630991,630994],{"style":435735},[34,630992],{"className":630993,"style":41508},[31813],[34,630995],{"className":630996,"style":435742},[41527,31821],[34,630998,630999,631002],{"style":589620},[34,631000],{"className":631001,"style":41508},[31813],[34,631003,631005],{"className":631004},[31818,43410,43411,31821],[34,631006,631008,631011,631014],{"className":631007},[31685,31821],[34,631009,640],{"className":631010},[31685,31821],[34,631012,223],{"className":631013},[32024,31821],[34,631015,217392],{"className":631016,"style":217496},[31685,31686,31821],[34,631018,31829],{"className":631019},[31828],[34,631021,631023],{"className":631022},[31801],[34,631024,631026],{"className":631025,"style":435782},[31805],[34,631027],{},[34,631029],{"className":631030},[31690,41488,31818,43410,435704],[34,631032,58442],{"className":631033,"style":32062},[31685,31686,31821],[34,631035,2557],{"className":631036},[31690,31821],[34,631038,631039,631042],{"style":41520},[34,631040],{"className":631041,"style":41508},[31813],[34,631043],{"className":631044,"style":41528},[41527],[34,631046,631047,631050],{"style":46771},[34,631048],{"className":631049,"style":41508},[31813],[34,631051,631053],{"className":631052},[31818,31819,31820,31821],[34,631054,631056],{"className":631055},[31685,31821],[34,631057,58442],{"className":631058,"style":32062},[31685,31686,31821],[34,631060,31829],{"className":631061},[31828],[34,631063,631065],{"className":631064},[31801],[34,631066,631069],{"className":631067,"style":631068},[31805],"height:0.6418em;",[34,631070],{},[34,631072],{"className":631073},[31690,41488],[108,631075,631076,347],{},[34,631077,631079,631137],{"className":631078},[31642],[34,631080,631082],{"className":631081},[31646],[31648,631083,631084],{"xmlns":31650},[31652,631085,631086,631134],{},[31655,631087,631088,631090,631092,631094,631100,631102,631112,631114,631116,631118,631120,631122,631132],{},[31658,631089,547853],{},[31662,631091,32682],{},[31662,631093,456],{"stretchy":702},[41453,631095,631096,631098],{},[31658,631097,58442],{},[31749,631099,768],{},[31662,631101,223],{},[41453,631103,631104,631110],{},[31655,631105,631106,631108],{},[31658,631107,217392],{},[31658,631109,58442],{},[31749,631111,768],{},[31662,631113,2557],{"stretchy":702},[31662,631115,202],{},[31658,631117,547860],{},[31662,631119,32682],{},[31662,631121,456],{"stretchy":702},[41453,631123,631124,631130],{},[31655,631125,631126,631128],{},[31658,631127,217392],{},[31658,631129,58442],{},[31749,631131,768],{},[31662,631133,2557],{"stretchy":702},[31665,631135,631136],{"encoding":31667},"\\sin(\\frac{\\pi}{2} + \\frac{\\alpha\\pi}{2}) = \\cos(\\frac{\\alpha\\pi}{2})",[34,631138,631140,631230,631319],{"className":631139,"ariaHidden":758},[31672],[34,631141,631143,631147,631150,631153,631221,631224,631227],{"className":631142},[31676],[34,631144],{"className":631145,"style":631146},[31680],"height:1.095em;vertical-align:-0.345em;",[34,631148,547853],{"className":631149},[32767],[34,631151,456],{"className":631152},[31782],[34,631154,631156,631159,631218],{"className":631155},[31685],[34,631157],{"className":631158},[31782,41488],[34,631160,631162],{"className":631161},[41453],[34,631163,631165,631210],{"className":631164},[31796,31797],[34,631166,631168,631207],{"className":631167},[31801],[34,631169,631171,631185,631193],{"className":631170,"style":244945},[31805],[34,631172,631173,631176],{"style":41504},[34,631174],{"className":631175,"style":41508},[31813],[34,631177,631179],{"className":631178},[31818,31819,31820,31821],[34,631180,631182],{"className":631181},[31685,31821],[34,631183,768],{"className":631184},[31685,31821],[34,631186,631187,631190],{"style":41520},[34,631188],{"className":631189,"style":41508},[31813],[34,631191],{"className":631192,"style":41528},[41527],[34,631194,631195,631198],{"style":46771},[34,631196],{"className":631197,"style":41508},[31813],[34,631199,631201],{"className":631200},[31818,31819,31820,31821],[34,631202,631204],{"className":631203},[31685,31821],[34,631205,58442],{"className":631206,"style":32062},[31685,31686,31821],[34,631208,31829],{"className":631209},[31828],[34,631211,631213],{"className":631212},[31801],[34,631214,631216],{"className":631215,"style":41619},[31805],[34,631217],{},[34,631219],{"className":631220},[31690,41488],[34,631222],{"className":631223,"style":32604},[31845],[34,631225,223],{"className":631226},[32024],[34,631228],{"className":631229,"style":32604},[31845],[34,631231,631233,631236,631307,631310,631313,631316],{"className":631232},[31676],[34,631234],{"className":631235,"style":631146},[31680],[34,631237,631239,631242,631304],{"className":631238},[31685],[34,631240],{"className":631241},[31782,41488],[34,631243,631245],{"className":631244},[41453],[34,631246,631248,631296],{"className":631247},[31796,31797],[34,631249,631251,631293],{"className":631250},[31801],[34,631252,631254,631268,631276],{"className":631253,"style":244945},[31805],[34,631255,631256,631259],{"style":41504},[34,631257],{"className":631258,"style":41508},[31813],[34,631260,631262],{"className":631261},[31818,31819,31820,31821],[34,631263,631265],{"className":631264},[31685,31821],[34,631266,768],{"className":631267},[31685,31821],[34,631269,631270,631273],{"style":41520},[34,631271],{"className":631272,"style":41508},[31813],[34,631274],{"className":631275,"style":41528},[41527],[34,631277,631278,631281],{"style":46771},[34,631279],{"className":631280,"style":41508},[31813],[34,631282,631284],{"className":631283},[31818,31819,31820,31821],[34,631285,631287,631290],{"className":631286},[31685,31821],[34,631288,217392],{"className":631289,"style":217496},[31685,31686,31821],[34,631291,58442],{"className":631292,"style":32062},[31685,31686,31821],[34,631294,31829],{"className":631295},[31828],[34,631297,631299],{"className":631298},[31801],[34,631300,631302],{"className":631301,"style":41619},[31805],[34,631303],{},[34,631305],{"className":631306},[31690,41488],[34,631308,2557],{"className":631309},[31690],[34,631311],{"className":631312,"style":31846},[31845],[34,631314,202],{"className":631315},[31850],[34,631317],{"className":631318,"style":31846},[31845],[34,631320,631322,631325,631328,631331,631402],{"className":631321},[31676],[34,631323],{"className":631324,"style":631146},[31680],[34,631326,547860],{"className":631327},[32767],[34,631329,456],{"className":631330},[31782],[34,631332,631334,631337,631399],{"className":631333},[31685],[34,631335],{"className":631336},[31782,41488],[34,631338,631340],{"className":631339},[41453],[34,631341,631343,631391],{"className":631342},[31796,31797],[34,631344,631346,631388],{"className":631345},[31801],[34,631347,631349,631363,631371],{"className":631348,"style":244945},[31805],[34,631350,631351,631354],{"style":41504},[34,631352],{"className":631353,"style":41508},[31813],[34,631355,631357],{"className":631356},[31818,31819,31820,31821],[34,631358,631360],{"className":631359},[31685,31821],[34,631361,768],{"className":631362},[31685,31821],[34,631364,631365,631368],{"style":41520},[34,631366],{"className":631367,"style":41508},[31813],[34,631369],{"className":631370,"style":41528},[41527],[34,631372,631373,631376],{"style":46771},[34,631374],{"className":631375,"style":41508},[31813],[34,631377,631379],{"className":631378},[31818,31819,31820,31821],[34,631380,631382,631385],{"className":631381},[31685,31821],[34,631383,217392],{"className":631384,"style":217496},[31685,31686,31821],[34,631386,58442],{"className":631387,"style":32062},[31685,31686,31821],[34,631389,31829],{"className":631390},[31828],[34,631392,631394],{"className":631393},[31801],[34,631395,631397],{"className":631396,"style":41619},[31805],[34,631398],{},[34,631400],{"className":631401},[31690,41488],[34,631403,2557],{"className":631404},[31690],[108,631406,631407,631408,347],{},"得证：",[34,631409,631411,631445],{"className":631410},[31642],[34,631412,631414],{"className":631413},[31646],[31648,631415,631416],{"xmlns":31650},[31652,631417,631418,631442],{},[31655,631419,631420],{},[41453,631421,631422,631424],{},[31658,631423,58442],{},[31655,631425,631426,631428,631430,631432],{},[31749,631427,768],{},[31658,631429,547860],{},[31662,631431,32682],{},[41453,631433,631434,631440],{},[31655,631435,631436,631438],{},[31658,631437,217392],{},[31658,631439,58442],{},[31749,631441,768],{},[31665,631443,631444],{"encoding":31667},"\\frac{\\pi}{2 \\cos \\frac{\\alpha \\pi}{2}}",[34,631446,631448],{"className":631447,"ariaHidden":758},[31672],[34,631449,631451,631454],{"className":631450},[31676],[34,631452],{"className":631453,"style":600976},[31680],[34,631455,631457,631460,631608],{"className":631456},[31685],[34,631458],{"className":631459},[31782,41488],[34,631461,631463],{"className":631462},[41453],[34,631464,631466,631600],{"className":631465},[31796,31797],[34,631467,631469,631597],{"className":631468},[31801],[34,631470,631472,631575,631583],{"className":631471,"style":244945},[31805],[34,631473,631474,631477],{"style":41504},[34,631475],{"className":631476,"style":41508},[31813],[34,631478,631480],{"className":631479},[31818,31819,31820,31821],[34,631481,631483,631486,631489,631501,631504],{"className":631482},[31685,31821],[34,631484,768],{"className":631485},[31685,31821],[34,631487],{"className":631488,"style":43430},[31845,31821],[34,631490,631492,631495,631498],{"className":631491},[32767,31821],[34,631493,31965],{"className":631494},[31821],[34,631496,43383],{"className":631497},[31821],[34,631499,31747],{"className":631500},[31821],[34,631502],{"className":631503,"style":43430},[31845,31821],[34,631505,631507,631510,631572],{"className":631506},[31685,31821],[34,631508],{"className":631509},[31782,41488,31818,43410,435704],[34,631511,631513],{"className":631512},[41453],[34,631514,631516,631564],{"className":631515},[31796,31797],[34,631517,631519,631561],{"className":631518},[31801],[34,631520,631522,631536,631544],{"className":631521,"style":589595},[31805],[34,631523,631524,631527],{"style":435720},[34,631525],{"className":631526,"style":41508},[31813],[34,631528,631530],{"className":631529},[31818,43410,43411,31821],[34,631531,631533],{"className":631532},[31685,31821],[34,631534,768],{"className":631535},[31685,31821],[34,631537,631538,631541],{"style":435735},[34,631539],{"className":631540,"style":41508},[31813],[34,631542],{"className":631543,"style":435742},[41527,31821],[34,631545,631546,631549],{"style":589620},[34,631547],{"className":631548,"style":41508},[31813],[34,631550,631552],{"className":631551},[31818,43410,43411,31821],[34,631553,631555,631558],{"className":631554},[31685,31821],[34,631556,217392],{"className":631557,"style":217496},[31685,31686,31821],[34,631559,58442],{"className":631560,"style":32062},[31685,31686,31821],[34,631562,31829],{"className":631563},[31828],[34,631565,631567],{"className":631566},[31801],[34,631568,631570],{"className":631569,"style":435782},[31805],[34,631571],{},[34,631573],{"className":631574},[31690,41488,31818,43410,435704],[34,631576,631577,631580],{"style":41520},[34,631578],{"className":631579,"style":41508},[31813],[34,631581],{"className":631582,"style":41528},[41527],[34,631584,631585,631588],{"style":46771},[34,631586],{"className":631587,"style":41508},[31813],[34,631589,631591],{"className":631590},[31818,31819,31820,31821],[34,631592,631594],{"className":631593},[31685,31821],[34,631595,58442],{"className":631596,"style":32062},[31685,31686,31821],[34,631598,31829],{"className":631599},[31828],[34,631601,631603],{"className":631602},[31801],[34,631604,631606],{"className":631605,"style":601123},[31805],[34,631607],{},[34,631609],{"className":631610},[31690,41488],[32188,631612],{},[253,631614,631616],{"id":631615},"例子-11-13-讲解","例子 1.1 - 1.3 讲解",[15,631618,631619],{},[2627,631620,631621,631622],{},"例子 1.1: ",[34,631623,631625,631675],{"className":631624},[31642],[34,631626,631628],{"className":631627},[31646],[31648,631629,631630],{"xmlns":31650},[31652,631631,631632,631672],{},[31655,631633,631634,631642,631648,631650,631652,631654,631660,631666,631668,631670],{},[38846,631635,631636,631638,631640],{},[31662,631637,546295],{},[31749,631639,459],{},[31749,631641,640],{},[37624,631643,631644,631646],{},[31658,631645,32936],{},[31658,631647,111],{},[31662,631649,456],{"stretchy":702},[31749,631651,640],{},[31662,631653,32226],{},[37624,631655,631656,631658],{},[31658,631657,32936],{},[31658,631659,1697],{},[37624,631661,631662,631664],{},[31662,631663,2557],{"stretchy":702},[31658,631665,31965],{},[39361,631667,39363],{},[31658,631669,46900],{},[31658,631671,32936],{},[31665,631673,631674],{"encoding":31667},"\\int_0^1 x^a (1 - x^b)^c \\, dx",[34,631676,631678,631782],{"className":631677,"ariaHidden":758},[31672],[34,631679,631681,631684,631735,631738,631767,631770,631773,631776,631779],{"className":631680},[31676],[34,631682],{"className":631683,"style":563309},[31680],[34,631685,631687,631690],{"className":631686},[32767],[34,631688,546295],{"className":631689,"style":546331},[32767,131249,156560],[34,631691,631693],{"className":631692},[31792],[34,631694,631696,631727],{"className":631695},[31796,31797],[34,631697,631699,631724],{"className":631698},[31801],[34,631700,631702,631713],{"className":631701,"style":563328},[31805],[34,631703,631704,631707],{"style":546347},[34,631705],{"className":631706,"style":31814},[31813],[34,631708,631710],{"className":631709},[31818,31819,31820,31821],[34,631711,459],{"className":631712},[31685,31821],[34,631714,631715,631718],{"style":546362},[34,631716],{"className":631717,"style":31814},[31813],[34,631719,631721],{"className":631720},[31818,31819,31820,31821],[34,631722,640],{"className":631723},[31685,31821],[34,631725,31829],{"className":631726},[31828],[34,631728,631730],{"className":631729},[31801],[34,631731,631733],{"className":631732,"style":546384},[31805],[34,631734],{},[34,631736],{"className":631737,"style":32123},[31845],[34,631739,631741,631744],{"className":631740},[31685],[34,631742,32936],{"className":631743},[31685,31686],[34,631745,631747],{"className":631746},[31792],[34,631748,631750],{"className":631749},[31796],[34,631751,631753],{"className":631752},[31801],[34,631754,631756],{"className":631755,"style":37959},[31805],[34,631757,631758,631761],{"style":37777},[34,631759],{"className":631760,"style":31814},[31813],[34,631762,631764],{"className":631763},[31818,31819,31820,31821],[34,631765,111],{"className":631766},[31685,31686,31821],[34,631768,456],{"className":631769},[31782],[34,631771,640],{"className":631772},[31685],[34,631774],{"className":631775,"style":32604},[31845],[34,631777,32226],{"className":631778},[32024],[34,631780],{"className":631781,"style":32604},[31845],[34,631783,631785,631788,631817,631846,631849,631852],{"className":631784},[31676],[34,631786],{"className":631787,"style":43651},[31680],[34,631789,631791,631794],{"className":631790},[31685],[34,631792,32936],{"className":631793},[31685,31686],[34,631795,631797],{"className":631796},[31792],[34,631798,631800],{"className":631799},[31796],[34,631801,631803],{"className":631802},[31801],[34,631804,631806],{"className":631805,"style":38486},[31805],[34,631807,631808,631811],{"style":37777},[34,631809],{"className":631810,"style":31814},[31813],[34,631812,631814],{"className":631813},[31818,31819,31820,31821],[34,631815,1697],{"className":631816},[31685,31686,31821],[34,631818,631820,631823],{"className":631819},[31690],[34,631821,2557],{"className":631822},[31690],[34,631824,631826],{"className":631825},[31792],[34,631827,631829],{"className":631828},[31796],[34,631830,631832],{"className":631831},[31801],[34,631833,631835],{"className":631834,"style":37959},[31805],[34,631836,631837,631840],{"style":37777},[34,631838],{"className":631839,"style":31814},[31813],[34,631841,631843],{"className":631842},[31818,31819,31820,31821],[34,631844,31965],{"className":631845},[31685,31686,31821],[34,631847],{"className":631848,"style":32123},[31845],[34,631850,46900],{"className":631851},[31685,31686],[34,631853,32936],{"className":631854},[31685,31686],[105,631856,631857,632026,632282,632992],{},[108,631858,396859,631859,347],{},[34,631860,631862,631902],{"className":631861},[31642],[34,631863,631865],{"className":631864},[31646],[31648,631866,631867],{"xmlns":31650},[31652,631868,631869,631899],{},[31655,631870,631871,631873,631875,631881,631883,631885,631887],{},[31658,631872,33249],{},[31662,631874,202],{},[37624,631876,631877,631879],{},[31658,631878,32936],{},[31658,631880,1697],{},[31662,631882,140094],{},[31658,631884,32936],{},[31662,631886,202],{},[37624,631888,631889,631891],{},[31658,631890,33249],{},[31655,631892,631893,631895,631897],{},[31749,631894,640],{},[31658,631896,38080],{"mathvariant":38079},[31658,631898,1697],{},[31665,631900,631901],{"encoding":31667},"t = x^b \\Rightarrow x = t^{1\u002Fb}",[34,631903,631905,631923,631967,631985],{"className":631904,"ariaHidden":758},[31672],[34,631906,631908,631911,631914,631917,631920],{"className":631907},[31676],[34,631909],{"className":631910,"style":33280},[31680],[34,631912,33249],{"className":631913},[31685,31686],[34,631915],{"className":631916,"style":31846},[31845],[34,631918,202],{"className":631919},[31850],[34,631921],{"className":631922,"style":31846},[31845],[34,631924,631926,631929,631958,631961,631964],{"className":631925},[31676],[34,631927],{"className":631928,"style":38486},[31680],[34,631930,631932,631935],{"className":631931},[31685],[34,631933,32936],{"className":631934},[31685,31686],[34,631936,631938],{"className":631937},[31792],[34,631939,631941],{"className":631940},[31796],[34,631942,631944],{"className":631943},[31801],[34,631945,631947],{"className":631946,"style":38486},[31805],[34,631948,631949,631952],{"style":37777},[34,631950],{"className":631951,"style":31814},[31813],[34,631953,631955],{"className":631954},[31818,31819,31820,31821],[34,631956,1697],{"className":631957},[31685,31686,31821],[34,631959],{"className":631960,"style":31846},[31845],[34,631962,140094],{"className":631963},[31850],[34,631965],{"className":631966,"style":31846},[31845],[34,631968,631970,631973,631976,631979,631982],{"className":631969},[31676],[34,631971],{"className":631972,"style":32527},[31680],[34,631974,32936],{"className":631975},[31685,31686],[34,631977],{"className":631978,"style":31846},[31845],[34,631980,202],{"className":631981},[31850],[34,631983],{"className":631984,"style":31846},[31845],[34,631986,631988,631991],{"className":631987},[31676],[34,631989],{"className":631990,"style":61037},[31680],[34,631992,631994,631997],{"className":631993},[31685],[34,631995,33249],{"className":631996},[31685,31686],[34,631998,632000],{"className":631999},[31792],[34,632001,632003],{"className":632002},[31796],[34,632004,632006],{"className":632005},[31801],[34,632007,632009],{"className":632008,"style":61037},[31805],[34,632010,632011,632014],{"style":37777},[34,632012],{"className":632013,"style":31814},[31813],[34,632015,632017],{"className":632016},[31818,31819,31820,31821],[34,632018,632020,632023],{"className":632019},[31685,31821],[34,632021,249219],{"className":632022},[31685,31821],[34,632024,1697],{"className":632025},[31685,31686,31821],[108,632027,632028,347],{},[34,632029,632031,632075],{"className":632030},[31642],[34,632032,632034],{"className":632033},[31646],[31648,632035,632036],{"xmlns":31650},[31652,632037,632038,632072],{},[31655,632039,632040,632042,632044,632046,632052,632068,632070],{},[31658,632041,46900],{},[31658,632043,32936],{},[31662,632045,202],{},[41453,632047,632048,632050],{},[31749,632049,640],{},[31658,632051,1697],{},[37624,632053,632054,632056],{},[31658,632055,33249],{},[31655,632057,632058,632064,632066],{},[41453,632059,632060,632062],{},[31749,632061,640],{},[31658,632063,1697],{},[31662,632065,32226],{},[31749,632067,640],{},[31658,632069,46900],{},[31658,632071,33249],{},[31665,632073,632074],{"encoding":31667},"dx = \\frac{1}{b} t^{\\frac{1}{b}-1} dt",[34,632076,632078,632099],{"className":632077,"ariaHidden":758},[31672],[34,632079,632081,632084,632087,632090,632093,632096],{"className":632080},[31676],[34,632082],{"className":632083,"style":31681},[31680],[34,632085,46900],{"className":632086},[31685,31686],[34,632088,32936],{"className":632089},[31685,31686],[34,632091],{"className":632092,"style":31846},[31845],[34,632094,202],{"className":632095},[31850],[34,632097],{"className":632098,"style":31846},[31845],[34,632100,632102,632105,632173,632276,632279],{"className":632101},[31676],[34,632103],{"className":632104,"style":598357},[31680],[34,632106,632108,632111,632170],{"className":632107},[31685],[34,632109],{"className":632110},[31782,41488],[34,632112,632114],{"className":632113},[41453],[34,632115,632117,632162],{"className":632116},[31796,31797],[34,632118,632120,632159],{"className":632119},[31801],[34,632121,632123,632137,632145],{"className":632122,"style":46746},[31805],[34,632124,632125,632128],{"style":41504},[34,632126],{"className":632127,"style":41508},[31813],[34,632129,632131],{"className":632130},[31818,31819,31820,31821],[34,632132,632134],{"className":632133},[31685,31821],[34,632135,1697],{"className":632136},[31685,31686,31821],[34,632138,632139,632142],{"style":41520},[34,632140],{"className":632141,"style":41508},[31813],[34,632143],{"className":632144,"style":41528},[41527],[34,632146,632147,632150],{"style":46771},[34,632148],{"className":632149,"style":41508},[31813],[34,632151,632153],{"className":632152},[31818,31819,31820,31821],[34,632154,632156],{"className":632155},[31685,31821],[34,632157,640],{"className":632158},[31685,31821],[34,632160,31829],{"className":632161},[31828],[34,632163,632165],{"className":632164},[31801],[34,632166,632168],{"className":632167,"style":41619},[31805],[34,632169],{},[34,632171],{"className":632172},[31690,41488],[34,632174,632176,632179],{"className":632175},[31685],[34,632177,33249],{"className":632178},[31685,31686],[34,632180,632182],{"className":632181},[31792],[34,632183,632185],{"className":632184},[31796],[34,632186,632188],{"className":632187},[31801],[34,632189,632191],{"className":632190,"style":598444},[31805],[34,632192,632193,632196],{"style":435688},[34,632194],{"className":632195,"style":41508},[31813],[34,632197,632199],{"className":632198},[31818,31819,31820,31821],[34,632200,632202,632270,632273],{"className":632201},[31685,31821],[34,632203,632205,632208,632267],{"className":632204},[31685,31821],[34,632206],{"className":632207},[31782,41488,31818,43410,435704],[34,632209,632211],{"className":632210},[41453],[34,632212,632214,632259],{"className":632213},[31796,31797],[34,632215,632217,632256],{"className":632216},[31801],[34,632218,632220,632234,632242],{"className":632219,"style":593176},[31805],[34,632221,632222,632225],{"style":435720},[34,632223],{"className":632224,"style":41508},[31813],[34,632226,632228],{"className":632227},[31818,43410,43411,31821],[34,632229,632231],{"className":632230},[31685,31821],[34,632232,1697],{"className":632233},[31685,31686,31821],[34,632235,632236,632239],{"style":435735},[34,632237],{"className":632238,"style":41508},[31813],[34,632240],{"className":632241,"style":435742},[41527,31821],[34,632243,632244,632247],{"style":589620},[34,632245],{"className":632246,"style":41508},[31813],[34,632248,632250],{"className":632249},[31818,43410,43411,31821],[34,632251,632253],{"className":632252},[31685,31821],[34,632254,640],{"className":632255},[31685,31821],[34,632257,31829],{"className":632258},[31828],[34,632260,632262],{"className":632261},[31801],[34,632263,632265],{"className":632264,"style":435782},[31805],[34,632266],{},[34,632268],{"className":632269},[31690,41488,31818,43410,435704],[34,632271,32226],{"className":632272},[32024,31821],[34,632274,640],{"className":632275},[31685,31821],[34,632277,46900],{"className":632278},[31685,31686],[34,632280,33249],{"className":632281},[31685,31686],[108,632283,168124,632284,347],{},[34,632285,632287,632415],{"className":632286},[31642],[34,632288,632290],{"className":632289},[31646],[31648,632291,632292],{"xmlns":31650},[31652,632293,632294,632412],{},[31655,632295,632296,632304,632316,632318,632320,632322,632324,632330,632336,632352,632354,632356,632358,632364,632372,632394,632396,632398,632400,632402,632408,632410],{},[38846,632297,632298,632300,632302],{},[31662,632299,546295],{},[31749,632301,459],{},[31749,632303,640],{},[37624,632305,632306,632308],{},[31658,632307,33249],{},[31655,632309,632310,632312,632314],{},[31658,632311,111],{},[31658,632313,38080],{"mathvariant":38079},[31658,632315,1697],{},[31662,632317,456],{"stretchy":702},[31749,632319,640],{},[31662,632321,32226],{},[31658,632323,33249],{},[37624,632325,632326,632328],{},[31662,632327,2557],{"stretchy":702},[31658,632329,31965],{},[41453,632331,632332,632334],{},[31749,632333,640],{},[31658,632335,1697],{},[37624,632337,632338,632340],{},[31658,632339,33249],{},[31655,632341,632342,632344,632346,632348,632350],{},[31749,632343,640],{},[31658,632345,38080],{"mathvariant":38079},[31658,632347,1697],{},[31662,632349,32226],{},[31749,632351,640],{},[31658,632353,46900],{},[31658,632355,33249],{},[31662,632357,202],{},[41453,632359,632360,632362],{},[31749,632361,640],{},[31658,632363,1697],{},[38846,632365,632366,632368,632370],{},[31662,632367,546295],{},[31749,632369,459],{},[31749,632371,640],{},[37624,632373,632374,632376],{},[31658,632375,33249],{},[31655,632377,632378,632390,632392],{},[41453,632379,632380,632388],{},[31655,632381,632382,632384,632386],{},[31658,632383,111],{},[31662,632385,223],{},[31749,632387,640],{},[31658,632389,1697],{},[31662,632391,32226],{},[31749,632393,640],{},[31662,632395,456],{"stretchy":702},[31749,632397,640],{},[31662,632399,32226],{},[31658,632401,33249],{},[37624,632403,632404,632406],{},[31662,632405,2557],{"stretchy":702},[31658,632407,31965],{},[31658,632409,46900],{},[31658,632411,33249],{},[31665,632413,632414],{"encoding":31667},"\\int_0^1 t^{a\u002Fb} (1-t)^c \\frac{1}{b} t^{1\u002Fb - 1} dt = \\frac{1}{b} \\int_0^1 t^{\\frac{a+1}{b}-1} (1-t)^c dt",[34,632416,632418,632531,632693,632948],{"className":632417,"ariaHidden":758},[31672],[34,632419,632421,632424,632475,632478,632516,632519,632522,632525,632528],{"className":632420},[31676],[34,632422],{"className":632423,"style":563309},[31680],[34,632425,632427,632430],{"className":632426},[32767],[34,632428,546295],{"className":632429,"style":546331},[32767,131249,156560],[34,632431,632433],{"className":632432},[31792],[34,632434,632436,632467],{"className":632435},[31796,31797],[34,632437,632439,632464],{"className":632438},[31801],[34,632440,632442,632453],{"className":632441,"style":563328},[31805],[34,632443,632444,632447],{"style":546347},[34,632445],{"className":632446,"style":31814},[31813],[34,632448,632450],{"className":632449},[31818,31819,31820,31821],[34,632451,459],{"className":632452},[31685,31821],[34,632454,632455,632458],{"style":546362},[34,632456],{"className":632457,"style":31814},[31813],[34,632459,632461],{"className":632460},[31818,31819,31820,31821],[34,632462,640],{"className":632463},[31685,31821],[34,632465,31829],{"className":632466},[31828],[34,632468,632470],{"className":632469},[31801],[34,632471,632473],{"className":632472,"style":546384},[31805],[34,632474],{},[34,632476],{"className":632477,"style":32123},[31845],[34,632479,632481,632484],{"className":632480},[31685],[34,632482,33249],{"className":632483},[31685,31686],[34,632485,632487],{"className":632486},[31792],[34,632488,632490],{"className":632489},[31796],[34,632491,632493],{"className":632492},[31801],[34,632494,632496],{"className":632495,"style":61037},[31805],[34,632497,632498,632501],{"style":37777},[34,632499],{"className":632500,"style":31814},[31813],[34,632502,632504],{"className":632503},[31818,31819,31820,31821],[34,632505,632507,632510,632513],{"className":632506},[31685,31821],[34,632508,111],{"className":632509},[31685,31686,31821],[34,632511,38080],{"className":632512},[31685,31821],[34,632514,1697],{"className":632515},[31685,31686,31821],[34,632517,456],{"className":632518},[31782],[34,632520,640],{"className":632521},[31685],[34,632523],{"className":632524,"style":32604},[31845],[34,632526,32226],{"className":632527},[32024],[34,632529],{"className":632530,"style":32604},[31845],[34,632532,632534,632537,632540,632569,632637,632678,632681,632684,632687,632690],{"className":632533},[31676],[34,632535],{"className":632536,"style":554457},[31680],[34,632538,33249],{"className":632539},[31685,31686],[34,632541,632543,632546],{"className":632542},[31690],[34,632544,2557],{"className":632545},[31690],[34,632547,632549],{"className":632548},[31792],[34,632550,632552],{"className":632551},[31796],[34,632553,632555],{"className":632554},[31801],[34,632556,632558],{"className":632557,"style":37959},[31805],[34,632559,632560,632563],{"style":37777},[34,632561],{"className":632562,"style":31814},[31813],[34,632564,632566],{"className":632565},[31818,31819,31820,31821],[34,632567,31965],{"className":632568},[31685,31686,31821],[34,632570,632572,632575,632634],{"className":632571},[31685],[34,632573],{"className":632574},[31782,41488],[34,632576,632578],{"className":632577},[41453],[34,632579,632581,632626],{"className":632580},[31796,31797],[34,632582,632584,632623],{"className":632583},[31801],[34,632585,632587,632601,632609],{"className":632586,"style":46746},[31805],[34,632588,632589,632592],{"style":41504},[34,632590],{"className":632591,"style":41508},[31813],[34,632593,632595],{"className":632594},[31818,31819,31820,31821],[34,632596,632598],{"className":632597},[31685,31821],[34,632599,1697],{"className":632600},[31685,31686,31821],[34,632602,632603,632606],{"style":41520},[34,632604],{"className":632605,"style":41508},[31813],[34,632607],{"className":632608,"style":41528},[41527],[34,632610,632611,632614],{"style":46771},[34,632612],{"className":632613,"style":41508},[31813],[34,632615,632617],{"className":632616},[31818,31819,31820,31821],[34,632618,632620],{"className":632619},[31685,31821],[34,632621,640],{"className":632622},[31685,31821],[34,632624,31829],{"className":632625},[31828],[34,632627,632629],{"className":632628},[31801],[34,632630,632632],{"className":632631,"style":41619},[31805],[34,632633],{},[34,632635],{"className":632636},[31690,41488],[34,632638,632640,632643],{"className":632639},[31685],[34,632641,33249],{"className":632642},[31685,31686],[34,632644,632646],{"className":632645},[31792],[34,632647,632649],{"className":632648},[31796],[34,632650,632652],{"className":632651},[31801],[34,632653,632655],{"className":632654,"style":61037},[31805],[34,632656,632657,632660],{"style":37777},[34,632658],{"className":632659,"style":31814},[31813],[34,632661,632663],{"className":632662},[31818,31819,31820,31821],[34,632664,632666,632669,632672,632675],{"className":632665},[31685,31821],[34,632667,249219],{"className":632668},[31685,31821],[34,632670,1697],{"className":632671},[31685,31686,31821],[34,632673,32226],{"className":632674},[32024,31821],[34,632676,640],{"className":632677},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B(\\frac{a+1}{b}, 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1.2: ",[34,633234,633236,633282],{"className":633235},[31642],[34,633237,633239],{"className":633238},[31646],[31648,633240,633241],{"xmlns":31650},[31652,633242,633243,633279],{},[31655,633244,633245,633253,633259,633273,633275,633277],{},[38846,633246,633247,633249,633251],{},[31662,633248,546295],{},[31749,633250,459],{},[31749,633252,640],{},[37624,633254,633255,633257],{},[31658,633256,32936],{},[31749,633258,60525],{},[41463,633260,633261],{},[31655,633262,633263,633265,633267],{},[31749,633264,640],{},[31662,633266,32226],{},[37624,633268,633269,633271],{},[31658,633270,32936],{},[31749,633272,786],{},[39361,633274,39363],{},[31658,633276,46900],{},[31658,633278,32936],{},[31665,633280,633281],{"encoding":31667},"\\int_0^1 x^8 \\sqrt{1 - x^3} \\, dx",[34,633283,633285],{"className":633284,"ariaHidden":758},[31672],[34,633286,633288,633291,633342,633345,633374,633459,633462,633465],{"className":633287},[31676],[34,633289],{"className":633290,"style":563309},[31680],[34,633292,633294,633297],{"className":633293},[32767],[34,633295,546295],{"className":633296,"style":546331},[32767,131249,156560],[34,633298,633300],{"className":633299},[31792],[34,633301,633303,633334],{"className":633302},[31796,31797],[34,633304,633306,633331],{"className":633305},[31801],[34,633307,633309,633320],{"className":633308,"style":563328},[31805],[34,633310,633311,633314],{"style":546347},[34,633312],{"className":633313,"style":31814},[31813],[34,633315,633317],{"className":633316},[31818,31819,31820,31821],[34,633318,459],{"className":633319},[31685,31821],[34,633321,633322,633325],{"style":546362},[34,633323],{"className":633324,"style":31814},[31813],[34,633326,633328],{"className":633327},[31818,31819,31820,31821],[34,633329,640],{"className":633330},[31685,31821],[34,633332,31829],{"className":633333},[31828],[34,633335,633337],{"className":633336},[31801],[34,633338,633340],{"className":633339,"style":546384},[31805],[34,633341],{},[34,633343],{"className":633344,"style":32123},[31845],[34,633346,633348,633351],{"className":633347},[31685],[34,633349,32936],{"className":633350},[31685,31686],[34,633352,633354],{"className":633353},[31792],[34,633355,633357],{"className":633356},[31796],[34,633358,633360],{"className":633359},[31801],[34,633361,633363],{"className":633362,"style":37774},[31805],[34,633364,633365,633368],{"style":37777},[34,633366],{"className":633367,"style":31814},[31813],[34,633369,633371],{"className":633370},[31818,31819,31820,31821],[34,633372,60525],{"className":633373},[31685,31821],[34,633375,633377],{"className":633376},[31685,41550],[34,633378,633380,633451],{"className":633379},[31796,31797],[34,633381,633383,633448],{"className":633382},[31801],[34,633384,633386,633436],{"className":633385,"style":591825},[31805],[34,633387,633389,633392],{"className":633388,"style":41565},[41564],[34,633390],{"className":633391,"style":41508},[31813],[34,633393,633395,633398,633401,633404,633407],{"className":633394,"style":41572},[31685],[34,633396,640],{"className":633397},[31685],[34,633399],{"className":633400,"style":32604},[31845],[34,633402,32226],{"className":633403},[32024],[34,633405],{"className":633406,"style":32604},[31845],[34,633408,633410,633413],{"className":633409},[31685],[34,633411,32936],{"className":633412},[31685,31686],[34,633414,633416],{"className":633415},[31792],[34,633417,633419],{"className":633418},[31796],[34,633420,633422],{"className":633421},[31801],[34,633423,633425],{"className":633424,"style":590807},[31805],[34,633426,633427,633430],{"style":590776},[34,633428],{"className":633429,"style":31814},[31813],[34,633431,633433],{"className":633432},[31818,31819,31820,31821],[34,633434,786],{"className":633435},[31685,31821],[34,633437,633438,633441],{"style":591878},[34,633439],{"className":633440,"style":41508},[31813],[34,633442,633444],{"className":633443,"style":41586},[41585],[41588,633445,633446],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,633447],{"d":41597},[34,633449,31829],{"className":633450},[31828],[34,633452,633454],{"className":633453},[31801],[34,633455,633457],{"className":633456,"style":591898},[31805],[34,633458],{},[34,633460],{"className":633461,"style":32123},[31845],[34,633463,46900],{"className":633464},[31685,31686],[34,633466,32936],{"className":633467},[31685,31686],[105,633469,633470,633598,634096],{},[108,633471,633472,633473,347],{},"对应上题，",[34,633474,633476,633514],{"className":633475},[31642],[34,633477,633479],{"className":633478},[31646],[31648,633480,633481],{"xmlns":31650},[31652,633482,633483,633511],{},[31655,633484,633485,633487,633489,633491,633493,633495,633497,633499,633501,633503,633505,633507,633509],{},[31658,633486,111],{},[31662,633488,202],{},[31749,633490,60525],{},[31662,633492,31960],{"separator":758},[31658,633494,1697],{},[31662,633496,202],{},[31749,633498,786],{},[31662,633500,31960],{"separator":758},[31658,633502,31965],{},[31662,633504,202],{},[31749,633506,640],{},[31658,633508,38080],{"mathvariant":38079},[31749,633510,768],{},[31665,633512,633513],{"encoding":31667},"a=8, b=3, c=1\u002F2",[34,633515,633517,633535,633562,633589],{"className":633516,"ariaHidden":758},[31672],[34,633518,633520,633523,633526,633529,633532],{"className":633519},[31676],[34,633521],{"className":633522,"style":32527},[31680],[34,633524,111],{"className":633525},[31685,31686],[34,633527],{"className":633528,"style":31846},[31845],[34,633530,202],{"className":633531},[31850],[34,633533],{"className":633534,"style":31846},[31845],[34,633536,633538,633541,633544,633547,633550,633553,633556,633559],{"className":633537},[31676],[34,633539],{"className":633540,"style":43217},[31680],[34,633542,60525],{"className":633543},[31685],[34,633545,31960],{"className":633546},[32119],[34,633548],{"className":633549,"style":32123},[31845],[34,633551,1697],{"className":633552},[31685,31686],[34,633554],{"className":633555,"style":31846},[31845],[34,633557,202],{"className":633558},[31850],[34,633560],{"className":633561,"style":31846},[31845],[34,633563,633565,633568,633571,633574,633577,633580,633583,633586],{"className":633564},[31676],[34,633566],{"className":633567,"style":40292},[31680],[34,633569,786],{"className":633570},[31685],[34,633572,31960],{"className":633573},[32119],[34,633575],{"className":633576,"style":32123},[31845],[34,633578,31965],{"className":633579},[31685,31686],[34,633581],{"className":633582,"style":31846},[31845],[34,633584,202],{"className":633585},[31850],[34,633587],{"className":633588,"style":31846},[31845],[34,633590,633592,633595],{"className":633591},[31676],[34,633593],{"className":633594,"style":31774},[31680],[34,633596,42407],{"className":633597},[31685],[108,633599,95791,633600,347],{},[34,633601,633603,633675],{"className":633602},[31642],[34,633604,633606],{"className":633605},[31646],[31648,633607,633608],{"xmlns":31650},[31652,633609,633610,633672],{},[31655,633611,633612,633618,633620,633622,633634,633636,633642,633644,633646,633648,633650,633656,633658,633660,633662,633664,633670],{},[41453,633613,633614,633616],{},[31749,633615,640],{},[31749,633617,786],{},[31658,633619,54224],{},[31662,633621,456],{"stretchy":702},[41453,633623,633624,633632],{},[31655,633625,633626,633628,633630],{},[31749,633627,60525],{},[31662,633629,223],{},[31749,633631,640],{},[31749,633633,786],{},[31662,633635,31960],{"separator":758},[41453,633637,633638,633640],{},[31749,633639,640],{},[31749,633641,768],{},[31662,633643,223],{},[31749,633645,640],{},[31662,633647,2557],{"stretchy":702},[31662,633649,202],{},[41453,633651,633652,633654],{},[31749,633653,640],{},[31749,633655,786],{},[31658,633657,54224],{},[31662,633659,456],{"stretchy":702},[31749,633661,786],{},[31662,633663,31960],{"separator":758},[41453,633665,633666,633668],{},[31749,633667,786],{},[31749,633669,768],{},[31662,633671,2557],{"stretchy":702},[31665,633673,633674],{"encoding":31667},"\\frac{1}{3} B(\\frac{8+1}{3}, \\frac{1}{2}+1) = \\frac{1}{3} B(3, \\frac{3}{2})",[34,633676,633678,633915,633936],{"className":633677,"ariaHidden":758},[31672],[34,633679,633681,633684,633752,633755,633758,633832,633835,633838,633906,633909,633912],{"className":633680},[31676],[34,633682],{"className":633683,"style":46727},[31680],[34,633685,633687,633690,633749],{"className":633686},[31685],[34,633688],{"className":633689},[31782,41488],[34,633691,633693],{"className":633692},[41453],[34,633694,633696,633741],{"className":633695},[31796,31797],[34,633697,633699,633738],{"className":633698},[31801],[34,633700,633702,633716,633724],{"className":633701,"style":46746},[31805],[34,633703,633704,633707],{"style":41504},[34,633705],{"className":633706,"style":41508},[31813],[34,633708,633710],{"className":633709},[31818,31819,31820,31821],[34,633711,633713],{"className":633712},[31685,31821],[34,633714,786],{"className":633715},[31685,31821],[34,633717,633718,633721],{"style":41520},[34,633719],{"className":633720,"style":41508},[31813],[34,633722],{"className":633723,"style":41528},[41527],[34,633725,633726,633729],{"style":46771},[34,633727],{"className":633728,"style":41508},[31813],[34,633730,633732],{"className":633731},[31818,31819,31820,31821],[34,633733,633735],{"className":633734},[31685,31821],[34,633736,640],{"className":633737},[31685,31821],[34,633739,31829],{"className":633740},[31828],[34,633742,633744],{"className":633743},[31801],[34,633745,633747],{"className":633746,"style":41619},[31805],[34,633748],{},[34,633750],{"className":633751},[31690,41488],[34,633753,54224],{"className":633754,"style":54252},[31685,31686],[34,633756,456],{"className":633757},[31782],[34,633759,633761,633764,633829],{"className":633760},[31685],[34,633762],{"className":633763},[31782,41488],[34,633765,633767],{"className":633766},[41453],[34,633768,633770,633821],{"className":633769},[31796,31797],[34,633771,633773,633818],{"className":633772},[31801],[34,633774,633776,633790,633798],{"className":633775,"style":46746},[31805],[34,633777,633778,633781],{"style":41504},[34,633779],{"className":633780,"style":41508},[31813],[34,633782,633784],{"className":633783},[31818,31819,31820,31821],[34,633785,633787],{"className":633786},[31685,31821],[34,633788,786],{"className":633789},[31685,31821],[34,633791,633792,633795],{"style":41520},[34,633793],{"className":633794,"style":41508},[31813],[34,633796],{"className":633797,"style":41528},[41527],[34,633799,633800,633803],{"style":46771},[34,633801],{"className":633802,"style":41508},[31813],[34,633804,633806],{"className":633805},[31818,31819,31820,31821],[34,633807,633809,633812,633815],{"className":633808},[31685,31821],[34,633810,60525],{"className":633811},[31685,31821],[34,633813,223],{"className":633814},[32024,31821],[34,633816,640],{"className":633817},[31685,31821],[34,633819,31829],{"className":633820},[31828],[34,633822,633824],{"className":633823},[31801],[34,633825,633827],{"className":633826,"style":41619},[31805],[34,633828],{},[34,633830],{"className":633831},[31690,41488],[34,633833,31960],{"className":633834},[32119],[34,633836],{"className":633837,"style":32123},[31845],[34,633839,633841,633844,633903],{"className":633840},[31685],[34,633842],{"className":633843},[31782,41488],[34,633845,633847],{"className":633846},[41453],[34,633848,633850,633895],{"className":633849},[31796,31797],[34,633851,633853,633892],{"className":633852},[31801],[34,633854,633856,633870,633878],{"className":633855,"style":46746},[31805],[34,633857,633858,633861],{"style":41504},[34,633859],{"className":633860,"style":41508},[31813],[34,633862,633864],{"className":633863},[31818,31819,31820,31821],[34,633865,633867],{"className":633866},[31685,31821],[34,633868,768],{"className":633869},[31685,31821],[34,633871,633872,633875],{"style":41520},[34,633873],{"className":633874,"style":41508},[31813],[34,633876],{"className":633877,"style":41528},[41527],[34,633879,633880,633883],{"style":46771},[34,633881],{"className":633882,"style":41508},[31813],[34,633884,633886],{"className":633885},[31818,31819,31820,31821],[34,633887,633889],{"className":633888},[31685,31821],[34,633890,640],{"className":633891},[31685,31821],[34,633893,31829],{"className":633894},[31828],[34,633896,633898],{"className":633897},[31801],[34,633899,633901],{"className":633900,"style":41619},[31805],[34,633902],{},[34,633904],{"className":633905},[31690,41488],[34,633907],{"className":633908,"style":32604},[31845],[34,633910,223],{"className":633911},[32024],[34,633913],{"className":633914,"style":32604},[31845],[34,633916,633918,633921,633924,633927,633930,633933],{"className":633917},[31676],[34,633919],{"className":633920,"style":31774},[31680],[34,633922,640],{"className":633923},[31685],[34,633925,2557],{"className":633926},[31690],[34,633928],{"className":633929,"style":31846},[31845],[34,633931,202],{"className":633932},[31850],[34,633934],{"className":633935,"style":31846},[31845],[34,633937,633939,633942,634010,634013,634016,634019,634022,634025,634093],{"className":633938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\\frac{\\Gamma(3)\\Gamma(3\u002F2)}{\\Gamma(9\u002F2)} = \\frac{1}{3} \\frac{2 \\cdot \\frac{1}{2}\\sqrt{\\pi}}{\\frac{7}{2}\\frac{5}{2}\\frac{3}{2}\\frac{1}{2}\\sqrt{\\pi}} = \\frac{1}{3} \\frac{1}{\\frac{105}{16}} = 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771},[34,635276],{"className":635277,"style":41508},[31813],[34,635279,635281],{"className":635280},[31818,31819,31820,31821],[34,635282,635284],{"className":635283},[31685,31821],[34,635285,177869],{"className":635286},[31685,31821],[34,635288,31829],{"className":635289},[31828],[34,635291,635293],{"className":635292},[31801],[34,635294,635296],{"className":635295,"style":41619},[31805],[34,635297],{},[34,635299],{"className":635300},[31690,41488],[15,635302,635303],{},[2627,635304,635305],{},"例子 1.3: 三叶玫瑰线面积",[105,635307,635308,635487,635729,635826,635892,636275,636560,636740],{},[108,635309,635310,635311,347],{},"面积公式（极坐标）：",[34,635312,635314,635348],{"className":635313},[31642],[34,635315,635317],{"className":635316},[31646],[31648,635318,635319],{"xmlns":31650},[31652,635320,635321,635345],{},[31655,635322,635323,635325,635327,635333,635335,635341,635343],{},[31658,635324,94675],{},[31662,635326,202],{},[41453,635328,635329,635331],{},[31749,635330,640],{},[31749,635332,768],{},[31662,635334,546295],{},[37624,635336,635337,635339],{},[31658,635338,220084],{},[31749,635340,768],{},[31658,635342,46900],{},[31658,635344,479697],{},[31665,635346,635347],{"encoding":31667},"S = \\frac{1}{2} \\int \\rho^2 d\\theta",[34,635349,635351,635369],{"className":635350,"ariaHidden":758},[31672],[34,635352,635354,635357,635360,635363,635366],{"className":635353},[31676],[34,635355],{"className":635356,"style":33218},[31680],[34,635358,94675],{"className":635359,"style":106455},[31685,31686],[34,635361],{"className":635362,"style":31846},[31845],[34,635364,202],{"className":635365},[31850],[34,635367],{"className":635368,"style":31846},[31845],[34,635370,635372,635375,635443,635446,635449,635452,635481,635484],{"className":635371},[31676],[34,635373],{"className":635374,"style":46727},[31680],[34,635376,635378,635381,635440],{"className":635377},[31685],[34,635379],{"className":635380},[31782,41488],[34,635382,635384],{"className":635383},[41453],[34,635385,635387,635432],{"className":635386},[31796,31797],[34,635388,635390,635429],{"className":635389},[31801],[34,635391,635393,635407,635415],{"className":635392,"style":46746},[31805],[34,635394,635395,635398],{"style":41504},[34,635396],{"className":635397,"style":41508},[31813],[34,635399,635401],{"className":635400},[31818,31819,31820,31821],[34,635402,635404],{"className":635403},[31685,31821],[34,635405,768],{"className":635406},[31685,31821],[34,635408,635409,635412],{"style":41520},[34,635410],{"className":635411,"style":41508},[31813],[34,635413],{"className":635414,"style":41528},[41527],[34,635416,635417,635420],{"style":46771},[34,635418],{"className":635419,"style":41508},[31813],[34,635421,635423],{"className":635422},[31818,31819,31820,31821],[34,635424,635426],{"className":635425},[31685,31821],[34,635427,640],{"className":635428},[31685,31821],[34,635430,31829],{"className":635431},[31828],[34,635433,635435],{"className":635434},[31801],[34,635436,635438],{"className":635437,"style":41619},[31805],[34,635439],{},[34,635441],{"className":635442},[31690,41488],[34,635444],{"className":635445,"style":32123},[31845],[34,635447,546295],{"className":635448,"style":546331},[32767,131249,156560],[34,635450],{"className":635451,"style":32123},[31845],[34,635453,635455,635458],{"className":635454},[31685],[34,635456,220084],{"className":635457},[31685,31686],[34,635459,635461],{"className":635460},[31792],[34,635462,635464],{"className":635463},[31796],[34,635465,635467],{"className":635466},[31801],[34,635468,635470],{"className":635469,"style":37774},[31805],[34,635471,635472,635475],{"style":37777},[34,635473],{"className":635474,"style":31814},[31813],[34,635476,635478],{"className":635477},[31818,31819,31820,31821],[34,635479,768],{"className":635480},[31685,31821],[34,635482,46900],{"className":635483},[31685,31686],[34,635485,479697],{"className":635486,"style":34452},[31685,31686],[108,635488,25496,635489,635531,635532,635654,635655,347],{},[34,635490,635492,635510],{"className":635491},[31642],[34,635493,635495],{"className":635494},[31646],[31648,635496,635497],{"xmlns":31650},[31652,635498,635499,635507],{},[31655,635500,635501,635503,635505],{},[31658,635502,32936],{},[31662,635504,31960],{"separator":758},[31658,635506,67764],{},[31665,635508,635509],{"encoding":31667},"x,y",[34,635511,635513],{"className":635512,"ariaHidden":758},[31672],[34,635514,635516,635519,635522,635525,635528],{"className":635515},[31676],[34,635517],{"className":635518,"style":38774},[31680],[34,635520,32936],{"className":635521},[31685,31686],[34,635523,31960],{"className":635524},[32119],[34,635526],{"className":635527,"style":32123},[31845],[34,635529,67764],{"className":635530,"style":32062},[31685,31686]," 给出的是参数方程，我们将其转为极坐标。注意到 ",[34,635533,635535,635573],{"className":635534},[31642],[34,635536,635538],{"className":635537},[31646],[31648,635539,635540],{"xmlns":31650},[31652,635541,635542,635570],{},[31655,635543,635544,635546,635548,635550,635552,635554,635556,635558,635560,635562,635564,635566,635568],{},[31658,635545,32936],{},[31662,635547,202],{},[31658,635549,220084],{},[31658,635551,547860],{},[31662,635553,32682],{},[31658,635555,479697],{},[31662,635557,31960],{"separator":758},[31658,635559,67764],{},[31662,635561,202],{},[31658,635563,220084],{},[31658,635565,547853],{},[31662,635567,32682],{},[31658,635569,479697],{},[31665,635571,635572],{"encoding":31667},"x = \\rho \\cos \\theta, y = \\rho \\sin \\theta",[34,635574,635576,635594,635633],{"className":635575,"ariaHidden":758},[31672],[34,635577,635579,635582,635585,635588,635591],{"className":635578},[31676],[34,635580],{"className":635581,"style":32527},[31680],[34,635583,32936],{"className":635584},[31685,31686],[34,635586],{"className":635587,"style":31846},[31845],[34,635589,202],{"className":635590},[31850],[34,635592],{"className":635593,"style":31846},[31845],[34,635595,635597,635600,635603,635606,635609,635612,635615,635618,635621,635624,635627,635630],{"className":635596},[31676],[34,635598],{"className":635599,"style":43217},[31680],[34,635601,220084],{"className":635602},[31685,31686],[34,635604],{"className":635605,"style":32123},[31845],[34,635607,547860],{"className":635608},[32767],[34,635610],{"className":635611,"style":32123},[31845],[34,635613,479697],{"className":635614,"style":34452},[31685,31686],[34,635616,31960],{"className":635617},[32119],[34,635619],{"className":635620,"style":32123},[31845],[34,635622,67764],{"className":635623,"style":32062},[31685,31686],[34,635625],{"className":635626,"style":31846},[31845],[34,635628,202],{"className":635629},[31850],[34,635631],{"className":635632,"style":31846},[31845],[34,635634,635636,635639,635642,635645,635648,635651],{"className":635635},[31676],[34,635637],{"className":635638,"style":43217},[31680],[34,635640,220084],{"className":635641},[31685,31686],[34,635643],{"className":635644,"style":32123},[31845],[34,635646,547853],{"className":635647},[32767],[34,635649],{"className":635650,"style":32123},[31845],[34,635652,479697],{"className":635653,"style":34452},[31685,31686],"，对比题目 ",[34,635656,635658,635684],{"className":635657},[31642],[34,635659,635661],{"className":635660},[31646],[31648,635662,635663],{"xmlns":31650},[31652,635664,635665,635681],{},[31655,635666,635667,635669,635671,635673,635675,635677,635679],{},[31658,635668,220084],{},[31662,635670,202],{},[31658,635672,111],{},[31658,635674,547860],{},[31662,635676,32682],{},[31749,635678,786],{},[31658,635680,479697],{},[31665,635682,635683],{"encoding":31667},"\\rho = a \\cos 3\\theta",[34,635685,635687,635705],{"className":635686,"ariaHidden":758},[31672],[34,635688,635690,635693,635696,635699,635702],{"className":635689},[31676],[34,635691],{"className":635692,"style":38774},[31680],[34,635694,220084],{"className":635695},[31685,31686],[34,635697],{"className":635698,"style":31846},[31845],[34,635700,202],{"className":635701},[31850],[34,635703],{"className":635704,"style":31846},[31845],[34,635706,635708,635711,635714,635717,635720,635723,635726],{"className":635707},[31676],[34,635709],{"className":635710,"style":31681},[31680],[34,635712,111],{"className":635713},[31685,31686],[34,635715],{"className":635716,"style":32123},[31845],[34,635718,547860],{"className":635719},[32767],[34,635721],{"className":635722,"style":32123},[31845],[34,635724,786],{"className":635725},[31685],[34,635727,479697],{"className":635728,"style":34452},[31685,31686],[108,635730,635731,635732,635825],{},"因为有三片叶子，我们可以算一片（",[34,635733,635735,635771],{"className":635734},[31642],[34,635736,635738],{"className":635737},[31646],[31648,635739,635740],{"xmlns":31650},[31652,635741,635742,635768],{},[31655,635743,635744,635746,635748,635750,635752,635754,635756,635758,635760,635762,635764,635766],{},[31658,635745,479697],{},[31662,635747,112296],{},[31662,635749,15972],{"stretchy":702},[31662,635751,32226],{},[31658,635753,58442],{},[31658,635755,38080],{"mathvariant":38079},[31749,635757,36656],{},[31662,635759,31960],{"separator":758},[31658,635761,58442],{},[31658,635763,38080],{"mathvariant":38079},[31749,635765,36656],{},[31662,635767,32712],{"stretchy":702},[31665,635769,635770],{"encoding":31667},"\\theta \\in [-\\pi\u002F6, \\pi\u002F6]",[34,635772,635774,635792],{"className":635773,"ariaHidden":758},[31672],[34,635775,635777,635780,635783,635786,635789],{"className":635776},[31676],[34,635778],{"className":635779,"style":46991},[31680],[34,635781,479697],{"className":635782,"style":34452},[31685,31686],[34,635784],{"className":635785,"style":31846},[31845],[34,635787,112296],{"className":635788},[31850],[34,635790],{"className":635791,"style":31846},[31845],[34,635793,635795,635798,635801,635804,635807,635810,635813,635816,635819,635822],{"className":635794},[31676],[34,635796],{"className":635797,"style":31774},[31680],[34,635799,15972],{"className":635800},[31782],[34,635802,32226],{"className":635803},[31685],[34,635805,58442],{"className":635806,"style":32062},[31685,31686],[34,635808,230802],{"className":635809},[31685],[34,635811,31960],{"className":635812},[32119],[34,635814],{"className":635815,"style":32123},[31845],[34,635817,58442],{"className":635818,"style":32062},[31685,31686],[34,635820,230802],{"className":635821},[31685],[34,635823,32712],{"className":635824},[31690],"）然后乘 3。或者直接利用周期性。",[108,635827,635828,635856,635857,635891],{},[34,635829,635831,635844],{"className":635830},[31642],[34,635832,635834],{"className":635833},[31646],[31648,635835,635836],{"xmlns":31650},[31652,635837,635838,635842],{},[31655,635839,635840],{},[31658,635841,479697],{},[31665,635843,481351],{"encoding":31667},[34,635845,635847],{"className":635846,"ariaHidden":758},[31672],[34,635848,635850,635853],{"className":635849},[31676],[34,635851],{"className":635852,"style":31681},[31680],[34,635854,479697],{"className":635855,"style":34452},[31685,31686]," 从 0 到 ",[34,635858,635860,635876],{"className":635859},[31642],[34,635861,635863],{"className":635862},[31646],[31648,635864,635865],{"xmlns":31650},[31652,635866,635867,635873],{},[31655,635868,635869,635871],{},[31749,635870,768],{},[31658,635872,58442],{},[31665,635874,635875],{"encoding":31667},"2\\pi",[34,635877,635879],{"className":635878,"ariaHidden":758},[31672],[34,635880,635882,635885,635888],{"className":635881},[31676],[34,635883],{"className":635884,"style":32617},[31680],[34,635886,768],{"className":635887},[31685],[34,635889,58442],{"className":635890,"style":32062},[31685,31686]," 会画出完整的花。",[108,635893,635894,636154,636155,636206,636207,636240,636241,636274],{},[34,635895,635897,635949],{"className":635896},[31642],[34,635898,635900],{"className":635899},[31646],[31648,635901,635902],{"xmlns":31650},[31652,635903,635904,635946],{},[31655,635905,635906,635908,635910,635916,635924,635926,635928,635930,635932,635934,635936,635942,635944],{},[31658,635907,94675],{},[31662,635909,202],{},[41453,635911,635912,635914],{},[31749,635913,640],{},[31749,635915,768],{},[38846,635917,635918,635920,635922],{},[31662,635919,546295],{},[31749,635921,459],{},[31658,635923,58442],{},[31662,635925,456],{"stretchy":702},[31658,635927,111],{},[31658,635929,547860],{},[31662,635931,32682],{},[31749,635933,786],{},[31658,635935,479697],{},[37624,635937,635938,635940],{},[31662,635939,2557],{"stretchy":702},[31749,635941,768],{},[31658,635943,46900],{},[31658,635945,479697],{},[31665,635947,635948],{"encoding":31667},"S = \\frac{1}{2} \\int_0^\\pi (a \\cos 3\\theta)^2 d\\theta",[34,635950,635952,635970],{"className":635951,"ariaHidden":758},[31672],[34,635953,635955,635958,635961,635964,635967],{"className":635954},[31676],[34,635956],{"className":635957,"style":33218},[31680],[34,635959,94675],{"className":635960,"style":106455},[31685,31686],[34,635962],{"className":635963,"style":31846},[31845],[34,635965,202],{"className":635966},[31850],[34,635968],{"className":635969,"style":31846},[31845],[34,635971,635973,635976,636044,636047,636098,636101,636104,636107,636110,636113,636116,636119,636148,636151],{"className":635972},[31676],[34,635974],{"className":635975,"style":613610},[31680],[34,635977,635979,635982,636041],{"className":635978},[31685],[34,635980],{"className":635981},[31782,41488],[34,635983,635985],{"className":635984},[41453],[34,635986,635988,636033],{"className":635987},[31796,31797],[34,635989,635991,636030],{"className":635990},[31801],[34,635992,635994,636008,636016],{"className":635993,"style":46746},[31805],[34,635995,635996,635999],{"style":41504},[34,635997],{"className":635998,"style":41508},[31813],[34,636000,636002],{"className":636001},[31818,31819,31820,31821],[34,636003,636005],{"className":636004},[31685,31821],[34,636006,768],{"className":636007},[31685,31821],[34,636009,636010,636013],{"style":41520},[34,636011],{"className":636012,"style":41508},[31813],[34,636014],{"className":636015,"style":41528},[41527],[34,636017,636018,636021],{"style":46771},[34,636019],{"className":636020,"style":41508},[31813],[34,636022,636024],{"className":636023},[31818,31819,31820,31821],[34,636025,636027],{"className":636026},[31685,31821],[34,636028,640],{"className":636029},[31685,31821],[34,636031,31829],{"className":636032},[31828],[34,636034,636036],{"className":636035},[31801],[34,636037,636039],{"className":636038,"style":41619},[31805],[34,636040],{},[34,636042],{"className":636043},[31690,41488],[34,636045],{"className":636046,"style":32123},[31845],[34,636048,636050,636053],{"className":636049},[32767],[34,636051,546295],{"className":636052,"style":546331},[32767,131249,156560],[34,636054,636056],{"className":636055},[31792],[34,636057,636059,636090],{"className":636058},[31796,31797],[34,636060,636062,636087],{"className":636061},[31801],[34,636063,636065,636076],{"className":636064,"style":567928},[31805],[34,636066,636067,636070],{"style":546347},[34,636068],{"className":636069,"style":31814},[31813],[34,636071,636073],{"className":636072},[31818,31819,31820,31821],[34,636074,459],{"className":636075},[31685,31821],[34,636077,636078,636081],{"style":546362},[34,636079],{"className":636080,"style":31814},[31813],[34,636082,636084],{"className":636083},[31818,31819,31820,31821],[34,636085,58442],{"className":636086,"style":32062},[31685,31686,31821],[34,636088,31829],{"className":636089},[31828],[34,636091,636093],{"className":636092},[31801],[34,636094,636096],{"className":636095,"style":546384},[31805],[34,636097],{},[34,636099,456],{"className":636100},[31782],[34,636102,111],{"className":636103},[31685,31686],[34,636105],{"className":636106,"style":32123},[31845],[34,636108,547860],{"className":636109},[32767],[34,636111],{"className":636112,"style":32123},[31845],[34,636114,786],{"className":636115},[31685],[34,636117,479697],{"className":636118,"style":34452},[31685,31686],[34,636120,636122,636125],{"className":636121},[31690],[34,636123,2557],{"className":636124},[31690],[34,636126,636128],{"className":636127},[31792],[34,636129,636131],{"className":636130},[31796],[34,636132,636134],{"className":636133},[31801],[34,636135,636137],{"className":636136,"style":37774},[31805],[34,636138,636139,636142],{"style":37777},[34,636140],{"className":636141,"style":31814},[31813],[34,636143,636145],{"className":636144},[31818,31819,31820,31821],[34,636146,768],{"className":636147},[31685,31821],[34,636149,46900],{"className":636150},[31685,31686],[34,636152,479697],{"className":636153,"style":34452},[31685,31686]," （注意区间重复性，通常三叶玫瑰线 ",[34,636156,636158,636176],{"className":636157},[31642],[34,636159,636161],{"className":636160},[31646],[31648,636162,636163],{"xmlns":31650},[31652,636164,636165,636173],{},[31655,636166,636167,636169,636171],{},[31749,636168,459],{},[31662,636170,33239],{},[31658,636172,58442],{},[31665,636174,636175],{"encoding":31667},"0 \\to \\pi",[34,636177,636179,636197],{"className":636178,"ariaHidden":758},[31672],[34,636180,636182,636185,636188,636191,636194],{"className":636181},[31676],[34,636183],{"className":636184,"style":32617},[31680],[34,636186,459],{"className":636187},[31685],[34,636189],{"className":636190,"style":31846},[31845],[34,636192,33239],{"className":636193},[31850],[34,636195],{"className":636196,"style":31846},[31845],[34,636198,636200,636203],{"className":636199},[31676],[34,636201],{"className":636202,"style":32527},[31680],[34,636204,58442],{"className":636205,"style":32062},[31685,31686]," 即可画完，再往后是重复。如果是 ",[34,636208,636210,636225],{"className":636209},[31642],[34,636211,636213],{"className":636212},[31646],[31648,636214,636215],{"xmlns":31650},[31652,636216,636217,636223],{},[31655,636218,636219,636221],{},[31749,636220,768],{},[31658,636222,58442],{},[31665,636224,635875],{"encoding":31667},[34,636226,636228],{"className":636227,"ariaHidden":758},[31672],[34,636229,636231,636234,636237],{"className":636230},[31676],[34,636232],{"className":636233,"style":32617},[31680],[34,636235,768],{"className":636236},[31685],[34,636238,58442],{"className":636239,"style":32062},[31685,31686],"，可能算了两次，题目给定 ",[34,636242,636244,636259],{"className":636243},[31642],[34,636245,636247],{"className":636246},[31646],[31648,636248,636249],{"xmlns":31650},[31652,636250,636251,636257],{},[31655,636252,636253,636255],{},[31749,636254,768],{},[31658,636256,58442],{},[31665,636258,635875],{"encoding":31667},[34,636260,636262],{"className":636261,"ariaHidden":758},[31672],[34,636263,636265,636268,636271],{"className":636264},[31676],[34,636266],{"className":636267,"style":32617},[31680],[34,636269,768],{"className":636270},[31685],[34,636272,58442],{"className":636273,"style":32062},[31685,31686]," 则需确认是否覆盖）。",[108,636276,636277,636278,347],{},"一般计算：",[34,636279,636281,636335],{"className":636280},[31642],[34,636282,636284],{"className":636283},[31646],[31648,636285,636286],{"xmlns":31650},[31652,636287,636288,636332],{},[31655,636289,636290,636292,636294,636300,636306,636314,636324,636326,636328,636330],{},[31658,636291,94675],{},[31662,636293,202],{},[41453,636295,636296,636298],{},[31749,636297,640],{},[31749,636299,768],{},[37624,636301,636302,636304],{},[31658,636303,111],{},[31749,636305,768],{},[38846,636307,636308,636310,636312],{},[31662,636309,546295],{},[31749,636311,459],{},[31658,636313,58442],{},[37624,636315,636316,636322],{},[31655,636317,636318,636320],{},[31658,636319,547860],{},[31662,636321,32682],{},[31749,636323,768],{},[31749,636325,786],{},[31658,636327,479697],{},[31658,636329,46900],{},[31658,636331,479697],{},[31665,636333,636334],{"encoding":31667},"S = \\frac{1}{2} a^2 \\int_0^\\pi \\cos^2 3\\theta d\\theta",[34,636336,636338,636356],{"className":636337,"ariaHidden":758},[31672],[34,636339,636341,636344,636347,636350,636353],{"className":636340},[31676],[34,636342],{"className":636343,"style":33218},[31680],[34,636345,94675],{"className":636346,"style":106455},[31685,31686],[34,636348],{"className":636349,"style":31846},[31845],[34,636351,202],{"className":636352},[31850],[34,636354],{"className":636355,"style":31846},[31845],[34,636357,636359,636362,636430,636459,636462,636513,636516,636545,636548,636551,636554,636557],{"className":636358},[31676],[34,636360],{"className":636361,"style":613610},[31680],[34,636363,636365,636368,636427],{"className":636364},[31685],[34,636366],{"className":636367},[31782,41488],[34,636369,636371],{"className":636370},[41453],[34,636372,636374,636419],{"className":636373},[31796,31797],[34,636375,636377,636416],{"className":636376},[31801],[34,636378,636380,636394,636402],{"className":636379,"style":46746},[31805],[34,636381,636382,636385],{"style":41504},[34,636383],{"className":636384,"style":41508},[31813],[34,636386,636388],{"className":636387},[31818,31819,31820,31821],[34,636389,636391],{"className":636390},[31685,31821],[34,636392,768],{"className":636393},[31685,31821],[34,636395,636396,636399],{"style":41520},[34,636397],{"className":636398,"style":41508},[31813],[34,636400],{"className":636401,"style":41528},[41527],[34,636403,636404,636407],{"style":46771},[34,636405],{"className":636406,"style":41508},[31813],[34,636408,636410],{"className":636409},[31818,31819,31820,31821],[34,636411,636413],{"className":636412},[31685,31821],[34,636414,640],{"className":636415},[31685,31821],[34,636417,31829],{"className":636418},[31828],[34,636420,636422],{"className":636421},[31801],[34,636423,636425],{"className":636424,"style":41619},[31805],[34,636426],{},[34,636428],{"className":636429},[31690,41488],[34,636431,636433,636436],{"className":636432},[31685],[34,636434,111],{"className":636435},[31685,31686],[34,636437,636439],{"className":636438},[31792],[34,636440,636442],{"className":636441},[31796],[34,636443,636445],{"className":636444},[31801],[34,636446,636448],{"className":636447,"style":37774},[31805],[34,636449,636450,636453],{"style":37777},[34,636451],{"className":636452,"style":31814},[31813],[34,636454,636456],{"className":636455},[31818,31819,31820,31821],[34,636457,768],{"className":636458},[31685,31821],[34,636460],{"className":636461,"style":32123},[31845],[34,636463,636465,636468],{"className":636464},[32767],[34,636466,546295],{"className":636467,"style":546331},[32767,131249,156560],[34,636469,636471],{"className":636470},[31792],[34,636472,636474,636505],{"className":636473},[31796,31797],[34,636475,636477,636502],{"className":636476},[31801],[34,636478,636480,636491],{"className":636479,"style":567928},[31805],[34,636481,636482,636485],{"style":546347},[34,636483],{"className":636484,"style":31814},[31813],[34,636486,636488],{"className":636487},[31818,31819,31820,31821],[34,636489,459],{"className":636490},[31685,31821],[34,636492,636493,636496],{"style":546362},[34,636494],{"className":636495,"style":31814},[31813],[34,636497,636499],{"className":636498},[31818,31819,31820,31821],[34,636500,58442],{"className":636501,"style":32062},[31685,31686,31821],[34,636503,31829],{"className":636504},[31828],[34,636506,636508],{"className":636507},[31801],[34,636509,636511],{"className":636510,"style":546384},[31805],[34,636512],{},[34,636514],{"className":636515,"style":32123},[31845],[34,636517,636519,636522],{"className":636518},[32767],[34,636520,547860],{"className":636521},[32767],[34,636523,636525],{"className":636524},[31792],[34,636526,636528],{"className":636527},[31796],[34,636529,636531],{"className":636530},[31801],[34,636532,636534],{"className":636533,"style":37774},[31805],[34,636535,636536,636539],{"style":37777},[34,636537],{"className":636538,"style":31814},[31813],[34,636540,636542],{"className":636541},[31818,31819,31820,31821],[34,636543,768],{"className":636544},[31685,31821],[34,636546],{"className":636547,"style":32123},[31845],[34,636549,786],{"className":636550},[31685],[34,636552,479697],{"className":636553,"style":34452},[31685,31686],[34,636555,46900],{"className":636556},[31685,31686],[34,636558,479697],{"className":636559,"style":34452},[31685,31686],[108,636561,636562,636563,347],{},"利用 Walli's 公式或倍角公式，",[34,636564,636566,636612],{"className":636565},[31642],[34,636567,636569],{"className":636568},[31646],[31648,636570,636571],{"xmlns":31650},[31652,636572,636573,636609],{},[31655,636574,636575,636583,636593,636595,636597,636599,636601,636603,636605,636607],{},[38846,636576,636577,636579,636581],{},[31662,636578,546295],{},[31749,636580,459],{},[31658,636582,58442],{},[37624,636584,636585,636591],{},[31655,636586,636587,636589],{},[31658,636588,547860],{},[31662,636590,32682],{},[31749,636592,768],{},[31658,636594,33759],{},[31658,636596,32936],{},[31658,636598,46900],{},[31658,636600,32936],{},[31662,636602,202],{},[31658,636604,58442],{},[31658,636606,38080],{"mathvariant":38079},[31749,636608,768],{},[31665,636610,636611],{"encoding":31667},"\\int_0^\\pi \\cos^2 kx dx = \\pi\u002F2",[34,636613,636615,636728],{"className":636614,"ariaHidden":758},[31672],[34,636616,636618,636621,636672,636675,636704,636707,636710,636713,636716,636719,636722,636725],{"className":636617},[31676],[34,636619],{"className":636620,"style":613610},[31680],[34,636622,636624,636627],{"className":636623},[32767],[34,636625,546295],{"className":636626,"style":546331},[32767,131249,156560],[34,636628,636630],{"className":636629},[31792],[34,636631,636633,636664],{"className":636632},[31796,31797],[34,636634,636636,636661],{"className":636635},[31801],[34,636637,636639,636650],{"className":636638,"style":567928},[31805],[34,636640,636641,636644],{"style":546347},[34,636642],{"className":636643,"style":31814},[31813],[34,636645,636647],{"className":636646},[31818,31819,31820,31821],[34,636648,459],{"className":636649},[31685,31821],[34,636651,636652,636655],{"style":546362},[34,636653],{"className":636654,"style":31814},[31813],[34,636656,636658],{"className":636657},[31818,31819,31820,31821],[34,636659,58442],{"className":636660,"style":32062},[31685,31686,31821],[34,636662,31829],{"className":636663},[31828],[34,636665,636667],{"className":636666},[31801],[34,636668,636670],{"className":636669,"style":546384},[31805],[34,636671],{},[34,636673],{"className":636674,"style":32123},[31845],[34,636676,636678,636681],{"className":636677},[32767],[34,636679,547860],{"className":636680},[32767],[34,636682,636684],{"className":636683},[31792],[34,636685,636687],{"className":636686},[31796],[34,636688,636690],{"className":636689},[31801],[34,636691,636693],{"className":636692,"style":37774},[31805],[34,636694,636695,636698],{"style":37777},[34,636696],{"className":636697,"style":31814},[31813],[34,636699,636701],{"className":636700},[31818,31819,31820,31821],[34,636702,768],{"className":636703},[31685,31821],[34,636705],{"className":636706,"style":32123},[31845],[34,636708,33759],{"className":636709,"style":33774},[31685,31686],[34,636711,32936],{"className":636712},[31685,31686],[34,636714,46900],{"className":636715},[31685,31686],[34,636717,32936],{"className":636718},[31685,31686],[34,636720],{"className":636721,"style":31846},[31845],[34,636723,202],{"className":636724},[31850],[34,636726],{"className":636727,"style":31846},[31845],[34,636729,636731,636734,636737],{"className":636730},[31676],[34,636732],{"className":636733,"style":31774},[31680],[34,636735,58442],{"className":636736,"style":32062},[31685,31686],[34,636738,38149],{"className":636739},[31685],[108,636741,600826,636742,347],{},[34,636743,636745,636797],{"className":636744},[31642],[34,636746,636748],{"className":636747},[31646],[31648,636749,636750],{"xmlns":31650},[31652,636751,636752,636794],{},[31655,636753,636754,636756,636758,636764,636770,636772,636778,636780],{},[31658,636755,94675],{},[31662,636757,202],{},[41453,636759,636760,636762],{},[31749,636761,640],{},[31749,636763,768],{},[37624,636765,636766,636768],{},[31658,636767,111],{},[31749,636769,768],{},[31662,636771,71191],{},[41453,636773,636774,636776],{},[31658,636775,58442],{},[31749,636777,768],{},[31662,636779,202],{},[41453,636781,636782,636792],{},[31655,636783,636784,636786],{},[31658,636785,58442],{},[37624,636787,636788,636790],{},[31658,636789,111],{},[31749,636791,768],{},[31749,636793,15932],{},[31665,636795,636796],{"encoding":31667},"S = \\frac{1}{2} a^2 \\cdot \\frac{\\pi}{2} = \\frac{\\pi a^2}{4}",[34,636798,636800,636818,636930,637013],{"className":636799,"ariaHidden":758},[31672],[34,636801,636803,636806,636809,636812,636815],{"className":636802},[31676],[34,636804],{"className":636805,"style":33218},[31680],[34,636807,94675],{"className":636808,"style":106455},[31685,31686],[34,636810],{"className":636811,"style":31846},[31845],[34,636813,202],{"className":636814},[31850],[34,636816],{"className":636817,"style":31846},[31845],[34,636819,636821,636824,636892,636921,636924,636927],{"className":636820},[31676],[34,636822],{"className":636823,"style":46727},[31680],[34,636825,636827,636830,636889],{"className":636826},[31685],[34,636828],{"className":636829},[31782,41488],[34,636831,636833],{"className":636832},[41453],[34,636834,636836,636881],{"className":636835},[31796,31797],[34,636837,636839,636878],{"className":636838},[31801],[34,636840,636842,636856,636864],{"className":636841,"style":46746},[31805],[34,636843,636844,636847],{"style":41504},[34,636845],{"className":636846,"style":41508},[31813],[34,636848,636850],{"className":636849},[31818,31819,31820,31821],[34,636851,636853],{"className":636852},[31685,31821],[34,636854,768],{"className":636855},[31685,31821],[34,636857,636858,636861],{"style":41520},[34,636859],{"className":636860,"style":41508},[31813],[34,636862],{"className":636863,"style":41528},[41527],[34,636865,636866,636869],{"style":46771},[34,636867],{"className":636868,"style":41508},[31813],[34,636870,636872],{"className":636871},[31818,31819,31820,31821],[34,636873,636875],{"className":636874},[31685,31821],[34,636876,640],{"className":636877},[31685,31821],[34,636879,31829],{"className":636880},[31828],[34,636882,636884],{"className":636883},[31801],[34,636885,636887],{"className":636886,"style":41619},[31805],[34,636888],{},[34,636890],{"className":636891},[31690,41488],[34,636893,636895,636898],{"className":636894},[31685],[34,636896,111],{"className":636897},[31685,31686],[34,636899,636901],{"className":636900},[31792],[34,636902,636904],{"className":636903},[31796],[34,636905,636907],{"className":636906},[31801],[34,636908,636910],{"className":636909,"style":37774},[31805],[34,636911,636912,636915],{"style":37777},[34,636913],{"className":636914,"style":31814},[31813],[34,636916,636918],{"className":636917},[31818,31819,31820,31821],[34,636919,768],{"className":636920},[31685,31821],[34,636922],{"className":636923,"style":32604},[31845],[34,636925,71191],{"className":636926},[32024],[34,636928],{"className":636929,"style":32604},[31845],[34,636931,636933,636936,637004,637007,637010],{"className":636932},[31676],[34,636934],{"className":636935,"style":244926},[31680],[34,636937,636939,636942,637001],{"className":636938},[31685],[34,636940],{"className":636941},[31782,41488],[34,636943,636945],{"className":636944},[41453],[34,636946,636948,636993],{"className":636947},[31796,31797],[34,636949,636951,636990],{"className":636950},[31801],[34,636952,636954,636968,636976],{"className":636953,"style":244945},[31805],[34,636955,636956,636959],{"style":41504},[34,636957],{"className":636958,"style":41508},[31813],[34,636960,636962],{"className":636961},[31818,31819,31820,31821],[34,636963,636965],{"className":636964},[31685,31821],[34,636966,768],{"className":636967},[31685,31821],[34,636969,636970,636973],{"style":41520},[34,636971],{"className":636972,"style":41508},[31813],[34,636974],{"className":636975,"style":41528},[41527],[34,636977,636978,636981],{"style":46771},[34,636979],{"className":636980,"style":41508},[31813],[34,636982,636984],{"className":636983},[31818,31819,31820,31821],[34,636985,636987],{"className":636986},[31685,31821],[34,636988,58442],{"className":636989,"style":32062},[31685,31686,31821],[34,636991,31829],{"className":636992},[31828],[34,636994,636996],{"className":636995},[31801],[34,636997,636999],{"className":636998,"style":41619},[31805],[34,637000],{},[34,637002],{"className":637003},[31690,41488],[34,637005],{"className":637006,"style":31846},[31845],[34,637008,202],{"className":637009},[31850],[34,637011],{"className":637012,"style":31846},[31845],[34,637014,637016,637020],{"className":637015},[31676],[34,637017],{"className":637018,"style":637019},[31680],"height:1.3629em;vertical-align:-0.345em;",[34,637021,637023,637026,637114],{"className":637022},[31685],[34,637024],{"className":637025},[31782,41488],[34,637027,637029],{"className":637028},[41453],[34,637030,637032,637106],{"className":637031},[31796,31797],[34,637033,637035,637103],{"className":637034},[31801],[34,637036,637038,637052,637060],{"className":637037,"style":603956},[31805],[34,637039,637040,637043],{"style":41504},[34,637041],{"className":637042,"style":41508},[31813],[34,637044,637046],{"className":637045},[31818,31819,31820,31821],[34,637047,637049],{"className":637048},[31685,31821],[34,637050,15932],{"className":637051},[31685,31821],[34,637053,637054,637057],{"style":41520},[34,637055],{"className":637056,"style":41508},[31813],[34,637058],{"className":637059,"style":41528},[41527],[34,637061,637062,637065],{"style":46771},[34,637063],{"className":637064,"style":41508},[31813],[34,637066,637068],{"className":637067},[31818,31819,31820,31821],[34,637069,637071,637074],{"className":637070},[31685,31821],[34,637072,58442],{"className":637073,"style":32062},[31685,31686,31821],[34,637075,637077,637080],{"className":637076},[31685,31821],[34,637078,111],{"className":637079},[31685,31686,31821],[34,637081,637083],{"className":637082},[31792],[34,637084,637086],{"className":637085},[31796],[34,637087,637089],{"className":637088},[31801],[34,637090,637092],{"className":637091,"style":144123},[31805],[34,637093,637094,637097],{"style":552391},[34,637095],{"className":637096,"style":43406},[31813],[34,637098,637100],{"className":637099},[31818,43410,43411,31821],[34,637101,768],{"className":637102},[31685,31821],[34,637104,31829],{"className":637105},[31828],[34,637107,637109],{"className":637108},[31801],[34,637110,637112],{"className":637111,"style":41619},[31805],[34,637113],{},[34,637115],{"className":637116},[31690,41488],[32188,637118],{},[253,637120,637122],{"id":637121},"第三部分p263-第一类曲线积分-t1-t2","第三部分：P263 第一类曲线积分 T1 & T2",[32490,637124,637126],{"id":637125},"t1-求下列第一类曲线积分","T1. 求下列第一类曲线积分",[15,637128,637129],{},[2627,637130,637131,637132],{},"(1) 三角形 ",[34,637133,637135,637187],{"className":637134},[31642],[34,637136,637138],{"className":637137},[31646],[31648,637139,637140],{"xmlns":31650},[31652,637141,637142,637184],{},[31655,637143,637144,637146,637148,637150,637152,637154,637156,637158,637160,637162,637164,637166,637168,637170,637172,637174,637176,637178,637180,637182],{},[31658,637145,37890],{},[31662,637147,456],{"stretchy":702},[31749,637149,459],{},[31662,637151,31960],{"separator":758},[31749,637153,459],{},[31662,637155,2557],{"stretchy":702},[31662,637157,31960],{"separator":758},[31658,637159,54151],{},[31662,637161,456],{"stretchy":702},[31749,637163,640],{},[31662,637165,31960],{"separator":758},[31749,637167,459],{},[31662,637169,2557],{"stretchy":702},[31662,637171,31960],{"separator":758},[31658,637173,54224],{},[31662,637175,456],{"stretchy":702},[31749,637177,459],{},[31662,637179,31960],{"separator":758},[31749,637181,640],{},[31662,637183,2557],{"stretchy":702},[31665,637185,637186],{"encoding":31667},"O(0,0), A(1,0), B(0,1)",[34,637188,637190],{"className":637189,"ariaHidden":758},[31672],[34,637191,637193,637196,637199,637202,637205,637208,637211,637214,637217,637220,637223,637226,637229,637232,637235,637238,637241,637244,637247,637250,637253,637256,637259,637262,637265,637268],{"className":637192},[31676],[34,637194],{"className":637195,"style":31774},[31680],[34,637197,37890],{"className":637198,"style":34452},[31685,31686],[34,637200,456],{"className":637201},[31782],[34,637203,459],{"className":637204},[31685],[34,637206,31960],{"className":637207},[32119],[34,637209],{"className":637210,"style":32123},[31845],[34,637212,459],{"className":637213},[31685],[34,637215,2557],{"className":637216},[31690],[34,637218,31960],{"className":637219},[32119],[34,637221],{"className":637222,"style":32123},[31845],[34,637224,54151],{"className":637225},[31685,31686],[34,637227,456],{"className":637228},[31782],[34,637230,640],{"className":637231},[31685],[34,637233,31960],{"className":637234},[32119],[34,637236],{"className":637237,"style":32123},[31845],[34,637239,459],{"className":637240},[31685],[34,637242,2557],{"className":637243},[31690],[34,637245,31960],{"className":637246},[32119],[34,637248],{"className":637249,"style":32123},[31845],[34,637251,54224],{"className":637252,"style":54252},[31685,31686],[34,637254,456],{"className":637255},[31782],[34,637257,459],{"className":637258},[31685],[34,637260,31960],{"className":637261},[32119],[34,637263],{"className":637264,"style":32123},[31845],[34,637266,640],{"className":637267},[31685],[34,637269,2557],{"className":637270},[31690],[105,637272,637273,639224],{},[108,637274,637275,637276,637395,637396],{},"积分 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ds",[34,637312,637314,637377],{"className":637313,"ariaHidden":758},[31672],[34,637315,637317,637321,637362,637365,637368,637371,637374],{"className":637316},[31676],[34,637318],{"className":637319,"style":637320},[31680],"height:1.1608em;vertical-align:-0.3558em;",[34,637322,637324,637327],{"className":637323},[32767],[34,637325,546295],{"className":637326,"style":546331},[32767,131249,156560],[34,637328,637330],{"className":637329},[31792],[34,637331,637333,637354],{"className":637332},[31796,31797],[34,637334,637336,637351],{"className":637335},[31801],[34,637337,637340],{"className":637338,"style":637339},[31805],"height:0.1225em;",[34,637341,637342,637345],{"style":546347},[34,637343],{"className":637344,"style":31814},[31813],[34,637346,637348],{"className":637347},[31818,31819,31820,31821],[34,637349,33206],{"className":637350},[31685,31686,31821],[34,637352,31829],{"className":637353},[31828],[34,637355,637357],{"className":637356},[31801],[34,637358,637360],{"className":637359,"style":546384},[31805],[34,637361],{},[34,637363,456],{"className":637364},[31782],[34,637366,32936],{"className":637367},[31685,31686],[34,637369],{"className":637370,"style":32604},[31845],[34,637372,223],{"className":637373},[32024],[34,637375],{"className":637376,"style":32604},[31845],[34,637378,637380,637383,637386,637389,637392],{"className":637379},[31676],[34,637381],{"className":637382,"style":31774},[31680],[34,637384,67764],{"className":637385,"style":32062},[31685,31686],[34,637387,2557],{"className":637388},[31690],[34,637390,46900],{"className":637391},[31685,31686],[34,637393,31747],{"className":637394},[31685,31686]," 分三段算。\n",[2230,637397,637398,637716,638832],{},[108,637399,637400,8471,637528,347,637589,347],{},[34,637401,637403,637441],{"className":637402},[31642],[34,637404,637406],{"className":637405},[31646],[31648,637407,637408],{"xmlns":31650},[31652,637409,637410,637438],{},[31655,637411,637412,637414,637416,637418,637420,637422,637424,637426,637428,637430,637432,637434,637436],{},[31658,637413,37890],{},[31658,637415,54151],{},[31662,637417,456],{"stretchy":702},[31658,637419,67764],{},[31662,637421,202],{},[31749,637423,459],{},[31662,637425,31960],{"separator":758},[31658,637427,32936],{},[31662,637429,5598],{},[31749,637431,459],{},[31662,637433,33239],{},[31749,637435,640],{},[31662,637437,2557],{"stretchy":702},[31665,637439,637440],{"encoding":31667},"OA (y=0, x: 0\\to 1)",[34,637442,637444,637471,637498,637516],{"className":637443,"ariaHidden":758},[31672],[34,637445,637447,637450,637453,637456,637459,637462,637465,637468],{"className":637446},[31676],[34,637448],{"className":637449,"style":31774},[31680],[34,637451,37890],{"className":637452,"style":34452},[31685,31686],[34,637454,54151],{"className":637455},[31685,31686],[34,637457,456],{"className":637458},[31782],[34,637460,67764],{"className":637461,"style":32062},[31685,31686],[34,637463],{"className":637464,"style":31846},[31845],[34,637466,202],{"className":637467},[31850],[34,637469],{"className":637470,"style":31846},[31845],[34,637472,637474,637477,637480,637483,637486,637489,637492,637495],{"className":637473},[31676],[34,637475],{"className":637476,"style":40292},[31680],[34,637478,459],{"className":637479},[31685],[34,637481,31960],{"className":637482},[32119],[34,637484],{"className":637485,"style":32123},[31845],[34,637487,32936],{"className":637488},[31685,31686],[34,637490],{"className":637491,"style":31846},[31845],[34,637493,5598],{"className":637494},[31850],[34,637496],{"className":637497,"style":31846},[31845],[34,637499,637501,637504,637507,637510,637513],{"className":637500},[31676],[34,637502],{"className":637503,"style":32617},[31680],[34,637505,459],{"className":637506},[31685],[34,637508],{"className":637509,"style":31846},[31845],[34,637511,33239],{"className":637512},[31850],[34,637514],{"className":637515,"style":31846},[31845],[34,637517,637519,637522,637525],{"className":637518},[31676],[34,637520],{"className":637521,"style":31774},[31680],[34,637523,640],{"className":637524},[31685],[34,637526,2557],{"className":637527},[31690],[34,637529,637531,637553],{"className":637530},[31642],[34,637532,637534],{"className":637533},[31646],[31648,637535,637536],{"xmlns":31650},[31652,637537,637538,637550],{},[31655,637539,637540,637542,637544,637546,637548],{},[31658,637541,46900],{},[31658,637543,31747],{},[31662,637545,202],{},[31658,637547,46900],{},[31658,637549,32936],{},[31665,637551,637552],{"encoding":31667},"ds=dx",[34,637554,637556,637577],{"className":637555,"ariaHidden":758},[31672],[34,637557,637559,637562,637565,637568,637571,637574],{"className":637558},[31676],[34,637560],{"className":637561,"style":31681},[31680],[34,637563,46900],{"className":637564},[31685,31686],[34,637566,31747],{"className":637567},[31685,31686],[34,637569],{"className":637570,"style":31846},[31845],[34,637572,202],{"className":637573},[31850],[34,637575],{"className":637576,"style":31846},[31845],[34,637578,637580,637583,637586],{"className":637579},[31676],[34,637581],{"className":637582,"style":31681},[31680],[34,637584,46900],{"className":637585},[31685,31686],[34,637587,32936],{"className":637588},[31685,31686],[34,637590,637592,637626],{"className":637591},[31642],[34,637593,637595],{"className":637594},[31646],[31648,637596,637597],{"xmlns":31650},[31652,637598,637599,637623],{},[31655,637600,637601,637609,637611,637613,637615,637617,637619,637621],{},[38846,637602,637603,637605,637607],{},[31662,637604,546295],{},[31749,637606,459],{},[31749,637608,640],{},[31658,637610,32936],{},[31658,637612,46900],{},[31658,637614,32936],{},[31662,637616,202],{},[31749,637618,640],{},[31658,637620,38080],{"mathvariant":38079},[31749,637622,768],{},[31665,637624,637625],{"encoding":31667},"\\int_0^1 x dx = 1\u002F2",[34,637627,637629,637707],{"className":637628,"ariaHidden":758},[31672],[34,637630,637632,637635,637686,637689,637692,637695,637698,637701,637704],{"className":637631},[31676],[34,637633],{"className":637634,"style":563309},[31680],[34,637636,637638,637641],{"className":637637},[32767],[34,637639,546295],{"className":637640,"style":546331},[32767,131249,156560],[34,637642,637644],{"className":637643},[31792],[34,637645,637647,637678],{"className":637646},[31796,31797],[34,637648,637650,637675],{"className":637649},[31801],[34,637651,637653,637664],{"className":637652,"style":563328},[31805],[34,637654,637655,637658],{"style":546347},[34,637656],{"className":637657,"style":31814},[31813],[34,637659,637661],{"className":637660},[31818,31819,31820,31821],[34,637662,459],{"className":637663},[31685,31821],[34,637665,637666,637669],{"style":546362},[34,637667],{"className":637668,"style":31814},[31813],[34,637670,637672],{"className":637671},[31818,31819,31820,31821],[34,637673,640],{"className":637674},[31685,31821],[34,637676,31829],{"className":637677},[31828],[34,637679,637681],{"className":637680},[31801],[34,637682,637684],{"className":637683,"style":546384},[31805],[34,637685],{},[34,637687],{"className":637688,"style":32123},[31845],[34,637690,32936],{"className":637691},[31685,31686],[34,637693,46900],{"className":637694},[31685,31686],[34,637696,32936],{"className":637697},[31685,31686],[34,637699],{"className":637700,"style":31846},[31845],[34,637702,202],{"className":637703},[31850],[34,637705],{"className":637706,"style":31846},[31845],[34,637708,637710,637713],{"className":637709},[31676],[34,637711],{"className":637712,"style":31774},[31680],[34,637714,42407],{"className":637715},[31685],[108,637717,637718,637868,637869,46528,637947,638240,638241,638406,638407,638440,638441,638590,638591,638666,638667,347],{},[34,637719,637721,637763],{"className":637720},[31642],[34,637722,637724],{"className":637723},[31646],[31648,637725,637726],{"xmlns":31650},[31652,637727,637728,637760],{},[31655,637729,637730,637732,637734,637736,637738,637740,637742,637744,637746,637748,637750,637752,637754,637756,637758],{},[31658,637731,54151],{},[31658,637733,54224],{},[31662,637735,456],{"stretchy":702},[31658,637737,67764],{},[31662,637739,202],{},[31749,637741,640],{},[31662,637743,32226],{},[31658,637745,32936],{},[31662,637747,31960],{"separator":758},[31658,637749,32936],{},[31662,637751,5598],{},[31749,637753,640],{},[31662,637755,33239],{},[31749,637757,459],{},[31662,637759,2557],{"stretchy":702},[31665,637761,637762],{"encoding":31667},"AB (y=1-x, x: 1\\to 0)",[34,637764,637766,637793,637811,637838,637856],{"className":637765,"ariaHidden":758},[31672],[34,637767,637769,637772,637775,637778,637781,637784,637787,637790],{"className":637768},[31676],[34,637770],{"className":637771,"style":31774},[31680],[34,637773,54151],{"className":637774},[31685,31686],[34,637776,54224],{"className":637777,"style":54252},[31685,31686],[34,637779,456],{"className":637780},[31782],[34,637782,67764],{"className":637783,"style":32062},[31685,31686],[34,637785],{"className":637786,"style":31846},[31845],[34,637788,202],{"className":637789},[31850],[34,637791],{"className":637792,"style":31846},[31845],[34,637794,637796,637799,637802,637805,637808],{"className":637795},[31676],[34,637797],{"className":637798,"style":35942},[31680],[34,637800,640],{"className":637801},[31685],[34,637803],{"className":637804,"style":32604},[31845],[34,637806,32226],{"className":637807},[32024],[34,637809],{"className":637810,"style":32604},[31845],[34,637812,637814,637817,637820,637823,637826,637829,637832,637835],{"className":637813},[31676],[34,637815],{"className":637816,"style":38774},[31680],[34,637818,32936],{"className":637819},[31685,31686],[34,637821,31960],{"className":637822},[32119],[34,637824],{"className":637825,"style":32123},[31845],[34,637827,32936],{"className":637828},[31685,31686],[34,637830],{"className":637831,"style":31846},[31845],[34,637833,5598],{"className":637834},[31850],[34,637836],{"className":637837,"style":31846},[31845],[34,637839,637841,637844,637847,637850,637853],{"className":637840},[31676],[34,637842],{"className":637843,"style":32617},[31680],[34,637845,640],{"className":637846},[31685],[34,637848],{"className":637849,"style":31846},[31845],[34,637851,33239],{"className":637852},[31850],[34,637854],{"className":637855,"style":31846},[31845],[34,637857,637859,637862,637865],{"className":637858},[31676],[34,637860],{"className":637861,"style":31774},[31680],[34,637863,459],{"className":637864},[31685],[34,637866,2557],{"className":637867},[31690],": 直线 ",[34,637870,637872,637896],{"className":637871},[31642],[34,637873,637875],{"className":637874},[31646],[31648,637876,637877],{"xmlns":31650},[31652,637878,637879,637893],{},[31655,637880,637881,637883,637885,637887,637889,637891],{},[31658,637882,67764],{},[31662,637884,202],{},[31662,637886,32226],{},[31658,637888,32936],{},[31662,637890,223],{},[31749,637892,640],{},[31665,637894,637895],{"encoding":31667},"y=-x+1",[34,637897,637899,637917,637938],{"className":637898,"ariaHidden":758},[31672],[34,637900,637902,637905,637908,637911,637914],{"className":637901},[31676],[34,637903],{"className":637904,"style":38774},[31680],[34,637906,67764],{"className":637907,"style":32062},[31685,31686],[34,637909],{"className":637910,"style":31846},[31845],[34,637912,202],{"className":637913},[31850],[34,637915],{"className":637916,"style":31846},[31845],[34,637918,637920,637923,637926,637929,637932,637935],{"className":637919},[31676],[34,637921],{"className":637922,"style":32597},[31680],[34,637924,32226],{"className":637925},[31685],[34,637927,32936],{"className":637928},[31685,31686],[34,637930],{"className":637931,"style":32604},[31845],[34,637933,223],{"className":637934},[32024],[34,637936],{"className":637937,"style":32604},[31845],[34,637939,637941,637944],{"className":637940},[31676],[34,637942],{"className":637943,"style":32617},[31680],[34,637945,640],{"className":637946},[31685],[34,637948,637950,638010],{"className":637949},[31642],[34,637951,637953],{"className":637952},[31646],[31648,637954,637955],{"xmlns":31650},[31652,637956,637957,638007],{},[31655,637958,637959,637961,637963,637965,637989,637991,637993,637995,637999,638001,638003,638005],{},[31658,637960,46900],{},[31658,637962,31747],{},[31662,637964,202],{},[41463,637966,637967],{},[31655,637968,637969,637975,637977,637979,637981,637983],{},[37624,637970,637971,637973],{},[31749,637972,640],{},[31749,637974,768],{},[31662,637976,223],{},[31662,637978,456],{"stretchy":702},[31662,637980,32226],{},[31749,637982,640],{},[37624,637984,637985,637987],{},[31662,637986,2557],{"stretchy":702},[31749,637988,768],{},[31658,637990,46900],{},[31658,637992,32936],{},[31662,637994,202],{},[41463,637996,637997],{},[31749,637998,768],{},[31658,638000,162086],{"mathvariant":38079},[31658,638002,46900],{},[31658,638004,32936],{},[31658,638006,162086],{"mathvariant":38079},[31665,638008,638009],{"encoding":31667},"ds = \\sqrt{1^2+(-1)^2}dx = \\sqrt{2}|dx|",[34,638011,638013,638034,638175],{"className":638012,"ariaHidden":758},[31672],[34,638014,638016,638019,638022,638025,638028,638031],{"className":638015},[31676],[34,638017],{"className":638018,"style":31681},[31680],[34,638020,46900],{"className":638021},[31685,31686],[34,638023,31747],{"className":638024},[31685,31686],[34,638026],{"className":638027,"style":31846},[31845],[34,638029,202],{"className":638030},[31850],[34,638032],{"className":638033,"style":31846},[31845],[34,638035,638037,638040,638160,638163,638166,638169,638172],{"className":638036},[31676],[34,638038],{"className":638039,"style":590727},[31680],[34,638041,638043],{"className":638042},[31685,41550],[34,638044,638046,638152],{"className":638045},[31796,31797],[34,638047,638049,638149],{"className":638048},[31801],[34,638050,638052,638137],{"className":638051,"style":590740},[31805],[34,638053,638055,638058],{"className":638054,"style":590744},[41564],[34,638056],{"className":638057,"style":552795},[31813],[34,638059,638061,638090,638093,638096,638099,638102,638105,638108],{"className":638060,"style":590751},[31685],[34,638062,638064,638067],{"className":638063},[31685],[34,638065,640],{"className":638066},[31685],[34,638068,638070],{"className":638069},[31792],[34,638071,638073],{"className":638072},[31796],[34,638074,638076],{"className":638075},[31801],[34,638077,638079],{"className":638078,"style":590807},[31805],[34,638080,638081,638084],{"style":590776},[34,638082],{"className":638083,"style":31814},[31813],[34,638085,638087],{"className":638086},[31818,31819,31820,31821],[34,638088,768],{"className":638089},[31685,31821],[34,638091],{"className":638092,"style":32604},[31845],[34,638094,223],{"className":638095},[32024],[34,638097],{"className":638098,"style":32604},[31845],[34,638100,456],{"className":638101},[31782],[34,638103,32226],{"className":638104},[31685],[34,638106,640],{"className":638107},[31685],[34,638109,638111,638114],{"className":638110},[31690],[34,638112,2557],{"className":638113},[31690],[34,638115,638117],{"className":638116},[31792],[34,638118,638120],{"className":638119},[31796],[34,638121,638123],{"className":638122},[31801],[34,638124,638126],{"className":638125,"style":590807},[31805],[34,638127,638128,638131],{"style":590776},[34,638129],{"className":638130,"style":31814},[31813],[34,638132,638134],{"className":638133},[31818,31819,31820,31821],[34,638135,768],{"className":638136},[31685,31821],[34,638138,638139,638142],{"style":590894},[34,638140],{"className":638141,"style":552795},[31813],[34,638143,638145],{"className":638144,"style":590901},[41585],[41588,638146,638147],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,638148],{"d":590908},[34,638150,31829],{"className":638151},[31828],[34,638153,638155],{"className":638154},[31801],[34,638156,638158],{"className":638157,"style":590918},[31805],[34,638159],{},[34,638161,46900],{"className":638162},[31685,31686],[34,638164,32936],{"className":638165},[31685,31686],[34,638167],{"className":638168,"style":31846},[31845],[34,638170,202],{"className":638171},[31850],[34,638173],{"className":638174,"style":31846},[31845],[34,638176,638178,638181,638228,638231,638234,638237],{"className":638177},[31676],[34,638179],{"className":638180,"style":571985},[31680],[34,638182,638184],{"className":638183},[31685,41550],[34,638185,638187,638220],{"className":638186},[31796,31797],[34,638188,638190,638217],{"className":638189},[31801],[34,638191,638193,638205],{"className":638192,"style":572001},[31805],[34,638194,638196,638199],{"className":638195,"style":41565},[41564],[34,638197],{"className":638198,"style":41508},[31813],[34,638200,638202],{"className":638201,"style":41572},[31685],[34,638203,768],{"className":638204},[31685],[34,638206,638207,638210],{"style":572016},[34,638208],{"className":638209,"style":41508},[31813],[34,638211,638213],{"className":638212,"style":41586},[41585],[41588,638214,638215],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,638216],{"d":41597},[34,638218,31829],{"className":638219},[31828],[34,638221,638223],{"className":638222},[31801],[34,638224,638226],{"className":638225,"style":572036},[31805],[34,638227],{},[34,638229,162086],{"className":638230},[31685],[34,638232,46900],{"className":638233},[31685,31686],[34,638235,32936],{"className":638236},[31685,31686],[34,638238,162086],{"className":638239},[31685],"。注意第一类曲线积分下限小上限大。或者直接参数化 ",[34,638242,638244,638286],{"className":638243},[31642],[34,638245,638247],{"className":638246},[31646],[31648,638248,638249],{"xmlns":31650},[31652,638250,638251,638283],{},[31655,638252,638253,638255,638257,638259,638261,638263,638265,638267,638269,638271,638273,638275,638277,638279,638281],{},[31658,638254,32936],{},[31662,638256,202],{},[31658,638258,33249],{},[31662,638260,31960],{"separator":758},[31658,638262,67764],{},[31662,638264,202],{},[31749,638266,640],{},[31662,638268,32226],{},[31658,638270,33249],{},[31662,638272,31960],{"separator":758},[31658,638274,33249],{},[31662,638276,5598],{},[31749,638278,640],{},[31662,638280,33239],{},[31749,638282,459],{},[31665,638284,638285],{"encoding":31667},"x=t, y=1-t, t: 1\\to 0",[34,638287,638289,638307,638334,638352,638379,638397],{"className":638288,"ariaHidden":758},[31672],[34,638290,638292,638295,638298,638301,638304],{"className":638291},[31676],[34,638293],{"className":638294,"style":32527},[31680],[34,638296,32936],{"className":638297},[31685,31686],[34,638299],{"className":638300,"style":31846},[31845],[34,638302,202],{"className":638303},[31850],[34,638305],{"className":638306,"style":31846},[31845],[34,638308,638310,638313,638316,638319,638322,638325,638328,638331],{"className":638309},[31676],[34,638311],{"className":638312,"style":223552},[31680],[34,638314,33249],{"className":638315},[31685,31686],[34,638317,31960],{"className":638318},[32119],[34,638320],{"className":638321,"style":32123},[31845],[34,638323,67764],{"className":638324,"style":32062},[31685,31686],[34,638326],{"className":638327,"style":31846},[31845],[34,638329,202],{"className":638330},[31850],[34,638332],{"className":638333,"style":31846},[31845],[34,638335,638337,638340,638343,638346,638349],{"className":638336},[31676],[34,638338],{"className":638339,"style":35942},[31680],[34,638341,640],{"className":638342},[31685],[34,638344],{"className":638345,"style":32604},[31845],[34,638347,32226],{"className":638348},[32024],[34,638350],{"className":638351,"style":32604},[31845],[34,638353,638355,638358,638361,638364,638367,638370,638373,638376],{"className":638354},[31676],[34,638356],{"className":638357,"style":223552},[31680],[34,638359,33249],{"className":638360},[31685,31686],[34,638362,31960],{"className":638363},[32119],[34,638365],{"className":638366,"style":32123},[31845],[34,638368,33249],{"className":638369},[31685,31686],[34,638371],{"className":638372,"style":31846},[31845],[34,638374,5598],{"className":638375},[31850],[34,638377],{"className":638378,"style":31846},[31845],[34,638380,638382,638385,638388,638391,638394],{"className":638381},[31676],[34,638383],{"className":638384,"style":32617},[31680],[34,638386,640],{"className":638387},[31685],[34,638389],{"className":638390,"style":31846},[31845],[34,638392,33239],{"className":638393},[31850],[34,638395],{"className":638396,"style":31846},[31845],[34,638398,638400,638403],{"className":638399},[31676],[34,638401],{"className":638402,"style":32617},[31680],[34,638404,459],{"className":638405},[31685]," 是错的，应该用长度参数。简单说，把线段拉直积分。\n在 ",[34,638408,638410,638425],{"className":638409},[31642],[34,638411,638413],{"className":638412},[31646],[31648,638414,638415],{"xmlns":31650},[31652,638416,638417,638423],{},[31655,638418,638419,638421],{},[31658,638420,54151],{},[31658,638422,54224],{},[31665,638424,93307],{"encoding":31667},[34,638426,638428],{"className":638427,"ariaHidden":758},[31672],[34,638429,638431,638434,638437],{"className":638430},[31676],[34,638432],{"className":638433,"style":33218},[31680],[34,638435,54151],{"className":638436},[31685,31686],[34,638438,54224],{"className":638439,"style":54252},[31685,31686]," 上，函数值 ",[34,638442,638444,638482],{"className":638443},[31642],[34,638445,638447],{"className":638446},[31646],[31648,638448,638449],{"xmlns":31650},[31652,638450,638451,638479],{},[31655,638452,638453,638455,638457,638459,638461,638463,638465,638467,638469,638471,638473,638475,638477],{},[31658,638454,32936],{},[31662,638456,223],{},[31658,638458,67764],{},[31662,638460,202],{},[31658,638462,32936],{},[31662,638464,223],{},[31662,638466,456],{"stretchy":702},[31749,638468,640],{},[31662,638470,32226],{},[31658,638472,32936],{},[31662,638474,2557],{"stretchy":702},[31662,638476,202],{},[31749,638478,640],{},[31665,638480,638481],{"encoding":31667},"x+y = x+(1-x) = 1",[34,638483,638485,638503,638521,638539,638560,638581],{"className":638484,"ariaHidden":758},[31672],[34,638486,638488,638491,638494,638497,638500],{"className":638487},[31676],[34,638489],{"className":638490,"style":32597},[31680],[34,638492,32936],{"className":638493},[31685,31686],[34,638495],{"className":638496,"style":32604},[31845],[34,638498,223],{"className":638499},[32024],[34,638501],{"className":638502,"style":32604},[31845],[34,638504,638506,638509,638512,638515,638518],{"className":638505},[31676],[34,638507],{"className":638508,"style":38774},[31680],[34,638510,67764],{"className":638511,"style":32062},[31685,31686],[34,638513],{"className":638514,"style":31846},[31845],[34,638516,202],{"className":638517},[31850],[34,638519],{"className":638520,"style":31846},[31845],[34,638522,638524,638527,638530,638533,638536],{"className":638523},[31676],[34,638525],{"className":638526,"style":32597},[31680],[34,638528,32936],{"className":638529},[31685,31686],[34,638531],{"className":638532,"style":32604},[31845],[34,638534,223],{"className":638535},[32024],[34,638537],{"className":638538,"style":32604},[31845],[34,638540,638542,638545,638548,638551,638554,638557],{"className":638541},[31676],[34,638543],{"className":638544,"style":31774},[31680],[34,638546,456],{"className":638547},[31782],[34,638549,640],{"className":638550},[31685],[34,638552],{"className":638553,"style":32604},[31845],[34,638555,32226],{"className":638556},[32024],[34,638558],{"className":638559,"style":32604},[31845],[34,638561,638563,638566,638569,638572,638575,638578],{"className":638562},[31676],[34,638564],{"className":638565,"style":31774},[31680],[34,638567,32936],{"className":638568},[31685,31686],[34,638570,2557],{"className":638571},[31690],[34,638573],{"className":638574,"style":31846},[31845],[34,638576,202],{"className":638577},[31850],[34,638579],{"className":638580,"style":31846},[31845],[34,638582,638584,638587],{"className":638583},[31676],[34,638585],{"className":638586,"style":32617},[31680],[34,638588,640],{"className":638589},[31685],"。线段长是 ",[34,638592,638594,638610],{"className":638593},[31642],[34,638595,638597],{"className":638596},[31646],[31648,638598,638599],{"xmlns":31650},[31652,638600,638601,638607],{},[31655,638602,638603],{},[41463,638604,638605],{},[31749,638606,768],{},[31665,638608,638609],{"encoding":31667},"\\sqrt{2}",[34,638611,638613],{"className":638612,"ariaHidden":758},[31672],[34,638614,638616,638619],{"className":638615},[31676],[34,638617],{"className":638618,"style":607702},[31680],[34,638620,638622],{"className":638621},[31685,41550],[34,638623,638625,638658],{"className":638624},[31796,31797],[34,638626,638628,638655],{"className":638627},[31801],[34,638629,638631,638643],{"className":638630,"style":572001},[31805],[34,638632,638634,638637],{"className":638633,"style":41565},[41564],[34,638635],{"className":638636,"style":41508},[31813],[34,638638,638640],{"className":638639,"style":41572},[31685],[34,638641,768],{"className":638642},[31685],[34,638644,638645,638648],{"style":572016},[34,638646],{"className":638647,"style":41508},[31813],[34,638649,638651],{"className":638650,"style":41586},[41585],[41588,638652,638653],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,638654],{"d":41597},[34,638656,31829],{"className":638657},[31828],[34,638659,638661],{"className":638660},[31801],[34,638662,638664],{"className":638663,"style":572036},[31805],[34,638665],{},"。积分 = ",[34,638668,638670,638696],{"className":638669},[31642],[34,638671,638673],{"className":638672},[31646],[31648,638674,638675],{"xmlns":31650},[31652,638676,638677,638693],{},[31655,638678,638679,638681,638683,638687,638689],{},[31749,638680,640],{},[31662,638682,42071],{},[41463,638684,638685],{},[31749,638686,768],{},[31662,638688,202],{},[41463,638690,638691],{},[31749,638692,768],{},[31665,638694,638695],{"encoding":31667},"1 \\times \\sqrt{2} = \\sqrt{2}",[34,638697,638699,638717,638779],{"className":638698,"ariaHidden":758},[31672],[34,638700,638702,638705,638708,638711,638714],{"className":638701},[31676],[34,638703],{"className":638704,"style":35942},[31680],[34,638706,640],{"className":638707},[31685],[34,638709],{"className":638710,"style":32604},[31845],[34,638712,42071],{"className":638713},[32024],[34,638715],{"className":638716,"style":32604},[31845],[34,638718,638720,638723,638770,638773,638776],{"className":638719},[31676],[34,638721],{"className":638722,"style":607702},[31680],[34,638724,638726],{"className":638725},[31685,41550],[34,638727,638729,638762],{"className":638728},[31796,31797],[34,638730,638732,638759],{"className":638731},[31801],[34,638733,638735,638747],{"className":638734,"style":572001},[31805],[34,638736,638738,638741],{"className":638737,"style":41565},[41564],[34,638739],{"className":638740,"style":41508},[31813],[34,638742,638744],{"className":638743,"style":41572},[31685],[34,638745,768],{"className":638746},[31685],[34,638748,638749,638752],{"style":572016},[34,638750],{"className":638751,"style":41508},[31813],[34,638753,638755],{"className":638754,"style":41586},[41585],[41588,638756,638757],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,638758],{"d":41597},[34,638760,31829],{"className":638761},[31828],[34,638763,638765],{"className":638764},[31801],[34,638766,638768],{"className":638767,"style":572036},[31805],[34,638769],{},[34,638771],{"className":638772,"style":31846},[31845],[34,638774,202],{"className":638775},[31850],[34,638777],{"className":638778,"style":31846},[31845],[34,638780,638782,638785],{"className":638781},[31676],[34,638783],{"className":638784,"style":607702},[31680],[34,638786,638788],{"className":638787},[31685,41550],[34,638789,638791,638824],{"className":638790},[31796,31797],[34,638792,638794,638821],{"className":638793},[31801],[34,638795,638797,638809],{"className":638796,"style":572001},[31805],[34,638798,638800,638803],{"className":638799,"style":41565},[41564],[34,638801],{"className":638802,"style":41508},[31813],[34,638804,638806],{"className":638805,"style":41572},[31685],[34,638807,768],{"className":638808},[31685],[34,638810,638811,638814],{"style":572016},[34,638812],{"className":638813,"style":41508},[31813],[34,638815,638817],{"className":638816,"style":41586},[41585],[41588,638818,638819],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,638820],{"d":41597},[34,638822,31829],{"className":638823},[31828],[34,638825,638827],{"className":638826},[31801],[34,638828,638830],{"className":638829,"style":572036},[31805],[34,638831],{},[108,638833,638834,8471,638962,639023,639024,347,639097,347],{},[34,638835,638837,638875],{"className":638836},[31642],[34,638838,638840],{"className":638839},[31646],[31648,638841,638842],{"xmlns":31650},[31652,638843,638844,638872],{},[31655,638845,638846,638848,638850,638852,638854,638856,638858,638860,638862,638864,638866,638868,638870],{},[31658,638847,54224],{},[31658,638849,37890],{},[31662,638851,456],{"stretchy":702},[31658,638853,32936],{},[31662,638855,202],{},[31749,638857,459],{},[31662,638859,31960],{"separator":758},[31658,638861,67764],{},[31662,638863,5598],{},[31749,638865,640],{},[31662,638867,33239],{},[31749,638869,459],{},[31662,638871,2557],{"stretchy":702},[31665,638873,638874],{"encoding":31667},"BO (x=0, y: 1\\to 0)",[34,638876,638878,638905,638932,638950],{"className":638877,"ariaHidden":758},[31672],[34,638879,638881,638884,638887,638890,638893,638896,638899,638902],{"className":638880},[31676],[34,638882],{"className":638883,"style":31774},[31680],[34,638885,54224],{"className":638886,"style":54252},[31685,31686],[34,638888,37890],{"className":638889,"style":34452},[31685,31686],[34,638891,456],{"className":638892},[31782],[34,638894,32936],{"className":638895},[31685,31686],[34,638897],{"className":638898,"style":31846},[31845],[34,638900,202],{"className":638901},[31850],[34,638903],{"className":638904,"style":31846},[31845],[34,638906,638908,638911,638914,638917,638920,638923,638926,638929],{"className":638907},[31676],[34,638909],{"className":638910,"style":40292},[31680],[34,638912,459],{"className":638913},[31685],[34,638915,31960],{"className":638916},[32119],[34,638918],{"className":638919,"style":32123},[31845],[34,638921,67764],{"className":638922,"style":32062},[31685,31686],[34,638924],{"className":638925,"style":31846},[31845],[34,638927,5598],{"className":638928},[31850],[34,638930],{"className":638931,"style":31846},[31845],[34,638933,638935,638938,638941,638944,638947],{"className":638934},[31676],[34,638936],{"className":638937,"style":32617},[31680],[34,638939,640],{"className":638940},[31685],[34,638942],{"className":638943,"style":31846},[31845],[34,638945,33239],{"className":638946},[31850],[34,638948],{"className":638949,"style":31846},[31845],[34,638951,638953,638956,638959],{"className":638952},[31676],[34,638954],{"className":638955,"style":31774},[31680],[34,638957,459],{"className":638958},[31685],[34,638960,2557],{"className":638961},[31690],[34,638963,638965,638987],{"className":638964},[31642],[34,638966,638968],{"className":638967},[31646],[31648,638969,638970],{"xmlns":31650},[31652,638971,638972,638984],{},[31655,638973,638974,638976,638978,638980,638982],{},[31658,638975,46900],{},[31658,638977,31747],{},[31662,638979,202],{},[31658,638981,46900],{},[31658,638983,67764],{},[31665,638985,638986],{"encoding":31667},"ds=dy",[34,638988,638990,639011],{"className":638989,"ariaHidden":758},[31672],[34,638991,638993,638996,638999,639002,639005,639008],{"className":638992},[31676],[34,638994],{"className":638995,"style":31681},[31680],[34,638997,46900],{"className":638998},[31685,31686],[34,639000,31747],{"className":639001},[31685,31686],[34,639003],{"className":639004,"style":31846},[31845],[34,639006,202],{"className":639007},[31850],[34,639009],{"className":639010,"style":31846},[31845],[34,639012,639014,639017,639020],{"className":639013},[31676],[34,639015],{"className":639016,"style":43217},[31680],[34,639018,46900],{"className":639019},[31685,31686],[34,639021,67764],{"className":639022,"style":32062},[31685,31686],"。函数值 ",[34,639025,639027,639049],{"className":639026},[31642],[34,639028,639030],{"className":639029},[31646],[31648,639031,639032],{"xmlns":31650},[31652,639033,639034,639046],{},[31655,639035,639036,639038,639040,639042,639044],{},[31658,639037,32936],{},[31662,639039,223],{},[31658,639041,67764],{},[31662,639043,202],{},[31658,639045,67764],{},[31665,639047,639048],{"encoding":31667},"x+y=y",[34,639050,639052,639070,639088],{"className":639051,"ariaHidden":758},[31672],[34,639053,639055,639058,639061,639064,639067],{"className":639054},[31676],[34,639056],{"className":639057,"style":32597},[31680],[34,639059,32936],{"className":639060},[31685,31686],[34,639062],{"className":639063,"style":32604},[31845],[34,639065,223],{"className":639066},[32024],[34,639068],{"className":639069,"style":32604},[31845],[34,639071,639073,639076,639079,639082,639085],{"className":639072},[31676],[34,639074],{"className":639075,"style":38774},[31680],[34,639077,67764],{"className":639078,"style":32062},[31685,31686],[34,639080],{"className":639081,"style":31846},[31845],[34,639083,202],{"className":639084},[31850],[34,639086],{"className":639087,"style":31846},[31845],[34,639089,639091,639094],{"className":639090},[31676],[34,639092],{"className":639093,"style":38774},[31680],[34,639095,67764],{"className":639096,"style":32062},[31685,31686],[34,639098,639100,639134],{"className":639099},[31642],[34,639101,639103],{"className":639102},[31646],[31648,639104,639105],{"xmlns":31650},[31652,639106,639107,639131],{},[31655,639108,639109,639117,639119,639121,639123,639125,639127,639129],{},[38846,639110,639111,639113,639115],{},[31662,639112,546295],{},[31749,639114,459],{},[31749,639116,640],{},[31658,639118,67764],{},[31658,639120,46900],{},[31658,639122,67764],{},[31662,639124,202],{},[31749,639126,640],{},[31658,639128,38080],{"mathvariant":38079},[31749,639130,768],{},[31665,639132,639133],{"encoding":31667},"\\int_0^1 y dy = 1\u002F2",[34,639135,639137,639215],{"className":639136,"ariaHidden":758},[31672],[34,639138,639140,639143,639194,639197,639200,639203,639206,639209,639212],{"className":639139},[31676],[34,639141],{"className":639142,"style":563309},[31680],[34,639144,639146,639149],{"className":639145},[32767],[34,639147,546295],{"className":639148,"style":546331},[32767,131249,156560],[34,639150,639152],{"className":639151},[31792],[34,639153,639155,639186],{"className":639154},[31796,31797],[34,639156,639158,639183],{"className":639157},[31801],[34,639159,639161,639172],{"className":639160,"style":563328},[31805],[34,639162,639163,639166],{"style":546347},[34,639164],{"className":639165,"style":31814},[31813],[34,639167,639169],{"className":639168},[31818,31819,31820,31821],[34,639170,459],{"className":639171},[31685,31821],[34,639173,639174,639177],{"style":546362},[34,639175],{"className":639176,"style":31814},[31813],[34,639178,639180],{"className":639179},[31818,31819,31820,31821],[34,639181,640],{"className":639182},[31685,31821],[34,639184,31829],{"className":639185},[31828],[34,639187,639189],{"className":639188},[31801],[34,639190,639192],{"className":639191,"style":546384},[31805],[34,639193],{},[34,639195],{"className":639196,"style":32123},[31845],[34,639198,67764],{"className":639199,"style":32062},[31685,31686],[34,639201,46900],{"className":639202},[31685,31686],[34,639204,67764],{"className":639205,"style":32062},[31685,31686],[34,639207],{"className":639208,"style":31846},[31845],[34,639210,202],{"className":639211},[31850],[34,639213],{"className":639214,"style":31846},[31845],[34,639216,639218,639221],{"className":639217},[31676],[34,639219],{"className":639220,"style":31774},[31680],[34,639222,42407],{"className":639223},[31685],[108,639225,639226,639227,347],{},"总和：",[34,639228,639230,639272],{"className":639229},[31642],[34,639231,639233],{"className":639232},[31646],[31648,639234,639235],{"xmlns":31650},[31652,639236,639237,639269],{},[31655,639238,639239,639241,639243,639245,639247,639251,639253,639255,639257,639259,639261,639263,639265],{},[31749,639240,640],{},[31658,639242,38080],{"mathvariant":38079},[31749,639244,768],{},[31662,639246,223],{},[41463,639248,639249],{},[31749,639250,768],{},[31662,639252,223],{},[31749,639254,640],{},[31658,639256,38080],{"mathvariant":38079},[31749,639258,768],{},[31662,639260,202],{},[31749,639262,640],{},[31662,639264,223],{},[41463,639266,639267],{},[31749,639268,768],{},[31665,639270,639271],{"encoding":31667},"1\u002F2 + \\sqrt{2} + 1\u002F2 = 1 + \\sqrt{2}",[34,639273,639275,639293,639355,639373,639391],{"className":639274,"ariaHidden":758},[31672],[34,639276,639278,639281,639284,639287,639290],{"className":639277},[31676],[34,639279],{"className":639280,"style":31774},[31680],[34,639282,42407],{"className":639283},[31685],[34,639285],{"className":639286,"style":32604},[31845],[34,639288,223],{"className":639289},[32024],[34,639291],{"className":639292,"style":32604},[31845],[34,639294,639296,639299,639346,639349,639352],{"className":639295},[31676],[34,639297],{"className":639298,"style":607702},[31680],[34,639300,639302],{"className":639301},[31685,41550],[34,639303,639305,639338],{"className":639304},[31796,31797],[34,639306,639308,639335],{"className":639307},[31801],[34,639309,639311,639323],{"className":639310,"style":572001},[31805],[34,639312,639314,639317],{"className":639313,"style":41565},[41564],[34,639315],{"className":639316,"style":41508},[31813],[34,639318,639320],{"className":639319,"style":41572},[31685],[34,639321,768],{"className":639322},[31685],[34,639324,639325,639328],{"style":572016},[34,639326],{"className":639327,"style":41508},[31813],[34,639329,639331],{"className":639330,"style":41586},[41585],[41588,639332,639333],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,639334],{"d":41597},[34,639336,31829],{"className":639337},[31828],[34,639339,639341],{"className":639340},[31801],[34,639342,639344],{"className":639343,"style":572036},[31805],[34,639345],{},[34,639347],{"className":639348,"style":32604},[31845],[34,639350,223],{"className":639351},[32024],[34,639353],{"className":639354,"style":32604},[31845],[34,639356,639358,639361,639364,639367,639370],{"className":639357},[31676],[34,639359],{"className":639360,"style":31774},[31680],[34,639362,42407],{"className":639363},[31685],[34,639365],{"className":639366,"style":31846},[31845],[34,639368,202],{"className":639369},[31850],[34,639371],{"className":639372,"style":31846},[31845],[34,639374,639376,639379,639382,639385,639388],{"className":639375},[31676],[34,639377],{"className":639378,"style":35942},[31680],[34,639380,640],{"className":639381},[31685],[34,639383],{"className":639384,"style":32604},[31845],[34,639386,223],{"className":639387},[32024],[34,639389],{"className":639390,"style":32604},[31845],[34,639392,639394,639397],{"className":639393},[31676],[34,639395],{"className":639396,"style":607702},[31680],[34,639398,639400],{"className":639399},[31685,41550],[34,639401,639403,639436],{"className":639402},[31796,31797],[34,639404,639406,639433],{"className":639405},[31801],[34,639407,639409,639421],{"className":639408,"style":572001},[31805],[34,639410,639412,639415],{"className":639411,"style":41565},[41564],[34,639413],{"className":639414,"style":41508},[31813],[34,639416,639418],{"className":639417,"style":41572},[31685],[34,639419,768],{"className":639420},[31685],[34,639422,639423,639426],{"style":572016},[34,639424],{"className":639425,"style":41508},[31813],[34,639427,639429],{"className":639428,"style":41586},[41585],[41588,639430,639431],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,639432],{"d":41597},[34,639434,31829],{"className":639435},[31828],[34,639437,639439],{"className":639438},[31801],[34,639440,639442],{"className":639441,"style":572036},[31805],[34,639443],{},[15,639445,639446],{},[2627,639447,639448,639449,639582,639583],{},"(2) 单位圆 ",[34,639450,639452,639482],{"className":639451},[31642],[34,639453,639455],{"className":639454},[31646],[31648,639456,639457],{"xmlns":31650},[31652,639458,639459,639479],{},[31655,639460,639461,639467,639469,639475,639477],{},[37624,639462,639463,639465],{},[31658,639464,32936],{},[31749,639466,768],{},[31662,639468,223],{},[37624,639470,639471,639473],{},[31658,639472,67764],{},[31749,639474,768],{},[31662,639476,202],{},[31749,639478,640],{},[31665,639480,639481],{"encoding":31667},"x^2+y^2=1",[34,639483,639485,639529,639573],{"className":639484,"ariaHidden":758},[31672],[34,639486,639488,639491,639520,639523,639526],{"className":639487},[31676],[34,639489],{"className":639490,"style":40044},[31680],[34,639492,639494,639497],{"className":639493},[31685],[34,639495,32936],{"className":639496},[31685,31686],[34,639498,639500],{"className":639499},[31792],[34,639501,639503],{"className":639502},[31796],[34,639504,639506],{"className":639505},[31801],[34,639507,639509],{"className":639508,"style":37774},[31805],[34,639510,639511,639514],{"style":37777},[34,639512],{"className":639513,"style":31814},[31813],[34,639515,639517],{"className":639516},[31818,31819,31820,31821],[34,639518,768],{"className":639519},[31685,31821],[34,639521],{"className":639522,"style":32604},[31845],[34,639524,223],{"className":639525},[32024],[34,639527],{"className":639528,"style":32604},[31845],[34,639530,639532,639535,639564,639567,639570],{"className":639531},[31676],[34,639533],{"className":639534,"style":315747},[31680],[34,639536,639538,639541],{"className":639537},[31685],[34,639539,67764],{"className":639540,"style":32062},[31685,31686],[34,639542,639544],{"className":639543},[31792],[34,639545,639547],{"className":639546},[31796],[34,639548,639550],{"className":639549},[31801],[34,639551,639553],{"className":639552,"style":37774},[31805],[34,639554,639555,639558],{"style":37777},[34,639556],{"className":639557,"style":31814},[31813],[34,639559,639561],{"className":639560},[31818,31819,31820,31821],[34,639562,768],{"className":639563},[31685,31821],[34,639565],{"className":639566,"style":31846},[31845],[34,639568,202],{"className":639569},[31850],[34,639571],{"className":639572,"style":31846},[31845],[34,639574,639576,639579],{"className":639575},[31676],[34,639577],{"className":639578,"style":32617},[31680],[34,639580,640],{"className":639581},[31685],"，求 ",[34,639584,639586,639614],{"className":639585},[31642],[34,639587,639589],{"className":639588},[31646],[31648,639590,639591],{"xmlns":31650},[31652,639592,639593,639611],{},[31655,639594,639595,639601,639603,639605,639607,639609],{},[31743,639596,639597,639599],{},[31662,639598,546295],{},[31658,639600,33206],{},[31658,639602,162086],{"mathvariant":38079},[31658,639604,67764],{},[31658,639606,162086],{"mathvariant":38079},[31658,639608,46900],{},[31658,639610,31747],{},[31665,639612,639613],{"encoding":31667},"\\int_L |y| ds",[34,639615,639617],{"className":639616,"ariaHidden":758},[31672],[34,639618,639620,639623,639663,639666,639669,639672,639675,639678],{"className":639619},[31676],[34,639621],{"className":639622,"style":637320},[31680],[34,639624,639626,639629],{"className":639625},[32767],[34,639627,546295],{"className":639628,"style":546331},[32767,131249,156560],[34,639630,639632],{"className":639631},[31792],[34,639633,639635,639655],{"className":639634},[31796,31797],[34,639636,639638,639652],{"className":639637},[31801],[34,639639,639641],{"className":639640,"style":637339},[31805],[34,639642,639643,639646],{"style":546347},[34,639644],{"className":639645,"style":31814},[31813],[34,639647,639649],{"className":639648},[31818,31819,31820,31821],[34,639650,33206],{"className":639651},[31685,31686,31821],[34,639653,31829],{"className":639654},[31828],[34,639656,639658],{"className":639657},[31801],[34,639659,639661],{"className":639660,"style":546384},[31805],[34,639662],{},[34,639664],{"className":639665,"style":32123},[31845],[34,639667,162086],{"className":639668},[31685],[34,639670,67764],{"className":639671,"style":32062},[31685,31686],[34,639673,162086],{"className":639674},[31685],[34,639676,46900],{"className":639677},[31685,31686],[34,639679,31747],{"className":639680},[31685,31686],[105,639682,639683,639855,639923,640073],{},[108,639684,639685,639686,347],{},"参数化：",[34,639687,639689,639741],{"className":639688},[31642],[34,639690,639692],{"className":639691},[31646],[31648,639693,639694],{"xmlns":31650},[31652,639695,639696,639738],{},[31655,639697,639698,639700,639702,639704,639706,639708,639710,639712,639714,639716,639718,639720,639722,639724,639726,639728,639730,639732,639734,639736],{},[31658,639699,32936],{},[31662,639701,202],{},[31658,639703,547860],{},[31662,639705,32682],{},[31658,639707,33249],{},[31662,639709,31960],{"separator":758},[31658,639711,67764],{},[31662,639713,202],{},[31658,639715,547853],{},[31662,639717,32682],{},[31658,639719,33249],{},[31662,639721,31960],{"separator":758},[31658,639723,33249],{},[31662,639725,112296],{},[31662,639727,15972],{"stretchy":702},[31749,639729,459],{},[31662,639731,31960],{"separator":758},[31749,639733,768],{},[31658,639735,58442],{},[31662,639737,32712],{"stretchy":702},[31665,639739,639740],{"encoding":31667},"x=\\cos t, y=\\sin t, t \\in [0, 2\\pi]",[34,639742,639744,639762,639795,639828],{"className":639743,"ariaHidden":758},[31672],[34,639745,639747,639750,639753,639756,639759],{"className":639746},[31676],[34,639748],{"className":639749,"style":32527},[31680],[34,639751,32936],{"className":639752},[31685,31686],[34,639754],{"className":639755,"style":31846},[31845],[34,639757,202],{"className":639758},[31850],[34,639760],{"className":639761,"style":31846},[31845],[34,639763,639765,639768,639771,639774,639777,639780,639783,639786,639789,639792],{"className":639764},[31676],[34,639766],{"className":639767,"style":223552},[31680],[34,639769,547860],{"className":639770},[32767],[34,639772],{"className":639773,"style":32123},[31845],[34,639775,33249],{"className":639776},[31685,31686],[34,639778,31960],{"className":639779},[32119],[34,639781],{"className":639782,"style":32123},[31845],[34,639784,67764],{"className":639785,"style":32062},[31685,31686],[34,639787],{"className":639788,"style":31846},[31845],[34,639790,202],{"className":639791},[31850],[34,639793],{"className":639794,"style":31846},[31845],[34,639796,639798,639801,639804,639807,639810,639813,639816,639819,639822,639825],{"className":639797},[31676],[34,639799],{"className":639800,"style":547875},[31680],[34,639802,547853],{"className":639803},[32767],[34,639805],{"className":639806,"style":32123},[31845],[34,639808,33249],{"className":639809},[31685,31686],[34,639811,31960],{"className":639812},[32119],[34,639814],{"className":639815,"style":32123},[31845],[34,639817,33249],{"className":639818},[31685,31686],[34,639820],{"className":639821,"style":31846},[31845],[34,639823,112296],{"className":639824},[31850],[34,639826],{"className":639827,"style":31846},[31845],[34,639829,639831,639834,639837,639840,639843,639846,639849,639852],{"className":639830},[31676],[34,639832],{"className":639833,"style":31774},[31680],[34,639835,15972],{"className":639836},[31782],[34,639838,459],{"className":639839},[31685],[34,639841,31960],{"className":639842},[32119],[34,639844],{"className":639845,"style":32123},[31845],[34,639847,768],{"className":639848},[31685],[34,639850,58442],{"className":639851,"style":32062},[31685,31686],[34,639853,32712],{"className":639854},[31690],[108,639856,639857,347],{},[34,639858,639860,639884],{"className":639859},[31642],[34,639861,639863],{"className":639862},[31646],[31648,639864,639865],{"xmlns":31650},[31652,639866,639867,639881],{},[31655,639868,639869,639871,639873,639875,639877,639879],{},[31658,639870,46900],{},[31658,639872,31747],{},[31662,639874,202],{},[31749,639876,640],{},[31658,639878,46900],{},[31658,639880,33249],{},[31665,639882,639883],{"encoding":31667},"ds = 1 dt",[34,639885,639887,639908],{"className":639886,"ariaHidden":758},[31672],[34,639888,639890,639893,639896,639899,639902,639905],{"className":639889},[31676],[34,639891],{"className":639892,"style":31681},[31680],[34,639894,46900],{"className":639895},[31685,31686],[34,639897,31747],{"className":639898},[31685,31686],[34,639900],{"className":639901,"style":31846},[31845],[34,639903,202],{"className":639904},[31850],[34,639906],{"className":639907,"style":31846},[31845],[34,639909,639911,639914,639917,639920],{"className":639910},[31676],[34,639912],{"className":639913,"style":31681},[31680],[34,639915,640],{"className":639916},[31685],[34,639918,46900],{"className":639919},[31685,31686],[34,639921,33249],{"className":639922},[31685,31686],[108,639924,592630,639925,347],{},[34,639926,639928,639968],{"className":639927},[31642],[34,639929,639931],{"className":639930},[31646],[31648,639932,639933],{"xmlns":31650},[31652,639934,639935,639965],{},[31655,639936,639937,639939,639951,639953,639955,639957,639959,639961,639963],{},[31662,639938,202],{},[38846,639940,639941,639943,639945],{},[31662,639942,546295],{},[31749,639944,459],{},[31655,639946,639947,639949],{},[31749,639948,768],{},[31658,639950,58442],{},[31658,639952,162086],{"mathvariant":38079},[31658,639954,547853],{},[31662,639956,32682],{},[31658,639958,33249],{},[31658,639960,162086],{"mathvariant":38079},[31658,639962,46900],{},[31658,639964,33249],{},[31665,639966,639967],{"encoding":31667},"= \\int_0^{2\\pi} |\\sin t| dt",[34,639969,639971,639983],{"className":639970,"ariaHidden":758},[31672],[34,639972,639974,639977,639980],{"className":639973},[31676],[34,639975],{"className":639976,"style":73089},[31680],[34,639978,202],{"className":639979},[31850],[34,639981],{"className":639982,"style":31846},[31845],[34,639984,639986,639989,640046,640049,640052,640055,640058,640061,640064,640067,640070],{"className":639985},[31676],[34,639987],{"className":639988,"style":563309},[31680],[34,639990,639992,639995],{"className":639991},[32767],[34,639993,546295],{"className":639994,"style":546331},[32767,131249,156560],[34,639996,639998],{"className":639997},[31792],[34,639999,640001,640038],{"className":640000},[31796,31797],[34,640002,640004,640035],{"className":640003},[31801],[34,640005,640007,640018],{"className":640006,"style":563328},[31805],[34,640008,640009,640012],{"style":546347},[34,640010],{"className":640011,"style":31814},[31813],[34,640013,640015],{"className":640014},[31818,31819,31820,31821],[34,640016,459],{"className":640017},[31685,31821],[34,640019,640020,640023],{"style":546362},[34,640021],{"className":640022,"style":31814},[31813],[34,640024,640026],{"className":640025},[31818,31819,31820,31821],[34,640027,640029,640032],{"className":640028},[31685,31821],[34,640030,768],{"className":640031},[31685,31821],[34,640033,58442],{"className":640034,"style":32062},[31685,31686,31821],[34,640036,31829],{"className":640037},[31828],[34,640039,640041],{"className":640040},[31801],[34,640042,640044],{"className":640043,"style":546384},[31805],[34,640045],{},[34,640047],{"className":640048,"style":32123},[31845],[34,640050,162086],{"className":640051},[31685],[34,640053],{"className":640054,"style":32123},[31845],[34,640056,547853],{"className":640057},[32767],[34,640059],{"className":640060,"style":32123},[31845],[34,640062,33249],{"className":640063},[31685,31686],[34,640065,162086],{"className":640066},[31685],[34,640068,46900],{"className":640069},[31685,31686],[34,640071,33249],{"className":640072},[31685,31686],[108,640074,640075,640076,347],{},"利用对称性，",[34,640077,640079,640171],{"className":640078},[31642],[34,640080,640082],{"className":640081},[31646],[31648,640083,640084],{"xmlns":31650},[31652,640085,640086,640168],{},[31655,640087,640088,640090,640092,640106,640108,640110,640112,640114,640116,640118,640120,640122,640124,640126,640128,640130,640144,640146,640148,640150,640152,640154,640156,640158,640160,640162,640164,640166],{},[31662,640089,202],{},[31749,640091,15932],{},[38846,640093,640094,640096,640098],{},[31662,640095,546295],{},[31749,640097,459],{},[31655,640099,640100,640102,640104],{},[31658,640101,58442],{},[31658,640103,38080],{"mathvariant":38079},[31749,640105,768],{},[31658,640107,547853],{},[31662,640109,32682],{},[31658,640111,33249],{},[31658,640113,46900],{},[31658,640115,33249],{},[31662,640117,202],{},[31749,640119,15932],{},[31662,640121,15972],{"stretchy":702},[31662,640123,32226],{},[31658,640125,547860],{},[31662,640127,32682],{},[31658,640129,33249],{},[38846,640131,640132,640134,640136],{},[31662,640133,32712],{"stretchy":702},[31749,640135,459],{},[31655,640137,640138,640140,640142],{},[31658,640139,58442],{},[31658,640141,38080],{"mathvariant":38079},[31749,640143,768],{},[31662,640145,202],{},[31749,640147,15932],{},[31662,640149,456],{"stretchy":702},[31749,640151,459],{},[31662,640153,32226],{},[31662,640155,456],{"stretchy":702},[31662,640157,32226],{},[31749,640159,640],{},[31662,640161,2557],{"stretchy":702},[31662,640163,2557],{"stretchy":702},[31662,640165,202],{},[31749,640167,15932],{},[31665,640169,640170],{"encoding":31667},"= 4 \\int_0^{\\pi\u002F2} \\sin t dt = 4 [-\\cos t]_0^{\\pi\u002F2} = 4(0 - (-1)) = 4",[34,640172,640174,640186,640282,640376,640400,640427],{"className":640173,"ariaHidden":758},[31672],[34,640175,640177,640180,640183],{"className":640176},[31676],[34,640178],{"className":640179,"style":73089},[31680],[34,640181,202],{"className":640182},[31850],[34,640184],{"className":640185,"style":31846},[31845],[34,640187,640189,640192,640195,640198,640255,640258,640261,640264,640267,640270,640273,640276,640279],{"className":640188},[31676],[34,640190],{"className":640191,"style":596892},[31680],[34,640193,15932],{"className":640194},[31685],[34,640196],{"className":640197,"style":32123},[31845],[34,640199,640201,640204],{"className":640200},[32767],[34,640202,546295],{"className":640203,"style":546331},[32767,131249,156560],[34,640205,640207],{"className":640206},[31792],[34,640208,640210,640247],{"className":640209},[31796,31797],[34,640211,640213,640244],{"className":640212},[31801],[34,640214,640216,640227],{"className":640215,"style":586480},[31805],[34,640217,640218,640221],{"style":546347},[34,640219],{"className":640220,"style":31814},[31813],[34,640222,640224],{"className":640223},[31818,31819,31820,31821],[34,640225,459],{"className":640226},[31685,31821],[34,640228,640229,640232],{"style":546362},[34,640230],{"className":640231,"style":31814},[31813],[34,640233,640235],{"className":640234},[31818,31819,31820,31821],[34,640236,640238,640241],{"className":640237},[31685,31821],[34,640239,58442],{"className":640240,"style":32062},[31685,31686,31821],[34,640242,38149],{"className":640243},[31685,31821],[34,640245,31829],{"className":640246},[31828],[34,640248,640250],{"className":640249},[31801],[34,640251,640253],{"className":640252,"style":546384},[31805],[34,640254],{},[34,640256],{"className":640257,"style":32123},[31845],[34,640259,547853],{"className":640260},[32767],[34,640262],{"className":640263,"style":32123},[31845],[34,640265,33249],{"className":640266},[31685,31686],[34,640268,46900],{"className":640269},[31685,31686],[34,640271,33249],{"className":640272},[31685,31686],[34,640274],{"className":640275,"style":31846},[31845],[34,640277,202],{"className":640278},[31850],[34,640280],{"className":640281,"style":31846},[31845],[34,640283,640285,640289,640292,640295,640298,640301,640304,640307,640310,640367,640370,640373],{"className":640284},[31676],[34,640286],{"className":640287,"style":640288},[31680],"height:1.3111em;vertical-align:-0.2663em;",[34,640290,15932],{"className":640291},[31685],[34,640293,15972],{"className":640294},[31782],[34,640296,32226],{"className":640297},[31685],[34,640299],{"className":640300,"style":32123},[31845],[34,640302,547860],{"className":640303},[32767],[34,640305],{"className":640306,"style":32123},[31845],[34,640308,33249],{"className":640309},[31685,31686],[34,640311,640313,640316],{"className":640312},[31690],[34,640314,32712],{"className":640315},[31690],[34,640317,640319],{"className":640318},[31792],[34,640320,640322,640359],{"className":640321},[31796,31797],[34,640323,640325,640356],{"className":640324},[31801],[34,640326,640328,640339],{"className":640327,"style":152022},[31805],[34,640329,640330,640333],{"style":317365},[34,640331],{"className":640332,"style":31814},[31813],[34,640334,640336],{"className":640335},[31818,31819,31820,31821],[34,640337,459],{"className":640338},[31685,31821],[34,640340,640341,640344],{"style":152040},[34,640342],{"className":640343,"style":31814},[31813],[34,640345,640347],{"className":640346},[31818,31819,31820,31821],[34,640348,640350,640353],{"className":640349},[31685,31821],[34,640351,58442],{"className":640352,"style":32062},[31685,31686,31821],[34,640354,38149],{"className":640355},[31685,31821],[34,640357,31829],{"className":640358},[31828],[34,640360,640362],{"className":640361},[31801],[34,640363,640365],{"className":640364,"style":317402},[31805],[34,640366],{},[34,640368],{"className":640369,"style":31846},[31845],[34,640371,202],{"className":640372},[31850],[34,640374],{"className":640375,"style":31846},[31845],[34,640377,640379,640382,640385,640388,640391,640394,640397],{"className":640378},[31676],[34,640380],{"className":640381,"style":31774},[31680],[34,640383,15932],{"className":640384},[31685],[34,640386,456],{"className":640387},[31782],[34,640389,459],{"className":640390},[31685],[34,640392],{"className":640393,"style":32604},[31845],[34,640395,32226],{"className":640396},[32024],[34,640398],{"className":640399,"style":32604},[31845],[34,640401,640403,640406,640409,640412,640415,640418,640421,640424],{"className":640402},[31676],[34,640404],{"className":640405,"style":31774},[31680],[34,640407,456],{"className":640408},[31782],[34,640410,32226],{"className":640411},[31685],[34,640413,640],{"className":640414},[31685],[34,640416,44253],{"className":640417},[31690],[34,640419],{"className":640420,"style":31846},[31845],[34,640422,202],{"className":640423},[31850],[34,640425],{"className":640426,"style":31846},[31845],[34,640428,640430,640433],{"className":640429},[31676],[34,640431],{"className":640432,"style":32617},[31680],[34,640434,15932],{"className":640435},[31685],[15,640437,640438],{},[2627,640439,640440,640441,639582,640633],{},"(3) 星形线 ",[34,640442,640444,640496],{"className":640443},[31642],[34,640445,640447],{"className":640446},[31646],[31648,640448,640449],{"xmlns":31650},[31652,640450,640451,640493],{},[31655,640452,640453,640465,640467,640479,640481],{},[37624,640454,640455,640457],{},[31658,640456,32936],{},[31655,640458,640459,640461,640463],{},[31749,640460,768],{},[31658,640462,38080],{"mathvariant":38079},[31749,640464,786],{},[31662,640466,223],{},[37624,640468,640469,640471],{},[31658,640470,67764],{},[31655,640472,640473,640475,640477],{},[31749,640474,768],{},[31658,640476,38080],{"mathvariant":38079},[31749,640478,786],{},[31662,640480,202],{},[37624,640482,640483,640485],{},[31658,640484,111],{},[31655,640486,640487,640489,640491],{},[31749,640488,768],{},[31658,640490,38080],{"mathvariant":38079},[31749,640492,786],{},[31665,640494,640495],{"encoding":31667},"x^{2\u002F3} + y^{2\u002F3} = a^{2\u002F3}",[34,640497,640499,640548,640595],{"className":640498,"ariaHidden":758},[31672],[34,640500,640502,640506,640539,640542,640545],{"className":640501},[31676],[34,640503],{"className":640504,"style":640505},[31680],"height:0.9713em;vertical-align:-0.0833em;",[34,640507,640509,640512],{"className":640508},[31685],[34,640510,32936],{"className":640511},[31685,31686],[34,640513,640515],{"className":640514},[31792],[34,640516,640518],{"className":640517},[31796],[34,640519,640521],{"className":640520},[31801],[34,640522,640524],{"className":640523,"style":61037},[31805],[34,640525,640526,640529],{"style":37777},[34,640527],{"className":640528,"style":31814},[31813],[34,640530,640532],{"className":640531},[31818,31819,31820,31821],[34,640533,640535],{"className":640534},[31685,31821],[34,640536,640538],{"className":640537},[31685,31821],"2\u002F3",[34,640540],{"className":640541,"style":32604},[31845],[34,640543,223],{"className":640544},[32024],[34,640546],{"className":640547,"style":32604},[31845],[34,640549,640551,640554,640586,640589,640592],{"className":640550},[31676],[34,640552],{"className":640553,"style":625707},[31680],[34,640555,640557,640560],{"className":640556},[31685],[34,640558,67764],{"className":640559,"style":32062},[31685,31686],[34,640561,640563],{"className":640562},[31792],[34,640564,640566],{"className":640565},[31796],[34,640567,640569],{"className":640568},[31801],[34,640570,640572],{"className":640571,"style":61037},[31805],[34,640573,640574,640577],{"style":37777},[34,640575],{"className":640576,"style":31814},[31813],[34,640578,640580],{"className":640579},[31818,31819,31820,31821],[34,640581,640583],{"className":640582},[31685,31821],[34,640584,640538],{"className":640585},[31685,31821],[34,640587],{"className":640588,"style":31846},[31845],[34,640590,202],{"className":640591},[31850],[34,640593],{"className":640594,"style":31846},[31845],[34,640596,640598,640601],{"className":640597},[31676],[34,640599],{"className":640600,"style":61037},[31680],[34,640602,640604,640607],{"className":640603},[31685],[34,640605,111],{"className":640606},[31685,31686],[34,640608,640610],{"className":640609},[31792],[34,640611,640613],{"className":640612},[31796],[34,640614,640616],{"className":640615},[31801],[34,640617,640619],{"className":640618,"style":61037},[31805],[34,640620,640621,640624],{"style":37777},[34,640622],{"className":640623,"style":31814},[31813],[34,640625,640627],{"className":640626},[31818,31819,31820,31821],[34,640628,640630],{"className":640629},[31685,31821],[34,640631,640538],{"className":640632},[31685,31821],[34,640634,640636,640674],{"className":640635},[31642],[34,640637,640639],{"className":640638},[31646],[31648,640640,640641],{"xmlns":31650},[31652,640642,640643,640671],{},[31655,640644,640645,640651,640653,640655,640667,640669],{},[31743,640646,640647,640649],{},[31662,640648,546295],{},[31658,640650,33206],{},[31658,640652,162086],{"mathvariant":38079},[31658,640654,32936],{},[37624,640656,640657,640659],{},[31658,640658,162086],{"mathvariant":38079},[31655,640660,640661,640663,640665],{},[31749,640662,640],{},[31658,640664,38080],{"mathvariant":38079},[31749,640666,786],{},[31658,640668,46900],{},[31658,640670,31747],{},[31665,640672,640673],{"encoding":31667},"\\int_L |x|^{1\u002F3} ds",[34,640675,640677],{"className":640676,"ariaHidden":758},[31672],[34,640678,640680,640684,640724,640727,640730,640733,640766,640769],{"className":640679},[31676],[34,640681],{"className":640682,"style":640683},[31680],"height:1.2438em;vertical-align:-0.3558em;",[34,640685,640687,640690],{"className":640686},[32767],[34,640688,546295],{"className":640689,"style":546331},[32767,131249,156560],[34,640691,640693],{"className":640692},[31792],[34,640694,640696,640716],{"className":640695},[31796,31797],[34,640697,640699,640713],{"className":640698},[31801],[34,640700,640702],{"className":640701,"style":637339},[31805],[34,640703,640704,640707],{"style":546347},[34,640705],{"className":640706,"style":31814},[31813],[34,640708,640710],{"className":640709},[31818,31819,31820,31821],[34,640711,33206],{"className":640712},[31685,31686,31821],[34,640714,31829],{"className":640715},[31828],[34,640717,640719],{"className":640718},[31801],[34,640720,640722],{"className":640721,"style":546384},[31805],[34,640723],{},[34,640725],{"className":640726,"style":32123},[31845],[34,640728,162086],{"className":640729},[31685],[34,640731,32936],{"className":640732},[31685,31686],[34,640734,640736,640739],{"className":640735},[31685],[34,640737,162086],{"className":640738},[31685],[34,640740,640742],{"className":640741},[31792],[34,640743,640745],{"className":640744},[31796],[34,640746,640748],{"className":640747},[31801],[34,640749,640751],{"className":640750,"style":61037},[31805],[34,640752,640753,640756],{"style":37777},[34,640754],{"className":640755,"style":31814},[31813],[34,640757,640759],{"className":640758},[31818,31819,31820,31821],[34,640760,640762],{"className":640761},[31685,31821],[34,640763,640765],{"className":640764},[31685,31821],"1\u002F3",[34,640767,46900],{"className":640768},[31685,31686],[34,640770,31747],{"className":640771},[31685,31686],[105,640773,640774,641026,641674,641758,642038,642524],{},[108,640775,639685,640776,347],{},[34,640777,640779,640847],{"className":640778},[31642],[34,640780,640782],{"className":640781},[31646],[31648,640783,640784],{"xmlns":31650},[31652,640785,640786,640844],{},[31655,640787,640788,640790,640792,640794,640804,640806,640808,640810,640812,640814,640824,640826,640828,640830,640832,640834,640836,640838,640840,640842],{},[31658,640789,32936],{},[31662,640791,202],{},[31658,640793,111],{},[37624,640795,640796,640802],{},[31655,640797,640798,640800],{},[31658,640799,547860],{},[31662,640801,32682],{},[31749,640803,786],{},[31658,640805,33249],{},[31662,640807,31960],{"separator":758},[31658,640809,67764],{},[31662,640811,202],{},[31658,640813,111],{},[37624,640815,640816,640822],{},[31655,640817,640818,640820],{},[31658,640819,547853],{},[31662,640821,32682],{},[31749,640823,786],{},[31658,640825,33249],{},[31662,640827,31960],{"separator":758},[31658,640829,33249],{},[31662,640831,112296],{},[31662,640833,15972],{"stretchy":702},[31749,640835,459],{},[31662,640837,31960],{"separator":758},[31749,640839,768],{},[31658,640841,58442],{},[31662,640843,32712],{"stretchy":702},[31665,640845,640846],{"encoding":31667},"x = a \\cos^3 t, y = a \\sin^3 t, t \\in [0, 2\\pi]",[34,640848,640850,640868,640933,640999],{"className":640849,"ariaHidden":758},[31672],[34,640851,640853,640856,640859,640862,640865],{"className":640852},[31676],[34,640854],{"className":640855,"style":32527},[31680],[34,640857,32936],{"className":640858},[31685,31686],[34,640860],{"className":640861,"style":31846},[31845],[34,640863,202],{"className":640864},[31850],[34,640866],{"className":640867,"style":31846},[31845],[34,640869,640871,640874,640877,640880,640909,640912,640915,640918,640921,640924,640927,640930],{"className":640870},[31676],[34,640872],{"className":640873,"style":315747},[31680],[34,640875,111],{"className":640876},[31685,31686],[34,640878],{"className":640879,"style":32123},[31845],[34,640881,640883,640886],{"className":640882},[32767],[34,640884,547860],{"className":640885},[32767],[34,640887,640889],{"className":640888},[31792],[34,640890,640892],{"className":640891},[31796],[34,640893,640895],{"className":640894},[31801],[34,640896,640898],{"className":640897,"style":37774},[31805],[34,640899,640900,640903],{"style":37777},[34,640901],{"className":640902,"style":31814},[31813],[34,640904,640906],{"className":640905},[31818,31819,31820,31821],[34,640907,786],{"className":640908},[31685,31821],[34,640910],{"className":640911,"style":32123},[31845],[34,640913,33249],{"className":640914},[31685,31686],[34,640916,31960],{"className":640917},[32119],[34,640919],{"className":640920,"style":32123},[31845],[34,640922,67764],{"className":640923,"style":32062},[31685,31686],[34,640925],{"className":640926,"style":31846},[31845],[34,640928,202],{"className":640929},[31850],[34,640931],{"className":640932,"style":31846},[31845],[34,640934,640936,640940,640943,640946,640975,640978,640981,640984,640987,640990,640993,640996],{"className":640935},[31676],[34,640937],{"className":640938,"style":640939},[31680],"height:1.0663em;vertical-align:-0.1944em;",[34,640941,111],{"className":640942},[31685,31686],[34,640944],{"className":640945,"style":32123},[31845],[34,640947,640949,640952],{"className":640948},[32767],[34,640950,547853],{"className":640951},[32767],[34,640953,640955],{"className":640954},[31792],[34,640956,640958],{"className":640957},[31796],[34,640959,640961],{"className":640960},[31801],[34,640962,640964],{"className":640963,"style":585884},[31805],[34,640965,640966,640969],{"style":586008},[34,640967],{"className":640968,"style":31814},[31813],[34,640970,640972],{"className":640971},[31818,31819,31820,31821],[34,640973,786],{"className":640974},[31685,31821],[34,640976],{"className":640977,"style":32123},[31845],[34,640979,33249],{"className":640980},[31685,31686],[34,640982,31960],{"className":640983},[32119],[34,640985],{"className":640986,"style":32123},[31845],[34,640988,33249],{"className":640989},[31685,31686],[34,640991],{"className":640992,"style":31846},[31845],[34,640994,112296],{"className":640995},[31850],[34,640997],{"className":640998,"style":31846},[31845],[34,641000,641002,641005,641008,641011,641014,641017,641020,641023],{"className":641001},[31676],[34,641003],{"className":641004,"style":31774},[31680],[34,641006,15972],{"className":641007},[31782],[34,641009,459],{"className":641010},[31685],[34,641012,31960],{"className":641013},[32119],[34,641015],{"className":641016,"style":32123},[31845],[34,641018,768],{"className":641019},[31685],[34,641021,58442],{"className":641022,"style":32062},[31685,31686],[34,641024,32712],{"className":641025},[31690],[108,641027,641028,641029,33183,641063,641225,641382,347],{},"算 ",[34,641030,641032,641048],{"className":641031},[31642],[34,641033,641035],{"className":641034},[31646],[31648,641036,641037],{"xmlns":31650},[31652,641038,641039,641045],{},[31655,641040,641041,641043],{},[31658,641042,46900],{},[31658,641044,31747],{},[31665,641046,641047],{"encoding":31667},"ds",[34,641049,641051],{"className":641050,"ariaHidden":758},[31672],[34,641052,641054,641057,641060],{"className":641053},[31676],[34,641055],{"className":641056,"style":31681},[31680],[34,641058,46900],{"className":641059},[31685,31686],[34,641061,31747],{"className":641062},[31685,31686],[34,641064,641066,641110],{"className":641065},[31642],[34,641067,641069],{"className":641068},[31646],[31648,641070,641071],{"xmlns":31650},[31652,641072,641073,641107],{},[31655,641074,641075,641081,641083,641085,641087,641089,641099,641101,641103,641105],{},[37624,641076,641077,641079],{},[31658,641078,32936],{},[31662,641080,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,641082,202],{},[31662,641084,32226],{},[31749,641086,786],{},[31658,641088,111],{},[37624,641090,641091,641097],{},[31655,641092,641093,641095],{},[31658,641094,547860],{},[31662,641096,32682],{},[31749,641098,768],{},[31658,641100,33249],{},[31658,641102,547853],{},[31662,641104,32682],{},[31658,641106,33249],{},[31665,641108,641109],{"encoding":31667},"x' = -3a \\cos^2 t \\sin t",[34,641111,641113,641160],{"className":641112,"ariaHidden":758},[31672],[34,641114,641116,641119,641151,641154,641157],{"className":641115},[31676],[34,641117],{"className":641118,"style":132027},[31680],[34,641120,641122,641125],{"className":641121},[31685],[34,641123,32936],{"className":641124},[31685,31686],[34,641126,641128],{"className":641127},[31792],[34,641129,641131],{"className":641130},[31796],[34,641132,641134],{"className":641133},[31801],[34,641135,641137],{"className":641136,"style":132027},[31805],[34,641138,641139,641142],{"style":37777},[34,641140],{"className":641141,"style":31814},[31813],[34,641143,641145],{"className":641144},[31818,31819,31820,31821],[34,641146,641148],{"className":641147},[31685,31821],[34,641149,116495],{"className":641150},[31685,31821],[34,641152],{"className":641153,"style":31846},[31845],[34,641155,202],{"className":641156},[31850],[34,641158],{"className":641159,"style":31846},[31845],[34,641161,641163,641166,641169,641172,641175,641178,641207,641210,641213,641216,641219,641222],{"className":641162},[31676],[34,641164],{"className":641165,"style":40044},[31680],[34,641167,32226],{"className":641168},[31685],[34,641170,786],{"className":641171},[31685],[34,641173,111],{"className":641174},[31685,31686],[34,641176],{"className":641177,"style":32123},[31845],[34,641179,641181,641184],{"className":641180},[32767],[34,641182,547860],{"className":641183},[32767],[34,641185,641187],{"className":641186},[31792],[34,641188,641190],{"className":641189},[31796],[34,641191,641193],{"className":641192},[31801],[34,641194,641196],{"className":641195,"style":37774},[31805],[34,641197,641198,641201],{"style":37777},[34,641199],{"className":641200,"style":31814},[31813],[34,641202,641204],{"className":641203},[31818,31819,31820,31821],[34,641205,768],{"className":641206},[31685,31821],[34,641208],{"className":641209,"style":32123},[31845],[34,641211,33249],{"className":641212},[31685,31686],[34,641214],{"className":641215,"style":32123},[31845],[34,641217,547853],{"className":641218},[32767],[34,641220],{"className":641221,"style":32123},[31845],[34,641223,33249],{"className":641224},[31685,31686],[34,641226,641228,641270],{"className":641227},[31642],[34,641229,641231],{"className":641230},[31646],[31648,641232,641233],{"xmlns":31650},[31652,641234,641235,641267],{},[31655,641236,641237,641243,641245,641247,641249,641259,641261,641263,641265],{},[37624,641238,641239,641241],{},[31658,641240,67764],{},[31662,641242,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,641244,202],{},[31749,641246,786],{},[31658,641248,111],{},[37624,641250,641251,641257],{},[31655,641252,641253,641255],{},[31658,641254,547853],{},[31662,641256,32682],{},[31749,641258,768],{},[31658,641260,33249],{},[31658,641262,547860],{},[31662,641264,32682],{},[31658,641266,33249],{},[31665,641268,641269],{"encoding":31667},"y' = 3a \\sin^2 t \\cos t",[34,641271,641273,641320],{"className":641272,"ariaHidden":758},[31672],[34,641274,641276,641279,641311,641314,641317],{"className":641275},[31676],[34,641277],{"className":641278,"style":395485},[31680],[34,641280,641282,641285],{"className":641281},[31685],[34,641283,67764],{"className":641284,"style":32062},[31685,31686],[34,641286,641288],{"className":641287},[31792],[34,641289,641291],{"className":641290},[31796],[34,641292,641294],{"className":641293},[31801],[34,641295,641297],{"className":641296,"style":132027},[31805],[34,641298,641299,641302],{"style":37777},[34,641300],{"className":641301,"style":31814},[31813],[34,641303,641305],{"className":641304},[31818,31819,31820,31821],[34,641306,641308],{"className":641307},[31685,31821],[34,641309,116495],{"className":641310},[31685,31821],[34,641312],{"className":641313,"style":31846},[31845],[34,641315,202],{"className":641316},[31850],[34,641318],{"className":641319,"style":31846},[31845],[34,641321,641323,641326,641329,641332,641335,641364,641367,641370,641373,641376,641379],{"className":641322},[31676],[34,641324],{"className":641325,"style":585884},[31680],[34,641327,786],{"className":641328},[31685],[34,641330,111],{"className":641331},[31685,31686],[34,641333],{"className":641334,"style":32123},[31845],[34,641336,641338,641341],{"className":641337},[32767],[34,641339,547853],{"className":641340},[32767],[34,641342,641344],{"className":641343},[31792],[34,641345,641347],{"className":641346},[31796],[34,641348,641350],{"className":641349},[31801],[34,641351,641353],{"className":641352,"style":585884},[31805],[34,641354,641355,641358],{"style":586008},[34,641356],{"className":641357,"style":31814},[31813],[34,641359,641361],{"className":641360},[31818,31819,31820,31821],[34,641362,768],{"className":641363},[31685,31821],[34,641365],{"className":641366,"style":32123},[31845],[34,641368,33249],{"className":641369},[31685,31686],[34,641371],{"className":641372,"style":32123},[31845],[34,641374,547860],{"className":641375},[32767],[34,641377],{"className":641378,"style":32123},[31845],[34,641380,33249],{"className":641381},[31685,31686],[34,641383,641385,641459],{"className":641384},[31642],[34,641386,641388],{"className":641387},[31646],[31648,641389,641390],{"xmlns":31650},[31652,641391,641392,641456],{},[31655,641393,641394,641396,641398,641400,641426,641428,641430,641432,641434,641436,641438,641440,641442,641444,641446,641448,641450,641452,641454],{},[31658,641395,46900],{},[31658,641397,31747],{},[31662,641399,202],{},[41463,641401,641402],{},[31655,641403,641404,641414,641416],{},[37624,641405,641406,641408],{},[31658,641407,32936],{},[31655,641409,641410,641412],{},[31662,641411,116495],{"mathvariant":38079},[31749,641413,768],{},[31662,641415,223],{},[37624,641417,641418,641420],{},[31658,641419,67764],{},[31655,641421,641422,641424],{},[31662,641423,116495],{"mathvariant":38079},[31749,641425,768],{},[31658,641427,46900],{},[31658,641429,33249],{},[31662,641431,202],{},[31749,641433,786],{},[31658,641435,111],{},[31658,641437,162086],{"mathvariant":38079},[31658,641439,547853],{},[31662,641441,32682],{},[31658,641443,33249],{},[31658,641445,547860],{},[31662,641447,32682],{},[31658,641449,33249],{},[31658,641451,162086],{"mathvariant":38079},[31658,641453,46900],{},[31658,641455,33249],{},[31665,641457,641458],{"encoding":31667},"ds = \\sqrt{x'^2 + y'^2} dt = 3a |\\sin t \\cos t| dt",[34,641460,641462,641483,641626],{"className":641461,"ariaHidden":758},[31672],[34,641463,641465,641468,641471,641474,641477,641480],{"className":641464},[31676],[34,641466],{"className":641467,"style":31681},[31680],[34,641469,46900],{"className":641470},[31685,31686],[34,641472,31747],{"className":641473},[31685,31686],[34,641475],{"className":641476,"style":31846},[31845],[34,641478,202],{"className":641479},[31850],[34,641481],{"className":641482,"style":31846},[31845],[34,641484,641486,641490,641611,641614,641617,641620,641623],{"className":641485},[31676],[34,641487],{"className":641488,"style":641489},[31680],"height:1.24em;vertical-align:-0.2822em;",[34,641491,641493],{"className":641492},[31685,41550],[34,641494,641496,641602],{"className":641495},[31796,31797],[34,641497,641499,641599],{"className":641498},[31801],[34,641500,641503,641586],{"className":641501,"style":641502},[31805],"height:0.9578em;",[34,641504,641506,641509],{"className":641505,"style":590744},[41564],[34,641507],{"className":641508,"style":552795},[31813],[34,641510,641512,641545,641548,641551,641554],{"className":641511,"style":590751},[31685],[34,641513,641515,641518],{"className":641514},[31685],[34,641516,32936],{"className":641517},[31685,31686],[34,641519,641521],{"className":641520},[31792],[34,641522,641524],{"className":641523},[31796],[34,641525,641527],{"className":641526},[31801],[34,641528,641530],{"className":641529,"style":590807},[31805],[34,641531,641532,641535],{"style":590776},[34,641533],{"className":641534,"style":31814},[31813],[34,641536,641538],{"className":641537},[31818,31819,31820,31821],[34,641539,641541],{"className":641540},[31685,31821],[34,641542,641544],{"className":641543},[31685,31821],"′2",[34,641546],{"className":641547,"style":32604},[31845],[34,641549,223],{"className":641550},[32024],[34,641552],{"className":641553,"style":32604},[31845],[34,641555,641557,641560],{"className":641556},[31685],[34,641558,67764],{"className":641559,"style":32062},[31685,31686],[34,641561,641563],{"className":641562},[31792],[34,641564,641566],{"className":641565},[31796],[34,641567,641569],{"className":641568},[31801],[34,641570,641572],{"className":641571,"style":590807},[31805],[34,641573,641574,641577],{"style":590776},[34,641575],{"className":641576,"style":31814},[31813],[34,641578,641580],{"className":641579},[31818,31819,31820,31821],[34,641581,641583],{"className":641582},[31685,31821],[34,641584,641544],{"className":641585},[31685,31821],[34,641587,641589,641592],{"style":641588},"top:-2.9178em;",[34,641590],{"className":641591,"style":552795},[31813],[34,641593,641595],{"className":641594,"style":590901},[41585],[41588,641596,641597],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,641598],{"d":590908},[34,641600,31829],{"className":641601},[31828],[34,641603,641605],{"className":641604},[31801],[34,641606,641609],{"className":641607,"style":641608},[31805],"height:0.2822em;",[34,641610],{},[34,641612,46900],{"className":641613},[31685,31686],[34,641615,33249],{"className":641616},[31685,31686],[34,641618],{"className":641619,"style":31846},[31845],[34,641621,202],{"className":641622},[31850],[34,641624],{"className":641625,"style":31846},[31845],[34,641627,641629,641632,641635,641638,641641,641644,641647,641650,641653,641656,641659,641662,641665,641668,641671],{"className":641628},[31676],[34,641630],{"className":641631,"style":31774},[31680],[34,641633,786],{"className":641634},[31685],[34,641636,111],{"className":641637},[31685,31686],[34,641639,162086],{"className":641640},[31685],[34,641642],{"className":641643,"style":32123},[31845],[34,641645,547853],{"className":641646},[32767],[34,641648],{"className":641649,"style":32123},[31845],[34,641651,33249],{"className":641652},[31685,31686],[34,641654],{"className":641655,"style":32123},[31845],[34,641657,547860],{"className":641658},[32767],[34,641660],{"className":641661,"style":32123},[31845],[34,641663,33249],{"className":641664},[31685,31686],[34,641666,162086],{"className":641667},[31685],[34,641669,46900],{"className":641670},[31685,31686],[34,641672,33249],{"className":641673},[31685,31686],[108,641675,641676,641677,641757],{},"利用对称性，只算第一象限 (",[34,641678,641680,641706],{"className":641679},[31642],[34,641681,641683],{"className":641682},[31646],[31648,641684,641685],{"xmlns":31650},[31652,641686,641687,641703],{},[31655,641688,641689,641691,641693,641695,641697,641699,641701],{},[31658,641690,33249],{},[31662,641692,5598],{},[31749,641694,459],{},[31662,641696,33239],{},[31658,641698,58442],{},[31658,641700,38080],{"mathvariant":38079},[31749,641702,768],{},[31665,641704,641705],{"encoding":31667},"t: 0 \\to \\pi\u002F2",[34,641707,641709,641727,641745],{"className":641708,"ariaHidden":758},[31672],[34,641710,641712,641715,641718,641721,641724],{"className":641711},[31676],[34,641713],{"className":641714,"style":33280},[31680],[34,641716,33249],{"className":641717},[31685,31686],[34,641719],{"className":641720,"style":31846},[31845],[34,641722,5598],{"className":641723},[31850],[34,641725],{"className":641726,"style":31846},[31845],[34,641728,641730,641733,641736,641739,641742],{"className":641729},[31676],[34,641731],{"className":641732,"style":32617},[31680],[34,641734,459],{"className":641735},[31685],[34,641737],{"className":641738,"style":31846},[31845],[34,641740,33239],{"className":641741},[31850],[34,641743],{"className":641744,"style":31846},[31845],[34,641746,641748,641751,641754],{"className":641747},[31676],[34,641749],{"className":641750,"style":31774},[31680],[34,641752,58442],{"className":641753,"style":32062},[31685,31686],[34,641755,38149],{"className":641756},[31685],") 乘 4。",[108,641759,65840,641760,347],{},[34,641761,641763,641841],{"className":641762},[31642],[34,641764,641766],{"className":641765},[31646],[31648,641767,641768],{"xmlns":31650},[31652,641769,641770,641838],{},[31655,641771,641772,641774,641776,641788,641790,641792,641794,641804,641806,641818,641820,641832,641834,641836],{},[31658,641773,162086],{"mathvariant":38079},[31658,641775,32936],{},[37624,641777,641778,641780],{},[31658,641779,162086],{"mathvariant":38079},[31655,641781,641782,641784,641786],{},[31749,641783,640],{},[31658,641785,38080],{"mathvariant":38079},[31749,641787,786],{},[31662,641789,202],{},[31662,641791,456],{"stretchy":702},[31658,641793,111],{},[37624,641795,641796,641802],{},[31655,641797,641798,641800],{},[31658,641799,547860],{},[31662,641801,32682],{},[31749,641803,786],{},[31658,641805,33249],{},[37624,641807,641808,641810],{},[31662,641809,2557],{"stretchy":702},[31655,641811,641812,641814,641816],{},[31749,641813,640],{},[31658,641815,38080],{"mathvariant":38079},[31749,641817,786],{},[31662,641819,202],{},[37624,641821,641822,641824],{},[31658,641823,111],{},[31655,641825,641826,641828,641830],{},[31749,641827,640],{},[31658,641829,38080],{"mathvariant":38079},[31749,641831,786],{},[31658,641833,547860],{},[31662,641835,32682],{},[31658,641837,33249],{},[31665,641839,641840],{"encoding":31667},"|x|^{1\u002F3} = (a \\cos^3 t)^{1\u002F3} = a^{1\u002F3} \\cos 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},[31658,642119,111],{},[31655,642121,642122,642124,642126],{},[31749,642123,15932],{},[31658,642125,38080],{"mathvariant":38079},[31749,642127,786],{},[38846,642129,642130,642132,642134],{},[31662,642131,546295],{},[31749,642133,459],{},[31655,642135,642136,642138,642140],{},[31658,642137,58442],{},[31658,642139,38080],{"mathvariant":38079},[31749,642141,768],{},[37624,642143,642144,642150],{},[31655,642145,642146,642148],{},[31658,642147,547860],{},[31662,642149,32682],{},[31749,642151,768],{},[31658,642153,33249],{},[31658,642155,547853],{},[31662,642157,32682],{},[31658,642159,33249],{},[39361,642161,39363],{},[31658,642163,46900],{},[31658,642165,33249],{},[31665,642167,642168],{"encoding":31667},"= 4 \\int_0^{\\pi\u002F2} a^{1\u002F3} \\cos t \\cdot 3a \\sin t \\cos t \\, dt = 12 a^{4\u002F3} \\int_0^{\\pi\u002F2} \\cos^2 t \\sin t \\, dt",[34,642170,642172,642184,642309,642363],{"className":642171,"ariaHidden":758},[31672],[34,642173,642175,642178,642181],{"className":642174},[31676],[34,642176],{"className":642177,"style":73089},[31680],[34,642179,202],{"className":642180},[31850],[34,642182],{"className":642183,"style":31846},[31845],[34,642185,642187,642190,642193,642196,642253,642256,642288,642291,642294,642297,642300,642303,642306],{"className":642186},[31676],[34,642188],{"className":642189,"style":596892},[31680],[34,642191,15932],{"className":642192},[31685],[34,642194],{"className":642195,"style":32123},[31845],[34,642197,642199,642202],{"className":642198},[32767],[34,642200,546295],{"className":642201,"style":546331},[32767,131249,156560],[34,642203,642205],{"className":642204},[31792],[34,642206,642208,642245],{"className":642207},[31796,31797],[34,642209,642211,642242],{"className":642210},[31801],[34,642212,642214,642225],{"className":642213,"style":586480},[31805],[34,642215,642216,642219],{"style":546347},[34,642217],{"className":642218,"style":31814},[31813],[34,642220,642222],{"className":642221},[31818,31819,31820,31821],[34,642223,459],{"className":642224},[31685,31821],[34,642226,642227,642230],{"style":546362},[34,642228],{"className":642229,"style":31814},[31813],[34,642231,642233],{"className":642232},[31818,31819,31820,31821],[34,642234,642236,642239],{"className":642235},[31685,31821],[34,642237,58442],{"className":642238,"style":32062},[31685,31686,31821],[34,642240,38149],{"className":642241},[31685,31821],[34,642243,31829],{"className":642244},[31828],[34,642246,642248],{"className":642247},[31801],[34,642249,642251],{"className":642250,"style":546384},[31805],[34,642252],{},[34,642254],{"className":642255,"style":32123},[31845],[34,642257,642259,642262],{"className":642258},[31685],[34,642260,111],{"className":642261},[31685,31686],[34,642263,642265],{"className":642264},[31792],[34,642266,642268],{"className":642267},[31796],[34,642269,642271],{"className":642270},[31801],[34,642272,642274],{"className":642273,"style":61037},[31805],[34,642275,642276,642279],{"style":37777},[34,642277],{"className":642278,"style":31814},[31813],[34,642280,642282],{"className":642281},[31818,31819,31820,31821],[34,642283,642285],{"className":642284},[31685,31821],[34,642286,640765],{"className":642287},[31685,31821],[34,642289],{"className":642290,"style":32123},[31845],[34,642292,547860],{"className":642293},[32767],[34,642295],{"className":642296,"style":32123},[31845],[34,642298,33249],{"className":642299},[31685,31686],[34,642301],{"className":642302,"style":32604},[31845],[34,642304,71191],{"className":642305},[32024],[34,642307],{"className":642308,"style":32604},[31845],[34,642310,642312,642315,642318,642321,642324,642327,642330,642333,642336,642339,642342,642345,642348,642351,642354,642357,642360],{"className":642311},[31676],[34,642313],{"className":642314,"style":31681},[31680],[34,642316,786],{"className":642317},[31685],[34,642319,111],{"className":642320},[31685,31686],[34,642322],{"className":642323,"style":32123},[31845],[34,642325,547853],{"className":642326},[32767],[34,642328],{"className":642329,"style":32123},[31845],[34,642331,33249],{"className":642332},[31685,31686],[34,642334],{"className":642335,"style":32123},[31845],[34,642337,547860],{"className":642338},[32767],[34,642340],{"className":642341,"style":32123},[31845],[34,642343,33249],{"className":642344},[31685,31686],[34,642346],{"className":642347,"style":32123},[31845],[34,642349,46900],{"className":642350},[31685,31686],[34,642352,33249],{"className":642353},[31685,31686],[34,642355],{"className":642356,"style":31846},[31845],[34,642358,202],{"className":642359},[31850],[34,642361],{"className":642362,"style":31846},[31845],[34,642364,642366,642369,642372,642405,642408,642465,642468,642497,642500,642503,642506,642509,642512,642515,642518,642521],{"className":642365},[31676],[34,642367],{"className":642368,"style":596892},[31680],[34,642370,71869],{"className":642371},[31685],[34,642373,642375,642378],{"className":642374},[31685],[34,642376,111],{"className":642377},[31685,31686],[34,642379,642381],{"className":642380},[31792],[34,642382,642384],{"className":642383},[31796],[34,642385,642387],{"className":642386},[31801],[34,642388,642390],{"className":642389,"style":61037},[31805],[34,642391,642392,642395],{"style":37777},[34,642393],{"className":642394,"style":31814},[31813],[34,642396,642398],{"className":642397},[31818,31819,31820,31821],[34,642399,642401],{"className":642400},[31685,31821],[34,642402,642404],{"className":642403},[31685,31821],"4\u002F3",[34,642406],{"className":642407,"style":32123},[31845],[34,642409,642411,642414],{"className":642410},[32767],[34,642412,546295],{"className":642413,"style":546331},[32767,131249,156560],[34,642415,642417],{"className":642416},[31792],[34,642418,642420,642457],{"className":642419},[31796,31797],[34,642421,642423,642454],{"className":642422},[31801],[34,642424,642426,642437],{"className":642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= \\cos t",[34,642552,642554,642572],{"className":642553,"ariaHidden":758},[31672],[34,642555,642557,642560,642563,642566,642569],{"className":642556},[31676],[34,642558],{"className":642559,"style":32527},[31680],[34,642561,94628],{"className":642562},[31685,31686],[34,642564],{"className":642565,"style":31846},[31845],[34,642567,202],{"className":642568},[31850],[34,642570],{"className":642571,"style":31846},[31845],[34,642573,642575,642578,642581,642584],{"className":642574},[31676],[34,642576],{"className":642577,"style":33280},[31680],[34,642579,547860],{"className":642580},[32767],[34,642582],{"className":642583,"style":32123},[31845],[34,642585,33249],{"className":642586},[31685,31686],[34,642588,642590,642676],{"className":642589},[31642],[34,642591,642593],{"className":642592},[31646],[31648,642594,642595],{"xmlns":31650},[31652,642596,642597,642673],{},[31655,642598,642599,642601,642603,642615,642623,642629,642631,642633,642635,642637,642649,642651,642657,642659,642661],{},[31662,642600,202],{},[31749,642602,71869],{},[37624,642604,642605,642607],{},[31658,642606,111],{},[31655,642608,642609,642611,642613],{},[31749,642610,15932],{},[31658,642612,38080],{"mathvariant":38079},[31749,642614,786],{},[38846,642616,642617,642619,642621],{},[31662,642618,546295],{},[31749,642620,459],{},[31749,642622,640],{},[37624,642624,642625,642627],{},[31658,642626,94628],{},[31749,642628,768],{},[31658,642630,46900],{},[31658,642632,94628],{},[31662,642634,202],{},[31749,642636,71869],{},[37624,642638,642639,642641],{},[31658,642640,111],{},[31655,642642,642643,642645,642647],{},[31749,642644,15932],{},[31658,642646,38080],{"mathvariant":38079},[31749,642648,786],{},[31662,642650,71191],{},[41453,642652,642653,642655],{},[31749,642654,640],{},[31749,642656,786],{},[31662,642658,202],{},[31749,642660,15932],{},[37624,642662,642663,642665],{},[31658,642664,111],{},[31655,642666,642667,642669,642671],{},[31749,642668,15932],{},[31658,642670,38080],{"mathvariant":38079},[31749,642672,786],{},[31665,642674,642675],{"encoding":31667},"= 12 a^{4\u002F3} \\int_0^1 u^2 du = 12 a^{4\u002F3} \\cdot \\frac{1}{3} = 4 a^{4\u002F3}",[34,642677,642679,642691,642833,642883,642966],{"className":642678,"ariaHidden":758},[31672],[34,642680,642682,642685,642688],{"className":642681},[31676],[34,642683],{"className":642684,"style":73089},[31680],[34,642686,202],{"className":642687},[31850],[34,642689],{"className":642690,"style":31846},[31845],[34,642692,642694,642697,642700,642732,642735,642786,642789,642818,642821,642824,642827,642830],{"className":642693},[31676],[34,642695],{"className":642696,"style":563309},[31680],[34,642698,71869],{"className":642699},[31685],[34,642701,642703,642706],{"className":642702},[31685],[34,642704,111],{"className":642705},[31685,31686],[34,642707,642709],{"className":642708},[31792],[34,642710,642712],{"className":642711},[31796],[34,642713,642715],{"className":642714},[31801],[34,642716,642718],{"className":642717,"style":61037},[31805],[34,642719,642720,642723],{"style":37777},[34,642721],{"className":642722,"style":31814},[31813],[34,642724,642726],{"className":642725},[31818,31819,31820,31821],[34,642727,642729],{"className":642728},[31685,31821],[34,642730,642404],{"className":642731},[31685,31821],[34,642733],{"className":642734,"style":32123},[31845],[34,642736,642738,642741],{"className":642737},[32767],[34,642739,546295],{"className":642740,"style":546331},[32767,131249,156560],[34,642742,642744],{"className":642743},[31792],[34,642745,642747,642778],{"className":642746},[31796,31797],[34,642748,642750,642775],{"className":642749},[31801],[34,642751,642753,642764],{"className":642752,"style":563328},[31805],[34,642754,642755,642758],{"style":546347},[34,642756],{"className":642757,"style":31814},[31813],[34,642759,642761],{"className":642760},[31818,31819,31820,31821],[34,642762,459],{"className":642763},[31685,31821],[34,642765,642766,642769],{"style":546362},[34,642767],{"className":642768,"style":31814},[31813],[34,642770,642772],{"className":642771},[31818,31819,31820,31821],[34,642773,640],{"className":642774},[31685,31821],[34,642776,31829],{"className":642777},[31828],[34,642779,642781],{"className":642780},[31801],[34,642782,642784],{"className":642783,"style":546384},[31805],[34,642785],{},[34,642787],{"className":642788,"style":32123},[31845],[34,642790,642792,642795],{"className":642791},[31685],[34,642793,94628],{"className":642794},[31685,31686],[34,642796,642798],{"className":642797},[31792],[34,642799,642801],{"className":642800},[31796],[34,642802,642804],{"className":642803},[31801],[34,642805,642807],{"className":642806,"style":37774},[31805],[34,642808,642809,642812],{"style":37777},[34,642810],{"className":642811,"style":31814},[31813],[34,642813,642815],{"className":642814},[31818,31819,31820,31821],[34,642816,768],{"className":642817},[31685,31821],[34,642819,46900],{"className":642820},[31685,31686],[34,642822,94628],{"className":642823},[31685,31686],[34,642825],{"className":642826,"style":31846},[31845],[34,642828,202],{"className":642829},[31850],[34,642831],{"className":642832,"style":31846},[31845],[34,642834,642836,642839,642842,642874,642877,642880],{"className":642835},[31676],[34,642837],{"className":642838,"style":61037},[31680],[34,642840,71869],{"className":642841},[31685],[34,642843,642845,642848],{"className":642844},[31685],[34,642846,111],{"className":642847},[31685,31686],[34,642849,642851],{"className":642850},[31792],[34,642852,642854],{"className":642853},[31796],[34,642855,642857],{"className":642856},[31801],[34,642858,642860],{"className":642859,"style":61037},[31805],[34,642861,642862,642865],{"style":37777},[34,642863],{"className":642864,"style":31814},[31813],[34,642866,642868],{"className":642867},[31818,31819,31820,31821],[34,642869,642871],{"className":642870},[31685,31821],[34,642872,642404],{"className":642873},[31685,31821],[34,642875],{"className":642876,"style":32604},[31845],[34,642878,71191],{"className":642879},[32024],[34,642881],{"className":642882,"style":32604},[31845],[34,642884,642886,642889,642957,642960,642963],{"className":642885},[31676],[34,642887],{"className":642888,"style":46727},[31680],[34,642890,642892,642895,642954],{"className":642891},[31685],[34,642893],{"className":642894},[31782,41488],[34,642896,642898],{"className":642897},[41453],[34,642899,642901,642946],{"className":642900},[31796,31797],[34,642902,642904,642943],{"className":642903},[31801],[34,642905,642907,642921,642929],{"className":642906,"style":46746},[31805],[34,642908,642909,642912],{"style":41504},[34,642910],{"className":642911,"style":41508},[31813],[34,642913,642915],{"className":642914},[31818,31819,31820,31821],[34,642916,642918],{"className":642917},[31685,31821],[34,642919,786],{"className":642920},[31685,31821],[34,642922,642923,642926],{"style":41520},[34,642924],{"className":642925,"style":41508},[31813],[34,642927],{"className":642928,"style":41528},[41527],[34,642930,642931,642934],{"style":46771},[34,642932],{"className":642933,"style":41508},[31813],[34,642935,642937],{"className":642936},[31818,31819,31820,31821],[34,64293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双纽线 ",[34,643013,643015,643065],{"className":643014},[31642],[34,643016,643018],{"className":643017},[31646],[31648,643019,643020],{"xmlns":31650},[31652,643021,643022,643062],{},[31655,643023,643024,643026,643032,643034,643040,643046,643048,643054,643056],{},[31662,643025,456],{"stretchy":702},[37624,643027,643028,643030],{},[31658,643029,32936],{},[31749,643031,768],{},[31662,643033,223],{},[37624,643035,643036,643038],{},[31658,643037,67764],{},[31749,643039,768],{},[37624,643041,643042,643044],{},[31662,643043,2557],{"stretchy":702},[31749,643045,768],{},[31662,643047,202],{},[37624,643049,643050,643052],{},[31658,643051,32936],{},[31749,643053,768],{},[31662,643055,32226],{},[37624,643057,643058,643060],{},[31658,643059,67764],{},[31749,643061,768],{},[31665,643063,643064],{"encoding":31667},"(x^2+y^2)^2 = x^2-y^2",[34,643066,643068,643115,643188,643232],{"className":643067,"ariaHidden":758},[31672],[34,643069,643071,643074,643077,643106,643109,643112],{"className":643070},[31676],[34,643072],{"className":643073,"style":37755},[31680],[34,643075,456],{"className":643076},[31782],[34,643078,643080,643083],{"className":643079},[31685],[34,643081,32936],{"className":643082},[31685,31686],[34,643084,643086],{"className":643085},[31792],[34,643087,643089],{"className":643088},[31796],[34,643090,643092],{"className":643091},[31801],[34,643093,643095],{"className":643094,"style":37774},[31805],[34,643096,643097,643100],{"style":37777},[34,643098],{"className":643099,"style":31814},[31813],[34,643101,643103],{"className":643102},[31818,31819,31820,31821],[34,643104,768],{"className":643105},[31685,31821],[34,643107],{"className":643108,"style":32604},[31845],[34,643110,223],{"className":643111},[32024],[34,643113],{"className":643114,"style":32604},[31845],[34,643116,643118,643121,643150,643179,643182,643185],{"className":643117},[31676],[34,643119],{"className":643120,"style":37755},[31680],[34,643122,643124,643127],{"className":643123},[31685],[34,643125,67764],{"className":643126,"style":32062},[31685,31686],[34,643128,643130],{"className":643129},[31792],[34,643131,643133],{"className":643132},[31796],[34,643134,643136],{"className":643135},[31801],[34,643137,643139],{"className":643138,"style":37774},[31805],[34,643140,643141,643144],{"style":37777},[34,643142],{"className":643143,"style":31814},[31813],[34,643145,643147],{"className":643146},[31818,31819,31820,31821],[34,643148,768],{"className":643149},[31685,31821],[34,643151,643153,643156],{"className":643152},[31690],[34,643154,2557],{"className":643155},[31690],[34,643157,643159],{"className":643158},[31792],[34,643160,643162],{"className":643161},[31796],[34,643163,643165],{"className":643164},[31801],[34,643166,643168],{"className":643167,"style":37774},[31805],[34,643169,643170,643173],{"style":37777},[34,643171],{"className":643172,"style":31814},[31813],[34,643174,643176],{"className":643175},[31818,31819,31820,31821],[34,643177,768],{"className":643178},[31685,31821],[34,643180],{"className":643181,"style":31846},[31845],[34,643183,202],{"className":643184},[31850],[34,643186],{"className":643187,"style":31846},[31845],[34,643189,643191,643194,643223,643226,643229],{"className":643190},[31676],[34,643192],{"className":643193,"style":40044},[31680],[34,643195,643197,643200],{"className":643196},[31685],[34,643198,32936],{"className":643199},[31685,31686],[34,643201,643203],{"className":643202},[31792],[34,643204,643206],{"className":643205},[31796],[34,643207,643209],{"className":643208},[31801],[34,643210,643212],{"className":643211,"style":37774},[31805],[34,643213,643214,643217],{"style":37777},[34,643215],{"className":643216,"style":31814},[31813],[34,643218,643220],{"className":643219},[31818,31819,31820,31821],[34,643221,768],{"className":643222},[31685,31821],[34,643224],{"className":643225,"style":32604},[31845],[34,643227,32226],{"className":643228},[32024],[34,643230],{"className":643231,"style":32604},[31845],[34,643233,643235,643238],{"className":643234},[31676],[34,643236],{"className":643237,"style":315747},[31680],[34,643239,643241,643244],{"className":643240},[31685],[34,643242,67764],{"className":643243,"style":32062},[31685,31686],[34,643245,643247],{"className":643246},[31792],[34,643248,643250],{"className":643249},[31796],[34,643251,643253],{"className":643252},[31801],[34,643254,643256],{"className":643255,"style":37774},[31805],[34,643257,643258,643261],{"style":37777},[34,643259],{"className":643260,"style":31814},[31813],[34,643262,643264],{"className":643263},[31818,31819,31820,31821],[34,643265,768],{"className":643266},[31685,31821],[34,643268,643270,643298],{"className":643269},[31642],[34,643271,643273],{"className":643272},[31646],[31648,643274,643275],{"xmlns":31650},[31652,643276,643277,643295],{},[31655,643278,643279,643285,643287,643289,643291,643293],{},[31743,643280,643281,643283],{},[31662,643282,546295],{},[31658,643284,33206],{},[31658,643286,162086],{"mathvariant":38079},[31658,643288,32936],{},[31658,643290,162086],{"mathvariant":38079},[31658,643292,46900],{},[31658,643294,31747],{},[31665,643296,643297],{"encoding":31667},"\\int_L |x| ds",[34,643299,643301],{"className":643300,"ariaHidden":758},[31672],[34,643302,643304,643307,643347,643350,643353,643356,643359,643362],{"className":643303},[31676],[34,643305],{"className":643306,"style":637320},[31680],[34,643308,643310,643313],{"className":643309},[32767],[34,643311,546295],{"className":643312,"style":546331},[32767,131249,156560],[34,643314,643316],{"className":643315},[31792],[34,643317,643319,643339],{"className":643318},[31796,31797],[34,643320,643322,643336],{"className":643321},[31801],[34,643323,643325],{"className":643324,"style":637339},[31805],[34,643326,643327,643330],{"style":546347},[34,643328],{"className":643329,"style":31814},[31813],[34,643331,643333],{"className":643332},[31818,31819,31820,31821],[34,643334,33206],{"className":643335},[31685,31686,31821],[34,643337,31829],{"className":643338},[31828],[34,643340,643342],{"className":643341},[31801],[34,643343,643345],{"className":643344,"style":546384},[31805],[34,643346],{},[34,643348],{"className":643349,"style":32123},[31845],[34,643351,162086],{"className":643352},[31685],[34,643354,32936],{"className":643355},[31685,31686],[34,643357,162086],{"className":643358},[31685],[34,643360,46900],{"className":643361},[31685,31686],[34,643363,31747],{"className":643364},[31685,31686],[105,643366,643367,643547,645299,645486,645998],{},[108,643368,643369,643370,643466,643467,641757],{},"极坐标方程：",[34,643371,643373,643401],{"className":643372},[31642],[34,643374,643376],{"className":643375},[31646],[31648,643377,643378],{"xmlns":31650},[31652,643379,643380,643398],{},[31655,643381,643382,643388,643390,643392,643394,643396],{},[37624,643383,643384,643386],{},[31658,643385,34395],{},[31749,643387,768],{},[31662,643389,202],{},[31658,643391,547860],{},[31662,643393,32682],{},[31749,643395,768],{},[31658,643397,479697],{},[31665,643399,643400],{"encoding":31667},"r^2 = \\cos 2\\theta",[34,643402,643404,643448],{"className":643403,"ariaHidden":758},[31672],[34,643405,643407,643410,643439,643442,643445],{"className":643406},[31676],[34,643408],{"className":643409,"style":37774},[31680],[34,643411,643413,643416],{"className":643412},[31685],[34,643414,34395],{"className":643415,"style":34452},[31685,31686],[34,643417,643419],{"className":643418},[31792],[34,643420,643422],{"className":643421},[31796],[34,643423,643425],{"className":643424},[31801],[34,643426,643428],{"className":643427,"style":37774},[31805],[34,643429,643430,643433],{"style":37777},[34,643431],{"className":643432,"style":31814},[31813],[34,643434,643436],{"className":643435},[31818,31819,31820,31821],[34,643437,768],{"className":643438},[31685,31821],[34,643440],{"className":643441,"style":31846},[31845],[34,643443,202],{"className":643444},[31850],[34,643446],{"className":643447,"style":31846},[31845],[34,643449,643451,643454,643457,643460,643463],{"className":643450},[31676],[34,643452],{"className":643453,"style":31681},[31680],[34,643455,547860],{"className":643456},[32767],[34,643458],{"className":643459,"style":32123},[31845],[34,643461,768],{"className":643462},[31685],[34,643464,479697],{"className":643465,"style":34452},[31685,31686],"。对称性，只算第一象限半个叶子 (",[34,643468,643470,643496],{"className":643469},[31642],[34,643471,643473],{"className":643472},[31646],[31648,643474,643475],{"xmlns":31650},[31652,643476,643477,643493],{},[31655,643478,643479,643481,643483,643485,643487,643489,643491],{},[31658,643480,479697],{},[31662,643482,5598],{},[31749,643484,459],{},[31662,643486,33239],{},[31658,643488,58442],{},[31658,643490,38080],{"mathvariant":38079},[31749,643492,15932],{},[31665,643494,643495],{"encoding":31667},"\\theta: 0 \\to \\pi\u002F4",[34,643497,643499,643517,643535],{"className":643498,"ariaHidden":758},[31672],[34,643500,643502,643505,643508,643511,643514],{"className":643501},[31676],[34,643503],{"className":643504,"style":31681},[31680],[34,643506,479697],{"className":643507,"style":34452},[31685,31686],[34,643509],{"className":643510,"style":31846},[31845],[34,643512,5598],{"className":643513},[31850],[34,643515],{"className":643516,"style":31846},[31845],[34,643518,643520,643523,643526,643529,643532],{"className":643519},[31676],[34,643521],{"className":643522,"style":32617},[31680],[34,643524,459],{"className":643525},[31685],[34,643527],{"className":643528,"style":31846},[31845],[34,643530,33239],{"className":643531},[31850],[34,643533],{"className":643534,"style":31846},[31845],[34,643536,643538,643541,643544],{"className":643537},[31676],[34,643539],{"className":643540,"style":31774},[31680],[34,643542,58442],{"className":643543,"style":32062},[31685,31686],[34,643545,42280],{"className":643546},[31685],[108,643548,643549,643582,643583,39984,643789,39984,644088,645087,645088,347],{},[34,643550,643552,643567],{"className":643551},[31642],[34,643553,643555],{"className":643554},[31646],[31648,643556,643557],{"xmlns":31650},[31652,643558,643559,643565],{},[31655,643560,643561,643563],{},[31658,643562,46900],{},[31658,643564,31747],{},[31665,643566,641047],{"encoding":31667},[34,643568,643570],{"className":643569,"ariaHidden":758},[31672],[34,643571,643573,643576,643579],{"className":643572},[31676],[34,643574],{"className":643575,"style":31681},[31680],[34,643577,46900],{"className":643578},[31685,31686],[34,643580,31747],{"className":643581},[31685,31686]," 公式（极坐标）：",[34,643584,643586,643628],{"className":643585},[31642],[34,643587,643589],{"className":643588},[31646],[31648,643590,643591],{"xmlns":31650},[31652,643592,643593,643625],{},[31655,643594,643595,643621,643623],{},[41463,643596,643597],{},[31655,643598,643599,643605,643607,643609,643615],{},[37624,643600,643601,643603],{},[31658,643602,34395],{},[31749,643604,768],{},[31662,643606,223],{},[31662,643608,456],{"stretchy":702},[37624,643610,643611,643613],{},[31658,643612,34395],{},[31662,643614,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,643616,643617,643619],{},[31662,643618,2557],{"stretchy":702},[31749,643620,768],{},[31658,643622,46900],{},[31658,643624,479697],{},[31665,643626,643627],{"encoding":31667},"\\sqrt{r^2 + (r')^2} d\\theta",[34,643629,643631],{"className":643630,"ariaHidden":758},[31672],[34,643632,643634,643637,643783,643786],{"className":643633},[31676],[34,643635],{"className":643636,"style":590727},[31680],[34,643638,643640],{"className":643639},[31685,41550],[34,643641,643643,643775],{"className":643642},[31796,31797],[34,643644,643646,643772],{"className":643645},[31801],[34,643647,643649,643760],{"className":643648,"style":590740},[31805],[34,643650,643652,643655],{"className":643651,"style":590744},[41564],[34,643653],{"className":643654,"style":552795},[31813],[34,643656,643658,643687,643690,643693,643696,643699,643731],{"className":643657,"style":590751},[31685],[34,643659,643661,643664],{"className":643660},[31685],[34,643662,34395],{"className":643663,"style":34452},[31685,31686],[34,643665,643667],{"className":643666},[31792],[34,643668,643670],{"className":643669},[31796],[34,643671,643673],{"className":643672},[31801],[34,643674,643676],{"className":643675,"style":590807},[31805],[34,643677,643678,643681],{"style":590776},[34,643679],{"className":643680,"style":31814},[31813],[34,643682,643684],{"className":643683},[31818,31819,31820,31821],[34,643685,768],{"className":643686},[31685,31821],[34,643688],{"className":643689,"style":32604},[31845],[34,643691,223],{"className":643692},[32024],[34,643694],{"className":643695,"style":32604},[31845],[34,643697,456],{"className":643698},[31782],[34,643700,643702,643705],{"className":643701},[31685],[34,643703,34395],{"className":643704,"style":34452},[31685,31686],[34,643706,643708],{"className":643707},[31792],[34,643709,643711],{"className":643710},[31796],[34,643712,643714],{"className":643713},[31801],[34,643715,643717],{"className":643716,"style":590773},[31805],[34,643718,643719,643722],{"style":590776},[34,643720],{"className":643721,"style":31814},[31813],[34,643723,643725],{"className":643724},[31818,31819,31820,31821],[34,643726,643728],{"className":643727},[31685,31821],[34,643729,116495],{"className":643730},[31685,31821],[34,643732,643734,643737],{"className":643733},[31690],[34,643735,2557],{"className":643736},[31690],[34,643738,643740],{"className":643739},[31792],[34,643741,643743],{"className":643742},[31796],[34,643744,643746],{"className":643745},[31801],[34,643747,643749],{"className":643748,"style":590807},[31805],[34,643750,643751,643754],{"style":590776},[34,643752],{"className":643753,"style":31814},[31813],[34,643755,643757],{"className":643756},[31818,31819,31820,31821],[34,643758,768],{"className":643759},[31685,31821],[34,643761,643762,643765],{"style":590894},[34,643763],{"className":643764,"style":552795},[31813],[34,643766,643768],{"className":643767,"style":590901},[41585],[41588,643769,643770],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,643771],{"d":590908},[34,643773,31829],{"className":643774},[31828],[34,643776,643778],{"className":643777},[31801],[34,643779,643781],{"className":643780,"style":590918},[31805],[34,643782],{},[34,643784,46900],{"className":643785},[31685,31686],[34,643787,479697],{"className":643788,"style":34452},[31685,31686],[34,643790,643792,643854],{"className":643791},[31642],[34,643793,643795],{"className":643794},[31646],[31648,643796,643797],{"xmlns":31650},[31652,643798,643799,643851],{},[31655,643800,643801,643803,643805,643811,643813,643815,643817,643819,643821,643823,643825,643827,643833,643835,643837],{},[31749,643802,768],{},[31658,643804,34395],{},[37624,643806,643807,643809],{},[31658,643808,34395],{},[31662,643810,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,643812,202],{},[31662,643814,32226],{},[31749,643816,768],{},[31658,643818,547853],{},[31662,643820,32682],{},[31749,643822,768],{},[31658,643824,479697],{},[31662,643826,140094],{},[37624,643828,643829,643831],{},[31658,643830,34395],{},[31662,643832,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,643834,202],{},[31662,643836,32226],{},[41453,643838,643839,643849],{},[31655,643840,643841,643843,643845,643847],{},[31658,643842,547853],{},[31662,643844,32682],{},[31749,643846,768],{},[31658,643848,479697],{},[31658,643850,34395],{},[31665,643852,643853],{"encoding":31667},"2r r' = -2\\sin 2\\theta \\Rightarrow r' = -\\frac{\\sin 2\\theta}{r}",[34,643855,643857,643910,643946,643993],{"className":643856,"ariaHidden":758},[31672],[34,643858,643860,643863,643866,643869,643901,643904,643907],{"className":643859},[31676],[34,643861],{"className":643862,"style":132027},[31680],[34,643864,768],{"className":643865},[31685],[34,643867,34395],{"className":643868,"style":34452},[31685,31686],[34,643870,643872,643875],{"className":643871},[31685],[34,643873,34395],{"className":643874,"style":34452},[31685,31686],[34,643876,643878],{"className":643877},[31792],[34,643879,643881],{"className":643880},[31796],[34,643882,643884],{"className":643883},[31801],[34,643885,643887],{"className":643886,"style":132027},[31805],[34,643888,643889,643892],{"style":37777},[34,643890],{"className":643891,"style":31814},[31813],[34,643893,643895],{"className":643894},[31818,31819,31820,31821],[34,643896,643898],{"className":643897},[31685,31821],[34,643899,116495],{"className":643900},[31685,31821],[34,643902],{"className":643903,"style":31846},[31845],[34,643905,202],{"className":643906},[31850],[34,643908],{"className":643909,"style":31846},[31845],[34,643911,643913,643916,643919,643922,643925,643928,643931,643934,643937,643940,643943],{"className":643912},[31676],[34,643914],{"className":643915,"style":34299},[31680],[34,643917,32226],{"className":643918},[31685],[34,643920,768],{"className":643921},[31685],[34,643923],{"className":643924,"style":32123},[31845],[34,643926,547853],{"className":643927},[32767],[34,643929],{"className":643930,"style":32123},[31845],[34,643932,768],{"className":643933},[31685],[34,643935,479697],{"className":643936,"style":34452},[31685,31686],[34,643938],{"className":643939,"style":31846},[31845],[34,643941,140094],{"className":643942},[31850],[34,643944],{"className":643945,"style":31846},[31845],[34,643947,643949,643952,643984,643987,643990],{"className":643948},[31676],[34,643950],{"className":643951,"style":132027},[31680],[34,643953,643955,643958],{"className":643954},[31685],[34,643956,34395],{"className":643957,"style":34452},[31685,31686],[34,643959,643961],{"className":643960},[31792],[34,643962,643964],{"className":643963},[31796],[34,643965,643967],{"className":643966},[31801],[34,643968,643970],{"className":643969,"style":132027},[31805],[34,643971,643972,643975],{"style":37777},[34,643973],{"className":643974,"style":31814},[31813],[34,643976,643978],{"className":643977},[31818,31819,31820,31821],[34,643979,643981],{"className":643980},[31685,31821],[34,643982,116495],{"className":643983},[31685,31821],[34,643985],{"className":643986,"style":31846},[31845],[34,643988,202],{"className":643989},[31850],[34,643991],{"className":643992,"style":31846},[31845],[34,643994,643996,643999,644002],{"className":643995},[31676],[34,643997],{"className":643998,"style":232492},[31680],[34,644000,32226],{"className":644001},[31685],[34,644003,644005,644008,644085],{"className":644004},[31685],[34,644006],{"className":644007},[31782,41488],[34,644009,644011],{"className":644010},[41453],[34,644012,644014,644077],{"className":644013},[31796,31797],[34,644015,644017,644074],{"className":644016},[31801],[34,644018,644020,644034,644042],{"className":644019,"style":232511},[31805],[34,644021,644022,644025],{"style":41504},[34,644023],{"className":644024,"style":41508},[31813],[34,644026,644028],{"className":644027},[31818,31819,31820,31821],[34,644029,644031],{"className":644030},[31685,31821],[34,644032,34395],{"className":644033,"style":34452},[31685,31686,31821],[34,644035,644036,644039],{"style":41520},[34,644037],{"className":644038,"style":41508},[31813],[34,644040],{"className":644041,"style":41528},[41527],[34,644043,644044,644047],{"style":46771},[34,644045],{"className":644046,"style":41508},[31813],[34,644048,644050],{"className":644049},[31818,31819,31820,31821],[34,644051,644053,644065,644068,644071],{"className":644052},[31685,31821],[34,644054,644056,644059,644062],{"className":644055},[32767,31821],[34,644057,31747],{"className":644058},[31821],[34,644060,31955],{"className":644061},[31821],[34,644063,31660],{"className":644064},[31821],[34,644066],{"className":644067,"style":43430},[31845,31821],[34,644069,768],{"className":644070},[31685,31821],[34,644072,479697],{"className":644073,"style":34452},[31685,31686,31821],[34,644075,31829],{"className":644076},[31828],[34,644078,644080],{"className":644079},[31801],[34,644081,644083],{"className":644082,"style":41619},[31805],[34,644084],{},[34,644086],{"className":644087},[31690,41488],[34,644089,644091,644249],{"className":644090},[31642],[34,644092,644094],{"className":644093},[31646],[31648,644095,644096],{"xmlns":31650},[31652,644097,644098,644246],{},[31655,644099,644100,644102,644104,644106,644142,644144,644146,644148,644182,644184,644186,644188,644230,644232,644234,644236,644242,644244],{},[31658,644101,46900],{},[31658,644103,31747],{},[31662,644105,202],{},[41463,644107,644108],{},[31655,644109,644110,644116,644118],{},[37624,644111,644112,644114],{},[31658,644113,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= \\sqrt{r^2 + \\frac{\\sin^2 2\\theta}{r^2}} d\\theta = \\sqrt{\\frac{r^4 + \\sin^2 2\\theta}{r^2}} d\\theta = \\sqrt{\\frac{\\cos^2 2\\theta + \\sin^2 2\\theta}{r^2}} d\\theta = \\frac{1}{r} 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ssName":645021},[41453],[34,645023,645025,645070],{"className":645024},[31796,31797],[34,645026,645028,645067],{"className":645027},[31801],[34,645029,645031,645045,645053],{"className":645030,"style":46746},[31805],[34,645032,645033,645036],{"style":41504},[34,645034],{"className":645035,"style":41508},[31813],[34,645037,645039],{"className":645038},[31818,31819,31820,31821],[34,645040,645042],{"className":645041},[31685,31821],[34,645043,34395],{"className":645044,"style":34452},[31685,31686,31821],[34,645046,645047,645050],{"style":41520},[34,645048],{"className":645049,"style":41508},[31813],[34,645051],{"className":645052,"style":41528},[41527],[34,645054,645055,645058],{"style":46771},[34,645056],{"className":645057,"style":41508},[31813],[34,645059,645061],{"className":645060},[31818,31819,31820,31821],[34,645062,645064],{"className":645063},[31685,31821],[34,645065,640],{"className":645066},[31685,31821],[34,645068,31829],{"className":645069},[31828],[34,645071,645073],{"className":645072},[31801],[34,645074,645076],{"className":645075,"style":41619},[31805],[34,645077],{},[34,645079],{"className":645080},[31690,41488],[34,645082,46900],{"className":645083},[31685,31686],[34,645085,479697],{"className":645086,"style":34452},[31685,31686],"。\n或者更简单的：双纽线弧长微分 ",[34,645089,645091,645129],{"className":645090},[31642],[34,645092,645094],{"className":645093},[31646],[31648,645095,645096],{"xmlns":31650},[31652,645097,645098,645126],{},[31655,645099,645100,645102,645104,645106,645122,645124],{},[31658,645101,46900],{},[31658,645103,31747],{},[31662,645105,202],{},[41453,645107,645108,645110],{},[31749,645109,640],{},[41463,645111,645112],{},[31655,645113,645114,645116,645118,645120],{},[31658,645115,547860],{},[31662,645117,32682],{},[31749,645119,768],{},[31658,645121,479697],{},[31658,645123,46900],{},[31658,645125,479697],{},[31665,645127,645128],{"encoding":31667},"ds = \\frac{1}{\\sqrt{\\cos 2\\theta}} d\\theta",[34,645130,645132,645153],{"className":645131,"ariaHidden":758},[31672],[34,645133,645135,645138,645141,645144,645147,645150],{"className":645134},[31676],[34,645136],{"className":645137,"style":31681},[31680],[34,645139,46900],{"className":645140},[31685,31686],[34,645142,31747],{"className":645143},[31685,31686],[34,645145],{"className":645146,"style":31846},[31845],[34,645148,202],{"className":645149},[31850],[34,645151],{"className":645152,"style":31846},[31845],[34,645154,645156,645159,645293,645296],{"className":645155},[31676],[34,645157],{"className":645158,"style":558996},[31680],[34,645160,645162,645165,645290],{"className":645161},[31685],[34,645163],{"className":645164},[31782,41488],[34,645166,645168],{"className":645167},[41453],[34,645169,645171,645282],{"className":645170},[31796,31797],[34,645172,645174,645279],{"className":645173},[31801],[34,645175,645177,645257,645265],{"className":645176,"style":46746},[31805],[34,645178,645180,645183],{"style":645179},"top:-2.5335em;",[34,645181],{"className":645182,"style":41508},[31813],[34,645184,645186],{"className":645185},[31818,31819,31820,31821],[34,645187,645189],{"className":645188},[31685,31821],[34,645190,645192],{"className":645191},[31685,41550,31821],[34,645193,645195,645248],{"className":645194},[31796,31797],[34,645196,645198,645245],{"className":645197},[31801],[34,645199,645202,645232],{"className":645200,"style":645201},[31805],"height:0.9378em;",[34,645203,645205,645208],{"className":645204,"style":41565},[41564],[34,645206],{"className":645207,"style":41508},[31813],[34,645209,645211,645223,645226,645229],{"className":645210,"style":41572},[31685,31821],[34,645212,645214,645217,645220],{"className":645213},[32767,31821],[34,645215,31965],{"className":645216},[31821],[34,645218,43383],{"className":645219},[31821],[34,645221,31747],{"className":645222},[31821],[34,645224],{"className":645225,"style":43430},[31845,31821],[34,645227,768],{"className":645228},[31685,31821],[34,645230,479697],{"className":645231,"style":34452},[31685,31686,31821],[34,645233,645235,645238],{"style":645234},"top:-2.8978em;",[34,645236],{"className":645237,"style":41508},[31813],[34,645239,645241],{"className":645240,"style":41586},[41585,31821],[41588,645242,645243],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,645244],{"d":41597},[34,645246,31829],{"className":645247},[31828],[34,645249,645251],{"className":645250},[31801],[34,645252,645255],{"className":645253,"style":645254},[31805],"height:0.1022em;",[34,645256],{},[34,645258,645259,645262],{"style":41520},[34,645260],{"className":645261,"style":41508},[31813],[34,645263],{"className":645264,"style":41528},[41527],[34,645266,645267,645270],{"style":46771},[34,645268],{"className":645269,"style":41508},[31813],[34,645271,645273],{"className":645272},[31818,31819,31820,31821],[34,645274,645276],{"className":645275},[31685,31821],[34,645277,640],{"className":645278},[31685,31821],[34,645280,31829],{"className":645281},[31828],[34,645283,645285],{"className":645284},[31801],[34,645286,645288],{"className":645287,"style":555424},[31805],[34,645289],{},[34,645291],{"className":645292},[31690,41488],[34,645294,46900],{"className":645295},[31685,31686],[34,645297,479697],{"className":645298,"style":34452},[31685,31686],[108,645300,645301,347],{},[34,645302,645304,645352],{"className":645303},[31642],[34,645305,645307],{"className":645306},[31646],[31648,645308,645309],{"xmlns":31650},[31652,645310,645311,645349],{},[31655,645312,645313,645315,645317,645319,645321,645323,645325,645327,645329,645331,645343,645345,645347],{},[31658,645314,162086],{"mathvariant":38079},[31658,645316,32936],{},[31658,645318,162086],{"mathvariant":38079},[31662,645320,202],{},[31658,645322,34395],{},[31658,645324,547860],{},[31662,645326,32682],{},[31658,645328,479697],{},[31662,645330,202],{},[41463,645332,645333],{},[31655,645334,645335,645337,645339,645341],{},[31658,645336,547860],{},[31662,645338,32682],{},[31749,645340,768],{},[31658,645342,479697],{},[31658,645344,547860],{},[31662,645346,32682],{},[31658,645348,479697],{},[31665,645350,645351],{"encoding":31667},"|x| = r \\cos \\theta = \\sqrt{\\cos 2\\theta} \\cos \\theta",[34,645353,645355,645379,645409],{"className":645354,"ariaHidden":758},[31672],[34,645356,645358,645361,645364,645367,645370,645373,645376],{"className":645357},[31676],[34,645359],{"className":645360,"style":31774},[31680],[34,645362,162086],{"className":645363},[31685],[34,645365,32936],{"className":645366},[31685,31686],[34,645368,162086],{"className":645369},[31685],[34,645371],{"className":645372,"style":31846},[31845],[34,645374,202],{"className":645375},[31850],[34,645377],{"className":645378,"style":31846},[31845],[34,645380,645382,645385,645388,645391,645394,645397,645400,645403,645406],{"className":645381},[31676],[34,645383],{"className":645384,"style":31681},[31680],[34,645386,34395],{"className":645387,"style":34452},[31685,31686],[34,645389],{"className":645390,"style":32123},[31845],[34,645392,547860],{"className":645393},[32767],[34,645395],{"className":645396,"style":32123},[31845],[34,645398,479697],{"className":645399,"style":34452},[31685,31686],[34,645401],{"className":645402,"style":31846},[31845],[34,645404,202],{"className":645405},[31850],[34,645407],{"className":645408,"style":31846},[31845],[34,645410,645412,645416,645474,645477,645480,645483],{"className":645411},[31676],[34,645413],{"className":645414,"style":645415},[31680],"height:1.04em;vertical-align:-0.1078em;",[34,645417,645419],{"className":645418},[31685,41550],[34,645420,645422,645465],{"className":645421},[31796,31797],[34,645423,645425,645462],{"className":645424},[31801],[34,645426,645428,645449],{"className":645427,"style":608935},[31805],[34,645429,645431,645434],{"className":645430,"style":41565},[41564],[34,645432],{"className":645433,"style":41508},[31813],[34,645435,645437,645440,645443,645446],{"className":645436,"style":41572},[31685],[34,645438,547860],{"className":645439},[32767],[34,645441],{"className":645442,"style":32123},[31845],[34,645444,768],{"className":645445},[31685],[34,645447,479697],{"className":645448,"style":34452},[31685,31686],[34,645450,645452,645455],{"style":645451},"top:-2.8922em;",[34,645453],{"className":645454,"style":41508},[31813],[34,645456,645458],{"className":645457,"style":41586},[41585],[41588,645459,645460],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,645461],{"d":41597},[34,645463,31829],{"className":645464},[31828],[34,645466,645468],{"className":645467},[31801],[34,645469,645472],{"className":645470,"style":645471},[31805],"height:0.1078em;",[34,645473],{},[34,645475],{"className":645476,"style":32123},[31845],[34,645478,547860],{"className":645479},[32767],[34,645481],{"className":645482,"style":32123},[31845],[34,645484,479697],{"className":645485,"style":34452},[31685,31686],[108,645487,637275,645488,347],{},[34,645489,645491,645593],{"className":645490},[31642],[34,645492,645494],{"className":645493},[31646],[31648,645495,645496],{"xmlns":31650},[31652,645497,645498,645590],{},[31655,645499,645500,645502,645504,645518,645520,645532,645534,645536,645538,645540,645542,645558,645560,645562,645564,645566,645580,645582,645584,645586,645588],{},[31662,645501,202],{},[31749,645503,15932],{},[38846,645505,645506,645508,645510],{},[31662,645507,546295],{},[31749,645509,459],{},[31655,645511,645512,645514,645516],{},[31658,645513,58442],{},[31658,645515,38080],{"mathvariant":38079},[31749,645517,15932],{},[31662,645519,456],{"stretchy":702},[41463,645521,645522],{},[31655,645523,645524,645526,645528,645530],{},[31658,645525,547860],{},[31662,645527,32682],{},[31749,645529,768],{},[31658,645531,479697],{},[31658,645533,547860],{},[31662,645535,32682],{},[31658,645537,479697],{},[31662,645539,2557],{"stretchy":702},[31662,645541,71191],{},[41453,645543,645544,645546],{},[31749,645545,640],{},[41463,645547,645548],{},[31655,645549,645550,645552,645554,645556],{},[31658,645551,547860],{},[31662,645553,32682],{},[31749,645555,768],{},[31658,645557,479697],{},[31658,645559,46900],{},[31658,645561,479697],{},[31662,645563,202],{},[31749,645565,15932],{},[38846,645567,645568,645570,645572],{},[31662,645569,546295],{},[31749,645571,459],{},[31655,645573,645574,645576,645578],{},[31658,645575,58442],{},[31658,645577,38080],{"mathvariant":38079},[31749,645579,15932],{},[31658,645581,547860],{},[31662,645583,32682],{},[31658,645585,479697],{},[31658,645587,46900],{},[31658,645589,479697],{},[31665,645591,645592],{"encoding":31667},"= 4 \\int_0^{\\pi\u002F4} (\\sqrt{\\cos 2\\theta} \\cos \\theta) \\cdot \\frac{1}{\\sqrt{\\cos 2\\theta}} d\\theta = 4 \\int_0^{\\pi\u002F4} \\cos \\theta 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= t \\cdot \\frac{2\\sqrt{2}}{3}t^{3\u002F2} \\cdot \\frac{1}{2}t^2 = \\frac{\\sqrt{2}}{3} t^{9\u002F2}",[34,649682,649684,649708,649726,649888,650000],{"className":649683,"ariaHidden":758},[31672],[34,649685,649687,649690,649693,649696,649699,649702,649705],{"className":649686},[31676],[34,649688],{"className":649689,"style":38774},[31680],[34,649691,32936],{"className":649692},[31685,31686],[34,649694,67764],{"className":649695,"style":32062},[31685,31686],[34,649697,382333],{"className":649698,"style":382406},[31685,31686],[34,649700],{"className":649701,"style":31846},[31845],[34,649703,202],{"className":649704},[31850],[34,649706],{"className":649707,"style":31846},[31845],[34,649709,649711,649714,649717,649720,649723],{"className":649710},[31676],[34,649712],{"className":649713,"style":33280},[31680],[34,649715,33249],{"className":649716},[31685,31686],[34,649718],{"className":649719,"style":32604},[31845],[34,649721,71191],{"className":649722},[32024],[34,649724],{"className":649725,"style":32604},[31845],[34,649727,649729,649732,649847,649879,649882,649885],{"className":649728},[31676],[34,649730],{"className":649731,"style":41481},[31680],[34,649733,649735,649738,649844],{"className":649734},[31685],[34,649736],{"className":649737},[31782,41488],[34,649739,649741],{"className":649740},[41453],[34,649742,649744,649836],{"className":649743},[31796,31797],[34,649745,649747,649833],{"className":649746},[31801],[34,649748,649750,649764,649772],{"className":649749,"style":41501},[31805],[34,649751,649752,649755],{"style":41504},[34,649753],{"className":649754,"style":41508},[31813],[34,649756,649758],{"className":649757},[31818,31819,31820,31821],[34,649759,649761],{"className":649760},[31685,31821],[34,649762,786],{"className":649763},[31685,31821],[34,649765,649766,649769],{"style":41520},[34,649767],{"className":649768,"style":41508},[31813],[34,649770],{"className":649771,"style":41528},[41527],[34,649773,649774,649777],{"style":41531},[34,649775],{"className":649776,"style":41508},[31813],[34,649778,649780],{"className":649779},[31818,31819,31820,31821],[34,649781,649783,649786],{"className":649782},[31685,31821],[34,649784,768],{"className":649785},[31685,31821],[34,649787,649789],{"className":649788},[31685,41550,31821],[34,649790,649792,649825],{"className":649791},[31796,31797],[34,649793,649795,649822],{"className":649794},[31801],[34,649796,649798,649810],{"className":649797,"style":41560},[31805],[34,649799,649801,649804],{"className":649800,"style":41565},[41564],[34,649802],{"className":649803,"style":41508},[31813],[34,649805,649807],{"className":649806,"style":41572},[31685,31821],[34,649808,768],{"className":649809},[31685,31821],[34,649811,649812,649815],{"style":41578},[34,649813],{"className":649814,"style":41508},[31813],[34,649816,649818],{"className":649817,"style":41586},[41585,31821],[41588,649819,649820],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,649821],{"d":41597},[34,649823,31829],{"className":649824},[31828],[34,649826,649828],{"className":649827},[31801],[34,649829,649831],{"className":649830,"style":41607},[31805],[34,649832],{},[34,649834,31829],{"className":649835},[31828],[34,649837,649839],{"className":649838},[31801],[34,649840,649842],{"className":649841,"style":41619},[31805],[34,649843],{},[34,649845],{"className":649846},[31690,41488],[34,649848,649850,649853],{"className":649849},[31685],[34,649851,33249],{"className":649852},[31685,31686],[34,649854,649856],{"className":649855},[31792],[34,649857,649859],{"className":649858},[31796],[34,649860,649862],{"className":649861},[31801],[34,649863,649865],{"className":649864,"style":61037},[31805],[34,649866,649867,649870],{"style":37777},[34,649868],{"className":649869,"style":31814},[31813],[34,649871,649873],{"className":649872},[31818,31819,31820,31821],[34,649874,649876],{"className":649875},[31685,31821],[34,649877,42777],{"className":649878},[31685,31821],[34,649880],{"className":649881,"style":32604},[31845],[34,649883,71191],{"className":649884},[32024],[34,649886],{"className":649887,"style":32604},[31845],[34,649889,649891,649894,649962,649991,649994,649997],{"className":649890},[31676],[34,649892],{"className":649893,"style":46727},[31680],[34,649895,649897,649900,649959],{"className":649896},[31685],[34,649898],{"className":649899},[31782,41488],[34,649901,649903],{"className":649902},[41453],[34,649904,649906,649951],{"className":649905},[31796,31797],[34,649907,649909,649948],{"className":649908},[31801],[34,649910,649912,649926,649934],{"className":649911,"style":46746},[31805],[34,649913,649914,649917],{"style":41504},[34,649915],{"className":649916,"style":41508},[31813],[34,649918,649920],{"className":649919},[31818,31819,31820,31821],[34,649921,649923],{"className":649922},[31685,31821],[34,649924,768],{"className":649925},[31685,31821],[34,649927,649928,649931],{"style":41520},[34,649929],{"className":649930,"style":41508},[31813],[34,649932],{"className":649933,"style":41528},[41527],[34,649935,649936,649939],{"style":46771},[34,64993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211,650212],{},[31749,650213,768],{},[31749,650215,786],{},[38846,650217,650218,650220,650222],{},[31662,650219,546295],{},[31749,650221,459],{},[31749,650223,640],{},[31662,650225,456],{"stretchy":702},[37624,650227,650228,650230],{},[31658,650229,33249],{},[31655,650231,650232,650234,650236],{},[31749,650233,36732],{},[31658,650235,38080],{"mathvariant":38079},[31749,650237,768],{},[31662,650239,223],{},[37624,650241,650242,650244],{},[31658,650243,33249],{},[31655,650245,650246,650248,650250],{},[31749,650247,39620],{},[31658,650249,38080],{"mathvariant":38079},[31749,650251,768],{},[31662,650253,2557],{"stretchy":702},[31658,650255,46900],{},[31658,650257,33249],{},[31665,650259,650260],{"encoding":31667},"\\int_0^1 \\frac{\\sqrt{2}}{3} t^{9\u002F2} (1+t) dt = \\frac{\\sqrt{2}}{3} \\int_0^1 (t^{9\u002F2} + t^{11\u002F2}) dt",[34,650262,650264,650484,650511,650727],{"className":650263,"ariaHidden":758},[31672],[34,650265,650267,650271,650322,650325,650437,650469,650472,650475,650478,650481],{"className":650266},[31676],[34,650268],{"className":650269,"style":650270},[31680],"height:1.3938em;vertical-align:-0.3558em;",[34,650272,650274,650277],{"className":650273},[32767],[34,650275,546295],{"className":650276,"style":546331},[32767,131249,156560],[34,650278,650280],{"className":650279},[31792],[34,650281,650283,650314],{"className":650282},[31796,31797],[34,650284,650286,650311],{"className":650285},[31801],[34,650287,650289,650300],{"className":650288,"style":563328},[31805],[34,650290,650291,650294],{"style":546347},[34,650292],{"className":650293,"style":31814},[31813],[34,650295,650297],{"className":650296},[31818,31819,31820,31821],[34,650298,459],{"className":650299},[31685,31821],[34,650301,650302,650305],{"style":546362},[34,650303],{"className":650304,"style":31814},[31813],[34,650306,650308],{"className":650307},[31818,31819,31820,31821],[34,650309,640],{"className":650310},[31685,31821],[34,650312,31829],{"className":650313},[31828],[34,650315,650317],{"className":650316},[31801],[34,650318,650320],{"className":650319,"style":546384},[31805],[34,650321],{},[34,650323],{"className":650324,"style":32123},[31845],[34,650326,650328,650331,650434],{"className":650327},[31685],[34,650329],{"className":650330},[31782,41488],[34,650332,650334],{"className":650333},[41453],[34,650335,650337,650426],{"className":650336},[31796,31797],[34,650338,650340,650423],{"className":650339},[31801],[34,650341,650343,650357,650365],{"className":650342,"style":41501},[31805],[34,650344,650345,650348],{"style":41504},[34,650346],{"className":650347,"style":41508},[31813],[34,650349,650351],{"className":650350},[31818,31819,31820,31821],[34,650352,650354],{"className":650353},[31685,31821],[34,650355,786],{"className":650356},[31685,31821],[34,650358,650359,650362],{"style":41520},[34,650360],{"className":650361,"style":41508},[31813],[34,650363],{"className":650364,"style":41528},[41527],[34,650366,650367,650370],{"style":41531},[34,650368],{"className":650369,"style":41508},[31813],[34,650371,650373],{"className":650372},[31818,31819,31820,31821],[34,650374,650376],{"className":650375},[31685,31821],[34,650377,650379],{"className":650378},[31685,41550,31821],[34,650380,650382,650415],{"className":650381},[31796,31797],[34,650383,650385,650412],{"className":650384},[31801],[34,650386,650388,650400],{"className":650387,"style":41560},[31805],[34,650389,650391,650394],{"className":650390,"style":41565},[41564],[34,650392],{"className":650393,"style":41508},[31813],[34,650395,650397],{"className":650396,"style":41572},[31685,31821],[34,650398,768],{"className":650399},[31685,31821],[34,650401,650402,650405],{"style":41578},[34,650403],{"className":650404,"style":41508},[31813],[34,650406,650408],{"className":650407,"style":41586},[41585,31821],[41588,650409,650410],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,650411],{"d":41597},[34,650413,31829],{"className":650414},[31828],[34,650416,650418],{"className":650417},[31801],[34,650419,650421],{"className":650420,"style":41607},[31805],[34,650422],{},[34,650424,31829],{"className":650425},[31828],[34,650427,650429],{"className":650428},[31801],[34,650430,650432],{"className":650431,"style":41619},[31805],[34,650433],{},[34,650435],{"className":650436},[31690,41488],[34,650438,650440,650443],{"className":650439},[31685],[34,650441,33249],{"className":650442},[31685,31686],[34,650444,650446],{"className":650445},[31792],[34,650447,650449],{"className":650448},[31796],[34,650450,650452],{"className":650451},[31801],[34,650453,650455],{"className":650454,"style":61037},[31805],[34,650456,650457,650460],{"style":37777},[34,650458],{"className":650459,"style":31814},[31813],[34,650461,650463],{"className":650462},[31818,31819,31820,31821],[34,650464,650466],{"className":650465},[31685,31821],[34,650467,634354],{"className":650468},[31685,31821],[34,650470,456],{"className":650471},[31782],[34,650473,640],{"className":650474},[31685],[34,650476],{"className":650477,"style":32604},[31845],[34,650479,223],{"className":650480},[32024],[34,650482],{"className":650483,"style":32604},[31845],[34,650485,650487,650490,650493,650496,650499,650502,650505,650508],{"className":650486},[31676],[34,650488],{"className":650489,"style":31774},[31680],[34,650491,33249],{"className":650492},[31685,31686],[34,650494,2557],{"className":650495},[31690],[34,650497,46900],{"className":650498},[31685,31686],[34,650500,33249],{"className":650501},[31685,31686],[34,650503],{"className":650504,"style":31846},[31845],[34,650506,202],{"className":650507},[31850],[34,650509],{"className":650510,"style":31846},[31845],[34,650512,650514,650517,650629,650632,650683,650686,650718,650721,650724],{"className":650513},[31676],[34,650515],{"className":650516,"style":650270},[31680],[34,6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\\frac{\\sqrt{2}}{3} (\\frac{2}{11} + \\frac{2}{13}) = \\frac{2\\sqrt{2}}{3} \\frac{13+11}{143} = \\frac{2\\sqrt{2}}{3} \\frac{24}{143} = 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球面 ",[34,651709,651711,651753],{"className":651710},[31642],[34,651712,651714],{"className":651713},[31646],[31648,651715,651716],{"xmlns":31650},[31652,651717,651718,651750],{},[31655,651719,651720,651726,651728,651734,651736,651742,651744],{},[37624,651721,651722,651724],{},[31658,651723,32936],{},[31749,651725,768],{},[31662,651727,223],{},[37624,651729,651730,651732],{},[31658,651731,67764],{},[31749,651733,768],{},[31662,651735,223],{},[37624,651737,651738,651740],{},[31658,651739,382333],{},[31749,651741,768],{},[31662,651743,202],{},[37624,651745,651746,651748],{},[31658,651747,111],{},[31749,651749,768],{},[31665,651751,651752],{"encoding":31667},"x^2+y^2+z^2=a^2",[34,651754,651756,651800,651844,651888],{"className":651755,"ariaHidden":758},[31672],[34,651757,651759,651762,651791,651794,651797],{"className":651758},[31676],[34,651760],{"className":651761,"style":40044},[31680],[34,651763,651765,651768],{"className":651764},[31685],[34,651766,32936],{"className":651767},[31685,31686],[34,651769,651771],{"className":651770},[31792],[34,651772,651774],{"className":651773},[31796],[34,651775,651777],{"className":651776},[31801],[34,651778,651780],{"className":651779,"style":37774},[31805],[34,651781,651782,651785],{"style":37777},[34,651783],{"className":651784,"style":31814},[31813],[34,651786,651788],{"className":651787},[31818,31819,31820,31821],[34,651789,768],{"className":651790},[31685,31821],[34,651792],{"className":651793,"style":32604},[31845],[34,651795,223],{"className":651796},[32024],[34,651798],{"className":651799,"style":32604},[31845],[34,651801,651803,651806,651835,651838,651841],{"className":651802},[31676],[34,651804],{"className":651805,"style":315747},[31680],[34,651807,651809,651812],{"className":651808},[31685],[34,651810,67764],{"className":651811,"style":32062},[31685,31686],[34,651813,651815],{"className":651814},[31792],[34,651816,651818],{"className":651817},[31796],[34,651819,651821],{"className":651820},[31801],[34,651822,651824],{"className":651823,"style":37774},[31805],[34,651825,651826,651829],{"style":37777},[34,651827],{"className":651828,"style":31814},[31813],[34,651830,651832],{"className":651831},[31818,31819,31820,31821],[34,651833,768],{"className":651834},[31685,31821],[34,651836],{"className":651837,"style":32604},[31845],[34,651839,223],{"className":651840},[32024],[34,651842],{"className":651843,"style":32604},[31845],[34,651845,651847,651850,651879,651882,651885],{"className":651846},[31676],[34,651848],{"className":651849,"style":37774},[31680],[34,651851,651853,651856],{"className":651852},[31685],[34,651854,382333],{"className":651855,"style":382406},[31685,31686],[34,651857,651859],{"className":651858},[31792],[34,651860,651862],{"className":651861},[31796],[34,651863,651865],{"className":651864},[31801],[34,651866,651868],{"className":651867,"style":37774},[31805],[34,651869,651870,651873],{"style":37777},[34,651871],{"className":651872,"style":31814},[31813],[34,651874,651876],{"className":651875},[31818,31819,31820,31821],[34,651877,768],{"className":651878},[31685,31821],[34,651880],{"className":651881,"style":31846},[31845],[34,651883,202],{"className":651884},[31850],[34,651886],{"className":651887,"style":31846},[31845],[34,651889,651891,651894],{"className":651890},[31676],[34,651892],{"className":651893,"style":37774},[31680],[34,651895,651897,651900],{"className":651896},[31685],[34,651898,111],{"className":651899},[31685,31686],[34,651901,651903],{"className":651902},[31792],[34,651904,651906],{"className":651905},[31796],[34,651907,651909],{"className":651908},[31801],[34,651910,651912],{"className":651911,"style":37774},[31805],[34,651913,651914,651917],{"style":37777},[34,651915],{"className":651916,"style":31814},[31813],[34,651918,651920],{"className":651919},[31818,31819,31820,31821],[34,651921,768],{"className":651922},[31685,31821]," 与平面 ",[34,651925,651927,651953],{"className":651926},[31642],[34,651928,651930],{"className":651929},[31646],[31648,651931,651932],{"xmlns":31650},[31652,651933,651934,651950],{},[31655,651935,651936,651938,651940,651942,651944,651946,651948],{},[31658,651937,32936],{},[31662,651939,223],{},[31658,651941,67764],{},[31662,651943,223],{},[31658,651945,382333],{},[31662,651947,202],{},[31749,651949,459],{},[31665,651951,651952],{"encoding":31667},"x+y+z=0",[34,651954,651956,651974,651992,652010],{"className":651955,"ariaHidden":758},[31672],[34,651957,651959,651962,651965,651968,651971],{"className":651958},[31676],[34,651960],{"className":651961,"style":32597},[31680],[34,651963,32936],{"className":651964},[31685,31686],[34,651966],{"className":651967,"style":32604},[31845],[34,651969,223],{"className":651970},[32024],[34,651972],{"className":651973,"style":32604},[31845],[34,651975,651977,651980,651983,651986,651989],{"className":651976},[31676],[34,651978],{"className":651979,"style":55949},[31680],[34,651981,67764],{"className":651982,"style":32062},[31685,31686],[34,651984],{"className":651985,"style":32604},[31845],[34,651987,223],{"className":651988},[32024],[34,651990],{"className":651991,"style":32604},[31845],[34,651993,651995,651998,652001,652004,652007],{"className":651994},[31676],[34,651996],{"className":651997,"style":32527},[31680],[34,651999,382333],{"className":652000,"style":382406},[31685,31686],[34,652002],{"className":652003,"style":31846},[31845],[34,652005,202],{"className":652006},[31850],[34,652008],{"className":652009,"style":31846},[31845],[34,652011,652013,652016],{"className":652012},[31676],[34,652014],{"className":652015,"style":32617},[31680],[34,652017,459],{"className":652018},[31685]," 交线",[105,652021,652022,652057,652157,652525,652836,653015,653233,653611,653667],{},[108,652023,652024,652025,652028,652029,347],{},"这是一个过原点的平面截球面，得到的是",[2627,652026,652027],{},"大圆","，半径为 ",[34,652030,652032,652045],{"className":652031},[31642],[34,652033,652035],{"className":652034},[31646],[31648,652036,652037],{"xmlns":31650},[31652,652038,652039,652043],{},[31655,652040,652041],{},[31658,652042,111],{},[31665,652044,111],{"encoding":31667},[34,652046,652048],{"className":652047,"ariaHidden":758},[31672],[34,652049,652051,652054],{"className":652050},[31676],[34,652052],{"className":652053,"style":32527},[31680],[34,652055,111],{"className":652056},[31685,31686],[108,652058,598682,652059,347],{},[34,652060,652062,652094],{"className":652061},[31642],[34,652063,652065],{"className":652064},[31646],[31648,652066,652067],{"xmlns":31650},[31652,652068,652069,652091],{},[31655,652070,652071,652073,652075,652077,652079,652081,652083,652085,652087,652089],{},[31662,652072,456],{"stretchy":702},[31658,652074,32936],{},[31658,652076,67764],{},[31662,652078,223],{},[31658,652080,67764],{},[31658,652082,382333],{},[31662,652084,223],{},[31658,652086,382333],{},[31658,652088,32936],{},[31662,652090,2557],{"stretchy":702},[31665,652092,652093],{"encoding":31667},"(xy+yz+zx)",[34,652095,652097,652121,652142],{"className":652096,"ariaHidden":758},[31672],[34,652098,652100,652103,652106,652109,652112,652115,652118],{"className":652099},[31676],[34,652101],{"className":652102,"style":31774},[31680],[34,652104,456],{"className":652105},[31782],[34,652107,32936],{"className":652108},[31685,31686],[34,652110,67764],{"className":652111,"style":32062},[31685,31686],[34,652113],{"className":652114,"style":32604},[31845],[34,652116,223],{"className":652117},[32024],[34,652119],{"className":652120,"style":32604},[31845],[34,652122,652124,652127,652130,652133,652136,652139],{"className":652123},[31676],[34,652125],{"className":652126,"style":55949},[31680],[34,652128,67764],{"className":652129,"style":32062},[31685,31686],[34,652131,382333],{"className":652132,"style":382406},[31685,31686],[34,652134],{"className":652135,"style":32604},[31845],[34,652137,223],{"className":652138},[32024],[34,652140],{"className":652141,"style":32604},[31845],[34,652143,652145,652148,652151,652154],{"className":652144},[31676],[34,652146],{"className":652147,"style":31774},[31680],[34,652149,382333],{"className":652150,"style":382406},[31685,31686],[34,652152,32936],{"className":652153},[31685,31686],[34,652155,2557],{"className":652156},[31690],[108,652158,652159,652160,347],{},"利用代数恒等式：",[34,652161,652163,652241],{"className":652162},[31642],[34,652164,652166],{"className":652165},[31646],[31648,652167,652168],{"xmlns":31650},[31652,652169,652170,652238],{},[31655,652171,652172,652174,652176,652178,652180,652182,652184,652190,652192,652198,652200,652206,652208,652214,652216,652218,652220,652222,652224,652226,652228,652230,652232,652234,652236],{},[31662,652173,456],{"stretchy":702},[31658,652175,32936],{},[31662,652177,223],{},[31658,652179,67764],{},[31662,652181,223],{},[31658,652183,382333],{},[37624,652185,652186,652188],{},[31662,652187,2557],{"stretchy":702},[31749,652189,768],{},[31662,652191,202],{},[37624,652193,652194,652196],{},[31658,652195,32936],{},[31749,652197,768],{},[31662,652199,223],{},[37624,652201,652202,652204],{},[31658,652203,67764],{},[31749,652205,768],{},[31662,652207,223],{},[37624,652209,652210,652212],{},[31658,652211,382333],{},[31749,652213,768],{},[31662,652215,223],{},[31749,652217,768],{},[31662,652219,456],{"stretchy":702},[31658,652221,32936],{},[31658,652223,67764],{},[31662,652225,223],{},[31658,652227,67764],{},[31658,652229,382333],{},[31662,652231,223],{},[31658,652233,382333],{},[31658,652235,32936],{},[31662,652237,2557],{"stretchy":702},[31665,652239,652240],{"encoding":31667},"(x+y+z)^2 = x^2+y^2+z^2 + 2(xy+yz+zx)",[34,652242,652244,652265,652283,652330,652374,652418,652462,652489,652510],{"className":652243,"ariaHidden":758},[31672],[34,652245,652247,652250,652253,652256,652259,652262],{"className":652246},[31676],[34,652248],{"className":652249,"style":31774},[31680],[34,652251,456],{"className":652252},[31782],[34,652254,32936],{"className":652255},[31685,31686],[34,652257],{"className":652258,"style":32604},[31845],[34,652260,223],{"className":652261},[32024],[34,652263],{"className":652264,"style":32604},[31845],[34,652266,652268,652271,652274,652277,652280],{"className":652267},[31676],[34,652269],{"className":652270,"style":55949},[31680],[34,652272,67764],{"className":652273,"style":32062},[31685,31686],[34,652275],{"className":652276,"style":32604},[31845],[34,652278,223],{"className":652279},[32024],[34,652281],{"className":652282,"style":32604},[31845],[34,652284,652286,652289,652292,652321,652324,652327],{"className":652285},[31676],[34,652287],{"className":652288,"style":37755},[31680],[34,652290,382333],{"className":652291,"style":382406},[31685,31686],[34,652293,652295,652298],{"className":652294},[31690],[34,652296,2557],{"className":652297},[31690],[34,652299,652301],{"className":652300},[31792],[34,652302,652304],{"className":652303},[31796],[34,652305,652307],{"className":652306},[31801],[34,652308,652310],{"className":652309,"style":37774},[31805],[34,652311,652312,652315],{"style":37777},[34,652313],{"className":652314,"style":31814},[31813],[34,652316,652318],{"className":652317},[31818,31819,31820,31821],[34,652319,768],{"className":652320},[31685,31821],[34,652322],{"className":652323,"style":31846},[31845],[34,652325,202],{"className":652326},[31850],[34,652328],{"className":652329,"style":31846},[31845],[34,652331,652333,652336,652365,652368,652371],{"className":652332},[31676],[34,652334],{"className":652335,"style":40044},[31680],[34,652337,652339,652342],{"className":652338},[31685],[34,652340,32936],{"className":652341},[31685,31686],[34,652343,652345],{"className":652344},[31792],[34,652346,652348],{"className":652347},[31796],[34,652349,652351],{"className":652350},[31801],[34,652352,652354],{"className":652353,"style":37774},[31805],[34,652355,652356,652359],{"style":37777},[34,652357],{"className":652358,"style":31814},[31813],[34,652360,652362],{"className":652361},[31818,31819,31820,31821],[34,652363,768],{"className":652364},[31685,31821],[34,652366],{"className":652367,"style":32604},[31845],[34,652369,223],{"className":652370},[32024],[34,652372],{"className":652373,"style":32604},[31845],[34,652375,652377,652380,652409,652412,652415],{"className":652376},[31676],[34,652378],{"className":652379,"style":315747},[31680],[34,652381,652383,652386],{"className":652382},[31685],[34,652384,67764],{"className":652385,"style":32062},[31685,31686],[34,652387,652389],{"className":652388},[31792],[34,652390,652392],{"className":652391},[31796],[34,652393,652395],{"className":652394},[31801],[34,652396,652398],{"className":652397,"style":37774},[31805],[34,652399,652400,652403],{"style":37777},[34,652401],{"className":652402,"style":31814},[31813],[34,652404,652406],{"className":652405},[31818,31819,31820,31821],[34,652407,768],{"className":652408},[31685,31821],[34,652410],{"className":652411,"style":32604},[31845],[34,652413,223],{"className":652414},[32024],[34,652416],{"className":652417,"style":32604},[31845],[34,652419,652421,652424,652453,652456,652459],{"className":652420},[31676],[34,652422],{"className":652423,"style":40044},[31680],[34,652425,652427,652430],{"className":652426},[31685],[34,652428,382333],{"className":652429,"style":382406},[31685,31686],[34,652431,652433],{"className":652432},[31792],[34,652434,652436],{"className":652435},[31796],[34,652437,652439],{"className":652438},[31801],[34,652440,652442],{"className":652441,"style":37774},[31805],[34,652443,652444,652447],{"style":37777},[34,652445],{"className":652446,"style":31814},[31813],[34,652448,652450],{"className":652449},[31818,31819,31820,31821],[34,652451,768],{"className":652452},[31685,31821],[34,652454],{"className":652455,"style":32604},[31845],[34,652457,223],{"className":652458},[32024],[34,652460],{"className":652461,"style":32604},[31845],[34,652463,652465,652468,652471,652474,652477,652480,652483,652486],{"className":652464},[31676],[34,652466],{"className":652467,"style":31774},[31680],[34,652469,768],{"className":652470},[31685],[34,652472,456],{"className":652473},[31782],[34,652475,32936],{"className":652476},[31685,31686],[34,652478,67764],{"className":652479,"style":32062},[31685,31686],[34,652481],{"className":652482,"style":32604},[31845],[34,652484,223],{"className":652485},[32024],[34,652487],{"className":652488,"style":32604},[31845],[34,652490,652492,652495,652498,652501,652504,652507],{"className":652491},[31676],[34,652493],{"className":652494,"style":55949},[31680],[34,652496,67764],{"className":652497,"style":32062},[31685,31686],[34,652499,382333],{"className":652500,"style":382406},[31685,31686],[34,652502],{"className":652503,"style":32604},[31845],[34,652505,223],{"className":652506},[32024],[34,652508],{"className":652509,"style":32604},[31845],[34,652511,652513,652516,652519,652522],{"className":652512},[31676],[34,652514],{"className":652515,"style":31774},[31680],[34,652517,382333],{"className":652518,"style":382406},[31685,31686],[34,652520,32936],{"className":652521},[31685,31686],[34,652523,2557],{"className":652524},[31690],[108,652526,652527,652528,258,652622,347],{},"在曲线上，满足 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= a^2 + 2(xy+yz+zx)",[34,652888,652890,652908,652952,652979,653000],{"className":652889,"ariaHidden":758},[31672],[34,652891,652893,652896,652899,652902,652905],{"className":652892},[31676],[34,652894],{"className":652895,"style":32617},[31680],[34,652897,459],{"className":652898},[31685],[34,652900],{"className":652901,"style":31846},[31845],[34,652903,202],{"className":652904},[31850],[34,652906],{"className":652907,"style":31846},[31845],[34,652909,652911,652914,652943,652946,652949],{"className":652910},[31676],[34,652912],{"className":652913,"style":40044},[31680],[34,652915,652917,652920],{"className":652916},[31685],[34,652918,111],{"className":652919},[31685,31686],[34,652921,652923],{"className":652922},[31792],[34,652924,652926],{"className":652925},[31796],[34,652927,652929],{"className":652928},[31801],[34,652930,652932],{"className":652931,"style":37774},[31805],[34,652933,652934,652937],{"style":37777},[34,652935],{"className":652936,"style":31814},[31813],[34,652938,652940],{"className":652939},[31818,31819,31820,31821],[34,652941,768],{"className":652942},[31685,31821],[34,652944],{"className":652945,"style":32604},[31845],[34,652947,223],{"className":652948},[32024],[34,652950],{"className":652951,"style":32604},[31845],[34,652953,652955,652958,652961,652964,652967,652970,652973,652976],{"className":652954},[31676],[34,652956],{"className":652957,"style":31774},[31680],[34,652959,768],{"className":652960},[31685],[34,652962,456],{"className":652963},[31782],[34,652965,32936],{"className":652966},[31685,31686],[34,652968,67764],{"className":652969,"style":32062},[31685,31686],[34,652971],{"className":652972,"style":32604},[31845],[34,652974,223],{"className":652975},[32024],[34,652977],{"className":652978,"style":32604},[31845],[34,652980,652982,652985,652988,652991,652994,652997],{"className":652981},[31676],[34,652983],{"className":652984,"style":55949},[31680],[34,652986,67764],{"className":652987,"style":32062},[31685,31686],[34,652989,382333],{"className":652990,"style":382406},[31685,31686],[34,652992],{"className":652993,"style":32604},[31845],[34,652995,223],{"className":652996},[32024],[34,652998],{"className":652999,"style":32604},[31845],[34,653001,653003,653006,653009,653012],{"className":653002},[31676],[34,653004],{"className":653005,"style":31774},[31680],[34,653007,382333],{"className":653008,"style":382406},[31685,31686],[34,653010,32936],{"className":653011},[31685,31686],[34,653013,2557],{"className":653014},[31690],[108,653016,653017,653018,653232],{},"所以被积函数 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（常数！）。",[108,653234,637275,653235,347],{},[34,653236,653238,653301],{"className":653237},[31642],[34,653239,653241],{"className":653240},[31646],[31648,653242,653243],{"xmlns":31650},[31652,653244,653245,653298],{},[31655,653246,653247,653249,653255,653257,653259,653269,653271,653273,653275,653277,653279,653289,653291,653293,653296],{},[31662,653248,202],{},[31743,653250,653251,653253],{},[31662,653252,546295],{},[31658,653254,33206],{},[31662,653256,456],{"stretchy":702},[31662,653258,32226],{},[41453,653260,653261,653267],{},[37624,653262,653263,653265],{},[31658,653264,111],{},[31749,653266,768],{},[31749,653268,768],{},[31662,653270,2557],{"stretchy":702},[31658,653272,46900],{},[31658,653274,31747],{},[31662,653276,202],{},[31662,653278,32226],{},[41453,653280,653281,653287],{},[37624,653282,653283,653285],{},[31658,653284,111],{},[31749,653286,768],{},[31749,653288,768],{},[31662,653290,71191],{},[31662,653292,456],{"stretchy":702},[39361,653294,653295],{},"大圆周长",[31662,653297,2557],{"stretchy":702},[31665,653299,653300],{"encoding":31667},"= \\int_L (-\\frac{a^2}{2}) ds = -\\frac{a^2}{2} \\cdot 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",[34,653615,653617,653637],{"className":653616},[31642],[34,653618,653620],{"className":653619},[31646],[31648,653621,653622],{"xmlns":31650},[31652,653623,653624,653634],{},[31655,653625,653626,653628,653630,653632],{},[31662,653627,202],{},[31749,653629,768],{},[31658,653631,58442],{},[31658,653633,111],{},[31665,653635,653636],{"encoding":31667},"= 2\\pi a",[34,653638,653640,653652],{"className":653639,"ariaHidden":758},[31672],[34,653641,653643,653646,653649],{"className":653642},[31676],[34,653644],{"className":653645,"style":73089},[31680],[34,653647,202],{"className":653648},[31850],[34,653650],{"className":653651,"style":31846},[31845],[34,653653,653655,653658,653661,653664],{"className":653654},[31676],[34,653656],{"className":653657,"style":32617},[31680],[34,653659,768],{"className":653660},[31685],[34,653662,58442],{"className":653663,"style":32062},[31685,31686],[34,653665,111],{"className":653666},[31685,31686],[108,653668,600826,653669,347],{},[34,653670,653672,653718],{"className":653671},[31642],[34,653673,653675],{"className":653674},[31646],[31648,653676,653677],{"xmlns":31650},[31652,653678,653679,653715],{},[31655,653680,653681,653683,653685,653695,653697,653699,653701,653703,653705,653707,653709],{},[31662,653682,202],{},[31662,653684,32226],{},[41453,653686,653687,653693],{},[37624,653688,653689,653691],{},[31658,653690,111],{},[31749,653692,768],{},[31749,653694,768],{},[31662,653696,71191],{},[31749,653698,768],{},[31658,653700,58442],{},[31658,653702,111],{},[31662,653704,202],{},[31662,653706,32226],{},[31658,653708,58442],{},[37624,653710,653711,653713],{},[31658,653712,111],{},[31749,653714,786],{},[31665,653716,653717],{"encoding":31667},"= -\\frac{a^2}{2} \\cdot 2\\pi a = -\\pi 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椭圆质量，线密度 ",[34,653915,653917,653939],{"className":653916},[31642],[34,653918,653920],{"className":653919},[31646],[31648,653921,653922],{"xmlns":31650},[31652,653923,653924,653936],{},[31655,653925,653926,653928,653930,653932,653934],{},[31658,653927,220084],{},[31662,653929,202],{},[31658,653931,162086],{"mathvariant":38079},[31658,653933,67764],{},[31658,653935,162086],{"mathvariant":38079},[31665,653937,653938],{"encoding":31667},"\\rho = |y|",[34,653940,653942,653960],{"className":653941,"ariaHidden":758},[31672],[34,653943,653945,653948,653951,653954,653957],{"className":653944},[31676],[34,653946],{"className":653947,"style":38774},[31680],[34,653949,220084],{"className":653950},[31685,31686],[34,653952],{"className":653953,"style":31846},[31845],[34,653955,202],{"className":653956},[31850],[34,653958],{"className":653959,"style":31846},[31845],[34,653961,653963,653966,653969,653972],{"className":653962},[31676],[34,653964],{"className":653965,"style":31774},[31680],[34,653967,162086],{"className":653968},[31685],[34,653970,67764],{"className":653971,"style":32062},[31685,31686],[34,653973,162086],{"className":653974},[31685],[105,653976,653977,654101,654397,654791,654852,655244,655797,656271,656696,657300,657471],{},[108,653978,639685,653979,347],{},[34,653980,653982,654020],{"className":653981},[31642],[34,653983,653985],{"className":653984},[31646],[31648,653986,653987],{"xmlns":31650},[31652,653988,653989,654017],{},[31655,653990,653991,653993,653995,653997,653999,654001,654003,654005,654007,654009,654011,654013,654015],{},[31658,653992,32936],{},[31662,653994,202],{},[31658,653996,111],{},[31658,653998,547860],{},[31662,654000,32682],{},[31658,654002,33249],{},[31662,654004,31960],{"separator":758},[31658,654006,67764],{},[31662,654008,202],{},[31658,654010,1697],{},[31658,654012,547853],{},[31662,654014,32682],{},[31658,654016,33249],{},[31665,654018,654019],{"encoding":31667},"x=a \\cos t, y=b \\sin t",[34,654021,654023,654041,654080],{"className":654022,"ariaHidden":758},[31672],[34,654024,654026,654029,654032,654035,654038],{"className":654025},[31676],[34,654027],{"className":654028,"style":32527},[31680],[34,654030,32936],{"className":654031},[31685,31686],[34,654033],{"className":654034,"style":31846},[31845],[34,654036,202],{"className":654037},[31850],[34,654039],{"className":654040,"style":31846},[31845],[34,654042,654044,654047,654050,654053,654056,654059,654062,654065,654068,654071,654074,654077],{"className":654043},[31676],[34,654045],{"className":654046,"style":223552},[31680],[34,654048,111],{"className":654049},[31685,31686],[34,654051],{"className":654052,"style":32123},[31845],[34,654054,547860],{"className":654055},[32767],[34,654057],{"className":654058,"style":32123},[31845],[34,654060,33249],{"className":654061},[31685,31686],[34,654063,31960],{"className":654064},[32119],[34,654066],{"className":654067,"style":32123},[31845],[34,654069,67764],{"className":654070,"style":32062},[31685,31686],[34,654072],{"className":654073,"style":31846},[31845],[34,654075,202],{"className":654076},[31850],[34,654078],{"className":654079,"style":31846},[31845],[34,654081,654083,654086,654089,654092,654095,654098],{"className":654082},[31676],[34,654084],{"className":654085,"style":31681},[31680],[34,654087,1697],{"className":654088},[31685,31686],[34,654090],{"className":654091,"style":32123},[31845],[34,654093,547853],{"className":654094},[32767],[34,654096],{"className":654097,"style":32123},[31845],[34,654099,33249],{"className":654100},[31685,31686],[108,654102,654103,347],{},[34,654104,654106,654170],{"className":654105},[31642],[34,654107,654109],{"className":654108},[31646],[31648,654110,654111],{"xmlns":31650},[31652,654112,654113,654167],{},[31655,654114,654115,654117,654119,654121,654163,654165],{},[31658,654116,46900],{},[31658,654118,31747],{},[31662,654120,202],{},[41463,654122,654123],{},[31655,654124,654125,654131,654141,654143,654145,654151,654161],{},[37624,654126,654127,654129],{},[31658,654128,111],{},[31749,654130,768],{},[37624,654132,654133,654139],{},[31655,654134,654135,654137],{},[31658,654136,547853],{},[31662,654138,32682],{},[31749,654140,768],{},[31658,654142,33249],{},[31662,654144,223],{},[37624,654146,654147,654149],{},[31658,654148,1697],{},[31749,654150,768],{},[37624,654152,654153,654159],{},[31655,654154,654155,654157],{},[31658,654156,547860],{},[31662,654158,32682],{},[31749,654160,768],{},[31658,654162,33249],{},[31658,654164,46900],{},[31658,654166,33249],{},[31665,654168,654169],{"encoding":31667},"ds = \\sqrt{a^2 \\sin^2 t + b^2 \\cos^2 t} dt",[34,654171,654173,654194],{"className":654172,"ariaHidden":758},[31672],[34,654174,654176,654179,654182,654185,654188,654191],{"className":654175},[31676],[34,654177],{"className":654178,"style":31681},[31680],[34,654180,46900],{"className":654181},[31685,31686],[34,654183,31747],{"className":654184},[31685,31686],[34,654186],{"className":654187,"style":31846},[31845],[34,654189,202],{"className":654190},[31850],[34,654192],{"className":654193,"style":31846},[31845],[34,654195,654197,654201,654391,654394],{"className":654196},[31676],[34,654198],{"className":654199,"style":654200},[31680],"height:1.24em;vertical-align:-0.1607em;",[34,654202,654204],{"className":654203},[31685,41550],[34,654205,654207,654382],{"className":654206},[31796,31797],[34,654208,654210,654379],{"className":654209},[31801],[34,654211,654214,654366],{"className":654212,"style":654213},[31805],"height:1.0793em;",[34,654215,654217,654220],{"className":654216,"style":590744},[41564],[34,654218],{"className":654219,"style":552795},[31813],[34,654221,654223,654252,654255,654284,654287,654290,654293,654296,654299,654328,654331,654360,654363],{"className":654222,"style":590751},[31685],[34,654224,654226,654229],{"className":654225},[31685],[34,654227,111],{"className":654228},[31685,31686],[34,654230,654232],{"className":654231},[31792],[34,654233,654235],{"className":654234},[31796],[34,654236,654238],{"className":654237},[31801],[34,654239,654241],{"className":654240,"style":590807},[31805],[34,654242,654243,654246],{"style":590776},[34,654244],{"className":654245,"style":31814},[31813],[34,654247,654249],{"className":654248},[31818,31819,31820,31821],[34,654250,768],{"className":654251},[31685,31821],[34,654253],{"className":654254,"style":32123},[31845],[34,654256,654258,654261],{"className":654257},[32767],[34,654259,547853],{"className":654260},[32767],[34,654262,654264],{"className":654263},[31792],[34,654265,654267],{"className":654266},[31796],[34,654268,654270],{"className":654269},[31801],[34,654271,654273],{"className":654272,"style":585884},[31805],[34,654274,654275,654278],{"style":586008},[34,654276],{"className":654277,"style":31814},[31813],[34,654279,654281],{"className":654280},[31818,31819,31820,31821],[34,654282,768],{"className":654283},[31685,31821],[34,654285],{"className":654286,"style":32123},[31845],[34,654288,33249],{"className":654289},[31685,31686],[34,654291],{"className":654292,"style":32604},[31845],[34,654294,223],{"className":654295},[32024],[34,654297],{"className":654298,"style":32604},[31845],[34,654300,654302,654305],{"className":654301},[31685],[34,654303,1697],{"className":654304},[31685,31686],[34,654306,654308],{"className":654307},[31792],[34,654309,654311],{"className":654310},[31796],[34,654312,654314],{"className":654313},[31801],[34,654315,654317],{"className":654316,"style":590807},[31805],[34,654318,654319,654322],{"style":590776},[34,654320],{"className":654321,"style":31814},[31813],[34,654323,654325],{"className":654324},[31818,31819,31820,31821],[34,654326,768],{"className":654327},[31685,31821],[34,654329],{"className":654330,"style":32123},[31845],[34,654332,654334,654337],{"className":654333},[32767],[34,654335,547860],{"className":654336},[32767],[34,654338,654340],{"className":654339},[31792],[34,654341,654343],{"className":654342},[31796],[34,654344,654346],{"className":654345},[31801],[34,654347,654349],{"className":654348,"style":590807},[31805],[34,654350,654351,654354],{"style":590776},[34,654352],{"className":654353,"style":31814},[31813],[34,654355,654357],{"className":654356},[31818,31819,31820,31821],[34,654358,768],{"className":654359},[31685,31821],[34,654361],{"className":654362,"style":32123},[31845],[34,654364,33249],{"className":654365},[31685,31686],[34,654367,654369,654372],{"style":654368},"top:-3.0393em;",[34,654370],{"className":654371,"style":552795},[31813],[34,654373,654375],{"className":654374,"style":590901},[41585],[41588,654376,654377],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,654378],{"d":590908},[34,654380,31829],{"className":654381},[31828],[34,654383,654385],{"className":654384},[31801],[34,654386,654389],{"className":654387,"style":654388},[31805],"height:0.1607em;",[34,654390],{},[34,654392,46900],{"className":654393},[31685,31686],[34,654395,33249],{"className":654396},[31685,31686],[108,654398,654399,654400,347],{},"质量 ",[34,654401,654403,654489],{"className":654402},[31642],[34,654404,654406],{"className":654405},[31646],[31648,654407,654408],{"xmlns":31650},[31652,654409,654410,654486],{},[31655,654411,654412,654414,654416,654428,654430,654432,654434,654436,654438,654440,654482,654484],{},[31658,654413,98360],{},[31662,654415,202],{},[38846,654417,654418,654420,654422],{},[31662,654419,546295],{},[31749,654421,459],{},[31655,654423,654424,654426],{},[31749,654425,768],{},[31658,654427,58442],{},[31658,654429,162086],{"mathvariant":38079},[31658,654431,1697],{},[31658,654433,547853],{},[31662,654435,32682],{},[31658,654437,33249],{},[31658,654439,162086],{"mathvariant":38079},[41463,654441,654442],{},[31655,654443,654444,654450,654460,654462,654464,654470,654480],{},[37624,654445,654446,654448],{},[31658,654447,111],{},[31749,654449,768],{},[37624,654451,654452,654458],{},[31655,654453,654454,654456],{},[31658,654455,547853],{},[31662,654457,32682],{},[31749,654459,768],{},[31658,654461,33249],{},[31662,654463,223],{},[37624,654465,654466,654468],{},[31658,654467,1697],{},[31749,654469,768],{},[37624,654471,654472,654478],{},[31655,654473,654474,654476],{},[31658,654475,547860],{},[31662,654477,32682],{},[31749,654479,768],{},[31658,654481,33249],{},[31658,654483,46900],{},[31658,654485,33249],{},[31665,654487,654488],{"encoding":31667},"M = \\int_0^{2\\pi} |b \\sin t| \\sqrt{a^2 \\sin^2 t + b^2 \\cos^2 t} dt",[34,654490,654492,654510],{"className":654491,"ariaHidden":758},[31672],[34,654493,654495,654498,654501,654504,654507],{"className":654494},[31676],[34,654496],{"className":654497,"style":33218},[31680],[34,654499,98360],{"className":654500,"style":33389},[31685,31686],[34,654502],{"className":654503,"style":31846},[31845],[34,654505,202],{"className":654506},[31850],[34,654508],{"className":654509,"style":31846},[31845],[34,654511,654513,654517,654574,654577,654580,654583,654586,654589,654592,654595,654598,654785,654788],{"className":654512},[31676],[34,654514],{"className":654515,"style":654516},[31680],"height:1.4351em;vertical-align:-0.3558em;",[34,654518,654520,654523],{"className":654519},[32767],[34,654521,546295],{"className":654522,"style":546331},[32767,131249,156560],[34,654524,654526],{"className":654525},[31792],[34,654527,654529,654566],{"className":654528},[31796,31797],[34,654530,654532,654563],{"className":654531},[31801],[34,654533,654535,654546],{"className":654534,"style":563328},[31805],[34,654536,654537,654540],{"style":546347},[34,654538],{"className":654539,"style":31814},[31813],[34,654541,654543],{"className":654542},[31818,31819,31820,31821],[34,654544,459],{"className":654545},[31685,31821],[34,654547,654548,654551],{"style":546362},[34,654549],{"className":654550,"style":31814},[31813],[34,654552,654554],{"className":654553},[31818,31819,31820,31821],[34,654555,654557,654560],{"className":654556},[31685,31821],[34,654558,768],{"className":654559},[31685,31821],[34,654561,58442],{"className":654562,"style":32062},[31685,31686,31821],[34,654564,31829],{"className":654565},[31828],[34,654567,654569],{"className":654568},[31801],[34,654570,654572],{"className":654571,"style":546384},[31805],[34,654573],{},[34,654575],{"className":654576,"style":32123},[31845],[34,654578,162086],{"className":654579},[31685],[34,654581,1697],{"className":654582},[31685,31686],[34,654584],{"className":654585,"style":32123},[31845],[34,654587,547853],{"className":654588},[32767],[34,654590],{"className":654591,"style":32123},[31845],[34,654593,33249],{"className":654594},[31685,31686],[34,654596,162086],{"className":654597},[31685],[34,654599,654601],{"className":654600},[31685,41550],[34,654602,654604,654777],{"className":654603},[31796,31797],[34,654605,654607,654774],{"className":654606},[31801],[34,654608,654610,654762],{"className":654609,"style":654213},[31805],[34,654611,654613,654616],{"className":654612,"style":590744},[41564],[34,654614],{"className":654615,"style":552795},[31813],[34,654617,654619,654648,654651,654680,654683,654686,654689,654692,654695,654724,654727,654756,654759],{"className":654618,"style":590751},[31685],[34,654620,654622,654625],{"className":654621},[31685],[34,654623,111],{"className":654624},[31685,31686],[34,654626,654628],{"className":654627},[31792],[34,654629,654631],{"className":654630},[31796],[34,654632,654634],{"className":654633},[31801],[34,654635,654637],{"className":654636,"style":590807},[31805],[34,654638,654639,654642],{"style":590776},[34,654640],{"className":654641,"style":31814},[31813],[34,654643,654645],{"className":654644},[31818,31819,31820,31821],[34,654646,768],{"className":654647},[31685,31821],[34,654649],{"className":654650,"style":32123},[31845],[34,654652,654654,654657],{"className":654653},[32767],[34,654655,547853],{"className":654656},[32767],[34,654658,654660],{"className":654659},[31792],[34,654661,654663],{"className":654662},[31796],[34,654664,654666],{"className":654665},[31801],[34,654667,654669],{"className":654668,"style":585884},[31805],[34,654670,654671,654674],{"style":586008},[34,654672],{"className":654673,"style":31814},[31813],[34,654675,654677],{"className":654676},[31818,31819,31820,31821],[34,654678,768],{"className":654679},[31685,31821],[34,654681],{"className":654682,"style":32123},[31845],[34,654684,33249],{"className":654685},[31685,31686],[34,654687],{"className":654688,"style":32604},[31845],[34,654690,223],{"className":654691},[32024],[34,654693],{"className":654694,"style":32604},[31845],[34,654696,654698,654701],{"className":654697},[31685],[34,654699,1697],{"className":654700},[31685,31686],[34,654702,654704],{"className":654703},[31792],[34,654705,654707],{"className":654706},[31796],[34,654708,654710],{"className":654709},[31801],[34,654711,654713],{"className":654712,"style":590807},[31805],[34,654714,654715,654718],{"style":590776},[34,654716],{"className":654717,"style":31814},[31813],[34,654719,654721],{"className":654720},[31818,31819,31820,31821],[34,654722,768],{"className":654723},[31685,31821],[34,654725],{"className":654726,"style":32123},[31845],[34,654728,654730,654733],{"className":654729},[32767],[34,654731,547860],{"className":654732},[32767],[34,654734,654736],{"className":654735},[31792],[34,654737,654739],{"className":654738},[31796],[34,654740,654742],{"className":654741},[31801],[34,654743,654745],{"className":654744,"style":590807},[31805],[34,654746,654747,654750],{"style":590776},[34,654748],{"className":654749,"style":31814},[31813],[34,654751,654753],{"className":654752},[31818,31819,31820,31821],[34,654754,768],{"className":654755},[31685,31821],[34,654757],{"className":654758,"style":32123},[31845],[34,654760,33249],{"className":654761},[31685,31686],[34,654763,654764,654767],{"style":654368},[34,654765],{"className":654766,"style":552795},[31813],[34,654768,654770],{"className":654769,"style":590901},[41585],[41588,654771,654772],{"xmlns":41590,"width":41591,"height":590904,"viewBox":590905,"preserveAspectRatio":41594},[20398,654773],{"d":590908},[34,654775,31829],{"className":654776},[31828],[34,654778,654780],{"className":654779},[31801],[34,654781,654783],{"className":654782,"style":654388},[31805],[34,654784],{},[34,654786,46900],{"className":654787},[31685,31686],[34,654789,33249],{"className":654790},[31685,31686],[108,654792,654793,654794,641757],{},"利用对称性，上下左右对称，算第一象限 (",[34,654795,654797,654819],{"className":654796},[31642],[34,654798,654800],{"className":654799},[31646],[31648,654801,654802],{"xmlns":31650},[31652,654803,654804,654816],{},[31655,654805,654806,654808,654810,654812,654814],{},[31749,654807,459],{},[31662,654809,33239],{},[31658,654811,58442],{},[31658,654813,38080],{"mathvariant":38079},[31749,654815,768],{},[31665,654817,654818],{"encoding":31667},"0 \\to \\pi\u002F2",[34,654820,654822,654840],{"className":654821,"ariaHidden":758},[31672],[34,654823,654825,654828,654831,654834,654837],{"className":654824},[31676],[34,654826],{"className":654827,"style":32617},[31680],[34,654829,459],{"className":654830},[31685],[34,654832],{"className":654833,"style":31846},[31845],[34,654835,33239],{"className":654836},[31850],[34,654838],{"className":654839,"style":31846},[31845],[34,654841,654843,654846,654849],{"className":654842},[31676],[34,654844],{"className":654845,"style":31774},[31680],[34,654847,58442],{"className":654848,"style":32062},[31685,31686],[34,654850,38149],{"className":654851},[31685],[108,654853,654854,347],{},[34,654855,654857,654943],{"className":654856},[31642],[34,654858,654860],{"className":654859},[31646],[31648,654861,654862],{"xmlns":31650},[31652,654863,654864,654940],{},[31655,654865,654866,654868,654870,654872,654886,654888,654890,654892,654894,654936,654938],{},[31658,654867,98360],{},[31662,654869,202],{},[31749,654871,15932],{},[38846,654873,654874,654876,654878],{},[31662,654875,546295],{},[31749,654877,459],{},[31655,654879,654880,654882,654884],{},[31658,654881,58442],{},[31658,654883,38080],{"mathvariant":38079},[31749,654885,768],{},[31658,654887,1697],{},[31658,654889,547853],{},[31662,654891,32682],{},[31658,654893,33249],{},[41463,654895,654896],{},[31655,654897,654898,654904,654914,654916,654918,654924,654934],{},[37624,654899,654900,654902],{},[31658,654901,111],{},[31749,654903,768],{},[37624,654905,654906,654912],{},[31655,654907,654908,654910],{},[31658,654909,547853],{},[31662,654911,32682],{},[31749,654913,768],{},[31658,654915,33249],{},[31662,654917,223],{},[37624,654919,654920,654922],{},[31658,654921,1697],{},[31749,654923,768],{},[37624,654925,654926,654932],{},[31655,654927,654928,654930],{},[31658,654929,547860],{},[31662,654931,32682],{},[31749,654933,768],{},[31658,654935,33249],{},[31658,654937,46900],{},[31658,654939,33249],{},[31665,654941,654942],{"encoding":31667},"M = 4 \\int_0^{\\pi\u002F2} b \\sin t \\sqrt{a^2 \\sin^2 t + b^2 \\cos^2 t} dt",[34,654944,654946,654964],{"className":654945,"ariaHidden":758},[31672],[34,654947,654949,654952,654955,654958,654961],{"className":654948},[31676],[34,654950],{"className":654951,"style":33218},[31680],[34,654953,98360],{"className":654954,"style":33389},[31685,31686],[34,654956],{"className":654957,"style":31846},[31845],[34,654959,202],{"className":654960},[31850],[34,654962],{"className":654963,"style":31846},[31845],[34,654965,654967,654970,654973,654976,655033,655036,655039,655042,655045,655048,655051,655238,655241],{"className":654966},[31676],[34,654968],{"className":654969,"style":596892},[31680],[34,654971,15932],{"className":654972},[31685],[34,654974],{"className":654975,"style":32123},[31845],[34,654977,654979,654982],{"className":654978},[32767],[34,654980,546295],{"className":654981,"style":546331},[32767,131249,156560],[34,654983,654985],{"className":654984},[31792],[34,654986,654988,655025],{"className":654987},[31796,31797],[34,654989,654991,655022],{"className":654990},[31801],[34,654992,654994,655005],{"className":654993,"style":586480},[31805],[34,654995,654996,654999],{"style":546347},[34,654997],{"className":654998,"style":31814},[31813],[34,655000,655002],{"className":655001},[31818,31819,31820,31821],[34,655003,459],{"className":655004},[31685,31821],[34,655006,655007,655010],{"style":546362},[34,655008],{"className":655009,"style":31814},[31813],[34,655011,655013],{"className":655012},[31818,31819,31820,31821],[34,655014,655016,655019],{"className":655015},[31685,31821],[34,655017,58442],{"className":655018,"style":32062},[31685,31686,31821],[34,655020,38149],{"className":655021},[31685,31821],[34,655023,31829],{"className":655024},[31828],[34,655026,655028],{"className":655027},[31801],[34,655029,655031],{"className":655030,"style":546384},[31805],[34,655032],{},[34,655034],{"className":655035,"style":32123},[31845],[34,655037,1697],{"className":655038},[31685,31686],[34,655040],{"className":655041,"style":32123},[31845],[34,655043,547853],{"className":655044},[32767],[34,655046],{"className":655047,"style":32123},[31845],[34,655049,33249],{"className":655050},[31685,31686],[34,655052,655054],{"className":655053},[31685,41550],[34,655055,655057,655230],{"className":655056},[31796,31797],[34,655058,655060,655227],{"className":655059},[31801],[34,655061,655063,655215],{"className":655062,"style":654213},[31805],[34,655064,655066,655069],{"className":655065,"style":590744},[41564],[34,655067],{"className":655068,"style":552795},[31813],[34,655070,655072,655101,655104,655133,655136,655139,655142,655145,655148,655177,655180,655209,655212],{"className":655071,"style":590751},[31685],[34,655073,655075,655078],{"className":655074},[31685],[34,655076,111],{"className":655077},[31685,31686],[34,655079,655081],{"className":655080},[31792],[34,655082,655084],{"className":655083},[31796],[34,655085,655087],{"className":655086},[31801],[34,655088,655090],{"className":655089,"style":590807},[31805],[34,655091,655092,655095],{"style":590776},[34,655093],{"className":655094,"style":31814},[31813],[34,655096,655098],{"className":655097},[31818,31819,31820,31821],[34,655099,768],{"className":655100},[31685,31821],[34,655102],{"className":655103,"style":32123},[31845],[34,655105,655107,655110],{"className":655106},[32767],[34,655108,547853],{"className":655109},[32767],[34,655111,655113],{"className":655112},[31792],[34,655114,655116],{"className":655115},[31796],[34,655117,655119],{"className":655118},[31801],[34,655120,655122],{"className":655121,"style":585884},[31805],[34,655123,655124,655127],{"style":586008},[34,655125],{"className":655126,"style":31814},[31813],[34,655128,655130],{"className":655129},[31818,31819,31820,31821],[34,655131,768],{"className":655132},[31685,31821],[34,655134],{"className":655135,"style":32123},[31845],[34,655137,33249],{"className":655138},[31685,31686],[34,655140],{"className":655141,"style":32604},[31845],[34,655143,223],{"className":655144},[32024],[34,655146],{"className":655147,"style":32604},[31845],[34,655149,655151,655154],{"className":655150},[31685],[34,655152,1697],{"className":655153},[31685,31686],[34,655155,655157],{"className":655156},[31792],[34,655158,655160],{"className":655159},[31796],[34,655161,655163],{"className":655162},[31801],[34,655164,655166],{"className":655165,"style":590807},[31805],[34,655167,655168,655171],{"style":590776},[34,655169],{"className":655170,"style":31814},[31813],[34,655172,655174],{"className":655173},[31818,31819,31820,31821],[34,655175,768],{"className":655176},[31685,31821],[34,655178],{"className":655179,"style":32123},[31845],[34,655181,655183,655186],{"className":655182},[32767],[34,655184,547860],{"className":655185},[32767],[34,655187,655189],{"className":655188},[31792],[34,655190,655192],{"className":655191},[31796],[34,655193,655195],{"className":655194},[31801],[34,655196,655198],{"className":655197,"style":590807},[31805],[34,655199,655200,655203],{"style":590776},[34,655201],{"className":655202,"styl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a^2-b^2",[34,656311,656313,656357,656401],{"className":656312,"ariaHidden":758},[31672],[34,656314,656316,656319,656348,656351,656354],{"className":656315},[31676],[34,656317],{"className":656318,"style":37774},[31680],[34,656320,656322,656325],{"className":656321},[31685],[34,656323,31965],{"className":656324},[31685,31686],[34,656326,656328],{"className":656327},[31792],[34,656329,656331],{"className":656330},[31796],[34,656332,656334],{"className":656333},[31801],[34,656335,656337],{"className":656336,"style":37774},[31805],[34,656338,656339,656342],{"style":37777},[34,656340],{"className":656341,"style":31814},[31813],[34,656343,656345],{"className":656344},[31818,31819,31820,31821],[34,656346,768],{"className":656347},[31685,31821],[34,656349],{"className":656350,"style":31846},[31845],[34,656352,202],{"className":656353},[31850],[34,656355],{"className":656356,"style":31846},[31845],[34,656358,656360,656363,656392,656395,656398],{"className":656359},[31676],[34,656361],{"className":656362,"style":40044},[31680],[34,656364,656366,656369],{"className":656365},[31685],[34,656367,111],{"className":656368},[31685,31686],[34,656370,656372],{"className":656371},[31792],[34,656373,656375],{"className":656374},[31796],[34,656376,656378],{"className":656377},[31801],[34,656379,656381],{"className":656380,"style":37774},[31805],[34,656382,656383,656386],{"style":37777},[34,656384],{"className":656385,"style":31814},[31813],[34,656387,656389],{"className":656388},[31818,31819,31820,31821],[34,656390,768],{"className":656391},[31685,31821],[34,656393],{"className":656394,"style":32604},[31845],[34,656396,32226],{"className":656397},[32024],[34,656399],{"className":656400,"style":32604},[31845],[34,656402,656404,656407],{"className":656403},[31676],[34,656405],{"className":656406,"style":37774},[31680],[34,656408,656410,656413],{"className":656409},[31685],[34,656411,1697],{"className":656412},[31685,31686],[34,656414,656416],{"className":656415},[31792],[34,656417,656419],{"className":656418},[31796],[34,656420,656422],{"className":656421},[31801],[34,656423,656425],{"className":656424,"style":37774},[31805],[34,656426,656427,656430],{"style":37777},[34,656428],{"className":656429,"style":31814},[31813],[34,656431,656433],{"className":656432},[31818,31819,31820,31821],[34,656434,768],{"className":656435},[31685,31821]," (若 ",[34,656438,656440,656458],{"className":656439},[31642],[34,656441,656443],{"className":656442},[31646],[31648,656444,656445],{"xmlns":31650},[31652,656446,656447,656455],{},[31655,656448,656449,656451,656453],{},[31658,656450,111],{},[31662,656452,29622],{},[31658,656454,1697],{},[31665,656456,656457],{"encoding":31667},"a>b",[34,656459,656461,656479],{"className":656460,"ariaHidden":758},[31672],[34,656462,656464,656467,656470,656473,656476],{"className":656463},[31676],[34,656465],{"className":656466,"style":34812},[31680],[34,656468,111],{"className":656469},[31685,31686],[34,656471],{"className":656472,"style":31846},[31845],[34,656474,29622],{"className":656475},[31850],[34,656477],{"className":656478,"style":31846},[31845],[34,656480,656482,656485],{"className":656481},[31676],[34,656483],{"className":656484,"style":31681},[31680],[34,656486,1697],{"className":656487},[31685,31686],")。积分形式 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du",[34,656535,656537],{"className":656536,"ariaHidden":758},[31672],[34,656538,656540,656544,656547,656550,656690,656693],{"className":656539},[31676],[34,656541],{"className":656542,"style":656543},[31680],"height:1.2195em;vertical-align:-0.3061em;",[34,656545,546295],{"className":656546,"style":546331},[32767,131249,156560],[34,656548],{"className":656549,"style":32123},[31845],[34,656551,656553],{"className":656552},[31685,41550],[34,656554,656556,656682],{"className":656555},[31796,31797],[34,656557,656559,656679],{"className":656558},[31801],[34,656560,656562,656667],{"className":656561,"style":591825},[31805],[34,656563,656565,656568],{"className":656564,"style":41565},[41564],[34,656566],{"className":656567,"style":41508},[31813],[34,656569,656571,656600,656603,656606,656609,656638],{"className":656570,"style":41572},[31685],[34,656572,656574,656577],{"className":656573},[31685],[34,656575,111],{"className":656576},[31685,31686],[34,656578,656580],{"className":656579},[31792],[34,656581,656583],{"className":656582},[31796],[34,656584,656586],{"className":656585},[31801],[34,656587,656589],{"className":656588,"style":590807},[31805],[34,656590,656591,656594],{"style":590776},[34,656592],{"className":656593,"style":31814},[31813],[34,656595,656597],{"className":656596},[31818,31819,31820,31821],[34,656598,768],{"className":656599},[31685,31821],[34,656601],{"className":656602,"style":32604},[31845],[34,656604,32226],{"className":656605},[32024],[34,656607],{"className":656608,"style":32604},[31845],[34,656610,656612,656615],{"className":656611},[31685],[34,656613,31965],{"className":656614},[31685,31686],[34,656616,656618],{"className":656617},[31792],[34,656619,656621],{"className":656620},[31796],[34,656622,656624],{"className":656623},[31801],[34,656625,656627],{"className":656626,"style":590807},[31805],[34,656628,656629,656632],{"style":590776},[34,656630],{"className":656631,"style":31814},[31813],[34,656633,656635],{"className":656634},[31818,31819,31820,31821],[34,656636,768],{"className":656637},[31685,31821],[34,656639,656641,656644],{"className":656640},[31685],[34,656642,94628],{"className":656643},[31685,31686],[34,656645,656647],{"className":656646},[31792],[34,656648,656650],{"className":656649},[31796],[34,656651,656653],{"className":656652},[31801],[34,656654,656656],{"className":656655,"style":590807},[31805],[34,656657,656658,656661],{"style":590776},[34,656659],{"className":656660,"style":31814},[31813],[34,656662,656664],{"className":656663},[31818,31819,31820,31821],[34,656665,768],{"className":656666},[31685,31821],[34,656668,656669,656672],{"style":591878},[34,656670],{"className":656671,"style":41508},[31813],[34,656673,656675],{"className":656674,"style":41586},[41585],[41588,656676,656677],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,656678],{"d":41597},[34,656680,31829],{"className":656681},[31828],[34,656683,656685],{"className":656684},[31801],[34,656686,656688],{"className":656687,"style":591898},[31805],[34,656689],{},[34,656691,46900],{"className":656692},[31685,31686],[34,656694,94628],{"className":656695},[31685,31686],[108,656697,656698,656699,347],{},"这是一个标准积分：",[34,656700,656702,656787],{"className":656701},[31642],[34,656703,656705],{"className":656704},[31646],[31648,656706,656707],{"xmlns":31650},[31652,656708,656709,656784],{},[31655,656710,656711,656713,656731,656733,656735,656737,656743,656761,656763,656773,656776,656778],{},[31662,656712,546295],{},[41463,656714,656715],{},[31655,656716,656717,656723,656725],{},[37624,656718,656719,656721],{},[31658,656720,54151],{},[31749,656722,768],{},[31662,656724,32226],{},[37624,656726,656727,656729],{},[31658,656728,32936],{},[31749,656730,768],{},[31658,656732,46900],{},[31658,656734,32936],{},[31662,656736,202],{},[41453,656738,656739,656741],{},[31658,656740,32936],{},[31749,656742,768],{},[41463,656744,656745],{},[31655,656746,656747,656753,656755],{},[37624,656748,656749,656751],{},[31658,656750,54151],{},[31749,656752,768],{},[31662,656754,32226],{},[37624,656756,656757,656759],{},[31658,656758,32936],{},[31749,656760,768],{},[31662,656762,223],{},[41453,656764,656765,656771],{},[37624,656766,656767,656769],{},[31658,656768,54151],{},[31749,656770,768],{},[31749,656772,768],{},[31658,656774,656775],{},"arcsin",[31662,656777,32682],{},[41453,656779,656780,656782],{},[31658,656781,32936],{},[31658,656783,54151],{},[31665,656785,656786],{"encoding":31667},"\\int \\sqrt{A^2-x^2} dx = \\frac{x}{2}\\sqrt{A^2-x^2} + \\frac{A^2}{2}\\arcsin\\frac{x}{A}",[34,656788,656790,656928,657123],{"className":656789,"ariaHidden":758},[31672],[34,656791,656793,656796,656799,656802,656913,656916,656919,656922,656925],{"className":656792},[31676],[34,656794],{"className":656795,"style":656543},[31680],[34,656797,546295],{"className":656798,"style":546331},[32767,131249,156560],[34,656800],{"className":656801,"style":32123},[31845],[34,656803,656805],{"className":656804},[31685,41550],[34,656806,656808,656905],{"className":656807},[31796,31797],[34,656809,656811,656902],{"className":656810},[31801],[34,656812,656814,656890],{"className":656813,"style":591825},[31805],[34,656815,656817,656820],{"className":656816,"style":41565},[41564],[34,656818],{"className":656819,"style":41508},[31813],[34,656821,656823,656852,656855,656858,656861],{"className":656822,"style":41572},[31685],[34,656824,656826,656829],{"className":656825},[31685],[34,656827,54151],{"className":656828},[31685,31686],[34,656830,656832],{"className":656831},[31792],[34,656833,656835],{"className":656834},[31796],[34,656836,656838],{"className":656837},[31801],[34,656839,656841],{"className":656840,"style":590807},[31805],[34,656842,656843,656846],{"style":590776},[34,656844],{"className":656845,"style":31814},[31813],[34,656847,656849],{"className":656848},[31818,31819,31820,31821],[34,656850,768],{"className":656851},[31685,31821],[34,656853],{"className":656854,"style":32604},[31845],[34,656856,32226],{"className":656857},[32024],[34,656859],{"className":656860,"style":32604},[31845],[34,656862,656864,656867],{"className":656863},[31685],[34,656865,32936],{"className":656866},[31685,31686],[34,656868,656870],{"className":656869},[31792],[34,656871,656873],{"className":656872},[31796],[34,656874,656876],{"className":656875},[31801],[34,656877,656879],{"className":656878,"style":590807},[31805],[34,656880,656881,656884],{"style":590776},[34,656882],{"className":656883,"style":31814},[31813],[34,656885,656887],{"className":656886},[31818,31819,31820,31821],[34,656888,768],{"className":656889},[31685,31821],[34,656891,656892,656895],{"style":591878},[34,656893],{"className":656894,"style":41508},[31813],[34,656896,656898],{"className":656897,"style":41586},[41585],[41588,656899,656900],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,656901],{"d":41597},[34,656903,31829],{"className":656904},[31828],[34,656906,656908],{"className":656907},[31801],[34,656909,656911],{"className":656910,"style":591898},[31805],[34,656912],{},[34,656914,46900],{"className":656915},[31685,31686],[34,656917,32936],{"className":656918},[31685,31686],[34,656920],{"className":656921,"style":31846},[31845],[34,656923,202],{"className":656924},[31850],[34,656926],{"className":656927,"style":31846},[31845],[34,656929,656931,656935,657003,657114,657117,657120],{"className":656930},[31676],[34,656932],{"className":656933,"style":656934},[31680],"height:1.2584em;vertical-align:-0.345em;",[34,656936,656938,656941,657000],{"className":656937},[31685],[34,656939],{"className":656940},[31782,41488],[34,656942,656944],{"className":656943},[41453],[34,656945,656947,656992],{"className":656946},[31796,31797],[34,656948,656950,656989],{"className":656949},[31801],[34,656951,656953,656967,656975],{"className":656952,"style":244945},[31805],[34,656954,656955,656958],{"style":41504},[34,656956],{"className":656957,"style":41508},[31813],[34,656959,656961],{"className":656960},[31818,31819,31820,31821],[34,656962,656964],{"className":656963},[31685,31821],[34,656965,768],{"className":656966},[31685,31821],[34,656968,656969,656972],{"style":41520},[34,656970],{"className":656971,"style":41508},[31813],[34,656973],{"className":656974,"style":41528},[41527],[34,656976,656977,656980],{"style":46771},[34,656978],{"className":656979,"style":41508},[31813],[34,656981,656983],{"className":656982},[31818,31819,31820,31821],[34,656984,656986],{"className":656985},[31685,31821],[34,656987,32936],{"className":656988},[31685,31686,31821],[34,656990,31829],{"className":656991},[31828],[34,656993,656995],{"className":656994},[31801],[34,656996,656998],{"className":656997,"style":41619},[31805],[34,656999],{},[34,657001],{"className":657002},[31690,41488],[34,657004,657006],{"className":657005},[31685,41550],[34,657007,657009,657106],{"className":657008},[31796,31797],[34,657010,657012,657103],{"className":657011},[31801],[34,657013,657015,657091],{"className":657014,"style":591825},[31805],[34,657016,657018,657021],{"className":657017,"style":41565},[41564],[34,657019],{"className":657020,"style":41508},[31813],[34,657022,657024,657053,657056,657059,657062],{"className":657023,"style":41572},[31685],[34,657025,657027,657030],{"className":657026},[31685],[34,657028,54151],{"className":657029},[31685,31686],[34,657031,657033],{"className":657032},[31792],[34,657034,657036],{"className":657035},[31796],[34,657037,657039],{"className":657038},[31801],[34,657040,657042],{"className":657041,"style":590807},[31805],[34,657043,657044,657047],{"style":590776},[34,657045],{"className":657046,"style":31814},[31813],[34,657048,657050],{"className":657049},[31818,31819,31820,31821],[34,657051,768],{"className":657052},[31685,31821],[34,657054],{"className":657055,"style":32604},[31845],[34,657057,32226],{"className":657058},[32024],[34,657060],{"className":657061,"style":32604},[31845],[34,657063,657065,657068],{"className":657064},[31685],[34,657066,32936],{"className":657067},[31685,31686],[34,657069,657071],{"className":657070},[31792],[34,657072,657074],{"className":657073},[31796],[34,657075,657077],{"className":657076},[31801],[34,657078,657080],{"className":657079,"style":590807},[31805],[34,657081,657082,657085],{"style":590776},[34,657083],{"className":657084,"style":31814},[31813],[34,657086,657088],{"className":657087},[31818,31819,31820,31821],[34,657089,768],{"className":657090},[31685,31821],[34,657092,657093,657096],{"style":591878},[34,657094],{"className":657095,"style":41508},[31813],[34,657097,657099],{"className":657098,"style":41586},[41585],[41588,657100,657101],{"xmlns":41590,"width":41591,"height":41592,"viewBox":41593,"preserveAspectRatio":41594},[20398,657102],{"d":41597},[34,657104,31829],{"className":657105},[31828],[34,657107,657109],{"className":657108},[31801],[34,657110,657112],{"className":657111,"style":591898},[31805],[34,657113],{},[34,657115],{"className":657116,"style":32604},[31845],[34,657118,223],{"className":657119},[32024],[34,657121],{"className":657122,"style":32604},[31845],[34,657124,657126,657129,657223,657226,657229,657232],{"className":657125},[31676],[34,657127],{"className":657128,"style":637019},[31680],[34,657130,657132,657135,657220],{"className":657131},[31685],[34,657133],{"className":657134},[31782,41488],[34,657136,657138],{"className":657137},[41453],[34,657139,657141,657212],{"className":657140},[31796,31797],[34,657142,657144,657209],{"className":657143},[31801],[34,657145,657147,657161,657169],{"className":657146,"style":603956},[31805],[34,657148,657149,657152],{"style":41504},[34,657150],{"className":657151,"style":41508},[31813],[34,657153,657155],{"className":657154},[31818,31819,31820,31821],[34,657156,657158],{"className":657157},[31685,31821],[34,657159,768],{"className":657160},[31685,31821],[34,657162,657163,657166],{"style":41520},[34,657164],{"className":657165,"style":41508},[31813],[34,657167],{"className":657168,"style":41528},[41527],[34,657170,657171,657174],{"style":46771},[34,657172],{"className":657173,"style":41508},[31813],[34,657175,657177],{"className":657176},[31818,31819,31820,31821],[34,657178,657180],{"className":657179},[31685,31821],[34,657181,657183,657186],{"className":657182},[31685,31821],[34,657184,54151],{"className":657185},[31685,31686,31821],[34,657187,657189],{"className":657188},[31792],[34,657190,657192],{"className":657191},[31796],[34,657193,657195],{"className":657194},[31801],[34,657196,657198],{"className":657197,"style":144123},[31805],[34,657199,657200,657203],{"style":552391},[34,657201],{"className":657202,"style":43406},[31813],[34,657204,657206],{"className":657205},[31818,43410,43411,31821],[34,657207,768],{"className":657208},[31685,31821],[34,657210,31829],{"className":657211},[31828],[34,657213,657215],{"className":657214},[31801],[34,657216,657218],{"className":657217,"style":41619},[31805],[34,657219],{},[34,657221],{"className":657222},[31690,41488],[34,657224],{"className":657225,"style":32123},[31845],[34,657227,656775],{"className":657228},[32767],[34,657230],{"className":657231,"style":32123},[31845],[34,657233,657235,657238,657297],{"className":657234},[31685],[34,657236],{"className":657237},[31782,41488],[34,657239,657241],{"className":657240},[41453],[34,657242,657244,657289],{"className":657243},[31796,31797],[34,657245,657247,657286],{"className":657246},[31801],[34,657248,657250,657264,657272],{"className":657249,"style":244945},[31805],[34,657251,657252,657255],{"style":41504},[34,657253],{"className":657254,"style":41508},[31813],[34,657256,657258],{"className":657257},[31818,31819,31820,31821],[34,657259,657261],{"className":657260},[31685,31821],[34,657262,54151],{"className":657263},[31685,31686,31821],[34,657265,657266,657269],{"style":41520},[34,657267],{"className":657268,"style":41508},[31813],[34,657270],{"className":657271,"style":41528},[41527],[34,657273,657274,657277],{"style":46771},[34,657275],{"className":657276,"style":41508},[31813],[34,657278,657280],{"className":657279},[31818,31819,31820,31821],[34,657281,657283],{"className":657282},[31685,31821],[34,657284,32936],{"className":657285},[31685,31686,31821],[34,657287,31829],{"className":657288},[31828],[34,657290,657292],{"className":657291},[31801],[34,657293,657295],{"className":657294,"style":41619},[31805],[34,657296],{},[34,657298],{"className":657299},[31690,41488],[108,657301,657302,657303,657353,657354,657438,657439,657470],{},"稍微复杂，但这道题如果 ",[34,657304,657306,657323],{"className":657305},[31642],[34,657307,657309],{"className":657308},[31646],[31648,657310,657311],{"xmlns":31650},[31652,657312,657313,657321],{},[31655,657314,657315,657317,657319],{},[31658,657316,111],{},[31662,657318,202],{},[31658,657320,1697],{},[31665,657322,439513],{"encoding":31667},[34,657324,657326,657344],{"className":657325,"ariaHidden":758},[31672],[34,657327,657329,657332,657335,657338,657341],{"className":657328},[31676],[34,657330],{"className":657331,"style":32527},[31680],[34,657333,111],{"className":657334},[31685,31686],[34,657336],{"className":657337,"style":31846},[31845],[34,657339,202],{"className":657340},[31850],[34,657342],{"className":657343,"style":31846},[31845],[34,657345,657347,657350],{"className":657346},[31676],[34,657348],{"className":657349,"style":31681},[31680],[34,657351,1697],{"className":657352},[31685,31686]," (圆) 就很简单。如果 ",[34,657355,657357,657375],{"className":657356},[31642],[34,657358,657360],{"className":657359},[31646],[31648,657361,657362],{"xmlns":31650},[31652,657363,657364,657372],{},[31655,657365,657366,657368,657370],{},[31658,657367,111],{},[31662,657369,54707],{"mathvariant":38079},[31658,657371,1697],{},[31665,657373,657374],{"encoding":31667},"a \\neq b",[34,657376,657378,657429],{"className":657377,"ariaHidden":758},[31672],[34,657379,657381,657384,657387,657390,657426],{"className":657380},[31676],[34,657382],{"className":657383,"style":43217},[31680],[34,657385,111],{"className":657386},[31685,31686],[34,657388],{"className":657389,"style":31846},[31845],[34,657391,657393,657420,657423],{"className":657392},[31850],[34,657394,657396],{"className":657395},[31850],[34,657397,657399],{"className":657398},[31685,54774],[34,657400,657402],{"className":657401},[54778],[34,657403,657405,657408,657417],{"className":657404},[54782],[34,657406],{"className":657407,"style":43217},[31680],[34,657409,657411],{"className":657410},[54789],[34,657412,657414],{"className":657413},[31685],[34,657415,54796],{"className":657416},[31850],[34,657418],{"className":657419},[54800],[34,657421],{"className":657422},[31845,54804],[34,657424,202],{"className":657425},[31850],[34,657427],{"className":657428,"style":31846},[31845],[34,657430,657432,657435],{"className":657431},[31676],[34,657433],{"className":657434,"style":31681},[31680],[34,657436,1697],{"className":657437},[31685,31686],"，结果含 ",[34,657440,657442,657458],{"className":657441},[31642],[34,657443,657445],{"className":657444},[31646],[31648,657446,657447],{"xmlns":31650},[31652,657448,657449,657455],{},[31655,657450,657451,657453],{},[31658,657452,656775],{},[31662,657454,32682],{},[31665,657456,657457],{"encoding":31667},"\\arcsin",[34,657459,657461],{"className":657460,"ariaHidden":758},[31672],[34,657462,657464,657467],{"className":657463},[31676],[34,657465],{"className":657466,"style":71430},[31680],[34,657468,656775],{"className":657469},[32767]," 或对数。",[108,657472,657473,657474,43434,657524,39984,658072,347],{},"计算结果（假设 ",[34,657475,657477,657494],{"className":657476},[31642],[34,657478,657480],{"className":657479},[31646],[31648,657481,657482],{"xmlns":31650},[31652,657483,657484,657492],{},[31655,657485,657486,657488,657490],{},[31658,657487,111],{},[31662,657489,29622],{},[31658,657491,1697],{},[31665,657493,656457],{"encoding":31667},[34,657495,657497,657515],{"className":657496,"ariaHidden":758},[31672],[34,657498,657500,657503,657506,657509,657512],{"className":657499},[31676],[34,657501],{"className":657502,"style":34812},[31680],[34,657504,111],{"className":657505},[31685,31686],[34,657507],{"className":657508,"style":31846},[31845],[34,657510,29622],{"className":657511},[31850],[34,657513],{"className":657514,"style":31846},[31845],[34,657516,657518,657521],{"className":657517},[31676],[34,657519],{"className":657520,"style":31681},[31680],[34,657522,1697],{"className":657523},[31685,31686],[34,657525,657527,657611],{"className":657526},[31642],[34,657528,657530],{"className":657529},[31646],[31648,657531,657532],{"xmlns":31650},[31652,657533,657534,657608],{},[31655,657535,657536,657538,657540,657542,657548,657550,657582,657584,657586],{},[31658,657537,98360],{},[31662,657539,202],{},[31749,657541,768],{},[37624,657543,657544,657546],{},[31658,657545,1697],{},[31749,657547,768],{},[31662,657549,223],{},[41453,657551,657552,657564],{},[31655,657553,657554,657556,657558],{},[31749,657555,768],{},[31658,657557,111],{},[37624,657559,657560,657562],{},[31658,657561,1697],{},[31749,657563,768],{},[41463,657565,657566],{},[31655,657567,657568,657574,657576],{},[37624,657569,657570,657572],{},[31658,657571,111],{},[31749,657573,768],{},[31662,657575,32226],{},[37624,657577,657578,657580],{},[31658,657579,1697],{},[31749,657581,768],{},[31658,657583,656775],{},[31662,657585,32682],{},[41453,657587,657588,657606],{},[41463,657589,657590],{},[31655,657591,657592,657598,657600],{},[37624,657593,657594,657596],{},[31658,657595,111],{},[31749,657597,768],{},[31662,657599,32226],{},[37624,657601,657602,657604],{},[31658,657603,1697],{},[31749,657605,768],{},[31658,657607,111],{},[31665,657609,657610],{"encoding":31667},"M = 2b^2 + \\frac{2ab^2}{\\sqrt{a^2-b^2}} \\arcsin 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作业",{"type":8,"value":658090,"toc":693465},[658091,658095,658099,658103,658352,658909,658911,658914,658917,658992,663696,663770,665044,665048,665227,665229,665233,665383,665386,665389,667790,668068,668070,668074,668757,668760,668857,669866,670087,670092,670200,673119,673226,674832,674939,677912,678480,678482,678486,678489,679039,679042,679045,679050,679190,680326,680468,681634,681775,683301,683442,684604,685559,685564,685913,685915,685918,686430,686433,686552,689544,689587,689589,689593,689988,689991,689997,693160,693165,693167,693170,693175],[11,658092,658094],{"id":658093},"_129-作业","12.9 作业",[253,658096,658098],{"id":658097},"p268-例题讲解","P268 例题讲解",[32490,658100,658102],{"id":658101},"题目例-1421","题目：例 14.2.1",[15,658104,658105,658108,658321],{},[2627,658106,658107],{},"计算积分：",[34,658109,658111,658159],{"className":658110},[31642],[34,658112,658114],{"className":658113},[31646],[31648,658115,658116],{"xmlns":31650},[31652,658117,658118,658156],{},[31655,658119,658120,658122,658124,658130,658136,658138,658140,658142,658144,658150,658152,658154],{},[31658,658121,106063],{},[31662,658123,202],{},[31743,658125,658126,658128],{},[31662,658127,546295],{},[31658,658129,33206],{},[37624,658131,658132,658134],{},[31658,658133,67764],{},[31749,658135,768],{},[39361,658137,39363],{},[31658,658139,46900],{"mathvariant":38079},[31658,658141,32936],{},[31662,658143,223],{},[37624,658145,658146,658148],{},[31658,658147,32936],{},[31749,658149,768],{},[39361,658151,39363],{},[31658,658153,46900],{"mathvariant":38079},[31658,658155,67764],{},[31665,658157,658158],{"encoding":31667},"I = \\int_L y^2 \\, \\mathrm{d}x + x^2 \\, \\mathrm{d}y",[34,658160,658162,658180,658277],{"className":658161,"ariaHidden":758},[31672],[34,658163,658165,658168,658171,658174,658177],{"className":658164},[31676],[34,658166],{"className":658167,"style":33218},[31680],[34,658169,106063],{"className":658170,"style":67482},[31685,31686],[34,658172],{"className":658173,"style":31846},[31845],[34,658175,202],{"className":658176},[31850],[34,658178],{"className":658179,"style":31846},[31845],[34,658181,658183,658187,658227,658230,658259,658262,658265,658268,658271,658274],{"className":658182},[31676],[34,658184],{"className":658185,"style":658186},[31680],"height:1.1699em;vertical-align:-0.3558em;",[34,658188,658190,658193],{"className":658189},[32767],[34,658191,546295],{"className":658192,"style":546331},[32767,131249,156560],[34,658194,658196],{"className":658195},[31792],[34,658197,658199,658219],{"className":658198},[31796,31797],[34,658200,658202,658216],{"className":658201},[31801],[34,658203,658205],{"className":658204,"style":637339},[31805],[34,658206,658207,658210],{"style":546347},[34,658208],{"className":658209,"style":31814},[31813],[34,658211,658213],{"className":658212},[31818,31819,31820,31821],[34,658214,33206],{"className":658215},[31685,31686,31821],[34,658217,31829],{"className":658218},[31828],[34,658220,658222],{"className":658221},[31801],[34,658223,658225],{"className":658224,"style":546384},[31805],[34,658226],{},[34,658228],{"className":658229,"style":32123},[31845],[34,658231,658233,658236],{"className":658232},[31685],[34,658234,67764],{"className":658235,"style":32062},[31685,31686],[34,658237,658239],{"className":658238},[31792],[34,658240,658242],{"className":658241},[31796],[34,658243,658245],{"className":658244},[31801],[34,658246,658248],{"className":658247,"style":37774},[31805],[34,658249,658250,658253],{"style":37777},[34,658251],{"className":658252,"style":31814},[31813],[34,658254,658256],{"className":658255},[31818,31819,31820,31821],[34,658257,768],{"className":658258},[31685,31821],[34,658260],{"className":658261,"style":32123},[31845],[34,658263,46900],{"className":658264},[31685,207923],[34,658266,32936],{"className":658267},[31685,31686],[34,658269],{"className":658270,"style":32604},[31845],[34,658272,223],{"className":658273},[32024],[34,658275],{"className":658276,"style":32604},[31845],[34,658278,658280,658283,658312,658315,658318],{"className":658279},[31676],[34,658281],{"className":658282,"style":315747},[31680],[34,658284,658286,658289],{"className":658285},[31685],[34,658287,32936],{"className":658288},[31685,31686],[34,658290,658292],{"className":658291},[31792],[34,658293,658295],{"className":658294},[31796],[34,658296,658298],{"className":658297},[31801],[34,658299,658301],{"className":658300,"style":37774},[31805],[34,658302,658303,658306],{"style":37777},[34,658304],{"className":658305,"style":31814},[31813],[34,658307,658309],{"className":658308},[31818,31819,31820,31821],[34,658310,768],{"className":658311},[31685,31821],[34,658313],{"className":658314,"style":32123},[31845],[34,658316,46900],{"className":658317},[31685,207923],[34,658319,67764],{"className":658320,"style":32062},[31685,31686],[2627,658322,115851,658323,658351],{},[34,658324,658326,658339],{"className":658325},[31642],[34,658327,658329],{"className":658328},[31646],[31648,658330,658331],{"xmlns":31650},[31652,658332,658333,658337],{},[31655,658334,658335],{},[31658,658336,33206],{},[31665,658338,33206],{"encoding":31667},[34,658340,658342],{"className":658341,"ariaHidden":758},[31672],[34,658343,658345,658348],{"className":658344},[31676],[34,658346],{"className":658347,"style":33218},[31680],[34,658349,33206],{"className":658350},[31685,31686]," 分为两种情况：",[2230,658353,658354,658714],{},[108,658355,658356,658429,658430,658593,658594,658651,658652,31901],{},[2627,658357,658358,658359],{},"路径 ",[34,658360,658362,658380],{"className":658361},[31642],[34,658363,658365],{"className":658364},[31646],[31648,658366,658367],{"xmlns":31650},[31652,658368,658369,658377],{},[31655,658370,658371],{},[31743,658372,658373,658375],{},[31658,658374,33206],{},[31749,658376,640],{},[31665,658378,658379],{"encoding":31667},"L_1",[34,658381,658383],{"className":658382,"ariaHidden":758},[31672],[34,658384,658386,658389],{"className":658385},[31676],[34,658387],{"className":658388,"style":70915},[31680],[34,658390,658392,658395],{"className":658391},[31685],[34,658393,33206],{"className":658394},[31685,31686],[34,658396,658398],{"className":658397},[31792],[34,658399,658401,658421],{"className":658400},[31796,31797],[34,658402,658404,658418],{"className":658403},[31801],[34,658405,658407],{"className":658406,"style":31806},[31805],[34,658408,658409,658412],{"style":31809},[34,658410],{"className":658411,"style":31814},[31813],[34,658413,658415],{"className":658414},[31818,31819,31820,31821],[34,658416,640],{"className":658417},[31685,31821],[34,658419,31829],{"className":658420},[31828],[34,658422,658424],{"className":658423},[31801],[34,658425,658427],{"className":658426,"style":31836},[31805],[34,658428],{},"：圆周 ",[34,658431,658433,658467],{"className":658432},[31642],[34,658434,658436],{"className":658435},[31646],[31648,658437,658438],{"xmlns":31650},[31652,658439,658440,658464],{},[31655,658441,658442,658448,658450,658456,658458],{},[37624,658443,658444,658446],{},[31658,658445,32936],{},[31749,658447,768],{},[31662,658449,223],{},[37624,658451,658452,658454],{},[31658,658453,67764],{},[31749,658455,768],{},[31662,658457,202],{},[37624,658459,658460,658462],{},[31658,658461,34386],{},[31749,658463,768],{},[31665,658465,658466],{"encoding":31667},"x^2 + y^2 = R^2",[34,658468,658470,658514,658558],{"className":658469,"ariaHidden":758},[31672],[34,658471,658473,658476,658505,658508,658511],{"className":658472},[31676],[34,658474],{"className":658475,"style":40044},[31680],[34,658477,658479,658482],{"className":658478},[31685],[34,658480,32936],{"className":658481},[31685,31686],[34,658483,658485],{"className":658484},[31792],[34,658486,658488],{"className":658487},[31796],[34,658489,658491],{"className":658490},[31801],[34,658492,658494],{"className":658493,"style":37774},[31805],[34,658495,658496,658499],{"style":37777},[34,658497],{"className":658498,"style":31814},[31813],[34,658500,658502],{"className":658501},[31818,31819,31820,31821],[34,658503,768],{"className":658504},[31685,31821],[34,658506],{"className":658507,"style":32604},[31845],[34,658509,223],{"className":658510},[32024],[34,658512],{"className":658513,"style":32604},[31845],[34,658515,658517,658520,658549,658552,658555],{"className":658516},[31676],[34,658518],{"className":658519,"style":315747},[31680],[34,658521,658523,658526],{"className":658522},[31685],[34,658524,67764],{"className":658525,"style":32062},[31685,31686],[34,658527,658529],{"className":658528},[31792],[34,658530,658532],{"className":658531},[31796],[34,658533,658535],{"className":658534},[31801],[34,658536,658538],{"className":658537,"style":37774},[31805],[34,658539,658540,658543],{"style":37777},[34,658541],{"className":658542,"style":31814},[31813],[34,658544,658546],{"className":658545},[31818,31819,31820,31821],[34,658547,768],{"className":658548},[31685,31821],[34,658550],{"className":658551,"style":31846},[31845],[34,658553,202],{"className":658554},[31850],[34,658556],{"className":658557,"style":31846},[31845],[34,658559,658561,658564],{"className":658560},[31676],[34,658562],{"className":658563,"style":37774},[31680],[34,658565,658567,658570],{"className":658566},[31685],[34,658568,34386],{"className":658569,"style":34427},[31685,31686],[34,658571,658573],{"className":658572},[31792],[34,658574,658576],{"className":658575},[31796],[34,658577,658579],{"className":658578},[31801],[34,658580,658582],{"className":658581,"style":37774},[31805],[34,658583,658584,658587],{"style":37777},[34,658585],{"className":658586,"style":31814},[31813],[34,658588,658590],{"className":658589},[31818,31819,31820,31821],[34,658591,768],{"className":658592},[31685,31821]," 的上半部分，方向为逆时针方向（从点 ",[34,658595,658597,658621],{"className":658596},[31642],[34,658598,658600],{"className":658599},[31646],[31648,658601,658602],{"xmlns":31650},[31652,658603,658604,658618],{},[31655,658605,658606,658608,658610,658612,658614,658616],{},[31658,658607,54151],{},[31662,658609,456],{"stretchy":702},[31658,658611,34386],{},[31662,658613,31960],{"separator":758},[31749,658615,459],{},[31662,658617,2557],{"stretchy":702},[31665,658619,658620],{"encoding":31667},"A(R, 0)",[34,658622,658624],{"className":658623,"ariaHidden":758},[31672],[34,658625,658627,658630,658633,658636,658639,658642,658645,658648],{"className":658626},[31676],[34,658628],{"className":658629,"style":31774},[31680],[34,658631,54151],{"className":658632},[31685,31686],[34,658634,456],{"className":658635},[31782],[34,658637,34386],{"className":658638,"style":34427},[31685,31686],[34,658640,31960],{"className":658641},[32119],[34,658643],{"className":658644,"style":32123},[31845],[34,658646,459],{"className":658647},[31685],[34,658649,2557],{"className":658650},[31690]," 到点 ",[34,658653,658655,658681],{"className":658654},[31642],[34,658656,658658],{"className":658657},[31646],[31648,658659,658660],{"xmlns":31650},[31652,658661,658662,658678],{},[31655,658663,658664,658666,658668,658670,658672,658674,658676],{},[31658,658665,54224],{},[31662,658667,456],{"stretchy":702},[31662,658669,32226],{},[31658,658671,34386],{},[31662,658673,31960],{"separator":758},[31749,658675,459],{},[31662,658677,2557],{"stretchy":702},[31665,658679,658680],{"encoding":31667},"B(-R, 0)",[34,658682,658684],{"className":658683,"ariaHidden":758},[31672],[34,658685,658687,658690,658693,658696,658699,658702,658705,658708,658711],{"className":658686},[31676],[34,658688],{"className":658689,"style":31774},[31680],[34,658691,54224],{"className":658692,"style":54252},[31685,31686],[34,658694,456],{"className":658695},[31782],[34,658697,32226],{"className":658698},[31685],[34,658700,34386],{"className":658701,"style":34427},[31685,31686],[34,658703,31960],{"className":658704},[32119],[34,658706],{"className":658707,"style":32123},[31845],[34,658709,459],{"className":658710},[31685],[34,658712,2557],{"className":658713},[31690],[108,658715,658716,658788,658789,658651,658846,658908],{},[2627,658717,658358,658718],{},[34,658719,658721,658739],{"className":658720},[31642],[34,658722,658724],{"className":658723},[31646],[31648,658725,658726],{"xmlns":31650},[31652,658727,658728,658736],{},[31655,658729,658730],{},[31743,658731,658732,658734],{},[31658,658733,33206],{},[31749,658735,768],{},[31665,658737,658738],{"encoding":31667},"L_2",[34,658740,658742],{"className":658741,"ariaHidden":758},[31672],[34,658743,658745,658748],{"className":658744},[31676],[34,658746],{"className":658747,"style":70915},[31680],[34,658749,658751,658754],{"className":658750},[31685],[34,658752,33206],{"className":658753},[31685,31686],[34,658755,658757],{"className":658756},[31792],[34,658758,658760,658780],{"className":658759},[31796,31797],[34,658761,658763,658777],{"className":658762},[31801],[34,658764,658766],{"className":658765,"style":31806},[31805],[34,658767,658768,658771],{"style":31809},[34,658769],{"className":658770,"style":31814},[31813],[34,658772,658774],{"className":658773},[31818,31819,31820,31821],[34,658775,768],{"className":658776},[31685,31821],[34,658778,31829],{"className":658779},[31828],[34,658781,658783],{"className":658782},[31801],[34,658784,658786],{"className":658785,"style":31836},[31805],[34,658787],{},"：从点 ",[34,658790,658792,658816],{"className":658791},[31642],[34,658793,658795],{"className":658794},[31646],[31648,658796,658797],{"xmlns":31650},[31652,658798,658799,658813],{},[31655,658800,658801,658803,658805,658807,658809,658811],{},[31658,658802,98360],{},[31662,658804,456],{"stretchy":702},[31658,658806,34386],{},[31662,658808,31960],{"separator":758},[31749,658810,459],{},[31662,658812,2557],{"stretchy":702},[31665,658814,658815],{"encoding":31667},"M(R, 0)",[34,658817,658819],{"className":658818,"ariaHidden":758},[31672],[34,658820,658822,658825,658828,658831,658834,658837,658840,658843],{"className":658821},[31676],[34,658823],{"className":658824,"style":31774},[31680],[34,658826,98360],{"className":658827,"style":33389},[31685,31686],[34,658829,456],{"className":658830},[31782],[34,658832,34386],{"className":658833,"style":34427},[31685,31686],[34,658835,31960],{"className":658836},[32119],[34,658838],{"className":658839,"style":32123},[31845],[34,658841,459],{"className":658842},[31685],[34,658844,2557],{"className":658845},[31690],[34,658847,658849,658875],{"className":658848},[31642],[34,658850,658852],{"className":658851},[31646],[31648,658853,658854],{"xmlns":31650},[31652,658855,658856,658872],{},[31655,658857,658858,658860,658862,658864,658866,658868,658870],{},[31658,658859,33347],{},[31662,658861,456],{"stretchy":702},[31662,658863,32226],{},[31658,658865,34386],{},[31662,658867,31960],{"separator":758},[31749,658869,459],{},[31662,658871,2557],{"stretchy":702},[31665,658873,658874],{"encoding":31667},"N(-R, 0)",[34,658876,658878],{"className":658877,"ariaHidden":758},[31672],[34,658879,658881,658884,658887,658890,658893,658896,658899,658902,658905],{"className":658880},[31676],[34,658882],{"className":658883,"style":31774},[31680],[34,658885,33347],{"className":658886,"style":33389},[31685,31686],[34,658888,456],{"className":658889},[31782],[34,658891,32226],{"className":658892},[31685],[34,658894,34386],{"className":658895,"style":34427},[31685,31686],[34,658897,31960],{"className":658898},[32119],[34,658900],{"className":658901,"style":32123},[31845],[34,658903,459],{"className":658904},[31685],[34,658906,2557],{"className":658907},[31690]," 的直线段。",[32188,658910],{},[32490,658912,658913],{"id":658913},"详细解答",[15,658915,658916],{},"我们分别计算这两条路径下的积分值，以此来观察积分结果是否与路径有关。",[15,658918,658919],{},[2627,658920,658921,658922,658991],{},"情形 1：沿上半圆周 ",[34,658923,658925,658942],{"className":658924},[31642],[34,658926,658928],{"className":658927},[31646],[31648,658929,658930],{"xmlns":31650},[31652,658931,658932,658940],{},[31655,658933,658934],{},[31743,658935,658936,658938],{},[31658,658937,33206],{},[31749,658939,640],{},[31665,658941,658379],{"encoding":31667},[34,658943,658945],{"className":658944,"ariaHidden":758},[31672],[34,658946,658948,658951],{"className":658947},[31676],[34,658949],{"className":658950,"style":70915},[31680],[34,658952,658954,658957],{"className":658953},[31685],[34,658955,33206],{"className":658956},[31685,31686],[34,658958,658960],{"className":658959},[31792],[34,658961,658963,658983],{"className":658962},[31796,31797],[34,658964,658966,658980],{"className":658965},[31801],[34,658967,658969],{"className":658968,"style":31806},[31805],[34,658970,658971,658974],{"style":31809},[34,658972],{"className":658973,"style":31814},[31813],[34,658975,658977],{"className":658976},[31818,31819,31820,31821],[34,658978,640],{"className":658979},[31685,31821],[34,658981,31829],{"className":658982},[31828],[34,658984,658986],{"className":658985},[31801],[34,658987,658989],{"className":658988,"style":31836},[31805],[34,658990],{}," 积分",[2230,658993,658994,659484,659973,661893],{},[108,658995,658996,658999,659000,347,659028],{},[2627,658997,658998],{},"建立参数方程","\n对于圆心在原点的圆，最常用的参数是极坐标角度 ",[34,659001,659003,659016],{"className":659002},[31642],[34,659004,659006],{"className":659005},[31646],[31648,659007,659008],{"xmlns":31650},[31652,659009,659010,659014],{},[31655,659011,659012],{},[31658,659013,479697],{},[31665,659015,481351],{"encoding":31667},[34,659017,659019],{"className":659018,"ariaHidden":758},[31672],[34,659020,659022,659025],{"className":659021},[31676],[34,659023],{"className":659024,"style":31681},[31680],[34,659026,479697],{"className":659027,"style":34452},[31685,31686],[105,659029,659030,659101,659266,659373],{},[108,659031,659032],{},[34,659033,659035,659059],{"className":659034},[31642],[34,659036,659038],{"className":659037},[31646],[31648,659039,659040],{"xmlns":31650},[31652,659041,659042,659056],{},[31655,659043,659044,659046,659048,659050,659052,659054],{},[31658,659045,32936],{},[31662,659047,202],{},[31658,659049,34386],{},[31658,659051,547860],{},[31662,659053,32682],{},[31658,659055,479697],{},[31665,659057,659058],{"encoding":31667},"x = R \\cos \\theta",[34,659060,659062,659080],{"className":659061,"ariaHidden":758},[31672],[34,659063,659065,659068,659071,659074,659077],{"className":659064},[31676],[34,659066],{"className":659067,"style":32527},[31680],[34,659069,32936],{"className":659070},[31685,31686],[34,659072],{"className":659073,"style":31846},[31845],[34,659075,202],{"className":659076},[31850],[34,659078],{"className":659079,"style":31846},[31845],[34,659081,659083,659086,659089,659092,659095,659098],{"className":659082},[31676],[34,659084],{"className":659085,"style":31681},[31680],[34,659087,34386],{"className":659088,"style":34427},[31685,31686],[34,659090],{"className":659091,"style":32123},[31845],[34,659093,547860],{"className":659094},[32767],[34,659096],{"className":659097,"style":32123},[31845],[34,659099,479697],{"className":659100,"style":34452},[31685,31686],[108,659102,659103,659172,659173,659176,659177,659180,659181,659209,659210,53834,659238,347],{},[34,659104,659106,659130],{"className":659105},[31642],[34,659107,659109],{"className":659108},[31646],[31648,659110,659111],{"xmlns":31650},[31652,659112,659113,659127],{},[31655,659114,659115,659117,659119,659121,659123,659125],{},[31658,659116,67764],{},[31662,659118,202],{},[31658,659120,34386],{},[31658,659122,547853],{},[31662,659124,32682],{},[31658,659126,479697],{},[31665,659128,659129],{"encoding":31667},"y = R \\sin \\theta",[34,659131,659133,659151],{"className":659132,"ariaHidden":758},[31672],[34,659134,659136,659139,659142,659145,659148],{"className":659135},[31676],[34,659137],{"className":659138,"style":38774},[31680],[34,659140,67764],{"className":659141,"style":32062},[31685,31686],[34,659143],{"className":659144,"style":31846},[31845],[34,659146,202],{"className":659147},[31850],[34,659149],{"className":659150,"style":31846},[31845],[34,659152,659154,659157,659160,659163,659166,659169],{"className":659153},[31676],[34,659155],{"className":659156,"style":31681},[31680],[34,659158,34386],{"className":659159,"style":34427},[31685,31686],[34,659161],{"className":659162,"style":32123},[31845],[34,659164,547853],{"className":659165},[32767],[34,659167],{"className":659168,"style":32123},[31845],[34,659170,479697],{"className":659171,"style":34452},[31685,31686],"\n由于是",[2627,659174,659175],{},"上半圆","，且方向是",[2627,659178,659179],{},"逆时针","（从右向左），角度 ",[34,659182,659184,659197],{"className":659183},[31642],[34,659185,659187],{"className":659186},[31646],[31648,659188,659189],{"xmlns":31650},[31652,659190,659191,659195],{},[31655,659192,659193],{},[31658,659194,479697],{},[31665,659196,481351],{"encoding":31667},[34,659198,659200],{"className":659199,"ariaHidden":758},[31672],[34,659201,659203,659206],{"className":659202},[31676],[34,659204],{"className":659205,"style":31681},[31680],[34,659207,479697],{"className":659208,"style":34452},[31685,31686]," 的变化范围是从 ",[34,659211,659213,659226],{"className":659212},[31642],[34,659214,659216],{"className":659215},[31646],[31648,659217,659218],{"xmlns":31650},[31652,659219,659220,659224],{},[31655,659221,659222],{},[31749,659223,459],{},[31665,659225,459],{"encoding":31667},[34,659227,659229],{"className":659228,"ariaHidden":758},[31672],[34,659230,659232,659235],{"className":659231},[31676],[34,659233],{"className":659234,"style":32617},[31680],[34,659236,459],{"className":659237},[31685],[34,659239,659241,659254],{"className":659240},[31642],[34,659242,659244],{"className":659243},[31646],[31648,659245,659246],{"xmlns":31650},[31652,659247,659248,659252],{},[31655,659249,659250],{},[31658,659251,58442],{},[31665,659253,58445],{"encoding":31667},[34,659255,659257],{"className":659256,"ariaHidden":758},[31672],[34,659258,659260,659263],{"className":659259},[31676],[34,659261],{"className":659262,"style":32527},[31680],[34,659264,58442],{"className":659265,"style":32062},[31685,31686],[108,659267,659268,659269,659321,659322,347],{},"起点 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全部代入原式：",[34,660058,660060],{"className":660059},[126385],[34,660061,660063,660409],{"className":660062},[31642],[34,660064,660066],{"className":660065},[31646],[31648,660067,660068],{"xmlns":31650,"display":126394},[31652,660069,660070,660406],{},[71078,660071,660072,660244,660320],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,660073,660074,660084],{},[71087,660075,660076],{},[71090,660077,660078],{"scriptlevel":459,"displaystyle":758},[31743,660079,660080,660082],{},[31658,660081,106063],{},[31749,660083,640],{},[71087,660085,660086],{},[71090,660087,660088],{"scriptlevel":459,"displaystyle":758},[31655,660089,660090,660092,660094,660102,660240,660242],{},[31655,660091],{},[31662,660093,202],{},[38846,660095,660096,660098,660100],{},[31662,660097,546295],{},[31749,660099,459],{},[31658,660101,58442],{},[31655,660103,660104,660106,660136,660138,660172,660174,660204,660206,660238],{},[31662,660105,15972],{"fence":758},[71115,660107,660108,660130],{},[71115,660109,660110,660128],{},[31655,660111,660112,660114,660116,660118,660120,660122],{},[31662,660113,456],{"stretchy":702},[31658,660115,34386],{},[31658,660117,547853],{},[31662,660119,32682],{},[31658,660121,479697],{},[37624,660123,660124,660126],{},[31662,660125,2557],{"stretchy":702},[31749,660127,768],{},[31662,660129,477736],{"stretchy":758},[37624,660131,660132,660134],{},[31658,660133,67764],{},[31749,660135,768],{},[31662,660137,71191],{},[71115,660139,660140,660160],{},[71115,660141,660142,660158],{},[31655,660143,660144,660146,660148,660150,660152,660154,660156],{},[31662,660145,456],{"stretchy":702},[31662,660147,32226],{},[31658,660149,34386],{},[31658,660151,547853],{},[31662,660153,32682],{},[31658,660155,479697],{},[31662,660157,2557],{"stretchy":702},[31662,660159,477736],{"stretchy":758},[31655,660161,660162,660164,660166,660168,660170],{},[31658,660163,46900],{},[31658,660165,32936],{},[31658,660167,38080],{"mathvariant":38079},[31658,660169,46900],{"mathvariant":38079},[31658,660171,479697],{},[31662,660173,223],{},[71115,660175,660176,660198],{},[71115,660177,660178,660196],{},[31655,660179,660180,660182,660184,660186,660188,660190],{},[31662,660181,456],{"stretchy":702},[31658,660183,34386],{},[31658,660185,547860],{},[31662,660187,32682],{},[31658,660189,479697],{},[37624,660191,660192,660194],{},[31662,660193,2557],{"stretchy":702},[31749,660195,768],{},[31662,660197,477736],{"stretchy":758},[37624,660199,660200,660202],{},[31658,660201,32936],{},[31749,660203,768],{},[31662,660205,71191],{},[71115,660207,660208,660226],{},[71115,660209,660210,660224],{},[31655,660211,660212,660214,660216,660218,660220,660222],{},[31662,660213,456],{"stretchy":702},[31658,660215,34386],{},[31658,660217,547860],{},[31662,660219,32682],{},[31658,660221,479697],{},[31662,660223,2557],{"stretchy":702},[31662,660225,477736],{"stretchy":758},[31655,660227,660228,660230,660232,660234,660236],{},[31658,660229,46900],{},[31658,660231,67764],{},[31658,660233,38080],{"mathvariant":38079},[31658,660235,46900],{"mathvariant":38079},[31658,660237,479697],{},[31662,660239,32712],{"fence":758},[31658,660241,46900],{"mathvariant":38079},[31658,660243,479697],{},[71084,660245,660246,660252],{},[71087,660247,660248],{},[71090,660249,660250],{"scriptlevel":459,"displaystyle":758},[31655,660251],{},[71087,660253,660254],{},[71090,660255,660256],{"scriptlevel":459,"displaystyle":758},[31655,660257,660258,660260,660262,660270,660316,660318],{},[31655,660259],{},[31662,660261,202],{},[38846,660263,660264,660266,660268],{},[31662,660265,546295],{},[31749,660267,459],{},[31658,660269,58442],{},[31655,660271,660272,660274,660276,660282,660292,660294,660296,660302,660312,660314],{},[31662,660273,15972],{"fence":758},[31662,660275,32226],{},[37624,660277,660278,660280],{},[31658,660279,34386],{},[31749,660281,786],{},[37624,660283,660284,660290],{},[31655,660285,660286,660288],{},[31658,660287,547853],{},[31662,660289,32682],{},[31749,660291,786],{},[31658,660293,479697],{},[31662,660295,223],{},[37624,660297,660298,660300],{},[31658,660299,34386],{},[31749,660301,786],{},[37624,660303,660304,660310],{},[31655,660305,660306,660308],{},[31658,660307,547860],{},[31662,660309,32682],{},[31749,660311,786],{},[31658,660313,479697],{},[31662,660315,32712],{"fence":758},[31658,660317,46900],{"mathvariant":38079},[31658,660319,479697],{},[71084,660321,660322,660328],{},[71087,660323,660324],{},[71090,660325,660326],{"scriptlevel":459,"displaystyle":758},[31655,660327],{},[71087,660329,660330],{},[71090,660331,660332],{"scriptlevel":459,"displaystyle":758},[31655,660333,660334,660336,660338,660344],{},[31655,660335],{},[31662,660337,202],{},[37624,660339,660340,660342],{},[31658,660341,34386],{},[31749,660343,786],{},[31655,660345,660346,660348,660350,660358,660368,660370,660372,660374,660376,660378,660386,660396,660398,660400,660402,660404],{},[31662,660347,456],{"fence":758},[31662,660349,32226],{},[38846,660351,660352,660354,660356],{},[31662,660353,546295],{},[31749,660355,459],{},[31658,660357,58442],{},[37624,660359,660360,660366],{},[31655,660361,660362,660364],{},[31658,660363,547853],{},[31662,660365,32682],{},[31749,660367,786],{},[31658,660369,479697],{},[39361,660371,39363],{},[31658,660373,46900],{"mathvariant":38079},[31658,660375,479697],{},[31662,660377,223],{},[38846,660379,660380,660382,660384],{},[31662,660381,546295],{},[31749,660383,459],{},[31658,660385,58442],{},[37624,660387,660388,660394],{},[31655,660389,660390,660392],{},[31658,660391,547860],{},[31662,660393,32682],{},[31749,660395,786],{},[31658,660397,479697],{},[39361,660399,39363],{},[31658,660401,46900],{"mathvariant":38079},[31658,660403,479697],{},[31662,660405,2557],{"fence":758},[31665,660407,660408],{"encoding":31667},"\\begin{aligned}\nI_1 &= \\int_0^\\pi \\left[ \\underbrace{(R \\sin \\theta)^2}_{y^2} \\cdot \\underbrace{(-R \\sin \\theta)}_{dx\u002F\\mathrm{d}\\theta} + \\underbrace{(R \\cos \\theta)^2}_{x^2} \\cdot \\underbrace{(R \\cos \\theta)}_{dy\u002F\\mathrm{d}\\theta} \\right] \\mathrm{d}\\theta \\\\\n&= \\int_0^\\pi \\left[ -R^3 \\sin^3 \\theta + R^3 \\cos^3 \\theta \\right] \\mathrm{d}\\theta \\\\\n&= R^3 \\left( - \\int_0^\\pi \\sin^3 \\theta \\, \\mathrm{d}\\theta + \\int_0^\\pi \\cos^3 \\theta \\, \\mathrm{d}\\theta 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},[31818,31819,31820,31821],[34,661859,786],{"className":661860},[31685,31821],[34,661862],{"className":661863,"style":32123},[31845],[34,661865,479697],{"className":661866,"style":34452},[31685,31686],[34,661868],{"className":661869,"style":32123},[31845],[34,661871,46900],{"className":661872},[31685,207923],[34,661874,479697],{"className":661875,"style":34452},[31685,31686],[34,661877,661879],{"className":661878,"style":75811},[31690,75810],[34,661880,2557],{"className":661881},[71281,31820],[34,661883,31829],{"className":661884},[31828],[34,661886,661888],{"className":661887},[31801],[34,661889,661891],{"className":661890,"style":660515},[31805],[34,661892],{},[108,661894,661895,661898,661899,663319,663321,663324],{},[2627,661896,661897],{},"计算定积分","\n我们分别计算这两个三角函数的积分：",[105,661900,661901,662961],{},[108,661902,661903,2664,661906,662055,662056,478555,662241,55016,662302,662386,662387,662390,662391,53834,662419,662447,662448,347],{},[2627,661904,661905],{},"第一部分",[34,661907,661909,661947],{"className":661908},[31642],[34,661910,661912],{"className":661911},[31646],[31648,661913,661914],{"xmlns":31650},[31652,661915,661916,661944],{},[31655,661917,661918,661926,661936,661938,661940,661942],{},[38846,661919,661920,661922,661924],{},[31662,661921,546295],{},[31749,661923,459],{},[31658,661925,58442],{},[37624,661927,661928,661934],{},[31655,661929,661930,661932],{},[31658,661931,547853],{},[31662,661933,32682],{},[31749,661935,786],{},[31658,661937,479697],{},[39361,661939,39363],{},[31658,661941,46900],{"mathvariant":38079},[31658,661943,479697],{},[31665,661945,661946],{"encoding":31667},"\\int_0^\\pi \\sin^3 \\theta \\, \\mathrm{d}\\theta",[34,661948,661950],{"className":661949,"ariaHidden":758},[31672],[34,661951,661953,661957,662008,662011,662040,662043,662046,662049,662052],{"className":661952},[31676],[34,661954],{"className":661955,"style":661956},[31680],"height:1.2277em;vertical-align:-0.3558em;",[34,661958,661960,661963],{"className":661959},[32767],[34,661961,546295],{"className":661962,"style":546331},[32767,131249,156560],[34,661964,661966],{"className":661965},[31792],[34,661967,661969,662000],{"className":661968},[31796,31797],[34,661970,661972,661997],{"className":661971},[31801],[34,661973,661975,661986],{"className":661974,"style":567928},[31805],[34,661976,661977,661980],{"style":546347},[34,661978],{"className":661979,"style":31814},[31813],[34,661981,661983],{"className":661982},[31818,31819,31820,31821],[34,661984,459],{"className":661985},[31685,31821],[34,661987,661988,661991],{"style":546362},[34,661989],{"className":661990,"style":31814},[31813],[34,661992,661994],{"className":661993},[31818,31819,31820,31821],[34,661995,58442],{"className":661996,"style":32062},[31685,31686,31821],[34,661998,31829],{"className":661999},[31828],[34,662001,662003],{"className":662002},[31801],[34,662004,662006],{"className":662005,"style":546384},[31805],[34,662007],{},[34,662009],{"className":662010,"style":32123},[31845],[34,662012,662014,662017],{"className":662013},[32767],[34,662015,547853],{"className":662016},[32767],[34,662018,662020],{"className":662019},[31792],[34,662021,662023],{"className":662022},[31796],[34,662024,662026],{"className":662025},[31801],[34,662027,662029],{"className":662028,"style":585884},[31805],[34,662030,662031,662034],{"style":586008},[34,662032],{"className":662033,"style":31814},[31813],[34,662035,662037],{"className":662036},[31818,31819,31820,31821],[34,662038,786],{"className":662039},[31685,31821],[34,662041],{"className":662042,"style":32123},[31845],[34,662044,479697],{"className":662045,"style":34452},[31685,31686],[34,662047],{"className":662048,"style":32123},[31845],[34,662050,46900],{"className":662051},[31685,207923],[34,662053,479697],{"className":662054,"style":34452},[31685,31686],"\n利用公式 ",[34,662057,662059,662111],{"className":662058},[31642],[34,662060,662062],{"className":662061},[31646],[31648,662063,662064],{"xmlns":31650},[31652,662065,662066,662108],{},[31655,662067,662068,662078,662080,662082,662084,662086,662088,662098,662100,662102,662104,662106],{},[37624,662069,662070,662076],{},[31655,662071,662072,662074],{},[31658,662073,547853],{},[31662,662075,32682],{},[31749,662077,786],{},[31658,662079,479697],{},[31662,662081,202],{},[31662,662083,456],{"stretchy":702},[31749,662085,640],{},[31662,662087,32226],{},[37624,662089,662090,662096],{},[31655,662091,662092,662094],{},[31658,662093,547860],{},[31662,662095,32682],{},[31749,662097,768],{},[31658,662099,479697],{},[31662,662101,2557],{"stretchy":702},[31658,662103,547853],{},[31662,662105,32682],{},[31658,662107,479697],{},[31665,662109,662110],{"encoding":31667},"\\sin^3 \\theta = (1-\\cos^2 \\theta)\\sin \\theta",[34,662112,662114,662164,662185],{"className":662113,"ariaHidden":758},[31672],[34,662115,662117,662120,662149,662152,662155,662158,662161],{"className":662116},[31676],[34,662118],{"className":662119,"style":585884},[31680],[34,662121,662123,662126],{"className":662122},[32767],[34,662124,547853],{"className":662125},[32767],[34,662127,662129],{"className":662128},[31792],[34,662130,662132],{"className":662131},[31796],[34,662133,662135],{"className":662134},[31801],[34,662136,662138],{"className":662137,"style":585884},[31805],[34,662139,662140,662143],{"style":586008},[34,662141],{"className":662142,"style":31814},[31813],[34,662144,662146],{"className":662145},[31818,31819,31820,31821],[34,662147,786],{"className":662148},[31685,31821],[34,662150],{"className":662151,"style":32123},[31845],[34,662153,479697],{"className":662154,"style":34452},[31685,31686],[34,662156],{"className":662157,"style":31846},[31845],[34,662159,202],{"className":662160},[31850],[34,662162],{"className":662163,"style":31846},[31845],[34,662165,662167,662170,662173,662176,662179,662182],{"className":662166},[31676],[34,662168],{"className":662169,"style":31774},[31680],[34,662171,456],{"className":662172},[31782],[34,662174,640],{"className":662175},[31685],[34,662177],{"className":662178,"style":32604},[31845],[34,662180,32226],{"className":662181},[32024],[34,662183],{"className":662184,"style":32604},[31845],[34,662186,662188,662191,662220,662223,662226,662229,662232,662235,662238],{"className":662187},[31676],[34,662189],{"className":662190,"style":37755},[31680],[34,662192,662194,662197],{"className":662193},[32767],[34,662195,547860],{"className":662196},[32767],[34,662198,662200],{"className":662199},[31792],[34,662201,662203],{"className":662202},[31796],[34,662204,662206],{"className":662205},[31801],[34,662207,662209],{"className":662208,"style":37774},[31805],[34,662210,662211,662214],{"style":37777},[34,662212],{"className":662213,"style":31814},[31813],[34,662215,662217],{"className":662216},[31818,31819,31820,31821],[34,662218,768],{"className":662219},[31685,31821],[34,662221],{"className":662222,"style":32123},[31845],[34,662224,479697],{"className":662225,"style":34452},[31685,31686],[34,662227,2557],{"className":662228},[31690],[34,662230],{"className":662231,"style":32123},[31845],[34,662233,547853],{"className":662234},[32767],[34,662236],{"className":662237,"style":32123},[31845],[34,662239,479697],{"className":662240,"style":34452},[31685,31686],[34,662242,662244,662266],{"className":662243},[31642],[34,662245,662247],{"className":662246},[31646],[31648,662248,662249],{"xmlns":31650},[31652,662250,662251,662263],{},[31655,662252,662253,662255,662257,662259,662261],{},[31658,662254,94628],{},[31662,662256,202],{},[31658,662258,547860],{},[31662,662260,32682],{},[31658,662262,479697],{},[31665,662264,662265],{"encoding":31667},"u = \\cos \\theta",[34,662267,662269,662287],{"className":662268,"ariaHidden":758},[31672],[34,662270,662272,662275,662278,662281,662284],{"className":662271},[31676],[34,662273],{"className":662274,"style":32527},[31680],[34,662276,94628],{"className":662277},[31685,31686],[34,662279],{"className":662280,"style":31846},[31845],[34,662282,202],{"className":662283},[31850],[34,662285],{"className":662286,"style":31846},[31845],[34,662288,662290,662293,662296,662299],{"className":662289},[31676],[34,662291],{"className":662292,"style":31681},[31680],[34,662294,547860],{"className":662295},[32767],[34,662297],{"className":662298,"style":32123},[31845],[34,662300,479697],{"className":662301,"style":34452},[31685,31686],[34,662303,662305,662335],{"className":662304},[31642],[34,662306,662308],{"className":662307},[31646],[31648,662309,662310],{"xmlns":31650},[31652,662311,662312,662332],{},[31655,662313,662314,662316,662318,662320,662322,662324,662326,662328,662330],{},[31658,662315,46900],{"mathvariant":38079},[31658,662317,94628],{},[31662,662319,202],{},[31662,662321,32226],{},[31658,662323,547853],{},[31662,662325,32682],{},[31658,662327,479697],{},[31658,662329,46900],{"mathvariant":38079},[31658,662331,479697],{},[31665,662333,662334],{"encoding":31667},"\\mathrm{d}u = -\\sin \\theta \\mathrm{d}\\theta",[34,662336,662338,662359],{"className":662337,"ariaHidden":758},[31672],[34,662339,662341,662344,662347,662350,662353,662356],{"className":662340},[31676],[34,662342],{"className":662343,"style":31681},[31680],[34,662345,46900],{"className":662346},[31685,207923],[34,662348,94628],{"className":662349},[31685,31686],[34,662351],{"className":662352,"style":31846},[31845],[34,662354,202],{"className":662355},[31850],[34,662357],{"className":662358,"style":31846},[31845],[34,662360,662362,662365,662368,662371,662374,662377,662380,662383],{"className":662361},[31676],[34,662363],{"className":662364,"style":34299},[31680],[34,662366,32226],{"className":662367},[31685],[34,662369],{"className":662370,"style":32123},[31845],[34,662372,547853],{"className":662373},[32767],[34,662375],{"className":662376,"style":32123},[31845],[34,662378,479697],{"className":662379,"style":34452},[31685,31686],[34,662381,46900],{"className":662382},[31685,207923],[34,662384,479697],{"className":662385,"style":34452},[31685,31686],"。\n或者直接使用",[2627,662388,662389],{},"瓦里斯公式 (Wallis Formula)"," 的推广（注意区间是 ",[34,662392,662394,662407],{"className":662393},[31642],[34,662395,662397],{"className":662396},[31646],[31648,662398,662399],{"xmlns":31650},[31652,662400,662401,662405],{},[31655,662402,662403],{},[31749,662404,459],{},[31665,662406,459],{"encoding":31667},[34,662408,662410],{"className":662409,"ariaHidden":758},[31672],[34,662411,662413,662416],{"className":662412},[31676],[34,662414],{"className":662415,"style":32617},[31680],[34,662417,459],{"className":662418},[31685],[34,662420,662422,662435],{"className":662421},[31642],[34,662423,662425],{"className":662424},[31646],[31648,662426,662427],{"xmlns":31650},[31652,662428,662429,662433],{},[31655,662430,662431],{},[31658,662432,58442],{},[31665,662434,58445],{"encoding":31667},[34,662436,662438],{"className":662437,"ariaHidden":758},[31672],[34,662439,662441,662444],{"className":662440},[31676],[34,662442],{"className":662443,"style":32527},[31680],[34,662445,58442],{"className":662446,"style":32062},[31685,31686],"，是对称的）：\n",[34,662449,662451,662545],{"className":662450},[31642],[34,662452,662454],{"className":662453},[31646],[31648,662455,662456],{"xmlns":31650},[31652,662457,662458,662542],{},[31655,662459,662460,662468,662478,662480,662482,662484,662486,662488,662490,662504,662514,662516,662518,662520,662522,662524,662526,662528,662534,662536],{},[38846,662461,662462,662464,662466],{},[31662,662463,546295],{},[31749,662465,459],{},[31658,662467,58442],{},[37624,662469,662470,662476],{},[31655,662471,662472,662474],{},[31658,662473,547853],{},[31662,662475,32682],{},[31749,662477,786],{},[31658,662479,479697],{},[39361,662481,39363],{},[31658,662483,46900],{"mathvariant":38079},[31658,662485,479697],{},[31662,662487,202],{},[31749,662489,768],{},[38846,662491,662492,662494,662496],{},[31662,662493,546295],{},[31749,662495,459],{},[31655,662497,662498,662500,662502],{},[31658,662499,58442],{},[31658,662501,38080],{"mathvariant":38079},[31749,662503,768],{},[37624,662505,662506,662512],{},[31655,662507,662508,662510],{},[31658,662509,547853],{},[31662,662511,32682],{},[31749,662513,786],{},[31658,662515,479697],{},[39361,662517,39363],{},[31658,662519,46900],{"mathvariant":38079},[31658,662521,479697],{},[31662,662523,202],{},[31749,662525,768],{},[31662,662527,42071],{},[41453,662529,662530,662532],{},[31749,662531,768],{},[31749,662533,786],{},[31662,662535,202],{},[41453,662537,662538,662540],{},[31749,662539,15932],{},[31749,662541,786],{},[31665,662543,662544],{"encoding":31667},"\\int_0^\\pi \\sin^3 \\theta \\, \\mathrm{d}\\theta = 2 \\int_0^{\\pi\u002F2} \\sin^3 \\theta \\, \\mathrm{d}\\theta = 2 \\times \\frac{2}{3} = \\frac{4}{3}",[34,662546,662548,662661,662786,662804,662887],{"className":662547,"ariaHidden":758},[31672],[34,662549,662551,662554,662605,662608,662637,662640,662643,662646,662649,662652,662655,662658],{"className":662550},[31676],[34,662552],{"className":662553,"style":661956},[31680],[34,662555,662557,662560],{"className":662556},[32767],[34,662558,546295],{"className":662559,"style":546331},[32767,131249,156560],[34,662561,662563],{"className":662562},[31792],[34,662564,662566,662597],{"className":662565},[31796,31797],[34,662567,662569,662594],{"className":662568},[31801],[34,662570,662572,662583],{"className":662571,"style":567928},[31805],[34,662573,662574,662577],{"style":546347},[34,662575],{"className":662576,"style":31814},[31813],[34,662578,662580],{"className":662579},[31818,31819,31820,31821],[34,662581,459],{"className":662582},[31685,31821],[34,662584,662585,662588],{"style":546362},[34,662586],{"className":662587,"style":31814},[31813],[34,662589,662591],{"className":662590},[31818,31819,31820,31821],[34,662592,58442],{"className":662593,"style":32062},[31685,31686,31821],[34,662595,31829],{"className":662596},[31828],[34,662598,662600],{"className":662599},[31801],[34,662601,662603],{"className":662602,"style":546384},[31805],[34,662604],{},[34,662606],{"className":662607,"style":32123},[31845],[34,662609,662611,662614],{"className":662610},[32767],[34,662612,547853],{"className":662613},[32767],[34,662615,662617],{"className":662616},[31792],[34,662618,662620],{"className":662619},[31796],[34,662621,662623],{"className":662622},[31801],[34,662624,662626],{"className":662625,"style":585884},[31805],[34,662627,662628,662631],{"style":586008},[34,662629],{"className":662630,"style":31814},[31813],[34,662632,662634],{"className":662633},[31818,31819,31820,31821],[34,662635,786],{"className":662636},[31685,31821],[34,662638],{"className":662639,"style":32123},[31845],[34,662641,479697],{"className":662642,"style":34452},[31685,31686],[34,662644],{"className":662645,"style":32123},[31845],[34,662647,46900],{"className":662648},[31685,207923],[34,662650,479697],{"className":662651,"style":34452},[31685,31686],[34,662653],{"className":662654,"style":31846},[31845],[34,662656,202],{"className":662657},[31850],[34,662659],{"className":662660,"style":31846},[31845],[34,662662,662664,662667,662670,662673,662730,662733,662762,662765,662768,662771,662774,662777,662780,662783],{"className":662663},[31676],[34,662665],{"className":662666,"style":596892},[31680],[34,662668,768],{"className":662669},[31685],[34,662671],{"className":662672,"style":32123},[31845],[34,662674,662676,662679],{"className":662675},[32767],[34,662677,546295],{"className":662678,"style":546331},[32767,131249,156560],[34,662680,662682],{"className":662681},[31792],[34,662683,662685,662722],{"className":662684},[31796,31797],[34,662686,662688,662719],{"className":662687},[31801],[34,662689,662691,662702],{"className":662690,"style":586480},[31805],[34,662692,662693,662696],{"style":546347},[34,662694],{"className":662695,"style":31814},[31813],[34,662697,662699],{"className":662698},[31818,31819,31820,31821],[34,662700,459],{"className":662701},[31685,31821],[34,662703,662704,662707],{"style":546362},[34,662705],{"className":662706,"style":31814},[31813],[34,662708,662710],{"className":662709},[31818,31819,31820,31821],[34,662711,662713,662716],{"className":662712},[31685,31821],[34,662714,58442],{"className":662715,"style":32062},[31685,31686,31821],[34,662717,38149],{"className":662718},[31685,31821],[34,662720,31829],{"className":662721},[31828],[34,662723,662725],{"className":662724},[31801],[34,662726,662728],{"className":662727,"style":546384},[31805],[34,662729],{},[34,662731],{"className":662732,"style":32123},[31845],[34,662734,662736,662739],{"className":662735},[32767],[34,662737,547853],{"className":662738},[32767],[34,662740,662742],{"className":662741},[31792],[34,662743,662745],{"className":662744},[31796],[34,662746,662748],{"className":662747},[31801],[34,662749,662751],{"className":662750,"style":585884},[31805],[34,662752,662753,662756],{"style":586008},[34,662754],{"className":662755,"style":31814},[31813],[34,662757,662759],{"className":662758},[31818,31819,31820,31821],[34,662760,786],{"className":662761},[31685,31821],[34,662763],{"className":662764,"style":32123},[31845],[34,662766,479697],{"className":662767,"style":34452},[31685,31686],[34,662769],{"className":662770,"style":32123},[31845],[34,662772,46900],{"className":662773},[31685,207923],[34,662775,479697],{"className":662776,"style":34452},[31685,31686],[34,662778],{"className":662779,"style":31846},[31845],[34,662781,202],{"className":662782},[31850],[34,662784],{"className":662785,"style":31846},[31845],[34,662787,662789,662792,662795,662798,662801],{"className":662788},[31676],[34,662790],{"className":662791,"style":35942},[31680],[34,662793,768],{"className":662794},[31685],[34,662796],{"className":662797,"style":32604},[31845],[34,662799,42071],{"className":662800},[32024],[34,662802],{"className":662803,"style":32604},[31845],[34,662805,662807,662810,662878,662881,662884],{"className":662806},[31676],[34,662808],{"className":662809,"style":46727},[31680],[34,662811,662813,662816,662875],{"className":662812},[31685],[34,662814],{"className":662815},[31782,41488],[34,662817,662819],{"className":662818},[41453],[34,662820,662822,662867],{"className":662821},[31796,31797],[34,662823,662825,662864],{"className":662824},[31801],[34,662826,662828,662842,662850],{"className":662827,"style":46746},[31805],[34,662829,662830,662833],{"style":41504},[34,662831],{"className":662832,"style":41508},[31813],[34,662834,662836],{"className":662835},[31818,31819,31820,31821],[34,662837,662839],{"className":662838},[31685,31821],[34,662840,786],{"className":662841},[31685,31821],[34,662843,662844,662847],{"style":41520},[34,662845],{"className":662846,"style":41508},[31813],[34,662848],{"className":662849,"style":41528},[41527],[34,662851,662852,662855],{"style":46771},[34,662853],{"className":662854,"style":41508},[31813],[34,662856,662858],{"className":662857},[31818,31819,31820,31821],[34,662859,662861],{"className":662860},[31685,31821],[34,662862,768],{"className":662863},[31685,31821],[34,662865,31829],{"className":662866},[31828],[34,662868,662870],{"className":662869},[31801],[34,662871,662873],{"className":662872,"style":41619},[31805],[34,662874],{},[34,662876],{"className":662877},[31690,41488],[34,662879],{"className":662880,"style":31846},[31845],[34,662882,202],{"className":662883},[31850],[34,662885],{"className":662886,"style":31846},[31845],[34,662888,662890,662893],{"className":662889},[31676],[34,662891],{"className":662892,"style":46727},[31680],[34,662894,662896,662899,662958],{"className":662895},[31685],[34,662897],{"className":662898},[31782,41488],[34,662900,662902],{"className":662901},[41453],[34,662903,662905,662950],{"className":662904},[31796,31797],[34,662906,662908,662947],{"className":662907},[31801],[34,662909,662911,662925,662933],{"className":662910,"style":46746},[31805],[34,662912,662913,662916],{"style":41504},[34,662914],{"className":662915,"style":41508},[31813],[34,662917,662919],{"className":662918},[31818,31819,31820,31821],[34,662920,662922],{"className":662921},[31685,31821],[34,662923,786],{"className":662924},[31685,31821],[34,662926,662927,662930],{"style":41520},[34,662928],{"className":662929,"style":41508},[31813],[34,662931],{"className":662932,"style":41528},[41527],[34,662934,662935,662938],{"style":46771},[34,662936],{"className":662937,"style":41508},[31813],[34,662939,662941],{"className":662940},[31818,31819,31820,31821],[34,662942,662944],{"className":662943},[31685,31821],[34,662945,15932],{"className":662946},[31685,31821],[34,662948,31829],{"className":662949},[31828],[34,662951,662953],{"className":662952},[31801],[34,662954,662956],{"className":662955,"style":41619},[31805],[34,662957],{},[34,662959],{"className":662960},[31690,41488],[108,662962,662963,2664,662966,663114,663115,663186,663187,663239,663240,663279,663280,663316,663317,347],{},[2627,662964,662965],{},"第二部分",[34,662967,662969,663007],{"className":662968},[31642],[34,662970,662972],{"className":662971},[31646],[31648,662973,662974],{"xmlns":31650},[31652,662975,662976,663004],{},[31655,662977,662978,662986,662996,662998,663000,663002],{},[38846,662979,662980,662982,662984],{},[31662,662981,546295],{},[31749,662983,459],{},[31658,662985,58442],{},[37624,662987,662988,662994],{},[31655,662989,662990,662992],{},[31658,662991,547860],{},[31662,662993,32682],{},[31749,662995,786],{},[31658,662997,479697],{},[39361,662999,39363],{},[31658,663001,46900],{"mathvariant":38079},[31658,663003,479697],{},[31665,663005,663006],{"encoding":31667},"\\int_0^\\pi \\cos^3 \\theta \\, \\mathrm{d}\\theta",[34,663008,663010],{"className":663009,"ariaHidden":758},[31672],[34,663011,663013,663016,663067,663070,663099,663102,663105,663108,663111],{"className":663012},[31676],[34,663014],{"className":663015,"style":613610},[31680],[34,663017,663019,663022],{"className":663018},[32767],[34,663020,546295],{"className":663021,"style":546331},[32767,131249,156560],[34,663023,663025],{"className":663024},[31792],[34,663026,663028,663059],{"className":663027},[31796,31797],[34,663029,663031,663056],{"className":663030},[31801],[34,663032,663034,663045],{"className":663033,"style":567928},[31805],[34,663035,663036,663039],{"style":546347},[34,663037],{"className":663038,"style":31814},[31813],[34,663040,663042],{"className":663041},[31818,31819,31820,31821],[34,663043,459],{"className":663044},[31685,31821],[34,663046,663047,663050],{"style":546362},[34,663048],{"className":663049,"style":31814},[31813],[34,663051,663053],{"className":663052},[31818,31819,31820,31821],[34,663054,58442],{"className":663055,"style":32062},[31685,31686,31821],[34,663057,31829],{"className":663058},[31828],[34,663060,663062],{"className":663061},[31801],[34,663063,663065],{"className":663064,"style":546384},[31805],[34,663066],{},[34,663068],{"className":663069,"style":32123},[31845],[34,663071,663073,663076],{"className":663072},[32767],[34,663074,547860],{"className":663075},[32767],[34,663077,663079],{"className":663078},[31792],[34,663080,663082],{"className":663081},[31796],[34,663083,663085],{"className":663084},[31801],[34,663086,663088],{"className":663087,"style":37774},[31805],[34,663089,663090,663093],{"style":37777},[34,663091],{"className":663092,"style":31814},[31813],[34,663094,663096],{"className":663095},[31818,31819,31820,31821],[34,663097,786],{"className":663098},[31685,31821],[34,663100],{"className":663101,"style":32123},[31845],[34,663103,479697],{"className":663104,"style":34452},[31685,31686],[34,663106],{"className":663107,"style":32123},[31845],[34,663109,46900],{"className":663110},[31685,207923],[34,663112,479697],{"className":663113,"style":34452},[31685,31686],"\n观察 ",[34,663116,663118,663142],{"className":663117},[31642],[34,663119,663121],{"className":663120},[31646],[31648,663122,663123],{"xmlns":31650},[31652,663124,663125,663139],{},[31655,663126,663127,663137],{},[37624,663128,663129,663135],{},[31655,663130,663131,663133],{},[31658,663132,547860],{},[31662,663134,32682],{},[31749,663136,786],{},[31658,663138,479697],{},[31665,663140,663141],{"encoding":31667},"\\cos^3 \\theta",[34,663143,663145],{"className":663144,"ariaHidden":758},[31672],[34,663146,663148,663151,663180,663183],{"className":663147},[31676],[34,663149],{"className":663150,"style":37774},[31680],[34,663152,663154,663157],{"className":663153},[32767],[34,663155,547860],{"className":663156},[32767],[34,663158,663160],{"className":663159},[31792],[34,663161,663163],{"className":663162},[31796],[34,663164,663166],{"className":663165},[31801],[34,663167,663169],{"className":663168,"style":37774},[31805],[34,663170,663171,663174],{"style":37777},[34,663172],{"className":663173,"style":31814},[31813],[34,663175,663177],{"className":663176},[31818,31819,31820,31821],[34,663178,786],{"className":663179},[31685,31821],[34,663181],{"className":663182,"style":32123},[31845],[34,663184,479697],{"className":663185,"style":34452},[31685,31686]," 的图像，在 ",[34,663188,663190,663212],{"className":663189},[31642],[34,663191,663193],{"className":663192},[31646],[31648,663194,663195],{"xmlns":31650},[31652,663196,663197,663209],{},[31655,663198,663199,663201,663203,663205,663207],{},[31662,663200,15972],{"stretchy":702},[31749,663202,459],{},[31662,663204,31960],{"separator":758},[31658,663206,58442],{},[31662,663208,32712],{"stretchy":702},[31665,663210,663211],{"encoding":31667},"[0, \\pi]",[34,663213,663215],{"className":663214,"ariaHidden":758},[31672],[34,663216,663218,663221,663224,663227,663230,663233,663236],{"className":663217},[31676],[34,663219],{"className":663220,"style":31774},[31680],[34,663222,15972],{"className":663223},[31782],[34,663225,459],{"className":663226},[31685],[34,663228,31960],{"className":663229},[32119],[34,663231],{"className":663232,"style":32123},[31845],[34,663234,58442],{"className":663235,"style":32062},[31685,31686],[34,663237,32712],{"className":663238},[31690]," 上，",[34,663241,663243,663261],{"className":663242},[31642],[34,663244,663246],{"className":663245},[31646],[31648,663247,663248],{"xmlns":31650},[31652,663249,663250,663258],{},[31655,663251,663252,663254,663256],{},[31658,663253,547860],{},[31662,663255,32682],{},[31658,663257,479697],{},[31665,663259,663260],{"encoding":31667},"\\cos \\theta",[34,663262,663264],{"className":663263,"ariaHidden":758},[31672],[34,663265,663267,663270,663273,663276],{"className":663266},[31676],[34,663268],{"className":663269,"style":31681},[31680],[34,663271,547860],{"className":663272},[32767],[34,663274],{"className":663275,"style":32123},[31845],[34,663277,479697],{"className":663278,"style":34452},[31685,31686]," 关于 ",[34,663281,663283,663301],{"className":663282},[31642],[34,663284,663286],{"className":663285},[31646],[31648,663287,663288],{"xmlns":31650},[31652,663289,663290,663298],{},[31655,663291,663292,663294,663296],{},[31658,663293,58442],{},[31658,663295,38080],{"mathvariant":38079},[31749,663297,768],{},[31665,663299,663300],{"encoding":31667},"\\pi\u002F2",[34,663302,663304],{"className":663303,"ariaHidden":758},[31672],[34,663305,663307,663310,663313],{"className":663306},[31676],[34,663308],{"className":663309,"style":31774},[31680],[34,663311,58442],{"className":663312,"style":32062},[31685,31686],[34,663314,38149],{"className":663315},[31685]," 中心对称（第一象限为正，第二象限为负，且绝对值相等）。\n因此，积分值为 ",[2627,663318,459],{},[58531,663320],{},[2627,663322,663323],{},"合并结果：",[34,663325,663327,663387],{"className":663326},[31642],[34,663328,663330],{"className":663329},[31646],[31648,663331,663332],{"xmlns":31650},[31652,663333,663334,663384],{},[31655,663335,663336,663342,663344,663350,663368,663370,663372,663378],{},[31743,663337,663338,663340],{},[31658,663339,106063],{},[31749,663341,640],{},[31662,663343,202],{},[37624,663345,663346,663348],{},[31658,663347,34386],{},[31749,663349,786],{},[31655,663351,663352,663354,663356,663362,663364,663366],{},[31662,663353,456],{"fence":758},[31662,663355,32226],{},[41453,663357,663358,663360],{},[31749,663359,15932],{},[31749,663361,786],{},[31662,663363,223],{},[31749,663365,459],{},[31662,663367,2557],{"fence":758},[31662,663369,202],{},[31662,663371,32226],{},[41453,663373,663374,663376],{},[31749,663375,15932],{},[31749,663377,786],{},[37624,663379,663380,663382],{},[31658,663381,34386],{},[31749,663383,786],{},[31665,663385,663386],{"encoding":31667},"I_1 = R^3 \\left( - \\frac{4}{3} + 0 \\right) = -\\frac{4}{3}R^3",[34,663388,663390,663445,663590],{"className":663389,"ariaHidden":758},[31672],[34,663391,663393,663396,663436,663439,663442],{"className":663392},[31676],[34,663394],{"className":663395,"style":70915},[31680],[34,663397,663399,663402],{"className":663398},[31685],[34,663400,106063],{"className":663401,"style":67482},[31685,31686],[34,663403,663405],{"className":663404},[31792],[34,663406,663408,663428],{"className":663407},[31796,31797],[34,663409,663411,663425],{"className":663410},[31801],[34,663412,663414],{"className":663413,"style":31806},[31805],[34,663415,663416,663419],{"style":217742},[34,663417],{"className":663418,"style":31814},[31813],[34,663420,663422],{"className":663421},[31818,31819,31820,31821],[34,663423,640],{"className":663424},[31685,31821],[34,663426,31829],{"className":663427},[31828],[34,663429,663431],{"className":663430},[31801],[34,663432,663434],{"className":663433,"style":31836},[31805],[34,663435],{},[34,663437],{"className":663438,"style":31846},[31845],[34,663440,202],{"className":663441},[31850],[34,663443],{"className":663444,"style":31846},[31845],[34,663446,663448,663451,663480,663483,663581,663584,663587],{"className":663447},[31676],[34,663449],{"className":663450,"style":345868},[31680],[34,663452,663454,663457],{"className":663453},[31685],[34,663455,34386],{"className":663456,"style":34427},[31685,31686],[34,663458,663460],{"className":663459},[31792],[34,663461,663463],{"className":663462},[31796],[34,663464,663466],{"className":663465},[31801],[34,663467,663469],{"className":663468,"style":37774},[31805],[34,663470,663471,663474],{"style":37777},[34,663472],{"className":663473,"style":31814},[31813],[34,663475,663477],{"className":663476},[31818,31819,31820,31821],[34,663478,786],{"className":663479},[31685,31821],[34,663481],{"className":663482,"style":32123},[31845],[34,663484,663486,663492,663495,663563,663566,663569,663572,663575],{"className":663485},[34006],[34,663487,663489],{"className":663488,"style":75811},[31782,75810],[34,663490,456],{"className":663491},[71281,43411],[34,663493,32226],{"className":663494},[31685],[34,663496,663498,663501,663560],{"className":663497},[31685],[34,663499],{"className":663500},[31782,41488],[34,663502,663504],{"className":663503},[41453],[34,663505,663507,663552],{"className":663506},[31796,31797],[34,663508,663510,663549],{"className":663509},[31801],[34,663511,663513,663527,663535],{"className":663512,"style":46746},[31805],[34,663514,663515,663518],{"style":41504},[34,663516],{"className":663517,"style":41508},[31813],[34,663519,663521],{"className":663520},[31818,31819,31820,31821],[34,663522,663524],{"className":663523},[31685,31821],[34,663525,786],{"className":663526},[31685,31821],[34,663528,663529,663532],{"style":41520},[34,663530],{"className":663531,"style":41508},[31813],[34,663533],{"className":663534,"style":41528},[41527],[34,663536,663537,663540],{"style":46771},[34,663538],{"className":663539,"style":41508},[31813],[34,663541,663543],{"className":663542},[31818,31819,31820,31821],[34,663544,663546],{"className":663545},[31685,31821],[34,663547,15932],{"className":663548},[31685,31821],[34,663550,31829],{"className":663551},[31828],[34,663553,663555],{"className":663554},[31801],[34,663556,663558],{"className":663557,"style":41619},[31805],[34,663559],{},[34,663561],{"className":663562},[31690,41488],[34,663564],{"className":663565,"style":32604},[31845],[34,663567,223],{"className":663568},[32024],[34,663570],{"className":663571,"style":32604},[31845],[34,663573,459],{"className":663574},[31685],[34,663576,663578],{"className":663577,"style":75811},[31690,75810],[34,663579,2557],{"className":663580},[71281,43411],[34,663582],{"className":663583,"style":31846},[31845],[34,663585,202],{"className":663586},[31850],[34,663588],{"className":663589,"style":31846},[31845],[34,663591,663593,663596,663599,663667],{"className":663592},[31676],[34,663594],{"className":663595,"style":46727},[31680],[34,663597,32226],{"className":663598},[31685],[34,663600,663602,663605,663664],{"className":663601},[31685],[34,663603],{"className":663604},[31782,41488],[34,663606,663608],{"className":663607},[41453],[34,663609,663611,663656],{"className":663610},[31796,31797],[34,663612,663614,663653],{"className":663613},[31801],[34,663615,663617,663631,663639],{"className":663616,"style":46746},[31805],[34,663618,663619,663622],{"style":41504},[34,663620],{"className":663621,"style":41508},[31813],[34,663623,663625],{"className":663624},[31818,31819,31820,31821],[34,663626,663628],{"className":663627},[31685,31821],[34,663629,786],{"className":663630},[31685,31821],[34,663632,663633,663636],{"style":41520},[34,663634],{"className":663635,"style":41508},[31813],[34,663637],{"className":663638,"style":41528},[41527],[34,663640,663641,663644],{"style":46771},[34,663642],{"className":663643,"style":41508},[31813],[34,663645,663647],{"className":663646},[31818,31819,31820,31821],[34,663648,663650],{"className":663649},[31685,31821],[34,663651,15932],{"className":663652},[31685,31821],[34,663654,31829],{"className":663655},[31828],[34,663657,663659],{"className":663658},[31801],[34,663660,663662],{"className":663661,"style":41619},[31805],[34,663663],{},[34,663665],{"className":663666},[31690,41488],[34,663668,663670,663673],{"className":663669},[31685],[34,663671,34386],{"className":663672,"style":34427},[31685,31686],[34,663674,663676],{"className":663675},[31792],[34,663677,663679],{"className":663678},[31796],[34,663680,663682],{"className":663681},[31801],[34,663683,663685],{"className":663684,"style":37774},[31805],[34,663686,663687,663690],{"style":37777},[34,663688],{"className":663689,"style":31814},[31813],[34,663691,663693],{"className":663692},[31818,31819,31820,31821],[34,663694,786],{"className":663695},[31685,31821],[15,663697,663698],{},[2627,663699,663700,663701,658991],{},"情形 2：沿直线段 ",[34,663702,663704,663721],{"className":663703},[31642],[34,663705,663707],{"className":663706},[31646],[31648,663708,663709],{"xmlns":31650},[31652,663710,663711,663719],{},[31655,663712,663713],{},[31743,663714,663715,663717],{},[31658,663716,33206],{},[31749,663718,768],{},[31665,663720,658738],{"encoding":31667},[34,663722,663724],{"className":663723,"ariaHidden":758},[31672],[34,663725,663727,663730],{"className":663726},[31676],[34,663728],{"className":663729,"style":70915},[31680],[34,663731,663733,663736],{"className":663732},[31685],[34,663734,33206],{"className":663735},[31685,31686],[34,663737,663739],{"className":663738},[31792],[34,663740,663742,663762],{"className":663741},[31796,31797],[34,663743,663745,663759],{"className":663744},[31801],[34,663746,663748],{"className":663747,"style":31806},[31805],[34,663749,663750,663753],{"style":31809},[34,663751],{"className":663752,"style":31814},[31813],[34,663754,663756],{"className":663755},[31818,31819,31820,31821],[34,663757,768],{"className":663758},[31685,31821],[34,663760,31829],{"className":663761},[31828],[34,663763,663765],{"className":663764},[31801],[34,663766,663768],{"className":663767,"style":31836},[31805],[34,663769],{},[2230,663771,663772,664063,664248],{},[108,663773,663774,663776,663777,663805,663806,663857,663858,347,663914],{},[2627,663775,658998],{},"\n路径是在 ",[34,663778,663780,663793],{"className":663779},[31642],[34,663781,663783],{"className":663782},[31646],[31648,663784,663785],{"xmlns":31650},[31652,663786,663787,663791],{},[31655,663788,663789],{},[31658,663790,32936],{},[31665,663792,32936],{"encoding":31667},[34,663794,663796],{"className":663795,"ariaHidden":758},[31672],[34,663797,663799,663802],{"className":663798},[31676],[34,663800],{"className":663801,"style":32527},[31680],[34,663803,32936],{"className":663804},[31685,31686]," 轴上，从 ",[34,663807,663809,663830],{"className":663808},[31642],[34,663810,663812],{"className":663811},[31646],[31648,663813,663814],{"xmlns":31650},[31652,663815,663816,663828],{},[31655,663817,663818,663820,663822,663824,663826],{},[31662,663819,456],{"stretchy":702},[31658,663821,34386],{},[31662,663823,31960],{"separator":758},[31749,663825,459],{},[31662,663827,2557],{"stretchy":702},[31665,663829,659293],{"encoding":31667},[34,663831,663833],{"className":663832,"ariaHidden":758},[31672],[34,663834,663836,663839,663842,663845,663848,663851,663854],{"className":663835},[31676],[34,663837],{"className":663838,"style":31774},[31680],[34,663840,456],{"className":663841},[31782],[34,663843,34386],{"className":663844,"style":34427},[31685,31686],[34,663846,31960],{"className":663847},[32119],[34,663849],{"className":663850,"style":32123},[31845],[34,663852,459],{"className":663853},[31685],[34,663855,2557],{"className":663856},[31690]," 走到 ",[34,663859,663861,663884],{"className":663860},[31642],[34,663862,663864],{"className":663863},[31646],[31648,663865,663866],{"xmlns":31650},[31652,663867,663868,663882],{},[31655,663869,663870,663872,663874,663876,663878,663880],{},[31662,663871,456],{"stretchy":702},[31662,663873,32226],{},[31658,663875,34386],{},[31662,663877,31960],{"separator":758},[31749,663879,459],{},[31662,663881,2557],{"stretchy":702},[31665,663883,659402],{"encoding":31667},[34,663885,663887],{"className":663886,"ariaHidden":758},[31672],[34,663888,663890,663893,663896,663899,663902,663905,663908,663911],{"className":663889},[31676],[34,663891],{"className":663892,"style":31774},[31680],[34,663894,456],{"className":663895},[31782],[34,663897,32226],{"className":663898},[31685],[34,663900,34386],{"className":663901,"style":34427},[31685,31686],[34,663903,31960],{"className":663904},[32119],[34,663906],{"className":663907,"style":32123},[31845],[34,663909,459],{"className":663910},[31685],[34,663912,2557],{"className":663913},[31690],[105,663915,663916,663970],{},[108,663917,663918,663969],{},[34,663919,663921,663939],{"className":663920},[31642],[34,663922,663924],{"className":663923},[31646],[31648,663925,663926],{"xmlns":31650},[31652,663927,663928,663936],{},[31655,663929,663930,663932,663934],{},[31658,663931,67764],{},[31662,663933,202],{},[31749,663935,459],{},[31665,663937,663938],{"encoding":31667},"y = 0",[34,663940,663942,663960],{"className":663941,"ariaHidden":758},[31672],[34,663943,663945,663948,663951,663954,663957],{"className":663944},[31676],[34,663946],{"className":663947,"style":38774},[31680],[34,663949,67764],{"className":663950,"style":32062},[31685,31686],[34,663952],{"className":663953,"style":31846},[31845],[34,663955,202],{"className":663956},[31850],[34,663958],{"className":663959,"style":31846},[31845],[34,663961,663963,663966],{"className":663962},[31676],[34,663964],{"className":663965,"style":32617},[31680],[34,663967,459],{"className":663968},[31685]," （恒成立）",[108,663971,663972,138176,664000,664028,664029,347],{},[34,663973,663975,663988],{"className":663974},[31642],[34,663976,663978],{"className":663977},[31646],[31648,663979,663980],{"xmlns":31650},[31652,663981,663982,663986],{},[31655,663983,663984],{},[31658,663985,32936],{},[31665,663987,32936],{"encoding":31667},[34,663989,663991],{"className":663990,"ariaHidden":758},[31672],[34,663992,663994,663997],{"className":663993},[31676],[34,663995],{"className":663996,"style":32527},[31680],[34,663998,32936],{"className":663999},[31685,31686],[34,664001,664003,664016],{"className":664002},[31642],[34,664004,664006],{"className":664005},[31646],[31648,664007,664008],{"xmlns":31650},[31652,664009,664010,664014],{},[31655,664011,664012],{},[31658,664013,34386],{},[31665,664015,34386],{"encoding":31667},[34,664017,664019],{"className":664018,"ariaHidden":758},[31672],[34,664020,664022,664025],{"className":664021},[31676],[34,664023],{"className":664024,"style":33218},[31680],[34,664026,34386],{"className":664027,"style":34427},[31685,31686]," 变到 ",[34,664030,664032,664048],{"className":664031},[31642],[34,664033,664035],{"className":664034},[31646],[31648,664036,664037],{"xmlns":31650},[31652,664038,664039,664045],{},[31655,664040,664041,664043],{},[31662,664042,32226],{},[31658,664044,34386],{},[31665,664046,664047],{"encoding":31667},"-R",[34,664049,664051],{"className":664050,"ariaHidden":758},[31672],[34,664052,664054,664057,664060],{"className":664053},[31676],[34,664055],{"className":664056,"style":34944},[31680],[34,664058,32226],{"className":664059},[31685],[34,664061,34386],{"className":664062,"style":34427},[31685,31686],[108,664064,664065,664067],{},[2627,664066,659488],{},[105,664068,664069,664179],{},[108,664070,664071,664072,664122,664123,347],{},"由于 ",[34,664073,664075,664092],{"className":664074},[31642],[34,664076,664078],{"className":664077},[31646],[31648,664079,664080],{"xmlns":31650},[31652,664081,664082,664090],{},[31655,664083,664084,664086,664088],{},[31658,664085,67764],{},[31662,664087,202],{},[31749,664089,459],{},[31665,664091,663938],{"encoding":31667},[34,664093,664095,664113],{"className":664094,"ariaHidden":758},[31672],[34,664096,664098,664101,664104,664107,664110],{"className":664097},[31676],[34,664099],{"className":664100,"style":38774},[31680],[34,664102,67764],{"className":664103,"style":32062},[31685,31686],[34,664105],{"className":664106,"style":31846},[31845],[34,664108,202],{"className":664109},[31850],[34,664111],{"className":664112,"style":31846},[31845],[34,664114,664116,664119],{"className":664115},[31676],[34,664117],{"className":664118,"style":32617},[31680],[34,664120,459],{"className":664121},[31685]," 是常数，所以 ",[34,664124,664126,664146],{"className":664125},[31642],[34,664127,664129],{"className":664128},[31646],[31648,664130,664131],{"xmlns":31650},[31652,664132,664133,664143],{},[31655,664134,664135,664137,664139,664141],{},[31658,664136,46900],{"mathvariant":38079},[31658,664138,67764],{},[31662,664140,202],{},[31749,664142,459],{},[31665,664144,664145],{"encoding":31667},"\\mathrm{d}y = 0",[34,664147,664149,664170],{"className":664148,"ariaHidden":758},[31672],[34,664150,664152,664155,664158,664161,664164,664167],{"className":664151},[31676],[34,664153],{"className":664154,"style":43217},[31680],[34,664156,46900],{"className":664157},[31685,207923],[34,664159,67764],{"className":664160,"style":32062},[31685,31686],[34,664162],{"className":664163,"style":31846},[31845],[34,664165,202],{"className":664166},[31850],[34,664168],{"className":664169,"style":31846},[31845],[34,664171,664173,664176],{"className":664172},[31676],[34,664174],{"className":664175,"style":32617},[31680],[34,664177,459],{"className":664178},[31685],[108,664180,664181,664214,664215,347],{},[34,664182,664184,664199],{"className":664183},[31642],[34,664185,664187],{"className":664186},[31646],[31648,664188,664189],{"xmlns":31650},[31652,664190,664191,664197],{},[31655,664192,664193,664195],{},[31658,664194,46900],{"mathvariant":38079},[31658,664196,32936],{},[31665,664198,659508],{"encoding":31667},[34,664200,664202],{"className":664201,"ariaHidden":758},[31672],[34,664203,664205,664208,664211],{"className":664204},[31676],[34,664206],{"className":664207,"style":31681},[31680],[34,664209,46900],{"className":664210},[31685,207923],[34,664212,32936],{"className":664213},[31685,31686]," 保持为 ",[34,664216,664218,664233],{"className":664217},[31642],[34,664219,664221],{"className":664220},[31646],[31648,664222,664223],{"xmlns":31650},[31652,664224,664225,664231],{},[31655,664226,664227,664229],{},[31658,664228,46900],{"mathvariant":38079},[31658,664230,32936],{},[31665,664232,659508],{"encoding":31667},[34,664234,664236],{"className":664235,"ariaHidden":758},[31672],[34,664237,664239,664242,664245],{"className":664238},[31676],[34,664240],{"className":664241,"style":31681},[31680],[34,664243,46900],{"className":664244},[31685,207923],[34,664246,32936],{"className":664247},[31685,31686],[108,664249,664250,664252,664505,664506,258,664557,33183,664613],{},[2627,664251,659977],{},[34,664253,664255,664307],{"className":664254},[31642],[34,664256,664258],{"className":664257},[31646],[31648,664259,664260],{"xmlns":31650},[31652,664261,664262,664304],{},[31655,664263,664264,664270,664272,664278,664284,664286,664288,664290,664292,664298,664300,664302],{},[31743,664265,664266,664268],{},[31658,664267,106063],{},[31749,664269,768],{},[31662,664271,202],{},[31743,664273,664274,664276],{},[31662,664275,546295],{},[31658,664277,33206],{},[37624,664279,664280,664282],{},[31658,664281,67764],{},[31749,664283,768],{},[39361,664285,39363],{},[31658,664287,46900],{"mathvariant":38079},[31658,664289,32936],{},[31662,664291,223],{},[37624,664293,664294,664296],{},[31658,664295,32936],{},[31749,664297,768],{},[39361,664299,39363],{},[31658,664301,46900],{"mathvariant":38079},[31658,664303,67764],{},[31665,664305,664306],{"encoding":31667},"I_2 = \\int_L y^2 \\, \\mathrm{d}x + x^2 \\, \\mathrm{d}y",[34,664308,664310,664365,664461],{"className":664309,"ariaHidden":758},[31672],[34,664311,664313,664316,664356,664359,664362],{"className":664312},[31676],[34,664314],{"className":664315,"style":70915},[31680],[34,664317,664319,664322],{"className":664318},[31685],[34,664320,106063],{"className":664321,"style":67482},[31685,31686],[34,664323,664325],{"className":664324},[31792],[34,664326,664328,664348],{"className":664327},[31796,31797],[34,664329,664331,664345],{"className":664330},[31801],[34,664332,664334],{"className":664333,"style":31806},[31805],[34,664335,664336,664339],{"style":217742},[34,664337],{"className":664338,"style":31814},[31813],[34,664340,664342],{"className":664341},[31818,31819,31820,31821],[34,664343,768],{"className":664344},[31685,31821],[34,664346,31829],{"className":664347},[31828],[34,664349,664351],{"className":664350},[31801],[34,664352,664354],{"className":664353,"style":31836},[31805],[34,664355],{},[34,664357],{"className":664358,"style":31846},[31845],[34,664360,202],{"className":664361},[31850],[34,664363],{"className":664364,"style":31846},[31845],[34,664366,664368,664371,664411,664414,664443,664446,664449,664452,664455,664458],{"className":664367},[31676],[34,664369],{"className":664370,"style":658186},[31680],[34,664372,664374,664377],{"className":664373},[32767],[34,664375,546295],{"className":664376,"style":546331},[32767,131249,156560],[34,664378,664380],{"className":664379},[31792],[34,664381,664383,664403],{"className":664382},[31796,31797],[34,664384,664386,664400],{"className":664385},[31801],[34,664387,664389],{"className":664388,"style":637339},[31805],[34,664390,664391,664394],{"style":546347},[34,664392],{"className":664393,"style":31814},[31813],[34,664395,664397],{"className":664396},[31818,31819,31820,31821],[34,664398,33206],{"className":664399},[31685,31686,31821],[34,664401,31829],{"className":664402},[31828],[34,664404,664406],{"className":664405},[31801],[34,664407,664409],{"className":664408,"style":546384},[31805],[34,664410],{},[34,664412],{"className":664413,"style":32123},[31845],[34,664415,664417,664420],{"className":664416},[31685],[34,664418,67764],{"className":664419,"style":32062},[31685,31686],[34,664421,664423],{"className":664422},[31792],[34,664424,664426],{"className":664425},[31796],[34,664427,664429],{"className":664428},[31801],[34,664430,664432],{"className":664431,"style":37774},[31805],[34,664433,664434,664437],{"style":37777},[34,664435],{"className":664436,"style":31814},[31813],[34,664438,664440],{"className":664439},[31818,31819,31820,31821],[34,664441,768],{"className":664442},[31685,31821],[34,664444],{"className":664445,"style":32123},[31845],[34,664447,46900],{"className":664448},[31685,207923],[34,664450,32936],{"className":664451},[31685,31686],[34,664453],{"className":664454,"style":32604},[31845],[34,664456,223],{"className":664457},[32024],[34,664459],{"className":664460,"style":32604},[31845],[34,664462,664464,664467,664496,664499,664502],{"className":664463},[31676],[34,664465],{"className":664466,"style":315747},[31680],[34,664468,664470,664473],{"className":664469},[31685],[34,664471,32936],{"className":664472},[31685,31686],[34,664474,664476],{"className":664475},[31792],[34,664477,664479],{"className":664478},[31796],[34,664480,664482],{"className":664481},[31801],[34,664483,664485],{"className":664484,"style":37774},[31805],[34,664486,664487,664490],{"style":37777},[34,664488],{"className":664489,"style":31814},[31813],[34,664491,664493],{"className":664492},[31818,31819,31820,31821],[34,664494,768],{"className":664495},[31685,31821],[34,664497],{"className":664498,"style":32123},[31845],[34,664500,46900],{"className":664501},[31685,207923],[34,664503,67764],{"className":664504,"style":32062},[31685,31686],"\n代入 ",[34,664507,664509,664527],{"className":664508},[31642],[34,664510,664512],{"className":664511},[31646],[31648,664513,664514],{"xmlns":31650},[31652,664515,664516,664524],{},[31655,664517,664518,664520,664522],{},[31658,664519,67764],{},[31662,664521,202],{},[31749,664523,459],{},[31665,664525,664526],{"encoding":31667},"y=0",[34,664528,664530,664548],{"className":664529,"ariaHidden":758},[31672],[34,664531,664533,664536,664539,664542,664545],{"className":664532},[31676],[34,664534],{"className":664535,"style":38774},[31680],[34,664537,67764],{"className":664538,"style":32062},[31685,31686],[34,664540],{"className":664541,"style":31846},[31845],[34,664543,202],{"className":664544},[31850],[34,664546],{"className":664547,"style":31846},[31845],[34,664549,664551,664554],{"className":664550},[31676],[34,664552],{"className":664553,"style":32617},[31680],[34,664555,459],{"className":664556},[31685],[34,664558,664560,664580],{"className":664559},[31642],[34,664561,664563],{"className":664562},[31646],[31648,664564,664565],{"xmlns":31650},[31652,664566,664567,664577],{},[31655,664568,664569,664571,664573,664575],{},[31658,664570,46900],{"mathvariant":38079},[31658,664572,67764],{},[31662,664574,202],{},[31749,664576,459],{},[31665,664578,664579],{"encoding":31667},"\\mathrm{d}y=0",[34,664581,664583,664604],{"className":664582,"ariaHidden":758},[31672],[34,664584,664586,664589,664592,664595,664598,664601],{"className":664585},[31676],[34,664587],{"className":664588,"style":43217},[31680],[34,664590,46900],{"className":664591},[31685,207923],[34,664593,67764],{"className":664594,"style":32062},[31685,31686],[34,664596],{"className":664597,"style":31846},[31845],[34,664599,202],{"className":664600},[31850],[34,664602],{"className":664603,"style":31846},[31845],[34,664605,664607,664610],{"className":664606},[31676],[34,664608],{"className":664609,"style":32617},[31680],[34,664611,459],{"className":664612},[31685],[34,664614,664616,664706],{"className":664615},[31642],[34,664617,664619],{"className":664618},[31646],[31648,664620,664621],{"xmlns":31650},[31652,664622,664623,664703],{},[31655,664624,664625,664631,664633,664645,664647,664649,664655,664657,664659,664661,664663,664669,664671,664673,664675,664677,664679,664691,664693,664695,664697,664699,664701],{},[31743,664626,664627,664629],{},[31658,664628,106063],{},[31749,664630,768],{},[31662,664632,202],{},[38846,664634,664635,664637,664639],{},[31662,664636,546295],{},[31658,664638,34386],{},[31655,664640,664641,664643],{},[31662,664642,32226],{},[31658,664644,34386],{},[31662,664646,456],{"stretchy":702},[31749,664648,459],{},[37624,664650,664651,664653],{},[31662,664652,2557],{"stretchy":702},[31749,664654,768],{},[39361,664656,39363],{},[31658,664658,46900],{"mathvariant":38079},[31658,664660,32936],{},[31662,664662,223],{},[37624,664664,664665,664667],{},[31658,664666,32936],{},[31749,664668,768],{},[31662,664670,71191],{},[31662,664672,456],{"stretchy":702},[31749,664674,459],{},[31662,664676,2557],{"stretchy":702},[31662,664678,202],{},[38846,664680,664681,664683,664685],{},[31662,664682,546295],{},[31658,664684,34386],{},[31655,664686,664687,664689],{},[31662,664688,32226],{},[31658,664690,34386],{},[31749,664692,459],{},[39361,664694,39363],{},[31658,664696,46900],{"mathvariant":38079},[31658,664698,32936],{},[31662,664700,202],{},[31749,664702,459],{},[31665,664704,664705],{"encoding":31667},"I_2 = \\int_R^{-R} (0)^2 \\, \\mathrm{d}x + x^2 \\cdot (0) = \\int_R^{-R} 0 \\, \\mathrm{d}x = 0",[34,664707,664709,664764,664880,664924,664948,665035],{"className":664708,"ariaHidden":758},[31672],[34,664710,664712,664715,664755,664758,664761],{"className":664711},[31676],[34,664713],{"className":664714,"style":70915},[31680],[34,664716,664718,664721],{"className":664717},[31685],[34,664719,106063],{"className":664720,"style":67482},[31685,31686],[34,664722,664724],{"className":664723},[31792],[34,664725,664727,664747],{"className":664726},[31796,31797],[34,664728,664730,664744],{"className":664729},[31801],[34,664731,664733],{"className":664732,"style":31806},[31805],[34,664734,664735,664738],{"style":217742},[34,664736],{"className":664737,"style":31814},[31813],[34,664739,664741],{"className":664740},[31818,31819,31820,31821],[34,664742,768],{"className":664743},[31685,31821],[34,664745,31829],{"className":664746},[31828],[34,664748,664750],{"className":664749},[31801],[34,664751,664753],{"className":664752,"style":31836},[31805],[34,664754],{},[34,664756],{"className":664757,"style":31846},[31845],[34,664759,202],{"className":664760},[31850],[34,664762],{"className":664763,"style":31846},[31845],[34,664765,664767,664770,664827,664830,664833,664862,664865,664868,664871,664874,664877],{"className":664766},[31676],[34,664768],{"className":664769,"style":566818},[31680],[34,664771,664773,664776],{"className":664772},[32767],[34,664774,546295],{"className":664775,"style":546331},[32767,131249,156560],[34,664777,664779],{"className":664778},[31792],[34,664780,664782,664819],{"className":664781},[31796,31797],[34,664783,664785,664816],{"className":664784},[31801],[34,664786,664788,664799],{"className":664787,"style":552657},[31805],[34,664789,664790,664793],{"style":546347},[34,664791],{"className":664792,"style":31814},[31813],[34,664794,664796],{"className":664795},[31818,31819,31820,31821],[34,664797,34386],{"className":664798,"style":34427},[31685,31686,31821],[34,664800,664801,664804],{"style":546362},[34,664802],{"className":664803,"style":31814},[31813],[34,664805,664807],{"className":664806},[31818,31819,31820,31821],[34,664808,664810,664813],{"className":664809},[31685,31821],[34,664811,32226],{"className":664812},[31685,31821],[34,664814,34386],{"className":664815,"style":34427},[31685,31686,31821],[34,664817,31829],{"className":664818},[31828],[34,664820,664822],{"className":664821},[31801],[34,664823,664825],{"className":664824,"style":546384},[31805],[34,664826],{},[34,664828,456],{"className":664829},[31782],[34,664831,459],{"className":664832},[31685],[34,664834,664836,664839],{"className":664835},[31690],[34,664837,2557],{"className":664838},[31690],[34,664840,664842],{"className":664841},[31792],[34,664843,664845],{"className":664844},[31796],[34,664846,664848],{"className":664847},[31801],[34,664849,664851],{"className":664850,"style":37774},[31805],[34,664852,664853,664856],{"style":37777},[34,664854],{"className":664855,"style":31814},[31813],[34,664857,664859],{"className":664858},[31818,31819,31820,31821],[34,664860,768],{"className":664861},[31685,31821],[34,664863],{"className":664864,"style":32123},[31845],[34,664866,46900],{"className":664867},[31685,207923],[34,664869,32936],{"className":664870},[31685,31686],[34,664872],{"className":664873,"style":32604},[31845],[34,664875,223],{"className":664876},[32024],[34,664878],{"className":664879,"style":32604},[31845],[34,664881,664883,664886,664915,664918,664921],{"className":664882},[31676],[34,664884],{"className":664885,"style":37774},[31680],[34,664887,664889,664892],{"className":664888},[31685],[34,664890,32936],{"className":664891},[31685,31686],[34,664893,664895],{"className":664894},[31792],[34,664896,664898],{"className":664897},[31796],[34,664899,664901],{"className":664900},[31801],[34,664902,664904],{"className":664903,"style":37774},[31805],[34,664905,664906,664909],{"style":37777},[34,664907],{"className":664908,"style":31814},[31813],[34,664910,664912],{"className":664911},[31818,31819,31820,31821],[34,664913,768],{"className":664914},[31685,31821],[34,664916],{"className":664917,"style":32604},[31845],[34,664919,71191],{"className":664920},[32024],[34,664922],{"className":664923,"style":32604},[31845],[34,664925,664927,664930,664933,664936,664939,664942,664945],{"className":664926},[31676],[34,664928],{"className":664929,"style":31774},[31680],[34,664931,456],{"className":664932},[31782],[34,664934,459],{"className":664935},[31685],[34,664937,2557],{"className":664938},[31690],[34,664940],{"className":664941,"style":31846},[31845],[34,664943,202],{"className":664944},[31850],[34,664946],{"className":664947,"style":31846},[31845],[34,664949,664951,664954,665011,665014,665017,665020,665023,665026,665029,665032],{"className":664950},[31676],[34,664952],{"className":664953,"style":566818},[31680],[34,664955,664957,664960],{"className":664956},[32767],[34,664958,546295],{"className":664959,"style":546331},[32767,131249,156560],[34,664961,664963],{"className":664962},[31792],[34,664964,664966,665003],{"className":664965},[31796,31797],[34,664967,664969,665000],{"className":664968},[31801],[34,664970,664972,664983],{"className":664971,"style":552657},[31805],[34,664973,664974,664977],{"style":546347},[34,664975],{"className":664976,"style":31814},[31813],[34,664978,664980],{"className":664979},[31818,31819,31820,31821],[34,664981,34386],{"className":664982,"style":34427},[31685,31686,31821],[34,664984,664985,664988],{"style":546362},[34,664986],{"className":664987,"style":31814},[31813],[34,664989,664991],{"className":664990},[31818,31819,31820,31821],[34,664992,664994,664997],{"className":664993},[31685,31821],[34,664995,32226],{"className":664996},[31685,31821],[34,664998,34386],{"className":664999,"style":34427},[31685,31686,31821],[34,665001,31829],{"className":665002},[31828],[34,665004,665006],{"className":665005},[31801],[34,665007,665009],{"className":665008,"style":546384},[31805],[34,665010],{},[34,665012],{"className":665013,"style":32123},[31845],[34,665015,459],{"className":665016},[31685],[34,665018],{"className":665019,"style":32123},[31845],[34,665021,46900],{"className":665022},[31685,207923],[34,665024,32936],{"className":665025},[31685,31686],[34,665027],{"className":665028,"style":31846},[31845],[34,665030,202],{"className":665031},[31850],[34,665033],{"className":665034,"style":31846},[31845],[34,665036,665038,665041],{"className":665037},[31676],[34,665039],{"className":665040,"style":32617},[31680],[34,665042,459],{"className":665043},[31685],[15,665045,665046],{},[2627,665047,62532],{},[105,665049,665050,665191,665222],{},[108,665051,665052,665053,347],{},"沿上半圆周积分结果为 ",[34,665054,665056,665082],{"className":665055},[31642],[34,665057,665059],{"className":665058},[31646],[31648,665060,665061],{"xmlns":31650},[31652,665062,665063,665079],{},[31655,665064,665065,665067,665073],{},[31662,665066,32226],{},[41453,665068,665069,665071],{},[31749,665070,15932],{},[31749,665072,786],{},[37624,665074,665075,665077],{},[31658,665076,34386],{},[31749,665078,786],{},[31665,665080,665081],{"encoding":31667},"-\\frac{4}{3}R^3",[34,665083,665085],{"className":665084,"ariaHidden":758},[31672],[34,665086,665088,665091,665094,665162],{"className":665087},[31676],[34,665089],{"className":665090,"style":46727},[31680],[34,665092,32226],{"className":665093},[31685],[34,665095,665097,665100,665159],{"className":665096},[31685],[34,665098],{"className":665099},[31782,41488],[34,665101,665103],{"className":665102},[41453],[34,665104,665106,665151],{"className":665105},[31796,31797],[34,665107,665109,665148],{"className":665108},[31801],[34,665110,665112,665126,665134],{"className":665111,"style":46746},[31805],[34,665113,665114,665117],{"style":41504},[34,665115],{"className":665116,"style":41508},[31813],[34,665118,665120],{"className":665119},[31818,31819,31820,31821],[34,665121,665123],{"className":665122},[31685,31821],[34,665124,786],{"className":665125},[31685,31821],[34,665127,665128,665131],{"style":41520},[34,665129],{"className":665130,"style":41508},[31813],[34,665132],{"className":665133,"style":41528},[41527],[34,665135,665136,665139],{"style":46771},[34,665137],{"className":665138,"style":41508},[31813],[34,665140,665142],{"className":665141},[31818,31819,31820,31821],[34,665143,665145],{"className":665144},[31685,31821],[34,665146,15932],{"className":665147},[31685,31821],[34,665149,31829],{"className":665150},[31828],[34,665152,665154],{"className":665153},[31801],[34,665155,665157],{"className":665156,"style":41619},[31805],[34,665158],{},[34,665160],{"className":665161},[31690,41488],[34,665163,665165,665168],{"className":665164},[31685],[34,665166,34386],{"className":665167,"style":34427},[31685,31686],[34,665169,665171],{"className":665170},[31792],[34,665172,665174],{"className":665173},[31796],[34,665175,665177],{"className":665176},[31801],[34,665178,665180],{"className":665179,"style":37774},[31805],[34,665181,665182,665185],{"style":37777},[34,665183],{"className":665184,"style":31814},[31813],[34,665186,665188],{"className":665187},[31818,31819,31820,31821],[34,665189,786],{"className":665190},[31685,31821],[108,665192,665193,665194,347],{},"沿直线段积分结果为 ",[34,665195,665197,665210],{"className":665196},[31642],[34,665198,665200],{"className":665199},[31646],[31648,665201,665202],{"xmlns":31650},[31652,665203,665204,665208],{},[31655,665205,665206],{},[31749,665207,459],{},[31665,665209,459],{"encoding":31667},[34,665211,665213],{"className":665212,"ariaHidden":758},[31672],[34,665214,665216,665219],{"className":665215},[31676],[34,665217],{"className":665218,"style":32617},[31680],[34,665220,459],{"className":665221},[31685],[108,665223,665224,665226],{},[2627,665225,496800],{},"：这也说明了该积分与路径有关。",[32188,665228],{},[32490,665230,665232],{"id":665231},"题目例-1422","题目：例 14.2.2",[15,665234,665235,665353,665354,665382],{},[2627,665236,665237,665238,665266,665267,138176,665295,665323,665324,665352],{},"求空间中一质量为 ",[34,665239,665241,665254],{"className":665240},[31642],[34,665242,665244],{"className":665243},[31646],[31648,665245,665246],{"xmlns":31650},[31652,665247,665248,665252],{},[31655,665249,665250],{},[31658,665251,31759],{},[31665,665253,31759],{"encoding":31667},[34,665255,665257],{"className":665256,"ariaHidden":758},[31672],[34,665258,665260,665263],{"className":665259},[31676],[34,665261],{"className":665262,"style":32527},[31680],[34,665264,31759],{"className":665265},[31685,31686]," 的物体沿某一光滑曲线 ",[34,665268,665270,665283],{"className":665269},[31642],[34,665271,665273],{"className":665272},[31646],[31648,665274,665275],{"xmlns":31650},[31652,665276,665277,665281],{},[31655,665278,665279],{},[31658,665280,33206],{},[31665,665282,33206],{"encoding":31667},[34,665284,665286],{"className":665285,"ariaHidden":758},[31672],[34,665287,665289,665292],{"className":665288},[31676],[34,665290],{"className":665291,"style":33218},[31680],[34,665293,33206],{"className":665294},[31685,31686],[34,665296,665298,665311],{"className":665297},[31642],[34,665299,665301],{"className":665300},[31646],[31648,665302,665303],{"xmlns":31650},[31652,665304,665305,665309],{},[31655,665306,665307],{},[31658,665308,54151],{},[31665,665310,54151],{"encoding":31667},[34,665312,665314],{"className":665313,"ariaHidden":758},[31672],[34,665315,665317,665320],{"className":665316},[31676],[34,665318],{"className":665319,"style":33218},[31680],[34,665321,54151],{"className":665322},[31685,31686]," 点移动到 ",[34,665325,665327,665340],{"className":665326},[31642],[34,665328,665330],{"className":665329},[31646],[31648,665331,665332],{"xmlns":31650},[31652,665333,665334,665338],{},[31655,665335,665336],{},[31658,665337,54224],{},[31665,665339,54224],{"encoding":31667},[34,665341,665343],{"className":665342,"ariaHidden":758},[31672],[34,665344,665346,665349],{"className":665345},[31676],[34,665347],{"className":665348,"style":33218},[31680],[34,665350,54224],{"className":665351,"style":54252},[31685,31686]," 点时，重力所做的功。","\n（假设 ",[34,665355,665357,665370],{"className":665356},[31642],[34,665358,665360],{"className":665359},[31646],[31648,665361,665362],{"xmlns":31650},[31652,665363,665364,665368],{},[31655,665365,665366],{},[31658,665367,382333],{},[31665,665369,382333],{"encoding":31667},[34,665371,665373],{"className":665372,"ariaHidden":758},[31672],[34,665374,665376,665379],{"className":665375},[31676],[34,665377],{"className":665378,"style":32527},[31680],[34,665380,382333],{"className":665381,"style":382406},[31685,31686]," 轴竖直向上，重力方向向下）",[32490,665384,658913],{"id":665385},"详细解答-1",[15,665387,665388],{},"这是一道物理应用题，实际上是考查第二类曲线积分的物理意义。",[2230,665390,665391,665770,666545],{},[108,665392,665393,665396,665397,665427,665428,665462,665463,665491,665492],{},[2627,665394,665395],{},"分析力的向量","\n重力 ",[34,665398,665400,665414],{"className":665399},[31642],[34,665401,665403],{"className":665402},[31646],[31648,665404,665405],{"xmlns":31650},[31652,665406,665407,665411],{},[31655,665408,665409],{},[31658,665410,51834],{"mathvariant":226163},[31665,665412,665413],{"encoding":31667},"\\mathbf{F}",[34,665415,665417],{"className":665416,"ariaHidden":758},[31672],[34,665418,665420,665424],{"className":665419},[31676],[34,665421],{"className":665422,"style":665423},[31680],"height:0.6861em;",[34,665425,51834],{"className":665426},[31685,226302]," 的大小是 ",[34,665429,665431,665447],{"className":665430},[31642],[34,665432,665434],{"className":665433},[31646],[31648,665435,665436],{"xmlns":31650},[31652,665437,665438,665444],{},[31655,665439,665440,665442],{},[31658,665441,31759],{},[31658,665443,31942],{},[31665,665445,665446],{"encoding":31667},"mg",[34,665448,665450],{"className":665449,"ariaHidden":758},[31672],[34,665451,665453,665456,665459],{"className":665452},[31676],[34,665454],{"className":665455,"style":38774},[31680],[34,665457,31759],{"className":665458},[31685,31686],[34,665460,31942],{"className":665461,"style":32062},[31685,31686],"，方向指向 ",[34,665464,665466,665479],{"className":665465},[31642],[34,665467,665469],{"className":665468},[31646],[31648,665470,665471],{"xmlns":31650},[31652,665472,665473,665477],{},[31655,665474,665475],{},[31658,665476,382333],{},[31665,665478,382333],{"encoding":31667},[34,665480,665482],{"className":665481,"ariaHidden":758},[31672],[34,665483,665485,665488],{"className":665484},[31676],[34,665486],{"className":665487,"style":32527},[31680],[34,665489,382333],{"className":665490,"style":382406},[31685,31686]," 轴负方向。\n写成向量坐标形式：\n",[34,665493,665495,665557],{"className":665494},[31642],[34,665496,665498],{"className":665497},[31646],[31648,665499,665500],{"xmlns":31650},[31652,665501,665502,665554],{},[31655,665503,665504,665506,665508,665510,665516,665518,665524,665526,665532,665534,665536,665538,665540,665542,665544,665546,665548,665550,665552],{},[31658,665505,51834],{"mathvariant":226163},[31662,665507,202],{},[31662,665509,456],{"stretchy":702},[31743,665511,665512,665514],{},[31658,665513,51834],{},[31658,665515,32936],{},[31662,665517,31960],{"separator":758},[31743,665519,665520,665522],{},[31658,665521,51834],{},[31658,665523,67764],{},[31662,665525,31960],{"separator":758},[31743,665527,665528,665530],{},[31658,665529,51834],{},[31658,665531,382333],{},[31662,665533,2557],{"stretchy":702},[31662,665535,202],{},[31662,665537,456],{"stretchy":702},[31749,665539,459],{},[31662,665541,31960],{"separator":758},[31749,665543,459],{},[31662,665545,31960],{"separator":758},[31662,665547,32226],{},[31658,665549,31759],{},[31658,665551,31942],{},[31662,665553,2557],{"stretchy":702},[31665,665555,665556],{"encoding":31667},"\\mathbf{F} = (F_x, F_y, F_z) = (0, 0, -mg)",[34,665558,665560,665578,665731],{"className":665559,"ariaHidden":758},[31672],[34,665561,665563,665566,665569,665572,665575],{"className":665562},[31676],[34,665564],{"className":665565,"style":665423},[31680],[34,665567,51834],{"className":665568},[31685,226302],[34,665570],{"className":665571,"style":31846},[31845],[34,665573,202],{"className":665574},[31850],[34,665576],{"className":665577,"style":31846},[31845],[34,665579,665581,665584,665587,665627,665630,665633,665673,665676,665679,665719,665722,665725,665728],{"className":665580},[31676],[34,665582],{"className":665583,"style":32058},[31680],[34,665585,456],{"className":665586},[31782],[34,665588,665590,665593],{"className":665589},[31685],[34,665591,51834],{"className":665592,"style":37422},[31685,31686],[34,665594,665596],{"className":665595},[31792],[34,665597,665599,665619],{"className":665598},[31796,31797],[34,665600,665602,665616],{"className":665601},[31801],[34,665603,665605],{"className":665604,"style":32362},[31805],[34,665606,665607,665610],{"style":214424},[34,665608],{"className":665609,"style":31814},[31813],[34,665611,665613],{"className":665612},[31818,31819,31820,31821],[34,665614,32936],{"className":665615},[31685,31686,31821],[34,665617,31829],{"className":665618},[31828],[34,665620,665622],{"className":665621},[31801],[34,665623,665625],{"className":665624,"style":31836},[31805],[34,665626],{},[34,665628,31960],{"className":665629},[32119],[34,665631],{"className":665632,"style":32123},[31845],[34,665634,665636,665639],{"className":665635},[31685],[34,665637,51834],{"className":665638,"style":37422},[31685,31686],[34,665640,665642],{"className":665641},[31792],[34,665643,665645,665665],{"className":665644},[31796,31797],[34,665646,665648,665662],{"className":665647},[31801],[34,665649,665651],{"className":665650,"style":32362},[31805],[34,665652,665653,665656],{"style":214424},[34,665654],{"className":665655,"style":31814},[31813],[34,665657,665659],{"className":665658},[31818,31819,31820,31821],[34,665660,67764],{"className":665661,"style":32062},[31685,31686,31821],[34,665663,31829],{"className":665664},[31828],[34,665666,665668],{"className":665667},[31801],[34,665669,665671],{"className":665670,"style":32163},[31805],[34,665672],{},[34,665674,31960],{"className":665675},[32119],[34,665677],{"className":665678,"style":32123},[31845],[34,665680,665682,665685],{"className":665681},[31685],[34,665683,51834],{"className":665684,"style":37422},[31685,31686],[34,665686,665688],{"className":665687},[31792],[34,665689,665691,665711],{"className":665690},[31796,31797],[34,665692,665694,665708],{"className":665693},[31801],[34,665695,665697],{"className":665696,"style":32362},[31805],[34,665698,665699,665702],{"style":214424},[34,665700],{"className":665701,"style":31814},[31813],[34,665703,665705],{"className":665704},[31818,31819,31820,31821],[34,665706,382333],{"className":665707,"style":382406},[31685,31686,31821],[34,665709,31829],{"className":665710},[31828],[34,665712,665714],{"className":665713},[31801],[34,665715,665717],{"className":665716,"style":31836},[31805],[34,665718],{},[34,665720,2557],{"className":665721},[31690],[34,665723],{"className":665724,"style":31846},[31845],[34,665726,202],{"className":665727},[31850],[34,665729],{"className":665730,"style":31846},[31845],[34,665732,665734,665737,665740,665743,665746,665749,665752,665755,665758,665761,665764,665767],{"className":665733},[31676],[34,665735],{"className":665736,"style":31774},[31680],[34,665738,456],{"className":665739},[31782],[34,665741,459],{"className":665742},[31685],[34,665744,31960],{"className":665745},[32119],[34,665747],{"className":665748,"style":32123},[31845],[34,665750,459],{"className":665751},[31685],[34,665753,31960],{"className":665754},[32119],[34,665756],{"className":665757,"style":32123},[31845],[34,665759,32226],{"className":665760},[31685],[34,665762,31759],{"className":665763},[31685,31686],[34,665765,31942],{"className":665766,"style":32062},[31685,31686],[34,665768,2557],{"className":665769},[31690],[108,665771,665772,665775,665776,665804,665805,666215,666216],{},[2627,665773,665774],{},"建立功的积分表达式","\n功 ",[34,665777,665779,665792],{"className":665778},[31642],[34,665780,665782],{"className":665781},[31646],[31648,665783,665784],{"xmlns":31650},[31652,665785,665786,665790],{},[31655,665787,665788],{},[31658,665789,56852],{},[31665,665791,56852],{"encoding":31667},[34,665793,665795],{"className":665794,"ariaHidden":758},[31672],[34,665796,665798,665801],{"className":665797},[31676],[34,665799],{"className":665800,"style":33218},[31680],[34,665802,56852],{"className":665803,"style":37422},[31685,31686]," 是力在路径上的线积分：\n",[34,665806,665808,665886],{"className":665807},[31642],[34,665809,665811],{"className":665810},[31646],[31648,665812,665813],{"xmlns":31650},[31652,665814,665815,665883],{},[31655,665816,665817,665819,665821,665827,665829,665831,665833,665835,665837,665843,665849,665851,665853,665855,665857,665863,665865,665867,665869,665871,665877,665879,665881],{},[31658,665818,56852],{},[31662,665820,202],{},[31743,665822,665823,665825],{},[31662,665824,546295],{},[31658,665826,33206],{},[31658,665828,51834],{"mathvariant":226163},[31662,665830,71191],{},[31658,665832,46900],{"mathvariant":38079},[31658,665834,34395],{"mathvariant":226163},[31662,665836,202],{},[31743,665838,665839,665841],{},[31662,665840,546295],{},[31658,665842,33206],{},[31743,665844,665845,665847],{},[31658,665846,51834],{},[31658,665848,32936],{},[39361,665850,39363],{},[31658,665852,46900],{"mathvariant":38079},[31658,665854,32936],{},[31662,665856,223],{},[31743,665858,665859,665861],{},[31658,665860,51834],{},[31658,665862,67764],{},[39361,665864,39363],{},[31658,665866,46900],{"mathvariant":38079},[31658,665868,67764],{},[31662,665870,223],{},[31743,665872,665873,665875],{},[31658,665874,51834],{},[31658,665876,382333],{},[39361,665878,39363],{},[31658,665880,46900],{"mathvariant":38079},[31658,665882,382333],{},[31665,665884,665885],{"encoding":31667},"W = \\int_L \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{r} = \\int_L F_x \\, \\mathrm{d}x + F_y \\, \\mathrm{d}y + F_z \\, \\mathrm{d}z",[34,665887,665889,665907,665968,665989,666096,666160],{"className":665888,"ariaHidden":758},[31672],[34,665890,665892,665895,665898,665901,665904],{"className":665891},[31676],[34,665893],{"className":665894,"style":33218},[31680],[34,665896,56852],{"className":665897,"style":37422},[31685,31686],[34,665899],{"className":665900,"style":31846},[31845],[34,665902,202],{"className":665903},[31850],[34,665905],{"className":665906,"style":31846},[31845],[34,665908,665910,665913,665953,665956,665959,665962,665965],{"className":665909},[31676],[34,665911],{"className":665912,"style":637320},[31680],[34,665914,665916,665919],{"className":665915},[32767],[34,665917,546295],{"className":665918,"style":546331},[32767,131249,156560],[34,665920,665922],{"className":665921},[31792],[34,665923,665925,665945],{"className":665924},[31796,31797],[34,665926,665928,665942],{"className":665927},[31801],[34,665929,665931],{"className":665930,"style":637339},[31805],[34,665932,665933,665936],{"style":546347},[34,665934],{"className":665935,"style":31814},[31813],[34,665937,665939],{"className":665938},[31818,31819,31820,31821],[34,665940,33206],{"className":665941},[31685,31686,31821],[34,665943,31829],{"className":665944},[31828],[34,665946,665948],{"className":665947},[31801],[34,665949,665951],{"className":665950,"style":546384},[31805],[34,665952],{},[34,665954],{"className":665955,"style":32123},[31845],[34,665957,51834],{"className":665958},[31685,226302],[34,665960],{"className":665961,"style":32604},[31845],[34,665963,71191],{"className":665964},[32024],[34,665966],{"className":665967,"style":32604},[31845],[34,665969,665971,665974,665977,665980,665983,665986],{"className":665970},[31676],[34,665972],{"className":665973,"style":31681},[31680],[34,665975,46900],{"className":665976},[31685,207923],[34,665978,34395],{"className":665979},[31685,226302],[34,665981],{"className":665982,"style":31846},[31845],[34,665984,202],{"className":665985},[31850],[34,665987],{"className":665988,"style":31846},[31845],[34,665990,665992,665995,666035,666038,666078,666081,666084,666087,666090,666093],{"className":665991},[31676],[34,665993],{"className":665994,"style":637320},[31680],[34,665996,665998,666001],{"className":665997},[32767],[34,665999,546295],{"className":666000,"style":546331},[32767,131249,156560],[34,666002,666004],{"className":666003},[31792],[34,666005,666007,666027],{"className":666006},[31796,31797],[34,666008,666010,666024],{"className":666009},[31801],[34,666011,666013],{"className":666012,"style":637339},[31805],[34,666014,666015,666018],{"style":546347},[34,666016],{"className":666017,"style":31814},[31813],[34,666019,666021],{"className":666020},[31818,31819,31820,31821],[34,666022,33206],{"className":666023},[31685,31686,31821],[34,666025,31829],{"className":666026},[31828],[34,666028,666030],{"className":666029},[31801],[34,666031,666033],{"className":666032,"style":546384},[31805],[34,666034],{},[34,666036],{"className":666037,"style":32123},[31845],[34,666039,666041,666044],{"className":666040},[31685],[34,666042,51834],{"className":666043,"style":37422},[31685,31686],[34,666045,666047],{"className":666046},[31792],[34,666048,666050,666070],{"className":666049},[31796,31797],[34,666051,666053,666067],{"className":666052},[31801],[34,666054,666056],{"className":666055,"style":32362},[31805],[34,666057,666058,666061],{"style":214424},[34,666059],{"className":666060,"style":31814},[31813],[34,666062,666064],{"className":666063},[31818,31819,31820,31821],[34,666065,32936],{"className":666066},[31685,31686,31821],[34,666068,31829],{"className":666069},[31828],[34,666071,666073],{"className":666072},[31801],[34,666074,666076],{"className":666075,"style":31836},[31805],[34,666077],{},[34,666079],{"className":666080,"style":32123},[31845],[34,666082,46900],{"className":666083},[31685,207923],[34,666085,32936],{"className":666086},[31685,31686],[34,666088],{"className":666089,"style":32604},[31845],[34,666091,223],{"className":666092},[32024],[34,666094],{"className":666095,"style":32604},[31845],[34,666097,666099,666102,666142,666145,666148,666151,666154,666157],{"className":666098},[31676],[34,666100],{"className":666101,"style":141223},[31680],[34,666103,666105,666108],{"className":666104},[31685],[34,666106,51834],{"className":666107,"style":37422},[31685,31686],[34,666109,666111],{"className":666110},[31792],[34,666112,666114,666134],{"className":666113},[31796,31797],[34,666115,666117,666131],{"className":666116},[31801],[34,666118,666120],{"className":666119,"style":32362},[31805],[34,666121,666122,666125],{"style":214424},[34,666123],{"className":666124,"style":31814},[31813],[34,666126,666128],{"className":666127},[31818,31819,31820,31821],[34,666129,67764],{"className":666130,"style":32062},[31685,31686,31821],[34,666132,31829],{"className":666133},[31828],[34,666135,666137],{"className":666136},[31801],[34,666138,666140],{"className":666139,"style":32163},[31805],[34,666141],{},[34,666143],{"className":666144,"style":32123},[31845],[34,666146,46900],{"className":666147},[31685,207923],[34,666149,67764],{"className":666150,"style":32062},[31685,31686],[34,666152],{"className":666153,"style":32604},[31845],[34,666155,223],{"className":666156},[32024],[34,666158],{"className":666159,"style":32604},[31845],[34,666161,666163,666166,666206,666209,666212],{"className":666162},[31676],[34,666164],{"className":666165,"style":33814},[31680],[34,666167,666169,666172],{"className":666168},[31685],[34,666170,51834],{"className":666171,"style":37422},[31685,31686],[34,666173,666175],{"className":666174},[31792],[34,666176,666178,666198],{"className":666177},[31796,31797],[34,666179,666181,666195],{"className":666180},[31801],[34,666182,666184],{"className":666183,"style":32362},[31805],[34,666185,666186,666189],{"style":214424},[34,666187],{"className":666188,"style":31814},[31813],[34,666190,666192],{"className":666191},[31818,31819,31820,31821],[34,666193,382333],{"className":666194,"style":382406},[31685,31686,31821],[34,666196,31829],{"className":666197},[31828],[34,666199,666201],{"className":666200},[31801],[34,666202,666204],{"className":666203,"style":31836},[31805],[34,666205],{},[34,666207],{"className":666208,"style":32123},[31845],[34,666210,46900],{"className":666211},[31685,207923],[34,666213,382333],{"className":666214,"style":382406},[31685,31686],"\n代入力的分量：\n",[34,666217,666219,666297],{"className":666218},[31642],[34,666220,666222],{"className":666221},[31646],[31648,666223,666224],{"xmlns":31650},[31652,666225,666226,666294],{},[31655,666227,666228,666230,666232,666238,666240,666242,666244,666246,666248,666250,666252,666254,666256,666258,666260,666262,666264,666266,666268,666270,666272,666274,666276,666282,666284,666286,666288,666290,666292],{},[31658,666229,56852],{},[31662,666231,202],{},[31743,666233,666234,666236],{},[31662,666235,546295],{},[31658,666237,33206],{},[31749,666239,459],{},[31662,666241,71191],{},[31658,666243,46900],{"mathvariant":38079},[31658,666245,32936],{},[31662,666247,223],{},[31749,666249,459],{},[31662,666251,71191],{},[31658,666253,46900],{"mathvariant":38079},[31658,666255,67764],{},[31662,666257,223],{},[31662,666259,456],{"stretchy":702},[31662,666261,32226],{},[31658,666263,31759],{},[31658,666265,31942],{},[31662,666267,2557],{"stretchy":702},[39361,666269,39363],{},[31658,666271,46900],{"mathvariant":38079},[31658,666273,382333],{},[31662,666275,202],{},[31743,666277,666278,666280],{},[31662,666279,546295],{},[31658,666281,33206],{},[31662,666283,32226],{},[31658,666285,31759],{},[31658,666287,31942],{},[39361,666289,39363],{},[31658,666291,46900],{"mathvariant":38079},[31658,666293,382333],{},[31665,666295,666296],{"encoding":31667},"W = \\int_L 0 \\cdot \\mathrm{d}x + 0 \\cdot \\mathrm{d}y + (-mg) \\, \\mathrm{d}z = \\int_L -mg \\, \\mathrm{d}z",[34,666298,666300,666318,666379,666400,666418,666439,666478],{"className":666299,"ariaHidden":758},[31672],[34,666301,666303,666306,666309,666312,666315],{"className":666302},[31676],[34,666304],{"className":666305,"style":33218},[31680],[34,666307,56852],{"className":666308,"style":37422},[31685,31686],[34,666310],{"className":666311,"style":31846},[31845],[34,666313,202],{"className":666314},[31850],[34,666316],{"className":666317,"style":31846},[31845],[34,666319,666321,666324,666364,666367,666370,666373,666376],{"className":666320},[31676],[34,666322],{"className":666323,"style":637320},[31680],[34,666325,666327,666330],{"className":666326},[32767],[34,666328,546295],{"className":666329,"style":546331},[32767,131249,156560],[34,666331,666333],{"className":666332},[31792],[34,666334,666336,666356],{"className":666335},[31796,31797],[34,666337,666339,666353],{"className":666338},[31801],[34,666340,666342],{"className":666341,"style":637339},[31805],[34,666343,666344,666347],{"style":546347},[34,666345],{"className":666346,"style":31814},[31813],[34,666348,666350],{"className":666349},[31818,31819,31820,31821],[34,666351,33206],{"className":666352},[31685,31686,31821],[34,666354,31829],{"className":666355},[31828],[34,666357,666359],{"className":666358},[31801],[34,666360,666362],{"className":666361,"style":546384},[31805],[34,666363],{},[34,666365],{"className":666366,"style":32123},[31845],[34,666368,459],{"className":666369},[31685],[34,666371],{"className":666372,"style":32604},[31845],[34,666374,71191],{"className":666375},[32024],[34,666377],{"className":666378,"style":32604},[31845],[34,666380,666382,666385,666388,666391,666394,666397],{"className":666381},[31676],[34,666383],{"className":666384,"style":34299},[31680],[34,666386,46900],{"className":666387},[31685,207923],[34,666389,32936],{"className":666390},[31685,31686],[34,666392],{"className":666393,"style":32604},[31845],[34,666395,223],{"className":666396},[32024],[34,666398],{"className":666399,"style":32604},[31845],[34,666401,666403,666406,666409,666412,666415],{"className":666402},[31676],[34,666404],{"className":666405,"style":32617},[31680],[34,666407,459],{"className":666408},[31685],[34,666410],{"className":666411,"style":32604},[31845],[34,666413,71191],{"className":666414},[32024],[34,666416],{"className":666417,"style":32604},[31845],[34,666419,666421,666424,666427,666430,666433,666436],{"className":666420},[31676],[34,666422],{"className":666423,"style":43217},[31680],[34,666425,46900],{"className":666426},[31685,207923],[34,666428,67764],{"className":666429,"style":32062},[31685,31686],[34,666431],{"className":666432,"style":32604},[31845],[34,666434,223],{"className":666435},[32024],[34,666437],{"className":666438,"style":32604},[31845],[34,666440,666442,666445,666448,666451,666454,666457,666460,666463,666466,666469,666472,666475],{"className":666441},[31676],[34,666443],{"className":666444,"style":31774},[31680],[34,666446,456],{"className":666447},[31782],[34,666449,32226],{"className":666450},[31685],[34,666452,31759],{"className":666453},[31685,31686],[34,666455,31942],{"className":666456,"style":32062},[31685,31686],[34,666458,2557],{"className":666459},[31690],[34,666461],{"className":666462,"style":32123},[31845],[34,666464,46900],{"className":666465},[31685,207923],[34,666467,382333],{"className":666468,"style":382406},[31685,31686],[34,666470],{"className":666471,"style":31846},[31845],[34,666473,202],{"className":666474},[31850],[34,666476],{"className":666477,"style":31846},[31845],[34,666479,666481,666484,666524,666527,666530,666533,666536,666539,666542],{"className":666480},[31676],[34,666482],{"className":666483,"style":637320},[31680],[34,666485,666487,666490],{"className":666486},[32767],[34,666488,546295],{"className":666489,"style":546331},[32767,131249,156560],[34,666491,666493],{"className":666492},[31792],[34,666494,666496,666516],{"className":666495},[31796,31797],[34,666497,666499,666513],{"className":666498},[31801],[34,666500,666502],{"className":666501,"style":637339},[31805],[34,666503,666504,666507],{"style":546347},[34,666505],{"className":666506,"style":31814},[31813],[34,666508,666510],{"className":666509},[31818,31819,31820,31821],[34,666511,33206],{"className":666512},[31685,31686,31821],[34,666514,31829],{"className":666515},[31828],[34,666517,666519],{"className":666518},[31801],[34,666520,666522],{"className":666521,"style":546384},[31805],[34,666523],{},[34,666525],{"className":666526,"style":32123},[31845],[34,666528,32226],{"className":666529},[31685],[34,666531,31759],{"className":666532},[31685,31686],[34,666534,31942],{"className":666535,"style":32062},[31685,31686],[34,666537],{"className":666538,"style":32123},[31845],[34,666540,46900],{"className":666541},[31685,207923],[34,666543,382333],{"className":666544,"style":382406},[31685,31686],[108,666546,666547,666550,666551,666579,666580,666769,666770,666579,666798,666986,666987,667015,667016,667243,667614,667615],{},[2627,6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",[34,666552,666554,666567],{"className":666553},[31642],[34,666555,666557],{"className":666556},[31646],[31648,666558,666559],{"xmlns":31650},[31652,666560,666561,666565],{},[31655,666562,666563],{},[31658,666564,54151],{},[31665,666566,54151],{"encoding":31667},[34,666568,666570],{"className":666569,"ariaHidden":758},[31672],[34,666571,666573,666576],{"className":666572},[31676],[34,666574],{"className":666575,"style":33218},[31680],[34,666577,54151],{"className":666578},[31685,31686]," 的坐标为 ",[34,666581,666583,666621],{"className":666582},[31642],[34,666584,666586],{"className":666585},[31646],[31648,666587,666588],{"xmlns":31650},[31652,666589,666590,666618],{},[31655,666591,666592,666594,666600,666602,666608,666610,666616],{},[31662,666593,456],{"stretchy":702},[31743,666595,666596,666598],{},[31658,666597,32936],{},[31749,666599,640],{},[31662,666601,31960],{"separator":758},[31743,666603,666604,666606],{},[31658,666605,67764],{},[31749,666607,640],{},[31662,666609,31960],{"separator":758},[31743,666611,666612,666614],{},[31658,666613,382333],{},[31749,666615,640],{},[31662,666617,2557],{"stretchy":702},[31665,666619,666620],{"encoding":31667},"(x_1, y_1, z_1)",[34,666622,666624],{"className":666623,"ariaHidden":758},[31672],[34,666625,666627,666630,666633,666673,666676,666679,666719,666722,666725,666766],{"className":666626},[31676],[34,666628],{"className":666629,"style":31774},[31680],[34,666631,456],{"className":666632},[31782],[34,666634,666636,666639],{"className":666635},[31685],[34,666637,32936],{"className":666638},[31685,31686],[34,666640,666642],{"className":666641},[31792],[34,666643,666645,666665],{"className":666644},[31796,31797],[34,666646,666648,666662],{"className":666647},[31801],[34,666649,666651],{"className":666650,"style":31806},[31805],[34,666652,666653,666656],{"style":31809},[34,666654],{"className":666655,"style":31814},[31813],[34,666657,666659],{"className":666658},[31818,31819,31820,31821],[34,666660,640],{"className":666661},[31685,31821],[34,666663,31829],{"className":666664},[31828],[34,666666,666668],{"className":666667},[31801],[34,666669,666671],{"className":666670,"style":31836},[31805],[34,666672],{},[34,666674,31960],{"className":666675},[32119],[34,666677],{"className":666678,"style":32123},[31845],[34,666680,666682,666685],{"className":666681},[31685],[34,666683,67764],{"className":666684,"style":32062},[31685,31686],[34,666686,666688],{"className":666687},[31792],[34,666689,666691,666711],{"className":666690},[31796,31797],[34,666692,666694,666708],{"className":666693},[31801],[34,666695,666697],{"className":666696,"style":31806},[31805],[34,666698,666699,666702],{"style":38795},[34,666700],{"className":666701,"style":31814},[31813],[34,666703,666705],{"className":666704},[31818,31819,31820,31821],[34,666706,640],{"className":666707},[31685,31821],[34,666709,31829],{"className":666710},[31828],[34,666712,666714],{"className":666713},[31801],[34,666715,666717],{"className":666716,"style":31836},[31805],[34,666718],{},[34,666720,31960],{"className":666721},[32119],[34,666723],{"className":666724,"style":32123},[31845],[34,666726,666728,666731],{"className":666727},[31685],[34,666729,382333],{"className":666730,"style":382406},[31685,31686],[34,666732,666734],{"className":666733},[31792],[34,666735,666737,666758],{"className":666736},[31796,31797],[34,666738,666740,666755],{"className":666739},[31801],[34,666741,666743],{"className":666742,"style":31806},[31805],[34,666744,666746,666749],{"style":666745},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[34,666747],{"className":666748,"style":31814},[31813],[34,666750,666752],{"className":666751},[31818,31819,31820,31821],[34,666753,640],{"className":666754},[31685,31821],[34,666756,31829],{"className":666757},[31828],[34,666759,666761],{"className":666760},[31801],[34,666762,666764],{"className":666763,"style":31836},[31805],[34,666765],{},[34,666767,2557],{"className":666768},[31690],"，终点 ",[34,666771,666773,666786],{"className":666772},[31642],[34,666774,666776],{"className":666775},[31646],[31648,666777,666778],{"xmlns":31650},[31652,666779,666780,666784],{},[31655,666781,666782],{},[31658,666783,54224],{},[31665,666785,54224],{"encoding":31667},[34,666787,666789],{"className":666788,"ariaHidden":758},[31672],[34,666790,666792,666795],{"className":666791},[31676],[34,666793],{"className":666794,"style":33218},[31680],[34,666796,54224],{"className":666797,"style":54252},[31685,31686],[34,666799,666801,666839],{"className":666800},[31642],[34,666802,666804],{"className":666803},[31646],[31648,666805,666806],{"xmlns":31650},[31652,666807,666808,666836],{},[31655,666809,666810,666812,666818,666820,666826,666828,666834],{},[31662,666811,456],{"stretchy":702},[31743,666813,666814,666816],{},[31658,666815,32936],{},[31749,666817,768],{},[31662,666819,31960],{"separator":758},[31743,666821,666822,666824],{},[31658,666823,67764],{},[31749,666825,768],{},[31662,666827,31960],{"separator":758},[31743,666829,666830,666832],{},[31658,666831,382333],{},[31749,666833,768],{},[31662,666835,2557],{"stretchy":702},[31665,666837,666838],{"encoding":31667},"(x_2, y_2, z_2)",[34,666840,666842],{"className":666841,"ariaHidden":758},[31672],[34,666843,666845,666848,666851,666891,666894,666897,666937,666940,666943,666983],{"className":666844},[31676],[34,666846],{"className":666847,"style":31774},[31680],[34,666849,456],{"className":666850},[31782],[34,666852,666854,666857],{"className":666853},[31685],[34,666855,32936],{"className":666856},[31685,31686],[34,666858,666860],{"className":666859},[31792],[34,666861,666863,666883],{"className":666862},[31796,31797],[34,666864,666866,666880],{"className":666865},[31801],[34,666867,666869],{"className":666868,"style":31806},[31805],[34,666870,666871,666874],{"style":31809},[34,666872],{"className":666873,"style":31814},[31813],[34,666875,666877],{"className":666876},[31818,31819,31820,31821],[34,666878,768],{"className":666879},[31685,31821],[34,666881,31829],{"className":666882},[31828],[34,666884,666886],{"className":666885},[31801],[34,666887,666889],{"className":666888,"style":31836},[31805],[34,666890],{},[34,666892,31960],{"className":666893},[32119],[34,666895],{"className":666896,"style":32123},[31845],[34,666898,666900,666903],{"className":666899},[31685],[34,666901,67764],{"className":666902,"style":32062},[31685,31686],[34,666904,666906],{"className":666905},[31792],[34,666907,666909,666929],{"className":666908},[31796,31797],[34,666910,666912,666926],{"className":666911},[31801],[34,666913,666915],{"className":666914,"style":31806},[31805],[34,666916,666917,666920],{"style":38795},[34,666918],{"className":666919,"style":31814},[31813],[34,666921,666923],{"className":666922},[31818,31819,31820,31821],[34,666924,768],{"className":666925},[31685,31821],[34,666927,31829],{"className":666928},[31828],[34,666930,666932],{"className":666931},[31801],[34,666933,666935],{"className":666934,"style":31836},[31805],[34,666936],{},[34,666938,31960],{"className":666939},[32119],[34,666941],{"className":666942,"style":32123},[31845],[34,666944,666946,666949],{"className":666945},[31685],[34,666947,382333],{"className":666948,"style":382406},[31685,31686],[34,666950,666952],{"className":666951},[31792],[34,666953,666955,666975],{"className":666954},[31796,31797],[34,666956,666958,666972],{"className":666957},[31801],[34,666959,666961],{"className":666960,"style":31806},[31805],[34,666962,666963,666966],{"style":666745},[34,666964],{"className":666965,"style":31814},[31813],[34,666967,666969],{"className":666968},[31818,31819,31820,31821],[34,666970,768],{"className":666971},[31685,31821],[34,666973,31829],{"className":666974},[31828],[34,666976,666978],{"className":666977},[31801],[34,666979,666981],{"className":666980,"style":31836},[31805],[34,666982],{},[34,666984,2557],{"className":666985},[31690],"。\n积分变量仅剩下 ",[34,666988,666990,667003],{"className":666989},[31642],[34,666991,666993],{"className":666992},[31646],[31648,666994,666995],{"xmlns":31650},[31652,666996,666997,667001],{},[31655,666998,666999],{},[31658,667000,382333],{},[31665,667002,382333],{"encoding":31667},[34,667004,667006],{"className":667005,"ariaHidden":758},[31672],[34,667007,667009,667012],{"className":667008},[31676],[34,667010],{"className":667011,"style":32527},[31680],[34,667013,382333],{"className":667014,"style":382406},[31685,31686],"，这变成了一个简单的定积分：\n",[34,667017,667019,667061],{"className":667018},[31642],[34,667020,667022],{"className":667021},[31646],[31648,667023,667024],{"xmlns":31650},[31652,667025,667026,667058],{},[31655,667027,667028,667030,667032,667034,667036,667038,667054,667056],{},[31658,667029,56852],{},[31662,667031,202],{},[31662,667033,32226],{},[31658,667035,31759],{},[31658,667037,31942],{},[38846,667039,667040,667042,667048],{},[31662,667041,546295],{},[31743,667043,667044,667046],{},[31658,667045,382333],{},[31749,667047,640],{},[31743,667049,667050,667052],{},[31658,667051,382333],{},[31749,667053,768],{},[31658,667055,46900],{"mathvariant":38079},[31658,667057,382333],{},[31665,667059,667060],{"encoding":31667},"W = -mg \\int_{z_1}^{z_2} \\mathrm{d}z",[34,667062,667064,667082],{"className":667063,"ariaHidden":758},[31672],[34,667065,667067,667070,667073,667076,667079],{"className":667066},[31676],[34,667068],{"className":667069,"style":33218},[31680],[34,667071,56852],{"className":667072,"style":37422},[31685,31686],[34,667074],{"className":667075,"style":31846},[31845],[34,667077,202],{"className":667078},[31850],[34,667080],{"className":667081,"style":31846},[31845],[34,667083,667085,667089,667092,667095,667098,667101,667234,667237,667240],{"className":667084},[31676],[34,667086],{"className":667087,"style":667088},[31680],"height:1.3152em;vertical-align:-0.4559em;",[34,667090,32226],{"className":667091},[31685],[34,667093,31759],{"className":667094},[31685,31686],[34,667096,31942],{"className":667097,"style":32062},[31685,31686],[34,667099],{"className":667100,"style":32123},[31845],[34,667102,667104,667107],{"className":667103},[32767],[34,667105,546295],{"className":667106,"style":546331},[32767,131249,156560],[34,667108,667110],{"className":667109},[31792],[34,667111,667113,667225],{"className":667112},[31796,31797],[34,667114,667116,667222],{"className":667115},[31801],[34,667117,667119,667171],{"className":667118,"style":567928},[31805],[34,667120,667121,667124],{"style":546347},[34,667122],{"className":667123,"style":31814},[31813],[34,667125,667127],{"className":667126},[31818,31819,31820,31821],[34,667128,667130],{"className":667129},[31685,31821],[34,667131,667133,667136],{"className":667132},[31685,31821],[34,667134,382333],{"className":667135,"style":382406},[31685,31686,31821],[34,667137,667139],{"className":667138},[31792],[34,667140,667142,667163],{"className":667141},[31796,31797],[34,667143,667145,667160],{"className":667144},[31801],[34,667146,667148],{"className":667147,"style":318658},[31805],[34,667149,667151,667154],{"style":667150},"top:-2.357em;margin-left:-0.044em;margin-right:0.0714em;",[34,667152],{"className":667153,"style":43406},[31813],[34,667155,667157],{"className":667156},[31818,43410,43411,31821],[34,667158,640],{"className":667159},[31685,31821],[34,667161,31829],{"className":667162},[31828],[34,667164,667166],{"className":667165},[31801],[34,667167,667169],{"className":667168,"style":220540},[31805],[34,667170],{},[34,667172,667173,667176],{"style":546362},[34,667174],{"className":667175,"style":31814},[31813],[34,667177,667179],{"className":667178},[31818,31819,31820,31821],[34,667180,667182],{"className":667181},[31685,31821],[34,667183,667185,667188],{"className":667184},[31685,31821],[34,667186,382333],{"className":667187,"style":382406},[31685,31686,31821],[34,667189,667191],{"className":667190},[31792],[34,667192,667194,667214],{"className":667193},[31796,31797],[34,667195,667197,667211],{"className":667196},[31801],[34,667198,667200],{"className":667199,"style":318658},[31805],[34,667201,667202,667205],{"style":667150},[34,667203],{"className":667204,"style":43406},[31813],[34,667206,667208],{"className":667207},[31818,43410,43411,31821],[34,667209,768],{"className":667210},[31685,31821],[34,667212,31829],{"className":667213},[31828],[34,667215,667217],{"className":667216},[31801],[34,667218,667220],{"className":667219,"style":220540},[31805],[34,667221],{},[34,667223,31829],{"className":667224},[31828],[34,667226,667228],{"className":667227},[31801],[34,667229,667232],{"className":667230,"style":667231},[31805],"height:0.4559em;",[34,667233],{},[34,667235],{"className":667236,"style":32123},[31845],[34,667238,46900],{"className":667239},[31685,207923],[34,667241,382333],{"className":667242,"style":382406},[31685,31686],[34,667244,667246,667314],{"className":667245},[31642],[34,667247,667249],{"className":667248},[31646],[31648,667250,667251],{"xmlns":31650},[31652,667252,667253,667311],{},[31655,667254,667255,667257,667259,667261,667263,667265,667267,667269,667285,667287,667289,667291,667293,667295,667301,667303,667309],{},[31658,667256,56852],{},[31662,667258,202],{},[31662,667260,32226],{},[31658,667262,31759],{},[31658,667264,31942],{},[31662,667266,15972],{"stretchy":702},[31658,667268,382333],{},[38846,667270,667271,667273,667279],{},[31662,667272,32712],{"stretchy":702},[31743,667274,667275,667277],{},[31658,667276,382333],{},[31749,667278,640],{},[31743,667280,667281,667283],{},[31658,667282,382333],{},[31749,667284,768],{},[31662,667286,202],{},[31662,667288,32226],{},[31658,667290,31759],{},[31658,667292,31942],{},[31662,667294,456],{"stretchy":702},[31743,667296,667297,667299],{},[31658,667298,382333],{},[31749,667300,768],{},[31662,667302,32226],{},[31743,667304,667305,667307],{},[31658,667306,382333],{},[31749,667308,640],{},[31662,667310,2557],{"stretchy":702},[31665,667312,667313],{"encoding":31667},"W = -mg [z]_{z_1}^{z_2} = -mg(z_2 - z_1)",[34,667315,667317,667335,667498,667565],{"className":667316,"ariaHidden":758},[31672],[34,667318,667320,667323,667326,667329,667332],{"className":667319},[31676],[34,667321],{"className":667322,"style":33218},[31680],[34,667324,56852],{"className":667325,"style":37422},[31685,31686],[34,667327],{"className":667328,"style":31846},[31845],[34,667330,202],{"className":667331},[31850],[34,667333],{"className":667334,"style":31846},[31845],[34,667336,667338,667342,667345,667348,667351,667354,667357,667489,667492,667495],{"className":667337},[31676],[34,667339],{"className":667340,"style":667341},[31680],"height:1.0971em;vertical-align:-0.3471em;",[34,667343,32226],{"className":667344},[31685],[34,667346,31759],{"className":667347},[31685,31686],[34,667349,31942],{"className":667350,"style":32062},[31685,31686],[34,667352,15972],{"className":667353},[31782],[34,667355,382333],{"className":667356,"style":382406},[31685,31686],[34,667358,667360,667363],{"className":667359},[31690],[34,667361,32712],{"className":667362},[31690],[34,667364,667366],{"className":667365},[31792],[34,667367,667369,667480],{"className":667368},[31796,31797],[34,667370,667372,667477],{"className":667371},[31801],[34,667373,667375,667426],{"className":667374,"style":37959},[31805],[34,667376,667377,667380],{"style":502694},[34,667378],{"className":667379,"style":31814},[31813],[34,667381,667383],{"className":667382},[31818,31819,31820,31821],[34,667384,667386],{"className":667385},[31685,31821],[34,667387,667389,667392],{"className":667388},[31685,31821],[34,667390,382333],{"className":667391,"style":382406},[31685,31686,31821],[34,667393,667395],{"className":667394},[31792],[34,667396,667398,667418],{"className":667397},[31796,31797],[34,667399,667401,667415],{"className":667400},[31801],[34,667402,667404],{"className":667403,"style":318658},[31805],[34,667405,667406,667409],{"style":667150},[34,667407],{"className":667408,"style":43406},[31813],[34,667410,667412],{"className":667411},[31818,43410,43411,31821],[34,667413,640],{"className":667414},[31685,31821],[34,667416,31829],{"className":667417},[31828],[34,667419,667421],{"className":667420},[31801],[34,667422,667424],{"className":667423,"style":220540},[31805],[34,667425],{},[34,667427,667428,667431],{"style":37777},[34,667429],{"className":667430,"style":31814},[31813],[34,667432,667434],{"className":667433},[31818,31819,31820,31821],[34,667435,667437],{"className":667436},[31685,31821],[34,667438,667440,667443],{"className":667439},[31685,31821],[34,667441,382333],{"className":667442,"style":382406},[31685,31686,31821],[34,667444,667446],{"className":667445},[31792],[34,667447,667449,667469],{"className":667448},[31796,31797],[34,667450,667452,667466],{"className":667451},[31801],[34,667453,667455],{"className":667454,"style":318658},[31805],[34,667456,667457,667460],{"style":667150},[34,667458],{"className":667459,"style":43406},[31813],[34,667461,667463],{"className":667462},[31818,43410,43411,31821],[34,667464,768],{"className":667465},[31685,31821],[34,667467,31829],{"className":667468},[31828],[34,667470,667472],{"className":667471},[31801],[34,667473,667475],{"className":667474,"style":220540},[31805],[34,667476],{},[34,667478,31829],{"className":667479},[31828],[34,667481,667483],{"className":667482},[31801],[34,667484,667487],{"className":667485,"style":667486},[31805],"height:0.3471em;",[34,667488],{},[34,667490],{"className":667491,"style":31846},[31845],[34,667493,202],{"className":667494},[31850],[34,667496],{"className":667497,"style":31846},[31845],[34,667499,667501,667504,667507,667510,667513,667516,667556,667559,667562],{"className":667500},[31676],[34,667502],{"className":667503,"style":31774},[31680],[34,667505,32226],{"className":667506},[31685],[34,667508,31759],{"className":667509},[31685,31686],[34,667511,31942],{"className":667512,"style":32062},[31685,31686],[34,667514,456],{"className":667515},[31782],[34,667517,667519,667522],{"className":667518},[31685],[34,667520,382333],{"className":667521,"style":382406},[31685,31686],[34,667523,667525],{"className":667524},[31792],[34,667526,667528,667548],{"className":667527},[31796,31797],[34,667529,667531,667545],{"className":667530},[31801],[34,667532,667534],{"className":667533,"style":31806},[31805],[34,667535,667536,667539],{"style":666745},[34,667537],{"className":667538,"style":31814},[31813],[34,667540,667542],{"className":667541},[31818,31819,31820,31821],[34,667543,768],{"className":667544},[31685,31821],[34,667546,31829],{"className":667547},[31828],[34,667549,667551],{"className":667550},[31801],[34,667552,667554],{"className":667553,"style":31836},[31805],[34,667555],{},[34,667557],{"className":667558,"style":32604},[31845],[34,667560,32226],{"className":667561},[32024],[34,667563],{"className":667564,"style":32604},[31845],[34,667566,667568,667571,667611],{"className":667567},[31676],[34,667569],{"className":667570,"style":31774},[31680],[34,667572,667574,667577],{"className":667573},[31685],[34,667575,382333],{"className":667576,"style":382406},[31685,31686],[34,667578,667580],{"className":667579},[31792],[34,667581,667583,667603],{"className":667582},[31796,31797],[34,667584,667586,667600],{"className":667585},[31801],[34,667587,667589],{"className":667588,"style":31806},[31805],[34,667590,667591,667594],{"style":666745},[34,667592],{"className":667593,"style":31814},[31813],[34,667595,667597],{"className":667596},[31818,31819,31820,31821],[34,667598,640],{"className":667599},[31685,31821],[34,667601,31829],{"className":667602},[31828],[34,667604,667606],{"className":667605},[31801],[34,667607,667609],{"className":667608,"style":31836},[31805],[34,667610],{},[34,667612,2557],{"className":667613},[31690],"\n或者写作：\n",[34,667616,667618,667656],{"className":667617},[31642],[34,667619,667621],{"className":667620},[31646],[31648,667622,667623],{"xmlns":31650},[31652,667624,667625,667653],{},[31655,667626,667627,667629,667631,667633,667635,667637,667643,667645,667651],{},[31658,667628,56852],{},[31662,667630,202],{},[31658,667632,31759],{},[31658,667634,31942],{},[31662,667636,456],{"stretchy":702},[31743,667638,667639,667641],{},[31658,667640,382333],{},[31749,667642,640],{},[31662,667644,32226],{},[31743,667646,667647,667649],{},[31658,667648,382333],{},[31749,667650,768],{},[31662,667652,2557],{"stretchy":702},[31665,667654,667655],{"encoding":31667},"W = mg(z_1 - z_2)",[34,667657,667659,667677,667741],{"className":667658,"ariaHidden":758},[31672],[34,667660,667662,667665,667668,667671,667674],{"className":667661},[31676],[34,667663],{"className":667664,"style":33218},[31680],[34,667666,56852],{"className":667667,"style":37422},[31685,31686],[34,667669],{"className":667670,"style":31846},[31845],[34,667672,202],{"className":667673},[31850],[34,667675],{"className":667676,"style":31846},[31845],[34,667678,667680,667683,667686,667689,667692,667732,667735,667738],{"className":667679},[31676],[34,667681],{"className":667682,"style":31774},[31680],[34,667684,31759],{"className":667685},[31685,31686],[34,667687,31942],{"className":667688,"style":32062},[31685,31686],[34,667690,456],{"className":667691},[31782],[34,667693,667695,667698],{"className":667694},[31685],[34,667696,382333],{"className":667697,"style":382406},[31685,31686],[34,667699,667701],{"className":667700},[31792],[34,667702,667704,667724],{"className":667703},[31796,31797],[34,667705,667707,667721],{"className":667706},[31801],[34,667708,667710],{"className":667709,"style":31806},[31805],[34,667711,667712,667715],{"style":666745},[34,667713],{"className":667714,"style":31814},[31813],[34,667716,667718],{"className":667717},[31818,31819,31820,31821],[34,667719,640],{"className":667720},[31685,31821],[34,667722,31829],{"className":667723},[31828],[34,667725,667727],{"className":667726},[31801],[34,667728,667730],{"className":667729,"style":31836},[31805],[34,667731],{},[34,667733],{"className":667734,"style":32604},[31845],[34,667736,32226],{"className":667737},[32024],[34,667739],{"className":667740,"style":32604},[31845],[34,667742,667744,667747,667787],{"className":667743},[31676],[34,667745],{"className":667746,"style":31774},[31680],[34,667748,667750,667753],{"className":667749},[31685],[34,667751,382333],{"className":667752,"style":382406},[31685,31686],[34,667754,667756],{"className":667755},[31792],[34,667757,667759,667779],{"className":667758},[31796,31797],[34,667760,667762,667776],{"className":667761},[31801],[34,667763,667765],{"className":667764,"style":31806},[31805],[34,667766,667767,667770],{"style":666745},[34,667768],{"className":667769,"style":31814},[31813],[34,667771,667773],{"className":667772},[31818,31819,31820,31821],[34,667774,768],{"className":667775},[31685,31821],[34,667777,31829],{"className":667778},[31828],[34,667780,667782],{"className":667781},[31801],[34,667783,667785],{"className":667784,"style":31836},[31805],[34,667786],{},[34,667788,2557],{"className":667789},[31690],[15,667791,667792,667795,667796,667799,667800,667933,667934,668067],{},[2627,667793,667794],{},"物理意义解读：","\n这说明重力做功只与",[2627,667797,667798],{},"起点和终点的高度差","有关，与具体的移动路径无关（因为重力场是保守场）。如果 ",[34,667801,667803,667829],{"className":667802},[31642],[34,667804,667806],{"className":667805},[31646],[31648,667807,667808],{"xmlns":31650},[31652,667809,667810,667826],{},[31655,667811,667812,667818,667820],{},[31743,667813,667814,667816],{},[31658,667815,382333],{},[31749,667817,640],{},[31662,667819,29622],{},[31743,667821,667822,667824],{},[31658,667823,382333],{},[31749,667825,768],{},[31665,667827,667828],{"encoding":31667},"z_1 > z_2",[34,667830,667832,667887],{"className":667831,"ariaHidden":758},[31672],[34,667833,667835,667838,667878,667881,667884],{"className":667834},[31676],[34,667836],{"className":667837,"style":225773},[31680],[34,667839,667841,667844],{"className":667840},[31685],[34,667842,382333],{"className":667843,"style":382406},[31685,31686],[34,667845,667847],{"className":667846},[31792],[34,667848,667850,667870],{"className":667849},[31796,31797],[34,667851,667853,667867],{"className":667852},[31801],[34,667854,667856],{"className":667855,"style":31806},[31805],[34,667857,667858,667861],{"style":666745},[34,667859],{"className":667860,"style":31814},[31813],[34,667862,667864],{"className":667863},[31818,31819,31820,31821],[34,667865,640],{"className":667866},[31685,31821],[34,667868,31829],{"className":667869},[31828],[34,667871,667873],{"className":667872},[31801],[34,667874,667876],{"className":667875,"style":31836},[31805],[34,667877],{},[34,667879],{"className":667880,"style":31846},[31845],[34,667882,29622],{"className":667883},[31850],[34,667885],{"className":667886,"style":31846},[31845],[34,667888,667890,667893],{"className":667889},[31676],[34,667891],{"className":667892,"style":31860},[31680],[34,667894,667896,667899],{"className":667895},[31685],[34,667897,382333],{"className":667898,"style":382406},[31685,31686],[34,667900,667902],{"className":667901},[31792],[34,667903,667905,667925],{"className":667904},[31796,31797],[34,667906,667908,667922],{"className":667907},[31801],[34,667909,667911],{"className":667910,"style":31806},[31805],[34,667912,667913,667916],{"style":666745},[34,667914],{"className":667915,"style":31814},[31813],[34,667917,667919],{"className":667918},[31818,31819,31820,31821],[34,667920,768],{"className":667921},[31685,31821],[34,667923,31829],{"className":667924},[31828],[34,667926,667928],{"className":667927},[31801],[34,667929,667931],{"className":667930,"style":31836},[31805],[34,667932],{},"（物体下降），功为正；如果 ",[34,667935,667937,667963],{"className":667936},[31642],[34,667938,667940],{"className":667939},[31646],[31648,667941,667942],{"xmlns":31650},[31652,667943,667944,667960],{},[31655,667945,667946,667952,667954],{},[31743,667947,667948,667950],{},[31658,667949,382333],{},[31749,667951,768],{},[31662,667953,29622],{},[31743,667955,667956,667958],{},[31658,667957,382333],{},[31749,667959,640],{},[31665,667961,667962],{"encoding":31667},"z_2 > z_1",[34,667964,667966,668021],{"className":667965,"ariaHidden":758},[31672],[34,667967,667969,667972,668012,668015,668018],{"className":667968},[31676],[34,667970],{"className":667971,"style":225773},[31680],[34,667973,667975,667978],{"className":667974},[31685],[34,667976,382333],{"className":667977,"style":382406},[31685,31686],[34,667979,667981],{"className":667980},[31792],[34,667982,667984,668004],{"className":667983},[31796,31797],[34,667985,667987,668001],{"className":667986},[31801],[34,667988,667990],{"className":667989,"style":31806},[31805],[34,667991,667992,667995],{"style":666745},[34,667993],{"className":667994,"style":31814},[31813],[34,667996,667998],{"className":667997},[31818,31819,31820,31821],[34,667999,768],{"className":668000},[31685,31821],[34,668002,31829],{"className":668003},[31828],[34,668005,668007],{"className":668006},[31801],[34,668008,668010],{"className":668009,"style":31836},[31805],[34,668011],{},[34,668013],{"className":668014,"style":31846},[31845],[34,668016,29622],{"className":668017},[31850],[34,668019],{"className":668020,"style":31846},[31845],[34,668022,668024,668027],{"className":668023},[31676],[34,668025],{"className":668026,"style":31860},[31680],[34,668028,668030,668033],{"className":668029},[31685],[34,668031,382333],{"className":668032,"style":382406},[31685,31686],[34,668034,668036],{"className":668035},[31792],[34,668037,668039,668059],{"className":668038},[31796,31797],[34,668040,668042,668056],{"className":668041},[31801],[34,668043,668045],{"className":668044,"style":31806},[31805],[34,668046,668047,668050],{"style":666745},[34,668048],{"className":668049,"style":31814},[31813],[34,668051,668053],{"className":668052},[31818,31819,31820,31821],[34,668054,640],{"className":668055},[31685,31821],[34,668057,31829],{"className":668058},[31828],[34,668060,668062],{"className":668061},[31801],[34,668063,668065],{"className":668064,"style":31836},[31805],[34,668066],{},"（物体上升），功为负。",[32188,668069],{},[32490,668071,668073],{"id":668072},"题目例-1423","题目：例 14.2.3",[15,668075,668076,668078,668540],{},[2627,668077,658107],{},[34,668079,668081,668179],{"className":668080},[31642],[34,668082,668084],{"className":668083},[31646],[31648,668085,668086],{"xmlns":31650},[31652,668087,668088,668176],{},[31655,668089,668090,668092,668094,668100,668102,668108,668110,668116,668118,668120,668122,668124,668126,668128,668134,668136,668142,668144,668146,668148,668150,668152,668154,668160,668162,668168,668170,668172,668174],{},[31658,668091,106063],{},[31662,668093,202],{},[31743,668095,668096,668098],{},[31662,668097,546295],{},[31658,668099,33206],{},[31662,668101,456],{"stretchy":702},[37624,668103,668104,668106],{},[31658,668105,67764],{},[31749,668107,768],{},[31662,668109,32226],{},[37624,668111,668112,668114],{},[31658,668113,382333],{},[31749,668115,768],{},[31662,668117,2557],{"stretchy":702},[39361,668119,39363],{},[31658,668121,46900],{"mathvariant":38079},[31658,668123,32936],{},[31662,668125,223],{},[31662,668127,456],{"stretchy":702},[37624,668129,668130,668132],{},[31658,668131,382333],{},[31749,668133,768],{},[31662,668135,32226],{},[37624,668137,668138,668140],{},[31658,668139,32936],{},[31749,668141,768],{},[31662,668143,2557],{"stretchy":702},[39361,668145,39363],{},[31658,668147,46900],{"mathvariant":38079},[31658,668149,67764],{},[31662,668151,223],{},[31662,668153,456],{"stretchy":702},[37624,668155,668156,668158],{},[31658,668157,32936],{},[31749,668159,768],{},[31662,668161,32226],{},[37624,668163,668164,668166],{},[31658,668165,67764],{},[31749,668167,768],{},[31662,668169,2557],{"stretchy":702},[39361,668171,39363],{},[31658,668173,46900],{"mathvariant":38079},[31658,668175,382333],{},[31665,668177,668178],{"encoding":31667},"I = \\int_L (y^2 - z^2) \\, \\mathrm{d}x + (z^2 - x^2) \\, \\mathrm{d}y + (x^2 - y^2) \\, \\mathrm{d}z",[34,668180,668182,668200,668287,668343,668390,668446,668493],{"className":668181,"ariaHidden":758},[31672],[34,668183,668185,668188,668191,668194,668197],{"className":668184},[31676],[34,668186],{"className":668187,"style":33218},[31680],[34,668189,106063],{"className":668190,"style":67482},[31685,31686],[34,668192],{"className":668193,"style":31846},[31845],[34,668195,202],{"className":668196},[31850],[34,668198],{"className":668199,"style":31846},[31845],[34,668201,668203,668206,668246,668249,668278,668281,668284],{"className":668202},[31676],[34,668204],{"className":668205,"style":658186},[31680],[34,668207,668209,668212],{"className":668208},[32767],[34,668210,546295],{"className":668211,"style":546331},[32767,131249,156560],[34,668213,668215],{"className":668214},[31792],[34,668216,668218,668238],{"className":668217},[31796,31797],[34,668219,668221,668235],{"className":668220},[31801],[34,668222,668224],{"className":668223,"style":637339},[31805],[34,668225,668226,668229],{"style":546347},[34,668227],{"className":668228,"style":31814},[31813],[34,668230,668232],{"className":668231},[31818,31819,31820,31821],[34,668233,33206],{"className":668234},[31685,31686,31821],[34,668236,31829],{"className":668237},[31828],[34,668239,668241],{"className":668240},[31801],[34,668242,668244],{"className":668243,"style":546384},[31805],[34,668245],{},[34,668247,456],{"className":668248},[31782],[34,668250,668252,668255],{"className":668251},[31685],[34,668253,67764],{"className":668254,"style":32062},[31685,31686],[34,668256,668258],{"className":668257},[31792],[34,668259,668261],{"className":668260},[31796],[34,668262,668264],{"className":668263},[31801],[34,668265,668267],{"className":668266,"style":37774},[31805],[34,668268,668269,668272],{"style":37777},[34,668270],{"className":668271,"style":31814},[31813],[34,668273,668275],{"className":668274},[31818,31819,31820,31821],[34,668276,768],{"className":668277},[31685,31821],[34,668279],{"className":668280,"style":32604},[31845],[34,668282,32226],{"className":668283},[32024],[34,668285],{"className":668286,"style":32604},[31845],[34,668288,668290,668293,668322,668325,668328,668331,668334,668337,668340],{"className":668289},[31676],[34,668291],{"className":668292,"style":37755},[31680],[34,668294,668296,668299],{"className":668295},[31685],[34,668297,382333],{"className":668298,"style":382406},[31685,31686],[34,668300,668302],{"className":668301},[31792],[34,668303,668305],{"className":668304},[31796],[34,668306,668308],{"className":668307},[31801],[34,668309,668311],{"className":668310,"style":37774},[31805],[34,668312,668313,668316],{"style":37777},[34,668314],{"className":668315,"style":31814},[31813],[34,668317,668319],{"className":668318},[31818,31819,31820,31821],[34,668320,768],{"className":668321},[31685,31821],[34,668323,2557],{"className":668324},[31690],[34,668326],{"className":668327,"style":32123},[31845],[34,668329,46900],{"className":668330},[31685,207923],[34,668332,32936],{"className":668333},[31685,31686],[34,668335],{"className":668336,"style":32604},[31845],[34,668338,223],{"className":668339},[32024],[34,668341],{"className":668342,"style":32604},[31845],[34,668344,668346,668349,668352,668381,668384,668387],{"className":668345},[31676],[34,668347],{"className":668348,"style":37755},[31680],[34,668350,456],{"className":668351},[31782],[34,668353,668355,668358],{"className":668354},[31685],[34,668356,382333],{"className":668357,"style":382406},[31685,31686],[34,668359,668361],{"className":668360},[31792],[34,668362,668364],{"className":668363},[31796],[34,668365,668367],{"className":668366},[31801],[34,668368,668370],{"className":668369,"style":37774},[31805],[34,668371,668372,668375],{"style":37777},[34,668373],{"className":668374,"style":31814},[31813],[34,668376,668378],{"className":668377},[31818,31819,31820,31821],[34,668379,768],{"className":668380},[31685,31821],[34,668382],{"className":668383,"style":32604},[31845],[34,668385,32226],{"className":668386},[32024],[34,668388],{"className":668389,"style":32604},[31845],[34,668391,668393,668396,668425,668428,668431,668434,668437,668440,668443],{"className":668392},[31676],[34,668394],{"className":668395,"style":37755},[31680],[34,668397,668399,668402],{"className":668398},[31685],[34,668400,32936],{"className":668401},[31685,31686],[34,668403,668405],{"className":668404},[31792],[34,668406,668408],{"className":668407},[31796],[34,668409,668411],{"className":668410},[31801],[34,668412,668414],{"className":668413,"style":37774},[31805],[34,668415,668416,668419],{"style":37777},[34,668417],{"className":668418,"style":31814},[31813],[34,668420,668422],{"className":668421},[31818,31819,31820,31821],[34,668423,768],{"className":668424},[31685,31821],[34,668426,2557],{"className":668427},[31690],[34,668429],{"className":668430,"style":32123},[31845],[34,668432,46900],{"className":668433},[31685,207923],[34,668435,67764],{"className":668436,"style":32062},[31685,31686],[34,668438],{"className":668439,"style":32604},[31845],[34,668441,223],{"className":668442},[32024],[34,668444],{"className":668445,"style":32604},[31845],[34,668447,668449,668452,668455,668484,668487,668490],{"className":668448},[31676],[34,668450],{"className":668451,"style":37755},[31680],[34,668453,456],{"className":668454},[31782],[34,668456,668458,668461],{"className":668457},[31685],[34,668459,32936],{"className":668460},[31685,31686],[34,668462,668464],{"className":668463},[31792],[34,668465,668467],{"className":668466},[31796],[34,668468,668470],{"className":668469},[31801],[34,668471,668473],{"className":668472,"style":37774},[31805],[34,668474,668475,668478],{"style":37777},[34,668476],{"className":668477,"style":31814},[31813],[34,668479,668481],{"className":668480},[31818,31819,31820,31821],[34,668482,768],{"className":668483},[31685,31821],[34,668485],{"className":668486,"style":32604},[31845],[34,668488,32226],{"className":668489},[32024],[34,668491],{"className":668492,"style":32604},[31845],[34,668494,668496,668499,668528,668531,668534,668537],{"className":668495},[31676],[34,668497],{"className":668498,"style":37755},[31680],[34,668500,668502,668505],{"className":668501},[31685],[34,668503,67764],{"className":668504,"style":32062},[31685,31686],[34,668506,668508],{"className":668507},[31792],[34,668509,668511],{"className":668510},[31796],[34,668512,668514],{"className":668513},[31801],[34,668515,668517],{"className":668516,"style":37774},[31805],[34,668518,668519,668522],{"style":37777},[34,668520],{"className":668521,"style":31814},[31813],[34,668523,668525],{"className":668524},[31818,31819,31820,31821],[34,668526,768],{"className":668527},[31685,31821],[34,668529,2557],{"className":668530},[31690],[34,668532],{"className":668533,"style":32123},[31845],[34,668535,46900],{"className":668536},[31685,207923],[34,668538,382333],{"className":668539,"style":382406},[31685,31686],[2627,668541,115851,668542,668570,668571,668756],{},[34,668543,668545,668558],{"className":668544},[31642],[34,668546,668548],{"className":668547},[31646],[31648,668549,668550],{"xmlns":31650},[31652,668551,668552,668556],{},[31655,668553,668554],{},[31658,668555,33206],{},[31665,668557,33206],{"encoding":31667},[34,668559,668561],{"className":668560,"ariaHidden":758},[31672],[34,668562,668564,668567],{"className":668563},[31676],[34,668565],{"className":668566,"style":33218},[31680],[34,668568,33206],{"className":668569},[31685,31686]," 为球面 ",[34,668572,668574,668612],{"className":668573},[31642],[34,668575,668577],{"className":668576},[31646],[31648,668578,668579],{"xmlns":31650},[31652,668580,668581,668609],{},[31655,668582,668583,668589,668591,668597,668599,668605,668607],{},[37624,668584,668585,668587],{},[31658,668586,32936],{},[31749,668588,768],{},[31662,668590,223],{},[37624,668592,668593,668595],{},[31658,668594,67764],{},[31749,668596,768],{},[31662,668598,223],{},[37624,668600,668601,668603],{},[31658,668602,382333],{},[31749,668604,768],{},[31662,668606,202],{},[31749,668608,640],{},[31665,668610,668611],{"encoding":31667},"x^2 + y^2 + z^2 = 1",[34,668613,668615,668659,668703,668747],{"className":668614,"ariaHidden":758},[31672],[34,668616,668618,668621,668650,668653,668656],{"className":668617},[31676],[34,668619],{"className":668620,"style":40044},[31680],[34,668622,668624,668627],{"className":668623},[31685],[34,668625,32936],{"className":668626},[31685,31686],[34,668628,668630],{"className":668629},[31792],[34,668631,668633],{"className":668632},[31796],[34,668634,668636],{"className":668635},[31801],[34,668637,668639],{"className":668638,"style":37774},[31805],[34,668640,668641,668644],{"style":37777},[34,668642],{"className":668643,"style":31814},[31813],[34,668645,668647],{"className":668646},[31818,31819,31820,31821],[34,668648,768],{"className":668649},[31685,31821],[34,668651],{"className":668652,"style":32604},[31845],[34,668654,223],{"className":668655},[32024],[34,668657],{"className":668658,"style":32604},[31845],[34,668660,668662,668665,668694,668697,668700],{"className":668661},[31676],[34,668663],{"className":668664,"style":315747},[31680],[34,668666,668668,668671],{"className":668667},[31685],[34,668669,67764],{"className":668670,"style":32062},[31685,31686],[34,668672,668674],{"className":668673},[31792],[34,668675,668677],{"className":668676},[31796],[34,668678,668680],{"className":668679},[31801],[34,668681,668683],{"className":668682,"style":37774},[31805],[34,668684,668685,668688],{"style":37777},[34,668686],{"className":668687,"style":31814},[31813],[34,668689,668691],{"className":668690},[31818,31819,31820,31821],[34,668692,768],{"className":668693},[31685,31821],[34,668695],{"className":668696,"style":32604},[31845],[34,668698,223],{"className":668699},[32024],[34,668701],{"className":668702,"style":32604},[31845],[34,668704,668706,668709,668738,668741,668744],{"className":668705},[31676],[34,668707],{"className":668708,"style":37774},[31680],[34,668710,668712,668715],{"className":668711},[31685],[34,668713,382333],{"className":668714,"style":382406},[31685,31686],[34,668716,668718],{"className":668717},[31792],[34,668719,668721],{"className":668720},[31796],[34,668722,668724],{"className":668723},[31801],[34,668725,668727],{"className":668726,"style":37774},[31805],[34,668728,668729,668732],{"style":37777},[34,668730],{"className":668731,"style":31814},[31813],[34,668733,668735],{"className":668734},[31818,31819,31820,31821],[34,668736,768],{"className":668737},[31685,31821],[34,668739],{"className":668740,"style":31846},[31845],[34,668742,202],{"className":668743},[31850],[34,668745],{"className":668746,"style":31846},[31845],[34,668748,668750,668753],{"className":668749},[31676],[34,668751],{"className":668752,"style":32617},[31680],[34,668754,640],{"className":668755},[31685]," 在第一象限部分的边界，当从球面外侧看时为顺时针方向。",[32490,668758,658913],{"id":668759},"详细解答-2",[15,668761,668762,668793,668794,668822,668823,668856],{},[2627,668763,668764,668765],{},"1. 分析路径 ",[34,668766,668768,668781],{"className":668767},[31642],[34,668769,668771],{"className":668770},[31646],[31648,668772,668773],{"xmlns":31650},[31652,668774,668775,668779],{},[31655,668776,668777],{},[31658,668778,33206],{},[31665,668780,33206],{"encoding":31667},[34,668782,668784],{"className":668783,"ariaHidden":758},[31672],[34,668785,668787,668790],{"className":668786},[31676],[34,668788],{"className":668789,"style":33218},[31680],[34,668791,33206],{"className":668792},[31685,31686],"\n题目描述的曲线 ",[34,668795,668797,668810],{"className":668796},[31642],[34,668798,668800],{"className":668799},[31646],[31648,668801,668802],{"xmlns":31650},[31652,668803,668804,668808],{},[31655,668805,668806],{},[31658,668807,33206],{},[31665,668809,33206],{"encoding":31667},[34,668811,668813],{"className":668812,"ariaHidden":758},[31672],[34,668814,668816,668819],{"className":668815},[31676],[34,668817],{"className":668818,"style":33218},[31680],[34,668820,33206],{"className":668821},[31685,31686]," 是第一象限球面（即 ",[34,668824,668826,668844],{"className":668825},[31642],[34,668827,668829],{"className":668828},[31646],[31648,668830,668831],{"xmlns":31650},[31652,668832,668833,668841],{},[31655,668834,668835,668837,668839],{},[31749,668836,640],{},[31658,668838,38080],{"mathvariant":38079},[31749,668840,60525],{},[31665,668842,668843],{"encoding":31667},"1\u002F8",[34,668845,668847],{"className":668846,"ariaHidden":758},[31672],[34,668848,668850,668853],{"className":668849},[31676],[34,668851],{"className":668852,"style":31774},[31680],[34,668854,668843],{"className":668855},[31685]," 球面）的边界线。这个边界由三段位于坐标平面上的圆弧组成。\n为了符合“从球面外侧看顺时针”的描述，我们可以画个图想象一下：",[105,668858,668859,668862],{},[108,668860,668861],{},"我们站在第一象限的球面上方往原点看。",[108,668863,668864,668865,39984,669124],{},"路径顺次经过三个点：",[34,668866,668868,668950],{"className":668867},[31642],[34,668869,668871],{"className":668870},[31646],[31648,668872,668873],{"xmlns":31650},[31652,668874,668875,668947],{},[31655,668876,668877,668879,668881,668883,668885,668887,668889,668891,668893,668895,668897,668899,668901,668903,668905,668907,668909,668911,668913,668915,668917,668919,668921,668923,668925,668927,668929,668931,668933,668935,668937,668939,668941,668943,668945],{},[31658,668878,54151],{},[31662,668880,456],{"stretchy":702},[31749,668882,640],{},[31662,668884,31960],{"separator":758},[31749,668886,459],{},[31662,668888,31960],{"separator":758},[31749,668890,459],{},[31662,668892,2557],{"stretchy":702},[31662,668894,33239],{},[31658,668896,54224],{},[31662,668898,456],{"stretchy":702},[31749,668900,459],{},[31662,668902,31960],{"separator":758},[31749,668904,640],{},[31662,668906,31960],{"separator":758},[31749,668908,459],{},[31662,668910,2557],{"stretchy":702},[31662,668912,33239],{},[31658,668914,47384],{},[31662,668916,456],{"stretchy":702},[31749,668918,459],{},[31662,668920,31960],{"separator":758},[31749,668922,459],{},[31662,668924,31960],{"separator":758},[31749,668926,640],{},[31662,668928,2557],{"stretchy":702},[31662,668930,33239],{},[31658,668932,54151],{},[31662,668934,456],{"stretchy":702},[31749,668936,640],{},[31662,668938,31960],{"separator":758},[31749,668940,459],{},[31662,668942,31960],{"separator":758},[31749,668944,459],{},[31662,668946,2557],{"stretchy":702},[31665,668948,668949],{"encoding":31667},"A(1,0,0) \\to B(0,1,0) \\to C(0,0,1) \\to 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平面上",[105,670201,670202,670314,670453,671158,671595],{},[108,670203,670204,2664,670206,670257,670258,347],{},[2627,670205,491730],{},[34,670207,670209,670227],{"className":670208},[31642],[34,670210,670212],{"className":670211},[31646],[31648,670213,670214],{"xmlns":31650},[31652,670215,670216,670224],{},[31655,670217,670218,670220,670222],{},[31658,670219,382333],{},[31662,670221,202],{},[31749,670223,459],{},[31665,670225,670226],{"encoding":31667},"z = 0",[34,670228,670230,670248],{"className":670229,"ariaHidden":758},[31672],[34,670231,670233,670236,670239,670242,670245],{"className":670232},[31676],[34,670234],{"className":670235,"style":32527},[31680],[34,670237,382333],{"className":670238,"style":382406},[31685,31686],[34,670240],{"className":670241,"style":31846},[31845],[34,670243,202],{"className":670244},[31850],[34,670246],{"className":670247,"style":31846},[31845],[34,670249,670251,670254],{"className":670250},[31676],[34,670252],{"className":670253,"style":32617},[31680],[34,670255,459],{"className":670256},[31685],"，因此 ",[34,670259,670261,670281],{"className":670260},[31642],[34,670262,670264],{"className":670263},[31646],[31648,670265,670266],{"xmlns":31650},[31652,670267,670268,670278],{},[31655,670269,670270,670272,670274,670276],{},[31658,670271,46900],{"mathvariant":38079},[31658,670273,382333],{},[31662,670275,202],{},[31749,670277,459],{},[31665,670279,670280],{"encoding":31667},"\\mathrm{d}z = 0",[34,670282,670284,670305],{"className":670283,"ariaHidden":758},[31672],[34,670285,670287,670290,670293,670296,670299,670302],{"className":670286},[31676],[34,670288],{"className":670289,"style":31681},[31680],[34,670291,46900],{"className":670292},[31685,207923],[34,670294,382333],{"className":670295,"style":382406},[31685,31686],[34,670297],{"className":670298,"style":31846},[31845],[34,670300,202],{"className":670301},[31850],[34,670303],{"className":670304,"style":31846},[31845],[34,670306,670308,670311],{"className":670307},[31676],[34,670309],{"className":670310,"style":32617},[31680],[34,670312,459],{"className":670313},[31685],[108,670315,670316,2664,670319,670452],{},[2627,670317,670318],{},"方程",[34,670320,670322,670352],{"className":670321},[31642],[34,670323,670325],{"className":670324},[31646],[31648,670326,670327],{"xmlns":31650},[31652,670328,670329,670349],{},[31655,670330,670331,670337,670339,670345,670347],{},[37624,670332,670333,670335],{},[31658,670334,32936],{},[31749,670336,768],{},[31662,670338,223],{},[37624,670340,670341,670343],{},[31658,670342,67764],{},[31749,670344,768],{},[31662,670346,202],{},[31749,670348,640],{},[31665,670350,670351],{"encoding":31667},"x^2 + y^2 = 1",[34,670353,670355,670399,670443],{"className":670354,"ariaHidden":758},[31672],[34,670356,670358,670361,670390,670393,670396],{"className":670357},[31676],[34,670359],{"className":670360,"style":40044},[31680],[34,670362,670364,670367],{"className":670363},[31685],[34,670365,32936],{"className":670366},[31685,31686],[34,670368,670370],{"className":670369},[31792],[34,670371,670373],{"className":670372},[31796],[34,670374,670376],{"className":670375},[31801],[34,670377,670379],{"className":670378,"style":37774},[31805],[34,670380,670381,670384],{"style":37777},[34,670382],{"className":670383,"style":31814},[31813],[34,670385,670387],{"className":670386},[31818,31819,31820,31821],[34,670388,768],{"className":670389},[31685,31821],[34,670391],{"className":670392,"style":32604},[31845],[34,670394,223],{"className":670395},[32024],[34,670397],{"className":670398,"style":32604},[31845],[34,670400,670402,670405,670434,670437,670440],{"className":670401},[31676],[34,670403],{"className":670404,"style":315747},[31680],[34,670406,670408,670411],{"className":670407},[31685],[34,670409,67764],{"className":670410,"style":32062},[31685,31686],[34,670412,670414],{"className":670413},[31792],[34,670415,670417],{"className":670416},[31796],[34,670418,670420],{"className":670419},[31801],[34,670421,670423],{"className":670422,"style":37774},[31805],[34,670424,670425,670428],{"style":37777},[34,670426],{"className":670427,"style":31814},[31813],[34,670429,670431],{"className":670430},[31818,31819,31820,31821],[34,670432,768],{"className":670433},[31685,31821],[34,670435],{"className":670436,"style":31846},[31845],[34,670438,202],{"className":670439},[31850],[34,670441],{"className":670442,"style":31846},[31845],[34,670444,670446,670449],{"className":670445},[31676],[34,670447],{"className":670448,"style":32617},[31680],[34,670450,640],{"className":670451},[31685]," (第一象限弧)。",[108,670454,670455,670458,670459,33183,670550],{},[2627,670456,670457],{},"原积分简化","：代入 ",[34,670460,670462,670490],{"className":670461},[31642],[34,670463,670465],{"className":670464},[31646],[31648,670466,670467],{"xmlns":31650},[31652,670468,670469,670487],{},[31655,670470,670471,670473,670475,670477,670479,670481,670483,670485],{},[31658,670472,382333],{},[31662,670474,202],{},[31749,670476,459],{},[31662,670478,31960],{"separator":758},[31658,670480,46900],{"mathvariant":38079},[31658,670482,382333],{},[31662,670484,202],{},[31749,670486,459],{},[31665,670488,670489],{"encoding":31667},"z=0, \\mathrm{d}z=0",[34,670491,670493,670511,670541],{"className":670492,"ariaHidden":758},[31672],[34,670494,670496,670499,670502,670505,670508],{"className":670495},[31676],[34,670497],{"className":670498,"style":32527},[31680],[34,670500,382333],{"className":670501,"style":382406},[31685,31686],[34,670503],{"className":670504,"style":31846},[31845],[34,670506,202],{"className":670507},[31850],[34,670509],{"className":670510,"style":31846},[31845],[34,670512,670514,670517,670520,670523,670526,670529,670532,670535,670538],{"className":670513},[31676],[34,670515],{"className":670516,"style":43217},[31680],[34,670518,459],{"className":670519},[31685],[34,670521,31960],{"className":670522},[32119],[34,670524],{"className":670525,"style":32123},[31845],[34,670527,46900],{"className":670528},[31685,207923],[34,670530,382333],{"className":670531,"style":382406},[31685,31686],[34,670533],{"className":670534,"style":31846},[31845],[34,670536,202],{"className":670537},[31850],[34,670539],{"className":670540,"style":31846},[31845],[34,670542,670544,670547],{"className":670543},[31676],[34,670545],{"className":670546,"style":32617},[31680],[34,670548,459],{"className":670549},[31685],[34,670551,670553,670667],{"className":670552},[31642],[34,670554,670556],{"className":670555},[31646],[31648,670557,670558],{"xmlns":31650},[31652,670559,670560,670664],{},[31655,670561,670562,670568,670570,670580,670582,670588,670590,670592,670594,670596,670598,670600,670602,670604,670606,670608,670614,670616,670618,670620,670622,670624,670626,670628,670638,670644,670646,670648,670650,670652,670658,670660,670662],{},[31743,670563,670564,670566],{},[31658,670565,106063],{},[31749,670567,640],{},[31662,670569,202],{},[31743,670571,670572,670574],{},[31662,670573,546295],{},[31743,670575,670576,670578],{},[31658,670577,33206],{},[31749,670579,640],{},[31662,670581,456],{"stretchy":702},[37624,670583,670584,670586],{},[31658,670585,67764],{},[31749,670587,768],{},[31662,670589,32226],{},[31749,670591,459],{},[31662,670593,2557],{"stretchy":702},[39361,670595,39363],{},[31658,670597,46900],{"mathvariant":38079},[31658,670599,32936],{},[31662,670601,223],{},[31662,670603,456],{"stretchy":702},[31749,670605,459],{},[31662,670607,32226],{},[37624,670609,670610,670612],{},[31658,670611,32936],{},[31749,670613,768],{},[31662,670615,2557],{"stretchy":702},[39361,670617,39363],{},[31658,670619,46900],{"mathvariant":38079},[31658,670621,67764],{},[31662,670623,223],{},[31749,670625,459],{},[31662,670627,202],{},[31743,670629,670630,670632],{},[31662,670631,546295],{},[31743,670633,670634,670636],{},[31658,670635,33206],{},[31749,670637,640],{},[37624,670639,670640,670642],{},[31658,670641,67764],{},[31749,670643,768],{},[39361,670645,39363],{},[31658,670647,46900],{"mathvariant":38079},[31658,670649,32936],{},[31662,670651,32226],{},[37624,670653,670654,670656],{},[31658,670655,32936],{},[31749,670657,768],{},[39361,670659,39363],{},[31658,670661,46900],{"mathvariant":38079},[31658,670663,67764],{},[31665,670665,670666],{"encoding":31667},"I_1 = \\int_{L_1} (y^2 - 0) \\, \\mathrm{d}x + (0 - x^2) \\, \\mathrm{d}y + 0 = \\int_{L_1} y^2 \\, \\mathrm{d}x - x^2 \\, \\mathrm{d}y",[34,670668,670670,670725,670853,670883,670904,670960,670978,671114],{"className":670669,"ariaHidden":758},[31672],[34,670671,670673,670676,670716,670719,670722],{"className":670672},[31676],[34,670674],{"className":670675,"style":70915},[31680],[34,670677,670679,670682],{"className":670678},[31685],[34,670680,106063],{"className":670681,"style":67482},[31685,31686],[34,670683,670685],{"className":670684},[31792],[34,670686,670688,670708],{"className":670687},[31796,31797],[34,670689,670691,670705],{"className":670690},[31801],[34,670692,670694],{"className":670693,"style":31806},[31805],[34,670695,670696,670699],{"style":217742},[34,670697],{"className":670698,"style":31814},[31813],[34,670700,670702],{"className":670701},[31818,31819,31820,31821],[34,670703,640],{"className":670704},[31685,31821],[34,670706,31829],{"className":670707},[31828],[34,670709,670711],{"className":670710},[31801],[34,670712,670714],{"className":670713,"style":31836},[31805],[34,670715],{},[34,670717],{"className":670718,"style":31846},[31845],[34,670720,202],{"className":670721},[31850],[34,670723],{"className":670724,"style":31846},[31845],[34,670726,670728,670732,670812,670815,670844,670847,670850],{"className":670727},[31676],[34,670729],{"className":670730,"style":670731},[31680],"height:1.27em;vertical-align:-0.4559em;",[34,670733,670735,670738],{"className":670734},[32767],[34,670736,546295],{"className":670737,"style":546331},[32767,131249,156560],[34,670739,670741],{"className":670740},[31792],[34,670742,670744,670804],{"className":670743},[31796,31797],[34,670745,670747,670801],{"className":670746},[31801],[34,670748,670750],{"className":670749,"style":637339},[31805],[34,670751,670752,670755],{"style":546347},[34,670753],{"className":670754,"style":31814},[31813],[34,670756,670758],{"className":670757},[31818,31819,31820,31821],[34,670759,670761],{"className":670760},[31685,31821],[34,670762,670764,670767],{"className":670763},[31685,31821],[34,670765,33206],{"className":670766},[31685,31686,31821],[34,670768,670770],{"className":670769},[31792],[34,670771,670773,670793],{"className":670772},[31796,31797],[34,670774,670776,670790],{"className":670775},[31801],[34,670777,670779],{"className":670778,"style":318658},[31805],[34,670780,670781,670784],{"style":318661},[34,670782],{"className":670783,"style":43406},[31813],[34,670785,670787],{"className":670786},[31818,43410,43411,31821],[34,670788,640],{"className":670789},[31685,31821],[34,670791,31829],{"className":670792},[31828],[34,670794,670796],{"className":670795},[31801],[34,670797,670799],{"className":670798,"style":220540},[31805],[34,670800],{},[34,670802,31829],{"className":670803},[31828],[34,670805,670807],{"className":670806},[31801],[34,670808,670810],{"className":670809,"style":667231},[31805],[34,670811],{},[34,670813,456],{"className":670814},[31782],[34,670816,670818,670821],{"className":670817},[31685],[34,670819,67764],{"className":670820,"style":32062},[31685,31686],[34,670822,670824],{"className":670823},[31792],[34,670825,670827],{"className":670826},[31796],[34,670828,670830],{"className":670829},[31801],[34,670831,670833],{"className":670832,"style":37774},[31805],[34,670834,670835,670838],{"style":37777},[34,670836],{"className":670837,"style":31814},[31813],[34,670839,670841],{"className":670840},[31818,31819,31820,31821],[34,670842,768],{"className":670843},[31685,31821],[34,670845],{"className":670846,"style":32604},[31845],[34,670848,32226],{"className":670849},[32024],[34,670851],{"className":670852,"style":32604},[31845],[34,670854,670856,670859,670862,670865,670868,670871,670874,670877,670880],{"className":670855},[31676],[34,670857],{"className":670858,"style":31774},[31680],[34,670860,459],{"className":670861},[31685],[34,670863,2557],{"className":670864},[31690],[34,670866],{"className":670867,"style":32123},[31845],[34,670869,46900],{"className":670870},[31685,207923],[34,670872,32936],{"className":670873},[31685,31686],[34,670875],{"className":670876,"style":32604},[31845],[34,670878,223],{"className":670879},[32024],[34,670881],{"className":670882,"style":32604},[31845],[34,670884,670886,670889,670892,670895,670898,670901],{"className":670885},[31676],[34,670887],{"className":670888,"style":31774},[31680],[34,670890,456],{"className":670891},[31782],[34,670893,459],{"className":670894},[31685],[34,670896],{"className":670897,"style":32604},[31845],[34,670899,32226],{"className":670900},[32024],[34,670902],{"className":670903,"style":32604},[31845],[34,670905,670907,670910,670939,670942,670945,670948,670951,670954,670957],{"className":670906},[31676],[34,670908],{"className":670909,"style":37755},[31680],[34,670911,670913,670916],{"className":670912},[31685],[34,670914,32936],{"className":670915},[31685,31686],[34,670917,670919],{"className":670918},[31792],[34,670920,670922],{"className":670921},[31796],[34,670923,670925],{"className":670924},[31801],[34,670926,670928],{"className":670927,"style":37774},[31805],[34,6709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71198,671199],{"encoding":31667},"x = \\cos t, y = \\sin t",[34,671201,671203,671221,671254],{"className":671202,"ariaHidden":758},[31672],[34,671204,671206,671209,671212,671215,671218],{"className":671205},[31676],[34,671207],{"className":671208,"style":32527},[31680],[34,671210,32936],{"className":671211},[31685,31686],[34,671213],{"className":671214,"style":31846},[31845],[34,671216,202],{"className":671217},[31850],[34,671219],{"className":671220,"style":31846},[31845],[34,671222,671224,671227,671230,671233,671236,671239,671242,671245,671248,671251],{"className":671223},[31676],[34,671225],{"className":671226,"style":223552},[31680],[34,671228,547860],{"className":671229},[32767],[34,671231],{"className":671232,"style":32123},[31845],[34,671234,33249],{"className":671235},[31685,31686],[34,671237,31960],{"className":671238},[32119],[34,671240],{"className":671241,"style":32123},[31845],[34,671243,67764],{"className":671244,"style":32062},[31685,31686],[34,671246],{"className":671247,"style":31846},[31845],[34,671249,202],{"className":671250},[31850],[34,671252],{"className":671253,"style":31846},[31845],[34,671255,671257,671260,671263,671266],{"className":671256},[31676],[34,671258],{"className":671259,"style":71430},[31680],[34,671261,547853],{"className":671262},[32767],[34,671264],{"className":671265,"style":32123},[31845],[34,671267,33249],{"className":671268},[31685,31686],[34,671270,671272,671285],{"className":671271},[31642],[34,671273,671275],{"className":671274},[31646],[31648,671276,671277],{"xmlns":31650},[31652,671278,671279,671283],{},[31655,671280,671281],{},[31658,671282,33249],{},[31665,671284,33249],{"encoding":31667},[34,671286,671288],{"className":671287,"ariaHidden":758},[31672],[34,671289,671291,671294],{"className":671290},[31676],[34,671292],{"className":671293,"style":33280},[31680],[34,671295,33249],{"className":671296},[31685,31686],[34,671298,671300,671313],{"className":671299},[31642],[34,671301,671303],{"className":671302},[31646],[31648,671304,671305],{"xmlns":31650},[31652,671306,671307,671311],{},[31655,671308,671309],{},[31749,671310,459],{},[31665,671312,459],{"encoding":31667},[34,671314,671316],{"className":671315,"ariaHidden":758},[31672],[34,671317,671319,671322],{"className":671318},[31676],[34,671320],{"className":671321,"style":32617},[31680],[34,671323,459],{"className":671324},[31685]," (点 A) 走到 ",[34,671327,671329,671347],{"className":671328},[31642],[34,671330,671332],{"className":671331},[31646],[31648,671333,671334],{"xmlns":31650},[31652,671335,671336,671344],{},[31655,671337,671338],{},[41453,671339,671340,671342],{},[31658,671341,58442],{},[31749,671343,768],{},[31665,671345,671346],{"encoding":31667},"\\frac{\\pi}{2}",[34,671348,671350],{"className":671349,"ariaHidden":758},[31672],[34,671351,671353,671356],{"className":671352},[31676],[34,671354],{"className":671355,"style":244926},[31680],[34,671357,671359,671362,671421],{"className":671358},[31685],[34,671360],{"className":671361},[31782,41488],[34,671363,671365],{"className":671364},[41453],[34,671366,671368,671413],{"className":671367},[31796,31797],[34,671369,671371,671410],{"className":671370},[31801],[34,671372,671374,671388,671396],{"className":671373,"style":244945},[31805],[34,671375,671376,671379],{"style":41504},[34,671377],{"className":671378,"style":41508},[31813],[34,671380,671382],{"className":671381},[31818,31819,31820,31821],[34,671383,671385],{"className":671384},[31685,31821],[34,671386,768],{"className":671387},[31685,31821],[34,671389,671390,671393],{"style":41520},[34,671391],{"className":671392,"style":41508},[31813],[34,671394],{"className":671395,"style":41528},[41527],[34,671397,671398,671401],{"style":46771},[34,671399],{"className":671400,"style":41508},[31813],[34,671402,671404],{"className":671403},[31818,31819,31820,31821],[34,671405,671407],{"className":671406},[31685,31821],[34,671408,58442],{"className":671409,"style":32062},[31685,31686,31821],[34,671411,31829],{"className":671412},[31828],[34,671414,671416],{"className":671415},[31801],[34,671417,671419],{"className":671418,"style":41619},[31805],[34,671420],{},[34,671422],{"className":671423},[31690,41488]," (点 B)。\n",[34,671426,671428,671460],{"className":671427},[31642],[34,671429,671431],{"className":671430},[31646],[31648,671432,671433],{"xmlns":31650},[31652,671434,671435,671457],{},[31655,671436,671437,671439,671441,671443,671445,671447,671449,671451,671453,671455],{},[31658,671438,46900],{"mathvariant":38079},[31658,671440,32936],{},[31662,671442,202],{},[31662,671444,32226],{},[31658,671446,547853],{},[31662,671448,32682],{},[31658,671450,33249],{},[39361,671452,39363],{},[31658,671454,46900],{"mathvariant":38079},[31658,671456,33249],{},[31665,671458,671459],{"encoding":31667},"\\mathrm{d}x = -\\sin t \\, \\mathrm{d}t",[34,671461,671463,671484],{"className":671462,"ariaHidden":758},[31672],[34,671464,671466,671469,671472,671475,671478,671481],{"className":671465},[31676],[34,671467],{"className":671468,"style":31681},[31680],[34,671470,46900],{"className":671471},[31685,207923],[34,671473,32936],{"className":671474},[31685,31686],[34,671476],{"className":671477,"style":31846},[31845],[34,671479,202],{"className":671480},[31850],[34,671482],{"className":671483,"style":31846},[31845],[34,671485,671487,671490,671493,671496,671499,671502,671505,671508,671511],{"className":671486},[31676],[34,671488],{"className":671489,"style":34299},[31680],[34,671491,32226],{"className":671492},[31685],[34,671494],{"className":671495,"style":32123},[31845],[34,671497,547853],{"className":671498},[32767],[34,671500],{"className":671501,"style":32123},[31845],[34,671503,33249],{"className":671504},[31685,31686],[34,671506],{"className":671507,"style":32123},[31845],[34,671509,46900],{"className":671510},[31685,207923],[34,671512,33249],{"className":671513},[31685,31686],[34,671515,671517,671547],{"className":671516},[31642],[34,671518,671520],{"className":671519},[31646],[31648,671521,671522],{"xmlns":31650},[31652,671523,671524,671544],{},[31655,671525,671526,671528,671530,671532,671534,671536,671538,671540,671542],{},[31658,671527,46900],{"mathvariant":38079},[31658,671529,67764],{},[31662,671531,202],{},[31658,671533,547860],{},[31662,671535,32682],{},[31658,671537,33249],{},[39361,671539,39363],{},[31658,671541,46900],{"mathvariant":38079},[31658,671543,33249],{},[31665,671545,671546],{"encoding":31667},"\\mathrm{d}y = \\cos t \\, \\mathrm{d}t",[34,671548,671550,671571],{"className":671549,"ariaHidden":758},[31672],[34,671551,671553,671556,671559,671562,671565,671568],{"className":671552},[31676],[34,671554],{"className":671555,"style":43217},[31680],[34,671557,46900],{"className":671558},[31685,207923],[34,671560,67764],{"className":671561,"style":32062},[31685,31686],[34,671563],{"className":671564,"style":31846},[31845],[34,671566,202],{"className":671567},[31850],[34,671569],{"className":671570,"style":31846},[31845],[34,671572,671574,671577,671580,671583,671586,671589,671592],{"className":671573},[31676],[34,671575],{"className":671576,"style":31681},[31680],[34,671578,547860],{"className":671579},[32767],[34,671581],{"className":671582,"style":32123},[31845],[34,671584,33249],{"className":671585},[31685,31686],[34,671587],{"className":671588,"style":32123},[31845],[34,671590,46900],{"className":671591},[31685,207923],[34,671593,33249],{"className":671594},[31685,31686],[108,671596,671597,33183,671600,662055,672303,617872,672606,672657,672658,39984,672756],{},[2627,671598,671599],{},"代入计算",[34,671601,671603],{"className":671602},[126385],[34,671604,671606,671790],{"className":671605},[31642],[34,671607,671609],{"className":671608},[31646],[31648,671610,671611],{"xmlns":31650,"display":126394},[31652,671612,671613,671787],{},[71078,671614,671615,671717],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,671616,671617,671627],{},[71087,671618,671619],{},[71090,671620,671621],{"scriptlevel":459,"displaystyle":758},[31743,671622,671623,671625],{},[31658,671624,106063],{},[31749,671626,640],{},[71087,671628,671629],{},[71090,671630,671631],{"scriptlevel":459,"displaystyle":758},[31655,671632,671633,671635,671637,671651,671653,671655,671665,671667,671669,671671,671673,671675,671677,671679,671681,671683,671685,671695,671697,671699,671701,671703,671705,671707,671709,671711,671713,671715],{},[31655,671634],{},[31662,671636,202],{},[38846,671638,671639,671641,671643],{},[31662,671640,546295],{},[31749,671642,459],{},[31655,671644,671645,671647,671649],{},[31658,671646,58442],{},[31658,671648,38080],{"mathvariant":38079},[31749,671650,768],{},[31662,671652,15972],{"stretchy":702},[31662,671654,456],{"stretchy":702},[37624,671656,671657,671663],{},[31655,671658,671659,671661],{},[31658,671660,547853],{},[31662,671662,32682],{},[31749,671664,768],{},[31658,671666,33249],{},[31662,671668,2557],{"stretchy":702},[31662,671670,456],{"stretchy":702},[31662,671672,32226],{},[31658,671674,547853],{},[31662,671676,32682],{},[31658,671678,33249],{},[31662,671680,2557],{"stretchy":702},[31662,671682,32226],{},[31662,671684,456],{"stretchy":702},[37624,671686,671687,671693],{},[31655,671688,671689,671691],{},[31658,671690,547860],{},[31662,671692,32682],{},[31749,671694,768],{},[31658,671696,33249],{},[31662,671698,2557],{"stretchy":702},[31662,671700,456],{"stretchy":702},[31658,671702,547860],{},[31662,671704,32682],{},[31658,671706,33249],{},[31662,671708,2557],{"stretchy":702},[31662,671710,32712],{"stretchy":702},[39361,671712,39363],{},[31658,671714,46900],{"mathvariant":38079},[31658,671716,33249],{},[71084,671718,671719,671725],{},[71087,671720,671721],{},[71090,671722,671723],{"scriptlevel":459,"displaystyle":758},[31655,671724],{},[71087,671726,671727],{},[71090,671728,671729],{"scriptlevel":459,"displaystyle":758},[31655,671730,671731,671733,671735,671737,671751,671753,671763,671765,671767,671777,671779,671781,671783,671785],{},[31655,671732],{},[31662,671734,202],{},[31662,671736,32226],{},[38846,671738,671739,671741,671743],{},[31662,671740,546295],{},[31749,671742,459],{},[31655,671744,671745,671747,671749],{},[31658,671746,58442],{},[31658,671748,38080],{"mathvariant":38079},[31749,671750,768],{},[31662,671752,456],{"stretchy":702},[37624,671754,671755,671761],{},[31655,671756,671757,671759],{},[31658,671758,547853],{},[31662,671760,32682],{},[31749,671762,786],{},[31658,671764,33249],{},[31662,671766,223],{},[37624,671768,671769,671775],{},[31655,671770,671771,671773],{},[31658,671772,547860],{},[31662,671774,32682],{},[31749,671776,786],{},[31658,671778,33249],{},[31662,671780,2557],{"stretchy":702},[39361,671782,39363],{},[31658,671784,46900],{"mathvariant":38079},[31658,671786,33249],{},[31665,671788,671789],{"encoding":31667},"\\begin{aligned}\nI_1 &= \\int_0^{\\pi\u002F2} [ (\\sin^2 t)(-\\sin t) - (\\cos^2 t)(\\cos t) ] \\, \\mathrm{d}t \\\\\n&= - \\int_0^{\\pi\u002F2} (\\sin^3 t + \\cos^3 t) \\, \\mathrm{d}t\n\\end{aligned}",[34,671791,671793],{"className":671792,"ariaHidden":758},[31672],[34,671794,671796,671800],{"className":671795},[31676],[34,671797],{"className":671798,"style":671799},[31680],"height:5.3997em;vertical-align:-2.4498em;",[34,671801,671803],{"className":671802},[31685],[34,671804,671806,671889],{"className":671805},[71078],[34,671807,671809],{"className":671808},[369003],[34,671810,671812,671880],{"className":671811},[31796,31797],[34,671813,671815,671877],{"className":671814},[31801],[34,671816,671819,671869],{"className":671817,"style":671818},[31805],"height:2.9498em;",[34,671820,671822,671826],{"style":671821},"top:-4.9498em;",[34,671823],{"className":671824,"style":671825},[31813],"height:3.6379em;",[34,671827,671829],{"className":671828},[31685],[34,671830,671832,671835],{"className":671831},[31685],[34,671833,106063],{"className":671834,"style":67482},[31685,31686],[34,671836,671838],{"className":671837},[31792],[34,671839,671841,671861],{"className":671840},[31796,31797],[34,671842,671844,671858],{"className":671843},[31801],[34,671845,671847],{"className":671846,"style":31806},[31805],[34,671848,671849,671852],{"style":217742},[34,671850],{"className":671851,"style":31814},[31813],[34,671853,671855],{"className":671854},[31818,31819,31820,31821],[34,671856,640],{"className":671857},[31685,31821],[34,671859,31829],{"className":671860},[31828],[34,671862,671864],{"className":671863},[31801],[34,671865,671867],{"className":671866,"style":31836},[31805],[34,671868],{},[34,671870,671871,671874],{"style":503106},[34,671872],{"className":671873,"style":671825},[31813],[34,671875],{"className":671876},[31685],[34,671878,31829],{"className":671879},[31828],[34,671881,671883],{"className":671882},[31801],[34,671884,671887],{"className":671885,"style":671886},[31805],"height:2.4498em;",[34,671888],{},[34,671890,671892],{"className":671891},[71384],[34,671893,671895,672295],{"className":671894},[31796,31797],[34,671896,671898,672292],{"className":671897},[31801],[34,671899,671901,672115],{"className":671900,"style":671818},[31805],[34,671902,671903,671906],{"style":671821},[34,671904],{"className":671905,"style":671825},[31813],[34,671907,671909,671912,671915,671918,671921,671979,671982,672011,672014,672017,672020,672023,672026,672029,672032,672035,672038,672041,672044,672047,672050,672053,672082,672085,672088,672091,672094,672097,672100,672103,672106,672109,672112],{"className":671908},[31685],[34,671910],{"className":671911},[31685],[34,671913],{"className":671914,"style":31846},[31845],[34,671916,202],{"className":671917},[31850],[34,671919],{"className":671920,"style":31846},[31845],[34,671922,671924,671927],{"className":671923},[32767],[34,671925,546295],{"className":671926,"style":591475},[32767,131249,131250],[34,671928,671930],{"className":671929},[31792],[34,671931,671933,671971],{"className":671932},[31796,31797],[34,671934,671936,671968],{"className":671935},[31801],[34,671937,671940,671951],{"className":671938,"style":671939},[31805],"height:1.6379em;",[34,671941,671942,671945],{"style":591491},[34,671943],{"className":671944,"style":31814},[31813],[34,671946,671948],{"className":671947},[31818,31819,31820,31821],[34,671949,459],{"className":671950},[31685,31821],[34,671952,671953,671956],{"style":591503},[34,671954],{"className":671955,"style":31814},[31813],[34,671957,671959],{"className":671958},[31818,31819,31820,31821],[34,671960,671962,671965],{"className":671961},[31685,31821],[34,671963,58442],{"className":671964,"style":32062},[31685,31686,31821],[34,671966,38149],{"className":671967},[31685,31821],[34,671969,31829],{"className":671970},[31828],[34,671972,671974],{"className":671973},[31801],[34,671975,671977],{"className":671976,"style":591522},[31805],[34,671978],{},[34,671980,86328],{"className":671981},[31782],[34,671983,671985,671988],{"className":671984},[32767],[34,671986,547853],{"className":671987},[32767],[34,671989,671991],{"className":671990},[31792],[34,671992,671994],{"className":671993},[31796],[34,671995,671997],{"className":671996},[31801],[34,671998,672000],{"className":671999,"style":585884},[31805],[34,672001,672002,672005],{"style":586008},[34,672003],{"className":672004,"style":31814},[31813],[34,672006,672008],{"className":672007},[31818,31819,31820,31821],[34,672009,768],{"className":672010},[31685,31821],[34,672012],{"className":672013,"style":32123},[31845],[34,672015,33249],{"className":672016},[31685,31686],[34,672018,2557],{"className":672019},[31690],[34,672021,456],{"className":672022},[31782],[34,672024,32226],{"className":672025},[31685],[34,672027],{"className":672028,"style":32123},[31845],[34,672030,547853],{"className":672031},[32767],[34,672033],{"className":672034,"style":32123},[31845],[34,672036,33249],{"className":672037},[31685,31686],[34,672039,2557],{"className":672040},[31690],[34,672042],{"className":672043,"style":32604},[31845],[34,672045,32226],{"className":672046},[32024],[34,672048],{"className":672049,"style":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\\sin^n x dx = \\int_0^{\\pi\u002F2} \\cos^n x 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时，值为 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= - (\\frac{2}{3} + \\frac{2}{3}) = 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= \\int_{L_2} 0 + (z^2 - 0) \\, \\mathrm{d}y + (0 - y^2) \\, \\mathrm{d}z = \\int_{L_2} z^2 \\, \\mathrm{d}y - y^2 \\, \\mathrm{d}z",[34,673689,673691,673746,673848,673895,673925,673946,674002,674138],{"className":673690,"ariaHidden":758},[31672],[34,673692,673694,673697,673737,673740,673743],{"className":673693},[31676],[34,673695],{"className":673696,"style":70915},[31680],[34,673698,673700,673703],{"className":673699},[31685],[34,673701,106063],{"className":673702,"style":67482},[31685,31686],[34,673704,673706],{"className":673705},[31792],[34,673707,673709,673729],{"className":673708},[31796,31797],[34,673710,673712,673726],{"className":673711},[31801],[34,673713,673715],{"className":673714,"style":31806},[31805],[34,673716,673717,673720],{"style":217742},[34,673718],{"className":673719,"style":31814},[31813],[34,673721,673723],{"className":673722},[31818,31819,31820,31821],[34,673724,768],{"className":673725},[31685,31821],[34,673727,31829],{"className":673728},[31828],[34,673730,673732],{"className":673731},[31801],[34,673733,673735],{"className":673734,"style":31836},[31805],[34,673736],{},[34,673738],{"className":673739,"style":31846},[31845],[34,673741,202],{"className":673742},[31850],[34,673744],{"className":673745,"style":31846},[31845],[34,673747,673749,673753,673833,673836,673839,673842,673845],{"className":673748},[31676],[34,673750],{"className":673751,"style":673752},[31680],"height:1.2609em;vertical-align:-0.4559em;",[34,673754,673756,673759],{"className":673755},[32767],[34,673757,546295],{"className":673758,"style":546331},[32767,131249,156560],[34,673760,673762],{"className":673761},[31792],[34,673763,673765,673825],{"className":673764},[31796,31797],[34,673766,673768,673822],{"className":673767},[31801],[34,673769,673771],{"className":673770,"style":637339},[31805],[34,673772,673773,673776],{"style":546347},[34,673774],{"className":673775,"style":31814},[31813],[34,673777,673779],{"className":673778},[31818,31819,31820,31821],[34,673780,673782],{"className":673781},[31685,31821],[34,673783,673785,673788],{"className":673784},[31685,31821],[34,673786,33206],{"className":673787},[31685,31686,31821],[34,673789,673791],{"className":673790},[31792],[34,673792,673794,673814],{"className":673793},[31796,31797],[34,673795,673797,673811],{"className":673796},[31801],[34,673798,673800],{"className":673799,"style":318658},[31805],[34,673801,673802,673805],{"style":318661},[34,673803],{"className":673804,"style":43406},[31813],[34,673806,673808],{"className":673807},[31818,43410,43411,31821],[34,673809,768],{"className":673810},[31685,31821],[34,673812,31829],{"className":673813},[31828],[34,673815,673817],{"className":673816},[31801],[34,673818,673820],{"className":673819,"style":220540},[31805],[34,673821],{},[34,673823,31829],{"className":673824},[31828],[34,673826,673828],{"className":673827},[31801],[34,673829,673831],{"className":673830,"style":667231},[31805],[34,673832],{},[34,673834],{"className":673835,"style":32123},[31845],[34,673837,459],{"className":673838},[31685],[34,673840],{"className":673841,"style":32604},[31845],[34,673843,223],{"className":673844},[32024],[34,673846],{"className":673847,"style":32604},[31845],[34,673849,673851,673854,673857,673886,673889,673892],{"className":673850},[31676],[34,673852],{"className":673853,"style":37755},[31680],[34,673855,456],{"className":673856},[31782],[34,673858,673860,673863],{"className":673859},[31685],[34,673861,382333],{"className":673862,"style":382406},[31685,31686],[34,673864,673866],{"className":673865},[31792],[34,673867,673869],{"className":673868},[31796],[34,673870,673872],{"className":673871},[31801],[34,673873,673875],{"className":673874,"style":37774},[31805],[34,673876,673877,673880],{"style":37777},[34,673878],{"className":673879,"style":31814},[31813],[34,673881,673883],{"className":673882},[31818,31819,31820,31821],[34,673884,768],{"className":673885},[31685,31821],[34,673887],{"className":673888,"style":32604},[31845],[34,673890,32226],{"className":673891},[32024],[34,673893],{"className":673894,"style":32604},[31845],[34,673896,673898,673901,673904,673907,673910,673913,673916,673919,673922],{"className":673897},[31676],[34,673899],{"className":673900,"style":31774},[31680],[34,673902,459],{"className":673903},[31685],[34,673905,2557],{"className":673906},[31690],[34,673908],{"className":673909,"style":32123},[31845],[34,673911,46900],{"className":673912},[31685,207923],[34,673914,67764],{"className":673915,"style":32062},[31685,31686],[34,673917],{"className":673918,"style":32604},[31845],[34,673920,223],{"className":673921},[32024],[34,673923],{"className":673924,"style":32604},[31845],[34,673926,673928,673931,673934,673937,673940,673943],{"className":673927},[31676],[34,673929],{"className":673930,"style":31774},[31680],[34,673932,456],{"className":673933},[31782],[34,673935,459],{"className":673936},[31685],[34,673938],{"className":673939,"style":32604},[31845],[34,673941,32226],{"className":673942},[32024],[34,673944],{"className":673945,"style":32604},[31845],[34,673947,673949,673952,673981,673984,673987,673990,673993,673996,673999],{"className":673948},[31676],[34,673950],{"className":673951,"style":37755},[31680],[34,673953,673955,673958],{"className":673954},[31685],[34,673956,67764],{"className":673957,"style":32062},[31685,31686],[34,673959,673961],{"className":673960},[31792],[34,673962,673964],{"className":673963},[31796],[34,673965,673967],{"className":673966},[31801],[34,673968,673970],{"className":673969,"style":37774},[31805],[34,673971,673972,673975],{"style":37777},[34,673973],{"className":673974,"style":31814},[31813],[34,673976,673978],{"className":673977},[31818,31819,31820,31821],[34,673979,768],{"className":673980},[31685,31821],[34,673982,2557],{"className":673983},[31690],[34,673985],{"className":673986,"style":32123},[31845],[34,673988,46900],{"className":673989},[31685,207923],[34,673991,382333],{"className":673992,"style":382406},[31685,31686],[34,673994],{"className":673995,"style":31846},[31845],[34,673997,202],{"className":673998},[31850],[34,674000],{"className":674001,"style":31846},[31845],[34,674003,674005,674008,674088,674091,674120,674123,674126,674129,674132,674135],{"className":674004},[31676],[34,674006],{"className":674007,"style":670731},[31680],[34,674009,674011,674014],{"className":674010},[32767],[34,674012,546295],{"className":674013,"style":546331},[32767,131249,156560],[34,674015,674017],{"className":674016},[31792],[34,674018,674020,674080],{"className":674019},[31796,31797],[34,674021,674023,674077],{"className":674022},[31801],[34,674024,674026],{"className":674025,"style":637339},[31805],[34,674027,674028,674031],{"style":546347},[34,674029],{"className":674030,"style":31814},[31813],[34,674032,674034],{"className":674033},[31818,31819,31820,31821],[34,674035,674037],{"className":674036},[31685,31821],[34,674038,674040,674043],{"className":674039},[31685,31821],[34,674041,33206],{"className":674042},[31685,31686,31821],[34,674044,674046],{"className":674045},[31792],[34,674047,674049,674069],{"className":674048},[31796,31797],[34,674050,674052,674066],{"className":674051},[31801],[34,674053,674055],{"className":674054,"style":318658},[31805],[34,674056,674057,674060],{"style":318661},[34,674058],{"className":674059,"style":43406},[31813],[34,674061,674063],{"className":674062},[31818,43410,43411,31821],[34,674064,768],{"className":674065},[31685,31821],[34,674067,31829],{"className":674068},[31828],[34,674070,674072],{"className":674071},[31801],[34,674073,674075],{"className":674074,"style":220540},[31805],[34,674076],{},[34,674078,31829],{"className":674079},[31828],[34,674081,674083],{"className":674082},[31801],[34,674084,674086],{"className":674085,"style":667231},[31805],[34,674087],{},[34,674089],{"className":674090,"style":32123},[31845],[34,674092,674094,674097],{"className":674093},[31685],[34,674095,382333],{"className":674096,"style":382406},[31685,31686],[34,674098,674100],{"className":674099},[31792],[34,674101,674103],{"className":674102},[31796],[34,674104,674106],{"className":674105},[31801],[34,674107,674109],{"className":674108,"style":37774},[31805],[34,674110,674111,674114],{"style":37777},[34,674112],{"className":674113,"style":31814},[31813],[34,674115,674117],{"className":674116},[31818,31819,31820,31821],[34,674118,768],{"className":674119},[31685,31821],[34,674121],{"className":674122,"style":32123},[31845],[34,674124,46900],{"className":674125},[31685,207923],[34,674127,67764],{"className":674128,"style":32062},[31685,31686],[34,674130],{"className":674131,"style":32604},[31845],[34,674133,32226],{"className":674134},[32024],[34,674136],{"className":674137,"style":32604},[31845],[34,674139,674141,674144,674173,674176,674179],{"className":674140},[31676],[34,674142],{"className":674143,"style":315747},[31680],[34,674145,674147,674150],{"className":674146},[31685],[34,674148,67764],{"className":674149,"style":32062},[31685,31686],[34,674151,674153],{"className":674152},[31792],[34,674154,674156],{"className":674155},[31796],[34,674157,674159],{"className":674158},[31801],[34,674160,674162],{"className":674161,"style":37774},[31805],[34,674163,674164,674167],{"style":37777},[34,674165],{"className":674166,"style":31814},[31813],[34,674168,674170],{"className":674169},[31818,31819,31820,31821],[34,674171,768],{"className":674172},[31685,31821],[34,674174],{"className":674175,"style":32123},[31845],[34,674177,46900],{"className":674178},[31685,207923],[34,674180,382333],{"className":674181,"style":382406},[31685,31686],[108,674183,674184,674187,674188,43298,674319,674388,674389,674520,674521,674607,674608,674636,674637,674665,674666],{},[2627,674185,674186],{},"观察对称性","：\n这里的形式 ",[34,674189,674191,674225],{"className":674190},[31642],[34,674192,674194],{"className":674193},[31646],[31648,674195,674196],{"xmlns":31650},[31652,674197,674198,674222],{},[31655,674199,674200,674206,674208,674210,674212,674218,674220],{},[37624,674201,674202,674204],{},[31658,674203,382333],{},[31749,674205,768],{},[31658,674207,46900],{"mathvariant":38079},[31658,674209,67764],{},[31662,674211,32226],{},[37624,674213,674214,674216],{},[31658,674215,67764],{},[31749,674217,768],{},[31658,674219,46900],{"mathvariant":38079},[31658,674221,382333],{},[31665,674223,674224],{"encoding":31667},"z^2 \\mathrm{d}y - y^2 \\mathrm{d}z",[34,674226,674228,674278],{"className":674227,"ariaHidden":758},[31672],[34,674229,674231,674234,674263,674266,674269,674272,674275],{"className":674230},[31676],[34,674232],{"className":674233,"style":315747},[31680],[34,674235,674237,674240],{"className":674236},[31685],[34,674238,382333],{"className":674239,"style":382406},[31685,31686],[34,674241,674243],{"className":674242},[31792],[34,674244,674246],{"className":674245},[31796],[34,674247,674249],{"className":674248},[31801],[34,674250,674252],{"className":674251,"style":37774},[31805],[34,674253,674254,674257],{"style":37777},[34,674255],{"className":674256,"style":31814},[31813],[34,674258,674260],{"className":674259},[31818,31819,31820,31821],[34,674261,768],{"className":674262},[31685,31821],[34,674264,46900],{"className":674265},[31685,207923],[34,674267,67764],{"className":674268,"style":32062},[31685,31686],[34,674270],{"className":674271,"style":32604},[31845],[34,674273,32226],{"className":674274},[32024],[34,674276],{"className":674277,"style":32604},[31845],[34,674279,674281,674284,674313,674316],{"className":674280},[31676],[34,674282],{"className":674283,"style":315747},[31680],[34,674285,674287,674290],{"className":674286},[31685],[34,674288,67764],{"className":674289,"style":32062},[31685,31686],[34,674291,674293],{"className":674292},[31792],[34,674294,674296],{"className":674295},[31796],[34,674297,674299],{"className":674298},[31801],[34,674300,674302],{"className":674301,"style":37774},[31805],[34,674303,674304,674307],{"style":37777},[34,674305],{"className":674306,"style":31814},[31813],[34,674308,674310],{"className":674309},[31818,31819,31820,31821],[34,674311,768],{"className":674312},[31685,31821],[34,674314,46900],{"className":674315},[31685,207923],[34,674317,382333],{"className":674318,"style":382406},[31685,31686],[34,674320,674322,674339],{"className":674321},[31642],[34,674323,674325],{"className":674324},[31646],[31648,674326,674327],{"xmlns":31650},[31652,674328,674329,674337],{},[31655,674330,674331],{},[31743,674332,674333,674335],{},[31658,674334,33206],{},[31749,674336,640],{},[31665,674338,658379],{"encoding":31667},[34,674340,674342],{"className":674341,"ariaHidden":758},[31672],[34,674343,674345,674348],{"className":674344},[31676],[34,674346],{"className":674347,"style":70915},[31680],[34,674349,674351,674354],{"className":674350},[31685],[34,674352,33206],{"className":674353},[31685,31686],[34,674355,674357],{"className":674356},[31792],[34,674358,674360,674380],{"className":674359},[31796,31797],[34,674361,674363,674377],{"className":674362},[31801],[34,674364,674366],{"className":674365,"style":31806},[31805],[34,674367,674368,674371],{"style":31809},[34,674369],{"className":674370,"style":31814},[31813],[34,674372,674374],{"className":674373},[31818,31819,31820,31821],[34,674375,640],{"className":674376},[31685,31821],[34,674378,31829],{"className":674379},[31828],[34,674381,674383],{"className":674382},[31801],[34,674384,674386],{"className":674385,"style":31836},[31805],[34,674387],{}," 中的 ",[34,674390,674392,674426],{"className":674391},[31642],[34,674393,674395],{"className":674394},[31646],[31648,674396,674397],{"xmlns":31650},[31652,674398,674399,674423],{},[31655,674400,674401,674407,674409,674411,674413,674419,674421],{},[37624,674402,674403,674405],{},[31658,674404,67764],{},[31749,674406,768],{},[31658,674408,46900],{"mathvariant":38079},[31658,674410,32936],{},[31662,674412,32226],{},[37624,674414,674415,674417],{},[31658,674416,32936],{},[31749,674418,768],{},[31658,674420,46900],{"mathvariant":38079},[31658,674422,67764],{},[31665,674424,674425],{"encoding":31667},"y^2 \\mathrm{d}x - x^2 \\mathrm{d}y",[34,674427,674429,674479],{"className":674428,"ariaHidden":758},[31672],[34,674430,674432,674435,674464,674467,674470,674473,674476],{"className":674431},[31676],[34,674433],{"className":674434,"style":315747},[31680],[34,674436,674438,674441],{"className":674437},[31685],[34,674439,67764],{"className":674440,"style":32062},[31685,31686],[34,674442,674444],{"className":674443},[31792],[34,674445,674447],{"className":674446},[31796],[34,674448,674450],{"className":674449},[31801],[34,674451,674453],{"className":674452,"style":37774},[31805],[34,674454,674455,674458],{"style":37777},[34,674456],{"className":674457,"style":31814},[31813],[34,674459,674461],{"className":674460},[31818,31819,31820,31821],[34,674462,768],{"className":674463},[31685,31821],[34,674465,46900],{"className":674466},[31685,207923],[34,674468,32936],{"className":674469},[31685,31686],[34,674471],{"className":674472,"style":32604},[31845],[34,674474,32226],{"className":674475},[32024],[34,674477],{"className":674478,"style":32604},[31845],[34,674480,674482,674485,674514,674517],{"className":674481},[31676],[34,674483],{"className":674484,"style":315747},[31680],[34,674486,674488,674491],{"className":674487},[31685],[34,674489,32936],{"className":674490},[31685,31686],[34,674492,674494],{"className":674493},[31792],[34,674495,674497],{"className":674496},[31796],[34,674498,674500],{"className":674499},[31801],[34,674501,674503],{"className":674502,"style":37774},[31805],[34,674504,674505,674508],{"style":37777},[34,674506],{"className":674507,"style":31814},[31813],[34,674509,674511],{"className":674510},[31818,31819,31820,31821],[34,674512,768],{"className":674513},[31685,31821],[34,674515,46900],{"className":674516},[31685,207923],[34,674518,67764],{"className":674519,"style":32062},[31685,31686]," 结构完全一致，只是变量轮换了 (",[34,674522,674524,674550],{"className":674523},[31642],[34,674525,674527],{"className":674526},[31646],[31648,674528,674529],{"xmlns":31650},[31652,674530,674531,674547],{},[31655,674532,674533,674535,674537,674539,674541,674543,674545],{},[31658,674534,32936],{},[31662,674536,33239],{},[31658,674538,67764],{},[31662,674540,31960],{"separator":758},[31658,674542,67764],{},[31662,674544,33239],{},[31658,674546,382333],{},[31665,674548,674549],{"encoding":31667},"x \\to y, y \\to z",[34,674551,674553,674571,674598],{"className":674552,"ariaHidden":758},[31672],[34,674554,674556,674559,674562,674565,674568],{"className":674555},[31676],[34,674557],{"className":674558,"style":32527},[31680],[34,674560,32936],{"className":674561},[31685,31686],[34,674563],{"className":674564,"style":31846},[31845],[34,674566,33239],{"className":674567},[31850],[34,674569],{"className":674570,"style":31846},[31845],[34,674572,674574,674577,674580,674583,674586,674589,674592,674595],{"className":674573},[31676],[34,674575],{"className":674576,"style":38774},[31680],[34,674578,67764],{"className":674579,"style":32062},[31685,31686],[34,674581,31960],{"className":674582},[32119],[34,674584],{"className":674585,"style":32123},[31845],[34,674587,67764],{"className":674588,"style":32062},[31685,31686],[34,674590],{"className":674591,"style":31846},[31845],[34,674593,33239],{"className":674594},[31850],[34,674596],{"className":674597,"style":31846},[31845],[34,674599,674601,674604],{"className":674600},[31676],[34,674602],{"className":674603,"style":32527},[31680],[34,674605,382333],{"className":674606,"style":382406},[31685,31686],")。\n路径也是从轴上的 ",[34,674609,674611,674624],{"className":674610},[31642],[34,674612,674614],{"className":674613},[31646],[31648,674615,674616],{"xmlns":31650},[31652,674617,674618,674622],{},[31655,674619,674620],{},[31749,674621,640],{},[31665,674623,640],{"encoding":31667},[34,674625,674627],{"className":674626,"ariaHidden":758},[31672],[34,674628,674630,674633],{"className":674629},[31676],[34,674631],{"className":674632,"style":32617},[31680],[34,674634,640],{"className":674635},[31685]," 到轴上的 ",[34,674638,674640,674653],{"className":674639},[31642],[34,674641,674643],{"className":674642},[31646],[31648,674644,674645],{"xmlns":31650},[31652,674646,674647,674651],{},[31655,674648,674649],{},[31749,674650,640],{},[31665,674652,640],{"encoding":31667},[34,674654,674656],{"className":674655,"ariaHidden":758},[31672],[34,674657,674659,674662],{"className":674658},[31676],[34,674660],{"className":674661,"style":32617},[31680],[34,674663,640],{"className":674664},[31685]," 的四分之一圆弧。\n因此，计算结果应当相同。\n",[34,674667,674669,674697],{"className":674668},[31642],[34,674670,674672],{"className":674671},[31646],[31648,674673,674674],{"xmlns":31650},[31652,674675,674676,674694],{},[31655,674677,674678,674684,674686,674688],{},[31743,674679,674680,674682],{},[31658,674681,106063],{},[31749,674683,768],{},[31662,674685,202],{},[31662,674687,32226],{},[41453,674689,674690,674692],{},[31749,674691,15932],{},[31749,674693,786],{},[31665,674695,674696],{"encoding":31667},"I_2 = -\\frac{4}{3}",[34,674698,674700,674755],{"className":674699,"ariaHidden":758},[31672],[34,674701,674703,674706,674746,674749,674752],{"className":674702},[31676],[34,674704],{"className":674705,"style":70915},[31680],[34,674707,674709,674712],{"className":674708},[31685],[34,674710,106063],{"className":674711,"style":67482},[31685,31686],[34,674713,674715],{"className":674714},[31792],[34,674716,674718,674738],{"className":674717},[31796,31797],[34,674719,674721,674735],{"className":674720},[31801],[34,674722,674724],{"className":674723,"style":31806},[31805],[34,674725,674726,674729],{"style":217742},[34,674727],{"className":674728,"style":31814},[31813],[34,674730,674732],{"className":674731},[31818,31819,31820,31821],[34,674733,768],{"className":674734},[31685,31821],[34,674736,31829],{"className":674737},[31828],[34,674739,674741],{"className":674740},[31801],[34,674742,674744],{"className":674743,"style":31836},[31805],[34,674745],{},[34,674747],{"className":674748,"style":31846},[31845],[34,674750,202],{"className":674751},[31850],[34,674753],{"className":674754,"style":31846},[31845],[34,674756,674758,674761,674764],{"className":674757},[31676],[34,674759],{"className":674760,"style":46727},[31680],[34,674762,32226],{"className":674763},[31685],[34,674765,674767,674770,674829],{"className":674766},[31685],[34,674768],{"className":674769},[31782,41488],[34,674771,674773],{"className":674772},[41453],[34,674774,674776,674821],{"className":674775},[31796,31797],[34,674777,674779,674818],{"className":674778},[31801],[34,674780,674782,674796,674804],{"className":674781,"style":46746},[31805],[34,674783,674784,674787],{"style":41504},[34,674785],{"className":674786,"style":41508},[31813],[34,674788,674790],{"className":674789},[31818,31819,31820,31821],[34,674791,674793],{"className":674792},[31685,31821],[34,674794,786],{"className":674795},[31685,31821],[34,674797,674798,674801],{"style":41520},[34,674799],{"className":674800,"style":41508},[31813],[34,674802],{"className":674803,"style":41528},[41527],[34,674805,674806,674809],{"style":46771},[34,674807],{"className":674808,"style":41508},[31813],[34,674810,674812],{"className":674811},[31818,31819,31820,31821],[34,674813,674815],{"className":674814},[31685,31821],[34,674816,15932],{"className":674817},[31685,31821],[34,674819,31829],{"className":674820},[31828],[34,674822,674824],{"className":674823},[31801],[34,674825,674827],{"className":674826,"style":41619},[31805],[34,674828],{},[34,674830],{"className":674831},[31690,41488],[15,674833,674834],{},[2627,674835,674836,674837,343532,674906,670199],{},"第三段 ",[34,674838,674840,674857],{"className":674839},[31642],[34,674841,674843],{"className":674842},[31646],[31648,674844,674845],{"xmlns":31650},[31652,674846,674847,674855],{},[31655,674848,674849],{},[31743,674850,674851,674853],{},[31658,674852,33206],{},[31749,674854,786],{},[31665,674856,669644],{"encoding":31667},[34,674858,674860],{"className":674859,"ariaHidden":758},[31672],[34,674861,674863,674866],{"className":674862},[31676],[34,674864],{"className":674865,"style":70915},[31680],[34,674867,674869,674872],{"className":674868},[31685],[34,674870,33206],{"className":674871},[31685,31686],[34,674873,674875],{"className":674874},[31792],[34,674876,674878,674898],{"className":674877},[31796,31797],[34,674879,674881,674895],{"className":674880},[31801],[34,674882,674884],{"className":674883,"style":31806},[31805],[34,674885,674886,674889],{"style":31809},[34,674887],{"className":674888,"style":31814},[31813],[34,674890,674892],{"className":674891},[31818,31819,31820,31821],[34,674893,786],{"className":674894},[31685,31821],[34,674896,31829],{"className":674897},[31828],[34,674899,674901],{"className":674900},[31801],[34,674902,674904],{"className":674903,"style":31836},[31805],[34,674905],{},[34,674907,674909,674924],{"className":674908},[31642],[34,674910,674912],{"className":674911},[31646],[31648,674913,674914],{"xmlns":31650},[31652,674915,674916,674922],{},[31655,674917,674918,674920],{},[31658,674919,382333],{},[31658,674921,32936],{},[31665,674923,669712],{"encoding":31667},[34,674925,674927],{"className":674926,"ariaHidden":758},[31672],[34,674928,674930,674933,674936],{"className":674929},[31676],[34,674931],{"className":674932,"style":32527},[31680],[34,674934,382333],{"className":674935,"style":382406},[31685,31686],[34,674937,32936],{"className":674938},[31685,31686],[105,674940,674941,675050,675187,675803,676146,676903],{},[108,674942,674943,2664,674945,670257,674995,347],{},[2627,674944,491730],{},[34,674946,674948,674965],{"className":674947},[31642],[34,674949,674951],{"className":674950},[31646],[31648,674952,674953],{"xmlns":31650},[31652,674954,674955,674963],{},[31655,674956,674957,674959,674961],{},[31658,674958,67764],{},[31662,674960,202],{},[31749,674962,459],{},[31665,674964,663938],{"encoding":31667},[34,674966,674968,674986],{"className":674967,"ariaHidden":758},[31672],[34,674969,674971,674974,674977,674980,674983],{"className":674970},[31676],[34,674972],{"className":674973,"style":38774},[31680],[34,674975,67764],{"className":674976,"style":32062},[31685,31686],[34,674978],{"className":674979,"style":31846},[31845],[34,674981,202],{"className":674982},[31850],[34,674984],{"className":674985,"style":31846},[31845],[34,674987,674989,674992],{"className":674988},[31676],[34,674990],{"className":674991,"style":32617},[31680],[34,674993,459],{"className":674994},[31685],[34,674996,674998,675017],{"className":674997},[31642],[34,674999,675001],{"className":675000},[31646],[31648,675002,675003],{"xmlns":31650},[31652,675004,675005,675015],{},[31655,675006,675007,675009,675011,675013],{},[31658,675008,46900],{"mathvariant":38079},[31658,675010,67764],{},[31662,675012,202],{},[31749,675014,459],{},[31665,675016,664145],{"encoding":31667},[34,675018,675020,675041],{"className":675019,"ariaHidden":758},[31672],[34,675021,675023,675026,675029,675032,675035,675038],{"className":675022},[31676],[34,675024],{"className":675025,"style":43217},[31680],[34,675027,46900],{"className":675028},[31685,207923],[34,675030,67764],{"className":675031,"style":32062},[31685,31686],[34,675033],{"className":675034,"style":31846},[31845],[34,675036,202],{"className":675037},[31850],[34,675039],{"className":675040,"style":31846},[31845],[34,675042,675044,675047],{"className":675043},[31676],[34,675045],{"className":675046,"style":32617},[31680],[34,675048,459],{"className":675049},[31685],[108,675051,675052,2664,675054,347],{},[2627,675053,670318],{},[34,675055,675057,675087],{"className":675056},[31642],[34,675058,675060],{"className":675059},[31646],[31648,675061,675062],{"xmlns":31650},[31652,675063,675064,675084],{},[31655,675065,675066,675072,675074,675080,675082],{},[37624,675067,675068,675070],{},[31658,675069,382333],{},[31749,675071,768],{},[31662,675073,223],{},[37624,675075,675076,675078],{},[31658,675077,32936],{},[31749,675079,768],{},[31662,675081,202],{},[31749,675083,640],{},[31665,675085,675086],{"encoding":31667},"z^2 + x^2 = 1",[34,675088,675090,675134,675178],{"className":675089,"ariaHidden":758},[31672],[34,675091,675093,675096,675125,675128,675131],{"className":675092},[31676],[34,675094],{"className":675095,"style":40044},[31680],[34,675097,675099,675102],{"className":675098},[31685],[34,675100,382333],{"className":675101,"style":382406},[31685,31686],[34,675103,675105],{"className":675104},[31792],[34,675106,675108],{"className":675107},[31796],[34,675109,675111],{"className":675110},[31801],[34,675112,675114],{"className":675113,"style":37774},[31805],[34,675115,675116,675119],{"style":37777},[34,675117],{"className":675118,"style":31814},[31813],[34,675120,675122],{"className":675121},[31818,31819,31820,31821],[34,675123,768],{"className":675124},[31685,31821],[34,675126],{"className":675127,"style":32604},[31845],[34,675129,223],{"className":675130},[32024],[34,675132],{"className":675133,"style":32604},[31845],[34,675135,675137,675140,675169,675172,675175],{"className":675136},[31676],[34,675138],{"className":675139,"style":37774},[31680],[34,675141,675143,675146],{"className":675142},[31685],[34,675144,32936],{"className":675145},[31685,31686],[34,675147,675149],{"className":675148},[31792],[34,675150,675152],{"className":675151},[31796],[34,675153,675155],{"className":675154},[31801],[34,675156,675158],{"className":675157,"style":37774},[31805],[34,675159,675160,675163],{"style":37777},[34,675161],{"className":675162,"style":31814},[31813],[34,675164,675166],{"className":675165},[31818,31819,31820,31821],[34,675167,768],{"className":675168},[31685,31821],[34,675170],{"className":675171,"style":31846},[31845],[34,675173,202],{"className":675174},[31850],[34,675176],{"className":675177,"style":31846},[31845],[34,675179,675181,675184],{"className":675180},[31676],[34,675182],{"className":675183,"style":32617},[31680],[34,675185,640],{"className":675186},[31685],[108,675188,675189,33183,675191],{},[2627,675190,670457],{},[34,675192,675194,675310],{"className":675193},[31642],[34,675195,675197],{"className":675196},[31646],[31648,675198,675199],{"xmlns":31650},[31652,675200,675201,675307],{},[31655,675202,675203,675209,675211,675221,675223,675225,675227,675233,675235,675237,675239,675241,675243,675245,675247,675249,675255,675257,675259,675261,675263,675265,675267,675269,675279,675281,675287,675289,675291,675293,675295,675301,675303,675305],{},[31743,675204,675205,675207],{},[31658,675206,106063],{},[31749,675208,786],{},[31662,675210,202],{},[31743,675212,675213,675215],{},[31662,675214,546295],{},[31743,675216,675217,675219],{},[31658,675218,33206],{},[31749,675220,786],{},[31662,675222,456],{"stretchy":702},[31749,675224,459],{},[31662,675226,32226],{},[37624,675228,675229,675231],{},[31658,675230,382333],{},[31749,675232,768],{},[31662,675234,2557],{"stretchy":702},[39361,675236,39363],{},[31658,675238,46900],{"mathvariant":38079},[31658,675240,32936],{},[31662,675242,223],{},[31749,675244,459],{},[31662,675246,223],{},[31662,675248,456],{"stretchy":702},[37624,675250,675251,675253],{},[31658,675252,32936],{},[31749,675254,768],{},[31662,675256,32226],{},[31749,675258,459],{},[31662,675260,2557],{"stretchy":702},[39361,675262,39363],{},[31658,675264,46900],{"mathvariant":38079},[31658,675266,382333],{},[31662,675268,202],{},[31743,675270,675271,675273],{},[31662,675272,546295],{},[31743,675274,675275,675277],{},[31658,675276,33206],{},[31749,675278,786],{},[31662,675280,32226],{},[37624,675282,675283,675285],{},[31658,675284,382333],{},[31749,675286,768],{},[39361,675288,39363],{},[31658,675290,46900],{"mathvariant":38079},[31658,675292,32936],{},[31662,675294,223],{},[37624,675296,675297,675299],{},[31658,675298,32936],{},[31749,675300,768],{},[39361,675302,39363],{},[31658,675304,46900],{"mathvariant":38079},[31658,675306,382333],{},[31665,675308,675309],{"encoding":31667},"I_3 = \\int_{L_3} (0 - z^2) \\, \\mathrm{d}x + 0 + (x^2 - 0) \\, \\mathrm{d}z = \\int_{L_3} -z^2 \\, \\mathrm{d}x + x^2 \\, \\mathrm{d}z",[34,675311,675313,675368,675469,675525,675543,675590,675620,675759],{"className":675312,"ariaHidden":758},[31672],[34,675314,675316,675319,675359,675362,675365],{"className":675315},[31676],[34,675317],{"className":675318,"style":70915},[31680],[34,675320,675322,675325],{"className":675321},[31685],[34,675323,106063],{"className":675324,"style":67482},[31685,31686],[34,675326,675328],{"className":675327},[31792],[34,675329,675331,675351],{"className":675330},[31796,31797],[34,675332,675334,675348],{"className":675333},[31801],[34,675335,675337],{"className":675336,"style":31806},[31805],[34,675338,675339,675342],{"style":217742},[34,675340],{"className":675341,"style":31814},[31813],[34,675343,675345],{"className":675344},[31818,31819,31820,31821],[34,675346,786],{"className":675347},[31685,31821],[34,675349,31829],{"className":675350},[31828],[34,675352,675354],{"className":675353},[31801],[34,675355,675357],{"className":675356,"style":31836},[31805],[34,675358],{},[34,675360],{"className":675361,"style":31846},[31845],[34,675363,202],{"className":675364},[31850],[34,675366],{"className":675367,"style":31846},[31845],[34,675369,675371,675374,675454,675457,675460,675463,675466],{"className":675370},[31676],[34,675372],{"className":675373,"style":673752},[31680],[34,675375,675377,675380],{"className":675376},[32767],[34,675378,546295],{"className":675379,"style":546331},[32767,131249,156560],[34,675381,675383],{"className":675382},[31792],[34,675384,675386,675446],{"className":675385},[31796,31797],[34,675387,675389,675443],{"className":675388},[31801],[34,675390,675392],{"className":675391,"style":637339},[31805],[34,675393,675394,675397],{"style":546347},[34,675395],{"className":675396,"style":31814},[31813],[34,675398,675400],{"className":675399},[31818,31819,31820,31821],[34,675401,675403],{"className":675402},[31685,31821],[34,675404,675406,675409],{"className":675405},[31685,31821],[34,675407,33206],{"className":675408},[31685,31686,31821],[34,675410,675412],{"className":675411},[31792],[34,675413,675415,675435],{"className":675414},[31796,31797],[34,675416,675418,675432],{"className":675417},[31801],[34,675419,675421],{"className":675420,"style":318658},[31805],[34,675422,675423,675426],{"style":318661},[34,675424],{"className":675425,"style":43406},[31813],[34,675427,675429],{"className":675428},[31818,43410,43411,31821],[34,675430,786],{"className":675431},[31685,31821],[34,675433,31829],{"className":675434},[31828],[34,675436,675438],{"className":675437},[31801],[34,675439,675441],{"className":675440,"style":220540},[31805],[34,675442],{},[34,675444,31829],{"className":675445},[31828],[34,675447,675449],{"className":675448},[31801],[34,675450,675452],{"className":675451,"style":667231},[31805],[34,675453],{},[34,675455,456],{"className":675456},[31782],[34,675458,459],{"className":675459},[31685],[34,675461],{"className":675462,"style":32604},[31845],[34,675464,32226],{"className":675465},[32024],[34,675467],{"className":675468,"style":32604},[31845],[34,675470,675472,675475,675504,675507,675510,675513,675516,675519,675522],{"className":675471},[31676],[34,675473],{"className":675474,"style":37755},[31680],[34,675476,675478,675481],{"className":675477},[31685],[34,675479,382333],{"className":675480,"style":382406},[31685,31686],[34,675482,675484],{"className":675483},[31792],[34,675485,675487],{"className":675486},[31796],[34,675488,675490],{"className":675489},[31801],[34,675491,675493],{"className":675492,"style":37774},[31805],[34,675494,675495,675498],{"style":37777},[34,675496],{"className":675497,"style":31814},[31813],[34,675499,675501],{"className":675500},[31818,31819,31820,31821],[34,675502,768],{"className":675503},[31685,31821],[34,675505,2557],{"className":675506},[31690],[34,675508],{"className":675509,"style":32123},[31845],[34,675511,46900],{"className":675512},[31685,207923],[34,675514,32936],{"className":675515},[31685,31686],[34,675517],{"className":675518,"style":32604},[31845],[34,675520,223],{"className":675521},[32024],[34,675523],{"className":675524,"style":32604},[31845],[34,675526,675528,675531,675534,675537,675540],{"className":675527},[31676],[34,675529],{"className":675530,"style":35942},[31680],[34,675532,459],{"className":675533},[31685],[34,675535],{"className":675536,"style":32604},[31845],[34,675538,223],{"className":675539},[32024],[34,675541],{"className":675542,"style":32604},[31845],[34,675544,675546,675549,675552,675581,675584,675587],{"className":675545},[31676],[34,675547],{"className":675548,"style":37755},[31680],[34,675550,456],{"className":675551},[31782],[34,675553,675555,675558],{"className":675554},[31685],[34,675556,32936],{"className":675557},[31685,31686],[34,675559,675561],{"className":675560},[31792],[34,675562,675564],{"className":675563},[31796],[34,675565,675567],{"className":675566},[31801],[34,675568,675570],{"className":675569,"style":37774},[31805],[34,675571,675572,675575],{"style":37777},[34,675573],{"className":675574,"sty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",[34,675810,675812,675848],{"className":675811},[31642],[34,675813,675815],{"className":675814},[31646],[31648,675816,675817],{"xmlns":31650},[31652,675818,675819,675845],{},[31655,675820,675821,675823,675829,675831,675833,675835,675841,675843],{},[31662,675822,32226],{},[37624,675824,675825,675827],{},[31658,675826,382333],{},[31749,675828,768],{},[31658,675830,46900],{"mathvariant":38079},[31658,675832,32936],{},[31662,675834,223],{},[37624,675836,675837,675839],{},[31658,675838,32936],{},[31749,675840,768],{},[31658,675842,46900],{"mathvariant":38079},[31658,675844,382333],{},[31665,675846,675847],{"encoding":31667},"-z^2 \\mathrm{d}x + x^2 \\mathrm{d}z",[34,675849,675851,675904],{"className":675850,"ariaHidden":758},[31672],[34,675852,675854,675857,675860,675889,675892,675895,675898,675901],{"className":675853},[31676],[34,675855],{"className":675856,"style":40044},[31680],[34,675858,32226],{"className":675859},[31685],[34,675861,675863,675866],{"className":675862},[31685],[34,675864,382333],{"className":675865,"style":382406},[31685,31686],[34,675867,675869],{"className":675868},[31792],[34,675870,675872],{"className":675871},[31796],[34,675873,675875],{"className":675874},[31801],[34,675876,675878],{"className":675877,"style":37774},[31805],[34,675879,675880,675883],{"style":37777},[34,675881],{"className":675882,"style":31814},[31813],[34,675884,675886],{"className":675885},[31818,31819,31820,31821],[34,675887,768],{"className":675888},[31685,31821],[34,675890,46900],{"className":675891},[31685,207923],[34,675893,32936],{"className":675894},[31685,31686],[34,675896],{"className":675897,"style":32604},[31845],[34,675899,223],{"className":675900},[32024],[34,675902],{"className":675903,"style":32604},[31845],[34,675905,675907,675910,675939,675942],{"className":675906},[31676],[34,675908],{"className":675909,"style":37774},[31680],[34,675911,675913,675916],{"className":675912},[31685],[34,675914,32936],{"className":675915},[31685,31686],[34,675917,675919],{"className":675918},[31792],[34,675920,675922],{"className":675921},[31796],[34,675923,675925],{"className":675924},[31801],[34,675926,675928],{"className":675927,"style":37774},[31805],[34,675929,675930,675933],{"style":37777},[34,675931],{"className":675932,"style":31814},[31813],[34,675934,675936],{"className":675935},[31818,31819,31820,31821],[34,675937,768],{"className":675938},[31685,31821],[34,675940,46900],{"className":675941},[31685,207923],[34,675943,382333],{"className":675944,"style":382406},[31685,31686],"。\n对比 ",[34,675947,675949,675966],{"className":675948},[31642],[34,675950,675952],{"className":675951},[31646],[31648,675953,675954],{"xmlns":31650},[31652,675955,675956,675964],{},[31655,675957,675958],{},[31743,675959,675960,675962],{},[31658,675961,33206],{},[31749,675963,640],{},[31665,675965,658379],{"encoding":31667},[34,675967,675969],{"className":675968,"ariaHidden":758},[31672],[34,675970,675972,675975],{"className":675971},[31676],[34,675973],{"className":675974,"style":70915},[31680],[34,675976,675978,675981],{"className":675977},[31685],[34,675979,33206],{"className":675980},[31685,31686],[34,675982,675984],{"className":675983},[31792],[34,675985,675987,676007],{"className":675986},[31796,31797],[34,675988,675990,676004],{"className":675989},[31801],[34,675991,675993],{"className":675992,"style":31806},[31805],[34,675994,675995,675998],{"style":31809},[34,675996],{"className":675997,"style":31814},[31813],[34,675999,676001],{"className":676000},[31818,31819,31820,31821],[34,676002,640],{"className":676003},[31685,31821],[34,676005,31829],{"className":676006},[31828],[34,676008,676010],{"className":676009},[31801],[34,676011,676013],{"className":676012,"style":31836},[31805],[34,676014],{},[34,676016,676018,676051],{"className":676017},[31642],[34,676019,676021],{"className":676020},[31646],[31648,676022,676023],{"xmlns":31650},[31652,676024,676025,676049],{},[31655,676026,676027,676033,676035,676037,676039,676045,676047],{},[37624,676028,676029,676031],{},[31658,676030,67764],{},[31749,676032,768],{},[31658,676034,46900],{"mathvariant":38079},[31658,676036,32936],{},[31662,676038,32226],{},[37624,676040,676041,676043],{},[31658,676042,32936],{},[31749,676044,768],{},[31658,676046,46900],{"mathvariant":38079},[31658,676048,67764],{},[31665,676050,674425],{"encoding":31667},[34,676052,676054,676104],{"className":676053,"ariaHidden":758},[31672],[34,676055,676057,676060,676089,676092,676095,676098,676101],{"className":676056},[31676],[34,676058],{"className":676059,"style":315747},[31680],[34,676061,676063,676066],{"className":676062},[31685],[34,676064,67764],{"className":676065,"style":32062},[31685,31686],[34,676067,676069],{"className":676068},[31792],[34,676070,676072],{"className":676071},[31796],[34,676073,676075],{"className":676074},[31801],[34,676076,676078],{"className":676077,"style":37774},[31805],[34,676079,676080,676083],{"style":37777},[34,676081],{"className":676082,"style":31814},[31813],[34,676084,676086],{"className":676085},[31818,31819,31820,31821],[34,676087,768],{"className":676088},[31685,31821],[34,676090,46900],{"className":676091},[31685,207923],[34,676093,32936],{"className":676094},[31685,31686],[34,676096],{"className":676097,"style":32604},[31845],[34,676099,32226],{"className":676100},[32024],[34,676102],{"className":676103,"style":32604},[31845],[34,676105,676107,676110,676139,676142],{"className":676106},[31676],[34,676108],{"className":676109,"style":315747},[31680],[34,676111,676113,676116],{"className":676112},[31685],[34,676114,32936],{"className":676115},[31685,31686],[34,676117,676119],{"className":676118},[31792],[34,676120,676122],{"className":676121},[31796],[34,676123,676125],{"className":676124},[31801],[34,676126,676128],{"className":676127,"style":37774},[31805],[34,676129,676130,676133],{"style":37777},[34,676131],{"className":676132,"style":31814},[31813],[34,676134,676136],{"className":676135},[31818,31819,31820,31821],[34,676137,768],{"className":676138},[31685,31821],[34,676140,46900],{"className":676141},[31685,207923],[34,676143,67764],{"className":676144,"style":32062},[31685,31686],"。\n这稍微有点不同（符号反了或者项的顺序变了）。让我们仔细参数化算一下以防万一。",[108,676147,676148,33183,676150,676256,676257,53834,676326,676395,676396,8471,676424,1840,676474,676560,676561,8471,676589,1840,676647,511020,676733,676822],{},[2627,676149,671162],{},[34,676151,676153,676187],{"className":676152},[31642],[34,676154,676156],{"className":676155},[31646],[31648,676157,676158],{"xmlns":31650},[31652,676159,676160,676184],{},[31655,676161,676162,676164,676166,676168,676170,676172,676174,676176,676178,676180,676182],{},[31658,676163,382333],{},[31662,676165,202],{},[31658,676167,547860],{},[31662,676169,32682],{},[31658,676171,33249],{},[31662,676173,31960],{"separator":758},[31658,676175,32936],{},[31662,676177,202],{},[31658,676179,547853],{},[31662,676181,32682],{},[31658,676183,33249],{},[31665,676185,676186],{"encoding":31667},"z = \\cos t, x = \\sin t",[34,676188,676190,676208,676241],{"className":676189,"ariaHidden":758},[31672],[34,676191,676193,676196,676199,676202,676205],{"className":676192},[31676],[34,676194],{"className":676195,"style":32527},[31680],[34,676197,382333],{"className":676198,"style":382406},[31685,31686],[34,676200],{"className":676201,"style":31846},[31845],[34,676203,202],{"className":676204},[31850],[34,676206],{"className":676207,"style":31846},[31845],[34,676209,676211,676214,676217,676220,676223,676226,676229,676232,676235,676238],{"className":676210},[31676],[34,676212],{"className":676213,"style":223552},[31680],[34,676215,547860],{"className":676216},[32767],[34,676218],{"className":676219,"style":32123},[31845],[34,676221,33249],{"className":676222},[31685,31686],[34,676224,31960],{"className":676225},[32119],[34,676227],{"className":676228,"style":32123},[31845],[34,676230,32936],{"className":676231},[31685,31686],[34,676233],{"className":676234,"style":31846},[31845],[34,676236,202],{"className":676237},[31850],[34,676239],{"className":676240,"style":31846},[31845],[34,676242,676244,676247,676250,676253],{"className":676243},[31676],[34,676245],{"className":676246,"style":71430},[31680],[34,676248,547853],{"className":676249},[32767],[34,676251],{"className":676252,"style":32123},[31845],[34,676254,33249],{"className":676255},[31685,31686]," （注意：是从 ",[34,676258,676260,676287],{"className":676259},[31642],[34,676261,676263],{"className":676262},[31646],[31648,676264,676265],{"xmlns":31650},[31652,676266,676267,676285],{},[31655,676268,676269,676271,676273,676275,676277,676279,676281,676283],{},[31658,676270,47384],{},[31662,676272,456],{"stretchy":702},[31749,676274,459],{},[31662,676276,31960],{"separator":758},[31749,676278,459],{},[31662,676280,31960],{"separator":758},[31749,676282,640],{},[31662,676284,2557],{"stretchy":702},[31665,676286,669580],{"encoding":31667},[34,676288,676290],{"className":676289,"ariaHidden":758},[31672],[34,676291,676293,676296,676299,676302,676305,676308,676311,676314,676317,676320,676323],{"className":676292},[31676],[34,676294],{"className":676295,"style":31774},[31680],[34,676297,47384],{"className":676298,"style":38706},[31685,31686],[34,676300,456],{"className":676301},[31782],[34,676303,459],{"className":676304},[31685],[34,676306,31960],{"className":676307},[32119],[34,676309],{"className":676310,"style":32123},[31845],[34,676312,459],{"className":676313},[31685],[34,676315,31960],{"className":676316},[32119],[34,676318],{"className":676319,"style":32123},[31845],[34,676321,640],{"className":676322},[31685],[34,676324,2557],{"className":676325},[31690],[34,676327,676329,676356],{"className":676328},[31642],[34,676330,676332],{"className":676331},[31646],[31648,676333,676334],{"xmlns":31650},[31652,676335,676336,676354],{},[31655,676337,676338,676340,676342,676344,676346,676348,676350,676352],{},[31658,676339,54151],{},[31662,676341,456],{"stretchy":702},[31749,676343,640],{},[31662,676345,31960],{"separator":758},[31749,676347,459],{},[31662,676349,31960],{"separator":758},[31749,676351,459],{},[31662,676353,2557],{"stretchy":702},[31665,676355,669264],{"encoding":31667},[34,676357,676359],{"className":676358,"ariaHidden":758},[31672],[34,676360,676362,676365,676368,676371,676374,676377,676380,676383,676386,676389,676392],{"className":676361},[31676],[34,676363],{"className":676364,"style":31774},[31680],[34,676366,54151],{"className":676367},[31685,31686],[34,676369,456],{"className":676370},[31782],[34,676372,640],{"className":676373},[31685],[34,676375,31960],{"className":676376},[32119],[34,676378],{"className":676379,"style":32123},[31845],[34,676381,459],{"className":676382},[31685],[34,676384,31960],{"className":676385},[32119],[34,676387],{"className":676388,"style":32123},[31845],[34,676390,459],{"className":676391},[31685],[34,676393,2557],{"className":676394},[31690],"）。\n起点 ",[34,676397,676399,676412],{"className":676398},[31642],[34,676400,676402],{"className":676401},[31646],[31648,676403,676404],{"xmlns":31650},[31652,676405,676406,676410],{},[31655,676407,676408],{},[31658,676409,47384],{},[31665,676411,47384],{"encoding":31667},[34,676413,676415],{"className":676414,"ariaHidden":758},[31672],[34,676416,676418,676421],{"className":676417},[31676],[34,676419],{"className":676420,"style":33218},[31680],[34,676422,47384],{"className":676423,"style":38706},[31685,31686],[34,676425,676427,676444],{"className":676426},[31642],[34,676428,676430],{"className":676429},[31646],[31648,676431,676432],{"xmlns":31650},[31652,676433,676434,676442],{},[31655,676435,676436,676438,676440],{},[31658,676437,33249],{},[31662,676439,202],{},[31749,676441,459],{},[31665,676443,570178],{"encoding":31667},[34,676445,676447,676465],{"className":676446,"ariaHidden":758},[31672],[34,676448,676450,676453,676456,676459,676462],{"className":676449},[31676],[34,676451],{"className":676452,"style":33280},[31680],[34,676454,33249],{"className":676455},[31685,31686],[34,676457],{"className":676458,"style":31846},[31845],[34,676460,202],{"className":676461},[31850],[34,676463],{"className":676464,"style":31846},[31845],[34,676466,676468,676471],{"className":676467},[31676],[34,676469],{"className":676470,"style":32617},[31680],[34,676472,459],{"className":676473},[31685],[34,676475,676477,676503],{"className":676476},[31642],[34,676478,676480],{"className":676479},[31646],[31648,676481,676482],{"xmlns":31650},[31652,676483,676484,676500],{},[31655,676485,676486,676488,676490,676492,676494,676496,676498],{},[31658,676487,382333],{},[31662,676489,202],{},[31749,676491,640],{},[31662,676493,31960],{"separator":758},[31658,676495,32936],{},[31662,676497,202],{},[31749,676499,459],{},[31665,676501,676502],{"encoding":31667},"z=1, x=0",[34,676504,676506,676524,676551],{"className":676505,"ariaHidden":758},[31672],[34,676507,676509,676512,676515,676518,676521],{"className":676508},[31676],[34,676510],{"className":676511,"style":32527},[31680],[34,676513,382333],{"className":676514,"style":382406},[31685,31686],[34,676516],{"className":676517,"style":31846},[31845],[34,676519,202],{"className":676520},[31850],[34,676522],{"className":676523,"style":31846},[31845],[34,676525,676527,676530,676533,676536,676539,676542,676545,676548],{"className":676526},[31676],[34,676528],{"className":676529,"style":40292},[31680],[34,676531,640],{"className":676532},[31685],[34,676534,31960],{"className":676535},[32119],[34,676537],{"className":676538,"style":32123},[31845],[34,676540,32936],{"className":676541},[31685,31686],[34,676543],{"className":676544,"style":31846},[31845],[34,676546,202],{"className":676547},[31850],[34,676549],{"className":676550,"style":31846},[31845],[34,676552,676554,676557],{"className":676553},[31676],[34,676555],{"className":676556,"style":32617},[31680],[34,676558,459],{"className":676559},[31685],")。\n终点 ",[34,676562,676564,676577],{"className":676563},[31642],[34,676565,676567],{"className":676566},[31646],[31648,676568,676569],{"xmlns":31650},[31652,676570,676571,676575],{},[31655,676572,676573],{},[31658,676574,54151],{},[31665,676576,54151],{"encoding":31667},[34,676578,676580],{"className":676579,"ariaHidden":758},[31672],[34,676581,676583,676586],{"className":676582},[31676],[34,676584],{"className":676585,"style":33218},[31680],[34,676587,54151],{"className":676588},[31685,31686],[34,676590,676592,676614],{"className":676591},[31642],[34,676593,676595],{"className":676594},[31646],[31648,676596,676597],{"xmlns":31650},[31652,676598,676599,676611],{},[31655,676600,676601,676603,676605,676607,676609],{},[31658,676602,33249],{},[31662,676604,202],{},[31658,676606,58442],{},[31658,676608,38080],{"mathvariant":38079},[31749,676610,768],{},[31665,676612,676613],{"encoding":31667},"t=\\pi\u002F2",[34,676615,676617,676635],{"className":676616,"ariaHidden":758},[31672],[34,676618,676620,676623,676626,676629,676632],{"className":676619},[31676],[34,676621],{"className":676622,"style":33280},[31680],[34,676624,33249],{"className":676625},[31685,31686],[34,676627],{"className":676628,"style":31846},[31845],[34,676630,202],{"className":676631},[31850],[34,676633],{"className":676634,"style":31846},[31845],[34,676636,676638,676641,676644],{"className":676637},[31676],[34,676639],{"className":676640,"style":31774},[31680],[34,676642,58442],{"className":676643,"style":32062},[31685,31686],[34,676645,38149],{"className":676646},[31685],[34,676648,676650,676676],{"className":676649},[31642],[34,676651,676653],{"className":676652},[31646],[31648,676654,676655],{"xmlns":31650},[31652,676656,676657,676673],{},[31655,676658,676659,676661,676663,676665,676667,676669,676671],{},[31658,676660,382333],{},[31662,676662,202],{},[31749,676664,459],{},[31662,676666,31960],{"separator":758},[31658,676668,32936],{},[31662,676670,202],{},[31749,676672,640],{},[31665,676674,676675],{"encoding":31667},"z=0, x=1",[34,676677,676679,676697,676724],{"className":676678,"ariaHidden":758},[31672],[34,676680,676682,676685,676688,676691,676694],{"className":676681},[31676],[34,676683],{"className":676684,"style":32527},[31680],[34,676686,382333],{"className":676687,"style":382406},[31685,31686],[34,676689],{"className":676690,"style":31846},[31845],[34,676692,202],{"className":676693},[31850],[34,676695],{"className":676696,"style":31846},[31845],[34,676698,676700,676703,676706,676709,676712,676715,676718,676721],{"className":676699},[31676],[34,676701],{"className":676702,"style":40292},[31680],[34,676704,459],{"className":676705},[31685],[34,676707,31960],{"className":676708},[32119],[34,676710],{"className":676711,"style":32123},[31845],[34,676713,32936],{"className":676714},[31685,31686],[34,676716],{"className":676717,"style":31846},[31845],[34,676719,202],{"className":676720},[31850],[34,676722],{"className":676723,"style":31846},[31845],[34,676725,676727,676730],{"className":676726},[31676],[34,676728],{"className":676729,"style":32617},[31680],[34,676731,640],{"className":676732},[31685],[34,676734,676736,676768],{"className":676735},[31642],[34,676737,676739],{"className":676738},[31646],[31648,676740,676741],{"xmlns":31650},[31652,676742,676743,676765],{},[31655,676744,676745,676747,676749,676751,676753,676755,676757,676759,676761,676763],{},[31658,676746,46900],{"mathvariant":38079},[31658,676748,382333],{},[31662,676750,202],{},[31662,676752,32226],{},[31658,676754,547853],{},[31662,676756,32682],{},[31658,676758,33249],{},[39361,676760,39363],{},[31658,676762,46900],{"mathvariant":38079},[31658,676764,33249],{},[31665,676766,676767],{"encoding":31667},"\\mathrm{d}z = -\\sin t \\, \\mathrm{d}t",[34,676769,676771,676792],{"className":676770,"ariaHidden":758},[31672],[34,676772,676774,676777,676780,676783,676786,676789],{"className":676773},[31676],[34,676775],{"className":676776,"style":31681},[31680],[34,676778,46900],{"className":676779},[31685,207923],[34,676781,382333],{"className":676782,"style":382406},[31685,31686],[34,676784],{"className":676785,"style":31846},[31845],[34,676787,202],{"className":676788},[31850],[34,676790],{"className":676791,"style":31846},[31845],[34,676793,676795,676798,676801,676804,676807,676810,676813,676816,676819],{"className":676794},[31676],[34,676796],{"className":676797,"style":34299},[31680],[34,676799,32226],{"className":676800},[31685],[34,676802],{"className":676803,"style":32123},[31845],[34,676805,547853],{"className":676806},[32767],[34,676808],{"className":676809,"style":32123},[31845],[34,676811,33249],{"className":676812},[31685,31686],[34,676814],{"className":676815,"style":32123},[31845],[34,676817,46900],{"className":676818},[31685,207923],[34,676820,33249],{"className":676821},[31685,31686],[34,676823,676825,676855],{"className":676824},[31642],[34,676826,676828],{"className":676827},[31646],[31648,676829,676830],{"xmlns":31650},[31652,676831,676832,676852],{},[31655,676833,676834,676836,676838,676840,676842,676844,676846,676848,676850],{},[31658,676835,46900],{"mathvariant":38079},[31658,676837,32936],{},[31662,676839,202],{},[31658,676841,547860],{},[31662,676843,32682],{},[31658,676845,33249],{},[39361,676847,39363],{},[31658,676849,46900],{"mathvariant":38079},[31658,676851,33249],{},[31665,676853,676854],{"encoding":31667},"\\mathrm{d}x = \\cos t \\, \\mathrm{d}t",[34,676856,676858,676879],{"className":676857,"ariaHidden":758},[31672],[34,676859,676861,676864,676867,676870,676873,676876],{"className":676860},[31676],[34,676862],{"className":676863,"style":31681},[31680],[34,676865,46900],{"className":676866},[31685,207923],[34,676868,32936],{"className":676869},[31685,31686],[34,676871],{"className":676872,"style":31846},[31845],[34,676874,202],{"className":676875},[31850],[34,676877],{"className":676878,"style":31846},[31845],[34,676880,676882,676885,676888,676891,676894,676897,676900],{"className":676881},[31676],[34,676883],{"className":676884,"style":31681},[31680],[34,676886,547860],{"className":676887},[32767],[34,676889],{"className":676890,"style":32123},[31845],[34,676892,33249],{"className":676893},[31685,31686],[34,676895],{"className":676896,"style":32123},[31845],[34,676898,46900],{"className":676899},[31685,207923],[34,676901,33249],{"className":676902},[31685,31686],[108,676904,676905,33183,676908,677911],{},[2627,676906,676907],{},"代入",[34,676909,676911],{"className":676910},[126385],[34,676912,676914,677148],{"className":676913},[31642],[34,676915,676917],{"className":676916},[31646],[31648,676918,676919],{"xmlns":31650,"display":126394},[31652,676920,676921,677145],{},[71078,676922,676923,677027,677097],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,676924,676925,676935],{},[71087,676926,676927],{},[71090,676928,676929],{"scriptlevel":459,"displaystyle":758},[31743,676930,676931,676933],{},[31658,676932,106063],{},[31749,676934,786],{},[71087,676936,676937],{},[71090,676938,676939],{"scriptlevel":459,"displaystyle":758},[31655,676940,676941,676943,676945,676959,676961,676963,676965,676975,676977,676979,676981,676983,676985,676987,676989,676991,676993,677003,677005,677007,677009,677011,677013,677015,677017,677019,677021,677023,677025],{},[31655,676942],{},[31662,676944,202],{},[38846,676946,676947,676949,676951],{},[31662,676948,546295],{},[31749,676950,459],{},[31655,676952,676953,676955,676957],{},[31658,676954,58442],{},[31658,676956,38080],{"mathvariant":38079},[31749,676958,768],{},[31662,676960,15972],{"stretchy":702},[31662,676962,32226],{},[31662,676964,456],{"stretchy":702},[37624,676966,676967,676973],{},[31655,676968,676969,676971],{},[31658,676970,547860],{},[31662,676972,32682],{},[31749,676974,768],{},[31658,676976,33249],{},[31662,676978,2557],{"stretchy":702},[31662,676980,456],{"stretchy":702},[31658,676982,547860],{},[31662,676984,32682],{},[31658,676986,33249],{},[31662,676988,2557],{"stretchy":702},[31662,676990,223],{},[31662,676992,456],{"stretchy":702},[37624,676994,676995,677001],{},[31655,676996,676997,676999],{},[31658,676998,547853],{},[31662,677000,32682],{},[31749,677002,768],{},[31658,677004,33249],{},[31662,677006,2557],{"stretchy":702},[31662,677008,456],{"stretchy":702},[31662,677010,32226],{},[31658,677012,547853],{},[31662,677014,32682],{},[31658,677016,33249],{},[31662,677018,2557],{"stretchy":702},[31662,677020,32712],{"stretchy":702},[39361,677022,39363],{},[31658,677024,46900],{"mathvariant":38079},[31658,677026,33249],{},[71084,677028,677029,677035],{},[71087,677030,677031],{},[71090,677032,677033],{"scriptlevel":459,"displaystyle":758},[31655,677034],{},[71087,677036,677037],{},[71090,677038,677039],{"scriptlevel":459,"displaystyle":758},[31655,677040,677041,677043,677045,677059,677061,677063,677073,677075,677077,677087,677089,677091,677093,677095],{},[31655,677042],{},[31662,677044,202],{},[38846,677046,677047,677049,677051],{},[31662,677048,546295],{},[31749,677050,459],{},[31655,677052,677053,677055,677057],{},[31658,677054,58442],{},[31658,677056,38080],{"mathvariant":38079},[31749,677058,768],{},[31662,677060,15972],{"stretchy":702},[31662,677062,32226],{},[37624,677064,677065,677071],{},[31655,677066,677067,677069],{},[31658,677068,547860],{},[31662,677070,32682],{},[31749,677072,786],{},[31658,677074,33249],{},[31662,677076,32226],{},[37624,677078,677079,677085],{},[31655,677080,677081,677083],{},[31658,677082,547853],{},[31662,677084,32682],{},[31749,677086,786],{},[31658,677088,33249],{},[31662,677090,32712],{"stretchy":702},[39361,677092,39363],{},[31658,677094,46900],{"mathvariant":38079},[31658,677096,33249],{},[71084,677098,677099,677105],{},[71087,677100,677101],{},[71090,677102,677103],{"scriptlevel":459,"displaystyle":758},[31655,677104],{},[71087,677106,677107],{},[71090,677108,677109],{"scriptlevel":459,"displaystyle":758},[31655,677110,677111,677113,677115,677117,677119,677125,677127,677133,677135,677137,677139],{},[31655,677112],{},[31662,677114,202],{},[31662,677116,32226],{},[31662,677118,456],{"stretchy":702},[41453,677120,677121,677123],{},[31749,677122,768],{},[31749,677124,786],{},[31662,677126,223],{},[41453,677128,677129,677131],{},[31749,677130,768],{},[31749,677132,786],{},[31662,677134,2557],{"stretchy":702},[31662,677136,202],{},[31662,677138,32226],{},[41453,677140,677141,677143],{},[31749,677142,15932],{},[31749,677144,786],{},[31665,677146,677147],{"encoding":31667},"\\begin{aligned}\nI_3 &= \\int_0^{\\pi\u002F2} [ -(\\cos^2 t)(\\cos t) + (\\sin^2 t)(-\\sin t) ] \\, \\mathrm{d}t \\\\\n&= \\int_0^{\\pi\u002F2} [ -\\cos^3 t - \\sin^3 t ] \\, \\mathrm{d}t \\\\\n&= - (\\frac{2}{3} + \\frac{2}{3}) = -\\frac{4}{3}\n\\end{aligned}",[34,677149,677151],{"className":677150,"ariaHidden":758},[31672],[34,677152,677154,677158],{"className":677153},[31676],[34,677155],{"className":677156,"style":677157},[31680],"height:7.7071em;vertical-align:-3.6036em;",[34,677159,677161],{"className":677160},[31685],[34,677162,677164,677256],{"className":677163},[71078],[34,677165,677167],{"className":677166},[369003],[34,677168,677170,677247],{"className":677169},[31796,31797],[34,677171,677173,677244],{"className":677172},[31801],[34,677174,677177,677226,677235],{"className":677175,"style":677176},[31805],"height:4.1036em;",[34,677178,677180,677183],{"style":677179},"top:-6.1036em;",[34,677181],{"className":677182,"style":671825},[31813],[34,677184,677186],{"className":677185},[31685],[34,677187,677189,677192],{"className":677188},[31685],[34,677190,106063],{"className":677191,"style":67482},[31685,31686],[34,677193,677195],{"className":677194},[31792],[34,677196,677198,677218],{"className":677197},[31796,31797],[34,677199,677201,677215],{"className":677200},[31801],[34,677202,677204],{"className":677203,"style":31806},[31805],[34,677205,677206,677209],{"style":217742},[34,677207],{"className":677208,"style":31814},[31813],[34,677210,677212],{"className":677211},[31818,31819,31820,31821],[34,677213,786],{"className":677214},[31685,31821],[34,677216,31829],{"className":677217},[31828],[34,677219,677221],{"className":677220},[31801],[34,677222,677224],{"className":677223,"style":31836},[31805],[34,677225],{},[34,677227,677229,677232],{"style":677228},"top:-3.2537em;",[34,677230],{"className":677231,"style":671825},[31813],[34,677233],{"className":677234},[31685],[34,677236,677238,677241],{"style":677237},"top:-0.7203em;",[34,677239],{"className":677240,"style":671825},[31813],[34,677242],{"className":677243},[31685],[34,677245,31829],{"className":677246},[31828],[34,677248,677250],{"className":677249},[31801],[34,677251,677254],{"className":677252,"style":677253},[31805],"height:3.6036em;",[34,677255],{},[34,677257,677259],{"className":677258},[71384],[34,677260,677262,677903],{"className":677261},[31796,31797],[34,677263,677265,677900],{"className":677264},[31801],[34,677266,677268,677487,677664],{"className":677267,"style":677176},[31805],[34,677269,677270,677273],{"style":677179},[34,677271],{"className":677272,"style":671825},[31813],[34,677274,677276,677279,677282,677285,677288,677345,677348,677351,677354,677383,677386,677389,677392,677395,677398,677401,677404,677407,677410,677413,677416,677419,677448,677451,677454,677457,677460,677463,677466,677469,677472,677475,677478,677481,677484],{"className":677275},[31685],[34,677277],{"className":677278},[31685],[34,677280],{"className":677281,"style":31846},[31845],[34,677283,202],{"className":677284},[31850],[34,677286],{"className":677287,"style":31846},[31845],[34,677289,677291,677294],{"className":677290},[32767],[34,677292,546295],{"className":677293,"style":591475},[32767,131249,131250],[34,677295,677297],{"className":677296},[31792],[34,677298,677300,677337],{"className":677299},[31796,31797],[34,677301,677303,677334],{"className":677302},[31801],[34,677304,677306,677317],{"className":677305,"style":671939},[31805],[34,677307,677308,677311],{"style":591491},[34,677309],{"className":677310,"style":31814},[31813],[34,677312,677314],{"className":677313},[31818,31819,31820,31821],[34,677315,459],{"className":677316},[31685,31821],[34,677318,677319,677322],{"style":591503},[34,677320],{"className":677321,"style":31814},[31813],[34,677323,677325],{"className":677324},[31818,31819,31820,31821],[34,677326,677328,677331],{"className":677327},[31685,31821],[34,677329,58442],{"className":677330,"style":32062},[31685,31686,31821],[34,677332,38149],{"className":677333},[31685,31821],[34,677335,31829],{"className":677336},[31828],[34,677338,677340],{"className":677339},[31801],[34,677341,677343],{"className":677342,"style":591522},[31805],[34,677344],{},[34,677346,15972],{"className":677347},[31782],[34,677349,32226],{"className":677350},[31685],[34,677352,456],{"className":677353},[31782],[34,677355,677357,677360],{"className":677356},[32767],[34,677358,547860],{"className":677359},[32767],[34,677361,677363],{"className":677362},[31792],[34,677364,677366],{"className":677365},[31796],[34,677367,677369],{"className":677368},[31801],[34,677370,677372],{"className":677371,"style":129832},[31805],[34,677373,677374,677377],{"style":129846},[34,677375],{"className":677376,"style":31814},[31813],[34,677378,677380],{"className":677379},[31818,31819,31820,31821],[34,677381,768],{"className":677382},[31685,31821],[34,677384],{"className":677385,"style":32123},[31845],[34,677387,33249],{"className":677388},[31685,31686],[34,677390,2557],{"className":677391},[31690],[34,677393,456],{"className":677394},[31782],[34,677396,547860],{"className":677397},[32767],[34,677399],{"className":677400,"style":32123},[31845],[34,677402,33249],{"className":677403},[31685,31686],[34,677405,2557],{"className":677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最终求和",[34,677918,677920,678006],{"className":677919},[31642],[34,677921,677923],{"className":677922},[31646],[31648,677924,677925],{"xmlns":31650},[31652,677926,677927,678003],{},[31655,677928,677929,677931,677933,677939,677941,677947,677949,677955,677957,677959,677961,677967,677969,677971,677973,677975,677981,677983,677985,677987,677989,677995,677997,677999,678001],{},[31658,677930,106063],{},[31662,677932,202],{},[31743,677934,677935,677937],{},[31658,677936,106063],{},[31749,677938,640],{},[31662,677940,223],{},[31743,677942,677943,677945],{},[31658,677944,106063],{},[31749,677946,768],{},[31662,677948,223],{},[31743,677950,677951,677953],{},[31658,677952,106063],{},[31749,677954,786],{},[31662,677956,202],{},[31662,677958,456],{"stretchy":702},[31662,677960,32226],{},[41453,677962,677963,677965],{},[31749,677964,15932],{},[31749,677966,786],{},[31662,677968,2557],{"stretchy":702},[31662,677970,223],{},[31662,677972,456],{"stretchy":702},[31662,677974,32226],{},[41453,677976,677977,677979],{},[31749,677978,15932],{},[31749,677980,786],{},[31662,677982,2557],{"stretchy":702},[31662,677984,223],{},[31662,677986,456],{"stretchy":702},[31662,677988,32226],{},[41453,677990,677991,677993],{},[31749,677992,15932],{},[31749,677994,786],{},[31662,677996,2557],{"stretchy":702},[31662,677998,202],{},[31662,678000,32226],{},[31749,678002,15932],{},[31665,678004,678005],{"encoding":31667},"I = I_1 + I_2 + I_3 = (-\\frac{4}{3}) + (-\\frac{4}{3}) + (-\\frac{4}{3}) = -4",[34,678007,678009,678027,678082,678137,678192,678284,678376,678468],{"className":678008,"ariaHidden":758},[31672],[34,678010,678012,678015,678018,678021,678024],{"className":678011},[31676],[34,678013],{"className":678014,"style":33218},[31680],[34,678016,106063],{"className":678017,"style":67482},[31685,31686],[34,678019],{"className":678020,"style":31846},[31845],[34,678022,202],{"className":678023},[31850],[34,678025],{"className":678026,"style":31846},[31845],[34,678028,678030,678033,678073,678076,678079],{"className":678029},[31676],[34,678031],{"className":678032,"style":70915},[31680],[34,678034,678036,678039],{"className":678035},[31685],[34,678037,106063],{"className":678038,"style":67482},[31685,31686],[34,678040,678042],{"className":678041},[31792],[34,678043,678045,678065],{"className":678044},[31796,31797],[34,678046,678048,678062],{"className":678047},[31801],[34,678049,678051],{"className":678050,"style":31806},[31805],[34,678052,678053,678056],{"style":217742},[34,678054],{"className":678055,"style":31814},[31813],[34,678057,678059],{"className":678058},[31818,31819,31820,31821],[34,678060,640],{"className":678061},[31685,31821],[34,678063,31829],{"className":678064},[31828],[34,678066,678068],{"className":678067},[31801],[34,678069,678071],{"className":678070,"style":31836},[31805],[34,678072],{},[34,678074],{"className":678075,"style":32604},[31845],[34,678077,223],{"className":678078},[32024],[34,678080],{"className":678081,"style":32604},[31845],[34,678083,678085,678088,678128,678131,678134],{"className":678084},[31676],[34,678086],{"className":678087,"style":70915},[31680],[34,678089,678091,678094],{"className":678090},[31685],[34,678092,106063],{"className":678093,"style":67482},[31685,31686],[34,678095,678097],{"className":678096},[31792],[34,678098,678100,678120],{"className":678099},[31796,31797],[34,678101,678103,678117],{"className":678102},[31801],[34,678104,678106],{"className":678105,"style":31806},[31805],[34,678107,678108,678111],{"style":217742},[34,678109],{"className":678110,"style":31814},[31813],[34,678112,678114],{"className":678113},[31818,31819,31820,31821],[34,678115,768],{"className":678116},[31685,31821],[34,678118,31829],{"className":678119},[31828],[34,678121,678123],{"className":678122},[31801],[34,678124,678126],{"className":678125,"style":31836},[31805],[34,678127],{},[34,678129],{"className":678130,"style":32604},[31845],[34,678132,223],{"className":678133},[32024],[34,678135],{"className":678136,"style":32604},[31845],[34,678138,678140,678143,678183,678186,678189],{"className":678139},[31676],[34,678141],{"className":678142,"style":70915},[31680],[34,678144,678146,678149],{"className":678145},[31685],[34,678147,106063],{"className":678148,"style":67482},[31685,31686],[34,678150,678152],{"className":678151},[31792],[34,678153,678155,678175],{"className":678154},[31796,31797],[34,678156,678158,678172],{"className":678157},[31801],[34,678159,678161],{"className":678160,"style":31806},[31805],[34,678162,678163,678166],{"style":217742},[34,678164],{"className":678165,"style":31814},[31813],[34,678167,678169],{"className":678168},[31818,31819,31820,31821],[34,678170,786],{"className":678171},[31685,31821],[34,678173,31829],{"className":678174},[31828],[34,678176,678178],{"className":678177},[31801],[34,678179,678181],{"className":678180,"style":31836},[31805],[34,678182],{},[34,678184],{"className":678185,"style":31846},[31845],[34,678187,202],{"className":678188},[31850],[34,678190],{"className":678191,"style":31846},[31845],[34,678193,678195,678198,678201,678204,678272,678275,678278,678281],{"className":678194},[31676],[34,678196],{"className":678197,"style":46727},[31680],[34,678199,456],{"className":678200},[31782],[34,678202,32226],{"className":678203},[31685],[34,678205,678207,678210,678269],{"className":678206},[31685],[34,678208],{"className":678209},[31782,41488],[34,678211,678213],{"className":678212},[41453],[34,678214,678216,678261],{"className":678215},[31796,31797],[34,678217,678219,678258],{"className":678218},[31801],[34,678220,678222,678236,678244],{"className":678221,"style":46746},[31805],[34,678223,678224,678227],{"style":41504},[34,678225],{"className":678226,"style":41508},[31813],[34,678228,678230],{"className":678229},[31818,31819,31820,31821],[34,678231,678233],{"className":678232},[31685,31821],[34,678234,786],{"className":678235},[31685,31821],[34,678237,678238,678241],{"style":41520},[34,678239],{"className":678240,"style":41508},[31813],[34,678242],{"className":678243,"style":41528},[41527],[34,678245,678246,678249],{"style":46771},[34,678247],{"className":678248,"style":41508},[31813],[34,678250,678252],{"className":678251},[31818,31819,31820,31821],[34,678253,678255],{"className":678254},[31685,31821],[34,678256,15932],{"className":678257},[31685,31821],[34,678259,31829],{"className":678260},[31828],[34,678262,678264],{"className":678263},[31801],[34,678265,678267],{"className":678266,"style":41619},[31805],[34,678268],{},[34,678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习题 1 讲解",[32490,678487,678488],{"id":214289},"题目 (1)",[15,678490,678491,678494,678827,678857,678858,679038],{},[2627,678492,678493],{},"求积分：",[34,678495,678497,678569],{"className":678496},[31642],[34,678498,678500],{"className":678499},[31646],[31648,678501,678502],{"xmlns":31650},[31652,678503,678504,678566],{},[31655,678505,678506,678508,678510,678516,678518,678524,678526,678532,678534,678536,678538,678540,678542,678544,678550,678552,678558,678560,678562,678564],{},[31658,678507,106063],{},[31662,678509,202],{},[31743,678511,678512,678514],{},[31662,678513,546295],{},[31658,678515,33206],{},[31662,678517,456],{"stretchy":702},[37624,678519,678520,678522],{},[31658,678521,32936],{},[31749,678523,768],{},[31662,678525,223],{},[37624,678527,678528,678530],{},[31658,678529,67764],{},[31749,678531,768],{},[31662,678533,2557],{"stretchy":702},[39361,678535,39363],{},[31658,678537,46900],{"mathvariant":38079},[31658,678539,32936],{},[31662,678541,223],{},[31662,678543,456],{"stretchy":702},[37624,678545,678546,678548],{},[31658,678547,32936],{},[31749,678549,768],{},[31662,678551,32226],{},[37624,678553,678554,678556],{},[31658,678555,67764],{},[31749,678557,768],{},[31662,678559,2557],{"stretchy":702},[39361,678561,39363],{},[31658,678563,46900],{"mathvariant":38079},[31658,678565,67764],{},[31665,678567,678568],{"encoding":31667},"I = \\int_L (x^2 + y^2) \\, \\mathrm{d}x + (x^2 - y^2) \\, \\mathrm{d}y",[34,678570,678572,678590,678677,678733,678780],{"className":678571,"ariaHidden":758},[31672],[34,678573,678575,678578,678581,678584,678587],{"className":678574},[31676],[34,678576],{"className":678577,"style":33218},[31680],[34,678579,106063],{"className":678580,"style":67482},[31685,31686],[34,678582],{"className":678583,"style":31846},[31845],[34,678585,202],{"className":678586},[31850],[34,678588],{"className":678589,"style":31846},[31845],[34,678591,678593,678596,678636,678639,678668,678671,678674],{"className":678592},[31676],[34,678594],{"className":678595,"style":658186},[31680],[34,678597,678599,678602],{"className":678598},[32767],[34,678600,546295],{"className":678601,"style":546331},[32767,131249,156560],[34,678603,678605],{"className":678604},[31792],[34,678606,678608,678628],{"className":678607},[31796,31797],[34,678609,678611,678625],{"className":678610},[31801],[34,678612,678614],{"className":678613,"style":637339},[31805],[34,678615,678616,678619],{"style":546347},[34,678617],{"className":678618,"style":31814},[31813],[34,678620,678622],{"className":678621},[31818,31819,31820,31821],[34,678623,33206],{"className":678624},[31685,31686,31821],[34,678626,31829],{"className":678627},[31828],[34,678629,678631],{"className":678630},[31801],[34,678632,678634],{"className":678633,"style":546384},[31805],[34,678635],{},[34,678637,456],{"className":678638},[31782],[34,678640,678642,678645],{"className":678641},[31685],[34,678643,32936],{"className":678644},[31685,31686],[34,678646,678648],{"className":678647},[31792],[34,678649,678651],{"className":678650},[31796],[34,678652,678654],{"className":678653},[31801],[34,678655,678657],{"className":678656,"style":37774},[31805],[34,678658,678659,678662],{"style":37777},[34,678660],{"className":678661,"style":31814},[31813],[34,678663,678665],{"className":678664},[31818,31819,31820,31821],[34,678666,768],{"className":678667},[31685,31821],[34,678669],{"className":678670,"style":32604},[31845],[34,678672,223],{"className":678673},[32024],[34,678675],{"className":678676,"style":32604},[31845],[34,678678,678680,678683,678712,678715,678718,678721,678724,678727,678730],{"className":678679},[31676],[34,678681],{"className":678682,"style":37755},[31680],[34,678684,678686,678689],{"className":678685},[31685],[34,678687,67764],{"className":678688,"style":32062},[31685,31686],[34,678690,678692],{"className":678691},[31792],[34,678693,678695],{"className":678694},[31796],[34,678696,678698],{"className":678697},[31801],[34,678699,678701],{"className":678700,"style":37774},[31805],[34,678702,678703,678706],{"style":37777},[34,678704],{"className":678705,"style":31814},[31813],[34,678707,678709],{"className":678708},[31818,31819,31820,31821],[34,678710,768],{"className":678711},[31685,31821],[34,678713,2557],{"className":678714},[31690],[34,678716],{"className":678717,"style":32123},[31845],[34,678719,46900],{"className":678720},[31685,207923],[34,678722,32936],{"className":678723},[31685,31686],[34,678725],{"className":678726,"style":32604},[31845],[34,678728,223],{"className":678729},[32024],[34,678731],{"className":678732,"style":32604},[31845],[34,678734,678736,678739,678742,678771,678774,678777],{"className":678735},[31676],[34,678737],{"className":678738,"style":37755},[31680],[34,678740,456],{"className":678741},[31782],[34,678743,678745,678748],{"className":678744},[31685],[34,678746,32936],{"className":678747},[31685,31686],[34,678749,678751],{"className":678750},[31792],[34,678752,678754],{"className":678753},[31796],[34,678755,678757],{"className":678756},[31801],[34,678758,678760],{"className":678759,"style":37774},[31805],[34,678761,678762,678765],{"style":37777},[34,678763],{"className":678764,"style":31814},[31813],[34,678766,678768],{"className":678767},[31818,31819,31820,31821],[34,678769,768],{"className":678770},[31685,31821],[34,678772],{"className":678773,"style":32604},[31845],[34,678775,32226],{"className":678776},[32024],[34,678778],{"className":678779,"style":32604},[31845],[34,678781,678783,678786,678815,678818,678821,678824],{"className":678782},[31676],[34,678784],{"className":678785,"style":37755},[31680],[34,678787,678789,678792],{"className":678788},[31685],[34,678790,67764],{"className":678791,"style":32062},[31685,31686],[34,678793,678795],{"className":678794},[31792],[34,678796,678798],{"className":678797},[31796],[34,678799,678801],{"className":678800},[31801],[34,678802,678804],{"className":678803,"style":37774},[31805],[34,678805,678806,678809],{"style":37777},[34,678807],{"className":678808,"style":31814},[31813],[34,678810,678812],{"className":678811},[31818,31819,31820,31821],[34,678813,768],{"className":678814},[31685,31821],[34,678816,2557],{"className":678817},[31690],[34,678819],{"className":678820,"style":32123},[31845],[34,678822,46900],{"className":678823},[31685,207923],[34,678825,67764],{"className":678826,"style":32062},[31685,31686],[2627,678828,658358,678829],{},[34,678830,678832,678845],{"className":678831},[31642],[34,678833,678835],{"className":678834},[31646],[31648,678836,678837],{"xmlns":31650},[31652,678838,678839,678843],{},[31655,678840,678841],{},[31658,678842,33206],{},[31665,678844,33206],{"encoding":31667},[34,678846,678848],{"className":678847,"ariaHidden":758},[31672],[34,678849,678851,678854],{"className":678850},[31676],[34,678852],{"className":678853,"style":33218},[31680],[34,678855,33206],{"className":678856},[31685,31686],"：顶点为 ",[34,678859,678861,678927],{"className":678860},[31642],[34,678862,678864],{"className":678863},[31646],[31648,678865,678866],{"xmlns":31650},[31652,678867,678868,678924],{},[31655,678869,678870,678872,678874,678876,678878,678880,678882,678884,678886,678888,678890,678892,678894,678896,678898,678900,678902,678904,678906,678908,678910,678912,678914,678916,678918,678920,678922],{},[31658,678871,54151],{},[31662,678873,456],{"stretchy":702},[31749,678875,640],{},[31662,678877,31960],{"separator":758},[31749,678879,459],{},[31662,678881,2557],{"stretchy":702},[31662,678883,31960],{"separator":758},[31658,678885,54224],{},[31662,678887,456],{"stretchy":702},[31749,678889,768],{},[31662,678891,31960],{"separator":758},[31749,678893,459],{},[31662,678895,2557],{"stretchy":702},[31662,678897,31960],{"separator":758},[31658,678899,47384],{},[31662,678901,456],{"stretchy":702},[31749,678903,768],{},[31662,678905,31960],{"separator":758},[31749,678907,640],{},[31662,678909,2557],{"stretchy":702},[31662,678911,31960],{"separator":758},[31658,678913,91994],{},[31662,678915,456],{"stretchy":702},[31749,678917,640],{},[31662,678919,31960],{"separator":758},[31749,678921,640],{},[31662,678923,2557],{"stretchy":702},[31665,678925,678926],{"encoding":31667},"A(1,0), B(2,0), C(2,1), D(1,1)",[34,678928,678930],{"className":678929,"ariaHidden":758},[31672],[34,678931,678933,678936,678939,678942,678945,678948,678951,678954,678957,678960,678963,678966,678969,678972,678975,678978,678981,678984,678987,678990,678993,678996,678999,679002,679005,679008,679011,679014,679017,679020,679023,679026,679029,679032,679035],{"className":678932},[31676],[34,678934],{"className":678935,"style":31774},[31680],[34,678937,54151],{"className":678938},[31685,31686],[34,678940,456],{"className":678941},[31782],[34,678943,640],{"className":678944},[31685],[34,678946,31960],{"className":678947},[32119],[34,678949],{"className":678950,"style":32123},[31845],[34,678952,459],{"className":678953},[31685],[34,678955,2557],{"className":678956},[31690],[34,678958,31960],{"className":678959},[32119],[34,678961],{"className":678962,"style":32123},[31845],[34,678964,54224],{"className":678965,"style":54252},[31685,31686],[34,678967,456],{"className":678968},[31782],[34,678970,768],{"className":678971},[31685],[34,678973,31960],{"className":678974},[32119],[34,678976],{"className":678977,"style":32123},[31845],[34,678979,459],{"className":678980},[31685],[34,678982,2557],{"className":678983},[31690],[34,678985,31960],{"className":678986},[32119],[34,678988],{"className":678989,"style":32123},[31845],[34,678991,47384],{"className":678992,"style":38706},[31685,31686],[34,678994,456],{"className":678995},[31782],[34,678997,768],{"className":678998},[31685],[34,679000,31960],{"className":679001},[32119],[34,679003],{"className":679004,"style":32123},[31845],[34,679006,640],{"className":679007},[31685],[34,679009,2557],{"className":679010},[31690],[34,679012,31960],{"className":679013},[32119],[34,679015],{"className":679016,"style":32123},[31845],[34,679018,91994],{"className":679019,"style":34452},[31685,31686],[34,679021,456],{"className":679022},[31782],[34,679024,640],{"className":679025},[31685],[34,679027,31960],{"className":679028},[32119],[34,679030],{"className":679031,"style":32123},[31845],[34,679033,640],{"className":679034},[31685],[34,679036,2557],{"className":679037},[31690]," 的正方形，方向为逆时针。",[32490,679040,658913],{"id":679041},"详细解答-3",[15,679043,679044],{},"正方形路径由四条直线段组成，我们必须分四段计算，然后相加。",[15,679046,679047],{},[71807,679048,679049],{},"(这里可以想象一个在第一象限的小正方形，底边在x轴上)",[15,679051,679052],{},[2627,679053,679054,679055,54852,679088,53834,679139],{},"第 1 段 ",[34,679056,679058,679073],{"className":679057},[31642],[34,679059,679061],{"className":679060},[31646],[31648,679062,679063],{"xmlns":31650},[31652,679064,679065,679071],{},[31655,679066,679067,679069],{},[31658,679068,54151],{},[31658,679070,54224],{},[31665,679072,93307],{"encoding":31667},[34,679074,679076],{"className":679075,"ariaHidden":758},[31672],[34,679077,679079,679082,679085],{"className":679078},[31676],[34,679080],{"className":679081,"style":33218},[31680],[34,679083,54151],{"className":679084},[31685,31686],[34,679086,54224],{"className":679087,"style":54252},[31685,31686],[34,679089,679091,679112],{"className":679090},[31642],[34,679092,679094],{"className":679093},[31646],[31648,679095,679096],{"xmlns":31650},[31652,679097,679098,679110],{},[31655,679099,679100,679102,679104,679106,679108],{},[31662,679101,456],{"stretchy":702},[31749,679103,640],{},[31662,679105,31960],{"separator":758},[31749,679107,459],{},[31662,679109,2557],{"stretchy":702},[31665,679111,381896],{"encoding":31667},[34,679113,679115],{"className":679114,"ariaHidden":758},[31672],[34,679116,679118,679121,679124,679127,679130,679133,679136],{"className":679117},[31676],[34,679119],{"className":679120,"style":31774},[31680],[34,679122,456],{"className":679123},[31782],[34,679125,640],{"className":679126},[31685],[34,679128,31960],{"className":679129},[32119],[34,679131],{"className":679132,"style":32123},[31845],[34,679134,459],{"className":679135},[31685],[34,679137,2557],{"className":679138},[31690],[34,679140,679142,679163],{"className":679141},[31642],[34,679143,679145],{"className":679144},[31646],[31648,679146,679147],{"xmlns":31650},[31652,679148,679149,679161],{},[31655,679150,679151,679153,679155,679157,679159],{},[31662,679152,456],{"stretchy":702},[31749,679154,768],{},[31662,679156,31960],{"separator":758},[31749,679158,459],{},[31662,679160,2557],{"stretchy":702},[31665,679162,389609],{"encoding":31667},[34,679164,679166],{"className":679165,"ariaHidden":758},[31672],[34,679167,679169,679172,679175,679178,679181,679184,679187],{"className":679168},[31676],[34,679170],{"className":679171,"style":31774},[31680],[34,679173,456],{"className":679174},[31782],[34,679176,768],{"className":679177},[31685],[34,679179,31960],{"className":679180},[32119],[34,679182],{"className":679183,"style":32123},[31845],[34,679185,459],{"className":679186},[31685],[34,679188,2557],{"className":679189},[31690],[105,679191,679192,679306,679395],{},[108,679193,679194,2664,679196,347],{},[2627,679195,670318],{},[34,679197,679199,679231],{"className":679198},[31642],[34,679200,679202],{"className":679201},[31646],[31648,679203,679204],{"xmlns":31650},[31652,679205,679206,679228],{},[31655,679207,679208,679210,679212,679214,679216,679218,679220,679222,679224,679226],{},[31658,679209,67764],{},[31662,679211,202],{},[31749,679213,459],{},[39361,679215,262622],{},[31662,679217,262625],{},[39361,679219,262622],{},[31658,679221,46900],{"mathvariant":38079},[31658,679223,67764],{},[31662,679225,202],{},[31749,679227,459],{},[31665,679229,679230],{"encoding":31667},"y = 0 \\implies \\mathrm{d}y = 0",[34,679232,679234,679252,679276,679297],{"className":679233,"ariaHidden":758},[31672],[34,679235,679237,679240,679243,679246,679249],{"className":679236},[31676],[34,679238],{"className":679239,"style":38774},[31680],[34,679241,67764],{"className":679242,"style":32062},[31685,31686],[34,679244],{"className":679245,"style":31846},[31845],[34,679247,202],{"className":679248},[31850],[34,679250],{"className":679251,"style":31846},[31845],[34,679253,679255,679258,679261,679264,679267,679270,679273],{"className":679254},[31676],[34,679256],{"className":679257,"style":262700},[31680],[34,679259,459],{"className":679260},[31685],[34,679262],{"className":679263,"style":31846},[31845],[34,679265],{"className":679266,"style":31846},[31845],[34,679268,262625],{"className":679269},[31850],[34,679271],{"className":679272,"style":31846},[31845],[34,679274],{"className":679275,"style":31846},[31845],[34,679277,679279,679282,679285,679288,679291,679294],{"className":679278},[31676],[34,679280],{"className":679281,"style":43217},[31680],[34,679283,46900],{"className":679284},[31685,207923],[34,679286,67764],{"className":679287,"style":32062},[31685,31686],[34,679289],{"className":679290,"style":31846},[31845],[34,679292,202],{"className":679293},[31850],[34,679295],{"className":679296,"style":31846},[31845],[34,679298,679300,679303],{"className":679299},[31676],[34,679301],{"className":679302,"style":32617},[31680],[34,679304,459],{"className":679305},[31685],[108,679307,679308,2664,679311,138176,679339,53834,679367,347],{},[2627,679309,679310],{},"范围",[34,679312,679314,679327],{"className":679313},[31642],[34,679315,679317],{"className":679316},[31646],[31648,679318,679319],{"xmlns":31650},[31652,679320,679321,679325],{},[31655,679322,679323],{},[31658,679324,32936],{},[31665,679326,32936],{"encoding":31667},[34,679328,679330],{"className":679329,"ariaHidden":758},[31672],[34,679331,679333,679336],{"className":679332},[31676],[34,679334],{"className":679335,"style":32527},[31680],[34,679337,32936],{"className":679338},[31685,31686],[34,679340,679342,679355],{"className":679341},[31642],[34,679343,679345],{"className":679344},[31646],[31648,679346,679347],{"xmlns":31650},[31652,679348,679349,679353],{},[31655,679350,679351],{},[31749,679352,640],{},[31665,679354,640],{"encoding":31667},[34,679356,679358],{"className":679357,"ariaHidden":758},[31672],[34,679359,679361,679364],{"className":679360},[31676],[34,679362],{"className":679363,"style":32617},[31680],[34,679365,640],{"className":679366},[31685],[34,679368,679370,679383],{"className":679369},[31642],[34,679371,679373],{"className":679372},[31646],[31648,679374,679375],{"xmlns":31650},[31652,679376,679377,679381],{},[31655,679378,679379],{},[31749,679380,768],{},[31665,679382,768],{"encoding":31667},[34,679384,679386],{"className":679385,"ariaHidden":758},[31672],[34,679387,679389,679392],{"className":679388},[31676],[34,679390],{"className":679391,"style":32617},[31680],[34,679393,768],{"className":679394},[31685],[108,679396,679397,33183,679399,679788],{},[2627,679398,676907],{},[34,679400,679402,679480],{"className":679401},[31642],[34,679403,679405],{"className":679404},[31646],[31648,679406,679407],{"xmlns":31650},[31652,679408,679409,679477],{},[31655,679410,679411,679421,679423,679431,679433,679439,679441,679443,679445,679447,679449,679451,679453,679455,679457,679465,679471,679473,679475],{},[31743,679412,679413,679415],{},[31658,679414,106063],{},[31655,679416,679417,679419],{},[31658,679418,54151],{},[31658,679420,54224],{},[31662,679422,202],{},[38846,679424,679425,679427,679429],{},[31662,679426,546295],{},[31749,679428,640],{},[31749,679430,768],{},[31662,679432,456],{"stretchy":702},[37624,679434,679435,679437],{},[31658,679436,32936],{},[31749,679438,768],{},[31662,679440,223],{},[31749,679442,459],{},[31662,679444,2557],{"stretchy":702},[39361,679446,39363],{},[31658,679448,46900],{"mathvariant":38079},[31658,679450,32936],{},[31662,679452,223],{},[31749,679454,459],{},[31662,679456,202],{},[38846,679458,679459,679461,679463],{},[31662,679460,546295],{},[31749,679462,640],{},[31749,679464,768],{},[37624,679466,679467,679469],{},[31658,679468,32936],{},[31749,679470,768],{},[39361,679472,39363],{},[31658,679474,46900],{"mathvariant":38079},[31658,679476,32936],{},[31665,679478,679479],{"encoding":31667},"I_{AB} = \\int_1^2 (x^2 + 0) \\, \\mathrm{d}x + 0 = \\int_1^2 x^2 \\, \\mathrm{d}x",[34,679481,679483,679544,679642,679672,679690],{"className":679482,"ariaHidden":758},[31672],[34,679484,679486,679489,679535,679538,679541],{"className":679485},[31676],[34,679487],{"className":679488,"style":70915},[31680],[34,679490,679492,679495],{"className":679491},[31685],[34,679493,106063],{"className":679494,"style":67482},[31685,31686],[34,679496,679498],{"className":679497},[31792],[34,679499,679501,679527],{"className":679500},[31796,31797],[34,679502,679504,679524],{"className":679503},[31801],[34,679505,679507],{"className":679506,"style":116558},[31805],[34,679508,679509,679512],{"style":217742},[34,679510],{"className":679511,"style":31814},[31813],[34,679513,679515],{"className":679514},[31818,31819,31820,31821],[34,679516,679518,679521],{"className":679517},[31685,31821],[34,679519,54151],{"className":679520},[31685,31686,31821],[34,679522,54224],{"className":679523,"style":54252},[31685,31686,31821],[34,679525,31829],{"className":679526},[31828],[34,679528,679530],{"className":679529},[31801],[34,679531,679533],{"className":679532,"style":31836},[31805],[34,679534],{},[34,679536],{"className":679537,"style":31846},[31845],[34,679539,202],{"className":679540},[31850],[34,679542],{"className":679543,"style":31846},[31845],[34,679545,679547,679550,679601,679604,679633,679636,679639],{"className":679546},[31676],[34,679548],{"className":679549,"style":563309},[31680],[34,679551,679553,679556],{"className":679552},[32767],[34,679554,546295],{"className":679555,"style":546331},[32767,131249,156560],[34,679557,679559],{"className":679558},[31792],[34,679560,679562,679593],{"className":679561},[31796,31797],[34,679563,679565,679590],{"className":679564},[31801],[34,679566,679568,679579],{"className":679567,"style":563328},[31805],[34,679569,679570,679573],{"style":546347},[34,679571],{"className":679572,"style":31814},[31813],[34,679574,679576],{"className":679575},[31818,31819,31820,31821],[34,679577,640],{"className":679578},[31685,31821],[34,679580,679581,679584],{"style":546362},[34,679582],{"className":679583,"style":31814},[31813],[34,679585,679587],{"className":679586},[31818,31819,31820,31821],[34,679588,768],{"className":679589},[31685,31821],[34,679591,31829],{"className":679592},[31828],[34,679594,679596],{"className":679595},[31801],[34,679597,679599],{"className":679598,"style":546384},[31805],[34,679600],{},[34,679602,456],{"className":679603},[31782],[34,679605,679607,679610],{"className":679606},[31685],[34,679608,32936],{"className":679609},[31685,31686],[34,679611,679613],{"className":679612},[31792],[34,679614,679616],{"className":679615},[31796],[34,679617,679619],{"className":679618},[31801],[34,679620,679622],{"className":679621,"style":37774},[31805],[34,679623,679624,679627],{"style":37777},[34,679625],{"className":679626,"style":31814},[31813],[34,679628,679630],{"className":679629},[31818,31819,31820,31821],[34,679631,768],{"className":679632},[31685,31821],[34,679634],{"className":679635,"style":32604},[31845],[34,679637,223],{"className":679638},[32024],[34,679640],{"className":679641,"style":32604},[31845],[34,679643,679645,679648,679651,679654,679657,679660,679663,679666,679669],{"className":679644},[31676],[34,679646],{"className":679647,"style":31774},[31680],[34,679649,459],{"className":679650},[31685],[34,679652,2557],{"className":679653},[31690],[34,679655],{"className":679656,"style":32123},[31845],[34,679658,46900],{"className":679659},[31685,207923],[34,679661,32936],{"className":679662},[31685,31686],[34,679664],{"className":679665,"style":32604},[31845],[34,679667,223],{"className":679668},[32024],[34,679670],{"className":679671,"style":32604},[31845],[34,679673,679675,679678,679681,679684,679687],{"className":679674},[31676],[34,679676],{"className":679677,"style":32617},[31680],[34,679679,459],{"className":679680},[31685],[34,679682],{"className":679683,"style":31846},[31845],[34,679685,202],{"className":679686},[31850],[34,679688],{"className":679689,"style":31846},[31845],[34,679691,679693,679696,679747,679750,679779,679782,679785],{"className":679692},[31676],[34,679694],{"className":679695,"style":563309},[31680],[34,679697,679699,679702],{"className":679698},[32767],[34,679700,546295],{"className":679701,"style":546331},[32767,131249,156560],[34,679703,679705],{"className":679704},[31792],[34,679706,679708,679739],{"className":679707},[31796,31797],[34,679709,679711,679736],{"className":679710},[31801],[34,679712,679714,679725],{"className":679713,"style":563328},[31805],[34,679715,679716,679719],{"style":546347},[34,679717],{"className":679718,"style":31814},[31813],[34,679720,679722],{"className":679721},[31818,31819,31820,31821],[34,679723,640],{"className":679724},[31685,31821],[34,679726,679727,679730],{"style":546362},[34,679728],{"className":679729,"style":31814},[31813],[34,679731,679733],{"className":679732},[31818,31819,31820,31821],[34,679734,768],{"className":679735},[31685,31821],[34,679737,31829],{"className":679738},[31828],[34,679740,679742],{"className":679741},[31801],[34,679743,679745],{"className":679744,"style":546384},[31805],[34,679746],{},[34,679748],{"className":679749,"style":32123},[31845],[34,679751,679753,679756],{"className":679752},[31685],[34,679754,32936],{"className":679755},[31685,31686],[34,679757,679759],{"className":679758},[31792],[34,679760,679762],{"className":679761},[31796],[34,679763,679765],{"className":679764},[31801],[34,679766,679768],{"className":679767,"style":37774},[31805],[34,679769,679770,679773],{"style":37777},[34,679771],{"className":679772,"style":31814},[31813],[34,679774,679776],{"className":679775},[31818,31819,31820,31821],[34,679777,768],{"className":679778},[31685,31821],[34,679780],{"className":679781,"style":32123},[31845],[34,679783,46900],{"className":679784},[31685,207923],[34,679786,32936],{"className":679787},[31685,31686],[34,679789,679791,679859],{"className":679790},[31642],[34,679792,679794],{"className":679793},[31646],[31648,679795,679796],{"xmlns":31650},[31652,679797,679798,679856],{},[31655,679799,679800,679810,679812,679814,679824,679832,679834,679840,679842,679848,679850],{},[31743,679801,679802,679804],{},[31658,679803,106063],{},[31655,679805,679806,679808],{},[31658,679807,54151],{},[31658,679809,54224],{},[31662,679811,202],{},[31662,679813,15972],{"stretchy":702},[41453,679815,679816,679822],{},[37624,679817,679818,679820],{},[31658,679819,32936],{},[31749,679821,786],{},[31749,679823,786],{},[38846,679825,679826,679828,679830],{},[31662,679827,32712],{"stretchy":702},[31749,679829,640],{},[31749,679831,768],{},[31662,679833,202],{},[41453,679835,679836,679838],{},[31749,679837,60525],{},[31749,679839,786],{},[31662,679841,32226],{},[41453,679843,679844,679846],{},[31749,679845,640],{},[31749,679847,786],{},[31662,679849,202],{},[41453,679851,679852,679854],{},[31749,679853,36723],{},[31749,679855,786],{},[31665,679857,679858],{"encoding":31667},"I_{AB} = [\\frac{x^3}{3}]_1^2 = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}",[34,679860,679862,679923,680086,680169,680252],{"className":679861,"ariaHidden":758},[31672],[34,679863,679865,679868,679914,679917,679920],{"className":679864},[31676],[34,679866],{"className":679867,"style":70915},[31680],[34,679869,679871,679874],{"className":679870},[31685],[34,679872,106063],{"className":679873,"style":67482},[31685,31686],[34,679875,679877],{"className":679876},[31792],[34,679878,679880,679906],{"className":679879},[31796,31797],[34,679881,679883,679903],{"className":679882},[31801],[34,679884,679886],{"className":679885,"style":116558},[31805],[34,679887,679888,679891],{"style":217742},[34,679889],{"className":679890,"style":31814},[31813],[34,679892,679894],{"className":679893},[31818,31819,31820,31821],[34,679895,679897,679900],{"className":679896},[31685,31821],[34,679898,54151],{"className":679899},[31685,31686,31821],[34,679901,54224],{"className":679902,"style":54252},[31685,31686,31821],[34,679904,31829],{"className":679905},[31828],[34,679907,679909],{"className":679908},[31801],[34,679910,679912],{"className":679911,"style":31836},[31805],[34,679913],{},[34,679915],{"className":679916,"style":31846},[31845],[34,679918,202],{"className":679919},[31850],[34,679921],{"className":679922,"style":31846},[31845],[34,679924,679926,679929,679932,680026,680077,680080,680083],{"className":679925},[31676],[34,679927],{"className":679928,"style":637019},[31680],[34,679930,15972],{"className":679931},[31782],[34,679933,679935,679938,680023],{"className":679934},[31685],[34,679936],{"className":679937},[31782,41488],[34,679939,679941],{"className":679940},[41453],[34,679942,679944,680015],{"className":679943},[31796,31797],[34,679945,679947,680012],{"className":679946},[31801],[34,679948,679950,679964,679972],{"className":679949,"style":603956},[31805],[34,679951,679952,679955],{"style":41504},[34,679953],{"className":679954,"style":41508},[31813],[34,679956,679958],{"className":679957},[31818,31819,31820,31821],[34,679959,679961],{"className":679960},[31685,31821],[34,679962,786],{"className":679963},[31685,31821],[34,679965,679966,679969],{"style":41520},[34,679967],{"className":679968,"style":41508},[31813],[34,679970],{"className":679971,"style":41528},[41527],[34,679973,679974,679977],{"style":46771},[34,679975],{"className":679976,"style":41508},[31813],[34,679978,679980],{"className":679979},[31818,31819,31820,31821],[34,679981,679983],{"className":679982},[31685,31821],[34,679984,679986,679989],{"className":679985},[31685,31821],[34,679987,32936],{"className":679988},[31685,31686,31821],[34,679990,679992],{"className":679991},[31792],[34,679993,679995],{"className":679994},[31796],[34,679996,679998],{"className":679997},[31801],[34,679999,680001],{"className":680000,"style":144123},[31805],[34,680002,680003,680006],{"style":552391},[34,680004],{"className":680005,"style":43406},[31813],[34,680007,680009],{"className":680008},[31818,43410,43411,31821],[34,680010,786],{"className":680011},[31685,31821],[34,680013,31829],{"className":680014},[31828],[34,680016,680018],{"className":680017},[31801],[34,680019,680021],{"className":680020,"style":41619},[31805],[34,680022],{},[34,680024],{"className":680025},[31690,41488],[34,680027,680029,680032],{"className":680028},[31690],[34,680030,32712],{"className":680031},[31690],[34,680033,680035],{"className":680034},[31792],[34,680036,680038,680069],{"className":680037},[31796,31797],[34,680039,680041,680066],{"className":680040},[31801],[34,680042,680044,680055],{"className":680043,"style":37774},[31805],[34,680045,680046,680049],{"style":316930},[34,680047],{"className":680048,"style":31814},[31813],[34,680050,680052],{"className":680051},[31818,31819,31820,31821],[34,680053,640],{"className":680054},[31685,31821],[34,680056,680057,680060],{"style":37777},[34,680058],{"className":680059,"style":31814},[31813],[34,680061,680063],{"className":680062},[31818,31819,31820,31821],[34,680064,768],{"className":680065},[31685,31821],[34,680067,31829],{"className":680068},[31828],[34,680070,680072],{"className":680071},[31801],[34,680073,680075],{"className":680074,"style":39021},[31805],[34,680076],{},[34,680078],{"className":680079,"style":31846},[31845],[34,680081,202],{"className":680082},[31850],[34,680084],{"className":680085,"style":31846},[31845],[34,680087,680089,680092,680160,680163,680166],{"className":680088},[31676],[34,680090],{"className":680091,"style":46727},[31680],[34,680093,680095,680098,680157],{"className":680094},[31685],[34,680096],{"className":680097},[31782,41488],[34,680099,680101],{"className":680100},[41453],[34,680102,680104,680149],{"className":680103},[31796,31797],[34,680105,680107,680146],{"className":680106},[31801],[34,680108,680110,680124,680132],{"className":680109,"style":46746},[31805],[34,680111,680112,680115],{"style":41504},[34,680113],{"className":680114,"style":41508},[31813],[34,680116,680118],{"className":680117},[31818,31819,31820,31821],[34,680119,680121],{"className":680120},[31685,31821],[34,680122,786],{"className":680123},[31685,31821],[34,680125,680126,680129],{"style":41520},[34,680127],{"className":680128,"style":41508},[31813],[34,680130],{"className":680131,"style":41528},[41527],[34,680133,680134,680137],{"style":46771},[34,680135],{"className":680136,"style":41508},[31813],[34,680138,680140],{"className":680139},[31818,31819,31820,31821],[34,680141,680143],{"className":680142},[31685,31821],[34,680144,60525],{"className":680145},[31685,31821],[34,680147,31829],{"className":680148},[31828],[34,680150,680152],{"className":680151},[31801],[34,680153,680155],{"className":680154,"style":41619},[31805],[34,680156],{},[34,680158],{"className":680159},[31690,41488],[34,680161],{"className":680162,"style":32604},[31845],[34,680164,32226],{"className":680165},[32024],[34,680167],{"className":680168,"style":32604},[31845],[34,680170,680172,680175,680243,680246,680249],{"className":680171},[31676],[34,680173],{"className":680174,"style":46727},[31680],[34,680176,680178,680181,680240],{"className":680177},[31685],[34,680179],{"className":680180},[31782,41488],[34,680182,680184],{"className":680183},[41453],[34,680185,680187,680232],{"className":680186},[31796,31797],[34,680188,680190,680229],{"className":680189},[31801],[34,680191,680193,680207,680215],{"className":680192,"style":46746},[31805],[34,680194,680195,680198],{"style":41504},[34,680196],{"className":680197,"style":41508},[31813],[34,680199,680201],{"className":680200},[31818,31819,31820,31821],[34,680202,680204],{"className":680203},[31685,31821],[34,680205,786],{"className":680206},[31685,31821],[34,680208,680209,680212],{"style":41520},[34,680210],{"className":680211,"style":41508},[31813],[34,680213],{"className":680214,"style":41528},[41527],[34,680216,680217,680220],{"style":46771},[34,680218],{"className":680219,"style":41508},[31813],[34,680221,680223],{"className":680222},[31818,31819,31820,31821],[34,680224,680226],{"className":680225},[31685,31821],[34,680227,640],{"className":680228},[31685,31821],[34,680230,31829],{"className":680231},[31828],[34,680233,680235],{"className":680234},[31801],[34,680236,680238],{"className":680237,"style":41619},[31805],[34,680239],{},[34,680241],{"className":680242},[31690,41488],[34,680244],{"className":680245,"style":31846},[31845],[34,680247,202],{"className":680248},[31850],[34,680250],{"className":680251,"style":31846},[31845],[34,680253,680255,680258],{"className":680254},[31676],[34,680256],{"className":680257,"style":46727},[31680],[34,680259,680261,680264,680323],{"className":680260},[31685],[34,680262],{"className":680263},[31782,41488],[34,680265,680267],{"className":680266},[41453],[34,680268,680270,680315],{"className":680269},[31796,31797],[34,680271,680273,680312],{"className":680272},[31801],[34,680274,680276,680290,680298],{"className":680275,"style":46746},[31805],[34,680277,680278,680281],{"style":41504},[34,680279],{"className":680280,"style":41508},[31813],[34,680282,680284],{"className":680283},[31818,31819,31820,31821],[34,680285,680287],{"className":680286},[31685,31821],[34,680288,786],{"className":680289},[31685,31821],[34,680291,680292,680295],{"style":41520},[34,680293],{"className":680294,"style":41508},[31813],[34,680296],{"className":680297,"style":41528},[41527],[34,680299,680300,680303],{"style":46771},[34,680301],{"className":680302,"style":41508},[31813],[34,680304,680306],{"className":680305},[31818,31819,31820,31821],[34,680307,680309],{"className":680308},[31685,31821],[34,680310,36723],{"className":680311},[31685,31821],[34,680313,31829],{"className":680314},[31828],[34,680316,680318],{"className":680317},[31801],[34,680319,680321],{"className":680320,"style":41619},[31805],[34,680322],{},[34,680324],{"className":680325},[31690,41488],[15,680327,680328],{},[2627,680329,680330,680331,54852,680365,53834,680416],{},"第 2 段 ",[34,680332,680334,680350],{"className":680333},[31642],[34,680335,680337],{"className":680336},[31646],[31648,680338,680339],{"xmlns":31650},[31652,680340,680341,680347],{},[31655,680342,680343,680345],{},[31658,680344,54224],{},[31658,680346,47384],{},[31665,680348,680349],{"encoding":31667},"BC",[34,680351,680353],{"className":680352,"ariaHidden":758},[31672],[34,680354,680356,680359,680362],{"className":680355},[31676],[34,680357],{"className":680358,"style":33218},[31680],[34,680360,54224],{"className":680361,"style":54252},[31685,31686],[34,680363,47384],{"className":680364,"style":38706},[31685,31686],[34,680366,680368,680389],{"className":680367},[31642],[34,680369,680371],{"className":680370},[31646],[31648,680372,680373],{"xmlns":31650},[31652,680374,680375,680387],{},[31655,680376,680377,680379,680381,680383,680385],{},[31662,680378,456],{"stretchy":702},[31749,680380,768],{},[31662,680382,31960],{"separator":758},[31749,680384,459],{},[31662,680386,2557],{"stretchy":702},[31665,680388,389609],{"encoding":31667},[34,680390,680392],{"className":680391,"ariaHidden":758},[31672],[34,680393,680395,680398,680401,680404,680407,680410,680413],{"className":680394},[31676],[34,680396],{"className":680397,"style":31774},[31680],[34,680399,456],{"className":680400},[31782],[34,680402,768],{"className":680403},[31685],[34,680405,31960],{"className":680406},[32119],[34,680408],{"className":680409,"style":32123},[31845],[34,680411,459],{"className":680412},[31685],[34,680414,2557],{"className":680415},[31690],[34,680417,680419,680441],{"className":680418},[31642],[34,680420,680422],{"className":680421},[31646],[31648,680423,680424],{"xmlns":31650},[31652,680425,680426,680438],{},[31655,680427,680428,680430,680432,680434,680436],{},[31662,680429,456],{"stretchy":702},[31749,680431,768],{},[31662,680433,31960],{"separator":758},[31749,680435,640],{},[31662,680437,2557],{"stretchy":702},[31665,680439,680440],{"encoding":31667},"(2,1)",[34,680442,680444],{"className":680443,"ariaHidden":758},[31672],[34,680445,680447,680450,680453,680456,680459,680462,680465],{"className":680446},[31676],[34,680448],{"className":680449,"style":31774},[31680],[34,680451,456],{"className":680452},[31782],[34,680454,768],{"className":680455},[31685],[34,680457,31960],{"className":680458},[32119],[34,680460],{"className":680461,"style":32123},[31845],[34,680463,640],{"className":680464},[31685],[34,680466,2557],{"className":680467},[31690],[105,680469,680470,680584,680672],{},[108,680471,680472,2664,680474,347],{},[2627,680473,670318],{},[34,680475,680477,680509],{"className":680476},[31642],[34,680478,680480],{"className":680479},[31646],[31648,680481,680482],{"xmlns":31650},[31652,680483,680484,680506],{},[31655,680485,680486,680488,680490,680492,680494,680496,680498,680500,680502,680504],{},[31658,680487,32936],{},[31662,680489,202],{},[31749,680491,768],{},[39361,680493,262622],{},[31662,680495,262625],{},[39361,680497,262622],{},[31658,680499,46900],{"mathvariant":38079},[31658,680501,32936],{},[31662,680503,202],{},[31749,680505,459],{},[31665,680507,680508],{"encoding":31667},"x = 2 \\implies \\mathrm{d}x = 0",[34,680510,680512,680530,680554,680575],{"className":680511,"ariaHidden":758},[31672],[34,680513,680515,680518,680521,680524,680527],{"className":680514},[31676],[34,680516],{"className":680517,"style":32527},[31680],[34,680519,32936],{"className":680520},[31685,31686],[34,680522],{"className":680523,"style":31846},[31845],[34,680525,202],{"className":680526},[31850],[34,680528],{"className":680529,"style":31846},[31845],[34,680531,680533,680536,680539,680542,680545,680548,680551],{"className":680532},[31676],[34,680534],{"className":680535,"style":262700},[31680],[34,680537,768],{"className":680538},[31685],[34,680540],{"className":680541,"style":31846},[31845],[34,680543],{"className":680544,"style":31846},[31845],[34,680546,262625],{"className":680547},[31850],[34,680549],{"className":680550,"style":31846},[31845],[34,680552],{"className":680553,"style":31846},[31845],[34,680555,680557,680560,680563,680566,680569,680572],{"className":680556},[31676],[34,680558],{"className":680559,"style":31681},[31680],[34,680561,46900],{"className":680562},[31685,207923],[34,680564,32936],{"className":680565},[31685,31686],[34,680567],{"className":680568,"style":31846},[31845],[34,680570,202],{"className":680571},[31850],[34,680573],{"className":680574,"style":31846},[31845],[34,680576,680578,680581],{"className":680577},[31676],[34,680579],{"className":680580,"style":32617},[31680],[34,680582,459],{"className":680583},[31685],[108,680585,680586,2664,680588,138176,680616,53834,680644,347],{},[2627,680587,679310],{},[34,680589,680591,680604],{"className":680590},[31642],[34,680592,680594],{"className":680593},[31646],[31648,680595,680596],{"xmlns":31650},[31652,680597,680598,680602],{},[31655,680599,680600],{},[31658,680601,67764],{},[31665,680603,67764],{"encoding":31667},[34,680605,680607],{"className":680606,"ariaHidden":758},[31672],[34,680608,680610,680613],{"className":680609},[31676],[34,680611],{"className":680612,"style":38774},[31680],[34,680614,67764],{"className":680615,"style":32062},[31685,31686],[34,680617,680619,680632],{"className":680618},[31642],[34,680620,680622],{"className":680621},[31646],[31648,680623,680624],{"xmlns":31650},[31652,680625,680626,680630],{},[31655,680627,680628],{},[31749,680629,459],{},[31665,680631,459],{"encoding":31667},[34,680633,680635],{"className":680634,"ariaHidden":758},[31672],[34,680636,680638,680641],{"className":680637},[31676],[34,680639],{"className":680640,"style":32617},[31680],[34,680642,459],{"className":680643},[31685],[34,680645,680647,680660],{"className":680646},[31642],[34,680648,680650],{"className":680649},[31646],[31648,680651,680652],{"xmlns":31650},[31652,680653,680654,680658],{},[31655,680655,680656],{},[31749,680657,640],{},[31665,680659,640],{"encoding":31667},[34,680661,680663],{"className":680662,"ariaHidden":758},[31672],[34,680664,680666,680669],{"className":680665},[31676],[34,680667],{"className":680668,"style":32617},[31680],[34,680670,640],{"className":680671},[31685],[108,680673,680674,680676,680677,680710,680711,681137],{},[2627,680675,676907],{},"：\n注意 ",[34,680678,680680,680695],{"className":680679},[31642],[34,680681,680683],{"className":680682},[31646],[31648,680684,680685],{"xmlns":31650},[31652,680686,680687,680693],{},[31655,680688,680689,680691],{},[31658,680690,46900],{"mathvariant":38079},[31658,680692,32936],{},[31665,680694,659508],{"encoding":31667},[34,680696,680698],{"className":680697,"ariaHidden":758},[31672],[34,680699,680701,680704,680707],{"className":680700},[31676],[34,680702],{"className":680703,"style":31681},[31680],[34,680705,46900],{"className":680706},[31685,207923],[34,680708,32936],{"className":680709},[31685,31686]," 项消失。\n",[34,680712,680714,680800],{"className":680713},[31642],[34,680715,680717],{"className":680716},[31646],[31648,680718,680719],{"xmlns":31650},[31652,680720,680721,680797],{},[31655,680722,680723,680733,680735,680743,680745,680751,680753,680759,680761,680763,680765,680767,680769,680777,680779,680781,680783,680789,680791,680793,680795],{},[31743,680724,680725,680727],{},[31658,680726,106063],{},[31655,680728,680729,680731],{},[31658,680730,54224],{},[31658,680732,47384],{},[31662,680734,202],{},[38846,680736,680737,680739,680741],{},[31662,680738,546295],{},[31749,680740,459],{},[31749,680742,640],{},[31662,680744,456],{"stretchy":702},[37624,680746,680747,680749],{},[31749,680748,768],{},[31749,680750,768],{},[31662,680752,32226],{},[37624,680754,680755,680757],{},[31658,680756,67764],{},[31749,680758,768],{},[31662,680760,2557],{"stretchy":702},[39361,680762,39363],{},[31658,680764,46900],{"mathvariant":38079},[31658,680766,67764],{},[31662,680768,202],{},[38846,680770,680771,680773,680775],{},[31662,680772,546295],{},[31749,680774,459],{},[31749,680776,640],{},[31662,680778,456],{"stretchy":702},[31749,680780,15932],{},[31662,680782,32226],{},[37624,680784,680785,680787],{},[31658,680786,67764],{},[31749,680788,768],{},[31662,680790,2557],{"stretchy":702},[39361,680792,39363],{},[31658,680794,46900],{"mathvariant":38079},[31658,680796,67764],{},[31665,680798,680799],{"encoding":31667},"I_{BC} = \\int_0^1 (2^2 - y^2) \\, \\mathrm{d}y = \\int_0^1 (4 - y^2) \\, \\mathrm{d}y",[34,680801,680803,680864,680962,681018,681090],{"className":680802,"ariaHidden":758},[31672],[34,680804,680806,680809,680855,680858,680861],{"className":680805},[31676],[34,680807],{"className":680808,"style":70915},[31680],[34,680810,680812,680815],{"className":680811},[31685],[34,680813,106063],{"className":680814,"style":67482},[31685,31686],[34,680816,680818],{"className":680817},[31792],[34,680819,680821,680847],{"className":680820},[31796,31797],[34,680822,680824,680844],{"className":680823},[31801],[34,680825,680827],{"className":680826,"style":116558},[31805],[34,680828,680829,680832],{"style":217742},[34,680830],{"className":680831,"style":31814},[31813],[34,680833,680835],{"className":680834},[31818,31819,31820,31821],[34,680836,680838,680841],{"className":680837},[31685,31821],[34,680839,54224],{"className":680840,"style":54252},[31685,31686,31821],[34,680842,47384],{"className":680843,"style":38706},[31685,31686,31821],[34,680845,31829],{"className":680846},[31828],[34,680848,680850],{"className":680849},[31801],[34,680851,680853],{"className":680852,"style":31836},[31805],[34,680854],{},[34,680856],{"className":680857,"style":31846},[31845],[34,680859,202],{"className":680860},[31850],[34,680862],{"className":680863,"style":31846},[31845],[34,680865,680867,680870,680921,680924,680953,680956,680959],{"className":680866},[31676],[34,680868],{"className":680869,"style":563309},[31680],[34,680871,680873,680876],{"className":680872},[32767],[34,680874,546295],{"className":680875,"style":546331},[32767,131249,156560],[34,680877,680879],{"className":680878},[31792],[34,680880,680882,680913],{"className":680881},[31796,31797],[34,680883,680885,680910],{"className":680884},[31801],[34,680886,680888,680899],{"className":680887,"style":563328},[31805],[34,680889,680890,680893],{"style":546347},[34,680891],{"className":680892,"style":31814},[31813],[34,680894,680896],{"className":680895},[31818,31819,31820,31821],[34,680897,459],{"className":680898},[31685,31821],[34,680900,680901,680904],{"style":546362},[34,680902],{"className":680903,"style":31814},[31813],[34,680905,680907],{"className":680906},[31818,31819,31820,31821],[34,680908,640],{"className":680909},[31685,31821],[34,680911,31829],{"className":680912},[31828],[34,680914,680916],{"className":680915},[31801],[34,680917,680919],{"className":680918,"style":546384},[31805],[34,680920],{},[34,680922,456],{"className":680923},[31782],[34,680925,680927,680930],{"className":680926},[31685],[34,680928,768],{"className":680929},[31685],[34,680931,680933],{"className":680932},[31792],[34,680934,680936],{"className":680935},[31796],[34,680937,680939],{"className":680938},[31801],[34,680940,680942],{"className":680941,"style":37774},[31805],[34,680943,680944,680947],{"style":37777},[34,680945],{"className":680946,"style":31814},[31813],[34,680948,680950],{"className":680949},[31818,31819,31820,31821],[34,680951,768],{"className":680952},[31685,31821],[34,680954],{"className":680955,"style":32604},[31845],[34,680957,32226],{"className":680958},[32024],[34,680960],{"className":680961,"style":32604},[31845],[34,680963,680965,680968,680997,681000,681003,681006,681009,681012,681015],{"className":680964},[31676],[34,680966],{"className":680967,"style":37755},[31680],[34,680969,680971,680974],{"className":680970},[31685],[34,680972,67764],{"className":680973,"style":32062},[31685,31686],[34,680975,680977],{"className":680976},[31792],[34,680978,680980],{"className":680979},[31796],[34,680981,680983],{"className":680982},[31801],[34,680984,680986],{"className":680985,"style":37774},[31805],[34,680987,680988,680991],{"style":37777},[34,680989],{"className":680990,"style":31814},[31813],[34,680992,680994],{"className":680993},[31818,31819,31820,31821],[34,680995,768],{"className":680996},[31685,31821],[34,680998,2557],{"className":680999},[31690],[34,681001],{"className":681002,"style":32123},[31845],[34,681004,46900],{"className":681005},[31685,207923],[34,681007,67764],{"className":681008,"style":32062},[31685,31686],[34,681010],{"className":681011,"style":31846},[31845],[34,681013,202],{"className":681014},[31850],[34,681016],{"className":681017,"style":31846},[31845],[34,681019,681021,681024,681075,681078,681081,681084,681087],{"className":681020},[31676],[34,681022],{"className":681023,"style":563309},[31680],[34,681025,681027,681030],{"className":681026},[32767],[34,681028,546295],{"className":681029,"style":546331},[32767,131249,156560],[34,681031,681033],{"className":681032},[31792],[34,681034,681036,681067],{"className":681035},[31796,31797],[34,681037,681039,681064],{"className":681038},[31801],[34,681040,681042,681053],{"className":681041,"style":563328},[31805],[34,681043,681044,681047],{"style":546347},[34,681045],{"className":681046,"style":31814},[31813],[34,681048,681050],{"className":681049},[31818,31819,31820,31821],[34,681051,459],{"className":681052},[31685,31821],[34,681054,681055,681058],{"style":546362},[34,681056],{"className":681057,"style":31814},[31813],[34,681059,681061],{"className":681060},[31818,31819,31820,31821],[34,681062,640],{"className":681063},[31685,31821],[34,681065,31829],{"className":681066},[31828],[34,681068,681070],{"className":681069},[31801],[34,681071,681073],{"className":681072,"style":546384},[31805],[34,681074],{},[34,681076,456],{"className":681077},[31782],[34,681079,15932],{"className":681080},[31685],[34,681082],{"className":681083,"style":32604},[31845],[34,681085,32226],{"className":681086},[32024],[34,681088],{"className":681089,"style":32604},[31845],[34,681091,681093,681096,681125,681128,681131,681134],{"className":681092},[31676],[34,681094],{"className":681095,"style":37755},[31680],[34,681097,681099,681102],{"className":681098},[31685],[34,681100,67764],{"className":681101,"style":32062},[31685,31686],[34,681103,681105],{"className":681104},[31792],[34,681106,681108],{"className":681107},[31796],[34,681109,681111],{"className":681110},[31801],[34,681112,681114],{"className":681113,"style":37774},[31805],[34,681115,681116,681119],{"style":37777},[34,681117],{"className":681118,"style":31814},[31813],[34,681120,681122],{"className":681121},[31818,31819,31820,31821],[34,681123,768],{"className":681124},[31685,31821],[34,681126,2557],{"className":681127},[31690],[34,681129],{"className":681130,"style":32123},[31845],[34,681132,46900],{"className":681133},[31685,207923],[34,681135,67764],{"className":681136,"style":32062},[31685,31686],[34,681138,681140,681210],{"className":681139},[31642],[34,681141,681143],{"className":681142},[31646],[31648,681144,681145],{"xmlns":31650},[31652,681146,681147,681207],{},[31655,681148,681149,681159,681161,681163,681165,681167,681169,681179,681187,681189,681191,681193,681199,681201],{},[31743,681150,681151,681153],{},[31658,681152,106063],{},[31655,681154,681155,681157],{},[31658,681156,54224],{},[31658,681158,47384],{},[31662,681160,202],{},[31662,681162,15972],{"stretchy":702},[31749,681164,15932],{},[31658,681166,67764],{},[31662,681168,32226],{},[41453,681170,681171,681177],{},[37624,681172,681173,681175],{},[31658,681174,67764],{},[31749,681176,786],{},[31749,681178,786],{},[38846,681180,681181,681183,681185],{},[31662,681182,32712],{"stretchy":702},[31749,681184,459],{},[31749,681186,640],{},[31662,681188,202],{},[31749,681190,15932],{},[31662,681192,32226],{},[41453,681194,681195,681197],{},[31749,681196,640],{},[31749,681198,786],{},[31662,681200,202],{},[41453,681202,681203,681205],{},[31749,681204,39620],{},[31749,681206,786],{},[31665,681208,681209],{"encoding":31667},"I_{BC} = [4y - \\frac{y^3}{3}]_0^1 = 4 - \\frac{1}{3} = \\frac{11}{3}",[34,681211,681213,681274,681298,681459,681477,681560],{"className":681212,"ariaHidden":758},[31672],[34,681214,681216,681219,681265,681268,681271],{"className":681215},[31676],[34,681217],{"className":681218,"style":70915},[31680],[34,681220,681222,681225],{"className":681221},[31685],[34,681223,106063],{"className":681224,"style":67482},[31685,31686],[34,681226,681228],{"className":681227},[31792],[34,681229,681231,681257],{"className":681230},[31796,31797],[34,681232,681234,681254],{"className":681233},[31801],[34,681235,681237],{"className":681236,"style":116558},[31805],[34,681238,681239,681242],{"style":217742},[34,681240],{"className":681241,"style":31814},[31813],[34,681243,681245],{"className":681244},[31818,31819,31820,31821],[34,681246,681248,681251],{"className":681247},[31685,31821],[34,681249,54224],{"className":681250,"style":54252},[31685,31686,31821],[34,681252,47384],{"className":681253,"style":38706},[31685,31686,31821],[34,681255,31829],{"className":681256},[31828],[34,681258,681260],{"className":681259},[31801],[34,681261,681263],{"className":681262,"style":31836},[31805],[34,681264],{},[34,681266],{"className":681267,"style":31846},[31845],[34,681269,202],{"className":681270},[31850],[34,681272],{"className":681273,"style":31846},[31845],[34,681275,681277,681280,681283,681286,681289,681292,681295],{"className":681276},[31676],[34,681278],{"className":681279,"style":31774},[31680],[34,681281,15972],{"className":681282},[31782],[34,681284,15932],{"className":681285},[31685],[34,681287,67764],{"className":681288,"style":32062},[31685,31686],[34,681290],{"className":681291,"style":32604},[31845],[34,681293,32226],{"className":681294},[32024],[34,681296],{"className":681297,"style":32604},[31845],[34,681299,681301,681305,681399,681450,681453,681456],{"className":681300},[31676],[34,681302],{"className":681303,"style":681304},[31680],"height:1.415em;vertical-align:-0.345em;",[34,681306,681308,681311,681396],{"className":681307},[31685],[34,681309],{"className":681310},[31782,41488],[34,681312,681314],{"className":681313},[41453],[34,681315,681317,681388],{"className":681316},[31796,31797],[34,681318,681320,681385],{"className":681319},[31801],[34,681321,681323,681337,681345],{"className":681322,"style":601858},[31805],[34,681324,681325,681328],{"style":41504},[34,681326],{"className":681327,"style":41508},[31813],[34,681329,681331],{"className":681330},[31818,31819,31820,31821],[34,681332,681334],{"className":681333},[31685,31821],[34,681335,786],{"className":681336},[31685,31821],[34,681338,681339,681342],{"style":41520},[34,681340],{"className":681341,"style":41508},[31813],[34,681343],{"className":681344,"style":41528},[41527],[34,681346,681347,681350],{"style":548368},[34,681348],{"className":681349,"style":41508},[31813],[34,681351,681353],{"className":681352},[31818,31819,31820,31821],[34,681354,681356],{"className":681355},[31685,31821],[34,681357,681359,681362],{"className":681358},[31685,31821],[34,681360,67764],{"className":681361,"style":32062},[31685,31686,31821],[34,681363,681365],{"className":681364},[31792],[34,681366,681368],{"className":681367},[31796],[34,681369,681371],{"className":681370},[31801],[34,681372,681374],{"className":681373,"style":144123},[31805],[34,681375,681376,681379],{"style":552391},[34,681377],{"className":681378,"style":43406},[31813],[34,681380,681382],{"className":681381},[31818,43410,43411,31821],[34,681383,786],{"className":681384},[31685,31821],[34,681386,31829],{"className":681387},[31828],[34,681389,681391],{"className":681390},[31801],[34,681392,681394],{"className":681393,"style":41619},[31805],[34,681395],{},[34,681397],{"className":681398},[31690,41488],[34,681400,681402,681405],{"className":681401},[31690],[34,681403,32712],{"className":681404},[31690],[34,681406,681408],{"className":681407},[31792],[34,681409,681411,681442],{"className":681410},[31796,31797],[34,681412,681414,681439],{"className":681413},[31801],[34,681415,681417,681428],{"className":681416,"style":37774},[31805],[34,681418,681419,681422],{"style":316930},[34,681420],{"className":681421,"style":31814},[31813],[34,681423,681425],{"className":681424},[31818,31819,31820,31821],[34,681426,459],{"className":681427},[31685,31821],[34,681429,681430,681433],{"style":37777},[34,681431],{"className":681432,"style":31814},[31813],[34,681434,681436],{"className":681435},[31818,31819,31820,31821],[34,681437,640],{"className":681438},[31685,31821],[34,681440,31829],{"className":681441},[31828],[34,681443,681445],{"className":681444},[31801],[34,681446,681448],{"className":681447,"style":39021},[31805],[34,681449],{},[34,681451],{"className":681452,"style":31846},[31845],[34,681454,202],{"className":681455},[31850],[34,681457],{"className":681458,"style":31846},[31845],[34,681460,681462,681465,681468,681471,681474],{"className":681461},[31676],[34,681463],{"className":681464,"style":35942},[31680],[34,681466,15932],{"className":681467},[31685],[34,681469],{"className":681470,"style":32604},[31845],[34,681472,32226],{"className":681473},[32024],[34,681475],{"className":681476,"style":32604},[31845],[34,681478,681480,681483,681551,681554,681557],{"className":681479},[31676],[34,681481],{"className":681482,"style":46727},[31680],[34,681484,681486,681489,681548],{"className":681485},[31685],[34,681487],{"className":681488},[31782,41488],[34,681490,681492],{"className":681491},[41453],[34,681493,681495,681540],{"className":681494},[31796,31797],[34,681496,681498,681537],{"className":681497},[31801],[34,681499,681501,681515,681523],{"className":681500,"style":46746},[31805],[34,681502,681503,681506],{"style":41504},[34,681504],{"className":681505,"style":41508},[31813],[34,681507,681509],{"className":681508},[31818,31819,31820,31821],[34,681510,681512],{"className":681511},[31685,31821],[34,681513,786],{"className":681514},[31685,31821],[34,681516,681517,681520],{"style":41520},[34,681518],{"className":681519,"style":41508},[31813],[34,681521],{"className":681522,"style":41528},[41527],[34,681524,681525,681528],{"style":46771},[34,681526],{"className":681527,"style":41508},[31813],[34,681529,681531],{"className":681530},[31818,31819,31820,31821],[34,681532,681534],{"className":681533},[31685,31821],[34,681535,640],{"className":681536},[31685,31821],[34,681538,31829],{"className":681539},[31828],[34,681541,681543],{"className":681542},[31801],[34,681544,681546],{"className":681545,"style":41619},[31805],[34,681547],{},[34,681549],{"className":681550},[31690,41488],[34,681552],{"className":681553,"style":31846},[31845],[34,681555,202],{"className":681556},[31850],[34,681558],{"className":681559,"style":31846},[31845],[34,681561,681563,681566],{"className":681562},[31676],[34,681564],{"className":681565,"style":46727},[31680],[34,681567,681569,681572,681631],{"className":681568},[31685],[34,681570],{"className":681571},[31782,41488],[34,681573,681575],{"className":681574},[41453],[34,681576,681578,681623],{"className":681577},[31796,31797],[34,681579,681581,681620],{"className":681580},[31801],[34,681582,681584,681598,681606],{"className":681583,"style":46746},[31805],[34,681585,681586,681589],{"style":41504},[34,681587],{"className":681588,"style":41508},[31813],[34,681590,681592],{"className":681591},[31818,31819,31820,31821],[34,681593,681595],{"className":681594},[31685,31821],[34,681596,786],{"className":681597},[31685,31821],[34,681599,681600,681603],{"style":41520},[34,681601],{"className":681602,"style":41508},[31813],[34,681604],{"className":681605,"style":41528},[41527],[34,681607,681608,681611],{"style":46771},[34,681609],{"className":681610,"style":41508},[31813],[34,681612,681614],{"className":681613},[31818,31819,31820,31821],[34,681615,681617],{"className":681616},[31685,31821],[34,681618,39620],{"className":681619},[31685,31821],[34,681621,31829],{"className":681622},[31828],[34,681624,681626],{"className":681625},[31801],[34,681627,681629],{"className":681628,"style":41619},[31805],[34,681630],{},[34,681632],{"className":681633},[31690,41488],[15,681635,681636],{},[2627,681637,681638,681639,54852,681673,53834,681724],{},"第 3 段 ",[34,681640,681642,681658],{"className":681641},[31642],[34,681643,681645],{"className":681644},[31646],[31648,681646,681647],{"xmlns":31650},[31652,681648,681649,681655],{},[31655,681650,681651,681653],{},[31658,681652,47384],{},[31658,681654,91994],{},[31665,681656,681657],{"encoding":31667},"CD",[34,681659,681661],{"className":681660,"ariaHidden":758},[31672],[34,681662,681664,681667,681670],{"className":681663},[31676],[34,681665],{"className":681666,"style":33218},[31680],[34,681668,47384],{"className":681669,"style":38706},[31685,31686],[34,681671,91994],{"className":681672,"style":34452},[31685,31686],[34,681674,681676,681697],{"className":681675},[31642],[34,681677,681679],{"className":681678},[31646],[31648,681680,681681],{"xmlns":31650},[31652,681682,681683,681695],{},[31655,681684,681685,681687,681689,681691,681693],{},[31662,681686,456],{"stretchy":702},[31749,681688,768],{},[31662,681690,31960],{"separator":758},[31749,681692,640],{},[31662,681694,2557],{"stretchy":702},[31665,681696,680440],{"encoding":31667},[34,681698,681700],{"className":681699,"ariaHidden":758},[31672],[34,681701,681703,681706,681709,681712,681715,681718,681721],{"className":681702},[31676],[34,681704],{"className":681705,"style":31774},[31680],[34,681707,456],{"className":681708},[31782],[34,681710,768],{"className":681711},[31685],[34,681713,31960],{"className":681714},[32119],[34,681716],{"className":681717,"style":32123},[31845],[34,681719,640],{"className":681720},[31685],[34,681722,2557],{"className":681723},[31690],[34,681725,681727,681748],{"className":681726},[31642],[34,681728,681730],{"className":681729},[31646],[31648,681731,681732],{"xmlns":31650},[31652,681733,681734,681746],{},[31655,681735,681736,681738,681740,681742,681744],{},[31662,681737,456],{"stretchy":702},[31749,681739,640],{},[31662,681741,31960],{"separator":758},[31749,681743,640],{},[31662,681745,2557],{"stretchy":702},[31665,681747,389256],{"encoding":31667},[34,681749,681751],{"className":681750,"ariaHidden":758},[31672],[34,681752,681754,681757,681760,681763,681766,681769,681772],{"className":681753},[31676],[34,681755],{"className":681756,"style":31774},[31680],[34,681758,456],{"className":681759},[31782],[34,681761,640],{"className":681762},[31685],[34,681764,31960],{"className":681765},[32119],[34,681767],{"className":681768,"style":32123},[31845],[34,681770,640],{"className":681771},[31685],[34,681773,2557],{"className":681774},[31690],[105,681776,681777,681891,681980],{},[108,681778,681779,2664,681781,347],{},[2627,681780,670318],{},[34,681782,681784,681816],{"className":681783},[31642],[34,681785,681787],{"className":681786},[31646],[31648,681788,681789],{"xmlns":31650},[31652,681790,681791,681813],{},[31655,681792,681793,681795,681797,681799,681801,681803,681805,681807,681809,681811],{},[31658,681794,67764],{},[31662,681796,202],{},[31749,681798,640],{},[39361,681800,262622],{},[31662,681802,262625],{},[39361,681804,262622],{},[31658,681806,46900],{"mathvariant":38079},[31658,681808,67764],{},[31662,681810,202],{},[31749,681812,459],{},[31665,681814,681815],{"encoding":31667},"y = 1 \\implies \\mathrm{d}y = 0",[34,681817,681819,681837,681861,681882],{"className":681818,"ariaHidden":758},[31672],[34,681820,681822,681825,681828,681831,681834],{"className":681821},[31676],[34,681823],{"className":681824,"style":38774},[31680],[34,681826,67764],{"className":681827,"style":32062},[31685,31686],[34,681829],{"className":681830,"style":31846},[31845],[34,681832,202],{"className":681833},[31850],[34,681835],{"className":681836,"style":31846},[31845],[34,681838,681840,681843,681846,681849,681852,681855,681858],{"className":681839},[31676],[34,681841],{"className":681842,"style":262700},[31680],[34,681844,640],{"className":681845},[31685],[34,681847],{"className":681848,"style":31846},[31845],[34,681850],{"className":681851,"style":31846},[31845],[34,681853,262625],{"className":681854},[31850],[34,681856],{"className":681857,"style":31846},[31845],[34,681859],{"className":681860,"style":31846},[31845],[34,681862,681864,681867,681870,681873,681876,681879],{"className":681863},[31676],[34,681865],{"className":681866,"style":43217},[31680],[34,681868,46900],{"className":681869},[31685,207923],[34,681871,67764],{"className":681872,"style":32062},[31685,31686],[34,681874],{"className":681875,"style":31846},[31845],[34,681877,202],{"className":681878},[31850],[34,681880],{"className":681881,"style":31846},[31845],[34,681883,681885,681888],{"className":681884},[31676],[34,681886],{"className":681887,"style":32617},[31680],[34,681889,459],{"className":681890},[31685],[108,681892,681893,2664,681895,138176,681923,53834,681951,681979],{},[2627,681894,679310],{},[34,681896,681898,681911],{"className":681897},[31642],[34,681899,681901],{"className":681900},[31646],[31648,681902,681903],{"xmlns":31650},[31652,681904,681905,681909],{},[31655,681906,681907],{},[31658,681908,32936],{},[31665,681910,32936],{"encoding":31667},[34,681912,681914],{"className":681913,"ariaHidden":758},[31672],[34,681915,681917,681920],{"className":681916},[31676],[34,681918],{"className":681919,"style":32527},[31680],[34,681921,32936],{"className":681922},[31685,31686],[34,681924,681926,681939],{"className":681925},[31642],[34,681927,681929],{"className":681928},[31646],[31648,681930,681931],{"xmlns":31650},[31652,681932,681933,681937],{},[31655,681934,681935],{},[31749,681936,768],{},[31665,681938,768],{"encoding":31667},[34,681940,681942],{"className":681941,"ariaHidden":758},[31672],[34,681943,681945,681948],{"className":681944},[31676],[34,681946],{"className":681947,"style":32617},[31680],[34,681949,768],{"className":681950},[31685],[34,681952,681954,681967],{"className":681953},[31642],[34,681955,681957],{"className":681956},[31646],[31648,681958,681959],{"xmlns":31650},[31652,681960,681961,681965],{},[31655,681962,681963],{},[31749,681964,640],{},[31665,681966,640],{"encoding":31667},[34,681968,681970],{"className":681969,"ariaHidden":758},[31672],[34,681971,681973,681976],{"className":681972},[31676],[34,681974],{"className":681975,"style":32617},[31680],[34,681977,640],{"className":681978},[31685]," （注意方向！下限是2，上限是1）。",[108,681981,681982,33183,681984,682410,682943],{},[2627,681983,676907],{},[34,681985,681987,682073],{"className":681986},[31642],[34,681988,681990],{"className":681989},[31646],[31648,681991,681992],{"xmlns":31650},[31652,681993,681994,682070],{},[31655,681995,681996,682006,682008,682016,682018,682024,682026,682032,682034,682036,682038,682040,682042,682050,682052,682058,682060,682062,682064,682066,682068],{},[31743,681997,681998,682000],{},[31658,681999,106063],{},[31655,682001,682002,682004],{},[31658,682003,47384],{},[31658,682005,91994],{},[31662,682007,202],{},[38846,682009,682010,682012,682014],{},[31662,682011,546295],{},[31749,682013,768],{},[31749,682015,640],{},[31662,682017,456],{"stretchy":702},[37624,682019,682020,682022],{},[31658,682021,32936],{},[31749,682023,768],{},[31662,682025,223],{},[37624,682027,682028,682030],{},[31749,682029,640],{},[31749,682031,768],{},[31662,682033,2557],{"stretchy":702},[39361,682035,39363],{},[31658,682037,46900],{"mathvariant":38079},[31658,682039,32936],{},[31662,682041,202],{},[38846,682043,682044,682046,682048],{},[31662,682045,546295],{},[31749,682047,768],{},[31749,682049,640],{},[31662,682051,456],{"stretchy":702},[37624,682053,682054,682056],{},[31658,682055,32936],{},[31749,682057,768],{},[31662,682059,223],{},[31749,682061,640],{},[31662,682063,2557],{"stretchy":702},[39361,682065,39363],{},[31658,682067,46900],{"mathvariant":38079},[31658,682069,32936],{},[31665,682071,682072],{"encoding":31667},"I_{CD} = \\int_2^1 (x^2 + 1^2) \\, \\mathrm{d}x = \\int_2^1 (x^2 + 1) \\, \\mathrm{d}x",[34,682074,682076,682137,682235,682291,682389],{"className":682075,"ariaHidden":758},[31672],[34,682077,682079,682082,682128,682131,682134],{"className":682078},[31676],[34,682080],{"className":682081,"style":70915},[31680],[34,682083,682085,682088],{"className":682084},[31685],[34,682086,106063],{"className":682087,"style":67482},[31685,31686],[34,682089,682091],{"className":682090},[31792],[34,682092,682094,682120],{"className":682093},[31796,31797],[34,682095,682097,682117],{"className":682096},[31801],[34,682098,682100],{"className":682099,"style":116558},[31805],[34,682101,682102,682105],{"style":217742},[34,682103],{"className":682104,"style":31814},[31813],[34,682106,682108],{"className":682107},[31818,31819,31820,31821],[34,682109,682111,682114],{"className":682110},[31685,31821],[34,682112,47384],{"className":682113,"style":38706},[31685,31686,31821],[34,682115,91994],{"className":682116,"style":34452},[31685,31686,31821],[34,682118,31829],{"className":682119},[31828],[34,682121,682123],{"className":682122},[31801],[34,682124,682126],{"className":682125,"style":31836},[31805],[34,682127],{},[34,682129],{"className":682130,"style":31846},[31845],[34,682132,202],{"className":682133},[31850],[34,682135],{"className":682136,"style":31846},[31845],[34,682138,682140,682143,682194,682197,682226,682229,682232],{"className":682139},[31676],[34,682141],{"className":682142,"style":563309},[31680],[34,682144,682146,682149],{"className":682145},[32767],[34,682147,546295],{"className":682148,"style":546331},[32767,131249,156560],[34,682150,682152],{"className":682151},[31792],[34,682153,682155,682186],{"className":682154},[31796,31797],[34,682156,682158,682183],{"className":682157},[31801],[34,682159,682161,682172],{"className":682160,"style":563328},[31805],[34,682162,682163,682166],{"style":546347},[34,682164],{"className":682165,"style":31814},[31813],[34,682167,682169],{"className":682168},[31818,31819,31820,31821],[34,682170,768],{"className":682171},[31685,31821],[34,682173,682174,682177],{"style":546362},[34,682175],{"className":682176,"style":31814},[31813],[34,682178,682180],{"className":682179},[31818,31819,31820,31821],[34,682181,640],{"className":682182},[31685,31821],[34,682184,31829],{"className":682185},[31828],[34,682187,682189],{"className":682188},[31801],[34,682190,682192],{"className":682191,"style":546384},[31805],[34,682193],{},[34,682195,456],{"className":682196},[31782],[34,682198,682200,682203],{"className":682199},[31685],[34,682201,32936],{"className":682202},[31685,31686],[34,682204,682206],{"className":682205},[31792],[34,682207,682209],{"className":682208},[31796],[34,682210,682212],{"className":682211},[31801],[34,682213,682215],{"className":682214,"style":37774},[31805],[34,682216,682217,682220],{"style":37777},[34,682218],{"className":682219,"style":31814},[31813],[34,682221,682223],{"className":682222},[31818,31819,31820,31821],[34,682224,768],{"className":682225},[31685,31821],[34,682227],{"className":682228,"style":32604},[31845],[34,682230,223],{"className":682231},[32024],[34,682233],{"className":682234,"style":32604},[31845],[34,682236,682238,682241,682270,682273,682276,682279,682282,682285,682288],{"className":682237},[31676],[34,682239],{"className":682240,"style":37755},[31680],[34,682242,682244,682247],{"className":682243},[31685],[34,682245,640],{"className":682246},[31685],[34,682248,682250],{"className":682249},[31792],[34,682251,682253],{"className":682252},[31796],[34,682254,682256],{"className":682255},[31801],[34,682257,682259],{"className":682258,"style":37774},[31805],[34,682260,682261,682264],{"style":37777},[34,682262],{"className":682263,"style":31814},[31813],[34,682265,682267],{"className":682266},[31818,31819,31820,31821],[34,682268,768],{"className":682269},[31685,31821],[34,682271,2557],{"className":682272},[31690],[34,682274],{"className":682275,"style":32123},[31845],[34,682277,46900],{"className":682278},[31685,207923],[34,682280,32936],{"className":682281},[31685,31686],[34,682283],{"className":682284,"style":31846},[31845],[34,682286,202],{"className":682287},[31850],[34,682289],{"className":682290,"style":31846},[31845],[34,682292,682294,682297,682348,682351,682380,682383,682386],{"className":682293},[31676],[34,682295],{"className":682296,"style":563309},[31680],[34,682298,682300,682303],{"className":682299},[32767],[34,682301,546295],{"className":682302,"style":546331},[32767,131249,156560],[34,682304,682306],{"className":682305},[31792],[34,682307,682309,682340],{"className":682308},[31796,31797],[34,682310,682312,682337],{"className":682311},[31801],[34,682313,682315,682326],{"className":682314,"style":563328},[31805],[34,682316,682317,682320],{"style":546347},[34,682318],{"className":682319,"style":31814},[31813],[34,682321,682323],{"className":682322},[31818,31819,31820,31821],[34,682324,768],{"className":682325},[31685,31821],[34,682327,682328,682331],{"style":546362},[34,682329],{"className":682330,"style":31814},[31813],[34,682332,682334],{"className":682333},[31818,31819,31820,31821],[34,682335,640],{"className":682336},[31685,31821],[34,682338,31829],{"className":682339},[31828],[34,682341,682343],{"className":682342},[31801],[34,682344,682346],{"className":682345,"style":546384},[31805],[34,682347],{},[34,682349,456],{"className":682350},[31782],[34,682352,682354,682357],{"className":682353},[31685],[34,682355,32936],{"className":682356},[31685,31686],[34,682358,682360],{"className":682359},[31792],[34,682361,682363],{"className":682362},[31796],[34,682364,682366],{"className":682365},[31801],[34,682367,682369],{"className":682368,"style":37774},[31805],[34,682370,682371,682374],{"style":37777},[34,682372],{"className":682373,"style":31814},[31813],[34,682375,682377],{"className":682376},[31818,31819,31820,31821],[34,682378,768],{"className":682379},[31685,31821],[34,682381],{"className":682382,"style":32604},[31845],[34,682384,223],{"className":682385},[32024],[34,682387],{"className":682388,"style":32604},[31845],[34,682390,682392,682395,682398,682401,682404,682407],{"className":682391},[31676],[34,682393],{"className":682394,"style":31774},[31680],[34,682396,640],{"className":682397},[31685],[34,682399,2557],{"className":682400},[31690],[34,682402],{"className":682403,"style":32123},[31845],[34,682405,46900],{"className":682406},[31685,207923],[34,682408,32936],{"className":682409},[31685,31686],[34,682411,682413,682493],{"className":682412},[31642],[34,682414,682416],{"className":682415},[31646],[31648,682417,682418],{"xmlns":31650},[31652,682419,682420,682490],{},[31655,682421,682422,682432,682434,682436,682446,682448,682450,682458,682460,682462,682468,682470,682472,682474,682476,682478,682484,682486,682488],{},[31743,682423,682424,682426],{},[31658,682425,106063],{},[31655,682427,682428,682430],{},[31658,682429,47384],{},[31658,682431,91994],{},[31662,682433,202],{},[31662,682435,15972],{"stretchy":702},[41453,682437,682438,682444],{},[37624,682439,682440,682442],{},[31658,682441,32936],{},[31749,682443,786],{},[31749,682445,786],{},[31662,682447,223],{},[31658,682449,32936],{},[38846,682451,682452,682454,682456],{},[31662,682453,32712],{"stretchy":702},[31749,682455,768],{},[31749,682457,640],{},[31662,682459,202],{},[31662,682461,456],{"stretchy":702},[41453,682463,682464,682466],{},[31749,682465,640],{},[31749,682467,786],{},[31662,682469,223],{},[31749,682471,640],{},[31662,682473,2557],{"stretchy":702},[31662,682475,32226],{},[31662,682477,456],{"stretchy":702},[41453,682479,682480,682482],{},[31749,682481,60525],{},[31749,682483,786],{},[31662,682485,223],{},[31749,682487,768],{},[31662,682489,2557],{"stretchy":702},[31665,682491,682492],{"encoding":31667},"I_{CD} = [\\frac{x^3}{3} + x]_2^1 = (\\frac{1}{3} + 1) - (\\frac{8}{3} + 2)",[34,682494,682496,682557,682669,682738,682824,682845,682931],{"className":682495,"ariaHidden":758},[31672],[34,682497,682499,682502,682548,682551,682554],{"className":682498},[31676],[34,682500],{"className":682501,"style":70915},[31680],[34,682503,682505,682508],{"className":682504},[31685],[34,682506,106063],{"className":682507,"style":67482},[31685,31686],[34,682509,682511],{"className":682510},[31792],[34,682512,682514,682540],{"className":682513},[31796,31797],[34,682515,682517,682537],{"className":682516},[31801],[34,682518,682520],{"className":682519,"style":116558},[31805],[34,682521,682522,682525],{"style":217742},[34,682523],{"className":682524,"style":31814},[31813],[34,682526,682528],{"className":682527},[31818,31819,31820,31821],[34,682529,682531,682534],{"className":682530},[31685,31821],[34,682532,47384],{"className":682533,"style":38706},[31685,31686,31821],[34,682535,91994],{"className":682536,"style":34452},[31685,31686,31821],[34,682538,31829],{"className":682539},[31828],[34,682541,682543],{"className":682542},[31801],[34,682544,682546],{"className":682545,"style":31836},[31805],[34,682547],{},[34,682549],{"className":682550,"style":31846},[31845],[34,682552,202],{"className":682553},[31850],[34,682555],{"className":682556,"style":31846},[31845],[34,682558,682560,682563,682566,682660,682663,682666],{"className":682559},[31676],[34,682561],{"className":682562,"style":637019},[31680],[34,682564,15972],{"className":682565},[31782],[34,682567,682569,682572,682657],{"className":682568},[31685],[34,682570],{"className":682571},[31782,41488],[34,682573,682575],{"className":682574},[41453],[34,682576,682578,682649],{"className":682577},[31796,31797],[34,682579,682581,682646],{"className":682580},[31801],[34,682582,682584,682598,682606],{"className":682583,"style":603956},[31805],[34,682585,682586,682589],{"style":41504},[34,682587],{"className":682588,"style":41508},[31813],[34,682590,682592],{"className":682591},[31818,31819,31820,31821],[34,682593,682595],{"className":682594},[31685,31821],[34,682596,786],{"className":682597},[31685,31821],[34,682599,682600,682603],{"style":41520},[34,682601],{"className":682602,"style":41508},[31813],[34,682604],{"className":682605,"style":41528},[41527],[34,682607,682608,682611],{"style":46771},[34,682609],{"className":682610,"style":41508},[31813],[34,682612,682614],{"className":682613},[31818,31819,31820,31821],[34,682615,682617],{"className":682616},[31685,31821],[34,682618,682620,682623],{"className":682619},[31685,31821],[34,682621,32936],{"className":682622},[31685,31686,31821],[34,682624,682626],{"className":682625},[31792],[34,682627,682629],{"className":682628},[31796],[34,682630,682632],{"className":682631},[31801],[34,682633,682635],{"className":682634,"style":144123},[31805],[34,682636,682637,682640],{"style":552391},[34,682638],{"className":682639,"style":43406},[31813],[34,682641,682643],{"className":682642},[31818,43410,43411,31821],[34,682644,786],{"className":682645},[31685,31821],[34,682647,31829],{"className":682648},[31828],[34,682650,682652],{"className":682651},[31801],[34,682653,682655],{"className":682654,"style":41619},[31805],[34,682656],{},[34,682658],{"className":682659},[31690,41488],[34,682661],{"className":682662,"style":32604},[31845],[34,682664,223],{"className":682665},[32024],[34,682667],{"className":682668,"style":32604},[31845],[34,682670,682672,682675,682678,682729,682732,682735],{"className":682671},[31676],[34,682673],{"className":682674,"style":37755},[31680],[34,682676,32936],{"className":682677},[31685,31686],[34,682679,682681,682684],{"className":682680},[31690],[34,682682,32712],{"className":682683},[31690],[34,682685,682687],{"className":682686},[31792],[34,682688,682690,682721],{"className":682689},[31796,31797],[34,682691,682693,682718],{"className":682692},[31801],[34,682694,682696,682707],{"className":682695,"style":37774},[31805],[34,682697,682698,682701],{"style":316930},[34,682699],{"className":682700,"style":31814},[31813],[34,682702,682704],{"className":682703},[31818,31819,31820,31821],[34,682705,768],{"className":682706},[31685,31821],[34,682708,682709,682712],{"style":37777},[34,682710],{"className":682711,"style":31814},[31813],[34,682713,682715],{"className":682714},[31818,31819,31820,31821],[34,682716,640],{"className":682717},[31685,31821],[34,682719,31829],{"className":682720},[31828],[34,682722,682724],{"className":682723},[31801],[34,682725,682727],{"className":682726,"style":39021},[31805],[34,682728],{},[34,682730],{"className":682731,"style":31846},[31845],[34,682733,202],{"className":682734},[31850],[34,682736],{"className":682737,"style":31846},[31845],[34,682739,682741,682744,682747,682815,682818,682821],{"className":682740},[31676],[34,682742],{"className":682743,"style":46727},[31680],[34,682745,456],{"className":682746},[31782],[34,682748,682750,682753,682812],{"className":682749},[31685],[34,682751],{"className":682752},[31782,41488],[34,682754,682756],{"className":682755},[41453],[34,682757,682759,682804],{"className":682758},[31796,31797],[34,682760,682762,682801],{"className":682761},[31801],[34,682763,682765,682779,682787],{"className":682764,"style":46746},[31805],[34,682766,682767,682770],{"style":41504},[34,682768],{"className":682769,"style":41508},[31813],[34,682771,682773],{"className":682772},[31818,31819,31820,31821],[34,682774,682776],{"className":682775},[31685,31821],[34,682777,786],{"className":682778},[31685,31821],[34,682780,682781,682784],{"style":41520},[34,682782],{"className":682783,"style":41508},[31813],[34,682785],{"className":682786,"style":41528},[41527],[34,682788,682789,682792],{"style":46771},[34,682790],{"className":682791,"style":41508},[31813],[34,682793,682795],{"className":682794},[31818,31819,31820,31821],[34,682796,682798],{"className":682797},[31685,31821],[34,682799,640],{"className":682800},[31685,31821],[34,682802,31829],{"className":682803},[31828],[34,682805,682807],{"className":682806},[31801],[34,682808,682810],{"className":682809,"style":41619},[31805],[34,682811],{},[34,682813],{"className":682814},[31690,41488],[34,682816],{"className":682817,"style":32604},[31845],[34,682819,223],{"className":682820},[32024],[34,682822],{"className":682823,"style":32604},[31845],[34,682825,682827,682830,682833,682836,682839,682842],{"className":682826},[31676],[34,682828],{"className":682829,"style":31774},[31680],[34,682831,640],{"className":682832},[31685],[34,682834,2557],{"className":682835},[31690],[34,682837],{"className":682838,"style":32604},[31845],[34,682840,32226],{"className":682841},[32024],[34,682843],{"className":682844,"style":32604},[31845],[34,682846,682848,682851,682854,682922,682925,682928],{"className":682847},[31676],[34,682849],{"className":682850,"style":46727},[31680],[34,682852,456],{"className":682853},[31782],[34,682855,682857,682860,682919],{"className":682856},[31685],[34,682858],{"className":682859},[31782,41488],[34,682861,682863],{"className":682862},[41453],[34,682864,682866,682911],{"className":682865},[31796,31797],[34,682867,682869,682908],{"className":682868},[31801],[34,682870,682872,682886,682894],{"className":682871,"style":46746},[31805],[34,682873,682874,682877],{"style":41504},[34,682875],{"className":682876,"style":41508},[31813],[34,682878,682880],{"className":682879},[31818,31819,31820,31821],[34,682881,682883],{"className":682882},[31685,31821],[34,682884,786],{"className":682885},[31685,31821],[34,682887,682888,682891],{"style":41520},[34,682889],{"className":682890,"style":41508},[31813],[34,682892],{"className":682893,"style":41528},[41527],[34,682895,682896,682899],{"style":46771},[34,682897],{"className":682898,"style":41508},[31813],[34,682900,682902],{"className":682901},[31818,31819,31820,31821],[34,682903,682905],{"className":682904},[31685,31821],[34,682906,60525],{"className":682907},[31685,31821],[34,682909,31829],{"className":682910},[31828],[34,682912,682914],{"className":682913},[31801],[34,682915,682917],{"className":682916,"style":41619},[31805],[34,682918],{},[34,682920],{"className":682921},[31690,41488],[34,682923],{"className":682924,"style":32604},[31845],[34,682926,223],{"className":682927},[32024],[34,682929],{"className":682930,"style":32604},[31845],[34,682932,682934,682937,682940],{"className":682933},[31676],[34,682935],{"className":682936,"style":31774},[31680],[34,682938,768],{"className":682939},[31685],[34,682941,2557],{"className":682942},[31690],[34,682944,682946,682994],{"className":682945},[31642],[34,682947,682949],{"className":682948},[31646],[31648,682950,682951],{"xmlns":31650},[31652,682952,682953,682991],{},[31655,682954,682955,682965,682967,682973,682975,682981,682983,682985],{},[31743,682956,682957,682959],{},[31658,682958,106063],{},[31655,682960,682961,682963],{},[31658,682962,47384],{},[31658,682964,91994],{},[31662,682966,202],{},[41453,682968,682969,682971],{},[31749,682970,15932],{},[31749,682972,786],{},[31662,682974,32226],{},[41453,682976,682977,682979],{},[31749,682978,164756],{},[31749,682980,786],{},[31662,682982,202],{},[31662,682984,32226],{},[41453,682986,682987,682989],{},[31749,682988,3244],{},[31749,682990,786],{},[31665,682992,682993],{"encoding":31667},"I_{CD} = \\frac{4}{3} - \\frac{14}{3} = -\\frac{10}{3}",[34,682995,682997,683058,683141,683224],{"className":682996,"ariaHidden":758},[31672],[34,682998,683000,683003,683049,683052,683055],{"className":682999},[31676],[34,683001],{"className":683002,"style":70915},[31680],[34,683004,683006,683009],{"className":683005},[31685],[34,683007,106063],{"className":683008,"style":67482},[31685,31686],[34,683010,683012],{"className":683011},[31792],[34,683013,683015,683041],{"className":683014},[31796,31797],[34,683016,683018,683038],{"className":683017},[31801],[34,683019,683021],{"className":683020,"style":116558},[31805],[34,683022,683023,683026],{"style":217742},[34,683024],{"className":683025,"style":31814},[31813],[34,683027,683029],{"className":683028},[31818,31819,31820,31821],[34,683030,683032,683035],{"className":683031},[31685,31821],[34,683033,47384],{"className":683034,"style":38706},[31685,31686,31821],[34,683036,91994],{"className":683037,"style":34452},[31685,31686,31821],[34,683039,31829],{"className":683040},[31828],[34,683042,683044],{"className":683043},[31801],[34,683045,683047],{"className":683046,"style":31836},[31805],[34,683048],{},[34,683050],{"className":683051,"style":31846},[31845],[34,683053,202],{"className":683054},[31850],[34,683056],{"className":683057,"style":31846},[31845],[34,683059,683061,683064,683132,683135,683138],{"className":683060},[31676],[34,683062],{"className":683063,"style":46727},[31680],[34,683065,683067,683070,683129],{"className":683066},[31685],[34,683068],{"className":683069},[31782,41488],[34,683071,683073],{"className":683072},[41453],[34,683074,683076,683121],{"className":683075},[31796,31797],[34,683077,683079,683118],{"className":683078},[31801],[34,683080,683082,683096,683104],{"className":683081,"style":46746},[31805],[34,683083,683084,683087],{"style":41504},[34,683085],{"className":683086,"style":41508},[31813],[34,683088,683090],{"className":683089},[31818,31819,31820,31821],[34,683091,683093],{"className":683092},[31685,31821],[34,683094,786],{"className":683095},[31685,31821],[34,683097,683098,683101],{"style":41520},[34,683099],{"className":683100,"style":41508},[31813],[34,683102],{"className":683103,"style":41528},[41527],[34,683105,683106,683109],{"style":46771},[34,683107],{"className":683108,"style":41508},[31813],[34,683110,683112],{"className":683111},[31818,31819,31820,31821],[34,683113,683115],{"className":683114},[31685,31821],[34,683116,15932],{"className":683117},[31685,31821],[34,683119,31829],{"className":683120},[31828],[34,683122,683124],{"className":683123},[31801],[34,683125,683127],{"className":683126,"style":41619},[31805],[34,683128],{},[34,683130],{"className":683131},[31690,41488],[34,683133],{"className":683134,"style":32604},[31845],[34,683136,32226],{"className":683137},[32024],[34,683139],{"className":683140,"style":32604},[31845],[34,683142,683144,683147,683215,683218,683221],{"className":683143},[31676],[34,683145],{"className":683146,"style":46727},[31680],[34,683148,683150,683153,683212],{"className":683149},[31685],[34,683151],{"className":683152},[31782,41488],[34,683154,683156],{"className":683155},[41453],[34,683157,683159,683204],{"className":683158},[31796,31797],[34,683160,683162,683201],{"className":683161},[31801],[34,683163,683165,683179,683187],{"className":683164,"style":46746},[31805],[34,683166,683167,683170],{"style":41504},[34,683168],{"className":683169,"style":41508},[31813],[34,683171,683173],{"className":683172},[31818,31819,31820,31821],[34,683174,683176],{"className":683175},[31685,31821],[34,683177,786],{"className":683178},[31685,31821],[34,683180,683181,683184],{"style":41520},[34,683182],{"className":683183,"style":41508},[31813],[34,683185],{"className":683186,"style":41528},[41527],[34,683188,683189,683192],{"style":46771},[34,683190],{"className":683191,"style":41508},[31813],[34,683193,683195],{"className":683194},[31818,31819,31820,31821],[34,683196,683198],{"className":683197},[31685,31821],[34,683199,164756],{"className":683200},[31685,31821],[34,683202,31829],{"className":683203},[31828],[34,683205,683207],{"className":683206},[31801],[34,683208,683210],{"className":683209,"style":41619},[31805],[34,683211],{},[34,683213],{"className":683214},[31690,41488],[34,683216],{"className":683217,"style":31846},[31845],[34,683219,202],{"className":683220},[31850],[34,683222],{"className":683223,"style":31846},[31845],[34,683225,683227,683230,683233],{"className":683226},[31676],[34,683228],{"className":683229,"style":46727},[31680],[34,683231,32226],{"className":683232},[31685],[34,683234,683236,683239,683298],{"className":683235},[31685],[34,683237],{"className":683238},[31782,41488],[34,683240,683242],{"className":683241},[41453],[34,683243,683245,683290],{"className":683244},[31796,31797],[34,683246,683248,683287],{"className":683247},[31801],[34,683249,683251,683265,683273],{"className":683250,"style":46746},[31805],[34,683252,683253,683256],{"style":41504},[34,683254],{"className":683255,"style":41508},[31813],[34,683257,683259],{"className":683258},[31818,31819,31820,31821],[34,683260,683262],{"className":683261},[31685,31821],[34,683263,786],{"className":683264},[31685,31821],[34,683266,683267,683270],{"style":41520},[34,683268],{"className":683269,"style":41508},[31813],[34,683271],{"className":683272,"style":41528},[41527],[34,683274,683275,683278],{"style":46771},[34,683276],{"className":683277,"style":41508},[31813],[34,683279,683281],{"className":683280},[31818,31819,31820,31821],[34,683282,683284],{"className":683283},[31685,31821],[34,683285,3244],{"className":683286},[31685,31821],[34,683288,31829],{"className":683289},[31828],[34,683291,683293],{"className":683292},[31801],[34,683294,683296],{"className":683295,"style":41619},[31805],[34,683297],{},[34,683299],{"className":683300},[31690,41488],[15,683302,683303],{},[2627,683304,683305,683306,54852,683340,53834,683391],{},"第 4 段 ",[34,683307,683309,683325],{"className":683308},[31642],[34,683310,683312],{"className":683311},[31646],[31648,683313,683314],{"xmlns":31650},[31652,683315,683316,683322],{},[31655,683317,683318,683320],{},[31658,683319,91994],{},[31658,683321,54151],{},[31665,683323,683324],{"encoding":31667},"DA",[34,683326,683328],{"className":683327,"ariaHidden":758},[31672],[34,683329,683331,683334,683337],{"className":683330},[31676],[34,683332],{"className":683333,"style":33218},[31680],[34,683335,91994],{"className":683336,"style":34452},[31685,31686],[34,683338,54151],{"className":683339},[31685,31686],[34,683341,683343,683364],{"className":683342},[31642],[34,683344,683346],{"className":683345},[31646],[31648,683347,683348],{"xmlns":31650},[31652,683349,683350,683362],{},[31655,683351,683352,683354,683356,683358,683360],{},[31662,683353,456],{"stretchy":702},[31749,683355,640],{},[31662,683357,31960],{"separator":758},[31749,683359,640],{},[31662,683361,2557],{"stretchy":702},[31665,683363,389256],{"encoding":31667},[34,683365,683367],{"className":683366,"ariaHidden":758},[31672],[34,683368,683370,683373,683376,683379,683382,683385,683388],{"className":683369},[31676],[34,683371],{"className":683372,"style":31774},[31680],[34,683374,456],{"className":683375},[31782],[34,683377,640],{"className":683378},[31685],[34,683380,31960],{"className":683381},[32119],[34,683383],{"className":683384,"style":32123},[31845],[34,683386,640],{"className":683387},[31685],[34,683389,2557],{"className":683390},[31690],[34,683392,683394,683415],{"className":683393},[31642],[34,683395,683397],{"className":683396},[31646],[31648,683398,683399],{"xmlns":31650},[31652,683400,683401,683413],{},[31655,683402,683403,683405,683407,683409,683411],{},[31662,683404,456],{"stretchy":702},[31749,683406,640],{},[31662,683408,31960],{"separator":758},[31749,683410,459],{},[31662,683412,2557],{"stretchy":702},[31665,683414,381896],{"encoding":31667},[34,683416,683418],{"className":683417,"ariaHidden":758},[31672],[34,683419,683421,683424,683427,683430,683433,683436,683439],{"className":683420},[31676],[34,683422],{"className":683423,"style":31774},[31680],[34,683425,456],{"className":683426},[31782],[34,683428,640],{"className":683429},[31685],[34,683431,31960],{"className":683432},[32119],[34,683434],{"className":683435,"style":32123},[31845],[34,683437,459],{"className":683438},[31685],[34,683440,2557],{"className":683441},[31690],[105,683443,683444,683558,683646],{},[108,683445,683446,2664,683448,347],{},[2627,683447,670318],{},[34,683449,683451,683483],{"className":683450},[31642],[34,683452,683454],{"className":683453},[31646],[31648,683455,683456],{"xmlns":31650},[31652,683457,683458,683480],{},[31655,683459,683460,683462,683464,683466,683468,683470,683472,683474,683476,683478],{},[31658,683461,32936],{},[31662,683463,202],{},[31749,683465,640],{},[39361,683467,262622],{},[31662,683469,262625],{},[39361,683471,262622],{},[31658,683473,46900],{"mathvariant":38079},[31658,683475,32936],{},[31662,683477,202],{},[31749,683479,459],{},[31665,683481,683482],{"encoding":31667},"x = 1 \\implies \\mathrm{d}x = 0",[34,683484,683486,683504,683528,683549],{"className":683485,"ariaHidden":758},[31672],[34,683487,683489,683492,683495,683498,683501],{"className":683488},[31676],[34,683490],{"className":683491,"style":32527},[31680],[34,683493,32936],{"className":683494},[31685,31686],[34,683496],{"className":683497,"style":31846},[31845],[34,683499,202],{"className":683500},[31850],[34,683502],{"className":683503,"style":31846},[31845],[34,683505,683507,683510,683513,683516,683519,683522,683525],{"className":683506},[31676],[34,683508],{"className":683509,"style":262700},[31680],[34,683511,640],{"className":683512},[31685],[34,683514],{"className":683515,"style":31846},[31845],[34,683517],{"className":683518,"style":31846},[31845],[34,683520,262625],{"className":683521},[31850],[34,683523],{"className":683524,"style":31846},[31845],[34,683526],{"className":683527,"style":31846},[31845],[34,683529,683531,683534,683537,683540,683543,683546],{"className":683530},[31676],[34,683532],{"className":683533,"style":31681},[31680],[34,683535,46900],{"className":683536},[31685,207923],[34,683538,32936],{"className":683539},[31685,31686],[34,683541],{"className":683542,"style":31846},[31845],[34,683544,202],{"className":683545},[31850],[34,683547],{"className":683548,"style":31846},[31845],[34,683550,683552,683555],{"className":683551},[31676],[34,683553],{"className":683554,"style":32617},[31680],[34,683556,459],{"className":683557},[31685],[108,683559,683560,2664,683562,138176,683590,53834,683618,347],{},[2627,683561,679310],{},[34,683563,683565,683578],{"className":683564},[31642],[34,683566,683568],{"className":683567},[31646],[31648,683569,683570],{"xmlns":31650},[31652,683571,683572,683576],{},[31655,683573,683574],{},[31658,683575,67764],{},[31665,683577,67764],{"encoding":31667},[34,683579,683581],{"className":683580,"ariaHidden":758},[31672],[34,683582,683584,683587],{"className":683583},[31676],[34,683585],{"className":683586,"style":38774},[31680],[34,683588,67764],{"className":683589,"style":32062},[31685,31686],[34,683591,683593,683606],{"className":683592},[31642],[34,683594,683596],{"className":683595},[31646],[31648,683597,683598],{"xmlns":31650},[31652,683599,683600,683604],{},[31655,683601,683602],{},[31749,683603,640],{},[31665,683605,640],{"encoding":31667},[34,683607,683609],{"className":683608,"ariaHidden":758},[31672],[34,683610,683612,683615],{"className":683611},[31676],[34,683613],{"className":683614,"style":32617},[31680],[34,683616,640],{"className":683617},[31685],[34,683619,683621,683634],{"className":683620},[31642],[34,683622,683624],{"className":683623},[31646],[31648,683625,683626],{"xmlns":31650},[31652,683627,683628,683632],{},[31655,683629,683630],{},[31749,683631,459],{},[31665,683633,459],{"encoding":31667},[34,683635,683637],{"className":683636,"ariaHidden":758},[31672],[34,683638,683640,683643],{"className":683639},[31676],[34,683641],{"className":683642,"style":32617},[31680],[34,683644,459],{"className":683645},[31685],[108,683647,683648,33183,683650,684076],{},[2627,683649,676907],{},[34,683651,683653,683739],{"className":683652},[31642],[34,683654,683656],{"className":683655},[31646],[31648,683657,683658],{"xmlns":31650},[31652,683659,683660,683736],{},[31655,683661,683662,683672,683674,683682,683684,683690,683692,683698,683700,683702,683704,683706,683708,683716,683718,683720,683722,683728,683730,683732,683734],{},[31743,683663,683664,683666],{},[31658,683665,106063],{},[31655,683667,683668,683670],{},[31658,683669,91994],{},[31658,683671,54151],{},[31662,683673,202],{},[38846,683675,683676,683678,683680],{},[31662,683677,546295],{},[31749,683679,640],{},[31749,683681,459],{},[31662,683683,456],{"stretchy":702},[37624,683685,683686,683688],{},[31749,683687,640],{},[31749,683689,768],{},[31662,683691,32226],{},[37624,683693,683694,683696],{},[31658,683695,67764],{},[31749,683697,768],{},[31662,683699,2557],{"stretchy":702},[39361,683701,39363],{},[31658,683703,46900],{"mathvariant":38079},[31658,683705,67764],{},[31662,683707,202],{},[38846,683709,683710,683712,683714],{},[31662,683711,546295],{},[31749,683713,640],{},[31749,683715,459],{},[31662,683717,456],{"stretchy":702},[31749,683719,640],{},[31662,683721,32226],{},[37624,683723,683724,683726],{},[31658,683725,67764],{},[31749,683727,768],{},[31662,683729,2557],{"stretchy":702},[39361,683731,39363],{},[31658,683733,46900],{"mathvariant":38079},[31658,683735,67764],{},[31665,683737,683738],{"encoding":31667},"I_{DA} = \\int_1^0 (1^2 - y^2) \\, \\mathrm{d}y = \\int_1^0 (1 - y^2) \\, \\mathrm{d}y",[34,683740,683742,683803,683901,683957,684029],{"className":683741,"ariaHidden":758},[31672],[34,683743,683745,683748,683794,683797,683800],{"className":683744},[31676],[34,683746],{"className":683747,"style":70915},[31680],[34,683749,683751,683754],{"className":683750},[31685],[34,683752,106063],{"className":683753,"style":67482},[31685,31686],[34,683755,683757],{"className":683756},[31792],[34,683758,683760,683786],{"className":683759},[31796,31797],[34,683761,683763,683783],{"className":683762},[31801],[34,683764,683766],{"className":683765,"style":116558},[31805],[34,683767,683768,683771],{"style":217742},[34,683769],{"className":683770,"style":31814},[31813],[34,683772,683774],{"className":683773},[31818,31819,31820,31821],[34,683775,683777,683780],{"className":683776},[31685,31821],[34,683778,91994],{"className":683779,"style":34452},[31685,31686,31821],[34,683781,54151],{"className":683782},[31685,31686,31821],[34,683784,31829],{"className":683785},[31828],[34,683787,683789],{"className":683788},[31801],[34,683790,683792],{"className":683791,"style":31836},[31805],[34,683793],{},[34,683795],{"className":683796,"style":31846},[31845],[34,683798,202],{"className":683799},[31850],[34,683801],{"className":683802,"style":31846},[31845],[34,683804,683806,683809,683860,683863,683892,683895,683898],{"className":683805},[31676],[34,683807],{"className":683808,"style":563309},[31680],[34,683810,683812,683815],{"className":683811},[32767],[34,683813,546295],{"className":683814,"style":546331},[32767,131249,156560],[34,683816,683818],{"className":683817},[31792],[34,683819,683821,683852],{"className":683820},[31796,31797],[34,683822,683824,683849],{"className":683823},[31801],[34,683825,683827,683838],{"className":683826,"style":563328},[31805],[34,683828,683829,683832],{"style":546347},[34,683830],{"className":683831,"style":31814},[31813],[34,683833,683835],{"className":683834},[31818,31819,31820,31821],[34,683836,640],{"className":683837},[31685,31821],[34,683839,683840,683843],{"style":546362},[34,683841],{"className":683842,"style":31814},[31813],[34,683844,683846],{"className":683845},[31818,31819,31820,31821],[34,683847,459],{"className":683848},[31685,31821],[34,683850,31829],{"className":683851},[31828],[34,683853,683855],{"className":683854},[31801],[34,683856,683858],{"className":683857,"style":546384},[31805],[34,683859],{},[34,683861,456],{"className":683862},[31782],[34,683864,683866,683869],{"className":683865},[31685],[34,683867,640],{"className":683868},[31685],[34,683870,683872],{"className":683871},[31792],[34,683873,683875],{"className":683874},[31796],[34,683876,683878],{"className":683877},[31801],[34,683879,683881],{"className":683880,"style":37774},[31805],[34,683882,683883,683886],{"style":37777},[34,683884],{"className":683885,"style":31814},[31813],[34,683887,683889],{"className":683888},[31818,31819,31820,31821],[34,683890,768],{"className":683891},[31685,31821],[34,683893],{"className":683894,"style":32604},[31845],[34,683896,32226],{"className":683897},[32024],[34,683899],{"className":683900,"style":32604},[31845],[34,683902,683904,683907,683936,683939,683942,683945,683948,683951,683954],{"className":683903},[31676],[34,683905],{"className":683906,"style":37755},[31680],[34,683908,683910,683913],{"className":683909},[31685],[34,683911,67764],{"className":683912,"style":32062},[31685,31686],[34,683914,683916],{"className":683915},[31792],[34,683917,683919],{"className":683918},[31796],[34,683920,683922],{"className":683921},[31801],[34,683923,683925],{"className":683924,"style":37774},[31805],[34,683926,683927,683930],{"style":37777},[34,683928],{"className":683929,"style":31814},[31813],[34,683931,683933],{"className":683932},[31818,31819,31820,31821],[34,683934,768],{"className":683935},[31685,31821],[34,683937,2557],{"className":683938},[31690],[34,683940],{"className":683941,"style":32123},[31845],[34,683943,46900],{"className":683944},[31685,207923],[34,683946,67764],{"className":683947,"style":32062},[31685,31686],[34,683949],{"className":683950,"style":31846},[31845],[34,683952,202],{"className":683953},[31850],[34,683955],{"className":683956,"style":31846},[31845],[34,683958,683960,683963,684014,684017,684020,684023,684026],{"className":683959},[31676],[34,683961],{"className":683962,"style":563309},[31680],[34,683964,683966,683969],{"className":683965},[32767],[34,683967,546295],{"className":683968,"style":546331},[32767,131249,156560],[34,683970,683972],{"className":683971},[31792],[34,683973,683975,684006],{"className":683974},[31796,31797],[34,683976,683978,684003],{"className":683977},[31801],[34,683979,683981,683992],{"className":683980,"style":563328},[31805],[34,683982,683983,683986],{"style":546347},[34,683984],{"className":683985,"style":31814},[31813],[34,683987,683989],{"className":683988},[31818,31819,31820,31821],[34,683990,640],{"className":683991},[31685,31821],[34,683993,683994,683997],{"style":546362},[34,683995],{"className":683996,"style":31814},[31813],[34,683998,684000],{"className":683999},[31818,31819,31820,31821],[34,684001,459],{"className":684002},[31685,31821],[34,684004,31829],{"className":684005},[31828],[34,684007,684009],{"className":684008},[31801],[34,684010,684012],{"className":684011,"style":546384},[31805],[34,684013],{},[34,684015,456],{"className":684016},[31782],[34,684018,640],{"className":684019},[31685],[34,684021],{"className":684022,"style":32604},[31845],[34,684024,32226],{"className":684025},[32024],[34,684027],{"className":684028,"style":32604},[31845],[34,684030,684032,684035,684064,684067,684070,684073],{"className":684031},[31676],[34,684033],{"className":684034,"style":37755},[31680],[34,684036,684038,684041],{"className":684037},[31685],[34,684039,67764],{"className":684040,"style":32062},[31685,31686],[34,684042,684044],{"className":684043},[31792],[34,684045,684047],{"className":684046},[31796],[34,684048,684050],{"className":684049},[31801],[34,684051,684053],{"className":684052,"style":37774},[31805],[34,684054,684055,684058],{"style":37777},[34,684056],{"className":684057,"style":31814},[31813],[34,684059,684061],{"className":684060},[31818,31819,31820,31821],[34,684062,768],{"className":684063},[31685,31821],[34,684065,2557],{"className":684066},[31690],[34,684068],{"className":684069,"style":32123},[31845],[34,684071,46900],{"className":684072},[31685,207923],[34,684074,67764],{"className":684075,"style":32062},[31685,31686],[34,684077,684079,684157],{"className":684078},[31642],[34,684080,684082],{"className":684081},[31646],[31648,684083,684084],{"xmlns":31650},[31652,684085,684086,684154],{},[31655,684087,684088,684098,684100,684102,684104,684106,684116,684124,684126,684128,684130,684132,684134,684136,684142,684144,684146,684148],{},[31743,684089,684090,684092],{},[31658,684091,106063],{},[31655,684093,684094,684096],{},[31658,684095,91994],{},[31658,684097,54151],{},[31662,684099,202],{},[31662,684101,15972],{"stretchy":702},[31658,684103,67764],{},[31662,684105,32226],{},[41453,684107,684108,684114],{},[37624,684109,684110,684112],{},[31658,684111,67764],{},[31749,684113,786],{},[31749,684115,786],{},[38846,684117,684118,684120,684122],{},[31662,684119,32712],{"stretchy":702},[31749,684121,640],{},[31749,684123,459],{},[31662,684125,202],{},[31749,684127,459],{},[31662,684129,32226],{},[31662,684131,456],{"stretchy":702},[31749,684133,640],{},[31662,684135,32226],{},[41453,684137,684138,684140],{},[31749,684139,640],{},[31749,684141,786],{},[31662,684143,2557],{"stretchy":702},[31662,684145,202],{},[31662,684147,32226],{},[41453,684149,684150,684152],{},[31749,684151,768],{},[31749,684153,786],{},[31665,684155,684156],{"encoding":31667},"I_{DA} = [y - \\frac{y^3}{3}]_1^0 = 0 - (1 - \\frac{1}{3}) = -\\frac{2}{3}",[34,684158,684160,684221,684242,684402,684420,684441,684527],{"className":684159,"ariaHidden":758},[31672],[34,684161,684163,684166,684212,684215,684218],{"className":684162},[31676],[34,684164],{"className":684165,"style":70915},[31680],[34,684167,684169,684172],{"className":684168},[31685],[34,684170,106063],{"className":684171,"style":67482},[31685,31686],[34,684173,684175],{"className":684174},[31792],[34,684176,684178,684204],{"className":684177},[31796,31797],[34,684179,684181,684201],{"className":684180},[31801],[34,684182,684184],{"className":684183,"style":116558},[31805],[34,684185,684186,684189],{"style":217742},[34,684187],{"className":684188,"style":31814},[31813],[34,684190,684192],{"className":684191},[31818,31819,31820,31821],[34,684193,684195,684198],{"className":684194},[31685,31821],[34,684196,91994],{"className":684197,"style":34452},[31685,31686,31821],[34,684199,54151],{"className":684200},[31685,31686,31821],[34,684202,31829],{"className":684203},[31828],[34,684205,684207],{"className":684206},[31801],[34,684208,684210],{"className":684209,"style":31836},[31805],[34,684211],{},[34,684213],{"className":684214,"style":31846},[31845],[34,684216,202],{"className":684217},[31850],[34,684219],{"className":684220,"style":31846},[31845],[34,684222,684224,684227,684230,684233,684236,684239],{"className":684223},[31676],[34,684225],{"className":684226,"style":31774},[31680],[34,684228,15972],{"className":684229},[31782],[34,684231,67764],{"className":684232,"style":32062},[31685,31686],[34,684234],{"className":684235,"style":32604},[31845],[34,684237,32226],{"className":684238},[32024],[34,684240],{"className":684241,"style":32604},[31845],[34,684243,684245,684248,684342,684393,684396,684399],{"className":684244},[31676],[34,684246],{"className":684247,"style":681304},[31680],[34,684249,684251,684254,684339],{"className":684250},[31685],[34,684252],{"className":684253},[31782,41488],[34,684255,684257],{"className":684256},[41453],[34,684258,684260,684331],{"className":684259},[31796,31797],[34,684261,684263,684328],{"className":684262},[31801],[34,684264,684266,684280,684288],{"className":684265,"style":601858},[31805],[34,684267,684268,684271],{"style":41504},[34,684269],{"className":684270,"style":41508},[31813],[34,684272,684274],{"className":684273},[31818,31819,31820,31821],[34,684275,684277],{"className":684276},[31685,31821],[34,684278,786],{"className":684279},[31685,31821],[34,684281,684282,684285],{"style":41520},[34,684283],{"className":684284,"style":41508},[31813],[34,684286],{"className":684287,"style":41528},[41527],[34,684289,684290,684293],{"style":548368},[34,684291],{"className":684292,"style":41508},[31813],[34,684294,684296],{"className":684295},[31818,31819,31820,31821],[34,684297,684299],{"className":684298},[31685,31821],[34,684300,684302,684305],{"className":684301},[31685,31821],[34,684303,67764],{"className":684304,"style":32062},[31685,31686,31821],[34,684306,684308],{"className":684307},[31792],[34,684309,684311],{"className":684310},[31796],[34,684312,684314],{"className":684313},[31801],[34,684315,684317],{"className":684316,"style":144123},[31805],[34,684318,684319,684322],{"style":552391},[34,684320],{"className":684321,"style":43406},[31813],[34,684323,684325],{"className":684324},[31818,43410,43411,31821],[34,684326,786],{"className":684327},[31685,31821],[34,684329,31829],{"className":684330},[31828],[34,684332,684334],{"className":684333},[31801],[34,684335,684337],{"className":684336,"style":41619},[31805],[34,684338],{},[34,684340],{"className":684341},[31690,41488],[34,684343,684345,684348],{"className":684344},[31690],[34,684346,32712],{"className":684347},[31690],[34,684349,684351],{"className":684350},[31792],[34,684352,684354,684385],{"className":684353},[31796,31797],[34,684355,684357,684382],{"className":684356},[31801],[34,684358,684360,684371],{"className":684359,"style":37774},[31805],[34,684361,684362,684365],{"style":316930},[34,684363],{"className":684364,"style":31814},[31813],[34,684366,684368],{"className":684367},[31818,31819,31820,31821],[34,684369,640],{"className":684370},[31685,31821],[34,684372,684373,684376],{"style":37777},[34,684374],{"className":684375,"style":31814},[31813],[34,684377,684379],{"className":684378},[31818,31819,31820,31821],[34,684380,459],{"className":684381},[31685,31821],[34,684383,31829],{"className":684384},[31828],[34,684386,684388],{"className":684387},[31801],[34,684389,684391],{"className":684390,"style":39021},[31805],[34,684392],{},[34,684394],{"className":684395,"style":31846},[31845],[34,684397,202],{"className":684398},[31850],[34,684400],{"className":684401,"style":31846},[31845],[34,684403,684405,684408,684411,684414,684417],{"className":684404},[31676],[34,684406],{"className":684407,"style":35942},[31680],[34,684409,459],{"className":684410},[31685],[34,684412],{"className":684413,"style":32604},[31845],[34,684415,32226],{"className":684416},[32024],[34,684418],{"className":684419,"style":32604},[31845],[34,684421,684423,684426,684429,684432,684435,684438],{"className":684422},[31676],[34,684424],{"className":684425,"style":31774},[31680],[34,684427,456],{"className":684428},[31782],[34,684430,640],{"className":684431},[31685],[34,684433],{"className":684434,"style":32604},[31845],[34,684436,32226],{"className":684437},[32024],[34,684439],{"className":684440,"style":32604},[31845],[34,684442,684444,684447,684515,684518,684521,684524],{"className":684443},[31676],[34,684445],{"className":684446,"style":46727},[31680],[34,684448,684450,684453,684512],{"className":684449},[31685],[34,684451],{"className":684452},[31782,41488],[34,684454,684456],{"className":684455},[41453],[34,684457,684459,684504],{"className":684458},[31796,31797],[34,684460,684462,684501],{"className":684461},[31801],[34,684463,684465,684479,684487],{"className":684464,"style":46746},[31805],[34,684466,684467,684470],{"style":41504},[34,684468],{"className":684469,"style":41508},[31813],[34,684471,684473],{"className":684472},[31818,31819,31820,31821],[34,684474,684476],{"className":684475},[31685,31821],[34,684477,786],{"className":684478},[31685,31821],[34,684480,684481,684484],{"style":41520},[34,684482],{"className":684483,"style":41508},[31813],[34,684485],{"className":684486,"style":41528},[41527],[34,684488,684489,684492],{"style":46771},[34,684490],{"className":684491,"style":41508},[31813],[34,684493,684495],{"className":684494},[31818,31819,31820,31821],[34,684496,684498],{"className":684497},[31685,31821],[34,684499,640],{"className":684500},[31685,31821],[34,684502,31829],{"className":684503},[31828],[34,684505,684507],{"className":684506},[31801],[34,684508,684510],{"className":684509,"style":41619},[31805],[34,684511],{},[34,684513],{"className":684514},[31690,41488],[34,684516,2557],{"className":684517},[31690],[34,684519],{"className":684520,"style":31846},[31845],[34,684522,202],{"className":684523},[31850],[34,684525],{"className":684526,"style":31846},[31845],[34,684528,684530,684533,684536],{"className":684529},[31676],[34,684531],{"className":684532,"style":46727},[31680],[34,684534,32226],{"className":684535},[31685],[34,684537,684539,684542,684601],{"className":684538},[31685],[34,684540],{"className":684541},[31782,41488],[34,684543,684545],{"className":684544},[41453],[34,684546,684548,684593],{"className":684547},[31796,31797],[34,684549,684551,684590],{"className":684550},[31801],[34,684552,684554,684568,684576],{"className":684553,"style":46746},[31805],[34,684555,684556,684559],{"style":41504},[34,684557],{"className":684558,"style":41508},[31813],[34,684560,684562],{"className":684561},[31818,31819,31820,31821],[34,684563,684565],{"className":684564},[31685,31821],[34,684566,786],{"className":684567},[31685,31821],[34,684569,684570,684573],{"style":41520},[34,684571],{"className":684572,"style":41508},[31813],[34,684574],{"className":684575,"style":41528},[41527],[34,684577,684578,684581],{"style":46771},[34,684579],{"className":684580,"style":41508},[31813],[34,684582,684584],{"className":684583},[31818,31819,31820,31821],[34,684585,684587],{"className":684586},[31685,31821],[34,684588,768],{"className":684589},[31685,31821],[34,684591,31829],{"className":684592},[31828],[34,684594,684596],{"className":684595},[31801],[34,684597,684599],{"className":684598,"style":41619},[31805],[34,684600],{},[34,684602],{"className":684603},[31690,41488],[15,684605,684606,684609,685294],{},[2627,684607,684608],{},"总和",[34,684610,684612,684706],{"className":684611},[31642],[34,684613,684615],{"className":684614},[31646],[31648,684616,684617],{"xmlns":31650},[31652,684618,684619,684703],{},[31655,684620,684621,684623,684625,684635,684637,684647,684649,684659,684661,684671,684673,684679,684681,684687,684689,684695,684697],{},[31658,684622,106063],{},[31662,684624,202],{},[31743,684626,684627,684629],{},[31658,684628,106063],{},[31655,684630,684631,684633],{},[31658,684632,54151],{},[31658,684634,54224],{},[31662,684636,223],{},[31743,684638,684639,684641],{},[31658,684640,106063],{},[31655,684642,684643,684645],{},[31658,684644,54224],{},[31658,684646,47384],{},[31662,684648,223],{},[31743,684650,684651,684653],{},[31658,684652,106063],{},[31655,684654,684655,684657],{},[31658,684656,47384],{},[31658,684658,91994],{},[31662,684660,223],{},[31743,684662,684663,684665],{},[31658,684664,106063],{},[31655,684666,684667,684669],{},[31658,684668,91994],{},[31658,684670,54151],{},[31662,684672,202],{},[41453,684674,684675,684677],{},[31749,684676,36723],{},[31749,684678,786],{},[31662,684680,223],{},[41453,684682,684683,684685],{},[31749,684684,39620],{},[31749,684686,786],{},[31662,684688,32226],{},[41453,684690,684691,684693],{},[31749,684692,3244],{},[31749,684694,786],{},[31662,684696,32226],{},[41453,684698,684699,684701],{},[31749,684700,768],{},[31749,684702,786],{},[31665,684704,684705],{"encoding":31667},"I = I_{AB} + I_{BC} + I_{CD} + I_{DA} = \\frac{7}{3} + \\frac{11}{3} - \\frac{10}{3} - \\frac{2}{3}",[34,684707,684709,684727,684788,684849,684910,684971,685054,685137,685220],{"className":684708,"ariaHidden":758},[31672],[34,684710,684712,684715,684718,684721,684724],{"className":684711},[31676],[34,684713],{"className":684714,"style":33218},[31680],[34,684716,106063],{"className":684717,"style":67482},[31685,31686],[34,684719],{"className":684720,"style":31846},[31845],[34,684722,202],{"className":684723},[31850],[34,684725],{"className":684726,"style":31846},[31845],[34,684728,684730,684733,684779,684782,684785],{"className":684729},[31676],[34,684731],{"className":684732,"style":70915},[31680],[34,684734,684736,684739],{"className":684735},[31685],[34,684737,106063],{"className":684738,"style":67482},[31685,31686],[34,684740,684742],{"className":684741},[31792],[34,684743,684745,684771],{"className":684744},[31796,31797],[34,684746,684748,684768],{"className":684747},[31801],[34,684749,684751],{"className":684750,"style":116558},[31805],[34,684752,684753,684756],{"style":217742},[34,684754],{"className":684755,"style":31814},[31813],[34,684757,684759],{"className":684758},[31818,31819,31820,31821],[34,684760,684762,684765],{"className":684761},[31685,31821],[34,684763,54151],{"className":684764},[31685,31686,31821],[34,684766,54224],{"className":684767,"style":54252},[31685,31686,31821],[34,684769,31829],{"className":684770},[31828],[34,684772,684774],{"className":684773},[31801],[34,684775,684777],{"className":684776,"style":31836},[31805],[34,684778],{},[34,684780],{"className":684781,"style":32604},[31845],[34,684783,223],{"className":684784},[32024],[34,684786],{"className":684787,"style":32604},[31845],[34,684789,684791,684794,684840,684843,684846],{"className":684790},[31676],[34,684792],{"className":684793,"style":70915},[31680],[34,684795,684797,684800],{"className":684796},[31685],[34,684798,106063],{"className":684799,"style":67482},[31685,31686],[34,684801,684803],{"className":684802},[31792],[34,684804,684806,684832],{"className":684805},[31796,31797],[34,684807,684809,684829],{"className":684808},[31801],[34,684810,684812],{"className":684811,"style":116558},[31805],[34,684813,684814,684817],{"style":217742},[34,684815],{"className":684816,"style":31814},[31813],[34,684818,684820],{"className":684819},[31818,31819,31820,31821],[34,684821,684823,684826],{"className":684822},[31685,31821],[34,684824,54224],{"className":684825,"style":54252},[31685,31686,31821],[34,684827,47384],{"className":684828,"style":38706},[31685,31686,31821],[34,684830,31829],{"className":684831},[31828],[34,684833,684835],{"className":684834},[31801],[34,684836,684838],{"className":684837,"style":31836},[31805],[34,684839],{},[34,684841],{"className":684842,"style":32604},[31845],[34,684844,223],{"className":684845},[32024],[34,684847],{"className":684848,"style":32604},[31845],[34,684850,684852,684855,684901,684904,684907],{"className":684851},[31676],[34,684853],{"className":684854,"style":70915},[31680],[34,684856,684858,684861],{"className":684857},[31685],[34,684859,106063],{"className":684860,"style":67482},[31685,31686],[34,684862,684864],{"className":684863},[31792],[34,684865,684867,684893],{"className":684866},[31796,31797],[34,684868,684870,684890],{"className":684869},[31801],[34,684871,684873],{"className":684872,"style":116558},[31805],[34,684874,684875,684878],{"style":217742},[34,684876],{"className":684877,"style":31814},[31813],[34,684879,684881],{"className":684880},[31818,31819,31820,31821],[34,684882,684884,684887],{"className":684883},[31685,31821],[34,684885,47384],{"className":684886,"style":38706},[31685,31686,31821],[34,684888,91994],{"className":684889,"style":34452},[31685,31686,31821],[34,684891,31829],{"className":684892},[31828],[34,684894,684896],{"className":684895},[31801],[34,684897,684899],{"className":684898,"style":31836},[31805],[34,684900],{},[34,684902],{"className":684903,"style":32604},[31845],[34,684905,223],{"className":684906},[32024],[34,684908],{"className":684909,"style":32604},[31845],[34,684911,684913,684916,684962,684965,684968],{"className":684912},[31676],[34,684914],{"className":684915,"style":70915},[31680],[34,684917,684919,684922],{"className":684918},[31685],[34,684920,106063],{"className":684921,"style":67482},[31685,31686],[34,684923,684925],{"className":684924},[31792],[34,684926,684928,684954],{"className":684927},[31796,31797],[34,684929,684931,684951],{"className":684930},[31801],[34,684932,684934],{"className":684933,"style":116558},[31805],[34,684935,684936,684939],{"style":217742},[34,684937],{"className":684938,"style":31814},[31813],[34,684940,684942],{"className":684941},[31818,31819,31820,31821],[34,684943,684945,684948],{"className":684944},[31685,31821],[34,684946,91994],{"className":684947,"style":34452},[31685,31686,31821],[34,684949,54151],{"className":684950},[31685,31686,31821],[34,684952,31829],{"className":684953},[31828],[34,684955,684957],{"className":684956},[31801],[34,684958,684960],{"className":684959,"style":31836},[31805],[34,684961],{},[34,684963],{"className":684964,"style":31846},[31845],[34,684966,202],{"className":684967},[31850],[34,684969],{"className":684970,"style":31846},[31845],[34,684972,684974,684977,685045,685048,685051],{"className":684973},[31676],[34,684975],{"className":684976,"style":46727},[31680],[34,684978,684980,684983,685042],{"className":684979},[31685],[34,684981],{"className":684982},[31782,41488],[34,684984,684986],{"className":684985},[41453],[34,684987,684989,685034],{"className":684988},[31796,31797],[34,684990,684992,685031],{"className":684991},[31801],[34,684993,684995,685009,685017],{"className":684994,"style":46746},[31805],[34,684996,684997,685000],{"style":41504},[34,684998],{"className":684999,"style":41508},[31813],[34,685001,685003],{"className":685002},[31818,31819,31820,31821],[34,685004,685006],{"className":685005},[31685,31821],[34,685007,786],{"className":685008},[31685,31821],[34,685010,685011,685014],{"style":41520},[34,685012],{"className":685013,"style":41508},[31813],[34,685015],{"className":685016,"style":41528},[41527],[34,685018,685019,685022],{"style":46771},[34,685020],{"className":685021,"style":41508},[31813],[34,685023,685025],{"className":685024},[31818,31819,31820,31821],[34,685026,685028],{"className":685027},[31685,31821],[34,685029,36723],{"className":685030},[31685,31821],[34,685032,31829],{"className":685033},[31828],[34,685035,685037],{"className":685036},[31801],[34,685038,685040],{"className":685039,"style":41619},[31805],[34,685041],{},[34,685043],{"className":685044},[31690,41488],[34,685046],{"className":685047,"style":32604},[31845],[34,685049,223],{"className":685050},[32024],[34,685052],{"className":685053,"style":32604},[31845],[34,685055,685057,685060,685128,685131,685134],{"className":685056},[31676],[34,685058],{"className":685059,"style":46727},[31680],[34,685061,685063,685066,685125],{"className":685062},[31685],[34,685064],{"className":685065},[31782,41488],[34,685067,685069],{"className":685068},[41453],[34,685070,685072,685117],{"className":685071},[31796,31797],[34,685073,685075,685114],{"className":685074},[31801],[34,685076,685078,685092,685100],{"className":685077,"style":46746},[31805],[34,685079,685080,685083],{"style":41504},[34,685081],{"className":685082,"style":41508},[31813],[34,685084,685086],{"className":685085},[31818,31819,31820,31821],[34,685087,685089],{"className":685088},[31685,31821],[34,685090,786],{"className":685091},[31685,31821],[34,685093,685094,685097],{"style":41520},[34,685095],{"className":685096,"style":41508},[31813],[34,685098],{"className":685099,"style":41528},[41527],[34,685101,685102,685105],{"style":46771},[34,685103],{"className":685104,"style":41508},[31813],[34,685106,685108],{"className":685107},[31818,31819,31820,31821],[34,685109,685111],{"className":685110},[31685,31821],[34,685112,39620],{"className":685113},[31685,31821],[34,685115,31829],{"className":685116},[31828],[34,685118,685120],{"className":685119},[31801],[34,685121,685123],{"className":685122,"style":41619},[31805],[34,685124],{},[34,685126],{"className":685127},[31690,41488],[34,685129],{"className":685130,"style":32604},[31845],[34,685132,32226],{"className":685133},[32024],[34,685135],{"className":685136,"style":32604},[31845],[34,685138,685140,685143,685211,685214,685217],{"className":685139},[31676],[34,685141],{"className":685142,"style":46727},[31680],[34,685144,685146,685149,685208],{"className":685145},[31685],[34,685147],{"className":685148},[31782,41488],[34,685150,685152],{"className":685151},[41453],[34,685153,685155,685200],{"className":685154},[31796,31797],[34,685156,685158,685197],{"className":685157},[31801],[34,685159,685161,685175,685183],{"className":685160,"style":46746},[31805],[34,685162,685163,685166],{"style":41504},[34,685164],{"className":685165,"style":41508},[31813],[34,685167,685169],{"className":685168},[31818,31819,31820,31821],[34,685170,685172],{"className":685171},[31685,31821],[34,685173,786],{"className":685174},[31685,31821],[34,685176,685177,685180],{"style":41520},[34,685178],{"className":685179,"style":41508},[31813],[34,685181],{"className":685182,"style":41528},[41527],[34,685184,685185,685188],{"style":46771},[34,685186],{"className":685187,"style":41508},[31813],[34,685189,685191],{"className":685190},[31818,31819,31820,31821],[34,685192,685194],{"className":685193},[31685,31821],[34,685195,3244],{"className":685196},[31685,31821],[34,685198,31829],{"className":685199},[31828],[34,685201,685203],{"className":685202},[31801],[34,685204,685206],{"className":685205,"style":41619},[31805],[34,685207],{},[34,685209],{"className":685210},[31690,41488],[34,685212],{"className":685213,"style":32604},[31845],[34,685215,32226],{"className":685216},[32024],[34,685218],{"className":685219,"style":32604},[31845],[34,685221,685223,685226],{"className":685222},[31676],[34,685224],{"className":685225,"style":46727},[31680],[34,685227,685229,685232,685291],{"className":685228},[31685],[34,685230],{"className":685231},[31782,41488],[34,685233,685235],{"className":685234},[41453],[34,685236,685238,685283],{"className":685237},[31796,31797],[34,685239,685241,685280],{"className":685240},[31801],[34,685242,685244,685258,685266],{"className":685243,"style":46746},[31805],[34,685245,685246,685249],{"style":41504},[34,685247],{"className":685248,"style":41508},[31813],[34,685250,685252],{"className":685251},[31818,31819,31820,31821],[34,685253,685255],{"className":685254},[31685,31821],[34,685256,786],{"className":685257},[31685,31821],[34,685259,685260,685263],{"style":41520},[34,685261],{"className":685262,"style":41508},[31813],[34,685264],{"className":685265,"style":41528},[41527],[34,685267,685268,685271],{"style":46771},[34,685269],{"className":685270,"style":41508},[31813],[34,685272,685274],{"className":685273},[31818,31819,31820,31821],[34,685275,685277],{"className":685276},[31685,31821],[34,685278,768],{"className":685279},[31685,31821],[34,685281,31829],{"className":685282},[31828],[34,685284,685286],{"className":685285},[31801],[34,685287,685289],{"className":685288,"style":41619},[31805],[34,685290],{},[34,685292],{"className":685293},[31690,41488],[34,685295,685297,685345],{"className":685296},[31642],[34,685298,685300],{"className":685299},[31646],[31648,685301,685302],{"xmlns":31650},[31652,685303,685304,685342],{},[31655,685305,685306,685308,685310,685330,685332,685338,685340],{},[31658,685307,106063],{},[31662,685309,202],{},[41453,685311,685312,685328],{},[31655,685313,685314,685316,685318,685320,685322,685324,685326],{},[31749,685315,36723],{},[31662,685317,223],{},[31749,685319,39620],{},[31662,685321,32226],{},[31749,685323,3244],{},[31662,685325,32226],{},[31749,685327,768],{},[31749,685329,786],{},[31662,685331,202],{},[41453,685333,685334,685336],{},[31749,685335,36656],{},[31749,685337,786],{},[31662,685339,202],{},[31749,685341,768],{},[31665,685343,685344],{"encoding":31667},"I = \\frac{7 + 11 - 10 - 2}{3} = \\frac{6}{3} = 2",[34,685346,685348,685366,685467,685550],{"className":685347,"ariaHidden":758},[31672],[34,685349,685351,685354,685357,685360,685363],{"className":685350},[31676],[34,685352],{"className":685353,"style":33218},[31680],[34,685355,106063],{"className":685356,"style":67482},[31685,31686],[34,685358],{"className":685359,"style":31846},[31845],[34,685361,202],{"className":685362},[31850],[34,685364],{"className":685365,"style":31846},[31845],[34,685367,685369,685372,685458,685461,685464],{"className":685368},[31676],[34,685370],{"className":685371,"style":46727},[31680],[34,685373,685375,685378,685455],{"className":685374},[31685],[34,685376],{"className":685377},[31782,41488],[34,685379,685381],{"className":685380},[41453],[34,685382,685384,685447],{"className":685383},[31796,31797],[34,685385,685387,685444],{"className":685386},[31801],[34,685388,685390,685404,685412],{"className":685389,"style":46746},[31805],[34,685391,685392,685395],{"style":41504},[34,685393],{"className":685394,"style":41508},[31813],[34,685396,685398],{"className":685397},[31818,31819,31820,31821],[34,685399,685401],{"className":685400},[31685,31821],[34,685402,786],{"className":685403},[31685,31821],[34,685405,685406,685409],{"style":41520},[34,685407],{"className":685408,"style":41508},[31813],[34,685410],{"className":685411,"style":41528},[41527],[34,685413,685414,685417],{"style":46771},[34,685415],{"className":685416,"style":41508},[31813],[34,685418,685420],{"className":685419},[31818,31819,31820,31821],[34,685421,685423,685426,685429,685432,685435,685438,685441],{"className":685422},[31685,31821],[34,685424,36723],{"className":685425},[31685,31821],[34,685427,223],{"className":685428},[32024,31821],[34,685430,39620],{"className":685431},[31685,31821],[34,685433,32226],{"className":685434},[32024,31821],[34,685436,3244],{"className":685437},[31685,31821],[34,685439,32226],{"className":685440},[32024,31821],[34,685442,768],{"className":685443},[31685,31821],[34,685445,31829],{"className":685446},[31828],[34,685448,685450],{"className":685449},[31801],[34,685451,685453],{"className":685452,"style":41619},[31805],[34,685454],{},[34,685456],{"className":685457},[31690,41488],[34,685459],{"className":685460,"style":31846},[31845],[34,685462,202],{"className":685463},[31850],[34,685465],{"className":685466,"style":31846},[31845],[34,685468,685470,685473,685541,685544,685547],{"className":685469},[31676],[34,685471],{"className":685472,"style":46727},[31680],[34,685474,685476,685479,685538],{"className":685475},[31685],[34,685477],{"className":685478},[31782,41488],[34,685480,685482],{"className":685481},[41453],[34,685483,685485,685530],{"className":685484},[31796,31797],[34,685486,685488,685527],{"className":685487},[31801],[34,685489,685491,685505,685513],{"className":685490,"style":46746},[31805],[34,685492,685493,685496],{"style":41504},[34,685494],{"className":685495,"style":41508},[31813],[34,685497,685499],{"className":685498},[31818,31819,31820,31821],[34,685500,685502],{"className":685501},[31685,31821],[34,685503,786],{"className":685504},[31685,31821],[34,685506,685507,685510],{"style":41520},[34,685508],{"className":685509,"style":41508},[31813],[34,685511],{"className":685512,"style":41528},[41527],[34,685514,685515,685518],{"style":46771},[34,685516],{"className":685517,"style":41508},[31813],[34,685519,685521],{"className":685520},[31818,31819,31820,31821],[34,685522,685524],{"className":685523},[31685,31821],[34,685525,36656],{"className":685526},[31685,31821],[34,685528,31829],{"className":685529},[31828],[34,685531,685533],{"className":685532},[31801],[34,685534,685536],{"className":685535,"style":41619},[31805],[34,685537],{},[34,685539],{"className":685540},[31690,41488],[34,685542],{"className":685543,"style":31846},[31845],[34,685545,202],{"className":685546},[31850],[34,685548],{"className":685549,"style":31846},[31845],[34,685551,685553,685556],{"className":685552},[31676],[34,685554],{"className":685555,"style":32617},[31680],[34,685557,768],{"className":685558},[31685],[15,685560,685561],{},[2627,685562,685563],{},"结果：2",[15,685565,685566],{},[71807,685567,685568,685569,685912],{},"(进阶验证：使用格林公式 ",[34,685570,685572,685655],{"className":685571},[31642],[34,685573,685575],{"className":685574},[31646],[31648,685576,685577],{"xmlns":31650},[31652,685578,685579,685652],{},[31655,685580,685581,685584,685586,685600,685602,685616,685618,685620,685622,685624,685626,685628,685630,685632,685634,685636,685638,685640,685642,685644,685646,685648,685650],{},[31662,685582,685583],{},"∬",[31662,685585,456],{"stretchy":702},[41453,685587,685588,685594],{},[31655,685589,685590,685592],{},[31658,685591,619473],{"mathvariant":38079},[31658,685593,116863],{},[31655,685595,685596,685598],{},[31658,685597,619473],{"mathvariant":38079},[31658,685599,32936],{},[31662,685601,32226],{},[41453,685603,685604,685610],{},[31655,685605,685606,685608],{},[31658,685607,619473],{"mathvariant":38079},[31658,685609,105852],{},[31655,685611,685612,685614],{},[31658,685613,619473],{"mathvariant":38079},[31658,685615,67764],{},[31662,685617,2557],{"stretchy":702},[31658,685619,46900],{},[31658,685621,32936],{},[31658,685623,46900],{},[31658,685625,67764],{},[31662,685627,202],{},[31662,685629,685583],{},[31662,685631,456],{"stretchy":702},[31749,685633,768],{},[31658,685635,32936],{},[31662,685637,32226],{},[31749,685639,768],{},[31658,685641,67764],{},[31662,685643,2557],{"stretchy":702},[31658,685645,46900],{},[31658,685647,32936],{},[31658,685649,46900],{},[31658,685651,67764],{},[31665,685653,685654],{"encoding":31667},"\\iint (\\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y}) dxdy = \\iint (2x - 2y) dxdy",[34,685656,685658,685753,685858,685885],{"className":685657,"ariaHidden":758},[31672],[34,685659,685661,685664,685667,685670,685744,685747,685750],{"className":685660},[31676],[34,685662],{"className":685663,"style":608762},[31680],[34,685665,685583],{"className":685666,"style":625171},[32767,131249,156560],[34,685668,456],{"className":685669},[31782],[34,685671,685673,685676,685741],{"className":685672},[31685],[34,685674],{"className":685675},[31782,41488],[34,685677,685679],{"className":685678},[41453],[34,685680,685682,685733],{"className":685681},[31796,31797],[34,685683,685685,685730],{"className":685684},[31801],[34,685686,685688,685705,685713],{"className":685687,"style":608935},[31805],[34,685689,685690,685693],{"style":41504},[34,685691],{"className":685692,"style":41508},[31813],[34,685694,685696],{"className":685695},[31818,31819,31820,31821],[34,685697,685699,685702],{"className":685698},[31685,31821],[34,685700,619473],{"className":685701,"style":112322},[31685,31821],[34,685703,32936],{"className":685704},[31685,31686,31821],[34,685706,685707,685710],{"style":41520},[34,685708],{"className":685709,"style":41508},[31813],[34,685711],{"className":685712,"style":41528},[41527],[34,685714,685715,685718],{"style":548368},[34,685716],{"className":685717,"style":41508},[31813],[34,685719,685721],{"className":685720},[31818,31819,31820,31821],[34,685722,685724,685727],{"className":685723},[31685,31821],[34,685725,619473],{"className":685726,"style":112322},[31685,31821],[34,685728,116863],{"className":685729},[31685,31686,31821],[34,685731,31829],{"className":685732},[31828],[34,685734,685736],{"className":685735},[31801],[34,685737,685739],{"className":685738,"style":41619},[31805],[34,685740],{},[34,685742],{"className":685743},[31690,41488],[34,685745],{"className":685746,"style":32604},[31845],[34,685748,32226],{"className":685749},[32024],[34,685751],{"className":685752,"style":32604},[31845],[34,685754,685756,685760,685834,685837,685840,685843,685846,685849,685852,685855],{"className":685755},[31676],[34,685757],{"className":685758,"style":685759},[31680],"height:1.3612em;vertical-align:-0.4811em;",[34,685761,685763,685766,685831],{"className":685762},[31685],[34,685764],{"className":685765},[31782,41488],[34,685767,685769],{"className":685768},[41453],[34,685770,685772,685823],{"className":685771},[31796,31797],[34,685773,685775,685820],{"className":685774},[31801],[34,685776,685778,685795,685803],{"className":685777,"style":232511},[31805],[34,685779,685780,685783],{"style":41504},[34,685781],{"className":685782,"style":41508},[31813],[34,685784,685786],{"className":685785},[31818,31819,31820,31821],[34,685787,685789,685792],{"className":685788},[31685,31821],[34,685790,619473],{"className":685791,"style":112322},[31685,31821],[34,685793,67764],{"className":685794,"style":32062},[31685,31686,31821],[34,685796,685797,685800],{"style":41520},[34,685798],{"className":685799,"style":41508},[31813],[34,685801],{"className":685802,"style":41528},[41527],[34,685804,685805,685808],{"style":46771},[34,685806],{"className":685807,"style":41508},[31813],[34,685809,685811],{"className":685810},[31818,31819,31820,31821],[34,685812,685814,685817],{"className":685813},[31685,31821],[34,685815,619473],{"className":685816,"style":112322},[31685,31821],[34,685818,105852],{"className":685819,"style":37422},[31685,31686,31821],[34,685821,31829],{"className":685822},[31828],[34,685824,685826],{"className":685825},[31801],[34,685827,685829],{"className":685828,"style":220191},[31805],[34,685830],{},[34,685832],{"className":685833},[31690,41488],[34,685835,2557],{"className":685836},[31690],[34,685838,46900],{"className":685839},[31685,31686],[34,685841,32936],{"className":685842},[31685,31686],[34,685844,46900],{"className":685845},[31685,31686],[34,685847,67764],{"className":685848,"style":32062},[31685,31686],[34,685850],{"className":685851,"style":31846},[31845],[34,685853,202],{"className":685854},[31850],[34,685856],{"className":685857,"style":31846},[31845],[34,685859,685861,685864,685867,685870,685873,685876,685879,685882],{"className":685860},[31676],[34,685862],{"className":685863,"style":627376},[31680],[34,685865,685583],{"className":685866,"style":625171},[32767,131249,156560],[34,685868,456],{"className":685869},[31782],[34,685871,768],{"className":685872},[31685],[34,685874,32936],{"className":685875},[31685,31686],[34,685877],{"className":685878,"style":32604},[31845],[34,685880,32226],{"className":685881},[32024],[34,685883],{"className":685884,"style":32604},[31845],[34,685886,685888,685891,685894,685897,685900,685903,685906,685909],{"className":685887},[31676],[34,685889],{"className":685890,"style":31774},[31680],[34,685892,768],{"className":685893},[31685],[34,685895,67764],{"className":685896,"style":32062},[31685,31686],[34,685898,2557],{"className":685899},[31690],[34,685901,46900],{"className":685902},[31685,31686],[34,685904,32936],{"className":685905},[31685,31686],[34,685907,46900],{"className":685908},[31685,31686],[34,685910,67764],{"className":685911,"style":32062},[31685,31686],"，计算结果也是2。但初学阶段分段计算是基本功)",[32188,685914],{},[32490,685916,685917],{"id":219969},"题目 (3)",[15,685919,685920,685922,686236,658429,686266,686429],{},[2627,685921,678493],{},[34,685923,685925,685997],{"className":685924},[31642],[34,685926,685928],{"className":685927},[31646],[31648,685929,685930],{"xmlns":31650},[31652,685931,685932,685994],{},[31655,685933,685934,685936,685938,685944],{},[31658,685935,106063],{},[31662,685937,202],{},[31743,685939,685940,685942],{},[31662,685941,546295],{},[31658,685943,33206],{},[41453,685945,685946,685978],{},[31655,685947,685948,685950,685952,685954,685956,685958,685960,685962,685964,685966,685968,685970,685972,685974,685976],{},[31662,685949,456],{"stretchy":702},[31658,685951,32936],{},[31662,685953,223],{},[31658,685955,67764],{},[31662,685957,2557],{"stretchy":702},[31658,685959,46900],{"mathvariant":38079},[31658,685961,32936],{},[31662,685963,32226],{},[31662,685965,456],{"stretchy":702},[31658,685967,32936],{},[31662,685969,32226],{},[31658,685971,67764],{},[31662,685973,2557],{"stretchy":702},[31658,685975,46900],{"mathvariant":38079},[31658,685977,67764],{},[31655,685979,685980,685986,685988],{},[37624,685981,685982,685984],{},[31658,685983,32936],{},[31749,685985,768],{},[31662,685987,223],{},[37624,685989,685990,685992],{},[31658,685991,67764],{},[31749,685993,768],{},[31665,685995,685996],{"encoding":31667},"I = \\int_L \\frac{(x+y)\\mathrm{d}x - (x-y)\\mathrm{d}y}{x^2 + y^2}",[34,685998,686000,686018],{"className":685999,"ariaHidden":758},[31672],[34,686001,686003,686006,686009,686012,686015],{"className":686002},[31676],[34,686004],{"className":686005,"style":33218},[31680],[34,686007,106063],{"className":686008,"style":67482},[31685,31686],[34,686010],{"className":686011,"style":31846},[31845],[34,686013,202],{"className":686014},[31850],[34,686016],{"className":686017,"style":31846},[31845],[34,686019,686021,686025,686065,686068],{"className":686020},[31676],[34,686022],{"className":686023,"style":686024},[31680],"height:1.4911em;vertical-align:-0.4811em;",[34,686026,686028,686031],{"className":686027},[32767],[34,686029,546295],{"className":686030,"style":546331},[32767,131249,156560],[34,686032,686034],{"className":686033},[31792],[34,686035,686037,686057],{"className":686036},[31796,31797],[34,686038,686040,686054],{"className":686039},[31801],[34,686041,686043],{"className":686042,"style":637339},[31805],[34,686044,686045,686048],{"style":546347},[34,686046],{"className":686047,"style":31814},[31813],[34,686049,686051],{"className":686050},[31818,31819,31820,31821],[34,686052,33206],{"className":686053},[31685,31686,31821],[34,686055,31829],{"className":686056},[31828],[34,686058,686060],{"className":686059},[31801],[34,686061,686063],{"className":686062,"style":546384},[31805],[34,686064],{},[34,686066],{"className":686067,"style":32123},[31845],[34,686069,686071,686074,686233],{"className":686070},[31685],[34,686072],{"className":686073},[31782,41488],[34,686075,686077],{"className":686076},[41453],[34,686078,686080,686225],{"className":686079},[31796,31797],[34,686081,686083,686222],{"className":686082},[31801],[34,686084,686086,686158,686166],{"className":686085,"style":70504},[31805],[34,686087,686088,686091],{"style":41504},[34,686089],{"className":686090,"style":41508},[31813],[34,686092,686094],{"className":686093},[31818,31819,31820,31821],[34,686095,686097,686126,686129],{"className":686096},[31685,31821],[34,686098,686100,686103],{"className":686099},[31685,31821],[34,686101,32936],{"className":686102},[31685,31686,31821],[34,686104,686106],{"className":686105},[31792],[34,686107,686109],{"className":686108},[31796],[34,686110,686112],{"className":686111},[31801],[34,686113,686115],{"className":686114,"style":548314},[31805],[34,686116,686117,686120],{"style":116668},[34,686118],{"className":686119,"style":43406},[31813],[34,686121,686123],{"className":686122},[31818,43410,43411,31821],[34,686124,768],{"className":686125},[31685,31821],[34,686127,223],{"className":686128},[32024,31821],[34,686130,686132,686135],{"className":686131},[31685,31821],[34,686133,67764],{"className":686134,"style":32062},[31685,31686,31821],[34,686136,686138],{"className":686137},[31792],[34,686139,686141],{"className":686140},[31796],[34,686142,686144],{"className":686143},[31801],[34,686145,686147],{"className":686146,"style":548314},[31805],[34,686148,686149,686152],{"style":116668},[34,686150],{"className":686151,"style":43406},[31813],[34,686153,686155],{"className":686154},[31818,43410,43411,31821],[34,686156,768],{"className":686157},[31685,31821],[34,686159,686160,686163],{"style":41520},[34,686161],{"className":686162,"style":41508},[31813],[34,686164],{"className":686165,"style":41528},[41527],[34,686167,686168,686171],{"style":70529},[34,686169],{"className":686170,"style":41508},[31813],[34,686172,686174],{"className":686173},[31818,31819,31820,31821],[34,686175,686177,686180,686183,686186,686189,686192,686195,686198,686201,686204,686207,686210,686213,686216,686219],{"className":686176},[31685,31821],[34,686178,456],{"className":686179},[31782,31821],[34,686181,32936],{"className":686182},[31685,31686,31821],[34,686184,223],{"className":686185},[32024,31821],[34,686187,67764],{"className":686188,"style":32062},[31685,31686,31821],[34,686190,2557],{"className":686191},[31690,31821],[34,686193,46900],{"className":686194},[31685,207923,31821],[34,686196,32936],{"className":686197},[31685,31686,31821],[34,686199,32226],{"className":686200},[32024,31821],[34,686202,456],{"className":686203},[31782,31821],[34,686205,32936],{"className":686206},[31685,31686,31821],[34,686208,32226],{"className":686209},[32024,31821],[34,686211,67764],{"className":686212,"style":32062},[31685,31686,31821],[34,686214,2557],{"className":686215},[31690,31821],[34,686217,46900],{"className":686218},[31685,207923,31821],[34,686220,67764],{"className":686221,"style":32062},[31685,31686,31821],[34,686223,31829],{"className":686224},[31828],[34,686226,686228],{"className":686227},[31801],[34,686229,686231],{"className":686230,"style":220191},[31805],[34,686232],{},[34,686234],{"className":686235},[31690,41488],[2627,686237,658358,686238],{},[34,686239,686241,686254],{"className":686240},[31642],[34,686242,686244],{"className":686243},[31646],[31648,686245,686246],{"xmlns":31650},[31652,686247,686248,686252],{},[31655,686249,686250],{},[31658,686251,33206],{},[31665,686253,33206],{"encoding":31667},[34,686255,686257],{"className":686256,"ariaHidden":758},[31672],[34,686258,686260,686263],{"className":686259},[31676],[34,686261],{"className":686262,"style":33218},[31680],[34,686264,33206],{"className":686265},[31685,31686],[34,686267,686269,686303],{"className":686268},[31642],[34,686270,686272],{"className":686271},[31646],[31648,686273,686274],{"xmlns":31650},[31652,686275,686276,686300],{},[31655,686277,686278,686284,686286,686292,686294],{},[37624,686279,686280,686282],{},[31658,686281,32936],{},[31749,686283,768],{},[31662,686285,223],{},[37624,686287,686288,686290],{},[31658,686289,67764],{},[31749,686291,768],{},[31662,686293,202],{},[37624,686295,686296,686298],{},[31658,686297,111],{},[31749,686299,768],{},[31665,686301,686302],{"encoding":31667},"x^2 + y^2 = a^2",[34,686304,686306,686350,686394],{"className":686305,"ariaHidden":758},[31672],[34,686307,686309,686312,686341,686344,686347],{"className":686308},[31676],[34,686310],{"className":686311,"style":40044},[31680],[34,686313,686315,686318],{"className":686314},[31685],[34,686316,32936],{"className":686317},[31685,31686],[34,686319,686321],{"className":686320},[31792],[34,686322,686324],{"className":686323},[31796],[34,686325,686327],{"className":686326},[31801],[34,686328,686330],{"className":686329,"style":37774},[31805],[34,686331,686332,686335],{"style":37777},[34,686333],{"className":686334,"style":31814},[31813],[34,686336,686338],{"className":686337},[31818,31819,31820,31821],[34,686339,768],{"className":686340},[31685,31821],[34,686342],{"className":686343,"style":32604},[31845],[34,686345,223],{"className":686346},[32024],[34,686348],{"className":686349,"style":32604},[31845],[34,686351,686353,686356,686385,686388,686391],{"className":686352},[31676],[34,686354],{"className":686355,"style":315747},[31680],[34,686357,686359,686362],{"className":686358},[31685],[34,686360,67764],{"className":686361,"style":32062},[31685,31686],[34,686363,686365],{"className":686364},[31792],[34,686366,686368],{"className":686367},[31796],[34,686369,686371],{"className":686370},[31801],[34,686372,686374],{"className":686373,"style":37774},[31805],[34,686375,686376,686379],{"style":37777},[34,686377],{"className":686378,"style":31814},[31813],[34,686380,686382],{"className":686381},[31818,31819,31820,31821],[34,686383,768],{"className":686384},[31685,31821],[34,686386],{"className":686387,"style":31846},[31845],[34,686389,202],{"className":686390},[31850],[34,686392],{"className":686393,"style":31846},[31845],[34,686395,686397,686400],{"className":686396},[31676],[34,686398],{"className":686399,"style":37774},[31680],[34,686401,686403,686406],{"className":686402},[31685],[34,686404,111],{"className":686405},[31685,31686],[34,686407,686409],{"className":686408},[31792],[34,686410,686412],{"className":686411},[31796],[34,686413,686415],{"className":686414},[31801],[34,686416,686418],{"className":686417,"style":37774},[31805],[34,686419,686420,686423],{"style":37777},[34,686421],{"className":686422,"style":31814},[31813],[34,686424,686426],{"className":686425},[31818,31819,31820,31821],[34,686427,768],{"className":686428},[31685,31821],"，逆时针方向。",[32490,686431,658913],{"id":686432},"详细解答-4",[15,686434,686435,686436,686547,686548,686551],{},"看到 ",[34,686437,686439,686465],{"className":686438},[31642],[34,686440,686442],{"className":686441},[31646],[31648,686443,686444],{"xmlns":31650},[31652,686445,686446,686462],{},[31655,686447,686448,686454,686456],{},[37624,686449,686450,686452],{},[31658,686451,32936],{},[31749,686453,768],{},[31662,686455,223],{},[37624,686457,686458,686460],{},[31658,686459,67764],{},[31749,686461,768],{},[31665,686463,686464],{"encoding":31667},"x^2 + y^2",[34,686466,686468,686512],{"className":686467,"ariaHidden":758},[31672],[34,686469,686471,686474,686503,686506,686509],{"className":686470},[31676],[34,686472],{"className":686473,"style":40044},[31680],[34,686475,686477,686480],{"className":686476},[31685],[34,686478,32936],{"className":686479},[31685,31686],[34,686481,686483],{"className":686482},[31792],[34,686484,686486],{"className":686485},[31796],[34,686487,686489],{"className":686488},[31801],[34,686490,686492],{"className":686491,"style":37774},[31805],[34,686493,686494,686497],{"style":37777},[34,686495],{"className":686496,"style":31814},[31813],[34,686498,686500],{"className":686499},[31818,31819,31820,31821],[34,686501,768],{"className":686502},[31685,31821],[34,686504],{"className":686505,"style":32604},[31845],[34,686507,223],{"className":686508},[32024],[34,686510],{"className":686511,"style":32604},[31845],[34,686513,686515,686518],{"className":686514},[31676],[34,686516],{"className":686517,"style":315747},[31680],[34,686519,686521,686524],{"className":686520},[31685],[34,686522,67764],{"className":686523,"style":32062},[31685,31686],[34,686525,686527],{"className":686526},[31792],[34,686528,686530],{"className":686529},[31796],[34,686531,686533],{"className":686532},[31801],[34,686534,686536],{"className":686535,"style":37774},[31805],[34,686537,686538,686541],{"style":37777},[34,686539],{"className":686540,"style":31814},[31813],[34,686542,686544],{"className":686543},[31818,31819,31820,31821],[34,686545,768],{"className":686546},[31685,31821],"，尤其是分母中出现，",[2627,686549,686550],{},"极坐标参数化","是最佳选择。",[2230,686553,686554,686786,687984,689090],{},[108,686555,686556,686559,686628,686697,138176,686725,53834,686753,347],{},[2627,686557,686558],{},"参数方程",[34,686560,686562,686586],{"className":686561},[31642],[34,686563,686565],{"className":686564},[31646],[31648,686566,686567],{"xmlns":31650},[31652,686568,686569,686583],{},[31655,686570,686571,686573,686575,686577,686579,686581],{},[31658,686572,32936],{},[31662,686574,202],{},[31658,686576,111],{},[31658,686578,547860],{},[31662,686580,32682],{},[31658,686582,33249],{},[31665,686584,686585],{"encoding":31667},"x = a \\cos t",[34,686587,686589,686607],{"className":686588,"ariaHidden":758},[31672],[34,686590,686592,686595,686598,686601,686604],{"className":686591},[31676],[34,686593],{"className":686594,"style":32527},[31680],[34,686596,32936],{"className":686597},[31685,31686],[34,686599],{"className":686600,"style":31846},[31845],[34,686602,202],{"className":686603},[31850],[34,686605],{"className":686606,"style":31846},[31845],[34,686608,686610,686613,686616,686619,686622,686625],{"className":686609},[31676],[34,686611],{"className":686612,"style":33280},[31680],[34,686614,111],{"className":686615},[31685,31686],[34,686617],{"className":686618,"style":32123},[31845],[34,686620,547860],{"className":686621},[32767],[34,686623],{"className":686624,"style":32123},[31845],[34,686626,33249],{"className":686627},[31685,31686],[34,686629,686631,686655],{"className":686630},[31642],[34,686632,686634],{"className":686633},[31646],[31648,686635,686636],{"xmlns":31650},[31652,686637,686638,686652],{},[31655,686639,686640,686642,686644,686646,686648,686650],{},[31658,686641,67764],{},[31662,686643,202],{},[31658,686645,111],{},[31658,686647,547853],{},[31662,686649,32682],{},[31658,686651,33249],{},[31665,686653,686654],{"encoding":31667},"y = a \\sin t",[34,686656,686658,686676],{"className":686657,"ariaHidden":758},[31672],[34,686659,686661,686664,686667,686670,686673],{"className":686660},[31676],[34,686662],{"className":686663,"style":38774},[31680],[34,686665,67764],{"className":686666,"style":32062},[31685,31686],[34,686668],{"className":686669,"style":31846},[31845],[34,686671,202],{"className":686672},[31850],[34,686674],{"className":686675,"style":31846},[31845],[34,686677,686679,686682,686685,686688,686691,686694],{"className":686678},[31676],[34,686680],{"className":686681,"style":71430},[31680],[34,686683,111],{"className":686684},[31685,31686],[34,686686],{"className":686687,"style":32123},[31845],[34,686689,547853],{"className":686690},[32767],[34,686692],{"className":686693,"style":32123},[31845],[34,686695,33249],{"className":686696},[31685,31686],[34,686698,686700,686713],{"className":686699},[31642],[34,686701,686703],{"className":686702},[31646],[31648,686704,686705],{"xmlns":31650},[31652,686706,686707,686711],{},[31655,686708,686709],{},[31658,686710,33249],{},[31665,686712,33249],{"encoding":31667},[34,686714,686716],{"className":686715,"ariaHidden":758},[31672],[34,686717,686719,686722],{"className":686718},[31676],[34,686720],{"className":686721,"style":33280},[31680],[34,686723,33249],{"className":686724},[31685,31686],[34,686726,686728,686741],{"className":686727},[31642],[34,686729,686731],{"className":686730},[31646],[31648,686732,686733],{"xmlns":31650},[31652,686734,686735,686739],{},[31655,686736,686737],{},[31749,686738,459],{},[31665,686740,459],{"encoding":31667},[34,686742,686744],{"className":686743,"ariaHidden":758},[31672],[34,686745,686747,686750],{"className":686746},[31676],[34,686748],{"className":686749,"style":32617},[31680],[34,686751,459],{"className":686752},[31685],[34,686754,686756,686771],{"className":686755},[31642],[34,686757,686759],{"className":686758},[31646],[31648,686760,686761],{"xmlns":31650},[31652,686762,686763,686769],{},[31655,686764,686765,686767],{},[31749,686766,768],{},[31658,686768,58442],{},[31665,686770,635875],{"encoding":31667},[34,686772,686774],{"className":686773,"ariaHidden":758},[31672],[34,686775,686777,686780,686783],{"className":686776},[31676],[34,686778],{"className":686779,"style":32617},[31680],[34,686781,768],{"className":686782},[31685],[34,686784,58442],{"className":686785,"style":32062},[31685,31686],[108,686787,686788,686791],{},[2627,686789,686790],{},"准备代入项",[105,686792,686793,686958,687144,687562],{},[108,686794,686795,686796],{},"分母：",[34,686797,686799,686832],{"className":686798},[31642],[34,686800,686802],{"className":686801},[31646],[31648,686803,686804],{"xmlns":31650},[31652,686805,686806,686830],{},[31655,686807,686808,686814,686816,686822,686824],{},[37624,686809,686810,686812],{},[31658,686811,32936],{},[31749,686813,768],{},[31662,686815,223],{},[37624,686817,686818,686820],{},[31658,686819,67764],{},[31749,686821,768],{},[31662,686823,202],{},[37624,686825,686826,686828],{},[31658,686827,111],{},[31749,686829,768],{},[31665,686831,686302],{"encoding":31667},[34,686833,686835,686879,686923],{"className":686834,"ariaHidden":758},[31672],[34,686836,686838,686841,686870,686873,686876],{"className":686837},[31676],[34,686839],{"className":686840,"style":40044},[31680],[34,686842,686844,686847],{"className":686843},[31685],[34,686845,32936],{"className":686846},[31685,31686],[34,686848,686850],{"className":686849},[31792],[34,686851,686853],{"className":686852},[31796],[34,686854,686856],{"className":686855},[31801],[34,686857,686859],{"className":686858,"style":37774},[31805],[34,686860,686861,686864],{"style":37777},[34,686862],{"className":686863,"style":31814},[31813],[34,686865,686867],{"className":686866},[31818,31819,31820,31821],[34,686868,768],{"className":686869},[31685,31821],[34,686871],{"className":686872,"style":32604},[31845],[34,686874,223],{"className":686875},[32024],[34,686877],{"className":686878,"style":32604},[31845],[34,686880,686882,686885,686914,686917,686920],{"className":686881},[31676],[34,686883],{"className":686884,"style":315747},[31680],[34,686886,686888,686891],{"className":686887},[31685],[34,686889,67764],{"className":686890,"style":32062},[31685,31686],[34,686892,686894],{"className":686893},[31792],[34,686895,686897],{"className":686896},[31796],[34,686898,686900],{"className":686899},[31801],[34,686901,686903],{"className":686902,"style":37774},[31805],[34,686904,686905,686908],{"style":37777},[34,686906],{"className":686907,"style":31814},[31813],[34,686909,686911],{"className":686910},[31818,31819,31820,31821],[34,686912,768],{"className":686913},[31685,31821],[34,686915],{"className":686916,"style":31846},[31845],[34,686918,202],{"className":686919},[31850],[34,686921],{"className":686922,"style":31846},[31845],[34,686924,686926,686929],{"className":686925},[31676],[34,686927],{"className":686928,"style":37774},[31680],[34,686930,686932,686935],{"className":686931},[31685],[34,686933,111],{"className":686934},[31685,31686],[34,686936,686938],{"className":686937},[31792],[34,686939,686941],{"className":686940},[31796],[34,686942,686944],{"className":686943},[31801],[34,686945,686947],{"className":686946,"style":37774},[31805],[34,686948,686949,686952],{"style":37777},[34,686950],{"className":686951,"style":31814},[31813],[34,686953,686955],{"className":686954},[31818,31819,31820,31821],[34,686956,768],{"className":686957},[31685,31821],[108,686959,686960,686961,687055],{},"微分：\n",[34,686962,686964,686998],{"className":686963},[31642],[34,686965,686967],{"className":686966},[31646],[31648,686968,686969],{"xmlns":31650},[31652,686970,686971,686995],{},[31655,686972,686973,686975,686977,686979,686981,686983,686985,686987,686989,686991,686993],{},[31658,686974,46900],{"mathvariant":38079},[31658,686976,32936],{},[31662,686978,202],{},[31662,686980,32226],{},[31658,686982,111],{},[31658,686984,547853],{},[31662,686986,32682],{},[31658,686988,33249],{},[39361,686990,39363],{},[31658,686992,46900],{"mathvariant":38079},[31658,686994,33249],{},[31665,686996,686997],{"encoding":31667},"\\mathrm{d}x = -a \\sin t \\, \\mathrm{d}t",[34,686999,687001,687022],{"className":687000,"ariaHidden":758},[31672],[34,687002,687004,687007,687010,687013,687016,687019],{"className":687003},[31676],[34,687005],{"className":687006,"style":31681},[31680],[34,687008,46900],{"className":687009},[31685,207923],[34,687011,32936],{"className":687012},[31685,31686],[34,687014],{"className":687015,"style":31846},[31845],[34,687017,202],{"className":687018},[31850],[34,687020],{"className":687021,"style":31846},[31845],[34,687023,687025,687028,687031,687034,687037,687040,687043,687046,687049,687052],{"className":687024},[31676],[34,687026],{"className":687027,"style":34299},[31680],[34,687029,32226],{"className":687030},[31685],[34,687032,111],{"className":687033},[31685,31686],[34,687035],{"className":687036,"style":32123},[31845],[34,687038,547853],{"className":687039},[32767],[34,687041],{"className":687042,"style":32123},[31845],[34,687044,33249],{"className":687045},[31685,31686],[34,687047],{"className":687048,"style":32123},[31845],[34,687050,46900],{"className":687051},[31685,207923],[34,687053,33249],{"className":687054},[31685,31686],[34,687056,687058,687090],{"className":687057},[31642],[34,687059,687061],{"className":687060},[31646],[31648,687062,687063],{"xmlns":31650},[31652,687064,687065,687087],{},[31655,687066,687067,687069,687071,687073,687075,687077,687079,687081,687083,687085],{},[31658,687068,46900],{"mathvariant":38079},[31658,687070,67764],{},[31662,687072,202],{},[31658,687074,111],{},[31658,687076,547860],{},[31662,687078,32682],{},[31658,687080,33249],{},[39361,687082,39363],{},[31658,687084,46900],{"mathvariant":38079},[31658,687086,33249],{},[31665,687088,687089],{"encoding":31667},"\\mathrm{d}y = a \\cos t \\, \\mathrm{d}t",[34,687091,687093,687114],{"className":687092,"ariaHidden":758},[31672],[34,687094,687096,687099,687102,687105,687108,687111],{"className":687095},[31676],[34,687097],{"className":687098,"style":43217},[31680],[34,687100,46900],{"className":687101},[31685,207923],[34,687103,67764],{"className":687104,"style":32062},[31685,31686],[34,687106],{"className":687107,"style":31846},[31845],[34,687109,202],{"className":687110},[31850],[34,687112],{"className":687113,"style":31846},[31845],[34,687115,687117,687120,687123,687126,687129,687132,687135,687138,687141],{"className":687116},[31676],[34,687118],{"className":687119,"style":31681},[31680],[34,687121,111],{"className":687122},[31685,31686],[34,687124],{"className":687125,"style":32123},[31845],[34,687127,547860],{"className":687128},[32767],[34,687130],{"className":687131,"style":32123},[31845],[34,687133,33249],{"className":687134},[31685,31686],[34,687136],{"className":687137,"style":32123},[31845],[34,687139,46900],{"className":687140},[31685,207923],[34,687142,33249],{"className":687143},[31685,31686],[108,687145,687146,687147,33183,687218],{},"分子第一部分 ",[34,687148,687150,687176],{"className":687149},[31642],[34,687151,687153],{"className":687152},[31646],[31648,687154,687155],{"xmlns":31650},[31652,687156,687157,687173],{},[31655,687158,687159,687161,687163,687165,687167,687169,687171],{},[31662,687160,456],{"stretchy":702},[31658,687162,32936],{},[31662,687164,223],{},[31658,687166,67764],{},[31662,687168,2557],{"stretchy":702},[31658,687170,46900],{"mathvariant":38079},[31658,687172,32936],{},[31665,687174,687175],{"encoding":31667},"(x+y)\\mathrm{d}x",[34,687177,687179,687200],{"className":687178,"ariaHidden":758},[31672],[34,687180,687182,687185,687188,687191,687194,687197],{"className":687181},[31676],[34,687183],{"className":687184,"style":31774},[31680],[34,687186,456],{"className":687187},[31782],[34,687189,32936],{"className":687190},[31685,31686],[34,687192],{"className":687193,"style":32604},[31845],[34,687195,223],{"className":687196},[32024],[34,687198],{"className":687199,"style":32604},[31845],[34,687201,687203,687206,687209,687212,687215],{"className":687202},[31676],[34,687204],{"className":687205,"style":31774},[31680],[34,687207,67764],{"className":687208,"style":32062},[31685,31686],[34,687210,2557],{"className":687211},[31690],[34,687213,46900],{"className":687214},[31685,207923],[34,687216,32936],{"className":687217},[31685,31686],[34,687219,687221,687323],{"className":687220},[31642],[34,687222,687224],{"className":687223},[31646],[31648,687225,687226],{"xmlns":31650},[31652,687227,687228,687320],{},[31655,687229,687230,687232,687234,687236,687238,687240,687242,687244,687246,687248,687250,687252,687254,687256,687258,687260,687262,687264,687266,687268,687270,687272,687274,687276,687278,687284,687286,687288,687290,687292,687294,687296,687298,687300,687310,687312,687314,687316,687318],{},[31662,687231,202],{},[31662,687233,456],{"stretchy":702},[31658,687235,111],{},[31658,687237,547860],{},[31662,687239,32682],{},[31658,687241,33249],{},[31662,687243,223],{},[31658,687245,111],{},[31658,687247,547853],{},[31662,687249,32682],{},[31658,687251,33249],{},[31662,687253,2557],{"stretchy":702},[31662,687255,456],{"stretchy":702},[31662,687257,32226],{},[31658,687259,111],{},[31658,687261,547853],{},[31662,687263,32682],{},[31658,687265,33249],{},[31662,687267,2557],{"stretchy":702},[39361,687269,39363],{},[31658,687271,46900],{"mathvariant":38079},[31658,687273,33249],{},[31662,687275,202],{},[31662,687277,32226],{},[37624,687279,687280,687282],{},[31658,687281,111],{},[31749,687283,768],{},[31662,687285,456],{"stretchy":702},[31658,687287,547853],{},[31662,687289,32682],{},[31658,687291,33249],{},[31658,687293,547860],{},[31662,687295,32682],{},[31658,687297,33249],{},[31662,687299,223],{},[37624,687301,687302,687308],{},[31655,687303,687304,687306],{},[31658,687305,547853],{},[31662,687307,32682],{},[31749,687309,768],{},[31658,687311,33249],{},[31662,687313,2557],{"stretchy":702},[39361,687315,39363],{},[31658,687317,46900],{"mathvariant":38079},[31658,687319,33249],{},[31665,687321,687322],{"encoding":31667},"= (a \\cos t + a \\sin t)(-a \\sin t) \\, \\mathrm{d}t = -a^2 (\\sin t \\cos t + \\sin^2 t) \\, 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",[34,687566,687568,687596],{"className":687567},[31642],[34,687569,687571],{"className":687570},[31646],[31648,687572,687573],{"xmlns":31650},[31652,687574,687575,687593],{},[31655,687576,687577,687579,687581,687583,687585,687587,687589,687591],{},[31662,687578,32226],{},[31662,687580,456],{"stretchy":702},[31658,687582,32936],{},[31662,687584,32226],{},[31658,687586,67764],{},[31662,687588,2557],{"stretchy":702},[31658,687590,46900],{"mathvariant":38079},[31658,687592,67764],{},[31665,687594,687595],{"encoding":31667},"-(x-y)\\mathrm{d}y",[34,687597,687599,687623],{"className":687598,"ariaHidden":758},[31672],[34,687600,687602,687605,687608,687611,687614,687617,687620],{"className":687601},[31676],[34,687603],{"className":687604,"style":31774},[31680],[34,687606,32226],{"className":687607},[31685],[34,687609,456],{"className":687610},[31782],[34,687612,32936],{"className":687613},[31685,31686],[34,687615],{"className":687616,"style":32604},[31845],[34,687618,32226],{"className":687619},[32024],[34,687621],{"className":687622,"style":32604},[31845],[34,687624,687626,687629,687632,687635,687638],{"className":687625},[31676],[34,687627],{"className":687628,"style":31774},[31680],[34,687630,67764],{"className":687631,"style":32062},[31685,31686],[34,687633,2557],{"className":687634},[31690],[34,687636,46900],{"className":687637},[31685,207923],[34,687639,67764],{"className":687640,"style":32062},[31685,31686],[34,687642,687644,687746],{"className":687643},[31642],[34,687645,687647],{"className":687646},[31646],[31648,687648,687649],{"xmlns":31650},[31652,687650,687651,687743],{},[31655,687652,687653,687655,687657,687659,687661,687663,687665,687667,687669,687671,687673,687675,687677,687679,687681,687683,687685,687687,687689,687691,687693,687695,687697,687699,687701,687707,687709,687719,687721,687723,687725,687727,687729,687731,687733,687735,687737,687739,687741],{},[31662,687654,202],{},[31662,687656,32226],{},[31662,687658,456],{"stretchy":702},[31658,687660,111],{},[31658,687662,547860],{},[31662,687664,32682],{},[31658,687666,33249],{},[31662,687668,32226],{},[31658,687670,111],{},[31658,687672,547853],{},[31662,687674,32682],{},[31658,687676,33249],{},[31662,687678,2557],{"stretchy":702},[31662,687680,456],{"stretchy":702},[31658,687682,111],{},[31658,687684,547860],{},[31662,687686,32682],{},[31658,687688,33249],{},[31662,687690,2557],{"stretchy":702},[39361,687692,39363],{},[31658,687694,46900],{"mathvariant":38079},[31658,687696,33249],{},[31662,687698,202],{},[31662,687700,32226],{},[37624,687702,687703,687705],{},[31658,687704,111],{},[31749,687706,768],{},[31662,687708,456],{"stretchy":702},[37624,687710,687711,687717],{},[31655,687712,687713,687715],{},[31658,687714,547860],{},[31662,687716,32682],{},[31749,687718,768],{},[31658,687720,33249],{},[31662,687722,32226],{},[31658,687724,547853],{},[31662,687726,32682],{},[31658,687728,33249],{},[31658,687730,547860],{},[31662,687732,32682],{},[31658,687734,33249],{},[31662,687736,2557],{"stretchy":702},[39361,687738,39363],{},[31658,687740,46900],{"mathvariant":38079},[31658,687742,33249],{},[31665,687744,687745],{"encoding":31667},"= -(a \\cos t - a \\sin t)(a \\cos t) \\, \\mathrm{d}t = -a^2 (\\cos^2 t - \\sin t \\cos t) \\, \\mathrm{d}t",[34,687747,687749,687761,687797,687860,687945],{"className":687748,"ariaHidden":758},[31672],[34,687750,687752,687755,687758],{"className":687751},[31676],[34,687753],{"className":687754,"style":73089},[31680],[34,687756,202],{"className":687757},[31850],[34,687759],{"className":687760,"style":31846},[31845],[34,687762,687764,687767,687770,687773,687776,687779,687782,687785,687788,687791,687794],{"className":687763},[31676],[34,687765],{"className":687766,"style":31774},[31680],[34,687768,32226],{"className":687769},[31685],[34,687771,456],{"className":687772},[31782],[34,687774,111],{"className":687775},[31685,31686],[34,687777],{"className":687778,"style":32123},[31845],[34,687780,547860],{"className":687781},[32767],[34,687783],{"className":687784,"style":32123},[31845],[34,687786,33249],{"className":687787},[31685,31686],[34,687789],{"className":687790,"style":32604},[31845],[34,687792,32226],{"className":687793},[32024],[34,687795],{"className":687796,"style":32604},[31845],[34,687798,687800,687803,687806,687809,687812,687815,687818,687821,687824,687827,687830,687833,687836,687839,687842,687845,687848,687851,687854,687857],{"className":687799},[31676],[34,687801],{"className":687802,"style":31774},[31680],[34,687804,111],{"className":687805},[31685,31686],[34,687807],{"className":687808,"style":32123},[31845],[34,687810,547853],{"className":687811},[32767],[34,687813],{"className":687814,"style":32123},[31845],[34,687816,33249],{"className":687817},[31685,31686],[34,687819,2557],{"className":687820},[31690],[34,687822,456],{"className":687823},[31782],[34,687825,111],{"className":687826},[31685,31686],[34,687828],{"className":687829,"style":32123},[31845],[34,687831,547860],{"className":687832},[32767],[34,687834],{"className":687835,"style":32123},[31845],[34,687837,33249],{"className":687838},[31685,31686],[34,687840,2557],{"className":687841},[31690],[34,687843],{"className":687844,"style":32123},[31845],[34,687846,46900],{"className":687847},[31685,207923],[34,687849,33249],{"className":687850},[31685,31686],[34,687852],{"className":687853,"style":31846},[31845],[34,687855,202],{"className":687856},[31850],[34,687858],{"className":687859,"style":31846},[31845],[34,687861,687863,687866,687869,687898,687901,687930,687933,687936,687939,687942],{"className":687862},[31676],[34,687864],{"className":687865,"style":37755},[31680],[34,687867,32226],{"className":687868},[31685],[34,687870,687872,687875],{"className":687871},[31685],[34,687873,111],{"className":687874},[31685,31686],[34,687876,687878],{"className":687877},[31792],[34,687879,687881],{"className":687880},[31796],[34,687882,687884],{"className":687883},[31801],[34,687885,687887],{"className":687886,"style":37774},[31805],[34,687888,687889,687892],{"style":37777},[34,687890],{"className":687891,"style":31814},[31813],[34,687893,687895],{"className":687894},[31818,31819,31820,31821],[34,687896,768],{"className":687897},[31685,31821],[34,687899,456],{"className":687900},[31782],[34,687902,687904,687907],{"className":687903},[32767],[34,687905,547860],{"className":687906},[32767],[34,687908,687910],{"className":687909},[31792],[34,687911,687913],{"className":687912},[31796],[34,687914,687916],{"className":687915},[31801],[34,687917,687919],{"className":687918,"style":37774},[31805],[34,687920,687921,687924],{"style":37777},[34,687922],{"className":687923,"style":31814},[31813],[34,687925,687927],{"className":687926},[31818,31819,31820,31821],[34,687928,768],{"className":687929},[31685,31821],[34,687931],{"className":687932,"style":32123},[31845],[34,687934,33249],{"className":687935},[31685,31686],[34,687937],{"className":687938,"style":32604},[31845],[34,687940,32226],{"className":687941},[32024],[34,687943],{"className":687944,"style":32604},[31845],[34,687946,687948,687951,687954,687957,687960,687963,687966,687969,687972,687975,687978,687981],{"className":687947},[31676],[34,687949],{"className":687950,"style":31774},[31680],[34,687952,547853],{"className":687953},[32767],[34,687955],{"className":687956,"style":32123},[31845],[34,687958,33249],{"className":687959},[31685,31686],[34,687961],{"className":687962,"style":32123},[31845],[34,687964,547860],{"className":687965},[32767],[34,687967],{"className":687968,"style":32123},[31845],[34,687970,33249],{"className":687971},[31685,31686],[34,687973,2557],{"className":687974},[31690],[34,687976],{"className":687977,"style":32123},[31845],[34,687979,46900],{"className":687980},[31685,207923],[34,687982,33249],{"className":687983},[31685,31686],[108,687985,687986,687989,687990,688349,688351,688352,2664,688416,688719,688721,688722,688779,688780,347,688937],{},[2627,687987,687988],{},"合并分子","\n把上面两部分加起来：",[34,687991,687993],{"className":687992},[126385],[34,687994,687996,688089],{"className":687995},[31642],[34,687997,687999],{"className":687998},[31646],[31648,688000,688001],{"xmlns":31650,"display":126394},[31652,688002,688003,688086],{},[31655,688004,688005,688008,688010,688012,688018,688020,688022,688024,688026,688028,688030,688032,688034,688044,688046,688048,688050,688056,688058,688068,688070,688072,688074,688076,688078,688080,688082,688084],{},[39361,688006,688007],{},"Numerator",[31662,688009,202],{},[31662,688011,32226],{},[37624,688013,688014,688016],{},[31658,688015,111],{},[31749,688017,768],{},[31662,688019,456],{"stretchy":702},[31658,688021,547853],{},[31662,688023,32682],{},[31658,688025,33249],{},[31658,688027,547860],{},[31662,688029,32682],{},[31658,688031,33249],{},[31662,688033,223],{},[37624,688035,688036,688042],{},[31655,688037,688038,688040],{},[31658,688039,547853],{},[31662,688041,32682],{},[31749,688043,768],{},[31658,688045,33249],{},[31662,688047,2557],{"stretchy":702},[31662,688049,32226],{},[37624,688051,688052,688054],{},[31658,688053,111],{},[31749,688055,768],{},[31662,688057,456],{"stretchy":702},[37624,688059,688060,688066],{},[31655,688061,688062,688064],{},[31658,688063,547860],{},[31662,688065,32682],{},[31749,688067,768],{},[31658,688069,33249],{},[31662,688071,32226],{},[31658,688073,547853],{},[31662,688075,32682],{},[31658,688077,33249],{},[31658,688079,547860],{},[31662,688081,32682],{},[31658,688083,33249],{},[31662,688085,2557],{"stretchy":702},[31665,688087,688088],{"encoding":31667},"\\text{Numerator} = -a^2 (\\sin t \\cos t + \\sin^2 t) - a^2 (\\cos^2 t - \\sin t \\cos t)",[34,688090,688092,688113,688184,688237,688319],{"className":688091,"ariaHidden":758},[31672],[34,688093,688095,688098,688104,688107,688110],{"className":688094},[31676],[34,688096],{"className":688097,"style":33218},[31680],[34,688099,688101],{"className":688100},[31685,2638],[34,688102,688007],{"className":688103},[31685],[34,688105],{"className":688106,"style":31846},[31845],[34,688108,202],{"className":688109},[31850],[34,688111],{"className":688112,"style":31846},[31845],[34,688114,688116,688119,688122,688151,688154,688157,688160,688163,688166,688169,688172,688175,688178,688181],{"className":688115},[31676],[34,688117],{"className":688118,"style":145763},[31680],[34,688120,32226],{"className":688121},[31685],[34,688123,688125,688128],{"className":688124},[31685],[34,688126,111],{"className":688127},[31685,31686],[34,688129,688131],{"className":688130},[31792],[34,688132,688134],{"className":688133},[31796],[34,688135,688137],{"className":688136},[31801],[34,688138,688140],{"className":688139,"style":129832},[31805],[34,688141,688142,688145],{"style":129846},[34,688143],{"className":688144,"style":31814},[31813],[34,688146,688148],{"className":688147},[31818,31819,31820,31821],[34,688149,768],{"className":688150},[31685,31821],[34,688152,456],{"className":688153},[31782],[34,688155,547853],{"className":688156},[32767],[34,688158],{"className":688159,"style":32123},[31845],[34,688161,33249],{"className":688162},[31685,31686],[34,688164],{"className":688165,"style":32123},[31845],[34,688167,547860],{"className":688168},[32767],[34,688170],{"className":688171,"style":32123},[31845],[34,688173,33249],{"className":688174},[31685,31686],[34,688176],{"className":688177,"style":32604},[31845],[34,688179,223],{"className":688180},[32024],[34,688182],{"className":688183,"style":32604},[31845],[34,688185,688187,688190,688219,688222,688225,688228,688231,688234],{"className":688186},[31676],[34,688188],{"className":688189,"style":687514},[31680],[34,688191,688193,688196],{"className":688192},[32767],[34,688194,547853],{"className":688195},[32767],[34,688197,688199],{"className":688198},[31792],[34,688200,688202],{"className":688201},[31796],[34,688203,688205],{"className":688204},[31801],[34,688206,688208],{"className":688207,"style":585884},[31805],[34,688209,688210,688213],{"style":586008},[34,688211],{"className":688212,"style":31814},[31813],[34,688214,688216],{"className":688215},[31818,31819,31820,31821],[34,688217,768],{"className":688218},[31685,31821],[34,688220],{"className":688221,"style":32123},[31845],[34,688223,33249],{"className":688224},[31685,31686],[34,688226,2557],{"className":688227},[31690],[34,688229],{"className":688230,"style":32604},[31845],[34,688232,32226],{"className":688233},[32024],[34,688235],{"className":688236,"style":32604},[31845],[34,688238,688240,688243,688272,688275,688304,688307,688310,688313,688316],{"className":688239},[31676],[34,688241],{"className":688242,"style":145763},[31680],[34,688244,688246,688249],{"className":688245},[31685],[34,688247,111],{"className":688248},[31685,31686],[34,688250,688252],{"className":688251},[31792],[34,688253,688255],{"className":688254},[31796],[34,688256,688258],{"className":688257},[31801],[34,688259,688261],{"className":688260,"style":129832},[31805],[34,688262,688263,688266],{"style":129846},[34,688264],{"className":688265,"style":31814},[31813],[34,688267,688269],{"className":688268},[31818,31819,31820,31821],[34,688270,768],{"className":688271},[31685,31821],[34,688273,456],{"className":688274},[31782],[34,688276,688278,688281],{"className":688277},[32767],[34,688279,547860],{"className":688280},[32767],[34,688282,688284],{"className":688283},[31792],[34,688285,688287],{"className":688286},[31796],[34,688288,688290],{"className":688289},[31801],[34,688291,688293],{"className":688292,"style":129832},[31805],[34,688294,688295,688298],{"style":129846},[34,688296],{"className":688297,"style":31814},[31813],[34,688299,688301],{"className":688300},[31818,31819,31820,31821],[34,688302,768],{"className":688303},[31685,31821],[34,688305],{"className":688306,"style":32123},[31845],[34,688308,33249],{"className":688309},[31685,31686],[34,688311],{"className":688312,"style":32604},[31845],[34,688314,32226],{"className":688315},[32024],[34,688317],{"className":688318,"style":32604},[31845],[34,688320,688322,688325,688328,688331,688334,688337,688340,688343,688346],{"className":688321},[31676],[34,688323],{"className":688324,"style":31774},[31680],[34,688326,547853],{"className":688327},[32767],[34,688329],{"className":688330,"style":32123},[31845],[34,688332,33249],{"className":688333},[31685,31686],[34,688335],{"className":688336,"style":32123},[31845],[34,688338,547860],{"className":688339},[32767],[34,688341],{"className":688342,"style":32123},[31845],[34,688344,33249],{"className":688345},[31685,31686],[34,688347,2557],{"className":688348},[31690],[58531,688350],{},"提取 ",[34,688353,688355,688375],{"className":688354},[31642],[34,688356,688358],{"className":688357},[31646],[31648,688359,688360],{"xmlns":31650},[31652,688361,688362,688372],{},[31655,688363,688364,688366],{},[31662,688365,32226],{},[37624,688367,688368,688370],{},[31658,688369,111],{},[31749,688371,768],{},[31665,688373,688374],{"encoding":31667},"-a^2",[34,688376,688378],{"className":688377,"ariaHidden":758},[31672],[34,688379,688381,688384,688387],{"className":688380},[31676],[34,688382],{"className":688383,"style":40044},[31680],[34,688385,32226],{"className":688386},[31685],[34,688388,688390,688393],{"className":688389},[31685],[34,688391,111],{"className":688392},[31685,31686],[34,688394,688396],{"className":688395},[31792],[34,688397,688399],{"className":688398},[31796],[34,688400,688402],{"className":688401},[31801],[34,688403,688405],{"className":688404,"style":37774},[31805],[34,688406,688407,688410],{"style":37777},[34,688408],{"className":688409,"style":31814},[31813],[34,688411,688413],{"className":688412},[31818,31819,31820,31821],[34,688414,768],{"className":688415},[31685,31821],[34,688417,688419],{"className":688418},[126385],[34,688420,688422,688502],{"className":688421},[31642],[34,688423,688425],{"className":688424},[31646],[31648,688426,688427],{"xmlns":31650,"display":126394},[31652,688428,688429,688499],{},[31655,688430,688431,688433,688435,688441,688443,688445,688447,688449,688451,688453,688455,688457,688467,688469,688471,688481,688483,688485,688487,688489,688491,688493,688495,688497],{},[31662,688432,202],{},[31662,688434,32226],{},[37624,688436,688437,688439],{},[31658,688438,111],{},[31749,688440,768],{},[31662,688442,456],{"stretchy":702},[31658,688444,547853],{},[31662,688446,32682],{},[31658,688448,33249],{},[31658,688450,547860],{},[31662,688452,32682],{},[31658,688454,33249],{},[31662,688456,223],{},[37624,688458,688459,688465],{},[31655,688460,688461,688463],{},[31658,688462,547853],{},[31662,688464,32682],{},[31749,688466,768],{},[31658,688468,33249],{},[31662,688470,223],{},[37624,688472,688473,688479],{},[31655,688474,688475,688477],{},[31658,688476,547860],{},[31662,688478,32682],{},[31749,688480,768],{},[31658,688482,33249],{},[31662,688484,32226],{},[31658,688486,547853],{},[31662,688488,32682],{},[31658,688490,33249],{},[31658,688492,547860],{},[31662,688494,32682],{},[31658,688496,33249],{},[31662,688498,2557],{"stretchy":702},[31665,688500,688501],{"encoding":31667},"= -a^2 (\\sin t \\cos t + \\sin^2 t + \\cos^2 t - \\sin t \\cos t)",[34,688503,688505,688517,688588,688639,688689],{"className":688504,"ariaHidden":758},[31672],[34,688506,688508,688511,688514],{"className":688507},[31676],[34,688509],{"className":688510,"style":73089},[31680],[34,688512,202],{"className":688513},[31850],[34,688515],{"className":688516,"style":31846},[31845],[34,688518,688520,688523,688526,688555,688558,688561,688564,688567,688570,688573,688576,688579,688582,688585],{"className":688519},[31676],[34,688521],{"className":688522,"style":145763},[31680],[34,688524,32226],{"className":688525},[31685],[34,688527,688529,688532],{"className":688528},[31685],[34,688530,111],{"className":688531},[31685,31686],[34,688533,688535],{"className":688534},[31792],[34,688536,688538],{"className":688537},[31796],[34,688539,688541],{"className":688540},[31801],[34,688542,688544],{"className":688543,"style":129832},[31805],[34,688545,688546,688549],{"style":129846},[34,688547],{"className":688548,"style":31814},[31813],[34,688550,688552],{"className":688551},[31818,31819,31820,31821],[34,688553,768],{"className":688554},[31685,31821],[34,688556,456],{"className":688557},[31782],[34,688559,547853],{"className":688560},[32767],[34,688562],{"className":688563,"style":32123},[31845],[34,688565,33249],{"className":688566},[31685,31686],[34,688568],{"className":688569,"style":32123},[31845],[34,688571,547860],{"className":688572},[32767],[34,688574],{"className":688575,"style":32123},[31845],[34,688577,33249],{"className":688578},[31685,31686],[34,688580],{"className":688581,"style":32604},[31845],[34,688583,223],{"className":688584},[32024],[34,688586],{"className":688587,"style":32604},[31845],[34,688589,688591,688595,688624,688627,688630,688633,688636],{"className":688590},[31676],[34,688592],{"className":688593,"style":688594},[31680],"height:0.9552em;vertical-align:-0.0833em;",[34,688596,688598,688601],{"className":688597},[32767],[34,688599,547853],{"className":688600},[32767],[34,688602,688604],{"className":688603},[31792],[34,688605,688607],{"className":688606},[31796],[34,688608,688610],{"className":688609},[31801],[34,688611,688613],{"className":688612,"style":585884},[31805],[34,688614,688615,688618],{"style":586008},[34,688616],{"className":688617,"style":31814},[31813],[34,688619,688621],{"className":688620},[31818,31819,31820,31821],[34,688622,768],{"className":688623},[31685,31821],[34,688625],{"className":688626,"style":32123},[31845],[34,688628,33249],{"className":688629},[31685,31686],[34,688631],{"className":688632,"style":32604},[31845],[34,688634,223],{"className":688635},[32024],[34,688637],{"className":688638,"style":32604},[31845],[34,688640,688642,688645,688674,688677,688680,688683,688686],{"className":688641},[31676],[34,688643],{"className":688644,"style":147646},[31680],[34,688646,688648,688651],{"className":688647},[32767],[34,688649,547860],{"className":688650},[32767],[34,688652,688654],{"className":688653},[31792],[34,688655,688657],{"className":688656},[31796],[34,688658,688660],{"className":688659},[31801],[34,688661,688663],{"className":688662,"style":129832},[31805],[34,688664,688665,688668],{"style":129846},[34,688666],{"className":688667,"style":31814},[31813],[34,688669,688671],{"className":688670},[31818,31819,31820,31821],[34,688672,768],{"className":688673},[31685,31821],[34,688675],{"className":688676,"style":32123},[31845],[34,688678,33249],{"className":688679},[31685,31686],[34,688681],{"className":688682,"style":32604},[31845],[34,688684,32226],{"className":688685},[32024],[34,688687],{"className":688688,"style":32604},[31845],[34,688690,688692,688695,688698,688701,688704,688707,688710,688713,688716],{"className":688691},[31676],[34,688693],{"className":688694,"style":31774},[31680],[34,688696,547853],{"className":688697},[32767],[34,688699],{"className":688700,"style":32123},[31845],[34,688702,33249],{"className":688703},[31685,31686],[34,688705],{"className":688706,"style":32123},[31845],[34,688708,547860],{"className":688709},[32767],[34,688711],{"className":688712,"style":32123},[31845],[34,688714,33249],{"className":688715},[31685,31686],[34,688717,2557],{"className":688718},[31690],[58531,688720],{},"中间的 ",[34,688723,688725,688749],{"className":688724},[31642],[34,688726,688728],{"className":688727},[31646],[31648,688729,688730],{"xmlns":31650},[31652,688731,688732,688746],{},[31655,688733,688734,688736,688738,688740,688742,688744],{},[31658,688735,547853],{},[31662,688737,32682],{},[31658,688739,33249],{},[31658,688741,547860],{},[31662,688743,32682],{},[31658,688745,33249],{},[31665,688747,688748],{"encoding":31667},"\\sin t \\cos t",[34,688750,688752],{"className":688751,"ariaHidden":758},[31672],[34,688753,688755,688758,688761,688764,688767,688770,688773,688776],{"className":688754},[31676],[34,688756],{"className":688757,"style":71430},[31680],[34,688759,547853],{"className":688760},[32767],[34,688762],{"className":688763,"style":32123},[31845],[34,688765,33249],{"className":688766},[31685,31686],[34,688768],{"className":688769,"style":32123},[31845],[34,688771,547860],{"className":688772},[32767],[34,688774],{"className":688775,"style":32123},[31845],[34,688777,33249],{"className":688778},[31685,31686]," 抵消了，",[34,688781,688783,688825],{"className":688782},[31642],[34,688784,688786],{"className":688785},[31646],[31648,688787,688788],{"xmlns":31650},[31652,688789,688790,688822],{},[31655,688791,688792,688802,688804,688806,688816,688818,688820],{},[37624,688793,688794,688800],{},[31655,688795,688796,688798],{},[31658,688797,547853],{},[31662,688799,32682],{},[31749,688801,768],{},[31658,688803,33249],{},[31662,688805,223],{},[37624,688807,688808,688814],{},[31655,688809,688810,688812],{},[31658,688811,547860],{},[31662,688813,32682],{},[31749,688815,768],{},[31658,688817,33249],{},[31662,688819,202],{},[31749,688821,640],{},[31665,688823,688824],{"encoding":31667},"\\sin^2 t + \\cos^2 t = 1",[34,688826,688828,688878,688928],{"className":688827,"ariaHidden":758},[31672],[34,688829,688831,688834,688863,688866,688869,688872,688875],{"className":688830},[31676],[34,688832],{"className":688833,"style":688594},[31680],[34,688835,688837,688840],{"className":688836},[32767],[34,688838,547853],{"className":688839},[32767],[34,688841,688843],{"className":688842},[31792],[34,688844,688846],{"className":688845},[31796],[34,688847,688849],{"className":688848},[31801],[34,688850,688852],{"className":688851,"style":585884},[31805],[34,688853,688854,688857],{"style":586008},[34,688855],{"className":688856,"style":31814},[31813],[34,688858,688860],{"className":688859},[31818,31819,31820,31821],[34,688861,768],{"className":688862},[31685,31821],[34,688864],{"className":688865,"style":32123},[31845],[34,688867,33249],{"className":688868},[31685,31686],[34,688870],{"className":688871,"style":32604},[31845],[34,688873,223],{"className":688874},[32024],[34,688876],{"className":688877,"style":32604},[31845],[34,688879,688881,688884,688913,688916,688919,688922,688925],{"className":688880},[31676],[34,688882],{"className":688883,"style":37774},[31680],[34,688885,688887,688890],{"className":688886},[32767],[34,688888,547860],{"className":688889},[32767],[34,688891,688893],{"className":688892},[31792],[34,688894,688896],{"className":688895},[31796],[34,688897,688899],{"className":688898},[31801],[34,688900,688902],{"className":688901,"style":37774},[31805],[34,688903,688904,688907],{"style":37777},[34,688905],{"className":688906,"style":31814},[31813],[34,688908,688910],{"className":688909},[31818,31819,31820,31821],[34,688911,768],{"className":688912},[31685,31821],[34,688914],{"className":688915,"style":32123},[31845],[34,688917,33249],{"className":688918},[31685,31686],[34,688920],{"className":688921,"style":31846},[31845],[34,688923,202],{"className":688924},[31850],[34,688926],{"className":688927,"style":31846},[31845],[34,688929,688931,688934],{"className":688930},[31676],[34,688932],{"className":688933,"style":32617},[31680],[34,688935,640],{"className":688936},[31685],[34,688938,688940],{"className":688939},[126385],[34,688941,688943,688981],{"className":688942},[31642],[34,688944,688946],{"className":688945},[31646],[31648,688947,688948],{"xmlns":31650,"display":126394},[31652,688949,688950,688978],{},[31655,688951,688952,688954,688956,688962,688964,688966,688968,688970,688972],{},[31662,688953,202],{},[31662,688955,32226],{},[37624,688957,688958,688960],{},[31658,688959,111],{},[31749,688961,768],{},[31662,688963,456],{"stretchy":702},[31749,688965,640],{},[31662,688967,2557],{"stretchy":702},[31662,688969,202],{},[31662,688971,32226],{},[37624,688973,688974,688976],{},[31658,688975,111],{},[31749,688977,768],{},[31665,688979,688980],{"encoding":31667},"= -a^2 (1) = 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x \\, \\mathrm{d}x + y \\, \\mathrm{d}y + (x + y - 1) \\, \\mathrm{d}z",[34,689659,689661,689731,689758,689779,689797],{"className":689660,"ariaHidden":758},[31672],[34,689662,689664,689667,689707,689710,689713,689716,689719,689722,689725,689728],{"className":689663},[31676],[34,689665],{"className":689666,"style":637320},[31680],[34,689668,689670,689673],{"className":689669},[32767],[34,689671,546295],{"className":689672,"style":546331},[32767,131249,156560],[34,689674,689676],{"className":689675},[31792],[34,689677,689679,689699],{"className":689678},[31796,31797],[34,689680,689682,689696],{"className":689681},[31801],[34,689683,689685],{"className":689684,"style":637339},[31805],[34,689686,689687,689690],{"style":546347},[34,689688],{"className":689689,"style":31814},[31813],[34,689691,689693],{"className":689692},[31818,31819,31820,31821],[34,689694,33206],{"className":689695},[31685,31686,31821],[34,689697,31829],{"className":689698},[31828],[34,689700,689702],{"className":689701},[31801],[34,689703,689705],{"className":689704,"style":546384},[31805],[34,689706],{},[34,689708],{"className":689709,"style":32123},[31845],[34,689711,32936],{"className":689712},[31685,31686],[34,689714],{"className":689715,"style":32123},[31845],[34,689717,46900],{"className":689718},[31685,207923],[34,689720,32936],{"className":689721},[31685,31686],[34,689723],{"className":689724,"style":32604},[31845],[34,689726,223],{"className":689727},[32024],[34,689729],{"className":689730,"style":32604},[31845],[34,689732,689734,689737,689740,689743,689746,689749,689752,689755],{"className":689733},[31676],[34,689735],{"className":689736,"style":43217},[31680],[34,689738,67764],{"className":689739,"style":32062},[31685,31686],[34,689741],{"className":689742,"style":32123},[31845],[34,689744,46900],{"className":689745},[31685,207923],[34,689747,67764],{"className":689748,"style":32062},[31685,31686],[34,689750],{"className":689751,"style":32604},[31845],[34,689753,223],{"className":689754},[32024],[34,689756],{"className":689757,"style":32604},[31845],[34,689759,689761,689764,689767,689770,689773,689776],{"className":689760},[31676],[34,689762],{"className":689763,"style":31774},[31680],[34,689765,456],{"className":689766},[31782],[34,689768,32936],{"className":689769},[31685,31686],[34,689771],{"className":689772,"style":32604},[31845],[34,689774,223],{"className":689775},[32024],[34,689777],{"className":689778,"style":32604},[31845],[34,689780,689782,689785,689788,689791,689794],{"className":689781},[31676],[34,689783],{"className":689784,"style":55949},[31680],[34,689786,67764],{"className":689787,"style":32062},[31685,31686],[34,689789],{"className":689790,"style":32604},[31845],[34,689792,32226],{"className":689793},[32024],[34,689795],{"className":689796,"style":32604},[31845],[34,689798,689800,689803,689806,689809,689812,689815],{"className":689799},[31676],[34,689801],{"className":689802,"style":31774},[31680],[34,689804,640],{"className":689805},[31685],[34,689807,2557],{"className":689808},[31690],[34,689810],{"className":689811,"style":32123},[31845],[34,689813,46900],{"className":689814},[31685,207923],[34,689816,382333],{"className":689817,"style":382406},[31685,31686],[2627,689819,658358,689820],{},[34,689821,689823,689836],{"className":689822},[31642],[34,689824,689826],{"className":689825},[31646],[31648,689827,689828],{"xmlns":31650},[31652,689829,689830,689834],{},[31655,689831,689832],{},[31658,689833,33206],{},[31665,689835,33206],{"encoding":31667},[34,689837,689839],{"className":689838,"ariaHidden":758},[31672],[34,689840,689842,689845],{"className":689841},[31676],[34,689843],{"className":689844,"style":33218},[31680],[34,689846,33206],{"className":689847},[31685,31686],[34,689849,689851,689879],{"className":689850},[31642],[34,689852,689854],{"className":689853},[31646],[31648,689855,689856],{"xmlns":31650},[31652,689857,689858,689876],{},[31655,689859,689860,689862,689864,689866,689868,689870,689872,689874],{},[31658,689861,54151],{},[31662,689863,456],{"stretchy":702},[31749,689865,640],{},[31662,689867,31960],{"separator":758},[31749,689869,640],{},[31662,689871,31960],{"separator":758},[31749,689873,640],{},[31662,689875,2557],{"stretchy":702},[31665,689877,689878],{"encoding":31667},"A(1,1,1)",[34,689880,689882],{"className":689881,"ariaHidden":758},[31672],[34,689883,689885,689888,689891,689894,689897,689900,689903,689906,689909,689912,689915],{"className":689884},[31676],[34,689886],{"className":689887,"style":31774},[31680],[34,689889,54151],{"className":689890},[31685,31686],[34,689892,456],{"className":689893},[31782],[34,689895,640],{"className":689896},[31685],[34,689898,31960],{"className":689899},[32119],[34,689901],{"className":689902,"style":32123},[31845],[34,689904,640],{"className":689905},[31685],[34,689907,31960],{"className":689908},[32119],[34,689910],{"className":689911,"style":32123},[31845],[34,689913,640],{"className":689914},[31685],[34,689916,2557],{"className":689917},[31690],[34,689919,689921,689949],{"className":689920},[31642],[34,68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",[34,690006,690008,690035],{"className":690007},[31642],[34,690009,690011],{"className":690010},[31646],[31648,690012,690013],{"xmlns":31650},[31652,690014,690015,690033],{},[31655,690016,690017,690019,690021,690023,690025,690027,690029,690031],{},[31658,690018,54151],{},[31662,690020,456],{"stretchy":702},[31749,690022,640],{},[31662,690024,31960],{"separator":758},[31749,690026,640],{},[31662,690028,31960],{"separator":758},[31749,690030,640],{},[31662,690032,2557],{"stretchy":702},[31665,690034,689878],{"encoding":31667},[34,690036,690038],{"className":690037,"ariaHidden":758},[31672],[34,690039,690041,690044,690047,690050,690053,690056,690059,690062,690065,690068,690071],{"className":690040},[31676],[34,690042],{"className":690043,"style":31774},[31680],[34,690045,54151],{"className":690046},[31685,31686],[34,690048,456],{"className":690049},[31782],[34,690051,640],{"className":690052},[31685],[34,690054,31960],{"className":690055},[32119],[34,690057],{"className":690058,"style":32123},[31845],[34,690060,640],{"className":690061},[31685],[34,690063,31960],{"className":690064},[32119],[34,690066],{"className":690067,"style":32123},[31845],[34,690069,640],{"className":690070},[31685],[34,690072,2557],{"className":690073},[31690],[34,690075,690077,690104],{"className":690076},[31642],[34,690078,690080],{"className":690079},[31646],[31648,690081,690082],{"xmlns":31650},[31652,690083,690084,690102],{},[31655,690085,690086,690088,690090,690092,690094,690096,690098,690100],{},[31658,690087,54224],{},[31662,690089,456],{"stretchy":702},[31749,690091,768],{},[31662,690093,31960],{"separator":758},[31749,690095,786],{},[31662,690097,31960],{"separator":758},[31749,690099,15932],{},[31662,690101,2557],{"stretchy":702},[31665,690103,689948],{"encoding":31667},[34,690105,690107],{"className":690106,"ariaHidden":758},[31672],[34,690108,690110,690113,690116,690119,690122,690125,690128,690131,690134,690137,690140],{"className":690109},[31676],[34,690111],{"className":690112,"style":31774},[31680],[34,690114,54224],{"className":690115,"style":54252},[31685,31686],[34,690117,456],{"className":690118},[31782],[34,690120,768],{"className":690121},[31685],[34,690123,31960],{"className":690124},[32119],[34,690126],{"className":690127,"style":32123},[31845],[34,690129,786],{"className":690130},[31685],[34,690132,31960],{"className":690133},[32119],[34,690135],{"className":690136,"style":32123},[31845],[34,690138,15932],{"className":690139},[31685],[34,690141,2557],{"className":690142},[31690],"。\n方向向量 ",[34,690145,690147,690218],{"className":690146},[31642],[34,690148,690150],{"className":690149},[31646],[31648,690151,690152],{"xmlns":31650},[31652,690153,690154,690215],{},[31655,690155,690156,690163,690165,690167,690169,690171,690173,690175,690177,690179,690181,690183,690185,690187,690189,690191,690193,690195,690197,690199,690201,690203,690205,690207,690209,690211,690213],{},[134014,690157,690158,690160],{"accent":758},[31658,690159,56782],{},[31662,690161,690162],{},"⃗",[31662,690164,202],{},[31658,690166,54224],{},[31662,690168,32226],{},[31658,690170,54151],{},[31662,690172,202],{},[31662,690174,456],{"stretchy":702},[31749,690176,768],{},[31662,690178,32226],{},[31749,690180,640],{},[31662,690182,31960],{"separator":758},[31749,690184,786],{},[31662,690186,32226],{},[31749,690188,640],{},[31662,690190,31960],{"separator":758},[31749,690192,15932],{},[31662,690194,32226],{},[31749,690196,640],{},[31662,690198,2557],{"stretchy":702},[31662,690200,202],{},[31662,690202,456],{"stretchy":702},[31749,690204,640],{},[31662,690206,31960],{"separator":758},[31749,690208,768],{},[31662,690210,31960],{"separator":758},[31749,690212,786],{},[31662,690214,2557],{"stretchy":702},[31665,690216,690217],{"encoding":31667},"\\vec{v} = B - A = (2-1, 3-1, 4-1) = (1, 2, 3)",[34,690219,690221,690280,690298,690316,690337,690364,690391,690412],{"className":690220,"ariaHidden":758},[31672],[34,690222,690224,690228,690271,690274,690277],{"className":690223},[31676],[34,690225],{"className":690226,"style":690227},[31680],"height:0.714em;",[34,690229,690231],{"className":690230},[31685,346171],[34,690232,690234],{"className":690233},[31796],[34,690235,690237],{"className":690236},[31801],[34,690238,690240,690248],{"className":690239,"style":690227},[31805],[34,690241,690242,690245],{"style":41565},[34,690243],{"className":690244,"style":41508},[31813],[34,690246,56782],{"className":690247,"style":32062},[31685,31686],[34,690249,690250,690253],{"style":41565},[34,690251],{"className":690252,"style":41508},[31813],[34,690254,690257],{"className":690255,"style":690256},[346198],"left:-0.2077em;",[34,690258,690262],{"className":690259,"style":690261},[690260],"overlay","height:0.714em;width:0.471em;",[41588,690263,690268],{"xmlns":41590,"width":690264,"height":690265,"style":690266,"viewBox":690267,"preserveAspectRatio":71323},"0.471em","0.714em","width:0.471em","0 0 471 714",[20398,690269],{"d":690270},"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",[34,690272],{"className":690273,"style":31846},[31845],[34,690275,202],{"className":690276},[31850],[34,690278],{"className":690279,"style":31846},[31845],[34,690281,690283,690286,690289,690292,690295],{"className":690282},[31676],[34,690284],{"className":690285,"style":34944},[31680],[34,690287,54224],{"className":690288,"style":54252},[31685,31686],[34,690290],{"className":690291,"style":32604},[31845],[34,690293,32226],{"className":690294},[32024],[34,690296],{"className":690297,"style":32604},[31845],[34,690299,690301,690304,690307,690310,690313],{"className":690300},[31676],[34,690302],{"className":690303,"style":33218},[31680],[34,690305,54151],{"className":690306},[31685,31686],[34,690308],{"className":690309,"style":31846},[31845],[34,690311,202],{"className":690312},[31850],[34,690314],{"className":690315,"style":31846},[31845],[34,690317,690319,690322,690325,690328,690331,690334],{"className":690318},[31676],[34,690320],{"className":690321,"style":31774},[31680],[34,690323,456],{"className":690324},[31782],[34,690326,768],{"className":690327},[31685],[34,690329],{"className":690330,"style":32604},[31845],[34,690332,32226],{"className":690333},[32024],[34,690335],{"className":690336,"style":32604},[31845],[34,690338,690340,690343,690346,690349,690352,690355,690358,690361],{"className":690339},[31676],[34,690341],{"className":690342,"style":40292},[31680],[34,690344,640],{"className":690345},[31685],[34,690347,31960],{"className":690348},[32119],[34,690350],{"className":690351,"style":32123},[31845],[34,690353,786],{"className":690354},[31685],[34,690356],{"className":690357,"style":32604},[31845],[34,690359,32226],{"className":690360},[32024],[34,690362],{"className":690363,"style":32604},[31845],[34,690365,690367,690370,690373,690376,690379,690382,690385,690388],{"className":690366},[31676],[34,690368],{"className":690369,"style":40292},[31680],[34,690371,640],{"className":690372},[31685],[34,690374,31960],{"className":690375},[32119],[34,690377],{"className":690378,"style":32123},[31845],[34,690380,15932],{"className":690381},[31685],[34,690383],{"className":690384,"style":32604},[31845],[34,690386,32226],{"className":690387},[32024],[34,690389],{"className":690390,"style":32604},[31845],[34,690392,690394,690397,690400,690403,690406,690409],{"className":690393},[31676],[34,690395],{"className":690396,"style":31774},[31680],[34,690398,640],{"className":690399},[31685],[34,690401,2557],{"className":690402},[31690],[34,690404],{"className":690405,"style":31846},[31845],[34,690407,202],{"className":690408},[31850],[34,690410],{"className":690411,"style":31846},[31845],[34,690413,690415,690418,690421,690424,690427,690430,690433,690436,690439,690442],{"className":690414},[31676],[34,690416],{"className":690417,"style":31774},[31680],[34,690419,456],{"className":690420},[31782],[34,690422,640],{"className":690423},[31685],[34,690425,31960],{"className":690426},[32119],[34,690428],{"className":690429,"style":32123},[31845],[34,690431,768],{"className":690432},[31685],[34,690434,31960],{"className":690435},[32119],[34,690437],{"className":690438,"style":32123},[31845],[34,690440,786],{"className":690441},[31685],[34,690443,2557],{"className":690444},[31690],"。\n令参数 ",[34,690447,690449,690462],{"className":690448},[31642],[34,690450,690452],{"className":690451},[31646],[31648,690453,690454],{"xmlns":31650},[31652,690455,690456,690460],{},[31655,690457,690458],{},[31658,690459,33249],{},[31665,690461,33249],{"encoding":31667},[34,690463,690465],{"className":690464,"ariaHidden":758},[31672],[34,690466,690468,690471],{"className":690467},[31676],[34,690469],{"className":690470,"style":33280},[31680],[34,690472,33249],{"className":690473},[31685,31686],[34,690475,690477,690490],{"className":690476},[31642],[34,690478,690480],{"className":690479},[31646],[31648,690481,690482],{"xmlns":31650},[31652,690483,690484,690488],{},[31655,690485,690486],{},[31749,690487,459],{},[31665,690489,459],{"encoding":31667},[34,690491,690493],{"className":690492,"ariaHidden":758},[31672],[34,690494,690496,690499],{"className":690495},[31676],[34,690497],{"className":690498,"style":32617},[31680],[34,690500,459],{"className":690501},[31685],[34,690503,690505,690518],{"className":690504},[31642],[34,690506,690508],{"className":690507},[31646],[31648,690509,690510],{"xmlns":31650},[31652,690511,690512,690516],{},[31655,690513,690514],{},[31749,690515,640],{},[31665,690517,640],{"encoding":31667},[34,690519,690521],{"className":690520,"ariaHidden":758},[31672],[34,690522,690524,690527],{"className":690523},[31676],[34,690525],{"className":690526,"style":32617},[31680],[34,690528,640],{"className":690529},[31685],[105,690531,690532,690673,690819],{},[108,690533,690534],{},[34,690535,690537,690571],{"className":690536},[31642],[34,690538,690540],{"className":690539},[31646],[31648,690541,690542],{"xmlns":31650},[31652,690543,690544,690568],{},[31655,690545,690546,690548,690550,690552,690554,690556,690558,690560,690562,690564,690566],{},[31658,690547,32936],{},[31662,690549,202],{},[31749,690551,640],{},[31662,690553,223],{},[31749,690555,640],{},[31662,690557,71191],{},[31658,690559,33249],{},[31662,690561,202],{},[31749,690563,640],{},[31662,690565,223],{},[31658,690567,33249],{},[31665,690569,690570],{"encoding":31667},"x = 1 + 1\\cdot t = 1 + t",[34,690572,690574,690592,690610,690628,690646,690664],{"className":690573,"ariaHidden":758},[31672],[34,690575,690577,690580,690583,690586,690589],{"className":690576},[31676],[34,690578],{"className":690579,"style":32527},[31680],[34,690581,32936],{"className":690582},[31685,31686],[34,690584],{"className":690585,"style":31846},[31845],[34,690587,202],{"className":690588},[31850],[34,690590],{"className":690591,"style":31846},[31845],[34,690593,690595,690598,690601,690604,690607],{"className":690594},[31676],[34,690596],{"className":690597,"style":35942},[31680],[34,690599,640],{"className":690600},[31685],[34,690602],{"className":690603,"style":32604},[31845],[34,690605,223],{"className":690606},[32024],[34,690608],{"className":690609,"style":32604},[31845],[34,690611,690613,690616,690619,690622,690625],{"className":690612},[31676],[34,690614],{"className":690615,"style":32617},[31680],[34,690617,640],{"className":690618},[31685],[34,690620],{"className":690621,"style":32604},[31845],[34,690623,71191],{"className":690624},[32024],[34,690626],{"className":690627,"style":32604},[31845],[34,690629,690631,690634,690637,690640,690643],{"className":690630},[31676],[34,690632],{"className":690633,"style":33280},[31680],[34,690635,33249],{"className":690636},[31685,31686],[34,690638],{"className":690639,"style":31846},[31845],[34,690641,202],{"className":690642},[31850],[34,690644],{"className":690645,"style":31846},[31845],[34,690647,690649,690652,690655,690658,690661],{"className":690648},[31676],[34,690650],{"className":690651,"style":35942},[31680],[34,690653,640],{"className":690654},[31685],[34,690656],{"className":690657,"style":32604},[31845],[34,690659,223],{"className":690660},[32024],[34,690662],{"className":690663,"style":32604},[31845],[34,690665,690667,690670],{"className":690666},[31676],[34,690668],{"className":690669,"style":33280},[31680],[34,690671,33249],{"className":690672},[31685,31686],[108,690674,690675],{},[34,690676,690678,690714],{"className":690677},[31642],[34,690679,690681],{"className":690680},[31646],[31648,690682,690683],{"xmlns":31650},[31652,690684,690685,690711],{},[31655,690686,690687,690689,690691,690693,690695,690697,690699,690701,690703,690705,690707,690709],{},[31658,690688,67764],{},[31662,690690,202],{},[31749,690692,640],{},[31662,690694,223],{},[31749,690696,768],{},[31662,690698,71191],{},[31658,690700,33249],{},[31662,690702,202],{},[31749,690704,640],{},[31662,690706,223],{},[31749,690708,768],{},[31658,690710,33249],{},[31665,690712,690713],{"encoding":31667},"y = 1 + 2\\cdot t = 1 + 2t",[34,690715,690717,690735,690753,690771,690789,690807],{"className":690716,"ariaHidden":758},[31672],[34,690718,690720,690723,690726,690729,690732],{"className":690719},[31676],[34,690721],{"className":690722,"style":38774},[31680],[34,690724,67764],{"className":690725,"style":32062},[31685,31686],[34,690727],{"className":690728,"style":31846},[31845],[34,690730,202],{"className":690731},[31850],[34,690733],{"className":690734,"style":31846},[31845],[34,690736,690738,690741,690744,690747,690750],{"className":690737},[31676],[34,690739],{"className":690740,"style":35942},[31680],[34,690742,640],{"className":690743},[31685],[34,690745],{"className":690746,"style":32604},[31845],[34,690748,223],{"className":690749},[32024],[34,690751],{"className":690752,"style":32604},[31845],[34,690754,690756,690759,690762,690765,690768],{"className":690755},[31676],[34,690757],{"className":690758,"style":32617},[31680],[34,690760,768],{"className":690761},[31685],[34,690763],{"className":690764,"style":32604},[31845],[34,690766,71191],{"className":690767},[32024],[34,690769],{"className":690770,"style":32604},[31845],[34,690772,690774,690777,690780,690783,690786],{"className":690773},[31676],[34,690775],{"className":690776,"style":33280},[31680],[34,690778,33249],{"className":690779},[31685,31686],[34,690781],{"className":690782,"style":31846},[31845],[34,690784,202],{"className":690785},[31850],[34,690787],{"className":690788,"style":31846},[31845],[34,690790,690792,690795,690798,690801,690804],{"className":690791},[31676],[34,690793],{"className":690794,"style":35942},[31680],[34,690796,640],{"className":690797},[31685],[34,690799],{"className":690800,"style":32604},[31845],[34,690802,223],{"className":690803},[32024],[34,690805],{"className":690806,"style":32604},[31845],[34,690808,690810,690813,690816],{"className":690809},[31676],[34,690811],{"className":690812,"style":32617},[31680],[34,690814,768],{"className":690815},[31685],[34,690817,33249],{"className":690818},[31685,31686],[108,690820,690821],{},[34,690822,690824,690860],{"className":690823},[31642],[34,690825,690827],{"className":690826},[31646],[31648,690828,690829],{"xmlns":31650},[31652,690830,690831,690857],{},[31655,690832,690833,690835,690837,690839,690841,690843,690845,690847,690849,690851,690853,690855],{},[31658,690834,382333],{},[31662,690836,202],{},[31749,690838,640],{},[31662,690840,223],{},[31749,690842,786],{},[31662,690844,71191],{},[31658,690846,33249],{},[31662,690848,202],{},[31749,690850,640],{},[31662,690852,223],{},[31749,690854,786],{},[31658,690856,33249],{},[31665,690858,690859],{"encoding":31667},"z = 1 + 3\\cdot t = 1 + 3t",[34,690861,690863,690881,690899,690917,690935,690953],{"className":690862,"ariaHidden":758},[31672],[34,690864,690866,690869,690872,690875,690878],{"className":690865},[31676],[34,690867],{"className":690868,"style":32527},[31680],[34,690870,382333],{"className":690871,"style":382406},[31685,31686],[34,690873],{"className":690874,"style":31846},[31845],[34,690876,202],{"className":690877},[31850],[34,690879],{"className":690880,"style":31846},[31845],[34,690882,690884,690887,690890,690893,690896],{"className":690883},[31676],[34,690885],{"className":690886,"style":35942},[31680],[34,690888,640],{"className":690889},[31685],[34,690891],{"className":690892,"style":32604},[31845],[34,690894,223],{"className":690895},[32024],[34,690897],{"className":690898,"style":32604},[31845],[34,690900,690902,690905,690908,690911,690914],{"className":690901},[31676],[34,690903],{"className":690904,"style":32617},[31680],[34,690906,786],{"className":690907},[31685],[34,690909],{"className":690910,"style":32604},[31845],[34,690912,71191],{"className":690913},[32024],[34,690915],{"className":690916,"style":32604},[31845],[34,690918,690920,690923,690926,690929,690932],{"className":690919},[31676],[34,690921],{"className":690922,"style":33280},[31680],[34,690924,33249],{"className":690925},[31685,31686],[34,690927],{"className":690928,"style":31846},[31845],[34,690930,202],{"className":690931},[31850],[34,690933],{"className":690934,"style":31846},[31845],[34,690936,690938,690941,690944,690947,690950],{"className":690937},[31676],[34,690939],{"className":690940,"style":35942},[31680],[34,690942,640],{"className":690943},[31685],[34,690945],{"className":690946,"style":32604},[31845],[34,690948,223],{"className":690949},[32024],[34,690951],{"className":690952,"style":32604},[31845],[34,690954,690956,690959,690962],{"className":690955},[31676],[34,690957],{"className":690958,"style":32617},[31680],[34,690960,786],{"className":690961},[31685],[34,690963,33249],{"className":690964},[31685,31686],[108,690966,690967,690969],{},[2627,690968,659488],{},[105,690970,690971,691034,691107],{},[108,690972,690973],{},[34,690974,690976,690998],{"className":690975},[31642],[34,690977,690979],{"className":690978},[31646],[31648,690980,690981],{"xmlns":31650},[31652,690982,690983,690995],{},[31655,690984,690985,690987,690989,690991,690993],{},[31658,690986,46900],{"mathvariant":38079},[31658,690988,32936],{},[31662,690990,202],{},[31658,690992,46900],{"mathvariant":38079},[31658,690994,33249],{},[31665,690996,690997],{"encoding":31667},"\\mathrm{d}x = \\mathrm{d}t",[34,690999,691001,691022],{"className":691000,"ariaHidden":758},[31672],[34,691002,691004,691007,691010,691013,691016,691019],{"className":691003},[31676],[34,691005],{"className":691006,"style":31681},[31680],[34,691008,46900],{"className":691009},[31685,207923],[34,691011,32936],{"className":691012},[31685,31686],[34,691014],{"className":691015,"style":31846},[31845],[34,691017,202],{"className":691018},[31850],[34,691020],{"className":691021,"style":31846},[31845],[34,691023,691025,691028,691031],{"className":691024},[31676],[34,691026],{"className":691027,"style":31681},[31680],[34,691029,46900],{"className":691030},[31685,207923],[34,691032,33249],{"className":691033},[31685,31686],[108,691035,691036],{},[34,691037,691039,691065],{"className":691038},[31642],[34,691040,691042],{"className":691041},[31646],[31648,691043,691044],{"xmlns":31650},[31652,691045,691046,691062],{},[31655,691047,691048,691050,691052,691054,691056,691058,691060],{},[31658,691049,46900],{"mathvariant":38079},[31658,691051,67764],{},[31662,691053,202],{},[31749,691055,768],{},[39361,691057,39363],{},[31658,691059,46900],{"mathvariant":38079},[31658,691061,33249],{},[31665,691063,691064],{"encoding":31667},"\\mathrm{d}y = 2 \\, \\mathrm{d}t",[34,691066,691068,691089],{"className":691067,"ariaHidden":758},[31672],[34,691069,691071,691074,691077,691080,691083,691086],{"className":691070},[31676],[34,691072],{"className":691073,"style":43217},[31680],[34,691075,46900],{"className":691076},[31685,207923],[34,691078,67764],{"className":691079,"style":32062},[31685,31686],[34,691081],{"className":691082,"style":31846},[31845],[34,691084,202],{"className":691085},[31850],[34,691087],{"className":691088,"style":31846},[31845],[34,691090,691092,691095,691098,691101,691104],{"className":691091},[31676],[34,691093],{"className":691094,"style":31681},[31680],[34,691096,768],{"className":691097},[31685],[34,691099],{"className":691100,"style":32123},[31845],[34,691102,46900],{"className":691103},[31685,207923],[34,691105,33249],{"className":691106},[31685,31686],[108,691108,691109],{},[34,691110,691112,691138],{"className":691111},[31642],[34,691113,691115],{"className":691114},[31646],[31648,691116,691117],{"xmlns":31650},[31652,691118,691119,691135],{},[31655,691120,691121,691123,691125,691127,691129,691131,691133],{},[31658,691122,46900],{"mathvariant":38079},[31658,691124,382333],{},[31662,691126,202],{},[31749,691128,786],{},[39361,691130,39363],{},[31658,691132,46900],{"mathvariant":38079},[31658,691134,33249],{},[31665,691136,691137],{"encoding":31667},"\\mathrm{d}z = 3 \\, \\mathrm{d}t",[34,691139,691141,691162],{"className":691140,"ariaHidden":758},[31672],[34,691142,691144,691147,691150,691153,691156,691159],{"className":691143},[31676],[34,691145],{"className":691146,"style":31681},[31680],[34,691148,46900],{"className":691149},[31685,207923],[34,691151,382333],{"className":691152,"style":382406},[31685,31686],[34,691154],{"className":691155,"style":31846},[31845],[34,691157,202],{"className":691158},[31850],[34,691160],{"className":691161,"style":31846},[31845],[34,691163,691165,691168,691171,691174,691177],{"className":691164},[31676],[34,691166],{"className":691167,"style":31681},[31680],[34,691169,786],{"className":691170},[31685],[34,691172],{"className":691173,"style":32123},[31845],[34,691175,46900],{"className":691176},[31685,207923],[34,691178,33249],{"className":691179},[31685,31686],[108,691181,691182,691184,691185],{},[2627,691183,659977],{},"\n我们要计算三项之和：",[105,691186,691187,691303,691520],{},[108,691188,691189,691190],{},"第1项 ",[34,691191,691193,691231],{"className":691192},[31642],[34,691194,691196],{"className":691195},[31646],[31648,691197,691198],{"xmlns":31650},[31652,691199,691200,691228],{},[31655,691201,691202,691204,691206,691208,691210,691212,691214,691216,691218,691220,691222,691224,691226],{},[31658,691203,32936],{},[39361,691205,39363],{},[31658,691207,46900],{"mathvariant":38079},[31658,691209,32936],{},[31662,691211,33239],{},[31662,691213,456],{"stretchy":702},[31749,691215,640],{},[31662,691217,223],{},[31658,691219,33249],{},[31662,691221,2557],{"stretchy":702},[39361,691223,39363],{},[31658,691225,46900],{"mathvariant":38079},[31658,691227,33249],{},[31665,691229,691230],{"encoding":31667},"x \\, \\mathrm{d}x \\to (1+t) \\, \\mathrm{d}t",[34,691232,691234,691261,691282],{"className":691233,"ariaHidden":758},[31672],[34,691235,691237,691240,691243,691246,691249,691252,691255,691258],{"className":691236},[31676],[34,691238],{"className":691239,"style":31681},[31680],[34,691241,32936],{"className":691242},[31685,31686],[34,691244],{"className":691245,"style":32123},[31845],[34,691247,46900],{"className":691248},[31685,207923],[34,691250,32936],{"className":691251},[31685,31686],[34,691253],{"className":691254,"style":31846},[31845],[34,691256,33239],{"className":691257},[31850],[34,691259],{"className":691260,"style":31846},[31845],[34,691262,691264,691267,691270,691273,691276,691279],{"className":691263},[31676],[34,691265],{"className":691266,"style":31774},[31680],[34,691268,456],{"className":691269},[31782],[34,691271,640],{"className":691272},[31685],[34,691274],{"className":691275,"style":32604},[31845],[34,691277,223],{"className":691278},[32024],[34,691280],{"className":691281,"style":32604},[31845],[34,691283,691285,691288,691291,691294,691297,691300],{"className":691284},[31676],[34,691286],{"className":691287,"style":31774},[31680],[34,691289,33249],{"className":691290},[31685,31686],[34,691292,2557],{"className":691293},[31690],[34,691295],{"className":691296,"style":32123},[31845],[34,691298,46900],{"className":691299},[31685,207923],[34,691301,33249],{"className":691302},[31685,31686],[108,691304,691305,691306],{},"第2项 ",[34,691307,691309,691373],{"className":691308},[31642],[34,691310,691312],{"className":691311},[31646],[31648,691313,691314],{"xmlns":31650},[31652,691315,691316,691370],{},[31655,691317,691318,691320,691322,691324,691326,691328,691330,691332,691334,691336,691338,691340,691342,691344,691346,691348,691350,691352,691354,691356,691358,691360,691362,691364,691366,691368],{},[31658,691319,67764],{},[39361,691321,39363],{},[31658,691323,46900],{"mathvariant":38079},[31658,691325,67764],{},[31662,691327,33239],{},[31662,691329,456],{"stretchy":702},[31749,691331,640],{},[31662,691333,223],{},[31749,691335,768],{},[31658,691337,33249],{},[31662,691339,2557],{"stretchy":702},[31662,691341,71191],{},[31749,691343,768],{},[39361,691345,39363],{},[31658,691347,46900],{"mathvariant":38079},[31658,691349,33249],{},[31662,691351,202],{},[31662,691353,456],{"stretchy":702},[31749,691355,768],{},[31662,691357,223],{},[31749,691359,15932],{},[31658,691361,33249],{},[31662,691363,2557],{"stretchy":702},[39361,691365,39363],{},[31658,691367,46900],{"mathvariant":38079},[31658,691369,33249],{},[31665,691371,691372],{"encoding":31667},"y \\, \\mathrm{d}y \\to (1+2t) \\cdot 2 \\, \\mathrm{d}t = (2+4t) \\, \\mathrm{d}t",[34,691374,691376,691403,691424,691448,691475,691496],{"className":691375,"ariaHidden":758},[31672],[34,691377,691379,691382,691385,691388,691391,691394,691397,691400],{"className":691378},[31676],[34,691380],{"className":691381,"style":43217},[31680],[34,691383,67764],{"className":691384,"style":32062},[31685,31686],[34,691386],{"className":691387,"style":32123},[31845],[34,691389,46900],{"className":691390},[31685,207923],[34,691392,67764],{"className":691393,"style":32062},[31685,31686],[34,691395],{"className":691396,"style":31846},[31845],[34,691398,33239],{"className":691399},[31850],[34,691401],{"className":691402,"style":31846},[31845],[34,691404,691406,691409,691412,691415,691418,691421],{"className":691405},[31676],[34,691407],{"className":691408,"style":31774},[31680],[34,691410,456],{"className":691411},[31782],[34,691413,640],{"className":691414},[31685],[34,691416],{"className":691417,"style":32604},[31845],[34,691419,223],{"className":691420},[32024],[34,691422],{"className":691423,"style":32604},[31845],[34,691425,691427,691430,691433,691436,691439,691442,691445],{"className":691426},[31676],[34,691428],{"className":691429,"style":31774},[31680],[34,691431,768],{"className":691432},[31685],[34,691434,33249],{"className":691435},[31685,31686],[34,691437,2557],{"className":691438},[31690],[34,691440],{"className":691441,"style":32604},[31845],[34,691443,71191],{"className":691444},[32024],[34,691446],{"className":691447,"style":32604},[31845],[34,691449,691451,691454,691457,691460,691463,691466,691469,691472],{"className":691450},[31676],[34,691452],{"className":691453,"style":31681},[31680],[34,691455,768],{"className":691456},[31685],[34,691458],{"className":691459,"style":32123},[31845],[34,691461,46900],{"className":691462},[31685,207923],[34,691464,33249],{"className":691465},[31685,31686],[34,691467],{"className":691468,"style":31846},[31845],[34,691470,202],{"className":691471},[31850],[34,691473],{"className":691474,"style":31846},[31845],[34,691476,691478,691481,691484,691487,691490,691493],{"className":691477},[31676],[34,691479],{"className":691480,"style":31774},[31680],[34,691482,456],{"className":691483},[31782],[34,691485,768],{"className":691486},[31685],[34,691488],{"className":691489,"style":32604},[31845],[34,691491,223],{"className":691492},[32024],[34,691494],{"className":691495,"style":32604},[31845],[34,691497,691499,691502,691505,691508,691511,691514,691517],{"className":691498},[31676],[34,691500],{"className":691501,"style":31774},[31680],[34,691503,15932],{"className":691504},[31685],[34,691506,33249],{"className":691507},[31685,31686],[34,691509,2557],{"className":691510},[31690],[34,691512],{"className":691513,"style":32123},[31845],[34,691515,46900],{"className":691516},[31685,207923],[34,691518,33249],{"className":691519},[31685,31686],[108,691521,691522,691523,691621,691622,691813,691814,2664,691848],{},"第3项 ",[34,691524,691526,691558],{"className":691525},[31642],[34,691527,691529],{"className":691528},[31646],[31648,691530,691531],{"xmlns":31650},[31652,691532,691533,691555],{},[31655,691534,691535,691537,691539,691541,691543,691545,691547,691549,691551,691553],{},[31662,691536,456],{"stretchy":702},[31658,691538,32936],{},[31662,691540,223],{},[31658,691542,67764],{},[31662,691544,32226],{},[31749,691546,640],{},[31662,691548,2557],{"stretchy":702},[39361,691550,39363],{},[31658,691552,46900],{"mathvariant":38079},[31658,691554,382333],{},[31665,691556,691557],{"encoding":31667},"(x+y-1) \\, \\mathrm{d}z",[34,691559,691561,691582,691600],{"className":691560,"ariaHidden":758},[31672],[34,691562,691564,691567,691570,691573,691576,691579],{"className":691563},[31676],[34,691565],{"className":691566,"style":31774},[31680],[34,691568,456],{"className":691569},[31782],[34,691571,32936],{"className":691572},[31685,31686],[34,691574],{"className":691575,"style":32604},[31845],[34,691577,223],{"className":691578},[32024],[34,691580],{"className":691581,"style":32604},[31845],[34,691583,691585,691588,691591,691594,691597],{"className":691584},[31676],[34,691586],{"className":691587,"style":55949},[31680],[34,691589,67764],{"className":691590,"style":32062},[31685,31686],[34,691592],{"className":691593,"style":32604},[31845],[34,691595,32226],{"className":691596},[32024],[34,691598],{"className":691599,"style":32604},[31845],[34,691601,691603,691606,691609,691612,691615,691618],{"className":691602},[31676],[34,691604],{"className":691605,"style":31774},[31680],[34,691607,640],{"className":691608},[31685],[34,691610,2557],{"className":691611},[31690],[34,691613],{"className":691614,"style":32123},[31845],[34,691616,46900],{"className":691617},[31685,207923],[34,691619,382333],{"className":691620,"style":382406},[31685,31686],"：\n先算括弧里的：",[34,691623,691625,691675],{"className":691624},[31642],[34,691626,691628],{"className":691627},[31646],[31648,691629,691630],{"xmlns":31650},[31652,691631,691632,691672],{},[31655,691633,691634,691636,691638,691640,691642,691644,691646,691648,691650,691652,691654,691656,691658,691660,691662,691664,691666,691668,691670],{},[31662,691635,456],{"stretchy":702},[31749,691637,640],{},[31662,691639,223],{},[31658,691641,33249],{},[31662,691643,2557],{"stretchy":702},[31662,691645,223],{},[31662,691647,456],{"stretchy":702},[31749,691649,640],{},[31662,691651,223],{},[31749,691653,768],{},[31658,691655,33249],{},[31662,691657,2557],{"stretchy":702},[31662,691659,32226],{},[31749,691661,640],{},[31662,691663,202],{},[31749,691665,640],{},[31662,691667,223],{},[31749,691669,786],{},[31658,691671,33249],{},[31665,691673,691674],{"encoding":31667},"(1+t) + (1+2t) - 1 = 1 + 3t",[34,691676,691678,691699,691720,691741,691765,691783,691801],{"className":691677,"ariaHidden":758},[31672],[34,691679,691681,691684,691687,691690,691693,691696],{"className":691680},[31676],[34,691682],{"className":691683,"style":31774},[31680],[34,691685,456],{"className":691686},[31782],[34,691688,640],{"className":691689},[31685],[34,691691],{"className":691692,"style":32604},[31845],[34,691694,223],{"className":691695},[32024],[34,691697],{"className":691698,"style":32604},[31845],[34,691700,691702,691705,691708,691711,691714,691717],{"className":691701},[31676],[34,691703],{"className":691704,"style":31774},[31680],[34,691706,33249],{"className":691707},[31685,31686],[34,691709,2557],{"className":691710},[31690],[34,691712],{"className":691713,"style":32604},[31845],[34,691715,223],{"className":691716},[32024],[34,691718],{"className":691719,"style":32604},[31845],[34,691721,691723,691726,691729,691732,691735,691738],{"className":691722},[31676],[34,691724],{"className":691725,"style":31774},[31680],[34,691727,456],{"className":691728},[31782],[34,691730,640],{"className":691731},[31685],[34,691733],{"className":691734,"style":32604},[31845],[34,691736,223],{"className":691737},[32024],[34,691739],{"className":691740,"style":32604},[31845],[34,691742,691744,691747,691750,691753,691756,691759,691762],{"className":691743},[31676],[34,691745],{"className":691746,"style":31774},[31680],[34,691748,768],{"className":691749},[31685],[34,691751,33249],{"className":691752},[31685,31686],[34,691754,2557],{"className":691755},[31690],[34,691757],{"className":691758,"style":32604},[31845],[34,691760,32226],{"className":691761},[32024],[34,691763],{"className":691764,"style":32604},[31845],[34,691766,691768,691771,691774,691777,691780],{"className":691767},[31676],[34,691769],{"className":691770,"style":32617},[31680],[34,691772,640],{"className":691773},[31685],[34,691775],{"className":691776,"style":31846},[31845],[34,691778,202],{"className":691779},[31850],[34,691781],{"className":691782,"style":31846},[31845],[34,691784,691786,691789,691792,691795,691798],{"className":691785},[31676],[34,691787],{"className":691788,"style":35942},[31680],[34,691790,640],{"className":691791},[31685],[34,691793],{"className":691794,"style":32604},[31845],[34,691796,223],{"className":691797},[32024],[34,691799],{"className":691800,"style":32604},[31845],[34,691802,691804,691807,691810],{"className":691803},[31676],[34,691805],{"className":691806,"style":32617},[31680],[34,691808,786],{"className":691809},[31685],[34,691811,33249],{"className":691812},[31685,31686],"\n再乘 ",[34,691815,691817,691833],{"className":691816},[31642],[34,691818,691820],{"className":691819},[31646],[31648,691821,691822],{"xmlns":31650},[31652,691823,691824,691830],{},[31655,691825,691826,691828],{},[31658,691827,46900],{"mathvariant":38079},[31658,691829,382333],{},[31665,691831,691832],{"encoding":31667},"\\mathrm{d}z",[34,691834,691836],{"className":691835,"ariaHidden":758},[31672],[34,691837,691839,691842,691845],{"className":691838},[31676],[34,691840],{"className":691841,"style":31681},[31680],[34,691843,46900],{"className":691844},[31685,207923],[34,691846,382333],{"className":691847,"style":382406},[31685,31686],[34,691849,691851,691905],{"className":691850},[31642],[34,691852,691854],{"className":691853},[31646],[31648,691855,691856],{"xmlns":31650},[31652,691857,691858,691902],{},[31655,691859,691860,691862,691864,691866,691868,691870,691872,691874,691876,691878,691880,691882,691884,691886,691888,691890,691892,691894,691896,691898,691900],{},[31662,691861,456],{"stretchy":702},[31749,691863,640],{},[31662,691865,223],{},[31749,691867,786],{},[31658,691869,33249],{},[31662,691871,2557],{"stretchy":702},[31662,691873,71191],{},[31749,691875,786],{},[39361,691877,39363],{},[31658,691879,46900],{"mathvariant":38079},[31658,691881,33249],{},[31662,691883,202],{},[31662,691885,456],{"stretchy":702},[31749,691887,786],{},[31662,691889,223],{},[31749,691891,36732],{},[31658,691893,33249],{},[31662,691895,2557],{"stretchy":702},[39361,691897,39363],{},[31658,691899,46900],{"mathvariant":38079},[31658,691901,33249],{},[31665,691903,691904],{"encoding":31667},"(1+3t) \\cdot 3 \\, \\mathrm{d}t = (3+9t) \\, \\mathrm{d}t",[34,691906,691908,691929,691953,691980,692001],{"className":691907,"ariaHidden":758},[31672],[34,691909,691911,691914,691917,691920,691923,691926],{"className":691910},[31676],[34,691912],{"className":691913,"style":31774},[31680],[34,691915,456],{"className":691916},[31782],[34,691918,640],{"className":691919},[31685],[34,691921],{"className":691922,"style":32604},[31845],[34,691924,223],{"className":691925},[32024],[34,691927],{"className":691928,"style":32604},[31845],[34,691930,691932,691935,691938,691941,691944,691947,691950],{"className":691931},[31676],[34,691933],{"className":691934,"style":31774},[31680],[34,691936,786],{"className":691937},[31685],[34,691939,33249],{"className":691940},[31685,31686],[34,691942,2557],{"className":691943},[31690],[34,691945],{"className":691946,"style":32604},[31845],[34,691948,71191],{"className":691949},[32024],[34,691951],{"className":691952,"style":32604},[31845],[34,691954,691956,691959,691962,691965,691968,691971,691974,691977],{"className":691955},[31676],[34,691957],{"className":691958,"style":31681},[31680],[34,691960,786],{"className":691961},[31685],[34,691963],{"className":691964,"style":32123},[31845],[34,691966,46900],{"className":691967},[31685,207923],[34,691969,33249],{"className":691970},[31685,31686],[34,691972],{"className":691973,"style":31846},[31845],[34,691975,202],{"className":691976},[31850],[34,691978],{"className":691979,"style":31846},[31845],[34,691981,691983,691986,691989,691992,691995,691998],{"className":691982},[31676],[34,691984],{"className":691985,"style":31774},[31680],[34,691987,456],{"className":691988},[31782],[34,691990,786],{"className":691991},[31685],[34,691993],{"className":691994,"style":32604},[31845],[34,691996,223],{"className":691997},[32024],[34,691999],{"className":692000,"style":32604},[31845],[34,692002,692004,692007,692010,692013,692016,692019,692022],{"className":692003},[31676],[34,692005],{"className":692006,"style":31774},[31680],[34,692008,36732],{"className":692009},[31685],[34,692011,33249],{"className":692012},[31685,31686],[34,692014,2557],{"className":692015},[31690],[34,692017],{"className":692018,"style":32123},[31845],[34,692020,46900],{"className":692021},[31685,207923],[34,692023,33249],{"className":692024},[31685,31686],[108,692026,692027,692030,692761,692763,692764],{},[2627,692028,692029],{},"合并并积分",[34,692031,692033],{"className":692032},[126385],[34,692034,692036,692234],{"className":692035},[31642],[34,692037,692039],{"className":692038},[31646],[31648,692040,692041],{"xmlns":31650,"display":126394},[31652,692042,692043,692231],{},[71078,692044,692045,692119,692187],{"rowspacing":368839,"columnalign":409990,"columnspacing":116494},[71084,692046,692047,692053],{},[71087,692048,692049],{},[71090,692050,692051],{"scriptlevel":459,"displaystyle":758},[31658,692052,106063],{},[71087,692054,692055],{},[71090,692056,692057],{"scriptlevel":459,"displaystyle":758},[31655,692058,692059,692061,692063,692071,692073,692075,692077,692079,692081,692083,692085,692087,692089,692091,692093,692095,692097,692099,692101,692103,692105,692107,692109,692111,692113,692115,692117],{},[31655,692060],{},[31662,692062,202],{},[38846,692064,692065,692067,692069],{},[31662,692066,546295],{},[31749,692068,459],{},[31749,692070,640],{},[31662,692072,15972],{"stretchy":702},[31662,692074,456],{"stretchy":702},[31749,692076,640],{},[31662,692078,223],{},[31658,692080,33249],{},[31662,692082,2557],{"stretchy":702},[31662,692084,223],{},[31662,692086,456],{"stretchy":702},[31749,692088,768],{},[31662,692090,223],{},[31749,692092,15932],{},[31658,692094,33249],{},[31662,692096,2557],{"stretchy":702},[31662,692098,223],{},[31662,692100,456],{"stretchy":702},[31749,692102,786],{},[31662,692104,223],{},[31749,692106,36732],{},[31658,692108,33249],{},[31662,692110,2557],{"stretchy":702},[31662,692112,32712],{"stretchy":702},[39361,692114,39363],{},[31658,692116,46900],{"mathvariant":38079},[31658,692118,33249],{},[71084,692120,692121,692127],{},[71087,692122,692123],{},[71090,692124,692125],{"scriptlevel":459,"displaystyle":758},[31655,692126],{},[71087,692128,692129],{},[71090,692130,692131],{"scriptlevel":459,"displaystyle":758},[31655,692132,692133,692135,692137,692145,692147,692149,692151,692153,692155,692157,692159,692161,692163,692165,692167,692169,692171,692173,692175,692177,692179,692181,692183,692185],{},[31655,692134],{},[31662,692136,202],{},[38846,692138,692139,692141,692143],{},[31662,692140,546295],{},[31749,692142,459],{},[31749,692144,640],{},[31662,692146,15972],{"stretchy":702},[31662,692148,456],{"stretchy":702},[31749,692150,640],{},[31662,692152,223],{},[31749,692154,768],{},[31662,692156,223],{},[31749,692158,786],{},[31662,692160,2557],{"stretchy":702},[31662,692162,223],{},[31662,692164,456],{"stretchy":702},[31749,692166,640],{},[31662,692168,223],{},[31749,692170,15932],{},[31662,692172,223],{},[31749,692174,36732],{},[31662,692176,2557],{"stretchy":702},[31658,692178,33249],{},[31662,692180,32712],{"stretchy":702},[39361,692182,39363],{},[31658,692184,46900],{"mathvariant":38079},[31658,692186,33249],{},[71084,692188,692189,692195],{},[71087,692190,692191],{},[71090,692192,692193],{"scriptlevel":459,"displaystyle":758},[31655,692194],{},[71087,692196,692197],{},[71090,692198,692199],{"scriptlevel":459,"displaystyle":758},[31655,692200,692201,692203,692205,692213,692215,692217,692219,692221,692223,692225,692227,692229],{},[31655,692202],{},[31662,692204,202],{},[38846,692206,692207,692209,692211],{},[31662,692208,546295],{},[31749,692210,459],{},[31749,692212,640],{},[31662,692214,456],{"stretchy":702},[31749,692216,36656],{},[31662,692218,223],{},[31749,692220,164756],{},[31658,692222,33249],{},[31662,692224,2557],{"stretchy":702},[39361,692226,39363],{},[31658,692228,46900],{"mathvariant":38079},[31658,692230,33249],{},[31665,692232,692233],{"encoding":31667},"\\begin{aligned}\nI &= \\int_0^1 [ (1+t) + (2+4t) + (3+9t) ] \\, \\mathrm{d}t \\\\\n&= \\int_0^1 [ (1+2+3) + (1+4+9)t ] \\, \\mathrm{d}t \\\\\n&= \\int_0^1 (6 + 14t) \\, \\mathrm{d}t\n\\end{aligned}",[34,692235,692237],{"className":692236,"ariaHidden":758},[31672],[34,692238,692240,692244],{"className":692239},[31676],[34,692241],{"className":692242,"style":692243},[31680],"height:8.0279em;vertical-align:-3.7639em;",[34,692245,692247],{"className":692246},[31685],[34,692248,692250,692306],{"className":692249},[71078],[34,692251,692253],{"className":692252},[369003],[34,692254,692256,692297],{"className":692255},[31796,31797],[34,692257,692259,692294],{"className":692258},[31801],[34,692260,692263,692276,692285],{"className":692261,"style":692262},[31805],"height:4.2639em;",[34,692264,692266,692270],{"style":692265},"top:-6.2639em;",[34,692267],{"className":692268,"style":692269},[31813],"height:3.564em;",[34,692271,692273],{"className":692272},[31685],[34,692274,106063],{"className":692275,"style":67482},[31685,31686],[34,692277,692279,692282],{"style":692278},"top:-3.488em;",[34,692280],{"className":692281,"style":692269},[31813],[34,692283],{"className":692284},[31685],[34,692286,692288,692291],{"style":692287},"top:-0.712em;",[34,692289],{"className":692290,"style":692269},[31813],[34,692292],{"className":692293},[31685],[34,692295,31829],{"className":692296},[31828],[34,692298,692300],{"className":692299},[31801],[34,692301,692304],{"className":692302,"style":692303},[31805],"height:3.7639em;",[34,692305],{},[34,692307,692309],{"className":692308},[71384],[34,692310,692312,692753],{"className":692311},[31796,31797],[34,692313,692315,692750],{"className":692314},[31801],[34,692316,692318,692485,692646],{"className":692317,"style":692262},[31805],[34,692319,692320,692323],{"style":692265},[34,692321],{"className":692322,"style":692269},[31813],[34,692324,692326,692329,692332,692335,692338,692389,692392,692395,692398,692401,692404,692407,692410,692413,692416,692419,692422,692425,692428,692431,692434,692437,692440,692443,692446,692449,692452,692455,692458,692461,692464,692467,692470,692473,692476,692479,692482],{"className":692325},[31685],[34,692327],{"className":692328},[31685],[34,692330],{"className":692331,"style":31846},[31845],[34,692333,202],{"className":692334},[31850],[34,692336],{"className":692337,"style":31846},[31845],[34,692339,692341,692344],{"className":692340},[32767],[34,692342,546295],{"className":692343,"style":591475},[32767,131249,131250],[34,692345,692347],{"className":692346},[31792],[34,692348,692350,692381],{"className":692349},[31796,31797],[34,692351,692353,692378],{"className":692352},[31801],[34,692354,692356,692367],{"className":692355,"style":591488},[31805],[34,692357,692358,692361],{"style":591491},[34,692359],{"className":692360,"style":31814},[31813],[34,692362,692364],{"className":692363},[31818,31819,31820,31821],[34,692365,459],{"className":692366},[31685,31821],[34,692368,692369,692372],{"style":591503},[34,692370],{"className":692371,"style":31814},[31813],[34,692373,692375],{"className":692374},[31818,31819,31820,31821],[34,692376,640],{"className":692377},[31685,31821],[34,692379,31829],{"className":692380},[31828],[34,692382,692384],{"className":692383},[31801],[34,692385,692387],{"className":692386,"style":591522},[31805],[34,692388],{},[34,692390,86328],{"className":692391},[31782],[34,692393,640],{"className":692394},[31685],[34,692396],{"className":692397,"style":32604},[31845],[34,692399,223],{"className":692400},[32024],[34,692402],{"className":692403,"style":32604},[31845],[34,692405,33249],{"className":692406},[31685,31686],[34,692408,2557],{"className":692409},[31690],[34,692411],{"className":692412,"style":32604},[31845],[34,692414,223],{"className":692415},[32024],[34,692417],{"className":692418,"style":32604},[31845],[34,692420,456],{"className":692421},[31782],[34,692423,768],{"className":692424},[31685],[34,692426],{"className":692427,"style":32604},[31845],[34,692429,223],{"className":692430},[32024],[34,692432],{"className":692433,"style":32604},[31845],[34,692435,15932],{"className":692436},[31685],[34,692438,33249],{"className":692439},[31685,31686],[34,692441,2557],{"className":692442},[31690],[34,692444],{"className":692445,"style":32604},[31845],[34,692447,223],{"className":692448},[32024],[34,692450],{"className":692451,"style":32604},[31845],[34,692453,456],{"className":692454},[31782],[34,692456,786],{"className":692457},[31685],[34,692459],{"className":692460,"style":32604},[31845],[34,692462,223],{"className":692463},[32024],[34,692465],{"className":692466,"style":32604},[31845],[34,692468,36732],{"className":692469},[31685],[34,692471,33249],{"className":692472},[31685,31686],[34,692474,481211],{"className":692475},[31690],[34,692477],{"className":692478,"style":32123},[31845],[34,692480,46900],{"className":692481},[31685,207923],[34,692483,33249],{"className":692484},[31685,31686],[34,692486,692487,692490],{"style":692278},[34,692488],{"className":692489,"style":692269},[31813],[34,692491,692493,692496,692499,692502,692505,692556,692559,692562,692565,692568,692571,692574,692577,692580,692583,692586,692589,692592,692595,692598,692601,692604,692607,692610,692613,692616,692619,692622,692625,692628,692631,692634,692637,692640,692643],{"className":692492},[31685],[34,692494],{"className":692495},[31685],[34,692497],{"className":692498,"style":31846},[31845],[34,692500,202],{"className":692501},[31850],[34,692503],{"className":692504,"style":31846},[31845],[34,692506,692508,692511],{"className":692507},[32767],[34,692509,546295],{"className":692510,"style":591475},[32767,131249,131250],[34,692512,692514],{"className":692513},[31792],[34,692515,692517,692548],{"className":692516},[31796,31797],[34,692518,692520,692545],{"className":692519},[31801],[34,692521,692523,692534],{"className":692522,"style":591488},[31805],[34,692524,692525,692528],{"style":591491},[34,692526],{"className":692527,"style":31814},[31813],[34,692529,692531],{"className":692530},[31818,31819,31820,31821],[34,692532,459],{"className":692533},[31685,31821],[34,692535,692536,692539],{"style":591503},[34,692537],{"className":692538,"style":31814},[31813],[34,692540,692542],{"className":692541},[31818,31819,31820,31821],[34,692543,640],{"className":692544},[31685,31821],[34,692546,31829],{"className":692547},[31828],[34,692549,692551],{"className":692550},[31801],[34,692552,692554],{"className":692553,"style":591522},[31805],[34,692555],{},[34,692557,86328],{"className":692558},[31782],[34,692560,640],{"className":692561},[31685],[34,692563],{"className":692564,"style":32604},[31845],[34,692566,223],{"className":692567},[32024],[34,692569],{"className":692570,"style":32604},[31845],[34,692572,768],{"className":692573},[31685],[34,692575],{"className":692576,"style":32604},[31845],[34,692578,223],{"className":692579},[32024],[34,692581],{"className":692582,"style":32604},[31845],[34,692584,786],{"className":692585},[31685],[34,692587,2557],{"className":692588},[31690],[34,692590],{"className":692591,"style":32604},[31845],[34,692593,223],{"className":692594},[32024],[34,692596],{"className":692597,"style":32604},[31845],[34,692599,456],{"className":692600},[31782],[34,692602,640],{"className":692603},[31685],[34,692605],{"className":692606,"style":32604},[31845],[34,692608,223],{"className":692609},[32024],[34,692611],{"className":692612,"style":32604},[31845],[34,692614,15932],{"className":692615},[31685],[34,692617],{"className":692618,"style":32604},[31845],[34,692620,223],{"className":692621},[32024],[34,692623],{"className":692624,"style":32604},[31845],[34,692626,36732],{"className":692627},[31685],[34,692629,2557],{"className":692630},[31690],[34,692632,33249],{"className":692633},[31685,31686],[34,692635,32712],{"className":692636},[31690],[34,692638],{"className":692639,"style":32123},[31845],[34,692641,46900],{"className":692642},[31685,207923],[34,692644,33249],{"className":692645},[31685,31686],[34,692647,692648,692651],{"style":692287},[34,692649],{"className":692650,"style":692269},[31813],[34,692652,692654,692657,692660,692663,692666,692717,692720,692723,692726,692729,692732,692735,692738,692741,692744,692747],{"className":692653},[31685],[34,692655],{"className":692656},[31685],[34,692658],{"className":692659,"style":31846},[31845],[34,692661,202],{"className":692662},[31850],[34,692664],{"className":692665,"style":31846},[31845],[34,692667,692669,692672],{"className":692668},[32767],[34,692670,546295],{"className":692671,"style":591475},[32767,131249,131250],[34,692673,692675],{"className":692674},[31792],[34,692676,692678,692709],{"className":692677},[31796,31797],[34,692679,692681,692706],{"className":692680},[31801],[34,692682,692684,692695],{"className":692683,"style":591488},[31805],[34,692685,692686,692689],{"style":591491},[34,692687],{"className":692688,"style":31814},[31813],[34,692690,692692],{"className":692691},[31818,31819,31820,31821],[34,692693,459],{"className":692694},[31685,31821],[34,692696,692697,692700],{"style":591503},[34,692698],{"className":692699,"style":31814},[31813],[34,692701,692703],{"className":692702},[31818,31819,31820,31821],[34,692704,640],{"className":692705},[31685,31821],[34,692707,31829],{"className":692708},[31828],[34,692710,692712],{"className":692711},[31801],[34,692713,692715],{"className":692714,"style":591522},[31805],[34,692716],{},[34,692718,456],{"className":692719},[31782],[34,692721,36656],{"className":692722},[31685],[34,692724],{"className":692725,"style":32604},[31845],[34,692727,223],{"className":692728},[32024],[34,692730],{"className":692731,"style":32604},[31845],[34,692733,164756],{"className":692734},[31685],[34,692736,33249],{"className":692737},[31685,31686],[34,692739,2557],{"className":692740},[31690],[34,692742],{"className":692743,"style":32123},[31845],[34,692745,46900],{"className":692746},[31685,207923],[34,692748,33249],{"className":692749},[31685,31686],[34,692751,31829],{"className":692752},[31828],[34,692754,692756],{"className":692755},[31801],[34,692757,692759],{"className":692758,"style":692303},[31805],[34,692760],{},[58531,692762],{},"计算定积分：",[34,692765,692767],{"className":692766},[126385],[34,692768,692770,692850],{"className":692769},[31642],[34,692771,692773],{"className":692772},[31646],[31648,692774,692775],{"xmlns":31650,"display":126394},[31652,692776,692777,692847],{},[31655,692778,692779,692781,692783,692785,692787,692789,692791,692793,692799,692807,692809,692811,692813,692815,692817,692819,692821,692823,692829,692831,692833,692835,692837,692839,692841,692843,692845],{},[31658,692780,106063],{},[31662,692782,202],{},[31662,692784,15972],{"stretchy":702},[31749,692786,36656],{},[31658,692788,33249],{},[31662,692790,223],{},[31749,692792,36723],{},[37624,692794,692795,692797],{},[31658,692796,33249],{},[31749,692798,768],{},[38846,692800,692801,692803,692805],{},[31662,692802,32712],{"stretchy":702},[31749,692804,459],{},[31749,692806,640],{},[31662,692808,202],{},[31662,692810,456],{"stretchy":702},[31749,692812,36656],{},[31662,692814,42071],{},[31749,692816,640],{},[31662,692818,223],{},[31749,692820,36723],{},[31662,692822,42071],{},[37624,692824,692825,692827],{},[31749,692826,640],{},[31749,692828,768],{},[31662,692830,2557],{"stretchy":702},[31662,692832,32226],{},[31749,692834,459],{},[31662,692836,202],{},[31749,692838,36656],{},[31662,692840,223],{},[31749,692842,36723],{},[31662,692844,202],{},[31749,692846,205234],{},[31665,692848,692849],{"encoding":31667},"I = [6t + 7t^2]_0^1 = (6 \\times 1 + 7 \\times 1^2) - 0 = 6 + 7 = 13",[34,692851,692853,692871,692895,692993,693014,693032,693050,693097,693115,693133,693151],{"className":692852,"ariaHidden":758},[31672],[34,692854,692856,692859,692862,692865,692868],{"className":692855},[31676],[34,692857],{"className":692858,"style":33218},[31680],[34,692860,106063],{"className":692861,"style":67482},[31685,31686],[34,692863],{"className":692864,"style":31846},[31845],[34,692866,202],{"className":692867},[31850],[34,692869],{"className":692870,"style":31846},[31845],[34,692872,692874,692877,692880,692883,692886,692889,692892],{"className":692873},[31676],[34,692875],{"className":692876,"style":31774},[31680],[34,692878,15972],{"className":692879},[31782],[34,692881,36656],{"className":692882},[31685],[34,692884,33249],{"className":692885},[31685,31686],[34,692887],{"className":692888,"style":32604},[31845],[34,692890,223],{"className":692891},[32024],[34,692893],{"className":692894,"style":32604},[31845],[34,692896,692898,692901,692904,692933,692984,692987,692990],{"className":692897},[31676],[34,692899],{"className":692900,"style":145763},[31680],[34,692902,36723],{"className":692903},[31685],[34,692905,692907,692910],{"className":692906},[31685],[34,692908,33249],{"className":692909},[31685,31686],[34,692911,692913],{"className":692912},[31792],[34,692914,692916],{"className":692915},[31796],[34,692917,692919],{"className":692918},[31801],[34,692920,692922],{"className":692921,"style":129832},[31805],[34,692923,692924,692927],{"style":129846},[34,692925],{"className":692926,"style":31814},[31813],[34,692928,692930],{"className":692929},[31818,31819,31820,31821],[34,692931,768],{"className":692932},[31685,31821],[34,692934,692936,692939],{"className":692935},[31690],[34,692937,32712],{"className":692938},[31690],[34,692940,692942],{"className":692941},[31792],[34,692943,692945,692976],{"className":692944},[31796,31797],[34,692946,692948,692973],{"className":692947},[31801],[34,692949,692951,692962],{"className":692950,"style":129832},[31805],[34,692952,692953,692956],{"style":502694},[34,692954],{"className":692955,"style":31814},[31813],[34,692957,692959],{"className":692958},[31818,31819,31820,31821],[34,692960,459],{"className":692961},[31685,31821],[34,692963,692964,692967],{"style":129846},[34,692965],{"className":692966,"style":31814},[31813],[34,692968,692970],{"className":692969},[31818,31819,31820,31821],[34,692971,640],{"className":692972},[31685,31821],[34,692974,31829],{"className":692975},[31828],[34,692977,692979],{"className":692978},[31801],[34,692980,692982],{"className":692981,"style":52354},[31805],[34,692983],{},[34,692985],{"className":692986,"style":31846},[31845],[34,692988,202],{"className":692989},[31850],[34,692991],{"className":692992,"style":31846},[31845],[34,692994,692996,692999,693002,693005,693008,693011],{"className":692995},[31676],[34,692997],{"className":692998,"style":31774},[31680],[34,693000,456],{"className":693001},[31782],[34,693003,36656],{"className":693004},[31685],[34,693006],{"className":693007,"style":32604},[31845],[34,693009,42071],{"className":693010},[32024],[34,693012],{"className":693013,"style":32604},[31845],[34,693015,693017,693020,693023,693026,693029],{"className":693016},[31676],[34,693018],{"className":693019,"style":35942},[31680],[34,693021,640],{"className":693022},[31685],[34,693024],{"className":693025,"style":32604},[31845],[34,693027,223],{"className":693028},[32024],[34,693030],{"className":693031,"style":32604},[31845],[34,693033,693035,693038,693041,693044,693047],{"className":693034},[31676],[34,693036],{"className":693037,"style":35942},[31680],[34,693039,36723],{"className":693040},[31685],[34,693042],{"className":693043,"style":32604},[31845],[34,693045,42071],{"className":693046},[32024],[34,693048],{"className":693049,"style":32604},[31845],[34,693051,693053,693056,693085,693088,693091,693094],{"className":693052},[31676],[34,693054],{"className":693055,"style":145763},[31680],[34,693057,693059,693062],{"className":693058},[31685],[34,693060,640],{"className":693061},[31685],[34,693063,693065],{"className":693064},[31792],[34,693066,693068],{"className":693067},[31796],[34,693069,693071],{"className":693070},[31801],[34,693072,693074],{"className":693073,"style":129832},[31805],[34,693075,693076,693079],{"style":129846},[34,693077],{"className":693078,"style":31814},[31813],[34,693080,693082],{"className":693081},[31818,31819,31820,31821],[34,693083,768],{"className":693084},[31685,31821],[34,693086,2557],{"className":693087},[31690],[34,693089],{"className":693090,"style":32604},[31845],[34,693092,32226],{"className":693093},[32024],[34,693095],{"className":693096,"style":32604},[31845],[34,693098,693100,693103,693106,693109,693112],{"className":693099},[31676],[34,693101],{"className":693102,"style":32617},[31680],[34,693104,459],{"className":693105},[31685],[34,693107],{"className":693108,"style":31846},[31845],[34,693110,202],{"className":693111},[31850],[34,693113],{"className":693114,"style":31846},[318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(",[34,693184,693186,693199],{"className":693185},[31642],[34,693187,693189],{"className":693188},[31646],[31648,693190,693191],{"xmlns":31650},[31652,693192,693193,693197],{},[31655,693194,693195],{},[31658,693196,33249],{},[31665,693198,33249],{"encoding":31667},[34,693200,693202],{"className":693201,"ariaHidden":758},[31672],[34,693203,693205,693208],{"className":693204},[31676],[34,693206],{"className":693207,"style":33280},[31680],[34,693209,33249],{"className":693210},[31685,31686],[34,693212,693214,693227],{"className":693213},[31642],[34,693215,693217],{"className":693216},[31646],[31648,693218,693219],{"xmlns":31650},[31652,693220,693221,693225],{},[31655,693222,693223],{},[31658,693224,479697],{},[31665,693226,481351],{"encoding":31667},[34,693228,693230],{"className":693229,"ariaHidden":758},[31672],[34,693231,693233,693236],{"className":693232},[31676],[34,693234],{"className":693235,"style":31681},[31680],[34,693237,479697],{"className":693238,"style":34452},[31685,31686],[108,693240,693241,693244,693245,214569],{},[2627,693242,693243],{},"直线段","：优先用向量式参数 (",[34,693246,693248,693276],{"className":693247},[31642],[34,693249,693251],{"className":693250},[31646],[31648,693252,693253],{"xmlns":31650},[31652,693254,693255,693273],{},[31655,693256,693257,693259,693261,693267,693269,693271],{},[31658,693258,32936],{},[31662,693260,202],{},[31743,693262,693263,693265],{},[31658,693264,32936],{},[31749,693266,459],{},[31662,693268,223],{},[31658,693270,111],{},[31658,693272,33249],{},[31665,693274,693275],{"encoding":31667},"x = x_0 + at",[34,693277,693279,693297,693352],{"className":693278,"ariaHidden":758},[31672],[34,693280,693282,693285,693288,693291,693294],{"className":693281},[31676],[34,693283],{"className":693284,"style":32527},[31680],[34,693286,32936],{"className":693287},[31685,31686],[34,693289],{"className":693290,"style":31846},[31845],[34,693292,202],{"className":693293},[31850],[34,693295],{"className":693296,"style":31846},[31845],[34,693298,693300,693303,693343,693346,693349],{"className":693299},[31676],[34,693301],{"className":693302,"style":40522},[31680],[34,693304,693306,693309],{"className":693305},[31685],[34,693307,32936],{"className":693308},[31685,31686],[34,693310,693312],{"className":693311},[31792],[34,693313,693315,693335],{"className":693314},[31796,31797],[34,693316,693318,693332],{"className":693317},[31801],[34,693319,693321],{"className":693320,"style":31806},[31805],[34,693322,693323,693326],{"style":31809},[34,693324],{"className":693325,"style":31814},[31813],[34,693327,693329],{"className":693328},[31818,31819,31820,31821],[34,693330,459],{"className":693331},[31685,31821],[34,693333,31829],{"className":693334},[31828],[34,693336,693338],{"className":693337},[31801],[34,693339,693341],{"className":693340,"style":31836},[31805],[34,693342],{},[34,693344],{"className":693345,"style":32604},[31845],[34,693347,223],{"className":693348},[32024],[34,693350],{"className":693351,"style":32604},[31845],[34,693353,693355,693358,693361],{"className":693354},[31676],[34,693356],{"className":693357,"style":33280},[31680],[34,693359,111],{"className":693360},[31685,31686],[34,693362,33249],{"className":693363},[31685,31686],[108,693365,693366,1840,693369,693435,693436,693464],{},[2627,693367,693368],{},"函数曲线",[34,693370,693372,693396],{"className":693371},[31642],[34,693373,693375],{"className":693374},[31646],[31648,693376,693377],{"xmlns":31650},[31652,693378,693379,693393],{},[31655,693380,693381,693383,693385,693387,693389,693391],{},[31658,693382,67764],{},[31662,693384,202],{},[31658,693386,31739],{},[31662,693388,456],{"stretchy":702},[31658,693390,32936],{},[31662,693392,2557],{"stretchy":702},[31665,693394,693395],{"encoding":31667},"y=f(x)",[34,693397,693399,693417],{"className":693398,"ariaHidden":758},[31672],[34,693400,693402,693405,693408,693411,693414],{"className":693401},[31676],[34,693403],{"className":693404,"style":38774},[31680],[34,693406,67764],{"className":693407,"style":32062},[31685,31686],[34,693409],{"className":693410,"style":31846},[31845],[34,693412,202],{"className":693413},[31850],[34,693415],{"className":693416,"style":31846},[31845],[34,693418,693420,693423,693426,693429,693432],{"className":693419},[31676],[34,693421],{"className":693422,"style":31774},[31680],[34,693424,31739],{"className":693425,"style":31778},[31685,31686],[34,693427,456],{"className":693428},[31782],[34,693430,32936],{"className":693431},[31685,31686],[34,693433,2557],{"className":693434},[31690],")：直接把 ",[34,693437,693439,693452],{"className":693438},[31642],[34,693440,693442],{"className":693441},[31646],[31648,693443,693444],{"xmlns":31650},[31652,693445,693446,693450],{},[31655,693447,693448],{},[31658,693449,32936],{},[31665,693451,32936],{"encoding":31667},[34,693453,693455],{"className":693454,"ariaHidden":758},[31672],[34,693456,693458,693461],{"className":693457},[31676],[34,693459],{"className":693460,"style":32527},[31680],[34,693462,32936],{"className":693463},[31685,31686]," 当参数。",{"title":29,"searchDepth":43,"depth":43,"links":693466},[693467,693468,693469],{"id":658097,"depth":49,"text":658098},{"id":678484,"depth":49,"text":678485},{"id":693169,"depth":49,"text":693169},{"order":56},"\u002Fsource\u002Fmath\u002Fmath_analysis\u002F12-9hw",{"title":658088,"description":29},"source\u002Fmath\u002Fmath_analysis\u002F12-9HW","WM8T-uY3oGGods4IMv8-i0Qzwu_hIISvDdd32kub3DE",{"id":693476,"title":693477,"body":693478,"description":29,"extension":85,"meta":699533,"navigation":52,"path":699534,"seo":699535,"stem":699536,"__hash__":699537},"notes\u002Fsource\u002Fmath\u002Fmath_analysis\u002FWeek13HW.md","Week 13 作业",{"type":8,"value":693479,"toc":699522},[693480,693484,695644,696827,696830,696833,697163,697167,697650,697654,698182,698186,698189,699146,699150],[19,693481,693483],{"id":693482},"_11月20日-p234","11月20日 P234",[15,693485,693486,65807,693489,548467,693815,46528,694134,93535,694281,83913,694414,694456,694457,70884,694859,694916,694917,695191,695192,695643],{},[2627,693487,693488],{},"T1（讨论反常重积分敛散性）(选择其中两道)",[34,693490,693492,693564],{"className":693491},[31642],[34,693493,693495],{"className":693494},[31646],[31648,693496,693497],{"xmlns":31650},[31652,693498,693499,693561],{},[31655,693500,693501,693511],{},[31743,693502,693503,693505],{},[31662,693504,685583],{},[37624,693506,693507,693509],{},[31658,693508,34386],{"mathvariant":226163},[31749,693510,768],{},[41453,693512,693513,693523],{},[31655,693514,693515,693517,693519,693521],{},[31658,693516,46900],{"mathvariant":38079},[31658,693518,32936],{},[31658,693520,46900],{"mathvariant":38079},[31658,693522,67764],{},[31655,693524,693525,693527,693529,693531,693533,693535,693541,693543,693545,693547,693549,693551,693553,693559],{},[31662,693526,456],{"stretchy":702},[31749,693528,640],{},[31662,693530,223],{},[31658,693532,162086],{"mathvariant":38079},[31658,693534,32936],{},[37624,693536,693537,693539],{},[31658,693538,162086],{"mathvariant":38079},[31658,693540,15],{},[31662,693542,2557],{"stretchy":702},[31662,693544,456],{"stretchy":702},[31749,693546,640],{},[31662,693548,223],{},[31658,693550,162086],{"mathvariant":38079},[31658,693552,67764],{},[37624,693554,693555,693557],{},[31658,693556,162086],{"mathvariant":38079},[31658,693558,38759],{},[31662,693560,2557],{"stretchy":702},[31665,693562,693563],{"encoding":31667},"\\iint_{\\mathbf{R}^2} 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\\leq |\\varphi(x,y)| \\leq M",[34,694325,694327,694345,694363,694405],{"className":694326,"ariaHidden":758},[31672],[34,694328,694330,694333,694336,694339,694342],{"className":694329},[31676],[34,694331],{"className":694332,"style":61814},[31680],[34,694334,459],{"className":694335},[31685],[34,694337],{"className":694338,"style":31846},[31845],[34,694340,932],{"className":694341},[31850],[34,694343],{"className":694344,"style":31846},[31845],[34,694346,694348,694351,694354,694357,694360],{"className":694347},[31676],[34,694349],{"className":694350,"style":43198},[31680],[34,694352,31759],{"className":694353},[31685,31686],[34,694355],{"className":694356,"style":31846},[31845],[34,694358,45972],{"className":694359},[31850],[34,694361],{"className":694362,"style":31846},[31845],[34,694364,694366,694369,694372,694375,694378,694381,694384,694387,694390,694393,694396,694399,694402],{"className":694365},[31676],[34,694367],{"className":694368,"style":31774},[31680],[34,694370,162086],{"className":694371},[31685],[34,694373,126541],{"className":694374},[31685,31686],[34,694376,456],{"className":694377},[31782],[34,694379,32936],{"className":694380},[31685,31686],[34,694382,31960],{"className":694383},[32119],[34,694385],{"className":694386,"style":32123},[31845],[34,694388,67764],{"className":694389,"style":32062},[31685,31686],[34,694391,2557],{"className":694392},[31690],[34,694394,162086],{"className":694395},[31685],[34,694397],{"className":694398,"style":31846},[31845],[34,694400,45972],{"className":694401},[31850],[34,694403],{"className":694404,"style":31846},[31845],[34,694406,694408,694411],{"className":694407},[31676],[34,694409],{"className":694410,"style":33218},[31680],[34,694412,98360],{"className":694413,"style":33389},[31685,31686],[34,694415,694417,694435],{"className":694416},[31642],[34,694418,694420],{"className":694419},[31646],[31648,694421,694422],{"xmlns":31650},[31652,694423,694424,694432],{},[31655,694425,694426,694428,694430],{},[31658,694427,31759],{},[31662,694429,31960],{"separator":758},[31658,694431,98360],{},[31665,694433,694434],{"encoding":31667},"m,M",[34,694436,694438],{"className":694437,"ariaHidden":758},[31672],[34,694439,694441,694444,694447,694450,694453],{"className":694440},[31676],[34,694442],{"className":694443,"style":70739},[31680],[34,694445,31759],{"className":694446},[31685,31686],[34,694448,31960],{"className":694449},[32119],[34,694451],{"className":694452,"style":32123},[31845],[34,694454,98360],{"className":694455,"style":33389},[31685,31686]," 为常数）；\n(3) ",[34,694458,694460,694548],{"className":694459},[31642],[34,694461,694463],{"className":694462},[31646],[31648,694464,694465],{"xmlns":31650},[31652,694466,694467,694545],{},[31655,694468,694469,694493,694537,694539,694541,694543],{},[31743,694470,694471,694473],{},[31662,694472,685583],{},[31655,694474,694475,694481,694483,694489,694491],{},[37624,694476,694477,694479],{},[31658,694478,32936],{},[31749,694480,768],{},[31662,694482,223],{},[37624,694484,694485,694487],{},[31658,694486,67764],{},[31749,694488,768],{},[31662,694490,45972],{},[31749,694492,640],{},[41453,694494,694495,694509],{},[31655,694496,694497,694499,694501,694503,694505,694507],{},[31658,694498,126541],{},[31662,694500,456],{"stretchy":702},[31658,694502,32936],{},[31662,694504,31960],{"separator":758},[31658,694506,67764],{},[31662,694508,2557],{"stretchy":702},[31655,694510,694511,694513,694515,694517,694523,694525,694531],{},[31662,694512,456],{"stretchy":702},[31749,694514,640],{},[31662,694516,32226],{},[37624,694518,694519,694521],{},[31658,694520,32936],{},[31749,694522,768],{},[31662,694524,32226],{},[37624,694526,694527,694529],{},[31658,694528,67764],{},[31749,694530,768],{},[37624,694532,694533,694535],{},[31662,694534,2557],{"stretchy":702},[31658,694536,15],{},[31658,694538,46900],{"mathvariant":38079},[31658,694540,32936],{},[31658,694542,46900],{"mathvariant":38079},[31658,694544,67764],{},[31665,694546,694547],{"encoding":31667},"\\iint_{x^2+y^2 \\leq 1} \\frac{\\varphi(x,y)}{(1-x^2-y^2)^p}\\mathrm{d}x\\mathrm{d}y",[34,694549,694551],{"className":694550,"ariaHidden":758},[31672],[34,694552,694554,694557,694665,694668,694847,694850,694853,694856],{"className":694553},[31676],[34,694555],{"className":694556,"style":327841},[31680],[34,694558,694560,694563],{"className":694559},[32767],[34,694561,685583],{"className":694562,"style":625171},[32767,131249,156560],[34,694564,694566],{"className":694565},[31792],[34,694567,694569,694656],{"className":694568},[31796,31797],[34,694570,694572,694653],{"className":694571},[31801],[34,694573,694575],{"className":694574,"style":693592},[31805],[34,694576,694577,694580],{"style":625187},[34,694578],{"className":694579,"style":31814},[31813],[34,694581,694583],{"className":694582},[31818,31819,31820,31821],[34,694584,694586,694615,694618,694647,694650],{"className":694585},[31685,31821],[34,694587,694589,694592],{"className":694588},[31685,31821],[34,694590,32936],{"className":694591},[31685,31686,31821],[34,694593,694595],{"className":694594},[31792],[34,694596,694598],{"className":694597},[31796],[34,694599,694601],{"className":694600},[31801],[34,694602,694604],{"className":694603,"style":548314},[31805],[34,694605,694606,694609],{"style":116668},[34,694607],{"className":694608,"style":43406},[31813],[34,694610,694612],{"className":694611},[31818,43410,43411,31821],[34,694613,768],{"className":694614},[31685,31821],[34,694616,223],{"className":694617},[32024,31821],[34,694619,694621,694624],{"className":694620},[31685,31821],[34,694622,67764],{"className":694623,"style":32062},[31685,31686,31821],[34,694625,694627],{"className":694626},[31792],[34,694628,694630],{"className":694629},[31796],[34,694631,694633],{"className":694632},[31801],[34,694634,694636],{"className":694635,"style":548314},[31805],[34,694637,694638,694641],{"style":116668},[34,694639],{"className":694640,"style":43406},[31813],[34,694642,694644],{"className":694643},[31818,43410,43411,31821],[34,694645,768],{"className":694646},[31685,31821],[34,694648,45972],{"className":694649},[31850,31821],[34,694651,640],{"className":694652},[31685,31821],[34,694654,31829],{"className":694655},[31828],[34,694657,694659],{"className":694658},[31801],[34,694660,694663],{"className":694661,"style":694662},[31805],"height:0.4918em;",[34,694664],{},[34,694666],{"className":694667,"style":32123},[31845],[34,694669,694671,694674,694844],{"className":694670},[31685],[34,694672],{"className":694673},[31782,41488],[34,694675,694677],{"className":694676},[41453],[34,694678,694680,694836],{"className":694679},[31796,31797],[34,694681,694683,694833],{"className":694682},[31801],[34,694684,694686,694796,694804],{"className":694685,"style":70504},[31805],[34,694687,694688,694691],{"style":41504},[34,694689],{"className":694690,"style":41508},[31813],[34,694692,694694],{"className":694693},[31818,31819,31820,31821],[34,694695,694697,694700,694703,694706,694735,694738,694767],{"className":694696},[31685,31821],[34,694698,456],{"className":694699},[31782,31821],[34,694701,640],{"className":694702},[31685,31821],[34,694704,32226],{"className":694705},[32024,31821],[34,694707,694709,694712],{"className":694708},[31685,31821],[34,694710,32936],{"className":694711},[31685,31686,31821],[34,694713,694715],{"className":694714},[31792],[34,694716,694718],{"className":694717},[31796],[34,694719,694721],{"className":694720},[31801],[34,694722,694724],{"className":694723,"style":548314},[31805],[34,694725,694726,694729],{"style":116668},[34,694727],{"className":694728,"style":43406},[31813],[34,694730,694732],{"className":694731},[31818,43410,43411,31821],[34,694733,768],{"className":694734},[31685,31821],[34,694736,32226],{"className":694737},[32024,31821],[34,694739,694741,694744],{"className":694740},[31685,31821],[34,694742,67764],{"className":694743,"style":32062},[31685,31686,31821],[34,694745,694747],{"className":694746},[31792],[34,694748,694750],{"className":694749},[31796],[34,694751,694753],{"className":694752},[31801],[34,694754,694756],{"className":694755,"style":548314},[31805],[34,694757,694758,694761],{"style":116668},[34,694759],{"className":694760,"style":43406},[31813],[34,694762,694764],{"className":694763},[31818,43410,43411,31821],[34,694765,768],{"className":694766},[31685,31821],[34,694768,694770,694773],{"className":694769},[31690,31821],[34,694771,2557],{"className":694772},[31690,31821],[34,694774,694776],{"className":694775},[31792],[34,694777,694779],{"className":694778},[31796],[34,694780,694782],{"className":694781},[31801],[34,694783,694785],{"className":694784,"style":566313},[31805],[34,694786,694787,694790],{"style":116668},[34,694788],{"className":694789,"style":43406},[31813],[34,694791,694793],{"className":694792},[31818,43410,43411,31821],[34,694794,15],{"className":694795},[31685,31686,31821],[34,694797,694798,694801],{"style":41520},[34,694799],{"className":694800,"style":41508},[31813],[34,694802],{"className":694803,"style":41528},[41527],[34,69480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满足与上题同样的条件；\n(4) 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\\times [0,a]} 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= \\iint_{\\mathbf{R}^2} \\frac{\\mathrm{d}x\\mathrm{d}y}{(1 + x^2)(1 + y^2)}",[34,696908,696910,696928],{"className":696909,"ariaHidden":758},[31672],[34,696911,696913,696916,696919,696922,696925],{"className":696912},[31676],[34,696914],{"className":696915,"style":33218},[31680],[34,696917,106063],{"className":696918,"style":67482},[31685,31686],[34,696920],{"className":696921,"style":31846},[31845],[34,696923,202],{"className":696924},[31850],[34,696926],{"className":696927,"style":31846},[31845],[34,696929,696931,696934,697003,697006],{"className":696930},[31676],[34,696932],{"className":696933,"style":693573},[31680],[34,696935,696937,696940],{"className":696936},[32767],[34,696938,685583],{"className":696939,"style":625171},[32767,131249,156560],[34,696941,696943],{"className":696942},[31792],[34,696944,696946,696995],{"className":696945},[31796,31797],[34,696947,696949,696992],{"className":696948},[31801],[34,696950,696952],{"className":696951,"style":693592},[31805],[34,696953,696954,696957],{"style":625187},[34,696955],{"className":696956,"style":31814},[31813],[34,696958,696960],{"className":696959},[31818,31819,31820,31821],[34,696961,696963],{"className":696962},[31685,31821],[34,696964,696966,696969],{"className":696965},[31685,31821],[34,696967,34386],{"className":696968},[31685,226302,31821],[34,696970,696972],{"className":696971},[31792],[34,696973,696975],{"className":696974},[31796],[34,696976,696978],{"className":696977},[31801],[34,696979,696981],{"className":696980,"style":548314},[31805],[34,696982,696983,696986],{"style":116668},[34,696984],{"className":696985,"style":43406},[31813],[34,696987,696989],{"className":696988},[31818,43410,43411,31821],[34,696990,768],{"className":696991},[31685,31821],[34,696993,31829],{"className":696994},[31828],[34,696996,696998],{"className":696997},[31801],[34,696999,697001],{"className":697000,"style":625206},[31805],[34,697002],{},[34,697004],{"className":697005,"style":32123},[31845],[34,697007,697009,697012,697159],{"className":697008},[31685],[34,697010],{"className":697011},[31782,41488],[34,697013,697015],{"className":697014},[41453],[34,697016,697018,697151],{"className":697017},[31796,31797],[34,697019,697021,697148],{"className":697020},[31801],[34,697022,697024,697117,697125],{"className":697023,"style":608935},[31805],[34,697025,697026,697029],{"style":41504},[34,697027],{"className":697028,"style":41508},[31813],[34,697030,697032],{"className":697031},[31818,31819,31820,31821],[34,697033,697035,697038,697041,697044,697073,697076,697079,697082,697085,697114],{"className":697034},[31685,31821],[34,697036,456],{"className":697037},[31782,31821],[34,697039,640],{"className":697040},[31685,31821],[34,697042,223],{"className":697043},[32024,31821],[34,697045,697047,697050],{"className":697046},[31685,31821],[34,697048,32936],{"className":697049},[31685,31686,31821],[34,697051,697053],{"className":697052},[31792],[34,697054,697056],{"className":697055},[31796],[34,697057,697059],{"className":697058},[31801],[34,697060,697062],{"className":697061,"style":548314},[31805],[34,697063,697064,697067],{"style":116668},[34,697065],{"className":697066,"style":43406},[31813],[34,697068,697070],{"className":697069},[31818,43410,43411,31821],[34,697071,768],{"className":697072},[31685,31821],[34,697074,2557],{"className":697075},[31690,31821],[34,697077,456],{"className":697078},[31782,31821],[34,697080,640],{"className":697081},[31685,31821],[34,697083,223],{"className":697084},[32024,31821],[34,697086,697088,697091],{"className":697087},[31685,31821],[34,697089,67764],{"className":697090,"style":32062},[31685,31686,31821],[34,697092,697094],{"className":697093},[31792],[34,697095,697097],{"className":697096},[31796],[34,697098,697100],{"className":697099},[31801],[34,697101,697103],{"className":697102,"style":548314},[31805],[34,697104,697105,697108],{"style":116668},[34,697106],{"className":697107,"style":43406},[31813],[34,697109,697111],{"className":697110},[31818,43410,43411,31821],[34,697112,768],{"className":697113},[31685,31821],[34,697115,2557],{"className":697116},[31690,31821],[34,697118,697119,697122],{"style":41520},[34,697120],{"className":697121,"style":41508},[31813],[34,697123],{"className":697124,"style":41528},[41527],[34,697126,697127,697130],{"style":548368},[34,697128],{"className":697129,"style":41508},[31813],[34,697131,697133],{"className":697132},[31818,31819,31820,31821],[34,697134,697136,697139,697142,697145],{"className":697135},[31685,31821],[34,697137,46900],{"className":697138},[31685,207923,31821],[34,697140,32936],{"className":697141},[31685,31686,31821],[34,697143,46900],{"className":697144},[31685,207923,31821],[34,697146,67764],{"className":697147,"style":32062},[31685,31686,31821],[34,697149,31829],{"className":697150},[31828],[34,697152,697154],{"className":697153},[31801],[34,697155,697157],{"className":697156,"style":324858},[31805],[34,697158],{},[34,697160],{"className":697161},[31690,41488],"\n是否收敛. 如果收敛, 求其值.",[253,697164,697166],{"id":697165},"t5","T5",[15,697168,697169,697170,697572,697573,2547],{},"设\n",[34,697171,697173,697268],{"className":697172},[31642],[34,697174,697176],{"className":697175},[31646],[31648,697177,697178],{"xmlns":31650},[31652,697179,697180,697265],{},[31655,697181,697182,697184,697186,697188,697190,697192,697237,697255,697257,697259,697261,697263],{},[31658,697183,51834],{},[31662,697185,456],{"stretchy":702},[31658,697187,33249],{},[31662,697189,2557],{"stretchy":702},[31662,697191,202],{},[31743,697193,697194,697196],{},[31662,697195,685583],{},[71090,697197,697198],{"scriptlevel":640},[71078,697199,697201,697219],{"rowspacing":697200,"columnalign":369740,"columnspacing":71082},"0.1em",[71084,697202,697203],{},[71087,697204,697205],{},[71090,697206,697207],{"scriptlevel":640,"displaystyle":702},[31655,697208,697209,697211,697213,697215,697217],{},[31749,697210,459],{},[31662,697212,45972],{},[31658,697214,32936],{},[31662,697216,45972],{},[31658,697218,33249],{},[71084,697220,697221],{},[71087,697222,697223],{},[71090,697224,697225],{"scriptlevel":640,"displaystyle":702},[31655,697226,697227,697229,697231,697233,697235],{},[31749,697228,459],{},[31662,697230,45972],{},[31658,697232,67764],{},[31662,697234,45972],{},[31658,697236,33249],{},[37624,697238,697239,697241],{},[31658,697240,33244],{"mathvariant":38079},[41453,697242,697243,697249],{},[31655,697244,697245,697247],{},[31658,697246,33249],{},[31658,697248,32936],{},[37624,697250,697251,697253],{},[31658,697252,67764],{},[31749,697254,768],{},[31658,697256,46900],{"mathvariant":38079},[31658,697258,32936],{},[31658,697260,46900],{"mathvariant":38079},[31658,697262,67764],{},[31662,697264,31960],{"separator":758},[31665,697266,697267],{"encoding":31667},"F(t) = \\iint_{\\substack{0 \\leq x \\leq t \\\\ 0 \\leq y \\leq t}} \\mathrm{e}^{\\frac{tx}{y^2}} \\mathrm{d}x\\mathrm{d}y,",[34,697269,697271,697298],{"className":697270,"ariaHidden":758},[31672],[34,697272,697274,697277,697280,697283,697286,697289,697292,697295],{"className":697273},[31676],[34,697275],{"className":697276,"style":31774},[31680],[34,697278,51834],{"className":697279,"style":37422},[31685,31686],[34,697281,456],{"className":697282},[31782],[34,697284,33249],{"className":697285},[31685,31686],[34,697287,2557],{"className":697288},[31690],[34,697290],{"className":697291,"style":31846},[31845],[34,697293,202],{"className":697294},[31850],[34,697296],{"className":697297,"style":31846},[31845],[34,697299,697301,697305,697427,697430,697557,697560,697563,697566,697569],{"className":697300},[31676],[34,697302],{"className":697303,"style":697304},[31680],"height:1.9465em;vertical-align:-0.8195em;",[34,697306,697308,697311],{"className":697307},[32767],[34,697309,685583],{"className":697310,"style":625171},[32767,131249,156560],[34,697312,697314],{"className":697313},[31792],[34,697315,697317,697418],{"className":697316},[31796,31797],[34,697318,697320,697415],{"className":697319},[31801],[34,697321,697323],{"className":697322,"style":332295},[31805],[34,697324,697326,697330],{"style":697325},"top:-2.3448em;margin-left:-0.1945em;margin-right:0.05em;",[34,697327],{"className":697328,"style":697329},[31813],"height:2.7572em;",[34,697331,697333],{"className":697332},[31818,31819,31820,31821],[34,697334,697336],{"className":697335},[31685,31821],[34,697337,697339],{"className":697338},[31685,31821],[34,697340,697342],{"className":697341},[71078],[34,697343,697345],{"className":697344},[141481],[34,697346,697348,697406],{"className":697347},[31796,31797],[34,697349,697351,697403],{"className":697350},[31801],[34,697352,697355,697379],{"className":697353,"style":697354},[31805],"height:1.0817em;",[34,697356,697358,697361],{"style":697357},"top:-3.1372em;",[34,697359],{"className":697360,"style":31814},[31813],[34,697362,697364,697367,697370,697373,697376],{"className":697363},[31685,31821],[34,697365,459],{"className":697366},[31685,31821],[34,697368,45972],{"className":697369},[31850,31821],[34,697371,32936],{"className":697372},[31685,31686,31821],[34,697374,45972],{"className":697375},[31850,31821],[34,697377,33249],{"className":697378},[31685,31686,31821],[34,697380,697382,697385],{"style":697381},"top:-2.3128em;",[34,697383],{"className":697384,"style":31814},[31813],[34,697386,697388,697391,697394,697397,697400],{"className":697387},[31685,31821],[34,697389,459],{"className":697390},[31685,31821],[34,697392,45972],{"className":697393},[31850,31821],[34,697395,67764],{"className":697396,"style":32062},[31685,31686,31821],[34,697398,45972],{"className":697399},[31850,31821],[34,697401,33249],{"className":697402},[31685,31686,31821],[34,697404,31829],{"className":697405},[31828],[34,697407,697409],{"className":697408},[31801],[34,697410,697413],{"className":697411,"style":697412},[31805],"height:0.5817em;",[34,697414],{},[34,697416,31829],{"className":697417},[31828],[34,697419,697421],{"className":697420},[31801],[34,697422,697425],{"className":697423,"style":697424},[31805],"height:0.8195em;",[34,697426],{},[34,697428],{"className":697429,"style":32123},[31845],[34,697431,697433,697436],{"className":697432},[31685],[34,697434,33244],{"className":697435},[31685,207923],[34,697437,697439],{"className":697438},[31792],[34,697440,697442],{"className":697441},[31796],[34,697443,697445],{"className":697444},[31801],[34,697446,697449],{"className":697447,"style":697448},[31805],"height:1.127em;",[34,697450,697452,697455],{"style":697451},"top:-3.5506em;margin-right:0.05em;",[34,697453],{"className":697454,"style":41508},[31813],[34,697456,697458],{"className":697457},[31818,31819,31820,31821],[34,697459,697461],{"className":697460},[31685,31821],[34,697462,697464,697467,697554],{"className":697463},[31685,31821],[34,697465],{"className":697466},[31782,41488,31818,43410,435704],[34,697468,697470],{"className":697469},[41453],[34,697471,697473,697545],{"className":697472},[31796,31797],[34,697474,697476,697542],{"className":697475},[31801],[34,697477,697480,697517,697525],{"className":697478,"style":697479},[31805],"height:0.8233em;",[34,697481,697482,697485],{"style":610874},[34,697483],{"className":697484,"style":41508},[31813],[34,697486,697488],{"className":697487},[31818,43410,43411,31821],[34,697489,697491],{"className":697490},[31685,31821],[34,697492,697494,697497],{"className":697493},[31685,31821],[34,697495,67764],{"className":697496,"style":32062},[31685,31686,31821],[34,697498,697500],{"className":697499},[31792],[34,697501,697503],{"className":697502},[31796],[34,697504,697506],{"className":697505},[31801],[34,697507,697509],{"className":697508,"style":610902},[31805],[34,697510,697511,697514],{"style":610905},[34,697512],{"className":697513,"style":499953},[31813],[34,697515,768],{"className":697516},[31685,31821],[34,697518,697519,697522],{"style":435735},[34,697520],{"className":697521,"st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\\mathrm{e}^{-(x_1^2 + x_2^2 + \\dots + x_n^2)} \\mathrm{d}x_1 \\mathrm{d}x_2 \\cdots 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x\\mathrm{d}x",[34,699207,699209],{"className":699208,"ariaHidden":758},[31672],[34,699210,699212,699215,699272,699275,699313,699316,699345,699348,699351,699354],{"className":699211},[31676],[34,699213],{"className":699214,"style":591468},[31680],[34,699216,699218,699221],{"className":699217},[32767],[34,699219,546295],{"className":699220,"style":591475},[32767,131249,131250],[34,699222,699224],{"className":699223},[31792],[34,699225,699227,699264],{"className":699226},[31796,31797],[34,699228,699230,699261],{"className":699229},[31801],[34,699231,699233,699247],{"className":699232,"style":591488},[31805],[34,699234,699235,699238],{"style":591491},[34,699236],{"className":699237,"style":31814},[31813],[34,699239,699241],{"className":699240},[31818,31819,31820,31821],[34,699242,699244],{"className":699243},[31685,31821],[34,699245,459],{"className":699246},[31685,31821],[34,699248,699249,699252],{"style":591503},[34,699250],{"className":699251,"style":31814},[31813],[34,699253,699255],{"className":699254},[31818,31819,31820,31821],[34,699256,699258],{"className":699257},[31685,31821],[34,699259,640],{"className":699260},[31685,31821],[34,699262,31829],{"className":699263},[31828],[34,699265,699267],{"className":699266},[31801],[34,699268,699270],{"className":699269,"style":591522},[31805],[34,699271],{},[34,699273],{"className":699274,"style":32123},[31845],[34,699276,699278,699281],{"className":699277},[31685],[34,699279,32936],{"className":699280},[31685,31686],[34,699282,699284],{"className":699283},[31792],[34,699285,699287],{"className":699286},[31796],[34,699288,699290],{"className":699289},[31801],[34,699291,699293],{"className":699292,"style":129832},[31805],[34,699294,699295,699298],{"style":129846},[34,699296],{"className":699297,"style":31814},[31813],[34,699299,699301],{"className":699300},[31818,31819,31820,31821],[34,699302,699304,699307,699310],{"className":699303},[31685,31821],[34,699305,15],{"className":699306},[31685,31686,31821],[34,699308,32226],{"className":699309},[32024,31821],[34,699311,640],{"className":699312},[31685,31821],[34,699314],{"className":699315,"style":32123},[31845],[34,699317,699319,699322],{"className":699318},[32767],[34,699320,563286],{"className":699321},[32767],[34,699323,699325],{"className":699324},[31792],[34,699326,699328],{"className":699327},[31796],[34,699329,699331],{"className":699330},[31801],[34,699332,699334],{"className":699333,"style":563430},[31805],[34,699335,699336,699339],{"style":563433},[34,699337],{"className":699338,"style":31814},[31813],[34,699340,699342],{"className":699341},[31818,31819,31820,31821],[34,699343,768],{"className":699344},[31685,31821],[34,699346],{"className":699347,"style":32123},[31845],[34,699349,32936],{"className":699350},[31685,31686],[34,699352,46900],{"className":699353},[31685,207923],[34,699355,32936],{"className":699356},[31685,31686],"，在 (i) ",[34,699359,699361,699386],{"className":699360},[31642],[34,699362,699364],{"className":699363},[31646],[31648,699365,699366],{"xmlns":31650},[31652,699367,699368,699384],{},[31655,699369,699370,699372,699374,699380,699382],{},[31658,699371,15],{},[31662,699373,43175],{},[31743,699375,699376,699378],{},[31658,699377,15],{},[31749,699379,459],{},[31662,699381,29622],{},[31749,699383,459],{},[31665,699385,563543],{"encoding":31667},[34,699387,699389,699407,699462],{"className":699388,"ariaHidden":758},[31672],[34,699390,699392,699395,699398,699401,699404],{"className":699391},[31676],[34,699393],{"className":699394,"style":549750},[31680],[34,699396,15],{"className":699397},[31685,31686],[34,699399],{"className":699400,"style":31846},[31845],[34,699402,43175],{"className":699403},[31850],[34,699405],{"className":699406,"style":31846},[31845],[34,699408,699410,699413,699453,699456,699459],{"className":699409},[31676],[34,699411],{"className":699412,"style":86652},[31680],[34,699414,699416,699419],{"className":699415},[31685],[34,699417,15],{"className":699418},[31685,31686],[34,699420,699422],{"className":699421},[31792],[34,699423,699425,699445],{"className":699424},[31796,31797],[34,699426,699428,699442],{"className":699427},[31801],[34,699429,699431],{"className":699430,"style":31806},[31805],[34,699432,699433,699436],{"style":31809},[34,699434],{"className":699435,"style":31814},[31813],[34,699437,699439],{"className":699438},[31818,31819,31820,31821],[34,699440,459],{"className":699441},[31685,31821],[34,699443,31829],{"className":699444},[31828],[34,699446,699448],{"className":699447},[31801],[34,699449,699451],{"className":699450,"style":31836},[31805],[34,699452],{},[34,699454],{"className":699455,"style":31846},[31845],[34,699457,29622],{"className":699458},[31850],[34,699460],{"className":699461,"style":31846},[31845],[34,699463,699465,699468],{"className":699464},[31676],[34,699466],{"className":699467,"style":32617},[31680],[34,699469,459],{"className":699470},[31685],"；(ii) 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HW.md","3-2 4-1 作业",{"type":8,"value":699542,"toc":742659},[699543,699547,699549,699553,699558,699572,699577,699940,699942,699946,699950,700097,700211,700216,701037,701041,701046,701249,701409,702882,702884,702888,702891,702893,703741,703745,704622,705807,707705,708372,708374,708376,709069,709073,709290,709680,710112,711614,713197,714582,714584,714587,715008,715014,715040,715042,715046,715598,715603,716450,717474,717480,718000,719109,719111,719115,719121,719484,719632,721063,721888,722256,723729,725334,725682,725684,726023,726182,726741,726746,729239,729245,729247,729704,730583,731043,731618,732433,732907,732912,733281,733594,733596,733600,733940,734376,734381,735738,737048,737866,737868,737872,737878,737887,737962,740741,740743,740747,740804,740815,740820],[11,699544,699546],{"id":699545},"习题-32-41","习题 3.2 & 4.1",[32188,699548],{},[253,699550,699552],{"id":699551},"第一步剖析问题-executive-summary","第一步：剖析问题 (Executive Summary)",[15,699554,699555],{},[2627,699556,699557],{},"核心任务：",[2230,699559,699560,699566],{},[108,699561,699562,699565],{},[2627,699563,699564],{},"理解理论："," 解释常微分方程解的“生命周期”（即能延伸多远——延拓定理）以及解对“起跑线”（初值）的敏感程度（连续性与可微性）。",[108,699567,699568,699571],{},[2627,699569,699570],{},"应用实战："," 利用定理5中的微分公式（公式3.17和3.18）解决两个具体的计算问题。",[15,699573,699574],{},[2627,699575,699576],{},"关键难点与隐含假设：",[105,699578,699579,699839],{},[108,699580,699581,699584,699585,699738,699739,699809,699810,699838],{},[2627,699582,699583],{},"难点："," 理解 ",[34,699586,699588,699618],{"className":699587},[31642],[34,699589,699591],{"className":699590},[31646],[31648,699592,699593],{"xmlns":31650},[31652,699594,699595,699615],{},[31655,699596,699597],{},[41453,699598,699599,699605],{},[31655,699600,699601,699603],{},[31658,699602,619473],{"mathvariant":38079},[31658,699604,126541],{},[31655,699606,699607,699609],{},[31658,699608,619473],{"mathvariant":38079},[31743,699610,699611,699613],{},[31658,699612,67764],{},[31749,699614,459],{},[31665,699616,699617],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial y_0}",[34,699619,699621],{"className":699620,"ariaHidden":758},[31672],[34,699622,699624,699627],{"className":699623},[31676],[34,699625],{"className":699626,"style":695705},[31680],[34,699628,699630,699633,699735],{"className":699629},[31685],[34,699631],{"className":699632},[31782,41488],[34,699634,699636],{"className":699635},[41453],[34,699637,699639,699727],{"className":699638},[31796,31797],[34,699640,699642,699724],{"className":699641},[31801],[34,699643,699645,699699,699707],{"className":699644,"style":608935},[31805],[34,699646,699647,699650],{"style":41504},[34,699648],{"className":699649,"style":41508},[31813],[34,699651,699653],{"className":699652},[31818,31819,31820,31821],[34,699654,699656,699659],{"className":699655},[31685,31821],[34,699657,619473],{"className":699658,"style":112322},[31685,31821],[34,699660,699662,699665],{"className":699661},[31685,31821],[34,699663,67764],{"className":699664,"style":32062},[31685,31686,31821],[34,699666,699668],{"className":699667},[31792],[34,699669,699671,699691],{"className":699670},[31796,31797],[34,699672,699674,699688],{"className":699673},[31801],[34,699675,699677],{"className":699676,"style":318658},[31805],[34,699678,699679,699682],{"style":561507},[34,699680],{"className":699681,"style":43406},[31813],[34,699683,699685],{"className":699684},[31818,43410,43411,31821],[34,699686,459],{"className":699687},[31685,31821],[34,699689,31829],{"className":699690},[31828],[34,699692,699694],{"className":699693},[31801],[34,699695,699697],{"className":699696,"style":220540},[31805],[34,699698],{},[34,699700,699701,699704],{"style":41520},[34,699702],{"className":699703,"style":41508},[31813],[34,699705],{"className":699706,"style":41528},[41527],[34,699708,699709,699712],{"style":548368},[34,699710],{"className":699711,"style":41508},[31813],[34,699713,699715],{"className":699714},[31818,31819,31820,31821],[34,699716,699718,699721],{"className":699717},[31685,31821],[34,699719,619473],{"className":699720,"style":112322},[31685,31821],[34,699722,126541],{"className":699723},[31685,31686,31821],[34,699725,31829],{"className":699726},[31828],[34,699728,699730],{"className":699729},[31801],[34,699731,699733],{"className":699732,"style":220191},[31805],[34,699734],{},[34,699736],{"className":699737},[31690,41488]," 这种符号的物理意义。它代表的是“当初值 ",[34,699740,699742,699760],{"className":699741},[31642],[34,699743,699745],{"className":699744},[31646],[31648,699746,699747],{"xmlns":31650},[31652,699748,699749,699757],{},[31655,699750,699751],{},[31743,699752,699753,699755],{},[31658,699754,67764],{},[31749,699756,459],{},[31665,699758,699759],{"encoding":31667},"y_0",[34,699761,699763],{"className":699762,"ariaHidden":758},[31672],[34,699764,699766,699769],{"className":699765},[31676],[34,699767],{"className":699768,"style":38774},[31680],[34,699770,699772,699775],{"className":699771},[31685],[34,699773,67764],{"className":699774,"style":32062},[31685,31686],[34,699776,699778],{"className":699777},[31792],[34,699779,699781,699801],{"className":699780},[31796,31797],[34,699782,699784,699798],{"className":699783},[31801],[34,699785,699787],{"className":699786,"style":31806},[31805],[34,699788,699789,699792],{"style":38795},[34,699790],{"className":699791,"style":31814},[31813],[34,699793,699795],{"className":699794},[31818,31819,31820,31821],[34,699796,459],{"className":699797},[31685,31821],[34,699799,31829],{"className":699800},[31828],[34,699802,699804],{"className":699803},[31801],[34,699805,699807],{"className":699806,"style":31836},[31805],[34,699808],{}," 变化一点点时，整个函数曲线在 ",[34,699811,699813,699826],{"className":699812},[31642],[34,699814,699816],{"className":699815},[31646],[31648,699817,699818],{"xmlns":31650},[31652,699819,699820,699824],{},[31655,699821,699822],{},[31658,699823,32936],{},[31665,699825,32936],{"encoding":31667},[34,699827,699829],{"className":699828,"ariaHidden":758},[31672],[34,699830,699832,699835],{"className":699831},[31676],[34,699833],{"className":699834,"style":32527},[31680],[34,699836,32936],{"className":699837},[31685,31686]," 处的变化率”。",[108,699840,699841,699844,699845,699889,699890,347],{},[2627,699842,699843],{},"假设："," 在计算题目时，必须先知道“原方程的解 ",[34,699846,699848,699868],{"className":699847},[31642],[34,699849,699851],{"className":699850},[31646],[31648,699852,699853],{"xmlns":31650},[31652,699854,699855,699865],{},[31655,699856,699857,699859,699861,699863],{},[31658,699858,126541],{},[31662,699860,456],{"stretchy":702},[31658,699862,32936],{},[31662,699864,2557],{"stretchy":702},[31665,699866,699867],{"encoding":31667},"\\varphi(x)",[34,699869,699871],{"className":699870,"ariaHidden":758},[31672],[34,699872,699874,699877,699880,699883,699886],{"className":699873},[31676],[34,699875],{"className":699876,"style":31774},[31680],[34,699878,126541],{"className":699879},[31685,31686],[34,699881,456],{"className":699882},[31782],[34,699884,32936],{"className":699885},[31685,31686],[34,699887,2557],{"className":699888},[31690]," 是什么”，才能将其代入积分公式中进行计算。特别是第2题，隐含了一个特解 ",[34,699891,699893,699910],{"className":699892},[31642],[34,699894,699896],{"className":699895},[31646],[31648,699897,699898],{"xmlns":31650},[31652,699899,699900,699908],{},[31655,699901,699902,699904,699906],{},[31658,699903,67764],{},[31662,699905,202],{},[31749,699907,459],{},[31665,699909,664526],{"encoding":31667},[34,699911,699913,699931],{"className":699912,"ariaHidden":758},[31672],[34,699914,699916,699919,699922,699925,699928],{"className":699915},[31676],[34,699917],{"className":699918,"style":38774},[31680],[34,699920,67764],{"className":699921,"style":32062},[31685,31686],[34,699923],{"className":699924,"style":31846},[31845],[34,699926,202],{"className":699927},[31850],[34,699929],{"className":699930,"style":31846},[31845],[34,699932,699934,699937],{"className":699933},[31676],[34,699935],{"className":699936,"style":32617},[31680],[34,699938,459],{"className":699939},[31685],[32188,699941],{},[253,699943,699945],{"id":699944},"第二步深入分析理论-deep-dive","第二步：深入分析理论 (Deep Dive)",[32490,699947,699949],{"id":699948},"_1-解的延拓定理-extension-theorem-解的生命力","1. 解的延拓定理 (Extension Theorem) —— 解的“生命力”",[15,699951,699952,699954,699955,699983,699984,700096],{},[2627,699953,139555],{},"\n想象你在一个大广场（区域 ",[34,699956,699958,699971],{"className":699957},[31642],[34,699959,699961],{"className":699960},[31646],[31648,699962,699963],{"xmlns":31650},[31652,699964,699965,699969],{},[31655,699966,699967],{},[31658,699968,97709],{},[31665,699970,97709],{"encoding":31667},[34,699972,699974],{"className":699973,"ariaHidden":758},[31672],[34,699975,699977,699980],{"className":699976},[31676],[34,699978],{"className":699979,"style":33218},[31680],[34,699981,97709],{"className":699982},[31685,31686],"）上按照一张藏宝图（方程 ",[34,699985,699987,700019],{"className":699986},[31642],[34,699988,699990],{"className":699989},[31646],[31648,699991,699992],{"xmlns":31650},[31652,699993,699994,700016],{},[31655,699995,699996,700002,700004,700006,700008,700010,700012,700014],{},[37624,699997,699998,700000],{},[31658,699999,67764],{},[31662,700001,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,700003,202],{},[31658,700005,31739],{},[31662,700007,456],{"stretchy":702},[31658,700009,32936],{},[31662,700011,31960],{"separator":758},[31658,700013,67764],{},[31662,700015,2557],{"stretchy":702},[31665,700017,700018],{"encoding":31667},"y'=f(x,y)",[34,700020,700022,700069],{"className":700021,"ariaHidden":758},[31672],[34,700023,700025,700028,700060,700063,700066],{"className":700024},[31676],[34,700026],{"className":700027,"style":395485},[31680],[34,700029,700031,700034],{"className":700030},[31685],[34,700032,67764],{"className":700033,"style":32062},[31685,31686],[34,700035,700037],{"className":700036},[31792],[34,700038,700040],{"className":700039},[31796],[34,700041,700043],{"className":700042},[31801],[34,700044,700046],{"className":700045,"style":132027},[31805],[34,700047,700048,700051],{"style":37777},[34,700049],{"className":700050,"style":31814},[31813],[34,700052,700054],{"className":700053},[31818,31819,31820,31821],[34,700055,700057],{"className":700056},[31685,31821],[34,700058,116495],{"className":700059},[31685,31821],[34,700061],{"className":700062,"style":31846},[31845],[34,700064,202],{"className":700065},[31850],[34,700067],{"className":700068,"style":31846},[31845],[34,700070,700072,700075,700078,700081,700084,700087,700090,700093],{"className":700071},[31676],[34,700073],{"className":700074,"style":31774},[31680],[34,700076,31739],{"className":700077,"style":31778},[31685,31686],[34,700079,456],{"className":700080},[31782],[34,700082,32936],{"className":700083},[31685,31686],[34,700085,31960],{"className":700086},[32119],[34,700088],{"className":700089,"style":32123},[31845],[34,700091,67764],{"className":700092,"style":32062},[31685,31686],[34,700094,2557],{"className":700095},[31690]," 指示的方向）行走。",[105,700098,700099,700134],{},[108,700100,700101,700104,700105,700133],{},[2627,700102,700103],{},"定理的核心含义："," 只要你还在广场内，且指示方向是清晰的（",[34,700106,700108,700121],{"className":700107},[31642],[34,700109,700111],{"className":700110},[31646],[31648,700112,700113],{"xmlns":31650},[31652,700114,700115,700119],{},[31655,700116,700117],{},[31658,700118,31739],{},[31665,700120,31739],{"encoding":31667},[34,700122,700124],{"className":700123,"ariaHidden":758},[31672],[34,700125,700127,700130],{"className":700126},[31676],[34,700128],{"className":700129,"style":43217},[31680],[34,700131,31739],{"className":700132,"style":31778},[31685,31686]," 连续且满足利普希茨条件），你就不会突然“卡住”或“消失”。你一定能一直走下去。",[108,700135,700136,700139],{},[2627,700137,700138],{},"停下来的原因只有两个：",[2230,700140,700141,700147],{},[108,700142,700143,700146],{},[2627,700144,700145],{},"撞墙了："," 你走到了广场的边缘（趋于边界）。",[108,700148,700149,700152,700153,700181,700182,700210],{},[2627,700150,700151],{},"飞出去了："," 你的速度变得无穷大，或者你走到了无穷远的地方（",[34,700154,700156,700169],{"className":700155},[31642],[34,700157,700159],{"className":700158},[31646],[31648,700160,700161],{"xmlns":31650},[31652,700162,700163,700167],{},[31655,700164,700165],{},[31658,700166,67764],{},[31665,700168,67764],{"encoding":31667},[34,700170,700172],{"className":700171,"ariaHidden":758},[31672],[34,700173,700175,700178],{"className":700174},[31676],[34,700176],{"className":700177,"style":38774},[31680],[34,700179,67764],{"className":700180,"style":32062},[31685,31686]," 无界或 ",[34,700183,700185,700198],{"className":700184},[31642],[34,700186,700188],{"className":700187},[31646],[31648,700189,700190],{"xmlns":31650},[31652,700191,700192,700196],{},[31655,700193,700194],{},[31658,700195,32936],{},[31665,700197,32936],{"encoding":31667},[34,700199,700201],{"className":700200,"ariaHidden":758},[31672],[34,700202,700204,700207],{"className":700203},[31676],[34,700205],{"className":700206,"style":32527},[31680],[34,700208,32936],{"className":700209},[31685,31686]," 无界）。",[15,700212,700213],{},[2627,700214,700215],{},"例1的深度解读：",[105,700217,700218,700374,700651],{},[108,700219,700220,700221,347],{},"方程 ",[34,700222,700224,700262],{"className":700223},[31642],[34,700225,700227],{"className":700226},[31646],[31648,700228,700229],{"xmlns":31650},[31652,700230,700231,700259],{},[31655,700232,700233,700239,700241,700243,700249,700251,700253,700255,700257],{},[37624,700234,700235,700237],{},[31658,700236,67764],{},[31662,700238,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,700240,202],{},[31662,700242,456],{"stretchy":702},[37624,700244,700245,700247],{},[31658,700246,67764],{},[31749,700248,768],{},[31662,700250,32226],{},[31749,700252,640],{},[31662,700254,2557],{"stretchy":702},[31658,700256,38080],{"mathvariant":38079},[31749,700258,768],{},[31665,700260,700261],{"encoding":31667},"y' = (y^2-1)\u002F2",[34,700263,700265,700312,700359],{"className":700264,"ariaHidden":758},[31672],[34,700266,700268,700271,700303,700306,700309],{"className":700267},[31676],[34,700269],{"className":700270,"style":395485},[31680],[34,700272,700274,700277],{"className":700273},[31685],[34,700275,67764],{"className":700276,"style":32062},[31685,31686],[34,700278,700280],{"className":700279},[31792],[34,700281,700283],{"className":700282},[31796],[34,700284,700286],{"className":700285},[31801],[34,700287,700289],{"className":700288,"style":132027},[31805],[34,700290,700291,700294],{"style":37777},[34,700292],{"className":700293,"style":31814},[31813],[34,700295,700297],{"className":700296},[31818,31819,31820,31821],[34,700298,700300],{"className":700299},[31685,31821],[34,700301,116495],{"className":700302},[31685,31821],[34,700304],{"className":700305,"style":31846},[31845],[34,700307,202],{"className":700308},[31850],[34,700310],{"className":700311,"style":31846},[31845],[34,700313,700315,700318,700321,700350,700353,700356],{"className":700314},[31676],[34,700316],{"className":700317,"style":37755},[31680],[34,700319,456],{"className":700320},[31782],[34,700322,700324,700327],{"className":700323},[31685],[34,700325,67764],{"className":700326,"style":32062},[31685,31686],[34,700328,700330],{"className":700329},[31792],[34,700331,700333],{"className":700332},[31796],[34,700334,700336],{"className":700335},[31801],[34,700337,700339],{"className":700338,"style":37774},[31805],[34,700340,700341,700344],{"style":37777},[34,700342],{"className":700343,"style":31814},[31813],[34,700345,700347],{"className":700346},[31818,31819,31820,31821],[34,700348,768],{"className":700349},[31685,31821],[34,700351],{"className":700352,"style":32604},[31845],[34,700354,32226],{"className":700355},[32024],[34,700357],{"className":700358,"style":32604},[31845],[34,700360,700362,700365,700368,700371],{"className":700361},[31676],[34,700363],{"className":700364,"style":31774},[31680],[34,700366,640],{"className":700367},[31685],[34,700369,2557],{"className":700370},[31690],[34,700372,38149],{"className":700373},[31685],[108,700375,700376,700586,700587,53834,700620,700648,700649,347],{},[2627,700377,700378,700379,700585],{},"解 ",[34,700380,700382,700424],{"className":700381},[31642],[34,700383,700385],{"className":700384},[31646],[31648,700386,700387],{"xmlns":31650},[31652,700388,700389,700421],{},[31655,700390,700391,700393,700395],{},[31658,700392,67764],{},[31662,700394,202],{},[41453,700396,700397,700409],{},[31655,700398,700399,700401,700403],{},[31749,700400,640],{},[31662,700402,32226],{},[37624,700404,700405,700407],{},[31658,700406,33244],{},[31658,700408,32936],{},[31655,700410,700411,700413,700415],{},[31749,700412,640],{},[31662,700414,223],{},[37624,700416,700417,700419],{},[31658,700418,33244],{},[31658,700420,32936],{},[31665,700422,700423],{"encoding":31667},"y = \\frac{1-e^x}{1+e^x}",[34,700425,700427,700445],{"className":700426,"ariaHidden":758},[31672],[34,700428,700430,700433,700436,700439,700442],{"className":700429},[31676],[34,700431],{"className":700432,"style":38774},[31680],[34,700434,67764],{"className":700435,"style":32062},[31685,31686],[34,700437],{"className":700438,"style":31846},[31845],[34,700440,202],{"className":700441},[31850],[34,700443],{"className":700444,"style":31846},[31845],[34,700446,700448,700452],{"className":700447},[31676],[34,700449],{"className":700450,"style":700451},[31680],"height:1.3143em;vertical-align:-0.4033em;",[34,700453,700455,700458,700582],{"className":700454},[31685],[34,700456],{"className":700457},[31782,41488],[34,700459,700461],{"className":700460},[41453],[34,700462,700464,700574],{"className":700463},[31796,31797],[34,700465,700467,700571],{"className":700466},[31801],[34,700468,700471,700517,700525],{"className":700469,"style":700470},[31805],"height:0.911em;",[34,700472,700473,700476],{"style":41504},[34,700474],{"className":700475,"style":41508},[31813],[34,700477,700479],{"className":700478},[31818,31819,31820,31821],[34,700480,700482,700485,700488],{"className":700481},[31685,31821],[34,700483,640],{"className":700484},[31685,31821],[34,700486,223],{"className":700487},[32024,31821],[34,700489,700491,700494],{"className":700490},[31685,31821],[34,700492,33244],{"className":700493},[31685,31686,31821],[34,700495,700497],{"className":700496},[31792],[34,700498,700500],{"className":700499},[31796],[34,700501,700503],{"className":700502},[31801],[34,700504,700506],{"className":700505,"style":566313},[31805],[34,700507,700508,700511],{"style":116668},[34,700509],{"className":700510,"style":43406},[31813],[34,700512,700514],{"className":700513},[31818,43410,43411,31821],[34,700515,32936],{"className":700516},[31685,31686,31821],[34,700518,700519,700522],{"style":41520},[34,700520],{"className":700521,"style":41508},[31813],[34,700523],{"className":700524,"style":41528},[41527],[34,700526,700527,700530],{"style":46771},[34,700528],{"className":700529,"style":41508},[31813],[34,700531,700533],{"className":700532},[31818,31819,31820,31821],[34,700534,700536,700539,700542],{"className":700535},[31685,31821],[34,700537,640],{"className":700538},[31685,31821],[34,700540,32226],{"className":700541},[32024,31821],[34,700543,700545,700548],{"className":700544},[31685,31821],[34,700546,33244],{"className":700547},[31685,31686,31821],[34,700549,700551],{"className":700550},[31792],[34,700552,700554],{"className":700553},[31796],[34,700555,700557],{"className":700556},[31801],[34,700558,700560],{"className":700559,"style":611529},[31805],[34,700561,700562,700565],{"style":552391},[34,700563],{"className":700564,"style":43406},[31813],[34,700566,700568],{"className":700567},[31818,43410,43411,31821],[34,700569,32936],{"className":700570},[31685,31686,31821],[34,700572,31829],{"className":700573},[31828],[34,700575,700577],{"className":700576},[31801],[34,700578,700580],{"className":700579,"style":400163},[31805],[34,700581],{},[34,700583],{"className":700584},[31690,41488]," (过原点)："," 这个解被限制在 ",[34,700588,700590,700605],{"className":700589},[31642],[34,700591,700593],{"className":700592},[31646],[31648,700594,700595],{"xmlns":31650},[31652,700596,700597,700603],{},[31655,700598,700599,700601],{},[31662,700600,32226],{},[31749,700602,640],{},[31665,700604,259653],{"encoding":31667},[34,700606,700608],{"className":700607,"ariaHidden":758},[31672],[34,700609,700611,700614,700617],{"className":700610},[31676],[34,700612],{"className":700613,"style":35942},[31680],[34,700615,32226],{"className":700616},[31685],[34,700618,640],{"className":700619},[31685],[34,700621,700623,700636],{"className":700622},[31642],[34,700624,700626],{"className":700625},[31646],[31648,700627,700628],{"xmlns":31650},[31652,700629,700630,700634],{},[31655,700631,700632],{},[31749,700633,640],{},[31665,700635,640],{"encoding":31667},[34,700637,700639],{"className":700638,"ariaHidden":758},[31672],[34,700640,700642,700645],{"className":700641},[31676],[34,700643],{"className":700644,"style":32617},[31680],[34,700646,640],{"className":700647},[31685]," 之间，它永远不会变成无穷大，所以它可以一直向左右延伸到无穷远。这就是推论中的",[2627,700650,47889],{},[108,700652,700653,561807,700927,700977,700978,701006,701007,83913,701009,700210],{},[2627,700654,700378,700655,700859,700860,245558],{},[34,700656,700658,700700],{"className":700657},[31642],[34,700659,700661],{"className":700660},[31646],[31648,700662,700663],{"xmlns":31650},[31652,700664,700665,700697],{},[31655,700666,700667,700669,700671],{},[31658,700668,67764],{},[31662,700670,202],{},[41453,700672,700673,700685],{},[31655,700674,700675,700677,700679],{},[31749,700676,640],{},[31662,700678,223],{},[37624,700680,700681,700683],{},[31658,700682,33244],{},[31658,700684,32936],{},[31655,700686,700687,700689,700691],{},[31749,700688,640],{},[31662,700690,32226],{},[37624,700692,700693,700695],{},[31658,700694,33244],{},[31658,700696,32936],{},[31665,700698,700699],{"encoding":31667},"y = \\frac{1+e^x}{1-e^x}",[34,700701,700703,700721],{"className":700702,"ariaHidden":758},[31672],[34,700704,700706,700709,700712,700715,700718],{"className":700705},[31676],[34,700707],{"className":700708,"style":38774},[31680],[34,700710,67764],{"className":700711,"style":32062},[31685,31686],[34,700713],{"className":700714,"style":31846},[31845],[34,700716,202],{"className":700717},[31850],[34,700719],{"className":700720,"style":31846},[31845],[34,700722,700724,700727],{"className":700723},[31676],[34,700725],{"className":700726,"style":700451},[31680],[34,700728,700730,700733,700856],{"className":700729},[31685],[34,700731],{"className":700732},[31782,41488],[34,700734,700736],{"className":700735},[41453],[34,700737,700739,700848],{"className":700738},[31796,31797],[34,700740,700742,700845],{"className":700741},[31801],[34,700743,700745,700791,700799],{"className":700744,"style":700470},[31805],[34,700746,700747,700750],{"style":41504},[34,700748],{"className":700749,"style":41508},[31813],[34,700751,700753],{"className":700752},[31818,31819,31820,31821],[34,700754,700756,700759,700762],{"className":700755},[31685,31821],[34,700757,640],{"className":700758},[31685,31821],[34,700760,32226],{"className":700761},[32024,31821],[34,700763,700765,700768],{"className":700764},[31685,31821],[34,700766,33244],{"className":700767},[31685,31686,31821],[34,700769,700771],{"className":700770},[31792],[34,700772,700774],{"className":700773},[31796],[34,700775,700777],{"className":700776},[31801],[34,700778,700780],{"className":700779,"style":566313},[31805],[34,700781,700782,700785],{"style":116668},[34,700783],{"className":700784,"style":43406},[31813],[34,700786,700788],{"className":700787},[31818,43410,43411,31821],[34,700789,32936],{"className":700790},[31685,31686,31821],[34,700792,700793,700796],{"style":41520},[34,700794],{"className":700795,"style":41508},[31813],[34,700797],{"className":700798,"style":41528},[41527],[34,700800,700801,700804],{"style":46771},[34,700802],{"className":700803,"style":41508},[31813],[34,700805,700807],{"className":700806},[31818,31819,31820,31821],[34,700808,700810,700813,700816],{"className":700809},[31685,31821],[34,700811,640],{"className":700812},[31685,31821],[34,700814,223],{"className":700815},[32024,31821],[34,700817,700819,700822],{"className":700818},[31685,31821],[34,700820,33244],{"className":700821},[31685,31686,31821],[34,700823,700825],{"className":700824},[31792],[34,700826,700828],{"className":700827},[31796],[34,700829,700831],{"className":700830},[31801],[34,700832,700834],{"className":700833,"style":611529},[31805],[34,700835,700836,700839],{"style":552391},[34,700837],{"className":700838,"style":43406},[31813],[34,700840,700842],{"className":700841},[31818,43410,43411,31821],[34,700843,32936],{"className":700844},[31685,31686,31821],[34,700846,31829],{"className":700847},[31828],[34,700849,700851],{"className":700850},[31801],[34,700852,700854],{"className":700853,"style":400163},[31805],[34,700855],{},[34,700857],{"className":700858},[31690,41488]," (过 ",[34,700861,700863,700891],{"className":700862},[31642],[34,700864,700866],{"className":700865},[31646],[31648,700867,700868],{"xmlns":31650},[31652,700869,700870,700888],{},[31655,700871,700872,700874,700876,700878,700880,700882,700884,700886],{},[31662,700873,456],{"stretchy":702},[31658,700875,563286],{},[31662,700877,32682],{},[31749,700879,768],{},[31662,700881,31960],{"separator":758},[31662,700883,32226],{},[31749,700885,786],{},[31662,700887,2557],{"stretchy":702},[31665,700889,700890],{"encoding":31667},"(\\ln 2, -3)",[34,700892,700894],{"className":700893,"ariaHidden":758},[31672],[34,700895,700897,700900,700903,700906,700909,700912,700915,700918,700921,700924],{"className":700896},[31676],[34,700898],{"className":700899,"style":31774},[31680],[34,700901,456],{"className":700902},[31782],[34,700904,563286],{"className":700905},[32767],[34,700907],{"className":700908,"style":32123},[31845],[34,700910,768],{"className":700911},[31685],[34,700913,31960],{"className":700914},[32119],[34,700916],{"className":700917,"style":32123},[31845],[34,700919,32226],{"className":700920},[31685],[34,700922,786],{"className":700923},[31685],[34,700925,2557],{"className":700926},[31690],[34,700928,700930,700947],{"className":700929},[31642],[34,700931,700933],{"className":700932},[31646],[31648,700934,700935],{"xmlns":31650},[31652,700936,700937,700945],{},[31655,700938,700939,700941,700943],{},[31658,700940,32936],{},[31662,700942,33239],{},[31749,700944,459],{},[31665,700946,568274],{"encoding":31667},[34,700948,700950,700968],{"className":700949,"ariaHidden":758},[31672],[34,700951,700953,700956,700959,700962,700965],{"className":700952},[31676],[34,700954],{"className":700955,"style":32527},[31680],[34,700957,32936],{"className":700958},[31685,31686],[34,700960],{"className":700961,"style":31846},[31845],[34,700963,33239],{"className":700964},[31850],[34,700966],{"className":700967,"style":31846},[31845],[34,700969,700971,700974],{"className":700970},[31676],[34,700972],{"className":700973,"style":32617},[31680],[34,700975,459],{"className":700976},[31685]," 时，分母趋近于0，整个 ",[34,700979,700981,700994],{"className":700980},[31642],[34,700982,700984],{"className":700983},[31646],[31648,700985,700986],{"xmlns":31650},[31652,700987,700988,700992],{},[31655,700989,700990],{},[31658,700991,67764],{},[31665,700993,67764],{"encoding":31667},[34,700995,700997],{"className":700996,"ariaHidden":758},[31672],[34,700998,701000,701003],{"className":700999},[31676],[34,701001],{"className":701002,"style":38774},[31680],[34,701004,67764],{"className":701005,"style":32062},[31685,31686]," 趋向于无穷大。这意味着解“爆炸”了。这就是推论中的",[2627,701008,48031],{},[34,701010,701012,701025],{"className":701011},[31642],[34,701013,701015],{"className":701014},[31646],[31648,701016,701017],{"xmlns":31650},[31652,701018,701019,701023],{},[31655,701020,701021],{},[31658,701022,67764],{},[31665,701024,67764],{"encoding":31667},[34,701026,701028],{"className":701027,"ariaHidden":758},[31672],[34,701029,701031,701034],{"className":701030},[31676],[34,701032],{"className":701033,"style":38774},[31680],[34,701035,67764],{"className":701036,"style":32062},[31685,31686],[32490,701038,701040],{"id":701039},"_2-解对初值的依赖-dependence-on-initial-values-失之毫厘谬以千里","2. 解对初值的依赖 (Dependence on Initial Values) —— “失之毫厘，谬以千里”",[15,701042,701043,701045],{},[2627,701044,139555],{},"\n想象射击。",[105,701047,701048,701054],{},[108,701049,701050,701053],{},[2627,701051,701052],{},"连续性 (定理4)："," 你的手抖了一下（初值变动一点点），子弹的落点只会偏一点点，不会突然飞到背后去。",[108,701055,701056,701059,701060,39984,701063],{},[2627,701057,701058],{},"可微性 (定理5)："," 我们不仅想知道偏了，还想知道",[2627,701061,701062],{},"偏了多少",[105,701064,701065],{},[108,701066,701067,701219,701220,701248],{},[34,701068,701070,701099],{"className":701069},[31642],[34,701071,701073],{"className":701072},[31646],[31648,701074,701075],{"xmlns":31650},[31652,701076,701077,701097],{},[31655,701078,701079],{},[41453,701080,701081,701087],{},[31655,701082,701083,701085],{},[31658,701084,619473],{"mathvariant":38079},[31658,701086,126541],{},[31655,701088,701089,701091],{},[31658,701090,619473],{"mathvariant":38079},[31743,701092,701093,701095],{},[31658,701094,67764],{},[31749,701096,459],{},[31665,701098,699617],{"encoding":31667},[34,701100,701102],{"className":701101,"ariaHidden":758},[31672],[34,701103,701105,701108],{"className":701104},[31676],[34,701106],{"className":701107,"style":695705},[31680],[34,701109,701111,701114,701216],{"className":701110},[31685],[34,701112],{"className":701113},[31782,41488],[34,701115,701117],{"className":701116},[41453],[34,701118,701120,701208],{"className":701119},[31796,31797],[34,701121,701123,701205],{"className":701122},[31801],[34,701124,701126,701180,701188],{"className":701125,"style":608935},[31805],[34,701127,701128,701131],{"style":41504},[34,701129],{"className":701130,"style":41508},[31813],[34,701132,701134],{"className":701133},[31818,31819,31820,31821],[34,701135,701137,701140],{"className":701136},[31685,31821],[34,701138,619473],{"className":701139,"style":112322},[31685,31821],[34,701141,701143,701146],{"className":701142},[31685,31821],[34,701144,67764],{"className":701145,"style":32062},[31685,31686,31821],[34,701147,701149],{"className":701148},[31792],[34,701150,701152,701172],{"className":701151},[31796,31797],[34,701153,701155,701169],{"className":701154},[31801],[34,701156,701158],{"className":701157,"style":318658},[31805],[34,701159,701160,701163],{"style":561507},[34,701161],{"className":701162,"style":43406},[31813],[34,701164,701166],{"className":701165},[31818,43410,43411,31821],[34,701167,459],{"className":701168},[31685,31821],[34,701170,31829],{"className":701171},[31828],[34,701173,701175],{"className":701174},[31801],[34,701176,701178],{"className":701177,"style":220540},[31805],[34,701179],{},[34,701181,701182,701185],{"style":41520},[34,701183],{"className":701184,"style":41508},[31813],[34,701186],{"className":701187,"style":41528},[41527],[34,701189,701190,701193],{"style":548368},[34,701191],{"className":701192,"style":41508},[31813],[34,701194,701196],{"className":701195},[31818,31819,31820,31821],[34,701197,701199,701202],{"className":701198},[31685,31821],[34,701200,619473],{"className":701201,"style":112322},[31685,31821],[34,701203,126541],{"className":701204},[31685,31686,31821],[34,701206,31829],{"className":701207},[31828],[34,701209,701211],{"className":701210},[31801],[34,701212,701214],{"className":701213,"style":220191},[31805],[34,701215],{},[34,701217],{"className":701218},[31690,41488],"：不仅是连续的，还是光滑的，我们可以精确计算出“初值改变 1 单位，解在 ",[34,701221,701223,701236],{"className":701222},[31642],[34,701224,701226],{"className":701225},[31646],[31648,701227,701228],{"xmlns":31650},[31652,701229,701230,701234],{},[31655,701231,701232],{},[31658,701233,32936],{},[31665,701235,32936],{"encoding":31667},[34,701237,701239],{"className":701238,"ariaHidden":758},[31672],[34,701240,701242,701245],{"className":701241},[31676],[34,701243],{"className":701244,"style":32527},[31680],[34,701246,32936],{"className":701247},[31685,31686]," 处改变多少单位”。",[15,701250,701251,701254,701255,701283,701284,701408],{},[2627,701252,701253],{},"公式解读（至关重要）：","\n我们要计算的是解 ",[34,701256,701258,701271],{"className":701257},[31642],[34,701259,701261],{"className":701260},[31646],[31648,701262,701263],{"xmlns":31650},[31652,701264,701265,701269],{},[31655,701266,701267],{},[31658,701268,126541],{},[31665,701270,126708],{"encoding":31667},[34,701272,701274],{"className":701273,"ariaHidden":758},[31672],[34,701275,701277,701280],{"className":701276},[31676],[34,701278],{"className":701279,"style":38774},[31680],[34,701281,126541],{"className":701282},[31685,31686]," 对初值 ",[34,701285,701287,701313],{"className":701286},[31642],[34,701288,701290],{"className":701289},[31646],[31648,701291,701292],{"xmlns":31650},[31652,701293,701294,701310],{},[31655,701295,701296,701302,701304],{},[31743,701297,701298,701300],{},[31658,701299,32936],{},[31749,701301,459],{},[31662,701303,31960],{"separator":758},[31743,701305,701306,701308],{},[31658,701307,67764],{},[31749,701309,459],{},[31665,701311,701312],{"encoding":31667},"x_0, y_0",[34,701314,701316],{"className":701315,"ariaHidden":758},[31672],[34,701317,701319,701322,701362,701365,701368],{"className":701318},[31676],[34,701320],{"className":701321,"style":38774},[31680],[34,701323,701325,701328],{"className":701324},[31685],[34,701326,32936],{"className":701327},[31685,31686],[34,701329,701331],{"className":701330},[31792],[34,701332,701334,701354],{"className":701333},[31796,31797],[34,701335,701337,701351],{"className":701336},[31801],[34,701338,701340],{"className":701339,"style":31806},[31805],[34,701341,701342,701345],{"style":31809},[34,701343],{"className":701344,"style":31814},[31813],[34,701346,701348],{"className":701347},[31818,31819,31820,31821],[34,701349,459],{"className":701350},[31685,31821],[34,701352,31829],{"className":701353},[31828],[34,701355,701357],{"className":701356},[31801],[34,701358,701360],{"className":701359,"style":31836},[31805],[34,701361],{},[34,701363,31960],{"className":701364},[32119],[34,701366],{"className":701367,"style":32123},[31845],[34,701369,701371,701374],{"className":701370},[31685],[34,701372,67764],{"className":701373,"style":32062},[31685,31686],[34,701375,701377],{"className":701376},[31792],[34,701378,701380,701400],{"className":701379},[31796,31797],[34,701381,701383,701397],{"className":701382},[31801],[34,701384,701386],{"className":701385,"style":31806},[31805],[34,701387,701388,701391],{"style":38795},[34,701389],{"className":701390,"style":31814},[31813],[34,701392,701394],{"className":701393},[31818,31819,31820,31821],[34,701395,459],{"className":701396},[31685,31821],[34,701398,31829],{"className":701399},[31828],[34,701401,701403],{"className":701402},[31801],[34,701404,701406],{"className":701405,"style":31836},[31805],[34,701407],{}," 的偏导数。这里有两个现成的公式（无需死记硬背，但要会用）：",[105,701410,701411,702078],{},[108,701412,701413,701486,701929],{},[2627,701414,701415,701416,701485],{},"公式 (3.18) 对 ",[34,701417,701419,701436],{"className":701418},[31642],[34,701420,701422],{"className":701421},[31646],[31648,701423,701424],{"xmlns":31650},[31652,701425,701426,701434],{},[31655,701427,701428],{},[31743,701429,701430,701432],{},[31658,701431,67764],{},[31749,701433,459],{},[31665,701435,699759],{"encoding":31667},[34,701437,701439],{"className":701438,"ariaHidden":758},[31672],[34,701440,701442,701445],{"className":701441},[31676],[34,701443],{"className":701444,"style":38774},[31680],[34,701446,701448,701451],{"className":701447},[31685],[34,701449,67764],{"className":701450,"style":32062},[31685,31686],[34,701452,701454],{"className":701453},[31792],[34,701455,701457,701477],{"className":701456},[31796,31797],[34,701458,701460,701474],{"className":701459},[31801],[34,701461,701463],{"className":701462,"style":31806},[31805],[34,701464,701465,701468],{"style":38795},[34,701466],{"className":701467,"style":31814},[31813],[34,701469,701471],{"className":701470},[31818,31819,31820,31821],[34,701472,459],{"className":701473},[31685,31821],[34,701475,31829],{"className":701476},[31828],[34,701478,701480],{"className":701479},[31801],[34,701481,701483],{"className":701482,"style":31836},[31805],[34,701484],{}," 的偏导：",[34,701487,701489,701578],{"className":701488},[31642],[34,701490,701492],{"className":701491},[31646],[31648,701493,701494],{"xmlns":31650},[31652,701495,701496,701575],{},[31655,701497,701498,701516,701518,701521,701523],{},[41453,701499,701500,701506],{},[31655,701501,701502,701504],{},[31658,701503,619473],{"mathvariant":38079},[31658,701505,126541],{},[31655,701507,701508,701510],{},[31658,701509,619473],{"mathvariant":38079},[31743,701511,701512,701514],{},[31658,701513,67764],{},[31749,701515,459],{},[31662,701517,202],{},[31658,701519,701520],{},"exp",[31662,701522,32682],{},[31655,701524,701525,701527,701539,701569,701571,701573],{},[31662,701526,456],{"fence":758},[38846,701528,701529,701531,701537],{},[31662,701530,546295],{},[31743,701532,701533,701535],{},[31658,701534,32936],{},[31749,701536,459],{},[31658,701538,32936],{},[41453,701540,701541,701563],{},[31655,701542,701543,701545,701547,701549,701551,701553,701555,701557,701559,701561],{},[31658,701544,619473],{"mathvariant":38079},[31658,701546,31739],{},[31662,701548,456],{"stretchy":702},[31658,701550,33249],{},[31662,701552,31960],{"separator":758},[31658,701554,126541],{},[31662,701556,456],{"stretchy":702},[31658,701558,33249],{},[31662,701560,2557],{"stretchy":702},[31662,701562,2557],{"stretchy":702},[31655,701564,701565,701567],{},[31658,701566,619473],{"mathvariant":38079},[31658,701568,67764],{},[31658,701570,46900],{},[31658,701572,33249],{},[31662,701574,2557],{"fence":758},[31665,701576,701577],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial y_0} = \\exp\\left( \\int_{x_0}^x \\frac{\\partial f(t,\\varphi(t))}{\\partial y} dt \\right)",[34,701579,701581,701707],{"className":701580,"ariaHidden":758},[31672],[34,701582,701584,701587,701698,701701,701704],{"className":701583},[31676],[34,701585],{"className":701586,"style":695705},[31680],[34,701588,701590,701593,701695],{"className":701589},[31685],[34,701591],{"className":701592},[31782,41488],[34,701594,701596],{"className":701595},[41453],[34,701597,701599,701687],{"className":701598},[31796,31797],[34,701600,701602,701684],{"className":701601},[31801],[34,701603,701605,701659,701667],{"className":701604,"style":608935},[31805],[34,701606,701607,701610],{"style":41504},[34,701608],{"className":701609,"style":41508},[31813],[34,701611,701613],{"className":701612},[31818,31819,31820,31821],[34,701614,701616,701619],{"className":701615},[31685,31821],[34,701617,619473],{"className":701618,"style":112322},[31685,31821],[34,701620,701622,701625],{"className":701621},[31685,31821],[34,701623,67764],{"className":701624,"style":32062},[31685,31686,31821],[34,701626,701628],{"className":701627},[31792],[34,701629,701631,701651],{"className":701630},[31796,31797],[34,701632,701634,701648],{"className":701633},[31801],[34,701635,701637],{"className":701636,"style":318658},[31805],[34,701638,701639,701642],{"style":561507},[34,701640],{"className":701641,"style":43406},[31813],[34,701643,701645],{"className":701644},[31818,43410,43411,31821],[34,701646,459],{"className":701647},[31685,31821],[34,701649,31829],{"className":701650},[31828],[34,701652,701654],{"className":701653},[31801],[34,701655,701657],{"className":701656,"style":220540},[31805],[34,701658],{},[34,701660,701661,701664],{"style":41520},[34,701662],{"className":701663,"style":41508},[31813],[34,701665],{"className":701666,"style":41528},[41527],[34,701668,701669,701672],{"style":548368},[34,701670],{"className":701671,"style":41508},[31813],[34,701673,701675],{"className":701674},[31818,31819,31820,31821],[34,701676,701678,701681],{"className":701677},[31685,31821],[34,701679,619473],{"className":701680,"style":112322},[31685,31821],[34,701682,126541],{"className":701683},[31685,31686,31821],[34,701685,31829],{"className":701686},[31828],[34,701688,701690],{"className":701689},[31801],[34,701691,701693],{"className":701692,"style":220191},[31805],[34,701694],{},[34,701696],{"className":701697},[31690,41488],[34,701699],{"className":701700,"style":31846},[31845],[34,701702,202],{"className":701703},[31850],[34,701705],{"className":701706,"style":31846},[31845],[34,701708,701710,701713,701716,701719],{"className":701709},[31676],[34,701711],{"className":701712,"style":141282},[31680],[34,701714,701520],{"className":701715},[32767],[34,701717],{"className":701718,"style":32123},[31845],[34,701720,701722,701728,701819,701822,701917,701920,701923],{"className":701721},[34006],[34,701723,701725],{"className":701724,"style":75811},[31782,75810],[34,701726,456],{"className":701727},[71281,141292],[34,701729,701731,701734],{"className":701730},[32767],[34,701732,546295],{"className":701733,"style":546331},[32767,131249,156560],[34,701735,701737],{"className":701736},[31792],[34,701738,701740,701811],{"className":701739},[31796,31797],[34,701741,701743,701808],{"className":701742},[31801],[34,701744,701746,701797],{"className":701745,"style":567928},[31805],[34,701747,701748,701751],{"style":546347},[34,701749],{"className":701750,"style":31814},[31813],[34,701752,701754],{"className":701753},[31818,31819,31820,31821],[34,701755,701757],{"className":701756},[31685,31821],[34,701758,701760,701763],{"className":701759},[31685,31821],[34,701761,32936],{"className":701762},[31685,31686,31821],[34,701764,701766],{"className":701765},[31792],[34,701767,701769,701789],{"className":701768},[31796,31797],[34,701770,701772,701786],{"className":701771},[31801],[34,701773,701775],{"className":701774,"style":318658},[31805],[34,701776,701777,701780],{"style":318661},[34,701778],{"className":701779,"style":43406},[31813],[34,701781,701783],{"className":701782},[31818,43410,43411,31821],[34,701784,459],{"className":701785},[31685,31821],[34,701787,31829],{"className":701788},[31828],[34,701790,701792],{"className":701791},[31801],[34,701793,701795],{"className":701794,"style":220540},[31805],[34,701796],{},[34,701798,701799,701802],{"style":546362},[34,701800],{"className":701801,"style":31814},[31813],[34,701803,701805],{"className":701804},[31818,31819,31820,31821],[34,701806,32936],{"className":701807},[31685,31686,31821],[34,701809,31829],{"className":701810},[31828],[34,701812,701814],{"className":701813},[31801],[34,701815,701817],{"className":701816,"style":667231},[31805],[34,701818],{},[34,701820],{"className":701821,"style":32123},[31845],[34,701823,701825,701828,701914],{"className":701824},[31685],[34,701826],{"className":701827},[31782,41488],[34,701829,701831],{"className":701830},[41453],[34,701832,701834,701906],{"className":701833},[31796,31797],[34,701835,701837,701903],{"className":701836},[31801],[34,701838,701840,701857,701865],{"className":701839,"style":70504},[31805],[34,701841,701842,701845],{"style":41504},[34,701843],{"className":701844,"style":41508},[31813],[34,701846,701848],{"className":701847},[31818,31819,31820,31821],[34,701849,701851,701854],{"className":701850},[31685,31821],[34,701852,619473],{"className":701853,"style":112322},[31685,31821],[34,701855,67764],{"className":701856,"style":32062},[31685,31686,31821],[34,701858,701859,701862],{"style":41520},[34,701860],{"className":701861,"style":41508},[31813],[34,701863],{"className":701864,"style":41528},[41527],[34,701866,701867,701870],{"style":70529},[34,701868],{"className":701869,"style":41508},[31813],[34,701871,701873],{"className":701872},[31818,31819,31820,31821],[34,701874,701876,701879,701882,701885,701888,701891,701894,701897,701900],{"className":701875},[31685,31821],[34,701877,619473],{"className":701878,"style":112322},[31685,31821],[34,701880,31739],{"className":701881,"style":31778},[31685,31686,31821],[34,701883,456],{"className":701884},[31782,31821],[34,701886,33249],{"className":701887},[31685,31686,31821],[34,701889,31960],{"className":701890},[32119,31821],[34,701892,126541],{"className":701893},[31685,31686,31821],[34,701895,456],{"className":701896},[31782,31821],[34,701898,33249],{"className":701899},[31685,31686,31821],[34,701901,44253],{"className":701902},[31690,31821],[34,701904,31829],{"className":701905},[31828],[34,701907,701909],{"className":701908},[31801],[34,701910,701912],{"className":701911,"style":220191},[31805],[34,701913],{},[34,701915],{"className":701916},[31690,41488],[34,701918,46900],{"className":701919},[31685,31686],[34,701921,33249],{"className":701922},[31685,31686],[34,701924,701926],{"className":701925,"style":75811},[31690,75810],[34,701927,2557],{"className":701928},[71281,141292],[105,701930,701931],{},[108,701932,701933,701935,701936,702048,702049,702077],{},[2627,701934,13689],{}," 这是一个指数累积的过程。",[34,701937,701939,701965],{"className":701938},[31642],[34,701940,701942],{"className":701941},[31646],[31648,701943,701944],{"xmlns":31650},[31652,701945,701946,701962],{},[31655,701947,701948],{},[41453,701949,701950,701956],{},[31655,701951,701952,701954],{},[31658,701953,619473],{"mathvariant":38079},[31658,701955,31739],{},[31655,701957,701958,701960],{},[31658,701959,619473],{"mathvariant":38079},[31658,701961,67764],{},[31665,701963,701964],{"encoding":31667},"\\frac{\\partial f}{\\partial y}",[34,701966,701968],{"className":701967,"ariaHidden":758},[31672],[34,701969,701971,701974],{"className":701970},[31676],[34,701972],{"className":701973,"style":695705},[31680],[34,701975,701977,701980,702045],{"className":701976},[31685],[34,701978],{"className":701979},[31782,41488],[34,701981,701983],{"className":701982},[41453],[34,701984,701986,702037],{"className":701985},[31796,31797],[34,701987,701989,702034],{"className":701988},[31801],[34,701990,701992,702009,702017],{"className":701991,"style":608935},[31805],[34,701993,701994,701997],{"style":41504},[34,701995],{"className":701996,"style":41508},[31813],[34,701998,702000],{"className":701999},[31818,31819,31820,31821],[34,702001,702003,702006],{"className":702002},[31685,31821],[34,702004,619473],{"className":702005,"style":112322},[31685,31821],[34,702007,67764],{"className":702008,"style":32062},[31685,31686,31821],[34,702010,702011,702014],{"style":41520},[34,702012],{"className":702013,"style":41508},[31813],[34,702015],{"className":702016,"style":41528},[41527],[34,702018,702019,702022],{"style":548368},[34,702020],{"className":702021,"style":41508},[31813],[34,702023,702025],{"className":702024},[31818,31819,31820,31821],[34,702026,702028,702031],{"className":702027},[31685,31821],[34,702029,619473],{"className":702030,"style":112322},[31685,31821],[34,702032,31739],{"className":702033,"style":31778},[31685,31686,31821],[34,702035,31829],{"className":702036},[31828],[34,702038,702040],{"className":702039},[31801],[34,702041,702043],{"className":702042,"style":220191},[31805],[34,702044],{},[34,702046],{"className":702047},[31690,41488]," 就像是一个“放大系数”，如果它大于0，初值的微小误差会随着 ",[34,702050,702052,702065],{"className":702051},[31642],[34,702053,702055],{"className":702054},[31646],[31648,702056,702057],{"xmlns":31650},[31652,702058,702059,702063],{},[31655,702060,702061],{},[31658,702062,32936],{},[31665,702064,32936],{"encoding":31667},[34,702066,702068],{"className":702067,"ariaHidden":758},[31672],[34,702069,702071,702074],{"className":702070},[31676],[34,702072],{"className":702073,"style":32527},[31680],[34,702075,32936],{"className":702076},[31685,31686]," 增大而指数级放大。",[108,702079,702080,702153,702590],{},[2627,702081,702082,702083,701485],{},"公式 (3.17) 对 ",[34,702084,702086,702104],{"className":702085},[31642],[34,702087,702089],{"className":702088},[31646],[31648,702090,702091],{"xmlns":31650},[31652,702092,702093,702101],{},[31655,702094,702095],{},[31743,702096,702097,702099],{},[31658,702098,32936],{},[31749,702100,459],{},[31665,702102,702103],{"encoding":31667},"x_0",[34,702105,702107],{"className":702106,"ariaHidden":758},[31672],[34,702108,702110,702113],{"className":702109},[31676],[34,702111],{"className":702112,"style":31860},[31680],[34,702114,702116,702119],{"className":702115},[31685],[34,702117,32936],{"className":702118},[31685,31686],[34,702120,702122],{"className":702121},[31792],[34,702123,702125,702145],{"className":702124},[31796,31797],[34,702126,702128,702142],{"className":702127},[31801],[34,702129,702131],{"className":702130,"style":31806},[31805],[34,702132,702133,702136],{"style":31809},[34,702134],{"className":702135,"style":31814},[31813],[34,702137,702139],{"className":702138},[31818,31819,31820,31821],[34,702140,459],{"className":702141},[31685,31821],[34,702143,31829],{"className":702144},[31828],[34,702146,702148],{"className":702147},[31801],[34,702149,702151],{"className":702150,"style":31836},[31805],[34,702152],{},[34,702154,702156,702230],{"className":702155},[31642],[34,702157,702159],{"className":702158},[31646],[31648,702160,702161],{"xmlns":31650},[31652,702162,702163,702227],{},[31655,702164,702165,702183,702185,702187,702189,702191,702197,702199,702205,702207,702209],{},[41453,702166,702167,702173],{},[31655,702168,702169,702171],{},[31658,702170,619473],{"mathvariant":38079},[31658,702172,126541],{},[31655,702174,702175,702177],{},[31658,702176,619473],{"mathvariant":38079},[31743,702178,702179,702181],{},[31658,702180,32936],{},[31749,702182,459],{},[31662,702184,202],{},[31662,702186,32226],{},[31658,702188,31739],{},[31662,702190,456],{"stretchy":702},[31743,702192,702193,702195],{},[31658,702194,32936],{},[31749,702196,459],{},[31662,702198,31960],{"separator":758},[31743,702200,702201,702203],{},[31658,702202,67764],{},[31749,702204,459],{},[31662,702206,2557],{"stretchy":702},[31662,702208,71191],{},[41453,702210,702211,702217],{},[31655,702212,702213,702215],{},[31658,702214,619473],{"mathvariant":38079},[31658,702216,126541],{},[31655,702218,702219,702221],{},[31658,702220,619473],{"mathvariant":38079},[31743,702222,702223,702225],{},[31658,702224,67764],{},[31749,702226,459],{},[31665,702228,702229],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial x_0} = -f(x_0,y_0) \\cdot \\frac{\\partial \\varphi}{\\partial y_0}",[34,702231,702233,702360,702473],{"className":702232,"ariaHidden":758},[31672],[34,702234,702236,702240,702351,702354,702357],{"className":702235},[31676],[34,702237],{"className":702238,"style":702239},[31680],"height:1.3773em;vertical-align:-0.4451em;",[34,702241,702243,702246,702348],{"className":702242},[31685],[34,702244],{"className":702245},[31782,41488],[34,702247,702249],{"className":702248},[41453],[34,702250,702252,702340],{"className":702251},[31796,31797],[34,702253,702255,702337],{"className":702254},[31801],[34,702256,702258,702312,702320],{"className":702257,"style":608935},[31805],[34,702259,702260,702263],{"style":41504},[34,702261],{"className":702262,"style":41508},[31813],[34,702264,702266],{"className":702265},[31818,31819,31820,31821],[34,702267,702269,702272],{"className":702268},[31685,31821],[34,702270,619473],{"className":702271,"style":112322},[31685,31821],[34,702273,702275,702278],{"className":702274},[31685,31821],[34,702276,32936],{"className":702277},[31685,31686,31821],[34,702279,702281],{"className":702280},[31792],[34,702282,702284,702304],{"className":702283},[31796,31797],[34,702285,702287,702301],{"className":702286},[31801],[34,702288,702290],{"className":702289,"style":318658},[31805],[34,702291,702292,702295],{"style":318661},[34,702293],{"className":702294,"style":43406},[31813],[34,702296,702298],{"className":702297},[31818,43410,43411,31821],[34,702299,459],{"className":702300},[31685,31821],[34,702302,31829],{"className":702303},[31828],[34,702305,702307],{"className":702306},[31801],[34,702308,702310],{"className":702309,"style":220540},[31805],[34,702311],{},[34,702313,702314,702317],{"style":41520},[34,702315],{"className":702316,"style":41508},[31813],[34,702318],{"className":702319,"style":41528},[41527],[34,702321,702322,702325],{"style":548368},[34,702323],{"className":702324,"style":41508},[31813],[34,702326,702328],{"className":702327},[31818,31819,31820,31821],[34,702329,702331,702334],{"className":702330},[31685,31821],[34,702332,619473],{"className":702333,"style":112322},[31685,31821],[34,702335,126541],{"className":702336},[31685,31686,31821],[34,702338,31829],{"className":702339},[31828],[34,702341,702343],{"className":702342},[31801],[34,702344,702346],{"className":702345,"style":231800},[31805],[34,702347],{},[34,702349],{"className":702350},[31690,41488],[34,702352],{"className":702353,"style":31846},[31845],[34,702355,202],{"className":702356},[31850],[34,702358],{"className":702359,"style":31846},[31845],[34,702361,702363,702366,702369,702372,702375,702415,702418,702421,702461,702464,702467,702470],{"className":702362},[31676],[34,702364],{"className":702365,"style":31774},[31680],[34,702367,32226],{"className":702368},[31685],[34,702370,31739],{"className":702371,"style":31778},[31685,31686],[34,702373,456],{"className":702374},[31782],[34,702376,702378,702381],{"className":702377},[31685],[34,702379,32936],{"className":702380},[31685,31686],[34,702382,702384],{"className":702383},[31792],[34,702385,702387,702407],{"className":702386},[31796,31797],[34,702388,702390,702404],{"className":702389},[31801],[34,702391,702393],{"className":702392,"style":31806},[31805],[34,702394,702395,702398],{"style":31809},[34,702396],{"className":702397,"style":31814},[31813],[34,702399,702401],{"className":702400},[31818,31819,31820,31821],[34,702402,459],{"className":702403},[31685,31821],[34,702405,31829],{"className":702406},[31828],[34,702408,702410],{"className":702409},[31801],[34,702411,702413],{"className":702412,"style":31836},[31805],[34,702414],{},[34,702416,31960],{"className":702417},[32119],[34,702419],{"className":702420,"style":32123},[31845],[34,702422,702424,702427],{"className":702423},[31685],[34,702425,67764],{"className":702426,"style":32062},[31685,31686],[34,702428,702430],{"className":702429},[31792],[34,702431,702433,702453],{"className":702432},[31796,31797],[34,702434,702436,702450],{"className":702435},[31801],[34,702437,702439],{"className":702438,"style":31806},[31805],[34,702440,702441,702444],{"style":38795},[34,702442],{"className":702443,"style":31814},[31813],[34,702445,702447],{"className":702446},[31818,31819,31820,31821],[34,702448,459],{"className":702449},[31685,31821],[34,702451,31829],{"className":702452},[31828],[34,702454,702456],{"className":702455},[31801],[34,702457,702459],{"className":702458,"style":31836},[31805],[34,702460],{},[34,702462,2557],{"className":702463},[31690],[34,702465],{"className":702466,"style":32604},[31845],[34,702468,71191],{"className":702469},[32024],[34,702471],{"className":702472,"style":32604},[31845],[34,702474,702476,702479],{"className":702475},[31676],[34,702477],{"className":702478,"style":695705},[31680],[34,702480,702482,702485,702587],{"className":702481},[31685],[34,702483],{"className":702484},[31782,41488],[34,702486,702488],{"className":702487},[41453],[34,702489,702491,702579],{"className":702490},[31796,31797],[34,702492,702494,702576],{"className":702493},[31801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这个公式很有趣，它其实告诉我们，对 ",[34,702598,702600,702617],{"className":702599},[31642],[34,702601,702603],{"className":702602},[31646],[31648,702604,702605],{"xmlns":31650},[31652,702606,702607,702615],{},[31655,702608,702609],{},[31743,702610,702611,702613],{},[31658,702612,32936],{},[31749,702614,459],{},[31665,702616,702103],{"encoding":31667},[34,702618,702620],{"className":702619,"ariaHidden":758},[31672],[34,702621,702623,702626],{"className":702622},[31676],[34,702624],{"className":702625,"style":31860},[31680],[34,702627,702629,702632],{"className":702628},[31685],[34,702630,32936],{"className":702631},[31685,31686],[34,702633,702635],{"className":702634},[31792],[34,702636,702638,702658],{"className":702637},[31796,31797],[34,702639,702641,702655],{"className":702640},[31801],[34,702642,702644],{"className":702643,"style":31806},[31805],[34,702645,702646,702649],{"style":31809},[34,702647],{"className":702648,"style":31814},[31813],[34,702650,702652],{"className":702651},[31818,31819,31820,31821],[34,702653,459],{"className":702654},[31685,31821],[34,702656,31829],{"className":702657},[31828],[34,702659,702661],{"className":702660},[31801],[34,702662,702664],{"className":702663,"style":31836},[31805],[34,702665],{}," 的导数可以通过对 ",[34,702668,702670,702687],{"className":702669},[31642],[34,702671,702673],{"className":702672},[31646],[31648,702674,702675],{"xmlns":31650},[31652,702676,702677,702685],{},[31655,702678,702679],{},[31743,702680,702681,702683],{},[31658,702682,67764],{},[31749,702684,459],{},[31665,702686,699759],{"encoding":31667},[34,702688,702690],{"className":702689,"ariaHidden":758},[31672],[34,702691,702693,702696],{"className":702692},[31676],[34,702694],{"className":702695,"style":38774},[31680],[34,702697,702699,702702],{"className":702698},[31685],[34,702700,67764],{"className":702701,"style":32062},[31685,31686],[34,702703,702705],{"className":702704},[31792],[34,702706,702708,702728],{"className":702707},[31796,31797],[34,702709,702711,702725],{"className":702710},[31801],[34,702712,702714],{"className":702713,"style":31806},[31805],[34,702715,702716,702719],{"style":38795},[34,702717],{"className":702718,"style":31814},[31813],[34,702720,702722],{"className":702721},[31818,31819,31820,31821],[34,702723,459],{"className":702724},[31685,31821],[34,702726,31829],{"className":702727},[31828],[34,702729,702731],{"className":702730},[31801],[34,702732,702734],{"className":702733,"style":31836},[31805],[34,702735],{}," 的导数算出来。它多乘了一个 ",[34,702738,702740,702774],{"className":702739},[31642],[34,702741,702743],{"className":702742},[31646],[31648,702744,702745],{"xmlns":31650},[31652,702746,702747,702771],{},[31655,702748,702749,702751,702753,702755,702761,702763,702769],{},[31662,702750,32226],{},[31658,702752,31739],{},[31662,702754,456],{"stretchy":702},[31743,702756,702757,702759],{},[31658,702758,32936],{},[31749,702760,459],{},[31662,702762,31960],{"separator":758},[31743,702764,702765,702767],{},[31658,702766,67764],{},[31749,702768,459],{},[31662,702770,2557],{"stretchy":702},[31665,702772,702773],{"encoding":31667},"-f(x_0,y_0)",[34,702775,702777],{"className":702776,"ariaHidden":758},[31672],[34,702778,702780,702783,702786,702789,702792,702832,702835,702838,702878],{"className":702779},[31676],[34,702781],{"className":702782,"style":31774},[31680],[34,702784,32226],{"className":702785},[31685],[34,702787,31739],{"className":702788,"style":31778},[31685,31686],[34,702790,456],{"className":702791},[31782],[34,702793,702795,702798],{"className":702794},[31685],[34,702796,32936],{"className":702797},[31685,31686],[34,702799,702801],{"className":702800},[31792],[34,702802,702804,702824],{"className":702803},[31796,31797],[34,702805,702807,702821],{"className":702806},[31801],[34,702808,702810],{"className":702809,"style":31806},[31805],[34,702811,702812,702815],{"style":31809},[34,702813],{"className":702814,"style":31814},[31813],[34,702816,702818],{"className":702817},[31818,31819,31820,31821],[34,702819,459],{"className":702820},[31685,31821],[34,702822,31829],{"className":702823},[31828],[34,702825,702827],{"className":702826},[31801],[34,702828,702830],{"className":702829,"style":31836},[31805],[34,702831],{},[34,702833,31960],{"className":702834},[32119],[34,702836],{"className":702837,"style":32123},[31845],[34,702839,702841,702844],{"className":702840},[31685],[34,702842,67764],{"className":702843,"style":32062},[31685,31686],[34,702845,702847],{"className":702846},[31792],[34,702848,702850,702870],{"className":702849},[31796,31797],[34,702851,702853,702867],{"className":702852},[31801],[34,702854,702856],{"className":702855,"style":31806},[31805],[34,702857,702858,702861],{"style":38795},[34,702859],{"className":702860,"style":31814},[31813],[34,702862,702864],{"className":702863},[31818,31819,31820,31821],[34,702865,459],{"className":702866},[31685,31821],[34,702868,31829],{"className":702869},[31828],[34,702871,702873],{"className":702872},[31801],[34,702874,702876],{"className":702875,"style":31836},[31805],[34,702877],{},[34,702879,2557],{"className":702880},[31690],"（即初始处的斜率）。",[32188,702883],{},[253,702885,702887],{"id":702886},"第三步构建答案-problem-solving","第三步：构建答案 (Problem Solving)",[15,702889,702890],{},"我将一步步为你讲解这两道题目。",[32490,702892,214292],{"id":214289},[15,702894,702895,702897,702898,702970,702971,703145,703146,703337,703338,703628,703629,347],{},[2627,702896,101191],{}," 假设 ",[34,702899,702901,702931],{"className":702900},[31642],[34,702902,702904],{"className":702903},[31646],[31648,702905,702906],{"xmlns":31650},[31652,702907,702908,702928],{},[31655,702909,702910,702912,702914,702916,702918,702920,702922,702924,702926],{},[31658,702911,105852],{},[31662,702913,456],{"stretchy":702},[31658,702915,32936],{},[31662,702917,2557],{"stretchy":702},[31662,702919,31960],{"separator":758},[31658,702921,116863],{},[31662,702923,456],{"stretchy":702},[31658,702925,32936],{},[31662,702927,2557],{"stretchy":702},[31665,702929,702930],{"encoding":31667},"P(x), Q(x)",[34,702932,702934],{"className":702933,"ariaHidden":758},[31672],[34,702935,702937,702940,702943,702946,702949,702952,702955,702958,702961,702964,702967],{"className":702936},[31676],[34,702938],{"className":702939,"style":31774},[31680],[34,702941,105852],{"className":702942,"style":37422},[31685,31686],[34,702944,456],{"className":702945},[31782],[34,702947,32936],{"className":702948},[31685,31686],[34,702950,2557],{"className":702951},[31690],[34,702953,31960],{"className":702954},[32119],[34,702956],{"className":702957,"style":32123},[31845],[34,702959,116863],{"className":702960},[31685,31686],[34,702962,456],{"className":702963},[31782],[34,702965,32936],{"className":702966},[31685,31686],[34,702968,2557],{"className":702969},[31690]," 连续，",[34,702972,702974,703014],{"className":702973},[31642],[34,702975,702977],{"className":702976},[31646],[31648,702978,702979],{"xmlns":31650},[31652,702980,702981,703011],{},[31655,702982,702983,702985,702987,702989,702991,702993,702995,703001,703003,703009],{},[31658,702984,67764],{},[31662,702986,202],{},[31658,702988,126541],{},[31662,702990,456],{"stretchy":702},[31658,702992,32936],{},[31662,702994,31960],{"separator":758},[31743,702996,702997,702999],{},[31658,702998,32936],{},[31749,703000,459],{},[31662,703002,31960],{"separator":758},[31743,703004,703005,703007],{},[31658,703006,67764],{},[31749,703008,459],{},[31662,703010,2557],{"stretchy":702},[31665,703012,703013],{"encoding":31667},"y = \\varphi(x,x_0,y_0)",[34,703015,703017,703035],{"className":703016,"ariaHidden":758},[31672],[34,703018,703020,703023,703026,703029,703032],{"className":703019},[31676],[34,703021],{"className":703022,"style":38774},[31680],[34,703024,67764],{"className":703025,"style":32062},[31685,31686],[34,703027],{"className":703028,"style":31846},[31845],[34,703030,202],{"className":703031},[31850],[34,703033],{"className":703034,"style":31846},[31845],[34,703036,703038,703041,703044,703047,703050,703053,703056,703096,703099,703102,703142],{"className":703037},[31676],[34,703039],{"className":703040,"style":31774},[31680],[34,703042,126541],{"className":703043},[31685,31686],[34,703045,456],{"className":703046},[31782],[34,703048,32936],{"className":703049},[31685,31686],[34,703051,31960],{"className":703052},[32119],[34,703054],{"className":703055,"style":32123},[31845],[34,703057,703059,703062],{"className":703058},[31685],[34,703060,32936],{"className":703061},[31685,31686],[34,703063,703065],{"className":703064},[31792],[34,703066,703068,703088],{"className":703067},[31796,31797],[34,703069,703071,703085],{"className":703070},[31801],[34,703072,703074],{"className":703073,"style":31806},[31805],[34,703075,703076,703079],{"style":31809},[34,703077],{"className":703078,"style":31814},[31813],[34,703080,703082],{"className":703081},[31818,31819,31820,31821],[34,703083,459],{"className":703084},[31685,31821],[34,703086,31829],{"className":703087},[31828],[34,703089,703091],{"className":703090},[31801],[34,703092,703094],{"className":703093,"style":31836},[31805],[34,703095],{},[34,703097,31960],{"className":703098},[32119],[34,703100],{"className":703101,"style":32123},[31845],[34,703103,703105,703108],{"className":703104},[31685],[34,703106,67764],{"className":703107,"style":32062},[31685,31686],[34,703109,703111],{"className":703110},[31792],[34,703112,703114,703134],{"className":703113},[31796,31797],[34,703115,703117,703131],{"className":703116},[31801],[34,703118,703120],{"className":703119,"style":31806},[31805],[34,703121,703122,703125],{"style":38795},[34,703123],{"className":703124,"style":31814},[31813],[34,703126,703128],{"className":703127},[31818,31819,31820,31821],[34,703129,459],{"className":703130},[31685,31821],[34,703132,31829],{"className":703133},[31828],[34,703135,703137],{"className":703136},[31801],[34,703138,703140],{"className":703139,"style":31836},[31805],[34,703141],{},[34,703143,2557],{"className":703144},[31690]," 是方程 ",[34,703147,703149,703197],{"className":703148},[31642],[34,703150,703152],{"className":703151},[31646],[31648,703153,703154],{"xmlns":31650},[31652,703155,703156,703194],{},[31655,703157,703158,703172,703174,703176,703178,703180,703182,703184,703186,703188,703190,703192],{},[41453,703159,703160,703166],{},[31655,703161,703162,703164],{},[31658,703163,46900],{"mathvariant":38079},[31658,703165,67764],{},[31655,703167,703168,703170],{},[31658,703169,46900],{"mathvariant":38079},[31658,703171,32936],{},[31662,703173,202],{},[31658,703175,105852],{},[31662,703177,456],{"stretchy":702},[31658,703179,32936],{},[31662,703181,2557],{"stretchy":702},[31658,703183,67764],{},[31662,703185,223],{},[31658,703187,116863],{},[31662,703189,456],{"stretchy":702},[31658,703191,32936],{},[31662,703193,2557],{"stretchy":702},[31665,703195,703196],{"encoding":31667},"\\frac{\\mathrm{d}y}{\\mathrm{d}x} = P(x)y + Q(x)",[34,703198,703200,703289,703319],{"className":703199,"ariaHidden":758},[31672],[34,703201,703203,703206,703280,703283,703286],{"className":703202},[31676],[34,703204],{"className":703205,"style":608762},[31680],[34,703207,703209,703212,703277],{"className":703208},[31685],[34,703210],{"className":703211},[31782,41488],[34,703213,703215],{"className":703214},[41453],[34,703216,703218,703269],{"className":703217},[31796,31797],[34,703219,703221,703266],{"className":703220},[31801],[34,703222,703224,703241,703249],{"className":703223,"style":608935},[31805],[34,703225,703226,703229],{"style":41504},[34,703227],{"className":703228,"style":41508},[31813],[34,703230,703232],{"className":703231},[31818,31819,31820,31821],[34,703233,703235,703238],{"className":703234},[31685,31821],[34,703236,46900],{"className":703237},[31685,207923,31821],[34,703239,32936],{"className":703240},[31685,31686,31821],[34,703242,703243,703246],{"style":41520},[34,703244],{"className":703245,"style":41508},[31813],[34,703247],{"className":703248,"style":41528},[41527],[34,703250,703251,703254],{"style":548368},[34,703252],{"className":703253,"style":41508},[31813],[34,703255,703257],{"className":703256},[31818,31819,31820,31821],[34,703258,703260,703263],{"className":703259},[31685,31821],[34,703261,46900],{"className":703262},[31685,207923,31821],[34,703264,67764],{"className":703265,"style":32062},[31685,31686,31821],[34,703267,31829],{"className":703268},[31828],[34,703270,703272],{"className":703271},[31801],[34,703273,703275],{"className":703274,"style":41619},[31805],[34,703276],{},[34,703278],{"className":703279},[31690,41488],[34,703281],{"className":703282,"style":31846},[31845],[34,703284,202],{"className":703285},[31850],[34,703287],{"className":703288,"style":31846},[31845],[34,703290,703292,703295,703298,703301,703304,703307,703310,703313,703316],{"className":703291},[31676],[34,703293],{"className":703294,"style":31774},[31680],[34,703296,105852],{"className":703297,"style":37422},[31685,31686],[34,703299,456],{"className":703300},[31782],[34,703302,32936],{"className":703303},[31685,31686],[34,703305,2557],{"className":703306},[31690],[34,703308,67764],{"className":703309,"style":32062},[31685,31686],[34,703311],{"className":703312,"style":32604},[31845],[34,703314,223],{"className":703315},[32024],[34,703317],{"className":703318,"style":32604},[31845],[34,703320,703322,703325,703328,703331,703334],{"className":703321},[31676],[34,703323],{"className":703324,"style":31774},[31680],[34,703326,116863],{"className":703327},[31685,31686],[34,703329,456],{"className":703330},[31782],[34,703332,32936],{"className":703333},[31685,31686],[34,703335,2557],{"className":703336},[31690]," 的解。求 ",[34,703339,703341,703391],{"className":703340},[31642],[34,703342,703344],{"className":703343},[31646],[31648,703345,703346],{"xmlns":31650},[31652,703347,703348,703388],{},[31655,703349,703350,703368,703370],{},[41453,703351,703352,703358],{},[31655,703353,703354,703356],{},[31658,703355,619473],{"mathvariant":38079},[31658,703357,126541],{},[31655,703359,703360,703362],{},[31658,703361,619473],{"mathvariant":38079},[31743,703363,703364,703366],{},[31658,703365,32936],{},[31749,703367,459],{},[31662,703369,31960],{"separator":758},[41453,703371,703372,703378],{},[31655,703373,703374,703376],{},[31658,703375,619473],{"mathvariant":38079},[31658,703377,126541],{},[31655,703379,703380,703382],{},[31658,703381,619473],{"mathvariant":38079},[31743,703383,703384,703386],{},[31658,703385,67764],{},[31749,703387,459],{},[31665,703389,703390],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial x_0}, \\frac{\\partial \\varphi}{\\partial y_0}",[34,703392,703394],{"className":703393,"ariaHidden":758},[31672],[34,703395,703397,703400,703511,703514,703517],{"className":703396},[31676],[34,703398],{"className":703399,"style":695705},[31680],[34,703401,703403,703406,703508],{"className":703402},[31685],[34,703404],{"className":703405},[31782,41488],[34,703407,703409],{"className":703408},[41453],[34,703410,703412,703500],{"className":703411},[31796,31797],[34,703413,703415,703497],{"className":703414},[31801],[34,703416,703418,703472,703480],{"className":703417,"style":608935},[31805],[34,703419,703420,703423],{"style":41504},[34,703421],{"className":703422,"style":41508},[31813],[34,703424,703426],{"className":703425},[31818,31819,31820,31821],[34,703427,703429,703432],{"className":703428},[31685,31821],[34,703430,619473],{"className":703431,"style":112322},[31685,31821],[34,703433,703435,703438],{"className":703434},[31685,31821],[34,703436,32936],{"className":703437},[31685,31686,31821],[34,703439,703441],{"className":703440},[31792],[34,703442,703444,703464],{"className":703443},[31796,31797],[34,703445,703447,703461],{"className":703446},[31801],[34,703448,703450],{"className":703449,"style":318658},[31805],[34,703451,703452,703455],{"style":318661},[34,703453],{"className":703454,"style":43406},[31813],[34,703456,703458],{"className":703457},[31818,43410,43411,31821],[34,703459,459],{"className":703460},[31685,31821],[34,703462,31829],{"className":703463},[31828],[34,703465,703467],{"className":703466},[31801],[34,703468,703470],{"className":703469,"style":220540},[31805],[34,703471],{},[34,703473,703474,703477],{"style":41520},[34,703475],{"className":703476,"style":41508},[31813],[34,703478],{"className":703479,"style":41528},[41527],[34,703481,703482,703485],{"style":548368},[34,703483],{"className":703484,"style":41508},[31813],[34,703486,703488],{"className":703487},[31818,31819,31820,31821],[34,703489,703491,703494],{"className":703490},[31685,31821],[34,703492,619473],{"className":703493,"style":112322},[31685,31821],[34,703495,126541],{"className":703496},[31685,31686,31821],[34,703498,31829],{"className":703499},[31828],[34,703501,703503],{"className":703502},[31801],[34,703504,703506],{"className":703505,"style":231800},[31805],[34,703507],{},[34,703509],{"className":703510},[31690,41488],[34,703512,31960],{"className":703513},[32119],[34,703515],{"className":703516,"style":32123},[31845],[34,703518,703520,703523,703625],{"className":703519},[31685],[34,703521],{"className":703522},[31782,41488],[34,703524,703526],{"className":703525},[41453],[34,703527,703529,703617],{"className":703528},[31796,31797],[34,703530,703532,703614],{"className":703531},[31801],[34,703533,703535,703589,703597],{"className":703534,"style":608935},[31805],[34,703536,703537,703540],{"style":41504},[34,703538],{"className":703539,"style":41508},[31813],[34,703541,703543],{"className":703542},[31818,31819,31820,31821],[34,703544,703546,703549],{"className":703545},[31685,31821],[34,703547,619473],{"className":703548,"style":112322},[31685,31821],[34,703550,703552,703555],{"className":703551},[31685,31821],[34,703553,67764],{"className":703554,"style":32062},[31685,31686,31821],[34,703556,703558],{"className":703557},[31792],[34,703559,703561,703581],{"className":703560},[31796,31797],[34,703562,703564,703578],{"className":703563},[31801],[34,703565,703567],{"className":703566,"style":318658},[31805],[34,703568,703569,703572],{"style":561507},[34,703570],{"className":703571,"style":43406},[31813],[34,703573,703575],{"className":703574},[31818,43410,43411,31821],[34,703576,459],{"className":703577},[31685,31821],[34,703579,31829],{"className":703580},[31828],[34,703582,703584],{"className":703583},[31801],[34,703585,703587],{"className":703586,"style":220540},[31805],[34,703588],{},[34,703590,703591,703594],{"style":41520},[34,703592],{"className":703593,"style":41508},[31813],[34,703595],{"className":703596,"style":41528},[41527],[34,703598,703599,703602],{"style":548368},[34,703600],{"className":703601,"style":41508},[31813],[34,703603,703605],{"className":703604},[31818,31819,31820,31821],[34,703606,703608,703611],{"className":703607},[31685,31821],[34,703609,619473],{"className":703610,"style":112322},[31685,31821],[34,703612,126541],{"className":703613},[31685,31686,31821],[34,703615,31829],{"className":703616},[31828],[34,703618,703620],{"className":703619},[31801],[34,703621,703623],{"className":703622,"style":220191},[31805],[34,703624],{},[34,703626],{"className":703627},[31690,41488]," 及 ",[34,703630,703632,703658],{"className":703631},[31642],[34,703633,703635],{"className":703634},[31646],[31648,703636,703637],{"xmlns":31650},[31652,703638,703639,703655],{},[31655,703640,703641],{},[41453,703642,703643,703649],{},[31655,703644,703645,703647],{},[31658,703646,619473],{"mathvariant":38079},[31658,703648,126541],{},[31655,703650,703651,703653],{},[31658,703652,619473],{"mathvariant":38079},[31658,703654,32936],{},[31665,703656,703657],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial x}",[34,703659,703661],{"className":703660,"ariaHidden":758},[31672],[34,703662,703664,703667],{"className":703663},[31676],[34,703665],{"className":703666,"style":608762},[31680],[34,703668,703670,703673,703738],{"className":703669},[31685],[34,703671],{"className":703672},[31782,41488],[34,703674,703676],{"className":703675},[41453],[34,703677,703679,703730],{"className":703678},[31796,31797],[34,703680,703682,703727],{"className":703681},[31801],[34,703683,703685,703702,703710],{"className":703684,"style":608935},[31805],[34,703686,703687,703690],{"style":41504},[34,703688],{"className":703689,"style":41508},[31813],[34,703691,703693],{"className":703692},[31818,31819,31820,31821],[34,703694,703696,703699],{"className":703695},[31685,31821],[34,703697,619473],{"className":703698,"style":112322},[31685,31821],[34,703700,32936],{"className":703701},[31685,31686,31821],[34,703703,703704,703707],{"style":41520},[34,703705],{"className":703706,"style":41508},[31813],[34,703708],{"className":703709,"style":41528},[41527],[34,703711,703712,703715],{"style":548368},[34,703713],{"className":703714,"style":41508},[31813],[34,703716,703718],{"className":703717},[31818,31819,31820,31821],[34,703719,703721,703724],{"className":703720},[31685,31821],[34,703722,619473],{"className":703723,"style":112322},[31685,31821],[34,703725,126541],{"className":703726},[31685,31686,31821],[34,703728,31829],{"className":703729},[31828],[34,703731,703733],{"className":703732},[31801],[34,703734,703736],{"className":703735,"style":41619},[31805],[34,703737],{},[34,703739],{"className":703740},[31690,41488],[15,703742,703743],{},[2627,703744,258022],{},[15,703746,703747,703808,703809,703999,704000,704136,704137,112074,704165,704193,704194,704305,704306],{},[2627,703748,703749,703750,703807],{},"Step 1: 识别 ",[34,703751,703753,703777],{"className":703752},[31642],[34,703754,703756],{"className":703755},[31646],[31648,703757,703758],{"xmlns":31650},[31652,703759,703760,703774],{},[31655,703761,703762,703764,703766,703768,703770,703772],{},[31658,703763,31739],{},[31662,703765,456],{"stretchy":702},[31658,703767,32936],{},[31662,703769,31960],{"separator":758},[31658,703771,67764],{},[31662,703773,2557],{"stretchy":702},[31665,703775,703776],{"encoding":31667},"f(x,y)",[34,703778,703780],{"className":703779,"ariaHidden":758},[31672],[34,703781,703783,703786,703789,703792,703795,703798,703801,703804],{"className":703782},[31676],[34,703784],{"className":703785,"style":31774},[31680],[34,703787,31739],{"className":703788,"style":31778},[31685,31686],[34,703790,456],{"className":703791},[31782],[34,703793,32936],{"className":703794},[31685,31686],[34,703796,31960],{"className":703797},[32119],[34,703799],{"className":703800,"style":32123},[31845],[34,703802,67764],{"className":703803,"style":32062},[31685,31686],[34,703805,2557],{"className":703806},[31690]," 及其偏导数","\n根据方程 ",[34,703810,703812,703859],{"className":703811},[31642],[34,703813,703815],{"className":703814},[31646],[31648,703816,703817],{"xmlns":31650},[31652,703818,703819,703857],{},[31655,703820,703821,703835,703837,703839,703841,703843,703845,703847,703849,703851,703853,703855],{},[41453,703822,703823,703829],{},[31655,703824,703825,703827],{},[31658,703826,46900],{"mathvariant":38079},[31658,703828,67764],{},[31655,703830,703831,703833],{},[31658,703832,46900],{"mathvariant":38079},[31658,703834,32936],{},[31662,703836,202],{},[31658,703838,105852],{},[31662,703840,456],{"stretchy":702},[31658,703842,32936],{},[31662,703844,2557],{"stretchy":702},[31658,703846,67764],{},[31662,703848,223],{},[31658,703850,116863],{},[31662,703852,456],{"stretchy":702},[31658,703854,32936],{},[31662,703856,2557],{"stretchy":702},[31665,703858,703196],{"encoding":31667},[34,703860,703862,703951,703981],{"className":703861,"ariaHidden":758},[31672],[34,703863,703865,703868,703942,703945,703948],{"className":703864},[31676],[34,703866],{"className":703867,"style":608762},[31680],[34,703869,703871,703874,703939],{"className":703870},[31685],[34,703872],{"className":703873},[31782,41488],[34,703875,703877],{"className":703876},[41453],[34,703878,703880,703931],{"className":703879},[31796,31797],[34,703881,703883,703928],{"className":703882},[31801],[34,703884,703886,703903,703911],{"className":703885,"style":608935},[31805],[34,703887,703888,703891],{"style":41504},[34,703889],{"className":703890,"style":41508},[31813],[34,703892,703894],{"className":703893},[31818,31819,31820,31821],[34,703895,703897,703900],{"className":703896},[31685,31821],[34,703898,46900],{"className":703899},[31685,207923,31821],[34,703901,32936],{"className":703902},[31685,31686,31821],[34,703904,703905,703908],{"style":41520},[34,703906],{"className":703907,"style":41508},[31813],[34,703909],{"className":703910,"style":41528},[41527],[34,703912,703913,703916],{"style":548368},[34,703914],{"className":703915,"style":41508},[31813],[34,703917,703919],{"className":703918},[31818,31819,31820,31821],[34,703920,703922,703925],{"className":703921},[31685,31821],[34,703923,46900],{"className":703924},[31685,207923,31821],[34,703926,67764],{"className":703927,"style":32062},[31685,31686,31821],[34,703929,31829],{"className":703930},[31828],[34,703932,703934],{"className":703933},[31801],[34,703935,703937],{"className":703936,"style":41619},[31805],[34,703938],{},[34,703940],{"className":703941},[31690,41488],[34,703943],{"className":703944,"style":31846},[31845],[34,703946,202],{"className":703947},[31850],[34,703949],{"className":703950,"style":31846},[31845],[34,703952,703954,703957,703960,703963,703966,703969,703972,703975,703978],{"className":703953},[31676],[34,703955],{"className":703956,"style":31774},[31680],[34,703958,105852],{"className":703959,"style":37422},[31685,31686],[34,703961,456],{"className":703962},[31782],[34,703964,32936],{"className":703965},[31685,31686],[34,703967,2557],{"className":703968},[31690],[34,703970,67764],{"className":703971,"style":32062},[31685,31686],[34,703973],{"className":703974,"style":32604},[31845],[34,703976,223],{"className":703977},[32024],[34,703979],{"className":703980,"style":32604},[31845],[34,703982,703984,703987,703990,703993,703996],{"className":703983},[31676],[34,703985],{"className":703986,"style":31774},[31680],[34,703988,116863],{"className":703989},[31685,31686],[34,703991,456],{"className":703992},[31782],[34,703994,32936],{"className":703995},[31685,31686],[34,703997,2557],{"className":703998},[31690],"，我们可以确定右端函数为：\n",[34,704001,704003,704049],{"className":704002},[31642],[34,704004,704006],{"className":704005},[31646],[31648,704007,704008],{"xmlns":31650},[31652,704009,704010,704046],{},[31655,704011,704012,704014,704016,704018,704020,704022,704024,704026,704028,704030,704032,704034,704036,704038,704040,704042,704044],{},[31658,704013,31739],{},[31662,704015,456],{"stretchy":702},[31658,704017,32936],{},[31662,704019,31960],{"separator":758},[31658,704021,67764],{},[31662,704023,2557],{"stretchy":702},[31662,704025,202],{},[31658,704027,105852],{},[31662,704029,456],{"stretchy":702},[31658,704031,32936],{},[31662,704033,2557],{"stretchy":702},[31658,704035,67764],{},[31662,704037,223],{},[31658,704039,116863],{},[31662,704041,456],{"stretchy":702},[31658,704043,32936],{},[31662,704045,2557],{"stretchy":702},[31665,704047,704048],{"encoding":31667},"f(x,y) = P(x)y + Q(x)",[34,704050,704052,704088,704118],{"className":704051,"ariaHidden":758},[31672],[34,704053,704055,704058,704061,704064,704067,704070,704073,704076,704079,704082,704085],{"className":704054},[31676],[34,704056],{"className":704057,"style":31774},[31680],[34,704059,31739],{"className":704060,"style":31778},[31685,31686],[34,704062,456],{"className":704063},[31782],[34,704065,32936],{"className":704066},[31685,31686],[34,704068,31960],{"className":704069},[32119],[34,704071],{"className":704072,"style":32123},[31845],[34,704074,67764],{"className":704075,"style":32062},[31685,31686],[34,704077,2557],{"className":704078},[31690],[34,704080],{"className":704081,"style":31846},[31845],[34,704083,202],{"className":704084},[31850],[34,704086],{"className":704087,"style":31846},[31845],[34,704089,704091,704094,704097,704100,704103,704106,704109,704112,704115],{"className":704090},[31676],[34,704092],{"className":704093,"style":31774},[31680],[34,704095,105852],{"className":704096,"style":37422},[31685,31686],[34,704098,456],{"className":704099},[31782],[34,704101,32936],{"className":704102},[31685,31686],[34,704104,2557],{"className":704105},[31690],[34,704107,67764],{"className":704108,"style":32062},[31685,31686],[34,704110],{"className":704111,"style":32604},[31845],[34,704113,223],{"className":704114},[32024],[34,704116],{"className":704117,"style":32604},[31845],[34,704119,704121,704124,704127,704130,704133],{"className":704120},[31676],[34,704122],{"className":704123,"style":31774},[31680],[34,704125,116863],{"className":704126},[31685,31686],[34,704128,456],{"className":704129},[31782],[34,704131,32936],{"className":704132},[31685,31686],[34,704134,2557],{"className":704135},[31690],"\n我们需要计算 ",[34,704138,704140,704153],{"className":704139},[31642],[34,704141,704143],{"className":704142},[31646],[31648,704144,704145],{"xmlns":31650},[31652,704146,704147,704151],{},[31655,704148,704149],{},[31658,704150,31739],{},[31665,704152,31739],{"encoding":31667},[34,704154,704156],{"className":704155,"ariaHidden":758},[31672],[34,704157,704159,704162],{"className":704158},[31676],[34,704160],{"className":704161,"style":43217},[31680],[34,704163,31739],{"className":704164,"style":31778},[31685,31686],[34,704166,704168,704181],{"className":704167},[31642],[34,704169,704171],{"className":704170},[31646],[31648,704172,704173],{"xmlns":31650},[31652,704174,704175,704179],{},[31655,704176,704177],{},[31658,704178,67764],{},[31665,704180,67764],{"encoding":31667},[34,704182,704184],{"className":704183,"ariaHidden":758},[31672],[34,704185,704187,704190],{"className":704186},[31676],[34,704188],{"className":704189,"style":38774},[31680],[34,704191,67764],{"className":704192,"style":32062},[31685,31686]," 的偏导数 ",[34,704195,704197,704222],{"className":704196},[31642],[34,704198,704200],{"className":704199},[31646],[31648,704201,704202],{"xmlns":31650},[31652,704203,704204,704220],{},[31655,704205,704206],{},[41453,704207,704208,704214],{},[31655,704209,704210,704212],{},[31658,704211,619473],{"mathvariant":38079},[31658,704213,31739],{},[31655,704215,704216,704218],{},[31658,704217,619473],{"mathvariant":38079},[31658,704219,67764],{},[31665,704221,701964],{"encoding":31667},[34,704223,704225],{"className":704224,"ariaHidden":758},[31672],[34,704226,704228,704231],{"className":704227},[31676],[34,704229],{"className":704230,"style":695705},[31680],[34,704232,704234,704237,704302],{"className":704233},[31685],[34,704235],{"className":704236},[31782,41488],[34,704238,704240],{"className":704239},[41453],[34,704241,704243,704294],{"className":704242},[31796,31797],[34,704244,704246,704291],{"className":704245},[31801],[34,704247,704249,704266,704274],{"className":704248,"style":608935},[31805],[34,704250,704251,704254],{"style":41504},[34,704252],{"className":704253,"style":41508},[31813],[34,704255,704257],{"className":704256},[31818,31819,31820,31821],[34,704258,704260,704263],{"className":704259},[31685,31821],[34,704261,619473],{"className":704262,"style":112322},[31685,31821],[34,704264,67764],{"className":704265,"style":32062},[31685,31686,31821],[34,704267,704268,704271],{"style":41520},[34,704269],{"className":704270,"style":41508},[31813],[34,704272],{"className":704273,"style":41528},[41527],[34,704275,704276,704279],{"style":548368},[34,704277],{"className":704278,"style":41508},[31813],[34,704280,704282],{"className":704281},[31818,31819,31820,31821],[34,704283,704285,704288],{"className":704284},[31685,31821],[34,704286,619473],{"className":704287,"style":112322},[31685,31821],[34,704289,31739],{"className":704290,"style":31778},[31685,31686,31821],[34,704292,31829],{"className":704293},[31828],[34,704295,704297],{"className":704296},[31801],[34,704298,704300],{"className":704299,"style":220191},[31805],[34,704301],{},[34,704303],{"className":704304},[31690,41488],"，这是使用定理5公式的关键：\n",[34,704307,704309,704381],{"className":704308},[31642],[34,704310,704312],{"className":704311},[31646],[31648,704313,704314],{"xmlns":31650},[31652,704315,704316,704378],{},[31655,704317,704318,704332,704334,704344,704346,704348,704350,704352,704354,704356,704358,704360,704362,704364,704366,704368,704370,704372,704374,704376],{},[41453,704319,704320,704326],{},[31655,704321,704322,704324],{},[31658,704323,619473],{"mathvariant":38079},[31658,704325,31739],{},[31655,704327,704328,704330],{},[31658,704329,619473],{"mathvariant":38079},[31658,704331,67764],{},[31662,704333,202],{},[41453,704335,704336,704338],{},[31658,704337,619473],{"mathvariant":38079},[31655,704339,704340,704342],{},[31658,704341,619473],{"mathvariant":38079},[31658,704343,67764],{},[31662,704345,15972],{"stretchy":702},[31658,704347,105852],{},[31662,704349,456],{"stretchy":702},[31658,704351,32936],{},[31662,704353,2557],{"stretchy":702},[31658,704355,67764],{},[31662,704357,223],{},[31658,704359,116863],{},[31662,704361,456],{"stretchy":702},[31658,704363,32936],{},[31662,704365,2557],{"stretchy":702},[31662,704367,32712],{"stretchy":702},[31662,704369,202],{},[31658,704371,105852],{},[31662,704373,456],{"stretchy":702},[31658,704375,32936],{},[31662,704377,2557],{"stretchy":702},[31665,704379,704380],{"encoding":31667},"\\frac{\\partial f}{\\partial y} = \\frac{\\partial}{\\partial y} [P(x)y + Q(x)] = P(x)",[34,704382,704384,704473,704577,704604],{"className":704383,"ariaHidden":758},[31672],[34,704385,704387,704390,704464,704467,704470],{"className":704386},[31676],[34,704388],{"className":704389,"style":695705},[31680],[34,704391,704393,704396,704461],{"className":704392},[31685],[34,704394],{"className":704395},[31782,41488],[34,704397,704399],{"className":704398},[41453],[34,704400,704402,704453],{"className":704401},[31796,31797],[34,704403,704405,704450],{"className":704404},[31801],[34,704406,704408,704425,704433],{"className":704407,"style":608935},[31805],[34,704409,704410,704413],{"style":41504},[34,704411],{"className":704412,"style":41508},[31813],[34,704414,704416],{"className":704415},[31818,31819,31820,31821],[34,704417,704419,704422],{"className":704418},[31685,31821],[34,704420,619473],{"className":704421,"style":112322},[31685,31821],[34,704423,67764],{"className":704424,"style":32062},[31685,31686,31821],[34,704426,704427,704430],{"style":41520},[34,704428],{"className":704429,"style":41508},[31813],[34,704431],{"className":704432,"style":41528},[41527],[34,704434,704435,704438],{"style":548368},[34,704436],{"className":704437,"style":41508},[31813],[34,704439,704441],{"className":704440},[31818,31819,31820,31821],[34,704442,704444,704447],{"className":704443},[31685,31821],[34,704445,619473],{"className":704446,"style":112322},[31685,31821],[34,704448,31739],{"className":704449,"style":31778},[31685,31686,31821],[34,704451,31829],{"className":704452},[31828],[34,704454,704456],{"className":704455},[31801],[34,704457,704459],{"className":704458,"style":220191},[31805],[34,704460],{},[34,704462],{"className":704463},[31690,41488],[34,704465],{"className":704466,"style":31846},[31845],[34,704468,202],{"className":704469},[31850],[34,704471],{"className":704472,"style":31846},[31845],[34,704474,704476,704479,704550,704553,704556,704559,704562,704565,704568,704571,704574],{"className":704475},[31676],[34,704477],{"className":704478,"style":685759},[31680],[34,704480,704482,704485,704547],{"className":704481},[31685],[34,704483],{"className":704484},[31782,41488],[34,704486,704488],{"className":704487},[41453],[34,704489,704491,704539],{"className":704490},[31796,31797],[34,704492,704494,704536],{"className":704493},[31801],[34,704495,704497,704514,704522],{"className":704496,"style":232511},[31805],[34,704498,704499,704502],{"style":41504},[34,704500],{"className":704501,"style":41508},[31813],[34,704503,704505],{"className":704504},[31818,31819,31820,31821],[34,704506,704508,704511],{"className":704507},[31685,31821],[34,704509,619473],{"className":704510,"style":112322},[31685,31821],[34,704512,67764],{"className":704513,"style":32062},[31685,31686,31821],[34,704515,704516,704519],{"style":41520},[34,704517],{"className":704518,"style":41508},[31813],[34,704520],{"className":704521,"style":41528},[41527],[34,704523,704524,704527],{"style":46771},[34,704525],{"className":704526,"style":41508},[31813],[34,704528,704530],{"className":704529},[31818,31819,31820,31821],[34,704531,704533],{"className":704532},[31685,31821],[34,704534,619473],{"className":704535,"style":112322},[31685,31821],[34,704537,31829],{"className":704538},[31828],[34,704540,704542],{"className":704541},[31801],[34,704543,704545],{"className":704544,"style":220191},[31805],[34,704546],{},[34,704548],{"className":704549},[31690,41488],[34,704551,15972],{"className":704552},[31782],[34,704554,105852],{"className":704555,"style":37422},[31685,31686],[34,704557,456],{"className":704558},[31782],[34,704560,32936],{"className":704561},[31685,31686],[34,704563,2557],{"className":704564},[31690],[34,704566,67764],{"className":704567,"style":32062},[31685,31686],[34,704569],{"className":704570,"style":32604},[31845],[34,704572,223],{"className":704573},[32024],[34,704575],{"className":704576,"style":32604},[31845],[34,704578,704580,704583,704586,704589,704592,704595,704598,704601],{"className":704579},[31676],[34,704581],{"className":704582,"style":31774},[31680],[34,704584,116863],{"className":704585},[31685,31686],[34,704587,456],{"className":704588},[31782],[34,704590,32936],{"className":704591},[31685,31686],[34,704593,481211],{"className":704594},[31690],[34,704596],{"className":704597,"style":31846},[31845],[34,704599,202],{"className":704600},[31850],[34,704602],{"className":704603,"style":31846},[31845],[34,704605,704607,704610,704613,704616,704619],{"className":704606},[31676],[34,704608],{"className":704609,"style":31774},[31680],[34,704611,105852],{"className":704612,"style":37422},[31685,31686],[34,704614,456],{"className":704615},[31782],[34,704617,32936],{"className":704618},[31685,31686],[34,704620,2557],{"className":704621},[31690],[15,704623,704624,704780,704781,221775,705186,705335,705336,705364,705365,705701],{},[2627,704625,704626,704627,704779],{},"Step 2: 计算 ",[34,704628,704630,704659],{"className":704629},[31642],[34,704631,704633],{"className":704632},[31646],[31648,704634,704635],{"xmlns":31650},[31652,704636,704637,704657],{},[31655,704638,704639],{},[41453,704640,704641,704647],{},[31655,704642,704643,704645],{},[31658,704644,619473],{"mathvariant":38079},[31658,704646,126541],{},[31655,704648,704649,704651],{},[31658,704650,619473],{"mathvariant":38079},[31743,704652,704653,704655],{},[31658,704654,67764],{},[31749,704656,459],{},[31665,704658,699617],{"encoding":31667},[34,704660,704662],{"className":704661,"ariaHidden":758},[31672],[34,704663,704665,704668],{"className":704664},[31676],[34,704666],{"className":704667,"style":695705},[31680],[34,704669,704671,704674,704776],{"className":704670},[31685],[34,704672],{"className":704673},[31782,41488],[34,704675,704677],{"className":704676},[41453],[34,704678,704680,704768],{"className":704679},[31796,31797],[34,704681,704683,704765],{"className":704682},[31801],[34,704684,704686,704740,704748],{"className":704685,"style":608935},[31805],[34,704687,704688,704691],{"style":41504},[34,704689],{"className":704690,"style":41508},[31813],[34,704692,704694],{"className":704693},[31818,31819,31820,31821],[34,704695,704697,704700],{"className":704696},[31685,31821],[34,704698,619473],{"className":704699,"style":112322},[31685,31821],[34,704701,704703,704706],{"className":704702},[31685,31821],[34,704704,67764],{"className":704705,"style":32062},[31685,31686,31821],[34,704707,704709],{"className":704708},[31792],[34,704710,704712,704732],{"className":704711},[31796,31797],[34,704713,704715,704729],{"className":704714},[31801],[34,704716,704718],{"className":704717,"style":318658},[31805],[34,704719,704720,704723],{"style":561507},[34,704721],{"className":704722,"style":43406},[31813],[34,704724,704726],{"className":704725},[31818,43410,43411,31821],[34,704727,459],{"className":704728},[31685,31821],[34,704730,31829],{"className":704731},[31828],[34,704733,704735],{"className":704734},[31801],[34,704736,704738],{"className":704737,"style":220540},[31805],[34,704739],{},[34,704741,704742,704745],{"style":41520},[34,704743],{"className":704744,"style":41508},[31813],[34,704746],{"className":704747,"style":41528},[41527],[34,704749,704750,704753],{"style":548368},[34,704751],{"className":704752,"style":41508},[31813],[34,704754,704756],{"className":704755},[31818,31819,31820,31821],[34,704757,704759,704762],{"className":704758},[31685,31821],[34,704760,619473],{"className":704761,"style":112322},[31685,31821],[34,704763,126541],{"className":704764},[31685,31686,31821],[34,704766,31829],{"className":704767},[31828],[34,704769,704771],{"className":704770},[31801],[34,704772,704774],{"className":704773,"style":220191},[31805],[34,704775],{},[34,704777],{"className":704778},[31690,41488]," (利用公式 3.18)","\n根据公式：\n",[34,704782,704784,704856],{"className":704783},[31642],[34,704785,704787],{"className":704786},[31646],[31648,704788,704789],{"xmlns":31650},[31652,704790,704791,704853],{},[31655,704792,704793,704811,704813,704815,704817],{},[41453,704794,704795,704801],{},[31655,704796,704797,704799],{},[31658,704798,619473],{"mathvariant":38079},[31658,704800,126541],{},[31655,704802,704803,704805],{},[31658,704804,619473],{"mathvariant":38079},[31743,704806,704807,704809],{},[31658,704808,67764],{},[31749,704810,459],{},[31662,704812,202],{},[31658,704814,701520],{},[31662,704816,32682],{},[31655,704818,704819,704821,704833,704847,704849,704851],{},[31662,704820,456],{"fence":758},[38846,704822,704823,704825,704831],{},[31662,704824,546295],{},[31743,704826,704827,704829],{},[31658,704828,32936],{},[31749,704830,459],{},[31658,704832,32936],{},[41453,704834,704835,704841],{},[31655,704836,704837,704839],{},[31658,704838,619473],{"mathvariant":38079},[31658,704840,31739],{},[31655,704842,704843,704845],{},[31658,704844,619473],{"mathvariant":38079},[31658,704846,67764],{},[31658,704848,46900],{},[31658,704850,33249],{},[31662,704852,2557],{"fence":758},[31665,704854,704855],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial y_0} = \\exp\\left( \\int_{x_0}^x \\frac{\\partial f}{\\partial y} dt \\right)",[34,704857,704859,704985],{"className":704858,"ariaHidden":758},[31672],[34,704860,704862,704865,704976,704979,704982],{"className":704861},[31676],[34,704863],{"className":704864,"style":695705},[31680],[34,704866,704868,704871,704973],{"className":704867},[31685],[34,704869],{"className":704870},[31782,41488],[34,704872,704874],{"className":704873},[41453],[34,704875,704877,704965],{"className":704876},[31796,31797],[34,704878,704880,704962],{"className":704879},[31801],[34,704881,704883,704937,704945],{"className":704882,"style":608935},[31805],[34,704884,704885,704888],{"style":41504},[34,704886],{"className":704887,"style":41508},[31813],[34,704889,704891],{"className":704890},[31818,31819,31820,31821],[34,704892,704894,704897],{"className":704893},[31685,31821],[34,704895,619473],{"className":704896,"style":112322},[31685,31821],[34,704898,704900,704903],{"className":704899},[31685,31821],[34,704901,67764],{"className":704902,"style":32062},[31685,31686,31821],[34,704904,704906],{"className":704905},[31792],[34,704907,704909,704929],{"className":704908},[31796,31797],[34,704910,704912,704926],{"className":704911},[31801],[34,704913,704915],{"className":704914,"style":318658},[31805],[34,704916,704917,704920],{"style":561507},[34,704918],{"className":704919,"style":43406},[31813],[34,704921,704923],{"className":704922},[31818,43410,43411,31821],[34,704924,459],{"className":704925},[31685,31821],[34,704927,31829],{"className":704928},[31828],[34,704930,704932],{"className":704931},[31801],[34,704933,704935],{"className":704934,"style":220540},[31805],[34,704936],{},[34,704938,704939,704942],{"style":41520},[34,704940],{"className":704941,"style":41508},[31813],[34,704943],{"className":704944,"style":41528},[41527],[34,704946,704947,704950],{"style":548368},[34,704948],{"className":704949,"style":41508},[31813],[34,704951,704953],{"className":704952},[31818,31819,31820,31821],[34,704954,704956,704959],{"className":704955},[31685,31821],[34,704957,619473],{"className":704958,"style":112322},[31685,31821],[34,704960,126541],{"className":704961},[31685,31686,31821],[34,704963,31829],{"className":704964},[31828],[34,704966,704968],{"className":704967},[31801],[34,704969,704971],{"className":704970,"style":220191},[31805],[34,704972],{},[34,704974],{"className":704975},[31690,41488],[34,704977],{"className":704978,"style":31846},[31845],[34,704980,202],{"className":704981},[31850],[34,704983],{"className":704984,"style":31846},[31845],[34,704986,704988,704991,704994,704997],{"className":704987},[31676],[34,704989],{"className":704990,"style":141282},[31680],[34,704992,701520],{"className":704993},[32767],[34,704995],{"className":704996,"style":32123},[31845],[34,704998,705000,705006,705097,705100,705174,705177,705180],{"className":704999},[34006],[34,705001,705003],{"className":705002,"style":75811},[31782,75810],[34,705004,456],{"className":705005},[71281,141292],[34,705007,705009,705012],{"className":705008},[32767],[34,705010,546295],{"className":705011,"style":546331},[32767,131249,156560],[34,705013,705015],{"className":705014},[31792],[34,705016,705018,705089],{"className":705017},[31796,31797],[34,705019,705021,705086],{"className":705020},[31801],[34,705022,705024,705075],{"className":705023,"style":567928},[31805],[34,705025,705026,705029],{"style":546347},[34,705027],{"className":705028,"style":31814},[31813],[34,705030,705032],{"className":705031},[31818,31819,31820,31821],[34,705033,705035],{"className":705034},[31685,31821],[34,705036,705038,705041],{"className":705037},[31685,31821],[34,705039,32936],{"className":705040},[31685,31686,31821],[34,705042,705044],{"className":705043},[31792],[34,705045,705047,705067],{"className":705046},[31796,31797],[34,705048,705050,705064],{"className":705049},[31801],[34,705051,705053],{"className":705052,"style":318658},[31805],[34,705054,705055,705058],{"style":318661},[34,705056],{"className":705057,"style":43406},[31813],[34,705059,705061],{"className":705060},[31818,43410,43411,31821],[34,705062,459],{"className":705063},[31685,31821],[34,705065,31829],{"className":705066},[31828],[34,705068,705070],{"className":705069},[31801],[34,705071,705073],{"className":705072,"style":220540},[31805],[34,705074],{},[34,705076,705077,705080],{"style":546362},[34,705078],{"className":705079,"style":31814},[31813],[34,705081,705083],{"className":705082},[31818,31819,31820,31821],[34,705084,32936],{"className":705085},[31685,31686,31821],[34,705087,31829],{"className":705088},[31828],[34,705090,705092],{"className":705091},[31801],[34,705093,705095],{"className":705094,"style":667231},[31805],[34,705096],{},[34,705098],{"className":705099,"style":32123},[31845],[34,705101,705103,705106,705171],{"className":705102},[31685],[34,705104],{"className":705105},[31782,41488],[34,705107,705109],{"className":705108},[41453],[34,705110,705112,705163],{"className":705111},[31796,31797],[34,705113,705115,705160],{"className":705114},[31801],[34,705116,705118,705135,705143],{"className":705117,"style":608935},[31805],[34,705119,705120,705123],{"style":41504},[34,705121],{"className":705122,"style":41508},[31813],[34,705124,705126],{"className":705125},[31818,31819,31820,31821],[34,705127,705129,705132],{"className":705128},[31685,31821],[34,705130,619473],{"className":705131,"style":112322},[31685,31821],[34,705133,67764],{"className":705134,"style":32062},[31685,31686,31821],[34,705136,705137,705140],{"style":41520},[34,705138],{"className":705139,"style":41508},[31813],[34,705141],{"className":705142,"style":41528},[41527],[34,705144,705145,705148],{"style":548368},[34,705146],{"className":705147,"style":41508},[31813],[34,705149,705151],{"className":705150},[31818,31819,31820,31821],[34,705152,705154,705157],{"className":705153},[31685,31821],[34,705155,619473],{"className":705156,"style":112322},[31685,31821],[34,705158,31739],{"className":705159,"style":31778},[31685,31686,31821],[34,705161,31829],{"className":705162},[31828],[34,705164,705166],{"className":705165},[31801],[34,705167,705169],{"className":705168,"style":220191},[31805],[34,705170],{},[34,705172],{"className":705173},[31690,41488],[34,705175,46900],{"className":705176},[31685,31686],[34,705178,33249],{"className":705179},[31685,31686],[34,705181,705183],{"className":705182,"style":75811},[31690,75810],[34,705184,2557],{"className":705185},[71281,141292],[34,705187,705189,705225],{"className":705188},[31642],[34,705190,705192],{"className":705191},[31646],[31648,705193,705194],{"xmlns":31650},[31652,705195,705196,705222],{},[31655,705197,705198,705212,705214,705216,705218,705220],{},[41453,705199,705200,705206],{},[31655,705201,705202,705204],{},[31658,705203,619473],{"mathvariant":38079},[31658,705205,31739],{},[31655,705207,705208,705210],{},[31658,705209,619473],{"mathvariant":38079},[31658,705211,67764],{},[31662,705213,202],{},[31658,705215,105852],{},[31662,705217,456],{"stretchy":702},[31658,705219,32936],{},[31662,705221,2557],{"stretchy":702},[31665,705223,705224],{"encoding":31667},"\\frac{\\partial f}{\\partial y} = P(x)",[34,705226,705228,705317],{"className":705227,"ariaHidden":758},[31672],[34,705229,705231,705234,705308,705311,705314],{"className":705230},[31676],[34,705232],{"className":705233,"style":695705},[31680],[34,705235,705237,705240,705305],{"className":705236},[31685],[34,705238],{"className":705239},[31782,41488],[34,705241,705243],{"className":705242},[41453],[34,705244,705246,705297],{"className":705245},[31796,31797],[34,705247,705249,705294],{"className":705248},[31801],[34,705250,705252,705269,705277],{"className":705251,"style":608935},[31805],[34,705253,705254,705257],{"style":41504},[34,705255],{"className":705256,"style":41508},[31813],[34,705258,705260],{"className":705259},[31818,31819,31820,31821],[34,705261,705263,705266],{"className":705262},[31685,31821],[34,705264,619473],{"className":705265,"style":112322},[31685,31821],[34,705267,67764],{"className":705268,"style":32062},[31685,31686,31821],[34,705270,705271,705274],{"style":41520},[34,705272],{"className":705273,"style":41508},[31813],[34,705275],{"className":705276,"style":41528},[41527],[34,705278,705279,705282],{"style":548368},[34,705280],{"className":705281,"style":41508},[31813],[34,705283,705285],{"className":705284},[31818,31819,31820,31821],[34,705286,705288,705291],{"className":705287},[31685,31821],[34,705289,619473],{"className":705290,"style":112322},[31685,31821],[34,705292,31739],{"className":705293,"style":31778},[31685,31686,31821],[34,705295,31829],{"className":705296},[31828],[34,705298,705300],{"className":705299},[31801],[34,705301,705303],{"className":705302,"style":220191},[31805],[34,705304],{},[34,705306],{"className":705307},[31690,41488],[34,705309],{"className":705310,"style":31846},[31845],[34,705312,202],{"className":705313},[31850],[34,705315],{"className":705316,"style":31846},[31845],[34,705318,705320,705323,705326,705329,705332],{"className":705319},[31676],[34,705321],{"className":705322,"style":31774},[31680],[34,705324,105852],{"className":705325,"style":37422},[31685,31686],[34,705327,456],{"className":705328},[31782],[34,705330,32936],{"className":705331},[31685,31686],[34,705333,2557],{"className":705334},[31690]," 代入（注意积分变量用 ",[34,705337,705339,705352],{"className":705338},[31642],[34,705340,705342],{"className":705341},[31646],[31648,705343,705344],{"xmlns":31650},[31652,705345,705346,705350],{},[31655,705347,705348],{},[31658,705349,33249],{},[31665,705351,33249],{"encoding":31667},[34,705353,705355],{"className":705354,"ariaHidden":758},[31672],[34,705356,705358,705361],{"className":705357},[31676],[34,705359],{"className":705360,"style":33280},[31680],[34,705362,33249],{"className":705363},[31685,31686]," 以示区别）：\n",[34,705366,705368,705434],{"className":705367},[31642],[34,705369,705371],{"className":705370},[31646],[31648,705372,705373],{"xmlns":31650},[31652,705374,705375,705431],{},[31655,705376,705377,705395,705397,705399,705401],{},[41453,705378,705379,705385],{},[31655,705380,705381,705383],{},[31658,705382,619473],{"mathvariant":226163},[31658,705384,126541],{"mathvariant":226163},[31655,705386,705387,705389],{},[31658,705388,619473],{"mathvariant":226163},[31743,705390,705391,705393],{},[31658,705392,67764],{"mathvariant":226163},[31749,705394,459],{"mathvariant":226163},[31662,705396,202],{},[31658,705398,701520],{},[31662,705400,32682],{},[31655,705402,705403,705405,705417,705419,705421,705423,705425,705427,705429],{},[31662,705404,456],{"fence":758},[38846,705406,705407,705409,705415],{},[31662,705408,546295],{},[31743,705410,705411,705413],{},[31658,705412,32936],{"mathvariant":226163},[31749,705414,459],{"mathvariant":226163},[31658,705416,32936],{"mathvariant":226163},[31658,705418,105852],{"mathvariant":226163},[31662,705420,456],{"stretchy":702},[31658,705422,33249],{"mathvariant":226163},[31662,705424,2557],{"stretchy":702},[31658,705426,46900],{"mathvariant":226163},[31658,705428,33249],{"mathvariant":226163},[31662,705430,2557],{"fence":758},[31665,705432,705433],{"encoding":31667},"\\mathbf{\\frac{\\partial \\varphi}{\\partial y_0} = \\exp\\left( \\int_{x_0}^x P(t) dt \\right)}",[34,705435,705437],{"className":705436,"ariaHidden":758},[31672],[34,705438,705440,705443],{"className":705439},[31676],[34,705441],{"className":705442,"style":141282},[31680],[34,705444,705446,705560,705563,705566,705569,705572,705575],{"className":705445},[31685],[34,705447,705449,705452,705557],{"className":705448},[31685],[34,705450],{"className":705451},[31782,41488],[34,705453,705455],{"className":705454},[41453],[34,705456,705458,705549],{"className":705457},[31796,31797],[34,705459,705461,705546],{"className":705460},[31801],[34,705462,705464,705521,705529],{"className":705463,"style":608935},[31805],[34,705465,705466,705469],{"style":41504},[34,705467],{"className":705468,"style":41508},[31813],[34,705470,705472],{"className":705471},[31818,31819,31820,31821],[34,705473,705475,705479],{"className":705474},[31685,31821],[34,705476,619473],{"className":705477,"style":705478},[31685,226302,31821],"margin-right:0.0639em;",[34,705480,705482,705486],{"className":705481},[31685,31821],[34,705483,67764],{"className":705484,"style":705485},[31685,226302,31821],"margin-right:0.016em;",[34,705487,705489],{"className":705488},[31792],[34,705490,705492,705513],{"className":705491},[31796,31797],[34,705493,705495,705510],{"className":705494},[31801],[34,705496,705498],{"className":705497,"style":318658},[31805],[34,705499,705501,705504],{"style":705500},"top:-2.357em;margin-left:-0.016em;margin-right:0.0714em;",[34,705502],{"className":705503,"style":43406},[31813],[34,705505,705507],{"className":705506},[31818,43410,43411,31821],[34,705508,459],{"className":705509},[31685,226302,31821],[34,705511,31829],{"className":705512},[31828],[34,705514,705516],{"className":705515},[31801],[34,705517,705519],{"className":705518,"style":220540},[31805],[34,705520],{},[34,705522,705523,705526],{"style":41520},[34,705524],{"className":705525,"style":41508},[31813],[34,705527],{"className":705528,"style":41528},[41527],[34,705530,705531,705534],{"style":548368},[34,705532],{"className":705533,"style":41508},[31813],[34,705535,705537],{"className":705536},[31818,31819,31820,31821],[34,705538,705540,705543],{"className":705539},[31685,31821],[34,705541,619473],{"className":705542,"style":705478},[31685,226302,31821],[34,705544,126541],{"className":705545},[31685,31686,31821],[34,705547,31829],{"className":705548},[31828],[34,705550,705552],{"className":705551},[31801],[34,705553,705555],{"className":705554,"style":220191},[31805],[34,705556],{},[34,705558],{"className":705559},[31690,41488],[34,705561],{"className":705562,"style":31846},[31845],[34,705564,202],{"className":705565},[31850],[34,705567],{"className":705568,"style":31846},[31845],[34,705570,701520],{"className":705571},[32767],[34,705573],{"className":705574,"style":32123},[31845],[34,705576,705578,705584,705676,705679,705682,705685,705688,705691,705695],{"className":705577},[34006],[34,705579,705581],{"className":705580,"style":75811},[31782,75810],[34,705582,456],{"className":705583},[71281,141292],[34,705585,705587,705590],{"className":705586},[32767],[34,705588,546295],{"className":705589,"style":546331},[32767,131249,156560],[34,705591,705593],{"className":705592},[31792],[34,705594,705596,705668],{"className":705595},[31796,31797],[34,705597,705599,705665],{"className":705598},[31801],[34,705600,705603,705654],{"className":705601,"style":705602},[31805],"height:0.869em;",[34,705604,705605,705608],{"style":546347},[34,705606],{"className":705607,"style":31814},[31813],[34,705609,705611],{"className":705610},[31818,31819,31820,31821],[34,705612,705614],{"className":705613},[31685,31821],[34,705615,705617,705620],{"className":705616},[31685,31821],[34,705618,32936],{"className":705619},[31685,226302,31821],[34,705621,705623],{"className":705622},[31792],[34,705624,705626,705646],{"className":705625},[31796,31797],[34,705627,705629,705643],{"className":705628},[31801],[34,705630,705632],{"className":705631,"style":318658},[31805],[34,705633,705634,705637],{"style":318661},[34,705635],{"className":705636,"style":43406},[31813],[34,705638,705640],{"className":705639},[31818,43410,43411,31821],[34,705641,459],{"className":705642},[31685,226302,31821],[34,705644,31829],{"className":705645},[31828],[34,705647,705649],{"className":705648},[31801],[34,705650,705652],{"className":705651,"style":220540},[31805],[34,705653],{},[34,705655,705656,705659],{"style":546362},[34,705657],{"className":705658,"style":31814},[31813],[34,705660,705662],{"className":705661},[31818,31819,31820,31821],[34,705663,32936],{"className":705664},[31685,226302,31821],[34,705666,31829],{"className":705667},[31828],[34,705669,705671],{"className":705670},[31801],[34,705672,705674],{"className":705673,"style":667231},[31805],[34,705675],{},[34,705677],{"className":705678,"style":32123},[31845],[34,705680,105852],{"className":705681},[31685,226302],[34,705683,456],{"className":705684},[31782],[34,705686,33249],{"className":705687},[31685,226302],[34,705689,2557],{"className":705690},[31690],[34,705692,705694],{"className":705693},[31685,226302],"dt",[34,705696,705698],{"className":705697,"style":75811},[31690,75810],[34,705699,2557],{"className":705700},[71281,141292],[71807,705702,705703,705704,705732,705733,705761,705762,705806],{},"(解释：这就是如果你的初始 ",[34,705705,705707,705720],{"className":705706},[31642],[34,705708,705710],{"className":705709},[31646],[31648,705711,705712],{"xmlns":31650},[31652,705713,705714,705718],{},[31655,705715,705716],{},[31658,705717,67764],{},[31665,705719,67764],{"encoding":31667},[34,705721,705723],{"className":705722,"ariaHidden":758},[31672],[34,705724,705726,705729],{"className":705725},[31676],[34,705727],{"className":705728,"style":38774},[31680],[34,705730,67764],{"className":705731,"style":32062},[31685,31686]," 值改变一点点，解在 ",[34,705734,705736,705749],{"className":705735},[31642],[34,705737,705739],{"className":705738},[31646],[31648,705740,705741],{"xmlns":31650},[31652,705742,705743,705747],{},[31655,705744,705745],{},[31658,705746,32936],{},[31665,705748,32936],{"encoding":31667},[34,705750,705752],{"className":705751,"ariaHidden":758},[31672],[34,705753,705755,705758],{"className":705754},[31676],[34,705756],{"className":705757,"style":32527},[31680],[34,705759,32936],{"className":705760},[31685,31686]," 处变化的大小。因为它是一阶线性方程，这个放大因子只与 ",[34,705763,705765,705785],{"className":705764},[31642],[34,705766,705768],{"className":705767},[31646],[31648,705769,705770],{"xmlns":31650},[31652,705771,705772,705782],{},[31655,705773,705774,705776,705778,705780],{},[31658,705775,105852],{},[31662,705777,456],{"stretchy":702},[31658,705779,32936],{},[31662,705781,2557],{"stretchy":702},[31665,705783,705784],{"encoding":31667},"P(x)",[34,705786,705788],{"className":705787,"ariaHidden":758},[31672],[34,705789,705791,705794,705797,705800,705803],{"className":705790},[31676],[34,705792],{"className":705793,"style":31774},[31680],[34,705795,105852],{"className":705796,"style":37422},[31685,31686],[34,705798,456],{"className":705799},[31782],[34,705801,32936],{"className":705802},[31685,31686],[34,705804,2557],{"className":705805},[31690]," 有关。)",[15,705808,705809,704780,705966,706508,706509,68939,706648,706771,706772,706828,706829,707170,707171],{},[2627,705810,705811,705812,705965],{},"Step 3: 计算 ",[34,705813,705815,705845],{"className":705814},[31642],[34,705816,705818],{"className":705817},[31646],[31648,705819,705820],{"xmlns":31650},[31652,705821,705822,705842],{},[31655,705823,705824],{},[41453,705825,705826,705832],{},[31655,705827,705828,705830],{},[31658,705829,619473],{"mathvariant":38079},[31658,705831,126541],{},[31655,705833,705834,705836],{},[31658,705835,619473],{"mathvariant":38079},[31743,705837,705838,705840],{},[31658,705839,32936],{},[31749,705841,459],{},[31665,705843,705844],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial x_0}",[34,705846,705848],{"className":705847,"ariaHidden":758},[31672],[34,705849,705851,705854],{"className":705850},[31676],[34,705852],{"className":705853,"style":702239},[31680],[34,705855,705857,705860,705962],{"className":705856},[31685],[34,705858],{"className":705859},[31782,41488],[34,705861,705863],{"className":705862},[41453],[34,705864,705866,705954],{"className":705865},[31796,31797],[34,705867,705869,705951],{"className":705868},[31801],[34,705870,705872,705926,705934],{"className":705871,"style":608935},[31805],[34,705873,705874,705877],{"style":41504},[34,705875],{"className":705876,"style":41508},[31813],[34,705878,705880],{"className":705879},[31818,31819,31820,31821],[34,705881,705883,705886],{"className":705882},[31685,31821],[34,705884,619473],{"className":705885,"style":112322},[31685,31821],[34,705887,705889,705892],{"className":705888},[31685,31821],[34,705890,32936],{"className":705891},[31685,31686,31821],[34,705893,705895],{"className":705894},[31792],[34,705896,705898,705918],{"className":705897},[31796,31797],[34,705899,705901,705915],{"className":705900},[31801],[34,705902,705904],{"className":705903,"style":318658},[31805],[34,705905,705906,705909],{"style":318661},[34,705907],{"className":705908,"style":43406},[31813],[34,705910,705912],{"className":705911},[31818,43410,43411,31821],[34,705913,459],{"className":705914},[31685,31821],[34,705916,31829],{"className":705917},[31828],[34,705919,705921],{"className":705920},[31801],[34,705922,705924],{"className":705923,"style":220540},[31805],[34,705925],{},[34,705927,705928,705931],{"style":41520},[34,705929],{"className":705930,"style":41508},[31813],[34,705932],{"className":705933,"style":41528},[41527],[34,705935,705936,705939],{"style":548368},[34,705937],{"className":705938,"style":41508},[31813],[34,705940,705942],{"className":705941},[31818,31819,31820,31821],[34,705943,705945,705948],{"className":705944},[31685,31821],[34,705946,619473],{"className":705947,"style":112322},[31685,31821],[34,705949,126541],{"className":705950},[31685,31686,31821],[34,705952,31829],{"className":705953},[31828],[34,705955,705957],{"className":705956},[31801],[34,705958,705960],{"className":705959,"style":231800},[31805],[34,705961],{},[34,705963],{"className":705964},[31690,41488]," (利用公式 3.17)",[34,705967,705969,706065],{"className":705968},[31642],[34,705970,705972],{"className":705971},[31646],[31648,705973,705974],{"xmlns":31650},[31652,705975,705976,706062],{},[31655,705977,705978,705996,705998,706000,706002,706004,706010,706012,706018,706020,706022,706024,706026],{},[41453,705979,705980,705986],{},[31655,705981,705982,705984],{},[31658,705983,619473],{"mathvariant":38079},[31658,705985,126541],{},[31655,705987,705988,705990],{},[31658,705989,619473],{"mathvariant":38079},[31743,705991,705992,705994],{},[31658,705993,32936],{},[31749,705995,459],{},[31662,705997,202],{},[31662,705999,32226],{},[31658,706001,31739],{},[31662,706003,456],{"stretchy":702},[31743,706005,706006,706008],{},[31658,706007,32936],{},[31749,706009,459],{},[31662,706011,31960],{"separator":758},[31743,706013,706014,706016],{},[31658,706015,67764],{},[31749,706017,459],{},[31662,706019,2557],{"stretchy":702},[31662,706021,71191],{},[31658,706023,701520],{},[31662,706025,32682],{},[31655,706027,706028,706030,706042,706056,706058,706060],{},[31662,706029,456],{"fence":758},[38846,706031,706032,706034,706040],{},[31662,706033,546295],{},[31743,706035,706036,706038],{},[31658,706037,32936],{},[31749,706039,459],{},[31658,706041,32936],{},[41453,706043,706044,706050],{},[31655,706045,706046,706048],{},[31658,706047,619473],{"mathvariant":38079},[31658,706049,31739],{},[31655,706051,706052,706054],{},[31658,706053,619473],{"mathvariant":38079},[31658,706055,67764],{},[31658,706057,46900],{},[31658,706059,33249],{},[31662,706061,2557],{"fence":758},[31665,706063,706064],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial x_0} = -f(x_0,y_0) \\cdot \\exp\\left( \\int_{x_0}^x \\frac{\\partial f}{\\partial y} dt \\right)",[34,706066,706068,706194,706307],{"className":706067,"ariaHidden":758},[31672],[34,706069,706071,706074,706185,706188,706191],{"className":706070},[31676],[34,706072],{"className":706073,"style":702239},[31680],[34,706075,706077,706080,706182],{"className":706076},[31685],[34,706078],{"className":706079},[31782,41488],[34,706081,706083],{"className":706082},[41453],[34,706084,706086,706174],{"className":706085},[31796,31797],[34,706087,706089,706171],{"className":706088},[31801],[34,706090,706092,706146,706154],{"className":706091,"style":608935},[31805],[34,706093,706094,706097],{"style":41504},[34,706095],{"className":706096,"style":41508},[31813],[34,706098,706100],{"className":706099},[31818,31819,31820,31821],[34,706101,706103,706106],{"className":706102},[31685,31821],[34,706104,619473],{"className":706105,"style":112322},[31685,31821],[34,706107,706109,706112],{"className":706108},[31685,31821],[34,706110,32936],{"className":706111},[31685,31686,31821],[34,706113,706115],{"className":706114},[31792],[34,706116,706118,706138],{"className":706117},[31796,31797],[34,706119,706121,706135],{"className":706120},[31801],[34,706122,706124],{"className":706123,"style":318658},[31805],[34,706125,706126,706129],{"style":318661},[34,706127],{"className":706128,"style":43406},[31813],[34,706130,706132],{"className":706131},[31818,43410,43411,31821],[34,706133,459],{"className":706134},[31685,31821],[34,706136,31829],{"className":706137},[31828],[34,706139,706141],{"className":706140},[31801],[34,706142,706144],{"className":706143,"style":220540},[31805],[34,706145],{},[34,706147,706148,706151],{"style":41520},[34,706149],{"className":706150,"style":41508},[31813],[34,706152],{"className":706153,"style":41528},[41527],[34,706155,706156,706159],{"style":548368},[34,706157],{"className":706158,"style":41508},[31813],[34,706160,706162],{"className":706161},[31818,31819,31820,31821],[34,706163,706165,706168],{"className":706164},[31685,31821],[34,706166,619473],{"className":706167,"style":112322},[31685,31821],[34,706169,126541],{"className":706170},[31685,31686,31821],[34,706172,31829],{"className":706173},[31828],[34,706175,706177],{"className":706176},[31801],[34,706178,706180],{"className":706179,"style":231800},[31805],[34,706181],{},[34,706183],{"className":706184},[31690,41488],[34,706186],{"className":706187,"style":31846},[31845],[34,706189,202],{"className":706190},[31850],[34,706192],{"className":706193,"style":31846},[31845],[34,706195,706197,706200,706203,706206,706209,706249,706252,706255,706295,706298,706301,706304],{"className":706196},[31676],[34,706198],{"className":706199,"style":31774},[31680],[34,706201,32226],{"className":706202},[31685],[34,706204,31739],{"className":706205,"style":31778},[31685,31686],[34,706207,456],{"className":706208},[31782],[34,706210,706212,706215],{"className":706211},[31685],[34,706213,32936],{"className":706214},[31685,31686],[34,706216,706218],{"className":706217},[31792],[34,706219,706221,706241],{"className":706220},[31796,31797],[34,706222,706224,706238],{"className":706223},[31801],[34,706225,706227],{"className":706226,"style":31806},[31805],[34,706228,706229,706232],{"style":31809},[34,706230],{"className":706231,"style":31814},[31813],[34,706233,706235],{"className":706234},[31818,31819,31820,31821],[34,706236,459],{"className":706237},[31685,31821],[34,706239,31829],{"className":706240},[31828],[34,706242,706244],{"className":706243},[31801],[34,706245,706247],{"className":706246,"style":31836},[31805],[34,706248],{},[34,706250,31960],{"className":706251},[32119],[34,706253],{"className":706254,"style":32123},[31845],[34,706256,706258,706261],{"className":706257},[31685],[34,706259,67764],{"className":706260,"style":32062},[31685,31686],[34,706262,706264],{"className":706263},[31792],[34,706265,706267,706287],{"className":706266},[31796,31797],[34,706268,706270,706284],{"className":706269},[31801],[34,706271,706273],{"className":706272,"style":31806},[31805],[34,706274,706275,706278],{"style":38795},[34,706276],{"className":706277,"style":31814},[31813],[34,706279,706281],{"className":706280},[31818,31819,31820,31821],[34,706282,459],{"className":706283},[31685,31821],[34,706285,31829],{"className":706286},[31828],[34,706288,706290],{"className":706289},[31801],[34,706291,706293],{"className":706292,"style":31836},[31805],[34,706294],{},[34,706296,2557],{"className":706297},[31690],[34,706299],{"className":706300,"style":32604},[31845],[34,706302,71191],{"className":706303},[32024],[34,706305],{"className":706306,"style":32604},[31845],[34,706308,706310,706313,706316,706319],{"className":706309},[31676],[34,706311],{"className":706312,"style":141282},[31680],[34,706314,701520],{"className":706315},[32767],[34,706317],{"className":706318,"style":32123},[31845],[34,706320,706322,706328,706419,706422,706496,706499,706502],{"className":706321},[34006],[34,706323,706325],{"className":706324,"style":75811},[31782,75810],[34,706326,456],{"className":706327}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",[34,706510,706512,706544],{"className":706511},[31642],[34,706513,706515],{"className":706514},[31646],[31648,706516,706517],{"xmlns":31650},[31652,706518,706519,706541],{},[31655,706520,706521,706523,706525,706531,706533,706539],{},[31658,706522,31739],{},[31662,706524,456],{"stretchy":702},[31743,706526,706527,706529],{},[31658,706528,32936],{},[31749,706530,459],{},[31662,706532,31960],{"separator":758},[31743,706534,706535,706537],{},[31658,706536,67764],{},[31749,706538,459],{},[31662,706540,2557],{"stretchy":702},[31665,706542,706543],{"encoding":31667},"f(x_0, y_0)",[34,706545,706547],{"className":706546,"ariaHidden":758},[31672],[34,706548,706550,706553,706556,706559,706599,706602,706605,706645],{"className":706549},[31676],[34,706551],{"className":706552,"style":31774},[31680],[34,706554,31739],{"className":706555,"style":31778},[31685,31686],[34,706557,456],{"className":706558},[31782],[34,706560,706562,706565],{"className":706561},[31685],[34,706563,32936],{"className":706564},[31685,31686],[34,706566,706568],{"className":706567},[31792],[34,706569,706571,706591],{"className":706570},[31796,31797],[34,706572,706574,706588],{"className":706573},[31801],[34,706575,706577],{"className":706576,"style":31806},[31805],[34,706578,706579,706582],{"style":31809},[34,706580],{"className":706581,"style":31814},[31813],[34,706583,706585],{"className":706584},[31818,31819,31820,31821],[34,706586,459],{"className":706587},[31685,31821],[34,706589,31829],{"className":706590},[31828],[34,706592,706594],{"className":706593},[31801],[34,706595,706597],{"className":706596,"style":31836},[31805],[34,706598],{},[34,706600,31960],{"className":706601},[32119],[34,706603],{"className":706604,"style":32123},[31845],[34,706606,706608,706611],{"className":706607},[31685],[34,706609,67764],{"className":706610,"style":32062},[31685,31686],[34,706612,706614],{"className":706613},[31792],[34,706615,706617,706637],{"className":706616},[31796,31797],[34,706618,706620,706634],{"className":706619},[31801],[34,706621,706623],{"className":706622,"style":31806},[31805],[34,706624,706625,706628],{"style":38795},[34,706626],{"className":706627,"style":31814},[31813],[34,706629,706631],{"className":706630},[31818,31819,31820,31821],[34,706632,459],{"className":706633},[31685,31821],[34,706635,31829],{"className":706636},[31828],[34,706638,706640],{"className":706639},[31801],[34,706641,706643],{"className":706642,"style":31836},[31805],[34,706644],{},[34,706646,2557],{"className":706647},[31690],[34,706649,706651,706676],{"className":706650},[31642],[34,706652,706654],{"className":706653},[31646],[31648,706655,706656],{"xmlns":31650},[31652,706657,706658,706674],{},[31655,706659,706660,706666,706668],{},[31743,706661,706662,706664],{},[31658,706663,32936],{},[31749,706665,459],{},[31662,706667,31960],{"separator":758},[31743,706669,706670,706672],{},[31658,706671,67764],{},[31749,706673,459],{},[31665,706675,701312],{"encoding":31667},[34,706677,706679],{"className":706678,"ariaHidden":758},[31672],[34,706680,706682,706685,706725,706728,706731],{"className":706681},[31676],[34,706683],{"className":706684,"style":38774},[31680],[34,706686,706688,706691],{"className":706687},[31685],[34,706689,32936],{"className":706690},[31685,31686],[34,706692,706694],{"className":706693},[31792],[34,706695,706697,706717],{"className":706696},[31796,31797],[34,706698,706700,706714],{"className":706699},[31801],[34,706701,706703],{"className":706702,"style":31806},[31805],[34,706704,706705,706708],{"style":31809},[34,706706],{"className":706707,"style":31814},[31813],[34,706709,706711],{"className":706710},[31818,31819,31820,31821],[34,706712,459],{"className":706713},[31685,31821],[34,706715,31829],{"className":706716},[31828],[34,706718,706720],{"className":706719},[31801],[34,706721,706723],{"className":706722,"style":31836},[31805],[34,706724],{},[34,706726,31960],{"className":706727},[32119],[34,706729],{"className":706730,"style":32123},[31845],[34,706732,706734,706737],{"className":706733},[31685],[34,706735,67764],{"className":706736,"style":32062},[31685,31686],[34,706738,706740],{"className":706739},[31792],[34,706741,706743,706763],{"className":706742},[31796,31797],[34,706744,706746,706760],{"className":706745},[31801],[34,706747,706749],{"className":706748,"style":31806},[31805],[34,706750,706751,706754],{"style":38795},[34,706752],{"className":706753,"style":31814},[31813],[34,706755,706757],{"className":706756},[31818,31819,31820,31821],[34,706758,459],{"className":706759},[31685,31821],[34,706761,31829],{"className":706762},[31828],[34,706764,706766],{"className":706765},[31801],[34,706767,706769],{"className":706768,"style":31836},[31805],[34,706770],{}," 代入 ",[34,706773,706775,706798],{"className":706774},[31642],[34,706776,706778],{"className":706777},[31646],[31648,706779,706780],{"xmlns":31650},[31652,706781,706782,706796],{},[31655,706783,706784,706786,706788,706790,706792,706794],{},[31658,706785,31739],{},[31662,706787,456],{"stretchy":702},[31658,706789,32936],{},[31662,706791,31960],{"separator":758},[31658,706793,67764],{},[31662,706795,2557],{"stretchy":702},[31665,706797,703776],{"encoding":31667},[34,706799,706801],{"className":706800,"ariaHidden":758},[31672],[34,706802,706804,706807,706810,706813,706816,706819,706822,706825],{"className":706803},[31676],[34,706805],{"className":706806,"style":31774},[31680],[34,706808,31739],{"className":706809,"style":31778},[31685,31686],[34,706811,456],{"className":706812},[31782],[34,706814,32936],{"className":706815},[31685,31686],[34,706817,31960],{"className":706818},[32119],[34,706820],{"className":706821,"style":32123},[31845],[34,706823,67764],{"className":706824,"style":32062},[31685,31686],[34,706826,2557],{"className":706827},[31690]," 的表达式：\n",[34,706830,706832,706898],{"className":706831},[31642],[34,706833,706835],{"className":706834},[31646],[31648,706836,706837],{"xmlns":31650},[31652,706838,706839,706895],{},[31655,706840,706841,706843,706845,706851,706853,706859,706861,706863,706865,706867,706873,706875,706881,706883,706885,706887,706893],{},[31658,706842,31739],{},[31662,706844,456],{"stretchy":702},[31743,706846,706847,706849],{},[31658,706848,32936],{},[31749,706850,459],{},[31662,706852,31960],{"separator":758},[31743,706854,706855,706857],{},[31658,706856,67764],{},[31749,706858,459],{},[31662,706860,2557],{"stretchy":702},[31662,706862,202],{},[31658,706864,105852],{},[31662,706866,456],{"stretchy":702},[31743,706868,706869,706871],{},[31658,706870,32936],{},[31749,706872,459],{},[31662,706874,2557],{"stretchy":702},[31743,706876,706877,706879],{},[31658,706878,67764],{},[31749,706880,459],{},[31662,706882,223],{},[31658,706884,116863],{},[31662,706886,456],{"stretchy":702},[31743,706888,706889,706891],{},[31658,706890,32936],{},[31749,706892,459],{},[31662,706894,2557],{"stretchy":702},[31665,706896,706897],{"encoding":31667},"f(x_0, y_0) = P(x_0)y_0 + Q(x_0)",[34,706899,706901,707011,707115],{"className":706900,"ariaHidden":758},[31672],[34,706902,706904,706907,706910,706913,706953,706956,706959,706999,707002,707005,707008],{"className":706903},[31676],[34,706905],{"className":706906,"style":31774},[31680],[34,706908,31739],{"className":706909,"style":31778},[31685,31686],[34,706911,456],{"className":706912},[31782],[34,706914,706916,706919],{"className":706915},[31685],[34,706917,32936],{"className":706918},[31685,31686],[34,706920,706922],{"className":706921},[31792],[34,706923,706925,706945],{"className":706924},[31796,31797],[34,706926,706928,706942],{"className":706927},[31801],[34,706929,706931],{"className":706930,"style":31806},[31805],[34,706932,706933,706936],{"style":31809},[34,706934],{"className":706935,"style":31814},[31813],[34,706937,706939],{"className":706938},[31818,31819,31820,31821],[34,706940,459],{"className":706941},[31685,31821],[34,706943,31829],{"className":706944},[31828],[34,706946,706948],{"className":706947},[31801],[34,706949,706951],{"className":706950,"style":31836},[31805],[34,706952],{},[34,706954,31960],{"className":706955},[32119],[34,706957],{"className":706958,"style":32123},[31845],[34,706960,706962,706965],{"className":706961},[31685],[34,706963,67764],{"className":706964,"style":32062},[31685,31686],[34,706966,706968],{"className":706967},[31792],[34,706969,706971,706991],{"className":706970},[31796,31797],[34,706972,706974,706988],{"className":706973},[31801],[34,706975,706977],{"className":706976,"style":31806},[31805],[34,706978,706979,706982],{"style":38795},[34,706980],{"className":706981,"style":31814},[31813],[34,706983,706985],{"className":706984},[31818,31819,31820,31821],[34,706986,459],{"className":706987},[31685,31821],[34,706989,31829],{"className":706990},[31828],[34,706992,706994],{"className":706993},[31801],[34,706995,706997],{"className":706996,"style":31836},[31805],[34,706998],{},[34,707000,2557],{"className":707001},[31690],[34,707003],{"className":707004,"style":31846},[31845],[34,707006,202],{"className":707007},[31850],[34,707009],{"className":707010,"style":31846},[31845],[34,707012,707014,707017,707020,707023,707063,707066,707106,707109,707112],{"className":707013},[31676],[34,707015],{"className":707016,"style":31774},[31680],[34,707018,105852],{"className":707019,"style":37422},[31685,31686],[34,707021,456],{"className":707022},[31782],[34,707024,707026,707029],{"className":707025},[31685],[34,707027,32936],{"className":707028},[31685,31686],[34,707030,707032],{"className":707031},[31792],[34,707033,707035,707055],{"className":707034},[31796,31797],[34,707036,707038,707052],{"className":707037},[31801],[34,707039,707041],{"className":707040,"style":31806},[31805],[34,707042,707043,707046],{"style":31809},[34,707044],{"className":707045,"style":31814},[31813],[34,707047,707049],{"className":707048},[31818,31819,31820,31821],[34,707050,459],{"className":707051},[31685,31821],[34,707053,31829],{"className":707054},[31828],[34,707056,707058],{"className":707057},[31801],[34,707059,707061],{"className":707060,"style":31836},[31805],[34,707062],{},[34,707064,2557],{"className":707065},[31690],[34,707067,707069,707072],{"className":707068},[31685],[34,707070,67764],{"className":707071,"style":32062},[31685,31686],[34,707073,707075],{"className":707074},[31792],[34,707076,707078,707098],{"className":707077},[31796,31797],[34,707079,707081,707095],{"className":707080},[31801],[34,707082,707084],{"className":707083,"style":31806},[31805],[34,707085,707086,707089],{"style":38795},[34,707087],{"className":707088,"style":31814},[31813],[34,707090,707092],{"className":707091},[31818,31819,31820,31821],[34,707093,459],{"className":707094},[31685,31821],[34,707096,31829],{"className":707097},[31828],[34,707099,707101],{"className":707100},[31801],[34,707102,707104],{"className":707103,"style":31836},[31805],[34,707105],{},[34,707107],{"className":707108,"style":32604},[31845],[34,707110,223],{"className":707111},[32024],[34,707113],{"className":707114,"style":32604},[31845],[34,707116,707118,707121,707124,707127,707167],{"className":707117},[31676],[34,707119],{"className":707120,"style":31774},[31680],[34,707122,116863],{"className":707123},[31685,31686],[34,707125,456],{"className":707126},[31782],[34,707128,707130,707133],{"className":707129},[31685],[34,707131,32936],{"className":707132},[31685,31686],[34,707134,707136],{"className":707135},[31792],[34,707137,707139,707159],{"className":707138},[31796,31797],[34,707140,707142,707156],{"className":707141},[31801],[34,707143,707145],{"className":707144,"style":31806},[31805],[34,707146,707147,707150],{"style":31809},[34,707148],{"className":707149,"style":31814},[31813],[34,707151,707153],{"className":707152},[31818,31819,31820,31821],[34,707154,459],{"className":707155},[31685,31821],[34,707157,31829],{"className":707158},[31828],[34,707160,707162],{"className":707161},[31801],[34,707163,707165],{"className":707164,"style":31836},[31805],[34,707166],{},[34,707168,2557],{"className":707169},[31690],"\n结合 Step 2 的结果，得到：\n",[34,707172,707174,707280],{"className":707173},[31642],[34,707175,707177],{"className":707176},[31646],[31648,707178,707179],{"xmlns":31650},[31652,707180,707181,707277],{},[31655,707182,707183,707201,707203,707205,707207,707209,707211,707217,707219,707225,707227,707229,707231,707237,707239,707241,707243,707245,707247],{},[41453,707184,707185,707191],{},[31655,707186,707187,707189],{},[31658,707188,619473],{"mathvariant":226163},[31658,707190,126541],{"mathvariant":226163},[31655,707192,707193,707195],{},[31658,707194,619473],{"mathvariant":226163},[31743,707196,707197,707199],{},[31658,707198,32936],{"mathvariant":226163},[31749,707200,459],{"mathvariant":226163},[31662,707202,202],{},[31662,707204,32226],{},[31662,707206,15972],{"stretchy":702},[31658,707208,105852],{"mathvariant":226163},[31662,707210,456],{"stretchy":702},[31743,707212,707213,707215],{},[31658,707214,32936],{"mathvariant":226163},[31749,707216,459],{"mathvariant":226163},[31662,707218,2557],{"stretchy":702},[31743,707220,707221,707223],{},[31658,707222,67764],{"mathvariant":226163},[31749,707224,459],{"mathvariant":226163},[31662,707226,223],{},[31658,707228,116863],{"mathvariant":226163},[31662,707230,456],{"stretchy":702},[31743,707232,707233,707235],{},[31658,707234,32936],{"mathvariant":226163},[31749,707236,459],{"mathvariant":226163},[31662,707238,2557],{"stretchy":702},[31662,707240,32712],{"stretchy":702},[31662,707242,71191],{},[31658,707244,701520],{},[31662,707246,32682],{},[31655,707248,707249,707251,707263,707265,707267,707269,707271,707273,707275],{},[31662,707250,456],{"fence":758},[38846,707252,707253,707255,707261],{},[31662,707254,546295],{},[31743,707256,707257,707259],{},[31658,707258,32936],{"mathvariant":226163},[31749,707260,459],{"mathvariant":226163},[31658,707262,32936],{"mathvariant":226163},[31658,707264,105852],{"mathvariant":226163},[31662,707266,456],{"stretchy":702},[31658,707268,33249],{"mathvariant":226163},[31662,707270,2557],{"stretchy":702},[31658,707272,46900],{"mathvariant":226163},[31658,707274,33249],{"mathvariant":226163},[31662,707276,2557],{"fence":758},[31665,707278,707279],{"encoding":31667},"\\mathbf{\\frac{\\partial \\varphi}{\\partial x_0} = -[P(x_0)y_0 + Q(x_0)] \\cdot \\exp\\left( \\int_{x_0}^x P(t) dt \\right)}",[34,707281,707283],{"className":707282,"ariaHidden":758},[31672],[34,707284,707286,707289],{"className":707285},[31676],[34,707287],{"className":707288,"style":141282},[31680],[34,707290,707292,707403,707406,707409,707412,707415,707418,707421,707424,707464,707467,707508,707511,707514,707517,707520,707523,707563,707566,707569,707572,707575,707578,707581],{"className":707291},[31685],[34,707293,707295,707298,707400],{"className":707294},[31685],[34,707296],{"className":707297},[31782,41488],[34,707299,707301],{"className":707300},[41453],[34,707302,707304,707392],{"className":707303},[31796,31797],[34,707305,707307,707389],{"className":707306},[31801],[34,707308,707310,707364,707372],{"className":707309,"style":608935},[31805],[34,707311,707312,707315],{"style":41504},[34,707313],{"className":707314,"style":41508},[31813],[34,707316,707318],{"className":707317},[31818,31819,31820,31821],[34,707319,707321,707324],{"className":707320},[31685,31821],[34,707322,619473],{"className":707323,"style":705478},[31685,226302,31821],[34,707325,707327,707330],{"className":707326},[31685,31821],[34,707328,32936],{"className":707329},[31685,226302,31821],[34,707331,707333],{"className":707332},[31792],[34,707334,707336,707356],{"className":707335},[31796,31797],[34,707337,707339,707353],{"className":707338},[31801],[34,707340,707342],{"className":707341,"style":318658},[31805],[34,707343,707344,707347],{"style":318661},[34,707345],{"className":707346,"style":43406},[31813],[34,707348,707350],{"className":707349},[31818,43410,43411,31821],[34,707351,459],{"className":707352},[31685,226302,31821],[34,707354,31829],{"className":707355},[31828],[34,707357,707359],{"className":707358},[31801],[34,707360,707362],{"className":707361,"style":220540},[31805],[34,707363],{},[34,707365,707366,707369],{"style":41520},[34,707367],{"className":707368,"style":41508},[31813],[34,707370],{"className":707371,"style":41528},[41527],[34,707373,707374,707377],{"style":548368},[34,707375],{"className":707376,"style":41508},[31813],[34,707378,707380],{"className":707379},[31818,31819,31820,31821],[34,707381,707383,707386],{"className":707382},[31685,31821],[34,707384,619473],{"className":707385,"style":705478},[31685,226302,31821],[34,707387,126541],{"className":707388},[31685,31686,31821],[34,707390,31829],{"className":707391},[31828],[34,707393,707395],{"className":707394},[31801],[34,707396,707398],{"className":707397,"style":231800},[31805],[34,707399],{},[34,707401],{"className":707402},[31690,41488],[34,707404],{"className":707405,"style":31846},[31845],[34,707407,202],{"className":707408},[31850],[34,707410],{"className":707411,"style":31846},[31845],[34,707413,32226],{"className":707414},[31685],[34,707416,15972],{"className":707417},[31782],[34,707419,105852],{"className":707420},[31685,226302],[34,707422,456],{"className":707423},[31782],[34,707425,707427,707430],{"className":707426},[31685],[34,707428,32936],{"className":707429},[31685,226302],[34,707431,707433],{"className":707432},[31792],[34,707434,707436,707456],{"className":707435},[31796,31797],[34,707437,707439,707453],{"className":707438},[31801],[34,707440,707442],{"className":707441,"style":31806},[31805],[34,707443,707444,707447],{"style":31809},[34,707445],{"className":707446,"style":31814},[31813],[34,707448,707450],{"className":707449},[31818,31819,31820,31821],[34,707451,459],{"className":707452},[31685,226302,31821],[34,707454,31829],{"className":707455},[31828],[34,707457,707459],{"className":707458},[31801],[34,707460,707462],{"className":707461,"style":31836},[31805],[34,707463],{},[34,707465,2557],{"className":707466},[31690],[34,707468,707470,707473],{"className":707469},[31685],[34,707471,67764],{"className":707472,"style":705485},[31685,226302],[34,707474,707476],{"className":707475},[31792],[34,707477,707479,707500],{"className":707478},[31796,31797],[34,707480,707482,707497],{"className":707481},[31801],[34,707483,707485],{"className":707484,"style":31806},[31805],[34,707486,707488,707491],{"style":707487},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[34,707489],{"className":707490,"style":31814},[31813],[34,707492,707494],{"className":707493},[31818,31819,31820,31821],[34,707495,459],{"className":707496},[31685,226302,31821],[34,707498,31829],{"className":707499},[31828],[34,707501,707503],{"className":707502},[31801],[34,707504,707506],{"className":707505,"style":31836},[31805],[34,707507],{},[34,707509],{"className":707510,"style":32604},[31845],[34,707512,223],{"className":707513},[32024],[34,707515],{"className":707516,"style":32604},[31845],[34,707518,116863],{"className":707519},[31685,226302],[34,707521,456],{"className":707522},[31782],[34,707524,707526,707529],{"className":707525},[31685],[34,707527,32936],{"className":707528},[31685,226302],[34,707530,707532],{"className":707531},[31792],[34,707533,707535,707555],{"className":707534},[31796,31797],[34,707536,707538,707552],{"className":707537},[31801],[34,707539,707541],{"className":707540,"style":31806},[31805],[34,707542,707543,707546],{"style":31809},[34,707544],{"className":707545,"style":31814},[31813],[34,707547,707549],{"className":707548},[31818,31819,31820,31821],[34,707550,459],{"className":707551},[31685,226302,31821],[34,707553,31829],{"className":707554},[31828],[34,707556,707558],{"className":707557},[31801],[34,707559,707561],{"className":707560,"style":31836},[31805],[34,707562],{},[34,707564,481211],{"className":707565},[31690],[34,707567],{"className":707568,"style":32604},[31845],[34,707570,71191],{"className":707571},[32024],[34,707573],{"className":707574,"style":32604},[31845],[34,707576,701520],{"className":707577},[32767],[34,707579],{"className":707580,"style":32123},[31845],[34,707582,707584,707590,707681,707684,707687,707690,707693,707696,707699],{"className":707583},[34006],[34,707585,707587],{"className":707586,"style":75811},[31782,75810],[34,707588,456],{"className":707589},[71281,141292],[34,707591,707593,707596],{"className":707592},[32767],[34,707594,546295],{"className":707595,"style":546331},[32767,131249,156560],[34,707597,707599],{"className":707598},[31792],[34,707600,707602,707673],{"className":707601},[31796,31797],[34,707603,707605,707670],{"className":707604},[31801],[34,707606,707608,707659],{"className":707607,"style":705602},[31805],[34,707609,707610,707613],{"style":546347},[34,707611],{"className":707612,"style":31814},[31813],[34,707614,707616],{"className":707615},[31818,31819,31820,31821],[34,707617,707619],{"className":707618},[31685,31821],[34,707620,707622,707625],{"className":707621},[31685,31821],[34,707623,32936],{"className":707624},[31685,226302,31821],[34,707626,707628],{"className":707627},[31792],[34,707629,707631,707651],{"className":707630},[31796,31797],[34,707632,707634,707648],{"className":707633},[31801],[34,707635,707637],{"className":707636,"style":318658},[31805],[34,707638,707639,707642],{"style":318661},[34,707640],{"className":707641,"style":43406},[31813],[34,707643,707645],{"className":707644},[31818,43410,43411,31821],[34,707646,459],{"className":707647},[31685,226302,31821],[34,707649,31829],{"className":707650},[31828],[34,707652,707654],{"className":707653},[31801],[34,707655,707657],{"className":707656,"style":220540},[31805],[34,707658],{},[34,707660,707661,707664],{"style":546362},[34,707662],{"className":707663,"style":31814},[31813],[34,707665,707667],{"className":707666},[31818,31819,31820,31821],[34,707668,32936],{"className":707669},[31685,226302,31821],[34,707671,31829],{"className":707672},[31828],[34,707674,707676],{"className":707675},[31801],[34,707677,707679],{"className":707678,"style":667231},[31805],[34,707680],{},[34,707682],{"className":707683,"style":32123},[31845],[34,707685,105852],{"className":707686},[31685,226302],[34,707688,456],{"className":707689},[31782],[34,707691,33249],{"className":707692},[31685,226302],[34,707694,2557],{"className":707695},[31690],[34,707697,705694],{"className":707698},[31685,226302],[34,707700,707702],{"className":707701,"style":75811},[31690,75810],[34,707703,2557],{"className":707704},[71281,141292],[15,707706,707707,707821,707822,707850,707851,384551,707962,708024,708025],{},[2627,707708,707709,707710],{},"Step 4: 计算 ",[34,707711,707713,707738],{"className":707712},[31642],[34,707714,707716],{"className":707715},[31646],[31648,707717,707718],{"xmlns":31650},[31652,707719,707720,707736],{},[31655,707721,707722],{},[41453,707723,707724,707730],{},[31655,707725,707726,707728],{},[31658,707727,619473],{"mathvariant":38079},[31658,707729,126541],{},[31655,707731,707732,707734],{},[31658,707733,619473],{"mathvariant":38079},[31658,707735,32936],{},[31665,707737,703657],{"encoding":31667},[34,707739,707741],{"className":707740,"ariaHidden":758},[31672],[34,707742,707744,707747],{"className":707743},[31676],[34,707745],{"className":707746,"style":608762},[31680],[34,707748,707750,707753,707818],{"className":707749},[31685],[34,707751],{"className":707752},[31782,41488],[34,707754,707756],{"className":707755},[41453],[34,707757,707759,707810],{"className":707758},[31796,31797],[34,707760,707762,707807],{"className":707761},[31801],[34,707763,707765,707782,707790],{"className":707764,"style":608935},[31805],[34,707766,707767,707770],{"style":41504},[34,707768],{"className":707769,"style":41508},[31813],[34,707771,707773],{"className":707772},[31818,31819,31820,31821],[34,707774,707776,707779],{"className":707775},[31685,31821],[34,707777,619473],{"className":707778,"style":112322},[31685,31821],[34,707780,32936],{"className":707781},[31685,31686,31821],[34,707783,707784,707787],{"style":41520},[34,707785],{"className":707786,"style":41508},[31813],[34,707788],{"className":707789,"style":41528},[41527],[34,707791,707792,707795],{"style":548368},[34,707793],{"className":707794,"style":41508},[31813],[34,707796,707798],{"className":707797},[31818,31819,31820,31821],[34,707799,707801,707804],{"className":707800},[31685,31821],[34,707802,619473],{"className":707803,"style":112322},[31685,31821],[34,707805,126541],{"className":707806},[31685,31686,31821],[34,707808,31829],{"className":707809},[31828],[34,707811,707813],{"className":707812},[31801],[34,707814,707816],{"className":707815,"style":41619},[31805],[34,707817],{},[34,707819],{"className":707820},[31690,41488],"\n这个最简单，但容易想复杂。",[34,707823,707825,707838],{"className":707824},[31642],[34,707826,707828],{"className":707827},[31646],[31648,707829,707830],{"xmlns":31650},[31652,707831,707832,707836],{},[31655,707833,707834],{},[31658,707835,126541],{},[31665,707837,126708],{"encoding":31667},[34,707839,707841],{"className":707840,"ariaHidden":758},[31672],[34,707842,707844,707847],{"className":707843},[31676],[34,707845],{"className":707846,"style":38774},[31680],[34,707848,126541],{"className":707849},[31685,31686]," 是原方程的解。\n解的定义就是：把它代入方程，方程成立。\n即 ",[34,707852,707854,707879],{"className":707853},[31642],[34,707855,707857],{"className":707856},[31646],[31648,707858,707859],{"xmlns":31650},[31652,707860,707861,707877],{},[31655,707862,707863],{},[41453,707864,707865,707871],{},[31655,707866,707867,707869],{},[31658,707868,619473],{"mathvariant":38079},[31658,707870,126541],{},[31655,707872,707873,707875],{},[31658,707874,619473],{"mathvariant":38079},[31658,707876,32936],{},[31665,707878,703657],{"encoding":31667},[34,707880,707882],{"className":707881,"ariaHidden":758},[31672],[34,707883,707885,707888],{"className":707884},[31676],[34,707886],{"className":707887,"style":608762},[31680],[34,707889,707891,707894,707959],{"className":707890},[31685],[34,707892],{"className":707893},[31782,41488],[34,707895,707897],{"className":707896},[41453],[34,707898,707900,707951],{"className":707899},[31796,31797],[34,707901,707903,707948],{"className":707902},[31801],[34,707904,707906,707923,707931],{"className":707905,"style":608935},[31805],[34,707907,707908,707911],{"style":41504},[34,707909],{"className":707910,"style":41508},[31813],[34,707912,707914],{"className":707913},[31818,31819,31820,31821],[34,707915,707917,707920],{"className":707916},[31685,31821],[34,707918,619473],{"className":707919,"style":112322},[31685,31821],[34,707921,32936],{"className":707922},[31685,31686,31821],[34,707924,707925,707928],{"style":41520},[34,707926],{"className":707927,"style":41508},[31813],[34,707929],{"className":707930,"style":41528},[41527],[34,707932,707933,707936],{"style":548368},[34,707934],{"className":707935,"style":41508},[31813],[34,707937,707939],{"className":707938},[31818,31819,31820,31821],[34,707940,707942,707945],{"className":707941},[31685,31821],[34,707943,619473],{"className":707944,"style":112322},[31685,31821],[34,707946,126541],{"className":707947},[31685,31686,31821],[34,707949,31829],{"className":707950},[31828],[34,707952,707954],{"className":707953},[31801],[34,707955,707957],{"className":707956,"style":41619},[31805],[34,707958],{},[34,707960],{"className":707961},[31690,41488],[34,707963,707965,707983],{"className":707964},[31642],[34,707966,707968],{"className":707967},[31646],[31648,707969,707970],{"xmlns":31650},[31652,707971,707972,707980],{},[31655,707973,707974],{},[37624,707975,707976,707978],{},[31658,707977,67764],{},[31662,707979,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,707981,707982],{"encoding":31667},"y'",[34,707984,707986],{"className":707985,"ariaHidden":758},[31672],[34,707987,707989,707992],{"className":707988},[31676],[34,707990],{"className":707991,"style":395485},[31680],[34,707993,707995,707998],{"className":707994},[31685],[34,707996,67764],{"className":707997,"style":32062},[31685,31686],[34,707999,708001],{"className":708000},[31792],[34,708002,708004],{"className":708003},[31796],[34,708005,708007],{"className":708006},[31801],[34,708008,708010],{"className":708009,"style":132027},[31805],[34,708011,708012,708015],{"style":37777},[34,708013],{"className":708014,"style":31814},[31813],[34,708016,708018],{"className":708017},[31818,31819,31820,31821],[34,708019,708021],{"className":708020},[31685,31821],[34,708022,116495],{"className":708023},[31685,31821],"，也就是方程的右端：\n",[34,708026,708028,708112],{"className":708027},[31642],[34,708029,708031],{"className":708030},[31646],[31648,708032,708033],{"xmlns":31650},[31652,708034,708035,708109],{},[31655,708036,708037,708051,708053,708055,708057,708059,708061,708063,708065,708067,708069,708071,708073,708075,708077,708079,708081,708083,708089,708091,708097,708099,708101,708103,708105,708107],{},[41453,708038,708039,708045],{},[31655,708040,708041,708043],{},[31658,708042,619473],{"mathvariant":226163},[31658,708044,126541],{"mathvariant":226163},[31655,708046,708047,708049],{},[31658,708048,619473],{"mathvariant":226163},[31658,708050,32936],{"mathvariant":226163},[31662,708052,202],{},[31658,708054,31739],{"mathvariant":226163},[31662,708056,456],{"stretchy":702},[31658,708058,32936],{"mathvariant":226163},[31662,708060,31960],{"separator":758},[31658,708062,126541],{"mathvariant":226163},[31662,708064,2557],{"stretchy":702},[31662,708066,202],{},[31658,708068,105852],{"mathvariant":226163},[31662,708070,456],{"stretchy":702},[31658,708072,32936],{"mathvariant":226163},[31662,708074,2557],{"stretchy":702},[31658,708076,126541],{"mathvariant":226163},[31662,708078,456],{"stretchy":702},[31658,708080,32936],{"mathvariant":226163},[31662,708082,31960],{"separator":758},[31743,708084,708085,708087],{},[31658,708086,32936],{"mathvariant":226163},[31749,708088,459],{"mathvariant":226163},[31662,708090,31960],{"separator":758},[31743,708092,708093,708095],{},[31658,708094,67764],{"mathvariant":226163},[31749,708096,459],{"mathvariant":226163},[31662,708098,2557],{"stretchy":702},[31662,708100,223],{},[31658,708102,116863],{"mathvariant":226163},[31662,708104,456],{"stretchy":702},[31658,708106,32936],{"mathvariant":226163},[31662,708108,2557],{"stretchy":702},[31665,708110,708111],{"encoding":31667},"\\mathbf{\\frac{\\partial \\varphi}{\\partial x} = f(x, \\varphi) = P(x)\\varphi(x,x_0,y_0) + Q(x)}",[34,708113,708115],{"className":708114,"ariaHidden":758},[31672],[34,708116,708118,708121],{"className":708117},[31676],[34,708119],{"className":708120,"style":608762},[31680],[34,708122,708124,708196,708199,708202,708205,708208,708211,708214,708217,708220,708223,708226,708229,708232,708235,708238,708241,708244,708247,708250,708253,708256,708259,708262,708302,708305,708308,708348,708351,708354,708357,708360,708363,708366,708369],{"className":708123},[31685],[34,708125,708127,708130,708193],{"className":708126},[31685],[34,708128],{"className":708129},[31782,41488],[34,708131,708133],{"className":708132},[41453],[34,708134,708136,708185],{"className":708135},[31796,31797],[34,708137,708139,708182],{"className":708138},[31801],[34,708140,708142,708157,708165],{"className":708141,"style":608935},[31805],[34,708143,708144,708147],{"style":41504},[34,708145],{"className":708146,"style":41508},[31813],[34,708148,708150],{"className":708149},[31818,31819,31820,31821],[34,708151,708153],{"className":708152},[31685,31821],[34,708154,708156],{"className":708155},[31685,226302,31821],"∂x",[34,708158,708159,708162],{"style":41520},[34,708160],{"className":708161,"style":41508},[31813],[34,708163],{"className":708164,"style":41528},[41527],[34,708166,708167,708170],{"style":548368},[34,708168],{"className":708169,"style":41508},[31813],[34,708171,708173],{"className":708172},[31818,31819,31820,31821],[34,708174,708176,708179],{"className":708175},[31685,31821],[34,708177,619473],{"className":708178,"style":705478},[31685,226302,31821],[34,708180,126541],{"className":708181},[31685,31686,31821],[34,708183,31829],{"className":708184},[31828],[34,708186,708188],{"className":708187},[31801],[34,708189,708191],{"className":708190,"style":41619},[31805],[34,708192],{},[34,708194],{"className":708195},[31690,41488],[34,708197],{"className":708198,"style":31846},[31845],[34,708200,202],{"className":708201},[31850],[34,708203],{"className":708204,"style":31846},[31845],[34,708206,31739],{"className":708207,"style":33389},[31685,226302],[34,708209,456],{"className":708210},[31782],[34,708212,32936],{"className":708213},[31685,226302],[34,708215,31960],{"className":708216},[32119],[34,708218],{"className":708219,"style":32123},[31845],[34,708221,126541],{"className":708222},[31685,31686],[34,708224,2557],{"className":708225},[31690],[34,708227],{"className":708228,"style":31846},[31845],[34,708230,202],{"className":708231},[31850],[34,708233],{"className":708234,"style":31846},[31845],[34,708236,105852],{"className":708237},[31685,226302],[34,708239,456],{"className":708240},[31782],[34,708242,32936],{"className":708243},[31685,226302],[34,708245,2557],{"className":708246},[31690],[34,708248,126541],{"className":708249},[31685,31686],[34,708251,456],{"className":708252},[31782],[34,708254,32936],{"className":708255},[31685,226302],[34,708257,31960],{"className":708258},[32119],[34,708260],{"className":708261,"style":32123},[31845],[34,708263,708265,708268],{"className":708264},[31685],[34,708266,32936],{"className":708267},[31685,226302],[34,708269,708271],{"className":708270},[31792],[34,708272,708274,708294],{"className":708273},[31796,31797],[34,708275,708277,708291],{"className":708276},[31801],[34,708278,708280],{"className":708279,"style":31806},[31805],[34,708281,708282,708285],{"style":31809},[34,708283],{"className":708284,"style":31814},[31813],[34,708286,708288],{"className":708287},[31818,31819,31820,31821],[34,708289,459],{"className":708290},[31685,226302,31821],[34,708292,31829],{"className":708293},[31828],[34,708295,708297],{"className":708296},[31801],[34,708298,708300],{"className":708299,"style":31836},[31805],[34,708301],{},[34,708303,31960],{"className":708304},[32119],[34,708306],{"className":708307,"style":32123},[31845],[34,708309,708311,708314],{"className":708310},[31685],[34,708312,67764],{"className":708313,"style":705485},[31685,226302],[34,708315,708317],{"className":708316},[31792],[34,708318,708320,708340],{"className":708319},[31796,31797],[34,708321,708323,708337],{"className":708322},[31801],[34,708324,708326],{"className":708325,"style":31806},[31805],[34,708327,708328,708331],{"style":707487},[34,708329],{"className":708330,"style":31814},[31813],[34,708332,708334],{"className":708333},[31818,31819,31820,31821],[34,708335,459],{"className":708336},[31685,226302,31821],[34,708338,31829],{"className":708339},[31828],[34,708341,708343],{"className":708342},[31801],[34,708344,708346],{"className":708345,"style":31836},[31805],[34,708347],{},[34,708349,2557],{"className":708350},[31690],[34,708352],{"className":708353,"style":32604},[31845],[34,708355,223],{"className":708356},[32024],[34,708358],{"className":708359,"style":32604},[31845],[34,708361,116863],{"className":708362},[31685,226302],[34,708364,456],{"className":708365},[31782],[34,708367,32936],{"className":708368},[31685,226302],[34,708370,2557],{"className":708371},[31690],[32188,708373],{},[32490,708375,217356],{"id":217353},[15,708377,708378,708380,708381,639582,708594,258,708747,314945,708900,709068],{},[2627,708379,101191],{}," 给定方程 ",[34,708382,708384,708422],{"className":708383},[31642],[34,708385,708387],{"className":708386},[31646],[31648,708388,708389],{"xmlns":31650},[31652,708390,708391,708419],{},[31655,708392,708393,708407,708409,708411,708413],{},[41453,708394,708395,708401],{},[31655,708396,708397,708399],{},[31658,708398,46900],{"mathvariant":38079},[31658,708400,67764],{},[31655,708402,708403,708405],{},[31658,708404,46900],{"mathvariant":38079},[31658,708406,32936],{},[31662,708408,202],{},[31658,708410,547853],{},[31662,708412,32682],{},[41453,708414,708415,708417],{},[31658,708416,67764],{},[31658,708418,32936],{},[31665,708420,708421],{"encoding":31667},"\\frac{\\mathrm{d}y}{\\mathrm{d}x} = \\sin \\frac{y}{x}",[34,708423,708425,708514],{"className":708424,"ariaHidden":758},[31672],[34,708426,708428,708431,708505,708508,708511],{"className":708427},[31676],[34,708429],{"className":708430,"style":608762},[31680],[34,708432,708434,708437,708502],{"className":708433},[31685],[34,708435],{"className":708436},[31782,41488],[34,708438,708440],{"className":708439},[41453],[34,708441,708443,708494],{"className":708442},[31796,31797],[34,708444,708446,708491],{"className":708445},[31801],[34,708447,708449,708466,708474],{"className":708448,"style":608935},[31805],[34,708450,708451,708454],{"style":41504},[34,708452],{"className":708453,"style":41508},[31813],[34,708455,708457],{"className":708456},[31818,31819,31820,31821],[34,708458,708460,708463],{"className":708459},[31685,31821],[34,708461,46900],{"className":708462},[31685,207923,31821],[34,708464,32936],{"className":708465},[31685,31686,31821],[34,708467,708468,708471],{"style":41520},[34,708469],{"className":708470,"style":41508},[31813],[34,708472],{"className":708473,"style":41528},[41527],[34,708475,708476,708479],{"style":548368},[34,708477],{"className":708478,"style":41508},[31813],[34,708480,708482],{"className":708481},[31818,31819,31820,31821],[34,708483,708485,708488],{"className":708484},[31685,31821],[34,708486,46900],{"className":708487},[31685,207923,31821],[34,708489,67764],{"className":708490,"style":32062},[31685,31686,31821],[34,708492,31829],{"className":708493},[31828],[34,708495,708497],{"className":708496},[31801],[34,708498,708500],{"className":708499,"style":41619},[31805],[34,708501],{},[34,708503],{"className":708504},[31690,41488],[34,708506],{"className":708507,"style":31846},[31845],[34,708509,202],{"className":708510},[31850],[34,708512],{"className":708513,"style":31846},[31845],[34,708515,708517,708520,708523,708526],{"className":708516},[31676],[34,708518],{"className":708519,"style":615850},[31680],[34,708521,547853],{"className":708522},[32767],[34,708524],{"className":708525,"style":32123},[31845],[34,708527,708529,708532,708591],{"className":708528},[31685],[34,708530],{"className":708531},[31782,41488],[34,708533,708535],{"className":708534},[41453],[34,708536,708538,708583],{"className":708537},[31796,31797],[34,708539,708541,708580],{"className":708540},[31801],[34,708542,708544,708558,708566],{"className":708543,"style":548284},[31805],[34,708545,708546,708549],{"style":41504},[34,708547],{"className":708548,"style":41508},[31813],[34,708550,708552],{"className":708551},[31818,31819,31820,31821],[34,708553,708555],{"className":708554},[31685,31821],[34,708556,32936],{"className":708557},[31685,31686,31821],[34,708559,708560,708563],{"style":41520},[34,708561],{"className":708562,"style":41508},[31813],[34,708564],{"className":708565,"style":41528},[41527],[34,708567,708568,708571],{"style":548368},[34,708569],{"className":708570,"style":41508},[31813],[34,708572,708574],{"className":708573},[31818,31819,31820,31821],[34,708575,708577],{"className":708576},[31685,31821],[34,708578,67764],{"className":708579,"style":32062},[31685,31686,31821],[34,708581,31829],{"className":708582},[31828],[34,708584,708586],{"className":708585},[31801],[34,708587,708589],{"className":708588,"style":41619},[31805],[34,708590],{},[34,708592],{"className":708593},[31690,41488],[34,708595,708597,708627],{"className":708596},[31642],[34,708598,708600],{"className":708599},[31646],[31648,708601,708602],{"xmlns":31650},[31652,708603,708604,708624],{},[31655,708605,708606],{},[41453,708607,708608,708614],{},[31655,708609,708610,708612],{},[31658,708611,619473],{"mathvariant":38079},[31658,708613,67764],{},[31655,708615,708616,708618],{},[31658,708617,619473],{"mathvariant":38079},[31743,708619,708620,708622],{},[31658,708621,32936],{},[31749,708623,459],{},[31665,708625,708626],{"encoding":31667},"\\frac{\\partial y}{\\partial x_0}",[34,708628,708630],{"className":708629,"ariaHidden":758},[31672],[34,708631,708633,708636],{"className":708632},[31676],[34,708634],{"className":708635,"style":702239},[31680],[34,708637,708639,708642,708744],{"className":708638},[31685],[34,708640],{"className":708641},[31782,41488],[34,708643,708645],{"className":708644},[41453],[34,708646,708648,708736],{"className":708647},[31796,31797],[34,708649,708651,708733],{"className":708650},[31801],[34,708652,708654,708708,708716],{"className":708653,"style":608935},[31805],[34,708655,708656,708659],{"style":41504},[34,708657],{"className":708658,"style":41508},[31813],[34,708660,708662],{"className":708661},[31818,31819,31820,31821],[34,708663,708665,708668],{"className":708664},[31685,31821],[34,708666,619473],{"className":708667,"style":112322},[31685,31821],[34,708669,708671,708674],{"className":708670},[31685,31821],[34,708672,32936],{"className":708673},[31685,31686,31821],[34,708675,708677],{"className":708676},[31792],[34,708678,708680,708700],{"className":708679},[31796,31797],[34,708681,708683,708697],{"className":708682},[31801],[34,708684,708686],{"className":708685,"style":318658},[31805],[34,708687,708688,708691],{"style":318661},[34,708689],{"className":708690,"style":43406},[31813],[34,708692,708694],{"className":708693},[31818,43410,43411,31821],[34,708695,459],{"className":708696},[31685,31821],[34,708698,31829],{"className":708699},[31828],[34,708701,708703],{"className":708702},[31801],[34,708704,708706],{"className":708705,"style":220540},[31805],[34,708707],{},[34,708709,708710,708713],{"style":41520},[34,708711],{"className":708712,"style":41508},[31813],[34,708714],{"className":708715,"style":41528},[41527],[34,708717,708718,708721],{"style":548368},[34,708719],{"className":708720,"style":41508},[31813],[34,708722,708724],{"className":708723},[31818,31819,31820,31821],[34,708725,708727,708730],{"className":708726},[31685,31821],[34,708728,619473],{"className":708729,"style":112322},[31685,31821],[34,708731,67764],{"className":708732,"style":32062},[31685,31686,31821],[34,708734,31829],{"className":708735},[31828],[34,708737,708739],{"className":708738},[31801],[34,708740,708742],{"className":708741,"style":231800},[31805],[34,708743],{},[34,708745],{"className":708746},[31690,41488],[34,708748,708750,708780],{"className":708749},[31642],[34,708751,708753],{"className":708752},[31646],[31648,708754,708755],{"xmlns":31650},[31652,708756,708757,708777],{},[31655,708758,708759],{},[41453,708760,708761,708767],{},[31655,708762,708763,708765],{},[31658,708764,619473],{"mathvariant":38079},[31658,708766,67764],{},[31655,708768,708769,708771],{},[31658,708770,619473],{"mathvariant":38079},[31743,708772,708773,708775],{},[31658,708774,67764],{},[31749,708776,459],{},[31665,708778,708779],{"encoding":31667},"\\frac{\\partial y}{\\partial y_0}",[34,708781,708783],{"className":708782,"ariaHidden":758},[31672],[34,708784,708786,708789],{"className":708785},[31676],[34,708787],{"className":708788,"style":695705},[31680],[34,708790,708792,708795,708897],{"className":708791},[31685],[34,708793],{"className":708794},[31782,41488],[34,708796,708798],{"className":708797},[41453],[34,708799,708801,708889],{"className":708800},[31796,31797],[34,708802,708804,708886],{"className":708803},[31801],[34,708805,708807,708861,708869],{"className":708806,"style":608935},[31805],[34,708808,708809,708812],{"style":41504},[34,708810],{"className":708811,"style":41508},[31813],[34,708813,708815],{"className":708814},[31818,31819,31820,31821],[34,708816,708818,708821],{"className":708817},[31685,31821],[34,708819,619473],{"className":708820,"style":112322},[31685,31821],[34,708822,708824,708827],{"className":708823},[31685,31821],[34,708825,67764],{"className":708826,"style":32062},[31685,31686,31821],[34,708828,708830],{"className":708829},[31792],[34,708831,708833,708853],{"className":708832},[31796,31797],[34,708834,708836,708850],{"className":708835},[31801],[34,708837,708839],{"className":708838,"style":318658},[31805],[34,708840,708841,708844],{"style":561507},[34,708842],{"className":708843,"style":43406},[31813],[34,708845,708847],{"className":708846},[31818,43410,43411,31821],[34,708848,459],{"className":708849},[31685,31821],[34,708851,31829],{"className":708852},[31828],[34,708854,708856],{"className":708855},[31801],[34,708857,708859],{"className":708858,"style":220540},[31805],[34,708860],{},[34,708862,708863,708866],{"style":41520},[34,708864],{"className":708865,"style":41508},[31813],[34,708867],{"className":708868,"style":41528},[41527],[34,708870,708871,708874],{"style":548368},[34,708872],{"className":708873,"style":41508},[31813],[34,708875,708877],{"className":708876},[31818,31819,31820,31821],[34,708878,708880,708883],{"className":708879},[31685,31821],[34,708881,619473],{"className":708882,"style":112322},[31685,31821],[34,708884,67764],{"className":708885,"style":32062},[31685,31686,31821],[34,708887,31829],{"className":708888},[31828],[34,708890,708892],{"className":708891},[31801],[34,708893,708895],{"className":708894,"style":220191},[31805],[34,708896],{},[34,708898],{"className":708899},[31690,41488],[34,708901,708903,708937],{"className":708902},[31642],[34,708904,708906],{"className":708905},[31646],[31648,708907,708908],{"xmlns":31650},[31652,708909,708910,708934],{},[31655,708911,708912,708918,708920,708922,708924,708930,708932],{},[31743,708913,708914,708916],{},[31658,708915,32936],{},[31749,708917,459],{},[31662,708919,202],{},[31749,708921,640],{},[31662,708923,31960],{"separator":758},[31743,708925,708926,708928],{},[31658,708927,67764],{},[31749,708929,459],{},[31662,708931,202],{},[31749,708933,459],{},[31665,708935,708936],{"encoding":31667},"x_0 = 1, y_0 = 0",[34,708938,708940,708995,709059],{"className":708939,"ariaHidden":758},[31672],[34,708941,708943,708946,708986,708989,708992],{"className":708942},[31676],[34,708944],{"className":708945,"style":31860},[31680],[34,708947,708949,708952],{"className":708948},[31685],[34,708950,32936],{"className":708951},[31685,31686],[34,708953,708955],{"className":708954},[31792],[34,708956,708958,708978],{"className":708957},[31796,31797],[34,708959,708961,708975],{"className":708960},[31801],[34,708962,708964],{"className":708963,"style":31806},[31805],[34,708965,708966,708969],{"style":31809},[34,708967],{"className":708968,"style":31814},[31813],[34,708970,708972],{"className":708971},[31818,31819,31820,31821],[34,708973,459],{"className":708974},[31685,31821],[34,708976,31829],{"className":708977},[31828],[34,708979,708981],{"className":708980},[31801],[34,708982,708984],{"className":708983,"style":31836},[31805],[34,708985],{},[34,708987],{"className":708988,"style":31846},[31845],[34,708990,202],{"className":708991},[31850],[34,708993],{"className":708994,"style":31846},[31845],[34,708996,708998,709001,709004,709007,709010,709050,709053,709056],{"className":708997},[31676],[34,708999],{"className":709000,"style":40292},[31680],[34,709002,640],{"className":709003},[31685],[34,709005,31960],{"className":709006},[32119],[34,709008],{"className":709009,"style":32123},[31845],[34,709011,709013,709016],{"className":709012},[31685],[34,709014,67764],{"className":709015,"style":32062},[31685,31686],[34,709017,709019],{"className":709018},[31792],[34,709020,709022,709042],{"className":709021},[31796,31797],[34,709023,709025,709039],{"className":709024},[31801],[34,709026,709028],{"className":709027,"style":31806},[31805],[34,709029,709030,709033],{"style":38795},[34,709031],{"className":709032,"style":31814},[31813],[34,709034,709036],{"className":709035},[31818,31819,31820,31821],[34,709037,459],{"className":709038},[31685,31821],[34,709040,31829],{"className":709041},[31828],[34,709043,709045],{"className":709044},[31801],[34,709046,709048],{"className":709047,"style":31836},[31805],[34,709049],{},[34,709051],{"className":709052,"style":31846},[31845],[34,709054,202],{"className":709055},[31850],[34,709057],{"className":709058,"style":31846},[31845],[34,709060,709062,709065],{"className":709061},[31676],[34,709063],{"className":709064,"style":32617},[31680],[34,709066,459],{"className":709067},[31685]," 时的表达式。",[15,709070,709071],{},[2627,709072,258022],{},[15,709074,709075,709078,709079,709190,709191,709219,709220,709289],{},[2627,709076,709077],{},"关键提示："," 对于非线性方程，积分里的 ",[34,709080,709082,709107],{"className":709081},[31642],[34,709083,709085],{"className":709084},[31646],[31648,709086,709087],{"xmlns":31650},[31652,709088,709089,709105],{},[31655,709090,709091],{},[41453,709092,709093,709099],{},[31655,709094,709095,709097],{},[31658,709096,619473],{"mathvariant":38079},[31658,709098,31739],{},[31655,709100,709101,709103],{},[31658,709102,619473],{"mathvariant":38079},[31658,709104,67764],{},[31665,709106,701964],{"encoding":31667},[34,709108,709110],{"className":709109,"ariaHidden":758},[31672],[34,709111,709113,709116],{"className":709112},[31676],[34,709114],{"className":709115,"style":695705},[31680],[34,709117,709119,709122,709187],{"className":709118},[31685],[34,709120],{"className":709121},[31782,41488],[34,709123,709125],{"className":709124},[41453],[34,709126,709128,709179],{"className":709127},[31796,31797],[34,709129,709131,709176],{"className":709130},[31801],[34,709132,709134,709151,709159],{"className":709133,"style":608935},[31805],[34,709135,709136,709139],{"style":41504},[34,709137],{"className":709138,"style":41508},[31813],[34,709140,709142],{"className":709141},[31818,31819,31820,31821],[34,709143,709145,709148],{"className":709144},[31685,31821],[34,709146,619473],{"className":709147,"style":112322},[31685,31821],[34,709149,67764],{"className":709150,"style":32062},[31685,31686,31821],[34,709152,709153,709156],{"style":41520},[34,709154],{"className":709155,"style":41508},[31813],[34,709157],{"className":709158,"style":41528},[41527],[34,709160,709161,709164],{"style":548368},[34,709162],{"className":709163,"style":41508},[31813],[34,709165,709167],{"className":709166},[31818,31819,31820,31821],[34,709168,709170,709173],{"className":709169},[31685,31821],[34,709171,619473],{"className":709172,"style":112322},[31685,31821],[34,709174,31739],{"className":709175,"style":31778},[31685,31686,31821],[34,709177,31829],{"className":709178},[31828],[34,709180,709182],{"className":709181},[31801],[34,709183,709185],{"className":709184,"style":220191},[31805],[34,709186],{},[34,709188],{"className":709189},[31690,41488]," 通常含有 ",[34,709192,709194,709207],{"className":709193},[31642],[34,709195,709197],{"className":709196},[31646],[31648,709198,709199],{"xmlns":31650},[31652,709200,709201,709205],{},[31655,709202,709203],{},[31658,709204,67764],{},[31665,709206,67764],{"encoding":31667},[34,709208,709210],{"className":709209,"ariaHidden":758},[31672],[34,709211,709213,709216],{"className":709212},[31676],[34,709214],{"className":709215,"style":38774},[31680],[34,709217,67764],{"className":709218,"style":32062},[31685,31686],"。要计算积分，我们必须先知道",[2627,709221,709222,709223],{},"具体的解 ",[34,709224,709226,709250],{"className":709225},[31642],[34,709227,709229],{"className":709228},[31646],[31648,709230,709231],{"xmlns":31650},[31652,709232,709233,709247],{},[31655,709234,709235,709237,709239,709241,709243,709245],{},[31658,709236,67764],{},[31662,709238,202],{},[31658,709240,126541],{},[31662,709242,456],{"stretchy":702},[31658,709244,32936],{},[31662,709246,2557],{"stretchy":702},[31665,709248,709249],{"encoding":31667},"y=\\varphi(x)",[34,709251,709253,709271],{"className":709252,"ariaHidden":758},[31672],[34,709254,709256,709259,709262,709265,709268],{"className":709255},[31676],[34,709257],{"className":709258,"style":38774},[31680],[34,709260,67764],{"className":709261,"style":32062},[31685,31686],[34,709263],{"className":709264,"style":31846},[31845],[34,709266,202],{"className":709267},[31850],[34,709269],{"className":709270,"style":31846},[31845],[34,709272,709274,709277,709280,709283,709286],{"className":709273},[31676],[34,709275],{"className":709276,"style":31774},[31680],[34,709278,126541],{"className":709279},[31685,31686],[34,709281,456],{"className":709282},[31782],[34,709284,32936],{"className":709285},[31685,31686],[34,709287,2557],{"className":709288},[31690]," 是什么。",[15,709291,709292,709347,709348,709560,709561,709627,709628,347],{},[2627,709293,709294,709295,709346],{},"Step 1: 寻找通过点 ",[34,709296,709298,709319],{"className":709297},[31642],[34,709299,709301],{"className":709300},[31646],[31648,709302,709303],{"xmlns":31650},[31652,709304,709305,709317],{},[31655,709306,709307,709309,709311,709313,709315],{},[31662,709308,456],{"stretchy":702},[31749,709310,640],{},[31662,709312,31960],{"separator":758},[31749,709314,459],{},[31662,709316,2557],{"stretchy":702},[31665,709318,381896],{"encoding":31667},[34,709320,709322],{"className":709321,"ariaHidden":758},[31672],[34,709323,709325,709328,709331,709334,709337,709340,709343],{"className":709324},[31676],[34,709326],{"className":709327,"style":31774},[31680],[34,709329,456],{"className":709330},[31782],[34,709332,640],{"className":709333},[31685],[34,709335,31960],{"className":709336},[32119],[34,709338],{"className":709339,"style":32123},[31845],[34,709341,459],{"className":709342},[31685],[34,709344,2557],{"className":709345},[31690]," 的特解","\n我们观察方程 ",[34,709349,709351,709388],{"className":709350},[31642],[34,709352,709354],{"className":709353},[31646],[31648,709355,709356],{"xmlns":31650},[31652,709357,709358,709386],{},[31655,709359,709360,709374,709376,709378,709380],{},[41453,709361,709362,709368],{},[31655,709363,709364,709366],{},[31658,709365,46900],{"mathvariant":38079},[31658,709367,67764],{},[31655,709369,709370,709372],{},[31658,709371,46900],{"mathvariant":38079},[31658,709373,32936],{},[31662,709375,202],{},[31658,709377,547853],{},[31662,709379,32682],{},[41453,709381,709382,709384],{},[31658,709383,67764],{},[31658,709385,32936],{},[31665,709387,708421],{"encoding":31667},[34,709389,709391,709480],{"className":709390,"ariaHidden":758},[31672],[34,709392,709394,709397,709471,709474,709477],{"className":709393},[31676],[34,709395],{"className":709396,"style":608762},[31680],[34,709398,709400,709403,709468],{"className":709399},[31685],[34,709401],{"className":709402},[31782,41488],[34,709404,709406],{"className":709405},[41453],[34,709407,709409,709460],{"className":709408},[31796,31797],[34,709410,709412,709457],{"className":709411},[31801],[34,709413,709415,709432,709440],{"className":709414,"style":608935},[31805],[34,709416,709417,709420],{"style":41504},[34,709418],{"className":709419,"style":41508},[31813],[34,709421,709423],{"className":709422},[31818,31819,31820,31821],[34,709424,709426,709429],{"className":709425},[31685,31821],[34,709427,46900],{"className":709428},[31685,207923,31821],[34,709430,32936],{"className":709431},[31685,31686,31821],[34,709433,709434,709437],{"style":41520},[34,709435],{"className":709436,"style":41508},[31813],[34,709438],{"className":709439,"style":41528},[41527],[34,709441,709442,709445],{"style":548368},[34,709443],{"className":709444,"style":41508},[31813],[34,709446,709448],{"className":709447},[31818,31819,31820,31821],[34,709449,709451,709454],{"className":709450},[31685,31821],[34,709452,46900],{"className":709453},[31685,207923,31821],[34,709455,67764],{"className":709456,"style":32062},[31685,31686,31821],[34,709458,31829],{"className":709459},[31828],[34,709461,709463],{"className":709462},[31801],[34,709464,709466],{"className":709465,"style":41619},[31805],[34,709467],{},[34,709469],{"className":709470},[31690,41488],[34,709472],{"className":709473,"style":31846},[31845],[34,709475,202],{"className":709476},[31850],[34,709478],{"className":709479,"style":31846},[31845],[34,709481,709483,709486,709489,709492],{"className":709482},[31676],[34,709484],{"className":709485,"style":615850},[31680],[34,709487,547853],{"className":709488},[32767],[34,709490],{"className":709491,"style":32123},[31845],[34,709493,709495,709498,709557],{"className":709494},[31685],[34,709496],{"className":709497},[31782,41488],[34,709499,709501],{"className":709500},[41453],[34,709502,709504,709549],{"className":709503},[31796,31797],[34,709505,709507,709546],{"className":709506},[31801],[34,709508,709510,709524,709532],{"className":709509,"style":548284},[31805],[34,709511,709512,709515],{"style":41504},[34,709513],{"className":709514,"style":41508},[31813],[34,709516,709518],{"className":709517},[31818,31819,31820,31821],[34,709519,709521],{"className":709520},[31685,31821],[34,709522,32936],{"className":709523},[31685,31686,31821],[34,709525,709526,709529],{"style":41520},[34,709527],{"className":709528,"style":41508},[31813],[34,709530],{"className":709531,"style":41528},[41527],[34,709533,709534,709537],{"style":548368},[34,709535],{"className":709536,"style":41508},[31813],[34,709538,709540],{"className":709539},[31818,31819,31820,31821],[34,709541,709543],{"className":709542},[31685,31821],[34,709544,67764],{"className":709545,"style":32062},[31685,31686,31821],[34,709547,31829],{"className":709548},[31828],[34,709550,709552],{"className":709551},[31801],[34,709553,709555],{"className":709554,"style":41619},[31805],[34,709556],{},[34,709558],{"className":709559},[31690,41488]," 和初值 ",[34,709562,709564,709588],{"className":709563},[31642],[34,709565,709567],{"className":709566},[31646],[31648,709568,709569],{"xmlns":31650},[31652,709570,709571,709585],{},[31655,709572,709573,709575,709577,709579,709581,709583],{},[31658,709574,67764],{},[31662,709576,456],{"stretchy":702},[31749,709578,640],{},[31662,709580,2557],{"stretchy":702},[31662,709582,202],{},[31749,709584,459],{},[31665,709586,709587],{"encoding":31667},"y(1)=0",[34,709589,709591,709618],{"className":709590,"ariaHidden":758},[31672],[34,709592,709594,709597,709600,709603,709606,709609,709612,709615],{"className":709593},[31676],[34,709595],{"className":709596,"style":31774},[31680],[34,709598,67764],{"className":709599,"style":32062},[31685,31686],[34,709601,456],{"className":709602},[31782],[34,709604,640],{"className":709605},[31685],[34,709607,2557],{"className":709608},[31690],[34,709610],{"className":709611,"style":31846},[31845],[34,709613,202],{"className":709614},[31850],[34,709616],{"className":709617,"style":31846},[31845],[34,709619,709621,709624],{"className":709620},[31676],[34,709622],{"className":709623,"style":32617},[31680],[34,709625,459],{"className":709626},[31685],"。\n让我们猜测一个最简单的解：恒为 0 的解，即 ",[34,709629,709631,709649],{"className":709630},[31642],[34,709632,709634],{"className":709633},[31646],[31648,709635,709636],{"xmlns":31650},[31652,709637,709638,709646],{},[31655,709639,709640,709642,709644],{},[31658,709641,67764],{},[31662,709643,341696],{},[31749,709645,459],{},[31665,709647,709648],{"encoding":31667},"y \\equiv 0",[34,709650,709652,709671],{"className":709651,"ariaHidden":758},[31672],[34,709653,709655,709659,709662,709665,709668],{"className":709654},[31676],[34,709656],{"className":709657,"style":709658},[31680],"height:0.6582em;vertical-align:-0.1944em;",[34,709660,67764],{"className":709661,"style":32062},[31685,31686],[34,709663],{"className":709664,"style":31846},[31845],[34,709666,341696],{"className":709667},[31850],[34,709669],{"className":709670,"style":31846},[31845],[34,709672,709674,709677],{"className":709673},[31676],[34,709675],{"className":709676,"style":32617},[31680],[34,709678,459],{"className":709679},[31685],[105,709681,709682,709833,709950],{},[108,709683,386318,709684,347],{},[34,709685,709687,709721],{"className":709686},[31642],[34,709688,709690],{"className":709689},[31646],[31648,709691,709692],{"xmlns":31650},[31652,709693,709694,709718],{},[31655,709695,709696,709702,709704,709706,709708,709714,709716],{},[37624,709697,709698,709700],{},[31658,709699,67764],{},[31662,709701,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,709703,202],{},[31662,709705,456],{"stretchy":702},[31749,709707,459],{},[37624,709709,709710,709712],{},[31662,709711,2557],{"stretchy":702},[31662,709713,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,709715,202],{},[31749,709717,459],{},[31665,709719,709720],{"encoding":31667},"y' = (0)' = 0",[34,709722,709724,709771,709824],{"className":709723,"ariaHidden":758},[31672],[34,709725,709727,709730,709762,709765,709768],{"className":709726},[31676],[34,709728],{"className":709729,"style":395485},[31680],[34,709731,709733,709736],{"className":709732},[31685],[34,709734,67764],{"className":709735,"style":32062},[31685,31686],[34,709737,709739],{"className":709738},[31792],[34,709740,709742],{"className":709741},[31796],[34,709743,709745],{"className":709744},[31801],[34,709746,709748],{"className":709747,"style":132027},[31805],[34,709749,709750,709753],{"style":37777},[34,709751],{"className":709752,"style":31814},[31813],[34,709754,709756],{"className":709755},[31818,31819,31820,31821],[34,709757,709759],{"className":709758},[31685,31821],[34,709760,116495],{"className":709761},[31685,31821],[34,709763],{"className":709764,"style":31846},[31845],[34,709766,202],{"className":709767},[31850],[34,709769],{"className":709770,"style":31846},[31845],[34,709772,709774,709777,709780,709783,709815,709818,709821],{"className":709773},[31676],[34,709775],{"className":709776,"style":570646},[31680],[34,709778,456],{"className":709779},[31782],[34,709781,459],{"className":709782},[31685],[34,709784,709786,709789],{"className":709785},[31690],[34,709787,2557],{"className":709788},[31690],[34,709790,709792],{"className":709791},[31792],[34,709793,709795],{"className":709794},[31796],[34,709796,709798],{"className":709797},[31801],[34,709799,709801],{"className":709800,"style":132027},[31805],[34,709802,709803,709806],{"style":37777},[34,709804],{"className":709805,"style":31814},[31813],[34,709807,709809],{"className":709808},[31818,31819,31820,31821],[34,709810,709812],{"className":709811},[31685,31821],[34,709813,116495],{"className":709814},[31685,31821],[34,709816],{"className":709817,"style":31846},[31845],[34,709819,202],{"className":709820},[31850],[34,709822],{"className":709823,"style":31846},[31845],[34,709825,709827,709830],{"className":709826},[31676],[34,709828],{"className":709829,"style":32617},[31680],[34,709831,459],{"className":709832},[31685],[108,709834,386924,709835,347],{},[34,709836,709838,709880],{"className":709837},[31642],[34,709839,709841],{"className":709840},[31646],[31648,709842,709843],{"xmlns":31650},[31652,709844,709845,709877],{},[31655,709846,709847,709849,709851,709853,709855,709857,709859,709861,709863,709865,709867,709869,709871,709873,709875],{},[31658,709848,547853],{},[31662,709850,32682],{},[31662,709852,456],{"stretchy":702},[31749,709854,459],{},[31658,709856,38080],{"mathvariant":38079},[31658,709858,32936],{},[31662,709860,2557],{"stretchy":702},[31662,709862,202],{},[31658,709864,547853],{},[31662,709866,32682],{},[31662,709868,456],{"stretchy":702},[31749,709870,459],{},[31662,709872,2557],{"stretchy":702},[31662,709874,202],{},[31749,709876,459],{},[31665,709878,709879],{"encoding":31667},"\\sin(0\u002Fx) = \\sin(0) = 0",[34,709881,709883,709914,709941],{"className":709882,"ariaHidden":758},[31672],[34,709884,709886,709889,709892,709895,709899,709902,709905,709908,709911],{"className":709885},[31676],[34,709887],{"className":709888,"style":31774},[31680],[34,709890,547853],{"className":709891},[32767],[34,709893,456],{"className":709894},[31782],[34,709896,709898],{"className":709897},[31685],"0\u002F",[34,709900,32936],{"className":709901},[31685,31686],[34,709903,2557],{"className":709904},[31690],[34,709906],{"className":709907,"style":31846},[31845],[34,709909,202],{"className":709910},[31850],[34,709912],{"className":709913,"style":31846},[31845],[34,709915,709917,709920,709923,709926,709929,709932,709935,709938],{"className":709916},[31676],[34,709918],{"className":709919,"style":31774},[31680],[34,709921,547853],{"className":709922},[32767],[34,709924,456],{"className":709925},[31782],[34,709927,459],{"className":709928},[31685],[34,709930,2557],{"className":709931},[31690],[34,709933],{"className":709934,"style":31846},[31845],[34,709936,202],{"className":709937},[31850],[34,709939],{"className":709940,"style":31846},[31845],[34,709942,709944,709947],{"className":709943},[31676],[34,709945],{"className":709946,"style":32617},[31680],[34,709948,459],{"className":709949},[31685],[108,709951,709952,709953,710018,710019,39984,710109],{},"左边 = 右边，且满足初值 ",[34,709954,709956,709979],{"className":709955},[31642],[34,709957,709959],{"className":709958},[31646],[31648,709960,709961],{"xmlns":31650},[31652,709962,709963,709977],{},[31655,709964,709965,709967,709969,709971,709973,709975],{},[31658,709966,67764],{},[31662,709968,456],{"stretchy":702},[31749,709970,640],{},[31662,709972,2557],{"stretchy":702},[31662,709974,202],{},[31749,709976,459],{},[31665,709978,709587],{"encoding":31667},[34,709980,709982,710009],{"className":709981,"ariaHidden":758},[31672],[34,709983,709985,709988,709991,709994,709997,710000,710003,710006],{"className":709984},[31676],[34,709986],{"className":709987,"style":31774},[31680],[34,709989,67764],{"className":709990,"style":32062},[31685,31686],[34,709992,456],{"className":709993},[31782],[34,709995,640],{"className":709996},[31685],[34,709998,2557],{"className":709999},[31690],[34,710001],{"className":710002,"style":31846},[31845],[34,710004,202],{"className":710005},[31850],[34,710007],{"className":710008,"style":31846},[31845],[34,710010,710012,710015],{"className":710011},[31676],[34,710013],{"className":710014,"style":32617},[31680],[34,710016,459],{"className":710017},[31685],"。\n因此，此初值问题的解就是 ",[2627,710020,710021],{},[34,710022,710024,710052],{"className":710023},[31642],[34,710025,710027],{"className":710026},[31646],[31648,710028,710029],{"xmlns":31650},[31652,710030,710031,710049],{},[31655,710032,710033,710035,710037,710039,710041,710043,710045,710047],{},[31658,710034,67764],{},[31662,710036,202],{},[31658,710038,126541],{},[31662,710040,456],{"stretchy":702},[31658,710042,32936],{},[31662,710044,2557],{"stretchy":702},[31662,710046,202],{},[31749,710048,459],{},[31665,710050,710051],{"encoding":31667},"y = \\varphi(x) = 0",[34,710053,710055,710073,710100],{"className":710054,"ariaHidden":758},[31672],[34,710056,710058,710061,710064,710067,710070],{"className":710057},[31676],[34,710059],{"className":710060,"style":38774},[31680],[34,710062,67764],{"className":710063,"style":32062},[31685,31686],[34,710065],{"className":710066,"style":31846},[31845],[34,710068,202],{"className":710069},[31850],[34,710071],{"className":710072,"style":31846},[31845],[34,710074,710076,710079,710082,710085,710088,710091,710094,710097],{"className":710075},[31676],[34,710077],{"className":710078,"style":31774},[31680],[34,710080,126541],{"className":710081},[31685,31686],[34,710083,456],{"className":710084},[31782],[34,710086,32936],{"className":710087},[31685,31686],[34,710089,2557],{"className":710090},[31690],[34,710092],{"className":710093,"style":31846},[31845],[34,710095,202],{"className":710096},[31850],[34,710098],{"className":710099,"style":31846},[31845],[34,710101,710103,710106],{"className":710102},[31676],[34,710104],{"className":710105,"style":32617},[31680],[34,710107,459],{"className":710108},[31685],[71807,710110,710111],{},"(这一步非常关键！如果不知道解是0，后面的积分就没法算。)",[15,710113,710114,710228,710229,710394,710395,711024,711025,711075,711076],{},[2627,710115,704626,710116,710227],{},[34,710117,710119,710144],{"className":710118},[31642],[34,710120,710122],{"className":710121},[31646],[31648,710123,710124],{"xmlns":31650},[31652,710125,710126,710142],{},[31655,710127,710128],{},[41453,710129,710130,710136],{},[31655,710131,710132,710134],{},[31658,710133,619473],{"mathvariant":38079},[31658,710135,31739],{},[31655,710137,710138,710140],{},[31658,710139,619473],{"mathvariant":38079},[31658,710141,67764],{},[31665,710143,701964],{"encoding":31667},[34,710145,710147],{"className":710146,"ariaHidden":758},[31672],[34,710148,710150,710153],{"className":710149},[31676],[34,710151],{"className":710152,"style":695705},[31680],[34,710154,710156,710159,710224],{"className":710155},[31685],[34,710157],{"className":710158},[31782,41488],[34,710160,710162],{"className":710161},[41453],[34,710163,710165,710216],{"className":710164},[31796,31797],[34,710166,710168,710213],{"className":710167},[31801],[34,710169,710171,710188,710196],{"className":710170,"style":608935},[31805],[34,710172,710173,710176],{"style":41504},[34,710174],{"className":710175,"style":41508},[31813],[34,710177,710179],{"className":710178},[31818,31819,31820,31821],[34,710180,710182,710185],{"className":710181},[31685,31821],[34,710183,619473],{"className":710184,"style":112322},[31685,31821],[34,710186,67764],{"className":710187,"style":32062},[31685,31686,31821],[34,710189,710190,710193],{"style":41520},[34,710191],{"className":710192,"style":41508},[31813],[34,710194],{"className":710195,"style":41528},[41527],[34,710197,710198,710201],{"style":548368},[34,710199],{"className":710200,"style":41508},[31813],[34,710202,710204],{"className":710203},[31818,31819,31820,31821],[34,710205,710207,710210],{"className":710206},[31685,31821],[34,710208,619473],{"className":710209,"style":112322},[31685,31821],[34,710211,31739],{"className":710212,"style":31778},[31685,31686,31821],[34,710214,31829],{"className":710215},[31828],[34,710217,710219],{"className":710218},[31801],[34,710220,710222],{"className":710221,"style":220191},[31805],[34,710223],{},[34,710225],{"className":710226},[31690,41488]," 并代入特解","\n由 ",[34,710230,710232,710272],{"className":710231},[31642],[34,710233,710235],{"className":710234},[31646],[31648,710236,710237],{"xmlns":31650},[31652,710238,710239,710269],{},[31655,710240,710241,710243,710245,710247,710249,710251,710253,710255,710257,710259,710261,710267],{},[31658,710242,31739],{},[31662,710244,456],{"stretchy":702},[31658,710246,32936],{},[31662,710248,31960],{"separator":758},[31658,710250,67764],{},[31662,710252,2557],{"stretchy":702},[31662,710254,202],{},[31658,710256,547853],{},[31662,710258,32682],{},[31662,710260,456],{"stretchy":702},[41453,710262,710263,710265],{},[31658,710264,67764],{},[31658,710266,32936],{},[31662,710268,2557],{"stretchy":702},[31665,710270,710271],{"encoding":31667},"f(x,y) = \\sin(\\frac{y}{x})",[34,710273,710275,710311],{"className":710274,"ariaHidden":758},[31672],[34,710276,710278,710281,710284,710287,710290,710293,710296,710299,710302,710305,710308],{"className":710277},[31676],[34,710279],{"className":710280,"style":31774},[31680],[34,710282,31739],{"className":710283,"style":31778},[31685,31686],[34,710285,456],{"className":710286},[31782],[34,710288,32936],{"className":710289},[31685,31686],[34,710291,31960],{"className":710292},[32119],[34,710294],{"className":710295,"style":32123},[31845],[34,710297,67764],{"className":710298,"style":32062},[31685,31686],[34,710300,2557],{"className":710301},[31690],[34,710303],{"className":710304,"style":31846},[31845],[34,710306,202],{"className":710307},[31850],[34,710309],{"className":710310,"style":31846},[31845],[34,710312,710314,710317,710320,710323,710391],{"className":710313},[31676],[34,710315],{"className":710316,"style":631146},[31680],[34,710318,547853],{"className":710319},[32767],[34,710321,456],{"className":710322},[31782],[34,710324,710326,710329,710388],{"className":710325},[31685],[34,710327],{"className":710328},[31782,41488],[34,710330,710332],{"className":710331},[41453],[34,710333,710335,710380],{"className":710334},[31796,31797],[34,710336,710338,710377],{"className":710337},[31801],[34,710339,710341,710355,710363],{"className":710340,"style":548284},[31805],[34,710342,710343,710346],{"style":41504},[34,710344],{"className":710345,"style":41508},[31813],[34,710347,710349],{"className":710348},[31818,31819,31820,31821],[34,710350,710352],{"className":710351},[31685,31821],[34,710353,32936],{"className":710354},[31685,31686,31821],[34,710356,710357,710360],{"style":41520},[34,710358],{"className":710359,"style":41508},[31813],[34,710361],{"className":710362,"style":41528},[41527],[34,710364,710365,710368],{"style":548368},[34,710366],{"className":710367,"style":41508},[31813],[34,710369,710371],{"className":710370},[31818,31819,31820,31821],[34,710372,710374],{"className":710373},[31685,31821],[34,710375,67764],{"className":710376,"style":32062},[31685,31686,31821],[34,710378,31829],{"className":710379},[31828],[34,710381,710383],{"className":710382},[31801],[34,710384,710386],{"className":710385,"style":41619},[31805],[34,710387],{},[34,710389],{"className":710390},[31690,41488],[34,710392,2557],{"className":710393},[31690],"，根据链式法则求导：\n",[34,710396,710398,710490],{"className":710397},[31642],[34,710399,710401],{"className":710400},[31646],[31648,710402,710403],{"xmlns":31650},[31652,710404,710405,710487],{},[31655,710406,710407,710421,710423,710425,710427,710439,710441,710451,710463,710465,710471,710473,710475],{},[41453,710408,710409,710415],{},[31655,710410,710411,710413],{},[31658,710412,619473],{"mathvariant":38079},[31658,710414,31739],{},[31655,710416,710417,710419],{},[31658,710418,619473],{"mathvariant":38079},[31658,710420,67764],{},[31662,710422,202],{},[31658,710424,547860],{},[31662,710426,32682],{},[31655,710428,710429,710431,710437],{},[31662,710430,456],{"fence":758},[41453,710432,710433,710435],{},[31658,710434,67764],{},[31658,710436,32936],{},[31662,710438,2557],{"fence":758},[31662,710440,71191],{},[41453,710442,710443,710445],{},[31658,710444,619473],{"mathvariant":38079},[31655,710446,710447,710449],{},[31658,710448,619473],{"mathvariant":38079},[31658,710450,67764],{},[31655,710452,710453,710455,710461],{},[31662,710454,456],{"fence":758},[41453,710456,710457,710459],{},[31658,710458,67764],{},[31658,710460,32936],{},[31662,710462,2557],{"fence":758},[31662,710464,202],{},[41453,710466,710467,710469],{},[31749,710468,640],{},[31658,710470,32936],{},[31658,710472,547860],{},[31662,710474,32682],{},[31655,710476,710477,710479,710485],{},[31662,710478,456],{"fence":758},[41453,710480,710481,710483],{},[31658,710482,67764],{},[31658,710484,32936],{},[31662,710486,2557],{"fence":758},[31665,710488,710489],{"encoding":31667},"\\frac{\\partial f}{\\partial y} = \\cos\\left(\\frac{y}{x}\\right) \\cdot \\frac{\\partial}{\\partial y}\\left(\\frac{y}{x}\\right) = \\frac{1}{x} \\cos\\left(\\frac{y}{x}\\right)",[34,710491,710493,710582,710686,710858],{"className":710492,"ariaHidden":758},[31672],[34,710494,710496,710499,710573,710576,710579],{"className":710495},[31676],[34,710497],{"className":710498,"style":695705},[31680],[34,710500,710502,710505,710570],{"className":710501},[31685],[34,710503],{"className":710504},[31782,41488],[34,710506,710508],{"className":710507},[41453],[34,710509,710511,710562],{"className":710510},[31796,31797],[34,710512,710514,710559],{"className":710513},[31801],[34,710515,710517,710534,710542],{"className":710516,"style":608935},[31805],[34,710518,710519,710522],{"style":41504},[34,710520],{"className":710521,"style":41508},[31813],[34,710523,710525],{"className":710524},[31818,31819,31820,31821],[34,710526,710528,710531],{"className":710527},[31685,31821],[34,710529,619473],{"className":710530,"style":112322},[31685,31821],[34,710532,67764],{"className":710533,"style":32062},[31685,31686,31821],[34,710535,710536,710539],{"style":41520},[34,710537],{"className":710538,"style":41508},[31813],[34,710540],{"className":710541,"style":41528},[41527],[34,710543,710544,710547],{"style":548368},[34,710545],{"className":710546,"style":41508},[31813],[34,710548,710550],{"className":710549},[31818,31819,31820,31821],[34,710551,710553,710556],{"className":710552},[31685,31821],[34,710554,619473],{"className":710555,"style":112322},[31685,31821],[34,710557,31739],{"className":710558,"style":31778},[31685,31686,31821],[34,710560,31829],{"className":710561},[31828],[34,710563,710565],{"className":710564},[31801],[34,710566,710568],{"className":710567,"style":220191},[31805],[34,710569],{},[34,710571],{"className":710572},[31690,41488],[34,710574],{"className":710575,"style":31846},[31845],[34,710577,202],{"className":710578},[31850],[34,710580],{"className":710581,"style":31846},[31845],[34,710583,710585,710588,710591,710594,710677,710680,710683],{"className":710584},[31676],[34,710586],{"className":710587,"style":345868},[31680],[34,710589,547860],{"className":710590},[32767],[34,710592],{"className":710593,"style":32123},[31845],[34,710595,710597,710603,710671],{"className":710596},[34006],[34,710598,710600],{"className":710599,"style":75811},[31782,75810],[34,710601,456],{"className":710602},[71281,43411],[34,710604,710606,710609,710668],{"className":710605},[31685],[34,710607],{"className":710608},[31782,41488],[34,710610,710612],{"className":710611},[41453],[34,710613,710615,710660],{"className":710614},[31796,31797],[34,710616,710618,710657],{"className":710617},[31801],[34,710619,710621,710635,710643],{"className":710620,"style":548284},[31805],[34,710622,710623,710626],{"style":41504},[34,710624],{"className":710625,"style":41508},[31813],[34,710627,710629],{"className":710628},[31818,31819,31820,31821],[34,710630,710632],{"className":710631},[31685,31821],[34,710633,32936],{"className":710634},[31685,31686,31821],[34,710636,710637,710640],{"style":41520},[34,710638],{"className":710639,"style":41508},[31813],[34,710641],{"className":710642,"style":41528},[41527],[34,710644,710645,710648],{"style":548368},[34,710646],{"className":710647,"style":41508},[31813],[34,710649,710651],{"className":710650},[31818,31819,31820,31821],[34,710652,710654],{"className":710653},[31685,31821],[34,710655,67764],{"className":710656,"style":32062},[31685,31686,31821],[34,710658,31829],{"className":710659},[31828],[34,710661,710663],{"className":710662},[31801],[34,710664,710666],{"className":710665,"style":41619},[31805],[34,710667],{},[34,710669],{"className":710670},[31690,41488],[34,710672,710674],{"className":710673,"style":75811},[31690,75810],[34,710675,2557],{"className":710676},[71281,43411],[34,710678],{"className":710679,"style":32604},[31845],[34,710681,71191],{"className":710682},[32024],[34,710684],{"className":710685,"style":32604},[31845],[34,710687,710689,710692,710763,710766,710849,710852,710855],{"className":710688},[31676],[34,710690],{"className":710691,"style":685759},[31680],[34,710693,710695,710698,710760],{"className":710694},[31685],[34,710696],{"className":710697},[31782,41488],[34,710699,710701],{"className":710700},[41453],[34,710702,710704,710752],{"className":710703},[31796,31797],[34,710705,710707,710749],{"className":710706},[31801],[34,710708,710710,710727,710735],{"className":710709,"style":232511},[31805],[34,710711,710712,710715],{"style":41504},[34,710713],{"className":710714,"style":41508},[31813],[34,710716,710718],{"className":710717},[31818,31819,31820,31821],[34,710719,710721,710724],{"className":710720},[31685,31821],[34,710722,619473],{"className":710723,"style":112322},[31685,31821],[34,710725,67764],{"className":710726,"style":32062},[31685,31686,31821],[34,710728,710729,710732],{"style":41520},[34,710730],{"className":710731,"style":41508},[31813],[34,710733],{"className":710734,"style":41528},[41527],[34,710736,710737,710740],{"style":46771},[34,710738],{"className":710739,"style":41508},[31813],[34,710741,710743],{"className":710742},[31818,31819,31820,31821],[34,710744,710746],{"className":710745},[31685,31821],[34,710747,619473],{"className":710748,"style":112322},[31685,31821],[34,710750,31829],{"className":710751},[31828],[34,710753,710755],{"className":710754},[31801],[34,710756,710758],{"className":710757,"style":220191},[31805],[34,710759],{},[34,710761],{"className":710762},[31690,41488],[34,710764],{"className":710765,"style":32123},[31845],[34,710767,710769,710775,710843],{"className":710768},[34006],[34,710770,710772],{"className":710771,"style":75811},[31782,75810],[34,710773,456],{"className":710774},[71281,43411],[34,710776,710778,710781,710840],{"className":710777},[31685],[34,710779],{"className":710780},[31782,41488],[34,710782,710784],{"className":710783},[41453],[34,710785,710787,710832],{"className":710786},[31796,31797],[34,710788,710790,710829],{"className":710789},[31801],[34,710791,710793,710807,710815],{"className":710792,"style":548284},[31805],[34,710794,710795,710798],{"style":41504},[34,710796],{"className":710797,"style":41508},[31813],[34,710799,710801],{"className":710800},[31818,31819,31820,31821],[34,710802,710804],{"className":710803},[31685,31821],[34,710805,32936],{"className":710806},[31685,31686,31821],[34,710808,710809,710812],{"style":41520},[34,710810],{"className":710811,"style":41508},[31813],[34,710813],{"className":710814,"style":41528},[41527],[34,710816,710817,710820],{"style":548368},[34,710818],{"className":710819,"style":41508},[31813],[34,710821,710823],{"className":710822},[31818,31819,31820,31821],[34,710824,710826],{"className":710825},[31685,31821],[34,710827,67764],{"className":710828,"style":32062},[31685,31686,31821],[34,710830,31829],{"className":710831},[31828],[34,710833,710835],{"className":710834},[31801],[34,710836,710838],{"className":710837,"style":41619},[31805],[34,710839],{},[34,710841],{"className":710842},[31690,41488],[34,710844,710846],{"className":710845,"style":75811},[31690,75810],[34,710847,2557],{"className":710848},[71281,43411],[34,710850],{"className":710851,"style":31846},[31845],[34,710853,202],{"className":710854},[31850],[34,710856],{"className":710857,"style":31846},[31845],[34,710859,710861,710864,710932,710935,710938,710941],{"className":710860},[31676],[34,710862],{"className":710863,"style":345868},[31680],[34,710865,710867,710870,710929],{"className":710866},[31685],[34,710868],{"className":710869},[31782,41488],[34,710871,710873],{"className":710872},[41453],[34,710874,710876,710921],{"className":710875},[31796,31797],[34,710877,710879,710918],{"className":710878},[31801],[34,710880,710882,710896,710904],{"className":710881,"style":46746},[31805],[34,710883,710884,710887],{"style":41504},[34,710885],{"className":710886,"style":41508},[31813],[34,710888,710890],{"className":710889},[31818,31819,31820,31821],[34,710891,710893],{"className":710892},[31685,31821],[34,710894,32936],{"className":710895},[31685,31686,31821],[34,710897,710898,710901],{"style":41520},[34,710899],{"className":710900,"style":41508},[31813],[34,710902],{"className":710903,"style":41528},[41527],[34,710905,710906,710909],{"style":46771},[34,710907],{"className":710908,"style":41508},[31813],[34,710910,710912],{"className":710911},[31818,31819,31820,31821],[34,710913,710915],{"className":710914},[31685,31821],[34,710916,640],{"className":710917},[31685,31821],[34,710919,31829],{"className":710920},[31828],[34,710922,710924],{"className":710923},[31801],[34,710925,710927],{"className":710926,"style":41619},[31805],[34,710928],{},[34,710930],{"className":710931},[31690,41488],[34,710933],{"className":710934,"style":32123},[31845],[34,710936,547860],{"className":710937},[32767],[34,710939],{"className":710940,"style":32123},[31845],[34,710942,710944,710950,711018],{"className":710943},[34006],[34,710945,710947],{"className":710946,"style":75811},[31782,75810],[34,710948,456],{"className":710949},[71281,43411],[34,710951,710953,710956,711015],{"className":710952},[31685],[34,710954],{"className":710955},[31782,41488],[34,710957,710959],{"className":710958},[41453],[34,710960,710962,711007],{"className":710961},[31796,31797],[34,710963,710965,711004],{"className":710964},[31801],[34,710966,710968,710982,710990],{"className":710967,"style":548284},[31805],[34,710969,710970,710973],{"style":41504},[34,710971],{"className":710972,"style":41508},[31813],[34,710974,710976],{"className":710975},[31818,31819,31820,31821],[34,710977,710979],{"className":710978},[31685,31821],[34,710980,32936],{"className":710981},[31685,31686,31821],[34,710983,710984,710987],{"style":41520},[34,710985],{"className":710986,"style":41508},[31813],[34,710988],{"className":710989,"style":41528},[41527],[34,710991,710992,710995],{"style":548368},[34,710993],{"className":710994,"style":41508},[31813],[34,710996,710998],{"className":710997},[31818,31819,31820,31821],[34,710999,711001],{"className":711000},[31685,31821],[34,711002,67764],{"className":711003,"style":32062},[31685,31686,31821],[34,711005,31829],{"className":711006},[31828],[34,711008,711010],{"className":711009},[31801],[34,711011,711013],{"className":711012,"style":41619},[31805],[34,711014],{},[34,711016],{"className":711017},[31690,41488],[34,711019,711021],{"className":711020,"style":75811},[31690,75810],[34,711022,2557],{"className":711023},[71281,43411],"\n现在，我们将具体的解 ",[34,711026,711028,711045],{"className":711027},[31642],[34,711029,711031],{"className":711030},[31646],[31648,711032,711033],{"xmlns":31650},[31652,711034,711035,711043],{},[31655,711036,711037,711039,711041],{},[31658,711038,67764],{},[31662,711040,202],{},[31749,711042,459],{},[31665,711044,664526],{"encoding":31667},[34,711046,711048,711066],{"className":711047,"ariaHidden":758},[31672],[34,711049,711051,711054,711057,711060,711063],{"className":711050},[31676],[34,711052],{"className":711053,"style":38774},[31680],[34,711055,67764],{"className":711056,"style":32062},[31685,31686],[34,711058],{"className":711059,"style":31846},[31845],[34,711061,202],{"className":711062},[31850],[34,711064],{"className":711065,"style":31846},[31845],[34,711067,711069,711072],{"className":711068},[31676],[34,711070],{"className":711071,"style":32617},[31680],[34,711073,459],{"className":711074},[31685]," 代入上式：\n",[34,711077,711079,711157],{"className":711078},[31642],[34,711080,711082],{"className":711081},[31646],[31648,711083,711084],{"xmlns":31650},[31652,711085,711086,711154],{},[31655,711087,711088,711116,711118,711124,711126,711128,711130,711132,711134,711136,711142,711144,711146,711148],{},[31743,711089,711090,711108],{},[31655,711091,711092,711106],{},[41453,711093,711094,711100],{},[31655,711095,711096,711098],{},[31658,711097,619473],{"mathvariant":38079},[31658,711099,31739],{},[31655,711101,711102,711104],{},[31658,711103,619473],{"mathvariant":38079},[31658,711105,67764],{},[31662,711107,162086],{"fence":758},[31655,711109,711110,711112,711114],{},[31658,711111,67764],{},[31662,711113,202],{},[31749,711115,459],{},[31662,711117,202],{},[41453,711119,711120,711122],{},[31749,711121,640],{},[31658,711123,32936],{},[31658,711125,547860],{},[31662,711127,32682],{},[31662,711129,456],{"stretchy":702},[31749,711131,459],{},[31662,711133,2557],{"stretchy":702},[31662,711135,202],{},[41453,711137,711138,711140],{},[31749,711139,640],{},[31658,711141,32936],{},[31662,711143,71191],{},[31749,711145,640],{},[31662,711147,202],{},[41453,711149,711150,711152],{},[31749,711151,640],{},[31658,711153,32936],{},[31665,711155,711156],{"encoding":31667},"\\left. \\frac{\\partial f}{\\partial y} \\right|_{y=0} = \\frac{1}{x} \\cos(0) = \\frac{1}{x} \\cdot 1 = \\frac{1}{x}",[34,711158,711160,711341,711439,711522,711540],{"className":711159,"ariaHidden":758},[31672],[34,711161,711163,711167,711332,711335,711338],{"className":711162},[31676],[34,711164],{"className":711165,"style":711166},[31680],"height:1.9858em;vertical-align:-0.8358em;",[34,711168,711170,711287],{"className":711169},[34006],[34,711171,711173,711176,711250],{"className":711172},[34006],[34,711174],{"className":711175},[31782,41488],[34,711177,711179,711182,711247],{"className":711178},[31685],[34,711180],{"className":711181},[31782,41488],[34,711183,711185],{"className":711184},[41453],[34,711186,711188,711239],{"className":711187},[31796,31797],[34,711189,711191,711236],{"className":711190},[31801],[34,711192,711194,711211,711219],{"className":711193,"style":608935},[31805],[34,711195,711196,711199],{"style":41504},[34,711197],{"className":711198,"style":41508},[31813],[34,711200,711202],{"className":711201},[31818,31819,31820,31821],[34,711203,711205,711208],{"className":711204},[31685,31821],[34,711206,619473],{"className":711207,"style":112322},[31685,31821],[34,711209,67764],{"className":711210,"style":32062},[31685,31686,31821],[34,711212,711213,711216],{"style":41520},[34,711214],{"className":711215,"style":41508},[31813],[34,711217],{"className":711218,"style":41528},[41527],[34,711220,711221,711224],{"style":548368},[34,711222],{"className":711223,"style":41508},[31813],[34,711225,711227],{"className":711226},[31818,31819,31820,31821],[34,711228,711230,711233],{"className":711229},[31685,31821],[34,711231,619473],{"className":711232,"style":112322},[31685,31821],[34,711234,31739],{"className":711235,"style":31778},[31685,31686,31821],[34,711237,31829],{"className":711238},[31828],[34,711240,711242],{"className":711241},[31801],[34,711243,711245],{"className":711244,"style":220191},[31805],[34,711246],{},[34,711248],{"className":711249},[31690,41488],[34,711251,711253],{"className":711252},[31690],[34,711254,711256],{"className":711255},[71281,71282],[34,711257,711259,711279],{"className":711258},[31796,31797],[34,711260,711262,711276],{"className":711261},[31801],[34,711263,711265],{"className":711264,"style":550104},[31805],[34,711266,711267,711270],{"style":71329},[34,711268],{"className":711269,"style":550110},[31813],[34,711271,711272],{"style":550113},[41588,711273,711274],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,711275],{"d":550120},[34,711277,31829],{"className":711278},[31828],[34,711280,711282],{"className":711281},[31801],[34,711283,711285],{"className":711284,"style":550130},[31805],[34,711286],{},[34,711288,711290],{"className":711289},[31792],[34,711291,711293,711323],{"className":711292},[31796,31797],[34,711294,711296,711320],{"className":711295},[31801],[34,711297,711300],{"className":711298,"style":711299},[31805],"height:-0.2486em;",[34,711301,711302,711305],{"style":575022},[34,711303],{"className":711304,"style":31814},[31813],[34,711306,711308],{"className":711307},[31818,31819,31820,31821],[34,711309,711311,711314,711317],{"className":711310},[31685,31821],[34,711312,67764],{"className":711313,"style":32062},[31685,31686,31821],[34,711315,202],{"className":711316},[31850,31821],[34,711318,459],{"className":711319},[31685,31821],[34,711321,31829],{"className":711322},[31828],[34,711324,711326],{"className":711325},[31801],[34,711327,711330],{"className":711328,"style":711329},[31805],"height:0.8358em;",[34,711331],{},[34,711333],{"className":711334,"style":31846},[31845],[34,711336,202],{"className":711337},[31850],[34,711339],{"className":711340,"style":31846},[31845],[34,711342,711344,711347,711415,711418,711421,711424,711427,711430,711433,711436],{"className":711343},[31676],[34,711345],{"className":711346,"style":46727},[31680],[34,711348,711350,711353,711412],{"className":711349},[31685],[34,711351],{"className":711352},[31782,41488],[34,711354,711356],{"className":711355},[41453],[34,711357,711359,711404],{"className":711358},[31796,31797],[34,711360,711362,711401],{"className":711361},[31801],[34,711363,711365,711379,711387],{"className":711364,"style":46746},[31805],[34,711366,711367,711370],{"style":41504},[34,711368],{"className":711369,"style":41508},[31813],[34,711371,711373],{"className":711372},[31818,31819,31820,31821],[34,711374,711376],{"className":711375},[31685,31821],[34,711377,32936],{"className":711378},[31685,31686,31821],[34,711380,711381,711384],{"style":41520},[34,711382],{"className":711383,"style":41508},[31813],[34,711385],{"className":711386,"style":41528},[41527],[34,711388,711389,711392],{"style":46771},[34,711390],{"className":711391,"style":41508},[31813],[34,711393,711395],{"className":711394},[31818,31819,31820,31821],[34,711396,711398],{"className":711397},[31685,31821],[34,711399,640],{"className":711400},[31685,31821],[34,711402,31829],{"className":711403},[31828],[34,711405,711407],{"className":711406},[31801],[34,711408,711410],{"className":711409,"style":41619},[31805],[34,711411],{},[34,711413],{"className":711414},[31690,41488],[34,711416],{"className":711417,"style":32123},[31845],[34,711419,547860],{"className":711420},[32767],[34,711422,456],{"className":711423},[31782],[34,711425,459],{"className":711426},[31685],[34,711428,2557],{"className":711429},[31690],[34,711431],{"className":711432,"style":31846},[31845],[34,711434,202],{"className":711435},[31850],[34,711437],{"className":711438,"style":31846},[31845],[34,711440,711442,711445,711513,711516,711519],{"className":711441},[31676],[34,711443],{"className":711444,"style":46727},[31680],[34,711446,711448,711451,711510],{"className":711447},[31685],[34,711449],{"className":711450},[31782,41488],[34,711452,711454],{"className":711453},[41453],[34,711455,711457,711502],{"className":711456},[31796,31797],[34,711458,711460,711499],{"className":711459},[31801],[34,711461,711463,711477,711485],{"className":711462,"style":46746},[31805],[34,711464,711465,711468],{"style":41504},[34,711466],{"className":711467,"style":41508},[31813],[34,711469,711471],{"className":711470},[31818,31819,31820,31821],[34,711472,711474],{"className":711473},[31685,31821],[34,711475,32936],{"className":711476},[31685,31686,31821],[34,711478,711479,711482],{"style":41520},[34,711480],{"className":711481,"style":41508},[31813],[34,711483],{"className":711484,"style":41528},[41527],[34,711486,711487,711490],{"style":46771},[34,711488],{"className":711489,"style":41508},[31813],[34,711491,711493],{"className":711492},[31818,31819,31820,31821],[34,711494,711496],{"className":711495},[31685,31821],[34,711497,640],{"className":711498},[31685,31821],[34,711500,31829],{"className":711501},[31828],[34,711503,711505],{"className":711504},[31801],[34,711506,711508],{"className":711507,"style":41619},[31805],[34,711509],{},[34,711511],{"className":711512},[31690,41488],[34,711514],{"className":711515,"style":32604},[31845],[34,711517,71191],{"className":711518},[32024],[34,711520],{"className":711521,"style":32604},[31845],[34,711523,711525,711528,711531,711534,711537],{"className":711524},[31676],[34,711526],{"className":711527,"style":32617},[31680],[34,711529,640],{"className":711530},[31685],[34,711532],{"className":711533,"style":31846},[31845],[34,711535,202],{"className":711536},[31850],[34,711538],{"className":711539,"style":31846},[31845],[34,711541,711543,711546],{"className":711542},[31676],[34,711544],{"className":711545,"style":46727},[31680],[34,711547,711549,711552,711611],{"className":711548},[31685],[34,711550],{"className":711551},[31782,41488],[34,711553,711555],{"className":711554},[41453],[34,711556,711558,711603],{"className":711557},[31796,31797],[34,711559,711561,711600],{"className":711560},[31801],[34,711562,711564,711578,711586],{"className":711563,"style":46746},[31805],[34,711565,711566,711569],{"style":41504},[34,711567],{"className":711568,"style":41508},[31813],[34,711570,711572],{"className":711571},[31818,31819,31820,31821],[34,711573,711575],{"className":711574},[31685,31821],[34,711576,32936],{"className":711577},[31685,31686,31821],[34,711579,711580,711583],{"style":41520},[34,711581],{"className":711582,"style":41508},[31813],[34,711584],{"className":711585,"style":41528},[41527],[34,711587,711588,711591],{"style":46771},[34,711589],{"className":711590,"style":41508},[31813],[34,711592,711594],{"className":711593},[31818,31819,31820,31821],[34,711595,711597],{"className":711596},[31685,31821],[34,711598,640],{"className":711599},[31685,31821],[34,711601,31829],{"className":711602},[31828],[34,711604,711606],{"className":711605},[31801],[34,711607,711609],{"className":711608,"style":41619},[31805],[34,711610],{},[34,711612],{"className":711613},[31690,41488],[15,711615,711616,711770,711771,33183,711863,712227,712228,712326,712327,349407,712801,713025,47649,713027],{},[2627,711617,705811,711618],{},[34,711619,711621,711650],{"className":711620},[31642],[34,711622,711624],{"className":711623},[31646],[31648,711625,711626],{"xmlns":31650},[31652,711627,711628,711648],{},[31655,711629,711630],{},[41453,711631,711632,711638],{},[31655,711633,711634,711636],{},[31658,711635,619473],{"mathvariant":38079},[31658,711637,67764],{},[31655,711639,711640,711642],{},[31658,711641,619473],{"mathvariant":38079},[31743,711643,711644,711646],{},[31658,711645,67764],{},[31749,711647,459],{},[31665,711649,708779],{"encoding":31667},[34,711651,711653],{"className":711652,"ariaHidden":758},[31672],[34,711654,711656,711659],{"className":711655},[31676],[34,711657],{"className":711658,"style":695705},[31680],[34,711660,711662,711665,711767],{"className":711661},[31685],[34,711663],{"className":711664},[31782,41488],[34,711666,711668],{"className":711667},[41453],[34,711669,711671,711759],{"className":711670},[31796,31797],[34,711672,711674,711756],{"className":711673},[31801],[34,711675,711677,711731,711739],{"className":711676,"style":608935},[31805],[34,711678,711679,711682],{"style":41504},[34,711680],{"className":711681,"style":41508},[31813],[34,711683,711685],{"className":711684},[31818,31819,31820,31821],[34,711686,711688,711691],{"className":711687},[31685,31821],[34,711689,619473],{"className":711690,"style":112322},[31685,31821],[34,711692,711694,711697],{"className":711693},[31685,31821],[34,711695,67764],{"className":711696,"style":32062},[31685,31686,31821],[34,711698,711700],{"className":711699},[31792],[34,711701,711703,711723],{"className":711702},[31796,31797],[34,711704,711706,711720],{"className":711705},[31801],[34,711707,711709],{"className":711708,"style":318658},[31805],[34,711710,711711,711714],{"style":561507},[34,711712],{"className":711713,"style":43406},[31813],[34,711715,711717],{"className":711716},[31818,43410,43411,31821],[34,711718,459],{"className":711719},[31685,31821],[34,711721,31829],{"className":711722},[31828],[34,711724,711726],{"className":711725},[31801],[34,711727,711729],{"className":711728,"style":220540},[31805],[34,711730],{},[34,711732,711733,711736],{"style":41520},[34,711734],{"className":711735,"style":41508},[31813],[34,711737],{"className":711738,"style":41528},[41527],[34,711740,711741,711744],{"style":548368},[34,711742],{"className":711743,"style":41508},[31813],[34,711745,711747],{"className":711746},[31818,31819,31820,31821],[34,711748,711750,711753],{"className":711749},[31685,31821],[34,711751,619473],{"className":711752,"style":112322},[31685,31821],[34,711754,67764],{"className":711755,"style":32062},[31685,31686,31821],[34,711757,31829],{"className":711758},[31828],[34,711760,711762],{"className":711761},[31801],[34,711763,711765],{"className":711764,"style":220191},[31805],[34,711766],{},[34,711768],{"className":711769},[31690,41488],"\n利用公式 (3.18)，注意下限 ",[34,711772,711774,711796],{"className":711773},[31642],[34,711775,711777],{"className":711776},[31646],[31648,711778,711779],{"xmlns":31650},[31652,711780,711781,711793],{},[31655,711782,711783,711789,711791],{},[31743,711784,711785,711787],{},[31658,711786,32936],{},[31749,711788,459],{},[31662,711790,202],{},[31749,711792,640],{},[31665,711794,711795],{"encoding":31667},"x_0 = 1",[34,711797,711799,711854],{"className":711798,"ariaHidden":758},[31672],[34,711800,711802,711805,711845,711848,711851],{"className":711801},[31676],[34,711803],{"className":711804,"style":31860},[31680],[34,711806,711808,711811],{"className":711807},[31685],[34,711809,32936],{"className":711810},[31685,31686],[34,711812,711814],{"className":711813},[31792],[34,711815,711817,711837],{"className":711816},[31796,31797],[34,711818,711820,711834],{"className":711819},[31801],[34,711821,711823],{"className":711822,"style":31806},[31805],[34,711824,711825,711828],{"style":31809},[34,711826],{"className":711827,"style":31814},[31813],[34,711829,711831],{"className":711830},[31818,31819,31820,31821],[34,711832,459],{"className":711833},[31685,31821],[34,711835,31829],{"className":711836},[31828],[34,711838,711840],{"className":711839},[31801],[34,711841,711843],{"className":711842,"style":31836},[31805],[34,711844],{},[34,711846],{"className":711847,"style":31846},[31845],[34,711849,202],{"className":711850},[31850],[34,711852],{"className":711853,"style":31846},[31845],[34,711855,711857,711860],{"className":711856},[31676],[34,711858],{"className":711859,"style":32617},[31680],[34,711861,640],{"className":711862},[31685],[34,711864,711866,711934],{"className":711865},[31642],[34,711867,711869],{"className":711868},[31646],[31648,711870,711871],{"xmlns":31650},[31652,711872,711873,711931],{},[31655,711874,711875,711893,711895,711897,711899],{},[41453,711876,711877,711883],{},[31655,711878,711879,711881],{},[31658,711880,619473],{"mathvariant":38079},[31658,711882,67764],{},[31655,711884,711885,711887],{},[31658,711886,619473],{"mathvariant":38079},[31743,711888,711889,711891],{},[31658,711890,67764],{},[31749,711892,459],{},[31662,711894,202],{},[31658,711896,701520],{},[31662,711898,32682],{},[31655,711900,711901,711903,711911,711925,711927,711929],{},[31662,711902,456],{"fence":758},[38846,711904,711905,711907,711909],{},[31662,711906,546295],{},[31749,711908,640],{},[31658,711910,32936],{},[41453,711912,711913,711919],{},[31655,711914,711915,711917],{},[31658,711916,619473],{"mathvariant":38079},[31658,711918,31739],{},[31655,711920,711921,711923],{},[31658,711922,619473],{"mathvariant":38079},[31658,711924,67764],{},[31658,711926,46900],{},[31658,711928,33249],{},[31662,711930,2557],{"fence":758},[31665,711932,711933],{"encoding":31667},"\\frac{\\partial y}{\\partial y_0} = \\exp\\left( \\int_{1}^x \\frac{\\partial f}{\\partial y} dt \\right)",[34,711935,711937,712063],{"className":711936,"ariaHidden":758},[31672],[34,711938,711940,711943,712054,712057,712060],{"className":711939},[31676],[34,711941],{"className":711942,"style":695705},[31680],[34,711944,711946,711949,712051],{"className":711945},[31685],[34,711947],{"className":711948},[31782,41488],[34,711950,711952],{"className":711951},[41453],[34,711953,711955,712043],{"className":711954},[31796,31797],[34,711956,711958,712040],{"className":711957},[31801],[34,711959,711961,712015,712023],{"className":711960,"style":608935},[31805],[34,711962,711963,711966],{"style":41504},[34,711964],{"className":711965,"style":41508},[31813],[34,711967,711969],{"className":711968},[31818,31819,31820,31821],[34,711970,711972,711975],{"className":711971},[31685,31821],[34,711973,619473],{"className":711974,"style":112322},[31685,31821],[34,711976,711978,711981],{"className":711977},[31685,31821],[34,711979,67764],{"className":711980,"style":32062},[31685,31686,31821],[34,711982,711984],{"className":711983},[31792],[34,711985,711987,712007],{"className":711986},[31796,31797],[34,711988,711990,712004],{"className":711989},[31801],[34,711991,711993],{"className":711992,"style":318658},[31805],[34,711994,711995,711998],{"style":561507},[34,711996],{"className":711997,"style":43406},[31813],[34,711999,712001],{"className":712000},[31818,43410,43411,31821],[34,712002,459],{"className":712003},[31685,31821],[34,712005,31829],{"className":712006},[31828],[34,712008,712010],{"className":712009},[31801],[34,712011,712013],{"className":712012,"style":220540},[31805],[34,712014],{},[34,712016,712017,712020],{"style":41520},[34,712018],{"className":712019,"style":41508},[31813],[34,712021],{"className":712022,"style":41528},[41527],[34,712024,712025,712028],{"style":548368},[34,712026],{"className":712027,"style":41508},[31813],[34,712029,712031],{"className":712030},[31818,31819,31820,31821],[34,712032,712034,712037],{"className":712033},[31685,31821],[34,712035,619473],{"className":712036,"style":112322},[31685,31821],[34,712038,67764],{"className":712039,"style":32062},[31685,31686,31821],[34,712041,31829],{"className":712042},[31828],[34,712044,712046],{"className":712045},[31801],[34,712047,712049],{"className":712048,"style":220191},[31805],[34,712050],{},[34,712052],{"className":712053},[31690,41488],[34,712055],{"className":712056,"style":31846},[31845],[34,712058,202],{"className":712059},[31850],[34,712061],{"className":712062,"style":31846},[31845],[34,712064,712066,712069,712072,712075],{"className":712065},[31676],[34,712067],{"className":712068,"style":141282},[31680],[34,712070,701520],{"className":712071},[32767],[34,712073],{"className":712074,"style":32123},[31845],[34,712076,712078,712084,712138,712141,712215,712218,712221],{"className":712077},[34006],[34,712079,712081],{"className":712080,"style":75811},[31782,75810],[34,712082,456],{"className":712083},[71281,141292],[34,712085,712087,712090],{"className":712086},[32767],[34,712088,546295],{"className":712089,"style":546331},[32767,131249,156560],[34,712091,712093],{"className":712092},[31792],[34,712094,712096,712130],{"className":712095},[31796,31797],[34,712097,712099,712127],{"className":712098},[31801],[34,712100,712102,712116],{"className":712101,"style":567928},[31805],[34,712103,712104,712107],{"style":546347},[34,712105],{"className":712106,"style":31814},[31813],[34,712108,712110],{"className":712109},[31818,31819,31820,31821],[34,712111,712113],{"className":712112},[31685,31821],[34,712114,640],{"className":712115},[31685,31821],[34,712117,712118,712121],{"style":546362},[34,712119],{"className":712120,"style":31814},[31813],[34,712122,712124],{"className":712123},[31818,31819,31820,31821],[34,712125,32936],{"className":712126},[31685,31686,31821],[34,712128,31829],{"className":712129},[31828],[34,712131,712133],{"className":712132},[31801],[34,712134,712136],{"className":712135,"style":546384},[31805],[34,712137],{},[34,712139],{"className":712140,"style":32123},[31845],[34,712142,712144,712147,712212],{"className":712143},[31685],[34,712145],{"className":712146},[31782,41488],[34,712148,712150],{"className":712149},[41453],[34,712151,712153,712204],{"className":712152},[31796,31797],[34,712154,712156,712201],{"className":712155},[31801],[34,712157,712159,712176,712184],{"className":712158,"style":608935},[31805],[34,712160,712161,712164],{"style":41504},[34,712162],{"className":712163,"style":41508},[31813],[34,712165,712167],{"className":712166},[31818,31819,31820,31821],[34,712168,712170,712173],{"className":712169},[31685,31821],[34,712171,619473],{"className":712172,"style":112322},[31685,31821],[34,712174,67764],{"className":712175,"style":32062},[31685,31686,31821],[34,712177,712178,712181],{"style":41520},[34,712179],{"className":712180,"style":41508},[31813],[34,712182],{"className":712183,"style":41528},[41527],[34,712185,712186,712189],{"style":548368},[34,712187],{"className":712188,"style":41508},[31813],[34,712190,712192],{"className":712191},[31818,31819,31820,31821],[34,712193,712195,712198],{"className":712194},[31685,31821],[34,712196,619473],{"className":712197,"style":112322},[31685,31821],[34,712199,31739],{"className":712200,"style":31778},[31685,31686,31821],[34,712202,31829],{"className":712203},[31828],[34,712205,712207],{"className":712206},[31801],[34,712208,712210],{"className":712209,"style":220191},[31805],[34,712211],{},[34,712213],{"className":712214},[31690,41488],[34,712216,46900],{"className":712217},[31685,31686],[34,712219,33249],{"className":712220},[31685,31686],[34,712222,712224],{"className":712223,"style":75811},[31690,75810],[34,712225,2557],{"className":712226},[71281,141292],"\n将 Step 2 算出的 ",[34,712229,712231,712249],{"className":712230},[31642],[34,712232,712234],{"className":712233},[31646],[31648,712235,712236],{"xmlns":31650},[31652,712237,712238,712246],{},[31655,712239,712240],{},[41453,712241,712242,712244],{},[31749,712243,640],{},[31658,712245,33249],{},[31665,712247,712248],{"encoding":31667},"\\frac{1}{t}",[34,712250,712252],{"className":712251,"ariaHidden":758},[31672],[34,712253,712255,712258],{"className":712254},[31676],[34,712256],{"className":712257,"style":46727},[31680],[34,712259,712261,712264,712323],{"className":712260},[31685],[34,712262],{"className":712263},[31782,41488],[34,712265,712267],{"className":712266},[41453],[34,712268,712270,712315],{"className":712269},[31796,31797],[34,712271,712273,712312],{"className":712272},[31801],[34,712274,712276,712290,712298],{"className":712275,"style":46746},[31805],[34,712277,712278,712281],{"style":41504},[34,712279],{"className":712280,"style":41508},[31813],[34,712282,712284],{"className":712283},[31818,31819,31820,31821],[34,712285,712287],{"className":712286},[31685,31821],[34,712288,33249],{"className":712289},[31685,31686,31821],[34,712291,712292,712295],{"style":41520},[34,712293],{"className":712294,"style":41508},[31813],[34,712296],{"className":712297,"style":41528},[41527],[34,712299,712300,712303],{"style":46771},[34,712301],{"className":712302,"style":41508},[31813],[34,712304,712306],{"className":712305},[31818,31819,31820,31821],[34,712307,712309],{"className":712308},[31685,31821],[34,712310,640],{"className":712311},[31685,31821],[34,712313,31829],{"className":712314},[31828],[34,712316,712318],{"className":712317},[31801],[34,712319,712321],{"className":712320,"style":41619},[31805],[34,712322],{},[34,712324],{"className":712325},[31690,41488]," 代入积分：\n",[34,712328,712330,712423],{"className":712329},[31642],[34,712331,712333],{"className":712332},[31646],[31648,712334,712335],{"xmlns":31650},[31652,712336,712337,712420],{},[31655,712338,712339,712347,712353,712355,712357,712359,712361,712363,712365,712367,712369,712377,712379,712381,712383,712385,712387,712389,712391,712393,712395,712397,712399,712401,712403,712405,712407,712409,712412,712414,712416,712418],{},[38846,712340,712341,712343,712345],{},[31662,712342,546295],{},[31749,712344,640],{},[31658,712346,32936],{},[41453,712348,712349,712351],{},[31749,712350,640],{},[31658,712352,33249],{},[31658,712354,46900],{},[31658,712356,33249],{},[31662,712358,202],{},[31658,712360,563286],{},[31662,712362,32682],{},[31658,712364,162086],{"mathvariant":38079},[31658,712366,33249],{},[31658,712368,162086],{"mathvariant":38079},[38846,712370,712371,712373,712375],{},[31662,712372,162086],{"fence":702,"stretchy":758,"minsize":550116,"maxsize":550116},[31749,712374,640],{},[31658,712376,32936],{},[31662,712378,202],{},[31658,712380,563286],{},[31662,712382,32682],{},[31658,712384,162086],{"mathvariant":38079},[31658,712386,32936],{},[31658,712388,162086],{"mathvariant":38079},[31662,712390,32226],{},[31658,712392,563286],{},[31662,712394,32682],{},[31749,712396,640],{},[31662,712398,202],{},[31658,712400,563286],{},[31662,712402,32682],{},[31658,712404,32936],{},[31845,712406],{"width":71082},[31662,712408,456],{"stretchy":702},[39361,712410,712411],{},"考虑 ",[31658,712413,32936],{},[31662,712415,29622],{},[31749,712417,459],{},[31662,712419,2557],{"stretchy":702},[31665,712421,712422],{"encoding":31667},"\\int_{1}^x \\frac{1}{t} dt = \\ln|t| \\Big|_{1}^x = \\ln|x| - \\ln 1 = \\ln x \\quad (\\text{考虑 } x>0)",[34,712424,712426,712572,712692,712722,712746,712789],{"className":712425,"ariaHidden":758},[31672],[34,712427,712429,712432,712486,712489,712557,712560,712563,712566,712569],{"className":712428},[31676],[34,712430],{"className":712431,"style":613610},[31680],[34,712433,712435,712438],{"className":712434},[32767],[34,712436,546295],{"className":712437,"style":546331},[32767,131249,156560],[34,712439,712441],{"className":712440},[31792],[34,712442,712444,712478],{"className":712443},[31796,31797],[34,712445,712447,712475],{"className":712446},[31801],[34,712448,712450,712464],{"className":712449,"style":567928},[31805],[34,712451,712452,712455],{"style":546347},[34,712453],{"className":712454,"style":31814},[31813],[34,712456,712458],{"className":712457},[31818,31819,31820,31821],[34,712459,712461],{"className":712460},[31685,31821],[34,712462,640],{"className":712463},[31685,31821],[34,712465,712466,712469],{"style":546362},[34,712467],{"className":712468,"style":31814},[31813],[34,712470,712472],{"className":712471},[31818,31819,31820,31821],[34,712473,32936],{"className":712474},[31685,31686,31821],[34,712476,31829],{"className":712477},[31828],[34,712479,712481],{"className":712480},[31801],[34,712482,712484],{"className":712483,"style":546384},[31805],[34,712485],{},[34,712487],{"className":712488,"style":32123},[31845],[34,712490,712492,712495,712554],{"className":712491},[31685],[34,712493],{"className":712494},[31782,41488],[34,712496,712498],{"className":712497},[41453],[34,712499,712501,712546],{"className":712500},[31796,31797],[34,712502,712504,712543],{"className":712503},[31801],[34,712505,712507,712521,712529],{"className":712506,"style":46746},[31805],[34,712508,712509,712512],{"style":41504},[34,712510],{"className":712511,"style":41508},[31813],[34,712513,712515],{"className":712514},[31818,31819,31820,31821],[34,712516,712518],{"className":712517},[31685,31821],[34,712519,33249],{"className":712520},[31685,31686,31821],[34,712522,712523,712526],{"style":41520},[34,712524],{"className":712525,"style":41508},[31813],[34,712527],{"className":712528,"style":41528},[41527],[34,712530,712531,712534],{"style":46771},[34,712532],{"className":712533,"style":41508},[31813],[34,712535,712537],{"className":712536},[31818,31819,31820,31821],[34,712538,712540],{"className":712539},[31685,31821],[34,712541,640],{"className":712542},[31685,31821],[34,712544,31829],{"className":712545},[31828],[34,712547,712549],{"className":712548},[31801],[34,712550,712552],{"className":712551,"style":41619},[31805],[34,712553],{},[34,712555],{"className":712556},[31690,41488],[34,712558,46900],{"className":712559},[31685,31686],[34,712561,33249],{"className":712562},[31685,31686],[34,712564],{"className":712565,"style":31846},[31845],[34,712567,202],{"className":712568},[31850],[34,712570],{"className":712571,"style":31846},[31845],[34,712573,712575,712579,712582,712585,712588,712591,712594,712683,712686,712689],{"className":712574},[31676],[34,712576],{"className":712577,"style":712578},[31680],"height:1.904em;vertical-align:-0.6997em;",[34,712580,563286],{"className":712581},[32767],[34,712583],{"className":712584,"style":32123},[31845],[34,712586,162086],{"className":712587},[31685],[34,712589,33249],{"className":712590},[31685,31686],[34,712592,162086],{"className":712593},[31685],[34,712595,712597,712634],{"className":712596},[31685],[34,712598,712600],{"className":712599},[31685],[34,712601,712603],{"className":712602},[71281,71282],[34,712604,712606,712626],{"className":712605},[31796,31797],[34,712607,712609,712623],{"className":712608},[31801],[34,712610,712612],{"className":712611,"style":550104},[31805],[34,712613,712614,712617],{"style":71329},[34,712615],{"className":712616,"style":550110},[31813],[34,712618,712619],{"style":550113},[41588,712620,712621],{"xmlns":41590,"width":503054,"height":550116,"viewBox":550117},[20398,712622],{"d":550120},[34,712624,31829],{"className":712625},[31828],[34,712627,712629],{"className":712628},[31801],[34,712630,712632],{"className":712631,"style":550130},[31805],[34,712633],{},[34,712635,712637],{"className":712636},[31792],[34,712638,712640,712675],{"className":712639},[31796,31797],[34,712641,712643,712672],{"className":712642},[31801],[34,712644,712647,712661],{"className":712645,"style":712646},[31805],"height:1.2043em;",[34,712648,712649,712652],{"style":575022},[34,712650],{"className":712651,"style":31814},[31813],[34,712653,712655],{"className":712654},[31818,31819,31820,31821],[34,712656,712658],{"className":712657},[31685,31821],[34,712659,640],{"className":712660},[31685,31821],[34,712662,712663,712666],{"style":575043},[34,712664],{"className":712665,"style":31814},[31813],[34,712667,712669],{"className":712668},[31818,31819,31820,31821],[34,712670,32936],{"className":712671},[31685,31686,31821],[34,712673,31829],{"className":712674},[31828],[34,712676,712678],{"className":712677},[31801],[34,712679,712681],{"className":712680,"style":575074},[31805],[34,712682],{},[34,712684],{"className":712685,"style":31846},[31845],[34,712687,202],{"className":712688},[31850],[34,712690],{"className":712691,"style":31846},[31845],[34,712693,712695,712698,712701,712704,712707,712710,712713,712716,712719],{"className":712694},[31676],[34,712696],{"className":712697,"style":31774},[31680],[34,712699,563286],{"className":712700},[32767],[34,712702],{"className":712703,"style":32123},[31845],[34,712705,162086],{"className":712706},[31685],[34,712708,32936],{"className":712709},[31685,31686],[34,712711,162086],{"className":712712},[31685],[34,712714],{"className":712715,"style":32604},[31845],[34,712717,32226],{"className":712718},[32024],[34,712720],{"className":712721,"style":32604},[31845],[34,712723,712725,712728,712731,712734,712737,712740,712743],{"className":712724},[31676],[34,712726],{"className":712727,"style":31681},[31680],[34,712729,563286],{"className":712730},[32767],[34,712732],{"className":712733,"style":32123},[31845],[34,712735,640],{"className":712736},[31685],[34,712738],{"className":712739,"style":31846},[31845],[34,712741,202],{"className":712742},[31850],[34,712744],{"className":712745,"style":31846},[31845],[34,712747,712749,712752,712755,712758,712761,712764,712767,712777,712780,712783,712786],{"className":712748},[31676],[34,712750],{"className":712751,"style":31774},[31680],[34,712753,563286],{"className":712754},[32767],[34,712756],{"className":712757,"style":32123},[31845],[34,712759,32936],{"className":712760},[31685,31686],[34,712762],{"className":712763,"style":128438},[31845],[34,712765,456],{"className":712766},[31782],[34,712768,712770,712774],{"className":712769},[31685,2638],[34,712771,712773],{"className":712772},[31685,105462],"考虑",[34,712775,130298],{"className":712776},[31685],[34,712778,32936],{"className":712779},[31685,31686],[34,712781],{"className":712782,"style":31846},[31845],[34,712784,29622],{"className":712785},[31850],[34,712787],{"className":712788,"style":31846},[31845],[34,712790,712792,712795,712798],{"className":712791},[31676],[34,712793],{"className":712794,"style":31774},[31680],[34,712796,459],{"className":712797},[31685],[34,712799,2557],{"className":712800},[31690],[34,712802,712804,712854],{"className":712803},[31642],[34,712805,712807],{"className":712806},[31646],[31648,712808,712809],{"xmlns":31650},[31652,712810,712811,712851],{},[31655,712812,712813,712831,712833,712835,712837,712839,712841,712843,712845,712847,712849],{},[41453,712814,712815,712821],{},[31655,712816,712817,712819],{},[31658,712818,619473],{"mathvariant":38079},[31658,712820,67764],{},[31655,712822,712823,712825],{},[31658,712824,619473],{"mathvariant":38079},[31743,712826,712827,712829],{},[31658,712828,67764],{},[31749,712830,459],{},[31662,712832,202],{},[31658,712834,701520],{},[31662,712836,32682],{},[31662,712838,456],{"stretchy":702},[31658,712840,563286],{},[31662,712842,32682],{},[31658,712844,32936],{},[31662,712846,2557],{"stretchy":702},[31662,712848,202],{},[31658,712850,32936],{},[31665,712852,712853],{"encoding":31667},"\\frac{\\partial y}{\\partial y_0} = \\exp(\\ln x) = x",[34,712855,712857,712983,713016],{"className":712856,"ariaHidden":758},[31672],[34,712858,712860,712863,712974,712977,712980],{"className":712859},[31676],[34,712861],{"className":712862,"style":695705},[31680],[34,712864,712866,712869,712971],{"className":712865},[31685],[34,712867],{"className":712868},[31782,41488],[34,712870,712872],{"className":712871},[41453],[34,712873,712875,712963],{"className":712874},[31796,31797],[34,712876,712878,712960],{"className":712877},[31801],[34,712879,712881,712935,712943],{"className":712880,"style":608935},[31805],[34,712882,712883,712886],{"style":41504},[34,712884],{"className":712885,"style":41508},[31813],[34,712887,712889],{"className":712888},[31818,31819,31820,31821],[34,712890,712892,712895],{"className":712891},[31685,31821],[34,712893,619473],{"className":712894,"style":112322},[31685,31821],[34,712896,712898,712901],{"className":712897},[31685,31821],[34,712899,67764],{"className":712900,"style":32062},[31685,31686,31821],[34,712902,712904],{"className":712903},[31792],[34,712905,712907,712927],{"className":712906},[31796,31797],[34,712908,712910,712924],{"className":712909},[31801],[34,712911,712913],{"className":712912,"style":318658},[31805],[34,712914,712915,712918],{"style":561507},[34,712916],{"className":712917,"style":43406},[31813],[34,712919,712921],{"className":712920},[31818,43410,43411,31821],[34,712922,459],{"className":712923},[31685,31821],[34,712925,31829],{"className":712926},[31828],[34,712928,712930],{"className":712929},[31801],[34,712931,712933],{"className":712932,"style":220540},[31805],[34,712934],{},[34,712936,712937,712940],{"style":41520},[34,712938],{"className":712939,"style":41508},[31813],[34,712941],{"className":712942,"style":41528},[41527],[34,712944,712945,712948],{"style":548368},[34,712946],{"className":712947,"style":41508},[31813],[34,712949,712951],{"className":712950},[31818,31819,31820,31821],[34,712952,712954,712957],{"className":712953},[31685,31821],[34,712955,619473],{"className":712956,"style":112322},[31685,31821],[34,712958,67764],{"className":712959,"style":32062},[31685,31686,31821],[34,712961,31829],{"className":712962},[31828],[34,712964,712966],{"className":712965},[31801],[34,712967,712969],{"className":712968,"style":220191},[31805],[34,712970],{},[34,712972],{"className":712973},[31690,41488],[34,712975],{"className":712976,"style":31846},[31845],[34,712978,202],{"className":712979},[31850],[34,712981],{"className":712982,"style":31846},[31845],[34,712984,712986,712989,712992,712995,712998,713001,713004,713007,713010,713013],{"className":712985},[31676],[34,712987],{"className":712988,"style":31774},[31680],[34,712990,701520],{"className":712991},[32767],[34,712993,456],{"className":712994},[31782],[34,712996,563286],{"className":712997},[32767],[34,712999],{"className":713000,"style":32123},[31845],[34,713002,32936],{"className":713003},[31685,31686],[34,713005,2557],{"className":713006},[31690],[34,713008],{"className":713009,"style":31846},[31845],[34,713011,202],{"className":713012},[31850],[34,713014],{"className":713015,"style":31846},[31845],[34,713017,713019,713022],{"className":713018},[31676],[34,713020],{"className":713021,"style":32527},[31680],[34,713023,32936],{"className":713024},[31685,31686],[2627,713026,62532],{},[34,713028,713030,713064],{"className":713029},[31642],[34,713031,713033],{"className":713032},[31646],[31648,713034,713035],{"xmlns":31650},[31652,713036,713037,713061],{},[31655,713038,713039,713057,713059],{},[41453,713040,713041,713047],{},[31655,713042,713043,713045],{},[31658,713044,619473],{"mathvariant":226163},[31658,713046,67764],{"mathvariant":226163},[31655,713048,713049,713051],{},[31658,713050,619473],{"mathvariant":226163},[31743,713052,713053,713055],{},[31658,713054,67764],{"mathvariant":226163},[31749,713056,459],{"mathvariant":226163},[31662,713058,202],{},[31658,713060,32936],{"mathvariant":226163},[31665,713062,713063],{"encoding":31667},"\\mathbf{\\frac{\\partial y}{\\partial y_0} = x}",[34,713065,713067],{"className":713066,"ariaHidden":758},[31672],[34,713068,713070,713073],{"className":713069},[31676],[34,713071],{"className":713072,"style":695705},[31680],[34,713074,713076,713185,713188,713191,713194],{"className":713075},[31685],[34,713077,713079,713082,713182],{"className":713078},[31685],[34,713080],{"className":713081},[31782,41488],[34,713083,713085],{"className":713084},[41453],[34,713086,713088,713174],{"className":713087},[31796,31797],[34,713089,713091,713171],{"className":713090},[31801],[34,713092,713094,713148,713156],{"className":713093,"style":608935},[31805],[34,713095,713096,713099],{"style":41504},[34,713097],{"className":713098,"style":41508},[31813],[34,713100,713102],{"className":713101},[31818,31819,31820,31821],[34,713103,713105,713108],{"className":713104},[31685,31821],[34,713106,619473],{"className":713107,"style":705478},[31685,226302,31821],[34,713109,713111,713114],{"className":713110},[31685,31821],[34,713112,67764],{"className":713113,"style":705485},[31685,226302,31821],[34,713115,713117],{"className":713116},[31792],[34,713118,713120,713140],{"className":713119},[31796,31797],[34,713121,713123,713137],{"className":713122},[31801],[34,713124,713126],{"className":713125,"style":318658},[31805],[34,713127,713128,713131],{"style":705500},[34,713129],{"className":713130,"style":43406},[31813],[34,713132,713134],{"className":713133},[31818,43410,43411,31821],[34,713135,459],{"className":713136},[31685,226302,31821],[34,713138,31829],{"className":713139},[31828],[34,713141,713143],{"className":713142},[31801],[34,713144,713146],{"className":713145,"style":220540},[31805],[34,713147],{},[34,713149,713150,713153],{"style":41520},[34,713151],{"className":713152,"style":41508},[31813],[34,713154],{"className":713155,"style":41528},[41527],[34,713157,713158,713161],{"style":548368},[34,713159],{"className":713160,"style":41508},[31813],[34,713162,713164],{"className":713163},[31818,31819,31820,31821],[34,713165,713167],{"className":713166},[31685,31821],[34,713168,713170],{"className":713169,"style":705485},[31685,226302,31821],"∂y",[34,713172,31829],{"className":713173},[31828],[34,713175,713177],{"className":713176},[31801],[34,713178,713180],{"className":713179,"style":220191},[31805],[34,713181],{},[34,713183],{"className":713184},[31690,41488],[34,713186],{"className":713187,"style":31846},[31845],[34,713189,202],{"className":713190},[31850],[34,713192],{"className":713193,"style":31846},[31845],[34,713195,32936],{"className":713196},[31685,226302],[15,713198,713199,713353,713354,704136,713790,317245,713928,33183,713985,714186,714187,714411,47649,714413],{},[2627,713200,707709,713201],{},[34,713202,713204,713233],{"className":713203},[31642],[34,713205,713207],{"className":713206},[31646],[31648,713208,713209],{"xmlns":31650},[31652,713210,713211,713231],{},[31655,713212,713213],{},[41453,713214,713215,713221],{},[31655,713216,713217,713219],{},[31658,713218,619473],{"mathvariant":38079},[31658,713220,67764],{},[31655,713222,713223,713225],{},[31658,713224,619473],{"mathvariant":38079},[31743,713226,713227,713229],{},[31658,713228,32936],{},[31749,713230,459],{},[31665,713232,708626],{"encoding":31667},[34,713234,713236],{"className":713235,"ariaHidden":758},[31672],[34,713237,713239,713242],{"className":713238},[31676],[34,713240],{"className":713241,"style":702239},[31680],[34,713243,713245,713248,713350],{"className":713244},[31685],[34,713246],{"className":713247},[31782,41488],[34,713249,713251],{"className":713250},[41453],[34,713252,713254,713342],{"className":713253},[31796,31797],[34,713255,713257,713339],{"className":713256},[31801],[34,713258,713260,713314,713322],{"className":713259,"style":608935},[31805],[34,713261,713262,713265],{"style":41504},[34,713263],{"className":713264,"style":41508},[31813],[34,713266,713268],{"className":713267},[31818,31819,31820,31821],[34,713269,713271,713274],{"className":713270},[31685,31821],[34,713272,619473],{"className":713273,"style":112322},[31685,31821],[34,713275,713277,713280],{"className":713276},[31685,31821],[34,713278,32936],{"className":713279},[31685,31686,31821],[34,713281,713283],{"className":713282},[31792],[34,713284,713286,713306],{"className":713285},[31796,31797],[34,713287,713289,713303],{"className":713288},[31801],[34,713290,713292],{"className":713291,"style":318658},[31805],[34,713293,713294,713297],{"style":318661},[34,713295],{"className":713296,"style":43406},[31813],[34,713298,713300],{"className":713299},[31818,43410,43411,31821],[34,713301,459],{"className":713302},[31685,31821],[34,713304,31829],{"className":713305},[31828],[34,713307,713309],{"className":713308},[31801],[34,713310,713312],{"className":713311,"style":220540},[31805],[34,713313],{},[34,713315,713316,713319],{"style":41520},[34,713317],{"className":713318,"style":41508},[31813],[34,713320],{"className":713321,"style":41528},[41527],[34,713323,713324,713327],{"style":548368},[34,713325],{"className":713326,"style":41508},[31813],[34,713328,713330],{"className":713329},[31818,31819,31820,31821],[34,713331,713333,713336],{"className":713332},[31685,31821],[34,713334,619473],{"className":713335,"style":112322},[31685,31821],[34,713337,67764],{"className":713338,"style":32062},[31685,31686,31821],[34,713340,31829],{"className":713341},[31828],[34,713343,713345],{"className":713344},[31801],[34,713346,713348],{"className":713347,"style":231800},[31805],[34,713349],{},[34,713351],{"className":713352},[31690,41488],"\n利用公式 (3.17)：\n",[34,713355,713357,713431],{"className":713356},[31642],[34,713358,713360],{"className":713359},[31646],[31648,713361,713362],{"xmlns":31650},[31652,713363,713364,713428],{},[31655,713365,713366,713384,713386,713388,713390,713392,713398,713400,713406,713408,713410],{},[41453,713367,713368,713374],{},[31655,713369,713370,713372],{},[31658,713371,619473],{"mathvariant":38079},[31658,713373,67764],{},[31655,713375,713376,713378],{},[31658,713377,619473],{"mathvariant":38079},[31743,713379,713380,713382],{},[31658,713381,32936],{},[31749,713383,459],{},[31662,713385,202],{},[31662,713387,32226],{},[31658,713389,31739],{},[31662,713391,456],{"stretchy":702},[31743,713393,713394,713396],{},[31658,713395,32936],{},[31749,713397,459],{},[31662,713399,31960],{"separator":758},[31743,713401,713402,713404],{},[31658,713403,67764],{},[31749,713405,459],{},[31662,713407,2557],{"stretchy":702},[31662,713409,71191],{},[41453,713411,713412,713418],{},[31655,713413,713414,713416],{},[31658,713415,619473],{"mathvariant":38079},[31658,713417,67764],{},[31655,713419,713420,713422],{},[31658,713421,619473],{"mathvariant":38079},[31743,713423,713424,713426],{},[31658,713425,67764],{},[31749,713427,459],{},[31665,713429,713430],{"encoding":31667},"\\frac{\\partial y}{\\partial x_0} = -f(x_0, y_0) \\cdot \\frac{\\partial y}{\\partial 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1680],[34,713965,31739],{"className":713966,"style":31778},[31685,31686],[34,713968,456],{"className":713969},[31782],[34,713971,640],{"className":713972},[31685],[34,713974,31960],{"className":713975},[32119],[34,713977],{"className":713978,"style":32123},[31845],[34,713980,459],{"className":713981},[31685],[34,713983,2557],{"className":713984},[31690],[34,713986,713988,714034],{"className":713987},[31642],[34,713989,713991],{"className":713990},[31646],[31648,713992,713993],{"xmlns":31650},[31652,713994,713995,714031],{},[31655,713996,713997,713999,714001,714003,714005,714007,714009,714011,714013,714015,714027,714029],{},[31658,713998,31739],{},[31662,714000,456],{"stretchy":702},[31749,714002,640],{},[31662,714004,31960],{"separator":758},[31749,714006,459],{},[31662,714008,2557],{"stretchy":702},[31662,714010,202],{},[31658,714012,547853],{},[31662,714014,32682],{},[31655,714016,714017,714019,714025],{},[31662,714018,456],{"fence":758},[41453,714020,714021,714023],{},[31749,714022,459],{},[31749,714024,640],{},[31662,714026,2557],{"fence":758},[31662,714028,202],{},[31749,714030,459],{},[31665,714032,714033],{"encoding":31667},"f(1,0) = \\sin\\left(\\frac{0}{1}\\right) = 0",[34,714035,714037,714073,714177],{"className":714036,"ariaHidden":758},[31672],[34,714038,714040,714043,714046,714049,714052,714055,714058,714061,714064,714067,714070],{"className":714039},[31676],[34,714041],{"className":714042,"style":31774},[31680],[34,714044,31739],{"className":714045,"style":31778},[31685,31686],[34,714047,456],{"className":714048},[31782],[34,714050,640],{"className":714051},[31685],[34,714053,31960],{"className":714054},[32119],[34,714056],{"className":714057,"style":32123},[31845],[34,714059,459],{"className":714060},[31685],[34,714062,2557],{"className":714063},[31690],[34,714065],{"className":714066,"style":31846},[31845],[34,714068,202],{"className":714069},[31850],[34,714071],{"className":714072,"style":31846},[31845],[34,714074,714076,714079,714082,714085,714168,714171,714174],{"className":714075},[31676],[34,714077],{"className":714078,"style":345868},[31680],[34,714080,547853],{"className":714081},[32767],[34,714083],{"className":714084,"style":32123},[31845],[34,714086,714088,714094,714162],{"className":714087},[34006],[34,714089,714091],{"className":714090,"style":75811},[31782,75810],[34,714092,456],{"className":714093},[71281,43411],[34,714095,714097,714100,714159],{"className":714096},[31685],[34,714098],{"className":714099},[31782,41488],[34,714101,714103],{"className":714102},[41453],[34,714104,714106,714151],{"className":714105},[31796,31797],[34,714107,714109,714148],{"className":714108},[31801],[34,714110,714112,714126,714134],{"className":714111,"style":46746},[31805],[34,714113,714114,714117],{"style":41504},[34,714115],{"className":714116,"style":41508},[31813],[34,714118,714120],{"className":714119},[31818,31819,31820,31821],[34,714121,714123],{"className":714122},[31685,31821],[34,714124,640],{"className":714125},[31685,31821],[34,714127,714128,714131],{"style":41520},[34,714129],{"className":714130,"style":41508},[31813],[34,714132],{"className":714133,"style":41528},[41527],[34,714135,714136,714139],{"style":46771},[34,714137],{"className":714138,"style":41508},[31813],[34,714140,714142],{"className":714141},[31818,31819,31820,31821],[34,714143,714145],{"className":714144},[31685,31821],[34,714146,459],{"className":714147},[31685,31821],[34,714149,31829],{"className":714150},[31828],[34,714152,714154],{"className":714153},[31801],[34,714155,714157],{"className":714156,"style":41619},[31805],[34,714158],{},[34,714160],{"className":714161},[31690,41488],[34,714163,714165],{"className":714164,"style":75811},[31690,75810],[34,714166,2557],{"className":714167},[71281,43411],[34,714169],{"className":714170,"style":31846},[31845],[34,714172,202],{"className":714173},[31850],[34,714175],{"className":714176,"style":31846},[31845],[34,714178,714180,714183],{"className":714179},[31676],[34,714181],{"className":714182,"style":32617},[31680],[34,714184,459],{"className":714185},[31685],"\n由于前面的系数是 0，无论后面是多少，结果都是 0。\n",[34,714188,714190,714234],{"className":714189},[31642],[34,714191,714193],{"className":714192},[31646],[31648,714194,714195],{"xmlns":31650},[31652,714196,714197,714231],{},[31655,714198,714199,714217,714219,714221,714223,714225,714227,714229],{},[41453,714200,714201,714207],{},[31655,714202,714203,714205],{},[31658,714204,619473],{"mathvariant":38079},[31658,714206,67764],{},[31655,714208,714209,714211],{},[31658,714210,619473],{"mathvariant":38079},[31743,714212,714213,714215],{},[31658,714214,32936],{},[31749,714216,459],{},[31662,714218,202],{},[31662,714220,32226],{},[31749,714222,459],{},[31662,714224,71191],{},[31658,714226,32936],{},[31662,714228,202],{},[31749,714230,459],{},[31665,714232,714233],{"encoding":31667},"\\frac{\\partial y}{\\partial x_0} = -0 \\cdot x = 0",[34,714235,714237,714363,714384,714402],{"className":714236,"ariaHidden":758},[31672],[34,714238,714240,714243,714354,714357,714360],{"className":714239},[31676],[34,714241],{"className":714242,"style":702239},[31680],[34,714244,714246,714249,714351],{"className":714245},[31685],[34,714247],{"className":714248},[31782,41488],[34,714250,714252],{"className":714251},[41453],[34,714253,714255,714343],{"className":714254},[31796,31797],[34,714256,714258,714340],{"className":714257},[31801],[34,714259,714261,714315,714323],{"className":714260,"style":608935},[31805],[34,714262,714263,714266],{"style":41504},[34,714264],{"className":714265,"style":41508},[31813],[34,714267,714269],{"className":714268},[31818,31819,31820,31821],[34,714270,714272,714275],{"className":714271},[31685,31821],[34,714273,619473],{"className":714274,"style":112322},[31685,31821],[34,714276,714278,714281],{"className":714277},[31685,31821],[34,714279,32936],{"className":714280},[31685,31686,31821],[34,714282,714284],{"className":714283},[31792],[34,714285,714287,714307],{"className":714286},[31796,31797],[34,714288,714290,714304],{"className":714289},[31801],[34,714291,714293],{"className":714292,"style":318658},[31805],[34,714294,714295,714298],{"style":318661},[34,714296],{"className":714297,"style":43406},[31813],[34,714299,714301],{"className":714300},[31818,43410,43411,31821],[34,714302,459],{"className":714303},[31685,31821],[34,714305,31829],{"className":714306},[31828],[34,714308,714310],{"className":714309},[31801],[34,714311,714313],{"className":714312,"style":220540},[31805],[34,714314],{},[34,714316,714317,714320],{"style":41520},[34,714318],{"className":714319,"style":41508},[31813],[34,714321],{"className":714322,"style":41528},[41527],[34,714324,714325,714328],{"style":548368},[34,714326],{"className":714327,"style":41508},[31813],[34,714329,714331],{"className":714330},[31818,31819,31820,31821],[34,714332,714334,714337],{"className":714333},[31685,31821],[34,714335,619473],{"className":714336,"style":112322},[31685,31821],[34,714338,67764],{"className":714339,"style":32062},[31685,31686,31821],[34,714341,31829],{"className":714342},[31828],[34,714344,714346],{"className":714345},[31801],[34,714347,714349],{"className":714348,"style":231800},[31805],[34,714350],{},[34,714352],{"className":714353},[31690,41488],[34,714355],{"className":714356,"style":31846},[31845],[34,714358,202],{"className":714359},[31850],[34,714361],{"className":714362,"style":31846},[31845],[34,714364,714366,714369,714372,714375,714378,714381],{"className":714365},[31676],[34,714367],{"className":714368,"style":35942},[31680],[34,714370,32226],{"className":714371},[31685],[34,714373,459],{"className":714374},[31685],[34,714376],{"className":714377,"style":32604},[31845],[34,714379,71191],{"className":714380},[32024],[34,714382],{"className":714383,"style":32604},[31845],[34,714385,714387,714390,714393,714396,714399],{"className":714386},[31676],[34,714388],{"className":714389,"style":32527},[31680],[34,714391,32936],{"className":714392},[31685,31686],[34,714394],{"className":714395,"style":31846},[31845],[34,714397,202],{"className":714398},[31850],[34,714400],{"className":714401,"style":31846},[31845],[34,714403,714405,714408],{"className":714404},[31676],[34,714406],{"className":714407,"style":32617},[31680],[34,714409,459],{"className":714410},[31685],[2627,714412,62532],{},[34,714414,714416,714450],{"className":714415},[31642],[34,714417,714419],{"className":714418},[31646],[31648,714420,714421],{"xmlns":31650},[31652,714422,714423,714447],{},[31655,714424,714425,714443,714445],{},[41453,714426,714427,714433],{},[31655,714428,714429,714431],{},[31658,714430,619473],{"mathvariant":226163},[31658,714432,67764],{"mathvariant":226163},[31655,714434,714435,714437],{},[31658,714436,619473],{"mathvariant":226163},[31743,714438,714439,714441],{},[31658,714440,32936],{"mathvariant":226163},[31749,714442,459],{"mathvariant":226163},[31662,714444,202],{},[31749,714446,459],{"mathvariant":226163},[31665,714448,714449],{"encoding":31667},"\\mathbf{\\frac{\\partial y}{\\partial x_0} = 0}",[34,714451,714453],{"className":714452,"ariaHidden":758},[31672],[34,714454,714456,714459],{"className":714455},[31676],[34,714457],{"className":714458,"style":702239},[31680],[34,714460,714462,714570,714573,714576,714579],{"className":714461},[31685],[34,714463,714465,714468,714567],{"className":714464},[31685],[34,714466],{"className":714467},[31782,41488],[34,714469,714471],{"className":714470},[41453],[34,714472,714474,714559],{"className":714473},[31796,31797],[34,714475,714477,714556],{"className":714476},[31801],[34,714478,714480,714534,714542],{"className":714479,"style":608935},[31805],[34,714481,714482,714485],{"style":41504},[34,714483],{"className":714484,"style":41508},[31813],[34,714486,714488],{"className":714487},[31818,31819,31820,31821],[34,714489,714491,714494],{"className":714490},[31685,31821],[34,714492,619473],{"className":714493,"style":705478},[31685,226302,31821],[34,714495,714497,714500],{"className":714496},[31685,31821],[34,714498,32936],{"className":714499},[31685,226302,31821],[34,714501,714503],{"className":714502},[31792],[34,714504,714506,714526],{"className":714505},[31796,31797],[34,714507,714509,714523],{"className":714508},[31801],[34,714510,714512],{"className":714511,"style":318658},[31805],[34,714513,714514,714517],{"style":318661},[34,714515],{"className":714516,"style":43406},[31813],[34,714518,714520],{"className":714519},[31818,43410,43411,31821],[34,714521,459],{"className":714522},[31685,226302,31821],[34,714524,31829],{"className":714525},[31828],[34,714527,714529],{"className":714528},[31801],[34,714530,714532],{"className":714531,"style":220540},[31805],[34,714533],{},[34,714535,714536,714539],{"style":41520},[34,714537],{"className":714538,"style":41508},[31813],[34,714540],{"className":714541,"style":41528},[41527],[34,714543,714544,714547],{"style":548368},[34,714545],{"className":714546,"style":41508},[31813],[34,714548,714550],{"className":714549},[31818,31819,31820,31821],[34,714551,714553],{"className":714552},[31685,31821],[34,714554,713170],{"className":714555,"style":705485},[31685,226302,31821],[34,714557,31829],{"className":714558},[31828],[34,714560,714562],{"className":714561},[31801],[34,714563,714565],{"className":714564,"style":231800},[31805],[34,714566],{},[34,714568],{"className":714569},[31690,41488],[34,714571],{"className":714572,"style":31846},[31845],[34,714574,202],{"className":714575},[31850],[34,714577],{"className":714578,"style":31846},[31845],[34,714580,459],{"className":714581},[31685,226302],[32188,714583],{},[253,714585,714586],{"id":714586},"总结与建议",[2230,714588,714589,714807],{},[108,714590,714591,714594,714595,714747,714748,214569],{},[2627,714592,714593],{},"理论核心："," 只要记住，解对初值的偏导数 ",[34,714596,714598,714627],{"className":714597},[31642],[34,714599,714601],{"className":714600},[31646],[31648,714602,714603],{"xmlns":31650},[31652,714604,714605,714625],{},[31655,714606,714607],{},[41453,714608,714609,714615],{},[31655,714610,714611,714613],{},[31658,714612,619473],{"mathvariant":38079},[31658,714614,126541],{},[31655,714616,714617,714619],{},[31658,714618,619473],{"mathvariant":38079},[31743,714620,714621,714623],{},[31658,714622,67764],{},[31749,714624,459],{},[31665,714626,699617],{"encoding":31667},[34,714628,714630],{"className":714629,"ariaHidden":758},[31672],[34,714631,714633,714636],{"className":714632},[31676],[34,714634],{"className":714635,"style":695705},[31680],[34,714637,714639,714642,714744],{"className":714638},[31685],[34,714640],{"className":714641},[31782,41488],[34,714643,714645],{"className":714644},[41453],[34,714646,714648,714736],{"className":714647},[31796,31797],[34,714649,714651,714733],{"className":714650},[31801],[34,714652,714654,714708,714716],{"className":714653,"style":608935},[31805],[34,714655,714656,714659],{"style":41504},[34,714657],{"className":714658,"style":41508},[31813],[34,714660,714662],{"className":714661},[31818,31819,31820,31821],[34,714663,714665,714668],{"className":714664},[31685,31821],[34,714666,619473],{"className":714667,"style":112322},[31685,31821],[34,714669,714671,714674],{"className":714670},[31685,31821],[34,714672,67764],{"className":714673,"style":32062},[31685,31686,31821],[34,714675,714677],{"className":714676},[31792],[34,714678,714680,714700],{"className":714679},[31796,31797],[34,714681,714683,714697],{"className":714682},[31801],[34,714684,714686],{"className":714685,"style":318658},[31805],[34,714687,714688,714691],{"style":561507},[34,714689],{"className":714690,"style":43406},[31813],[34,714692,714694],{"className":714693},[31818,43410,43411,31821],[34,714695,459],{"className":714696},[31685,31821],[34,714698,31829],{"className":714699},[31828],[34,714701,714703],{"className":714702},[31801],[34,714704,714706],{"className":714705,"style":220540},[31805],[34,714707],{},[34,714709,714710,714713],{"style":41520},[34,714711],{"className":714712,"style":41508},[31813],[34,714714],{"className":714715,"style":41528},[41527],[34,714717,714718,714721],{"style":548368},[34,714719],{"className":714720,"style":41508},[31813],[34,714722,714724],{"className":714723},[31818,31819,31820,31821],[34,714725,714727,714730],{"className":714726},[31685,31821],[34,714728,619473],{"className":714729,"style":112322},[31685,31821],[34,714731,126541],{"className":714732},[31685,31686,31821],[34,714734,31829],{"className":714735},[31828],[34,714737,714739],{"className":714738},[31801],[34,714740,714742],{"className":714741,"style":220191},[31805],[34,714743],{},[34,714745],{"className":714746},[31690,41488]," 本质上是一个解的“变分”，它服从一个线性的微分关系，所以结果总是一个指数积分形式 (",[34,714749,714751,714777],{"className":714750},[31642],[34,714752,714754],{"className":714753},[31646],[31648,714755,714756],{"xmlns":31650},[31652,714757,714758,714774],{},[31655,714759,714760,714762,714764,714766,714768,714770,714772],{},[31658,714761,701520],{},[31662,714763,32682],{},[31662,714765,456],{"stretchy":702},[31662,714767,546295],{},[31662,714769,33934],{},[39361,714771,39363],{},[31662,714773,2557],{"stretchy":702},[31665,714775,714776],{"encoding":31667},"\\exp(\\int \\dots)",[34,714778,714780],{"className":714779,"ariaHidden":758},[31672],[34,714781,714783,714786,714789,714792,714795,714798,714801,714804],{"className":714782},[31676],[34,714784],{"className":714785,"style":546896},[31680],[34,714787,701520],{"className":714788},[32767],[34,714790,456],{"className":714791},[31782],[34,714793,546295],{"className":714794,"style":546331},[32767,131249,156560],[34,714796],{"className":714797,"style":32123},[31845],[34,714799,33934],{"className":714800},[34006],[34,714802],{"className":714803,"style":32123},[31845],[34,714805,2557],{"className":714806},[31690],[108,714808,714809,714812],{},[2627,714810,714811],{},"做题技巧：",[105,714813,714814,714860],{},[108,714815,714816,714817,119453],{},"如果是线性方程（如题1），直接代入 ",[34,714818,714820,714839],{"className":714819},[31642],[34,714821,714823],{"className":714822},[31646],[31648,714824,714825],{"xmlns":31650},[31652,714826,714827,714837],{},[31655,714828,714829,714831,714833,714835],{},[31658,714830,105852],{},[31662,714832,456],{"stretchy":702},[31658,714834,32936],{},[31662,714836,2557],{"stretchy":702},[31665,714838,705784],{"encoding":31667},[34,714840,714842],{"className":714841,"ariaHidden":758},[31672],[34,714843,714845,714848,714851,714854,714857],{"className":714844},[31676],[34,714846],{"className":714847,"style":31774},[31680],[34,714849,105852],{"className":714850,"style":37422},[31685,31686],[34,714852,456],{"className":714853},[31782],[34,714855,32936],{"className":714856},[31685,31686],[34,714858,2557],{"className":714859},[31690],[108,714861,714862,714863,714866,714867,714978,714979,715007],{},"如果是非线性方程（如题2），",[2627,714864,714865],{},"必须先找出原方程的特解","，代入 ",[34,714868,714870,714895],{"className":714869},[31642],[34,714871,714873],{"className":714872},[31646],[31648,714874,714875],{"xmlns":31650},[31652,714876,714877,714893],{},[31655,714878,714879],{},[41453,714880,714881,714887],{},[31655,714882,714883,714885],{},[31658,714884,619473],{"mathvariant":38079},[31658,714886,31739],{},[31655,714888,714889,714891],{},[31658,714890,619473],{"mathvariant":38079},[31658,714892,67764],{},[31665,714894,701964],{"encoding":31667},[34,714896,714898],{"className":714897,"ariaHidden":758},[31672],[34,714899,714901,714904],{"className":714900},[31676],[34,714902],{"className":714903,"style":695705},[31680],[34,714905,714907,714910,714975],{"className":714906},[31685],[34,714908],{"className":714909},[31782,41488],[34,714911,714913],{"className":714912},[41453],[34,714914,714916,714967],{"className":714915},[31796,31797],[34,714917,714919,714964],{"className":714918},[31801],[34,714920,714922,714939,714947],{"className":714921,"style":608935},[31805],[34,714923,714924,714927],{"style":41504},[34,714925],{"className":714926,"style":41508},[31813],[34,714928,714930],{"className":714929},[31818,31819,31820,31821],[34,714931,714933,714936],{"className":714932},[31685,31821],[34,714934,619473],{"className":714935,"style":112322},[31685,31821],[34,714937,67764],{"className":714938,"style":32062},[31685,31686,31821],[34,714940,714941,714944],{"style":41520},[34,714942],{"className":714943,"style":41508},[31813],[34,714945],{"className":714946,"style":41528},[41527],[34,714948,714949,714952],{"style":548368},[34,714950],{"className":714951,"style":41508},[31813],[34,714953,714955],{"className":714954},[31818,31819,31820,31821],[34,714956,714958,714961],{"className":714957},[31685,31821],[34,714959,619473],{"className":714960,"style":112322},[31685,31821],[34,714962,31739],{"className":714963,"style":31778},[31685,31686,31821],[34,714965,31829],{"className":714966},[31828],[34,714968,714970],{"className":714969},[31801],[34,714971,714973],{"className":714972,"style":220191},[31805],[34,714974],{},[34,714976],{"className":714977},[31690,41488]," 把它变成只含 ",[34,714980,714982,714995],{"className":714981},[31642],[34,714983,714985],{"className":714984},[31646],[31648,714986,714987],{"xmlns":31650},[31652,714988,714989,714993],{},[31655,714990,714991],{},[31658,714992,32936],{},[31665,714994,32936],{"encoding":31667},[34,714996,714998],{"className":714997,"ariaHidden":758},[31672],[34,714999,715001,715004],{"className":715000},[31676],[34,715002],{"className":715003,"style":32527},[31680],[34,715005,32936],{"className":715006},[31685,31686]," 的函数，才能进行积分。",[15,715009,715010,715011,2664],{},"这些题目覆盖了",[2627,715012,715013],{},"线性微分方程的四大支柱",[2230,715015,715016,715022,715028,715034],{},[108,715017,715018,715021],{},[2627,715019,715020],{},"叠加原理","（解的性质）",[108,715023,715024,715027],{},[2627,715025,715026],{},"常数变易法","（解的计算，核心中的核心）",[108,715029,715030,715033],{},[2627,715031,715032],{},"初值问题","（归一化）",[108,715035,715036,715039],{},[2627,715037,715038],{},"理论证明","（朗斯基行列式与解的空间结构）",[32188,715041],{},[253,715043,715045],{"id":715044},"_2-证明非齐次线性微分方程的叠加原理","2. 证明非齐次线性微分方程的叠加原理",[15,715047,715048,715050,715051,715120,715121,322015,715191,715120,715260,715330,715331,715464,715465,347],{},[2627,715049,148881],{},"\n微分方程其实是一个“机器”（算子）。你把函数放进去，它就吐出一个结果。\n这道题是说：如果把 ",[34,715052,715054,715071],{"className":715053},[31642],[34,715055,715057],{"className":715056},[31646],[31648,715058,715059],{"xmlns":31650},[31652,715060,715061,715069],{},[31655,715062,715063],{},[31743,715064,715065,715067],{},[31658,715066,32936],{},[31749,715068,640],{},[31665,715070,224014],{"encoding":31667},[34,715072,715074],{"className":715073,"ariaHidden":758},[31672],[34,715075,715077,715080],{"className":715076},[31676],[34,715078],{"className":715079,"style":31860},[31680],[34,715081,715083,715086],{"className":715082},[31685],[34,715084,32936],{"className":715085},[31685,31686],[34,715087,715089],{"className":715088},[31792],[34,715090,715092,715112],{"className":715091},[31796,31797],[34,715093,715095,715109],{"className":715094},[31801],[34,715096,715098],{"className":715097,"style":31806},[31805],[34,715099,715100,715103],{"style":31809},[34,715101],{"className":715102,"style":31814},[31813],[34,715104,715106],{"className":715105},[31818,31819,31820,31821],[34,715107,640],{"className":715108},[31685,31821],[34,715110,31829],{"className":715111},[31828],[34,715113,715115],{"className":715114},[31801],[34,715116,715118],{"className":715117,"style":31836},[31805],[34,715119],{}," 放进去得到 ",[34,715122,715124,715142],{"className":715123},[31642],[34,715125,715127],{"className":715126},[31646],[31648,715128,715129],{"xmlns":31650},[31652,715130,715131,715139],{},[31655,715132,715133],{},[31743,715134,715135,715137],{},[31658,715136,31739],{},[31749,715138,640],{},[31665,715140,715141],{"encoding":31667},"f_1",[34,715143,715145],{"className":715144,"ariaHidden":758},[31672],[34,715146,715148,715151],{"className":715147},[31676],[34,715149],{"className":715150,"style":43217},[31680],[34,715152,715154,715157],{"className":715153},[31685],[34,715155,31739],{"className":715156,"style":31778},[31685,31686],[34,715158,715160],{"className":715159},[31792],[34,715161,715163,715183],{"className":715162},[31796,31797],[34,715164,715166,715180],{"className":715165},[31801],[34,715167,715169],{"className":715168,"style":31806},[31805],[34,715170,715171,715174],{"style":141014},[34,715172],{"className":715173,"style":31814},[31813],[34,715175,715177],{"className":715176},[31818,31819,31820,31821],[34,715178,640],{"className":715179},[31685,31821],[34,715181,31829],{"className":715182},[31828],[34,715184,715186],{"className":715185},[31801],[34,715187,715189],{"className":715188,"style":31836},[31805],[34,715190],{},[34,715192,715194,715211],{"className":715193},[31642],[34,715195,715197],{"className":715196},[31646],[31648,715198,715199],{"xmlns":31650},[31652,715200,715201,715209],{},[31655,715202,715203],{},[31743,715204,715205,715207],{},[31658,715206,32936],{},[31749,715208,768],{},[31665,715210,224085],{"encoding":31667},[34,715212,715214],{"className":715213,"ariaHidden":758},[31672],[34,715215,715217,715220],{"className":715216},[31676],[34,715218],{"className":715219,"style":31860},[31680],[34,715221,715223,715226],{"className":715222},[31685],[34,715224,32936],{"className":715225},[31685,31686],[34,715227,715229],{"className":715228},[31792],[34,715230,715232,715252],{"className":715231},[31796,31797],[34,715233,715235,715249],{"className":715234},[31801],[34,715236,715238],{"className":715237,"style":31806},[31805],[34,715239,715240,715243],{"style":31809},[34,715241],{"className":715242,"style":31814},[31813],[34,715244,715246],{"className":715245},[31818,31819,31820,31821],[34,715247,768],{"className":715248},[31685,31821],[34,715250,31829],{"className":715251},[31828],[34,715253,715255],{"className":715254},[31801],[34,715256,715258],{"className":715257,"style":31836},[31805],[34,715259],{},[34,715261,715263,715281],{"className":715262},[31642],[34,715264,715266],{"className":715265},[31646],[31648,715267,715268],{"xmlns":31650},[31652,715269,715270,715278],{},[31655,715271,715272],{},[31743,715273,715274,715276],{},[31658,715275,31739],{},[31749,715277,768],{},[31665,715279,715280],{"encoding":31667},"f_2",[34,715282,715284],{"className":715283,"ariaHidden":758},[31672],[34,715285,715287,715290],{"className":715286},[31676],[34,715288],{"className":715289,"style":43217},[31680],[34,715291,715293,715296],{"className":715292},[31685],[34,715294,31739],{"className":715295,"style":31778},[31685,31686],[34,715297,715299],{"className":715298},[31792],[34,715300,715302,715322],{"className":715301},[31796,31797],[34,715303,715305,715319],{"className":715304},[31801],[34,715306,715308],{"className":715307,"style":31806},[31805],[34,715309,715310,715313],{"style":141014},[34,715311],{"className":715312,"style":31814},[31813],[34,715314,715316],{"className":715315},[31818,31819,31820,31821],[34,715317,768],{"className":715318},[31685,31821],[34,715320,31829],{"className":715321},[31828],[34,715323,715325],{"className":715324},[31801],[34,715326,715328],{"className":715327,"style":31836},[31805],[34,715329],{},"，那么把 ",[34,715332,715334,715360],{"className":715333},[31642],[34,715335,715337],{"className":715336},[31646],[31648,715338,715339],{"xmlns":31650},[31652,715340,715341,715357],{},[31655,715342,715343,715349,715351],{},[31743,715344,715345,715347],{},[31658,715346,32936],{},[31749,715348,640],{},[31662,715350,223],{},[31743,715352,715353,715355],{},[31658,715354,32936],{},[31749,715356,768],{},[31665,715358,715359],{"encoding":31667},"x_1+x_2",[34,715361,715363,715418],{"className":715362,"ariaHidden":758},[31672],[34,715364,715366,715369,715409,715412,715415],{"className":715365},[31676],[34,715367],{"className":715368,"style":40522},[31680],[34,715370,715372,715375],{"className":715371},[31685],[34,715373,32936],{"className":715374},[31685,31686],[34,715376,715378],{"className":715377},[31792],[34,715379,715381,715401],{"className":715380},[31796,31797],[34,715382,715384,715398],{"className":715383},[31801],[34,715385,715387],{"className":715386,"style":31806},[31805],[34,715388,715389,715392],{"style":31809},[34,715390],{"className":715391,"style":31814},[31813],[34,715393,715395],{"className":715394},[31818,31819,31820,31821],[34,715396,640],{"className":715397},[31685,31821],[34,715399,31829],{"className":715400},[31828],[34,715402,715404],{"className":715403},[31801],[34,715405,715407],{"className":715406,"style":31836},[31805],[34,715408],{},[34,715410],{"className":715411,"style":32604},[31845],[34,715413,223],{"className":715414},[32024],[34,715416],{"className":715417,"style":32604},[31845],[34,715419,715421,715424],{"className":715420},[31676],[34,715422],{"className":715423,"style":31860},[31680],[34,715425,715427,715430],{"className":715426},[31685],[34,715428,32936],{"className":715429},[31685,31686],[34,715431,715433],{"className":715432},[31792],[34,715434,715436,715456],{"className":715435},[31796,31797],[34,715437,715439,715453],{"className":715438},[31801],[34,715440,715442],{"className":715441,"style":31806},[31805],[34,715443,715444,715447],{"style":31809},[34,715445],{"className":715446,"style":31814},[31813],[34,715448,715450],{"className":715449},[31818,31819,31820,31821],[34,715451,768],{"className":715452},[31685,31821],[34,715454,31829],{"className":715455},[31828],[34,715457,715459],{"className":715458},[31801],[34,715460,715462],{"className":715461,"style":31836},[31805],[34,715463],{}," 一起放进去，得到的就是 ",[34,715466,715468,715494],{"className":715467},[31642],[34,715469,715471],{"className":715470},[31646],[31648,715472,715473],{"xmlns":31650},[31652,715474,715475,715491],{},[31655,715476,715477,715483,715485],{},[31743,715478,715479,715481],{},[31658,715480,31739],{},[31749,715482,640],{},[31662,715484,223],{},[31743,715486,715487,715489],{},[31658,715488,31739],{},[31749,715490,768],{},[31665,715492,715493],{"encoding":31667},"f_1+f_2",[34,715495,715497,715552],{"className":715496,"ariaHidden":758},[31672],[34,715498,715500,715503,715543,715546,715549],{"className":715499},[31676],[34,715501],{"className":715502,"style":43217},[31680],[34,715504,715506,715509],{"className":715505},[31685],[34,715507,31739],{"className":715508,"style":31778},[31685,31686],[34,715510,715512],{"className":715511},[31792],[34,715513,715515,715535],{"className":715514},[31796,31797],[34,715516,715518,715532],{"className":715517},[31801],[34,715519,715521],{"className":715520,"style":31806},[31805],[34,715522,715523,715526],{"style":141014},[34,715524],{"className":715525,"style":31814},[31813],[34,715527,715529],{"className":715528},[31818,31819,31820,31821],[34,715530,640],{"className":715531},[31685,31821],[34,715533,31829],{"className":715534},[31828],[34,715536,715538],{"className":715537},[31801],[34,715539,715541],{"className":715540,"style":31836},[31805],[34,715542],{},[34,715544],{"className":715545,"style":32604},[31845],[34,715547,223],{"className":715548},[32024],[34,715550],{"className":715551,"style":32604},[31845],[34,715553,715555,715558],{"className":715554},[31676],[34,715556],{"className":715557,"style":43217},[31680],[34,715559,715561,715564],{"className":715560},[31685],[34,715562,31739],{"className":715563,"style":31778},[31685,31686],[34,715565,715567],{"className":715566},[31792],[34,715568,715570,715590],{"className":715569},[31796,31797],[34,715571,715573,715587],{"className":715572},[31801],[34,715574,715576],{"className":715575,"style":31806},[31805],[34,715577,715578,715581],{"style":141014},[34,715579],{"className":715580,"style":31814},[31813],[34,715582,715584],{"className":715583},[31818,31819,31820,31821],[34,715585,768],{"className":715586},[31685,31821],[34,715588,31829],{"className":715589},[31828],[34,715591,715593],{"className":715592},[31801],[34,715594,715596],{"className":715595,"style":31836},[31805],[34,715597],{},[15,715599,715600],{},[2627,715601,715602],{},"证明步骤：",[15,715604,715605,715608,715609,715805,715806,715850,715851,715879,715880],{},[2627,715606,715607],{},"第一步：简化符号（为了不晕车）","\n为了不每次都写那一长串 ",[34,715610,715612,715650],{"className":715611},[31642],[34,715613,715615],{"className":715614},[31646],[31648,715616,715617],{"xmlns":31650},[31652,715618,715619,715647],{},[31655,715620,715621,715643,715645],{},[41453,715622,715623,715633],{},[31655,715624,715625,715631],{},[37624,715626,715627,715629],{},[31658,715628,46900],{"mathvariant":38079},[31658,715630,31660],{},[31658,715632,32936],{},[31655,715634,715635,715637],{},[31658,715636,46900],{"mathvariant":38079},[37624,715638,715639,715641],{},[31658,715640,33249],{},[31658,715642,31660],{},[31662,715644,223],{},[31662,715646,38874],{},[31665,715648,715649],{"encoding":31667},"\\frac{\\mathrm{d}^n x}{\\mathrm{d}t^n} + \\cdots",[34,715651,715653,715795],{"className":715652,"ariaHidden":758},[31672],[34,715654,715656,715660,715786,715789,715792],{"className":715655},[31676],[34,715657],{"className":715658,"style":715659},[31680],"height:1.256em;vertical-align:-0.345em;",[34,715661,715663,715666,715783],{"className":715662},[31685],[34,715664],{"className":715665},[31782,41488],[34,715667,715669],{"className":715668},[41453],[34,715670,715672,715775],{"className":715671},[31796,31797],[34,715673,715675,715772],{"className":715674},[31801],[34,715676,715678,715721,715729],{"className":715677,"style":700470},[31805],[34,715679,715680,715683],{"style":41504},[34,715681],{"className":715682,"style":41508},[31813],[34,715684,715686],{"className":715685},[31818,31819,31820,31821],[34,715687,715689,715692],{"className":715688},[31685,31821],[34,715690,46900],{"className":715691},[31685,207923,31821],[34,715693,715695,715698],{"className":715694},[31685,31821],[34,715696,33249],{"className":715697},[31685,31686,31821],[34,715699,715701],{"className":715700},[31792],[34,715702,715704],{"className":715703},[31796],[34,715705,715707],{"className":715706},[31801],[34,715708,715710],{"className":715709,"style":566313},[31805],[34,715711,715712,715715],{"style":116668},[34,715713],{"className":715714,"style":43406},[31813],[34,715716,715718],{"className":715717},[31818,43410,43411,31821],[34,715719,31660],{"className":715720},[31685,31686,31821],[34,715722,715723,715726],{"style":41520},[34,715724],{"className":715725,"style":41508},[31813],[34,715727],{"className":715728,"style":41528},[41527],[34,715730,715731,715734],{"style":46771},[34,715732],{"className":715733,"style":41508},[31813],[34,715735,715737],{"className":715736},[31818,31819,31820,31821],[34,715738,715740,715769],{"className":715739},[31685,31821],[34,715741,715743,715746],{"className":715742},[31685,31821],[34,715744,46900],{"className":715745},[31685,207923,31821],[34,715747,715749],{"className":715748},[31792],[34,715750,715752],{"className":715751},[31796],[34,715753,715755],{"className":715754},[31801],[34,715756,715758],{"className":715757,"style":611529},[31805],[34,715759,715760,715763],{"style":552391},[34,715761],{"className":715762,"style":43406},[31813],[34,715764,715766],{"className":715765},[31818,43410,43411,31821],[34,715767,31660],{"className":715768},[31685,31686,31821],[34,715770,32936],{"className":715771},[31685,31686,31821],[34,715773,31829],{"className":715774},[31828],[34,715776,715778],{"className":715777},[31801],[34,715779,715781],{"className":715780,"style":41619},[31805],[34,715782],{},[34,715784],{"className":715785},[31690,41488],[34,715787],{"className":715788,"style":32604},[31845],[34,715790,223],{"className":715791},[32024],[34,715793],{"className":715794,"style":32604},[31845],[34,715796,715798,715802],{"className":715797},[31676],[34,715799],{"className":715800,"style":715801},[31680],"height:0.313em;",[34,715803,38874],{"className":715804},[34006],"，我们定义一个记号 ",[34,715807,715809,715829],{"className":715808},[31642],[34,715810,715812],{"className":715811},[31646],[31648,715813,715814],{"xmlns":31650},[31652,715815,715816,715826],{},[31655,715817,715818,715820,715822,715824],{},[31658,715819,33206],{},[31662,715821,15972],{"stretchy":702},[31658,715823,32936],{},[31662,715825,32712],{"stretchy":702},[31665,715827,715828],{"encoding":31667},"L[x]",[34,715830,715832],{"className":715831,"ariaHidden":758},[31672],[34,715833,715835,715838,715841,715844,715847],{"className":715834},[31676],[34,715836],{"className":715837,"style":31774},[31680],[34,715839,33206],{"className":715840},[31685,31686],[34,715842,15972],{"className":715843},[31782],[34,715845,32936],{"className":715846},[31685,31686],[34,715848,32712],{"className":715849},[31690],"。\n令 ",[34,715852,715854,715867],{"className":715853},[31642],[34,715855,715857],{"className":715856},[31646],[31648,715858,715859],{"xmlns":31650},[31652,715860,715861,715865],{},[31655,715862,715863],{},[31658,715864,33206],{},[31665,715866,33206],{"encoding":31667},[34,715868,715870],{"className":715869,"ariaHidden":758},[31672],[34,715871,715873,715876],{"className":715872},[31676],[34,715874],{"className":715875,"style":33218},[31680],[34,715877,33206],{"className":715878},[31685,31686]," 代表方程左边的运算：\n",[34,715881,715883,715995],{"className":715882},[31642],[34,715884,715886],{"className":715885},[31646],[31648,715887,715888],{"xmlns":31650},[31652,715889,715890,715992],{},[31655,715891,715892,715894,715896,715898,715900,715902,715924,715926,715932,715934,715936,715938,715972,715974,715976,715978,715984,715986,715988,715990],{},[31658,715893,33206],{},[31662,715895,15972],{"stretchy":702},[31658,715897,32936],{},[31662,715899,32712],{"stretchy":702},[31662,715901,202],{},[41453,715903,715904,715914],{},[31655,715905,715906,715912],{},[37624,715907,715908,715910],{},[31658,715909,46900],{"mathvariant":38079},[31658,715911,31660],{},[31658,715913,32936],{},[31655,715915,715916,715918],{},[31658,715917,46900],{"mathvariant":38079},[37624,715919,715920,715922],{},[31658,715921,33249],{},[31658,715923,31660],{},[31662,715925,223],{},[31743,715927,715928,715930],{},[31658,715929,111],{},[31749,715931,640],{},[31662,715933,456],{"stretchy":702},[31658,715935,33249],{},[31662,715937,2557],{"stretchy":702},[41453,715939,715940,715956],{},[31655,715941,715942,715954],{},[37624,715943,715944,715946],{},[31658,715945,46900],{"mathvariant":38079},[31655,715947,715948,715950,715952],{},[31658,715949,31660],{},[31662,715951,32226],{},[31749,715953,640],{},[31658,715955,32936],{},[31655,715957,715958,715960],{},[31658,715959,46900],{"mathvariant":38079},[37624,715961,715962,715964],{},[31658,715963,33249],{},[31655,715965,715966,715968,715970],{},[31658,715967,31660],{},[31662,715969,32226],{},[31749,715971,640],{},[31662,715973,223],{},[31662,715975,38874],{},[31662,715977,223],{},[31743,715979,715980,715982],{},[31658,715981,111],{},[31658,715983,31660],{},[31662,715985,456],{"stretchy":702},[31658,715987,33249],{},[31662,715989,2557],{"stretchy":702},[31658,715991,32936],{},[31665,715993,715994],{"encoding":31667},"L[x] = \\frac{\\mathrm{d}^n x}{\\mathrm{d}t^n} + a_1(t)\\frac{\\mathrm{d}^{n-1}x}{\\mathrm{d}t^{n-1}} + \\cdots + a_n(t)x",[34,715996,715998,716025,716166,716374,716392],{"className":715997,"ariaHidden":758},[31672],[34,715999,716001,716004,716007,716010,716013,716016,716019,716022],{"className":716000},[31676],[34,716002],{"className":716003,"style":31774},[31680],[34,716005,33206],{"className":716006},[31685,31686],[34,716008,15972],{"className":716009},[31782],[34,716011,32936],{"className":716012},[31685,31686],[34,716014,32712],{"className":716015},[31690],[34,716017],{"className":716018,"style":31846},[31845],[34,716020,202],{"className":716021},[31850],[34,716023],{"className":716024,"style":31846},[31845],[34,716026,716028,716031,716157,716160,716163],{"className":716027},[31676],[34,716029],{"className":716030,"style":715659},[31680],[34,716032,716034,716037,716154],{"className":716033},[31685],[34,716035],{"className":716036},[31782,41488],[34,716038,716040],{"className":716039},[41453],[34,716041,716043,716146],{"className":716042},[31796,31797],[34,716044,716046,716143],{"className":716045},[31801],[34,716047,716049,716092,716100],{"className":716048,"style":700470},[31805],[34,716050,716051,716054],{"style":41504},[34,716052],{"className":716053,"style":41508},[31813],[34,716055,716057],{"className":716056},[31818,31819,31820,31821],[34,716058,716060,716063],{"className":716059},[31685,31821],[34,716061,46900],{"className":716062},[31685,207923,31821],[34,716064,716066,716069],{"className":716065},[31685,31821],[34,716067,33249],{"className":716068},[31685,31686,31821],[34,716070,716072],{"className":716071},[31792],[34,716073,716075],{"className":716074},[31796],[34,716076,716078],{"className":716077},[31801],[34,716079,716081],{"className":716080,"style":566313},[31805],[34,716082,716083,716086],{"style":116668},[34,716084],{"className":716085,"style":43406},[31813],[34,716087,716089],{"className":716088},[31818,43410,43411,31821],[34,716090,31660],{"className":716091},[31685,31686,31821],[34,716093,716094,716097],{"style":41520},[34,716095],{"className":716096,"style":41508},[31813],[34,716098],{"className":716099,"style":41528},[41527],[34,716101,716102,716105],{"style":46771},[34,716103],{"className":716104,"style":41508},[31813],[34,716106,716108],{"className":716107},[31818,31819,31820,31821],[34,716109,716111,716140],{"className":716110},[31685,31821],[34,716112,716114,716117],{"className":716113},[31685,31821],[34,716115,46900],{"className":716116},[31685,207923,31821],[34,716118,716120],{"className":716119},[31792],[34,716121,716123],{"className":716122},[31796],[34,716124,716126],{"className":716125},[31801],[34,716127,716129],{"className":716128,"style":611529},[31805],[34,716130,716131,716134],{"style":552391},[34,716132],{"className":716133,"style":43406},[31813],[34,716135,716137],{"className":716136},[31818,43410,43411,31821],[34,716138,31660],{"className":716139},[31685,31686,31821],[34,716141,32936],{"className":716142},[31685,31686,31821],[34,716144,31829],{"className":716145},[31828],[34,716147,716149],{"className":716148},[31801],[34,716150,716152],{"className":716151,"style":41619},[31805],[34,716153],{},[34,716155],{"className":716156},[31690,41488],[34,716158],{"className":716159,"style":32604},[31845],[34,716161,223],{"className":716162},[32024],[34,716164],{"className":716165,"style":32604},[31845],[34,716167,716169,716172,716212,716215,716218,716221,716365,716368,716371],{"className":716168},[31676],[34,716170],{"className":716171,"style":637019},[31680],[34,716173,716175,716178],{"className":716174},[31685],[34,716176,111],{"className":716177},[31685,31686],[34,716179,716181],{"className":716180},[31792],[34,716182,716184,716204],{"className":716183},[31796,31797],[34,716185,716187,716201],{"className":716186},[31801],[34,716188,716190],{"className":716189,"style":31806},[31805],[34,716191,716192,716195],{"style":31809},[34,716193],{"className":716194,"style":31814},[31813],[34,716196,716198],{"className":716197},[31818,31819,31820,31821],[34,716199,640],{"className":716200},[31685,31821],[34,716202,31829],{"className":716203},[31828],[34,716205,716207],{"className":716206},[31801],[34,716208,716210],{"className":716209,"style":31836},[31805],[34,716211],{},[34,716213,456],{"className":716214},[31782],[34,716216,33249],{"className":716217},[31685,31686],[34,716219,2557],{"className":716220},[31690],[34,716222,716224,716227,716362],{"className":716223},[31685],[34,716225],{"className":716226},[31782,41488],[34,716228,716230],{"className":716229},[41453],[34,716231,716233,716354],{"className":716232},[31796,31797],[34,716234,716236,716351],{"className":716235},[31801],[34,716237,716239,716291,716299],{"className":716238,"style":603956},[31805],[34,716240,716241,716244],{"style":41504},[34,716242],{"className":716243,"style":41508},[31813],[34,716245,716247],{"className":716246},[31818,31819,31820,31821],[34,716248,716250,716253],{"className":716249},[31685,31821],[34,716251,46900],{"className":716252},[31685,207923,31821],[34,716254,716256,716259],{"className":716255},[31685,31821],[34,716257,33249],{"className":716258},[31685,31686,31821],[34,716260,716262],{"className":716261},[31792],[34,716263,716265],{"className":716264},[31796],[34,716266,716268],{"className":716267},[31801],[34,716269,716271],{"className":716270,"style":548314},[31805],[34,716272,716273,716276],{"style":116668},[34,716274],{"className":716275,"style":43406},[31813],[34,716277,716279],{"className":716278},[31818,43410,43411,31821],[34,716280,716282,716285,716288],{"className":716281},[31685,31821],[34,716283,31660],{"className":716284},[31685,31686,31821],[34,716286,32226],{"className":716287},[32024,31821],[34,716289,640],{"className":716290},[31685,31821],[34,716292,716293,716296],{"style":41520},[34,716294],{"className":716295,"style":41508},[31813],[34,716297],{"className":716298,"style":41528},[41527],[34,716300,716301,716304],{"style":46771},[34,716302],{"className":716303,"style":41508},[31813],[34,716305,716307],{"className":716306},[31818,31819,31820,31821],[34,716308,716310,716348],{"className":716309},[31685,31821],[34,716311,716313,716316],{"className":716312},[31685,31821],[34,716314,46900],{"className":716315},[31685,207923,31821],[34,716317,716319],{"className":716318},[31792],[34,716320,716322],{"className":716321},[31796],[34,716323,716325],{"className":716324},[31801],[34,716326,716328],{"className":716327,"style":144123},[31805],[34,716329,716330,716333],{"style":552391},[34,716331],{"className":716332,"style":43406},[31813],[34,716334,716336],{"className":716335},[31818,43410,43411,31821],[34,716337,716339,716342,716345],{"className":716338},[31685,31821],[34,716340,31660],{"className":716341},[31685,31686,31821],[34,716343,32226],{"className":716344},[32024,31821],[34,716346,640],{"className":716347},[31685,31821],[34,716349,32936],{"className":716350},[31685,31686,31821],[34,716352,31829],{"className":716353},[31828],[34,716355,716357],{"className":716356},[31801],[34,716358,716360],{"className":716359,"style":41619},[31805],[34,716361],{},[34,716363],{"className":716364},[31690,41488],[34,716366],{"className":716367,"style":32604},[31845],[34,716369,223],{"className":716370},[32024],[34,716372],{"className":716373,"style":32604},[31845],[34,716375,716377,716380,716383,716386,716389],{"className":716376},[31676],[34,716378],{"className":716379,"style":32597},[31680],[34,716381,38874],{"className":716382},[34006],[34,716384],{"className":716385,"style":32604},[31845],[34,716387,223],{"className":716388},[32024],[34,716390],{"className":716391,"style":32604},[31845],[34,716393,716395,716398,716438,716441,716444,716447],{"className":716394},[31676],[34,716396],{"className":716397,"style":31774},[31680],[34,716399,716401,716404],{"className":716400},[31685],[34,716402,111],{"className":716403},[31685,31686],[34,716405,716407],{"className":716406},[31792],[34,716408,716410,716430],{"className":716409},[31796,31797],[34,716411,716413,716427],{"className":716412},[31801],[34,716414,716416],{"className":716415,"style":32362},[31805],[34,716417,716418,716421],{"style":31809},[34,716419],{"className":716420,"style":31814},[31813],[34,716422,716424],{"className":716423},[31818,31819,31820,31821],[34,716425,31660],{"className":716426},[31685,31686,31821],[34,716428,31829],{"className":716429},[31828],[34,716431,716433],{"className":716432},[31801],[34,716434,716436],{"className":716435,"style":31836},[31805],[34,716437],{},[34,716439,456],{"className":716440},[31782],[34,716442,33249],{"className":716443},[31685,31686],[34,716445,2557],{"className":716446},[31690],[34,716448,32936],{"className":716449},[31685,31686],[15,716451,716452,716455,716456,452821,716459,46528,716794,717140,717141,717169,717170],{},[2627,716453,716454],{},"第二步：利用导数的“线性”性质","\n我们要知道一个常识：",[2627,716457,716458],{},"和的导数 = 导数的和",[34,716460,716462,716516],{"className":716461},[31642],[34,716463,716465],{"className":716464},[31646],[31648,716466,716467],{"xmlns":31650},[31652,716468,716469,716513],{},[31655,716470,716471,716473,716479,716481,716487,716493,716495,716503,716505],{},[31662,716472,456],{"stretchy":702},[31743,716474,716475,716477],{},[31658,716476,32936],{},[31749,716478,640],{},[31662,716480,223],{},[31743,716482,716483,716485],{},[31658,716484,32936],{},[31749,716486,768],{},[37624,716488,716489,716491],{},[31662,716490,2557],{"stretchy":702},[31662,716492,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,716494,202],{},[38846,716496,716497,716499,716501],{},[31658,716498,32936],{},[31749,716500,640],{},[31662,716502,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,716504,223],{},[38846,716506,716507,716509,716511],{},[31658,716508,32936],{},[31749,716510,768],{},[31662,716512,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,716514,716515],{"encoding":31667},"(x_1 + x_2)' = x_1' + x_2'",[34,716517,716519,716577,716664,716734],{"className":716518,"ariaHidden":758},[31672],[34,716520,716522,716525,716528,716568,716571,716574],{"className":716521},[31676],[34,716523],{"className":716524,"style":31774},[31680],[34,716526,456],{"className":716527},[31782],[34,716529,716531,716534],{"className":716530},[31685],[34,716532,32936],{"className":716533},[31685,31686],[34,716535,716537],{"className":716536},[31792],[34,716538,716540,716560],{"className":716539},[31796,31797],[34,716541,716543,716557],{"className":716542},[31801],[34,716544,716546],{"className":716545,"style":31806},[31805],[34,716547,716548,716551],{"style":31809},[34,716549],{"className":716550,"style":31814},[31813],[34,716552,716554],{"className":716553},[31818,31819,31820,31821],[34,716555,640],{"className":716556},[31685,31821],[34,716558,31829],{"className":716559},[31828],[34,716561,716563],{"className":716562},[31801],[34,716564,716566],{"className":716565,"style":31836},[31805],[34,716567],{},[34,716569],{"className":716570,"style":32604},[31845],[34,716572,223],{"className":716573},[32024],[34,716575],{"className":716576,"style":32604},[31845],[34,716578,716580,716583,716623,716655,716658,716661],{"className":716579},[31676],[34,716581],{"className":716582,"style":570646},[31680],[34,716584,716586,716589],{"className":716585},[31685],[34,716587,32936],{"className":716588},[31685,31686],[34,716590,716592],{"className":716591},[31792],[34,716593,716595,716615],{"className":716594},[31796,31797],[34,716596,716598,716612],{"className":716597},[31801],[34,716599,716601],{"className":716600,"style":31806},[31805],[34,716602,716603,716606],{"style":31809},[34,716604],{"className":716605,"style":31814},[31813],[34,716607,716609],{"className":716608},[31818,31819,31820,31821],[34,716610,768],{"className":716611},[31685,31821],[34,716613,31829],{"className":716614},[31828],[34,716616,716618],{"className":716617},[31801],[34,716619,716621],{"className":716620,"style":31836},[31805],[34,716622],{},[34,716624,716626,716629],{"className":716625},[31690],[34,716627,2557],{"className":716628},[31690],[34,716630,716632],{"className":716631},[31792],[34,716633,716635],{"className":716634},[31796],[34,716636,716638],{"className":716637},[31801],[34,716639,716641],{"className":716640,"style":132027},[31805],[34,716642,716643,716646],{"style":37777},[34,716644],{"className":716645,"style":31814},[31813],[34,716647,716649],{"className":716648},[31818,31819,31820,31821],[34,716650,716652],{"className":716651},[31685,31821],[34,716653,116495],{"className":716654},[31685,31821],[34,716656],{"className":716657,"style":31846},[31845],[34,716659,202],{"className":716660},[31850],[34,716662],{"className":716663,"style":31846},[31845],[34,716665,716667,716671,716725,716728,716731],{"className":716666},[31676],[34,716668],{"className":716669,"style":716670},[31680],"height:1em;vertical-align:-0.2481em;",[34,716672,716674,716677],{"className":716673},[31685],[34,716675,32936],{"className":716676},[31685,31686],[34,716678,716680],{"className":716679},[31792],[34,716681,716683,716717],{"className":716682},[31796,31797],[34,716684,716686,716714],{"className":716685},[31801],[34,716687,716689,716700],{"className":716688,"style":132027},[31805],[34,716690,716691,716694],{"style":316930},[34,716692],{"className":716693,"style":31814},[31813],[34,716695,716697],{"className":716696},[31818,31819,31820,31821],[34,716698,640],{"className":716699},[31685,31821],[34,716701,716702,716705],{"style":37777},[34,716703],{"className":716704,"style":31814},[31813],[34,716706,716708],{"className":716707},[31818,31819,31820,31821],[34,716709,716711],{"className":716710},[31685,31821],[34,716712,116495],{"className":716713},[31685,31821],[34,716715,31829],{"className":716716},[31828],[34,716718,716720],{"className":716719},[31801],[34,716721,716723],{"className":716722,"style":39021},[31805],[34,716724],{},[34,716726],{"className":716727,"style":32604},[31845],[34,716729,223],{"className":716730},[32024],[34,716732],{"className":716733,"style":32604},[31845],[34,716735,716737,716740],{"className":716736},[31676],[34,716738],{"className":716739,"style":716670},[31680],[34,716741,716743,716746],{"className":716742},[31685],[34,716744,32936],{"className":716745},[31685,31686],[34,716747,716749],{"className":716748},[31792],[34,716750,716752,716786],{"className":716751},[31796,31797],[34,716753,716755,716783],{"className":716754},[31801],[34,716756,716758,716769],{"className":716757,"style":132027},[31805],[34,716759,716760,716763],{"style":316930},[34,716761],{"className":716762,"style":31814},[31813],[34,716764,716766],{"className":716765},[31818,31819,31820,31821],[34,716767,768],{"className":716768},[31685,31821],[34,716770,716771,716774],{"style":37777},[34,716772],{"className":716773,"style":31814},[31813],[34,716775,716777],{"className":716776},[31818,31819,31820,31821],[34,716778,716780],{"className":716779},[31685,31821],[34,716781,116495],{"className":716782},[31685,31821],[34,716784,31829],{"className":716785},[31828],[34,716787,716789],{"className":716788},[31801],[34,716790,716792],{"className":716791,"style":39021},[31805],[34,716793],{},[34,716795,716797,716863],{"className":716796},[31642],[34,716798,716800],{"className":716799},[31646],[31648,716801,716802],{"xmlns":31650},[31652,716803,716804,716860],{},[31655,716805,716806,716808,716814,716816,716822,716832,716834,716846,716848],{},[31662,716807,456],{"stretchy":702},[31743,716809,716810,716812],{},[31658,716811,32936],{},[31749,716813,640],{},[31662,716815,223],{},[31743,716817,716818,716820],{},[31658,716819,32936],{},[31749,716821,768],{},[37624,716823,716824,716826],{},[31662,716825,2557],{"stretchy":702},[31655,716827,716828,716830],{},[31662,716829,116495],{"mathvariant":38079},[31662,716831,116495],{"mathvariant":38079},[31662,716833,202],{},[38846,716835,716836,716838,716840],{},[31658,716837,32936],{},[31749,716839,640],{},[31655,716841,716842,716844],{},[31662,716843,116495],{"mathvariant":38079},[31662,716845,116495],{"mathvariant":38079},[31662,716847,223],{},[38846,716849,716850,716852,716854],{},[31658,716851,32936],{},[31749,716853,768],{},[31655,716855,716856,716858],{},[31662,716857,116495],{"mathvariant":38079},[31662,716859,116495],{"mathvariant":38079},[31665,716861,716862],{"encoding":31667},"(x_1 + x_2)'' = x_1'' + x_2''",[34,716864,716866,716924,717011,717080],{"className":716865,"ariaHidden":758},[31672],[34,716867,716869,716872,716875,716915,716918,716921],{"className":716868},[31676],[34,716870],{"className":716871,"style":31774},[31680],[34,716873,456],{"className":716874},[31782],[34,716876,716878,716881],{"className":716877},[31685],[34,716879,32936],{"className":716880},[31685,31686],[34,716882,716884],{"className":716883},[31792],[34,716885,716887,716907],{"className":716886},[31796,31797],[34,716888,716890,716904],{"className":716889},[31801],[34,716891,716893],{"className":716892,"style":31806},[31805],[34,716894,716895,716898],{"style":31809},[34,716896],{"className":716897,"style":31814},[31813],[34,716899,716901],{"className":716900},[31818,31819,31820,31821],[34,716902,640],{"className":716903},[31685,31821],[34,716905,31829],{"className":716906},[31828],[34,716908,716910],{"className":716909},[31801],[34,716911,716913],{"className":716912,"style":31836},[31805],[34,716914],{},[34,716916],{"className":716917,"style":32604},[31845],[34,716919,223],{"className":716920},[32024],[34,716922],{"className":716923,"style":32604},[31845],[34,716925,716927,716930,716970,717002,717005,717008],{"className":716926},[31676],[34,716928],{"className":716929,"style":570646},[31680],[34,716931,716933,716936],{"className":716932},[31685],[34,716934,32936],{"className":716935},[31685,31686],[34,716937,716939],{"className":716938},[31792],[34,716940,716942,716962],{"className":716941},[31796,31797],[34,716943,716945,716959],{"className":716944},[31801],[34,716946,716948],{"className":716947,"style":31806},[31805],[34,716949,716950,716953],{"style":31809},[34,716951],{"className":716952,"style":31814},[31813],[34,716954,716956],{"className":716955},[31818,31819,31820,31821],[34,716957,768],{"className":716958},[31685,31821],[34,716960,31829],{"className":716961},[31828],[34,716963,716965],{"className":716964},[31801],[34,716966,716968],{"className":716967,"style":31836},[31805],[34,716969],{},[34,716971,716973,716976],{"className":716972},[31690],[34,716974,2557],{"className":716975},[31690],[34,716977,716979],{"className":716978},[31792],[34,716980,716982],{"className":716981},[31796],[34,716983,716985],{"className":716984},[31801],[34,716986,716988],{"className":716987,"style":132027},[31805],[34,716989,716990,716993],{"style":37777},[34,716991],{"className":716992,"style":31814},[31813],[34,716994,716996],{"className":716995},[31818,31819,31820,31821],[34,716997,716999],{"className":716998},[31685,31821],[34,717000,502782],{"className":717001},[31685,31821],[34,717003],{"className":717004,"style":31846},[31845],[34,717006,202],{"className":717007},[31850],[34,717009],{"className":717010,"style":31846},[31845],[34,717012,717014,717017,717071,717074,717077],{"className":717013},[31676],[34,717015],{"className":717016,"style":716670},[31680],[34,717018,717020,717023],{"className":717019},[31685],[34,717021,32936],{"className":717022},[31685,31686],[34,717024,717026],{"className":717025},[31792],[34,717027,717029,717063],{"className":717028},[31796,31797],[34,717030,717032,717060],{"className":717031},[31801],[34,717033,717035,717046],{"className":717034,"style":132027},[31805],[34,717036,717037,717040],{"style":316930},[34,717038],{"className":717039,"style":31814},[31813],[34,717041,717043],{"className":717042},[31818,31819,31820,31821],[34,717044,640],{"className":717045},[31685,31821],[34,717047,717048,717051],{"style":37777},[34,717049],{"className":717050,"style":31814},[31813],[34,717052,717054],{"className":717053},[31818,31819,31820,31821],[34,717055,717057],{"className":717056},[31685,31821],[34,717058,502782],{"className":717059},[31685,31821],[34,717061,31829],{"className":717062},[31828],[34,717064,717066],{"className":717065},[31801],[34,717067,717069],{"className":717068,"style":39021},[31805],[34,717070],{},[34,717072],{"className":717073,"style":32604},[31845],[34,717075,223],{"className":717076},[32024],[34,717078],{"className":717079,"style":32604},[31845],[34,717081,717083,717086],{"className":717082},[31676],[34,717084],{"className":717085,"style":716670},[31680],[34,717087,717089,717092],{"className":717088},[31685],[34,717090,32936],{"className":717091},[31685,31686],[34,717093,717095],{"className":717094},[31792],[34,717096,717098,717132],{"className":717097},[31796,31797],[34,717099,717101,717129],{"className":717100},[31801],[34,717102,717104,717115],{"className":717103,"style":132027},[31805],[34,717105,717106,717109],{"style":316930},[34,717107],{"className":717108,"style":31814},[31813],[34,717110,717112],{"className":717111},[31818,31819,31820,31821],[34,717113,768],{"className":717114},[31685,31821],[34,717116,717117,717120],{"style":37777},[34,717118],{"className":717119,"style":31814},[31813],[34,717121,717123],{"className":717122},[31818,31819,31820,31821],[34,717124,717126],{"className":717125},[31685,31821],[34,717127,502782],{"className":717128},[31685,31821],[34,717130,31829],{"className":717131},[31828],[34,717133,717135],{"className":717134},[31801],[34,717136,717138],{"className":717137,"style":39021},[31805],[34,717139],{},"，以此类推。\n这意味着运算 ",[34,717142,717144,717157],{"className":717143},[31642],[34,717145,717147],{"className":717146},[31646],[31648,717148,717149],{"xmlns":31650},[31652,717150,717151,717155],{},[31655,717152,717153],{},[31658,717154,33206],{},[31665,717156,33206],{"encoding":31667},[34,717158,717160],{"className":717159,"ariaHidden":758},[31672],[34,717161,717163,717166],{"className":717162},[31676],[34,717164],{"className":717165,"style":33218},[31680],[34,717167,33206],{"className":717168},[31685,31686]," 也是线性的：\n",[34,717171,717173,717233],{"className":717172},[31642],[34,717174,717176],{"className":717175},[31646],[31648,717177,717178],{"xmlns":31650},[31652,717179,717180,717230],{},[31655,717181,717182,717184,717186,717192,717194,717200,717202,717204,717206,717208,717214,717216,717218,717220,717222,717228],{},[31658,717183,33206],{},[31662,717185,15972],{"stretchy":702},[31743,717187,717188,717190],{},[31658,717189,32936],{},[31749,717191,640],{},[31662,717193,223],{},[31743,717195,717196,717198],{},[31658,717197,32936],{},[31749,717199,768],{},[31662,717201,32712],{"stretchy":702},[31662,717203,202],{},[31658,717205,33206],{},[31662,717207,15972],{"stretchy":702},[31743,717209,717210,717212],{},[31658,717211,32936],{},[31749,717213,640],{},[31662,717215,32712],{"stretchy":702},[31662,717217,223],{},[31658,717219,33206],{},[31662,717221,15972],{"stretchy":702},[31743,717223,717224,717226],{},[31658,717225,32936],{},[31749,717227,768],{},[31662,717229,32712],{"stretchy":702},[31665,717231,717232],{"encoding":31667},"L[x_1 + x_2] = L[x_1] + L[x_2]",[34,717234,717236,717297,717355,717419],{"className":717235,"ariaHidden":758},[31672],[34,717237,717239,717242,717245,717248,717288,717291,717294],{"className":717238},[31676],[34,717240],{"className":717241,"style":31774},[31680],[34,717243,33206],{"className":717244},[31685,31686],[34,717246,15972],{"className":717247},[31782],[34,717249,717251,717254],{"className":717250},[31685],[34,717252,32936],{"className":717253},[31685,31686],[34,717255,717257],{"className":717256},[31792],[34,717258,717260,717280],{"className":717259},[31796,31797],[34,717261,717263,717277],{"className":717262},[31801],[34,717264,717266],{"className":717265,"style":31806},[31805],[34,717267,717268,717271],{"style":31809},[34,717269],{"className":717270,"style":31814},[31813],[34,717272,717274],{"className":717273},[31818,31819,31820,31821],[34,717275,640],{"className":717276},[31685,31821],[34,717278,31829],{"className":717279},[31828],[34,717281,717283],{"className":717282},[31801],[34,717284,717286],{"className":717285,"style":31836},[31805],[34,717287],{},[34,717289],{"className":717290,"style":32604},[31845],[34,717292,223],{"className":717293},[32024],[34,717295],{"className":717296,"style":32604},[31845],[34,717298,717300,717303,717343,717346,717349,717352],{"className":717299},[31676],[34,717301],{"className":717302,"style":31774},[31680],[34,717304,717306,717309],{"className":717305},[31685],[34,717307,32936],{"className":717308},[31685,31686],[34,717310,717312],{"className":717311},[31792],[34,717313,717315,717335],{"className":717314},[31796,31797],[34,717316,717318,717332],{"className":717317},[31801],[34,717319,717321],{"className":717320,"style":31806},[31805],[34,717322,717323,717326],{"style":31809},[34,717324],{"className":717325,"style":31814},[31813],[34,717327,717329],{"className":717328},[31818,31819,31820,31821],[34,717330,768],{"className":717331},[31685,31821],[34,717333,31829],{"className":717334},[31828],[34,717336,717338],{"className":717337},[31801],[34,717339,717341],{"className":717340,"style":31836},[31805],[34,717342],{},[34,717344,32712],{"className":717345},[31690],[34,717347],{"className":717348,"style":31846},[31845],[34,717350,202],{"className":717351},[31850],[34,717353],{"className":717354,"style":31846},[31845],[34,717356,717358,717361,717364,717367,717407,717410,717413,717416],{"className":717357},[31676],[34,717359],{"className":717360,"style":31774},[31680],[34,717362,33206],{"className":717363},[31685,31686],[34,717365,15972],{"className":717366},[31782],[34,717368,717370,717373],{"className":717369},[31685],[34,717371,32936],{"className":717372},[31685,31686],[34,717374,717376],{"className":717375},[31792],[34,717377,717379,717399],{"className":717378},[31796,31797],[34,717380,717382,717396],{"className":717381},[31801],[34,717383,717385],{"className":717384,"style":31806},[31805],[34,717386,717387,717390],{"style":31809},[34,717388],{"className":717389,"style":31814},[31813],[34,717391,717393],{"className":717392},[31818,31819,31820,31821],[34,717394,640],{"className":717395},[31685,31821],[34,717397,31829],{"className":717398},[31828],[34,717400,717402],{"className":717401},[31801],[34,717403,717405],{"className":717404,"style":31836},[31805],[34,717406],{},[34,717408,32712],{"className":717409},[31690],[34,717411],{"className":717412,"style":32604},[31845],[34,717414,223],{"className":717415},[32024],[34,717417],{"className":717418,"style":32604},[31845],[34,717420,717422,717425,717428,717431,717471],{"className":717421},[31676],[34,717423],{"className":717424,"style":31774},[31680],[34,717426,33206],{"className":717427},[31685,31686],[34,717429,15972],{"className":717430},[31782],[34,717432,717434,717437],{"className":717433},[31685],[34,717435,32936],{"className":717436},[31685,31686],[34,717438,717440],{"className":717439},[31792],[34,717441,717443,717463],{"className":717442},[31796,31797],[34,717444,717446,717460],{"className":717445},[31801],[34,717447,717449],{"className":717448,"style":31806},[31805],[34,717450,717451,717454],{"style":31809},[34,717452],{"className":717453,"style":31814},[31813],[34,717455,717457],{"className":717456},[31818,31819,31820,31821],[34,717458,768],{"className":717459},[31685,31821],[34,717461,31829],{"className":717462},[31828],[34,717464,717466],{"className":717465},[31801],[34,717467,717469],{"className":717468,"style":31836},[31805],[34,717470],{},[34,717472,32712],{"className":717473},[31690],[15,717475,717476,717479],{},[2627,717477,717478],{},"第三步：代入题目条件","\n题目告诉我们：",[2230,717481,717482,717741],{},[108,717483,717484,717553,717554],{},[34,717485,717487,717504],{"className":717486},[31642],[34,717488,717490],{"className":717489},[31646],[31648,717491,717492],{"xmlns":31650},[31652,717493,717494,717502],{},[31655,717495,717496],{},[31743,717497,717498,717500],{},[31658,717499,32936],{},[31749,717501,640],{},[31665,717503,224014],{"encoding":31667},[34,717505,717507],{"className":717506,"ariaHidden":758},[31672],[34,717508,717510,717513],{"className":717509},[31676],[34,717511],{"className":717512,"style":31860},[31680],[34,717514,717516,717519],{"className":717515},[31685],[34,717517,32936],{"className":717518},[31685,31686],[34,717520,717522],{"className":717521},[31792],[34,717523,717525,717545],{"className":717524},[31796,31797],[34,717526,717528,717542],{"className":717527},[31801],[34,717529,717531],{"className":717530,"style":31806},[31805],[34,717532,717533,717536],{"style":31809},[34,717534],{"className":717535,"style":31814},[31813],[34,717537,717539],{"className":717538},[31818,31819,31820,31821],[34,717540,640],{"className":717541},[31685,31821],[34,717543,31829],{"className":717544},[31828],[34,717546,717548],{"className":717547},[31801],[34,717549,717551],{"className":717550,"style":31836},[31805],[34,717552],{}," 是第一个方程的解 ",[34,717555,717557,717601],{"className":717556},[31642],[34,717558,717560],{"className":717559},[31646],[31648,717561,717562],{"xmlns":31650},[31652,717563,717564,717598],{},[31655,717565,717566,717568,717570,717572,717574,717576,717582,717584,717586,717592,717594,717596],{},[39361,717567,262622],{},[31662,717569,262625],{},[39361,717571,262622],{},[31658,717573,33206],{},[31662,717575,15972],{"stretchy":702},[31743,717577,717578,717580],{},[31658,717579,32936],{},[31749,717581,640],{},[31662,717583,32712],{"stretchy":702},[31662,717585,202],{},[31743,717587,717588,717590],{},[31658,717589,31739],{},[31749,717591,640],{},[31662,717593,456],{"stretchy":702},[31658,717595,33249],{},[31662,717597,2557],{"stretchy":702},[31665,717599,717600],{"encoding":31667},"\\implies L[x_1] = f_1(t)",[34,717602,717604,717622,717686],{"className":717603,"ariaHidden":758},[31672],[34,717605,717607,717610,717613,717616,717619],{"className":717606},[31676],[34,717608],{"className":717609,"style":318264},[31680],[34,717611],{"className":717612,"style":31846},[31845],[34,717614,262625],{"className":717615},[31850],[34,717617],{"className":717618,"style":31846},[31845],[34,717620],{"className":717621,"style":31846},[31845],[34,717623,717625,717628,717631,717634,717674,717677,717680,717683],{"className":717624},[31676],[34,717626],{"className":717627,"style":31774},[31680],[34,717629,33206],{"className":717630},[31685,31686],[34,717632,15972],{"className":717633},[31782],[34,717635,717637,717640],{"className":717636},[31685],[34,717638,32936],{"className":717639},[31685,31686],[34,717641,717643],{"className":717642},[31792],[34,717644,717646,717666],{"className":717645},[31796,31797],[34,717647,717649,717663],{"className":717648},[31801],[34,717650,717652],{"className":717651,"style":31806},[31805],[34,717653,717654,717657],{"style":31809},[34,717655],{"className":717656,"style":31814},[31813],[34,717658,717660],{"className":717659},[31818,31819,31820,31821],[34,717661,640],{"className":717662},[31685,31821],[34,717664,31829],{"className":717665},[31828],[34,717667,717669],{"className":717668},[31801],[34,717670,717672],{"className":717671,"style":31836},[31805],[34,717673],{},[34,717675,32712],{"className":717676},[31690],[34,717678],{"className":717679,"style":31846},[31845],[34,717681,202],{"className":717682},[31850],[34,717684],{"className":717685,"style":31846},[31845],[34,717687,717689,717692,717732,717735,717738],{"className":717688},[31676],[34,717690],{"className":717691,"style":31774},[31680],[34,717693,717695,717698],{"className":717694},[31685],[34,717696,31739],{"className":717697,"style":31778},[31685,31686],[34,717699,717701],{"className":717700},[31792],[34,717702,717704,717724],{"className":717703},[31796,31797],[34,717705,717707,717721],{"className":717706},[31801],[34,717708,717710],{"className":717709,"style":31806},[31805],[34,717711,717712,717715],{"style":141014},[34,717713],{"className":717714,"style":31814},[31813],[34,717716,717718],{"className":717717},[31818,31819,31820,31821],[34,717719,640],{"className":717720},[31685,31821],[34,717722,31829],{"className":717723},[31828],[34,717725,717727],{"className":717726},[31801],[34,717728,717730],{"className":717729,"style":31836},[31805],[34,717731],{},[34,717733,456],{"className":717734},[31782],[34,717736,33249],{"className":717737},[31685,31686],[34,717739,2557],{"className":717740},[31690],[108,717742,717743,717812,717813],{},[34,717744,717746,717763],{"className":717745},[31642],[34,717747,717749],{"className":717748},[31646],[31648,717750,717751],{"xmlns":31650},[31652,717752,717753,717761],{},[31655,717754,717755],{},[31743,717756,717757,717759],{},[31658,717758,32936],{},[31749,717760,768],{},[31665,717762,224085],{"encoding":31667},[34,717764,717766],{"className":717765,"ariaHidden":758},[31672],[34,717767,717769,717772],{"className":717768},[31676],[34,717770],{"className":717771,"style":31860},[31680],[34,717773,717775,717778],{"className":717774},[31685],[34,717776,32936],{"className":717777},[31685,31686],[34,717779,717781],{"className":717780},[31792],[34,717782,717784,717804],{"className":717783},[31796,31797],[34,717785,717787,717801],{"className":717786},[31801],[34,717788,717790],{"className":717789,"style":31806},[31805],[34,717791,717792,717795],{"style":31809},[34,717793],{"className":717794,"style":31814},[31813],[34,717796,717798],{"className":717797},[31818,31819,31820,31821],[34,717799,768],{"className":717800},[31685,31821],[34,717802,31829],{"className":717803},[31828],[34,717805,717807],{"className":717806},[31801],[34,717808,717810],{"className":717809,"style":31836},[31805],[34,717811],{}," 是第二个方程的解 ",[34,717814,717816,717860],{"className":717815},[31642],[34,717817,717819],{"className":717818},[31646],[31648,717820,717821],{"xmlns":31650},[31652,717822,717823,717857],{},[31655,717824,717825,717827,717829,717831,717833,717835,717841,717843,717845,717851,717853,717855],{},[39361,717826,262622],{},[31662,717828,262625],{},[39361,717830,262622],{},[31658,717832,33206],{},[31662,717834,15972],{"stretchy":702},[31743,717836,717837,717839],{},[31658,717838,32936],{},[31749,717840,768],{},[31662,717842,32712],{"stretchy":702},[31662,717844,202],{},[31743,717846,717847,717849],{},[31658,717848,31739],{},[31749,717850,768],{},[31662,717852,456],{"stretchy":702},[31658,717854,33249],{},[31662,717856,2557],{"stretchy":702},[31665,717858,717859],{"encoding":31667},"\\implies L[x_2] = f_2(t)",[34,717861,717863,717881,717945],{"className":717862,"ariaHidden":758},[31672],[34,717864,717866,717869,717872,717875,717878],{"className":717865},[31676],[34,717867],{"className":717868,"style":318264},[31680],[34,717870],{"className":717871,"style":31846},[31845],[34,717873,262625],{"className":717874},[31850],[34,717876],{"className":717877,"style":31846},[31845],[34,717879],{"className":717880,"style":31846},[31845],[34,717882,717884,717887,717890,717893,717933,717936,717939,717942],{"className":717883},[31676],[34,717885],{"className":717886,"style":31774},[31680],[34,717888,33206],{"className":717889},[31685,31686],[34,717891,15972],{"className":717892},[31782],[34,717894,717896,717899],{"className":717895},[31685],[34,717897,32936],{"className":717898},[31685,31686],[34,717900,717902],{"className":717901},[31792],[34,717903,717905,717925],{"className":717904},[31796,31797],[34,717906,717908,717922],{"className":717907},[31801],[34,717909,717911],{"className":717910,"style":31806},[31805],[34,717912,717913,717916],{"style":31809},[34,717914],{"className":717915,"style":31814},[31813],[34,717917,717919],{"className":717918},[31818,31819,31820,31821],[34,717920,768],{"className":717921},[31685,31821],[34,717923,31829],{"className":717924},[31828],[34,717926,717928],{"className":717927},[31801],[34,717929,717931],{"className":717930,"style":31836},[31805],[34,717932],{},[34,717934,32712],{"className":717935},[31690],[34,717937],{"className":717938,"style":31846},[31845],[34,717940,202],{"className":717941},[31850],[34,717943],{"className":717944,"style":31846},[31845],[34,717946,717948,717951,717991,717994,717997],{"className":717947},[31676],[34,717949],{"className":717950,"style":31774},[31680],[34,717952,717954,717957],{"className":717953},[31685],[34,717955,31739],{"className":717956,"style":31778},[31685,31686],[34,717958,717960],{"className":717959},[31792],[34,717961,717963,717983],{"className":717962},[31796,31797],[34,717964,717966,717980],{"className":717965},[31801],[34,717967,717969],{"className":717968,"style":31806},[31805],[34,717970,717971,717974],{"style":141014},[34,717972],{"className":717973,"style":31814},[31813],[34,717975,717977],{"className":717976},[31818,31819,31820,31821],[34,717978,768],{"className":717979},[31685,31821],[34,717981,31829],{"className":717982},[31828],[34,717984,717986],{"className":717985},[31801],[34,717987,717989],{"className":717988,"style":31836},[31805],[34,717990],{},[34,717992,456],{"className":717993},[31782],[34,717995,33249],{"className":717996},[31685,31686],[34,717998,2557],{"className":717999},[31690],[15,718001,718002,718005,718006,718139,718140,718272,718273,718301,718302,719106],{},[2627,718003,718004],{},"第四步：得出结论","\n我们要验证 ",[34,718007,718009,718035],{"className":718008},[31642],[34,718010,718012],{"className":718011},[31646],[31648,718013,718014],{"xmlns":31650},[31652,718015,718016,718032],{},[31655,718017,718018,718024,718026],{},[31743,718019,718020,718022],{},[31658,718021,32936],{},[31749,718023,640],{},[31662,718025,223],{},[31743,718027,718028,718030],{},[31658,718029,32936],{},[31749,718031,768],{},[31665,718033,718034],{"encoding":31667},"x_1 + x_2",[34,718036,718038,718093],{"className":718037,"ariaHidden":758},[31672],[34,718039,718041,718044,718084,718087,718090],{"className":718040},[31676],[34,718042],{"className":718043,"style":40522},[31680],[34,718045,718047,718050],{"className":718046},[31685],[34,718048,32936],{"className":718049},[31685,31686],[34,718051,718053],{"className":718052},[31792],[34,718054,718056,718076],{"className":718055},[31796,31797],[34,718057,718059,718073],{"className":718058},[31801],[34,718060,718062],{"className":718061,"style":31806},[31805],[34,718063,718064,718067],{"style":31809},[34,718065],{"className":718066,"style":31814},[31813],[34,718068,718070],{"className":718069},[31818,31819,31820,31821],[34,718071,640],{"className":718072},[31685,31821],[34,718074,31829],{"className":718075},[31828],[34,718077,718079],{"className":718078},[31801],[34,718080,718082],{"className":718081,"style":31836},[31805],[34,718083],{},[34,718085],{"className":718086,"style":32604},[31845],[34,718088,223],{"className":718089},[32024],[34,718091],{"className":718092,"style":32604},[31845],[34,718094,718096,718099],{"className":718095},[31676],[34,718097],{"className":718098,"style":31860},[31680],[34,718100,718102,718105],{"className":718101},[31685],[34,718103,32936],{"className":718104},[31685,31686],[34,718106,718108],{"className":718107},[31792],[34,718109,718111,718131],{"className":718110},[31796,31797],[34,718112,718114,718128],{"className":718113},[31801],[34,718115,718117],{"className":718116,"style":31806},[31805],[34,718118,718119,718122],{"style":31809},[34,718120],{"className":718121,"style":31814},[31813],[34,718123,718125],{"className":718124},[31818,31819,31820,31821],[34,718126,768],{"className":718127},[31685,31821],[34,718129,31829],{"className":718130},[31828],[34,718132,718134],{"className":718133},[31801],[34,718135,718137],{"className":718136,"style":31836},[31805],[34,718138],{}," 是不是第三个方程的解，就是把 ",[34,718141,718143,718168],{"className":718142},[31642],[34,718144,718146],{"className":718145},[31646],[31648,718147,718148],{"xmlns":31650},[31652,718149,718150,718166],{},[31655,718151,718152,718158,718160],{},[31743,718153,718154,718156],{},[31658,718155,32936],{},[31749,718157,640],{},[31662,718159,223],{},[31743,718161,718162,718164],{},[31658,718163,32936],{},[31749,718165,768],{},[31665,718167,715359],{"encoding":31667},[34,718169,718171,718226],{"className":718170,"ariaHidden":758},[31672],[34,718172,718174,718177,718217,718220,718223],{"className":718173},[31676],[34,718175],{"className":718176,"style":40522},[31680],[34,718178,718180,718183],{"className":718179},[31685],[34,718181,32936],{"className":718182},[31685,31686],[34,718184,718186],{"className":718185},[31792],[34,718187,718189,718209],{"className":718188},[31796,31797],[34,718190,718192,718206],{"className":718191},[31801],[34,718193,718195],{"className":718194,"style":31806},[31805],[34,718196,718197,718200],{"style":31809},[34,718198],{"className":718199,"style":31814},[31813],[34,718201,718203],{"className":718202},[31818,31819,31820,31821],[34,718204,640],{"className":718205},[31685,31821],[34,718207,31829],{"className":718208},[31828],[34,718210,718212],{"className":718211},[31801],[34,718213,718215],{"className":718214,"style":31836},[31805],[34,718216],{},[34,718218],{"className":718219,"style":32604},[31845],[34,718221,223],{"className":718222},[32024],[34,718224],{"className":718225,"style":32604},[31845],[34,718227,718229,718232],{"className":718228},[31676],[34,718230],{"className":718231,"style":31860},[31680],[34,718233,718235,718238],{"className":718234},[31685],[34,718236,32936],{"className":718237},[31685,31686],[34,718239,718241],{"className":718240},[31792],[34,718242,718244,718264],{"className":718243},[31796,31797],[34,718245,718247,718261],{"className":718246},[31801],[34,718248,718250],{"className":718249,"style":31806},[31805],[34,718251,718252,718255],{"style":31809},[34,718253],{"className":718254,"style":31814},[31813],[34,718256,718258],{"className":718257},[31818,31819,31820,31821],[34,718259,768],{"className":718260},[31685,31821],[34,718262,31829],{"className":718263},[31828],[34,718265,718267],{"className":718266},[31801],[34,718268,718270],{"className":718269,"style":31836},[31805],[34,718271],{}," 塞进 ",[34,718274,718276,718289],{"className":718275},[31642],[34,718277,718279],{"className":718278},[31646],[31648,718280,718281],{"xmlns":31650},[31652,718282,718283,718287],{},[31655,718284,718285],{},[31658,718286,33206],{},[31665,718288,33206],{"encoding":31667},[34,718290,718292],{"className":718291,"ariaHidden":758},[31672],[34,718293,718295,718298],{"className":718294},[31676],[34,718296],{"className":718297,"style":33218},[31680],[34,718299,33206],{"className":718300},[31685,31686]," 里看看：\n",[34,718303,718305,718437],{"className":718304},[31642],[34,718306,718308],{"className":718307},[31646],[31648,718309,718310],{"xmlns":31650},[31652,718311,718312,718434],{},[31655,718313,718314,718316,718318,718324,718326,718332,718334,718336,718370,718372,718406,718408,718414,718416,718418,718420,718422,718428,718430,718432],{},[31658,718315,33206],{},[31662,718317,15972],{"stretchy":702},[31743,718319,718320,718322],{},[31658,718321,32936],{},[31749,718323,640],{},[31662,718325,223],{},[31743,718327,718328,718330],{},[31658,718329,32936],{},[31749,718331,768],{},[31662,718333,32712],{"stretchy":702},[31662,718335,202],{},[71115,718337,718338,718356],{},[71115,718339,718340,718354],{},[31655,718341,718342,718344,718346,718352],{},[31658,718343,33206],{},[31662,718345,15972],{"stretchy":702},[31743,718347,718348,718350],{},[31658,718349,32936],{},[31749,718351,640],{},[31662,718353,32712],{"stretchy":702},[31662,718355,477736],{"stretchy":758},[31655,718357,718358,718364,718366,718368],{},[31743,718359,718360,718362],{},[31658,718361,31739],{},[31749,718363,640],{},[31662,718365,456],{"stretchy":702},[31658,718367,33249],{},[31662,718369,2557],{"stretchy":702},[31662,718371,223],{},[71115,718373,718374,718392],{},[71115,718375,718376,718390],{},[31655,718377,718378,718380,718382,718388],{},[31658,718379,33206],{},[31662,718381,15972],{"stretchy":702},[31743,718383,718384,718386],{},[31658,718385,32936],{},[31749,718387,768],{},[31662,718389,32712],{"stretchy":702},[31662,718391,477736],{"stretchy":758},[31655,718393,718394,718400,718402,718404],{},[31743,718395,718396,718398],{},[31658,718397,31739],{},[31749,718399,768],{},[31662,718401,456],{"stretchy":702},[31658,718403,33249],{},[31662,718405,2557],{"stretchy":702},[31662,718407,202],{},[31743,718409,718410,718412],{},[31658,718411,31739],{},[31749,718413,640],{},[31662,718415,456],{"stretchy":702},[31658,718417,33249],{},[31662,718419,2557],{"stretchy":702},[31662,718421,223],{},[31743,718423,718424,718426],{},[31658,718425,31739],{},[31749,718427,768],{},[31662,718429,456],{"stretchy":702},[31658,718431,33249],{},[31662,718433,2557],{"stretchy":702},[31665,718435,718436],{"encoding":31667},"L[x_1 + x_2] = \\underbrace{L[x_1]}_{f_1(t)} + \\underbrace{L[x_2]}_{f_2(t)} = f_1(t) + f_2(t)",[34,718438,718440,718501,718559,718774,718987,719051],{"className":718439,"ariaHidden":758},[31672],[34,718441,718443,718446,718449,718452,718492,718495,718498],{"className":718442},[31676],[34,718444],{"className":718445,"style":31774},[31680],[34,718447,33206],{"className":718448},[31685,31686],[34,718450,15972],{"className":718451},[31782],[34,718453,718455,718458],{"className":718454},[31685],[34,718456,32936],{"className":718457},[31685,31686],[34,718459,718461],{"className":718460},[31792],[34,718462,718464,718484],{"className":718463},[31796,31797],[34,718465,718467,718481],{"className":718466},[31801],[34,718468,718470],{"className":718469,"style":31806},[31805],[34,718471,718472,718475],{"style":31809},[34,718473],{"className":718474,"style":31814},[31813],[34,718476,718478],{"className":718477},[31818,31819,31820,31821],[34,718479,640],{"className":718480},[31685,31821],[34,718482,31829],{"className":718483},[31828],[34,718485,718487],{"className":718486},[31801],[34,718488,718490],{"className":718489,"style":31836},[31805],[34,718491],{},[34,718493],{"className":718494,"style":32604},[31845],[34,718496,223],{"className":718497},[32024],[34,718499],{"className":718500,"style":32604},[31845],[34,718502,718504,718507,718547,718550,718553,718556],{"className":718503},[31676],[34,718505],{"className":718506,"style":31774},[31680],[34,718508,718510,718513],{"className":718509},[31685],[34,718511,32936],{"className":718512},[31685,31686],[34,718514,718516],{"className":718515},[31792],[34,718517,718519,718539],{"className":718518},[31796,31797],[34,718520,718522,718536],{"className":718521},[31801],[34,718523,718525],{"className":718524,"style":31806},[31805],[34,718526,718527,718530],{"style":31809},[34,718528],{"className":718529,"style":31814},[31813],[34,718531,718533],{"className":718532},[31818,31819,31820,31821],[34,718534,768],{"className":718535},[31685,31821],[34,718537,31829],{"className":718538},[31828],[34,718540,718542],{"className":718541},[31801],[34,718543,718545],{"className":718544,"style":31836},[31805],[34,718546],{},[34,718548,32712],{"className":718549},[31690],[34,718551],{"className":718552,"style":31846},[31845],[34,718554,202],{"className":718555},[31850],[34,718557],{"className":718558,"style":31846},[31845],[34,718560,718562,718566,718765,718768,718771],{"className":718561},[31676],[34,718563],{"className":718564,"style":718565},[31680],"height:2.548em;vertical-align:-1.798em;",[34,718567,718569],{"className":718568},[34006,71115],[34,718570,718572,718757],{"className":718571},[31796,31797],[34,718573,718575,718754],{"className":718574},[31801],[34,718576,718578,718639],{"className":718577,"style":660852},[31805],[34,718579,718580,718583],{"style":660855},[34,718581],{"className":718582,"style":41508},[31813],[34,718584,718586],{"className":718585},[31818,31819,31820,31821],[34,718587,718589,718630,718633,718636],{"className":718588},[31685,31821],[34,718590,718592,718595],{"className":718591},[31685,31821],[34,718593,31739],{"className":718594,"style":31778},[31685,31686,31821],[34,718596,718598],{"className":718597},[31792],[34,718599,718601,718622],{"className":718600},[31796,31797],[34,718602,718604,718619],{"className":718603},[31801],[34,718605,718607],{"className":718606,"style":318658},[31805],[34,718608,718610,718613],{"style":718609},"top:-2.357em;margin-left:-0.1076em;margin-right:0.0714em;",[34,718611],{"className":718612,"style":43406},[31813],[34,718614,718616],{"className":718615},[31818,43410,43411,31821],[34,718617,640],{"className":718618},[31685,31821],[34,718620,31829],{"className":718621},[31828],[34,718623,718625],{"className":718624},[31801],[34,718626,718628],{"className":718627,"style":220540},[31805],[34,718629],{},[34,718631,456],{"className":718632},[31782,31821],[34,718634,33249],{"className":718635},[31685,31686,31821],[34,718637,2557],{"className":718638},[31690,31821],[34,718640,718641,718644],{"style":41565},[34,718642],{"className":718643,"style":41508},[31813],[34,718645,718647],{"className":718646},[34006,71115],[34,718648,718650,718746],{"className":718649},[31796,31797],[34,718651,718653,718743],{"className":718652},[31801],[34,718654,718656,718686],{"className":718655,"style":660852},[31805],[34,718657,718659,718662],{"className":718658,"style":660724},[41564],[34,718660],{"className":718661,"style":41508},[31813],[34,718663,718665,718672,718679],{"className":718664,"style":477950},[126470],[34,718666,718668],{"className":718667,"style":477955},[477954],[41588,718669,718670],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},[20398,718671],{"d":477962},[34,718673,718675],{"className":718674,"style":477955},[477966],[41588,718676,718677],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},[20398,718678],{"d":477972},[34,718680,718682],{"className":718681,"style":477955},[477976],[41588,718683,718684],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},[20398,718685],{"d":477982},[34,718687,718688,718691],{"style":41565},[34,718689],{"className":718690,"style":41508},[31813],[34,718692,718694,718697,718700,718740],{"className":718693},[31685],[34,718695,33206],{"className":718696},[31685,31686],[34,718698,15972],{"className":718699},[31782],[34,718701,718703,718706],{"className":718702},[31685],[34,718704,32936],{"className":718705},[31685,31686],[34,718707,718709],{"className":718708},[31792],[34,718710,718712,718732],{"className":718711},[31796,31797],[34,718713,718715,718729],{"className":718714},[31801],[34,718716,718718],{"className":718717,"style":31806},[31805],[34,718719,718720,718723],{"style":31809},[34,718721],{"className":718722,"style":31814},[31813],[34,718724,718726],{"className":718725},[31818,31819,31820,31821],[34,718727,640],{"className":718728},[31685,31821],[34,718730,31829],{"className":718731},[31828],[34,718733,718735],{"className":718734},[31801],[34,718736,718738],{"className":718737,"style":31836},[31805],[34,718739],{},[34,718741,32712],{"className":718742},[31690],[34,718744,31829],{"className":718745},[31828],[34,718747,718749],{"className":718748},[31801],[34,718750,718752],{"className":718751,"style":660816},[31805],[34,718753],{},[34,718755,31829],{"className":718756},[31828],[34,718758,718760],{"className":718759},[31801],[34,718761,718763],{"className":718762,"style":660979},[31805],[34,718764],{},[34,718766],{"className":718767,"style":32604},[31845],[34,718769,223],{"className":718770},[32024],[34,718772],{"className":718773,"style":32604},[31845],[34,718775,718777,718780,718978,718981,718984],{"className":718776},[31676],[34,718778],{"className":718779,"style":718565},[31680],[34,718781,718783],{"className":718782},[34006,71115],[34,718784,718786,718970],{"className":718785},[31796,31797],[34,718787,718789,718967],{"className":718788},[31801],[34,718790,718792,718852],{"className":718791,"style":660852},[31805],[34,718793,718794,718797],{"style":660855},[34,718795],{"className":718796,"style":41508},[31813],[34,718798,718800],{"className":718799},[31818,31819,31820,31821],[34,718801,718803,718843,718846,718849],{"className":718802},[31685,31821],[34,718804,718806,718809],{"className":718805},[31685,31821],[34,718807,31739],{"className":718808,"style":31778},[31685,31686,31821],[34,718810,718812],{"className":718811},[31792],[34,718813,718815,718835],{"className":718814},[31796,31797],[34,718816,718818,718832],{"className":718817},[31801],[34,718819,718821],{"className":718820,"style":318658},[31805],[34,718822,718823,718826],{"style":718609},[34,718824],{"className":718825,"style":43406},[31813],[34,718827,718829],{"className":718828},[31818,43410,43411,31821],[34,718830,768],{"className":718831},[31685,31821],[34,718833,31829],{"className":718834},[31828],[34,718836,718838],{"className":718837},[31801],[34,718839,718841],{"className":718840,"style":220540},[31805],[34,718842],{},[34,718844,456],{"className":718845},[31782,31821],[34,718847,33249],{"className":718848},[31685,31686,31821],[34,718850,2557],{"className":718851},[31690,31821],[34,718853,718854,718857],{"style":41565},[34,718855],{"className":718856,"style":41508},[31813],[34,718858,718860],{"className":718859},[34006,71115],[34,718861,718863,718959],{"className":718862},[31796,31797],[34,718864,718866,718956],{"className":718865},[31801],[34,718867,718869,718899],{"className":718868,"style":660852},[31805],[34,718870,718872,718875],{"className":718871,"style":660724},[41564],[34,718873],{"className":718874,"style":41508},[31813],[34,718876,718878,718885,718892],{"className":718877,"style":477950},[126470],[34,718879,718881],{"className":718880,"style":477955},[477954],[41588,718882,718883],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":41594},[20398,718884],{"d":477962},[34,718886,718888],{"className":718887,"style":477955},[477966],[41588,718889,718890],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477969},[20398,718891],{"d":477972},[34,718893,718895],{"className":718894,"style":477955},[477976],[41588,718896,718897],{"xmlns":41590,"width":41591,"height":477958,"viewBox":477959,"preserveAspectRatio":477979},[20398,718898],{"d":477982},[34,718900,718901,718904],{"style":41565},[34,718902],{"className":718903,"style":41508},[31813],[34,718905,718907,718910,718913,718953],{"className":718906},[31685],[34,718908,33206],{"className":718909},[31685,31686],[34,718911,15972],{"className":718912},[31782],[34,718914,718916,718919],{"className":718915},[31685],[34,718917,32936],{"className":718918},[31685,31686],[34,718920,718922],{"className":718921},[31792],[34,718923,718925,718945],{"className":718924},[31796,31797],[34,718926,718928,718942],{"className":718927},[31801],[34,718929,718931],{"className":718930,"style":31806},[31805],[34,718932,718933,718936],{"style":31809},[34,718934],{"className":718935,"style":31814},[31813],[34,718937,718939],{"className":718938},[31818,31819,31820,31821],[34,718940,768],{"className":718941},[31685,31821],[34,718943,31829],{"className":718944},[31828],[34,718946,718948],{"className":718947},[31801],[34,718949,718951],{"className":718950,"style":31836},[31805],[34,718952],{},[34,718954,32712],{"className":718955},[31690],[34,718957,31829],{"className":718958},[31828],[34,718960,718962],{"className":718961},[31801],[34,718963,718965],{"className":718964,"style":660816},[31805],[34,718966],{},[34,718968,31829],{"className":718969},[31828],[34,718971,718973],{"className":718972},[31801],[34,718974,718976],{"className":718975,"style":660979},[31805],[34,718977],{},[34,718979],{"className":718980,"style":31846},[31845],[34,718982,202],{"className":718983},[31850],[34,718985],{"className":718986,"style":31846},[31845],[34,718988,718990,718993,719033,719036,719039,719042,719045,719048],{"className":718989},[31676],[34,718991],{"className":718992,"style":31774},[31680],[34,718994,718996,718999],{"className":718995},[31685],[34,718997,31739],{"className":718998,"style":31778},[31685,31686],[34,719000,719002],{"className":719001},[31792],[34,719003,719005,719025],{"className":719004},[31796,31797],[34,719006,719008,719022],{"className":719007},[31801],[34,719009,719011],{"className":719010,"style":31806},[31805],[34,719012,719013,719016],{"style":141014},[34,719014],{"className":719015,"style":31814},[31813],[34,719017,719019],{"className":719018},[31818,31819,31820,31821],[34,719020,640],{"className":719021},[31685,31821],[34,719023,31829],{"className":719024},[31828],[34,719026,719028],{"className":719027},[31801],[34,719029,719031],{"className":719030,"style":31836},[31805],[34,719032],{},[34,719034,456],{"className":719035},[31782],[34,719037,33249],{"className":719038},[31685,31686],[34,719040,2557],{"className":719041},[31690],[34,719043],{"className":719044,"style":32604},[31845],[34,719046,223],{"className":719047},[32024],[34,719049],{"className":719050,"style":32604},[31845],[34,719052,719054,719057,719097,719100,719103],{"className":719053},[31676],[34,719055],{"className":719056,"style":31774},[31680],[34,719058,719060,719063],{"className":719059},[31685],[34,719061,31739],{"className":719062,"style":31778},[31685,31686],[34,719064,719066],{"className":719065},[31792],[34,719067,719069,719089],{"className":719068},[31796,31797],[34,719070,719072,719086],{"className":719071},[31801],[34,719073,719075],{"className":719074,"style":31806},[31805],[34,719076,719077,719080],{"style":141014},[34,719078],{"className":719079,"style":31814},[31813],[34,719081,719083],{"className":719082},[31818,31819,31820,31821],[34,719084,768],{"className":719085},[31685,31821],[34,719087,31829],{"className":719088},[31828],[34,719090,719092],{"className":719091},[31801],[34,719093,719095],{"className":719094,"style":31836},[31805],[34,719096],{},[34,719098,456],{"className":719099},[31782],[34,719101,33249],{"className":719102},[31685,31686],[34,719104,2557],{"className":719105},[31690],[2627,719107,719108],{},"得证。",[32188,719110],{},[253,719112,719114],{"id":719113},"_3-求非齐次方程的通解","3. 求非齐次方程的通解",[15,719116,719117,719118,719120],{},"利用",[2627,719119,715026],{},"求解。",[32490,719122,387575,719124,719244,719245],{"id":719123},"_1-xxcostx-x-cos-txxcost已知-x1etx2etx_1-et-x_2-e-tx1etx2et",[34,719125,719127,719161],{"className":719126},[31642],[34,719128,719130],{"className":719129},[31646],[31648,719131,719132],{"xmlns":31650},[31652,719133,719134,719158],{},[31655,719135,719136,719146,719148,719150,719152,719154,719156],{},[37624,719137,719138,719140],{},[31658,719139,32936],{},[31655,719141,719142,719144],{},[31662,719143,116495],{"mathvariant":38079},[31662,719145,116495],{"mathvariant":38079},[31662,719147,32226],{},[31658,719149,32936],{},[31662,719151,202],{},[31658,719153,547860],{},[31662,719155,32682],{},[31658,719157,33249],{},[31665,719159,719160],{"encoding":31667},"x'' - x = \\cos t",[34,719162,719164,719211,719229],{"className":719163,"ariaHidden":758},[31672],[34,719165,719167,719170,719202,719205,719208],{"className":719166},[31676],[34,719168],{"className":719169,"style":384364},[31680],[34,719171,719173,719176],{"className":719172},[31685],[34,719174,32936],{"className":719175},[31685,31686],[34,719177,719179],{"className":719178},[31792],[34,719180,719182],{"className":719181},[31796],[34,719183,719185],{"className":719184},[31801],[34,719186,719188],{"className":719187,"style":132027},[31805],[34,719189,719190,719193],{"style":37777},[34,719191],{"className":719192,"style":31814},[31813],[34,719194,719196],{"className":719195},[31818,31819,31820,31821],[34,719197,719199],{"className":719198},[31685,31821],[34,719200,502782],{"className":719201},[31685,31821],[34,719203],{"className":719204,"style":32604},[31845],[34,719206,32226],{"className":719207},[32024],[34,719209],{"className":719210,"style":32604},[31845],[34,719212,719214,719217,719220,719223,719226],{"className":719213},[31676],[34,719215],{"className":719216,"style":32527},[31680],[34,719218,32936],{"className":719219},[31685,31686],[34,719221],{"className":719222,"style":31846},[31845],[34,719224,202],{"className":719225},[31850],[34,719227],{"className":719228,"style":31846},[31845],[34,719230,719232,719235,719238,719241],{"className":719231},[31676],[34,719233],{"className":719234,"style":33280},[31680],[34,719236,547860],{"className":719237},[32767],[34,719239],{"className":719240,"style":32123},[31845],[34,719242,33249],{"className":719243},[31685,31686],"，已知 ",[34,719246,719248,719294],{"className":719247},[31642],[34,719249,719251],{"className":719250},[31646],[31648,719252,719253],{"xmlns":31650},[31652,719254,719255,719291],{},[31655,719256,719257,719263,719265,719271,719273,719279,719281],{},[31743,719258,719259,719261],{},[31658,719260,32936],{},[31749,719262,640],{},[31662,719264,202],{},[37624,719266,719267,719269],{},[31658,719268,33244],{},[31658,719270,33249],{},[31662,719272,31960],{"separator":758},[31743,719274,719275,719277],{},[31658,719276,32936],{},[31749,719278,768],{},[31662,719280,202],{},[37624,719282,719283,719285],{},[31658,719284,33244],{},[31655,719286,719287,719289],{},[31662,719288,32226],{},[31658,719290,33249],{},[31665,719292,719293],{"encoding":31667},"x_1 = e^t, x_2 = e^{-t}",[34,719295,719297,719352,719443],{"className":719296,"ariaHidden":758},[31672],[34,719298,719300,719303,719343,719346,719349],{"className":719299},[31676],[34,719301],{"className":719302,"style":31860},[31680],[34,719304,719306,719309],{"className":719305},[31685],[34,719307,32936],{"className":719308},[31685,31686],[34,719310,719312],{"className":719311},[31792],[34,719313,719315,719335],{"className":719314},[31796,31797],[34,719316,719318,719332],{"className":719317},[31801],[34,719319,719321],{"className":719320,"style":31806},[31805],[34,719322,719323,719326],{"style":31809},[34,719324],{"className":719325,"style":31814},[31813],[34,719327,719329],{"className":719328},[31818,31819,31820,31821],[34,719330,640],{"className":719331},[31685,31821],[34,719333,31829],{"className":719334},[31828],[34,719336,719338],{"className":719337},[31801],[34,719339,719341],{"className":719340,"style":31836},[31805],[34,719342],{},[34,719344],{"className":719345,"style":31846},[31845],[34,719347,202],{"className":719348},[31850],[34,719350],{"className":719351,"style":31846},[31845],[34,719353,719355,719359,719388,719391,719394,719434,719437,719440],{"className":719354},[31676],[34,719356],{"className":719357,"style":719358},[31680],"height:0.988em;vertical-align:-0.1944em;",[34,719360,719362,719365],{"className":719361},[31685],[34,719363,33244],{"className":719364},[31685,31686],[34,719366,719368],{"className":719367},[31792],[34,719369,719371],{"className":719370},[31796],[34,719372,719374],{"className":719373},[31801],[34,719375,719377],{"className":719376,"style":611758},[31805],[34,719378,719379,719382],{"style":37777},[34,719380],{"className":719381,"style":31814},[31813],[34,719383,719385],{"className":719384},[31818,31819,31820,31821],[34,719386,33249],{"className":719387},[31685,31686,31821],[34,719389,31960],{"className":719390},[32119],[34,719392],{"className":719393,"style":32123},[31845],[34,719395,719397,719400],{"className":719396},[31685],[34,719398,32936],{"className":719399},[31685,31686],[34,719401,719403],{"className":719402},[31792],[34,719404,719406,719426],{"className":719405},[31796,31797],[34,719407,719409,719423],{"className":719408},[31801],[34,719410,719412],{"className":719411,"style":31806},[31805],[34,719413,719414,719417],{"style":31809},[34,719415],{"className":719416,"style":31814},[31813],[34,719418,719420],{"className":719419},[31818,31819,31820,31821],[34,719421,768],{"className":719422},[31685,31821],[34,719424,31829],{"className":719425},[31828],[34,719427,719429],{"className":719428},[31801],[34,719430,719432],{"className":719431,"style":31836},[31805],[34,719433],{},[34,719435],{"className":719436,"style":31846},[31845],[34,719438,202],{"className":719439},[31850],[34,719441],{"className":719442,"style":31846},[31845],[34,719444,719446,719449],{"className":719445},[31676],[34,719447],{"className":719448,"style":611758},[31680],[34,719450,719452,719455],{"className":719451},[31685],[34,719453,33244],{"className":719454},[31685,31686],[34,719456,719458],{"className":719457},[31792],[34,719459,719461],{"className":719460},[31796],[34,719462,719464],{"className":719463},[31801],[34,719465,719467],{"className":719466,"style":611758},[31805],[34,719468,719469,719472],{"style":37777},[34,719470],{"className":719471,"style":31814},[31813],[34,719473,719475],{"className":719474},[31818,31819,31820,31821],[34,719476,719478,719481],{"className":719477},[31685,31821],[34,719479,32226],{"className":719480},[31685,31821],[34,719482,33249],{"className":719483},[31685,31686,31821],[15,719485,719486,719489,719555,719556,347],{},[2627,719487,719488],{},"Step 1: 检查系数",[34,719490,719492,719514],{"className":719491},[31642],[34,719493,719495],{"className":719494},[31646],[31648,719496,719497],{"xmlns":31650},[31652,719498,719499,719511],{},[31655,719500,719501],{},[37624,719502,719503,719505],{},[31658,719504,32936],{},[31655,719506,719507,719509],{},[31662,719508,116495],{"mathvariant":38079},[31662,719510,116495],{"mathvariant":38079},[31665,719512,719513],{"encoding":31667},"x''",[34,719515,719517],{"className":719516,"ariaHidden":758},[31672],[34,719518,719520,719523],{"className":719519},[31676],[34,719521],{"className":719522,"style":132027},[31680],[34,719524,719526,719529],{"className":719525},[31685],[34,719527,32936],{"className":719528},[31685,31686],[34,719530,719532],{"className":719531},[31792],[34,719533,719535],{"className":719534},[31796],[34,719536,719538],{"className":719537},[31801],[34,719539,719541],{"className":719540,"style":132027},[31805],[34,719542,719543,719546],{"style":37777},[34,719544],{"className":719545,"style":31814},[31813],[34,719547,719549],{"className":719548},[31818,31819,31820,31821],[34,719550,719552],{"className":719551},[31685,31821],[34,719553,502782],{"className":719554},[31685,31821]," 前面系数是 1，合格。",[34,719557,719559,719587],{"className":719558},[31642],[34,719560,719562],{"className":719561},[31646],[31648,719563,719564],{"xmlns":31650},[31652,719565,719566,719584],{},[31655,719567,719568,719570,719572,719574,719576,719578,719580,719582],{},[31658,719569,31739],{},[31662,719571,456],{"stretchy":702},[31658,719573,33249],{},[31662,719575,2557],{"stretchy":702},[31662,719577,202],{},[31658,719579,547860],{},[31662,719581,32682],{},[31658,719583,33249],{},[31665,719585,719586],{"encoding":31667},"f(t) = \\cos t",[34,719588,719590,719617],{"className":719589,"ariaHidden":758},[31672],[34,719591,719593,719596,719599,719602,719605,719608,719611,719614],{"className":719592},[31676],[34,719594],{"className":719595,"style":31774},[31680],[34,719597,31739],{"className":719598,"style":31778},[31685,31686],[34,719600,456],{"className":719601},[31782],[34,719603,33249],{"className":719604},[31685,31686],[34,719606,2557],{"className":719607},[31690],[34,719609],{"className":719610,"style":31846},[31845],[34,719612,202],{"className":719613},[31850],[34,719615],{"className":719616,"style":31846},[31845],[34,719618,719620,719623,719626,719629],{"className":719619},[31676],[34,719621],{"className":719622,"style":33280},[31680],[34,719624,547860],{"className":719625},[32767],[34,719627],{"className":719628,"style":32123},[31845],[34,719630,33249],{"className":719631},[31685,31686],[15,719633,719634,719637,719638,719794,719795,721062],{},[2627,719635,719636],{},"Step 2: 列方程组（常数变易法模板）","\n我们需要求 ",[34,719639,719641,719671],{"className":719640},[31642],[34,719642,719644],{"className":719643},[31646],[31648,719645,719646],{"xmlns":31650},[31652,719647,719648,719668],{},[31655,719649,719650,719658,719660],{},[38846,719651,719652,719654,719656],{},[31658,719653,31965],{},[31749,719655,640],{},[31662,719657,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,719659,31960],{"separator":758},[38846,719661,719662,719664,719666],{},[31658,719663,31965],{},[31749,719665,768],{},[31662,719667,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,719669,719670],{"encoding":31667},"c_1', c_2'",[34,719672,719674],{"className":719673,"ariaHidden":758},[31672],[34,719675,719677,719680,719734,719737,719740],{"className":719676},[31676],[34,719678],{"className":719679,"style":716670},[31680],[34,719681,719683,719686],{"className":719682},[31685],[34,719684,31965],{"className":719685},[31685,31686],[34,719687,719689],{"className":719688},[31792],[34,719690,719692,719726],{"className":719691},[31796,31797],[34,719693,719695,719723],{"className":719694},[31801],[34,719696,719698,719709],{"className":719697,"style":132027},[31805],[34,719699,719700,719703],{"style":316930},[34,719701],{"className":719702,"style":31814},[31813],[34,719704,719706],{"className":719705},[31818,31819,31820,31821],[34,719707,640],{"className":719708},[31685,31821],[34,719710,719711,719714],{"style":37777},[34,719712],{"className":719713,"style":31814},[31813],[34,719715,719717],{"className":719716},[31818,31819,31820,31821],[34,719718,719720],{"className":719719},[31685,31821],[34,719721,116495],{"className":719722},[31685,31821],[34,719724,31829],{"className":719725},[31828],[34,719727,719729],{"className":719728},[31801],[34,719730,719732],{"className":719731,"style":39021},[31805],[34,719733],{},[34,719735,31960],{"className":719736},[32119],[34,719738],{"className":719739,"style":32123},[31845],[34,719741,719743,719746],{"className":719742},[31685],[34,719744,31965],{"className":719745},[31685,31686],[34,719747,719749],{"className":719748},[31792],[34,719750,719752,719786],{"className":719751},[31796,31797],[34,719753,719755,719783],{"className":719754},[31801],[34,719756,719758,719769],{"className":719757,"style":132027},[31805],[34,719759,719760,719763],{"style":316930},[34,719761],{"className":719762,"style":31814},[31813],[34,719764,719766],{"className":719765},[31818,31819,31820,31821],[34,719767,768],{"className":719768},[31685,31821],[34,719770,719771,719774],{"style":37777},[34,719772],{"className":719773,"style":31814},[31813],[34,719775,719777],{"className":719776},[31818,31819,31820,31821],[34,719778,719780],{"className":719779},[31685,31821],[34,719781,116495],{"className":719782},[31685,31821],[34,719784,31829],{"className":719785},[31828],[34,719787,719789],{"className":719788},[31801],[34,719790,719792],{"className":719791,"style":39021},[31805],[34,719793],{},"。\n[\n",[34,719796,719798,720046],{"className":719797},[31642],[34,719799,719801],{"className":719800},[31646],[31648,719802,719803],{"xmlns":31650},[31652,719804,719805,720043],{},[31655,719806,719807,719925,719927,719929,719931],{},[31655,719808,719809,719811],{},[31662,719810,32685],{"fence":758},[71078,719812,719813,719859],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,719814,719815],{},[71087,719816,719817],{},[71090,719818,719819],{"scriptlevel":459,"displaystyle":702},[31655,719820,719821,719829,719835,719837,719845,719855,719857],{},[38846,719822,719823,719825,719827],{},[31658,719824,31965],{},[31749,719826,640],{},[31662,719828,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,719830,719831,719833],{},[31658,719832,33244],{},[31658,719834,33249],{},[31662,719836,223],{},[38846,719838,719839,719841,719843],{},[31658,719840,31965],{},[31749,719842,768],{},[31662,719844,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,719846,719847,719849],{},[31658,719848,33244],{},[31655,719850,719851,719853],{},[31662,719852,32226],{},[31658,719854,33249],{},[31662,719856,202],{},[31749,719858,459],{},[71084,719860,719861],{},[71087,719862,719863],{},[71090,719864,719865],{"scriptlevel":459,"displaystyle":702},[31655,719866,719867,719875,719877,719883,719889,719891,719899,719901,719911,719917,719919,719921,719923],{},[38846,719868,719869,719871,719873],{},[31658,719870,31965],{},[31749,719872,640],{},[31662,719874,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,719876,456],{"stretchy":702},[37624,719878,719879,719881],{},[31658,719880,33244],{},[31658,719882,33249],{},[37624,719884,719885,719887],{},[31662,719886,2557],{"stretchy":702},[31662,719888,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,719890,223],{},[38846,719892,719893,719895,719897],{},[31658,719894,31965],{},[31749,719896,768],{},[31662,719898,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,719900,456],{"stretchy":702},[37624,719902,719903,719905],{},[31658,719904,33244],{},[31655,719906,719907,719909],{},[31662,719908,32226],{},[31658,719910,33249],{},[37624,719912,719913,719915],{},[31662,719914,2557],{"stretchy":702},[31662,719916,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,719918,202],{},[31658,719920,547860],{},[31662,719922,32682],{},[31658,719924,33249],{},[39361,719926,262622],{},[31662,719928,262625],{},[39361,719930,262622],{},[31655,719932,719933,719935],{},[31662,719934,32685],{"fence":758},[71078,719936,719937,719988],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,719938,719939],{},[71087,719940,719941],{},[71090,719942,719943],{"scriptlevel":459,"displaystyle":702},[31655,719944,719945,719953,719959,719961,719969,719979,719981,719983,719985],{},[38846,719946,719947,719949,719951],{},[31658,719948,31965],{},[31749,719950,640],{},[31662,719952,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,719954,719955,719957],{},[31658,719956,33244],{},[31658,719958,33249],{},[31662,719960,223],{},[38846,719962,719963,719965,719967],{},[31658,719964,31965],{},[31749,719966,768],{},[31662,719968,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,719970,719971,719973],{},[31658,719972,33244],{},[31655,719974,719975,719977],{},[31662,719976,32226],{},[31658,719978,33249],{},[31662,719980,202],{},[31749,719982,459],{},[31845,719984],{"width":71082},[39361,719986,719987],{},"—(1)",[71084,719989,719990],{},[71087,719991,719992],{},[71090,719993,719994],{"scriptlevel":459,"displaystyle":702},[31655,719995,719996,720004,720010,720012,720020,720030,720032,720034,720036,720038,720040],{},[38846,719997,719998,720000,720002],{},[31658,719999,31965],{},[31749,720001,640],{},[31662,720003,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,720005,720006,720008],{},[31658,720007,33244],{},[31658,720009,33249],{},[31662,720011,32226],{},[38846,720013,720014,720016,720018],{},[31658,720015,31965],{},[31749,720017,768],{},[31662,720019,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,720021,720022,720024],{},[31658,720023,33244],{},[31655,720025,720026,720028],{},[31662,720027,32226],{},[31658,720029,33249],{},[31662,720031,202],{},[31658,720033,547860],{},[31662,720035,32682],{},[31658,720037,33249],{},[31845,720039],{"width":71082},[39361,720041,720042],{},"—(2)",[31665,720044,720045],{"encoding":31667},"\\begin{cases}\nc_1' e^t + c_2' e^{-t} = 0 \\\\\nc_1' (e^t)' + c_2' (e^{-t})' = \\cos t\n\\end{cases}\n\\implies\n\\begin{cases}\nc_1' e^t + c_2' e^{-t} = 0 \\quad \\text{---(1)} \\\\\nc_1' e^t - c_2' e^{-t} = \\cos t \\quad \\text{---(2)}\n\\end{cases}",[34,720047,720049,720589],{"className":720048,"ariaHidden":758},[31672],[34,720050,720052,720055,720574,720577,720580,720583,720586],{"className":720051},[31676],[34,720053],{"className":720054,"style":75797},[31680],[34,720056,720058,720064,720571],{"className":720057},[34006],[34,720059,720061],{"className":720060,"style":75811},[31782,75810],[34,720062,32685],{"className":720063},[71281,75815],[34,720065,720067],{"className":720066},[31685],[34,720068,720070],{"className":720069},[71078],[34,720071,720073],{"className":720072},[71384],[34,720074,720076,720563],{"className":720075},[31796,31797],[34,720077,720079,720560],{"className":720078},[31801],[34,720080,720082,720283],{"className":720081,"style":75834},[31805],[34,720083,720084,720087],{"style":75837},[34,720085],{"className":720086,"style":71401},[31813],[34,720088,720090,720144,720173,720176,720179,720182,720236,720271,720274,720277,720280],{"className":720089},[31685],[34,720091,720093,720096],{"className":720092},[31685],[34,720094,31965],{"className":720095},[31685,31686],[34,720097,720099],{"className":720098},[31792],[34,720100,720102,720136],{"className":720101},[31796,31797],[34,720103,720105,720133],{"className":720104},[31801],[34,720106,720108,720119],{"className":720107,"style":132027},[31805],[34,720109,720110,720113],{"style":316930},[34,720111],{"className":720112,"style":31814},[31813],[34,720114,720116],{"className":720115},[31818,31819,31820,31821],[34,720117,640],{"className":720118},[31685,31821],[34,720120,720121,720124],{"style":37777},[34,720122],{"className":720123,"style":31814},[31813],[34,720125,720127],{"className":720126},[31818,31819,31820,31821],[34,720128,720130],{"className":720129},[31685,31821],[34,720131,116495],{"className":720132},[31685,31821],[34,720134,31829],{"className":720135},[31828],[34,720137,720139],{"className":720138},[31801],[34,720140,720142],{"className":720141,"style":39021},[31805],[34,720143],{},[34,720145,720147,720150],{"className":720146},[31685],[34,720148,33244],{"className":720149},[31685,31686],[34,720151,720153],{"className":720152},[31792],[34,720154,720156],{"className":720155},[31796],[34,720157,720159],{"className":720158},[31801],[34,720160,720162],{"className":720161,"style":611758},[31805],[34,720163,720164,720167],{"style":37777},[34,720165],{"className":720166,"style":31814},[31813],[34,720168,720170],{"className":720169},[31818,31819,31820,31821],[34,720171,33249],{"className":720172},[31685,31686,31821],[34,720174],{"className":720175,"style":32604},[31845],[34,720177,223],{"className":720178},[32024],[34,720180],{"className":720181,"style":32604},[31845],[34,720183,720185,720188],{"className":720184},[31685],[34,720186,31965],{"className":720187},[31685,31686],[34,720189,720191],{"className":720190},[31792],[34,720192,720194,720228],{"className":720193},[31796,31797],[34,720195,720197,720225],{"className":720196},[31801],[34,720198,720200,720211],{"className":720199,"style":132027},[31805],[34,720201,720202,720205],{"style":316930},[34,720203],{"className":720204,"style":31814},[31813],[34,720206,720208],{"className":720207},[31818,31819,31820,31821],[34,720209,768],{"className":720210},[31685,31821],[34,720212,720213,720216],{"style":37777},[34,720214],{"className":720215,"style":31814},[31813],[34,720217,720219],{"className":720218},[31818,31819,31820,31821],[34,720220,720222],{"className":720221},[31685,31821],[34,720223,116495],{"className":720224},[31685,31821],[34,720226,31829],{"className":720227},[31828],[34,720229,720231],{"className":720230},[31801],[34,720232,720234],{"className":720233,"style":39021},[31805],[34,720235],{},[34,720237,720239,720242],{"className":720238},[31685],[34,720240,33244],{"className":720241},[31685,31686],[34,720243,720245],{"className":720244},[31792],[34,720246,720248],{"className":720247},[31796],[34,720249,720251],{"className":720250},[31801],[34,720252,720254],{"className":720253,"style":611758},[31805],[34,720255,720256,720259],{"style":37777},[34,720257],{"className":720258,"style":31814},[31813],[34,720260,720262],{"className":720261},[31818,31819,31820,31821],[34,720263,720265,720268],{"className":720264},[31685,31821],[34,720266,32226],{"className":720267},[31685,31821],[34,720269,33249],{"className":720270},[31685,31686,31821],[34,720272],{"className":720273,"style":31846},[31845],[34,720275,202],{"className":720276},[31850],[34,720278],{"className":720279,"style":31846},[31845],[34,720281,459],{"className":720282},[31685],[34,720284,720285,720288],{"style":76011},[34,720286],{"className":720287,"style":71401},[31813],[34,720289,720291,720345,720348,720377,720409,720412,720415,720418,720472,720475,720510,720542,720545,720548,720551,720554,720557],{"className":720290},[31685],[34,720292,720294,720297],{"className":720293},[31685],[34,720295,31965],{"className":720296},[31685,31686],[34,720298,720300],{"className":720299},[31792],[34,720301,720303,720337],{"className":720302},[31796,31797],[34,72030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42},[31685],[34,720844,31965],{"className":720845},[31685,31686],[34,720847,720849],{"className":720848},[31792],[34,720850,720852,720886],{"className":720851},[31796,31797],[34,720853,720855,720883],{"className":720854},[31801],[34,720856,720858,720869],{"className":720857,"style":132027},[31805],[34,720859,720860,720863],{"style":316930},[34,720861],{"className":720862,"style":31814},[31813],[34,720864,720866],{"className":720865},[31818,31819,31820,31821],[34,720867,640],{"className":720868},[31685,31821],[34,720870,720871,720874],{"style":37777},[34,720872],{"className":720873,"style":31814},[31813],[34,720875,720877],{"className":720876},[31818,31819,31820,31821],[34,720878,720880],{"className":720879},[31685,31821],[34,720881,116495],{"className":720882},[31685,31821],[34,720884,31829],{"className":720885},[31828],[34,720887,720889],{"className":720888},[31801],[34,720890,720892],{"className":720891,"style":39021},[31805],[34,720893],{},[34,720895,720897,720900],{"className":720896},[31685],[34,720898,33244],{"className":720899},[31685,31686],[34,720901,720903],{"className":720902},[31792],[34,720904,720906],{"className":720905},[31796],[34,720907,720909],{"className":720908},[31801],[34,720910,720912],{"className":720911,"style":611758},[31805],[34,720913,720914,720917],{"style":37777},[34,720915],{"className":720916,"style":31814},[31813],[34,720918,720920],{"className":720919},[31818,31819,31820,31821],[34,720921,33249],{"className":720922},[31685,31686,31821],[34,720924],{"className":720925,"style":32604},[31845],[34,720927,32226],{"className":720928},[32024],[34,720930],{"className":720931,"style":32604},[31845],[34,720933,720935,720938],{"className":720934},[31685],[34,720936,31965],{"className":720937},[31685,31686],[34,720939,720941],{"className":720940},[31792],[34,720942,720944,720978],{"className":720943},[31796,31797],[34,720945,720947,720975],{"className":720946},[31801],[34,720948,720950,720961],{"className":720949,"style":132027},[31805],[34,720951,720952,720955],{"style":316930},[34,720953],{"className":720954,"style":31814},[31813],[34,720956,720958],{"className":720957},[31818,31819,31820,31821],[34,720959,768],{"className":720960},[31685,31821],[34,720962,720963,720966],{"style":37777},[34,720964],{"className":720965,"style":31814},[31813],[34,720967,720969],{"className":720968},[31818,31819,31820,31821],[34,720970,720972],{"className":720971},[31685,31821],[34,720973,116495],{"className":720974},[31685,31821],[34,720976,31829],{"className":720977},[31828],[34,720979,720981],{"className":720980},[31801],[34,720982,720984],{"className":720983,"style":39021},[31805],[34,720985],{},[34,720987,720989,720992],{"className":720988},[31685],[34,720990,33244],{"className":720991},[31685,31686],[34,720993,720995],{"className":720994},[31792],[34,720996,720998],{"className":720997},[31796],[34,720999,721001],{"className":721000},[31801],[34,721002,721004],{"className":721003,"style":611758},[31805],[34,721005,721006,721009],{"style":37777},[34,721007],{"className":721008,"style":31814},[31813],[34,721010,721012],{"className":721011},[31818,31819,31820,31821],[34,721013,721015,721018],{"className":721014},[31685,31821],[34,721016,32226],{"className":721017},[31685,31821],[34,721019,33249],{"className":721020},[31685,31686,31821],[34,721022],{"className":721023,"style":31846},[31845],[34,721025,202],{"className":721026},[31850],[34,721028],{"className":721029,"style":31846},[31845],[34,721031,547860],{"className":721032},[32767],[34,721034],{"className":721035,"style":32123},[31845],[34,721037,33249],{"className":721038},[31685,31686],[34,721040],{"className":721041,"style":128438},[31845],[34,721043,721045],{"className":721044},[31685,2638],[34,721046,720042],{"className":721047},[31685],[34,721049,31829],{"className":721050},[31828],[34,721052,721054],{"className":721053},[31801],[34,721055,721057],{"className":721056,"style":76192},[31805],[34,721058],{},[34,721060],{"className":721061},[31690,41488],"\n]",[15,721064,721065,721068,721069,721472,721473],{},[2627,721066,721067],{},"Step 3: 解方程组","\n(1) + (2) 得：",[34,721070,721072,721146],{"className":721071},[31642],[34,721073,721075],{"className":721074},[31646],[31648,721076,721077],{"xmlns":31650},[31652,721078,721079,721143],{},[31655,721080,721081,721083,721091,721097,721099,721101,721103,721105,721107,721109,721111,721119,721121,721127,721137,721139,721141],{},[31749,721082,768],{},[38846,721084,721085,721087,721089],{},[31658,721086,31965],{},[31749,721088,640],{},[31662,721090,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,721092,721093,721095],{},[31658,721094,33244],{},[31658,721096,33249],{},[31662,721098,202],{},[31658,721100,547860],{},[31662,721102,32682],{},[31658,721104,33249],{},[39361,721106,262622],{},[31662,721108,262625],{},[39361,721110,262622],{},[38846,721112,721113,721115,721117],{},[31658,721114,31965],{},[31749,721116,640],{},[31662,721118,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,721120,202],{},[41453,721122,721123,721125],{},[31749,721124,640],{},[31749,721126,768],{},[37624,721128,721129,721131],{},[31658,721130,33244],{},[31655,721132,721133,721135],{},[31662,721134,32226],{},[31658,721136,33249],{},[31658,721138,547860],{},[31662,721140,32682],{},[31658,721142,33249],{},[31665,721144,721145],{"encoding":31667},"2c_1' e^t = \\cos t \\implies c_1' = \\frac{1}{2}e^{-t}\\cos t",[34,721147,721149,721251,721282,721351],{"className":721148,"ariaHidden":758},[31672],[34,721150,721152,721156,721159,721213,721242,721245,721248],{"className":721151},[31676],[34,721153],{"className":721154,"style":721155},[31680],"height:1.0417em;vertical-align:-0.2481em;",[34,721157,768],{"className":721158},[31685],[34,721160,721162,721165],{"className":721161},[31685],[34,721163,31965],{"className":721164},[31685,31686],[34,721166,721168],{"className":721167},[31792],[34,721169,721171,721205],{"className":721170},[31796,31797],[34,721172,721174,721202],{"className":721173},[31801],[34,721175,721177,721188],{"className":721176,"style":132027},[31805],[34,721178,721179,721182],{"style":316930},[34,721180],{"className":721181,"style":31814},[31813],[34,721183,721185],{"className":721184},[31818,31819,31820,31821],[34,721186,640],{"className":721187},[31685,31821],[34,721189,721190,721193],{"style":37777},[34,721191],{"className":721192,"style":31814},[31813],[34,721194,721196],{"className":721195},[31818,31819,31820,31821],[34,721197,721199],{"className":721198},[31685,31821],[34,721200,116495],{"className":721201},[31685,31821],[34,721203,31829],{"className":721204},[31828],[34,721206,721208],{"className":721207},[31801],[34,721209,721211],{"className":721210,"style":39021},[31805],[34,721212],{},[34,721214,721216,721219],{"className":721215},[31685],[34,721217,33244],{"className":721218},[31685,31686],[34,721220,721222],{"className":721221},[31792],[34,721223,721225],{"className":721224},[31796],[34,721226,721228],{"className":721227},[31801],[34,721229,721231],{"className":721230,"style":611758},[31805],[34,721232,721233,721236],{"style":37777},[34,721234],{"className":721235,"style":31814},[31813],[34,721237,721239],{"className":721238},[31818,31819,31820,31821],[34,721240,33249],{"className":721241},[31685,31686,31821],[34,721243],{"className":721244,"style":31846},[31845],[34,721246,202],{"className":721247},[31850],[34,721249],{"className":721250,"style":31846},[31845],[34,721252,721254,721258,721261,721264,721267,721270,721273,721276,721279],{"className":721253},[31676],[34,721255],{"className":721256,"style":721257},[31680],"height:0.6391em;vertical-align:-0.024em;",[34,721259,547860],{"className":721260},[32767],[34,721262],{"className":721263,"style":32123},[31845],[34,721265,33249],{"className":721266},[31685,31686],[34,721268],{"className":721269,"style":31846},[31845],[34,721271],{"className":721272,"style":31846},[31845],[34,721274,262625],{"className":721275},[31850],[34,721277],{"className":721278,"style":31846},[31845],[34,721280],{"className":721281,"style":31846},[31845],[34,721283,721285,721288,721342,721345,721348],{"className":721284},[31676],[34,721286],{"className":721287,"style":716670},[31680],[34,721289,721291,721294],{"className":721290},[31685],[34,721292,31965],{"className":721293},[31685,31686],[34,721295,721297],{"className":721296},[31792],[34,721298,721300,721334],{"className":721299},[31796,31797],[34,721301,721303,721331],{"className":721302},[31801],[34,721304,721306,721317],{"className":721305,"style":132027},[31805],[34,721307,721308,721311],{"style":316930},[34,721309],{"className":721310,"style":31814},[31813],[34,721312,721314],{"className":721313},[31818,31819,31820,31821],[34,721315,640],{"className":721316},[31685,31821],[34,721318,721319,721322],{"style":37777},[34,721320],{"className":721321,"style":31814},[31813],[34,721323,721325],{"className":721324},[31818,31819,31820,31821],[34,721326,721328],{"className":721327},[31685,31821],[34,721329,116495],{"className":721330},[31685,31821],[34,721332,31829],{"className":721333},[31828],[34,721335,721337],{"className":721336},[31801],[34,721338,721340],{"className":721339,"style":39021},[31805],[34,721341],{},[34,721343],{"className":721344,"style":31846},[31845],[34,721346,202],{"className":721347},[31850],[34,721349],{"className":721350,"style":31846},[31845],[34,721352,721354,721357,721425,721460,721463,721466,721469],{"className":721353},[31676],[34,721355],{"className":721356,"style":46727},[31680],[34,721358,721360,721363,721422],{"className":721359},[31685],[34,721361],{"className":721362},[31782,41488],[34,721364,721366],{"className":721365},[41453],[34,721367,721369,721414],{"className":721368},[31796,31797],[34,721370,721372,721411],{"className":721371},[31801],[34,721373,721375,721389,721397],{"className":721374,"style":46746},[31805],[34,721376,721377,721380],{"style":41504},[34,721378],{"className":721379,"style":41508},[31813],[34,721381,721383],{"className":721382},[31818,31819,31820,31821],[34,721384,721386],{"className":721385},[31685,31821],[34,721387,768],{"className":721388},[31685,31821],[34,721390,721391,721394],{"style":41520},[34,721392],{"className":721393,"style":41508},[31813],[34,721395],{"className":721396,"style":41528},[41527],[34,721398,721399,721402],{"style":46771},[34,721400],{"className":721401,"style":41508},[31813],[34,721403,721405],{"className":721404},[31818,31819,31820,31821],[34,721406,721408],{"className":721407},[31685,31821],[34,721409,640],{"className":721410},[31685,31821],[34,721412,31829],{"className":721413},[31828],[34,721415,721417],{"className":721416},[31801],[34,721418,721420],{"className":721419,"style":41619},[31805],[34,721421],{},[34,721423],{"className":721424},[31690,41488],[34,721426,721428,721431],{"className":721427},[31685],[34,721429,33244],{"className":721430},[31685,31686],[34,721432,721434],{"className":721433},[31792],[34,721435,721437],{"className":721436},[31796],[34,721438,721440],{"className":721439},[31801],[34,721441,721443],{"className":721442,"style":611758},[31805],[34,721444,721445,721448],{"style":37777},[34,721446],{"className":721447,"style":31814},[31813],[34,721449,721451],{"className":721450},[31818,31819,31820,31821],[34,721452,721454,721457],{"className":721453},[31685,31821],[34,721455,32226],{"className":721456},[31685,31821],[34,721458,33249],{"className":721459},[31685,31686,31821],[34,721461],{"className":721462,"style":32123},[31845],[34,721464,547860],{"className":721465},[32767],[34,721467],{"className":721468,"style":32123},[31845],[34,721470,33249],{"className":721471},[31685,31686],"\n(1) - (2) 得：",[34,721474,721476,721554],{"className":721475},[31642],[34,721477,721479],{"className":721478},[31646],[31648,721480,721481],{"xmlns":31650},[31652,721482,721483,721551],{},[31655,721484,721485,721487,721495,721505,721507,721509,721511,721513,721515,721517,721519,721521,721529,721531,721533,721539,721545,721547,721549],{},[31749,721486,768],{},[38846,721488,721489,721491,721493],{},[31658,721490,31965],{},[31749,721492,768],{},[31662,721494,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[37624,721496,721497,721499],{},[31658,721498,33244],{},[31655,721500,721501,721503],{},[31662,721502,32226],{},[31658,721504,33249],{},[31662,721506,202],{},[31662,721508,32226],{},[31658,721510,547860],{},[31662,721512,32682],{},[31658,721514,33249],{},[39361,721516,262622],{},[31662,721518,262625],{},[39361,721520,262622],{},[38846,721522,721523,721525,721527],{},[31658,721524,31965],{},[31749,721526,768],{},[31662,721528,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,721530,202],{},[31662,721532,32226],{},[41453,721534,721535,721537],{},[31749,721536,640],{},[31749,721538,768],{},[37624,721540,721541,721543],{},[31658,721542,33244],{},[31658,721544,33249],{},[31658,721546,547860],{},[31662,721548,32682],{},[31658,721550,33249],{},[31665,721552,721553],{"encoding":31667},"2c_2' e^{-t} = -\\cos t \\implies c_2' = -\\frac{1}{2}e^t\\cos t",[34,721555,721557,721664,721701,721770],{"className":721556,"ariaHidden":758},[31672],[34,721558,721560,721563,721566,721620,721655,721658,721661],{"className":721559},[31676],[34,721561],{"className":721562,"style":721155},[31680],[34,721564,768],{"className":721565},[31685],[34,721567,721569,721572],{"className":721568},[31685],[34,721570,31965],{"className":721571},[31685,31686],[34,721573,721575],{"className":721574},[31792],[34,721576,721578,721612],{"className":721577},[31796,31797],[34,721579,721581,721609],{"className":721580},[31801],[34,721582,721584,721595],{"className":721583,"style":132027},[31805],[34,721585,721586,721589],{"style":316930},[34,721587],{"className":721588,"style":31814},[31813],[34,721590,721592],{"className":721591},[31818,31819,31820,31821],[34,721593,768],{"className":721594},[31685,31821],[34,721596,721597,721600],{"style":37777},[34,721598],{"className":721599,"style":31814},[31813],[34,721601,721603],{"className":721602},[31818,31819,31820,31821],[34,721604,721606],{"className":721605},[31685,31821],[34,721607,116495],{"className":721608},[31685,31821],[34,721610,31829],{"className":721611},[31828],[34,721613,721615],{"className":721614},[31801],[34,721616,721618],{"className":721617,"style":39021},[31805],[34,721619],{},[34,721621,721623,721626],{"className":721622},[31685],[34,721624,33244],{"className":721625},[31685,31686],[34,721627,721629],{"className":721628},[31792],[34,721630,721632],{"className":721631},[31796],[34,721633,721635],{"className":721634},[31801],[34,721636,721638],{"className":721637,"style":611758},[31805],[34,721639,721640,721643],{"style":37777},[34,721641],{"className":721642,"style":31814},[31813],[34,721644,721646],{"className":721645},[31818,31819,31820,31821],[34,721647,721649,721652],{"className":721648},[31685,31821],[34,721650,32226],{"className":721651},[31685,31821],[34,721653,33249],{"className":721654},[31685,31686,31821],[34,721656],{"className":721657,"style":31846},[31845],[34,721659,202],{"className":721660},[31850],[34,721662],{"className":721663,"style":31846},[31845],[34,721665,721667,721671,721674,721677,721680,721683,721686,721689,721692,721695,721698],{"className":721666},[31676],[34,721668],{"className":721669,"style":721670},[31680],"height:0.6984em;vertical-align:-0.0833em;",[34,721672,32226],{"className":721673},[31685],[34,721675],{"className":721676,"style":32123},[31845],[34,721678,547860],{"className":721679},[32767],[34,721681],{"className":721682,"style":32123},[31845],[34,721684,33249],{"className":721685},[31685,31686],[34,721687],{"className":721688,"style":31846},[31845],[34,721690],{"className":721691,"style":31846},[31845],[34,721693,262625],{"className":721694},[31850],[34,721696],{"className":721697,"style":31846},[31845],[34,721699],{"className":721700,"style":31846},[31845],[34,721702,721704,721707,721761,721764,721767],{"className":721703},[31676],[34,721705],{"className":721706,"style":716670},[31680],[34,721708,721710,721713],{"className":721709},[31685],[34,721711,31965],{"className":721712},[31685,31686],[34,721714,721716],{"className":721715},[31792],[34,721717,721719,721753],{"className":721718},[31796,31797],[34,721720,721722,721750],{"className":721721},[31801],[34,721723,721725,721736],{"className":721724,"style":132027},[31805],[34,721726,721727,721730],{"style":316930},[34,721728],{"className":721729,"style":31814},[31813],[34,721731,721733],{"className":721732},[31818,31819,31820,31821],[34,721734,768],{"className":721735},[31685,31821],[34,721737,721738,721741],{"style":37777},[34,721739],{"className":721740,"style":31814},[31813],[34,721742,721744],{"className":721743},[31818,31819,31820,31821],[34,721745,721747],{"className":721746},[31685,31821],[34,721748,116495],{"className":721749},[31685,31821],[34,721751,31829],{"className":721752},[31828],[34,721754,721756],{"className":721755},[31801],[34,721757,721759],{"className":721758,"style":39021},[31805],[34,721760],{},[34,721762],{"className":721763,"style":31846},[31845],[34,721765,202],{"className":721766},[31850],[34,721768],{"className":721769,"style":31846},[31845],[34,721771,721773,721776,721779,721847,721876,721879,721882,721885],{"className":721772},[31676],[34,721774],{"className":721775,"style":46727},[31680],[34,721777,32226],{"className":721778},[31685],[34,721780,721782,721785,721844],{"className":721781},[31685],[34,721783],{"className":721784},[31782,41488],[34,721786,721788],{"className":721787},[41453],[34,721789,721791,721836],{"className":721790},[31796,31797],[34,721792,721794,721833],{"className":721793},[31801],[34,721795,721797,721811,721819],{"className":721796,"style":46746},[31805],[34,721798,721799,721802],{"style":41504},[34,721800],{"className":721801,"style":41508},[31813],[34,721803,721805],{"className":721804},[31818,31819,31820,31821],[34,721806,721808],{"className":721807},[31685,31821],[34,721809,768],{"className":721810},[31685,31821],[34,721812,721813,721816],{"style":41520},[34,721814],{"className":721815,"style":41508},[31813],[34,721817],{"className":721818,"style":41528},[41527],[34,721820,721821,721824],{"style":46771},[34,721822],{"className":721823,"style":41508},[31813],[34,721825,721827],{"className":721826},[31818,31819,31820,31821],[34,721828,721830],{"className":721829},[31685,31821],[34,721831,640],{"className":721832},[31685,31821],[34,721834,31829],{"className":721835},[31828],[34,721837,721839],{"className":721838},[31801],[34,721840,721842],{"className":721841,"style":41619},[31805],[34,721843],{},[34,721845],{"className":721846},[31690,41488],[34,721848,721850,721853],{"className":721849},[31685],[34,721851,33244],{"className":721852},[31685,31686],[34,721854,721856],{"className":721855},[31792],[34,721857,721859],{"className":721858},[31796],[34,721860,721862],{"className":721861},[31801],[34,721863,721865],{"className":721864,"style":611758},[31805],[34,721866,721867,721870],{"style":37777},[34,721868],{"className":721869,"style":31814},[31813],[34,721871,721873],{"className":721872},[31818,31819,31820,31821],[34,721874,33249],{"className":721875},[31685,31686,31821],[34,721877],{"className":721878,"style":32123},[31845],[34,721880,547860],{"className":721881},[32767],[34,721883],{"className":721884,"style":32123},[31845],[34,721886,33249],{"className":721887},[31685,31686],[15,721889,721890,721893,721894,721999,722000,82346],{},[2627,721891,721892],{},"Step 4: 积分（这里考微积分基本功）","\n这两个积分需要用到“分部积分法”或者“欧拉公式”。\n（注：",[34,721895,721897,721931],{"className":721896},[31642],[34,721898,721900],{"className":721899},[31646],[31648,721901,721902],{"xmlns":31650},[31652,721903,721904,721928],{},[31655,721905,721906,721908,721918,721920,721922,721924,721926],{},[31662,721907,546295],{},[37624,721909,721910,721912],{},[31658,721911,33244],{},[31655,721913,721914,721916],{},[31658,721915,111],{},[31658,721917,33249],{},[31658,721919,547860],{},[31662,721921,32682],{},[31658,721923,33249],{},[31658,721925,46900],{},[31658,721927,33249],{},[31665,721929,721930],{"encoding":31667},"\\int e^{at}\\cos t dt",[34,721932,721934],{"className":721933,"ariaHidden":758},[31672],[34,721935,721937,721940,721943,721946,721981,721984,721987,721990,721993,721996],{"className":721936},[31676],[34,721938],{"className":721939,"style":546896},[31680],[34,721941,546295],{"className":721942,"style":546331},[32767,131249,156560],[34,721944],{"className":721945,"style":32123},[31845],[34,721947,721949,721952],{"className":721948},[31685],[34,721950,33244],{"className":721951},[31685,31686],[34,721953,721955],{"className":721954},[31792],[34,721956,721958],{"className":721957},[31796],[34,721959,721961],{"className":721960},[31801],[34,721962,721964],{"className":721963,"style":611758},[31805],[34,721965,721966,721969],{"style":37777},[34,721967],{"className":721968,"style":31814},[31813],[34,721970,721972],{"className":721971},[31818,31819,31820,31821],[34,721973,721975,721978],{"className":721974},[31685,31821],[34,721976,111],{"className":721977},[31685,31686,31821],[34,721979,33249],{"className":721980},[31685,31686,31821],[34,721982],{"className":721983,"style":32123},[31845],[34,721985,547860],{"className":721986},[32767],[34,721988],{"className":721989,"style":32123},[31845],[34,721991,33249],{"className":721992},[31685,31686],[34,721994,46900],{"className":721995},[31685,31686],[34,721997,33249],{"className":721998},[31685,31686]," 是经典循环积分，公式为 ",[34,722001,722003,722061],{"className":722002},[31642],[34,722004,722006],{"className":722005},[31646],[31648,722007,722008],{"xmlns":31650},[31652,722009,722010,722058],{},[31655,722011,722012],{},[41453,722013,722014,722046],{},[31655,722015,722016,722026,722028,722030,722032,722034,722036,722038,722040,722042,722044],{},[37624,722017,722018,722020],{},[31658,722019,33244],{},[31655,722021,722022,722024],{},[31658,722023,111],{},[31658,722025,33249],{},[31662,722027,456],{"stretchy":702},[31658,722029,111],{},[31658,722031,547860],{},[31662,722033,32682],{},[31658,722035,33249],{},[31662,722037,223],{},[31658,722039,547853],{},[31662,722041,32682],{},[31658,722043,33249],{},[31662,722045,2557],{"stretchy":702},[31655,722047,722048,722054,722056],{},[37624,722049,722050,722052],{},[31658,722051,111],{},[31749,722053,768],{},[31662,722055,223],{},[31749,722057,640],{},[31665,722059,722060],{"encoding":31667},"\\frac{e^{at}(a\\cos t + \\sin t)}{a^2+1}",[34,722062,722064],{"className":722063,"ariaHidden":758},[31672],[34,722065,722067,722071],{"className":722066},[31676],[34,722068],{"className":722069,"style":722070},[31680],"height:1.4976em;vertical-align:-0.4033em;",[34,722072,722074,722077,722253],{"className":722073},[31685],[34,722075],{"className":722076},[31782,41488],[34,722078,722080],{"className":722079},[41453],[34,722081,722083,722245],{"className":722082},[31796,31797],[34,722084,722086,722242],{"className":722085},[31801],[34,722087,722090,722136,722144],{"className":722088,"style":722089},[31805],"height:1.0942em;",[34,722091,722092,722095],{"style":41504},[34,722093],{"className":722094,"style":41508},[31813],[34,722096,722098],{"className":722097},[31818,31819,31820,31821],[34,722099,722101,722130,722133],{"className":722100},[31685,31821],[34,722102,722104,722107],{"className":722103},[31685,31821],[34,722105,111],{"className":722106},[31685,31686,31821],[34,722108,722110],{"className":722109},[31792],[34,722111,722113],{"className":722112},[31796],[34,722114,722116],{"className":722115},[31801],[34,722117,722119],{"className":722118,"style":548314},[31805],[34,722120,722121,722124],{"style":116668},[34,722122],{"className":722123,"style":43406},[31813],[34,722125,722127],{"className":722126},[31818,43410,43411,31821],[34,722128,768],{"className":722129},[31685,31821],[34,722131,223],{"className":722132},[32024,31821],[34,722134,640],{"className":722135},[31685,31821],[34,722137,722138,722141],{"style":41520},[34,722139],{"className":722140,"style":41508},[31813],[34,722142],{"className":722143,"style":41528},[41527],[34,722145,722146,722149],{"style":70529},[34,722147],{"className":722148,"style":41508},[31813],[34,722150,722152],{"className":722151},[31818,31819,31820,31821],[34,722153,722155,722191,722194,722197,722200,722212,722215,722218,722221,722233,722236,722239],{"className":722154},[31685,31821],[34,722156,722158,722161],{"className":722157},[31685,31821],[34,722159,33244],{"className":722160},[31685,31686,31821],[34,722162,722164],{"className":722163},[31792],[34,722165,722167],{"className":722166},[31796],[34,722168,722170],{"className":722169},[31801],[34,722171,722174],{"className":722172,"style":722173},[31805],"height:0.8703em;",[34,722175,722176,722179],{"style":552391},[34,722177],{"className":722178,"style":43406},[31813],[34,722180,722182],{"className":722181},[31818,43410,43411,31821],[34,722183,722185,722188],{"className":722184},[31685,31821],[34,722186,111],{"className":722187},[31685,31686,31821],[34,722189,33249],{"className":722190},[31685,31686,31821],[34,722192,456],{"className":722193},[31782,31821],[34,722195,111],{"className":722196},[31685,31686,31821],[34,722198],{"className":722199,"style":43430},[31845,31821],[34,722201,722203,722206,722209],{"className":722202},[32767,31821],[34,722204,31965],{"className":722205},[31821],[34,722207,43383],{"className":722208},[31821],[34,722210,31747],{"className":722211},[31821],[34,722213],{"className":722214,"style":43430},[31845,31821],[34,722216,33249],{"className":722217},[31685,31686,31821],[34,722219,223],{"className":722220},[32024,31821],[34,722222,722224,722227,722230],{"className":722223},[32767,31821],[34,722225,31747],{"className":722226},[31821],[34,722228,31955],{"className":722229},[31821],[34,722231,31660],{"className":722232},[31821],[34,722234],{"className":722235,"style":43430},[31845,31821],[34,722237,33249],{"className":722238},[31685,31686,31821],[34,722240,2557],{"className":722241},[31690,31821],[34,722243,31829],{"className":722244},[31828],[34,722246,722248],{"className":722247},[31801],[34,722249,722251],{"className":722250,"style":400163},[31805],[34,722252],{},[34,722254],{"className":722255},[31690,41488],[105,722257,722258,723005],{},[108,722259,722260],{},[34,722261,722263,722399],{"className":722262},[31642],[34,722264,722266],{"className":722265},[31646],[31648,722267,722268],{"xmlns":31650},[31652,722269,722270,722396],{},[31655,722271,722272,722278,722280,722286,722288,722298,722300,722302,722304,722306,722308,722310,722312,722318,722360,722362,722368,722378,722380,722382,722384,722386,722388,722390,722392,722394],{},[31743,722273,722274,722276],{},[31658,722275,31965],{},[31749,722277,640],{},[31662,722279,202],{},[41453,722281,722282,722284],{},[31749,722283,640],{},[31749,722285,768],{},[31662,722287,546295],{},[37624,722289,722290,722292],{},[31658,722291,33244],{},[31655,722293,722294,722296],{},[31662,722295,32226],{},[31658,722297,33249],{},[31658,722299,547860],{},[31662,722301,32682],{},[31658,722303,33249],{},[39361,722305,39363],{},[31658,722307,46900],{},[31658,722309,33249],{},[31662,722311,202],{},[41453,722313,722314,722316],{},[31749,722315,640],{},[31749,722317,768],{},[31655,722319,722320,722322,722358],{},[31662,722321,15972],{"fence":758},[41453,722323,722324,722356],{},[31655,722325,722326,722336,722338,722340,722342,722344,722346,722348,722350,722352,722354],{},[37624,722327,722328,722330],{},[31658,722329,33244],{},[31655,722331,722332,722334],{},[31662,722333,32226],{},[31658,722335,33249],{},[31662,722337,456],{"stretchy":702},[31662,722339,32226],{},[31658,722341,547860],{},[31662,722343,32682],{},[31658,722345,33249],{},[31662,722347,223],{},[31658,722349,547853],{},[31662,722351,32682],{},[31658,722353,33249],{},[31662,722355,2557],{"stretchy":702},[31749,722357,768],{},[31662,722359,32712],{"fence":758},[31662,722361,202],{},[41453,722363,722364,722366],{},[31749,722365,640],{},[31749,722367,15932],{},[37624,722369,722370,722372],{},[31658,722371,33244],{},[31655,722373,722374,722376],{},[31662,722375,32226],{},[31658,722377,33249],{},[31662,722379,456],{"stretchy":702},[31658,722381,547853],{},[31662,722383,32682],{},[31658,722385,33249],{},[31662,722387,32226],{},[31658,722389,547860],{},[31662,722391,32682],{},[31658,722393,33249],{},[31662,722395,2557],{"stretchy":702},[31665,722397,722398],{"encoding":31667},"c_1 = \\frac{1}{2} \\int e^{-t}\\cos t \\, dt = \\frac{1}{2} \\left[ \\frac{e^{-t}(-\\cos t + \\sin t)}{2} \\right] = \\frac{1}{4}e^{-t}(\\sin t - \\cos t)",[34,722400,722402,722457,722605,722857,722987],{"className":722401,"ariaHidden":758},[31672],[34,722403,722405,722408,722448,722451,722454],{"className":722404},[31676],[34,722406],{"className":722407,"style":31860},[31680],[34,722409,722411,722414],{"className":722410},[31685],[34,722412,31965],{"className":722413},[31685,31686],[34,722415,722417],{"className":722416},[31792],[34,722418,722420,722440],{"className":722419},[31796,31797],[34,722421,722423,722437],{"className":722422},[31801],[34,722424,722426],{"className":722425,"style":31806},[31805],[34,722427,722428,722431],{"style":31809},[34,722429],{"className":722430,"style":31814},[31813],[34,722432,722434],{"className":722433},[31818,31819,31820,31821],[34,722435,640],{"className":722436},[31685,31821],[34,722438,31829],{"className":722439},[31828],[34,722441,722443],{"className":722442},[31801],[34,722444,722446],{"className":722445,"style":31836},[31805],[34,722447],{},[34,722449],{"className":722450,"style":31846},[31845],[34,722452,202],{"className":722453},[31850],[34,722455],{"className":722456,"style":31846},[31845],[34,722458,722460,722463,722531,722534,722537,722540,722575,722578,722581,722584,722587,722590,722593,722596,722599,722602],{"className":722459},[31676],[34,722461],{"className":722462,"style":46727},[31680],[34,722464,722466,722469,722528],{"className":722465},[31685],[34,722467],{"className":722468},[31782,41488],[34,722470,722472],{"className":722471},[41453],[34,722473,722475,722520],{"className":722474},[31796,31797],[34,722476,722478,722517],{"className":722477},[31801],[34,722479,722481,722495,722503],{"className":722480,"style":46746},[31805],[34,722482,722483,722486],{"style":41504},[34,722484],{"className":722485,"style":41508},[31813],[34,722487,722489],{"className":722488},[31818,31819,31820,31821],[34,722490,722492],{"className":722491},[31685,31821],[34,722493,768],{"className":722494},[31685,31821],[34,722496,722497,722500],{"style":41520},[34,722498],{"className":722499,"style":41508},[31813],[34,722501],{"className":722502,"style":41528},[41527],[34,722504,722505,722508],{"style":46771},[34,722506],{"className":722507,"style":41508},[31813],[34,722509,722511],{"className":722510},[31818,31819,31820,31821],[34,722512,722514],{"className":722513},[31685,31821],[34,722515,640],{"className":722516},[31685,31821],[34,722518,31829],{"className":722519},[31828],[34,722521,722523],{"className":722522},[31801],[34,722524,722526],{"className":722525,"style":41619},[31805],[34,722527],{},[34,722529],{"className":722530},[31690,41488],[34,722532],{"className":722533,"style":32123},[31845],[34,722535,546295],{"className":722536,"style":546331},[32767,131249,156560],[34,722538],{"className":722539,"style":32123},[31845],[34,722541,722543,722546],{"className":722542},[31685],[34,722544,33244],{"className":722545},[31685,31686],[34,722547,722549],{"className":722548},[31792],[34,722550,722552],{"className":722551},[31796],[34,722553,722555],{"className":722554},[31801],[34,722556,722558],{"className":722557,"style":611758},[31805],[34,722559,722560,722563],{"style":37777},[34,722561],{"className":722562,"style":31814},[31813],[34,722564,722566],{"className":722565},[31818,31819,31820,31821],[34,722567,722569,722572],{"className":722568},[31685,31821],[34,722570,32226],{"className":722571},[31685,31821],[34,722573,33249],{"className":722574},[31685,31686,31821],[34,722576],{"className":722577,"style":32123},[31845],[34,722579,547860],{"className":722580},[32767],[34,722582],{"className":722583,"style":32123},[31845],[34,722585,33249],{"className":722586},[31685,31686],[34,722588],{"className":722589,"style":32123},[31845],[34,722591,46900],{"className":722592},[31685,31686],[34,722594,33249],{"className":722595},[31685,31686],[34,722597],{"className":722598,"style":31846},[31845],[34,722600,202],{"className":722601},[31850],[34,722603],{"className":722604,"style":31846},[31845],[34,722606,722608,722611,722679,722682,722848,722851,722854],{"className":722607},[31676],[34,722609],{"className":722610,"style":141282},[31680],[34,722612,722614,722617,722676],{"className":722613},[31685],[34,722615],{"className":722616},[31782,41488],[34,722618,722620],{"className":722619},[41453],[34,722621,722623,722668],{"className":722622},[31796,31797],[34,722624,722626,722665],{"className":722625},[31801],[34,722627,722629,722643,722651],{"className":722628,"style":46746},[31805],[34,722630,722631,722634],{"style":41504},[34,722632],{"className":722633,"style":41508},[31813],[34,722635,722637],{"className":722636},[31818,31819,31820,31821],[34,722638,722640],{"className":722639},[31685,31821],[34,722641,768],{"className":722642},[31685,31821],[34,722644,722645,722648],{"style":41520},[34,722646],{"className":722647,"style":41508},[31813],[34,722649],{"className":722650,"style":41528},[41527],[34,722652,722653,722656],{"style":46771},[34,722654],{"className":722655,"style":41508},[31813],[34,722657,722659],{"className":722658},[31818,31819,318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= -\\frac{1}{2} \\int e^t\\cos t \\, dt = -\\frac{1}{2} \\left[ \\frac{e^t(\\cos t + \\sin t)}{2} \\right] = -\\frac{1}{4}e^t(\\sin t + \\cos t)",[34,723139,723141,723196,723341,723584,723711],{"className":723140,"ariaHidden":758},[31672],[34,723142,723144,723147,723187,723190,723193],{"className":723143},[31676],[34,723145],{"className":723146,"style":31860},[31680],[34,723148,723150,723153],{"className":723149},[31685],[34,723151,31965],{"className":723152},[31685,31686],[34,723154,723156],{"className":723155},[31792],[34,723157,723159,723179],{"className":723158},[31796,31797],[34,723160,723162,723176],{"className":723161},[31801],[34,723163,723165],{"className":723164,"style":31806},[31805],[34,723166,723167,723170],{"style":31809},[34,723168],{"className":723169,"style":31814},[31813],[34,723171,723173],{"className":723172},[31818,31819,31820,31821],[34,723174,768],{"className":723175},[31685,31821],[34,723177,31829],{"className":723178},[31828],[34,723180,723182],{"className":723181},[31801],[34,723183,723185],{"className":723184,"style":31836},[31805],[34,723186],{},[34,723188],{"className":723189,"style":31846},[31845],[34,723191,202],{"className":723192},[31850],[34,723194],{"className":723195,"style":31846},[31845],[34,723197,723199,723202,723205,723273,723276,723279,723282,723311,723314,723317,723320,723323,723326,723329,723332,723335,723338],{"className":723198},[31676],[34,723200],{"className":723201,"style":46727},[31680],[34,723203,32226],{"className":723204},[31685],[34,723206,723208,723211,723270],{"className":723207},[31685],[34,723209],{"className":723210},[31782,41488],[34,723212,723214],{"className":723213},[41453],[34,723215,723217,723262],{"className":723216},[31796,31797],[34,723218,723220,723259],{"className":723219},[31801],[34,723221,723223,723237,723245],{"className":723222,"style":46746},[31805],[34,723224,723225,723228],{"style":41504},[34,723226],{"className":723227,"style":41508},[31813],[34,723229,723231],{"className":723230},[31818,31819,31820,31821],[34,723232,723234],{"className":723233},[31685,31821],[34,723235,768],{"className":723236},[31685,31821],[34,723238,723239,723242],{"style":41520},[34,723240],{"className":723241,"style":41508},[31813],[34,723243],{"className":723244,"style":41528},[41527],[34,723246,723247,723250],{"style":46771},[34,723248],{"className":723249,"style":41508},[31813],[34,723251,723253],{"className":723252},[31818,31819,31820,31821],[34,723254,723256],{"className":723255},[31685,31821],[34,723257,640],{"className":723258},[31685,31821],[34,723260,31829],{"className":723261},[31828],[34,723263,723265],{"className":723264},[31801],[34,723266,723268],{"className":723267,"style":41619},[31805],[34,723269],{},[34,723271],{"className":723272},[31690,41488],[34,723274],{"className":723275,"style":32123},[31845],[34,723277,546295],{"className":723278,"style":546331},[32767,131249,156560],[34,723280],{"className":723281,"style":32123},[31845],[34,723283,723285,723288],{"className":723284},[31685],[34,723286,33244],{"className":723287},[31685,31686],[34,723289,723291],{"className":723290},[31792],[34,723292,723294],{"className":723293},[31796],[34,723295,723297],{"className":723296},[31801],[34,723298,723300],{"className":723299,"style":611758},[31805],[34,723301,723302,723305],{"style":37777},[34,723303],{"className":723304,"style":31814},[31813],[34,723306,723308],{"className":723307},[31818,31819,31820,31821],[34,723309,33249],{"className":723310},[31685,31686,31821],[34,723312],{"className":723313,"style":32123},[31845],[34,723315,547860],{"className":723316},[32767],[34,723318],{"className":723319,"style":32123},[31845],[34,723321,33249],{"className":723322},[31685,31686],[34,723324],{"className":723325,"style":32123},[31845],[34,723327,46900],{"className":723328},[31685,31686],[34,723330,33249],{"className":723331},[31685,31686],[34,723333],{"className":723334,"style":31846},[31845],[34,723336,202],{"className":723337},[31850],[34,723339],{"className":723340,"style":31846},[31845],[34,723342,723344,723347,723350,723418,723421,723575,723578,723581],{"className":723343},[31676],[34,723345],{"className":723346,"style":141282},[31680],[34,723348,32226],{"className":723349},[31685],[34,723351,723353,723356,723415],{"className":723352},[31685],[34,723354],{"className":723355},[31782,41488],[34,723357,723359],{"className":723358},[41453],[34,723360,723362,723407],{"className":723361},[31796,31797],[34,723363,723365,723404],{"className":723364},[31801],[34,723366,723368,723382,723390],{"className":723367,"style":46746},[31805],[34,723369,723370,723373],{"style":41504},[34,723371],{"className":723372,"style":41508},[31813],[34,723374,723376],{"className":723375},[31818,31819,31820,31821],[34,723377,723379],{"className":723378},[31685,31821],[34,723380,768],{"className":723381},[31685,31821],[34,723383,723384,723387],{"style":41520},[34,723385],{"className":723386,"style":41508},[31813],[34,723388],{"className":723389,"style":41528},[41527],[34,723391,723392,723395],{"style":46771},[34,723393],{"className":723394,"style":41508},[31813],[34,723396,723398],{"className":723397},[31818,31819,31820,318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5: 写通解","\n通解 ",[34,723736,723738,723816],{"className":723737},[31642],[34,723739,723741],{"className":723740},[31646],[31648,723742,723743],{"xmlns":31650},[31652,723744,723745,723813],{},[31655,723746,723747,723749,723751,723757,723763,723765,723771,723781,723783,723785,723791,723797,723799,723805,723811],{},[31658,723748,32936],{},[31662,723750,202],{},[31743,723752,723753,723755],{},[31658,723754,47384],{},[31749,723756,640],{},[37624,723758,723759,723761],{},[31658,723760,33244],{},[31658,723762,33249],{},[31662,723764,223],{},[31743,723766,723767,723769],{},[31658,723768,47384],{},[31749,723770,768],{},[37624,723772,723773,723775],{},[31658,723774,33244],{},[31655,723776,723777,723779],{},[31662,723778,32226],{},[31658,723780,33249],{},[31662,723782,223],{},[31662,723784,456],{"stretchy":702},[31743,723786,723787,723789],{},[31658,723788,31965],{},[31749,723790,640],{},[31743,723792,723793,723795],{},[31658,723794,32936],{},[31749,723796,640],{},[31662,723798,223],{},[31743,723800,723801,723803],{},[31658,723802,31965],{},[31749,723804,768],{},[31743,723806,723807,723809],{},[31658,723808,32936],{},[31749,723810,768],{},[31662,723812,2557],{"stretchy":702},[31665,723814,723815],{"encoding":31667},"x = C_1 e^t + C_2 e^{-t} + (c_1 x_1 + c_2 x_2)",[34,723817,723819,723837,723922,724012,724110],{"className":723818,"ariaHidden":758},[31672],[34,723820,723822,723825,723828,723831,723834],{"className":723821},[31676],[34,723823],{"className":723824,"style":32527},[31680],[34,723826,32936],{"className":723827},[31685,31686],[34,723829],{"className":723830,"style":31846},[31845],[34,723832,202],{"className":723833},[31850],[34,723835],{"className":723836,"style":31846},[31845],[34,723838,723840,723844,723884,723913,723916,723919],{"className":723839},[31676],[34,723841],{"className":723842,"style":723843},[31680],"height:0.9436em;vertical-align:-0.15em;",[34,723845,723847,723850],{"className":723846},[31685],[34,723848,47384],{"className":723849,"style":38706},[31685,31686],[34,723851,723853],{"className":723852},[31792],[34,723854,723856,723876],{"className":723855},[31796,31797],[34,723857,723859,723873],{"className":723858},[31801],[34,723860,723862],{"className":723861,"style":31806},[31805],[34,723863,723864,723867],{"style":129511},[34,723865],{"className":723866,"style":31814},[31813],[34,723868,723870],{"className":723869},[31818,31819,31820,31821],[34,723871,640],{"className":723872},[31685,31821],[34,723874,31829],{"className":723875},[31828],[34,723877,723879],{"className":723878},[31801],[34,723880,723882],{"className":723881,"style":31836},[31805],[34,723883],{},[34,723885,723887,723890],{"className":723886},[31685],[34,723888,33244],{"className":723889},[31685,31686],[34,723891,723893],{"className":723892},[31792],[34,723894,723896],{"className":723895},[31796],[34,723897,723899],{"className":723898},[31801],[34,723900,723902],{"className":723901,"style":611758},[31805],[34,723903,723904,723907],{"style":37777},[34,723905],{"className":723906,"style":31814},[31813],[34,723908,723910],{"className":723909},[31818,31819,31820,31821],[34,723911,33249],{"className":723912},[31685,31686,31821],[34,723914],{"className":723915,"style":32604},[31845],[34,723917,223],{"className":723918},[32024],[34,723920],{"className":723921,"style":32604},[31845],[34,723923,723925,723928,723968,724003,724006,724009],{"className":723924},[31676],[34,723926],{"className":723927,"style":723843},[31680],[34,723929,723931,723934],{"className":723930},[31685],[34,723932,47384],{"className":723933,"style":38706},[31685,31686],[34,723935,723937],{"className":723936},[31792],[34,723938,723940,723960],{"className":723939},[31796,31797],[34,723941,723943,723957],{"className":723942},[31801],[34,723944,723946],{"className":723945,"style":31806},[31805],[34,723947,723948,723951],{"style":129511},[34,723949],{"className":723950,"style":31814},[31813],[34,723952,723954],{"className":723953},[31818,31819,31820,31821],[34,723955,768],{"className":723956},[31685,31821],[34,723958,31829],{"className":723959},[31828],[34,723961,723963],{"className":723962},[31801],[34,723964,723966],{"className":723965,"style":31836},[31805],[34,723967],{},[34,723969,723971,723974],{"className":723970},[31685],[34,723972,33244],{"className":723973},[31685,31686],[34,723975,723977],{"className":723976},[31792],[34,723978,723980],{"className":723979},[31796],[34,723981,723983],{"className":723982},[31801],[34,723984,723986],{"className":723985,"style":611758},[31805],[34,723987,723988,723991],{"style":37777},[34,723989],{"className":723990,"style":31814},[31813],[34,723992,723994],{"className":723993},[31818,31819,31820,31821],[34,723995,723997,724000],{"className":723996},[31685,31821],[34,723998,32226],{"className":723999},[31685,31821],[34,724001,33249],{"className":724002},[31685,31686,31821],[34,724004],{"className":724005,"style":32604},[31845],[34,724007,223],{"className":724008},[32024],[34,724010],{"className":724011,"style":32604},[31845],[34,724013,724015,724018,724021,724061,724101,724104,724107],{"className":724014},[31676],[34,724016],{"className":724017,"style":31774},[31680],[34,724019,456],{"className":724020},[31782],[34,724022,724024,724027],{"className":724023},[31685],[34,724025,31965],{"className":724026},[31685,31686],[34,724028,724030],{"className":724029},[31792],[34,724031,724033,724053],{"className":724032},[31796,31797],[34,724034,724036,724050],{"className":724035},[31801],[34,724037,724039],{"className":724038,"style":31806},[31805],[34,724040,724041,724044],{"style":31809},[34,724042],{"className":724043,"style":31814},[31813],[34,724045,724047],{"className":724046},[31818,31819,31820,31821],[34,724048,640],{"className":724049},[31685,31821],[34,724051,31829],{"className":724052},[31828],[34,724054,724056],{"className":724055},[31801],[34,724057,724059],{"className":724058,"style":31836},[31805],[34,724060],{},[34,724062,724064,724067],{"className":724063},[31685],[34,724065,32936],{"className":724066},[31685,31686],[34,724068,724070],{"className":724069},[31792],[34,724071,724073,724093],{"className":724072},[31796,31797],[34,724074,724076,724090],{"className":724075},[31801],[34,724077,724079],{"className":724078,"style":31806},[31805],[34,724080,724081,724084],{"style":31809},[34,724082],{"className":724083,"style":31814},[31813],[34,724085,724087],{"className":724086},[31818,31819,31820,31821],[34,724088,640],{"className":724089},[31685,31821],[34,724091,31829],{"className":724092},[31828],[34,724094,724096],{"className":724095},[31801],[34,724097,724099],{"className":724098,"style":31836},[31805],[34,724100],{},[34,724102],{"className":724103,"style":32604},[31845],[34,724105,223],{"className":724106},[32024],[34,724108],{"className":724109,"style":32604},[31845],[34,724111,724113,724116,724156,724196],{"className":724112},[31676],[34,724114],{"className":724115,"style":31774},[31680],[34,724117,724119,724122],{"className":724118},[31685],[34,724120,31965],{"className":724121},[31685,31686],[34,724123,724125],{"className":724124},[31792],[34,724126,724128,724148],{"className":724127},[31796,31797],[34,724129,724131,724145],{"className":724130},[31801],[34,724132,724134],{"className":724133,"style":31806},[31805],[34,724135,724136,724139],{"style":31809},[34,724137],{"className":724138,"style":31814},[31813],[34,724140,724142],{"className":724141},[31818,31819,31820,31821],[34,724143,768],{"className":724144},[31685,31821],[34,724146,31829],{"className":724147},[31828],[34,724149,724151],{"className":724150},[31801],[34,724152,724154],{"className":724153,"style":31836},[31805],[34,724155],{},[34,724157,724159,724162],{"className":724158},[31685],[34,724160,32936],{"className":724161},[31685,31686],[34,724163,724165],{"className":724164},[31792],[34,724166,724168,724188],{"className":724167},[31796,31797],[34,724169,724171,724185],{"className":724170},[31801],[34,724172,724174],{"className":724173,"style":31806},[31805],[34,724175,724176,724179],{"style":31809},[34,724177],{"className":724178,"style":31814},[31813],[34,724180,724182],{"className":724181},[31818,31819,31820,31821],[34,724183,768],{"className":724184},[31685,31821],[34,724186,31829],{"className":724187},[31828],[34,724189,724191],{"className":724190},[31801],[34,724192,724194],{"className":724193,"style":31836},[31805],[34,724195],{},[34,724197,2557],{"className":724198},[31690],"。\n特解部分：\n",[34,724201,724203,724253],{"className":724202},[31642],[34,724204,724206],{"className":724205},[31646],[31648,724207,724208],{"xmlns":31650},[31652,724209,724210,724250],{},[31655,724211,724212,724218,724224,724226,724232,724234,724236,724238,724240,724242,724244,724246,724248],{},[31743,724213,724214,724216],{},[31658,724215,31965],{},[31749,724217,640],{},[37624,724219,724220,724222],{},[31658,724221,33244],{},[31658,724223,33249],{},[31662,724225,202],{},[41453,724227,724228,724230],{},[31749,724229,640],{},[31749,724231,15932],{},[31662,724233,456],{"stretchy":702},[31658,724235,547853],{},[31662,724237,32682],{},[31658,724239,33249],{},[31662,724241,32226],{},[31658,724243,547860],{},[31662,724245,32682],{},[31658,724247,33249],{},[31662,724249,2557],{"stretchy":702},[31665,724251,724252],{"encoding":31667},"c_1 e^t = \\frac{1}{4}(\\sin t - \\cos t)",[34,724254,724256,724340,724435],{"className":724255,"ariaHidden":758},[31672],[34,724257,724259,724262,724302,724331,724334,724337],{"className":724258},[31676],[34,724260],{"className":724261,"style":723843},[31680],[34,724263,724265,724268],{"className":724264},[31685],[34,724266,31965],{"className":724267},[31685,31686],[34,724269,724271],{"className":724270},[31792],[34,724272,724274,724294],{"className":724273},[31796,31797],[34,724275,724277,724291],{"className":724276},[31801],[34,724278,724280],{"className":724279,"style":31806},[31805],[34,724281,724282,724285],{"style":31809},[34,724283],{"className":724284,"style":31814},[31813],[34,724286,724288],{"className":724287},[31818,31819,31820,31821],[34,724289,640],{"className":724290},[31685,31821],[34,724292,31829],{"className":724293},[31828],[34,724295,724297],{"className":724296},[31801],[34,724298,724300],{"className":724299,"style":31836},[31805],[34,724301],{},[34,724303,724305,724308],{"className":724304},[31685],[34,724306,33244],{"className":724307},[31685,31686],[34,724309,724311],{"className":724310},[31792],[34,724312,724314],{"className":724313},[31796],[34,724315,724317],{"className":724316},[31801],[34,724318,724320],{"className":724319,"style":611758},[31805],[34,724321,724322,724325],{"style":37777},[34,724323],{"className":724324,"style":31814},[31813],[34,724326,724328],{"className":724327},[31818,31819,31820,31821],[34,724329,33249],{"className":724330},[31685,31686,31821],[34,724332],{"className":724333,"style":31846},[31845],[34,724335,202],{"className":724336},[31850],[34,724338],{"className":724339,"style":31846},[31845],[34,724341,724343,724346,724414,724417,724420,724423,724426,724429,724432],{"className":724342},[31676],[34,724344],{"className":724345,"style":46727},[31680],[34,724347,724349,724352,724411],{"className":724348},[31685],[34,724350],{"className":724351},[31782,41488],[34,724353,724355],{"className":724354},[41453],[34,724356,724358,724403],{"className":724357},[31796,31797],[34,724359,724361,724400],{"className":724360},[31801],[34,724362,724364,724378,724386],{"className":724363,"style":46746},[31805],[34,724365,724366,724369],{"style":41504},[34,724367],{"className":724368,"style":41508},[31813],[34,724370,724372],{"className":724371},[31818,31819,31820,31821],[34,724373,724375],{"className":724374},[31685,31821],[34,724376,15932],{"className":724377},[31685,31821],[34,724379,724380,724383],{"style":41520},[34,724381],{"className":724382,"style":41508},[31813],[34,724384],{"className":724385,"style":41528},[41527],[34,724387,724388,724391],{"style":46771},[34,724389],{"className":724390,"style":41508},[31813],[34,724392,724394],{"className":724393},[31818,31819,31820,31821],[34,724395,724397],{"className":724396},[31685,31821],[34,724398,640],{"className":724399},[31685,31821],[34,724401,31829],{"className":724402},[31828],[34,724404,724406],{"className":724405},[31801],[34,724407,724409],{"className":724408,"style":41619},[31805],[34,724410],{},[34,724412],{"className":724413},[31690,41488],[34,724415,456],{"className":724416},[31782],[34,724418,547853],{"className":724419},[32767],[34,724421],{"className":724422,"style":32123},[31845],[34,724424,33249],{"className":724425},[31685,31686],[34,724427],{"className":724428,"style":32604},[31845],[34,724430,32226],{"className":724431},[32024],[34,724433],{"className":724434,"style":32604},[31845],[34,724436,724438,724441,724444,724447,724450],{"className":724437},[31676],[34,724439],{"className":724440,"style":31774},[31680],[34,724442,547860],{"className":724443},[32767],[34,724445],{"className":724446,"style":32123},[31845],[34,724448,33249],{"className":724449},[31685,31686],[34,724451,2557],{"className":724452},[31690],[34,724454,724456,724512],{"className":724455},[31642],[34,724457,724459],{"className":724458},[31646],[31648,724460,724461],{"xmlns":31650},[31652,724462,724463,724509],{},[31655,724464,724465,724471,724481,724483,724485,724491,724493,724495,724497,724499,724501,724503,724505,724507],{},[31743,724466,724467,724469],{},[31658,724468,31965],{},[31749,724470,768],{},[37624,724472,724473,724475],{},[31658,724474,33244],{},[31655,724476,724477,724479],{},[31662,724478,32226],{},[31658,724480,33249],{},[31662,724482,202],{},[31662,724484,32226],{},[41453,724486,724487,724489],{},[31749,724488,640],{},[31749,724490,15932],{},[31662,724492,456],{"stretchy":702},[31658,724494,547853],{},[31662,724496,32682],{},[31658,724498,33249],{},[31662,724500,223],{},[31658,724502,547860],{},[31662,724504,32682],{},[31658,724506,33249],{},[31662,724508,2557],{"stretchy":702},[31665,724510,724511],{"encoding":31667},"c_2 e^{-t} = -\\frac{1}{4}(\\sin t + \\cos t)",[34,724513,724515,724605,724703],{"className":724514,"ariaHidden":758},[31672],[34,724516,724518,724521,724561,724596,724599,724602],{"className":724517},[31676],[34,724519],{"className":724520,"style":723843},[31680],[34,724522,724524,724527],{"className":724523},[31685],[34,724525,31965],{"className":724526},[31685,31686],[34,724528,724530],{"className":724529},[31792],[34,724531,724533,724553],{"className":724532},[31796,31797],[34,724534,724536,724550],{"className":724535},[31801],[34,724537,724539],{"className":724538,"style":31806},[31805],[34,724540,724541,724544],{"style":31809},[34,724542],{"className":724543,"style":31814},[31813],[34,724545,724547],{"className":724546},[31818,31819,31820,31821],[34,724548,768],{"className":724549},[31685,31821],[34,724551,31829],{"className":724552},[31828],[34,724554,724556],{"className":724555},[31801],[34,724557,724559],{"className":724558,"style":31836},[31805],[34,724560],{},[34,724562,724564,724567],{"className":724563},[31685],[34,724565,33244],{"className":724566},[31685,31686],[34,724568,724570],{"className":724569},[31792],[34,724571,724573],{"className":724572},[31796],[34,724574,724576],{"className":724575},[31801],[34,724577,724579],{"className":724578,"style":611758},[31805],[34,724580,724581,724584],{"style":37777},[34,724582],{"className":724583,"style":31814},[31813],[34,724585,724587],{"className":724586},[31818,31819,31820,31821],[34,724588,724590,724593],{"className":724589},[31685,31821],[34,724591,32226],{"className":724592},[31685,31821],[34,724594,33249],{"className":724595},[31685,31686,31821],[34,724597],{"className":724598,"style":31846},[31845],[34,724600,202],{"className":724601},[31850],[34,724603],{"className":724604,"style":31846},[31845],[34,724606,724608,724611,724614,724682,724685,724688,724691,724694,724697,724700],{"className":724607},[31676],[34,724609],{"className":724610,"style":46727},[31680],[34,724612,32226],{"className":724613},[31685],[34,724615,724617,724620,724679],{"className":724616},[31685],[34,724618],{"className":724619},[31782,41488],[34,724621,724623],{"className":724622},[41453],[34,724624,724626,724671],{"className":724625},[31796,31797],[34,724627,724629,724668],{"className":724628},[31801],[34,724630,724632,724646,724654],{"className":724631,"style":46746},[31805],[34,724633,724634,724637],{"style":41504},[34,724635],{"className":724636,"style":41508},[31813],[34,724638,724640],{"className":724639},[31818,31819,31820,31821],[34,724641,724643],{"className":724642},[31685,31821],[34,724644,15932],{"className":724645},[31685,31821],[34,724647,724648,724651],{"style":41520},[34,724649],{"className":724650,"style":41508},[31813],[34,724652],{"className":724653,"style":41528},[41527],[34,724655,724656,724659],{"style":46771},[34,724657],{"className":724658,"style":41508},[31813],[34,724660,724662],{"className":724661},[31818,31819,31820,31821],[34,724663,724665],{"className":724664},[31685,31821],[34,724666,640],{"className":724667},[31685,31821],[34,724669,31829],{"className":724670},[31828],[34,724672,724674],{"className":724673},[31801],[34,724675,724677],{"className":724676,"style":41619},[31805],[34,724678],{},[34,724680],{"className":724681},[31690,41488],[34,724683,456],{"className":724684},[31782],[34,724686,547853],{"className":724687},[32767],[34,724689],{"className":724690,"style":32123},[31845],[34,724692,33249],{"className":724693},[31685,31686],[34,724695],{"className":724696,"style":32604},[31845],[34,724698,223],{"className":724699},[32024],[34,724701],{"className":724702,"style":32604},[31845],[34,724704,724706,724709,724712,724715,724718],{"className":724705},[31676],[34,724707],{"className":724708,"style":31774},[31680],[34,724710,547860],{"className":724711},[32767],[34,724713],{"className":724714,"style":32123},[31845],[34,724716,33249],{"className":724717},[31685,31686],[34,724719,2557],{"className":724720},[31690],"\n特解 ",[34,724723,724725,724815],{"className":724724},[31642],[34,724726,724728],{"className":724727},[31646],[31648,724729,724730],{"xmlns":31650},[31652,724731,724732,724812],{},[31655,724733,724734,724740,724742,724748,724750,724752,724754,724756,724762,724764,724766,724768,724770,724776,724778,724780,724782,724784,724790,724792,724794,724796,724798,724800,724806,724808,724810],{},[134014,724735,724736,724738],{"accent":758},[31658,724737,32936],{},[31662,724739,346142],{},[31662,724741,202],{},[41453,724743,724744,724746],{},[31749,724745,640],{},[31749,724747,15932],{},[31658,724749,547853],{},[31662,724751,32682],{},[31658,724753,33249],{},[31662,724755,32226],{},[41453,724757,724758,724760],{},[31749,724759,640],{},[31749,724761,15932],{},[31658,724763,547860],{},[31662,724765,32682],{},[31658,724767,33249],{},[31662,724769,32226],{},[41453,724771,724772,724774],{},[31749,724773,640],{},[31749,724775,15932],{},[31658,724777,547853],{},[31662,724779,32682],{},[31658,724781,33249],{},[31662,724783,32226],{},[41453,724785,724786,724788],{},[31749,724787,640],{},[31749,724789,15932],{},[31658,724791,547860],{},[31662,724793,32682],{},[31658,724795,33249],{},[31662,724797,202],{},[31662,724799,32226],{},[41453,724801,724802,724804],{},[31749,724803,640],{},[31749,724805,768],{},[31658,724807,547860],{},[31662,724809,32682],{},[31658,724811,33249],{},[31665,724813,724814],{"encoding":31667},"\\bar{x} = \\frac{1}{4}\\sin t - \\frac{1}{4}\\cos t - \\frac{1}{4}\\sin t - \\frac{1}{4}\\cos t = -\\frac{1}{2}\\cos t",[34,724816,724818,724865,724960,725055,725150,725245],{"className":724817,"ariaHidden":758},[31672],[34,724819,724821,724824,724856,724859,724862],{"className":724820},[31676],[34,724822],{"className":724823,"style":346181},[31680],[34,724825,724827],{"className":724826},[31685,346171],[34,724828,724830],{"className":724829},[31796],[34,724831,724833],{"className":724832},[31801],[34,724834,724836,724844],{"className":724835,"style":346181},[31805],[34,724837,724838,724841],{"style":41565},[34,724839],{"className":724840,"style":41508},[31813],[34,724842,32936],{"className":724843},[31685,31686],[34,724845,724846,724849],{"style":41565},[34,724847],{"className":724848,"style":41508},[31813],[34,724850,724853],{"className":724851,"style":724852},[346198],"left:-0.2222em;",[34,724854,346142],{"className":724855},[31685],[34,724857],{"className":724858,"style":31846},[31845],[34,724860,202],{"className":724861},[31850],[34,724863],{"className":724864,"style":31846},[31845],[34,724866,724868,724871,724939,724942,724945,724948,724951,724954,724957],{"className":724867},[31676],[34,724869],{"className":724870,"style":46727},[31680],[34,724872,724874,724877,724936],{"className":724873},[31685],[34,724875],{"className":724876},[31782,41488],[34,724878,724880],{"className":724879},[41453],[34,724881,724883,724928],{"className":724882},[31796,31797],[34,724884,724886,724925],{"className":724885},[31801],[34,724887,724889,724903,724911],{"className":724888,"style":46746},[31805],[34,724890,724891,724894],{"style":41504},[34,724892],{"className":724893,"style":41508},[31813],[34,724895,724897],{"className":724896},[31818,31819,31820,31821],[34,724898,724900],{"className":724899},[31685,31821],[34,724901,15932],{"className":724902},[31685,31821],[34,724904,724905,724908],{"style":41520},[34,724906],{"className":724907,"style":41508},[31813],[34,724909],{"className":724910,"style":41528},[41527],[34,724912,724913,724916],{"style":46771},[34,724914],{"className":724915,"style":41508},[31813],[34,724917,724919],{"className":724918},[31818,31819,31820,31821],[34,724920,724922],{"className":724921},[31685,31821],[34,724923,640],{"className":724924},[31685,31821],[34,724926,31829],{"className":724927},[31828],[34,724929,724931],{"className":724930},[31801],[34,724932,724934],{"className":724933,"style":41619},[31805],[34,724935],{},[34,724937],{"className":724938},[31690,41488],[34,724940],{"className":724941,"style":32123},[31845],[34,724943,547853],{"className":724944},[32767],[34,724946],{"className":724947,"style":32123},[31845],[34,724949,33249],{"className":724950},[31685,31686],[34,724952],{"className":724953,"style":32604},[31845],[34,724955,32226],{"className":724956},[32024],[34,724958],{"className":724959,"style":32604},[31845],[34,724961,724963,724966,725034,725037,725040,725043,725046,725049,725052],{"className":724962},[31676],[34,724964],{"className":724965,"style":46727},[31680],[34,724967,724969,724972,725031],{"className":724968},[31685],[34,724970],{"className":724971},[31782,41488],[34,724973,724975],{"className":724974},[41453],[34,724976,724978,725023],{"className":724977},[31796,31797],[34,724979,724981,725020],{"className":724980},[31801],[34,724982,724984,724998,725006],{"className":724983,"style":46746},[31805],[34,724985,724986,724989],{"style":41504},[34,724987],{"className":724988,"style":41508},[31813],[34,724990,724992],{"className":724991},[31818,31819,31820,31821],[34,724993,724995],{"className":724994},[31685,31821],[34,724996,15932],{"className":724997},[31685,31821],[34,724999,725000,725003],{"style":41520},[34,725001],{"className":725002,"style":41508},[31813],[34,725004],{"className":725005,"style":41528},[41527],[34,725007,725008,725011],{"style":46771},[34,725009],{"className":725010,"style":41508},[31813],[34,725012,725014],{"className":725013},[31818,31819,31820,31821],[34,725015,725017],{"className":725016},[31685,31821],[34,725018,640],{"className":725019},[31685,31821],[34,725021,31829],{"className":725022},[31828],[34,725024,725026],{"className":725025},[31801],[34,725027,725029],{"className":725028,"style":41619},[31805],[34,725030],{},[34,725032],{"className":725033},[31690,41488],[34,725035],{"className":725036,"style":32123},[31845],[34,725038,547860],{"className":725039},[32767],[34,725041],{"className":725042,"style":32123},[31845],[34,725044,33249],{"className":725045},[31685,31686],[34,725047],{"className":725048,"style":32604},[31845],[34,725050,32226],{"className":725051},[32024],[34,725053],{"className":725054,"style":32604},[31845],[34,725056,725058,725061,725129,725132,725135,725138,725141,725144,725147],{"className":725057},[31676],[34,725059],{"className":725060,"style":46727},[31680],[34,725062,725064,725067,725126],{"className":725063},[31685],[34,725065],{"className":725066},[31782,41488],[34,725068,725070],{"className":725069},[41453],[34,725071,725073,725118],{"className":725072},[31796,31797],[34,725074,725076,725115],{"className":725075},[31801],[34,725077,725079,725093,725101],{"className":725078,"style":46746},[31805],[34,725080,725081,725084],{"style":41504},[34,725082],{"className":725083,"style":41508},[31813],[34,725085,725087],{"className":725086},[31818,31819,31820,31821],[34,725088,725090],{"className":725089},[31685,31821],[34,725091,15932],{"className":725092},[31685,31821],[34,725094,725095,725098],{"style":41520},[34,725096],{"className":725097,"style":41508},[31813],[34,725099],{"className":725100,"style":41528},[41527],[34,725102,725103,725106],{"style":46771},[34,725104],{"className":725105,"style":41508},[31813],[34,725107,725109],{"className":725108},[31818,31819,31820,31821],[34,725110,725112],{"className":725111},[31685,31821],[34,725113,640],{"className":725114},[31685,31821],[34,725116,31829],{"className":725117},[31828],[34,725119,725121],{"className":725120},[31801],[34,725122,725124],{"className":725123,"style":41619},[31805],[34,725125],{},[34,725127],{"className":725128},[31690,41488],[34,725130],{"className":725131,"style":32123},[31845],[34,725133,547853],{"className":725134},[32767],[34,725136],{"className":725137,"style":32123},[31845],[34,725139,33249],{"className":725140},[31685,31686],[34,725142],{"className":725143,"style":32604},[31845],[34,725145,32226],{"className":725146},[32024],[34,725148],{"className":725149,"style":32604},[31845],[34,725151,725153,725156,725224,725227,725230,725233,725236,725239,725242],{"className":725152},[31676],[34,725154],{"className":725155,"style":46727},[31680],[34,725157,725159,725162,725221],{"className":725158},[31685],[34,725160],{"className":725161},[31782,41488],[34,725163,725165],{"className":725164},[41453],[34,725166,725168,725213],{"className":725167},[31796,31797],[34,725169,725171,725210],{"className":725170},[31801],[34,725172,725174,725188,725196],{"className":725173,"style":46746},[31805],[34,725175,725176,725179],{"style":41504},[34,725177],{"className":725178,"style":41508},[31813],[34,725180,725182],{"className":725181},[31818,31819,31820,31821],[34,725183,725185],{"className":725184},[31685,31821],[34,725186,15932],{"className":725187},[31685,31821],[34,725189,725190,725193],{"style":41520},[34,725191],{"className":725192,"style":41508},[31813],[34,725194],{"className":725195,"style":41528},[41527],[34,725197,725198,725201],{"style":46771},[34,725199],{"className":725200,"style":41508},[31813],[34,725202,725204],{"className":725203},[31818,31819,31820,31821],[34,725205,725207],{"className":725206},[31685,31821],[34,725208,640],{"className":725209},[31685,31821],[34,725211,31829],{"className":725212},[31828],[34,725214,725216],{"className":725215},[31801],[34,725217,725219],{"className":725218,"style":41619},[31805],[34,725220],{},[34,725222],{"className":725223},[31690,41488],[34,725225],{"className":725226,"style":32123},[31845],[34,725228,547860],{"className":725229},[32767],[34,725231],{"className":725232,"style":32123},[31845],[34,725234,33249],{"className":725235},[31685,31686],[34,725237],{"className":725238,"style":31846},[31845],[34,725240,202],{"className":725241},[31850],[34,725243],{"className":725244,"style":31846},[31845],[34,725246,725248,725251,725254,725322,725325,725328,725331],{"className":725247},[31676],[34,725249],{"className":725250,"style":46727},[31680],[34,725252,32226],{"className":725253},[31685],[34,725255,725257,725260,725319],{"className":725256},[31685],[34,725258],{"className":725259},[31782,41488],[34,725261,725263],{"className":725262},[41453],[34,725264,725266,725311],{"className":725265},[31796,31797],[34,725267,725269,725308],{"className":725268},[31801],[34,725270,725272,725286,725294],{"className":725271,"style":46746},[31805],[34,725273,725274,725277],{"style":41504},[34,725275],{"className":725276,"style":41508},[31813],[34,725278,725280],{"className":725279},[31818,31819,31820,31821],[34,725281,725283],{"className":725282},[31685,31821],[34,725284,768],{"className":725285},[31685,31821],[34,725287,725288,725291],{"style":41520},[34,725289],{"className":725290,"style":41508},[31813],[34,725292],{"className":725293,"style":41528},[41527],[34,725295,725296,725299],{"style":46771},[34,725297],{"className":725298,"style":41508},[31813],[34,725300,725302],{"className":725301},[31818,31819,31820,31821],[34,725303,725305],{"className":725304},[31685,31821],[34,725306,640],{"className":725307},[31685,31821],[34,725309,31829],{"className":725310},[31828],[34,725312,725314],{"className":725313},[31801],[34,725315,725317],{"className":725316,"style":41619},[31805],[34,725318],{},[34,725320],{"className":725321},[31690,41488],[34,725323],{"className":725324,"style":32123},[31845],[34,725326,547860],{"className":725327},[32767],[34,725329],{"className":725330,"style":32123},[31845],[34,725332,33249],{"className":725333},[31685,31686],[15,725335,725336,47649,725338],{},[2627,725337,69098],{},[34,725339,725341,725401],{"className":725340},[31642],[34,725342,725344],{"className":725343},[31646],[31648,725345,725346],{"xmlns":31650},[31652,725347,725348,725398],{},[31655,725349,725350,725352,725354,725360,725366,725368,725374,725384,725386,725392,725394,725396],{},[31658,725351,32936],{},[31662,725353,202],{},[31743,725355,725356,725358],{},[31658,725357,47384],{},[31749,725359,640],{},[37624,725361,725362,725364],{},[31658,725363,33244],{},[31658,725365,33249],{},[31662,725367,223],{},[31743,725369,725370,725372],{},[31658,725371,47384],{},[31749,725373,768],{},[37624,725375,725376,725378],{},[31658,725377,33244],{},[31655,725379,725380,725382],{},[31662,725381,32226],{},[31658,725383,33249],{},[31662,725385,32226],{},[41453,725387,725388,725390],{},[31749,725389,640],{},[31749,725391,768],{},[31658,725393,547860],{},[31662,725395,32682],{},[31658,725397,33249],{},[31665,725399,725400],{"encoding":31667},"x = C_1 e^t + C_2 e^{-t} - \\frac{1}{2}\\cos t",[34,725402,725404,725422,725506,725596],{"className":725403,"ariaHidden":758},[31672],[34,725405,725407,725410,725413,725416,725419],{"className":725406},[31676],[34,725408],{"className":725409,"style":32527},[31680],[34,725411,32936],{"className":725412},[31685,31686],[34,725414],{"className":725415,"style":31846},[31845],[34,725417,202],{"className":725418},[31850],[34,725420],{"className":725421,"style":31846},[31845],[34,725423,725425,725428,725468,725497,725500,725503],{"className":725424},[31676],[34,725426],{"className":725427,"style":723843},[31680],[34,725429,725431,725434],{"className":725430},[31685],[34,725432,47384],{"className":725433,"style":38706},[31685,31686],[34,725435,725437],{"className":725436},[31792],[34,725438,725440,725460],{"className":725439},[31796,31797],[34,725441,725443,725457],{"className":725442},[31801],[34,725444,725446],{"className":725445,"style":31806},[31805],[34,725447,725448,725451],{"style":129511},[34,725449],{"className":725450,"style":31814},[31813],[34,725452,725454],{"className":725453},[31818,31819,31820,31821],[34,725455,640],{"className":725456},[31685,31821],[34,725458,31829],{"className":725459},[31828],[34,725461,725463],{"className":725462},[31801],[34,725464,725466],{"className":725465,"style":31836},[31805],[34,725467],{},[34,725469,725471,725474],{"className":725470},[31685],[34,725472,33244],{"className":725473},[31685,31686],[34,725475,725477],{"className":725476},[31792],[34,725478,725480],{"className":725479},[31796],[34,725481,725483],{"className":725482},[31801],[34,725484,725486],{"className":725485,"style":611758},[31805],[34,725487,725488,725491],{"style":37777},[34,725489],{"className":725490,"style":31814},[31813],[34,725492,725494],{"className":725493},[31818,31819,31820,31821],[34,725495,33249],{"className":725496},[31685,31686,31821],[34,725498],{"className":725499,"style":32604},[31845],[34,725501,223],{"className":725502},[32024],[34,725504],{"className":725505,"style":32604},[31845],[34,725507,725509,725512,725552,725587,725590,725593],{"className":725508},[31676],[34,725510],{"className":725511,"style":723843},[31680],[34,725513,725515,725518],{"className":725514},[31685],[34,725516,47384],{"className":725517,"style":38706},[31685,31686],[34,725519,725521],{"className":725520},[31792],[34,725522,725524,725544],{"className":725523},[31796,31797],[34,725525,725527,725541],{"className":725526},[31801],[34,725528,725530],{"className":725529,"style":31806},[31805],[34,725531,725532,725535],{"style":129511},[34,725533],{"className":725534,"style":31814},[31813],[34,725536,725538],{"className":725537},[31818,31819,31820,31821],[34,725539,768],{"className":725540},[31685,31821],[34,725542,31829],{"className":725543},[31828],[34,725545,725547],{"className":725546},[31801],[34,725548,725550],{"className":725549,"style":31836},[31805],[34,725551],{},[34,725553,725555,725558],{"className":725554},[31685],[34,725556,33244],{"className":725557},[31685,31686],[34,725559,725561],{"className":725560},[31792],[34,725562,725564],{"className":725563},[31796],[34,725565,725567],{"className":725566},[31801],[34,725568,725570],{"className":725569,"style":611758},[31805],[34,725571,725572,725575],{"style":37777},[34,725573],{"className":725574,"style":31814},[31813],[34,725576,725578],{"className":725577},[31818,31819,31820,31821],[34,725579,725581,725584],{"className":725580},[31685,31821],[34,725582,32226],{"className":725583},[31685,31821],[34,725585,33249],{"className":725586},[31685,31686,31821],[34,725588],{"className":725589,"style":32604},[31845],[34,725591,32226],{"className":725592},[32024],[34,725594],{"className":725595,"style":32604},[31845],[34,725597,725599,725602,725670,725673,725676,725679],{"className":725598},[31676],[34,725600],{"className":725601,"style":46727},[31680],[34,725603,725605,725608,725667],{"className":725604},[31685],[34,725606],{"className":725607},[31782,41488],[34,725609,725611],{"className":725610},[41453],[34,725612,725614,725659],{"className":725613},[31796,31797],[34,725615,725617,725656],{"className":725616},[31801],[34,725618,725620,725634,725642],{"className":725619,"style":46746},[31805],[34,725621,725622,725625],{"style":41504},[34,725623],{"className":725624,"style":41508},[31813],[34,725626,725628],{"className":725627},[31818,31819,31820,31821],[34,725629,725631],{"className":725630},[31685,31821],[34,725632,768],{"className":725633},[31685,31821],[34,725635,725636,725639],{"style":41520},[34,725637],{"className":725638,"style":41508},[31813],[34,725640],{"className":725641,"style":41528},[41527],[34,725643,725644,725647],{"style":46771},[34,725645],{"className":725646,"style":41508},[31813],[34,725648,725650],{"className":725649},[31818,31819,31820,31821],[34,725651,725653],{"className":725652},[31685,31821],[34,725654,640],{"className":725655},[31685,31821],[34,725657,31829],{"className":725658},[31828],[34,725660,725662],{"className":725661},[31801],[34,725663,725665],{"className":725664,"style":41619},[31805],[34,725666],{},[34,725668],{"className":725669},[31690,41488],[34,725671],{"className":725672,"style":32123},[31845],[34,725674,547860],{"className":725675},[32767],[34,725677],{"className":725678,"style":32123},[31845],[34,725680,33249],{"className":725681},[31685,31686],[32188,725683],{},[32490,725685,389060,725687,719244,725825],{"id":725686},"_3-x4xtsin2tx-4x-tsin-2tx4xtsin2t已知-x1cos2tx2sin2tx_1-cos-2t-x_2-sin-2tx1cos2tx2sin2t",[34,725688,725690,725730],{"className":725689},[31642],[34,725691,725693],{"className":725692},[31646],[31648,725694,725695],{"xmlns":31650},[31652,725696,725697,725727],{},[31655,725698,725699,725709,725711,725713,725715,725717,725719,725721,725723,725725],{},[37624,725700,725701,725703],{},[31658,725702,32936],{},[31655,725704,725705,725707],{},[31662,725706,116495],{"mathvariant":38079},[31662,725708,116495],{"mathvariant":38079},[31662,725710,223],{},[31749,725712,15932],{},[31658,725714,32936],{},[31662,725716,202],{},[31658,725718,33249],{},[31658,725720,547853],{},[31662,725722,32682],{},[31749,725724,768],{},[31658,725726,33249],{},[31665,725728,725729],{"encoding":31667},"x'' + 4x = t\\sin 2t",[34,725731,725733,725780,725801],{"className":725732,"ariaHidden":758},[31672],[34,725734,725736,725739,725771,725774,725777],{"className":725735},[31676],[34,725737],{"className":725738,"style":384364},[31680],[34,725740,725742,725745],{"className":725741},[31685],[34,725743,32936],{"className":725744},[31685,31686],[34,725746,725748],{"className":725747},[31792],[34,725749,725751],{"className":725750},[31796],[34,725752,725754],{"className":725753},[31801],[34,725755,725757],{"className":725756,"style":132027},[31805],[34,725758,725759,725762],{"style":37777},[34,725760],{"className":725761,"style":31814},[31813],[34,725763,725765],{"className":725764},[31818,31819,31820,31821],[34,725766,725768],{"className":725767},[31685,31821],[34,725769,502782],{"className":725770},[31685,31821],[34,725772],{"className":725773,"style":32604},[31845],[34,725775,223],{"className":725776},[32024],[34,725778],{"className":725779,"style":32604},[31845],[34,725781,725783,725786,725789,725792,725795,725798],{"className":725782},[31676],[34,725784],{"className":725785,"style":32617},[31680],[34,725787,15932],{"className":725788},[31685],[34,725790,32936],{"className":725791},[31685,31686],[34,725793],{"className":725794,"style":31846},[31845],[34,725796,202],{"className":725797},[31850],[34,725799],{"className":725800,"style":31846},[31845],[34,725802,725804,725807,725810,725813,725816,725819,725822],{"className":725803},[31676],[34,725805],{"className":725806,"style":71430},[31680],[34,725808,33249],{"className":725809},[31685,31686],[34,725811],{"className":725812,"style":32123},[31845],[34,725814,547853],{"className":725815},[32767],[34,725817],{"className":725818,"style":32123},[31845],[34,725820,768],{"className":725821},[31685],[34,725823,33249],{"className":725824},[31685,31686],[34,725826,725828,725874],{"className":725827},[31642],[34,725829,725831],{"className":725830},[31646],[31648,725832,725833],{"xmlns":31650},[31652,725834,725835,725871],{},[31655,725836,725837,725843,725845,725847,725849,725851,725853,725855,725861,725863,725865,725867,725869],{},[31743,725838,725839,725841],{},[31658,725840,32936],{},[31749,725842,640],{},[31662,725844,202],{},[31658,725846,547860],{},[31662,725848,32682],{},[31749,725850,768],{},[31658,725852,33249],{},[31662,725854,31960],{"separator":758},[31743,725856,725857,725859],{},[31658,725858,32936],{},[31749,725860,768],{},[31662,725862,202],{},[31658,725864,547853],{},[31662,725866,32682],{},[31749,725868,768],{},[31658,725870,33249],{},[31665,725872,725873],{"encoding":31667},"x_1 = \\cos 2t, x_2 = \\sin 2t",[34,725875,725877,725932,726005],{"className":725876,"ariaHidden":758},[31672],[34,725878,725880,725883,725923,725926,725929],{"className":725879},[31676],[34,725881],{"className":725882,"style":31860},[31680],[34,725884,725886,725889],{"className":725885},[31685],[34,725887,32936],{"className":725888},[31685,31686],[34,725890,725892],{"className":725891},[31792],[34,725893,725895,725915],{"className":725894},[31796,31797],[34,725896,725898,725912],{"className":725897},[31801],[34,725899,725901],{"className":725900,"style":31806},[31805],[34,725902,725903,725906],{"style":31809},[34,725904],{"className":725905,"style":31814},[31813],[34,725907,725909],{"className":725908},[31818,31819,31820,31821],[34,725910,640],{"className":725911},[31685,31821],[34,725913,31829],{"className":725914},[31828],[34,725916,725918],{"className":725917},[31801],[34,725919,725921],{"className":725920,"style":31836},[31805],[34,725922],{},[34,725924],{"className":725925,"style":31846},[31845],[34,725927,202],{"className":725928},[31850],[34,725930],{"className":725931,"style":31846},[31845],[34,725933,725935,725938,725941,725944,725947,725950,725953,725956,725996,725999,726002],{"className":725934},[31676],[34,725936],{"className":725937,"style":40292},[31680],[34,725939,547860],{"className":725940},[32767],[34,725942],{"className":725943,"style":32123},[31845],[34,725945,768],{"className":725946},[31685],[34,725948,33249],{"className":725949},[31685,31686],[34,725951,31960],{"className":725952},[32119],[34,725954],{"className":725955,"style":32123},[31845],[34,725957,725959,725962],{"className":725958},[31685],[34,725960,32936],{"className":725961},[31685,31686],[34,725963,725965],{"className":725964},[31792],[34,725966,725968,725988],{"className":725967},[31796,31797],[34,725969,725971,725985],{"className":725970},[31801],[34,725972,725974],{"className":725973,"style":31806},[31805],[34,725975,725976,725979],{"style":31809},[34,725977],{"className":725978,"style":31814},[31813],[34,725980,725982],{"className":725981},[31818,31819,31820,31821],[34,725983,768],{"className":725984},[31685,31821],[34,725986,31829],{"className":725987},[31828],[34,725989,725991],{"className":725990},[31801],[34,725992,725994],{"className":725993,"style":31836},[31805],[34,725995],{},[34,725997],{"className":725998,"style":31846},[31845],[34,726000,202],{"className":726001},[31850],[34,726003],{"className":726004,"style":31846},[31845],[34,726006,726008,726011,726014,726017,726020],{"className":726007},[31676],[34,726009],{"className":726010,"style":71430},[31680],[34,726012,547853],{"className":726013},[32767],[34,726015],{"className":726016,"style":32123},[31845],[34,726018,768],{"className":726019},[31685],[34,726021,33249],{"className":726022},[31685,31686],[15,726024,726025,726027,726092,726093,347],{},[2627,726026,719488],{},[34,726028,726030,726051],{"className":726029},[31642],[34,726031,726033],{"className":726032},[31646],[31648,726034,726035],{"xmlns":31650},[31652,726036,726037,726049],{},[31655,726038,726039],{},[37624,726040,726041,726043],{},[31658,726042,32936],{},[31655,726044,726045,726047],{},[31662,726046,116495],{"mathvariant":38079},[31662,726048,116495],{"mathvariant":38079},[31665,726050,719513],{"encoding":31667},[34,726052,726054],{"className":726053,"ariaHidden":758},[31672],[34,726055,726057,726060],{"className":726056},[31676],[34,726058],{"className":726059,"style":132027},[31680],[34,726061,726063,726066],{"className":726062},[31685],[34,726064,32936],{"className":726065},[31685,31686],[34,726067,726069],{"className":726068},[31792],[34,726070,726072],{"className":726071},[31796],[34,726073,726075],{"className":726074},[31801],[34,726076,726078],{"className":726077,"style":132027},[31805],[34,726079,726080,726083],{"style":37777},[34,726081],{"className":726082,"style":31814},[31813],[34,726084,726086],{"className":726085},[31818,31819,31820,31821],[34,726087,726089],{"className":726088},[31685,31821],[34,726090,502782],{"className":726091},[31685,31821]," 前系数为 1，合格。",[34,726094,726096,726128],{"className":726095},[31642],[34,726097,726099],{"className":726098},[31646],[31648,726100,726101],{"xmlns":31650},[31652,726102,726103,726125],{},[31655,726104,726105,726107,726109,726111,726113,726115,726117,726119,726121,726123],{},[31658,726106,31739],{},[31662,726108,456],{"stretchy":702},[31658,726110,33249],{},[31662,726112,2557],{"stretchy":702},[31662,726114,202],{},[31658,726116,33249],{},[31658,726118,547853],{},[31662,726120,32682],{},[31749,726122,768],{},[31658,726124,33249],{},[31665,726126,726127],{"encoding":31667},"f(t) = t\\sin 2t",[34,726129,726131,726158],{"className":726130,"ariaHidden":758},[31672],[34,726132,726134,726137,726140,726143,726146,726149,726152,726155],{"className":726133},[31676],[34,726135],{"className":726136,"style":31774},[31680],[34,726138,31739],{"className":726139,"style":31778},[31685,31686],[34,726141,456],{"className":726142},[31782],[34,726144,33249],{"className":726145},[31685,31686],[34,726147,2557],{"className":726148},[31690],[34,726150],{"className":726151,"style":31846},[31845],[34,726153,202],{"className":726154},[31850],[34,726156],{"className":726157,"style":31846},[31845],[34,726159,726161,726164,726167,726170,726173,726176,726179],{"className":726160},[31676],[34,726162],{"className":726163,"style":71430},[31680],[34,726165,33249],{"className":726166},[31685,31686],[34,726168],{"className":726169,"style":32123},[31845],[34,726171,547853],{"className":726172},[32767],[34,726174],{"className":726175,"style":32123},[31845],[34,726177,768],{"className":726178},[31685],[34,726180,33249],{"className":726181},[31685,31686],[15,726183,726184,726187,726188,721062],{},[2627,726185,726186],{},"Step 2: 列方程组","\n[\n",[34,726189,726191,726321],{"className":726190},[31642],[34,726192,726194],{"className":726193},[31646],[31648,726195,726196],{"xmlns":31650},[31652,726197,726198,726318],{},[31655,726199,726200,726202],{},[31662,726201,32685],{"fence":758},[71078,726203,726204,726250],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,726205,726206],{},[71087,726207,726208],{},[71090,726209,726210],{"scriptlevel":459,"displaystyle":702},[31655,726211,726212,726220,726222,726224,726226,726228,726230,726238,726240,726242,726244,726246,726248],{},[38846,726213,726214,726216,726218],{},[31658,726215,31965],{},[31749,726217,640],{},[31662,726219,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,726221,547860],{},[31662,726223,32682],{},[31749,726225,768],{},[31658,726227,33249],{},[31662,726229,223],{},[38846,726231,726232,726234,726236],{},[31658,726233,31965],{},[31749,726235,768],{},[31662,726237,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,726239,547853],{},[31662,726241,32682],{},[31749,726243,768],{},[31658,726245,33249],{},[31662,726247,202],{},[31749,726249,459],{},[71084,726251,726252],{},[71087,726253,726254],{},[71090,726255,726256],{"scriptlevel":459,"displaystyle":702},[31655,726257,726258,726266,726268,726270,726272,726274,726276,726278,726280,726282,726284,726292,726294,726296,726298,726300,726302,726304,726306,726308,726310,726312,726314,726316],{},[38846,726259,726260,726262,726264],{},[31658,726261,31965],{},[31749,726263,640],{},[31662,726265,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,726267,456],{"stretchy":702},[31662,726269,32226],{},[31749,726271,768],{},[31658,726273,547853],{},[31662,726275,32682],{},[31749,726277,768],{},[31658,726279,33249],{},[31662,726281,2557],{"stretchy":702},[31662,726283,223],{},[38846,726285,726286,726288,726290],{},[31658,726287,31965],{},[31749,726289,768],{},[31662,726291,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,726293,456],{"stretchy":702},[31749,726295,768],{},[31658,726297,547860],{},[31662,726299,32682],{},[31749,726301,768],{},[31658,726303,33249],{},[31662,726305,2557],{"stretchy":702},[31662,726307,202],{},[31658,726309,33249],{},[31658,726311,547853],{},[31662,726313,32682],{},[31749,726315,768],{},[31658,726317,33249],{},[31665,726319,726320],{"encoding":31667},"\\begin{cases}\nc_1' \\cos 2t + c_2' \\sin 2t = 0 \\\\\nc_1' (-2\\sin 2t) + c_2' (2\\cos 2t) = t\\sin 2t\n\\end{cases}",[34,726322,726324],{"className":726323,"ariaHidden":758},[31672],[34,726325,726327,726330],{"className":726326},[31676],[34,726328],{"className":726329,"style":75797},[31680],[34,726331,726333,726339,726738],{"className":726332},[34006],[34,726334,726336],{"className":726335,"style":75811},[31782,75810],[34,726337,32685],{"className":726338},[71281,75815],[34,726340,726342],{"className":726341},[31685],[34,726343,726345],{"className":726344},[71078],[34,726346,726348],{"className":726347},[71384],[34,726349,726351,726730],{"className":726350},[31796,31797],[34,726352,726354,726727],{"className":726353},[31801],[34,726355,726357,726524],{"className":726356,"style":75834},[31805],[34,726358,726359,726362],{"style":75837},[34,726360],{"className":726361,"style":71401},[31813],[34,726363,726365,726419,726422,726425,726428,726431,726434,726437,726440,726443,726497,726500,726503,726506,726509,726512,726515,726518,726521],{"className":726364},[31685],[34,726366,726368,726371],{"className":726367},[31685],[34,726369,31965],{"className":726370},[31685,31686],[34,726372,726374],{"className":726373},[31792],[34,726375,726377,726411],{"className":726376},[31796,31797],[34,726378,726380,726408],{"className":726379},[31801],[34,726381,726383,726394],{"className":726382,"style":132027},[31805],[34,726384,726385,726388],{"style":316930},[34,726386],{"className":726387,"style":31814},[31813],[34,726389,726391],{"className":726390},[31818,31819,31820,31821],[34,726392,640],{"className":726393},[31685,31821],[34,726395,726396,726399],{"style":37777},[34,726397],{"className":726398,"style":31814},[31813],[34,726400,726402],{"className":726401},[31818,31819,31820,31821],[34,726403,726405],{"className":726404},[31685,31821],[34,726406,116495],{"className":726407},[31685,31821],[34,726409,31829],{"className":726410},[31828],[34,726412,726414],{"className":726413},[31801],[34,726415,726417],{"className":726416,"style":39021},[31805],[34,726418],{},[34,726420],{"className":726421,"style":32123},[31845],[34,726423,547860],{"className":726424},[32767],[34,726426],{"className":726427,"style":32123},[31845],[34,726429,768],{"className":726430},[31685],[34,726432,33249],{"className":726433},[31685,31686],[34,726435],{"className":726436,"style":32604},[31845],[34,726438,223],{"className":726439},[32024],[34,726441],{"className":726442,"style":32604},[31845],[34,726444,726446,726449],{"className":726445},[31685],[34,726447,31965],{"className":726448},[31685,31686],[34,726450,726452],{"className":726451},[31792],[34,726453,726455,726489],{"className":726454},[31796,31797],[34,726456,726458,726486],{"className":726457},[31801],[34,726459,726461,726472],{"className":726460,"style":132027},[31805],[34,726462,726463,726466],{"style":316930},[34,726464],{"className":726465,"style":31814},[31813],[34,726467,726469],{"className":726468},[31818,31819,31820,31821],[34,726470,768],{"className":726471},[31685,31821],[34,726473,726474,726477],{"style":37777},[34,726475],{"className":726476,"style":31814},[31813],[34,726478,726480],{"className":726479},[31818,31819,31820,31821],[34,726481,726483],{"className":726482},[31685,31821],[34,726484,116495],{"className":726485},[31685,31821],[34,726487,31829],{"className":726488},[31828],[34,726490,726492],{"className":726491},[31801],[34,726493,726495],{"className":726494,"style":39021},[31805],[34,726496],{},[34,726498],{"className":726499,"style":32123},[31845],[34,726501,547853],{"className":726502},[32767],[34,726504],{"className":726505,"style":32123},[31845],[34,726507,768],{"className":726508},[31685],[34,726510,33249],{"className":726511},[31685,31686],[34,726513],{"className":726514,"style":31846},[31845],[34,726516,202],{"className":726517},[31850],[34,726519],{"className":726520,"style":31846},[31845],[34,726522,459],{"className":726523},[31685],[34,726525,726526,726529],{"style":76011},[34,726527],{"className":726528,"style":71401},[31813],[34,726530,726532,726586,726589,726592,726595,726598,726601,726604,726607,726610,726613,726616,726619,726622,726676,726679,726682,726685,726688,726691,726694,726697,726700,726703,726706,726709,726712,726715,726718,726721,726724],{"className":726531},[31685],[34,726533,726535,726538],{"className":726534},[31685],[34,726536,31965],{"className":726537},[31685,31686],[34,726539,726541],{"className":726540},[31792],[34,726542,726544,726578],{"className":726543},[31796,31797],[34,726545,726547,726575],{"className":726546},[31801],[34,726548,726550,726561],{"className":726549,"style":132027},[31805],[34,726551,726552,726555],{"style":316930},[34,726553],{"className":726554,"style":31814},[31813],[34,726556,726558],{"className":726557},[31818,31819,31820,31821],[34,726559,640],{"className":726560},[31685,31821],[34,726562,726563,726566],{"style":37777},[34,726564],{"className":726565,"style":31814},[31813],[34,726567,726569],{"className":726568},[31818,31819,31820,31821],[34,726570,726572],{"className":726571},[31685,31821],[34,726573,116495],{"className":726574},[31685,31821],[34,726576,31829],{"className":726577},[31828],[34,726579,726581],{"className":726580},[31801],[34,726582,726584],{"className":726583,"style":39021},[31805],[34,726585],{},[34,726587,456],{"className":726588},[31782],[34,726590,32226],{"className":726591},[31685],[34,726593,768],{"className":726594},[31685],[34,726596],{"className":726597,"style":32123},[31845],[34,726599,547853],{"className":726600},[32767],[34,726602],{"className":726603,"style":32123},[31845],[34,726605,768],{"className":726606},[31685],[34,726608,33249],{"className":726609},[31685,31686],[34,726611,2557],{"className":726612},[31690],[34,726614],{"className":726615,"style":32604},[31845],[34,726617,223],{"className":726618},[32024],[34,726620],{"className":726621,"style":32604},[31845],[34,726623,726625,726628],{"className":726624},[31685],[34,726626,31965],{"className":726627},[31685,31686],[34,726629,726631],{"className":726630},[31792],[34,726632,726634,726668],{"className":726633},[31796,31797],[34,726635,726637,726665],{"className":726636},[31801],[34,726638,726640,726651],{"className":726639,"style":132027},[31805],[34,726641,726642,726645],{"style":316930},[34,726643],{"className":726644,"style":31814},[31813],[34,726646,726648],{"className":726647},[31818,31819,31820,31821],[34,726649,768],{"className":726650},[31685,31821],[34,726652,726653,726656],{"style":37777},[34,726654],{"className":726655,"style":31814},[31813],[34,726657,726659],{"className":726658},[31818,31819,31820,31821],[34,726660,726662],{"className":726661},[31685,31821],[34,726663,116495],{"className":726664},[31685,31821],[34,726666,31829],{"className":726667},[31828],[34,726669,726671],{"className":726670},[31801],[34,726672,726674],{"className":726673,"style":39021},[31805],[34,726675],{},[34,726677,456],{"className":726678},[31782],[34,726680,768],{"className":726681},[31685],[34,726683],{"className":726684,"style":32123},[31845],[34,726686,547860],{"className":726687},[32767],[34,726689],{"className":726690,"style":32123},[31845],[34,726692,768],{"className":726693},[31685],[34,726695,33249],{"className":726696},[31685,31686],[34,726698,2557],{"className":726699},[31690],[34,726701],{"className":726702,"style":31846},[31845],[34,726704,202],{"className":726705},[31850],[34,726707],{"className":726708,"style":31846},[31845],[34,726710,33249],{"className":726711},[31685,31686],[34,726713],{"className":726714,"style":32123},[31845],[34,726716,547853],{"className":726717},[32767],[34,726719],{"className":726720,"style":32123},[31845],[34,726722,768],{"className":726723},[31685],[34,726725,33249],{"className":726726},[31685,31686],[34,726728,31829],{"className":726729},[31828],[34,726731,726733],{"className":726732},[31801],[34,726734,726736],{"className":726735,"style":76192},[31805],[34,726737],{},[34,726739],{"className":726740},[31690,41488],[15,726742,726743,726745],{},[2627,726744,721067],{},"\n这是一个二元一次方程组。",[105,726747,726748,727760],{},[108,726749,726750,726751,726803,726804,726848,726849,33183,726935,727253,727531],{},"将第一式乘以 ",[34,726752,726754,726776],{"className":726753},[31642],[34,726755,726757],{"className":726756},[31646],[31648,726758,726759],{"xmlns":31650},[31652,726760,726761,726773],{},[31655,726762,726763,726765,726767,726769,726771],{},[31749,726764,768],{},[31658,726766,547853],{},[31662,726768,32682],{},[31749,726770,768],{},[31658,726772,33249],{},[31665,726774,726775],{"encoding":31667},"2\\sin 2t",[34,726777,726779],{"className":726778,"ariaHidden":758},[31672],[34,726780,726782,726785,726788,726791,726794,726797,726800],{"className":726781},[31676],[34,726783],{"className":726784,"style":71430},[31680],[34,726786,768],{"className":726787},[31685],[34,726789],{"className":726790,"style":32123},[31845],[34,726792,547853],{"className":726793},[32767],[34,726795],{"className":726796,"style":32123},[31845],[34,726798,768],{"className":726799},[31685],[34,726801,33249],{"className":726802},[31685,31686],"，第二式乘以 ",[34,726805,726807,726827],{"className":726806},[31642],[34,726808,726810],{"className":726809},[31646],[31648,726811,726812],{"xmlns":31650},[31652,726813,726814,726824],{},[31655,726815,726816,726818,726820,726822],{},[31658,726817,547860],{},[31662,726819,32682],{},[31749,726821,768],{},[31658,726823,33249],{},[31665,726825,726826],{"encoding":31667},"\\cos 2t",[34,726828,726830],{"className":726829,"ariaHidden":758},[31672],[34,726831,726833,726836,726839,726842,726845],{"className":726832},[31676],[34,726834],{"className":726835,"style":32617},[31680],[34,726837,547860],{"className":726838},[32767],[34,726840],{"className":726841,"style":32123},[31845],[34,726843,768],{"className":726844},[31685],[34,726846,33249],{"className":726847},[31685,31686],"，相加消去 ",[34,726850,726852,726872],{"className":726851},[31642],[34,726853,726855],{"className":726854},[31646],[31648,726856,726857],{"xmlns":31650},[31652,726858,726859,726869],{},[31655,726860,726861],{},[38846,726862,726863,726865,726867],{},[31658,726864,31965],{},[31749,726866,640],{},[31662,726868,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,726870,726871],{"encoding":31667},"c_1'",[34,726873,726875],{"className":726874,"ariaHidden":758},[31672],[34,726876,726878,726881],{"className":726877},[31676],[34,726879],{"className":726880,"style":716670},[31680],[34,726882,726884,726887],{"className":726883},[31685],[34,726885,31965],{"className":726886},[31685,31686],[34,726888,726890],{"className":726889},[31792],[34,726891,726893,726927],{"className":726892},[31796,31797],[34,726894,726896,726924],{"className":726895},[31801],[34,726897,726899,726910],{"className":726898,"style":132027},[31805],[34,726900,726901,726904],{"style":316930},[34,726902],{"className":726903,"style":31814},[31813],[34,726905,726907],{"className":726906},[31818,31819,31820,31821],[34,726908,640],{"className":726909},[31685,31821],[34,726911,726912,726915],{"style":37777},[34,726913],{"className":726914,"style":31814},[31813],[34,726916,726918],{"className":726917},[31818,31819,31820,31821],[34,726919,726921],{"className":726920},[31685,31821],[34,726922,116495],{"className":726923},[31685,31821],[34,726925,31829],{"className":726926},[31828],[34,726928,726930],{"className":726929},[31801],[34,726931,726933],{"className":726932,"style":39021},[31805],[34,726934],{},[34,726936,726938,727024],{"className":726937},[31642],[34,726939,726941],{"className":726940},[31646],[31648,726942,726943],{"xmlns":31650},[31652,726944,726945,727021],{},[31655,726946,726947,726955,726957,726959,726969,726971,726973,726975,726977,726987,726989,726991,726993,726995,726997,726999,727001,727003,727005,727007,727009,727011,727013,727015,727017,727019],{},[38846,726948,726949,726951,726953],{},[31658,726950,31965],{},[31749,726952,768],{},[31662,726954,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,726956,456],{"stretchy":702},[31749,726958,768],{},[37624,726960,726961,726967],{},[31655,726962,726963,726965],{},[31658,726964,547853],{},[31662,726966,32682],{},[31749,726968,768],{},[31749,726970,768],{},[31658,726972,33249],{},[31662,726974,223],{},[31749,726976,768],{},[37624,726978,726979,726985],{},[31655,726980,726981,726983],{},[31658,726982,547860],{},[31662,726984,32682],{},[31749,726986,768],{},[31749,726988,768],{},[31658,726990,33249],{},[31662,726992,2557],{"stretchy":702},[31662,726994,202],{},[31662,726996,456],{"stretchy":702},[31658,726998,33249],{},[31658,727000,547853],{},[31662,727002,32682],{},[31749,727004,768],{},[31658,727006,33249],{},[31662,727008,2557],{"stretchy":702},[31662,727010,456],{"stretchy":702},[31658,727012,547860],{},[31662,727014,32682],{},[31749,727016,768],{},[31658,727018,33249],{},[31662,727020,2557],{"stretchy":702},[31665,727022,727023],{"encoding":31667},"c_2'(2\\sin^2 2t + 2\\cos^2 2t) = (t\\sin 2t)(\\cos 2t)",[34,727025,727027,727143,727205],{"className":727026,"ariaHidden":758},[31672],[34,727028,727030,727033,727087,727090,727093,727096,727125,727128,727131,727134,727137,727140],{"className":727029},[31676],[34,727031],{"className":727032,"style":687514},[31680],[34,727034,727036,727039],{"className":727035},[31685],[34,727037,31965],{"className":727038},[31685,31686],[34,727040,727042],{"className":727041},[31792],[34,727043,727045,727079],{"className":727044},[31796,31797],[34,727046,727048,727076],{"className":727047},[31801],[34,727049,727051,727062],{"className":727050,"style":132027},[31805],[34,727052,727053,727056],{"style":316930},[34,727054],{"className":727055,"style":31814},[31813],[34,727057,727059],{"className":727058},[31818,31819,31820,31821],[34,727060,768],{"className":727061},[31685,31821],[34,727063,727064,727067],{"style":37777},[34,727065],{"className":727066,"style":31814},[31813],[34,727068,727070],{"className":727069},[31818,31819,31820,31821],[34,727071,727073],{"className":727072},[31685,31821],[34,727074,116495],{"className":727075},[31685,31821],[34,727077,31829],{"className":727078},[31828],[34,727080,727082],{"className":727081},[31801],[34,727083,727085],{"className":727084,"style":39021},[31805],[34,727086],{},[34,727088,456],{"className":727089},[31782],[34,727091,768],{"className":727092},[31685],[34,727094],{"className":727095,"style":32123},[31845],[34,727097,727099,727102],{"className":727098},[32767],[34,727100,547853],{"className":727101},[32767],[34,727103,727105],{"className":727104},[31792],[34,727106,727108],{"className":727107},[31796],[34,727109,727111],{"className":727110},[31801],[34,727112,727114],{"className":727113,"style":585884},[31805],[34,727115,727116,727119],{"style":586008},[34,727117],{"className":727118,"style":31814},[31813],[34,727120,727122],{"className":727121},[31818,31819,31820,31821],[34,727123,768],{"className":727124},[31685,31821],[34,727126],{"className":727127,"style":32123},[31845],[34,727129,768],{"className":727130},[31685],[34,727132,33249],{"className":727133},[31685,31686],[34,727135],{"className":727136,"style":32604},[31845],[34,727138,223],{"className":727139},[32024],[34,727141],{"className":727142,"style":32604},[31845],[34,727144,727146,727149,727152,727155,727184,727187,727190,727193,727196,727199,727202],{"className":727145},[31676],[34,727147],{"className":727148,"style":37755},[31680],[34,727150,768],{"className":727151},[31685],[34,727153],{"className":727154,"style":32123},[31845],[34,727156,727158,727161],{"className":727157},[32767],[34,727159,547860],{"className":727160},[32767],[34,727162,727164],{"className":727163},[31792],[34,727165,727167],{"className":727166},[31796],[34,727168,727170],{"className":727169},[31801],[34,727171,727173],{"className":727172,"style":37774},[31805],[34,727174,727175,727178],{"style":37777},[34,727176],{"className":727177,"style":31814},[31813],[34,727179,727181],{"className":727180},[31818,31819,31820,31821],[34,727182,768],{"className":727183},[31685,31821],[34,727185],{"className":727186,"style":32123},[31845],[34,727188,768],{"className":727189},[31685],[34,727191,33249],{"className":727192},[31685,31686],[34,727194,2557],{"className":727195},[31690],[34,727197],{"className":727198,"style":31846},[31845],[34,727200,202],{"className":727201},[31850],[34,727203],{"className":727204,"style":31846},[31845],[34,727206,727208,727211,727214,727217,727220,727223,727226,727229,727232,727235,727238,727241,727244,727247,727250],{"className":727207},[31676],[34,727209],{"className":727210,"style":31774},[31680],[34,727212,456],{"className":727213},[31782],[34,727215,33249],{"className":727216},[31685,31686],[34,727218],{"className":727219,"style":32123},[31845],[34,727221,547853],{"className":727222},[32767],[34,727224],{"className":727225,"style":32123},[31845],[34,727227,768],{"className":727228},[31685],[34,727230,33249],{"className":727231},[31685,31686],[34,727233,2557],{"className":727234},[31690],[34,727236,456],{"className":727237},[31782],[34,727239,547860],{"className":727240},[32767],[34,727242],{"className":727243,"style":32123},[31845],[34,727245,768],{"className":727246},[31685],[34,727248,33249],{"className":727249},[31685,31686],[34,727251,2557],{"className":727252},[31690],[34,727254,727256,727316],{"className":727255},[31642],[34,727257,727259],{"className":727258},[31646],[31648,727260,727261],{"xmlns":31650},[31652,727262,727263,727313],{},[31655,727264,727265,727267,727275,727277,727279,727281,727283,727285,727287,727289,727291,727293,727295,727297,727303,727305,727307,727309,727311],{},[31749,727266,768],{},[38846,727268,727269,727271,727273],{},[31658,727270,31965],{},[31749,727272,768],{},[31662,727274,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,727276,202],{},[31658,727278,33249],{},[31658,727280,547853],{},[31662,727282,32682],{},[31749,727284,768],{},[31658,727286,33249],{},[31658,727288,547860],{},[31662,727290,32682],{},[31749,727292,768],{},[31658,727294,33249],{},[31662,727296,202],{},[41453,727298,727299,727301],{},[31749,727300,640],{},[31749,727302,768],{},[31658,727304,33249],{},[31658,727306,547853],{},[31662,727308,32682],{},[31749,727310,15932],{},[31658,727312,33249],{},[31665,727314,727315],{"encoding":31667},"2c_2' = t\\sin 2t \\cos 2t = \\frac{1}{2}t \\sin 4t",[34,727317,727319,727391,727439],{"className":727318,"ariaHidden":758},[31672],[34,727320,727322,727325,727328,727382,727385,727388],{"className":727321},[31676],[34,727323],{"className":727324,"style":716670},[31680],[34,727326,768],{"className":727327},[31685],[34,727329,727331,727334],{"className":727330},[31685],[34,727332,31965],{"className":727333},[31685,31686],[34,727335,727337],{"className":727336},[31792],[34,727338,727340,727374],{"className":727339},[31796,31797],[34,727341,727343,727371],{"className":727342},[31801],[34,727344,727346,727357],{"className":727345,"style":132027},[31805],[34,727347,727348,727351],{"style":316930},[34,727349],{"className":727350,"style":31814},[31813],[34,727352,727354],{"className":727353},[31818,31819,31820,31821],[34,727355,768],{"className":727356},[31685,31821],[34,727358,727359,727362],{"style":37777},[34,727360],{"className":727361,"style":31814},[31813],[34,727363,727365],{"className":727364},[31818,31819,31820,31821],[34,727366,727368],{"className":727367},[31685,31821],[34,727369,116495],{"className":727370},[31685,31821],[34,727372,31829],{"className":727373},[31828],[34,727375,727377],{"className":727376},[31801],[34,727378,727380],{"className":727379,"style":39021},[31805],[34,727381],{},[34,727383],{"className":727384,"style":31846},[31845],[34,727386,202],{"className":727387},[31850],[34,727389],{"className":727390,"style":31846},[31845],[34,727392,727394,727397,727400,727403,727406,727409,727412,727415,727418,727421,727424,727427,727430,727433,727436],{"className":727393},[31676],[34,727395],{"className":727396,"style":71430},[31680],[34,727398,33249],{"className":727399},[31685,31686],[34,727401],{"className":727402,"style":32123},[31845],[34,727404,547853],{"className":727405},[32767],[34,727407],{"className":727408,"style":32123},[31845],[34,727410,768],{"className":727411},[31685],[34,727413,33249],{"className":727414},[31685,31686],[34,727416],{"className":727417,"style":32123},[31845],[34,727419,547860],{"className":727420},[32767],[34,727422],{"className":727423,"style":32123},[31845],[34,727425,768],{"className":727426},[31685],[34,727428,33249],{"className":727429},[31685,31686],[34,727431],{"className":727432,"style":31846},[31845],[34,727434,202],{"className":727435},[31850],[34,727437],{"className":727438,"style":31846},[31845],[34,727440,727442,727445,727513,727516,727519,727522,727525,727528],{"className":727441},[31676],[34,727443],{"className":727444,"style":46727},[31680],[34,727446,727448,727451,727510],{"className":727447},[31685],[34,727449],{"className":727450},[31782,41488],[34,727452,727454],{"className":727453},[41453],[34,727455,727457,727502],{"className":727456},[31796,31797],[34,727458,727460,727499],{"className":727459},[31801],[34,727461,727463,727477,727485],{"className":727462,"style":46746},[31805],[34,727464,727465,727468],{"style":41504},[34,727466],{"className":727467,"style":41508},[31813],[34,727469,727471],{"className":727470},[31818,31819,31820,31821],[34,727472,727474],{"className":727473},[31685,31821],[34,727475,768],{"className":727476},[31685,31821],[34,727478,727479,727482],{"style":41520},[34,727480],{"className":727481,"style":41508},[31813],[34,727483],{"className":727484,"style":41528},[41527],[34,727486,727487,727490],{"style":46771},[34,727488],{"className":727489,"style":41508},[31813],[34,727491,727493],{"className":727492},[31818,31819,31820,31821],[34,727494,727496],{"className":727495},[31685,31821],[34,727497,640],{"className":727498},[31685,31821],[34,727500,31829],{"className":727501},[31828],[34,727503,727505],{"className":727504},[31801],[34,727506,727508],{"className":727507,"style":41619},[31805],[34,727509],{},[34,727511],{"className":727512},[31690,41488],[34,727514,33249],{"className":727515},[31685,31686],[34,727517],{"className":727518,"style":32123},[31845],[34,727520,547853],{"className":727521},[32767],[34,727523],{"className":727524,"style":32123},[31845],[34,727526,15932],{"className":727527},[31685],[34,727529,33249],{"className":727530},[31685,31686],[34,727532,727534,727578],{"className":727533},[31642],[34,727535,727537],{"className":727536},[31646],[31648,727538,727539],{"xmlns":31650},[31652,727540,727541,727575],{},[31655,727542,727543,727545,727547,727549,727557,727559,727565,727567,727569,727571,727573],{},[39361,727544,262622],{},[31662,727546,262625],{},[39361,727548,262622],{},[38846,727550,727551,727553,727555],{},[31658,727552,31965],{},[31749,727554,768],{},[31662,727556,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,727558,202],{},[41453,727560,727561,727563],{},[31749,727562,640],{},[31749,727564,15932],{},[31658,727566,33249],{},[31658,727568,547853],{},[31662,727570,32682],{},[31749,727572,15932],{},[31658,727574,33249],{},[31665,727576,727577],{"encoding":31667},"\\implies c_2' = \\frac{1}{4}t \\sin 4t",[34,727579,727581,727599,727668],{"className":727580,"ariaHidden":758},[31672],[34,727582,727584,727587,727590,727593,727596],{"className":727583},[31676],[34,727585],{"className":727586,"style":318264},[31680],[34,727588],{"className":727589,"style":31846},[31845],[34,727591,262625],{"className":727592},[31850],[34,727594],{"className":727595,"style":31846},[31845],[34,727597],{"className":727598,"style":31846},[31845],[34,727600,727602,727605,727659,727662,727665],{"className":727601},[31676],[34,727603],{"className":727604,"style":716670},[31680],[34,727606,727608,727611],{"className":727607},[31685],[34,727609,31965],{"className":727610},[31685,31686],[34,727612,727614],{"className":727613},[31792],[34,727615,727617,727651],{"className":727616},[31796,31797],[34,727618,727620,727648],{"className":727619},[31801],[34,727621,727623,727634],{"className":727622,"style":132027},[31805],[34,727624,727625,727628],{"style":316930},[34,727626],{"className":727627,"style":31814},[31813],[34,727629,727631],{"className":727630},[31818,31819,31820,31821],[34,727632,768],{"className":727633},[31685,31821],[34,727635,727636,727639],{"style":37777},[34,727637],{"className":727638,"style":31814},[31813],[34,727640,727642],{"className":727641},[31818,31819,31820,31821],[34,727643,727645],{"className":727644},[31685,31821],[34,727646,116495],{"className":727647},[31685,31821],[34,727649,31829],{"className":727650},[31828],[34,727652,727654],{"className":727653},[31801],[34,727655,727657],{"className":727656,"style":39021},[31805],[34,727658],{},[34,727660],{"className":727661,"style":31846},[31845],[34,727663,202],{"className":727664},[31850],[34,727666],{"className":727667,"style":31846},[31845],[34,727669,727671,727674,727742,727745,727748,727751,727754,727757],{"className":727670},[31676],[34,727672],{"className":727673,"style":46727},[31680],[34,727675,727677,727680,727739],{"className":727676},[31685],[34,727678],{"className":727679},[31782,41488],[34,727681,727683],{"className":727682},[41453],[34,727684,727686,727731],{"className":727685},[31796,31797],[34,727687,727689,727728],{"className":727688},[31801],[34,727690,727692,727706,727714],{"className":727691,"style":46746},[31805],[34,727693,727694,727697],{"style":41504},[34,727695],{"className":727696,"style":41508},[31813],[34,727698,727700],{"className":727699},[31818,31819,31820,31821],[34,727701,727703],{"className":727702},[31685,31821],[34,727704,15932],{"className":727705},[31685,31821],[34,727707,727708,727711],{"style":41520},[34,727709],{"className":727710,"style":41508},[31813],[34,727712],{"className":727713,"style":41528},[41527],[34,727715,727716,727719],{"style":46771},[34,727717],{"className":727718,"style":41508},[31813],[34,727720,727722],{"className":727721},[31818,31819,31820,31821],[34,727723,727725],{"className":727724},[31685,31821],[34,727726,640],{"className":727727},[31685,31821],[34,727729,31829],{"className":727730},[31828],[34,727732,727734],{"className":727733},[31801],[34,727735,727737],{"className":727736,"style":41619},[31805],[34,727738],{},[34,727740],{"className":727741},[31690,41488],[34,727743,33249],{"className":727744},[31685,31686],[34,727746],{"className":727747,"style":32123},[31845],[34,727749,547853],{"className":727750},[32767],[34,727752],{"className":727753,"style":32123},[31845],[34,727755,15932],{"className":727756},[31685],[34,727758,33249],{"className":727759},[31685,31686],[108,727761,727762,727763,727848,727849,728552,728553,6111,728756],{},"求 ",[34,727764,727766,727785],{"className":727765},[31642],[34,727767,727769],{"className":727768},[31646],[31648,727770,727771],{"xmlns":31650},[31652,727772,727773,727783],{},[31655,727774,727775],{},[38846,727776,727777,727779,727781],{},[31658,727778,31965],{},[31749,727780,640],{},[31662,727782,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,727784,726871],{"encoding":31667},[34,727786,727788],{"className":727787,"ariaHidden":758},[31672],[34,727789,727791,727794],{"className":727790},[31676],[34,727792],{"className":727793,"style":716670},[31680],[34,727795,727797,727800],{"className":727796},[31685],[34,727798,31965],{"className":727799},[31685,31686],[34,727801,727803],{"className":727802},[31792],[34,727804,727806,727840],{"className":727805},[31796,31797],[34,727807,727809,727837],{"className":727808},[31801],[34,727810,727812,727823],{"className":727811,"style":132027},[31805],[34,727813,727814,727817],{"style":316930},[34,727815],{"className":727816,"style":31814},[31813],[34,727818,727820],{"className":727819},[31818,31819,31820,31821],[34,727821,640],{"className":727822},[31685,31821],[34,727824,727825,727828],{"style":37777},[34,727826],{"className":727827,"style":31814},[31813],[34,727829,727831],{"className":727830},[31818,31819,31820,31821],[34,727832,727834],{"className":727833},[31685,31821],[34,727835,116495],{"className":727836},[31685,31821],[34,727838,31829],{"className":727839},[31828],[34,727841,727843],{"className":727842},[31801],[34,727844,727846],{"className":727845,"style":39021},[31805],[34,727847],{},"：由第一式 ",[34,727850,727852,727974],{"className":727851},[31642],[34,727853,727855],{"className":727854},[31646],[31648,727856,727857],{"xmlns":31650},[31652,727858,727859,727971],{},[31655,727860,727861,727869,727871,727873,727881,727903,727905,727907,727913,727915,727917,727919,727921,727923,727945,727947,727949,727955,727957,727967,727969],{},[38846,727862,727863,727865,727867],{},[31658,727864,31965],{},[31749,727866,640],{},[31662,727868,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,727870,202],{},[31662,727872,32226],{},[38846,727874,727875,727877,727879],{},[31658,727876,31965],{},[31749,727878,768],{},[31662,727880,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[41453,727882,727883,727893],{},[31655,727884,727885,727887,727889,727891],{},[31658,727886,547853],{},[31662,727888,32682],{},[31749,727890,768],{},[31658,727892,33249],{},[31655,727894,727895,727897,727899,727901],{},[31658,727896,547860],{},[31662,727898,32682],{},[31749,727900,768],{},[31658,727902,33249],{},[31662,727904,202],{},[31662,727906,32226],{},[41453,727908,727909,727911],{},[31749,727910,640],{},[31749,727912,15932],{},[31658,727914,33249],{},[31658,727916,547853],{},[31662,727918,32682],{},[31749,727920,15932],{},[31658,727922,33249],{},[41453,727924,727925,727935],{},[31655,727926,727927,727929,727931,727933],{},[31658,727928,547853],{},[31662,727930,32682],{},[31749,727932,768],{},[31658,727934,33249],{},[31655,727936,727937,727939,727941,727943],{},[31658,727938,547860],{},[31662,727940,32682],{},[31749,727942,768],{},[31658,727944,33249],{},[31662,727946,202],{},[31662,727948,32226],{},[41453,727950,727951,727953],{},[31749,727952,640],{},[31749,727954,768],{},[31658,727956,33249],{},[37624,727958,727959,727965],{},[31655,727960,727961,727963],{},[31658,727962,547853],{},[31662,727964,32682],{},[31749,727966,768],{},[31749,727968,768],{},[31658,727970,33249],{},[31665,727972,727973],{"encoding":31667},"c_1' = -c_2' \\frac{\\sin 2t}{\\cos 2t} = -\\frac{1}{4}t \\sin 4t \\frac{\\sin 2t}{\\cos 2t} = -\\frac{1}{2}t \\sin^2 2t",[34,727975,727977,728046,728223,728431],{"className":727976,"ariaHidden":758},[31672],[34,727978,727980,727983,728037,728040,728043],{"className":727979},[31676],[34,727981],{"className":727982,"style":716670},[31680],[34,727984,727986,727989],{"className":727985},[31685],[34,727987,31965],{"className":727988},[31685,31686],[34,727990,727992],{"className":727991},[31792],[34,727993,727995,728029],{"className":727994},[31796,31797],[34,727996,727998,728026],{"className":727997},[31801],[34,727999,728001,728012],{"className":728000,"style":132027},[31805],[34,728002,728003,728006],{"style":316930},[34,728004],{"className":728005,"style":31814},[31813],[34,728007,728009],{"className":728008},[31818,31819,31820,31821],[34,728010,640],{"className":728011},[31685,31821],[34,728013,728014,728017],{"style":37777},[34,728015],{"className":728016,"style":31814},[31813],[34,728018,728020],{"className":728019},[31818,31819,31820,31821],[34,728021,728023],{"className":728022},[31685,31821],[34,728024,116495],{"className":728025},[31685,31821],[34,728027,31829],{"className":728028},[31828],[34,728030,728032],{"className":728031},[31801],[34,728033,728035],{"className":728034,"style":39021},[31805],[34,728036],{},[34,728038],{"className":728039,"style":31846},[31845],[34,728041,202],{"className":728042},[31850],[34,728044],{"className":728045,"style":31846},[31845],[34,728047,728049,728053,728056,728110,728214,728217,728220],{"className":728048},[31676],[34,728050],{"className":728051,"style":728052},[31680],"height:1.2065em;vertical-align:-0.345em;",[34,728054,32226],{"className":728055},[31685],[34,728057,728059,728062],{"className":728058},[31685],[34,728060,31965],{"className":728061},[31685,31686],[34,728063,728065],{"className":728064},[31792],[34,728066,728068,728102],{"className":728067},[31796,31797],[34,728069,728071,728099],{"className":728070},[31801],[34,728072,728074,728085],{"className":728073,"style":132027},[31805],[34,728075,728076,728079],{"style":316930},[34,728077],{"className":728078,"style":31814},[31813],[34,728080,728082],{"className":728081},[31818,31819,31820,31821],[34,728083,768],{"className":728084},[31685,31821],[34,728086,728087,728090],{"style":37777},[34,728088],{"className":728089,"style":31814},[31813],[34,728091,728093],{"className":728092},[31818,31819,31820,31821],[34,728094,728096],{"className":728095},[31685,31821],[34,728097,116495],{"className":728098},[31685,31821],[34,728100,31829],{"className":728101},[31828],[34,728103,728105],{"className":728104},[31801],[34,728106,728108],{"className":728107,"style":39021},[31805],[34,728109],{},[34,728111,728113,728116,728211],{"className":728112},[31685],[34,728114],{"className":728115},[31782,41488],[34,728117,728119],{"className":728118},[41453],[34,728120,728122,728203],{"className":728121},[31796,31797],[34,728123,728125,728200],{"className":728124},[31801],[34,728126,728128,728160,728168],{"className":728127,"style":548640},[31805],[34,728129,728130,728133],{"style":41504},[34,728131],{"className":728132,"style":41508},[31813],[34,728134,728136],{"className":728135},[31818,31819,31820,31821],[34,728137,728139,728151,728154,728157],{"className":728138},[31685,31821],[34,728140,728142,728145,728148],{"className":728141},[32767,31821],[34,728143,31965],{"className":728144},[31821],[34,728146,43383],{"className":728147},[31821],[34,728149,31747],{"className":728150},[31821],[34,728152],{"className":728153,"style":43430},[31845,31821],[34,728155,768],{"className":728156},[31685,31821],[34,728158,33249],{"className":728159},[31685,31686,31821],[34,728161,728162,728165],{"style":41520},[34,728163],{"className":728164,"style":41508},[31813],[34,728166],{"className":728167,"style":41528},[41527],[34,728169,728170,728173],{"style":46771},[34,728171],{"className":728172,"style":41508},[31813],[34,728174,728176],{"className":728175},[31818,31819,31820,31821],[34,728177,728179,728191,728194,728197],{"className":728178},[31685,31821],[34,728180,728182,728185,728188],{"className":728181},[32767,31821],[34,728183,31747],{"className":728184},[31821],[34,728186,31955],{"className":728187},[31821],[34,728189,31660],{"className":728190},[31821],[34,728192],{"className":728193,"style":43430},[31845,31821],[34,728195,768],{"className":728196},[31685,31821],[34,728198,33249],{"className":728199},[31685,31686,31821],[34,728201,31829],{"className":728202},[31828],[34,728204,728206],{"className":728205},[31801],[34,728207,728209],{"className":728208,"style":41619},[31805],[34,728210],{},[34,728212],{"className":728213},[31690,41488],[34,728215],{"className":728216,"style":31846},[31845],[34,728218,202],{"className":728219},[31850],[34,728221],{"className":728222,"style":31846},[31845],[34,728224,728226,728229,728232,728300,728303,728306,728309,728312,728315,728318,728422,728425,728428],{"className":728225},[31676],[34,728227],{"className":728228,"style":728052},[31680],[34,728230,32226],{"className":728231},[31685],[34,728233,728235,728238,728297],{"className":728234},[31685],[34,728236],{"className":728237},[31782,41488],[34,728239,728241],{"className":728240},[41453],[34,728242,728244,728289],{"className":728243},[31796,31797],[34,728245,728247,728286],{"className":728246},[31801],[34,728248,728250,728264,728272],{"className":728249,"style":46746},[31805],[34,728251,728252,728255],{"style":41504},[34,728253],{"className":728254,"style":41508},[31813],[34,728256,728258],{"className":728257},[31818,31819,31820,31821],[34,728259,728261],{"className":728260},[31685,31821],[34,728262,15932],{"className":728263},[31685,31821],[34,728265,728266,728269],{"style":41520},[34,728267],{"className":728268,"style":41508},[31813],[34,728270],{"className":728271,"style":41528},[41527],[34,728273,728274,728277],{"style":46771},[34,728275],{"className":728276,"style":41508},[31813],[34,728278,728280],{"className":728279},[31818,31819,31820,31821],[34,728281,728283],{"className":728282},[31685,31821],[34,728284,640],{"className":728285},[31685,31821],[34,728287,31829],{"className":728288},[31828],[34,728290,728292],{"className":728291},[31801],[34,728293,728295],{"className":728294,"style":41619},[31805],[34,728296],{},[34,728298],{"className":728299},[31690,41488],[34,728301,33249],{"className":728302},[31685,31686],[34,728304],{"className":728305,"style":32123},[31845],[34,728307,547853],{"className":728308},[32767],[34,728310],{"className":728311,"style":32123},[31845],[34,728313,15932],{"className":728314},[31685],[34,728316,33249],{"className":728317},[31685,31686],[34,728319,728321,728324,728419],{"className":728320},[31685],[34,728322],{"className":728323},[31782,41488],[34,728325,728327],{"className":728326},[41453],[34,728328,728330,728411],{"className":728329},[31796,31797],[34,728331,728333,728408],{"className":728332},[31801],[34,728334,728336,728368,728376],{"className":728335,"style":548640},[31805],[34,728337,728338,728341],{"style":41504},[34,728339],{"className":728340,"style":41508},[31813],[34,728342,728344],{"className":728343},[31818,31819,31820,31821],[34,728345,728347,728359,728362,728365],{"className":728346},[31685,31821],[34,728348,728350,728353,728356],{"className":728349},[32767,31821],[34,728351,31965],{"className":728352},[31821],[34,728354,43383],{"className":728355},[31821],[34,728357,31747],{"className":728358},[31821],[34,728360],{"className":728361,"style":43430},[31845,31821],[34,728363,768],{"className":728364},[31685,31821],[34,728366,33249],{"className":728367},[31685,31686,31821],[34,728369,728370,728373],{"style":41520},[34,728371],{"className":728372,"style":41508},[31813],[34,728374],{"className":728375,"style":41528},[41527],[34,728377,728378,728381],{"style":46771},[34,728379],{"className":728380,"style":41508},[31813],[34,728382,728384],{"className":728383},[31818,31819,31820,31821],[34,728385,728387,728399,728402,728405],{"className":728386},[31685,31821],[34,728388,728390,728393,728396],{"className":728389},[32767,31821],[34,728391,31747],{"className":728392},[31821],[34,728394,31955],{"className":728395},[31821],[34,728397,31660],{"className":728398},[31821],[34,728400],{"className":728401,"style":43430},[31845,31821],[34,728403,768],{"className":728404},[31685,31821],[34,728406,33249],{"className":728407},[31685,31686,31821],[34,728409,31829],{"className":728410},[31828],[34,728412,728414],{"className":728413},[31801],[34,728415,728417],{"className":728416,"style":41619},[31805],[34,728418],{},[34,728420],{"className":728421},[31690,41488],[34,728423],{"className":728424,"style":31846},[31845],[34,728426,202],{"className":728427},[31850],[34,728429],{"className":728430,"style":31846},[31845],[34,728432,728434,728437,728440,728508,728511,728514,728543,728546,728549],{"className":728433},[31676],[34,728435],{"className":728436,"style":594576},[31680],[34,728438,32226],{"className":728439},[31685],[34,728441,728443,728446,728505],{"className":728442},[31685],[34,728444],{"className":728445},[31782,41488],[34,728447,728449],{"className":728448},[41453],[34,728450,728452,728497],{"className":728451},[31796,31797],[34,728453,728455,728494],{"className":728454},[31801],[34,728456,728458,728472,728480],{"className":728457,"style":46746},[31805],[34,728459,728460,728463],{"style":41504},[34,728461],{"className":728462,"style":41508},[31813],[34,728464,728466],{"className":728465},[31818,31819,31820,31821],[34,728467,728469],{"className":728468},[31685,31821],[34,728470,768],{"className":728471},[31685,31821],[34,728473,728474,728477],{"style":41520},[34,728475],{"className":728476,"style":41508},[31813],[34,728478],{"className":728479,"style":41528},[41527],[34,728481,728482,728485],{"style":46771},[34,728483],{"className":728484,"style":41508},[31813],[34,728486,728488],{"className":728487},[31818,31819,31820,31821],[34,728489,728491],{"className":728490},[31685,31821],[34,728492,640],{"className":728493},[31685,31821],[34,728495,31829],{"className":728496},[31828],[34,728498,728500],{"className":728499},[31801],[34,728501,728503],{"className":728502,"style":41619},[31805],[34,728504],{},[34,728506],{"className":728507},[31690,41488],[34,728509,33249],{"className":728510},[31685,31686],[34,728512],{"className":728513,"style":32123},[31845],[34,728515,728517,728520],{"className":728516},[32767],[34,728518,547853],{"className":728519},[32767],[34,728521,728523],{"className":728522},[31792],[34,728524,728526],{"className":728525},[31796],[34,728527,728529],{"className":728528},[31801],[34,728530,728532],{"className":728531,"style":585884},[31805],[34,728533,728534,728537],{"style":586008},[34,728535],{"className":728536,"style":31814},[31813],[34,728538,728540],{"className":728539},[31818,31819,31820,31821],[34,728541,768],{"className":728542},[31685,31821],[34,728544],{"className":728545,"style":32123},[31845],[34,728547,768],{"className":728548},[31685],[34,728550,33249],{"className":728551},[31685,31686],"\n(利用三角公式降幂：",[34,728554,728556,728602],{"className":728555},[31642],[34,728557,728559],{"className":728558},[31646],[31648,728560,728561],{"xmlns":31650},[31652,728562,728563,728599],{},[31655,728564,728565,728575,728577,728579,728581],{},[37624,728566,728567,728573],{},[31655,728568,728569,728571],{},[31658,728570,547853],{},[31662,728572,32682],{},[31749,728574,768],{},[31749,728576,768],{},[31658,728578,33249],{},[31662,728580,202],{},[41453,728582,728583,728597],{},[31655,728584,728585,728587,728589,728591,728593,728595],{},[31749,728586,640],{},[31662,728588,32226],{},[31658,728590,547860],{},[31662,728592,32682],{},[31749,728594,15932],{},[31658,728596,33249],{},[31749,728598,768],{},[31665,728600,728601],{"encoding":31667},"\\sin^2 2t = \\frac{1-\\cos 4t}{2}",[34,728603,728605,728658],{"className":728604,"ariaHidden":758},[31672],[34,728606,728608,728611,728640,728643,728646,728649,728652,728655],{"className":728607},[31676],[34,728609],{"className":728610,"style":585884},[31680],[34,728612,728614,728617],{"className":728613},[32767],[34,728615,547853],{"className":728616},[32767],[34,728618,728620],{"className":728619},[31792],[34,728621,728623],{"className":728622},[31796],[34,728624,728626],{"className":728625},[31801],[34,728627,728629],{"className":728628,"style":585884},[31805],[34,728630,728631,728634],{"style":586008},[34,728632],{"className":728633,"style":31814},[31813],[34,728635,728637],{"className":728636},[31818,31819,31820,31821],[34,728638,768],{"className":728639},[31685,31821],[34,728641],{"className":728642,"style":32123},[31845],[34,728644,768],{"className":728645},[31685],[34,728647,33249],{"className":728648},[31685,31686],[34,728650],{"className":728651,"style":31846},[31845],[34,728653,202],{"className":728654},[31850],[34,728656],{"className":728657,"style":31846},[31845],[34,728659,728661,728664],{"className":728660},[31676],[34,728662],{"className":728663,"style":46727},[31680],[34,728665,728667,728670,728753],{"className":728666},[31685],[34,728668],{"className":728669},[31782,41488],[34,728671,728673],{"className":728672},[41453],[34,728674,728676,728745],{"className":728675},[31796,31797],[34,728677,728679,728742],{"className":728678},[31801],[34,728680,728682,728696,728704],{"className":728681,"style":46746},[31805],[34,728683,728684,728687],{"style":41504},[34,728685],{"className":728686,"style":41508},[31813],[34,728688,728690],{"className":728689},[31818,31819,31820,31821],[34,728691,728693],{"className":728692},[31685,31821],[34,728694,768],{"className":728695},[31685,31821],[34,728697,728698,728701],{"style":41520},[34,728699],{"className":728700,"style":41508},[31813],[34,728702],{"className":728703,"style":41528},[41527],[34,728705,728706,728709],{"style":46771},[34,728707],{"className":728708,"style":41508},[31813],[34,728710,728712],{"className":728711},[31818,31819,31820,31821],[34,728713,728715,728718,728721,728733,728736,728739],{"className":728714},[31685,31821],[34,728716,640],{"className":728717},[31685,31821],[34,728719,32226],{"className":728720},[32024,31821],[34,728722,728724,728727,728730],{"className":728723},[32767,31821],[34,728725,31965],{"className":728726},[31821],[34,728728,43383],{"className":728729},[31821],[34,728731,31747],{"className":728732},[31821],[34,728734],{"className":728735,"style":43430},[31845,31821],[34,728737,15932],{"className":728738},[31685,31821],[34,728740,33249],{"className":728741},[31685,31686,31821],[34,728743,31829],{"className":728744},[31828],[34,728746,728748],{"className":728747},[31801],[34,728749,728751],{"className":728750,"style":41619},[31805],[34,728752],{},[34,728754],{"className":728755},[31690,41488],[34,728757,728759,728843],{"className":728758},[31642],[34,728760,728762],{"className":728761},[31646],[31648,728763,728764],{"xmlns":31650},[31652,728765,728766,728840],{},[31655,728767,728768,728770,728772,728774,728782,728784,728786,728792,728794,728796,728798,728800,728802,728804,728806,728808,728810,728812,728814,728820,728822,728824,728830,728832,728834,728836,728838],{},[39361,728769,262622],{},[31662,728771,262625],{},[39361,728773,262622],{},[38846,728775,728776,728778,728780],{},[31658,728777,31965],{},[31749,728779,640],{},[31662,728781,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,728783,202],{},[31662,728785,32226],{},[41453,728787,728788,728790],{},[31749,728789,640],{},[31749,728791,15932],{},[31658,728793,33249],{},[31662,728795,456],{"stretchy":702},[31749,728797,640],{},[31662,728799,32226],{},[31658,728801,547860],{},[31662,728803,32682],{},[31749,728805,15932],{},[31658,728807,33249],{},[31662,728809,2557],{"stretchy":702},[31662,728811,202],{},[31662,728813,32226],{},[41453,728815,728816,728818],{},[31749,728817,640],{},[31749,728819,15932],{},[31658,728821,33249],{},[31662,728823,223],{},[41453,728825,728826,728828],{},[31749,728827,640],{},[31749,728829,15932],{},[31658,728831,33249],{},[31658,728833,547860],{},[31662,728835,32682],{},[31749,728837,15932],{},[31658,728839,33249],{},[31665,728841,728842],{"encoding":31667},"\\implies c_1' = -\\frac{1}{4}t(1-\\cos 4t) = -\\frac{1}{4}t + \\frac{1}{4}t \\cos 4t",[34,728844,728846,728864,728933,729028,729058,729147],{"className":728845,"ariaHidden":758},[31672],[34,728847,728849,728852,728855,728858,728861],{"className":728848},[31676],[34,728850],{"className":728851,"style":318264},[31680],[34,728853],{"className":728854,"style":31846},[31845],[34,728856,262625],{"className":728857},[31850],[34,728859],{"className":728860,"style":31846},[31845],[34,728862],{"className":728863,"style":31846},[31845],[34,728865,728867,728870,728924,728927,728930],{"className":728866},[31676],[34,728868],{"className":728869,"style":716670},[31680],[34,728871,728873,728876],{"className":728872},[31685],[34,728874,31965],{"className":728875},[31685,31686],[34,728877,728879],{"className":728878},[31792],[34,728880,728882,728916],{"className":728881},[31796,31797],[34,728883,728885,728913],{"className":728884},[31801],[34,728886,728888,728899],{"className":728887,"style":132027},[31805],[34,728889,728890,728893],{"style":316930},[34,728891],{"className":728892,"style":31814},[31813],[34,728894,728896],{"className":728895},[31818,31819,31820,31821],[34,728897,640],{"className":728898},[31685,31821],[34,728900,728901,728904],{"style":37777},[34,728902],{"className":728903,"style":31814},[31813],[34,728905,728907],{"className":728906},[31818,31819,31820,31821],[34,728908,728910],{"className":728909},[31685,31821],[34,728911,116495],{"className":728912},[31685,31821],[34,728914,31829],{"className":728915},[31828],[34,728917,728919],{"className":728918},[31801],[34,728920,728922],{"className":728921,"style":39021},[31805],[34,728923],{},[34,728925],{"className":728926,"style":31846},[31845],[34,728928,202],{"className":728929},[31850],[34,728931],{"className":728932,"style":31846},[31845],[34,728934,728936,728939,728942,729010,729013,729016,729019,729022,729025],{"className":728935},[31676],[34,728937],{"className":728938,"style":46727},[31680],[34,728940,32226],{"className":728941},[31685],[34,728943,728945,728948,729007],{"className":728944},[31685],[34,728946],{"className":728947},[31782,41488],[34,728949,728951],{"className":728950},[41453],[34,728952,728954,728999],{"className":728953},[31796,31797],[34,728955,728957,728996],{"className":728956},[31801],[34,728958,728960,728974,728982],{"className":728959,"style":46746},[31805],[34,728961,728962,728965],{"style":41504},[34,728963],{"className":728964,"style":41508},[31813],[34,728966,728968],{"className":728967},[31818,31819,31820,31821],[34,728969,728971],{"className":728970},[31685,31821],[34,728972,15932],{"className":728973},[31685,31821],[34,728975,728976,728979],{"style":41520},[34,728977],{"className":728978,"style":41508},[31813],[34,728980],{"className":728981,"style":41528},[41527],[34,728983,728984,728987],{"style":46771},[34,728985],{"className":728986,"style":41508},[31813],[34,728988,728990],{"className":728989},[31818,31819,31820,31821],[34,728991,728993],{"className":728992},[31685,31821],[34,728994,640],{"className":728995},[31685,31821],[34,728997,31829],{"className":728998},[31828],[34,729000,729002],{"className":729001},[31801],[34,729003,729005],{"className":729004,"style":41619},[31805],[34,729006],{},[34,729008],{"className":729009},[31690,41488],[34,729011,33249],{"className":729012},[31685,31686],[34,729014,456],{"className":729015},[31782],[34,729017,640],{"className":729018},[31685],[34,729020],{"className":729021,"style":32604},[31845],[34,729023,32226],{"className":729024},[32024],[34,729026],{"className":729027,"style":32604},[31845],[34,729029,729031,729034,729037,729040,729043,729046,729049,729052,729055],{"className":729030},[31676],[34,729032],{"className":729033,"style":31774},[31680],[34,729035,547860],{"className":729036},[32767],[34,729038],{"className":729039,"style":32123},[31845],[34,729041,15932],{"className":729042},[31685],[34,729044,33249],{"className":729045},[31685,31686],[34,729047,2557],{"className":729048},[31690],[34,729050],{"className":729051,"style":31846},[31845],[34,729053,202],{"className":729054},[31850],[34,729056],{"className":729057,"style":31846},[31845],[34,729059,729061,729064,729067,729135,729138,729141,729144],{"className":729060},[31676],[34,729062],{"className":729063,"style":46727},[31680],[34,729065,32226],{"className":729066},[31685],[34,729068,729070,729073,729132],{"className":729069},[31685],[34,729071],{"className":729072},[31782,41488],[34,729074,729076],{"className":729075},[41453],[34,729077,729079,729124],{"className":729078},[31796,31797],[34,729080,729082,729121],{"className":729081},[31801],[34,729083,729085,729099,729107],{"className":729084,"style":46746},[31805],[34,729086,729087,729090],{"style":41504},[34,729088],{"className":729089,"style":41508},[31813],[34,729091,729093],{"className":729092},[31818,31819,31820,31821],[34,729094,729096],{"className":729095},[31685,31821],[34,729097,15932],{"className":729098},[31685,31821],[34,729100,729101,729104],{"style":41520},[34,729102],{"className":729103,"style":41508},[31813],[34,729105],{"className":729106,"style":41528},[41527],[34,729108,729109,729112],{"style":46771},[34,729110],{"className":729111,"style":41508},[31813],[34,729113,729115],{"className":729114},[31818,31819,31820,31821],[34,729116,729118],{"className":729117},[31685,31821],[34,729119,640],{"className":729120},[31685,31821],[34,729122,31829],{"className":729123},[31828],[34,729125,729127],{"className":729126},[31801],[34,729128,729130],{"className":729129,"style":41619},[31805],[34,729131],{},[34,729133],{"className":729134},[31690,41488],[34,729136,33249],{"className":729137},[31685,31686],[34,729139],{"className":729140,"style":32604},[31845],[34,729142,223],{"className":729143},[32024],[34,729145],{"className":729146,"style":32604},[31845],[34,729148,729150,729153,729221,729224,729227,729230,729233,729236],{"className":729149},[31676],[34,729151],{"className":729152,"style":46727},[31680],[34,729154,729156,729159,729218],{"className":729155},[31685],[34,729157],{"className":729158},[31782,41488],[34,729160,729162],{"className":729161},[41453],[34,729163,729165,729210],{"className":729164},[31796,31797],[34,729166,729168,729207],{"className":729167},[31801],[34,729169,729171,729185,729193],{"className":729170,"style":46746},[31805],[34,729172,729173,729176],{"style":41504},[34,729174],{"className":729175,"style":41508},[31813],[34,729177,729179],{"className":729178},[31818,31819,31820,31821],[34,729180,729182],{"className":729181},[31685,31821],[34,729183,15932],{"className":729184},[31685,31821],[34,729186,729187,729190],{"style":41520},[34,729188],{"className":729189,"style":41508},[31813],[34,729191],{"className":729192,"style":41528},[41527],[34,729194,729195,729198],{"style":46771},[34,729196],{"className":729197,"style":41508},[31813],[34,729199,729201],{"className":729200},[31818,31819,31820,31821],[34,729202,729204],{"className":729203},[31685,31821],[34,729205,640],{"className":729206},[31685,31821],[34,729208,31829],{"className":729209},[31828],[34,729211,729213],{"className":729212},[31801],[34,729214,729216],{"className":729215,"style":41619},[31805],[34,729217],{},[34,729219],{"className":729220},[31690,41488],[34,729222,33249],{"className":729223},[31685,31686],[34,729225],{"className":729226,"style":32123},[31845],[34,729228,547860],{"className":729229},[32767],[34,729231],{"className":729232,"style":32123},[31845],[34,729234,15932],{"className":729235},[31685],[34,729237,33249],{"className":729238},[31685,31686],[15,729240,729241,729244],{},[2627,729242,729243],{},"Step 4: 积分（略去繁琐步骤，你需要自己练习分部积分）","\n这里积出来会比较长，但逻辑就是简单的积分。",[32188,729246],{},[32490,729248,485488,729250,719244,729518],{"id":729249},"_5-t2xtxx6t34t2t2-x-tx-x-6t-34t2t2xtxx6t34t2已知-x1tx2tlntx_1-t-x_2-tln-tx1tx2tlnt",[34,729251,729253,729312],{"className":729252},[31642],[34,729254,729256],{"className":729255},[31646],[31648,729257,729258],{"xmlns":31650},[31652,729259,729260,729309],{},[31655,729261,729262,729268,729278,729280,729282,729288,729290,729292,729294,729296,729298,729300,729303],{},[37624,729263,729264,729266],{},[31658,729265,33249],{},[31749,729267,768],{},[37624,729269,729270,729272],{},[31658,729271,32936],{},[31655,729273,729274,729276],{},[31662,729275,116495],{"mathvariant":38079},[31662,729277,116495],{"mathvariant":38079},[31662,729279,32226],{},[31658,729281,33249],{},[37624,729283,729284,729286],{},[31658,729285,32936],{},[31662,729287,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,729289,223],{},[31658,729291,32936],{},[31662,729293,202],{},[31749,729295,36656],{},[31658,729297,33249],{},[31662,729299,223],{},[31749,729301,729302],{},"34",[37624,729304,729305,729307],{},[31658,729306,33249],{},[31749,729308,768],{},[31665,729310,729311],{"encoding":31667},"t^2 x'' - tx' + x = 6t + 34t^2",[34,729313,729315,729391,729441,729459,729480],{"className":729314,"ariaHidden":758},[31672],[34,729316,729318,729321,729350,729382,729385,729388],{"className":729317},[31676],[34,729319],{"className":729320,"style":40044},[31680],[34,729322,729324,729327],{"className":729323},[31685],[34,729325,33249],{"className":729326},[31685,31686],[34,729328,729330],{"className":729329},[31792],[34,729331,729333],{"className":729332},[31796],[34,729334,729336],{"className":729335},[31801],[34,729337,729339],{"className":729338,"style":37774},[31805],[34,729340,729341,729344],{"style":37777},[34,729342],{"className":729343,"style":31814},[31813],[34,729345,729347],{"className":729346},[31818,31819,31820,31821],[34,729348,768],{"className":729349},[31685,31821],[34,729351,729353,729356],{"className":729352},[31685],[34,729354,32936],{"className":729355},[31685,31686],[34,729357,729359],{"className":729358},[31792],[34,729360,729362],{"className":729361},[31796],[34,729363,729365],{"className":729364},[31801],[34,729366,729368],{"className":729367,"style":132027},[31805],[34,729369,729370,729373],{"style":37777},[34,729371],{"className":729372,"style":31814},[31813],[34,729374,729376],{"className":729375},[31818,31819,31820,31821],[34,729377,729379],{"className":729378},[31685,31821],[34,729380,502782],{"className":729381},[31685,31821],[34,729383],{"className":729384,"style":32604},[31845],[34,729386,32226],{"className":729387},[32024],[34,729389],{"className":729390,"style":32604},[31845],[34,729392,729394,729397,729400,729432,729435,729438],{"className":729393},[31676],[34,729395],{"className":729396,"style":384364},[31680],[34,729398,33249],{"className":729399},[31685,31686],[34,729401,729403,729406],{"className":729402},[31685],[34,729404,32936],{"className":729405},[31685,31686],[34,729407,729409],{"className":729408},[31792],[34,729410,729412],{"className":729411},[31796],[34,729413,729415],{"className":729414},[31801],[34,729416,729418],{"className":729417,"style":132027},[31805],[34,729419,729420,729423],{"style":37777},[34,729421],{"className":729422,"style":31814},[31813],[34,729424,729426],{"className":729425},[31818,31819,31820,31821],[34,729427,729429],{"className":729428},[31685,31821],[34,729430,116495],{"className":729431},[31685,31821],[34,729433],{"className":729434,"style":32604},[31845],[34,729436,223],{"className":729437},[32024],[34,729439],{"className":729440,"style":32604},[31845],[34,729442,729444,729447,729450,729453,729456],{"className":729443},[31676],[34,729445],{"className":729446,"style":32527},[31680],[34,729448,32936],{"className":729449},[31685,31686],[34,729451],{"className":729452,"style":31846},[31845],[34,729454,202],{"className":729455},[31850],[34,729457],{"className":729458,"style":31846},[31845],[34,729460,729462,729465,729468,729471,729474,729477],{"className":729461},[31676],[34,729463],{"className":729464,"style":35942},[31680],[34,729466,36656],{"className":729467},[31685],[34,729469,33249],{"className":729470},[31685,31686],[34,729472],{"className":729473,"style":32604},[31845],[34,729475,223],{"className":729476},[32024],[34,729478],{"className":729479,"style":32604},[31845],[34,729481,729483,729486,729489],{"className":729482},[31676],[34,729484],{"className":729485,"style":37774},[31680],[34,729487,729302],{"className":729488},[31685],[34,729490,729492,729495],{"className":729491},[31685],[34,729493,33249],{"className":729494},[31685,31686],[34,729496,729498],{"className":729497},[31792],[34,729499,729501],{"className":729500},[31796],[34,729502,729504],{"className":729503},[31801],[34,729505,729507],{"className":729506,"style":37774},[31805],[34,729508,729509,729512],{"style":37777},[34,729510],{"className":729511,"style":31814},[31813],[34,729513,729515],{"className":729514},[31818,31819,31820,31821],[34,729516,768],{"className":729517},[31685,31821],[34,729519,729521,729561],{"className":729520},[31642],[34,729522,729524],{"className":729523},[31646],[31648,729525,729526],{"xmlns":31650},[31652,729527,729528,729558],{},[31655,729529,729530,729536,729538,729540,729542,729548,729550,729552,729554,729556],{},[31743,729531,729532,729534],{},[31658,729533,32936],{},[31749,729535,640],{},[31662,729537,202],{},[31658,729539,33249],{},[31662,729541,31960],{"separator":758},[31743,729543,729544,729546],{},[31658,729545,32936],{},[31749,729547,768],{},[31662,729549,202],{},[31658,729551,33249],{},[31658,729553,563286],{},[31662,729555,32682],{},[31658,729557,33249],{},[31665,729559,729560],{"encoding":31667},"x_1 = t, x_2 = t\\ln t",[34,729562,729564,729619,729683],{"className":729563,"ariaHidden":758},[31672],[34,729565,729567,729570,729610,729613,729616],{"className":729566},[31676],[34,729568],{"className":729569,"style":31860},[31680],[34,729571,729573,729576],{"className":729572},[31685],[34,729574,32936],{"className":729575},[31685,31686],[34,729577,729579],{"className":729578},[31792],[34,729580,729582,729602],{"className":729581},[31796,31797],[34,729583,729585,729599],{"className":729584},[31801],[34,729586,729588],{"className":729587,"style":31806},[31805],[34,729589,729590,729593],{"style":31809},[34,729591],{"className":729592,"style":31814},[31813],[34,729594,729596],{"className":729595},[31818,31819,31820,31821],[34,729597,640],{"className":729598},[31685,31821],[34,729600,31829],{"className":729601},[31828],[34,729603,729605],{"className":729604},[31801],[34,729606,729608],{"className":729607,"style":31836},[31805],[34,729609],{},[34,729611],{"className":729612,"style":31846},[31845],[34,729614,202],{"className":729615},[31850],[34,729617],{"className":729618,"style":31846},[31845],[34,729620,729622,729625,729628,729631,729634,729674,729677,729680],{"className":729621},[31676],[34,729623],{"className":729624,"style":223552},[31680],[34,729626,33249],{"className":729627},[31685,31686],[34,729629,31960],{"className":729630},[32119],[34,729632],{"className":729633,"style":32123},[31845],[34,729635,729637,729640],{"className":729636},[31685],[34,729638,32936],{"className":729639},[31685,31686],[34,729641,729643],{"className":729642},[31792],[34,729644,729646,729666],{"className":729645},[31796,31797],[34,729647,729649,729663],{"className":729648},[31801],[34,729650,729652],{"className":729651,"style":31806},[31805],[34,729653,729654,729657],{"style":31809},[34,729655],{"className":729656,"style":31814},[31813],[34,729658,729660],{"className":729659},[31818,31819,31820,31821],[34,729661,768],{"className":729662},[31685,31821],[34,729664,31829],{"className":729665},[31828],[34,729667,729669],{"className":729668},[31801],[34,729670,729672],{"className":729671,"style":31836},[31805],[34,729673],{},[34,729675],{"className":729676,"style":31846},[31845],[34,729678,202],{"className":729679},[31850],[34,729681],{"className":729682,"style":31846},[31845],[34,729684,729686,729689,729692,729695,729698,729701],{"className":729685},[31676],[34,729687],{"className":729688,"style":31681},[31680],[34,729690,33249],{"className":729691},[31685,31686],[34,729693],{"className":729694,"style":32123},[31845],[34,729696,563286],{"className":729697},[32767],[34,729699],{"className":729700,"style":32123},[31845],[34,729702,33249],{"className":729703},[31685,31686],[15,729705,729706,729709,729774,729775,46528,729834,729895,729896,730330,730331,730490,730491,730582],{},[2627,729707,729708],{},"Step 1: 检查系数（陷阱！！）",[34,729710,729712,729733],{"className":729711},[31642],[34,729713,729715],{"className":729714},[31646],[31648,729716,729717],{"xmlns":31650},[31652,729718,729719,729731],{},[31655,729720,729721],{},[37624,729722,729723,729725],{},[31658,729724,32936],{},[31655,729726,729727,729729],{},[31662,729728,116495],{"mathvariant":38079},[31662,729730,116495],{"mathvariant":38079},[31665,729732,719513],{"encoding":31667},[34,729734,729736],{"className":729735,"ariaHidden":758},[31672],[34,729737,729739,729742],{"className":729738},[31676],[34,729740],{"className":729741,"style":132027},[31680],[34,729743,729745,729748],{"className":729744},[31685],[34,729746,32936],{"className":729747},[31685,31686],[34,729749,729751],{"className":729750},[31792],[34,729752,729754],{"className":729753},[31796],[34,729755,729757],{"className":729756},[31801],[34,729758,729760],{"className":729759,"style":132027},[31805],[34,729761,729762,729765],{"style":37777},[34,729763],{"className":729764,"style":31814},[31813],[34,729766,729768],{"className":729767},[31818,31819,31820,31821],[34,729769,729771],{"className":729770},[31685,31821],[34,729772,502782],{"className":729773},[31685,31821]," 前面是 ",[34,729776,729778,729796],{"className":729777},[31642],[34,729779,729781],{"className":729780},[31646],[31648,729782,729783],{"xmlns":31650},[31652,729784,729785,729793],{},[31655,729786,729787],{},[37624,729788,729789,729791],{},[31658,729790,33249],{},[31749,729792,768],{},[31665,729794,729795],{"encoding":31667},"t^2",[34,729797,729799],{"className":729798,"ariaHidden":758},[31672],[34,729800,729802,729805],{"className":729801},[31676],[34,729803],{"className":729804,"style":37774},[31680],[34,729806,729808,729811],{"className":729807},[31685],[34,729809,33249],{"className":729810},[31685,31686],[34,729812,729814],{"className":729813},[31792],[34,729815,729817],{"className":729816},[31796],[34,729818,729820],{"className":729819},[31801],[34,729821,729823],{"className":729822,"style":37774},[31805],[34,729824,729825,729828],{"style":37777},[34,729826],{"className":729827,"style":31814},[31813],[34,729829,729831],{"className":729830},[31818,31819,31820,31821],[34,729832,768],{"className":729833},[31685,31821],[2627,729835,729836,729837],{},"必须两边同时除以 ",[34,729838,729840,729857],{"className":729839},[31642],[34,729841,729843],{"className":729842},[31646],[31648,729844,729845],{"xmlns":31650},[31652,729846,729847,729855],{},[31655,729848,729849],{},[37624,729850,729851,729853],{},[31658,729852,33249],{},[31749,729854,768],{},[31665,729856,729795],{"encoding":31667},[34,729858,729860],{"className":729859,"ariaHidden":758},[31672],[34,729861,729863,729866],{"className":729862},[31676],[34,729864],{"className":729865,"style":37774},[31680],[34,729867,729869,729872],{"className":729868},[31685],[34,729870,33249],{"className":729871},[31685,31686],[34,729873,729875],{"className":729874},[31792],[34,729876,729878],{"className":729877},[31796],[34,729879,729881],{"className":729880},[31801],[34,729882,729884],{"className":729883,"style":37774},[31805],[34,729885,729886,729889],{"style":37777},[34,729887],{"className":729888,"style":31814},[31813],[34,729890,729892],{"className":729891},[31818,31819,31820,31821],[34,729893,768],{"className":729894},[31685,31821],"！\n标准化方程：\n",[34,729897,729899,729961],{"className":729898},[31642],[34,729900,729902],{"className":729901},[31646],[31648,729903,729904],{"xmlns":31650},[31652,729905,729906,729958],{},[31655,729907,729908,729918,729920,729926,729932,729934,729944,729946,729948,729954,729956],{},[37624,729909,729910,729912],{},[31658,729911,32936],{},[31655,729913,729914,729916],{},[31662,729915,116495],{"mathvariant":38079},[31662,729917,116495],{"mathvariant":38079},[31662,729919,32226],{},[41453,729921,729922,729924],{},[31749,729923,640],{},[31658,729925,33249],{},[37624,729927,729928,729930],{},[31658,729929,32936],{},[31662,729931,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,729933,223],{},[41453,729935,729936,729938],{},[31749,729937,640],{},[37624,729939,729940,729942],{},[31658,729941,33249],{},[31749,729943,768],{},[31658,729945,32936],{},[31662,729947,202],{},[41453,729949,729950,729952],{},[31749,729951,36656],{},[31658,729953,33249],{},[31662,729955,223],{},[31749,729957,729302],{},[31665,729959,729960],{"encoding":31667},"x'' - \\frac{1}{t}x' + \\frac{1}{t^2}x = \\frac{6}{t} + 34",[34,729962,729964,730011,730126,730238,730321],{"className":729963,"ariaHidden":758},[31672],[34,729965,729967,729970,730002,730005,730008],{"className":729966},[31676],[34,729968],{"className":729969,"style":384364},[31680],[34,729971,729973,729976],{"className":729972},[31685],[34,729974,32936],{"className":729975},[31685,31686],[34,729977,729979],{"className":729978},[31792],[34,729980,729982],{"className":729981},[31796],[34,729983,729985],{"className":729984},[31801],[34,729986,729988],{"className":729987,"style":132027},[31805],[34,729989,729990,729993],{"style":37777},[34,729991],{"className":729992,"style":31814},[31813],[34,729994,729996],{"className":729995},[31818,31819,31820,31821],[34,729997,729999],{"className":729998},[31685,31821],[34,730000,502782],{"className":730001},[31685,31821],[34,730003],{"className":730004,"style":32604},[31845],[34,730006,32226],{"className":730007},[32024],[34,730009],{"className":730010,"style":32604},[31845],[34,730012,730014,730017,730085,730117,730120,730123],{"className":730013},[31676],[34,730015],{"className":730016,"style":46727},[31680],[34,730018,730020,730023,730082],{"className":730019},[31685],[34,730021],{"className":730022},[31782,41488],[34,730024,730026],{"className":730025},[41453],[34,730027,730029,730074],{"className":730028},[31796,31797],[34,730030,730032,730071],{"className":730031},[31801],[34,730033,730035,730049,730057],{"className":730034,"style":46746},[31805],[34,730036,730037,730040],{"style":41504},[34,730038],{"className":730039,"style":41508},[31813],[34,730041,730043],{"className":730042},[31818,31819,31820,31821],[34,730044,730046],{"className":730045},[31685,31821],[34,730047,33249],{"className":730048},[31685,31686,31821],[34,730050,730051,730054],{"style":41520},[34,730052],{"className":730053,"style":41508},[31813],[34,730055],{"className":730056,"style":41528},[41527],[34,730058,730059,730062],{"style":46771},[34,730060],{"className":730061,"style":41508},[31813],[34,730063,730065],{"className":730064},[31818,31819,31820,31821],[34,730066,730068],{"className":730067},[31685,31821],[34,730069,640],{"className":730070},[31685,31821],[34,730072,31829],{"className":730073},[31828],[34,730075,730077],{"className":730076},[31801],[34,730078,730080],{"className":730079,"style":41619},[31805],[34,730081],{},[34,730083],{"className":730084},[31690,41488],[34,730086,730088,730091],{"className":730087},[31685],[34,730089,32936],{"className":730090},[31685,31686],[34,730092,730094],{"className":730093},[31792],[34,730095,730097],{"className":730096},[31796],[34,730098,730100],{"className":730099},[31801],[34,730101,730103],{"className":730102,"style":132027},[31805],[34,730104,730105,730108],{"style":37777},[34,730106],{"className":730107,"style":31814},[31813],[34,730109,730111],{"className":730110},[31818,31819,31820,31821],[34,730112,730114],{"className":730113},[31685,31821],[34,730115,116495],{"className":730116},[31685,31821],[34,730118],{"className":730119,"style":32604},[31845],[34,730121,223],{"className":730122},[32024],[34,730124],{"className":730125,"style":32604},[31845],[34,730127,730129,730132,730226,730229,730232,730235],{"className":730128},[31676],[34,730130],{"className":730131,"style":46727},[31680],[34,730133,730135,730138,730223],{"className":730134},[31685],[34,730136],{"className":730137},[31782,41488],[34,730139,730141],{"className":730140},[41453],[34,730142,730144,730215],{"className":730143},[31796,31797],[34,730145,730147,730212],{"className":730146},[31801],[34,730148,730150,730190,730198],{"className":730149,"style":46746},[31805],[34,730151,730152,730155],{"style":41504},[34,730153],{"className":730154,"style":41508},[31813],[34,730156,730158],{"className":730157},[31818,31819,31820,31821],[34,730159,730161],{"className":730160},[31685,31821],[34,730162,730164,730167],{"className":730163},[31685,31821],[34,730165,33249],{"className":730166},[31685,31686,31821],[34,730168,730170],{"className":730169},[31792],[34,730171,730173],{"className":730172},[31796],[34,730174,730176],{"className":730175},[31801],[34,730177,730179],{"className":730178,"style":548314},[31805],[34,730180,730181,730184],{"style":116668},[34,730182],{"className":730183,"style":43406},[31813],[34,730185,730187],{"className":730186},[31818,43410,43411,31821],[34,730188,768],{"className":730189},[31685,31821],[34,730191,730192,730195],{"style":41520},[34,730193],{"className":730194,"style":41508},[31813],[34,730196],{"className":730197,"style":41528},[41527],[34,730199,730200,730203],{"style":46771},[34,730201],{"className":730202,"style":41508},[31813],[34,730204,730206],{"className":730205},[31818,31819,31820,31821],[34,730207,730209],{"className":730208},[31685,31821],[34,730210,640],{"className":730211},[31685,31821],[34,730213,31829],{"className":730214},[31828],[34,730216,730218],{"className":730217},[31801],[34,730219,730221],{"className":730220,"style":41619},[31805],[34,730222],{},[34,730224],{"className":730225},[31690,41488],[34,730227,32936],{"className":730228},[31685,31686],[34,730230],{"className":730231,"style":31846},[31845],[34,730233,202],{"className":730234},[31850],[34,730236],{"className":730237,"style":31846},[31845],[34,730239,730241,730244,730312,730315,730318],{"className":730240},[31676],[34,730242],{"className":730243,"style":46727},[31680],[34,730245,730247,730250,730309],{"className":730246},[31685],[34,730248],{"className":730249},[31782,41488],[34,730251,730253],{"className":730252},[41453],[34,730254,730256,730301],{"className":730255},[31796,31797],[34,730257,730259,730298],{"className":730258},[31801],[34,730260,730262,730276,730284],{"className":730261,"style":46746},[31805],[34,730263,730264,730267],{"style":41504},[34,730265],{"className":730266,"style":41508},[31813],[34,730268,730270],{"className":730269},[31818,31819,31820,31821],[34,730271,730273],{"className":730272},[31685,31821],[34,730274,33249],{"className":730275},[31685,31686,31821],[34,730277,730278,730281],{"style":41520},[34,730279],{"className":730280,"style":41508},[31813],[34,730282],{"className":730283,"style":41528},[41527],[34,730285,730286,730289],{"style":46771},[34,730287],{"className":730288,"style":41508},[31813],[34,730290,730292],{"className":730291},[31818,31819,31820,31821],[34,730293,730295],{"className":730294},[31685,31821],[34,730296,36656],{"className":730297},[31685,31821],[34,730299,31829],{"className":730300},[31828],[34,730302,730304],{"className":730303},[31801],[34,730305,730307],{"className":730306,"style":41619},[31805],[34,730308],{},[34,730310],{"className":730311},[31690,41488],[34,730313],{"className":730314,"style":32604},[31845],[34,730316,223],{"className":730317},[32024],[34,730319],{"className":730320,"style":32604},[31845],[34,730322,730324,730327],{"className":730323},[31676],[34,730325],{"className":730326,"style":32617},[31680],[34,730328,729302],{"className":730329},[31685],"\n所以，",[2627,730332,730333],{},[34,730334,730336,730368],{"className":730335},[31642],[34,730337,730339],{"className":730338},[31646],[31648,730340,730341],{"xmlns":31650},[31652,730342,730343,730365],{},[31655,730344,730345,730347,730349,730351,730353,730355,730361,730363],{},[31658,730346,31739],{},[31662,730348,456],{"stretchy":702},[31658,730350,33249],{},[31662,730352,2557],{"stretchy":702},[31662,730354,202],{},[41453,730356,730357,730359],{},[31749,730358,36656],{},[31658,730360,33249],{},[31662,730362,223],{},[31749,730364,729302],{},[31665,730366,730367],{"encoding":31667},"f(t) = \\frac{6}{t} + 34",[34,730369,730371,730398,730481],{"className":730370,"ariaHidden":758},[31672],[34,730372,730374,730377,730380,730383,730386,730389,730392,730395],{"className":730373},[31676],[34,730375],{"className":730376,"style":31774},[31680],[34,730378,31739],{"className":730379,"style":31778},[31685,31686],[34,730381,456],{"className":730382},[31782],[34,730384,33249],{"className":730385},[31685,31686],[34,730387,2557],{"className":730388},[31690],[34,730390],{"className":730391,"style":31846},[31845],[34,730393,202],{"className":730394},[31850],[34,730396],{"className":730397,"style":31846},[31845],[34,730399,730401,730404,730472,730475,730478],{"className":730400},[31676],[34,730402],{"className":730403,"style":46727},[31680],[34,730405,730407,730410,730469],{"className":730406},[31685],[34,730408],{"className":730409},[31782,41488],[34,730411,730413],{"className":730412},[41453],[34,730414,730416,730461],{"className":730415},[31796,31797],[34,730417,730419,730458],{"className":730418},[31801],[34,730420,730422,730436,730444],{"className":730421,"style":46746},[31805],[34,730423,730424,730427],{"style":41504},[34,730425],{"className":730426,"style":41508},[31813],[34,730428,730430],{"className":730429},[31818,31819,31820,31821],[34,730431,730433],{"className":730432},[31685,31821],[34,730434,33249],{"className":730435},[31685,31686,31821],[34,730437,730438,730441],{"style":41520},[34,730439],{"className":730440,"style":41508},[31813],[34,730442],{"className":730443,"style":41528},[41527],[34,730445,730446,730449],{"style":46771},[34,730447],{"className":730448,"style":41508},[31813],[34,730450,730452],{"className":730451},[31818,31819,31820,31821],[34,730453,730455],{"className":730454},[31685,31821],[34,730456,36656],{"className":730457},[31685,31821],[34,730459,31829],{"className":730460},[31828],[34,730462,730464],{"className":730463},[31801],[34,730465,730467],{"className":730466,"style":41619},[31805],[34,730468],{},[34,730470],{"className":730471},[31690,41488],[34,730473],{"className":730474,"style":32604},[31845],[34,730476,223],{"className":730477},[32024],[34,730479],{"className":730480,"style":32604},[31845],[34,730482,730484,730487],{"className":730483},[31676],[34,730485],{"className":730486,"style":32617},[31680],[34,730488,729302],{"className":730489},[31685],"。如果你用 ",[34,730492,730494,730520],{"className":730493},[31642],[34,730495,730497],{"className":730496},[31646],[31648,730498,730499],{"xmlns":31650},[31652,730500,730501,730517],{},[31655,730502,730503,730505,730507,730509,730511],{},[31749,730504,36656],{},[31658,730506,33249],{},[31662,730508,223],{},[31749,730510,729302],{},[37624,730512,730513,730515],{},[31658,730514,33249],{},[31749,730516,768],{},[31665,730518,730519],{"encoding":31667},"6t+34t^2",[34,730521,730523,730544],{"className":730522,"ariaHidden":758},[31672],[34,730524,730526,730529,730532,730535,730538,730541],{"className":730525},[31676],[34,730527],{"className":730528,"style":35942},[31680],[34,730530,36656],{"className":730531},[31685],[34,730533,33249],{"className":730534},[31685,31686],[34,730536],{"className":730537,"style":32604},[31845],[34,730539,223],{"className":730540},[32024],[34,730542],{"className":730543,"style":32604},[31845],[34,730545,730547,730550,730553],{"className":730546},[31676],[34,730548],{"className":730549,"style":37774},[31680],[34,730551,729302],{"className":730552},[31685],[34,730554,730556,730559],{"className":730555},[31685],[34,730557,33249],{"className":730558},[31685,31686],[34,730560,730562],{"className":730561},[31792],[34,730563,730565],{"className":730564},[31796],[34,730566,730568],{"className":730567},[31801],[34,730569,730571],{"className":730570,"style":37774},[31805],[34,730572,730573,730576],{"style":37777},[34,730574],{"className":730575,"style":31814},[31813],[34,730577,730579],{"className":730578},[31818,31819,31820,31821],[34,730580,768],{"className":730581},[31685,31821]," 算，就全错了。",[15,730584,730585,730587,730790],{},[2627,730586,726186],{},[34,730588,730590,730630],{"className":730589},[31642],[34,730591,730593],{"className":730592},[31646],[31648,730594,730595],{"xmlns":31650},[31652,730596,730597,730627],{},[31655,730598,730599,730605,730607,730609,730611,730613,730615,730623,730625],{},[31743,730600,730601,730603],{},[31658,730602,32936],{},[31749,730604,640],{},[31662,730606,202],{},[31658,730608,33249],{},[39361,730610,262622],{},[31662,730612,262625],{},[39361,730614,262622],{},[38846,730616,730617,730619,730621],{},[31658,730618,32936],{},[31749,730620,640],{},[31662,730622,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,730624,202],{},[31749,730626,640],{},[31665,730628,730629],{"encoding":31667},"x_1 = t \\implies x_1' = 1",[34,730631,730633,730688,730712,730781],{"className":730632,"ariaHidden":758},[31672],[34,730634,730636,730639,730679,730682,730685],{"className":730635},[31676],[34,730637],{"className":730638,"style":31860},[31680],[34,730640,730642,730645],{"className":730641},[31685],[34,730643,32936],{"className":730644},[31685,31686],[34,730646,730648],{"className":730647},[31792],[34,730649,730651,730671],{"className":730650},[31796,31797],[34,730652,730654,730668],{"className":730653},[31801],[34,730655,730657],{"className":730656,"style":31806},[31805],[34,730658,730659,730662],{"style":31809},[34,730660],{"className":730661,"style":31814},[31813],[34,730663,730665],{"className":730664},[31818,31819,31820,31821],[34,730666,640],{"className":730667},[31685,31821],[34,730669,31829],{"className":730670},[31828],[34,730672,730674],{"className":730673},[31801],[34,730675,730677],{"className":730676,"style":31836},[31805],[34,730678],{},[34,730680],{"className":730681,"style":31846},[31845],[34,730683,202],{"className":730684},[31850],[34,730686],{"className":730687,"style":31846},[31845],[34,730689,730691,730694,730697,730700,730703,730706,730709],{"className":730690},[31676],[34,730692],{"className":730693,"style":721257},[31680],[34,730695,33249],{"className":730696},[31685,31686],[34,730698],{"className":730699,"style":31846},[31845],[34,730701],{"className":730702,"style":31846},[31845],[34,730704,262625],{"className":730705},[31850],[34,730707],{"className":730708,"style":31846},[31845],[34,730710],{"className":730711,"style":31846},[31845],[34,730713,730715,730718,730772,730775,730778],{"className":730714},[31676],[34,730716],{"className":730717,"style":716670},[31680],[34,730719,730721,730724],{"className":730720},[31685],[34,730722,32936],{"className":730723},[31685,31686],[34,730725,730727],{"className":730726},[31792],[34,730728,730730,730764],{"className":730729},[31796,31797],[34,730731,730733,730761],{"className":730732},[31801],[34,730734,730736,730747],{"className":730735,"style":132027},[31805],[34,730737,730738,730741],{"style":316930},[34,730739],{"className":730740,"style":31814},[31813],[34,730742,730744],{"className":730743},[31818,31819,31820,31821],[34,730745,640],{"className":730746},[31685,31821],[34,730748,730749,730752],{"style":37777},[34,730750],{"className":730751,"style":31814},[31813],[34,730753,730755],{"className":730754},[31818,31819,31820,31821],[34,730756,730758],{"className":730757},[31685,31821],[34,730759,116495],{"className":730760},[31685,31821],[34,730762,31829],{"className":730763},[31828],[34,730765,730767],{"className":730766},[31801],[34,730768,730770],{"className":730769,"style":39021},[31805],[34,730771],{},[34,730773],{"className":730774,"style":31846},[31845],[34,730776,202],{"className":730777},[31850],[34,730779],{"className":730780,"style":31846},[31845],[34,730782,730784,730787],{"className":730783},[31676],[34,730785],{"className":730786,"style":32617},[31680],[34,730788,640],{"className":730789},[31685],[34,730791,730793,730847],{"className":730792},[31642],[34,730794,730796],{"className":730795},[31646],[31648,730797,730798],{"xmlns":31650},[31652,730799,730800,730844],{},[31655,730801,730802,730808,730810,730812,730814,730816,730818,730820,730822,730824,730832,730834,730836,730838,730840,730842],{},[31743,730803,730804,730806],{},[31658,730805,32936],{},[31749,730807,768],{},[31662,730809,202],{},[31658,730811,33249],{},[31658,730813,563286],{},[31662,730815,32682],{},[31658,730817,33249],{},[39361,730819,262622],{},[31662,730821,262625],{},[39361,730823,262622],{},[38846,730825,730826,730828,730830],{},[31658,730827,32936],{},[31749,730829,768],{},[31662,730831,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,730833,202],{},[31658,730835,563286],{},[31662,730837,32682],{},[31658,730839,33249],{},[31662,730841,223],{},[31749,730843,640],{},[31665,730845,730846],{"encoding":31667},"x_2 = t\\ln t \\implies x_2' = \\ln t + 1",[34,730848,730850,730905,730941,731010,731034],{"className":730849,"ariaHidden":758},[31672],[34,730851,730853,730856,730896,730899,730902],{"className":730852},[31676],[34,730854],{"className":730855,"style":31860},[31680],[34,730857,730859,730862],{"className":730858},[31685],[34,730860,32936],{"className":730861},[31685,31686],[34,730863,730865],{"className":730864},[31792],[34,730866,730868,730888],{"className":730867},[31796,31797],[34,730869,730871,730885],{"className":730870},[31801],[34,730872,730874],{"className":730873,"style":31806},[31805],[34,730875,730876,730879],{"style":31809},[34,730877],{"className":730878,"style":31814},[31813],[34,730880,730882],{"className":730881},[31818,31819,31820,31821],[34,730883,768],{"className":730884},[31685,31821],[34,730886,31829],{"className":730887},[31828],[34,730889,730891],{"className":730890},[31801],[34,730892,730894],{"className":730893,"style":31836},[31805],[34,730895],{},[34,730897],{"className":730898,"style":31846},[31845],[34,730900,202],{"className":730901},[31850],[34,730903],{"className":730904,"style":31846},[31845],[34,730906,730908,730911,730914,730917,730920,730923,730926,730929,730932,730935,730938],{"className":730907},[31676],[34,730909],{"className":730910,"style":474142},[31680],[34,730912,33249],{"className":730913},[31685,31686],[34,730915],{"className":730916,"style":32123},[31845],[34,730918,563286],{"className":730919},[32767],[34,730921],{"className":730922,"style":32123},[31845],[34,730924,33249],{"className":730925},[31685,31686],[34,730927],{"className":730928,"style":31846},[31845],[34,730930],{"className":730931,"style":31846},[31845],[34,730933,262625],{"className":730934},[31850],[34,730936],{"className":730937,"style":31846},[31845],[34,730939],{"className":730940,"style":31846},[31845],[34,730942,730944,730947,731001,731004,731007],{"className":730943},[31676],[34,730945],{"className":730946,"style":716670},[31680],[34,730948,730950,730953],{"className":730949},[31685],[34,730951,32936],{"className":730952},[31685,31686],[34,730954,730956],{"className":730955},[31792],[34,730957,730959,730993],{"className":730958},[31796,31797],[34,730960,730962,730990],{"className":730961},[31801],[34,730963,730965,730976],{"className":730964,"style":132027},[31805],[34,730966,730967,730970],{"style":316930},[34,730968],{"className":730969,"style":31814},[31813],[34,730971,730973],{"className":730972},[31818,31819,31820,31821],[34,730974,768],{"className":730975},[31685,31821],[34,730977,730978,730981],{"style":37777},[34,730979],{"className":730980,"style":31814},[31813],[34,730982,730984],{"className":730983},[31818,31819,31820,31821],[34,730985,730987],{"className":730986},[31685,31821],[34,730988,116495],{"className":730989},[31685,31821],[34,730991,31829],{"className":730992},[31828],[34,730994,730996],{"className":730995},[31801],[34,730997,730999],{"className":730998,"style":39021},[31805],[34,731000],{},[34,731002],{"className":731003,"style":31846},[31845],[34,731005,202],{"className":731006},[31850],[34,731008],{"className":731009,"style":31846},[31845],[34,731011,731013,731016,731019,731022,731025,731028,731031],{"className":731012},[31676],[34,731014],{"className":731015,"style":34299},[31680],[34,731017,563286],{"className":731018},[32767],[34,731020],{"className":731021,"style":32123},[31845],[34,731023,33249],{"className":731024},[31685,31686],[34,731026],{"className":731027,"style":32604},[31845],[34,731029,223],{"className":731030},[32024],[34,731032],{"className":731033,"style":32604},[31845],[34,731035,731037,731040],{"className":731036},[31676],[34,731038],{"className":731039,"style":32617},[31680],[34,731041,640],{"className":731042},[31685],[15,731044,731045,731046,721062],{},"[\n",[34,731047,731049,731163],{"className":731048},[31642],[34,731050,731052],{"className":731051},[31646],[31648,731053,731054],{"xmlns":31650},[31652,731055,731056,731160],{},[31655,731057,731058,731060],{},[31662,731059,32685],{"fence":758},[71078,731061,731062,731102],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,731063,731064],{},[71087,731065,731066],{},[71090,731067,731068],{"scriptlevel":459,"displaystyle":702},[31655,731069,731070,731078,731080,731082,731090,731092,731094,731096,731098,731100],{},[38846,731071,731072,731074,731076],{},[31658,731073,31965],{},[31749,731075,640],{},[31662,731077,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,731079,33249],{},[31662,731081,223],{},[38846,731083,731084,731086,731088],{},[31658,731085,31965],{},[31749,731087,768],{},[31662,731089,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,731091,33249],{},[31658,731093,563286],{},[31662,731095,32682],{},[31658,731097,33249],{},[31662,731099,202],{},[31749,731101,459],{},[71084,731103,731104],{},[71087,731105,731106],{},[71090,731107,731108],{"scriptlevel":459,"displaystyle":702},[31655,731109,731110,731118,731120,731122,731124,731126,731134,731136,731138,731140,731142,731144,731146,731148,731150,731156,731158],{},[38846,731111,731112,731114,731116],{},[31658,731113,31965],{},[31749,731115,640],{},[31662,731117,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,731119,456],{"stretchy":702},[31749,731121,640],{},[31662,731123,2557],{"stretchy":702},[31662,731125,223],{},[38846,731127,731128,731130,731132],{},[31658,731129,31965],{},[31749,731131,768],{},[31662,731133,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,731135,456],{"stretchy":702},[31658,731137,563286],{},[31662,731139,32682],{},[31658,731141,33249],{},[31662,731143,223],{},[31749,731145,640],{},[31662,731147,2557],{"stretchy":702},[31662,731149,202],{},[41453,731151,731152,731154],{},[31749,731153,36656],{},[31658,731155,33249],{},[31662,731157,223],{},[31749,731159,729302],{},[31665,731161,731162],{"encoding":31667},"\\begin{cases}\nc_1' t + c_2' t\\ln t = 0 \\\\\nc_1' (1) + c_2' (\\ln t + 1) = \\frac{6}{t} + 34\n\\end{cases}",[34,731164,731166],{"className":731165,"ariaHidden":758},[31672],[34,731167,731169,731172],{"className":731168},[31676],[34,731170],{"className":731171,"style":75797},[31680],[34,731173,731175,731181,731615],{"className":731174},[34006],[34,731176,731178],{"className":731177,"style":75811},[31782,75810],[34,731179,32685],{"className":731180},[71281,75815],[34,731182,731184],{"className":731183},[31685],[34,731185,731187],{"className":731186},[71078],[34,731188,731190],{"className":731189},[71384],[34,731191,731193,731607],{"className":731192},[31796,31797],[34,731194,731196,731604],{"className":731195},[31801],[34,731197,731199,731354],{"className":731198,"style":75834},[31805],[34,731200,731201,731204],{"style":75837},[34,731202],{"className":731203,"style":71401},[31813],[34,731205,731207,731261,731264,731267,731270,731273,731327,731330,731333,731336,731339,731342,731345,731348,731351],{"className":731206},[31685],[34,731208,731210,731213],{"className":731209},[31685],[34,731211,31965],{"className":731212},[31685,31686],[34,731214,731216],{"className":731215},[31792],[34,731217,731219,731253],{"className":731218},[31796,31797],[34,731220,731222,731250],{"className":731221},[31801],[34,731223,731225,731236],{"className":731224,"style":132027},[31805],[34,731226,731227,731230],{"style":316930},[34,731228],{"className":731229,"style":31814},[31813],[34,731231,731233],{"className":731232},[31818,31819,31820,31821],[34,731234,640],{"className":731235},[31685,31821],[34,731237,731238,731241],{"style":37777},[34,731239],{"className":731240,"style":31814},[31813],[34,731242,731244],{"className":731243},[31818,31819,31820,31821],[34,731245,731247],{"className":731246},[31685,31821],[34,731248,116495],{"className":731249},[31685,31821],[34,731251,31829],{"className":731252},[31828],[34,731254,731256],{"className":731255},[31801],[34,731257,731259],{"className":731258,"style":39021},[31805],[34,731260],{},[34,731262,33249],{"className":731263},[31685,31686],[34,731265],{"className":731266,"style":32604},[31845],[34,731268,223],{"className":731269},[32024],[34,731271],{"className":731272,"style":32604},[31845],[34,731274,731276,731279],{"className":731275},[31685],[34,731277,31965],{"className":731278},[31685,31686],[34,731280,731282],{"className":731281},[31792],[34,731283,731285,731319],{"className":731284},[31796,31797],[34,731286,731288,731316],{"className":731287},[31801],[34,731289,731291,731302],{"className":731290,"style":132027},[31805],[34,731292,731293,731296],{"style":316930},[34,731294],{"className":731295,"style":31814},[31813],[34,731297,731299],{"className":731298},[31818,31819,31820,31821],[34,731300,768],{"className":731301},[31685,31821],[34,731303,731304,731307],{"style":37777},[34,731305],{"className":731306,"style":31814},[31813],[34,731308,731310],{"className":731309},[31818,31819,31820,31821],[34,731311,731313],{"className":731312},[31685,31821],[34,731314,116495],{"className":731315},[31685,31821],[34,731317,31829],{"className":731318},[31828],[34,731320,731322],{"className":731321},[31801],[34,731323,731325],{"className":731324,"style":39021},[31805],[34,731326],{},[34,731328,33249],{"className":731329},[31685,31686],[34,731331],{"className":731332,"style":32123},[31845],[34,731334,563286],{"className":731335},[32767],[34,731337],{"className":731338,"style":32123},[31845],[34,731340,33249],{"className":731341},[31685,31686],[34,731343],{"className":731344,"style":31846},[31845],[34,731346,202],{"className":731347},[31850],[34,731349],{"className":731350,"style":31846},[31845],[34,731352,459],{"className":731353},[31685],[34,731355,731356,731359],{"style":76011},[34,731357],{"className":731358,"style":71401},[31813],[34,731360,731362,731416,731419,731422,731425,731428,731431,731434,731488,731491,731494,731497,731500,731503,731506,731509,731512,731515,731518,731521,731524,731592,731595,731598,731601],{"className":731361},[31685],[34,731363,731365,731368],{"className":731364},[31685],[34,731366,31965],{"className":731367},[31685,31686],[34,731369,731371],{"className":731370},[31792],[34,731372,731374,731408],{"className":731373},[31796,31797],[34,731375,731377,731405],{"className":731376},[31801],[34,731378,731380,731391],{"className":731379,"style":132027},[31805],[34,731381,731382,731385],{"style":316930},[34,731383],{"className":731384,"style":31814},[31813],[34,731386,731388],{"className":731387},[31818,31819,31820,31821],[34,731389,640],{"className":731390},[31685,31821],[34,731392,731393,731396],{"style":37777},[34,731394],{"className":731395,"style":31814},[31813],[34,731397,731399],{"className":731398},[31818,31819,31820,31821],[34,731400,731402],{"className":731401},[31685,31821],[34,731403,116495],{"className":731404},[31685,31821],[34,731406,31829],{"className":731407},[31828],[34,731409,731411],{"className":731410},[31801],[34,731412,731414],{"className":731413,"style":39021},[31805],[34,731415],{},[34,731417,456],{"className":731418},[31782],[34,731420,640],{"className":731421},[31685],[34,731423,2557],{"className":731424},[31690],[34,731426],{"className":731427,"style":32604},[31845],[34,731429,223],{"className":731430},[32024],[34,731432],{"className":731433,"style":32604},[31845],[34,731435,731437,731440],{"className":731436},[31685],[34,731438,31965],{"className":731439},[31685,31686],[34,731441,731443],{"className":731442},[31792],[34,731444,731446,731480],{"className":731445},[31796,31797],[34,731447,731449,731477],{"className":731448},[31801],[34,731450,731452,731463],{"className":731451,"style":132027},[31805],[34,731453,731454,731457],{"style":316930},[34,731455],{"className":731456,"style":31814},[31813],[34,731458,731460],{"className":731459},[31818,31819,31820,31821],[34,731461,768],{"className":731462},[31685,31821],[34,731464,731465,731468],{"style":37777},[34,731466],{"className":731467,"style":31814},[31813],[34,731469,731471],{"className":731470},[31818,31819,31820,31821],[34,731472,731474],{"className":731473},[31685,31821],[34,731475,116495],{"className":731476},[31685,31821],[34,731478,31829],{"className":731479},[31828],[34,731481,731483],{"className":731482},[31801],[34,731484,731486],{"className":731485,"style":39021},[31805],[34,731487],{},[34,731489,456],{"className":731490},[31782],[34,731492,563286],{"className":731493},[32767],[34,731495],{"className":731496,"style":32123},[31845],[34,731498,33249],{"className":731499},[31685,31686],[34,731501],{"className":731502,"style":32604},[31845],[34,731504,223],{"className":731505},[32024],[34,731507],{"className":731508,"style":32604},[31845],[34,731510,640],{"className":731511},[31685],[34,731513,2557],{"className":731514},[31690],[34,731516],{"className":731517,"style":31846},[31845],[34,731519,202],{"className":731520},[31850],[34,731522],{"className":731523,"style":31846},[31845],[34,731525,731527,731530,731589],{"className":731526},[31685],[34,731528],{"className":731529},[31782,41488],[34,731531,731533],{"className":731532},[41453],[34,731534,731536,731581],{"className":731535},[31796,31797],[34,731537,731539,731578],{"className":731538},[31801],[34,731540,731542,731556,731564],{"className":731541,"style":46746},[31805],[34,731543,731544,731547],{"style":41504},[34,731545],{"className":731546,"style":41508},[31813],[34,731548,731550],{"className":731549},[31818,31819,31820,31821],[34,731551,731553],{"className":731552},[31685,31821],[34,731554,33249],{"className":731555},[31685,31686,31821],[34,731557,731558,731561],{"style":41520},[34,731559],{"className":731560,"style":41508},[31813],[34,731562],{"className":731563,"style":41528},[41527],[34,731565,731566,731569],{"style":46771},[34,731567],{"className":731568,"style":41508},[31813],[34,731570,731572],{"className":731571},[31818,31819,31820,31821],[34,731573,731575],{"className":731574},[31685,31821],[34,731576,36656],{"className":731577},[31685,31821],[34,731579,31829],{"className":731580},[31828],[34,731582,731584],{"className":731583},[31801],[34,731585,731587],{"className":731586,"style":41619},[31805],[34,731588],{},[34,731590],{"className":731591},[31690,41488],[34,731593],{"className":731594,"style":32604},[31845],[34,731596,223],{"className":731597},[32024],[34,731599],{"className":731600,"style":32604},[31845],[34,731602,729302],{"className":731603},[31685],[34,731605,31829],{"className":731606},[31828],[34,731608,731610],{"className":731609},[31801],[34,731611,731613],{"className":731612,"style":76192},[31805],[34,731614],{},[34,731616],{"className":731617},[31690,41488],[15,731619,731620,731622,731623,731811,731812,732210],{},[2627,731621,721067],{},"\n由第一式：",[34,731624,731626,731664],{"className":731625},[31642],[34,731627,731629],{"className":731628},[31646],[31648,731630,731631],{"xmlns":31650},[31652,731632,731633,731661],{},[31655,731634,731635,731643,731645,731647,731655,731657,731659],{},[38846,731636,731637,731639,731641],{},[31658,731638,31965],{},[31749,731640,640],{},[31662,731642,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,731644,202],{},[31662,731646,32226],{},[38846,731648,731649,731651,731653],{},[31658,731650,31965],{},[31749,731652,768],{},[31662,731654,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,731656,563286],{},[31662,731658,32682],{},[31658,731660,33249],{},[31665,731662,731663],{"encoding":31667},"c_1' = -c_2' \\ln t",[34,731665,731667,731736],{"className":731666,"ariaHidden":758},[31672],[34,731668,731670,731673,731727,731730,731733],{"className":731669},[31676],[34,731671],{"className":731672,"style":716670},[31680],[34,731674,731676,731679],{"className":731675},[31685],[34,731677,31965],{"className":731678},[31685,31686],[34,731680,731682],{"className":731681},[31792],[34,731683,731685,731719],{"className":731684},[31796,31797],[34,731686,731688,731716],{"className":731687},[31801],[34,731689,731691,731702],{"className":731690,"style":132027},[31805],[34,731692,731693,731696],{"style":316930},[34,731694],{"className":731695,"style":31814},[31813],[34,731697,731699],{"className":731698},[31818,31819,31820,31821],[34,731700,640],{"className":731701},[31685,31821],[34,731703,731704,731707],{"style":37777},[34,731705],{"className":731706,"style":31814},[31813],[34,731708,731710],{"className":731709},[31818,31819,31820,31821],[34,731711,731713],{"className":731712},[31685,31821],[34,731714,116495],{"className":731715},[31685,31821],[34,731717,31829],{"className":731718},[31828],[34,731720,731722],{"className":731721},[31801],[34,731723,731725],{"className":731724,"style":39021},[31805],[34,731726],{},[34,731728],{"className":731729,"style":31846},[31845],[34,731731,202],{"className":731732},[31850],[34,731734],{"className":731735,"style":31846},[31845],[34,731737,731739,731742,731745,731799,731802,731805,731808],{"className":731738},[31676],[34,731740],{"className":731741,"style":716670},[31680],[34,731743,32226],{"className":731744},[31685],[34,731746,731748,731751],{"className":731747},[31685],[34,731749,31965],{"className":731750},[31685,31686],[34,731752,731754],{"className":731753},[31792],[34,731755,731757,731791],{"className":731756},[31796,31797],[34,731758,731760,731788],{"className":731759},[31801],[34,731761,731763,731774],{"className":731762,"style":132027},[31805],[34,731764,731765,731768],{"style":316930},[34,731766],{"className":731767,"style":31814},[31813],[34,731769,731771],{"className":731770},[31818,31819,31820,31821],[34,731772,768],{"className":731773},[31685,31821],[34,731775,731776,731779],{"style":37777},[34,731777],{"className":731778,"style":31814},[31813],[34,731780,731782],{"className":731781},[31818,31819,31820,31821],[34,731783,731785],{"className":731784},[31685,31821],[34,731786,116495],{"className":731787},[31685,31821],[34,731789,31829],{"className":731790},[31828],[34,731792,731794],{"className":731793},[31801],[34,731795,731797],{"className":731796,"style":39021},[31805],[34,731798],{},[34,731800],{"className":731801,"style":32123},[31845],[34,731803,563286],{"className":731804},[32767],[34,731806],{"className":731807,"style":32123},[31845],[34,731809,33249],{"className":731810},[31685,31686],"。\n代入第二式：\n",[34,731813,731815,731881],{"className":731814},[31642],[34,731816,731818],{"className":731817},[31646],[31648,731819,731820],{"xmlns":31650},[31652,731821,731822,731878],{},[31655,731823,731824,731826,731834,731836,731838,731840,731842,731850,731852,731854,731856,731858,731866,731868,731874,731876],{},[31662,731825,32226],{},[38846,731827,731828,731830,731832],{},[31658,731829,31965],{},[31749,731831,768],{},[31662,731833,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,731835,563286],{},[31662,731837,32682],{},[31658,731839,33249],{},[31662,731841,223],{},[38846,731843,731844,731846,731848],{},[31658,731845,31965],{},[31749,731847,768],{},[31662,731849,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,731851,563286],{},[31662,731853,32682],{},[31658,731855,33249],{},[31662,731857,223],{},[38846,731859,731860,731862,731864],{},[31658,731861,31965],{},[31749,731863,768],{},[31662,731865,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,731867,202],{},[41453,731869,731870,731872],{},[31749,731871,36656],{},[31658,731873,33249],{},[31662,731875,223],{},[31749,731877,729302],{},[31665,731879,731880],{"encoding":31667},"-c_2' \\ln t + c_2' \\ln t + c_2' = \\frac{6}{t} + 34",[34,731882,731884,731968,732049,732118,732201],{"className":731883,"ariaHidden":758},[31672],[34,731885,731887,731890,731893,731947,731950,731953,731956,731959,731962,731965],{"className":731886},[31676],[34,731888],{"className":731889,"style":716670},[31680],[34,731891,32226],{"className":731892},[31685],[34,731894,731896,731899],{"className":731895},[31685],[34,731897,31965],{"className":731898},[31685,31686],[34,731900,731902],{"className":731901},[31792],[34,731903,731905,731939],{"className":731904},[31796,31797],[34,731906,731908,731936],{"className":731907},[31801],[34,731909,731911,731922],{"className":731910,"style":132027},[31805],[34,731912,731913,731916],{"style":316930},[34,731914],{"className":731915,"style":31814},[31813],[34,731917,731919],{"className":731918},[31818,31819,31820,31821],[34,731920,768],{"className":731921},[31685,31821],[34,731923,731924,731927],{"style":37777},[34,731925],{"className":731926,"style":31814},[31813],[34,731928,731930],{"className":731929},[31818,31819,31820,31821],[34,731931,731933],{"className":731932},[31685,31821],[34,731934,116495],{"className":731935},[31685,31821],[34,731937,31829],{"className":731938},[31828],[34,731940,731942],{"className":731941},[31801],[34,731943,731945],{"className":731944,"style":39021},[31805],[34,731946],{},[34,731948],{"className":731949,"style":32123},[31845],[34,731951,563286],{"className":731952},[32767],[34,731954],{"className":731955,"style":32123},[31845],[34,731957,33249],{"className":731958},[31685,31686],[34,731960],{"className":731961,"style":32604},[31845],[34,731963,223],{"className":731964},[32024],[34,731966],{"className":731967,"style":32604},[31845],[34,731969,731971,731974,732028,732031,732034,732037,732040,732043,732046],{"className":731970},[31676],[34,731972],{"className":731973,"style":716670},[31680],[34,731975,731977,731980],{"className":731976},[31685],[34,731978,31965],{"className":731979},[31685,31686],[34,731981,731983],{"className":731982},[31792],[34,731984,731986,732020],{"className":731985},[31796,31797],[34,731987,731989,732017],{"className":731988},[31801],[34,731990,731992,732003],{"className":731991,"style":132027},[31805],[34,731993,731994,731997],{"style":316930},[34,731995],{"className":731996,"style":31814},[31813],[34,731998,732000],{"className":731999},[31818,31819,31820,31821],[34,732001,768],{"className":732002},[31685,31821],[34,732004,732005,732008],{"style":37777},[34,732006],{"className":732007,"style":31814},[31813],[34,732009,732011],{"className":732010},[31818,31819,31820,31821],[34,732012,732014],{"className":732013},[31685,31821],[34,732015,116495],{"className":732016},[31685,31821],[34,732018,31829],{"className":732019},[31828],[34,732021,732023],{"className":732022},[31801],[34,732024,732026],{"className":732025,"style":39021},[31805],[34,732027],{},[34,732029],{"className":732030,"style":32123},[31845],[34,732032,563286],{"className":732033},[32767],[34,732035],{"className":732036,"style":32123},[31845],[34,732038,33249],{"className":732039},[31685,31686],[34,732041],{"className":732042,"style":32604},[31845],[34,732044,223],{"className":732045},[32024],[34,732047],{"className":732048,"style":32604},[31845],[34,732050,732052,732055,732109,732112,732115],{"className":732051},[31676],[34,732053],{"className":732054,"style":716670},[31680],[34,732056,732058,732061],{"className":732057},[31685],[34,732059,31965],{"className":732060},[31685,31686],[34,732062,732064],{"className":732063},[31792],[34,732065,732067,732101],{"className":732066},[31796,31797],[34,732068,732070,732098],{"className":732069},[31801],[34,732071,732073,732084],{"className":732072,"style":132027},[31805],[34,732074,732075,732078],{"style":316930},[34,732076],{"className":732077,"style":31814},[31813],[34,732079,732081],{"className":732080},[31818,31819,31820,31821],[34,732082,768],{"className":732083},[31685,31821],[34,732085,732086,732089],{"style":37777},[34,732087],{"className":732088,"style":31814},[31813],[34,732090,732092],{"className":732091},[31818,31819,31820,31821],[34,732093,732095],{"className":732094},[31685,31821],[34,732096,116495],{"className":732097},[31685,31821],[34,732099,31829],{"className":732100},[31828],[34,732102,732104],{"className":732103},[31801],[34,732105,732107],{"className":732106,"style":39021},[31805],[34,732108],{},[34,732110],{"className":732111,"style":31846},[31845],[34,732113,202],{"className":732114},[31850],[34,732116],{"className":732117,"style":31846},[31845],[34,732119,732121,732124,732192,732195,732198],{"className":732120},[31676],[34,732122],{"className":732123,"style":46727},[31680],[34,732125,732127,732130,732189],{"className":732126},[31685],[34,732128],{"className":732129},[31782,41488],[34,732131,732133],{"className":732132},[41453],[34,732134,732136,732181],{"className":732135},[31796,31797],[34,732137,732139,732178],{"className":732138},[31801],[34,732140,732142,732156,732164],{"className":732141,"style":46746},[31805],[34,732143,732144,732147],{"style":41504},[34,732145],{"className":732146,"style":41508},[31813],[34,732148,732150],{"className":732149},[31818,31819,31820,31821],[34,732151,732153],{"className":732152},[31685,31821],[34,732154,33249],{"className":732155},[31685,31686,31821],[34,732157,732158,732161],{"style":41520},[34,732159],{"className":732160,"style":41508},[31813],[34,732162],{"className":732163,"style":41528},[41527],[34,732165,732166,732169],{"style":46771},[34,732167],{"className":732168,"style":41508},[31813],[34,732170,732172],{"className":732171},[31818,31819,31820,31821],[34,732173,732175],{"className":732174},[31685,31821],[34,732176,36656],{"className":732177},[31685,31821],[34,732179,31829],{"className":732180},[31828],[34,732182,732184],{"className":732183},[31801],[34,732185,732187],{"className":732186,"style":41619},[31805],[34,732188],{},[34,732190],{"className":732191},[31690,41488],[34,732193],{"className":732194,"style":32604},[31845],[34,732196,223],{"className":732197},[32024],[34,732199],{"className":732200,"style":32604},[31845],[34,732202,732204,732207],{"className":732203},[31676],[34,732205],{"className":732206,"style":32617},[31680],[34,732208,729302],{"className":732209},[31685],[34,732211,732213,732251],{"className":732212},[31642],[34,732214,732216],{"className":732215},[31646],[31648,732217,732218],{"xmlns":31650},[31652,732219,732220,732248],{},[31655,732221,732222,732224,732226,732228,732236,732238,732244,732246],{},[39361,732223,262622],{},[31662,732225,262625],{},[39361,732227,262622],{},[38846,732229,732230,732232,732234],{},[31658,732231,31965],{},[31749,732233,768],{},[31662,732235,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,732237,202],{},[41453,732239,732240,732242],{},[31749,732241,36656],{},[31658,732243,33249],{},[31662,732245,223],{},[31749,732247,729302],{},[31665,732249,732250],{"encoding":31667},"\\implies c_2' = \\frac{6}{t} + 34",[34,732252,732254,732272,732341,732424],{"className":732253,"ariaHidden":758},[31672],[34,732255,732257,732260,732263,732266,732269],{"className":732256},[31676],[34,732258],{"className":732259,"style":318264},[31680],[34,732261],{"className":732262,"style":31846},[31845],[34,732264,262625],{"className":732265},[31850],[34,732267],{"className":732268,"style":31846},[31845],[34,732270],{"className":732271,"style":31846},[31845],[34,732273,732275,732278,732332,732335,732338],{"className":732274},[31676],[34,732276],{"className":732277,"style":716670},[31680],[34,732279,732281,732284],{"className":732280},[31685],[34,732282,31965],{"className":732283},[31685,31686],[34,732285,732287],{"className":732286},[31792],[34,732288,732290,732324],{"className":732289},[31796,31797],[34,732291,732293,732321],{"className":732292},[31801],[34,732294,732296,732307],{"className":732295,"style":132027},[31805],[34,732297,732298,732301],{"style":316930},[34,732299],{"className":732300,"style":31814},[31813],[34,732302,732304],{"className":732303},[31818,31819,31820,31821],[34,732305,768],{"className":732306},[31685,31821],[34,732308,732309,732312],{"style":37777},[34,732310],{"className":732311,"style":31814},[31813],[34,732313,732315],{"className":732314},[31818,31819,31820,31821],[34,732316,732318],{"className":732317},[31685,31821],[34,732319,116495],{"className":732320},[31685,31821],[34,732322,31829],{"className":732323},[31828],[34,732325,732327],{"className":732326},[31801],[34,732328,732330],{"className":732329,"style":39021},[31805],[34,732331],{},[34,732333],{"className":732334,"style":31846},[31845],[34,732336,202],{"className":732337},[31850],[34,732339],{"className":732340,"style":31846},[31845],[34,732342,732344,732347,732415,732418,732421],{"className":732343},[31676],[34,732345],{"className":732346,"style":46727},[31680],[34,732348,732350,732353,732412],{"className":732349},[31685],[34,732351],{"className":732352},[31782,41488],[34,732354,732356],{"className":732355},[41453],[34,732357,732359,732404],{"className":732358},[31796,31797],[34,732360,732362,732401],{"className":732361},[31801],[34,732363,732365,732379,732387],{"className":732364,"style":46746},[31805],[34,732366,732367,732370],{"style":41504},[34,732368],{"className":732369,"style":41508},[31813],[34,732371,732373],{"className":732372},[31818,31819,31820,31821],[34,732374,732376],{"className":732375},[31685,31821],[34,732377,33249],{"className":732378},[31685,31686,31821],[34,732380,732381,732384],{"style":41520},[34,732382],{"className":732383,"style":41508},[31813],[34,732385],{"className":732386,"style":41528},[41527],[34,732388,732389,732392],{"style":46771},[34,732390],{"className":732391,"style":41508},[31813],[34,732393,732395],{"className":732394},[31818,31819,31820,31821],[34,732396,732398],{"className":732397},[31685,31821],[34,732399,36656],{"className":732400},[31685,31821],[34,732402,31829],{"className":732403},[31828],[34,732405,732407],{"className":732406},[31801],[34,732408,732410],{"className":732409,"style":41619},[31805],[34,732411],{},[34,732413],{"className":732414},[31690,41488],[34,732416],{"className":732417,"style":32604},[31845],[34,732419,223],{"className":732420},[32024],[34,732422],{"className":732423,"style":32604},[31845],[34,732425,732427,732430],{"className":732426},[31676],[34,732428],{"className":732429,"style":32617},[31680],[34,732431,729302],{"className":732432},[31685],[15,732434,732435,732436,33183,732521],{},"回代求 ",[34,732437,732439,732458],{"className":732438},[31642],[34,732440,732442],{"className":732441},[31646],[31648,732443,732444],{"xmlns":31650},[31652,732445,732446,732456],{},[31655,732447,732448],{},[38846,732449,732450,732452,732454],{},[31658,732451,31965],{},[31749,732453,640],{},[31662,732455,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,732457,726871],{"encoding":31667},[34,732459,732461],{"className":732460,"ariaHidden":758},[31672],[34,732462,732464,732467],{"className":732463},[31676],[34,732465],{"className":732466,"style":716670},[31680],[34,732468,732470,732473],{"className":732469},[31685],[34,732471,31965],{"className":732472},[31685,31686],[34,732474,732476],{"className":732475},[31792],[34,732477,732479,732513],{"className":732478},[31796,31797],[34,732480,732482,732510],{"className":732481},[31801],[34,732483,732485,732496],{"className":732484,"style":132027},[31805],[34,732486,732487,732490],{"style":316930},[34,732488],{"className":732489,"style":31814},[31813],[34,732491,732493],{"className":732492},[31818,31819,31820,31821],[34,732494,640],{"className":732495},[31685,31821],[34,732497,732498,732501],{"style":37777},[34,732499],{"className":732500,"style":31814},[31813],[34,732502,732504],{"className":732503},[31818,31819,31820,31821],[34,732505,732507],{"className":732506},[31685,31821],[34,732508,116495],{"className":732509},[31685,31821],[34,732511,31829],{"className":732512},[31828],[34,732514,732516],{"className":732515},[31801],[34,732517,732519],{"className":732518,"style":39021},[31805],[34,732520],{},[34,732522,732524,732594],{"className":732523},[31642],[34,732525,732527],{"className":732526},[31646],[31648,732528,732529],{"xmlns":31650},[31652,732530,732531,732591],{},[31655,732532,732533,732541,732543,732545,732547,732553,732555,732557,732559,732561,732563,732565,732567,732569,732575,732577,732579,732581,732583,732585,732587,732589],{},[38846,732534,732535,732537,732539],{},[31658,732536,31965],{},[31749,732538,640],{},[31662,732540,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,732542,202],{},[31662,732544,32226],{},[31662,732546,456],{"stretchy":702},[41453,732548,732549,732551],{},[31749,732550,36656],{},[31658,732552,33249],{},[31662,732554,223],{},[31749,732556,729302],{},[31662,732558,2557],{"stretchy":702},[31658,732560,563286],{},[31662,732562,32682],{},[31658,732564,33249],{},[31662,732566,202],{},[31662,732568,32226],{},[41453,732570,732571,732573],{},[31749,732572,36656],{},[31658,732574,33249],{},[31658,732576,563286],{},[31662,732578,32682],{},[31658,732580,33249],{},[31662,732582,32226],{},[31749,732584,729302],{},[31658,732586,563286],{},[31662,732588,32682],{},[31658,732590,33249],{},[31665,732592,732593],{"encoding":31667},"c_1' = -(\\frac{6}{t} + 34)\\ln t = -\\frac{6}{t}\\ln t - 34\\ln t",[34,732595,732597,732666,732755,732788,732886],{"className":732596,"ariaHidden":758},[31672],[34,732598,732600,732603,732657,732660,732663],{"className":732599},[31676],[34,732601],{"className":732602,"style":716670},[31680],[34,732604,732606,732609],{"className":732605},[31685],[34,732607,31965],{"className":732608},[31685,31686],[34,732610,732612],{"className":732611},[31792],[34,732613,732615,732649],{"className":732614},[31796,31797],[34,732616,732618,732646],{"className":732617},[31801],[34,732619,732621,732632],{"className":732620,"style":132027},[31805],[34,732622,732623,732626],{"style":316930},[34,732624],{"className":732625,"style":31814},[31813],[34,732627,732629],{"className":732628},[31818,31819,31820,31821],[34,732630,640],{"className":732631},[31685,31821],[34,732633,732634,732637],{"style":37777},[34,732635],{"className":732636,"style":31814},[31813],[34,732638,732640],{"className":732639},[31818,31819,31820,31821],[34,732641,732643],{"className":732642},[31685,31821],[34,732644,116495],{"className":732645},[31685,31821],[34,732647,31829],{"className":732648},[31828],[34,732650,732652],{"className":732651},[31801],[34,732653,732655],{"className":732654,"style":39021},[31805],[34,732656],{},[34,732658],{"className":732659,"style":31846},[31845],[34,732661,202],{"className":732662},[31850],[34,732664],{"className":732665,"style":31846},[31845],[34,732667,732669,732672,732675,732678,732746,732749,732752],{"className":732668},[31676],[34,732670],{"className":732671,"style":46727},[31680],[34,732673,32226],{"className":732674},[31685],[34,732676,456],{"className":732677},[31782],[34,732679,732681,732684,732743],{"className":732680},[31685],[34,732682],{"className":732683},[31782,41488],[34,732685,732687],{"className":732686},[41453],[34,732688,732690,732735],{"className":732689},[31796,31797],[34,732691,732693,732732],{"className":732692},[31801],[34,732694,732696,732710,732718],{"className":732695,"style":46746},[31805],[34,732697,732698,732701],{"style":41504},[34,732699],{"className":732700,"style":41508},[31813],[34,732702,732704],{"className":732703},[31818,31819,31820,31821],[34,732705,732707],{"className":732706},[31685,31821],[34,732708,33249],{"className":732709},[31685,31686,31821],[34,732711,732712,732715],{"style":41520},[34,732713],{"className":732714,"style":41508},[31813],[34,732716],{"className":732717,"style":41528},[41527],[34,732719,732720,732723],{"style":46771},[34,732721],{"className":732722,"style":41508},[31813],[34,732724,732726],{"className":732725},[31818,31819,31820,31821],[34,732727,732729],{"className":732728},[31685,31821],[34,732730,36656],{"className":732731},[31685,31821],[34,732733,31829],{"className":732734},[31828],[34,732736,732738],{"className":732737},[31801],[34,732739,732741],{"className":732740,"style":41619},[31805],[34,732742],{},[34,732744],{"className":732745},[31690,41488],[34,732747],{"className":732748,"style":32604},[31845],[34,732750,223],{"className":732751},[32024],[34,732753],{"className":732754,"style":32604},[31845],[34,732756,732758,732761,732764,732767,732770,732773,732776,732779,732782,732785],{"className":732757},[31676],[34,732759],{"className":732760,"style":31774},[31680],[34,732762,729302],{"className":732763},[31685],[34,732765,2557],{"className":732766},[31690],[34,732768],{"className":732769,"style":32123},[31845],[34,732771,563286],{"className":732772},[32767],[34,732774],{"className":732775,"style":32123},[31845],[34,732777,33249],{"className":732778},[31685,31686],[34,732780],{"className":732781,"style":31846},[31845],[34,732783,202],{"className":732784},[31850],[34,732786],{"className":732787,"style":31846},[31845],[34,732789,732791,732794,732797,732865,732868,732871,732874,732877,732880,732883],{"className":732790},[31676],[34,732792],{"className":732793,"style":46727},[31680],[34,732795,32226],{"className":732796},[31685],[34,732798,732800,732803,732862],{"className":732799},[31685],[34,732801],{"className":732802},[31782,41488],[34,732804,732806],{"className":732805},[41453],[34,732807,732809,732854],{"className":732808},[31796,31797],[34,732810,732812,732851],{"className":732811},[31801],[34,732813,732815,732829,732837],{"className":732814,"style":46746},[31805],[34,732816,732817,732820],{"style":41504},[34,732818],{"className":732819,"style":41508},[31813],[34,732821,732823],{"className":732822},[31818,31819,31820,31821],[34,732824,732826],{"className":732825},[31685,31821],[34,732827,33249],{"className":732828},[31685,31686,31821],[34,732830,732831,732834],{"style":41520},[34,732832],{"className":732833,"style":41508},[31813],[34,732835],{"className":732836,"style":41528},[41527],[34,732838,732839,732842],{"style":46771},[34,732840],{"className":732841,"style":41508},[31813],[34,732843,732845],{"className":732844},[31818,31819,31820,31821],[34,732846,732848],{"className":732847},[31685,31821],[34,732849,36656],{"className":732850},[31685,31821],[34,732852,31829],{"className":732853},[31828],[34,732855,732857],{"className":732856},[31801],[34,732858,732860],{"className":732859,"style":41619},[31805],[34,732861],{},[34,732863],{"className":732864},[31690,41488],[34,732866],{"className":732867,"style":32123},[31845],[34,732869,563286],{"className":732870},[32767],[34,732872],{"className":732873,"style":32123},[31845],[34,732875,33249],{"className":732876},[31685,31686],[34,732878],{"className":732879,"style":32604},[31845],[34,732881,32226],{"className":732882},[32024],[34,732884],{"className":732885,"style":32604},[31845],[34,732887,732889,732892,732895,732898,732901,732904],{"className":732888},[31676],[34,732890],{"className":732891,"style":31681},[31680],[34,732893,729302],{"className":732894},[31685],[34,732896],{"className":732897,"style":32123},[31845],[34,732899,563286],{"className":732900},[32767],[34,732902],{"className":732903,"style":32123},[31845],[34,732905,33249],{"className":732906},[31685,31686],[15,732908,732909],{},[2627,732910,732911],{},"Step 4: 积分",[105,732913,732914,733053],{},[108,732915,732916],{},[34,732917,732919,732953],{"className":732918},[31642],[34,732920,732922],{"className":732921},[31646],[31648,732923,732924],{"xmlns":31650},[31652,732925,732926,732950],{},[31655,732927,732928,732934,732936,732938,732940,732942,732944,732946,732948],{},[31743,732929,732930,732932],{},[31658,732931,31965],{},[31749,732933,768],{},[31662,732935,202],{},[31749,732937,36656],{},[31658,732939,563286],{},[31662,732941,32682],{},[31658,732943,33249],{},[31662,732945,223],{},[31749,732947,729302],{},[31658,732949,33249],{},[31665,732951,732952],{"encoding":31667},"c_2 = 6\\ln t + 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= -3\\ln^2 t - 34(t\\ln t - t)",[34,733115,733117,733172,733232,733268],{"className":733116,"ariaHidden":758},[31672],[34,733118,733120,733123,733163,733166,733169],{"className":733119},[31676],[34,733121],{"className":733122,"style":31860},[31680],[34,733124,733126,733129],{"className":733125},[31685],[34,733127,31965],{"className":733128},[31685,31686],[34,733130,733132],{"className":733131},[31792],[34,733133,733135,733155],{"className":733134},[31796,31797],[34,733136,733138,733152],{"className":733137},[31801],[34,733139,733141],{"className":733140,"style":31806},[31805],[34,733142,733143,733146],{"style":31809},[34,733144],{"className":733145,"style":31814},[31813],[34,733147,733149],{"className":733148},[31818,31819,31820,31821],[34,733150,640],{"className":733151},[31685,31821],[34,733153,31829],{"className":733154},[31828],[34,733156,733158],{"className":733157},[31801],[34,733159,733161],{"className":733160,"style":31836},[31805],[34,733162],{},[34,733164],{"className":733165,"style":31846},[31845],[34,733167,202],{"className":733168},[31850],[34,733170],{"className":733171,"style":31846},[31845],[34,733173,733175,733179,733182,733185,733188,733217,733220,733223,733226,733229],{"className":733174},[31676],[34,733176],{"className":733177,"style":733178},[31680],"height:0.9818em;vertical-align:-0.0833em;",[34,733180,32226],{"className":733181},[31685],[34,733183,786],{"className":733184},[31685],[34,733186],{"className":733187,"style":32123},[31845],[34,733189,733191,733194],{"className":733190},[32767],[34,733192,563286],{"className":733193},[32767],[34,733195,733197],{"className":733196},[31792],[34,733198,733200],{"className":733199},[31796],[34,733201,733203],{"className":733202},[31801],[34,733204,733206],{"className":733205,"style":563430},[31805],[34,733207,733208,733211],{"style":563433},[34,733209],{"className":733210,"style":31814},[31813],[34,733212,733214],{"className":733213},[31818,31819,31820,31821],[34,733215,768],{"className":733216},[31685,31821],[34,733218],{"className":733219,"style":32123},[31845],[34,733221,33249],{"className":733222},[31685,31686],[34,733224],{"className":733225,"style":32604},[31845],[34,733227,32226],{"className":733228},[32024],[34,733230],{"className":733231,"style":32604},[31845],[34,733233,733235,733238,733241,733244,733247,733250,733253,733256,733259,733262,733265],{"className":733234},[31676],[34,733236],{"className":733237,"style":31774},[31680],[34,733239,729302],{"className":733240},[31685],[34,733242,456],{"className":733243},[31782],[34,733245,33249],{"className":733246},[31685,31686],[34,733248],{"className":733249,"style":32123},[31845],[34,733251,563286],{"className":733252},[32767],[34,733254],{"className":733255,"style":32123},[31845],[34,733257,33249],{"className":733258},[31685,31686],[34,733260],{"className":733261,"style":32604},[31845],[34,733263,32226],{"className":733264},[32024],[34,733266],{"className":733267,"style":32604},[31845],[34,733269,733271,733274,733277],{"className":733270},[31676],[34,733272],{"className":733273,"style":31774},[31680],[34,733275,33249],{"className":733276},[31685,31686],[34,733278,2557],{"className":733279},[31690]," （分部积分）",[15,733282,733283,221775,733286,706771,733410,733593],{},[2627,733284,733285],{},"Step 5: 组合通解",[34,733287,733289,733315],{"className":733288},[31642],[34,733290,733292],{"className":733291},[31646],[31648,733293,733294],{"xmlns":31650},[31652,733295,733296,733312],{},[31655,733297,733298,733304,733306],{},[31743,733299,733300,733302],{},[31658,733301,31965],{},[31749,733303,640],{},[31662,733305,31960],{"separator":758},[31743,733307,733308,733310],{},[31658,733309,31965],{},[31749,733311,768],{},[31665,733313,733314],{"encoding":31667},"c_1, c_2",[34,733316,733318],{"className":733317,"ariaHidden":758},[31672],[34,733319,733321,733324,733364,733367,733370],{"className":733320},[31676],[34,733322],{"className":733323,"style":38774},[31680],[34,733325,733327,733330],{"className":733326},[31685],[34,733328,31965],{"className":733329},[31685,31686],[34,733331,733333],{"className":733332},[31792],[34,733334,733336,733356],{"className":733335},[31796,31797],[34,733337,733339,733353],{"className":733338},[31801],[34,733340,733342],{"className":733341,"style":31806},[31805],[34,733343,733344,733347],{"style":31809},[34,733345],{"className":733346,"style":31814},[31813],[34,733348,733350],{"className":733349},[31818,31819,31820,31821],[34,733351,640],{"className":733352},[31685,31821],[34,733354,31829],{"className":733355},[31828],[34,733357,733359],{"className":733358},[31801],[34,733360,733362],{"className":733361,"style":31836},[31805],[34,733363],{},[34,733365,31960],{"className":733366},[32119],[34,733368],{"className":733369,"style":32123},[31845],[34,733371,733373,733376],{"className":733372},[31685],[34,733374,31965],{"className":733375},[31685,31686],[34,733377,733379],{"className":733378},[31792],[34,733380,733382,733402],{"className":733381},[31796,31797],[34,733383,733385,733399],{"className":733384},[31801],[34,733386,733388],{"className":733387,"style":31806},[31805],[34,733389,733390,733393],{"style":31809},[34,733391],{"className":733392,"style":31814},[31813],[34,733394,733396],{"className":733395},[31818,31819,31820,31821],[34,733397,768],{"className":733398},[31685,31821],[34,733400,31829],{"className":733401},[31828],[34,733403,733405],{"className":733404},[31801],[34,733406,733408],{"className":733407,"style":31836},[31805],[34,733409],{},[34,733411,733413,733453],{"className":733412},[31642],[34,733414,733416],{"className":733415},[31646],[31648,733417,733418],{"xmlns":31650},[31652,733419,733420,733450],{},[31655,733421,733422,733424,733426,733432,733434,733436,733442,733444,733446,733448],{},[31658,733423,32936],{},[31662,733425,202],{},[31743,733427,733428,733430],{},[31658,733429,31965],{},[31749,733431,640],{},[31658,733433,33249],{},[31662,733435,223],{},[31743,733437,733438,733440],{},[31658,733439,31965],{},[31749,733441,768],{},[31658,733443,33249],{},[31658,733445,563286],{},[31662,733447,32682],{},[31658,733449,33249],{},[31665,733451,733452],{"encoding":31667},"x = c_1 t + c_2 t\\ln t",[34,733454,733456,733474,733532],{"className":733455,"ariaHidden":758},[31672],[34,733457,733459,733462,733465,733468,733471],{"className":733458},[31676],[34,733460],{"className":733461,"style":32527},[31680],[34,733463,32936],{"className":733464},[31685,31686],[34,733466],{"className":733467,"style":31846},[31845],[34,733469,202],{"className":733470},[31850],[34,733472],{"className":733473,"style":31846},[31845],[34,733475,733477,733480,733520,733523,733526,733529],{"className":733476},[31676],[34,733478],{"className":733479,"style":83872},[31680],[34,733481,733483,733486],{"className":733482},[31685],[34,733484,31965],{"className":733485},[31685,31686],[34,733487,733489],{"className":733488},[31792],[34,733490,733492,733512],{"className":733491},[31796,31797],[34,733493,733495,733509],{"className":733494},[31801],[34,733496,733498],{"className":733497,"style":31806},[31805],[34,733499,733500,733503],{"style":31809},[34,733501],{"className":733502,"style":31814},[31813],[34,733504,733506],{"className":733505},[31818,31819,31820,31821],[34,733507,640],{"className":733508},[31685,31821],[34,733510,31829],{"className":733511},[31828],[34,733513,733515],{"className":733514},[31801],[34,733516,733518],{"className":733517,"style":31836},[31805],[34,733519],{},[34,733521,33249],{"className":733522},[31685,31686],[34,733524],{"className":733525,"style":32604},[31845],[34,733527,223],{"className":733528},[32024],[34,733530],{"className":733531,"style":32604},[31845],[34,733533,733535,733538,733578,733581,733584,733587,733590],{"className":733534},[31676],[34,733536],{"className":733537,"style":33814},[31680],[34,733539,733541,733544],{"className":733540},[31685],[34,733542,31965],{"className":733543},[31685,31686],[34,733545,733547],{"className":733546},[31792],[34,733548,733550,733570],{"className":733549},[31796,31797],[34,733551,733553,733567],{"className":733552},[31801],[34,733554,733556],{"className":733555,"style":31806},[31805],[34,733557,733558,733561],{"style":31809},[34,733559],{"className":733560,"style":31814},[31813],[34,733562,733564],{"className":733563},[31818,31819,31820,31821],[34,733565,768],{"className":733566},[31685,31821],[34,733568,31829],{"className":733569},[31828],[34,733571,733573],{"className":733572},[31801],[34,733574,733576],{"className":733575,"style":31836},[31805],[34,733577],{},[34,733579,33249],{"className":733580},[31685,31686],[34,733582],{"className":733583,"style":32123},[31845],[34,733585,563286],{"className":733586},[32767],[34,733588],{"className":733589,"style":32123},[31845],[34,733591,33249],{"className":733592},[31685,31686]," 整理即可。",[32188,733595],{},[253,733597,733599],{"id":733598},"_4-归一化基本解组标准基本解组","4. 归一化基本解组（标准基本解组）",[15,733601,733602,733605,733606,314945,733718,733768,733769,258,733854,733939],{},[2627,733603,733604],{},"题目意思：","\n现在的基本解组 ",[34,733607,733609,733639],{"className":733608},[31642],[34,733610,733612],{"className":733611},[31646],[31648,733613,733614],{"xmlns":31650},[31652,733615,733616,733636],{},[31655,733617,733618,733624,733626],{},[37624,733619,733620,733622],{},[31658,733621,33244],{},[31658,733623,33249],{},[31662,733625,31960],{"separator":758},[37624,733627,733628,733630],{},[31658,733629,33244],{},[31655,733631,733632,733634],{},[31662,733633,32226],{},[31658,733635,33249],{},[31665,733637,733638],{"encoding":31667},"e^t, e^{-t}",[34,733640,733642],{"className":733641,"ariaHidden":758},[31672],[34,733643,733645,733648,733677,733680,733683],{"className":733644},[31676],[34,733646],{"className":733647,"style":719358},[31680],[34,733649,733651,733654],{"className":733650},[31685],[34,733652,33244],{"className":733653},[31685,31686],[34,733655,733657],{"className":733656},[31792],[34,733658,733660],{"className":733659},[31796],[34,733661,733663],{"className":733662},[31801],[34,733664,733666],{"className":733665,"style":611758},[31805],[34,733667,733668,733671],{"style":37777},[34,733669],{"className":733670,"style":31814},[31813],[34,733672,733674],{"className":733673},[31818,31819,31820,31821],[34,733675,33249],{"className":733676},[31685,31686,31821],[34,733678,31960],{"className":733679},[32119],[34,733681],{"className":733682,"style":32123},[31845],[34,733684,733686,733689],{"className":733685},[31685],[34,733687,33244],{"className":733688},[31685,31686],[34,733690,733692],{"className":733691},[31792],[34,733693,733695],{"className":733694},[31796],[34,733696,733698],{"className":733697},[31801],[34,733699,733701],{"className":733700,"style":611758},[31805],[34,733702,733703,733706],{"style":37777},[34,733704],{"className":733705,"style":31814},[31813],[34,733707,733709],{"className":733708},[31818,31819,31820,31821],[34,733710,733712,733715],{"className":733711},[31685,31821],[34,733713,32226],{"className":733714},[31685,31821],[34,733716,33249],{"className":733717},[31685,31686,31821],[34,733719,733721,733738],{"className":733720},[31642],[34,733722,733724],{"className":733723},[31646],[31648,733725,733726],{"xmlns":31650},[31652,733727,733728,733736],{},[31655,733729,733730,733732,733734],{},[31658,733731,33249],{},[31662,733733,202],{},[31749,733735,459],{},[31665,733737,570178],{"encoding":31667},[34,733739,733741,733759],{"className":733740,"ariaHidden":758},[31672],[34,733742,733744,733747,733750,733753,733756],{"className":733743},[31676],[34,733745],{"className":733746,"style":33280},[31680],[34,733748,33249],{"className":733749},[31685,31686],[34,733751],{"className":733752,"style":31846},[31845],[34,733754,202],{"className":733755},[31850],[34,733757],{"className":733758,"style":31846},[31845],[34,733760,733762,733765],{"className":733761},[31676],[34,733763],{"className":733764,"style":32617},[31680],[34,733766,459],{"className":733767},[31685]," 时的数值乱七八糟的。\n我们想要两个新的“完美”解 ",[34,733770,733772,733796],{"className":733771},[31642],[34,733773,733775],{"className":733774},[31646],[31648,733776,733777],{"xmlns":31650},[31652,733778,733779,733793],{},[31655,733780,733781,733787,733789,733791],{},[31743,733782,733783,733785],{},[31658,733784,329201],{},[31749,733786,640],{},[31662,733788,456],{"stretchy":702},[31658,733790,33249],{},[31662,733792,2557],{"stretchy":702},[31665,733794,733795],{"encoding":31667},"\\phi_1(t)",[34,733797,733799],{"className":733798,"ariaHidden":758},[31672],[34,733800,733802,733805,733845,733848,733851],{"className":733801},[31676],[34,733803],{"className":733804,"style":31774},[31680],[34,733806,733808,733811],{"className":733807},[31685],[34,733809,329201],{"className":733810},[31685,31686],[34,733812,733814],{"className":733813},[31792],[34,733815,733817,733837],{"className":733816},[31796,31797],[34,733818,733820,733834],{"className":733819},[31801],[34,733821,733823],{"className":733822,"style":31806},[31805],[34,733824,733825,733828],{"style":31809},[34,733826],{"className":733827,"style":31814},[31813],[34,733829,733831],{"className":733830},[31818,31819,31820,31821],[34,733832,640],{"className":733833},[31685,31821],[34,733835,31829],{"className":733836},[31828],[34,733838,733840],{"className":733839},[31801],[34,733841,733843],{"className":733842,"style":31836},[31805],[34,733844],{},[34,733846,456],{"className":733847},[31782],[34,733849,33249],{"className":733850},[31685,31686],[34,733852,2557],{"className":733853},[31690],[34,733855,733857,733881],{"className":733856},[31642],[34,733858,733860],{"className":733859},[31646],[31648,733861,733862],{"xmlns":31650},[31652,733863,733864,733878],{},[31655,733865,733866,733872,733874,733876],{},[31743,733867,733868,733870],{},[31658,733869,329201],{},[31749,733871,768],{},[31662,733873,456],{"stretchy":702},[31658,733875,33249],{},[31662,733877,2557],{"stretchy":702},[31665,733879,733880],{"encoding":31667},"\\phi_2(t)",[34,733882,733884],{"className":733883,"ariaHidden":758},[31672],[34,733885,733887,733890,733930,733933,733936],{"className":733886},[31676],[34,733888],{"className":733889,"style":31774},[31680],[34,733891,733893,733896],{"className":733892},[31685],[34,733894,329201],{"className":733895},[31685,31686],[34,733897,733899],{"className":733898},[31792],[34,733900,733902,733922],{"className":733901},[31796,31797],[34,733903,733905,733919],{"className":733904},[31801],[34,733906,733908],{"className":733907,"style":31806},[31805],[34,733909,733910,733913],{"style":31809},[34,733911],{"className":733912,"style":31814},[31813],[34,733914,733916],{"className":733915},[31818,31819,31820,31821],[34,733917,768],{"className":733918},[31685,31821],[34,733920,31829],{"className":733921},[31828],[34,733923,733925],{"className":733924},[31801],[34,733926,733928],{"className":733927,"style":31836},[31805],[34,733929],{},[34,733931,456],{"className":733932},[31782],[34,733934,33249],{"className":733935},[31685,31686],[34,733937,2557],{"className":733938},[31690],"，它们满足：",[105,733941,733942,734159],{},[108,733943,733944,734158],{},[34,733945,733947,733995],{"className":733946},[31642],[34,733948,733950],{"className":733949},[31646],[31648,733951,733952],{"xmlns":31650},[31652,733953,733954,733992],{},[31655,733955,733956,733962,733964,733966,733968,733970,733972,733974,733982,733984,733986,733988,733990],{},[31743,733957,733958,733960],{},[31658,733959,329201],{},[31749,733961,640],{},[31662,733963,456],{"stretchy":702},[31749,733965,459],{},[31662,733967,2557],{"stretchy":702},[31662,733969,202],{},[31749,733971,640],{},[31662,733973,31960],{"separator":758},[38846,733975,733976,733978,733980],{},[31658,733977,329201],{},[31749,733979,640],{},[31662,733981,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,733983,456],{"stretchy":702},[31749,733985,459],{},[31662,733987,2557],{"stretchy":702},[31662,733989,202],{},[31749,733991,459],{},[31665,733993,733994],{"encoding":31667},"\\phi_1(0)=1, \\phi_1'(0)=0",[34,733996,733998,734062,734149],{"className":733997,"ariaHidden":758},[31672],[34,733999,734001,734004,734044,734047,734050,734053,734056,734059],{"className":734000},[31676],[34,734002],{"className":734003,"style":31774},[31680],[34,734005,734007,734010],{"className":734006},[31685],[34,734008,329201],{"className":734009},[31685,31686],[34,734011,734013],{"className":734012},[31792],[34,734014,734016,734036],{"className":734015},[31796,31797],[34,734017,734019,734033],{"className":734018},[31801],[34,734020,734022],{"className":734021,"style":31806},[31805],[34,734023,734024,734027],{"style":31809},[34,734025],{"className":734026,"style":31814},[31813],[34,734028,734030],{"className":734029},[31818,31819,31820,31821],[34,734031,640],{"className":734032},[31685,31821],[34,734034,31829],{"className":734035},[31828],[34,734037,734039],{"className":734038},[31801],[34,734040,734042],{"className":734041,"style":31836},[31805],[34,734043],{},[34,734045,456],{"className":734046},[31782],[34,734048,459],{"className":734049},[31685],[34,734051,2557],{"className":734052},[31690],[34,734054],{"className":734055,"style":31846},[31845],[34,734057,202],{"className":734058},[31850],[34,734060],{"className":734061,"style":31846},[31845],[34,734063,734065,734068,734071,734074,734077,734131,734134,734137,734140,734143,734146],{"className":734064},[31676],[34,734066],{"className":734067,"style":570646},[31680],[34,734069,640],{"className":734070},[31685],[34,734072,31960],{"className":734073},[32119],[34,734075],{"className":734076,"style":32123},[31845],[34,734078,734080,734083],{"className":734079},[31685],[34,734081,329201],{"className":734082},[31685,31686],[34,734084,734086],{"className":734085},[31792],[34,734087,734089,734123],{"className":734088},[31796,31797],[34,734090,734092,734120],{"className":734091},[31801],[34,734093,734095,734106],{"className":734094,"style":132027},[31805],[34,734096,734097,734100],{"style":316930},[34,734098],{"className":734099,"style":31814},[31813],[34,734101,734103],{"className":734102},[31818,31819,31820,31821],[34,734104,640],{"className":734105},[31685,31821],[34,734107,734108,734111],{"style":37777},[34,734109],{"className":734110,"style":31814},[31813],[34,734112,734114],{"className":734113},[31818,31819,31820,31821],[34,734115,734117],{"className":734116},[31685,31821],[34,734118,116495],{"className":734119},[31685,31821],[34,734121,31829],{"className":734122},[31828],[34,734124,734126],{"className":734125},[31801],[34,734127,734129],{"className":734128,"style":39021},[31805],[34,734130],{},[34,734132,456],{"className":734133},[31782],[34,734135,459],{"className":734136},[31685],[34,734138,2557],{"className":734139},[31690],[34,734141],{"className":734142,"style":31846},[31845],[34,734144,202],{"className":734145},[31850],[34,734147],{"className":734148,"style":31846},[31845],[34,734150,734152,734155],{"className":734151},[31676],[34,734153],{"className":734154,"style":32617},[31680],[34,734156,459],{"className":734157},[31685]," （这叫“位置解”，只管初始位置）",[108,734160,734161,734375],{},[34,734162,734164,734212],{"className":734163},[31642],[34,734165,734167],{"className":734166},[31646],[31648,734168,734169],{"xmlns":31650},[31652,734170,734171,734209],{},[31655,734172,734173,734179,734181,734183,734185,734187,734189,734191,734199,734201,734203,734205,734207],{},[31743,734174,734175,734177],{},[31658,734176,329201],{},[31749,734178,768],{},[31662,734180,456],{"stretchy":702},[31749,734182,459],{},[31662,734184,2557],{"stretchy":702},[31662,734186,202],{},[31749,734188,459],{},[31662,734190,31960],{"separator":758},[38846,734192,734193,734195,734197],{},[31658,734194,329201],{},[31749,734196,768],{},[31662,734198,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,734200,456],{"stretchy":702},[31749,734202,459],{},[31662,734204,2557],{"stretchy":702},[31662,734206,202],{},[31749,734208,640],{},[31665,734210,734211],{"encoding":31667},"\\phi_2(0)=0, \\phi_2'(0)=1",[34,734213,734215,734279,734366],{"className":734214,"ariaHidden":758},[31672],[34,734216,734218,734221,734261,734264,734267,734270,734273,734276],{"className":734217},[31676],[34,734219],{"className":734220,"style":31774},[31680],[34,734222,734224,734227],{"className":734223},[31685],[34,734225,329201],{"className":734226},[31685,31686],[34,734228,734230],{"className":734229},[31792],[34,734231,734233,734253],{"className":734232},[31796,31797],[34,734234,734236,734250],{"className":734235},[31801],[34,734237,734239],{"className":734238,"style":31806},[31805],[34,734240,734241,734244],{"style":31809},[34,734242],{"className":734243,"style":31814},[31813],[34,734245,734247],{"className":734246},[31818,31819,31820,31821],[34,734248,768],{"className":734249},[31685,31821],[34,734251,31829],{"className":734252},[31828],[34,734254,734256],{"className":734255},[31801],[34,734257,734259],{"className":734258,"style":31836},[31805],[34,734260],{},[34,734262,456],{"className":734263},[31782],[34,734265,459],{"className":734266},[31685],[34,734268,2557],{"className":734269},[31690],[34,734271],{"className":734272,"style":31846},[31845],[34,734274,202],{"className":734275},[31850],[34,734277],{"className":734278,"style":31846},[31845],[34,734280,734282,734285,734288,734291,734294,734348,734351,734354,734357,734360,734363],{"className":734281},[31676],[34,734283],{"className":734284,"style":570646},[31680],[34,734286,459],{"className":734287},[31685],[34,734289,31960],{"className":734290},[32119],[34,734292],{"className":734293,"style":32123},[31845],[34,734295,734297,734300],{"className":734296},[31685],[34,734298,329201],{"className":734299},[31685,31686],[34,734301,734303],{"className":734302},[31792],[34,734304,734306,734340],{"className":734305},[31796,31797],[34,734307,734309,734337],{"className":734308},[31801],[34,734310,734312,734323],{"className":734311,"style":132027},[31805],[34,734313,734314,734317],{"style":316930},[34,734315],{"className":734316,"style":31814},[31813],[34,734318,734320],{"className":734319},[31818,31819,31820,31821],[34,734321,768],{"className":734322},[31685,31821],[34,734324,734325,734328],{"style":37777},[34,734326],{"className":734327,"style":31814},[31813],[34,734329,734331],{"className":734330},[31818,31819,31820,31821],[34,734332,734334],{"className":734333},[31685,31821],[34,734335,116495],{"className":734336},[31685,31821],[34,734338,31829],{"className":734339},[31828],[34,734341,734343],{"className":734342},[31801],[34,734344,734346],{"className":734345,"style":39021},[31805],[34,734347],{},[34,734349,456],{"className":734350},[31782],[34,734352,459],{"className":734353},[31685],[34,734355,2557],{"className":734356},[31690],[34,734358],{"className":734359,"style":31846},[31845],[34,734361,202],{"className":734362},[31850],[34,734364],{"className":734365,"style":31846},[31845],[34,734367,734369,734372],{"className":734368},[31676],[34,734370],{"className":734371,"style":32617},[31680],[34,734373,640],{"className":734374},[31685]," （这叫“速度解”，只管初始速度）",[15,734377,734378],{},[2627,734379,734380],{},"解法：",[15,734382,734383,362914,734456,734747,734748,735135,735136,355408,735442,735737],{},[2627,734384,734385,734386],{},"第一步：构造 ",[34,734387,734389,734407],{"className":734388},[31642],[34,734390,734392],{"className":734391},[31646],[31648,734393,734394],{"xmlns":31650},[31652,734395,734396,734404],{},[31655,734397,734398],{},[31743,734399,734400,734402],{},[31658,734401,329201],{},[31749,734403,640],{},[31665,734405,734406],{"encoding":31667},"\\phi_1",[34,734408,734410],{"className":734409,"ariaHidden":758},[31672],[34,734411,734413,734416],{"className":734412},[31676],[34,734414],{"className":734415,"style":43217},[31680],[34,734417,734419,734422],{"className":734418},[31685],[34,734420,329201],{"className":734421},[31685,31686],[34,734423,734425],{"className":734424},[31792],[34,734426,734428,734448],{"className":734427},[31796,31797],[34,734429,734431,734445],{"className":734430},[31801],[34,734432,734434],{"className":734433,"style":31806},[31805],[34,734435,734436,734439],{"style":31809},[34,734437],{"className":734438,"style":31814},[31813],[34,734440,734442],{"className":734441},[31818,31819,31820,31821],[34,734443,640],{"className":734444},[31685,31821],[34,734446,31829],{"className":734447},[31828],[34,734449,734451],{"className":734450},[31801],[34,734452,734454],{"className":734453,"style":31836},[31805],[34,734455],{},[34,734457,734459,734515],{"className":734458},[31642],[34,734460,734462],{"className":734461},[31646],[31648,734463,734464],{"xmlns":31650},[31652,734465,734466,734512],{},[31655,734467,734468,734474,734476,734478,734480,734482,734488,734494,734496,734502],{},[31743,734469,734470,734472],{},[31658,734471,329201],{},[31749,734473,640],{},[31662,734475,456],{"stretchy":702},[31658,734477,33249],{},[31662,734479,2557],{"stretchy":702},[31662,734481,202],{},[31743,734483,734484,734486],{},[31658,734485,31965],{},[31749,734487,640],{},[37624,734489,734490,734492],{},[31658,734491,33244],{},[31658,734493,33249],{},[31662,734495,223],{},[31743,734497,734498,734500],{},[31658,734499,31965],{},[31749,734501,768],{},[37624,734503,734504,734506],{},[31658,734505,33244],{},[31655,734507,734508,734510],{},[31662,734509,32226],{},[31658,734511,33249],{},[31665,734513,734514],{"encoding":31667},"\\phi_1(t) = c_1 e^t + c_2 e^{-t}",[34,734516,734518,734582,734666],{"className":734517,"ariaHidden":758},[31672],[34,734519,734521,734524,734564,734567,734570,734573,734576,734579],{"className":734520},[31676],[34,734522],{"className":734523,"style":31774},[31680],[34,734525,734527,734530],{"className":734526},[31685],[34,734528,329201],{"className":734529},[31685,31686],[34,734531,734533],{"className":734532},[31792],[34,734534,734536,734556],{"className":734535},[31796,31797],[34,734537,734539,734553],{"className":734538},[31801],[34,734540,734542],{"className":734541,"style":31806},[31805],[34,734543,734544,734547],{"style":31809},[34,734545],{"className":734546,"style":31814},[31813],[34,734548,734550],{"className":734549},[31818,31819,31820,31821],[34,734551,640],{"className":734552},[31685,31821],[34,734554,31829],{"className":734555},[31828],[34,734557,734559],{"className":734558},[31801],[34,734560,734562],{"className":734561,"style":31836},[31805],[34,734563],{},[34,734565,456],{"className":734566},[31782],[34,734568,33249],{"className":734569},[31685,31686],[34,734571,2557],{"className":734572},[31690],[34,734574],{"className":734575,"style":31846},[31845],[34,734577,202],{"className":734578},[31850],[34,734580],{"className":734581,"style":31846},[31845],[34,734583,734585,734588,734628,734657,734660,734663],{"className":734584},[31676],[34,734586],{"className":734587,"style":723843},[31680],[34,734589,734591,734594],{"className":734590},[31685],[34,734592,31965],{"className":734593},[31685,31686],[34,734595,734597],{"className":734596},[31792],[34,734598,734600,734620],{"className":734599},[31796,31797],[34,734601,734603,734617],{"className":734602},[31801],[34,734604,734606],{"className":734605,"style":31806},[31805],[34,734607,734608,734611],{"style":31809},[34,734609],{"className":734610,"style":31814},[31813],[34,734612,734614],{"className":734613},[31818,31819,31820,31821],[34,734615,640],{"className":734616},[31685,31821],[34,734618,31829],{"className":734619},[31828],[34,734621,734623],{"className":734622},[31801],[34,734624,734626],{"className":734625,"style":31836},[31805],[34,734627],{},[34,734629,734631,734634],{"className":734630},[31685],[34,734632,33244],{"className":734633},[31685,31686],[34,734635,734637],{"className":734636},[31792],[34,734638,734640],{"className":734639},[31796],[34,734641,734643],{"className":734642},[31801],[34,734644,734646],{"className":734645,"style":611758},[31805],[34,734647,734648,734651],{"style":37777},[34,734649],{"className":734650,"style":31814},[31813],[34,734652,734654],{"className":734653},[31818,31819,31820,31821],[34,734655,33249],{"className":734656},[31685,31686,31821],[34,734658],{"className":734659,"style":32604},[31845],[34,734661,223],{"className":734662},[32024],[34,734664],{"className":734665,"style":32604},[31845],[34,734667,734669,734672,734712],{"className":734668},[31676],[34,734670],{"className":734671,"style":723843},[31680],[34,734673,734675,734678],{"className":734674},[31685],[34,734676,31965],{"className":734677},[31685,31686],[34,734679,734681],{"className":734680},[31792],[34,734682,734684,734704],{"className":734683},[31796,31797],[34,734685,734687,734701],{"className":734686},[31801],[34,734688,734690],{"className":734689,"style":31806},[31805],[34,734691,734692,734695],{"style":31809},[34,734693],{"className":734694,"style":31814},[31813],[34,734696,734698],{"className":734697},[31818,31819,31820,31821],[34,734699,768],{"className":734700},[31685,31821],[34,734702,31829],{"className":734703},[31828],[34,734705,734707],{"className":734706},[31801],[34,734708,734710],{"className":734709,"style":31836},[31805],[34,734711],{},[34,734713,734715,734718],{"className":734714},[31685],[34,734716,33244],{"className":734717},[31685,31686],[34,734719,734721],{"className":734720},[31792],[34,734722,734724],{"className":734723},[31796],[34,734725,734727],{"className":734726},[31801],[34,734728,734730],{"className":734729,"style":611758},[31805],[34,734731,734732,734735],{"style":37777},[34,734733],{"className":734734,"style":31814},[31813],[34,734736,734738],{"className":734737},[31818,31819,31820,31821],[34,734739,734741,734744],{"className":734740},[31685,31821],[34,734742,32226],{"className":734743},[31685,31821],[34,734745,33249],{"className":734746},[31685,31686,31821],"。\n代入初值条件：\n[\n",[34,734749,734751,734837],{"className":734750},[31642],[34,734752,734754],{"className":734753},[31646],[31648,734755,734756],{"xmlns":31650},[31652,734757,734758,734834],{},[31655,734759,734760,734762],{},[31662,734761,32685],{"fence":758},[71078,734763,734764,734799],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,734765,734766],{},[71087,734767,734768],{},[71090,734769,734770],{"scriptlevel":459,"displaystyle":702},[31655,734771,734772,734778,734780,734786,734788,734790,734792,734794,734797],{},[31743,734773,734774,734776],{},[31658,734775,31965],{},[31749,734777,640],{},[31662,734779,223],{},[31743,734781,734782,734784],{},[31658,734783,31965],{},[31749,734785,768],{},[31662,734787,202],{},[31749,734789,640],{},[31845,734791],{"width":71082},[31662,734793,456],{"stretchy":702},[39361,734795,734796],{},"函数值",[31662,734798,2557],{"stretchy":702},[71084,734800,734801],{},[71087,734802,734803],{},[71090,734804,734805],{"scriptlevel":459,"displaystyle":702},[31655,734806,734807,734813,734815,734821,734823,734825,734827,734829,734832],{},[31743,734808,734809,734811],{},[31658,734810,31965],{},[31749,734812,640],{},[31662,734814,32226],{},[31743,734816,734817,734819],{},[31658,734818,31965],{},[31749,734820,768],{},[31662,734822,202],{},[31749,734824,459],{},[31845,734826],{"width":71082},[31662,734828,456],{"stretchy":702},[39361,734830,734831],{},"导数值",[31662,734833,2557],{"stretchy":702},[31665,734835,734836],{"encoding":31667},"\\begin{cases}\nc_1 + c_2 = 1 \\quad (\\text{函数值}) \\\\\nc_1 - c_2 = 0 \\quad (\\text{导数值})\n\\end{cases}",[34,734838,734840],{"className":734839,"ariaHidden":758},[31672],[34,734841,734843,734846],{"className":734842},[31676],[34,734844],{"className":734845,"style":75797},[31680],[34,734847,734849,734855,735132],{"className":734848},[34006],[34,734850,734852],{"className":734851,"style":75811},[31782,75810],[34,734853,32685],{"className":734854},[71281,75815],[34,734856,734858],{"className":734857},[31685],[34,734859,734861],{"className":734860},[71078],[34,734862,734864],{"className":734863},[71384],[34,734865,734867,735124],{"className":734866},[31796,31797],[34,734868,734870,735121],{"className":734869},[31801],[34,734871,734873,734997],{"className":734872,"style":75834},[31805],[34,734874,734875,734878],{"style":75837},[34,734876],{"className":734877,"style":71401},[31813],[34,734879,734881,734921,734924,734927,734930,734970,734973,734976,734979,734982,734985,734988,734994],{"className":734880},[31685],[34,734882,734884,734887],{"className":734883},[31685],[34,734885,31965],{"className":734886},[31685,31686],[34,734888,734890],{"className":734889},[31792],[34,734891,734893,734913],{"className":734892},[31796,31797],[34,734894,734896,734910],{"className":734895},[31801],[34,734897,734899],{"className":734898,"style":31806},[31805],[34,734900,734901,734904],{"style":31809},[34,734902],{"className":734903,"style":31814},[31813],[34,734905,734907],{"className":734906},[31818,31819,31820,31821],[34,734908,640],{"className":734909},[31685,31821],[34,734911,31829],{"className":734912},[31828],[34,734914,734916],{"className":734915},[31801],[34,734917,734919],{"className":734918,"style":31836},[31805],[34,734920],{},[34,734922],{"className":734923,"style":32604},[31845],[34,734925,223],{"className":734926},[32024],[34,734928],{"className":734929,"style":32604},[31845],[34,734931,734933,734936],{"className":734932},[31685],[34,734934,31965],{"className":734935},[31685,31686],[34,734937,734939],{"className":734938},[31792],[34,734940,734942,734962],{"className":734941},[31796,31797],[34,734943,734945,734959],{"className":734944},[31801],[34,734946,734948],{"className":734947,"style":31806},[31805],[34,734949,734950,734953],{"style":31809},[34,734951],{"className":734952,"style":31814},[31813],[34,734954,734956],{"className":734955},[31818,31819,31820,31821],[34,734957,768],{"className":734958},[31685,31821],[34,734960,31829],{"className":734961},[31828],[34,734963,734965],{"className":734964},[31801],[34,734966,734968],{"className":734967,"style":31836},[31805],[34,734969],{},[34,734971],{"className":734972,"style":31846},[31845],[34,734974,202],{"className":734975},[31850],[34,734977],{"className":734978,"style":31846},[31845],[34,734980,640],{"className":734981},[31685],[34,734983],{"className":734984,"style":128438},[31845],[34,734986,456],{"className":734987},[31782],[34,734989,734991],{"className":734990},[31685,2638],[34,734992,734796],{"className":734993},[31685,105462],[34,734995,2557],{"className":734996},[31690],[34,734998,734999,735002],{"style":76011},[34,735000],{"className":735001,"style":71401},[31813],[34,735003,735005,735045,735048,735051,735054,735094,735097,735100,735103,735106,735109,735112,735118],{"className":735004},[31685],[34,735006,735008,735011],{"className":735007},[31685],[34,735009,31965],{"className":735010},[31685,31686],[34,735012,735014],{"className":735013},[31792],[34,735015,735017,735037],{"className":735016},[31796,31797],[34,735018,735020,735034],{"className":735019},[31801],[34,735021,735023],{"className":735022,"style":31806},[31805],[34,735024,735025,735028],{"style":31809},[34,735026],{"className":735027,"style":31814},[31813],[34,735029,735031],{"className":735030},[31818,31819,31820,31821],[34,735032,640],{"className":735033},[31685,31821],[34,735035,31829],{"className":735036},[31828],[34,735038,735040],{"className":735039},[31801],[34,735041,735043],{"className":735042,"style":31836},[31805],[34,735044],{},[34,735046],{"className":735047,"style":32604},[31845],[34,735049,32226],{"className":735050},[32024],[34,735052],{"className":735053,"style":32604},[31845],[34,735055,735057,735060],{"className":735056},[31685],[34,735058,31965],{"className":735059},[31685,31686],[34,735061,735063],{"className":735062},[31792],[34,735064,735066,735086],{"className":735065},[31796,31797],[34,735067,735069,735083],{"className":735068},[31801],[34,735070,735072],{"className":735071,"style":31806},[31805],[34,735073,735074,735077],{"style":31809},[34,735075],{"className":735076,"style":31814},[31813],[34,735078,735080],{"className":735079},[31818,31819,31820,31821],[34,735081,768],{"className":735082},[31685,31821],[34,735084,31829],{"className":735085},[31828],[34,735087,735089],{"className":735088},[31801],[34,735090,735092],{"className":735091,"style":31836},[31805],[34,735093],{},[34,735095],{"className":735096,"style":31846},[31845],[34,735098,202],{"className":735099},[31850],[34,735101],{"className":735102,"style":31846},[31845],[34,735104,459],{"className":735105},[31685],[34,735107],{"className":735108,"style":128438},[31845],[34,735110,456],{"className":735111},[31782],[34,735113,735115],{"className":735114},[31685,2638],[34,735116,734831],{"className":735117},[31685,105462],[34,735119,2557],{"className":735120},[31690],[34,735122,31829],{"className":735123},[31828],[34,735125,735127],{"className":735126},[31801],[34,735128,735130],{"className":735129,"style":76192},[31805],[34,735131],{},[34,735133],{"className":735134},[31690,41488],"\n]\n解得 ",[34,735137,735139,735181],{"className":735138},[31642],[34,735140,735142],{"className":735141},[31646],[31648,735143,735144],{"xmlns":31650},[31652,735145,735146,735178],{},[31655,735147,735148,735154,735156,735162,735164,735170,735172],{},[31743,735149,735150,735152],{},[31658,735151,31965],{},[31749,735153,640],{},[31662,735155,202],{},[41453,735157,735158,735160],{},[31749,735159,640],{},[31749,735161,768],{},[31662,735163,31960],{"separator":758},[31743,735165,735166,735168],{},[31658,735167,31965],{},[31749,735169,768],{},[31662,735171,202],{},[41453,735173,735174,735176],{},[31749,735175,640],{},[31749,735177,768],{},[31665,735179,735180],{"encoding":31667},"c_1 = \\frac{1}{2}, c_2 = \\frac{1}{2}",[34,735182,735184,735239,735368],{"className":735183,"ariaHidden":758},[31672],[34,735185,735187,735190,735230,735233,735236],{"className":735186},[31676],[34,735188],{"className":735189,"style":31860},[31680],[34,735191,735193,735196],{"className":735192},[31685],[34,735194,31965],{"className":735195},[31685,31686],[34,735197,735199],{"className":735198},[31792],[34,735200,735202,735222],{"className":735201},[31796,31797],[34,735203,735205,735219],{"className":735204},[31801],[34,735206,735208],{"className":735207,"style":31806},[31805],[34,735209,735210,735213],{"style":31809},[34,735211],{"className":735212,"style":31814},[31813],[34,735214,735216],{"className":735215},[31818,31819,31820,31821],[34,735217,640],{"className":735218},[31685,31821],[34,735220,31829],{"className":735221},[31828],[34,735223,735225],{"className":735224},[31801],[34,735226,735228],{"className":735227,"style":31836},[31805],[34,735229],{},[34,735231],{"className":735232,"style":31846},[31845],[34,735234,202],{"className":735235},[31850],[34,735237],{"className":735238,"style":31846},[31845],[34,735240,735242,735245,735313,735316,735319,735359,735362,735365],{"className":735241},[31676],[34,735243],{"className":735244,"style":46727},[31680],[34,735246,735248,735251,735310],{"className":735247},[31685],[34,735249],{"className":735250},[31782,41488],[34,735252,735254],{"className":735253},[41453],[34,735255,735257,735302],{"className":735256},[31796,31797],[34,735258,735260,735299],{"className":735259},[31801],[34,735261,735263,735277,735285],{"className":735262,"style":46746},[31805],[34,735264,735265,735268],{"style":41504},[34,735266],{"className":735267,"style":41508},[31813],[34,735269,735271],{"className":735270},[31818,31819,31820,31821],[34,735272,735274],{"className":735273},[31685,31821],[34,735275,768],{"className":735276},[31685,31821],[34,735278,735279,735282],{"style":41520},[34,735280],{"className":735281,"style":41508},[31813],[34,735283],{"className":735284,"style":41528},[41527],[34,735286,735287,735290],{"style":46771},[34,735288],{"className":735289,"style":41508},[31813],[34,735291,735293],{"className":735292},[31818,31819,31820,31821],[34,735294,735296],{"className":735295},[31685,31821],[34,735297,640],{"className":735298},[31685,31821],[34,735300,31829],{"className":735301},[31828],[34,735303,735305],{"className":735304},[31801],[34,735306,735308],{"className":735307,"style":41619},[31805],[34,735309],{},[34,735311],{"className":735312},[31690,41488],[34,735314,31960],{"className":735315},[32119],[34,735317],{"className":735318,"style":32123},[31845],[34,735320,735322,735325],{"className":735321},[31685],[34,735323,31965],{"className":735324},[31685,31686],[34,735326,735328],{"className":735327},[31792],[34,735329,735331,735351],{"className":735330},[31796,31797],[34,735332,735334,735348],{"className":735333},[31801],[34,735335,735337],{"className":735336,"style":31806},[31805],[34,735338,735339,735342],{"style":31809},[34,735340],{"className":735341,"style":31814},[31813],[34,735343,735345],{"className":735344},[31818,31819,31820,31821],[34,735346,768],{"className":735347},[31685,31821],[34,735349,31829],{"className":735350},[31828],[34,735352,735354],{"className":735353},[31801],[34,735355,735357],{"className":735356,"style":31836},[31805],[34,735358],{},[34,735360],{"className":735361,"style":31846},[31845],[34,735363,202],{"className":735364},[31850],[34,735366],{"className":735367,"style":31846},[31845],[34,735369,735371,735374],{"className":735370},[31676],[34,735372],{"className":735373,"style":46727},[31680],[34,735375,735377,735380,735439],{"className":735376},[31685],[34,735378],{"className":735379},[31782,41488],[34,735381,735383],{"className":735382},[41453],[34,735384,735386,735431],{"className":735385},[31796,31797],[34,735387,735389,735428],{"className":735388},[31801],[34,735390,735392,735406,735414],{"className":735391,"style":46746},[31805],[34,735393,735394,735397],{"style":41504},[34,735395],{"className":735396,"style":41508},[31813],[34,735398,735400],{"className":735399},[31818,31819,31820,31821],[34,735401,735403],{"className":735402},[31685,31821],[34,735404,768],{"className":735405},[31685,31821],[34,735407,735408,735411],{"style":41520},[34,735409],{"className":735410,"style":41508},[31813],[34,735412],{"className":735413,"style":41528},[41527],[34,735415,735416,735419],{"style":46771},[34,735417],{"className":735418,"style":41508},[31813],[34,735420,735422],{"className":735421},[31818,31819,31820,31821],[34,735423,735425],{"className":735424},[31685,31821],[34,735426,640],{"className":735427},[31685,31821],[34,735429,31829],{"className":735430},[31828],[34,735432,735434],{"className":735433},[31801],[34,735435,735437],{"className":735436,"style":41619},[31805],[34,735438],{},[34,735440],{"className":735441},[31690,41488],[2627,735443,735444],{},[34,735445,735447,735506],{"className":735446},[31642],[34,735448,735450],{"className":735449},[31646],[31648,735451,735452],{"xmlns":31650},[31652,735453,735454,735503],{},[31655,735455,735456,735462,735464,735466,735468,735470,735494,735496,735499,735501],{},[31743,735457,735458,735460],{},[31658,735459,329201],{},[31749,735461,640],{},[31662,735463,456],{"stretchy":702},[31658,735465,33249],{},[31662,735467,2557],{"stretchy":702},[31662,735469,202],{},[41453,735471,735472,735492],{},[31655,735473,735474,735480,735482],{},[37624,735475,735476,735478],{},[31658,735477,33244],{},[31658,735479,33249],{},[31662,735481,223],{},[37624,735483,735484,735486],{},[31658,735485,33244],{},[31655,735487,735488,735490],{},[31662,735489,32226],{},[31658,735491,33249],{},[31749,735493,768],{},[31662,735495,202],{},[31658,735497,735498],{},"cosh",[31662,735500,32682],{},[31658,735502,33249],{},[31665,735504,735505],{"encoding":31667},"\\phi_1(t) = \\frac{e^t + e^{-t}}{2} = \\cosh t",[34,735507,735509,735573,735722],{"className":735508,"ariaHidden":758},[31672],[34,735510,735512,735515,735555,735558,735561,735564,735567,735570],{"className":735511},[31676],[34,735513],{"className":735514,"style":31774},[31680],[34,735516,735518,735521],{"className":735517},[31685],[34,735519,329201],{"className":735520},[31685,31686],[34,735522,735524],{"className":735523},[31792],[34,735525,735527,735547],{"className":735526},[31796,31797],[34,735528,735530,735544],{"className":735529},[31801],[34,735531,735533],{"className":735532,"style":31806},[31805],[34,735534,735535,735538],{"style":31809},[34,735536],{"className":735537,"style":31814},[31813],[34,735539,735541],{"className":735540},[31818,31819,31820,31821],[34,735542,640],{"className":735543},[31685,31821],[34,735545,31829],{"className":735546},[31828],[34,735548,735550],{"className":735549},[31801],[34,735551,735553],{"className":735552,"style":31836},[31805],[34,735554],{},[34,735556,456],{"className":735557},[31782],[34,735559,33249],{"className":735560},[31685,31686],[34,735562,2557],{"className":735563},[31690],[34,735565],{"className":735566,"style":31846},[31845],[34,735568,202],{"className":735569},[31850],[34,735571],{"className":735572,"style":31846},[31845],[34,735574,735576,735580,735713,735716,735719],{"className":735575},[31676],[34,735577],{"className":735578,"style":735579},[31680],"height:1.3482em;vertical-align:-0.345em;",[34,735581,735583,735586,735710],{"className":735582},[31685],[34,735584],{"className":735585},[31782,41488],[34,735587,735589],{"className":735588},[41453],[34,735590,735592,735702],{"className":735591},[31796,31797],[34,735593,735595,735699],{"className":735594},[31801],[34,735596,735599,735613,735621],{"className":735597,"style":735598},[31805],"height:1.0032em;",[34,735600,735601,735604],{"style":41504},[34,735602],{"className":735603,"style":41508},[31813],[34,735605,735607],{"className":735606},[31818,31819,31820,31821],[34,735608,735610],{"className":735609},[31685,31821],[34,735611,768],{"className":735612},[31685,31821],[34,735614,735615,735618],{"style":41520},[34,735616],{"className":735617,"style":41508},[31813],[34,735619],{"className":735620,"style":41528},[41527],[34,735622,735623,735626],{"style":46771},[34,735624],{"className":735625,"style":41508},[31813],[34,735627,735629],{"className":735628},[31818,31819,31820,31821],[34,735630,735632,735661,735664],{"className":735631},[31685,31821],[34,735633,735635,735638],{"className":735634},[31685,31821],[34,735636,33244],{"className":735637},[31685,31686,31821],[34,735639,735641],{"className":735640},[31792],[34,735642,735644],{"className":735643},[31796],[34,735645,735647],{"className":735646},[31801],[34,735648,735650],{"className":735649,"style":722173},[31805],[34,735651,735652,735655],{"style":552391},[34,735653],{"className":735654,"style":43406},[31813],[34,735656,735658],{"className":735657},[31818,43410,43411,31821],[34,735659,33249],{"className":735660},[31685,31686,31821],[34,735662,223],{"className":735663},[32024,31821],[34,735665,735667,735670],{"className":735666},[31685,31821],[34,735668,33244],{"className":735669},[31685,31686,31821],[34,735671,735673],{"className":735672},[31792],[34,735674,735676],{"className":735675},[31796],[34,735677,735679],{"className":735678},[31801],[34,735680,735682],{"className":735681,"style":722173},[31805],[34,735683,735684,735687],{"style":552391},[34,735685],{"className":735686,"style":43406},[31813],[34,735688,735690],{"className":735689},[31818,43410,43411,31821],[34,735691,735693,735696],{"className":735692},[31685,31821],[34,735694,32226],{"className":735695},[31685,31821],[34,735697,33249],{"className":735698},[31685,31686,31821],[34,735700,31829],{"className":735701},[31828],[34,735703,735705],{"className":735704},[31801],[34,735706,735708],{"className":735707,"style":41619},[31805],[34,735709],{},[34,735711],{"className":735712},[31690,41488],[34,735714],{"className":735715,"style":31846},[31845],[34,735717,202],{"className":735718},[31850],[34,735720],{"className":735721,"style":31846},[31845],[34,735723,735725,735728,735731,735734],{"className":735724},[31676],[34,735726],{"className":735727,"style":31681},[31680],[34,735729,735498],{"className":735730},[32767],[34,735732],{"className":735733,"style":32123},[31845],[34,735735,33249],{"className":735736},[31685,31686]," (双曲余弦)。",[15,735739,735740,362914,735813,734747,736104,735135,736443,355408,736754,737047],{},[2627,735741,735742,735743],{},"第二步：构造 ",[34,735744,735746,735764],{"className":735745},[31642],[34,735747,735749],{"className":735748},[31646],[31648,735750,735751],{"xmlns":31650},[31652,735752,735753,735761],{},[31655,735754,735755],{},[31743,735756,735757,735759],{},[31658,735758,329201],{},[31749,735760,768],{},[31665,735762,735763],{"encoding":31667},"\\phi_2",[34,735765,735767],{"className":735766,"ariaHidden":758},[31672],[34,735768,735770,735773],{"className":735769},[31676],[34,735771],{"className":735772,"style":43217},[31680],[34,735774,735776,735779],{"className":735775},[31685],[34,735777,329201],{"className":735778},[31685,31686],[34,735780,735782],{"className":735781},[31792],[34,735783,735785,735805],{"className":735784},[31796,31797],[34,735786,735788,735802],{"className":735787},[31801],[34,735789,735791],{"className":735790,"style":31806},[31805],[34,735792,735793,735796],{"style":31809},[34,735794],{"className":735795,"style":31814},[31813],[34,735797,735799],{"className":735798},[31818,31819,31820,31821],[34,735800,768],{"className":735801},[31685,31821],[34,735803,31829],{"className":735804},[31828],[34,735806,735808],{"className":735807},[31801],[34,735809,735811],{"className":735810,"style":31836},[31805],[34,735812],{},[34,735814,735816,735872],{"className":735815},[31642],[34,735817,735819],{"className":735818},[31646],[31648,735820,735821],{"xmlns":31650},[31652,735822,735823,735869],{},[31655,735824,735825,735831,735833,735835,735837,735839,735845,735851,735853,735859],{},[31743,735826,735827,735829],{},[31658,735828,329201],{},[31749,735830,768],{},[31662,735832,456],{"stretchy":702},[31658,735834,33249],{},[31662,735836,2557],{"stretchy":702},[31662,735838,202],{},[31743,735840,735841,735843],{},[31658,735842,33759],{},[31749,735844,640],{},[37624,735846,735847,735849],{},[31658,735848,33244],{},[31658,735850,33249],{},[31662,735852,223],{},[31743,735854,735855,735857],{},[31658,735856,33759],{},[31749,735858,768],{},[37624,735860,735861,735863],{},[31658,735862,33244],{},[31655,735864,735865,735867],{},[31662,735866,32226],{},[31658,735868,33249],{},[31665,735870,735871],{"encoding":31667},"\\phi_2(t) = k_1 e^t + k_2 e^{-t}",[34,735873,735875,735939,736023],{"className":735874,"ariaHidden":758},[31672],[34,735876,735878,735881,735921,735924,735927,735930,735933,735936],{"className":735877},[31676],[34,735879],{"className":735880,"style":31774},[31680],[34,735882,735884,735887],{"className":735883},[31685],[34,735885,329201],{"className":735886},[31685,31686],[34,735888,735890],{"className":735889},[31792],[34,735891,735893,735913],{"className":735892},[31796,31797],[34,735894,735896,735910],{"className":735895},[31801],[34,735897,735899],{"className":735898,"style":31806},[31805],[34,735900,735901,735904],{"style":31809},[34,735902],{"className":735903,"style":31814},[31813],[34,735905,735907],{"className":735906},[31818,31819,31820,31821],[34,735908,768],{"className":735909},[31685,31821],[34,735911,31829],{"className":735912},[31828],[34,735914,735916],{"className":735915},[31801],[34,735917,735919],{"className":735918,"style":31836},[31805],[34,735920],{},[34,735922,456],{"className":735923},[31782],[34,735925,33249],{"className":735926},[31685,31686],[34,735928,2557],{"className":735929},[31690],[34,735931],{"className":735932,"style":31846},[31845],[34,735934,202],{"className":735935},[31850],[34,735937],{"className":735938,"style":31846},[31845],[34,735940,735942,735945,735985,736014,736017,736020],{"className":735941},[31676],[34,735943],{"className":735944,"style":723843},[31680],[34,735946,735948,735951],{"className":735947},[31685],[34,735949,33759],{"className":735950,"style":33774},[31685,31686],[34,735952,735954],{"className":735953},[31792],[34,735955,735957,735977],{"className":735956},[31796,31797],[34,735958,735960,735974],{"className":735959},[31801],[34,735961,735963],{"className":735962,"style":31806},[31805],[34,735964,735965,735968],{"style":319623},[34,735966],{"className":735967,"style":31814},[31813],[34,735969,735971],{"className":735970},[31818,31819,31820,31821],[34,735972,640],{"className":735973},[31685,31821],[34,735975,31829],{"className":735976},[31828],[34,735978,735980],{"className":735979},[31801],[34,735981,735983],{"className":735982,"style":31836},[31805],[34,735984],{},[34,735986,735988,735991],{"className":735987},[31685],[34,735989,33244],{"className":735990},[31685,31686],[34,735992,735994],{"className":735993},[31792],[34,735995,735997],{"className":735996},[31796],[34,735998,736000],{"className":735999},[31801],[34,736001,736003],{"className":736002,"style":611758},[31805],[34,736004,736005,736008],{"style":37777},[34,736006],{"className":736007,"style":31814},[31813],[34,736009,736011],{"className":736010},[31818,31819,31820,31821],[34,736012,33249],{"className":736013},[31685,31686,31821],[34,736015],{"className":736016,"style":32604},[31845],[34,736018,223],{"className":736019},[32024],[34,736021],{"className":736022,"style":32604},[31845],[34,736024,736026,736029,736069],{"className":736025},[31676],[34,736027],{"className":736028,"style":723843},[31680],[34,736030,736032,736035],{"className":736031},[31685],[34,736033,33759],{"className":736034,"style":33774},[31685,31686],[34,736036,736038],{"className":736037},[31792],[34,736039,736041,736061],{"className":736040},[31796,31797],[34,736042,736044,736058],{"className":736043},[31801],[34,736045,736047],{"className":736046,"style":31806},[31805],[34,736048,736049,736052],{"style":319623},[34,736050],{"className":736051,"style":31814},[31813],[34,736053,736055],{"className":736054},[31818,31819,31820,31821],[34,736056,768],{"className":736057},[31685,31821],[34,736059,31829],{"className":736060},[31828],[34,736062,736064],{"className":736063},[31801],[34,736065,736067],{"className":736066,"style":31836},[31805],[34,736068],{},[34,736070,736072,736075],{"className":736071},[31685],[34,736073,33244],{"className":736074},[31685,31686],[34,736076,736078],{"className":736077},[31792],[34,736079,736081],{"className":736080},[31796],[34,736082,736084],{"className":736083},[31801],[34,736085,736087],{"className":736086,"style":611758},[31805],[34,736088,736089,736092],{"style":37777},[34,736090],{"className":736091,"style":31814},[31813],[34,736093,736095],{"className":736094},[31818,31819,31820,31821],[34,736096,736098,736101],{"className":736097},[31685,31821],[34,736099,32226],{"className":736100},[31685,31821],[34,736102,33249],{"className":736103},[31685,31686,31821],[34,736105,736107,736175],{"className":736106},[31642],[34,736108,736110],{"className":736109},[31646],[31648,736111,736112],{"xmlns":31650},[31652,736113,736114,736172],{},[31655,736115,736116,736118],{},[31662,736117,32685],{"fence":758},[71078,736119,736120,736146],{"rowspacing":71080,"columnalign":71081,"columnspacing":71082},[71084,736121,736122],{},[71087,736123,736124],{},[71090,736125,736126],{"scriptlevel":459,"displaystyle":702},[31655,736127,736128,736134,736136,736142,736144],{},[31743,736129,736130,736132],{},[31658,736131,33759],{},[31749,736133,640],{},[31662,736135,223],{},[31743,736137,736138,736140],{},[31658,736139,33759],{},[31749,736141,768],{},[31662,736143,202],{},[31749,736145,459],{},[71084,736147,736148],{},[71087,736149,736150],{},[71090,736151,736152],{"scriptlevel":459,"displaystyle":702},[31655,736153,736154,736160,736162,736168,736170],{},[31743,736155,736156,736158],{},[31658,736157,33759],{},[31749,736159,640],{},[31662,736161,32226],{},[31743,736163,736164,736166],{},[31658,736165,33759],{},[31749,736167,768],{},[31662,736169,202],{},[31749,736171,640],{},[31665,736173,736174],{"encoding":31667},"\\begin{cases}\nk_1 + k_2 = 0 \\\\\nk_1 - k_2 = 1\n\\end{cases}",[34,736176,736178],{"className":736177,"ariaHidden":758},[31672],[34,736179,736181,736184],{"className":736180},[31676],[34,736182],{"className":736183,"style":75797},[31680],[34,736185,736187,736193,736440],{"className":736186},[34006],[34,736188,736190],{"className":736189,"style":75811},[31782,75810],[34,736191,32685],{"className":736192},[71281,75815],[34,736194,736196],{"className":736195},[31685],[34,736197,736199],{"className":736198},[71078],[34,736200,736202],{"className":736201},[71384],[34,736203,736205,736432],{"className":736204},[31796,31797],[34,736206,736208,736429],{"className":736207},[31801],[34,736209,736211,736320],{"className":736210,"style":75834},[31805],[34,736212,736213,736216],{"style":75837},[34,736214],{"className":736215,"style":71401},[31813],[34,736217,736219,736259,736262,736265,736268,736308,736311,736314,736317],{"className":736218},[31685],[34,736220,736222,736225],{"className":736221},[31685],[34,736223,33759],{"className":736224,"style":33774},[31685,31686],[34,736226,736228],{"className":736227},[31792],[34,736229,736231,736251],{"className":736230},[31796,31797],[34,736232,736234,736248],{"className":736233},[31801],[34,736235,736237],{"className":736236,"style":31806},[31805],[34,736238,736239,736242],{"style":319623},[34,736240],{"className":736241,"style":31814},[31813],[34,736243,736245],{"className":736244},[31818,31819,31820,31821],[34,736246,640],{"className":736247},[31685,31821],[34,736249,31829],{"className":736250},[31828],[34,736252,736254],{"className":736253},[31801],[34,736255,736257],{"className":736256,"style":31836},[31805],[34,736258],{},[34,736260],{"className":736261,"style":32604},[31845],[34,736263,223],{"className":736264},[32024],[34,736266],{"className":736267,"style":32604},[31845],[34,736269,736271,736274],{"className":736270},[31685],[34,736272,33759],{"className":736273,"style":33774},[31685,31686],[34,736275,736277],{"className":736276},[31792],[34,736278,736280,736300],{"className":736279},[31796,31797],[34,736281,736283,736297],{"className":736282},[31801],[34,736284,736286],{"className":736285,"style":31806},[31805],[34,736287,736288,736291],{"style":319623},[34,736289],{"className":736290,"style":31814},[31813],[34,736292,736294],{"className":736293},[31818,31819,31820,31821],[34,736295,768],{"className":736296},[31685,31821],[34,736298,31829],{"className":736299},[31828],[34,736301,736303],{"className":736302},[31801],[34,736304,736306],{"className":736305,"style":31836},[31805],[34,736307],{},[34,736309],{"className":736310,"style":31846},[31845],[34,736312,202],{"className":736313},[31850],[34,736315],{"className":736316,"style":31846},[31845],[34,736318,459],{"className":736319},[31685],[34,736321,736322,736325],{"style":76011},[34,736323],{"className":736324,"style":71401},[31813],[34,736326,736328,736368,736371,736374,736377,736417,736420,736423,736426],{"className":736327},[31685],[34,736329,736331,736334],{"className":736330},[31685],[34,736332,33759],{"className":736333,"style":33774},[31685,31686],[34,736335,736337],{"className":736336},[31792],[34,736338,736340,736360],{"className":736339},[31796,31797],[34,736341,736343,736357],{"className":736342},[31801],[34,736344,736346],{"className":736345,"style":31806},[31805],[34,736347,736348,736351],{"style":319623},[34,736349],{"className":736350,"style":31814},[31813],[34,736352,736354],{"className":736353},[31818,31819,31820,31821],[34,736355,640],{"className":736356},[31685,31821],[34,736358,31829],{"className":736359},[31828],[34,736361,736363],{"className":736362},[31801],[34,736364,736366],{"className":736365,"style":31836},[31805],[34,736367],{},[34,736369],{"className":736370,"style":32604},[31845],[34,736372,32226],{"className":736373},[32024],[34,736375],{"className":736376,"style":32604},[31845],[34,736378,736380,736383],{"className":736379},[31685],[34,736381,33759],{"className":736382,"style":33774},[31685,31686],[34,736384,736386],{"className":736385},[31792],[34,736387,736389,736409],{"className":736388},[31796,31797],[34,736390,736392,736406],{"className":736391},[31801],[34,736393,736395],{"className":736394,"style":31806},[31805],[34,736396,736397,736400],{"style":319623},[34,736398],{"className":736399,"style":31814},[31813],[34,736401,736403],{"className":736402},[31818,31819,31820,31821],[34,736404,768],{"className":736405},[31685,31821],[34,736407,31829],{"className":736408},[31828],[34,736410,736412],{"className":736411},[31801],[34,736413,736415],{"className":736414,"style":31836},[31805],[34,736416],{},[34,736418],{"className":736419,"style":31846},[31845],[34,736421,202],{"className":736422},[31850],[34,736424],{"className":736425,"style":31846},[31845],[34,736427,640],{"className":736428},[31685],[34,736430,31829],{"className":736431},[31828],[34,736433,736435],{"className":736434},[31801],[34,736436,736438],{"className":736437,"style":76192},[31805],[34,736439],{},[34,736441],{"className":736442},[31690,41488],[34,736444,736446,73649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= \\frac{1}{2}, k_2 = -\\frac{1}{2}",[34,736491,736493,736548,736677],{"className":736492,"ariaHidden":758},[31672],[34,736494,736496,736499,736539,736542,736545],{"className":736495},[31676],[34,736497],{"className":736498,"style":33814},[31680],[34,736500,736502,736505],{"className":736501},[31685],[34,736503,33759],{"className":736504,"style":33774},[31685,31686],[34,736506,736508],{"className":736507},[31792],[34,736509,736511,736531],{"className":736510},[31796,31797],[34,736512,736514,736528],{"className":736513},[31801],[34,736515,736517],{"className":736516,"style":31806},[31805],[34,736518,736519,736522],{"style":319623},[34,736520],{"className":736521,"style":31814},[31813],[34,736523,736525],{"className":736524},[31818,31819,31820,31821],[34,736526,640],{"className":736527},[31685,31821],[34,736529,31829],{"className":736530},[31828],[34,736532,736534],{"className":736533},[31801],[34,736535,736537],{"className":736536,"style":31836},[31805],[34,736538],{},[34,736540],{"className":736541,"style":31846},[31845],[34,736543,202],{"className":736544},[31850],[34,736546],{"className":736547,"style":31846},[31845],[34,736549,736551,736554,736622,736625,736628,736668,736671,736674],{"className":736550},[31676],[34,736552],{"className":736553,"style":46727},[31680],[34,736555,736557,736560,736619],{"className":736556},[31685],[34,736558],{"className":736559},[31782,41488],[34,736561,736563],{"className":736562},[41453],[34,736564,736566,736611],{"className":736565},[31796,31797],[34,736567,736569,736608],{"className":736568},[31801],[34,736570,736572,736586,736594],{"className":736571,"style":46746},[31805],[34,736573,736574,736577],{"style":41504},[34,736575],{"className":736576,"style":41508},[31813],[34,736578,736580],{"className":736579},[31818,31819,31820,31821],[34,736581,736583],{"className":736582},[31685,31821],[34,736584,768],{"className":736585},[31685,31821],[34,736587,736588,736591],{"style":41520},[34,736589],{"className":736590,"style":41508},[31813],[34,736592],{"className":736593,"style":41528},[41527],[34,736595,736596,736599],{"style":46771},[34,736597],{"className":736598,"style":41508},[31813],[34,736600,736602],{"className":736601},[31818,31819,31820,31821],[34,736603,736605],{"className":736604},[31685,31821],[34,736606,640],{"className":736607},[31685,31821],[34,736609,31829],{"className":736610},[31828],[34,736612,736614],{"className":736613},[31801],[34,736615,736617],{"className":736616,"style":41619},[31805],[34,736618],{},[34,736620],{"className":736621},[31690,41488],[34,736623,31960],{"className":736624},[32119],[34,736626],{"className":736627,"style":32123},[31845],[34,736629,736631,736634],{"className":736630},[31685],[34,736632,33759],{"className":736633,"style":33774},[31685,31686],[34,736635,736637],{"className":736636},[31792],[34,736638,736640,736660],{"className":736639},[31796,31797],[34,736641,736643,736657],{"className":736642},[31801],[34,736644,736646],{"className":736645,"style":31806},[31805],[34,736647,736648,736651],{"style":319623},[34,736649],{"className":736650,"style":31814},[31813],[34,736652,736654],{"className":736653},[31818,31819,31820,31821],[34,736655,768],{"className":736656},[31685,31821],[34,736658,31829],{"className":736659},[31828],[34,736661,736663],{"className":736662},[31801],[34,736664,736666],{"className":736665,"style":31836},[31805],[34,736667],{},[34,736669],{"className":736670,"style":31846},[31845],[34,736672,202],{"className":736673},[31850],[34,736675],{"className":736676,"style":31846},[31845],[34,736678,736680,736683,736686],{"className":736679},[31676],[34,736681],{"className":736682,"style":46727},[31680],[34,736684,32226],{"className":736685},[31685],[34,736687,736689,736692,736751],{"className":736688},[31685],[34,736690],{"className":736691},[31782,41488],[34,736693,736695],{"className":736694},[41453],[34,736696,736698,736743],{"className":736697},[31796,31797],[34,736699,736701,736740],{"className":736700},[31801],[34,736702,736704,736718,736726],{"className":736703,"style":46746},[31805],[34,736705,736706,736709],{"style":41504},[34,736707],{"className":736708,"style":41508},[31813],[34,736710,736712],{"className":736711},[31818,31819,31820,31821],[34,736713,736715],{"className":736714},[31685,31821],[34,736716,768],{"className":736717},[31685,31821],[34,736719,736720,736723],{"style":41520},[34,736721],{"className":736722,"style":41508},[31813],[34,736724],{"className":736725,"style":41528},[41527],[34,736727,736728,736731],{"style":46771},[34,736729],{"className":736730,"style":41508},[31813],[34,736732,736734],{"className":736733},[31818,31819,31820,31821],[34,736735,736737],{"className":736736},[31685,31821],[34,736738,640],{"className":736739},[31685,31821],[34,736741,31829],{"className":736742},[31828],[34,736744,736746],{"className":736745},[31801],[34,736747,736749],{"className":736748,"style":41619},[31805],[34,736750],{},[34,736752],{"className":736753},[31690,41488],[2627,736755,736756],{},[34,736757,736759,736818],{"className":736758},[31642],[34,736760,736762],{"className":736761},[31646],[31648,736763,736764],{"xmlns":31650},[31652,736765,736766,736815],{},[31655,736767,736768,736774,736776,736778,736780,736782,736806,736808,736811,736813],{},[31743,736769,736770,736772],{},[31658,736771,329201],{},[31749,736773,768],{},[31662,736775,456],{"stretchy":702},[31658,736777,33249],{},[31662,736779,2557],{"stretchy":702},[31662,736781,202],{},[41453,736783,736784,736804],{},[31655,736785,736786,736792,736794],{},[37624,736787,736788,736790],{},[31658,736789,33244],{},[31658,736791,33249],{},[31662,736793,32226],{},[37624,736795,736796,736798],{},[31658,736797,33244],{},[31655,736799,736800,736802],{},[31662,736801,32226],{},[31658,736803,33249],{},[31749,736805,768],{},[31662,736807,202],{},[31658,736809,736810],{},"sinh",[31662,736812,32682],{},[31658,736814,33249],{},[31665,736816,736817],{"encoding":31667},"\\phi_2(t) = \\frac{e^t - e^{-t}}{2} = \\sinh t",[34,736819,736821,736885,737032],{"className":736820,"ariaHidden":758},[31672],[34,736822,736824,736827,736867,736870,736873,736876,736879,736882],{"className":736823},[31676],[34,736825],{"className":736826,"style":31774},[31680],[34,736828,736830,736833],{"className":736829},[31685],[34,736831,329201],{"className":736832},[31685,31686],[34,736834,736836],{"className":736835},[31792],[34,736837,736839,736859],{"className":736838},[31796,31797],[34,736840,736842,736856],{"className":736841},[31801],[34,736843,736845],{"className":736844,"style":31806},[31805],[34,736846,736847,736850],{"style":31809},[34,736848],{"className":736849,"style":31814},[31813],[34,736851,736853],{"className":736852},[31818,31819,31820,31821],[34,736854,768],{"className":736855},[31685,31821],[34,736857,31829],{"className":736858},[31828],[34,736860,736862],{"className":736861},[31801],[34,736863,736865],{"className":736864,"style":31836},[31805],[34,736866],{},[34,736868,456],{"className":736869},[31782],[34,736871,33249],{"className":736872},[31685,31686],[34,736874,2557],{"className":736875},[31690],[34,736877],{"className":736878,"style":31846},[31845],[34,736880,202],{"className":736881},[31850],[34,736883],{"className":736884,"style":31846},[31845],[34,736886,736888,736891,737023,737026,737029],{"className":736887},[31676],[34,736889],{"className":736890,"style":735579},[31680],[34,736892,736894,736897,737020],{"className":736893},[31685],[34,736895],{"className":736896},[31782,41488],[34,736898,736900],{"className":736899},[41453],[34,736901,736903,737012],{"className":736902},[31796,31797],[34,736904,736906,737009],{"className":736905},[31801],[34,736907,736909,736923,736931],{"className":736908,"style":735598},[31805],[34,736910,736911,736914],{"style":41504},[34,736912],{"className":736913,"style":41508},[31813],[34,736915,736917],{"className":736916},[31818,31819,31820,31821],[34,736918,736920],{"className":736919},[31685,31821],[34,736921,768],{"className":736922},[31685,31821],[34,736924,736925,736928],{"style":41520},[34,736926],{"className":736927,"style":41508},[31813],[34,736929],{"className":736930,"style":41528},[41527],[34,736932,736933,736936],{"style":46771},[34,736934],{"className":736935,"style":41508},[31813],[34,736937,736939],{"className":736938},[31818,31819,31820,31821],[34,736940,736942,736971,736974],{"className":736941},[31685,31821],[34,736943,736945,736948],{"className":736944},[31685,31821],[34,736946,33244],{"className":736947},[31685,31686,31821],[34,736949,736951],{"className":736950},[31792],[34,736952,736954],{"className":736953},[31796],[34,736955,736957],{"className":736956},[31801],[34,736958,736960],{"className":736959,"style":722173},[31805],[34,736961,736962,736965],{"style":552391},[34,736963],{"className":736964,"style":43406},[31813],[34,736966,736968],{"className":736967},[31818,43410,43411,31821],[34,736969,33249],{"className":736970},[31685,31686,31821],[34,736972,32226],{"className":736973},[32024,31821],[34,736975,736977,736980],{"className":736976},[31685,31821],[34,736978,33244],{"className":736979},[31685,31686,31821],[34,736981,736983],{"className":736982},[31792],[34,736984,736986],{"className":736985},[31796],[34,736987,736989],{"className":736988},[31801],[34,736990,736992],{"className":736991,"style":722173},[31805],[34,736993,736994,736997],{"style":552391},[34,736995],{"className":736996,"style":43406},[31813],[34,736998,737000],{"className":736999},[31818,43410,43411,31821],[34,737001,737003,737006],{"className":737002},[31685,31821],[34,737004,32226],{"className":737005},[31685,31821],[34,737007,33249],{"className":737008},[31685,31686,31821],[34,737010,31829],{"className":737011},[31828],[34,737013,737015],{"className":737014},[31801],[34,737016,737018],{"className":737017,"style":41619},[31805],[34,737019],{},[34,737021],{"className":737022},[31690,41488],[34,737024],{"className":737025,"style":31846},[31845],[34,737027,202],{"className":737028},[31850],[34,737030],{"className":737031,"style":31846},[31845],[34,737033,737035,737038,737041,737044],{"className":737034},[31676],[34,737036],{"className":737037,"style":31681},[31680],[34,737039,736810],{"className":737040},[32767],[34,737042],{"className":737043,"style":32123},[31845],[34,737045,33249],{"className":737046},[31685,31686]," (双曲正弦)。",[15,737049,737050,737053,737054,737301,737302,737644],{},[2627,737051,737052],{},"第三步：利用标准解组写任意解","\n有了这两个标准解，任意初值 ",[34,737055,737057,737109],{"className":737056},[31642],[34,737058,737060],{"className":737059},[31646],[31648,737061,737062],{"xmlns":31650},[31652,737063,737064,737106],{},[31655,737065,737066,737068,737070,737072,737074,737076,737082,737084,737090,737092,737094,737096,737098],{},[31658,737067,32936],{},[31662,737069,456],{"stretchy":702},[31749,737071,459],{},[31662,737073,2557],{"stretchy":702},[31662,737075,202],{},[31743,737077,737078,737080],{},[31658,737079,32936],{},[31749,737081,459],{},[31662,737083,31960],{"separator":758},[37624,737085,737086,737088],{},[31658,737087,32936],{},[31662,737089,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,737091,456],{"stretchy":702},[31749,737093,459],{},[31662,737095,2557],{"stretchy":702},[31662,737097,202],{},[38846,737099,737100,737102,737104],{},[31658,737101,32936],{},[31749,737103,459],{},[31662,737105,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,737107,737108],{"encoding":31667},"x(0)=x_0, x'(0)=x_0'",[34,737110,737112,737139,737241],{"className":737111,"ariaHidden":758},[31672],[34,737113,737115,737118,737121,737124,737127,737130,737133,737136],{"className":737114},[31676],[34,737116],{"className":737117,"style":31774},[31680],[34,737119,32936],{"className":737120},[31685,31686],[34,737122,456],{"className":737123},[31782],[34,737125,459],{"className":737126},[31685],[34,737128,2557],{"className":737129},[31690],[34,737131],{"className":737132,"style":31846},[31845],[34,737134,202],{"className":737135},[31850],[34,737137],{"className":737138,"style":31846},[31845],[34,737140,737142,737145,737185,737188,737191,737223,737226,737229,737232,737235,737238],{"className":737141},[31676],[34,737143],{"className":737144,"style":570646},[31680],[34,737146,737148,737151],{"className":737147},[31685],[34,737149,32936],{"className":737150},[31685,31686],[34,737152,737154],{"className":737153},[31792],[34,737155,737157,737177],{"className":737156},[31796,31797],[34,737158,737160,737174],{"className":737159},[31801],[34,737161,737163],{"className":737162,"style":31806},[31805],[34,737164,737165,737168],{"style":31809},[34,737166],{"className":737167,"style":31814},[31813],[34,737169,737171],{"className":737170},[31818,31819,31820,31821],[34,737172,459],{"className":737173},[31685,31821],[34,737175,31829],{"className":737176},[31828],[34,737178,737180],{"className":737179},[31801],[34,737181,737183],{"className":737182,"style":31836},[31805],[34,737184],{},[34,737186,31960],{"className":737187},[32119],[34,737189],{"className":737190,"style":32123},[31845],[34,737192,737194,737197],{"className":737193},[31685],[34,737195,32936],{"className":737196},[31685,31686],[34,737198,737200],{"className":737199},[31792],[34,737201,737203],{"className":737202},[31796],[34,737204,737206],{"className":737205},[31801],[34,737207,737209],{"className":737208,"style":132027},[31805],[34,737210,737211,737214],{"style":37777},[34,737212],{"className":737213,"style":31814},[31813],[34,737215,737217],{"className":737216},[31818,31819,31820,31821],[34,737218,737220],{"className":737219},[31685,31821],[34,737221,116495],{"className":737222},[31685,31821],[34,737224,456],{"className":737225},[31782],[34,737227,459],{"className":737228},[31685],[34,737230,2557],{"className":737231},[31690],[34,737233],{"className":737234,"style":31846},[31845],[34,737236,202],{"className":737237},[31850],[34,737239],{"className":737240,"style":31846},[31845],[34,737242,737244,737247],{"className":737243},[31676],[34,737245],{"className":737246,"style":716670},[31680],[34,737248,737250,737253],{"className":737249},[31685],[34,737251,32936],{"className":737252},[31685,31686],[34,737254,737256],{"className":737255},[31792],[34,737257,737259,737293],{"className":737258},[31796,31797],[34,737260,737262,737290],{"className":737261},[31801],[34,737263,737265,737276],{"className":737264,"style":132027},[31805],[34,737266,737267,737270],{"style":316930},[34,737268],{"className":737269,"style":31814},[31813],[34,737271,737273],{"className":737272},[31818,31819,31820,31821],[34,737274,459],{"className":737275},[31685,31821],[34,737277,737278,737281],{"style":37777},[34,737279],{"className":737280,"style":31814},[31813],[34,737282,737284],{"className":737283},[31818,31819,31820,31821],[34,737285,737287],{"className":737286},[31685,31821],[34,737288,116495],{"className":737289},[31685,31821],[34,737291,31829],{"className":737292},[31828],[34,737294,737296],{"className":737295},[31801],[34,737297,737299],{"className":737298,"style":39021},[31805],[34,737300],{}," 的解直接写出来，不用计算：\n",[34,737303,737305,737371],{"className":737304},[31642],[34,737306,737308],{"className":737307},[31646],[31648,737309,737310],{"xmlns":31650},[31652,737311,737312,737368],{},[31655,737313,737314,737316,737318,737320,737322,737324,737330,737332,737338,737340,737342,737344,737346,737354,737356,737362,737364,737366],{},[31658,737315,32936],{},[31662,737317,456],{"stretchy":702},[31658,737319,33249],{},[31662,737321,2557],{"stretchy":702},[31662,737323,202],{},[31743,737325,737326,737328],{},[31658,737327,32936],{},[31749,737329,459],{},[31662,737331,71191],{},[31743,737333,737334,737336],{},[31658,737335,329201],{},[31749,737337,640],{},[31662,737339,456],{"stretchy":702},[31658,737341,33249],{},[31662,737343,2557],{"stretchy":702},[31662,737345,223],{},[38846,737347,737348,737350,737352],{},[31658,737349,32936],{},[31749,737351,459],{},[31662,737353,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,737355,71191],{},[31743,737357,737358,737360],{},[31658,737359,329201],{},[31749,737361,768],{},[31662,737363,456],{"stretchy":702},[31658,737365,33249],{},[31662,737367,2557],{"stretchy":702},[31665,737369,737370],{"encoding":31667},"x(t) = x_0 \\cdot \\phi_1(t) + x_0' \\cdot \\phi_2(t)",[34,737372,737374,737401,737456,737520,737589],{"className":737373,"ariaHidden":758},[31672],[34,737375,737377,737380,737383,737386,737389,737392,737395,737398],{"className":737376},[31676],[34,737378],{"className":737379,"style":31774},[31680],[34,737381,32936],{"className":737382},[31685,31686],[34,737384,456],{"className":737385},[31782],[34,737387,33249],{"className":737388},[31685,31686],[34,737390,2557],{"className":737391},[31690],[34,737393],{"className":737394,"style":31846},[31845],[34,737396,202],{"className":737397},[31850],[34,737399],{"className":737400,"style":31846},[31845],[34,737402,737404,737407,737447,737450,737453],{"className":737403},[31676],[34,737405],{"className":737406,"style":466434},[31680],[34,737408,737410,737413],{"className":737409},[31685],[34,737411,32936],{"className":737412},[31685,31686],[34,737414,737416],{"className":737415},[31792],[34,737417,737419,737439],{"className":737418},[31796,31797],[34,737420,737422,737436],{"className":737421},[31801],[34,737423,737425],{"className":737424,"style":31806},[31805],[34,737426,737427,737430],{"style":31809},[34,737428],{"className":737429,"style":31814},[31813],[34,737431,737433],{"className":737432},[31818,31819,31820,31821],[34,737434,459],{"className":737435},[31685,31821],[34,737437,31829],{"className":737438},[31828],[34,737440,737442],{"className":737441},[31801],[34,737443,737445],{"className":737444,"style":31836},[31805],[34,737446],{},[34,737448],{"className":737449,"style":32604},[31845],[34,737451,71191],{"className":737452},[32024],[34,737454],{"className":737455,"style":32604},[31845],[34,737457,737459,737462,737502,737505,737508,737511,737514,737517],{"className":737458},[31676],[34,737460],{"className":737461,"style":31774},[31680],[34,737463,737465,737468],{"className":737464},[31685],[34,737466,329201],{"className":737467},[31685,31686],[34,737469,737471],{"className":737470},[31792],[34,737472,737474,737494],{"className":737473},[31796,31797],[34,737475,737477,737491],{"className":737476},[31801],[34,737478,737480],{"className":737479,"style":31806},[31805],[34,737481,737482,737485],{"style":31809},[34,737483],{"className":737484,"style":31814},[31813],[34,737486,737488],{"className":737487},[31818,31819,31820,31821],[34,737489,640],{"className":737490},[31685,31821],[34,737492,31829],{"className":737493},[31828],[34,737495,737497],{"className":737496},[31801],[34,737498,737500],{"className":737499,"style":31836},[31805],[34,737501],{},[34,737503,456],{"className":737504},[31782],[34,737506,33249],{"className":737507},[31685,31686],[34,737509,2557],{"className":737510},[31690],[34,737512],{"className":737513,"style":32604},[31845],[34,737515,223],{"className":737516},[32024],[34,737518],{"className":737519,"style":32604},[31845],[34,737521,737523,737526,737580,737583,737586],{"className":737522},[31676],[34,737524],{"className":737525,"style":716670},[31680],[34,737527,737529,737532],{"className":737528},[31685],[34,737530,32936],{"className":737531},[31685,31686],[34,737533,737535],{"className":737534},[31792],[34,737536,737538,737572],{"className":737537},[31796,31797],[34,737539,737541,737569],{"className":737540},[31801],[34,737542,737544,737555],{"className":737543,"style":132027},[31805],[34,737545,737546,737549],{"style":316930},[34,737547],{"className":737548,"style":31814},[31813],[34,737550,737552],{"className":737551},[31818,31819,31820,31821],[34,737553,459],{"className":737554},[31685,31821],[34,737556,737557,737560],{"style":37777},[34,737558],{"className":737559,"style":31814},[31813],[34,737561,737563],{"className":737562},[31818,31819,31820,31821],[34,737564,737566],{"className":737565},[31685,31821],[34,737567,116495],{"className":737568},[31685,31821],[34,737570,31829],{"className":737571},[31828],[34,737573,737575],{"className":737574},[31801],[34,737576,737578],{"className":737577,"style":39021},[31805],[34,737579],{},[34,737581],{"className":737582,"style":32604},[31845],[34,737584,71191],{"className":737585},[32024],[34,737587],{"className":737588,"style":32604},[31845],[34,737590,737592,737595,737635,737638,737641],{"className":737591},[31676],[34,737593],{"className":737594,"style":31774},[31680],[34,737596,737598,737601],{"className":737597},[31685],[34,737599,329201],{"className":737600},[31685,31686],[34,737602,737604],{"className":737603},[31792],[34,737605,737607,737627],{"className":737606},[31796,31797],[34,737608,737610,737624],{"className":737609},[31801],[34,737611,737613],{"className":737612,"style":31806},[31805],[34,737614,737615,737618],{"style":31809},[34,737616],{"className":737617,"style":31814},[31813],[34,737619,737621],{"className":737620},[31818,31819,31820,31821],[34,737622,768],{"className":737623},[31685,31821],[34,737625,31829],{"className":737626},[31828],[34,737628,737630],{"className":737629},[31801],[34,737631,737633],{"className":737632,"style":31836},[31805],[34,737634],{},[34,737636,456],{"className":737637},[31782],[34,737639,33249],{"className":737640},[31685,31686],[34,737642,2557],{"className":737643},[31690],[34,737645,737647,737697],{"className":737646},[31642],[34,737648,737650],{"className":737649},[31646],[31648,737651,737652],{"xmlns":31650},[31652,737653,737654,737694],{},[31655,737655,737656,737658,737660,737662,737664,737666,737672,737674,737676,737678,737680,737688,737690,737692],{},[31658,737657,32936],{},[31662,737659,456],{"stretchy":702},[31658,737661,33249],{},[31662,737663,2557],{"stretchy":702},[31662,737665,202],{},[31743,737667,737668,737670],{},[31658,737669,32936],{},[31749,737671,459],{},[31658,737673,735498],{},[31662,737675,32682],{},[31658,737677,33249],{},[31662,737679,223],{},[38846,737681,737682,737684,737686],{},[31658,737683,32936],{},[31749,737685,459],{},[31662,737687,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,737689,736810],{},[31662,737691,32682],{},[31658,737693,33249],{},[31665,737695,737696],{"encoding":31667},"x(t) = x_0 \\cosh t + x_0' \\sinh t",[34,737698,737700,737727,737794],{"className":737699,"ariaHidden":758},[31672],[34,737701,737703,737706,737709,737712,737715,737718,737721,737724],{"className":737702},[31676],[34,737704],{"className":737705,"style":31774},[31680],[34,737707,32936],{"className":737708},[31685,31686],[34,737710,456],{"className":737711},[31782],[34,737713,33249],{"className":737714},[31685,31686],[34,737716,2557],{"className":737717},[31690],[34,737719],{"className":737720,"style":31846},[31845],[34,737722,202],{"className":737723},[31850],[34,737725],{"className":737726,"style":31846},[31845],[34,737728,737730,737733,737773,737776,737779,737782,737785,737788,737791],{"className":737729},[31676],[34,737731],{"className":737732,"style":33814},[31680],[34,737734,737736,737739],{"className":737735},[31685],[34,737737,32936],{"className":737738},[31685,31686],[34,737740,737742],{"className":737741},[31792],[34,737743,737745,737765],{"className":737744},[31796,31797],[34,737746,737748,737762],{"className":737747},[31801],[34,737749,737751],{"className":737750,"style":31806},[31805],[34,737752,737753,737756],{"style":31809},[34,737754],{"className":737755,"style":31814},[31813],[34,737757,737759],{"className":737758},[31818,31819,31820,31821],[34,737760,459],{"className":737761},[31685,31821],[34,737763,31829],{"className":737764},[31828],[34,737766,737768],{"className":737767},[31801],[34,737769,737771],{"className":737770,"style":31836},[31805],[34,737772],{},[34,737774],{"className":737775,"style":32123},[31845],[34,737777,735498],{"className":737778},[32767],[34,737780],{"className":737781,"style":32123},[31845],[34,737783,33249],{"className":737784},[31685,31686],[34,737786],{"className":737787,"style":32604},[31845],[34,737789,223],{"className":737790},[32024],[34,737792],{"className":737793,"style":32604},[31845],[34,737795,737797,737800,737854,737857,737860,737863],{"className":737796},[31676],[34,737798],{"className":737799,"style":716670},[31680],[34,737801,737803,737806],{"className":737802},[31685],[34,737804,32936],{"className":737805},[31685,31686],[34,737807,737809],{"className":737808},[31792],[34,737810,737812,737846],{"className":737811},[31796,31797],[34,737813,737815,737843],{"className":737814},[31801],[34,737816,737818,737829],{"className":737817,"style":132027},[31805],[34,737819,737820,737823],{"style":316930},[34,737821],{"className":737822,"style":31814},[31813],[34,737824,737826],{"className":737825},[31818,31819,31820,31821],[34,737827,459],{"className":737828},[31685,31821],[34,737830,737831,737834],{"style":37777},[34,737832],{"className":737833,"style":31814},[31813],[34,737835,737837],{"className":737836},[31818,31819,31820,31821],[34,737838,737840],{"className":737839},[31685,31821],[34,737841,116495],{"className":737842},[31685,31821],[34,737844,31829],{"className":737845},[31828],[34,737847,737849],{"className":737848},[31801],[34,737850,737852],{"className":737851,"style":39021},[31805],[34,737853],{},[34,737855],{"className":737856,"style":32123},[31845],[34,737858,736810],{"className":737859},[32767],[34,737861],{"className":737862,"style":32123},[31845],[34,737864,33249],{"className":737865},[31685,31686],[32188,737867],{},[253,737869,737871],{"id":737870},"_5-朗斯基行列式的导数刘维尔公式","5. 朗斯基行列式的导数（刘维尔公式）",[15,737873,737874,737875,347],{},"这是一个理论证明题，核心在于",[2627,737876,737877],{},"行列式怎么求导",[15,737879,737880,737883,737884],{},[2627,737881,737882],{},"证明思路：","\n行列式求导法则：",[2627,737885,737886],{},"等于“每次只对一行求导，其他行不变”的行列式之和。",[15,737888,59432,737889,737917,737918,2664],{},[34,737890,737892,737905],{"className":737891},[31642],[34,737893,737895],{"className":737894},[31646],[31648,737896,737897],{"xmlns":31650},[31652,737898,737899,737903],{},[31655,737900,737901],{},[31658,737902,31660],{},[31665,737904,31660],{"encoding":31667},[34,737906,737908],{"className":737907,"ariaHidden":758},[31672],[34,737909,737911,737914],{"className":737910},[31676],[34,737912],{"className":737913,"style":32527},[31680],[34,737915,31660],{"className":737916},[31685,31686]," 阶朗斯基行列式 ",[34,737919,737921,737941],{"className":737920},[31642],[34,737922,737924],{"className":737923},[31646],[31648,737925,737926],{"xmlns":31650},[31652,737927,737928,737938],{},[31655,737929,737930,737932,737934,737936],{},[31658,737931,56852],{},[31662,737933,456],{"stretchy":702},[31658,737935,33249],{},[31662,737937,2557],{"stretchy":702},[31665,737939,737940],{"encoding":31667},"W(t)",[34,737942,737944],{"className":737943,"ariaHidden":758},[31672],[34,737945,737947,737950,737953,737956,737959],{"className":737946},[31676],[34,737948],{"className":737949,"style":31774},[31680],[34,737951,56852],{"className":737952,"style":37422},[31685,31686],[34,737954,456],{"className":737955},[31782],[34,737957,33249],{"className":737958},[31685,31686],[34,737960,2557],{"className":737961},[31690],[2230,737963,737964,738233,738308,738456,739350,740116,740464,740586],{},[108,737965,737966,737967,738037,738038,738124,738125,347],{},"第一行是 ",[34,737968,737970,737988],{"className":737969},[31642],[34,737971,737973],{"className":737972},[31646],[31648,737974,737975],{"xmlns":31650},[31652,737976,737977,737985],{},[31655,737978,737979],{},[31743,737980,737981,737983],{},[31658,737982,32936],{},[31658,737984,31955],{},[31665,737986,737987],{"encoding":31667},"x_i",[34,737989,737991],{"className":737990,"ariaHidden":758},[31672],[34,737992,737994,737997],{"className":737993},[31676],[34,737995],{"className":737996,"style":31860},[31680],[34,737998,738000,738003],{"className":737999},[31685],[34,738001,32936],{"className":738002},[31685,31686],[34,738004,738006],{"className":738005},[31792],[34,738007,738009,738029],{"className":738008},[31796,31797],[34,738010,738012,738026],{"className":738011},[31801],[34,738013,738015],{"className":738014,"style":32090},[31805],[34,738016,738017,738020],{"style":31809},[34,738018],{"className":738019,"style":31814},[31813],[34,738021,738023],{"className":738022},[31818,31819,31820,31821],[34,738024,31955],{"className":738025},[31685,31686,31821],[34,738027,31829],{"className":738028},[31828],[34,738030,738032],{"className":738031},[31801],[34,738033,738035],{"className":738034,"style":31836},[31805],[34,738036],{},"，第二行是 ",[34,738039,738041,738061],{"className":738040},[31642],[34,738042,738044],{"className":738043},[31646],[31648,738045,738046],{"xmlns":31650},[31652,738047,738048,738058],{},[31655,738049,738050],{},[38846,738051,738052,738054,738056],{},[31658,738053,32936],{},[31658,738055,31955],{},[31662,738057,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,738059,738060],{"encoding":31667},"x_i'",[34,738062,738064],{"className":738063,"ariaHidden":758},[31672],[34,738065,738067,738070],{"className":738066},[31676],[34,738068],{"className":738069,"style":420875},[31680],[34,738071,738073,738076],{"className":738072},[31685],[34,738074,32936],{"className":738075},[31685,31686],[34,738077,738079],{"className":738078},[31792],[34,738080,738082,738116],{"className":738081},[31796,31797],[34,738083,738085,738113],{"className":738084},[31801],[34,738086,738088,738099],{"className":738087,"style":132027},[31805],[34,738089,738090,738093],{"style":503473},[34,738091],{"className":738092,"style":31814},[31813],[34,738094,738096],{"className":738095},[31818,31819,31820,31821],[34,738097,31955],{"className":738098},[31685,31686,31821],[34,738100,738101,738104],{"style":37777},[34,738102],{"className":738103,"style":31814},[31813],[34,738105,738107],{"className":738106},[31818,31819,31820,31821],[34,738108,738110],{"className":738109},[31685,31821],[34,738111,116495],{"className":738112},[31685,31821],[34,738114,31829],{"className":738115},[31828],[34,738117,738119],{"className":738118},[31801],[34,738120,738122],{"className":738121,"style":39572},[31805],[34,738123],{},"，……，最后一行是 ",[34,738126,738128,738158],{"className":738127},[31642],[34,738129,738131],{"className":738130},[31646],[31648,738132,738133],{"xmlns":31650},[31652,738134,738135,738155],{},[31655,738136,738137],{},[38846,738138,738139,738141,738143],{},[31658,738140,32936],{},[31658,738142,31955],{},[31655,738144,738145,738147,738149,738151,738153],{},[31662,738146,456],{"stretchy":702},[31658,738148,31660],{},[31662,738150,32226],{},[31749,738152,640],{},[31662,738154,2557],{"stretchy":702},[31665,738156,738157],{"encoding":31667},"x_i^{(n-1)}",[34,738159,738161],{"className":738160,"ariaHidden":758},[31672],[34,738162,738164,738167],{"className":738163},[31676],[34,738165],{"className":738166,"style":152559},[31680],[34,738168,738170,738173],{"className":738169},[31685],[34,738171,32936],{"className":738172},[31685,31686],[34,738174,738176],{"className":738175},[31792],[34,738177,738179,738225],{"className":738178},[31796,31797],[34,738180,738182,738222],{"className":738181},[31801],[34,738183,738185,738196],{"className":738184,"style":152022},[31805],[34,738186,738187,738190],{"style":152025},[34,738188],{"className":738189,"style":31814},[31813],[34,738191,738193],{"className":738192},[31818,31819,31820,31821],[34,738194,31955],{"className":738195},[31685,31686,31821],[34,738197,738198,738201],{"style":152040},[34,738199],{"className":738200,"style":31814},[31813],[34,738202,738204],{"className":738203},[31818,31819,31820,31821],[34,738205,738207,738210,738213,738216,738219],{"className":738206},[31685,31821],[34,738208,456],{"className":738209},[31782,31821],[34,738211,31660],{"className":738212},[31685,31686,31821],[34,738214,32226],{"className":738215},[32024,31821],[34,738217,640],{"className":738218},[31685,31821],[34,738220,2557],{"className":738221},[31690,31821],[34,738223,31829],{"className":738224},[31828],[34,738226,738228],{"className":738227},[31801],[34,738229,738231],{"className":738230,"style":152621},[31805],[34,738232],{},[108,738234,3101,738235,738278,738279,738307],{},[34,738236,738238,738257],{"className":738237},[31642],[34,738239,738241],{"className":738240},[31646],[31648,738242,738243],{"xmlns":31650},[31652,738244,738245,738255],{},[31655,738246,738247,738249,738251,738253],{},[31658,738248,56852],{},[31662,738250,456],{"stretchy":702},[31658,738252,33249],{},[31662,738254,2557],{"stretchy":702},[31665,738256,737940],{"encoding":31667},[34,738258,738260],{"className":738259,"ariaHidden":758},[31672],[34,738261,738263,738266,738269,738272,738275],{"className":738262},[31676],[34,738264],{"className":738265,"style":31774},[31680],[34,738267,56852],{"className":738268,"style":37422},[31685,31686],[34,738270,456],{"className":738271},[31782],[34,738273,33249],{"className":738274},[31685,31686],[34,738276,2557],{"className":738277},[31690]," 求导，会得到 ",[34,738280,738282,738295],{"className":738281},[31642],[34,738283,738285],{"className":738284},[31646],[31648,738286,738287],{"xmlns":31650},[31652,738288,738289,738293],{},[31655,738290,738291],{},[31658,738292,31660],{},[31665,738294,31660],{"encoding":31667},[34,738296,738298],{"className":738297,"ariaHidden":758},[31672],[34,738299,738301,738304],{"className":738300},[31676],[34,738302],{"className":738303,"style":32527},[31680],[34,738305,31660],{"className":738306},[31685,31686]," 个行列式的和。",[108,738309,738310,738312,738313,738363,738364,738367,738368,738453,738454,347],{},[2627,738311,533838],{},"：前 ",[34,738314,738316,738333],{"className":738315},[31642],[34,738317,738319],{"className":738318},[31646],[31648,738320,738321],{"xmlns":31650},[31652,738322,738323,738331],{},[31655,738324,738325,738327,738329],{},[31658,738326,31660],{},[31662,738328,32226],{},[31749,738330,640],{},[31665,738332,32587],{"encoding":31667},[34,738334,738336,738354],{"className":738335,"ariaHidden":758},[31672],[34,738337,738339,738342,738345,738348,738351],{"className":738338},[31676],[34,738340],{"className":738341,"style":32597},[31680],[34,738343,31660],{"className":738344},[31685,31686],[34,738346],{"className":738347,"style":32604},[31845],[34,738349,32226],{"className":738350},[32024],[34,738352],{"className":738353,"style":32604},[31845],[34,738355,738357,738360],{"className":738356},[31676],[34,738358],{"className":738359,"style":32617},[31680],[34,738361,640],{"className":738362},[31685]," 个行列式中，都会出现",[2627,738365,738366],{},"两行完全一样","的情况（比如对第一行求导变成 ",[34,738369,738371,738390],{"className":738370},[31642],[34,738372,738374],{"className":738373},[31646],[31648,738375,738376],{"xmlns":31650},[31652,738377,738378,738388],{},[31655,738379,738380],{},[38846,738381,738382,738384,738386],{},[31658,738383,32936],{},[31658,738385,31955],{},[31662,738387,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,738389,738060],{"encoding":31667},[34,738391,738393],{"className":738392,"ariaHidden":758},[31672],[34,738394,738396,738399],{"className":738395},[31676],[34,738397],{"className":738398,"style":420875},[31680],[34,738400,738402,738405],{"className":738401},[31685],[34,738403,32936],{"className":738404},[31685,31686],[34,738406,738408],{"className":738407},[31792],[34,738409,738411,738445],{"className":738410},[31796,31797],[34,738412,738414,738442],{"className":738413},[31801],[34,738415,738417,738428],{"className":738416,"style":132027},[31805],[34,738418,738419,738422],{"style":503473},[34,738420],{"className":738421,"style":31814},[31813],[34,738423,738425],{"className":738424},[31818,31819,31820,31821],[34,738426,31955],{"className":738427},[31685,31686,31821],[34,738429,738430,738433],{"style":37777},[34,738431],{"className":738432,"style":31814},[31813],[34,738434,738436],{"className":738435},[31818,31819,31820,31821],[34,738437,738439],{"className":738438},[31685,31821],[34,738440,116495],{"className":738441},[31685,31821],[34,738443,31829],{"className":738444},[31828],[34,738446,738448],{"className":738447},[31801],[34,738449,738451],{"className":738450,"style":39572},[31805],[34,738452],{},"，这就和原来的第二行一样了）。两行一样的行列式等于 ",[2627,738455,459],{},[108,738457,738458,738461,738462,738465,738466,721062],{},[2627,738459,738460],{},"唯一剩下的","：只有对",[2627,738463,738464],{},"最后一行","求导的那个行列式不为0。\n[\n",[34,738467,738469,738654],{"className":738468},[31642],[34,738470,738472],{"className":738471},[31646],[31648,738473,738474],{"xmlns":31650},[31652,738475,738476,738651],{},[31655,738477,738478,738484,738486],{},[37624,738479,738480,738482],{},[31658,738481,56852],{},[31662,738483,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,738485,202],{},[31655,738487,738488,738490,738649],{},[31662,738489,162086],{"fence":758},[71078,738491,738493,738521,738553,738605],{"rowspacing":370541,"columnalign":738492,"columnspacing":71082},"center center 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= \\begin{vmatrix} x_1 & \\cdots & x_n \\\\ \\vdots & & \\vdots \\\\ x_1^{(n-2)} & \\cdots & x_n^{(n-2)} \\\\ \\mathbf{x_1^{(n)}} & \\cdots & \\mathbf{x_n^{(n)}} \\end{vmatrix}",[34,738655,738657,738704],{"className":738656,"ariaHidden":758},[31672],[34,738658,738660,738663,738695,738698,738701],{"className":738659},[31676],[34,738661],{"className":738662,"style":132027},[31680],[34,738664,738666,738669],{"className":738665},[31685],[34,738667,56852],{"className":738668,"style":37422},[31685,31686],[34,738670,738672],{"className":738671},[31792],[34,738673,738675],{"className":738674},[31796],[34,738676,738678],{"className":738677},[31801],[34,738679,738681],{"className":738680,"style":132027},[31805],[34,738682,738683,738686],{"style":37777},[34,738684],{"className":738685,"style":31814},[31813],[34,738687,738689],{"className":738688},[31818,31819,31820,31821],[34,738690,738692],{"className":738691},[31685,31821],[34,738693,116495],{"className":738694},[31685,31821],[34,738696],{"className":738697,"style":31846},[31845],[34,738699,202],{"className":738700},[31850],[34,738702],{"className":738703,"style":31846},[31845],[34,738705,738707,738711],{"className":738706},[31676],[34,738708],{"className":738709,"style":738710},[31680],"height:5.8696em;vertical-align:-2.6848em;",[34,738712,738714,738757,739313],{"className":738713},[34006],[34,738715,738717],{"className":738716},[31782],[34,738718,738720],{"className":738719},[71281,71282],[34,738721,738723,738749],{"className":738722},[31796,31797],[34,738724,738726,738746],{"className":738725},[31801],[34,738727,738729],{"className":738728,"style":80117},[31805],[34,738730,738732,738736],{"style":738731},"top:-4.95em;",[34,738733],{"className":738734,"style":738735},[31813],"height:7.4em;",[34,738737,738739],{"style":738738},"width:0.333em;height:5.4em;",[41588,738740,738743],{"xmlns":41590,"width":503054,"height":738741,"viewBox":738742},"5.4em","0 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lassName":739280},[31685,31821],[34,739282,456],{"className":739283},[31782,31821],[34,739285,31660],{"className":739286},[31685,226302,31821],[34,739288,2557],{"className":739289},[31690,31821],[34,739291,31829],{"className":739292},[31828],[34,739294,739296],{"className":739295},[31801],[34,739297,739300],{"className":739298,"style":739299},[31805],"height:0.1263em;",[34,739301],{},[34,739303,31829],{"className":739304},[31828],[34,739306,739308],{"className":739307},[31801],[34,739309,739311],{"className":739310,"style":738996},[31805],[34,739312],{},[34,739314,739316],{"className":739315},[31690],[34,739317,739319],{"className":739318},[71281,71282],[34,739320,739322,739342],{"className":739321},[31796,31797],[34,739323,739325,739339],{"className":739324},[31801],[34,739326,739328],{"className":739327,"style":80117},[31805],[34,739329,739330,739333],{"style":738731},[34,739331],{"className":739332,"style":738735},[31813],[34,739334,739335],{"style":738738},[41588,739336,739337],{"xmlns":41590,"width":503054,"height":738741,"viewBox":738742},[20398,739338],{"d":738745},[34,739340,31829],{"className":739341},[31828],[34,739343,739345],{"className":739344},[31801],[34,739346,739348],{"className":739347,"style":80186},[31805],[34,739349],{},[108,739351,739352,332541,739355,703145,739424,739734,739735],{},[2627,739353,739354],{},"代入方程",[34,739356,739358,739375],{"className":739357},[31642],[34,739359,739361],{"className":739360},[31646],[31648,739362,739363],{"xmlns":31650},[31652,739364,739365,739373],{},[31655,739366,739367],{},[31743,739368,739369,739371],{},[31658,739370,32936],{},[31658,739372,31955],{},[31665,739374,737987],{"encoding":31667},[34,739376,739378],{"className":739377,"ariaHidden":758},[31672],[34,739379,739381,739384],{"className":739380},[31676],[34,739382],{"className":739383,"style":31860},[31680],[34,739385,739387,739390],{"className":739386},[31685],[34,739388,32936],{"className":739389},[31685,31686],[34,739391,739393],{"className":739392},[31792],[34,739394,739396,739416],{"className":739395},[31796,31797],[34,739397,739399,739413],{"className":739398},[31801],[34,739400,739402],{"className":739401,"style":32090},[31805],[34,739403,739404,739407],{"style":31809},[34,739405],{"className":739406,"style":31814},[31813],[34,739408,739410],{"className":739409},[31818,31819,31820,31821],[34,739411,31955],{"className":739412},[31685,31686,31821],[34,739414,31829],{"className":739415},[31828],[34,739417,739419],{"className":739418},[31801],[34,739420,739422],{"className":739421,"style":31836},[31805],[34,739423],{},[34,739425,739427,739493],{"className":739426},[31642],[34,739428,739430],{"className":739429},[31646],[31648,739431,739432],{"xmlns":31650},[31652,739433,739434,739490],{},[31655,739435,739436,739448,739450,739456,739472,739474,739476,739478,739484,739486,739488],{},[37624,739437,739438,739440],{},[31658,739439,32936],{},[31655,739441,739442,739444,739446],{},[31662,739443,456],{"stretchy":702},[31658,739445,31660],{},[31662,739447,2557],{"stretchy":702},[31662,739449,223],{},[31743,739451,739452,739454],{},[31658,739453,111],{},[31749,739455,640],{},[37624,739457,739458,739460],{},[31658,739459,32936],{},[31655,739461,739462,739464,739466,739468,739470],{},[31662,739463,456],{"stretchy":702},[31658,739465,31660],{},[31662,739467,32226],{},[31749,739469,640],{},[31662,739471,2557],{"stretchy":702},[31662,739473,223],{},[31662,739475,38874],{},[31662,739477,223],{},[31743,739479,739480,739482],{},[31658,739481,111],{},[31658,739483,31660],{},[31658,739485,32936],{},[31662,739487,202],{},[31749,739489,459],{},[31665,739491,739492],{"encoding":31667},"x^{(n)} + a_1 x^{(n-1)} + \\cdots + a_n x = 0",[34,739494,739496,739549,739649,739667,739725],{"className":739495,"ariaHidden":758},[31672],[34,739497,739499,739502,739540,739543,739546],{"className":739498},[31676],[34,739500],{"className":739501,"style":640505},[31680],[34,739503,739505,739508],{"className":739504},[31685],[34,739506,32936],{"className":739507},[31685,31686],[34,739509,739511],{"className":739510},[31792],[34,739512,739514],{"className":739513},[31796],[34,739515,739517],{"className":739516},[31801],[34,739518,739520],{"className":739519,"style":61037},[31805],[34,739521,739522,739525],{"style":37777},[34,739523],{"className":739524,"style":31814},[31813],[34,739526,739528],{"className":739527},[31818,31819,31820,31821],[34,739529,739531,739534,739537],{"className":739530},[31685,31821],[34,739532,456],{"className":739533},[31782,31821],[34,739535,31660],{"className":739536},[31685,31686,31821],[34,739538,2557],{"className":739539},[31690,31821],[34,739541],{"className":739542,"style":32604},[31845],[34,739544,223],{"className":739545},[32024],[34,739547],{"className":739548,"style":32604},[31845],[34,739550,739552,739556,739596,739640,739643,739646],{"className":739551},[31676],[34,739553],{"className":739554,"style":739555},[31680],"height:1.038em;vertical-align:-0.15em;",[34,739557,739559,739562],{"className":739558},[31685],[34,739560,111],{"className":739561},[31685,31686],[34,739563,739565],{"className":739564},[31792],[34,739566,739568,739588],{"className":739567},[31796,31797],[34,739569,739571,739585],{"className":739570},[31801],[34,739572,739574],{"className":739573,"style":31806},[31805],[34,739575,739576,739579],{"style":31809},[34,739577],{"className":739578,"style":31814},[31813],[34,739580,739582],{"className":739581},[31818,31819,31820,31821],[34,739583,640],{"className":739584},[31685,31821],[34,739586,31829],{"className":739587},[31828],[34,739589,739591],{"className":739590},[31801],[34,739592,739594],{"className":739593,"style":31836},[31805],[34,739595],{},[34,739597,739599,739602],{"className":739598},[31685],[34,739600,32936],{"className":739601},[31685,31686],[34,739603,739605],{"className":739604},[31792],[34,739606,739608],{"className":739607},[31796],[34,739609,739611],{"className":739610},[31801],[34,739612,739614],{"className":739613,"style":61037},[31805],[34,739615,739616,739619],{"style":37777},[34,739617],{"className":739618,"style":31814},[31813],[34,739620,739622],{"className":739621},[31818,31819,31820,31821],[34,739623,739625,739628,739631,739634,739637],{"className":739624},[31685,31821],[34,739626,456],{"className":739627},[31782,31821],[34,739629,31660],{"className":739630},[31685,31686,31821],[34,739632,32226],{"className":739633},[32024,31821],[34,739635,640],{"className":739636},[31685,31821],[34,739638,2557],{"className":739639},[31690,31821],[34,739641],{"className":739642,"style":32604},[31845],[34,739644,223],{"className":739645},[32024],[34,739647],{"className":739648,"style":32604},[31845],[34,739650,739652,739655,739658,739661,739664],{"className":739651},[31676],[34,739653],{"className":739654,"style":32597},[31680],[34,739656,38874],{"className":739657},[34006],[34,739659],{"className":739660,"style":32604},[31845],[34,739662,223],{"className":739663},[32024],[34,739665],{"className":739666,"style":32604},[31845],[34,739668,739670,739673,739713,739716,739719,739722],{"className":739669},[31676],[34,739671],{"className":739672,"style":31860},[31680],[34,739674,739676,739679],{"className":739675},[31685],[34,739677,111],{"className":739678},[31685,31686],[34,739680,739682],{"className":739681},[31792],[34,739683,739685,739705],{"className":739684},[31796,31797],[34,739686,739688,739702],{"className":739687},[31801],[34,739689,739691],{"className":739690,"style":32362},[31805],[34,739692,739693,739696],{"style":31809},[34,739694],{"className":739695,"style":31814},[31813],[34,739697,739699],{"className":739698},[31818,31819,31820,31821],[34,739700,31660],{"className":739701},[31685,31686,31821],[34,739703,31829],{"className":739704},[31828],[34,739706,739708],{"className":739707},[31801],[34,739709,739711],{"className":739710,"style":31836},[31805],[34,739712],{},[34,739714,32936],{"className":739715},[31685,31686],[34,739717],{"className":739718,"style":31846},[31845],[34,739720,202],{"className":739721},[31850],[34,739723],{"className":739724,"style":31846},[31845],[34,739726,739728,739731],{"className":739727},[31676],[34,739729],{"className":739730,"style":32617},[31680],[34,739732,459],{"className":739733},[31685]," 的解，所以：\n",[34,739736,739738,739810],{"className":739737},[31642],[34,739739,739741],{"className":739740},[31646],[31648,739742,739743],{"xmlns":31650},[31652,739744,739745,739807],{},[31655,739746,739747,739761,739763,739765,739771,739789,739791,739793,739795,739801],{},[38846,739748,739749,739751,739753],{},[31658,739750,32936],{},[31658,739752,31955],{},[31655,739754,739755,739757,739759],{},[31662,739756,456],{"stretchy":702},[31658,739758,31660],{},[31662,739760,2557],{"stretchy":702},[31662,739762,202],{},[31662,739764,32226],{},[31743,739766,739767,739769],{},[31658,739768,111],{},[31749,739770,640],{},[38846,739772,739773,739775,739777],{},[31658,739774,32936],{},[31658,739776,31955],{},[31655,739778,739779,739781,739783,739785,739787],{},[31662,739780,456],{"stretchy":702},[31658,739782,31660],{},[31662,739784,32226],{},[31749,739786,640],{},[31662,739788,2557],{"stretchy":702},[31662,739790,32226],{},[31662,739792,38874],{},[31662,739794,32226],{},[31743,739796,739797,739799],{},[31658,739798,111],{},[31658,739800,31660],{},[31743,739802,739803,739805],{},[31658,739804,32936],{},[31658,739806,31955],{},[31665,739808,739809],{"encoding":31667},"x_i^{(n)} = -a_1 x_i^{(n-1)} - \\cdots - a_n x_i",[34,739811,739813,739888,740012,740030],{"className":739812,"ariaHidden":758},[31672],[34,739814,739816,739819,739879,739882,739885],{"className":739815},[31676],[34,739817],{"className":739818,"style":152559},[31680],[34,739820,739822,739825],{"className":739821},[31685],[34,739823,32936],{"className":739824},[31685,31686],[34,739826,739828],{"className":739827},[31792],[34,739829,739831,739871],{"className":739830},[31796,31797],[34,739832,739834,739868],{"className":739833},[31801],[34,739835,739837,739848],{"className":739836,"style":152022},[31805],[34,739838,739839,739842],{"style":152025},[34,739840],{"className":739841,"style":31814},[31813],[34,739843,739845],{"className":739844},[31818,31819,31820,31821],[34,739846,31955],{"className":739847},[31685,31686,31821],[34,739849,739850,739853],{"style":152040},[34,739851],{"className":739852,"style":31814},[31813],[34,739854,739856],{"className":739855},[31818,31819,31820,31821],[34,739857,739859,739862,739865],{"className":739858},[31685,31821],[34,739860,456],{"className":739861},[31782,31821],[34,739863,31660],{"className":739864},[31685,31686,31821],[34,739866,2557],{"className":739867},[31690,31821],[34,739869,31829],{"className":739870},[31828],[34,739872,739874],{"className":739873},[31801],[34,739875,739877],{"className":739876,"style":152621},[31805],[34,739878],{},[34,739880],{"className":739881,"style":31846},[31845],[34,739883,202],{"className":739884},[31850],[34,739886],{"className":739887,"style":31846},[31845],[34,739889,739891,739894,739897,739937,740003,740006,740009],{"className":739890},[31676],[34,739892],{"className":739893,"style":152559},[31680],[34,739895,32226],{"className":739896},[31685],[34,739898,739900,739903],{"className":739899},[31685],[34,739901,111],{"className":739902},[31685,31686],[34,739904,739906],{"className":739905},[31792],[34,739907,739909,739929],{"className":739908},[31796,31797],[34,739910,739912,739926],{"className":739911},[31801],[34,739913,739915],{"className":739914,"style":31806},[31805],[34,739916,739917,739920],{"style":31809},[34,739918],{"className":739919,"style":31814},[31813],[34,739921,739923],{"className":739922},[31818,31819,31820,31821],[34,739924,640],{"className":739925},[31685,31821],[34,739927,31829],{"className":739928},[31828],[34,739930,739932],{"className":739931},[31801],[34,739933,739935],{"className":739934,"style":31836},[31805],[34,739936],{},[34,739938,739940,739943],{"className":739939},[31685],[34,739941,32936],{"className":739942},[31685,31686],[34,739944,739946],{"className":739945},[31792],[34,739947,739949,739995],{"className":739948},[31796,31797],[34,739950,739952,739992],{"className":739951},[31801],[34,739953,739955,739966],{"className":739954,"style":152022},[31805],[34,739956,739957,739960],{"style":152025},[34,739958],{"className":739959,"style":31814},[31813],[34,739961,739963],{"className":739962},[31818,31819,31820,31821],[34,739964,31955],{"className":739965},[31685,31686,31821],[34,739967,739968,739971],{"style":152040},[34,739969],{"className":739970,"style":31814},[31813],[34,739972,739974],{"className":739973},[31818,31819,31820,31821],[34,739975,739977,739980,739983,739986,739989],{"className":739976},[31685,31821],[34,739978,456],{"className":739979},[31782,31821],[34,739981,31660],{"className":739982},[31685,31686,31821],[34,739984,32226],{"className":739985},[32024,31821],[34,739987,640],{"className":739988},[31685,31821],[34,739990,2557],{"className":739991},[31690,31821],[34,739993,31829],{"className":739994},[31828],[34,739996,739998],{"className":739997},[31801],[34,739999,740001],{"className":740000,"style":152621},[31805],[34,740002],{},[34,740004],{"className":740005,"style":32604},[31845],[34,740007,32226],{"className":740008},[32024],[34,740010],{"className":740011,"style":32604},[31845],[34,740013,740015,740018,740021,740024,740027],{"className":740014},[31676],[34,740016],{"className":740017,"style":32597},[31680],[34,740019,38874],{"className":740020},[34006],[34,740022],{"className":740023,"style":32604},[31845],[34,740025,32226],{"className":740026},[32024],[34,740028],{"className":740029,"style":32604},[31845],[34,740031,740033,740036,740076],{"className":740032},[31676],[34,740034],{"className":740035,"style":31860},[31680],[34,740037,740039,740042],{"className":740038},[31685],[34,740040,111],{"className":740041},[31685,31686],[34,740043,740045],{"className":740044},[31792],[34,740046,740048,740068],{"className":740047},[31796,31797],[34,740049,740051,740065],{"className":740050},[31801],[34,740052,740054],{"className":740053,"style":32362},[31805],[34,740055,740056,740059],{"style":31809},[34,740057],{"className":740058,"style":31814},[31813],[34,740060,740062],{"className":740061},[31818,31819,31820,31821],[34,740063,31660],{"className":740064},[31685,31686,31821],[34,740066,31829],{"className":740067},[31828],[34,740069,740071],{"className":740070},[31801],[34,740072,740074],{"className":740073,"style":31836},[31805],[34,740075],{},[34,740077,740079,740082],{"className":740078},[31685],[34,740080,32936],{"className":740081},[31685,31686],[34,740083,740085],{"className":740084},[31792],[34,740086,740088,740108],{"className":740087},[31796,31797],[34,740089,740091,740105],{"className":740090},[31801],[34,740092,740094],{"className":740093,"style":32090},[31805],[34,740095,740096,740099],{"style":31809},[34,740097],{"className":740098,"style":31814},[31813],[34,740100,740102],{"className":740101},[31818,31819,31820,31821],[34,740103,31955],{"className":740104},[31685,31686,31821],[34,740106,31829],{"className":740107},[31828],[34,740109,740111],{"className":740110},[31801],[34,740112,740114],{"className":740113,"style":31836},[31805],[34,740115],{},[108,740117,740118,740119,740181,740182,740341,740342,740463],{},"把这个代入到 ",[34,740120,740122,740140],{"className":740121},[31642],[34,740123,740125],{"className":740124},[31646],[31648,740126,740127],{"xmlns":31650},[31652,740128,740129,740137],{},[31655,740130,740131],{},[37624,740132,740133,740135],{},[31658,740134,56852],{},[31662,740136,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,740138,740139],{"encoding":31667},"W'",[34,740141,740143],{"className":740142,"ariaHidden":758},[31672],[34,740144,740146,740149],{"className":740145},[31676],[34,740147],{"className":740148,"style":132027},[31680],[34,740150,740152,740155],{"className":740151},[31685],[34,740153,56852],{"className":740154,"style":37422},[31685,31686],[34,740156,740158],{"className":740157},[31792],[34,740159,740161],{"className":740160},[31796],[34,740162,740164],{"className":740163},[31801],[34,740165,740167],{"className":740166,"style":132027},[31805],[34,740168,740169,740172],{"style":37777},[34,740170],{"className":740171,"style":31814},[31813],[34,740173,740175],{"className":740174},[31818,31819,31820,31821],[34,740176,740178],{"className":740177},[31685,31821],[34,740179,116495],{"className":740180},[31685,31821]," 的最后一行。利用行列式拆分性质，只有包含 ",[34,740183,740185,740223],{"className":740184},[31642],[34,740186,740188],{"className":740187},[31646],[31648,740189,740190],{"xmlns":31650},[31652,740191,740192,740220],{},[31655,740193,740194,740196,740202],{},[31662,740195,32226],{},[31743,740197,740198,740200],{},[31658,740199,111],{},[31749,740201,640],{},[38846,740203,740204,740206,740208],{},[31658,740205,32936],{},[31658,740207,31955],{},[31655,740209,740210,740212,740214,740216,740218],{},[31662,740211,456],{"stretchy":702},[31658,740213,31660],{},[31662,740215,32226],{},[31749,740217,640],{},[31662,740219,2557],{"stretchy":702},[31665,740221,740222],{"encoding":31667},"-a_1 x_i^{(n-1)}",[34,740224,740226],{"className":740225,"ariaHidden":758},[31672],[34,740227,740229,740232,740235,740275],{"className":740228},[31676],[34,740230],{"className":740231,"style":152559},[31680],[34,740233,32226],{"className":740234},[31685],[34,740236,740238,740241],{"className":740237},[31685],[34,740239,111],{"className":740240},[31685,31686],[34,740242,740244],{"className":740243},[31792],[34,740245,740247,740267],{"className":740246},[31796,31797],[34,740248,740250,740264],{"className":740249},[31801],[34,740251,740253],{"className":740252,"style":31806},[31805],[34,740254,740255,740258],{"style":31809},[34,740256],{"className":740257,"style":31814},[31813],[34,740259,740261],{"className":740260},[31818,31819,31820,31821],[34,740262,640],{"className":740263},[31685,31821],[34,740265,31829],{"className":740266},[31828],[34,740268,740270],{"className":740269},[31801],[34,740271,740273],{"className":740272,"style":31836},[31805],[34,740274],{},[34,740276,740278,740281],{"className":740277},[31685],[34,740279,32936],{"className":740280},[31685,31686],[34,740282,740284],{"className":740283},[31792],[34,740285,740287,740333],{"className":740286},[31796,31797],[34,740288,740290,740330],{"className":740289},[31801],[34,740291,740293,740304],{"className":740292,"style":152022},[31805],[34,740294,740295,740298],{"style":152025},[34,740296],{"className":740297,"style":31814},[31813],[34,740299,740301],{"className":740300},[31818,31819,31820,31821],[34,740302,31955],{"className":740303},[31685,31686,31821],[34,740305,740306,740309],{"style":152040},[34,740307],{"className":740308,"style":31814},[31813],[34,740310,740312],{"className":740311},[31818,31819,31820,31821],[34,740313,740315,740318,740321,740324,740327],{"className":740314},[31685,31821],[34,740316,456],{"className":740317},[31782,31821],[34,740319,31660],{"className":740320},[31685,31686,31821],[34,740322,32226],{"className":740323},[32024,31821],[34,740325,640],{"className":740326},[31685,31821],[34,740328,2557],{"className":740329},[31690,31821],[34,740331,31829],{"className":740332},[31828],[34,740334,740336],{"className":740335},[31801],[34,740337,740339],{"className":740338,"style":152621},[31805],[34,740340],{}," 的那一项是非零的（因为其他的项如 ",[34,740343,740345,740371],{"className":740344},[31642],[34,740346,740348],{"className":740347},[31646],[31648,740349,740350],{"xmlns":31650},[31652,740351,740352,740368],{},[31655,740353,740354,740356,740362],{},[31662,740355,32226],{},[31743,740357,740358,740360],{},[31658,740359,111],{},[31658,740361,31660],{},[31743,740363,740364,740366],{},[31658,740365,32936],{},[31658,740367,31955],{},[31665,740369,740370],{"encoding":31667},"-a_n x_i",[34,740372,740374],{"className":740373,"ariaHidden":758},[31672],[34,740375,740377,740380,740383,740423],{"className":740376},[31676],[34,740378],{"className":740379,"style":40522},[31680],[34,740381,32226],{"className":740382},[31685],[34,740384,740386,740389],{"className":740385},[31685],[34,740387,111],{"className":740388},[31685,31686],[34,740390,740392],{"className":740391},[31792],[34,740393,740395,740415],{"className":740394},[31796,31797],[34,740396,740398,740412],{"className":740397},[31801],[34,740399,740401],{"className":740400,"style":32362},[31805],[34,740402,740403,740406],{"style":31809},[34,740404],{"className":740405,"style":31814},[31813],[34,740407,740409],{"className":740408},[31818,31819,31820,31821],[34,740410,31660],{"className":740411},[31685,31686,31821],[34,740413,31829],{"className":740414},[31828],[34,740416,740418],{"className":740417},[31801],[34,740419,740421],{"className":740420,"style":31836},[31805],[34,740422],{},[34,740424,740426,740429],{"className":740425},[31685],[34,740427,32936],{"className":740428},[31685,31686],[34,740430,740432],{"className":740431},[31792],[34,740433,740435,740455],{"className":740434},[31796,31797],[34,740436,740438,740452],{"className":740437},[31801],[34,740439,740441],{"className":740440,"style":32090},[31805],[34,740442,740443,740446],{"style":31809},[34,740444],{"className":740445,"style":31814},[31813],[34,740447,740449],{"className":740448},[31818,31819,31820,31821],[34,740450,31955],{"className":740451},[31685,31686,31821],[34,740453,31829],{"className":740454},[31828],[34,740456,740458],{"className":740457},[31801],[34,740459,740461],{"className":740460,"style":31836},[31805],[34,740462],{}," 会让最后一行和第一行成比例，行列式为0）。",[108,740465,740466,740467,740542,740543,347],{},"提取公因子 ",[34,740468,740470,740490],{"className":740469},[31642],[34,740471,740473],{"className":740472},[31646],[31648,740474,740475],{"xmlns":31650},[31652,740476,740477,740487],{},[31655,740478,740479,740481],{},[31662,740480,32226],{},[31743,740482,740483,740485],{},[31658,740484,111],{},[31749,740486,640],{},[31665,740488,740489],{"encoding":31667},"-a_1",[34,740491,740493],{"className":740492,"ariaHidden":758},[31672],[34,740494,740496,740499,740502],{"className":740495},[31676],[34,740497],{"className":740498,"style":40522},[31680],[34,740500,32226],{"className":740501},[31685],[34,740503,740505,740508],{"className":740504},[31685],[34,740506,111],{"className":740507},[31685,31686],[34,740509,740511],{"className":740510},[31792],[34,740512,740514,740534],{"className":740513},[31796,31797],[34,740515,740517,740531],{"className":740516},[31801],[34,740518,740520],{"className":740519,"style":31806},[31805],[34,740521,740522,740525],{"style":31809},[34,740523],{"className":740524,"style":31814},[31813],[34,740526,740528],{"className":740527},[31818,31819,31820,31821],[34,740529,640],{"className":740530},[31685,31821],[34,740532,31829],{"className":740533},[31828],[34,740535,740537],{"className":740536},[31801],[34,740538,740540],{"className":740539,"style":31836},[31805],[34,740541],{},"，剩下的正是原行列式 ",[34,740544,740546,740565],{"className":740545},[31642],[34,740547,740549],{"className":740548},[31646],[31648,740550,740551],{"xmlns":31650},[31652,740552,740553,740563],{},[31655,740554,740555,740557,740559,740561],{},[31658,740556,56852],{},[31662,740558,456],{"stretchy":702},[31658,740560,33249],{},[31662,740562,2557],{"stretchy":702},[31665,740564,737940],{"encoding":31667},[34,740566,740568],{"className":740567,"ariaHidden":758},[31672],[34,740569,740571,740574,740577,740580,740583],{"className":740570},[31676],[34,740572],{"className":740573,"style":31774},[31680],[34,740575,56852],{"className":740576,"style":37422},[31685,31686],[34,740578,456],{"className":740579},[31782],[34,740581,33249],{"className":740582},[31685,31686],[34,740584,2557],{"className":740585},[31690],[108,740587,740588,2664,740590,740740],{},[2627,740589,114043],{},[34,740591,740593,740629],{"className":740592},[31642],[34,740594,740596],{"className":740595},[31646],[31648,740597,740598],{"xmlns":31650},[31652,740599,740600,740626],{},[31655,740601,740602,740608,740610,740612,740618,740620,740622,740624],{},[37624,740603,740604,740606],{},[31658,740605,56852],{},[31662,740607,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,740609,202],{},[31662,740611,32226],{},[31743,740613,740614,740616],{},[31658,740615,111],{},[31749,740617,640],{},[31662,740619,456],{"stretchy":702},[31658,740621,33249],{},[31662,740623,2557],{"stretchy":702},[31658,740625,56852],{},[31665,740627,740628],{"encoding":31667},"W' = -a_1(t) W",[34,740630,740632,740679],{"className":740631,"ariaHidden":758},[31672],[34,740633,740635,740638,740670,740673,740676],{"className":740634},[31676],[34,740636],{"className":740637,"style":132027},[31680],[34,740639,740641,740644],{"className":740640},[31685],[34,740642,56852],{"className":740643,"style":37422},[31685,31686],[34,740645,740647],{"className":740646},[31792],[34,740648,740650],{"className":740649},[31796],[34,740651,740653],{"className":740652},[31801],[34,740654,740656],{"className":740655,"style":132027},[31805],[34,740657,740658,740661],{"style":37777},[34,740659],{"className":740660,"style":31814},[31813],[34,740662,740664],{"className":740663},[31818,31819,31820,31821],[34,740665,740667],{"className":740666},[31685,31821],[34,740668,116495],{"className":740669},[31685,31821],[34,740671],{"className":740672,"style":31846},[31845],[34,740674,202],{"className":740675},[31850],[34,740677],{"className":740678,"style":31846},[31845],[34,740680,740682,740685,740688,740728,740731,740734,740737],{"className":740681},[31676],[34,740683],{"className":740684,"style":31774},[31680],[34,740686,32226],{"className":740687},[31685],[34,740689,740691,740694],{"className":740690},[31685],[34,740692,111],{"className":740693},[31685,31686],[34,740695,740697],{"className":740696},[31792],[34,740698,740700,740720],{"className":740699},[31796,31797],[34,740701,740703,740717],{"className":740702},[31801],[34,740704,740706],{"className":740705,"style":31806},[31805],[34,740707,740708,740711],{"style":31809},[34,740709],{"className":740710,"style":31814},[31813],[34,740712,740714],{"className":740713},[31818,31819,31820,31821],[34,740715,640],{"className":740716},[31685,31821],[34,740718,31829],{"className":740719},[31828],[34,740721,740723],{"className":740722},[31801],[34,740724,740726],{"className":740725,"style":31836},[31805],[34,740727],{},[34,740729,456],{"className":740730},[31782],[34,740732,33249],{"className":740733},[31685,31686],[34,740735,2557],{"className":740736},[31690],[34,740738,56852],{"className":740739,"style":37422},[31685,31686],"。这是一个简单的一阶方程，解它就得到公式。",[32188,740742],{},[253,740744,740746],{"id":740745},"_7-证明-n-阶非齐次方程最多有-n1-个线性无关解","7. 证明 n 阶非齐次方程最多有 n+1 个线性无关解",[15,740748,740749,740752,740753,740803],{},[2627,740750,740751],{},"直觉理解：","\n想象三维空间（",[34,740754,740756,740773],{"className":740755},[31642],[34,740757,740759],{"className":740758},[31646],[31648,740760,740761],{"xmlns":31650},[31652,740762,740763,740771],{},[31655,740764,740765,740767,740769],{},[31658,740766,31660],{},[31662,740768,202],{},[31749,740770,786],{},[31665,740772,672626],{"encoding":31667},[34,740774,740776,740794],{"className":740775,"ariaHidden":758},[31672],[34,740777,740779,740782,740785,740788,740791],{"className":740778},[31676],[34,740780],{"className":740781,"style":32527},[31680],[34,740783,31660],{"className":740784},[31685,31686],[34,740786],{"className":740787,"style":31846},[31845],[34,740789,202],{"className":740790},[31850],[34,740792],{"className":740793,"style":31846},[31845],[34,740795,740797,740800],{"className":740796},[31676],[34,740798],{"className":740799,"style":32617},[31680],[34,740801,786],{"className":740802},[31685],"）里的一个平面。",[105,740805,740806,740809,740812],{},[108,740807,740808],{},"齐次方程的解通过原点，构成一个平面（3维线性空间）。",[108,740810,740811],{},"非齐次方程的解，就是这个平面平移了一下（不过原点）。",[108,740813,740814],{},"在这个平移后的平面上找点，你最多能找几个点，使得它们不“共面\u002F共线”？",[15,740816,740817],{},[2627,740818,740819],{},"证明逻辑：",[2230,740821,740822,741608],{},[108,740823,740824,740827,740828,740856,740857,740997,740998,394037,741059,741556,741557,741607],{},[2627,740825,740826],{},"存在性","：取齐次方程的 ",[34,740829,740831,740844],{"className":740830},[31642],[34,740832,740834],{"className":740833},[31646],[31648,740835,740836],{"xmlns":31650},[31652,740837,740838,740842],{},[31655,740839,740840],{},[31658,740841,31660],{},[31665,740843,31660],{"encoding":31667},[34,740845,740847],{"className":740846,"ariaHidden":758},[31672],[34,740848,740850,740853],{"className":740849},[31676],[34,740851],{"className":740852,"style":32527},[31680],[34,740854,31660],{"className":740855},[31685,31686]," 个线性无关解 ",[34,740858,740860,740890],{"className":740859},[31642],[34,740861,740863],{"className":740862},[31646],[31648,740864,740865],{"xmlns":31650},[31652,740866,740867,740887],{},[31655,740868,740869,740875,740877,740879,740881],{},[31743,740870,740871,740873],{},[31658,740872,32936],{},[31749,740874,640],{},[31662,740876,31960],{"separator":758},[31662,740878,33934],{},[31662,740880,31960],{"separator":758},[31743,740882,740883,740885],{},[31658,740884,32936],{},[31658,740886,31660],{},[31665,740888,740889],{"encoding":31667},"x_1, \\dots, x_n",[34,740891,740893],{"className":740892,"ariaHidden":758},[31672],[34,740894,740896,740899,740939,740942,740945,740948,740951,740954,740957],{"className":740895},[31676],[34,740897],{"className":740898,"style":38774},[31680],[34,740900,740902,740905],{"className":740901},[31685],[34,740903,32936],{"className":740904},[31685,31686],[34,740906,740908],{"className":740907},[31792],[34,740909,740911,740931],{"className":740910},[31796,31797],[34,740912,740914,740928],{"className":740913},[31801],[34,740915,740917],{"className":740916,"style":31806},[31805],[34,740918,740919,740922],{"style":31809},[34,740920],{"className":740921,"style":31814},[31813],[34,740923,740925],{"className":740924},[31818,31819,31820,31821],[34,740926,640],{"className":740927},[31685,31821],[34,740929,31829],{"className":740930},[31828],[34,740932,740934],{"className":740933},[31801],[34,740935,740937],{"className":740936,"style":31836},[31805],[34,740938],{},[34,740940,31960],{"className":740941},[32119],[34,740943],{"className":740944,"style":32123},[31845],[34,740946,33934],{"className":740947},[34006],[34,740949],{"className":740950,"style":32123},[31845],[34,740952,31960],{"className":740953},[32119],[34,740955],{"className":740956,"style":32123},[31845],[34,740958,740960,740963],{"className":740959},[31685],[34,740961,32936],{"className":740962},[31685,31686],[34,740964,740966],{"className":740965},[31792],[34,740967,740969,740989],{"className":740968},[31796,31797],[34,740970,740972,740986],{"className":740971},[31801],[34,740973,740975],{"className":740974,"style":32362},[31805],[34,740976,740977,740980],{"style":31809},[34,740978],{"className":740979,"style":31814},[31813],[34,740981,740983],{"className":740982},[31818,31819,31820,31821],[34,740984,31660],{"className":740985},[31685,31686,31821],[34,740987,31829],{"className":740988},[31828],[34,740990,740992],{"className":740991},[31801],[34,740993,740995],{"className":740994,"style":31836},[31805],[34,740996],{}," 和非齐次方程的一个特解 ",[34,740999,741001,741019],{"className":741000},[31642],[34,741002,741004],{"className":741003},[31646],[31648,741005,741006],{"xmlns":31650},[31652,741007,741008,741016],{},[31655,741009,741010],{},[134014,741011,741012,741014],{"accent":758},[31658,741013,32936],{},[31662,741015,346142],{},[31665,741017,741018],{"encoding":31667},"\\bar{x}",[34,741020,741022],{"className":741021,"ariaHidden":758},[31672],[34,741023,741025,741028],{"className":741024},[31676],[34,741026],{"className":741027,"style":346181},[31680],[34,741029,741031],{"className":741030},[31685,346171],[34,741032,741034],{"className":741033},[31796],[34,741035,741037],{"className":741036},[31801],[34,741038,741040,741048],{"className":741039,"style":346181},[31805],[34,741041,741042,741045],{"style":41565},[34,741043],{"className":741044,"style":41508},[31813],[34,741046,32936],{"className":741047},[31685,31686],[34,741049,741050,741053],{"style":41565},[34,741051],{"className":741052,"style":41508},[31813],[34,741054,741056],{"className":741055,"style":724852},[346198],[34,741057,346142],{"className":741058},[31685],[34,741060,741062,741146],{"className":741061},[31642],[34,741063,741065],{"className":741064},[31646],[31648,741066,741067],{"xmlns":31650},[31652,741068,741069,741143],{},[31655,741070,741071,741077,741079,741085,741087,741089,741095,741097,741103,741105,741111,741113,741115,741117,741119,741121,741127,741129,741135,741137],{},[31743,741072,741073,741075],{},[31658,741074,67764],{},[31749,741076,459],{},[31662,741078,202],{},[134014,741080,741081,741083],{"accent":758},[31658,741082,32936],{},[31662,741084,346142],{},[31662,741086,31960],{"separator":758},[39361,741088,130298],{},[31743,741090,741091,741093],{},[31658,741092,67764],{},[31749,741094,640],{},[31662,741096,202],{},[134014,741098,741099,741101],{"accent":758},[31658,741100,32936],{},[31662,741102,346142],{},[31662,741104,223],{},[31743,741106,741107,741109],{},[31658,741108,32936],{},[31749,741110,640],{},[31662,741112,31960],{"separator":758},[39361,741114,130298],{},[31662,741116,33934],{},[31662,741118,31960],{"separator":758},[39361,741120,130298],{},[31743,741122,741123,741125],{},[31658,741124,67764],{},[31658,741126,31660],{},[31662,741128,202],{},[134014,741130,741131,741133],{"accent":758},[31658,741132,32936],{},[31662,741134,346142],{},[31662,741136,223],{},[31743,741138,741139,741141],{},[31658,741140,32936],{},[31658,741142,31660],{},[31665,741144,741145],{"encoding":31667},"y_0 = \\bar{x}, \\ y_1 = \\bar{x}+x_1, \\ \\dots, \\ y_n = \\bar{x}+x_n",[34,741147,741149,741204,741299,741345,741464,741510],{"className":741148,"ariaHidden":758},[31672],[34,741150,741152,741155,741195,741198,741201],{"className":741151},[31676],[34,741153],{"className":741154,"style":38774},[31680],[34,741156,741158,741161],{"className":741157},[31685],[34,741159,67764],{"className":741160,"style":32062},[31685,31686],[34,741162,741164],{"className":741163},[31792],[34,741165,741167,741187],{"className":741166},[31796,31797],[34,741168,741170,741184],{"className":741169},[31801],[34,741171,741173],{"className":741172,"style":31806},[31805],[34,741174,741175,741178],{"style":38795},[34,741176],{"className":741177,"style":31814},[31813],[34,741179,741181],{"className":741180},[31818,31819,31820,31821],[34,741182,459],{"className":741183},[31685,31821],[34,741185,31829],{"className":741186},[31828],[34,741188,741190],{"className":741189},[31801],[34,741191,741193],{"className":741192,"style":31836},[31805],[34,741194],{},[34,741196],{"className":741197,"style":31846},[31845],[34,741199,202],{"className":741200},[31850],[34,741202],{"className":741203,"style":31846},[31845],[34,741205,741207,741210,741241,741244,741247,741250,741290,741293,741296],{"className":741206},[31676],[34,741208],{"className":741209,"style":375433},[31680],[34,741211,741213],{"className":741212},[31685,346171],[34,741214,741216],{"className":741215},[31796],[34,741217,741219],{"className":741218},[31801],[34,741220,741222,741230],{"className":741221,"style":346181},[31805],[34,741223,741224,741227],{"style":41565},[34,741225],{"className":741226,"style":41508},[31813],[34,741228,32936],{"className":741229},[31685,31686],[34,741231,741232,741235],{"style":41565},[34,741233],{"className":741234,"style":41508},[31813],[34,741236,741238],{"className":741237,"style":724852},[346198],[34,741239,346142],{"className":741240},[31685],[34,741242,31960],{"className":741243},[32119],[34,741245,130298],{"className":741246},[31845],[34,741248],{"className":741249,"style":32123},[31845],[34,741251,741253,741256],{"className":741252},[31685],[34,741254,67764],{"className":741255,"style":32062},[31685,31686],[34,741257,741259],{"className":741258},[31792],[34,741260,741262,741282],{"className":741261},[31796,31797],[34,741263,741265,741279],{"className":741264},[31801],[34,741266,741268],{"className":741267,"style":31806},[31805],[34,741269,741270,741273],{"style":38795},[34,741271],{"className":741272,"style":31814},[31813],[34,741274,741276],{"className":741275},[31818,31819,31820,31821],[34,741277,640],{"className":741278},[31685,31821],[34,741280,31829],{"className":741281},[31828],[34,741283,741285],{"className":741284},[31801],[34,741286,741288],{"className":741287,"style":31836},[31805],[34,741289],{},[34,741291],{"className":741292,"style":31846},[31845],[34,741294,202],{"className":741295},[31850],[34,741297],{"className":741298,"style":31846},[31845],[34,741300,741302,741305,741336,741339,741342],{"className":741301},[31676],[34,741303],{"className":741304,"style":32597},[31680],[34,741306,741308],{"className":741307},[31685,346171],[34,741309,741311],{"className":741310},[31796],[34,741312,741314],{"className":741313},[31801],[34,741315,741317,741325],{"className":741316,"style":346181},[31805],[34,741318,741319,741322],{"style":41565},[34,741320],{"className":741321,"style":41508},[31813],[34,741323,32936],{"className":741324},[31685,31686],[34,741326,741327,741330],{"style":41565},[34,741328],{"className":741329,"style":41508},[31813],[34,741331,741333],{"className":741332,"style":724852},[346198],[34,741334,346142],{"className":741335},[31685],[34,741337],{"className":741338,"style":32604},[31845],[34,741340,223],{"className":741341},[32024],[34,741343],{"className":741344,"style":32604},[31845],[34,741346,741348,741351,741391,741394,741397,741400,741403,741406,741409,741412,741415,741455,741458,741461],{"className":741347},[31676],[34,741349],{"className":741350,"style":38774},[31680],[34,741352,741354,741357],{"className":741353},[31685],[34,741355,32936],{"className":741356},[31685,31686],[34,741358,741360],{"className":741359},[31792],[34,741361,741363,741383],{"className":741362},[31796,31797],[34,741364,741366,741380],{"className":741365},[31801],[34,741367,741369],{"className":741368,"style":31806},[31805],[34,741370,741371,741374],{"style":31809},[34,741372],{"className":741373,"style":31814},[31813],[34,741375,741377],{"className":741376},[31818,31819,31820,31821],[34,741378,640],{"className":741379},[31685,31821],[34,741381,31829],{"className":741382},[31828],[34,741384,741386],{"className":741385},[31801],[34,741387,741389],{"className":741388,"style":31836},[31805],[34,741390],{},[34,741392,31960],{"className":741393},[32119],[34,741395,130298],{"className":741396},[31845],[34,741398],{"className":741399,"style":32123},[31845],[34,741401,33934],{"className":741402},[34006],[34,741404],{"className":741405,"style":32123},[31845],[34,741407,31960],{"className":741408},[32119],[34,7414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",[34,741558,741560,741577],{"className":741559},[31642],[34,741561,741563],{"className":741562},[31646],[31648,741564,741565],{"xmlns":31650},[31652,741566,741567,741575],{},[31655,741568,741569,741571,741573],{},[31658,741570,31660],{},[31662,741572,223],{},[31749,741574,640],{},[31665,741576,87807],{"encoding":31667},[34,741578,741580,741598],{"className":741579,"ariaHidden":758},[31672],[34,741581,741583,741586,741589,741592,741595],{"className":741582},[31676],[34,741584],{"className":741585,"style":32597},[31680],[34,741587,31660],{"className":741588},[31685,31686],[34,741590],{"className":741591,"style":32604},[31845],[34,741593,223],{"className":741594},[32024],[34,741596],{"className":741597,"style":32604},[31845],[34,741599,741601,741604],{"className":741600},[31676],[34,741602],{"className":741603,"style":32617},[31680],[34,741605,640],{"className":741606},[31685]," 个解是线性无关的。（可以用定义证）。",[108,741609,741610,741613,741614,741665,741666,741875,741876,616,742072,616,742268,742297,742298,742368,742369,742372,742373,742401,742402,742452,742453,742503,742504,742532,742533,742603,742604,742658],{},[2627,741611,741612],{},"最多 n+1 个","：假设有 ",[34,741615,741617,741635],{"className":741616},[31642],[34,741618,741620],{"className":741619},[31646],[31648,741621,741622],{"xmlns":31650},[31652,741623,741624,741632],{},[31655,741625,741626,741628,741630],{},[31658,741627,31660],{},[31662,741629,223],{},[31749,741631,768],{},[31665,741633,741634],{"encoding":31667},"n+2",[34,741636,741638,741656],{"className":741637,"ariaHidden":758},[31672],[34,741639,741641,741644,741647,741650,741653],{"className":741640},[31676],[34,741642],{"className":741643,"style":32597},[31680],[34,741645,31660],{"className":741646},[31685,31686],[34,741648],{"className":741649,"style":32604},[31845],[34,741651,223],{"className":741652},[32024],[34,741654],{"className":741655,"style":32604},[31845],[34,741657,741659,741662],{"className":741658},[31676],[34,741660],{"className":741661,"style":32617},[31680],[34,741663,768],{"className":741664},[31685]," 个解 ",[34,741667,741669,741713],{"className":741668},[31642],[34,741670,741672],{"className":741671},[31646],[31648,741673,741674],{"xmlns":31650},[31652,741675,741676,741710],{},[31655,741677,741678,741684,741686,741692,741694,741696,741698],{},[31743,741679,741680,741682],{},[31658,741681,67764],{},[31749,741683,640],{},[31662,741685,31960],{"separator":758},[31743,741687,741688,741690],{},[31658,741689,67764],{},[31749,741691,768],{},[31662,741693,31960],{"separator":758},[31662,741695,33934],{},[31662,741697,31960],{"separator":758},[31743,741699,741700,741702],{},[31658,741701,67764],{},[31655,741703,741704,741706,741708],{},[31658,741705,31660],{},[31662,741707,223],{},[31749,741709,768],{},[31665,741711,741712],{"encoding":31667},"y_1, y_2, \\dots, y_{n+2}",[34,741714,741716],{"className":741715,"ariaHidden":758},[31672],[34,741717,741719,741722,741762,741765,741768,741808,741811,741814,741817,741820,741823,741826],{"className":741718},[31676],[34,741720],{"className":741721,"style":73210},[31680],[34,741723,741725,741728],{"className":741724},[31685],[34,741726,67764],{"className":741727,"style":32062},[31685,31686],[34,741729,741731],{"className":741730},[31792],[34,741732,741734,741754],{"className":741733},[31796,31797],[34,741735,741737,741751],{"className":741736},[31801],[34,741738,741740],{"className":741739,"style":31806},[31805],[34,741741,741742,741745],{"style":38795},[34,741743],{"className":741744,"style":31814},[31813],[34,741746,741748],{"className":741747},[31818,31819,31820,31821],[34,741749,640],{"className":741750},[31685,31821],[34,741752,31829],{"className":741753},[31828],[34,741755,741757],{"className":741756},[31801],[34,741758,741760],{"className":741759,"style":31836},[31805],[34,741761],{},[34,741763,31960],{"className":741764},[32119],[34,741766],{"className":741767,"style":32123},[31845],[34,741769,741771,741774],{"className":741770},[31685],[34,741772,67764],{"className":741773,"style":32062},[31685,31686],[34,741775,741777],{"className":741776},[31792],[34,741778,741780,741800],{"className":741779},[31796,31797],[34,741781,741783,741797],{"className":741782},[31801],[34,741784,741786],{"className":741785,"style":31806},[31805],[34,741787,741788,741791],{"style":38795},[34,741789],{"className":741790,"style":31814},[31813],[34,741792,741794],{"className":741793},[31818,31819,31820,31821],[34,741795,768],{"className":741796},[31685,31821],[34,741798,31829],{"className":741799},[31828],[34,741801,741803],{"className":741802},[31801],[34,741804,741806],{"className":741805,"style":31836},[31805],[34,741807],{},[34,741809,31960],{"className":741810},[32119],[34,741812],{"className":741813,"style":32123},[31845],[34,741815,33934],{"className":741816},[34006],[34,741818],{"className":741819,"style":32123},[31845],[34,741821,31960],{"className":741822},[32119],[34,741824],{"className":741825,"style":32123},[31845],[34,741827,741829,741832],{"className":741828},[31685],[34,741830,67764],{"className":741831,"style":32062},[31685,31686],[34,741833,741835],{"className":741834},[31792],[34,741836,741838,741867],{"className":741837},[31796,31797],[34,741839,741841,741864],{"className":741840},[31801],[34,741842,741844],{"className":741843,"style":31806},[31805],[34,741845,741846,741849],{"style":38795},[34,741847],{"className":741848,"style":31814},[31813],[34,741850,741852],{"className":741851},[31818,31819,31820,31821],[34,741853,741855,741858,741861],{"className":741854},[31685,31821],[34,741856,31660],{"className":741857},[31685,31686,31821],[34,741859,223],{"className":741860},[32024,31821],[34,741862,768],{"className":741863},[31685,31821],[34,741865,31829],{"className":741866},[31828],[34,741868,741870],{"className":741869},[31801],[34,741871,741873],{"className":741872,"style":32037},[31805],[34,741874],{},"。\n考虑它们的差：",[34,741877,741879,741913],{"className":741878},[31642],[34,741880,741882],{"className":741881},[31646],[31648,741883,741884],{"xmlns":31650},[31652,741885,741886,741910],{},[31655,741887,741888,741894,741896,741902,741904],{},[31743,741889,741890,741892],{},[31658,741891,382333],{},[31749,741893,640],{},[31662,741895,202],{},[31743,741897,741898,741900],{},[31658,741899,67764],{},[31749,741901,768],{},[31662,741903,32226],{},[31743,741905,741906,741908],{},[31658,741907,67764],{},[31749,741909,640],{},[31665,741911,741912],{"encoding":31667},"z_1 = y_2 - y_1",[34,741914,741916,741971,742026],{"className":741915,"ariaHidden":758},[31672],[34,741917,741919,741922,741962,741965,741968],{"className":741918},[31676],[34,741920],{"className":741921,"style":31860},[31680],[34,741923,741925,741928],{"className":741924},[31685],[34,741926,382333],{"className":741927,"style":382406},[31685,31686],[34,741929,741931],{"className":741930},[31792],[34,741932,741934,741954],{"className":741933},[31796,31797],[34,741935,741937,741951],{"className":741936},[31801],[34,741938,741940],{"className":741939,"style":31806},[31805],[34,741941,741942,741945],{"style":666745},[34,741943],{"className":741944,"style":31814},[31813],[34,741946,741948],{"className":741947},[31818,31819,31820,31821],[34,741949,640],{"className":741950},[31685,31821],[34,741952,31829],{"className":741953},[31828],[34,741955,741957],{"className":741956},[31801],[34,741958,741960],{"className":741959,"style":31836},[31805],[34,741961],{},[34,741963],{"className":741964,"style":31846},[31845],[34,741966,202],{"className":741967},[31850],[34,741969],{"className":741970,"style":31846},[31845],[34,741972,741974,741977,742017,742020,742023],{"className":741973},[31676],[34,741975],{"className":741976,"style":55949},[31680],[34,741978,741980,741983],{"className":741979},[31685],[34,741981,67764],{"className":741982,"style":32062},[31685,31686],[34,741984,741986],{"className":741985},[31792],[34,741987,741989,742009],{"className":741988},[31796,31797],[34,741990,741992,742006],{"className":741991},[31801],[34,741993,741995],{"className":741994,"style":31806},[31805],[34,741996,741997,742000],{"style":38795},[34,741998],{"className":741999,"style":31814},[31813],[34,742001,742003],{"className":742002},[31818,31819,31820,31821],[34,742004,768],{"className":742005},[31685,31821],[34,742007,31829],{"className":742008},[31828],[34,742010,742012],{"className":742011},[31801],[34,742013,742015],{"className":742014,"style":31836},[31805],[34,742016],{},[34,742018],{"className":742019,"style":32604},[31845],[34,742021,32226],{"className":742022},[32024],[34,742024],{"className":742025,"style":32604},[31845],[34,742027,742029,742032],{"className":742028},[31676],[34,742030],{"className":742031,"style":38774},[31680],[34,742033,742035,742038],{"className":742034},[31685],[34,742036,67764],{"className":742037,"style":32062},[31685,31686],[34,742039,742041],{"className":742040},[31792],[34,742042,742044,742064],{"className":742043},[31796,31797],[34,742045,742047,742061],{"className":742046},[31801],[34,742048,742050],{"className":742049,"style":31806},[31805],[34,742051,742052,742055],{"style":38795},[34,742053],{"className":742054,"style":31814},[31813],[34,742056,742058],{"className":742057},[31818,31819,31820,31821],[34,742059,640],{"className":742060},[31685,31821],[34,742062,31829],{"className":742063},[31828],[34,742065,742067],{"className":742066},[31801],[34,742068,742070],{"className":742069,"style":31836},[31805],[34,742071],{},[34,742073,742075,742109],{"className":742074},[31642],[34,742076,742078],{"className":742077},[31646],[31648,742079,742080],{"xmlns":31650},[31652,742081,742082,742106],{},[31655,742083,742084,742090,742092,742098,742100],{},[31743,742085,742086,742088],{},[31658,742087,382333],{},[31749,742089,768],{},[31662,742091,202],{},[31743,742093,742094,742096],{},[31658,742095,67764],{},[31749,742097,786],{},[31662,742099,32226],{},[31743,742101,742102,742104],{},[31658,742103,67764],{},[31749,742105,640],{},[31665,742107,742108],{"encoding":31667},"z_2 = y_3 - y_1",[34,742110,742112,742167,742222],{"className":742111,"ariaHidden":758},[31672],[34,742113,742115,742118,742158,742161,742164],{"className":742114},[31676],[34,742116],{"className":742117,"style":31860},[31680],[34,742119,742121,742124],{"className":742120},[31685],[34,742122,382333],{"className":742123,"style":382406},[31685,31686],[34,742125,742127],{"className":742126},[31792],[34,742128,742130,742150],{"className":742129},[31796,31797],[34,742131,742133,742147],{"className":742132},[31801],[34,742134,742136],{"className":742135,"style":31806},[31805],[34,742137,742138,742141],{"style":666745},[34,742139],{"className":742140,"style":31814},[31813],[34,742142,742144],{"className":742143},[31818,31819,31820,31821],[34,742145,768],{"className":742146},[31685,31821],[34,742148,31829],{"className":742149},[31828],[34,742151,742153],{"className":742152},[31801],[34,742154,742156],{"className":742155,"style":31836},[31805],[34,742157],{},[34,742159],{"className":742160,"style":31846},[31845],[34,742162,202],{"className":742163},[31850],[34,742165],{"className":742166,"style":31846},[31845],[34,742168,742170,742173,742213,742216,742219],{"className":742169},[31676],[34,742171],{"className":742172,"style":55949},[31680],[34,742174,742176,742179],{"className":742175},[31685],[34,742177,67764],{"className":742178,"style":32062},[31685,31686],[34,742180,742182],{"className":742181},[31792],[34,742183,742185,742205],{"className":742184},[31796,31797],[34,742186,742188,742202],{"className":742187},[31801],[34,742189,742191],{"className":742190,"style":31806},[31805],[34,742192,742193,742196],{"style":38795},[34,742194],{"className":742195,"style":31814},[31813],[34,742197,742199],{"className":742198},[31818,31819,31820,31821],[34,742200,786],{"className":742201},[31685,31821],[34,742203,31829],{"className":742204},[31828],[34,742206,742208],{"className":742207},[31801],[34,742209,742211],{"className":742210,"style":31836},[31805],[34,742212],{},[34,742214],{"className":742215,"style":32604},[31845],[34,742217,32226],{"className":742218},[32024],[34,742220],{"className":742221,"style":32604},[31845],[34,742223,742225,742228],{"className":742224},[31676],[34,742226],{"className":742227,"style":38774},[31680],[34,742229,742231,742234],{"className":742230},[31685],[34,742232,67764],{"className":742233,"style":32062},[31685,31686],[34,742235,742237],{"className":742236},[31792],[34,742238,742240,742260],{"className":742239},[31796,31797],[34,742241,742243,742257],{"className":742242},[31801],[34,742244,742246],{"className":742245,"style":31806},[31805],[34,742247,742248,742251],{"style":38795},[34,742249],{"className":742250,"style":31814},[31813],[34,742252,742254],{"className":742253},[31818,31819,31820,31821],[34,742255,640],{"className":742256},[31685,31821],[34,742258,31829],{"className":742259},[31828],[34,742261,742263],{"className":742262},[31801],[34,742264,742266],{"className":742265,"style":31836},[31805],[34,742267],{},[34,742269,742271,742285],{"className":742270},[31642],[34,742272,742274],{"className":742273},[31646],[31648,742275,742276],{"xmlns":31650},[31652,742277,742278,742282],{},[31655,742279,742280],{},[31662,742281,33934],{},[31665,742283,742284],{"encoding":31667},"\\dots",[34,742286,742288],{"className":742287,"ariaHidden":758},[31672],[34,742289,742291,742294],{"className":742290},[31676],[34,742292],{"className":742293,"style":400308},[31680],[34,742295,33934],{"className":742296},[34006],".\n根据非齐次解的性质2，这些差 ",[34,742299,742301,742319],{"className":742300},[31642],[34,742302,742304],{"className":742303},[31646],[31648,742305,742306],{"xmlns":31650},[31652,742307,742308,742316],{},[31655,742309,742310],{},[31743,742311,742312,742314],{},[31658,742313,382333],{},[31658,742315,31955],{},[31665,742317,742318],{"encoding":31667},"z_i",[34,742320,742322],{"className":742321,"ariaHidden":758},[31672],[34,742323,742325,742328],{"className":742324},[31676],[34,742326],{"className":742327,"style":31860},[31680],[34,742329,742331,742334],{"className":742330},[31685],[34,742332,382333],{"className":742333,"style":382406},[31685,31686],[34,742335,742337],{"className":742336},[31792],[34,742338,742340,742360],{"className":742339},[31796,31797],[34,742341,742343,742357],{"className":742342},[31801],[34,742344,742346],{"className":742345,"style":32090},[31805],[34,742347,742348,742351],{"style":666745},[34,742349],{"className":742350,"style":31814},[31813],[34,742352,742354],{"className":742353},[31818,31819,31820,31821],[34,742355,31955],{"className":742356},[31685,31686,31821],[34,742358,31829],{"className":742359},[31828],[34,742361,742363],{"className":742362},[31801],[34,742364,742366],{"className":742365,"style":31836},[31805],[34,742367],{}," 都是",[2627,742370,742371],{},"齐次方程的解","。\n齐次方程的解空间维度是 ",[34,742374,742376,742389],{"className":742375},[31642],[34,742377,742379],{"className":742378},[31646],[31648,742380,742381],{"xmlns":31650},[31652,742382,742383,742387],{},[31655,742384,742385],{},[31658,742386,31660],{},[31665,742388,31660],{"encoding":31667},[34,742390,742392],{"className":742391,"ariaHidden":758},[31672],[34,742393,742395,742398],{"className":742394},[31676],[34,742396],{"className":742397,"style":32527},[31680],[34,742399,31660],{"className":742400},[31685,31686],"。\n如果你有 ",[34,742403,742405,742422],{"className":742404},[31642],[34,742406,742408],{"className":742407},[31646],[31648,742409,742410],{"xmlns":31650},[31652,742411,742412,742420],{},[31655,742413,742414,742416,742418],{},[31658,742415,31660],{},[31662,742417,223],{},[31749,742419,640],{},[31665,742421,87807],{"encoding":31667},[34,742423,742425,742443],{"className":742424,"ariaHidden":758},[31672],[34,742426,742428,742431,742434,742437,742440],{"className":742427},[31676],[34,742429],{"className":742430,"style":32597},[31680],[34,742432,31660],{"className":742433},[31685,31686],[34,742435],{"className":742436,"style":32604},[31845],[34,742438,223],{"className":742439},[32024],[34,742441],{"className":742442,"style":32604},[31845],[34,742444,742446,742449],{"className":742445},[31676],[34,742447],{"className":742448,"style":32617},[31680],[34,742450,640],{"className":742451},[31685]," 个差向量（由 ",[34,742454,742456,742473],{"className":742455},[31642],[34,742457,742459],{"className":742458},[31646],[31648,742460,742461],{"xmlns":31650},[31652,742462,742463,742471],{},[31655,742464,742465,742467,742469],{},[31658,742466,31660],{},[31662,742468,223],{},[31749,742470,768],{},[31665,742472,741634],{"encoding":31667},[34,742474,742476,742494],{"className":742475,"ariaHidden":758},[31672],[34,742477,742479,742482,742485,742488,742491],{"className":742478},[31676],[34,742480],{"className":742481,"style":32597},[31680],[34,742483,31660],{"className":742484},[31685,31686],[34,742486],{"className":742487,"style":32604},[31845],[34,742489,223],{"className":742490},[32024],[34,742492],{"className":742493,"style":32604},[31845],[34,742495,742497,742500],{"className":742496},[31676],[34,742498],{"className":742499,"style":32617},[31680],[34,742501,768],{"className":742502},[31685]," 个解产生），它们在 ",[34,742505,742507,742520],{"className":742506},[31642],[34,742508,742510],{"className":742509},[31646],[31648,742511,742512],{"xmlns":31650},[31652,742513,742514,742518],{},[31655,742515,742516],{},[31658,742517,31660],{},[31665,742519,31660],{"encoding":31667},[34,742521,742523],{"className":742522,"ariaHidden":758},[31672],[34,742524,742526,742529],{"className":742525},[31676],[34,742527],{"className":742528,"style":32527},[31680],[34,742530,31660],{"className":742531},[31685,31686]," 维空间里必然线性相关。\n既然差向量线性相关，那么原解 ",[34,742534,742536,742554],{"className":742535},[31642],[34,742537,742539],{"className":742538},[31646],[31648,742540,742541],{"xmlns":31650},[31652,742542,742543,742551],{},[31655,742544,742545],{},[31743,742546,742547,742549],{},[31658,742548,67764],{},[31658,742550,31955],{},[31665,742552,742553],{"encoding":31667},"y_i",[34,742555,742557],{"className":742556,"ariaHidden":758},[31672],[34,742558,742560,742563],{"className":742559},[31676],[34,742561],{"className":742562,"style":38774},[31680],[34,742564,742566,742569],{"className":742565},[31685],[34,742567,67764],{"className":742568,"style":32062},[31685,31686],[34,742570,742572],{"className":742571},[31792],[34,742573,742575,742595],{"className":742574},[31796,31797],[34,742576,742578,742592],{"className":742577},[31801],[34,742579,742581],{"className":742580,"style":32090},[31805],[34,742582,742583,742586],{"style":38795},[34,742584],{"className":742585,"style":31814},[31813],[34,742587,742589],{"className":742588},[31818,31819,31820,31821],[34,742590,31955],{"className":742591},[31685,31686,31821],[34,742593,31829],{"className":742594},[31828],[34,742596,742598],{"className":742597},[31801],[34,742599,742601],{"className":742600,"style":31836},[31805],[34,742602],{}," 也就线性相关了。\n",[2627,742605,742606,742607,742657],{},"所以，不可能有 ",[34,742608,742610,742627],{"className":742609},[31642],[34,742611,742613],{"className":742612},[31646],[31648,742614,742615],{"xmlns":31650},[31652,742616,742617,742625],{},[31655,742618,742619,742621,742623],{},[31658,742620,31660],{},[31662,742622,223],{},[31749,742624,768],{},[31665,742626,741634],{"encoding":31667},[34,742628,742630,742648],{"className":742629,"ariaHidden":758},[31672],[34,742631,742633,742636,742639,742642,742645],{"className":742632},[31676],[34,742634],{"className":742635,"style":32597},[31680],[34,742637,31660],{"className":742638},[31685,31686],[34,742640],{"className":742641,"style":32604},[31845],[34,742643,223],{"className":742644},[32024],[34,742646],{"className":742647,"style":32604},[31845],[34,742649,742651,742654],{"className":742650},[31676],[34,742652],{"className":742653,"style":32617},[31680],[34,742655,768],{"className":742656},[31685]," 个线性无关解。","\n希望这套详细的讲解能帮你打通任督二脉！如果哪个积分算不出来，或者哪个步骤卡住了，继续问！",{"title":29,"searchDepth":43,"depth":43,"links":742660},[742661,742662,742663,742664,742665,742666,742667,742668,742669],{"id":699551,"depth":49,"text":699552},{"id":699944,"depth":49,"text":699945},{"id":702886,"depth":49,"text":702887},{"id":714586,"depth":49,"text":714586},{"id":715044,"depth":49,"text":715045},{"id":719113,"depth":49,"text":719114},{"id":733598,"depth":49,"text":733599},{"id":737870,"depth":49,"text":737871},{"id":740745,"depth":49,"text":740746},{"order":43},"\u002Fsource\u002Fmath\u002Fode\u002F3-2and4-1-hw",{"title":699540,"description":29},"source\u002Fmath\u002Fode\u002F3-2&4-1 HW","HSTljBz4eFiceU5D5lrN4uFxWvDWdNGBhdUjMx5IIwY",{"id":742676,"title":742677,"body":742678,"description":29,"extension":85,"meta":768257,"navigation":52,"path":768258,"seo":768259,"stem":768260,"__hash__":768261},"notes\u002Fsource\u002Fmath\u002Fode\u002F4-2HW.md","4-2 作业",{"type":8,"value":742679,"toc":768255},[742680,742684,742686,742690,742881,744610,744612,744927,746463,746465,746738,751227,751229,751233,751834,756129,756131,756135,756142,756632,756920,761687,761779,761781,762197,762761,762969,764687,764689,764908,768253],[11,742681,742683],{"id":742682},"习题42","习题4.2",[32188,742685],{},[32490,742687,742689],{"id":742688},"_2-求解下列常系数线性微分方程","2. 求解下列常系数线性微分方程",[15,742691,742692],{},[2627,742693,387575,742694],{},[34,742695,742697,742745],{"className":742696},[31642],[34,742698,742700],{"className":742699},[31646],[31648,742701,742702],{"xmlns":31650},[31652,742703,742704,742742],{},[31655,742705,742706,742718,742720,742722,742732,742734,742736,742738,742740],{},[37624,742707,742708,742710],{},[31658,742709,32936],{},[31655,742711,742712,742714,742716],{},[31662,742713,456],{"stretchy":702},[31749,742715,15932],{},[31662,742717,2557],{"stretchy":702},[31662,742719,32226],{},[31749,742721,18136],{},[37624,742723,742724,742726],{},[31658,742725,32936],{},[31655,742727,742728,742730],{},[31662,742729,116495],{"mathvariant":38079},[31662,742731,116495],{"mathvariant":38079},[31662,742733,223],{},[31749,742735,15932],{},[31658,742737,32936],{},[31662,742739,202],{},[31749,742741,459],{},[31665,742743,742744],{"encoding":31667},"x^{(4)} - 5x'' + 4x = 0",[34,742746,742748,742801,742851,742872],{"className":742747,"ariaHidden":758},[31672],[34,742749,742751,742754,742792,742795,742798],{"className":742750},[31676],[34,742752],{"className":742753,"style":640505},[31680],[34,742755,742757,742760],{"className":742756},[31685],[34,742758,32936],{"className":742759},[31685,31686],[34,742761,742763],{"className":742762},[31792],[34,742764,742766],{"className":742765},[31796],[34,742767,742769],{"className":742768},[31801],[34,742770,742772],{"className":742771,"style":61037},[31805],[34,742773,742774,742777],{"style":37777},[34,742775],{"className":742776,"style":31814},[31813],[34,742778,742780],{"className":742779},[31818,31819,31820,31821],[34,742781,742783,742786,742789],{"className":742782},[31685,31821],[34,742784,456],{"className":742785},[31782,31821],[34,742787,15932],{"className":742788},[31685,31821],[34,742790,2557],{"className":742791},[31690,31821],[34,742793],{"className":742794,"style":32604},[31845],[34,742796,32226],{"className":742797},[32024],[34,742799],{"className":742800,"style":32604},[31845],[34,742802,742804,742807,742810,742842,742845,742848],{"className":742803},[31676],[34,742805],{"className":742806,"style":384364},[31680],[34,742808,18136],{"className":742809},[31685],[34,742811,742813,742816],{"className":742812},[31685],[34,742814,32936],{"className":742815},[31685,31686],[34,742817,742819],{"className":742818},[31792],[34,742820,742822],{"className":742821},[31796],[34,742823,742825],{"className":742824},[31801],[34,742826,742828],{"className":742827,"style":132027},[31805],[34,742829,742830,742833],{"style":37777},[34,742831],{"className":742832,"style":31814},[31813],[34,742834,742836],{"className":742835},[31818,31819,31820,31821],[34,742837,742839],{"className":742838},[31685,31821],[34,742840,502782],{"className":742841},[31685,31821],[34,742843],{"className":742844,"style":32604},[31845],[34,742846,223],{"className":742847},[32024],[34,742849],{"className":742850,"style":32604},[31845],[34,742852,742854,742857,742860,742863,742866,742869],{"className":742853},[31676],[34,742855],{"className":742856,"style":32617},[31680],[34,742858,15932],{"className":742859},[31685],[34,742861,32936],{"className":742862},[31685,31686],[34,742864],{"className":742865,"style":31846},[31845],[34,742867,202],{"className":742868},[31850],[34,742870],{"className":742871,"style":31846},[31845],[34,742873,742875,742878],{"className":742874},[31676],[34,742876],{"className":742877,"style":32617},[31680],[34,742879,459],{"className":742880},[31685],[105,742882,742883,742888],{},[108,742884,742885,742887],{},[2627,742886,58554],{}," 这是一个4阶齐次方程。虽然阶数高，但因为它没有一阶和三阶导数项，实际上解起来像解二元一次方程一样简单。",[108,742889,742890,742893],{},[2627,742891,742892],{},"解题步骤：",[2230,742894,742895,743193,744147],{},[108,742896,742897,222256,742900,655318,742974,33183,743033],{},[2627,742898,742899],{},"写出特征方程：",[34,742901,742903,742927],{"className":742902},[31642],[34,742904,742906],{"className":742905},[31646],[31648,742907,742908],{"xmlns":31650},[31652,742909,742910,742924],{},[31655,742911,742912],{},[37624,742913,742914,742916],{},[31658,742915,32936],{},[31655,742917,742918,742920,742922],{},[31662,742919,456],{"stretchy":702},[31658,742921,31660],{},[31662,742923,2557],{"stretchy":702},[31665,742925,742926],{"encoding":31667},"x^{(n)}",[34,742928,742930],{"className":742929,"ariaHidden":758},[31672],[34,742931,742933,742936],{"className":742932},[31676],[34,742934],{"className":742935,"style":61037},[31680],[34,742937,742939,742942],{"className":742938},[31685],[34,742940,32936],{"className":742941},[31685,31686],[34,742943,742945],{"className":742944},[31792],[34,742946,742948],{"className":742947},[31796],[34,742949,742951],{"className":742950},[31801],[34,742952,742954],{"className":742953,"style":61037},[31805],[34,742955,742956,742959],{"style":37777},[34,742957],{"className":742958,"style":31814},[31813],[34,742960,742962],{"className":742961},[31818,31819,31820,31821],[34,742963,742965,742968,742971],{"className":742964},[31685,31821],[34,742966,456],{"className":742967},[31782,31821],[34,742969,31660],{"className":742970},[31685,31686,31821],[34,742972,2557],{"className":742973},[31690,31821],[34,742975,742977,742995],{"className":742976},[31642],[34,742978,742980],{"className":742979},[31646],[31648,742981,742982],{"xmlns":31650},[31652,742983,742984,742992],{},[31655,742985,742986],{},[37624,742987,742988,742990],{},[31658,742989,220792],{},[31658,742991,31660],{},[31665,742993,742994],{"encoding":31667},"\\lambda^n",[34,742996,742998],{"className":742997,"ariaHidden":758},[31672],[34,742999,743001,743004],{"className":743000},[31676],[34,743002],{"className":743003,"style":31681},[31680],[34,743005,743007,743010],{"className":743006},[31685],[34,743008,220792],{"className":743009},[31685,31686],[34,743011,743013],{"className":743012},[31792],[34,743014,743016],{"className":743015},[31796],[34,743017,743019],{"className":743018},[31801],[34,743020,743022],{"className":743021,"style":37959},[31805],[34,743023,743024,743027],{"style":37777},[34,743025],{"className":743026,"style":31814},[31813],[34,743028,743030],{"className":743029},[31818,31819,31820,31821],[34,743031,31660],{"className":743032},[31685,31686,31821],[34,743034,743036,743072],{"className":743035},[31642],[34,743037,743039],{"className":743038},[31646],[31648,743040,743041],{"xmlns":31650},[31652,743042,743043,743069],{},[31655,743044,743045,743051,743053,743055,743061,743063,743065,743067],{},[37624,743046,743047,743049],{},[31658,743048,220792],{},[31749,743050,15932],{},[31662,743052,32226],{},[31749,743054,18136],{},[37624,743056,743057,743059],{},[31658,743058,220792],{},[31749,743060,768],{},[31662,743062,223],{},[31749,743064,15932],{},[31662,743066,202],{},[31749,743068,459],{},[31665,743070,743071],{"encoding":31667},"\\lambda^4 - 5\\lambda^2 + 4 = 0",[34,743073,743075,743119,743166,743184],{"className":743074,"ariaHidden":758},[31672],[34,743076,743078,743081,743110,743113,743116],{"className":743077},[31676],[34,743079],{"className":743080,"style":40044},[31680],[34,743082,743084,743087],{"className":743083},[31685],[34,743085,220792],{"className":743086},[31685,31686],[34,743088,743090],{"className":743089},[31792],[34,743091,743093],{"className":743092},[31796],[34,743094,743096],{"className":743095},[31801],[34,743097,743099],{"className":743098,"style":37774},[31805],[34,743100,743101,743104],{"style":37777},[34,743102],{"className":743103,"style":31814},[31813],[34,743105,743107],{"className":743106},[31818,31819,31820,31821],[34,743108,15932],{"className":743109},[31685,31821],[34,743111],{"className":743112,"style":32604},[31845],[34,743114,32226],{"className":743115},[32024],[34,743117],{"className":743118,"style":32604},[31845],[34,743120,743122,743125,743128,743157,743160,743163],{"className":743121},[31676],[34,743123],{"className":743124,"style":40044},[31680],[34,743126,18136],{"className":743127},[31685],[34,743129,743131,743134],{"className":743130},[31685],[34,743132,220792],{"className":743133},[31685,31686],[34,743135,743137],{"className":743136},[31792],[34,743138,743140],{"className":743139},[31796],[34,743141,743143],{"className":743142},[31801],[34,743144,743146],{"className":743145,"style":37774},[31805],[34,743147,743148,743151],{"style":37777},[34,743149],{"className":743150,"style":31814},[31813],[34,743152,743154],{"className":743153},[31818,31819,31820,31821],[34,743155,768],{"className":743156},[31685,31821],[34,743158],{"className":743159,"style":32604},[31845],[34,743161,223],{"className":743162},[32024],[34,743164],{"className":743165,"style":32604},[31845],[34,743167,743169,743172,743175,743178,743181],{"className":743168},[31676],[34,743170],{"className":743171,"style":32617},[31680],[34,743173,15932],{"className":743174},[31685],[34,743176],{"className":743177,"style":31846},[31845],[34,743179,202],{"className":743180},[31850],[34,743182],{"className":743183,"style":31846},[31845],[34,743185,743187,743190],{"className":743186},[31676],[34,743188],{"className":743189,"style":32617},[31680],[34,743191,459],{"className":743192},[31685],[108,743194,743195,743198,743199,743258,743259,743458,743459,341819,743639,4374,743720,743801,743802],{},[2627,743196,743197],{},"求解特征根：","\n这看起来是个4次方程，但我们可以把它看作关于 ",[34,743200,743202,743220],{"className":743201},[31642],[34,743203,743205],{"className":743204},[31646],[31648,743206,743207],{"xmlns":31650},[31652,743208,743209,743217],{},[31655,743210,743211],{},[37624,743212,743213,743215],{},[31658,743214,220792],{},[31749,743216,768],{},[31665,743218,743219],{"encoding":31667},"\\lambda^2",[34,743221,743223],{"className":743222,"ariaHidden":758},[31672],[34,743224,743226,743229],{"className":743225},[31676],[34,743227],{"className":743228,"style":37774},[31680],[34,743230,743232,743235],{"className":743231},[31685],[34,743233,220792],{"className":743234},[31685,31686],[34,743236,743238],{"className":743237},[31792],[34,743239,743241],{"className":743240},[31796],[34,743242,743244],{"className":743243},[31801],[34,743245,743247],{"className":743246,"style":37774},[31805],[34,743248,743249,743252],{"style":37777},[34,743250],{"className":743251,"style":31814},[31813],[34,743253,743255],{"className":743254},[31818,31819,31820,31821],[34,743256,768],{"className":743257},[31685,31821]," 的二次方程。\n利用十字相乘法（或者求根公式）：",[34,743260,743262,743312],{"className":743261},[31642],[34,743263,743265],{"className":743264},[31646],[31648,743266,743267],{"xmlns":31650},[31652,743268,743269,743309],{},[31655,743270,743271,743277,743279,743281,743283,743285,743287,743289,743291,743293,743295,743297,743299,743301,743303,743305,743307],{},[37624,743272,743273,743275],{},[31658,743274,94628],{},[31749,743276,768],{},[31662,743278,32226],{},[31749,743280,18136],{},[31658,743282,94628],{},[31662,743284,223],{},[31749,743286,15932],{},[31662,743288,202],{},[31662,743290,456],{"stretchy":702},[31658,743292,94628],{},[31662,743294,32226],{},[31749,743296,640],{},[31662,743298,2557],{"stretchy":702},[31662,743300,456],{"stretchy":702},[31658,743302,94628],{},[31662,743304,32226],{},[31749,743306,15932],{},[31662,743308,2557],{"stretchy":702},[31665,743310,743311],{"encoding":31667},"u^2 - 5u + 4 = (u-1)(u-4)",[34,743313,743315,743359,743380,743398,743419,743446],{"className":743314,"ariaHidden":758},[31672],[34,743316,743318,743321,743350,743353,743356],{"className":743317},[31676],[34,743319],{"className":743320,"style":40044},[31680],[34,743322,743324,743327],{"className":743323},[31685],[34,743325,94628],{"className":743326},[31685,31686],[34,743328,743330],{"className":743329},[31792],[34,743331,743333],{"className":743332},[31796],[34,743334,743336],{"className":743335},[31801],[34,743337,743339],{"className":743338,"style":37774},[31805],[34,743340,743341,743344],{"style":37777},[34,743342],{"className":743343,"style":31814},[31813],[34,743345,743347],{"className":743346},[31818,31819,31820,31821],[34,743348,768],{"className":743349},[31685,31821],[34,743351],{"className":743352,"style":32604},[31845],[34,743354,32226],{"className":743355},[32024],[34,743357],{"className":743358,"style":32604},[31845],[34,743360,743362,743365,743368,743371,743374,743377],{"className":743361},[31676],[34,743363],{"className":743364,"style":35942},[31680],[34,743366,18136],{"className":743367},[31685],[34,743369,94628],{"className":743370},[31685,31686],[34,743372],{"className":743373,"style":32604},[31845],[34,743375,223],{"className":743376},[32024],[34,743378],{"className":743379,"style":32604},[31845],[34,743381,743383,743386,743389,743392,743395],{"className":743382},[31676],[34,743384],{"className":743385,"style":32617},[31680],[34,743387,15932],{"className":743388},[31685],[34,743390],{"className":743391,"style":31846},[31845],[34,743393,202],{"className":743394},[31850],[34,743396],{"className":743397,"style":31846},[31845],[34,743399,743401,743404,743407,743410,743413,743416],{"className":743400},[31676],[34,743402],{"className":743403,"style":31774},[31680],[34,743405,456],{"className":743406},[31782],[34,743408,94628],{"className":743409},[31685,31686],[34,743411],{"className":743412,"style":32604},[31845],[34,743414,32226],{"className":743415},[32024],[34,743417],{"className":743418,"style":32604},[31845],[34,743420,743422,743425,743428,743431,743434,743437,743440,743443],{"className":743421},[31676],[34,743423],{"className":743424,"style":31774},[31680],[34,743426,640],{"className":743427},[31685],[34,743429,2557],{"className":743430},[31690],[34,743432,456],{"className":743433},[31782],[34,743435,94628],{"className":743436},[31685,31686],[34,743438],{"className":743439,"style":32604},[31845],[34,743441,32226],{"className":743442},[32024],[34,743444],{"className":743445,"style":32604},[31845],[34,743447,743449,743452,743455],{"className":743448},[31676],[34,743450],{"className":743451,"style":31774},[31680],[34,743453,15932],{"className":743454},[31685],[34,743456,2557],{"className":743457},[31690],"。\n所以：\n",[34,743460,743462,743506],{"className":743461},[31642],[34,743463,743465],{"className":743464},[31646],[31648,743466,743467],{"xmlns":31650},[31652,743468,743469,743503],{},[31655,743470,743471,743473,743479,743481,743483,743485,743487,743493,743495,743497,743499,743501],{},[31662,743472,456],{"stretchy":702},[37624,743474,743475,743477],{},[31658,743476,220792],{},[31749,743478,768],{},[31662,743480,32226],{},[31749,743482,640],{},[31662,743484,2557],{"stretchy":702},[31662,743486,456],{"stretchy":702},[37624,743488,743489,743491],{},[31658,743490,220792],{},[31749,743492,768],{},[31662,743494,32226],{},[31749,743496,15932],{},[31662,743498,2557],{"stretchy":702},[31662,743500,202],{},[31749,743502,459],{},[31665,743504,743505],{"encoding":31667},"(\\lambda^2 - 1)(\\lambda^2 - 4) = 0",[34,743507,743509,743556,743609,743630],{"className":743508,"ariaHidden":758},[31672],[34,743510,743512,743515,743518,743547,743550,743553],{"className":743511},[31676],[34,743513],{"className":743514,"style":37755},[31680],[34,743516,456],{"className":743517},[31782],[34,743519,743521,743524],{"className":743520},[31685],[34,743522,220792],{"className":743523},[31685,31686],[34,743525,743527],{"className":743526},[31792],[34,743528,743530],{"className":743529},[31796],[34,743531,743533],{"className":743532},[31801],[34,743534,743536],{"className":743535,"style":37774},[31805],[34,743537,743538,743541],{"style":37777},[34,743539],{"className":743540,"style":31814},[31813],[34,743542,743544],{"className":743543},[31818,31819,31820,31821],[34,743545,768],{"className":743546},[31685,31821],[34,743548],{"className":743549,"style":32604},[31845],[34,743551,32226],{"className":743552},[32024],[34,743554],{"className":743555,"style":32604},[31845],[34,743557,743559,743562,743565,743568,743571,743600,743603,743606],{"className":743558},[31676],[34,743560],{"className":743561,"style":37755},[31680],[34,743563,640],{"className":743564},[31685],[34,743566,2557],{"className":743567},[31690],[34,743569,456],{"className":743570},[31782],[34,743572,743574,743577],{"className":743573},[31685],[34,743575,220792],{"className":743576},[31685,31686],[34,743578,743580],{"className":743579},[31792],[34,743581,743583],{"className":743582},[31796],[34,743584,743586],{"className":743585},[31801],[34,743587,743589],{"className":743588,"style":37774},[31805],[34,743590,743591,743594],{"style":37777},[34,743592],{"className":743593,"style":31814},[31813],[34,743595,743597],{"className":743596},[31818,31819,31820,31821],[34,743598,768],{"className":743599},[31685,31821],[34,743601],{"className":743602,"style":32604},[31845],[34,743604,32226],{"className":743605},[32024],[34,743607],{"className":743608,"style":32604},[31845],[34,743610,743612,743615,743618,743621,743624,743627],{"className":743611},[31676],[34,743613],{"className":743614,"style":31774},[31680],[34,743616,15932],{"className":743617},[31685],[34,743619,2557],{"className":743620},[31690],[34,743622],{"className":743623,"style":31846},[31845],[34,743625,202],{"className":743626},[31850],[34,743628],{"className":743629,"style":31846},[31845],[34,743631,743633,743636],{"className":743632},[31676],[34,743634],{"className":743635,"style":32617},[31680],[34,743637,459],{"className":743638},[31685],[34,743640,743642,743664],{"className":743641},[31642],[34,743643,743645],{"className":743644},[31646],[31648,743646,743647],{"xmlns":31650},[31652,743648,743649,743661],{},[31655,743650,743651,743657,743659],{},[37624,743652,743653,743655],{},[31658,743654,220792],{},[31749,743656,768],{},[31662,743658,202],{},[31749,743660,640],{},[31665,743662,743663],{"encoding":31667},"\\lambda^2 = 1",[34,743665,743667,743711],{"className":743666,"ariaHidden":758},[31672],[34,743668,743670,743673,743702,743705,743708],{"className":743669},[31676],[34,743671],{"className":743672,"style":37774},[31680],[34,743674,743676,743679],{"className":743675},[31685],[34,743677,220792],{"className":743678},[31685,31686],[34,743680,743682],{"className":743681},[31792],[34,743683,743685],{"className":743684},[31796],[34,743686,743688],{"className":743687},[31801],[34,743689,743691],{"className":743690,"style":37774},[31805],[34,743692,743693,743696],{"style":37777},[34,743694],{"className":743695,"style":31814},[31813],[34,743697,743699],{"className":743698},[31818,31819,31820,31821],[34,743700,768],{"className":743701},[31685,31821],[34,743703],{"className":743704,"style":31846},[31845],[34,743706,202],{"className":743707},[31850],[34,743709],{"className":743710,"style":31846},[31845],[34,743712,743714,743717],{"className":743713},[31676],[34,743715],{"className":743716,"style":32617},[31680],[34,743718,640],{"className":743719},[31685],[34,743721,743723,743745],{"className":743722},[31642],[34,743724,743726],{"className":743725},[31646],[31648,743727,743728],{"xmlns":31650},[31652,743729,743730,743742],{},[31655,743731,743732,743738,743740],{},[37624,743733,743734,743736],{},[31658,743735,220792],{},[31749,743737,768],{},[31662,743739,202],{},[31749,743741,15932],{},[31665,743743,743744],{"encoding":31667},"\\lambda^2 = 4",[34,743746,743748,743792],{"className":743747,"ariaHidden":758},[31672],[34,743749,743751,743754,743783,743786,743789],{"className":743750},[31676],[34,743752],{"className":743753,"style":37774},[31680],[34,743755,743757,743760],{"className":743756},[31685],[34,743758,220792],{"className":743759},[31685,31686],[34,743761,743763],{"className":743762},[31792],[34,743764,743766],{"className":743765},[31796],[34,743767,743769],{"className":743768},[31801],[34,743770,743772],{"className":743771,"style":37774},[31805],[34,743773,743774,743777],{"style":37777},[34,743775],{"className":743776,"style":31814},[31813],[34,743778,743780],{"className":743779},[31818,31819,31820,31821],[34,743781,768],{"className":743782},[31685,31821],[34,743784],{"className":743785,"style":31846},[31845],[34,743787,202],{"className":743788},[31850],[34,743790],{"className":743791,"style":31846},[31845],[34,743793,743795,743798],{"className":743794},[31676],[34,743796],{"className":743797,"style":32617},[31680],[34,743799,15932],{"className":743800},[31685],"。\n解得四个互不相等的实根：\n",[34,743803,743805,743873],{"className":743804},[31642],[34,743806,743808],{"className":743807},[31646],[31648,743809,743810],{"xmlns":31650},[31652,743811,743812,743870],{},[31655,743813,743814,743820,743822,743824,743826,743828,743834,743836,743838,743840,743842,743844,743850,743852,743854,743856,743858,743864,743866,743868],{},[31743,743815,743816,743818],{},[31658,743817,220792],{},[31749,743819,640],{},[31662,743821,202],{},[31749,743823,640],{},[31662,743825,31960],{"separator":758},[31845,743827],{"width":71082},[31743,743829,743830,743832],{},[31658,743831,220792],{},[31749,743833,768],{},[31662,743835,202],{},[31662,743837,32226],{},[31749,743839,640],{},[31662,743841,31960],{"separator":758},[31845,743843],{"width":71082},[31743,743845,743846,743848],{},[31658,743847,220792],{},[31749,743849,786],{},[31662,743851,202],{},[31749,743853,768],{},[31662,743855,31960],{"separator":758},[31845,743857],{"width":71082},[31743,743859,743860,743862],{},[31658,743861,220792],{},[31749,743863,15932],{},[31662,743865,202],{},[31662,743867,32226],{},[31749,743869,768],{},[31665,743871,743872],{"encoding":31667},"\\lambda_1 = 1, \\quad \\lambda_2 = -1, \\quad \\lambda_3 = 2, \\quad \\lambda_4 = -2",[34,743874,743876,743931,743998,744068,744135],{"className":743875,"ariaHidden":758},[31672],[34,743877,743879,743882,743922,743925,743928],{"className":743878},[31676],[34,743880],{"className":743881,"style":33814},[31680],[34,743883,743885,743888],{"className":743884},[31685],[34,743886,220792],{"className":743887},[31685,31686],[34,743889,743891],{"className":743890},[31792],[34,743892,743894,743914],{"className":743893},[31796,31797],[34,743895,743897,743911],{"className":743896},[31801],[34,743898,743900],{"className":743899,"style":31806},[31805],[34,743901,743902,743905],{"style":31809},[34,743903],{"className":743904,"style":31814},[31813],[34,743906,743908],{"className":743907},[31818,31819,31820,31821],[34,743909,640],{"className":743910},[31685,31821],[34,743912,31829],{"className":743913},[31828],[34,743915,743917],{"className":743916},[31801],[34,743918,743920],{"className":743919,"style":31836},[31805],[34,743921],{},[34,743923],{"className":743924,"style":31846},[31845],[34,743926,202],{"className":743927},[31850],[34,743929],{"className":743930,"style":31846},[31845],[34,743932,743934,743937,743940,743943,743946,743949,743989,743992,743995],{"className":743933},[31676],[34,743935],{"className":743936,"style":43217},[31680],[34,743938,640],{"className":743939},[31685],[34,743941,31960],{"className":743942},[32119],[34,743944],{"className":743945,"style":128438},[31845],[34,743947],{"className":743948,"style":32123},[31845],[34,743950,743952,743955],{"className":743951},[31685],[34,743953,220792],{"className":743954},[31685,31686],[34,743956,743958],{"className":743957},[31792],[34,743959,743961,743981],{"className":743960},[31796,31797],[34,743962,743964,743978],{"className":743963},[31801],[34,743965,743967],{"className":743966,"style":31806},[31805],[34,743968,743969,743972],{"style":31809},[34,743970],{"className":743971,"style":31814},[31813],[34,743973,743975],{"className":743974},[31818,31819,31820,31821],[34,743976,768],{"className":743977},[31685,31821],[34,743979,31829],{"className":743980},[31828],[34,743982,743984],{"className":743983},[31801],[34,743985,743987],{"className":743986,"style":31836},[31805],[34,743988],{},[34,743990],{"className":743991,"style":31846},[31845],[34,743993,202],{"className":743994},[31850],[34,743996],{"className":743997,"style":31846},[31845],[34,743999,744001,744004,744007,744010,744013,744016,744019,744059,744062,744065],{"className":744000},[31676],[34,744002],{"className":744003,"style":43217},[31680],[34,744005,32226],{"className":744006},[31685],[34,744008,640],{"className":744009},[31685],[34,744011,31960],{"className":744012},[32119],[34,744014],{"className":744015,"style":128438},[31845],[34,744017],{"className":744018,"style":32123},[31845],[34,744020,744022,744025],{"className":744021},[31685],[34,744023,220792],{"className":744024},[31685,31686],[34,744026,744028],{"className":744027},[31792],[34,744029,744031,744051],{"className":744030},[31796,31797],[34,744032,744034,744048],{"className":744033},[31801],[34,744035,744037],{"className":744036,"style":31806},[31805],[34,744038,744039,744042],{"style":31809},[34,744040],{"className":744041,"style":31814},[31813],[34,744043,744045],{"className":744044},[31818,31819,31820,31821],[34,744046,786],{"className":744047},[31685,31821],[34,744049,31829],{"className":744050},[31828],[34,744052,744054],{"className":744053},[31801],[34,744055,744057],{"className":744056,"style":31836},[31805],[34,744058],{},[34,744060],{"className":744061,"style":31846},[31845],[34,744063,202],{"className":744064},[31850],[34,744066],{"className":744067,"style":31846},[31845],[34,744069,744071,744074,744077,744080,744083,744086,744126,744129,744132],{"className":744070},[31676],[34,744072],{"className":744073,"style":43217},[31680],[34,744075,768],{"className":744076},[31685],[34,744078,31960],{"className":744079},[32119],[34,744081],{"className":744082,"style":128438},[31845],[34,744084],{"className":744085,"style":32123},[31845],[34,744087,744089,744092],{"className":744088},[31685],[34,744090,220792],{"className":744091},[31685,31686],[34,744093,744095],{"className":744094},[31792],[34,744096,744098,744118],{"className":744097},[31796,31797],[34,744099,744101,744115],{"className":744100},[31801],[34,744102,744104],{"className":744103,"style":31806},[31805],[34,744105,744106,744109],{"style":31809},[34,744107],{"className":744108,"style":31814},[31813],[34,744110,744112],{"className":744111},[31818,31819,31820,31821],[34,744113,15932],{"className":744114},[31685,31821],[34,744116,31829],{"className":744117},[31828],[34,744119,744121],{"className":744120},[31801],[34,744122,744124],{"className":744123,"style":31836},[31805],[34,744125],{},[34,744127],{"className":744128,"style":31846},[31845],[34,744130,202],{"className":744131},[31850],[34,744133],{"className":744134,"style":31846},[31845],[34,744136,744138,744141,744144],{"className":744137},[31676],[34,744139],{"className":744140,"style":35942},[31680],[34,744142,32226],{"className":744143},[31685],[34,744145,768],{"className":744146},[31685],[108,744148,744149,744152,744153],{},[2627,744150,744151],{},"写出通解：","\n因为是四个不同的实根，根据规则，直接线性叠加。\n",[34,744154,744156,744240],{"className":744155},[31642],[34,744157,744159],{"className":744158},[31646],[31648,744160,744161],{"xmlns":31650},[31652,744162,744163,744237],{},[31655,744164,744165,744167,744169,744175,744181,744183,744189,744199,744201,744207,744217,744219,744225],{},[31658,744166,32936],{},[31662,744168,202],{},[31743,744170,744171,744173],{},[31658,744172,31965],{},[31749,744174,640],{},[37624,744176,744177,744179],{},[31658,744178,33244],{"mathvariant":38079},[31658,744180,33249],{},[31662,744182,223],{},[31743,744184,744185,744187],{},[31658,744186,31965],{},[31749,744188,768],{},[37624,744190,744191,744193],{},[31658,744192,33244],{"mathvariant":38079},[31655,744194,744195,744197],{},[31662,744196,32226],{},[31658,744198,33249],{},[31662,744200,223],{},[31743,744202,744203,744205],{},[31658,744204,31965],{},[31749,744206,786],{},[37624,744208,744209,744211],{},[31658,744210,33244],{"mathvariant":38079},[31655,744212,744213,744215],{},[31749,744214,768],{},[31658,744216,33249],{},[31662,744218,223],{},[31743,744220,744221,744223],{},[31658,744222,31965],{},[31749,744224,15932],{},[37624,744226,744227,744229],{},[31658,744228,33244],{"mathvariant":38079},[31655,744230,744231,744233,744235],{},[31662,744232,32226],{},[31749,744234,768],{},[31658,744236,33249],{},[31665,744238,744239],{"encoding":31667},"x = c_1 \\mathrm{e}^t + c_2 \\mathrm{e}^{-t} + c_3 \\mathrm{e}^{2t} + c_4 \\mathrm{e}^{-2t}",[34,744241,744243,744261,744345,744435,744526],{"className":744242,"ariaHidden":758},[31672],[34,744244,744246,744249,744252,744255,744258],{"className":744245},[31676],[34,744247],{"className":744248,"style":32527},[31680],[34,744250,32936],{"className":744251},[31685,31686],[34,744253],{"className":744254,"style":31846},[31845],[34,744256,202],{"className":744257},[31850],[34,744259],{"className":744260,"style":31846},[31845],[34,744262,744264,744267,744307,744336,744339,744342],{"className":744263},[31676],[34,744265],{"className":744266,"style":723843},[31680],[34,744268,744270,744273],{"className":744269},[31685],[34,744271,31965],{"className":744272},[31685,31686],[34,744274,744276],{"className":744275},[31792],[34,744277,744279,744299],{"className":744278},[31796,31797],[34,744280,744282,744296],{"className":744281},[31801],[34,744283,744285],{"className":744284,"style":31806},[31805],[34,744286,744287,744290],{"style":31809},[34,744288],{"className":744289,"style":31814},[31813],[34,744291,744293],{"className":744292},[31818,31819,31820,31821],[34,744294,640],{"className":744295},[31685,31821],[34,744297,31829],{"className":744298},[31828],[34,744300,744302],{"className":744301},[31801],[34,744303,744305],{"className":744304,"style":31836},[31805],[34,744306],{},[34,744308,744310,744313],{"className":744309},[31685],[34,744311,33244],{"className":744312},[31685,207923],[34,744314,744316],{"className":744315},[31792],[34,744317,744319],{"className":744318},[31796],[34,744320,744322],{"className":744321},[31801],[34,744323,744325],{"className":744324,"style":611758},[31805],[34,744326,744327,744330],{"style":37777},[34,744328],{"className":744329,"style":31814},[31813],[34,744331,744333],{"className":744332},[31818,31819,31820,31821],[34,744334,33249],{"className":744335},[31685,31686,31821],[34,744337],{"className":744338,"style":32604},[31845],[34,744340,223],{"className":744341},[32024],[34,744343],{"className":744344,"style":32604},[31845],[34,744346,744348,744351,744391,744426,744429,744432],{"className":744347},[31676],[34,744349],{"className":744350,"style":723843},[31680],[34,744352,744354,744357],{"className":744353},[31685],[34,744355,31965],{"className":744356},[31685,31686],[34,744358,744360],{"className":744359},[31792],[34,744361,744363,744383],{"className":744362},[31796,31797],[34,744364,744366,744380],{"className":744365},[31801],[34,744367,744369],{"className":744368,"style":31806},[31805],[34,744370,744371,744374],{"style":31809},[34,744372],{"className":744373,"style":31814},[31813],[34,744375,744377],{"className":744376},[31818,31819,31820,31821],[34,744378,768],{"className":744379},[31685,31821],[34,744381,31829],{"className":744382},[31828],[34,744384,744386],{"className":744385},[31801],[34,744387,744389],{"className":744388,"style":31836},[31805],[34,744390],{},[34,744392,744394,744397],{"className":744393},[31685],[34,744395,33244],{"className":744396},[31685,207923],[34,744398,744400],{"className":744399},[31792],[34,744401,744403],{"className":744402},[31796],[34,744404,744406],{"className":744405},[31801],[34,744407,744409],{"className":744408,"style":611758},[31805],[34,744410,744411,744414],{"style":37777},[34,744412],{"className":744413,"style":31814},[31813],[34,744415,744417],{"className":744416},[31818,31819,31820,31821],[34,744418,744420,744423],{"className":744419},[31685,31821],[34,744421,32226],{"className":744422},[31685,31821],[34,744424,33249],{"className":744425},[31685,31686,31821],[34,744427],{"className":744428,"style":32604},[31845],[34,744430,223],{"className":744431},[32024],[34,744433],{"className":744434,"style":32604},[31845],[34,744436,744438,744442,744482,744517,744520,744523],{"className":744437},[31676],[34,744439],{"className":744440,"style":744441},[31680],"height:0.9641em;vertical-align:-0.15em;",[34,744443,744445,744448],{"className":744444},[31685],[34,744446,31965],{"className":744447},[31685,31686],[34,744449,744451],{"className":744450},[31792],[34,744452,744454,744474],{"className":744453},[31796,31797],[34,744455,744457,744471],{"className":744456},[31801],[34,744458,744460],{"className":744459,"style":31806},[31805],[34,744461,744462,744465],{"style":31809},[34,744463],{"className":744464,"style":31814},[31813],[34,744466,744468],{"className":744467},[31818,31819,31820,31821],[34,744469,786],{"className":744470},[31685,31821],[34,744472,31829],{"className":744473},[31828],[34,744475,744477],{"className":744476},[31801],[34,744478,744480],{"className":744479,"style":31836},[31805],[34,744481],{},[34,744483,744485,744488],{"className":744484},[31685],[34,744486,33244],{"className":744487},[31685,207923],[34,744489,744491],{"className":744490},[31792],[34,744492,744494],{"className":744493},[31796],[34,744495,744497],{"className":744496},[31801],[34,744498,744500],{"className":744499,"style":37774},[31805],[34,744501,744502,744505],{"style":37777},[34,744503],{"className":744504,"style":31814},[31813],[34,744506,744508],{"className":744507},[31818,31819,31820,31821],[34,744509,744511,744514],{"className":744510},[31685,31821],[34,744512,768],{"className":744513},[31685,31821],[34,744515,33249],{"className":744516},[31685,31686,31821],[34,744518],{"className":744519,"style":32604},[31845],[34,744521,223],{"className":744522},[32024],[34,744524],{"className":744525,"style":32604},[31845],[34,744527,744529,744532,744572],{"className":744528},[31676],[34,744530],{"className":744531,"style":744441},[31680],[34,744533,744535,744538],{"className":744534},[31685],[34,744536,31965],{"className":744537},[31685,31686],[34,744539,744541],{"className":744540},[31792],[34,744542,744544,744564],{"className":744543},[31796,31797],[34,744545,744547,744561],{"className":744546},[31801],[34,744548,744550],{"className":744549,"style":31806},[31805],[34,744551,744552,744555],{"style":31809},[34,744553],{"className":744554,"style":31814},[31813],[34,744556,744558],{"className":744557},[31818,31819,31820,31821],[34,744559,15932],{"className":744560},[31685,31821],[34,744562,31829],{"className":744563},[31828],[34,744565,744567],{"className":744566},[31801],[34,744568,744570],{"className":744569,"style":31836},[31805],[34,744571],{},[34,744573,744575,744578],{"className":744574},[31685],[34,744576,33244],{"className":744577},[31685,207923],[34,744579,744581],{"className":744580},[31792],[34,744582,744584],{"className":744583},[31796],[34,744585,744587],{"className":744586},[31801],[34,744588,744590],{"className":744589,"style":37774},[31805],[34,744591,744592,744595],{"style":37777},[34,744593],{"className":744594,"style":31814},[31813],[34,744596,744598],{"className":744597},[31818,31819,31820,31821],[34,744599,744601,744604,744607],{"className":744600},[31685,31821],[34,744602,32226],{"className":744603},[31685,31821],[34,744605,768],{"className":744606},[31685,31821],[34,744608,33249],{"className":744609},[31685,31686,31821],[32188,744611],{},[15,744613,744614],{},[2627,744615,388194,744616],{},[34,744617,744619,744689],{"className":744618},[31642],[34,744620,744622],{"className":744621},[31646],[31648,744623,744624],{"xmlns":31650},[31652,744625,744626,744686],{},[31655,744627,744628,744640,744642,744644,744646,744656,744658,744660,744666,744672,744674,744680,744682,744684],{},[37624,744629,744630,744632],{},[31658,744631,32936],{},[31655,744633,744634,744636,744638],{},[31662,744635,116495],{"mathvariant":38079},[31662,744637,116495],{"mathvariant":38079},[31662,744639,116495],{"mathvariant":38079},[31662,744641,32226],{},[31749,744643,786],{},[31658,744645,111],{},[37624,744647,744648,744650],{},[31658,744649,32936],{},[31655,744651,744652,744654],{},[31662,744653,116495],{"mathvariant":38079},[31662,744655,116495],{"mathvariant":38079},[31662,744657,223],{},[31749,744659,786],{},[37624,744661,744662,744664],{},[31658,744663,111],{},[31749,744665,768],{},[37624,744667,744668,744670],{},[31658,744669,32936],{},[31662,744671,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,744673,32226],{},[37624,744675,744676,744678],{},[31658,744677,111],{},[31749,744679,786],{},[31658,744681,32936],{},[31662,744683,202],{},[31749,744685,459],{},[31665,744687,744688],{"encoding":31667},"x''' - 3a x'' + 3a^2 x' - a^3 x = 0",[34,744690,744692,744739,744792,744871,744918],{"className":744691,"ariaHidden":758},[31672],[34,744693,744695,744698,744730,744733,744736],{"className":744694},[31676],[34,744696],{"className":744697,"style":384364},[31680],[34,744699,744701,744704],{"className":744700},[31685],[34,744702,32936],{"className":744703},[31685,31686],[34,744705,744707],{"className":744706},[31792],[34,744708,744710],{"className":744709},[31796],[34,744711,744713],{"className":744712},[31801],[34,744714,744716],{"className":744715,"style":132027},[31805],[34,744717,744718,744721],{"style":37777},[34,744719],{"className":744720,"style":31814},[31813],[34,744722,744724],{"className":744723},[31818,31819,31820,31821],[34,744725,744727],{"className":744726},[31685,31821],[34,744728,502955],{"className":744729},[31685,31821],[34,744731],{"className":744732,"style":32604},[31845],[34,744734,32226],{"className":744735},[32024],[34,744737],{"className":744738,"style":32604},[31845],[34,744740,744742,744745,744748,744751,744783,744786,744789],{"className":744741},[31676],[34,744743],{"className":744744,"style":384364},[31680],[34,744746,786],{"className":744747},[31685],[34,744749,111],{"className":744750},[31685,31686],[34,744752,744754,744757],{"className":744753},[31685],[34,744755,32936],{"className":744756},[31685,31686],[34,744758,744760],{"className":744759},[31792],[34,744761,744763],{"className":744762},[31796],[34,744764,744766],{"className":744765},[31801],[34,744767,744769],{"className":744768,"style":132027},[31805],[34,744770,744771,744774],{"style":37777},[34,744772],{"className":744773,"style":31814},[31813],[34,744775,744777],{"className":744776},[31818,31819,31820,31821],[34,744778,744780],{"className":744779},[31685,31821],[34,744781,502782],{"className":744782},[31685,31821],[34,744784],{"className":744785,"style":32604},[31845],[34,744787,223],{"className":744788},[32024],[34,744790],{"className":744791,"style":32604},[31845],[34,744793,744795,744798,744801,744830,744862,744865,744868],{"className":744794},[31676],[34,744796],{"className":744797,"style":40044},[31680],[34,744799,786],{"className":744800},[31685],[34,744802,744804,744807],{"className":744803},[31685],[34,744805,111],{"className":744806},[31685,31686],[34,744808,744810],{"className":744809},[31792],[34,744811,744813],{"className":744812},[31796],[34,744814,744816],{"className":744815},[31801],[34,744817,744819],{"className":744818,"style":37774},[31805],[34,744820,744821,744824],{"style":37777},[34,744822],{"className":744823,"style":31814},[31813],[34,744825,744827],{"className":744826},[31818,31819,31820,31821],[34,744828,768],{"className":744829},[31685,31821],[34,744831,744833,744836],{"className":744832},[31685],[34,744834,32936],{"className":744835},[31685,31686],[34,744837,744839],{"className":744838},[31792],[34,744840,744842],{"className":744841},[31796],[34,744843,744845],{"className":744844},[31801],[34,744846,744848],{"className":744847,"style":132027},[31805],[34,744849,744850,744853],{"style":37777},[34,744851],{"className":744852,"style":31814},[31813],[34,744854,744856],{"className":744855},[31818,31819,31820,31821],[34,744857,744859],{"className":744858},[31685,31821],[34,744860,116495],{"className":744861},[31685,31821],[34,744863],{"className":744864,"style":32604},[31845],[34,744866,32226],{"className":744867},[32024],[34,744869],{"className":744870,"style":32604},[31845],[34,744872,744874,744877,744906,744909,744912,744915],{"className":744873},[31676],[34,744875],{"className":744876,"style":37774},[31680],[34,744878,744880,744883],{"className":744879},[31685],[34,744881,111],{"className":744882},[31685,31686],[34,744884,744886],{"className":744885},[31792],[34,744887,744889],{"className":744888},[31796],[34,744890,744892],{"className":744891},[31801],[34,744893,744895],{"className":744894,"style":37774},[31805],[34,744896,744897,744900],{"style":37777},[34,744898],{"className":744899,"style":31814},[31813],[34,744901,744903],{"className":744902},[31818,31819,31820,31821],[34,744904,786],{"className":744905},[31685,31821],[34,744907,32936],{"className":744908},[31685,31686],[34,744910],{"className":744911,"style":31846},[31845],[34,744913,202],{"className":744914},[31850],[34,744916],{"className":744917,"style":31846},[31845],[34,744919,744921,744924],{"className":744920},[31676],[34,744922],{"className":744923,"style":32617},[31680],[34,744925,459],{"className":744926},[31685],[105,744928,744929,744963],{},[108,744930,744931,744933,744934,744962],{},[2627,744932,58554],{}," 这里系数变成了字母 ",[34,744935,744937,744950],{"className":744936},[31642],[34,744938,744940],{"className":744939},[31646],[31648,744941,744942],{"xmlns":31650},[31652,744943,744944,744948],{},[31655,744945,744946],{},[31658,744947,111],{},[31665,744949,111],{"encoding":31667},[34,744951,744953],{"className":744952,"ariaHidden":758},[31672],[34,744954,744956,744959],{"className":744955},[31676],[34,744957],{"className":744958,"style":32527},[31680],[34,744960,111],{"className":744961},[31685,31686],"，把它当成普通数字处理即可。",[108,744964,744965,744967],{},[2627,744966,742892],{},[2230,744968,744969,745230,745806],{},[108,744970,744971,744973],{},[2627,744972,742899],{},[34,744974,744976,745030],{"className":744975},[31642],[34,744977,744979],{"className":744978},[31646],[31648,744980,744981],{"xmlns":31650},[31652,744982,744983,745027],{},[31655,744984,744985,744991,744993,744995,744997,745003,745005,745007,745013,745015,745017,745023,745025],{},[37624,744986,744987,744989],{},[31658,744988,220792],{},[31749,744990,786],{},[31662,744992,32226],{},[31749,744994,786],{},[31658,744996,111],{},[37624,744998,744999,745001],{},[31658,745000,220792],{},[31749,745002,768],{},[31662,745004,223],{},[31749,745006,786],{},[37624,745008,745009,745011],{},[31658,745010,111],{},[31749,745012,768],{},[31658,745014,220792],{},[31662,745016,32226],{},[37624,745018,745019,745021],{},[31658,745020,111],{},[31749,745022,786],{},[31662,745024,202],{},[31749,745026,459],{},[31665,745028,745029],{"encoding":31667},"\\lambda^3 - 3a\\lambda^2 + 3a^2\\lambda - a^3 = 0",[34,745031,745033,745077,745127,745177,745221],{"className":745032,"ariaHidden":758},[31672],[34,745034,745036,745039,745068,745071,745074],{"className":745035},[31676],[34,745037],{"className":745038,"style":40044},[31680],[34,745040,745042,745045],{"className":745041},[31685],[34,745043,220792],{"className":745044},[31685,31686],[34,745046,745048],{"className":745047},[31792],[34,745049,745051],{"className":745050},[31796],[34,745052,745054],{"className":745053},[31801],[34,745055,745057],{"className":745056,"style":37774},[31805],[34,745058,745059,745062],{"style":37777},[34,745060],{"className":745061,"style":31814},[31813],[34,745063,745065],{"className":745064},[31818,31819,31820,31821],[34,745066,786],{"className":745067},[31685,31821],[34,745069],{"className":745070,"style":32604},[31845],[34,745072,32226],{"className":745073},[32024],[34,745075],{"className":745076,"style":32604},[31845],[34,745078,745080,745083,745086,745089,745118,745121,745124],{"className":745079},[31676],[34,745081],{"className":745082,"style":40044},[31680],[34,745084,786],{"className":745085},[31685],[34,745087,111],{"className":745088},[31685,31686],[34,745090,745092,745095],{"className":745091},[31685],[34,745093,220792],{"className":745094},[31685,31686],[34,745096,745098],{"className":745097},[31792],[34,745099,745101],{"className":745100},[31796],[34,745102,745104],{"className":745103},[31801],[34,745105,745107],{"className":745106,"style":37774},[31805],[34,745108,745109,745112],{"style":37777},[34,745110],{"className":745111,"style":31814},[31813],[34,745113,745115],{"className":745114},[31818,31819,31820,31821],[34,745116,768],{"className":745117},[31685,31821],[34,745119],{"className":745120,"style":32604},[31845],[34,745122,223],{"className":745123},[32024],[34,745125],{"className":745126,"style":32604},[31845],[34,745128,745130,745133,745136,745165,745168,745171,745174],{"className":745129},[31676],[34,745131],{"className":745132,"style":40044},[31680],[34,745134,786],{"className":745135},[31685],[34,745137,745139,745142],{"className":745138},[31685],[34,745140,111],{"className":745141},[31685,31686],[34,745143,745145],{"className":745144},[31792],[34,745146,745148],{"className":745147},[31796],[34,745149,745151],{"className":745150},[31801],[34,745152,745154],{"className":745153,"style":37774},[31805],[34,745155,745156,745159],{"style":37777},[34,745157],{"className":745158,"style":31814},[31813],[34,745160,745162],{"className":745161},[31818,31819,31820,31821],[34,745163,768],{"className":745164},[31685,31821],[34,745166,220792],{"className":745167},[31685,31686],[34,745169],{"className":745170,"style":32604},[31845],[34,745172,32226],{"className":745173},[32024],[34,745175],{"className":745176,"style":32604},[31845],[34,745178,745180,745183,745212,745215,745218],{"className":745179},[31676],[34,745181],{"className":745182,"style":37774},[31680],[34,745184,745186,745189],{"className":745185},[31685],[34,745187,111],{"className":745188},[31685,31686],[34,745190,745192],{"className":745191},[31792],[34,745193,745195],{"className":745194},[31796],[34,745196,745198],{"className":745197},[31801],[34,745199,745201],{"className":745200,"style":37774},[31805],[34,745202,745203,745206],{"style":37777},[34,745204],{"className":745205,"style":31814},[31813],[34,745207,745209],{"className":745208},[31818,31819,31820,31821],[34,745210,786],{"className":745211},[31685,31821],[34,745213],{"className":745214,"style":31846},[31845],[34,745216,202],{"className":745217},[31850],[34,745219],{"className":745220,"style":31846},[31845],[34,745222,745224,745227],{"className":745223},[31676],[34,745225],{"className":745226,"style":32617},[31680],[34,745228,459],{"className":745229},[31685],[108,745231,745232,745234,745235,745554,745555,745668,745669,39984,745720,745722,745723,745751,745752,745755,745756,31901],{},[2627,745233,743197],{},"\n观察这个式子，系数是 1, -3, 3, -1。这正是二项式展开公式 ",[34,745236,745238,745304],{"className":745237},[31642],[34,745239,745241],{"className":745240},[31646],[31648,745242,745243],{"xmlns":31650},[31652,745244,745245,745301],{},[31655,745246,745247,745249,745251,745253,745255,745261,745263,745269,745271,745273,745279,745281,745283,745285,745287,745293,745295],{},[31662,745248,456],{"stretchy":702},[31658,745250,54151],{},[31662,745252,32226],{},[31658,745254,54224],{},[37624,745256,745257,745259],{},[31662,745258,2557],{"stretchy":702},[31749,745260,786],{},[31662,745262,202],{},[37624,745264,745265,745267],{},[31658,745266,54151],{},[31749,745268,786],{},[31662,745270,32226],{},[31749,745272,786],{},[37624,745274,745275,745277],{},[31658,745276,54151],{},[31749,745278,768],{},[31658,745280,54224],{},[31662,745282,223],{},[31749,745284,786],{},[31658,745286,54151],{},[37624,745288,745289,745291],{},[31658,745290,54224],{},[31749,745292,768],{},[31662,745294,32226],{},[37624,745296,745297,745299],{},[31658,745298,54224],{},[31749,745300,786],{},[31665,745302,745303],{"encoding":31667},"(A-B)^3 = A^3 - 3A^2B + 3AB^2 - B^3",[34,745305,745307,745328,745375,745419,745469,745519],{"className":745306,"ariaHidden":758},[31672],[34,745308,745310,745313,745316,745319,745322,745325],{"className":745309},[31676],[34,745311],{"className":745312,"style":31774},[31680],[34,745314,456],{"className":745315},[31782],[34,745317,54151],{"className":745318},[31685,31686],[34,745320],{"className":745321,"style":32604},[31845],[34,745323,32226],{"className":745324},[32024],[34,745326],{"className":745327,"style":32604},[31845],[34,745329,745331,745334,745337,745366,745369,745372],{"className":745330},[31676],[34,745332],{"className":745333,"style":37755},[31680],[34,745335,54224],{"className":745336,"style":54252},[31685,31686],[34,745338,745340,745343],{"className":745339},[31690],[34,745341,2557],{"className":745342},[31690],[34,745344,745346],{"className":745345},[31792],[34,745347,745349],{"className":745348},[31796],[34,745350,745352],{"className":745351},[31801],[34,745353,745355],{"className":745354,"style":37774},[31805],[34,745356,745357,745360],{"style":37777},[34,745358],{"className":745359,"style":31814},[31813],[34,745361,745363],{"className":745362},[31818,31819,31820,31821],[34,745364,786],{"className":745365},[31685,31821],[34,745367],{"className":745368,"style":31846},[31845],[34,745370,202],{"className":745371},[31850],[34,745373],{"className":745374,"style":31846},[31845],[34,745376,745378,745381,745410,745413,745416],{"className":745377},[31676],[34,745379],{"className":745380,"style":40044},[31680],[34,745382,745384,745387],{"className":745383},[31685],[34,745385,54151],{"className":745386},[31685,31686],[34,745388,745390],{"className":745389},[31792],[34,745391,745393],{"className":745392},[31796],[34,745394,745396],{"className":745395},[31801],[34,745397,745399],{"className":745398,"style":37774},[31805],[34,745400,745401,745404],{"style":37777},[34,745402],{"className":745403,"style":31814},[31813],[34,745405,745407],{"className":745406},[31818,31819,31820,31821],[34,745408,786],{"className":745409},[31685,31821],[34,745411],{"className":745412,"style":32604},[31845],[34,745414,32226],{"className":745415},[32024],[34,745417],{"className":745418,"style":32604},[31845],[34,745420,745422,745425,745428,745457,745460,745463,745466],{"className":745421},[31676],[34,745423],{"className":745424,"style":40044},[31680],[34,745426,786],{"className":745427},[31685],[34,745429,745431,745434],{"className":745430},[31685],[34,745432,54151],{"className":745433},[31685,31686],[34,745435,745437],{"className":745436},[31792],[34,745438,745440],{"className":745439},[31796],[34,745441,745443],{"className":745442},[31801],[34,745444,745446],{"className":745445,"style":37774},[31805],[34,745447,745448,745451],{"style":37777},[34,745449],{"className":745450,"style":31814},[31813],[34,745452,745454],{"className":745453},[31818,31819,31820,31821],[34,745455,768],{"className":745456},[31685,31821],[34,745458,54224],{"className":745459,"style":54252},[31685,31686],[34,745461],{"className":745462,"style":32604},[31845],[34,745464,223],{"className":745465},[32024],[34,745467],{"className":745468,"style":32604},[31845],[34,745470,745472,745475,745478,745481,745510,745513,745516],{"className":745471},[31676],[34,745473],{"className":745474,"style":40044},[31680],[34,745476,786],{"className":745477},[31685],[34,745479,54151],{"className":745480},[31685,31686],[34,745482,745484,745487],{"className":745483},[31685],[34,745485,54224],{"className":745486,"style":54252},[31685,31686],[34,745488,745490],{"className":745489},[31792],[34,745491,745493],{"className":745492},[31796],[34,745494,745496],{"className":745495},[31801],[34,745497,745499],{"className":745498,"style":37774},[31805],[34,745500,745501,745504],{"style":37777},[34,745502],{"className":745503,"style":31814},[31813],[34,745505,745507],{"className":745506},[31818,31819,31820,31821],[34,745508,768],{"className":745509},[31685,31821],[34,745511],{"className":745512,"style":32604},[31845],[34,745514,32226],{"className":745515},[32024],[34,745517],{"className":745518,"style":32604},[31845],[34,745520,745522,745525],{"className":745521},[31676],[34,745523],{"className":745524,"style":37774},[31680],[34,745526,745528,745531],{"className":745527},[31685],[34,745529,54224],{"className":745530,"style":54252},[31685,31686],[34,745532,745534],{"className":745533},[31792],[34,745535,745537],{"className":745536},[31796],[34,745538,745540],{"className":745539},[31801],[34,745541,745543],{"className":745542,"style":37774},[31805],[34,745544,745545,745548],{"style":37777},[34,745546],{"className":745547,"style":31814},[31813],[34,745549,745551],{"className":745550},[31818,31819,31820,31821],[34,745552,786],{"className":745553},[31685,31821]," 的形式。\n所以方程可以因式分解为：\n",[34,745556,745558,745588],{"className":745557},[31642],[34,745559,745561],{"className":745560},[31646],[31648,745562,745563],{"xmlns":31650},[31652,745564,745565,745585],{},[31655,745566,745567,745569,745571,745573,745575,745581,745583],{},[31662,745568,456],{"stretchy":702},[31658,745570,220792],{},[31662,745572,32226],{},[31658,745574,111],{},[37624,745576,745577,745579],{},[31662,745578,2557],{"stretchy":702},[31749,745580,786],{},[31662,745582,202],{},[31749,745584,459],{},[31665,745586,745587],{"encoding":31667},"(\\lambda - a)^3 = 0",[34,745589,745591,745612,745659],{"className":745590,"ariaHidden":758},[31672],[34,745592,745594,745597,745600,745603,745606,745609],{"className":745593},[31676],[34,745595],{"className":745596,"style":31774},[31680],[34,745598,456],{"className":745599},[31782],[34,745601,220792],{"className":745602},[31685,31686],[34,745604],{"className":745605,"style":32604},[31845],[34,745607,32226],{"className":745608},[32024],[34,745610],{"className":745611,"style":32604},[31845],[34,745613,745615,745618,745621,745650,745653,745656],{"className":745614},[31676],[34,745616],{"className":745617,"style":37755},[31680],[34,745619,111],{"className":745620},[31685,31686],[34,745622,745624,745627],{"className":745623},[31690],[34,745625,2557],{"className":745626},[31690],[34,745628,745630],{"className":745629},[31792],[34,745631,745633],{"className":745632},[31796],[34,745634,745636],{"className":745635},[31801],[34,745637,745639],{"className":745638,"style":37774},[31805],[34,745640,745641,745644],{"style":37777},[34,745642],{"className":745643,"style":31814},[31813],[34,745645,745647],{"className":745646},[31818,31819,31820,31821],[34,745648,786],{"className":745649},[31685,31821],[34,745651],{"className":745652,"style":31846},[31845],[34,745654,202],{"className":745655},[31850],[34,745657],{"className":745658,"style":31846},[31845],[34,745660,745662,745665],{"className":745661},[31676],[34,745663],{"className":745664,"style":32617},[31680],[34,745666,459],{"className":745667},[31685],"\n这里得出特征值 ",[34,745670,745672,745690],{"className":745671},[31642],[34,745673,745675],{"className":745674},[31646],[31648,745676,745677],{"xmlns":31650},[31652,745678,745679,745687],{},[31655,745680,745681,745683,745685],{},[31658,745682,220792],{},[31662,745684,202],{},[31658,745686,111],{},[31665,745688,745689],{"encoding":31667},"\\lambda = a",[34,745691,745693,745711],{"className":745692,"ariaHidden":758},[31672],[34,745694,745696,745699,745702,745705,745708],{"className":745695},[31676],[34,745697],{"className":745698,"style":31681},[31680],[34,745700,220792],{"className":745701},[31685,31686],[34,745703],{"className":745704,"style":31846},[31845],[34,745706,202],{"className":745707},[31850],[34,745709],{"className":745710,"style":31846},[31845],[34,745712,745714,745717],{"className":745713},[31676],[34,745715],{"className":745716,"style":32527},[31680],[34,745718,111],{"className":745719},[31685,31686],[2627,745721,8882],{}," 这里的 ",[34,745724,745726,745739],{"className":745725},[31642],[34,745727,745729],{"className":745728},[31646],[31648,745730,745731],{"xmlns":31650},[31652,745732,745733,745737],{},[31655,745734,745735],{},[31658,745736,111],{},[31665,745738,111],{"encoding":31667},[34,745740,745742],{"className":745741,"ariaHidden":758},[31672],[34,745743,745745,745748],{"className":745744},[31676],[34,745746],{"className":745747,"style":32527},[31680],[34,745749,111],{"className":745750},[31685,31686]," 出现了3次，所以它是",[2627,745753,745754],{},"三重根","（重数 ",[34,745757,745759,745776],{"className":745758},[31642],[34,745760,745762],{"className":745761},[31646],[31648,745763,745764],{"xmlns":31650},[31652,745765,745766,745774],{},[31655,745767,745768,745770,745772],{},[31658,745769,33759],{},[31662,745771,202],{},[31749,745773,786],{},[31665,745775,68938],{"encoding":31667},[34,745777,745779,745797],{"className":745778,"ariaHidden":758},[31672],[34,745780,745782,745785,745788,745791,745794],{"className":745781},[31676],[34,745783],{"className":745784,"style":31681},[31680],[34,745786,33759],{"className":745787,"style":33774},[31685,31686],[34,745789],{"className":745790,"style":31846},[31845],[34,745792,202],{"className":745793},[31850],[34,745795],{"className":745796,"style":31846},[31845],[34,745798,745800,745803],{"className":745799},[31676],[34,745801],{"className":745802,"style":32617},[31680],[34,745804,786],{"className":745805},[31685],[108,745807,745808,745810,745811,258,745839,745897,745898,745967,745968,746042,746043,746147,746150],{},[2627,745809,744151],{},"\n对于三重根，我们需要补充乘上 ",[34,745812,745814,745827],{"className":745813},[31642],[34,745815,745817],{"className":745816},[31646],[31648,745818,745819],{"xmlns":31650},[31652,745820,745821,745825],{},[31655,745822,745823],{},[31658,745824,33249],{},[31665,745826,33249],{"encoding":31667},[34,745828,745830],{"className":745829,"ariaHidden":758},[31672],[34,745831,745833,745836],{"className":745832},[31676],[34,745834],{"className":745835,"style":33280},[31680],[34,745837,33249],{"className":745838},[31685,31686],[34,745840,745842,745859],{"className":745841},[31642],[34,745843,745845],{"className":745844},[31646],[31648,745846,745847],{"xmlns":31650},[31652,745848,745849,745857],{},[31655,745850,745851],{},[37624,745852,745853,745855],{},[31658,745854,33249],{},[31749,745856,768],{},[31665,745858,729795],{"encoding":31667},[34,745860,745862],{"className":745861,"ariaHidden":758},[31672],[34,745863,745865,745868],{"className":745864},[31676],[34,745866],{"className":745867,"style":37774},[31680],[34,745869,745871,745874],{"className":745870},[31685],[34,745872,33249],{"className":745873},[31685,31686],[34,745875,745877],{"className":745876},[31792],[34,745878,745880],{"className":745879},[31796],[34,745881,745883],{"className":745882},[31801],[34,745884,745886],{"className":745885,"style":37774},[31805],[34,745887,745888,745891],{"style":37777},[34,745889],{"className":745890,"style":31814},[31813],[34,745892,745894],{"className":745893},[31818,31819,31820,31821],[34,745895,768],{"className":745896},[31685,31821]," 才能凑够3个独立的解。\n基础解：",[34,745899,745901,745923],{"className":745900},[31642],[34,745902,745904],{"className":745903},[31646],[31648,745905,745906],{"xmlns":31650},[31652,745907,745908,745920],{},[31655,745909,745910],{},[37624,745911,745912,745914],{},[31658,745913,33244],{"mathvariant":38079},[31655,745915,745916,745918],{},[31658,745917,111],{},[31658,745919,33249],{},[31665,745921,745922],{"encoding":31667},"\\mathrm{e}^{at}",[34,745924,745926],{"className":745925,"ariaHidden":758},[31672],[34,745927,745929,745932],{"className":745928},[31676],[34,745930],{"className":745931,"style":611758},[31680],[34,745933,745935,745938],{"className":745934},[31685],[34,745936,33244],{"className":745937},[31685,207923],[34,745939,745941],{"className":745940},[31792],[34,745942,745944],{"className":745943},[31796],[34,745945,745947],{"className":745946},[31801],[34,745948,745950],{"className":745949,"style":611758},[31805],[34,745951,745952,745955],{"style":37777},[34,745953],{"className":745954,"style":31814},[31813],[34,745956,745958],{"className":745957},[31818,31819,31820,31821],[34,745959,745961,745964],{"className":745960},[31685,31821],[34,745962,111],{"className":745963},[31685,31686,31821],[34,745965,33249],{"className":745966},[31685,31686,31821],"\n补乘一次：",[34,745969,745971,745995],{"className":745970},[31642],[34,745972,745974],{"className":745973},[31646],[31648,745975,745976],{"xmlns":31650},[31652,745977,745978,745992],{},[31655,745979,745980,745982],{},[31658,745981,33249],{},[37624,745983,745984,745986],{},[31658,745985,33244],{"mathvariant":38079},[31655,745987,745988,745990],{},[31658,745989,111],{},[31658,745991,33249],{},[31665,745993,745994],{"encoding":31667},"t\\mathrm{e}^{at}",[34,745996,745998],{"className":745997,"ariaHidden":758},[31672],[34,745999,746001,746004,746007],{"className":746000},[31676],[34,746002],{"className":746003,"style":611758},[31680],[34,746005,33249],{"className":746006},[31685,31686],[34,746008,746010,746013],{"className":746009},[31685],[34,746011,33244],{"className":746012},[31685,207923],[34,746014,746016],{"className":746015},[31792],[34,746017,746019],{"className":746018},[31796],[34,746020,746022],{"className":746021},[31801],[34,746023,746025],{"className":746024,"style":611758},[31805],[34,746026,746027,746030],{"style":37777},[34,746028],{"className":746029,"style":31814},[31813],[34,746031,746033],{"className":746032},[31818,31819,31820,31821],[34,746034,746036,746039],{"className":746035},[31685,31821],[34,746037,111],{"className":746038},[31685,31686,31821],[34,746040,33249],{"className":746041},[31685,31686,31821],"\n补乘两次：",[34,746044,746046,746074],{"className":746045},[31642],[34,746047,746049],{"className":746048},[31646],[31648,746050,746051],{"xmlns":31650},[31652,746052,746053,746071],{},[31655,746054,746055,746061],{},[37624,746056,746057,746059],{},[31658,746058,33249],{},[31749,746060,768],{},[37624,746062,746063,746065],{},[31658,746064,33244],{"mathvariant":38079},[31655,746066,746067,746069],{},[31658,746068,111],{},[31658,746070,33249],{},[31665,746072,746073],{"encoding":31667},"t^2\\mathrm{e}^{at}",[34,746075,746077],{"className":746076,"ariaHidden":758},[31672],[34,746078,746080,746083,746112],{"className":746079},[31676],[34,746081],{"className":746082,"style":37774},[31680],[34,746084,746086,746089],{"className":746085},[31685],[34,746087,33249],{"className":746088},[31685,31686],[34,746090,746092],{"className":746091},[31792],[34,746093,746095],{"className":746094},[31796],[34,746096,746098],{"className":746097},[31801],[34,746099,746101],{"className":746100,"style":37774},[31805],[34,746102,746103,746106],{"style":37777},[34,746104],{"className":746105,"style":31814},[31813],[34,746107,746109],{"className":746108},[31818,31819,31820,31821],[34,746110,768],{"className":746111},[31685,31821],[34,746113,746115,746118],{"className":746114},[31685],[34,746116,33244],{"className":746117},[31685,207923],[34,746119,746121],{"className":746120},[31792],[34,746122,746124],{"className":746123},[31796],[34,746125,746127],{"className":746126},[31801],[34,746128,746130],{"className":746129,"style":611758},[31805],[34,746131,746132,746135],{"style":37777},[34,746133],{"className":746134,"style":31814},[31813],[34,746136,746138],{"className":746137},[31818,31819,31820,31821],[34,746139,746141,746144],{"className":746140},[31685,31821],[34,746142,111],{"className":746143},[31685,31686,31821],[34,746145,33249],{"className":746146},[31685,31686,31821],[2627,746148,746149],{},"通解：",[34,746151,746153,746213],{"className":746152},[31642],[34,746154,746156],{"className":746155},[31646],[31648,746157,746158],{"xmlns":31650},[31652,746159,746160,746210],{},[31655,746161,746162,746164,746166,746168,746174,746176,746182,746184,746186,746192,746198,746200],{},[31658,746163,32936],{},[31662,746165,202],{},[31662,746167,456],{"stretchy":702},[31743,746169,746170,746172],{},[31658,746171,31965],{},[31749,746173,640],{},[31662,746175,223],{},[31743,746177,746178,746180],{},[31658,746179,31965],{},[31749,746181,768],{},[31658,746183,33249],{},[31662,746185,223],{},[31743,746187,746188,746190],{},[31658,746189,31965],{},[31749,746191,786],{},[37624,746193,746194,746196],{},[31658,746195,33249],{},[31749,746197,768],{},[31662,746199,2557],{"stretchy":702},[37624,746201,746202,746204],{},[31658,746203,33244],{"mathvariant":38079},[31655,746205,746206,746208],{},[31658,746207,111],{},[31658,746209,33249],{},[31665,746211,746212],{"encoding":31667},"x = (c_1 + c_2 t + c_3 t^2) \\mathrm{e}^{at}",[34,746214,746216,746234,746292,746350],{"className":746215,"ariaHidden":758},[31672],[34,746217,746219,746222,746225,746228,746231],{"className":746218},[31676],[34,746220],{"className":746221,"style":32527},[31680],[34,746223,32936],{"className":746224},[31685,31686],[34,746226],{"className":746227,"style":31846},[31845],[34,746229,202],{"className":746230},[31850],[34,746232],{"className":746233,"style":31846},[31845],[34,746235,746237,746240,746243,746283,746286,746289],{"className":746236},[31676],[34,746238],{"className":746239,"style":31774},[31680],[34,746241,456],{"className":746242},[31782],[34,746244,746246,746249],{"className":746245},[31685],[34,746247,31965],{"className":746248},[31685,31686],[34,746250,746252],{"className":746251},[31792],[34,746253,746255,746275],{"className":746254},[31796,31797],[34,746256,746258,746272],{"className":746257},[31801],[34,746259,746261],{"className":746260,"style":31806},[31805],[34,746262,746263,746266],{"style":31809},[34,746264],{"className":746265,"style":31814},[31813],[34,746267,746269],{"className":746268},[31818,31819,31820,31821],[34,746270,640],{"className":746271},[31685,31821],[34,746273,31829],{"className":746274},[31828],[34,746276,746278],{"className":746277},[31801],[34,746279,746281],{"className":746280,"style":31836},[31805],[34,746282],{},[34,746284],{"className":746285,"style":32604},[31845],[34,746287,223],{"className":746288},[32024],[34,746290],{"className":746291,"style":32604},[31845],[34,746293,746295,746298,746338,746341,746344,746347],{"className":746294},[31676],[34,746296],{"className":746297,"style":83872},[31680],[34,746299,746301,746304],{"className":746300},[31685],[34,746302,31965],{"className":746303},[31685,31686],[34,746305,746307],{"className":746306},[31792],[34,746308,746310,746330],{"className":746309},[31796,31797],[34,746311,746313,746327],{"className":746312},[31801],[34,746314,746316],{"className":746315,"style":31806},[31805],[34,746317,746318,746321],{"style":31809},[34,746319],{"className":746320,"style":31814},[31813],[34,746322,746324],{"className":746323},[31818,31819,31820,31821],[34,746325,768],{"className":746326},[31685,31821],[34,746328,31829],{"className":746329},[31828],[34,746331,746333],{"className":746332},[31801],[34,746334,746336],{"className":746335,"style":31836},[31805],[34,746337],{},[34,746339,33249],{"className":746340},[31685,31686],[34,746342],{"className":746343,"style":32604},[31845],[34,746345,223],{"className":746346},[32024],[34,746348],{"className":746349,"style":32604},[31845],[34,746351,746353,746356,746396,746425,746428],{"className":746352},[31676],[34,746354],{"className":746355,"style":37755},[31680],[34,746357,746359,746362],{"className":746358},[31685],[34,746360,31965],{"className":746361},[31685,31686],[34,746363,746365],{"className":746364},[31792],[34,746366,746368,746388],{"className":746367},[31796,31797],[34,746369,746371,746385],{"className":746370},[31801],[34,746372,746374],{"className":746373,"style":31806},[31805],[34,746375,746376,746379],{"style":31809},[34,746377],{"className":746378,"style":31814},[31813],[34,746380,746382],{"className":746381},[31818,31819,31820,31821],[34,746383,786],{"className":746384},[31685,31821],[34,746386,31829],{"className":746387},[31828],[34,746389,746391],{"className":746390},[31801],[34,746392,746394],{"className":746393,"style":31836},[31805],[34,746395],{},[34,746397,746399,746402],{"className":746398},[31685],[34,746400,33249],{"className":746401},[31685,31686],[34,746403,746405],{"className":746404},[31792],[34,746406,746408],{"className":746407},[31796],[34,746409,746411],{"className":746410},[31801],[34,746412,746414],{"className":746413,"style":37774},[31805],[34,746415,746416,746419],{"style":37777},[34,746417],{"className":746418,"style":31814},[31813],[34,746420,746422],{"className":746421},[31818,31819,31820,31821],[34,746423,768],{"className":746424},[31685,31821],[34,746426,2557],{"className":746427},[31690],[34,746429,746431,746434],{"className":746430},[31685],[34,746432,33244],{"className":746433},[31685,207923],[34,746435,746437],{"className":746436},[31792],[34,746438,746440],{"className":746439},[31796],[34,746441,746443],{"className":746442},[31801],[34,746444,746446],{"className":746445,"style":611758},[31805],[34,746447,746448,746451],{"style":37777},[34,746449],{"className":746450,"style":31814},[31813],[34,746452,746454],{"className":746453},[31818,31819,31820,31821],[34,746455,746457,746460],{"className":746456},[31685,31821],[34,746458,111],{"className":746459},[31685,31686,31821],[34,746461,33249],{"className":746462},[31685,31686,31821],[32188,746464],{},[15,746466,746467],{},[2627,746468,746469,746470],{},"(6) ",[34,746471,746473,746537],{"className":746472},[31642],[34,746474,746476],{"className":746475},[31646],[31648,746477,746478],{"xmlns":31650},[31652,746479,746480,746534],{},[31655,746481,746482,746494,746496,746498,746508,746510,746512,746518,746520,746522,746524,746526,746528,746530,746532],{},[37624,746483,746484,746486],{},[31658,746485,32936],{},[31655,746487,746488,746490,746492],{},[31662,746489,116495],{"mathvariant":38079},[31662,746491,116495],{"mathvariant":38079},[31662,746493,116495],{"mathvariant":38079},[31662,746495,32226],{},[31749,746497,15932],{},[37624,746499,746500,746502],{},[31658,746501,32936],{},[31655,746503,746504,746506],{},[31662,746505,116495],{"mathvariant":38079},[31662,746507,116495],{"mathvariant":38079},[31662,746509,223],{},[31749,746511,18136],{},[37624,746513,746514,746516],{},[31658,746515,32936],{},[31662,746517,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,746519,32226],{},[31749,746521,768],{},[31658,746523,32936],{},[31662,746525,202],{},[31749,746527,768],{},[31658,746529,33249],{},[31662,746531,223],{},[31749,746533,786],{},[31665,746535,746536],{"encoding":31667},"x''' - 4x'' + 5x' - 2x = 2t + 3",[34,746538,746540,746587,746637,746687,746708,746729],{"className":746539,"ariaHidden":758},[31672],[34,746541,746543,746546,746578,746581,746584],{"className":746542},[31676],[34,746544],{"className":746545,"style":384364},[31680],[34,746547,746549,746552],{"className":746548},[31685],[34,746550,32936],{"className":746551},[31685,31686],[34,746553,746555],{"className":746554},[31792],[34,746556,746558],{"className":746557},[31796],[34,746559,746561],{"className":746560},[31801],[34,746562,746564],{"className":746563,"style":132027},[31805],[34,746565,746566,746569],{"style":37777},[34,746567],{"className":746568,"style":31814},[31813],[34,746570,746572],{"className":746571},[31818,31819,31820,31821],[34,746573,746575],{"className":746574},[31685,31821],[34,746576,502955],{"className":746577},[31685,31821],[34,746579],{"className":746580,"style":32604},[31845],[34,746582,32226],{"className":746583},[32024],[34,746585],{"className":746586,"style":32604},[31845],[34,746588,746590,746593,746596,746628,746631,746634],{"className":746589},[31676],[34,746591],{"className":746592,"style":384364},[31680],[34,746594,15932],{"className":746595},[31685],[34,746597,746599,746602],{"className":746598},[31685],[34,746600,32936],{"className":746601},[31685,31686],[34,746603,746605],{"className":746604},[31792],[34,746606,746608],{"className":746607},[31796],[34,746609,746611],{"className":746610},[31801],[34,746612,746614],{"className":746613,"style":132027},[31805],[34,746615,746616,746619],{"style":37777},[34,746617],{"className":746618,"style":31814},[31813],[34,746620,746622],{"className":746621},[31818,31819,31820,31821],[34,746623,746625],{"className":746624},[31685,31821],[34,746626,502782],{"className":746627},[31685,31821],[34,746629],{"className":746630,"style":32604},[31845],[34,746632,223],{"className":746633},[32024],[34,746635],{"className":746636,"style":32604},[31845],[34,746638,746640,746643,746646,746678,746681,746684],{"className":746639},[31676],[34,746641],{"className":746642,"style":384364},[31680],[34,746644,18136],{"className":746645},[31685],[34,746647,746649,746652],{"className":746648},[31685],[34,746650,32936],{"className":746651},[31685,31686],[34,746653,746655],{"className":746654},[31792],[34,746656,746658],{"className":746657},[31796],[34,746659,746661],{"className":746660},[31801],[34,746662,746664],{"className":746663,"style":132027},[31805],[34,746665,746666,746669],{"style":37777},[34,746667],{"className":746668,"style":31814},[31813],[34,746670,746672],{"className":746671},[31818,31819,31820,31821],[34,746673,746675],{"className":746674},[31685,31821],[34,746676,116495],{"className":746677},[31685,31821],[34,746679],{"className":746680,"style":32604},[31845],[34,746682,32226],{"className":746683},[32024],[34,746685],{"className":746686,"style":32604},[31845],[34,746688,746690,746693,746696,746699,746702,746705],{"className":746689},[31676],[34,746691],{"className":746692,"style":32617},[31680],[34,746694,768],{"className":746695},[31685],[34,746697,32936],{"className":746698},[31685,31686],[34,746700],{"className":746701,"style":31846},[31845],[34,746703,202],{"className":746704},[31850],[34,746706],{"className":746707,"style":31846},[31845],[34,746709,746711,746714,746717,746720,746723,746726],{"className":746710},[31676],[34,746712],{"className":746713,"style":35942},[31680],[34,746715,768],{"className":746716},[31685],[34,746718,33249],{"className":746719},[31685,31686],[34,746721],{"className":746722,"style":32604},[31845],[34,746724,223],{"className":746725},[32024],[34,746727],{"className":746728,"style":32604},[31845],[34,746730,746732,746735],{"className":746731},[31676],[34,746733],{"className":746734,"style":32617},[31680],[34,746736,786],{"className":746737},[31685],[105,746739,746740,746900,748773,750865],{},[108,746741,746742,746744,746745,746748,746749],{},[2627,746743,58554],{}," 这是一个",[2627,746746,746747],{},"非齐次","方程。",[105,746750,746751,746824,746897],{},[108,746752,746753,746754,31901],{},"第一步：先算左边等于0的通解（齐次解 ",[34,746755,746757,746775],{"className":746756},[31642],[34,746758,746760],{"className":746759},[31646],[31648,746761,746762],{"xmlns":31650},[31652,746763,746764,746772],{},[31655,746765,746766],{},[31743,746767,746768,746770],{},[31658,746769,32936],{},[31658,746771,47292],{},[31665,746773,746774],{"encoding":31667},"x_h",[34,746776,746778],{"className":746777,"ariaHidden":758},[31672],[34,746779,746781,746784],{"className":746780},[31676],[34,746782],{"className":746783,"style":31860},[31680],[34,746785,746787,746790],{"className":746786},[31685],[34,746788,32936],{"className":746789},[31685,31686],[34,746791,746793],{"className":746792},[31792],[34,746794,746796,746816],{"className":746795},[31796,31797],[34,746797,746799,746813],{"className":746798},[31801],[34,746800,746802],{"className":746801,"style":33833},[31805],[34,746803,746804,746807],{"style":31809},[34,746805],{"className":746806,"style":31814},[31813],[34,746808,746810],{"className":746809},[31818,31819,31820,31821],[34,746811,47292],{"className":746812},[31685,31686,31821],[34,746814,31829],{"className":746815},[31828],[34,746817,746819],{"className":746818},[31801],[34,746820,746822],{"className":746821,"style":31836},[31805],[34,746823],{},[108,746825,746826,746827,31901],{},"第二步：利用待定系数法算右边的特解（",[34,746828,746830,746848],{"className":746829},[31642],[34,746831,746833],{"className":746832},[31646],[31648,746834,746835],{"xmlns":31650},[31652,746836,746837,746845],{},[31655,746838,746839],{},[31743,746840,746841,746843],{},[31658,746842,32936],{},[31658,746844,15],{},[31665,746846,746847],{"encoding":31667},"x_p",[34,746849,746851],{"className":746850,"ariaHidden":758},[31672],[34,746852,746854,746857],{"className":746853},[31676],[34,746855],{"className":746856,"style":56005},[31680],[34,746858,746860,746863],{"className":746859},[31685],[34,746861,32936],{"className":746862},[31685,31686],[34,746864,746866],{"className":746865},[31792],[34,746867,746869,746889],{"className":746868},[31796,31797],[34,746870,746872,746886],{"className":746871},[31801],[34,746873,746875],{"className":746874,"style":32362},[31805],[34,746876,746877,746880],{"style":31809},[34,746878],{"className":746879,"style":31814},[31813],[34,746881,746883],{"className":746882},[31818,31819,31820,31821],[34,746884,15],{"className":746885},[31685,31686,31821],[34,746887,31829],{"className":746888},[31828],[34,746890,746892],{"className":746891},[31801],[34,746893,746895],{"className":746894,"style":32163},[31805],[34,746896],{},[108,746898,746899],{},"第三步：加起来。",[108,746901,746902,746974],{},[2627,746903,746904,746905],{},"步骤一：求齐次通解 ",[34,746906,746908,746925],{"className":746907},[31642],[34,746909,746911],{"className":746910},[31646],[31648,746912,746913],{"xmlns":31650},[31652,746914,746915,746923],{},[31655,746916,746917],{},[31743,746918,746919,746921],{},[31658,746920,32936],{},[31658,746922,47292],{},[31665,746924,746774],{"encoding":31667},[34,746926,746928],{"className":746927,"ariaHidden":758},[31672],[34,746929,746931,746934],{"className":746930},[31676],[34,746932],{"className":746933,"style":31860},[31680],[34,746935,746937,746940],{"className":746936},[31685],[34,746938,32936],{"className":746939},[31685,31686],[34,746941,746943],{"className":746942},[31792],[34,746944,746946,746966],{"className":746945},[31796,31797],[34,746947,746949,746963],{"className":746948},[31801],[34,746950,746952],{"className":746951,"style":33833},[31805],[34,746953,746954,746957],{"style":31809},[34,746955],{"className":746956,"style":31814},[31813],[34,746958,746960],{"className":746959},[31818,31819,31820,31821],[34,746961,47292],{"className":746962},[31685,31686,31821],[34,746964,31829],{"className":746965},[31828],[34,746967,746969],{"className":746968},[31801],[34,746970,746972],{"className":746971,"style":31836},[31805],[34,746973],{},[2230,746975,746976,747166,747456,747806,748008,748413],{},[108,746977,746978,746979,347],{},"特征方程：",[34,746980,746982,747024],{"className":746981},[31642],[34,746983,746985],{"className":746984},[31646],[31648,746986,746987],{"xmlns":31650},[31652,746988,746989,747021],{},[31655,746990,746991,746997,746999,747001,747007,747009,747011,747013,747015,747017,747019],{},[37624,746992,746993,746995],{},[31658,746994,220792],{},[31749,746996,786],{},[31662,746998,32226],{},[31749,747000,15932],{},[37624,747002,747003,747005],{},[31658,747004,220792],{},[31749,747006,768],{},[31662,747008,223],{},[31749,747010,18136],{},[31658,747012,220792],{},[31662,747014,32226],{},[31749,747016,768],{},[31662,747018,202],{},[31749,747020,459],{},[31665,747022,747023],{"encoding":31667},"\\lambda^3 - 4\\lambda^2 + 5\\lambda - 2 = 0",[34,747025,747027,747071,747118,747139,747157],{"className":747026,"ariaHidden":758},[31672],[34,747028,747030,747033,747062,747065,747068],{"className":747029},[31676],[34,747031],{"className":747032,"style":40044},[31680],[34,747034,747036,747039],{"className":747035},[31685],[34,747037,220792],{"className":747038},[31685,31686],[34,747040,747042],{"className":747041},[31792],[34,747043,747045],{"className":747044},[31796],[34,747046,747048],{"className":747047},[31801],[34,747049,747051],{"className":747050,"style":37774},[31805],[34,747052,747053,747056],{"style":37777},[34,747054],{"className":747055,"style":31814},[31813],[34,747057,747059],{"className":747058},[31818,31819,31820,31821],[34,747060,786],{"className":747061},[31685,31821],[34,747063],{"className":747064,"style":32604},[31845],[34,747066,32226],{"className":747067},[32024],[34,747069],{"className":747070,"style":32604},[31845],[34,747072,747074,747077,747080,747109,747112,747115],{"className":747073},[31676],[34,747075],{"className":747076,"style":40044},[31680],[34,747078,15932],{"className":747079},[31685],[34,747081,747083,747086],{"className":747082},[31685],[34,747084,220792],{"className":747085},[31685,31686],[34,747087,747089],{"className":747088},[31792],[34,747090,747092],{"className":747091},[31796],[34,747093,747095],{"className":747094},[31801],[34,747096,747098],{"className":747097,"style":37774},[31805],[34,747099,747100,747103],{"style":37777},[34,747101],{"className":747102,"style":31814},[31813],[34,747104,747106],{"className":747105},[31818,31819,31820,31821],[34,747107,768],{"className":747108},[31685,31821],[34,747110],{"className":747111,"style":32604},[31845],[34,747113,223],{"className":747114},[32024],[34,747116],{"className":747117,"style":32604},[31845],[34,747119,747121,747124,747127,747130,747133,747136],{"className":747120},[31676],[34,747122],{"className":747123,"style":34299},[31680],[34,747125,18136],{"className":747126},[31685],[34,747128,220792],{"className":747129},[31685,31686],[34,747131],{"className":747132,"style":32604},[31845],[34,747134,32226],{"className":747135},[32024],[34,747137],{"className":747138,"style":32604},[31845],[34,747140,747142,747145,747148,747151,747154],{"className":747141},[31676],[34,747143],{"className":747144,"style":32617},[31680],[34,747146,768],{"className":747147},[31685],[34,747149],{"className":747150,"style":31846},[31845],[34,747152,202],{"className":747153},[31850],[34,747155],{"className":747156,"style":31846},[31845],[34,747158,747160,747163],{"className":747159},[31676],[34,747161],{"className":747162,"style":32617},[31680],[34,747164,459],{"className":747165},[31685],[108,747167,747168,747171,747172,747224,747225,2664,747276,747393,747394,747455],{},[2627,747169,747170],{},"试根法："," 这种高次方程通常试一下 ",[34,747173,747175,747197],{"className":747174},[31642],[34,747176,747178],{"className":747177},[31646],[31648,747179,747180],{"xmlns":31650},[31652,747181,747182,747194],{},[31655,747183,747184,747186,747188,747190,747192],{},[31662,747185,41461],{},[31749,747187,640],{},[31662,747189,31960],{"separator":758},[31662,747191,41461],{},[31749,747193,768],{},[31665,747195,747196],{"encoding":31667},"\\pm 1, \\pm 2",[34,747198,747200],{"className":747199,"ariaHidden":758},[31672],[34,747201,747203,747206,747209,747212,747215,747218,747221],{"className":747202},[31676],[34,747204],{"className":747205,"style":40292},[31680],[34,747207,41461],{"className":747208},[31685],[34,747210,640],{"className":747211},[31685],[34,747213,31960],{"className":747214},[32119],[34,747216],{"className":747217,"style":32123},[31845],[34,747219,41461],{"className":747220},[31685],[34,747222,768],{"className":747223},[31685],"。\n试 ",[34,747226,747228,747246],{"className":747227},[31642],[34,747229,747231],{"className":747230},[31646],[31648,747232,747233],{"xmlns":31650},[31652,747234,747235,747243],{},[31655,747236,747237,747239,747241],{},[31658,747238,220792],{},[31662,747240,202],{},[31749,747242,640],{},[31665,747244,747245],{"encoding":31667},"\\lambda = 1",[34,747247,747249,747267],{"className":747248,"ariaHidden":758},[31672],[34,747250,747252,747255,747258,747261,747264],{"className":747251},[31676],[34,747253],{"className":747254,"style":31681},[31680],[34,747256,220792],{"className":747257},[31685,31686],[34,747259],{"className":747260,"style":31846},[31845],[34,747262,202],{"className":747263},[31850],[34,747265],{"className":747266,"style":31846},[31845],[34,747268,747270,747273],{"className":747269},[31676],[34,747271],{"className":747272,"style":32617},[31680],[34,747274,640],{"className":747275},[31685],[34,747277,747279,747309],{"className":747278},[31642],[34,747280,747282],{"className":747281},[31646],[31648,747283,747284],{"xmlns":31650},[31652,747285,747286,747306],{},[31655,747287,747288,747290,747292,747294,747296,747298,747300,747302,747304],{},[31749,747289,640],{},[31662,747291,32226],{},[31749,747293,15932],{},[31662,747295,223],{},[31749,747297,18136],{},[31662,747299,32226],{},[31749,747301,768],{},[31662,747303,202],{},[31749,747305,459],{},[31665,747307,747308],{"encoding":31667},"1 - 4 + 5 - 2 = 0",[34,747310,747312,747330,747348,747366,747384],{"className":747311,"ariaHidden":758},[31672],[34,747313,747315,747318,747321,747324,747327],{"className":747314},[31676],[34,747316],{"className":747317,"style":35942},[31680],[34,747319,640],{"className":747320},[31685],[34,747322],{"className":747323,"style":32604},[31845],[34,747325,32226],{"className":747326},[32024],[34,747328],{"className":747329,"style":32604},[31845],[34,747331,747333,747336,747339,747342,747345],{"className":747332},[31676],[34,747334],{"className":747335,"style":35942},[31680],[34,747337,15932],{"className":747338},[31685],[34,747340],{"className":747341,"style":32604},[31845],[34,747343,223],{"className":747344},[32024],[34,747346],{"className":747347,"style":32604},[31845],[34,747349,747351,747354,747357,747360,747363],{"className":747350},[31676],[34,747352],{"className":747353,"style":35942},[31680],[34,747355,18136],{"className":747356},[31685],[34,747358],{"className":747359,"style":32604},[31845],[34,747361,32226],{"className":747362},[32024],[34,747364],{"className":747365,"style":32604},[31845],[34,747367,747369,747372,747375,747378,747381],{"className":747368},[31676],[34,747370],{"className":747371,"style":32617},[31680],[34,747373,768],{"className":747374},[31685],[34,747376],{"className":747377,"style":31846},[31845],[34,747379,202],{"className":747380},[31850],[34,747382],{"className":747383,"style":31846},[31845],[34,747385,747387,747390],{"className":747386},[31676],[34,747388],{"className":747389,"style":32617},[31680],[34,747391,459],{"className":747392},[31685],"。成立！说明 ",[34,747395,747397,747419],{"className":747396},[31642],[34,747398,747400],{"className":747399},[31646],[31648,747401,747402],{"xmlns":31650},[31652,747403,747404,747416],{},[31655,747405,747406,747408,747410,747412,747414],{},[31662,747407,456],{"stretchy":702},[31658,747409,220792],{},[31662,747411,32226],{},[31749,747413,640],{},[31662,747415,2557],{"stretchy":702},[31665,747417,747418],{"encoding":31667},"(\\lambda - 1)",[34,747420,747422,747443],{"className":747421,"ariaHidden":758},[31672],[34,747423,747425,747428,747431,747434,747437,747440],{"className":747424},[31676],[34,747426],{"className":747427,"style":31774},[31680],[34,747429,456],{"className":747430},[31782],[34,747432,220792],{"className":747433},[31685,31686],[34,747435],{"className":747436,"style":32604},[31845],[34,747438,32226],{"className":747439},[32024],[34,747441],{"className":747442,"style":32604},[31845],[34,747444,747446,747449,747452],{"className":747445},[31676],[34,747447],{"className":747448,"style":31774},[31680],[34,747450,640],{"className":747451},[31685],[34,747453,2557],{"className":747454},[31690]," 是一个因子。",[108,747457,747458,747461,747462,373177,747637,747697,747698,347],{},[2627,747459,747460],{},"多项式除法（或综合除法）：","\n用 ",[34,747463,747465,747507],{"className":747464},[31642],[34,747466,747468],{"className":747467},[31646],[31648,747469,747470],{"xmlns":31650},[31652,747471,747472,747504],{},[31655,747473,747474,747476,747482,747484,747486,747492,747494,747496,747498,747500,747502],{},[31662,747475,456],{"stretchy":702},[37624,747477,747478,747480],{},[31658,747479,220792],{},[31749,747481,786],{},[31662,747483,32226],{},[31749,747485,15932],{},[37624,747487,747488,747490],{},[31658,747489,220792],{},[31749,747491,768],{},[31662,747493,223],{},[31749,747495,18136],{},[31658,747497,220792],{},[31662,747499,32226],{},[31749,747501,768],{},[31662,747503,2557],{"stretchy":702},[31665,747505,747506],{"encoding":31667},"(\\lambda^3 - 4\\lambda^2 + 5\\lambda - 2)",[34,747508,747510,747557,747604,747625],{"className":747509,"ariaHidden":758},[31672],[34,747511,747513,747516,747519,747548,747551,747554],{"className":747512},[31676],[34,747514],{"className":747515,"style":37755},[31680],[34,747517,456],{"className":747518},[31782],[34,747520,747522,747525],{"className":747521},[31685],[34,747523,220792],{"className":747524},[31685,31686],[34,747526,747528],{"className":747527},[31792],[34,747529,747531],{"className":747530},[31796],[34,747532,747534],{"className":747533},[31801],[34,747535,747537],{"className":747536,"style":37774},[31805],[34,747538,747539,747542],{"style":37777},[34,747540],{"className":747541,"style":31814},[31813],[34,747543,747545],{"className":747544},[31818,31819,31820,31821],[34,747546,786],{"className":747547},[31685,31821],[34,747549],{"className":747550,"style":32604},[31845],[34,747552,32226],{"className":747553},[32024],[34,747555],{"className":747556,"style":32604},[31845],[34,747558,747560,747563,747566,747595,747598,747601],{"className":747559},[31676],[34,747561],{"className":747562,"style":40044},[31680],[34,747564,15932],{"className":747565},[31685],[34,747567,747569,747572],{"className":747568},[31685],[34,747570,220792],{"className":747571},[31685,31686],[34,747573,747575],{"className":747574},[31792],[34,747576,747578],{"className":747577},[31796],[34,747579,747581],{"className":747580},[31801],[34,747582,747584],{"className":747583,"style":37774},[31805],[34,747585,747586,747589],{"style":37777},[34,747587],{"className":747588,"style":31814},[31813],[34,747590,747592],{"className":747591},[31818,31819,31820,31821],[34,747593,768],{"className":747594},[31685,31821],[34,747596],{"className":747597,"style":32604},[31845],[34,747599,223],{"className":747600},[32024],[34,747602],{"className":747603,"style":32604},[31845],[34,747605,747607,747610,747613,747616,747619,747622],{"className":747606},[31676],[34,747608],{"className":747609,"style":34299},[31680],[34,747611,18136],{"className":747612},[31685],[34,747614,220792],{"className":747615},[31685,31686],[34,747617],{"className":747618,"style":32604},[31845],[34,747620,32226],{"className":747621},[32024],[34,747623],{"className":747624,"style":32604},[31845],[34,747626,747628,747631,747634],{"className":747627},[31676],[34,747629],{"className":747630,"style":31774},[31680],[34,747632,768],{"className":747633},[31685],[34,747635,2557],{"className":747636},[31690],[34,747638,747640,747661],{"className":747639},[31642],[34,747641,747643],{"className":747642},[31646],[31648,747644,747645],{"xmlns":31650},[31652,747646,747647,747659],{},[31655,747648,747649,747651,747653,747655,747657],{},[31662,747650,456],{"stretchy":702},[31658,747652,220792],{},[31662,747654,32226],{},[31749,747656,640],{},[31662,747658,2557],{"stretchy":702},[31665,747660,747418],{"encoding":31667},[34,747662,747664,747685],{"className":747663,"ariaHidden":758},[31672],[34,747665,747667,747670,747673,747676,747679,747682],{"className":747666},[31676],[34,747668],{"className":747669,"style":31774},[31680],[34,747671,456],{"className":747672},[31782],[34,747674,220792],{"className":747675},[31685,31686],[34,747677],{"className":747678,"style":32604},[31845],[34,747680,32226],{"className":747681},[32024],[34,747683],{"className":747684,"style":32604},[31845],[34,747686,747688,747691,747694],{"className":747687},[31676],[34,747689],{"className":747690,"style":31774},[31680],[34,747692,640],{"className":747693},[31685],[34,747695,2557],{"className":747696},[31690],"，得到 ",[34,747699,747701,747729],{"className":747700},[31642],[34,747702,747704],{"className":747703},[31646],[31648,747705,747706],{"xmlns":31650},[31652,747707,747708,747726],{},[31655,747709,747710,747716,747718,747720,747722,747724],{},[37624,747711,747712,747714],{},[31658,747713,220792],{},[31749,747715,768],{},[31662,747717,32226],{},[31749,747719,786],{},[31658,747721,220792],{},[31662,747723,223],{},[31749,747725,768],{},[31665,747727,747728],{"encoding":31667},"\\lambda^2 - 3\\lambda + 2",[34,747730,747732,747776,747797],{"className":747731,"ariaHidden":758},[31672],[34,747733,747735,747738,747767,747770,747773],{"className":747734},[31676],[34,747736],{"className":747737,"style":40044},[31680],[34,747739,747741,747744],{"className":747740},[31685],[34,747742,220792],{"className":747743},[31685,31686],[34,747745,747747],{"className":747746},[31792],[34,747748,747750],{"className":747749},[31796],[34,747751,747753],{"className":747752},[31801],[34,747754,747756],{"className":747755,"style":37774},[31805],[34,747757,747758,747761],{"style":37777},[34,747759],{"className":747760,"style":31814},[31813],[34,747762,747764],{"className":747763},[31818,31819,31820,31821],[34,747765,768],{"className":747766},[31685,31821],[34,747768],{"className":747769,"style":32604},[31845],[34,747771,32226],{"className":747772},[32024],[34,747774],{"className":747775,"style":32604},[31845],[34,747777,747779,747782,747785,747788,747791,747794],{"className":747778},[31676],[34,747780],{"className":747781,"style":34299},[31680],[34,747783,786],{"className":747784},[31685],[34,747786,220792],{"className":747787},[31685,31686],[34,747789],{"className":747790,"style":32604},[31845],[34,747792,223],{"className":747793},[32024],[34,747795],{"className":747796,"style":32604},[31845],[34,747798,747800,747803],{"className":747799},[31676],[34,747801],{"className":747802,"style":32617},[31680],[34,747804,768],{"className":747805},[31685],[108,747807,747808,747809,347],{},"继续分解 ",[34,747810,747812,747862],{"className":747811},[31642],[34,747813,747815],{"className":747814},[31646],[31648,747816,747817],{"xmlns":31650},[31652,747818,747819,747859],{},[31655,747820,747821,747827,747829,747831,747833,747835,747837,747839,747841,747843,747845,747847,747849,747851,747853,747855,747857],{},[37624,747822,747823,747825],{},[31658,747824,220792],{},[31749,747826,768],{},[31662,747828,32226],{},[31749,747830,786],{},[31658,747832,220792],{},[31662,747834,223],{},[31749,747836,768],{},[31662,747838,202],{},[31662,747840,456],{"stretchy":702},[31658,747842,220792],{},[31662,747844,32226],{},[31749,747846,640],{},[31662,747848,2557],{"stretchy":702},[31662,747850,456],{"stretchy":702},[31658,747852,220792],{},[31662,747854,32226],{},[31749,747856,768],{},[31662,747858,2557],{"stretchy":702},[31665,747860,747861],{"encoding":31667},"\\lambda^2 - 3\\lambda + 2 = (\\lambda - 1)(\\lambda - 2)",[34,747863,747865,747909,747930,747948,747969,747996],{"className":747864,"ariaHidden":758},[31672],[34,747866,747868,747871,747900,747903,747906],{"className":747867},[31676],[34,747869],{"className":747870,"style":40044},[31680],[34,747872,747874,747877],{"className":747873},[31685],[34,747875,220792],{"className":747876},[31685,31686],[34,747878,747880],{"className":747879},[31792],[34,747881,747883],{"className":747882},[31796],[34,747884,747886],{"className":747885},[31801],[34,747887,747889],{"className":747888,"style":37774},[31805],[34,747890,747891,747894],{"style":37777},[34,747892],{"className":747893,"style":31814},[31813],[34,747895,747897],{"className":747896},[31818,31819,31820,31821],[34,747898,768],{"className":747899},[31685,31821],[34,747901],{"className":747902,"style":32604},[31845],[34,747904,32226],{"className":747905},[32024],[34,747907],{"className":747908,"style":32604},[31845],[34,747910,747912,747915,747918,747921,747924,747927],{"className":747911},[31676],[34,747913],{"className":747914,"style":34299},[31680],[34,747916,786],{"className":747917},[31685],[34,747919,220792],{"className":747920},[31685,31686],[34,747922],{"className":747923,"style":32604},[31845],[34,747925,223],{"className":747926},[32024],[34,747928],{"className":747929,"style":32604},[31845],[34,747931,747933,747936,747939,747942,747945],{"className":747932},[31676],[34,747934],{"className":747935,"style":32617},[31680],[34,747937,768],{"className":747938},[31685],[34,747940],{"className":747941,"style":31846},[31845],[34,747943,202],{"className":747944},[31850],[34,747946],{"className":747947,"style":31846},[31845],[34,747949,747951,747954,747957,747960,747963,747966],{"className":747950},[31676],[34,747952],{"className":747953,"style":31774},[31680],[34,747955,456],{"className":747956},[31782],[34,747958,220792],{"className":747959},[31685,31686],[34,747961],{"className":747962,"style":32604},[31845],[34,747964,32226],{"className":747965},[32024],[34,747967],{"className":747968,"style":32604},[31845],[34,747970,747972,747975,747978,747981,747984,747987,747990,747993],{"className":747971},[31676],[34,747973],{"className":747974,"style":31774},[31680],[34,747976,640],{"className":747977},[31685],[34,747979,2557],{"className":747980},[31690],[34,747982,456],{"className":747983},[31782],[34,747985,220792],{"className":747986},[31685,31686],[34,747988],{"className":747989,"style":32604},[31845],[34,747991,32226],{"className":747992},[32024],[34,747994],{"className":747995,"style":32604},[31845],[34,747997,747999,748002,748005],{"className":747998},[31676],[34,748000],{"className":748001,"style":31774},[31680],[34,748003,768],{"className":748004},[31685],[34,748006,2557],{"className":748007},[31690],[108,748009,748010,748011,748161,748162,1840,748317,432343,748320,748412],{},"综合起来，特征方程为：\n",[34,748012,748014,748054],{"className":748013},[31642],[34,748015,748017],{"className":748016},[31646],[31648,748018,748019],{"xmlns":31650},[31652,748020,748021,748051],{},[31655,748022,748023,748025,748027,748029,748031,748037,748039,748041,748043,748045,748047,748049],{},[31662,748024,456],{"stretchy":702},[31658,748026,220792],{},[31662,748028,32226],{},[31749,748030,640],{},[37624,748032,748033,748035],{},[31662,748034,2557],{"stretchy":702},[31749,748036,768],{},[31662,748038,456],{"stretchy":702},[31658,748040,220792],{},[31662,748042,32226],{},[31749,748044,768],{},[31662,748046,2557],{"stretchy":702},[31662,748048,202],{},[31749,748050,459],{},[31665,748052,748053],{"encoding":31667},"(\\lambda - 1)^2 (\\lambda - 2) = 0",[34,748055,748057,748078,748131,748152],{"className":748056,"ariaHidden":758},[31672],[34,748058,748060,748063,748066,748069,748072,748075],{"className":748059},[31676],[34,748061],{"className":748062,"style":31774},[31680],[34,748064,456],{"className":748065},[31782],[34,748067,220792],{"className":748068},[31685,31686],[34,748070],{"className":748071,"style":32604},[31845],[34,748073,32226],{"className":748074},[32024],[34,748076],{"className":748077,"style":32604},[31845],[34,748079,748081,748084,748087,748116,748119,748122,748125,748128],{"className":748080},[31676],[34,748082],{"className":748083,"style":37755},[31680],[34,748085,640],{"className":748086},[31685],[34,748088,748090,748093],{"className":748089},[31690],[34,748091,2557],{"className":748092},[31690],[34,748094,748096],{"className":748095},[31792],[34,748097,748099],{"className":748098},[31796],[34,748100,748102],{"className":748101},[31801],[34,748103,748105],{"className":748104,"style":37774},[31805],[34,748106,748107,748110],{"style":37777},[34,748108],{"className":748109,"style":31814},[31813],[34,748111,748113],{"className":748112},[31818,31819,31820,31821],[34,748114,768],{"className":748115},[31685,31821],[34,748117,456],{"className":748118},[31782],[34,748120,220792],{"className":748121},[31685,31686],[34,748123],{"className":748124,"style":32604},[31845],[34,748126,32226],{"className":748127},[32024],[34,748129],{"className":748130,"style":32604},[31845],[34,748132,748134,748137,748140,748143,748146,748149],{"className":748133},[31676],[34,748135],{"className":748136,"style":31774},[31680],[34,748138,768],{"className":748139},[31685],[34,748141,2557],{"className":748142},[31690],[34,748144],{"className":748145,"style":31846},[31845],[34,748147,202],{"className":748148},[31850],[34,748150],{"className":748151,"style":31846},[31845],[34,748153,748155,748158],{"className":748154},[31676],[34,748156],{"className":748157,"style":32617},[31680],[34,748159,459],{"className":748160},[31685],"\n特征根为：",[34,748163,748165,748195],{"className":748164},[31642],[34,748166,748168],{"className":748167},[31646],[31648,748169,748170],{"xmlns":31650},[31652,748171,748172,748192],{},[31655,748173,748174,748180,748182,748188,748190],{},[31743,748175,748176,748178],{},[31658,748177,220792],{},[31749,748179,640],{},[31662,748181,202],{},[31743,748183,748184,748186],{},[31658,748185,220792],{},[31749,748187,768],{},[31662,748189,202],{},[31749,748191,640],{},[31665,748193,748194],{"encoding":31667},"\\lambda_1 = \\lambda_2 = 1",[34,748196,748198,748253,748308],{"className":748197,"ariaHidden":758},[31672],[34,748199,748201,748204,748244,748247,748250],{"className":748200},[31676],[34,748202],{"className":748203,"style":33814},[31680],[34,748205,748207,748210],{"className":748206},[31685],[34,748208,220792],{"className":748209},[31685,31686],[34,748211,748213],{"className":748212},[31792],[34,748214,748216,748236],{"className":748215},[31796,31797],[34,748217,748219,748233],{"className":748218},[31801],[34,748220,748222],{"className":748221,"style":31806},[31805],[34,748223,748224,748227],{"style":31809},[34,748225],{"className":748226,"style":31814},[31813],[34,748228,748230],{"className":748229},[31818,31819,31820,31821],[34,748231,640],{"className":748232},[31685,31821],[34,748234,31829],{"className":748235},[31828],[34,748237,748239],{"className":748238},[31801],[34,748240,748242],{"className":748241,"style":31836},[31805],[34,748243],{},[34,748245],{"className":748246,"style":31846},[31845],[34,748248,202],{"className":748249},[31850],[34,748251],{"className":748252,"style":31846},[31845],[34,748254,748256,748259,748299,748302,748305],{"className":748255},[31676],[34,748257],{"className":748258,"style":33814},[31680],[34,748260,748262,748265],{"className":748261},[31685],[34,748263,220792],{"className":748264},[31685,31686],[34,748266,748268],{"className":748267},[31792],[34,748269,748271,748291],{"className":748270},[31796,31797],[34,748272,748274,748288],{"className":748273},[31801],[34,748275,748277],{"className":748276,"style":31806},[31805],[34,748278,748279,748282],{"style":31809},[34,748280],{"className":748281,"style":31814},[31813],[34,748283,748285],{"className":748284},[31818,31819,31820,31821],[34,748286,768],{"className":748287},[31685,31821],[34,748289,31829],{"className":748290},[31828],[34,748292,748294],{"className":748293},[31801],[34,748295,748297],{"className":748296,"style":31836},[31805],[34,748298],{},[34,748300],{"className":748301,"style":31846},[31845],[34,748303,202],{"className":748304},[31850],[34,748306],{"className":748307,"style":31846},[31845],[34,748309,748311,748314],{"className":748310},[31676],[34,748312],{"className":748313,"style":32617},[31680],[34,748315,640],{"className":748316},[31685],[2627,748318,748319],{},"二重根",[34,748321,748323,748345],{"className":748322},[31642],[34,748324,748326],{"className":748325},[31646],[31648,748327,748328],{"xmlns":31650},[31652,748329,748330,748342],{},[31655,748331,748332,748338,748340],{},[31743,748333,748334,748336],{},[31658,748335,220792],{},[31749,748337,786],{},[31662,748339,202],{},[31749,748341,768],{},[31665,748343,748344],{"encoding":31667},"\\lambda_3 = 2",[34,748346,748348,748403],{"className":748347,"ariaHidden":758},[31672],[34,748349,748351,748354,748394,748397,748400],{"className":748350},[31676],[34,748352],{"className":748353,"style":33814},[31680],[34,748355,748357,748360],{"className":748356},[31685],[34,748358,220792],{"className":748359},[31685,31686],[34,748361,748363],{"className":748362},[31792],[34,748364,748366,748386],{"className":748365},[31796,31797],[34,748367,748369,748383],{"className":748368},[31801],[34,748370,748372],{"className":748371,"style":31806},[31805],[34,748373,748374,748377],{"style":31809},[34,748375],{"className":748376,"style":31814},[31813],[34,748378,748380],{"className":748379},[31818,31819,31820,31821],[34,748381,786],{"className":748382},[31685,31821],[34,748384,31829],{"className":748385},[31828],[34,748387,748389],{"className":748388},[31801],[34,748390,748392],{"className":748391,"style":31836},[31805],[34,748393],{},[34,748395],{"className":748396,"style":31846},[31845],[34,748398,202],{"className":748399},[31850],[34,748401],{"className":748402,"style":31846},[31845],[34,748404,748406,748409],{"className":748405},[31676],[34,748407],{"className":748408,"style":32617},[31680],[34,748410,768],{"className":748411},[31685]," (单根)。",[108,748414,748415,748418],{},[2627,748416,748417],{},"齐次解：",[34,748419,748421,748485],{"className":748420},[31642],[34,748422,748424],{"className":748423},[31646],[31648,748425,748426],{"xmlns":31650},[31652,748427,748428,748482],{},[31655,748429,748430,748436,748438,748440,748446,748448,748454,748456,748458,748464,748466,748472],{},[31743,748431,748432,748434],{},[31658,748433,32936],{},[31658,748435,47292],{},[31662,748437,202],{},[31662,748439,456],{"stretchy":702},[31743,748441,748442,748444],{},[31658,748443,31965],{},[31749,748445,640],{},[31662,748447,223],{},[31743,748449,748450,748452],{},[31658,748451,31965],{},[31749,748453,768],{},[31658,748455,33249],{},[31662,748457,2557],{"stretchy":702},[37624,748459,748460,748462],{},[31658,748461,33244],{"mathvariant":38079},[31658,748463,33249],{},[31662,748465,223],{},[31743,748467,748468,748470],{},[31658,748469,31965],{},[31749,748471,786],{},[37624,748473,748474,748476],{},[31658,748475,33244],{"mathvariant":38079},[31655,748477,748478,748480],{},[31749,748479,768],{},[31658,748481,33249],{},[31665,748483,748484],{"encoding":31667},"x_h = (c_1 + c_2 t)\\mathrm{e}^t + c_3 \\mathrm{e}^{2t}",[34,748486,748488,748543,748601,748692],{"className":748487,"ariaHidden":758},[31672],[34,748489,748491,748494,748534,748537,748540],{"className":748490},[31676],[34,748492],{"className":748493,"style":31860},[31680],[34,748495,748497,748500],{"className":748496},[31685],[34,748498,32936],{"className":748499},[31685,31686],[34,748501,748503],{"className":748502},[31792],[34,748504,748506,748526],{"className":748505},[31796,31797],[34,748507,748509,748523],{"className":748508},[31801],[34,748510,748512],{"className":748511,"style":33833},[31805],[34,748513,748514,748517],{"style":31809},[34,748515],{"className":748516,"style":31814},[31813],[34,748518,748520],{"className":748519},[31818,31819,31820,31821],[34,748521,47292],{"className":748522},[31685,31686,31821],[34,748524,31829],{"className":748525},[31828],[34,748527,748529],{"className":748528},[31801],[34,748530,748532],{"className":748531,"style":31836},[31805],[34,748533],{},[34,748535],{"className":748536,"style":31846},[31845],[34,748538,202],{"className":748539},[31850],[34,748541],{"className":748542,"style":31846},[31845],[34,748544,748546,748549,748552,748592,748595,748598],{"className":748545},[31676],[34,748547],{"className":748548,"style":31774},[31680],[34,748550,456],{"className":748551},[31782],[34,748553,748555,748558],{"className":748554},[31685],[34,748556,31965],{"className":748557},[31685,31686],[34,748559,748561],{"className":748560},[31792],[34,748562,748564,748584],{"className":748563},[31796,31797],[34,748565,748567,748581],{"className":748566},[31801],[34,748568,748570],{"className":748569,"style":31806},[31805],[34,748571,748572,748575],{"style":31809},[34,748573],{"className":748574,"style":31814},[31813],[34,748576,748578],{"className":748577},[31818,31819,31820,31821],[34,748579,640],{"className":748580},[31685,31821],[34,748582,31829],{"className":748583},[31828],[34,748585,748587],{"className":748586},[31801],[34,748588,748590],{"className":748589,"style":31836},[31805],[34,748591],{},[34,748593],{"className":748594,"style":32604},[31845],[34,748596,223],{"className":748597},[32024],[34,748599],{"className":748600,"style":32604},[31845],[34,748602,748604,748608,748648,748651,748654,748683,748686,748689],{"className":748603},[31676],[34,748605],{"className":748606,"style":748607},[31680],"height:1.0436em;vertical-align:-0.25em;",[34,748609,748611,748614],{"className":748610},[31685],[34,748612,31965],{"className":748613},[31685,31686],[34,748615,748617],{"className":748616},[31792],[34,748618,748620,748640],{"className":748619},[31796,31797],[34,748621,748623,748637],{"className":748622},[31801],[34,748624,748626],{"className":748625,"style":31806},[31805],[34,748627,748628,748631],{"style":31809},[34,748629],{"className":748630,"style":31814},[31813],[34,748632,748634],{"className":748633},[31818,31819,31820,31821],[34,748635,768],{"className":748636},[31685,31821],[34,748638,31829],{"className":748639},[31828],[34,748641,748643],{"className":748642},[31801],[34,748644,748646],{"className":748645,"style":31836},[31805],[34,748647],{},[34,748649,33249],{"className":748650},[31685,31686],[34,748652,2557],{"className":748653},[31690],[34,748655,748657,748660],{"className":748656},[31685],[34,748658,33244],{"className":748659},[31685,207923],[34,748661,748663],{"className":748662},[31792],[34,748664,748666],{"className":748665},[31796],[34,748667,748669],{"className":748668},[31801],[34,748670,748672],{"className":748671,"style":611758},[31805],[34,748673,748674,748677],{"style":37777},[34,748675],{"className":748676,"style":31814},[31813],[34,748678,748680],{"className":748679},[31818,31819,31820,31821],[34,748681,33249],{"className":748682},[31685,31686,31821],[34,748684],{"className":748685,"style":32604},[31845],[34,748687,223],{"className":748688},[32024],[34,748690],{"className":748691,"style":32604},[31845],[34,748693,748695,748698,748738],{"className":748694},[31676],[34,748696],{"className":748697,"style":744441},[31680],[34,748699,748701,748704],{"className":748700},[31685],[34,748702,31965],{"className":748703},[31685,31686],[34,748705,748707],{"className":748706},[31792],[34,748708,748710,748730],{"className":748709},[31796,31797],[34,748711,748713,748727],{"className":748712},[31801],[34,748714,748716],{"className":748715,"style":31806},[31805],[34,748717,748718,748721],{"style":31809},[34,748719],{"className":748720,"style":31814},[31813],[34,748722,748724],{"className":748723},[31818,31819,31820,31821],[34,748725,786],{"className":748726},[31685,31821],[34,748728,31829],{"className":748729},[31828],[34,748731,748733],{"className":748732},[31801],[34,748734,748736],{"className":748735,"style":31836},[31805],[34,748737],{},[34,748739,748741,748744],{"className":748740},[31685],[34,748742,33244],{"className":748743},[31685,207923],[34,748745,748747],{"className":748746},[31792],[34,748748,748750],{"className":748749},[31796],[34,748751,748753],{"className":748752},[31801],[34,748754,748756],{"className":748755,"style":37774},[31805],[34,748757,748758,748761],{"style":37777},[34,748759],{"className":748760,"style":31814},[31813],[34,748762,748764],{"className":748763},[31818,31819,31820,31821],[34,748765,748767,748770],{"className":748766},[31685,31821],[34,748768,768],{"className":748769},[31685,31821],[34,748771,33249],{"className":748772},[31685,31686,31821],[108,748774,748775,748847],{},[2627,748776,748777,748778],{},"步骤二：求特解 ",[34,748779,748781,748798],{"className":748780},[31642],[34,748782,748784],{"className":748783},[31646],[31648,748785,748786],{"xmlns":31650},[31652,748787,748788,748796],{},[31655,748789,748790],{},[31743,748791,748792,748794],{},[31658,748793,32936],{},[31658,748795,15],{},[31665,748797,746847],{"encoding":31667},[34,748799,748801],{"className":748800,"ariaHidden":758},[31672],[34,748802,748804,748807],{"className":748803},[31676],[34,748805],{"className":748806,"style":56005},[31680],[34,748808,748810,748813],{"className":748809},[31685],[34,748811,32936],{"className":748812},[31685,31686],[34,748814,748816],{"className":748815},[31792],[34,748817,748819,748839],{"className":748818},[31796,31797],[34,748820,748822,748836],{"className":748821},[31801],[34,748823,748825],{"className":748824,"style":32362},[31805],[34,748826,748827,748830],{"style":31809},[34,748828],{"className":748829,"style":31814},[31813],[34,748831,748833],{"className":748832},[31818,31819,31820,31821],[34,748834,15],{"className":748835},[31685,31686,31821],[34,748837,31829],{"className":748838},[31828],[34,748840,748842],{"className":748841},[31801],[34,748843,748845],{"className":748844,"style":32163},[31805],[34,748846],{},[2230,748848,748849,749139,749257,749381,749725,750246],{},[108,748850,748851,47649,748854,39984,748947],{},[2627,748852,748853],{},"观察右边：",[34,748855,748857,748887],{"className":748856},[31642],[34,748858,748860],{"className":748859},[31646],[31648,748861,748862],{"xmlns":31650},[31652,748863,748864,748884],{},[31655,748865,748866,748868,748870,748872,748874,748876,748878,748880,748882],{},[31658,748867,31739],{},[31662,748869,456],{"stretchy":702},[31658,748871,33249],{},[31662,748873,2557],{"stretchy":702},[31662,748875,202],{},[31749,748877,768],{},[31658,748879,33249],{},[31662,748881,223],{},[31749,748883,786],{},[31665,748885,748886],{"encoding":31667},"f(t) = 2t + 3",[34,748888,748890,748917,748938],{"className":748889,"ariaHidden":758},[31672],[34,748891,748893,748896,748899,748902,748905,748908,748911,748914],{"className":748892},[31676],[34,748894],{"className":748895,"style":31774},[31680],[34,748897,31739],{"className":748898,"style":31778},[31685,31686],[34,748900,456],{"className":748901},[31782],[34,748903,33249],{"className":748904},[31685,31686],[34,748906,2557],{"className":748907},[31690],[34,748909],{"className":748910,"style":31846},[31845],[34,748912,202],{"className":748913},[31850],[34,748915],{"className":748916,"style":31846},[31845],[34,748918,748920,748923,748926,748929,748932,748935],{"className":748919},[31676],[34,748921],{"className":748922,"style":35942},[31680],[34,748924,768],{"className":748925},[31685],[34,748927,33249],{"className":748928},[31685,31686],[34,748930],{"className":748931,"style":32604},[31845],[34,748933,223],{"className":748934},[32024],[34,748936],{"className":748937,"style":32604},[31845],[34,748939,748941,748944],{"className":748940},[31676],[34,748942],{"className":748943,"style":32617},[31680],[34,748945,786],{"className":748946},[31685],[105,748948,748949,749037],{},[108,748950,748951,748952,347],{},"这是一个一次多项式 ",[34,748953,748955,748979],{"className":748954},[31642],[34,748956,748958],{"className":748957},[31646],[31648,748959,748960],{"xmlns":31650},[31652,748961,748962,748976],{},[31655,748963,748964,748970,748972,748974],{},[31743,748965,748966,748968],{},[31658,748967,105852],{},[31749,748969,640],{},[31662,748971,456],{"stretchy":702},[31658,748973,33249],{},[31662,748975,2557],{"stretchy":702},[31665,748977,748978],{"encoding":31667},"P_1(t)",[34,748980,748982],{"className":748981,"ariaHidden":758},[31672],[34,748983,748985,748988,749028,749031,749034],{"className":748984},[31676],[34,748986],{"className":748987,"style":31774},[31680],[34,748989,748991,748994],{"className":748990},[31685],[34,748992,105852],{"className":748993,"style":37422},[31685,31686],[34,748995,748997],{"className":748996},[31792],[34,748998,749000,749020],{"className":748999},[31796,31797],[34,749001,749003,749017],{"className":749002},[31801],[34,749004,749006],{"className":749005,"style":31806},[31805],[34,749007,749008,749011],{"style":214424},[34,749009],{"className":749010,"style":31814},[31813],[34,749012,749014],{"className":749013},[31818,31819,31820,31821],[34,749015,640],{"className":749016},[31685,31821],[34,749018,31829],{"className":749019},[31828],[34,749021,749023],{"className":749022},[31801],[34,749024,749026],{"className":749025,"style":31836},[31805],[34,749027],{},[34,749029,456],{"className":749030},[31782],[34,749032,33249],{"className":749033},[31685,31686],[34,749035,2557],{"className":749036},[31690],[108,749038,749039,749040,749109,749110,749138],{},"对应指数形式是 ",[34,749041,749043,749065],{"className":749042},[31642],[34,749044,749046],{"className":749045},[31646],[31648,749047,749048],{"xmlns":31650},[31652,749049,749050,749062],{},[31655,749051,749052],{},[37624,749053,749054,749056],{},[31658,749055,33244],{"mathvariant":38079},[31655,749057,749058,749060],{},[31749,749059,459],{},[31658,749061,33249],{},[31665,749063,749064],{"encoding":31667},"\\mathrm{e}^{0t}",[34,749066,749068],{"className":749067,"ariaHidden":758},[31672],[34,749069,749071,749074],{"className":749070},[31676],[34,749072],{"className":749073,"style":37774},[31680],[34,749075,749077,749080],{"className":749076},[31685],[34,749078,33244],{"className":749079},[31685,207923],[34,749081,749083],{"className":749082},[31792],[34,749084,749086],{"className":749085},[31796],[34,749087,749089],{"className":749088},[31801],[34,749090,749092],{"className":749091,"style":37774},[31805],[34,749093,749094,749097],{"style":37777},[34,749095],{"className":749096,"style":31814},[31813],[34,749098,749100],{"className":749099},[31818,31819,31820,31821],[34,749101,749103,749106],{"className":749102},[31685,31821],[34,749104,459],{"className":749105},[31685,31821],[34,749107,33249],{"className":749108},[31685,31686,31821],"（因为没有显式的 ",[34,749111,749113,749126],{"className":749112},[31642],[34,749114,749116],{"className":749115},[31646],[31648,749117,749118],{"xmlns":31650},[31652,749119,749120,749124],{},[31655,749121,749122],{},[31658,749123,33244],{},[31665,749125,33244],{"encoding":31667},[34,749127,749129],{"className":749128,"ariaHidden":758},[31672],[34,749130,749132,749135],{"className":749131},[31676],[34,749133],{"className":749134,"style":32527},[31680],[34,749136,33244],{"className":749137},[31685,31686],"，指数就是0）。",[108,749140,749141,749172,749173,749176,749177,749228,749229,31901],{},[2627,749142,749143,749144,414527],{},"检查共振（确定 ",[34,749145,749147,749160],{"className":749146},[31642],[34,749148,749150],{"className":749149},[31646],[31648,749151,749152],{"xmlns":31650},[31652,749153,749154,749158],{},[31655,749155,749156],{},[31658,749157,33759],{},[31665,749159,33759],{"encoding":31667},[34,749161,749163],{"className":749162,"ariaHidden":758},[31672],[34,749164,749166,749169],{"className":749165},[31676],[34,749167],{"className":749168,"style":31681},[31680],[34,749170,33759],{"className":749171,"style":33774},[31685,31686],"\n右边的指数是 0。我们检查特征根（1和2），里面",[2627,749174,749175],{},"没有0","。\n所以，不是特征根，",[34,749178,749180,749198],{"className":749179},[31642],[34,749181,749183],{"className":749182},[31646],[31648,749184,749185],{"xmlns":31650},[31652,749186,749187,749195],{},[31655,749188,749189,749191,749193],{},[31658,749190,33759],{},[31662,749192,202],{},[31749,749194,459],{},[31665,749196,749197],{"encoding":31667},"k=0",[34,749199,749201,749219],{"className":749200,"ariaHidden":758},[31672],[34,749202,749204,749207,749210,749213,749216],{"className":749203},[31676],[34,749205],{"className":749206,"style":31681},[31680],[34,749208,33759],{"className":749209,"style":33774},[31685,31686],[34,749211],{"className":749212,"style":31846},[31845],[34,749214,202],{"className":749215},[31850],[34,749217],{"className":749218,"style":31846},[31845],[34,749220,749222,749225],{"className":749221},[31676],[34,749223],{"className":749224,"style":32617},[31680],[34,749226,459],{"className":749227},[31685],"（不需要补乘 ",[34,749230,749232,749245],{"className":749231},[31642],[34,749233,749235],{"className":749234},[31646],[31648,749236,749237],{"xmlns":31650},[31652,749238,749239,749243],{},[31655,749240,749241],{},[31658,749242,33249],{},[31665,749244,33249],{"encoding":31667},[34,749246,749248],{"className":749247,"ariaHidden":758},[31672],[34,749249,749251,749254],{"className":749250},[31676],[34,749252],{"className":749253,"style":33280},[31680],[34,749255,33249],{"className":749256},[31685,31686],[108,749258,749259,749262,749263],{},[2627,749260,749261],{},"设特解：","\n因为右边是一次多项式，我们设特解也是一次多项式：\n",[34,749264,749266,749295],{"className":749265},[31642],[34,749267,749269],{"className":749268},[31646],[31648,749270,749271],{"xmlns":31650},[31652,749272,749273,749292],{},[31655,749274,749275,749282,749284,749286,749288,749290],{},[134014,749276,749277,749279],{"accent":758},[31658,749278,32936],{},[31662,749280,749281],{"stretchy":758},"~",[31662,749283,202],{},[31658,749285,54151],{},[31658,749287,33249],{},[31662,749289,223],{},[31658,749291,54224],{},[31665,749293,749294],{"encoding":31667},"\\widetilde{x} = At + B",[34,749296,749298,749351,749372],{"className":749297,"ariaHidden":758},[31672],[34,749299,749301,749305,749342,749345,749348],{"className":749300},[31676],[34,749302],{"className":749303,"style":749304},[31680],"height:0.6906em;",[34,749306,749308],{"className":749307},[31685,346171],[34,749309,749311],{"className":749310},[31796],[34,749312,749314],{"className":749313},[31801],[34,749315,749317,749325],{"className":749316,"style":749304},[31805],[34,749318,749319,749322],{"style":41565},[34,749320],{"className":749321,"style":41508},[31813],[34,749323,32936],{"className":749324},[31685,31686],[34,749326,749329,749332],{"className":749327,"style":749328},[41564],"width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4306em;",[34,749330],{"className":749331,"style":41508},[31813],[34,749333,749335],{"style":749334},"height:0.26em;",[41588,749336,749339],{"xmlns":41590,"width":168121,"height":749337,"viewBox":749338,"preserveAspectRatio":29503},"0.26em","0 0 600 260",[20398,749340],{"d":749341},"M200 55.538c-77 0-168 73.953-177 73.953-3 0-7\n-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0\n 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0\n 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128\n-68.267.847-113-73.952-191-73.952z",[34,749343],{"className":749344,"style":31846},[31845],[34,749346,202],{"className":749347},[31850],[34,749349],{"className":749350,"style":31846},[31845],[34,749352,749354,749357,749360,749363,749366,749369],{"className":749353},[31676],[34,749355],{"className":749356,"style":34944},[31680],[34,749358,54151],{"className":749359},[31685,31686],[34,749361,33249],{"className":749362},[31685,31686],[34,749364],{"className":749365,"style":32604},[31845],[34,749367,223],{"className":749368},[32024],[34,749370],{"className":749371,"style":32604},[31845],[34,749373,749375,749378],{"className":749374},[31676],[34,749376],{"className":749377,"style":33218},[31680],[34,749379,54224],{"className":749380,"style":54252},[31685,31686],[108,749382,749383,749386],{},[2627,749384,749385],{},"代入求导：",[34,749387,749389,749461],{"className":749388},[31642],[34,749390,749392],{"className":749391},[31646],[31648,749393,749394],{"xmlns":31650},[31652,749395,749396,749458],{},[31655,749397,749398,749408,749410,749412,749414,749416,749430,749432,749434,749436,749438,749454,749456],{},[37624,749399,749400,749406],{},[134014,749401,749402,749404],{"accent":758},[31658,749403,32936],{},[31662,749405,749281],{"stretchy":758},[31662,749407,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,749409,202],{},[31658,749411,54151],{},[31662,749413,31960],{"separator":758},[31845,749415],{"width":71082},[37624,749417,749418,749424],{},[134014,749419,749420,749422],{"accent":758},[31658,749421,32936],{},[31662,749423,749281],{"stretchy":758},[31655,749425,749426,749428],{},[31662,749427,116495],{"mathvariant":38079},[31662,749429,116495],{"mathvariant":38079},[31662,749431,202],{},[31749,749433,459],{},[31662,749435,31960],{"separator":758},[31845,749437],{"width":71082},[37624,749439,749440,749446],{},[134014,749441,749442,749444],{"accent":758},[31658,749443,32936],{},[31662,749445,749281],{"stretchy":758},[31655,749447,749448,749450,749452],{},[31662,749449,116495],{"mathvariant":38079},[31662,749451,116495],{"mathvariant":38079},[31662,749453,116495],{"mathvariant":38079},[31662,749455,202],{},[31749,749457,459],{},[31665,749459,749460],{"encoding":31667},"\\widetilde{x}' = A, \\quad \\widetilde{x}'' = 0, \\quad \\widetilde{x}''' = 0",[34,749462,749464,749540,749628,749716],{"className":749463,"ariaHidden":758},[31672],[34,749465,749467,749470,749531,749534,749537],{"className":749466},[31676],[34,749468],{"className":749469,"style":132027},[31680],[34,749471,749473,749505],{"className":749472},[31685],[34,749474,749476],{"className":749475},[31685,346171],[34,749477,749479],{"className":749478},[31796],[34,749480,749482],{"className":749481},[31801],[34,749483,749485,749493],{"className":749484,"style":749304},[31805],[34,749486,749487,749490],{"style":41565},[34,749488],{"className":749489,"style":41508},[31813],[34,749491,32936],{"className":749492},[31685,31686],[34,749494,749496,749499],{"className":749495,"style":749328},[41564],[34,749497],{"className":749498,"style":41508},[31813],[34,749500,749501],{"style":749334},[41588,749502,749503],{"xmlns":41590,"width":168121,"height":749337,"viewBox":749338,"preserveAspectRatio":29503},[20398,749504],{"d":749341},[34,749506,749508],{"className":749507},[31792],[34,749509,749511],{"className":749510},[31796],[34,749512,749514],{"className":749513},[31801],[34,749515,749517],{"className":749516,"style":132027},[31805],[34,749518,749519,749522],{"style":37777},[34,749520],{"className":749521,"style":31814},[31813],[34,749523,749525],{"className":749524},[31818,31819,31820,31821],[34,749526,749528],{"className":749527},[31685,31821],[34,749529,116495],{"className":749530},[31685,31821],[34,749532],{"className":749533,"style":31846},[31845],[34,749535,202],{"className":749536},[31850],[34,749538],{"className":749539,"style":31846},[31845],[34,749541,749543,749546,749549,749552,749555,749558,749619,749622,749625],{"className":749542},[31676],[34,749544],{"className":749545,"style":395485},[31680],[34,749547,54151],{"className":749548},[31685,31686],[34,749550,31960],{"className":749551},[32119],[34,749553],{"className":749554,"style":128438},[31845],[34,749556],{"className":749557,"style":32123},[31845],[34,749559,749561,749593],{"className":749560},[31685],[34,749562,749564],{"className":749563},[31685,346171],[34,749565,749567],{"className":749566},[31796],[34,749568,749570],{"className":749569},[31801],[34,749571,749573,749581],{"className":749572,"style":749304},[31805],[34,749574,749575,749578],{"style":41565},[34,749576],{"className":749577,"style":41508},[31813],[34,749579,32936],{"className":749580},[31685,31686],[34,749582,749584,749587],{"className":749583,"style":749328},[41564],[34,749585],{"className":749586,"style":41508},[31813],[34,749588,749589],{"style":749334},[41588,749590,749591],{"xmlns":41590,"width":168121,"height":749337,"viewBox":749338,"preserveAspectRatio":29503},[20398,749592],{"d":749341},[34,749594,749596],{"className":749595},[31792],[34,749597,749599],{"className":749598},[31796],[34,749600,749602],{"className":749601},[31801],[34,749603,749605],{"className":749604,"style":132027},[31805],[34,749606,749607,749610],{"style":37777},[34,749608],{"className":749609,"style":31814},[31813],[34,749611,749613],{"className":749612},[31818,31819,31820,31821],[34,749614,749616],{"className":749615},[31685,31821],[34,749617,502782],{"className":749618},[31685,31821],[34,749620],{"className":749621,"style":31846},[31845],[34,749623,202],{"className":749624},[31850],[34,749626],{"className":749627,"style":31846},[31845],[34,749629,749631,749634,749637,749640,749643,749646,749707,749710,749713],{"className":749630},[31676],[34,749632],{"className":749633,"style":395485},[31680],[34,749635,459],{"className":749636},[31685],[34,749638,31960],{"className":749639},[32119],[34,749641],{"className":749642,"style":128438},[31845],[34,749644],{"className":749645,"style":32123},[31845],[34,749647,749649,749681],{"className":749648},[31685],[34,749650,749652],{"className":749651},[31685,346171],[34,749653,749655],{"className":749654},[31796],[34,749656,749658],{"className":749657},[31801],[34,749659,749661,749669],{"className":749660,"style":749304},[31805],[34,749662,749663,749666],{"style":41565},[34,749664],{"className":749665,"style":41508},[31813],[34,749667,32936],{"className":749668},[31685,31686],[34,749670,749672,749675],{"className":749671,"style":749328},[41564],[34,749673],{"className":749674,"style":41508},[31813],[34,749676,749677],{"style":749334},[41588,749678,749679],{"xmlns":41590,"width":168121,"height":749337,"viewBox":749338,"preserveAspectRatio":29503},[20398,749680],{"d":749341},[34,749682,749684],{"className":749683},[31792],[34,749685,749687],{"className":749686},[31796],[34,749688,749690],{"className":749689},[31801],[34,749691,749693],{"className":749692,"style":132027},[31805],[34,749694,749695,749698],{"style":37777},[34,749696],{"className":749697,"style":31814},[31813],[34,749699,749701],{"className":749700},[31818,31819,31820,31821],[34,749702,749704],{"className":749703},[31685,31821],[34,749705,502955],{"className":749706},[31685,31821],[34,749708],{"className":749709,"style":31846},[31845],[34,749711,202],{"className":749712},[31850],[34,749714],{"className":749715,"style":31846},[31845],[34,749717,749719,749722],{"className":749718},[31676],[34,749720],{"className":749721,"style":32617},[31680],[34,749723,459],{"className":749724},[31685],[108,749726,749727,749730,749946,750088,750089],{},[2627,749728,749729],{},"代入原方程：",[34,749731,749733,749793],{"className":749732},[31642],[34,749734,749736],{"className":749735},[31646],[31648,749737,749738],{"xmlns":31650},[31652,749739,749740,749790],{},[31655,749741,749742,749744,749746,749748,749750,749752,749754,749756,749758,749760,749762,749764,749766,749768,749770,749772,749774,749776,749778,749780,749782,749784,749786,749788],{},[31749,749743,459],{},[31662,749745,32226],{},[31749,749747,15932],{},[31662,749749,456],{"stretchy":702},[31749,749751,459],{},[31662,749753,2557],{"stretchy":702},[31662,749755,223],{},[31749,749757,18136],{},[31662,749759,456],{"stretchy":702},[31658,749761,54151],{},[31662,749763,2557],{"stretchy":702},[31662,749765,32226],{},[31749,749767,768],{},[31662,749769,456],{"stretchy":702},[31658,749771,54151],{},[31658,749773,33249],{},[31662,749775,223],{},[31658,749777,54224],{},[31662,749779,2557],{"stretchy":702},[31662,749781,202],{},[31749,749783,768],{},[31658,749785,33249],{},[31662,749787,223],{},[31749,749789,786],{},[31665,749791,749792],{"encoding":31667},"0 - 4(0) + 5(A) - 2(At + B) = 2t + 3",[34,749794,749796,749814,749841,749868,749895,749916,749937],{"className":749795,"ariaHidden":758},[31672],[34,749797,749799,749802,749805,749808,749811],{"className":749798},[31676],[34,749800],{"className":749801,"style":35942},[31680],[34,749803,459],{"className":749804},[31685],[34,749806],{"className":749807,"style":32604},[31845],[34,749809,32226],{"className":749810},[32024],[34,749812],{"className":749813,"style":32604},[31845],[34,749815,749817,749820,749823,749826,749829,749832,749835,749838],{"className":749816},[31676],[34,749818],{"className":749819,"style":31774},[31680],[34,749821,15932],{"className":749822},[31685],[34,749824,456],{"className":749825},[31782],[34,749827,459],{"className":749828},[31685],[34,749830,2557],{"className":749831},[31690],[34,749833],{"className":749834,"style":32604},[31845],[34,749836,223],{"className":749837},[32024],[34,749839],{"className":749840,"style":32604},[31845],[34,749842,749844,749847,749850,749853,749856,749859,749862,749865],{"className":749843},[31676],[34,749845],{"className":749846,"style":31774},[31680],[34,749848,18136],{"className":749849},[31685],[34,749851,456],{"className":749852},[31782],[34,749854,54151],{"className":749855},[31685,31686],[34,749857,2557],{"className":749858},[31690],[34,749860],{"className":749861,"style":32604},[31845],[34,749863,32226],{"className":749864},[32024],[34,749866],{"className":749867,"style":32604},[31845],[34,749869,749871,749874,749877,749880,749883,749886,749889,749892],{"className":749870},[31676],[34,749872],{"className":749873,"style":31774},[31680],[34,749875,768],{"className":749876},[31685],[34,749878,456],{"className":749879},[31782],[34,749881,54151],{"className":749882},[31685,31686],[34,749884,33249],{"className":749885},[31685,31686],[34,749887],{"className":749888,"style":32604},[31845],[34,749890,223],{"className":749891},[32024],[34,749893],{"className":749894,"style":32604},[31845],[34,749896,749898,749901,749904,749907,749910,749913],{"className":749897},[31676],[34,749899],{"className":749900,"style":31774},[31680],[34,749902,54224],{"className":749903,"style":54252},[31685,31686],[34,749905,2557],{"className":749906},[31690],[34,749908],{"className":749909,"style":31846},[31845],[34,749911,202],{"className":749912},[31850],[34,749914],{"className":749915,"style":31846},[31845],[34,749917,749919,749922,749925,749928,749931,749934],{"className":749918},[31676],[34,749920],{"className":749921,"style":35942},[31680],[34,749923,768],{"className":749924},[31685],[34,749926,33249],{"className":749927},[31685,31686],[34,749929],{"className":749930,"style":32604},[31845],[34,749932,223],{"className":749933},[32024],[34,749935],{"className":749936,"style":32604},[31845],[34,749938,749940,749943],{"className":749939},[31676],[34,749941],{"className":749942,"style":32617},[31680],[34,749944,786],{"className":749945},[31685],[34,749947,749949,749989],{"className":749948},[31642],[34,749950,749952],{"className":749951},[31646],[31648,749953,749954],{"xmlns":31650},[31652,749955,749956,749986],{},[31655,749957,749958,749960,749962,749964,749966,749968,749970,749972,749974,749976,749978,749980,749982,749984],{},[31749,749959,18136],{},[31658,749961,54151],{},[31662,749963,32226],{},[31749,749965,768],{},[31658,749967,54151],{},[31658,749969,33249],{},[31662,749971,32226],{},[31749,749973,768],{},[31658,749975,54224],{},[31662,749977,202],{},[31749,749979,768],{},[31658,749981,33249],{},[31662,749983,223],{},[31749,749985,786],{},[31665,749987,749988],{"encoding":31667},"5A - 2At - 2B = 2t + 3",[34,749990,749992,750013,750037,750058,750079],{"className":749991,"ariaHidden":758},[31672],[34,749993,749995,749998,750001,750004,750007,750010],{"className":749994},[31676],[34,749996],{"className":749997,"style":34944},[31680],[34,749999,18136],{"className":750000},[31685],[34,750002,54151],{"className":750003},[31685,31686],[34,750005],{"className":750006,"style":32604},[31845],[34,750008,32226],{"className":750009},[32024],[34,750011],{"className":750012,"style":32604},[31845],[34,750014,750016,750019,750022,750025,750028,750031,750034],{"className":750015},[31676],[34,750017],{"className":750018,"style":34944},[31680],[34,750020,768],{"className":750021},[31685],[34,750023,54151],{"className":750024},[31685,31686],[34,750026,33249],{"className":750027},[31685,31686],[34,750029],{"className":750030,"style":32604},[31845],[34,750032,32226],{"className":750033},[32024],[34,750035],{"className":750036,"style":32604},[31845],[34,750038,750040,750043,750046,750049,750052,750055],{"className":750039},[31676],[34,750041],{"className":750042,"style":33218},[31680],[34,750044,768],{"className":750045},[31685],[34,750047,54224],{"className":750048,"style":54252},[31685,31686],[34,750050],{"className":750051,"style":31846},[31845],[34,750053,202],{"className":750054},[31850],[34,750056],{"className":750057,"style":31846},[31845],[34,750059,750061,750064,750067,750070,750073,750076],{"className":750060},[31676],[34,750062],{"className":750063,"style":35942},[31680],[34,750065,768],{"className":750066},[31685],[34,750068,33249],{"className":750069},[31685,31686],[34,750071],{"className":750072,"style":32604},[31845],[34,750074,223],{"className":750075},[32024],[34,750077],{"className":750078,"style":32604},[31845],[34,750080,750082,750085],{"className":750081},[31676],[34,750083],{"className":750084,"style":32617},[31680],[34,750086,786],{"className":750087},[31685],"\n整理得到：\n",[34,750090,750092,750138],{"className":750091},[31642],[34,750093,750095],{"className":750094},[31646],[31648,750096,750097],{"xmlns":31650},[31652,750098,750099,750135],{},[31655,750100,750101,750103,750105,750107,750109,750111,750113,750115,750117,750119,750121,750123,750125,750127,750129,750131,750133],{},[31662,750102,32226],{},[31749,750104,768],{},[31658,750106,54151],{},[31658,750108,33249],{},[31662,750110,223],{},[31662,750112,456],{"stretchy":702},[31749,750114,18136],{},[31658,750116,54151],{},[31662,750118,32226],{},[31749,750120,768],{},[31658,750122,54224],{},[31662,750124,2557],{"stretchy":702},[31662,750126,202],{},[31749,750128,768],{},[31658,750130,33249],{},[31662,750132,223],{},[31749,750134,786],{},[31665,750136,750137],{"encoding":31667},"-2A t + (5A - 2B) = 2t + 3",[34,750139,750141,750168,750192,750216,750237],{"className":750140,"ariaHidden":758},[31672],[34,750142,750144,750147,750150,750153,750156,750159,750162,750165],{"className":750143},[31676],[34,750145],{"className":750146,"style":34944},[31680],[34,750148,32226],{"className":750149},[31685],[34,750151,768],{"className":750152},[31685],[34,750154,54151],{"className":750155},[31685,31686],[34,750157,33249],{"className":750158},[31685,31686],[34,750160],{"className":750161,"style":32604},[31845],[34,750163,223],{"className":750164},[32024],[34,750166],{"className":750167,"style":32604},[31845],[34,750169,750171,750174,750177,750180,750183,750186,750189],{"className":750170},[31676],[34,750172],{"className":750173,"style":31774},[31680],[34,750175,456],{"className":750176},[31782],[34,750178,18136],{"className":750179},[31685],[34,750181,54151],{"className":750182},[31685,31686],[34,750184],{"className":750185,"style":32604},[31845],[34,750187,32226],{"className":750188},[32024],[34,750190],{"className":750191,"style":32604},[31845],[34,750193,750195,750198,750201,750204,750207,750210,750213],{"className":750194},[31676],[34,750196],{"className":750197,"style":31774},[31680],[34,750199,768],{"className":750200},[31685],[34,750202,54224],{"className":750203,"style":54252},[31685,31686],[34,750205,2557],{"className":750206},[31690],[34,750208],{"className":750209,"style":31846},[31845],[34,750211,202],{"className":750212},[31850],[34,750214],{"className":750215,"style":31846},[31845],[34,750217,750219,750222,750225,750228,750231,750234],{"className":750218},[31676],[34,750220],{"className":750221,"style":35942},[31680],[34,750223,768],{"className":750224},[31685],[34,750226,33249],{"className":750227},[31685,31686],[34,750229],{"className":750230,"style":32604},[31845],[34,750232,223],{"className":750233},[32024],[34,750235],{"className":750236,"style":32604},[31845],[34,750238,750240,750243],{"className":750239},[31676],[34,750241],{"className":750242,"style":32617},[31680],[34,750244,786],{"className":750245},[31685],[108,750247,750248,750251],{},[2627,750249,750250],{},"比较系数：",[105,750252,750253,750404],{},[108,750254,750255,750283,750284],{},[34,750256,750258,750271],{"className":750257},[31642],[34,750259,750261],{"className":750260},[31646],[31648,750262,750263],{"xmlns":31650},[31652,750264,750265,750269],{},[31655,750266,750267],{},[31658,750268,33249],{},[31665,750270,33249],{"encoding":31667},[34,750272,750274],{"className":750273,"ariaHidden":758},[31672],[34,750275,750277,750280],{"className":750276},[31676],[34,750278],{"className":750279,"style":33280},[31680],[34,750281,33249],{"className":750282},[31685,31686]," 的系数：",[34,750285,750287,750323],{"className":750286},[31642],[34,750288,750290],{"className":750289},[31646],[31648,750291,750292],{"xmlns":31650},[31652,750293,750294,750320],{},[31655,750295,750296,750298,750300,750302,750304,750306,750308,750310,750312,750314,750316,750318],{},[31662,750297,32226],{},[31749,750299,768],{},[31658,750301,54151],{},[31662,750303,202],{},[31749,750305,768],{},[39361,750307,262622],{},[31662,750309,262625],{},[39361,750311,262622],{},[31658,750313,54151],{},[31662,750315,202],{},[31662,750317,32226],{},[31749,750319,640],{},[31665,750321,750322],{"encoding":31667},"-2A = 2 \\implies A = -1",[34,750324,750326,750350,750374,750392],{"className":750325,"ariaHidden":758},[31672],[34,750327,750329,750332,750335,750338,750341,750344,750347],{"className":750328},[31676],[34,750330],{"className":750331,"style":34944},[31680],[34,750333,32226],{"className":750334},[31685],[34,750336,768],{"className":750337},[31685],[34,750339,54151],{"className":750340},[31685,31686],[34,750342],{"className":750343,"style":31846},[31845],[34,750345,202],{"className":750346},[31850],[34,750348],{"className":750349,"style":31846},[31845],[34,750351,750353,750356,750359,750362,750365,750368,750371],{"className":750352},[31676],[34,750354],{"className":750355,"style":262700},[31680],[34,750357,768],{"className":750358},[31685],[34,750360],{"className":750361,"style":31846},[31845],[34,750363],{"className":750364,"style":31846},[31845],[34,750366,262625],{"className":750367},[31850],[34,750369],{"className":750370,"style":31846},[31845],[34,750372],{"className":750373,"style":31846},[31845],[34,750375,750377,750380,750383,750386,750389],{"className":750376},[31676],[34,750378],{"className":750379,"style":33218},[31680],[34,750381,54151],{"className":750382},[31685,31686],[34,750384],{"className":750385,"style":31846},[31845],[34,750387,202],{"className":750388},[31850],[34,750390],{"className":750391,"style":31846},[31845],[34,750393,750395,750398,750401],{"className":750394},[31676],[34,750396],{"className":750397,"style":35942},[31680],[34,750399,32226],{"className":750400},[31685],[34,750402,640],{"className":750403},[31685],[108,750405,750406,750407,68939,750490,750546,750547,750753,750754],{},"常数项：",[34,750408,750410,750436],{"className":750409},[31642],[34,750411,750413],{"className":750412},[31646],[31648,750414,750415],{"xmlns":31650},[31652,750416,750417,750433],{},[31655,750418,750419,750421,750423,750425,750427,750429,750431],{},[31749,750420,18136],{},[31658,750422,54151],{},[31662,750424,32226],{},[31749,750426,768],{},[31658,750428,54224],{},[31662,750430,202],{},[31749,750432,786],{},[31665,750434,750435],{"encoding":31667},"5A - 2B = 3",[34,750437,750439,750460,750481],{"className":750438,"ariaHidden":758},[31672],[34,750440,750442,750445,750448,750451,750454,750457],{"className":750441},[31676],[34,750443],{"className":750444,"style":34944},[31680],[34,750446,18136],{"className":750447},[31685],[34,750449,54151],{"className":750450},[31685,31686],[34,750452],{"className":750453,"style":32604},[31845],[34,750455,32226],{"className":750456},[32024],[34,750458],{"className":750459,"style":32604},[31845],[34,750461,750463,750466,750469,750472,750475,750478],{"className":750462},[31676],[34,750464],{"className":750465,"style":33218},[31680],[34,750467,768],{"className":750468},[31685],[34,750470,54224],{"className":750471,"style":54252},[31685,31686],[34,750473],{"className":750474,"style":31846},[31845],[34,750476,202],{"className":750477},[31850],[34,750479],{"className":750480,"style":31846},[31845],[34,750482,750484,750487],{"className":750483},[31676],[34,750485],{"className":750486,"style":32617},[31680],[34,750488,786],{"className":750489},[31685],[34,750491,750493,750513],{"className":750492},[31642],[34,750494,750496],{"className":750495},[31646],[31648,750497,750498],{"xmlns":31650},[31652,750499,750500,750510],{},[31655,750501,750502,750504,750506,750508],{},[31658,750503,54151],{},[31662,750505,202],{},[31662,750507,32226],{},[31749,750509,640],{},[31665,750511,750512],{"encoding":31667},"A=-1",[34,750514,750516,750534],{"className":750515,"ariaHidden":758},[31672],[34,750517,750519,750522,750525,750528,750531],{"className":750518},[31676],[34,750520],{"className":750521,"style":33218},[31680],[34,750523,54151],{"className":750524},[31685,31686],[34,750526],{"className":750527,"style":31846},[31845],[34,750529,202],{"className":750530},[31850],[34,750532],{"className":750533,"style":31846},[31845],[34,750535,750537,750540,750543],{"className":750536},[31676],[34,750538],{"className":750539,"style":35942},[31680],[34,750541,32226],{"className":750542},[31685],[34,750544,640],{"className":750545},[31685]," 代入：",[34,750548,750550,750606],{"className":750549},[31642],[34,750551,750553],{"className":750552},[31646],[31648,750554,750555],{"xmlns":31650},[31652,750556,750557,750603],{},[31655,750558,750559,750561,750563,750565,750567,750569,750571,750573,750575,750577,750579,750581,750583,750585,750587,750589,750591,750593,750595,750597,750599,750601],{},[31662,750560,32226],{},[31749,750562,18136],{},[31662,750564,32226],{},[31749,750566,768],{},[31658,750568,54224],{},[31662,750570,202],{},[31749,750572,786],{},[39361,750574,262622],{},[31662,750576,262625],{},[39361,750578,262622],{},[31662,750580,32226],{},[31749,750582,768],{},[31658,750584,54224],{},[31662,750586,202],{},[31749,750588,60525],{},[39361,750590,262622],{},[31662,750592,262625],{},[39361,750594,262622],{},[31658,750596,54224],{},[31662,750598,202],{},[31662,750600,32226],{},[31749,750602,15932],{},[31665,750604,750605],{"encoding":31667},"-5 - 2B = 3 \\implies -2B = 8 \\implies B = -4",[34,750607,750609,750630,750651,750675,750699,750723,750741],{"className":750608,"ariaHidden":758},[31672],[34,750610,750612,750615,750618,750621,750624,750627],{"className":750611},[31676],[34,750613],{"className":750614,"style":35942},[31680],[34,750616,32226],{"className":750617},[31685],[34,750619,18136],{"className":750620},[31685],[34,750622],{"className":750623,"style":32604},[31845],[34,750625,32226],{"className":750626},[32024],[34,750628],{"className":750629,"style":32604},[31845],[34,750631,750633,750636,750639,750642,750645,750648],{"className":750632},[31676],[34,750634],{"className":750635,"style":33218},[31680],[34,750637,768],{"className":750638},[31685],[34,750640,54224],{"className":750641,"style":54252},[31685,31686],[34,750643],{"className":750644,"style":31846},[31845],[34,750646,202],{"className":750647},[31850],[34,750649],{"className":750650,"style":31846},[31845],[34,750652,750654,750657,750660,750663,750666,750669,750672],{"className":750653},[31676],[34,750655],{"className":750656,"style":262700},[31680],[34,750658,786],{"className":750659},[31685],[34,750661],{"className":750662,"style":31846},[31845],[34,750664],{"className":750665,"style":31846},[31845],[34,750667,262625],{"className":750668},[31850],[34,750670],{"className":750671,"style":31846},[31845],[34,750673],{"className":750674,"style":31846},[31845],[34,750676,750678,750681,750684,750687,750690,750693,750696],{"className":750677},[31676],[34,750679],{"className":750680,"style":34944},[31680],[34,750682,32226],{"className":750683},[31685],[34,750685,768],{"className":750686},[31685],[34,750688,54224],{"className":750689,"style":54252},[31685,31686],[34,750691],{"className":750692,"style":31846},[31845],[34,750694,202],{"className":750695},[31850],[34,750697],{"className":750698,"style":31846},[31845],[34,750700,750702,750705,750708,750711,750714,750717,750720],{"className":750701},[31676],[34,750703],{"className":750704,"style":262700},[31680],[34,750706,60525],{"className":750707},[31685],[34,750709],{"className":750710,"style":31846},[31845],[34,750712],{"className":750713,"style":31846},[31845],[34,750715,262625],{"className":750716},[31850],[34,750718],{"className":750719,"style":31846},[31845],[34,750721],{"className":750722,"style":31846},[31845],[34,750724,750726,750729,750732,750735,750738],{"className":750725},[31676],[34,750727],{"className":750728,"style":33218},[31680],[34,750730,54224],{"className":750731,"style":54252},[31685,31686],[34,750733],{"className":750734,"style":31846},[31845],[34,750736,202],{"className":750737},[31850],[34,750739],{"className":750740,"style":31846},[31845],[34,750742,750744,750747,750750],{"className":750743},[31676],[34,750745],{"className":750746,"style":35942},[31680],[34,750748,32226],{"className":750749},[31685],[34,750751,15932],{"className":750752},[31685],"。\n由此得到特解：\n",[34,750755,750757,750785],{"className":750756},[31642],[34,750758,750760],{"className":750759},[31646],[31648,750761,750762],{"xmlns":31650},[31652,750763,750764,750782],{},[31655,750765,750766,750772,750774,750776,750778,750780],{},[134014,750767,750768,750770],{"accent":758},[31658,750769,32936],{},[31662,750771,749281],{"stretchy":758},[31662,750773,202],{},[31662,750775,32226],{},[31658,750777,33249],{},[31662,750779,32226],{},[31749,750781,15932],{},[31665,750783,750784],{"encoding":31667},"\\widetilde{x} = -t - 4",[34,750786,750788,750835,750856],{"className":750787,"ariaHidden":758},[31672],[34,750789,750791,750794,750826,750829,750832],{"className":750790},[31676],[34,750792],{"className":750793,"style":749304},[31680],[34,750795,750797],{"className":750796},[31685,346171],[34,750798,750800],{"className":750799},[31796],[34,750801,750803],{"className":750802},[31801],[34,750804,750806,750814],{"className":750805,"style":749304},[31805],[34,750807,750808,750811],{"style":41565},[34,750809],{"className":750810,"style":41508},[31813],[34,750812,32936],{"className":750813},[31685,31686],[34,750815,750817,750820],{"className":750816,"style":749328},[41564],[34,750818],{"className":750819,"style":41508},[31813],[34,750821,750822],{"style":749334},[41588,750823,750824],{"xmlns":41590,"width":168121,"height":749337,"viewBox":749338,"preserveAspectRatio":29503},[20398,750825],{"d":749341},[34,750827],{"className":750828,"style":31846},[31845],[34,750830,202],{"className":750831},[31850],[34,750833],{"className":750834,"style":31846},[31845],[34,750836,750838,750841,750844,750847,750850,750853],{"className":750837},[31676],[34,750839],{"className":750840,"style":721670},[31680],[34,750842,32226],{"className":750843},[31685],[34,750845,33249],{"className":750846},[31685,31686],[34,750848],{"className":750849,"style":32604},[31845],[34,750851,32226],{"className":750852},[32024],[34,750854],{"className":750855,"style":32604},[31845],[34,750857,750859,750862],{"className":750858},[31676],[34,750860],{"className":750861,"style":32617},[31680],[34,750863,15932],{"className":750864},[31685],[108,750866,750867,750870],{},[2627,750868,750869],{},"步骤三：合成通解",[34,750871,750873,750941],{"className":750872},[31642],[34,750874,750876],{"className":750875},[31646],[31648,750877,750878],{"xmlns":31650},[31652,750879,750880,750938],{},[31655,750881,750882,750884,750886,750888,750894,750896,750902,750904,750906,750912,750914,750920,750930,750932,750934,750936],{},[31658,750883,32936],{},[31662,750885,202],{},[31662,750887,456],{"stretchy":702},[31743,750889,750890,750892],{},[31658,750891,31965],{},[31749,750893,640],{},[31662,750895,223],{},[31743,750897,750898,750900],{},[31658,750899,31965],{},[31749,750901,768],{},[31658,750903,33249],{},[31662,750905,2557],{"stretchy":702},[37624,750907,750908,750910],{},[31658,750909,33244],{"mathvariant":38079},[31658,750911,33249],{},[31662,750913,223],{},[31743,750915,750916,750918],{},[31658,750917,31965],{},[31749,750919,786],{},[37624,750921,750922,750924],{},[31658,750923,33244],{"mathvariant":38079},[31655,750925,750926,750928],{},[31749,750927,768],{},[31658,750929,33249],{},[31662,750931,32226],{},[31658,750933,33249],{},[31662,750935,32226],{},[31749,750937,15932],{},[31665,750939,750940],{"encoding":31667},"x = (c_1 + c_2 t)\\mathrm{e}^t + c_3 \\mathrm{e}^{2t} - t - 4",[34,750942,750944,750962,751020,751110,751200,751218],{"className":750943,"ariaHidden":758},[31672],[34,750945,750947,750950,750953,750956,750959],{"className":750946},[31676],[34,750948],{"className":750949,"style":32527},[31680],[34,750951,32936],{"className":750952},[31685,31686],[34,750954],{"className":750955,"style":31846},[31845],[34,750957,202],{"className":750958},[31850],[34,750960],{"className":750961,"style":31846},[31845],[34,750963,750965,750968,750971,751011,751014,751017],{"className":750964},[31676],[34,750966],{"className":750967,"style":31774},[31680],[34,750969,456],{"className":750970},[31782],[34,750972,750974,750977],{"className":750973},[31685],[34,750975,31965],{"className":750976},[31685,31686],[34,750978,750980],{"className":750979},[31792],[34,750981,750983,751003],{"className":750982},[31796,31797],[34,750984,750986,751000],{"className":750985},[31801],[34,750987,750989],{"className":750988,"style":31806},[31805],[34,750990,750991,750994],{"style":31809},[34,750992],{"className":750993,"style":31814},[31813],[34,750995,750997],{"className":750996},[31818,31819,31820,31821],[34,750998,640],{"className":750999},[31685,31821],[34,751001,31829],{"className":751002},[31828],[34,751004,751006],{"className":751005},[31801],[34,751007,751009],{"className":751008,"style":31836},[31805],[34,751010],{},[34,751012],{"className":751013,"style":32604},[31845],[34,751015,223],{"className":751016},[32024],[34,751018],{"className":751019,"style":32604},[31845],[34,751021,751023,751026,751066,751069,751072,751101,751104,751107],{"className":751022},[31676],[34,751024],{"className":751025,"style":748607},[31680],[34,751027,751029,751032],{"className":751028},[31685],[34,751030,31965],{"className":751031},[31685,31686],[34,751033,751035],{"className":751034},[31792],[34,751036,751038,751058],{"className":751037},[31796,31797],[34,751039,751041,751055],{"className":751040},[31801],[34,751042,751044],{"className":751043,"style":31806},[31805],[34,751045,751046,751049],{"style":31809},[34,751047],{"className":751048,"style":31814},[31813],[34,751050,751052],{"className":751051},[31818,31819,31820,31821],[34,751053,768],{"className":751054},[31685,31821],[34,751056,31829],{"className":751057},[31828],[34,751059,751061],{"className":751060},[31801],[34,751062,751064],{"className":751063,"style":31836},[31805],[34,751065],{},[34,751067,33249],{"className":751068},[31685,31686],[34,751070,2557],{"className":751071},[31690],[34,751073,751075,751078],{"className":751074},[31685],[34,751076,33244],{"className":751077},[31685,207923],[34,751079,751081],{"className":751080},[31792],[34,751082,751084],{"className":751083},[31796],[34,751085,751087],{"className":751086},[31801],[34,751088,751090],{"className":751089,"style":611758},[31805],[34,751091,751092,751095],{"style":37777},[34,751093],{"className":751094,"style":31814},[31813],[34,751096,751098],{"className":751097},[31818,31819,31820,31821],[34,751099,33249],{"className":751100},[31685,31686,31821],[34,751102],{"className":751103,"style":32604},[31845],[34,751105,223],{"className":751106},[32024],[34,751108],{"className":751109,"style":32604},[31845],[34,751111,751113,751116,751156,751191,751194,751197],{"className":751112},[31676],[34,751114],{"className":751115,"style":744441},[31680],[34,751117,751119,751122],{"className":751118},[31685],[34,751120,31965],{"className":751121},[31685,31686],[34,751123,751125],{"className":751124},[31792],[34,751126,751128,751148],{"className":751127},[31796,31797],[34,751129,751131,751145],{"className":751130},[31801],[34,751132,751134],{"className":751133,"style":31806},[31805],[34,751135,751136,751139],{"style":31809},[34,751137],{"className":751138,"style":31814},[31813],[34,751140,751142],{"className":751141},[31818,31819,31820,31821],[34,751143,786],{"className":751144},[31685,31821],[34,751146,31829],{"className":751147},[31828],[34,751149,751151],{"className":751150},[31801],[34,751152,751154],{"className":751153,"style":31836},[31805],[34,751155],{},[34,751157,751159,751162],{"className":751158},[31685],[34,751160,33244],{"className":751161},[31685,207923],[34,751163,751165],{"className":751164},[31792],[34,751166,751168],{"className":751167},[31796],[34,751169,751171],{"className":751170},[31801],[34,751172,751174],{"className":751173,"style":37774},[31805],[34,751175,751176,751179],{"style":37777},[34,751177],{"className":751178,"style":31814},[31813],[34,751180,751182],{"className":751181},[31818,31819,31820,31821],[34,751183,751185,751188],{"className":751184},[31685,31821],[34,751186,768],{"className":751187},[31685,31821],[34,751189,33249],{"className":751190},[31685,31686,31821],[34,751192],{"className":751193,"style":32604},[31845],[34,751195,32226],{"className":751196},[32024],[34,751198],{"className":751199,"style":32604},[31845],[34,751201,751203,751206,751209,751212,751215],{"className":751202},[31676],[34,751204],{"className":751205,"style":721670},[31680],[34,751207,33249],{"className":751208},[31685,31686],[34,751210],{"className":751211,"style":32604},[31845],[34,751213,32226],{"className":751214},[32024],[34,751216],{"className":751217,"style":32604},[31845],[34,751219,751221,751224],{"className":751220},[31676],[34,751222],{"className":751223,"style":32617},[31680],[34,751225,15932],{"className":751226},[31685],[32188,751228],{},[32490,751230,751232],{"id":751231},"_4-证明题","4. 证明题",[15,751234,751235,751237,751238,46528,751467,751591,751592,751833],{},[2627,751236,101191],{}," 给定 ",[34,751239,751241,751299],{"className":751240},[31642],[34,751242,751244],{"className":751243},[31646],[31648,751245,751246],{"xmlns":31650},[31652,751247,751248,751296],{},[31655,751249,751250,751262,751264,751266,751276,751278,751280,751286,751288,751290,751292,751294],{},[37624,751251,751252,751254],{},[31658,751253,32936],{},[31655,751255,751256,751258,751260],{},[31662,751257,116495],{"mathvariant":38079},[31662,751259,116495],{"mathvariant":38079},[31662,751261,116495],{"mathvariant":38079},[31662,751263,223],{},[31749,751265,18136],{},[37624,751267,751268,751270],{},[31658,751269,32936],{},[31655,751271,751272,751274],{},[31662,751273,116495],{"mathvariant":38079},[31662,751275,116495],{"mathvariant":38079},[31662,751277,223],{},[31749,751279,36656],{},[37624,751281,751282,751284],{},[31658,751283,32936],{},[31662,751285,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,751287,202],{},[31658,751289,31739],{},[31662,751291,456],{"stretchy":702},[31658,751293,33249],{},[31662,751295,2557],{"stretchy":702},[31665,751297,751298],{"encoding":31667},"x''' + 5x'' + 6x' = f(t)",[34,751300,751302,751349,751399,751449],{"className":751301,"ariaHidden":758},[31672],[34,751303,751305,751308,751340,751343,751346],{"className":751304},[31676],[34,751306],{"className":751307,"style":384364},[31680],[34,751309,751311,751314],{"className":751310},[31685],[34,751312,32936],{"className":751313},[31685,31686],[34,751315,751317],{"className":751316},[31792],[34,751318,751320],{"className":751319},[31796],[34,751321,751323],{"className":751322},[31801],[34,751324,751326],{"className":751325,"style":132027},[31805],[34,751327,751328,751331],{"style":37777},[34,751329],{"className":751330,"style":31814},[31813],[34,751332,751334],{"className":751333},[31818,31819,31820,31821],[34,751335,751337],{"className":751336},[31685,31821],[34,751338,502955],{"className":751339},[31685,31821],[34,751341],{"className":751342,"style":32604},[31845],[34,751344,223],{"className":751345},[32024],[34,751347],{"className":751348,"style":32604},[31845],[34,751350,751352,751355,751358,751390,751393,751396],{"className":751351},[31676],[34,751353],{"className":751354,"style":384364},[31680],[34,751356,18136],{"className":751357},[31685],[34,751359,751361,751364],{"className":751360},[31685],[34,751362,32936],{"className":751363},[31685,31686],[34,751365,751367],{"className":751366},[31792],[34,751368,751370],{"className":751369},[31796],[34,751371,751373],{"className":751372},[31801],[34,751374,751376],{"className":751375,"style":132027},[31805],[34,751377,751378,751381],{"style":37777},[34,751379],{"className":751380,"style":31814},[31813],[34,751382,751384],{"className":751383},[31818,31819,31820,31821],[34,751385,751387],{"className":751386},[31685,31821],[34,751388,502782],{"className":751389},[31685,31821],[34,751391],{"className":751392,"style":32604},[31845],[34,751394,223],{"className":751395},[32024],[34,751397],{"className":751398,"style":32604},[31845],[34,751400,751402,751405,751408,751440,751443,751446],{"className":751401},[31676],[34,751403],{"className":751404,"style":132027},[31680],[34,751406,36656],{"className":751407},[31685],[34,751409,751411,751414],{"className":751410},[31685],[34,751412,32936],{"className":751413},[31685,31686],[34,751415,751417],{"className":751416},[31792],[34,751418,751420],{"className":751419},[31796],[34,751421,751423],{"className":751422},[31801],[34,751424,751426],{"className":751425,"style":132027},[31805],[34,751427,751428,751431],{"style":37777},[34,751429],{"className":751430,"style":31814},[31813],[34,751432,751434],{"className":751433},[31818,31819,31820,31821],[34,751435,751437],{"className":751436},[31685,31821],[34,751438,116495],{"className":751439},[31685,31821],[34,751441],{"className":751442,"style":31846},[31845],[34,751444,202],{"className":751445},[31850],[34,751447],{"className":751448,"style":31846},[31845],[34,751450,751452,751455,751458,751461,751464],{"className":751451},[31676],[34,751453],{"className":751454,"style":31774},[31680],[34,751456,31739],{"className":751457,"style":31778},[31685,31686],[34,751459,456],{"className":751460},[31782],[34,751462,33249],{"className":751463},[31685,31686],[34,751465,2557],{"className":751466},[31690],[34,751468,751470,751496],{"className":751469},[31642],[34,751471,751473],{"className":751472},[31646],[31648,751474,751475],{"xmlns":31650},[31652,751476,751477,751493],{},[31655,751478,751479,751485,751487],{},[31743,751480,751481,751483],{},[31658,751482,126541],{},[31749,751484,640],{},[31662,751486,31960],{"separator":758},[31743,751488,751489,751491],{},[31658,751490,126541],{},[31749,751492,768],{},[31665,751494,751495],{"encoding":31667},"\\varphi_1, \\varphi_2",[34,751497,751499],{"className":751498,"ariaHidden":758},[31672],[34,751500,751502,751505,751545,751548,751551],{"className":751501},[31676],[34,751503],{"className":751504,"style":38774},[31680],[34,751506,751508,751511],{"className":751507},[31685],[34,751509,126541],{"className":751510},[31685,31686],[34,751512,751514],{"className":751513},[31792],[34,751515,751517,751537],{"className":751516},[31796,31797],[34,751518,751520,751534],{"className":751519},[31801],[34,751521,751523],{"className":751522,"style":31806},[31805],[34,751524,751525,751528],{"style":31809},[34,751526],{"className":751527,"style":31814},[31813],[34,751529,751531],{"className":751530},[31818,31819,31820,31821],[34,751532,640],{"className":751533},[31685,31821],[34,751535,31829],{"className":751536},[31828],[34,751538,751540],{"className":751539},[31801],[34,751541,751543],{"className":751542,"style":31836},[31805],[34,751544],{},[34,751546,31960],{"className":751547},[32119],[34,751549],{"className":751550,"style":32123},[31845],[34,751552,751554,751557],{"className":751553},[31685],[34,751555,126541],{"className":751556},[31685,31686],[34,751558,751560],{"className":751559},[31792],[34,751561,751563,751583],{"className":751562},[31796,31797],[34,751564,751566,751580],{"className":751565},[31801],[34,751567,751569],{"className":751568,"style":31806},[31805],[34,751570,751571,751574],{"style":31809},[34,751572],{"className":751573,"style":31814},[31813],[34,751575,751577],{"className":751576},[31818,31819,31820,31821],[34,751578,768],{"className":751579},[31685,31821],[34,751581,31829],{"className":751582},[31828],[34,751584,751586],{"className":751585},[31801],[34,751587,751589],{"className":751588,"style":31836},[31805],[34,751590],{}," 是解，证明 ",[34,751593,751595,751653],{"className":751594},[31642],[34,751596,751598],{"className":751597},[31646],[31648,751599,751600],{"xmlns":31650},[31652,751601,751602,751650],{},[31655,751603,751604,751620,751622,751628,751630,751632,751634,751636,751642,751644,751646,751648],{},[71115,751605,751606,751612],{},[31655,751607,751608,751610],{},[31658,751609,546745],{},[31662,751611,32682],{},[31655,751613,751614,751616,751618],{},[31658,751615,33249],{},[31662,751617,33239],{},[31658,751619,138286],{"mathvariant":38079},[31662,751621,15972],{"stretchy":702},[31743,751623,751624,751626],{},[31658,751625,126541],{},[31749,751627,640],{},[31662,751629,456],{"stretchy":702},[31658,751631,33249],{},[31662,751633,2557],{"stretchy":702},[31662,751635,32226],{},[31743,751637,751638,751640],{},[31658,751639,126541],{},[31749,751641,768],{},[31662,751643,456],{"stretchy":702},[31658,751645,33249],{},[31662,751647,2557],{"stretchy":702},[31662,751649,32712],{"stretchy":702},[31665,751651,751652],{"encoding":31667},"\\lim\\limits_{t \\to \\infty} [\\varphi_1(t) - \\varphi_2(t)]",[34,751654,751656,751778],{"className":751655,"ariaHidden":758},[31672],[34,751657,751659,751663,751717,751720,751760,751763,751766,751769,751772,751775],{"className":751658},[31676],[34,751660],{"className":751661,"style":751662},[31680],"height:1.45em;vertical-align:-0.7em;",[34,751664,751666],{"className":751665},[32767,71420],[34,751667,751669,751708],{"className":751668},[31796,31797],[34,751670,751672,751705],{"className":751671},[31801],[34,751673,751675,751695],{"className":751674,"style":31681},[31805],[34,751676,751677,751680],{"style":576058},[34,751678],{"className":751679,"style":41508},[31813],[34,751681,751683],{"className":751682},[31818,31819,31820,31821],[34,751684,751686,751689,751692],{"className":751685},[31685,31821],[34,751687,33249],{"className":751688},[31685,31686,31821],[34,751690,33239],{"className":751691},[31850,31821],[34,751693,138286],{"className":751694},[31685,31821],[34,751696,751697,751700],{"style":41565},[34,751698],{"className":751699,"style":41508},[31813],[34,751701,751702],{},[34,751703,546745],{"className":751704},[32767],[34,751706,31829],{"className":751707},[31828],[34,751709,751711],{"className":751710},[31801],[34,751712,751715],{"className":751713,"style":751714},[31805],"height:0.7em;",[34,751716],{},[34,751718,15972],{"className":751719},[31782],[34,751721,751723,751726],{"className":751722},[31685],[34,751724,126541],{"className":751725},[31685,31686],[34,751727,751729],{"className":751728},[31792],[34,751730,751732,751752],{"className":751731},[31796,31797],[34,751733,751735,751749],{"className":751734},[31801],[34,751736,751738],{"className":751737,"style":31806},[31805],[34,751739,751740,751743],{"style":31809},[34,751741],{"className":751742,"style":31814},[31813],[34,751744,751746],{"className":751745},[31818,31819,31820,31821],[34,751747,640],{"className":751748},[31685,31821],[34,751750,31829],{"className":751751},[31828],[34,751753,751755],{"className":751754},[31801],[34,751756,751758],{"className":751757,"style":31836},[31805],[34,751759],{},[34,751761,456],{"className":751762},[31782],[34,751764,33249],{"className":751765},[31685,31686],[34,751767,2557],{"className":751768},[31690],[34,751770],{"className":751771,"style":32604},[31845],[34,751773,32226],{"className":751774},[32024],[34,751776],{"className":751777,"style":32604},[31845],[34,751779,751781,751784,751824,751827,751830],{"className":751780},[31676],[34,751782],{"className":751783,"style":31774},[31680],[34,751785,751787,751790],{"className":751786},[31685],[34,751788,126541],{"className":751789},[31685,31686],[34,751791,751793],{"className":751792},[31792],[34,751794,751796,751816],{"className":751795},[31796,31797],[34,751797,751799,751813],{"className":751798},[31801],[34,751800,751802],{"className":751801,"style":31806},[31805],[34,751803,751804,751807],{"style":31809},[34,751805],{"className":751806,"style":31814},[31813],[34,751808,751810],{"className":751809},[31818,31819,31820,31821],[34,751811,768],{"className":751812},[31685,31821],[34,751814,31829],{"className":751815},[31828],[34,751817,751819],{"className":751818},[31801],[34,751820,751822],{"className":751821,"style":31836},[31805],[34,751823],{},[34,751825,456],{"className":751826},[31782],[34,751828,33249],{"className":751829},[31685,31686],[34,751831,481211],{"className":751832},[31690]," 存在.",[105,751835,751836,751850],{},[108,751837,751838,751841,751842,751845,751846,751849],{},[2627,751839,751840],{},"思路引导："," 这道题看起来很抽象，其实是在考查非齐次方程通解的结构。\n",[2627,751843,751844],{},"核心定理："," 非齐次方程的任意两个解之差，一定是对应",[2627,751847,751848],{},"齐次方程","的解。",[108,751851,751852,751854],{},[2627,751853,715602],{},[2230,751855,751856,752058,753085,753354,754085,754933],{},[108,751857,133910,751858,347],{},[34,751859,751861,751909],{"className":751860},[31642],[34,751862,751864],{"className":751863},[31646],[31648,751865,751866],{"xmlns":31650},[31652,751867,751868,751906],{},[31655,751869,751870,751872,751874,751876,751878,751880,751886,751888,751890,751892,751894,751900,751902,751904],{},[31658,751871,67764],{},[31662,751873,456],{"stretchy":702},[31658,751875,33249],{},[31662,751877,2557],{"stretchy":702},[31662,751879,202],{},[31743,751881,751882,751884],{},[31658,751883,126541],{},[31749,751885,640],{},[31662,751887,456],{"stretchy":702},[31658,751889,33249],{},[31662,751891,2557],{"stretchy":702},[31662,751893,32226],{},[31743,751895,751896,751898],{},[31658,751897,126541],{},[31749,751899,768],{},[31662,751901,456],{"stretchy":702},[31658,751903,33249],{},[31662,751905,2557],{"stretchy":702},[31665,751907,751908],{"encoding":31667},"y(t) = \\varphi_1(t) - \\varphi_2(t)",[34,751910,751912,751939,752003],{"className":751911,"ariaHidden":758},[31672],[34,751913,751915,751918,751921,751924,751927,751930,751933,751936],{"className":751914},[31676],[34,751916],{"className":751917,"style":31774},[31680],[34,751919,67764],{"className":751920,"style":32062},[31685,31686],[34,751922,456],{"className":751923},[31782],[34,751925,33249],{"className":751926},[31685,31686],[34,751928,2557],{"className":751929},[31690],[34,751931],{"className":751932,"style":31846},[31845],[34,751934,202],{"className":751935},[31850],[34,751937],{"className":751938,"style":31846},[31845],[34,751940,751942,751945,751985,751988,751991,751994,751997,752000],{"className":751941},[31676],[34,751943],{"className":751944,"style":31774},[31680],[34,751946,751948,751951],{"className":751947},[31685],[34,751949,126541],{"className":751950},[31685,31686],[34,751952,751954],{"className":751953},[31792],[34,751955,751957,751977],{"className":751956},[31796,31797],[34,751958,751960,751974],{"className":751959},[31801],[34,751961,751963],{"className":751962,"style":31806},[31805],[34,751964,751965,751968],{"style":31809},[34,751966],{"className":751967,"style":31814},[31813],[34,751969,751971],{"className":751970},[31818,31819,31820,31821],[34,751972,640],{"className":751973},[31685,31821],[34,751975,31829],{"className":751976},[31828],[34,751978,751980],{"className":751979},[31801],[34,751981,751983],{"className":751982,"style":31836},[31805],[34,751984],{},[34,751986,456],{"className":751987},[31782],[34,751989,33249],{"className":751990},[31685,31686],[34,751992,2557],{"className":751993},[31690],[34,751995],{"className":751996,"style":32604},[31845],[34,751998,32226],{"className":751999},[32024],[34,752001],{"className":752002,"style":32604},[31845],[34,752004,752006,752009,752049,752052,752055],{"className":752005},[31676],[34,752007],{"className":752008,"style":31774},[31680],[34,752010,752012,752015],{"className":752011},[31685],[34,752013,126541],{"className":752014},[31685,31686],[34,752016,752018],{"className":752017},[31792],[34,752019,752021,752041],{"className":752020},[31796,31797],[34,752022,752024,752038],{"className":752023},[31801],[34,752025,752027],{"className":752026,"style":31806},[31805],[34,752028,752029,752032],{"style":31809},[34,752030],{"className":752031,"style":31814},[31813],[34,752033,752035],{"className":752034},[31818,31819,31820,31821],[34,752036,768],{"className":752037},[31685,31821],[34,752039,31829],{"className":752040},[31828],[34,752042,752044],{"className":752043},[31801],[34,752045,752047],{"className":752046,"style":31836},[31805],[34,752048],{},[34,752050,456],{"className":752051},[31782],[34,752053,33249],{"className":752054},[31685,31686],[34,752056,2557],{"className":752057},[31690],[108,752059,93310,752060,752183,752184,752821,752822,753036,753037,347],{},[34,752061,752063,752088],{"className":752062},[31642],[34,752064,752066],{"className":752065},[31646],[31648,752067,752068],{"xmlns":31650},[31652,752069,752070,752086],{},[31655,752071,752072,752078,752080],{},[31743,752073,752074,752076],{},[31658,752075,126541],{},[31749,752077,640],{},[31662,752079,31960],{"separator":758},[31743,752081,752082,752084],{},[31658,752083,126541],{},[31749,752085,768],{},[31665,752087,751495],{"encoding":31667},[34,752089,752091],{"className":752090,"ariaHidden":758},[31672],[34,752092,752094,752097,752137,752140,752143],{"className":752093},[31676],[34,752095],{"className":752096,"style":38774},[31680],[34,752098,752100,752103],{"className":752099},[31685],[34,752101,126541],{"className":752102},[31685,31686],[34,752104,752106],{"className":752105},[31792],[34,752107,752109,752129],{"className":752108},[31796,31797],[34,752110,752112,752126],{"className":752111},[31801],[34,752113,752115],{"className":752114,"style":31806},[31805],[34,752116,752117,752120],{"style":31809},[34,752118],{"className":752119,"style":31814},[31813],[34,752121,752123],{"className":752122},[31818,31819,31820,31821],[34,752124,640],{"className":752125},[31685,31821],[34,752127,31829],{"className":752128},[31828],[34,752130,752132],{"className":752131},[31801],[34,752133,752135],{"className":752134,"style":31836},[31805],[34,752136],{},[34,752138,31960],{"className":752139},[32119],[34,752141],{"className":752142,"style":32123},[31845],[34,752144,752146,752149],{"className":752145},[31685],[34,752147,126541],{"className":752148},[31685,31686],[34,752150,752152],{"className":752151},[31792],[34,752153,752155,752175],{"className":752154},[31796,31797],[34,752156,752158,752172],{"className":752157},[31801],[34,752159,752161],{"className":752160,"style":31806},[31805],[34,752162,752163,752166],{"style":31809},[34,752164],{"className":752165,"style":31814},[31813],[34,752167,752169],{"className":752168},[31818,31819,31820,31821],[34,752170,768],{"className":752171},[31685,31821],[34,752173,31829],{"className":752174},[31828],[34,752176,752178],{"className":752177},[31801],[34,752179,752181],{"className":752180,"style":31836},[31805],[34,752182],{}," 都是原方程的解，代入方程相减：\n",[34,752185,752187,752317],{"className":752186},[31642],[34,752188,752190],{"className":752189},[31646],[31648,752191,752192],{"xmlns":31650},[31652,752193,752194,752314],{},[31655,752195,752196,752198,752212,752214,752216,752228,752230,752232,752240,752242,752244,752246,752260,752262,752264,752276,752278,752280,752288,752290,752292,752294,752296,752298,752300,752302,752304,752306,752308,752310,752312],{},[31662,752197,456],{"stretchy":702},[38846,752199,752200,752202,752204],{},[31658,752201,126541],{},[31749,752203,640],{},[31655,752205,752206,752208,752210],{},[31662,752207,116495],{"mathvariant":38079},[31662,752209,116495],{"mathvariant":38079},[31662,752211,116495],{"mathvariant":38079},[31662,752213,223],{},[31749,752215,18136],{},[38846,752217,752218,752220,752222],{},[31658,752219,126541],{},[31749,752221,640],{},[31655,752223,752224,752226],{},[31662,752225,116495],{"mathvariant":38079},[31662,752227,116495],{"mathvariant":38079},[31662,752229,223],{},[31749,752231,36656],{},[38846,752233,752234,752236,752238],{},[31658,752235,126541],{},[31749,752237,640],{},[31662,752239,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,752241,2557],{"stretchy":702},[31662,752243,32226],{},[31662,752245,456],{"stretchy":702},[38846,752247,752248,752250,752252],{},[31658,752249,126541],{},[31749,752251,768],{},[31655,752253,752254,752256,752258],{},[31662,752255,116495],{"mathvariant":38079},[31662,752257,116495],{"mathvariant":38079},[31662,752259,116495],{"mathvariant":38079},[31662,752261,223],{},[31749,752263,18136],{},[38846,752265,752266,752268,752270],{},[31658,752267,126541],{},[31749,752269,768],{},[31655,752271,752272,752274],{},[31662,752273,116495],{"mathvariant":38079},[31662,752275,116495],{"mathvariant":38079},[31662,752277,223],{},[31749,752279,36656],{},[38846,752281,752282,752284,752286],{},[31658,752283,126541],{},[31749,752285,768],{},[31662,752287,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,752289,2557],{"stretchy":702},[31662,752291,202],{},[31658,752293,31739],{},[31662,752295,456],{"stretchy":702},[31658,752297,33249],{},[31662,752299,2557],{"stretchy":702},[31662,752301,32226],{},[31658,752303,31739],{},[31662,752305,456],{"stretchy":702},[31658,752307,33249],{},[31662,752309,2557],{"stretchy":702},[31662,752311,202],{},[31749,752313,459],{},[31665,752315,752316],{"encoding":31667},"(\\varphi_1''' + 5\\varphi_1'' + 6\\varphi_1') - (\\varphi_2''' + 5\\varphi_2'' + 6\\varphi_2') = f(t) - f(t) = 0",[34,752318,752320,752392,752464,752539,752611,752683,752758,752785,752812],{"className":752319,"ariaHidden":758},[31672],[34,752321,752323,752326,752329,752383,752386,752389],{"className":752322},[31676],[34,752324],{"className":752325,"style":570646},[31680],[34,752327,456],{"className":752328},[31782],[34,752330,752332,752335],{"className":752331},[31685],[34,752333,126541],{"className":752334},[31685,31686],[34,752336,752338],{"className":752337},[31792],[34,752339,752341,752375],{"className":752340},[31796,31797],[34,752342,752344,752372],{"className":752343},[31801],[34,752345,752347,752358],{"className":752346,"style":132027},[31805],[34,752348,752349,752352],{"style":316930},[34,752350],{"className":752351,"style":31814},[31813],[34,752353,752355],{"className":752354},[31818,31819,31820,31821],[34,752356,640],{"className":752357},[31685,31821],[34,752359,752360,752363],{"style":37777},[34,752361],{"className":752362,"style":31814},[31813],[34,752364,752366],{"className":752365},[31818,31819,31820,31821],[34,752367,752369],{"className":752368},[31685,31821],[34,752370,502955],{"className":752371},[31685,31821],[34,752373,31829],{"className":752374},[31828],[34,752376,752378],{"className":752377},[31801],[34,752379,752381],{"className":752380,"style":39021},[31805],[34,752382],{},[34,752384],{"className":752385,"style":32604},[31845],[34,752387,223],{"className":752388},[32024],[34,752390],{"className":752391,"style":32604},[31845],[34,752393,752395,752398,752401,752455,752458,752461],{"className":752394},[31676],[34,752396],{"className":752397,"style":716670},[31680],[34,752399,18136],{"className":752400},[31685],[34,752402,752404,752407],{"className":752403},[31685],[34,752405,126541],{"className":752406},[31685,31686],[34,752408,752410],{"className":752409},[31792],[34,752411,752413,752447],{"className":752412},[31796,31797],[34,752414,752416,752444],{"className":752415},[31801],[34,752417,752419,752430],{"className":752418,"style":132027},[31805],[34,752420,752421,752424],{"style":316930},[34,752422],{"className":752423,"style":31814},[31813],[34,752425,752427],{"className":752426},[31818,31819,31820,31821],[34,752428,640],{"className":752429},[31685,31821],[34,752431,752432,752435],{"style":37777},[34,752433],{"className":752434,"style":31814},[31813],[34,752436,752438],{"className":752437},[31818,31819,31820,31821],[34,752439,752441],{"className":752440},[31685,31821],[34,752442,502782],{"className":752443},[31685,31821],[34,752445,31829],{"className":752446},[31828],[34,752448,752450],{"className":752449},[31801],[34,752451,752453],{"className":752452,"style":39021},[31805],[34,752454],{},[34,752456],{"className":752457,"style":32604},[31845],[34,752459,223],{"className":752460},[32024],[34,752462],{"className":752463,"style":32604},[31845],[34,752465,752467,752470,752473,752527,752530,752533,752536],{"className":752466},[31676],[34,752468],{"className":752469,"style":570646},[31680],[34,752471,36656],{"className":752472},[31685],[34,752474,752476,752479],{"className":752475},[31685],[34,752477,126541],{"className":752478},[31685,31686],[34,752480,752482],{"className":752481},[31792],[34,752483,752485,752519],{"className":752484},[31796,31797],[34,752486,752488,752516],{"className":752487},[31801],[34,752489,752491,752502],{"className":752490,"style":132027},[31805],[34,752492,752493,752496],{"style":316930},[34,752494],{"className":752495,"style":31814},[31813],[34,752497,752499],{"className":752498},[31818,31819,31820,31821],[34,752500,640],{"className":752501},[31685,31821],[34,752503,752504,752507],{"style":37777},[34,752505],{"className":752506,"style":31814},[31813],[34,752508,752510],{"className":752509},[31818,31819,31820,31821],[34,752511,752513],{"className":752512},[31685,31821],[34,752514,116495],{"className":752515},[31685,31821],[34,752517,31829],{"className":752518},[31828],[34,752520,752522],{"className":752521},[31801],[34,752523,752525],{"className":752524,"style":39021},[31805],[34,752526],{},[34,752528,2557],{"className":752529},[31690],[34,752531],{"className":752532,"style":32604},[31845],[34,752534,32226],{"className":752535},[32024],[34,752537],{"className":752538,"style":32604},[31845],[34,752540,752542,752545,752548,752602,752605,752608],{"className":752541},[31676],[34,752543],{"className":752544,"style":570646},[31680],[34,752546,456],{"className":752547},[31782],[34,752549,752551,752554],{"className":752550},[31685],[34,752552,126541],{"className":752553},[31685,31686],[34,752555,752557],{"className":752556},[31792],[34,752558,752560,752594],{"className":752559},[31796,31797],[34,752561,752563,752591],{"className":752562},[31801],[34,752564,752566,752577],{"className":752565,"style":132027},[31805],[34,752567,752568,752571],{"style":316930},[34,752569],{"className":752570,"style":31814},[31813],[34,752572,752574],{"className":752573},[31818,31819,31820,31821],[34,752575,768],{"className":752576},[31685,31821],[34,752578,752579,752582],{"style":37777},[34,752580],{"className":752581,"style":31814},[31813],[34,752583,752585],{"className":752584},[31818,31819,31820,31821],[34,752586,752588],{"className":752587},[31685,31821],[34,752589,502955],{"className":752590},[31685,31821],[34,752592,31829],{"className":752593},[31828],[34,752595,752597],{"className":752596},[31801],[34,752598,752600],{"className":752599,"style":39021},[31805],[34,752601],{},[34,752603],{"className":752604,"style":32604},[31845],[34,752606,223],{"className":752607},[32024],[34,752609],{"className":752610,"style":32604},[31845],[34,752612,752614,752617,752620,752674,752677,752680],{"className":752613},[31676],[34,752615],{"className":752616,"style":716670},[31680],[34,752618,18136],{"className":752619},[31685],[34,752621,752623,752626],{"className":752622},[31685],[34,752624,126541],{"className":752625},[31685,31686],[34,752627,752629],{"className":752628},[31792],[34,752630,752632,752666],{"className":752631},[31796,31797],[34,752633,752635,752663],{"className":752634},[31801],[34,752636,752638,752649],{"className":752637,"style":132027},[31805],[34,752639,752640,752643],{"style":316930},[34,752641],{"className":752642,"style":31814},[31813],[34,752644,752646],{"className":752645},[31818,31819,31820,31821],[34,752647,768],{"className":752648},[31685,31821],[34,752650,752651,752654],{"style":37777},[34,752652],{"className":752653,"style":31814},[31813],[34,752655,752657],{"className":752656},[31818,31819,31820,31821],[34,752658,752660],{"className":752659},[31685,31821],[34,752661,502782],{"className":752662},[31685,31821],[34,752664,31829],{"className":752665},[31828],[34,752667,752669],{"className":752668},[31801],[34,752670,752672],{"className":752671,"style":39021},[31805],[34,752673],{},[34,752675],{"className":752676,"style":32604},[31845],[34,752678,223],{"className":752679},[32024],[34,752681],{"className":752682,"style":32604},[31845],[34,752684,752686,752689,752692,752746,752749,752752,752755],{"className":752685},[31676],[34,752687],{"className":752688,"style":570646},[31680],[34,752690,36656],{"className":752691},[31685],[34,752693,752695,752698],{"className":752694},[31685],[34,752696,126541],{"className":752697},[31685,31686],[34,752699,752701],{"className":752700},[31792],[34,752702,752704,752738],{"className":752703},[31796,31797],[34,752705,752707,752735],{"className":752706},[31801],[34,752708,752710,752721],{"className":752709,"style":132027},[31805],[34,752711,752712,752715],{"style":316930},[34,752713],{"className":752714,"style":31814},[31813],[34,752716,752718],{"className":752717},[31818,31819,31820,31821],[34,752719,768],{"className":752720},[31685,31821],[34,752722,752723,752726],{"style":37777},[34,752724],{"className":752725,"style":31814},[31813],[34,752727,752729],{"className":752728},[31818,31819,31820,31821],[34,752730,752732],{"className":752731},[31685,31821],[34,752733,116495],{"className":752734},[31685,31821],[34,752736,31829],{"className":752737},[31828],[34,752739,752741],{"className":752740},[31801],[34,752742,752744],{"className":752743,"style":39021},[31805],[34,752745],{},[34,752747,2557],{"className":752748},[31690],[34,752750],{"className":752751,"style":31846},[31845],[34,752753,202],{"className":752754},[31850],[34,752756],{"className":752757,"style":31846},[31845],[34,752759,752761,752764,752767,752770,752773,752776,752779,752782],{"className":752760},[31676],[34,752762],{"className":752763,"style":31774},[31680],[34,752765,31739],{"className":752766,"style":31778},[31685,31686],[34,752768,456],{"className":752769},[31782],[34,752771,33249],{"className":752772},[31685,31686],[34,752774,2557],{"className":752775},[31690],[34,752777],{"className":752778,"style":32604},[31845],[34,752780,32226],{"className":752781},[32024],[34,752783],{"className":752784,"style":32604},[31845],[34,752786,752788,752791,752794,752797,752800,752803,752806,752809],{"className":752787},[31676],[34,752789],{"className":752790,"style":31774},[31680],[34,752792,31739],{"className":752793,"style":31778},[31685,31686],[34,752795,456],{"className":752796},[31782],[34,752798,33249],{"className":752799},[31685,31686],[34,752801,2557],{"className":752802},[31690],[34,752804],{"className":752805,"style":31846},[31845],[34,752807,202],{"className":752808},[31850],[34,752810],{"className":752811,"style":31846},[31845],[34,752813,752815,752818],{"className":752814},[31676],[34,752816],{"className":752817,"style":32617},[31680],[34,752819,459],{"className":752820},[31685],"\n整理得：\n",[34,752823,752825,752877],{"className":752824},[31642],[34,752826,752828],{"className":752827},[31646],[31648,752829,752830],{"xmlns":31650},[31652,752831,752832,752874],{},[31655,752833,752834,752846,752848,752850,752860,752862,752864,752870,752872],{},[37624,752835,752836,752838],{},[31658,752837,67764],{},[31655,752839,752840,752842,752844],{},[31662,752841,116495],{"mathvariant":38079},[31662,752843,116495],{"mathvariant":38079},[31662,752845,116495],{"mathvariant":38079},[31662,752847,223],{},[31749,752849,18136],{},[37624,752851,752852,752854],{},[31658,752853,67764],{},[31655,752855,752856,752858],{},[31662,752857,116495],{"mathvariant":38079},[31662,752859,116495],{"mathvariant":38079},[31662,752861,223],{},[31749,752863,36656],{},[37624,752865,752866,752868],{},[31658,752867,67764],{},[31662,752869,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,752871,202],{},[31749,752873,459],{},[31665,752875,752876],{"encoding":31667},"y''' + 5y'' + 6y' = 0",[34,752878,752880,752927,752977,753027],{"className":752879,"ariaHidden":758},[31672],[34,752881,752883,752886,752918,752921,752924],{"className":752882},[31676],[34,752884],{"className":752885,"style":395485},[31680],[34,752887,752889,752892],{"className":752888},[31685],[34,752890,67764],{"className":752891,"style":32062},[31685,31686],[34,752893,752895],{"className":752894},[31792],[34,752896,752898],{"className":752897},[31796],[34,752899,752901],{"className":752900},[31801],[34,752902,752904],{"className":752903,"style":132027},[31805],[34,752905,752906,752909],{"style":37777},[34,752907],{"className":752908,"style":31814},[31813],[34,752910,752912],{"className":752911},[31818,31819,31820,31821],[34,752913,752915],{"className":752914},[31685,31821],[34,752916,502955],{"className":752917},[31685,31821],[34,752919],{"className":752920,"style":32604},[31845],[34,752922,223],{"className":752923},[32024],[34,752925],{"className":752926,"style":32604},[31845],[34,752928,752930,752933,752936,752968,752971,752974],{"className":752929},[31676],[34,752931],{"className":752932,"style":395485},[31680],[34,752934,18136],{"className":752935},[31685],[34,752937,752939,752942],{"className":752938},[31685],[34,752940,67764],{"className":752941,"style":32062},[31685,31686],[34,752943,752945],{"className":752944},[31792],[34,752946,752948],{"className":752947},[31796],[34,752949,752951],{"className":752950},[31801],[34,752952,752954],{"className":752953,"style":132027},[31805],[34,752955,752956,752959],{"style":37777},[34,752957],{"className":752958,"style":31814},[31813],[34,752960,752962],{"className":752961},[31818,31819,31820,31821],[34,752963,752965],{"className":752964},[31685,31821],[34,752966,502782],{"className":752967},[31685,31821],[34,752969],{"className":752970,"style":32604},[31845],[34,752972,223],{"className":752973},[32024],[34,752975],{"className":752976,"style":32604},[31845],[34,752978,752980,752983,752986,753018,753021,753024],{"className":752979},[31676],[34,752981],{"className":752982,"style":395485},[31680],[34,752984,36656],{"className":752985},[31685],[34,752987,752989,752992],{"className":752988},[31685],[34,752990,67764],{"className":752991,"style":32062},[31685,31686],[34,752993,752995],{"className":752994},[31792],[34,752996,752998],{"className":752997},[31796],[34,752999,753001],{"className":753000},[31801],[34,753002,753004],{"className":753003,"style":132027},[31805],[34,753005,753006,753009],{"style":37777},[34,753007],{"className":753008,"style":31814},[31813],[34,753010,753012],{"className":753011},[31818,31819,31820,31821],[34,753013,753015],{"className":753014},[31685,31821],[34,753016,116495],{"className":753017},[31685,31821],[34,753019],{"className":753020,"style":31846},[31845],[34,753022,202],{"className":753023},[31850],[34,753025],{"className":753026,"style":31846},[31845],[34,753028,753030,753033],{"className":753029},[31676],[34,753031],{"className":753032,"style":32617},[31680],[34,753034,459],{"className":753035},[31685],"\n这说明 ",[2627,753038,753039,753040,753084],{},"差函数 ",[34,753041,753043,753063],{"className":753042},[31642],[34,753044,753046],{"className":753045},[31646],[31648,753047,753048],{"xmlns":31650},[31652,753049,753050,753060],{},[31655,753051,753052,753054,753056,753058],{},[31658,753053,67764],{},[31662,753055,456],{"stretchy":702},[31658,753057,33249],{},[31662,753059,2557],{"stretchy":702},[31665,753061,753062],{"encoding":31667},"y(t)",[34,753064,753066],{"className":753065,"ariaHidden":758},[31672],[34,753067,753069,753072,753075,753078,753081],{"className":753068},[31676],[34,753070],{"className":753071,"style":31774},[31680],[34,753073,67764],{"className":753074,"style":32062},[31685,31686],[34,753076,456],{"className":753077},[31782],[34,753079,33249],{"className":753080},[31685,31686],[34,753082,2557],{"className":753083},[31690]," 满足对应的齐次方程",[108,753086,753087,753088,753301,753302,753353],{},"现在问题转化为：求齐次方程 ",[34,753089,753091,753142],{"className":753090},[31642],[34,753092,753094],{"className":753093},[31646],[31648,753095,753096],{"xmlns":31650},[31652,753097,753098,753140],{},[31655,753099,753100,753112,753114,753116,753126,753128,753130,753136,753138],{},[37624,753101,753102,753104],{},[31658,753103,67764],{},[31655,753105,753106,753108,753110],{},[31662,753107,116495],{"mathvariant":38079},[31662,753109,116495],{"mathvariant":38079},[31662,753111,116495],{"mathvariant":38079},[31662,753113,223],{},[31749,753115,18136],{},[37624,753117,753118,753120],{},[31658,753119,67764],{},[31655,753121,753122,753124],{},[31662,753123,116495],{"mathvariant":38079},[31662,753125,116495],{"mathvariant":38079},[31662,753127,223],{},[31749,753129,36656],{},[37624,753131,753132,753134],{},[31658,753133,67764],{},[31662,753135,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,753137,202],{},[31749,753139,459],{},[31665,753141,752876],{"encoding":31667},[34,753143,753145,753192,753242,753292],{"className":753144,"ariaHidden":758},[31672],[34,753146,753148,753151,753183,753186,753189],{"className":753147},[31676],[34,753149],{"className":753150,"style":395485},[31680],[34,753152,753154,753157],{"className":753153},[31685],[34,753155,67764],{"className":753156,"style":32062},[31685,31686],[34,753158,753160],{"className":753159},[31792],[34,753161,753163],{"className":753162},[31796],[34,753164,753166],{"className":753165},[31801],[34,753167,753169],{"className":753168,"style":132027},[31805],[34,753170,753171,753174],{"style":37777},[34,753172],{"className":753173,"style":31814},[31813],[34,753175,753177],{"className":753176},[31818,31819,31820,31821],[34,753178,753180],{"className":753179},[31685,31821],[34,753181,502955],{"className":753182},[31685,31821],[34,753184],{"className":753185,"style":32604},[31845],[34,753187,223],{"className":753188},[32024],[34,753190],{"className":753191,"style":32604},[31845],[34,753193,753195,753198,753201,753233,753236,753239],{"className":753194},[31676],[34,753196],{"className":753197,"style":395485},[31680],[34,753199,18136],{"className":753200},[31685],[34,753202,753204,753207],{"className":753203},[31685],[34,753205,67764],{"className":753206,"style":32062},[31685,31686],[34,753208,753210],{"className":753209},[31792],[34,753211,753213],{"className":753212},[31796],[34,753214,753216],{"className":753215},[31801],[34,753217,753219],{"className":753218,"style":132027},[31805],[34,753220,753221,753224],{"style":37777},[34,753222],{"className":753223,"style":31814},[31813],[34,753225,753227],{"className":753226},[31818,31819,31820,31821],[34,753228,753230],{"className":753229},[31685,31821],[34,753231,502782],{"className":753232},[31685,31821],[34,753234],{"className":753235,"style":32604},[31845],[34,753237,223],{"className":753238},[32024],[34,753240],{"className":753241,"style":32604},[31845],[34,753243,753245,753248,753251,753283,753286,753289],{"className":753244},[31676],[34,753246],{"className":753247,"style":395485},[31680],[34,753249,36656],{"className":753250},[31685],[34,753252,753254,753257],{"className":753253},[31685],[34,753255,67764],{"className":753256,"style":32062},[31685,31686],[34,753258,753260],{"className":753259},[31792],[34,753261,753263],{"className":753262},[31796],[34,753264,753266],{"className":753265},[31801],[34,753267,753269],{"className":753268,"style":132027},[31805],[34,753270,753271,753274],{"style":37777},[34,753272],{"className":753273,"style":31814},[31813],[34,753275,753277],{"className":753276},[31818,31819,31820,31821],[34,753278,753280],{"className":753279},[31685,31821],[34,753281,116495],{"className":753282},[31685,31821],[34,753284],{"className":753285,"style":31846},[31845],[34,753287,202],{"className":753288},[31850],[34,753290],{"className":753291,"style":31846},[31845],[34,753293,753295,753298],{"className":753294},[31676],[34,753296],{"className":753297,"style":32617},[31680],[34,753299,459],{"className":753300},[31685]," 的通解，并看它在 ",[34,753303,753305,753323],{"className":753304},[31642],[34,753306,753308],{"className":753307},[31646],[31648,753309,753310],{"xmlns":31650},[31652,753311,753312,753320],{},[31655,753313,753314,753316,753318],{},[31658,753315,33249],{},[31662,753317,33239],{},[31658,753319,138286],{"mathvariant":38079},[31665,753321,753322],{"encoding":31667},"t \\to \\infty",[34,753324,753326,753344],{"className":753325,"ariaHidden":758},[31672],[34,753327,753329,753332,753335,753338,753341],{"className":753328},[31676],[34,753330],{"className":753331,"style":33280},[31680],[34,753333,33249],{"className":753334},[31685,31686],[34,753336],{"className":753337,"style":31846},[31845],[34,753339,33239],{"className":753340},[31850],[34,753342],{"className":753343,"style":31846},[31845],[34,753345,753347,753350],{"className":753346},[31676],[34,753348],{"className":753349,"style":32527},[31680],[34,753351,138286],{"className":753352},[31685]," 时的极限。",[108,753355,753356,753357,753522,753523,33183,753551,753696,748161,753821,347],{},"特征方程：\n",[34,753358,753360,753398],{"className":753359},[31642],[34,753361,753363],{"className":753362},[31646],[31648,753364,753365],{"xmlns":31650},[31652,753366,753367,753395],{},[31655,753368,753369,753375,753377,753379,753385,753387,753389,753391,753393],{},[37624,753370,753371,753373],{},[31658,753372,220792],{},[31749,753374,786],{},[31662,753376,223],{},[31749,753378,18136],{},[37624,753380,753381,753383],{},[31658,753382,220792],{},[31749,753384,768],{},[31662,753386,223],{},[31749,753388,36656],{},[31658,753390,220792],{},[31662,753392,202],{},[31749,753394,459],{},[31665,753396,753397],{"encoding":31667},"\\lambda^3 + 5\\lambda^2 + 6\\lambda = 0",[34,753399,753401,753445,753492,753513],{"className":753400,"ariaHidden":758},[31672],[34,753402,753404,753407,753436,753439,753442],{"className":753403},[31676],[34,753405],{"className":753406,"style":40044},[31680],[34,753408,753410,753413],{"className":753409},[31685],[34,753411,220792],{"className":753412},[31685,31686],[34,753414,753416],{"className":753415},[31792],[34,753417,753419],{"className":753418},[31796],[34,753420,753422],{"className":753421},[31801],[34,753423,753425],{"className":753424,"style":37774},[31805],[34,753426,753427,753430],{"style":37777},[34,753428],{"className":753429,"style":31814},[31813],[34,753431,753433],{"className":753432},[31818,31819,31820,31821],[34,753434,786],{"className":753435},[31685,31821],[34,753437],{"className":753438,"style":32604},[31845],[34,753440,223],{"className":753441},[32024],[34,753443],{"className":753444,"style":32604},[31845],[34,753446,753448,753451,753454,753483,753486,753489],{"className":753447},[31676],[34,753449],{"className":753450,"style":40044},[31680],[34,753452,18136],{"className":753453},[31685],[34,753455,753457,753460],{"className":753456},[31685],[34,753458,220792],{"className":753459},[31685,31686],[34,753461,753463],{"className":753462},[31792],[34,753464,753466],{"className":753465},[31796],[34,753467,753469],{"className":753468},[31801],[34,753470,753472],{"className":753471,"style":37774},[31805],[34,753473,753474,753477],{"style":37777},[34,753475],{"className":753476,"style":31814},[31813],[34,753478,753480],{"className":753479},[31818,31819,31820,31821],[34,753481,768],{"className":753482},[31685,31821],[34,753484],{"className":753485,"style":32604},[31845],[34,753487,223],{"className":753488},[32024],[34,753490],{"className":753491,"style":32604},[31845],[34,753493,753495,753498,753501,753504,753507,753510],{"className":753494},[31676],[34,753496],{"className":753497,"style":31681},[31680],[34,753499,36656],{"className":753500},[31685],[34,753502,220792],{"className":753503},[31685,31686],[34,753505],{"className":753506,"style":31846},[31845],[34,753508,202],{"className":753509},[31850],[34,753511],{"className":753512,"style":31846},[31845],[34,753514,753516,753519],{"className":753515},[31676],[34,753517],{"className":753518,"style":32617},[31680],[34,753520,459],{"className":753521},[31685],"\n提取公因式 ",[34,753524,753526,753539],{"className":753525},[31642],[34,753527,753529],{"className":753528},[31646],[31648,753530,753531],{"xmlns":31650},[31652,753532,753533,753537],{},[31655,753534,753535],{},[31658,753536,220792],{},[31665,753538,220795],{"encoding":31667},[34,753540,753542],{"className":753541,"ariaHidden":758},[31672],[34,753543,753545,753548],{"className":753544},[31676],[34,753546],{"className":753547,"style":31681},[31680],[34,753549,220792],{"className":753550},[31685,31686],[34,753552,753554,753592],{"className":753553},[31642],[34,753555,753557],{"className":753556},[31646],[31648,753558,753559],{"xmlns":31650},[31652,753560,753561,753589],{},[31655,753562,753563,753565,753567,753573,753575,753577,753579,753581,753583,753585,753587],{},[31658,753564,220792],{},[31662,753566,456],{"stretchy":702},[37624,753568,753569,753571],{},[31658,753570,220792],{},[31749,753572,768],{},[31662,753574,223],{},[31749,753576,18136],{},[31658,753578,220792],{},[31662,753580,223],{},[31749,753582,36656],{},[31662,753584,2557],{"stretchy":702},[31662,753586,202],{},[31749,753588,459],{},[31665,753590,753591],{"encoding":31667},"\\lambda(\\lambda^2 + 5\\lambda + 6) = 0",[34,753593,753595,753645,753666,753687],{"className":753594,"ariaHidden":758},[31672],[34,753596,753598,753601,753604,753607,753636,753639,753642],{"className":753597},[31676],[34,753599],{"className":753600,"style":37755},[31680],[34,753602,220792],{"className":753603},[31685,31686],[34,753605,456],{"className":753606},[31782],[34,753608,753610,753613],{"className":753609},[31685],[34,753611,220792],{"className":753612},[31685,31686],[34,753614,753616],{"className":753615},[31792],[34,753617,753619],{"className":753618},[31796],[34,753620,753622],{"className":753621},[31801],[34,753623,753625],{"className":753624,"style":37774},[31805],[34,753626,753627,753630],{"style":37777},[34,753628],{"className":753629,"style":31814},[31813],[34,753631,753633],{"className":753632},[31818,31819,31820,31821],[34,753634,768],{"className":753635},[31685,31821],[34,753637],{"className":753638,"style":32604},[31845],[34,753640,223],{"className":753641},[32024],[34,753643],{"className":753644,"style":32604},[31845],[34,753646,753648,753651,753654,753657,753660,753663],{"className":753647},[31676],[34,753649],{"className":753650,"style":34299},[31680],[34,753652,18136],{"className":753653},[31685],[34,753655,220792],{"className":753656},[31685,31686],[34,753658],{"className":753659,"style":32604},[31845],[34,753661,223],{"className":753662},[32024],[34,753664],{"className":753665,"style":32604},[31845],[34,753667,753669,753672,753675,753678,753681,753684],{"className":753668},[31676],[34,753670],{"className":753671,"style":31774},[31680],[34,753673,36656],{"className":753674},[31685],[34,753676,2557],{"className":753677},[31690],[34,753679],{"className":753680,"style":31846},[31845],[34,753682,202],{"className":753683},[31850],[34,753685],{"className":753686,"style":31846},[31845],[34,753688,753690,753693],{"className":753689},[31676],[34,753691],{"className":753692,"style":32617},[31680],[34,753694,459],{"className":753695},[31685],[34,753697,753699,753737],{"className":753698},[31642],[34,753700,753702],{"className":753701},[31646],[31648,753703,753704],{"xmlns":31650},[31652,753705,753706,753734],{},[31655,753707,753708,753710,753712,753714,753716,753718,753720,753722,753724,753726,753728,753730,753732],{},[31658,753709,220792],{},[31662,753711,456],{"stretchy":702},[31658,753713,220792],{},[31662,753715,223],{},[31749,753717,768],{},[31662,753719,2557],{"stretchy":702},[31662,753721,456],{"stretchy":702},[31658,753723,220792],{},[31662,753725,223],{},[31749,753727,786],{},[31662,753729,2557],{"stretchy":702},[31662,753731,202],{},[31749,753733,459],{},[31665,753735,753736],{"encoding":31667},"\\lambda(\\lambda + 2)(\\lambda + 3) = 0",[34,753738,753740,753764,753791,753812],{"className":753739,"ariaHidden":758},[31672],[34,753741,753743,753746,753749,753752,753755,753758,753761],{"className":753742},[31676],[34,753744],{"className":753745,"style":31774},[31680],[34,753747,220792],{"className":753748},[31685,31686],[34,753750,456],{"className":753751},[31782],[34,753753,220792],{"className":753754},[31685,31686],[34,753756],{"className":753757,"style":32604},[31845],[34,753759,223],{"className":753760},[32024],[34,753762],{"className":753763,"style":32604},[31845],[34,753765,753767,753770,753773,753776,753779,753782,753785,753788],{"className":753766},[31676],[34,753768],{"className":753769,"style":31774},[31680],[34,753771,768],{"className":753772},[31685],[34,753774,2557],{"className":753775},[31690],[34,753777,456],{"className":753778},[31782],[34,753780,220792],{"className":753781},[31685,31686],[34,753783],{"className":753784,"style":32604},[31845],[34,753786,223],{"className":753787},[32024],[34,753789],{"className":753790,"style":32604},[31845],[34,753792,753794,753797,753800,753803,753806,753809],{"className":753793},[31676],[34,753795],{"className":753796,"style":31774},[31680],[34,753798,786],{"className":753799},[31685],[34,753801,2557],{"className":753802},[31690],[34,753804],{"className":753805,"style":31846},[31845],[34,753807,202],{"className":753808},[31850],[34,753810],{"className":753811,"style":31846},[31845],[34,753813,753815,753818],{"className":753814},[31676],[34,753816],{"className":753817,"style":32617},[31680],[34,753819,459],{"className":753820},[31685],[34,753822,753824,753878],{"className":753823},[31642],[34,753825,753827],{"className":753826},[31646],[31648,753828,753829],{"xmlns":31650},[31652,753830,753831,753875],{},[31655,753832,753833,753839,753841,753843,753845,753847,753853,753855,753857,753859,753861,753863,753869,753871,753873],{},[31743,753834,753835,753837],{},[31658,753836,220792],{},[31749,753838,640],{},[31662,753840,202],{},[31749,753842,459],{},[31662,753844,31960],{"separator":758},[31845,753846],{"width":71082},[31743,753848,753849,753851],{},[31658,753850,220792],{},[31749,753852,768],{},[31662,753854,202],{},[31662,753856,32226],{},[31749,753858,768],{},[31662,753860,31960],{"separator":758},[31845,753862],{"width":71082},[31743,753864,753865,753867],{},[31658,753866,220792],{},[31749,753868,786],{},[31662,753870,202],{},[31662,753872,32226],{},[31749,753874,786],{},[31665,753876,753877],{"encoding":31667},"\\lambda_1 = 0, \\quad \\lambda_2 = -2, \\quad \\lambda_3 = -3",[34,753879,753881,753936,754003,754073],{"className":753880,"ariaHidden":758},[31672],[34,753882,753884,753887,753927,753930,753933],{"className":753883},[31676],[34,753885],{"className":753886,"style":33814},[31680],[34,753888,753890,753893],{"className":753889},[31685],[34,753891,220792],{"className":753892},[31685,31686],[34,753894,753896],{"className":753895},[31792],[34,753897,753899,753919],{"className":753898},[31796,31797],[34,753900,753902,753916],{"className":753901},[31801],[34,753903,753905],{"className":753904,"style":31806},[31805],[34,753906,753907,753910],{"style":31809},[34,753908],{"className":753909,"style":31814},[31813],[34,753911,753913],{"className":753912},[31818,31819,31820,31821],[34,753914,640],{"className":753915},[31685,31821],[34,753917,31829],{"className":753918},[31828],[34,753920,753922],{"className":753921},[31801],[34,753923,753925],{"className":753924,"style":31836},[31805],[34,753926],{},[34,753928],{"className":753929,"style":31846},[31845],[34,753931,202],{"className":753932},[31850],[34,753934],{"className":753935,"style":31846},[31845],[34,753937,753939,753942,753945,753948,753951,753954,753994,753997,754000],{"className":753938},[31676],[34,753940],{"className":753941,"style":43217},[31680],[34,753943,459],{"className":753944},[31685],[34,753946,31960],{"className":753947},[32119],[34,753949],{"className":753950,"style":128438},[31845],[34,753952],{"className":753953,"style":32123},[31845],[34,753955,753957,753960],{"className":753956},[31685],[34,753958,220792],{"className":753959},[31685,31686],[34,753961,753963],{"className":753962},[31792],[34,753964,753966,753986],{"className":753965},[31796,31797],[34,753967,753969,753983],{"className":753968},[31801],[34,753970,753972],{"className":753971,"style":31806},[31805],[34,753973,753974,753977],{"style":31809},[34,753975],{"className":753976,"style":31814},[31813],[34,753978,753980],{"className":753979},[31818,31819,31820,31821],[34,753981,768],{"className":753982},[31685,31821],[34,753984,31829],{"className":753985},[31828],[34,753987,753989],{"className":753988},[31801],[34,753990,753992],{"className":753991,"style":31836},[31805],[34,753993],{},[34,753995],{"className":753996,"style":31846},[31845],[34,753998,202],{"className":753999},[31850],[34,754001],{"className":754002,"style":31846},[31845],[34,754004,754006,754009,754012,754015,754018,754021,754024,754064,754067,754070],{"className":754005},[31676],[34,754007],{"className":754008,"style":43217},[31680],[34,754010,32226],{"className":754011},[31685],[34,754013,768],{"className":754014},[31685],[34,754016,31960],{"className":754017},[32119],[34,754019],{"className":754020,"style":128438},[31845],[34,754022],{"className":754023,"style":32123},[31845],[34,754025,754027,754030],{"className":754026},[31685],[34,754028,220792],{"className":754029},[31685,31686],[34,754031,754033],{"className":754032},[31792],[34,754034,754036,754056],{"className":754035},[31796,31797],[34,754037,754039,754053],{"className":754038},[31801],[34,754040,754042],{"className":754041,"style":31806},[31805],[34,754043,754044,754047],{"style":31809},[34,754045],{"className":754046,"style":31814},[31813],[34,754048,754050],{"className":754049},[31818,31819,31820,31821],[34,754051,786],{"className":754052},[31685,31821],[34,754054,31829],{"className":754055},[31828],[34,754057,754059],{"className":754058},[31801],[34,754060,754062],{"className":754061,"style":31836},[31805],[34,754063],{},[34,754065],{"className":754066,"style":31846},[31845],[34,754068,202],{"className":754069},[31850],[34,754071],{"className":754072,"style":31846},[31845],[34,754074,754076,754079,754082],{"className":754075},[31676],[34,754077],{"className":754078,"style":35942},[31680],[34,754080,32226],{"className":754081},[31685],[34,754083,786],{"className":754084},[31685],[108,754086,754087,754088,754221,754222,754600],{},"齐次方程的解（即 ",[34,754089,754091,754117],{"className":754090},[31642],[34,754092,754094],{"className":754093},[31646],[31648,754095,754096],{"xmlns":31650},[31652,754097,754098,754114],{},[31655,754099,754100,754106,754108],{},[31743,754101,754102,754104],{},[31658,754103,126541],{},[31749,754105,640],{},[31662,754107,32226],{},[31743,754109,754110,754112],{},[31658,754111,126541],{},[31749,754113,768],{},[31665,754115,754116],{"encoding":31667},"\\varphi_1 - \\varphi_2",[34,754118,754120,754175],{"className":754119,"ariaHidden":758},[31672],[34,754121,754123,754126,754166,754169,754172],{"className":754122},[31676],[34,754124],{"className":754125,"style":55949},[31680],[34,754127,754129,754132],{"className":754128},[31685],[34,754130,126541],{"className":754131},[31685,31686],[34,754133,754135],{"className":754134},[31792],[34,754136,754138,754158],{"className":754137},[31796,31797],[34,754139,754141,754155],{"className":754140},[31801],[34,754142,754144],{"className":754143,"style":31806},[31805],[34,754145,754146,754149],{"style":31809},[34,754147],{"className":754148,"style":31814},[31813],[34,754150,754152],{"className":754151},[31818,31819,31820,31821],[34,754153,640],{"className":754154},[31685,31821],[34,754156,31829],{"className":754157},[31828],[34,754159,754161],{"className":754160},[31801],[34,754162,754164],{"className":754163,"style":31836},[31805],[34,754165],{},[34,754167],{"className":754168,"style":32604},[31845],[34,754170,32226],{"className":754171},[32024],[34,754173],{"className":754174,"style":32604},[31845],[34,754176,754178,754181],{"className":754177},[31676],[34,754179],{"className":754180,"style":38774},[31680],[34,754182,754184,754187],{"className":754183},[31685],[34,754185,126541],{"className":754186},[31685,31686],[34,754188,754190],{"className":754189},[31792],[34,754191,754193,754213],{"className":754192},[31796,31797],[34,754194,754196,754210],{"className":754195},[31801],[34,754197,754199],{"className":754198,"style":31806},[31805],[34,754200,754201,754204],{"style":31809},[34,754202],{"className":754203,"style":31814},[31813],[34,754205,754207],{"className":754206},[31818,31819,31820,31821],[34,754208,768],{"className":754209},[31685,31821],[34,754211,31829],{"className":754212},[31828],[34,754214,754216],{"className":754215},[31801],[34,754217,754219],{"className":754218,"style":31836},[31805],[34,754220],{}," 的表达式）：\n",[34,754223,754225,754303],{"className":754224},[31642],[34,754226,754228],{"className":754227},[31646],[31648,754229,754230],{"xmlns":31650},[31652,754231,754232,754300],{},[31655,754233,754234,754236,754238,754240,754242,754244,754250,754260,754262,754268,754280,754282,754288],{},[31658,754235,67764],{},[31662,754237,456],{"stretchy":702},[31658,754239,33249],{},[31662,754241,2557],{"stretchy":702},[31662,754243,202],{},[31743,754245,754246,754248],{},[31658,754247,31965],{},[31749,754249,640],{},[37624,754251,754252,754254],{},[31658,754253,33244],{"mathvariant":38079},[31655,754255,754256,754258],{},[31749,754257,459],{},[31658,754259,33249],{},[31662,754261,223],{},[31743,754263,754264,754266],{},[31658,754265,31965],{},[31749,754267,768],{},[37624,754269,754270,754272],{},[31658,754271,33244],{"mathvariant":38079},[31655,754273,754274,754276,754278],{},[31662,754275,32226],{},[31749,754277,768],{},[31658,754279,33249],{},[31662,754281,223],{},[31743,754283,754284,754286],{},[31658,754285,31965],{},[31749,754287,786],{},[37624,754289,754290,754292],{},[31658,754291,33244],{"mathvariant":38079},[31655,754293,754294,754296,754298],{},[31662,754295,32226],{},[31749,754297,786],{},[31658,754299,33249],{},[31665,754301,754302],{"encoding":31667},"y(t) = c_1 \\mathrm{e}^{0t} + c_2 \\mathrm{e}^{-2t} + c_3 \\mathrm{e}^{-3t}",[34,754304,754306,754333,754423,754516],{"className":754305,"ariaHidden":758},[31672],[34,754307,754309,754312,754315,754318,754321,754324,754327,754330],{"className":754308},[31676],[34,754310],{"className":754311,"style":31774},[31680],[34,754313,67764],{"className":754314,"style":32062},[31685,31686],[34,754316,456],{"className":754317},[31782],[34,754319,33249],{"className":754320},[31685,31686],[34,754322,2557],{"className":754323},[31690],[34,754325],{"className":754326,"style":31846},[31845],[34,754328,202],{"className":754329},[31850],[34,754331],{"className":754332,"style":31846},[31845],[34,754334,754336,754339,754379,754414,754417,754420],{"className":754335},[31676],[34,754337],{"className":754338,"style":744441},[31680],[34,754340,754342,754345],{"className":754341},[31685],[34,754343,31965],{"className":754344},[31685,31686],[34,754346,754348],{"className":754347},[31792],[34,754349,754351,754371],{"className":754350},[31796,31797],[34,754352,754354,754368],{"className":754353},[31801],[34,754355,754357],{"className":754356,"style":31806},[31805],[34,754358,754359,754362],{"style":31809},[34,754360],{"className":754361,"style":31814},[31813],[34,754363,754365],{"className":754364},[31818,31819,31820,31821],[34,754366,640],{"className":754367},[31685,31821],[34,754369,31829],{"className":754370},[31828],[34,754372,754374],{"className":754373},[31801],[34,754375,754377],{"className":754376,"style":31836},[31805],[34,754378],{},[34,754380,754382,754385],{"className":754381},[31685],[34,754383,33244],{"className":754384},[31685,207923],[34,754386,754388],{"className":754387},[31792],[34,754389,754391],{"className":754390},[31796],[34,754392,754394],{"className":754393},[31801],[34,754395,754397],{"className":754396,"style":37774},[31805],[34,754398,754399,754402],{"style":37777},[34,754400],{"className":754401,"style":31814},[31813],[34,754403,754405],{"className":754404},[31818,31819,31820,31821],[34,754406,754408,754411],{"className":754407},[31685,31821],[34,754409,459],{"className":754410},[31685,31821],[34,754412,33249],{"className":754413},[31685,31686,31821],[34,754415],{"className":754416,"style":32604},[31845],[34,754418,223],{"className":754419},[32024],[34,754421],{"className":754422,"style":32604},[31845],[34,754424,754426,754429,754469,754507,754510,754513],{"className":754425},[31676],[34,754427],{"className":754428,"style":744441},[31680],[34,754430,754432,754435],{"className":754431},[31685],[34,754433,31965],{"className":754434},[31685,31686],[34,754436,754438],{"className":754437},[31792],[34,754439,754441,754461],{"className":754440},[31796,31797],[34,754442,754444,754458],{"className":754443},[31801],[34,754445,754447],{"className":754446,"style":31806},[31805],[34,754448,754449,754452],{"style":31809},[34,754450],{"className":754451,"style":31814},[31813],[34,754453,754455],{"className":754454},[31818,31819,31820,31821],[34,754456,768],{"className":754457},[31685,31821],[34,754459,31829],{"className":754460},[31828],[34,754462,754464],{"className":754463},[31801],[34,754465,754467],{"className":754466,"style":31836},[31805],[34,754468],{},[34,754470,754472,754475],{"className":754471},[31685],[34,754473,33244],{"className":754474},[31685,207923],[34,754476,754478],{"className":754477},[31792],[34,754479,754481],{"className":754480},[31796],[34,754482,754484],{"className":754483},[31801],[34,754485,754487],{"className":754486,"style":37774},[31805],[34,754488,754489,754492],{"style":37777},[34,754490],{"className":754491,"style":31814},[31813],[34,754493,754495],{"className":754494},[31818,31819,31820,31821],[34,754496,754498,754501,754504],{"className":754497},[31685,31821],[34,754499,32226],{"className":754500},[31685,31821],[34,754502,768],{"className":754503},[31685,31821],[34,754505,33249],{"className":754506},[31685,31686,31821],[34,754508],{"className":754509,"style":32604},[31845],[34,754511,223],{"className":754512},[32024],[34,754514],{"className":754515,"style":32604},[31845],[34,754517,754519,754522,754562],{"className":754518},[31676],[34,754520],{"className":754521,"style":744441},[31680],[34,754523,754525,754528],{"className":754524},[31685],[34,754526,31965],{"className":754527},[31685,31686],[34,754529,754531],{"className":754530},[31792],[34,754532,754534,754554],{"className":754533},[31796,31797],[34,754535,754537,754551],{"className":754536},[31801],[34,754538,754540],{"className":754539,"style":31806},[31805],[34,754541,754542,754545],{"style":31809},[34,754543],{"className":754544,"style":31814},[31813],[34,754546,754548],{"className":754547},[31818,31819,31820,31821],[34,754549,786],{"className":754550},[31685,31821],[34,754552,31829],{"className":754553},[31828],[34,754555,754557],{"className":754556},[31801],[34,754558,754560],{"className":754559,"style":31836},[31805],[34,754561],{},[34,754563,754565,754568],{"className":754564},[31685],[34,754566,33244],{"className":754567},[31685,207923],[34,754569,754571],{"className":754570},[31792],[34,754572,754574],{"className":754573},[31796],[34,754575,754577],{"className":754576},[31801],[34,754578,754580],{"className":754579,"style":37774},[31805],[34,754581,754582,754585],{"style":37777},[34,754583],{"className":754584,"style":31814},[31813],[34,754586,754588],{"className":754587},[31818,31819,31820,31821],[34,754589,754591,754594,754597],{"className":754590},[31685,31821],[34,754592,32226],{"className":754593},[31685,31821],[34,754595,786],{"className":754596},[31685,31821],[34,754598,33249],{"className":754599},[31685,31686,31821],[34,754601,754603,754671],{"className":754602},[31642],[34,754604,754606],{"className":754605},[31646],[31648,754607,754608],{"xmlns":31650},[31652,754609,754610,754668],{},[31655,754611,754612,754614,754616,754618,754620,754622,754628,754630,754636,754648,754650,754656],{},[31658,754613,67764],{},[31662,754615,456],{"stretchy":702},[31658,754617,33249],{},[31662,754619,2557],{"stretchy":702},[31662,754621,202],{},[31743,754623,754624,754626],{},[31658,754625,31965],{},[31749,754627,640],{},[31662,754629,223],{},[31743,754631,754632,754634],{},[31658,754633,31965],{},[31749,754635,768],{},[37624,754637,754638,754640],{},[31658,754639,33244],{"mathvariant":38079},[31655,754641,754642,754644,754646],{},[31662,754643,32226],{},[31749,754645,768],{},[31658,754647,33249],{},[31662,754649,223],{},[31743,754651,754652,754654],{},[31658,754653,31965],{},[31749,754655,786],{},[37624,754657,754658,754660],{},[31658,754659,33244],{"mathvariant":38079},[31655,754661,754662,754664,754666],{},[31662,754663,32226],{},[31749,754665,786],{},[31658,754667,33249],{},[31665,754669,754670],{"encoding":31667},"y(t) = c_1 + c_2 \\mathrm{e}^{-2t} + c_3 \\mathrm{e}^{-3t}",[34,754672,754674,754701,754756,754849],{"className":754673,"ariaHidden":758},[31672],[34,754675,754677,754680,754683,754686,754689,754692,754695,754698],{"className":754676},[31676],[34,754678],{"className":754679,"style":31774},[31680],[34,754681,67764],{"className":754682,"style":32062},[31685,31686],[34,754684,456],{"className":754685},[31782],[34,754687,33249],{"className":754688},[31685,31686],[34,754690,2557],{"className":754691},[31690],[34,754693],{"className":754694,"style":31846},[31845],[34,754696,202],{"className":754697},[31850],[34,754699],{"className":754700,"style":31846},[31845],[34,754702,754704,754707,754747,754750,754753],{"className":754703},[31676],[34,754705],{"className":754706,"style":40522},[31680],[34,754708,754710,754713],{"className":754709},[31685],[34,754711,31965],{"className":754712},[31685,31686],[34,754714,754716],{"className":754715},[31792],[34,754717,754719,754739],{"className":754718},[31796,31797],[34,754720,754722,754736],{"className":754721},[31801],[34,754723,754725],{"className":754724,"style":31806},[31805],[34,754726,754727,754730],{"style":31809},[34,754728],{"className":754729,"style":31814},[31813],[34,754731,754733],{"className":754732},[31818,31819,31820,31821],[34,754734,640],{"className":754735},[31685,31821],[34,754737,31829],{"className":754738},[31828],[34,754740,754742],{"className":754741},[31801],[34,754743,754745],{"className":754744,"style":31836},[31805],[34,754746],{},[34,754748],{"className":754749,"style":32604},[31845],[34,754751,223],{"className":754752},[32024],[34,754754],{"className":754755,"style":32604},[31845],[34,754757,754759,754762,754802,754840,754843,754846],{"className":754758},[31676],[34,754760],{"className":754761,"style":744441},[31680],[34,754763,754765,754768],{"className":754764},[31685],[34,754766,31965],{"className":754767},[31685,31686],[34,754769,754771],{"className":754770},[31792],[34,754772,754774,754794],{"className":754773},[31796,31797],[34,754775,754777,754791],{"className":754776},[31801],[34,754778,754780],{"className":754779,"style":31806},[31805],[34,754781,754782,754785],{"style":31809},[34,754783],{"className":754784,"style":31814},[31813],[34,754786,754788],{"className":754787},[31818,31819,31820,31821],[34,754789,768],{"className":754790},[31685,31821],[34,754792,31829],{"className":754793},[31828],[34,754795,754797],{"className":754796},[31801],[34,754798,754800],{"className":754799,"style":31836},[31805],[34,754801],{},[34,754803,754805,754808],{"className":754804},[31685],[34,754806,33244],{"className":754807},[31685,207923],[34,754809,754811],{"className":754810},[31792],[34,754812,754814],{"className":754813},[31796],[34,754815,754817],{"className":754816},[31801],[34,754818,754820],{"className":754819,"style":37774},[31805],[34,754821,754822,754825],{"style":37777},[34,754823],{"className":754824,"style":31814},[31813],[34,754826,754828],{"className":754827},[31818,31819,31820,31821],[34,754829,754831,754834,754837],{"className":754830},[31685,31821],[34,754832,32226],{"className":754833},[31685,31821],[34,754835,768],{"className":754836},[31685,31821],[34,754838,33249],{"className":754839},[31685,31686,31821],[34,754841],{"className":754842,"style":32604},[31845],[34,754844,223],{"className":754845},[32024],[34,754847],{"className":754848,"style":32604},[31845],[34,754850,754852,754855,754895],{"className":754851},[31676],[34,754853],{"className":754854,"style":744441},[31680],[34,754856,754858,754861],{"className":754857},[31685],[34,754859,31965],{"className":754860},[31685,31686],[34,754862,754864],{"className":754863},[31792],[34,754865,754867,754887],{"className":754866},[31796,31797],[34,754868,754870,754884],{"className":754869},[31801],[34,754871,754873],{"className":754872,"style":31806},[31805],[34,754874,754875,754878],{"style":31809},[34,754876],{"className":754877,"style":31814},[31813],[34,754879,754881],{"className":754880},[31818,31819,31820,31821],[34,754882,786],{"className":754883},[31685,31821],[34,754885,31829],{"className":754886},[31828],[34,754888,754890],{"className":754889},[31801],[34,754891,754893],{"className":754892,"style":31836},[31805],[34,754894],{},[34,754896,754898,754901],{"className":754897},[31685],[34,754899,33244],{"className":754900},[31685,207923],[34,754902,754904],{"className":754903},[31792],[34,754905,754907],{"className":754906},[31796],[34,754908,754910],{"className":754909},[31801],[34,754911,754913],{"className":754912,"style":37774},[31805],[34,754914,754915,754918],{"style":37777},[34,754916],{"className":754917,"style":31814},[31813],[34,754919,754921],{"className":754920},[31818,31819,31820,31821],[34,754922,754924,754927,754930],{"className":754923},[31685,31821],[34,754925,32226],{"className":754926},[31685,31821],[34,754928,786],{"className":754929},[31685,31821],[34,754931,33249],{"className":754932},[31685,31686,31821],[108,754934,754935,754936,553331,755531,4504,755581,258,755650,755719,755720,756125,756126,351462],{},"求极限：\n",[34,754937,754939,755043],{"className":754938},[31642],[34,754940,754942],{"className":754941},[31646],[31648,754943,754944],{"xmlns":31650},[31652,754945,754946,755040],{},[31655,754947,754948,754964,754966,754968,754970,754972,754974,754990,754992,754998,755000,755018,755020,755038],{},[31743,754949,754950,754956],{},[31655,754951,754952,754954],{},[31658,754953,546745],{},[31662,754955,32682],{},[31655,754957,754958,754960,754962],{},[31658,754959,33249],{},[31662,754961,33239],{},[31658,754963,138286],{"mathvariant":38079},[31658,754965,67764],{},[31662,754967,456],{"stretchy":702},[31658,754969,33249],{},[31662,754971,2557],{"stretchy":702},[31662,754973,202],{},[31743,754975,754976,754982],{},[31655,754977,754978,754980],{},[31658,754979,546745],{},[31662,754981,32682],{},[31655,754983,754984,754986,754988],{},[31658,754985,33249],{},[31662,754987,33239],{},[31658,754989,138286],{"mathvariant":38079},[31662,754991,456],{"stretchy":702},[31743,754993,754994,754996],{},[31658,754995,31965],{},[31749,754997,640],{},[31662,754999,223],{},[41453,755001,755002,755008],{},[31743,755003,755004,755006],{},[31658,755005,31965],{},[31749,755007,768],{},[37624,755009,755010,755012],{},[31658,755011,33244],{"mathvariant":38079},[31655,755013,755014,755016],{},[31749,755015,768],{},[31658,755017,33249],{},[31662,755019,223],{},[41453,755021,755022,755028],{},[31743,755023,755024,755026],{},[31658,755025,31965],{},[31749,755027,786],{},[37624,755029,755030,755032],{},[31658,755031,33244],{"mathvariant":38079},[31655,755033,755034,755036],{},[31749,755035,786],{},[31658,755037,33249],{},[31662,755039,2557],{"stretchy":702},[31665,755041,755042],{"encoding":31667},"\\lim_{t \\to \\infty} y(t) = \\lim_{t \\to \\infty} (c_1 + \\frac{c_2}{\\mathrm{e}^{2t}} + \\frac{c_3}{\\mathrm{e}^{3t}})",[34,755044,755046,755125,755232,755385],{"className":755045,"ariaHidden":758},[31672],[34,755047,755049,755052,755101,755104,755107,755110,755113,755116,755119,755122],{"className":755048},[31676],[34,755050],{"className":755051,"style":31774},[31680],[34,755053,755055,755058],{"className":755054},[32767],[34,755056,546745],{"className":755057},[32767],[34,755059,755061],{"className":755060},[31792],[34,755062,755064,755093],{"className":755063},[31796,31797],[34,755065,755067,755090],{"className":755066},[31801],[34,755068,755070],{"className":755069,"style":206419},[31805],[34,755071,755072,755075],{"style":56235},[34,755073],{"className":755074,"style":31814},[31813],[34,755076,755078],{"className":755077},[31818,31819,31820,31821],[34,755079,755081,755084,755087],{"className":755080},[31685,31821],[34,755082,33249],{"className":755083},[31685,31686,31821],[34,755085,33239],{"className":755086},[31850,31821],[34,755088,138286],{"className":755089},[31685,31821],[34,755091,31829],{"className":755092},[31828],[34,755094,755096],{"className":755095},[31801],[34,755097,755099],{"className":755098,"style":31836},[31805],[34,755100],{},[34,755102],{"className":755103,"style":32123},[31845],[34,755105,67764],{"className":755106,"style":32062},[31685,31686],[34,755108,456],{"className":755109},[31782],[34,755111,33249],{"className":755112},[31685,31686],[34,755114,2557],{"className":755115},[31690],[34,755117],{"className":755118,"style":31846},[31845],[34,755120,202],{"className":755121},[31850],[34,755123],{"className":755124,"style":31846},[31845],[34,755126,755128,755131,755180,755183,755223,755226,755229],{"className":755127},[31676],[34,755129],{"className":755130,"style":31774},[31680],[34,755132,755134,755137],{"className":755133},[32767],[34,755135,546745],{"className":755136},[32767],[34,755138,755140],{"className":755139},[31792],[34,755141,755143,755172],{"className":755142},[31796,31797],[34,755144,755146,755169],{"className":755145},[31801],[34,755147,755149],{"className":755148,"style":206419},[31805],[34,755150,755151,755154],{"style":56235},[34,755152],{"className":755153,"style":31814},[31813],[34,755155,755157],{"className":755156},[31818,31819,31820,31821],[34,755158,755160,755163,755166],{"className":755159},[31685,31821],[34,755161,33249],{"className":755162},[31685,31686,31821],[34,755164,33239],{"className":755165},[31850,31821],[34,755167,138286],{"className":755168},[31685,31821],[34,755170,31829],{"className":755171},[31828],[34,755173,755175],{"className":755174},[31801],[34,755176,755178],{"className":755177,"style":31836},[31805],[34,755179],{},[34,755181,456],{"className":755182},[31782],[34,755184,755186,755189],{"className":755185},[31685],[34,755187,31965],{"className":755188},[31685,31686],[34,755190,755192],{"className":755191},[31792],[34,755193,755195,755215],{"className":755194},[31796,31797],[34,755196,755198,755212],{"className":755197},[31801],[34,755199,755201],{"className":755200,"style":31806},[31805],[34,755202,755203,755206],{"style":31809},[34,755204],{"className":755205,"style":31814},[31813],[34,755207,755209],{"className":755208},[31818,31819,31820,31821],[34,755210,640],{"className":755211},[31685,31821],[34,755213,31829],{"className":755214},[31828],[34,755216,755218],{"className":755217},[31801],[34,755219,755221],{"className":755220,"style":31836},[31805],[34,755222],{},[34,755224],{"className":755225,"style":32604},[31845],[34,755227,223],{"className":755228},[32024],[34,755230],{"className":755231,"style":32604},[31845],[34,755233,755235,755239,755376,755379,755382],{"className":755234},[31676],[34,755236],{"className":755237,"style":755238},[31680],"height:1.0565em;vertical-align:-0.345em;",[34,755240,755242,755245,755373],{"className":755241},[31685],[34,755243],{"className":755244},[31782,41488],[34,755246,755248],{"className":755247},[41453],[34,755249,755251,755365],{"className":755250},[31796,31797],[34,755252,755254,755362],{"className":755253},[31801],[34,755255,755257,755303,755311],{"className":755256,"style":318628},[31805],[34,755258,755259,755262],{"style":41504},[34,755260],{"className":755261,"style":41508},[31813],[34,755263,755265],{"className":755264},[31818,31819,31820,31821],[34,755266,755268],{"className":755267},[31685,31821],[34,755269,755271,755274],{"className":755270},[31685,31821],[34,755272,33244],{"className":755273},[31685,207923,31821],[34,755275,755277],{"className":755276},[31792],[34,755278,755280],{"className":755279},[31796],[34,755281,755283],{"className":755282},[31801],[34,755284,755286],{"className":755285,"style":548314},[31805],[34,755287,755288,755291],{"style":116668},[34,755289],{"className":755290,"style":43406},[31813],[34,755292,755294],{"className":755293},[31818,43410,43411,31821],[34,755295,755297,755300],{"className":755296},[31685,31821],[34,755298,768],{"className":755299},[31685,31821],[34,755301,33249],{"className":755302},[31685,31686,31821],[34,755304,755305,755308],{"style":41520},[34,755306],{"className":755307,"style":41508},[31813],[34,755309],{"className":755310,"style":41528},[41527],[34,755312,755313,755316],{"style":220490},[34,755314],{"className":755315,"style":41508},[31813],[34,755317,755319],{"className":755318},[31818,31819,31820,31821],[34,755320,755322],{"className":755321},[31685,31821],[34,755323,755325,755328],{"className":755324},[31685,31821],[34,755326,31965],{"className":755327},[31685,31686,31821],[34,755329,755331],{"className":755330},[31792],[34,755332,755334,755354],{"className":755333},[31796,31797],[34,755335,755337,755351],{"className":755336},[31801],[34,755338,755340],{"className":755339,"style":318658},[31805],[34,755341,755342,755345],{"style":318661},[34,755343],{"className":755344,"style":43406},[31813],[34,755346,755348],{"className":755347},[31818,43410,43411,31821],[34,755349,768],{"className":755350},[31685,31821],[34,755352,31829],{"className":755353},[31828],[34,755355,755357],{"className":755356},[31801],[34,755358,755360],{"className":755359,"style":220540},[31805],[34,755361],{},[34,755363,31829],{"className":755364},[31828],[34,755366,755368],{"className":755367},[31801],[34,755369,755371],{"className":755370,"style":41619},[31805],[34,755372],{},[34,755374],{"className":755375},[31690,41488],[34,755377],{"className":755378,"style":32604},[31845],[34,755380,223],{"className":755381},[32024],[34,755383],{"className":755384,"style":32604},[31845],[34,755386,755388,755391,755528],{"className":755387},[31676],[34,755389],{"className":755390,"style":631146},[31680],[34,755392,755394,755397,755525],{"className":755393},[31685],[34,755395],{"className":755396},[31782,41488],[34,755398,755400],{"className":755399},[41453],[34,755401,755403,755517],{"className":755402},[31796,31797],[34,755404,755406,755514],{"className":755405},[31801],[34,755407,755409,755455,755463],{"className":755408,"style":318628},[31805],[34,755410,755411,755414],{"style":41504},[34,755412],{"className":755413,"style":41508},[31813],[34,755415,755417],{"className":755416},[31818,31819,31820,31821],[34,755418,755420],{"className":755419},[31685,31821],[34,755421,755423,755426],{"className":755422},[31685,31821],[34,755424,33244],{"className":755425},[31685,207923,31821],[34,755427,755429],{"className":755428},[31792],[34,755430,755432],{"className":755431},[31796],[34,755433,755435],{"className":755434},[31801],[34,755436,755438],{"className":755437,"style":548314},[31805],[34,755439,755440,755443],{"style":116668},[34,755441],{"className":755442,"style":43406},[31813],[34,755444,755446],{"className":755445},[31818,43410,43411,31821],[34,755447,755449,755452],{"className":755448},[31685,31821],[34,755450,786],{"className":755451},[31685,31821],[34,755453,33249],{"className":755454},[31685,31686,31821],[34,755456,755457,755460],{"style":41520},[34,755458],{"className":755459,"style":41508},[31813],[34,755461],{"className":755462,"style":41528},[41527],[34,755464,755465,755468],{"style":220490},[34,755466],{"className":755467,"style":41508},[31813],[34,755469,755471],{"className":755470},[31818,31819,31820,31821],[34,755472,755474],{"className":755473},[31685,31821],[34,755475,755477,755480],{"className":755476},[31685,31821],[34,755478,31965],{"className":755479},[31685,31686,31821],[34,755481,755483],{"className":755482},[31792],[34,755484,755486,755506],{"className":755485},[31796,31797],[34,755487,755489,755503],{"className":755488},[31801],[34,755490,755492],{"className":755491,"style":318658},[31805],[34,755493,755494,755497],{"style":318661},[34,755495],{"className":755496,"style":43406},[31813],[34,755498,755500],{"className":755499},[31818,43410,43411,31821],[34,755501,786],{"className":755502},[31685,31821],[34,755504,31829],{"className":755505},[31828],[34,755507,755509],{"className":755508},[31801],[34,755510,755512],{"className":755511,"style":220540},[31805],[34,755513],{},[34,755515,31829],{"className":755516},[31828],[34,755518,755520],{"className":755519},[31801],[34,755521,755523],{"className":755522,"style":41619},[31805],[34,755524],{},[34,755526],{"className":755527},[31690,41488],[34,755529,2557],{"className":755530},[31690],[34,755532,755534,755551],{"className":755533},[31642],[34,755535,755537],{"className":755536},[31646],[31648,755538,755539],{"xmlns":31650},[31652,755540,755541,755549],{},[31655,755542,755543,755545,755547],{},[31658,755544,33249],{},[31662,755546,33239],{},[31658,755548,138286],{"mathvariant":38079},[31665,755550,753322],{"encoding":31667},[34,755552,755554,755572],{"className":755553,"ariaHidden":758},[31672],[34,755555,755557,755560,755563,755566,755569],{"className":755556},[31676],[34,755558],{"className":755559,"style":33280},[31680],[34,755561,33249],{"className":755562},[31685,31686],[34,755564],{"className":755565,"style":31846},[31845],[34,755567,33239],{"className":755568},[31850],[34,755570],{"className":755571,"style":31846},[31845],[34,755573,755575,755578],{"className":755574},[31676],[34,755576],{"className":755577,"style":32527},[31680],[34,755579,138286],{"className":755580},[31685],[34,755582,755584,755606],{"className":755583},[31642],[34,755585,755587],{"className":755586},[31646],[31648,755588,755589],{"xmlns":31650},[31652,755590,755591,755603],{},[31655,755592,755593],{},[37624,755594,755595,755597],{},[31658,755596,33244],{"mathvariant":38079},[31655,755598,755599,755601],{},[31749,755600,768],{},[31658,755602,33249],{},[31665,755604,755605],{"encoding":31667},"\\mathrm{e}^{2t}",[34,755607,755609],{"className":755608,"ariaHidden":758},[31672],[34,755610,755612,755615],{"className":755611},[31676],[34,755613],{"className":755614,"style":37774},[31680],[34,755616,755618,755621],{"className":755617},[31685],[34,755619,33244],{"className":755620},[31685,207923],[34,755622,755624],{"className":755623},[31792],[34,755625,755627],{"className":755626},[31796],[34,755628,755630],{"className":755629},[31801],[34,755631,755633],{"className":755632,"style":37774},[31805],[34,755634,755635,755638],{"style":37777},[34,755636],{"className":755637,"style":31814},[31813],[34,755639,755641],{"className":755640},[31818,31819,31820,31821],[34,755642,755644,755647],{"className":755643},[31685,31821],[34,755645,768],{"className":755646},[31685,31821],[34,755648,33249],{"className":755649},[31685,31686,31821],[34,755651,755653,755675],{"className":755652},[31642],[34,755654,755656],{"className":755655},[31646],[31648,755657,755658],{"xmlns":31650},[31652,755659,755660,755672],{},[31655,755661,755662],{},[37624,755663,755664,755666],{},[31658,755665,33244],{"mathvariant":38079},[31655,755667,755668,755670],{},[31749,755669,786],{},[31658,755671,33249],{},[31665,755673,755674],{"encoding":31667},"\\mathrm{e}^{3t}",[34,755676,755678],{"className":755677,"ariaHidden":758},[31672],[34,755679,755681,755684],{"className":755680},[31676],[34,755682],{"className":755683,"style":37774},[31680],[34,755685,755687,755690],{"className":755686},[31685],[34,755688,33244],{"className":755689},[31685,207923],[34,755691,755693],{"className":755692},[31792],[34,755694,755696],{"className":755695},[31796],[34,755697,755699],{"className":755698},[31801],[34,755700,755702],{"className":755701,"style":37774},[31805],[34,755703,755704,755707],{"style":37777},[34,755705],{"className":755706,"style":31814},[31813],[34,755708,755710],{"className":755709},[31818,31819,31820,31821],[34,755711,755713,755716],{"className":755712},[31685,31821],[34,755714,786],{"className":755715},[31685,31821],[34,755717,33249],{"className":755718},[31685,31686,31821]," 趋向于无穷大，倒数趋向于0。\n",[34,755721,755723,755805],{"className":755722},[31642],[34,755724,755726],{"className":755725},[31646],[31648,755727,755728],{"xmlns":31650},[31652,755729,755730,755802],{},[31655,755731,755732,755748,755750,755756,755758,755760,755762,755764,755770,755772,755774,755776,755778,755780,755786,755788,755790,755792,755794,755796],{},[31743,755733,755734,755740],{},[31655,755735,755736,755738],{},[31658,755737,546745],{},[31662,755739,32682],{},[31655,755741,755742,755744,755746],{},[31658,755743,33249],{},[31662,755745,33239],{},[31658,755747,138286],{"mathvariant":38079},[31662,755749,15972],{"stretchy":702},[31743,755751,755752,755754],{},[31658,755753,126541],{},[31749,755755,640],{},[31662,755757,456],{"stretchy":702},[31658,755759,33249],{},[31662,755761,2557],{"stretchy":702},[31662,755763,32226],{},[31743,755765,755766,755768],{},[31658,755767,126541],{},[31749,755769,768],{},[31662,755771,456],{"stretchy":702},[31658,755773,33249],{},[31662,755775,2557],{"stretchy":702},[31662,755777,32712],{"stretchy":702},[31662,755779,202],{},[31743,755781,755782,755784],{},[31658,755783,31965],{},[31749,755785,640],{},[31662,755787,223],{},[31749,755789,459],{},[31662,755791,223],{},[31749,755793,459],{},[31662,755795,202],{},[31743,755797,755798,755800],{},[31658,755799,31965],{},[31749,755801,640],{},[31665,755803,755804],{"encoding":31667},"\\lim_{t \\to \\infty} [\\varphi_1(t) - \\varphi_2(t)] = c_1 + 0 + 0 = c_1",[34,755806,755808,755924,755988,756043,756061,756079],{"className":755807,"ariaHidden":758},[31672],[34,755809,755811,755814,755863,755866,755906,755909,755912,755915,755918,755921],{"className":755810},[31676],[34,755812],{"className":755813,"style":31774},[31680],[34,755815,755817,755820],{"className":755816},[32767],[34,755818,546745],{"className":755819},[32767],[34,755821,755823],{"className":755822},[31792],[34,755824,755826,755855],{"className":755825},[31796,31797],[34,755827,755829,755852],{"className":755828},[31801],[34,755830,755832],{"className":755831,"style":206419},[31805],[34,755833,755834,755837],{"style":56235},[34,755835],{"className":755836,"style":31814},[31813],[34,755838,755840],{"className":755839},[31818,31819,31820,31821],[34,755841,755843,755846,755849],{"className":755842},[31685,31821],[34,755844,33249],{"className":755845},[31685,31686,31821],[34,755847,33239],{"className":755848},[31850,31821],[34,755850,138286],{"className":755851},[31685,31821],[34,755853,31829],{"className":755854},[31828],[34,755856,755858],{"className":755857},[31801],[34,755859,755861],{"className":755860,"style":31836},[31805],[34,755862],{},[34,755864,15972],{"className":755865},[31782],[34,755867,755869,755872],{"className":755868},[31685],[34,755870,126541],{"className":755871},[31685,31686],[34,755873,755875],{"className":755874},[31792],[34,755876,755878,755898],{"className":755877},[31796,31797],[34,755879,755881,755895],{"className":755880},[31801],[34,755882,755884],{"className":755883,"style":31806},[31805],[34,755885,755886,755889],{"style":31809},[34,755887],{"className":755888,"style":31814},[31813],[34,755890,755892],{"className":755891},[31818,31819,31820,31821],[34,755893,640],{"className":755894},[31685,31821],[34,755896,31829],{"className":755897},[31828],[34,755899,755901],{"className":755900},[31801],[34,755902,755904],{"className":755903,"style":31836},[31805],[34,755905],{},[34,755907,456],{"className":755908},[31782],[34,755910,33249],{"className":755911},[31685,31686],[34,755913,2557],{"className":755914},[31690],[34,755916],{"className":755917,"style":32604},[31845],[34,755919,32226],{"className":755920},[32024],[34,755922],{"className":755923,"style":32604},[31845],[34,755925,755927,755930,755970,755973,755976,755979,755982,755985],{"className":755926},[31676],[34,755928],{"className":755929,"style":31774},[31680],[34,755931,755933,755936],{"className":755932},[31685],[34,755934,126541],{"className":755935},[31685,31686],[34,755937,755939],{"className":755938},[31792],[34,755940,755942,755962],{"className":755941},[31796,31797],[34,755943,755945,755959],{"className":755944},[31801],[34,755946,755948],{"className":755947,"style":31806},[31805],[34,755949,755950,755953],{"style":31809},[34,755951],{"className":755952,"style":31814},[31813],[34,755954,755956],{"className":755955},[31818,31819,31820,31821],[34,755957,768],{"className":755958},[31685,31821],[34,755960,31829],{"className":755961},[31828],[34,755963,755965],{"className":755964},[31801],[34,755966,755968],{"className":755967,"style":31836},[31805],[34,755969],{},[34,755971,456],{"className":755972},[31782],[34,755974,33249],{"className":755975},[31685,31686],[34,755977,481211],{"className":755978},[31690],[34,755980],{"className":755981,"style":31846},[31845],[34,755983,202],{"className":755984},[31850],[34,755986],{"className":755987,"style":31846},[31845],[34,755989,755991,755994,756034,756037,756040],{"className":755990},[31676],[34,755992],{"className":755993,"style":40522},[31680],[34,755995,755997,756000],{"className":755996},[31685],[34,755998,31965],{"className":755999},[31685,31686],[34,756001,756003],{"className":756002},[31792],[34,756004,756006,756026],{"className":756005},[31796,31797],[34,756007,756009,756023],{"className":756008},[31801],[34,756010,756012],{"className":756011,"style":31806},[31805],[34,756013,756014,756017],{"style":31809},[34,756015],{"className":756016,"style":31814},[31813],[34,756018,756020],{"className":756019},[31818,31819,31820,31821],[34,756021,640],{"className":756022},[31685,31821],[34,756024,31829],{"className":756025},[31828],[34,756027,756029],{"className":756028},[31801],[34,756030,756032],{"className":756031,"style":31836},[31805],[34,756033],{},[34,756035],{"className":756036,"style":32604},[31845],[34,756038,223],{"className":756039},[32024],[34,756041],{"className":756042,"style":32604},[31845],[34,756044,756046,756049,756052,756055,756058],{"className":756045},[31676],[34,756047],{"className":756048,"style":35942},[31680],[34,756050,459],{"className":756051},[31685],[34,756053],{"className":756054,"style":32604},[31845],[34,756056,223],{"className":756057},[32024],[34,756059],{"className":756060,"style":32604},[31845],[34,756062,756064,756067,756070,756073,756076],{"className":756063},[31676],[34,756065],{"className":756066,"style":32617},[31680],[34,756068,459],{"className":756069},[31685],[34,756071],{"className":756072,"style":31846},[31845],[34,756074,202],{"className":756075},[31850],[34,756077],{"className":756078,"style":31846},[31845],[34,756080,756082,756085],{"className":756081},[31676],[34,756083],{"className":756084,"style":31860},[31680],[34,756086,756088,756091],{"className":756087},[31685],[34,756089,31965],{"className":756090},[31685,31686],[34,756092,756094],{"className":756093},[31792],[34,756095,756097,756117],{"className":756096},[31796,31797],[34,756098,756100,756114],{"className":756099},[31801],[34,756101,756103],{"className":756102,"style":31806},[31805],[34,756104,756105,756108],{"style":31809},[34,756106],{"className":756107,"style":31814},[31813],[34,756109,756111],{"className":756110},[31818,31819,31820,31821],[34,756112,640],{"className":756113},[31685,31821],[34,756115,31829],{"className":756116},[31828],[34,756118,756120],{"className":756119},[31801],[34,756121,756123],{"className":756122,"style":31836},[31805],[34,756124],{},"\n这是一个常数，说明",[2627,756127,756128],{},"极限存在",[32188,756130],{},[32490,756132,756134],{"id":756133},"_5-欧拉方程-euler-equation","5. 欧拉方程 (Euler Equation)",[15,756136,756137,756138,756141],{},"这部分是全新的知识点，我们先搞懂它的原理。\n",[2627,756139,756140],{},"核心难点："," 变量代换中的求导法则（链式法则）。",[15,756143,756144,756147,756148,756401,756402,756430,756431,756459,756460,756541,756542,70884,756603,756631],{},[2627,756145,756146],{},"定义与证明","\n题目给出的定义是：",[34,756149,756151,756199],{"className":756150},[31642],[34,756152,756154],{"className":756153},[31646],[31648,756155,756156],{"xmlns":31650},[31652,756157,756158,756196],{},[31655,756159,756160,756166,756188,756190,756192,756194],{},[37624,756161,756162,756164],{},[31658,756163,32936],{},[31658,756165,31660],{},[41453,756167,756168,756178],{},[31655,756169,756170,756176],{},[37624,756171,756172,756174],{},[31658,756173,46900],{"mathvariant":38079},[31658,756175,31660],{},[31658,756177,67764],{},[31655,756179,756180,756182],{},[31658,756181,46900],{"mathvariant":38079},[37624,756183,756184,756186],{},[31658,756185,32936],{},[31658,756187,31660],{},[31662,756189,223],{},[31662,756191,38874],{},[31662,756193,202],{},[31749,756195,459],{},[31665,756197,756198],{"encoding":31667},"x^n \\frac{\\mathrm{d}^n y}{\\mathrm{d} x^n} + \\cdots = 0",[34,756200,756202,756374,756392],{"className":756201,"ariaHidden":758},[31672],[34,756203,756205,756209,756238,756365,756368,756371],{"className":756204},[31676],[34,756206],{"className":756207,"style":756208},[31680],"height:1.3081em;vertical-align:-0.345em;",[34,756210,756212,756215],{"className":756211},[31685],[34,756213,32936],{"className":756214},[31685,31686],[34,756216,756218],{"className":756217},[31792],[34,756219,756221],{"className":756220},[31796],[34,756222,756224],{"className":756223},[31801],[34,756225,756227],{"className":756226,"style":37959},[31805],[34,756228,756229,756232],{"style":37777},[34,756230],{"className":756231,"style":31814},[31813],[34,756233,756235],{"className":756234},[31818,31819,31820,31821],[34,756236,31660],{"className":756237},[31685,31686,31821],[34,756239,756241,756244,756362],{"className":756240},[31685],[34,756242],{"className":756243},[31782,41488],[34,756245,756247],{"className":756246},[41453],[34,756248,756250,756354],{"className":756249},[31796,31797],[34,756251,756253,756351],{"className":756252},[31801],[34,756254,756257,756300,756308],{"className":756255,"style":756256},[31805],"height:0.9631em;",[34,756258,756259,756262],{"style":41504},[34,756260],{"className":756261,"style":41508},[31813],[34,756263,756265],{"className":756264},[31818,31819,31820,31821],[34,756266,756268,756271],{"className":756267},[31685,31821],[34,756269,46900],{"className":756270},[31685,207923,31821],[34,756272,756274,756277],{"className":756273},[31685,31821],[34,756275,32936],{"className":756276},[31685,31686,31821],[34,756278,756280],{"className":756279},[31792],[34,756281,756283],{"className":756282},[31796],[34,756284,756286],{"className":756285},[31801],[34,756287,756289],{"className":756288,"style":566313},[31805],[34,756290,756291,756294],{"style":116668},[34,756292],{"className":756293,"style":43406},[31813],[34,756295,756297],{"className":756296},[31818,43410,43411,31821],[34,756298,31660],{"className":756299},[31685,31686,31821],[34,756301,756302,756305],{"style":41520},[34,756303],{"className":756304,"style":41508},[31813],[34,756306],{"className":756307,"style":41528},[41527],[34,756309,756310,756313],{"style":548368},[34,756311],{"className":756312,"style":41508},[31813],[34,756314,756316],{"className":756315},[31818,31819,31820,31821],[34,756317,756319,756348],{"className":756318},[31685,31821],[34,756320,756322,756325],{"className":756321},[31685,31821],[34,756323,46900],{"className":756324},[31685,207923,31821],[34,756326,756328],{"className":756327},[31792],[34,756329,756331],{"className":756330},[31796],[34,756332,756334],{"className":756333},[31801],[34,756335,756337],{"className":756336,"style":611529},[31805],[34,756338,756339,756342],{"style":552391},[34,756340],{"className":756341,"style":43406},[31813],[34,756343,756345],{"className":756344},[31818,43410,43411,31821],[34,756346,31660],{"className":756347},[31685,31686,31821],[34,756349,67764],{"className":756350,"style":32062},[31685,31686,31821],[34,756352,31829],{"className":756353},[31828],[34,756355,756357],{"className":756356},[31801],[34,756358,756360],{"className":756359,"style":41619},[31805],[34,756361],{},[34,756363],{"className":756364},[31690,41488],[34,756366],{"className":756367,"style":32604},[31845],[34,756369,223],{"className":756370},[32024],[34,756372],{"className":756373,"style":32604},[31845],[34,756375,756377,756380,756383,756386,756389],{"className":756376},[31676],[34,756378],{"className":756379,"style":73089},[31680],[34,756381,38874],{"className":756382},[34006],[34,756384],{"className":756385,"style":31846},[31845],[34,756387,202],{"className":756388},[31850],[34,756390],{"className":756391,"style":31846},[31845],[34,756393,756395,756398],{"className":756394},[31676],[34,756396],{"className":756397,"style":32617},[31680],[34,756399,459],{"className":756400},[31685],"。这里的自变量是 ",[34,756403,756405,756418],{"className":756404},[31642],[34,756406,756408],{"className":756407},[31646],[31648,756409,756410],{"xmlns":31650},[31652,756411,756412,756416],{},[31655,756413,756414],{},[31658,756415,32936],{},[31665,756417,32936],{"encoding":31667},[34,756419,756421],{"className":756420,"ariaHidden":758},[31672],[34,756422,756424,756427],{"className":756423},[31676],[34,756425],{"className":756426,"style":32527},[31680],[34,756428,32936],{"className":756429},[31685,31686],"，因变量是 ",[34,756432,756434,756447],{"className":756433},[31642],[34,756435,756437],{"className":756436},[31646],[31648,756438,756439],{"xmlns":31650},[31652,756440,756441,756445],{},[31655,756442,756443],{},[31658,756444,67764],{},[31665,756446,67764],{"encoding":31667},[34,756448,756450],{"className":756449,"ariaHidden":758},[31672],[34,756451,756453,756456],{"className":756452},[31676],[34,756454],{"className":756455,"style":38774},[31680],[34,756457,67764],{"className":756458,"style":32062},[31685,31686],"。\n变换公式：",[34,756461,756463,756485],{"className":756462},[31642],[34,756464,756466],{"className":756465},[31646],[31648,756467,756468],{"xmlns":31650},[31652,756469,756470,756482],{},[31655,756471,756472,756474,756476],{},[31658,756473,32936],{},[31662,756475,202],{},[37624,756477,756478,756480],{},[31658,756479,33244],{"mathvariant":38079},[31658,756481,33249],{},[31665,756483,756484],{"encoding":31667},"x = \\mathrm{e}^t",[34,756486,756488,756506],{"className":756487,"ariaHidden":758},[31672],[34,756489,756491,756494,756497,756500,756503],{"className":756490},[31676],[34,756492],{"className":756493,"style":32527},[31680],[34,756495,32936],{"className":756496},[31685,31686],[34,756498],{"className":756499,"style":31846},[31845],[34,756501,202],{"className":756502},[31850],[34,756504],{"className":756505,"style":31846},[31845],[34,756507,756509,756512],{"className":756508},[31676],[34,756510],{"className":756511,"style":611758},[31680],[34,756513,756515,756518],{"className":756514},[31685],[34,756516,33244],{"className":756517},[31685,207923],[34,756519,756521],{"className":756520},[31792],[34,756522,756524],{"className":756523},[31796],[34,756525,756527],{"className":756526},[31801],[34,756528,756530],{"className":756529,"style":611758},[31805],[34,756531,756532,756535],{"style":37777},[34,756533],{"className":756534,"style":31814},[31813],[34,756536,756538],{"className":756537},[31818,31819,31820,31821],[34,756539,33249],{"className":756540},[31685,31686,31821]," （这意味着 ",[34,756543,756545,756567],{"className":756544},[31642],[34,756546,756548],{"className":756547},[31646],[31648,756549,756550],{"xmlns":31650},[31652,756551,756552,756564],{},[31655,756553,756554,756556,756558,756560,756562],{},[31658,756555,33249],{},[31662,756557,202],{},[31658,756559,563286],{},[31662,756561,32682],{},[31658,756563,32936],{},[31665,756565,756566],{"encoding":31667},"t = \\ln x",[34,756568,756570,756588],{"className":756569,"ariaHidden":758},[31672],[34,756571,756573,756576,756579,756582,756585],{"className":756572},[31676],[34,756574],{"className":756575,"style":33280},[31680],[34,756577,33249],{"className":756578},[31685,31686],[34,756580],{"className":756581,"style":31846},[31845],[34,756583,202],{"className":756584},[31850],[34,756586],{"className":756587,"style":31846},[31845],[34,756589,756591,756594,756597,756600],{"className":756590},[31676],[34,756592],{"className":756593,"style":31681},[31680],[34,756595,563286],{"className":756596},[32767],[34,756598],{"className":756599,"style":32123},[31845],[34,756601,32936],{"className":756602},[31685,31686],[34,756604,756606,756619],{"className":756605},[31642],[34,756607,756609],{"className":756608},[31646],[31648,756610,756611],{"xmlns":31650},[31652,756612,756613,756617],{},[31655,756614,756615],{},[31658,756616,33249],{},[31665,756618,33249],{"encoding":31667},[34,756620,756622],{"className":756621,"ariaHidden":758},[31672],[34,756623,756625,756628],{"className":756624},[31676],[34,756626],{"className":756627,"style":33280},[31680],[34,756629,33249],{"className":756630},[31685,31686]," 是新的自变量）。",[15,756633,756634,756637,756638,756666,756667,756779,756780,756666,756808,347],{},[2627,756635,756636],{},"证明变换过程（从零推导）：","\n我们要把关于 ",[34,756639,756641,756654],{"className":756640},[31642],[34,756642,756644],{"className":756643},[31646],[31648,756645,756646],{"xmlns":31650},[31652,756647,756648,756652],{},[31655,756649,756650],{},[31658,756651,32936],{},[31665,756653,32936],{"encoding":31667},[34,756655,756657],{"className":756656,"ariaHidden":758},[31672],[34,756658,756660,756663],{"className":756659},[31676],[34,756661],{"className":756662,"style":32527},[31680],[34,756664,32936],{"className":756665},[31685,31686]," 的导数 ",[34,756668,756670,756696],{"className":756669},[31642],[34,756671,756673],{"className":756672},[31646],[31648,756674,756675],{"xmlns":31650},[31652,756676,756677,756693],{},[31655,756678,756679],{},[41453,756680,756681,756687],{},[31655,756682,756683,756685],{},[31658,756684,46900],{"mathvariant":38079},[31658,756686,67764],{},[31655,756688,756689,756691],{},[31658,756690,46900],{"mathvariant":38079},[31658,756692,32936],{},[31665,756694,756695],{"encoding":31667},"\\frac{\\mathrm{d}y}{\\mathrm{d}x}",[34,756697,756699],{"className":756698,"ariaHidden":758},[31672],[34,756700,756702,756705],{"className":756701},[31676],[34,756703],{"className":756704,"style":608762},[31680],[34,756706,756708,756711,756776],{"className":756707},[31685],[34,756709],{"className":756710},[31782,41488],[34,756712,756714],{"className":756713},[41453],[34,756715,756717,756768],{"className":756716},[31796,31797],[34,756718,756720,756765],{"className":756719},[31801],[34,756721,756723,756740,756748],{"className":756722,"style":608935},[31805],[34,756724,756725,756728],{"style":41504},[34,756726],{"className":756727,"style":41508},[31813],[34,756729,756731],{"className":756730},[31818,31819,31820,31821],[34,756732,756734,756737],{"className":756733},[31685,31821],[34,756735,46900],{"className":756736},[31685,207923,31821],[34,756738,32936],{"className":756739},[31685,31686,31821],[34,756741,756742,756745],{"style":41520},[34,756743],{"className":756744,"style":41508},[31813],[34,756746],{"className":756747,"style":41528},[41527],[34,756749,756750,756753],{"style":548368},[34,756751],{"className":756752,"style":41508},[31813],[34,756754,756756],{"className":756755},[31818,31819,31820,31821],[34,756757,756759,756762],{"className":756758},[31685,31821],[34,756760,46900],{"className":756761},[31685,207923,31821],[34,756763,67764],{"className":756764,"style":32062},[31685,31686,31821],[34,756766,31829],{"className":756767},[31828],[34,756769,756771],{"className":756770},[31801],[34,756772,756774],{"className":756773,"style":41619},[31805],[34,756775],{},[34,756777],{"className":756778},[31690,41488]," 变成关于 ",[34,756781,756783,756796],{"className":756782},[31642],[34,756784,756786],{"className":756785},[31646],[31648,756787,756788],{"xmlns":31650},[31652,756789,756790,756794],{},[31655,756791,756792],{},[31658,756793,33249],{},[31665,756795,33249],{"encoding":31667},[34,756797,756799],{"className":756798,"ariaHidden":758},[31672],[34,756800,756802,756805],{"className":756801},[31676],[34,756803],{"className":756804,"style":33280},[31680],[34,756806,33249],{"className":756807},[31685,31686],[34,756809,756811,756837],{"className":756810},[31642],[34,756812,756814],{"className":756813},[31646],[31648,756815,756816],{"xmlns":31650},[31652,756817,756818,756834],{},[31655,756819,756820],{},[41453,756821,756822,756828],{},[31655,756823,756824,756826],{},[31658,756825,46900],{"mathvariant":38079},[31658,756827,67764],{},[31655,756829,756830,756832],{},[31658,756831,46900],{"mathvariant":38079},[31658,756833,33249],{},[31665,756835,756836],{"encoding":31667},"\\frac{\\mathrm{d}y}{\\mathrm{d}t}",[34,756838,756840],{"className":756839,"ariaHidden":758},[31672],[34,756841,756843,756846],{"className":756842},[31676],[34,756844],{"className":756845,"style":608762},[31680],[34,756847,756849,756852,756917],{"className":756848},[31685],[34,756850],{"className":756851},[31782,41488],[34,756853,756855],{"className":756854},[41453],[34,756856,756858,756909],{"className":756857},[31796,31797],[34,756859,756861,756906],{"className":756860},[31801],[34,756862,756864,756881,756889],{"className":756863,"style":608935},[31805],[34,756865,756866,756869],{"style":41504},[34,756867],{"className":756868,"style":41508},[31813],[34,756870,756872],{"className":756871},[31818,31819,31820,31821],[34,756873,756875,756878],{"className":756874},[31685,31821],[34,756876,46900],{"className":756877},[31685,207923,31821],[34,756879,33249],{"className":756880},[31685,31686,31821],[34,756882,756883,756886],{"style":41520},[34,756884],{"className":756885,"style":41508},[31813],[34,756887],{"className":756888,"style":41528},[41527],[34,756890,756891,756894],{"style":548368},[34,756892],{"className":756893,"style":41508},[31813],[34,756895,756897],{"className":756896},[31818,31819,31820,31821],[34,756898,756900,756903],{"className":756899},[31685,31821],[34,756901,46900],{"className":756902},[31685,207923,31821],[34,756904,67764],{"className":756905,"style":32062},[31685,31686,31821],[34,756907,31829],{"className":756908},[31828],[34,756910,756912],{"className":756911},[31801],[34,756913,756915],{"className":756914,"style":41619},[31805],[34,756916],{},[34,756918],{"className":756919},[31690,41488],[2230,756921,756922,758088],{},[108,756923,756924,756927,756928,315440,757250,13977,757310,39984,757513,757804,757805,757833,757834],{},[2627,756925,756926],{},"一阶导数转换：","\n根据链式法则：\n",[34,756929,756931,756989],{"className":756930},[31642],[34,756932,756934],{"className":756933},[31646],[31648,756935,756936],{"xmlns":31650},[31652,756937,756938,756986],{},[31655,756939,756940,756954,756956,756970,756972],{},[41453,756941,756942,756948],{},[31655,756943,756944,756946],{},[31658,756945,46900],{"mathvariant":38079},[31658,756947,67764],{},[31655,756949,756950,756952],{},[31658,756951,46900],{"mathvariant":38079},[31658,756953,32936],{},[31662,756955,202],{},[41453,756957,756958,756964],{},[31655,756959,756960,756962],{},[31658,756961,46900],{"mathvariant":38079},[31658,756963,67764],{},[31655,756965,756966,756968],{},[31658,756967,46900],{"mathvariant":38079},[31658,756969,33249],{},[31662,756971,71191],{},[41453,756973,756974,756980],{},[31655,756975,756976,756978],{},[31658,756977,46900],{"mathvariant":38079},[31658,756979,33249],{},[31655,756981,756982,756984],{},[31658,756983,46900],{"mathvariant":38079},[31658,756985,32936],{},[31665,756987,756988],{"encoding":31667},"\\frac{\\mathrm{d} y}{\\mathrm{d} x} = \\frac{\\mathrm{d} y}{\\mathrm{d} t} \\cdot \\frac{\\mathrm{d} t}{\\mathrm{d} x}",[34,756990,756992,757081,757170],{"className":756991,"ariaHidden":758},[31672],[34,756993,756995,756998,757072,757075,757078],{"className":756994},[31676],[34,756996],{"className":756997,"style":608762},[31680],[34,756999,757001,757004,757069],{"className":757000},[31685],[34,757002],{"className":757003},[31782,41488],[34,757005,757007],{"className":757006},[41453],[34,757008,757010,757061],{"className":757009},[31796,31797],[34,757011,757013,757058],{"className":757012},[31801],[34,757014,757016,757033,757041],{"className":757015,"style":608935},[31805],[34,757017,757018,757021],{"style":41504},[34,757019],{"className":757020,"style":41508},[31813],[34,757022,757024],{"className":757023},[31818,31819,31820,31821],[34,757025,757027,757030],{"className":757026},[31685,31821],[34,757028,46900],{"className":757029},[31685,207923,31821],[34,757031,32936],{"className":757032},[31685,31686,31821],[34,757034,757035,757038],{"style":41520},[34,757036],{"className":757037,"style":41508},[31813],[34,757039],{"className":757040,"style":41528},[41527],[34,757042,757043,757046],{"style":548368},[34,757044],{"className":757045,"style":41508},[31813],[34,757047,757049],{"className":757048},[31818,31819,31820,31821],[34,757050,757052,757055],{"className":757051},[31685,31821],[34,757053,46900],{"className":757054},[31685,207923,31821],[34,757056,67764],{"className":757057,"style":32062},[31685,31686,31821],[34,757059,31829],{"className":757060},[31828],[34,757062,757064],{"className":757063},[31801],[34,757065,757067],{"className":757066,"style":41619},[31805],[34,757068],{},[34,757070],{"className":757071},[31690,41488],[34,757073],{"className":757074,"style":31846},[31845],[34,757076,202],{"className":757077},[31850],[34,757079],{"className":757080,"style":31846},[31845],[34,757082,757084,757087,757161,757164,757167],{"className":757083},[31676],[34,757085],{"className":757086,"style":608762},[31680],[34,757088,757090,757093,757158],{"className":757089},[31685],[34,757091],{"className":757092},[31782,41488],[34,757094,757096],{"className":757095},[41453],[34,757097,757099,757150],{"className":757098},[31796,31797],[34,757100,757102,757147],{"className":757101},[31801],[34,757103,757105,757122,757130],{"className":757104,"style":608935},[31805],[34,757106,757107,757110],{"style":41504},[34,757108],{"className":757109,"style":41508},[31813],[34,757111,757113],{"className":757112},[31818,31819,31820,31821],[34,757114,757116,757119],{"className":757115},[31685,31821],[34,757117,46900],{"className":757118},[31685,207923,31821],[34,757120,33249],{"className":757121},[31685,31686,31821],[34,757123,757124,757127],{"style":41520},[34,757125],{"className":757126,"style":41508},[31813],[34,757128],{"className":757129,"style":41528},[41527],[34,757131,757132,757135],{"style":548368},[34,757133],{"className":757134,"style":41508},[31813],[34,757136,757138],{"className":757137},[31818,31819,31820,31821],[34,757139,757141,757144],{"className":757140},[31685,31821],[34,757142,46900],{"className":757143},[31685,207923,31821],[34,757145,67764],{"className":757146,"style":32062},[31685,31686,31821],[34,757148,31829],{"className":757149},[31828],[34,757151,757153],{"className":757152},[31801],[34,757154,757156],{"className":757155,"style":41619},[31805],[34,757157],{},[34,757159],{"className":757160},[31690,41488],[34,757162],{"className":757163,"style":32604},[31845],[34,757165,71191],{"className":757166},[32024],[34,757168],{"className":757169,"style":32604},[31845],[34,757171,757173,757176],{"className":757172},[31676],[34,757174],{"className":757175,"style":232492},[31680],[34,757177,757179,757182,757247],{"className":757178},[31685],[34,757180],{"className":757181},[31782,41488],[34,757183,757185],{"className":757184},[41453],[34,757186,757188,757239],{"className":757187},[31796,31797],[34,757189,757191,757236],{"className":757190},[31801],[34,757192,757194,757211,757219],{"className":757193,"style":232511},[31805],[34,757195,757196,757199],{"style":41504},[34,757197],{"className":757198,"style":41508},[31813],[34,757200,757202],{"className":757201},[31818,31819,31820,31821],[34,757203,757205,757208],{"className":757204},[31685,31821],[34,757206,46900],{"className":757207},[31685,207923,31821],[34,757209,32936],{"className":757210},[31685,31686,31821],[34,757212,757213,757216],{"style":41520},[34,757214],{"className":757215,"style":41508},[31813],[34,757217],{"className":757218,"style":41528},[41527],[34,757220,757221,757224],{"style":46771},[34,757222],{"className":757223,"style":41508},[31813],[34,757225,757227],{"className":757226},[31818,31819,31820,31821],[34,757228,757230,757233],{"className":757229},[31685,31821],[34,757231,46900],{"className":757232},[31685,207923,31821],[34,757234,33249],{"className":757235},[31685,31686,31821],[34,757237,31829],{"className":757238},[31828],[34,757240,757242],{"className":757241},[31801],[34,757243,757245],{"className":757244,"style":41619},[31805],[34,757246],{},[34,757248],{"className":757249},[31690,41488],[34,757251,757253,757274],{"className":757252},[31642],[34,757254,757256],{"className":757255},[31646],[31648,757257,757258],{"xmlns":31650},[31652,757259,757260,757272],{},[31655,757261,757262,757264,757266,757268,757270],{},[31658,757263,33249],{},[31662,757265,202],{},[31658,757267,563286],{},[31662,757269,32682],{},[31658,757271,32936],{},[31665,757273,756566],{"encoding":31667},[34,757275,757277,757295],{"className":757276,"ariaHidden":758},[31672],[34,757278,757280,757283,757286,757289,757292],{"className":757279},[31676],[34,757281],{"className":757282,"style":33280},[31680],[34,757284,33249],{"className":757285},[31685,31686],[34,757287],{"className":757288,"style":31846},[31845],[34,757290,202],{"className":757291},[31850],[34,757293],{"className":757294,"style":31846},[31845],[34,757296,757298,757301,757304,757307],{"className":757297},[31676],[34,757299],{"className":757300,"style":31681},[31680],[34,757302,563286],{"className":757303},[32767],[34,757305],{"className":757306,"style":32123},[31845],[34,757308,32936],{"className":757309},[31685,31686],[34,757311,757313,757347],{"className":757312},[31642],[34,757314,757316],{"className":757315},[31646],[31648,757317,757318],{"xmlns":31650},[31652,757319,757320,757344],{},[31655,757321,757322,757336,757338],{},[41453,757323,757324,757330],{},[31655,757325,757326,757328],{},[31658,757327,46900],{"mathvariant":38079},[31658,757329,33249],{},[31655,757331,757332,757334],{},[31658,757333,46900],{"mathvariant":38079},[31658,757335,32936],{},[31662,757337,202],{},[41453,757339,757340,757342],{},[31749,757341,640],{},[31658,757343,32936],{},[31665,757345,757346],{"encoding":31667},"\\frac{\\mathrm{d} t}{\\mathrm{d} x} = \\frac{1}{x}",[34,757348,757350,757439],{"className":757349,"ariaHidden":758},[31672],[34,757351,757353,757356,757430,757433,757436],{"className":757352},[31676],[34,757354],{"className":757355,"style":232492},[31680],[34,757357,757359,757362,757427],{"className":757358},[31685],[34,757360],{"className":757361},[31782,41488],[34,757363,757365],{"className":757364},[41453],[34,757366,757368,757419],{"className":757367},[31796,31797],[34,757369,757371,757416],{"className":757370},[31801],[34,757372,757374,757391,757399],{"className":757373,"style":232511},[31805],[34,757375,757376,757379],{"style":41504},[34,757377],{"className":757378,"style":41508},[31813],[34,757380,757382],{"className":757381},[31818,31819,31820,31821],[34,757383,757385,757388],{"className":757384},[31685,31821],[34,757386,46900],{"className":757387},[31685,207923,31821],[34,757389,32936],{"className":757390},[31685,31686,31821],[34,757392,757393,757396],{"style":41520},[34,757394],{"className":757395,"style":41508},[31813],[34,757397],{"className":757398,"style":41528},[41527],[34,757400,757401,757404],{"style":46771},[34,757402],{"className":757403,"style":41508},[31813],[34,757405,757407],{"className":757406},[31818,31819,31820,31821],[34,757408,757410,757413],{"className":757409},[31685,31821],[34,757411,46900],{"className":757412},[31685,207923,31821],[34,757414,33249],{"className":757415},[31685,31686,31821],[34,757417,31829],{"className":757418},[31828],[34,757420,757422],{"className":757421},[31801],[34,757423,757425],{"className":757424,"style":41619},[31805],[34,757426],{},[34,757428],{"className":757429},[31690,41488],[34,757431],{"className":757432,"style":31846},[31845],[34,757434,202],{"className":757435},[31850],[34,757437],{"className":757438,"style":31846},[31845],[34,757440,757442,757445],{"className":757441},[31676],[34,757443],{"className":757444,"style":46727},[31680],[34,757446,757448,757451,757510],{"className":757447},[31685],[34,757449],{"className":757450},[31782,41488],[34,757452,757454],{"className":757453},[41453],[34,757455,757457,757502],{"className":757456},[31796,31797],[34,757458,757460,757499],{"className":757459},[31801],[34,757461,757463,757477,757485],{"className":757462,"style":46746},[31805],[34,757464,757465,757468],{"style":41504},[34,757466],{"className":757467,"style":41508},[31813],[34,757469,757471],{"className":757470},[31818,31819,31820,31821],[34,757472,757474],{"className":757473},[31685,31821],[34,757475,32936],{"className":757476},[31685,31686,31821],[34,757478,757479,757482],{"style":41520},[34,757480],{"className":757481,"style":41508},[31813],[34,757483],{"className":757484,"style":41528},[41527],[34,757486,757487,757490],{"style":46771},[34,757488],{"className":757489,"style":41508},[31813],[34,757491,757493],{"className":757492},[31818,31819,31820,31821],[34,757494,757496],{"className":757495},[31685,31821],[34,757497,640],{"className":757498},[31685,31821],[34,757500,31829],{"className":757501},[31828],[34,757503,757505],{"className":757504},[31801],[34,757506,757508],{"className":757507,"style":41619},[31805],[34,757509],{},[34,757511],{"className":757512},[31690,41488],[34,757514,757516,757564],{"className":757515},[31642],[34,757517,757519],{"className":757518},[31646],[31648,757520,757521],{"xmlns":31650},[31652,757522,757523,757561],{},[31655,757524,757525,757539,757541,757547],{},[41453,757526,757527,757533],{},[31655,757528,757529,757531],{},[31658,757530,46900],{"mathvariant":38079},[31658,757532,67764],{},[31655,757534,757535,757537],{},[31658,757536,46900],{"mathvariant":38079},[31658,757538,32936],{},[31662,757540,202],{},[41453,757542,757543,757545],{},[31749,757544,640],{},[31658,757546,32936],{},[41453,757548,757549,757555],{},[31655,757550,757551,757553],{},[31658,757552,46900],{"mathvariant":38079},[31658,757554,67764],{},[31655,757556,757557,757559],{},[31658,757558,46900],{"mathvariant":38079},[31658,757560,33249],{},[31665,757562,757563],{"encoding":31667},"\\frac{\\mathrm{d} y}{\\mathrm{d} x} = \\frac{1}{x} \\frac{\\mathrm{d} y}{\\mathrm{d} t}",[34,757565,757567,757656],{"className":757566,"ariaHidden":758},[31672],[34,757568,757570,757573,757647,757650,757653],{"className":757569},[31676],[34,757571],{"className":757572,"style":608762},[31680],[34,757574,757576,757579,757644],{"className":757575},[31685],[34,757577],{"className":757578},[31782,41488],[34,757580,757582],{"className":757581},[41453],[34,757583,757585,757636],{"className":757584},[31796,31797],[34,757586,757588,757633],{"className":757587},[31801],[34,757589,757591,757608,757616],{"className":757590,"style":608935},[31805],[34,757592,757593,757596],{"style":41504},[34,757594],{"className":757595,"style":41508},[31813],[34,757597,757599],{"className":757598},[31818,31819,31820,31821],[34,757600,757602,757605],{"className":757601},[31685,31821],[34,757603,46900],{"className":757604},[31685,207923,31821],[34,757606,32936],{"className":757607},[31685,31686,31821],[34,757609,757610,757613],{"style":41520},[34,757611],{"className":757612,"style":41508},[31813],[34,757614],{"className":757615,"style":41528},[41527],[34,757617,757618,757621],{"style":548368},[34,757619],{"className":757620,"style":41508},[31813],[34,757622,757624],{"className":757623},[31818,31819,31820,31821],[34,757625,757627,757630],{"className":757626},[31685,31821],[34,757628,46900],{"className":757629},[31685,207923,31821],[34,757631,67764],{"className":757632,"style":32062},[31685,31686,31821],[34,757634,31829],{"className":757635},[31828],[34,757637,757639],{"className":757638},[31801],[34,757640,757642],{"className":757641,"style":41619},[31805],[34,757643],{},[34,757645],{"className":757646},[31690,41488],[34,757648],{"className":757649,"style":31846},[31845],[34,757651,202],{"className":757652},[31850],[34,757654],{"className":757655,"style":31846},[31845],[34,757657,757659,757662,757730],{"className":757658},[31676],[34,757660],{"className":757661,"style":608762},[31680],[34,757663,757665,757668,757727],{"className":757664},[31685],[34,757666],{"className":757667},[31782,41488],[34,757669,757671],{"className":757670},[41453],[34,757672,757674,757719],{"className":757673},[31796,31797],[34,757675,757677,757716],{"className":757676},[31801],[34,757678,757680,757694,757702],{"className":757679,"style":46746},[31805],[34,757681,757682,757685],{"style":41504},[34,757683],{"className":757684,"style":41508},[31813],[34,757686,757688],{"className":757687},[31818,31819,31820,31821],[34,757689,757691],{"className":757690},[31685,31821],[34,757692,32936],{"className":757693},[31685,31686,31821],[34,757695,757696,757699],{"style":41520},[34,757697],{"className":757698,"style":41508},[31813],[34,757700],{"className":757701,"style":41528},[41527],[34,757703,757704,757707],{"style":46771},[34,757705],{"className":757706,"style":41508},[31813],[34,757708,757710],{"className":757709},[31818,31819,31820,31821],[34,757711,757713],{"className":757712},[31685,31821],[34,757714,640],{"className":757715},[31685,31821],[34,757717,31829],{"className":757718},[31828],[34,757720,757722],{"className":757721},[31801],[34,757723,757725],{"className":757724,"style":41619},[31805],[34,757726],{},[34,757728],{"className":757729},[31690,41488],[34,757731,757733,757736,757801],{"className":757732},[31685],[34,757734],{"className":757735},[31782,41488],[34,757737,757739],{"className":757738},[41453],[34,757740,757742,757793],{"className":757741},[31796,31797],[34,757743,757745,757790],{"className":757744},[31801],[34,757746,757748,757765,757773],{"className":757747,"style":608935},[31805],[34,757749,757750,757753],{"style":41504},[34,757751],{"className":757752,"style":41508},[31813],[34,757754,757756],{"className":757755},[31818,31819,31820,31821],[34,757757,757759,757762],{"className":757758},[31685,31821],[34,757760,46900],{"className":757761},[31685,207923,31821],[34,757763,33249],{"className":757764},[31685,31686,31821],[34,757766,757767,757770],{"style":41520},[34,757768],{"className":757769,"style":41508},[31813],[34,757771],{"className":757772,"style":41528},[41527],[34,757774,757775,757778],{"style":548368},[34,757776],{"className":757777,"style":41508},[31813],[34,757779,757781],{"className":757780},[31818,31819,31820,31821],[34,757782,757784,757787],{"className":757783},[31685,31821],[34,757785,46900],{"className":757786},[31685,207923,31821],[34,757788,67764],{"className":757789,"style":32062},[31685,31686,31821],[34,757791,31829],{"className":757792},[31828],[34,757794,757796],{"className":757795},[31801],[34,757797,757799],{"className":757798,"style":41619},[31805],[34,757800],{},[34,757802],{"className":757803},[31690,41488],"\n两边同乘 ",[34,757806,757808,757821],{"className":757807},[31642],[34,757809,757811],{"className":757810},[31646],[31648,757812,757813],{"xmlns":31650},[31652,757814,757815,757819],{},[31655,757816,757817],{},[31658,757818,32936],{},[31665,757820,32936],{"encoding":31667},[34,757822,757824],{"className":757823,"ariaHidden":758},[31672],[34,757825,757827,757830],{"className":757826},[31676],[34,757828],{"className":757829,"style":32527},[31680],[34,757831,32936],{"className":757832},[31685,31686],"，得出一个重要结论：\n",[34,757835,757837,757890],{"className":757836},[31642],[34,757838,757840],{"className":757839},[31646],[31648,757841,757842],{"xmlns":31650},[31652,757843,757844,757887],{},[31655,757845,757846,757848,757862,757864,757878,757880,757883,757885],{},[31658,757847,32936],{},[41453,757849,757850,757856],{},[31655,757851,757852,757854],{},[31658,757853,46900],{"mathvariant":38079},[31658,757855,67764],{},[31655,757857,757858,757860],{},[31658,757859,46900],{"mathvariant":38079},[31658,757861,32936],{},[31662,757863,202],{},[41453,757865,757866,757872],{},[31655,757867,757868,757870],{},[31658,757869,46900],{"mathvariant":38079},[31658,757871,67764],{},[31655,757873,757874,757876],{},[31658,757875,46900],{"mathvariant":38079},[31658,757877,33249],{},[31845,757879],{"width":71082},[39361,757881,757882],{},"(记作 D",[31658,757884,67764],{},[31662,757886,2557],{"stretchy":702},[31665,757888,757889],{"encoding":31667},"x \\frac{\\mathrm{d} y}{\\mathrm{d} x} = \\frac{\\mathrm{d} y}{\\mathrm{d} t} \\quad \\text{(记作 D}y)",[34,757891,757893,757985],{"className":757892,"ariaHidden":758},[31672],[34,757894,757896,757899,757902,757976,757979,757982],{"className":757895},[31676],[34,757897],{"className":757898,"style":608762},[31680],[34,757900,32936],{"className":757901},[31685,31686],[34,757903,757905,757908,757973],{"className":757904},[31685],[34,757906],{"className":757907},[31782,41488],[34,757909,757911],{"className":757910},[41453],[34,757912,757914,757965],{"className":757913},[31796,31797],[34,757915,757917,757962],{"className":757916},[31801],[34,757918,757920,757937,757945],{"className":757919,"style":608935},[31805],[34,757921,757922,757925],{"style":41504},[34,757923],{"className":757924,"style":41508},[31813],[34,757926,757928],{"className":757927},[31818,31819,31820,31821],[34,757929,757931,757934],{"className":757930},[31685,31821],[34,757932,46900],{"className":757933},[31685,207923,31821],[34,757935,32936],{"className":757936},[31685,31686,31821],[34,757938,757939,757942],{"style":41520},[34,757940],{"className":757941,"style":41508},[31813],[34,757943],{"className":757944,"style":41528},[41527],[34,757946,757947,757950],{"style":548368},[34,757948],{"className":757949,"style":41508},[31813],[34,757951,757953],{"className":757952},[31818,31819,31820,31821],[34,757954,757956,757959],{"className":757955},[31685,31821],[34,757957,46900],{"className":757958},[31685,207923,31821],[34,757960,67764],{"className":757961,"style":32062},[31685,31686,31821],[34,757963,31829],{"className":757964},[31828],[34,757966,757968],{"className":757967},[31801],[34,757969,757971],{"className":757970,"style":41619},[31805],[34,757972],{},[34,757974],{"className":757975},[31690,41488],[34,757977],{"className":757978,"style":31846},[31845],[34,757980,202],{"className":757981},[31850],[34,757983],{"className":757984,"style":31846},[31845],[34,757986,757988,757991,758065,758068,758082,758085],{"className":757987},[31676],[34,757989],{"className":757990,"style":608762},[31680],[34,757992,757994,757997,758062],{"className":757993},[31685],[34,757995],{"className":757996},[31782,41488],[34,757998,758000],{"className":757999},[41453],[34,758001,758003,758054],{"className":758002},[31796,31797],[34,758004,758006,758051],{"className":758005},[31801],[34,758007,758009,758026,758034],{"className":758008,"style":608935},[31805],[34,758010,758011,758014],{"style":41504},[34,758012],{"className":758013,"style":41508},[31813],[34,758015,758017],{"className":758016},[31818,31819,31820,31821],[34,758018,758020,758023],{"className":758019},[31685,31821],[34,758021,46900],{"className":758022},[31685,207923,31821],[34,758024,33249],{"className":758025},[31685,31686,31821],[34,758027,758028,758031],{"style":41520},[34,758029],{"className":758030,"style":41508},[31813],[34,758032],{"className":758033,"style":41528},[41527],[34,758035,758036,758039],{"style":548368},[34,758037],{"className":758038,"style":41508},[31813],[34,758040,758042],{"className":758041},[31818,31819,31820,31821],[34,758043,758045,758048],{"className":758044},[31685,31821],[34,758046,46900],{"className":758047},[31685,207923,31821],[34,758049,67764],{"className":758050,"style":32062},[31685,31686,31821],[34,758052,31829],{"className":758053},[31828],[34,758055,758057],{"className":758056},[31801],[34,758058,758060],{"className":758059,"style":41619},[31805],[34,758061],{},[34,758063],{"className":758064},[31690,41488],[34,758066],{"className":758067,"style":128438},[31845],[34,758069,758071,758074,758078],{"className":758070},[31685,2638],[34,758072,456],{"className":758073},[31685],[34,758075,758077],{"className":758076},[31685,105462],"记作",[34,758079,758081],{"className":758080},[31685]," D",[34,758083,67764],{"className":758084,"style":32062},[31685,31686],[34,758086,2557],{"className":758087},[31690],[108,758089,758090,758093,758094,758476,758477,758768,758769,33183,758963,759507,760173,760738,757804,761151,761209,761210],{},[2627,758091,758092],{},"二阶导数转换：","\n我们需要算 ",[34,758095,758097,758163],{"className":758096},[31642],[34,758098,758100],{"className":758099},[31646],[31648,758101,758102],{"xmlns":31650},[31652,758103,758104,758160],{},[31655,758105,758106,758128,758130,758140],{},[41453,758107,758108,758118],{},[31655,758109,758110,758116],{},[37624,758111,758112,758114],{},[31658,758113,46900],{"mathvariant":38079},[31749,758115,768],{},[31658,758117,67764],{},[31655,758119,758120,758122],{},[31658,758121,46900],{"mathvariant":38079},[37624,758123,758124,758126],{},[31658,758125,32936],{},[31749,758127,768],{},[31662,758129,202],{},[41453,758131,758132,758134],{},[31658,758133,46900],{"mathvariant":38079},[31655,758135,758136,758138],{},[31658,758137,46900],{"mathvariant":38079},[31658,758139,32936],{},[31655,758141,758142,758144,758158],{},[31662,758143,456],{"fence":758},[41453,758145,758146,758152],{},[31655,758147,758148,758150],{},[31658,758149,46900],{"mathvariant":38079},[31658,758151,67764],{},[31655,758153,758154,758156],{},[31658,758155,46900],{"mathvariant":38079},[31658,758157,32936],{},[31662,758159,2557],{"fence":758},[31665,758161,758162],{"encoding":31667},"\\frac{\\mathrm{d}^2 y}{\\mathrm{d} x^2} = \\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( \\frac{\\mathrm{d} y}{\\mathrm{d} x} \\right)",[34,758164,758166,758307],{"className":758165,"ariaHidden":758},[31672],[34,758167,758169,758172,758298,758301,758304],{"className":758168},[31676],[34,758170],{"className":758171,"style":681304},[31680],[34,758173,758175,758178,758295],{"className":758174},[31685],[34,758176],{"className":758177},[31782,41488],[34,758179,758181],{"className":758180},[41453],[34,758182,758184,758287],{"className":758183},[31796,31797],[34,758185,758187,758284],{"className":758186},[31801],[34,758188,758190,758233,758241],{"className":758189,"style":601858},[31805],[34,758191,758192,758195],{"style":41504},[34,758193],{"className":758194,"style":41508},[31813],[34,758196,758198],{"className":758197},[31818,31819,31820,31821],[34,758199,758201,758204],{"className":758200},[31685,31821],[34,758202,46900],{"className":758203},[31685,207923,31821],[34,758205,758207,758210],{"className":758206},[31685,31821],[34,758208,32936],{"className":758209},[31685,31686,31821],[34,758211,758213],{"className":758212},[31792],[34,758214,758216],{"className":758215},[31796],[34,758217,758219],{"className":758218},[31801],[34,758220,758222],{"className":758221,"style":548314},[31805],[34,758223,758224,758227],{"style":116668},[34,758225],{"className":758226,"style":43406},[31813],[34,758228,758230],{"className":758229},[31818,43410,43411,31821],[34,758231,768],{"className":758232},[31685,31821],[34,758234,758235,758238],{"style":41520},[34,758236],{"className":758237,"style":41508},[31813],[34,758239],{"className":758240,"style":41528},[41527],[34,758242,758243,758246],{"style":548368},[34,758244],{"className":758245,"style":41508},[31813],[34,758247,758249],{"className":758248},[31818,31819,31820,31821],[34,758250,758252,758281],{"className":758251},[31685,31821],[34,758253,758255,758258],{"className":758254},[31685,31821],[34,758256,46900],{"className":758257},[31685,207923,31821],[34,758259,758261],{"className":758260},[31792],[34,758262,758264],{"className":758263},[31796],[34,758265,758267],{"className":758266},[31801],[34,758268,758270],{"className":758269,"style":144123},[31805],[34,758271,758272,758275],{"style":552391},[34,758273],{"className":758274,"style":43406},[31813],[34,758276,758278],{"className":758277},[31818,43410,43411,31821],[34,758279,768],{"className":758280},[31685,31821],[34,758282,67764],{"className":758283,"style":32062},[31685,31686,31821],[34,758285,31829],{"className":758286},[31828],[34,758288,758290],{"className":758289},[31801],[34,758291,758293],{"className":758292,"style":41619},[31805],[34,758294],{},[34,758296],{"className":758297},[31690,41488],[34,758299],{"className":758300,"style":31846},[31845],[34,758302,202],{"className":758303},[31850],[34,758305],{"className":758306,"style":31846},[31845],[34,758308,758310,758313,758384,758387],{"className":758309},[31676],[34,758311],{"className":758312,"style":141282},[31680],[34,758314,758316,758319,758381],{"className":758315},[31685],[34,758317],{"className":758318},[31782,41488],[34,758320,758322],{"className":758321},[41453],[34,758323,758325,758373],{"className":758324},[31796,31797],[34,758326,758328,758370],{"className":758327},[31801],[34,758329,758331,758348,758356],{"className":758330,"style":232511},[31805],[34,758332,758333,758336],{"style":41504},[34,758334],{"className":758335,"style":41508},[31813],[34,758337,758339],{"className":758338},[31818,31819,31820,31821],[34,758340,758342,758345],{"className":758341},[31685,31821],[34,758343,46900],{"className":758344},[31685,207923,31821],[34,758346,32936],{"className":758347},[31685,31686,31821],[34,758349,758350,758353],{"style":41520},[34,758351],{"className":758352,"style":41508},[31813],[34,758354],{"className":758355,"style":41528},[41527],[34,758357,758358,758361],{"style":46771},[34,758359],{"className":758360,"style":41508},[31813],[34,758362,758364],{"className":758363},[31818,31819,31820,31821],[34,758365,758367],{"className":758366},[31685,31821],[34,758368,46900],{"className":758369},[31685,207923,31821],[34,758371,31829],{"className":758372},[31828],[34,758374,758376],{"className":758375},[31801],[34,758377,758379],{"className":758378,"style":41619},[31805],[34,758380],{},[34,758382],{"className":758383},[31690,41488],[34,758385],{"className":758386,"style":32123},[31845],[34,758388,758390,758396,758470],{"className":758389},[34006],[34,758391,758393],{"className":758392,"style":75811},[31782,75810],[34,758394,456],{"className":758395},[71281,141292],[34,758397,758399,758402,758467],{"className":758398},[31685],[34,758400],{"className":758401},[31782,41488],[34,758403,758405],{"className":758404},[41453],[34,758406,758408,758459],{"className":758407},[31796,31797],[34,758409,758411,758456],{"className":758410},[31801],[34,758412,758414,758431,758439],{"className":758413,"style":608935},[31805],[34,758415,758416,758419],{"style":41504},[34,758417],{"className":758418,"style":41508},[31813],[34,758420,758422],{"className":758421},[31818,31819,31820,31821],[34,758423,758425,758428],{"className":758424},[31685,31821],[34,758426,46900],{"className":758427},[31685,207923,31821],[34,758429,32936],{"className":758430},[31685,31686,31821],[34,758432,758433,758436],{"style":41520},[34,758434],{"className":758435,"style":41508},[31813],[34,758437],{"className":758438,"style":41528},[41527],[34,758440,758441,758444],{"style":548368},[34,758442],{"className":758443,"style":41508},[31813],[34,758445,758447],{"className":758446},[31818,31819,31820,31821],[34,758448,758450,758453],{"className":758449},[31685,31821],[34,758451,46900],{"className":758452},[31685,207923,31821],[34,758454,67764],{"className":758455,"style":32062},[31685,31686,31821],[34,758457,31829],{"className":758458},[31828],[34,758460,758462],{"className":758461},[31801],[34,758463,758465],{"className":758464,"style":41619},[31805],[34,758466],{},[34,758468],{"className":758469},[31690,41488],[34,758471,758473],{"className":758472,"style":75811},[31690,75810],[34,758474,2557],{"className":758475},[71281,141292],"。\n把刚才的一阶结果代进去：\n",[34,758478,758480,758528],{"className":758479},[31642],[34,758481,758483],{"className":758482},[31646],[31648,758484,758485],{"xmlns":31650},[31652,758486,758487,758525],{},[31655,758488,758489,758499],{},[41453,758490,758491,758493],{},[31658,758492,46900],{"mathvariant":38079},[31655,758494,758495,758497],{},[31658,758496,46900],{"mathvariant":38079},[31658,758498,32936],{},[31655,758500,758501,758503,758509,758523],{},[31662,758502,456],{"fence":758},[41453,758504,758505,758507],{},[31749,758506,640],{},[31658,758508,32936],{},[41453,758510,758511,758517],{},[31655,758512,758513,758515],{},[31658,758514,46900],{"mathvariant":38079},[31658,758516,67764],{},[31655,758518,758519,758521],{},[31658,758520,46900],{"mathvariant":38079},[31658,758522,33249],{},[31662,758524,2557],{"fence":758},[31665,758526,758527],{"encoding":31667},"\\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( \\frac{1}{x} \\frac{\\mathrm{d} y}{\\mathrm{d} t} \\right)",[34,758529,758531],{"className":758530,"ariaHidden":758},[31672],[34,758532,758534,758537,758608,758611],{"className":758533},[31676],[34,758535],{"className":758536,"style":141282},[31680],[34,758538,758540,758543,758605],{"className":758539},[31685],[34,758541],{"className":758542},[31782,41488],[34,758544,758546],{"className":758545},[41453],[34,758547,758549,758597],{"className":758548},[31796,31797],[34,758550,758552,758594],{"className":758551},[31801],[34,758553,758555,758572,758580],{"className":758554,"style":232511},[31805],[34,758556,758557,758560],{"style":41504},[34,758558],{"className":758559,"style":41508},[31813],[34,758561,758563],{"className":758562},[31818,31819,31820,31821],[34,758564,758566,758569],{"className":758565},[31685,31821],[34,758567,46900],{"className":758568},[31685,207923,31821],[34,758570,32936],{"className":758571},[31685,31686,31821],[34,758573,758574,758577],{"style":41520},[34,758575],{"className":758576,"style":41508},[31813],[34,758578],{"className":758579,"style":41528},[41527],[34,758581,758582,758585],{"style":46771},[34,758583],{"className":758584,"style":41508},[31813],[34,758586,758588],{"className":758587},[31818,31819,31820,31821],[34,758589,758591],{"className":758590},[31685,31821],[34,758592,46900],{"className":758593},[31685,207923,31821],[34,758595,31829],{"className":758596},[31828],[34,758598,758600],{"className":758599},[31801],[34,758601,758603],{"className":758602,"style":41619},[31805],[34,758604],{},[34,758606],{"className":758607},[31690,41488],[34,758609],{"className":758610,"style":32123},[31845],[34,758612,758614,758620,758688,758762],{"className":758613},[34006],[34,758615,758617],{"className":758616,"style":75811},[31782,75810],[34,758618,456],{"className":758619},[71281,141292],[34,758621,758623,758626,758685],{"className":758622},[31685],[34,758624],{"className":758625},[31782,41488],[34,758627,758629],{"className":758628},[41453],[34,758630,758632,758677],{"className":758631},[31796,31797],[34,758633,758635,758674],{"className":758634},[31801],[34,758636,758638,758652,758660],{"className":758637,"style":46746},[31805],[34,758639,758640,758643],{"style":41504},[34,758641],{"className":758642,"style":41508},[31813],[34,758644,758646],{"className":758645},[31818,31819,31820,31821],[34,758647,758649],{"className":758648},[31685,31821],[34,758650,32936],{"className":758651},[31685,31686,31821],[34,758653,758654,758657],{"style":41520},[34,758655],{"className":758656,"style":41508},[31813],[34,758658],{"className":758659,"style":41528},[41527],[34,758661,758662,758665],{"style":46771},[34,758663],{"className":758664,"style":41508},[31813],[34,758666,758668],{"className":758667},[31818,31819,31820,31821],[34,758669,758671],{"className":758670},[31685,31821],[34,758672,640],{"className":758673},[31685,31821],[34,758675,31829],{"className":758676},[31828],[34,758678,758680],{"className":758679},[31801],[34,758681,758683],{"className":758682,"style":41619},[31805],[34,758684],{},[34,758686],{"className":758687},[31690,41488],[34,758689,758691,758694,758759],{"className":758690},[31685],[34,758692],{"className":758693},[31782,41488],[34,758695,758697],{"className":758696},[41453],[34,758698,758700,758751],{"className":758699},[31796,31797],[34,758701,758703,758748],{"className":758702},[31801],[34,758704,758706,758723,758731],{"className":758705,"style":608935},[31805],[34,758707,758708,758711],{"style":41504},[34,758709],{"className":758710,"style":41508},[31813],[34,758712,758714],{"className":758713},[31818,31819,31820,31821],[34,758715,758717,758720],{"className":758716},[31685,31821],[34,758718,46900],{"className":758719},[31685,207923,31821],[34,758721,33249],{"className":758722},[31685,31686,31821],[34,758724,758725,758728],{"style":41520},[34,758726],{"className":758727,"style":41508},[31813],[34,758729],{"className":758730,"style":41528},[41527],[34,758732,758733,758736],{"style":548368},[34,758734],{"className":758735,"style":41508},[31813],[34,758737,758739],{"className":758738},[31818,31819,31820,31821],[34,758740,758742,758745],{"className":758741},[31685,31821],[34,758743,46900],{"className":758744},[31685,207923,31821],[34,758746,67764],{"className":758747,"style":32062},[31685,31686,31821],[34,758749,31829],{"className":758750},[31828],[34,758752,758754],{"className":758753},[31801],[34,758755,758757],{"className":758756,"style":41619},[31805],[34,758758],{},[34,758760],{"className":758761},[31690,41488],[34,758763,758765],{"className":758764,"style":75811},[31690,75810],[34,758766,2557],{"className":758767},[71281,141292],"\n利用乘法求导法则 ",[34,758770,758772,758816],{"className":758771},[31642],[34,758773,758775],{"className":758774},[31646],[31648,758776,758777],{"xmlns":31650},[31652,758778,758779,758813],{},[31655,758780,758781,758783,758785,758787,758793,758795,758801,758803,758805,758807],{},[31662,758782,456],{"stretchy":702},[31658,758784,94628],{},[31658,758786,56782],{},[37624,758788,758789,758791],{},[31662,758790,2557],{"stretchy":702},[31662,758792,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,758794,202],{},[37624,758796,758797,758799],{},[31658,758798,94628],{},[31662,758800,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31658,758802,56782],{},[31662,758804,223],{},[31658,758806,94628],{},[37624,758808,758809,758811],{},[31658,758810,56782],{},[31662,758812,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,758814,758815],{"encoding":31667},"(uv)' = u'v + 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\\left( \\frac{1}{x} \\right)' \\frac{\\mathrm{d} y}{\\mathrm{d} t} + \\frac{1}{x} \\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( \\frac{\\mathrm{d} y}{\\mathrm{d} t} 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\\frac{1}{x^2} \\left( \\frac{\\mathrm{d}^2 y}{\\mathrm{d} t^2} - \\frac{\\mathrm{d} y}{\\mathrm{d} t} \\right)",[34,760810,760812,760824],{"className":760811,"ariaHidden":758},[31672],[34,760813,760815,760818,760821],{"className":760814},[31676],[34,760816],{"className":760817,"style":73089},[31680],[34,760819,202],{"className":760820},[31850],[34,760822],{"className":760823,"style":31846},[31845],[34,760825,760827,760830,760924,760927],{"className":760826},[31676],[34,760828],{"className":760829,"style":141282},[31680],[34,760831,760833,760836,760921],{"className":760832},[31685],[34,760834],{"className":760835},[31782,41488],[34,760837,760839],{"className":760838},[41453],[34,760840,760842,760913],{"className":760841},[31796,31797],[34,760843,760845,760910],{"className":760844},[31801],[34,760846,760848,760888,760896],{"className":760847,"style":46746},[31805],[34,760849,760850,760853],{"style":41504},[34,760851],{"className":760852,"style":41508},[31813],[34,760854,760856],{"className":760855},[31818,31819,31820,31821],[34,760857,760859],{"className":760858},[31685,31821],[34,760860,760862,760865],{"className":760861},[31685,31821],[34,760863,32936],{"className":760864},[31685,31686,31821],[34,760866,760868],{"className":760867},[31792],[34,760869,760871],{"className":760870},[31796],[34,760872,760874],{"className":760873},[31801],[34,760875,760877],{"className":760876,"style":548314},[31805],[34,760878,760879,760882],{"style":116668},[34,760880],{"className":760881,"style":43406},[31813],[34,760883,760885],{"className":760884},[31818,43410,43411,31821],[34,760886,768],{"className":760887},[31685,31821],[34,760889,760890,760893],{"style":41520},[34,760891],{"className":760892,"style":41508},[31813],[34,760894],{"className":760895,"style":41528},[41527],[34,760897,760898,760901],{"style":46771},[34,760899],{"className":760900,"style":41508},[31813],[34,760902,760904],{"className":760903},[31818,31819,31820,31821],[34,760905,760907],{"className":760906},[31685,31821],[34,760908,640],{"className":760909},[31685,31821],[34,760911,31829],{"className":760912},[31828],[34,760914,760916],{"className":760915},[31801],[34,760917,760919],{"className":760918,"style":41619},[31805],[34,760920],{},[34,760922],{"className":760923},[31690,41488],[34,760925],{"className":760926,"style":32123},[31845],[34,760928,760930,760936,761062,761065,761068,761071,761145],{"className":760929},[34006],[34,760931,760933],{"className":760932,"style":75811},[31782,75810],[34,760934,456],{"className":760935},[71281,141292],[34,760937,760939,760942,761059],{"className":760938},[31685],[34,760940],{"className":760941},[31782,41488],[34,760943,760945],{"className":760944},[41453],[34,760946,760948,761051],{"className":760947},[31796,31797],[34,760949,760951,761048],{"className":760950},[31801],[34,760952,760954,760997,761005],{"className":760953,"style":601858},[31805],[34,760955,760956,760959],{"style":41504},[34,760957],{"className":760958,"style":41508},[31813],[34,760960,760962],{"className":760961},[31818,31819,31820,31821],[34,760963,760965,760968],{"className":760964},[31685,31821],[34,760966,46900],{"className":760967},[31685,207923,31821],[34,760969,760971,760974],{"className":760970},[31685,31821],[34,760972,33249],{"className":760973},[31685,31686,31821],[34,760975,760977],{"className":760976},[31792],[34,760978,760980],{"className":760979},[31796],[34,760981,760983],{"className":760982},[31801],[34,760984,760986],{"className":760985,"style":548314},[31805],[34,760987,760988,760991],{"style":116668},[34,760989],{"className":760990,"style":43406},[31813],[34,760992,760994],{"className":760993},[31818,43410,43411,31821],[34,760995,768],{"className":760996},[31685,31821],[34,760998,760999,761002],{"style":41520},[34,761000],{"className":761001,"style":41508},[31813],[34,761003],{"className":761004,"style":41528},[41527],[34,761006,761007,761010],{"style":548368},[34,761008],{"className":761009,"style":41508},[31813],[34,761011,761013],{"className":761012},[31818,31819,31820,31821],[34,761014,761016,761045],{"className":761015},[31685,31821],[34,761017,761019,761022],{"className":761018},[31685,31821],[34,761020,46900],{"className":761021},[31685,207923,31821],[34,761023,761025],{"className":761024},[31792],[34,761026,761028],{"className":761027},[31796],[34,761029,761031],{"className":761030},[31801],[34,761032,761034],{"className":761033,"style":144123},[31805],[34,761035,761036,761039],{"style":552391},[34,761037],{"className":761038,"style":43406},[31813],[34,761040,761042],{"className":761041},[31818,43410,43411,31821],[34,761043,768],{"className":761044},[31685,31821],[34,761046,67764],{"className":761047,"style":32062},[31685,31686,31821],[34,761049,31829],{"className":761050},[31828],[34,761052,761054],{"className":761053},[31801],[34,761055,761057],{"className":761056,"style":41619},[31805],[34,761058],{},[34,761060],{"className":761061},[31690,41488],[34,761063],{"className":761064,"style":32604},[31845],[34,761066,32226],{"className":761067},[32024],[34,761069],{"className":761070,"style":32604},[31845],[34,761072,761074,761077,761142],{"className":761073},[31685],[34,761075],{"className":761076},[31782,41488],[34,761078,761080],{"className":761079},[41453],[34,761081,761083,761134],{"className":761082},[31796,31797],[34,761084,761086,761131],{"className":761085},[31801],[34,761087,761089,761106,761114],{"className":761088,"style":608935},[31805],[34,761090,761091,761094],{"style":41504},[34,761092],{"className":761093,"style":41508},[31813],[34,761095,761097],{"className":761096},[31818,31819,31820,31821],[34,761098,761100,761103],{"className":761099},[31685,31821],[34,761101,46900],{"className":761102},[31685,207923,31821],[34,761104,33249],{"className":761105},[31685,31686,31821],[34,761107,761108,761111],{"style":41520},[34,761109],{"className":761110,"style":41508},[31813],[34,761112],{"className":761113,"style":41528},[41527],[34,761115,761116,761119],{"style":548368},[34,761117],{"className":761118,"style":41508},[31813],[34,761120,761122],{"className":761121},[31818,31819,31820,31821],[34,761123,761125,761128],{"className":761124},[31685,31821],[34,761126,46900],{"className":761127},[31685,207923,31821],[34,761129,67764],{"className":761130,"style":32062},[31685,31686,31821],[34,761132,31829],{"className":761133},[31828],[34,761135,761137],{"className":761136},[31801],[34,761138,761140],{"className":761139,"style":41619},[31805],[34,761141],{},[34,761143],{"className":761144},[31690,41488],[34,761146,761148],{"className":761147,"style":75811},[31690,75810],[34,761149,2557],{"className":761150},[71281,141292],[34,761152,761154,761171],{"className":761153},[31642],[34,761155,761157],{"className":761156},[31646],[31648,761158,761159],{"xmlns":31650},[31652,761160,761161,761169],{},[31655,761162,761163],{},[37624,761164,761165,761167],{},[31658,761166,32936],{},[31749,761168,768],{},[31665,761170,583583],{"encoding":31667},[34,761172,761174],{"className":761173,"ariaHidden":758},[31672],[34,761175,761177,761180],{"className":761176},[31676],[34,761178],{"className":761179,"style":37774},[31680],[34,761181,761183,761186],{"className":761182},[31685],[34,761184,32936],{"className":761185},[31685,31686],[34,761187,761189],{"className":761188},[31792],[34,761190,761192],{"className":761191},[31796],[34,761193,761195],{"className":761194},[31801],[34,761196,761198],{"className":761197,"style":37774},[31805],[34,761199,761200,761203],{"style":37777},[34,761201],{"className":761202,"style":31814},[31813],[34,761204,761206],{"className":761205},[31818,31819,31820,31821],[34,761207,768],{"className":761208},[31685,31821],"，得出第二个重要结论：\n",[34,761211,761213,761293],{"className":761212},[31642],[34,761214,761216],{"className":761215},[31646],[31648,761217,761218],{"xmlns":31650},[31652,761219,761220,761290],{},[31655,761221,761222,761228,761250,761252,761274,761276],{},[37624,761223,761224,761226],{},[31658,761225,32936],{},[31749,761227,768],{},[41453,761229,761230,761240],{},[31655,761231,761232,761238],{},[37624,761233,761234,761236],{},[31658,761235,46900],{"mathvariant":38079},[31749,761237,768],{},[31658,761239,67764],{},[31655,761241,761242,761244],{},[31658,761243,46900],{"mathvariant":38079},[37624,761245,761246,761248],{},[31658,761247,32936],{},[31749,761249,768],{},[31662,761251,202],{},[41453,761253,761254,761264],{},[31655,761255,761256,761262],{},[37624,761257,761258,761260],{},[31658,761259,46900],{"mathvariant":38079},[31749,761261,768],{},[31658,761263,67764],{},[31655,761265,761266,761268],{},[31658,761267,46900],{"mathvariant":38079},[37624,761269,761270,761272],{},[31658,761271,33249],{},[31749,761273,768],{},[31662,761275,32226],{},[41453,761277,761278,761284],{},[31655,761279,761280,761282],{},[31658,761281,46900],{"mathvariant":38079},[31658,761283,67764],{},[31655,761285,761286,761288],{},[31658,761287,46900],{"mathvariant":38079},[31658,761289,33249],{},[31665,761291,761292],{"encoding":31667},"x^2 \\frac{\\mathrm{d}^2 y}{\\mathrm{d} x^2} = \\frac{\\mathrm{d}^2 y}{\\mathrm{d} t^2} - \\frac{\\mathrm{d} y}{\\mathrm{d} t}",[34,761294,761296,761466,761607],{"className":761295,"ariaHidden":758},[31672],[34,761297,761299,761302,761331,761457,761460,761463],{"className":761298},[31676],[34,761300],{"className":761301,"style":681304},[31680],[34,761303,761305,761308],{"className":761304},[31685],[34,761306,32936],{"className":761307},[31685,31686],[34,761309,761311],{"className":761310},[31792],[34,761312,761314],{"className":761313},[31796],[34,761315,761317],{"className":761316},[31801],[34,761318,761320],{"className":761319,"style":37774},[31805],[34,761321,761322,761325],{"style":37777},[34,761323],{"className":761324,"style":31814},[31813],[34,761326,761328],{"className":761327},[31818,31819,31820,31821],[34,761329,768],{"className":761330},[31685,31821],[34,761332,761334,761337,761454],{"className":761333},[31685],[34,761335],{"className":761336},[31782,41488],[34,761338,761340],{"className":761339},[41453],[34,761341,761343,761446],{"className":761342},[31796,31797],[34,761344,761346,761443],{"className":761345},[31801],[34,761347,761349,761392,761400],{"className":761348,"style":601858},[31805],[34,761350,761351,761354],{"style":41504},[34,761352],{"className":761353,"style":41508},[31813],[34,761355,761357],{"className":761356},[31818,31819,31820,31821],[34,761358,761360,761363],{"className":761359},[31685,31821],[34,761361,46900],{"className":761362},[31685,207923,31821],[34,761364,761366,761369],{"className":761365},[31685,31821],[34,761367,32936],{"className":761368},[31685,31686,31821],[34,761370,761372],{"className":761371},[31792],[34,761373,761375],{"className":761374},[31796],[34,761376,761378],{"className":761377},[31801],[34,761379,761381],{"className":761380,"style":548314},[31805],[34,761382,761383,761386],{"style":116668},[34,761384],{"className":761385,"style":43406},[31813],[34,761387,761389],{"className":761388},[31818,43410,43411,31821],[34,761390,768],{"className":761391},[31685,31821],[34,761393,761394,761397],{"style":41520},[34,761395],{"className":761396,"style":41508},[31813],[34,761398],{"className":761399,"style":41528},[41527],[34,761401,761402,761405],{"style":548368},[34,761403],{"className":761404,"style":41508},[31813],[34,761406,761408],{"className":761407},[31818,31819,31820,31821],[34,761409,761411,761440],{"className":761410},[31685,31821],[34,761412,761414,761417],{"className":761413},[31685,31821],[34,761415,46900],{"className":761416},[31685,207923,31821],[34,761418,761420],{"className":761419},[31792],[34,761421,761423],{"className":761422},[31796],[34,761424,761426],{"className":761425},[31801],[34,761427,761429],{"className":761428,"style":144123},[31805],[34,761430,761431,761434],{"style":552391},[34,761432],{"className":761433,"style":43406},[31813],[34,761435,761437],{"className":761436},[31818,43410,43411,31821],[34,761438,768],{"className":761439},[31685,31821],[34,761441,67764],{"className":761442,"style":32062},[31685,31686,31821],[34,761444,31829],{"className":761445},[31828],[34,761447,761449],{"className":761448},[31801],[34,761450,761452],{"className":761451,"style":41619},[31805],[34,761453],{},[34,761455],{"className":761456},[31690,41488],[34,761458],{"className":761459,"style":31846},[31845],[34,761461,202],{"className":761462},[31850],[34,761464],{"className":761465,"style":31846},[31845],[34,761467,761469,761472,761598,761601,761604],{"className":761468},[31676],[34,761470],{"className":761471,"style":681304},[31680],[34,761473,761475,761478,761595],{"className":761474},[31685],[34,761476],{"className":761477},[31782,41488],[34,761479,761481],{"className":761480},[41453],[34,761482,761484,761587],{"className":761483},[31796,31797],[34,761485,761487,761584],{"className":761486},[31801],[34,761488,761490,761533,761541],{"className":761489,"style":601858},[31805],[34,761491,761492,761495],{"style":41504},[34,761493],{"className":761494,"style":41508},[31813],[34,761496,761498],{"className":761497},[31818,31819,31820,31821],[34,761499,761501,761504],{"className":761500},[31685,31821],[34,761502,46900],{"className":761503},[31685,207923,31821],[34,761505,761507,761510],{"className":761506},[31685,31821],[34,761508,33249],{"className":761509},[31685,31686,31821],[34,761511,761513],{"className":761512},[31792],[34,761514,761516],{"className":761515},[31796],[34,761517,761519],{"className":761518},[31801],[34,761520,761522],{"className":761521,"style":548314},[31805],[34,761523,761524,761527],{"style":116668},[34,761525],{"className":761526,"style":43406},[31813],[34,761528,761530],{"className":761529},[31818,43410,43411,31821],[34,761531,768],{"className":761532},[31685,31821],[34,761534,761535,761538],{"style":41520},[34,761536],{"className":761537,"style":41508},[31813],[34,761539],{"className":761540,"style":41528},[41527],[34,761542,761543,761546],{"style":548368},[34,761544],{"className":761545,"style":41508},[31813],[34,761547,761549],{"className":761548},[31818,31819,31820,31821],[34,761550,761552,761581],{"className":761551},[31685,31821],[34,761553,761555,761558],{"className":761554},[31685,31821],[34,761556,46900],{"className":761557},[31685,207923,31821],[34,761559,761561],{"className":761560},[31792],[34,761562,761564],{"className":761563},[31796],[34,761565,761567],{"className":761566},[31801],[34,761568,761570],{"className":761569,"style":144123},[31805],[34,761571,761572,761575],{"style":552391},[34,761573],{"className":761574,"style":43406},[31813],[34,761576,761578],{"className":761577},[31818,43410,43411,31821],[34,761579,768],{"className":761580},[31685,31821],[34,761582,67764],{"className":761583,"style":32062},[31685,31686,31821],[34,761585,31829],{"className":761586},[31828],[34,761588,761590],{"className":761589},[31801],[34,761591,761593],{"className":761592,"style":41619},[31805],[34,761594],{},[34,761596],{"className":761597},[31690,41488],[34,761599],{"className":761600,"style":32604},[31845],[34,761602,32226],{"className":761603},[32024],[34,761605],{"className":761606,"style":32604},[31845],[34,761608,761610,761613],{"className":761609},[31676],[34,761611],{"className":761612,"style":608762},[31680],[34,761614,761616,761619,761684],{"className":761615},[31685],[34,761617],{"className":761618},[31782,41488],[34,761620,761622],{"className":761621},[41453],[34,761623,761625,761676],{"className":761624},[31796,31797],[34,761626,761628,761673],{"className":761627},[31801],[34,761629,761631,761648,761656],{"className":761630,"style":608935},[31805],[34,761632,761633,761636],{"style":41504},[34,761634],{"className":761635,"style":41508},[31813],[34,761637,761639],{"className":761638},[31818,31819,31820,31821],[34,761640,761642,761645],{"className":761641},[31685,31821],[34,761643,46900],{"className":761644},[31685,207923,31821],[34,761646,33249],{"className":761647},[31685,31686,31821],[34,761649,761650,761653],{"style":41520},[34,761651],{"className":761652,"style":41508},[31813],[34,761654],{"className":761655,"style":41528},[41527],[34,761657,761658,761661],{"style":548368},[34,761659],{"className":761660,"style":41508},[31813],[34,761662,761664],{"className":761663},[31818,31819,31820,31821],[34,761665,761667,761670],{"className":761666},[31685,31821],[34,761668,46900],{"className":761669},[31685,207923,31821],[34,761671,67764],{"className":761672,"style":32062},[31685,31686,31821],[34,761674,31829],{"className":761675},[31828],[34,761677,761679],{"className":761678},[31801],[34,761680,761682],{"className":761681,"style":41619},[31805],[34,761683],{},[34,761685],{"className":761686},[31690,41488],[15,761688,761689,761691,761692,258,761720,761778],{},[2627,761690,62532],{}," 这个变换把前面的变系数 ",[34,761693,761695,761708],{"className":761694},[31642],[34,761696,761698],{"className":761697},[31646],[31648,761699,761700],{"xmlns":31650},[31652,761701,761702,761706],{},[31655,761703,761704],{},[31658,761705,32936],{},[31665,761707,32936],{"encoding":31667},[34,761709,761711],{"className":761710,"ariaHidden":758},[31672],[34,761712,761714,761717],{"className":761713},[31676],[34,761715],{"className":761716,"style":32527},[31680],[34,761718,32936],{"className":761719},[31685,31686],[34,761721,761723,761740],{"className":761722},[31642],[34,761724,761726],{"className":761725},[31646],[31648,761727,761728],{"xmlns":31650},[31652,761729,761730,761738],{},[31655,761731,761732],{},[37624,761733,761734,761736],{},[31658,761735,32936],{},[31749,761737,768],{},[31665,761739,583583],{"encoding":31667},[34,761741,761743],{"className":761742,"ariaHidden":758},[31672],[34,761744,761746,761749],{"className":761745},[31676],[34,761747],{"className":761748,"style":37774},[31680],[34,761750,761752,761755],{"className":761751},[31685],[34,761753,32936],{"className":761754},[31685,31686],[34,761756,761758],{"className":761757},[31792],[34,761759,761761],{"className":761760},[31796],[34,761762,761764],{"className":761763},[31801],[34,761765,761767],{"className":761766,"style":37774},[31805],[34,761768,761769,761772],{"style":37777},[34,761770],{"className":761771,"style":31814},[31813],[34,761773,761775],{"className":761774},[31818,31819,31820,31821],[34,761776,768],{"className":761777},[31685,31821]," 完美消掉了，剩下的全是常数系数！这就变成了我们学过的常系数方程。",[32188,761780],{},[15,761782,761783,761786,761787,761992,761993,762052,762053,762134,762135,762196],{},[2627,761784,761785],{},"特别注意："," 在下面的两道小题中，题目把字母换了！\n题目写的是：",[34,761788,761790,761836],{"className":761789},[31642],[34,761791,761793],{"className":761792},[31646],[31648,761794,761795],{"xmlns":31650},[31652,761796,761797,761833],{},[31655,761798,761799,761805,761815,761817,761819,761825,761827,761829,761831],{},[37624,761800,761801,761803],{},[31658,761802,33249],{},[31749,761804,768],{},[37624,761806,761807,761809],{},[31658,761808,32936],{},[31655,761810,761811,761813],{},[31662,761812,116495],{"mathvariant":38079},[31662,761814,116495],{"mathvariant":38079},[31662,761816,223],{},[31658,761818,33249],{},[37624,761820,761821,761823],{},[31658,761822,32936],{},[31662,761824,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,761826,32226],{},[31658,761828,32936],{},[31662,761830,202],{},[31749,761832,459],{},[31665,761834,761835],{"encoding":31667},"t^2 x'' + t x' - x = 0",[34,761837,761839,761915,761965,761983],{"className":761838,"ariaHidden":758},[31672],[34,761840,761842,761845,761874,761906,761909,761912],{"className":761841},[31676],[34,761843],{"className":761844,"style":40044},[31680],[34,761846,761848,761851],{"className":761847},[31685],[34,761849,33249],{"className":761850},[31685,31686],[34,761852,761854],{"className":761853},[31792],[34,761855,761857],{"className":761856},[31796],[34,761858,761860],{"className":761859},[31801],[34,761861,761863],{"className":761862,"style":37774},[31805],[34,761864,761865,761868],{"style":37777},[34,761866],{"className":761867,"style":31814},[31813],[34,761869,761871],{"className":761870},[31818,31819,31820,31821],[34,761872,768],{"className":761873},[31685,31821],[34,761875,761877,761880],{"className":761876},[31685],[34,761878,32936],{"className":761879},[31685,31686],[34,761881,761883],{"className":761882},[31792],[34,761884,761886],{"className":761885},[31796],[34,761887,761889],{"className":761888},[31801],[34,761890,761892],{"className":761891,"style":132027},[31805],[34,761893,761894,761897],{"style":37777},[34,761895],{"className":761896,"style":31814},[31813],[34,761898,761900],{"className":761899},[31818,31819,31820,31821],[34,761901,761903],{"className":761902},[31685,31821],[34,761904,502782],{"className":761905},[31685,31821],[34,761907],{"className":761908,"style":32604},[31845],[34,761910,223],{"className":761911},[32024],[34,761913],{"className":761914,"style":32604},[31845],[34,761916,761918,761921,761924,761956,761959,761962],{"className":761917},[31676],[34,761919],{"className":761920,"style":384364},[31680],[34,761922,33249],{"className":761923},[31685,31686],[34,761925,761927,761930],{"className":761926},[31685],[34,761928,32936],{"className":761929},[31685,31686],[34,761931,761933],{"className":761932},[31792],[34,761934,761936],{"className":761935},[31796],[34,761937,761939],{"className":761938},[31801],[34,761940,761942],{"className":761941,"style":132027},[31805],[34,761943,761944,761947],{"style":37777},[34,761945],{"className":761946,"style":31814},[31813],[34,761948,761950],{"className":761949},[31818,31819,31820,31821],[34,761951,761953],{"className":761952},[31685,31821],[34,761954,116495],{"className":761955},[31685,31821],[34,761957],{"className":761958,"style":32604},[31845],[34,761960,32226],{"className":761961},[32024],[34,761963],{"className":761964,"style":32604},[31845],[34,761966,761968,761971,761974,761977,761980],{"className":761967},[31676],[34,761969],{"className":761970,"style":32527},[31680],[34,761972,32936],{"className":761973},[31685,31686],[34,761975],{"className":761976,"style":31846},[31845],[34,761978,202],{"className":761979},[31850],[34,761981],{"className":761982,"style":31846},[31845],[34,761984,761986,761989],{"className":761985},[31676],[34,761987],{"className":761988,"style":32617},[31680],[34,761990,459],{"className":761991},[31685],"。\n这里：",[2627,761994,761995,761996,756430,762024],{},"自变量是 ",[34,761997,761999,762012],{"className":761998},[31642],[34,762000,762002],{"className":762001},[31646],[31648,762003,762004],{"xmlns":31650},[31652,762005,762006,762010],{},[31655,762007,762008],{},[31658,762009,33249],{},[31665,762011,33249],{"encoding":31667},[34,762013,762015],{"className":762014,"ariaHidden":758},[31672],[34,762016,762018,762021],{"className":762017},[31676],[34,762019],{"className":762020,"style":33280},[31680],[34,762022,33249],{"className":762023},[31685,31686],[34,762025,762027,762040],{"className":762026},[31642],[34,762028,762030],{"className":762029},[31646],[31648,762031,762032],{"xmlns":31650},[31652,762033,762034,762038],{},[31655,762035,762036],{},[31658,762037,32936],{},[31665,762039,32936],{"encoding":31667},[34,762041,762043],{"className":762042,"ariaHidden":758},[31672],[34,762044,762046,762049],{"className":762045},[31676],[34,762047],{"className":762048,"style":32527},[31680],[34,762050,32936],{"className":762051},[31685,31686],"。\n为了不混淆，我们令变换为 ",[34,762054,762056,762078],{"className":762055},[31642],[34,762057,762059],{"className":762058},[31646],[31648,762060,762061],{"xmlns":31650},[31652,762062,762063,762075],{},[31655,762064,762065,762067,762069],{},[31658,762066,33249],{},[31662,762068,202],{},[37624,762070,762071,762073],{},[31658,762072,33244],{"mathvariant":38079},[31658,762074,31747],{},[31665,762076,762077],{"encoding":31667},"t = \\mathrm{e}^s",[34,762079,762081,762099],{"className":762080,"ariaHidden":758},[31672],[34,762082,762084,762087,762090,762093,762096],{"className":762083},[31676],[34,762085],{"className":762086,"style":33280},[31680],[34,762088,33249],{"className":762089},[31685,31686],[34,762091],{"className":762092,"style":31846},[31845],[34,762094,202],{"className":762095},[31850],[34,762097],{"className":762098,"style":31846},[31845],[34,762100,762102,762105],{"className":762101},[31676],[34,762103],{"className":762104,"style":37959},[31680],[34,762106,762108,762111],{"className":762107},[31685],[34,762109,33244],{"className":762110},[31685,207923],[34,762112,762114],{"className":762113},[31792],[34,762115,762117],{"className":762116},[31796],[34,762118,762120],{"className":762119},[31801],[34,762121,762123],{"className":762122,"style":37959},[31805],[34,762124,762125,762128],{"style":37777},[34,762126],{"className":762127,"style":31814},[31813],[34,762129,762131],{"className":762130},[31818,31819,31820,31821],[34,762132,31747],{"className":762133},[31685,31686,31821]," （也就是引进新变量 ",[34,762136,762138,762160],{"className":762137},[31642],[34,762139,762141],{"className":762140},[31646],[31648,762142,762143],{"xmlns":31650},[31652,762144,762145,762157],{},[31655,762146,762147,762149,762151,762153,762155],{},[31658,762148,31747],{},[31662,762150,202],{},[31658,762152,563286],{},[31662,762154,32682],{},[31658,762156,33249],{},[31665,762158,762159],{"encoding":31667},"s = \\ln t",[34,762161,762163,762181],{"className":762162,"ariaHidden":758},[31672],[34,762164,762166,762169,762172,762175,762178],{"className":762165},[31676],[34,762167],{"className":762168,"style":32527},[31680],[34,762170,31747],{"className":762171},[31685,31686],[34,762173],{"className":762174,"style":31846},[31845],[34,762176,202],{"className":762177},[31850],[34,762179],{"className":762180,"style":31846},[31845],[34,762182,762184,762187,762190,762193],{"className":762183},[31676],[34,762185],{"className":762186,"style":31681},[31680],[34,762188,563286],{"className":762189},[32767],[34,762191],{"className":762192,"style":32123},[31845],[34,762194,33249],{"className":762195},[31685,31686],"）。\n根据刚才的推导，我们有直接替换公式：",[105,762198,762199,762381],{},[108,762200,762201,762268,762269],{},[34,762202,762204,762224],{"className":762203},[31642],[34,762205,762207],{"className":762206},[31646],[31648,762208,762209],{"xmlns":31650},[31652,762210,762211,762221],{},[31655,762212,762213,762215],{},[31658,762214,33249],{},[37624,762216,762217,762219],{},[31658,762218,32936],{},[31662,762220,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,762222,762223],{"encoding":31667},"t x'",[34,762225,762227],{"className":762226,"ariaHidden":758},[31672],[34,762228,762230,762233,762236],{"className":762229},[31676],[34,762231],{"className":762232,"style":132027},[31680],[34,762234,33249],{"className":762235},[31685,31686],[34,762237,762239,762242],{"className":762238},[31685],[34,762240,32936],{"className":762241},[31685,31686],[34,762243,762245],{"className":762244},[31792],[34,762246,762248],{"className":762247},[31796],[34,762249,762251],{"className":762250},[31801],[34,762252,762254],{"className":762253,"style":132027},[31805],[34,762255,762256,762259],{"style":37777},[34,762257],{"className":762258,"style":31814},[31813],[34,762260,762262],{"className":762261},[31818,31819,31820,31821],[34,762263,762265],{"className":762264},[31685,31821],[34,762266,116495],{"className":762267},[31685,31821]," 替换为 ",[34,762270,762272,762298],{"className":762271},[31642],[34,762273,762275],{"className":762274},[31646],[31648,762276,762277],{"xmlns":31650},[31652,762278,762279,762295],{},[31655,762280,762281],{},[41453,762282,762283,762289],{},[31655,762284,762285,762287],{},[31658,762286,46900],{"mathvariant":38079},[31658,762288,32936],{},[31655,762290,762291,762293],{},[31658,762292,46900],{"mathvariant":38079},[31658,762294,31747],{},[31665,762296,762297],{"encoding":31667},"\\frac{\\mathrm{d}x}{\\mathrm{d}s}",[34,762299,762301],{"className":762300,"ariaHidden":758},[31672],[34,762302,762304,762307],{"className":762303},[31676],[34,762305],{"className":762306,"style":232492},[31680],[34,762308,762310,762313,762378],{"className":762309},[31685],[34,762311],{"className":762312},[31782,41488],[34,762314,762316],{"className":762315},[41453],[34,762317,762319,762370],{"className":762318},[31796,31797],[34,762320,762322,762367],{"className":762321},[31801],[34,762323,762325,762342,762350],{"className":762324,"style":232511},[31805],[34,762326,762327,762330],{"style":41504},[34,762328],{"className":762329,"style":41508},[31813],[34,762331,762333],{"className":762332},[31818,31819,31820,31821],[34,762334,762336,762339],{"className":762335},[31685,31821],[34,762337,46900],{"className":762338},[31685,207923,31821],[34,762340,31747],{"className":762341},[31685,31686,31821],[34,762343,762344,762347],{"style":41520},[34,762345],{"className":762346,"style":41508},[31813],[34,762348],{"className":762349,"style":41528},[41527],[34,762351,762352,762355],{"style":46771},[34,762353],{"className":762354,"style":41508},[31813],[34,762356,762358],{"className":762357},[31818,31819,31820,31821],[34,762359,762361,762364],{"className":762360},[31685,31821],[34,762362,46900],{"className":762363},[31685,207923,31821],[34,762365,32936],{"className":762366},[31685,31686,31821],[34,762368,31829],{"className":762369},[31828],[34,762371,762373],{"className":762372},[31801],[34,762374,762376],{"className":762375,"style":41619},[31805],[34,762377],{},[34,762379],{"className":762380},[31690,41488],[108,762382,762383,762268,762484],{},[34,762384,762386,762414],{"className":762385},[31642],[34,762387,762389],{"className":762388},[31646],[31648,762390,762391],{"xmlns":31650},[31652,762392,762393,762411],{},[31655,762394,762395,762401],{},[37624,762396,762397,762399],{},[31658,762398,33249],{},[31749,762400,768],{},[37624,762402,762403,762405],{},[31658,762404,32936],{},[31655,762406,762407,762409],{},[31662,762408,116495],{"mathvariant":38079},[31662,762410,116495],{"mathvariant":38079},[31665,762412,762413],{"encoding":31667},"t^2 x''",[34,762415,762417],{"className":762416,"ariaHidden":758},[31672],[34,762418,762420,762423,762452],{"className":762419},[31676],[34,762421],{"className":762422,"style":37774},[31680],[34,762424,762426,762429],{"className":762425},[31685],[34,762427,33249],{"className":762428},[31685,31686],[34,762430,762432],{"className":762431},[31792],[34,762433,762435],{"className":762434},[31796],[34,762436,762438],{"className":762437},[31801],[34,762439,762441],{"className":762440,"style":37774},[31805],[34,762442,762443,762446],{"style":37777},[34,762444],{"className":762445,"style":31814},[31813],[34,762447,762449],{"className":762448},[31818,31819,31820,31821],[34,762450,768],{"className":762451},[31685,31821],[34,762453,762455,762458],{"className":762454},[31685],[34,762456,32936],{"className":762457},[31685,31686],[34,762459,762461],{"className":762460},[31792],[34,762462,762464],{"className":762463},[31796],[34,762465,762467],{"className":762466},[31801],[34,762468,762470],{"className":762469,"style":132027},[31805],[34,762471,762472,762475],{"style":37777},[34,762473],{"className":762474,"style":31814},[31813],[34,762476,762478],{"className":762477},[31818,31819,31820,31821],[34,762479,762481],{"className":762480},[31685,31821],[34,762482,502782],{"className":762483},[31685,31821],[34,762485,762487,762537],{"className":762486},[31642],[34,762488,762490],{"className":762489},[31646],[31648,762491,762492],{"xmlns":31650},[31652,762493,762494,762534],{},[31655,762495,762496,762518,762520],{},[41453,762497,762498,762508],{},[31655,762499,762500,762506],{},[37624,762501,762502,762504],{},[31658,762503,46900],{"mathvariant":38079},[31749,762505,768],{},[31658,762507,32936],{},[31655,762509,762510,762512],{},[31658,762511,46900],{"mathvariant":38079},[37624,762513,762514,762516],{},[31658,762515,31747],{},[31749,762517,768],{},[31662,762519,32226],{},[41453,762521,762522,762528],{},[31655,762523,762524,762526],{},[31658,762525,46900],{"mathvariant":38079},[31658,762527,32936],{},[31655,762529,762530,762532],{},[31658,762531,46900],{"mathvariant":38079},[31658,762533,31747],{},[31665,762535,762536],{"encoding":31667},"\\frac{\\mathrm{d}^2x}{\\mathrm{d}s^2} - 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\\frac{\\mathrm{d}^2x}{\\mathrm{d}s^2} - \\frac{\\mathrm{d}x}{\\mathrm{d}s} \\right) + \\frac{\\mathrm{d}x}{\\mathrm{d}s} - x = 0",[34,763061,763063,763302,763391,763409],{"className":763062,"ariaHidden":758},[31672],[34,763064,763066,763069,763293,763296,763299],{"className":763065},[31676],[34,763067],{"className":763068,"style":141282},[31680],[34,763070,763072,763078,763204,763207,763210,763213,763287],{"className":763071},[34006],[34,763073,763075],{"className":763074,"style":75811},[31782,75810],[34,763076,456],{"className":763077},[71281,141292],[34,763079,763081,763084,763201],{"className":763080},[31685],[34,763082],{"className":763083},[31782,41488],[34,763085,763087],{"className":763086},[41453],[34,763088,763090,763193],{"className":763089},[31796,31797],[34,763091,763093,763190],{"className":763092},[31801],[34,763094,763096,763139,763147],{"className":763095,"style":603956},[31805],[34,763097,763098,763101],{"style":41504},[34,763099],{"className":763100,"style":41508},[31813],[34,763102,763104],{"className":763103},[31818,31819,31820,31821],[34,763105,763107,763110],{"className":763106},[31685,31821],[34,763108,46900],{"className":763109},[31685,207923,31821],[34,763111,763113,763116],{"className":763112},[31685,31821],[34,763114,31747],{"className":763115},[31685,31686,31821],[34,763117,763119],{"className":763118},[31792],[34,763120,763122],{"className":763121},[31796],[34,763123,763125],{"className":763124},[31801],[34,763126,763128],{"className":763127,"style":548314},[31805],[34,763129,763130,763133],{"style":116668},[34,763131],{"className":763132,"style":43406},[31813],[34,763134,763136],{"className":763135},[31818,43410,43411,31821],[34,763137,768],{"className":763138},[31685,31821],[34,763140,763141,763144],{"style":41520},[34,763142],{"className":763143,"style":41508},[31813],[34,763145],{"className":763146,"style":41528},[41527],[34,763148,763149,763152],{"style":46771},[34,763150],{"className":763151,"style":41508},[31813],[34,763153,763155],{"className":763154},[31818,31819,31820,31821],[34,763156,763158,763187],{"className":763157},[31685,31821],[34,763159,763161,763164],{"className":763160},[31685,31821],[34,763162,46900],{"className":763163},[31685,207923,31821],[34,763165,763167],{"className":763166},[31792],[34,763168,763170],{"className":763169},[31796],[34,763171,763173],{"className":763172},[31801],[34,763174,763176],{"className":763175,"style":144123},[31805],[34,763177,763178,763181],{"style":552391},[34,763179],{"className":763180,"style":43406},[31813],[34,763182,763184],{"className":763183},[31818,43410,43411,31821],[34,763185,768],{"className":763186},[31685,31821],[34,763188,32936],{"className":763189},[31685,31686,31821],[34,763191,31829],{"className":763192},[31828],[34,763194,763196],{"className":763195},[31801],[34,763197,763199],{"className":763198,"style":41619},[31805],[34,763200],{},[34,763202],{"className":763203},[31690,41488],[34,763205],{"className":763206,"style":32604},[31845],[34,763208,32226],{"className":763209},[32024],[34,763211],{"className":763212,"style":32604},[31845],[34,763214,763216,763219,763284],{"className":763215},[31685],[34,763217],{"className":763218},[31782,41488],[34,763220,763222],{"className":763221},[41453],[34,763223,763225,763276],{"className":763224},[31796,31797],[34,763226,763228,763273],{"className":763227},[31801],[34,763229,763231,763248,763256],{"className":763230,"style":232511},[31805],[34,763232,763233,763236],{"style":41504},[34,763234],{"className":763235,"style":41508},[31813],[34,763237,763239],{"className":763238},[31818,31819,31820,31821],[34,763240,763242,763245],{"className":763241},[31685,31821],[34,763243,46900],{"className":763244},[31685,207923,31821],[34,763246,31747],{"className":763247},[31685,31686,31821],[34,763249,763250,763253],{"style":41520},[34,763251],{"className":763252,"style":41508},[31813],[34,763254],{"className":763255,"style":41528},[41527],[34,763257,763258,763261],{"style":46771},[34,763259],{"className":763260,"style":41508},[31813],[34,763262,763264],{"className":763263},[31818,31819,31820,31821],[34,763265,763267,763270],{"className":763266},[31685,31821],[34,763268,46900],{"className":763269},[31685,207923,31821],[34,763271,32936],{"className":763272},[31685,31686,31821],[34,763274,31829],{"className":763275},[31828],[34,763277,763279],{"className":763278},[31801],[34,763280,763282],{"className":763281,"style":41619},[31805],[34,763283],{},[34,763285],{"className":763286},[31690,41488],[34,763288,763290],{"className":763289,"style":75811},[31690,75810],[34,763291,2557],{"className":763292},[71281,141292],[34,763294],{"className":763295,"style":32604},[31845],[34,763297,223],{"className":763298},[32024],[34,763300],{"className":763301,"style":32604},[31845],[34,763303,763305,763308,763382,763385,763388],{"className":763304},[31676],[34,763306],{"className":763307,"style":232492},[31680],[34,763309,763311,763314,763379],{"className":763310},[31685],[34,763312],{"className":763313},[31782,41488],[34,763315,763317],{"className":763316},[41453],[34,763318,763320,763371],{"className":763319},[31796,31797],[34,763321,763323,763368],{"className":763322},[31801],[34,763324,763326,763343,763351],{"className":763325,"style":232511},[31805],[34,763327,763328,763331],{"style":41504},[34,763329],{"className":763330,"style":41508},[31813],[34,763332,763334],{"className":763333},[31818,31819,31820,31821],[34,763335,763337,763340],{"className":763336},[31685,31821],[34,763338,46900],{"className":763339},[31685,207923,31821],[34,763341,31747],{"className":763342},[31685,31686,31821],[34,763344,763345,763348],{"style":41520},[34,763346],{"className":763347,"style":41508},[31813],[34,763349],{"className":763350,"style":41528},[41527],[34,763352,763353,763356],{"style":46771},[34,763354],{"className":763355,"style":41508},[31813],[34,763357,763359],{"className":763358},[31818,31819,31820,31821],[34,763360,763362,763365],{"className":763361},[31685,31821],[34,763363,46900],{"className":763364},[31685,207923,31821],[34,763366,32936],{"className":763367},[31685,31686,31821],[34,763369,31829],{"className":763370},[31828],[34,763372,763374],{"className":763373},[31801],[34,763375,763377],{"className":763376,"style":41619},[31805],[34,763378],{},[34,763380],{"className":763381},[31690,41488],[34,763383],{"className":763384,"style":32604},[31845],[34,763386,32226],{"className":763387},[32024],[34,763389],{"className":763390,"style":32604},[31845],[34,763392,763394,763397,763400,763403,763406],{"className":763393},[31676],[34,763395],{"className":763396,"style":32527},[31680],[34,763398,32936],{"className":763399},[31685,31686],[34,763401],{"className":763402,"style":31846},[31845],[34,763404,202],{"className":763405},[31850],[34,763407],{"className":763408,"style":31846},[31845],[34,763410,763412,763415],{"className":763411},[31676],[34,763413],{"className":763414,"style":32617},[31680],[34,763416,459],{"className":763417},[31685],[108,763419,763420,763423,763639],{},[2627,763421,763422],{},"化简：",[34,763424,763426,763468],{"className":763425},[31642],[34,763427,763429],{"className":763428},[31646],[31648,763430,763431],{"xmlns":31650},[31652,763432,763433,763465],{},[31655,763434,763435,763457,763459,763461,763463],{},[41453,763436,763437,763447],{},[31655,763438,763439,763445],{},[37624,763440,763441,763443],{},[31658,763442,46900],{"mathvariant":38079},[31749,763444,768],{},[31658,763446,32936],{},[31655,763448,763449,763451],{},[31658,763450,46900],{"mathvariant":38079},[37624,763452,763453,763455],{},[31658,763454,31747],{},[31749,763456,768],{},[31662,763458,32226],{},[31658,763460,32936],{},[31662,763462,202],{},[31749,763464,459],{},[31665,763466,763467],{"encoding":31667},"\\frac{\\mathrm{d}^2x}{\\mathrm{d}s^2} - x = 0",[34,763469,763471,763612,763630],{"className":763470,"ariaHidden":758},[31672],[34,763472,763474,763477,763603,763606,763609],{"className":763473},[31676],[34,763475],{"className":763476,"style":637019},[31680],[34,763478,763480,763483,763600],{"className":763479},[31685],[34,763481],{"className":763482},[31782,41488],[34,763484,763486],{"className":763485},[41453],[34,763487,763489,763592],{"className":763488},[31796,31797],[34,763490,763492,763589],{"className":763491},[31801],[34,763493,763495,763538,763546],{"className":763494,"style":603956},[31805],[34,763496,763497,763500],{"style":41504},[34,763498],{"className":763499,"style":41508},[31813],[34,763501,763503],{"className":763502},[31818,31819,31820,31821],[34,763504,763506,763509],{"className":763505},[31685,31821],[34,763507,46900],{"className":763508},[31685,207923,31821],[34,763510,763512,763515],{"className":763511},[31685,31821],[34,763513,31747],{"className":763514},[31685,31686,31821],[34,763516,763518],{"className":763517},[31792],[34,763519,763521],{"className":763520},[31796],[34,763522,763524],{"className":763523},[31801],[34,763525,763527],{"className":763526,"style":548314},[31805],[34,763528,763529,763532],{"style":116668},[34,763530],{"className":763531,"style":43406},[31813],[34,763533,763535],{"className":763534},[31818,43410,43411,31821],[34,763536,768],{"className":763537},[31685,31821],[34,763539,763540,763543],{"style":41520},[34,763541],{"className":763542,"style":41508},[31813],[34,763544],{"className":763545,"style":41528},[41527],[34,763547,763548,763551],{"style":46771},[34,763549],{"className":763550,"style":41508},[31813],[34,763552,763554],{"className":763553},[31818,31819,31820,31821],[34,763555,763557,763586],{"className":763556},[31685,31821],[34,763558,763560,763563],{"className":763559},[31685,31821],[34,763561,46900],{"className":763562},[31685,207923,31821],[34,763564,763566],{"className":763565},[31792],[34,763567,763569],{"className":763568},[31796],[34,763570,763572],{"className":763571},[31801],[34,763573,763575],{"className":763574,"style":144123},[31805],[34,763576,763577,763580],{"style":552391},[34,763578],{"className":763579,"style":43406},[31813],[34,763581,763583],{"className":763582},[31818,43410,43411,31821],[34,763584,768],{"className":763585},[31685,31821],[34,763587,32936],{"className":763588},[31685,31686,31821],[34,763590,31829],{"className":763591},[31828],[34,763593,763595],{"className":763594},[31801],[34,763596,763598],{"className":763597,"style":41619},[31805],[34,763599],{},[34,763601],{"className":763602},[31690,41488],[34,763604],{"className":763605,"style":32604},[31845],[34,763607,32226],{"className":763608},[32024],[34,763610],{"className":763611,"style":32604},[31845],[34,763613,763615,763618,763621,763624,763627],{"className":763614},[31676],[34,763616],{"className":763617,"style":32527},[31680],[34,763619,32936],{"className":763620},[31685,31686],[34,763622],{"className":763623,"style":31846},[31845],[34,763625,202],{"className":763626},[31850],[34,763628],{"className":763629,"style":31846},[31845],[34,763631,763633,763636],{"className":763632},[31676],[34,763634],{"className":763635,"style":32617},[31680],[34,763637,459],{"className":763638},[31685],"\n化简后变成了一个简单的常系数方程。",[108,763641,763642,763673,763674,763836,763837,347],{},[2627,763643,763644,763645,414527],{},"求解（关于 ",[34,763646,763648,763661],{"className":763647},[31642],[34,763649,763651],{"className":763650},[31646],[31648,763652,763653],{"xmlns":31650},[31652,763654,763655,763659],{},[31655,763656,763657],{},[31658,763658,31747],{},[31665,763660,31747],{"encoding":31667},[34,763662,763664],{"className":763663,"ariaHidden":758},[31672],[34,763665,763667,763670],{"className":763666},[31676],[34,763668],{"className":763669,"style":32527},[31680],[34,763671,31747],{"className":763672},[31685,31686],"\n特征方程：",[34,763675,763677,763717],{"className":763676},[31642],[34,763678,763680],{"className":763679},[31646],[31648,763681,763682],{"xmlns":31650},[31652,763683,763684,763714],{},[31655,763685,763686,763692,763694,763696,763698,763700,763702,763704,763706,763708,763710,763712],{},[37624,763687,763688,763690],{},[31658,763689,220792],{},[31749,763691,768],{},[31662,763693,32226],{},[31749,763695,640],{},[31662,763697,202],{},[31749,763699,459],{},[39361,763701,262622],{},[31662,763703,262625],{},[39361,763705,262622],{},[31658,763707,220792],{},[31662,763709,202],{},[31662,763711,41461],{},[31749,763713,640],{},[31665,763715,763716],{"encoding":31667},"\\lambda^2 - 1 = 0 \\implies \\lambda = \\pm 1",[34,763718,763720,763764,763782,763806,763824],{"className":763719,"ariaHidden":758},[31672],[34,763721,763723,763726,763755,763758,763761],{"className":763722},[31676],[34,763724],{"className":763725,"style":40044},[31680],[34,763727,763729,763732],{"className":763728},[31685],[34,763730,220792],{"className":763731},[31685,31686],[34,763733,763735],{"className":763734},[31792],[34,763736,763738],{"className":763737},[31796],[34,763739,763741],{"className":763740},[31801],[34,763742,763744],{"className":763743,"style":37774},[31805],[34,763745,763746,763749],{"style":37777},[34,763747],{"className":763748,"style":31814},[31813],[34,763750,763752],{"className":763751},[31818,31819,31820,31821],[34,763753,768],{"className":763754},[31685,31821],[34,763756],{"className":763757,"style":32604},[31845],[34,763759,32226],{"className":763760},[32024],[34,763762],{"className":763763,"style":32604},[31845],[34,763765,763767,763770,763773,763776,763779],{"className":763766},[31676],[34,763768],{"className":763769,"style":32617},[31680],[34,763771,640],{"className":763772},[31685],[34,763774],{"className":763775,"style":31846},[31845],[34,763777,202],{"className":763778},[31850],[34,763780],{"className":763781,"style":31846},[31845],[34,763783,763785,763788,763791,763794,763797,763800,763803],{"className":763784},[31676],[34,763786],{"className":763787,"style":262700},[31680],[34,763789,459],{"className":763790},[31685],[34,763792],{"className":763793,"style":31846},[31845],[34,763795],{"className":763796,"style":31846},[31845],[34,763798,262625],{"className":763799},[31850],[34,763801],{"className":763802,"style":31846},[31845],[34,763804],{"className":763805,"style":31846},[31845],[34,763807,763809,763812,763815,763818,763821],{"className":763808},[31676],[34,763810],{"className":763811,"style":31681},[31680],[34,763813,220792],{"className":763814},[31685,31686],[34,763816],{"className":763817,"style":31846},[31845],[34,763819,202],{"className":763820},[31850],[34,763822],{"className":763823,"style":31846},[31845],[34,763825,763827,763830,763833],{"className":763826},[31676],[34,763828],{"className":763829,"style":35942},[31680],[34,763831,41461],{"className":763832},[31685],[34,763834,640],{"className":763835},[31685],"。\n解为：",[34,763838,763840,763886],{"className":763839},[31642],[34,763841,763843],{"className":763842},[31646],[31648,763844,763845],{"xmlns":31650},[31652,763846,763847,763883],{},[31655,763848,763849,763851,763853,763859,763865,763867,763873],{},[31658,763850,32936],{},[31662,763852,202],{},[31743,763854,763855,763857],{},[31658,763856,31965],{},[31749,763858,640],{},[37624,763860,763861,763863],{},[31658,763862,33244],{"mathvariant":38079},[31658,763864,31747],{},[31662,763866,223],{},[31743,763868,763869,763871],{},[31658,763870,31965],{},[31749,763872,768],{},[37624,763874,763875,763877],{},[31658,763876,33244],{"mathvariant":38079},[31655,763878,763879,763881],{},[31662,763880,32226],{},[31658,763882,31747],{},[31665,763884,763885],{"encoding":31667},"x = c_1 \\mathrm{e}^s + c_2 \\mathrm{e}^{-s}",[34,763887,763889,763907,763991],{"className":763888,"ariaHidden":758},[31672],[34,763890,763892,763895,763898,763901,763904],{"className":763891},[31676],[34,763893],{"className":763894,"style":32527},[31680],[34,763896,32936],{"className":763897},[31685,31686],[34,763899],{"className":763900,"style":31846},[31845],[34,763902,202],{"className":763903},[31850],[34,763905],{"className":763906,"style":31846},[31845],[34,763908,763910,763913,763953,763982,763985,763988],{"className":763909},[31676],[34,763911],{"className":763912,"style":40578},[31680],[34,763914,763916,763919],{"className":763915},[31685],[34,763917,31965],{"className":763918},[31685,31686],[34,763920,763922],{"className":763921},[31792],[34,763923,763925,763945],{"className":763924},[31796,31797],[34,763926,763928,763942],{"className":763927},[31801],[34,763929,763931],{"className":763930,"style":31806},[31805],[34,763932,763933,763936],{"style":31809},[34,763934],{"className":763935,"style":31814},[31813],[34,763937,763939],{"className":763938},[31818,31819,31820,31821],[34,763940,640],{"className":763941},[31685,31821],[34,763943,31829],{"className":763944},[31828],[34,763946,763948],{"className":763947},[31801],[34,763949,763951],{"className":763950,"style":31836},[31805],[34,763952],{},[34,763954,763956,763959],{"className":763955},[31685],[34,763957,33244],{"className":763958},[31685,207923],[34,763960,763962],{"className":763961},[31792],[34,763963,763965],{"className":763964},[31796],[34,763966,763968],{"className":763967},[31801],[34,763969,763971],{"className":763970,"style":37959},[31805],[34,763972,763973,763976],{"style":37777},[34,763974],{"className":763975,"style":31814},[31813],[34,763977,763979],{"className":763978},[31818,31819,31820,31821],[34,763980,31747],{"className":763981},[31685,31686,31821],[34,763983],{"className":763984,"style":32604},[31845],[34,763986,223],{"className":763987},[32024],[34,763989],{"className":763990,"style":32604},[31845],[34,763992,763994,763998,764038],{"className":763993},[31676],[34,763995],{"className":763996,"style":763997},[31680],"height:0.9213em;vertical-align:-0.15em;",[34,763999,764001,764004],{"className":764000},[31685],[34,764002,31965],{"className":764003},[31685,31686],[34,764005,764007],{"className":764006},[31792],[34,764008,764010,764030],{"className":764009},[31796,31797],[34,764011,764013,764027],{"className":764012},[31801],[34,764014,764016],{"className":764015,"style":31806},[31805],[34,764017,764018,764021],{"style":31809},[34,764019],{"className":764020,"style":31814},[31813],[34,764022,764024],{"className":764023},[31818,31819,31820,31821],[34,764025,768],{"className":764026},[31685,31821],[34,764028,31829],{"className":764029},[31828],[34,764031,764033],{"className":764032},[31801],[34,764034,764036],{"className":764035,"style":31836},[31805],[34,764037],{},[34,764039,764041,764044],{"className":764040},[31685],[34,764042,33244],{"className":764043},[31685,207923],[34,764045,764047],{"className":764046},[31792],[34,764048,764050],{"className":764049},[31796],[34,764051,764053],{"className":764052},[31801],[34,764054,764056],{"className":764055,"style":156828},[31805],[34,764057,764058,764061],{"style":37777},[34,764059],{"className":764060,"style":31814},[31813],[34,764062,764064],{"className":764063},[31818,31819,31820,31821],[34,764065,764067,764070],{"className":764066},[31685,31821],[34,764068,32226],{"className":764069},[31685,31821],[34,764071,31747],{"className":764072},[31685,31686,31821],[108,764074,764075,315440,764135,13977,764215,46528,764296,39984,764456,764458],{},[2627,764076,764077,764078,764106,764107,414527],{},"回代变量（从 ",[34,764079,764081,764094],{"className":764080},[31642],[34,764082,764084],{"className":764083},[31646],[31648,764085,764086],{"xmlns":31650},[31652,764087,764088,764092],{},[31655,764089,764090],{},[31658,764091,31747],{},[31665,764093,31747],{"encoding":31667},[34,764095,764097],{"className":764096,"ariaHidden":758},[31672],[34,764098,764100,764103],{"className":764099},[31676],[34,764101],{"className":764102,"style":32527},[31680],[34,764104,31747],{"className":764105},[31685,31686]," 变回 ",[34,764108,764110,764123],{"className":764109},[31642],[34,764111,764113],{"className":764112},[31646],[31648,764114,764115],{"xmlns":31650},[31652,764116,764117,764121],{},[31655,764118,764119],{},[31658,764120,33249],{},[31665,764122,33249],{"encoding":31667},[34,764124,764126],{"className":764125,"ariaHidden":758},[31672],[34,764127,764129,764132],{"className":764128},[31676],[34,764130],{"className":764131,"style":33280},[31680],[34,764133,33249],{"className":764134},[31685,31686],[34,764136,764138,764159],{"className":764137},[31642],[34,764139,764141],{"className":764140},[31646],[31648,764142,764143],{"xmlns":31650},[31652,764144,764145,764157],{},[31655,764146,764147,764149,764151],{},[31658,764148,33249],{},[31662,764150,202],{},[37624,764152,764153,764155],{},[31658,764154,33244],{"mathvariant":38079},[31658,764156,31747],{},[31665,764158,762077],{"encoding":31667},[34,764160,764162,764180],{"className":764161,"ariaHidden":758},[31672],[34,764163,764165,764168,764171,764174,764177],{"className":764164},[31676],[34,764166],{"className":764167,"style":33280},[31680],[34,764169,33249],{"className":764170},[31685,31686],[34,764172],{"className":764173,"style":31846},[31845],[34,764175,202],{"className":764176},[31850],[34,764178],{"className":764179,"style":31846},[31845],[34,764181,764183,764186],{"className":764182},[31676],[34,764184],{"className":764185,"style":37959},[31680],[34,764187,764189,764192],{"className":764188},[31685],[34,764190,33244],{"className":764191},[31685,207923],[34,764193,764195],{"className":764194},[31792],[34,764196,764198],{"className":764197},[31796],[34,764199,764201],{"className":764200},[31801],[34,764202,764204],{"className":764203,"style":37959},[31805],[34,764205,764206,764209],{"style":37777},[34,764207],{"className":764208,"style":31814},[31813],[34,764210,764212],{"className":764211},[31818,31819,31820,31821],[34,764213,31747],{"className":764214},[31685,31686,31821],[34,764216,764218,764240],{"className":764217},[31642],[34,764219,764221],{"className":764220},[31646],[31648,764222,764223],{"xmlns":31650},[31652,764224,764225,764237],{},[31655,764226,764227,764233,764235],{},[37624,764228,764229,764231],{},[31658,764230,33244],{"mathvariant":38079},[31658,764232,31747],{},[31662,764234,202],{},[31658,764236,33249],{},[31665,764238,764239],{"encoding":31667},"\\mathrm{e}^s = t",[34,764241,764243,764287],{"className":764242,"ariaHidden":758},[31672],[34,764244,764246,764249,764278,764281,764284],{"className":764245},[31676],[34,764247],{"className":764248,"style":37959},[31680],[34,764250,764252,764255],{"className":764251},[31685],[34,764253,33244],{"className":764254},[31685,207923],[34,764256,764258],{"className":764257},[31792],[34,764259,764261],{"className":764260},[31796],[34,764262,764264],{"className":764263},[31801],[34,764265,764267],{"className":764266,"style":37959},[31805],[34,764268,764269,764272],{"style":37777},[34,764270],{"className":764271,"style":31814},[31813],[34,764273,764275],{"className":764274},[31818,31819,31820,31821],[34,764276,31747],{"className":764277},[31685,31686,31821],[34,764279],{"className":764280,"style":31846},[31845],[34,764282,202],{"className":764283},[31850],[34,764285],{"className":764286,"style":31846},[31845],[34,764288,764290,764293],{"className":764289},[31676],[34,764291],{"className":764292,"style":33280},[31680],[34,764294,33249],{"className":764295},[31685,31686],[34,764297,764299,764329],{"className":764298},[31642],[34,764300,764302],{"className":764301},[31646],[31648,764303,764304],{"xmlns":31650},[31652,764305,764306,764326],{},[31655,764307,764308,764318,764320],{},[37624,764309,764310,764312],{},[31658,764311,33244],{"mathvariant":38079},[31655,764313,764314,764316],{},[31662,764315,32226],{},[31658,764317,31747],{},[31662,764319,202],{},[41453,764321,764322,764324],{},[31749,764323,640],{},[31658,764325,33249],{},[31665,764327,764328],{"encoding":31667},"\\mathrm{e}^{-s} = \\frac{1}{t}",[34,764330,764332,764382],{"className":764331,"ariaHidden":758},[31672],[34,764333,764335,764338,764373,764376,764379],{"className":764334},[31676],[34,764336],{"className":764337,"style":156828},[31680],[34,764339,764341,764344],{"className":764340},[31685],[34,764342,33244],{"className":764343},[31685,207923],[34,764345,764347],{"className":764346},[31792],[34,764348,764350],{"className":764349},[31796],[34,764351,764353],{"className":764352},[31801],[34,764354,764356],{"className":764355,"style":156828},[31805],[34,764357,764358,764361],{"style":37777},[34,764359],{"className":764360,"style":31814},[31813],[34,764362,764364],{"className":764363},[31818,31819,31820,31821],[34,764365,764367,764370],{"className":764366},[31685,31821],[34,764368,32226],{"className":764369},[31685,31821],[34,764371,31747],{"className":764372},[31685,31686,31821],[34,764374],{"className":764375,"style":31846},[31845],[34,764377,202],{"className":764378},[31850],[34,764380],{"className":764381,"style":31846},[31845],[34,764383,764385,764388],{"className":764384},[31676],[34,764386],{"className":764387,"style":46727},[31680],[34,764389,764391,764394,764453],{"className":764390},[31685],[34,764392],{"className":764393},[31782,41488],[34,764395,764397],{"className":764396},[41453],[34,764398,764400,764445],{"className":764399},[31796,31797],[34,764401,764403,764442],{"className":764402},[31801],[34,764404,764406,764420,764428],{"className":764405,"style":46746},[31805],[34,764407,764408,764411],{"style":41504},[34,764409],{"className":764410,"style":41508},[31813],[34,764412,764414],{"className":764413},[31818,31819,31820,31821],[34,764415,764417],{"className":764416},[31685,31821],[34,764418,33249],{"className":764419},[31685,31686,31821],[34,764421,764422,764425],{"style":41520},[34,764423],{"className":764424,"style":41508},[31813],[34,764426],{"className":764427,"style":41528},[41527],[34,764429,764430,764433],{"style":46771},[34,764431],{"className":764432,"style":41508},[31813],[34,764434,764436],{"className":764435},[31818,31819,31820,31821],[34,764437,764439],{"className":764438},[31685,31821],[34,764440,640],{"className":764441},[31685,31821],[34,764443,31829],{"className":764444},[31828],[34,764446,764448],{"className":764447},[31801],[34,764449,764451],{"className":764450,"style":41619},[31805],[34,764452],{},[34,764454],{"className":764455},[31690,41488],[2627,764457,177510],{},[34,764459,764461,764497],{"className":764460},[31642],[34,764462,764464],{"className":764463},[31646],[31648,764465,764466],{"xmlns":31650},[31652,764467,764468,764494],{},[31655,764469,764470,764472,764474,764480,764482,764484],{},[31658,764471,32936],{},[31662,764473,202],{},[31743,764475,764476,764478],{},[31658,764477,31965],{},[31749,764479,640],{},[31658,764481,33249],{},[31662,764483,223],{},[41453,764485,764486,764492],{},[31743,764487,764488,764490],{},[31658,764489,31965],{},[31749,764491,768],{},[31658,764493,33249],{},[31665,764495,764496],{"encoding":31667},"x = c_1 t + \\frac{c_2}{t}",[34,764498,764500,764518,764576],{"className":764499,"ariaHidden":758},[31672],[34,764501,764503,764506,764509,764512,764515],{"className":764502},[31676],[34,764504],{"className":764505,"style":32527},[31680],[34,764507,32936],{"className":764508},[31685,31686],[34,764510],{"className":764511,"style":31846},[31845],[34,764513,202],{"className":764514},[31850],[34,764516],{"className":764517,"style":31846},[31845],[34,764519,764521,764524,764564,764567,764570,764573],{"className":764520},[31676],[34,764522],{"className":764523,"style":83872},[31680],[34,764525,764527,764530],{"className":764526},[31685],[34,764528,31965],{"className":764529},[31685,31686],[34,764531,764533],{"className":764532},[31792],[34,764534,764536,764556],{"className":764535},[31796,31797],[34,764537,764539,764553],{"className":764538},[31801],[34,764540,764542],{"className":764541,"style":31806},[31805],[34,764543,764544,764547],{"style":31809},[34,764545],{"className":764546,"style":31814},[31813],[34,764548,764550],{"className":764549},[31818,31819,31820,31821],[34,764551,640],{"className":764552},[31685,31821],[34,764554,31829],{"className":764555},[31828],[34,764557,764559],{"className":764558},[31801],[34,764560,764562],{"className":764561,"style":31836},[31805],[34,764563],{},[34,764565,33249],{"className":764566},[31685,31686],[34,764568],{"className":764569,"style":32604},[31845],[34,764571,223],{"className":764572},[32024],[34,764574],{"className":764575,"style":32604},[31845],[34,764577,764579,764582],{"className":764578},[31676],[34,764580],{"className":764581,"style":755238},[31680],[34,764583,764585,764588,764684],{"className":764584},[31685],[34,764586],{"className":764587},[31782,41488],[34,764589,764591],{"className":764590},[41453],[34,764592,764594,764676],{"className":764593},[31796,31797],[34,764595,764597,764673],{"className":764596},[31801],[34,764598,764600,764614,764622],{"className":764599,"style":318628},[31805],[34,764601,764602,764605],{"style":41504},[34,764603],{"className":764604,"style":41508},[31813],[34,764606,764608],{"className":764607},[31818,31819,31820,31821],[34,764609,764611],{"className":764610},[31685,31821],[34,764612,33249],{"className":764613},[31685,31686,31821],[34,764615,764616,764619],{"style":41520},[34,764617],{"className":764618,"style":41508},[31813],[34,764620],{"className":764621,"style":41528},[41527],[34,764623,764624,764627],{"style":220490},[34,764625],{"className":764626,"style":41508},[31813],[34,764628,764630],{"className":764629},[31818,31819,31820,31821],[34,764631,764633],{"className":764632},[31685,31821],[34,764634,764636,764639],{"className":764635},[31685,31821],[34,764637,31965],{"className":764638},[31685,31686,31821],[34,764640,764642],{"className":764641},[31792],[34,764643,764645,764665],{"className":764644},[31796,31797],[34,764646,764648,764662],{"className":764647},[31801],[34,764649,764651],{"className":764650,"style":318658},[31805],[34,764652,764653,764656],{"style":318661},[34,764654],{"className":764655,"style":43406},[31813],[34,764657,764659],{"className":764658},[31818,43410,43411,31821],[34,764660,768],{"className":764661},[31685,31821],[34,764663,31829],{"className":764664},[31828],[34,764666,764668],{"className":764667},[31801],[34,764669,764671],{"className":764670,"style":220540},[31805],[34,764672],{},[34,764674,31829],{"className":764675},[31828],[34,764677,764679],{"className":764678},[31801],[34,764680,764682],{"className":764681,"style":41619},[31805],[34,764683],{},[34,764685],{"className":764686},[31690,41488],[32188,764688],{},[15,764690,764691],{},[2627,764692,388194,764693],{},[34,764694,764696,764746],{"className":764695},[31642],[34,764697,764699],{"className":764698},[31646],[31648,764700,764701],{"xmlns":31650},[31652,764702,764703,764743],{},[31655,764704,764705,764711,764721,764723,764725,764727,764733,764735,764737,764739,764741],{},[37624,764706,764707,764709],{},[31658,764708,33249],{},[31749,764710,768],{},[37624,764712,764713,764715],{},[31658,764714,32936],{},[31655,764716,764717,764719],{},[31662,764718,116495],{"mathvariant":38079},[31662,764720,116495],{"mathvariant":38079},[31662,764722,32226],{},[31749,764724,15932],{},[31658,764726,33249],{},[37624,764728,764729,764731],{},[31658,764730,32936],{},[31662,764732,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,764734,223],{},[31749,764736,36656],{},[31658,764738,32936],{},[31662,764740,202],{},[31658,764742,33249],{},[31665,764744,764745],{"encoding":31667},"t^2 x'' - 4t x' + 6x = t",[34,764747,764749,764825,764878,764899],{"className":764748,"ariaHidden":758},[31672],[34,764750,764752,764755,764784,764816,764819,764822],{"className":764751},[31676],[34,764753],{"className":764754,"style":40044},[31680],[34,764756,764758,764761],{"className":764757},[31685],[34,764759,33249],{"className":764760},[31685,31686],[34,764762,764764],{"className":764763},[31792],[34,764765,764767],{"className":764766},[31796],[34,764768,764770],{"className":764769},[31801],[34,764771,764773],{"className":764772,"style":37774},[31805],[34,764774,764775,764778],{"style":37777},[34,764776],{"className":764777,"style":31814},[31813],[34,764779,764781],{"className":764780},[31818,31819,31820,31821],[34,764782,768],{"className":764783},[31685,31821],[34,764785,764787,764790],{"className":764786},[31685],[34,764788,32936],{"className":764789},[31685,31686],[34,764791,764793],{"className":764792},[31792],[34,764794,764796],{"className":764795},[31796],[34,764797,764799],{"className":764798},[31801],[34,764800,764802],{"className":764801,"style":132027},[31805],[34,764803,764804,764807],{"style":37777},[34,764805],{"className":764806,"style":31814},[31813],[34,764808,764810],{"className":764809},[31818,31819,31820,31821],[34,764811,764813],{"className":764812},[31685,31821],[34,764814,502782],{"className":764815},[31685,31821],[34,764817],{"className":764818,"style":32604},[31845],[34,764820,32226],{"className":764821},[32024],[34,764823],{"className":764824,"style":32604},[31845],[34,764826,764828,764831,764834,764837,764869,764872,764875],{"className":764827},[31676],[34,764829],{"className":764830,"style":384364},[31680],[34,764832,15932],{"className":764833},[31685],[34,764835,33249],{"className":764836},[31685,31686],[34,764838,764840,764843],{"className":764839},[31685],[34,764841,32936],{"className":764842},[31685,31686],[34,764844,764846],{"className":764845},[31792],[34,764847,764849],{"className":764848},[31796],[34,764850,764852],{"className":764851},[31801],[34,764853,764855],{"className":764854,"style":132027},[31805],[34,764856,764857,764860],{"style":37777},[34,764858],{"className":764859,"style":31814},[31813],[34,764861,764863],{"className":764862},[31818,31819,31820,31821],[34,764864,764866],{"className":764865},[31685,31821],[34,764867,116495],{"className":764868},[31685,31821],[34,764870],{"className":764871,"style":32604},[31845],[34,764873,223],{"className":764874},[32024],[34,764876],{"className":764877,"style":32604},[31845],[34,764879,764881,764884,764887,764890,764893,764896],{"className":764880},[31676],[34,764882],{"className":764883,"style":32617},[31680],[34,764885,36656],{"className":764886},[31685],[34,764888,32936],{"className":764889},[31685,31686],[34,764891],{"className":764892,"style":31846},[31845],[34,764894,202],{"className":764895},[31850],[34,764897],{"className":764898,"style":31846},[31845],[34,764900,764902,764905],{"className":764901},[31676],[34,764903],{"className":764904,"style":33280},[31680],[34,764906,33249],{"className":764907},[31685,31686],[2230,764909,764910,765514,765881,766426,767170,767575],{},[108,764911,764912,764995,764996,765425,765426,765454,765455,347],{},[2627,764913,764914,764915,414527],{},"代入变换（令 ",[34,764916,764918,764939],{"className":764917},[31642],[34,764919,764921],{"className":764920},[31646],[31648,764922,764923],{"xmlns":31650},[31652,764924,764925,764937],{},[31655,764926,764927,764929,764931],{},[31658,764928,33249],{},[31662,764930,202],{},[37624,764932,764933,764935],{},[31658,764934,33244],{"mathvariant":38079},[31658,764936,31747],{},[31665,764938,762077],{"encoding":31667},[34,764940,764942,764960],{"className":764941,"ariaHidden":758},[31672],[34,764943,764945,764948,764951,764954,764957],{"className":764944},[31676],[34,764946],{"className":764947,"style":33280},[31680],[34,764949,33249],{"className":764950},[31685,31686],[34,764952],{"className":764953,"style":31846},[31845],[34,764955,202],{"className":764956},[31850],[34,764958],{"className":764959,"style":31846},[31845],[34,764961,764963,764966],{"className":764962},[31676],[34,764964],{"className":764965,"style":37959},[31680],[34,764967,764969,764972],{"className":764968},[31685],[34,764970,33244],{"className":764971},[31685,207923],[34,764973,764975],{"className":764974},[31792],[34,764976,764978],{"className":764977},[31796],[34,764979,764981],{"className":764980},[31801],[34,764982,764984],{"className":764983,"style":37959},[31805],[34,764985,764986,764989],{"style":37777},[34,764987],{"className":764988,"style":31814},[31813],[34,764990,764992],{"className":764991},[31818,31819,31820,31821],[34,764993,31747],{"className":764994},[31685,31686,31821],"\n左边：\n",[34,764997,764999,765079],{"className":764998},[31642],[34,765000,765002],{"className":765001},[31646],[31648,765003,765004],{"xmlns":31650},[31652,765005,765006,765076],{},[31655,765007,765008,765052,765054,765056,765070,765072,765074],{},[31655,765009,765010,765012,765034,765036,765050],{},[31662,765011,456],{"fence":758},[41453,765013,765014,765024],{},[31655,765015,765016,765022],{},[37624,765017,765018,765020],{},[31658,765019,46900],{"mathvariant":38079},[31749,765021,768],{},[31658,765023,32936],{},[31655,765025,765026,765028],{},[31658,765027,46900],{"mathvariant":38079},[37624,765029,765030,765032],{},[31658,765031,31747],{},[31749,765033,768],{},[31662,765035,32226],{},[41453,765037,765038,765044],{},[31655,765039,765040,765042],{},[31658,765041,46900],{"mathvariant":38079},[31658,765043,32936],{},[31655,765045,765046,765048],{},[31658,765047,46900],{"mathvariant":38079},[31658,765049,31747],{},[31662,765051,2557],{"fence":758},[31662,765053,32226],{},[31749,765055,15932],{},[41453,765057,765058,765064],{},[31655,765059,765060,765062],{},[31658,765061,46900],{"mathvariant":38079},[31658,765063,32936],{},[31655,765065,765066,765068],{},[31658,765067,46900],{"mathvariant":38079},[31658,765069,31747],{},[31662,765071,223],{},[31749,765073,36656],{},[31658,765075,32936],{},[31665,765077,765078],{"encoding":31667},"\\left( \\frac{\\mathrm{d}^2x}{\\mathrm{d}s^2} - \\frac{\\mathrm{d}x}{\\mathrm{d}s} \\right) - 4\\frac{\\mathrm{d}x}{\\mathrm{d}s} + 6x",[34,765080,765082,765321,765413],{"className":765081,"ariaHidden":758},[31672],[34,765083,765085,765088,765312,765315,765318],{"className":765084},[31676],[34,765086],{"className":765087,"style":141282},[31680],[34,765089,765091,765097,765223,765226,765229,765232,765306],{"className":765090},[34006],[34,765092,765094],{"className":765093,"style":75811},[31782,75810],[34,765095,456],{"className":765096},[71281,141292],[34,765098,765100,765103,765220],{"className":765099},[31685],[34,765101],{"className":765102},[31782,41488],[34,765104,765106],{"className":765105},[41453],[34,765107,765109,765212],{"className":765108},[31796,31797],[34,765110,765112,765209],{"className":765111},[31801],[34,765113,765115,765158,765166],{"className":765114,"style":603956},[31805],[34,765116,765117,765120],{"style":41504},[34,765118],{"className":765119,"style":41508},[31813],[34,765121,765123],{"className":765122},[31818,31819,31820,31821],[34,765124,765126,765129],{"className":765125},[31685,31821],[34,765127,46900],{"className":765128},[31685,207923,31821],[34,765130,765132,765135],{"className":765131},[31685,31821],[34,765133,31747],{"className":765134},[31685,31686,31821],[34,765136,765138],{"className":765137},[31792],[34,765139,765141],{"className":765140},[31796],[34,765142,765144],{"className":765143},[31801],[34,765145,765147],{"className":765146,"style":548314},[31805],[34,765148,765149,765152],{"style":116668},[34,765150],{"className":765151,"style":43406},[31813],[34,765153,765155],{"className":765154},[31818,43410,43411,31821],[34,765156,768],{"className":765157},[31685,31821],[34,765159,765160,765163],{"style":41520},[34,765161],{"className":765162,"style":41508},[31813],[34,765164],{"className":765165,"style":41528},[41527],[34,765167,765168,765171],{"style":46771},[34,765169],{"className":765170,"style":41508},[31813],[34,765172,765174],{"className":765173},[31818,31819,31820,31821],[34,765175,765177,765206],{"className":765176},[31685,31821],[34,765178,765180,765183],{"className":765179},[31685,31821],[34,765181,46900],{"className":765182},[31685,207923,31821],[34,765184,765186],{"className":765185},[31792],[34,765187,765189],{"className":765188},[31796],[34,765190,765192],{"className":765191},[31801],[34,765193,765195],{"className":765194,"style":144123},[31805],[34,765196,765197,765200],{"style":552391},[34,765198],{"className":765199,"style":43406},[31813],[34,765201,765203],{"className":765202},[31818,43410,43411,31821],[34,765204,768],{"className":765205},[31685,31821],[34,765207,32936],{"className":765208},[31685,31686,31821],[34,765210,31829],{"className":765211},[31828],[34,765213,765215],{"className":765214},[31801],[34,765216,765218],{"className":765217,"style":41619},[31805],[34,765219],{},[34,765221],{"className":765222},[31690,41488],[34,765224],{"className":765225,"style":32604},[31845],[34,765227,32226],{"className":765228},[32024],[34,765230],{"className":765231,"style":32604},[31845],[34,765233,765235,765238,765303],{"className":765234},[31685],[34,765236],{"className":765237},[31782,41488],[34,765239,765241],{"className":765240},[41453],[34,765242,765244,765295],{"className":765243},[31796,31797],[34,765245,765247,765292],{"className":765246},[31801],[34,765248,765250,765267,765275],{"className":765249,"style":232511},[31805],[34,765251,765252,765255],{"style":41504},[34,765253],{"className":765254,"style":41508},[31813],[34,765256,765258],{"className":765257},[31818,31819,31820,31821],[34,765259,765261,765264],{"className":765260},[31685,31821],[34,765262,46900],{"className":765263},[31685,207923,31821],[34,765265,31747],{"className":765266},[31685,31686,31821],[34,765268,765269,765272],{"style":41520},[34,765270],{"className":765271,"style":41508},[31813],[34,765273],{"className":765274,"style":41528},[41527],[34,765276,765277,765280],{"style":46771},[34,765278],{"className":765279,"style":41508},[31813],[34,765281,765283],{"className":765282},[31818,31819,31820,31821],[34,765284,765286,765289],{"className":765285},[31685,31821],[34,765287,46900],{"className":765288},[31685,207923,31821],[34,765290,32936],{"className":765291},[31685,31686,31821],[34,765293,31829],{"className":765294},[31828],[34,765296,765298],{"className":765297},[31801],[34,765299,765301],{"className":765300,"style":41619},[31805],[34,765302],{},[34,765304],{"className":765305},[31690,41488],[34,765307,765309],{"className":765308,"style":75811},[31690,75810],[34,765310,2557],{"className":765311},[71281,141292],[34,765313],{"className":765314,"style":32604},[31845],[34,765316,32226],{"className":765317},[32024],[34,765319],{"className":765320,"style":32604},[31845],[34,765322,765324,765327,765330,765404,765407,765410],{"className":765323},[31676],[34,765325],{"className":765326,"style":232492},[31680],[34,765328,15932],{"className":765329},[31685],[34,765331,765333,765336,765401],{"className":765332},[31685],[34,765334],{"className":765335},[31782,41488],[34,765337,765339],{"className":765338},[41453],[34,765340,765342,765393],{"className":765341},[31796,31797],[34,765343,765345,765390],{"className":765344},[31801],[34,765346,765348,765365,765373],{"className":765347,"style":232511},[31805],[34,765349,765350,765353],{"style":41504},[34,765351],{"className":765352,"style":41508},[31813],[34,765354,765356],{"className":765355},[31818,31819,31820,31821],[34,765357,765359,765362],{"className":765358},[31685,31821],[34,765360,46900],{"className":765361},[31685,207923,31821],[34,765363,31747],{"className":765364},[31685,31686,31821],[34,765366,765367,765370],{"style":41520},[34,765368],{"className":765369,"style":41508},[31813],[34,765371],{"className":765372,"style":41528},[41527],[34,765374,765375,765378],{"style":46771},[34,765376],{"className":765377,"style":41508},[31813],[34,765379,765381],{"className":765380},[31818,31819,31820,31821],[34,765382,765384,765387],{"className":765383},[31685,31821],[34,765385,46900],{"className":765386},[31685,207923,31821],[34,765388,32936],{"className":765389},[31685,31686,31821],[34,765391,31829],{"className":765392},[31828],[34,765394,765396],{"className":765395},[31801],[34,765397,765399],{"className":765398,"style":41619},[31805],[34,765400],{},[34,765402],{"className":765403},[31690,41488],[34,765405],{"className":765406,"style":32604},[31845],[34,765408,223],{"className":765409},[32024],[34,765411],{"className":765412,"style":32604},[31845],[34,765414,765416,765419,765422],{"className":765415},[31676],[34,765417],{"className":765418,"style":32617},[31680],[34,765420,36656],{"className":765421},[31685],[34,765423,32936],{"className":765424},[31685,31686],"\n右边：",[34,765427,765429,765442],{"className":765428},[31642],[34,765430,765432],{"className":765431},[31646],[31648,765433,765434],{"xmlns":31650},[31652,765435,765436,765440],{},[31655,765437,765438],{},[31658,765439,33249],{},[31665,765441,33249],{"encoding":31667},[34,765443,765445],{"className":765444,"ariaHidden":758},[31672],[34,765446,765448,765451],{"className":765447},[31676],[34,765449],{"className":765450,"style":33280},[31680],[34,765452,33249],{"className":765453},[31685,31686]," 变成了 ",[34,765456,765458,765476],{"className":765457},[31642],[34,765459,765461],{"className":765460},[31646],[31648,765462,765463],{"xmlns":31650},[31652,765464,765465,765473],{},[31655,765466,765467],{},[37624,765468,765469,765471],{},[31658,765470,33244],{"mathvariant":38079},[31658,765472,31747],{},[31665,765474,765475],{"encoding":31667},"\\mathrm{e}^s",[34,765477,765479],{"className":765478,"ariaHidden":758},[31672],[34,765480,765482,765485],{"className":765481},[31676],[34,765483],{"className":765484,"style":37959},[31680],[34,765486,765488,765491],{"className":765487},[31685],[34,765489,33244],{"className":765490},[31685,207923],[34,765492,765494],{"className":765493},[31792],[34,765495,765497],{"className":765496},[31796],[34,765498,765500],{"className":765499},[31801],[34,765501,765503],{"className":765502,"style":37959},[31805],[34,765504,765505,765508],{"style":37777},[34,765506],{"className":765507,"style":31814},[31813],[34,765509,765511],{"className":765510},[31818,31819,31820,31821],[34,765512,31747],{"className":765513},[31685,31686,31821],[108,765515,765516,765519,765880],{},[2627,765517,765518],{},"整理方程：",[34,765520,765522,765588],{"className":765521},[31642],[34,765523,765525],{"className":765524},[31646],[31648,765526,765527],{"xmlns":31650},[31652,765528,765529,765585],{},[31655,765530,765531,765553,765555,765557,765571,765573,765575,765577,765579],{},[41453,765532,765533,765543],{},[31655,765534,765535,765541],{},[37624,765536,765537,765539],{},[31658,765538,46900],{"mathvariant":38079},[31749,765540,768],{},[31658,765542,32936],{},[31655,765544,765545,765547],{},[31658,765546,46900],{"mathvariant":38079},[37624,765548,765549,765551],{},[31658,765550,31747],{},[31749,765552,768],{},[31662,765554,32226],{},[31749,765556,18136],{},[41453,765558,765559,765565],{},[31655,765560,765561,765563],{},[31658,765562,46900],{"mathvariant":38079},[31658,765564,32936],{},[31655,765566,765567,765569],{},[31658,765568,46900],{"mathvariant":38079},[31658,765570,31747],{},[31662,765572,223],{},[31749,765574,36656],{},[31658,765576,32936],{},[31662,765578,202],{},[37624,765580,765581,765583],{},[31658,765582,33244],{"mathvariant":38079},[31658,765584,31747],{},[31665,765586,765587],{"encoding":31667},"\\frac{\\mathrm{d}^2x}{\\mathrm{d}s^2} - 5\\frac{\\mathrm{d}x}{\\mathrm{d}s} + 6x = \\mathrm{e}^s",[34,765589,765591,765732,765824,765845],{"className":765590,"ariaHidden":758},[31672],[34,765592,765594,765597,765723,765726,765729],{"className":765593},[31676],[34,765595],{"className":765596,"style":637019},[31680],[34,765598,765600,765603,765720],{"className":765599},[31685],[34,765601],{"className":765602},[31782,41488],[34,765604,765606],{"className":765605},[41453],[34,765607,765609,765712],{"className":765608},[31796,31797],[34,765610,765612,765709],{"className":765611},[31801],[34,765613,765615,765658,765666],{"className":765614,"style":603956},[31805],[34,765616,765617,765620],{"style":41504},[34,765618],{"className":765619,"style":41508},[31813],[34,765621,765623],{"className":765622},[31818,31819,31820,31821],[34,765624,765626,765629],{"className":765625},[31685,31821],[34,765627,46900],{"className":765628},[31685,207923,31821],[34,765630,765632,765635],{"className":765631},[31685,31821],[34,765633,31747],{"className":765634},[31685,31686,31821],[34,765636,765638],{"className":765637},[31792],[34,765639,765641],{"className":765640},[31796],[34,765642,765644],{"className":765643},[31801],[34,765645,765647],{"className":765646,"style":548314},[31805],[34,765648,765649,765652],{"style":116668},[34,765650],{"className":765651,"style":43406},[31813],[34,765653,765655],{"className":765654},[31818,43410,43411,31821],[34,765656,768],{"className":765657},[31685,31821],[34,765659,765660,765663],{"style":41520},[34,765661],{"className":765662,"style":41508},[31813],[34,765664],{"className":765665,"style":41528},[41527],[34,765667,765668,765671],{"style":46771},[34,765669],{"className":765670,"style":41508},[31813],[34,765672,765674],{"className":765673},[31818,31819,31820,31821],[34,765675,765677,765706],{"className":765676},[31685,31821],[34,765678,765680,765683],{"className":765679},[31685,31821],[34,765681,46900],{"className":765682},[31685,207923,31821],[34,765684,765686],{"className":765685},[31792],[34,765687,765689],{"className":765688},[31796],[34,765690,765692],{"className":765691},[31801],[34,765693,765695],{"className":765694,"style":144123},[31805],[34,765696,765697,765700],{"style":552391},[34,765698],{"className":765699,"style":43406},[31813],[34,765701,765703],{"className":765702},[31818,43410,43411,31821],[34,765704,768],{"className":765705},[31685,31821],[34,765707,32936],{"className":765708},[31685,31686,31821],[34,765710,31829],{"className":765711},[31828],[34,765713,765715],{"className":765714},[31801],[34,765716,765718],{"className":765717,"style":41619},[31805],[34,765719],{},[34,765721],{"className":765722},[31690,41488],[34,765724],{"className":765725,"style":32604},[31845],[34,765727,32226],{"className":765728},[32024],[34,765730],{"className":765731,"style":32604},[31845],[34,765733,765735,765738,765741,765815,765818,765821],{"className":765734},[31676],[34,765736],{"className":765737,"style":232492},[31680],[34,765739,18136],{"className":765740},[31685],[34,765742,765744,765747,765812],{"className":765743},[31685],[34,765745],{"className":765746},[31782,41488],[34,765748,765750],{"className":765749},[41453],[34,765751,765753,765804],{"className":765752},[31796,31797],[34,765754,765756,765801],{"className":765755},[31801],[34,765757,765759,765776,765784],{"className":765758,"style":232511},[31805],[34,765760,765761,765764],{"style":41504},[34,765762],{"className":765763,"style":41508},[31813],[34,765765,765767],{"className":765766},[31818,31819,31820,31821],[34,765768,765770,765773],{"className":765769},[31685,31821],[34,765771,46900],{"className":765772},[31685,207923,31821],[34,765774,31747],{"className":765775},[31685,31686,31821],[34,765777,765778,765781],{"style":41520},[34,765779],{"className":765780,"style":41508},[31813],[34,765782],{"className":765783,"style":41528},[41527],[34,765785,765786,765789],{"style":46771},[34,765787],{"className":765788,"style":41508},[31813],[34,765790,765792],{"className":765791},[31818,31819,31820,31821],[34,765793,765795,765798],{"className":765794},[31685,31821],[34,765796,46900],{"className":765797},[31685,207923,31821],[34,765799,32936],{"className":765800},[31685,31686,31821],[34,765802,31829],{"className":765803},[31828],[34,765805,765807],{"className":765806},[31801],[34,765808,765810],{"className":765809,"style":41619},[31805],[34,765811],{},[34,765813],{"className":765814},[31690,41488],[34,765816],{"className":765817,"style":32604},[31845],[34,765819,223],{"className":765820},[32024],[34,765822],{"className":765823,"style":32604},[31845],[34,765825,765827,765830,765833,765836,765839,765842],{"className":765826},[31676],[34,765828],{"className":765829,"style":32617},[31680],[34,765831,36656],{"className":765832},[31685],[34,765834,32936],{"className":765835},[31685,31686],[34,765837],{"className":765838,"style":31846},[31845],[34,765840,202],{"className":765841},[31850],[34,765843],{"className":765844,"style":31846},[31845],[34,765846,765848,765851],{"className":765847},[31676],[34,765849],{"className":765850,"style":37959},[31680],[34,765852,765854,765857],{"className":765853},[31685],[34,765855,33244],{"className":765856},[31685,207923],[34,765858,765860],{"className":765859},[31792],[34,765861,765863],{"className":765862},[31796],[34,765864,765866],{"className":765865},[31801],[34,765867,765869],{"className":765868,"style":37959},[31805],[34,765870,765871,765874],{"style":37777},[34,765872],{"className":765873,"style":31814},[31813],[34,765875,765877],{"className":765876},[31818,31819,31820,31821],[34,765878,31747],{"className":765879},[31685,31686,31821],"\n这是一个非齐次常系数方程。",[108,765882,765883,763673,765886,766139,766140,347],{},[2627,765884,765885],{},"解齐次部分：",[34,765887,765889,765951],{"className":765888},[31642],[34,765890,765892],{"className":765891},[31646],[31648,765893,765894],{"xmlns":31650},[31652,765895,765896,765948],{},[31655,765897,765898,765904,765906,765908,765910,765912,765914,765916,765918,765920,765922,765924,765926,765928,765930,765932,765934,765936,765938,765940,765942,765944,765946],{},[37624,765899,765900,765902],{},[31658,765901,220792],{},[31749,765903,768],{},[31662,765905,32226],{},[31749,765907,18136],{},[31658,765909,220792],{},[31662,765911,223],{},[31749,765913,36656],{},[31662,765915,202],{},[31749,765917,459],{},[39361,765919,262622],{},[31662,765921,262625],{},[39361,765923,262622],{},[31662,765925,456],{"stretchy":702},[31658,765927,220792],{},[31662,765929,32226],{},[31749,765931,768],{},[31662,765933,2557],{"stretchy":702},[31662,765935,456],{"stretchy":702},[31658,765937,220792],{},[31662,765939,32226],{},[31749,765941,786],{},[31662,765943,2557],{"stretchy":702},[31662,765945,202],{},[31749,765947,459],{},[31665,765949,765950],{"encoding":31667},"\\lambda^2 - 5\\lambda + 6 = 0 \\implies (\\lambda-2)(\\lambda-3)=0",[34,765952,765954,765998,766019,766037,766061,766082,766109,766130],{"className":765953,"ariaHidden":758},[31672],[34,765955,765957,765960,765989,765992,765995],{"className":765956},[31676],[34,765958],{"className":765959,"style":40044},[31680],[34,765961,765963,765966],{"className":765962},[31685],[34,765964,220792],{"className":765965},[31685,31686],[34,765967,765969],{"className":765968},[31792],[34,765970,765972],{"className":765971},[31796],[34,765973,765975],{"className":765974},[31801],[34,765976,765978],{"className":765977,"style":37774},[31805],[34,765979,765980,765983],{"style":37777},[34,765981],{"className":765982,"style":31814},[31813],[34,765984,765986],{"className":765985},[31818,31819,31820,31821],[34,765987,768],{"className":765988},[31685,31821],[34,765990],{"className":765991,"style":32604},[31845],[34,765993,32226],{"className":765994},[32024],[34,765996],{"className":765997,"style":32604},[31845],[34,765999,766001,766004,766007,766010,766013,766016],{"className":766000},[31676],[34,766002],{"className":766003,"style":34299},[31680],[34,766005,18136],{"className":766006},[31685],[34,766008,220792],{"className":766009},[31685,31686],[34,766011],{"className":766012,"style":32604},[31845],[34,766014,223],{"className":766015},[32024],[34,766017],{"className":766018,"style":32604},[31845],[34,766020,766022,766025,766028,766031,766034],{"className":766021},[31676],[34,766023],{"className":766024,"style":32617},[31680],[34,766026,36656],{"className":766027},[31685],[34,766029],{"className":766030,"style":31846},[31845],[34,766032,202],{"className":766033},[31850],[34,766035],{"className":766036,"style":31846},[31845],[34,766038,766040,766043,766046,766049,766052,766055,766058],{"className":766039},[31676],[34,766041],{"className":766042,"style":262700},[31680],[34,766044,459],{"className":766045},[31685],[34,766047],{"className":766048,"style":31846},[31845],[34,766050],{"className":766051,"style":31846},[31845],[34,766053,262625],{"className":766054},[31850],[34,766056],{"className":766057,"style":31846},[31845],[34,766059],{"className":766060,"style":31846},[31845],[34,766062,766064,766067,766070,766073,766076,766079],{"className":766063},[31676],[34,766065],{"className":766066,"style":31774},[31680],[34,766068,456],{"className":766069},[31782],[34,766071,220792],{"className":766072},[31685,31686],[34,766074],{"className":766075,"style":32604},[31845],[34,766077,32226],{"className":766078},[32024],[34,766080],{"className":766081,"style":32604},[31845],[34,766083,766085,766088,766091,766094,766097,766100,766103,766106],{"className":766084},[31676],[34,766086],{"className":766087,"style":31774},[31680],[34,766089,768],{"className":766090},[31685],[34,766092,2557],{"className":766093},[31690],[34,766095,456],{"className":766096},[31782],[34,766098,220792],{"className":766099},[31685,31686],[34,766101],{"className":766102,"style":32604},[31845],[34,766104,32226],{"className":766105},[32024],[34,766107],{"className":766108,"style":32604},[31845],[34,766110,766112,766115,766118,766121,766124,766127],{"className":766111},[31676],[34,766113],{"className":766114,"style":31774},[31680],[34,766116,786],{"className":766117},[31685],[34,766119,2557],{"className":766120},[31690],[34,766122],{"className":766123,"style":31846},[31845],[34,766125,202],{"className":766126},[31850],[34,766128],{"className":766129,"style":31846},[31845],[34,766131,766133,766136],{"className":766132},[31676],[34,766134],{"className":766135,"style":32617},[31680],[34,766137,459],{"className":766138},[31685],"。\n齐次解：",[34,766141,766143,766197],{"className":766142},[31642],[34,766144,766146],{"className":766145},[31646],[31648,766147,766148],{"xmlns":31650},[31652,766149,766150,766194],{},[31655,766151,766152,766158,766160,766166,766176,766178,766184],{},[31743,766153,766154,766156],{},[31658,766155,32936],{},[31658,766157,47292],{},[31662,766159,202],{},[31743,766161,766162,766164],{},[31658,766163,31965],{},[31749,766165,640],{},[37624,766167,766168,766170],{},[31658,766169,33244],{"mathvariant":38079},[31655,766171,766172,766174],{},[31749,766173,768],{},[31658,766175,31747],{},[31662,766177,223],{},[31743,766179,766180,766182],{},[31658,766181,31965],{},[31749,766183,768],{},[37624,766185,766186,766188],{},[31658,766187,33244],{"mathvariant":38079},[31655,766189,766190,766192],{},[31749,766191,786],{},[31658,766193,31747],{},[31665,766195,766196],{"encoding":31667},"x_h = c_1 \\mathrm{e}^{2s} + c_2 \\mathrm{e}^{3s}",[34,766198,766200,766255,766345],{"className":766199,"ariaHidden":758},[31672],[34,766201,766203,766206,766246,766249,766252],{"className":766202},[31676],[34,766204],{"className":766205,"style":31860},[31680],[34,766207,766209,766212],{"className":766208},[31685],[34,766210,32936],{"className":766211},[31685,31686],[34,766213,766215],{"className":766214},[31792],[34,766216,766218,766238],{"className":766217},[31796,31797],[34,766219,766221,766235],{"className":766220},[31801],[34,766222,766224],{"className":766223,"style":33833},[31805],[34,766225,766226,766229],{"style":31809},[34,766227],{"className":766228,"style":31814},[31813],[34,766230,766232],{"className":766231},[31818,31819,31820,31821],[34,766233,47292],{"className":766234},[31685,31686,31821],[34,766236,31829],{"className":766237},[31828],[34,766239,766241],{"className":766240},[31801],[34,766242,766244],{"className":766243,"style":31836},[31805],[34,766245],{},[34,766247],{"className":766248,"style":31846},[31845],[34,766250,202],{"className":766251},[31850],[34,766253],{"className":766254,"style":31846},[31845],[34,766256,766258,766261,766301,766336,766339,766342],{"className":766257},[31676],[34,766259],{"className":766260,"style":744441},[31680],[34,766262,766264,766267],{"className":766263},[31685],[34,766265,31965],{"className":766266},[31685,31686],[34,766268,766270],{"className":766269},[31792],[34,766271,766273,766293],{"className":766272},[31796,31797],[34,766274,766276,766290],{"className":766275},[31801],[34,766277,766279],{"className":766278,"style":31806},[31805],[34,766280,766281,766284],{"style":31809},[34,766282],{"className":766283,"style":31814},[31813],[34,766285,766287],{"className":766286},[31818,31819,31820,31821],[34,766288,640],{"className":766289},[31685,31821],[34,766291,31829],{"className":766292},[31828],[34,766294,766296],{"className":766295},[31801],[34,766297,766299],{"className":766298,"style":31836},[31805],[34,766300],{},[34,766302,766304,766307],{"className":766303},[31685],[34,766305,33244],{"className":766306},[31685,207923],[34,766308,766310],{"className":766309},[31792],[34,766311,766313],{"className":766312},[31796],[34,766314,766316],{"className":766315},[31801],[34,766317,766319],{"className":766318,"style":37774},[31805],[34,766320,766321,766324],{"style":37777},[34,766322],{"className":766323,"style":31814},[31813],[34,766325,766327],{"className":766326},[31818,31819,31820,31821],[34,766328,766330,766333],{"className":766329},[31685,31821],[34,766331,768],{"className":766332},[31685,31821],[34,766334,31747],{"className":766335},[31685,31686,31821],[34,766337],{"className":766338,"style":32604},[31845],[34,766340,223],{"className":766341},[32024],[34,766343],{"className":766344,"style":32604},[31845],[34,766346,766348,766351,766391],{"className":766347},[31676],[34,766349],{"className":766350,"style":744441},[31680],[34,766352,766354,766357],{"className":766353},[31685],[34,766355,31965],{"className":766356},[31685,31686],[34,766358,766360],{"className":766359},[31792],[34,766361,766363,766383],{"className":766362},[31796,31797],[34,766364,766366,766380],{"className":766365},[31801],[34,766367,766369],{"className":766368,"style":31806},[31805],[34,766370,766371,766374],{"style":31809},[34,766372],{"className":766373,"style":31814},[31813],[34,766375,766377],{"className":766376},[31818,31819,31820,31821],[34,766378,768],{"className":766379},[31685,31821],[34,766381,31829],{"className":766382},[31828],[34,766384,766386],{"className":766385},[31801],[34,766387,766389],{"className":766388,"style":31836},[31805],[34,766390],{},[34,766392,766394,766397],{"className":766393},[31685],[34,766395,33244],{"className":766396},[31685,207923],[34,766398,766400],{"className":766399},[31792],[34,766401,766403],{"className":766402},[31796],[34,766404,766406],{"className":766405},[31801],[34,766407,766409],{"className":766408,"style":37774},[31805],[34,766410,766411,766414],{"style":37777},[34,766412],{"className":766413,"style":31814},[31813],[34,766415,766417],{"className":766416},[31818,31819,31820,31821],[34,766418,766420,766423],{"className":766419},[31685,31821],[34,766421,786],{"className":766422},[31685,31821],[34,766424,31747],{"className":766425},[31685,31686,31821],[108,766427,766428,766431,766432,766490,766491,766610,766611,766791,767036,767037,347],{},[2627,766429,766430],{},"解特解部分：","\n右边是 ",[34,766433,766435,766452],{"className":766434},[31642],[34,766436,766438],{"className":766437},[31646],[31648,766439,766440],{"xmlns":31650},[31652,766441,766442,766450],{},[31655,766443,766444],{},[37624,766445,766446,766448],{},[31658,766447,33244],{"mathvariant":38079},[31658,766449,31747],{},[31665,766451,765475],{"encoding":31667},[34,766453,766455],{"className":766454,"ariaHidden":758},[31672],[34,766456,766458,766461],{"className":766457},[31676],[34,766459],{"className":766460,"style":37959},[31680],[34,766462,766464,766467],{"className":766463},[31685],[34,766465,33244],{"className":766466},[31685,207923],[34,766468,766470],{"className":766469},[31792],[34,766471,766473],{"className":766472},[31796],[34,766474,766476],{"className":766475},[31801],[34,766477,766479],{"className":766478,"style":37959},[31805],[34,766480,766481,766484],{"style":37777},[34,766482],{"className":766483,"style":31814},[31813],[34,766485,766487],{"className":766486},[31818,31819,31820,31821],[34,766488,31747],{"className":766489},[31685,31686,31821],"（指数是1）。\n检查：1 不是特征根（特征根是2和3）。\n设特解 ",[34,766492,766494,766522],{"className":766493},[31642],[34,766495,766497],{"className":766496},[31646],[31648,766498,766499],{"xmlns":31650},[31652,766500,766501,766519],{},[31655,766502,766503,766509,766511,766513],{},[134014,766504,766505,766507],{"accent":758},[31658,766506,32936],{},[31662,766508,749281],{"stretchy":758},[31662,766510,202],{},[31658,766512,54151],{},[37624,766514,766515,766517],{},[31658,766516,33244],{"mathvariant":38079},[31658,766518,31747],{},[31665,766520,766521],{"encoding":31667},"\\widetilde{x} = A \\mathrm{e}^s",[34,766523,766525,766572],{"className":766524,"ariaHidden":758},[31672],[34,766526,766528,766531,766563,766566,766569],{"className":766527},[31676],[34,766529],{"className":766530,"style":749304},[31680],[34,766532,766534],{"className":766533},[31685,346171],[34,766535,766537],{"className":766536},[31796],[34,766538,766540],{"className":766539},[31801],[34,766541,766543,766551],{"className":766542,"style":749304},[31805],[34,766544,766545,766548],{"style":41565},[34,766546],{"className":766547,"style":41508},[31813],[34,766549,32936],{"className":766550},[31685,31686],[34,766552,766554,766557],{"className":766553,"style":749328},[41564],[34,766555],{"className":766556,"style":41508},[31813],[34,766558,766559],{"style":749334},[41588,766560,766561],{"xmlns":41590,"width":168121,"height":749337,"viewBox":749338,"preserveAspectRatio":29503},[20398,766562],{"d":749341},[34,766564],{"className":766565,"style":31846},[31845],[34,766567,202],{"className":766568},[31850],[34,766570],{"className":766571,"style":31846},[31845],[34,766573,766575,766578,766581],{"className":766574},[31676],[34,766576],{"className":766577,"style":33218},[31680],[34,766579,54151],{"className":766580},[31685,31686],[34,766582,766584,766587],{"className":766583},[31685],[34,766585,33244],{"className":766586},[31685,207923],[34,766588,766590],{"className":766589},[31792],[34,766591,766593],{"className":766592},[31796],[34,766594,766596],{"className":766595},[31801],[34,766597,766599],{"className":766598,"style":37959},[31805],[34,766600,766601,766604],{"style":37777},[34,766602],{"className":766603,"style":31814},[31813],[34,766605,766607],{"className":766606},[31818,31819,31820,31821],[34,766608,31747],{"className":766609},[31685,31686,31821],"。\n代入方程：\n",[34,766612,766614,766658],{"className":766613},[31642],[34,766615,766617],{"className":766616},[31646],[31648,766618,766619],{"xmlns":31650},[31652,766620,766621,766655],{},[31655,766622,766623,766625,766627,766629,766631,766633,766635,766637,766639,766641,766647,766649],{},[31662,766624,456],{"stretchy":702},[31658,766626,54151],{},[31662,766628,32226],{},[31749,766630,18136],{},[31658,766632,54151],{},[31662,766634,223],{},[31749,766636,36656],{},[31658,766638,54151],{},[31662,766640,2557],{"stretchy":702},[37624,766642,766643,766645],{},[31658,766644,33244],{"mathvariant":38079},[31658,766646,31747],{},[31662,766648,202],{},[37624,766650,766651,766653],{},[31658,766652,33244],{"mathvariant":38079},[31658,766654,31747],{},[31665,766656,766657],{"encoding":31667},"(A - 5A + 6A)\\mathrm{e}^s = \\mathrm{e}^s",[34,766659,766661,766682,766703,766756],{"className":766660,"ariaHidden":758},[31672],[34,766662,766664,766667,766670,766673,766676,766679],{"className":766663},[31676],[34,766665],{"className":766666,"style":31774},[31680],[34,766668,456],{"className":766669},[31782],[34,766671,54151],{"className":766672},[31685,31686],[34,766674],{"className":766675,"style":32604},[31845],[34,766677,32226],{"className":766678},[32024],[34,766680],{"className":766681,"style":32604},[31845],[34,766683,766685,766688,766691,766694,766697,766700],{"className":766684},[31676],[34,766686],{"className":766687,"style":34944},[31680],[34,766689,18136],{"className":766690},[31685],[34,766692,54151],{"className":766693},[31685,31686],[34,766695],{"className":766696,"style":32604},[31845],[34,766698,223],{"className":766699},[32024],[34,766701],{"className":766702,"style":32604},[31845],[34,766704,766706,766709,766712,766715,766718,766747,766750,766753],{"className":766705},[31676],[34,766707],{"className":766708,"style":31774},[31680],[34,766710,36656],{"className":766711},[31685],[34,766713,54151],{"className":766714},[31685,31686],[34,766716,2557],{"className":766717},[31690],[34,766719,766721,766724],{"className":766720},[31685],[34,766722,33244],{"className":766723},[31685,207923],[34,766725,766727],{"className":766726},[31792],[34,766728,766730],{"className":766729},[31796],[34,766731,766733],{"className":766732},[31801],[34,766734,766736],{"className":766735,"style":37959},[31805],[34,766737,766738,766741],{"style":37777},[34,766739],{"className":766740,"style":31814},[31813],[34,766742,766744],{"className":766743},[31818,31819,31820,31821],[34,766745,31747],{"className":766746},[31685,31686,31821],[34,766748],{"className":766749,"style":31846},[31845],[34,766751,202],{"className":766752},[31850],[34,766754],{"className":766755,"style":31846},[31845],[34,766757,766759,766762],{"className":766758},[31676],[34,766760],{"className":766761,"style":37959},[31680],[34,766763,766765,766768],{"className":766764},[31685],[34,766766,33244],{"className":766767},[31685,207923],[34,766769,766771],{"className":766770},[31792],[34,766772,766774],{"className":766773},[31796],[34,766775,766777],{"className":766776},[31801],[34,766778,766780],{"className":766779,"style":37959},[31805],[34,766781,766782,766785],{"style":37777},[34,766783],{"className":766784,"style":31814},[31813],[34,766786,766788],{"className":766787},[31818,31819,31820,31821],[34,766789,31747],{"className":766790},[31685,31686,31821],[34,766792,766794,766840],{"className":766793},[31642],[34,766795,766797],{"className":766796},[31646],[31648,766798,766799],{"xmlns":31650},[31652,766800,766801,766837],{},[31655,766802,766803,766805,766807,766813,766815,766821,766823,766825,766827,766829,766831],{},[31749,766804,768],{},[31658,766806,54151],{},[37624,766808,766809,766811],{},[31658,766810,33244],{"mathvariant":38079},[31658,766812,31747],{},[31662,766814,202],{},[37624,766816,766817,766819],{},[31658,766818,33244],{"mathvariant":38079},[31658,766820,31747],{},[39361,766822,262622],{},[31662,766824,262625],{},[39361,766826,262622],{},[31658,766828,54151],{},[31662,766830,202],{},[41453,766832,766833,766835],{},[31749,766834,640],{},[31749,766836,768],{},[31665,766838,766839],{"encoding":31667},"2A \\mathrm{e}^s = \\mathrm{e}^s \\implies A = \\frac{1}{2}",[34,766841,766843,766893,766944,766962],{"className":766842,"ariaHidden":758},[31672],[34,766844,766846,766849,766852,766855,766884,766887,766890],{"className":766845},[31676],[34,766847],{"className":766848,"style":33218},[31680],[34,766850,768],{"className":766851},[31685],[34,766853,54151],{"className":766854},[31685,31686],[34,766856,766858,766861],{"className":766857},[31685],[34,766859,33244],{"className":766860},[31685,207923],[34,766862,766864],{"className":766863},[31792],[34,766865,766867],{"className":766866},[31796],[34,766868,766870],{"className":766869},[31801],[34,766871,766873],{"className":766872,"style":37959},[31805],[34,766874,766875,766878],{"style":37777},[34,766876],{"className":766877,"style":31814},[31813],[34,766879,766881],{"className":766880},[31818,31819,31820,31821],[34,766882,31747],{"className":766883},[31685,31686,31821],[34,766885],{"className":766886,"style":31846},[31845],[34,766888,202],{"className":766889},[31850],[34,766891],{"className":766892,"style":31846},[31845],[34,766894,766896,766900,766929,766932,766935,766938,766941],{"className":766895},[31676],[34,766897],{"className":766898,"style":766899},[31680],"height:0.6884em;vertical-align:-0.024em;",[34,766901,766903,766906],{"className":766902},[31685],[34,766904,33244],{"className":766905},[31685,207923],[34,766907,766909],{"className":766908},[31792],[34,766910,766912],{"className":766911},[31796],[34,766913,766915],{"className":766914},[31801],[34,766916,766918],{"className":766917,"style":37959},[31805],[34,766919,766920,766923],{"style":37777},[34,766921],{"className":766922,"style":31814},[31813],[34,766924,766926],{"className":766925},[31818,31819,31820,31821],[34,766927,31747],{"className":766928},[31685,31686,31821],[34,766930],{"className":766931,"style":31846},[31845],[34,766933],{"className":766934,"style":31846},[31845],[34,766936,262625],{"className":766937},[31850],[34,766939],{"className":766940,"style":31846},[31845],[34,766942],{"className":766943,"style":31846},[31845],[34,766945,766947,766950,766953,766956,766959],{"className":766946},[31676],[34,766948],{"className":766949,"style":33218},[31680],[34,766951,54151],{"className":766952},[31685,31686],[34,766954],{"className":766955,"style":31846},[31845],[34,766957,202],{"className":766958},[31850],[34,766960],{"className":766961,"style":31846},[31845],[34,766963,766965,766968],{"className":766964},[31676],[34,766966],{"className":766967,"style":46727},[31680],[34,766969,766971,766974,767033],{"className":766970},[31685],[34,766972],{"className":766973},[31782,41488],[34,766975,766977],{"className":766976},[41453],[34,766978,766980,767025],{"className":766979},[31796,31797],[34,766981,766983,767022],{"className":766982},[31801],[34,766984,766986,767000,767008],{"className":766985,"style":46746},[31805],[34,766987,766988,766991],{"style":41504},[34,766989],{"className":766990,"style":41508},[31813],[34,766992,766994],{"className":766993},[31818,31819,31820,31821],[34,766995,766997],{"className":766996},[31685,31821],[34,766998,768],{"className":766999},[31685,31821],[34,767001,767002,767005],{"style":41520},[34,767003],{"className":767004,"style":41508},[31813],[34,767006],{"className":767007,"style":41528},[41527],[34,767009,767010,767013],{"style":46771},[34,767011],{"className":767012,"style":41508},[31813],[34,767014,767016],{"className":767015},[31818,31819,31820,31821],[34,767017,767019],{"className":767018},[31685,31821],[34,767020,640],{"className":767021},[31685,31821],[34,767023,31829],{"className":767024},[31828],[34,767026,767028],{"className":767027},[31801],[34,767029,767031],{"className":767030,"style":41619},[31805],[34,767032],{},[34,767034],{"className":767035},[31690,41488],"\n所以特解是 ",[34,767038,767040,767064],{"className":767039},[31642],[34,767041,767043],{"className":767042},[31646],[31648,767044,767045],{"xmlns":31650},[31652,767046,767047,767061],{},[31655,767048,767049,767055],{},[41453,767050,767051,767053],{},[31749,767052,640],{},[31749,767054,768],{},[37624,767056,767057,767059],{},[31658,767058,33244],{"mathvariant":38079},[31658,767060,31747],{},[31665,767062,767063],{"encoding":31667},"\\frac{1}{2}\\mathrm{e}^s",[34,767065,767067],{"className":767066,"ariaHidden":758},[31672],[34,767068,767070,767073,767141],{"className":767069},[31676],[34,767071],{"className":767072,"style":46727},[31680],[34,767074,767076,767079,767138],{"className":767075},[31685],[34,767077],{"className":767078},[31782,41488],[34,767080,767082],{"className":767081},[41453],[34,767083,767085,767130],{"className":767084},[31796,31797],[34,767086,767088,767127],{"className":767087},[31801],[34,767089,767091,767105,767113],{"className":767090,"style":46746},[31805],[34,767092,767093,767096],{"style":41504},[34,767094],{"className":767095,"style":41508},[31813],[34,767097,767099],{"className":767098},[31818,31819,31820,31821],[34,767100,767102],{"className":767101},[31685,31821],[34,767103,768],{"className":767104},[31685,31821],[34,767106,767107,767110],{"style":41520},[34,767108],{"className":767109,"style":41508},[31813],[34,767111],{"className":767112,"style":41528},[41527],[34,767114,767115,767118],{"style":46771},[34,767116],{"className":767117,"style":41508},[31813],[34,767119,767121],{"className":767120},[31818,31819,31820,31821],[34,767122,767124],{"className":767123},[31685,31821],[34,767125,640],{"className":767126},[31685,31821],[34,767128,31829],{"className":767129},[31828],[34,767131,767133],{"className":767132},[31801],[34,767134,767136],{"className":767135,"style":41619},[31805],[34,767137],{},[34,767139],{"className":767140},[31690,41488],[34,767142,767144,767147],{"className":767143},[31685],[34,767145,33244],{"className":767146},[31685,207923],[34,767148,767150],{"className":767149},[31792],[34,767151,767153],{"className":767152},[31796],[34,767154,767156],{"className":767155},[31801],[34,767157,767159],{"className":767158,"style":37959},[31805],[34,767160,767161,767164],{"style":37777},[34,767162],{"className":767163,"style":31814},[31813],[34,767165,767167],{"className":767166},[31818,31819,31820,31821],[34,767168,31747],{"className":767169},[31685,31686,31821],[108,767171,767172,767204],{},[2627,767173,767174,767175,767203],{},"写出关于 ",[34,767176,767178,767191],{"className":767177},[31642],[34,767179,767181],{"className":767180},[31646],[31648,767182,767183],{"xmlns":31650},[31652,767184,767185,767189],{},[31655,767186,767187],{},[31658,767188,31747],{},[31665,767190,31747],{"encoding":31667},[34,767192,767194],{"className":767193,"ariaHidden":758},[31672],[34,767195,767197,767200],{"className":767196},[31676],[34,767198],{"className":767199,"style":32527},[31680],[34,767201,31747],{"className":767202},[31685,31686]," 的通解：",[34,767205,767207,767271],{"className":767206},[31642],[34,767208,767210],{"className":767209},[31646],[31648,767211,767212],{"xmlns":31650},[31652,767213,767214,767268],{},[31655,767215,767216,767218,767220,767226,767236,767238,767244,767254,767256,767262],{},[31658,767217,32936],{},[31662,767219,202],{},[31743,767221,767222,767224],{},[31658,767223,31965],{},[31749,767225,640],{},[37624,767227,767228,767230],{},[31658,767229,33244],{"mathvariant":38079},[31655,767231,767232,767234],{},[31749,767233,768],{},[31658,767235,31747],{},[31662,767237,223],{},[31743,767239,767240,767242],{},[31658,767241,31965],{},[31749,767243,768],{},[37624,767245,767246,767248],{},[31658,767247,33244],{"mathvariant":38079},[31655,767249,767250,767252],{},[31749,767251,786],{},[31658,767253,31747],{},[31662,767255,223],{},[41453,767257,767258,767260],{},[31749,767259,640],{},[31749,767261,768],{},[37624,767263,767264,767266],{},[31658,767265,33244],{"mathvariant":38079},[31658,767267,31747],{},[31665,767269,767270],{"encoding":31667},"x = c_1 \\mathrm{e}^{2s} + c_2 \\mathrm{e}^{3s} + \\frac{1}{2}\\mathrm{e}^s",[34,767272,767274,767292,767382,767472],{"className":767273,"ariaHidden":758},[31672],[34,767275,767277,767280,767283,767286,767289],{"className":767276},[31676],[34,767278],{"className":767279,"style":32527},[31680],[34,767281,32936],{"className":767282},[31685,31686],[34,767284],{"className":767285,"style":31846},[31845],[34,767287,202],{"className":767288},[31850],[34,767290],{"className":767291,"style":31846},[31845],[34,767293,767295,767298,767338,767373,767376,767379],{"className":767294},[31676],[34,767296],{"className":767297,"style":744441},[31680],[34,767299,767301,767304],{"className":767300},[31685],[34,767302,31965],{"className":767303},[31685,31686],[34,767305,767307],{"className":767306},[31792],[34,767308,767310,767330],{"className":767309},[31796,31797],[34,767311,767313,767327],{"className":767312},[31801],[34,767314,767316],{"className":767315,"style":31806},[31805],[34,767317,767318,767321],{"style":31809},[34,767319],{"className":767320,"style":31814},[31813],[34,767322,767324],{"className":767323},[31818,31819,31820,31821],[34,767325,640],{"className":767326},[31685,31821],[34,767328,31829],{"className":767329},[31828],[34,767331,767333],{"className":767332},[31801],[34,767334,767336],{"className":767335,"style":31836},[31805],[34,767337],{},[34,767339,767341,767344],{"className":767340},[31685],[34,767342,33244],{"className":767343},[31685,207923],[34,767345,767347],{"className":767346},[31792],[34,767348,767350],{"className":767349},[31796],[34,767351,767353],{"className":767352},[31801],[34,767354,767356],{"className":767355,"style":37774},[31805],[34,767357,767358,767361],{"style":37777},[34,767359],{"className":767360,"style":31814},[31813],[34,767362,767364],{"className":767363},[31818,31819,31820,31821],[34,767365,767367,767370],{"className":767366},[31685,31821],[34,767368,768],{"className":767369},[31685,31821],[34,767371,31747],{"className":767372},[31685,31686,31821],[34,767374],{"className":767375,"style":32604},[31845],[34,767377,223],{"className":767378},[32024],[34,767380],{"className":767381,"style":32604},[31845],[34,767383,767385,767388,767428,767463,767466,767469],{"className":767384},[31676],[34,767386],{"className":767387,"style":744441},[31680],[34,767389,767391,767394],{"className":767390},[31685],[34,767392,31965],{"className":767393},[31685,31686],[34,767395,767397],{"className":767396},[31792],[34,767398,767400,767420],{"className":767399},[31796,31797],[34,767401,767403,767417],{"className":767402},[31801],[34,767404,767406],{"className":767405,"style":31806},[31805],[34,767407,767408,767411],{"style":31809},[34,767409],{"className":767410,"style":31814},[31813],[34,767412,767414],{"className":767413},[31818,31819,31820,31821],[34,767415,768],{"className":767416},[31685,31821],[34,767418,31829],{"className":767419},[31828],[34,767421,767423],{"className":767422},[31801],[34,767424,767426],{"className":767425,"style":31836},[31805],[34,767427],{},[34,767429,767431,767434],{"className":767430},[31685],[34,767432,33244],{"className":767433},[31685,207923],[34,767435,767437],{"className":767436},[31792],[34,767438,767440],{"className":767439},[31796],[34,767441,767443],{"className":767442},[31801],[34,767444,767446],{"className":767445,"style":37774},[31805],[34,767447,767448,767451],{"style":37777},[34,767449],{"className":767450,"style":31814},[31813],[34,767452,767454],{"className":767453},[31818,31819,31820,31821],[34,767455,767457,767460],{"className":767456},[31685,31821],[34,767458,786],{"className":767459},[31685,31821],[34,767461,31747],{"className":767462},[31685,31686,31821],[34,767464],{"className":767465,"style":32604},[31845],[34,767467,223],{"className":767468},[32024],[34,767470],{"className":767471,"style":32604},[31845],[34,767473,767475,767478,767546],{"className":767474},[31676],[34,767476],{"className":767477,"style":46727},[31680],[34,767479,767481,767484,767543],{"className":767480},[31685],[34,767482],{"className":767483},[31782,41488],[34,767485,767487],{"className":767486},[41453],[34,767488,767490,767535],{"className":767489},[31796,31797],[34,767491,767493,767532],{"className":767492},[31801],[34,767494,767496,767510,767518],{"className":767495,"style":46746},[31805],[34,767497,767498,767501],{"style":41504},[34,767499],{"className":767500,"style":41508},[31813],[34,767502,767504],{"className":767503},[31818,31819,31820,31821],[34,767505,767507],{"className":767506},[31685,31821],[34,767508,768],{"className":767509},[31685,31821],[34,767511,767512,767515],{"style":41520},[34,767513],{"className":767514,"style":41508},[31813],[34,767516],{"className":767517,"style":41528},[41527],[34,767519,767520,767523],{"style":46771},[34,767521],{"className":767522,"style":41508},[31813],[34,767524,767526],{"className":767525},[31818,31819,31820,31821],[34,767527,767529],{"className":767528},[31685,31821],[34,767530,640],{"className":767531},[31685,31821],[34,767533,31829],{"className":767534},[31828],[34,767536,767538],{"className":767537},[31801],[34,767539,767541],{"className":767540,"style":41619},[31805],[34,767542],{},[34,767544],{"className":767545},[31690,41488],[34,767547,767549,767552],{"className":767548},[31685],[34,767550,33244],{"className":767551},[31685,207923],[34,767553,767555],{"className":767554},[31792],[34,767556,767558],{"className":767557},[31796],[34,767559,767561],{"className":767560},[31801],[34,767562,767564],{"className":767563,"style":37959},[31805],[34,767565,767566,767569],{"style":37777},[34,767567],{"className":767568,"style":31814},[31813],[34,767570,767572],{"className":767571},[31818,31819,31820,31821],[34,767573,31747],{"className":767574},[31685,31686,31821],[108,767576,767577,525700,767608,55016,767688,46528,767809,39984,767930,767932],{},[2627,767578,767579,767580,414527],{},"回代变量（变回 ",[34,767581,767583,767596],{"className":767582},[31642],[34,767584,767586],{"className":767585},[31646],[31648,767587,767588],{"xmlns":31650},[31652,767589,767590,767594],{},[31655,767591,767592],{},[31658,767593,33249],{},[31665,767595,33249],{"encoding":31667},[34,767597,767599],{"className":767598,"ariaHidden":758},[31672],[34,767600,767602,767605],{"className":767601},[31676],[34,767603],{"className":767604,"style":33280},[31680],[34,767606,33249],{"className":767607},[31685,31686],[34,767609,767611,767632],{"className":767610},[31642],[34,767612,767614],{"className":767613},[31646],[31648,767615,767616],{"xmlns":31650},[31652,767617,767618,767630],{},[31655,767619,767620,767626,767628],{},[37624,767621,767622,767624],{},[31658,767623,33244],{"mathvariant":38079},[31658,767625,31747],{},[31662,767627,202],{},[31658,767629,33249],{},[31665,767631,764239],{"encoding":31667},[34,767633,767635,767679],{"className":767634,"ariaHidden":758},[31672],[34,767636,767638,767641,767670,767673,767676],{"className":767637},[31676],[34,767639],{"className":767640,"style":37959},[31680],[34,767642,767644,767647],{"className":767643},[31685],[34,767645,33244],{"className":767646},[31685,207923],[34,767648,767650],{"className":767649},[31792],[34,767651,767653],{"className":767652},[31796],[34,767654,767656],{"className":767655},[31801],[34,767657,767659],{"className":767658,"style":37959},[31805],[34,767660,767661,767664],{"style":37777},[34,767662],{"className":767663,"style":31814},[31813],[34,767665,767667],{"className":767666},[31818,31819,31820,31821],[34,767668,31747],{"className":767669},[31685,31686,31821],[34,767671],{"className":767672,"style":31846},[31845],[34,767674,202],{"className":767675},[31850],[34,767677],{"className":767678,"style":31846},[31845],[34,767680,767682,767685],{"className":767681},[31676],[34,767683],{"className":767684,"style":33280},[31680],[34,767686,33249],{"className":767687},[31685,31686],[34,767689,767691,767721],{"className":767690},[31642],[34,767692,767694],{"className":767693},[31646],[31648,767695,767696],{"xmlns":31650},[31652,767697,767698,767718],{},[31655,767699,767700,767710,767712],{},[37624,767701,767702,767704],{},[31658,767703,33244],{"mathvariant":38079},[31655,767705,767706,767708],{},[31749,767707,768],{},[31658,767709,31747],{},[31662,767711,202],{},[37624,767713,767714,767716],{},[31658,767715,33249],{},[31749,767717,768],{},[31665,767719,767720],{"encoding":31667},"\\mathrm{e}^{2s} = t^2",[34,767722,767724,767774],{"className":767723,"ariaHidden":758},[31672],[34,767725,767727,767730,767765,767768,767771],{"className":767726},[31676],[34,767728],{"className":767729,"style":37774},[31680],[34,767731,767733,767736],{"className":767732},[31685],[34,767734,33244],{"className":767735},[31685,207923],[34,767737,767739],{"className":767738},[31792],[34,767740,767742],{"className":767741},[31796],[34,767743,767745],{"className":767744},[31801],[34,767746,767748],{"className":767747,"style":37774},[31805],[34,767749,767750,767753],{"style":37777},[34,767751],{"className":767752,"style":31814},[31813],[34,767754,767756],{"className":767755},[31818,31819,31820,31821],[34,767757,767759,767762],{"className":767758},[31685,31821],[34,767760,768],{"className":767761},[31685,31821],[34,767763,31747],{"className":767764},[31685,31686,31821],[34,767766],{"className":767767,"style":31846},[31845],[34,767769,202],{"className":767770},[31850],[34,767772],{"className":767773,"style":31846},[31845],[34,767775,767777,767780],{"className":767776},[31676],[34,767778],{"className":767779,"style":37774},[31680],[34,767781,767783,767786],{"className":767782},[31685],[34,767784,33249],{"className":767785},[31685,31686],[34,767787,767789],{"className":767788},[31792],[34,767790,767792],{"className":767791},[31796],[34,767793,767795],{"className":767794},[31801],[34,767796,767798],{"className":767797,"style":37774},[31805],[34,767799,767800,767803],{"style":37777},[34,767801],{"className":767802,"style":31814},[31813],[34,767804,767806],{"className":767805},[31818,31819,31820,31821],[34,767807,768],{"className":767808},[31685,31821],[34,767810,767812,767842],{"className":767811},[31642],[34,767813,767815],{"className":767814},[31646],[31648,767816,767817],{"xmlns":31650},[31652,767818,767819,767839],{},[31655,767820,767821,767831,767833],{},[37624,767822,767823,767825],{},[31658,767824,33244],{"mathvariant":38079},[31655,767826,767827,767829],{},[31749,767828,786],{},[31658,767830,31747],{},[31662,767832,202],{},[37624,767834,767835,767837],{},[31658,767836,33249],{},[31749,767838,786],{},[31665,767840,767841],{"encoding":31667},"\\mathrm{e}^{3s} = t^3",[34,767843,767845,767895],{"className":767844,"ariaHidden":758},[31672],[34,767846,767848,767851,767886,767889,767892],{"className":767847},[31676],[34,767849],{"className":767850,"style":37774},[31680],[34,767852,767854,767857],{"className":767853},[31685],[34,767855,33244],{"className":767856},[31685,207923],[34,767858,767860],{"className":767859},[31792],[34,767861,767863],{"className":767862},[31796],[34,767864,767866],{"className":767865},[31801],[34,767867,767869],{"className":767868,"style":37774},[31805],[34,767870,767871,767874],{"style":37777},[34,767872],{"className":767873,"style":31814},[31813],[34,767875,767877],{"className":767876},[31818,31819,31820,31821],[34,767878,767880,767883],{"className":767879},[31685,31821],[34,767881,786],{"className":767882},[31685,31821],[34,767884,31747],{"className":767885},[31685,31686,31821],[34,767887],{"className":767888,"style":31846},[31845],[34,767890,202],{"className":767891},[31850],[34,767893],{"className":767894,"style":31846},[31845],[34,767896,767898,767901],{"className":767897},[31676],[34,767899],{"className":767900,"style":37774},[31680],[34,767902,767904,767907],{"className":767903},[31685],[34,767905,33249],{"className":767906},[31685,31686],[34,767908,767910],{"className":767909},[31792],[34,767911,767913],{"className":767912},[31796],[34,767914,767916],{"className":767915},[31801],[34,767917,767919],{"className":767918,"style":37774},[31805],[34,767920,767921,767924],{"style":37777},[34,767922],{"className":767923,"style":31814},[31813],[34,767925,767927],{"className":767926},[31818,31819,31820,31821],[34,767928,786],{"className":767929},[31685,31821],[2627,767931,177510],{},[34,767933,767935,767987],{"className":767934},[31642],[34,767936,767938],{"className":767937},[31646],[31648,767939,767940],{"xmlns":31650},[31652,767941,767942,767984],{},[31655,767943,767944,767946,767948,767954,767960,767962,767968,767974,767976,767982],{},[31658,767945,32936],{},[31662,767947,202],{},[31743,767949,767950,767952],{},[31658,767951,31965],{},[31749,767953,640],{},[37624,767955,767956,767958],{},[31658,767957,33249],{},[31749,767959,768],{},[31662,767961,223],{},[31743,767963,767964,767966],{},[31658,767965,31965],{},[31749,767967,768],{},[37624,767969,767970,767972],{},[31658,767971,33249],{},[31749,767973,786],{},[31662,767975,223],{},[41453,767977,767978,767980],{},[31749,767979,640],{},[31749,767981,768],{},[31658,767983,33249],{},[31665,767985,767986],{"encoding":31667},"x = c_1 t^2 + c_2 t^3 + 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于区间 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的解，",[34,768781,768783,768831],{"className":768782},[31642],[34,768784,768786],{"className":768785},[31646],[31648,768787,768788],{"xmlns":31650},[31652,768789,768790,768828],{},[31655,768791,768792,768798,768800,768802,768804,768810,768812,768818,768820,768826],{},[31743,768793,768794,768796],{},[31658,768795,67764],{},[31749,768797,459],{},[31662,768799,202],{},[31658,768801,126541],{},[31662,768803,456],{"stretchy":702},[31743,768805,768806,768808],{},[31658,768807,32936],{},[31749,768809,459],{},[31662,768811,31960],{"separator":758},[31743,768813,768814,768816],{},[31658,768815,32936],{},[31749,768817,459],{},[31662,768819,31960],{"separator":758},[31743,768821,768822,768824],{},[31658,768823,67764],{},[31749,768825,459],{},[31662,768827,2557],{"stretchy":702},[31665,768829,768830],{"encoding":31667},"y_0 = \\varphi(x_0,x_0,y_0)",[34,768832,768834,768889],{"className":768833,"ariaHidden":758},[31672],[34,768835,768837,768840,768880,768883,768886],{"className":768836},[31676],[34,768838],{"className":768839,"style":38774},[31680],[34,768841,768843,768846],{"className":768842},[31685],[34,768844,67764],{"className":768845,"style":32062},[31685,31686],[34,768847,768849],{"className":768848},[31792],[34,768850,768852,768872],{"className":768851},[31796,31797],[34,768853,768855,768869],{"className":768854},[31801],[34,768856,768858],{"className":768857,"style":31806},[31805],[34,768859,768860,768863],{"style":38795},[34,768861],{"className":768862,"style":31814},[31813],[34,768864,768866],{"className":768865},[31818,31819,31820,31821],[34,768867,459],{"className":768868},[31685,31821],[34,768870,31829],{"className":768871},[31828],[34,768873,768875],{"className":768874},[31801],[34,768876,768878],{"className":768877,"style":31836},[31805],[34,768879],{},[34,768881],{"className":768882,"style":31846},[31845],[34,768884,202],{"className":768885},[31850],[34,768887],{"className":768888,"style":31846},[31845],[34,768890,768892,768895,768898,768901,768941,768944,768947,768987,768990,768993,769033],{"className":768891},[31676],[34,768893],{"className":768894,"style":31774},[31680],[34,768896,126541],{"className":768897},[31685,31686],[34,768899,456],{"className":768900},[31782],[34,768902,768904,768907],{"className":768903},[31685],[34,768905,32936],{"className":768906},[31685,31686],[34,768908,768910],{"className":768909},[31792],[34,768911,768913,768933],{"className":768912},[31796,31797],[34,768914,768916,768930],{"className":768915},[31801],[34,768917,768919],{"className":768918,"style":31806},[31805],[34,768920,768921,768924],{"style":31809},[34,768922],{"className":768923,"style":31814},[31813],[34,768925,768927],{"className":768926},[31818,31819,31820,31821],[34,768928,459],{"className":768929},[31685,31821],[34,768931,31829],{"className":768932},[31828],[34,768934,768936],{"className":768935},[31801],[34,768937,768939],{"className":768938,"style":31836},[31805],[34,768940],{},[34,768942,31960],{"className":768943},[32119],[34,768945],{"className":768946,"style":32123},[31845],[34,768948,768950,768953],{"className":768949},[31685],[34,768951,32936],{"className":768952},[31685,31686],[34,768954,768956],{"className":768955},[31792],[34,768957,768959,768979],{"className":768958},[31796,31797],[34,768960,768962,768976],{"className":768961},[31801],[34,768963,768965],{"className":768964,"style":31806},[31805],[34,768966,768967,768970],{"style":31809},[34,768968],{"className":768969,"style":31814},[31813],[34,768971,768973],{"className":768972},[31818,31819,31820,31821],[34,768974,459],{"className":768975},[31685,31821],[34,768977,31829],{"className":768978},[31828],[34,768980,768982],{"className":768981},[31801],[34,768983,768985],{"className":768984,"style":31836},[31805],[34,768986],{},[34,768988,31960],{"className":768989},[32119],[34,768991],{"className":768992,"style":32123},[31845],[34,768994,768996,768999],{"className":768995},[31685],[34,768997,67764],{"className":768998,"style":32062},[31685,31686],[34,769000,769002],{"className":769001},[31792],[34,769003,769005,769025],{"className":769004},[31796,31797],[34,769006,769008,769022],{"className":769007},[31801],[34,769009,769011],{"className":769010,"style":31806},[31805],[34,769012,769013,769016],{"style":38795},[34,769014],{"className":769015,"style":31814},[31813],[34,769017,769019],{"className":769018},[31818,31819,31820,31821],[34,769020,459],{"className":769021},[31685,31821],[34,769023,31829],{"className":769024},[31828],[34,769026,769028],{"className":769027},[31801],[34,769029,769031],{"className":769030,"style":31836},[31805],[34,769032],{},[34,769034,2557],{"className":769035},[31690],"，试求 ",[34,769038,769040,769090],{"className":769039},[31642],[34,769041,769043],{"className":769042},[31646],[31648,769044,769045],{"xmlns":31650},[31652,769046,769047,769087],{},[31655,769048,769049,769067,769069],{},[41453,769050,769051,769057],{},[31655,769052,769053,769055],{},[31658,769054,619473],{"mathvariant":38079},[31658,769056,126541],{},[31655,769058,769059,769061],{},[31658,769060,619473],{"mathvariant":38079},[31743,769062,769063,769065],{},[31658,769064,32936],{},[31749,769066,459],{},[31662,769068,31960],{"separator":758},[41453,769070,769071,769077],{},[31655,769072,769073,769075],{},[31658,769074,619473],{"mathvariant":38079},[31658,769076,126541],{},[31655,769078,769079,769081],{},[31658,769080,619473],{"mathvariant":38079},[31743,769082,769083,769085],{},[31658,769084,67764],{},[31749,769086,459],{},[31665,769088,769089],{"encoding":31667},"\\frac{\\partial \\varphi}{\\partial x_0},\\frac{\\partial \\varphi}{\\partial 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x}{\\mathrm{d}t^n} + a_1(t)\\frac{\\mathrm{d}^{n-1}x}{\\mathrm{d}t^{n-1}} + \\cdots + a_n(t)x = 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2,771178,223],{},[31743,771180,771181,771183],{},[31658,771182,111],{},[31749,771184,640],{},[31662,771186,456],{"stretchy":702},[31658,771188,33249],{},[31662,771190,2557],{"stretchy":702},[41453,771192,771193,771209],{},[31655,771194,771195,771207],{},[37624,771196,771197,771199],{},[31658,771198,46900],{"mathvariant":38079},[31655,771200,771201,771203,771205],{},[31658,771202,31660],{},[31662,771204,32226],{},[31749,771206,640],{},[31658,771208,32936],{},[31655,771210,771211,771213],{},[31658,771212,46900],{"mathvariant":38079},[37624,771214,771215,771217],{},[31658,771216,33249],{},[31655,771218,771219,771221,771223],{},[31658,771220,31660],{},[31662,771222,32226],{},[31749,771224,640],{},[31662,771226,223],{},[31662,771228,38874],{},[31662,771230,223],{},[31743,771232,771233,771235],{},[31658,771234,111],{},[31658,771236,31660],{},[31662,771238,456],{"stretchy":702},[31658,771240,33249],{},[31662,771242,2557],{"stretchy":702},[31658,771244,32936],{},[31662,771246,202],{},[31743,771248,771249,771251],{},[31658,771250,31739],{},[31749,771252,768],{},[31662,771254,456],{"stretchy":702},[31658,771256,33249],{},[31662,771258,2557],{"stretchy":702},[31665,771260,771261],{"encoding":31667},"\\frac{\\mathrm{d}^n x}{\\mathrm{d}t^n} + a_1(t)\\frac{\\mathrm{d}^{n-1}x}{\\mathrm{d}t^{n-1}} + \\cdots + a_n(t)x = f_2(t)",[34,771263,771265,771406,771614,771632,771699],{"className":771264,"ariaHidden":758},[31672],[34,771266,771268,771271,771397,771400,771403],{"className":771267},[31676],[34,771269],{"className":771270,"style":715659},[31680],[34,771272,771274,771277,771394],{"className":771273},[31685],[34,771275],{"className":771276},[31782,41488],[34,771278,771280],{"className":771279},[41453],[34,771281,771283,771386],{"className":771282},[31796,31797],[34,771284,771286,771383],{"className":771285},[31801],[34,771287,771289,771332,771340],{"className":771288,"style":700470},[31805],[34,771290,771291,771294],{"style":41504},[34,771292],{"className":771293,"style":41508},[31813],[34,771295,771297],{"className":771296},[31818,31819,31820,31821],[34,771298,771300,771303],{"className":771299},[31685,31821],[34,771301,46900],{"className":771302},[31685,207923,31821],[34,771304,771306,771309],{"className":771305},[31685,31821],[34,771307,33249],{"className":771308},[31685,31686,31821],[34,771310,771312],{"className":771311},[31792],[34,771313,771315],{"className":771314},[31796],[34,771316,771318],{"className":771317},[31801],[34,771319,771321],{"className":771320,"style":566313},[31805],[34,771322,771323,771326],{"style":116668},[34,771324],{"className":771325,"style":43406},[31813],[34,771327,771329],{"className":771328},[31818,43410,43411,31821],[34,771330,31660],{"className":771331},[31685,31686,31821],[34,771333,771334,771337],{"style":41520},[34,771335],{"className":771336,"style":41508},[31813],[34,771338],{"className":771339,"style":41528},[41527],[34,771341,771342,771345],{"style":46771},[34,771343],{"className":771344,"style":41508},[31813],[34,771346,771348],{"className":771347},[31818,31819,31820,31821],[34,771349,771351,771380],{"className":771350},[31685,31821],[34,771352,771354,771357],{"className":771353},[31685,31821],[34,771355,46900],{"className":771356},[31685,207923,31821],[34,771358,771360],{"className":771359},[31792],[34,771361,771363],{"className":771362},[31796],[34,771364,771366],{"className":771365},[31801],[34,771367,771369],{"className":771368,"style":611529},[31805],[34,771370,771371,771374],{"style":552391},[34,771372],{"className":771373,"style":43406},[31813],[34,771375,771377],{"className":771376},[31818,43410,43411,31821],[34,771378,31660],{"className":771379},[31685,31686,31821],[34,771381,32936],{"className":771382},[31685,31686,31821],[34,771384,31829],{"className":771385},[31828],[34,771387,771389],{"className":771388},[31801],[34,771390,771392],{"className":771391,"style":41619},[31805],[34,771393],{},[34,771395],{"className":771396},[31690,41488],[34,771398],{"className":771399,"style":32604},[31845],[34,771401,223],{"className":771402},[32024],[34,771404],{"className":771405,"style":32604},[31845],[34,771407,771409,771412,771452,771455,771458,771461,771605,771608,771611],{"className":771408},[31676],[34,771410],{"className":771411,"style":637019},[31680],[34,771413,771415,771418],{"className":771414},[31685],[34,771416,111],{"className":771417},[31685,31686],[34,771419,771421],{"className":771420},[31792],[34,771422,771424,771444],{"className":771423},[31796,31797],[34,771425,771427,771441],{"className":771426},[31801],[34,771428,771430],{"className":771429,"style":31806},[31805],[34,771431,771432,771435],{"style":31809},[34,771433],{"className":771434,"style":31814},[31813],[34,771436,771438],{"className":771437},[31818,31819,31820,31821],[34,771439,640],{"className":771440},[31685,31821],[34,771442,31829],{"className":771443},[31828],[34,771445,771447],{"className":771446},[31801],[34,771448,771450],{"className":771449,"style":31836},[31805],[34,771451],{},[34,771453,456],{"className":771454},[31782],[34,771456,33249],{"className":771457},[31685,31686],[34,771459,2557],{"className":771460},[31690],[34,771462,771464,771467,771602],{"className":771463},[31685],[34,771465],{"className":771466},[31782,41488],[34,771468,771470],{"className":771469},[41453],[34,771471,771473,771594],{"className":771472},[31796,31797],[34,771474,771476,771591],{"className":771475},[31801],[34,771477,771479,771531,771539],{"className":771478,"style":603956},[31805],[34,771480,771481,771484],{"style":41504},[34,771482],{"className":771483,"style":41508},[31813],[34,771485,771487],{"className":771486},[31818,31819,31820,31821],[34,771488,771490,771493],{"className":771489},[31685,31821],[34,771491,46900],{"className":771492},[31685,207923,31821],[34,771494,771496,771499],{"className":771495},[31685,31821],[34,771497,33249],{"className":771498},[31685,31686,31821],[34,771500,771502],{"className":771501},[31792],[34,771503,771505],{"className":771504},[31796],[34,771506,771508],{"className":771507},[31801],[34,771509,771511],{"className":771510,"style":548314},[31805],[34,771512,771513,771516],{"style":116668},[34,771514],{"className":771515,"style":43406},[31813],[34,771517,771519],{"className":771518},[31818,43410,43411,31821],[34,771520,771522,771525,771528],{"className":771521},[31685,31821],[34,771523,31660],{"className":771524},[31685,31686,31821],[34,771526,32226],{"className":771527},[32024,31821],[34,771529,640],{"className":771530},[31685,31821],[34,771532,771533,771536],{"style":41520},[34,771534],{"className":771535,"style":41508},[31813],[34,771537],{"className":771538,"style":41528},[41527],[34,771540,771541,771544],{"style":46771},[34,771542],{"className":771543,"style":41508},[31813],[34,771545,771547],{"className":771546},[31818,31819,31820,31821],[34,771548,771550,771588],{"className":771549},[31685,31821],[34,771551,771553,771556],{"className":771552},[31685,31821],[34,771554,46900],{"className":771555},[31685,207923,31821],[34,771557,771559],{"className":771558},[31792],[34,771560,771562],{"className":771561},[31796],[34,771563,771565],{"className":771564},[31801],[34,771566,771568],{"className":771567,"style":144123},[31805],[34,771569,771570,771573],{"style":552391},[34,771571],{"className":771572,"style":43406},[31813],[34,771574,771576],{"className":771575},[31818,43410,43411,31821],[34,771577,771579,771582,771585],{"className":771578},[31685,31821],[34,771580,31660],{"className":771581},[31685,31686,31821],[34,771583,32226],{"className":771584},[32024,31821],[34,771586,640],{"className":771587},[31685,31821],[34,771589,32936],{"className":771590},[31685,31686,31821],[34,771592,31829],{"className":771593},[31828],[34,771595,771597],{"className":771596},[31801],[34,771598,771600],{"className":771599,"style":41619},[31805],[34,771601],{},[34,771603],{"className":771604},[31690,41488],[34,771606],{"className":771607,"style":32604},[31845],[34,771609,223],{"className":771610},[32024],[34,771612],{"className":771613,"style":32604},[31845],[34,771615,771617,771620,771623,771626,771629],{"className":771616},[31676],[34,771618],{"className":771619,"style":32597},[31680],[34,771621,38874],{"className":771622},[34006],[34,771624],{"className":771625,"style":32604},[31845],[34,771627,223],{"className":771628},[32024],[34,771630],{"className":771631,"style":32604},[31845],[34,771633,771635,771638,771678,771681,771684,771687,771690,771693,771696],{"className":771634},[31676],[34,771636],{"className":771637,"style":31774},[31680],[34,771639,771641,771644],{"className":771640},[31685],[34,771642,111],{"className":771643},[31685,31686],[34,771645,771647],{"className":771646},[31792],[34,771648,771650,771670],{"className":771649},[31796,31797],[34,771651,771653,771667],{"className":771652},[31801],[34,771654,771656],{"className":771655,"style":32362},[31805],[34,771657,771658,771661],{"style":31809},[34,771659],{"className":771660,"style":31814},[31813],[34,771662,771664],{"className":771663},[31818,31819,31820,31821],[34,771665,31660],{"className":771666},[31685,31686,31821],[34,771668,31829],{"className":771669},[31828],[34,771671,771673],{"className":771672},[31801],[34,771674,771676],{"className":771675,"style":31836},[31805],[34,771677],{},[34,771679,456],{"className":771680},[31782],[34,771682,33249],{"className":771683},[31685,31686],[34,771685,2557],{"className":771686},[31690],[34,771688,32936],{"className":771689},[31685,31686],[34,771691],{"className":771692,"style":31846},[31845],[34,771694,202],{"className":771695},[31850],[34,771697],{"className":771698,"style":31846},[31845],[34,771700,771702,771705,771745,771748,771751],{"className":771701},[31676],[34,771703],{"className":771704,"style":31774},[31680],[34,771706,771708,771711],{"className":771707},[31685],[34,771709,31739],{"className":771710,"style":31778},[31685,31686],[34,771712,771714],{"className":771713},[31792],[34,771715,771717,771737],{"className":771716},[31796,31797],[34,771718,771720,771734],{"className":771719},[31801],[34,771721,771723],{"className":771722,"style":31806},[31805],[34,771724,771725,771728],{"style":141014},[34,771726],{"className":771727,"style":31814},[31813],[34,771729,771731],{"className":771730},[31818,31819,31820,31821],[34,771732,768],{"className":771733},[31685,31821],[34,771735,31829],{"className":771736},[31828],[34,771738,771740],{"className":771739},[31801],[34,771741,771743],{"className":771742,"style":31836},[31805],[34,771744],{},[34,771746,456],{"className":771747},[31782],[34,771749,33249],{"className":771750},[31685,31686],[34,771752,2557],{"className":771753},[31690],"\n的解，则 ",[34,771756,771758,771796],{"className":771757},[31642],[34,771759,771761],{"className":771760},[31646],[31648,771762,771763],{"xmlns":31650},[31652,771764,771765,771793],{},[31655,771766,771767,771773,771775,771777,771779,771781,771787,771789,771791],{},[31743,771768,771769,771771],{},[31658,771770,32936],{},[31749,771772,640],{},[31662,771774,456],{"stretchy":702},[31658,771776,33249],{},[31662,771778,2557],{"stretchy":702},[31662,771780,223],{},[31743,771782,771783,771785],{},[31658,771784,32936],{},[31749,771786,768],{},[31662,771788,456],{"stretchy":702},[31658,771790,33249],{},[31662,771792,2557],{"stretchy":702},[31665,771794,771795],{"encoding":31667},"x_1(t) + x_2(t)",[34,771797,771799,771863],{"className":771798,"ariaHidden":758},[31672],[34,771800,771802,771805,771845,771848,771851,771854,771857,771860],{"className":771801},[31676],[34,771803],{"className":771804,"style":31774},[31680],[34,771806,771808,771811],{"className":771807},[31685],[34,771809,32936],{"className":771810},[31685,31686],[34,771812,771814],{"className":771813},[31792],[34,771815,771817,771837],{"className":771816},[31796,31797],[34,771818,771820,771834],{"className":771819},[31801],[34,771821,771823],{"className":771822,"style":31806},[31805],[34,771824,771825,771828],{"style":31809},[34,771826],{"className":771827,"style":31814},[31813],[34,771829,771831],{"className":771830},[31818,31819,31820,31821],[34,771832,640],{"className":771833},[31685,31821],[34,771835,31829],{"className":771836},[31828],[34,771838,771840],{"className":771839},[31801],[34,771841,771843],{"className":771842,"style":31836},[31805],[34,771844],{},[34,771846,456],{"className":771847},[31782],[34,771849,33249],{"className":771850},[31685,31686],[34,771852,2557],{"className":771853},[31690],[34,771855],{"className":771856,"style":32604},[31845],[34,771858,223],{"className":771859},[32024],[34,771861],{"className":771862,"style":32604},[31845],[34,771864,771866,771869,771909,771912,771915],{"className":771865},[31676],[34,771867],{"className":771868,"style":31774},[31680],[34,771870,771872,771875],{"className":771871},[31685],[34,771873,32936],{"className":771874},[31685,31686],[34,771876,771878],{"className":771877},[31792],[34,771879,771881,771901],{"className":771880},[31796,31797],[34,771882,771884,771898],{"className":771883},[31801],[34,771885,771887],{"className":771886,"style":31806},[31805],[34,771888,771889,771892],{"style":31809},[34,771890],{"className":771891,"style":31814},[31813],[34,771893,771895],{"className":771894},[31818,31819,31820,31821],[34,771896,768],{"className":771897},[31685,31821],[34,771899,31829],{"className":771900},[31828],[34,771902,771904],{"className":771903},[31801],[34,771905,771907],{"className":771906,"style":31836},[31805],[34,771908],{},[34,771910,456],{"className":771911},[31782],[34,771913,33249],{"className":771914},[31685,31686],[34,771916,2557],{"className":771917},[31690]," 是方程\n",[34,771920,771922,772052],{"className":771921},[31642],[34,771923,771925],{"className":771924},[31646],[31648,771926,771927],{"xmlns":31650},[31652,771928,771929,772049],{},[31655,771930,771931,771953,771955,771961,771963,771965,771967,772001,772003,772005,772007,772013,772015,772017,772019,772021,772023,772029,772031,772033,772035,772037,772043,772045,772047],{},[41453,771932,771933,771943],{},[31655,771934,771935,771941],{},[37624,771936,771937,771939],{},[31658,771938,46900],{"mathvariant":38079},[31658,771940,31660],{},[31658,771942,32936],{},[31655,771944,771945,771947],{},[31658,771946,46900],{"mathvariant":38079},[37624,771948,771949,771951],{},[31658,771950,33249],{},[31658,771952,31660],{},[31662,771954,223],{},[31743,771956,771957,771959],{},[31658,771958,111],{},[31749,771960,640],{},[31662,771962,456],{"stretchy":702},[31658,771964,33249],{},[31662,771966,2557],{"stretchy":702},[41453,771968,771969,771985],{},[31655,771970,771971,771983],{},[37624,771972,771973,771975],{},[31658,771974,46900],{"mathvariant":38079},[31655,771976,771977,771979,771981],{},[31658,771978,31660],{},[31662,771980,32226],{},[31749,771982,640],{},[31658,771984,32936],{},[31655,771986,771987,771989],{},[31658,771988,46900],{"mathvariant":38079},[37624,771990,771991,771993],{},[31658,771992,33249],{},[31655,771994,771995,771997,771999],{},[31658,771996,31660],{},[31662,771998,32226],{},[31749,772000,640],{},[31662,772002,223],{},[31662,772004,38874],{},[31662,772006,223],{},[31743,772008,772009,772011],{},[31658,772010,111],{},[31658,772012,31660],{},[31662,772014,456],{"stretchy":702},[31658,772016,33249],{},[31662,772018,2557],{"stretchy":702},[31658,772020,32936],{},[31662,772022,202],{},[31743,772024,772025,772027],{},[31658,772026,31739],{},[31749,772028,640],{},[31662,772030,456],{"stretchy":702},[31658,772032,33249],{},[31662,772034,2557],{"stretchy":702},[31662,772036,223],{},[31743,772038,772039,772041],{},[31658,772040,31739],{},[31749,772042,768],{},[31662,772044,456],{"stretchy":702},[31658,772046,33249],{},[31662,772048,2557],{"stretchy":702},[31665,772050,772051],{"encoding":31667},"\\frac{\\mathrm{d}^n x}{\\mathrm{d}t^n} + a_1(t)\\frac{\\mathrm{d}^{n-1}x}{\\mathrm{d}t^{n-1}} + \\cdots + a_n(t)x = f_1(t) + f_2(t)",[34,772053,772055,772196,772404,772422,772489,772553],{"className":772054,"ariaHidden":758},[31672],[34,772056,772058,772061,772187,772190,772193],{"className":772057},[31676],[34,772059],{"className":772060,"style":715659},[31680],[34,772062,772064,772067,772184],{"className":772063},[31685],[34,772065],{"className":772066},[31782,41488],[34,772068,772070],{"className":772069},[41453],[34,772071,772073,772176],{"className":772072},[31796,31797],[34,772074,772076,772173],{"className":772075},[31801],[34,772077,772079,772122,772130],{"className":772078,"style":700470},[31805],[34,772080,772081,772084],{"style":41504},[34,772082],{"className":772083,"style":41508},[31813],[34,772085,772087],{"className":772086},[31818,31819,31820,31821],[34,772088,772090,772093],{"className":772089},[31685,31821],[34,772091,46900],{"className":772092},[31685,207923,31821],[34,772094,772096,772099],{"className":772095},[31685,31821],[34,772097,33249],{"className":772098},[31685,31686,31821],[34,772100,772102],{"className":772101},[31792],[34,772103,772105],{"className":772104},[31796],[34,772106,772108],{"className":772107},[31801],[34,772109,772111],{"className":772110,"style":566313},[31805],[34,772112,772113,772116],{"style":116668},[34,772114],{"className":772115,"style":43406},[31813],[34,772117,772119],{"className":772118},[31818,43410,43411,31821],[34,772120,31660],{"className":772121},[31685,31686,31821],[34,772123,772124,772127],{"style":41520},[34,772125],{"className":772126,"style":41508},[31813],[34,772128],{"className":772129,"style":41528},[41527],[34,772131,772132,772135],{"style":46771},[34,772133],{"className":772134,"style":41508},[31813],[34,772136,772138],{"className":772137},[31818,31819,31820,31821],[34,772139,772141,772170],{"className":772140},[31685,31821],[34,772142,772144,772147],{"className":772143},[31685,31821],[34,772145,46900],{"className":772146},[31685,207923,31821],[34,772148,772150],{"className":772149},[31792],[34,772151,772153],{"className":772152},[31796],[34,772154,772156],{"className":772155},[31801],[34,772157,772159],{"className":772158,"style":611529},[31805],[34,772160,772161,772164],{"style":552391},[34,772162],{"className":772163,"style":43406},[31813],[34,772165,772167],{"className":772166},[31818,43410,43411,31821],[34,772168,31660],{"className":772169},[31685,31686,31821],[34,772171,32936],{"className":772172},[31685,31686,31821],[34,772174,31829],{"className":772175},[31828],[34,772177,772179],{"className":772178},[31801],[34,772180,772182],{"className":772181,"style":41619},[31805],[34,772183],{},[34,772185],{"className":772186},[31690,41488],[34,772188],{"className":772189,"style":32604},[31845],[34,772191,223],{"className":772192},[32024],[34,772194],{"className":772195,"style":32604},[31845],[34,772197,772199,772202,772242,772245,772248,772251,772395,772398,772401],{"className":772198},[31676],[34,772200],{"className":772201,"style":637019},[31680],[34,772203,772205,772208],{"className":772204},[31685],[34,772206,111],{"className":772207},[31685,31686],[34,772209,772211],{"className":772210},[31792],[34,772212,772214,772234],{"className":772213},[31796,31797],[34,772215,772217,772231],{"className":772216},[31801],[34,772218,772220],{"className":772219,"style":31806},[31805],[34,772221,772222,772225],{"style":31809},[34,772223],{"className":772224,"style":31814},[31813],[34,772226,772228],{"className":772227},[31818,31819,31820,31821],[34,772229,640],{"className":772230},[31685,31821],[34,772232,31829],{"className":772233},[31828],[34,772235,772237],{"className":772236},[31801],[34,772238,772240],{"className":772239,"style":31836},[31805],[34,772241],{},[34,772243,456],{"className":772244},[31782],[34,772246,33249],{"className":772247},[31685,31686],[34,772249,2557],{"className":772250},[31690],[34,772252,772254,772257,772392],{"className":772253},[31685],[34,772255],{"className":772256},[31782,41488],[34,772258,772260],{"className":772259},[41453],[34,772261,772263,772384],{"className":772262},[31796,31797],[34,772264,772266,772381],{"className":772265},[31801],[34,772267,772269,772321,772329],{"className":772268,"style":603956},[31805],[34,772270,772271,772274],{"style":41504},[34,772272],{"className":772273,"style":41508},[31813],[34,772275,772277],{"className":772276},[31818,31819,31820,31821],[34,772278,772280,772283],{"className":772279},[31685,31821],[34,772281,46900],{"className":772282},[31685,207923,31821],[34,772284,772286,772289],{"className":772285},[31685,31821],[34,772287,33249],{"className":772288},[31685,31686,31821],[34,772290,772292],{"className":772291},[31792],[34,772293,772295],{"className":772294},[31796],[34,772296,772298],{"className":772297},[31801],[34,772299,772301],{"className":772300,"style":548314},[31805],[34,772302,772303,772306],{"style":116668},[34,772304],{"className":772305,"style":43406},[31813],[34,772307,772309],{"className":772308},[31818,43410,43411,31821],[34,772310,772312,772315,772318],{"className":772311},[31685,31821],[34,772313,31660],{"className":772314},[31685,31686,31821],[34,772316,32226],{"className":772317},[32024,31821],[34,772319,640],{"className":772320},[31685,31821],[34,772322,772323,772326],{"style":41520},[34,772324],{"className":772325,"style":41508},[31813],[34,772327],{"className":772328,"style":41528},[41527],[34,772330,772331,772334],{"style":46771},[34,772332],{"className":772333,"style":41508},[31813],[34,772335,772337],{"className":772336},[31818,31819,31820,31821],[34,772338,772340,772378],{"className":772339},[31685,31821],[34,772341,772343,772346],{"className":772342},[31685,31821],[34,772344,46900],{"className":772345},[31685,207923,31821],[34,772347,772349],{"className":772348},[31792],[34,772350,772352],{"className":772351},[31796],[34,772353,772355],{"className":772354},[31801],[34,772356,772358],{"className":772357,"style":144123},[31805],[34,772359,772360,772363],{"style":552391},[34,772361],{"className":772362,"style":43406},[31813],[34,772364,772366],{"className":772365},[31818,43410,43411,31821],[34,772367,772369,772372,772375],{"className":772368},[31685,31821],[34,772370,31660],{"className":772371},[31685,31686,31821],[34,772373,32226],{"className":772374},[32024,31821],[34,772376,640],{"className":772377},[31685,31821],[34,772379,32936],{"className":772380},[31685,31686,31821],[34,772382,31829],{"className":772383},[31828],[34,772385,772387],{"className":772386},[31801],[34,772388,772390],{"className":772389,"style":41619},[31805],[34,772391],{},[34,772393],{"className":772394},[31690,41488],[34,772396],{"className":772397,"style":32604},[31845],[34,772399,223],{"className":772400},[32024],[34,772402],{"className":772403,"style":32604},[31845],[34,772405,772407,772410,772413,772416,772419],{"className":772406},[31676],[34,772408],{"className":772409,"style":32597},[31680],[34,772411,38874],{"className":772412},[34006],[34,772414],{"className":772415,"style":32604},[31845],[34,772417,223],{"className":772418},[32024],[34,772420],{"className":772421,"style":32604},[31845],[34,772423,772425,772428,772468,772471,772474,772477,772480,772483,772486],{"className":772424},[31676],[34,772426],{"className":772427,"style":31774},[31680],[34,772429,772431,772434],{"className":772430},[31685],[34,772432,111],{"className":772433},[31685,31686],[34,772435,772437],{"className":772436},[31792],[34,772438,772440,772460],{"className":772439},[31796,31797],[34,772441,772443,772457],{"className":772442},[31801],[34,772444,772446],{"className":772445,"style":32362},[31805],[34,772447,772448,772451],{"style":31809},[34,772449],{"className":772450,"style":31814},[31813],[34,772452,772454],{"className":772453},[31818,31819,31820,31821],[34,772455,31660],{"className":772456},[31685,31686,31821],[34,772458,31829],{"className":772459},[31828],[34,772461,772463],{"className":772462},[31801],[34,772464,772466],{"className":772465,"style":31836},[31805],[34,772467],{},[34,772469,456],{"className":772470},[31782],[34,772472,33249],{"className":772473},[31685,31686],[34,772475,2557],{"className":772476},[31690],[34,772478,32936],{"className":772479},[31685,31686],[34,772481],{"className":772482,"style":31846},[31845],[34,772484,202],{"className":772485},[31850],[34,772487],{"className":772488,"style":31846},[31845],[34,772490,772492,772495,772535,772538,772541,772544,772547,772550],{"className":772491},[31676],[34,772493],{"className":772494,"style":31774},[31680],[34,772496,772498,772501],{"className":772497},[31685],[34,772499,31739],{"className":772500,"style":31778},[31685,31686],[34,772502,772504],{"className":772503},[31792],[34,772505,772507,772527],{"className":772506},[31796,31797],[34,772508,772510,772524],{"className":772509},[31801],[34,772511,772513],{"className":772512,"style":31806},[31805],[34,772514,772515,772518],{"style":141014},[34,772516],{"className":772517,"style":31814},[31813],[34,772519,772521],{"className":772520},[31818,31819,31820,31821],[34,772522,640],{"className":772523},[31685,31821],[34,772525,31829],{"className":772526},[31828],[34,772528,772530],{"className":772529},[31801],[34,772531,772533],{"className":772532,"style":31836},[31805],[34,772534],{},[34,772536,456],{"className":772537},[31782],[34,772539,33249],{"className":772540},[31685,31686],[34,772542,2557],{"className":772543},[31690],[34,772545],{"className":772546,"style":32604},[31845],[34,772548,223],{"className":772549},[32024],[34,772551],{"className":772552,"style":32604},[31845],[34,772554,772556,772559,772599,772602,772605],{"className":772555},[31676],[34,772557],{"className":772558,"style":31774},[31680],[34,772560,772562,772565],{"className":772561},[31685],[34,772563,31739],{"className":772564,"style":31778},[31685,31686],[34,772566,772568],{"className":772567},[31792],[34,772569,772571,772591],{"className":772570},[31796,31797],[34,772572,772574,772588],{"className":772573},[31801],[34,772575,772577],{"className":772576,"style":31806},[31805],[34,772578,772579,772582],{"style":141014},[34,772580],{"className":772581,"style":31814},[31813],[34,772583,772585],{"className":772584},[31818,31819,31820,31821],[34,772586,768],{"className":772587},[31685,31821],[34,772589,31829],{"className":772590},[31828],[34,772592,772594],{"className":772593},[31801],[34,772595,772597],{"className":772596,"style":31836},[31805],[34,772598],{},[34,772600,456],{"className":772601},[31782],[34,772603,33249],{"className":772604},[31685,31686],[34,772606,2557],{"className":772607},[31690],"\n的解.",[108,772610,772611,772612,772736,772737,773089,773090,773420,773421,199477],{},"已知齐次线性微分方程的基本解组 ",[34,772613,772615,772641],{"className":772614},[31642],[34,772616,772618],{"className":772617},[31646],[31648,772619,772620],{"xmlns":31650},[31652,772621,772622,772638],{},[31655,772623,772624,772630,772632],{},[31743,772625,772626,772628],{},[31658,772627,32936],{},[31749,772629,640],{},[31662,772631,31960],{"separator":758},[31743,772633,772634,772636],{},[31658,772635,32936],{},[31749,772637,768],{},[31665,772639,772640],{"encoding":31667},"x_1,x_2",[34,772642,772644],{"className":772643,"ariaHidden":758},[31672],[34,772645,772647,772650,772690,772693,772696],{"className":772646},[31676],[34,772648],{"className":772649,"style":38774},[31680],[34,772651,772653,772656],{"className":772652},[31685],[34,772654,32936],{"className":772655},[31685,31686],[34,772657,772659],{"className":772658},[31792],[34,772660,772662,772682],{"className":772661},[31796,31797],[34,772663,772665,772679],{"className":772664},[31801],[34,772666,772668],{"className":772667,"style":31806},[31805],[34,772669,772670,772673],{"style":31809},[34,772671],{"className":772672,"style":31814},[31813],[34,772674,772676],{"className":772675},[31818,31819,31820,31821],[34,772677,640],{"className":772678},[31685,31821],[34,772680,31829],{"className":772681},[31828],[34,772683,772685],{"className":772684},[31801],[34,772686,772688],{"className":772687,"style":31836},[31805],[34,772689],{},[34,772691,31960],{"className":772692},[32119],[34,772694],{"className":772695,"style":32123},[31845],[34,772697,772699,772702],{"className":772698},[31685],[34,772700,32936],{"className":772701},[31685,31686],[34,772703,772705],{"className":772704},[31792],[34,772706,772708,772728],{"className":772707},[31796,31797],[34,772709,772711,772725],{"className":772710},[31801],[34,772712,772714],{"className":772713,"style":31806},[31805],[34,772715,772716,772719],{"style":31809},[34,772717],{"className":772718,"style":31814},[31813],[34,772720,772722],{"className":772721},[31818,31819,31820,31821],[34,772723,768],{"className":772724},[31685,31821],[34,772726,31829],{"className":772727},[31828],[34,772729,772731],{"className":772730},[31801],[34,772732,772734],{"className":772733,"style":31836},[31805],[34,772735],{},"，求下列方程对应的非齐次线性微分方程的通解:\n(1) ",[34,772738,772740,772814],{"className":772739},[31642],[34,772741,772743],{"className":772742},[31646],[31648,772744,772745],{"xmlns":31650},[31652,772746,772747,772811],{},[31655,772748,772749,772759,772761,772763,772765,772767,772769,772771,772773,772775,772781,772783,772789,772791,772793,772799,772801],{},[37624,772750,772751,772753],{},[31658,772752,32936],{},[31655,772754,772755,772757],{},[31662,772756,116495],{"mathvariant":38079},[31662,772758,116495],{"mathvariant":38079},[31662,772760,32226],{},[31658,772762,32936],{},[31662,772764,202],{},[31658,772766,547860],{},[31662,772768,32682],{},[31658,772770,33249],{},[31662,772772,31960],{"separator":758},[39361,772774,130298],{},[31743,772776,772777,772779],{},[31658,772778,32936],{},[31749,772780,640],{},[31662,772782,202],{},[37624,772784,772785,772787],{},[31658,772786,33244],{"mathvariant":38079},[31658,772788,33249],{},[31662,772790,31960],{"separator":758},[39361,772792,130298],{},[31743,772794,772795,772797],{},[31658,772796,32936],{},[31749,772798,768],{},[31662,772800,202],{},[37624,772802,772803,772805],{},[31658,772804,33244],{"mathvariant":38079},[31655,772806,772807,772809],{},[31662,772808,32226],{},[31658,772810,33249],{},[31665,772812,772813],{"encoding":31667},"x'' - x = \\cos t, \\ x_1 = \\mathrm{e}^t, \\ x_2 = \\mathrm{e}^{-t}",[34,772815,772817,772864,772882,772955,773048],{"className":772816,"ariaHidden":758},[31672],[34,772818,772820,772823,772855,772858,772861],{"className":772819},[31676],[34,772821],{"className":772822,"style":384364},[31680],[34,772824,772826,772829],{"className":772825},[31685],[34,772827,32936],{"className":772828},[31685,31686],[34,772830,772832],{"className":772831},[31792],[34,772833,772835],{"className":772834},[31796],[34,772836,772838],{"className":772837},[31801],[34,772839,772841],{"className":772840,"style":132027},[31805],[34,772842,772843,772846],{"style":37777},[34,772844],{"className":772845,"style":31814},[31813],[34,772847,772849],{"className":772848},[31818,31819,31820,31821],[34,772850,772852],{"className":772851},[31685,31821],[34,772853,502782],{"className":772854},[31685,31821],[34,772856],{"className":772857,"style":32604},[31845],[34,772859,32226],{"className":772860},[32024],[34,772862],{"className":772863,"style":32604},[31845],[34,772865,772867,772870,772873,772876,772879],{"className":772866},[31676],[34,772868],{"className":772869,"style":32527},[31680],[34,772871,32936],{"className":772872},[31685,31686],[34,772874],{"className":772875,"style":31846},[31845],[34,772877,202],{"className":772878},[31850],[34,772880],{"className":772881,"style":31846},[31845],[34,772883,772885,772888,772891,772894,772897,772900,772903,772906,772946,772949,772952],{"className":772884},[31676],[34,772886],{"className":772887,"style":223552},[31680],[34,772889,547860],{"className":772890},[32767],[34,772892],{"className":772893,"style":32123},[31845],[34,772895,33249],{"className":772896},[31685,31686],[34,772898,31960],{"className":772899},[32119],[34,772901,130298],{"className":772902},[31845],[34,772904],{"className":772905,"style":32123},[31845],[34,772907,772909,772912],{"className":772908},[31685],[34,772910,32936],{"className":772911},[31685,31686],[34,772913,772915],{"className":772914},[31792],[34,772916,772918,772938],{"className":772917},[31796,31797],[34,772919,772921,772935],{"className":772920},[31801],[34,772922,772924],{"className":772923,"style":31806},[31805],[34,772925,772926,772929],{"style":31809},[34,772927],{"className":772928,"style":31814},[31813],[34,772930,772932],{"className":772931},[31818,31819,31820,31821],[34,772933,640],{"className":772934},[31685,31821],[34,772936,31829],{"className":772937},[31828],[34,772939,772941],{"className":772940},[31801],[34,772942,772944],{"className":772943,"style":31836},[31805],[34,772945],{},[34,772947],{"className":772948,"style":31846},[31845],[34,772950,202],{"className":772951},[31850],[34,772953],{"className":772954,"style":31846},[31845],[34,772956,772958,772961,772990,772993,772996,772999,773039,773042,773045],{"className":772957},[31676],[34,772959],{"className":772960,"style":719358},[31680],[34,772962,772964,772967],{"className":772963},[31685],[34,772965,33244],{"className":772966},[31685,207923],[34,772968,772970],{"className":772969},[31792],[34,772971,772973],{"className":772972},[31796],[34,772974,772976],{"className":772975},[31801],[34,772977,772979],{"className":772978,"style":611758},[31805],[34,772980,772981,772984],{"style":37777},[34,772982],{"className":772983,"style":31814},[31813],[34,772985,772987],{"className":772986},[31818,31819,31820,31821],[34,772988,33249],{"className":772989},[31685,31686,31821],[34,772991,31960],{"className":772992},[32119],[34,772994,130298],{"className":772995},[31845],[34,772997],{"className":772998,"style":32123},[31845],[34,773000,773002,773005],{"className":773001},[31685],[34,773003,32936],{"className":773004},[31685,31686],[34,773006,773008],{"className":773007},[31792],[34,773009,773011,773031],{"className":773010},[31796,31797],[34,773012,773014,773028],{"className":773013},[31801],[34,773015,773017],{"className":773016,"style":31806},[31805],[34,773018,773019,773022],{"style":31809},[34,773020],{"className":773021,"style":31814},[31813],[34,773023,773025],{"className":773024},[31818,31819,31820,31821],[34,773026,768],{"className":773027},[31685,31821],[34,773029,31829],{"className":773030},[31828],[34,773032,773034],{"className":773033},[31801],[34,773035,773037],{"className":773036,"style":31836},[31805],[34,773038],{},[34,773040],{"className":773041,"style":31846},[31845],[34,773043,202],{"className":773044},[31850],[34,773046],{"className":773047,"style":31846},[31845],[34,773049,773051,773054],{"className":773050},[31676],[34,773052],{"className":773053,"style":611758},[31680],[34,773055,773057,773060],{"className":773056},[31685],[34,773058,33244],{"className":773059},[31685,207923],[34,773061,773063],{"className":773062},[31792],[34,773064,773066],{"className":773065},[31796],[34,773067,773069],{"className":773068},[31801],[34,773070,773072],{"className":773071,"style":611758},[31805],[34,773073,773074,773077],{"style":37777},[34,773075],{"className":773076,"style":31814},[31813],[34,773078,773080],{"className":773079},[31818,31819,31820,31821],[34,773081,773083,773086],{"className":773082},[31685,31821],[34,773084,32226],{"className":773085},[31685,31821],[34,773087,33249],{"className":773088},[31685,31686,31821],";\n(3) ",[34,773091,773093,773173],{"className":773092},[31642],[34,773094,773096],{"className":773095},[31646],[31648,773097,773098],{"xmlns":31650},[31652,773099,773100,773170],{},[31655,773101,773102,773112,773114,773116,773118,773120,773122,773124,773126,773128,773130,773132,773134,773140,773142,773144,773146,773148,773150,773152,773154,773160,773162,773164,773166,773168],{},[37624,773103,773104,773106],{},[31658,773105,32936],{},[31655,773107,773108,773110],{},[31662,773109,116495],{"mathvariant":38079},[31662,773111,116495],{"mathvariant":38079},[31662,773113,223],{},[31749,773115,15932],{},[31658,773117,32936],{},[31662,773119,202],{},[31658,773121,33249],{},[31658,773123,547853],{},[31662,773125,32682],{},[31749,773127,768],{},[31658,773129,33249],{},[31662,773131,31960],{"separator":758},[39361,773133,130298],{},[31743,773135,773136,773138],{},[31658,773137,32936],{},[31749,773139,640],{},[31662,773141,202],{},[31658,773143,547860],{},[31662,773145,32682],{},[31749,773147,768],{},[31658,773149,33249],{},[31662,773151,31960],{"separator":758},[39361,773153,130298],{},[31743,773155,773156,773158],{},[31658,773157,32936],{},[31749,773159,768],{},[31662,773161,202],{},[31658,773163,547853],{},[31662,773165,32682],{},[31749,773167,768],{},[31658,773169,33249],{},[31665,773171,773172],{"encoding":31667},"x'' + 4x = t\\sin 2t, \\ x_1 = \\cos 2t, \\ x_2 = \\sin 2t",[34,773174,773176,773223,773244,773326,773402],{"className":773175,"ariaHidden":758},[31672],[34,773177,773179,773182,773214,773217,773220],{"className":773178},[31676],[34,773180],{"className":773181,"style":384364},[31680],[34,773183,773185,773188],{"className":773184},[31685],[34,773186,32936],{"className":773187},[31685,31686],[34,773189,773191],{"className":773190},[31792],[34,773192,773194],{"className":773193},[31796],[34,773195,773197],{"className":773196},[31801],[34,773198,773200],{"className":773199,"style":132027},[31805],[34,773201,773202,773205],{"style":37777},[34,773203],{"className":773204,"style":31814},[31813],[34,773206,773208],{"className":773207},[31818,31819,31820,31821],[34,773209,773211],{"className":773210},[31685,31821],[34,773212,502782],{"className":773213},[31685,31821],[34,773215],{"className":773216,"style":32604},[31845],[34,773218,223],{"className":773219},[32024],[34,773221],{"className":773222,"style":32604},[31845],[34,773224,773226,773229,773232,773235,773238,773241],{"className":773225},[31676],[34,773227],{"className":773228,"style":32617},[31680],[34,773230,15932],{"className":773231},[31685],[34,773233,32936],{"className":773234},[31685,31686],[34,773236],{"className":773237,"style":31846},[31845],[34,773239,202],{"className":773240},[31850],[34,773242],{"className":773243,"style":31846},[31845],[34,773245,773247,773250,773253,773256,773259,773262,773265,773268,773271,773274,773277,773317,773320,773323],{"className":773246},[31676],[34,773248],{"className":773249,"style":547875},[31680],[34,773251,33249],{"className":773252},[31685,31686],[34,773254],{"className":773255,"style":32123},[31845],[34,773257,547853],{"className":773258},[32767],[34,773260],{"className":773261,"style":32123},[31845],[34,773263,768],{"className":773264},[31685],[34,773266,33249],{"className":773267},[31685,31686],[34,773269,31960],{"className":773270},[32119],[34,773272,130298],{"className":773273},[31845],[34,773275],{"className":773276,"style":32123},[31845],[34,773278,773280,773283],{"className":773279},[31685],[34,773281,32936],{"className":773282},[31685,31686],[34,773284,773286],{"className":773285},[31792],[34,773287,773289,773309],{"className":773288},[31796,31797],[34,773290,773292,773306],{"className":773291},[31801],[34,773293,773295],{"className":773294,"style":31806},[31805],[34,773296,773297,773300],{"style":31809},[34,773298],{"className":773299,"style":31814},[31813],[34,773301,773303],{"className":773302},[31818,31819,31820,31821],[34,773304,640],{"className":773305},[31685,31821],[34,773307,31829],{"className":773308},[31828],[34,773310,773312],{"className":773311},[31801],[34,773313,773315],{"className":773314,"style":31836},[31805],[34,773316],{},[34,773318],{"className":773319,"style":31846},[31845],[34,773321,202],{"className":773322},[31850],[34,773324],{"className":773325,"style":31846},[31845],[34,773327,773329,773332,773335,773338,773341,773344,773347,773350,773353,773393,773396,773399],{"className":773328},[31676],[34,773330],{"className":773331,"style":40292},[31680],[34,773333,547860],{"className":773334},[32767],[34,773336],{"className":773337,"style":32123},[31845],[34,773339,768],{"className":773340},[31685],[34,773342,33249],{"className":773343},[31685,31686],[34,773345,31960],{"className":773346},[32119],[34,773348,130298],{"className":773349},[31845],[34,773351],{"className":773352,"style":32123},[31845],[34,773354,773356,773359],{"className":773355},[31685],[34,773357,32936],{"className":773358},[31685,31686],[34,773360,773362],{"className":773361},[31792],[34,773363,773365,773385],{"className":773364},[31796,31797],[34,773366,773368,773382],{"className":773367},[31801],[34,773369,773371],{"className":773370,"style":31806},[31805],[34,773372,773373,773376],{"style":31809},[34,773374],{"className":773375,"style":31814},[31813],[34,773377,773379],{"className":773378},[31818,31819,31820,31821],[34,773380,768],{"className":773381},[31685,31821],[34,773383,31829],{"className":773384},[31828],[34,773386,773388],{"className":773387},[31801],[34,773389,773391],{"className":773390,"style":31836},[31805],[34,773392],{},[34,773394],{"className":773395,"style":31846},[31845],[34,773397,202],{"className":773398},[31850],[34,773400],{"className":773401,"style":31846},[31845],[34,773403,773405,773408,773411,773414,773417],{"className":773404},[31676],[34,773406],{"className":773407,"style":71430},[31680],[34,773409,547853],{"className":773410},[32767],[34,773412],{"className":773413,"style":32123},[31845],[34,773415,768],{"className":773416},[31685],[34,773418,33249],{"className":773419},[31685,31686],";\n(5) ",[34,773422,773424,773516],{"className":773423},[31642],[34,773425,773427],{"className":773426},[31646],[31648,773428,773429],{"xmlns":31650},[31652,773430,773431,773513],{},[31655,773432,773433,773439,773449,773451,773453,773459,773461,773463,773465,773467,773469,773471,773473,773479,773481,773483,773489,773491,773493,773495,773497,773503,773505,773507,773509,773511],{},[37624,773434,773435,773437],{},[31658,773436,33249],{},[31749,773438,768],{},[37624,773440,773441,773443],{},[31658,773442,32936],{},[31655,773444,773445,773447],{},[31662,773446,116495],{"mathvariant":38079},[31662,773448,116495],{"mathvariant":38079},[31662,773450,32226],{},[31658,773452,33249],{},[37624,773454,773455,773457],{},[31658,773456,32936],{},[31662,773458,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,773460,223],{},[31658,773462,32936],{},[31662,773464,202],{},[31749,773466,36656],{},[31658,773468,33249],{},[31662,773470,223],{},[31749,773472,729302],{},[37624,773474,773475,773477],{},[31658,773476,33249],{},[31749,773478,768],{},[31662,773480,31960],{"separator":758},[39361,773482,130298],{},[31743,773484,773485,773487],{},[31658,773486,32936],{},[31749,773488,640],{},[31662,773490,202],{},[31658,773492,33249],{},[31662,773494,31960],{"separator":758},[39361,773496,130298],{},[31743,773498,773499,773501],{},[31658,773500,32936],{},[31749,773502,768],{},[31662,773504,202],{},[31658,773506,33249],{},[31658,773508,563286],{},[31662,773510,32682],{},[31658,773512,33249],{},[31665,773514,773515],{"encoding":31667},"t^2 x'' - tx' + x = 6t + 34t^2, \\ x_1 = t, \\ x_2 = t\\ln t",[34,773517,773519,773595,773645,773663,773684,773780,773847],{"className":773518,"ariaHidden":758},[31672],[34,773520,773522,773525,773554,773586,773589,773592],{"className":773521},[31676],[34,773523],{"className":773524,"style":40044},[31680],[34,773526,773528,773531],{"className":773527},[31685],[34,773529,33249],{"className":773530},[31685,31686],[34,773532,773534],{"className":773533},[31792],[34,773535,773537],{"className":773536},[31796],[34,773538,773540],{"className":773539},[31801],[34,773541,773543],{"className":773542,"style":37774},[31805],[34,773544,773545,773548],{"style":37777},[34,773546],{"className":773547,"style":31814},[31813],[34,773549,773551],{"className":773550},[31818,31819,31820,31821],[34,773552,768],{"className":773553},[31685,31821],[34,773555,773557,773560],{"className":773556},[31685],[34,773558,32936],{"className":773559},[31685,31686],[34,773561,773563],{"className":773562},[31792],[34,773564,773566],{"className":773565},[31796],[34,773567,773569],{"className":773568},[31801],[34,773570,773572],{"className":773571,"style":132027},[31805],[34,773573,773574,773577],{"style":37777},[34,773575],{"className":773576,"style":31814},[31813],[34,773578,773580],{"className":773579},[31818,31819,31820,31821],[34,773581,773583],{"className":773582},[31685,31821],[34,773584,502782],{"className":773585},[31685,31821],[34,773587],{"className":773588,"style":32604},[31845],[34,773590,32226],{"className":773591},[32024],[34,773593],{"className":773594,"style":32604},[31845],[34,773596,773598,773601,773604,773636,773639,773642],{"className":773597},[31676],[34,773599],{"className":773600,"style":384364},[31680],[34,773602,33249],{"className":773603},[31685,31686],[34,773605,773607,773610],{"className":773606},[31685],[34,773608,32936],{"className":773609},[31685,31686],[34,773611,773613],{"className":773612},[31792],[34,773614,773616],{"className":773615},[31796],[34,773617,773619],{"className":773618},[31801],[34,773620,773622],{"className":773621,"style":132027},[31805],[34,773623,773624,773627],{"style":37777},[34,773625],{"className":773626,"style":31814},[31813],[34,773628,773630],{"className":773629},[31818,31819,31820,31821],[34,773631,773633],{"className":773632},[31685,31821],[34,773634,116495],{"className":773635},[31685,31821],[34,773637],{"className":773638,"style":32604},[31845],[34,773640,223],{"className":773641},[32024],[34,773643],{"className":773644,"style":32604},[31845],[34,773646,773648,773651,773654,773657,773660],{"className":773647},[31676],[34,773649],{"className":773650,"style":32527},[31680],[34,773652,32936],{"className":773653},[31685,31686],[34,773655],{"className":773656,"style":31846},[31845],[34,773658,202],{"className":773659},[31850],[34,773661],{"className":773662,"style":31846},[31845],[34,773664,773666,773669,773672,773675,773678,773681],{"className":773665},[31676],[34,773667],{"className":773668,"style":35942},[31680],[34,773670,36656],{"className":773671},[31685],[34,773673,33249],{"className":773674},[31685,31686],[34,773676],{"className":773677,"style":32604},[31845],[34,773679,223],{"className":773680},[32024],[34,773682],{"className":773683,"style":32604},[31845],[34,773685,773687,773690,773693,773722,773725,773728,773731,773771,773774,773777],{"className":773686},[31676],[34,773688],{"className":773689,"style":315747},[31680],[34,773691,729302],{"className":773692},[31685],[34,773694,773696,773699],{"className":773695},[31685],[34,773697,33249],{"className":773698},[31685,31686],[34,773700,773702],{"className":773701},[31792],[34,773703,773705],{"className":773704},[31796],[34,773706,773708],{"className":773707},[31801],[34,773709,773711],{"className":773710,"style":37774},[31805],[34,773712,773713,773716],{"style":37777},[34,773714],{"className":773715,"style":31814},[31813],[34,773717,773719],{"className":773718},[31818,31819,31820,31821],[34,773720,768],{"className":773721},[31685,31821],[34,773723,31960],{"className":773724},[32119],[34,773726,130298],{"className":773727},[31845],[34,773729],{"className":773730,"style":32123},[31845],[34,773732,773734,773737],{"className":773733},[31685],[34,773735,32936],{"className":773736},[31685,31686],[34,773738,773740],{"className":773739},[31792],[34,773741,773743,773763],{"className":773742},[31796,31797],[34,773744,773746,773760],{"className":773745},[31801],[34,773747,773749],{"className":773748,"style":31806},[31805],[34,773750,773751,773754],{"style":31809},[34,773752],{"className":773753,"style":31814},[31813],[34,773755,773757],{"className":773756},[31818,31819,31820,31821],[34,773758,640],{"className":773759},[31685,31821],[34,773761,31829],{"className":773762},[31828],[34,773764,773766],{"className":773765},[31801],[34,773767,773769],{"className":773768,"style":31836},[31805],[34,773770],{},[34,773772],{"className":773773,"style":31846},[31845],[34,773775,202],{"className":773776},[31850],[34,773778],{"className":773779,"style":31846},[31845],[34,773781,773783,773786,773789,773792,773795,773798,773838,773841,773844],{"className":773782},[31676],[34,773784],{"className":773785,"style":223552},[31680],[34,773787,33249],{"className":773788},[31685,31686],[34,773790,31960],{"className":773791},[32119],[34,773793,130298],{"className":773794},[31845],[34,773796],{"className":773797,"style":32123},[31845],[34,773799,773801,773804],{"className":773800},[31685],[34,773802,32936],{"className":773803},[31685,31686],[34,773805,773807],{"className":773806},[31792],[34,773808,773810,773830],{"className":773809},[31796,31797],[34,773811,773813,773827],{"className":773812},[31801],[34,773814,773816],{"className":773815,"style":31806},[31805],[34,773817,773818,773821],{"style":31809},[34,773819],{"className":773820,"style":31814},[31813],[34,773822,773824],{"className":773823},[31818,31819,31820,31821],[34,773825,768],{"className":773826},[31685,31821],[34,773828,31829],{"className":773829},[31828],[34,773831,773833],{"className":773832},[31801],[34,773834,773836],{"className":773835,"style":31836},[31805],[34,773837],{},[34,773839],{"className":773840,"style":31846},[31845],[34,773842,202],{"className":773843},[31850],[34,773845],{"className":773846,"style":31846},[31845],[34,773848,773850,773853,773856,773859,773862,773865],{"className":773849},[31676],[34,773851],{"className":773852,"style":31681},[31680],[34,773854,33249],{"className":773855},[31685,31686],[34,773857],{"className":773858,"style":32123},[31845],[34,773860,563286],{"className":773861},[32767],[34,773863],{"className":773864,"style":32123},[31845],[34,773866,33249],{"className":773867},[31685,31686],[108,773869,773870,773871,774087,774088,774200,774201,774355,774356,774510,774511,774577,774578,772608],{},"已知方程 ",[34,773872,773874,773916],{"className":773873},[31642],[34,773875,773877],{"className":773876},[31646],[31648,773878,773879],{"xmlns":31650},[31652,773880,773881,773913],{},[31655,773882,773883,773905,773907,773909,773911],{},[41453,773884,773885,773895],{},[31655,773886,773887,773893],{},[37624,773888,773889,773891],{},[31658,773890,46900],{"mathvariant":38079},[31749,773892,768],{},[31658,773894,32936],{},[31655,773896,773897,773899],{},[31658,773898,46900],{"mathvariant":38079},[37624,773900,773901,773903],{},[31658,773902,33249],{},[31749,773904,768],{},[31662,773906,32226],{},[31658,773908,32936],{},[31662,773910,202],{},[31749,773912,459],{},[31665,773914,773915],{"encoding":31667},"\\frac{\\mathrm{d}^2 x}{\\mathrm{d}t^2} - x = 0",[34,773917,773919,774060,774078],{"className":773918,"ariaHidden":758},[31672],[34,773920,773922,773925,774051,774054,774057],{"className":773921},[31676],[34,773923],{"className":773924,"style":637019},[31680],[34,773926,773928,773931,774048],{"className":773927},[31685],[34,773929],{"className":773930},[31782,41488],[34,773932,773934],{"className":773933},[41453],[34,773935,773937,774040],{"className":773936},[31796,31797],[34,773938,773940,774037],{"className":773939},[31801],[34,773941,773943,773986,773994],{"className":773942,"style":603956},[31805],[34,773944,773945,773948],{"style":41504},[34,773946],{"className":773947,"style":41508},[31813],[34,773949,773951],{"className":773950},[31818,31819,31820,31821],[34,773952,773954,773957],{"className":773953},[31685,31821],[34,773955,46900],{"className":773956},[31685,207923,31821],[34,773958,773960,773963],{"className":773959},[31685,31821],[34,773961,33249],{"className":773962},[31685,31686,31821],[34,773964,773966],{"className":773965},[31792],[34,773967,773969],{"className":773968},[31796],[34,773970,773972],{"className":773971},[31801],[34,773973,773975],{"className":773974,"style":548314},[31805],[34,773976,773977,773980],{"style":116668},[34,773978],{"className":773979,"style":43406},[31813],[34,773981,773983],{"className":773982},[31818,43410,43411,31821],[34,773984,768],{"className":773985},[31685,31821],[34,773987,773988,773991],{"style":41520},[34,773989],{"className":773990,"style":41508},[31813],[34,773992],{"className":773993,"style":41528},[41527],[34,773995,773996,773999],{"style":46771},[34,773997],{"className":773998,"style":41508},[31813],[34,774000,774002],{"className":774001},[31818,31819,31820,31821],[34,774003,774005,774034],{"className":774004},[31685,31821],[34,774006,774008,774011],{"className":774007},[31685,31821],[34,774009,46900],{"className":774010},[31685,207923,31821],[34,774012,774014],{"className":774013},[31792],[34,774015,774017],{"className":774016},[31796],[34,774018,774020],{"className":774019},[31801],[34,774021,774023],{"className":774022,"style":144123},[31805],[34,774024,774025,774028],{"style":552391},[34,774026],{"className":774027,"style":43406},[31813],[34,774029,774031],{"className":774030},[31818,43410,43411,31821],[34,774032,768],{"className":774033},[31685,31821],[34,774035,32936],{"className":774036},[31685,31686,31821],[34,774038,31829],{"className":774039},[31828],[34,774041,774043],{"className":774042},[31801],[34,774044,774046],{"className":774045,"style":41619},[31805],[34,774047],{},[34,774049],{"className":774050},[31690,41488],[34,774052],{"className":774053,"style":32604},[31845],[34,774055,32226],{"className":774056},[32024],[34,774058],{"className":774059,"style":32604},[31845],[34,774061,774063,774066,774069,774072,774075],{"className":774062},[31676],[34,774064],{"className":774065,"style":32527},[31680],[34,774067,32936],{"className":774068},[31685,31686],[34,774070],{"className":774071,"style":31846},[31845],[34,774073,202],{"className":774074},[31850],[34,774076],{"className":774077,"style":31846},[31845],[34,774079,774081,774084],{"className":774080},[31676],[34,774082],{"className":774083,"style":32617},[31680],[34,774085,459],{"className":774086},[31685]," 有基本解组 ",[34,774089,774091,774121],{"className":774090},[31642],[34,774092,774094],{"className":774093},[31646],[31648,774095,774096],{"xmlns":31650},[31652,774097,774098,774118],{},[31655,774099,774100,774106,774108],{},[37624,774101,774102,774104],{},[31658,774103,33244],{"mathvariant":38079},[31658,774105,33249],{},[31662,774107,31960],{"separator":758},[37624,774109,774110,774112],{},[31658,774111,33244],{"mathvariant":38079},[31655,774113,774114,774116],{},[31662,774115,32226],{},[31658,774117,33249],{},[31665,774119,774120],{"encoding":31667},"\\mathrm{e}^t,\\mathrm{e}^{-t}",[34,774122,774124],{"className":774123,"ariaHidden":758},[31672],[34,774125,774127,774130,774159,774162,774165],{"className":774126},[31676],[34,774128],{"className":774129,"style":719358},[31680],[34,774131,774133,774136],{"className":774132},[31685],[34,774134,33244],{"className":774135},[31685,207923],[34,774137,774139],{"className":774138},[31792],[34,774140,774142],{"className":774141},[31796],[34,774143,774145],{"className":774144},[31801],[34,774146,774148],{"className":774147,"style":611758},[31805],[34,774149,774150,774153],{"style":37777},[34,774151],{"className":774152,"style":31814},[31813],[34,774154,774156],{"className":774155},[31818,31819,31820,31821],[34,774157,33249],{"className":774158},[31685,31686,31821],[34,774160,31960],{"className":774161},[32119],[34,774163],{"className":774164,"style":32123},[31845],[34,774166,774168,774171],{"className":774167},[31685],[34,774169,33244],{"className":774170},[31685,207923],[34,774172,774174],{"className":774173},[31792],[34,774175,774177],{"className":774176},[31796],[34,774178,774180],{"className":774179},[31801],[34,774181,774183],{"className":774182,"style":611758},[31805],[34,774184,774185,774188],{"style":37777},[34,774186],{"className":774187,"style":31814},[31813],[34,774189,774191],{"className":774190},[31818,31819,31820,31821],[34,774192,774194,774197],{"className":774193},[31685,31821],[34,774195,32226],{"className":774196},[31685,31821],[34,774198,33249],{"className":774199},[31685,31686,31821],"，试求此方程适合初值条件\n",[34,774202,774204,774248],{"className":774203},[31642],[34,774205,774207],{"className":774206},[31646],[31648,774208,774209],{"xmlns":31650},[31652,774210,774211,774245],{},[31655,774212,774213,774215,774217,774219,774221,774223,774225,774227,774229,774235,774237,774239,774241,774243],{},[31658,774214,32936],{},[31662,774216,456],{"stretchy":702},[31749,774218,459],{},[31662,774220,2557],{"stretchy":702},[31662,774222,202],{},[31749,774224,640],{},[31662,774226,31960],{"separator":758},[39361,774228,130298],{},[37624,774230,774231,774233],{},[31658,774232,32936],{},[31662,774234,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,774236,456],{"stretchy":702},[31749,774238,459],{},[31662,774240,2557],{"stretchy":702},[31662,774242,202],{},[31749,774244,459],{},[31665,774246,774247],{"encoding":31667},"x(0) = 1, \\ x'(0) = 0",[34,774249,774251,774278,774346],{"className":774250,"ariaHidden":758},[31672],[34,774252,774254,774257,774260,774263,774266,774269,774272,774275],{"className":774253},[31676],[34,774255],{"className":774256,"style":31774},[31680],[34,774258,32936],{"className":774259},[31685,31686],[34,774261,456],{"className":774262},[31782],[34,774264,459],{"className":774265},[31685],[34,774267,2557],{"className":774268},[31690],[34,774270],{"className":774271,"style":31846},[31845],[34,774273,202],{"className":774274},[31850],[34,774276],{"className":774277,"style":31846},[31845],[34,774279,774281,774284,774287,774290,774293,774296,774328,774331,774334,774337,774340,774343],{"className":774280},[31676],[34,774282],{"className":774283,"style":570646},[31680],[34,774285,640],{"className":774286},[31685],[34,774288,31960],{"className":774289},[32119],[34,774291,130298],{"className":774292},[31845],[34,774294],{"className":774295,"style":32123},[31845],[34,774297,774299,774302],{"className":774298},[31685],[34,774300,32936],{"className":774301},[31685,31686],[34,774303,774305],{"className":774304},[31792],[34,774306,774308],{"className":774307},[31796],[34,774309,774311],{"className":774310},[31801],[34,774312,774314],{"className":774313,"style":132027},[31805],[34,774315,774316,774319],{"style":37777},[34,774317],{"className":774318,"style":31814},[31813],[34,774320,774322],{"className":774321},[31818,31819,31820,31821],[34,774323,774325],{"className":774324},[31685,31821],[34,774326,116495],{"className":774327},[31685,31821],[34,774329,456],{"className":774330},[31782],[34,774332,459],{"className":774333},[31685],[34,774335,2557],{"className":774336},[31690],[34,774338],{"className":774339,"style":31846},[31845],[34,774341,202],{"className":774342},[31850],[34,774344],{"className":774345,"style":31846},[31845],[34,774347,774349,774352],{"className":774348},[31676],[34,774350],{"className":774351,"style":32617},[31680],[34,774353,459],{"className":774354},[31685],"\n及\n",[34,774357,774359,774403],{"className":774358},[31642],[34,774360,774362],{"className":774361},[31646],[31648,774363,774364],{"xmlns":31650},[31652,774365,774366,774400],{},[31655,774367,774368,774370,774372,774374,774376,774378,774380,774382,774384,774390,774392,774394,774396,774398],{},[31658,774369,32936],{},[31662,774371,456],{"stretchy":702},[31749,774373,459],{},[31662,774375,2557],{"stretchy":702},[31662,774377,202],{},[31749,774379,459],{},[31662,774381,31960],{"separator":758},[39361,774383,130298],{},[37624,774385,774386,774388],{},[31658,774387,32936],{},[31662,774389,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,774391,456],{"stretchy":702},[31749,774393,459],{},[31662,774395,2557],{"stretchy":702},[31662,774397,202],{},[31749,774399,640],{},[31665,774401,774402],{"encoding":31667},"x(0) = 0, \\ x'(0) = 1",[34,774404,774406,774433,774501],{"className":774405,"ariaHidden":758},[31672],[34,774407,774409,774412,774415,774418,774421,774424,774427,774430],{"className":774408},[31676],[34,774410],{"className":774411,"style":31774},[31680],[34,774413,32936],{"className":774414},[31685,31686],[34,774416,456],{"className":774417},[31782],[34,774419,459],{"className":774420},[31685],[34,774422,2557],{"className":774423},[31690],[34,774425],{"className":774426,"style":31846},[31845],[34,774428,202],{"className":774429},[31850],[34,774431],{"className":774432,"style":31846},[31845],[34,774434,774436,774439,774442,774445,774448,774451,774483,774486,774489,774492,774495,774498],{"className":774435},[31676],[34,774437],{"className":774438,"style":570646},[31680],[34,774440,459],{"className":774441},[31685],[34,774443,31960],{"className":774444},[32119],[34,774446,130298],{"className":774447},[31845],[34,774449],{"className":774450,"style":32123},[31845],[34,774452,774454,774457],{"className":774453},[31685],[34,774455,32936],{"className":774456},[31685,31686],[34,774458,774460],{"className":774459},[31792],[34,774461,774463],{"className":774462},[31796],[34,774464,774466],{"className":774465},[31801],[34,774467,774469],{"className":774468,"style":132027},[31805],[34,774470,774471,774474],{"style":37777},[34,774472],{"className":774473,"style":31814},[31813],[34,774475,774477],{"className":774476},[31818,31819,31820,31821],[34,774478,774480],{"className":774479},[31685,31821],[34,774481,116495],{"className":774482},[31685,31821],[34,774484,456],{"className":774485},[31782],[34,774487,459],{"className":774488},[31685],[34,774490,2557],{"className":774491},[31690],[34,774493],{"className":774494,"style":31846},[31845],[34,774496,202],{"className":774497},[31850],[34,774499],{"className":774500,"style":31846},[31845],[34,774502,774504,774507],{"className":774503},[31676],[34,774505],{"className":774506,"style":32617},[31680],[34,774508,640],{"className":774509},[31685],"\n的基本解组（称为标准基本解组，即有 ",[34,774512,774514,774538],{"className":774513},[31642],[34,774515,774517],{"className":774516},[31646],[31648,774518,774519],{"xmlns":31650},[31652,774520,774521,774535],{},[31655,774522,774523,774525,774527,774529,774531,774533],{},[31658,774524,56852],{},[31662,774526,456],{"stretchy":702},[31749,774528,459],{},[31662,774530,2557],{"stretchy":702},[31662,774532,202],{},[31749,774534,640],{},[31665,774536,774537],{"encoding":31667},"W(0) = 1",[34,774539,774541,774568],{"className":774540,"ariaHidden":758},[31672],[34,774542,774544,774547,774550,774553,774556,774559,774562,774565],{"className":774543},[31676],[34,774545],{"className":774546,"style":31774},[31680],[34,774548,56852],{"className":774549,"style":37422},[31685,31686],[34,774551,456],{"className":774552},[31782],[34,774554,459],{"className":774555},[31685],[34,774557,2557],{"className":774558},[31690],[34,774560],{"className":774561,"style":31846},[31845],[34,774563,202],{"className":774564},[31850],[34,774566],{"className":774567,"style":31846},[31845],[34,774569,774571,774574],{"className":774570},[31676],[34,774572],{"className":774573,"style":32617},[31680],[34,774575,640],{"className":774576},[31685],"），并由此求出方程适合初值条件\n",[34,774579,774581,774635],{"className":774580},[31642],[34,774582,774584],{"className":774583},[31646],[31648,774585,774586],{"xmlns":31650},[31652,774587,774588,774632],{},[31655,774589,774590,774592,774594,774596,774598,774600,774606,774608,774610,774616,774618,774620,774622,774624],{},[31658,774591,32936],{},[31662,774593,456],{"stretchy":702},[31749,774595,459],{},[31662,774597,2557],{"stretchy":702},[31662,774599,202],{},[31743,774601,774602,774604],{},[31658,774603,32936],{},[31749,774605,459],{},[31662,774607,31960],{"separator":758},[39361,774609,130298],{},[37624,774611,774612,774614],{},[31658,774613,32936],{},[31662,774615,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,774617,456],{"stretchy":702},[31749,774619,459],{},[31662,774621,2557],{"stretchy":702},[31662,774623,202],{},[38846,774625,774626,774628,774630],{},[31658,774627,32936],{},[31749,774629,459],{},[31662,774631,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31665,774633,774634],{"encoding":31667},"x(0) = x_0, \\ x'(0) = x'_0",[34,774636,774638,774665,774770],{"className":774637,"ariaHidden":758},[31672],[34,774639,774641,774644,774647,774650,774653,774656,774659,774662],{"className":774640},[31676],[34,774642],{"className":774643,"style":31774},[31680],[34,774645,32936],{"className":774646},[31685,31686],[34,774648,456],{"className":774649},[31782],[34,774651,459],{"className":774652},[31685],[34,774654,2557],{"className":774655},[31690],[34,774657],{"className":774658,"style":31846},[31845],[34,774660,202],{"className":774661},[31850],[34,774663],{"className":774664,"style":31846},[31845],[34,774666,774668,774671,774711,774714,774717,774720,774752,774755,774758,774761,774764,774767],{"className":774667},[31676],[34,774669],{"className":774670,"style":570646},[31680],[34,774672,774674,774677],{"className":774673},[31685],[34,774675,32936],{"className":774676},[31685,31686],[34,774678,774680],{"className":774679},[31792],[34,774681,774683,774703],{"className":774682},[31796,31797],[34,774684,774686,774700],{"className":774685},[31801],[34,774687,774689],{"className":774688,"style":31806},[31805],[34,774690,774691,774694],{"style":31809},[34,774692],{"className":774693,"style":31814},[31813],[34,774695,774697],{"className":774696},[31818,31819,31820,31821],[34,774698,459],{"className":774699},[31685,31821],[34,774701,31829],{"className":774702},[31828],[34,774704,774706],{"className":774705},[31801],[34,774707,774709],{"className":774708,"style":31836},[31805],[34,774710],{},[34,774712,31960],{"className":774713},[32119],[34,774715,130298],{"className":774716},[31845],[34,774718],{"className":774719,"style":32123},[31845],[34,774721,774723,774726],{"className":774722},[31685],[34,774724,32936],{"className":774725},[31685,31686],[34,774727,774729],{"className":774728},[31792],[34,774730,774732],{"className":774731},[31796],[34,774733,774735],{"className":774734},[31801],[34,774736,774738],{"className":774737,"style":132027},[31805],[34,774739,774740,774743],{"style":37777},[34,774741],{"className":774742,"style":31814},[31813],[34,774744,774746],{"className":774745},[31818,31819,31820,31821],[34,774747,774749],{"className":774748},[31685,31821],[34,774750,116495],{"className":774751},[31685,31821],[34,774753,456],{"className":774754},[31782],[34,774756,459],{"className":774757},[31685],[34,774759,2557],{"className":774760},[31690],[34,774762],{"className":774763,"style":31846},[31845],[34,774765,202],{"className":774766},[31850],[34,774768],{"className":774769,"style":31846},[31845],[34,774771,774773,774776],{"className":774772},[31676],[34,774774],{"className":774775,"style":716670},[31680],[34,774777,774779,774782],{"className":774778},[31685],[34,774780,32936],{"className":774781},[31685,31686],[34,774783,774785],{"className":774784},[31792],[34,774786,774788,774822],{"className":774787},[31796,31797],[34,774789,774791,774819],{"className":774790},[31801],[34,774792,774794,774805],{"className":774793,"style":132027},[31805],[34,774795,774796,774799],{"style":316930},[34,774797],{"className":774798,"style":31814},[31813],[34,774800,774802],{"className":774801},[31818,31819,31820,31821],[34,774803,459],{"className":774804},[31685,31821],[34,774806,774807,774810],{"style":37777},[34,774808],{"className":774809,"style":31814},[31813],[34,774811,774813],{"className":774812},[31818,31819,31820,31821],[34,774814,774816],{"className":774815},[31685,31821],[34,774817,116495],{"className":774818},[31685,31821],[34,774820,31829],{"className":774821},[31828],[34,774823,774825],{"className":774824},[31801],[34,774826,774828],{"className":774827,"style":39021},[31805],[34,774829],{},[108,774831,133910,774832,775006,775007,775035,775036,775079,775080,775123,775124,775296,775297],{},[34,774833,774835,774885],{"className":774834},[31642],[34,774836,774838],{"className":774837},[31646],[31648,774839,774840],{"xmlns":31650},[31652,774841,774842,774882],{},[31655,774843,774844,774850,774852,774854,774856,774858,774860,774862,774864,774866,774868,774870,774872,774874,774876,774878,774880],{},[31743,774845,774846,774848],{},[31658,774847,32936],{},[31658,774849,31955],{},[31662,774851,456],{"stretchy":702},[31658,774853,33249],{},[31662,774855,2557],{"stretchy":702},[39361,774857,130298],{},[31662,774859,456],{"stretchy":702},[31658,774861,31955],{},[31662,774863,202],{},[31749,774865,640],{},[31662,774867,31960],{"separator":758},[31749,774869,768],{},[31662,774871,31960],{"separator":758},[31662,774873,38874],{},[39361,774875,39363],{},[31662,774877,31960],{"separator":758},[31658,774879,31660],{},[31662,774881,2557],{"stretchy":702},[31665,774883,774884],{"encoding":31667},"x_i(t) \\ (i = 1,2,\\cdots,n)",[34,774886,774888,774961],{"className":774887,"ariaHidden":758},[31672],[34,774889,774891,774894,774934,774937,774940,774943,774946,774949,774952,774955,774958],{"className":774890},[31676],[34,774892],{"className":774893,"style":31774},[31680],[34,774895,774897,774900],{"className":774896},[31685],[34,774898,32936],{"className":774899},[31685,31686],[34,774901,774903],{"className":774902},[31792],[34,774904,774906,774926],{"className":774905},[31796,31797],[34,774907,774909,774923],{"className":774908},[31801],[34,774910,774912],{"className":774911,"style":32090},[31805],[34,774913,774914,774917],{"style":31809},[34,774915],{"className":774916,"style":31814},[31813],[34,774918,774920],{"className":774919},[31818,31819,31820,31821],[34,774921,31955],{"className":774922},[31685,31686,31821],[34,774924,31829],{"className":774925},[31828],[34,774927,774929],{"className":774928},[31801],[34,774930,774932],{"className":774931,"style":31836},[31805],[34,774933],{},[34,774935,456],{"className":774936},[31782],[34,774938,33249],{"className":774939},[31685,31686],[34,774941,2557],{"className":774942},[31690],[34,774944,130298],{"className":774945},[31845],[34,774947,456],{"className":774948},[31782],[34,774950,31955],{"className":774951},[31685,31686],[34,774953],{"className":774954,"style":31846},[31845],[34,774956,202],{"className":774957},[31850],[34,774959],{"className":774960,"style":31846},[31845],[34,774962,774964,774967,774970,774973,774976,774979,774982,774985,774988,774991,774994,774997,775000,775003],{"className":774963},[31676],[34,774965],{"className":774966,"style":31774},[31680],[34,774968,640],{"className":774969},[31685],[34,774971,31960],{"className":774972},[32119],[34,774974],{"className":774975,"style":32123},[31845],[34,774977,768],{"className":774978},[31685],[34,774980,31960],{"className":774981},[32119],[34,774983],{"className":774984,"style":32123},[31845],[34,774986,38874],{"className":774987},[34006],[34,774989],{"className":774990,"style":32123},[31845],[34,774992],{"className":774993,"style":32123},[31845],[34,774995,31960],{"className":774996},[32119],[34,774998],{"className":774999,"style":32123},[31845],[34,775001,31660],{"className":775002},[31685,31686],[34,775004,2557],{"className":775005},[31690]," 是齐次线性微分方程 (4.2) 的任意 ",[34,775008,775010,775023],{"className":775009},[31642],[34,775011,775013],{"className":775012},[31646],[31648,775014,775015],{"xmlns":31650},[31652,775016,775017,775021],{},[31655,775018,775019],{},[31658,775020,31660],{},[31665,775022,31660],{"encoding":31667},[34,775024,775026],{"className":775025,"ariaHidden":758},[31672],[34,775027,775029,775032],{"className":775028},[31676],[34,775030],{"className":775031,"style":32527},[31680],[34,775033,31660],{"className":775034},[31685,31686]," 个解，它们所构成的朗斯基行列式记为 ",[34,775037,775039,775058],{"className":775038},[31642],[34,775040,775042],{"className":775041},[31646],[31648,775043,775044],{"xmlns":31650},[31652,775045,775046,775056],{},[31655,775047,775048,775050,775052,775054],{},[31658,775049,56852],{},[31662,775051,456],{"stretchy":702},[31658,775053,33249],{},[31662,775055,2557],{"stretchy":702},[31665,775057,737940],{"encoding":31667},[34,775059,775061],{"className":775060,"ariaHidden":758},[31672],[34,775062,775064,775067,775070,775073,775076],{"className":775063},[31676],[34,775065],{"className":775066,"style":31774},[31680],[34,775068,56852],{"className":775069,"style":37422},[31685,31686],[34,775071,456],{"className":775072},[31782],[34,775074,33249],{"className":775075},[31685,31686],[34,775077,2557],{"className":775078},[31690],". 试证明 ",[34,775081,775083,775102],{"className":775082},[31642],[34,775084,775086],{"className":775085},[31646],[31648,775087,775088],{"xmlns":31650},[31652,775089,775090,775100],{},[31655,775091,775092,775094,775096,775098],{},[31658,775093,56852],{},[31662,775095,456],{"stretchy":702},[31658,775097,33249],{},[31662,775099,2557],{"stretchy":702},[31665,775101,737940],{"encoding":31667},[34,775103,775105],{"className":775104,"ariaHidden":758},[31672],[34,775106,775108,775111,775114,775117,775120],{"className":775107},[31676],[34,775109],{"className":775110,"style":31774},[31680],[34,775112,56852],{"className":775113,"style":37422},[31685,31686],[34,775115,456],{"className":775116},[31782],[34,775118,33249],{"className":775119},[31685,31686],[34,775121,2557],{"className":775122},[31690]," 满足一阶线性微分方程\n",[34,775125,775127,775167],{"className":775126},[31642],[34,775128,775130],{"className":775129},[31646],[31648,775131,775132],{"xmlns":31650},[31652,775133,775134,775164],{},[31655,775135,775136,775142,775144,775150,775152,775154,775156,775158,775160,775162],{},[37624,775137,775138,775140],{},[31658,775139,56852],{},[31662,775141,116495],{"mathvariant":38079,"lspace":116494,"rspace":116494},[31662,775143,223],{},[31743,775145,775146,775148],{},[31658,775147,111],{},[31749,775149,640],{},[31662,775151,456],{"stretchy":702},[31658,775153,33249],{},[31662,775155,2557],{"stretchy":702},[31658,775157,56852],{},[31662,775159,202],{},[31749,775161,459],{},[31662,775163,31960],{"separator":758},[31665,775165,775166],{"encoding":31667},"W' + a_1(t)W = 0,",[34,775168,775170,775217,775284],{"className":775169,"ariaHidden":758},[31672],[34,775171,775173,775176,775208,775211,775214],{"className":775172},[31676],[34,775174],{"className":775175,"style":384364},[31680],[34,775177,775179,775182],{"className":775178},[31685],[34,775180,56852],{"className":775181,"style":37422},[31685,31686],[34,775183,775185],{"className":775184},[31792],[34,775186,775188],{"className":775187},[31796],[34,775189,775191],{"className":775190},[31801],[34,775192,775194],{"className":775193,"style":132027},[31805],[34,775195,775196,775199],{"style":37777},[34,775197],{"className":775198,"style":31814},[31813],[34,775200,775202],{"className":775201},[31818,31819,31820,31821],[34,775203,775205],{"className":775204},[31685,31821],[34,775206,116495],{"className":775207},[31685,31821],[34,775209],{"className":775210,"style":32604},[31845],[34,775212,223],{"className":775213},[32024],[34,775215],{"className":775216,"style":32604},[31845],[34,775218,775220,775223,775263,775266,775269,775272,775275,775278,775281],{"className":775219},[31676],[34,775221],{"className":775222,"style":31774},[31680],[34,775224,775226,775229],{"className":775225},[31685],[34,775227,111],{"className":775228},[31685,31686],[34,775230,775232],{"className":775231},[31792],[34,775233,775235,775255],{"className":775234},[31796,31797],[34,775236,775238,775252],{"className":775237},[31801],[34,775239,775241],{"className":775240,"style":31806},[31805],[34,775242,775243,775246],{"style":31809},[34,775244],{"className":775245,"style":31814},[31813],[34,775247,775249],{"className":775248},[31818,31819,31820,31821],[34,775250,640],{"className":775251},[31685,31821],[34,775253,31829],{"className":775254},[31828],[34,775256,775258],{"className":775257},[31801],[34,775259,775261],{"className":775260,"style":31836},[31805],[34,775262],{},[34,775264,456],{"className":775265},[31782],[34,775267,33249],{"className":775268},[31685,31686],[34,775270,2557],{"className":775271},[31690],[34,775273,56852],{"className":775274,"style":37422},[31685,31686],[34,775276],{"className":775277,"style":31846},[31845],[34,775279,202],{"className":775280},[31850],[34,775282],{"className":775283,"style":31846},[31845],[34,775285,775287,775290,775293],{"className":775286},[31676],[34,775288],{"className":775289,"style":40292},[31680],[34,775291,459],{"className":775292},[31685],[34,775294,31960],{"className":775295},[32119],"\n因而有\n",[34,775298,775300,775396],{"className":775299},[31642],[34,775301,775303],{"className":775302},[31646],[31648,775304,775305],{"xmlns":31650},[31652,775306,775307,775393],{},[31655,775308,775309,775311,775313,775315,775317,775319,775321,775323,775329,775331,775367,775373,775375,775377,775379,775381,775383,775385,775387,775389,775391],{},[31658,775310,56852],{},[31662,775312,456],{"stretchy":702},[31658,775314,33249],{},[31662,775316,2557],{"stretchy":702},[31662,775318,202],{},[31658,775320,56852],{},[31662,775322,456],{"stretchy":702},[31743,775324,775325,775327],{},[31658,775326,33249],{},[31749,775328,459],{},[31662,775330,2557],{"stretchy":702},[37624,775332,775333,775335],{},[31658,775334,33244],{"mathvariant":38079},[31655,775336,775337,775339,775351,775357,775359,775361,775363,775365],{},[31662,775338,32226],{},[38846,775340,775341,775343,775349],{},[31662,775342,546295],{},[31743,775344,775345,775347],{},[31658,775346,33249],{},[31749,775348,459],{},[31658,775350,33249],{},[31743,775352,775353,775355],{},[31658,775354,111],{},[31749,775356,640],{},[31662,775358,456],{"stretchy":702},[31658,775360,31747],{},[31662,775362,2557],{"stretchy":702},[31658,775364,46900],{"mathvariant":38079},[31658,775366,31747],{},[31743,775368,775369,775371],{},[31658,775370,33249],{},[31749,775372,459],{},[31662,775374,31960],{"separator":758},[39361,775376,130298],{},[31658,775378,33249],{},[31662,775380,112296],{},[31662,775382,456],{"stretchy":702},[31658,775384,111],{},[31662,775386,31960],{"separator":758},[31658,775388,1697],{},[31662,775390,2557],{"stretchy":702},[31658,775392,2547],{"mathvariant":38079},[31665,775394,775395],{"encoding":31667},"W(t) = W(t_0)\\mathrm{e}^{-\\int_{t_0}^t a_1(s)\\mathrm{d}s}t_0, \\ t \\in (a,b).",[34,775397,775399,775426,775729],{"className":775398,"ariaHidden":758},[31672],[34,775400,775402,775405,775408,775411,775414,775417,775420,775423],{"className":775401},[31676],[34,775403],{"className":775404,"style":31774},[31680],[34,775406,56852],{"className":775407,"style":37422},[31685,31686],[34,775409,456],{"className":775410},[31782],[34,775412,33249],{"className":775413},[31685,31686],[34,775415,2557],{"className":775416},[31690],[34,775418],{"className":775419,"style":31846},[31845],[34,775421,202],{"className":775422},[31850],[34,775424],{"className":775425,"style":31846},[31845],[34,775427,775429,775433,775436,775439,775479,775482,775668,775708,775711,775714,775717,775720,775723,775726],{"className":775428},[31676],[34,775430],{"className":775431,"style":775432},[31680],"height:1.3959em;vertical-align:-0.25em;",[34,775434,56852],{"className":775435,"style":37422},[31685,31686],[34,775437,456],{"className":775438},[31782],[34,775440,775442,775445],{"className":775441},[31685],[34,775443,33249],{"className":775444},[31685,31686],[34,775446,775448],{"className":775447},[31792],[34,775449,775451,775471],{"className":775450},[31796,31797],[34,775452,775454,775468],{"className":775453},[31801],[34,775455,775457],{"className":775456,"style":31806},[31805],[34,775458,775459,775462],{"style":31809},[34,775460],{"className":775461,"style":31814},[31813],[34,775463,775465],{"className":775464},[31818,31819,31820,31821],[34,775466,459],{"className":775467},[31685,31821],[34,775469,31829],{"className":775470},[31828],[34,775472,775474],{"className":775473},[31801],[34,775475,775477],{"className":775476,"style":31836},[31805],[34,775478],{},[34,775480,2557],{"className":775481},[31690],[34,775483,775485,775488],{"className":775484},[31685],[34,775486,33244],{"className":775487},[31685,207923],[34,775489,775491],{"className":775490},[31792],[34,775492,775494],{"className":775493},[31796],[34,775495,775497],{"className":775496},[31801],[34,775498,775501],{"className":775499,"style":775500},[31805],"height:1.1459em;",[34,775502,775504,775507],{"style":775503},"top:-3.2219em;margin-right:0.05em;",[34,775505],{"className":775506,"style":31814},[31813],[34,775508,775510],{"className":775509},[31818,31819,31820,31821],[34,775511,775513,775516,775519,775610,775613,775653,775656,775659,775662,775665],{"className":775512},[31685,31821],[34,775514,32226],{"className":775515},[31685,31821],[34,775517],{"className":775518,"style":43430},[31845,31821],[34,775520,775522,775525],{"className":775521},[32767,31821],[34,775523,546295],{"className":775524,"style":546331},[32767,131249,156560,31821],[34,775526,775528],{"className":775527},[31792],[34,775529,775531,775602],{"className":775530},[31796,31797],[34,775532,775534,775599],{"className":775533},[31801],[34,775535,775538,775587],{"className":775536,"style":775537},[31805],"height:0.8915em;",[34,775539,775541,775544],{"style":775540},"top:-2.1225em;margin-left:-0.1945em;margin-right:0.0714em;",[34,775542],{"className":775543,"style":43406},[31813],[34,775545,775547],{"className":775546},[31818,43410,43411,31821],[34,775548,775550],{"className":775549},[31685,31821],[34,775551,775553,775556],{"className":775552},[31685,31821],[34,775554,33249],{"className":775555},[31685,31686,31821],[34,775557,775559],{"className":775558},[31792],[34,775560,775562,775579],{"className":775561},[31796,31797],[34,775563,775565,775576],{"className":775564},[31801],[34,775566,775568],{"className":775567,"style":332295},[31805],[34,775569,775570,775573],{"style":499949},[34,775571],{"className":775572,"style":499953},[31813],[34,775574,459],{"className":775575},[31685,31821],[34,775577,31829],{"className":775578},[31828],[34,775580,775582],{"className":775581},[31801],[34,775583,775585],{"className":775584,"style":499966},[31805],[34,775586],{},[34,775588,775590,775593],{"style":775589},"top:-2.9521em;margin-right:0.0714em;",[34,775591],{"className":775592,"style":43406},[31813],[34,775594,775596],{"className":775595},[31818,43410,43411,31821],[34,775597,33249],{"className":775598},[31685,31686,31821],[34,775600,31829],{"className":775601},[31828],[34,775603,775605],{"className":775604},[31801],[34,775606,775608],{"className":775607,"style":594988},[31805],[34,775609],{},[34,775611],{"className":775612,"style":43430},[31845,31821],[34,775614,775616,775619],{"className":775615},[31685,31821],[34,775617,111],{"className":775618},[31685,31686,31821],[34,775620,775622],{"className":775621},[31792],[34,775623,775625,775645],{"className":775624},[31796,31797],[34,775626,775628,775642],{"className":775627},[31801],[34,775629,775631],{"className":775630,"style":318658},[31805],[34,775632,775633,775636],{"style":318661},[34,775634],{"className":775635,"style":43406},[31813],[34,775637,775639],{"className":775638},[31818,43410,43411,31821],[34,775640,640],{"className":775641},[31685,31821],[34,775643,31829],{"className":775644},[31828],[34,775646,775648],{"className":775647},[31801],[34,775649,77565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",[34,775760,775762,775775],{"className":775761},[31642],[34,775763,775765],{"className":775764},[31646],[31648,775766,775767],{"xmlns":31650},[31652,775768,775769,775773],{},[31655,775770,775771],{},[31658,775772,31660],{},[31665,775774,31660],{"encoding":31667},[34,775776,775778],{"className":775777,"ariaHidden":758},[31672],[34,775779,775781,775784],{"className":775780},[31676],[34,775782],{"className":775783,"style":32527},[31680],[34,775785,31660],{"className":775786},[31685,31686]," 阶非齐次线性微分方程 (4.1) 存在且最多存在 ",[34,775789,775791,775808],{"className":775790},[31642],[34,775792,775794],{"className":775793},[31646],[31648,775795,775796],{"xmlns":31650},[31652,775797,775798,775806],{},[31655,775799,775800,775802,775804],{},[31658,775801,31660],{},[31662,775803,223],{},[31749,775805,640],{},[31665,775807,87807],{"encoding":31667},[34,775809,775811,775829],{"className":775810,"ariaHidden":758},[31672],[34,775812,775814,775817,775820,775823,775826],{"className":775813},[31676],[34,775815],{"className":775816,"style":32597},[31680],[34,775818,31660],{"className":775819},[31685,31686],[34,775821],{"className":775822,"style":32604},[31845],[34,775824,223],{"className":775825},[32024],[34,775827],{"className":775828,"style":32604},[31845],[34,775830,775832,775835],{"className":775831},[31676],[34,775833],{"className":775834,"style":32617},[31680],[34,775836,640],{"className":775837},[31685]," 个线性无关解.",{"title":29,"searchDepth":43,"depth":43,"links":775840},[775841,775842],{"id":768268,"depth":49,"text":768269},{"id":770365,"depth":49,"text":770366},{"order":37},"\u002Fsource\u002Fmath\u002Fode\u002Fprob",{"title":162043,"description":29},"source\u002Fmath\u002Fode\u002Fprob","6SMX9Ca3dm_rT482zYUWRLpORk0vQbEAy57J-Hx0wrE",{"id":775849,"title":775850,"body":775851,"description":29,"extension":85,"meta":782680,"navigation":52,"path":782681,"seo":782682,"stem":782683,"__hash__":782684},"notes\u002Fsource\u002Fmath\u002Fode\u002Freview.md","复习总结",{"type":8,"value":775852,"toc":782670},[775853,775857,775861,776054,776057,776353,776357,776361,776365,777104,777109,779508,779512,779516,780270,780274,781392,781396,781400,781701,781706,781856,781861,782238,782272,782664,782667],[19,775854,775856],{"id":775855},"一阶线性ode求解公式","一阶线性ODE求解公式",[15,775858,775859],{},[2627,775860,128684],{},[15,775862,775863],{},[34,775864,775866,775914],{"className":775865},[31642],[34,775867,775869],{"className":775868},[31646],[31648,775870,775871],{"xmlns":31650},[31652,775872,775873,775911],{},[31655,775874,775875,775889,775891,775893,775895,775897,775899,775901,775903,775905,775907,775909],{},[41453,775876,775877,775883],{},[31655,775878,775879,775881],{},[31658,775880,46900],{},[31658,775882,67764],{},[31655,775884,775885,775887],{},[31658,775886,46900],{},[31658,775888,32936],{},[31662,775890,223],{},[31658,775892,105852],{},[31662,775894,456],{"stretchy":702},[31658,775896,32936],{},[31662,775898,2557],{"stretchy":702},[31658,775900,67764],{},[31662,775902,202],{},[31658,775904,116863],{},[31662,775906,456],{"stretchy":702},[31658,775908,32936],{},[31662,775910,2557],{"stretchy":702},[31665,775912,775913],{"encoding":31667},"\\frac{dy}{dx} + P(x)y = Q(x)",[34,775915,775917,776006,776036],{"className":775916,"ariaHidden":758},[31672],[34,775918,775920,775923,775997,776000,776003],{"className":775919},[31676],[34,775921],{"className":775922,"style":608762},[31680],[34,775924,775926,775929,775994],{"className":775925},[31685],[34,775927],{"className":775928},[31782,41488],[34,775930,775932],{"className":775931},[41453],[34,775933,775935,775986],{"className":775934},[31796,31797],[34,775936,775938,775983],{"className":775937},[31801],[34,775939,775941,775958,775966],{"className":775940,"style":608935},[31805],[34,775942,775943,775946],{"style":41504},[34,775944],{"className":775945,"style":41508},[31813],[34,775947,775949],{"className":775948},[31818,31819,31820,31821],[34,775950,775952,775955],{"className":775951},[31685,31821],[34,775953,46900],{"className":775954},[31685,31686,31821],[34,775956,32936],{"className":775957},[31685,31686,31821],[34,775959,775960,775963],{"style":41520},[34,775961],{"className":775962,"style":41508},[31813],[34,775964],{"className":775965,"style":41528},[41527],[34,775967,775968,775971],{"style":548368},[34,775969],{"className":775970,"style":41508},[31813],[34,775972,775974],{"className":775973},[31818,31819,31820,31821],[34,775975,775977,775980],{"className":775976},[31685,31821],[34,775978,46900],{"className":775979},[31685,31686,31821],[34,775981,67764],{"className":775982,"style":32062},[31685,31686,31821],[34,775984,31829],{"className":775985},[31828],[34,775987,775989],{"className":775988},[31801],[34,775990,775992],{"className":775991,"style":41619},[31805],[34,775993],{},[34,775995],{"className":775996},[31690,41488],[34,775998],{"className":775999,"style":32604},[31845],[34,776001,223],{"className":776002},[32024],[34,776004],{"className":776005,"style":32604},[31845],[34,776007,776009,776012,776015,776018,776021,776024,776027,776030,776033],{"className":776008},[31676],[34,776010],{"className":776011,"style":31774},[31680],[34,776013,105852],{"className":776014,"style":37422},[31685,31686],[34,776016,456],{"className":776017},[31782],[34,776019,32936],{"className":776020},[31685,31686],[34,776022,2557],{"className":776023},[31690],[34,776025,67764],{"className":776026,"style":32062},[31685,31686],[34,776028],{"className":776029,"style":31846},[31845],[34,776031,202],{"className":776032},[31850],[34,776034],{"className":776035,"style":31846},[31845],[34,776037,776039,776042,776045,776048,776051],{"className":776038},[31676],[34,776040],{"className":776041,"style":31774},[31680],[34,776043,116863],{"className":776044},[31685,31686],[34,776046,456],{"className":776047},[31782],[34,776049,32936],{"className":776050},[31685,31686],[34,776052,2557],{"className":776053},[31690],[15,776055,776056],{},"求解公式：",[15,776058,776059],{},[34,776060,776062,776150],{"className":776061},[31642],[34,776063,776065],{"className":776064},[31646],[31648,776066,776067],{"xmlns":31650},[31652,776068,776069,776147],{},[31655,776070,776071,776073,776075,776099],{},[31658,776072,67764],{},[31662,776074,202],{},[37624,776076,776077,776079],{},[31658,776078,33244],{},[31655,776080,776081,776083,776085,776087,776089,776091,776093,776095,776097],{},[31662,776082,32226],{},[31662,776084,546295],{},[31658,776086,105852],{},[31662,776088,456],{"stretchy":702},[31658,776090,32936],{},[31662,776092,2557],{"stretchy":702},[39361,776094,39363],{},[31658,776096,46900],{},[31658,776098,32936],{},[31655,776100,776101,776103,776105,776107,776109,776111,776113,776135,776137,776139,776141,776143,776145],{},[31662,776102,456],{"fence":758},[31662,776104,546295],{},[31658,776106,116863],{},[31662,776108,456],{"stretchy":702},[31658,776110,32936],{},[31662,776112,2557],{"stretchy":702},[37624,776114,776115,776117],{},[31658,776116,33244],{},[31655,776118,776119,776121,776123,776125,776127,776129,776131,776133],{},[31662,776120,546295],{},[31658,776122,105852],{},[31662,776124,456],{"stretchy":702},[31658,776126,32936],{},[31662,776128,2557],{"stretchy":702},[39361,776130,39363],{},[31658,776132,46900],{},[31658,776134,32936],{},[39361,776136,39363],{},[31658,776138,46900],{},[31658,776140,32936],{},[31662,776142,223],{},[31658,776144,47384],{},[31662,776146,2557],{"fence":758},[31665,776148,776149],{"encoding":31667},"y = e^{-\\int P(x) \\, dx} \\left( \\int Q(x) e^{\\int P(x) \\, dx} \\, dx + C \\right)",[34,776151,776153,776171],{"className":776152,"ariaHidden":758},[31672],[34,776154,776156,776159,776162,776165,776168],{"className":776155},[31676],[34,776157],{"className":776158,"style":38774},[31680],[34,776160,67764],{"className":776161,"style":32062},[31685,31686],[34,776163],{"className":776164,"style":31846},[31845],[34,776166,202],{"className":776167},[31850],[34,776169],{"className":776170,"style":31846},[31845],[34,776172,776174,776177,776240,776243],{"className":776173},[31676],[34,776175],{"className":776176,"style":141282},[31680],[34,776178,776180,776183],{"className":776179},[31685],[34,776181,33244],{"className":776182},[31685,31686],[34,776184,776186],{"className":776185},[31792],[34,776187,776189],{"className":776188},[31796],[34,776190,776192],{"className":776191},[31801],[34,776193,776196],{"className":776194,"style":776195},[31805],"height:0.9265em;",[34,776197,776198,776201],{"style":37777},[34,776199],{"className":776200,"style":31814},[31813],[34,776202,776204],{"className":776203},[31818,31819,31820,31821],[34,776205,776207,776210,776213,776216,776219,776222,776225,776228,776231,776234,776237],{"className":776206},[31685,31821],[34,776208,32226],{"className":776209},[31685,31821],[34,776211],{"className":776212,"style":43430},[31845,31821],[34,776214,546295],{"className":776215,"style":546331},[32767,131249,156560,31821],[34,776217],{"className":776218,"style":43430},[31845,31821],[34,776220,105852],{"className":776221,"style":37422},[31685,31686,31821],[34,776223,456],{"className":776224},[31782,31821],[34,776226,32936],{"className":776227},[31685,31686,31821],[34,776229,2557],{"className":776230},[31690,31821],[34,776232],{"className":776233,"style":43430},[31845,31821],[34,776235,46900],{"className":776236},[31685,31686,31821],[34,776238,32936],{"className":776239},[31685,31686,31821],[34,776241],{"className":776242,"style":32123},[31845],[34,776244,776246,776252,776255,776258,776261,776264,776267,776270,776326,776329,776332,776335,776338,776341,776344,776347],{"className":776245},[34006],[34,776247,776249],{"className":776248,"style":75811},[31782,75810],[34,776250,456],{"className":776251},[71281,141292],[34,776253,546295],{"className":776254,"style":546331},[32767,131249,156560],[34,776256],{"className":776257,"style":32123},[31845],[34,776259,116863],{"className":776260},[31685,31686],[34,776262,456],{"className":776263},[31782],[34,776265,32936],{"className":776266},[31685,31686],[34,776268,2557],{"className":776269},[31690],[34,776271,776273,776276],{"className":776272},[31685],[34,776274,33244],{"className":776275},[31685,31686],[34,776277,776279],{"className":776278},[31792],[34,776280,776282],{"className":776281},[31796],[34,776283,776285],{"className":776284},[31801],[34,776286,776288],{"className":776287,"style":776195},[31805],[34,776289,776290,776293],{"style":37777},[34,776291],{"className":776292,"style":31814},[31813],[34,776294,776296],{"className":776295},[31818,31819,31820,31821],[34,776297,776299,776302,776305,776308,776311,776314,776317,776320,776323],{"className":776298},[31685,31821],[34,776300,546295],{"className":776301,"style":546331},[32767,131249,156560,31821],[34,776303],{"className":776304,"style":43430},[31845,31821],[34,776306,105852],{"className":776307,"style":37422},[31685,31686,31821],[34,776309,456],{"className":776310},[31782,31821],[34,776312,32936],{"className":776313},[31685,31686,31821],[34,776315,2557],{"className":776316},[31690,31821],[34,776318],{"className":776319,"style":43430},[31845,31821],[34,776321,46900],{"className":776322},[31685,31686,31821],[34,776324,32936],{"className":776325},[31685,31686,31821],[34,776327],{"className":776328,"style":32123},[31845],[34,776330,46900],{"className":776331},[31685,31686],[34,776333,32936],{"className":776334},[31685,31686],[34,776336],{"className":776337,"style":32604},[31845],[34,776339,223],{"className":776340},[32024],[34,776342],{"className":776343,"style":32604},[31845],[34,776345,47384],{"className":776346,"style":38706},[31685,31686],[34,776348,776350],{"className":776349,"style":75811},[31690,75810],[34,776351,2557],{"className":776352},[71281,141292],[19,776354,776356],{"id":776355},"高阶线性ode求解方法","高阶线性ODE求解方法",[253,776358,776360],{"id":776359},"齐次线性ode","齐次线性ODE",[15,776362,776363],{},[2627,776364,128684],{},[15,776366,776367],{},[34,776368,776370,776498],{"className":776369},[31642],[34,776371,776373],{"className":776372},[31646],[31648,776374,776375],{"xmlns":31650},[31652,776376,776377,776495],{},[31655,776378,776379,776385,776407,776409,776421,776455,776457,776459,776461,776467,776481,776483,776489,776491,776493],{},[31743,776380,776381,776383],{},[31658,776382,111],{},[31658,776384,31660],{},[41453,776386,776387,776397],{},[31655,776388,776389,776395],{},[37624,776390,776391,776393],{},[31658,776392,46900],{},[31658,776394,31660],{},[31658,776396,67764],{},[31655,776398,776399,776401],{},[31658,776400,46900],{},[37624,776402,776403,776405],{},[31658,776404,32936],{},[31658,776406,31660],{},[31662,776408,223],{},[31743,776410,776411,776413],{},[31658,776412,111],{},[31655,776414,776415,776417,776419],{},[31658,776416,31660],{},[31662,776418,32226],{},[31749,776420,640],{},[41453,776422,776423,776439],{},[31655,776424,776425,776437],{},[37624,776426,776427,776429],{},[31658,776428,46900],{},[31655,776430,776431,776433,776435],{},[31658,776432,31660],{},[31662,776434,32226],{},[31749,776436,640],{},[31658,776438,67764],{},[31655,776440,776441,776443],{},[31658,776442,46900],{},[37624,776444,776445,776447],{},[31658,776446,32936],{},[31655,776448,776449,776451,776453],{},[31658,776450,31660],{},[31662,776452,32226],{},[31749,776454,640],{},[31662,776456,223],{},[31662,776458,38874],{},[31662,776460,223],{},[31743,776462,776463,776465],{},[31658,776464,111],{},[31749,776466,640],{},[41453,776468,776469,776475],{},[31655,776470,776471,776473],{},[31658,776472,46900],{},[31658,776474,67764],{},[31655,776476,776477,776479],{},[31658,776478,46900],{},[31658,776480,32936],{},[31662,776482,223],{},[31743,776484,776485,776487],{},[31658,776486,111],{},[31749,776488,459],{},[31658,776490,67764],{},[31662,776492,202],{},[31749,776494,459],{},[31665,776496,776497],{"encoding":31667},"a_n \\frac{d^n y}{dx^n} + a_{n-1} \\frac{d^{n-1} y}{dx^{n-1}} + \\dots + a_1 \\frac{dy}{dx} + a_0 y = 0",[34,776499,776501,776682,776890,776908,777037,777095],{"className":776500,"ariaHidden":758},[31672],[34,776502,776504,776507,776547,776673,776676,776679],{"className":776503},[31676],[34,776505],{"className":776506,"style":756208},[31680],[34,776508,776510,776513],{"className":776509},[31685],[34,776511,111],{"className":776512},[31685,31686],[34,776514,776516],{"className":776515},[31792],[34,776517,776519,776539],{"className":776518},[31796,31797],[34,776520,776522,776536],{"className":776521},[31801],[34,776523,776525],{"className":776524,"style":32362},[31805],[34,776526,776527,776530],{"style":31809},[34,776528],{"className":776529,"style":31814},[31813],[34,776531,776533],{"className":776532},[31818,31819,31820,31821],[34,776534,31660],{"className":776535},[31685,31686,31821],[34,776537,31829],{"className":776538},[31828],[34,776540,776542],{"className":776541},[31801],[34,776543,776545],{"className":776544,"style":31836},[31805],[34,776546],{},[34,776548,776550,776553,776670],{"className":776549},[31685],[34,776551],{"className":776552},[31782,41488],[34,776554,776556],{"className":776555},[41453],[34,776557,776559,776662],{"className":776558},[31796,31797],[34,776560,776562,776659],{"className":776561},[31801],[34,776563,776565,776608,776616],{"className":776564,"style":756256},[31805],[34,776566,776567,776570],{"style":41504},[34,776568],{"className":776569,"style":41508},[31813],[34,776571,776573],{"className":776572},[31818,31819,31820,31821],[34,776574,776576,776579],{"className":776575},[31685,31821],[34,776577,46900],{"className":776578},[31685,31686,31821],[34,776580,776582,776585],{"className":776581},[31685,31821],[34,776583,32936],{"className":776584},[31685,31686,31821],[34,776586,776588],{"className":776587},[31792],[34,776589,776591],{"className":776590},[31796],[34,776592,776594],{"className":776593},[31801],[34,776595,776597],{"className":776596,"style":566313},[31805],[34,776598,776599,776602],{"style":116668},[34,776600],{"className":776601,"style":43406},[31813],[34,776603,776605],{"className":776604},[31818,43410,43411,31821],[34,776606,31660],{"className":776607},[31685,31686,31821],[34,776609,776610,776613],{"style":41520},[34,776611],{"className":776612,"style":41508},[31813],[34,776614],{"className":776615,"style":41528},[41527],[34,776617,776618,776621],{"style":548368},[34,776619],{"className":776620,"style":41508},[31813],[34,776622,776624],{"className":776623},[31818,31819,31820,31821],[34,776625,776627,776656],{"className":776626},[31685,31821],[34,776628,776630,776633],{"className":776629},[31685,31821],[34,776631,46900],{"className":776632},[31685,31686,31821],[34,776634,776636],{"className":776635},[31792],[34,776637,776639],{"className":776638},[31796],[34,776640,776642],{"className":776641},[31801],[34,776643,776645],{"className":776644,"style":611529},[31805],[34,776646,776647,776650],{"style":552391},[34,776648],{"className":776649,"style":43406},[31813],[34,776651,776653],{"className":776652},[31818,43410,43411,31821],[34,776654,31660],{"className":776655},[31685,31686,31821],[34,776657,67764],{"className":776658,"style":32062},[31685,31686,31821],[34,776660,31829],{"className":776661},[31828],[34,776663,776665],{"className":776664},[31801],[34,776666,776668],{"className":776667,"style":41619},[31805],[34,776669],{},[34,776671],{"className":776672},[31690,41488],[34,776674],{"className":776675,"style":32604},[31845],[34,776677,223],{"className":776678},[32024],[34,776680],{"className":776681,"style":32604},[31845],[34,776683,776685,776688,776737,776881,776884,776887],{"className":776684},[31676],[34,776686],{"className":776687,"style":681304},[31680],[34,776689,776691,776694],{"className":776690},[31685],[34,776692,111],{"className":776693},[31685,31686],[34,776695,776697],{"className":776696},[31792],[34,776698,776700,776729],{"className":776699},[31796,31797],[34,776701,776703,776726],{"className":776702},[31801],[34,776704,776706],{"className":776705,"style":31806},[31805],[34,776707,776708,776711],{"style":31809},[34,776709],{"className":776710,"style":31814},[31813],[34,776712,776714],{"className":776713},[31818,31819,31820,31821],[34,776715,776717,776720,776723],{"className":776716},[31685,31821],[34,776718,31660],{"className":776719},[31685,31686,31821],[34,776721,32226],{"className":776722},[32024,31821],[34,776724,640],{"className":776725},[31685,31821],[34,776727,31829],{"className":776728},[31828],[34,776730,776732],{"className":776731},[31801],[34,776733,776735],{"className":776734,"style":32037},[31805],[34,776736],{},[34,776738,776740,776743,776878],{"className":776739},[31685],[34,776741],{"className":776742},[31782,41488],[34,776744,776746],{"className":776745},[41453],[34,776747,776749,776870],{"className":776748},[31796,31797],[34,776750,776752,776867],{"className":776751},[31801],[34,776753,776755,776807,776815],{"className":776754,"style":601858},[31805],[34,776756,776757,776760],{"style":41504},[34,776758],{"className":776759,"style":41508},[31813],[34,776761,776763],{"className":776762},[31818,31819,31820,31821],[34,776764,776766,776769],{"className":776765},[31685,31821],[34,776767,46900],{"className":776768},[31685,31686,31821],[34,776770,776772,776775],{"className":776771},[31685,31821],[34,776773,32936],{"className":776774},[31685,31686,31821],[34,776776,776778],{"className":776777},[31792],[34,776779,776781],{"className":776780},[31796],[34,776782,776784],{"className":776783},[31801],[34,776785,776787],{"className":776786,"style":548314},[31805],[34,776788,776789,776792],{"style":116668},[34,776790],{"className":776791,"style":43406},[31813],[34,776793,776795],{"className":776794},[31818,43410,43411,31821],[34,776796,776798,776801,776804],{"className":776797},[31685,31821],[34,776799,31660],{"className":776800},[31685,31686,31821],[34,776802,32226],{"className":776803},[32024,31821],[34,776805,640],{"className":776806},[31685,31821],[34,776808,776809,776812],{"style":41520},[34,776810],{"className":776811,"style":41508},[31813],[34,776813],{"className":776814,"style":41528},[41527],[34,776816,776817,776820],{"style":548368},[34,776818],{"className":776819,"style":41508},[31813],[34,776821,776823],{"className":776822},[31818,31819,31820,31821],[34,776824,776826,776864],{"className":776825},[31685,31821],[34,776827,776829,776832],{"className":776828},[31685,31821],[34,776830,46900],{"className":776831},[31685,31686,31821],[34,776833,776835],{"className":776834},[31792],[34,776836,776838],{"className":776837},[31796],[34,776839,776841],{"className":776840},[31801],[34,776842,776844],{"className":776843,"style":144123},[31805],[34,776845,776846,776849],{"style":552391},[34,776847],{"className":776848,"style":43406},[31813],[34,776850,776852],{"className":776851},[31818,43410,43411,31821],[34,776853,776855,776858,776861],{"className":776854},[31685,31821],[34,776856,31660],{"className":776857},[31685,31686,31821],[34,776859,32226],{"className":776860},[32024,31821],[34,776862,640],{"className":776863},[31685,31821],[34,776865,67764],{"className":776866,"style":32062},[31685,31686,31821],[34,776868,31829],{"className":776869},[31828],[34,776871,776873],{"className":776872},[31801],[34,776874,776876],{"className":776875,"style":41619},[31805],[34,776877],{},[34,776879],{"className":776880},[31690,41488],[34,776882],{"className":776883,"style":32604},[31845],[34,776885,223],{"className":776886},[32024],[34,776888],{"className":776889,"style":32604},[31845],[34,776891,776893,776896,776899,776902,776905],{"className":776892},[31676],[34,776894],{"className":776895,"style":32597},[31680],[34,776897,38874],{"className":776898},[34006],[34,776900],{"className":776901,"style":32604},[31845],[34,776903,223],{"className":776904},[32024],[34,776906],{"className":776907,"style":32604},[31845],[34,776909,776911,776914,776954,777028,777031,777034],{"className":776910},[31676],[34,776912],{"className":776913,"style":608762},[31680],[34,776915,776917,776920],{"className":776916},[31685],[34,776918,111],{"className":776919},[31685,31686],[34,776921,776923],{"className":776922},[31792],[34,776924,776926,776946],{"className":776925},[31796,31797],[34,776927,776929,776943],{"className":776928},[31801],[34,776930,776932],{"className":776931,"style":31806},[31805],[34,776933,776934,776937],{"style":31809},[34,776935],{"className":776936,"style":31814},[31813],[34,776938,776940],{"className":776939},[31818,31819,31820,31821],[34,776941,640],{"className":776942},[31685,31821],[34,776944,31829],{"className":776945},[31828],[34,776947,776949],{"className":776948},[31801],[34,776950,776952],{"className":776951,"style":31836},[31805],[34,776953],{},[34,776955,776957,776960,777025],{"className":776956},[31685],[34,776958],{"className":776959},[31782,41488],[34,776961,776963],{"className":776962},[41453],[34,776964,776966,777017],{"className":776965},[31796,31797],[34,776967,776969,777014],{"className":776968},[31801],[34,776970,776972,776989,776997],{"className":776971,"style":608935},[31805],[34,776973,776974,776977],{"style":41504},[34,776975],{"className":776976,"style":41508},[31813],[34,776978,776980],{"className":776979},[31818,31819,31820,31821],[34,776981,776983,776986],{"className":776982},[31685,31821],[34,776984,46900],{"className":776985},[31685,31686,31821],[34,776987,32936],{"className":776988},[31685,31686,31821],[34,776990,776991,776994],{"style":41520},[34,776992],{"className":776993,"style":41508},[31813],[34,776995],{"className":776996,"style":41528},[41527],[34,776998,776999,777002],{"style":548368},[34,777000],{"className":777001,"style":41508},[31813],[34,777003,777005],{"className":777004},[31818,31819,31820,31821],[34,777006,777008,777011],{"className":777007},[31685,31821],[34,777009,46900],{"className":777010},[31685,31686,31821],[34,777012,67764],{"className":777013,"style":32062},[31685,31686,31821],[34,777015,31829],{"className":777016},[31828],[34,777018,777020],{"className":777019},[31801],[34,777021,777023],{"className":777022,"style":41619},[31805],[34,777024],{},[34,777026],{"className":777027},[31690,41488],[34,777029],{"className":777030,"style":32604},[31845],[34,777032,223],{"className":777033},[32024],[34,777035],{"className":777036,"style":32604},[31845],[34,777038,777040,777043,777083,777086,777089,777092],{"className":777039},[31676],[34,777041],{"className":777042,"style":38774},[31680],[34,777044,777046,777049],{"className":777045},[31685],[34,777047,111],{"className":777048},[31685,31686],[34,777050,777052],{"className":777051},[31792],[34,777053,777055,777075],{"className":777054},[31796,31797],[34,777056,777058,777072],{"className":777057},[31801],[34,777059,777061],{"className":777060,"style":31806},[31805],[34,777062,777063,777066],{"style":31809},[34,777064],{"className":777065,"style":31814},[31813],[34,777067,777069],{"className":777068},[31818,31819,31820,31821],[34,777070,459],{"className":777071},[31685,31821],[34,777073,31829],{"className":777074},[31828],[34,777076,777078],{"className":777077},[31801],[34,777079,777081],{"className":777080,"style":31836},[31805],[34,777082],{},[34,777084,67764],{"className":777085,"style":32062},[31685,31686],[34,777087],{"className":777088,"style":31846},[31845],[34,777090,202],{"className":777091},[31850],[34,777093],{"className":777094,"style":31846},[31845],[34,777096,777098,777101],{"className":777097},[31676],[34,777099],{"className":777100,"style":32617},[31680],[34,777102,459],{"className":777103},[31685],[15,777105,777106],{},[2627,777107,777108],{},"求解步骤：",[2230,777110,777111,777524,777721,779505],{},[108,777112,742899,777113,777115],{},[58531,777114],{},[34,777116,777118,777194],{"className":777117},[31642],[34,777119,777121],{"className":777120},[31646],[31648,777122,777123],{"xmlns":31650},[31652,777124,777125,777191],{},[31655,777126,777127,777133,777139,777141,777153,777165,777167,777169,777171,777177,777179,777181,777187,777189],{},[31743,777128,777129,777131],{},[31658,777130,111],{},[31658,777132,31660],{},[37624,777134,777135,777137],{},[31658,777136,220792],{},[31658,777138,31660],{},[31662,777140,223],{},[31743,777142,777143,777145],{},[31658,777144,111],{},[31655,777146,777147,777149,777151],{},[31658,777148,31660],{},[31662,777150,32226],{},[31749,777152,640],{},[37624,777154,777155,777157],{},[31658,777156,220792],{},[31655,777158,777159,777161,777163],{},[31658,777160,31660],{},[31662,777162,32226],{},[31749,777164,640],{},[31662,777166,223],{},[31662,777168,38874],{},[31662,777170,223],{},[31743,777172,777173,777175],{},[31658,777174,111],{},[31749,777176,640],{},[31658,777178,220792],{},[31662,777180,223],{},[31743,777182,777183,777185],{},[31658,777184,111],{},[31749,777186,459],{},[31662,777188,202],{},[31749,777190,459],{},[31665,777192,777193],{"encoding":31667},"a_n \\lambda^n + a_{n-1} \\lambda^{n-1} + \\dots + a_1 \\lambda + a_0 = 0",[34,777195,777197,777281,777384,777402,777460,777515],{"className":777196,"ariaHidden":758},[31672],[34,777198,777200,777203,777243,777272,777275,777278],{"className":777199},[31676],[34,777201],{"className":777202,"style":33814},[31680],[34,777204,777206,777209],{"className":777205},[31685],[34,777207,111],{"className":777208},[31685,31686],[34,777210,777212],{"className":777211},[31792],[34,777213,777215,777235],{"className":777214},[31796,31797],[34,777216,777218,777232],{"className":777217},[31801],[34,777219,777221],{"className":777220,"style":32362},[31805],[34,777222,777223,777226],{"style":31809},[34,777224],{"className":777225,"style":31814},[31813],[34,777227,777229],{"className":777228},[31818,31819,31820,31821],[34,777230,31660],{"className":777231},[31685,31686,31821],[34,777233,31829],{"className":777234},[31828],[34,777236,777238],{"className":777237},[31801],[34,777239,777241],{"className":777240,"style":31836},[31805],[34,777242],{},[34,777244,777246,777249],{"className":777245},[31685],[34,777247,220792],{"className":777248},[31685,31686],[34,777250,777252],{"className":777251},[31792],[34,777253,777255],{"className":777254},[31796],[34,777256,777258],{"className":777257},[31801],[34,777259,777261],{"className":777260,"style":37959},[31805],[34,777262,777263,777266],{"style":37777},[34,777264],{"className":777265,"style":31814},[31813],[34,777267,777269],{"className":777268},[31818,31819,31820,31821],[34,777270,31660],{"className":777271},[31685,31686,31821],[34,777273],{"className":777274,"style":32604},[31845],[34,777276,223],{"className":777277},[32024],[34,777279],{"className":777280,"style":32604},[31845],[34,777282,777284,777288,777337,777375,777378,777381],{"className":777283},[31676],[34,777285],{"className":777286,"style":777287},[31680],"height:1.0224em;vertical-align:-0.2083em;",[34,777289,777291,777294],{"className":777290},[31685],[34,777292,111],{"className":777293},[31685,31686],[34,777295,777297],{"className":777296},[31792],[34,777298,777300,777329],{"className":777299},[31796,31797],[34,777301,777303,777326],{"className":777302},[31801],[34,777304,777306],{"className":777305,"style":31806},[31805],[34,777307,777308,777311],{"style":31809},[34,777309],{"className":777310,"style":31814},[31813],[34,777312,777314],{"className":777313},[31818,31819,31820,31821],[34,777315,777317,777320,777323],{"className":777316},[31685,31821],[34,777318,31660],{"className":777319},[31685,31686,31821],[34,777321,32226],{"className":777322},[32024,31821],[34,777324,640],{"className":777325},[31685,31821],[34,777327,31829],{"className":777328},[31828],[34,777330,777332],{"className":777331},[31801],[34,777333,777335],{"className":777334,"style":32037},[31805],[34,777336],{},[34,777338,777340,777343],{"className":777339},[31685],[34,777341,220792],{"className":777342},[31685,31686],[34,777344,777346],{"className":777345},[31792],[34,777347,777349],{"className":777348},[31796],[34,777350,777352],{"className":777351},[31801],[34,777353,777355],{"className":777354,"style":37774},[31805],[34,777356,777357,777360],{"style":37777},[34,777358],{"className":777359,"style":31814},[31813],[34,777361,777363],{"className":777362},[31818,31819,31820,31821],[34,777364,777366,777369,777372],{"className":777365},[31685,31821],[34,777367,31660],{"className":777368},[31685,31686,31821],[34,777370,32226],{"className":777371},[32024,31821],[34,777373,640],{"className":777374},[31685,31821],[34,777376],{"className":777377,"style":32604},[31845],[34,777379,223],{"className":777380},[32024],[34,777382],{"className":777383,"style":32604},[31845],[34,777385,777387,777390,777393,777396,777399],{"className":777386},[31676],[34,777388],{"className":777389,"style":32597},[31680],[34,777391,38874],{"className":777392},[34006],[34,777394],{"className":777395,"style":32604},[31845],[34,777397,223],{"className":777398},[32024],[34,777400],{"className":777401,"style":32604},[31845],[34,777403,777405,777408,777448,777451,777454,777457],{"className":777404},[31676],[34,777406],{"className":777407,"style":33814},[31680],[34,777409,777411,777414],{"className":777410},[31685],[34,777412,111],{"className":777413},[31685,31686],[34,777415,777417],{"className":777416},[31792],[34,777418,777420,777440],{"className":777419},[31796,31797],[34,777421,777423,777437],{"className":777422},[31801],[34,777424,777426],{"className":777425,"style":31806},[31805],[34,777427,777428,777431],{"style":31809},[34,777429],{"className":777430,"style":31814},[31813],[34,777432,777434],{"className":777433},[31818,31819,31820,31821],[34,777435,640],{"className":777436},[31685,31821],[34,777438,31829],{"className":777439},[31828],[34,777441,777443],{"className":777442},[31801],[34,777444,777446],{"className":777445,"style":31836},[31805],[34,777447],{},[34,777449,220792],{"className":777450},[31685,31686],[34,777452],{"className":777453,"style":32604},[31845],[34,777455,223],{"className":777456},[32024],[34,777458],{"className":777459,"style":32604},[31845],[34,777461,777463,777466,777506,777509,777512],{"className":777462},[31676],[34,777464],{"className":777465,"style":31860},[31680],[34,777467,777469,777472],{"className":777468},[31685],[34,777470,111],{"className":777471},[31685,31686],[34,777473,777475],{"className":777474},[31792],[34,777476,777478,777498],{"className":777477},[31796,31797],[34,777479,777481,777495],{"className":777480},[31801],[34,777482,777484],{"className":777483,"style":31806},[31805],[34,777485,777486,777489],{"style":31809},[34,777487],{"className":777488,"style":31814},[31813],[34,777490,777492],{"className":777491},[31818,31819,31820,31821],[34,777493,459],{"className":777494},[31685,31821],[34,777496,31829],{"className":777497},[31828],[34,777499,777501],{"className":777500},[31801],[34,777502,777504],{"className":777503,"style":31836},[31805],[34,777505],{},[34,777507],{"className":777508,"style":31846},[31845],[34,777510,202],{"className":777511},[31850],[34,777513],{"className":777514,"style":31846},[31845],[34,777516,777518,777521],{"className":777517},[31676],[34,777519],{"className":777520,"style":32617},[31680],[34,777522,459],{"className":777523},[31685],[108,777525,777526,777527,347],{},"求解得到特征根",[34,777528,777530,777568],{"className":777529},[31642],[34,777531,777533],{"className":777532},[31646],[31648,777534,777535],{"xmlns":31650},[31652,777536,777537,777565],{},[31655,777538,777539,777545,777547,777553,777555,777557,777559],{},[31743,777540,777541,777543],{},[31658,777542,220792],{},[31749,777544,640],{},[31662,777546,31960],{"separator":758},[31743,777548,777549,777551],{},[31658,777550,220792],{},[31749,777552,768],{},[31662,777554,31960],{"separator":758},[31662,777556,33934],{},[31662,777558,31960],{"separator":758},[31743,777560,777561,777563],{},[31658,777562,220792],{},[31658,777564,33759],{},[31665,777566,777567],{"encoding":31667},"\\lambda_1, \\lambda_2, \\dots, \\lambda_k",[34,777569,777571],{"className":777570,"ariaHidden":758},[31672],[34,777572,777574,777577,777617,777620,777623,777663,777666,777669,777672,777675,777678,777681],{"className":777573},[31676],[34,777575],{"className":777576,"style":43217},[31680],[34,777578,777580,777583],{"className":777579},[31685],[34,777581,220792],{"className":777582},[31685,31686],[34,777584,777586],{"className":777585},[31792],[34,777587,777589,777609],{"className":777588},[31796,31797],[34,777590,777592,777606],{"className":777591},[31801],[34,777593,777595],{"className":777594,"style":31806},[31805],[34,777596,777597,777600],{"style":31809},[34,777598],{"className":777599,"style":31814},[31813],[34,777601,777603],{"className":777602},[31818,31819,31820,31821],[34,777604,640],{"className":777605},[31685,31821],[34,777607,31829],{"className":777608},[31828],[34,777610,777612],{"className":777611},[31801],[34,777613,777615],{"className":777614,"style":31836},[31805],[34,777616],{},[34,777618,31960],{"className":777619},[32119],[34,777621],{"className":777622,"style":32123},[31845],[34,777624,777626,777629],{"className":777625},[31685],[34,777627,220792],{"className":777628},[31685,31686],[34,777630,777632],{"className":777631},[31792],[34,777633,777635,777655],{"className":777634},[31796,31797],[34,777636,777638,777652],{"className":777637},[31801],[34,777639,777641],{"className":777640,"style":31806},[31805],[34,777642,777643,777646],{"style":31809},[34,777644],{"className":777645,"style":31814},[31813],[34,777647,777649],{"className":777648},[31818,31819,31820,31821],[34,777650,768],{"className":777651},[31685,31821],[34,777653,31829],{"className":777654},[31828],[34,777656,777658],{"className":777657},[31801],[34,777659,777661],{"className":777660,"style":31836},[31805],[34,777662],{},[34,777664,31960],{"className":777665},[32119],[34,777667],{"className":777668,"style":32123},[31845],[34,777670,33934],{"className":777671},[34006],[34,777673],{"className":777674,"style":32123},[31845],[34,777676,31960],{"className":777677},[32119],[34,777679],{"className":777680,"style":32123},[31845],[34,777682,777684,777687],{"className":777683},[31685],[34,777685,220792],{"className":777686},[31685,31686],[34,777688,777690],{"className":777689},[31792],[34,777691,777693,777713],{"className":777692},[31796,31797],[34,777694,777696,777710],{"className":777695},[31801],[34,777697,777699],{"className":777698,"style":33833},[31805],[34,777700,777701,777704],{"style":31809},[34,777702],{"className":777703,"style":31814},[31813],[34,777705,777707],{"className":777706},[31818,31819,31820,31821],[34,777708,33759],{"className":777709,"style":33774},[31685,31686,31821],[34,777711,31829],{"className":777712},[31828],[34,777714,777716],{"className":777715},[31801],[34,777717,777719],{"className":777718,"style":31836},[31805],[34,777720],{},[108,777722,777723,777724],{},"根据特征根的类型构建通解：",[105,777725,777726,777910,778379,778605,779055],{},[108,777727,777728,777729,777799,777800],{},"单实根 ",[34,777730,777732,777750],{"className":777731},[31642],[34,777733,777735],{"className":777734},[31646],[31648,777736,777737],{"xmlns":31650},[31652,777738,777739,777747],{},[31655,777740,777741],{},[31743,777742,777743,777745],{},[31658,777744,220792],{},[31658,777746,31955],{},[31665,777748,777749],{"encoding":31667},"\\lambda_i",[34,777751,777753],{"className":777752,"ariaHidden":758},[31672],[34,777754,777756,777759],{"className":777755},[31676],[34,777757],{"className":777758,"style":33814},[31680],[34,777760,777762,777765],{"className":777761},[31685],[34,777763,220792],{"className":777764},[31685,31686],[34,777766,777768],{"className":777767},[31792],[34,777769,777771,777791],{"className":777770},[31796,31797],[34,777772,777774,777788],{"className":777773},[31801],[34,777775,777777],{"className":777776,"style":32090},[31805],[34,777778,777779,777782],{"style":31809},[34,777780],{"className":777781,"style":31814},[31813],[34,777783,777785],{"className":777784},[31818,31819,31820,31821],[34,777786,31955],{"className":777787},[31685,31686,31821],[34,777789,31829],{"className":777790},[31828],[34,777792,777794],{"className":777793},[31801],[34,777795,777797],{"className":777796,"style":31836},[31805],[34,777798],{},"：对应解为 ",[34,777801,777803,777829],{"className":777802},[31642],[34,777804,777806],{"className":777805},[31646],[31648,777807,777808],{"xmlns":31650},[31652,777809,777810,777826],{},[31655,777811,777812],{},[37624,777813,777814,777816],{},[31658,777815,33244],{},[31655,777817,777818,777824],{},[31743,777819,777820,777822],{},[31658,777821,220792],{},[31658,777823,31955],{},[31658,777825,32936],{},[31665,777827,777828],{"encoding":31667},"e^{\\lambda_i x}",[34,777830,777832],{"className":777831,"ariaHidden":758},[31672],[34,777833,777835,777838],{"className":777834},[31676],[34,777836],{"className":777837,"style":38486},[31680],[34,777839,777841,777844],{"className":777840},[31685],[34,777842,33244],{"className":777843},[31685,31686],[34,777845,777847],{"className":777846},[31792],[34,777848,777850],{"className":777849},[31796],[34,777851,777853],{"className":777852},[31801],[34,777854,777856],{"className":777855,"style":38486},[31805],[34,777857,777858,777861],{"style":37777},[34,777859],{"className":777860,"style":31814},[31813],[34,777862,777864],{"className":777863},[31818,31819,31820,31821],[34,777865,777867,777907],{"className":777866},[31685,31821],[34,777868,777870,777873],{"className":777869},[31685,31821],[34,777871,220792],{"className":777872},[31685,31686,31821],[34,777874,777876],{"className":777875},[31792],[34,777877,777879,777899],{"className":777878},[31796,31797],[34,777880,777882,777896],{"className":777881},[31801],[34,777883,777885],{"className":777884,"style":231708},[31805],[34,777886,777887,777890],{"style":318661},[34,777888],{"className":777889,"style":43406},[31813],[34,777891,777893],{"className":777892},[31818,43410,43411,31821],[34,777894,31955],{"className":777895},[31685,31686,31821],[34,777897,31829],{"className":777898},[31828],[34,777900,777902],{"className":777901},[31801],[34,777903,777905],{"className":777904,"style":220540},[31805],[34,777906],{},[34,777908,32936],{"className":777909},[31685,31686,31821],[108,777911,777912,777940,777941,777799,778010],{},[34,777913,777915,777928],{"className":777914},[31642],[34,777916,777918],{"className":777917},[31646],[31648,777919,777920],{"xmlns":31650},[31652,777921,777922,777926],{},[31655,777923,777924],{},[31658,777925,33759],{},[31665,777927,33759],{"encoding":31667},[34,777929,777931],{"className":777930,"ariaHidden":758},[31672],[34,777932,777934,777937],{"className":777933},[31676],[34,777935],{"className":777936,"style":31681},[31680],[34,777938,33759],{"className":777939,"style":33774},[31685,31686]," 重实根 ",[34,777942,777944,777961],{"className":777943},[31642],[34,777945,777947],{"className":777946},[31646],[31648,777948,777949],{"xmlns":31650},[31652,777950,777951,777959],{},[31655,777952,777953],{},[31743,777954,777955,777957],{},[31658,777956,220792],{},[31658,777958,31955],{},[31665,777960,777749],{"encoding":31667},[34,777962,777964],{"className":777963,"ariaHidden":758},[31672],[34,777965,777967,777970],{"className":777966},[31676],[34,777968],{"className":777969,"style":33814},[31680],[34,777971,777973,777976],{"className":777972},[31685],[34,777974,220792],{"className":777975},[31685,31686],[34,777977,777979],{"className":777978},[31792],[34,777980,777982,778002],{"className":777981},[31796,31797],[34,777983,777985,777999],{"className":777984},[31801],[34,777986,777988],{"className":777987,"style":32090},[31805],[34,777989,777990,777993],{"style":31809},[34,777991],{"className":777992,"style":31814},[31813],[34,777994,777996],{"className":777995},[31818,31819,31820,31821],[34,777997,31955],{"className":777998},[31685,31686,31821],[34,778000,31829],{"className":778001},[31828],[34,778003,778005],{"className":778004},[31801],[34,778006,778008],{"className":778007,"style":31836},[31805],[34,778009],{},[34,778011,778013,778083],{"className":778012},[31642],[34,778014,778016],{"className":778015},[31646],[31648,778017,778018],{"xmlns":31650},[31652,778019,778020,778080],{},[31655,778021,778022,778024,778030,778032,778038,778040,778042,778044,778046,778052,778064,778066],{},[31662,778023,456],{"stretchy":702},[31743,778025,778026,778028],{},[31658,778027,47384],{},[31749,778029,640],{},[31662,778031,223],{},[31743,778033,778034,778036],{},[31658,778035,47384],{},[31749,778037,768],{},[31658,778039,32936],{},[31662,778041,223],{},[31662,778043,38874],{},[31662,778045,223],{},[31743,778047,778048,778050],{},[31658,778049,47384],{},[31658,778051,33759],{},[37624,778053,778054,778056],{},[31658,778055,32936],{},[31655,778057,778058,778060,778062],{},[31658,778059,33759],{},[31662,778061,32226],{},[31749,778063,640],{},[31662,778065,2557],{"stretchy":702},[37624,778067,778068,778070],{},[31658,778069,33244],{},[31655,778071,778072,778078],{},[31743,778073,778074,778076],{},[31658,778075,220792],{},[31658,778077,31955],{},[31658,778079,32936],{},[31665,778081,778082],{"encoding":31667},"(C_1 + C_2 x + \\dots + C_k x^{k-1}) e^{\\lambda_i 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\\frac{d^n y}{dx^n} + a_{n-1} \\frac{d^{n-1} y}{dx^{n-1}} + \\dots + a_1 \\frac{dy}{dx} + a_0 y = 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",[34,780353,780355,780374],{"className":780354},[31642],[34,780356,780358],{"className":780357},[31646],[31648,780359,780360],{"xmlns":31650},[31652,780361,780362,780372],{},[31655,780363,780364,780366,780368,780370],{},[31658,780365,31942],{},[31662,780367,456],{"stretchy":702},[31658,780369,32936],{},[31662,780371,2557],{"stretchy":702},[31665,780373,547596],{"encoding":31667},[34,780375,780377],{"className":780376,"ariaHidden":758},[31672],[34,780378,780380,780383,780386,780389,780392],{"className":780379},[31676],[34,780381],{"className":780382,"style":31774},[31680],[34,780384,31942],{"className":780385,"style":32062},[31685,31686],[34,780387,456],{"className":780388},[31782],[34,780390,32936],{"className":780391},[31685,31686],[34,780393,2557],{"className":780394},[31690]," 的形式，一般是多项式、指数函数、三角函数等，尝试构造特解 ",[34,780397,780399,780417],{"className":780398},[31642],[34,780400,780402],{"className":780401},[31646],[31648,780403,780404],{"xmlns":31650},[31652,780405,780406,780414],{},[31655,780407,780408],{},[31743,780409,780410,780412],{},[31658,780411,67764],{},[31658,780413,15],{},[31665,780415,780416],{"encoding":31667},"y_p",[34,780418,780420],{"className":780419,"ariaHidden":758},[31672],[34,780421,780423,780426],{"className":780422},[31676],[34,780424],{"className":780425,"style":56005},[31680],[34,780427,780429,780432],{"className":780428},[31685],[34,780430,67764],{"className":780431,"style":32062},[31685,31686],[34,780433,780435],{"className":780434},[31792],[34,780436,780438,780458],{"className":780437},[31796,31797],[34,780439,780441,780455],{"className":780440},[31801],[34,780442,780444],{"className":780443,"style":32362},[31805],[34,780445,780446,780449],{"style":38795},[34,780447],{"className":780448,"style":31814},[31813],[34,780450,780452],{"className":780451},[31818,31819,31820,31821],[34,780453,15],{"className":780454},[31685,31686,31821],[34,780456,31829],{"className":780457},[31828],[34,780459,780461],{"className":780460},[31801],[34,780462,780464],{"className":780463,"style":32163},[31805],[34,780465],{},[105,780467,780468,780758,780941],{},[108,780469,2234,780470,780513,780514,780757],{},[34,780471,780473,780492],{"className":780472},[31642],[34,780474,780476],{"className":780475},[31646],[31648,780477,780478],{"xmlns":31650},[31652,780479,780480,780490],{},[31655,780481,780482,780484,780486,780488],{},[31658,780483,31942],{},[31662,780485,456],{"stretchy":702},[31658,780487,32936],{},[31662,780489,2557],{"stretchy":702},[31665,780491,547596],{"encoding":31667},[34,780493,780495],{"className":780494,"ariaHidden":758},[31672],[34,780496,780498,780501,780504,780507,780510],{"className":780497},[31676],[34,780499],{"className":780500,"style":31774},[31680],[34,780502,31942],{"className":780503,"style":32062},[31685,31686],[34,780505,456],{"className":780506},[31782],[34,780508,32936],{"className":780509},[31685,31686],[34,780511,2557],{"className":780512},[31690]," 是多项式 ，尝试 ",[34,780515,780517,780569],{"className":780516},[31642],[34,780518,780520],{"className":780519},[31646],[31648,780521,780522],{"xmlns":31650},[31652,780523,780524,780566],{},[31655,780525,780526,780532,780534,780536,780542,780544,780546,780558,780560,780562,780564],{},[31743,780527,780528,780530],{},[31658,780529,67764],{},[31658,780531,15],{},[31662,780533,202],{},[31658,780535,54151],{},[37624,780537,780538,780540],{},[31658,780539,32936],{},[31658,780541,31660],{},[31662,780543,223],{},[31658,780545,54224],{},[37624,780547,780548,780550],{},[31658,780549,32936],{},[31655,780551,780552,780554,780556],{},[31658,780553,31660],{},[31662,780555,32226],{},[31749,780557,640],{},[31662,780559,223],{},[31662,780561,38874],{},[31662,780563,223],{},[31658,780565,47384],{},[31665,780567,780568],{"encoding":31667},"y_p = Ax^n + Bx^{n-1} + \\dots + C",[34,780570,780572,780627,780674,780730,780748],{"className":780571,"ariaHidden":758},[31672],[34,780573,780575,780578,780618,780621,780624],{"className":780574},[31676],[34,780576],{"className":780577,"style":56005},[31680],[34,780579,780581,780584],{"className":780580},[31685],[34,780582,67764],{"className":780583,"style":32062},[31685,31686],[34,780585,780587],{"className":780586},[31792],[34,780588,780590,780610],{"className":780589},[31796,31797],[34,780591,780593,780607],{"className":780592},[31801],[34,780594,780596],{"className":780595,"style":32362},[31805],[34,780597,780598,780601],{"style":38795},[34,780599],{"className":780600,"style":31814},[31813],[34,780602,780604],{"className":780603},[31818,31819,31820,31821],[34,780605,15],{"className":780606},[31685,31686,31821],[34,780608,31829],{"className":780609},[31828],[34,780611,780613],{"className":780612},[31801],[34,780614,780616],{"className":780615,"style":32163},[31805],[34,780617],{},[34,780619],{"className":780620,"style":31846},[31845],[34,780622,202],{"className":780623},[31850],[34,780625],{"className":780626,"style":31846},[31845],[34,780628,780630,780633,780636,780665,780668,780671],{"className":780629},[31676],[34,780631],{"className":780632,"style":34944},[31680],[34,780634,54151],{"className":780635},[31685,31686],[34,780637,780639,780642],{"className":780638},[31685],[34,780640,32936],{"className":780641},[31685,31686],[34,780643,780645],{"className":780644},[31792],[34,780646,780648],{"className":780647},[31796],[34,780649,780651],{"className":780650},[31801],[34,780652,780654],{"className":780653,"style":37959},[31805],[34,780655,780656,780659],{"style":37777},[34,780657],{"className":780658,"style":31814},[31813],[34,780660,780662],{"className":780661},[31818,31819,31820,31821],[34,780663,31660],{"className":780664},[31685,31686,31821],[34,780666],{"className":780667,"style":32604},[31845],[34,780669,223],{"className":780670},[32024],[34,780672],{"className":780673,"style":32604},[31845],[34,780675,780677,780680,780683,780721,780724,780727],{"className":780676},[31676],[34,780678],{"className":780679,"style":40044},[31680],[34,780681,54224],{"className":780682,"style":54252},[31685,31686],[34,780684,780686,780689],{"className":780685},[31685],[34,780687,32936],{"className":780688},[31685,31686],[34,780690,780692],{"className":780691},[31792],[34,780693,780695],{"className":780694},[31796],[34,780696,780698],{"className":780697},[31801],[34,780699,780701],{"className":780700,"style":37774},[31805],[34,780702,780703,780706],{"style":37777},[34,780704],{"className":780705,"style":31814},[31813],[34,780707,780709],{"className":780708},[31818,31819,31820,31821],[34,780710,780712,780715,780718],{"className":780711},[31685,31821],[34,780713,31660],{"className":780714},[31685,31686,31821],[34,780716,32226],{"className":780717},[32024,31821],[34,780719,640],{"className":780720},[31685,31821],[34,780722],{"className":780723,"style":32604},[31845],[34,780725,223],{"className":780726},[32024],[34,780728],{"className":780729,"style":32604},[31845],[34,780731,780733,780736,780739,780742,780745],{"className":780732},[31676],[34,780734],{"className":780735,"style":32597},[31680],[34,780737,38874],{"className":780738},[34006],[34,780740],{"className":780741,"style":32604},[31845],[34,780743,223],{"className":780744},[32024],[34,780746],{"className":780747,"style":32604},[31845],[34,780749,780751,780754],{"className":780750},[31676],[34,780752],{"className":780753,"style":33218},[31680],[34,780755,47384],{"className":780756,"style":38706},[31685,31686]," 的形式。",[108,780759,2234,780760,780803,780804,780757],{},[34,780761,780763,780782],{"className":780762},[31642],[34,780764,780766],{"className":780765},[31646],[31648,780767,780768],{"xmlns":31650},[31652,780769,780770,780780],{},[31655,780771,780772,780774,780776,780778],{},[31658,780773,31942],{},[31662,780775,456],{"stretchy":702},[31658,780777,32936],{},[31662,780779,2557],{"stretchy":702},[31665,780781,547596],{"encoding":31667},[34,780783,780785],{"className":780784,"ariaHidden":758},[31672],[34,780786,780788,780791,780794,780797,780800],{"className":780787},[31676],[34,780789],{"className":780790,"style":31774},[31680],[34,780792,31942],{"className":780793,"style":32062},[31685,31686],[34,780795,456],{"className":780796},[31782],[34,780798,32936],{"className":780799},[31685,31686],[34,780801,2557],{"className":780802},[31690]," 是指数函数，尝试 ",[34,780805,780807,780839],{"className":780806},[31642],[34,780808,780810],{"className":780809},[31646],[31648,780811,780812],{"xmlns":31650},[31652,780813,780814,780836],{},[31655,780815,780816,780822,780824,780826],{},[31743,780817,780818,780820],{},[31658,780819,67764],{},[31658,780821,15],{},[31662,780823,202],{},[31658,780825,54151],{},[37624,780827,780828,780830],{},[31658,780829,33244],{},[31655,780831,780832,780834],{},[31658,780833,33759],{},[31658,780835,32936],{},[31665,780837,780838],{"encoding":31667},"y_p = Ae^{kx}",[34,780840,780842,780897],{"className":780841,"ariaHidden":758},[31672],[34,780843,780845,780848,780888,780891,780894],{"className":780844},[31676],[34,780846],{"className":780847,"style":56005},[31680],[34,780849,780851,780854],{"className":780850},[31685],[34,780852,67764],{"className":780853,"style":32062},[31685,31686],[34,780855,780857],{"className":780856},[31792],[34,780858,780860,780880],{"className":780859},[31796,31797],[34,780861,780863,780877],{"className":780862},[31801],[34,780864,780866],{"className":780865,"style":32362},[31805],[34,780867,780868,780871],{"style":38795},[34,780869],{"className":780870,"style":31814},[31813],[34,780872,780874],{"className":780873},[31818,31819,31820,31821],[34,780875,15],{"className":780876},[31685,31686,31821],[34,780878,31829],{"className":780879},[31828],[34,780881,780883],{"className":780882},[31801],[34,780884,780886],{"className":780885,"style":32163},[31805],[34,780887],{},[34,780889],{"className":780890,"style":31846},[31845],[34,780892,202],{"className":780893},[31850],[34,780895],{"className":780896,"style":31846},[31845],[34,780898,780900,780903,780906],{"className":780899},[31676],[34,780901],{"className":780902,"style":38486},[31680],[34,780904,54151],{"className":780905},[31685,31686],[34,780907,780909,780912],{"className":780908},[31685],[34,780910,33244],{"className":780911},[31685,31686],[34,780913,780915],{"className":780914},[31792],[34,780916,780918],{"className":780917},[31796],[34,780919,780921],{"className":780920},[31801],[34,780922,780924],{"className":780923,"style":38486},[31805],[34,780925,780926,780929],{"style":37777},[34,780927],{"className":780928,"style":31814},[31813],[34,780930,780932],{"className":780931},[31818,31819,31820,31821],[34,780933,780935,780938],{"className":780934},[31685,31821],[34,780936,33759],{"className":780937,"style":33774},[31685,31686,31821],[34,780939,32936],{"className":780940},[31685,31686,31821],[108,780942,2234,780943,780986,780987,780757],{},[34,780944,780946,780965],{"className":780945},[31642],[34,780947,780949],{"className":780948},[31646],[31648,780950,780951],{"xmlns":31650},[31652,780952,780953,780963],{},[31655,780954,780955,780957,780959,780961],{},[31658,780956,31942],{},[31662,780958,456],{"stretchy":702},[31658,780960,32936],{},[31662,780962,2557],{"stretchy":702},[31665,780964,547596],{"encoding":31667},[34,780966,780968],{"className":780967,"ariaHidden":758},[31672],[34,780969,780971,780974,780977,780980,780983],{"className":780970},[31676],[34,780972],{"className":780973,"style":31774},[31680],[34,780975,31942],{"className":780976,"style":32062},[31685,31686],[34,780978,456],{"className":780979},[31782],[34,780981,32936],{"className":780982},[31685,31686],[34,780984,2557],{"className":780985},[31690]," 是三角函数，尝试 ",[34,780988,780990,781040],{"className":780989},[31642],[34,780991,780993],{"className":780992},[31646],[31648,780994,780995],{"xmlns":31650},[31652,780996,780997,781037],{},[31655,780998,780999,781005,781007,781009,781011,781013,781015,781017,781019,781021,781023,781025,781027,781029,781031,781033,781035],{},[31743,781000,781001,781003],{},[31658,781002,67764],{},[31658,781004,15],{},[31662,781006,202],{},[31658,781008,54151],{},[31658,781010,547860],{},[31662,781012,32682],{},[31662,781014,456],{"stretchy":702},[31658,781016,33759],{},[31658,781018,32936],{},[31662,781020,2557],{"stretchy":702},[31662,781022,223],{},[31658,781024,54224],{},[31658,781026,547853],{},[31662,781028,32682],{},[31662,781030,456],{"stretchy":702},[31658,781032,33759],{},[31658,781034,32936],{},[31662,781036,2557],{"stretchy":702},[31665,781038,781039],{"encoding":31667},"y_p = A\\cos(kx) + B\\sin(kx)",[34,781041,781043,781098,781134],{"className":781042,"ariaHidden":758},[31672],[34,781044,781046,781049,781089,781092,781095],{"className":781045},[31676],[34,781047],{"className":781048,"style":56005},[31680],[34,781050,781052,781055],{"className":781051},[31685],[34,781053,67764],{"className":781054,"style":32062},[31685,31686],[34,781056,781058],{"className":781057},[31792],[34,781059,781061,781081],{"className":781060},[31796,31797],[34,781062,781064,781078],{"className":781063},[31801],[34,781065,781067],{"className":781066,"style":32362},[31805],[34,781068,781069,781072],{"style":38795},[34,781070],{"className":781071,"style":31814},[31813],[34,781073,781075],{"className":781074},[31818,31819,31820,31821],[34,781076,15],{"className":781077},[31685,31686,31821],[34,781079,31829],{"className":781080},[31828],[34,781082,781084],{"className":781083},[31801],[34,781085,781087],{"className":781086,"style":32163},[31805],[34,781088],{},[34,781090],{"className":781091,"style":31846},[31845],[34,781093,202],{"className":781094},[31850],[34,781096],{"className":781097,"style":31846},[31845],[34,781099,781101,781104,781107,781110,781113,781116,781119,781122,781125,781128,781131],{"className":781100},[31676],[34,781102],{"className":781103,"style":31774},[31680],[34,781105,54151],{"className":781106},[31685,31686],[34,781108],{"className":781109,"style":32123},[31845],[34,781111,547860],{"className":781112},[32767],[34,781114,456],{"className":781115},[31782],[34,781117,33759],{"className":781118,"style":33774},[31685,31686],[34,781120,32936],{"className":781121},[31685,31686],[34,781123,2557],{"className":781124},[31690],[34,781126],{"className":781127,"style":32604},[31845],[34,781129,223],{"className":781130},[32024],[34,781132],{"className":781133,"style":32604},[31845],[34,781135,781137,781140,781143,781146,781149,781152,781155,781158],{"className":781136},[31676],[34,781138],{"className":781139,"style":31774},[31680],[34,781141,54224],{"className":781142,"style":54252},[31685,31686],[34,781144],{"className":781145,"style":32123},[31845],[34,781147,547853],{"className":781148},[32767],[34,781150,456],{"className":781151},[31782],[34,781153,33759],{"className":781154,"style":33774},[31685,31686],[34,781156,32936],{"className":781157},[31685,31686],[34,781159,2557],{"className":781160},[31690],[108,781162,781163,781164,781233],{},"将 ",[34,781165,781167,781184],{"className":781166},[31642],[34,781168,781170],{"className":781169},[31646],[31648,781171,781172],{"xmlns":31650},[31652,781173,781174,781182],{},[31655,781175,781176],{},[31743,781177,781178,781180],{},[31658,781179,67764],{},[31658,781181,15],{},[31665,781183,780416],{"encoding":31667},[34,781185,781187],{"className":781186,"ariaHidden":758},[31672],[34,781188,781190,781193],{"className":781189},[31676],[34,781191],{"className":781192,"style":56005},[31680],[34,781194,781196,781199],{"className":781195},[31685],[34,781197,67764],{"className":781198,"style":32062},[31685,31686],[34,781200,781202],{"className":781201},[31792],[34,781203,781205,781225],{"className":781204},[31796,31797],[34,781206,781208,781222],{"className":781207},[31801],[34,781209,781211],{"className":781210,"style":32362},[31805],[34,781212,781213,781216],{"style":38795},[34,781214],{"className":781215,"style":31814},[31813],[34,781217,781219],{"className":781218},[31818,31819,31820,31821],[34,781220,15],{"className":781221},[31685,31686,31821],[34,781223,31829],{"className":781224},[31828],[34,781226,781228],{"className":781227},[31801],[34,781229,781231],{"className":781230,"style":32163},[31805],[34,781232],{}," 代入原方程，求解系数。",[108,781235,781236,781237],{},"最终通解为 ",[34,781238,781240,781270],{"className":781239},[31642],[34,781241,781243],{"className":781242},[31646],[31648,781244,781245],{"xmlns":31650},[31652,781246,781247,781267],{},[31655,781248,781249,781251,781253,781259,781261],{},[31658,781250,67764],{},[31662,781252,202],{},[31743,781254,781255,781257],{},[31658,781256,67764],{},[31658,781258,47292],{},[31662,781260,223],{},[31743,781262,781263,781265],{},[31658,781264,67764],{},[31658,781266,15],{},[31665,781268,781269],{"encoding":31667},"y = y_h + y_p",[34,781271,781273,781291,781346],{"className":781272,"ariaHidden":758},[31672],[34,781274,781276,781279,781282,781285,781288],{"className":781275},[31676],[34,781277],{"className":781278,"style":38774},[31680],[34,781280,67764],{"className":781281,"style":32062},[31685,31686],[34,781283],{"className":781284,"style":31846},[31845],[34,781286,202],{"className":781287},[31850],[34,781289],{"className":781290,"style":31846},[31845],[34,781292,781294,781297,781337,781340,781343],{"className":781293},[31676],[34,781295],{"className":781296,"style":55949},[31680],[34,781298,781300,781303],{"className":781299},[31685],[34,781301,67764],{"className":781302,"style":32062},[31685,31686],[34,781304,781306],{"className":781305},[31792],[34,781307,781309,781329],{"className":781308},[31796,31797],[34,781310,781312,781326],{"className":781311},[31801],[34,781313,781315],{"className":781314,"style":33833},[31805],[34,781316,781317,781320],{"style":38795},[34,781318],{"className":781319,"style":31814},[31813],[34,781321,781323],{"className":781322},[31818,31819,31820,31821],[34,781324,47292],{"className":781325},[31685,31686,31821],[34,781327,31829],{"className":781328},[31828],[34,781330,781332],{"className":781331},[31801],[34,781333,781335],{"className":781334,"style":31836},[31805],[34,781336],{},[34,781338],{"className":781339,"style":32604},[31845],[34,781341,223],{"className":781342},[32024],[34,781344],{"className":781345,"style":32604},[31845],[34,781347,781349,781352],{"className":781348},[31676],[34,781350],{"className":781351,"style":56005},[31680],[34,781353,781355,781358],{"className":781354},[31685],[34,781356,67764],{"className":781357,"style":32062},[31685,31686],[34,781359,781361],{"className":781360},[31792],[34,781362,781364,781384],{"className":781363},[31796,31797],[34,781365,781367,781381],{"className":781366},[31801],[34,781368,781370],{"className":781369,"style":32362},[31805],[34,781371,781372,781375],{"style":38795},[34,781373],{"className":781374,"style":31814},[31813],[34,781376,781378],{"className":781377},[31818,31819,31820,31821],[34,781379,15],{"className":781380},[31685,31686,31821],[34,781382,31829],{"className":781383},[31828],[34,781385,781387],{"className":781386},[31801],[34,781388,781390],{"className":781389,"style":32163},[31805],[34,781391],{},[19,781393,781395],{"id":781394},"利用皮卡逐步逼近序列求解ode的近似解","利用皮卡逐步逼近序列求解ODE的近似解",[15,781397,781398],{},[2627,781399,128684],{},[15,781401,781402],{},[34,781403,781405,781469],{"className":781404},[31642],[34,781406,781408],{"className":781407},[31646],[31648,781409,781410],{"xmlns":31650},[31652,781411,781412,781466],{},[31655,781413,781414,781428,781430,781432,781434,781436,781438,781440,781442,781444,781446,781448,781450,781456,781458,781460],{},[41453,781415,781416,781422],{},[31655,781417,781418,781420],{},[31658,781419,46900],{},[31658,781421,67764],{},[31655,781423,781424,781426],{},[31658,781425,46900],{},[31658,781427,32936],{},[31662,781429,202],{},[31658,781431,31739],{},[31662,781433,456],{"stretchy":702},[31658,781435,32936],{},[31662,781437,31960],{"separator":758},[31658,781439,67764],{},[31662,781441,2557],{"stretchy":702},[31662,781443,31960],{"separator":758},[31845,781445],{"width":71082},[31658,781447,67764],{},[31662,781449,456],{"stretchy":702},[31743,781451,781452,781454],{},[31658,781453,32936],{},[31749,781455,459],{},[31662,781457,2557],{"stretchy":702},[31662,781459,202],{},[31743,781461,781462,781464],{},[31658,781463,67764],{},[31749,781465,459],{},[31665,781467,781468],{"encoding":31667},"\\frac{dy}{dx} = f(x, y), \\quad y(x_0) = y_0",[34,781470,781472,781561,781655],{"className":781471,"ariaHidden":758},[31672],[34,781473,781475,781478,781552,781555,781558],{"className":781474},[31676],[34,781476],{"className":781477,"style":608762},[31680],[34,781479,781481,781484,781549],{"className":781480},[31685],[34,781482],{"className":781483},[31782,41488],[34,781485,781487],{"className":781486},[41453],[34,781488,781490,781541],{"className":781489},[31796,31797],[34,781491,781493,781538],{"className":781492},[31801],[34,781494,781496,781513,781521],{"className":781495,"style":608935},[31805],[34,781497,781498,781501],{"style":41504},[34,781499],{"className":781500,"style":41508},[31813],[34,781502,781504],{"className":781503},[31818,31819,31820,31821],[34,781505,781507,781510],{"className":781506},[31685,31821],[34,781508,46900],{"className":781509},[31685,31686,31821],[34,781511,32936],{"className":781512},[31685,31686,31821],[34,781514,781515,781518],{"style":41520},[34,781516],{"className":781517,"style":41508},[31813],[34,781519],{"className":781520,"style":41528},[41527],[34,781522,781523,781526],{"style":548368},[34,781524],{"className":781525,"style":41508},[31813],[34,781527,781529],{"className":781528},[31818,31819,31820,31821],[34,781530,781532,781535],{"className":781531},[31685,31821],[34,781533,46900],{"className":781534},[31685,31686,31821],[34,781536,67764],{"className":781537,"style":32062},[31685,31686,31821],[34,781539,31829],{"className":781540},[31828],[34,781542,781544],{"className":781543},[31801],[34,781545,781547],{"className":781546,"style":41619},[31805],[34,781548],{},[34,781550],{"className":781551},[31690,41488],[34,781553],{"className":781554,"style":31846},[31845],[34,781556,202],{"className":781557},[31850],[34,781559],{"className":781560,"style":31846},[31845],[34,781562,781564,781567,781570,781573,781576,781579,781582,781585,781588,781591,781594,781597,781600,781603,781643,781646,781649,781652],{"className":781563},[31676],[34,781565],{"className":781566,"style":31774},[31680],[34,781568,31739],{"className":781569,"style":31778},[31685,31686],[34,781571,456],{"className":781572},[31782],[34,781574,32936],{"className":781575},[31685,31686],[34,781577,31960],{"className":781578},[32119],[34,781580],{"className":781581,"style":32123},[31845],[34,781583,67764],{"className":781584,"style":32062},[31685,31686],[34,781586,2557],{"className":781587},[31690],[34,781589,31960],{"className":781590},[32119],[34,781592],{"className":781593,"style":128438},[31845],[34,781595],{"className":781596,"style":32123},[31845],[34,781598,67764],{"className":781599,"style":32062},[31685,31686],[34,781601,456],{"className":781602},[31782],[34,781604,781606,781609],{"className":781605},[31685],[34,781607,32936],{"className":781608},[31685,31686],[34,781610,781612],{"className":781611},[31792],[34,781613,781615,781635],{"className":781614},[31796,31797],[34,781616,781618,781632],{"className":781617},[31801],[34,781619,781621],{"className":781620,"style":31806},[31805],[34,781622,781623,781626],{"style":31809},[34,781624],{"className":781625,"style":31814},[31813],[34,781627,781629],{"className":781628},[31818,31819,31820,31821],[34,781630,459],{"className":781631},[31685,31821],[34,781633,31829],{"className":781634},[31828],[34,781636,781638],{"className":781637},[31801],[34,781639,781641],{"className":781640,"style":31836},[31805],[34,781642],{},[34,781644,2557],{"className":781645},[31690],[34,781647],{"className":781648,"style":31846},[31845],[34,781650,202],{"className":781651},[31850],[34,781653],{"className":781654,"style":31846},[31845],[34,781656,781658,781661],{"className":781657},[31676],[34,781659],{"className":781660,"style":38774},[31680],[34,781662,781664,781667],{"className":781663},[31685],[34,781665,67764],{"className":781666,"style":32062},[31685,31686],[34,781668,781670],{"className":781669},[31792],[34,781671,781673,781693],{"className":781672},[31796,31797],[34,781674,781676,781690],{"className":781675},[31801],[34,781677,781679],{"className":781678,"style":31806},[31805],[34,781680,781681,781684],{"style":38795},[34,781682],{"className":781683,"style":31814},[31813],[34,781685,781687],{"className":781686},[31818,31819,31820,31821],[34,781688,459],{"className":781689},[31685,31821],[34,781691,31829],{"className":781692},[31828],[34,781694,781696],{"className":781695},[31801],[34,781697,781699],{"className":781698,"style":31836},[31805],[34,781700],{},[15,781702,781703],{},[2627,781704,781705],{},"第零次近似：",[15,781707,781708],{},[34,781709,781711,781743],{"className":781710},[31642],[34,781712,781714],{"className":781713},[31646],[31648,781715,781716],{"xmlns":31650},[31652,781717,781718,781740],{},[31655,781719,781720,781726,781728,781730,781732,781734],{},[31743,781721,781722,781724],{},[31658,781723,329201],{},[31749,781725,459],{},[31662,781727,456],{"stretchy":702},[31658,781729,32936],{},[31662,781731,2557],{"stretchy":702},[31662,781733,202],{},[31743,781735,781736,781738],{},[31658,781737,67764],{},[31749,781739,459],{},[31665,781741,781742],{"encoding":31667},"\\phi_0(x) = y_0",[34,781744,781746,781810],{"className":781745,"ariaHidden":758},[31672],[34,781747,781749,781752,781792,781795,781798,781801,781804,781807],{"className":781748},[31676],[34,781750],{"className":781751,"style":31774},[31680],[34,781753,781755,781758],{"className":781754},[31685],[34,781756,329201],{"className":781757},[31685,31686],[34,781759,781761],{"className":781760},[31792],[34,781762,781764,781784],{"className":781763},[31796,31797],[34,781765,781767,781781],{"className":781766},[31801],[34,781768,781770],{"className":781769,"style":31806},[31805],[34,781771,781772,781775],{"style":31809},[34,781773],{"className":781774,"style":31814},[31813],[34,781776,781778],{"className":781777},[31818,31819,31820,31821],[34,781779,459],{"className":781780},[31685,31821],[34,781782,31829],{"className":781783},[31828],[34,781785,781787],{"className":781786},[31801],[34,781788,781790],{"className":781789,"style":31836},[31805],[34,781791],{},[34,781793,456],{"className":781794},[31782],[34,781796,32936],{"className":781797},[31685,31686],[34,781799,2557],{"className":781800},[31690],[34,781802],{"className":781803,"style":31846},[31845],[34,781805,202],{"className":781806},[31850],[34,781808],{"className":781809,"style":31846},[31845],[34,781811,781813,781816],{"className":781812},[31676],[34,781814],{"className":781815,"style":38774},[31680],[34,781817,781819,781822],{"className":781818},[31685],[34,781820,67764],{"className":781821,"style":32062},[31685,31686],[34,781823,781825],{"className":781824},[31792],[34,781826,781828,781848],{"className":781827},[31796,31797],[34,781829,781831,781845],{"className":781830},[31801],[34,781832,781834],{"className":781833,"style":31806},[31805],[34,781835,781836,781839],{"style":38795},[34,781837],{"className":781838,"style":31814},[31813],[34,781840,781842],{"className":781841},[31818,31819,31820,31821],[34,781843,459],{"className":781844},[31685,31821],[34,781846,31829],{"className":781847},[31828],[34,781849,781851],{"className":781850},[31801],[34,781852,781854],{"className":781853,"style":31836},[31805],[34,781855],{},[15,781857,781858],{},[2627,781859,781860],{},"第一次近似：",[15,781862,781863],{},[34,781864,781866,781940],{"className":781865},[31642],[34,781867,781869],{"className":781868},[31646],[31648,781870,781871],{"xmlns":31650},[31652,781872,781873,781937],{},[31655,781874,781875,781881,781883,781885,781887,781889,781895,781897,781909,781911,781913,781915,781917,781923,781925,781927,781929,781931,781933,781935],{},[31743,781876,781877,781879],{},[31658,781878,329201],{},[31749,781880,640],{},[31662,781882,456],{"stretchy":702},[31658,781884,32936],{},[31662,781886,2557],{"stretchy":702},[31662,781888,202],{},[31743,781890,781891,781893],{},[31658,781892,67764],{},[31749,781894,459],{},[31662,781896,223],{},[38846,781898,781899,781901,781907],{},[31662,781900,546295],{},[31743,781902,781903,781905],{},[31658,781904,32936],{},[31749,781906,459],{},[31658,781908,32936],{},[31658,781910,31739],{},[31662,781912,456],{"stretchy":702},[31658,781914,33249],{},[31662,781916,31960],{"separator":758},[31743,781918,781919,781921],{},[31658,781920,329201],{},[31749,781922,459],{},[31662,781924,456],{"stretchy":702},[31658,781926,33249],{},[31662,781928,2557],{"stretchy":702},[31662,781930,2557],{"stretchy":702},[39361,781932,39363],{},[31658,781934,46900],{},[31658,781936,33249],{},[31665,781938,781939],{"encoding":31667},"\\phi_1(x) = y_0 + \\int_{x_0}^{x} f(t, \\phi_0(t)) \\, dt",[34,781941,781943,782007,782062],{"className":781942,"ariaHidden":758},[31672],[34,781944,781946,781949,781989,781992,781995,781998,782001,782004],{"className":781945},[31676],[34,781947],{"className":781948,"style":31774},[31680],[34,781950,781952,781955],{"className":781951},[31685],[34,781953,329201],{"className":781954},[31685,31686],[34,781956,781958],{"className":781957},[31792],[34,781959,781961,781981],{"className":781960},[31796,31797],[34,781962,781964,781978],{"className":781963},[31801],[34,781965,781967],{"className":781966,"style":31806},[31805],[34,781968,781969,781972],{"style":31809},[34,781970],{"className":781971,"style":31814},[31813],[34,781973,781975],{"className":781974},[31818,31819,31820,31821],[34,781976,640],{"className":781977},[31685,31821],[34,781979,31829],{"className":781980},[31828],[34,781982,781984],{"className":781983},[31801],[34,781985,781987],{"className":781986,"style":31836},[31805],[34,781988],{},[34,781990,456],{"className":781991},[31782],[34,781993,32936],{"className":781994},[31685,31686],[34,781996,2557],{"className":781997},[31690],[34,781999],{"className":782000,"style":31846},[31845],[34,782002,202],{"className":782003},[31850],[34,782005],{"className":782006,"style":31846},[31845],[34,782008,782010,782013,782053,782056,782059],{"className":782009},[31676],[34,782011],{"className":782012,"style":55949},[31680],[34,782014,782016,782019],{"className":782015},[31685],[34,782017,67764],{"className":782018,"style":32062},[31685,31686],[34,782020,782022],{"className":782021},[31792],[34,782023,782025,782045],{"className":782024},[31796,31797],[34,782026,782028,782042],{"className":782027},[31801],[34,782029,782031],{"className":782030,"style":31806},[31805],[34,782032,782033,782036],{"style":38795},[34,782034],{"className":782035,"style":31814},[31813],[34,782037,782039],{"className":782038},[31818,31819,31820,31821],[34,782040,459],{"className":782041},[31685,31821],[34,782043,31829],{"className":782044},[31828],[34,782046,782048],{"className":782047},[31801],[34,782049,782051],{"className":782050,"style":31836},[31805],[34,782052],{},[34,782054],{"className":782055,"style":32604},[31845],[34,782057,223],{"className":782058},[32024],[34,782060],{"className":782061,"style":32604},[31845],[34,782063,782065,782068,782162,782165,782168,782171,782174,782177,782180,782220,782223,782226,782229,782232,782235],{"className":782064},[31676],[34,782066],{"className":782067,"style":667088},[31680],[34,782069,782071,782074],{"className":782070},[32767],[34,782072,546295],{"className":782073,"style":546331},[32767,131249,156560],[34,782075,782077],{"className":782076},[31792],[34,782078,782080,782154],{"className":782079},[31796,31797],[34,782081,782083,782151],{"className":782082},[31801],[34,782084,782086,782137],{"className":782085,"style":567928},[31805],[34,782087,782088,782091],{"style":546347},[34,782089],{"className":782090,"style":31814},[31813],[34,782092,782094],{"className":782093},[31818,31819,31820,31821],[34,782095,782097],{"className":782096},[31685,31821],[34,782098,782100,782103],{"className":782099},[31685,31821],[34,782101,32936],{"className":782102},[31685,31686,31821],[34,782104,782106],{"className":782105},[31792],[34,782107,782109,782129],{"className":782108},[31796,31797],[34,782110,782112,782126],{"className":782111},[31801],[34,782113,782115],{"className":782114,"style":318658},[31805],[34,782116,782117,782120],{"style":318661},[34,782118],{"className":782119,"style":43406},[31813],[34,782121,782123],{"className":782122},[31818,43410,43411,31821],[34,782124,459],{"className":782125},[31685,31821],[34,782127,31829],{"className":782128},[31828],[34,782130,782132],{"className":782131},[31801],[34,782133,782135],{"className":782134,"style":220540},[31805],[34,782136],{},[34,782138,782139,782142],{"style":546362},[34,782140],{"className":782141,"style":31814},[31813],[34,782143,782145],{"className":782144},[31818,31819,31820,31821],[34,782146,782148],{"className":782147},[31685,31821],[34,782149,32936],{"className":782150},[31685,31686,31821],[34,782152,31829],{"className":782153},[31828],[34,782155,782157],{"className":782156},[31801],[34,782158,782160],{"className":782159,"style":667231},[31805],[34,782161],{},[34,782163],{"className":782164,"style":32123},[31845],[34,782166,31739],{"className":782167,"style":31778},[31685,31686],[34,782169,456],{"className":782170},[31782],[34,782172,33249],{"className":782173},[31685,31686],[34,782175,31960],{"className":782176},[32119],[34,782178],{"className":782179,"style":32123},[31845],[34,782181,782183,782186],{"className":782182},[31685],[34,782184,329201],{"className":782185},[31685,31686],[34,782187,782189],{"className":782188},[31792],[34,782190,782192,782212],{"className":782191},[31796,31797],[34,782193,782195,782209],{"className":782194},[31801],[34,782196,782198],{"className":782197,"style":31806},[31805],[34,782199,782200,782203],{"style":31809},[34,782201],{"className":782202,"style":31814},[31813],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= y_0 + \\int_{x_0}^{x} f(t, \\phi_{n-1}(t)) \\, dt",[34,782358,782360,782424,782479],{"className":782359,"ariaHidden":758},[31672],[34,782361,782363,782366,782406,782409,782412,782415,782418,782421],{"className":782362},[31676],[34,782364],{"className":782365,"style":31774},[31680],[34,782367,782369,782372],{"className":782368},[31685],[34,782370,329201],{"className":782371},[31685,31686],[34,782373,782375],{"className":782374},[31792],[34,782376,782378,782398],{"className":782377},[31796,31797],[34,782379,782381,782395],{"className":782380},[31801],[34,782382,782384],{"className":782383,"style":32362},[31805],[34,782385,782386,782389],{"style":31809},[34,782387],{"className":782388,"style":31814},[31813],[34,782390,782392],{"className":782391},[31818,31819,31820,31821],[34,782393,31660],{"className":782394},[31685,31686,31821],[34,782396,31829],{"className":782397},[31828],[34,782399,782401],{"className":782400},[31801],[34,782402,782404],{"className":782403,"style":31836},[31805],[34,782405],{},[34,782407,456],{"className":782408},[31782],[34,782410,32936],{"className":782411},[31685,31686],[34,782413,2557],{"className":782414},[31690],[34,782416],{"className":782417,"style":31846},[31845],[34,782419,202],{"className":782420},[31850],[34,782422],{"className":782423,"style":31846},[31845],[34,782425,782427,782430,782470,782473,782476],{"className":782426},[31676],[34,782428],{"className":782429,"style":55949},[31680],[34,782431,782433,782436],{"className":782432},[31685],[34,782434,67764],{"className":782435,"style":32062},[31685,31686],[34,782437,782439],{"className":782438},[31792],[34,782440,782442,782462],{"className":782441},[31796,31797],[34,782443,782445,782459],{"className":782444},[31801],[34,782446,782448],{"className":782447,"style":31806},[31805],[34,782449,782450,782453],{"style":38795},[34,782451],{"className":782452,"style":31814},[31813],[34,782454,782456],{"className":782455},[31818,31819,31820,31821],[34,782457,459],{"className":782458},[31685,31821],[34,782460,31829],{"className":782461},[31828],[34,782463,782465],{"className":782464},[31801],[34,782466,782468],{"className":782467,"style":31836},[31805],[34,782469],{},[34,782471],{"className":782472,"style":32604},[31845],[34,782474,223],{"className":782475},[32024],[34,782477],{"className":782478,"style":32604},[31845],[34,782480,782482,782485,782579,782582,782585,782588,782591,782594,782597,782646,782649,782652,782655,782658,782661],{"className":782481},[31676],[34,782483],{"className":782484,"style":667088},[31680],[34,782486,782488,782491],{"className":782487},[32767],[34,782489,546295],{"className":782490,"style":546331},[32767,131249,156560],[34,782492,782494],{"className":782493},[31792],[34,782495,782497,782571],{"className":782496},[31796,31797],[34,782498,782500,782568],{"className":782499},[31801],[34,782501,782503,782554],{"className":782502,"style":567928},[31805],[34,782504,782505,782508],{"style":546347},[34,782506],{"className":782507,"style":31814},[31813],[34,782509,782511],{"className":782510},[31818,31819,31820,31821],[34,782512,782514],{"className":782513},[31685,31821],[34,782515,782517,782520],{"className":782516},[31685,31821],[34,782518,32936],{"className":782519},[31685,31686,31821],[34,782521,782523],{"className":782522},[31792],[34,782524,782526,782546],{"className":782525},[31796,31797],[34,782527,782529,782543],{"className":782528},[31801],[34,782530,782532],{"className":782531,"style":318658},[31805],[34,782533,782534,782537],{"style":318661},[34,782535],{"className":782536,"style":43406},[31813],[34,782538,782540],{"className":782539},[31818,43410,43411,31821],[34,782541,459],{"className":782542},[31685,31821],[34,782544,31829],{"className":782545},[31828],[34,782547,782549],{"className":782548},[31801],[34,782550,782552],{"className":782551,"style":220540},[31805],[34,782553],{},[34,782555,782556,782559],{"style":546362},[34,782557],{"className":782558,"style":31814},[31813],[34,782560,782562],{"className":782561},[31818,31819,31820,31821],[34,782563,782565],{"className":782564},[31685,31821],[34,782566,32936],{"className":782567},[31685,31686,31821],[34,782569,31829],{"className":782570},[31828],[34,782572,782574],{"className":782573},[31801],[34,782575,782577],{"className":782576,"style":667231},[31805],[34,782578],{},[34,782580],{"className":782581,"style":32123},[31845],[34,782583,31739],{"className":782584,"style":31778},[31685,31686],[34,782586,456],{"className":782587},[31782],[34,782589,33249],{"className":782590},[31685,31686],[34,782592,31960],{"className":782593},[32119],[34,782595],{"className":782596,"style":32123},[31845],[34,782598,782600,782603],{"className":782599},[31685],[34,782601,329201],{"className":782602},[31685,31686],[34,782604,782606],{"className":782605},[31792],[34,782607,782609,782638],{"className":782608},[31796,31797],[34,782610,782612,782635],{"className":782611},[31801],[34,782613,782615],{"className":782614,"style":31806},[31805],[34,782616,782617,782620],{"style":31809},[34,782618],{"className":782619,"style":31814},[31813],[34,782621,782623],{"className":782622},[31818,31819,31820,31821],[34,782624,782626,782629,782632],{"className":782625},[31685,31821],[34,782627,31660],{"className":782628},[31685,31686,31821],[34,782630,32226],{"className":782631},[32024,31821],[34,782633,640],{"className":782634},[31685,31821],[34,782636,31829],{"className":782637},[31828],[34,782639,782641],{"className":782640},[31801],[34,782642,782644],{"className":782643,"style":32037},[31805],[34,782645],{},[34,782647,456],{"className":782648},[31782],[34,782650,33249],{"className":782651},[31685,31686],[34,782653,44253],{"className":782654},[31690],[34,782656],{"className":782657,"style":32123},[31845],[34,782659,46900],{"className":782660},[31685,31686],[34,782662,33249],{"className":782663},[31685,31686],[253,782665,782666],{"id":782666},"积分求导公式",[15,782668,782669],{},"TBD",{"title":29,"searchDepth":43,"depth":43,"links":782671},[782672,782673,782677],{"id":775855,"depth":43,"text":775856},{"id":776355,"depth":43,"text":776356,"children":782674},[782675,782676],{"id":776359,"depth":49,"text":776360},{"id":779510,"depth":49,"text":779511},{"id":781394,"depth":43,"text":781395,"children":782678},[782679],{"id":782666,"depth":49,"text":782666},{"order":56},"\u002Fsource\u002Fmath\u002Fode\u002Freview",{"title":775850,"description":29},"source\u002Fmath\u002Fode\u002Freview","ZiZNYC3wZTXOZz3V2htRvPGIMnhO0PAmHBxfsoQtsDY",1784032895067]